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Multi-objective optimization problems with Fuzzy relation equation constraints regarding max-average composition
Mathematical and Computer Modelling 49 (2009) 856–867
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Multi-objective optimization problems with Fuzzy relation equation constraints regarding max-average composition E. Khorram ∗ , H. Zarei Faculty of Mathematics and computer Science, Amirkabir University of Technology, 424, Hafez Ave, 15914, Tehran, Iran
article
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Article history: Received 30 October 2006 Received in revised form 5 October 2008 Accepted 27 October 2008 Keywords: Fuzzy relation equations Max–average composition Multi-objective optimization Genetic algorithm
a b s t r a c t A multiple objective optimization model subject to a system of fuzzy relation equations with max-average composition and a reduction procedure in order to reduce problem dimension are presented. Furthermore, a genetic algorithm is reviewed and some of its components are modified to solve the problem. Finally, the algorithm is performed on different problems and its major components role in detecting the solution is examined. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction Let X = [0, 1]m , I = {1, 2, . . . , m}, J = (1, 2, . . . , n}, and A = (aij ) be an m × n dimensional fuzzy matrix with aij ∈ [0, 1] and also b be an n-dimensional vector [bj ]1×n , so that, bj ∈ [0, 1] for all i ∈ I and j ∈ J. Given A and b, a system of fuzzy relation equations is defined by xoA=b av
(1)
where ‘‘ o ’’ is max-average composition [1]. A solution is a vector x = (x1 , . . . , xm ); 0 ≤ xi ≤ 1 so that maxi=1,...,m (xi + aij ) = av
2bj , j = 1, . . . , n. The solution set of (1) is usually not a singleton set. When the solution set is nonempty, it is in general a non-convex set which can be completely determined by a unique maximum solution and a finite number of minimal solutions [2]. Each input of (1) may require certain amount of resources which can be considered cost, and a decision maker may wish to achieve certain objectives. Considering an optimization problem with a single linear objective function subjected to the fuzzy relational equations based on the max–min composition, Fang and Li [3] converted this optimization problem into a 0-1 integer programming and solved it by Branch-and Bounded method. Wu et al. [4] improved Fang and Li,s method by providing an upper bound for Branch and Bounded procedure. Lee and Guu [5] proposed a fuzzy relational optimization model for the streaming media provider seeking a minimum cost while fulfilled the requirements assumed by a three-tier framework. Khorram and Ghodousian [1] dealt with solving fuzzy relation equations with max-average composition with a linear objective function. Lu and Fang [6] proposed a genetic algorithm to solve nonlinear single objective problem with fuzzy relation equation constraints regarding max–min composition. The fundamental result for fuzzy relation equations with max-product composition goes back to Pedrycz [7]. Recent study in this regard can be found in Bourk and Fisher [8]. 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 solution
∗
Corresponding author. E-mail address: [email protected] (E. Khorram).
0895-7177/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2008.10.018
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sets, as well as the conditions for the existence of resolutions. Their results showed that such complete solution sets can be characterized by one maximum solution and a number of minimal solutions. Furthermore, the monograph by Di Nola et al. [9] contains a thorough discussion of this class of equations. The optimization problem with max-product composition and with max–min composition was first considered by Loetamonphong and Fang [10] and Fang and Li [3]. For multiple objective linear programming problems (MOLP), in general, the objective functions usually conflict with each other. The notation of efficiency or Pareto optimality associated with feasible region has been introduced to instead of the optimality concept for single objective optimization [11]. Many solution methods for MLOP model have been studied in the literature since the early 1970,s. The simplex method extended by Zeleng [12]. MOLP can be converted in the goal programming problems, when the decision maker sets the satisfying goals and priority to each objective. Ignizio and Cavalier [13] mentioned three common forms of the goal programming. Deciding the relative deviation of the objective functions from the ideal objective value instead of prior preference information, Yu and Leitman [14] presented a compromise programming method to deal with MOLP. Benayound et al. [15] proposed a step method that seems to be known as one the first interactive techniques for dealing with MOLP. Sakawa [16] considered a fruitful survey regarding to interactive methods for MOLP. Gardiner and Steuer [17] have shown a number of procedures of interactive multiple objectives programming which include the satisfying trade-off method, the ε -constraint method and so forth. To the author’s best knowledge, Wang [18] was the first paper to explore the fuzzy relational equations based on the max–min composition yet with multiple objective linear functions. Zimmermann [19,20] proposed the min operator model to solve MOLP using a linearing membership function. The solution generated by min operator does not guarantee compensatory and efficient. Lee and Li [21] proposed a two-phase approach to overcome this difficulty. Chen and Chou [22] proposed a fuzzy to integrate the min operator, average operator and two-phase method. Extending two phase approach to the case of the fuzzy multi-objectives linear programming problem, Wu and Guu [23] presented a compromise model to provide different efficient solutions for the decision maker. Loetamonphong and el al used the genetic algorithm and solved the multi-objective optimization Problem with fuzzy relation equation constraints with max–min composition [24]. Now, we have imposed necessary changes to their algorithm and we have used it to solve multi-objective optimization problem subject to a set of fuzzy relation equations with max-average composition. In other words, we have attempted to solve the optimization problem in the following form: min f1 (x), . . . , fp (x)
s.t x o A = b
(2)
av
0 ≤ xi ≤ 1 where fk (x) is an objective function, k ∈ K = {1, 2, . . . , p}. The rest of the paper is outlined as follows: Section 2 provides the background of multi-objective optimization and describes a reduction procedure for transforming the original problem into a reduced form. In Section 3 some components of the proposed genetic algorithm in [6] are reviewed and then refined to solve the problem. Section 4 reports some test results of the genetic algorithm. Concluding remarks are given in Section 5. 2. Basic definitions and problem reduction 2.1. Basic definitions Some background and the theory of multiobjective optimization are studied here [18].Let the feasible domain of (2) denotes by X = {x ∈ Rm |x o A = b, 0 ≤ xi ≤ 1, ∀i}. Each point x ∈ X is said a solution vector and also z = (f1 (x), . . . , fp (x)) av
is defined its criterion vector. Moreover, we define Z = {z ∈ Rp |z = (f1 (x), . . . , fp (x)), ∃x ∈ X }. Definition 1. A point x¯ ∈ X is an efficient or a Pareto optimal solution to problem (2) if and only if there is no any x ∈ X such that fk (x) ≤ fk (¯x), ∀k = 1, . . . , p, and fk (x) < fk (¯x) for at least one k. Otherwise, x¯ is an inefficient solution. Definition 2. Let z 1 , z 2 ∈ Z be two criterion vectors. Then, z 1 dominates z 2 if and only if z 1 ≤ z 2 and z 1 6= z 2 That is z 1 ≤ z 2 , ∀k ∈ K , and zk1 < zk2 for at least one k. Definition 3. Let z¯ ∈ Z , then, z¯ is non-dominated if and only if there does not exist any z ∈ Z that dominates z¯ . Otherwise, z¯ is a dominated criterion vector. The set of all efficient points is called the efficient set or Pareto optimal set. In the absence of a mathematical specification of the decision maker’s utility function, we can only provide the decision maker with the Pareto optimal set for further analysis. 2.2. Problem reduction Due to the requirement of x o A = b some components of every solution vector may have to assume a specific value. Some av
of these components can therefore be set aside from the problem. The genetic operators are then applied to this reduced problem.
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We define x1 ≤ x2 ⇔ x1i ≤ x2i ∀x1 , x2 ∈ X in this way, the operator ‘‘ ≤’’ forms a partial order relation on X and (X , ≤) becomes a lattice. xˆ ∈ X is called a maximum solution if xˆ ≥ x for all x ∈ X . Also x˘ ∈ X , is called a minimal solution if, for any x ∈ X , x ≤ x˘ implies x = x˘ . Notice that the maximum solution xˆ can be obtained by the following formula [25]: xˆ = AΘ b = [Λnj=1 (aij Θ bj )]i∈I Where ‘‘Λ’’ is the min operator that is aΛb = min(a, b) and
aij Θ bj =
1 2bj − aij
2bj ≥ aij + 1 2bj < aij + 1.
Denote the set of all minimal solutions by X˘ . The complete set of solutions, X , is obtained by X =
S
x˘ ∈X˘
x ∈ X |˘x ≤ x ≤ xˆ .
Lemma 1. If x ∈ X , then for each j ∈ J there exists i0 ∈ I such that xi0 + ai0 j = 2bj and xi + aij ≤ 2bj , ∀i ∈ I. Proof is analog to Lemma 1 in [1]. Let X 6= φ and define Ij = {i ∈ I |ˆxi + aij = 2bj }, ∀j ∈ J.
Lemma 2. If Ij = 1 then xˆ i = x˘ i = 2bj − aij , for i ∈ Ij . Proof is similar to Lemma
3 in [1]. Define I 0 = {i ∈ Ij | Ij = 1} and J 0 = {j ∈ J |Ij ∩ I 0 6= φ}.
Now, if there exist q ∈ J such that Iq ≥ 2 and Iq ∩ I 0 6= φ then for each component from Iq ∩ I 0 , say i0 , we remove i0 ∈ Ij0 and j0 from I 0 and J 0 ,respectivly. This that we update I 0 by I 0 − {i0 } and J 0 by J 0 − {j0 }.Therefore, it is assumed
means
0
that for each q ∈ J , Iq ∩ I = φ whenever Iq ≥ 2. Assume that X 0 is the solution space of the problem that obtained from 0 elimination i’th row and j’th column of A and the jth corresponding
element of column vector b and that x ∈ X and j ∈ J − J .
Since x is a feasible solution then maxi=1,...,m (xi + aij ) = 2bj . If Ij = 1 and i0 ∈ Ij then by Lemma 2 xi0 = xˆ i0 = 2bj − ai0 j . Condition j 6∈ J 0 implies that i0 6∈ I 0 . Therefore maxi∈I −I 0 (xi + aij ) = 2bj . Assume Ij ≥ 2, so by Lemma 1 there exists p ∈ I such that xp + apj = 2bj . Obviously xp ≤ xˆ p and xˆ p + apj ≤ maxi∈I (ˆxi + aij ) = 2bj = xp + apj , therefore, xp = xˆ p and p ∈ Ij . n
But Ij ∩ I 0 = φ thus p 6∈ I 0 . Therefore maxi∈I −I 0 (xi + aij ) = 2bj . Conversely, if x0 ∈ X 0 , set xi =
x0i xˆ i
i ∈ I − I0 . i ∈ I0
So it is clear
that x ∈ X . This proves that reduction procedure dose not impact feasible domain of original problem. Thus we can solve the reduced problem. Example 1. Consider a system of fuzzy relation equations with following A and b: 0.8 0.45 0.5
" A=
0.1 0.44 , 0.89
0 0.2 0.65
#
b = ( 0.85
0.5
0.62 ).
After solution xˆ = ( 0.9 0.8 0.35 ) and I1 = {1}, I2 = {2, 3}, I3 = {2, 3}. Thus I 0 = I1 and I 0 ∩ I2 = I 0 ∩ I3 = φ . Therefore we can eliminate the first row and first column of A and the first element of b. The reduced matrix A0 and reduced vector b0 become A0 =
0.2 0.65
0.44 , 0.89
b0 = ( 0.5
0.62 ).
3. Genetic algorithm to solve our problem We assume that the problem has been reduced by applying the procedures discussed in Section 2 before the genetic algorithm is performed. The major components of the proposed genetic algorithm in [24] including the crossover operator, the mutation operator, and the selection/reproduction procedure are reviewed and modified in this section. 3.1. Representation If we use binary string [26] for each individual in our problem with high desired precision of solutions it requires a huge search space and results in a substantial amount of computational time. Furthermore, in the binary representation, two points that are close to each other in the representation space might not be close at all in the problem space, and vice versa. In order to prevent of these problems the floating point representation [27] is used in which each gene or variable xi in an individual x = (x1 , . . . , xm ) is a real number from interval [0, 1]. 3.2. Initial population generation An initial population should be generated such that it provides a significant amount of coverage for the search space. For this proposal we can apply an approach which is used in [24] and generate an initial population as follows:
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Algorithm 1 (Initial Population Generation). Let d be the number of individuals to be generated for each component. 1. If b0 is empty, generate 2md individuals with xi , ∀i ∈ I, taking a value in the range of [0, xˆ i ]. 2. If b0 is not empty, the first half of population is generated as follows. For each i ∈ I,
• • • •
Generate d individuals with x0i = xˆ i0 ∀i0 6= i and randomly assign xi value in the range of [0, xˆ i ]. If the solution is infeasible; modify it using Algorithm 2. For the second half of the population, Generate md individuals with xi0 taking a value in the range of [0, xˆ i0 ]. If the solution is infeasible; modify it using Algorithm 1.
Algorithm 2 (Solutions Modification for Feasibility). Let Ej be the eligible set of indices of components that can satisfy constraint j. Ej = {i ∈ I |aij ≤ 2bj }. 1. Choose a violated constraint, j. Calculate it’s corresponding Ej . 2. Randomly choose k; k ∈ Ej , assign xk = xˆ k . 3. If the solution is still infeasible, then go to Step 1. Otherwise, stop. 3.3. Selection/reproduction procedure To avoid the problems such as the signs of the objective values and ‘‘super-individual’’ problem [27] that causes the premature convergence the ranking scheme [27] is applied for multi-objective optimization problems. Analogue to [24] the rank of each individual rp determined by the number of dominators or solutions whose criterion vectors dominate this individual’s criterion vector as follows: rp = pop-size − number of dominators;
p = 1, 2, . . . , pop-size,
and the probability of an individual with rank rp being selected for next generations is P[individual with rank rp being selected] =
rp
.
pop-size
P
(3)
ri
i=1
The roulette wheel [27] method is then applied to reproduce the next generation according to the probability of selection from (3). During the run we maintain non-dominated set as follows: At each iteration in the selection procedure after calculating the fitness each individual in population is checked to see whether its criterion vector can dominate any of individuals in E. At end E is updated by taking out the dominated individuals from E and adding dominating individuals in population. During the run usually the size of E grows rapidly and this causes the problem of computational efficiency. In order to overcome this problem the method which used in [24] i.e. fuzzy clustering method is applied. This method is outlined as follows: Firstly, the fuzzy C -partition matrix U˜ = µcp C ×P is calculated using C -means algorithm [28]. Here, C denotes number of clusters and P denotes the size of efficient set E. moreover, µcp represents the membership value to cluster c of solution p. Now, C = 5 is set. Then a solution is considered an object in a particular cluster if its membership value with respect to that cluster is the maximum value as compared to its membership values with respect to other clusters. Finally, the same number of solutions from each cluster is selected. If the size of some clusters is smaller than the amount required, then the shortage is picked up evently by other clusters that have excess capacity. 3.4. Mutation and crossover Similarity the feasible domain of fuzzy relation equation of max–min and max-average composition leads us to use the proposed mutation and crossover in [24]. These are designed specifically for the solution space of fuzzy relation equations. These operations are as follows: Algorithm 3 (Mutation Operation). Let ξ = 0.1 be a predefined threshold value for mutation. For each individual in the population, 1. Generate an m-vector of random numbers; r ∈ [0, 1]m . 2. For i = 1, . . . , m if ri ≤ ξ , then randomly assign to xi a number in the range of 0, xˆ i . 3. If mutated individual is infeasible, modify it using Algorithm 2. Algorithm 4 (Crossover Operation). 1. Set p = 1. 2. Randomly choose two different individuals, x1 and x2 , from the current population. 3. Generate a random number r ∈ [0, 1]. If r ≥ 0.5, go to Step 5.
860
4. 5. 6. 7. 8. 9.
10.
11. 12. 13. 14.
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Generate a random number λ ∈ [0.5, 1]. Compute x¯ 1 = λx1 + (1 − λ)x2 . Go to Step 6. Generate a random number γ ∈ [1, 0.5]. Compute x¯ 1 = γ x1 − (γ − 1)x2 . Generate a random number r ∈ [0, 1]. If r ≥ 0.5, then go to Step 8. Generate a random number λ ∈ [0.5, 1]. Compute x¯ 2 = λx1 + (1 − λ)x2 . Go to Step 9. Generate a random number λ ∈ [0.5, 1]. Compute x¯ 2 = γ x1 − (γ − 1)x2 . Generate a random number r1 ∈ [0, 1]. If r1 ≥ δ , go to next step. Otherwise, generate a random number r2 ∈ [0, 1]. If r2 < 0.5, Generate a new λ ∈ [0.5, 1]. Compute x¯ 1 = λx1 + (1 − λ)ˆx otherwise, generate a new γ ∈ [1, 0.5]. Compute x¯ 1 = γ x1 − (γ − 1)ˆx. Generate a random number r1 ∈ [0, 1]. If r1 ≥ δ , go to next step. Otherwise, generate another random number r2 ∈ [0, 1]. If r2 < 0.5, generate a new λ ∈ [0.5, 1]. Compute x¯ 1 = λx1 + (1 − λ)x2 . Otherwise, generate a new γ ∈ [1, 0.5]. Compute x¯ 2 = γ x¯ 2 − (γ − 1)ˆx. If for some i ∈ I , x¯ 1i (or x¯ 2i ) is out of the range of [0, xˆ i ], substitute the out-of-range value, x¯ i (or x¯ 2i ) with a random number in the range of [0, xˆ i ]. If x¯ 1 or x¯ 2 are infeasible, modify them using Algorithm 2. Insert x¯ 1 and x¯ 2 into the new population. Set p = p + 1. If p ≤ pop-size/2, then go to Step 2. Otherwise, stop.
3.5. Local improvement procedure In some cases, especially when the efficient set is finite and contains more than one solution, genetic algorithm can not exactly determine efficient sets. In this case we probably obtain a lot of point between efficient solutions. In order to prevent this problem and reduce the noise in the solution set as well as bring set to its local optimum the local improvement procedures [29] is applied as follows: We add an m-vector which its components are selected from a small neighborhood of zero (for example [−0.05, 0.05]) randomly to one of the solutions obtained by the algorithm. If this new solution is infeasible we project it back to the feasible domain. If this new solution can dominate some solutions in E we update E by eliminating these dominated solutions from E and adding new point to E. In our implementation, this procedure is performed for 100 times for each individual. Algorithm 5 (The Genetic Algorithm). Let max-size be the maximum size of the efficient set and max-gun is the maximum number of generations. 1. Set k = 1. 2. Generate initial population using Algorithm 4. 3. Perform the selection/reproduction procedure. Establish the efficient set E by putting all efficient solutions of the initial population into it. 4. Perform the mutation operation. 5. Perform the crossover operation. 6. Perform the selection/reproduction procedure and update the efficient set E. For every certain number of generations, say 5, if the size of E ≥ max-size, apply the fuzzy clustering method to reduce its size to max-size. 7. If k < max − gen, set k = k + 1 and go to step 3. 8. Perform the local improvement procedure on the efficient set E. It is noticeable that what we did so far is based on d = 15. 4. Results When the solution space is construted by two connected lines in two dimensional case, the effectness of the local improvement procedure will be observed,where,the efficient points occure in two extreme points which are not adjacent. So, the effectiveness of the procedure will be only considered in I-4. Futhermore, the effectiveness of the local improvement procedure in large search region such as three dimentional problems is more than other cases. Therefore, for more detail the procedure will be provided in II-1 and II-3 examples. The Branch and Bounded method [3] and also the tableau method [1] are usual approches among many, such as [30], which have presented in order to solve the single linear opotimization problems subject to fuzzy relation equation, but, these methods cannot be used to manage nonlinear and also multi-objective optimization problems.The genetic algorithm has applied for solving the later problems regarding max–min composition by few numbers of researcher [6]. In this paper, we attempt to use the genetic algorithm with some modification so as to solve multiobjective obtimization problem subject to max –average fuzzy relation equation problem. To the author’s best knowledge there is no other alternative approches which can solve the problem; perhaps other methods such as simulation can be used to solve the problem, but it certainly needs more research that we are doing now. Such as [24] some optimization problems with both linear and nonlinear objective functions are tested. For the linear case, we can theoretically obtain the Pareto optimal set given that the feasible domain is known. For this proposal the following definition and theorem is stated: Definition ([31]). Let ak = ∇ Zk (x) be the gradient of the kth objective function, with Zk (x) = a semi-positive polar cone is defined by C ≥ = {c ∈ Rm |aTk .c ≥ 0∀k, and ∃ aTk c > 0}.
Pm
i=1
aki xi , ak = [aki ]1×m ; then
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Fig. 1. The solution space for the two-dimensional problem.
For the multiobjective maximization problem the following problem are considered: Theorem. If x ∈ X then x is an efficient point iff ({x} + C ≥ ) ∩ X = {x}. For the proof see [31]. Remark: Since our problem is multiobjective minimization problem, so, negative gradient of the objective functions is considered instead of ak = ∇ Zk (x) in above definition. Theoretical Pareto optimal set is then used to compare with the results from the genetic algorithm. 4.1. Linear objective functions 4.1.1. Two variables and two linear objective functions We tested a three-dimensional problem from the example illustrated in Section 3. Since the reduction operation have been performed on matrix A in Section 3 the updated matrix A0 and b0 as follows: A0 =
0.2 0.65
0.44 , 0.89
b0 = ( 0.5
0.62 ).
For this problem, the values of x1 of all solution vectors have to be fixed at xˆ 1 = 0.9 Therefore we can focus on the values of x2 and x3 . The feasible domain is shown in Fig. 1. It composed of two connected lines. For illustrative purposes, the problem with a set of two linear objective functions was tested. Different settings of the objective functions were repeated. The semi-positive polar cones generated by the negative gradients of two objective functions for those four cases are shown in Fig. 6. Futhermore, v 1 and v 2 correspond to the negative gradients of the first objective and the second objective, respectively.Also, y1 and y2 are the vectors that are perpendicular to v 1 and v 2 , correspondingly. We set max-size = 50. Case I-1: Only one efficient solution at the maximum solution.
f1 (x) = 0.8x1 − 0.1x2 − 0.3x3 f2 (x) = 0.7x1 − 0.2x2 − 0.3x3 .
Theoretically, with respect to the above theorem and Fig. 6 this problem has only one Pareto optimal solution which is the maximum solution. The optimal solution which was obtained from the genetic algorithm is exactly this Point, i.e., x2 = 0.8, x3 = 0.35 as shown in Fig. 2. Case I-2: Efficient solutions are located on one edge.
f1 (x) = x1 − x2 − x3 f2 (x) = x1 − x2 + x3 .
Theoretical Pareto optimal solutions are {x1 = 0.9, x2 = 0.8, 0 ≤ x3 ≤ 0.35}. As the result shown in Fig. 3, we were able to obtain this set. Case I-3: Efficient solutions are located on two connected edges.
f1 (x) = 0.6x1 + 0.1x2 − 0.3x3 f2 (x) = 0.4x1 − 0.2x2 + 0.3x3 .
The theoretical Pareto optimal set is
{x1 = 0.9, x2 = 0.8, 0 ≤ x3 ≤ 0.35} ∪ {x1 = 0.9, 0 ≤ x2 ≤ 0.8, x3 = 0.35}.
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Fig. 2. The efficient set for case I-1.
Fig. 3. The efficient set for case I-2.
If the efficient set E is not controlled,then this set will get larger, consequently the size of the efficient set E is likely to increase uncontrollably as the program evolves. This can cause the efficiency problem. Fig. 4(a) shows the result when we do not apply the fuzzy clustering to control the size of the efficient set. On the other hand, when we limit the size of the efficient set, the result as shown in Fig. 4(b) is quite satisfactory. It approximates the efficient region and also helps improve the efficiency of the algorithm. Case I-4: Efficient solutions are located at two extreme points. f1 (x) = 0.3x1 + x2 f2 (x) = −0.5x1 + x3 .
For case I-4, theoretically there are exactly two Pareto optimal solutions which are located at (0.9, 0, 0.35) and (0.9, 0.8, 0). With the contribution from the local improvement procedure, these two points can achieved as shown in Fig. 5(b). If this local improvement technique is not applied, then, we probably obtain a lot of points in between. Fig. 5(a) shows the result of this case when the local improvement technique is not utilized. Note that the program written in Matlab was run on a Pentium(R) 4 CPU 2.80 GHz machine. Table 1 compares the CPU times required and the sizes of the final efficient sets of all four cases. 4.1.2. Three variables and three linear objective functions The following problem with a set of three linear objective functions was tested. Its maximum solution was as xˆ = (1, 0.8, 0.9, 0.5) 0.8 1 A= 0.5 0
0.2 0.6 0.5 0.3
0.3 0.5 0.4 0.28
0.3 0.25 , 0.127 1
b = (0.9, 0.7, 0.65, 0.75) .
E. Khorram, H. Zarei / Mathematical and Computer Modelling 49 (2009) 856–867
(a) Uncontrolled efficient set.
863
(b) Controlled efficient set.
Fig. 4. The efficient set for case I-3 with/without the control on the size of efficient set.
(a) Without local improvement.
(b) With local improvement.
Fig. 5. The efficient set for case I-4 with/without local improvement. Table 1 CPU time spent and size of the final efficient set for problem1. Case
CPU time (s)
Size of the final efficient set
I-1 I-2 I-3 I-3 With no control on size E and without local improvement with 30 iteration I-4 I-4 with no local Improvement
199/152 1060/799 1146/319 2063/829 242/052 241/083
1 50 50 1142 2 12
In this problem, the values of x4 for all solution vectors have to be fixed at 0.5. The problem can be reduced to a threedimensional problem. Therefore, the solution space is only considered with respect to x1 , x2 , and x3 . The solution space is presented in Fig. 7. It is composed of connected surface and line and five vertices. For this problem max − size = 80 is set. Case II-1: Efficient solutions lie on one edge. f1 (x) = x1 + 2x2 + 2x3 + 4x4 Min f2 (x) = 2x1 − 3x2 − x3 − 9x4 f3 (x) = 3x1 + 8x2 − 5x3 + x4 .
(
Theoretically, the efficient set of this case contains points that lie on the edge between vertices x1 and x3 . In other words, any points in {x1 = 0, x2 = 0.8, 0 ≤ x3 ≤ 1, x4 = 0.5} are efficient solutions. Fig. 8(b) represent the results in a three-
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(a) Case I-1.
(b) Case I-2.
(c) Case I-3.
(d) Case I-4. Fig. 6. The semi-positive polar cones of four cases.
Fig. 7. The solution space for three-dimensional problem.
dimensional space of x1 , x2 and x3 , Fig. 8(a) shows the result of this case when the local improvement technique is not utilized. Case II-2: each feasible point is efficient solution. f1 (x) = −2x1 − 3x2 + x3 − 9x4 Min f2 (x) = 2x1 + 3x2 − x3 + 9x4 f3 (x) = 3x1 + 8x2 − 5x3 + x4 .
(
Here f1 (x) = −f2 (x). This condition implies that every point in feasible domain is efficient solution. The results from the genetic algorithm shown in Fig. 9. Case II-3: Efficient solutions are located at two vertices. f1 (x) = x1 + x4 Min f2 (x) = x2 − x4 f3 (x) = x3 + 2x4 .
(
Theoretically, the efficient set contains only two points, i.e., the vertices x4 and x1 which are located at (0, 0.8, 0) and (1, 0, 0.9),respectively. The results from the genetic algorithm shown in Fig. 10(b) verify this fact. Fig. 10(a) shows the result of
E. Khorram, H. Zarei / Mathematical and Computer Modelling 49 (2009) 856–867
(a) Without local improvement.
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(b) With local improvement. Fig. 8. The efficient set for case II-1.
Fig. 9. The efficient set for case II-2.
(a) Without local improvement.
(b) With local improvement. Fig. 10. The efficient set for case II-3.
this case when the local improvement technique is not utilized. Table 2 reports the CPU time required to finish the run and the size of the final efficient set. 4.2. Non-linear objective functions We have tested genetic algorithm on problem, with two-dimensional space. Two variables and two non-linear objective functions f1 (x) = 10(x1 − 0.4)2 + 10(x2 − 0.35)2 f2 (x) = 10(x1 − 0.7)2 + 10(x2 − 0.35)2
Min
and A0 =
0.2 0.65
0.44 , 0.89
b0 = 0.5
0.62 .
866
E. Khorram, H. Zarei / Mathematical and Computer Modelling 49 (2009) 856–867
Table 2 CPU time spent and the size of the final efficient set for problem II. Case
CPU time (s)
Size of the final efficient set
II-1 II-1 With no local improvement II-2 With no control on size E and without local improvement with 20 iteration II-3 II-3 with no local Improvement
987/185 975/.83 1383/39 666/238 661/125
61 78 1173 2 18
(a) Surface plots.
(b) Contour plots.
(c) The efficient set. Fig. 11. The efficient set for non-linear two-variable case.
The system of fuzzy relation equations is exactly the same as problem I in Section 4.1. The feasible domain is presented in Fig. 1. Fig. 11(a) presents the surfaces of the two objective functions. Consider the contours of the objective functions as shown in Fig. 11(b). Those contours represent the solutions that yield the objective values at 0.3, 0.6, and 0.9 referencing from the inner contour to the outer contour, respectively. Using the same argument which presented in [24] p.161, we find that all of points between p1 and p1 inclusively are efficient solutions. Therefore, the efficient set should contain points in {x2 = 0.35, 0.4 ≤ x1 ≤ 0.7}. The CPU time consumed is 884/934 s. The size of the final set of efficient solutions is 50. Fig. 11(c) shows the result from the genetic algorithm. 5. Conclusion We have solved multi-objective optimization problems with fuzzy relation equations regarding max-average composition. Firstly, the dimension of the problem was reduced by determining the set of components whose values must be fixed for all solution vectors in order to satisfy the fuzzy relation equations. The result obtained from testing the algorithm shows that the genetic algorithm can determine the efficient set in two- and three-dimensional spaces regardless of linearity objective functions. Furthermore, it can identify the location of efficient sets, either at some vertices, edges, or the interior
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points. In order to improve the efficiency of the algorithm and improve the final efficient set it is necessary to impose control on the size of E and apply local improvement procedures. Acknowledgements The authors would like to thank the anonymous referees for their valuable comments that improved the quality of this paper. References [1] E. Khorram, A. Ghodousian, Linear objective function optimization with fuzzy relation equation constraints regarding max–av composition, Applied Mathematics and Computation 173 (2006) 872–886. [2] E. Sanchez, Resolution of composite fuzzy relation equations, Information and Control 30 (1976) 38–48. [3] S.-C. Fang, G. Li, Solving fuzzy relations equations with a linear objective function, Fuzzy Set and Systems 103 (1999) 107–113. [4] 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. [5] H.-C. Lee, S.-M. Guu, On the optimal three-tier multimedid streaming services, Fuzzy n Optimization and Decision Making 2 (2003) 31–39. [6] J. Lu, S.C. Fang, Solving nonlinear optimization problems with fuzzy relation equation constraints, Fuzzy Sets and Systems 119 (2001) 1–20. [7] W. Pedrycz, Proceeding in relational structures: Fuzzy relational equations, Fuzzy Set and Systems 40 (1991) 77–106. [8] M.M. Brouke, D.G. Fisher, Solution algorithms for fuzzy relation equations with max-product composition, Fuzzy sets and Systems 94 (1998) 61–69. [9] A. Di Nola, S. Sessa, W. Pedrycz, E. Sanchez, Fuzzy Relational Equations and their Applications in Knowledge Engineering, Kluwer Academic Press, Dordrecht, 1989. [10] J. Loetamonphong, S.C. Fang, Optimization of fuzzy relation equations with max-product composition, Fuzzy Sets and Systems 118 (2001) 509–517. [11] R.E. Steuer, Multiple Criteria Optimization: Theory; Complication and Application, Wiley, New York, 1998. [12] M. Zeleng, Linear Multiple Objective Programming, Spring-Verlag, New York, 1974. [13] J.P. Ignizio, T.M. Cavalier, Linear Programming, Prentice-Hall, Singapore, 1994. [14] P.L. Yu, G. Leitman, Compromise solutions, domination structure and Solukvakzemn, s solutions, Journal of Optimization Theory and Applications 13 (3) (1974) 362–378. [15] R.J. Benayoun, J. de Montgolfie,Tergny, O. Laritchev, Linear programming with multiple objective functions: Step method(STEM), Mathematical Programming (3) (1971) 366–375. [16] M. Sakawa, Fuzzy Sets and Interactive Multi-Objective Optimization, Plenum, New York, 1993. [17] L.R. Gardinerand, R.E. Steuer, Unified intractive multiple objective programming: An open architecture for accommodating new procedures, Journal of Operational Research Society 45 (12) (1994) 1456–1466. [18] H.-F. Wang, A multi-objective mathematical programming with fuzzy relation Constraint, Journal of Multi-Criteria Decision Analysis 4 (1995) 23–35. [19] H.J. Zimmermann, Fuzzy programming and linear programming with several objective functions, Fuzzy Sets and Systems 1 (1978) 45–55. [20] H.J Zimmermann, Fuzzy mathematical programming, Computer and Operations Research 10 (4) (1983) 45–55. [21] E.-S. Lee, R.-L. Li, Fuzzy multiobjective programming and compromise programming with Pareto optimum, Fuzzy Sets and Systems 53 (1993) 275–288. [22] H.K. Chen, H.W. Chou, Solving multi-objective linear programming problem- a genetic approach, Fuzzy Set and Systems 82 (1996) 5–38. [23] Y.-K. Wu, S.-M. Guu, A Compromise model for solving fuzzy multiple objective linear programming problems, Journal of the Chinese Institute Industrial engineering 18 (2001) 87–93. [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] G.J. Klir, T.A. Folger, Fuzzy sets, Uncertainty and Information, Prentice-Hall, NJ, 1988. [26] J.H. Holland, Adaptation in natural and artificial systems, University of Michigan, 1975. [27] Z. Michalewicz, Genetic Algorithms +Data Structures = Evolution Program, Springer, Berlin, Heidelberg, New York, 1996. [28] J.C. Bezdek, Pattern Recognition with Fuzzy Objective Function Algorithms, Plenum Press, New York, 1981. [29] C.R. Reeves, Modern Heuristic Techniques for Combinatorial Problems, Wiley, New York, 1993. [30] S.M. Guu, Y.-K. Wu, Minimizing a linear objective function with fuzzy relation equation constraints, Fuzzy Optimization and Decision Making 1 (4) (2002) 347–360. [31] R.E. Steuer, Multiple Criteria Optimization: Theory, Computation, and Application, Wiley, New York, 1986.
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Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm
Multi-objective optimization problems with Fuzzy relation equation constraints regarding max-average composition E. Khorram ∗ , H. Zarei Faculty of Mathematics and computer Science, Amirkabir University of Technology, 424, Hafez Ave, 15914, Tehran, Iran
article
info
Article history: Received 30 October 2006 Received in revised form 5 October 2008 Accepted 27 October 2008 Keywords: Fuzzy relation equations Max–average composition Multi-objective optimization Genetic algorithm
a b s t r a c t A multiple objective optimization model subject to a system of fuzzy relation equations with max-average composition and a reduction procedure in order to reduce problem dimension are presented. Furthermore, a genetic algorithm is reviewed and some of its components are modified to solve the problem. Finally, the algorithm is performed on different problems and its major components role in detecting the solution is examined. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction Let X = [0, 1]m , I = {1, 2, . . . , m}, J = (1, 2, . . . , n}, and A = (aij ) be an m × n dimensional fuzzy matrix with aij ∈ [0, 1] and also b be an n-dimensional vector [bj ]1×n , so that, bj ∈ [0, 1] for all i ∈ I and j ∈ J. Given A and b, a system of fuzzy relation equations is defined by xoA=b av
(1)
where ‘‘ o ’’ is max-average composition [1]. A solution is a vector x = (x1 , . . . , xm ); 0 ≤ xi ≤ 1 so that maxi=1,...,m (xi + aij ) = av
2bj , j = 1, . . . , n. The solution set of (1) is usually not a singleton set. When the solution set is nonempty, it is in general a non-convex set which can be completely determined by a unique maximum solution and a finite number of minimal solutions [2]. Each input of (1) may require certain amount of resources which can be considered cost, and a decision maker may wish to achieve certain objectives. Considering an optimization problem with a single linear objective function subjected to the fuzzy relational equations based on the max–min composition, Fang and Li [3] converted this optimization problem into a 0-1 integer programming and solved it by Branch-and Bounded method. Wu et al. [4] improved Fang and Li,s method by providing an upper bound for Branch and Bounded procedure. Lee and Guu [5] proposed a fuzzy relational optimization model for the streaming media provider seeking a minimum cost while fulfilled the requirements assumed by a three-tier framework. Khorram and Ghodousian [1] dealt with solving fuzzy relation equations with max-average composition with a linear objective function. Lu and Fang [6] proposed a genetic algorithm to solve nonlinear single objective problem with fuzzy relation equation constraints regarding max–min composition. The fundamental result for fuzzy relation equations with max-product composition goes back to Pedrycz [7]. Recent study in this regard can be found in Bourk and Fisher [8]. 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 solution
∗
Corresponding author. E-mail address: [email protected] (E. Khorram).
0895-7177/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2008.10.018
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sets, as well as the conditions for the existence of resolutions. Their results showed that such complete solution sets can be characterized by one maximum solution and a number of minimal solutions. Furthermore, the monograph by Di Nola et al. [9] contains a thorough discussion of this class of equations. The optimization problem with max-product composition and with max–min composition was first considered by Loetamonphong and Fang [10] and Fang and Li [3]. For multiple objective linear programming problems (MOLP), in general, the objective functions usually conflict with each other. The notation of efficiency or Pareto optimality associated with feasible region has been introduced to instead of the optimality concept for single objective optimization [11]. Many solution methods for MLOP model have been studied in the literature since the early 1970,s. The simplex method extended by Zeleng [12]. MOLP can be converted in the goal programming problems, when the decision maker sets the satisfying goals and priority to each objective. Ignizio and Cavalier [13] mentioned three common forms of the goal programming. Deciding the relative deviation of the objective functions from the ideal objective value instead of prior preference information, Yu and Leitman [14] presented a compromise programming method to deal with MOLP. Benayound et al. [15] proposed a step method that seems to be known as one the first interactive techniques for dealing with MOLP. Sakawa [16] considered a fruitful survey regarding to interactive methods for MOLP. Gardiner and Steuer [17] have shown a number of procedures of interactive multiple objectives programming which include the satisfying trade-off method, the ε -constraint method and so forth. To the author’s best knowledge, Wang [18] was the first paper to explore the fuzzy relational equations based on the max–min composition yet with multiple objective linear functions. Zimmermann [19,20] proposed the min operator model to solve MOLP using a linearing membership function. The solution generated by min operator does not guarantee compensatory and efficient. Lee and Li [21] proposed a two-phase approach to overcome this difficulty. Chen and Chou [22] proposed a fuzzy to integrate the min operator, average operator and two-phase method. Extending two phase approach to the case of the fuzzy multi-objectives linear programming problem, Wu and Guu [23] presented a compromise model to provide different efficient solutions for the decision maker. Loetamonphong and el al used the genetic algorithm and solved the multi-objective optimization Problem with fuzzy relation equation constraints with max–min composition [24]. Now, we have imposed necessary changes to their algorithm and we have used it to solve multi-objective optimization problem subject to a set of fuzzy relation equations with max-average composition. In other words, we have attempted to solve the optimization problem in the following form: min f1 (x), . . . , fp (x)
s.t x o A = b
(2)
av
0 ≤ xi ≤ 1 where fk (x) is an objective function, k ∈ K = {1, 2, . . . , p}. The rest of the paper is outlined as follows: Section 2 provides the background of multi-objective optimization and describes a reduction procedure for transforming the original problem into a reduced form. In Section 3 some components of the proposed genetic algorithm in [6] are reviewed and then refined to solve the problem. Section 4 reports some test results of the genetic algorithm. Concluding remarks are given in Section 5. 2. Basic definitions and problem reduction 2.1. Basic definitions Some background and the theory of multiobjective optimization are studied here [18].Let the feasible domain of (2) denotes by X = {x ∈ Rm |x o A = b, 0 ≤ xi ≤ 1, ∀i}. Each point x ∈ X is said a solution vector and also z = (f1 (x), . . . , fp (x)) av
is defined its criterion vector. Moreover, we define Z = {z ∈ Rp |z = (f1 (x), . . . , fp (x)), ∃x ∈ X }. Definition 1. A point x¯ ∈ X is an efficient or a Pareto optimal solution to problem (2) if and only if there is no any x ∈ X such that fk (x) ≤ fk (¯x), ∀k = 1, . . . , p, and fk (x) < fk (¯x) for at least one k. Otherwise, x¯ is an inefficient solution. Definition 2. Let z 1 , z 2 ∈ Z be two criterion vectors. Then, z 1 dominates z 2 if and only if z 1 ≤ z 2 and z 1 6= z 2 That is z 1 ≤ z 2 , ∀k ∈ K , and zk1 < zk2 for at least one k. Definition 3. Let z¯ ∈ Z , then, z¯ is non-dominated if and only if there does not exist any z ∈ Z that dominates z¯ . Otherwise, z¯ is a dominated criterion vector. The set of all efficient points is called the efficient set or Pareto optimal set. In the absence of a mathematical specification of the decision maker’s utility function, we can only provide the decision maker with the Pareto optimal set for further analysis. 2.2. Problem reduction Due to the requirement of x o A = b some components of every solution vector may have to assume a specific value. Some av
of these components can therefore be set aside from the problem. The genetic operators are then applied to this reduced problem.
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We define x1 ≤ x2 ⇔ x1i ≤ x2i ∀x1 , x2 ∈ X in this way, the operator ‘‘ ≤’’ forms a partial order relation on X and (X , ≤) becomes a lattice. xˆ ∈ X is called a maximum solution if xˆ ≥ x for all x ∈ X . Also x˘ ∈ X , is called a minimal solution if, for any x ∈ X , x ≤ x˘ implies x = x˘ . Notice that the maximum solution xˆ can be obtained by the following formula [25]: xˆ = AΘ b = [Λnj=1 (aij Θ bj )]i∈I Where ‘‘Λ’’ is the min operator that is aΛb = min(a, b) and
aij Θ bj =
1 2bj − aij
2bj ≥ aij + 1 2bj < aij + 1.
Denote the set of all minimal solutions by X˘ . The complete set of solutions, X , is obtained by X =
S
x˘ ∈X˘
x ∈ X |˘x ≤ x ≤ xˆ .
Lemma 1. If x ∈ X , then for each j ∈ J there exists i0 ∈ I such that xi0 + ai0 j = 2bj and xi + aij ≤ 2bj , ∀i ∈ I. Proof is analog to Lemma 1 in [1]. Let X 6= φ and define Ij = {i ∈ I |ˆxi + aij = 2bj }, ∀j ∈ J.
Lemma 2. If Ij = 1 then xˆ i = x˘ i = 2bj − aij , for i ∈ Ij . Proof is similar to Lemma
3 in [1]. Define I 0 = {i ∈ Ij | Ij = 1} and J 0 = {j ∈ J |Ij ∩ I 0 6= φ}.
Now, if there exist q ∈ J such that Iq ≥ 2 and Iq ∩ I 0 6= φ then for each component from Iq ∩ I 0 , say i0 , we remove i0 ∈ Ij0 and j0 from I 0 and J 0 ,respectivly. This that we update I 0 by I 0 − {i0 } and J 0 by J 0 − {j0 }.Therefore, it is assumed
means
0
that for each q ∈ J , Iq ∩ I = φ whenever Iq ≥ 2. Assume that X 0 is the solution space of the problem that obtained from 0 elimination i’th row and j’th column of A and the jth corresponding
element of column vector b and that x ∈ X and j ∈ J − J .
Since x is a feasible solution then maxi=1,...,m (xi + aij ) = 2bj . If Ij = 1 and i0 ∈ Ij then by Lemma 2 xi0 = xˆ i0 = 2bj − ai0 j . Condition j 6∈ J 0 implies that i0 6∈ I 0 . Therefore maxi∈I −I 0 (xi + aij ) = 2bj . Assume Ij ≥ 2, so by Lemma 1 there exists p ∈ I such that xp + apj = 2bj . Obviously xp ≤ xˆ p and xˆ p + apj ≤ maxi∈I (ˆxi + aij ) = 2bj = xp + apj , therefore, xp = xˆ p and p ∈ Ij . n
But Ij ∩ I 0 = φ thus p 6∈ I 0 . Therefore maxi∈I −I 0 (xi + aij ) = 2bj . Conversely, if x0 ∈ X 0 , set xi =
x0i xˆ i
i ∈ I − I0 . i ∈ I0
So it is clear
that x ∈ X . This proves that reduction procedure dose not impact feasible domain of original problem. Thus we can solve the reduced problem. Example 1. Consider a system of fuzzy relation equations with following A and b: 0.8 0.45 0.5
" A=
0.1 0.44 , 0.89
0 0.2 0.65
#
b = ( 0.85
0.5
0.62 ).
After solution xˆ = ( 0.9 0.8 0.35 ) and I1 = {1}, I2 = {2, 3}, I3 = {2, 3}. Thus I 0 = I1 and I 0 ∩ I2 = I 0 ∩ I3 = φ . Therefore we can eliminate the first row and first column of A and the first element of b. The reduced matrix A0 and reduced vector b0 become A0 =
0.2 0.65
0.44 , 0.89
b0 = ( 0.5
0.62 ).
3. Genetic algorithm to solve our problem We assume that the problem has been reduced by applying the procedures discussed in Section 2 before the genetic algorithm is performed. The major components of the proposed genetic algorithm in [24] including the crossover operator, the mutation operator, and the selection/reproduction procedure are reviewed and modified in this section. 3.1. Representation If we use binary string [26] for each individual in our problem with high desired precision of solutions it requires a huge search space and results in a substantial amount of computational time. Furthermore, in the binary representation, two points that are close to each other in the representation space might not be close at all in the problem space, and vice versa. In order to prevent of these problems the floating point representation [27] is used in which each gene or variable xi in an individual x = (x1 , . . . , xm ) is a real number from interval [0, 1]. 3.2. Initial population generation An initial population should be generated such that it provides a significant amount of coverage for the search space. For this proposal we can apply an approach which is used in [24] and generate an initial population as follows:
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Algorithm 1 (Initial Population Generation). Let d be the number of individuals to be generated for each component. 1. If b0 is empty, generate 2md individuals with xi , ∀i ∈ I, taking a value in the range of [0, xˆ i ]. 2. If b0 is not empty, the first half of population is generated as follows. For each i ∈ I,
• • • •
Generate d individuals with x0i = xˆ i0 ∀i0 6= i and randomly assign xi value in the range of [0, xˆ i ]. If the solution is infeasible; modify it using Algorithm 2. For the second half of the population, Generate md individuals with xi0 taking a value in the range of [0, xˆ i0 ]. If the solution is infeasible; modify it using Algorithm 1.
Algorithm 2 (Solutions Modification for Feasibility). Let Ej be the eligible set of indices of components that can satisfy constraint j. Ej = {i ∈ I |aij ≤ 2bj }. 1. Choose a violated constraint, j. Calculate it’s corresponding Ej . 2. Randomly choose k; k ∈ Ej , assign xk = xˆ k . 3. If the solution is still infeasible, then go to Step 1. Otherwise, stop. 3.3. Selection/reproduction procedure To avoid the problems such as the signs of the objective values and ‘‘super-individual’’ problem [27] that causes the premature convergence the ranking scheme [27] is applied for multi-objective optimization problems. Analogue to [24] the rank of each individual rp determined by the number of dominators or solutions whose criterion vectors dominate this individual’s criterion vector as follows: rp = pop-size − number of dominators;
p = 1, 2, . . . , pop-size,
and the probability of an individual with rank rp being selected for next generations is P[individual with rank rp being selected] =
rp
.
pop-size
P
(3)
ri
i=1
The roulette wheel [27] method is then applied to reproduce the next generation according to the probability of selection from (3). During the run we maintain non-dominated set as follows: At each iteration in the selection procedure after calculating the fitness each individual in population is checked to see whether its criterion vector can dominate any of individuals in E. At end E is updated by taking out the dominated individuals from E and adding dominating individuals in population. During the run usually the size of E grows rapidly and this causes the problem of computational efficiency. In order to overcome this problem the method which used in [24] i.e. fuzzy clustering method is applied. This method is outlined as follows: Firstly, the fuzzy C -partition matrix U˜ = µcp C ×P is calculated using C -means algorithm [28]. Here, C denotes number of clusters and P denotes the size of efficient set E. moreover, µcp represents the membership value to cluster c of solution p. Now, C = 5 is set. Then a solution is considered an object in a particular cluster if its membership value with respect to that cluster is the maximum value as compared to its membership values with respect to other clusters. Finally, the same number of solutions from each cluster is selected. If the size of some clusters is smaller than the amount required, then the shortage is picked up evently by other clusters that have excess capacity. 3.4. Mutation and crossover Similarity the feasible domain of fuzzy relation equation of max–min and max-average composition leads us to use the proposed mutation and crossover in [24]. These are designed specifically for the solution space of fuzzy relation equations. These operations are as follows: Algorithm 3 (Mutation Operation). Let ξ = 0.1 be a predefined threshold value for mutation. For each individual in the population, 1. Generate an m-vector of random numbers; r ∈ [0, 1]m . 2. For i = 1, . . . , m if ri ≤ ξ , then randomly assign to xi a number in the range of 0, xˆ i . 3. If mutated individual is infeasible, modify it using Algorithm 2. Algorithm 4 (Crossover Operation). 1. Set p = 1. 2. Randomly choose two different individuals, x1 and x2 , from the current population. 3. Generate a random number r ∈ [0, 1]. If r ≥ 0.5, go to Step 5.
860
4. 5. 6. 7. 8. 9.
10.
11. 12. 13. 14.
E. Khorram, H. Zarei / Mathematical and Computer Modelling 49 (2009) 856–867
Generate a random number λ ∈ [0.5, 1]. Compute x¯ 1 = λx1 + (1 − λ)x2 . Go to Step 6. Generate a random number γ ∈ [1, 0.5]. Compute x¯ 1 = γ x1 − (γ − 1)x2 . Generate a random number r ∈ [0, 1]. If r ≥ 0.5, then go to Step 8. Generate a random number λ ∈ [0.5, 1]. Compute x¯ 2 = λx1 + (1 − λ)x2 . Go to Step 9. Generate a random number λ ∈ [0.5, 1]. Compute x¯ 2 = γ x1 − (γ − 1)x2 . Generate a random number r1 ∈ [0, 1]. If r1 ≥ δ , go to next step. Otherwise, generate a random number r2 ∈ [0, 1]. If r2 < 0.5, Generate a new λ ∈ [0.5, 1]. Compute x¯ 1 = λx1 + (1 − λ)ˆx otherwise, generate a new γ ∈ [1, 0.5]. Compute x¯ 1 = γ x1 − (γ − 1)ˆx. Generate a random number r1 ∈ [0, 1]. If r1 ≥ δ , go to next step. Otherwise, generate another random number r2 ∈ [0, 1]. If r2 < 0.5, generate a new λ ∈ [0.5, 1]. Compute x¯ 1 = λx1 + (1 − λ)x2 . Otherwise, generate a new γ ∈ [1, 0.5]. Compute x¯ 2 = γ x¯ 2 − (γ − 1)ˆx. If for some i ∈ I , x¯ 1i (or x¯ 2i ) is out of the range of [0, xˆ i ], substitute the out-of-range value, x¯ i (or x¯ 2i ) with a random number in the range of [0, xˆ i ]. If x¯ 1 or x¯ 2 are infeasible, modify them using Algorithm 2. Insert x¯ 1 and x¯ 2 into the new population. Set p = p + 1. If p ≤ pop-size/2, then go to Step 2. Otherwise, stop.
3.5. Local improvement procedure In some cases, especially when the efficient set is finite and contains more than one solution, genetic algorithm can not exactly determine efficient sets. In this case we probably obtain a lot of point between efficient solutions. In order to prevent this problem and reduce the noise in the solution set as well as bring set to its local optimum the local improvement procedures [29] is applied as follows: We add an m-vector which its components are selected from a small neighborhood of zero (for example [−0.05, 0.05]) randomly to one of the solutions obtained by the algorithm. If this new solution is infeasible we project it back to the feasible domain. If this new solution can dominate some solutions in E we update E by eliminating these dominated solutions from E and adding new point to E. In our implementation, this procedure is performed for 100 times for each individual. Algorithm 5 (The Genetic Algorithm). Let max-size be the maximum size of the efficient set and max-gun is the maximum number of generations. 1. Set k = 1. 2. Generate initial population using Algorithm 4. 3. Perform the selection/reproduction procedure. Establish the efficient set E by putting all efficient solutions of the initial population into it. 4. Perform the mutation operation. 5. Perform the crossover operation. 6. Perform the selection/reproduction procedure and update the efficient set E. For every certain number of generations, say 5, if the size of E ≥ max-size, apply the fuzzy clustering method to reduce its size to max-size. 7. If k < max − gen, set k = k + 1 and go to step 3. 8. Perform the local improvement procedure on the efficient set E. It is noticeable that what we did so far is based on d = 15. 4. Results When the solution space is construted by two connected lines in two dimensional case, the effectness of the local improvement procedure will be observed,where,the efficient points occure in two extreme points which are not adjacent. So, the effectiveness of the procedure will be only considered in I-4. Futhermore, the effectiveness of the local improvement procedure in large search region such as three dimentional problems is more than other cases. Therefore, for more detail the procedure will be provided in II-1 and II-3 examples. The Branch and Bounded method [3] and also the tableau method [1] are usual approches among many, such as [30], which have presented in order to solve the single linear opotimization problems subject to fuzzy relation equation, but, these methods cannot be used to manage nonlinear and also multi-objective optimization problems.The genetic algorithm has applied for solving the later problems regarding max–min composition by few numbers of researcher [6]. In this paper, we attempt to use the genetic algorithm with some modification so as to solve multiobjective obtimization problem subject to max –average fuzzy relation equation problem. To the author’s best knowledge there is no other alternative approches which can solve the problem; perhaps other methods such as simulation can be used to solve the problem, but it certainly needs more research that we are doing now. Such as [24] some optimization problems with both linear and nonlinear objective functions are tested. For the linear case, we can theoretically obtain the Pareto optimal set given that the feasible domain is known. For this proposal the following definition and theorem is stated: Definition ([31]). Let ak = ∇ Zk (x) be the gradient of the kth objective function, with Zk (x) = a semi-positive polar cone is defined by C ≥ = {c ∈ Rm |aTk .c ≥ 0∀k, and ∃ aTk c > 0}.
Pm
i=1
aki xi , ak = [aki ]1×m ; then
E. Khorram, H. Zarei / Mathematical and Computer Modelling 49 (2009) 856–867
861
Fig. 1. The solution space for the two-dimensional problem.
For the multiobjective maximization problem the following problem are considered: Theorem. If x ∈ X then x is an efficient point iff ({x} + C ≥ ) ∩ X = {x}. For the proof see [31]. Remark: Since our problem is multiobjective minimization problem, so, negative gradient of the objective functions is considered instead of ak = ∇ Zk (x) in above definition. Theoretical Pareto optimal set is then used to compare with the results from the genetic algorithm. 4.1. Linear objective functions 4.1.1. Two variables and two linear objective functions We tested a three-dimensional problem from the example illustrated in Section 3. Since the reduction operation have been performed on matrix A in Section 3 the updated matrix A0 and b0 as follows: A0 =
0.2 0.65
0.44 , 0.89
b0 = ( 0.5
0.62 ).
For this problem, the values of x1 of all solution vectors have to be fixed at xˆ 1 = 0.9 Therefore we can focus on the values of x2 and x3 . The feasible domain is shown in Fig. 1. It composed of two connected lines. For illustrative purposes, the problem with a set of two linear objective functions was tested. Different settings of the objective functions were repeated. The semi-positive polar cones generated by the negative gradients of two objective functions for those four cases are shown in Fig. 6. Futhermore, v 1 and v 2 correspond to the negative gradients of the first objective and the second objective, respectively.Also, y1 and y2 are the vectors that are perpendicular to v 1 and v 2 , correspondingly. We set max-size = 50. Case I-1: Only one efficient solution at the maximum solution.
f1 (x) = 0.8x1 − 0.1x2 − 0.3x3 f2 (x) = 0.7x1 − 0.2x2 − 0.3x3 .
Theoretically, with respect to the above theorem and Fig. 6 this problem has only one Pareto optimal solution which is the maximum solution. The optimal solution which was obtained from the genetic algorithm is exactly this Point, i.e., x2 = 0.8, x3 = 0.35 as shown in Fig. 2. Case I-2: Efficient solutions are located on one edge.
f1 (x) = x1 − x2 − x3 f2 (x) = x1 − x2 + x3 .
Theoretical Pareto optimal solutions are {x1 = 0.9, x2 = 0.8, 0 ≤ x3 ≤ 0.35}. As the result shown in Fig. 3, we were able to obtain this set. Case I-3: Efficient solutions are located on two connected edges.
f1 (x) = 0.6x1 + 0.1x2 − 0.3x3 f2 (x) = 0.4x1 − 0.2x2 + 0.3x3 .
The theoretical Pareto optimal set is
{x1 = 0.9, x2 = 0.8, 0 ≤ x3 ≤ 0.35} ∪ {x1 = 0.9, 0 ≤ x2 ≤ 0.8, x3 = 0.35}.
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Fig. 2. The efficient set for case I-1.
Fig. 3. The efficient set for case I-2.
If the efficient set E is not controlled,then this set will get larger, consequently the size of the efficient set E is likely to increase uncontrollably as the program evolves. This can cause the efficiency problem. Fig. 4(a) shows the result when we do not apply the fuzzy clustering to control the size of the efficient set. On the other hand, when we limit the size of the efficient set, the result as shown in Fig. 4(b) is quite satisfactory. It approximates the efficient region and also helps improve the efficiency of the algorithm. Case I-4: Efficient solutions are located at two extreme points. f1 (x) = 0.3x1 + x2 f2 (x) = −0.5x1 + x3 .
For case I-4, theoretically there are exactly two Pareto optimal solutions which are located at (0.9, 0, 0.35) and (0.9, 0.8, 0). With the contribution from the local improvement procedure, these two points can achieved as shown in Fig. 5(b). If this local improvement technique is not applied, then, we probably obtain a lot of points in between. Fig. 5(a) shows the result of this case when the local improvement technique is not utilized. Note that the program written in Matlab was run on a Pentium(R) 4 CPU 2.80 GHz machine. Table 1 compares the CPU times required and the sizes of the final efficient sets of all four cases. 4.1.2. Three variables and three linear objective functions The following problem with a set of three linear objective functions was tested. Its maximum solution was as xˆ = (1, 0.8, 0.9, 0.5) 0.8 1 A= 0.5 0
0.2 0.6 0.5 0.3
0.3 0.5 0.4 0.28
0.3 0.25 , 0.127 1
b = (0.9, 0.7, 0.65, 0.75) .
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(a) Uncontrolled efficient set.
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(b) Controlled efficient set.
Fig. 4. The efficient set for case I-3 with/without the control on the size of efficient set.
(a) Without local improvement.
(b) With local improvement.
Fig. 5. The efficient set for case I-4 with/without local improvement. Table 1 CPU time spent and size of the final efficient set for problem1. Case
CPU time (s)
Size of the final efficient set
I-1 I-2 I-3 I-3 With no control on size E and without local improvement with 30 iteration I-4 I-4 with no local Improvement
199/152 1060/799 1146/319 2063/829 242/052 241/083
1 50 50 1142 2 12
In this problem, the values of x4 for all solution vectors have to be fixed at 0.5. The problem can be reduced to a threedimensional problem. Therefore, the solution space is only considered with respect to x1 , x2 , and x3 . The solution space is presented in Fig. 7. It is composed of connected surface and line and five vertices. For this problem max − size = 80 is set. Case II-1: Efficient solutions lie on one edge. f1 (x) = x1 + 2x2 + 2x3 + 4x4 Min f2 (x) = 2x1 − 3x2 − x3 − 9x4 f3 (x) = 3x1 + 8x2 − 5x3 + x4 .
(
Theoretically, the efficient set of this case contains points that lie on the edge between vertices x1 and x3 . In other words, any points in {x1 = 0, x2 = 0.8, 0 ≤ x3 ≤ 1, x4 = 0.5} are efficient solutions. Fig. 8(b) represent the results in a three-
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(a) Case I-1.
(b) Case I-2.
(c) Case I-3.
(d) Case I-4. Fig. 6. The semi-positive polar cones of four cases.
Fig. 7. The solution space for three-dimensional problem.
dimensional space of x1 , x2 and x3 , Fig. 8(a) shows the result of this case when the local improvement technique is not utilized. Case II-2: each feasible point is efficient solution. f1 (x) = −2x1 − 3x2 + x3 − 9x4 Min f2 (x) = 2x1 + 3x2 − x3 + 9x4 f3 (x) = 3x1 + 8x2 − 5x3 + x4 .
(
Here f1 (x) = −f2 (x). This condition implies that every point in feasible domain is efficient solution. The results from the genetic algorithm shown in Fig. 9. Case II-3: Efficient solutions are located at two vertices. f1 (x) = x1 + x4 Min f2 (x) = x2 − x4 f3 (x) = x3 + 2x4 .
(
Theoretically, the efficient set contains only two points, i.e., the vertices x4 and x1 which are located at (0, 0.8, 0) and (1, 0, 0.9),respectively. The results from the genetic algorithm shown in Fig. 10(b) verify this fact. Fig. 10(a) shows the result of
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(a) Without local improvement.
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(b) With local improvement. Fig. 8. The efficient set for case II-1.
Fig. 9. The efficient set for case II-2.
(a) Without local improvement.
(b) With local improvement. Fig. 10. The efficient set for case II-3.
this case when the local improvement technique is not utilized. Table 2 reports the CPU time required to finish the run and the size of the final efficient set. 4.2. Non-linear objective functions We have tested genetic algorithm on problem, with two-dimensional space. Two variables and two non-linear objective functions f1 (x) = 10(x1 − 0.4)2 + 10(x2 − 0.35)2 f2 (x) = 10(x1 − 0.7)2 + 10(x2 − 0.35)2
Min
and A0 =
0.2 0.65
0.44 , 0.89
b0 = 0.5
0.62 .
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Table 2 CPU time spent and the size of the final efficient set for problem II. Case
CPU time (s)
Size of the final efficient set
II-1 II-1 With no local improvement II-2 With no control on size E and without local improvement with 20 iteration II-3 II-3 with no local Improvement
987/185 975/.83 1383/39 666/238 661/125
61 78 1173 2 18
(a) Surface plots.
(b) Contour plots.
(c) The efficient set. Fig. 11. The efficient set for non-linear two-variable case.
The system of fuzzy relation equations is exactly the same as problem I in Section 4.1. The feasible domain is presented in Fig. 1. Fig. 11(a) presents the surfaces of the two objective functions. Consider the contours of the objective functions as shown in Fig. 11(b). Those contours represent the solutions that yield the objective values at 0.3, 0.6, and 0.9 referencing from the inner contour to the outer contour, respectively. Using the same argument which presented in [24] p.161, we find that all of points between p1 and p1 inclusively are efficient solutions. Therefore, the efficient set should contain points in {x2 = 0.35, 0.4 ≤ x1 ≤ 0.7}. The CPU time consumed is 884/934 s. The size of the final set of efficient solutions is 50. Fig. 11(c) shows the result from the genetic algorithm. 5. Conclusion We have solved multi-objective optimization problems with fuzzy relation equations regarding max-average composition. Firstly, the dimension of the problem was reduced by determining the set of components whose values must be fixed for all solution vectors in order to satisfy the fuzzy relation equations. The result obtained from testing the algorithm shows that the genetic algorithm can determine the efficient set in two- and three-dimensional spaces regardless of linearity objective functions. Furthermore, it can identify the location of efficient sets, either at some vertices, edges, or the interior
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points. In order to improve the efficiency of the algorithm and improve the final efficient set it is necessary to impose control on the size of E and apply local improvement procedures. Acknowledgements The authors would like to thank the anonymous referees for their valuable comments that improved the quality of this paper. References [1] E. Khorram, A. Ghodousian, Linear objective function optimization with fuzzy relation equation constraints regarding max–av composition, Applied Mathematics and Computation 173 (2006) 872–886. [2] E. Sanchez, Resolution of composite fuzzy relation equations, Information and Control 30 (1976) 38–48. [3] S.-C. Fang, G. Li, Solving fuzzy relations equations with a linear objective function, Fuzzy Set and Systems 103 (1999) 107–113. [4] 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. [5] H.-C. Lee, S.-M. Guu, On the optimal three-tier multimedid streaming services, Fuzzy n Optimization and Decision Making 2 (2003) 31–39. [6] J. Lu, S.C. Fang, Solving nonlinear optimization problems with fuzzy relation equation constraints, Fuzzy Sets and Systems 119 (2001) 1–20. [7] W. Pedrycz, Proceeding in relational structures: Fuzzy relational equations, Fuzzy Set and Systems 40 (1991) 77–106. [8] M.M. Brouke, D.G. Fisher, Solution algorithms for fuzzy relation equations with max-product composition, Fuzzy sets and Systems 94 (1998) 61–69. [9] A. Di Nola, S. Sessa, W. Pedrycz, E. Sanchez, Fuzzy Relational Equations and their Applications in Knowledge Engineering, Kluwer Academic Press, Dordrecht, 1989. [10] J. Loetamonphong, S.C. Fang, Optimization of fuzzy relation equations with max-product composition, Fuzzy Sets and Systems 118 (2001) 509–517. [11] R.E. Steuer, Multiple Criteria Optimization: Theory; Complication and Application, Wiley, New York, 1998. [12] M. Zeleng, Linear Multiple Objective Programming, Spring-Verlag, New York, 1974. [13] J.P. Ignizio, T.M. Cavalier, Linear Programming, Prentice-Hall, Singapore, 1994. [14] P.L. Yu, G. Leitman, Compromise solutions, domination structure and Solukvakzemn, s solutions, Journal of Optimization Theory and Applications 13 (3) (1974) 362–378. [15] R.J. Benayoun, J. de Montgolfie,Tergny, O. Laritchev, Linear programming with multiple objective functions: Step method(STEM), Mathematical Programming (3) (1971) 366–375. [16] M. Sakawa, Fuzzy Sets and Interactive Multi-Objective Optimization, Plenum, New York, 1993. [17] L.R. Gardinerand, R.E. Steuer, Unified intractive multiple objective programming: An open architecture for accommodating new procedures, Journal of Operational Research Society 45 (12) (1994) 1456–1466. [18] H.-F. Wang, A multi-objective mathematical programming with fuzzy relation Constraint, Journal of Multi-Criteria Decision Analysis 4 (1995) 23–35. [19] H.J. Zimmermann, Fuzzy programming and linear programming with several objective functions, Fuzzy Sets and Systems 1 (1978) 45–55. [20] H.J Zimmermann, Fuzzy mathematical programming, Computer and Operations Research 10 (4) (1983) 45–55. [21] E.-S. Lee, R.-L. Li, Fuzzy multiobjective programming and compromise programming with Pareto optimum, Fuzzy Sets and Systems 53 (1993) 275–288. [22] H.K. Chen, H.W. Chou, Solving multi-objective linear programming problem- a genetic approach, Fuzzy Set and Systems 82 (1996) 5–38. [23] Y.-K. Wu, S.-M. Guu, A Compromise model for solving fuzzy multiple objective linear programming problems, Journal of the Chinese Institute Industrial engineering 18 (2001) 87–93. [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] G.J. Klir, T.A. Folger, Fuzzy sets, Uncertainty and Information, Prentice-Hall, NJ, 1988. [26] J.H. Holland, Adaptation in natural and artificial systems, University of Michigan, 1975. [27] Z. Michalewicz, Genetic Algorithms +Data Structures = Evolution Program, Springer, Berlin, Heidelberg, New York, 1996. [28] J.C. Bezdek, Pattern Recognition with Fuzzy Objective Function Algorithms, Plenum Press, New York, 1981. [29] C.R. Reeves, Modern Heuristic Techniques for Combinatorial Problems, Wiley, New York, 1993. [30] S.M. Guu, Y.-K. Wu, Minimizing a linear objective function with fuzzy relation equation constraints, Fuzzy Optimization and Decision Making 1 (4) (2002) 347–360. [31] R.E. Steuer, Multiple Criteria Optimization: Theory, Computation, and Application, Wiley, New York, 1986.