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

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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 Tabu Search-Based Interactive Fuzzy Stochastic Multi-Level 0-1 Programming Masatoshi Sakawa∗, Takeshi Matsui Faculty of Engineering, Hiroshima University, Higashi-Hiroshima, 739-8527, Hiroshima, Japan ∗Corresponding Author: [email protected] Copyright c 2013 Horizon Research Publishing All rights reserved. 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 (!)x + ··· + c (!)x 9 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 (!)x + ··· + c (!)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(!) > n1 nK > rive a satisficing solution for the DM as a generalization x1 2 f0; 1g ;:::; xK 2 f0; 1g ; 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(!), 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 (!); clh (!)] , 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(!) 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 9 simultaneously considering various concepts such as hi- minimize z1(x) = c11(!)x1 + ··· + c1K (!)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(!)x1 + ··· + cKK (!)xK DMK (Level K) part of them. > subject to Prfai1x1 + ··· + aiK xK ≤ bi(!)g > By employing chance constrained programming [31], > ≥ βi; i = 1; : : : ; m > stochastic constraints are transformed into determinis- > x 2 f0; 1gn1 ;:::; x 2 f0; 1gnK ; 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 (!) is the ith element of b(!). 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- Prfai1x1 + ··· + aiK xK ≤ bi(!)g ≥ βi bution functions, the formulated stochastic multi-level , 1 − Prfai1x1 + ··· + aiK xK ≥ bi(!)g ≥ 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. ^ ∗ 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
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