تعاونی برنامه ریزی خطی تعاملی فازی دوسطحی تصادفی از طریق حداکثرسازی احتمال وقوع مبتنی بر مجموعه های سطحی

In this paper, assuming cooperative behavior of the decision makers, two-level linear programming problems under fuzzy random environments are considered. To deal with the formulated fuzzy random two-level linear programming problems, α-level sets of fuzzy random variables are introduced and an α-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 α-stochastic two-level linear programming problem is transformed into the problem to maximize the satisfaction degree for each fuzzy goal. Through 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 is provided to demonstrate the feasibility and efficiency of the proposed method.

In the real world, 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, Nishizaki and Sakawa, 2000, 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). Assuming 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, Lai, and Lee (1996) proposed solution concepts for two-level linear programming problems. 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., 1998 and Sakawa et al., 2000 have proposed interactive fuzzy programming for two-level or multi-level linear programming problems to obtain a satisfactory solution for decision makers. Extensions to two-level linear fractional programming problems (Sakawa, Nishizaki, & Uemura, 2001), decentralized two-level linear programming problems (Sakawa and Nishizaki, 2002 and Sakawa et al., 2002), two-level linear fractional programming problems with fuzzy parameters (Sakawa et al., 2000), and two-level nonconvex programming problems with fuzzy parameters (Sakawa & Nishizaki, 2002) were provided. Further extensions to two-level linear programming problems with random variables, called stochastic two-level linear programming problems (Sakawa & Katagiri, 2010) and two-level integer programming problems (Sakawa, Katagiri, & Matsui, 2012) have also been considered. 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. However, to utilize two-level programming for resolution of conflict in decision making problems in real-world decentralized organizations, it is important to realize that simultaneous considerations of both fuzziness (Sakawa, 1993, Sakawa, 2000 and Sakawa, 2001) and randomness (Birge and Louveaux, 1997, Sakawa and Kato, 2008 and Stancu-Minasian, 1984) would be required. Fuzzy random variables, first introduced by Kwakernaak (1978), have been developing (Kruse and Meyer, 1987, Liu and Liu, 2003 and Puri and Ralescu, 1986), and an overview of the developments of fuzzy random variables was found in Gil, Lopez-Diaz, and Ralescu (2006). 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, Zhang, and Wang (1994) as seeking the probability distribution of the optimal solution and optimal value. Optimization models for fuzzy random linear programming problems were first developed by Luhandjula (1996) and Luhandjula and Gupta (1996) and further developed by Liu, 2001a and Liu, 2001b and Rommelfanger (2007). A brief survey of major fuzzy stochastic programming models including fuzzy random programming was found in the paper by Luhandjula (2006). 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, α-level sets of fuzzy random variables are introduced and an α-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 α-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, α-level sets of fuzzy random variables were introduced and an α-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 α-stochastic two-level linear programming problem was transformed into the problem to maximize the satisfaction degree for each fuzzy goal. Through the probability 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 should be emphasized here that all 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. An illustrative numerical example demonstrated the feasibility and efficiency of the proposed method. However, as a subject of future work, applications of the proposed method to the real world decision making situations will be required in the near future. Extensions to other stochastic programming models will be considered elsewhere. Considerations from the view point of fuzzy random two-level linear programming problems with two decision makers under noncooperative environments will be reported elsewhere.