برنامه ریزی فازی تعاملی برای مشکلات برنامه ریزی خطی دو سطحی تصادفی فازی از طریق حداکثرسازی احتمال وقوع با احتمال

This paper considers two-level linear programming problems involving fuzzy random variables under cooperative behavior of the decision makers. Through the introduction of fuzzy goals together with possibility measures, the formulated fuzzy random two-level linear programming problem is transformed into the problem to maximize the satisfaction degree for each fuzzy goal. By adopting probability maximization, the transformed stochastic two-level programming problem can be reduced to a deterministic one. Interactive fuzzy programming to derive a satisfactory solution for the decision maker at the upper level in consideration of the cooperative relation between decision makers is presented. An illustrative numerical example demonstrates the feasibility and efficiency of the proposed method.

In actual decision making situations, we must often make a decision on the basis of vague information or uncertain data. For such decision making problems involving uncertainty, there exist two typical approaches: probability-theoretic approach (Charnes and Cooper, 1963, Stancu-Minasian, 1984, Stancu-Minasian, 1990, Birge and Louveaux, 1997 and Kall and Mayer, 2011) and fuzzy-theoretic one (Zimmermann, 1978, Zimmermann, 1987 and Sakawa, 1993). However, in practice, decision makers are faced with the situations where both fuzziness and randomness exist. In the case where some expert estimates coefficients of objective functions or constraints with uncertainty, they do not always represent the coefficients as random variables or fuzzy sets but as the values including both fuzziness and randomness. In such a case, it is important to realize that simultaneous considerations of both fuzziness and randomness would be required. A fuzzy random variable (Kwakernaak, 1978) is one of the mathematical concepts dealing with fuzziness and randomness simultaneously. Studies on linear programming problems with fuzzy random variable coefficients, called fuzzy random linear programming problems, were initiated by Wang and Qiao, 1993 and Qiao et al., 1994 as seeking the probability distribution of the optimal solution and optimal value. A brief survey of major fuzzy stochastic programming models was found in the paper by Luhandjula (2006). On the other hand, we often encounter situations where there are two or more decision makers in an organization with a hierarchical structure, and they make decisions in turn or at the same time so as to optimize their objective functions. In particular, consider a case where there are two decision makers; one of the decision makers first makes a decision, and then the other who knows the decision of the opponent makes a decision. Such a situation is formulated as a two-level programming problem (Sakawa & Nishizaki, 2009). In the context of two-level programming, the decision maker at the upper level first specifies a strategy, and then the decision maker at the lower level specifies a strategy so as to optimize the objective with full knowledge of the action of the decision maker at the upper level. In conventional multi-level mathematical programming models employing the solution concept of Stackelberg equilibrium, it is assumed that there is no communication among decision makers, or they do not make any binding agreement even if there exists such communication (Bialas and Karwan, 1984, Shimizu et al., 1997 and Simaan and Cruz, 1973). Compared with this, for decision making problems in such as decentralized large firms with divisional independence, it is quite natural to suppose that there exists communication and some cooperative relationship among the decision makers (Sakawa & Nishizaki, 2009). For two-level linear programming problems or multi-level ones such that decisions of decision makers in all levels are sequential and all of the decision makers essentially cooperate with each other, Lai, 1996 and Shih et al., 1996 proposed fuzzy interactive approaches. In their methods, the decision makers identify membership functions of the fuzzy goals for their objective functions, and in particular, the decision maker at the upper level also specifies those of the fuzzy goals for the decision variables. The decision maker at the lower level solves a fuzzy programming problem with a constraint with respect to a satisfactory degree of the decision maker at the upper level. Unfortunately, there is a possibility that their method leads a final solution to an undesirable one because of inconsistency between the fuzzy goals of the objective function and those of the decision variables. In order to overcome the problem in their methods, by eliminating the fuzzy goals for the decision variables, Sakawa et al. have proposed interactive fuzzy programming for two-level or multi-level linear programming problems to obtain a satisfactory solution for decision makers (Sakawa et al., 1998 and Sakawa et al., 2000a). Extensions to two-level linear fractional programming problems (Sakawa, Nishizaki, & Uemura, 2001), decentralized two-level linear programming problems (Sakawa and Nishizaki, 2002a and Sakawa et al., 2002), two-level linear fractional programming problems with fuzzy parameters (Sakawa, Nishizaki, & Uemura, 2000b), and two-level nonconvex programming problems with fuzzy parameters (Sakawa & Nishizaki, 2002b) were provided. Further extensions to two-level linear programming problems with random variables, called stochastic two-level linear programming problems (Sakawa and Katagiri, 2010 and Sakawa and Kato, 2009), two-level integer programming problems (Sakawa, Katagiri, & Matsui, 2010), and two-level linear programming problems involving fuzzy random variables, called fuzzy random two-level programming problems (Sakawa, Katagiri, & Matsui, 2011 and Sakawa, Nishizaki, & Katagiri, 2011), have also been considered. It should be noted here that fuzzy random variables (Kwakernaak, 1978, Puri and Ralescu, 1986 and Wang and Qiao, 1993) are considered to be random variables whose realized values are not real values but fuzzy numbers or fuzzy sets. A recent survey paper of Sakawa and Nishizaki (2012) is devoted to reviewing and classifying the numerous major papers in the area of so-called cooperative multi-level programming. Under these circumstances, in this paper, assuming cooperative behavior of the decision makers, we consider solution methods for decision making problems in hierarchical organizations under fuzzy random environments. To deal with the formulated two-level linear programming problems involving fuzzy random variables, possibility measures are introduced and a stochastic two-level linear programming problem is defined for guaranteeing the degree of realization of the problem. Taking into account vagueness of judgments of decision makers, fuzzy goals are introduced and the stochastic two-level linear programming problem is transformed into the problem to maximize the satisfaction degree for each fuzzy goal. Following probability maximization, the transformed stochastic two-level programming problem can be reduced to a deterministic one. Interactive fuzzy programming to obtain a satisfactory solution for the decision maker at the upper level in consideration of the cooperative relation between decision makers is presented. It is shown that all of the problems to be solved in the proposed interactive fuzzy programming can be easily solved by the simplex method or the combined use of the bisection method and the simplex method.

In this paper, assuming cooperative behavior of the decision makers, interactive decision making methods in hierarchical organizations under fuzzy random environments were considered. For the formulated fuzzy random two-level linear programming problems, possibility measures were introduced and a stochastic two-level linear programming problem was defined for guaranteeing the degree of realization of the problem. Considering the vague natures of decision makers’ judgments, fuzzy goals were introduced and the stochastic two-level linear programming problem was transformed into the problem to maximize the satisfaction degree for each fuzzy goal. Through the possibility maximization model, the transformed stochastic two-level programming problem was reduced to a deterministic one. Interactive fuzzy programming to obtain a satisfactory solution for the decision maker at the upper level in consideration of the cooperative relation between decision makers was presented. It was shown that all of the problems to be solved in the proposed interactive fuzzy programming can be solved through the combined use of the bisection method and the simplex method. An illustrative numerical example was provide to demonstrate the feasibility and efficiency of the proposed method. Extensions to other stochastic programming models will be considered elsewhere.