chapter 5 approximating nonlinear optimization problem...

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CHAPTER 5

Approximating nonlinear optimization problem with

fuzzy relation equations

______________________________________________________________________________

5.1 Introduction

In general, an abstract system is defined as the relation among its various possible inputs

and outputs. So their behavior can be described as the collection of facts and if-then rules

that in turn can be represented in the form of fuzzy relations. The inference process from

such systems ends up with solving a system of fuzzy relational equations. Fuzzy relation

equations (FRE) offer an appropriate tool to handle and model imprecision present in

relational structures. The notion of fuzzy relation equations (FRE) and fuzzy relational

calculus lie in the centre of the fuzzy set theory and its applications particularly in the

area of fuzzy modeling, diagnostic, fuzzy control etc. For this, several resolution

strategies of the fuzzy inverse problems based on various heuristics and metaheuristics

have been proposed in the past and the search is still on [35,73,79,124,157]. However

solving a system of fuzzy relational equations is not straightforward. It has been well

established that the solution set of a consistent system of sup-t FRE comprises of a

unique maximal solution and several minimal solutions [4,49].When the solution set of

the system of fuzzy relation equations is empty the system is said to be consistent and

inconsistent otherwise. The consistency of a system in this case can be easily verified by

checking the potential maximum solution. As the complexity of the system increases or

perturbed systems occur; it becomes difficult to find the exact solutions. This combats the

application of FRE for solving many practical problems. Many times the solution set of

FRE is an empty set. In such situation when no exact solution exists, the notion of

“Approximate solution” of FRE is addressed, as the solution is essential from practical

aspect. Hence, the field of searching methods of determining approximate solutions of

FRE needs to be explored.

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Numerous researchers [35,36,73,124,128] have investigated the issue of approximate

solutions of FRE. But despite of wide applicability, the stream for investigation of

methods to find the approximate solutions is still not rich enough.

Firstly, Pedrycz [124] came up with a numerical method and proposed quasi-Newton

method for solving fuzzy relation equations. Perfilieva and Gottwald [138] studied

solvability and approximate solvability of fuzzy relation equations. There are also

methods dealing with removal of one or more equations of system of fuzzy relation

equations in case of no solution. Further Pedrycz [128] showed that approximate

solutions for fuzzy relation equations of type X A B=� are those fuzzy relations in

which X has minimum distance of X A� from .B But the structure of the approximate

solution set obtained by this method is not clear. Statistical approach to solve the system

of FRE is presented in [130]. Gottwald and Pedrycz [35] discussed the solvability indices

of fuzzy relation equations based on the equality index introduced for fuzzy sets by

Gottwald [34]; and found that the degree of difficulty to solve the FRE system depends

upon the solvability index. Yuan and Klir [204] introduced a method based on goodness

measure of the performance of approximate solutions and derived a lower bound and an

upper bound of solvability of systems of fuzzy relation equations.

Later on various metaheuristics were applied to solve FRE. Different neural network

based approaches have been suggested to solve the system of fuzzy relation equations

[79,131,153,181]. Wu [185] described approximate solutions of fuzzy relation equations

based on approximate reasoning. Liu, Lur and Wu [88] studied the fuzzy relational

equations with max-Łukaseiwicz composition and proposed an algorithm that yields the

best approximate solution of the considered system.

The credit to apply genetic algorithms to solve fuzzy relation equations goes to Sanchez

[157]. Further work and results in this direction can be found in Negoita�

et al.

[108].Recently, Luoh and Liaw [94] gave a novel genetic algorithm to find approximate

solutions of a system of fuzzy relational equations based on max-product composition.

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Optimization problems with consistent fuzzy relation equations as constraints have

extensively been studied by numerous researchers [27,89,121,166,167,174]. But little

attention is paid to optimization problems when the system of FRE is inconsistent.

Thapar, Pandey and Gaur [176] studied a nonlinear optimization problem with an

inconsistent system of fuzzy relation equations based on max-min composition. The

method proceeds in two steps. Firstly, the search space is reduced. Then, a random jump

method is applied that results in a good approximate solution of the optimization

problem.

This chapter considers two nonlinear optimization problems subjected to system of max-

� fuzzy relational equations, when the system has no unique solution and � is any

continuous t- norm. Two different approaches have been proposed to solve the nonlinear

optimization problems presented.

5.2 A system of fuzzy relational equations

Consider the following system of fuzzy relational equations

(5.1)x A b=�

where [ ], 0 1,ij ijA a a= ≤ ≤ be a m n× dimensional fuzzy matrix and 1 2[ , , , ],n

b b b b= …

0 1,jb≤ ≤ be a -n dimensional vector, � is a continuous t-norm operator from the

residuated lattice [0,1], , , , ,0,1t

L = ∧ ∨ Θ� and “ �” denotes the max-� composition of

x and .A The resolution problem of FRE is to determine a solution vector

1 2[ , , , ],m

x x x x= … with 0 1,i

x≤ ≤ such that

1

max ( ) , 1, 2, , (5.2)m

i ij ji

x a b j n=

= ∀ =� …

Let {1, 2, , }I m= … and {1, 2, , }J n= … be the index sets. The maximum solution of (5.1)

can be computed explicitly by the residual implicator (pseudo complement) as follows:

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min ( ) (5.3)t ij t jj J i I

x A b a b∈ ∈

= Θ = Θ

�

where sup{ [0,1] | ( ) }ij t j i i ij ja b x x a bΘ = ∈ ≤�

The solution set of a consistent system of FRE is given by one unique maximal solution

and possibly finite number of minimal solutions say .L Let pX denotes thp sub-feasible

region given as { | }p pX x X x x x= ∈ ≤ ≤� �

which is a lattice. The entire solution set

( , )X A b of system (5.1) is given as:

1 2

1( , ) [ , ] [ , ] [ , ] [ , ].

Lp p L

pX A b X x x x x x x x x

== ∪ = ∪ ∪ ∪ ∪ ∪

� � � � � � � �… …

Lemma 5.2.1. If in the thj equation of system (5.1) we have , ,ij ja b i I< ∀ ∈ then the

solution set ( , ) .X A b φ=

Proof. If in the thj equation ij ja b< holds for all ,i I∈ then for ,i ijx a≠

( ) (1 )i ij ij ij jx a a a b≤ = <� � and for , ( ) ( ) .i ij i ij ij ij ij jx a x a a a a b= = ≤ <� � Thus, for

both cases , .ij ja b i I< ∀ ∈ Hence, max ( )i ij j

i Ix a b

∈<� and there exists no solution for the

thj equation. Thus, ( , ) .X A b φ=

Lemma 5.2.1 just represents only a sufficient condition. If this is not true, it does not

imply that some solution exists.

Lemma 5.2.2.[49]. Let ( , )X A b�

be the set of all minimal solutions of (5.1) then

( , ) ( , ) ( , ).X A b X A b x X A bφ φ≠ ⇔ ≠ ⇔ ∈� �

Lemma 5.2.3 gives a necessary and sufficient condition for the existence of ( , ).X A b

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Lemma 5.2.3. If ( , )x X A b∈ then for each j J∈ there exists i I∈�

such that

( )i i j jx a b=� �

� and ( ) , .i ij jx a b i I≤ ∀ ∈�

Proof. For ( , ),x X A b∈ max ( ) , .i ij j

i Ix a b j J

∈= ∀ ∈� This implies ( ) , .i ij jx a b j J≤ ∀ ∈�

Therefore, in order to satisfy the equality constraint, there must exist at least one i I∈�

such that ( ) .i i j jx a b=� �

�

We first consider the nonlinear optimization problem with a general nonlinear objective

function constrained to a system of continuous t-norm based FRE. A well structured real

coded genetic algorithm is suggested that is designed keeping the structure of the domain

under consideration.

5.3 The problem I

We are interested in solving the following nonlinear optimization problem

Min ( ) (5.4)

( , )

f x

x X A b∈

where ( )f x is a nonlinear function and 1 2[ , , , ]m

x x x x= … is the solution vector. We

assume the case when the solution set of (5.1) is empty i.e. ( , ) .X A b φ= Then, a solution

1 2[ , , , ]m

x x x x= … is said to be an approximate solution of (5.1) satisfying x A b b′= ≠� .

The goodness of the solution x is measured on the basis of Euclidean distance of x A�

from ,b also called as the error associated with that solution. The error of a particular

solution x with [0,1], 1,2, , ,i

x i m∈ = … is calculated as follows:

2( ) ( , ) ( ) (5.5)j j

j J

e x d b b b b∈

′ ′= = −∑

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5.4 Implementation of the genetic algorithm

We propose a real valued genetic algorithm (RVGA) that is designed specifically for the

considered optimization problem. The genetic operators are designed such that they

accelerate the procedure and help the algorithm to converge easily.

In the considered fuzzy optimization problem the feasible domain given by fuzzy relation

equations has no exact solutions. So an error based genetic algorithm is applied that leads

to the good and convergent approximate solutions of the considered optimization

problem. The design of the proposed algorithm is described as follows:

The procedure starts with generating a population of finite size with each chromosome as

a string of random values in the unit interval (0, 1). Once the population has been created,

the individuals are evaluated using some fitness criterion (or fitness function). In general

real valued genetic algorithm, the objective function is itself used as the fitness function.

In the considered case, the feasible domain has no exact solution so the goal is not only

just optimization but also the exotic exploration of the search space so as to find good

approximate and converging solutions of the optimization problem. For this, a pre-fixed

threshold error value maxε is set and the aim is to find solutions having distance (or error)

lesser than this threshold error maxε and optimizing the objective function as well. To

serve the purpose, a modified version of combined objective is formulated as follows that

optimizes the objective and minimizes the error function in parallel:

max

max 2

( ), if

( ) (5.6)( ) ( max ( ( )) , otherwisej i ij

i Ij J

f x

f x f x b x a

ε ε

∈∈

≤

′ = + −

∑ �

where max ( )f x is the value of the original objective function for the solution giving the

maximum error in the interval max(0, ]ε in that population(i.e. the most unfit individual

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for that run). This design of the objective lowers the possibility of transmission of the

unwanted solutions from the current population to the subsequent generations.

5.4.1 Selection

After the fitness function, the design of the selection scheme is the second factor

responsible for the faster and efficient operation of the proposed algorithm. Keeping the

insight of the considered problem, a binary tournament selection operator is designed that

has a dual criterion of selection. Two solutions are selected at random from the

population and compared at a time. The comparison of two solutions is performed

according to the following conditions:

1. Out of any two solutions having errors lesser than the threshold error maxε , the

solution giving the lesser value of the original objective function is selected.

2. Among two solutions having errors greater than the pre-specified threshold maxε ,

the solution giving lesser error value is selected.

3. The solution giving the error value in the range max(0, ]ε is preferred to the

solution giving error value greater than the threshold maxε .

Using the above selection scheme a finite strength of good population is selected that

undergoes the further cycle of genetic algorithm.

5.4.2 Crossover and mutation

The crossover operator is the main search tool and the major exploratory mechanism of

the genetic algorithms. The arithmetic crossover operation [63] is used. The operator

pushes the solutions towards the feasible region, as it is based on the ideas of linear

contraction and linear extraction (as discussed in section 4.5.2 of chapter 4) of two

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individuals. In arithmetic crossover, two parents, say 1 2, ,x x are randomly selected from

population to produce two offsprings. The two offsprings 1 2,y y are generated as follows:

1 1 2

2 1 2

(1 )

(1 )

i i i i i

i i i i i

y x x

y x x

λ λ

λ λ

← + −

← − +

where (0,1)i

λ ∈ 1, 2, , i m∀ = … are uniform random numbers in the unit interval. As a

result there might be a case when the variable components attain values outside their

obvious range (0,1). Therefore, if the value of the variable is larger than 1, it is set to 1

and if the value is negative then it is set to 0. If the value is between 0 and 1, it does not

change. After the crossover operation, the resulted crossed population undergoes the

mutation operation.

Mutation is a genetic operator responsible to introduce new genetic information in the

generation and prevents premature convergence of the algorithm. To meet the purpose,

feasible mutation has been adopted that applies to a solution probabilistically. For the

mutation operation, a random element is selected for thi variable { 1, 2, , }i i m∀ ∈ = … of a

chromosome, according to the mutation probability that is replaced with a feasible

random value lying in the range (0, ).i

x�

For example, let x be the chromosome and the

mutation probability be (0,1)δ ∈ .The mutation is applied to x probabilistically as:

1. For each 1, 2, , ,i m= … generate random number (0,1).i

r ∈

2. For 1, 2, , ,i m= … if ,i

r δ≤ then randomly assign ix a number in the range of (0, ).

ix�

The obtained new population again undergoes the cycle of RVGA and the algorithm

keeps running until some termination criterion is not met.

110

The whole cycle of applied GA can be summarized in the following steps:

Algorithm 1: Procedure to solve optimization problem (5.4)

Step 1: Get the matrices ,A b and the nonlinear objective function .f

Step 2: Find the maximum solution x�

using (5.3).

Step 3: If system of FRE is not solvable i.e. ,x A b≠�� go to step 4, else stop.

Step 4: Initialize population of fixed size, say ,k and set the threshold error value as

max | |x A bε = −�� and set generations counter gen=1.

Step 5: Evaluate population using the fitness function defined in equation (5.6).

Step 6: Select fixed no. of good solutions by the binary tournament selection operator

described in section 5.4.1 and the best fit individual for that generation say x′ and

determine its corresponding error | |x A bε ′ ′= −� by (5.5).

Step 7: Apply crossover and mutation operators as described in section 5.4.2.

Step 8: If maxε ε′ < , update the threshold error as max .ε ε ′=

Step 9: If the termination criterion is meet stop, otherwise set gen=gen+1and go to step 5.

5.5 Illustrative examples

For illustration of the proposed procedure, we consider some optimization problems with

nonlinear objective functions subjected to max-� composition based fuzzy relation

equations having no unique solution i.e. .x A b≠�� Nonlinear objective functions are

111

considered from well known source [52]. Algorithm 1 is used to obtain the best

converging solution of the considered optimization problem. Due to the large search

space, generally a large population size results in faster convergence of the algorithm. In

our case we set the threshold as 2

max ( max ( ( )) .j i iji I

j J

b x aε∈

∈

= −∑�� For our algorithm, we

take the following parameters settings:

Mutation probabilityδ : 0.1

Population size k : 200

The results obtained are presented numerically in tables 5.1-5.3 and graphically by

figures 5.1-5.3.

Example 5.1. 3 3

1 1 2 2Min ( ) 3000 1000 2000 666.667f x x x x x= + + +

ith (s.t., w ) wherexx A b a x a= ⋅= ��

[ ]0.4350 0.0128 0.5065

0.4352 0.4229 0.5323 , 0.5000 0.4092 0.6159

0.3440 0.2057 0.4385

A b

= =

The maximum solution of the system comes out to be [ ] 1.0000 0.9676 1.0000

giving max 0.0144ε = .

Table 5.1: Objective values by iterations - Example 5.1

Iterations 1x 2x 3x

310f × ( )e x

1

5

7

13

51

66

0.2490 0.9829 0.4491

0.0588 0.9847 0.3426

0.0185 0.9852 0.2225

0.0438 0.9867 0.7883

0.0254 0.9885 0.6087

0.0000 0.9834 0.5994

3.3613

2.7826

2.7451

2.6974

2.6631

2.6207

0.0139

0.0136

0.0135

0.0133

0.0130

0.0130

Figure

Figure 5.2: N

02600

2800

3000

3200

3400

Obje

cti

ve v

alu

e

0.810

2000

4000

6000

8000

Obje

cti

ve f

uncti

on

112

Figure 5.1: Performance of GA for Example 5.1

Nonlinear function with optimal point in Example

20 40 60 80Iterations

00.2

0.40.6

0.81

00.2

0.40.6

0.8

x1x2

Example 5.1

Example 5.2. Min ( )f x x x x x x=

ith ( ) min( , ) wheres.t., w xAx b= ��

0.5 0.7 0.5 0.8

0.6 0.3 0.6 0.9

0.1 0.9 1 0

0.8 0.5 0.9 0.6

0.1 0.4 0.7 0.9

A b

= =

The maximum solution is obtained as

max 0.1000ε = .

Table 5.

Figure

0

0

2

4

6

8

10

12

14x 10

Obje

cti

ve v

alu

e

Iteration 1x

1

3

8

14

17

0.6501 0.2210 0.0401 0.7244 0.2987

0.3044

0.0365 0.0266 0.6947 0.7148 0.3362

0.0407 0.0091 0.6942 0.7104 0.3039

0.0595 0.3228 0.6940 0.7090 0.0036

113

1 2 3 4 5Min ( )f x x x x x x=

ith ( ) min( , ) wherex a x a=�

0.5 0.7 0.5 0.8

0.6 0.3 0.6 0.9

, [0.8 0.7 0.6 0.5]0.1 0.9 1 0

0.8 0.5 0.9 0.6

0.1 0.4 0.7 0.9

A b

= =

The maximum solution is obtained as [ ]0.5000 0.5000 0.6000 0.5000 0.50

5.2: Objective values by iterations - Example 5.2


5 10 15 20

x 10-4

Iterations

1x 2x 3x 4x 5x ( )f x

0.6501 0.2210 0.0401 0.7244 0.2987

0.3044 0.6373 0.6476 0.7088 0.0026

0.0365 0.0266 0.6947 0.7148 0.3362

0.0407 0.0091 0.6942 0.7104 0.3039

0.0595 0.3228 0.6940 0.7090 0.0036

0.0012

0.0002

0.0002

0.0001

0.0000

[ ]0.5000 0.5000 0.6000 0.5000 0.5000 giving

( )e x

0.0462

0.0417

0.0305

0.0303

0.0302

Example 5.3. Min ( ) 10( 0.5) 10( 0.5) 5f x x x= − + − +

s.t., where ( ) min( , )x A b x a x a= =� �

0.9 0.8 0.8

0.8 0.7 0.8

0.9 0.7 0.6

A

=

b =

The maximum solution is

Table 5.

Figure

04.99

5

5.01

5.02

5.03

5.04

Obje

cti

ve v

alu

e

Iterations

1

2

3

4

6

31

114

2 2

1 2Min ( ) 10( 0.5) 10( 0.5) 5f x x x= − + − +

s.t., where ( ) min( , )x A b x a x a= =� �

[ ]0.7 0.6 0.5=

The maximum solution is obtained as [0.5000 0.5000 0.5000] givingε

5.3: Objective values by iterations - Example 5.3


10 20 30 40Iterations

1x 2x 3x ( )f x ( )e x

0.4423 0.4704 0.6688

0.4631 0.4853 0.6334

0.4784 0.5164 0.6617

0.4802 0.4933 0.6586

0.5027 0.4976 0.6506

0.5002 0.4990 0.6461

5.0421

5.0158

5.0074

5.0044

5.0001

5.0000

0.0157

0.0156

0.0153

0.0151

0.0150

0.0150

max 0.05.ε =

Figure 5.5: N

The following section considers

kind of geometric objective functions called generalized monomials subjected to a system

of continuous t-norm based FRE

5.6 The problem II

We consider the following nonlinear optimization problem

Min ( ) (5.7)

( , )

Z x

x X A b∈

where ( )Z x is the generalized monomial function defined as:

( ) max{ ( )} max{ }k k i

k K k KZ x f x c x

∈ ∈= =

where ( )k

f x is a monomial geometric objective function in

0k

c > and ,ikr R∈ (1 ,1 )k K i n≤ ≤ ≤ ≤

monomial and 1 2[ , , , ]x x x x=

is generally not differentiable

0.814

6

8

10

Obje

ctiv

e function

115

Nonlinear function with optimal point in Example

considers the second nonlinear optimization problem with special


norm based FRE, when the system has no unique solution.

the following nonlinear optimization problem:

Min ( ) (5.7)

is the generalized monomial function defined as:

1

( ) max{ ( )} max{ }ik

mr

k k ik K k K

i

Z x f x c x=

∏

is a monomial geometric objective function in x with

(1 ,1 )k K i n≤ ≤ ≤ ≤ are corresponding exponents of variable

1 2[ , , , ]m

x x x x… is the solution vector. The maximum of two

is generally not differentiable (when the two monomials are same), whereas a

00.2

0.40.6 0.8

1

00.2

0.40.6

0.8

x1x2

Example 5.3

nonlinear optimization problem with special


when the system has no unique solution.

Min ( ) (5.7)

each coefficient

are corresponding exponents of variables ini

x thk

he maximum of two monomials

, whereas a monomial

116

is everywhere differentiable. More detailed discussion on behavior and extensions of GP

can be viewed in [12].

5.7 Approximate solutions of FRE

The problem of approximating fuzzy relation equations (5.1) is to find one or more

vectors 1 2[ , , , ]m

x x x x= … having the least distance (error) between the left and right

parts of system of (5.1) i.e. finding the approximate solutions that have minimum

distance of x A� from .b

To get such solutions a real coded genetic algorithm (RCGA) is designed that finds a

vector min min min min

1 2[ , , , ]m

x x x x= … which provides the least distance between the left and

right parts of system (5.1) among all the solution vectors i.e. min

min ( ( )).x

e e x= Once the

solution giving least error value is obtained, the uncertainty interval providing the

essential range of each decision variable is determined. For this, the algorithm is operated

till a set of solution vectors having the same distance as the solution vector minx is

obtained. When a set of such equivalent solutions and vector minx has been obtained,

upper and lower bounds of the individual components of the obtained vectors are found.

Let 1 2{ , , , }, 1,2, ,l l l

mx x x l L=… … be the collection of such L equivalent vectors obtained.

The uncertainty interval [ , ]i ix x representing the essential range for each component ix is

determined by selecting the lower bound ix and upper bound i

x of thi component

respectively determined as 1

min { }L

l

i il

x x=

= and 1

max { }.L

l

i il

x x=

=

RCGA starts by randomly generating initial population of several solutions, where every

component , 1,2, , ,i

x i m= … of all the solution vectors is a random number in the unit

interval (0,1).

117

The chromosomes are evaluated using the distance function (error) as the fitness function

given as follows:

2( ) | | [ max ( ( )] (5.8)j i ij

i Ij J

e x x A b b x a∈

∈

= − = −∑� �

After evaluation of chromosomes, the selection is performed according to rank based

selection procedure discussed in section 4.5.1 of chapter 4. The lower fitness value (error

or distance) of chromosome represents the good candidature of the chromosome to

perform the criteria of optimization. For the speedy convergence of the algorithm and to

avoid unnecessary exploration, we set the upper bound of the error value (distance

function) as a real numberε .

After selecting good individuals, we apply the arithmetic crossover and feasible mutation

described in section 5.4.1 above. The newly generated solutions are processed until some

termination criterion is met. The proposed procedure to solve system (5.1) can be

summarized in algorithm 2.

Algorithm 2: Procedure to solve system (5.1)

1. Get the matrices and .A b

2. Find the maximum solution x�

using (5.3).

3. Check whether system (5.1) is solvable i.e. x A b=�� if yes, stop.

4. Run real coded genetic algorithm (RCGA) for finding vectors minx s.t. min min( )e x e= .

Determine L such equivalent solutions having the same error value mine .

5. Obtain uncertainty intervals [ , ],i ix x for each of the decision variable ix

1, 2, , .i m∀ = …

118

Once the essential range of decision variables have been determined by Algorithm 2, the

modified problem is formed as follows:

Min ( ) (5.9)

[ , ] [0,1], 1,2, , i i i

Z x

x x x i m∈ ⊆ ∀ = …

Again the genetic procedure RCGA as described above for finding the essential range is

used to solve the problem (5.9). The design of RCGA remains same except the fitness

function that is the objective function ( )Z x itself here. To improvise the convergence of

the genetic algorithm, the elitism criterion [18] is used in which a fraction of the best

chromosomes from the previous population is placed to the new population so that the

best chromosomes never disappear from the population through crossover or mutation.

The procedure to solve the considered optimization problem (5.9) is described in

algorithm 3.

Algorithm 3: Procedure to solve optimization problem (5.9)

1. Find the uncertainty intervals for each of the decision variable using Algorithm 2.

2. Solve the optimization problem with considered objectives using the RCGA.

5.8 Illustrative examples

We consider some optimization problems with generalized monomial objectives subject

to max-� composition based fuzzy relation equations having no unique solution.

Algorithm 2 is applied to obtain the approximate solutions of fuzzy relation equations

within their uncertainty intervals and then the modified optimization problem is solved.

The results obtained are presented with the help of tables 5.4-5.9. The behaviour of the

algorithm is presented graphically via figures 5.6-5.11.

119

Example 5.4. 1 2 3Min ( ) max{ ( ) ( ) ( )}Z x f x f x f x=

where0.2 0.3 2 3 0.2 1.5 0.33 2 1.5

1 1 2 3 2 1 2 3 3 1 2 3( ) 10 , ( ) 0.2 , ( ) 0.3f x x x x f x x x x f x x x x− − − − −= = =

s.t., where ( ) min( , ) andx A b x a x a= =� �

0.9 0.8 0.8

0.8 0.7 0.8

0.9 0.7 0.6

A

=

[ ]0.7 0.6 0.5b =

The maximum solution comes out to be [0.5000 0.5000 0.5000] .Since x A b≠�� , the

system is inconsistent. After running the Algorithm 2, the solution of fuzzy relation

equations can be represented in the form of uncertainty intervals as 1 [0,0.6],x ∈

2 [0,0.6],x ∈ 3 [0.6450,0.6550].x ∈

Table 5.4: Error values by iterations from RCGA - Example 5.4

Iteration 1x 2x 3x ( )e x

1

98

194

300

409

467

600

635

801

914

935

1025

1200

0.5000 0.5000 0.5000

0.5615 0.6800 0.3486

0.2399 0.5276 0.1762

0.0732 0.6105 0.8094

0.6128 0.0901 0.2476

0.1586 0.0380 0.6173

0.0516 0.5711 0.6061

0.2367 0.3015 0.6088

0.1955 0.4756 0.6754

0.1454 0.5591 0.6394

0.5705 0.5158 0.6648

0.3444 0.3877 0.6400

0.1957 0.3804 0.6481

0.0500

0.0392

0.0357

0.0342

0.0205

0.0189

0.0184

0.0171

0.0163

0.0157

0.0154

0.0152

0.0150

Figure 5.6: Performance

Iteration

20

27

45

86

110

165

204

224

303

309

378

500

536

840

2196

2490

9730

12637

14935

Table 5.

00.01

0.02

0.03

0.04

0.05

Error v

alu

e

120

Performance of GA for solving FRE in Example

Iterations 1x 2x 3x ( )Z x

1

20

27

45

86

110

165

204

224

303

309

378

500

536

840

2196

2490

9730

12637

14935

0.5949 0.5801 0.6503

0.5607 0.5964 0.6466

0.5812 0.5969 0.6488

0.5715 0.5936 0.6454

0.5885 0.5914 0.6463

0.5993 0.5890 0.6459

0.5841 0.5998 0.6456

0.5975 0.5926 0.6453

0.5932 0.5997 0.6455

0.5947 0.5980 0.6453

0.5937 0.5995 0.6453

0.5959 0.5988 0.6453

0.5973 0.5985 0.6450

0.5989 0.5980 0.6450

0.5996 0.5993 0.6450

0.5989 0.5999 0.6450

0.5998 0.5999 0.6450

0.5999 0.5999 0.6450

0.5999 0.5999 0.6450

0.6000 0.6000 0.6450

5.5244

5.4802

5.4767

5.4477

5.4363

5.4171

5.4096

5.3999

5.3920

5.3905

5.3884

5.3865

5.3806

5.3791

5.3741

5.3734

5.3716

5.3714

5.3712

5.3710


500 1000 1500Iterations

Example 5.4

Figure 5.

Example 5.5. Min ( ) max{ ( ) ( ) ( )}Z x f x f x f x

where0.5

11 2 3 2 1 2 3 3 1 2 3

2

( ) 2 , ( ) 3 , ( ) 0.5x

f x x x f x x x x f x x x xx

= = =

s.t., where ( ) max (0, 1)x A b x a x a= = + −� �

0.5000 0.6000 0.2000 0.3000

0.7000 0.2000 0.6000 0.4000 ,

0.8000 0.1000 0.2000 0.4000

A

=

The maximum solution of the considered FRE

min ( ) where min(1,1 ) ij t j ij t j ij jj J i I

x a b a b a b∈ ∈

= Θ Θ = − +

�

The maximum solution is obtained

system does not have a unique solution.

fuzzy relation equations can be represented in the form of uncertainty intervals as

1 2 3[0.0011,0.4084], [0.0021,0.4683], [0.8714,x x x∈ ∈ ∈

05.35

5.4

5.45

5.5

5.55

Obje

cti

ve v

alu

e

121


1 2 3Min ( ) max{ ( ) ( ) ( )}Z x f x f x f x=

0.5

0.5 2 0.3 0.2 1.5 0.5 2 2.5

1 2 3 2 1 2 3 3 1 2 3( ) 2 , ( ) 3 , ( ) 0.5f x x x f x x x x f x x x x− − −= = =

s.t., where ( ) max (0, 1)x A b x a x a= = + −� � and

0.5000 0.6000 0.2000 0.3000

0.7000 0.2000 0.6000 0.4000 ,

0.8000 0.1000 0.2000 0.4000

[ ]1.000 0 0 0b =

The maximum solution of the considered FRE is computed by assigning:

min ( ) where min(1,1 ) ij t j ij t j ij jx a b a b a b= Θ Θ = − +

The maximum solution is obtained as[0.4000 0.4000 0.6000] . Here

a unique solution. After running the algorithm 2


1 2 3[0.0011,0.4084], [0.0021,0.4683], [0.8714,0.8719].x x x∈ ∈ ∈

5000 10000 15000Iterations

0.5 2 0.3 0.2 1.5 0.5 2 2.5

1 2 3 2 1 2 3 3 1 2 3f x x x f x x x x f x x x x− − −

:

x A b≠�� i.e. the

2 the solution of


Table 5.6: Error value

Figure 5.8: Performance

Iterations

1

4

22

54

74

186

570

792

1073

2412

2426

7467

Table 5.

100

0.2

0.25

0.3

0.35

Err

or v

alu

e

122

Iterations 1x 2x 3x ( )e x

1

2

18

94

113

149

800

1000

0.4000 0.4000 0.6000

0.3800 0.4274 0.7971

0.3432 0.3264 0.8064

0.2261 0.1064 0.8358

0.4154 0.3050 0.8744

0.1709 0.1044 0.8577

0.4028 0.0347 0.8603

0.1400 0.2418 0.8692

0.3600

0.2019

0.1976

0.1895

0.1871

0.1869

0.1868

0.1867

: Error value by iterations from RCGA - Example

Performance of GA for solving FRE in Example

Iterations 1x 2x 3x ( )Z x

1

4

22

54

74

186

570

792

1073

2412

2426

7467

0.0110 0.3969 0.8717

0.0071 0.3933 0.8715

0.0050 0.2845 0.8716

0.0032 0.4622 0.8715

0.0015 0.4489 0.8716

0.0014 0.4422 0.8717

0.0012 0.4676 0.8714

0.0012 0.4682 0.8714

0.0011 0.4619 0.8714

0.0011 0.4668 0.8714

0.0011 0.4676 0.8714

0.0011 0.4682 0.8714

0.7592

0.6667

0.6401

0.5081

0.4104

0.3974

0.3795

0.3787

0.3722

0.3720

0.3685

0.3680


010

110

210

3

Iterations

Example 5.5

Example 5.5

123


Example 5.6. 1 2 3Min ( ) max{ ( ) ( ) ( )}Z x f x f x f x=

where0.2 3.9 1.7 0.7 0.5 2.2 0.5

1 2 3 2 1 2 3 3 1 2 3( ) 2 , ( ) 0.1 , ( ) 0.2f x x x f x x x x f x x x x− − −= = =

s.t., where ( ) andx A b x a x a= = ⋅� �

0.2 0.5 0.1

0.6 0.8 0.1 ,

0.3 0.1 0.5

A

=

[ ]0.7 0.3 0.54b =

The maximum solution comes out to be [0.5000 0.5000 0.5000] . Since x A b≠�� , the

system is inconsistent. After running the algorithm 2, the solution of fuzzy relation

equations can be represented in the form of uncertainty intervals as

1 2 3[0.0151, 0.9238], [0.6534, 0.6653], [0.9988,1].x x x∈ ∈ ∈

0 2000 4000 6000 8000

0.4

0.5

0.6

0.7

0.8

Iterations

Obje

ctive v

alu

e

Table 5.8: Error value

Figure 5.10:

Iterations

1

10

30

53

76

93

109

258

460

764

908

1132

4797

5304

0.33150000000000 0.65460000000000

0.05359783524843 0.65497163207262 0.99997144103778

0.37525811320337 0.65429688731845 0.99992731972972

0.56627835452936 0.65478799778839 0.99998623143182

0.17900064235879 0.65422067161832 0.99994951632460

0.33293273955089

0.28431221943608 0.65428387117017 0.99998899203319

0.21421775187278 0.65423073084255 0.99998727173686

0.45424581988486 0.65434033833548 0.99999930512376

0.58563359959818 0.65424282346368 0.99999505619264

0.31251051189700 0.65423061585757 0.99999875990685

0.34095105249210 0.65420464530495 0.99999931816709

0.25349123838667 0.65420530804149 0.99999990118377

0.48401520815171 0.65420181545464 0.99999984605373

Table 5.

0

0.146

0.148

0.15

0.152

0.154

0.156

Erro

r valu

e

124

Iterations 1x 2x 3x ( )e x

1 2 4 18

103 170 711

0.0178 0.7341 0.9449

0.4151 0.6194 0.9202

0.1542 0.6346 0.9825

0.5204 0.6388 0.9891

0.3204 0.6717 0.9877

0.6145 0.6597 0.9943

0.3281 0.6566 0.9998

0.1545

0.1524

0.1474

0.1469

0.1467

0.1462

0.1460

: Error value by iterations from RCGA-Example 5.

: Performance of GA for solving FRE in Example

1x 2x 3x

0.33150000000000 0.65460000000000 0.99990000000000

0.05359783524843 0.65497163207262 0.99997144103778

0.37525811320337 0.65429688731845 0.99992731972972

0.56627835452936 0.65478799778839 0.99998623143182

0.17900064235879 0.65422067161832 0.99994951632460

0.33293273955089 0.65471758308899 0.99999864362696

0.28431221943608 0.65428387117017 0.99998899203319

0.21421775187278 0.65423073084255 0.99998727173686

0.45424581988486 0.65434033833548 0.99999930512376

0.58563359959818 0.65424282346368 0.99999505619264

0.31251051189700 0.65423061585757 0.99999875990685

0.34095105249210 0.65420464530495 0.99999931816709

0.25349123838667 0.65420530804149 0.99999990118377

0.48401520815171 0.65420181545464 0.99999984605373


0 200 400 600 800Iterations

5.6

Example 5.6

( )Z x

1.838208

1.837904

1.837842

1.837695

1.837640

1.837567

1.837392

1.837375

1.837350

1.837326

1.837292

1.837274

1.837270

1.837268

Figure 5.11

5.9 Conclusion

This chapter considers two

norm based system of max

unsolvable, the approximate solutions

optimizing the objective simultaneously. Such optimization models are common in

practical situations when we face perturbed systems

The first problem states a nonlinear optimization problem with a general nonlinear

objective function. A well structured genetic al

converging solutions of nonlinear programming problem

the selection operator and fitness function are key features of the algorithm responsible

for the faster convergence. The generation wi

dual optimization of objective function and error function simultaneously. Experiment

results for optimization problems with different compositions based FRE systems are

presented that validate the capability o

The second problem considers a

inconsistent system of constraints

1.8372

1.8374

1.8376

1.8378

1.838

1.8382

Obje

cti

ve v

alu

e

125


two nonlinear optimization problems subjected to a continuous t

norm based system of max-� fuzzy relational equations. As the system of constraints is

the approximate solutions are determined leading to the least error and

the objective simultaneously. Such optimization models are common in

practical situations when we face perturbed systems.


A well structured genetic algorithm is applied to get the

solutions of nonlinear programming problem. The problem specific design of


for the faster convergence. The generation wise update of threshold error value results in



presented that validate the capability of the proposed algorithm.

The second problem considers a generalized geometric optimization problem

system of constraints. A method is proposed to find approximate solutions of

0 1000 2000 3000 4000 5000 6000

1.8372

1.8374

1.8376

1.8378

1.838

1.8382

Iterations

subjected to a continuous t-

fuzzy relational equations. As the system of constraints is

are determined leading to the least error and

the objective simultaneously. Such optimization models are common in


applied to get the good

. The problem specific design of


se update of threshold error value results in



generalized geometric optimization problem with an

method is proposed to find approximate solutions of

126

such fuzzy relation equations within uncertainty intervals described by lower and upper

bounds of each component of solution vector. Then the problem is modified to an

optimization problem with reduced search space. A well structured genetic algorithm is

applied to get the good converging solutions of nonlinear programming problem where

the value of each component of solution vector lies within their respective uncertainty

intervals. Experiment results for optimization problems with different compositions based

FRE systems are presented to demonstrate the working of the proposed procedure.

**********

chapter 5 approximating nonlinear optimization problem...

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