Tabu Search-Based Interactive Fuzzy Stochastic Multi-Level 0-1 Programming

Computational Research 1(1): 10-17, 2013 DOI: 10.13189/cr.2013.010102 http://www.hrpub.org

Masatoshi Sakawa∗, Takeshi Matsui

Faculty of Engineering, Hiroshima University, Higashi-Hiroshima, 739-8527, Hiroshima, Japan ∗Corresponding Author: [email protected]

Abstract This paper considers interactive fuzzy follower because it is supposed that there exists cooper- programming for multi-level 0-1 programming problems ative relationship between them. involving random variable coefficients both in objective For two-level linear programming problems or multi- functions and constraints. Following the concept of level ones such that decisions of decision makers in all fractile criterion optimization together with chance levels are sequential and all of the decision makers es- constrained programming, the formulated stochastic sentially cooperate with each other, Lai [2] and Shih multi-level 0-1 programming problems are transformed et al. [3] proposed fuzzy interactive approaches. In their into deterministic ones. Taking into account vagueness methods, the decision makers identify membership func- of judgments of the decision makers, interactive fuzzy tions of the fuzzy goals for their objective functions, and programming is presented. In the proposed interactive in particular, the decision maker at the upper level also method, after determining the fuzzy goals of the specifies those of the fuzzy goals for the decision vari- decision makers at all levels, a satisfactory solution ables. The decision maker at the lower level solves a is derived efficiently by updating satisfactory levels fuzzy programming problem with a constraint with re- of the decision makers with considerations of overall spect to a satisfactory degree of the decision maker at satisfactory balance among all levels. For solving the upper level. the transformed deterministic problems efficiently, Unfortunately, however, there is a possibility that the tabu search for general 0-1 programming problems is methods of Lai [2] and Shih et al. [3] lead a final solution introduced. An illustrative numerical example for a to an undesirable one because of inconsistency between three-level 0-1 programming problem is provided to the fuzzy goals of the objective function and those of the clarify the proposed method. decision variables. In order to overcome the problem in their methods, by eliminating the fuzzy goals for the Keywords Multi-level 0-1 programming, Random decision variables, Sakawa et al. have proposed interac- variables, Interactive fuzzy programming, Fractile tive fuzzy programming for two-level or multi-level linear criterion optimization, Tabu search programming problems to obtain a satisfactory solution for decision makers [4,5]. Extensions to two-level linear fractional programming problems [5] and decentralized two-level linear programming problems [6–8] have also 1 Introduction been considered. The subsequent works on two-level or multi-level programming have been appearing [9–12]. A In the real world, we often encounter situations where recent survey paper of Sakawa and Nishizaki [13] is de- there are two or more decision makers (DMs) in an or- voted to reviewing and classifying the numerous major ganization with a hierarchical structure, and they make papers in the area of so-called cooperative multi-level decisions in turn or at the same time so as to optimize programming. their objective functions. Such situations are formulated In actual decision making situations, however, we as multi-level programming problems, and the Stackel- must often make a decision on the basis of vague in- berg solution has been usually employed as a solution formation or uncertain data. For such decision making concept [1]. problems involving uncertainty, there exist two typical When the Stackelberg solution is employed, it is as- approaches: probability theoretic approach [14–16] and sumed that there is no communication between the two fuzzy-theoretic one [16–19]. Stochastic programming, as DMs, or they do not make any binding agreement even an optimization method based on the probability theory, if there exists such communication. However, the above have been developing in various ways [14–16], includ- assumption is not always reasonable when we model de- ing two stage problems considered by Dantzig [20] and cision making problems in a decentralized firm as a two- chance constrained programming proposed by Charnes level programming problem in which top management et al. [21]. Fuzzy mathematical programming represent- is a leader and an operation division of the firm is a ing the vagueness in decision making situations by fuzzy Computational Research 1(1): 10-17, 2013 11 concepts have been studied by many researchers [17,18]. tion and tries to optimize each objective function. Such Fuzzy multiobjective linear programming, first proposed a stochastic multi-level 0-1 programming problem can by Zimmermann [17], have been also developed by nu- be formulated as merous researchers, and an increasing number of suc- minimize z (x) = c (ω)x + ··· + c (ω)x  cessful applications has been appearing [18,19,22,23]. In 1 11 1 1K K  DM1 (Level 1)  particular, after reformulating stochastic multiobjective . .  . .  linear programming problems using several models for  minimize z (x) = c (ω)x + ··· + c (ω)x chance constrained programming, Sakawa et al. [24–26] K K1 1 KK K DMK (Level K)  presented an interactive fuzzy satisficing method to de-  subject to A1x1 + ··· + AK xK ≤ b(ω)  n1 nK  rive a satisficing solution for the DM as a generalization x1 ∈ {0, 1} ,..., xK ∈ {0, 1}  of their previous results [18, 22, 23, 27–30]. (1) Furthermore, in real world decision making situations, where xl, l = 1,...,K, is an nl-dimensional 0-1 decision it is often found that decision variables in a multiob- variable column vector, Al, l = 1,...,K are m × nl co- jective stochastic programming problem the formulated efficient matrices, and clj(ω), l = 1,...,K, j = 1,...,K programming problems are not continuous but rather are nj-dimensional Gaussian random variable row vec- discrete. To deal with practical sizes of the stochas- tors with mean vectors c¯lj, and covariance matrices tic multi-level 0-1 programming problems formulated for l Vlpq = v = Cov[clh (ω), clh (ω)] , p = 1,...,K, decision making problems in the real world, an efficient hphq p q q = 1,...,K, and they are independent of each other, tabu search method for general 0-1 programming prob- and b(ω) is a random variable vector whose joint distri- lems is introduced. bution function is F (·). Under these circumstances, in this paper, through Since (1) contains random variable coefficients, solu- a simultaneous consideration of both random variables tion methods for ordinary mathematical programming and integer decision variables involved in the real world problems cannot be applied directly. Consequently, we hierarchical decision making problems, we first formu- first deal with the constraints in (1) as chance con- late multi-level 0-1 programming problems with random straints [21] which mean that the constraints need to variable coefficients in both objective functions and con- be satisfied with a certain probability (satisficing level) straints. The main contribution of this paper is to pro- and over. Namely, replacing constraints in (1) by chance vide a novel decision making methodology including a constraints with a satisficing level β, the problem can be new model, solution concept and solution algorithm to transformed as: deal with more realistic problems in the real world, by  simultaneously considering various concepts such as hi- minimize z1(x) = c11(ω)x1 + ··· + c1K (ω)xK DM1 (Level 1)  erarchy structure, fuzziness, randomness, 0-1 decision  . .  variables and interactive fuzzy programming, while most . .   of previous papers dealt with either of the concepts or a minimize zK (x) = cK1(ω)x1 + ··· + cKK (ω)xK DMK (Level K) part of them.  subject to Pr{ai1x1 + ··· + aiK xK ≤ bi(ω)}  By employing chance constrained programming [31],  ≥ βi, i = 1, . . . , m  stochastic constraints are transformed into determinis-  x ∈ {0, 1}n1 ,..., x ∈ {0, 1}nK  tic ones. Adopting the fractile criterion optimization 1 K (2) model [32,33], the minimization of each stochastic objec- where a is the ith row vector of A , l = 1,...,K, and tive function is replaced with the optimization of the tar- ij l b (ω) is the ith element of b(ω). get variables under the condition that the attained prob- i The first constraint in (2) is rewritten as: abilities are greater than or equal to certain permissible levels. Under some appropriate assumptions for distri- Pr{ai1x1 + ··· + aiK xK ≤ bi(ω)} ≥ βi bution functions, the formulated stochastic multi-level ⇔ 1 − Pr{ai1x1 + ··· + aiK xK ≥ bi(ω)} ≥ βi 0-1 programming problems are transformed into deter- ⇔ 1 − Fi(ai1x1 + ··· + aiK xK ) ≥ βi ministic ones. In our interactive method, after determin- ⇔ Fi(ai1x1 + ··· + aiK xK ) ≤ 1 − βi ing the fuzzy goals of the DMs at all levels, a satisfactory ∗ ⇔ ai1x1 + ··· + aiK xK ≤ Fi (1 − βi) solution is derived efficiently by updating the satisfac- ∗ tory degrees of the DMs at the upper level with consid- where Fi is a pseudo-inverse function of Fi. ˆ ∗ erations of overall satisfactory balance among all levels. Letting bi = Fi (1 − βi), problem (2) can be rewritten For solving the transformed deterministic problems effi- as: ciently, we also propose a novel tabu search method by minimize z (x) = c (ω)x + ··· + c (ω)x  extending tabu search based on strategic oscillation for 1 11 1 1K K  DM1 (Level 1)  multidimensional 0-1 knapsack problems [34] into gen- . .  . .  eral 0-1 programming problems. . .  minimize zK (x) = cK1(ω)x1 + ··· + cKK (ω)xK DMK (Level K)  ˆ  subject to A1x1 + ··· + AK xK ≤ b  2 Problem formulation with frac- n1 nK  x1 ∈ {0, 1} ,..., xK ∈ {0, 1}  tile models (3) ˆ ˆ ˆ T where ˆb = (b1, b2,..., bm) . Consider stochastic multi-level 0-1 programming It should be noted here that the constraint of (3) is problems where each of the DMs at all levels takes over- no longer stochastic but becomes deterministic through all satisfactory balance among all levels into considera- the idea of chance constraint. 12 Tabu Search-Based Interactive Fuzzy Stochastic Multi-Level 0-1 Programming

Charnes and Cooper [31] also considered three types objective functions in (4) as follows. of decision rules for optimizing objective functions with random variables: (i) the minimum or maximum ex- Pr {zl(x1,..., xK , ω) ≤ fl} = Pr {cl1(ω)x1 + ··· + clK (ω)xK ≤ fl} pected value model, (ii) the minimum variance model, ( c (ω)x + ··· + c (ω)x − (¯c x + ··· +c ¯ x ) and (iii) the maximum probability model, which are re- = Pr l1 1 lK K l1 1 lK K p T T T T T ferred to as the expectation model, the variance model, (x1 ,..., x )Vl(x1 ,..., x ) K ) K and the probability model, respectively. Moreover, f − (¯c x + ··· +c ¯ x ) ≤ l l1 1 lK K Kataoka [32] and Geoffrion [33] individually proposed p T T T T T (x1 ,..., xK )Vl(x1 ,..., xK ) the fractile model. ! fl − (¯cl1x1 + ··· +c ¯lK xK ) = φl ≥ αl, p T T T T T For instance, let the objective function represent a (x1 ,..., xK )Vl(x1 ,..., xK ) profit. If the DM wishes to simply maximize the ex- where φ (·) is the distribution function of the standard pected profit without caring about the fluctuation of the l Gaussian random variable. From the monotonicity of profit, the expectation model to optimize the expecta- the distribution function, we can define the inverse function of the objective function is appropriate. On the tion φ−1(·) of φ (·). Then, the above inequalities are other hand, if the DM hopes to decrease the fluctuation l l expressed as: of the profit as little as possible from the viewpoint of the stability of the profit, the variance model to mini- f − (c¯ x + ··· + c¯ x ) l l1 1 lK K ≥ K mize the variance of the objective function is useful. In q αl T T T T T contrast to these two types of optimizing approaches, (x1 ,..., xK )Vl(x1 ,..., xK ) as satisficing approaches, the probability model and the ⇔ fl ≥ (c¯l1x1 + ··· + c¯lK xK ) q fractile model have been proposed. T T T T T +Kαl (x1 ,..., xK )Vl(x1 ,..., xK ) When the DM wants to maximize the probability that Letting K = φ−1(α ), l = 1, ··· ,K and noting that the profit is greater than or equal to a certain permissible αl l l level, probability model is recommended. In contrast, the equality when the DM wishes to optimize such a permissible level fl = (c¯l1x1 + ··· + c¯lK xK ) under a given threshold probability with respect to the q T T T T T achieved profit, the fractile model will be appropriate. +Kαl (x1 ,..., xK )Vl(x1 ,..., xK ) In this paper, assuming that the DM is interested in the probability that each objective function attains a goal holds at the minimum of fl, problem (4) is equivalent value rather than the expectation or variance of each to the following deterministic multi-level programming membership function, we adopt the fractile criterion op- problem: timization as a decision making model. F  minimize Z1 (x1,..., xK ) DM1iLevel 1j  In the fractile criterion optimization model, the min-  = (c¯11x1 + ··· + c¯1K xK )  p T T T T T  imization of each of objective function zl(x) in (3) is +Kα (x ,..., x )V1(x ,..., x )  1 1 K 1 K  substituted with the optimization of the target value . .  . . that zl(x) is less than or equal to a certain permissible F minimize ZK (x1,..., xK )  level fl under the chance constraints. Through fractile DMKiLevel Kj   criteria optimization, problem (3) can be rewritten as: = (c¯K1x1 + ··· + c¯KK xK )  p T T T T T  +Kα (x ,..., x )VK (x ,..., x )  K 1 K 1 K  subject to x ∈ X 

minimize f1  (5) DM1iLevel 1j   where X denotes the feasible region satisfying the con- . .  . .  straints of (4).  minimize fK  DMKiLevel Kj   subject to Pr [c11(ω)x1 + ··· + c1K (ω)xK ≤ f1]  3 Interactive fuzzy programming ≥ α1 .  In the deterministic multi-level 0-1 programming .   problem (5), considering the vague nature of the DMs’ Pr [cK1(ω)x1 + ··· + cKK (ω)xK ≤ fK ]   judgments for the target values ZF (x), l = 1, 2,...,K it ≥ αK  l ˆ  seems natural to assume that the DMs have fuzzy goals A1x1 + ··· + AK xK ≤ b  n1 nK  F x1 ∈ {0, 1} ,..., xK ∈ {0, 1} such as “Zl (x) should be substantially greater than or (4) equal to some speciﬁc value.” Since all elements of clj(ω), l = 1,...,K, j = 1,...,K Through the introduction of a membership function are assumed to be Gaussian random variable with mean µl(·) for quantifying a fuzzy goal for the l th objective c¯lj and covariance matrices Vlpq, p = 1,...,K, q = function in (5), (5) can be rewritten as: 1,...,K. F  maximize µ1(Z1 (x1,..., xK )) DM1iLevel 1j  Then, (hl − (¯cl1x1 + ··· +  q . .  T T T T T . . c¯lK xK ))/ (x ,..., x )Vl(x ,..., x ) is a random . . (6) 1 K 1 K F variable following the standard Gaussian distribution maximize µK (ZK (x1,..., xK ))  DMKiLevel Kj  with mean 0 and variance 1. Hence, we can rewrite the subject to x ∈ X  Computational Research 1(1): 10-17, 2013 13

where µl(·) is assumed to be nondecreasing. is solved for obtaining a solution which maximizes the Although the membership function does not always smaller degree of satisfaction between those of all DMs: need to be linear, for the sake of simplicity, we adopt a F linear membership function. To be more specific, if the maximize min µl(Zl (x)) (9) x∈X l=1,...,K F DM feels that Zl (x) should be greater than or equal to F F F F at least Zl,0 and Zl (x) ≥ Zl,1(> Zl,0) is satisfactory, Solving problem (9), we can obtain a solution which F the linear membership function µl(Zl (x)) is defined as: maximizes the smaller satisfactory degree between those F of all DMs. µl(Zl (x))  F F  0, µl(Zl (x)) < Zl,0 Termination conditions of the interactive process  F F  µl(Z (x)) − Z = l l,0 ,ZF ≤ µ (ZF (x)) ≤ ZF F F l,0 l l l,1 (1) For all λ = 1,...,K − 1, DMλ’s satisfactory degree  Zl,1 − Zl,0  F F is larger than or equal to the minimal satisfactory  1, µl(Zl (x)) > Zl,1 level δˆ specified by DMλ. (7) ∆λ and it is depicted in Figure 1. (2) For all λ = 1,...,K − 1, the ratio δλ of satisfactory

F degrees is in the closed interval the lower and the µl (Zl (x)) upper bounds of which are specified by DMλ. 1.0 Condition (1) means DMλ’s required condition for solutions proposed by DM(λ + 1). Condition (2) is provided in order to keep overal satisfactory balance among all the level. Unless the conditions are satisfied simultaneously, F F F 0 Zl, 1 Zl, 0 Zl (x) DMλ, λ = 1,...,K, needs to update the minimal sat- ˆ isfactory level δ∆λ . Suppose that the DMs from at the Figure 1. Linear membership function. (q + 1)th level to at the (K − 1)th level, i.e., DM(q + 1), DM(q + 2), ..., and DM(K − 1), satisfy the proposed Zimmermann [17] suggested a method for assessing solution but DMq does not satisfy it. Then DMq, the parameter values of the linear membership function. DM(q + 1), ..., and DM(K − 1) need to update their F In his method, the parameter values Zl,1, l = 1,...,K ˆ minimal satisfactory levels δ∆λ , λ = q, q + 1,...,K − 1. are determined as For any two levels adjacent to each other, giving a DM F F Z1,1 = Z1,max at an upper level serious consideration, a DM at a lower F 1 1 = Z1 (x1,max,..., xK,max) level should update the minimal satisfactory level. F = max Z1 (x1,..., xK ) Now we are ready to propose interactive fuzzy pro- (xT ,...,xT )T ∈X 1 K gramming for deriving a satisfactory solution by up- . . dating the satisfactory degree of the DM at the upper F F level with considerations of overall satisfactory balance ZK,1 = ZK,max F K K among all the levels. = ZK (x1,max,..., xK,max) F = max ZK (x1,..., xK ) T T T (x1 ,...,x ) ∈X K Computational procedure of interactive fuzzy programming F and the parameter values Xl,0, l = 1,...,K are specified as Step 1: Ask the decision maker at the upper level, F F 1 1 Z1,0 = Z1 (x1,min,..., xK,min) DM1, to subjectively determine a satisficing levels . . αl ∈ (0, 1), l = 1, 2,...,K for objective functions ZF = ZF (xK ,..., xK ) and βi ∈ (0, 1), i = 1, 2, . . . , m for constraints. Go K,0 K 1,min K,min to Step 2. 1 K where xl,min = (xl,min,..., xl,min), l = 1,...,K is an optimal solution to the following problem Step 2: The following problems are solved to find the minimum values ZF = ZF (xl,min) and ZF = maximize ZF (x ,..., x )  l,min l l,M l 1 K  F j,min F    max {Zl (x )} of objective functions Zl (x)  j=1,...,K fl − (¯cl1x1 + ··· +c ¯lK xK )  = φl q  . under the chance constraints with satisficing levels T T T T T (x1 ,..., xK )Vl(x1 ,..., xK )  αl, l = 1, 2,...,K and βi, i = 1, 2, . . . , m.  subject to x ∈ X  minimize ZF (x ,..., x ) (8) l 1 K , l = 1,...,K Since (8) is a 0-1 programming problem, it can be subject to x ∈ X solved by tabu search based on strategic oscillation [34]. (10) Then, by setting the parameters as described above, If the set of feasible solutions to these problems the linear membership functions (7) is identified. is empty, the satisficing levels αl ∈ (0, 1), l = To derive an overall satisfactory solution to the mem- 1, 2,...,K and βi, i = 1, 2, . . . , m must be re- F bership function maximization 6, we first find the max- assessed and return to step 1. Otherwise, let zl,min imizing decision of the fuzzy decision proposed by Bell- be optimal objective function values to (10). Since man and Zadeh [35]. Namely, the following problem (10) are 0-1 programming problems, they can be 14 Tabu Search-Based Interactive Fuzzy Stochastic Multi-Level 0-1 Programming

solved by tabu search based on strategic oscilla- 4 Tabu search for general 0-1 tion. Then, identify the linear membership function F programming problems µl(Zl (x)), l = 1, 2,...,K of the fuzzy goal for the corresponding objective function. Go to Step 3. For solving the 0-1 programming problems in the pro- Step 3: Solve the following corresponding maxmin posed interactive fuzzy programming method, it is con- problem. structive to extend tabu search based on strategic oscillation for multidimensional 0-1 knapsack problems [34] F maximize min µl(Zl (x)) (11) x∈X l=1,...,K into general 0-1 programming problems. With this observation, consider a general 0-1 program- Go to step 4. ming problem formulated as:

Step 4: Ask DM1 to subjectively set the minimal satis- minimize f(x)  factory level δˆ . Then, solve the following maxmin  1 subject to gi(x) ≤ 0, i = 1, . . . , m (15) problem. x ∈ {0, 1}n  F ) maximize min µl(Z (x)) x∈X l l=2,...,K (12) where f(·) and gi(·), i = 1, . . . , m are convex or noncon- ˆ T subject to µ1(p1(x)) ≥ δ1 vex real-valued functions and x = (x1, . . . , xn) is an n-dimensional column vector of 0-1 decision variables. 0 Set λ := 2Cλ := 1. Go to step 5. The tabu search proposed in [34] made use of the prop- Step 5: Ask DMλ to set the membership erty of multidimensional 0-1 knapsack problems that the improvement or disimprovement of the objective func- function µ∆λ (∆λ(x)) for the ratio ∆λ = F F tion value corresponds with the decrease or increase of (µλ+1(Z (x)))/(µλ(Z (x))) of satisfactory λ+1 λ the degree of feasibility. From the property, it is clear degrees and the minimal satisfactory level δˆ . ∆λ that the optimal solution to multidimensional 0-1 knap- Solve the following maxmin problem. sack problems exists in the area near the boundary of F  maximize min µl(Zl (x)) the feasible region which is called the promising zone. x∈X l=λ+1,...,K   Thus, the search direction in multidimensional 0-1 knap- subject to µ (ZF (x)) ≥ δˆ  1 1 1  sack problems can be controlled by checking the change µ (∆ (x)) ≥ δˆ (13) ∆2 2 ∆2 of the objective function value. In the case of general 0- .  .  1 programming problems, observing that the monotone .  ˆ  relation between the objective function value and the µ∆ (∆λ(x)) ≥ δ∆ λ λ degree of feasibility no longer holds, the promising zone Repeat this step until λ = K − 1. does not always exist near the boundary of the feasible Step 6: If the current solution satisfies the termination region. Considering that the promising zone originally conditions, DMK − λ0 accepts it, and K − λ0 = 1, means the area which include an optimal solution, we then the procedure stops and the current solution is define the promising zone for general 0-1 programming determined to be a satisfactory solution. Otherwise, problems as neighborhoods of local optimal solutions. ask DMK − λ0 to update the minimal satisfactory Thus, in order to use not only the change of the ob- ˆ 0 jective function value but the degree of feasibility, we level δ∆K−λ0 . If K − λ = 1, ask DM1 to update the ˆ introduce the index of surplus of constraints δ(x) and minimal satisfactory level δ1. Go to step 7. that of slackness of constraints (x) defined as: Step 7: Solve the following problem, and return to step X X 5. δ(x) = δi(x) = gi(x) + + maximize v  i∈I i∈I  subject to x ∈ X  X X  (x) = (x) = −g (x) 0 ≤ v ≤ 1  i i  − − F ˆ  i∈I i∈I µ1(Z1 (x)) ≥ δ1  ˆ  where I+ = {i | g (x) > 0, i ∈ {1, . . . , m}} and µ∆ (∆2(x)) ≥ δ∆  i 2 2  − .  I = {i | gi(x) < 0, i ∈ {1, . . . , m}}, Furthermore, .  . let ∆jf(x) denote the change of f(x) by setting µ (∆ (x)) ≥ δˆ ∆K−1 K−1 ∆K−1  xj := 1 − xj. Similarly, ∆jδ(x), ∆jδi(x), ∆j(x) and K−1 ˆ F  Π 0 ∆ µ 0 (Z 0 (x)) ≥ v ∆ (x) are defined for x := 1 − x . In addition, we l=K−λ +1 l K−λ +1 K−λ +1  j i j j .  assign the feasible solution to x†, and update x† when .   the feasible solution is updated. ˆ ˆ F  ∆K−1∆K−2µK−2(ZK−2(x)) ≥ v  ˆ F  ∆K−1µK−1(ZK−1(x)) ≥ v  F  µK (ZK (x)) ≥ v Computational procedure of tabu search (14) for general 0-1 programming problems It should be noted here that, in the proposed in- Step 0: INITIALIZATION teractive fuzzy programming method, it is required Generate an initial solution x at random, and initialize to solve the 0-1 programming problems (10), (11), the tabu list (TL). Set the tabu term (TT), the depth (12), (13) and (14), which is apparently difficult to (D), the maximum number of oscillation (Omax) and the solve. oscillation counter O := 1. If x is feasible, go to step 4. Computational Research 1(1): 10-17, 2013 15

Otherwise, go to step 1. the surplus of constraints δ(x) and gives the greatest im- Step 1: TS PROJECT provement (if exist) or the lest disimprovement of the ob- The aim of this step is to move the current solution in jective function value when its value would be changed, the infeasible region to the promising zone in the gen- changing the value of the decision variable actually and tlest ascent (disimproving) direction about the objective adding the decision variable to TL. If there does not ex- function with decreasing the surplus of constraints δ(x) ist such a decision variable or the number of repetitions While δ(x) is positive, i.e., the current solution is of the above procedure exceeds Omax, return to step 1. infeasible, repeat finding a non-tabu decision variable The search procedure of the proposed tabu search for which decreases δ(x) and gives the lest disimprovement general 0-1 programming problems is illustrated in Fig- of the objective function value when its value would be ure 2 changed, changing the value of the decision variable ac- infeasible region tually and adding the decision variable to TL. If there [TS_COMPLEMENT] does not exist any non-tabu decision variable that de- promising zone creases δ(x), select a decision variable randomly, change its value even if δ(x) increases and add the decision vari- [TS_ADD] [TS_INFEASIBLE_ADD] able to TL. If δ(x) = 0 and there does not exist any initial solution decision variable which improves the objective function [TS_PROJECT] feasible region value by changing its value, go to step 2. [TS_DROP] Step 2: TS COMPLEMENT promising zone The aim of this step is to search the promising zone intensively. Let x0 := x and x00 := x0. Then, select several Figure 2. Tabu search for general 0-1 programming. tabu decision variables of x00 and change their values. If δ(x00) = 0, then carry out step 4. Otherwise, carry out step 1. If f(x00) < f(x) for the solution δ(x00) obtained by step 4 or step 1, let x := x00. This procedure 5 Numerical example is repeated D times. If the previous step of this step is step 1, then go to step 3. If the previous step of this As an illustrative numerical example, consider the fol- step is step 4, then go to step 5. lowing stochastic three-level 0-1 programming problem: Step 3: TS DROP  minimize c11(ω)x1 + c12(ω)x2 + c13(ω)x3 The aim of this step is to move the current solution in DM1 (Level 1)   the promising zone to the inside of the feasible region minimize c21(ω)x1 + c22(ω)x2 + c23(ω)x3  DM2 (Level 2)  in the gentlest ascent direction of the objective function (16) with increasing the slackness of constraints (x) minimize c31(ω)x1 + c32(ω)x2 + c33(ω)x3 DM3 (Level 3)   Repeat finding a non-tabu decision variable which in- subject to A1x1 + A2x2 + A3x3 ≤ b(ω)  n1 n3  creases (x) and gives the lest disimprovement of the x1 ∈ {0, 1} ,..., x3 ∈ {0, 1}  objective function when its value would be changed, T T changing the value of the decision variable actually and where x1 = (x1, . . . , x15) , x2 = (x16, . . . , x30) , x3 = T adding the decision variable to TL. If there does not (x31, . . . , x45) ; each entry of 15-dimensional row con- exist any non-tabu decision variable that increases (x) stant vectors cij, i, j = 1, 2, 3, and each entry of 3 × 15 or the number of repetitions of the above procedure ex- coefficient matrices A1,A2, and A3 are random ceeds TT, go to step 4. Through the use of this numerical example, it is now Step 4: TS ADD appropriate to illustrate the proposed interactive fuzzy The aim of this step is to move the current solution in programming method. the feasible region to the promising zone in the steepest In step 1 of the interactive fuzzy programming, DM1 descent (improving) direction about the objective func- specifies satisficing levels αl, l = 1, 2, 3 and βi, i = tion with keeping δ(x) = 0. 1, 2,..., 9 as:

While δ(x) = 0, i.e., the current solution is feasible, T T repeat finding a non-tabu decision variable which keeps (α1, α2, α3) = (0.75, 0.75, 0.75) , δ(x) = 0 and gives the greatest improvement of the ob- (β , β , β , β , β , β , β , β , β )T = jective function value when its value would be changed, 1 2 3 4 5 6 7 8 9 (0.95, 0.80, 0.85, 0.90, 0.90, 0.85, 0.85, 0.95, 0.80)T . changing the value of the decision variable actually and adding the decision variable to TL. If there does not ex- For the specified satisficing levels αl, l = 1, 2, 3 and F ist such a decision variable, go to step 2. βi, i = 1, 2,..., 9, in step 2, minimal values Zl,min and F F Step 5: TS INFEASIBLE ADD Zl,M of objective functions Zl (x1, x2, x3) under the The aim of this step is to move the current solution in the chance constraints are calculated. By considering these promising zone to the infeasible region in the steepest values, the DMs identify the linear membership function descent (if exist) or the gentlest ascent direction about whose parameter values are determined by the Zimmer- the objective function with decreasing the slackness of mann method [17]. constraints (x) or increasing the surplus of constraints In step 3, the maxmin problem is solved. The obtain δ(x). result is shown at the column labeled “1st” in Table 1. Repeat finding a non-tabu decision variable which de- In step 4, for the obtained optimal solution, then, creasing the slackness of constraints (x) or increasing the ratio of satisfactory degrees ∆1 is equal to 0.9659. 16 Tabu Search-Based Interactive Fuzzy Stochastic Multi-Level 0-1 Programming

Table 1. Interaction process the proposed interactive method, after determining the fuzzy goals of the DMs at all levels, a satisfactory so- Interaction 1st 2nd 3rd 4th lution is derived efficiently by updating the satisfactory ˆ δ1 0.8000 0.8000 0.8500 degree of the DM at the 1st level with considerations of ˆ δ∆1 0.8000 overall satisfactory balance among all levels. It is sig- ˆ δ∆2 0.8000 nificant to note here that the transformed deterministic F problems to derive an overall satisfactory solution can be µ1(Z1 (x)) 0.7495 0.8247 0.8010 0.8680 F effectively solved through the proposed tabu search for µ2(Z2 (x)) 0.7239 0.6677 0.7107 0.6778 F general 0-1 programming problems. An illustrative nu- µ3(Z3 (x)) 0.7065 0.6925 0.6311 0.6162 merical example for a three-level 0-1 programming prob- ∆1(x) 0.9659 0.8096 0.8872 0.7809 lem was provided to demonstrate the feasibility of the ∆2(x) 0.9760 1.0372 0.8880 0.9091 proposed method. However, further computational ex- µ (∆ (x)) 0.9682 ∆1 1 periences should be carried out for several types of nu- µ (∆ (x)) 0.9829 0.9094 ∆2 2 merical examples. From such experiences the proposed method must be revised. As a subject of future work, applications of the proposed method to the real world Since DM1 is not satisfied with this solution, DM1 decision making situations should be considered in the ˆ near future. Extensions to other stochastic program- sets the minimal satisfactory level δ1 to 0.80. (12) For ˆ ming models will be considered elsewhere. Also exten- δ1 = 0.80 (12) is solved, and the obtained optimal solu- F F sions to multi-level 0-1 programming problems involving tion is µ1(Z1 (x)) = 0.8247, µ2(Z2 (x)) = 0.6677, and F fuzzy random variable coefficients and/or random fuzzy µ3(Z3 (x)) = 0.6925, as is shown at the column labeled “2nd” in Table 1. coefficients will be required in the near future. In step 5, DM2 sets the membership function

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