A genetic algorithm for optimization problems with fuzzy relation constraints using max-product composition

Applied Soft Computing 11 (2011) 551–560

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Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc

A genetic algorithm for optimization problems with fuzzy relation constraints using max-product composition Reza Hassanzadeh a , Esmaile Khorram b , Iraj Mahdavi a,∗ , Nezam Mahdavi-Amiri c a b c

Mazandaran University of Science and Technology, Babol, Iran Faculty of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran Faculty of Mathematical Sciences, Sharif University of Technology, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 18 December 2008 Received in revised form 15 November 2009 Accepted 6 December 2009 Available online 16 December 2009 Keywords: Fuzzy relation equations Genetic algorithms Nonlinear optimization Max-product composition

a b s t r a c t We consider nonlinear optimization problems constrained by a system of fuzzy relation equations. The solution set of the fuzzy relation equations being nonconvex, in general, conventional nonlinear programming methods are not practical. Here, we propose a genetic algorithm with max-product composition to obtain a near optimal solution for convex or nonconvex solution set. Test problems are constructed to evaluate the performance of the proposed algorithm showing alternative solutions obtained by our proposed model. © 2009 Elsevier B.V. All rights reserved.

1. Introduction Consider the following fuzzy relation equations, xoA = b,

(1)

where A = (aij )m×n , 0 ≤ aij ≤ 1, is a fuzzy matrix, b = (b1 ,b2 , . . ., bn ), 0 ≤ bj ≤ 1, is an n-dimensional vector and “o” stands for the maxproduct composition [1], that is, max (xi aij ) = bj ,

i=1,...,m

j = 1, 2, . . . , n.

(2)

Given the fuzzy relation matrix A and output vector b, the resolution problem is to determine all input vectors x = (x1 , . . ., xm ), 0 ≤ xi ≤ 1, satisfying (1). A nonempty solution set of the fuzzy relation equations is generally a nonconvex set determined in terms of the maximum solution and the finite number of minimal solutions [1,2,4–6]. The theory of fuzzy relational equations (FRE) forms a generalization of Boolean relation equations [14]. In [15], Sanchez investigated the notion of fuzzy relation equations based upon the max–min composition. He considered some theoretical methods

∗ Corresponding author. E-mail address: [email protected] (I. Mahdavi). 1568-4946/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2009.12.014

and conditions to resolve the fuzzy relations. He also presented some results for the determination and existence of solutions of certain basic fuzzy relation equations. The set of solutions of (1) is not usually a singleton. However, he showed that, when the set of solutions is nonempty, it is a nonconvex set, in general, and it can be completely determined by a unique maximum solution and a finite number of minimal solutions. In [16], Sanchez initiated a development of the theory and applications of FRE treated as a formalized model for imprecise notions. Fang and Li [2] converted an optimization problem with a single linear objective function subject to the fuzzy relation equations based on the max–min composition to a 0–1 integer programming problem and solved it by a branch and bound method. Wu et al. [17] improved Fang and Li’s method by providing an upper bound for the branch and bound procedure. Lee and Guu [34] proposed a fuzzy relational optimization model for the streaming media provider seeking a minimum cost while fulfilling the requirements assumed by a three-tier framework. The max–min composition is normally applied when a system involves conservative solutions in the sense that the goodness of one value cannot compensate the badness of another value [13]. Other compositions can also be used depending on the applications. Yager [18] gives some guidelines for selecting a proper composition. The fundamental result for the fuzzy relation equations with max-product composition having the conservative property goes

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back to Pedrycz [19]. Another study in this regard can be found in Brouke and Fisher [20]. They extended the study of an inverse solution of a system of fuzzy relation equations with max-product composition. They provided theoretical results for determining the complete set of solutions as well as the conditions for the existence of solutions. Their results showed that such complete set of solutions can be characterized by one maximum solution and a number of minimal solutions. Furthermore, the monograph by Nola et al. [21] contains a thorough discussion of this class of equations. Markovskii [22] showed that solving maxproduct FRE is closely related to the covering problem, and hence is NP-hard. Chen and Wang [23] designed an algorithm for obtaining the logical representation of all minimal solutions. They showed that a polynomial-time algorithm to find all minimal solutions of an FRE with max–min composition may not exist. Peeva and Kyosev [26] developed an exact method and a universal algorithm for solving max-product fuzzy linear system of equations and max-product fuzzy relational equations. Markovskii [27] described methods for reducing the dimension of the covering problem and methods for solving fuzzy relation equations with max-product composition. Linear optimization problems with max-product approach was investigated by Loetamonphong et al. [24]. They defined two subproblems by separating the negative and nonnegative coefficients in the objective function, and then obtained the optimal solution by combining the two subproblems. The subproblem with a negative coefficient is easily optimized by the maximum solution of the set of solutions. The other subproblem was converted to a 0–1 integer programming problem and solved by the branch and bound method. Guu and Wu [25] gave a necessary condition for an optimal solution in terms of the maximum solution derived from the fuzzy relation equations. Guu and Wu [28] studied the optimization problem subject to fuzzy relation equations with max-product composition. Yang and Cao [29] studied the fuzzy relation geometric programming problems with monomial objective function and fuzzy relation equations as constraints using max–min composition. Guo and Xia [30] presented a new approach for solving optimization problems with one linear objective function and finitely many fuzzy relation inequality constraints. Tao et al. [31] developed methods for solving the global optimization problem of max–min systems and established the criteria for the existence and uniqueness of global optimal solutions. Lu and Fang [11] proposed a genetic algorithm to solve a nonlinear single objective problem with fuzzy relation equations as constraints considering the max–min composition. Here, we consider minimizing a nonlinear objective function constrained by max-product fuzzy relation equations. The set of feasible solutions being nonconvex and the problem having a special structure, we propose to apply a genetic algorithm for finding a solution. The nonlinear programming model with fuzzy relation constraints is formally defined to be: minf (x),

s.t. xoA = b.

(3)

In Section 2, we describe our genetic algorithm, in detail. We devote Section 3 to the effective construction of test problems and numerical experimentation. Finally, we conclude in Section 4. 2. The proposed genetic algorithm Genetic algorithms (GAs) are built upon the mechanism of natural evolution of genetics. GAs emulate the biological evolutionary

theory to solve optimization problems. In general, GAs start with a randomly generated population and progress to improve solutions by using genetic operators such as crossover and mutation. In each iteration (a generation), based on the performance (fitness) and some selection criteria, the relatively good solutions are retained and the relatively bad solutions are replaced by some newly generated offsprings. An evaluation criterion (objective) usually guides the selection. Our proposed GA is designed specifically for solving nonlinear optimization problems with fuzzy relation constraints as specified by (3).

2.1. Representation In our algurithm, since the solutions of fuzzy relation equations are comprised of nonnegative real numbers not bigger than one then we use the floating point [12] representation in which each gene or variable xi in an individual x = (x1 , x2 , . . ., xm ) is a real number in the interval [0, 1].

2.2. Initialization In general, a GA initializes the population randomly. This works well when dealing with unconstrained optimization problems. However, for a constrained optimization problem, randomly generated solutions may not be feasible. Since GA intends to keep the solutions (chromosomes) feasible, we present an initialization module to initialize a population by randomly generating the individuals inside the feasible domain. Since some elements will never play a role in determining a solution of the fuzzy relation equations, then we can modify the fuzzy relation matrix by identifying these elements and setting their values to 0 hoping to accelerate the procedure for finding a new solution. To make it clear, we define the “equivalence operation”. Definition 1. If nullifying (setting to zero) some elements of a given fuzzy relation matrix A has no effect on the solutions of fuzzy relation Eq. (1), then nullifying is called an “equivalence operation”. The following lemma given in [11] can be useful. Lemma 1. For j1 , j2 ∈ {1, 2, . . ., n}, if bj1 /aij1 > bj2 /aij2 , aij1 ≥ bj1 , and aij2 ≥ bj2 , for some i, then an equivalence operation can be performed by “setting aij1 to zero”. We now give an example to illustrate an equivalence operation. Example 0.

⎛

Consider the matrix A and the vector b below:

0.3765

⎜ 0.8595 ⎜ ⎜ ⎜ 0.7939 A=⎜ ⎜ 0.6095 ⎜ ⎜ ⎝ 0.3318 b=



0.6240 0.6254

0.5858

⎞

0.6539

0.6423

0.6044

0.5603

0.2591

0.3769

0.4836 ⎟

0.2077

0.1377

0.3626

0.5429 ⎟ ⎟

⎟ ⎟, 0.0260 0.1207 0.8866 ⎟ ⎟ ⎟ 0.9870 0.4491 0.0816 ⎠

0.6198

0.6010



0.8521 .

Since b2 /a12 > b3 /a13 , a12 > b2 , and a13 > b3 , then the operation “setting a12 to 0” is an equivalence operation. Based on this idea, the initialization module originates a population consisting of a given number of randomly generated feasible solutions. An algorithm for initializing a population is described as having the following steps.

R. Hassanzadeh et al. / Applied Soft Computing 11 (2011) 551–560

Algorithm 1.

Population initiation.

Note that the set of random numbers generated as the matrix pop forms a population of Psize as solutions to the fuzzy relation equations. 2.3. Fitness function and selection strategy It has been observed in [12] that the Roulette Wheel selection stratege has some limitations. To avoid convergence to local optimal points, we modify the approach by a ranking strategy. Let the rank of pth person in current population be denoted by rp and defined to be: rp = Psize − the number of dominants,

p = 1, . . . , Psize ,

where the number of dominants denotes number of persons in population with objective function value smaller than the objective value corresponding to pth person. Then, the Roulette Wheel [12] procedure for producing next new generation is used with the probability of selecting pth person calculated as: r

p

Psize . i=1

553

ri

We can simply choose the objective function to be the fitness function. 2.4. Mutation operator To stay feasible, we can not mutate the chromosomes randomly. Although various mutation operators handling constrained

optimization problems have been proposed in the literature [3,8,9], but they have mostly been designed for convex problems. Mutation operators offered for nonconvex problems are rare. In the following, we present a mutation operator for GA whose feasible domain is considered to be nonconvex. Note that a chromosome in GA is represented by a 1 × m vector 1 ), as x = (x1 , x2 , . . ., xm ). For a given chromosome x1 = (x11 , x21 , . . . , xm we define a feasible mutation operator that mutates the chromosome by randomly choosing i0 from {1, 2, . . ., m} and decreasing xi1 0

to a random number in [0, xi1 ]. Since this operation may make the 0

/ i0 , to make chromosome x1 infeasible, we can adjust other xi1 , i = x1 feasible. We present a feasible mutation operation as follows. Algorithm 2.

Mutation operation.

(1) Get the simplified matrix A, b and x = (x1 , x2 , . . ., xm ). (2) Identify the decreasing set D, a subset of {1, 2, . . ., m}, such that there are more than one aij in column j of A satisfying bj /aij ≤ 1, for i ∈ D. (3) Randomly choose an index k in D. Generate a random number xk from the interval [0, xk ], and set x = (x1 , x2 , . . . , xk , . . . , xm ). (4) If ∨m (x a ) = bj , ∀j, then go to (7). i=1 i ij (5) Let the increasing set be N = D − {k}. (6) For an equation j not satisfied by x, randomly choose an element xl from the increasing set N such that xl < bj /alj and bj /alj ≤ 1. Set xl = bj /alj and go to (4). (7) Apply crossover operation.

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R. Hassanzadeh et al. / Applied Soft Computing 11 (2011) 551–560

Table 1 Elements of each column such that nj /aij ≤ 1. Column 1

Column 2

Column 3

Column 4

a21 , a31

a52

a13

a44

Note that this mutation operator only decreases one element of the vector x at a time. However, the number of increased elements is not necessarily equal to one. It could be any number from 1 to m − 1. This means that the mutation operator might change the chromosomes a great deal. For further insight, we use Example 0 discussed before to illustrate the mutation operator. First, we look at A and b. For each column j, the elements aij of A that satisfy bj /aij ≤ 1 are shown in Table 1. Since column 2 has only one eligible element, then a52 , i.e., 5 (corresponding to x5 ), is not a legitimate element of the decreasing set D. The single eligible element in column 3, that is a13 , suggests that the element 1 (corresponding to x1 ) is not in D. Furthermore, since column 4 has only one eligible element, a44 , then 4 (corresponding to x4 ) is not a legitimate element for the decreasing set D either. Taking all the indices 1, 4 and 5 out of D, the decreasing set becomes D = {2, 3, 6}. For the next step, we consider two cases. Case 1. x=



one might consider some linear combinations of two individuals. However, since the feasible domain of fuzzy relation equations is nonconvex, a linear combination of two feasible individuals will very likely result to be infeasible. Notice that the feasible domain of fuzzy relation equations is comprised of several connected convex sets that have a common maximal point (solution). We can take advantage of this special structure and call the maximal solution a “superpoint”. Definition 2. If a nonconvex set is a union of a number of connected convex sets and the intersection S0 of these sets is not empty, then any point s of S0 is called a superpoint [10]. By this definition, a maximal point of the feasible domain of fuzzy relation equations is a superpoint. In a connected set S, for any two points of S, linear contractions and extractions can be defined. Definition 3. Given a connected set S and two points x1 , x2 of S, 0 ≤  ≤ 1 and  ≥ 1, (i) a linear contraction of x1 supervised by x2 is defined to be: x1 ← x1 + (1 − )x2 , and

In Section 2.2, we obtained an (initial) solution,

0.9357

0.3152

0.7878

0.9611

0.6280



(ii) a linear extraction of x1 supervised by x2 is defined to be:

0.3111 , x1 ← x1 − ( − 1)x2 .

for the fuzzy relation Eq. (1). Let us see how the feasible mutation operator works for the example. Suppose we randomly choose 2 (corresponding to x2 ) from the decreasing set D and generate a random number in the interval [0, 0.3152], say 0.1111. The new solution would then be:

Now that the linear contraction and linear extraction are defined, we can present a “three-point” crossover operator. The algorithm is stated as follows [10].

x=

Algorithm 3.



0.9357

0.1111

0.7878

0.9611

0.6280



0.3111 .

By substituting the new solution into the fuzzy relation equations, it is found that this new x is still a feasible solution. We then go to the crossover operation. Case 2. x=



Assume we have another selection,

0.9357

0.3152

0.7878

0.9611

0.6280



0.3111 ,

and the element we randomly choose from D is 3 (corresponding to x3 ). We then generate a random number from [0, 7878], say 0.5211. The new solution is: x=



0.9357

0.3152

0.5211

0.9611

0.6280



0.3111 .

By substituting the new solution into the fuzzy relation Eq. (1), it is found that the forth constraint in Eq. (1) is not satisfied. Since the increasing set is N = {2, 6}, then we randomly choose 2 (corresponding to x2 ) from N and set x2 = b4 /a24 = 0.7276. The new solution is: x=



0.9357

0.7276

0.5211

0.9611

0.6280



0.3111 .

By substituting this solution into the fuzzy relation equations, we find it to be a feasible solution. This terminates the mutation operation. 2.5. Crossover operator Like the mutation operator, most crossover operators in the literature are proposed for unconstrained or convex optimization problems [3,8,9]. Although there are some creative operators available for handling constrained optimization problems, none explicitly deals with nonconvex problems. To keep the individuals in the interior of the feasible domain after the crossover operation,

Crossover operation.

(1) Get the maximum solution xˆ (a superpoint) by formula (4), two parents x1 , x2 ∈ X(A,b), 0 ≤  ≤ 1, 0 ≤  ≤ 1,  ≥ 1 and 0 ≤  ≤ 1. For i = 1,2 do (2) Generate a random number rs in [0, 1]. If rs <  then Perform a linear contraction of xi supervised by xˆ else Perform a linear extraction of xi supervised by xˆ . (3) Generate a random number rx in [0,1]. If rx <  then 1 ← x1 ; set xold Perform a linear contraction of x1 supervised by x2 else go to (6) (4) If x1 is not feasible then 1 ; set x1 ← xold Perform a linear extraction of x1 supervised by x2 else go to (6). (5) If x1 is not feasible then 1 . set x1 ← xold (6) Call the evaluation procedure to compute fitness function.

Remarks. (1) The parameter  is usually taken to be a very small positive number, which implies that the linear contraction or extraction supervised by another parent is carried out with a very small probability. (2) The parameters  and  are set to be positive numbers very close to 1. It means that the step lengths of linear contraction and extraction supervised by another parent are very small. This is necessary for preserving the feasibility of the solution, since the large movement may pull the solution point out of the feasible region.

R. Hassanzadeh et al. / Applied Soft Computing 11 (2011) 551–560

(3) The parameter  for linear contraction supervised by the superpoint is a random number in [0, 1]. This makes the step length of the linear contraction supervised by the superpoint a nonfixed number. We illustrate the crossover operator using Example 0. We have two solutions from the solution space: x1 = x2 =

 



0.9357

0.2525

0.7878

0.9611

0.6280

0.5211 ,

0.9357

0.7276

0.3121

0.9611

0.6280

0.7921 .



Step 1. The maximum solution of the fuzzy relation equations is: xˆ =



0.9357

0.7276

0.7878

0.9611

0.6280



1 .

The parameters are set to be: •  = 0.5: the probability of performing a linear contraction and extraction supervised by xˆ . •  = 0.01: the probability of linear contraction or extraction supervised by a parent as x1 or x2 . •  = 0.995,  = 1.005: the step lengths of linear contraction or extraction supervised by a parent as x1 or x2 . Step 2. For i = 1, we generate a random number rs = 0.6291. Since rs >  (=0.5), then we perform a linear extraction of x1 supervised by xˆ : x1 ← x1 − ( − 1)ˆx =



0.9357

0.7276

0.3145

0.9611

0.6280



Algorithm 4. generator.

555

Feasible fuzzy relation equations test problem

(1) Generate an m × n matrix A whose elements are random numbers from the interval [0, 1]. (2) Generate avector b with its jth element being a random number from (a ), maxm (a ) . the interval 3/4maxm i=1 ij i=1 ij

(3) For i = 1, . . ., m do For j = 1, . . ., n do If aij > bj and there exists j = / j such that bj /aij < bj /aij and aij > bj then Change aij to any number less than bj .

It can be shown that the fuzzy relation equations generated by Algorithm 4 have feasible solutions. Put together with an objective function from Hock and Schittkowski’s books [7], we obtain a feasible problem. We will prove that the fuzzy relation equations are feasible by giving a sufficient condition for the existence of the solution of the fuzzy relation equations. To characterize the system of fuzzy relation Eq. (1), we introduce a pseudo-characteristic matrix P. Definition 4. Given a system of fuzzy relation Eq. (1), a pseudo-characteristic matrix (P-matrix), P = (pij )m×n , is defined to be:

Pij =

1 0 −1

if if if

aij > bj aij = bj aij < bj .

The following result can now be proved.

0.7931 . Theorem 1. (sufficient conditions for existence of solutions). For each column of matrix A, if

For i = 2, we then generate another random number rs = 0.1256. Since rs <  then we perform a linear contraction of x2 supervised by xˆ . Therefore, where S = 0.995, x2 ← s x2 + (1 − s )ˆx =



0.9357

0.7276

0.3145

0.9611

0.6280



0.7931 .

Step 3. We generate a random number rs = 0.0014. Since rs < , then we perform a linear contraction of x1 supervised by x2 : x1 ← x1 + (1 − )x2 =



0.9357

0.2525

0.7854

0.9611

0.6280



0.5201 .

Since x1 is infeasible, then we perform a linear extraction of supervised by x2 .We observe that x1 is infeasible and then x1 and x2 obtained by Step 2 are considered as the crossover operation. x1

3. Test problem construction and numerical experiments Since the existing theory of genetic algorithms cannot provide for the effective measurement of the performance, empirical computational testing is necessary. As for the fuzzy relation constraints, we will construct a feasible problem by randomly generating a fuzzy matrix A and then a vector b using A according to some criteria and prove that the solution set X(A, b) of the constructed fuzzy relation equations is not empty. The algorithm for generating feasible fuzzy relation equations is given next.

/ −1, and (i) there is at least one pij = (ii) pij = −1 and bk /aik > bj /aij impliy that pik = −1, then X(A,b) = / . Proof. Assame that the conditions (i) and (ii) are satisfied and X(A,b) = , that is, xoA = / b, ∀x ∈ X. Then, there exists at least one constraint i such that xi aij < bj , or there exist at least two constraints / i, k and j such that ∃i, xi aij = bj or xi aij < bj and for constraint k, ∀i = xi ai k < bk . The above two cases indicate that ∃j such that ∀i, aij < bj , or pij = 1, xi = bj /aij < bk /aik , pi k = −1, ∀i = / i. These cases imply that ∃j, ∀i, pij = −1 or pij = 1, xi = bj /aij < bk /aik , pi k = −1, ∀i = / i, yielding a contradiction with the assumptions (i) and (ii).  3.1. Generated test problems We construct four test problems by picking up four objective functions from Hock and Schittkowski’s book [7] and the fuzzy relation constraints as follows. Example 1. f (x) = (x1 + 10x2 )2 + 5(x3 − x4 )2 + (x2 − 2x3 )4 + 10(x1 − x4 )4

⎛

⎞

0.5176 0.1370 0.4093 ⎜ 0.2278 0.4585 0.7399 ⎟ A=⎝ , 0.8993 0.6334 0.0313 ⎠ 0.9858 0.2790 0.3039 b=



0.7208

0.6334



0.4725 .

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R. Hassanzadeh et al. / Applied Soft Computing 11 (2011) 551–560

Fig. 1. The feasible region and the objective function of Example 3 with max-product composition.

Fig. 4. The performance curve of the run of fitness value by generations of Example 4 with max-product composition. Fig. 2. The performance curve of the run of fitness value by generations of Example 3 with max-product composition.

Example 3. f (x) = x1 x2 x3 x4 x5

Example 2. f (x) = 3000x1 + 1000x13 + 2000x2 + 666.667x23



A= b=



0.3381 0.7189 0.1895 0.6089 0.7431 0.0729

0.4523

0.6089

0.7101 0.8791 0.3594





0.7452 .

,

⎛ 0.3391 0.3682 0.6702 0.8195 ⎞ ⎜ 0.4757 0.3823 0.9954 0.6934 ⎟ A = ⎜ 0.4403 0.6001 0.6981 0.3742 ⎟ , ⎝ ⎠ b=



0.5857 0.4295 0.4329 0.1488

0.5456

0.5244

0.4027 0.8493

0.8987

0.0096 0.7798



0.7544 .

Fig. 3. The feasible region and the objective function of Example 4 with max-product composition.

R. Hassanzadeh et al. / Applied Soft Computing 11 (2011) 551–560

557

Fig. 5. The feasible region and the objective function of Example 1 with max–min composition.

Table 2 Solution for Example 3 with max-prodct composition.

Fig. 6. The performance curve of the run of fitness value by generations of Example 1 with max–min composition.

Example 4. f (x) =

5 

⎛

2

[100(xk+1 − xk2 ) + (1 − xk )2 ]

k=1

0.3765 0.6539 ⎜ 0.8595 0.6044 ⎜ 0.7939 0.2591 A=⎜ ⎜ 0.6095 0.0260 ⎝ 0.3318 0.9870 0.6240 0.2077 b=



0.6254

0.6198

0.642 0.5603 0.3769 0.1207 0.4491 0.1377 0.6010

⎞

0.5858 0.5426 ⎟ 0.4836 ⎟ ⎟, 0.8866 ⎟ ⎠ 0.0816 0.3626



0.8521 .

Example 1 is not feasible using max-product composition. Moreover, the feasible region of Example 2 with max-product composition is only one point as follows: 

xˆ = x =



0.8470

0.8477

0.6087



with

f (x) = 5250.1.

Generation

x1

x2

x3

x4

x5

f(x)

1 3 4 6 29 81 262 377 621

0.9206 0.9206 0.0064 0.0009 0.9206 0.9206 0.9206 0.0001 0.0000

0.9029 0.9029 0.9029 0.9029 0.9029 0.9029 0.9029 0.9029 0.9029

0.8739 0.8739 0.8739 0.8739 0.8739 0.8739 0.8739 0.8739 0.8739

0.9315 0.9315 0.9315 0.9315 0.9315 0.9315 0.9315 0.9315 0.9315

0.0270 0.0101 0.9674 0.9674 0.0006 0.0003 0.0002 0.9674 0.9674

0.0183 0.0069 0.0045 0.000610 0.000432 0.000186 0.000119 0.000055 0.000002

Table 3 Solution for Example 4 with max-product composition. Generation

x1

x2

x3

x4

x5

x6

f(x)

1 2 3 8 50 208 247 427

0.9357 0.9357 0.9357 0.9357 0.9357 0.9357 0.9357 0.9357

0.6699 0.7100 0.7276 0.7276 0.7276 0.7276 0.7276 0.7276

0.7878 0.7878 0.7838 0.7844 0.7854 0.7875 0.7865 0.7878

0.9611 0.9611 0.9611 0.9611 0.9611 0.9611 0.9611 0.9611

0.6280 0.6280 0.6280 0.6280 0.6280 0.6280 0.6280 0.6280

0.4525 0.3322 0.4263 0.4109 0.4019 0.4282 0.3855 0.3915

36.6891 31.7862 29.7974 29.6870 29.6083 29.5969 29.5480 29.4628

In Tables 2 and 3, the solutions for Examples 3 and 4 are shown. The results were obtained using genetic algorithm with max-product composition. Tables 5–8 give the results for all the examples using max–min composition [10]. Note that Examples 1–4 with max-average composition are infeasible [32,33]. The feasible region of Example 3 is shown in Fig. 1 corresponding to variables x1 and x5 and the objective function is shown with optimal point x1 . Also, Fig. 2 shows the performance curve of the run for fitness value by generations.

Fig. 7. The feasible region and the objective function of Example 2 with max–min composition.

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R. Hassanzadeh et al. / Applied Soft Computing 11 (2011) 551–560 Table 6 Solution for Example 2 with max–min composition. Generation

x1

x2

x3

f(x)

1 4 6 102

0.0011 0.0005 0.0001 0.0000

0.7452 0.7452 0.7452 0.7452

0.4523 0.4523 0.4523 0.4523

1769.6 1767.7 1766.6 1766.3

Fig. 8. The performance curve of the run of fitness value by generations of Example 2 with max–min composition.

The feasible region of Example 4 is shown in Fig. 3 corresponding to variables x2 , x3 and x6 and the optimal point of objective function with respect to x3 and x6 . The performance curve of the run for fitness value by generations is also shown in Fig. 4. A comparison of GA solution and optimum solution is given in Table 4 using max-product composition. The optimal solutions are acquired by: (1) Computing all the convex branches of the fuzzy relation equations. (2) Searching for the optimal solution for each branch. (3) Comparison of the local optimal solutions obtained to identify a global optimal solution.

Table 4 Comparison of the real optimal point with the GA results for Examples 1–4 using max-product composition.

GA solution Optimal solution

Example 1

Example 2

Example 3

Example 4

Infeasible Infeasible

5250.1 5250.1

0.000002 0

29.4628 29.4543

Fig. 10. The performance curve of the run of fitness value by generations of Example 3 with max–min composition.

Tables 5–8 give the solutions of the above four examples using max–min composition. Moreover, figures 5, 7, 9 and 11 show the solution space and objective function values on optimal points. Also figures 6, 8, 10 and 12 show the performance curves of the runs for fitness value by generations. A comparison of GA solutions and optimal solutions are provided in Table 9 for the above problems using max–min composition.

Table 7 Solution for Example 3 with max–min composition. Table 5 Solution for Example 1 with max–min composition. Generation

x1

x2

x3

x4

f(x)

1 2 4 34 163 331

0.0362 0.0369 0.0020 0.0047 0.0027 0.0012

0.4725 0.4725 0.4725 0.4725 0.4725 0.4725

0.4708 0.4708 0.4708 0.4708 0.4708 0.4708

0.5777 0.4707 0.4829 0.4432 0.4223 0.4279

24.5132 24.4025 24.0440 24.0073 23.9885 23.9797

Generation

x1

x2

x3

x4

x5

f(x)

1 2 8 15 26 149 220 450

0.0187 0.0091 0.0049 0.0019 0.0004 0.7544 0.7544 0.7544

0.8987 0.8987 0.8987 0.8987 0.8987 0.8987 0.8987 0.8987

0.5244 0.5244 0.5244 0.5244 0.5244 0.5244 0.5244 0.5244

0.5456 0.5456 0.5456 0.5456 0.5456 0.5456 0.5456 0.5456

0.7544 0.7544 0.7544 0.7544 0.7544 0.0001 0.0001 0.0000

0.0036 0.0018 0.000952 0.000374 0.0000687 0.0000189 0.000009 0.000001

Fig. 9. The feasible region and the objective function of Example 3 with max–min composition.

R. Hassanzadeh et al. / Applied Soft Computing 11 (2011) 551–560

559

Fig. 11. The feasible region and the objective function of Example 4 with max–min composition.

Table 8 Solution for Example 4 with max–min composition.

Acknowledgements

Generation

x1

x2

x3

x4

x5

x6

f(x)

1 2 3 60 97 115 164 181 221 246 284 415 622

0.6010 0.6010 0.6010 0.6010 0.6010 0.6010 0.6010 0.6010 0.6010 0.6010 0.6010 0.6010 0.6010

0.6254 0.5856 0.6199 0.6140 0.6254 0.6228 0.6228 0.6228 0.6252 0.6253 0.6253 0.6253 0.6253

0.6218 0.6254 0.6254 0.6254 0.6237 0.6254 0.6254 0.6254 0.6254 0.6254 0.6254 0.6254 0.6254

0.8521 0.8521 0.8521 0.8521 0.8521 0.8521 0.8521 0.8521 0.8521 0.8521 0.8521 0.8521 0.8521

0.6198 0.6198 0.6198 0.6198 0.6198 0.6198 0.6198 0.6198 0.6198 0.6198 0.6198 0.6198 0.6198

0.5119 0.3312 0.4197 0.3655 0.3794 0.3903 0.3872 0.3842 0.3957 0.3927 0.3792 0.3823 0.3854

36.3365 36.3116 35.6229 35.5904 35.5768 35.4760 35.4733 35.4725 35.4686 35.4625 35.4771 35.4550 35.4548

Fig. 12. The performance curve of the run of fitness value by generations of Example 4 with max–min composition.

Table 9 Comparing the real optimal point with the GA results for Examples 1–4 with max–min composition.

GA solution Optimum solution

Example 1

Example 2

Example 3

Example 4

23.9797 23.9712

1766.3 1766.3

0.000001 0

35.4548 35.4531

4. Conclusions We considered solving nonlinear optimization problems constrained by fuzzy relation equations. A genetic algorithm was applied to solve such problems using max-product composition for the constraints. We presented a method for generating test problems to evaluate the performance of the proposed algorithm. Max-product, max–min and max-average compositions were used to solve the generated test problems. Comparative analyses of optimal solutions were given to demonstrate the alternative solution obtained by the proposed model using the max-product composition.

The first and third authors thank Mazandaran University of Science and Technology, the second author thanks Amirkabir University of Technology and the last author thanks Sharif University of Technology for supporting this work. The authors are also grateful to the anonymous referee and the editor in chief for their constructive comments and suggestions leading to an improved presentation.

References [1] E. Czogala, J. Drewniak, W. Pedrycz, Fuzzy relation equations on a finite set, Fuzzy Sets and Systems 7 (1982) 89–101. [2] S.C. Fang, G. Li, Solving fuzzy relations equations with a linear objective function, Fuzzy Sets and Systems 103 (1999) 107–113. [3] D.B. Fogel, Evolving artificial intelligence, Ph.D. Thesis, University of California, 1992. [4] S.Z. Guo, P.Z. Wang, A. Di Nola, S. Sessa, Further contributions to the study of finite fuzzy relation equations, Fuzzy Sets and Systems 26 (1988) 93– 104. [5] A. Ghodousian, E. Khorram, An algorithm for optimizing the linear function with fuzzy relation equation constraints regarding max-prod composition, Applied Mathematics and Computation 178 (2006) 502–509. [6] M. Higashi, G.J. Klir, Resolution of finite fuzzy relation equations, Fuzzy Sets and Systems 13 (1984) 65–82. [7] W. Hock, K. Schittkowski, Test examples for nonlinear programming codes Lecture Notes in Economics and Mathematical Systems, vol. 187, Springer, New York, 1981. [8] A. Homaifar, S. Lai, X. Qi, Constrained optimization via genetic algorithms, Simulation 62 (1994) 242–254. [9] J.A. Joines, C. Houck, On the use of non-stationary penalty function to solve nonlinear constrained optimization problem with GAs, in: Z. Michalewicz (Ed.), Proceedings of the 1st IEEE International Conference on Evolutionary Computation, vol. 2, IEEE Service Center, Piscataway, NJ, 1994, pp. 579– 584. [10] J. Lu, Sh. Fang, Solving nonlinear optimization problems with fuzzy relation equation constraints, Fuzzy Set and Systems 119 (2001) 1–20. [11] J. Lu, S.C. Fang, Solving nonlinear optimization problems with fuzzy relation constraints, Fuzzy Sets and Systems 119 (2001) 1–20. [12] Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Program, Springer, Berlin/Heidelbeg/New York, 1996. [13] H.J. Zimmermann, Fuzzy Set Theory and its Application, Kluwer Academic Pubkishers, Boston/Dordrecht/London, 1999. [14] G.J. Klir, T.A. Folger, Fuzzy Sets, Uncertainty and Information, Prentice-Hall, NJ, 1988. [15] E. Sanchez, Resolution of composite fuzzy relation equations, Information and Control 30 (1976) 38–48. [16] E. Sanchez, Solution in composite fuzzy relation equations: application to medical diagnosis in Brouwerian logic, in: M.M. Gupta, G.N. Saridis, B.R. Games (Eds.), Fuzzy Automata and Decision Processes, North-Holland, New York, 1977, pp. 221–234. [17] Y.K. Wu, S.M. Guu, J.Y.C. Liu, An accelerated approach for solving fuzzy relation equations with a linear objective function, IEEE Transactions on Fuzzy Systems 10 (4) (2002) 552–558. [18] R. Yager, Some procedures for selecting fuzzy set—theoretic operations, International Journal of General Systems 8 (1982) 115–124. [19] W. Pedrycz, Proceeding in Relational Structures: Fuzzy Relational Equations, Fuzzy Sets and Systems, vol. 40, 1991, pp. 77–106. [20] M.M. Brouke, D.G. Fisher, Solution algorithms for fuzzy relation equations with max-product composition, Fuzzy Sets and System 94 (1998) 61–69. [21] A. Nola, S. Di Sessa, W. Pedrycz, E. Sanchez, Fuzzy Relational Equations and their Applications in Knowledge Engineering, Kluwer Academic Press, Dordrecht, 1989.

560

R. Hassanzadeh et al. / Applied Soft Computing 11 (2011) 551–560

[22] A.V. Markovskii, On the relation between equations with max-product composition and the covering problem, Fuzzy Sets and Systems 153 (2005) 261–273. [23] L. Chen, P.P. Wang, Fuzzy relation equations. (I). The general and specialized solving algorithms, Soft Computing 6 (2002) 428–435. [24] J. Loetamonphong, S.C. Fang, R.E. Young, Multi-objective optimization problems with fuzzy relation equation constraints, Fuzzy Sets and Systems 127 (2002) 141–164. [25] S.M. Guu, Y.K. Wu, Minimizing a linear objective function with fuzzy relation equation constraints, Fuzzy Optimization and Decision Making (2002) 121568–124539. [26] K. Peeva, Y. Kyosev, Algorithm for solving max-product fuzzy relational equations, Soft Computing 11 (2007) 593–605. [27] A.V. Markovskii, Solution of fuzzy equations with max-product composition in inverse control and decision making problems, Automation and Remote Control 65 (9) (2004) 1486–1495. [28] S.M. Guu, Y.K. Wu, Minimizing a linear objective function with fuzzy relation equation constraints, Fuzzy Optimization and Decision Making 1 (2002) 347–360.

[29] J. Yang, B. Cao, Monomial geometric programming with fuzzy relation equation constraints, Fuzzy Optimization and Decision Making 6 (2007) 337– 349. [30] F.F. Guo, Z.Q. Xia, An algorithm for solving optimization problems with one linear objective function and finitely many constraints of fuzzy relation inequalities, Fuzzy Optimization and Decision Making 5 (2006) 33–47. [31] Y. Tao, G.P. Liu, W. Chen, Globally optimal solutions of max–min systems, Journal of Global Optimization 39 (2007) 347–363 (Springer Science + Business Media, B.V.). [32] E. Khorram, R. Hassanzadeh, Solving nonlinear optimization problems subjected to fuzzy relation equation constraints with max-average composition using a modified genetic algorithm, Computers & Industrial Engineering 55 (2008) 1–14. [33] E. Khorram, A. Ghodousian, Linear objective function optimization with fuzzy relation equation constraints regarding max-average composition, Applied Mathematics and Computation 173 (2006) 872–886. [34] H.C. Lee, S.M. Guu, On the optimal three-tier multimedia streaming services, Fuzzy Optimization and Decision Making 2 (2003) 31–39.