ICSC 2008 - Evropský polytechnický institut, sro

Transkript

EUROPEAN POLYTECHNICAL INSTITUTE, KUNOVICE
PROCEEDINGS
SIXTH INTERNATIONAL CONFERENCE ON SOFT
COMPUTING APPLIED IN COMPUTER AND
ECONOMIC ENVIRONMENTS
ICSC 2008
January 25, Kunovice, Czech Republic
Edited by:
Prof. Ing. Imrich Rukovanský, CSc., Doc. Ing. Pavel Omera, CSc.
Prepared for print by:
Bc. Andrea imonová, DiS.
Printed by:
© European Polytechnical Institute Kunovice, 2008
ISBN : 978-80-7314-132-5
SIXTH INTERNATIONAL CONFERENCE ON SOFT COMPUTING
APPLIED IN COMPUTER AND ECONOMIC ENVIRONMENTS
ICSC 2008
Organized by
THE EUROPEAN POLYTECHNICAL INSTITUTE, KUNOVICE
THE CZECH REPUBLIC
Conference Chairman
H. prof. Ing. Old ich Kratochvíl, Dr.h.c.
rector
Conference Co-Chairmen
Prof. Ing. Imrich Rukovanský, CSc.
Assoc. Prof. Ing. Pavel Omera, CSc.
INTERNATIONAL PROGRAMME COMMITEE
O. Kratochvíl Chairman (CZ)
M. Bara ski (Poland)
J. Batinec (Czech Republic)
J. Brzobohatý (Czech Republic)
J. a o (Slovak Republic)
J. Diblík (Czech Republic)
P. Dostál (Czech Republic)
U. K. Chakraborthy (USA)
M. Kubát (USA)
B. Kulcsár (Hungary)
P. Omera (Czech Republic)
J. Petrucha (Czech Republic)
K. Rais (Czech Republic)
I. Rukovanský (Czech Republic)
G. N. Smirnov (Russian)
J. Stri (Slovak Republic)
G. Vértesy (Hungary)
W. Zamojski (Poland)
J. Zapletal (Czech Republic)
T. Walkowiak (Poland)
ORGANIZING COMMITEE
I. Rukovanský (Chairman)
P. Omera
J. áchová
A. imonová
I. Poláková
J. Kavka
Z. Omelková
M. Záleák
T. Chmela
J. Míek
M. Balus
Session 1: ICSC Soft Computing a jeho uplatn ní v managementu,
marketingu a i v moderních finan ních jako i technických systémech
Chairman: Doc. Ing. Petr Dostál, CSc.
Session 2: ICSC Soft Computing tvorba moderních po íta ových
nástroj pro optimalizaci proces
Chairman: Prof. RNDr. Jan Chvalina, DrSc., Ing. Jind ich Petrucha, Ph.D.
Oponentní rada
Doc. RNDr. Jaromír Batinec, CSc. Vysoké u ení technické v Brn
Dr. Gábor Vertésy, Dr.Sc. Ma arská akademie v d, Budape
Doc. Ing. Pavel Omera, CSc. Vysoké u ení technické v Brn
Prof. Ing. Wlodzimier M. Baranski, Ph.D. Ústav kybernetiky Wroclawské University
OBSAH
A MESSAGE FROM THE GENERAL CHAIRMAN OF THE CONFERENCE............................................................ 7
SESSION 1
REMARK ON AN APPLICATION OF THE LINEAR PROGRAMMING ................................................ 11
JAROMÍR BATINEC ........................................................................................................................................... 11
TEORIE HER V ROZHODOVACÍCH PROCESECH .................................................................................. 19
V
RA MATUTÍKOVÁ
........................................................................................................................................ 19
MODELING EXPORT PRICE INDEXES OF CZECH REPUBLIC BY ARIMA METHODOLOGY .... 25
ING. ZUZANA ME
IAROVÁ
................................................................................................................................ 25
MODERN MATHEMATICAL METHODS AND THEIR USE IN ECONOMY AND FINANCE............ 31
DOSTÁL PETR, OLD
ICH KRATOCHVÍL, KAREL RAIS ........................................................................................ 31
THE PREDICTION BY MEANS OF ARTIFICIAL NEURAL NETWORK............................................... 39
RADEK DOSKO
IL.............................................................................................................................................. 39
OPTIMALIZOVANÉ STAVEBNÍ BLOKY PRO NÁVRH MODERNÍCH INTEGROVANÝCH
OBVOD ............................................................................................................................................................. 45
JAROMÍR BRZOBOHATÝ, ROMAN PROKOP, VLADISLAV MUSIL ......................................................................... 45
RELIABILITY AND SECURITY IN LARGE COMPUTER SYSTEMS..................................................... 53
WOJCIECH ZAMOJSKI, MAREK BARA
SKI, KATARZYNA MICHALSKA ............................................................... 53
DISCRETE TRANSPORT SYSTEM - MODELING AND RELIABILITY ANALYSIS ........................... 57
JACEK MAZURKIEWICZ, TOMASZ WALKOWIAK................................................................................................. 57
QUALITY EXPERT ESTIMATE IN SOFT COMPUTING PROGRAMMES OF INTELLIGENT
AUTOMATION .................................................................................................................................................. 65
BRANISLAV LACKO............................................................................................................................................ 65
IMPLEMENTATION OF MOMENT METHOD FOR OBJECT RECOGNITION................................... 71
JI Í TASTNÝ, PETR LUDÍK ................................................................................................................................ 71
HYSTERETIC PROPERTIES OF A TWO DIMENSIONAL ARRAY OF SMALL MAGNETIC
PARTICLES: A TEST-BED FOR THE PREISACH MODEL...................................................................... 79
GÁBOR VÉRTESY1, MARTHA PARDA VI-HORVÁTH2 .......................................................................................... 79
INFORMATION SYSTEM EPI AND USING NEURAL NETWORK FOR ANALYZE OF TESTS ....... 87
JIND
ICH PETRUCHA .......................................................................................................................................... 87
FUZZY LOGIC AND GRANULAR RBF NEURAL NETWORKS: AN APLICATION TO THE INPUTOUTPUT FUNCTION ESTIMATION OF SALES PROCESSES................................................................. 91
MILAN MAR
EK ................................................................................................................................................ 91
WAGES FORECASTING USING TIME SERIES MODELS VERSUS SVM METHODS ....................... 97
DUAN MAR
1,2
EK
............................................................................................................................................. 97
APPLICATION OF LAPLACE TRANSFORM TO FRACTIONAL CALCULUS .................................. 105
B
ETISLAV
FAJMON, ZDEN
K MARDA .......................................................................................................... 105
VOLTERRA INTEGRODIFFERENTIAL EQUATIONS WITH DIFFERENCE KERNEL AND THEIR
APPLICATIONS IN THE THEORY OF ELECTRICAL CIRCUITS ....................................................... 111
OLGA FILIPOVÁ ............................................................................................................................................... 111
ICSC SIXTH INTERNATIONAL CONFERENCE ON SOFT COMPUTING APPLIED IN COMPUTER AND ECONOMIC
ENVIRONMENTS EPI Kunovice, Czech Republic. January 25, 2008.
5
SESSION 2
MATICOVÉ HRY A LINEÁRNÍ PROGRAMOVÁNÍ ................................................................................ 117
JITKA JABLONICKÁ .......................................................................................................................................... 117
NELINEÁRNÍ PROGRAMOVÁNÍ................................................................................................................ 123
MARIE TOMOVÁ ............................................................................................................................................ 123
METODY ROZHODOVÁNÍ ZA RIZIKA A NEJISTOTY ......................................................................... 131
JOSEF ZAPLETAL .............................................................................................................................................. 131
TEST OF OPTIMUM AND SOLUTION IMPROVEMENT OF THE BALANCED
TRANSFORMATION PROBLE..................................................................................................................... 141
JOSEF ZAPLETAL .............................................................................................................................................. 141
UNBALANCED TRANSPORTATION PROBLEM..................................................................................... 147
JOSEF ZAPLETAL .............................................................................................................................................. 147
VORTEX-FRACTAL-RING STRUCTURE OF ELECTRON .................................................................... 151
PAVEL OMERA ............................................................................................................................................... 151
VORTEX-FRACTAL STRUCTURE OF HYDROGEN............................................................................... 159
PAVEL OMERA ............................................................................................................................................... 159
THE USE OF FUZZY SETS IN MULTICRITERIAL OPTIMIZATION.................................................. 169
VÍT
ZSLAV EV ÍK, PAVEL KREJ Í .................................................................................................................
169
INCREASING OF COMPUTER NETWORKS PERFORMANCE VIA NODE TO NODE
THROUGHPUT OPTIMIZATION ................................................................................................................ 171
IMRICH RUKOVANSKÝ1, OND
EJ POPELKA
2
..................................................................................................... 171
NETWORK ELEMENT PROJECT BY MEANS OF NEURAL NETWORK........................................... 177
JI Í LIKA, JAN
ASTNÝ, MIROSLAV CEPL, M. TENCL .................................................................................
177
DETERMINING GENETIC ALGORITHM OPERATORS IN THE PROGRAM FOR OPTIMIZATION
OF PROGRESSIVE DISTRIBUTORS .......................................................................................................... 183
JI Í VEP
EK .....................................................................................................................................................
183
A BRIEF INTRODUCTION TO RECOGNITION OF DEFORMED OBJECTS ..................................... 191
MARTIN MINA
ÍK, JI Í ASTNÝ , OND EJ POPELKA .......................................................................................
191
SOME THOUGHTS ON INNER SYMMETRIES OF PROBABILITY THEORY AND EMERGENCE
OF KLEIN'S QUARTIC IN FUNDAMENTAL PHYSICS .......................................................................... 199
ALE GOTTVALD ............................................................................................................................................. 199
EXISTENCE OF BOUNDED SOLUTIONS FOR LINEAR DISCRETE SYSTEMS ............................... 205
JAROMÍR BATINEC, JOSEF DIBLÍK .................................................................................................................. 205
BINARYMULTISTRUCTURES OF PREFERENCE RELATIONS.......................................................... 211
JAN CHVALINA, JI Í MOU
KA, MICHAL NOVÁK .............................................................................................
211
PERIODICS SOLUTIONS OF VOLTERRA INTEGRODIFFERENTIAL EQUATIONS WITH
DIFFERENCE KERENEL .............................................................................................................................. 217
VLASTA KRUPKOVÁ, ZDEN
K MARDA...........................................................................................................
217
SUBSPACES OF THE SPACE OF SOLUTIONS OF ONE DIFFERENTIAL EQUATION WITH
CONSTANT DELAY ....................................................................................................................................... 223
ZDEN
K SVOBODA ...........................................................................................................................................
223
SOME PROPERTIES OF FRACTIONAL INTEGRALS AND DERIVATIVES...................................... 229
ZDEN
K MARDA
............................................................................................................................................ 229
6
A MESSAGE FROM THE GENERAL CHAIRMAN OF THE CONFERENCE
Dear guests and participants at this conference.
H. prof.., Ing. Old ich
Kratochvíl, Dr.h.c.
Prof. Ing. Imrich
Rukovanský, CSc.
Let me welcome you at the 6th International Conference on Soft
Computing Applied in Computer and Economic Environment ICSC
2008. During last six years the every year conference has appeared to
become an important meeting for introduction the latest knowledge and
results of collaborating universities and work places involved in modern
optimizing methods and tools of soft computing such as fuzzy control,
evolutional algorithms, usage of neuron web etc. We witness that the
papers from this conference are cited at many international conferences
abroad which make our school penetrate into the subconscious of
broader professional public.
Due to this reality the writers not only from the Czech Republic, but
also from Russia, the Slovak republic, Poland, Hungary, USA, read
papers at our conference.
As in every year the papers are divided into two groups. In the first one the focus is on the soft
computing and its application in marketing management and in the modern financial systems the
other one concentrates on modern computer tools for the process optimizing.
Dear guests, I believe that this anniversary sixth ICSC 2008 will support the further depth of
contacts and information exchange among the collaborating universities and other institutions both
at home and abroad in the area of the development of the modern optimizing methods and
application opportunities of soft computing.
Kunovice, January 25, 2008
Old ich Kratochvíl
Honorary professor, Ing., Dr.h.c.
rector
7
8
SESSION 1
9
10
REMARK ON AN APPLICATION OF THE LINEAR PROGRAMMING
Jaromír Batinec
Faculty of Electrical Engineering and Communication, Brno University of Technology
Abstract: In this paper we describe the solving of cutting plans - one application of the linear
programming.
Keywords: Linear programming, application, cutting plans.
1 REPETITION OF THE PROPERTIES OF SYSTEMS OF N LINEAR EQUATIONS IN N
UNKNOWNS
Let us put in remembrance the Frobenius theorem on the systems of linear equations.
Theorem 1. The system of n linear equations in n unknowns has the solution if and only if the rank of the matrix
of the system is equal to the rank of the expanded matrix of the system. If the rank of both the matrices is the
same and equal to n, there exists just one solution. If the rank of both the matrices is the same and less than n,
there exists infinite number of solutions.
Example 1: Find the solution of following system of equations:
3 x1 + 2 x2 + x3 = 6,
x1 + 4 x2 + 4 x3 = 5,
(x1 = 1, x2 = 2, x3 = -1)
2 x1 + 2 x2 - 4 x3 = 10.
This solution can be wrote also in the form of a three dimensional vector x
x1 , x 2 , x3
1, 2, 1 .
Remark: The rank of a matrix is given by the maximal order of non-zero determinant constructed from this
matrix.
3 2
We write the matrix of our system. It is: A = 1 4
2 2
1
3 2
4 the maximal determinant is D = 2 4
4
1 2
1
4 =4
54. We see that the determinant is non-zero, it is of the third order. We write the expanded matrix:
3 2
1 4
2 2
1 6
4 5 , the maximal determinant, constructed from this matrix is maximally of the order three.
4 10
Hence both the matrices have the same rank and as we see, the system has one and only one solution.
Example 2: We study another system of equations now.
x1 + 2 x2 + x3 = 45,
2 x1 + 5 x2 + 4 x3 = 100.
The matrix of this system is A =
1 2 1
2 5 4
expanded matrix of this system is
1 2 1 45
2 5 4 100
with non-zero determinant of the order two. Similarly the
and we see that a determinant of the maximal order is
determinant of the order two again. Hence a solution of then given system exists. But we see also that the rank of
11
both studied matrices is less than three. From the Frobenius theorem we can find nota bene infinite number of
solutions of this system.
2 LINEAR PROGRAMMING
Definition 1: The fundamental (primary) problem of linear programming is to find an n dimensional vector
x
which is non - negative, that is xi
x1 , x 2 , . . . , x n ,
(1)
0 , for i
(2)
1, 2, . . . , n .
satisfies the following linear independent conditions:
a11 x1
a 21 x 2
.
a m1 x1
a12 x 2
a 22 x2
.
am2 x2
.
.
.
.
.
.
.
.
. a1n x n
. a2 n xn
.
.
. a mn x n
optimises, it is maximises or minimises the objective function: z
b1
b2
.
bm
C1 x1
(3)
C2 x2
. . . , Cn xn .
Definition 2: A solution of (3) satisfying (2) is called admissible solution. We know from the upper repetition,
that the system of equations have the infinite number of solutions. This fact follows from the independence of
the system (3) and the Frobenius theorem.
Definition 3: An admissible solution which vector of solution has just m non - zero components is called
fundamental solution.
When we take the Example 2, we see that the number of fundamental solutions is just three. In the concrete:
x
x1 , x 2 , 0 , x
x1 , 0, x3 , x
0, x 2 , x 2 .
Remark: In full generality, for n unknowns (variables) and m conditions we obtain exactly
n
fundamental
m
solutions (vectors). Hence we obtain always only finite number of fundamental solutions.
Remark. It is necessary to say that we want to find such a solution of the system (3), which maximises or
minimises the objective function and as we have just seen the number of solutions is infinite. But following
theorem, the most important theorem of linear algebra, gives us the answer for our problem.
Theorem 2. If there exists a generally admissible solution of linear conditions which optimises (maximises or
minimises) the objective function then there exists the fundamental admissible solution which gives the same
optimal value of the objective function as that general one.
We return to the system of equations from the Example 2 and we eke out it with the objective function of the
form z = 3.x1 + 4.x2 + 3.x3. We want to find the maximal value of this objective function. We solve this system
using Gauss-Seidel method. We search only for fundamental admissible solutions. As the first, we calculate the
solution x
x1 , x 2 , 0 ,
x1 + 2 x2 + x3 = 45,
2 x1 + 5 x2 + 4 x3 = 100.
We multiply the first equation by (-2) and we add it to the second. We obtain:
x1 + 2 x2 + x3 = 45,
x2 + 2 x3 = 10 .
Now we multiply the second equation by (-2) and we add it to the first. Hence:
1.x1
- 3 x3 = 25
12
1.x2 + 2 x3 = 10 .
The fundamental (and also admissible) solution is: x1=25, x2= 10, x3= 0. We can write this result in the vector
form x
25, 10, 0 . Simultaneously the value of the objective function is: z = 3.25 + 4.10 + 3.0 = 115.
We see that the fundamental solution is determined by unit column vectors, which are standing at according
unknowns (variables) especially by the number one of this unit vector and the right side of coefficients. The
unknown (variable) at which there is not unit column vector is equal to zero. In our case it is the unknown x3.
The column vector is:
3
.
2
We calculate the second fundamental solution:
x1 + 2 x2 + x3 = 45
2 x1 + 5 x2 + 4 x3 = 100,
we multiply the first equation by (-4) and add it to the second one:
x1 + 2 x2 + x3 = 45
- 2 x1 - 3 x2
= -80.
Multiplying the second one by (-1/2) we have:
x1 + 2 x2 + x3 = 45
x1 +
3
x2
2
= 40.
We subtract the second equation from the first-one and we have:
1
x2 + x3 = 5
2
x1 +
3
x2
2
= 40 .
The second fundamental and also admissible solution is x1 = 40, x2 = 0 and x3 = 5. In other words,
x
40, 0, 5 . Simultaneously the value of the objective function is: z = 3.40 + 4.0 + 3.5 = 120 + 0 + 15 =
135.
Finally we calculate the last possible fundamental solution x
system we obtain
1
x1
+ x3 =
3
2
x1 + x2
=
3
0, x 2 , x2 . After rearrangement of our
25
3
80
3
Hence we have the solution x1 = 0, x2 = (80/3), x3 = (-25/3).
Watch! The last solution is fundamental but not admissible. We see that the optimal solution which gives the
maximal value of the objective function is x1 = 40, x2 = 0 , x3 = 5.
We have found this maximising solution without any difficulties. It is necessary to put in remembrance that we
had only three unknowns (variables) and only two conditions and hence with respect to the formula for
fundamental solutions only
3
2
3 solutions. For the greater systems of conditions (restrictions) on the vector
of optimal solution for given disjunctive function more complied algorithms are given. Especially the one called
13
Simplex method is used. It is necessary to say, that in general practice we meat with problems which are
described by mathematical models containing inequalities and not equations. We must convert such problem on
an equivalent problem with equations We do it by using additional and also dummy variables. We use as a
demonstration our second example, with partial modification. Also our modification of the second example will
be the following:
Example modification: Find a three dimensional vector maximising the objective function z = 3.x1 + 4.x2 +
3.x3 satisfying following conditions:
x1 + 2 x2 + x3
45
2 x1 + 5 x2 + 4 x3
100.
We can formulate this problem in full generality in a vector notation:
To find the vector x
x1 , x 2 , . . . , x n
satisfying following conditions:
A xT
bT
where A is the matrix of coefficients aij for i
(4)
1, 2, . . . , n , x T and b T transposed
1, 2, . . . , m and j
vectors, x such a vector which maximises the objective function z
C1 x1
C2 x2
. . . , Cn xn .
We convert (transform) this problem on the basic problem of linear programming. We do it with the aid of
additional variables. We obtain the system of linear equations:
x 1 + 2 x 2 + x 3 + x4
= 45
2 x1 + 5 x2 + 4 x3
+ x5 = 100.
We will simultaneously calculate and prospect the optimal solution with respect to maximisation of the objective
function. For this reason we choose such an algorithm which accomplishes this request. We do it so that we
ascribe to the system equations the objective function. We have: x1 + 2 x2 + x3 + x4
= 45,
2 x1 + 5 x2 + 4 x3
+ x5
= 100,
z = 3x1 + 4x2 + 3x3,
we transport the right side of the last equation on the left side and we do not write the letter z. That is a current
notation used in operation research literature. Using this instructions we have:
x1 + 2 x2 + x3 + x4
2 x1 + 5 x2 + 4 x3
= 45
+ x5
= 100
- 3x1 - 4x2 - 3x3
=
(5)
0
Looking at the system (5) we see that it contains two unit vectors - in the fourth and fifth column.
Simultaneously in the first, second and third columns there are not unit vectors of coefficients. We obtain an
0
enter (input) solution: x
0, 0, 0, 45, 100 in other words x1 = 0, x2 = 0, x3 = 0, x4 = 45, x5 =
100. This solution has only it's enter sense and no other. After substitution of this enter solution in the objective
function this one has the value equal to zero. For a simplification we rewrite our system into Simplex table:
Nonzero var.
x4
x5
x1
1
2
x2
2
5
x3
1
4
x4
Capacity
1
0
x5
0
1
-3
-4
-3
0
0
0
45
100
The maximisation is divided into some analogous parts. The first step in every of this parts is the determination
of the key column and of the key row. We describe these two operations.
1) Determination of the key column. We go through the last row - containing the negative coefficients of the
objective function and zeros. We construct the set of absolute values of this negative constants and find the
14
max
maximal value of them. Mathematically written:
i 1, 2, . . . ,n
abs C i
. In our concrete problem max {abs(-3),
abs(-4), abs(-3)}.
It is obvious that the maximal value is 4 and hence the column designated by x2 is the key column..
2) Determination of the key row. We go through all the positive coefficients in the key column over the
coefficient in the last row. We divide by them the relevant constant from the column of capacities. We obtain the
following results in our example: {(45/2), (100/5)}. We find the minimal value of this set of fractions and it will
determine the key row. Let us calculate (45/2) =22,5 and (100/5)= 20. The minimal value is obtained in the
second row, hence the second row is the key row and the intersection of key column and key row determines the
key element. In our example the key element is equal to five.
The second step consists of a new representation of vector-base. We calculate the new basic unit vector with the
unit on the place of key element. We do it so that we use all the allowed operations among equations of given
system of equations. We begin by dividing all the coefficients of the second row by 5. The other numbers of keycolumn must be transformed on zeros. We do it such that we multiply the new second row by -2 and we add it to
the first row. After this we multiply the new second row by 4 and add it to the third one. The table changes as
follows:
Nonzero var.
x4
x2
x1
1/5
2/5
x2
0
1
x3
-3/5
4/5
-7/5
0
1/5
x4
Capacity
1
0
x5
-2/5
1/5
0
4/5
80
5
20
We have the first approximation of optimal solution, which maximises the objective function z: x1 = 0, x2 =
20, x3 = 0, x4 = 5, x5 = 0 . The vector form: x
= 3.0 + 4.20 + 3.0 = 80.
1
0, 20, 0, 5, 0 . The value of the objective function is z
We go through the last row again. We do this step till all the coefficients in the new last row are positive or equal
to zero. We take all the negative coefficients and calculate their absolute values. From these we choose the
maximal absolute value which defines the new key-column. We see that in our case there exists only one
negative coefficient and therefore the key-column is defined. It is the first column
We calculate the key-row. We divide the capacities according to lying positive coefficients of the key column
and we have:
5/(1/5)=25 and 20/(2/5)=50. We obtained the minimal value for the first row and hence the key element is 1/5.
At this position will be the unit of the unit vector defined by key-column. We must multiply all the first row by
five. After this we multiply the new first row by -2/5 and add it to the second row and similarly then by 7/5 and
add it to the third row. After this operations we have received following table:
Nonzero var.
X1
x2
x1
1
x2
0
0
0
1
0
x3
-3
x4
5
x5
Capacity
-2
25
2
-4
-2
7
1
-2
10
115
We have the second approximation of optimal solution, which optimises the objective function x1 = 25, x2 =
2
10, x3 = 0, x4 = 0, x5 = 0 . The vector form: x
25, 10, 0, 0, 0 . The value of the objective function z
= 3.25 + 4.10 + 3.0 = 115. We see that the value of objective function for the given solution is in the last row
and last column of the table.
The last table has in its last row two negative numbers and it is the starting signal for iteration of looking for new
key column and key row. The key column is defined by abs (-4) which is greater then abs (-2). the key-row is
uniquely determined because the third column has only one positive member.
15
Such the new key-element is the number two in the third column and second row. We divide all members of this
second row by two and such obtained row is multiplied by three and added to the first row and then by four and
added to the last row. The resultant table is:
Nonzero var.
x1
x3
x1
1
0
x2
1,5
0,5
x3
0
1
x4
2
-1
x5
Capacity
-0,5
0,5
40
5
0
2
0
3
0
135
All the coefficients in the last row are positive. It means that we have obtained the optimal solution which
optimises the objective function. This optimal solution in vector form is: x
maximal value of the objective function is z = 3.40 + 4.0 + 3.5 = 135 .
40, 0, 5, 0, 0
and the
Remark. The calculus of optimising is usually done continuously in the form of one table as it follows on the
next side.
The second practical form of problems which can be solved using linear programming are problems of
minimising of objective function by vector x which satisfies following conditions:
Nonzero var.
x4
x1
1
x2
2
x3
1
x4
1
x5
0
Capacity
45
x5
2
-3
5
-4
4
-3
0
0
1
0
100
0
x4
1/5
0
-3/5
1
-2/5
5
x2
2/5
-7/5
1
0
4/5
1/5
0
0
1/5
4/5
20
80
x1
x2
1
0
-3
5
-2
25
0
0
1
0
2
-4
-2
7
1
-2
10
115
x1
1
1,5
0
2
-0,5
40
x3
0
0
0,5
2
1
0
-1
3
0,5
0
5
135
A xT
bT .
(6)
This problem is similar to the problem described in (4) only the inequality is opposite. The solution of this family
of problems is more complicated then the problem (4). We demonstrate one concrete example.
Find the vector minimising the objective function z = 2 x1 + 3 x2 + 4 x3, satisfying following conditions
x1 + 2 x2 + x3
120,
4 x1 + x2 + 2 x3
160.
We use the additional variables again. To transform the system of conditions on a system of equations we must
subtract the variables. We obtain: x1 + 2 x2 + x3 - x4
= 120
4 x1 + x2 + 2 x3
- x5 = 160.
In the next step we enlarge the system such that we write as the last row of it the objective function. We obtain:
x1 + 2 x2 + x3 - x4
= 120
16
4 x1 +
x2 + 2 x3
- x5 = 160
-2 x1 - 3 x2 - 4 x3
=
0.
When we return to our first problem (4) we see that we obtained after the adding of new variables starting
solution which was admissible solution. After the subtraction of new variables is the situation other,
unfortunately for the solving of problem worse. Here the initial solution is not admissible. This fact leads to the
other methods of solution and we show one of them. We extend the system of equations such that we put into it
new variables which are called dummy variables. We denote them by letter u and corresponding indices.
Simultaneously with these dummy variables we will implement into the row with the objective function new
negative values which are called prohibitive rates. After this the system of equations will have the following
form
x1 + 2 x2 + x3 - x4
+ u1
= 120
4 x1 + x2 + 2 x3
- x5
-2 x1 - 3 x2 - 4 x3
+
u2 = 160
-40 u1 - 40 u2 =
0 .
The prohibitive rates are negative products of a constant and a dummy variable with the same index as has the
dummy variable appropriate for given column. A very important question is: How to obtain the constant? We do
it by taking the greatest coefficient of the objective function in absolute value (it is in our case equal to 4) and we
multiply it by 10. Hence the constant is equal to 40. We have seen that the forth and fifth columns are not
applicable for the acquisition of an admissible solution therefore we prepare conditions for an admissible
solution obtained from the sixth and seventh column. For this reason we need unit vectors in this both columns
with the units at dummy variables. We get these unit vectors so that we multiply all of the first equation by forty
and we add this product to the equation in the last row. Similarly we multiply the second equation by forty and
we add this product to the last equation (to the last row). After this operations the system has following form:
x1 +
2 x2 +
x3 - x 4
+ u1
4 x1 +
x2 + 2 x3
- x5
198 x1 + 117 x2 + 116 x3 - 40 x4 - 40 x5
=
+
120
u2 = 160
= 3280.
0
The vector of the admissible introductory solution is: x
0, 0, 0, 0, 0,. 120, 160 .The optimising, in this
case the minimising, will go in a very similarly way as in the case of maximising only with that difference, that
we look the maximal coefficient in the last row. This one determines the key column. The key row is determined
in the same way as in the problem of maximising. we see that it is the second row. The fraction (160/4) is less
than (120/1). The process of minimising will be brought when all the coefficients in the last row will be negative
or equal to zero. We rewrite the problem into Simplex table.
Nonzero var.
x1
x3
x1
0
1
x2
x3
x4
7/4
1/4
½
½
1
0
0
135/2
17
-40
x5
1/4
-1/4
38/
4
u1
u2
Capacity
1
0
-1/4
1/4
80
40
0
-198/4
3280
We see that the table contains in the last row the positive numbers yet. Therefore we choose the maximal of
them, in our case 135/2 = 67,5 which assigns the new key column. The key row is given by the smallest fraction
of the right side of table (Capacity) and according lying positive coefficient from key column. In our case both
the coefficients from key column are positive again and 80/(7/4) = 45.7142 and 40/(1/4)= 160. Hence the first
row is the key row and 7/4 is the key element.
Nonzero var.
x1
x3
x1
0
x2
1
x3
x4
u2
Capacity
-4/7
x5
1/7
u1
2/7
4/7
-1/7
320/7
1
0
0
0
3/7
-16/7
-1/7
-10/7
-2/7
-1/7
-1/7
-270/7
2/7
-274/7
200/7
1360/7
17
The optimal solution for which the objective function is minimal is x
1360/7.
{200/7, 320/7, 0, 0, 0, 0, 0}and z =
3. APPLICATION
Lastly we introduce an application of utilizing of linear programming on a concrete problem from practice.
We have to solve the following enter. It is to cut the rod linkage material of the given length on various lengths,
every length of given amount. The concrete problem is this: From the given material of the length 12m cut four
different lengths of given amount.
a} 6,35 m
24 bits
c} 3,15 m 128 bits
b} 4,95 m
64 bits
d} 2,7 m 256 bits
such that the total drain will be minimal.
As the first step we create such called cutting plans using a program fully fashioned on department of
mathematics of the Faculty of Electrical Engineering and Communication, Brno University of Technology. We
obtain the following system of cutting plans and multiplicity of their usage:
Plan
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
Length
6,35
4.95
3,15
2,7
1
1
1
0
0
0
0
0
0
0
0
1
0
0
2
1
1
1
0
0
0
0
0
1
0
0
2
1
0
3
2
1
0
0
0
2
0
0
1
2
0
2
3
4
Multiplicity
of usage
24
38
26
52
Total
drain
0,7
2,5
0,2
2,1
0,7
1,2
1,6
2,5
0,3
0,7
1,2
109,4
The finite result is this: We apply 24-times the plan x3 then 38-times the plan x2 similarly 26-times the plan x7
and finally 52-times the plan x10. This is the way that guarantees .the minimal drain.
Acknowledgement: This work supported by the Council of Czech Government MSM 0021630529.
REFERENCES:
[1]
CHURCHMAN, C. W.; ACKOFF, R. L.; ARNOFF, E. L. Úvod do opera ného výskumu. Bratislava :
ALFA, 1968.
[2]
LA IAK, A. a kol. Dynamické modely. Bratislava : ALFA, 1985.
[3]
NOVÁK, M. Probability theory in combined form of study at FEEC BUT. Kunovice : Mezinárodní
konference EPI, s.r.o. 2006.
[4]
TYC, O. Opera ní analýza. Brno : MZLU, 2002.
[5]
ZAPLETAL, J. Opera ní analýza. Kunovice : SKRIPTORIUM VO, 1995.
[6]
DUDORKIN, N. Opera ní analýzy. Praha : FEL VUT, 1997.
ADRESS:
Doc. RNDr. Jaromír Batinec, CSc.
Department of Mathematics,
Faculty of Electrical Engineering and Communication,
Brno University of Technology,
Technická 8, 616 00 Brno, Czech Republic
[email protected]
18
TEORIE HER V ROZHODOVACÍCH PROCESECH
V ra Matutíková
Evropský polytechnický institut, s.r.o., Kunovice
Abstrakt. Mezi jedny z nejvýznamn jích metod p i rozhodování za rizika a nejistoty pat í metody
postavené na bázi teorie her. Jsou to metody aplikující v rámci rozhodovacího procesu metody
lineárního programování, ale také numerické itera ní metody eení maticových her, která vyuívá
principu postupného "u ení" hrá na základ zkueností z p edcházejících realizací hry. Výpo et
probíhá v ad iterací tak dlouho, dokud není stanovena cena hry s jistou, p edem zadanou
p esností a dokud nejsou získány smíené strategie dostate n blízké optimálním. Jednou takovou
metodou je Bronova metoda..
Keywords. Základní princip teorie her, maticové hry, sedlový bod hry, princip minimaxu, Bayes v
princip rozhodování, Laplace v princip nedostate né evidence, Wald v pesimistický princip
maximinu, Savage v princip minimaxu ztráty, Hurwicz v princip ukazatele optimismu.
1. BROWNOVA METODA
V této ásti uvedeme p iblinou Brownovu metodu eení maticových her jako p íklad jednoduchého
adaptivního algoritmu. Jedná se o numerickou itera ní metodu eení maticových her. Postup eení problému
Brownovou metodou objasníme na p íkladu
P íklad. Má se eit maticová hra s výplatní maticí
A=
4 1 5 2
2 3 1 4
1 6 4 3
Postup eení: Hrá i sehrají výchozí fiktivní partii tak, e zvolí své isté strategie zcela libovoln , protoe jet
nemají ádné informace o p edchozím pr b hu hry. Nech jsou to strategie i = 1 a j = 1. V dalí partii první
hrá p edpokládá, e druhý hrá zvolí op t strategii j = 1 a volí nejlepí odpov dle max (4, 2, 1) = 4 to
znamená zvolí strategii i = 1. Analogicky druhý hrá p edpokládá, e první hrá op t zvolí i = 1 a volí nejlepí
odpov dle min (4, 1, 5, 2) = 1, to znamená, e volí strategii j = 2. P ed následující partií první hrá ji ví, e
druhý hrá volil jednou j = 1 a jednou j = 2. To odpovídá smíené strategii druhého hrá e y = (0,5 ; 0,5 ; 0 ; 0)T.
P ísluná st ední hodnota výhry prvního hrá e pro jeho isté strategie i = 1, 2, 3 je
T
f ( i , y ) = A y = = ( 2,5 ; 2,5 ; 3,5 ) .
První hrá volí svoji dalí strategii dle maxima takto zjit né st ední hodnoty, tj. Max (2,5; 2,5; 3,5) = 3,5, tj. i =
3. Místo výpo tu sou inu A y vak z ejm posta í se íst ty sloupce matice A, jejich isté strategie druhý hrá
volil a rozhodnout se dle maxima t chto sou t . Je tedy
4
2
1
j
1
1
3
6
j
2
5
5
7
a max (5; 5; 7) = 7 vede na volbu i = 3. Analogicky první hrá volil dvakrát strategii i = 1, co odpovídá smíené
strategii x = (1; 0; 0), a p ítí optimální volb druhého hrá e dle min [(4; 1; 5; 2) + (4; 1; 5; 2)] = min (8; 2; 10;
4) = 2 to znamená j = 2. Výsledky výe popsaného výpo tu jsou dále uvedeny v následující tabulce
Je mono dokázat, e hodnota hry je zdola p ípadn shora ohrani ena mezemi
tyto meze rovny
v
= (min (8; 2; 10; 4)) / 2 = 2 / 2, co je jedna a
v p ípadn . Po druhé h e jsou
v = (max (5; 5; 7)) / 2 = 7 / 2 = 3,5. Tedy
19
v v , kde t = 1, 2, . . . udává po adí partie (iterace). Výsledek srovnání velikosti
obecn platí relace v
rozdílu s p edem zadanou tolerancí poskytuje pravidlo pro ukon ení itera ního výpo tu. V uvedeném p íkladu je
mono omezit cenu hry v nerovností 25 / 10
v
32 / 10 to znamená 2,5
v
3,2 .Tomu odpovídá
smíená strategie prvního hrá e x10 = (0,5; 0; 0,5) a smíená strategie druhého hrá e popsaná vektorem y10 =
T
(0,7; 0,2; 0; 0,1) . Pro srovnání uvádíme smíené strategie obou hrá po p ti tisících iterací x0= (0,512; 0,233;
0,255), y0= (0,5; 0,253; 0; 0,247) a cena hry v = 2,75. Nevýhodou popsaného algoritmu Brownovy metody je
jeho relativn pomalá konvergence. Krom této metody existuje ada dalích metod k eení maticových her,
jako je nap íklad metoda dvojího opisu.
1. hrá
P
o
a
d
í
p
a
r
t
i
e
2. hrá
1
1
4
1
2
2
3
3
1
6
1
4
1
2
8
2
3
9
8
4
10 14
5
11 20
6 12 26
7
13 32
8
17 33
9
21 34
10 25 35
2
W10 7
Y10 0,7 0,2
5
1
4
5
10
14
18
22
26
30
35
40
45
0
0
Po adí partie
1
4
5
6
8
12 16 20 24 28
32
2
5
8
12 14 16 18 20 22
24
1
7
13
16 17 18 19 20 21
22
Sloupcové sou ty
Po et voleb ryzích strategií 1. hrá e
Smíené strategie 1. hrá e po 10. partii
2
4
3
2
4
7
10
13
16
19
21
23
25 Po et voleb ryzích strategií 2. hrá e
1 Smíené strategie 2. hrá e po 10. partii
0,1
Z10
X10
5
0
5
0,5
0
0,5
Nejsou-li mnoiny istých strategií obou hrá v antagonistické h e kone né, jedná se o tzv. nekone né
antagonistické hry. Pro tyto hry v obecné formulaci neexistuje universální teoretický aparát, který by dával
normativní návod k jejich eení. Jsou známy metody eení pouze jistých speciálních her, u nich mnoiny
strategií, p ípadn výplatní funkce mají ur ité speciální vlastnosti (nap íklad mnoiny strategií jsou uzav ené
konvexní polyedry).
2. HRY PROTI PIROD
Hrou proti p írod rozumíme hru dvou hrá s nulovým sou tem, z nich první je racionální a druhý je
neracionální, nazývaný p írodou i sv tem. Neracionální ú astník je lhostejný k výsledk m rozhodování.
P edpokládáme, e neracionální ú astník se projevuje jako náhodný mechanismus, který volí stavy s S dle
n jakého rozd lení pravd podobnosti p(s). Je-li toto rozd lení racionálnímu ú astníkovi známé, hovo íme o
rozhodování za rizika; není-li mu známé, hovo íme o rozhodování za neur itosti. P sobení neracionálního
ú astníka m eme také povaovat za p sobení okolí rozhodovací situace, za p sobení okolního sv ta (p írody) a
j
Stavy p írody (sv ta)
i
S
T
R
A
T
E
G
I
E
1
2
.
.
.
m
1
f11
f21
.
.
.
f m1
2
f12
f22
.
.
.
fm2
............
.
.
.
.
.
.
n
.
.
.
.
.
.
f1n
f2n
.
.
.
fmn
Tabulka 1.
20
podle pot eby interpretovat nastoupení stavu s S jako volbu strategie s S okolním sv tem (p írodou). Je-li
mnoina stav (strategií) p írody S = {1, . . . , n} také kone nou mnoinou, je výplatní funkce racionálního hrá e
zadána v podob výplatní matice F, která je tvaru a kterou je kadému výsledku rozhodování (i, j) X x S
p i azena odpovídající výplata. Výplatní maticí F je tato rozhodovací situace pln popsána. Viz tab1.
Kadému stavu p írody j S m e být p i azena pravd podobnost jeho nastoupení pj [0 , 1]; j = 1 , 2 , . . . ;
= 1. Tyto pravd podobnosti mohou být zjit ny objektivn statisticky (v p ípad , e stavy sv ta p edstavují
hromadn se opakující jevy) nebo subjektivn odhadem.
Pro v tí názornost uvedeme uvedeme dalí p íklad. Výrobce vyrábí ur ité výrobky, z nich 10% je vadných.
Servisní organi- zace ú tuje výrobci za záru ní opravu jednoho výrobku pr m rn 8 PJ/kus.Výrobce vak m e
p ed expedicí výrobk provést jejich kontrolu kvality (stoprocentn ú innou) a vadné výrobky jet p ed
expedicí opravit. Pr m rné náklady na opravu jednoho vadného výrobku p ed expedicí iní 4 PJ/kus a náklady
na kontrolu jednoho výrobku iní 3 PJ/kus. Zisk z prodeje jednoho výrobku je 9 PJ/kus. Výrobce se rozhoduje,
zda má výrobky expedovat bez výstupní kontroly a platit servisní organizaci za záru ní opravy. Výrobce jako
racionální hrá má dv strategie:
1 ) Nekontrolovat výrobek p ed expedicí.
2 ) Kontrolovat výrobek p ed expedicí.
Na druhé stran p íroda volí dva stavy:
1 ) Dobrý výrobek - to nastává s pravd podobností 0,9.
2 ) Vadný výrobek - to nastává s pravd podobností 0,1.
Výplatní matice p ísluné hry je uvedena v následující tabulce 2.
Alternativy
i
popis
1
2
nekontrolovat
kontrolovat
Stavy p írody
j
1
popis dobrý výrobek
pj
0,9
9
9-3 = 6
2
vadný výrobek
0,1
9-8 = 1
9-3-4 = 2
Tabulka 2.
P i rozhodování za rizika a neur itosti naráí definice optimální strategie racionálního hrá e na adu potíí. Je
známo n kolik p edpis pro volbu optimální strategie, které je obvyklé nazývat principy rozhodování (kritéria
rozhodování). Je velmi nep íjemné, e v rozhodovacích situacích za rizika a neur itosti m e vést pouití
r zných princip rozhodování k obecn r zným optimálním rozhodnutím. V tomto p ípad je vhodn jí hovo it
o suboptimálních rozhodnutích - eeních. Je potom subjektivní záleitostí kadého rozhodovatele, s jakým
principem se ztotoní a jaké bude nejlépe odpovídat jeho osobním sklon m (nap . ochot riskovat) a vnit nímu
modelu rozhodovací situace.
Aby se co v nejv tí mí e vylou il subjektivismus p i rozhodování i v i vícekrite- riálním ú elovým funkcím,
aplikují se dnes jako podpora programové systémy, umo ující zobjektivizovat rozhodovací procesy. Jeden
takový systém byl vytvo en doc. RNDr. Jind ichem Klapkou CSs. na ústavu automatizace FS. VUT v Brn . Jeho
pouitelnost je maximální p i rozhodování v rámci konkursních ízení.
3. ZÁV R
V záv re né ásti uvedeme nej ast ji uívané principy rozhodování.
1) Bayes v princip rozhodování. V p ípad rozhodování za rizika p edpokládáme, e racionální subjekt zná
pravd podobnosti p1 , p2 , . . . , pn nastoupení stav p írody 1, . . . , n. Obvykle pouívaným principem
rozhodování za rizika je Bayes v princip, spo ívající ve výb ru strategie nejv tí st ední hodnoty výplat. Dle
Bayesova principu je optimálním rozhodnutím strategie k , pro ni platí
n
max
i
p j fi j
p j fk j
j 1
Kritika nedostatk Bayesova kritéria spo ívá na výhradách k reálnosti hypotéz o chování p írody a na zjit ních,
e v ad praktických rozhodovacích situacích se ú astníci vdy ne ídí st ední hodnotou výplat. P íkladem m e
být tak zvaný petrohradský paradox, zformulovaný v roce 1738 Bernoullim: Dva hrá i musí sehrát hru, v ní
první hrá hází mincí tak dlouho, dokud nepadne hlava. Padne-li hlava p i n-tém hodu (n = 1, 2 , . . . ), obdrí
21
první hrá od svého protivníka ástku 2n K , ím hra kon í. Pokud se hra rozb hne, druhý hrá z ejm vdy
prohrává ástku alespo 2 K . Proto vyzve prvního hrá e, aby si ur il ástku odstupného, kterou mu druhý hrá
zaplatí, vzdá-li se první hrá hry. St ední hodnota výhry prvního hrá e je :
2
1
2
2
1
2
... 2
n
1
n
...
2
2
2
Podle ji zmín né knihy M. Ma ase volí reální rozhodovatelé obvykle ástku odstupného v rozmezí 2 K a 40
K (s pravd podobností p = 1) p ed ú astí ve h e s moností získat 2 K s pravd podobností 1/2, 4 K s
pravd podobností 1/4 atd. To ukazuje, e vyplácené pen ní ástky jet nemusí udávat p ísluný uitek
p íjemce. Vyjad uje-li vak výplatní funkce uitek hrá , rozhodují se tito dle st ední hodnoty uitku.
V p ípad rozhodování za neur itosti p edpokládáme, e racionální subjekt nezná rozloení pravd podobností
stav p írody. Uvedeme nejznám jí principy rozhodování:
2) Laplace v princip nedostate né evidence. Vychází z p edpokladu rovnocennosti vech stav sv ta tudí z
jejich stejné pravd podobnosti. Za optimální se povauje strategie k, pro ni platí
1
n
max
i
n
j 1
f ij
1
n
n
1
j
f kj
Tento princip je ekvivalentní s Bayesovým principem maxima st ední hodnoty výplat p i stejných
pravd podobnostech stav p írody pj = 1/n, j = 1 , 2 , . . . , n. P edpoklad stejných pravd podobností stav
p írody m e být asto nep ijatelný.
3) Wald v pesimistický princip maximinu. Tato metoda zaru uje rozhodovateli získání nejv tí z minimálních
výplat volbou strategie k, pro ni platí
max min f i j
i
min f k j
j
j
Pesimistický princip p edstavuje "pojistku" proti nejhorímu. Zaru uje sice jistou výplatu, ale z druhé strany
vede k p ehnan konzervativním, "opatrnickým" rozhodnutím. Princip minimaxu v podstat povauje p írodu za
aktivního racionálního hrá e, který se snaí minimalizovat svoji prohru.
4) Savage v princip minimaxu ztráty. Vychází z matice ztrát Z =
zi j
fi j
max fi j
zi j
n
m
, v ní
i = 1,...,m ; j = 1,...,n
i
p edstavuje ztrátu z neznalosti skute ného stavu p írody, to znamená ztrátu ve srovnání s rozhodnutím, které
bychom volili p i znalosti skute né volby stavu p írody. Optimální strategie k je ur ena vztahem
min max z i j
i
j
zk j
Tato volba zabezpe uje rozhodovatele proti p ehnan velkým ztrátám oproti rozhodnutí zaloenému na znalosti
volby p írody. Charakter principu minimaxu ztráty je také konzervativní, pesimistický a odpovídá také
p edpokladu aktivního racionálního chování p írody. Je vak pojistkou proti námitkám t ch, kdo jsou "po bitv
generály".
5) Hurwicz v princip ukazatele optimismu. Umo uje volbou tak zvaného ukazatele optimismu a
respektovat subjektivní "zaloení" rozhodovatele. Za optimální strategii k povaujeme tu, pro ni platí
a max
f ij (1
max
i
j
a) min
f ij
j
a max
f kj
j
(1
[0, 1]
a) min
f kj
j
kde a
Pro a = 0 dostáváme pesimistický princip. Pro a = 1 obdríme optimistický maximový princip vedoucí na volbu
strategie s absolutn nejvyí výplatou. Hurwicz v princip je mono modifikovat zavedením ukazatele
optimismu pro kadou alternativu, to znamená stanovením ai [0, 1] pro vechna i = 1, . . . , m, Pro ur ení
problematického ukazatele optimismu je mono vyuít nap íklad expertních metod.
Uvedené principy jsou zobecnitelné i na hry s nekone nou mnoinou stav p írody.
Pro rozhodovací situaci z p edchozího p íkladu je v tabulce 3 uvedena matice výplat F a ztrát Z. V tée tabulce
jsou uvedeny výsledky výpo t dle jednotlivých princip a vyzna eny p ísluné optimální strategie (podtrené
hodnoty).
22
V p ípad rozhodovací situace za rizika je dle Bayesova principu optimální strategie 1 to znamená
nekontrolovat. Tato strategie je také nejlepí dle v tiny pouitých princip , které nejsou p ehnan pesimistické.
P i pesimistickém náhledu, charakterizovaným ukazatelem optimismu nejvýe 0,25, je vak vhodn jí volit
strategii 2 a uskute ovat výstupní kontrolu.
Krom popsaných nejuívan jích princip rozhodování za rizika a neur itosti je známa ada dalích princip s
omezeným pouitím, konstruovaných na pom rn speciálních p edpokladech. ádný ze známých princip
rozhodování není natolik evidentní, aby mohl být prohláen za jediný a nejlepí. Vechny dávají jakési
suboptimální výsledky. Výb r konkrétního principu je subjektivní záleitost a závisí na charakteru eené
situace. Ur itý návod pro výb r principu rozhodování m e dát axiomatický p ístup teorie rozhodování
formulací ne vdy konzistentního souboru poadavk , kladených n a optimální rozhodování. Doporu uje se volit
i
j
1
2
1 2
Bayes
pi f i j
j
Laplace
1/n f i j
j
Wald
min f i j
j
Hurwicz
Savage
max z i j
j
a
pj 0,9 0,1 x x
1
9
1 0 1
2
6
2 3 0
0,2 0,5
8,2
5,6
5
4
1
2
2,6
2,8
0,8
5
4
6,6 1
4,8 3
Výplaty Ztráty
fij
z ij
Tabulka 3.
ten princip, který poruuje pouze pro rozhodovatele málo d leité poadavky (axiomy). I kdy se rozhodovatel
zprvu nerozhodne jednozna n pro n který z uvedených princip rozhodování, umoní mu jejich aplikace
vylou it zcela chybná rozhodnutí a vybrat podmnoinu strategií, které jsou vícemén racionální.
REFERENCES
[1]
BATA, A. Plánové rozhodovací procesy a jejich systém. Praha : Academia, 1977.
[2]
BATINEC, J.; BATINEC, J. Structural interbranch system of dynamic model. Proceedings of
International conference "University as Facilities for Advancement of Community and Region."
Kunovice: EPI, Kunovice, 2005, s. 317 - 322, ISBN 80-7314-052-7.
[3]
BATINEC, J.; DIBLÍK, J. Solution of Structural Interbranch Systém of a Dynamic Model.
Proceedings Fourth international conference on soft computing applied in computer and economic
enviroment. European Polytechnical Institute Kunovice, 2006, s. 35 - 41, ISBN 80-7314-084-5.
[4]
BATINEC, J.; NOVÁK, M. Numerické metody v navazujícím magisterském studiu. DIDZA 2006
(3nd Didactic Conference in ilina with international participation), CD-ROM. ilina, ilinská
univerzita, 2006, 1 - 9, ISBN 80-8070-557-7.
[5]
ERNÝ, J.; GLÜCKAUFOVÁ, D. Vícekriteriální vyhodnocování v praxi. Praha : SNTL, 1982.
[6]
FOTR, J. P íprava a hodnocení podnikatelských projekt . Praha : VE, 1993.
[7]
MOORE, P. G. The Business of Risk. Cambridge. University Press, 1983.
[8]
NOVÁK M. Examples of using concepts of probability theory in managementdecision makinng.
Kunovice : Mezinárodní konference EPI, 2006.
[9]
STUCHLE, W. H. Management. München : Verlag Franz Valen, 1989.
[10]
VACULÍK, J.; ZAPLETAL, J. Podp rné metody rozhodovacích proces . Masarykova univerzita
v Brn , 1998.
[11]
VL EK, R. Hodnotový management. Praha : Management Press, 1992.
[12]
WATSON, S. R.; BUDGE, J. R. Decision Synthesi. Cambridge : Cambridge University Press, 1987.
[13]
ZAPLETAL, J. Opera ní analýza. Kunovice : Skriptorium VO, 1995.
[14]
ZAPLETAL. J. Poznámka k rozhodování za rizika a nejistoty. Mezinárodní v decká konference
Kunovice, leden, 2006.
[15]
ZAPLETAL J. Poznámka k rozhodování za rizika a nejistoty. Mezinárodní v decká konference
[16]
ZAPLETAL. J. Metody rozhodování za rizika a nejistoty-t etí ást. Mezinárodní v decká konference
[17]
ZÁRUBA, P. aj. Základy podnikového managementu. Praha : Aleko, 1991.
23
ADRESA:
Mgr. V ra Matu íková
Evropský polytechnický institut, s.r.o.,
Osvobození 699,
686 04 Kunovice
e-mail: [email protected]
tel.: 572 549 018
24
MODELING EXPORT PRICE INDEXES OF CZECH REPUBLIC BY ARIMA
METHODOLOGY
Ing. Zuzana Me iarová
ilinská univerzita v iline
Abstract: The following article will discuss analysis and modeling of export price indexes of Czech
Republic on revised external trade structure of the year 2005 by ARIMA methodology. This method
is used for modeling and forecasting of monthly data per years 1998 and 2006 (average of the year
2005=100).
Keywords: ARIMA methodology, autocorrelation function, Box-Jenkins methodology, forecasting,
time series, modeling, partial autocorrelation function
1. ARIMA METHODOLOGY
ARIMA methodology or otherwise called as Box-Jenkins methodology is one of the methods of time series
analysis. This methodology is developed by Box and Jenkins and arose in 1970s years of 20th century.
According to the authors time series is sequence of random variable values, those realization we can mark as
{yt} [2]. The ARIMA method is appropriate only for a time series that is stationary and it is recommended that
there are at least 50 observations in the input data. In general, time series is stationary, if its probability
distribution does not change through time but practically, we consider time series as stationary, if its mean,
variance, and autocorrelation function should be approximately constant through time [2]. The most widely used
methods of modifying time series to stationary are differentiation, logarithmic calculation and trend removal.
Process of time series analysis and forecasting by Box-Jenkins methodology has five phases - data preparation,
model identification, parameter estimation, statistical verification of model and parameters and forecasting.
2. APPLICATION OF BOX-JENKINS METHODOLOGY ON TIME SERIES OF EXPORT PRICE
INDEXES
There is illustrated the time series of monthly export price indexes of Czech republic [3] in the figure 2.1. From
graphics representation of the data we can see the time series is no stationary. This fact confirms also
autocorrelation function (ACF) and partial autocorrelation function (PACF), that are illustrated in the figures 2.2
and 2.3.
Fig. 2.1 The time series of export price indexes of Czech republic
25
Fig. 2.2 ACF of original time series
Fig. 2.3 PACF of original time series
ACF values go down slowly and the first value of autocorrelation and partial autocorrelation coefficient is very
close to value one. There is illustrated periodogram of time series in the figure 2.4. It has significant peak in null
frequency (frequency equals 0,018). On the basis of these facts simple differences of order one were taken.
Fig. 2.4 Periodogram of original time series
There is illustrated the residual autocorrelation and partial autocorrelation function after first difference in the
figure 2.5 and 2.6.
Fig. 2.5 Residual ACF of ARIMA(0,1,0) model
Fig. 2.6 Resdidual PACF of ARIMA(0,1,0) model
We can see from the pictures above the first value of residual autocorrelation and partial autocorrelation
coefficient exceeds confidence bands markedly. The type of ARIMA model is not explicit. It is possible to
extend the model about one autoregressive parameter (AR (1)), one moving average parameter (MA (1)) or both
together.
We make selection of optimal model by using statistical software Statgraphics Plus 5.1. This program can
choose optimal model, change model parameters and calculate values of these parameters, test statistical
significance of parameters and model, calculate measure of ex-post forecast and test suitable of model for
prognostic application.
The data cover 108 time period. In this case, the model was estimated from the first 54 data values. 54 data
values at the end of time series were withheld to validate the model.
An autoregressive integrated moving average model has been selected in form ARIMA(1,1,0)C. The figure 2.7
illustrates output from the statistical software Statgraphics and also summarizes the results of model
ARIMA(1,1,0)C.
26
Fig. 2.7 Output from statistical software Statgraphics for model ARIMA(1,1,0)C
As we can see from the t-test of autoregressive parameter and constant, they do not belong to the model. The Pvalue for the AR (1) term is greater than significance level =0.05, so it is not statistically significant. We should
therefore consider reducing the order of the AR term to 0. The P-value for the constant term is also greater than
0.05, so it is not statistically significant. We should therefore consider removing the constant from the model.
There are illustrated the residual ACF and PACF in the figures 2.8 and 2.9. The plots of residual ACF and PACF
confirm that the residuals are not white noise because of first autocorrelation and partial autocorrelation
coefficient exceeds confidence bands.
Fig. 2.8 Residual ACF of model ARIMA(1,1,0)C
Fig. 2.9 Residual PACF of model ARIMA(1,1,0)C
Next, we selected forecasting model in form ARIMA(0,1,1) without constant. The figure 2.10 illustrates output
from Statgraphics for model ARIMA(0,1,1).
Fig. 2.10 Output from Statgraphics for model ARIMA(0,1,1) without constant
The P-value for the MA (1) term (P-value = 0.000400) is less than statistical level =0.05, so it is significantly
different from zero. We reject null hypothesis and accept alternate hypothesis that moving average parameter is
statistically significant.
We estimated also model in form ARIMA (0,1,1) with constant, but it was not confirmed statistical significance
of this constant. It was removed from the model.
27
Residual ACF and PACF coefficients of estimated model ARIMA (0,1,1) without constant do not exceed
confidence bands, as we can see in the figures 2.11 an 2.12. It means the residuals are white noise drawings from
a fixed distribution with a constant mean and variance. Selected model is good model for the data because of the
residuals satisfy above mentioned assumptions.
Fig. 2.11 Residual ACF of ARIMA(0,1,1)
Fig. 2.12 Residual PACF of ARIMA(0,1,1)
Adequacy of selected model was validated by Ljung-Box statistic. The Ljung-Box test statistic is calculated as
re (k ) 2
2)
,
k
k 1 N
h
Qh
N (N
N is number of observations, h is length of coefficients to test autocorrelation, re (k ) is autocorrelation
coefficient. It usually recommends to set up h as number
with h-p-q degrees of freedom.
N [1]. Test statistic Qh has chi-square distribution
We applied the Ljung-Box test to the h=11 ( N = 108 11) residual autocorrelation coefficients of fitted
ARIMA model. Concretely, we tested null hypothesis that none of the autocorrelation coefficients up to lag h are
different from zero
H0: re(1) = re(2) = = re(11) = 0 against the alternate hypothesis that the residual autocorrelation coefficients are
not random H1: re(1) re(2) re(11) 0.
The calculated Ljung-Box statistic is Qh = 17.135. The critical value of chi-square distribution with 10 degrees of
freedom at the confidence level =0.05 is 18.307. The test statistic Qh is less than critical value 18.307 so we
accept null hypothesis that the residuals are random or white noise and reject alternative hypothesis.
The
is yt
estimated
yt
1
t
ARIMA
0, 464299
(0,1,1)
t 1.
model
is
appropriate
for
forecasting
and
its
form
We calculated prediction for next three month by using statistical software Statgraphics. There are illustrated
point and interval forecast in the figure 2.13.
Fig. 2.13 Point and interval forecast of ARIMA(0,1,1) model
The original time series with fitted values and point and interval forecast is illustrated in the figure 2.14.
28
Fig. 2.14 Original time series with fitted values and forecast
In our case, an autoregressive integrated moving average model ARIMA (0,1,1) without constant was estimated.
The data cover 108 time periods. We divided the time series into two parts. Model was estimated from the first
54 data values and 54 data values at the end of the time series were withheld to validate the model. The Root
Mean Square Error for validation periods was calculated and its value is 0,590863. RMSE is measure of the
differences between values predicted by a model and the values actually observed from the thing being modeled.
A better model will give a smaller value of RMSE. We compared calculated RMSE with the data value that are
from the interval (95 - 105) and can conclude that prediction error is accepted for this model and does not show a
high value.
3. CONCLUSION
We have discussed modeling and forecasting of monthly export price indexes of Czech republic by Box-Jenkins
methodology in this article. The optimal model ARIMA (0,1,1) without constant was chosen by the aid of
statistical software Statgraphics Plus 5.1. Next, we tested and confirmed statistical significance of moving
average parameter and randomness of model residuals. On the basis of RMSE calculated for validation period
we can conclude the ARIMA model is appropriate for modeling and forecasting of export price indexes of Czech
republic.
ACKNOWLEDGEMENTS
This work was supported by Slovak grant foundation under the grant VEGA No. 1/2628/05.
REFERENCES
[1]
CIPRA, T. Analýza asových ad s aplikacemi v ekonomii. Praha : SNTL/Alfa, 1986.
[2]
MAR EK, D.; MAR EK, M. Analýza, modelovanie a prognózovanie asových radov s aplikáciami
v praxi. ilina : ES U, 2001.
[3]
http://www.czso.cz/csu/redakce.nsf/i/izc_cr, 2. 1. 2008.
ADDRESS
Ing. Zuzana Me iarová
ilinská univerzita v iline
Univerzitná 8215/1
010 26 ilina
tel.: +421 (0) 41/51 34 430
29
30
MODERN MATHEMATICAL METHODS AND THEIR USE IN ECONOMY AND
FINANCE
Dostál Petr, Old ich Kratochvíl, Karel Rais
Abstract: New mathematical methods have begun to be used in the branch of economy and finance.
The methods such as fuzzy logic, neural network and genetic algorithm rank among them. The
article shortly describes these methods and represents their possible applications. These methods
can contribute to higher quality and more objective decision making that can increase the profit,
reduce costs but also total satisfaction of customers, staff, owners of firms and stock holders.
Keywords: mathematical methods, economy and finance, fuzzy logic, neural networks, genetic
algorithm
1. INTRODUCTION
The new mathematical methods that include the findings from the theory of fuzzy logic, neural networks and
genetic algorithm are dynamically developing. It is possible to find this knowledge even in the field of economy
and finance. The following can be mentioned: the choice of the best loan or credit, the choice of bank, insurance
company or fund, search for the best supplier or investor, the prediction of time series, the evaluation of audit
and financial report, the optimization of traffic problems, the optimization of investment, the optimization of
pick-up and delivery etc. It is possible to contribute to higher quality and more objective decision making by the
use of elements of modern mathematical methods that can lead to the increase of profit, decrease of costs and
also to increase of the total satisfaction of customers, employees, owners of companies and stock holders.
2. SOME MODERN MATHEMATICAL METHODS
Among significant modern mathematical methods it is possible to rank fuzzy logic, neural networks and genetic
algorithms. Further it is suitable to mention the parallel genetic, swarm particles, ant colonies, hill climbing
algorithms, tabu search, simulated annealing, methods based on artificial immune systems, multi-agent systems
that use agents or holons etc. A lot of methods were used or they are successfully applied in technical systems.
Their applications start also in the areas of economic and financial field.
2.1 FUZZY LOGIC
Fuzzy logic uses the fuzzy sets that measure the certainty and uncertainty of the pertinence of element to the set.
Likewise people make a decision during the mental and physical activities that is not easy to describe by
algorithm. It determines "how much" the element belongs to the set or not. For example, the risk of investment
can be described instead of numbers by scale: very high risk, high risk, medium risk, low risk, very low risk, no
risk. In such a way it is possible to describe the inputs (fuzzification), further it transfers them by means of fuzzy
logic (fuzzy inference) and the results convert to real values (defuzzification), for example very profitable,
profitable, non profitable investment etc.
2.2 NEURAL NETWORKS
Neural network rises from the findings of biology and it uses the principles which are controlled by human brain.
In a simplified version the biological neuron consists of more inputs (dendrites), body cell and one output (axon).
The inputs are processed by neuron and its output information is spread by axon to their endings (synapses).
These synapses influence the dendrites of other neurons. The activity of the human brain is enabled by means of
huge quantity of connections of neurons that create the life of man and include the process of learning. By means
of simplified model of activities of the human brain it is possible to set up a neural network model. Such a builtup computer program allows us to solve various problems that we find in economy and finance. Similarly a man
defines the process of learning, testing and realization.
31
2.3 GENETIC ALGORITHM
Genetic algorithm describes the development of population of animal kind. The surviving selected individuals
and the offspring of parents of other chosen individuals create further generation, while the old generation dies.
Such built-up program for computer allows us to do optimization of different problems which we can meet in
economy and finance. In the same way as the man we meet the process of selection, crossover and mutation (this
process is repeated continuously and creates one generation). Similarly work the build-up programs that do the
optimization of economic or financial problem.
3. POSSIBLE FIELDS OF THE USE OF MODERN MATHEMATICAL METHODS
3.1. DATA MINING
Data mining is a branch that covers a wide range of techniques that are used for obtaining information from great
amount of economic and financial data. It is used in various branches such as insurance companies, direct
mailing, retail, wholesale etc. For example one of the aims of companies is to decrease the risk of losses. For this
purpose we use the data mining that gets data and evaluates them. It is possible to search on these bases not only
the new customers but also those customers that could be lost or to reveal risk customers. For these purposes the
data mining is used to solve these problems by means of modern mathematical methods. The following figure
presents the realization of cluster analysis with the use of data obtained from the process of data mining. The
cluster S1 presents the group of people with low and medium income whose savings are low and medium, the
cluster S2 presents the groups of people with low and medium income and medium and high savings, the cluster
S3 presents the groups of people with high income and low, medium and high savings.
1,0
S2
0,8
0,6
S3
0,4
0,2
S1
0,0
0,0
0,2
0,4
0,6
0,8
1,0
Incom e
Figure 1: Data mining
3.2. PREDICTION
It is very important to know the future behaviour of various variables in many branches of human activities. A
lot of various methodologies are developed. Some of them are based on the principle of algorithms, other
methods are heuristic and the next uses the abilities of learning. During previous years a lot of different and very
powerful prediction methods have been developed and practically used.
Business and management processes can be compared to the most complicated ones because the society creates
the phenomena with considerable rate of chaotic behaviour. In these cases it is suitable to use the modern
mathematical methods. The following figure presents the time series with past and predicted values.
32
1665
1655
Prediction
1645
1635
1625
1
11
21
31
41
51
Order
Figure 2: Prediction
The largest use in business and management have the future development of financial and economic indicators,
prices of shares, commodities, values of indexes, currency ratio etc.
3.3 STOCK MARKET
The prices of shares, commodities, currency ratios and values of indexes on the stock market create the time
series. The time series are presented by the close, open, maximum, minimum prices or indexes, eventually by
volume of trade. As the courses of time series are influenced by complex economic and psychological
phenomena that contain high rate of chaos, the use of modern mathematical methods belongs to the best what at
present exists for processing and evaluation of information and data from this area. In this area a lot of analyses
and technical, fundamental and psychological methods were designed whose aim is to determine the sell and buy
signals to get a maximum profit. As well it is suitable to set up a portfolio and its percentage composition of
single items in the portfolio to be optimum as it is presented on a following figure.
40
30
20
10
0
1
2
3
4
5
6
7
Order
Figure 3: Stock market
3.4 RISK MANAGEMENT
Risk management is a branch that covers a wide range of methods used in many branches. The risk is connected
with the hope of achievement of the best economic results, that is the decrease of danger of failures or losses that
can affect the stability of the firm or to evoke the bankruptcy. This branch has become inevitable to keep the
competitiveness. This branch penetrates to many fields such as banking, insurance, direct mailing etc. The aim of
the firm is to achieve the best economic results that means to decrease failures and/or loses. The risk rises from
the reasons of lack of information and insufficient understanding of phenomena, the use of unsuitable and
unreliable data, and the use of unsuitable methods or by the influence of random processes. The following figure
presents the realization of cluster analysis in risk management. The cluster S1 presents the group of people with
low income and high risk of payment, the cluster S2 and S3 presents the groups with medium income and low
risk of payment and the cluster S4 presents the groups with high income and low risk of payment.
33
1,0
S1
0,8
0,6
0,4
S2
S3
0,2
S4
0,0
0,0
0,2
0,4
Income
0,6
0,8
1,0
Figure 4: Risk management
4. CASE STUDY
4.1 FUZZY LOGIC
The application of fuzzy logic can be demonstrated in the field of data mining on the case of direct mailing, it
means whether to visit the client personally, to send him a letter or not to speak to him. The solving of this case
on the computer requires enter input variables: Salary (low, medium, high), Loan (none, small, medium, high),
Children (no, a few, many), State (single, married, other), Age (young, medium, old, very old), Place (big city,
city, village). As a membership function we can use the functions in the shape of , , Z, S. The logical rules
are in the form <When> <Then> and it is necessary to set up on the basis of experience. The output variables is
Marketing with the attributes whether the customer will be visited personally, a letter will be sent to him or we
will not be interested in him.
The model for prediction can be presented by several inputs that present the trends of sections of time series that
vary but they are going in sequences. The inputs with five attributes were set up in this case when it was
considered the sign and the size of difference of neighbouring values (high positive, positive, zero, negative,
high negative difference. As a membership function we can use the functions in the shape of , , Z, S. It was
necessary to set up the logical rules in the form <When> <Then>. The block of rules must be set up on the basis
of experience by experts who are solving this problem. The output variable has five attributes that evaluate the
future course of time series (high increase, increase, stagnation, decrease, high decrease).
The model for stock market solves the problem whether to realize the trade on the stock or not. The inputs and
their attributes are as follows: Prediction of the trend of time series (high increase, increase, stagnation, decrease,
high decrease), Margin (insignificant, significant), Interest rate (low, medium, high), Strength of market (low,
medium, high) and Behaviour of time series (deterministic, random). As a membership function we can use
membership functions in the shape of , , Z, S. The logical rules are created by operators <When> <Then>.
The rules must be set up on the basis of experience, the best way is to do it by the experts that are solving this
problem. The output is represented by the variable Trade with two attributes: if to trade or not with the share,
index of bonds on terminal markets, commodities or currencies.
The application of fuzzy logic in risk management can be demonstrated on the case of evaluation of rate of risk
of payment of active debt. The application is solved with eleven input variables, three rule blocks and one output
variable with three attributes. The inputs and their attributes are as follows: Sex (man, woman), Age (young,
middle, old), Marital status (married, single, other), Children (none, one, more), Income (low, medium, high),
Account (none, medium, high), Debt (none, medium, high), Employment (short, medium, long term), Contact
with client (short, medium, long term), Order (first, few, more), Delayed payment (none, few, more). It presents
eleven inputs where from two to three attributes are selected according to the demand of realization of project.
The output from the rule box Personal data evaluates the personality of the client (excellent, good, bad), the rule
box Financial data evaluates the financial situation of client (excellent, good, bad), the rule box Quality of a
client evaluates the client from the point of view of the relation consumer - supplier (excellent, good, bad). The
output variables is the Risk of payment of active debt with three attributes (low, medium, high). It is necessary to
set up the membership function for all inputs and outputs. We can use the functions in the shape of , , Z, S.
The rule box must be set up with rules and their weight (DoS = Degree of Support) among inputs and outputs.
The weight of rules can be changed during the process of optimization. The build up model can be used for the
evaluation of the rate of risk of payment of active debt. On the basis of input values we obtain the information
whether the risk of payment of active debt is low, medium or high. The course of membership function and the
weight of rules DoS can be set up by means of neural network in case we have data at disposal.
34
4.2 NEURAL NETWORK
The application of neural network in the field of data mining can be demonstrated on the case of evaluation of
rate of risk of the client that can be evaluated by the scale from 0 to 100%. The input of neural network is a
matrix of values whose lines represent single customers and columns presents their characteristics: Sex, Age,
Marital status, Children, Income, Account, Debt, Employment, Contact with client, Order, Delayed payment.
The verbal notions of some inputs must be replaced by numbers, e.g. Sex (man = 1, woman = 0) and Marital
status (free = 0, married = 1, other = 2). The output is the risk of customer in the range from 0 to 100 per cent
where 0 (100) means that the customer is not (is) a risk client. It is necessary to determine the topology of neural
network, to choose the inputs and output of variables, the type of transfer function, the number of layers of
neural network. Further the data that will be used for learning and which for testing. The neural network can be
used for the estimation of risk of customer of payment after the successful process of learning and testing.
The input of neural network for prediction is a matrix of values whose lines represent the single time intervals
and the columns present the values of single time series, for example the values used on the stock market are
presented by open, close, minimum, maximum price and volume of trades. It is necessary to determine the
topology of neural network, to choose the input and output variables, the type of transfer function, the number of
layers of neural network. Further the data that will be used for learning and for testing. After the successful
process of learning and testing it is possible to use the learned neural network for prediction of future values.
The model of neural network in the field of stock market is used for the prediction of future values of time series.
The input of neural network is a matrix of values whose lines represent the single time intervals and the columns
presents the values of single time series created by open, close, minimum, maximum price and volume of trades.
It is possible to choose further various time series that have relation to predicted variable e.g. index related to
share etc. Further we set up the topology of neural network, choose the inputs and output variables, the type of
transfer function, the number of layers of neural network. Further the data that will be used for learning and for
testing. After the successful process of learning and testing it is possible to use the learned neural network for
prediction of future value of share.
The application of neural network in risk management can be demonstrated on the case of evaluation of the risk
of payment of active debt. The input of neural network is a matrix of values which characterize single
parameters. The output and inputs variables are Risk (from 0 to 100%), Age (years), Sex (man = 1, woman = 0),
Marital status (free = 0, married = 1, other = 2), Children (number), Income (USD), Account (USD), Debt USD),
Employment (years), Contact with client (months), Order (number), Delayed payment (number). The single rows
present single cases of the clients. If we do not know and we want to know the risk of payment of active debts
we mark it with symbol ?. If we determine the input matrix we set up the inputs and outputs, the type of transfer
function, the number of layers of neural network, the range of data for learning and testing. During the process of
testing and learning it is possible to trace the error of testing and learning. If the error is small the process is
terminated. The result is the suggested value of risk of payment of debt by client for new searched case.
4.3 Genetic algorithm
The use of genetic algorithms can be presented in the area of data mining on the case that detects mutual
dependencies of more variables. It is possible to use so called cluster analyses. It consists in the fact of division
of data into clusters. The inputs can be any number of values and it is possible to choose any number of areas of
division. The mentioned case can be multidimensional. In our case the input data include information about
single persons. Sex (man = 1, woman = 0) and Marital status (free = 0, married = 1, other = 2), age, Place (big
city =1, town = 2, village = 3, settlement = 4), Number of children, Height of income, Height of account, Height
of debt. The purposes are to find the dependencies among single input variables. The program needs to be set up
the number of clusters, set up the values and determination of parameters for calculation of genetics algorithm.
During the process of calculation the program set up randomly the centers of clusters and it assigns the points to
the nearest centre. The fitness function expresses the sum of distances among points and centers of
corresponding clusters. This function is optimized. The created clusters determine us the dependences that can be
used for further analyses, for example the dependence of height of savings on the height of income etc.
The model for prediction is based on the principle when the rule is set up that is optimized by genetic algorithm
on the maximum profit. For the description of this rule are used mainly the logical operators: <When> <Then>,
<And>, <Or>. In this way optimized rule is used for the determination of future trend of time series rise,
decrease or stagnate. The concrete optimized rule can be in a form <When> (close price in time t1) > (minimum
price in time t2) <And> (close price in time t3) < (close price in time t4) <Then> the future price will rise.
35
The genetic algorithms can be used on the stock market for optimization of portfolio. The method of the use is
based on the task of set up of portfolio from other bonds that are not included in the index or from the smaller
number of bonds from which the index is build up. If we are persuaded about growth of index, the portfolio is
optimized so that the growth of prices of shares included in portfolio will be in ratio the same. The number of
shares can be arbitrary. Because of the process of optimization it is necessary to define the fitness function. In
this case the fitness function is represented by the sum of square deviation of standardized values of index and
standardized portfolio created by chosen shares in this case. The fitness function presents the error that is
minimized during the process of optimization. The result of calculation is the values of percentage representation
of shares in portfolio.
The application of genetic algorithms in the field of risk management can be presented in the area of risk of
payment of active debt. It is possible to use the genetic algorithms in the process called cluster analyses. It is the
way when we try to divide the data into clusters and find their centers. The inputs can be any number of
variables and many numbers of values and it is possible to choose any number of areas of division. The
mentioned case is multidimensional. It is presented by twelve vectors that expresses dependences of risk (from 0
to 1) on the input variables: Sex (man = 1, woman = 0), Age (years), Marital status (free = 0, married = 1, other
= 2), %), Age (years), Children (number), Income (USD), Account (USD), Debt USD), Employment (years),
Contact with client (months), Order (number), Delayed payment (number). It is possible to run the calculation
after the determination of number of clusters, set up of table and parameters of calculation.
During the process of calculation the program set up randomly the centers of clusters and it assigns the points to
the nearest centers. The whole process is repeated as long as the position of centers fulfils the condition of
optimum of fitness function. The fitness function presents the sum of distances among points and centers of
corresponding clusters. This function is optimized. After the calculation it is possible to obtain the clusters that
present for example customers with high income and low risk of payment of active debt etc.
5. CONCLUSIONS
The economic and financial processes can be ranked among the most complicated processes because the society
creates the phenomena with significant rate of chaotic behaviour. In cases when it is necessary to make a
decision it is appropriate to use the modern mathematic methods that give promising results. These methods to
which fuzzy logic, neural network and genetic algorithms belong can be used separately but also in their
combination. These methods have wide range of utilization.
LITERATURE
[1]
ALIEV, A.; ALIEV, R. Soft Computing and Its Applications. World Scientific Publishing Ltd, UK2002,
444p., ISBN 981-02-4700-1.
[2]
ALTROCK, C. Fuzzy Logic & Neurofuzzy Applications in Business & Finance. Prentice Hall, USA,
1996, 375p., ISBN 0-13-591512-0.
[3]
BOSE, K.; LIANG, P. Neural Network, Fundamental with Graphs, Algorithm and Applications. Mc
Graw-Hill, USA, 1996, 478p., ISBN 0-07-114064-6.
[4]
DAVIS, L. Handbook of Genetic Algorithms. Int. Thomson Com. Press, USA, 1991, 385p., ISBN 1-85032825-0.
[5]
DOSTÁL, P.; RAIS, K. Methods of Large Investment Unit Modeling, In Transformation of CEEC
Economies to EU Standards. University of Trento, Italy, 2001, pp. 84-89, ISBN 80-86510-27-1.
[6]
DOSTÁL, P. Moderní metody ekonomických analýz Finan ní kybernetika. Zlín : UTB Zlín, 2002,
110p., ISBN 80-7318-075-8.
[7]
DOSTÁL P.; RAIS, K. Risk Management and Artificial Neural Network, In Word of Information
Systems. Zlín : 2005, pp. 292-297, ISBN 80-7318-276-9, ISSN 1214-9489.
[8]
DOSTÁL, P. The Use of Fuzzy Logic at Support of Manager Decision Making, In Management,
Economics and Business Development in the New European Conditions. Brno : 2005, p.44, 5p., ISBN
80-214-2953-4.
[9]
DOSTÁL, P.; RAIS, K.; SOJKA, Z. Pokro ilé metody manaerského rozhodování. Grada, 2005, 168p.,
ISBN 80-247-1338-1.
[10] DOSTÁL, P. Risk Management and Fuzzy Logic, In Word of Information Systems. Zlín : 2006, pp. 115120, ISBN 80-7318-400-1, ISSN 1214-9489.
[11] DOSTÁL, P. Pokro ilé metody analýz a modelování v ekonomii. Brno : VUT Brno, 2006, 61p., ISBN 80214-3324-8.
[12] HAGAN, T.; DEMUTH, B. Neural Network Design. PWS Publishing Comp., USA, 1996, 702p., ISBN
0-534-94332-2.
36
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
KAZABOV, K.; KOZMA, R. Neuro - Fuzzy Techniques for Intelligent Information System. PhysicaVerlag, Germany, 1998, 427p., ISBN 3-7908-1187-4.
KLIR, G. J.; YUAN, B. Fuzzy Sets and Fuzzy Logic, Theory and Applications. Prentice Hall, New Jersey,
USA, 1995, 279p., ISBN 0-13-101171-5.
MA ÍK, V.; T PÁNKOVÁ, O.; LAANSKÝ, J. Um lá inteligence (1). ACADEMIA, 1993, 264p.,
ISBN 80-200-0496-3.
ISBN 80-200-0504-8.
ISBN 80-200-0472-6.
ISBN 80-200-1044-0.
NOVÁK, N. Um lé neuronové sít teorie a aplikace. Praha : C. H. BECK, 1998, 382p., ISBN 80-7179132-6.
NOVOTNÝ, O.; POUR, J.; SLÁNSKÝ D. Business Inteligence. Praha : GRADA, 2005, 254p., ISBN 80247-1094-3.
RIBEIRO, R.; YAGER, R. Soft Computing in Financial Engineering. A Springer Verlag Company, 1999,
590p., ISBN 3-7908-1173-4.
SMEJKAL V.; RAIS, K. ízení rizik ve firmách a jiných organizacích. Grada, 2006, 296p., ISBN 80247-1667-4.
TICHÝ, M. Ovládání rizika. C. H. BECK, 2006, 396p., ISBN 80-7179-415-5.
ADDRESS
Ass. Prof . Petr Dostál, MSc, Ph.D.
Brno University of Technology
Faculty of Business and Management,
Department of Informatics
Kolejní 4, 612 00 Brno
Private European Polytechnic Institute
Osvobození 699
686 04 Kunovice
Tel. +420 541 143714,
Fax. +420 541 142 692
[email protected]
[email protected]
www.iqnet.cz/dostal
Hon. Prof. Ing. Old ich Kratochvíl, Dr.h.c.
Private European Polytechnic Institute
Osvobození 699
686 04 Kunovice
[email protected]
Karel Rais, Prof., MSc, Ph.D., MBA
Faculty of Business and Management,
Department of Informatics
[email protected]
37
38
THE PREDICTION BY MEANS OF ARTIFICIAL NEURAL NETWORK
Radek Dosko il
Abstract: The article deals with the use of soft computing as a support of prediction. For this
purpose the artificial neural network is used. The brief description of artificial neural network and
the process of calculation are mentioned. The use is demonstrated on the problem of prediction of
sales. The scheme of models, selection of data for testing, learning and forecasting, process of
calculation is mentioned. The use of artificial neural network is the advantage especially at decision
making processes where the description by algorithms is very difficult and criteria are multiplied.
Keywords: Artificial neural network, prediction, sales, soft computing.
INTRODUCTION
A lot of manager problems originate as an effect ignorance of future position. Creating predictions is one of the
possibilities to make a better decision. This possibility is also used in the financial branch to create prediction of
shares, commodities, currencies-rates, etc. The best method used for prediction of the time series, nowadays is
the artificial neural network. The artificial neural network enables us to describe non-linear processes created by
the most complicated psychological and social phenomena. This article presents the prediction of sale by the use
of artificial neural network.
THE ARTIFICIAL NEURAL NETWORK
The artificial neural network model represents the thinking of the human brains. The model is marked as a
black box. It is not possible to know the inside structure of system in detail. The neural network is suitable to
use in the cases, where the influences on searched phenomena are random and deterministic relations are very
complicated.
The simplest artificial neuronal network is called perceptron. It is possible to present it as an input of R variables
p1, p2, p3, . , pR. These variables are multiplied by weight coefficients w1, w2, w3, , wR. The threshold
value b influences the output; it increases the value of sum just about this value. The formula is a = w1 * p1 + w2
* p2 + w3 * p3 + + wR *pR + b. The following figure shows the single-layer neural network.
The single-layer neural network
Further the formula is, that n = f (a), when we use the transfer functions. The most important transfer functions
are hardlim, purelin, logsig and tansig. The graphs of these functions are following.
39
Transfer functions hardlim, purelin, logsig, tansig
hardlim
n
0 for a
0
n
1 for a
0
,
purelin n
a,
logsig n
1
1 e
a
,
tansig n
ea
ea
e
e
a
a
,
The simplest transfer function is hardlim, when output value is equal 1 or 0, according to the state if the value is
less or equal and bigger than 0. The sense of the using of this function is the fact, that the transformation gives
reasonable values. The function sigmoid has the values in the interval from 0 to 1. The function logsig has the
values in the interval from -1 to 1. The output value could achieve high values in case of not using of such type
or similar transfer functions.
The perceptron solves only linear separable tasks. The process of calculation of artificial neural network starts
with the initialization and continues by the process of learning.
The complicated tasks are necessary to solve only with multi-layer networks. Then the formula n = f (w * p + b)
is in matrix form. Generally the artificial network has one input layer, one or more hidden and one output layer.
The scheme of multi-layer network with input, hidden and output layers is following.
The scheme of multi-layer network
EXAMPLE THE APLICATION OF ARTIFICIAL NEURAL NETWORK FOR PREDICTION
The example solves the prediction of sales. The solution of this task is presented by means
of commercially selling software NeuroForecaster of NIBS Ltd. At first it is necessary to prepare the input data
in the form of table (matrix). Matrix of input values is following.
40
Part of the input table sales
The searched time series has 287 values that represent daily sales of concrete hotel in the period from 1. 1. 2007
to 14. 10. 2007. The aim of this search is to build up a model for prediction of total sales.
At first the data were analyzed on tendency, periodical cycle and outliners. The data were clarified in the case
when the occasional firm actions took place. This fact makes the prediction very complicated. The clarified time
series was used for build up and tuning of neural network model for prediction.
When the program runs it is necessary to load the prepared table. Then the data must be marked, if they are
inputs or outputs. The input data are only the time und sale, output data is prediction. Further the parameters of
artificial neural network such as transfer function must be selected (the program NeuroForecaster contains 12
functions - Hyperbolic Tangent, Mixed Functions, Basic, Hyperbolic Tanh and Sine, Competitive, Radial Basic
Function, Fast Prop Hyperbolic tangent, Fast Prop Sigmoid, Fast Prop Linear, Fast Prop Radial Basis Function,
Neuro fuzzy) or Genetics. (The Genetics means that the genetic algorithms are used for pruning of network). See
the following figures.
41
Set up of inputs and output, load of data and set up of network model
The following figure shows the table of the selection of the data for testing, learning and forecasting.
The selection of the data for testing, learning and forecasting
The following figure shows the table of the process of learning, testing and calculation.
42
The process of learning, testing and calculation
The following figure shows the table of part of past values, future values and prediction error. Empty future
value and prediction error is unknown.
Part of table of sales with suggestion
The result of the process of calculation can be presented in the following figure. A part of past values are
presented on the left and predicted values on the right.
43
Total sales
30000
25000
20000
15000
10000
5000
0
1
date
The part of time series with prediction
CONCLUSION
The various types of neural network models were built up. One of the best ones was presented in this paper. The
value of MAPE (Mean Absolute Percentage Error) reached 71 %. The results is possible to evaluate: the
prediction describe the tendency very good, the accuracy of single predictions is not very good, because of some
rate of randomness of time series.
Generally we can say that the artificial neural network can help us in the field of prediction of tendency. When
we have the correct data and we tuned the artificial neural network very well, the results of calculation can help
us in the decision making process. The artificial neural networks cannot replace the traditional deterministic
models, but they can be used as a tool for the support of decision making process, when conventional methods
fail.
LITERATURE
[1]
DOSTÁL, P.; RAIS, K.; SOJKA, Z. Pokro ilé metody manaerského rozhodování. 1. vyd., Praha : Grada
Publishing, a.s., 2005, 168 s, ISBN 80-247-1338-1.
[2]
RAIS, K.; DOSTÁL, P.; DOSKO IL, R. Opera ní a systémová analýza II. Brno : Akademické
nakladatelství CERM, 2007, 153 s., ISBN: 978-80-214-3371-7.
[3]
ALIEV, A.; ALIEV, R. Soft Computing and Its Applications. World Scientific Pub. Ltd, UK2002, 444 p.,
ISBN 981-02-4700-1.
[4]
ALTROCK, C. Fuzzy Logic & Neurofuzzy Applications in Business & Finance. Prentice Hall, USA,
1996, 375p., ISBN 0-13-591512-0.
[5]
RAIS, K.; SMEJKAL, V. ízení rizik. Praha : Grada, 2003, 270p., ISBN 80-247-0198-7.
[6]
RIBEIRO, R.; YAGER, R. Soft Computing in Financial Engineering. A Springer Verlag Copany, 1999,
590p., ISBN 3-7908-1173-4.
ADDRESS:
Radek Dosko il, MSc.
Faculty of Business and Management, Department of Informatics
Tel. +420 541 143 722, Fax. +420 541 142 692
[email protected]
www.fbm.vutbr.cz
44
OPTIMALIZOVANÉ STAVEBNÍ BLOKY PRO NÁVRH MODERNÍCH INTEGROVANÝCH
OBVOD
Jaromír Brzobohatý, Roman Prokop, Vladislav Musil
Vysoké u ení technické v Brn
Abstrakt: Byly navreny a realizovány speciální stavební bloky pro snadný blokový návrh nových
integrovaných obvod pracujících v proudovém módu. Spojováním t chto blok lze snadno
navrhnout a realizovat r zná funk ní zapojení a p edevím moderní proudové aktivní funk ní bloky
CDTA [2] a CCTA [4] odpovídající irokým aplika ním poadavk m. Cílem je navrhnout knihovnu
t chto základních stavebních blok , jako jsou nap íklad proudové konvejory, transkonduktan ní a
transimpedan ní zesilova e, proudové zesilova e, v technologii CMOS.
Klí ová slova: Optimalizace IO, stavební bloky IO, zpracování analogového signálu v proudovém
módu, proudový konvejor
1. ÚVOD
Stále rostoucí stupe integrace a monost navrhovat a pouívat nízkop íkonové obvody vede k velké popularit
SoC (System on Chip) obvod , kde je tém celý systém realizován na jednom ipu. P eván se jedná o
smíené analogov -digitální zákaznické obvody (ASICs). Zmenování minimálních rozm r polovodi ových
prvk vede k dominantnímu uití digitálního zpracování signálu. To je zp sobeno mnohými výhodami
digitálních obvod v porovnání s analogovými, nap íklad v tí odolnost proti ruení, umu a vlivu
technologického procesu.
S ohledem na tyto skute nosti se hlavní úsilí technologického výzkumu zam ilo na digitální zpracování signálu
a proto je v tina moderních technologií optimalizována pro digitální obvody. Oproti tomu stále existují d vody
pro jsou analogové obvody stále nezbytn vyadovány. Hlavním d vodem je to, e skute ný sv t kolem nás je
analogový. Tém vechny signály jsou na po átku analogové a vyadují tudí analogové p edzpracování p ed
hlavním digitálním procesem. To znamená, e p esnost kompletního zpracování signálu je dána p edevím
kvalitou analogových obvod .
Pro SoC musí analogové i digitální obvody pracovat ve stejné technologii (p eván CMOS). Z toho plyne, e od
moderních analogových prvk poadujeme schopnost pracovat v digitálních nízkonap ových a
nízkop íkonových technologiích. Pro spln ní vysokých nárok na analogové obvody byly vyvinuty nové
p ístupy a obvodové principy, jako nap íklad spínané kapacitory, spínané proudy nebo obvody v proudovém
módu. Návrh posledn jmenovaných aktivních obvod pomocí modulárních stavebních blok je uveden
v následujících kapitolách.
2. MODERNÍ PROUDOVÉ AKTIVNÍ PRVKY
K moderním aktivním prvk m pracujícím v proudovém módu adíme p edevím takové sou ástky jako COA
(Proudový opera ní zesilova ) [1], CDTA (Current Differencing Transconductance amplifier) [2] a CCTA
(Current Conveyor Transconductance amplifier) [4,5]. Tato práce je zam ena p edevím na návrh obvod
CDTA a CCTA, nebo COA m e být snadno realizován vhodným zapojením t chto univerzáln jích obvod .
Výe uvedené sou ástky jsou realizovány p eván jako 2-stup ové obvody. To umo uje navrhnout tyto
stupn jako nezávislé moduly navrené s ohledem na snadné vzájemné propojení. Díky tomuto p ístupu mohou
být potom CDTA a CCTA (a mnoho dalích moných obvod ) navreny jednodue jako kombinace t chto
základních blok s ohledem na aplikací poadované vlastnosti (impedance, výkon, rychlost atd.)
2.1 STRU NÝ POPIS CDTA
CDTA má 2 rozdílové proudové vstupy p a n. Rozdíl vstupních proud t chto svorek te e do uzlu Z
(p ípadn vyvedené svorky Z zatíené externí impedancí). Nap tí na svorce Z je konvertováno pomocí
výstupního transkonduktan ního stupn gm na proud, který je vyveden na vysokoimpedan ní výstupní
komplementární svorky. Transkonduktance m e být bu pevná nebo nastavitelná elektronicky i externí
admitancí.
45
2.2 STRU NÝ POPIS CCTA
CCTA byl navren p eván pro pouití v proudovém módu, ale je také výbornou volbou pro realizaci aplikací
pracujících ve smíeném (nap ov -proudovém) módu. CCTA se skládá ze 2 základních blok . První z nich je
reprezentován proudovým konvejorem CCIII, následovaným transkonduktan ním stupn m se dv ma
proudovými výstupy. Vstupní chování prvku je dáno p eván vlastnostmi konvejoru CCIII. Proudový výstup
konvejoru je p ipojen do svorky Z a dále zpracován podobn jako u p edcházejícího obvodu.
Jak ji bylo uvedeno v [4], CDTA obvod m e být jednodue realizován spojením proudového rozdílového
zesilova e se ziskem B=1 a transkonduktan ního stupn stejn tak jako je obvod CCTA vybudován
z proudového konvejoru CCIII a výstupního transkonduktan ního stupn gm. Náplní tohoto lánku je popis a
srovnání r zných topologií vhodných k realizaci t chto základních stavebních blok .
3. PRINCIPIÁLNÍ ZAPOJENÍ VYBRANÝCH STAVEBNÍCH BLOK
Jako p íklady vstupních stup jsou na obr.1 uvedeny r zné topologie proudového konvejoru CCIII a na obr.2
proudového rozdílového (current differencing stage) stupn . Ve snaze vyhov t rozli ným nárok m na výstupní
stupe je také vhodné navrhnout r zné typy transkonduktan ních stup . N které z nich jsou uvedeny na obr.3.
Obr. 1: a) CCIII zaloený na proudových zrcadlech
Obr. 2: Proudový rozdílový stupe
b) CCIII zaloený na opera ním zesilova i
a) na principu proudových zrcadel
b) na principu opera ního zesilova e
VDD
Ibias
M3
M4
z_in
out+
M1
outM2
Ibias
VSS
Obr.3: a) Transkonduktan ní stupe
s pevným parametrem gm
b) Transkonduktan ní stupe s gm nastavitelným
pomocí externího Ym.
4. REALIZACE UVEDENÝCH ZÁKLADNÍCH STAVEBNÍCH BLOK
V této kapitole jsou krátce p edstaveny jednotlivé bloky v etn obvodové topologie, realizovaného schématu a
m ených výsledk . Minimální poadavky pro zpracování signálu jsou maximální proud Imax=±100uA a nap tí
Vmax=±0.5V. P ehled dosaených parametr a srovnání jednotlivých bun k jsou uvedeny v Záv ru.
46
4.1 PROUDOVÝ ROZDÍLOVÝ ZESILOVA ZALOENÝ NA PRINCIPU PROUDOVÝCH ZRCADEL
Omezení této topologie je p eván v amplitud zpracovávaného proudu kv li kaskodovému zapojení
proudových zrcadel, které je nezbytné z d vodu vysoké výstupní impedance a p esnosti obvodu. Výhodou
tohoto obvodu je vysoká rychlost, nevýhodou potom vyí vstupní impedance. Realizované schéma je uvedeno
na obr. 4a. Zm ené charakteristiky jsou na obr. 4b a obr. 4c.
a)
150
100
I(z)=f(I(pin))
I(z)=f(I(nin))
50
0
-150
-100
-50
0
50
100
150
-50
-100
-150
I(IN) [uA]
b)
25
20
15
I(z)=f(V(z)
10
5
0
-2,5
-5
-2
-1,5
-1
-0,5
0
0,5
1
1,5
2
-10
-15
-20
V(z) [V]
c)
Obr. 4: Proudový rozdílový zesilova zaloený na zrcadlech proudu
a) realizované schema
b) graf výstupního proudu v závislosti na vstupním proudu
c) graf výstupního proudu v závislosti na výstupním nap tí (výstupní impedance)
ZALOENÝ NA PRINCIPU
4.2 PROUDOVÝ ROZDÍLOVÝ ZESILOVA
ZESILOVA E
Výhodou této topologie je nízká vstupní impedance, avak za cenu niího GBW.
OPERA NÍHO
47
a)
I(z)=f(I(in))
150
100
I(z)=f(I(pin))
I(z)=f(I(nin))
50
0
-150
-100
-50
0
50
100
150
-50
-100
-150
I(in) [uA]
b)
10
5
I(z)=f(V(z))
0
-2,5
-2
-1,5
-1
-0,5
0
0,5
1
1,5
2
-5
-10
-15
V(z) [V]
c)
Obr. 5: Proudový rozdílový zesilova zaloený na struktu e opera ního zesilova e
a) realizované schéma,
b) graf I(out)=f(I(pin),I(nin)),
c) graf I(out)=f(V(out)) -> Zout
4.3 PROUDOVÝ KONVEJOR CCIII VYBUDOVANÝ NA TOPOLOGII PROUDOVÝCH ZRCADEL
Topologie vstupní ásti tohoto obvodu je podobná jako u proudového zesilova e z kap. 4.1 a z tohoto d vodu má
i podobná omezení, avak mnohem ván jí limitací je pom rn velké omezení vstupního nap tí na uzlu Y
vzhledem k prahovému nap tí MOS tranzistoru, zejména pro návrh nízkonap ových obvod . Pro napájecí
nap tí 5V se poda ilo splnit poadovanou amplitudu vstupního signálu V(Y)=±0.5V.
a)
48
150
100
I(z) = f (I(x))
I(y) = f (I(x))
50
0
-150
-100
-50
0
50
100
150
-50
-100
-150
I(x) [uA]
b)
4
3
2
I(z) = f(V(z))
1
0
-2
-1,5
-1
-0,5
0
0,5
1
1,5
-1
-2
-3
V(z) [V]
c)
Obr. 6: Proudový konvejor CCIII zaloený na principu proudových zrcadel
b) graf I(z,y)=f(I(x))
c) graf I(z)=f(V(z)) -> Zout(z)
4.4 PROUDOVÝ KONVEJOR CCIII ZALOENÝ NA STRUKTU E OPERA NÍHO ZESILOVA E
a)
4000
3000
2000
I(z) = f(I(x))
I(y) = f(I(x))
1000
0
-4000
-3000
-2000
-1000
0
1000
2000
3000
4000
-1000
-2000
-3000
-4000
I(x) [uA]
b)
49
30
20
I(z) = f(V(z))
10
0
-3
-2
-1
0
1
2
3
-10
-20
-30
-40
V(z) [V]
c)
Obr. 7: Proudový konvejor CCIII zaloený na opera ním zesilova i
b) graf I(z,y)=f(I(x))
c) graf I(z)=f(V(z)) -> Zout(z)
Realizovaný konvejor CCIII byl postaven na základ robustního Rail-to-Rail opera ního zesilova e s výstupem
ve t íd AB. Díky této topologii je jeho vstup schopen pracovat v celém rozsahu napájecího nap tí a obvod m e
zpracovávat proudy do ±3.5mA zatímco klidová spot eba obvodu bez signálu je okolo 1mA. Tyto výhody jsou
zaplaceny niím mezním kmito tem.
4.5 TRANSKONDUKTAN NÍ STUPE GM
Realizovaný transkonduktan ní stupe se dv ma opa nými proudovými výstupy má pevný (ne ízený) parametr
gm. Ten je pro tento obvod zhruba 1,24mA/V.
a)
500
400
300
I(iop) = f(V(z))
I(ion) = f(V(z))
200
100
0
-800
-100
-600
-400
-200
0
200
400
600
800
-200
-300
-400
-500
V(z) [mV]
b)
50
20
15
10
I(iop) = f(V(iop))
5
0
-2,5
-5
-2
-1,5
-1
-0,5
0
0,5
1
1,5
2
2,5
-10
-15
-20
-25
-30
-35
V(iop) [V]
c)
Obr. 8: Transkonduktan ní stupe s pevným gm
b) p enosová charakterisatika I(iop),(ion) = f(V(in)) -> gm
c) výstupní charakteristika I(iop) = f(V(iop)) -> Zout
5. ZÁV R
Pomocí uvedeného principu lze navrhnout velké mnoství dalích stavebních blok . Jejich kombinací lze získat
obrovské mnoství r zných aktivních analogových obvod pracujících jak v klasickém nap ovém i proudovém
módu. Na realizovaném testovacím ipu byly navreny 4 nové obvody sestavené z výe uvedených stavebních
blok pomocí kaskádního zapojení 4 rozdílných vstupních blok a výstupního transkonduktan ního stupn .
Tímto zp sobem byly postaveny 2 r zné obvody CDTA a 2 obvody CCTA. Bohuel byl prozatím vyroben
pouze transkonduktan ní stupe s pevným parametrem gm. Výstupní stupe s gm nastavitelným pomocí externí
admitance lze snadno získat vhodným zapojením dalího bloku CCIII.
Tabulka 1: M ené hodnoty stejnosm rných parametr ; Napájecí nap tí Vdd = 5V
Rozsah Vin [V]
Rozsah Vout [V] Iin (max) [µA]
Vofset [mV]
Iofset [µA]
Diff I_mirr
N/A
-1,2 ÷ 0,8
-150 ÷ +150
5
1,4
Diff I_opa
N/A
-0,8 ÷ 0,7
-200 ÷ +200
7
0,6
CCIII_mirr
-0,7 ÷ 0,6
-1,1 ÷ 0,6
-150 ÷ +150
2,4
0,8
CCIII_opa
VSS ÷ VDD
-2 ÷ +2
-3600 ÷ +3600
0,36
Gm stage
-0,2 ÷ +0,2
-1 ÷ +1
N/A
N/A
1,6
P(out)=-2,1;
N(out)=1,1
Tabulka 2: M ené hodnoty st ídavých parametr ; Napájecí nap tí Vdd = 5V
Zin (proudové vstupy)
Zout
Zisk (I)
BW(-3dB)/GBW
Diff I_mirr
380
3,4 M
B0 = 1
BW = 180MHz
Diff I_opa
20
2,1 M
B0 = 1
BW = 100 MHz
CCIII_mirr
240
2.5 M
B0 = 1
BW = 100MHz
CCIII_opa
10
5.2 M
B0 = 1
BW = 50 MHz
2.4 M
gm = 1.24 mA/V
BW = 220 MHz
2.4 M
B0 = 4200
GBW = 35 MHz
Gm stage
CDTA_mirr
380
CCTA_mirr
240
2.4 M
B0 = 3100
GBW = 28 MHz
Pozn: a) BW(-3dB) je m ena jako 3dB í ka pásma p enosu, kdy proudové výstupy jsou p ipojeny do uzlu
s nízkou impedancí.
b) Uvedený tranzitní kmito et GBW je uveden pro kompenzovaný (stabilní) obvod s minimální fázovou
rezervou PM = 60 stup .
51
Vechny navrené stavební bloky jsou schopny
zpracovávat proud do hodnoty 150 µA a nap tí o
amplitud ± 0,5V. P ehled stejnosm rných
parametr navrených obvod je uveden v tab.1
a jejich st ídavé chování je patrno z tab.2. Ofsety
uvedené v dané tabulce odpovídají m enému
obvodu ale ze simulací soub hu (matching
analysis) je patrno, e nap ový ofset m e být
a 8mV a proudový okolo 8 uA, oba pro 6
normálového rozloení ofsetu sou ástek.
Z d vodu pouitelnosti obvodu vn pouzdra ipu
bylo nezbytné p idat p ísluné ESD ochrany na
externí piny obvodu.Tyto ochrany, v etn
kontaktovacích vodi a pouzdra, p idávají do
obvodu nezanedbatelnou parazitní kapacitu.
Pokud daný obvod bude pouze ve vnit ní
struktu e ipu, jeho tranzitní kmito et se výrazn
zvýí.
REFERENCE
[1]
MUCHA, I. Towards a true current operational amplifier. London : Proc. of ISCAS, 1994.
[2]
BIOLEK, D.; GUBEK, T. New Circuit Elements for Current-Mode Signal Processing.
www.elektrorevue.cz.
[3]
KUMAR, U. Current Conveyors: A review of the State of the Art. IEEE Circuits and Systems Magazine,
Vol. 3, No. 1, pp. 10-13.
[4]
PROKOP, R.; MUSIL, V. Modular approach to design of modern circuit blocks for current signal
processing and new device CCTA. In Proc. IASTED Signal and image processing , Hawaii, Honolulu,
ISBN:0-88986-516-7, 2005, 494-499.
[5]
PROKOP, R.; MUSIL, V. CCTA a new modern circuit block and its internal realization, In Electronic
Devices and Systems 05-Proceedings. The 12th Electronic Devices and Systems Conference, Brno, 2005.
[6]
PROKOP, R.; MUSIL, V. New modern circuit block CCTA and some its applications. In The Fourteenth
International Scientific and Applied Science Conference - Electronics ET'2005, Book 5, Sofia: TU Sofia,
2005, s. 93 - 98, ISBN 954-438-521-5.
ADRESA
Prof. Ing. Jaromír Brzobohatý, CSc.
FEKT, Ústav mikroelektroniky
Údolní 53, 602 00 Brno
http: http://www.umel.feec.vutbr.cz
Telefon: +420 541 146 160
e-mail: [email protected],
Ing. Roman Prokop
Telefon: +420 541 146 160
e-mail: [email protected],
Prof. Ing. Vladislav Musil, CSc.
Telefon: +420 541 146 160
52
RELIABILITY AND SECURITY IN LARGE COMPUTER SYSTEMS
Wojciech Zamojski, Marek Bara ski, Katarzyna Michalska
Wroclaw University of Technology
Abstract. Today, many services relay on large computer systems, that are more and more open and
interconnected. Every day a world society is depended on ICT-based large computer systems,
operating in various fields such as traffic management, energy distribution, transport and defence.
This dependence increases the consequences of accidents, failures, attacks and will imply a higher
susceptibility vulnerability. This paper is proposition of the architecture that could be used to
improve reliability and security for this kind of systems proposing several levels of decompositions.
Keywords: reliability, computer systems, simulation, modeling, reconfiguration, security
INTRODUCTION
In last few years extraordinarily strong trend of rapid development of dedicated large computer systems, is
observed. Mostly these are Information Computer System ones, where growing number of users as well as
continuously rising needs and requirements regarding business services delivered by these systems, implies
increasing complexity and large scale of solutions being build. Unfortunately at the same time a matter of safety
and reliability [14] is extremely important. These problems bring necessity for introducing new methods and
tools for managing such structures.
This issue is presently addressed by several disciplines such as: modelling, simulation, incident detection, fault
monitoring, reconfiguration. Still most of the time these problems are treated independently. This paper proposes
a joint step forward for these advanced techniques to dramatically improve the information and communication
systems supporting the critical services such as: traffic management, energy distribution, transport, defence.
GENERIC OVERALL ARCHITECTURE
Proposed architecture is based on a multi-disciplinary method approach. Main idea of proposed framework is to
ensure reliability and security of large complex computer system, that relay on an information network by
providing solutions on the three domains (see Fig. 1):
Planning: Modelling, simulation, and utility tools with a suitable approach to plan optimal operational
configurations, detection and reactions scenarios through modelling and simulation of critical system and
their potential threats. They allow to define coherent and homogeneous operational mode and define the
efficient response to anticipated incident, the process to face unexpected ones and methods to restore
optimal usage of the system after switching to a degraded configuration.
Detection: Distributed, multi-technology sensors and a set of detection mechanisms to detect all kinds of
incidents that can occur in the system. They ensure fast detection of elementary incidents and in addition,
elaborate the detection of distributed incident from a combination of (apparently) unrelated events or from
an abnormal behaviour in the system.
Response: a framework for computer-aided and automated measures initiatives in order to respond in a
quick and appropriate way to a large range of incidents. These responses include the identification of the
scope of a given incident, the best approach to isolate the suspected devices to avoid propagation of
threats or a cascading effect.
The methods, tools and utilities suppose to provide notifications, provisions, self-learning and human-aided rules
optimization and share a common repository with topology, planned configurations, and rules for activities
precedence, etc.
53
Fig. 1 Architecture for improving reliability and security in large computer systems
In order to achieve major objectives of proposed architecture it was necessary to construct several building
blocks which implement functionalities that are required for a full and efficient fulfilment of the given mission.
Fig. 2 presents four crucial high-level functionalities (Monitoring, Detection, Reconfiguration and Planning) as
well as general workflow within the system.
Fig. 2 High-level functionalities and workflow
2.1 MONITORING
As mentioned before there is a serious need for collecting different information from a real system, which
underlies proposed infrastructure. Framework gathers all kinds of data that might be useful for building
infrastructure, services and security views. It helps to determine many features that influence system state. Any
incident affecting a systems safety attributes causes changes in the system state. It is important to keep in mind
that not all events should be considered as incidents. Collected events are piece of information about the system
that is deemed to contain material for further analysis.
Data collection in proposed Framework is mainly focused on known solutions (RMON/SNMP, CIM/WMI,
sniffing software, etc.) and upon the sensors technology (sensors are the lowest level event collectors added to
specified real time system).
54
2.2 DETECTION
Framework integrates multi-technology sensors and a set of various detection mechanisms to detect all kind of
incidents that may occur in the system. They ensure fast detection of elementary incidents and in addition,
elaborate the detection of distributed incidents from a combination of apparently unrelated events or from an
abnormal behaviour in the system. Since different threats/violations affect different system components at
different abstraction level, and moreover malicious events may be recognized at different levels. This means that
Framework also provides the ability to detect more and sophisticated threats, like distributed attacks (e.g.
DDoS). Therefore, collection of information on the overall system status, leads to some centralized decision
modules.
2.3 PLANNING
Framework uses this functionality in different situations to achieve diverse aims. First of all the responsibility of
this functionality is to provide solutions and tools for building models of a real infrastructure. These models
describe the system from the Framework perspective. This is extremely important since these models will drive
the whole Framework.
Planning capability which is strongly correlated with long term reaction has to supply Framework with system
modes and configurations. These descriptions have to be prepared and ready for applying when needed, in order
to improve efficiency of framework reaction to an incident. To fulfill this duty each mode of the system has to be
planned, and at the end stored in the repository.
2.4 RECONFIGURATION
This functionality is the issue of proposed architecture for increasing system reliability. The basic idea is that by
this functionality the Framework or operator/administrator that is able to utilize the available resources in the
most efficient way.
PROBLEMS
Proposed architecture shown on Fig. 1, has as many advantages as also many open issues. These are mostly
implementation (software) problems though. For example Planning module should be considered as a set of
many system description languages concerning topology of the system and service description languages (i.e.
WSDL [12], WS-CDL [6][11], WS-BPEL [10], WSFL [13], BPML [1]) that this system realize. That is very
difficult task, because of multiple languages that is used in todays technology. Almost every week we can
observe new definition of XML-based language to describe some aspect of a system. However none of them has
been stated as a standard yet. Also neither of them fulfil all requirements when we focus on particular system,
because the granularity of the model obtained may be insufficient to produce realistic simulations.
The same problems we have to handle for Analysis module. It has to be analysed what the service aim and what
simulation or emulation techniques it requires. In case of a growing number of simulators (OPnet [5], NS-2 [3],
QualNet [7], OMNet++[4], SSFNet [9]) is seems to be a nontrivial challenge [2].
To sum up all inconvenience during creation process of proposed framework must be overtaken. The
architecture must be succeeded at:
Designing, developing and validating tools for incident detection and decision support. The tools should
span different time scales and provide solutions for survivability that range from immediate reaction to
global and smooth reconfiguration through policy based management for an improved resilience.
Enhancing the self-healing properties of critical infrastructures by planning, designing and simulating
optimised architectures tested against several realistic scenarios.
Improving risk management, crisis management in critical infrastructures with the design of new models,
measures, and incident management tools as well as a thorough analysis of several situations. Devises,
characterizes, models and designs mechanisms to mitigate the cascading and escalading effects induced
by inter and intra dependencies
Developing decision support tools for critical infrastructures, validated by scenarios for several case
studies on infrastructures.
CONCLUSIONS
We first highlighted the necessity of a understanding of large critical infrastructures and of their security and
reliability issues. We proposed some architecture that can be used to solve some of referred problems mainly in a
55
matter of security. We present features and gaps of existing tools to achieve this objectives.
Future works are related to overcome talked about problems and later on to test proposed method on the various
testbeds, accurately speaking empirical large infrastructure tests.
REFERENCES
[1]
Business Process Modeling Language; http://xml.coverpages.org/BPML-2002.pdf
[2]
MASCAL, C. M.; NORTH, M. J. Tutorial on agent-based modeling and simulation. Winter Simulation
Conference, December 2005.
[3]
NS-2 Simulator Home Page: www.isi.edu.nsnam/nas/
[4]
OMNeT++ Simulator Home Page: www.omnetpp.org
[5]
OPNet Modeler Home Page: http://www.opnet.com/
[6]
ORTIZ, G.; HERNANDEZ, J. Preparing and Re-using Web services for choreography. International
Journal of Web Engineering and Technology 2006 - Vol. 2, No.4 pp. 307 - 334
[7]
QualNet Simulator Home Page: www.scaleble-networks.com
[8]
SOA Modelling Page: : http://www.innoq.com/
[9]
SSFNet Simulator Home Page: www.ssfnet.org
[10] Web Services Business Process Execution Language; http://docs.oasis-open.org/wsbpel/2.0/
wsbpel-specification-draft.html
[11] Web Services Choreography Description Language; http://www.w3.org/TR/ws-cdl-10/
[12] Web Services Description Language; http://www.w3.org/TR/wsdl20/
[13] Web Services Flow Language; www-3.ibm.com/software/solutions/webservices/pdf/WSFL.pdf
[14] Zamojski W.: Remarks on reliability of future computer systems, ICITNS 2003, Amman 2003.
ADDRESS
Wojciech Zamojski
Institute of Computer Engineering, Control and Robotics,
Wroclaw University of Technology,
ul. Janiszewskiego 11/17, 50-372 Wroclaw, Poland
wojciech.zamojski @pwr.wroc.pl
Marek Bara ski
marek.baranski @pwr.wroc.pl
Katarzyna Michalska
[email protected]
56
DISCRETE TRANSPORT SYSTEM - MODELING AND RELIABILITY ANALYSIS
Jacek Mazurkiewicz, Tomasz Walkowiak
Abstract. The paper describes a novel approach to analysis of discrete transport systems realized
using Scalable Simulation Framework (SSF). The proposed method is based on modeling and
simulating of the system behavior. Monte Carlo simulation is a tool for proper reliability and
functional parameters calculation. No restriction on the system structure and on a kind of
distribution is the main advantage of the method. The paper presents some exemplar system
modeling. The authors stress the problem of influence of the reliability parameters for final system
functional measures (required time of delivery). The proposal of the economic quality measure
related to the discrete transport system is also presented. The problem described in the paper is
practically essential for defining an organization of vehicle maintenance and transport system
logistics.
Keywords: discrete transport system, Monte-Carlo simulation, reliability, reliability analysis
1 INTRODUCTION
Decisions related to technical systems ought to be taken based on different and sometimes contradictory
conditions. The reliability maybe is not the most important factor but is of a great weight as a support criterion.
So a quantitative information related to the reliability characteristics is important and can be used as a decisionaided system if it is necessary to discuss different economic aspects. But reliability criteria calculation is not
trivial in general and specially in a case of modern transportation systems which often have a complex network
of connections. From the reliability point of view [1] the transport systems are characterized by a very complex
structure. The performance of the network can be impaired by various types of faults related to the transport
vehicles, communication infrastructure or even by traffic congestion [8]. This analysis can only be done if there
is a formal model of the transport logistics. The classical models used for reliability analysis are mainly based on
Markov or Semi-Markov processes [1] which are idealized and it is hard to reconcile them with practice. We
suggest the Monte Carlo simulation [4] for proper reliability and functional parameters calculation. No
restriction on the system structure and on a kind of distribution is the main advantage of the method [9]. We
propose to use the SSF (Scalable Simulation Framework) [2] instead of dedicated system elaboration. Our
previous works perfectly show that is very hard to prepare the simulator which includes all aspects of discrete
transport. The SSF is a base for SSFNet [3] a popular simulator of computer networks. We developed an
extension to SSF allowing to simulate transport systems. We propose the formal model of discrete transport
system to analyze functional aspects of complex systems. The presented in the next section discrete transport
system model is based on the Polish Post regional centre of mail distribution.
2 DISCRETE TRANSPORT SYSTEM WITH CENTRAL NODE AND TIME-TABLE (DTSCNTT)
The model can be described as follow:
DTSCN
CN , N , R, V , T , M , TT
(1)
where: CN - central node, N - set of ordinary nodes; R - set of routes; V - set of vehicles; T - set of tasks, M - set
of maintenance crews and TT vehicles time-table.
Commodities: We can discuss several kinds of a commodity transported in the system. Single kind commodity is
placed in a unified container, and containers are transported by vehicles. The commodities are addressed and
there are no other parameters describing them.
Nodes: We have single central node in the system. The central node is the destination of all commodities taken
from other ordinary nodes. The central node is also the global generator of commodities driven to the nodes of
the system. The generation of containers is described by Poisson process. In case of central node there are
separate process for each ordinary node. Whereas, for ordinary nodes there is one process, since commodities are
57
transported from ordinary nodes to the central node or in other direction. Ordinary nodes are described by
intensity of container generation (routed to central node) and central node is described be a table of intensities of
containers for each ordinary node. Moreover the length between each two nodes is given.
Vehicles: We assumed that all vehicles or of the same type and is described by following functional and
reliability parameters: mean speed of a journey, capacity number of containers which can be loaded, reliability
function and time of vehicle maintenance. Central node is the base place for vehicles. They start from the central
node and the central node is the destination of their travel. The temporary state of each vehicle is characterized
by following data: vehicle state, distance travelled from the begin of the route, capacity of the commodity. The
vehicle running to the end of the route is able to take different kinds of commodity (located in unified containers,
each container includes single-kind commodity). The vehicle hauling a commodity is always fully loaded or
taking the last part of the commodity if it is less than its capacity.
Routes: Each route describes possible trip of vehicles. The set of routes we can describe as series of nodes:
R
c, v1 ,..., v n , c
vi
N
and
c
CN
(2)
Maintenance Crews: Maintenance crews are identical and unrecognized. The crews are not combined to any
node, are not combined to any route, they operate in the whole system and are described only by the number of
them. The temporary state of maintenance crew is characterized by: number of crews which are not involved into
maintenance procedures and queue of vehicle waiting for the maintenance.
Time-Table: Vehicles operate according to the time-table exactly as city buses or intercity coaches. The timetable is prepared by discrete transport system owner. The number of used vehicles, or the capacity of vehicles
does not depend on temporary situation described by number of transportation tasks or by the task amount for
example. It means that it is possible to realize the journey by completely empty vehicle or the vehicle cannot
load the available amount of commodity (the vehicle is to small). Yes it is possible to use different time-tables
for different seasons or months of the year, but in general time-table is fixed element of the system in
observable time horizon.
The system operates - in fact - with no dispatcher. The vehicles starts from central node and are assigned by
random to one of available routes. The vehicle is loaded in central node with containers addressed to each
ordinary nodes included in a given route. This is done in a proportional way. Next, after approaching given node
(it takes some time according to vehicle speed random process and road length) and the vehicle is waiting in an
input queue if there is any other vehicle being loaded/unload at the same time. There is only one handling point
in each node. The time of loading/unloading vehicle is described by a random distribution. The containers
addressed to given node are unloaded and empty space in the vehicle is filled by containers addressed to a central
node. The operation is repeated in each node on the route and finally the vehicle is approaching the central node
when is fully unloaded and after it is available for the next route. The process of vehicle operation could be
stopped at any moment due to a failure (described by a random process). After the failure, the vehicle waits for a
maintenance crew (if there are no available due to repairing other vehicles), is being repaired (random time) and
after it continues its journey.
3 SIMULATION METHODOLOGY
Discrete transport system described in the above section is very hard to analyze by formal model. It does not lay
in the Markov process framework. A common way of analyzing that kind of systems is a computer simulation.
To analyze the system we must first build a model, which was done based on description presented in the
previous section, and then operate the model. The process of system modelling requires to defined the level of
details. Increasing the system details causes the simulation becoming useless due to the computational
complexity and a large number of required parameter values to be given. On the other hand a high level of
modelling could not allow to record required data for system measure calculation. Therefore, the level of system
model details should be defined by requirements of system measure calculations.
The system model needed for simulation, has to encourage the system elements behavior and interaction
between elements. In case of dependability we have to include system element reliability model. Except the
system functionality model we have to model the traffic in the system. The data for simulation of a given real
exemplar system consists of system element model (described in the system functionality meta-model
formalism) and a given traffic configuration.
58
Once a model has been developed, it is executed on a computer. It is done by a computer program which steps
through time. One way of doing it is so called event-simulation. Which is based on a idea of event, which could
is described by time of event occurring, type of event (in case of DTSCNTT it could be vehicle failure) and
element or set of elements of the system on which event has its influence. The simulation is done by analyzing a
queue of event (sorted by time of event occurring) while updating the states of system elements according to
rules related to a proper type of event.
The event-simulation program could be written in general purpose programming language (like C++), in fast
prototyping environment (like Matlab) or special purpose discrete-event simulation kernels. One of such kernels,
is the Scalable Simulation Framework (SSF) [2] which is a used for SSFNet [3] computer network simulator.
SSF is an object-oriented API - a collection of class interfaces with prototype implementations. It is available in
C++ and Java. SSFAPI defines just five base classes: Entity, inChannel, outChannel, Process, and Event. The
communication between entities and delivery of events is done by channels (channel mappings connects
entities). [3]
For the purpose of simulating DTSCNTT we have used Parallel Real-time Immersive Modelling Environment
(PRIME) [6] implementation of SSF due to much better documentation then available for original SSF.
We have developed a generic class (named DTSObject) derived from SSF Entity which is a base of classes
modelling DTSCNTT objects like: scheduler, node, truck and crew which models the behaviour of presented in
section 2 discrete transport system.
The effectiveness of simulation done in PRIME environment is very promising. The tests done on one batch of
simulation of DTSCNTT exemplar described in the next section needed from 3.9 to 9 seconds on Pentium 2 GHz
computer. The time needed to perform one simulation depends on the number of events presented in the system,
which is a result DTSCNTT configuration.
Due to a presence of randomness in the DTSCNTT model the analysis of it has to be done based on Monte-Carlo
approach. What requires a large number of repeated simulation. The SSF is not a Monte-Carlo framework but by
simple re-execution of the same code (of course we have to start from different values of random number seed)
the statistical analysis of system behaviour could be realized.
4 FUNCTIONAL AVAILABILITY OF DTSCNTT
We define the availability of the system as a ability to realize the transportation task at required time. The
availability is the probability. Lets use the following values:
T - time measured from the moment when the container was introduced to the system to the moment when the
container was transferred to the destination (random value),
Tg - guaranteed time of delivery, if exceeded the container is delayed.
Ì
¿
ð
Ì
¾
Ìä
ó Ì¹
¬
Ì¹
Ìâ
Ì Ì¹
â Ì¹
¬
ð
±°-¦²·»²·»
¼»´¿§
Fig. 1. The delivery in guaranteed time (a) and delayed delivery (b)
At Fig. 1. we can observe two possible situations:
(a) - delivery was realised before guaranteed time T - there is no delay,
(b) - delivery was delayed - time of delay: Tg.
If N(t) is stochastic process describing the number of delayed containers at time t, the functional availability Ak(t)
we can define as probability that the number of delayed containers at time t does not exceed k. The value k is the
level of acceptable delay.
59
Ak t
Pr N t
k
(3)
5 DTSCNTT CASE STUDY
For testing purposes of presented DTSCNTT system (chapter 2) and developed extension of SSF (chapter 3) we
have developed an exemplar transport system. It consists of one central node (Wroclaw) and three ordinary
nodes (cites nearby Wroclaw: Rawicz, Olesnica and Nysa). The distances between nodes has been set according
to real distances between used cities and they equal to: 85, 60 and 30km. There were 5 trucks (two with capacity
set to 10 and three with capacity 15). The trucks travel with mean speed 50km/h. The vehicles realize 19 trips a
day: from central node to ordinary node and the return trip. MTTF of trucks is described by exponential
distribution with mean value 1000h. The repair time is related to normal distribution with mean value equal to 2h
and variance equal to 0.5h. The containers addressed to ordinary nodes are available in central node at every 0,5,
0,4 and 0,3 of hour respectively. Containers addressed to the central node are generated at every 0,6, 0,4, 0,3 of
hour in following ordinary nodes. There is single maintenance crew. The availability of the system Ak(t) was
calculated with guaranteed time Tg =24h and parameter k=20. Time-table as well as other functional parameters
were described in a DML file (see example in Fig. 2.). The Domain Modeling Language (DML) [6] is a SSF
specific text-based language which includes a hierarchical list of attributes used to describe the topology of the
model and model attributes values.
NET [
Vertex [ID Nys MTTB 0.6] Vertex [ID Raw MTTB 0.4]
Vertex [ID Ole MTTB 0.3]
CeVertex [ID Wro MTTB [Nys 0.5 Raw 0.4 Ole 0.3] ]
Truck [No 2 Speed 50 Size 10 MTTF 1000]
Truck [No 3 Speed 50 Size 15 MTTF 1000]
Trip [Size 10 Start 8.00 Dest[ID Ole Time 8.40]]]
Trip [Size 10 Start 9.30 Dest[ID Ole Time 10.10]]
…
Fig. 2. Exemplar DTSCNTT description in DML file
In the presented system we have observed the percentage of containers transported longer then a given threshold
value. The simulation time was set to 100 days and each simulation was repeated 10.000 times. The achieved
results are presented in Fig. 3. The 98th percentile (upper dashed line) could be understood as a guaranteed (with
0.98 probability), minimum quality of service. In other words the percentage of containers transported longer
then the value on X axis will no to be higher with probability 0.98.
The results of presented experiments could be used for example for selection the optimum value for SLA
(service level agreement). The developed software allows to analyze the other dependencies for example
influence of a number of vehicles or the vehicle reliability parameters on the percentage of containers
transported longer then given threshold value.
60
45
Mean value
2nd and 98th percentile
40
35
30
25
20
15
10
5
0
5
10
15
20
25
30
35
40
Time x
Fig. 3. The percentage of containers transported longer then given threshold time (solid line mean value, dashed 2nd and 98 th percentile)
The availability of system (Fig. 4.) is described with periodic changes. The situation is a result of the time-tables
and the way of containers generation. The containers are generated during all day (exponential distribution of
time between two following containers) and trucks do not operate in the night. The probability of delay increases
in the night, but the number of trucks (5) is satisfactory for the system. Before the second set of simulation we
reduced one vehicle offering 15 containers of capacity. The new shape of the availability of system is presented
at Fig. 5. The availability of the system decreases, the delivery delays step by step increase. The number of tucks
is insufficient.
ßîðø¬÷
ïòðð
ðòçë
ðòç
ðòèë
ðòè
ðòéë
ðòé
ð
ïðð
îðð
íðð
ìðð
ëðð
êðð
éðð
èðð
çðð
ïððð
¬
Fig. 4. Functional availability of the DTSCNTT, 5 trucks operate
61
ßîðø¬÷
ïòððï
ðòçë
ðòç
ðòèë
ðòè
ðòéë
ðòé
ð
ïðð
îðð
íðð
ìðð
ëðð
êðð
éðð
èðð
çðð
ïððð
¬
Fig. 5. Functional availability of the DTSCNTT, 4 trucks operate
6 CONCLUSION
We have presented a simulation approach to functional analysis of Discrete Transport System with Central Node
and Time-Table (DTSCNTT). The DTSCNTT models the Polish Post regional centre of mail distribution
behavior. Developed simulation software allows to analyze the effectiveness (understood in given exemplar as a
percentage of containers with delivery time exceeding given threshold) of a given time-table. Changes in a timetable or in a number of used trucks can be easily verified. Also, some economic analysis could be done
following the idea presented in [5], [11], [12]. The implementation of DTSCNTT simulator done based on SSF
allows to apply in a simple and fast way changes in the transport system model. Also the time performance of
SSF kernel results in a very effective simulator of discrete transport system.
Therefore, we think, that introduced exemplar analysis shows, that the described method of transport system
modeling can serve for practical solving of essential decision problems related to an organization and parameters
of a real transport system. The proposed analysis seems to be very useful for mail distribution centre
organization.
Work reported in this paper was sponsored by a grant No. 4 T12C 058 30, (years: 2006-2009) from the Polish
Committee for Scientific Research (KBN).
REFERENCES
[1]
BARLOW, R.; PROSCHAN, F. Mathematical Theory of Reliability. Society for Industrial and Applied
Mathematics, Philadelphia (1996).
[2]
COWIE, J. H. Scalable Simulation Framework API reference manual. [Online]. Available:
http://www.ssfnet.org/SSFdocs/ssfapiManual.pdf (1999).
[3]
COWIE, J. H.; NICOL, D. M.; OGIELSKI, A. T. Modeling the Global Internet. Computing in Science
and Engineering Vol. 1, no. 1, (1999) pp. 42-50.
[4]
FISHMAN: MONTE CARLO: Concepts, Algorithms, and Applications. Springer-Verlag, New York
(1996).
[5]
KAPLON, K.; MAZURKIEWICZ, J.; WALKOWIAK, T. Economic Analysis of Discrete Transport
Systems. Risk Decision and Policy, Vol. 8, No. 3. Taylor & Francis Inc. (2003) pp. 179-190
[6]
LIU, J. Parallel Real-time Immersive Modeling Environment (PRIME). Scalable Simulation Framework
(SSF), User's manual, Colorado School of Mines Department of Mathematical and Computer Sciences,
[Online]. Available: http://prime.mines.edu/ {2006).
[7]
MAZURKIEWICZ, J.; WALKOWIAK, T. Fuzzy Economic Analysis of Simulated Discrete Transport
System. Artificial Intelligence and Soft Computing - ICAISC 2004, Springer-Verlag, LNAI 3070 (2004)
pp. 1161-1167.
[8]
SANSO, B.; MILOT, L. Performability of a Congested Urban-Transportation Network when Accident
62
[9]
[10]
[11]
[12]
Information is Available. Transportation Science (1999), 1, vol. 33.
WALKOWIAK, T.; MAZURKIEWICZ, J. Hybrid Approach to Reliability and Functional Analysis of
Discrete Transport System. Computational Science - ICCS 2004, Springer-Verlag, LNCS 3037 (2004)
part II pp. 236-243.
WALKOWIAK, T.; MAZURKIEWICZ, J. Reliability and Functional Analysis of Discrete Transport
System with Dispatcher. Advances in Safety and Reliability, European Safety and Reliability Conference
- ESREL 2005, Taylor & Francis Group London (2005) pp. 2017-2023.
WALKOWIAK, T.; MAZURKIEWICZ, J. Simulation Based Management and Risk Analysis of Discrete
Transport Systems. IEEE TEHOSS 2005 Conference (2005) pp. 431-436.
WALKOWIAK, T.; MAZURKIEWICZ, J. Discrete transport system simulated by SSF for reliability and
functional analysis. International Conference on Dependability of Computer Systems. DepCoS RELCOMEX 2007, IEEE Computer Society [Press] (2007) pp. 352-359.
ADDRESS:
Jacek Mazurkiewicz
Institute of Computer Engineering, Control and Robotics
[email protected]
Tomasz Walkowiak
Institute of Computer Engineering, Control and Robotics
[email protected]
63
64
QUALITY EXPERT ESTIMATE IN SOFT COMPUTING PROGRAMMES OF
INTELLIGENT AUTOMATION
Branislav Lacko
Abstract: The paper focuses on expert estimates needed for the planning of soft computing
application projects. It analyses methods applicable in expert estimates and introduces rules
recommended in expert estimates. It defines the typical features of good quality expert estimates.
Keywords: automation, project, expert estimate, Team Delphi method
1 ISSUES UNDER DISCUSSION
The current demands on high quality planning and implementation of projects is increasing, as documented in
the standard SN/ISO 10 006 Project Quality Management Guidelines [7] that define important project
management processes. These include processes such as:
Time dependent processes.
Resource related processes.
Cost related processes.
Risk related processes.
If a project planning is to be useful for efficient management, individual activities have to be assessed correctly;
otherwise they are merely accurate calculations with inaccurate figures and cannot serve as the means for high
quality project management.
Software development projects, projects for introducing new information technology or information system
projects count among projects where prescriptive methods, benchmarking, statistical methods or modelling and
simulation can hardly be applied. They are applicable on different type of projects, so-called soft projects.
An expert estimate is frequently the only method that can be applied in such cases. Thus the quality of projects
planning and their efficient management aimed at a successful completion depends heavily on the quality of
estimates.
2 DEFINITION OF EXPERT ESTIMATE
An expert estimate means the definition of a certain value based on the knowledge and experience of experts. An
expert estimate is meant to replace other, often more accurate, methods of defining the required value
(measurement, calculation, etc.) that cannot be applied for various reasons.
Most commonly, an expert estimate is expressed as a certain interval that is most likely to contain a quantity that
otherwise would have been obtained through an accurate measurement.
The essence of expert estimate lies in the fact that experts perform an intuitively logical analysis of the problem
at hand and express their opinions assessment, which specifies the problem. The basic presumption is that a
group opinion by experts in well defined positions, assuming that the panel of experts has been properly selected,
is usually very close to the correct solution. [5]
The quality of expert estimates depends mainly on the quality of experts involved in the process. Of course, it
also depends on the correct process of experts answers evaluation. However, the selection of appropriate experts
is a key factor of success!
In order to define the term expert it is possible to apply the definition of our legislation of experts appointed by
court (expert witness): an expert is a person with an appropriate specialist knowledge listed in the register of
experts maintained by a regional court. The definition puts an emphasis on specialist knowledge, which also
includes relevant experience. The word relevant emphasises that an expert in real estate appraisals does not
necessarily have to be an expert in estimating a persons time of death. It is important to bear this in mind in
65
nominating experts for the assessment of network charts to ensure that experts in the relevant field be addressed
at all times. The note referring to a registration with regional court means that the person has to enjoy a generally
recognised expert status (in case of co-operation with court, only persons who have passed relevant tests and
have been recommended by the relevant chamber of expert witnesses can become registered). No person should
become an expert through a self-appointment or a mere declaration by another person without satisfying the
defined conditions and demonstrating their knowledge and experience. It is assumed under normal
circumstances that on request an expert can document and prove his or her education, experience and references
from previous appointments to estimation processes. We should always ask renowned experts!
The above suggests that for our purposes we can define an expert as follows:
An expert is a person widely recognised for relevant and properly documented knowledge and experience.
Let us be more precise in defining the term expert estimate, expanding on the brief definition at the beginning of
the paragraph.
An expert estimate means an acceptable interval of values containing the real value with a sufficiently high
probability (often pre-defined) that cannot be obtained through direct measurement.
A few important notes to the definition: an estimate is a substitute for a more accurate method. We do it either
because we cannot apply a more accurate method for various reasons (we have not the relevant prescriptions, we
have not the relevant data for a static analysis, we have not a model and simulation programme, we have not a
special measuring equipment, etc.) or simply because we are carrying out a forecast and there is nothing to be
measured yet.
The expert estimate interval can be either defined explicitly or is perceived implicitly as a tolerated estimate
inaccuracy. In the former case an expert directly defines the lower and upper limits of the value (for example, an
activity will consume 50 to 60 days) or defines the mean value and spread (for example, an activity will consume
55 days +/-5 days). In the latter case the expert defines only the mean value because it has been agreed that an
estimate with an inaccuracy interval of +/- x% is required (then such an interval applies even if it is not explicitly
stated). For the sake of accurate planning in project management when applying a network analysis we ought to
require a rather narrow interval. Let us try to realise the consequences of a statement that the cost of work will be
for sure somewhere between CZK 10,000 and 2 million. For good planning such an interval is useless, even
though the statement may be true for an actual cost of CZK 95,000.
The probability of conformity of expert estimate with the actual situation is normally expected to exceed 80%. In
truly good expert estimates it is explicitly stated. Nevertheless, sometimes it is perceived as implicit.
An example of a good quality estimate result:
In the defined situation it is expected at 95% that an activity will take 120 days (+/- 5%) and the cost will be
CZK 320,000 (+/- 10%).
If the activity eventually takes 300 days and the cost is CZK 800,000 or if it takes only 20 days and costs CZK
80,000, after the project completion it is necessary to perform an assessment to establish whether the expert
estimate was of poor quality or the conditions for the activity were different from information submitted to the
experts. On the contrary, if the activity takes 124 days and the cost is CZK 305,000, it is a case of a good quality
estimate.
We may, however, encounter circumstances in which an estimate +/- 20% will not be considered as sufficient,
for example in the initial project stages (opportunity study).
It is essential to be aware of the difference between an expert estimate and other things, such as the needs for
example! If an expert estimate states that an activity will take 120 days (+/-5%) and some say that in order to
ascertain a continuity it has to be completed in 50 days, then 50 days is not an expert estimate! 50 days in this
case is a requirement! We have found out that the obtained expert estimate is different from the need and thus we
have identified a problem to be solved: What do we have to do to make sure that the activity does not take more
than 50 days? This is where project teams often make mistakes.
Let us summarise the most frequently used expert estimate methods without describing them in detail.
66
3 METHODS UNDER DISCUSSION
3.1 DELPHI METHOD
The Delphi method [1] is suitable in cases where a numerical value can be an answer to a clearly defined
question. It is not appropriate in situations where the result of expert estimates are various statements, verbal
answers, descriptive scenarios, etc. It is based on two main principles:
Larger number of anonymous answers by experts
Resulting estimate is obtained through an iterative process
The Delphi method has a number of advantages for which it is considered as one of the best and most
elaborate methods of obtaining expert estimates.
Its disadvantages include relatively high costs of organisation, processing and time needed for obtaining
the resulting experts opinions.
3.2 TEAM DELPHI ALTERNATIVE
An alternative frequently referred to as TEAM Delphi is suitable for application in project teams conditions
where it is necessary to establish certain estimates during team meetings. This concerns for example the time
needed for the completion of certain activities, costs of implementing certain activities, need for sources for a
project, etc. This alternative is also known as the team iteration version of the Delphi method.
Its advantage is the simplicity and possibility of immediate application.
Its disadvantage is the possibility of influencing the estimate result by a larger number of erroneous estimates.
3.3 PROGRESSIVE COMPARISON METHOD
The progressive comparison method [9] (in literature also referred to as the sum organisation method) first
appeared in 1957. Its authors describe it as a systematic verification of relative judgements on the basis of
progressive comparison. First, preliminary judgements are established in several alternatives. In the following
stages these judgements are systematically corrected and verified. The results are arranged along a scale and
usually they are standardised.
This method can be successfully used if we need to obtain a relative, not absolute, assessment of several
alternatives.
3.4 EMPIRICAL RULES FOR ESTIMATES
Empirical rules are often defined to ensure a good quality of expert estimates and to avoid a distortion of
obtained values. For example Tom de Marco 10 , a software project expert, recommends the following set of
empirical rules to be generally respected in preparing expert estimates:
a)
Estimating is not parroting. Do you think you can do the job in 3 months? In my estimation you can.
b)
Estimating is not negotiating. I cant manage this, I need two more months!
c)
Estimate is not theft. It will take 3 months (because competitors estimated it at 3.5 months).
d)
Estimate is not a simple division of given time into smaller portions. If we are supposed to do it in 6
months, the analysis will take 2 months, proposal 1 month
e)
Estimate is not a change taking into account a previous delay. The previous activity was delayed by a
month, so we will cut the time for the following two by two weeks each.
f)
If I want a qualified estimate, I cannot offer an answer. Do you think it is going to take 6, 8, 12 or 14
days?
g)
An estimate applicable to planning must contain equal portions of pessimism and optimism. According to
this estimate it is equally likely that we will complete the project one week earlier and that it will be
delayed by the same time!
h)
The share between an individual estimate and an estimate applicable to planning is usually constant. Frank
estimated it at 8 months. He is a well-known optimist, so we will increase it to 12 months.
i)
Estimating does not mean using a single individual piece of information. It is necessary to ask several
independent experts. If information is obtained from a single expert, we should present it as Mr. XYs
opinion.
j)
The quality of estimate needs to be evaluated in the post-implementation stage.
Empirical rules based on experience, especially in the Team Delphi method, make it possible to avoid mistakes
in the estimation process and to reduce the number of inaccurate individual estimates by team members.
Note: for the sake keeping this paper reasonably short detailed descriptions of individual methods are not
provided. They are available in the recommended literature.
67
4 SUGGESTED FURTHER RESEARCH
The paper discussed the procedure in expert estimation.
It needs to be emphasised, however, that the system modelling principle is an efficient tool supporting expert
estimates. The accuracy of estimates and thus their quality can be improved significantly if a quality data model
is prepared (see the publication by Merunka, ZU Praha 8 ), which will enable the assessment of an
information system from the point of view of the data needed for the information system construction. Expert
estimates of IS laboriousness, estimates of the necessary memory capacity for data storage etc. will be
significantly more accurate, if a data model has been created.
The research within the MSM 0021630529 research project will focus on defining a procedure for obtaining
mathematic and logical models suitable for simulations on digital computers.
5 CONCLUSION
Very often in our project teams we can see the application of so-called qualified estimates, which are performed
in a very incompetent way and therefore in poor quality. This is why the attributes of a quality expert estimate
have to be reiterated:
It is performed by experts whose competence as well as reputation can be documented.
It is based on the judgements by more experts (at least 3 to 5).
Its calculation based on the judgements by inquired experts is performed according to a recognised
method.
Its value is an interval complemented with a definition of the expected probability of fulfilment.
The estimate is assessed retrospectively after the project completion.
Delphi is the most frequently used method experiencing a certain renaissance at the
moment. The internet [4] is used to support this method and it is also being elaborated into further modifications
to be used in other, less typical, applications, such as for elaborating scenarios [3], and it is applied on wellstructured problems (see the paper by FAST VUT Brno [5]).
Experts assume that the present conditions in software and other projects will call for a substantial expansion of
the computer modelling application and simulation in assessing network charts both for classic methods CPM
and PERT and for the critical chain method. It is expected that after 2020 the modelling and simulation will be
the prevailing methods of assessing network charts. This applies mainly to expert estimates concerning software
development for complex systems, such as mechatronic systems, robotic systems and control programmes for
robot communities and various other intelligent automatic control systems. This is why the research in estimates
for the project proposal and management in complex systems will focus on perfecting the modelling and
simulation procedures for obtaining more accurate values in constructing the required network charts with the
aim to achieve a higher quality of expert estimates.
Until then it will be necessary in many complex cases to use the method of expert estimates using specialised
software engineering methods for determining the laboriousness of information systems, for example the Use
Case Points (UCP) or Function Points (FP) methods [6]. The UCP method can be efficiently supported by an
expert system based on neuron networks 11 .
REFERENCES:
[1]
K OVÁK, J.; ZAMRAZILOVÁ, E. Expertní odhady (Expert Estimates). Praha : SNTL, 1989.
[2]
www. iit. edu/~it{delphi.html).
[3]
KATOLICKÝ, A. Aplikace Delftské metody v N mecku Delphi 98 (Application of Delphi Method in
Germany Delphi 98). (on line www.iqnet.cz/katolicky).
[4]
VLK, M. Po íta ová podpora metody DELFY v prost edí Internet (Computer Support to the Delphi
Method in the Internet Environment), thesis. Brno : VUT FS, 1998, 60 p.
[5]
IM NEK, P. Rizikové faktory hurdiskových stropních konstrukcí (Risk Factors in Hollow Clay Block
Ceiling Structures). In: Proceedings from the 5th International PhD Conference.. VUT Brno 2003, pages
57 62.
[6]
VAHALÍK, T. Odhadujete pracnost projektu? (Estimating a Project Laborousness?). KOMIX pages 6
7.
[7]
SN/ISO 10 006 ed.2 Systémy managementu jakosti Sm rnice pro management jakosti projekt
(Quality Management Systems Project Quality Management Guidelines). eský normaliza ní institut
2004 Praha, 42 p.
[8]
MERUNKA, V. Datové modelování (Data Modelling). Praha : ALFA Publishing 2006, 180 p.
68
[9]
[10]
[11]
CHURCHMAN, C. W.; ACKOFF, R. L.; ARNOFF. E. L. Introduction to Operations Research. New
York : John Wiley, 1957.
DEMARCO, T. Structured Analysis and System Specification. New York : Yourdon Press, 1978.
PAVLÍ EK, J. Odhad manaerských charakteristik vývoje IS v etap specifikace poadavk (IS
Development Management Characteristics Estimate at the Specification Stage). Thesis. Praha : PEF
ZU, 2006, 144 p.
This paper was supported by the MSM 0021630529 research project on Intelligent Systems in Automation.
ADDRESS:
Doc. Ing. Branislav Lacko, CSc.
Faculty of Mechanical Engineering
University of Technology
Technicka 2
616 69 Brno
Czech Republic
[email protected]
69
70
IMPLEMENTATION OF MOMENT METHOD FOR OBJECT RECOGNITION
Ji í tastný, Petr Ludík
Abstract: The paper describes the application of algorithms for object classification by using
moment method. In this paper were used invariant moments. The real technological scene for the
object classification was simulated with the digitization of two-dimensional pictures. The algorithm
given below have been used in an application that was developed at Brno University of Technology
Key-Words: Image Processing, Moment method, Invariant moments
1 INTRODUCTION
The pattern recognition consists in sorting objects into classes. The class is a subset of objects whose elements
have common features from the classification standtpoint. The object has a physical character, which in
computer vision is most frequently taken to mean a part of segmented image. Among many well known modern
methods (such as genetic algorithm, back propagation algorithm and others) we can also include moment
methods. As a good example might be used invariant moments. Invariant moments belong to the group of
statistical methods. Moment descriptions of image region interprete normalised brightness function as density of
probability of two dimensional random quantity. This interpretation can be used for either binary or grey scale
regions.
2 INVARIANT MOMENTS
Problems of pattern recognition demand solving of three basic conditions to make object invariant for scale,
translation and rotation. For this purpose must be performed the calculation of moments in following order.
2.1 GENERAL GEOMETRIC MOMENTS
General geometric moment M(m,n) of object G is defined as :
x n y m dx dy
M (n, m )
G
where n+m means grade of moment. Computer image has the discrete pattern and therefore the integration is
replaced by summation:
M (n, m)
xi
i
n
yj
m
p
j
where xi, yj go through center of pixels and the area of one pixel is marked as p. These general moments are used
for objects description which relates to general system of coordinates (see fig. 1) can not be used for object
description.
71
Fig.1 Main system of coordinates
2.2 CENTRAL GEOMETRIC MOMENTS
After calculation of general moments follows central moments CM(m,n) to obtain center of analysis object. The
system of coordinates is situated in the center of gravity. The center of gravity belongs among proper parameters
for object description. Coordinates of center of gravity are defined as:
xc
M (1,0)
M (0,0)
yc
M ( 0,1)
M (0,0)
where M(1,0) and M(0,1) are general moments defined above and M(0,0) is the area of the object in pixels. The
new system of coordinates translated into the center of gravity is illustrated on figure 2.
Fig. 2 center of coordinates after translation
Central geometric moments for object G are defined as:
CM (n, m)
Cx) n ( y j
( xi
Cy )m dx dy
G
and in discrete formula:
CM (n, m)
( xi
i
Cx) n ( y j
Cy ) m p
j
Central geometric moments are invariant for translation.
72
2.3
NORMALIZED CENTRAL GEOMETRIC MOMENTS
Normalized central moments NCM(n,m) relates to the same system of coordinates as the central moments. The
scale of this system is chosen to be equal 1, it means NCM(0,0)=1.
Normal central moments are defined as:
NCM (n, m)
CM ( n, m )
M ( 0,0)
n m 2
2
We have invariant moments for scale change and translation but we have to solve invariant rotation.
2.4 PRINCIPAL GEOMETRIC MOMENTS
Principal moments PM(n,m) are related to the main system of the coordinates of the object. The axes of this
system are marked according to conventions X,Y. This system of coordinates is chosen to follow these
conditions:
0)
1)
2)
3)
4)
5)
6)
system of coordinates X,Y is clockwise.
PM(0,0) = 1
PM(1,0) = 0
PM(0,1) = 0
PM(1,1) = 0
PM(2,0)
PM(0,2)
PM(3,0)
0
Conditions 0), 1), 2), 3) were accomplished by system defined earlier and mean that the center of the system is
situated in the center of gravity and scale of both axes is chosen in way where the area of the object is equal to 1.
Unfortunately these conditions say that we are able to find infinite number of solutions. Therefore it is necessary
to accomplish condition 4) which means performing axes rotation till principal moment PM(1,1)=0 (see fig. 3).
Fig. 3 Behaviour of M(1,1) according to angle
This picture represents the object itself and mixed moment of the second grade according to angle of rotation .
Conditions 0), 1), 2), 3), 4) also set the axes of the main system of coordinates but we are not able to make
decision which one is X and Y and their orientation as well. Now we have four different systems. If we pass
condition 5) (see fig.4) we can make decision which axe is X and Y.
73
Fig. 4 Finding axes X and Y
The last step is condition 6) which will make clear the orientation of axes. Due to perfect symmetry of objects
we have to deal with this problem by using two other conditions:
1.
2.
If PM(1,1)=0 for each then the axes X will be horizontal and Y vertical.
If we can exclude case 7) there still exists four systems according to conditions 0) - 4), it means we have
four angles 1, 2, 3, 4. Formula for angle rotation is:
k
1
2 NCM (1,1)
arctg
2
NCM (2,0) NCM (0,2)
(k 1)
2
If we are not able to decide which angle is correct we will choose the one with lowest angle. Once we have got
the proper angle we are able to calculate principal moments which are invariant. From this point we have set of
nine principal moments which are used for right object identification. We are ready to perform the classification
of objects into classes. These classes were created during learning process by using principal moments. Learning
process is based on simple idea-putting objects into user defined classes. For testing were used knowledges of
statistical and probability methods.
Object belongs to class if each of nine principal moments fits into interval defined as:
x t1
s
/2
n 1
; x t1
s
/2
n 1
where x is arithmetic average, t1- /2 is fractile s is dispersion and n 30 [1] [3]. Each class was created by
using at least thirty objects representing the same object. User has the full control of number and shape of
patterns used for learning. It is essential to use wide spectrum of the same object but with different scales and
angles of rotation (see fig. 5)
Fig. 5 Examples of ideal learning objects
74
3 PROBLEM SOLUTION
Learning/recognizing algorithm is possible to describe in this way:
Repeat
Browse_the_entry_image_pixel_after_ pixel;
Find_the_first_pixel_belonging_to_object;
Floodfill_the_area_of_object;
Perform_computing_of_all_moments;
Perform_object_classification;
Until end_of_entry_image;
This algorithm was used during software implementation (Fig. 6 and Fig. 7).
Fig. 6 Example of learning process
Fig. 7 Example of recognizing process
Because moment method is very sensitive even for small object changes there must be applied coefficients to
extend intervals. This procedure decreased the sensitivity but on the other hand brought errors during object
classification. We know two kinds of errors. Error of the first kind hypotesis says that object belongs to the
right class but we deny it. Error of the second kind-hypotesis is invalid but we accept it [2].
To verify ability for pattern recognizing of this method were executed experiments with real technological
scene(fig.8) and with common text (fig. 9 and fig.10).
A real technological scene for object classification was simulated by digitizing five selected objects. For this
purpose, two-dimensional images of three-dimensional objects were prepared. The aim was to test such objects
that resemble two-dimensional images of real objects. The choice of objects of similar shape was also
intentional.
75
Fig. 8 Example of technological scene
Fig. 9 Scanned text 300 dpi
Fig. 10 Recognizing of letter e in document
For experiment of real technological scene were also used common parts from manufacturing with no image
modification and taken by standard camera( Fig. 11)
76
Fig. 11 Example of tested real technological objects
In the table 1 are results of recognizing technological objects with no image modification with the same light
conditions and camera position.
Object
settings
99%
99%
1.kind
error
99%
2.kind
99,5%
error
30%
6,66%
99,5%
1.kind
error
99,5%
2.kind
error
20%
13,33%
Cogwheel
70%
Key
73,33% 26,67% 0%
83,33% 16,67% 6,66%
Screw
70%
80%
Nut big
73,33% 26,67% 6,66%
83,33% 16,67% 13,33%
Nut small
73,33% 26,67% 6,66%
83,33% 16,67% 13,33%
30%
0%
80%
20%
6,6%
Tab. 1 Results of object recognizing
Object
e (300 dpi)
18 %
error
(1. kind)
82 %
e (400 dpi)
20 %
80 %
99%
error
(2.kind)
0%
0%
Tab. 2 Results of text recognizing
Where can be imagined as reliability of proper object classification in %.
Table 2 shows results of text recognizing. Now we can see big differences at results between text and
technological objects.
4 CONCLUSION
Results received from tests proved that this method is very sensitive to entry image quality. This property can be
considered as at once an advantage and disadvantage. Moment methods are statistical fitting to be used for
objects represented by lots of pixels. It helps to decrease sensitivity caused during picture capturing. Recognizing
of texts was less sucessful because of used type of fonts and font size. We can say this method is not usable for
small objects. Results of experiments were affected by suitable set of learning objects. It is necessary to
77
guarantee the same light conditions during picture capturing. Recognition with the aid of moment method is
suitable where the speed of classification of randomly rotated objects is not required but we need to obtain high
level of correct object classification.
ACKNOWLEDGEMENT
This research was supported by the grants:
MSM 0021630529 Intelligent Systems in Automation (Research design of Brno University of Technology)
No 102/07/1503 Advanced Optimisation of Communications Systems Design by Means of Neural Networks.
The Grant Agency of the Czech Republic (GACR)
MSM 6215648904/03 Development of relationships in the business sphere as connected with changes in the
life style of purchasing behaviour of the Czech population and in the business environment in the course of
processes of integration and globalization (Research design of Mendel University of Agriculture and
Forestry in Brno)
5 REFERENCES
[1]
DRUCKMÜLLER, M.; HERIBAN, P. Digital Image Processing System 5.0. Brno : SOFO, 1996.
[2]
LUDÍK, P. Implementation of moment method of pattern recognition. Diploma thesis, Brno : VUT Brno,
2007.
[3]
KARPÍEK, Z. Matematika IV. Statistika a pravd podobnost. Brno : CERM, 2003, ISBN 80-214-25229.
[4]
ASTNÝ, J.; KORPIL, V. Comparison Methods for Pattern Recognition. International Journal
WSEAS Transactions on Circuits and Systems. Volume 3, 2004, ISSN 1109-2734.
[5]
ASTNÝ, J.; MINA ÍK, M. Object Recognition by Means of New Algorithms. In: Fourth
International Conference on Soft Computing Applied in Computer and Economic Environments ICSC
2006, Kunovice, 2006.
ADDRESS:
Doc. RNDr. Ing. Ji í tastný, CSc.
Department of Automation and Computer Science
Technicka 2
616 69 Brno
CZECH REPUBLIC,
Email: [email protected]
http://www.vutbr.cz/
Ing. Petr Ludík
Department of Automation and Computer Science,
Technicka 2
616 69 Brno
CZECH REPUBLIC,
78
HYSTERETIC PROPERTIES OF A TWO DIMENSIONAL ARRAY OF SMALL
MAGNETIC PARTICLES: A TEST-BED FOR THE PREISACH MODEL
Gábor Vértesy1, Martha Parda Vi-Horváth2
1
Hungarian Academy of Sciences
The George Washington University
2
Abstract: The magnetization process of a regular two-dimensional array of small, strongly uniaxial
single domain magnetic garnet particles, groups of particles, and major loop properties of a
"macroscopic" sample, has been investigated experimentally and simulated numerically. These
particles correspond to the assumptions of a simple Preisach model. The switching mode is by
rotation. Each particle has a square hysteresis loop, with no reversible or apparent reversible
component. Requirements of wiping-out and congruency properties are satisfied. From
measurements of the up- and down switching fields on individual particles, the major loop can be
reconstructed, and it is shown to be in in excellent agreement with the measured one. The transition
from individual to collective behavior is smooth and the properties of a system, consisting of 100
particles, correspond to the major loop behavior. The numerically simulated major hysteresis loops
agree very well with the measured loops, the switching sequence and the magnetization curve for
particle assembly was derived from the calculated interaction fields and found to be in a very good
agreement with the measured values, demonstrating the reliability of numerical modeling. A new
property, not included into the existing models, is the magnetization dependence of the standard
deviation of the interaction field.
INTRODUCTION
The predictive power of a theory is especially important for hysteretic phenomena, where the state of the system
depends on its history. Among the models, describing hysteretic phenomena, the Preisach model (PM) is one of
the earliest and best studied [1]. The original PM assumes the magnetization proceeding by switching of
individual two-state particles; and the statistics of this system determines the shape of the macroscopic major and
minor loop behavior. It is assumed that each particle switches by coherent rotation, i.e. by the Stoner-Wohlfarth
mechanism [2], and there is no reversible or apparent reversible component to the magnetization. The particles
might interact magnetostatically, leading to the shift of the individual loops by an "interaction field" Hi. The
coercivity of the individual particles, (or critical field) Hc, is determined by physical parameters, as the
anisotropy energy, the magnetic moment, and the defect structure. It is assumed that the switching units
(particles) are confined to the 4th quadrant, i.e. for the individual rectangular hysteresis loops H+ >0 and H--<0,
where H+ and H- are the up-switching and the down-switching field of a particle. It was shown in [3, 4] that the
congruency and the wiping-out properties of minor loops are necessary and sufficient conditions for a hysteretic
system to be described by a classical PM. For an assembly of particles both Hc and Hi have a statistical
distribution, assumed to be Gaussian, with a mean value H and standard deviation , leading to the Preisach
function in the form of:
P( H i , H c )
A exp[ ( H i
H i )2 / 2
2
i
(H c
Hc )2 / 2
2
c
].
(1)
The total irreversible magnetization for a system of such particles is given in terms of the experimentally
measured parameters H+ and H- by
M
(2)
D ( H , H ) P ( H , H )dH dH ,
where D = 1 gives the direction of magnetization at saturation and the Preisach density function, P(H+,H-)
corresponds to the number of particles at any given point on the (H+,H-) plane. The measured H+ and H- are
related to Hc and Hi as Hc = (H+ + |H-|)/2 and Hi= (H+ - |H-|)/2.
The description and identification of a magnetic materials hysteretic properties in terms of the PM is an ongoing
effort. However, most of the important magnetic materials do not conform with the assumptions of the original
PM. Therefore significant effort has been devoted to the development of the original model [4,5,6]. At the same
time, the experimental study of a hysteretic system, corresponding closely to the assumptions of the classical PM
79
is interesting from the point of view of testing those assumptions and, possibly, describing some properties, not
taken into account even in the modified PMs. Magnetic recording is moving toward tremendous recording
densities. A possible candidate system for the high density recording media, might be a regular two-dimensional
array of very small magnetic particles. The properties of such a system are very close to the original PM, thus
giving a direct significance to the investigation of such systems.
The motivation for the present research was to study the properties and the validity of the original PM on a
simple model system, corresponding to the assumptions of the simple PM. The hysteretic properties of a simple,
two-dimensional system of small single domain, particles are studied and compared to the predictions of the
original PM.
THE INVESTIGATED SYSTEM OF MAGNETIC PARTICLES
A 3 m thick single crystalline magnetic garnet film of magnetically diluted Y3-xBixFe5-yMyO12 (YIG),
(M=nonmagnetic ion) grown by liquid phase epitaxy on a [111] oriented non-magnetic garnet substrate, has been
etched into a 2D array of 42 m square pixels, separated by 12 m wide grooves. Each pixel corresponds to a
single particle. The excellent topographical uniformity of the particles, i.e. the perfections of corners and edges is
evident from SEM investigations (Fig. 1). Epitaxial garnets are known to have the lowest crystalline defect
densities; and optical and scanning electron microscopy observations also revealed the lack of any visible defect
in the material.
Fig. 1. Detail of the 2D array of garnet particles (SEM).
Experiments were performed on an assembly of up to several thousand garnet "particles". These particles are
small magnetically, despite of their relative large physical size, because their magnetic properties correspond to
the conditions necessary for single domain particles. The magnetooptically active transparent epitaxial garnet
film, grown on a transparent substrate, permits direct visual observation via the magnetooptical Faraday effect,
with simultaneous electrooptical recording of the state of the pixels (picture elements in optical readout).
The mean values of magnetic parameters of the whole sample were determined in a vibrating sample
magnetometer (VSM). The magnetization of the sample, 4 Ms = 160 G, is very low, while the uniaxial
anisotropy field Hu = 2 kOe. As a result, the particles have high Q = Hu/4 Ms > 10, ensuring that there are only
two stable magnetic states, either up or down along the easy axis, normal to the film plane (black and white
magnetooptic contrast). Each particle has a rectangular hysteresis loop, and the switching of the whole system
proceeds by consecutive switching of particles. The squareness of the major hysteresis loop Mr/Ms = 1, the
average switching field, Hc=280 Oe. Group of pixels and their gradual switching ina magnetic field, applied
along the easy axis, i.e. normal to the film, are shown in Fig. 2.
Hysteresis loops of individual pixels and groups of pixels have been measured with an optical magnetometer
operating on the principle of Faraday effect. The light from a halogen lamp after a polarizer is focused onto the
sample. After passing through the transparent magnetooptic material, the plane of polarization of the light is
rotated, depending on the magnitude and the direction of the magnetization. The intensity of the transmitted light
after an analyzer is proportional to the magnetization and it is measured by a photomultiplier. The sample is
placed in the magnetic field of magnetizing coils, and/or of an electromagnet. The field is oriented along the film
normal. Placing a microscope objective after the analyzer, the enlarged picture of the sample is obtained in the
image plane of the objective. By masking the picture in the image plane, any individual pixel of any group of
pixels can be chosen for the measurement of the hysteresis loops. If the sample is projected onto a screen in the
image plane, the actual magnetic state of the environment can be monitored simultaneously with the
measurement. The light, after passing through the mask is focused onto the photodetector. The hysteresis curve
of individual pixels (or groups of pixels) is obtained by measuring the intensity of the light after the analyzer as a
function of the external magnetic field.
80
The major hysteresis loop of several hundred pixels, measured in the optical magnetometer, is shown in Fig. 3.
This curve fully corresponds to the major loop, measured in the VSM, however, the large paramagnetic
contribution from the substrate is eliminated in this case. The measured hysteresis loop of an individual pixel is
also shown in the same figure, as an insert. The loop is square, there is no reversible contribution to the
magnetization.
Fig. 2. Microphotographs (Faraday effect, polarized light) of the sample in different states of magnetization.
Pixels magnetized "down" are black. Detail of the structured epitaxial garnet film (upper left), and the
magnetization process of an assembly of pixels in increasing magnetic field, starting from negative saturation
with all pixels dark; in H=0 the remanence is 1; pixels switched to the state of opposite magnetization in the
given field have a light contrast.
Fig. 3. Major hysteresis loop of several hundred pixels, and the hysteresis loop of a single pixel (insert) of the structured garnet sample.
INTERACTION EFFECTS
The magnetostatic interaction between the elements of the two-dimensional array can be investigated by taking
similar assumptions as described in the previous section. The effect of the magnetic state of the first 5
coordination shells (24 pixels) was investigated experimentally and numerically [7]. The demagnetization tensor
for an individual pixel was computed, followed by calculation of the interaction field acting on the center pixel
from its neighbors. The relationship between the magnetization state of the neighbors and the interaction field on
the central pixel was also calculated using a statistical model.
81
The model is illustrated in Fig. 4, where r is the distance of the test pixel (field point) from the nth neighbor
(source point). In the model, five neighbors are considered, at distances from the center pixel of d, 2d, 2d, 5d,
2 2d. The external field is applied normal to the sample plane. The test pixel is at the center of the sample. The
total magnetization is given by the difference of the number of up and down magnetized pixels. The
effective field acting on given pixels is the sum of the external field plus the vector sum of the interaction fields
from the surrounding pixels.
Fig. 4. The 2-D array of 25 pixels, corresponding to 5 coordination shells.
The demagnetizing tensor D describes the interaction between any two pixels. It can be calculated by a surface
integral or using the dipole approximation [8,9,10]. The field at pixel i due to pixel j, Hij, is expressed as:
Hij = D(rij) Mj
(3)
where rij= ri- rj is the relative distance between pixels , and Mj is the magnetization of pixel j. In the two
dimensional case, the only relevant term is Dzz(i,j). That means that the direction of magnetization of pixels and
the applied field are normal to the sample plane. The demagnetization tensor element is defined by the surface
integral. Expanding the integral into a Taylor series of 1/rij for large rij the first term gives the dipole
approximation:
Dzz (i , j )
(4)
d 3 / 4 rij3
where d is the size of the rectangular pixels. Table I shows the values of Dzz calculated by these methods. The
dipole approximation overestimates the exact surface integral calculation Dzz by 17.4% for the first neighbor, and
by 1.4% for the second neighbor; the deviation being negligible at larger distances.
Table I: Comparison between the dipole approximation and surface integral for demagnetizing tensor element Dzz(x,y,z).

Shell
1 st
2 nd
3 rd
4 th
5 th
Distance
d
2d
2d
5d
2 2d
Dipole -79.58
-28.13
-9.947
-7.118
-3.517
Surf. Int.
-67.79
-27.73
-9.848
-7.103
-3.513
Error, %
17.4
1.4
1.0
0.21
0.11

The interaction field at the center pixel is equal to the sum of demagnetizing fields from all neighbors. The
demagnetizing field from pixel j at pixel i depends on the magnetization:
HD ( i, j ) = Dzz (i,j) 4 Mj
(5)
The magnetization of the garnet sample is 4 Mj =160 G. The field, acting on the pixel, vs. distance of the
neighbors is shown in Fig. 5. The interaction field at a given pixel, originating from all the other pixels is given
by
82
Hi = 4 Ms Nzz
(6)
Fig. 5. The distance dependence of the demagnetizing fields at the central pixel, calculated by the surface integral and dipole approximation.
(The numbers on the curve are the dipole approximation.)
where N zz
Dzz (i , j ) , and the sign of Dzz (i,j) is determined by the state of pixel j. In fact, Nzz is
(de)magnetizing factor. The measured up-switching field of the pixel is equal to:
H+ = Hc0 - Ht
(7)
where Hc0 is the coercivity of central pixel. Hc0 could only be measured on an isolated pixel, however, it can be
determined from the measured H+ and calculated Hi. For all the 24 neighbors switched up, in other words, no
neighbors in the down state, Hi = -86 Oe, and H+ = 398 Oe, resulting in Hc0 = 312 Oe for that pixel.
Comparing and subtracting H+ for different configurations, one can determine the contribution of each neighbor
surrounding the central pixel. The slopes of the experimental curves are in good agreement with the numerical
model, proving that the contribution from the 4th, 5th and further coordination shells is negligible.
It can be assumed that the distribution of the up and down pixels around the central pixel is random. For the
system of 5x5 pixels of Fig. 4, there are 24 neighbors in five shells. Each pixel might be in 2 states. The first
shell has 4 pixels; their states may be 0, 1, 2, 3, 4 up, i.e. five cases. For whole system, there are 5625 possible
states of the 24 pixels. The probability of each individual state can be calculated, and the value of the interaction
field can be determined. It is trivial, that there is only one state when all 24 pixels are up, their demagnetizing
effect is maximal and equal to the sum from all pixels Hi = -86 Oe. In a similar way, when all the 24 pixels are
down, they have a magnetizing effect with , Hi = +86 Oe, added to the applied field. The number of cases is
the maximum in the demagnetized state, when 12 pixels are up, and the average interaction field is zero,
although the standard deviation of Ht is the maximum, i = 38.81 Oe. Fig. 6 shows , Hi and i and the calculated
probabilities of the statistical distribution.
83
Fig. 6. The mean value and standard deviation of the interaction field at the center pixel vs the number of up neighbors, and the probability
of having N neighbors magnetized up.
CONCLUSION
The magnetization process of a two dimensional regular array of small, single domain, uniaxial magnetic garnet
particles, groups of particles, and minor and major loop properties of a "macroscopic" sample have been
investigated experimentally in an optical magnetometer. This assembly of particles, interacting
magnetostatically, corresponds to the assumptions of a classical PM: the switching mode is by rotation; each
particle has a square hysteresis loop; there is no reversible or apparent reversible magnetization. The system
possesses the wiping-out and congruency properties, and the Preisach function is confined to the 4th quadrant of
the Preisach plane.
The macroscopic major hysteresis loop develops for an assembly of approximately 100 particles, although the
major loop coercivity develops much earlier. The distribution of the critical field for switching of the individual
particles follows a Gaussian, with a mean value equal to the coercivity of the major loop. The interaction fields
are distributed according to a Lorentzian function, with standard deviation strongly depending on the
magnetization.
A numerical model has been built to reconstruct the major loop for these assemblies. The simulation results
agree very well with the measurements. This demonstrates the efficiency and reliability of numerical modeling.
Thermal fluctuations, leading to fluctuating values of individual switching fields, are very unlikely for this
strongly uniaxial system. This is the reason that the calculated switching sequence is always the same for the
same group of pixels. Repeated measurements of major loops consistently show the same switching sequence,
indicating that for the given system the dominant factor, governing the shape of the major loop, is the
distribution of the coercivities of the individual particles. This coercivity, in turn, is determined by the
microstructure and the defects of the particles.
The interaction effects in this 2-D array between a central pixel and the surrounding coordination shells have
also been measured. The interaction field and it standard deviation was calculated for all possible random
distribution of 24 pixels around the test pixel. The measured and calculated interaction fields for different
configurations are in excellent agreement.
Besides the fact that the described system serves as a fine model material for the experimental verification of the
validity of the PM, the results are also applicable to very high density magnetic recording, because a promising
medium for future extreme high density magnetic storage consists of regular two-dimensional arrays of single
domain particle bits in the shape of rectangular platelets or cylinders. Similar treatment can be applied to
describe other magnetic devices, such as magnetic random access memories (MRAM) and sensor arrays, are also
based on small magnetic particles.
REFERENCES
[1]
PREISACH, F.; PHYS, Z. 94 (1935) 277.
[2]
STONER, E.C.; WOHLFARTH, E.P. Phil. Trans. Roy. Soc. London A 420 (1948) 599.
[3]
MAYERGOYZ, I.D. Phys. Rev. Len. 56 (1986) 1518.
84
[4]
[5]
[6]
[7]
[8]
[9]
[10]
MAYERGOYZ, I.D. Mathematical Models of Hysteresis. Springer Verlag, 1991.
BERTOTTI, G. Hysteresis in Magnetism. Academic Press, 1998.
DELLA TORRE, E. IEEE Trans. Audio Electroacoust., 14 (1966) 86.
PARDAVI-HORVATH, M.; ZHENG, G.; VÉRTESY, G. AND MAGNI, A. IEEE Trans. Magn. 32
(1996) 4469.
VAN KOOTEN, M.; DE HAAN, S.; LODDER, J.C.; LYBERATOS, A.; CHANTRELL, R. W.; AND
MILES, J. J.; MAGN, J. Magn. Mater. 120 (1993) 145.
PARDAVI-HORVATH, M.; VÉRTESY, G. IEEE Trans. Magn. 33 (1997) 3975.
PARDAVI-HORVATH, M. IEEE Trans. Magn. 32 (1996) 4458.
ADRESS:
Dr. Gábor Vértesy, Dr.Sc.
Research Institute for Technical Physics and Materials Sciences
Hungarian Academy of Sciences
H-1525 Budapest
P.O.B. 49
Hungary
Martha Pardavi-Horváth
Department of Electrical and Computer Engineering
The George Washington University
Washington
DC 20052
USA
85
86
INFORMATION SYSTEM EPI AND USING NEURAL NETWORK FOR ANALYZE OF
TESTS
Jind ich Petrucha
Evropský polytechnický institut, s.r.o.
Abstract: The paper deals about possibility of using neural networks for finding hidden
information's in data of tests. There some hidden information's about difficulties of every tests,
because every tests is generate by random function. The part of the paper describes information's
about neural networks and the special approach for learning neural networks. Main reason is to
explain using three level neural network for data-mining with data sources EPI tests by using Joone
modular system..
Keywords: Neural network, test, data-mining, JOONE, java application, simulator
1. INTRODUCTION
Artificial neural networks are based on behavior of biological neural networks and used mathematical model for
computing their results. The mathematical unit is artificial neuron that is connected to the group of artificial
neurons that represents neural network. There are simulators of neural networks that can modeling by using data
patterns as a input, real results as a output for decision process. There are many types of artificial neural
networks and various types of simulators. Very important is to understand learning phase of using neural
networks because we need to interpreted results of output to real life.
2. SIMULATOR OF ARTIFICIAL NEURAL NETWORK
Now I explain the process of building architecture of artificial neural network that we use in our experiments.
We use the three level architecture of neural network that is most using architecture in practice. We need to set
up number on artificial neurons in each level and create numbers of connections between the levels of networks.
fig.1 Architecture of mathematical neuron that use JOONE engine
We use the simulator on the base Joone core engine JOONE / Java Object Oriented Neural Engine / that is able
to create various types of neural networks. This simulator is on principles of GPL license and there is no problem
to obtain java program to make some experiments. There is visual editor for simple creating of basic model three
level architecture of neural network.
Each neural network is composed of a number of components /layers/ connected together by connections.
Depending on how these components are connected, several neural network architectures can be created on
concept feed forward signal method . Each component has its own pre built mechanism to adjust the weights and
biases according to the chosen learning algorithm. The main characteristic of Joone is that each layer runs on its
87
own thread, representing the unique active element of neural network based on Joone's core engine (modular
components).
fig. 2. Screen of the Joone Neural Net Editor with panel and components
We can see boxes on the panel that contain information about type of layer and specification for each component
of network. We can change type or specification by click mouse on the boxes an fill new information to the edit
box.
fig. 3. Concept of training mode of Joone simulator description is from Complete guide of Joone.
There is object matrix inside of simulator that contains matrix of doubles to store the value of the weights of the
connections and the biases. Each element of matrix contains two values: actual values represented weight, and
corresponding delta value, The delta value is the difference between the actual value and value of the previous
cycle. This is back-propagation system for changing of weights during the learning phase.
The component for creating of hidden levels has various type of function, sigmoidal function or tang hyperbolic
function, radial bias functions and more. The type of function depends on our approach to the network as a
system behavior and our experience with simulators.
2.1 EXTRACT DATE FOR SIMULATOR
Most important phase of project is to preparing data from testing system EPI where we can find many various
form testing results. For our case we use data of subject Macroeconomics because the number of tests is about
300 results. These results represented the patterns for input layer of neural networks. This data must be
processing by normalize components that change interval for input data from 0 -max questions to interval 0-1.
Another phase of normalize must change results of test 0 -100 to interval 0-1. All this can be done by program
PHP language that get data from database of test by using SQL command.
88
2.2 LEARNING PHASE OF SIMULATOR
The first we need to prepare the data for learning phase of simulator to be able to find right patterns in the data.
From the data table we are able to separate data and recognize the test number for all students that used this test.
There problem to create good patterns because system on server generate for each student random set of
questions from bank of tests. This data will be use for learning phase as input pattern to input layer of neural
networks. During the learning phase we split set of pattern for learning and valid phase of this experiment. For
learning phase we have to set up important parameter's.
fig. 4. Window of parameter's that we can set up by using control panel in simulator.
3. TESTING OF THE PROJECT AND RESULTS
During the test of simulator we can see on the graphical panel the curve of MSE that is decreasing global error
between output layer and desire output. The simulator worked about 5000 epoch and during learning phase was
able to teach desires patterns from the input file that represented tests of EPI. For learning phase and good
convergence is important distribution of questions in each test. We need some small part of tests for evaluation
phase of experiment. All this experiments with JOONE environments was run on computer PC CPU 1,5 MHz
and 256 Mb memory size. The JOONE system neural network was running under web explorer with JVM java
virtual machine system and during the creating architecture network there was no problem for simulation part.
fig. 5 Architecture of components JOONE simulator.
89
The basic components are on the figure 5 with input and output layers represented by input and output file. The
chart components allow us to see procedure of learning phase.
4. CONCLUSION
This article describes project of using neural network simulator JOONE and explain main features of this
program. We have got good result for experiments with EPI tests that provide RMSE under five percent. This
simulator can be used for learning neural network with desired outputs. There are some hidden informations that
represented difficulties of test. The simulator of neural network is able to predict result of test on basis questions
pattern.
LITERATURE:
[1]
LACKO, L. Datové sklady analýza OLAP a dolování dat s p íklady v Microsoft SQL Serveru a Oracle. 1.
vyd. Brno : Computer Press, 2003. s. 486. ISBN 80-7226-969-0.
[2]
PETRUCHA, J. Technologie analýzy dat OLAP systémy v prost edí DBPROVE. ACTA
UNIVERSITATIS AGRICULTURAE ET SILVICULTURAE MENDELIANAE BRUNENSIS, 2000,
ro ník XLVIII, íslo 2, s. 149-155. ISSN 1211-8516.
[3]
MARRONE, P. Java Object Orinted Engine, Complete guide [online]. 2007 [cit. 2008-01-20]. Dostupný
z WWW: <www.joone.org>.
[4]
Http://en.wikipedia.org/wiki/JOONE [online]. 2007 [cit. 2008-01-20]. Dostupný z WWW:
<http://en.wikipedia.org>.
Address:
Ing. Jind ich Petrucha, Ph.D.
Osvobození 699
686 04 Kunovice
Tel.: +420572549018
E-mail: [email protected].
90
FUZZY LOGIC AND GRANULAR RBF NEURAL NETWORKS: AN APLICATION TO
THE INPUT-OUTPUT FUNCTION ESTIMATION OF SALES PROCESSES
Milan Mar ek
Silesian University
Abstract: In this paper the outputs of RBF neurons of an RBF neural network are normalized and
the RBF function is modified by Gaussian cloud concept. In this case the activation (membership)
function values of hidden layer neurons are not known precisely, so are fuzzy, so are a Gaussian
distribution, and so are equivalence classes in rough set theory [14]. These RBF neural networks, i.
e. soft or granular neural networks are able to handle a state of the imprecise membership function
value about the true expectation value. In this study we concern with learning aspects of soft and
granular RBF networks. We also compare the results from the soft and granular network with that
from classic RBF network. As an illustration, we consider the case study related to task of time
series approximation of the sales process.
Keywords: Probabilistic time-series models, fuzzy system, classic and soft RBF network, cloud
models, granular computing, Mean Squares Error.
1 INTRODUCTION
The input-output function identification plays pivotal role in both fuzzy logic and neural network. In neural
network this task is performed as an approximation framework where the neural network weights are mostly
modified with backpropagation gradient-descent algorithm. In the case of fuzzy systems it is usually assumed,
the input-output pairs have the structure of fuzzy if-then rules or fuzzy relations, which must be known in
advance. Since finding the exact solution of fuzzy relations is very difficult and in practice unrealistic, more
sophisticated approaches are considered very frequently. For instance in [8, 9] is shown how to combine selforganizing network to cluster input-output data-pairs to obtain the structure of fuzzy if-then rules. In [13] is
illustrated how directly to compute the crisp output values from the adaptive fuzzy systems by soft RBF neural
network proposed by [5]. This approach was applied to sales forecasting of a company [12]. A new model of
neural networks have been studied with complex (granules) activation (membership) functions which finally
resulted in the methodology of granular computing [11].
The paper is organized as follows. Section 2 briefly present basic notions of soft networks for representation of
fuzzy additive systems. Section 3 introduces the architecture of granular network with the aim to highlight its
learning algorithm. Section 4 compares the adaptive neural systems for sales time series modelling and
benchmarks adaptive neural systems against a statistical approach. Section 4 concludes the paper.
2 SOFT (FUZZY LOGIC) NEURAL FUNCTION ESTIMATORS
In function estimation of complex input-output systems the fuzy systems and neural networks estimate a function
without requiring a mathematical description of how output functionally depends on the output. As mentioned in
[13] soft neural networks are able represent fuzzy systems to estimate sample input-output fuctions.
In function estimation of complex input-output systems the fuzy systems and neural networks estimate a function
without requiring a mathematical description of how output functionally depends on the output. As mentioned in
[13], [8] but neural networks and fuzzy systems differ in how their estimate sampled functions. Fuzzy systems
estimate functions with fuzzy sets samples. The fuzzy system maps input fuzzy sets to output fuzzy sets. The
fuzzy inference computes the output fuzzy sets. Centroidal output converts fuzzy sets vector B to a scalar. The
most popular centroidal defuzzification technique uses all the information in the fuzzy output distribution to
compute the crisp y value as the centroid ~
y , i.e.
~
y
n
j 1
yj
j
( x) /
n
j 1
j
( x)
(1)
where y j stands for the centre of gravidity of the jth output singleton, the notation
is used for a membership
91
function and n denotes the number of rules.
In [13] is shown, how to obtain fuzzy rules and how to determine the weights wi for fuzzy system using RBF
networks.
(a)
(b)
Fig. 1 Classic (a) and fuzzy logic (soft) (b) RBF neural network architecture
To show the similarity of the RBF neural network and the fuzzy system, consider again the scalar output y . The
classic RBF network (see Fig. 1a) computes the output data set as
y t =
s
j 1
v j ,t
2
(x t , c j ) =
s
j 1
v j o j ,t ,
t = 1, 2, ..., N
(2)
where N is the size of data samples, s denotes the number of the hidden layer neurons, and. cj represent the
centres of activation functions 2 . The hidden layer neurons receive the Euclidian distances ( x c j ) and
compute the scalar values o j ,t of the Gaussian function
2
(xt , c j ) that form the hidden layer output vector
ot .
Finally, the single linear output layer neuron computes the weighted sum of the Gaussian functions that form the
output value of y t .
If the scalar output values o j ,t from the hidden layer will be normalised, where the normalisation means that the
sum of the outputs from the hidden layer is equal to 1, then the RBF network will compute the normalised
output data set y t as follows
y t =
s
j 1
v j ,t
o j ,t
s
j 1
o j ,t
=
s
v j ,t
j 1
2
( xt , c j )
s
2
, t = 1, 2, ..., N.
(3)
( xt , c j )
j 1
The graphical representation of the form of RBF neural networks which produce the output values according to
the formula (3) is shown in Fig. 1 (b).
The similarity of approximation schemes (3) and (1) is obvious. From these schemes is shown that the weights
v j ,t in Eq. (3) to be learned correspond to wi in Eq. (1), and
2
(. / .) to
B
( y ) in Eq. (1). Thus, the
adaptive fuzzy system uses neural techniques to abstract fuzzy principles and to choose the weights wi , and
gradually refine those principles as the system samples new cases. These properties were firstly recognised by V.
Kecman [5]. In Fig. 1 (b), the network with one hidden layer and normalised output values o j ,t is the fuzzy
logic model or the soft RBF network.
The frequently used learning technique uses clustering to find a set of centers wich more accurately reflect the
distribution of the data points. For example by using K-means clustering algorithm, the member of K centers
92
must be decided in advances. After choosing the centers w, the standard deviations j can be selected as j ~ cj,
where cj denotes the average distance among the centers wj. To train the weights vj, the firs-order gradient
procedure is used. These weights can be adapted by the error back-propagation algorithm. In this case, the
weight update is particularly simple. If the estimated output for the single output neuron is y t , and the correct
output should be yt , then the error et is given by et = yt - y t and learning rule has the form
v j ,t
o j ,t et , j = 1, 2, ..., s; t = 1, 2, N
v j ,t +
where the term
(4)
is a constant called the learning rate, o j ,t is the normalised output signal from the hidden
layer. Typically, the updating process is divided into epochs. Each epoch involves updating all the weights for all
the examples.
3 GRANULAR RBF NETWORK
Next, to improve the abstraction ability of soft RBF neural networks with architecture depicted in Fig. 1, we
replaced the standard Gaussian activation (membership) function of RBF neurons with functions based on the
normal cloud concept [3, 4].
Cloud models are described by three numerical characteristics [3] (see Fig. 2): Expectation (Ex) as most typical
sample which represents a qualitative concept, Entropy (En) as the uncertainty measurement of the qualitative
concept and Hyper Entropy (He) which represents the uncertain degree of entropy. En and He represent the
granularity of the concept, because both the En and He not only represent fuzziness of the concept, but also
randomness and their relations. This is very important, because in economics there are processes where the
inherent uncertainty and randomness are associated with different time. Then, in the case of soft RBF network,
the Gaussian membership function
2
(x t , c j ) = exp (x t
2
(. / .) in Eq. (6) has the form
E( x j ) / 2( En ) 2 = exp (x t
c j ) / 2( En ) 2
(5)
where En is a normally distributed random number with mean En and standard deviation He , E is the
expectation operator.
Fig. 2 Illustration of the three numerical characteristics of a normal cloud concept
4 AN APPLICATION
We illustrate the classic, fuzzy logic (soft) and cloud (granular) RBF neural networks on the input output
function estimation of a sales process. The time plot of the data set used in this application (the 724 daily sales
for Hansa Flex company, 2004-2005) are shown in [13].
y t
0.1248 yt
(6)
7
and
yt
yt
7
0.93868
(7)
t 7
93
In the RBF neural network framework, the non-linear function f(x) was estimated according to the expressions
(2). In the case of RBF fuzzy logic network, the non-linear input output approximation function was estimated
according to the formula (3). Next, the fuzzy logic RBF neural network was extended towards estimation with (a
priori known) noise levels of the entropy. Noise levels are indicated by hyper entropy. It is assumed that the
noise level is constant over time. We select, for practical reasons, that the noise level is a multiple, say 0.015, of
entropy. In Table 1, we give the achieved results of approximation ability in dependence on various number of
RBF neurons. The mean square error (MSE) was used to measure the approximation ability.
Gaussian
with
Neural network Gaussian (classic normalised
architecture
RBF network)
outputs
(soft
RBF network)
Number of RBF MSE
neurons
RBF network representations for model (6)
3
1.439
0.698
5
0.729
0.693
10
0.687
0.675
15
0.697
0.681
RBF network representations for model (7)
3
0.783
0.646
5
0.810
0.632
10
0.607
0.571
15
0.582
0.563
Gaussian (classic
RBF
network
with
normal
cloud concept)
Gaussian
soft
RBF (with normal cloud concept granular
network)
1.503
0.817
0.671
0.681
0.729
0.716
0.678
0.678
0.786
0.803
0.607
0.582
0.647
0.630
0.571
0.563
Table 1 Approximation results of various RBF´s networks related to the different number of clusters (RBF neurons)
The mean (centre), standard deviation of the clusters (RBF neurons) are computed using K-means algorithm.
Comparing both approaches, i.e. models based on the Box-Jenkins methodology (the MSE for model expressed
by Eq. (6) is 0.7793 and by Eq. (7) is 0.74660 respectively), and models based on RBF network approaches, we
clearly see that models based on RBF networks are better approximation models because the estimated values
are close to the actual values. As shown in Table 1, models that generate the best MSE´s are soft RBF
networks.
5 CONCLUSION
In this article, we have extended RBF neural network methodology to approximate the non-linear time series
data using normal cloud models in the role of standard Gaussian activation (membership) function for RBF
neurons. This was done by formulating a hyper entropy of standard deviation (entropy) of the Gaussian cloud
model.
To approximate the input-output function of a business process, the RBF neural network approach was applied
on the daily sales data of the Hansa Flex company and compared with an approach based on the statistical
procedures. For the sake of approximation abilities we evaluated 34 models. Two models are based on the BoxJenkins time series analysis approach, and 32 models are based on the neural (fuzzy logic) methodology. Using
the disposable data a very appropriate model is the soft RBF network with activation functions based on the
granular concept. It is also interesting to note that the most computationally intensive models, the model based
on the Box-Jenkins methodology, is newer considered best.
ACKNOWLEDGEMENT
This work was supported by the grants VEGA 1/0024/08 and GA R 402/08/0022.
REFERENCES:
[1]
AN, P. E.; BROWN, M.; HARRIS, C. J.; CHEN, S. Comparative aspects of neural network algorithms
for online modelling of dynamic processes. J. of Institute of Mechanical Engineering, 207, 1993, 223-241
[2]
BOX, G. E. P. AND JENKINS, G. M.: Time Series Analysis, Forecasting and Control. San Francisco,
CA : Holden-Day, (1970).
[3]
CHANGYU, LIU; DEYI, LI; YI, DU; XU, HAN. Normal Cloud Models and Their Interpretation. The
94
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
11th World Congress of International Fuzzy Systems Association (IFSA 2005). Beijing China, July 2831, 2005, Springer, Volume III, 1540-1543.
CHANGYU, LIU; DEYI, LI; YI, DU; XU, HAN. Normal Cloud Models and Their Interpretation. The
11th World Congress of International Fuzzy Systems Association (IFSA 2005). Beijing China, July 2831, 2005, Springer, Volume III, 1540-1543.
KECMAN, V. Learning and Soft Computing: Support Vector Machines, Neural Networks, and Fuzzy
logic Models. Massachusetts Institute of Technology, 2001, The MIT Press.
KELLER, J. M.; YAGER, R. R.; TAHANI, H. Neural network implementation of fuzzy logic. Fuzzy Sets
and Systems 45 (1992), 1-12.
KOCVARA, B. Time Seres Modelling Using Statistial (Econometric) Methods and Machine Learning.
Diploma work, U. of ilina, Fac. Informatics and Management Sience, June 2007.
KOSKO, B. Neural networks and fuzzy systems a dynamic approach to machine intelligence. Prentice
Hall, Inc., 1992.
MARCEK, D. Determination of Fuzzy Relations for Economic Fuzzy Time Series models by Neural
Networks. Computing and Informatics, Vol. 22, 2003, 457-471.
MARCEK, D. Stock Price Forecasting: Autoregressive Modelling and Fuzzy Neural Network. Mathware
& Soft Computing, Vol. 7, No. 2-3: 139-148 (2000).
MARCEK, M.; MAR EK, D. RBF Neural Network Implementation of Fuzzy Systems: Application to
Time Series Modeling. In. Aijun An et al Eds.:RSFDGrC 2007, LNAI 4482, Springer Verlag Berlin,
Heidelberg (2007), 500-507.
MARCEK, M.; MAR EK, D. Granular RBF Neural Network Implementation of Fuzzy Systems:
Application to Time Series Modeling. Bryam.
MARCEK, D. Econometric and RBF Neural Network Modelling of Economic Time Series. ICSC 2007,
January 26, European Polytechnical Institute Kunovice, Czech Republik, 17-22.
ZADEH, L. A. Granular Computing and Rough Sets Theory. JRS´07. Infobright, York University
Toronto, May 14-16, 2007, Keynote Talk.
ADDRESS
Ing. Milan Mar ek
Faculty of Philosophy and Science
Silesian University949 01
746 01 Opava
Czech Republic
[email protected]
MEDIS Nitra, Ltd.
Pri Dobrotke 659/81
Nitra-Draovce
Slovak Republic
95
96
WAGES FORECASTING USING TIME SERIES MODELS VERSUS SVM METHODS
Duan Mar ek1,2
1
Institute of Computer Science, Faculty of Philosophy and Science, The Silesian University Opava
2
The Faculty of Management Science and Informatics, University of ilina
Abstract: SVM´s modelling approaches are used for automated specification of a functional form of
the mode and for training polynomial models. We provide the fit of the average nominal wages time
series by SVM (Support Vector Machine) model over the period January 1,1991 to December 31,
2006 in the Slovak Republic, and use them as a tool to compare their forecasting abilities with those
obtained using Box- Jenkins methodology [1]. Some methodological contributions are made to
dynamic and SVM´s modelling approaches in economics and to their use in time series modelling.
Keywords: Support vector machines, learning machines, time series analysis and forecasting,
1 INTRODUCTION
The wages time series are in fact stochastic in which successive observations are dependent and can be
represented by a linear combination of independent random variables t , t 1 , ... . If the successive observations
are highly dependent, we should use in model past values of the time series variable and (or) current and past
values of the error terms { t }. There are available techniques which are designed to exploit this dependency and
which will generally produce superior forecasts. Many of these techniques are based on developments in time
series analysis recently presented by Box and Jenkins [1].
In Support Vector Machines (SVM´s), a non-linear model is estimated based on solving a Quadratic
Programming (QP) problem. In the next section of this article, we briefly describe the framework of SVM´s
methods and support vector (SV) regressions within which our empirical investigation is conducted. In the next
section of this article, we briefly describe the framework of SVM´s methods and support vector (SV) regressions
within which our empirical investigation is conducted. Section 3 provides a fit of the SV regression model using
the Mathlab program, discusses the circumstances under which SV regression outputs are conditioned and
corresponding interpretation of SV regression results is also considered. In Section 4 attention is confined to the
application of Box-Jenkins steps for time series modeling of wages. Central to the interest of the ARMA model
will be the basic concept of last squares estimation when applied to the linear model and testing of its adequacy.
A section of conclusions will close the paper.
2 SUPPORT VECTOR MACHINE FOR FUNCTIONAL APPROXIMATION
This section presents quickly a relatively new type of learning machine the SVM´s applied in the regression
(functional approximation) problems. For details we refer to [2, 5, 10, 11, 13, 14, 15]. The general regression
learning task is set as follows. The learning machine is given n training data, from which it attempts to learn the
n
n
input-output relationship y f (x) , where { xi , yi
, i 1,2,..., n } consists of n pairs { yi , xi } i 1 . The xi
denotes the ith input and yi is the ith output. The SVM´s considers regression functions of two forms [6]. The
first one is
n
f (x )
(
i
*
i
(1)
) (x i , x j ) b
i 1
j 1
where
, i* are positive real constants (Lagrange multipliers), b is a real constant (see formulas (3)-(5)), (. / .)
is the kernel function. Admissible kernels have the following forms: ( x i , x j ) x Ti x j (linear SVM´s)
i
(x i , x j )
( x Ti x j 1) d (polynomial SVM´s of degree d),
(xi , x j )
exp
xi
xj
2
2
(radial basis SVM´s),
where is a positive real constant and other (spline, b-spline, exponential RBF, etc.).
The second approximation function is of the form [6]
97
n
f ( x, w )
wi
i
(2)
( x) b
i 1
where (.) is a non-linear function (kernel) which maps the input space into a high dimensional feature space. In
contrast to Eq. (1), the regression function f (x, w ) is explicitly written as a function of the weights w that are
subject of learning.
The SV regression approach is based on defining a loss function that ignores errors that are within a certain
distance of the true value. This type of function is referred to as an -insensitive loss function (see Fig. 1 and Fig.
2).
Fig. 1: The insensitive band for one dimensional linear (left), non-linear (right) function
Fig. 1 shows an example of an one dimensional function with an -insensitive band. The variables , * measure
the cost of the errors on the training points. These are zero for all points inside the -insensitive band, and only
the points outside the -tube are penalised by the so called Vapnik´s -insensitive loss function.
In regression, there are different error (loss) functions in use and that each one results from a different final
model. Fig. 2 shows the typical shapes of three loss functions [2, 6]. Left: quadratic 2- norm. Middle: absolute
error 1-norm. Right: Vapnik´s -insensitive loss function.
Fig. 2: Error (loss) functions
Formally this results from solving the following Quadratic Programming problem
min * R( w, ,
w ,b , ,
*
n
) = 1 wT w C (
2
i 1
i
*
i
)
(3)
where C is the value of capacity degrees
yi
subject to w
T
w T (x ) b
i
i 1,2, ..., n,
( x) b
*
i
i 1,2, ..., n,
yi
i
,
*
i
0
i 1,2, ..., n.
To solve (3), (4) one constructs the Lagrangian
98
(4)
L p (w, b, i ,
*
i
,
*
i
i,
,
*
i
i,
n
*
i
n
1 T
w w
2
)=
*
i
(
C
(
*
i
i
n
)
i
i 1
(
* *
i i
i i
w T ( x i ) b)
yi
i
i 1
n
w T ( x i ) b)
yi
(
(5)
)
i 1
i 1
by introducing Lagrange multipliers
*
i
,
i
0,
0 , i = 1, 2, ..., n. The solution is given by the saddle
*
i
,
i
point of the Lagrangian [3]
max
min * L p (w , b, i ,
*
*
,
i
i
,
w , b, i ,
i
*
i
,
i
*
i
,
,
i
*
i
,
(6)
)
i
subject to constrains
Lp
n
0
w
Lp
*
i
(
w
i
) (x i ),
i 1
n
0
b
Lp
(
*
i
i
) 0,
(7)
i 1
Lp
0,
i
0
0
0
0
i
C , i 1,..., n,
*
i
C , i 1,..., n,
i
Lp
Lp
0,
*
i
*
i
which leads to the solution of the QP problem:
max*
,
i
1 n
(
2 i, j 1
i
*
i
)(
*
j
j
n
) (x Ti x j )
(
i
i 1
subject to (7).
After computing Lagrange multipliers
i
*
i ,
and
n
f ( x)
(
*
i
i
*
i
n
)
yi (
*
i
i
)
(8)
i 1
one obtains of the form of (1), i.e.
) (x i , x j ) b .
(9)
i 1
j 1
By substituting the first equality constraint of (7) in (2), one obtains the regression hyperplane as
f (x , w )
(10)
w T ( x) b
Finally, b is computed by exploiting the Karush-Kuhn-Trucker (KKT) conditions [3]. These conditions state that
at the optimal solution the product between the dual variables and constrains has to vanish. For the SVM´s, we
*
*
obtain from, i , i 0 and i* , i* 0 , i = 1, 2, ..., n that (C
0 , respectively.
0 and (C
i ) i
i ) i
Hence,
i
0 (
*
i
0) if and only if C
n
b
yk
(
*
i
(C
i
) . From this and KKT conditions, we obtain
i
*
i
) (x i , x k )
for
i
*
i
) (x i , x k )
for
k
(0, C ),
*
k
(0, C ).
(11)
i 1
b
yk
n
(
i 1
When solving the QP-problem with an interior point method [11, 12], it is more convenient to exploit primaldual properties and to obtain b as a by-product of the optimization algorithm. Also note that for all samples
inside the
- band,
i
and
*
i
are zero. Therefore, we have a sparse expansion (1) of w in terms of xi. The
99
samples that come with non-vanishing coefficients are called Support Vectors.
3 APPLICATION OF ARMA MODELING IN WAGES PREDICTION PROBLEM
Most models for the time series of wages have centered about autoregressive (AR) processes. Central to the
interest of the ARMA model will be the basic concept of last squares estimation when applied to the linear
model and testing of its adequacy.
To illustrate of the Box-Jenkins methodology, consider the wages time readings {yt} of the Slovak economy. We
would like to develop a time series model for this process so that a predictor for the process output can be
developed. The quarterly data were collected for the period January 1, 1991 to December 31, 2006 which
provides total of 64 observations (displayed in Fig. 1). To build a forecast model we define the sample period for
analysis x1, ..., x60, i.e. the period over which we estimate the forecasting model and the ex post forecast period
(validation data set), x 61, ..., x 64 as the time period from the first observation after the end of the sample period to
the most recent observation. By using only the actual and forecast values within ex post forecasting period only,
the accuracy of the model can be calculated.
Fig. 1. Nominal wages (January 1991 - December 2006)
Fig. 2. The wages data after transformation to
stationary ARMA type process
To determine appropriate Box-Jenkins model, a tentative ARMA model in identification step is identified. In
order to fit a time series to data, first the data were transformed to a stationary ARMA type process, i.e. the data
must be modeled by a zero-mean and constant variability. After eliminating trend and seasonal component, the
natural logarithms of the once differenced data y t = xt - x t 4 are shown in Fig. 2. There are various methods and
criteria for selecting of an ARMA model. In this section we concentrate on model identification by HannanRissanen procedure [4]. The Matlab program developed in [7] selects as well as estimates the model. Using this
program the model for {yt} time series was tentatively identified as ARMA(1,3) with preliminary estimates of
the model parameters as follows
y t
0.0017 0.46 y t
1
0.905
t 1
0.588
t 2
0.365
t 3
(12)
After fitting a model to a given data set, the goodness of fit of the model is usually examined to see if it is indeed
an appropriate model. There are various ways of checking if a model is satisfactory. A good way to check the
adequacy of an overall Box-Jenkins model is to analyze the residuals y t - y t . If the residuals are truly random,
the autocorrelations and partial autocorrelations calculated using the residuals should be statistically equal to
zero. Since the residuals are also ordered in time, we can treat them as a time series and calculate the sample
correlation function of the residuals and see if it behaves to be a stationary random sequence.
Instead of looking at the correlation function we used the portmanteau test based on the LjungBox statistic. The test statistic is [8, 9]
Qh
which has an asymptotic chi square (
2
N (N
2)
K
k
re ( k ) 2
k
1 N
) distribution with K-p-q degrees of freedom if the model is appropriate.
2
1
If Q K
. The chi-square statistic applied to these
, ( K p q ) the adequacy of the model is reject at the level
autocorrelation is 13.79, and so we have no evidence to reject the model.
100
Because the model is written in terms of a stationary time series to obtain a point forecast, the final model must
be rewritten in terms of the original data and then solved algebraically for x t . The forecasts obtained from this
model for time t 61,62,63,64 are shown in Fig. 3.
Fig. 3 Forecasts of wages data ARMA(1,3) model
4 EXPERIMENTING WITH NON-LINEAR SV REGRESSION
The average nominal wages Wt can be described by the following regression equation
Wt = b + a W t
4
+
t
(13)
where a, b are the parameters, t is the disturbance term. We demonstrate here the use of the SV regression
framework for estimating the model given by Eq. (1). If Wt exhibits a curvilinear trend, one important approach
for generating an appropriate functional non-linear form of the model is to use the SV regression in which the Wt
is regressed either against Wt-4 or the time by the form
W t
where x t
n
i 1
wi
i
(x t ) b
(14)
(Wt 1 , Wt 4 , ...) is the vector of time sequence (regressor variable), or
W t
n
i 1
wi
i
(x t ) b
(15)
where x t (1,2, ..., 60) is the vector of time sequence (regressor variable). Our next step is the evaluation of the
goodness of last three regression equations to the data insite the estimation period expressed by the Mean Square
Error MSEA, and the forecast summary statistics the MSEEfor each of the models out of the estimation period.
One crucial design choice in constructing an SV machine is to decide on a kernel. The choosing of good kernels
often requires lateral thinking: many measures of similarity between inputs have been developed in different
contexts, and understanding which of them can provide good kernels depends on the insight into the
application´s domains. Tab 1 and corresponding Fig. 3 shows SVM´s learning of the historical period illustrating
the actual and the fitted values by using various kernels.
Tab. 1 presents the results for finding the proper model by using the quantities MSE. As shown in Tab. 1, the
model that generates the best forecasts is the model with MSEE = 62894 (Fig. 4a).
101
LOSS
FUNCTION
MSEA
MSEE
1150
Quadratic
15590
62894
RBF
600
Quadratic
10251
58900
causal (13)
Exp. RBF
600
Quadratic
3315
70331
4d
Time S. (14)
RBF
1.0
Quadratic
0.421
none
3
ARMA(1,3)
25289
104830
Causal (2)
44181
44549
MODEL
KERNEL
4a
causal (13)
RBF
4b
causal (13)
4c
Fig.
DEGREE-d
Tab. 1 SV regression results of three different choice of the kernels and the results of the dynamic model on the training set (1993Q1 to
2003Q4). In two last columns the fit to the data and forecasting performance respectively are analysed. See text for details.
The results shown in Tab.1 were obtained using degrees of capacity C = 104. The insensivity zone
and the
degrees of capacity are most relevant coefficients. To learn the SV regression machine we used partly modified
software developed by Steve R. Gunn [5]. The use of an SV machine is a powerful tool to the solution many
economic problems. It can provide extremely accurate approximating functions for time series models, the
solution to the problem is global and unique.
5 CONCLUSION
In this paper, we have examined the SVM´s approach to study linear and non-linear models on the time series of
wages in the Slovak Republic. For the sake of calculating the measure of the goodness of fit of the regression
model to the data we evaluated five models. One model was based on causal regression and for models on the
Support Vector Machines methodology. The benchmarking was performed among an ARMA model, by
traditional statistical technique in regression tasks and SVM´s method. The SVM´s approach was illustrated on
the conventional regression function. As it visually is clear from Fig. 4, this problem was readily solved by a SV
regression with excellent fit of the SV regression models to the data.
Acknowledgement: This work was supported by Slovak grant foundation under the grant No. VEGA 1/0024/08
and from the Grant Agency of the Czech Republic under the grant No. GA R 402/08/0022. I thank B. Ko vara
for computational support.
REFERENCES:
[1]
BOX, G. E.; JENKINS, G. M. Time Series Analysis, Forecasting and Control. Holden-Day, San
Francisco, CA 1976.
[2]
CRISTIANINI, N.; SHAWE-TAYLOR, J. An introduction to support vector machines. Cambridge
University Press, 2000.
[3]
FLETCHER, R. Practical methods of optimization. John Wiley and Sons, Chichester and New York,
1987.
[4]
GRANGER, C. W. J; AND NEWBOLD, P. Forecasting Economic Time Series. Academic Press, NY,
1986.
[5]
GUNN, S. R. Support Vector Machines for Classification and Regression. Technical Report, Image
Speech and Intelligent Systems Research Group, University of Southampton, 1997.
[6]
KECMAN, V. Learning and Soft Computing Support Vector Machines, Neural Networks and Fuzzy
logic Models. Massachusetts Institute of Technology, 2001.
[7]
KO VARA, B. Time Series Modeling Using Statistical (Economeric) Methods and Machine Learning.
Diploma work, Faculty of Management Science and Informatics, University of ilina, June 2007.
[8]
MAR EK, D.; MAR EK, M. Time Series Analysis, Modelling and Forecasting with Applications in
Economics. EDIS University of ilina, 2001.
[9]
MONGOMERY, D. C.; JOHNSTON, L. A.; GARDINER, J. S. Forecasting and Time Series Analysis.
McGraw-Hill, Inc., 1990.
[10] SUYKENS, J. A. K. Least squares support vector machines for classification and nonlinear modelling,
Neural Network World 1-2, (2000): 29-47.
[11] VAN GESTEL, T.; SUYKENS, J.; BART DE MOOR, DIRK-EMMA BAESTAENS. Volatility tube
support vector machines. Neural Network World, (2000), No.1-2: 287-297.
[12] VANDERBEI, R. J.; LOQO. An interior point code for quadratic programming. TRSOR-94-15,
Statistics and Operation Research, Princeton Univ., NY, 1994.
102
[13]
[14]
[15]
VAPNIK, V. Statistical learning theory. John-Wiley, New-York, 1998.
VAPNIK, V. The nature of statistical learning theory. Springer Verlag, New-York, 1995.
VAPNIK, V. The support vector method of function. In: Nonlinear Modeling: Advanced Black-Box
Techniques, Suykens, J.A.K., Vondewalle, J. (Eds.), Kluwer Academic Publishers, Boston, 1998: 55-85.
a)
b)
c)
d)
Fig. 4 Training results for different kernels, loss functions and
of the SV regression (see Tab 1).
ADDRESS
Prof. Ing. Duan Mar ek, CSc.
Institute of Computer Science
Faculty of Philosophy and Science
The Silesian University Opava
The Faculty of Management Science and Informatics
University of ilina
Tel.: 041/5134409
103
104
ßÐÐÔ×ÝßÌ×ÑÒ ÑÚ ÔßÐÔßÝÛ ÌÎßÒÍÚÑÎÓ ÌÑ
ÚÎßÝÌ×ÑÒßÔ ÝßÔÝËÔËÍ
Þ(»¬·-´¿ª Ú¿¶³±²ô Æ¼»²4µ w³¿®¼¿
Ú¿½«´¬§ ±º Û´»½¬®·½¿´ Û²¹·²»»®·²¹ ¿²¼ Ý±³³«²·½¿¬·±²ô Þ®²± Ë²·ª»®-·¬§ ±º Ì»½¸²±´±¹§
ß¾-¬®¿½¬æ Ì¸» °¿°»® ¼»¿´- ©·¬¸ Ô¿°´¿½» ¬®¿²-º±®³- ±º º®¿½¬·±²¿´ ·²¬»¹®¿´- ¿²¼ º®¿½¬·±²¿´ ¼»®·ó
ª¿¬·ª»-ò Þ§ ³»¿²- ±º Ô¿°´¿½» ¬®¿²-º±®³ -±³» ¾¿-·½ º±®³«´¿» ±º º®¿½¬·±²¿´ ½¿´½«´«- ¿®» ¼»®·ª»¼
¿- ©»´´ò
Õ»§ ©±®¼-æ Ô¿°´¿½» ¬®¿²-º±®³ô º®¿½¬·±²¿´ ·²¬»¹®¿´ ¿²¼ ¼»®·ª¿¬·ª»ò
ïò ×ÒÌÎÑÜËÝÌ×ÑÒ
Ì¸» ¬¸»±®§ ±º Ô¿°´¿½» ¬®¿²-º±®³ ·- ¿² ·³°±®¬¿²¬ °¿®¬ ±º ¬¸» ³¿¬¸»³¿¬·½¿´ ¾¿½µ®±«²¼ ®»¯«·®»¼ º±® »²¹·ó
×² ¬¸·- °¿°»® ©» ¼»®·ª» -±³» ¾¿-·½ º±®³«´¿» ±º º®¿½¬·±²¿´ ½¿´½«´«- ¾§ ³»¿²- ±º Ô¿°´¿½» ¬®¿²º±®³ò Í»ª»®¿´
³»¬¸±¼- ·²½´«¼·²¹ Ô¿°´¿½» ¬®¿²-º±®³ ¿®» ¼·-½«--»¼ ¬± ·²¬®±¼«½» Î·»³¿²²óÔ·±«ª·´´» º®¿½¬·±²¿´ ·²¬»¹®¿´-ò
Ú·®-¬ ©» ®»½¿´´ -±³» ·³°±®¬¿²¬ ²±¬·±²- º®±³ ¬¸» ¬¸»±®§ ±º Ô¿°´¿½» ¬®¿²-º±®³ò É» -¬¿®¬ ©·¬¸ Ú±«®·»® ·²¬»¹®¿´
º±®³«´¿ ©¸·½¸ »¨°®»--»- ¬¸» ®»°®»-»²¬¿¬·±² ±º ¿ º«²½¬·±² ºï ø¨
ä ¨ ä ï ·² ¬¸» º±®³
Æ ï
Æ ï
ï
ºï ø¨÷ ã
»¶ÿ¨ ¼ÿ
» ¶ÿ¬ ºï ø¬÷¼¬æ
øï÷
î
É» ²»¨¬ -»¬ ºï ø¨÷
ð ±²
ä ¨ ä ð ¿²¼ ©®·¬»
ºï ø¨÷ ã »
½¨
º ø¨÷Øø¨÷ ã »
½¨
º ø¨÷å
¨ â ðå
øî÷
º ø¬÷¼¬æ
øí÷
º ø¬÷¼¬æ
øì÷
©¸»®» Øø¨÷ ·- Ø»¿ª·-·¼» º«²½¬·±² ¿²¼ ½
º ø¨÷ ã
»½¨
î
Æ
ï
Æ
»¶ÿ¨ ¼ÿ
ï
»
¬ø½õ¶ÿ÷
ð
É·¬¸ ¿ ½¸¿²¹» ½ õ ¶ÿ ã ° ô ¶¼ÿ ã ¼° ©» ®»©®·¬» øí÷ ¿Æ ï
Æ
»½¨ ½õ¶ï ø° ½÷¨
¼°
»
º ø¨÷ ã
»
î
ð
½ ¶ï
°¬
Ì¸«- Ô¿°´¿½» ¬®¿²-º±®³ ±º º ø¬
Ôºº ø¬÷¹ ã Ú ø°÷ ã
©¸»®» »
°¬
Æ
ï
»
°¬
ð
¼¬å
Î» ° â ðå
øë÷
·- ¬¸» µ»®²»´ ±º ¬¸» ¬®¿²-º±®³ ¿²¼ ° ·- ¬¸» ¬®¿²-º±®³ ª¿®·¿¾´» ©¸·½¸ ·- ¿ ½±³°´»¨ ²«³¾»®ò Î»-«´¬
Ô
Ñ¾ª·±«-´§ Ô ¿²¼ Ô
±º º ø¬÷ æ
ï
ï
ºº ø¬÷¹ ã º ø¬÷ ã
ï
î
Æ
½õ¶ÿ
»°¬ Ú ø°÷¼°å
½ â ðæ
øê÷
½ ¶ÿ
¿®» ´·²»¿® ±°»®¿¬±®- ò Ò±© ´»¬ «- ·²¬®±¼«½» ½±²¼·¬·±²- ±º »¨·-¬»²½» Ô¿°´¿½» ¬®¿²-º±®³
ß º«²½¬·±² º ø¬÷ ·- -¿·¼ ¬± ¾» ±º »¨°±²»²¬·¿´ ±®¼»® ¿ â ð ±² Ö ã Åðå ï÷ ·º ¬¸»®» »¨·-¬- ¿
°±-·¬·ª» ½±²-¬¿²¬ Õ -«½¸ ¬¸¿¬ º±® ¬ î Ö
¶º ø¬÷
Õ»¿¬ æ
øé÷
"×ÝÍÝóÍ×ÈÌØ ×ÒÌÛÎÒßÌ×ÑÒßÔ ÝÑÒÚÛÎÛÒÝÛ ÑÒ ÍÑÚÌ ÝÑÓÐËÌ×ÒÙ ßÐÐÔ×ÛÜ ×Ò ÝÑÓÐËÌÛÎ ßÒÜ ÛÝÑÒÑÓ×Ý ÛÒÊ×ÎÑÒÓÛÒÌÍ!
ÛÐ× Õ«²±ª·½»ô Ý¦»½¸ Î»°«¾´·½ô Ö¿²«¿®§ îëô îððè
ïðë
É» ©®·¬» -§³¾±´·½¿´´§ ¿º ø¬÷ ã Ñø»¿¬ ÷ ¿- ¬ ÿ ïæ
Ì¸»±®»³ ïò ×º ¿ º«²½¬·±² º ø¬÷
±º »¨°±²»²¬·¿´ ±®¼»® »¿¬ ¬¸»² Ô¿°´¿½» ¬®¿²-º±®³ ±º º ø¬÷ »¨·-¬- º±® ¿´´ ° °®±ª·¼»¼ Î» ° â ¿æ
øðå Ì ÷å ¿²¼
îòÓß×Ò ÎÛÍËÔÌÍ
Æ
ï
º ø¬÷ ãð Ü¬ º ø¬÷ ã
Ü
÷
¬
ï÷
¨÷ø
ø¬
º ø¨÷¼¨å
ðæ
Î
ð
øè÷
·- ¿´·²»¿® ·²¬»¹®¿´ ±°»®¿¬±®ò ß -·³°´» ½¸¿²¹» ±º ª¿®·¿¾´» ø¬ ¨÷ ã « ·² øè÷ ¿´´±©- «- ¬± °®±ª»
Ý´»¿®´§ô Ü
¬¸» º±´´±©·²¹ ®»-«´¬ ø-»»ÅîÃ÷
¬ ï
ÜÅÜ º ø¬÷Ã ã Ü ÅÜº ø¬÷Ã õ º øð÷
æ
øç÷
÷
Ø»²½» ¬¸» ·²¬»¹®¿´ ·² øè÷ ·- ¿ ½±²ª±´«¬·±² ¿²¼ ¬¸» Ô¿°´¿½» ¬®¿²-º±®³ ±º øè÷ ¹·ª»ÔºÜ
º ø¬÷¹ ã Ôºº ø¬÷ ¹ø¬÷¹ ã Ôºº ø¬÷¹Ôº¹ø¬÷¹ ã °
©¸»®»
¹ø¬÷ ã
Î»-«´¬ øïð÷ ·- ¿´-± ª¿´·¼ º±®
ï
¬
÷
Ùø°÷ ã °
å
Ú ø°÷
ðå
øïð÷
æ
ã ð ô ¿²¼
ï
¬
´·³ Ô
Ü
ÅÜ
ã ´·³ °
÷
ÿð
ø õ ÷
º ø¬÷Ã ã Ü
ã ïæ
øïï÷
ÿð
º ø¬÷ ã Ü
ÅÜ
º ø¬÷Ãæ
øïî÷
Ú±®³«´¿ øïð÷ ½¿² ¾» «-»¼ º±® »ª¿´«¿¬·²¹ ¬¸» º®¿½¬·±²¿´ ·²¬»¹®¿´ ±º ¿ ¹·ª»² º«²½¬·±² «-·²¹ ¬¸» ·²ª»®-» Ô¿°´¿½»
¬®¿²-º±®³ò Ì¸» º±´´±©·²¹ »¨¿³°´»- ·´´«-¬®¿¬» ¬¸·- °±·²¬ò
Ñ®ô »¯«·ª¿´»²¬´§
Ü
×² °¿®¬·½«´¿®ô ·º
ã
ï
î
¬ ãÔ
õ ï÷
¬ ¹ã
ÔºÜ
°
õ ï÷
ï
°
ï
î
ÔºÜ
Ñ®
ï
ï
° ø°
ãÔ
÷
»¿¬ ¹ ã
Ûø
Ü
ï
î
ã
»¿¬ ã Ô
ï
î
¬¸»²
ï
°
ï
°ø°
¿÷
ï
ã °
ï
° ø°
ï
ï
°
©¸»®»
×² °¿®¬·½«´¿® ô ·º
õ
² õ ï÷ ²õ ï
¬ îå
² õ ïî õ ï÷
¬² ã
×¬ ¿´-± º±´´±©- º®±³ øïð÷ ¬¸¿¬
»¿¬ ã Ô
ã
õ õï
øïí÷
õ ï÷
¬
õ ï÷
õ
øïì÷
æ
ã ² ·- ¿² ·²¬»¹»® ¬¸»² øïì÷ ¹·ª»Ü
Ü
ïæ
õ õï
÷ã
÷
Æ
øïë÷
å ¿ â ðæ
¿
ïõ
õï
ï
÷
ïæ
²â
°
¿
øïê÷
¬
ã
õ ï÷
õ ¿Ûø
õ ïå ¿÷å
øïé÷
¬
ï ¿ø¬ ¨÷
¨
»
øïè÷
¼¨æ
ð
ï
»¿¬ ã °
Æ
ð
¬
ï
° »¿ø¬
÷
ã ø°«¬¬·²¹
°
ã ¨÷ ã
ïðê
øïç÷
î»¿¬
ã °
Æ
°
¿¬
»
°
»¿¬
¼¨ ã ° »®º ø ¿¬÷å
¿
¨î
ð
©¸»®» »®º ø¬
Æ
î
»®º ø¬÷ ã °
¬
»
¨î
¬ î Îæ
¼¨å
ð
Ì¸» º±´´±©·²¹ ®»-«´¬- º±´´±© ®»¿¼·´§ º®±³ øïð÷æ
ÔºÜ
-·² ¿¬¹ ã
¿
° ø°î õ ¿î ÷
ðæ
ÔºÜ
½±- ¿¬¹ ã
°
° ø°î õ ¿î ÷
ðæ
ÔºÜ
ï
»¿¬ ¬
¹ã
÷
¿÷
° ø°
ð
ðæ
Ô»¬ «- ²±© »ª¿´«¿¬» ¬¸» Ô¿°´¿½» ¬®¿²-º±®³ ±º ¬¸» º®¿½¬·±²¿´ ·²¬»¹®¿´ ±º ¬¸» ¼»®·ª¿¬·ª» ¿²¼ ¬¸»² ¬¸» Ô¿°´¿½»
¬®¿²-º±®³ ±º ¬¸» ¼»®·ª¿¬·ª» ±º ¬¸» º®¿½¬·±²¿´ ·²¬»¹®¿´ò ×² ª·»© ±º øïð÷ ·¬ º±´´±©- ¬¸¿¬
ÔºÜ
ÅÜº ø¬÷Ã¹ ã °
ß´¬¸±«¹¸ ¬¸·- ®»-«´¬ ·- °®±ª»¼ º±®
±º øç÷ ¹·ª»ÔºÜÅÜ
÷
º ø¬÷Ã¹ ã ÔºÜ
Ôº¼º ø¬÷¹ ã °
Å°Ú ø°÷
ð ·¬ ·- ª¿´·¼ »ª»² ·º
ï
¬
ÅÜº ø¬÷Ã¹ õ º øð÷Ô
ðæ
øîð÷
ã ðò Ñ² ¬¸» ±¬¸»® ¸¿²¼ ¬¸» Ô¿°´¿½» ¬®¿²-º±®³
ã°
÷
º øð÷Ã
Å°Ú ø°÷
º øð÷Ã õ °
º øð÷ ã °ï
Ñ¾ª·±«-´§ô ·º ã ð ô ¬¸·- ®»-«´¬ ¼±»- ²±¬ ¿¹®»» ©·¬¸ ¬¸¿¬ ±¾¬¿·²»¼ º®±³ øîð÷ ¿·- ¼«» ¬± º¿½¬ ¬¸¿¬ þÔþ ¿²¼ þ ´·³ þ ¼± ²±¬ ½±³³«¬» ¿- ·¬ ½¿² ¾» -»»² º®±³ øïï÷ò
ß²±¬¸»® ½±²-»¯«»²½» ±º øè÷ ·- ¬¸¿¬ ¬¸» º®¿½¬·±²¿´ ¼»®·ª¿¬·ª» Ü º ø¬
±º ¬¸» ·²¬»¹®¿´ »¯«¿¬·±²
Ü §ø¬÷ ã º ø¬÷æ
Ú ø°÷
ðæ
øîï÷
ÿ ðæ Ì¸·- ¼·-¿¹®»»³»²¬
§ø¬÷
øîî÷
Ì¸» Ô¿°´¿½» ¬®¿²-º±®³ ±º ¬¸·- ®»-«´¬ ¹·ª»- ¬¸» -±´«¬·±² º±® Ç ø°÷
Ç ø°÷ ã ° Ú ø°÷æ
øîí÷
Ì¸» ·²ª»®-·±² ¹·ª»- ¬¸» º®¿½¬·±²¿´ ¼»®·ª¿¬·ª» ±º º ø¬÷ ¿§ø¬÷ ã Ü º ø¬÷ ã Ô
´»¿¼·²¹ ¬± ¬¸» ®»-«´¬
ï
Ü º ø¬÷ ã
÷
Æ
ï
º° Ú ø°÷¹å
øîì÷
¬
ø¬
ï
¨÷
º ø¨÷
ðæ
øîë÷
ð
º ø¬÷ ã ¬ ·¬ ·±¾ª·±«- º®±³ øîì÷ ¬¸¿¬
Ü ¬ ãÔ
×² °¿®¬·½«´¿®ô ·º
ã
ï
î
¿²¼
°
õï
õ ï÷
¬
õ ï÷
ã
øîê÷
æ
ã ² ·- ¿² ·²¬»¹»®ô
ï
Ü î ¬² ã
×º ¿ â
õ ï÷
ï
² õ ï÷ ²
¬
² õ ïî ÷
ï
î
å
ïæ
²â
øîé÷
ï ©» ¸¿ª»
Ôº¬ ¹ ã
Æ
ï
¬ »
°¬
¼¬ ã ø°«¬¬·²¹ °¬ ã ¨÷ ã
ð
ï
°¿õï
Æ
ï
¨¿ »
ð
¨
¼¨ ã
¿ õ ï÷
æ
°¿õï
øîè÷
ïðé
×² °¿®¬·½«´¿®ô ©¸»² ¿ ã
ï
î
®»-«´¬ øîè÷ ¹·ª»ï
°
¬
Ô
×¬ ¿´-± º±´´±©- º®±³ øïé÷ôøïç÷ôøîç÷ ¬¸¿¬
°
°
ï
¿¬
ï
î
Ü » ãÔ
ãÔ
° ¿
ï
ï
÷
ã °î ã
°
¿
ï
° õ°
°
°ø°
®
¿÷
°
øîç÷
æ
°
°
ï
ã ° õ ¿»¿¬ »®º ø ¿¬÷æ
Í·³·´¿®´§ô «-·²¹ ±º Ô¿°´¿½» ¬®¿²-º±®³ ©» ½¿² ¼»®·ª» ±¬¸»® ¾¿-·½ º±®³«´¿» º±® ¬¸» º®¿½¬·±²¿´ ·²¬»¹®¿´ ¿²¼
¼»®·ª¿¬·ª»ò
ßÝÕÒÑÉÔÛÜÙÛÓÛÒÌ
Ì¸·- ®»-»¿®½¸ ¸¿- ¾»»² -«°°±®¬»¼ ¾§ ¬¸» Ý¦»½¸ Ó·²·-¬®§ ±º Û¼«½¿¬·±² ·² ¬¸» º®¿³»- ±º ÓÍÓððîïêðëðí Î»ó
-»¿®½¸ ×²¬»²¬·±² Ó×ÕÎÑÍÇÒ Ò»© Ì®»²¼- ·² Ó·½®±»´»½¬®±²·½ Í§-¬»³- ¿²¼ Ò¿²±¬»½¸²±´±¹·»- ¿²¼ ÓÍÓððîïêíðëîç
Î»-»¿®½¸ ×²¬»²¬·±² ×²¬»´·¹»²¬ Í§-¬»³- ·² ß«¬±³¿¬·¦¿¬·±²ò
Î»º»®»²½»
ÅïÃ ÞØßÌÌßôÜòÜòô ÜÛÞÒßÌØôÔòæ
Ú®¿½¬·±²¿´ ½¿´½«´«- ¿²¼ ß°°´·»¼ ß²¿´§-·- éô îððìô îïóíêò
ÅîÃ ÞØßÌÌßôÜòÜòô ÜÛÞÒßÌØôÔòæ ×²¬»¹®¿´ ¬®¿²-º±®³- ¿²¼ ¬¸»·® ¿°°´·½¿¬·±²-ô Ý¸¿°³¿² Ø¿´´ñÝÎÝô îððéô
×ÍÞÒ ïóëèìèèóëéëóðò
ÅíÃ ÜßÊ×ÛÍôÞòæ ×²¬»¹®¿´ ¬®¿²-º±®³- ¿²¼ ¬¸»·® ß°°´·½¿¬·±²ô Í°®·²¹»®óÊ»®´¿¹ô Ò»© Ç±®µô ïçéè ò
ÅìÃ ÜÛÞÒßÌØôÔòæ
½¿´½«´«- ¿²¼ ß°°´·»¼ ß²¿´§-·- êô îððíô ïïçóïëëò
Ú®¿½¬·±²¿´
ÅëÃ ÜÛÞÒßÌØôÔòæ Î»½»²¬ ¼»ª»´±°³»²¬- ·² º®¿½¬·±²¿´ ½¿´½«´«- ¿²¼ ·¬- ¿°°´·½¿¬·±²- ¬± -½·»²½» ¿²¼ »²¹·²»»ó
®·²¹ô ×²¬»®²¿¬ò ÖòÓ¿¬¸ò ¿²¼ Ó¿¬¸ò Í½·òô îððíô ïóíðò
ÅêÃ ÙßËÌÛÒÍÛÒôßòÕòô ÜËÒÜÛÎÍôÖòæ Ñ² ¬¸» -±´«¬·±² ¬± ¿ Ý¿«½¸§ °®·²½·°¿´ ª¿´«» ·²¬»¹®¿´ »¯«¿¬·±² ©¸·½¸
¿®·-» ·² º®¿½¬«®» ³»½¸¿²·½-ô Í×ßÓ Öòß°°´ò Ó¿¬¸òô ìéô ïðçóïïêô ïçèéò
ÅéÃ ÔßÕÍØÓ×ÕßÒÌØßÓôÊòôÎßÑô ÓòÎòÓòæ
Í½·»²½» Ð«¾´·-¸»®-ô ïççëô ×ÍÞÒ îóèèììçóðððóðò
ô ¹±®¼±² ¿²¼ Þ®»¿½¸
ÅèÃ ÔßÕÍØÓ×ÕßÒÌØßÓôÊòæ
×²¬»¹®¿´ Û¯«¿¬·±²- ïðô ïçèëô ïíéóïìêò
Ö±«®²¿´
ÅçÃ Ð×ÐÕ×ÒôßòÝòæ ß ½±«®-» ±² ·²¬»¹®¿´ »¯«¿¬·±²-ô Í°®·²¹»®óÊ»®´¿¹ô Ò»© Ç±®µô ïççïô ×ÍÞÒ ðóíèéóçéëëéóèò
ß½¿¼»³·½ Ð®»--ô Í¿² Ü·»¹±ô ïçççò
ÅïðÃ ÐÑÜÔËÞÒÇô ×òæ
ÅïïÃ ÍßÕØÒÑÊ×ÝØô Ôòßòæ
ô Ñ°»®¿¬±® Ì¸»±®§
ß¼ª¿²½»- ¿²¼ ß°°´·½¿¬·±²-ô Ê±´òèìô Þ·®µ¸¿«-»® Ê»®´¿¹ô Þ¿-»´óÞ±-¬±²óÞ»®´·²ô ïççêô ×ÍÞÒ íóéêìíóëîêéóïò
ÅïîÃ wÓßÎÜßôÆòô ÕÎËÐÕÑÊ_ô Êòæ Ü·-½®»¬·¦¿¬·±² ±º ¬¸» Ù®«²©¿´¼óÔ»¬²·µ±ª º®¿½¬·±²¿´ Ü»®·ª¿¬·ª» ¿²¼ ×²ó
¬»¹®¿´ô ×² Ð®±½»»¼·²¹- ±º ß¾-¬®¿½¬- ¿²¼ ®»ª·»©»¼ ½±²¬®·¾«¬·±²- ±º ÈÈÊò ×²¬»®²¿¬·±²¿´ Ý±´´±¯«·«³ ±²
Û¼«½¿¬·±² Ð®±½»-- ±² ÝÜÎÑÓô Þ®²±ô îððêô ï ó êô ×ÍÞÒ èðóéîíïïíçóëò
ÅïíÃ wÓßÎÜßôÆò ôÕÎËÐÕÑÊ_ôÊòæ Ñ² Í±³» Ð®±°»®¬·»- ±º Ú®¿½¬·±²¿´ Ý¿´½«´«-ô ×² Ð®±½»»¼·²¹- ±º ×Êò ×²ó
¬»®²¿¬·±²¿´ Ý±²º»®»²½» ±² Í±º¬ Ý±³°«¬·²¹ ß°°´·»¼ ·² Ý±³°«¬»® ¿²¼ Û½±²±³·½ Û²ª·®±³»²¬ô Õ«²±ª·½»ô
Û«®±°»¿² Ð±´§¬»½¸²·½¿´ ×²-¬·¬«¬» Õ«²±ª·½»ô îððêô ïëé ó ïêîô ×ÍÞÒ èðóéíïìóðèìóëò
ÅïìÃ ÉÛÇÔôØòæ
Ò¿¬«®º±®-½¸ò Ù»-»´´-½¸ò Æ«®·½¸ô êîô ïçïéô îçêóíðîò
Ê·»®¬»´¶-½¸®ò
ïðè
ßÜÜÎÛÍÍæ
ÎÒÜ®ò Þ(»¬·-´¿ª Ú¿¶³±²ô Ð¸òÜò
Ü»°¿®¬³»²¬ ±º Ó¿¬¸»³¿¬·½Ú¿½«´¬§ ±º Û´»½¬®·½¿´ Û²¹·²»»®·²¹ ¿²¼ Ý±³³«²·½¿¬·±²ô
Þ®²± Ë²·ª»®-·¬§ ±º Ì»½¸²±´±¹§
Ì»½¸²·½µ? è ô êïêðð ÞÎÒÑ ô Ý¦»½¸ Î»°«¾´·½
º¿¶³±²àº»»½òª«¬¾®ò½¦
Ü±½ò ÎÒÜ®ò Æ¼»²4µ w³¿®¼¿ ô ÝÍ½ò
Ü»°¿®¬³»²¬ ±º Ó¿¬¸»³¿¬·½Ú¿½«´¬§ ±º Û´»½¬®·½¿´ Û²¹·²»»®·²¹ ¿²¼ Ý±³³«²·½¿¬·±²ô
Þ®²± Ë²·ª»®-·¬§ ±º Ì»½¸²±´±¹§
-³¿®¼¿àº»»½òª«¬¾®ò½¦
ïðç
ïïð
ÊÑÔÌÛÎÎß ×ÒÌÛÙÎÑÜ×ÚÚÛÎÛÒÌ×ßÔ ÛÏËßÌ×ÑÒÍ É×ÌØ
Ü×ÚÚÛÎÛÒÝÛ ÕÛÎÒÛÔ ßÒÜ ÌØÛ×Î ßÐÐÔ×ÝßÌ×ÑÒÍ ×Ò ÌØÛ
ÌØÛÑÎÇ ÑÚ ÛÔÛÝÌÎ×ÝßÔ Ý×ÎÝË×ÌÍ
Ñ´¹¿ Ú·´·°°±ª¿
Ú¿½«´¬§ ±º Û´»½¬®·½¿´ Û²¹·²»»®·²¹ ¿²¼ Ý±³³«²·½¿¬·±² Þ®²± Ë²·ª»®-·¬§ ±º Ì»½¸²±´±¹§
ß¾-¬®¿½¬æ Ì¸·- °¿°»® ·- ¼»ª±¬»¼ ¬± ·²ª»-¬·¹¿¬·±² ÎÔÝ ½·®½«·¬-ò Î»´¿¬·±²- ¾»¬©»»² ¬¸» ³¿¹²»¬·½
½¸¿®¿½¬»®·-¬·½ ·- ¼»-½®·¾»¼ ¾§ ³»¿²- ±º ò
Õ»§ ©±®¼-æ
ÅéôèÃ ±² °®±¾´»³- ·² ³»½¸¿²·½-ô »½±²±³·½- ¿²¼ ¬¸»±®§ ±º »´»½¬®·½¿´ ½·®½«·¬-ò Ì¸» ©±®µ ±º Ê±´¬»®®¿ ±² ¬¸»
°®±¾´»³ ±º ½±³°»¬·²¹ -°»½·»- ·- ±º º«²¼¿³»²¬¿´ ·³°±®¬¿²½» º±® ¼»ª»´±°³»²¬ ±º ³¿¬¸»³¿¬·½¿´ ³±¼»´·²¹
»¯«¿¬·±²- ©·¬¸ ¾±«²¼»¼ ¿²¼ «²¾±«²¼»¼ ¼»´¿§- ¸¿ª» »³»®¹»¼ ¿- ²»© ¿®»¿- ±º ·²ª»-¬·¹¿¬·±²ò Í±³» °®±°»®¬·»×² ¬¸·- °¿°»® ©» ·²ª»-¬·¹¿¬» -±³» ÎÔÝ »´»½¬®·½¿´ ½·®½«·¬- «-·²¹ ¬¸» µ²±©² ´¿©- ¼»-½®·¾·²¹ ¾»¸¿ª·±«® ±º
»´»½¬®·½¿´ ½·®½«·¬-ò Ó±®»±ª»®ô ©» ¬®§ ¬± ¼»-½®·¾» ¾»¸¿ª·±«® ±º »´»½¬®·½ ½·®½«·¬- ©¸·½¸ ¼»°»²¼- ±² ¬¸» °®»½»²¼·²¹
-¬¿¬» ©·¬¸ ®»-°»½¬ ¬± ¬·³» ª¿®·¿¾´» ©¸·½¸ ¹»²»®¿´´§ ½±²ª»®¹»- ¬± ³·²«- ·²º¬§ò
îò Óß×Ò ÎÛÍËÔÌÍ
É» ©·´´ ¼»¿´ ©·¬¸ ¬¸» »´»½¬®·½ ½·®½«·¬ ø-»» Ú·¹òï÷
É» ½¿² ©®·¬» ¾¿-·½ »¯«¿¬·±²- ±º »´»½¬®±¼§²¿³·½- ©¸·½¸ ¼»-½®·¾» ¬¸» ¹·ª»² ½·®½«·¬ ·² ¬¸» º±®³
ËÎï ã Îï ·ï å
Æ ¬
ï
ËÝï ã
·ï ø-÷¼-å
Ýï ð
¼·ï
¼·î
ËÔï ã Ôïï
õ Ôïî
å
¼¬
¼¬
ËÎî ã Îî ·î å
Æ ¬
ï
ËÝî ã
·î ø-÷¼-å
Ýî ð
¼·ï
¼·î
ËÔî ã Ôîï
õ Ôîî
æ
¼¬
¼¬
î
È
Ë· ã ðæ
øï÷
øî÷
øí÷
øì÷
·ãï
×ÝÍÝóÍ×ÈÌØ ×ÒÌÛÎÒßÌ×ÑÒßÔ ÝÑÒÚÛÎÛÒÝÛ ÑÒ ÍÑÚÌ ÝÑÓÐËÌ×ÒÙ ßÐÐÔ×ÛÜ ×Ò ÝÑÓÐËÌÛÎ ßÒÜ ÛÝÑÒÑÓ×Ý ÛÒÊ×ÎÑÒÓÛÒÌÍÄ
þ
ïïï
×º ¾±¬¸ ½·®½«·¬- ¿®» ·² ½±³³±² ®»¿½¬¿²½» ©·¬¸ ¿ ½±²-¬¿²¬ ª±´¬¿¹» ¬¸»² -«¾-¬·¬«¬·²¹ øï÷ôøî÷ôøí÷ ·²¬± øì÷ ©»
¹»¬ º±® ·ï ø¬÷å ·î ø¬
Æ ¬
ï
¼·î
¼·ï
Ëï ã Îï ·ï õ
õ Ôïî
·ï ø-÷¼- õ Ôïï
Ýï ð
¼¬
¼¬
Æ ¬
ï
¼·î
¼·ï
Ëî ã Îî ·î õ
õ Ôîî
æ
·î ø-÷¼- õ Ôîï
Ýî ð
¼¬
¼¬
øë÷
Ô»¬ù- ½±²-·¼»® ¬¸» ½¿-» ©¸»² »²»®¹§ ·- ¬®¿²-º»®®»¼ ¾§ ¿½¬·²¹ ±º ½±³³±² ®»¿½¬¿²½»ò ×² ¬¸·- ½¿-» ½±·´- ©·²¼
Æ
¬
ï ø¬÷ ã Ôïï ·ï õ Ôïî ·î õ
Æ
¬
î ø¬÷
ã Ôîï ·ï õ Ôîî ·î õ
øÕïï ø¬
-÷·ï ø-÷ õ Õïî ø¬
-÷·î ø-÷÷ ¼-å
øê÷
øÕîï ø¬
-÷·ï ø-÷ õ Õîî ø¬
-÷·î ø-÷÷ ¼-å
øé÷
©¸»®» Õ·¶ ø¬ -÷ · ã ïå îå ¶ ã ïå îæ
·µ å µ ã ïå î ¿¬ ¬¸»
³±³»²¬ ¬ò Ì¸» ´±©»® ´·³·¬
³»¿²- ¬¸¿¬ ¾»¸¿ª·±«® ±º ¹·ª»² ½·®½«·¬
¼»°»²¼- ±² °®»½»²¼·²¹ -¬¿¬» ±º ·²ª»-¬·¹¿¬»¼ ½¸¿®¿½¬»®·-¬·½- ò ×² ¿ ¹»²»®¿´ ½¿-» ¬¸» -§-¬»³ ©·´´ ¾» ²±²´·²»¿®
ø-»» Ú·¹òî÷ò
Ü»°»²¼·²¹ ±² ²±²´·²»¿® ½¸¿®¿½¬»®·-¬·½- ±º ·®±² ½±®»- ª¿®·±«- °¸»²±³»²¿ ½¿² ¾» »²½±«²¬»®»¼ò
Ô»¬
Õø¬
Ôã
Ôïï
Ôîï
-÷ ã
Õïï ø¬
Õîï ø¬
¬÷ ã Ô º ø·÷ õ
Ôïî
å
Ôîî
Æ
-÷
-÷
øè÷
Õïî ø¬
Õîî ø¬
-÷
æ
-÷
øç÷
-÷º ø·ø-÷÷ ¼-
øïð÷
¬
Õø¬
©¸»®»
ºï ø·ïå ·î ÷
ºî ø·ïå ·î ÷
ºø·÷ ã
øïï÷
¿²¼ º ø·÷ ·- ¬¸» ²±²´·²»¿® ª»½¬±® º«²½¬·±² ¼»°»²¼·²¹ ±² ¬¸» ½¸¿®¿½¬»® ±º ·®±² ½±®»- ¿²¼ ¬¸» ª»½¬±® º«²½¬·±²
ºø·÷ ·- -»¬ ±® ¼»¬»®³·²»¼ »¨°»®·³»²¬¿´´§ò
ß½½±®¼·²¹ ¬± ¬¸» Ú¿®¿¼¿§ ´¿© ¬¸» ·²¼«½»¼ »´»½¬®±³±¬·ª» ª±´¬¿¹» ·- »¯«¿´ ¬± ¬¸» ²»¹¿¬·ª» ¬¿µ»² ½¸¿²¹» ±º
·
Ë· ã
¼¬
Ë-·²¹ ±º ¬¸» Ú¿®¿¼¿§ ´¿© º±® ¬¸» -§-¬»³ øë÷ ©» ¹»¬
å
¼ ï
¯ï ø¬÷
õ Îï ·ï ø¬÷ õ
ã ðå
¼¬
Ýï
· ã ïå îæ
¼ î
¯î ø¬÷
õ Îî ·î ø¬÷ õ
ãð
¼¬
Ýî
øïî÷
þ
ïïî
©¸»®»
¼¯µ
¼¬
ã ·µ ø¬÷ô µ ã ïå
¼î î
¼·î ·î ø¬÷
õ Îî
õ
ãð
¼¬î
¼¬
Ýî
¼·ï
·ï ø¬÷
¼î ï
õ Îï
õ
ã ðå
¼¬î
¼¬
Ýï
Ô»¬ Î±ø¬ -÷ ¾» ¬¸» ®»¦±´ª»²¬ ±º ¬¸» ³¿¬®·¨ µ»®²»´ Ô
·ø¬÷ ã ø·ï ø¬÷å ·î ø¬÷÷ ©» ¹»¬
Æ ¬
·ø¬÷ ã
Îð ø¬ -÷Ô ï
ï
øïí÷
-÷ ò ¬¸»² -±´ª·²¹ ±º øë÷ ©·¬¸ ®»-°»½¬ ¬±
Õø¬
-÷¼- õ Ô
ï
øïì÷
¬÷æ
Ý·®½«·¬ ½¿°¿½·¬·»- Ýï ¿²¼ Ýî
-³¿´´ øÝï Ýî þ
Ýï ¿²¼ Ýî ¿®»
Û
¼
ã ÔË ø¬÷å
¼¬
øïë÷
¼Ë
õ ÛÔ
¼¬
ï
ï î
ÎË ø¬÷ õ ÅÛ ÎÎð øð÷ õ ×ÃøÔ
©¸»®» × ·- ¬¸» «²·¬ ³¿¬®·¨ ¿²¼
Îã
÷
¬÷ õ
Îï
ð
ð
å
Îî
Æ
¬
ÅÛÎ
Ûã
àÎð ø¬ -÷
õ Îð ø¬
à¬
þ
ð
-÷ÃøÔ
ï î
÷
-÷¼- ã ðå
ð
æ
þ
Í§-¬»³ øïë÷ ¼»°»²¼- ±² ¿ -³¿´´ °¿®¿³»¬»® ©¸±-» ¿ -±´«¬·±² ·- «-«¿´´§ ½±²-¬®«½¬»¼ ¾§ ³»¿²- ±º ¿-§³°¬±¬·½
»¨°¿²-·±²- ¿- ÿ ð ò ×² ½¿-» ¬¸¿¬ ½¿°¿½·¬§ ¸¿- ¬¸» º±®³ ±º ¿ º«²½¬·±² ¹ø¬÷ -«½¸ ¬¸¿¬ ¹ø¬÷ â ðå ¹øðõ ÷ ã
ðå ¬ î øðå ¬ð Ãå ¬ð â ð ¿-§³°¬±¬·½ »¨°¿²-·±²- ¿®» ½±²-¬®«½¬»¼ ¿- ¬ ÿ ðõ ø-»» ÅìóêÃ÷ ò
Ì¸·- ®»-»¿®½¸ ¸¿- ¾»»² -«°°±®¬»¼ ¾§ ¬¸» Ý¦»½¸ Ó·²·-¬®§ ±º Û¼«½¿¬·±² ·² ¬¸» º®¿³» ±º ÓÍÓððîïêðëðí Î»ó
-»¿®½¸ ×²¬»²¬·±² Ó×ÕÎÑÍÇÒ Ò»© Ì®»²¼- ·² Ó·½®±»´»½¬®±²·½ Í§-¬»³- ¿²¼ Ò¿²±¬»½¸²±´±¹·»-ò
Î»º»®»²½»
ÅïÃ Ú×Ô×ÐÐÑÊßô Ñò
Ð®±½»»¼·²¹- ±º ÈÈÊ×ò Ý±´´±¯«·«³ ±² Û¼«½¿¬·±² Ð®±½»--ô ËÒÑÞ Þ®²± îððéô ïóëò
ÅîÃ ×ÓßÒßÔ×ÛÊô Óò
¬·±²-ô Ú®«²¦» ô ïçéìò
ÅíÃ ÔÑÌÕßô ßòÖò Ó¿¬¸»³¿¬·½¿´ Þ·±´±¹§ô Ò»© Ç±®µô ïçëêò
ÅìÃ ÍÓßÎÜßô Æò Ñ² ¬¸» «²·¯«»²»- ±º -±´«¬·±² ±º ¬¸» -·²¹«´¿® °®±¾´»³ º±® ½»®¬¿·² ½´¿-- ±º ·²¬»¹®±ó
Ü»³±²-¬®¿¬·± Ó¿¬¸»³¿¬·½¿ôîëøì÷ô ïççî ôèíë ó èìïò
ÅëÃ ÍÓßÎÜßô Æò Ì¸» »¨·-¬»²½» ¿²¼ ¿-§³°¬±¬·½ ¾»¸¿ª·±«® ±º -±´«¬·±²- ±º ½»®¬¿·² ½´¿-- ±º ¬¸» ·²¬»¹®±ó
ß®½¸·ª«³ Ó¿¬¸»³¿¬·½«³ô îêøï÷ôïççðô é ó ïèò
ÅêÃ ÍÓßÎÜßô Æò
Ü»³±²-¬®¿¬·± Ó¿¬¸»³¿¬·½¿ô îîøî÷ô ïçèçô îçí ó íðéò
ÅéÃ ÊÑÔÌÛÎÎßô Êò
Ç±®µô ïçëçò
Ü±ª»®ô Ò»©
»
»³¿¬·¯«» ¼» ´¿ ´«¬¬´» °±«® ´¿ ª·» Ù¿«¬¸·»®óÊ·´´¿®-ô Ð¿®·-ô
ïçíïò
þ
ïïí
ßÜÜÎÛÍÍæ
×²¹ò Ñ´¹¿ Ú·´·°°±ª¿
Ü»°¿®¬³»²¬ ±º Ó¿¬¸»³¿¬·½Ú¿½«´¬§ ±º Û´»½¬®·½¿´ Û²¹·²»»®·²¹ ¿²¼ Ý±³³«²·½¿¬·±²ô
Þ®²± Ë²·ª»®-·¬§ ±º Ì»½¸²±´±¹§
þ
ïïì
SESSION 2
115
116
MATICOVÉ HRY A LINEÁRNÍ PROGRAMOVÁNÍ
Jitka Jablonická
Evropský polytechnický institut, s.r.o., Osvobození 699, 686 04 Kunovice
Abstrakt. Matematické modely konfliktních rozhodovacích situací jsou studovány v rámci tzv.
teorie her , zabývající se studiem t chto situací alespo pro dva ú astníky s íselným ohodnocením
výsledk , v nich d sledky ur itého rozhodnutí jednoho ú astníka závisí na rozhodnutí ostatních
ú astník . V lánku se uvádí aplikace lineárního programování na eení proces s prvky rizika a
nejistoty.
Keywords. Rozhodovací proces za rizika a nejistoty, antagonistické hry, maticové hry, výplatní
matice, lineární a nelineární programování, duální eení.
1. ÚVOD
Rozhodování p edstavuje jednu z nejvýznamn jích aktivit, které manae i v organizacích realizují (n kdy se
dokonce chápe jako ur ité jádro ízení). Rozhodování je nedílnou slokou sekven ních manaerských funkcí,
nejvýrazn ji se vak uplat uje v plánování. Rozhodovací procesy probíhají na r zných úrovních ízení. Mají dv
stránky a to stránku meritorní (v cnou, obsahovou) a stránku formáln -logickou (procedurální). Meritorní
stránka odráí odlinost jednotlivých rozhodovacích proces , p ípadn jejich typ . Kadý typ rozhodovacího
procesu má své specifické rysy, které jsou zdrojem odlinosti t chto proces .
Na druhé stran mají vak jednotlivé rozhodovací procesy, p ípadn jejich typy ur ité spole né rysy a vlastnosti,
a to bez ohledu na jejich odlinou obsahovou nápl . To co jednotlivé rozhodovací procesy spojuje, je ur itý
rámcový postup (procedura) eení. Tato skute nost se vyuívá v kvalitativn orientovaných teoriích
rozhodování, zaloených na aplikaci matematických model a metod opera ní analýzy. Zejména je vyuívána,
krom lineárního a obecného programování dalí struktura opera ní analýzy teorie her studující konfliktní
rozhodovací procesy s významnými prvky rizika a nejistoty. V dalí ásti svého p ísp vku se budu zejména
v novat aplikaci lineárního programování p i eení problém pomocí maticových model konfliktních her.
K lepímu osv tlení uvedené problematiky uvedu n kolik p íklad .
2. LINEÁRNÍ PROGRAMOVÁNÍ V MATICOVÝCH HRÁCH
Uvedeme si dva konkrétní p íklady aplikace lineárního programování p i zji ování optimálního eení úloh,
zadaných platebními maticemi.
Jak ji bylo e eno, maticovou hru je mimo jiné mono eit metodami lineárního programování. Budeme
studovat maticovou hru s platební maticí A typu (m x n), která má vechny prvky kladné, co symbolicky
zapisujeme A > 0. Tento p edpoklad neubírá na obecnosti dalích úvah, nebo p i tením dostate n velkého
kladného ísla ke vem prvk m matice A m eme vdy zajistit jejich kladnost. Lze dokázat, e eení smíeného
rozí ení maticové hry se nezm ní, jestlie ke kadému prvku její matice p i teme stejnou konstantu w nebo
jestlie kadý její prvek násobíme kladnou konstantou w > 0. Je-li A > 0, je cena hry vdy kladná.
Musí proto vdy existovat v > 0 takové, e pro libovolné pevné x0 (X) a vechna y (Y) platí v
x0 A y,
nebo funkce f (y) = f (x0, y) je na (Y) spojitá a kladná a (Y) je kompaktní mnoina. Posta í vak platnost
nerovnosti v
x0 A y pouze pro vechny ryzí strategie y ve tvaru
T
T
T
( 1, 0, . . . , 0 ) , ( 0, 1, 0, . . . , 0 ) , . . . , ( 0, 0, . . . , 0, 1) ,
to jest v rozepsaném tvaru
a11 x1 + . . . + am1 xm
v
pro
y = (1, 0, 0, . . . , 0 )
a12 x1 + . . . + am2 xm
v
pro
y = (0, 1, 0, . . . , 0 )
.....................
T
T
117
a1n x1 + . . . + amn xm
v
pro
yj v
kde
y = (0, 0, 0, . . . , 1)
T
co implikuje spln ní následující nerovnosti
n
m
yj
n
aij xi
j 1
i 1
n
j 1
n
yj v
j 1
v
yj
v
j 1
i pro smíené strategie. Pokud vektor x0 = ( x1, . . . , xm) spl uje uvedené nerovnosti a sou asn platí x1 + x2
+ . . . + xm = 1 a xi
0 pro i = 1, 2, . . . , m, potom dle výe uvedeného je x0 optimální strategie prvního
hrá e. Uvedená omezení povaujeme za omezení úlohy lineárního programování s tím, e zavedeme nové
prom nné vztahem xi´ = xi / v , i = 1, 2, . . . , m, v 0, a kriteriální funkci x1´ + x2´ + . . . + xm´ = 1 / v. První
hrá se snaí maximalizovat svoji výhru v, to znamená minimalizovat 1 / v. Podobn se odvodí odpovídající
vztahy pro druhého hrá e. Tak lze maticové h e s maticí A > 0 typu (m x n) p i adit dvojici duáln sdruených
úloh lineárního programování:
x1´+ x2´+ . . . + xm´ = 1 / v ! MIN
a11 x1´ + . . . + am1 xm´
y1´+ y2´+ . . . + ym´ = 1 / v ! MAX
1
a11 y1´ + . . . + a1n yn´
a12 x1´ + . . . + am2 xm´ 1
.....................
a1n x1´ + . . . + amn xm´
1
------------------------------------x1, . . . , xm
0
1
a21 y1´ + . . . + a2n yn´
1
.....................
am1 y1´+ . . . + amn yn´
1
-------------------------------------y1, . . . , yn
0
kde xi ´ = xi / v
pro i = 1, 2, . . . , m
yj´ = yj / v
pro j = 1, 2, . . . , n
Poznámka. Je-li vektor x0´ = (x10´, . . . , xm0´) libovolným optimálním eením prvé úlohy lineárního
programování potom vektor x0 = (x10, . . . , xm0), pro n j platí x0 = v x0´, kde je optimální smíenou strategií
prvního hrá e a íslo v je cenou p ísluné maticové hry.
Je-li vektor y0´ = (y10´, . . . , yn0´) libovolným optimálním eením prvé úlohy lineárního programování potom
vektor y0 = (y10, . . . , yn0), pro n j platí y0 = v y0´, kde, je optimální strategií druhého hrá e a íslo v je cenou
p ísluné maticové hry.
Jako konkrétní problém eení rozhodovacího procesu uvedeme proces popsaný platební maticí A. Budeme jej
op t eit pomocí metody lineárního programování a to metody duální.
A
5 0
2
0 6
3
.
P ísluné duáln sdruené úlohy lineárního programování mají tvar :
x1´ + x2´ ! MIN
5 x1 ´ +
6 x2 ´
y1´ + y2´ + y3´ ! MAX
1
x2 ´
5 y1´
1
2 x1 ´ + 3 x2´
x1´ ; x2´
+ 2 y3´
1
y1´ + 6 y2 + 3 y3´
1
1
0
y1´ ; y2´ ; y3´
0
Optimální eení první úlohy je x0´ = (0,154 ; 0,231), v = 1 / (0,154 + 0,231) = 2,6 a tedy optimální strategie
prvního hrá e je x0 = v x0´ = (0,4 ; 0,6). Podobn z eení duální úlohy y0´= (0,077 ; 0 ; 0,308) vypo teme
optimální strategii druhého hrá e. Dostaneme y0 = v y0´ = y0 = (0,2 ; 0 ; 0,8). Cena hry je v = 2,6. Tyto výsledky
jsou shodné s výsledky získanými graficky v p íkladu 11.3.2. První hrá bude volit své strategie 1 a 2 náhodn se
118
zákonem rozloení pravd podobností (0,4 ; 0,6) a druhý hrá bude náhodn volit své strategie v souladu se
zákonem rozloení pravd podobností (0,2 ; 0 ; 0,8). O ekávaná hodnota výplat prvního hrá e bude druhý zp sob
ekvivalentního zápisu maticové hry jako úlohy lineárního programování je následující:
- xm + 1 ! MIN
a11 x1+ . . . + am1 xm - xm+1
0
- yn + 1 ! MAX
a11 y1 + . . . + a1n y n - yn+1
a12 x1 + . . . + am2 xm - xm+1
0
0
a21 y1 + . . . + a2n yn - yn+1
0
.....................
a1n x1 + . . . + amn xm - xm+1
0
.....................
am1 y1 + . . . + amn yn - yn+1
0
x1 + . . . ¨ + xm
= 1
------------------------------------x1, . . . , xm
0
y1 + . . .
+ yn
= 1
-------------------------------------y1, . . . , yn
0
Sou adnice optimálního eení x0 = (x10, . . . , xm0) p edstavují optimální smíenou strategii prvního hrá e.
Sou adnice optimálního eení duální úlohy y0 = (y10, . . . , yn0) optimální smíenou strategii druhého hrá e a xm+1,
= yn+1, 0 = v cenu hry.
0
11.4.5 P íklad. K maticové h e z p íkladu na stran 2. zapite podle p edchozí úpravy duáln sdruené úlohy
lineárního programování.
- x3 ! MIN
5x1 +
x2 - x3
0
6 x2 - x3
2x1 + 3 x2 - x3
x1 +
- y4 ! MAX
0
5 y1
+ 2 y3 - y4
y1 + 6 y2 + 3 y3 - y4
0
y1 +
0
y2 + y3
=1
y1
x2
x1
0
0
x2
0
y2
0
0
y3
0
Jejich eením dostáváme optimální eení x0 = (0,4; 0,6; 2,6) a y0 = (0,2; 0; 0,8; 2,6)
z n ho jsou okamit z ejmé optimální strategie obou hrá a cena hry.
3. ZÁKLADNÍ V TA MATICOVÝCH HER
Základní v ta teorie maticových her tvrdí, e smíené rozí ení kadé maticové hry má eení. K d kazu této
v ty posta í ukázat, e ekvivalentní sdruené problémy lineárního programování mají p ípustná a tedy i
optimální eení. Vyjd me z prvního zp sobu zápisu ekvivalentních problém lineárního programování.
Vzhledem k tomu, e vechny prvky výplatní matice A jsou kladné, mají z ejm ob úlohy p ípustné a tedy i
optimální eení.
Naopak lze úlohu lineárního programování zapsat ve tvaru maticové hry. P edpokládejme, e máme dvojici
duálních úloh lineárního programování ve tvaru :
min (c.x : A x
max (y.b : y A
b; x
a
c;y
0; x
0; y
En )
Em) .
Ekvivalentní hra k této dvojici problém lineárního programování je symetrická maticová s nulovým sou tem
T
(pro matici symetrické maticové hry platí A = - A ) se smíenými strategiemi prvního resp. druhého hrá e (x1, .
. . , xn, y1, . . . , ym, z) a s výplatní maticí
119
0 AT cT
A 0 b
c -b T 0
W
kde
xj = z xj , j = 1, . . . , n
yi = z yi , i = 1, . . . , m
xj
j
yi
z
1.
i
Jestlie existuje íslo z0 > 0 takové, e vektor (x10, . . . , xn0 , y10 , . . . , ym0 , z0) je eením smíeného rozí ení
ekvivalentní hry. Potom oba problémy lineárního programování mají optimální eení se sou adnicemi xj0 = xj0 /
z0 , j = 1, . . . , n ; yi0 = yi0 / z0 , i = 1, . . . , m a naopak. Podle základní v ty teorie maticových her musí
mít ekvivalentní hra vdy eení ve smíených strategiích, ale p ísluné problémy lineárního programování
nikoliv. Tento p ípad nastává, jestlie z0 = 0.
18 x1 + 4x2 + x3 ! MIN
3 x1 +
x2
3 y1 + 5 y2
3
3 y1 +
+ x3
5
y1
2 x1 + 3 x2 - x3
0
2 x1
2 y2
! MAX
18
4
y2
6
Ekvivalentní maticová hra je symetrická maticová hra s výplatní maticí
W
0
0
0
.
3
2
.
18
0
0
0
.
1
0
.
4
0
0
0
.
0
1
.
6
.
.
.
.
.
.
.
.
3
1
0
.
0
0
.
3
2
0
1
.
0
0
.
5
.
.
.
.
.
.
18
4
6
.
3
5
.
0
eení této hry bylo získáno numerickou Brownovou metodou, cena hry je v = 0 a vektor eení je (0,083; 0;
0,25; 0,167; 0,5; 0,083). Zp tnou transformací získáme eení obou problém lineárního programování x0 = (1;
0; 3) a y0 = (2; 6).
Lze tedy pomocí eení p i azené ekvivalentní hry získat eení úlohy lineárního programování. Tento zp sob
eení úloh lineárního programování je ve srovnání se simplexovými algoritmy neefektivní.
Poznámka. V p edchozích úvahách jsme vyuívali n které vlastnosti optimálních strategií smíeného rozí ení
maticových her. V dalí ásti budou popsány n které d leité dokazatelné vlastnosti optimálních strategií,
pouitelné p i analýze maticových her.
P i eení dalího procesu vyuijeme následujícího tvrzení.
M jme maticovou hru se smíeným rozí ením G = [(X), (Y), f (x, y)] s výplatní maticí. Ozna me indexem i itou ryzí strategii prvního hrá e a xi = (0, 0, . . . , 1, . . . , 0) odpovídající jednotkový vektor s jedni kou na i-tém
míst . Podobn ozna me indexem j j-tou ryzí strategii druhého hrá e. Odpovídající jednotkový vektor druhého
T
hrá e je yj = (0, 0, . . , 1, . . , 0) s jedni kou na j-tém míst . Potom z ejm platí:
n
f (i , y) = f (x i , y) = x i A y =
aij y j
(i = 1, . . . , m)
j 1
120
m
f (x , j) = f (x , y j) = x i A y =
(j = 1, . . . , n)
aij xi
i 1
m
f (x , y) =
n
f i, y xi
f x, j y j
i 1
j 1
Potom vektory x0 a y0 jsou optimálními smíenými strategiemi prvního a druhého hrá e práv tehdy, platí-li pro
vechna i = 1, . . . , m a vechna j = 1, . . . , n :
f (i , y0)
f (x0 , y0)
f (x0 , j),
f (x0 , j)
v,
f (i , y0)
v,
yj 0 (f (x0 , j) - v) = 0
xi 0 ( f (i , y0) - v) = 0, kde v je cena hry.
Pro cenu hry dále platí
v = min f (x0 , j) = max f (i , y0).
Optimální strategie hrá se nezm ní, p i teme-li k výplatní matici libovolnou konstantu k nebo vynásobíme-li
ji libovolnou kladnou konstantou q > 0. V prvém p ípad se cena hry zm ní na v + k , ve druhém na q.v.
P edchozí teoretickou úvahu aplikujeme nyní k výpo tu ceny hry zadané výplatní maticí A
A=
3 7 1
1 1 7
3 7 1
je známa optimální strategie prvního hrá e x0 = (0,6 ; 0,4 ; 0).
Cenu hry ur íme ze vztahu
v
min
f x0 , j
j
min (( 3.0, 6
Protoe f (x0 , 2) > v, tj. 3,8 > 2,2 bude
= v . Tedy eením soustavy rovnic
min( f ( x 0 , 1 ) , f ( x 0 , 2 ) , f ( x 0 , 3 ))
1.0, 4 ) , (7.0, 6 1.0, 4 ), ( 1.0.6
min( 2, 2 ; 3, 8 ; 2, 2 ) 2, 2 .
y2 0
= 0. Protoe x1 0 > 0 a
7.0, 4 ))
x2 0
> 0, bude f (1 , y0) = v a f (2 , y0)
3 y1 0 - y3 0 = 2,2
y1 0 + 7 y3 0 = 2,2
y1 0 + y3 0 = 1
získáme jednozna né eení y1 0 = 0,8 a y3 0 = 0,2 . Optimální smíenou strategií druhého hrá e je vektor y0 =
(0,8 ; 0 ; 0,2).
REFERENCES
[1]
[2]
BATINEC, J. Structural interbranch system of dynamic model. Proceedings of International conference
"University as Facilities for Advancement of Community and Region." Kunovice: EPI Kunovice, 2005, s.
121
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
317 - 322, ISBN 80-7314-052-7.
BATINEC, J.; DIBLÍK, J. Solution of Structural Interbranch Systém of a Dynamic Model. Proceedings
Fourth international conference on soft computing applied in computer and economic enviroment.
European Polytechnical Institute Kunovice, 2006, s. 35 - 41, ISBN 80-7314-084-5.
BATINEC, J.; NOVÁK, M. Numerické metody v navazujícím magisterském studiu. DIDZA 2006 (3nd
Didactic Conference in ilina with international participation), CD-ROM. ilina, ilinská univerzita,
2006, 1 - 9, ISBN 80-8070-557-7.
MOORE, P. G. The Business of Risk. Cambridge : University Press, 1983.
NOVÁK, M. Examples of using concepts of probability theory in managementdecision makinng.
Kunovice : Mezinárodní konference EPI, 2006.
STUCHLE, W. H. Management. München : Verlag Franz Valen, 1989.
VACULÍK, J.; ZAPLETAL, J. Podp rné metody rozhodovacích proces . Bron : Masarykova univerzita
v Brn , 1998.
WATSON, S. R.; BUDGE, J. R. Decision Synthesi. Cambridge : Cambridge University Press, 1987.
ZAPLETAL. J. Poznámka k rozhodování za rizika a nejistoty. Kunovice : Mezinárodní v decká
konference leden, 2006.
ZAPLETAL, J. Poznámka k rozhodování za rizika a nejistoty. Kunovice : Mezinárodní v decká
konference, leden, 2007.
ZAPLETAL, J. Metody rozhodování za rizika a nejistoty-t etí ást. Kunovice : Mezinárodní v decká
konference, leden, 2008.
ADRESS:
RNDr. Jitka Jablonická
Evropský polytechnický institut, s.r.o.,
Osvobození 699,
686 04 Kunovice
122
NELINEÁRNÍ PROGRAMOVÁNÍ
Marie Tomová
Fakulta elektrotechniky a komunika ních technologií, Vysoké u ení technické Brno
Abstrakt: V navazujícím magisterském studiu na Fakult elektrotechniky a komunika ních
technologií VUT v Brn se v p edm tu MPSO (Magisterské studium, Pravd podobnost, Statistika,
Opera ní výzkum) nebudou u it jen vybrané statistické metody, ale i elementy opera ní analýzy.
P ísp vek je v nován jedné její ásti nelineárnímu programování.
Klí ová slova: Opera ní analýza, lineární programování, nelineární programování, aplikace.
MATEMATICKÉ POJMY
Jak plyne z názvu, jde o p ípad, kdy jsou omezení nebo ú elová funkce (nebo obojí) nelineární.
Zna ení: symbolem
x (apod.) budeme zna it sloupcový vektor, x T k n mu transponovaný vektor ádkový:
Definice 1: Funkce f(x,y) má v bod [x0,y0] (lokální) minimum, existuje-li takové okolí tohoto bodu, e pro
vechny jeho body [x,y] platí f ( x, y ) f ( x0 , y0 ) .
Jestlie platí taková podmínka pro (n jakou) oblast M (to znamená: existuje takový bod [x0,y0] M, e pro
vechny jeho body [x,y] M platí f ( x, y ) f ( x0 , y0 ) ), íkáme, e funkce f nabývá v bod [x0,y0]
absolutního minima na oblasti M. Obdobn se definuje maximum. Ostré extrémy: místo neostrých nerovností
ostré.
Volný extrém: lokální extrém, který není vázán dalími podmínkami (zejména se p itom uvauje celý defini ní
obor funkce). M e jich být víc nap . sin(x) má nekone n mnoho maxim v bodech /2+2k , k=0, 1, 2,
Konvexní (a konkávní) mnoiny a funkce: Lze-li kadé dva body A,B dané mnoiny M spojit úse kou leící
v M, potom nazýváme takovou mnoinu konvexní. Libovolný bod X úse ky AB lze vyjád it na p íklad
takovouto konvexní lineární kombinací X= A+(1- )B,
0,1 .
Definice 2: Funkce f se nazývá konvexní na oblasti M, platí-li pro libovolné dva body A,B této oblasti:
f( A+(1- )B) f(A)+(1- )f(B).
Stacionární body: Uvame spojit diferencovatelnou funkci f. Z matematické analýzy víme, e nutnou
podmínkou pro volný extrém je nulová derivace (u funkcí více prom nných nulovost vech parciálních derivací).
Bod, ve kterém jsou vechny (první) parciální derivace nulové se nazývá stacionární. Platí tedy:
Lemma 1: Bu f spojit diferencovatelná funkce (jedné nebo více prom nných). Nabývá-li v bod A volného
extrému, je A stacionárním bodem, tj. vechny první parciální derivace jsou zde nulové.
Gradient (gradientní vektor) funkce f (x ) v bod x0 je vektor parciálních derivací podle jednotlivých
prom nných :
f ( x0 )
f ( x)
x1
...
f ( x)
xn
Tento vektor je kolmý na nadplochu
soustav nadploch
x
T
x0 ,
f ( x0 ) (
f (x)
,...,
x1
f (x)
) x
xn
x0 .
f ( x ) c0 v bod x0 a má sm r maximálního r stu funk ní hodnoty v
f ( x) c .
Záv rem: gradient funkce f v daném bod x je vektor parciálních derivací funkce f v tomto bod . Je zde kolmý
na vrstevnici (hladinu) a má sm r nejv tího r stu funkce f .
123
HLEDÁNÍ STACIONÁRNÍHO BODU
Omezíme se na diferencovatelné funkce jedné a dvou prom nných. P ípad více prom nných si tená snadno
rozváí zobecn ním úvah pro dv prom nné.
Funkce jedné prom nné:
Hledáme bod x, ve kterém je
F (x)
0 . Ozna íme-li si F ( x )
f ( x ) , je t eba eit obecnou rovnici
f ( x) 0 .
Z ady metod, které nabízí numerická matematika, je patrn nejlepí metoda Newtonova. Postupuje se takto:
zvolíme výchozí iteraci x0. V bod x0 vedeme k funkci g (x ) te nu (tj. aproximujeme g (x ) te nou). Pr se ík
této te ny s osou x je dalí iterací ko ene; ozna me x1. Postup opakujeme, a se po sob jdoucí iterace tém
nelií.
y
Prakticky: Rovnice te ny v bod x0 je
y
Její pr se ík s osou x
f ( x0 )
f ( x0 )( x x0 )
0
f ( x0 )
Dosadíme z druhé rovnice do první:
f ( x0 )( x x0 ) a odtud
x
x0
f ( x0 )
f ( x0 )
x1 .
Newtonova metoda je metoda itera ní: dalí p iblíení eení se po ítá z p edchozího podle stále stejného
p edpisu. Místo geometrického odvození m eme tentý výsledek dostat z Taylorovy ady. Je toti
f ( x0 )
( x x0 ) 2 ...
2!
Omezíme se na lineární ást rozvoje, tedy f ( x ) f ( x0 ) f ( x0 )( x
f ( x)
f ( x0 )
( x x0 )
1!
f ( x0 )
x0 ) a poloíme jej roven 0 (tj.
spo teme pr se ík této lineární ásti s osou x), co samoz ejm vede ke stejnému výsledku jako výe.
Funkce dvou a více prom nných
Hledejme stacionární bod funkce
Ozna me pro p ehlednost
Fx
F ( x, y ) , tj. bod, ve kterém je
f ( x, y ), Fy
f ( x, y )
1 2f
[
(x
2! x 2
0,
F ( x, y )
y
Fy
0.
g ( x, y ) 0
( x0 , y 0 ) . Taylorovy ady se st edem ( x0 , y0 ) pro f, g jsou
f ( x0 y 0 )
(x
x
f ( x0 y 0 )
Fx
g ( x, y ) .
f ( x, y ) 0
Máme tedy eit soustavu (nelineárních) rovnic
Vyjdeme op t z po áte ní iterace
F ( x, y )
x
f ( x0 y 0 )
(y
y
x0 )
2
x0 )
f
2
(x
x y
2
x 0 )( y
y0 )
y0 )
2
f
y
2
(y
y 0 ) 2 ] ...
( podobn pro g(x,y) )
Ozna me p ír stky
h
( x x0 ) [tj. x
x0
h]
k
(y
y0
k]
y 0 ) [tj. y
, omezme se na lineární ásti Taylorových ad a polome
je rovny nule:
f ( x0
h, y0
k)
f ( x0 , y0 )
f x ( x0 , y0 ) h
f y ( x0 , y0 ) k
0
g ( x0
h , y0
k)
g ( x0 , y0 ) g x ( x0 , y0 )h
g y ( x0 , y0 )k
0
Stru n ozna eno:
f xh
fyk
f
g xh
gyk
g
, ve v bod
( x0 , y 0 )
124
Jde o soustavu dvou lineárních rovnic pro p ír stky h, k. Vy eíme a dostaneme dalí iteraci
x1
x0
h , y1
y0
k
a celý postup opakujeme, a se po sob jdoucí iterace lií o mén , ne zvolená p esnost.
(Geometricky: nahradíme ob funkce te nými rovinami. Ty protnou rovinu xy ve dvou p ímkách a tyto p ímky
se protínají v dalí iteraci.)
Newtonova metoda je pro obecnou soustavu nelineárních rovnic s diferencovatelnými funkcemi nejpouívan jí.
Její slabinou je, e n kdy za ne itera ní proces oscilovat mezi dv ma body blízko ko ene a ko en se pak
v p ípad pot eby musí up esnit ru n (na p íklad níe popsanou gradientní metodou).
Gradientní metoda: Vyuívá toho, e vektor gradientu udává sm r nejv tího r stu funkce. ekn me, e
hledáme lokální minimum funkce dvou prom nných F(x,y). Vyjdeme tedy op t z n jakého výchozího bodu
( x0 , y0 ) . Vypo teme v n m gradient a postupujeme v opa ném sm ru (hledáme minimum, p i hledání maxima
postupujeme ve sm ru gradientu) o jistý násobek gradientu. Práv volba tohoto tzv. kroku je slabinou metody.
Empiricky zjit ný vhodný krok je n kde mezi 0,05 a 0,25 gradientu. Na po íta i je pomoc snadná: zvolíme
n kolik r zných krok a zvolíme ten, pro který poklesne gradient (jeho absolutní hodnota) co nejvíce. Postup
s novou iterací opakujeme, a se dostaneme do bodu, kde je gradient (a na zvolenou p esnost) nulový.
OBECNÁ ÚLOHA NLP (NELINEÁRNÍHO PROGRAMOVÁNÍ):
Nalézt globální extrém ú elové funkce f (x ) v oblasti, ur ené soustavou omezení
gi ( x )
0 . N které t ídy
úloh se ozna ují speciálními názvy:
kvadratické programování: kvadratická ú elová funkce, lineární omezení
konvexní programování: ú elová funkce i omezení jsou konvexní funkce
separabilní programování: bez smíených len (na p . pro f:
f (x)
f j (x j ) )
lomené programování: ú elová funkce a omezení jsou podíly lineárních funkcí.
P íklad 1 :
f ( x)
a) 0
b) 0
c) 0
( x1
2) 2
(x 2
2) 2
min p i omezeních
x1 4 , 0 x2 1
x1 1 , 0 x2 1
x1 4,5 , 0 x2 3,5
Snadno se nahlédne, e ú elová funkce má absolutní minimum 0 v bod [2,2]. Toto minimum není vázáno
ádnými omezujícími podmínkami; jde tedy o minimum volné.
Vázaný extrém je extrém v oblasti p ípustných eení M, je je ur ena omezeními. V naich p ípadech jde o
body:
a) bod (2,1) - leí na hranici M
b) bod (1,1) - jde o vrchol oblasti M
c) bod (2,2) - vnit ní bod M
Ji z této pom rn jednoduché úlohy kvadratického programování je vid t, e metodika eení bude podstatn
jiná (a daleko sloit jí) ne u úloh lineárního programování.
HLEDÁNÍ VÁZANÉHO EXTRÉMU (= EENÍ ÚLOHY NLP) PRO KONVEXNÍ PROBLÉM:
Úloha: minimalizujte
f (x ) ( x
( x1 , x2 ,..x n )) za podmínek g i (x ) 0 ( i=1,2,..m ), x 0 ,
kde ú elová funkce f i vekerá omezení jsou konvexní, spojit diferencovatelné funkce.
Uve me základní fakta o konvexních funkcích a konvexních mnoinách:
Pr nik konvexních mnoin je konvexní mnoina.
V ta 1: Bu
mnoina bod v En, spl ující omezení
g i ( x ) konvexní, je
x
0 , g i (x ) 0 , i=1,2,..,m. Jsou-li vechny funkce
konvexní mnoina (není-li prázdná).
V ta 2: Funkce konvexní na konvexní mnoin
zde má nejvýe jedno lokální minimum. Existuje-li, pak je
minimem globálním a dosahuje se na konvexní mnoin .
125
V ta 3: Funkce
kdy
f ( x1 )
f (x ) , definovaná a diferencovatelná na otev ené konvexní mnoin
f ( x2 )
T
( x1
x2 )
f ( x2 ) pro vechna x1 , x2
.
V ta 4: Konvexní, spojit diferencovatelná funkce na konvexní mnoin
x0
práv kdy
(x
x0 )
T
f ( x0 )
0 pro vechna x
je zde konvexní práv
nabývá globálního minima v bod
.
V ty 1 a 2 ukazují, e úloha konvexního programování má smysl, e ú elová funkce bude nabývat jediného
minima, a to bu v jediném bod , nebo v nekone né konvexní mnoin (podobn jak je tomu u lineárního
programování).
Základní mylenka eení: úloha se p evede na hledání speciálního volného extrému, tzv. sedlového bodu
Lagrangeovy funkce. Ta je definována takto:
m
F ( x, l )
f ( x)
li g i ( x )
(N1)
i 1
li ( asto se zna í i) jsou tzv. Lagrangeovy multiplikátory (Lagrangeovy neur ité koeficienty).
F ( x , l ) najd te nezáporné vektory x0 ,l 0 tak, e
Úloha o sedlovém bod : Pro Lagrangeovu funkci
F ( x0 , l )
F ( x0 , l0 )
F ( x, l0 ) .
Hledá se tedy maximum vzhledem k
slokách nezáporné).
V ta 5: Sedlový bod
V ta 6: Je-li
(N2)
l a minimum vzhledem k x (oba výsledné vektory jsou p itom ve vech
( x0 , l0 ) Lagrangeovy funkce je eením úlohy konvexního programování.
x0 eením úlohy konvexního programování, pak existuje vektor l 0 tak, e ( x 0 , l 0 ) je sedlový
bod Lagrangeovy funkce.
Langange vyvinul metodu multiplikátor pro varia ní po et. Tam je vysoce efektivní, zatímco v NLP je hledání
sedlového bodu obtíné. Východiskem je následující v ta:
V ta 7: (Kuhn-Tuckerova) Vektor
x0 je eením úlohy konvexního programování kdy a jen kdy existuje vektor
l 0 takový, e je spln no následujících est (Kuhn-Tuckerových) podmínek:
F ( x0 , l 0 )
xj
(KT1)
(KT2)
(KT3)
(KT4)
(KT5)
(KT6)
x0T
F ( x0 , l0 )
x
x0
0 pro vechna j
n
x0 j (
j 1
f ( x0 )
xj
m
li
i 1
g i ( x0 )
)
xj
0
0 (standardní podmínka nezápornosti eení)
F ( x0 , l 0 )
li
l 0T
g i ( x0 )
F ( x0 , l0 )
li
l0
0
(omezení)
m
l 0i g i ( x 0 )
0
i 1
0
126
P íklad 2: Ilustrujme si tuto ne práv jednoduchou v tu na praktickém p íkladu. Máme hledat minimum ú elové
funkce
f ( x1 , x2 ) ( x1 5) 2
mnoin
, dané nerovnostmi
( x2 5) 2 50
x12
x22 10 x1 10 x2 na
x1 1,
x1
x2
x1 , x2
4,
0.
eení: Funkce f nabývá absolutního minima v bod [5 , 5]. Vrstevnice jsou soust edné krunice se st edem
v tomto bod . Hledaný vázaný extrém ( eení naí úlohy) je z ejm bod, ve kterém se dotýká vrstevnice
s nejmení funk ní hodnotou oblasti . Je jím z ejm bod [2 , 2]. Ov me si te Kuhn-Tuckerovy podmínky:
g1 ( x )
x1
0
g2 ( x )
x1
x2 4 0
F ( x,l )
f
l1 g1 l 2 g 2
Parciální derivace Lagrangeovy funkce dle
v1
2 x1 10 l1 l 2
v2
2 x2 10 l2
Jak víme, musí být v optimu
x1
( x1
0,
2, x2
x2
0,
2, v1
v2
li g i
bude
0, g1
x12
x22 10 x1 10 x2 l1 ( x1 ) l2 ( x1
x2
4)
x1 , x2 ozna me v1, v2:
0 (také KT5), odtud l1
v1=v2=0.
2, g 2
0, l1
Odtud
0, l2
0 . Podle (KT2) je také x j v j
0 , a protoe
l2=10-2x2=10-4=6.
Uvedené
hodnoty
6 ) vyhovují Kuhn-Tuckerovým podmínkám.
Je-li hledané minimum konvexní funkce uvnit , jde o stacionární bod, v n m jsou derivace ú elové funkce
nulové. Ten bychom um li najít i bez Kuhn-Tuckerovy v ty, proto si tohoto p ípadu nebudeme vímat.
Je-li hledané minimum na hranici , uplatní se v plné mí e Lagrangeova idea hledání vázaného extrému
metodou neur itých koeficient . Pokusme se tuto ideu vyloit na následujícím problému:
2
Má se najít minimum funkce f ( x 2)
na oblasti , vyzna ené na obrázku, tedy pro
2 x y 6 0, x 0, y 0 .
Oblast je ohrani ena t emi p ímkami:
g1 ( x, y ) 2 x
y 6 0
g 2 ( x, y )
x
0
g 3 ( x, y )
y
0
( y 2) 2
Vrstevnice jsou krunice se st edem [ 2,2], minimum je
v hrani ním bod [0,2] na hrani ní p ímce x 0 .
Úlohu lze eit takto:
Najdeme volné minimum. Vidíme, e neleí v , proto
pokra ujeme dál.
Najdeme minimum f na kadé hrani ní p ímce zvlá ( eení práv takové úlohy podal Lagrange). Z t chto
minim najdeme nejmení.
V bodu 2 bychom tedy eili t i úlohy. Náhodou bychom dostali p ípustné eení, nebo bod
[0,2], kde je nejmení z t chto hodnot, leí v . Kdyby vak lo o ú elovou funkci f ( x 3)
( y 9) ,
nevedla by tato metoda k cíli, nebo nejmení z nejmeních hodnot nastává sice v bod p ímky
2 x y 6 0 , ale mimo (pro záporné x). Je tedy z ejm t eba uvaovat vechny hrani ní úse ky sou asn .
2
P ísluná Lagrangeova funkce by byla
F ( x, y )
f ( x, y ) l1 g1 l2 g 2
l3 g 3 .
2
íkejme jí t eba úplná
(Lagrangeova funkce). eením úlohy konvexního programování je sedlový bod práv této funkce, to jest bod,
ve kterém jsou vechny derivace (podle x,y,l1,l2,l3) rovny nule.
Z lineárního programování je známo, e je moné podmínky nezápornosti z modelu vyt snit, to jest za ídit, aby
se v po etním modelu p ímo nevyskytovaly. Samoz ejm se vak p i výpo tu respektují. Kuhn s Tuckerem se
pokusili, a to se zdarem, o podobné. Vylou ili z Lagrangeovy funkce podmínky nezápornosti. Tím si zna n
127
zkomplikovali odvození, tedy d kaz svých podmínek, ovem výsledkem bylo zjednoduení po etního modelu.
Ten se v kvadratickém p ípad nápadn podobá modelu z lineárního programování (viz p ítí odstavec), a také se
podobn eí.
Pro ná praktický p íklad to znamená, e se místo
F ( x, y)
x
2
f ( x, y ) l1 g1 l2 g 2
y
2
4 x 4 y l1 (2 x
2
l3 g 3
y 6) l2 ( x) l3 ( y )
2
pracuje s redukovanou funkcí F ( x, y ) f ( x, y ) l1 * g1 x
y 4 x 4 y l1 (2 x y 6 )
Cenou za toto zjednoduení pak je fakt, e derivace této funkce mohou být v optimu i kladné (nejen rovny nule).
Formulace Kuhn-Tuckerových podmínek, ur ená pro po etní model:
Pro p esné a zárove srozumitelné zn ní v ty uijeme této notace:
ozna me vj parciální derivaci Lagrangeovy funkce dle prom nné xj , v0j pak hodnotu této derivace
v bodu
( x0 , l0 ) , tedy v j
F ( x, l )
,
xj
F ( x0 , l0 )
xj
v0 j
ozna me di záporn vzatou derivaci Lagrangeovy funkce dle li,
di
di
F ( x, l )
li
g i (x ) , d 0i
Kuhn-Tuckerova v ta: Vektor
g i ( x ) , d0i pak hodnotu di v bodu ( x0 , l0 ) , tedy
g i ( x0 )
x0 je eením úlohy konvexního programování kdy a jen kdy existuje takový
vektor l0 , e je spln no následujících est (Kuhn-Tuckerových) podmínek:
(KT1)
v0
0 , to znamená v0 j
x0T v0
(KT2)
(KT3)
(KT4)
0 pro vechna j (j=1..n)
x0
0
0
(standardní podmínka nezápornosti eení)
d0
0
(omezení)
(KT5)
l0T d 0
(KT6)
l0
0
0
Podmínky 1,3,4,6 jsou podmínky nezápornosti vechny prom nné jsou v optimu nezáporné
Podmínky 2 a 5 jsou vylu ovací: vdy aspo jedna ze stejnolehlých párových prom nných
x 0 j , v0 j ,
respektive l0i , d 0i je (v optimu) rovna nule
Podmínky jsou nutné a posta ující. Jakmile se nám poda í najít bod, ve kterém jsou spln ny, máme eení. Jako
v lineárním p ípad nemusí být eení jediné.
LITERATURA:
[1]
CHURCHMAN, C. W.; ACKOFF, R. L.; ARNOFF, E. L. Úvod do opera ného výskumu. Bratislava :
ALFA, 1968.
[2]
LA IAK, A. a kol. Dynamické modely. Bratislava : ALFA, 1985.
[3]
NOVÁK, M. Probability theory in combined form of study at FEEC BUT. Kunovice : Mezinárodní
konference EPI, 2006.
[4]
TYC, O. Opera ní analýza. Brno : MZLU, 2002.
[5]
ZAPLETAL, J. Opera ní analýza. Kunovice : SKRIPTORIUM VO, 1995.
[6]
DUDORKIN, N. Opera ní analýzy. Praha : FEL VUT, 1997.
128
ADRESA:
Mgr. Marie Tomová
Department of Mathematics,
Faculty of Electrical Engineering and Communication,
Technická 8, 616 00 Brno, Czech Republic
[email protected]
129
130
METODY ROZHODOVÁNÍ ZA RIZIKA A NEJISTOTY
t etí ást
Josef Zapletal
Evropský polytechnický institut, s.r.o., Kunovice
Abstrakt. V lánku se popisují metody rozhodovací analýzy, které mají pon kud odliné rysy od
rozhodovacích metod opera ní analýzy, které p edstavují najm optimaliza ní nástroje vhodné pro
eení jednoduích, dob e strukturovaných rozhodovacích problém . Naopak charakteristickým
rysem rozhodovací analýzy je to, e snaí skloubit exaktní postupy a modelové nástroje se znalostmi
a zkuenostmi eitel t chto problém . Heuristické metody významn ovliv ují postupy a výsledné
eení problém . Uvedeme základní pojmy , metody a nástroje rozhodovací analýzy, resp.
rozhodování za rizika a nejistoty. Mezi n bude pat it pojem subjektivní pravd podobnost, funkce
utility za rizika a n které grafické nástroje podpory eení rozhodovacích problém za rizika a
nejistoty. Ve svém p ísp vku se budu v novat práv jednomu z hlavních grafických nástroj
zobrazení d sledk rizikových variant, ovlivn ných faktory rizika, které se realizují v ur itém
asovém sledu a to pravd podobnostním strom m.
Klí ová slova. Riziková varianta, faktory rizika, pravd podobnostní strom, rozd lení
pravd podobnosti d sledk rizikových variant, ohodnocený pravd podobnostní strom, význam hran
a uzl rozhodovacího stromu, optimální strategie rozhodování, posloupnost optimálních rozhodnutí
v jednotlivých etapách rozhodovacího procesu.
1 PRAVD PODOBNOSTNÍ STROMY
1. 1 ÚVOD
Tento p ísp vek má být jakýmsi metodickým návodem pro studenty EPI, kte í se zabývají ve svých projektech
problematikou rozhodování a to zejména rozhodování ízení nedeterministických proces . Výchozím materiálem
se mn stala skripta Ji ího Fotra a Ji ího D diny Manaérské rozhodování a dále práce [2], [4], [5], [7], [13].
Pravd podobnostní stromy p edstavují grafický nástroj zobrazení d sledk rizikových variant, ovlivn ných
faktory rizika, které se realizují v ur itém asovém sledu. Jednotlivé faktory rizika se zobrazují ppomocí uzl
pravd podobnostního stromu. Hrany, vycházející z t chto uzl , zobrazují jednotlivé moné hodnoty faktor
rizika (p ípadn moné výsledky rizikových aktivit), v etn jejich pravd podobností. Moné hodnoty d sledk
dané rizikové varianty jsou zobrazeny konci v tví pravd podobnostního stromu; jejich pravd podobnosti se
stanoví na základ pravd podobností hran stromu, leících na této v tvi.
P edností uití pravd podobnostního stromu pro stanovení rozd lení pravd podobnostních d sledk rizikových
variant je jednoduchost jeho konstrukce, p ehlednost a srozumitelnost. Sou asn m e tento strom slouit jako
významný nástroj komunikace, nebo kadá jeho v tev je zobrazením ur itého moného budoucího vývoje
sv ta; p edstavuje tudí ur itý scéná .
Z povahy pravd podobnostního stromu vak vyplývá, e m e slouit pouze pro zobrazení diskrétních faktor
rizika, p ípadn diskrétních d sledk variant. To vede k tomu, e v p ípad , jsou-li mezi faktory rizika spojité
náhodné veli iny musíme tyto veli iny aproximovat pomocí diskrétních náhodných veli in s omezeným po tem
hodnot (velmi asto t emi). Tato aproximace se obvykle realizuje tak, e spojitou distribu ní funkci nahradíme
stup ovitou funkcí, která se dané spojité distribu ní funkci co nejvíce p ibliuje. Po et stup ur uje po et
hodnot aproximující diskrétní veli iny a výka stup odpovídající pravd podobnosti jednotlivých hodnot této
diskrétní veli iny.
Pravd podobnostní stromy lze s výhodou uít k zobrazení d sledk rizikových variant zvlát tehdy, kdy tyto
varianty tvo í ur itý soubor asov uspo ádaných díl ích aktivit zatíených rizikem. Podstatu a monost
uplatn ní pravd podobnostních strom si ukáeme na jednoduchém p íkladu.
131
1.2 P ÍKLAD PRAVD PODOBNOSTNÍHO STROMU
P edpokládejme, e podnik chce rozí it sv j výrobní program o ur itý nový výrobek,který byl vytipován
pomocí lineárního programování. P edpokladem rozí ení výrobního programu je vak úsp ný vývoj výrobku,
jeho úsp né poloprovozní ov ení a zavedení do hromadné výroby. Komer ní úsp ch výrobk na trhu pak bude
záviset na velikosti poptávky a dosaené prodejní cen . P i aplikaci lineárního programování, vechny tyto
poloky byly dodány experty p i tvorb modelu na který byl aplikován princip lineárního programování, nyní
bude nutno provést ov ení, jeho výsledek ponese s sebou jisté riziko.
Kone né hospodá ské výsledky (zisk) tohoto podnikatelského projektu (rizikové varianty) budou tedy záviset na
úsp nosti a výsledcích jednotlivých, vzájemn navazujících inností, které jsou tvo eny
vývojem výrobku;
poloprovozním ov ením výrobku;
zahájením hromadné výroby;
uvedením výrobku na trh.
Je z ejmé, e výsledky kadé této innosti (operace) jsou nejisté a závisí na mnoha faktorech (nap . novost,
technická a technologická náro nost výrobku, zkuenosti, kvalifikace a p ístrojové vybavení týmu
zabezpe ujícího vývoj, marketingová strategie, pr zkum trhu a jiné).
Pro jednoduchost budeme p edpokládat, e kadá innost (operace) m e skon it bu úsp n nebo neúsp n .
V p ípad jejich úsp chu se uskute ní dalí navazující operace (tak nap íklad v p ípad úsp chu výzkumu a
vývoje se zahájí poloprovozní ov ování), pokud vak ur itá innost bude neúsp ná, podnikatelský projekt se
zastaví. Podnikatelský projekt a jeho moné výsledky m eme nyní zobrazit pravd podobnostním stromem (viz
obr. 1).
Jednotlivé innosti (operace) jsou na obr.1 zobrazeny krouky a jejich moné výsledky (úsp ch i neúsp ch)
hranami, které vycházejí z t chto krouk .
Situace
Neúsp ch
1
Neúsp ch
Úsp ch
2
3
1
2
3
4
Výzkum a vývoj
Poloprovoz
Zavedení do hromadné výroby
Uvedení na trh
Neúsp ch
Úsp ch
Neúsp ch
Úsp ch
A
B
C
D
4
Úsp ch
E
Obr.1 Pravd podobnostní strom
Z obrázku 1 je z ejmé, e pokud jde o výsledky podnikatelského projektu, m e nastat p t r zných budoucích
situací (rizikových situací), a to:
situace A: neúsp ch výzkumu a vývoje výrobku;
situace B: úsp ch výzkumu a vývoje a neúsp ch poloprovozního ov ení výrobku;
situace C: úsp ch výzkumu a vývoje i poloprovozního ov ení výrobku, neúsp ch jeho zavedení do
hromadné výroby;
situace D: úsp ch výzkumu a vývoje, poloprovozního ov ení výrobku i jeho zavedení do hromadné výroby,
neúsp ch na trhu;
situace E: úsp ch výzkumu a vývoje, poloprovozního ov ení výrobku i jeho zavedení do hromadné výroby,
v etn úsp chu na trhu;
ty i situace (A, B, C, D) se vytahují k neúsp chu podnikatelského projektu a pouze poslední situace (E)
odpovídá jeho úsp chu.
Je z ejmé, e pro rozhodnutí o p ijetí i zamítnutí podnikatelského projektu je t eba získat dalí informace, které
132
by nám pomohly blíe specifikovat jednotlivé výe uvedené rizikové situace. Pot ebné informace se týkají
jednak hospodá ského výsledku podnikatelského projektu (zisku i ztráty) p i jednotlivých rizikových situacích,
jednak nebezpe í (u prvních ty situací), respektive nad jnosti (poslední situace), se kterými mohou jednotlivé
situace nastat. Výchozí informace1 pro íselné ohodnocení rizikových situací shrnuje tabulka 1.
Uvedení na trh
innost (operace)
Výzkum
a vývoj
Poloprovozní
ov ení
5
0.7
3
0.9
Odhadované náklady (mil. K )
isté výnosy (mil. K )
Subjektivní pravd podobnost
úsp chu
Zavedení do
hromadné
výroby
302)
0.98
Üsp ch
2
100
0.8
Neúsp
ch
2
100
0.2
Tabulka 1 Odhady náklad , výnos a jejich pravd podobnosti
Propo et hospodá ských výsledk (zisku i ztráty) podnikatelského projektu p i jednotlivých rizikových
situacích i jejich pravd podobností ilustruje obrázek 2. U jednotlivých inností (operací) jsou uvedeny jim
odpovídající náklady, u hran zobrazujících výsledky t chto operací jsou uvedeny odpovídající subjektivní
pravd podobnosti úsp chu i neúsp chu dané operace a u hran zobrazujících výsledky uvedení na trh (viz hrany
vzcházející na obr. 2 z uzlu íslo 4) jsou uvedeny té isté výnosy z prodeje výrobku v p ípad jeho trní
úsp nosti i neúsp nosti.
Situace
Neúsp ch
N=5
1
0,7
2
0,9
Pravd podobnost
-5
0,30
B
-8
0,07
C
-38
0,01
D
-30
0,12
E
60
0,50
A
0,3
Úsp ch
Zisk
(ztráta)
Neúsp ch
0,1
Úsp ch
Neúsp ch
0,02
Neúsp ch
Úsp ch
V = 10
0,98
0,2
4
1 Výzkum a vývoj
2 Poloprovoz
3 Zavedení do hromadné výroby
4 Uvedení na trh
V Výnosy
3
N Náklady
0,8
Úsp ch
V = 100
Obr.2 Ohodnocený pravd podobnostní strom
Riziková situace A (neúsp ch výzkumu a vývoje) vede ke ztrát p ti milion K (odpovídá náklad m na
výzkum a vývoj viz tab.1), p i em pravd podobnost této ztráty je 0.3 (pravd podobnost úsp chu výzkumu a
vývoje byla odhadnuta na 0.7.
Ztrátu rizikové situace B stanovíme jednodue jako sou et náklad na výzkum a vývoj výrobku a na jeho
poloprovozní ov ení, co je 5 milion a 3 miliony K , tudí 8 milion K . Pravd podobnost této ztráty
(rizikové situace B) stanovíme touto úvahou: Tato ztráta nastane v tom p ípad , e bude výzkum a vývoj
úsp ný (tato událost nastává s pravd podobností 0.7) a sou asn bude neúsp né jeho poloprovozní ov ení
(druhá událost s pravd podobností 0.1. Pravd podobnost ztráty velikosti osni milion K stanovíme proto jako
sou in pravd podobností obou událostí, které nastávají sou asn . Dostaneme tedy 0.7 x 0.1 = 0.07, co je
pravd podobnost dané ztráty a tudí i pravd podobnost rizikové situace B.
1
íselné údaje uvedené v tabulce 1 týkající se náklad a výnos byly získány z podkladových materiál a technicko-ekonomické studie
podnikatelského projektu, odhady subjektivních pravd podobností jsou výsledkem diskuse analytika s odborníky z odpovídajících oblastí
(výzkum, vývoj, výroba, marketing). Tyto náklady jsou v etn náklad na nákup a instalaci výrobní aparatury.
133
Stejným zp sobem stanovíme risk, p ípadn ztrátu i pravd podobnosti dalích rizikových situací. Ztráta rizikové
situace C je rovna p ti +t em + t iceti milion m K co je dohromady 38 milion K . Pravd podobnost této
ztráty stanovíme op t jako sou in pravd podobností jednotlivých událostí vytvá ejících danou rizikovou situaci
(tj. sou in pravd podobnosti úsp chu výzkumu a vývoje velikosti 0.7, pravd podobnosti úsp chu
poloprovozního ov ení velikosti 0.9 a pravd podobnosti neúsp chu zavedení výrobku do hromadné výroby,
která je 1 - 0.98 = 0.02). Dostaneme tedy 0.7 x 0.9 x 0.02 = 0.01. Ztráta velikosti 38 milion K nastane proto
s pravd podobností 0.01.
Ztráta rizikové situace D je p t + t i + t icet + dva deset milion K , co je dohromady t icet milion K . Tato
ztráta nastane s pravd podobností 0.7x0.9x0.98x0.2 = 0.12.
Jedinou rizikovou situací, p i které vede daný podnikatelský projekt k zisku, je poslední situace E. Její zisk je
roven rozdílu istých výnos získaných prodejem výrobku v p ípad jeho trní úsp nosti (sto milion K ) a
souhrných náklad na výzkum a vývoj výrobku, jeho poloprovozní ov ení a jeho zavedení do hromadné výroby
co iní 5+3+30+2 = 40 milion K . Dostáváme tak ástku 100 40 = 60 milion K . Tohoto zisku dosáhneme
s pravd podobností 0.7 x 0.9 x 0.98 x 0.8 = 0.5. P ehledn jsou shrnuty moné hospodá ské výsledky daného
podnikatelského projektu v tabulce 2. Tyto výsledky mají podobu tabulkového zápisu rozd lení
pravd podobnosti zisku (ztráty) a pro lepí názornost je zobrazíme na obr. 3
Pravd podobnost
60
0,5
0,4
0,3
Zisk (mil. K )
-5
0,2
-30
-8
0,1
-38
A
B
C
D
E
Riziková situace
Obrázek 3 Rozd lení pravd podobnosti zisku (ztráty)
íselné údaje uvedené v tabulce 2. sou asn charakterizují riziko daného podnikatelského projektu. Z ejm jde o
zna n rizikový projekt. Pouze jediná situace (situace E s pravd podobností 0.5) vede k zisku, p i em vechny
zbývající situace jsou spojeny se ztrátou. Nejnií ztráta 5 milion K nastane v p ípad neúsp chu výzkumu a
vývoje (viz situace A) a naopak nejv tí ztráta hrozí podniku p i neúsp chu zavedení výrobku do hromadné
výroby (viz situace C). Tato ztráta velikosti 38 milion K je vak relativn málo pravd podobná (odpovídající
pravd podobnost iní pouze 0.01).
Riziková
A
B
C
D
E
situace
Pravd podobn
ost
0.3
0.07
0.01
0.12
..5
Z i s k (z t r á t a ) v mil
K
-5
-8
-38
-30
60
Tabulka 2
Z výe uvedeného p íkladu plyne, e pravd podobnostní strom je uite ným a názorným nástrojem zobrazení
nejistých d sledk rizikových variant. Sou asn je vak t eba míti na pam ti, e uplatn ní pravd podobnostního
stromu vyaduje v n kterých p ípadech ur ité zjednoduení eených problém . V naem p ípad k t mto
zjednoduením p edevím pat í
134
nerespektování nejistoty náklad vývoje, poloprovozního ov ení a zavedení produktu do hromadné výroby
(aproximace t chto nejistých veli in jejich deterministickými odhady.
Nahrazení výnosu daného produktu (v p ípad úsp chu vývoje, poloprovozního ov ení a zavedení do
hromadné výroby) dvouhodnotovou náhodnou veli inou (první hodnotou je výnos v p ípad komer ního
úspchu produktu a druhou hodnotou je výnos v p ípad jeho komer ního neúsp chu).
Zvlát druhý p edpoklad je zna n zjednoduující. Zatímco je dosti oprávn né respektovat pouze dva moné
výsledky vývoje, poloprovozního ov ení a zavedení do hromadné výroby (tj. jejich úsp ch a neúsp ch), lze
stejný postup u výsledk uvedení výrobku na trh snadno napadnout. D vod spo ívá v tom, e výnos z prodeje
daného výrobku je spojitá náhodná veli ina, která m e nabýt libovolné hodnoty mezi nulou a jedni kou a m la
by proto být aproximována více (alespo t emi) bodovými odhady. Pokud bychom cht li pracovat výnosem
produktu po jeho uvedení na trh jako se spojitou náhodnou veli inou, p icházejí v úvahu dv monosti. První z
nich je p ímé expertní stanovení rozd lení pravd podobnosti výnosu a druhou je stanovení tohoto rozd lení
simulací metodou Monte Carlo. P ímé expertní ohodnocení rozd lení pravd podobnosti je vak pro experty
zna n obtíné, nebo výnos ovliv uje v tí po et faktor , z nich n které (nap . výe poptávky, dosahovaná
prodejní cena, doba ivotnosti výrobku, ale i n které nákladové sloky, nap . investi ní náklady) jsou nejisté a
tvo í faktory rizika. Vhodn jím p ístupem ke stanovení rozd lení pravd podobnosti výnosu je proto v tomto
p ípad uplatn ní simulace metodou Monte Carlo (blíe viz [4 ].
1. 3 ROZHODOVACÍ STROMY
Rozhodovací stromy p edstavují jeden z nevýznamn jích nástroj rozhodovací analýzy. Umo ují nejen
zobrazení d sledk rizikových variant vzhledem ke zvolenému kritériu hodnocení, ale slouí té ke stanovení
optimální rozhodovací strategie ve víceetapových rozhodovacích procesech.
Rozhodovací stromy p edstavují ur itý grafický nástroj zobrazení rozhodovacích proces , vyuívající pojmový
aparát teorie graf . Lze je realizovat jako posloupnost uzl a hran orientovaného grafu. Uzly rozhodovacího
stromu mají povahu bud uzl rozhodovacích, nebo situa ních.
Situa ní uzly mají stejný charakter jako uzly pravd podobnostních strom , p i em hrany vycházející z t chto
uzl zobrazují tzv. Situa ní varianty. Rozhodovací uzly (oby ejn se zna í koso tvere ky) jsou zobrazením té
fáze rozhodovacího procesu, kdy má rozhodovatel monost volby ur ité varianty ze souboru navrených variant.
Tyto varianty jsou zobrazeny hranami, které vycházejí z rozhodovacích uzl . Uplatn ní rozhodovacího stromu
pro stanovení optimální strategie ve víceetapovém rozhodovacím procesu ukáeme na p íklad .
P íklad rozhodovacího stromu.
P edpokládejme, e hospodá ská jednotka nakupuje dosud ze zahrani í jeden z významných polotovar , na
kterém je zaloena ást nosného výrobního programu této jednotky. V sou asné dob zvauje vedení
hospodá ské jednotky zavést vlastní výrobu daného polotovaru, co vak p edpokládá nejprve zpracovat výrobní
postup v laboratorním m ítku a v p ípad úsp chu jeho poloprovozní ov ení.
Vedení hospodá ské jednotky stojí tedy p ed rozhodnutím, zda vynaloit prost edky na laboratorní výzkum a
vývoj daného polotovaru a v p ípad jeho úsp chu dalí prost edky na vybudování poloprovozu a ov ení
technologického postupu daného polotovaru v poloprovozním m ítku, nebo zda pokra ovat v nákupu ze
zahrani í.P itom je z ejmé, e úsp ch laboratorního výzkumu a vývoje není zaru en (stejn tak to platí o
výsledku poloprovozního ov ení), take v p ípad jejich neúsp chu je t eba pokra ovat v nákupu polotovaru ze
zahrani í (pro jednoduchost vylou íme nákup licence). Prost edky na laboratorní výzkum a vývoj i poloprovozní
ov ení zvýí náklady hospodá ské jednotky, ani by vedly k n jakému efektu.
135
1. etapa
2. etapa
3. etapa
Krytí vlastní pot eby
6
P = 0,9
PO
P = 0,7
LVV
1
2
N=5
3
4
N=3
5
I1
I2
Úsp ch
Neúsp ch
N = 32
N = 20
P = 0,1
Úsp ch
Neúsp ch
P = 0,2
P = 0,8
V=0
N = 72
Krytí vlastní pot eby a export
V = 90
N = 102
52
V=0
N = 58
86
Nákup polotovaru
N = 90
Nákup polotovaru
N = 90
Nákup polotovaru
N = 90
Nákup polotovaru
N = 90
P=0,3
112
98
95
95
90
Vysv tlivky:
LVV laboratorní výzkum a vývoj polotovaru
PO poloprovozní ov ení
I1 instalace výrobní aparatury 100 tis. kg/rok
I2 instalace výrobní aparatury 50 tis. kg/rok
V výnosy
N náklady
P pravd podobnost
Obr. 4. Rozhodovací strom problému zavedení výroby polotovaru
Budou-li výsledky poloprovozního ov ení úsp né (tj. vyrobený polotovar bude svojí kvalitou vyhovovat
p edepsaným poadavk m a bude tak moci nahradit provoz dováeného výrobku), bude t eba zakoupit a
instalovat výrobní aparaturu na výrobu daného polotovaru. Pokud by cht la hospodá ská jednotka vyráb t
polotovar pouze pro svoji vlastní pot ebu, byla by velikost výrobní aparatury dána jednozna n pouze výí této
pot eby (s p ípadnou rezervou na její moný r st). Vzhledem k tomu, e vak existuje ur itá monost vývozu
daného polotovaru, je t eba zvaovat rovn variantu instalace v tí výrobní aparatury, která by umonila nejen
uspokojit vlastní pot ebu, ale i pokrýt moný vývoz.
Rozhodnutí o realizaci laboratorního výzkumu a vývoje daného polotovaru má v naem p ípad charakter
t íetapového rozhodovacího procesu (viz. obr. 4). V první etap p icházejí v úvahu dv varianty, z nich první
tvo í výzkum a vývoj a duhou pokra ování v nákupu polotovaru (tyto varianty jsou zobrazeny hranami
vycházejícími z rozhodovacího uzlu íslo 1 na obrázku 4). V p ípad neúsp ch laboratorního výzkumu a vývoje
je nutné pokra ovat v nákupu polotovaru ze zahrani í (viz dolní hrana vycházející ze situa ního uzlu íslo 2 na
obrázku 4). V p ípad úsp chu tohoto výzkumu vstupuje rozhodovací proces do druhé etapy (viz rozhodovací
uzel íslo 3 na obrázku 4), kdy varianta rozhodování tvo í jednak vybudování poloprovozu a a poloprovozní
ov ení polotovaru (horní hrana vycházející z rozhodovacího uzlu íslo 3 na obrázku 4), jednak pokra ování
v nákupu polotovaru.
Úsp ch poloprovozního ov ení výroby polotovaru vede ke t etí etap rozhodovacího procesu (viz rozhodovací
uzel íslo 5 na obrázku 4). V této etap p ichází v úvahu instalace v tí výrobní aparatury pro pokrytí vlastní
pot eby daného polotovaru i jeho moného vývozu (horní hrana vycházející z rozhodovacího uzlu íslo 5),
pop ípad instalace mení výrobní aparatury pouze pro vlastní pot ebu (dolní hrana z rozhodovacího uzlu íslo
5)
Dosaené ekonomické výsledky v tí výrobní aparatury závisí na realizaci vývozu. To zobrazují hrany,
vycházející ze situa ního uzlu íslo 6 na obrázku 4. (U mení aparatury ur ené pouze pro vlastní pt ebu nejsou
ekonomické efekty na nejistém exportu závislé).
136
efektu 12 mil. K (vytvo eného rozdílem výnos z exportu a náklad produkce pro vlastní pot ebu a export) se
dosáhne s pravd podobností 0.8 (Pomocnou hodnotu 24 mil K uvádíme na obrázku íslo 5 nad situa ním uzlem
íslo 6).
O ekávané náklady varianty spo ívající v instalaci v tí výrobní aparatury, tedy iní
32 + 24 = 56 mil. K ,
Kde 32 mil. K jsou investi ní náklady aparatury velikosti 100 tisíc kg / rok. Vzhledem k tomu, e tyto
o ekávané náklady jsou nií ne o ekávané náklady nerizikové varianty, spo ívající v instalaci mení výrobní
aparatury (a ty jsou rovny souhrnu investi ních náklad velikosti 20 milion K a výrobních náklad velikosti
58 milion K , co je celkem 78 milion K ), vylou íme tuto nerizikovou variantu z dalích úvah. To ozna íme
dvojím krtem této hrany rozhodovacího stromu na obrázku íslo 5 (o ekávané náklady preferované varianty I1
zapíeme nad rozhodovací uzel íslo 5 v podob tak zvané pozi ní hodnoty).
Dalí postup stanovení optimální strategie je ji obdobný, a proto jej uvedeme ve stru né form . Ve druhé etap
je t eba volit mezi dv ma variantami, z nich první spo ívá v realizaci poloprovozního ov ení výroby
polotovaru (v tev ozna ená PO vycházející z rozhodovacího uzlu íslo 3) a druhá v nákupu polotovaru. Riziková
varianta poloprovozní ov ení vede s pravd podobností 0.1 k neúsp chu a vlivem toho k pokra ování
v nákupu polotovaru. S pravd podobností 0.9 vede k úsp chu, a tím k instalaci v tí výrobní aparatury (co je
optimální varianta ve t etí etap ). O ekávané náklady rizikové varianty poloprovozní ov ení stanovíme jako
3 + 0.9 x 56 + 0.1 x 90 = 62.4 mil. K .
Protoe jsou nií ne náklady varianty nákup polotovaru, je op t preferovanou variantou ve 2. etap
rozhodovacího procesu varianta !poloprovozní ov ení (její o ekávané náklady uvedeme op t nad
rozhodovacím uzlem íslo 3 obrázku íslo 5 jako jeho pozi ní hodnotu a zamítnutou variantu nákup
polotovaru ozna íme dvojím krtem).
V první etap daného rozhodovacího procesu je t eba volit mezi rizikovou variantou (laboratorní výzkum a
vývoj (v tev s ozna ením LVV vycházející z rozhodovacího uzlu íslo 1) a nerizikovou variantou nákup
materiálu. O ekávané náklady varianty laboratorní výzkum a vývoj stanovíme op t jako
5 + 0.7 x 62.4 + 0.3 x 90 = 75.7 mil. K .
Vzhledem k tomu, e o ekávané náklady této varianty jsou nií ne o ekávané náklady varianty nákup
polotovaru, která vychází z rozhodovacího uzlu íslo 1 (ty iní 9*0 milion K ), je optimální variantou v 1.
etap daného rozhodovacího procesu varianta laboratorní výzkum a vývoj s o ekávanými náklady 75.7
milion K (pozi ní hodnota rozhodovacího uzlu íslo 1).
Optimální rozhodovací strategie tvo í tedy následující posloupnost rozhodnutí:
laboratorní výzkum a vývoj v 1. etap ;
poloprovozní ov ení ve 2. etap ;
instalace aparatury velikosti 100 tisíc kg / rok ve 3. etap .
Jak je z ejmé, optimální strategie je závislá na subjektivním ohodnocení rozhodovacího stromu (p edevím na
subjektivních pravd podobnostech situa ních variant i odhadech náklad a výnos ), a proto je uite né ur it její
citlivost na zm ny vstupních veli in subjektivní povahy pomocí analýzy citlivosti. Tato analýza umo uje
stanovit veli iny, na jejich zm ny je optimální strategie málo citlivá a naopak specifikovat veli iny (kritické
faktory), s jejich malými zm nami dochází ke zm n této strategie. Dalí pozornost se pak soust e uje na tyto
kritické faktory a monosti jejich spolehliv jího stanovení.
138
24
56
P = 0,9
59,4 4
PO
62,4
70,7
75,5 LVV
1
P = 0,7
2
P=0,3
3
I1
5
P = 0,2 V = 0
6
N = 32
P = 0,8
I2 N = 20
Úsp ch
Neúsp ch
P = 0,1
N=3
Úsp ch
Neúsp ch
N = 72
112
Krytí vlastní pot eby a export
V = 90
N = 102
52
V=0
N = 58
86
Nákup polotovaru
N = 90
Nákup polotovaru
N = 90
Nákup polotovaru
N = 90
Nákup polotovaru
N = 90
98
95
95
90
Obr. 5. Stanovení optimální strategie pomocí rozhodovacího stromu
Analýza citlivosti rozhodovacího stromu je vak pro ru ní výpo et numericky velmi náro ná a proto je t eba
pouít po íta . Jako ilustraci uplatn ní po íta ové analýzy citlivosti uvádíme grafické zobrazení o ekávaných
náklad obou rozhodovacích variant z 1. etapy daného rozhodovacího procesu v závislosti na zm nách ceny
polotovaru (viz obrázek 5.). Z tohoto obrázku je z ejmé, e varianta laboratorního výzkumu a vývoje je
optimální pro cenu vyí ne asi 155 K . P i nií cen polotovaru je naopak výhodn jí pokra ovat v nákupu
tohoto polotovaru. Stejným zp sobem lze ur it citlivost optimální strategie na dalí parametry rozhodovacího
stromu.
O ekávané
náklady
(mil. K )
100
90
Nákup polotovaru
80
Výzkum a vývoj
70
60
50
120
140
160
180
200
Cena ( K /kg)
Obr. 6. Závislost o ekávaných náklad na cen polotovaru
LITERATURA
[1]
[2]
[3]
EDEN, C.; ONES, S.; SIMS, D. Messing About in Probléme. Oxford : Pergamon Press, 1983.
[4]
[5]
FOTR, J. Manaérská rozhodovací analýza. Praha : VE, 1992.
[6]
FOTR, J.; D DINA, J. Manaérské rozhodování. Praha : VE.
[7]
FOTR, J.; PÍEK, M. Exaktní metody ekonomického rozhodování. Praha : Academia, 1986.
139
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
IVANCEVICH, J. M.; DONESLY, J. H.; GIBBON, J. L. Management. Principles and Functions.
Homewood R. D. Irvin, 1989.
MOORE, P. G. The Business of Risk. Cambridge. University Press, 1983.
NOVÁK M. Examples of using concepts of probability theory in managementdecision makinng.
Mezinárodní konference Kunovice : EPI, 2006.
NOVÁK, M. Probability theorz in combined form of study at FEEC BUT. Mezinárodní konference
Kunovice : EPI, 2006.
PÍEK, M.; VOBO IL, J. Vybrané metody dlouhodobého prognózování a jejich vyuití. Praha :
Ekonomický ústav SAV, 1981.
STCHLE, W. H. Management. München : Verlag Franz Valen, 1989.
VL EK, R. P íru ka hodnotové analýzy. Praha : SNTL, 1983.
WATSON, S. R.; BUDGE, J. R. Decision Synthesi. Cambridge. Cambridge University Press, 1987.
ZAPLETAL, J. Poznámka k rozhodování za rizika a nejistoty. Mezinárodní v decká konference
Kunovice : leden 2007.
ADRESA:
Doc. RNDr. Josef Zapletal, CSc.
Evorpský polytechnický institut, s.r.o.
Osvobození 699
686 04 Kunovice
140
TEST OF OPTIMUM AND SOLUTION IMPROVEMENT OF THE BALANCED
TRANSFORMATION PROBLE
Josef Zapletal
Abstract. The transportation problem is one of problems of linear programming but very often it
solved by other special methods as for example by NW-corner method or VAM-method. The reason
of this consist in a very big number of zeros in the system constraining conditions. After the first
step of these and also others methods we do not receive the optimal value of the objective function.
It is necessary to do a betterment of solution. Such a method is given in this paper.
Abstrakt. Dopravní úloha je jednou ze základních úloh lineárního programování. P i klasickém
zpracování dochází k situaci, kdy se v rozsáhlé matici omezení vyskytují v tinou nuly. Z tohoto
d vodu byly vyvinuty metody, které nezabírají v po íta ích tolik místa a jsou i po stránce rychlosti
výpo tu efektivn jí. Mezi n pat í metoda severozápadního rohu, která nepracuje a nezískává
optimální eení na bázi ekonomických vstupních parametr , ale její princip je odvozen ist
pomocí geografických v domostí. Dalí metody mají sv j podklad v ekonomických vlastnostech
problému a základní p i azení se provádí pomocí p ímých nebo pom rných náklad . Sem pat í
indexová metoda a
Key words. Supplier, costumer, stone, occupied cell, waters, row and column numbers, differences
among costs, polygon, vertices of polygon, the signs of differences for optimal solution.
1) CALCULUS OF THE OPTIMIZING
In the second phase of solving of transportation problem we ask how to recognise that our solution is optimal i.e.
minimal optionally that the solution can be improved and how to receive the better solution (its objective
function is smaller than the previous one. We give an algorithm for the conciliation of a better solution.
We suppose non-degenerated transportation problem. At the solution of non- degenerated transportation problem
with m suppliers and n customers (m+n-1) cells are occupied. Simultaneously at least one stone (occupied cell)
is lying in every row and every column.
We introduce row numbers ui , i = 1, 2, ... , m and column numbers vj , j = 1, 2, ... , n satisfying the following
equations for every stone (occupied cell):
ui + vj = cij ,
(1)
where cij are the original costs of a given transportation problem. Let us solve this system (1). We stand ahead
of a problem. The number of unknowns i.e. the number of row and column numbers is m + n and we have only
m + n 1 equations. We see that there exists one degree of freedom and that is not a problem but on contrary a
preference. We can choose an arbitrary row and column numbers and assign to it an arbitrary value, preferably
zero. We do this choice in that row or column which contains maximal number of stones. After the calculus of
all the row and column numbers with the aid of (1) we work with waters (non occupied cells). We count new
'
parameters cij for waters using the following system of equations:
ui v j
cij'
(2)
We take down these parameters into waters as that we write them into the left bottom corner. The system of
identities holds for stones. Therefore we do not write the new parameters into the cells which are stones. We
exemplify computational procedure and assigning of values of row and column numbers of our example. We
come out from the Table 2 which was obtained using the North West corner method. We enlarge the table by
addition of one row and one column. Into the column and row headings there we write variables vj and ui and
141
into cells of columns and rows, concrete values of these numbers. We eke out the waters with differences
cíj cij. We write these differences into the left upper corner of waters. When all these differences are negative
or equal to zero then we have the optimal solution. When at least one of these differences is positive there exists
a water P ij for which cíj - cij > 0 we have not optimal solution yet. In the case of greater number of positive
differences we choose the water (cell) with maximal value of difference. We see that in our example the water
with this property is P 21. Now the process of optimising (minimising) is starting. We explain the method of
optimising in the next section.
Suppliers
Customers
Capacity
ui
Kj
20
S1
250
S3
Demands
vj
6
11
15
6
40
21
12
29
250
21
2
12
310
-1
200
0
190
190
8
200
15
700
14
17
15
S2
14
60
17
11
23
100
15
18
150
12
7
15
10
23
19
26
150
18
Table 1
2 ) PATTERNS OF CHANGE METHOD
The process of optimising starts as that we mark the cell with the greatest difference cíj cij by the sign + and
we do the following table search: We look for a polygon created by apexes which are represented by stones and
the apex assigned by the sign +. Simultaneously its sides are required to be horizontal or vertical only. Anyway
we start in the cell with sign + and continue downstream the row or the column so on down to stone which will
be the new apex of our polygon. This stone must satisfy the following condition. When we start from it, now in
vertical direction (when we came to it in the horizontal way) we must find an other stone from which there exists
a path to the next stone now in horizontal sense and so on till we come back into our starting point with the sign
+. Simultaneously the starting cell is assigned by + , the next cell (stone) which is simultaneously the apex of
the polygon will get the sign -. The next stone (apex) gets the sign + and so one. When we are going through the
polygon we are coming to apex which changes its signs. We construct the polygon for our example.
- 20
250
+ 14
60
+ 6
-= 15
40
Figure 1
In the figure the polygon for the situation on Table 1. The picture idealises the concrete situation in the fact that
the sides are of the zero length in our polygon. The apexes of our polygon are neighbouring stones in the table
in the concrete.
We return to the sequence of signs which is constructed as follows: As we know first apex which has sometimes
the notation H has the sign +. The second one has the sign -. The third one has the sign + and so on. After some
steps we come back to the apex H.
Very important part is played by the subset of that apex which are denoted by sign -. We study this subset, more
better said the subset of date content of this stones. The set of values in negative apexes is:
142
{ 40, 250}
The minimal value in this set is 40. We chose this smallest value and we add it to the content of apexes (stones)
which were denoted by sign + and we subtract this smallest value from the content in apexes (stones) with sign -.
The result of optimising rearrangements is given at the Figure 2.
Figure 2
We obtain after the first optimising the following Table 2
Suppliers
S1
Costumers
K1
K2
20
210
100
S2
40
6
Capacity
K3
14
11
15
18
150
17
250
12
100
12
310
15
10
19
S3
Demands
K4
150
200
23
190
190
200
700
Table 2
We must enumerate if it is the optimal solution. For this reason we extend the table rather one number row and
one number column.
We see that the process of optimising is not finished and that the water cell with maximal difference is the cell
P13. Now the apexes of the according polygon are the cells : P11 , P 21 , P 23 . The set of values in negative apexes
is:
{ 210, 150 }.
The minimal value in this optimising step is 150. Hence we add the value 150 into apexes with sign + and we
subtract this value in apexes with the sign -. We obtain:
143
Suppliers
Customers
K1
K2
20
S1
210
S2
40
Capacity
K4
K3
14
12
21 11
100
6
-3
10
150
17
0
-4
190
8
100
0
14
250
6
Demands
vj
200
0
190
8
23
19
12
14
5
18
-15 15
310
29
32
S3
ui
26
150
18
200
15
K3
Capacity
K4
Table 3
Suppliers
Customers
K1
K2
20
S1
60
S2
190
12
11
14
ui
150
100
310
0
200
-14
190
-6
29
6
-3
S3
-15 15
18
15
10
17
0
-4
12
-14
19
23
190
14
250
20
Demands
vj
8
100
14
5
150
11
200
29
Table 4
We see that we did not obtain the optimal (minimal) solution. We calculated new row and column numbers and
Suppliers
Customers
Capacity
ui
Kj
-11 20
S1
-10 14
11
12
110
200
-14 15
-10 18
-6
1
8
9
-3
310
-8
200
-11
190
0
9
6
S2
200
S3
50
Demands
vj
250
17
17
12
100
100
12
19
40
150
19
20
200
20
15
23
700
Table 5
also the new differences and we see that maximal value of the differences is in the water cell P14 is 17 and the
polygon contains apexes (stones) P 11 , P21 , P24 or of apexes P13 , P23, P24 and the initial water cell. For both the
polygons the value which will be recounting is equal to 10.
After a finite number of optimising steps we receive the stage described in the following Table 5.
We see that all the differences cíj cij calculated for water cells with the aid of the last row and column
numbers are negative. It would be sufficient for optimality to receive non positive differences (we admit also
differences equal to zero).
144
REFERENCES
[1]
ACKOFF, RUSSELL, L. Progress in Operation Research. New York, John Wiley & Sons, Inc. 1961.
[2]
CHURCHMAN, CH. W.; ACKOFF, R. L.; ARNOFF, L. Introduction to Operations Research. New
York : John Wiley & Sons, Inc. 1957.
[3]
HABR, J.; VEP EK, J. Systémová analýza a syntéza. Praha : SNTL, 1972 .
[4]
BECK, J.; LAGOVÁ, M.; ZELINKA, J. Lineární modely v ekonomii. Praha : SNTL, 1982.
[5]
KLAPKA, J.; DVO ÁK, J.; POPELA, P. Metody opera ního výzkumu. Brno : VUTIUM, 2001.
[6]
RAIS, K. Vybrané kapitoly z opera ní analýzy. Brno : PGS, 1985.
[7]
ROCCAFERRERA, G. M. F. Operation Research Models for Business and Indusry. New York : S.W
publishing company , Chicago, 1964.
[8]
TER-MANUELIANC, A. Modelování problém ízení. Praha : Institut ízení , 1977.
[9]
VACULÍK, J.; ZAPLETAL, J. Podp rné metody rozhodovacích proces . Brno : Masarykova univerzita
1998.
[10] WALTER, J. a kol. Opera ní výzkum. Praha : SNTL, 1973.
[11] WALTER, J. Stochastické modely v ekonomii. Praha : SNTL, 1970.
[12] ZAPLETAL, J.; ZÁST RA, B. Vybrané kapitoly z opera ního výzkumu. Zlín : VUTFT, 1983.
[13] ZAPLETAL, J. Opera ní analýza. Kunovice : Skriptorium VO, 1995.
Address:
1.soukromá vysoká kola na Morav
Osvobození 699
686 04 Kunovice
E-mail [email protected]
145
146
UNBALANCED TRANSPORTATION PROBLEM
Josef Zapletal
Abstract. The transportation problem is one of problems of linear programming but very often it
solved by other special methods as for example by NW-corner method or VAM-method. The reason
of this consist in a very big number of zeros in the system constraining conditions. After the first
step of these and also others methods we do not receive the optimal value of the objective
function.Often it is necessary tosolve unbalced transformation problem. A method for the solution of
such problem is in this paper given.
Abstrakt. Dopravní úloha je jednou ze základních úloh lineárního programování. P i klasickém
zpracování dochází k situaci, kdy se v rozsáhlé matici omezení vyskytují v tinou nuly. Z tohoto
d vodu byly vyvinuty metody, které nezabírají v po íta ích tolik místa a jsou i po stránce rychlosti
výpo tu efektivn jí. Mezi n pat í metoda severozápadního rohu, která nepracuje a nezískává
optimální eení na bázi ekonomických vstupních parametr , ale její princip je odvozen ist
pomocí geografických v domostí. Dalí metody mají sv j podklad v ekonomických vlastnostech
problému a základní p i azení se provádí pomocí p ímých nebo pom rných náklad . asto je nutné
eit také problémy, kdy nejsou poadavky vyrovnané s kapacitami. Tento problém se eí
v p edkládaném lánku.
Key words. Supplier, costumer, stone, occupied cell, waters, row and column numbers, differences
among costs, unbalanced problem, polygon, vertices of polygon, the signs of differences for optimal
solution.
1. A REMARK ON DEGENERATED TRANSPORTATION PROBLEM.
The degenerated transportation problem is characterised by fact that the number of occupied cells (stones) is
smaller than (m + n - 1). The concrete situation of a degenerated transportation problem can be not evaluated in
usual way with the aid of row and column numbers. We are not able to calculate all these numbers. The
consequence is obvious, we are not able to determine on the optimality of the received solution.
We do away with this deficiency as we extend the number of stones (occupied cells) to the necessary amount (m
+ n - 1) as the required number of water cells will be occupied by so called auxiliary zeros or by very small
coefficients which are usually denoted by . The problem is by this adding of the new auxiliary coefficients not
yet solved. New question comes up. How to occupy any cells with these auxiliary zeros and epsilons so that
these cells after the constructions of polygons will not be apexes denoted by the sign - .
The uniqueness of optimal solution of the transportation problem is qualified by the matrix of costs and also on
the system of suppliers and their inventories likewise on the set of customers and their demands
2 UNBALANCED TRANSPORTATION PROBLEM
Up to now we supposed that the inventories of the suppliers are equal to the demands of the customers. This
condition is not restrictive for the solving of transportation problem.
Unbalancing can arise by two ways. The sum of inventories of suppliers is greater the sum of all demands. For
the solving of such a transportation problem we can suppose the existence of another (fictive) customer whose
demand is just equal to the excess of inventories over the aggregate of all the demands. This means practically to
adjoined one column in the table of transporting problem. The costs in the cells of this column (fictive customer)
will be equal to zero. It is obvious the commodity is anywhere transported and it remains at the suppliers and no
transportation charges occur.
Analogously the transformation problem is solved when the sum of all the demands of customers is greater than
the sum of inventories. We define a fictive supplier whose inventories are defined by the analogous method as at
the fictive customers.
2.1 Example. Solve the following transportation problem. There are three suppliers with inventories 1800, 1200 a
2000 units of commodity and two customers with demands 2900 a 1500. The matrix of costs is at the Table 1.
147
Supplier
cij
from Si to Kj
K1
K2
53
51
S1
S2
48
S3
Demands
inven
tories
1800
53
1200
62
42
2000
2900
1500
Table 1
We se that the inventories are greater. We transform this problem on balanced transportation problem with the
addition 0f a third (fictive) customer K3 with the demand 600 units of commodity. We solve the problem using
the North West corner method and we receive the situation described in the Table 2 :
We see from the table 13 that the allocation is not optimal. The maximal positive difference cíj - cij is in the cell
P13 , namely 16. The according polygon for equivalent transformation is given by stone P11, P21, P22, P32, P33
and by the +water cell P13. The stones with sign are lying on the diagonal. The minimal value is 100. After the
completion of the whole operation of adding and subtraction there we receive the optimal solution yet. The
optimality of the last solution is obvious from the table 14.We see that all the differences cíj - ci in water cells
are negative.
Suppliers
Customers
K1
K2
K3
51 16
53 7
S1
1800
S2
1100
58
Demands
vj
1800
16
1200
11
0
0
100
11
-25
S3
ui
0
16
53 11
48
Inventories
62
37
2900
37
42
0
1400
600
2000
1500
42
600
0
5000
Table 2.
The result of the last optimal solution is the following. 100 units of commodity remain at the first supplier and
the further 500 units at the third supplier.
Analogously the following problem will be solved : The total sum of demands is larger than the sum of
inventories at all the suppliers. In this case there it is necessary to introduce a fictive supplier. The costs will be
equal to zero in the cells of the whole row of this supplier.
In the last example we were looking for the optimality only from the aspect of transportation freights.
Unfortunately in practice the situation is different other. These clear problems are sporadic. We show one
more complicated problem.
Suppliers
Customers
Inventories
ui
1800
0
1200
-5
2000
0
Kj
S1
1700
S2
1200
-9
48
42
-16
51
53
37
-9
S3
53
62
16
100
16
-5
0
0
-5
42
1500
0
500
148
Demands
vj
53
2900
53
1500
42
600
0
5000
Table 3
2.2 Example. Three producers deliver the same product to four customers. The traffic rates, the product
capacities and the demands of customers are given in the table 4.
Producers
Customers
K1
K2
Capacities
K3
70
K4
150
Pr1
60
20
Pr2
120
150
120
140
1500
30
130
160
2500
2500
1800
Pr3
80
Demands
120
1000
2000
Table 4
Production charges are 150 financial units for a unit of production at the first producer Pr1, analogously 160 fin.
units at the second producer Pr2 and finally 180 fin. units at the third producer Pr3 . The customer K2 must be
provided preferentially. The customer K1 had been provided from the producer Pr1 and hence he can penalty this
producer for no delivering by the financial amount 5 units for any unit of non delivered unit of demanded
commodity. We have to determine the optimal program of transportation under these conditions.
The amounts of customers reach the whole capacities of producers over 500 units of commodities. We set up the
fictive producer Pr4 with the capacity 500 units. But we do not put all the cost equal to zero in the row of this
fictive producer. Above all it is necessary to meat the delivery obligation to the second customer. Hence the cell
P42 may be not occupy (may be not a stone) We achieve it by introduction of prohibitive rate (cost) M which
must be large enough. The demands for the customer K1 are penalised therefore we predicate to the cell P41 the
cost in the high of penalty it is 5 fin. units. The costs in the cells P43 and P44 will be equal to zero. In the course
of solution of this problem there ewe can not suppose the traffic tariffs but we must entertain the production
charges. The costs are the sums of these traffic tariffs and production charges of individual producers Pr1, Pr2 a
Pr3. It is necessary to remember that the production charges of the fictive producer Pr4 are equal to zero.
The Table 5 on the following page contains finally solution.
In conformity with the table 16 the customer K1 leaves off the supply of the producer Pr1. The producer Pr
assumes this supply simultaneously the penalty will not occur as all the demand of the customer K1 will be
satisfied. Also the demand of the second customer K2 will be fully satisfied. The same situation is at the third
customer K3. Only the forth customer K4 will not be fully provided.
It is necessary to enjoy the following remark to the solution of this example:
Producers
Customers
Kj
-40
Pr1
Pr2
210 -50
170
-50
120
80 -180
230
1200
ui
2000
-90
1500
-30
2500
0
500
-330
300
240
310
280
200
300
1300
180
210
1000
310 -10
340
300
330
-75
Pr4
220 -60
2000
260
Pr3
170
Capacities
5 -120-M
M -20
0
0
500
-70
-120
-20
149
Demands
vj
1200
260
1000
210
2500
310
1800
330
6500
x
x
x
Table 5
2.3 Remark The solution is optimal under the assumption that all the producers Pr1, Pr2, Pr3 form one whole
(unit) and the products of all the three producers are interchangeable. This solution can be supposed as optimal
only from the view of greater organisational units (units with respect to the production and also to the
consumption) and not from the point of view of individual producers and isolated customers
REFERENCES
[1]
ACKOFF, R. L. Progress in Operation Research. New York : John Wiley & Sons, Inc. 1961.
[2]
CHURCHMAN, CH. W.; ACKOFF, R. L.; ARNOFF, L. Introduction to Operations Research. New
York John Wiley & Sons, Inc. 1957.
[3]
HABR, J.; VEP5EK, J. Systémová analýza a syntéza. Praha : SNTL, 1972.
[4]
BECK, J.; LAGOVÁ, M.; ZELINKA, J. Lineární modely v ekonomii. Praha : SNTL, 1982.
[5]
KLAPKA, J.; DVO ÁK, J.; POPELA, P. Metody opera ního výzkumu. Brno : VUTIUM, 2001.
[6]
RAIS, K. Vybrané kapitoly z opera ní analýzy. Brno : PGS, 1985.
[7]
ROCCAFERRERA, G. M. F. Operation Research Models for Business and Indusry. New York : S.W
publishing company , Chicago, 1964.
[8]
TER-MANUELIANC, A. Modelování problém ízení. Praha : Institut ízení , 1977.
[9]
VACULÍK, J.; ZAPLETAL, J. Podp rné metody rozhodovacích proces . Brno : Masarykova univerzita
1998.
[10] WALTER, J. a kol. Opera ní výzkum. Praha : SNTL, 1973.
[11] WALTER, J. Stochastické modely v ekonomii. Praha : SNTL, 1970.
[12] ZAPLETAL, J.; ZÁST RA, B. Vybrané kapitoly z opera ního výzkumu. Zlín : VUTFT, 1983.
[13] ZAPLETAL, J. Opera ní analýza. Kunovice : Skriptorium VO, 1995.
Address:
1.soukromá vysoká kola na Morav
Osvobození 699, 686 04 Kunovice
E-mail [email protected]
150
VORTEX-FRACTAL-RING STRUCTURE OF ELECTRON
Pavel Omera
Abstract: We would like to find some plausible structure of the electron as vortex-fractal-ring
structure. But it is in contradiction with general accepted knowledge, where the electron has not a
structure. This paper is an attempt to calculate the size of the electron. The vortex-fractal theory
could possibly explain what the charge, the electron, the proton, the electromagnetic field, etc
actually are.
Keywords: vortex-fractal physics, structure of electron, electric charge, size of electron.
1. INTRODUCTION
The discovery of the electron was a landmark in physics and led to great technological advances. The electron
emission is the process when negative charges in the form of electron, escape for example from the hot filament.
Streams of electrons moving at high speed are called cathode rays or electron rays. The rays are deflected by a
magnetic field too. If the N pole of a magnet is brought up to the neck of the tube, the rays move upwards, using
Flemings left-hand rule. The ratio of the charge q of an electron e to its mass me is called its specific charge and
can be found from experiments in which cathode rays are deflected by electric and magnetic fields. It was first
done by J. J. Thomson in 1897 using a deflection-type tube. His work is regarded as proving the existence of the
electron as a negatively charged particle of very small mass and not, as some scientists thought a form of
electromagnetic radiation like light.
In 1819 Oersted accidentally discovered the magnetic effect of an electric current. His experiment can be
repeated by holding a wire over and parallel to a compass needle, which is pointing N and S. A needle moves
when the current is switched on. Reversing current causes the needle to point in the opposite direction. A
solenoid is a long cylindrical coil. It produces a field similar to that of a bar magnet. The polarity is found as
before by applying the Right-hand screw rule to a short length of one turn of the solenoid.
When there is a current in a wire, the wire itself generates a magnetic field. Moving charges, then, produce a
magnetic field. We would like to use the laws that determine how such fields are created. The answer to this
question was determined experimentally by three critical experiments and brilliant theoretical argument given by
Ampere [1]. A wire carrying a current in magnetic field experiences a force.
In an insulator all electrons are bound firmly to their atoms; in a conductor some electrons can move freely from
atom to atom. An electric current creates a magnetic field. The reverse effect of producing electricity from
magnetism was discovered in 1831 by Faraday and is called electromagnetic induction.
Whenever magnetic forces between two sets of currents are computed, the result is invariant with respect to a
charge in the hand convention. The end result is that parallel currents attract, or that currents in opposite
directions repel. Faraday discovered in 1840 the essential feature that had been missed that electric effects exist
only when there is something changing. If one of a pair of wires has a changing current, a current is induced in
the other, or if a magnet is moved near an electric circuit, there is a current too.
Electron is defined as a fundamental particle of matter, with negative electric charge, which populates the outer
region of atoms.
2. THE ELECTRON WITH VORTEX-FRACTAL-RING STRUCTURE
The electrical force decreases inversely with the square of distance between charges. This relationship is called
Coulombs law. There are two kinds of matter, which we can call positive and negative. Like kinds repel each
other, while unlike kinds attract unlike gravity, where only attraction occurs [5]. When charges are moving the
electrical forces depend also on the motion of charges in a complicated way [1].
Fractals seem to be very powerful in describing natural objects on all scales. Fractal dimension and fractal
151
measure are crucial parameters for such description. Many natural objects have self-similarity or partial-selfsimilarity of the whole object and its part [10].
The structure of the electron in Fig. 1 presents the electron as pure ring fractal structure. Electrons 0e (or e) in
the electron ray 0r hold together by photons vortex structure 0f (a pair of vortices) [5], [11]. Generally, in the
fractal structure of the electron, the number n defines the level of substructure ne. The name osmeron we derived
from the name Osmera of Egyptian deity with 4 pairs of gods as primary creative forces (from a chaos
beginning). Osmerons are too small that is why have unmeasurable size and mass. Osmerons on osmerons
trajectory creates an osmeron ray.
We know that the apparent mass of a particle changes by 1/ (1 v2/ c2). Does its charge do something similar?
No charges are always the same, moving or not [1]. If the charge of a particle depended on the speed of the
particle carrying it, in the heated block the charge of the electrons and protons would no longer balance. A block
would become charged when heated. If the charge on an electron charged with speed, the net charge in piece of
material would be charged in a chemical reaction. Even a very small dependence of charge on speed would give
enormous fields from the simplest chemical reactions. No such effect has been observed [1], and we conclude
that the electric charge of a single particle is independent of its state of motion.
Fig. 1 The vortex-fractal structure of the electron ray
For calculation of fractal-ring electron structure we will use the structure that is shown in Fig.1 and Fig2. In the
ring electron structure (see Fig.2) the subelectrons eo rotate with a velocity ve and subsubelectrons e1 with a
velocity vo. The radius of the electron is Re and a radius of axes of subelectrons eo is re. A rough estimation of
number of subrings N and number of subsubring N2 in the electron structure are determined by the mass mp of
the proton structure [11].
152
Fig. 2 The fractal-ring structure of the electron
Let us calculate properties of the electron with a vortex-fractal theory [4-14]. This requires that subelectrons are
accelerated towards the center of the electron ring. The amount of the acceleration force Fa has to be in balance
with two coulomb forces Fo. A whole force of attraction FA can be calculated by Amperes law:
Fa FA
(2.1)
The fundamental physical law for an acceleration force Fa of mass m with velocity v and distance r is :
Fa
m
v2
r
(2.2)
The mass moe of the subelectron eo for the fractal structure of the electron is:
moe
me
N
(2.3)
where N is number of subelectrons. To cover creation of the proton structure and electron structure from the
same very small rings ( N2 subsubelectrons e1) [14 ]:
mp
N
42
me
(2.4)
where mp is the mass of the proton and me is the mass of the electron. From the fractal structure of the electron
on Fig.2:
Fa
Fao
2
N
2
me ve2
N re
2
N
2
The normalized values of forces Fao for N/2 subelectrons are on Fig.3. Their average value 2/
following way:
sin xdx
(2.5)
is calculated
2
(2.6)
0
153
sin xdx
2
0
(2.7)
Fig.3 Average value of the acceleration force Fao
The fundamental physical law for attraction force FA (Ampers law) between two wires with a current I, a length
l , a distance d , and a permeability o of vacuum:
o
FA
2
I2
l
d
(2.8)
Electric charges q1 in the subring e1 create the current I:
dQ q1 N
dt
T
2 roe voT
e
q1
N2
I
I
e
N
N2
T
d
Fa
FA
e
NT
evo
N 2 roe
(2.9)
(2.10)
(2.11)
evo
N2
re
N
evo
2 re
2 re
N
(2.13)
r
2 e
1 o e 2vo2
N
2
2 2 4 2 re2 2 re
N
2 Fo
(2.12)
(2.14)
The constant ½ was added because the magnetic field for two rings is half when it is compared to two parallel
line wires. From equation (2.5) and (2.14):
me ve2 2 N
N re
2
r
2 e
1 o e 2 vo2
N
2
2 2 4 2 re2 2 re
N
(2.15)
vo2
ve2
(2.16)
the solution being:
re
e2
4 me
o
2
When velocity of subrings and subsubrings have velocity c of the light:
154
vo
ve
c
(2.17)
2
re
e
4 me
o
2
0.89 10
15
m
(2.18)
Fig.4 Geometry of the electron and the subelectron
From the geometry of the electron on Fig.2 and Fig.4 we obtain:
Re
Reh
re
roe
r1e
r2 e
ee roe r1e
Te
Doe
r2 e
1
N2
re
re
re
1
1
re 1
2
3
N N
N
N N2
Deh 2 Reh 1.736 10 15 m
re
re
N
2
re
re
re
1
re 1
2
3
N N
N
N
15
De 2 Re 1.82 10 m
re
re
N2
re
N3
2 re
1
N
1
N2
1
N3
1
N3
0.91 10
15
m
(2.19)
(2.20)
1
N3
0.8682 10
15
m
(2.21)
(2.22)
0.0434 10
15
m
(2.23)
From the geometry of the subelectron on Fig.4 we obtain:
Roe
re
N
re
N2
Doe
2
Toe
2 re
re
N3
re
N
1
N2
re
re
N2
1
N
re
N3
1
N3
1
N2
2re
2 re 1
N N
re
1
1
N
N
1
N3
1
N
1
N2
1
N2
1
N2
0.0217 10
1
0.0434 10 15 m
N3
2re
1
1
0.001033 10
2
N
N
15
m
(2.24)
(2.25)
15
m
(2.26)
Energy Eo of a free quiet electron with velocity v, kinetic energies of subelectrons and subsubelectrons:
v
Eo
ve vo c
0
1 me 2
1 me 2 2
ve N
vo N
2 N
2 N2
(2.27)
me c 2
155
(2.28)
me c 2
Eo
(2.29)
This result is in coincidence with the well-known Einsteins law. Mass is a measure of the amount of matter in
an object. The objects inertia is proportional to its mass, and Einstein showed that mass is actually a very
compact form of energy.
To compare attraction forces calculated by Amperes law and Coulombs law between two subrings eo :
FoA
re
2 2 2
1 o q1 vo
N
2 2
r
2
2 2 4 roe
e
N
1 o
22
e
N
4
2
2
vo2 2 re
N
2
r
2
e
re
N
N
(2.30)
2
e
N
2 re
4 o
N
FoC
where
o
(2.31)
2
is permitivity of vacuum. It is defined as exactly10-7 times the speed of light squared.
When the Amperes force FoA is the same as Coulombs force FoC for the fractal-ring electron structure?
Comparison follows:
FoA
1 o
22
FoC
(2.32)
e
N
4
2
2
2
vo2 2 re
N
2
2
r
e
re
N
N
4
e
N
2 re
o
N
2
1
2
o o
v
(2.33)
(2.34)
o
1
vo2
c2
(2.35)
o o
vo
c
(2.35)
It is in coincidence with equation (2.27).
3. CONCLUSIONS
Our science makes terrific demands on the imagination. To understand the vortex-fractal-ring structure of the
electron requires a high degree of imagination. The degree of imagination that is required is much more extreme
than that required for some of the ancient ideas. The modern ideas are much harder to imagine. We use
mathematical equations and rules, and make a lot of pictures. We cant allow ourselves to seriously imagine
things, which are obviously in contradiction to the known laws of nature. And so our kind of imagination is quite
a difficult game (or a puzzle). One has to have the imagination to think of something that has never seen before,
never been heard before. At the same time the thoughts are restricted or limited by the conditions that come from
our knowledge of the way nature really is. The problem of creating something which is new, but which is
consistent with everything, which has been seen before, is one of extreme difficulty [1]. It has to be consistent
not only with knowledge of physics [1], [2] but with knowledge of chemistry [15], and topological
transformations too [3].
It is very interesting that we received the same result if we calculated forces in a fractal electron structure by
Amperes (2.30) or Coulombs law (2.31) where vo=c. Very important result is how depends the size (diameter)
156
of the electron on the velocities of subrings and subsubrings (2.16). If the value of velocity ve oscillates around
mean value, the diameter of the electron oscillates too. Perhaps this could explain how the electron could release
photons.
Acknowledgment: This work has been supported by the Ministry of education; Grant No: MSM21630529.
REFERENCES
[1]
FEYNMAN, R.P., LEIGHTON, R.B., SANDS, M.: The Feynman Lectures on Physics, volume I, II, III
Addison-Wesley publishing company, 1977
[2]
DUNCAN, T.: Physics for today and tomorrow, Butler & Tanner Ltd., London, 1978
[3]
HUGGETT, S.A., JORDAN, D.: A Topological Aperitif, Springer-Verlag, 2001
[4]
OMERA, P.: Evolution of universe structures, Proceedings of MENDEL 2005, Brno, Czech Republic
(2005) 1-6.
[5]
OMERA, P.: The Vortex-fractal Theory of the Gravitation, Proceedings of MENDEL2005, Brno,
Czech Republic (2005) 7-14.
[6]
OMERA, P.: The Vortex-fractal Theory of Universe Structures, Proceedings of the 4th International
Conference on Soft Computing ICSC2006, January 27, 2006, Kunovice, Czech Republic, 111-122
[7]
OMERA, P.: Vortex-fractal Physics, Proceedings of the 4th International Conference on Soft Computing
ICSC2006, January 27, 2006, Kunovice, Czech Republic, 123-129
[8]
OMERA, P.: Evolution of Complexity in Li Z., Halang W. A., Chen G.: Integration of Fuzzy Logic and
Chaos Theory; Springer, 2006 (ISBN: 3-540-26899-5)
[9]
OMERA, P.: The Vortex-fractal Theory of Universe Structures, CD Proceedings of MENDEL 2006,
Brno, Czech Republic (2006) 12 pages.
[10] OMERA P.: Vortex-fractal Physics, CD Proceedings of MENDEL 2006, Brno, Czech Republic (2006)
14 pages.
[11] OMERA, P.: Electromagnetic field of Electron in Vortex-fractal Structures, CD Proceedings of
MENDEL 2006, Brno, Czech Republic (2006) 10 pages.
[12] OMERA, P.: Vortex-ring Modelling of Complex Systems and Mendeleevs Table, WCECS2007,
proceedings of World Congress on Engineering and Computer Science, San Francisco, 2007, 152-157
[13] OMERA, P.: From Quantum Foam to Vortex-ring Fractal Structures and Mendeleevs Table, New
Trends in Physics, NTF 2007,Brno Czech Republic, 2007, 179-182
[14] OMERA, P.: Vortex-fractal Structure of Hydrogen, Proceedings of the 6th International Conference on
Soft Computing ICSC2008, January 25, 2008, Kunovice, Czech Republic
[15] PAULING, L.: General Chemistry, Dover publication, Inc, New York, 1988
ADDRESS
Doc. Ing. Pavel Omera, CSc.
Technická 2
616 69 Brno
Czech Republic
osmera @fme.vutbr.cz
157
158
VORTEX-FRACTAL STRUCTURE OF HYDROGEN
Pavel Omera
Abstract: We would like to find some acceptable structure of the hydrogen as vortex-fractal-coil
structure of the proton and vortex-fractal-ring structure of the electron. It is known that planetary
model of hydrogen is not right, the classical quantum model is too abstract. Our imagination is that
the hydrogen is a levitation system of the proton and the electron.
Keywords: structure of hydrogen, structure and size of proton, structure of electromagnetic field,
covalent bond
1. INTRODUCTION
Most of our knowledge of the electronic structure of atoms has been obtained by the study of the light given out
by atoms when they are exited. The light that is emitted by atoms of given substance can be refracted or
diffracted into a distinctive pattern of lines of certain frequencies and create the line spectrum of the atom. The
careful study of line spectra began about 1880. The regularity is evident in the spectrum of the hydrogen atom.
The interpretation of the spectrum of hydrogen was not achieved until 1913. In that year the Danish physicist
Niels Bohr successfully applied the quantum theory to this problem and created a model of hydrogen. Bohr also
discovered a method of calculation of the energy of the stationary states of the hydrogen atom, with use of
Plancks constant h. Later in 1923 it was recognized that Bohrs formulation of the theory of the electronic
structure of atoms to be improved and extended. The Bohr theory did not give correct values for the energy
levels of helium atom or the hydrogen molecule-ion, H2+, or of any other atom with more than one electron or
any molecule. During the two-year period 1924 to 1926 the Bohr description of electron orbits in atoms was
replaced by the greatly improved description of wave mechanics, which is still in use and seems to be
satisfactory. The discovery by de Broglie in 1924 that an electron moving with velocity v has a wavelength
=h/mev [15]. The theory of quantum mechanics was developed in 1925 by the German physicist Werner
Heisenberg. An equivalent theory, called wave mechanics, was independently developed early in 1926 by
Austrian physicist Ervin Schroedinger. Important contribution to the theory were also made by the English
physicist Paul Adrien Maurice Dirac. The most probable distance of the electron from the nucleus is thus just the
Bohr radius ro; the electron is, however, not restricted to this distance. The electron is not to be thought of as
going around the nucleus, but rather as going in and out, in varying directions, so as to make the electron
distribution spherically symmetrical [15].
Fig. 1 The vortex-ring structure of the hydrogen 11 H [13]
159
2. THE MODEL OF HYDROGEN WITH A LEVITATING ELECTRON
Fig. 2 The levitating electron in the field of the proton (the fractal structure model of hydrogen H is simplified to compare with the
hydrogen atom on Fig.1
In a new model of the hydrogen atom with a levitating electron there is attractive force F+ and repellent force F- :
F
F
e2
F
4
1
r2
o
e2
A
r exp
4
1
r2
o
e2 ro2
4 o r4
(2.1)
The hydrogen atom can have the electron on left side or on right side (see Fig.9a, 9b), thus the difference in
exponents must be 2 then exp=4. The attractive force F+ is Coulombs force. A distance between the electron an
the proton is r .
5.29 10 11 m and F
For Bohr distance ro
ro2
A
F
F
0:
(2.2)
e2
F
4
ro2
r4
1
r2
o
(2.3)
To find the distance where F has maximum (see Fig.3):
dF
dr
e2
4 o
4ro2
r5
2
r3
e2
2
o
1
r3
2ro2
r2
(2.4)
for ro :
dF
dr
e2
4
3
o o
r
Ko
(2.5)
It is line ko in Fig. 3
dF
dr
Fmax for
r1, 2
E
e2
2
o
r3
1
2ro2
r2
0
(2.6)
2 ro in Fig. 3. Energy E of the electron in the distance r:
e2
4
o
1
r2
ro2
dr
r4
e2
4
o
1
r2
ro2
dr
r4
e2
4
o
1
r
ro2
3r 3
The graph of E is on Fig. 4. Energy Eo which must be added to the electron to be free:
160
(2.7)
Eo
e2
4 o
ro2
dr
r4
1
r2
r
e2
4
ro2
3r 3
1
r
o
e2
r
4
o
1
r
ro2
3r 3
(2.8)
For E o = 0
ro
3
rEo
3.05 10 11 m
(2.9)
For ro is Eo:
Eo
e2
4
o
Eo
ro2
3ro3
1
ro
e2
4
o
e2
4
1 2
ro 3
e2
1 2
ro 3
o
6
2
27.2 eV
3
o
1
ro
18.13eV
(2.10)
18.13eV
(2.11)
We can calculate frequency fo and period To of oscillation of the electron in the hydrogen atom around ro if we
insert Ko defined in equation (2.5) into (2.12):
fo
1
2
Ko
me
To
1
fo
1.075 10
9.3 1014 Hz
15
(2.12)
s
(2.13)
To calculate quantum model of hydrogen we use radius re which was derived in [14]:
re
4
e 2 v o2
m e v e2
o
2
(2.14)
2 re
2
e2 vo2
4 me ve2
o
2
(2.15)
From paper [14] we know that
vo2
1
c2
o
(2.16)
o
On a circumference of a circle with re have to be n of a half-wavelength /2=h/2mev (n is quantum number):
2 re
e 2 vo2
2
4 me ve2
o
2
e2 1
o ve
e2 c 2
2 me ve2
e2
o
2
1
2
o me ve
n
2
n
1 h
2 me ve
nh
(2.17)
(2.18)
The kinetic energy of subelectrons is transformed to kinetic energy of the electron with mean velocity of v:
1
meve2
2
1
me v 2
2
161
(2.19)
ve
2
v
vm
(2.20)
where vm is maximum velocity of the electron if the electron has distance ro and minimum energy E:
e2
o
nh
2vm
(2.21)
vm
1 e2
n 2 oh
Eq
1
me vm2
2
1 mee 4
n 2 8 o2 h 2
(2.23)
Eqo
me e4
8 o2 h 2
13.6eV
(2.24)
Eqo
Eo
3
4
e2 1 3
6 o ro 4
Eqo
me e 4
8 o2 h 2
e2 1
8 o ro
(2.26)
5.29 10 11 m
(2.27)
(2.22)
For quantum number n=1
ro
r
h2
me e2
o
n2
e2 1
8 o ro
13.6eV
h2
mee 2
o
(2.25)
(2.28)
It is the same result as Borh obtained [15] but with quite different hydrogen model.
To find the size of r where Eo = 13.6eV = Eqo in (2.24) and (2.25) we must solve cubic equation:
3r 3
6r 2 ro
2ro3
0
The roots of equation (2.29) are: r1 = 0.7223517245ro ~ 0.382Å,
0.5148689384ro
(2.29)
r2 = 1.792517214ro ~ 0.948Å,
r3 = -
The values of r1 and r2 are the distances where the electron has velocity v=0 (see Fig.4).
Eq
1
Eqo
n2
(2.30)
For quantum number n=1 we calculate the maximum velocity vm from (2.22) and the couple constant :
vm
c
vm
e2
2 oh
1
137.6
In the hydrogen molecule H2 the covalent bond has ne = 2, np = 1 (see Fig.9):
162
(2.31)
(2.32)
F
F
rc
d cp
F
e2
4
o
ne n p
e2
F
4
r2
o
ro2
r4
2
r2
ro2
r4
(2.33)
0
(2.34)
ro
3.75 10 11 m
2
2 rc 7.5 10 11 m
(2.35)
(2.36)
It is in coincidence with the distance between two protons for their covalent bond [14].
For the hydrogen molecule-ion H2+ is ne = 1, np = 1 then dp+ (see Fig.9):
dp
2 ro
10.6 10
11
m
(2.37)
To calculate the size of the proton structure we use fractal-coil geometry on Fig.5:
2 p1
p1
NToe
re
1
N
2re N 1
1
1
2re
2
N N N
N
1
0.00693 10 15 m
2
N
1
N2
(2.38)
(2.39)
From the paper [14] we use the size of the subelectron:
DC1
2 p1
2 p2
NDC1
NDC1
0.666 10 15 m
2
2 p2 DC1 1.232 10 15 m
p2
D ph
R ph
2 p3
p3
DC 2
D ph
2
2 Doe Toe
0.0996 10
0.616 10
15
2 DC 2
Rp
Dp
2
D ph
2.70 10
1.35 10
15
m
(2.40)
(2.41)
(2.42)
(2.43)
m
(2.44)
NDoe
NDoe
0.290 10 15 m
2
2 p3 2 DC1 Doe 0.736 10
Dp
15
15
(2.45)
(2.46)
15
m
m
m
(2.47)
(2.48)
(2.49)
We received diameter Dp of the proton, which is in coincidence with an experiment. The calculated sizes of the
electron and the proton in hydrogen are shown on Fig.6.
163
Fig.3 Forces in the hydrogen atom
Fig.4 Energy E and the force F in the hydrogen atom
164
Fig.5 Vortex-fractal-coil structure of the proton
Fig.6 Calculated size of the electron and the proton in the hydrogen atom
165
Fig.7 A vortex structure of the electromagnetic field (photons) as two complementary vortex structures Vel and Vmg
Fig.8 Estimated forces between two subelectrons with ve = 0 to create a subelectron ray
Fig.9 Calculated distances between the proton and the electron
d) left side orientation of hydrogen
e) right side orientation of hydrogen
f)
the hydrogen molecule-ion H2+
g) the hydrogen molecule H2 with covalent bond
166
3. CONCLUSIONS
The exact analysis of real physical problems is usually quite complicated, and any particular physical situation
may be too complicated to analyze directly by solving the differential equations. Ideas as the field lines
(magnetic and electric lines) are for such purposes very useful. We think they are created from organized
subparts of vacuum [6], [9], [10]. A physical understanding is a completely unmathematical, imprecise, and
inexact, but it is absolutely necessary for a physicist [1]. It is necessary combine an imagination with a
calculation. Our approach is given by developing gradually the physical ideas by starting with simple situations
and going on more and more complicated situations. But the subject of physics has been developed over the past
200 years by some very ingenious people, and it is not easy to add something new that is not in discrepancy with
them. We used the knowledge from experiments. Now we realize that the phenomena of chemical interaction
and, ultimately, of life itself are to be understood in terms of electromagnetism. Maxwells discovery of the laws
of electrodynamics will be judged as very significant event of the 19th century.
The annular vortex structure of the hydrogen H could be imagined as the smallest electric motor with a
magnetic bearing in Nature (see Fig. 2). The proton creating rotary electromagnetic field is a stator and the
rotating electron is a rotor, which levitates in the electromagnetic field of the standing proton. The difference
between the particle and the antiparticle is in the direction of the vortex and ring rotation.
The electron structure is a pure fractal-ring structure with a vortex bond between rings. This vortex bond is
electromagnetic field with two complementary vortex structures; electric and magnetic vortex structures. The
proton structure is pure fractal-coil structure. Both are created from the same subsubrings.
In the covalent bond a pair of electrons rotate around a common axis of bond protons. There are two
arrangements of hydrogen: with a left and a right side orientation of the electron in their structure.
Acknowledgment: This work has been supported by the Czech Grant Agency; Grant No: MSM21630529.
REFERENCES
[1]
FEYNMAN, R.P., LEIGHTON, R.B., SANDS, M.: The Feynman Lectures on Physics, volume I, II, III
Addison-Wesley publishing company, 1977
[2]
DUNCAN, T.: Physics for today and tomorrow, Butler & Tanner Ltd., London, 1978
[3]
HUGGETT, S.A., JORDAN D.: A Topological Aperitif, Springer-Verlag, 2001
[4]
OMERA, P.: Evolution of universe structures, Proceedings of MENDEL 2005, Brno, Czech Republic
(2005) 1-6.
[5]
OMERA, P.: The Vortex-fractal Theory of the Gravitation, Proceedings of MENDEL2005, Brno,
Czech Republic (2005) 7-14.
[6]
OMERA, P.: The Vortex-fractal Theory of Universe Structures, Proceedings of the 4th International
Conference on Soft Computing ICSC2006, January 27, 2006, Kunovice, Czech Republic, 111-122
[7]
OMERA, P.: Vortex-fractal Physics, Proceedings of the 4th International Conference on Soft Computing
ICSC2006, January 27, 2006, Kunovice, Czech Republic, 123-129
[8]
OMERA, P.: Evolution of Complexity in Li Z., Halang W. A., Chen G.: Integration of Fuzzy Logic and
Chaos Theory; Springer, 2006 (ISBN: 3-540-26899-5)
[9]
OMERA, P.: The Vortex-fractal Theory of Universe Structures, CD Proceedings of MENDEL 2006,
Brno, Czech Republic (2006) 12 pages.
[10] OMERA, P.: Vortex-fractal Physics, CD Proceedings of MENDEL 2006, Brno, Czech Republic (2006)
14 pages.
[11] OMERA, P.: Electromagnetic field of Electron in Vortex-fractal Structures, CD Proceedings of
MENDEL 2006, Brno, Czech Republic (2006) 10 pages.
[12] OMERA, P.: Vortex-ring Modelling of Complex Systems and Mendeleevs Table, WCECS2007,
proceedings of World Congress on Engineering and Computer Science, San Francisco, 2007, 152-157
[13] OMERA, P.: From Quantum Foam to Vortex-ring Fractal Structures and Mendeleevs Table, New
Trends in Physics, NTF 2007, Brno Czech Republic, 2007, 179-182
[14] OMERA, P.: Vortex-fractal-ring Structure of Electron, Proceedings of the 6th International Conference
on Soft Computing ICSC2008, January 25, 2008, Kunovice, Czech Republic
[15] PAULING, L.: General Chemistry, Dover publication, Inc, New York, 1988
167
ADDRESS:
Doc. Ing. Pavel Omera , CSc.
Technická 2
616 69 B rno
Czech Republic
osmera @fme.vutbr.cz
168
THE USE OF FUZZY SETS IN MULTICRITERIAL OPTIMIZATION
Vít zslav ev ík, Pavel Krej í
Abstract: This article reviews fuzzy set theory influenced techniques of multicriterial optimisation.
In article is explained, how and why to introduce uncertainty to various parts of the multicriterial
optimisation. Different approaches are classified and benefit of the fuzzyfication in the
multicriterial optimisation is shown.
Keywords: Fuzzy Sets, Fuzzy Programming, Fuzzy multiple objective decision making, degree of
satisfaction, multiple criteria decision making (MCDM), Scalarizing function
INTRODUCTION
Decision-making becomes field of science at the beginning of the 20th century. Most of the real life decision
problems requires consideration of conflicting multiple criteria or viewpoints, therefore in the early 1970s was
introduced a new area of study to Operations Research, Management Science and Decision Science
Multicriterial Optimisation. This area aims to give the decision-maker some tools in order to enable him to
advance in solving a decision problem where severaloften contradictorypoints of view must be taken into
account. Multi-criteria analysis (MCA), Multicriteria decision-aid, multiple criteria decision-making (MCDM),
multiple criteria decision-aid (MCDA) or Multi Objective Optimisation (MOO)even if there are some
distinctions between these sub fields of decision theory and operations research, the overall objective is the
same: to help decision makers solve complex decision problems in a systematic, consistent and more productive
way [4].
DECISION-MAKING WITH HELP OF FUZZY SET THEORY
It is widely accepted now, that fuzzy extensions and new fuzzy based decision techniques can move
multicriterial decision methodology over limitations of the crisp methods. Fuzzy sets are introduced to
multicriterial and multiobjective methods mainly for these reasons: (i) input data are imprecise and judgements
are subjective, formulated in linguistic terms (ii) fuzzy approaches in difficult problems often outperform
nonfuzzy multicriterial methods [6] (iii) there exists problems, which was inaccessible to and unsolvable with
standard multicriterial decision support techniques.
Authors are distinguishing various families of traditional methods [3] and the fuzzy multicriterial decision
method has been developed along the same lines [2].
First family of method is based on multiple attribute utility theory, these methods are based on idea, that
decision-maker attempts to maximize utility function aggregating all the different points of view (criteria).
Aggregation can be additive, multiplicative or mixed. Well-known methods are The Analytic Hierarchy Process
(AHP) and Simple Multi-attribute Rating Technique (SMART). These techniques can be fuzzyfied how is shown
in [1] and [2]. Fuzzy approach is used for values of criteria, and for aggregation of difficult comparable criteria.
Fuzzy variant of SMART technique can also be easily used for sensitivity analysis of the results as described in
[2].
Second family of multicriteria decision methods are outranking methods. In these methods is complete or
incomplete rank order of the alternatives built up via outranking relations under the individual criteria by
pairwise comparison between two alternatives. This family includes Electre (ELimination Et Choix Traduisant la
Realité) and Promethee (Preference Ranking Organisation METHod for Enrichment Evaluations). In Electre III
method are two fuzzy concepts: the degree of agreement with the statement that the first alternative in the pair is
at least as good as the second, and the degree of disagreement with the above statement.
Third group of methods are type of Multi-Objective Optimization. This methodology was first developed in the
frame of linear programming. Important concepts are single objective vectors of best (ideal) and worst (nadir)
solutions. Compromise solution is computed by minimizing distance from the ideal point. To bring multicriterial
problem to optimization problem some type of scalarizing function is often used [8]. One of used method is
169
Fuzzy Linear Programming (FLP). This technique does not need human interaction to compute compromise
solution. Benefit of this method is described in [9]. However interactive methods exists and are described in [2]
namely the minimization of weighted Chebychev-norm distance and the maximization of the weighted degrees
of satisfaction. Interactive Multi Objective Bivalent Programming and Reference Point method for project
selection is presented for example in [7]. In interactive methods the decision-maker brings a direct contribution
towards the elaboration of a solution. This last trend follows line of reintroducing observer into processes as one
of the factors. As there does not exist real objective optimum an human expert her/his understanding of process
and knowledge can not be separated from the model to follow reality.
In the field of multiple criteria decision-making was traditionally an assumption of criteria independence.
Authors in [4] points at fact, that interdependence is part of the economic theory and all market economies. They
are introducing novel approachinterdependence in MCDM by definition of the grade of interdependency that
modifies membership function of objective to fuzzy set good solutions. Authors are presenting also case, when
objective functions are also fuzzyfied.
CONCLUSION
In these days fuzzy concepts are successfully used in practice and this fact justifies researchers and visionaries to
deal with them in future. Underlying concepts and theories of decision-making in a fuzzy environment is still in
rapid development and by far not closed. There is still need of further education at schools and in praxis about
these techniques.
REFERENCES
[1]
TALAOVÁ, J. Fuzzy metody vícekriteriálního hodnocení a rozhodování. Olomouc : Vydavatelství UP,
2003. 180 p., ISBN 80-244-0614-4.
[2]
LOOTSMA, F. A. Fuzzy Logic for Planning and Decision Making. Dordrecht : Kluwer, 1997. x, 195 p.
Applied Optimization; vol. 8. ISBN 0-7923-4681-5.
[3]
VINCKE, Ph. Multicriteria Decision-Aid. New York : J. Wiley, 1992. x, 154 p. ISBN 0-471-93184-5.
[4]
FULLÉR, R.; CARLSSON, C. Fuzzy multiple criteria decision making : Recent developments. Fuzzy
Sets and Systems. 1996, vol. 78, no. 0, pp. 139153.
[5]
HERRERA, F.; VERDEGAY, J. L. Fuzzy sets and operations research : Perspectives. Fuzzy Sets and
Systems. 1997, vol. 90, is. 2, pp. 207218.
[6]
PETROVIC-LAZAREVIC, S.; ABRAHAM, A. Hybrid-Fuzzy Linear Programming Approach for Multi
Criteria Decision Making Problems, International Journal of Neural, Parallel & Scientific Computations.
2003, vol. 11 is. 1/2, pp. 5368.
[7]
KLAPKA, J.; PI OS, P. Decision support system for multicriterial R&D and information systems
projects selection. European Journal of Operational Research. 2002, vol. 140, is. 2, pp. 434446.
[8]
MIETTINEN, K.; MÄKELÄ, M. M. On scalarizing functions in multiobjective optimization. OR
Spectrum. 2002, vol. 24, no. 2, pp. 193213.
[9]
DePORTER, E. L.; ELLIS, K P. Optimization of Project Networks with Goal Programming and Fuzzy
Linear Programming. Computer & Ind. Engng. 1990, vol. 19, no. 14, p. 500504.
The paper was supported by project from MSMT of the Czech Republic No. 1M06047 Center for Quality and
Reliability of Production.
ADRRESS:
Ing. Vít zslav ev ík
Fakulta strojního inenýrství, Ústav automatizace a informatiky
Technická 2
616 69 Brno
tel. +420 541 143 334, fax +420 541 142 330
Ing. Pavel Krej í
Fakulta strojního inenýrství, Ústav automatizace a informatiky
Technická 2
616 69 Brno
tel. +420 541 143 334, fax +420 541 142 330
170
INCREASING OF COMPUTER NETWORKS PERFORMANCE VIA NODE TO NODE
THROUGHPUT OPTIMIZATION
Imrich Rukovanský1, Ond ej Popelka2
1
2
Evropský polytechnický institut, s.r.o
Ústav informatiky PEF MZLU v Brn
Abstract: In order to find out demonstrably that a computer net is working efficiently, it is essential
to monitor its activity via measuring. This process can be carried out by means of monitors (net
analyzers). They are able to observe tens of various output parameters (permeability, CPU
utilization, utilization of communication lines between nodes/knots, narrow spots) at different net
levels. However, the nature of measuring varies from case to case. It depends on what output
parameters are being observed, whether we observe only a specific part of the net, or the net as a
whole, whether we use hardware or software monitor, or a combination of both. The significance of
computer net monitoring as well as the diversity of practical measuring is demonstrated in the
following contribution. For optimization of computer net can be used evolutionary algorithms.
Parallel grammatical evolution can be use for optimization of networks. This paper describes the
possibilities of using genetic algorithms for network design and optimization. The described
algorithm grammatical evolution has strong generalizing capabilities, which allow it to
generate symbolic representation of mathematics models, graphs and other structures. The
implementation requirements and the issues, which may arise is discussed together with several
methods addressing those issues.
Key words: Grammatical evolution, evolving structures, network optimization, genetic algorithms,
grammar, communication lines, nodes, throughput, performance
1. INTRODUCTION
Computer network generally consists of a number of interconnected switching nodes. A transmission from one
device is routed through these internal nodes to the specified destination device. These nodes (including the
boundary nodes) are not concerned with the contents of data; rather their purpose is to provide a switching
facility that will move the data from node to node until they reach their destination. Therefore performance
improvements may be obtained via node-to-node throughput optimization. The described method grammatical
evolution [1] has strong generalization capabilities, which enable it to evolve and optimize LAN, MAN and
other network structures [10]. It is also capable of evolving graphs and other structures which can be used to
represent the network structure. To describe the features of grammatical evolution a simple example of
generating a mathematical function will be used.
2. GRAMMATICAL EVOLUTION
Grammatical evolution GE is based on classic genetic algorithm extended with a context-free grammar. The
grammar forms an interface between the data representation of actual solution and the underlying genetic
algorithm. Each individual in the population is represented by a sequence of rules of the defined grammar. The
particular solution is then generated by translating the chromosome to a sequence of rules, which are then
applied in specified order. Therefore the main task of grammatical evolution is the translation of a chromosome
to a symbolic representation of a solution, also known as genotype to phenotype translation [2]. After the
translation is complete, an individual can be evaluated and its fitness can be computed. Then the standard
operators (selection, crossover [3] [4], mutation) of genetic algorithms are applied and a new population is
generated and a new population cycle is started.
Rewriting grammar is defined as a tuple G = (N, T, S, P) where N is a set of non-terminals, T is a set of
terminals, S is a starting symbol and P is a set of production rules. The non-terminals are items, which appear in
the individuals body (the solution) only before or during the translation. After the translation is finished all nonterminals are translated to terminals. Terminals are all symbols which may appear in the generated language,
thus they represent the solution. Start symbol is one non-terminal from the non-terminals set, which is used to
initialize the translation process. Production rules define the laws under which non-terminals are translated to
terminals. Production rules are key part of the grammar definition as they actually define the structure of the
171
generated solution. The grammar, which was used to generate mathematic functions [5][6] was defined as
follows: N = {expr, fnc, num}, where expr can be expanded to any expression; fnc can be translated to a function
terminal; num can be translated into a numeric constant; var can be translated into a variable; T = {sin, cos, +, -,
/, *, x, 1, 2, 3, 4, 5, 6, 7, 8, 9} and S = <expr>. The start symbol may be interpreted as a general rule of what
may appear in the final solution. With the given definition there are no constraints and any expression may
appear in the solution. of rules generating mathematical functions are shown in prefix notation, which simplifies
the grammar as no parentheses are necessary.
3. PARALLEL GRAMMATICAL EVOLUTION
The PGE is based on the grammatical evolution GE [1], where BNF grammars consist of terminals and nonterminals. Terminals are items, which can appear in the language. Non-terminals can be expanded into one or
more terminals and non-terminals. Grammar is represented by the tuple {N,T,P,S}, where N is the set of nonterminals, T the set of terminals, P a set of production rules which map the elements of N to T, and S is a start
symbol which is a member of N. For example, below is the BNF used for our problem:
N = {expr, fnc}
T = {sin, cos, +, -, /, *, X, 1, 2, 3, 4, 5, 6, 7, 8, 9}
S = <expr>
and P can be represented as 4 production rules:
1. <expr> := <fnc><expr>
<fnc><expr><expr>
<fnc><num><expr>
<var>
2. <fnc> :=
sin
cos
+
*
U3. <var> := X
4. <num> := 0,1,2,3,4,5,6,7,8,9
The symbol U- denotes an unary minus operation. There are notable differences when compared with [1]. We
dont use two elements <pre_op> and <op>, but only one element <fnc> for all functions with n arguments.
There are not rules for parentheses; they are substituted by a tree representation of the function. The element
<num> and the rule <fnc><num><expr> were added to cover generating numbers. The rule <fnc><num><expr>
is derived from the rule <fnc><expr><expr>. Using this approach we can generate the expressions more easily.
For example when one argument is a number, then +(4,x) can be produced, which is equivalent to (4 + x) in an
infix notation. The same result can be received if one of <expr> in the rule <fnc><expr><expr> is substituted
with <var> and then with a number, but it would need more genes. There are not any rules with parentheses
because all information is included in the tree representation of an individual. Parentheses are automatically
added during the creation of the text output.
If in the GE is not restricted anyhow, the search space can have infinite number of solutions. For example the
function cos(2x), can be expressed as cos(x+x); cos(x+x+1-1); cos(x+x+x-x); cos(x+x+0+0+0...) etc. It is desired
to limit the number of elements in the expression and the number of repetitions of the same terminals and nonterminals.
4. BACKWARD PROCESSING OF THE PGE
The chromosome is represented by a set of integers filled with random values in the initial population. Gene
values are used during chromosome translation to decide which terminal or nonterminal to pick from the set.
When selecting a production rule there are four possibilities, we use gene value mod 4 to select a rule. However
the list of variables has only one member (variable X) and gene value mod 1 always returns 0. A gene is always
read; no matter if a decision is to be made, this approach makes some genes in the chromosome somehow
redundant. Values of such genes can be randomly created, but genes must be present [9].
Body of the individual is s a linear structure, but in fact it is stored as a one-way tree (child objects have no links
to parent objects). In the diagram we use abbreviated notations for nonterminal symbols: f - <fnc>, e - <expr>, n
- <num>, v - <var>.
172
5. GENOTYPE TO PHENOTYPE TRANSLATION
.At the beginning of the process the body of an individual contains the starting symbol expr, which will be
rewritten to an expression. A gene pointer is created and initialized before the first gene. In the first step, the expr
symbol is to be translated, thus the according table of productions rules is selected. The selected table has four
entries so a decision has to be made on which one to choose. The gene pointer is incremented to the first gene
and index of the rule is computed using the modulo-operation. The selected rule is then applied to the individual
body and the non-terminal is rewritten with the right side of the rule. These steps are repeated until no more nonterminals are present in the individuals body.
However this basic principle can be further expanded by additional algorithms of translation which improve the
overall performance of the process. The backward-processing algorithm translates the non-terminals in the
individuals body from the end of the body to the beginning. The algorithm produces positive side effects in the
operations of crossover and mutation, which can be used to process the population cycles more efficiently.
Although this algorithm is very general it requires some modifications of the grammar, which might introduce
new issues when a grammar for optimal network generation is introduced.
6. LOGICAL FUNCTION XOR AS A TEST FUNCTION
Input values are two integer numbers a and b; a, b 2< 0, 1 >. Output number c is the value of logical function
XOR. Training data is a set of triples (a, b, c):
P = {(0, 0, 0); (0, 1, 1); (1, 0, 1); (1, 1, 0)}.
Thus the training set represents the truth table of the XOR function. The function XOR (+) can be expressed
using OR (_), AND (^), NOT (¬) functions:
a + b = (a ^ ¬b) _ (¬a ^ b) = (a _ b) ^ (¬a _ ¬b) = (a _ b) ^ ¬(a ^ b)
The grammar was simplified so that it does not contain conditional statement and numeric constants, on the other
hand three new terminals were added to generate functions _, ^, ¬. Thus the grammar generates representations
of the XOR functions using other logical functions with notation ( |, .&, ~ ):
1a)
function xxor($a,$b) {
$result = "no_value";
$result = ($result) |
((((~(~(~(~(~($result)))))) | (($a) & (($a) & (~($b))))) & ($a))
| ((~($a)) & ($b)));
return $result;
1b)
function xxor($a,$b) {
$result = "no_value";
$result = ($result) | (((~$result | ($a & ($a & ~$b))) & $a) | (~$a & $b));
return $result;
7. TWO-LEVEL OPTIMISATION
Optimization of networks can be solved using a two-level optimization. The first level of the optimization is
performed using grammatical evolution [11]. The output can be a function containing several symbolic
constants. Such function therefore cannot be evaluated and assigned a fitness value. In order to evaluate the
generated function a secondary optimization has to be performed. For secondary optimization differential
evolution is used. Input for the second level of optimization is the function with symbolic constants and the
output is vector of the values of those constants. A simplified flowchart diagram of the two-level optimization is
shown on Fig. 1. Basically it consists of two nested population loops. The inner loop is using standard
differential evolution with DE/rand/1 scheme. The outer loop is a single population of parallel grammatical
evolution. The resulting Grammatical Differential Evolution (GDE) takes advantage of both the original
methods.
8. CONCLUSION
We have described the Parallel Grammatical Evolution (PGE) that can map an integer genotype onto a
phenotype with the backward processing. PGE has proved successful for creating trigonometric functions. This
algorithm can be used for optimisation of computer networks. Our goal was to find very efficient algorithm for
173
it. We first used test task to find suitable structure of PGE. It seems that in future this optimisation tool can be
used for such problems as computer networks are.
Fig.1. The two-level PGE optimization
Parallel GEs with hierarchical structure can increase the efficiency and robustness of systems, and thus they can
track better optimal parameters in a changing environment. From the experimental session it can be concluded
that modified standard GEs with only two sub-populations can create PGE much better than classical versions of
GEs.
The parallel grammatical evolution can be used for the automatic generation of programs. We are far from
supposing that all difficulties are removed but first results with PGEs are very promising.
Both Grammatical evolution [1] and Differential Evolution [8] algorithms are evolutionary computation methods
based on genetic algorithms [4], [5]. Using these methods is especially suitable for optimization problems, which
are very difficult to solve using classical mathematical methods or where using deterministic computation
methods would require unacceptable simplification of the problem. These problems include nonlinear regression
and prediction. Most of the methods found in statistical analysis are not eligible since many of the real-world
problems have nonlinear character. Linear regression might be used only on short intervals of the measured data
or under restricted conditions. These tasks are therefore suitable for solving using genetic algorithms and other
evolutions methods [6].
The second task of the optimization is to find parameters for the model chosen in the first step so that the
resulting function describes the empirical data optimally. After solving the second step of regression a complete
statistical model of empirical data is found.
174
REFERENCES
[1]
ONEILL, M.; RYAN, C. Grammatical Evolution: Evolutionary Automatic Programming in an Arbitrary
Language Kluwer Academic Publishers 2003.
[2]
ONEILL, M.; BRABAZON, A.; ADLEY, C. The Automatic Generation of Programs for Classification
Problems with Grammatical Swarm. Proceedings of CEC 2004, Portland, Oregon (2004) 104 110.
[3]
PIASECZNY, W.; SUZUKI, H.; SAWAI, H. Chemical Genetic Programming Evolution of Amino Acid
Rewriting Rules Used for Genotype-Phenotype Translation. Proceedings of CEC 2004, Portland, Oregon
(2004) 1639 - 1646.
[4]
OMERA, P.; IMONÍK, I.; ROUPEC, J. Multilevel distributed genetic algorithms. In Proceedings of
the International Conference IEE/IEEE on Genetic Algorithms, Sheffield (1995) 505510.
[5]
OMERA, P.; ROUPEC, J. Limited Lifetime Genetic Algorithms in Comparison with Sexual
Reproduction Based Gas. Proceedings of MENDEL2000, Brno, Czech Republic (2000) 118 126.
[6]
LI, Z.; HALANG, W. A.; CHEN, G. Integration of Fuzzy Logic and Chaos Theory. paragraph: Osmera
P.: Evolution of Complexity, Springer, 2006 (ISBN: 3-540-26899-5) 527 578.
[7]
WALDROP, M. M. Complexity The Emerging Science at Edge of Order and Chaos. Viking, 1993.
[8]
PRICE, K. 1996. Differential evolution: a fast and simple numerical optimizer. 1996 Biennial Conference
of the North American Fuzzy Information Processing Society, NAFIPS, pp. 524-527, IEEE Press, New
York, NY, 1996.
[9]
POPELKA, O.; RUKOVANSKÝ, I. Optimization of Computer Network with Grammatical Evolution. In
ICSC 2007, Kunovice: EPI, s.r.o., 2007, pp. 201- 205.
[10] RUKOVANSKÝ, I. The Significance of Computer Networks Monitoring. In ICSC 2006 Kunovice: EPI,
s.r.o , 2006, pp.
[11] RUKOVANSKÝ, I. Evolution of Complex Systems. 8th Joint Conference on Information Sciences. Salt
Lake City, Utah, USA. July 21-25, 2005.
ADDRESS:
Prof. Ing. Imrich Rukovanský, CSc.
Evropský polytechnický institute, s.r.o,
Osvobození 699
686 04 Kunovice
[email protected]
Bc. Ond ej Popelka
Ústav informatiky PEF MZLU v Brn
Zem d lská 1
613 00 Brno
[email protected]
175
176
NETWORK ELEMENT PROJECT BY MEANS OF NEURAL NETWORK
Ji í Lika, Jan astný, Miroslav Cepl, M. tencl
Abstract: This paper deals with a project and simulation of high-speed active network element
controlled by neural network. It describes neural networks in general and Hopfield network in
particular. By means of mathematical software Matlab/Simulink it also describes a simulation of
data transferring through this active network element.
1. INTRODUCTION
Active elements form the basis of all communication networks. These elements process transferred data and on
the basis of result they transfer data units from sender to receiver. At present the most difficult task of active
elements is to determine which data unit should be processed in a particular moment so that it would correspond
to priorities assigned to particular data units.
Classic sequential data processing is limited by the speed of central processing unit. Increasing demands on the
speed of processors make higher and higher quality requirements on production technology. Parallel data
processing offers a different solution. When more complex functional elements are operating in parallel,
requirements on control their cooperation are increased substantially and the increase of effectiveness of the
system as a whole is therefore not so noticeable. More convenient possibility is to use simpler functional blocks
with individual memory operating in parallel. It is easier to coordinate their mutual cooperation and overall
effectiveness is bigger than in the case of more complex elements.
Artificial neural networks could be considered as an example of the second method, i.e. a great number of
comparatively simple functional blocks operating in parallel. Artificial neural network is a field of
interconnected elementary functional blocks with special architecture and in many cases it offers a very
perspective solution to data processing.
This article focuses on a special type of artificial neural networks, so called Hopfield network which can be used
for the solving of the optimization problem.
On the basis of active element architecture and the tasks that these elements are able to implement it is possible
to say that artificial neural networks can be used effectively to control these elements.
2. NEURAL NETWORKS
Neural networks are composed of simple elements operating in parallel. These elements are inspired by
biological nervous systems. As in nature, the network function is determined largely by the connections between
elements. We can train a neural network to perform a particular function by adjusting the values of the
connections (weights) between elements.
2.2 HOPFIELD NETWORK
A Hopfield net is a form of recurrent artificial neural network invented by John Hopfield. Hopfield nets serve as
content-addressable memory systems with binary threshold units. They are guaranteed to converge to a local
minimum, but convergence to one of the stored patterns is not guaranteed.
The units in Hopfield nets are binary threshold units, i.e. the units only take on two different values for their
states and the value is determined by whether or not the units' input exceeds their threshold. Hopfield nets can
either have units that take on values of 1 or -1, or units that take on values of 1 or 0. So, the two possible
definitions for unit i's activation, ai, are:
177
Where:
wij is the strength of the connection weight from unit j to unit i (the weight of the connection).
sj is the state of unit j.
i is the threshold of unit i.
The connections in a Hopfield net typically have the following restrictions:
(no unit has a connection with itself)
(connections are symmetric)
Pic.1
Architecture of Hopfield network [1]
The Hopfield network uses the saturated linear transfer function satlins (called in Matlab).
Pic.2
Satlins transfer function [1]
3. ARCHITECTURE OF NETWORK ELEMENT
Elementary model of active network element consists of several partial blocks. Block scheme of an active
element is described in picture 3. Receive data units are first saved in input buffers. Since data units contain
request address, it is possible to say through which output port it will leave the connection system. Except for
information concerning the request port, it is also necessary to know information concerning the priority of data
178
flow, which the data unit belongs to. Given information is gathered from all data units saved in input buffers and
it serves as input data for neural network.
Pic.3
Block scheme of active element
N-dimensional vector made from priorities particular requests can be made. The first data of the vector contains
the priority of the frame directed to the first output, the second term of the vector contains the priority of the
frames directed to the second output etc. N x N dimensional matrix can be made from the priority vectors
acquired from particular input ports, and this matrix contains input conditions which must be optimised. As an
answer to these input data a particular configuration of switches is generated, on the basis of which the switches
will be set in array of switches. The generated configuration will have the shape of a N x N dimensional
matrix, so called configuration matrix.
As for the generated result, it is required that one input and one output are interconnected. This condition is
reflected in the configuration matrix generated by neural network and in each line and in each column there will
be one particular active output, i.e.1 and the rest of them will be inactive, i.e. their value will be 0. When the
switches are set the data transfer can be carried out from input buffers to the system outputs. Set of the switches
should be optimal from the point of view of priority matrix. A configuration is chosen where just one element is
marked in every column and every line and the sum of thus chosen priorities is minimal provided that lower
value corresponds to higher priority.
4. SIMULATION (MATLAB)
Simulation of neural network functionality is made in mathematical software Matlab/Simulink. The simulation is
based on chapter 3 Architecture of network element. The simulation is implemented on an active network
element with four ports.
179
Pic.6
Overall scheme of simulation in Matlab/Simulink
The network is simulated by a block called net. This block generates random data, which contain a request port
and the priority value. These values are saving in a buffer. A matrix is made from these values and it serves as
input data for neural network.
Pic.7
Scheme of computer network simulation in Matlab/Simulink
This neural network optimises input data. As a result of this a matrix is created which serves as an input into
array of switches. This matrix consists only of 0 or 1 values and in every line and every column there is just one
1. Array of switches interconnects given inputs and outputs and the data transfer itself in carried out.
180
Pic.8
Scheme of Hopfield neural network in Matlab/Simulink
5. CONCLUSION
Creating various types of neural networks and their simulation could be the next step of this problem research. It
means comparing neural network algorithms according to their speed, finding optimal solution, size of neural
network etc. Then the chosen type of neural network can be implemented on a special card Netfpga 2.1. The
speed of switching and the speed of data transfer can be analysed on real data.
ACKNOWLEDGEMENT
MSM 0021630529 Intelligent Systems in Automation (Research design of Brno University of
Technology)
No 102/07/1503 Advanced Optimisation of Communications Systems Design by Means of Neural
Networks. The Grant Agency of the Czech Republic (GACR)
MSM 6215648904/03 Development of relationships in the business sphere as connected with changes
in the life style of purchasing behaviour of the Czech population and in the business environment in the
course of processes of integration and globalization (Research design of Mendel University of
Agriculture and Forestry in Brno)
6. REFERENCES
[1]
MATLAB/HELP: Neural Network Toolbox, Recurrent Networks, Hopfield Network.
[2]
Molnár, K.: Application of artificial neural network in high-speed active network elements, PhD Thesis,
VUT, 2002.
[3]
Hopfield, J. J., Tank, D. W: Neural Computation of Decisions. In: Optimization Problems, Biological
Cybernetics, Springer-Verlag, 1985
[4]
Hopfield net. [Online]. <http://en.wikipedia.org/wiki/Hopfield_network>.
[5]
norek, M., Ji ina, M.: Neuronové sít a po íta e. Praha, Vydavatelství VUT, 1996.
[6]
astný, J., korpil, V.: Neural Networks Learning Methods Comparison. International Journal WSEAS
Transactions on on Circuits and Systems, Issue 4, Volume 4, 2005, ISSN 1109-2734, pp. 325-330.
181
[7]
astný, J., Mina ík, M.: Object Recognition by Means of New Algorithms. In: Fourth International
Conference on Soft Computing Applied in Computer and Economic Environments ICSC 2006, Kunovice,
Czech Republic, 2006,
[8]
Sarle, W. S.: Neural Networks and Statistical Models. Proceedings of the Nineteenth Annual
SAS Users Group International Conference, Cary, NC: SAS Institute, 1994.
ADDRESS:
Ing. Ji í Lika
Technická 2
616 69 Brno
Czech Republic
[email protected],
Doc. RNDr. Ing. Jan astný
Technická 2
616 69 Brno
Czech Republic
[email protected]
Ing. Miroslav Cepl
Technická 2
616 69 Brno
Czech Republic
M. tencl
Technická 2
616 69 Brno
Czech Republic
182
DETERMINING GENETIC ALGORITHM OPERATORS IN THE PROGRAM FOR
OPTIMIZATION OF PROGRESSIVE DISTRIBUTORS
Ji í Vep ek
Abstract: This paper relates to [1] and [5] and it discusses the problem of determining suitable
operators of the genetic algorithm that is a part of the program for the design of ZP-A and ZP-B
progressive distributors. Among the operators to be determined are: the number of individuals
entering the tournament selection t and the mutation probability Pm. The values of the operators
affect the effectiveness of finding the best structures of ZP-A and ZP-B distributors, components of
lubrication systems. The crossover operator was not implemented in the program because the
number of individuals created by the mutation operator was sufficient. Both types of the ZP-A and
ZP-B distributors are prefabricated modules, which are used to distribute lubricant to a given
number of points in a proportional and progressive manner. ZP-A and ZP-B distributors can be
composed of 3 types of sections of different sizes. Three sections is the minimum for both types of
distributors. The following factors affect the way in which lubricant is divided in a distributor: the
number of sections, section sizes, the interconnections of cross-section channels, the opening
(closure) of or the interconnections of certain outlets in a section. The number and the organization
of lubrication points determine the number and the hierarchical organization of progressive
distributors in a lubrication system. With the aid of the optimization program, numerous structures
of ZP-A and ZP-B distributors, as a result of a great variety of the interconnections of cross-section
channels and many possible variations of section sizes, can be evaluated and a more suitable
solution can be found much faster. The program was created in Java.
Keywords: genetic algorithms, tournament selection, mutation, lubrication systems, progressive
distributors, optimization of flow rates
INTRODUCTION
ZP-A and ZP-B progressive distributors (Delimon) have three types of sections: input, middle and end section
(see Fig.1). Each section has two outlets and a piston that discharges a certain amount of lubricant in a stroke.
The amount of the discharged lubricant (lubricant volume) is determined by the section size. There are 4 section
sizes for ZP-A distributors (0.07, 0.1, 0.2 and 0.3 cm3 stroke-1 ) and 3 section sizes for ZP-B distributors (0.5, 1.2
and 2 cm3 stroke-1 ). The principle of operation of ZP-A and ZP-B distributors can be found in [10] and also in [1]
and [5]. The number of sections, section sizes, the way of interconnecting cross-section channels and the opening
(closure) of or the interconnections of specific progressive distributor outlets in a section affect the ratios of the
discharged lubricant volumes between distributor outlets.
Centralized lubrication systems with progressive distributors [7], [10] are used to lubricate small or mediumsized machines, e.g. machine tools, presses, punching presses, cutting and woodworking machines. This type of
the system is suitable for distributing grease and oil.
Fig. 1 shows a centralized lubrication system with the main distributor I and three secondary distributors A, B
and C. Two of the distributors (I, A) have three sections and the other two (B, C) have six sections. This
lubrication system distributes oil or grease to 16 lubrication points using a pump whose 3 outlets are joined to the
only output line. The numbers next to individual distributor outlets represent the lubricant volumes that are
discharged during one working cycle (cm3 cycle-1). The duration of the working phase of the lubrication system
is defined as the number of cycles performed by the secondary distributor C. The input section of the distributor
C is monitored by an indicator that registers every piston stroke. Two piston strokes represent one working cycle
of the distributor. Some of the results of the design of a distributor C in the lubrication system, obtained with the
optimal values of the genetic algorithm operators, are shown in Fig. 7-10.
183
Fig. 1: A schema of the lubrication system with the distributors A, B, C and I
The use of genetic algorithms in finding suitable structures of ZP-A and ZP-B progressive distributors
As it was already mentioned, a Java optimization program implementing a genetic algorithm [3] has
been developed to find suitable structures of ZP-A and ZP-B progressive distributors. The reasons for
employing a genetic algorithm are the following: a huge number of possible distributor structures,
mutual dependence between the interconnections of cross-section channels and the opening of
distributor outlets, the possibility to change fitness etc. The algorithm and the detailed description of the
optimization program was given in [1] and [5]. The number of structures of ZP-A and ZP-B progressive
distributors can be calculated based on the formula below [1]:
N DC
nk
2k
2k
l
2
l
,
(1)
where: n the number of section sizes, k the number of distributor sections, 2k the total number of
distributor outlets and l the number of open outlets. This formula is valid if 2k-2 l.
For ZP-A or ZP-B distributors with at most one closed outlet the following formula is used:
N DC
nk
2k
l
.
184
(2)
The meanings of the quantities are the same as in the previous case. The formula is valid if 2k l. A distributor
structure is a distributor configuration in which some outlets are open, some outlets in a section and some crosssection channels interconnected. The fact that there are several possibilities how to interconnect cross-section
channels and outlets in a section for some open outlets is not taken into account in formulas (1) and (2). The total
number of possible interconnections of cross-section channels would be higher. Tab. 1 gives an overview of the
numbers of structures of the ZP-A progressive distributor. The number of structures of the ZP-B distributor is
lower and can be determined analogously based on (1) and (2).
Number of Number of distributor sections
structures 3
4
5
6
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
5.76E+02
1.02E+03
8.96E+02
3.84E+02
6.40E+01
3.33E+03
9.22E+03
1.41E+04
1.28E+04
6.91E+03
2.05E+03
2.56E+02
1.74E+04
6.55E+04
1.43E+05
2.01E+05
1.86E+05
1.15E+05
4.51E+04
1.02E+04
1.02E+03
8.60E+04
4.10E+05
1.17E+06
2.21E+06
2.92E+06
2.75E+06
1.84E+06
8.60E+05
2.66E+05
4.92E+04
4.10E+03
7
8
9
10
4.10E+05
2.36E+06
8.29E+06
1.98E+07
3.41E+07
4.33E+07
4.11E+07
2.92E+07
1.53E+07
5.77E+06
1.47E+06
2.29E+05
1.64E+04
1.90E+06
1.28E+07
5.37E+07
1.55E+08
3.28E+08
5.25E+08
6.47E+08
6.19E+08
4.59E+08
2.62E+08
1.13E+08
3.58E+07
7.80E+06
1.05E+06
6.55E+04
8.65E+06
6.71E+07
3.25E+08
1.10E+09
2.77E+09
5.34E+09
8.10E+09
9.75E+09
9.37E+09
7.20E+09
4.39E+09
2.10E+09
7.71E+08
2.10E+08
3.98E+07
4.72E+06
2.62E+05
3.88E+07
3.40E+08
1.87E+09
7.27E+09
2.12E+10
4.79E+10
8.62E+10
1.25E+11
1.48E+11
1.43E+11
1.13E+11
7.23E+10
3.74E+10
1.54E+10
4.92E+09
1.18E+09
1.98E+08
2.10E+07
1.05E+06
Tab. 1: The number of structures of the ZP-A progressive distributor [10]
ENCODING AND EVALUATION OF THE ZP-A DISTRIBUTOR STRUCTURE
Each section of the distributor is encoded by 6 digits. It means that a three-section ZP-A or ZP-B distributor will
be encoded by an 18-digit string. The way of encoding the structures of ZP-A and ZP-B progressive distributors
and the meaning of individual positions in the code were described in [1] and [5]. Before the fitness function is
determined, the calculated and requested lubricant volumes are sorted in ascending order. Based on the fitness
function, which was modified compared to the fitness in [1] and [5], the ratios of calculated lubricant volumes to
requested lubricant volumes, i.e. VVVi,j+1/VVVij and VVPi,j+1/VVPij, are calculated. The j index identifies
a particular position of a calculated or requested lubricant volume. The sum of the absolute values of differences
between the ratios then defines the fitness value. The selection mechanism chooses distributor structures for the
next generation. The definition of the evaluation function (fitness) is given below:
n m 1
HFij
abs
i 0 j 0
VVVi , j
VVVij
1
VVPi , j
VVPij
1
,
(3)
where: VVPij the requested lubrication volume during one working cycle of distributor i (an individual of a
population) (cm3 cycle-1) from open outlet j, VVVij the calculated lubricant volume discharged during one
working cycle of distributor i from open outlet j, i <1;n> a j <1;m>; n the number of individuals in a
population, m the number of open distributor outlets. The requested and calculated volumes are ordered before
the fitness is applied. Therefore, the j index in (3) does not denote a particular outlet of the progressive
distributor. This ensures that all possible progressive distributor structures are evaluated equally, regardless of
the positions of open outlets.
185
DETERMINING GENETIC ALGORITHM OPERATORS IN THE PROGRAM FOR OPTIMIZATION
OF PROGRESSIVE DISTRIBUTORS ZP-AND ZP-B
The determination of suitable genetic algorithm operators in the optimization program was done by the following
procedure:
First, the input data of the program was chosen:
Genetic algorithm operators and calculation parameters (population size, the number of program
iterations, the number of individuals t entering tournament selection, mutation probability Pm)
Requirements for the ZP-A distributor (the number of sections, the number of open outlets, the ratios of
lubricant volumes between distributor outlets (cm3 cycle-1 ))
Then, a series of calculations (further referred to as a set of calculations) were done repeatedly. The program
searched for a structure of a 10-section ZP-A distributor with six open outlets and the ratios of lubricant volumes
between outlets set at 0.3:0.3:0.3:0.3:1.2:1.2 cm3 cycle-1 whose fitness value equals 0.
Fig. 2 shows the 6-section ZP-A distributor C that was found by the optimization program with the genetic
algorithm operators set to optimal values, namely t = 3, Pm = 1/30. The calculation was done for the population
size N = 500. The number of program iterations was 200. For clarity, descriptions were added to the schema. The
code of the distributor is the following: 211110100100200010201111300000200011. Because the distributor has
six sections, the code is composed of 36 digits.
Fig. 2: A selected structure of the ZP-A distributor C
186
For certain setting of genetic algorithm operators, i.e. the number of individuals (t) entering the tournament
selection (selection pressure) and mutation probability Pm, 50 calculations were done. Namely, for population
size N = 500, 200 iterations and t = 2, 15 different sets of calculations for Pm = 1/10, 1/20,, 1/150 were
performed. Another 15 sets of calculations were performed for t = 3 and the same values of Pm. Altogether, the
results for 30 different settings of genetic algorithm operators were obtained. For each set of calculations two
statistical parameters were determined, namely the average number of the best individuals per calculation (the
maximum value is defined as the number of iterations performed during a calculation) and the number of
calculations in which no best individual was found (the fitness value of the structures of the ZP-A distributor was
different from 0). The graphical representation of the statistical parameters for t = 2 and t = 3 are shown in
Fig. 3, Fig. 4 and in Fig. 5, Fig. 6 respectively.
90
50
80
70
40
60
30
50
40
20
30
20
10
10
0
10
30
50
70 90
Pm-1
0
110 130 150
10
Fig. 3: Average number of the best individuals
per population
30
50
70 90
Pm-1
110 130 150
Fig. 4: The number of calculations without best individuals
90
50
80
70
40
60
30
50
40
20
30
20
10
10
0
10
30
50
70 90
Pm-1
0
110 130 150
10
Fig. 5: Average number of the best individuals
per population
30
50
70 90
Pm-1
110 130 150
Fig. 6: The number of calculations without best individuals
It is obvious that for t = 2 the ideal value of mutation probability P m is 1/40. For this setting of the genetic
algorithm operators the greatest number of the best individuals (fitness value = 0) is found and the probability
that no best individual is found is low. For t = 3 it is convenient to set mutation probability Pm to 1/30. Fig. 7-10
show a selection of the calculation results (distributor C, see fig. 1) for the given requirements and for optimal
187
values of genetic algorithm operators, namely Pm = 1/30, selection pressure t = 3. As can be seen, the evolution
process converges to the best solutions. The average fitness value of a population settles around 1.2, the fitness
variance of a population oscillates between 1.7 and 2.5. After completing 8 iterations, the best individuals occur
in a population constantly (see Fig. 9). As a result of selection, the number of the best individuals increases and
oscillates between 80 and 250 of 300 individuals in a population (see Fig. 10).
3,0
5,0
2,5
4,0
2,0
3,0
1,5
2,0
1,0
1,0
0,5
0,0
0,0
0
20
40
60
iteration
80
0
100
Fig. 7: Average fitness values for the progressive distributor C
20
40
60
iteration
80
100
Fig. 8: Fitness variance values for progressive distributor C
350
1,0
300
0,8
250
0,6
200
150
0,4
100
0,2
50
0
0,0
0
20
40
60
iteration
80
0
100
20
40
60
iteration
80
100
Fig. 10 The number of best fitness values for progressive
distributor C
Fig. 9 The best fitness values for progressive distributor C
CONCLUSION
As seen from the example above, appropriate setting of the genetic algorithm operators in the optimization
program is important. The operators affect the quality of the results, which means finding the best individuals
(progressive distributors fulfilling specific requirements) during the evolutionary process. The better the results
are, the greater lubricant reduction in the designed lubrication system can be achieved. According to [11],
mutation probability should be in the range of 0.05 to 0.15. The obtained calculation results (see Fig. 3-6)
indicate that the optimal values of mutation probability, P m = 1/30 a P m = 1/40, are nearer to the lower value of
the mutation probability range. It is also obvious that higher mutation probability must correspond to higher
selection pressure (see Fig. 3-6). The granularity of the observed statistical parameters in Fig. 3-6 could be
improved by increasing the degree of the calculation.
188
REFERENCES
[1]
VEP EK, J. Optimization of Flow Rates in Lubrication Systems with Progressive Distributors by Genetic
Algorithms. In Proceedings of the Fifth International Conference on Soft Computing Applied in
Computer and Economic Environment. Kunovice: European Polytechnical Institute Kunovice, 2007. p.
71-79. ISBN 80-7314-108-6.
[2]
OMERA, P. Genetic Algorithms and Their Applications. (Genetické algoritmy a jejich aplikace). Brno,
2001. 108 p. Inaugural dissertation. Brno University of Technology (in Czech).
[3]
GEN, M. CHENG, Runwei. Genetic Algorithms and Engineering Design. New York: John Wiley &
Sons, Inc., 1997. 411 p. ISBN 0-471-12741-8.
[4]
LI, Z.; HALANG, A. W.; CHEN, G. (Eds.). Integration of Fuzzy Logic and Chaos Theory. BerlinHeidelberg-New York: Springer-Verlag, 2006. 625 p. ISBN 3-540-26899-5.
[5]
VEP EK, J. Optimisation of flow rates in progressive distributors by genetic algorithms. In Proceedings
of International Scientific-Technical conference Hydraulics and Pneumatics ´2007. Wroclaw: Polish
Society Mechanical Engineers and Technicians, 2007. p. 64-73. ISBN 978-83-87982-27-0.
[6]
VEP EK, J. Optimisation of Flow Rates in Lubrication Systems with Progressive Distributors by Genetic
Algorithms. (Optimalizace pr tokových pom r v mazacích systémech s progresivními rozd lova i
pomocí genetických algoritm ). Brno, 2006. 43 p. Thesis. Brno University of Technology (in Czech).
[7]
pondr CMS [online]. [citation 11. 1. 2008]. Available from URL <http://www.spondrcms.cz/contcms.htm>.
[8]
HEROUT, P. Java Language Textbook. (U ebnice jazyka Java). eské Bud jovice: Kopp, 2000. 349 p.
ISBN 80-7232-115-3 (in Czech).
[9]
NEVRLÝ, J.; PAVLOK, B. Methodics of Branched Lubrication Circuits Design with Aid of Modern
Computing Systems. (Metodika návrhu v tvených mazacích obvod s podporou moderních výpo etních
systém ). Brno, 2000. 267 p. Research report. GA R 101/98/0946. Brno University of Technology. (in
Czech).
[10] BIJURDELIMON Zentralschmiersysteme Für reibungslose Bewegung Produkte - Übersicht [online].
[citation 11. 1. 2008]. Available from URL < http://www.delimon.de/deutsch/produkte.html >.
[11] MA ÍK, V.; T PÁNKOVÁ, O.; LAANSKÝ, J. et al. Artificial intelligence. (Um lá intelligence).
Praha: Academia, 2001. 328 p. ISBN 80-200-0472-6 (in Czech).
Address:
Ing. Ji í Vep ek
Technická 2896/2
616 69 Brno
E-mail: [email protected]
189
190
A BRIEF INTRODUCTION TO RECOGNITION OF DEFORMED OBJECTS
Martin Mina ík, Ji í astný, Ond ej Popelka
Abstract: The paper describes methods of deformed object classification. First, methods of syntactic
analysis are described. Second, methods of recognition of deformed objects are described, i.e. use
of string metric and deformation grammar. The principles and algorithms given below have been
used in applications that were developed at Brno University of Technology.
Key-Words: pattern recognition, top-down parsing, bottom-up parsing, string metric, Levenshtein,
Earleys parser, deformation grammar
1. INTRODUCTION
Depending on the used description and the method of evaluation it is possible to divide pattern recognition
methods in two groups.
The first group describes objects with set of numeric characteristic flag methods. Flag methods are not proper
in applications where structural properties of objects are important, because they are lost with transformation in
to the flag spaces. The solution is to describe objects with elementary descriptive properties, so called primitive
and relations between them.
These methods are called structural (syntactic), objects are described with strings (sequences or structures
created with relation follow in). Every symbol matches one descriptive primitive from a set of primitives. The
set can be understood as alphabet of formal language. The task then consists in determination if the word
generated by given grammar matches the string, which describes given object.
Proper course of processing and recognition of digitalized image can be divided into several basic steps:
Image preprocessing
Image segmentation
Objects description
Pattern recognition
In the comparison of pattern recognition algorithms their memory and time requirement are taken in
consideration, which are key to real time application.
2. STRUCTURAL METHODS OF PATTERN RECOGNITION
The simplest way of recognition is template matching. String which represents image is compared with items
of string sets which represents particular template image. A full or partial match (based on a customizing
criterion) is required. This way is simple, efficient and fast. If full description of image is required for
recognition, syntactic analysis is necessary.
2.1 SYNTACTIC ANALYSIS - RECOGNITION WITH THE AID OF GRAMMAR
Syntactic analysis is a process which decides if the string belongs to a language generated by given grammar,
thus object recognition.
regular grammar deterministic finite state automaton is sufficient to analyze regular grammar. This
automaton is usually very simple in hardware and software realization
context-free grammar to analyze context-free grammar it is generally required nondeterministic finite
state automaton with stack
context grammar Useful and sensible syntactic analysis can be done with context-free grammar
with controlled re-writing
There are two basic methods of syntactic analysis:
bottom-up parsing we begin from analyzed string to initial symbol. The analysis begins with empty
191
stack. In case of successful acceptance only initial symbol remains in the stack, e.g. Cocke-YoungerKasami algorithm, which grants that the time of analysis is proportional to third power of string length
top-down parsing we begin from initial symbol and we are trying to generate analyzed string. String
generated so far is saved in the stack. Every time a terminal symbol appears on the top of the stack, it is
compared to actual input symbol of the analyzed string. If symbols are identical, the terminal symbol is
removed from the top of the stack. If not, the algorithm returns to a point where a different rule can be
chosen (e.g. with help of backtracking). Example of top down parser is Earleys Parser [1], [2], which
executes all ways of analysis to combine gained partial results. The time of analysis is proportional to
third power of string length, in case of unambiguous grammars the time is only quadratic. This
algorithm was used in used simulation environment.
2.2 EARLEY PARSER
The Earley parser is a type of chart parser mainly used for parsing in computational linguistics, named after its
inventor, Jay Earley. The algorithm uses dynamic programming.
Earley parsers are appealing because they can parse all context-free languages. The Earley parser executes in
cubic time ( (n3), where n is the length of the parsed string) in the general case, and quadratic time ( (n2)) for
unambiguous grammars. It performs particularly well when the rules are written left-recursively.
In the following descriptions, , , and represent any string of terminals/nonterminals (including the empty
string), X, Y, and Z represent single nonterminals, and a represents a terminal symbol.
Earley's algorithm is a top-down dynamic programming algorithm. In the following, we use Earley's dot
notation: given a production X
, the notation X
represents a condition in which has already been
parsed and is expected.
For every input position (which represents a position between tokens), the parser generates an ordered state set.
, i), consisting of
Each state is a tuple (X
the production currently being matched (X
)
our current position in that production (represented by the dot)
the position i in the input at which the matching of this production began: the origin position
Earley's original algorithm included a look-ahead in the state; later research showed this to have little practical
effect on the parsing efficiency, and it has subsequently been dropped from most implementations.
The state set at input position k is called S(k). The parser is seeded with S(0) consisting of only the top-level
rule. The parser then iteratively operates in three stages: prediction, scanning, and completion.
Prediction: For every state in S(k) of the form (X
Y , j) (where j is the origin position as above),
add (Y
, j+1) to S(k) for every production with Y on the left-hand side.
Scanning: If a is the next symbol in the input stream, for every state in S(k) of the form (X
a , j),
add (X
a , j) to S(k+1).
Completion: For every state in S(k) of the form (X
, j), find states in S(j) of the form (Y
X
, i) and add (Y
X , i) to S(k).
These steps are repeated until no more states can be added to the set. The set is generally implemented as a
queue of states to process (though a given state must appear in one place only), and performing the
corresponding operation depending on what kind of state it is.
2.3 THE WHOLE ALGORITHM OF SYNTACTIC ANALYSIS
a) If not all the strings have been analyzed, read a new string and proceed with step 2, otherwise proceed
with step 7
b) Perform the bottom-to-top analysis for the class selected
c) If the string belongs to the language of the grammar of selected class, proceed with step 6
d) If the number of string rotations is less than the string length, rotate the string and proceed with step 2,
otherwise proceed with step 5
e) If the number of string rotations is less than (360 / angle step), rotate the object by the angle step given
and proceed with step 2
f) enter the result and proceed with step 1
g) write the message about pattern recognition
192
The syntactical analyzer was designed for the left linear grammar.
String rotation: This mean shifting the last terminal symbol to the beginning
abcde
1.rotace
eabcd
Object rotation: This means rotating the object by a given angle and thus obtaining a different string.
If we need to classify N objects, we must create N classes, N grammars for them, and the respective languages
L(G1), L(G2), ..., L(GN). For example, if grammar Gx generates words containing only one terminal symbol b,
then all the objects containing just this symbol b will belong to class X pertaining to this grammar. Objects
containing more than one symbol b will be further analyzed using the remaining grammars. In case that no
grammar which corresponds to the given string is found, the object will be suppressed.
In case that primitives marking single edge segments occur, the grammar is very sensitive to small mistakes in
edge detection. This disadvantage is resolved by described methods below (string metric and deformation
grammar).
3. METHODS OF RECOGNITION OF DEFORMED OBJECTS
When designing a syntactic analyzer it is useful to assume random influences, e.g. image deformation.
3.1 METHODS FOR DETERMINATION OF DISTANCE BETWEEN ATTRIBUTE DESCRIPTION OF
IMAGES (STRING METRIC)
One method, which is used for recognition of structure given unknown image, consists in comparison of
structural representation of string, tree and relation graph with representation of class image pattern. This method
is necessary in tasks where number of training sample is not sufficient for grammar inference, or when every
image can be considered as class image prototype. For determining distance between two strings, specific
methods can be used to determine the distance between attribute descriptions of image (string metric).
3.2 DESCRIPTION OF SOME METHODS
From available string metric methods for strings distance determination, following methods were analyzed:
Hammings distance Hd(s,t) [6], Levenshtein distance Ld(s, t) [6], Daemer distance Dd(s, t) [6],, Jaccard distance
Jd [6], Minkow distance Md(s,t,power) [6], and Needleman-Wunsch method [6].
Hamming and Jaccard methods are unsuitable for considered application area because they are not able to
compare different lengths of strings, which is an indispensable requirement for this application (length of strings
can be very different from strings etalons owning to assorted image errors, which originate in image
preprocessing).
The use of Minkow method is very problematic because dynamically computed rating of price for operation
according to distance of compared signs (ASCII table distance) can be the cause of significant fluctuation of
final counted distance.
The remaining three methods Levenshtein, Damerau and Needleman-Wunsch are usable for determining the
distance between attribute descriptions of image. But in Damerau method is adjacent sign switch operation for
considered application useless and it complicates implementation, too. In Needleman-Wunsch methods is the
possibility of different operation price also almost useless but it does not complicate implementation either.
Based on the analysis of the methods mentioned above, Levenshtein [5] and Needleman-Wunsch [4] methods
were chosen as the most suitable for determining the distance between attribute descriptions of image methods.
None of the mentioned algorithms are invariable towards rotation, thereby it is necessary to rotate characters of
the string, whereas in each step the string is rotated by one sign. Then the shortest detected distance is chosen.
The time complexity of these algorithms is increased depending on the length of rotating string.
3.3 LEVENSHTEIN DISTANCE
Levenshtein distance Ld(s, t) [6], is defined as the rate of similarity between two strings s and t. Levenshtein
distance Ld(s, t) is a number of insertion, deletion and substitution required for transforming string s to string t.
193
3.4 NEEDLEMAN-WUNSCH METHOD
Needleman-Wunsch method [6], calculates the distance between two strings by using operation price matrix.
The result depends on prices of operations which can vary for different operations. Allowed operations are
insertion, deletion and substitution. For each operation only one price matrix exists. We can define different
evaluation for each cell inside the price matrix.
Needleman-Wunsch method offers similar representation of distance between two strings like the Levenshtein
distance. It can be used for different length of compared strings; the main difference is in different operation
rating but this difference is insignificant for considered application. Time complexity of this method is similar to
Levenshtein method.
4 SYNTACTIC ANALYSES WITH ERROR CORRECTION
If we conduct syntactic analysis of a string which describes structural deformed object, it will apparently not be
classified into given class because of its structural deformation. The solution is to enhance the original grammar
with rules which describes errors deformation rules [3] that cover up every possible deformation of object.
Then the task is changed to find a non-deformed string, whose distance from analyzed string is minimal.
Compared to the previous method, which is using only some metric, is this way more informed, because it uses
all available knowledge about the classification targets it uses grammar.
4.1 ENHANCED DEFORMATION GRAMMAR
Input: context free or regular grammar G = (VN, VT, P, S)
Output: Enhanced deformation grammar G = (VN, VT, P, S), where P is set of weighted rules
Step 1:
V N'
VN
VT
'
T
S'
E B | b VT
V
Step 2:
if is in P rule
A
b
0 1
b ...
1 2
b
m 1 m
m
;m
0;
VN'
l
bi
VT ; i 1,2,..., m; l
0,1,..., m
then into P add rule
A
0
Eb1 1 Eb 2 ...
m 1
Ebm
m
With weight = 0
Step 3:
Into P add following rules with weight according to chosen metric, used Levenshtein distance L, weighted
Levenshtein distance - w, weighted metric - W
Rule
'
S
S'
Ea
S
Sa
a
Ea
b
Ea
Ea
bEa
L
0
w
0
W
0
For
-
1
wl
I' a
a VT'
0
0
0
a VT
1
wS
S a, b
a VT , b VT' , a
1
wD
wl
Da
I a, b
a VT
1
b
a VT , b VT'
Rules of types b), d), e) and f) are called deformation rules. Syntactic analyzer with error correction works with
enhanced deformation grammar. This analyzer seeks out such deformation of input string, which is linked with
the smallest sum of weight of deformation rules. G is ambiguous grammar, i.e. its syntactic analysis is more
complicated. A modified Earley parser was used for syntactic analyses with error correction. Moreover, this
parser accumulates appropriate weight of rules which were used in deformed string derivation according to the
grammar G.
194
4.2 MODIFIED EARLEY PARSER
Input:
Enhanced deformation grammar G
Input string w b1b2 ...bm
Output:
Lists I0, I1,..., Im for string w
Distance d of input string from template string
Step 1: Create I0.
For every rule S
a)
P ' add into I0 field S '
'
,0, x
Execute until it is possible to add fields into I0. If it is in I0 field
A
B ,0, y then add for every
Z
,0, z into I0.
rule B
field B
Step 2: Repeat for j = 1, 2,..., m
a)
For every field in Ij-1 in form of B
a , i, x such that a = bj, add into Ij
a , i, x
Field B
Execute B and C until no more fields can be added into Ij
If field A
, i, x is in Ij and field B
A , k , y in Ii, then
If exists field in form of B
A
value x + y
If does not exist, then add new field B
For every field in form of
A
, k , z in Ij, then if x + y < z replace in this field value z with
A
, k, x
y
B , i , x in Ij add for every rule B
Z
field
B
, j, z
Step 3:
'
If the field S
,0, x is in Im, then string w is accepted with distance weight x
String w (or if you like his derivation tree) is gained by omitting all deformation rules from derivation of string
w.
5. RESULTS
5.1 RESULT OF STRING METRIC METHODS
In table 1 and table 2 there are total results for Levenshtein and Needleman-Wunsch method [6]. Final times of
implemented algorithms are approximate only for results comparison. In gradual computation of distance too
much excessive information is stored, which is important only for simulation environment. This excessive
information leads to greater memory requirements and higher CPU load. If this excessive information is omitted,
the effectiveness of these recognition methods would be higher.
Levenshtein
scene
time[s]
Classification
output file
correct incorrect nonidentif.
depth
percentile
1
2
3
4
5
6
7
8
9
10
11
12
0,13
0,15
0,13
0,11
0,1
0,05
0,1
0,06
0,23
0,11
0,251
0,09
3
4
3
4
2
3
2
3
4
3
4
4
0
0
0
0
0
0
0
0
0
0
0
0
50
50
50
50
50
50
50
50
50
50
50
50
2
1
2
0
0
0
0
0
1
0
1
0
0
0
0
1
1
0
1
0
0
2
0
1
Scene_01_Lev_0_50.bmp
195
1
2
3
4
5
6
7
8
9
10
11
12
0,13
0,191
0,16
0,13
0,11
0,05
0,11
0,06
0,251
0,11
0,25
0,08
3
3
3
4
2
3
2
3
2
3
2
3
1
1
1
1
0
0
0
0
0
0
0
0
1
1
1
0
1
0
1
0
3
2
3
2
0
0
0
0
0
0
0
0
0
0
0
0
60
60
60
60
60
60
60
60
60
60
60
60
1
2
3
4
5
6
7
8
9
10
11
12
0,13
0,16
0,12
0,15
0,141
0,05
0,11
0,05
0,241
0,11
0,221
0,09
3
3
3
3
2
3
2
3
2
3
2
3
1
1
1
1
0
0
0
0
0
0
0
0
1
1
1
1
1
0
1
0
3
2
3
2
0
0
0
0
0
0
0
0
0
0
0
0
70
70
70
70
70
70
70
70
70
70
70
70
Table 1. Test results of Levenshtein algorithm
Needleman-Wunsch
scene
time [s]
classification
correct
incorrect
nonidentif.
1
2
3
4
5
6
7
8
9
10
11
0,14
0,18
0,17
0,131
0,14
0,05
0,12
0,05
0,28
0,121
0,28
4
4
4
4
3
3
3
3
2
3
1
1
1
1
1
0
0
0
0
2
1
3
12
0,1
3
2
output file
depth
percentile
0
0
0
0
0
0
0
0
1
1
1
Scene_01_NW_0_90.bmp
0
0
0
0
0
0
0
0
0
0
0
90
90
90
90
90
90
90
90
90
90
90
0
0
90
Table 2. Test results of Needleman-Wunsch algorithm
5.2 RESULT OF ENHANCED DEFORMATION GRAMMAR
Results are in table 3. With regard to imperfect debugging of this algorithm, the time aspect is not accurate;
numbers are here only for orientation. But the final time will be longer then the time of syntactic analysis of
simple grammar due to ambiguity of deformation grammar and larger set of deformation rules. The speed of
analysis can be improved by uses of different parser (e.g. hybrid LRE parser).
196
Enhanced deformation grammar
Scene and number of
objects
#001, 2 objects
#002, 2 objects
#003, 3 objects
Classification
correct
2
2
3
Incorrect
0
0
0
nonidentif.
0
0
0
time [ms]
112
158
193
Table 3. Test results of Enhanced deformation grammar
6. CONCLUSION
String metric methods, if their parameters are adjusted correctly (depth, percentile), offer good speed of
classification and relatively good rate of successfully identified objects; however, false object identification can
occur.
Enhanced deformation grammar provides higher rate of successfully identified objects, and almost prevents false
object identification, therefore removes the biggest disadvantage of recognition by string metric methods at the
cost of slightly greater computation time.
ACKNOWLEDGEMENT
MSM 0021630529 Intelligent Systems in Automation (Research design of Brno University of
Technology)
No 102/07/1503 Advanced Optimisation of Communications Systems Design by Means of Neural
Networks. The Grant Agency of the Czech Republic (GACR)
MSM 6215648904/03 Development of relationships in the business sphere as connected with changes in
the life style of purchasing behaviour of the Czech population and in the business environment in the
course of processes of integration and globalization (Research design of Mendel University of
Agriculture and Forestry in Brno)
7. REFERENCES
[1]
Wikipedia, Earley parser, available from: <http://en.wikipedia.org/wiki/Earley_parser>
[2]
Wikipedia, Left recursion, available from: <http://en.wikipedia.org/wiki/Left_recursion>
[3]
elezný, M.: Strukturální metody rozpoznávání, available from: <http://artin.zcu.cz/courses/smr/>
[4]
Wikipedia, Needleman-Wunsch algorithm, available from:
<http://en.wikipedia.org/wiki/NeedlemanWunsch_algorithm>
[5]
Wikipedia, Levenshtein distance, available from <http://en.wikipedia.org/wiki/Levenshtein_distance>
[6]
ASTNÝ, J.: Nontraditional Methods and Algorithms for Object Recognition of Technological
Scene, VUT Brno, 2005, No.198, pp.1-30, ISSN 1213-418X
[7]
ASTNÝ, J., Mina ík, M.: Object Recognition by Means of New Algorithms, article in conference
proceedings, ICSC - International Conference on Soft Computing Applied in Computer and Economic
Environment, Kunovice, 2006. pp.99-104, ISBN 80-7314-084-5,
[8]
CHAPMAN, S.: String Metrics, University of Sheffield, [online]. 2005. available from:
<http://www.dcs.shef.ac.uk/~sam/stringmetrics.html>.
[9]
ASTNÝ, J., MINA ÍK, M.: A Brief Introduction to Image Pre-Processing for Object Recognition.
article in conference proceedings, ICSC - International Conference on Soft Computing Applied in
Computer and Economic Environment. Kunovice, 2007. pp. 55-64. ISBN: 80-7314-084-5.
ADDRESS:
Ing. Martin Mina ík
Technická 2, 616 69 Brno,
Czech Republic,
http://www.fme.vutbr.cz/
197
Doc. RNDr. Ing. Ji í astný, CSc.
Czech Republic,
Ond ej Popelka
Czech Republic
198
SOME THOUGHTS ON INNER SYMMETRIES OF PROBABILITY THEORY AND
EMERGENCE OF KLEIN'S QUARTIC IN FUNDAMENTAL PHYSICS
Ale Gottvald
Academy of Sciences of the CR
Abstract: Probability theory features as the internal symmetries of physical laws, acting in an
intrinsically 6-dimensional hyperspace. Concerning symmetries, classical thermodynamics and
Klein's Erlangen program involve the same underlying idea. Probability theory is an exceptional
structure, closely linked to a unique Triality symmetry and other exceptional structures in nature
(symmetric group S6, Platonic solids, (2, 3, 7)-triangular group and a correspondin tessellation of
a hyperbolic space, exceptional Lie groups, etc.). Exponential mapping of statistical physics is
associated with Klein's quartic curve, an extremal Hurwitz surface whose 168 automorphisms may
be related to Standard model of particle physics and to a highly composite number (42) of special
importance for fundamental physics.
Key-words: Probability theory, inner symmetries, Möbius group, Triality, hyperbolicity, Poincaré
disc, modularity, triangular groups, (2, 3, 7)-tessellation, platonicity, Hurwitz automorphism
theorem, Klein quartic curve, standard model, unification of physical theories.
1. INTRODUCTION: SYMMETRIES, ERLANGEN PROGRAM, THERMODYNAMICS
The concept of symmetries is fundamental to modern physics and mathematics. In the spirit of Klein's Erlangen
program, to characterize a geometry, we shall investigate those operations (groups of symmetries, automorphism
groups), which preserve some inner incidence structure (topological structure) of the system in question.
However, deeply in the wisdom of classical thermodynamics is also lurking an idea of those operations that
preserve the inner incidence structure of the system. This puts a clear demarkation line between the reversible
and irreversible processes in thermodynamics. Recall that if we integrate the variation of a conserved quantity A
(work) along a closed thermodynamic cycle, the result of the integration generally depends on the form of the
cycle. This is because some changes of the inner structure of the system are taking place, a phenomenon
atributed to heat variation. Mathematically, the Pfaffian form which is integrated along the cycle is not a total
differential. Fortunately, there always exists an integrating factor (1/ T ) of the heat variation which converts the
Pfaffian form to a total differential. Consequently, the result of the integration becomes independent of the form
of the thermodynamic cycle. This fundamental trick allows us to distinguish the work and the heat in
thermodynamics. The system is conservative and the integration along a close cycle of a total differential is zero,
as need be. All of this is well known from classical thermodynamics. Thus, we can imagine the symmetry
operations as some geometric objects, basically of the same type as those closed thermodynamic cycles
associated with a conservative system. This image should be emphasized: both the Klein Erlangen program and
classical thermodynamics actually involve the same fundamental underlying idea. [1]
2. PROBABILITY VS INNER SYMMETRIES
Ubiquity and importance of probability theory in physics is well recognized. We conjecture however that even
much more physics is hidden in the very structure of the probability theory than we usually presume. What is not
generally recognized and appreciated are the inner symmetries of the probability theory itself and their deep
impact on possible structure of the physical laws, - as they appear from our human's perspective. Probability
theory is an exceptional structure which features as the inner symmetries of the underlying theories (statistical
physics, quantum physics, fractal physics, information physics, chaos theory, etc.). In other words, we postulate
that a tremendeous success and ubiquity of probability theory is based on something deeper: its inner structure
(associated symmetries) encodes the very structure of reality as we can logically percept it. On the ultimate
gnoseologic level, our brains are evolutionary equipped with a probabilistic inferential system which is
instrumental in creating our picture of the Universe as we percept it.
Consequently, what are the inner symmetries of the probability theory itself? More specifically, can we associate
the probability theory with some particular symmetries, in terms of some specific symmetry or automorphism
groups? And what is the prominent dimension of such a probabilistic space on which the symmetry group acts?
199
3. PROBABILISTIC HYPERSPACE IS SIX-DIMENSIONAL
It shows that the fundamental arena (hyperspace) of physics takes the form of a cartesian product of two spaces,
M P . Here, M is basically a common 4-dimensional Minkowski space-time. The symmetries of M are those
of special relativity, associated with the Lorentz group (equivalently Möbius group). Statistical physics has it
that the knowledge of conserved quantities on the system uniquely determines the most-entropic probability
distributions (and their characteristic partition functions).
Conventionally, 7 conserved quantities are attributed to the symmetries of the space-time M (energy, 3impulse, 3-momentum). These 7 conserved quantities feature in the exponent of corresponding
probabilistic distributions (Gibbs method, Jaynes MaxEnt method). What conserved quantities correspond
to the three boosts (three hyperbolic rotations in the t-x, t-y and t-z planes)? The answer is -- the
relativistic center of rest-mass at time zero. However, the center of rest-mass is an intrinsically
probabilistic quantity.
The fundamental conserved quantity (invariant) with respect to Möbius group (Lorentz group) is the
cross-ratio, which is an expression of Bayes' rule [1]. Both these non-Abelian groups are 6-parametric and
lead to conservation of 6 quantities associated with 6 rotations in the four-dimensional Minkowski
space-time M, and also equivalent to three rotations and three boosts in 3-dim Euclidean space.
The full symmetry group of the Minkowski space-time M is the Poincaré group. It is a 10-parametric nonsimple group, actually a semi-direct product of the Lorentz group with the abelian group of four
translations along the orthogonal axes in 4-dim M.
4. PROBABILITY THEORY IS AN EXCEPTIONAL STRUCTURE
The structure of the internal symmetry space P is exceptional. It takes its exceptionality from probability theory,
which in turn corresponds to exceptionality of many other objects. Which fundamental features define this
probabilistic structure? First of all, its natural dimensionality is six. Why just the number 6 is so exceptional?
One view-angle stems directly from the fact that just 6 conditional probabilities are involved in the product rule
and the sum rule of probability theory. This in turn is a consequance of the associative and commutative laws
which govern Boolean algebra given some three propositions, say A, B, C. (Recall a proof by R. T. Cox and E. T.
Jaynes, see [2].) However, it also shows that six is the only dimensionality where unique features of some
symmetry groups meet: First, there is something very special behind the symmetric group S 6 , actually the
permutation group of six objects (of the order 6!=720). It is the only symmetric group which shows a special
phenomenon called the "outer automorphism" [3]. Second, we also need a set of 3 labels (say, A, B, C) for
labeling our six conditional probabilities. And a symmetric group S 3 , the permutation group of 3 objects (of
order 3!=6), is actually behind the Triality symmetry which is a remarkable symmetry of the group Spin(8).
Recall that the group Spin(8) is a double-cover of the Lie group SO(8)=D(4), which stands just at the cross-road
between a regular and an exceptional family of the Lie groups [3]. The group Spin(8) is of the order 28, but its
outer automorphism group is exceptionally large it is isomorphic to S 3 (of the order 6). As it shows, the outer
S 6 and the Triality symmetry of Spin(8) just remarkably meet like in a lock. We conjecture
that just the unique "locking phenomenon", where the exceptionality of S 6 meets the Triality symmetry
automorphisms of
associated with Spin(8) and
S 3 , supports the deep reasons why the number 6 is so special. Recall also that just 6
parameters define the moduli space of the Möbius group, the Lorentz group and the complex projective
line CP1 1. Summarizing the above,
Fundamental to probability theory is a projective space of dimension 6 (a complex projective line).
Fundamental to probability theory and its inner symmetries is the concept of Triality which is beyond the
unique symmetry of the Lie group D(4), its double-cover Spin(8) and the enigmatic exceptional family of
the Lie groups (G2, F4, E6, E7, E8).
The concept of Triality is basically an expression of the associative and commutative laws which govern
the Aristotle logic of 3 propositions (A, B, C) in six conditional probabilities involved in the governing
equations of probability theory (the product rule and the sum rule).
The Triality symmetry is actually the hidden way how discretness and quantization enters the stage.
Probability theory is the "glue" which attach a discrete nature of particle physics (Standard model) to the
intrinsically continuous Minkowski space-time M, via the probabilistic space P of dimensionality six.
The Triality acts "orthogonaly" at every point of the Minkowski space-time M, in agreement with
Coleman-Mandula theorem. The discreteness is associated with a modular group acting in P a discrete
subgroup of the Möbius group, actually a discrete version of the Lorentz group.
200
5. ENTERING HYPERBOLIC SPACE
The probabilistic space is an intrinsically hyperbolic space (see C. C. Rodríguez [4]). Its fundamental invariant
quantity, the cross-ratio, may be interpreted as the Bayes formula in disguise [1]. A logarithm of this invariant
quantity is additive and defines the hyperbolic distance. The hyperbolic distance may be associated with a
generalized potential, which (after averaging in the spirit of the MaxEnt method) may be interpreted as an
entropy. We employ a Poincaré disc model of the hyperbolic geometry, which is isomorphic to the upper-half
plane of the complex plane C .
6. TRIANGULAR GROUPS, (2, 3, 7)-TESSELLATION
The inner symmetries of the probability theory may be investigated using a tangible geometric model - a
triangulal group of the type (p, q, r), where the angles / p , .
/ q , / r define a basis triangle (a
fundamental domain of the tessellation). In general, every triangular group corresponds to a particular
tessellation (tiling) of a euclidean, spherical or hyperbolic plane. For most of the indices p, q, r, the triangular
group naturally lives in a hyperbolic plane. This is the case of the triangular group (2, 3, 7) which is particularly
important in our context.
Fig. 1: Poincaré disc as a model of hyperbolic geometry, and a tessellation associated with a triangular group (2, 3, 7). On the right, a portion
of the tessellation is indicated which is used for constructing the Klein quartic curve. For details, see [5, 7].
7. A PHYSICAL MOTIVATION
The tessellation (2, 3, 7) may physically be motivated as follows: Imagine all probabilities involved as some
geometric probabilities. Consider a probability distribution, taken at a point of the Minkowski space-time. This
probability may be factorized into three conditional probabilities using Bayes' formula. We can associate these
three conditional probabilities with three complex numbers (three angles or three Euclidean distances) in a
Poincaré disc. A particular Möbius transformation always allows us to map these three points to vertices of an
equilateral triangle and introduce a tessellation made of equilateral triangles. Then we may baricentrically define
a dual tessellation made of regular heptagons, which correspond to 7 conserved quantities in the Minkowski
space-time. Thus, the exponential mapping of the statistical physics and space-time symmetries naturally
translates to a (2, 3, 7) or (3, 7) tessellation of a hyperbolic model.
8. EMERGENCE OF THE KLEIN QUARTIC.
The tessellation of the type (2, 3, 7) is exceptional not only from a point of view of statistical physics and
corresponding exponential family. It is further singled out by the Hurwitz automorphism theorem. It also gives
rise to a "magic" number 42, which appears very important in quantum gravity and fundamental forces. Let X be
a hyperbolic space of a genus g. The area of X is (see Gauss-Bonnet theorem):
A( X )
4 ( g 1)
(1)
201
Let D be a fundamental domain in X (a triangle defined by the integers p, q, r). The area of D is
A( D )
(1
1
p
1
q
1
)
r
(2)
Let us vary the integers p, q, r, and seek the minimum of A(D). The fundamental domain takes its minimum
value for the numbers (p, q, r) = (2, 3, 7), and namely
1
1
2
1
3
1
7
1
42
(3)
Consequently, the hyperbolic surface allows the maximum number of triangular domains for the tessellation of
the type (2, 3, 7), and its extremal number is:
A( X )
168 ( g 1)
A( D )
(4)
The extremal surface which reaches this maximum possible number of symmetries is called the Hurwitz surface.
As (4) implies, a surface of genus g=3 can possess a maximum possible number of 336 automorphisms
(including reflexions). When we exclude the reflexions, the maximum number of automorphisms is 168. The
Klein quartic is just the surface of genus g=3 which shows the extremal number of 168 automorphisms (336
automorphisms including reflections). [2, 5]. Notice natural emergence of the important number 42 in the theory,
which features here as the so-called hyperbolic deficiency (3). Remarkably,
42 = 2.3.7 is the second sphenic number, and every sphenic number has just 8 divisors, in particular, for
42 the divisors
1, p, q, r , pq, pr , qr , pqr
1, 2, 3, 7, 6, 14 , 21, 42 .
The Klein quartic is a generalized Platonic solid, associated with the (2, 3, 7)-tiling. The faces of the Klein
quartic are regular heptagons. Thus, similarly to five classical Platonic solids, it is the object with the utmost
symmetry which is possible for the constraints imposed (faces of regural p-gons). Concerning properties,
importance, ubiquity and beauty of the Klein quartic in mathematics, see esp. [5].
Starting from Heisenberg's insight and his concept of isospin, it is more and more obvious that all elementary
particles are just different transformation states of an individual fundamental particle. Moreover, the group of the
transformations involved is a modular group. Analogically, all 168 triangular domains of the Klein quartic are
just different modular transformations of the fundamental domain.
9. CONCLUSION
In summary, when we adopt validity of probability theory and statistical physics, the geometric picture of the
internal symmetries involved leads to a hyperbolic space and culminates with the Klein quartic curve. We
conjecture that the Standard Model od particle physics is lurking just in the incidence structure, modularity and
hyperbolic geometry of the Klein quartic. Various aspects of this conjecture has recently and independently been
elaborated also by Mohammed El Naschie [6], John Baez [7] and F. "Tony" Smith [8]. Our approach emphasizes
that the probability theory itself is behind the rationale of the emergence of the Klein quartic in fundamental
physics.
REFERENCES:
[1]
GOTTVALD, A. "Physics from Probability Theory". In: Proc. of the NTF 2007Conf. (P. Dobis & J.
Brüstlová, eds.), Brno, 2007, pp. 161-166
[2]
JAYNES, E. T. "Probability Theory - The Logic of Science". Cambridge University Press, Cambridge,
2003
[3]
See "Wikipedia, The Free Encyclopedia". On/line: http://en.wikipedia.org. Articles on: Symmetries in
physics, Klein's Erlangen program, Lie groups, Möbius group, Cross-ratio, Projectice spaces, Hyperbolic
spaces, Poincaré disc, Triangular groups, Tesselations, PSL(2, 7), PSL(3, 2), Klein quartic.
[4]
RODRÍGUEZ, C. C. "From Euclid to Entropy". In: In Maximum Entropy and Bayesian Methods (W. T.
Grandy & L. H. Schick, eds)), Laramie, WY, 1990; Kluwer, Dordrecht, 1991, pp. 343-348. On-line:
http://omega.albany.edu:8008/
202
[5]
[6]
[7]
[8]
[9]
The Eightfold Way: The Beauty of Klein's Quartic Curve (Silvio Levy, ed.). MRSI Book, Vol. 35,
Cambridge University Press, Cambridge, 1999. On/line:
http://www.msri.org/publications/books/Book35/contents.html
EL NASCHIE, M. S. "An elementary model based method for determining the number of possible Higg
bosons in the standard model". Chaos, Solitons & Fractals 26 (2005), 3, pp. 701-706
Klein's quartic curve - see also John Baez - http://math.ucr.edu/home/baez/klein.html
SMITH,
F.
"TONY":
"Physics
of
the
Klein
Quartic".On/line:
http://www.valdostamuseum.org/hamsmith/KQphys.html
Ing. Ale Gottvald, CSc.
Institute of Scientific Instruments, v.v.i.
Academy of Sciences of the CR
Královopolská 147
612 64 Brno
E-mail: [email protected]
203
204
ÛÈ×ÍÌÛÒÝÛ ÑÚ ÞÑËÒÜÛÜ ÍÑÔËÌ×ÑÒÍ ÚÑÎ Ô×ÒÛßÎ
Ü×ÍÝÎÛÌÛ ÍÇÍÌÛÓÍ
Ö¿®±³3® Þ¿g¬·²»½ô Ö±-»º Ü·¾´3µ
Þ®²± Ë²·ª»®-·¬§ ±º Ì»½¸²±´±¹§
ß¾-¬®¿½¬æ ß ´±¬ ±º °¿°»®- ¿®» ¼»ª±¬»¼ ¬± ¬¸» ·²ª»-¬·¹¿¬·±² ±º ¬¸» °®±¾´»³ ±º °®»-½®·¾»¼ ¾»¸¿ª·±®
¿¬ ´»¿-¬ ±²» -±´«¬·±² ±º ¼·-½®»¬» »¯«¿¬·±²- ¸¿ª·²¹ °®»-½®·¾»¼ ¿-§³°¬±¬·½ ¾»¸¿ª·±® ¸¿ª» ¾»»² ·²¼·ó
½¿¬»¼ò Ò±¬ -± ³«½¸ ¿¬¬»²¬·±² ¸¿- ¾»»² °¿·¼ ¬± ¬¸» °®±¾´»³ ±º ¼»¬»®³·²·²¹ ½±®®»-°±²¼·²¹ ·²·¬·¿´
Ì¸» ·²·¬·¿´ ¼¿¬¿ ³»²¬·±²»¼ ¿®» ½±²-¬®«½¬»¼ ©·¬¸ ¬¸» «-» ±º ½±²ª»®¹»²¬ ³±²±¬±²» -»¯«»²½»-ò
Õ»§ ©±®¼-æ Í§-¬»³ ±º ´·²»¿® ¼·-½®»¬» »¯«¿¬·±²ô Þ±«²¼»¼ -±´«¬·±²-ô ×²·¬·¿´ ¼¿¬¿ò
ß ´±¬ ±º °¿°»®- ¿®» ¼»ª±¬»¼ ¬± ¬¸» ·²ª»-¬·¹¿¬·±² ±º ¬¸» °®±¾´»³ ±º °®»-½®·¾»¼ ¾»¸¿ª·±® ±º -±´«¬·±²- ±º
¼·-½®»¬» »¯«¿¬·±²- ¸¿ª·²¹ °®»-½®·¾»¼ ¿-§³°¬±¬·½ ¾»¸¿ª·±® ¸¿ª» ¾»»² ·²¼·½¿¬»¼ò Ò±¬ -± ³«½¸ ¿¬¬»²¬·±² ¸¿¹¿° º±® ¬¸» ½¿-» ±º ¬©± ´·²»¿® »¯«¿¬·±²- ·² ¬¸·- °¿°»®ò Ì¸» ·²·¬·¿´ ¼¿¬¿ ³»²¬·±²»¼ ¿®» ½±²-¬®«½¬»¼ ©·¬¸ ¬¸»
«-» ±º ½±²ª»®¹»²¬ ³±²±¬±²» -»¯«»²½»-ò
Ô»¬ «- ½±²-·¼»® ¿ -§-¬»³ ±º ¬©± ¼·-½®»¬» »¯«¿¬·±²«øµ÷ ã Ú øµå «øµ÷÷
øï÷
©·¬¸ Ú æ Ò ø¿÷ Îî ÿ Îî ô « ã ø«ï å «î
«øµ÷ ã «øµ õ ï÷ «øµ÷ô µ î Òø¿÷ ã º¿å ¿ õ ïå æ æ æ ¹ô ¿ î Ò
¿²¼ Ò ã ºðå ïå æ æ æ ¹æ
Ì±¹»¬¸»® ©·¬¸ ¼·-½®»¬» »¯«¿¬·±² øï÷ ©» ½±²-·¼»® ¿² ·²·¬·¿´ °®±¾´»³æ º±® ¿ ¹·ª»² - î Ò ¼»¬»®³·²» ¿ -±´«¬·±²
« ã «øµ÷ ±º øï÷ -¿¬·-º§·²¹ ¬¸» ·²·¬·¿´ ½±²¼·¬·±²
«ø¿ õ -÷ ã «- î Îî
øî÷
©·¬¸ ¿ °®»-½®·¾»¼ ½±²-¬¿²¬ «µ
º«µ ¹ï
µãð ©·¬¸ « ã «ø¿ õ - õ µ÷ô ·ò»òô
«ð ã «- ã «ø¿ õ -÷å «ï ã «ø¿ õ - õ ï÷å æ æ æ å «² ã «ø¿ õ - õ ²÷å æ æ æ
-«½¸ ¬¸¿¬ º±® ¿²§ µ î Òø¿ õ -÷ »¯«¿´·¬§ øï÷ ¸±´¼-ò Ì¸» »¨·-¬»²½» ¿²¼ «²·¯«»²»-- ±º ¬¸» -±´«¬·±² ±º ¬¸» ·²·¬·¿´
°®±¾´»³ øï÷ô øî÷ ·- ±¾ª·±«- ¿²¼ ·- ¿ ½±²-»¯«»²½» ±º °®±°»®¬·»- ±º ¬¸» º«²½¬·±² Ú ò ×º Ú ·- ½±²¬·²«±«- ©·¬¸
®»-°»½¬ ¬± ·¬- -»½±²¼ ¿®¹«³»²¬ô ¬¸»² ·²·¬·¿´ °®±¾´»³ øï÷ô øî÷ ¼»°»²¼- ½±²¬·²«±«-´§ ±² ·²·¬·¿´ ¼¿¬¿ò
Ô»¬ ¾· øµ÷ô ½· øµ÷ô · ã ïå
Ò ø¿÷ô -«½¸ ¬¸¿¬ ¾· øµ÷ ä ½· øµ÷ º±® »ª»®§ µ î Ò ø¿÷ò
Ò ø¿÷ Îî ©¸»®»
ºøµå «÷ æ µ î Ò ø¿÷å ¾· øµ÷ ä «· ä ½· øµ÷å · ã ïå î¹æ
¿²¼
ÿøµ÷ ã º« æ ¾· øµ÷ ä « ä ½· øµ÷å · ã ïå î¹å
©¸»®» µ î Ò ø¿
Þ· øµå «÷
Ý· øµå «÷
¾· øµ÷å
½· øµ÷å
«·
«·
· ã ïå îå
· ã ïå î
îðë
±² Ò ø¿÷
Îî ô ¿²¼ ¿«¨·´·¿®§ -»¬-
·
Þ
ã ºøµå «÷ æ µ î Ò ø¿÷å Þ· øµå «÷ ã ðå Þ¶ øµå «÷
ðå Ý¸ øµå «÷
ðå º±® ¿´´ ¶å ¸ ã ïå îå ¿²¼ ¶ êã ·¹å
·
Ý
ã ºøµå «÷ æ µ î Òø¿÷å Ý· øµå «÷ ã ðå Þ¶ øµå «÷
ðå Ý¸ øµå «÷
ðå º±® ¿´´ ¶å ¸ ã ïå îå ¿²¼ ¸ êã ·¹å
º±® · ã ïå
ºøµå «÷ æ µ î Òø¿÷å Þ· øµå «÷ â ðå Ý¶ øµå «÷ ä ðå ·å ¶ ã ïå î¹
¿²¼ º±® à
ï
Þ
à
Å
î
Þ
Å
ï
Ý
Å
î
Ýæ
Ì¸» º±´´±©·²¹ ¬¸»±®»³ ½±²½»®²·²¹ ¿-§³°¬±¬·½ ¾»¸¿ª·±® ±º -±´«¬·±²- ±º ¬¸» »¯«¿¬·±² øï÷ ·- ¿ °¿®¬·½«´¿® ½¿-»
±º ¿ ³±®» ¹»²»®¿´ ®»-«´¬ ·² Åçô Ì¸»±®»³ îÃò
Ì¸»±®»³ ïòï Ô»¬ ¾· øµ÷å ½· øµ÷å ¾· øµ÷ ä ½· øµ÷ô · ã ïå î
Îî ò ×ºô ³±®»±ª»®ô Ú ·- ½±²¬·²«±«- ©·¬¸ ®»-°»½¬ ¬± ·¬- -»½±²¼ ¿®¹«³»²¬ô
Ú· øµå «÷
º±® »ª»®§ · ã ïå î ¿²¼ »ª»®§ øµå «÷ î
·
Þô
·
Ýô
¾· øµ õ ï÷ õ ¾· øµ÷ ä ð
¿²¼
Ú· øµå «÷
º±® »ª»®§ · ã ïå î ¿²¼ »ª»®§ øµå «÷ î
Ò ø¿÷ ¿²¼ Ú æ Ò ø¿÷ Îî ÿ
½· øµ õ ï÷ õ ½· øµ÷ â ð
¬¸»² ¬¸»®» »¨·-¬- ¿² ·²·¬·¿´ °®±¾´»³
«ø¿÷ ã « î ÿø¿÷å
©·¬¸ «· î ø¾· ø¿÷å ½· ø¿÷÷ô · ã ïå î -«½¸ ¬¸¿¬ ¬¸» ½±®®»-°±²¼·²¹ -±´«¬·±² « ã « øµ÷ ±º ¬¸» -§-¬»³ øï÷
¬¸» ·²»¯«¿´·¬·»¾· øµ÷ ä «· øµ÷ ä ½· øµ÷
øí÷
º±® »ª»®§ µ î Ò ø¿÷ ¿²¼ · ã ïå îò
ß²¿´§-·²¹ ¬¸» ®»-«´¬ ¹·ª»² ¾§ Ì¸»±®»³ ïòïô ©» ½±²½´«¼» ¬¸¿¬ ¬¸» »¨·-¬»²½» ±º ø¿¬ ´»¿-¬ ±²»÷ -±´«¬·±² ±º ¬¸»
°®±¾´»³ øï÷ô øî÷ ¸¿ª·²¹ ¬¸» ·²¼·½¿¬»¼ ø¿-§³°¬±¬·½÷ ¾»¸¿ª·±® ½¸¿®¿½¬»®·¦»¼ ¾§ ·²»¯«¿´·¬·»- øí÷ ·- -¬¿¬»¼ ©·¬¸
« ¹·ª»²
±º ¬©± »¯«¿¬·±²-ò ×² ¬¸» ½¿-» ±º ±²» -½¿´¿® ´·²»¿® »¯«¿¬·±² ¿ -·³·´¿® °®±¾´»³ ©¿- ¬®»¿¬»¼ ·² ÅìÃò
îò Ô×ÒÛßÎ ÝßÍÛ
ß--«³» ¬¸¿¬ -§-¬»³ øï÷ ®»¼«½»- ¬± ¬¸» º±®³
«øµ÷ ã ß «øµ÷ õ ¹øµ÷å µ î Òø¿÷
øì÷
©¸»®» ß ã ø ·¶ ÷î·å¶ãï ·- î î ³¿¬®·¨ô ·¶ î Îô «øµ÷ ã ø«ï øµ÷å «î øµ÷÷Ì ¿²¼ ¹øµ÷ ã ø¹ï øµ÷å ¹î øµ÷÷Ì ô ¹ æ Òø¿÷ ÿ
Îî ò É» ©®·¬» »¯«¿¬·±² øì÷ ·² ¬¸» º±®³
«øµ õ ï÷ ã øß õ ×÷«øµ÷ õ ¹øµ÷å
©¸»®» × ·- ¬¸» ·¼»²¬·¬§ î
øë÷
î ³¿¬®·¨ô ±® ·² ¬¸» -½¿´¿® º±®³
«ï øµ õ ï÷
«î øµ õ ï÷
ã ø
ã
ïï
õ ï÷ «ï øµ÷ õ
«ï øµ÷ õ ø
îï
ïî
îî
õ ï÷
«î øµ÷ õ
«î øµ÷ õ
¹ï øµ÷å
¹î øµ÷å
¬±¹»¬¸»® ©·¬¸ ¿² ·²·¬·¿´ °®±¾´»³
«ø¿÷ ã « æ
øê÷
Ú«®¬¸»® ©» -¸¿´´ -«°°±-» ¬¸¿¬ ¼»¬øß õ ×÷ êã ðò Ì¸»±®»³ ·² ¬¸» ½¿-» ±º -§-¬»³ øì÷ ¬¿µ»- ¬¸» º±®³æ
îðê
Ì¸»±®»³ îòï Ô»¬ ¾· øµ÷å ½· øµ÷å ¾· øµ÷ ä ½· øµ÷ô · ã ïå î
Ò ø¿÷ò Ô»¬ ·²»¯«¿´·¬·»-
ø
õ ï÷¾ï øµ÷ õ ïî «î øµ÷ õ ¹ï øµ÷
îï «ï øµ÷ õ ø îî õ ï÷¾î øµ÷ õ ¹î øµ÷
¾ï øµ õ ï÷ ä
¾î øµ õ ï÷ ä
ðå
ðå
øé÷
øè÷
ø
õ ï÷½ï øµ÷ õ ïî «î øµ÷ õ ¹ï øµ÷
îï «ï øµ÷ õ ø îî õ ï÷½î øµ÷ õ ¹î øµ÷
½ï øµ õ ï÷ â
½î øµ õ ï÷ â
ðå
ðå
øç÷
øïð÷
ïï
ïï
¸±´¼ º±® »ª»®§ µ î Òø¿÷ ¿²¼ ¾· øµ÷ ä «· øµ÷ ä ½· øµ÷ô · ã ïå îò Ì¸»² ¬¸»®» »¨·-¬- ¿² ·²·¬·¿´ ½±²¼·¬·±²
« ø¿÷ ã « î ÿø¿÷å
øïï÷
-«½¸ ¬¸¿¬ ¬¸» ½±®®»-°±²¼·²¹ -±´«¬·±² « ã « øµ÷ ±º »¯«¿¬·±² øì÷
·²»¯«¿´·¬·»¾· øµ÷ ä «· øµ÷ ä ½· øµ÷æ
µ î Ò ø¿÷ ¿²¼ · ã ïå î ¬¸»
øïî÷
Ý±®±´´¿®§ îòî Ú±® »¨·-¬»²½» ¿ °±-·¬·ª» -±´«¬·±² «¢ ã ø¢
«ï å «
¢î ÷ ±º »¯«¿¬·±² øì÷
¢î øµ÷
ïî «
õ ¹ï øµ÷ ä ðå
¢ï øµ÷
îï «
õ ¹î øµ÷ ä ðå
¿²¼ ·²»¯«¿´·¬·»- øç÷ô øïð÷ ¸±´¼ô ©¸»®»
ð ä «¢· øµ÷ ä ½· øµ÷å · ã ïå îæ
Ò±©ô ´»¬ «- º±®³«´¿¬» ¬¸» °®±¾´»³ «²¼»® ½±²-·¼»®¿¬·±²ò
Ð®±¾´»³ò Ü»¬»®³·²» ¿¬ ´»¿-¬ ±²» ª¿´«» « -«½¸ ¬¸¿¬ ¬¸» ½±®®»-°±²¼·²¹ -±´«¬·±² « ã « øµ÷ ±º ¬¸» ´·²»¿®
°®±¾´»³ øì÷ô øê÷
øµå « øµ÷÷ î
µ î Ò ø¿
µ î Òø¿÷ò
íò ÎÛÍËÔÌÍ
ï
ï
ï
º«ï½- ¹ï
-ãð ô º«ï¾- ¹-ãð ô º«î½- ¹-ãð ô º«î¾- ¹-ãð ¿-
½ï ø¿ õ -÷
ø
ïï
õ ï÷-
ï ·
Å
ïî «î ø¿
õ ·÷ õ ¹ï ø¿ õ ·÷Ã
·ãð
«ï½- æã
¾ï ø¿ õ -÷
-Ðï
ø
ø
ïï
õ ï÷-
ïï
õ ï÷-
ï ·
Å
ïî «î ø¿
½î ø¿ õ -÷
-Ðï
ø
ø
îî
õ ï÷-
ïï
õ ï÷-
ï ·
Å
îï «ï ø¿
¾î ø¿ õ -÷
-Ðï
ø
ø
îî
õ ï÷-
îî
Å
îï «ï ø¿
îî
øïì÷
õ ·÷ õ ¹î ø¿ õ ·÷Ã
·ãð
ø
å
å
õ ï÷-
ï ·
øïí÷
õ ·÷ õ ¹î ø¿ õ ·÷Ã
·ãð
«î½- æã
å
õ ·÷ õ ¹ï ø¿ õ ·÷Ã
·ãð
«ï¾- æã
«î¾- æã
-Ðï
õ ï÷-
å
©¸»®» - î Ò ô ¿²¼ «ï ø¿ õ ·÷ î Å¾ï ø¿ õ ·÷å ½ï ø¿ õ ·÷Ãô «î ø¿ õ ·÷ î Å¾î ø¿ õ ·÷å ½î ø¿ õ ·
Ô»³³¿ íòï Ô»¬ ¾· øµ÷å ½· øµ÷å ¾· øµ÷ ä ½· øµ÷ô · ã ïå î
Òø¿÷ ¿²¼ ·· õ ï â ðô
· ã ïå îò Ô»¬ º±® »ª»®§ µ î Òø¿÷ ¬¸» ·²»¯«¿´·¬·»- øé÷¥øïð÷ ¸±´¼ò Ì¸»² ¬¸» -»¯«»²½» º«ï½- ¹ï
-ãð ·- ¼»½®»¿-·²¹
½±²ª»®¹»²¬ ¿²¼ ¬¸» -»¯«»²½» º«ï¾- ¹ï
-ãð ·- ·²½®»¿-·²¹ ½±²ª»®¹»²¬ -»¯«»²½»ò Ó±®»±ª»® «ï½- â «ï¾- ¸±´¼- º±®
»ª»®§ - î Ò ¿²¼ º±® ´·³·¬- ¾ï å ½ï ©¸»®»
¾ï ã ´·³ «ï¾- å
-ÿï
½ï ã ´·³ «ï½-ÿï
îðé
¬¸» ·²»¯«¿´·¬§ ½ï ¾ï ¸±´¼-ò
ï
Ì¸» -»¯«»²½» º«î½- ¹ï
-ãð ·- ¼»½®»¿-·²¹ ½±²ª»®¹»²¬ -»¯«»²½» ¿²¼ ¬¸» -»¯«»²½» º«î¾- ¹-ãð ·- ·²½®»¿-·²¹ ½±²ª»®ó
¹»²¬ -»¯«»²½»ò Ó±®»±ª»® «î½- â «î¾- ¸±´¼- º±® »ª»®§ - î Ò ¿²¼ º±® ´·³·¬- ¾î å ½î ©¸»®»
¾î ã ´·³ «î¾- å
½î ã ´·³ «î½-
-ÿï
¬¸» ·²»¯«¿´·¬§ ½î
-ÿï
¾î ¸±´¼-ò
Ð®±±ºæ Ô»¬ «- -¸±© ¬¸¿¬ «ï½- â «ï¾- º±® »ª»®§ - î Ò ò Ì¸·- º±´´±©- º®±³ øïí÷ ¿²¼ øïì÷ô -·²½» ½ï ø¿ õ -÷ â
¾ï ø¿ õ -÷ º±® »ª»®§ - î Ò ò Ò±© ©» -¸±© ¬¸¿¬ ¬¸» -»¯«»²½» º«ï½- ¹ï
-ãð ·- ¿ ¼»½®»¿-·²¹ -»¯«»²½»ò ×º - ã ð ¿²¼
- ã ï ¬¸»² øïí÷ ¹·ª»«ï½ð ã ½ï ø¿÷
¿²¼
½ï ø¿ õ ï÷
«ï½ï ã
Å¿ïî «î ø¿÷ õ ¹ï ø¿÷Ã
æ
¿ïï õ ï
Ì¸» ·²»¯«¿´·¬§ «ï½ð â «ï½ï ·- ¿ ½±²-»¯«»²½» ±º øç÷ ©·¬¸ µ ã ¿ -·²½»
½ï ø¿÷ â
½ï ø¿ õ ï÷
Å¿ïî «î ø¿÷ õ ¹ï ø¿÷Ã
å
¿ïï õ ï
Ú±® µ ã ¿ õ -ô - î Ò ¬¸» ·²»¯«¿´·¬§ øç÷ ¹·ª»ø
ïï
õ ï÷½ï ø¿ õ -÷ õ
ïî «î ø¿
õ -÷ õ ¹ï ø¿ õ -÷
½ï ø¿ õ - õ ï÷ â ðå
±®
½ï ø¿ õ - õ ï÷ ä ø
õ ï÷½ï ø¿ õ -÷ õ
ïï
ïî «î ø¿
õ -÷ õ ¹ï ø¿ õ -÷æ
øïë÷
Ú®±³ øïí÷ ¿²¼ øïë÷ ©» ¹»¬
Ð
½ï ø¿ õ - õ ï÷
ø
ïï
õ ï÷- · ø
ïî «î ø¿
õ ·÷ õ ¹ï ø¿ õ ·÷÷
·ãð
«ï½å-õï ã
þ
ï
ø ïï õ ï÷-õï
ø
È
ø
ïï
ø
õ ï÷½ï ø¿ õ -÷ õ
ïï
õ ï÷
- ·
ø
ä
õ ï÷-õï
ïï
ïî «î ø¿
ïî «î ø¿
õ -÷ õ ¹ï ø¿ õ -÷
õ ·÷ õ ¹ï ø¿ õ ·÷÷
·ãð
ø
ïï
õ ï÷½ï ø¿ õ -÷
-Ðï
ø
ïï
õ ï÷- · ø
ïï
õ ï÷-õï
ïî «î ø¿
ý
ã
õ ·÷ õ ¹ï ø¿ õ ·÷÷
·ãð
ø
ã «ï½- æ
Í± ¬¸» ·²»¯«¿´·¬§ «ï½- â «ï½å-õï ¸±´¼- º±® »ª»®§ - î Ò ò Ì¸» ®»-¬ ±º ¬¸» °®±±º ½¿² ¾» ³¿¼» ·² ¿ -·³·´¿® ©¿§
ø¬¸» »¨·-¬»²½» ±º ´·³·¬- ·- ¿² »´»³»²¬¿®§ ½±²-»¯«»²½» ±º ¬¸» ¬¸»±®§ ±º ²«³¾»® -»¯«»²½»-÷ò
Ì¸» -±´«¬·±² ±º øë÷ô øê÷ ·- ¹·ª»² ¾§ º±®³«´¿
«øµ÷ ã øß õ ×÷µ
¿
« õ
µ
Èï
øß õ ×÷µ
· ï
øïê÷
¹ø·÷
·ã¿
ø-»»ô »ò¹òô ÅïðÃ÷ò Þ§ ¬¸» -¿³» -½¸»³» ¿- ·² ÅìÃ Ì¸»±®»³ íòî ½¿² ¾» °®±ª»¼ò
Ì¸»±®»³ íòî Ô»¬ ¿´´ ½±²¼·¬·±²- ±º Ì¸»±®»³ îòï ¸±´¼ò Ì¸»² ¬¸» »¨°®»--·±² øïê÷ ©·¬¸ « ã ø«ï å «î ÷Ì î
øÅ¾ï å ½ï Ãå Å¾î å ½î Ã÷Ì ·- ¿ -±´«¬·±² ±º ¬¸» °®±¾´»³ øì÷ô øïï÷ -¿¬·-º§·²¹ ·²»¯«¿´·¬·»- øïî÷ò
Ý±®±´´¿®§ íòí Í±´«¬·±² « ã « øµ÷ ±º ·²·¬·¿´ °®±¾´»³ øë÷ô øê÷
« ø¿÷ ã øß õ ×÷
ï -
« ø¿ õ -÷
È
øß õ ×÷
ï - ·õï
¹ø-
·÷å
øïé÷
·ãï
îðè
©¸»®»
¾ï ø¿ õ µ÷ ä «· ø¿ õ µ÷ ä ½·ø¿ õ µ÷å µ ã ðå ïå æ æ æ å -å · ã ïå îæ
Ý±®±´´¿®§ íòì ×º ·²·¬·¿´ ½±²¼·¬·±²- øê÷ ¿®» °±-·¬·ª»ô ¬¸»² ¬¸»®» »¨·-¬- ¿ °±-·¬·ª» -±´«¬·±² ±º øë÷ò
ìò ÛÈßÓÐÔÛÍ
±º Ô»³³¿ íòï ¿®» ª·±´¿¬»¼ô ·ò»òô ±«® ®»-«´¬- ²±¬ ½±ª»® ¿´´ ¬¸» ½¿-»- ±º »¨·-¬»²½» ±º °±-·¬·ª» -±´«¬·±²-ò Ì¸»
-»½±²¼ »¨¿³°´» ½¿² ¾» ½±ª»® ¾§ ±«® ®»-«´¬-ò
Û¨¿³°´» ìòï Ý±²-·¼»® ¿ -§-¬»³ ±º ¼·-½®»¬» »¯«¿¬·±²-æ
¨ï øµ õ ï÷
¨î øµ õ ï÷
ã
¨ï øµ÷ õ ¨î øµ÷å
ã î¨î øµ÷å
µ ã ðå ïå æ æ æ ô ¬±¹»¬¸»® ©·¬¸ ·²·¬·¿´ ½±²¼·¬·±²¨ï øð÷ ã ïå ¨î øð÷ ã îæ
×¬ ·- »¿-§ ¬± ª»®·º§ ¬¸¿¬ ¬¸·- -§-¬»³ ¿¼³·¬- ¿ °±-·¬·ª» -±´«¬·±²
¨ï øµ÷ ã
ï µõï
î
õ ø ï÷µ å ¨î øµ÷ ã îµõï æ
í
ïï
õïã
ïò
Û¨¿³°´» ìòî Ý±²-·¼»® ¿ -§-¬»³ ±º ¼·-½®»¬» »¯«¿¬·±²§ï øµ õ ï÷
ã §ï øµ÷
§î øµ÷å
ë
§ï øµ÷ õ §î øµ÷
î
§î øµ õ ï÷
ã
µ ã ðå ïå æ æ æ ô ¬±¹»¬¸»® ©·¬¸ ·²·¬·¿´ ½±²¼·¬·±²- §ï øð÷ ã ïå §î øð÷ ã ïãîæ
×² ¿½½±®¼¿²½» ©·¬¸ ®»-«´¬- °®»-»²¬»¼ ¬¸»®» »¨·-¬- ¿ °±-·¬·ª» -±´«¬·±² ±º ¬¸·- -§-¬»³ò ×²¼»»¼ô ·¬ ·- »¿-§ ¬±
ª»®·º§ ¬¸¿¬
ï
ï
§ï øµ÷ ã µ å §î øµ÷ ã µõï
î
î
·- -«½¸ ¿ -±´«¬·±²ò
Ý¦»½¸ Ù±ª»®²³»²¬ ÓÍÓ ððîïêíðëîçò Ì¸» -»½±²¼ ¿«¬¸±® ©¿- -«°°±®¬»¼ ¾§ ¬¸» Ý±«²½·´ ±º Ý¦»½¸ Ù±ª»®²ó
³»²¬ ÓÍÓ ððîïê íðëðí ¿²¼ ÓÍÓ ððîïê íðëïçò
Î»º»®»²½»
ÅïÃ ÞßwÌ×ÒÛÝô Öòô Ü×ÞÔSÕô Öòô ÆØßÒÙ Þ×ÒÙÙÛÒæ Û¨·-¬»²½» ±º ¾±«²¼»¼ -±´«¬·±²- ±º ¼·-½®»¬» ¼»´¿§»¼
»¯«¿¬·±²-ô ÝÎÝ Ð®»-- ÔÝ îððìô íêðóíêêò
ÅîÃ ÞßwÌ×ÒÛÝô Öòô Ü×ÞÔSÕô Öòæ Ñ²» ½¿-» ±º ¿°°»¿®¿²½» ±º °±-·¬·ª» -±´«¬·±²- ±º ¼»´¿§»¼ ¼·-½®»¬» »¯«¿¬·±²-ô
ß°°´·½¿¬·±²- ±º Ó¿¬¸»³¿¬·½-ô ìè øîððí÷ô ìîçóìíêò
ÅíÃ ÞßwÌ×ÒÛÝô Öòô Ü×ÞÔSÕô Öòæ Í«¾¼±³·²¿²¬ °±-·¬·ª» -±´«¬·±²- ±º ¬¸» ¼·-½®»¬» »¯«¿¬·±²
°øµ÷«øµ÷ô ß¾-¬®¿½¬ ¿²¼ ß°°´·»¼ ß²¿´§-·-ô îððìæê øîððì÷ô ìêïóìéðò
«øµ õ ²÷ ã
ÅìÃ ÞßwÌ×ÒÛÝô Öòô Ü×ÞÔSÕô Öòô ÎËr×XÕÑÊ_ô Óòæ ×²·¬·¿´ ¼¿¬¿ ¹»²»®¿¬·²¹ ¾±«²¼»¼ -±´«¬·±²- ±º ´·²»¿®
¼·-½®»¬» »¯«¿¬·±²-ô Ñ°«-½«´¿ Ó¿¬¸»³¿¬·½¿ô îêô Ò± íô øîððê÷ô íçë¥ìðêò
îðç
ÅëÃ Ü×ÞÔSÕô Öòæ
±º Ý±³°«¬¿¬·±²¿´ ¿²¼ ß°°´·»¼ Ó¿¬¸»³¿¬·½-ô èè øïççè÷ô ïèëóîðîò
ÅêÃ Ü×ÞÔSÕô Öòæ
Ö±«®²¿´
Ð®±½»»¼·²¹- ±º ¬¸» Ú±«®¬¸ ×²¬»®²¿¬·±²¿´ Ý±²º»ó
ïðéóïïìô îðððò
ÅéÃ Ü×ÞÔSÕô Öòô ÎËr×XÕÑÊ_ô Óòæ Û¨·-¬»²½» ±º °±-·¬·ª» -±´«¬·±²- ±º ²ó¼·³»²-·±²¿´ -§-¬»³ ±º ²±²´·²»¿®
ß®½¸ò Ó¿¬ò íê øîððð÷ô ìíëóììêò
ÅèÃ Ü×ÞÔSÕô Öòæ Ü·-½®»¬» ®»¬®¿½¬ °®·²½·°´» º±® -§-¬»³- ±º ¼·-½®»¬» »¯«¿¬·±²-ô Ý±³°«¬»®- ¿²¼ Ó¿¬¸»³¿¬·½©·¬¸ ß°°´·½¿¬·±²-òô ìî øîððï÷ô ëïëóëîèò
ÅçÃ Ü×ÞÔSÕô Öòæ ß-§³°¬±¬·½ ¾»¸¿ª·±«® ±º -±´«¬·±²- ±º ¼·-½®»¬» »¯«¿¬·±²-ô
íé¥ìèò
ÅïðÃ ÛÔßÇÜ×ô ÍòÒòæ
ïï øîððì÷ô
Í°®·²¹»®ô îððëò Ì¸·®¼ Û¼·¬·±²ò
Á
ÅïïÃ ÉßÆÛÉÍÕ×ô
Ìòæ Í«® «² °®·²½·°» ¬±°±´±¹·¯«» ¼» ´ù»¨¿³»² ¼» ´ù¿´´«®» ¿-§³°¬±¬·¯«» ¼»- ·²¬¹®¿´»- ¼»ß²²ò Í±½ò Ð±´±²ò Ó¿¬¸ò îð øïçìé÷ô îéç¥íïíò
ßÜÜÎÛÍÍæ
¼±½ò ÎÒÜ®ò Ö¿®±³3® Þ¿g¬·²»½ô ÝÍ½ò
Ü»°¿®¬³»²¬ ±º Ó¿¬¸»³¿¬·½Ú¿½«´¬§ ±º Û´»½¬®·½¿´ Û²¹·²»»®·²¹ ¿²¼ Ý±³³«²·½¿¬·±²ô
Þ®²± Ë²·ª»®-·¬§ ±º Ì»½¸²±´±¹§
¾¿-¬·²»½àº»»½òª«¬¾®ò½¦
°®±ºò ÎÒÜ®ò Ö±-»º Ü·¾´3µô Ü®Í½ò
Þ®²± Ë²·ª»®-·¬§ ±º Ì»½¸²±´±¹§
ÞÎÒÑ ô Ý¦»½¸ Î»°«¾´·½
¼·¾´·µàº»»½òª«¬¾®ò½¦ô ¼·¾´·µò¶àº½»òª«¬¾®ò½¦
îïð
Þ×ÒßÎÇ ÓËÔÌ×ÍÌÎËÝÌËÎÛÍ ÑÚ ÐÎÛÚÛÎÛÒÝÛ ÎÛÔßÌ×ÑÒÍ
Ö¿² Ý¸ª¿´·²¿ô Ö·(3 Ó±«8µ¿ô Ó·½¸¿´ Ò±ª?µ
Ú¿½«´¬§ ±º Û´»½¬®·½¿´ Û²¹·²»»®·²¹ ¿²¼ Ý±³³«²·½¿¬·±²ô Þ®²± Ë²·ª»®-·¬§ ±º Ì»½¸²±´±¹§
ß¾-¬®¿½¬æ Þ·²¿®§ ¸§°»®-¬®«½¬«®»- øº±®³»®´§ ½¿´´»¼ ³«´¬·-¬®«½¬«®»-÷ ±² -§-¬»³- ±º °®»º»®»²½»
®»´¿¬·±²- º±® -¬®«½¬«®»¼ -»¬- ±º ½¸±·½» ¿´¬»®²¿¬·ª»- ¿²¼ ±² -°¿½»- ±º ¬¸»·® ³«´¬·ó«¬·´·¬§ ®»°®»-»²ó
¬¿¬·±²- ¿®» ½±²-¬®«½¬»¼ ¿²¼ ¼·-½«--»¼ò
Õ»§ ©±®¼-æ °®»º»®»²½» ®»´¿¬·±²-ô «¬·´·¬§ º«²½¬·±²-ô ³«´¬·ó«¬·´·¬§ ®»°®»-»²¬¿¬·±² ±º ¿ °®»º»®»²½»
®»´¿¬·±²ô ½±³³«¬¿¬·ª» ¸§°»®¹®±«°ô ¶±·² -°¿½»ô ·²½´«-·±² ¸±³±³±®°¸·-³ ±º ¸§°»®-¬®«½¬«®»-ò
Ì¸» ³¿·² ½±²-¬®«½¬·±² °®»-»²¬»¼ ·² ¬¸·- ½±²¬®·¾«¬·±² ·- ¾¿-»¼ ±² ¬©± ³±¬·ª¿¬·±² ´·²»-ò Ì¸» ¯«·¬» ²¿¬«®¿´´§
Û²¼´»³³¿ ¥ ½º »ò¹ò ÅîðÃ ¥ -¬¿¬»- ¬¸¿¬ ·º ø
¸§°»®±°»®¿¬·±² æ
ÿ Ðø ÷ ±² ¬¸» -»¬
¾§
ãÅ
÷ ãº å
¹ ©» ±¾¬¿·² ¿ º«²½¬±®·¿´
¬®¿²-º»® º®±³ ¬¸» ½¿¬»¹±®§ ±º ¿´´ ¯«¿-·ó±®¼»®»¼ -»³·¹®±«°- ¿²¼ ¬¸»·® ±®¼»® ¸±³±³±®°¸·-³- ·²¬± ¬¸» ½¿¬»¹±®§
±º ø·² ¹»²»®¿´ ²±²ó½±³³«¬¿¬·ª»÷ -»³·¸§°»®¹®±«°- ¿²¼ ¬¸»·® ·²½´«-·±² ¸±³±³±®°¸·-³-ò Ë²¼»® ¿ ½»®¬¿·²
-·³°´» ½±²¼·¬·±² ½±²½»®²·²¹ ¬¸» ±®¼»®·²¹ -¬®«½¬«®» ø
÷ ¥ ¬¸¿¬ ø
÷ ·- ¿ ¹®±«°ô ©» ±¾¬¿·² ¿ ¸§°»®¹®±«°
¿ ¸§°»®¹®±«°ò Ì¸» ¶«-¬ ³»²¬·±²»¼ ¿°°®±¿½¸ ·- ¹»²»®¿´·¦»¼ ·² ÅîðÃô ©¸»®» ¬¸» °®»-»²¬»¼ ½±²-¬®«½¬·±² ·² º¿½¬
§·»´¼- ¿ º«²½¬±®·¿´ ¬®¿²-º»® º®±³ ¬¸» ½¿¬»¹±®§ ±º ±®¼»®»¼ ´·²»¿® -°¿½»- ±º ®»¿´ º«²½¬·±² ±º ¬¸» ½´¿-- Ýµ ô ·ò»ò
¬¸» ½´¿-- ±º ®»¿´ ª¿´«»¼ º«²½¬·±²- ©·¬¸ ½±²¬·²«±«- ¼»®·ª¿¬·ª»- «° ¬± ¬¸» ±®¼»® ÷ò
«¬·´·¬§ º«²½¬·±² ·² ¬¸» º±®³
ø
÷ã
õ ò Ì¸» ½±²¬®·¾«¬·±² ·- ¼»ª±¬»¼ ¬± ¾·²¿®§ ¿´¹»¾®¿·½ -¬®«½¬«®»- ±º °®»º»®»²½» ®»´¿¬·±²- ¿²¼
®»´¿¬»¼ -¬®«½¬«®»- ±º «¬·´·¬§ º«²½¬·±²-ò Ì¸» ½±²½»°¬ ±º °®»º»®»²½» ·- ¬¸» º±«²¼¿¬·±² ±º ¿´´ ½¸±·½» ¬¸»±®§
·² »½±²±³·½-ò ×² °¿®¬·½«´¿® ·² ³·½®±»½±²±³·½-ô °®»º»®»²½»- ±º ½±²-«³»®- ¿²¼ ±¬¸»® »²¬·¬·»- ¿®» ³±¼»´´»¼
©·¬¸ °®»º»®»²½» ®»´¿¬·±²-ò Ó«½¸ ±º ³±¼»®² ³·½®±»½±²±³·½ ¬¸»±®§ ¿®·-»- º®±³ ¬¸·²µ·²¹ ¿¾±«¬ °®»º»®»²½»±ª»® ¬¸·²¹- ´·µ» °±´·¬·½¿´ °¿®¬·»-ô »²ª·®±²³»²¬¿´ °±´·½·»-ô ¾«-·²»-- -¬®¿¬»¹·»-ô ´±½¿´ ¼»½·-·±²- »¬½òô ©¸»®» ¬¸»
«²¼»®´§·²¹ -»¬
±º °±--·¾´» ¿´¬»®²¿¬·ª»- ½±«´¼ ¾» ª»®§ ¹»²»®¿´ ø½º »ò¹ ÅîéÃ÷ò Ò±¬·½» ¬¸¿¬ ¿½½±®¼·²¹ ¬± ÅîéÃô
°ò íô þß ½±²-«³°¬·±² ¾«²¼´» ·- ¿ °¿·® ø
§±« ½±²-«³» ±º ±²» ¹±±¼ ø¶«-¬ ½¿´´ ·¬ º±® -¸±®¬÷ô ¿²¼ ¬¸» -»½±²¼ ½±³°±²»²¬ ·- ¬¸» ¯«¿²¬·¬§ §±« ½±²-«³»
±º ¬¸» -»½±²¼ ¹±±¼ò Åò ò ò Ã ×º ©» Åò ò ò Ã ·³¿¹·²» ¬¸¿¬ ¹±±¼ ¸¿- ¿ °®·½» ¿²¼ ¹±±¼ ¸¿- ¿ °®·½» ô ¿²¼ ¬¸¿¬ §±«
¸¿ª»
¬± -°»²¼ô ¬¸»² ¬¸» ½±²-«³»® º¿½»- ¿ -»¬ ±º ¿´¬»®²¿¬·ª»©¸·½¸ ½±²-·-¬- ±º ¿´´ °¿·®- ø
÷ ©¸±-»
½±-¬ ·- ´»-- ¬¸¿² ±® »¯«¿´ ¬± ô ·ò»ò
ø
÷ î Îîõ æ
õ
¹
Ø»®» Îîõ ·- ¬¸» -»¬ ±º ¿´´ ª»½¬±®- ©·¬¸ ¬©± ²±²ó²»¹¿¬·ª» ½±³°±²»²¬-òþ Ú«®¬¸»®ô ©¸»² ³»²¬·±²·²¹ »½±²±³·½
³±¼»´´·²¹ô ¬¸» ¿«¬¸±® -¬¿¬»-æ þÌ¸» ³¿¬¸»³¿¬·¦¿¬·±² ±º »½±²±³·½- ±½½«®®»¼ ·² ¬¸» ´¿¬» ÅïçÃëðù- ¿²¼ ¸¿- ¸¿¼
¿ ®»³¿®µ¿¾´» ·³°¿½¬ ±² ¬¸» ©¿§ »½±²±³·-¬- ·²¬»®¿½¬ò Ì± «-» ³¿¬¸»³¿¬·½-ô ·¬ ·- ²»½»--¿®§ ¬¸¿¬ ¬¸» ½±²½»°¬-ô
Ô»¬ «- ®»½¿´´ ø½º »ò¹ò ÅïðÃ÷ ¬¸¿¬ ¿ °®»º»®»²½» ®»´¿¬·±² ±² ¿ ²±²»³°¬§ -»¬
±º ½¸±·½» ¿´¬»®²¿¬·ª»- ·- ¬®¿¼·¬·±²¿´´§
ò Î»½»²¬´§ô ¸±©»ª»®ô ¬¸»®» ¸¿- ¾»»²
¿ ½»®¬¿·² ·²¬»®»-¬ ·² ·²¼·ª·¼«¿´ ¼»½»-·±²ó³¿µ·²¹ ³±¼»´-ô ·² ©¸·½¸ ±²»ù- °®»º»®»²½» ®»´¿¬·±² ·- ¿´´±©»¼ ¬± ¾»
·²½±³°´»¬»ò Ð®»º»®»²½» ®»´¿¬·±²- ¿®» -±³»¬·³»- ¥ ½º »ò¹ò ÅîÃ ¥ ½±²-·¼»®»¼ ·² ¿ ®¿¬¸»® ¹»²»®¿´ ¾¿½µ¹®±«²¼ ¿¿®¾·¬®¿®§ ¾·²¿®§ ®»´¿¬·±²- ±² -»¬- ±º ¿´¬»®²¿¬·ª»-ò ß ´·-¬ ±º ·²¬»®»-¬·²¹ ©±®µ- ±² ¬¸·- ¬±°·½ ·²½´«¼»- ÅïÃô ÅíÃô ÅçÃô
ÅïðÃô ÅïïÃô ÅïëÃô ÅîïÃô Åîí¥îçÃò ß- ¸¿- ¾»»² ³»²¬·±²»¼ ·² ¬¸» ·²¬®±¼«½¬·±² ±º ÅïðÃô ¬¸»®» ¿®» ¬©± ³¿·² ²±¬·±²±º «¬·´·¬§ ®»°®»-»²¬¿¬·±² º±® ¿² ·²½±³°´»¬» °®»º»®»²½» ®»´¿¬·±² ó ±²
¿ º«²½¬·±² æ ÿ Î -«½¸ ¬¸¿¬ º±® ¿²§
î
ó
·³°´·»- ø ÷ ë ø ÷ ¿²¼
·³°´·»- ø ÷
ø ÷
îïï
Ì¸·- ²±¬·±² ±º ®»°®»-»²¬¿¬·±² ©¿- »¨°´±®»¼ ¾§ Óò Î·½¸¬»® øïçêê÷ ¿²¼ Þò Ð»´»¹ øïçéð÷å ·¬ ¸¿- ¬¸«- ±º¬»²
¾»»² ®»º»®®»¼ ¬± ¿- Î·½¸¬»® ¥ Ð»´»¹ «¬·´·¬§ ®»°®»-»²¬¿¬·±²ò Ò»ª»®¬¸»´»--ô Îò ß«³¿²² øïçêî÷ ©¿- ³±-¬ ´·µ»´§
¾»»² ·²ª»-¬·¹¿¬»¼ ¿¬ ´¿®¹»ò ß ´·-¬ ±º °¿°»®- ±² ¬¸·- ¬±°·½ ·- ·²½´«¼»¼ ·² °¿°»® ÅïðÃò Ë²º±®¬«²¿¬»´§ô ¬¸» «-»
±º Î·½¸¬»® ¥ Ð»´»¹ ®»°®»-»²¬¿¬·±² ±º ¿ °®»º»®»²½» ®»´¿¬·±² ·- ´·³·¬»¼ ¿- ·¬ ¼±»- ²±¬ ½¸¿®¿½¬»®·¦» ¬¸» ±®·¹·²¿´
®»´¿¬·±² ¾«¬ ®¿¬¸»® »¨¬»²¼- ·¬ ¬± ¿ ½±³°´»¬» °®»±®¼»® ©¸·½¸ ·- ®»°®»-»²¬¿¾´» ·² ¬¸» ±®·¹·²¿´ -»²-»ò Ì¸«- -«½¸
²±® ¬± ¼»¬»®³·²» ¬¸» ½¸±·½» ½±®®»-°±²¼»²½» ±º ¬¸» ·²¼·ª·¼«¿´ ·² ·¬- »²¬·®»¬§ò
Ë ±º ®»¿´ º«²½¬·±²- ±²
Ì¸» ±¬¸»® ¿°°®±¿½¸ ±º ®»°®»-»²¬·²¹ °®»º»®»²½» ®»´¿¬·±² ó
º±® ¿²§
î
ó ·º ¿²¼ ±²´§ ·º ø ÷ ë ø ÷ º±® ¿´´ î Ë
-«½¸ ¬¸¿¬
Ì¸·- ¿°°®±¿½¸ô ©¸·½¸ ·- ¿°¬´§ ½¿´´»¼ ¬¸» ³«´¬·ó«¬·´·¬§ ®»°®»-»²¬¿¬·±² ±º óô »²¿¾´»- º«´´ ½¸¿®¿½¬»®·¦¿¬·±² ±º
¬¸» °®»º»®»²½» ®»´¿¬·±² óò ß- ¸¿- ¾»»² ²±¬»¼ ·² ÅïðÃô °ò íô ¬¸·- ²±¬·±² ±º ®»°®»-»²¬¿¬·±² ¿®·-»- ²¿¬«®¿´´§ ·²
¬¸» ½´¿--·½¿´ º®¿³»©±®µ- ±º Í¿ª¿¹» ¿²¼ ª±² Ò»«³¿²² ¿²¼ Ó±®¹»²-¬»®²ô ®»-°»½¬·ª»´§ô «°±² ¬¸» ®»´¿¨¿¬·±² ±º
¬¸» ½±³°´»¬»²»-- ¿¨·±³ò
ß ¸§°»®¹®«°±·¼ ·- ¿ °¿·® ø
÷ô ©¸»®»
êã ð ¿²¼ æ
ÿ Ð ø ÷ ·- ¿ ¾·²¿®§ ¸§°»®±°»®¿¬·±² ±² ò
Í§³¾±´ Ð ø ÷ ¼»²±¬»- ¬¸» -§-¬»³ ±º ¿´´ ²±²»³°¬§ -«¾-»¬- ±º ò ×º ¬¸» ¿--±½·¿¬·ª·¬§ ¿¨·±³ ø ÷ ã ø
÷
¸±´¼- º±® ¿´´
î ô ¬¸»² ¬¸» °¿·® ø
÷ ·- ½¿´´»¼ ¿ -»³·¸§°»®¹®±«°ò ×º ³±®»±ª»® ¬¸» ®»°®±¼«½¬·±² ¿¨·±³
ã
ã
î ô ¬¸»² ¬¸» °¿·® ø
÷ ·- ½¿´´»¼ ¿ ¸§°»®¹®±«°ò ß
¸§°»®¹®±«°±·¼ ø
÷ ©¸»®» ¬¸» ®»°®±¼«½¬·±² ¿¨·±³ ¸±´¼- ·- ½¿´´»¼ ¿ ¯«¿-·ó¸§°»®¹®±«°ò ß ¸§°»®¹®±«° ø
÷
·- ½¿´´»¼ ¿ ¬®¿²-°±-·¬·±² ¸§°»®¹®±«° ±® ¿ ¶±·² -°¿½»
î ¬¸» ®»´¿¬·±²
·³°´·»ô ©¸»®»
º±®
³»¿²- Ä êã åò Í»¬ãº î å î
¹ ¿²¼
ãº î å î
¹ ¿®» ½¿´´»¼ »¨¬»²-·±²-ô ±® º®¿½¬·±²-ò Þ§ ¿ ¯«¿-·ó±®¼»®»¼
-»³·¹®±«° ©» ³»¿² ¿ ¬®·°´» ø
÷ô ©¸»®» ø
÷ ·- ¿ -»³·¹®±«° ¿²¼ ¾·²¿®§ ®»´¿¬·±² ·- ¿ ¯«¿-·ó±®¼»®·²¹
-«½¸ ¬¸¿¬ º±® ¿²§ ¬®·°´»
î
©·¬¸ ¬¸» °®±°»®¬§
¬¸»®» ¸±´¼- ¿´-±
¿²¼
ò Ì¸» °®·²½·°¿´ »²¼ô ±º¬»² ½¿´´»¼ ¿´-± °®·²½·°¿´ ´±©»® ½±²»
¹»²»®¿¬»¼ ¾§ î ·- ¿ -»¬ Å ÷ ã º î å
¹å ¬¸» °®·²½·°¿´ «°°»® ½±²»
Ú«®¬¸»®ô ´»¬
¾» ¿ ²±²»³°¬§ -»¬ ¿²¼ ó
ò
Ì¸» -»¬
·- «-«¿´´§ ®»¹¿®¼»¼ ¿- ¬¸» «²·ª»®-¿´ -»¬ ±º ³«¬«¿´´§ »¨½´«-·ª» ½¸±·½» ¿´¬»®²¿¬·ª»-ô ©¸·´» ó ·- ½¿´´»¼
¿ °®»º»®»²½» ®»´¿¬·±² ±º ¿² ·²¼·ª·¼«¿´ ±® ¹®±«° ±º ·²¼·ª·¼«¿´- ±ª»® ¬¸·- -»¬ò Ì¸» -¬®·½¬ °®»±®¼»®ô ¼»²±¬»¼ ¾§
ô ·- ¿ ¾·²¿®§ ®»´¿¬·±² ±²
·º ó ¿²¼ ²±¬ ó ò Ì¸» °®»±®¼»® ó ·- ½¿´´»¼ ½±³°´»¬» ·º
º±® ¿² ¿®¾·¬®¿®§ °¿·® »·¬¸»® ó ±® ó ò ß- «-«¿´ô ©» -¿§ ¬¸¿¬ ¿ ®»¿´ º«²½¬·±² æ ÿ Î ·- ó¥·²½®»¿-·²¹
·º ø ÷ ë ø ÷ º±® »ª»®§
î
©·¬¸ ó ò ×² ¬«®²ô ·- ½¿´´»¼ -¬®·½¬´§ ó¥·²½®»¿-·²¹ ·º ø ÷
ø ÷ º±®
î
©·¬¸
»ª»®§
Ô»¬
®»°®»-»²¬- ó ·º
¾» ¿ ²±²»³°¬§ -»¬ ¿²¼
ó
¿ °®»±®¼»® ±²
ò É» -¿§ ¬¸¿¬ ¿ ²±²»³°¬§ -«¾-»¬ Ë ±º ÎÓ
·º ¿²¼ ±²´§ ·º ø ÷ ë ø ÷ º±® ¿´´
îË
º±® »ª»®§
î ò ×º -«½¸ ¿ -»¬ Ë »¨·-¬-ô ¬¸»² ©» -¿§ ¬¸¿¬ ¬¸»®» »¨·-¬- ¿ ³«´¬·ó«¬·´·¬§ ®»°®»-»²¬¿¬·±² º±®
¬¸» °®»±®¼»® ø°®»º»®»²½» ®»´¿¬·±²÷ óò ×º Ë
³«´¬·ó«¬·´·¬§ ®»°®»-»²¬¿¬·±² º±® óò
ÎÓ ®»°®»-»²¬- ¿ °®»±®¼»® ó ±² ô ¬¸»² »ª»®§ ³»³¾»® ±º Ë
×¬ ·- ©±®¬¸ -¬®»--·²¹ ¬¸¿¬ ·º ¿ ²±²»³°¬§
¸¿- ¬± ¾» ó¥·²½®»¿-·²¹ ¾«¬ ²± ³»³¾»® ±º Ë ²»»¼- ¬± ¾» -¬®·½¬´§ ó¥·²½®»¿-·²¹ò
Ë-·²¹ ½¸¿®¿½¬»®·-¬·½ º«²½¬·±²- ±º °®·²½·°¿´ ´±©»® ½±²»- ¹»²»®¿¬»¼ ¾§ »´»³»²¬- ±º ¿ °®»±®¼»®»¼ -»¬ ¬¸» «¬¸±®±º ÅïðÃ °®±ª» ¬¸» º±´´±©·²¹æ
Ð®±°±-·¬·±² ï øÅïðÃô Ð®±°±-·¬·±² ïòô °ò ê÷òÌ¸»®» »¨·-¬- ¿ ³«´¬·ó«¬·´·¬§ ®»°®»-»²¬¿¬·±² º±® »ª»®§ °®»±®¼»®ò
®»°®»-»²¬¿¬·±²- ±º -»³·½±²¬·²«±«- ø·² º¿½¬ °±·²¬ó½´±-»¼ ¿²¼ ·²ª»®-»´§ °±·²¬ó½´±-»¼÷ °®»±®¼»®- ©¸»®» ¬¸»
«²¼»®´§·²¹ -»¬
¬±°±´±¹§ ±® ©·¬¸ ¿ -¬®«½¬«®» ±º ²±®³»¼ ´·²»¿® -°¿½» ±® ·² ¹»²»®¿´ ±º ¿ ´·²»¿® ¬±°±´±¹·½¿´ -°¿½»ò
×² ½±²²»¬·±² ©·¬¸ ¬¸» ¶«-¬ ³»²¬·±²»¼ ½±²-·¼»®¿¬·±²- ©» ©·´´ ¿°°´§ ¿ ½»®¬¿·² º«²½¬±®·¿´ ¬®¿²-º»® ½±²-¬®«½¬»¼
·² ÅîðÃ ·² ±®¼»® ¬± ¬®¿²-º±®³ ¬¸» ®»°®»-»²¬·²¹ -»¬ÎÓ ·²¬± ·³°±®¬¿²¬ ³«´¬·-¬®«½¬«®»- ½¿´´»¼ ¶±·²
-°¿½»-ò Ì¸» º±´´±©·²¹ ´»³³¿ ¿´´±©- «- ¬± -·³°´·º§ ¬¸» ¬»-¬ ±º ¿--±½·¿¬·ª·¬§ ±º ¿ ¾·²¿®§ ¸§°»®±°»®¿¬·±² ±² ¿
½±³³«¬¿¬·ª» ¸§°»®¹®±«°±·¼ò ×¬ ¸¿- ¾»»² °®±ª»¼ ·² ÅîðÃò
Ô»³³¿ ïò Ô»¬ ¿ ¬®·°´» ø
÷ ¾» ¿ ½±³³«¬¿¬·ª» ¸§°»®¹®±«°±·¼ò Ì¸»² ¬¸» º±´´±©·²¹ ¿--»®¬·±²- ¿®» »¯«·ª¿´»²¬
º±® ¿²§ ¬®·°´»
î æ ïð ø
÷ ã
ø
÷ô îð ø
÷
ø
÷ô íð
ø
÷ ø
÷ ò
îïî
×²½®»¿-·²¹ ¾·¶»½¬·ª» ¬®¿²-º±®³¿¬·±² ±º ¯«¿-·ó±®¼»®»¼ -»¬- ©·¬¸ ¬¸» ±°»®¿¬·±² ±º ½±³°±-·¬·±² ±º ³¿°°·²¹- ¿®»
±¾ª·±«-´§ ²±²½±³³«¬¿¬·ª» ¹®±«°-ò Ø±©»ª»®ô «¬·´·¬§ º«²½¬·±²- ©·¬¸ ¿ ¾·²¿®§ ±°»®¿¬·±² ±º «-«¿´ ¿¼¼·¬·±² ±º
º«²½¬·±²- ·- ¿ ½±³³«¬¿¬·ª»ô ±® ¿¾»´·¿²ô ¹®±«°ò Ó±®»±ª»®ô ·¬ ·- ±®¼»®»¼ ·² ¬¸» «-«¿´ ©¿§ô ·ò»ò º±® ¿²§ °¿·® ±º
º«²½¬·±²î ©» ¸¿ª» ¬¸¿¬
·º ø ÷ ë ø ÷ º±® ¿´´
î ò Î»-«´¬- ±¾¬¿·²»¼ ·² °¿°»® ÅîðÃ »²¿¾´»
«- ¬± ·²½´«¼» ¿ ½±²-¬®«½¬·±² ±º ¿ ½±³³«¬¿¬·ª» ¸§°»®¹®±«° ±² ¿²§ ±®¼»®»¼ ¿¾»´·¿² ¹®±«°ò ×²¼»»¼ô ®»¹¿®¼ ¿²
±®¼»®»¼ ¿¾»´·¿² ¹®±«° ø õ
æ
ÿ Ð ø ÷ ·² ¬¸» º±´´±©·²¹ ©¿§æ
Å
Å
ã
Å
õ ÷ ã
º î å
õ
¹
Å³å²ÃîÒð
º±® ¿²§
ò Ûª·¼»²¬´§ ø
î
Å³å²ÃîÒð
Òð
÷ ·- ¿ ½±³³«¬¿¬·ª» ¸§°»®¹®±«°±·¼ò
ø
Ð®±°±-·¬·±² îò
Ð®±±ºò
î
¬¸»®» ¸±´¼-
ø
Ý¸±±-» ¿² ¿®¾·¬®¿®§
÷ãø
ø
î
÷
ã
Í
Í
Å
õ
·- ¿--±½·¿¬·ª»ô ·ò»ò ¬¸¿¬ º±® ¿²§ ¬®·°´»
÷ ·- ¿ ½±³³«¬¿¬·ª» ¸§°»®¹®±«°±·¼ô ±²´§ ±²» ±º ¬¸»
ø
÷
ø
÷
º±® ¿²§ ¬®·°´»
î ò
Å
Å
Å
õ
÷
¸î¾ ½ Å³å²ÃîÒð Òð
¸î¾ ½
î
÷ ·- ¿ ¸§°»®¹®±«°ò
ò Í·²½» ø
÷ò Í·²½»
Å
÷ã
ø
©» ¸¿ª» ¬¸¿¬
Òð
÷ ô ©¸·½¸ ³»¿²- ¬¸¿¬ ¬¸»®» »¨·-¬ -«½¸
ð
î
ôÅ
ð
ðÃ
î Òð Òð
¸î¾ ½ Å³å²ÃîÒð Òð
¿²¼ Å
ðÃ
ð
î Òð
ø·÷
ð
õ
ð ð
ø··÷
ð ð
õ
ð
Òð ¬¸¿¬
îÅ
õ
ð
ð ð÷
ô ©¸·½¸ ·³°´·»- ¬¸¿¬
ðô
©¸·½¸ ·³°´·»- ¬¸¿¬
¿²¼
ð
îÅ
ð ð
ð ð
ð
õ
ð
÷ ò Ø±©»ª»®ô ¬¸·- ·³°´·»- ¬¸¿¬
ð
õ
ð ð
ð ð
âÚ®±³ ¬®¿²-·¬·ª·¬§ ±º ¬¸» ®»´¿¬·±² ô ©» ¸¿ª» ¬¸¿¬ ð ð õ ð ð
Ü»²±¬» ð ã
ò Ç»¬ -·²½»
ð ð ò Ì¸»² ð ð õ
ð
ð ô ·ò»ò ð î
Å
Å
Å
îÅ ðõ ð ð ÷
Å ðõ ÷
Å
õ
Å³å²ÃîÒð
ð
ô ·ò»ò ð ð õ ð
ð
ã ð õ ð ð ô ©» ¸¿ª» ¬¸¿¬
Å
÷ ã
ãø
÷
µî¿ ¾ Å³å²ÃîÒð Òð
Òð
ð
ò
µî¿ ¾
Ì¸»®»º±®»
ø
÷ ø
÷ ô ©¸·½¸ ¿½½±®¼·²¹ ¬± Ô»³³¿ ï ³»¿²- ¬¸¿¬ ¬¸» ¸§°»®±°»®¿¬·±² ·- ¿--±½·¿¬·ª»ò
Í»½±²¼ô ©» ¸¿ª» ¬± °®±±º ¬¸¿¬ ·² ø
÷ ¬¸»®» ¸±´¼- ¬¸» ®»°®±¼«½¬·±² ¿¨·±³
ã
ã
º±® ¿²§
î ò Í·²½» ø
÷ ·- ¿ ½±³³«¬¿¬·ª» ¸§°»®¹®±«°±·¼ô ¬¸»®» ¸±´¼ã
º±® ¿²§ î ò Ì¸»®»º±®»ô
©» ¸¿ª» ¬± -¸±© ¬¸¿¬
ã
º±® ¿²§ î ò Ç»¬ ¬¸» ·²½´«-·±²
·- ±¾ª·±«-ô ©¸·½¸ ´»¿ª»- ¬¸»
±°°±-·¬» ·²½´«-·±²
¬± ¾» °®±ª»¼ò Í·²½» ø õ÷ ·- ¿ ¹®±«°ô ¿² ¿®¾·¬®¿®§ î ½¿² ¾» ©®·¬¬»² ¿ã õ
ò Ü»²±¬» ã
ò Ì¸»²
Å
Å
ã õ îÅ õ ÷
Å
õ ÷ ã
ã
¹îÙ
Å³å²ÃîÒð Òð
Ì¸»®»º±®»ô ø
÷ ·- ¿ ½±³³«¬¿¬·ª» ¸§°»®¹®±«°ò
Ô»¬ ø ï õ ï ÷ ø î õ î ÷ ¾» ±®¼»®»¼ ¹®±«°- ¿²¼ æ ï ÿ î ¾» ¿² ·-±¬±²» ¸±³±³±®°¸·-³ô
·ò»ò ¿ ³¿°°·²¹ -«½¸ ¬¸¿¬ º±® ¿²§ °¿·® ±º »´»³»²¬î ïô
¬¸»®» ¸±´¼- ø ÷ î ø ÷ ¿²¼ ø õ ÷ ã
ï
ø ÷ õ ø ÷ò Ì¸»² ¬¸» ³¿°°·²¹ ·- ½¿´´»¼ ¿² ±®¼»® ¸±³±³±®°¸·-³ò
Ð®±°±-·¬·±² íò Ô»¬ ø ï õ ï ÷ ø î õ î ÷ ¾» ±®¼»®»¼ ¹®±«°- ¿²¼ æ ï ÿ î ¾» ¿² ±®¼»® ¸±³±³±®°¸·-³ò
Ô»¬ ø ï ï ÷ ø î î ÷ ¾» ½±³³«¬¿¬·ª» ¸§°»®¹®±«°- ½±²-¬®«½¬»¼ ¿- -«¹¹»-¬»¼ ·² Ð®±°±-·¬·±² îò Ì¸»² æ ï ÿ
î ·- ¿² ·²½´«-·±² ¸±³±³±®°¸·-³ò
Ð®±±ºò Ô»¬ æ ø ï õ ï ÷ ÿ ø î õ î ÷ ¾» ¿² ¿®¾·¬®¿®§ ±®¼»® ¸±³±³±®°¸·-³ ¿²¼
î ï
»´»³»²¬-ò Ì¸»² ©» ¸¿ª» øÅ õ ÷ ï ÷
Å ø ÷ õ ø ÷÷ î ò ×²¼»»¼ô î øÅ õ ÷ ï ÷ ·³°´·»- ¬¸»
±º
î Å õ ÷ ï ô ·ò»ò õ
-«½¸ ¬¸¿¬ ã ø ÷ò Í·²½» ø ÷ õ ø ÷ ã ø õ ÷ î ø ÷
ï
î Å ø ÷ õ ø ÷÷ î ô ©» ±¾¬¿·² ¬¸» ¿¾±ª» ·²½´«-·±²ò Ú«®¬¸»®
Í
Í
Í
ø ï ÷ã ø
õ ÷ ï÷ ã
øÅ
õ ÷ ï÷
Å
Å ø ÷õ ø
Å³å²ÃîÒð
Òð
Í
ã
Å³å²ÃîÒð
Å
Å³å²ÃîÒð
Òð
ø ÷õ
ø ÷÷
î
ã ø ÷
î
¿®¾·¬®¿®§
»¨·-¬»²½»
ã ô ·ò»ò
÷÷
î
ã
Òð
ø ÷
Å³å²ÃîÒð Òð
Ý±²-»¯«»²¬´§ô
æø
ï
ï÷
ÿø
î
î÷
·- ¿² ·²½´«-·±² ¸±³±³±®°¸·-³ò
îïí
ß- º¿® ¿- ³«´¬·ó«¬·´·¬§ ®»°®»-»²¬¿¬·±² ±º ¿ °®»±®¼»® ó ±² ¬¸» ½±³³±¼·¬§ -»¬
½¿®®§·²¹ -±³» ½±²¬·²«·¬§
³«´¬·ó«¬·´·¬§ ®»°®»-»²¬¿¬·±²- º±® ½±²¬·²«±«- °®»±®¼»® ±²
¿®» µ²±©² ó ½º »ò¹ò ÅïðÃô ¨ ìò Ø»®» ¿ °®»±®¼»®
®»´¿¬·±² ó ±² ¿ ¬±°±´±¹·½¿´ -°¿½»
·- ½¿´´»¼ ½±²¬·²«±«- ·º ·¬ ·- ½´±-»¼ ¿- ¿ -«¾-»¬ ±º ¬¸» °®±¼«½¬ -°¿½»
¿²¼ ¿ ³«´¬·ó«¬·´·¬§ ®»°®»-»²¬¿¬·±² Ë ±º ó ·- ½¿´´»¼ ½±²¬·²«±«- ©¸»²»ª»® ¿²§ º«²½¬·±² æ
ÿ ô îË
©» ±¾¬¿·² ¿ ¶±·² -°¿½» ±² ¿² »¨¬»²-·±² ø±² ¿ ½»®¬¿·² »¨°±²»²¬·¿´ó¿´¹»¾®¿·½ ½´±-«®» ÷ ±º ¿ ½±²¬·²«±«- ³«´¬·ó
«¬·´·¬§ ®»°®»-»²¬¿¬·±² Ë ±º ¿ ½±²¬·²«±«- °®»±®¼»® ó ±² ¬¸» ¬±°±´±¹·½¿´ -°¿½»
±º ³«¬«¿´´§ »¨½´«-·ª»
½¸±·½» ¿´¬»®²¿¬·ª»-ò Ú®±³ ÅïðÃô ¨
½´±-»¼ ²±²»³°¬§ -«¾-»¬- ±º ¬¸» Û«½´·¼»¿² -°¿½» ¸¿ª» ¬¸» ¶«-¬ ³»²¬·±²»¼ °®±°»®¬§ò
Ô»¬ Ýø ÷ ¾» ¿ -§-¬»³ ±º ¿´´ ½±²¬·²«±«- º«²½¬·±²- æ
ÿ Î ½±²-·¼»®»¼ ¿- ¿² ±®¼»®»¼ ´·²»¿® -°¿½»ò ×º
Ýø ÷ô ©¸»®»
·- ¿ ¬±°±´±¹·½¿´ -°¿½»ô ©» ¼»²±¬» ¾§ ÅËÃ ¬¸» »¨°±²»²¬·¿´ó¿´¹»¾®¿·½ »¨¬»²-·±² ±º ¬¸»
-§-¬»³ Ëô ·ò»ò ¿ -»¬
ÅËÃ ã º
õ
å
îË
î Î¹ Å º»¨°ø
õ
÷å
îË
î Î¹
©¸»®» »¨° æ ÿ Î ·- ¬¸» º±´´±©·²¹ ½±³°±-»¼ º«²½¬·±²æ ø»¨° ÷ø ÷ ã «ø¨÷
î
î ÅËÃô ¬¸» ²±¬¿¬·±²
ë
³»¿²- ¬¸¿¬ ø ÷ ë ø ÷ º±® ¿²§
¿®¾·¬®¿®§ °¿·®
³»²¬·±²»¼ ½±²-¬®«½¬·±² ©» ®»¿½¸ ¬¸» º±´´±©·²¹ ®»-«´¬æ
ò Ò±¬·½» ¬¸¿¬ º±® ¿²
î ò Ë-·²¹ ¬¸» ¶«-¬
Ì¸»±®»³ò Ô»¬
Ýø ÷ ¾» ¿ ½±²¬·²«±«- ³«´¬·ó«¬·´·¬§ ®»°®»-»²¬¿¬·±² ±º ¿ ½±²¬·²«±«- °®»±®¼»® ±² ¿ ¬±°±´±ó
¹·½¿´ -°¿½»
±º ¿´¬»®²¿¬·ª»- ¿²¼ ÅËÃ
Ë
¿ ¾·²¿®§ ¸§°»®±°»®¿¬·±²
æ ÅË Ã ÅËÃ ÿ ÐøÅËÃ÷
¾§
Å
ã
º å
î ÅËÃ
õ
ë
¹
Îõ
Å¿å¾ÃîÎõ
ð
ð
¬¸»² øÅËÃ ÷ ·- ¿ ø½±³³«¬¿¬·ª»÷ ¶±·² -°¿½»ò Ó±®»±ª»®ô ·º ÅË· Ã ·- ¬¸» »¨°±²»²¬·¿´ó¿´¹»¾®¿·½ »¨¬»²-·±² ±º ¿
½±²¬·²«±«- ³«´¬·ó«¬·´·¬§ ®»°®»-»²¬¿¬·±² Ë· ±º ¿ ½±²¬·²«±«- °®»±®¼»® ó· ô ã ï îô ±² ¬¸» ¬±°±´±¹·½¿´ -°¿½»
¿²¼ æ øÅËï Ã ëï ÷ ÿ øÅËî Ã ëî ÷ ·- ¿ ´·²»¿® ·²½®»¿-·²¹ ³¿°°·²¹ô ¬¸»² ·- ¿² ·²½´«-·±² ¸±³±³±®°¸·-³ ±º ¬¸»
¶±·² -°¿½» øÅËï Ã ï ÷ ·²¬± ¬¸» ¶±·² -°¿½» øÅËî Ã î ÷ò
Ð®±±ºò Ü»²±¬» ¬¸» -»¬ ±º º«²½¬·±²- º å î ÅËÃ
õ ë ¹ ¾§ Å õ ÷ë ò ß½½±®¼·²¹ ¬± ÅîðÃô Ì¸»±®»³ íòîô
©» ¹»¬ º±® ã ð ¬¸¿¬ ¬¸» ¸§°»®¹®±«°±·¼ øÅËÃ
«²¼»®´§·²¹ -¬®«½¬«®» ±º Ì¸»±®»³ íòî ·- Ýø ÷ô ½¿®»º«´ ®»¿¼·²¹ ±º ¬¸» °®±±º ´»¿¼- ¬± ¬¸» ½±²½´«-·±² ¬¸¿¬ ·º ¬¸»
½±²-·¼»®»¼ -§-¬»³ ±º º«²½¬·±²- ÅË
±º º«²½¬·±²-ô ¬¸» ¿--»®¬·±² ±º ¬¸» ¿¾±ª» ³»²¬·±²»¼ Ì¸»±®»³ íòî ®»³¿·²- ª¿´·¼ ©¸»² Ýø ÷ ·- -«¾-¬·¬«¬»¼ ¾§
ÅËÃò Ì¸»®»º±®»ô øÅËÃ ÷ ·- ¿ ¶±·² -°¿½»ò
Ú«®¬¸»®ô ·º æ øÅËï Ã ëï ÷ ÿ øÅËî Ã ëî ÷ ·- ¿² ·²½®»¿-·²¹ ´·²»¿® ³¿°°·²¹ô ¬¸»² ·² ¿ -·³·´¿® ©¿§ ¿- ·² ¬¸» °®±±º
±º Ð®±°±-·¬·±² î ©» ¸¿ª» º±® ¿² ¿®¾·¬®¿®§ °¿·® ±º º«²½¬·±²î ÅËï Ã ¬¸¿¬
Í
Í
Í
ø ï ÷ã ø
Å õ ÷ëï ÷ ã
øÅ õ ÷ëï ÷
Å ø õ ÷÷ëî ã
Å¿å¾ÃîÎõ
Îõ
ð
ð
Í
Å
Å¿å¾ÃîÎõ
ð
Îõ
ð
ø ÷õ
ø ÷÷ëî ã ø ÷
Å¿å¾ÃîÎõ
Îõ
ð
ð
ø ÷
î
Å¿å¾ÃîÎõ
Îõ
ð
ð
Ì¸«- ¬¸» ³¿°°·²¹
æ øÅËï Ã
ï÷
ÿ øÅËî Ã
î÷
·- ¿² ·²½´«-·±² ¸±³±³±®°¸·-³ ±º ¬¸» ®»-°»½¬·ª» ¶±·² -°¿½»-ò
Ò±© «-·²¹ ¬¸» ¾¿-·½ ·¼»¿ ±º ¬¸» ¿¾±ª» ½±²-¬®«½¬·±²-ô ±® ®¿¬¸»® ±º ¬¸» ½±²-¬®«½¬·±²- ·²½´«¼»¼ ·² ÅîðÃô ©» ½¿²
Ð
ø ÷ ±º ¿´´ °®»º»®»²½» ®»´¿¬·±² ±² ¬¸» -°¿½»
ã Ýµ ø ÷
µ
Îô î ºð ï î
¹ ·² ¿ -·³·´¿® ©¿§ò Í«°°±-» ¬¸¿¬ º±® ¿²§ ®»´¿¬·±²
î Ð
øÝ ø ÷÷ ¿²¼ ¿²§ º«²½¬·±²
î Ýµ ø ÷ ¬¸»®» »¨·-¬- ¿ º«²½¬·±² î Ýµ ø ÷ ©·¬¸ ¬¸» °®±°»®¬§ Å
Ã î ô ·ò»ò ¼±³ ã Ýµ ø ÷ò Í«½¸ ¿
îÐ
øÝµ ø
îÐ
øÝµ ø ÷÷ ©» °«¬
·º »·¬¸»® ã ±® º±® »¿½¸ º«²½¬·±² î
¿²¼ º±® ¿²§ °¿·® ±º º«²½¬·±²ø ÷ô ©¸»®» ø ÷ ã º î
æÅ
Ã î ¹ô ¿²¼ -·³·´¿®´§ º±® ô ¿²¼ º±® ¿²§ ®»¿´ ²«³¾»®
Å ï îÃ î ø ÷
î Î ¬¸» ®»´¿¬·±²
ã ºÅ
Ãå Å
Ã î ¹ò ×º ©» ½±²-·¼»® ¿ ¾·²¿®§ ¸§°»®±°»®¿¬·±²
æÐ
øÝµ ø ÷÷
øÝµ ø ÷÷ ÿ ÐøÐ
øÝµ ø ÷÷÷
îïì
ã
Í
Å¿å¾ÃîÎõ
ð
º å
îÐ
øÝµ ø ÷÷
õ
ë ¹ º±® ¿²§ °¿·® ±º °®»º»®»²½» ®»´¿¬·±²
î
Îõ
ð
øÝµ ø
Ð
øÝµ ø ÷÷ ÷ ·- ¿ ½±³³«¬¿¬·ª» ¯«¿-·ó¸§°»®¹®±«°ô ·ò»ò
Ð
¿ ½±³³«¬¿¬·ª» ¸§°»®¹®±«°±·¼ -¿¬·-º§·²¹ ¬¸» ®»°®±¼«½¬·±² ¿¨·±³ò É¸»¬¸»® ¬¸» ¸§°»®±°»®¿¬·±² þ þ ·- ¿´-±
Î»³¿®µò Ô»¬ «- ½±²½´«¼» ©·¬¸ ¿ ®»³¿®µ ¬¸¿¬ ¬¸» ¿®¾·¬®¿®§ -»¬ Îø ÷
ø
÷ ±º °®»º»®»²½» ®»´¿¬·±²±² ¿ ½±³³±¼·¬§ -»¬
½¿² ¾» »²¼±©»¼ ©·¬¸ ¿ ½±³³«¬¿¬·ª» ¸§°»®¹®±«°ò Þ»·²¹ ¿ -«¾-»¬ ±º ¿ °±©»® -»¬
Ðø
÷ ±º ¿´´ ¾·²¿®§ ®»´¿¬·±²- ±² ô ¬¸» -§-¬»³ Îø ÷ ±º °®»º»®»²½» ®»´¿¬·±²- ·- ²¿¬«®¿´´§ °¿®¬·¿´´§
±®¼»®»¼ ¾§ -»¬ ·²½´«-·±² þ
ãº å
î Îø
÷
±®
¹
©» ¹»¬ »¿-·´§ ¬¸¿¬ ¬¸» °¿·® øÎø ÷ ÷ ·- ¿ ½±³³«¬¿¬·ª» ¸§°»®¹®±«° ø½º ¿´-± ÅïìÃ÷ò ×º ¿ -§-¬»³ ±º °®»º»®»²½»
®»´¿¬·±²- ±²
º±®³- ¿ ´¿¬¬·½»ô ·ò»ò ¿² ±®¼»®»¼ -»¬ ·² ©¸·½¸ ¿²§ ¬©±ó»´»³»²¬ -«¾-»¬ ¸¿- ¬¸» ´»¿-¬ «°°»®
¾±«²¼ ø½¿´´»¼ -«°®»³«³ ±® ¶±·²÷ ¿²¼ ¬¸» ¹®»¿¬»-¬ ´±©»® ¾±«²¼ ø½¿´´»¼
±® ³»»¬ ÷ô ¬¸»² ¬¸»®» ½¿² ¾»
½±²-¬®«½¬»¼ º«®¬¸»® ¾·²¿®§ ¸§°»®±°»®¿¬·±²- ±² ¬¸» ½±®®»-°±²¼·²¹ ´¿¬¬·½» ±º °®»º»®»²½» ®»´¿¬·±²-ò ß ½»®¬¿·²
-«®ª»§ ±º °±--·¾´» ¾·²¿®§ ¸§°»®±°»®¿¬·±²- ±² ´¿¬¬·½»- ·- ·²½´«¼»¼ ·² °¿°»® ÅïëÃô °ò îìïò
ßÝÕÒÑÉÔÛÜÙÛÓÛÒÌ Ì¸» ¿«¬¸±®- ©»®» -«°°±®¬»¼ ¾§ ¬¸» Ý±«²½·´ ±º Ý¦»½¸ Ù±ª»®²³»²¬ ¹®¿²¬ ÓÍÓ
ððîïêíðëîçò
Î»º»®»²½»
ÅïÃ ßÞÞßÍô Óòå Ê×ÒÝÕÛô Ðò Ð®»º»®»²½» -¬®«½¬«®»- ¿²¼ ¬¸®»-¸±´¼ ³±¼»´-ò Ö±«®²¿´ ±º Ó«´¬·½®·¬»®·¿ Ü»½·-·±²
ß²¿´§-·-ô î øïççí÷ô ïéï¥ïéèò
ÅîÃ ßÒÜÎWÕßô Øòå ÎÇßÒô Óòå ÍÝØÑÞÞÛÒÍô ÐòóÇòæ Ñ°»®¿¬±®- ¿²¼ Ô¿©- º±® Ý±³¾·²·²¹ Ð®»º»®»²½» Î»ó
´¿¬·±²-ò Ö±«®²¿´ ±º Ô±¹·½ ¿²¼ Ý±³°«¬¿¬·±² îððî ïîøï÷æïí¥ëíå ¼±·æïðòïðçíñ´±¹½±³ñïîòïòïíò
ÅíÃ ßÎÎÑÉô Õò Öò Î¿¬·±²¿´ ½¸±·½» º«²½¬·±²- ¿²¼ ±®¼»®·²¹-ò Û½±²±³»¬®·½¿ îê øïçëç÷ô îï¥ïîéò
ÅìÃ ÞÛÎ_ÒÛÕô Öòå ÝØÊßÔ×Òßô Öò
Ü»°¬ò Ó¿¬¸ò Î»°±®¬ Í»®·»- Ê±´ò ïðô Ë²·ªò Í±«¬¸ò Þ±¸ò X»-µ7 Þ«¼4¶±ª·½»ô îððîô ïë¥îîò
ÅëÃ Þ×ÎÕØÑÚÚô Ùò Ô¿¬¬·½» Ì¸»±®§ò ßÓÍ Ý±´´±¯ò Ð«¾´ò Ê±´ò îëô Ð®±ª·¼»²½»ô Î¸±¼» ×-´¿²¼ô ïçéçò
ÅêÃ ÝÑÎÍ×Ò×ô Ðò Ð®±´»¹±³»²¿ ±º Ø§°»®¹®±«° Ì¸»±®§ò Ì®·½»-·³±ô ßª·¿²· Û¼·¬±®»ô ïççíò
ÅéÃ ÝÑÎÍ×Ò×ô Ðòå ÔÛÑÎÛßÒËô Êò ß°°´·½¿¬·±²- ±º Ø§°»®-¬®«½¬«®» Ì¸»±®§ò Ü±®¼®»½¸¬ô Õ´«©»® ß½¿¼»³·½
Ð«¾´·-¸»®-ô îððíò ×ÍÞÒ ïóìðîðóïîîîóëò
ÅèÃ ÜÎÛÍØÛÎô Óòå ÑÎÛô Ñò Ì¸»±®§ ±º ³«´¬·¹®±«°-ò ß³»®ò Öò Ó¿¬¸ò êð øïçíè÷ô éðë¥éííò
ÅçÃ Ú×ßÔßô Ðòô ÖßÞÔÑÒÍÕCô Öòå ÓßNßÍô Óò Ó«´¬·µ®·¬»®·?´²3 ®±¦¸±¼±ª?²3ò ÊwÛ ª Ð®¿¦»ô Ú¿µ«´¬¿ ·²º±®ó
³¿¬·µ§ ¿ -¬¿¬·-¬·µ§ô Ð®¿¸¿ ïççìò ×ÍÞÒ èðóéðéçóéìèóéò
ÅïðÃ ÛÊÎÛÒô Ñòå Ñµô Ûòßò Ñ² ¬¸» Ó«´¬·óË¬·´·¬§ Î»°®»-»²¬¿¬·±² ±º Ð®»º»®»²½» Î»´¿¬·±²-ò Ó·³»±¹®¿°¸»¼ô
Ü»°¬ò ±º Û½±²±³·½-ô Ò»© Ç±®µ Ë²·ª»®-·¬§ô Ó¿®½¸ îððëô îï°°ò
ÅïïÃ ÚÑÜÑÎô Öò Ð®»º»®»²½» ®»´¿¬·±²- ·² ¼»½·-·±² ³±¼»´-ò ÑÌÕß ÌÑ ìêéêî ¿²¼ ÞÍÌÝ Ú´¿²¼»®»- ¥ Ø«²¹¿®§
Þ×Ô ððñëïò îðððô é°°ò Ð®»°®·²¬ò
ÅïîÃ ØÑÎÌô Üòå ÝØÊßÔ×Òßô Öòå ÓÑËXÕßô Öò Ý¸¿®¿½¬»®·¦¿¬·±²- ±º ¬±¬¿´´§ ±®¼»®»¼ -»¬- ¾§ ¬¸»·® ª¿®·±«»²¼±³±®°¸·-³-ò Ý¦»½¸ò Ó¿¬¸ò Öò èîøïîé÷ô îððîô îí¥íîò
ÅïíÃ ØÑwÕÑÊ_ô wòå ÝØÊßÔ×Òßô Öò
®¿¬±®-ò Öò Þ¿-·½ Í½·»²½» íô Ò± ï øîððê÷ô ïï¥ïéò Ë²·ª»®-·¬§ ±º Ó¿¦¿²¼¿®¿²ô ×®¿²ò
ÅïìÃ ÝØÊßÔ×Òßô Öò Ú«²µ½·±²?´²3 ¹®¿º§ô µª¿¦·«-°±(?¼¿²7 ³²±b·²§ ¿ µ±³«¬¿¬·ª²·3 ¸§°»®¹®«°§ò Ó¿-¿®§µ±ª¿
«²·ª»®¦·¬¿ Þ®²±ô ïççëò
ÅïëÃ ÝØÊßÔ×Òßô Öòå ÝØÊßÔ×ÒÑÊ_ô Ôò Þ±¬¬´»²»½µ ¿´¹»¾®¿- ±º °®»º»®»²½» ®»´¿¬·±²-ò ×²æ ×ÝÍÝó Ú·º¬¸ ×²¬»®ó
²¿¬·±²¿´ Ý±²º»®»²½» ±² Í±º¬ Ý±³°«¬·²¹ ß°°´·»¼ ·² Ý±³°«¬»® ¿²¼ Û½±²±³·½ Û²ª·®±²³»²¬-ò Õ«²±ª·½»æ
ÛÐ× Õ«²±ª·½»ô îððéò °° îïí¥îïèò ×ÍÞÒ èðóéíïìóïðèóêò
îïë
ÅïêÃ ÝØÊßÔ×Òßô Öòå ÝØÊßÔ×ÒÑÊ_ô Ôò
±®¼»®ò
Þ®«²»²-·-ô Ó¿¬¸ò îððíô éé¥èêò ×ÍÞÒ èðóîïðóíïìçóîò
ÅïéÃ ÝØÊßÔ×Òßô Öòå ÝØÊßÔ×ÒÑÊ_ô Ôò Ì®¿²-°±-·¬·±² ¸§°»®¹®±«°- º±®³»¼ ¾§ ¬®¿²-º±®³¿¬·±² ±°»®¿¬±®- ±²
×²æ ×¬¿´·¿² Öò Ð«®» ß°°´ò Ó¿¬¸ò Ö¿²«¿®§ îððìô çíóïðêò
ÅïèÃ ÝØÊßÔ×Òßô Öòå ÓÑËXÕßô Öò Ð®±-¬±®§ °®»º»®»²8²3½¸ ®»´¿½3 ¿ ¸§°»®-¬®«µ¬«®§ò ×²æ ÈÈ× ×²¬»®²¿¬ò Ý±´´±¯ò
±² ß½¯«·-·¬·±² Ð®±½»-- Ó¿²¿¹»³»²¬ô Ê§gµ±ªæ ÊÊw ÐÊ îððí ÅÝÜóÎÑÓÃò
ÅïçÃ ÝØÊßÔ×Òßô Öòå ÓÑËXÕßô Öò É¸»² ½»®¬¿·² ¾¿-·½ ¸§°»®±°»®¿¬·±²- ±² ´¿¬¬·½»- ½±·²½·¼»ò ×²æ Ó¿¬»³ò
É±®µ-¸±° - ³»¦·²?®±¼²3 &8¿-¬3ô ÚßÍÌ ÊËÌ Þ®²± îððêò ÅÝÜóÎÑÓÃò ×ÍÞÒ èðóîïìóíîèîóçò
ÅîðÃ ÝØÊßÔ×Òßô Öòå ÎßXÕÑÊ_ô Ðò Ö±·² Í°¿½»- ±º -³±±¬¸ º«²½¬·±²- ¿²¼ ¬¸»·® ¿½¬·±²- ±² ¬®¿²-°±-·¬·±²
×²æ Ð®±½ò é¬¸ ×²¬»®²¿¬·±²¿´ Ý±²º»®»²½» ßÐÔ×ó
ÓßÌ îððè ø·² °®·²¬÷ò
ÅîïÃ ÝØÊßÔ×ÒÑÊ_ô Ôò Í»¬ ·²¬»®°®»¬¿¬·±² ±º Ð¿®»¬± °®·²½·°´»ò ×²æ Ð®±½ò ÈÈ×××ò ×²¬»®²¿¬ò Ý±´´±¯ò ±² ß½¯«·ó
-·¬·±² Ð®±½»-- Ó¿²¿¹»³»²¬ô Ë²·ªò ±º Ü»º»²½» Þ®²±ô Ó¿§ îððëò ÅÝÜóÎÑÓÃò ×ÍÞÒ èðóèëçêðóçíóíò
ÅîîÃ ÖßÕËÞSÕÑÊ_óÍÌËÜÛÒÑÊÍÕ_ô Üò Ó±²±«²¿®§ ¿´¹»¾®¿ ¿²¼ ¾±¬¬´»²»½µ ¿´¹»¾®¿-ò ß´¹»¾®¿ Ë²·ª»®-¿´·ìð øïççè÷ô ëç¥éîò
ÅîíÃ ÕËÆÓ×Òô Êò Þò Ð±-¬®±¶»²·¶» ¹®«°°±ª§½¸ ®»g»²·¶ ª °®±-¬®¿²-¬ª¿½¸ 8±¬µ·½¸ · ²»8±¬µ·½¸ ®»g»²·¶ò Ó±-µª¿
Ò¿«µ¿ ïçèîò
ÅîìÃ Ó×ÕËÔßô Êò Û¨°´±·¬¿¬·±² ±º º«¦¦§ ´±¹·½ ·² ½±²¬®±´ ¿²¼ ¼»½·-·±² °®±½»--»-ò ×²æ Ê§-±µ? gµ±´¿ ¶¿µ± º¿½·´·ó
¬?¬±® ®±¦ª±¶» -°±´»8²±-¬·ò ×²¬»®²¿¬·±²¿´ Ý±²ºòô ÛÐ× Õ«²±ª·½»ô Ý¦»½¸ Î»°«¾´·½ô Ö¿²«¿®§ îððëô ííí¥íìðò
ÅîëÃ Ó×ÎÕ×Òô Þò Ùò Ù®±«° Ý¸±·½»ò Öò É·´»§ ú Í±²-ô Ì±®±²¬±ô Ô±²¼±²ô Í§¼²»§ ïçéçò
ÅîêÃ JÌÆÌDÎÕô Óòå ÌÍÑËÕ×ßÍô ßòô Ê×ÒÝÕÛô Ðò Ð®»º»®»²½» ³±¼»´´·²¹ò Ü×ÓßÝÍ Ì»½¸ò Î»°±®¬ îððíóíìô
Ñ½¬±¾»® íðô îððíô ìí°°ò
ÅîéÃ ÐÛÌÛÎÍô Óò Ð®»º»®»²½»-ò îððëô ç°°ò Ð®»°®·²¬ò
ÅîèÃ ÎÑÍÛÒÍÌÛ×Òô Öò Ô·²»¿® Ñ®¼»®·²¹-ò ß½¿¼»³·½ Ð®»--ô Ò»© Ç±®µô ïçèîò
ÅîçÃ ÍÛÒô ßò Õò Ý±´´»½¬·ª» Ý¸±·½» ¿²¼ Í±½·¿´ É»´º¿®»ò Ñ´·ª»® ¿²¼ Þ±§¼ô Û¼·²¾±«®¹¸ô ïçéðò
ßÜÜÎÛÍÍæ
Ð®±ºò ÎÒÜ®ò Ö¿² Ý¸ª¿´·²¿ô Ü®Í½ò
Ü»°¿®¬³»²¬ ±º Ó¿¬¸»³¿¬·½Ú¿½«´¬§ ±º Û´»½¬®·½¿´ Û²¹·²»»®·²¹ ¿²¼ Ý±³³«²·½¿¬·±²ô
Þ®²± Ë²·ª»®-·¬§ ±º Ì»½¸²±´±¹§
Ì»½¸²·½µ? èô êïêðð ÞÎÒÑô Ý¦»½¸ Î»°«¾´·½
½¸ª¿´·²¿àº»»½òª«¬¾®ò½¦
¼±½ò ÎÒÜ®ò Ö·(3 Ó±«8µ¿ô Ð¸òÜò
Ü»°¿®¬³»²¬ ±º Û½±²±³»¬®·½Ú¿½«´¬§ ±º Û½±²±³·½- ¿²¼ Ó¿²¿¹»³»²¬ô
Ë²·ª»®-·¬§ ±º Ü»º»²½»
Õ±«²·½±ª¿ êëô êðîðð ÞÎÒÑô Ý¦»½¸ Î»°«¾´·½
Ö·®·òÓ±«½µ¿à«²±¾ò½¦
Ó¹®ò Ó·½¸¿´ Ò±ª?µô Ð¸òÜò
Ü»°¿®¬³»²¬ ±º Ó¿¬¸»³¿¬·½Ú¿½«´¬§ ±º Û´»½¬®·½¿´ Û²¹·²»»®·²¹ ¿²¼ Ý±³³«²·½¿¬·±²ô
Þ®²± Ë²·ª»®-·¬§ ±º Ì»½¸²±´±¹§
Ì»½¸²·½µ? èô êïêðð ÞÎÒÑô Ý¦»½¸ Î»°«¾´·½
²±ª¿µ³àº»»½òª«¬¾®ò½¦
îïê
ÐÛÎ×ÑÜ×Ý ÍÑÔËÌ×ÑÒÍ ÑÚ ÊÑÔÌÛÎÎß
×ÒÌÛÙÎÑÜ×ÚÚÛÎÛÒÌ×ßÔ ÛÏËßÌ×ÑÒÍ É×ÌØ Ü×ÚÚÛÎÛÒÝÛ
ÕÛÎÛÒÛÔ
Ê´¿-¬¿ Õ®«°µ±ª?ô Æ¼»²4µ w³¿®¼¿
Ú¿½«´¬§ ±º Û´»½¬®·½¿´ Û²¹·²»»®·²¹ ¿²¼ Ý±³³«²·½¿¬·±²ô Þ®²± Ë²·ª»®-·¬§ ±º Ì»½¸²±´±¹§
ß¾-¬®¿½¬æ
¿´´ -±´«¬·±²- ±º ¬¸» ¹·ª»² »¯«¿¬·±²- ¬»²¼ ¬± Ìó°»®·±¼·½ -±´«¬·±² ¿- ¬ ÿ ïò Ý¿´½«´«- ¿®» ¹·ª»² ¿©»´´ò
Õ»§ ©±®¼-æ
¼»´¿§ô °»®·±¼·½ -±´«¬·±²- ±½½«® ©¸»² -±´«¬·±²- ¿®» «²·º±®³´§ ¾±«²¼»¼ ø-»» ÅïóêÃ÷ ò ×¬ ·- ¿´-± µ²±©² ¬¸¿¬
¬
¨ð ø¬÷ ã ß¨ø¬÷ õ
Õø¬
-÷¨ø-÷¼- õ Ú ø¬÷
øï÷
·² ©¸·½¸ ß ·- ¿² ² ² ½±²-¬¿²¬ ³¿¬®·¨ô Õ ·¿ ¿² ² ² ³¿¬®·¨ º«²½¬·±² ½±²¬·²«±«- ±² Î ô Ú î ÅÎå Î² Ã ¿²¼
Ú ø¬õÌ ÷ ã º ø¬÷ º±® ¿´´ ¬ î Î ¿²¼ -±³» Ì â ðò ×² º¿½¬ º±® øï÷ ·¬ ·- »¿-§ ¬± ª»®·º§ ¬¸¿¬ ·º ¨ø¬÷ ·- ¿²§ -±´«¬·±² ¬¸»²
-± ·- ¨ø¬ õ Ì
ø-»»ÅéóïîÃ÷
¬
§ð ø¬÷ ã ß§ø¬÷ õ
-÷§ø-÷¼- õ Ú ø¬÷æ
Õø¬
øî÷
ð
§ø¬÷ ã ½±- ¬ õ -·² ¬ ·- ¿ -±´«¬·±² ±º
¬
§ ð ø¬÷ ã §ø¬÷ õ
»
ø¬ -÷
§ø-÷¼-
í -·² ¬å
ð
¿²¼ ½±²-»¯«»²¬´§ ¬¸» -¬«¼¦ ±º °»®·±¼·½ -±´«¬·±²- ±º øî÷ ·- ·²¬»®»-¬·²¹ ò ×² º¿½¬ô ©» -¸¿´´ -¸±© ¬¸¿¬ ¿-§³°¬±ó
¬·½¿´´§ °»®·±¼·½ -±´«¬·±²- »¨·-¬ º±® øî÷ò
ß² ¿-§³°¬±¬·½¿´´§ Ì ó°»®·±¼·½ º«²½¬·±² º ·- ¿ ¾±«²¼»¼ ½±²¬·²«±«- º«²½¬·±² º±® ©¸·½¸ ¬¸»®»
»¨·-¬- ¿ ½±²¬·²«±«- Ì ó°»®·±¼·½ º«²½¬·±² ¯ ¿²¼ ¿² ·²¬»¹»® ² -«½¸ ¬¸¿¬
º ø¬ õ ²Ì ÷
¯ø¬÷ ÿ ð
«²·º±®³´§ ·² Åðå Ì Ã ¿- ² ÿ ïò Ì¸·- ·- »¯«·ª¿´»²¬ ¬± ¬¸» ½±²¼·¬·±² º ø¬÷
¯ø¬÷ ÿ ð ¿- ¬ ÿ ïò
Î»½¿´´ ¬¸¿¬ ¾§ ª¿®·¿¬·±² ±º °¿®¿³»¬»®- º±®³«´¿ ¬¸» -±´«¬·±² §ø¬÷ ±º øî÷ ·- ¹·ª»² ¾§
¬
§ø¬÷ ã Æø¬÷§øð÷ õ
Æø¬
-÷Ú ø-÷¼-å
øí÷
ð
©¸»®» Æø¬
-¿¬·-º§·²¹ ¬¸» ·²·¬·¿´ ª¿´«» °®±¾´»³
Õø¬÷ ¿²¼ ·- ¿² ²
² ³¿¬®·¨ º«²½¬·±²
¬
Æ ð ø¬÷ ã ßÆø¬÷ õ
Õø¬
-÷Æø-÷¼-å
Æøð÷ ã ×æ
øì÷
ð
îïé
îò Óß×Ò ÎÛÍËÔÌÍ
Í§-¬»³ øî÷ ·- ½´±-»´§ ®»´¿¬»¼ ¬± ·¬- °»®¬«®¾¿¬·±² ø±® ´·³·¬·²¹÷ -§-¬»³ øï÷ô ¿²¼ ¸»²½» ·¬- ·²ª»-¬·¹¿¬·±² ¼»°»²¼±² °®±°»®¬·»- ±º -±´«¬·±²- ±º øï÷ ©·¬¸ ¾±«²¼»¼ ½±²¬·²«±«- ·²·¬·¿´ º«²½¬·±²- æ ø
å ðÃ ÿ Î² æ
É» «-» ¬¸» º±´´±©·²¹ ®»-«´¬- ·² ±«® -«¾-»¯«»²¬ ¼·-½«--·±²-ò
Ô»³³¿ îòïò ×º Õø¬÷å Æø¬÷ î Ôï Åðå ï÷ ¬¸»² Æø¬÷ ÿ ð ¿- ¬ ÿ ïò
Ð®±±ºò Í·²½» ß ·- ¿ ½±²-¬¿²¬ ³¿¬®·¨ ¿²¼ Æø¬÷ î Ôï Åðå ï÷ ·¬ ·- ½´»¿® ¬¸¿¬ ßÆø¬÷ î Ôï Åðå ï÷æ ß´-± ¬¸»
½±²ª±´«¬·±² ±º ¬©± º«²½¬·±²- ·² Ôï Åðå ï÷ ·- ·² Ôï Åðå ï÷ ¿²¼ ¸»²½»
¬
-÷Æø-÷¼- î Ôï Åðå ï÷æ
Õø¬
ð
Ì¸»®»º±®» º®±³ øì÷ Æ ð ø¬÷ î Ôï Åðå ï÷ ò Ì¸·- ·³°´·»- ¬¸¿¬ Æø¬÷ ¸¿- ¿ ´·³·¬ ¿- ¬ ÿ ïò Þ«¬ Æø¬÷ î Ôï Åðå ï÷ò
Ø»²½» Æø¬÷ ÿ ð ¿- ¬ ÿ ï ¿²¼ ¬¸·- ½±³°´»¬»- ¬¸» °®±±ºò
Ô»³³¿ îòîò ß--«³» ¬¸¿¬
ø·÷ Õ î Ôï Åðå ï ¿²¼ Ú ø¬÷ ·- ½±²¬·²«±«- º±® ¬ î Î ©·¬¸ Ú ø¬ õ Ì ÷ ã Ú ø¬÷ º±® ¿´´ ¬ î Î ¿²¼ -±³» ½±²-¬¿²¬
Ì â ðò
ø··÷ Ì¸»®» »¨·-¬- ¿ -±´«¬·±² §ø¬÷ ±º øî÷ ¿²¼ ·- ¾±«²¼»¼ ±² Îõ ò
Ì¸»² ¬¸»®» »¨·-¬- ¿ -»¯«»²½» ø ¶ ÷ ©·¬¸ ¶ ÿ ï ¿- ¶ ÿ ï -«½¸ ¬¸¿¬ §ø¬ õ ¶ Ì ÷ ÿ ¨ø¬÷ ©¸»®» ¨ø¬÷ ·- ¿
-±´«¬·±² ±º øï÷ ±² Î ¿²¼ ¬¸» ½±²ª»®¹»²½» ·- «²·º±®³ ±² ¿²§ ½±³°¿½¬ -«¾-»¬ ±º Îò
Ð®±±ºò Ô»¬ Ó ¾» ¿ ¾±«²¼ º±® ¶§ø¬÷¶ ±² Åðå ï÷æ Ì¸»² º®±³ øî÷ ©» ¸¿ª»
¬
ß¶¶§ø¬÷¶ õ
¶§ ð ø¬÷
¬
Ó ¶ß¶ õ Ó
-÷¶¶§ø-÷¶¼-
¶Õø¬
¶Õø-÷¶¼-æ
ð
ð
Í·²½» Õ î Ôï Åðå ï÷ ·¬ ·- ½´»¿® ¬¸¿¬ ¶§ð ø¬÷¶ ·- ¾±«²¼»¼ ±² Îõ ò Ì¸»®»º±®» §ø¬÷ ·- «²·º±®³´§ ½±²¬·²«±«- ±² Îõ ò
Ô»¬ ø ¶ ÷ ¾» ¿ -»¯«»²½» ±º ·²¬»¹»®- -«½¸ ¬¸¿¬ ¶ ÿ ï ¿- ¶ ÿ ïæ Ì¸»² ¾§ ß-½±´·ù- ¬¸»±®»³ ¬¸» -»¯«»²½»
ø§ø¬ õ ¶ Ì ÷÷ ½±²¬¿·²- ¿ -«¾-»¯«»²½» -«½¸ ¬¸¿¬ §ø¬ õ ¶³ Ì ÷ ÿ ¨ø¬÷ ¿- ³ ÿ ï ©·¬¸ «²·º±®³ ½±²ª»®¹»²½» ±²
½±³°¿½¬ -«¾-»¬- ±º Îò Ú«¬¸»® º®±³ ¬¸» ¿--«³°¬·±² ø··÷ ·¬ ·- ±¾ª·±«- ¬¸¿¬ Ú ø¬ õ ¶³ Ì ÷ ã Ú ø¬÷ º±® ¿´´ ¬ î Îò
Ô»¬ §³ ø¬÷ ã §ø¬ õ ¶³ Ì ÷ ¿²¼ Ú³ ø¬÷ ã Ú ø¬ õ ¶³ Ì ÷ò Ì¸»² º®±³ øî÷ º±® ¿²§ ³ ¿²¼ ¬
¶³ ©» ±¾¬¿·²
¬
ð
§³
ø¬÷ ã ß§³ ø¬÷ õ
-÷§³ ø-÷¼- õ Ú³ ø¬÷æ
Õø¬
¶³
Ì¸·- ·³°´·»- ¬¸¿¬
¬
§³ ø¬÷ ã §³ øð÷ õ
¬
ß§³ ø-÷¼- õ
ð
¬
-
Ú³ ø-÷¼- õ
Õø-
ð
ð
÷§³ ø ÷
¼-æ
øë÷
¶³
Ô»¬
³¿¨º¶§³ ø-÷
Ì¸»² º±® ¿²§ ®»¿´ ²«³¾»® -ô ·º - õ
¶³
¨ø-÷¶ æ
³
-
ïã³æ
³
â ð ô ·¬ º±´´±©- ¬¸¿¬
-
-
÷¨ø ÷
Õø-
÷§³ ø ÷
Õø¶³
³
Ó
³
Õø-
÷
õÓ
-
÷
Õø-
õ
ï
îÓ
÷¶¶¨ø ÷
¶Õø-
§³ ø ÷¶
³
¶³
¶Õø ÷¶
-õ³
õ
ï
³
-õ³
¶µø ÷¶
ÿð
ð
¿- ³ ÿ ï ò Ì¸»®»º±®» ©» ¬¿µ» ¬¸» ´·³·¬ ·² øë÷ ¿- ³ ÿ ï ¿²¼ ±¾¬¿·²
¬
¨ø¬÷ ã ¨øð÷ õ
¬
ß¨ø-÷¼- õ
ð
¬
Ú ø-÷¼- õ
ð
-
Õø-
÷¨ø ÷
¼-æ
ð
îïè
¬ ©» ¹»¬
¨ð ø¬÷ ã ß¨ø¬÷ õ
-÷¨ø-÷¼- õ Ú ø¬÷
Õø¬
¿²¼ ¸»²½» ¨ø¬÷ ·- ¿ -±´«¬·±² ±º øï÷ ò Ì¸·- ½±³°´»¬»- ¬¸» °®±±ºò
Ì¸»±®»³ îòïò Ô»¬ ¬¸» ¿--«³°¬·±²- ±º Ô»³³¿- îòïò ¿²¼ îòîò ¸±´¼ ò Ì¸»² ¬¸» »¯«¿¬·±² øï÷ ¸¿- ¿ Ì ó°»®·±¼·½
-±´«¬·±²
¬
-÷Ú ø-÷¼-å
Æø¬
ð ©·¬¸ ¾±«²¼»¼ ½±²¬·²«±«- ·²·¬·¿´ º«²½¬·±²- ±² ø
¬
å ðÃ ¬»²¼ ¬± Ì ó¬¸·-
°»®·±¼·½ -±´«¬·±² ¿- ¬ ÿ ïò
Ð®±±ºò Í·²½» Õ ¿²¼ Æ î Ôï Åðå ï÷ ·¬ º±´´±©- ¾§ Ô»³³¿ îòïò ¬¸¿¬
´·³ Æø¬÷ ã ðæ
¬ÿï
Ì¸»®»º±®» ¾§ øí÷ ¿´´ -±´«¬·±²- ±º øî÷ ¿®» ¾±«²¼»¼ò Ñ² ¬¸» ±¬¸»® ¸¿²¼ º®±³ øí÷ ©» ¹»¬
¬õ
§ø¬ õ
¶Ì ÷
ã Æø¬ õ
¶Ì
¶ Ì ÷§øð÷ õ
Æø¬ õ
¶Ì
-÷Ú ø-÷¼- ã
ð
¬
¶ Ì ÷§øð÷ õ
ã Æø¬ õ
¶ Ì ÷¼«æ
«÷Ú ø« õ
Æø¬
¶Ì
Í·²½» Ú ø« õ
¶Ì ÷
ã Ú ø«÷ ©» ±¾¬¿·²
¬
§ø¬ õ
¶Ì÷
ã Æø¬ õ
¶ Ì ÷§øð÷
«÷Ú ø«÷¼«
Æø¬
¶Ì
¿²¼ ¬¸«-
¬
§ø¬ õ
¿- ¶ ÿ ïæ Þ§ Ô»³³¿ îòîò
Ì ò Í«°°±-» ¬¸¿¬ ¨ø¬
©» ¸¿ª»
¬
Æø¬
¶Ì ÷
ÿ
«÷Ú ø«÷¼-
Æø¬
-÷Ú ø-÷¼- ·- ¿ -±´«¬·±² ±º øï÷ ¿²¼ ¬¸·- ·- ¿ °»®·±¼·½ º«²½¬·±² ©·¬¸ °»®·±¼
¬ ð ©·¬¸ ¾±«²¼»¼ ·²·¬·¿´ º«²½¬·±² ±² ø
å ðÃæ Ì¸»²
ð
¬
¨ð ø¬÷ ã ß¨ø¬÷
-÷¨ø-÷¼- õ
Õø¬
-÷ ø-÷¼- õ Ú ø¬÷æ
Õø¬
ð
Ú®±³ ¬¸» ª¿®·¿¬·±² ±º °¿®¿³»¬»®- º±®³«´¿ ©» ±¾¬¿·²
ð
¬
¨ø¬÷ ã Æø¬÷¨øð÷ õ
Æø¬
-÷ Ú ø-÷ õ
Õø-
÷ ø ÷
¼-æ
øê÷
ð
×² ª·»© ±º ¬¸» ¾±«¼»¼²»-- ±º
¿²¼ Õ î Ôï Åðå ï÷ ·¬ º±´´±©- ¬¸¿¬
ð
ï
Õø-
÷ ø ÷
-«° ¶ ø ÷¶
¶Õø«÷¶¼«å
-
©¸·½¸ ¬»²¼- ¬± ¦»®± ¿- - ÿ ïò Ì¸«- ¬¸» ´¿-¬ ¬»®³ ±² ¬¸» ®·¹¸¬ó¸¿²¼ -·¼» ±º øê÷ ·- ¬¸» ½±²ª±´«¬·±² ±º ¿²
Ôï óº«²½¬·±² ¿²¼ ¿ º«²½¬·±² ¬»²¼·²¹ ¬± ¦»®± ¿²¼ ¬¸»®»º±®» ¬¸» ½±²ª±´«¬·±² ¬»²¼- ¬± ¦»®± ¿- ¬ ÿ ï ò Ú«®¬¸»®
¬
Æø¬÷ ÿ ð ¿- ¬ ÿ ï ¾§ Ô»³³¿ îòïò Ø»²½» ·¬ º±´´±©- º®±³ øê÷ ¬¸¿¬ ¨ø¬÷ ÿ
Æø¬ -÷Ú ø-÷¼- ¿- ¬ ÿ ï
¿²¼ ¬¸·- ½±³°´»¬»- ¬¸» °®±±ºò
øî÷ ¸¿- ¿ °»®·±¼·½ -±´«¬·±²
Ì¸»±®»³ îòîò
«ø¬÷ ¬¸»² «ø¬÷ ·- ¬¸» «²·¯«» °»®·±¼·½ -±´«¬·±² ±º øï÷ò
Ð®±±ºò Þ§ Ì¸»±®»³ îòïò ¬¸» »¯«¿¬·±² øï÷ ¸¿- ¿ °»®·±¼·½ -±´«¬·±² ªø¬÷ ò Ð«¬ ©ø¬÷ ã ªø¬÷
øï÷ ¿²¼ øî÷ ©» ¹»¬
ð
¬
©ð ø¬÷ ã ß©ø¬÷ õ
«ø¬÷ò Ì¸»² º®±³
Õø¬
-÷©ø-÷¼- õ
Õø¬
-÷ªø-÷¼-æ
øé÷
ð
îïç
Þ§ ¬¸» ª¿®·¿¬·±² ±º °¿®¿³»¬»®- ¿²§ -±´«¬·±² ©ø¬÷ ±º øé÷ ½¿² ¾» »¨°®»--»¼ ¿ð
¬
©ø¬÷ ã Æø¬÷©øð÷ õ
Æø¬
-÷
÷ªø ÷
Õø-
¼-æ
øè÷
ð
×² ª·»© ±º ¬¸» ¾±«¼»¼²»-- ±º ªø¬÷ ¿²¼ Õ î Ôï Åðå ï÷ ·¬ º±´´±©ð
ï
Õø-
÷ªø ÷
-«° ¶ªø ÷¶
¶Õø«÷¶¼«å
-
©¸·½¸ ¬»²¼- ¬± ¦»®± ¿- - ÿ ï ò Ì¸«- ¬¸» ´¿-¬ ¬»®³ ±² ¬¸» ®·¹¸¬ó¸¿²¼ -·¼» ±º øè÷ ·- ¬¸» ½±²ª±´«¬·±² ±º ¿²
Ôï óº«²½¬·±² ©·¬¸ ¿ º«²½¬·±² ¬»²¼·²¹ ¬± ¦»®±ô ¿²¼ ¸»²½» ¬¸» ½±²ª±´«¬·±² ¬»²¼- ¬± ¦»®±ò Ó±®»±ª»®ô ¿°°´·½¿¬·±²
±º Ô»³³¿ îòïò §·»´¼- ¬¸¿¬ Æø¬÷ ÿ ð ¿- ¬ ÿ ïò Ì¸·- -¸±©- ¬¸¿¬ ©ø¬÷ ÿ ð ¿- ¬ ÿ ïò Ì¸«- ©» ¹»¬ ªø¬÷ «ø¬÷ò
Ì¸» °®±±º ·- ½±³°´»¬»ò
Ì¸·- ®»-»¿®½¸ ¸¿- ¾»»² -«°°±®¬»¼ ¾§ ¬¸» Ý¦»½¸ Ó·²·-¬®§ ±º Û¼«½¿¬·±² ·² ¬¸» º®¿³»- ±º ÓÍÓððîïêðëðí Î»ó
-»¿®½¸ ×²¬»²¬·±² Ó×ÕÎÑÍÇÒ Ò»© Ì®»²¼- ·² Ó·½®±»´»½¬®±²·½ Í§-¬»³- ¿²¼ Ò¿²±¬»½¸²±´±¹·»- ¿²¼ ÓÍÓððîïêíðëîç
Î»-»¿®½¸ ×²¬»²¬·±² ×²¬»´·¹»²¬ Í§-¬»³- ·² ß«¬±³¿¬·¦¿¬·±²ò
Î»º»®»²½»
ÅïÃ ÞËÎÌÑÒôÌòßòæ
Ð«¾´·½¿¬·±²-ô ×²½ò Ó·²»±´¿ô Ò»© Ç±®µô îððëô ×ÍÞÒ ðóìèêóììîëìóíò
ô Ü±ª»®
ÅîÃ ÞËÎÌÑÒô Ìòßòæ
×²½ò Ó·²»±´¿ô Ò»© Ç±®µô îððêô ×ÍÞÒ ðóìèêóìëííðóè
Ü±ª»® Ð«¾´·½¿¬·±²-ô
ÅíÃ ÚËÎËÓÑÝØ×ô Ìòæ
ïçèïô îëïóîêïò
Ú«²µ½·¿´ò Ûµª¿½ò îëô
Ôï óµ»®²»´-ô
ÅëÃ ØËôÍòô ÔßÕÍØÓ×ÕßÒÌØßÓôÊòæ
Ê±´¬»®®¿ ¬§°»ô Ò±²´·²»¿® ß²¿´§-·- ïðô ïçèêô ïîðíóïîðèò
ÅêÃ ÔßÙÛÒØÑÐôÝòÛòæ
ÅéÃ ÓËÎßÓßÕ×ôÍòæ
Ý±´´»¹» îïô ´çèëô ïíïóïííò
Î»-òÎ»°òØ¿½¸·²±¸» Ò¿¬ò
ÅèÃ ÍßÕØÒÑÊ×ÝØô Ôòßòæ
ô Ñ°»®¿¬±® Ì¸»±®§
ß¼ª¿²½»- ¿²¼ ß°°´·½¿¬·±²-ô Ê±´òèìô Þ·®µ¸¿«-»® Ê»®´¿¹ô Þ¿-»´óÞ±-¬±²óÞ»®´·²ô ïççêô ×ÍÞÒ íóéêìíóëîêéóïò
ÅçÃ wÓßÎÜßô Æòæ
Ü»°¿®¬³»²¬ ±º Ó¿¬¸»³¿¬·½- Î»°±®¬ Í»®·»-ô Ê±´òïî ô îððìô êïóêèô ×ÍÍÒ ïîïìóìêèïò
ô
ÅïðÃ wÓßÎÜßôÆòæ Ý±²ª±´«¬·±² Û¯«¿¬·±²- ±º Ê±´¬»®®¿ ¬§°»ô ×² Ð®±½»»¼·²¹- ±º ×ÝÍÝó Ú·º¬¸ ×²¬»®²¿¬·±²¿´
Ý±²º»®»²½» ±² Í±º¬ Ý±³°«¬·²¹ ß°°´·»¼ ·² Ý±³°«¬»® ¿²¼ Û½±²±³·½ Û²ª·®±³»²¬-ô ÛÐ× Õ«²±ª·½»ô îððé
ô ïïç ó ïîíò ×ÍÞÒ èðóéíïìóïðèóêò
ÅïïÃ wÓßÎÜßô Æò
°¿®¬³»²¬ ±º Ó¿¬¸»³¿¬·½- Î»°±®¬ Í»®·»-ô Ê±´òïìô îððêô îìë ó îìçò ×ÍÍÒ ïîïìóìêèïò
ÅïîÃ ÉÛÇÛÔôØòæ
ô Ü»ó
Ê·»®¬»´¶-½¸®ò
îîð
ßÜÜÎÛÍÍæ
ÎÒÜ®ò Ê´¿-¬¿ Õ®«°µ±ª?ô ÝÍ½ò
Ü»°¿®¬³»²¬ ±º Ó¿¬¸»³¿¬·½Ú¿½«´¬§ ±º Û´»½¬®·½¿´ Û²¹·²»»®·²¹ ¿²¼ Ý±³³«²·½¿¬·±²ô
Þ®²± Ë²·ª»®-·¬§ ±º Ì»½¸²±´±¹§
µ®«°µ±ª¿àº»»½òª«¬¾®ò½¦
Ü±½ò ÎÒÜ®ò Æ¼»²4µ w³¿®¼¿ ô ÝÍ½ò
Ü»°¿®¬³»²¬ ±º Ó¿¬¸»³¿¬·½Ú¿½«´¬§ ±º Û´»½¬®·½¿´ Û²¹·²»»®·²¹ ¿²¼ Ý±³³«²·½¿¬·±²ô
Þ®²± Ë²·ª»®-·¬§ ±º Ì»½¸²±´±¹§
-³¿®¼¿àº»»½òª«¬¾®ò½¦
îîï
îîî
ÍËÞÍÐßÝÛÍ ÑÚ ÌØÛ ÍÐßÝÛ ÑÚ ÍÑÔËÌ×ÑÒÍ ÑÚ ÑÒÛ
Ü×ÚÚÛÎÛÒÌ×ßÔ ÛÏËßÌ×ÑÒ É×ÌØ ÝÑÒÍÌßÒÌ ÜÛÔßÇ
Æ¼»²4µ Íª±¾±¼¿
Ú¿½«´¬§ ±º Û´»½¬®·½¿´ Û²¹·²»»®·²¹ ¿²¼ Ý±³³«²·½¿¬·±²ô Þ®²± Ë²·ª»®-·¬§ ±º Ì»½¸²±´±¹§
ß¾-¬®¿½¬æ
©·¬¸ ½±²-¬¿²¬ ¼»´¿§ ·- -¬«¼·»¼ò Ì¸·- »¯«¿¬·±² ·- °±--·¾´» ¬± ±¾¬¿·² ¾§ -«·¬¿¾´» ¬®¿²-º±®³¿¬·±² ±º
»¯«¿¬·±²- ·² ¿ ³±®» ¹»²»®¿´ º±®³ ¬¸»®»º±®» ³¿§ ¾» ®»°®»-»²¬ ¿- ¿ ½¿²±²·½ º±®³ ±º ¿ ½»®¬¿·² ½´¿-±º ¼»´¿§»¼ »¯«¿¬·±²-ò
Õ»§ ©±®¼-æ
ïòÓÑÌ×ÊßÌ×ÑÒ
½±²-¬¿²¬ ¼»´¿§ ® â ð
Á̈ ø¬÷ ã ¨ø¬
®÷å
øï÷
¿®» -¬«¼·»¼ò Û-°»½·¿´´§ ¿®» -¬«¼·»¼ ¬¸» -°¿½»- ©¸·½¸ ½±²¬¿·²- ¬¸» º«²½¬·±²¨ø¬÷ ã »¨°ø ÷å
øî÷
©¸»®»
ã »¨°ø
÷å
øí÷
¬¸» º±®³ ¨ø¬÷ ã » ¿²¼ ¬¸» »¯«¿¬·±² øí÷ ·- ¿²¿´±¹±«- ¬± ¬¸» ½¸¿®¿½¬»®·-¬·½ »¯«¿¬·±²ò Ì¸·- ½±²-·¼»®¿¬·±²
·- ³±¬·ª¿¬»¼ ¾§ ©»´´óµ²±©² ®»-«´¬- ¼»-½®·¾·²¹ ¬¸» ®»´¿¬·±² ¾»¬©»»² ¬¸» º«²½¬·±²¿´ -°¿½» ©¸·½¸ ·- ¹»²»®¿¬»
¾§ ¬¸» -»¬ ±º º«²½¬·±²- »¨°ø ÷ô ©¸»®» ·- ®±±¬ ±º ¬¸» »¯«¿¬·±² »¨°ø ÷ ã ïô ¿²¼ ¬¸» -°¿½» ±º ®ó°»®·±¼·½
º«²½¬·±²- ·ò»ò ¬¸» -°¿½» ±º -±´«¬·±²- ±º »¯«¿¬·±²
¨ø¬÷ ã ¨ø¬
î
Ì¸» »¯«¿¬·±² »¨°ø ÷ ã ï ¸¿- ®±±¬¹»²»®¿¬» ¬¸» º«²½¬·±²-
·
®
®÷æ
øì÷
ô ©¸»®» µ ·- ¿² ·²¬»¹»® ¿²¼ · ·- ·³¿¹·²¿®§ «²·¬ò Ì¸»-» ®±±¬-
î
· -·²
æ
®
®
Ì¸» ®»¿´ ¿²¼ ·³¿¹·²¿®§ °¿®¬- ±º ¬¸·- º«²½¬·±²- ¹»²»®¿¬» ¬¸» -°¿½» ©¸·½¸ ¼»-½®·¾»- ¿²§ ® °»®·±¼·½ º«²½¬·±²
-¿¬·-º§·²¹ ¬¸» Ü·®·½¸´»¬ù- ½±²¼·¬·±² ±² ¿²§ ·²¬»®ª¿´ ©·¬¸ ´»²¹¬¸ ® ¿- ¬¸» -«³ ±º ¬¸·- º«²½¬·±²òÌ¸»-» ®»-«´¬¿®» ½±®±´´¿®·»- ±º ´·²»¿® ·²¼»°»²¼»²½» ±º ¬¸»-» º«²½¬·±²-ò
½±-
î
îòÝØßÎßÕÌÛÎ×ÆßÌ×ÑÒ ÑÚ ÎÑÑÌÍ
õ · ±º ¬¸» ½¸¿®¿½¬»®·-¬·½ »¯«¿¬·±² ¸±´¼-æ
õ· ã»
®ø õ· ÷
÷»
®ø
· ÷
ã»
ø½±-
· -·²
÷ã
·
Ì¸·- ³»¿²- ¬¸¿¬ ¬¸» -±´«¬·±²- ±º »¯«¿¬·±² ã »¨°ø
÷ ¿®» ½±³°´»¨ ¿--±½·¿¬»¼ ²«³¾»®- ¿²¼ ¬¸» ¿--«³°¬·±²
ð ·- ¿¼³·--·¾´»ò Ì¸» ½¸¿®¿½¬»®·-¬·½ »¯«¿¬·±² ·- °±--·¾´» ·²¬»®°®»¬» ¿- ¬¸» -§-¬»³ ±º ¬©± »¯«¿¬·±²- ¿²¼
¬©± ®»¿´ ª¿®·¿¾´» ô æ
®ø õ· ÷
õ· ã»
ã»
Ú±®
êã ð êã
½±-ø
ã»
ø½±-
÷
· -·²
ã»
÷
-·²ø
·ò »ò
øë÷
÷
øê÷
©» ¸¿ª» »¯«·ª¿´»²¬ -§-¬»³ ·² ¬¸» º±®³æ
¿÷
î
õ
î
ã»
î
¾÷
ã
½±¬¹
øé÷
Ì¸·- -§-¬»³ ±º ¬©± ®»¿´ »¯«¿¬·±²- ±º ¬©± ®»¿´ ª¿®·¿¾´»ð ¿²¼ ® ã îò
îîí
5
4
3
y
2
1
0
-3
-2
-1
0
1
x
ð ©» ³¿§ ®»©®·¬» ¬¸» -±´«¬·±² øé¿÷÷ ¿- ¿ ¼»½®»¿-·²¹ º«²½¬·±²æ
Ú±®
®ø
÷ã
°
»
î
î
¿²¼
ð
®ø
÷ã
î
®»
°
»
î
õ
î
ä ðæ
Ñ² ¬¸» ±¬¸»® ¸¿²¼ô ¬¸» º«²½¬·±² ® ø ÷ ã
½±¬¹
¾÷÷ ·- ¿² ¼»½®»¿-·²¹ ·²
¬¸» ·²¬»®ª¿´- ×² ã ø² ï÷ å ²
º±® ²
ïò Ì¸»®» ¿®» ·²ª»®-» º«²½¬·±²- º² ø ÷ æ Î ÿ ×² º±® ² â ï
® ®
°
î ã ïô ¬¸»®» ·- ±²´§
¿²¼ ºï ø ÷ æ Å ïã®å ï÷ ÿ ×ï ò Ì¸»-» º«²½¬·±²- ¿®» ·²½®»¿-·²¹ò ß- ´·³
» î
¶«-¬ ±²» ·²¬»®-»½¬·±² Å
²Ã
±º º«²½¬·±²- º² ø ÷ ©·¬¸ ¬¸» º«²½¬·±²
ø ÷ô ©¸»®»
°
ø ÷ ã » î
²
ï
ï ·- ·²ª»®-» º«²½¬·±² º±®
²
¿²¼
®
®
ï
»-¬·³¿¬·±² ±º ¬¸» ·²ª»®-» º«²½¬·±²
º±´´±©- º®±³ ¬¸» º¿½¬ò Ú±® ¿²§ þ â ð ¿²¼
±² þô ¬¸» º«²½¬·±² ® ø
°
øï þ÷» ®¨ ä » î®¨ ¨î ä » ®¨ æ
²
î
ï
²
ø² õ ï÷
Ó±®»±ª»® º±® ·²¬»®-»½¬·±²- ø
º«²½¬·±² ® ï ¿²¼ ²«³¾»®´²
®
¨
®
ï
þ
ï÷ å ² ô
® ®
î ò Ì¸» ¿-§³°¬±¬·½
î
þô
ø²
©¸»®» ¨þ ¼»°»²¼-
ð ¿²¼ ¿´-± ² ø² õ ïãî÷ ò Ì¸»² ¬¸»
² ÷ -«½¸ ²
® øð÷ ©» ¸¿ª» ²
º±® »²±«¹¸ ´¿®¹» ² ø ²
® øð÷ ¿²¼ ² ä ¨þ ÷ -¿¬·-º§æ
®
°
°
ïãî÷
® ø²
ï
®
ø¨÷ ä ´² ¨
´²
´² ® ø² ï÷
² ä
ï þ
²
²
É» ²±¬» ¬¸¿¬ º±® ã ð ¬¸» »¯«·ª¿´»²¬ -§-¬»³ ¾»½±³»- ±²´§ ±²» »¯«¿¬·±² ã »
¿²¼ ¬¸» »¯«¿¬·±² øí÷
¸¿- ±²» ®»¿´ -±´«¬·±² ð ò Ú±® ® â ïã» ¬¸·- -±´«¬·±² ð ½¿² ¾» »¨°®»--»¼ ¿- ¿ º«²½¬·±² ±º ¬¸» ª¿®·¿¾´» ® ·²
¬¸» º±®³æ
ï
°
²
ß² ø®÷
å
ð ø®÷ ã ´·³ -«°
²ÿï
©¸»®» ß² ø®÷ ã
²
È
ø²
³ãð
¬¸» °±©»® -»®·»-
³
³÷ ³
® â ðò Ó±®»±ª»® ·¬ ·- °±--·¾´» ¬± »¨°®»-- ¬¸» º«²½¬·±²
³ÿ
ð ø®÷
º±® ®
ã
ï
È
ø ï÷²ø² õ ï÷²
²ÿ
²ãð
ï
ð ø®÷
·² ¬¸» º±®³ ±º
®² æ
ïã»ô -»» ÅïÃò
îîì
íòßÍÍÑÝ×ßÌÛÜ ÍÑÔËÌ×ÑÒÍ
Ì¸» º«²½¬·±² øî÷ ¨ð ø¬÷ ã »¨°ø ð ¬÷ ¿--±½·¿¬»¼ ©·¬¸ ®»¿´ ®±±¬ ð â ð ·- ·- ·²½®»¿-·²¹ò Ú±® ¬¸» -»¯«»²½» ½±³°´»¨
¿--±½·¿¬»¼ ²«³¾»®- ² õ ² ô ²
ïå ©» ¸¿ª» ¬¸» -»¯«»²½» ±º ½±«°´»- ±º º«²½¬·±²² ô º±® ²
¨½±² ø¬÷ ã »
²¬
½±-
¨-·²
² ø¬÷ ã »
²¬
²¬
-·²
² ¬å
øè÷
©¸·½¸ ¿®» -±´«¬·±²- ¬¸» ±º øï÷ò Ì¸·- ¼»-½®·°¬·±² ·- ¿½½»°¬¿¾´» º±® ®»¿´ ®±±¬ ¬±±ô ¨ð ø¬÷ ã ¨½±ð ø¬÷ ¿²¼ ð ã
¨-·²
ð ø¬
©¸·½¸ ¿®» ¼»¬»®³·²»¼ ¾§ ¬¸»-» ®±±¬-ò
Ô»³³¿ ïò Ì¸» º«²½¬·±²- øè÷ ¿²¼ ¨ð ø¬÷ ¿®» ´·²»¿® ·²¼»°»²¼»²¬ò
Ð®±±ºò É» ¿--«³» ¬¸¿¬ ¬¸» ´·²»¿® ½±³¾·²¿¬·±² ±º º«²½¬·±²- øè÷ ¿²¼ ¨ð ø¬
ºø¬÷ ã
²
È
· »¨°ø · ¬÷
ã ðæ
·ãï
Ì¸» º«²½¬·±² ª¿´«»- ±º ¬¸» ½±³¾·²¿¬·±² º ø¬÷ º±® ¬ ã ðå ®å æ æ æ å ø² ï÷® ¿®» ¦»®± ¬±±ò É» ¸¿ª» ¬¸» -§-¬»³
±º ´·²»¿® »¯«¿¬·±²- ©·¬¸ ®»-°»½¬ ¬± ª¿®·¿¾´»- · ò Ë-·²¹ ¬¸» ®»´¿¬·±² µ ã »¨°ø
³¿¬®·¨ -«½¸ ¬¸¿¬ ¬¸» ¼»¬»®³·²¿²¬ ±º ¬¸·- ³¿¬®·¨ ·- ¿ ©»´´óµ²±©² É¿²¼»®³±²¼ ¼»¬»®³·²¿²¬æ
ï
ï
æææ
ï
î
² ï
ï
² ï
î
ï
²
æææ
·ò
ã
È
ø
¶÷
·
·â¶
² ï
²
êã ðå
Ú®±³ ¬¸·- º¿½¬ º±´´±©- ¬¸¿¬ º«²½¬·±²- øî÷ ¿®» ´·²»¿® ·²¼»°»²¼»²¬ò Ì¸» ´·²»¿® ¿²¼
ìò ÌØÛ ÍÐßÝÛ ÑÚ ÍÑÔËÌ×ÑÒÍ ÜÛÚ×ÒÛÜ ÑÒ Î
Ûª»®§ ¿--±½·¿¬»¼ -±´«¬·±² ©·¬¸ ®±±¬ ±º ½¸¿®¿½¬»®·-¬·½ »¯«¿¬·±² øí÷ ¾»´±²¹- ·² ¬¸·- -°¿½»ô ¬¸»®»º±®» ¬¸» -°¿½» ±º
-«³
¨ø¬÷ ã ßð »
ð¬
õ
ï
È
-·²
ß² ¨½±² ø¬÷ õ Þ² ¨² ø¬÷ ã ßð »
ð¬
õ
²ãï
ï
È
ß² »
²¬
½±-
²¬
õ Þ² »
²¬
-·²
² ¬å
øç÷
²ãï
»
²¬
º±® ¬ ÿ ïò
Ì¸»±®»³ ïò Ô»¬ þ ¾» ½±²-¬¿²¬ -«½¸ ¬¸¿¬ ¬¸» -»¯«»²½»
øßî² õ Þ²î ÷
²þ
øïð÷
·- ¾±«²¼»¼ º±® ² ÿ ïò Ì¸»² ¬¸»®» »¨·-¬- ¿ ½±²-¬¿²¬ Ì -«½¸ ¬¸¿¬ º±® ¬
½±²ª»®¹»²¬ò
Ì ¬¸» -«³ ¨ø¬÷ ·- ¿¾-±´«¬»´§
Ð®±±ºò ß¾-±´«¬» ª¿´«» ±º ²ó¬¸ ¬»®³ ±º ¿ -»®·»- ¨ø¬
-·²
¶ß² ¨½±² ø¬÷ õ Þ² ¨² ø¬÷
Ë-·²¹ ¬¸» »-¬·³¿¬·±² º±®
²
øßî² õ Þ²î ÷»
²¬
¿¾±ª» ©» ¸¿ª»
-·²
¶ß² ¨½±² ø¬÷ õ Þ² ¨² ø¬÷
ßî² õ Þ²î
²¬ã®
Ò±© ©» «-» ¬¸» ¿--«³°¬·±² ±º ¬¸» Ì¸»±®»³ ¿²¼ º±® »²±«¹¸ ´¿®¹» Ì ä ¬ ©» ¸¿ª» ¬¸» ·²»¯«¿´·¬§
-·²
¶ß² ¨½±² ø¬÷ õ Þ² ¨² ø¬÷
ï
²î
Ë-·²¹ ¬¸» ½±³°¿®·-±² ¬¸»±®»³ ©» ¸¿ª» ¬¸» ¿¾-±´«¬» ½±²ª»®¹»²½»ò
îîë
Î»³¿®µ ïò
¿- ß¾»´ù- Ì¸»±®»³ º±® °±©»® -»®·»- ¿®» ²±¬ ª¿´·¼ò Ì¸·- ·- -¸±©² ·² ¬¸» ²»¨¬ »¨¿³°´»ò
Û¨¿³°´» ïò É» ¼»¿´ ©·¬¸ ¬¸» -«³ ¨ø¬÷ ã
ï
È
¨-·²
² ø¬÷ ã »
²¬
-·²ø
² ¬÷ò
²ãï
¨ø¬÷
¬
·- ¿¾-±´«¬» ½±²ª»®¹»²¬ò
Ú±® ¬ ã
® ©» ¸¿ª»æ
»¨°ø
² ®÷ -·²ø
² ®÷
ã
²å
»¨°ø
² ®÷ -·²ø ² ®÷
¿²¼ ¬¸» -«³- ¨ø ®÷ ¿®» ¼·ª»®¹»²¬ò
ã ãøï
²÷
ã
°
²ã
î
²
î
²
õ
Ú±® ¬ ã ð ·- ¨ø¬÷ ã ð »ª·¼»²¬ò
Ì¸»-» º¿½¬- -¸±© ¬¸¿¬ ¬¸» -»¬ ±º ½±²ª»®¹»²½» ¿²¼ ¬¸» -»¬ ±º ¼·ª»®¹»²½» ±º ¬¸» -«³ ¨ø¬÷ ¿®» ¼·-½±²¬·²«±«-ò
Ì¸»±®»³ îò Ô»¬ º±® »ª»®§ þ â ð
ßî² õ Þ²î
ãð
¬ÿï
²þ
´·³
øïï÷
Ì¸»² ¬¸» -«³ øç÷ ½±²ª»®¹» º±® ¿´´ ¬ò
Ð®±±ºò Ì¸·- ¬¸»±®»³ ·³³»¼·¿¬»´§ º±´´±©- º®±³ ¬¸» °®±±º ±º ¬¸» ¬¸»±®»³ ïò Ú®±³ ½±²¼·¬·±² øïï÷ º±´´±©- øïð÷
¿²¼ ¬¸» ¬¸»±®»³ î ·- ¿ ½±²-»¯«»²½» ±º ¬¸» ¬¸»±®»³ îò
Î»³¿®µ îò
-¸±© ¬¸¿¬ ¬¸·- ½±²¼·¬·±² ·- ²»½»--¿®§ô ¬±±ò
¬ î Î °´¿§- ¬¸» -¿³» ®±´» ¿- ¬¸» -°¿½» ±º
½±²¬·²«±«- ®ó°»®·±¼·½ º«²½¬·±²- º±® ¿²¿´±¹±«- »¯«¿¬·±² øì÷ò Þ«¬ «²º±®¬«²¿¬»´§ ¬¸» º«²½¬·±²- øè÷ ¿®» ²±¬
±®¬¸±¹±²¿´ ©·¬¸ ®»-°»½¬ ¬¸» «-«¿´´§ «-»¼ °®±¼«½¬ ·² ¬¸» -°¿½» ±º ½±²¬·²«±«- º«²½¬·±²- ±² ¿²§ ·²¬»®ª¿´ô
·³°±--·¾´» ·² ¬¸·- ½¿-»ò
ëò ÌØÛ ÝÑÛÚÚ×Ý×ÛÒÌ ÑÚ ÌØÛ ÍËÓ
«¬·´·¦» ¬¸» º¿½¬ ¬¸¿¬ ¸±´¼-·²
½±-·²
î
î
ß² ¨½±² ø¬÷ õ Þ² ¨² ø¬÷ø¬÷ ã ± ß³ ¨³ ø¬÷ õ Þ³ ¨³ ø¬÷ º±® ¬ ÿ ïå ³ ä ²å ß³ õ Þ³ êã ðæ
-ø¬÷ º±® ¬ ÿ ï ½±²¬¿·²- ¬¸» º«²½¬·±² ßð »
ð¬
©·¬¸ ßð êã ð
-ø¬÷
æ
» ð¬
Ì¸» º«²½¬·±² -ï ø¬÷ ã -ø¬÷ ßð » ð ¬ ·- ¾±«²¼»¼ º±® ¬ ÿ ïò Ú±® -¬¿¬»³»²¬ ¬¸» ±¬¸»® ½±²-¬¿²¬- ß² ô Þ² ·-«·¬¿¾´» °®»º»® ¬¸» ¿²±¬¸»® º±®³ ±º -«³
ßð ã ´·³
¬ÿï
ï
È
ß² Ý±-² ¬ õ Þ² Í·²² ¬ ã
²ãð
ï
È
Û² »¨°ø
² ¬÷ -·²ø ² ¬
õ Ú² ÷å
øïî÷
ß²
ßî² õ Þ²î
øïí÷
²ãð
©¸»®» ¬¸» ½±²-¬¿²¬- ¿®» »¨°´·½·¬ ¹·ª»² ¾§ ¬¸» ®»´¿¬·±²
Û² ã
°
ßî² õ Þ²î
½±- Ú² ã °
Þ²
ßî² õ Þ²î
-·² Ú² ã °
Ì¸» ¿--«³°¬·±² º±® «-·²¹ ±º ¬¸» ®»½«®®»²¬ ®±«¬·²» ±º -¬¿¬»³»²¬ ¬¸» ½±²-¬¿²¬ Û²ô Ú² øß² ô Þ² ¬±±÷ ·- ¬¸»
º¿½¬ ¬¸¿¬ ¬¸» -»¯«»²½» ±º ®»¿´ °¿®¬- ±º ®±±¬- ±º ¬¸» ½¸¿®¿½¬»®·-¬·½ »¯«¿¬·±² øí÷ ·- ¼»½®»¿-·²¹ò Ú±® ¬¸» -«³
-³ ø¬÷ ã
ï
È
Û² »¨°ø
² ¬÷ -·²ø ² ¬
õ Ú² ÷
øïì÷
²ã³
îîê
·- °±--·¾´» »ª¿´«¿¬»
´·³ -«° -³ ø¬÷ »ø
³ ¬÷
¬ÿï
ã ´·³ -«°
¬ÿï
ï
È
Û· »ø
·
³ ÷¬
-·²ø · ¬ õ Ú· ÷ ã ´·³ -«° Û³ -·²ø
¬ÿï
·ã³
³¬
õ Ú³ ÷ ã Û³ æ
Ì¸» -»½±²¼ -»¯«»´ ·- ¬¸» º¿½¬ ¬¸¿¬ ¬¸» ¦»®±- ±º ¬¸» -«³ -³ø¬÷ ½±·²½·¼» ©·¬¸ ¬¸» ¦»®±- ±º ¬¸» º«²½¬·±²
-·²ø ² ¬ õ Ú²
Ú² ò Ì¸» ½´±-·²¹ °¿®¬ ±º ¬¸·- ®»½«®®»²¬ ®±«¬·²» ·¹·ª»² ®»´¿¬·±²
-³õï ø¬÷ ã -³ ø¬÷ Û³ -·²ø ³ ¬ õ Ú³÷æ
-ø¬÷
Î»³¿®µ íò
·²¬»®ª¿´ Å¬ð å ï÷
Å¬ð å ¬ð õ ®Ã
ßð ò
Ì¸»±®»³ íò Ô»¬ ø¬÷
Å ®å ðÃ -«½¸ ¬¸¿¬ ¬¸» -»®·»-æ
ï
È
×µ ø ÷
µ
µãï
·- ½±²ª»®¹»²¬ò Ú±® ¬¸» -±´«¬·±² ¨ ø¬÷ ±º ¬¸» »¯«¿¬·±² øï÷ ·¬ ¸±´¼-æ
ï
È
¨ ø²®÷
ã øð÷ õ
×µ ø ÷
²ÿï ¨ð ø²®÷
´·³
µ
å
øïë÷
µãï
©¸»®» ¨ð ø¬÷ ·- ¬¸» -±´«¬·±² ±º ¬¸» »¯«¿¬·±² øï÷ ¿--±½·¿¬»¼ ©·¬¸ ®»¿´ ®±±¬ ±º ¬¸» ½¸¿®¿½¬»®·-¬·½ »¯«¿¬·±² øí÷
¿²¼
Æð «Æµõï Æ«î
Æð
ï
æææ
ø«ï ÷¼«ï æ æ æ ¼«µõï ã
ø «ï ÷µ ø«÷¼«æ
×µ ø ÷ ã
µÿ
®
®
®
®
Ú±® ³±®» ¼»¬¿·´- -»» ÅíÃ
Ì¸·- ®»-»¿®½¸ ¸¿- ¾»»² -«°°±®¬»¼ ¾§ ¬¸» Ý¦»½¸ Ó·²·-¬®§ ±º Û¼«½¿¬·±² ·² ¬¸» º®¿³» ±º ÓÍÓ ððîïêðëðí
Î»-»¿®½¸ ×²¬»²¬·±² Ó×ÕÎÑÍÇÒ Ò»© Ì®»²¼- ·² Ó·½®±»´»½¬®±²·½ Í§-¬»³- ¿²¼ Ò¿²±¬»½¸²±´±¹·»-ò
Î»º»®»²½»
ÅïÃ ÍÊÑÞÑÜß Æò
×² Ð®±½»»¼·²¹- ±º
ë¬¸ ·²¬»®¿²¬·±²¿´ ½±º»®»²½» ß°´·³¿¬ò ë¬¸ ×²¬»®²¿¬·±²¿´ Ý±²º»®»²½» ßÐÔ×ÓßÌò Þ®¿¬·-´¿ª¿æ Ú¿½«´¬§ ±º Ó»½¸¿²·½¿´
Û²¹·²»»®·²¹ Í´±ª¿µ Ë²·ª»®-·¬§ ±º Ì»½¸²±´±¹§ ·² Þ®¿¬·-´¿ª¿ô îððêô -ò ííë ó íìðô ×ÍÞÒ èðóçêéíðëóëóÈ
ÅîÃ ÍÊÑÞÑÜß Æò
×² Ð®±½»»¼·²¹ÈÈÊò ×²¬»®²¿¬·±²¿´ Ý±´´±¯«·«³ ±² ¬¸» ß½¯«·-·¬·±² Ð®±½»-- ³¿²¿¹»³»²¬æ ×ÍÞÒ èðóçêéíðëóëóÈ
ÅíÃ ÍÊÑÞÑÜß Æò
ëò Õ±²º»®»²½» ± ³¿¬»³¿¬·½» ¿ º§¦·½» ²¿ ª§-±µ#½¸ gµ±´?½¸ ¬»½¸²·½µ#½¸æ ×ÍÞÒ çéèóèðóéîíïóîéìóð
ßÜÜÎÛÍÍæ
ÎÒÜ®ò Æ¼»²4µ Íª±¾±¼¿ô ÝÍ½ò
Ü»°¿®¬³»²¬ ±º Ó¿¬¸»³¿¬·½Ú¿½«´¬§ ±º Û´»½¬®·½¿´ Û²¹·²»»®·²¹ ¿²¼ Ý±³³«²·½¿¬·±²ô
Þ®²± Ë²·ª»®-·¬§ ±º Ì»½¸²±´±¹§
-ª±¾±¼¿¦àº»»½òª«¬¾®ò½¦
îîé
îîè
ÍÑÓÛ ÐÎÑÐÛÎÌ×ÛÍ ÑÚ ÚÎßÝÌ×ÑÒßÔ ×ÒÌÛÙÎßÔÍ ßÒÜ
ÜÛÎ×ÊßÌ×ÊÛÍ
Æ¼»²4µ w³¿®¼¿
Ú¿½«´¬§ ±º Û´»½¬®·½¿´ Û²¹·²»»®·²¹ ¿²¼ Ý±³³«²·½¿¬·±²ô Þ®²± Ë²·ª»®-·¬§ ±º Ì»½¸²±´±¹§
ß¾-¬®¿½¬æ
Õ»§ ©±®¼-æ Ú®¿½¬·±²¿´ ·²¬»¹®¿´ ¿²¼ ¼»®·ª¿¬·ª»ò
²
²
ã
²
º±® ¬¸» ¬¸ ¼»®·ª¿¬·ª»ô ©¸»®» ·- ¿ ²±²ó²»¹¿¬·ª» ·²¬»¹»®ò Ô¿½®±·¨ ÅçÃøïèïç÷ ¼»ª»´±°»¼ ¬¸» º±®³«´¿ º±® ¬¸»
¬¸ ¼»®·ª¿¬·ª» ±º ã ³ ô
·- ¿ °±-·¬·ª» ·²¬»¹»® ô
²
ã
ÿ
ø
³ ²
øï÷
÷ÿ
©¸»®»
·- ¿² ·²¬»¹»®ò Î»°´¿½·²¹ ¬¸» º¿½¬±®·¿´ -§³¾±´ ¾§ ¬¸» ¹¿³³¿ º«²½¬·±²ô ¸» º«®¬¸»® ±¾¬¿·²»¼ ¬¸»
º±®³«´¿ º±® ¬¸» º®¿½¬·±²¿´ ¼»®·ª¿¬·ª»
õ ï÷
õ ï÷
ã
©¸»®»
¿²¼
øî÷
¿®» º®¿½¬·±²¿´ ²«³¾»®-ò ×² °¿®¬·½«´¿®ô ¸» ½¿´½«´»¼
®
ï
ï
î
î ã î
ã
í
÷
î
øí÷
Ñ² ¬¸» ±¬¸»® ¸¿²¼ ·² ïèíî Ö±-»°¸ Ô·±«ª·´´» ÅïïÃ º±®³¿´´§ »¨¬»²¼»¼ ¬¸» º±®³«´¿ º±® ¬¸» ¼»®·ª¿¬·ª» ±º ·²¬»¹®¿´
±®¼»®
² ¿¨
ã ² ¿¨
¬± ¬¸» ¼»®·ª¿¬·ª» ±º ¿®¾·¬®¿®§ ±®¼»®
¿¨
ã
øì÷
Ë-·²¹ ¬¸» -»®·»- »¨°¿²-·±² ±º ¿ º«²½¬·±² ø ÷ ô Ô·±«ª·´´» ¼»®·ª»¼ ¬¸» º±®³«´¿
ø ÷ã
ï
È
² ²
¿² ¨
øë÷
²ãð
©¸»®»
ø ÷ã
ï
È
²
¿² ¨
²ãð
Ú±®³«´¿ øë÷ ·- ®»º»®®»¼ ¬± ¿¼»®·ª¿¬·ª» ±º ¿®¾·¬®¿®§ ±®¼»®
Î
²
ð
øê÷
º±® º®¿½¬·±²¿´ ¼»®·ª¿¬·ª»ò ×¬ ½¿² ¾» «-»¼ ¿- ¿ º±®³«´¿ º±®
ô ©¸·½¸ ³¿§ ¾» ®¿¬·±²¿´ô ·®®¿¬·±²¿´ ±® ½±³°´»¨ò Ø±©»ª»®ô ·¬ ½¿² ±²´§ ¾» «-»¼
±º ¿ º®¿½¬·±²¿´ ¼»®·ª¿¬·ª» ¾¿-»¼ ±² ¬¸» ¹¿³³¿ º«²½¬·±² ø-»» Ü»¾²¿¬¸ ¿²¼ Í°»·¹¸¬ ÅëÃøïçéï÷÷
Æ ï
ï
¨¬
÷
ã
ð
øé÷
ð
îîç
õ ÷
÷
ã ø ï÷
ð
øè÷
¬¸» ¼»®·ª¿¬·ª» ±º ¿ ½±²-¬¿²¬ º«²½¬·±² ø
º®¿½¬·±²¿´ ¼»®·ª¿¬·ª» ±º ¿ ½±²-¬¿²¬ º«²½¬·±² ·² ¬¸» º±®³
ïã
÷
êã ð
×² ïèîîô Ú±«®·»® ÅêÃ ±¾¬¿·²»¼ ¬¸» º±´´±©·²¹ ·²¬»¹®¿´ ®»°®»-»²¬¿¬·±²- º±® ø ÷ ¿²¼ ·¬- ¼»®·ª¿¬·ª»-ò
Æ ï
Æ ï
ï
ø ÷ã
ø ÷
½±- ø
÷
î
¿²¼
²
Î»°´¿½·²¹ ·²¬»¹»®
ø ÷ã
Æ
ï
î
Æ
ï
ø ÷
øç÷
ï
²
§·»´¼- º±®³¿´´§
Æ ï
Æ ï
ï
ø ÷ã
ø ÷
î
½±-º ø
÷õ
½±-º ø
÷õ
î
¹
øïð÷
î
¹
øïï÷
¾§ ¿®¾·¬®¿®§ ®»¿´
Ù®»»® ÅéÃøïèëè÷ ¼»®·ª»¼ º±®³«´¿- º±® ¬¸» º®¿½¬·±²¿´ ¼»®·ª¿¬·ª»- ±º ¬®·¹±²±³»¬®·½ º«²½¬·±²- ¾¿-»¼ ±² øì÷ ·² ¬¸»
º±®³
·¿¨
ã
½±õ -·²
ø½±- õ -·²
î
î
-± ¬¸¿¬ ¬¸» º®¿½¬·±²¿´ ¼»®·ª¿¬·ª»- ±º ¬®·¹±²±³»¬®·½ º«²½¬·±²- ¿®» ¹·ª»² ¾§
É¸»²
ã
ï
î
ø½±-
÷ã
½±-
ø-·²
÷ã
-·²
î
î
½±-
-·²
½±-
õ ½±-
î
î
-·²
ã
½±-
õ
-·²
ã
-·²
õ
î
î
ã ïô Ù®»»®ù- º±®³«´¿- ¿®» ¿- º±´´±©-æ
ï
î
½±-
ã ½±-
õ
ï
î
ì
-·²
ã -·²
õ
ì
Í·³·´¿®´§ô º®¿½¬·±²¿´ ¼»®·ª¿¬·ª»- º±® ¸§°»®¾±´·½ º«²½¬·±²- ½¿² ¾» ±¾¬¿·²»¼ò
îò ÚÎßÝÌ×ÑÒßÔ ÜÛÎ×ÊßÌ×ÊÛÍ ßÒÜ ×ÒÌÛÙÎßÔÍ
¿ ´·²»¿® ²±²¸±³±¹»²»±«- ¬¸ ±®¼»® ±®¼·²¿®§ »¯«¿¬·±²
²
î
²
ã ø ÷
øïî÷
î
² ï
ã Å Ã Ì¸»² ºï
¹ ·- ¿ º«²¼¿³»²¬¿´ -»¬ ±º ¬¸» ½±®®»-°±²¼·²¹ ¸±³±¹»²»±«- »¯«¿¬·±²
ã ð ×º ø ÷ ·- ¿²§ ½±²¬·²«±«- ±² × ¬¸»² º±® ¿²§ î
ø ÷ã
Æ
¨
÷ø² ï÷
ø÷
ï÷ÿ
ø
ø
¿
øïí÷
·- ¬¸» «²·¯«» -±´«¬·±² ±º ¬¸» »¯«¿¬·±² øïî÷ ©·¬¸ ¬¸» ·²·¬·¿´ ½±²¼·¬·±²øµ÷
ø ÷
ð
Ñ® ô »¯«·ª¿´»²¬´§
ã
²
¿
¨
ø ÷ã
ãð ï
ï
÷
Æ
ï
¨
ø
÷ø²
ï÷
ø÷
øïì÷
¿
îíð
Î»°´¿½·²¹ ¾§ ô ©¸»®» Î
ð ©» ±¾¬¿·² ¬¸»
©¿- ®»°±®¬»¼ ¾§ Ô·±«ª·´´» ÅïïÃ ·² ïèíî ¿²¼ ¾§ Î·»³¿²² ÅïìÃ ·² ïèéê ¿Æ ¨
ï
ø
÷
ã
ø
÷
ã
ø
÷
¿ ¨
¿ ¨
÷ ¿
¬¸¿¬
ï
ø÷
øïë÷
©¸»®» ¿ ¨ ã¿ ¨ ·- ¬¸» Î·»³¿²²óÔ·±«ª·´´» ·²¬»¹®¿´ ±°»®¿¬±®ò É¸»²
±º º®¿½¬·±²¿´ ·²¬»¹®¿´ ¿²¼ ·º ã
×º ã ð ¿²¼
ð ¬¸»² ¬¸» ´¿°´¿½» ¬®¿²-º±®³ -±´«¬·±² ±º ¬¸» ·²·¬·¿´ ª¿´«» °®±¾´»³
²
·-
ø ÷ã
²
øµ÷
ø ÷ã ø ÷
ãð ï
øð÷ ã ð
ï
ø ÷ ò Ì¸» ·²ª»®-» Ô¿°´¿½» ¬®¿²-º±®³ ¹·ª»- ¬¸» -±´«¬·±² ±º ¬¸» ·²·¬·¿´ ª¿´«» °®±¾´»³
Æ ¨
ï
ø
ø ÷ ãð ¨ ² ø ÷ ã Ô ï º ² ø ÷¹ ã
÷² ï ø ÷
÷ ð
Ì¸·- ·- ¬¸» Î·»³¿²²óÔ·±«ª·´´» ·²¬»¹®¿´ º±®³«´¿ º±® ¿² ·²¬»¹»®
Î·»³¿²²óÔ·±«ª·´´» º®¿½¬·±²¿´ ·²¬»¹®¿´ øïë÷ ©·¬¸ ã ðò
ô ¿- ©»´´ò Î»°´¿½·²¹
¾§ ®»¿´
¹·ª»- ¬¸»
±º º®¿½¬·±²¿´ ·²¬»¹®¿´ ©¸·½¸ ©¿·²¬®±¼«½»¼ ¾§ É»§´ ÅïéÃ øïçïé÷ ¾§
¨
ï
Æ
ï
ã
ï
ø
×º ² ã ø ÷ ·- ¬¸»
ø ÷ ©¸±-» -±´«¬·±² ©·¬¸ ¬¸» ·²·¬·¿´ ½±²¼·¬·±²²
ø ÷ ã¨
½
÷
ï
ø÷
ð
Î
¨
øïê÷
ï÷²
ø ÷ãð
ãð ï
Æ ½
ï
ø ÷ã
÷²
ø
÷ ¨
ï
ø÷
²
ã
øïé÷
®»°´¿½·²¹ ¾§ -± ¬¸¿¬ Î
ðô ¿²¼ ã ï
Ú±® ¿ ½´¿-- ±º ¹±±¼ º«²½¬·±²½±²-·-¬·²¹ ±º º«²½¬·±²±º ¬·³»- ¿²¼ ¿´´ ±º ·¬- ¼»®·ª¿¬·ª»- ¿®» ø Ò ÷ ¿- ÿ ï º±® ¿´´ Ò ø-»» Ô·¹¸¬¸·´´ ÅïðÃô øïçëè÷÷ô
ã ·² øïê÷ ¹·ª»Æ ï
ï
ï
ø
÷
ã
ø õ ÷
¨ ï
÷ ð
ß°°´·½¿¬·±² ±º ¬¸» ±°»®¿¬±®
¹·ª»-
²
ï ·- ¹·ª»² ¾§
µ
¨
ï
ø ÷
øïè÷
¿²¼ ï ·² ¬¸» É»§´ ±°»®¿¬±®ô
¬± ¾±¬¸ -·¼»- ±º øïè÷ô ¼®±°°·²¹ ¬¸» -«¾-½®·°¬²
ø ÷ã
²
ø ÷
øïç÷
²
ø ÷ã
²
ø ÷
øîð÷
Í·³·´¿®´§ ©» ½¿² °®±ª»
©¸»®»
×º î
ã ø ï÷² ² ò
¿² óº±´¼ ·²¬»¹®¿¬·±² ±º ø÷ ¾§ °¿®¬- ¹·ª»-
²
ø õ²÷
ø ÷ã
¬¸¿²
-± ¬¸¿¬
ã
Å
²
ø
ã
²
¸
ø²
÷
²÷
·
ø ÷ ã
ø õ²÷
Å
Ã
øîï÷
ð ¿²¼ ·- ¬¸» -³¿´´»-¬ ·²¬»¹»® ¹®»¿¬»®
»¨·-¬- ¿²¼ ¸¿- ½±²¬·²«±«- ¼»®·ª¿¬·ª»-ò É» ¬¸»²
ðò ×º º±® ¿²§ º«²½¬·±²
±®¼»® ¾§
ø ÷ã
²
ø ÷Ã ã
ø ÷ã
²
¸
ø
ï
²
÷
Æ
²õ²÷
·
ø ÷ ã
ï
ø
÷²
ï
¨
×¬ ³¿§ ¾» ®»´»ª¿²¬ ¬± ³»²¬·±² ¬¸¿¬ Ô·±«ª·´´»ù- ½´¿--·½¿´ °®±¾´»³ ±º °±¬»²¬·¿´ ¬¸»±®§ ½¿² ¾» ¼»-½®·¾»¼ ¾§ ¿²
·²¬»¹®¿´ »¯«¿¬·±² ·²ª±´ª·²¹ ¬¸» É»§´ º®¿½¬·±²¿´ ·²¬»¹®¿´ ¿ï
ø ÷î
ï
î
ø ÷
ï
ø ÷
îíï
°
ø ÷ ã ø î÷
ã
Ù®¡²©¿´¼ ÅèÃ øïèêé÷ ·²¬®±¼«½»¼ ¬¸» ·¼»¿ ±º º®¿½¬·±²¿´ ¼»®·ª¿¬·ª» ¿- ¬¸» ´·³·¬ ±º ¿ -«³ ¹·ª»² ¾§
ø ÷
´·³
¸ÿð
²
ï È
õ ï÷ ø
õ ï÷ ø
ø ï÷®
®ãð
÷
õ ï÷
øîî÷
°®±ª·¼»¼ ¬¸» ´·³·¬ »¨·-¬-ò Ë-·²¹ ¬¸» ·¼»²¬·¬§
õ ï÷
ã
õ ï÷
ø ï÷®
÷
÷
¬¸» ®»-«´¬ øîî÷ ¾»½±³»ø ÷ ã ´·³
÷
¸ÿð
É¸»²
²
È
÷
ø
õ ï÷
®ãð
÷
·- »¯«¿´ ¬± ¿² ·²¬»¹»®
¿³
²
ï È
ø ï÷®
³
ø ÷ ã ´·³
¸ÿð
ø
÷
øîí÷
®ãð
Ñ² ¬¸» ±¬¸»® ¸¿²¼ Ó¿®½¸¿«¼ ÅïîÃøïçîé÷ º±®³«´¿¬»¼ ¬¸» º®¿½¬·±²¿´ ¼»®·ª¿¬·ª» ±º ¿®¾·¬®¿®§ ±®¼»®
º±®³
Æ ¨
ø ÷
ø ÷
ø÷
ø ÷ã
õ
÷
÷ ð ø
÷ õï
©¸»®» ð
ï ×¬ ¸¿- ¾»»² -¸±©² ¾§ Í¿³µ± øïçèé÷ ¬¸¿¬ øîî÷ ¿²¼ øîì÷ ¿®» »¯«·ª¿´»²¬ò Î»°´¿½·²¹
·² øîí÷ ·¬ ½¿² ¾» -¸±©² ·²¼«½¬·ª»´§ ¬¸¿¬
³
ð
¨
³
ø ÷ ã ´·³
¸ÿð
² ¸
È
®ãð
©¸»®»
¸
·
·
ø
ø
õ ï÷
÷ã
ï
÷
Æ
·² ¬¸»
øîì÷
¾§
¨
÷ø³
ø
ï÷
ø÷
ð
ø õ õ ï÷
ÿ
×¬ ·- ·³°±®¬¿²¬ ¬± °±·²¬ ±«¬ ¬¸¿¬ Ø¿®¹®»¿ª» øïèìè÷ »¨¬»²¼»¼ ¬¸» Ô»·¾²·¦ °®±¼«½¬ ®«´» º±® ¬¸» ¬¸ ¼»®·ó
ª¿¬·ª» ¬± ¬¸» º®¿½¬·±²¿´ ±®¼»® ·² ¬¸» º±®³
ã
Å ø ÷ ø ÷Ã ã
ï
È
õ ï÷
õ ï÷
®ãð
ø
®÷
ø ÷
®
ø ÷
øîë÷
®
¿²¼
°®±ª·¼»¼ ¬¸» -»®·»- ½±²ª»®¹»- ô ©¸»®» ®
·- ¿
º®¿½¬·±²¿´ ±°»®¿¬±®ò Ñ-´»® ÅïíÃøïçéð÷ ¬¸±®±«¹¸´§ -¬«¼·»¼ ¬¸» Ô»·¾²·¦ º±®³«´¿ ±º ¿®¾·¬®¿®§ ±®¼»® ¿²¼ °®±ª»¼
¿ ¹»²»®¿´ ®»-«´¬
Å ø ÷ ø ÷Ã ã
ï
È
õ ï÷
®÷
õ õ ï÷
®ã
ø ÷
ø õ®÷
ø ÷
øîê÷
©¸»®» ·- ¿®¾·¬®¿®§ò É¸»² ã ð Ñ-´»®ù- ®»-«´¬ øîê÷ ®»¼«½»- ¬± øîë÷ ò Ñ-´»® ¿´-± °®±ª»¼ ¿ ¹»²»®¿´·¦¿¬·±² ±º
¬¸» Ô»·¾²·¦ °®±¼«½¬ ®«´» ø÷ ·² ¬¸» ·²¬»¹®¿´ º±®³
Æ ï
õ ï÷
®÷
Å ø ÷ ø ÷Ã ã
ø ÷ ø õ®÷ ø ÷
õ õ ï÷
®ã
-»ª¿´ù- º±®³«´¿ ·² Ú±«®·»® ¿²¿´§-·- ò
·² ¬¸» -±´«¬·±² ±º ¿² ·²¬»¹®¿´ »¯«¿¬·±² ¬¸¿¬ ¿®·-»- ·² ¬¸» º±®³«´¿¬·±² ±º ¬¸» ¬¿«¬±½¸®±²±«- °®±¾´»³ò Ì¸·°®±¾´»³ ¼»¿´- ©·¬¸ ¬¸» ¼»¬»®³·²¿¬·±² ±º ¬¸» -¸¿°» ±º ¿ º®·½¬·±²´»-- °´¿²» ½«®ª» ¬¸®±«¹¸ ¬¸» ±®·¹·² ·² ¿ ª»®·½¿´
°´¿²» ¿´±²¹ ©¸·½¸ ¿ °¿®¬·½´» ±º ³¿-½¿² º¿´´ ·² ¿ ¬·³» ¬¸¿¬ ·- ·²¼»°»²¼»²¬ ±º ¬¸» -¬¿®¬·²¹ °±-·¬·±²ò ×º ¬¸»
-´·¼·²¹ ¬·³» ·- ½±²-¬¿²¬ ¬¸»² ¬¸» ß¾»´ ·²¬»¹®¿´ »¯«¿¬·±² ·Æ
°
ï
î ã
ø
÷ î ðø ÷
øîé÷
ð
îíî
©¸»®» ·- ¬¸» ¿½½»»®¿¬·±² ¼«» ¬± ¹®¿ª·¬§ ô ø ÷ ·- ¬¸» ·²·¬·¿´ °±-·¬·±² ¿²¼ ã ø ÷ ·- ¬¸» »¯«¿¬·±² ±º ¬¸»
-´·¼·²¹ ½«®ª»ò ×¬ ¬«®²- ±«¬ ¬¸¿¬ øîé÷ ·- »¯«·ª¿´»²¬ ¬± ¬¸» º®¿½¬·±²¿´ ·²¬»¹®¿´ »¯«¿¬·±²
°
Ñ®ô »¯«·ª¿´»²¬´§
ð
ï
÷ð
î
î
ø ÷ã
®
ï
î
î
ï
î
ð
©¸»®»
ïã
ð
ø ÷
®
øîè÷
î
î
ã
Ú·²¿´´§ ô ¬¸» -±´«¬·±² ·ø ÷ã
î
°
è
ã ì -·²
©¸»®» ¼- ã -·² ò Ì¸·- ½«®ª» ·- ¬¸» ½§½´±·¼ ©·¬¸ ¬¸» ª»®¬»¨ ¿¬ ¬¸» ±®·¹·²ò ¬¸» -±´«¬·±² ±º ¬¸» ß¾»´ °®±¾´»³
·- ¾¿-»¼ ±² ¬¸» º¿½¬ ¬¸¿¬ ¬¸» ¼»®·ª¿¬·ª» ±º ¿ ½±²-¬¿²¬ ·- ²±¬ ¿´©¿§- »¯«¿´ ¬± ¦»®±ò
Ü«®·²¹ ¬¸» -»½±²¼ ¸¿´º ±º ¬¸» ¬©»²¬·»¬¸ ½»²¬«®§ô ½±²-·¼»®¿¾´» ¿³±«²¬ ±º ®»-»¿®½¸ ·² º®¿½¬·±²¿´ ½¿´½«´«©¿- °«¾´·-¸»¼ ·² »²¹·²»»®·²¹ ´·¬»®¿¬«®»ò ×²¼»»¼ô ®»½»²¬ ¿¼ª¿²½»- ±º º®¿½¬·±²¿´ ½¿´½«´«- ¿®» ¼±³·²¿¬»¼ ¾§
½¸¿²·½-ô ª·-½»±»´¿-¬·½·¬§ô ³¿¬¸»³¿¬·½¿´ ¾·±´±¹§ ¿²¼ »´»½¬®±½¸»³·-¬®§ ø-»»ÅîôíôìôëôïëôïêÃ÷ò
Ì¸·- ®»-»¿®½¸ ¸¿- ¾»»² -«°°±®¬»¼ ¾§ ¬¸» Ý¦»½¸ Ó·²·-¬®§ ±º Û¼«½¿¬·±² ·² ¬¸» º®¿³» ±º ÓÍÓððîïêðëðí Î»ó
-»¿®½¸ ×²¬»²¬·±² Ó×ÕÎÑÍÇÒ Ò»© Ì®»²¼- ·² Ó·½®±»´»½¬®±²·½ Í§-¬»³- ¿²¼ Ò¿²±¬»½¸²±´±¹·»-ò
Î»º»®»²½»
ÅïÃ ßÞÛÔ´ôÒòØòæ Í±´«¬·±² ¼» Ï«»´¹«»- Ð®±¾´ ³»ïèîíô ïêóïè ò
Ôù¿·¼» Üù·²¬ ¹®¿´»- Ü
¬»- ïô
ÅîÃ ÞØßÌÌßôÜòÜòô ÜÛÞÒßÌØô Ôòæ
Ú®¿½¬·±²¿´ ½¿´½«´«- ¿²¼ ß°°´·»¼ ß²¿´§-·- éô îððìô îïóíêò
ÅíÃ ÜÛÞÒßÌØô Ôòæ
½¿´½«´«- ¿²¼ ß°°´·»¼ ß²¿´§-·- êô îððíô ïïçóïëëò
Ú®¿½¬·±²¿´
ÅìÃ ÜÛÞÒßÌØô Ôòæ Î»½»²¬ ¼»ª»´±°³»²¬- ·² º®¿½¬·±²¿´½¿´½«´«- ¿²¼ ·¬- ¿°°´·½¿¬·±²- ¬± -½·»²½» ¿²¼ »²¹·²»»ó
®·²¹ô ×²¬»®²¿¬ò ÖòÓ¿¬¸ò ¿²¼ Ó¿¬¸ò Í½·òô îððíô ïóíðò
ÅëÃ ÜÛÞÒßÌØôÔòô ÍÐÛ×ÙØÌôÌòæ Ñ² ¹»²»®¿´·¦»¼ ¼»®·ª¿¬·ª»-ô Ð· Ó« Û°-·´±² Ö±«®²¿´ ëô ïçéïô îïéóîîðò
ÅêÃ ÚÑËÎ×ÛÎôÖòæ Ô¿ Ì¸ ±®·» ß²¿´§¬·¯«» ¼» ´¿ Ý¸¿´»«®ô Û²¹´·-¸ ¬®¿²-´¿¬·±²¾§ ÛòÚ®»»³¿² ô Ü±ª»® °«¾´·ó
½¿¬·±²-ô ïçëëò
ÅéÃ ÙÎÛÛÎôØòÎòæ
Ï«¿®¬ò ÖòÓ¿¬¸ò íô ïèëèô íîéóííðò
ÅèÃ ÙÎ ¡ ÒÉßÔÜôßòÕòæ Ë»¾»®ô Þ»¹®»²¦¬»ô ¼»®·ª¿¬·±²»² «²¼ Ü»®»² ß²©»²¼«²¹ô ÆòÓ¿¬¸òÐ¸§-òôïîô ììïóìèðò
ÅçÃ ÔßÝÎÑ×ÈôÍòÚòæ Ì®¿·¬
Ð¿®·-÷ò
®»²¬·¿´ »¬ ¼« Ý¿´½«´ ×²¬ ¹®¿´ô î²¼ »¼òô íô ïèïçô ìðçóìïð øÝ±«®½·»®
ÅïðÃ Ô×ÙØÌØ×ÔÔôÓòÖòæ ×²¬®±¼«½¬·±² ¬± Ú±«®·»® ß²¿´§-·- ¿²¼ Ù»²»®¿´·-»¼ Ú«²½¬·±²-ô Ý¿³¾®·¼¹» Ë²·ª»®-·¬§
Ð®»--ô Ý¿³¾®·¼¹»ô ïçëèò
ÅïïÃ Ô×ÑËÊ×ÔÔÛôÖòæ
ÅïîÃ ÓßÎÝØßËÜôßòæ -«® ´» ¼
ß°°´òô êô ïçîéô ííéóìîëò
·²¼·½»- ¯«»´½±²¯«»-ô Öò ½±´» Ð±´§¬»½¸òô ïíô ïèíîô ïóêçò
®»²½»- ¼» º±²½¬·±²- ¼»- ª¿®·¿¾´»- Î ÛÔÛÍô ÖòÓ¿¬¸ò Ð«®»-
îíí
ÅïíÃ ÑÍÔÛÎôÌòÖòæ
Í×ßÓô Öòß°°´ò Ó¿¬¸òô ïêô ïçéðô êëèóêéìò
ÅïìÃ Î×ÛÓßÒÒôÞòæ ¡ ¾»® ¼·» ß²¦¿¸´ ¼»® Ð®·³¦¿¸´»² «²¬»® »·²» ¹»¹»¾»²»² Ù®¡--»ô Ù»-¿³³³»´¬» Ó¿¬¸ò
É»®µ»ô ïèéêô ïíêóïììò
ÅïëÃ wÓßÎÜßôÆòô ÕÎËÐÕÑÊ_ô Êòæ Ü·-½®»¬·¦¿¬·±² ±º ¬¸» Ù®«²©¿´¼óÔ»¬²·µ±ª º®¿½¬·±²¿´ Ü»®·ª¿¬·ª» ¿²¼ ×²ó
¬»¹®¿´ô ×² Ð®±½»»¼·²¹- ±º ß¾-¬®¿½¬- ¿²¼ ®»ª·»©»¼ ½±²¬®·¾«¬·±²- ±² ÝÜÎÑÓ ô Þ®²±ô îððêô ï ó êô ×ÍÞÒ
èðóéîíïïíçóëò
ÅïêÃ wÓßÎÜßôÆòô ÕÎËÐÕÑÊ_ôÊòæ Ñ² Í±³» Ð®±°»®¬·»- ±º Ú®¿½¬·±²¿´ Ý¿´½«´«-ô ×² Ð®±½»»¼·²¹- ±º ×Êò ×²ó
¬»®²¿¬·±²¿´ Ý±²º»®»²½» ±² Í±º¬ Ý±³°«¬·²¹ ß°°´·»¼ ·² Ý±³°«¬»® ¿²¼ Û½±²±³·½ Û²ª·®±³»²¬ô Õ«²±ª·½»ô
Û«®±°»¿² Ð±´§¬»½¸²·½¿´ ×²-¬·¬«¬» Õ«²±ª·½»ô îððêô ïëé ó ïêîô ×ÍÞÒ èðóéíïìóðèìóëò
ÅïéÃ ÉÛÇÛÔôØòæ
Ê·»®¬»´¶-½¸®ò
ßÜÜÎÛÍÍæ
Ü±½ò ÎÒÜ®ò Æ¼»²4µ w³¿®¼¿ô ÝÍ½ò
Ü»°¿®¬³»²¬ ±º Ó¿¬¸»³¿¬·½Ú¿½«´¬§ ±º Û´»½¬®·½¿´ Û²¹·²»»®·²¹ ¿²¼ Ý±³³«²·½¿¬·±²ô
Þ®²± Ë²·ª»®-·¬§ ±º Ì»½¸²±´±¹§
-³¿®¼¿àº»»½òª«¬¾®ò½¦
îíì
INDUCTION IN MULTI-LABEL TEXT CATEGORIZATION DOMAINS
Sareewan Dendamrongvit, Miroslav Kubat and Zeynel Sendur
University of Miami
Abstract: Modern approaches to automated text categorization often employ machine learning
techniques for the induction of the classifiers from preclassified examples. The distinguishing aspect
of this application field is that each example-document can fall into two or more classes (sometimes many classes) at the same time. This circumstance calls for specific induction algorithms and
necessitates the use of less traditional performance criteria in the course of their evaluation.
Another idiosyncrasy is that induction of text categorizers tends to be computationally very expensive because text documents are often described by thousands of features. In this paper, we
report our recent experience with these issues.
1 Introduction
The goal of the work reported here was to choose a mechanism to support a large multilingual thesaurus that
contained thousands of documents from many diverse fields and written in several different languages. To assist
the users search for relevant documents, an indexing mechanism was needed. However, the sheer size of the
database effectively precluded the possibility of creating such indexing scheme manually. By way of an
alternative, we decided to look into ways to automate the process by machine learning techniques.
The circumstance that the same document may simultaneously belong to two or more categories is in the
machine learning not typical, although some initial work has already been done. The simplest solution is to
induce a binary classifier separately for each classmechanisms based on Bayesian classifiers were studied by
[1], [2], and [3], the behavior of the instance-based rule was explored by [4], and the currently popular support
vector machines were employed by [5] and [6]. The main defect of solutions that induce a separate classifier for
each class is that the mutual relations between classes are thus ignored, which can impair classification
performance. By way of improvement, [7] modified the methodology of decision trees to make induction from
multi-label examples more natural, while [8] and [9] developed several algorithms that handle the multi-label
domains in the framework of the boosting technique originally developed by [10].
This little survey indicates that the relevant literature has so far approached the problem of induction from multilabel examples from two alternative perspectives: either by a binary classifier for each class or by a general
classifier that handles all combinations of classes. Either way, what all these approaches share is the extreme
computational intensity, especially in text categorization domains marked by thousands of features needed to
describe each document. This makes computational costs a critical performance criterion; before significant
reduction of these costs is achieved, full-fledged applications in real-world settings is hard to imagine. The other
critical criterion requires that the induced classifiers achieve high accuracy when identifying documents from a
given class. In multi-label domains, this calls for specific performance criteria.
In the next section, we will formally define the research task and the performance criteria to be used. Then, in
Section 3, we report experiments in which we compare two programs that have recently become quite popular.
We were interested both in classification performance and in computational costs. Based on the results, the same
section offers a brief discussion of which of the two techniques is to be preferred in concrete circumstances.
2 Problem Statement and Performance Criteria
Let
be an instance space, let
be a finite set of documents, and let be a finite set of classes such that
each
belongs to its subset,
. The features describing the documents have been obtained from the
relative frequencies of words or terms. Given a set of training examples,
, the goal
is to find a classifier to carry out the mapping
in a way that optimizes classification performance.
Moreover, the induction of the classifier has to be accomplished in realistic time.
Having gained some experience with the development of such classifiers in the work reported in [9], we wanted
to take a closer look at the practical behavior of some other approaches known from machine-learning literature.
ENVIRONMENT EPI Kunovice, Czech Republic, January 25, 2008
235
In particular, we wanted to investigate the behaviorin our specific domainof two programs: the multi-label
C4.5 proposed by [7] and BoosTexter developed by [8]. We found it surprising that no such comparison has so
far been made.
To establish the criteria to measure classification performance, let us start with those employed by the field of
information retrieval for domains where only two class labels are permitted: positive examples and negative
examples. Let us denote by TP (true positives) the number of correctly classified positive examples, by FN (false
negatives) the number of positive examples misclassified as negative, by FP (false positives) the number of
negative examples misclassified as positive ones, and by TN (true negatives) the number of correctly classified
negative examples. These four quantities define precision and recall as follows:
(1)
Observing that the user often wants to maximize both of them, while balancing their values, [11] proposed to
combine precision and recall in a single formula,
, parameterized by the user-specified
that
quantifies the relative importance ascribed to either criterion:
(2)
Here,
gives more weight to recall and
and to precision if
marked by
, in which case
gives more weight to precision;
converges to recall if
. The situation where precision and recall are deemed equally relevant is
degenerates to the following:
(3)
Based on these preliminaries, [12] proposed two alternative ways these criteria can be generalized for domains
with multi-label examples: (1) macro-averaging, where precision and recall are first computed for each category
and then averaged; and (2) micro-averaging, where precision and recall are obtained by summing over all
and
stand for
individual decisions. The formulas are summarized in Table 1 where
the precision, recall and the four basic variables for the i-th class.
Precision
Macro
Micro
Recall
=
=
=
=
Table 1: The macro-averaging and micro-averaging versions of the precision and recall performance criteria for
domains with multi-label examples.
3 Experimental Results
In the experiments, we wanted to compare BoosTexter with Multi-Label C4.5 along two performance criteria:
classification accuracy, and induction time; the former was evaluated in terms of precision- and-recall related
criteria, the latter was measured in minutes of CPU induction. Moreover, we wanted to find out how the
accuracy and computational costs depend on the number of features used to describe the individual documents.
Originally, we intended to apply the programs to the EUROVOC database, a large thesaurus where almost a
hundred thousands documentseach described by about 100,000 featureshave been preclassified into
thousands of hierarchically ordered classes. However, the extreme computational intensity of these programs
236
made it necessary to simplify the data, at least for the experimental purposes reported here. To be more specific,
we experimented with a reduced database containing 10,000 documents described by 4,000 features, and we
used only the 30 top-level class labels. To select the features, we used the document frequency criterion, an
unsupervised feature selection method recommended for the needs of text categorization by [13]: we picked the
4,000 random features from those that appeared in more than 50 documents.
To achieve acceptable statistical reliability of the results, we followed the methodology of 5-fold cross
validation. This means that in each run a classifier was induced from 8,000 documents and then tested on the
remaining 2,000 documents. We repeated this experiment for different numbers of features, running from 500 to
all 4,000 features used in the reduced database.
Figure 1 summarizes the experimental results. The first thing to observe is that BoosTexter systematically
outperformed Multi-Label C4.5 in terms of both
- and
- , especially when a larger number of
features was employed. This said, a closer look reveals that each of these methods displayed a somewhat
different behavior along the component criteria of : BoosTexter turned out to be better in terms of recall (both
micro and macro), whereas Multi-Label C4.5 turned out to be better in terms of precision, especially in situations
where only a relatively small subset of features was used. The experiments seem to indicate that the decision-tree
based Multi-Label C4.5 is able to get the most from even a very small feature set. This may be due to the fact
that BoosTexter considers only isolated features or linear combinations of features, whereas decision trees allow
for more flexible representation.
The disparate behavior of the two techniques along precision and recall needs to be properly understood before
choosing the induction method. For instance, users of automated recommender systems are discouraged when
offered a wrong document, even if this happens very rarely. Ability to minimize such cases is measured by
precision, and this is why the decision-tree based system will in domains of this kind be preferred. On the other
hand, recall is important when we want to make sure that all (or almost all) documents of the requested class
have been returned. Then, the recall criterion will be critical, which means that BoosTexter will be given
preference. At the same time, we have to be aware of the circumstance that our experiments indicate that a
growing number of features seems to mitigate the difference between the two systems' behavior.
237
Figure 1: Classification performance of the two approaches along different criteria, as measured on independent
testing data. Multi-Label C4.5 is better along precision, whereas BoosTexter is better along recall and .
BoosTexter seems to gain an edge with the growing number of features employed.
Figure 2: Induction time measured in minutes. The time indicated in the graph is always the sum total of all five
runs of the 5-fold cross validation procedure.
Another important aspect to consider, the computational complexity of the algorithms, was the subject of a next
round of experiments whose results are summarized by Figure 2 that plots the CPU times consumed by the two
programs for growing numbers of features. The reader can see that Multi-Label C4.5 is clearly more expensive
than the competing program: note how fast the costs grow with the increasing number of features. We conclude
that the practical utility of multi-label decision trees in the complete EUROVOC domain (with hundreds of
thousand of documents and tens of thousands of features) is limited.
4 Conclusion
In this brief communication, we shared with the readers our recent experience with two popular induction
algorithms that have been designed for text-categorization applications where each document can belong to two
or more classes. We were primarily interested in learning more about their classification performance and
computational costs as observed in a concrete real-world application domain. A closer look at the results reveals
that BoosTexter is probably the better choice. The exception is the case when the user wants to make sure that
the vast majority of the returned documents are relevant to the query, even if many other relevant documents
have been overlooked. Then, Multi-Label C4.5 might be preferable. Even so, the high computational costs
incurred in similar domains by decision-tree induction are a reason for major concern. More research is needed
to speed up the induction process if these techniques are to become practical for the use in the development of
large multilabel document thesauri.
5 Acknowledgment
The research was partly supported by the NSF grant IIS-0513702.
238
References
[1]
P. LANGLEY, W. IBA, and K. THOMPSON, An analysis of Bayesian classifiers, in Natl. Conf. on
Artificial Intelligence, 1992, pp. 223-228. [Online]. Available: citeseer.ist.psu.edu/article
/langley92analysis.html
[2]
N. FRIEDMAN, D. GEIGER, and M. GOLDSZMIDT, Bayesian network classifiers, Machine
Learning,
vol. 29, no. 2-3, pp. 131-163, 1997. [Online]. Available: citeseer.ist.psu.edu/
friedman97bayesian.html
[3]
A. MCCALLUM and K. NIGAM, A comparison of event models for naive Bayes text classification, in
Proc. Workshop on Learning for Text Categorization (AAAI'98), 1998. [Online]. Available:
citeseer.ist.psu.edu/mccallum98comparison.html
[4]
B. LI, Q. LU, and S. YU, \An adaptive k-nearest neighbor text categorization strategy," ACM Trans. on
Asian Language Information Processing (TALIP), vol. 3, pp. 215-226, 2004.
[5]
T. JOACHIMS, Text categorization with support vector machines: learning with many relevant
features, in Proc. European Conf. on Machine Learning (ECML'98), C. Nedellec and C. Rouveirol,
Eds., no. 1398. Chemnitz, DE: Springer Verlag, Heidelberg, DE, 1998, pp. 137-142. [Online]. Available:
citeseer.ist.psu.edu/article/joachims98text.html
[6]
J. T. KWOK, Automated text categorization using support vector machine, in Proc. Int'l Conf. on
Neural Information Processing (ICONIP'98), Kitakyushu, JP, 1998, pp. 347-351. [Online]. Available:
citeseer.ist.psu.edu/kwok98automated.html
[7]
A. CLARE and R. D. KING, Knowledge discovery in multi-label phenotype data, in Proceedings of
the 5th European Conference on Principles of Data Mining and Knowledge Discovery (PKDD'01),
Freiburg, Germany, 2001.
[8]
R. E. SCHAPIRE and Y. SINGER, Improved boosting using confidence-rated predictions, Machine
Learning, vol. 37, no. 3, pp. 297-336, 1999. [Online]. Available: citeseer.ist.psu.edu
/schapire99improved.html
[9]
K. SARINNAPAKORN and M. KUBAT, Combining subclassifiers in text categorization: A dst-based
solution and a case study, IEEE Transactions on Knowledge and Data Engineering, vol. 19, no. 12, pp.
1638-1651, 2007.
[10] R. E. SCHAPIRE, The strength of weak learnability, Machine Learning, vol. 5, no. 2, pp. 197227,1990.
[11] C. J. VAN RIJSBERGEN, Information Retrieval, 2nd ed. London: Butterworths, 1979. [Online].
Available: http://www.dcs.gla.ac.uk/iain/keith/index.htm
[12] Y. YANG, An evaluation of statistical approaches to text categorization, Information Retrieval, vol. 1,
no. 1/2, pp. 69-90, 1999. [Online]. Available: citeseer.ist.psu.edu/article/ yang98evaluation.html
[13] Y. YANG and J. O. PEDERSEN, A comparative study on feature selection in text categorization, in
Proceedings of ICML-97, 14th International Conference on Machine Learning, D. H. Fisher, Ed.
Nashville, US: Morgan Kaufmann Publishers, San Francisco, US, 1997, pp. 412-420. [Online]. Available:
citeseer.ist.psu.edu/yang97comparative.html
Address
Sareewan Dendamrongvit,
[email protected],
Department of Electrical & Computer Engineering
University of Miami
Coral Gables
FL 33146, U.S.A.
Miroslav Kubat
[email protected]
University of Miami
Coral Gables
FL 33146, U.S.A.
Zeynel Sendur
[email protected]
University of Miami
Coral Gables
FL 33146, U.S.A.
239
240
AUTOR INDEX
B
BARA SKI, M.............................................53
BATINEC, J........................................ 11, 205
BRZOBOHATÝ, J. .......................................45
P
PETRUCHA, J. .............................................87
POPELKA, O...................................... 171, 191
PROKOP, R.. ................................................45
C
CEPL, M.. ...................................................177
R
RAIS, K.........................................................31
RUKOVANSKÝ, I.. ...................................171
D
DIBLÍK, J. ..................................................205
DOSKO IL, R.. ............................................39
DOSTÁL, P...................................................31
S, Š
SVOBODA, Z. ............................................223
EV ÍK, V.. ...............................................169
MARDA, Z.. ............................. 105, 217, 229
TENCL, M.. ..............................................177
ASTNÝ, J. ................................ 71, 177, 191
F
FAJMON, B.. ..............................................105
FILIPOVÁ, O..............................................111
T
TOMOVÁ, M. ..........................................123
G
GOTTVALD, A. .........................................199
V
VEP EK, J..................................................183
VERTÉSY, G................................................79
VI-HORVÁTH, M. P.. ..................................79
CH
CHVALINA, J. ...........................................211
J
JABLONICKÁ, J. .......................................117
W
WALKOVIAK, T. ........................................57
K
KRATOCHVÍL, O.. ......................................31
KREJ Í, P.. .................................................169
KRUPKOVÁ,V.. ........................................217
Z
ZAMOJSKI, W.. ...........................................53
ZAPLETAL, J. .................................... 141, 147
L
LACKO, B.. ..................................................65
LIKA, J. ....................................................177
LUDÍK, P.. ....................................................71
M
MAR EK, D.................................................97
MAR EK, M.. ..............................................91
MATUTÍKOVÁ, V.....................................19
MAZURKIEWICZ, J, ...................................57
ME IAROVÁ, Z.. ........................................25
MICHALSKA, K. .........................................53
MINA ÍK, M..............................................191
MOU KA, J. ..............................................211
MUSIL, V.. ...................................................45
N
NOVÁK, M.................................................211
O
OMERA, P........................................ 151, 159
241
242
Název:
ICSC 2008 Fifth International Conference on Soft Computing Applied in
Computer and Economic Enviroment
Autor:
Kolektiv autor
Vydavatel, nositel autorských práv, vyrobil:
Evropský polytechnický institute, s.r.o.
Osvobození 699, 686 04 Kunovice
Náklad:
100 ks
Po et stran:
242
Vydání:
první
Rok vydání:
2008
ISBN 978-80-7314-132-5

ICSC 2008 - Evropský polytechnický institut, sro

Transkript

Podobné dokumenty

Dear business partners, We would like to inform

Map