برنامه ریزی خطی تصادفی دو سطحی تعاملی فازی بر اساس مجموعه سطح و بهینه سازی معیار فراکتال

In this paper, assuming cooperative behavior of the decision makers, two-level linear programming problems under fuzzy random environments are considered by introducing α-level sets of fuzzy random variables and defining an α-stochastic two-level linear programming problem for guaranteeing the degree of realization of the problem. Through a new decision making model proposed based on the fractile criterion optimization model considering fuzzy goals, the α-stochastic two-level programming problem can be equivalently transformed into a deterministic problem. 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 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. Such decision making situations can be formulated as a two-level programming problem [46]; one of the decision makers first makes a decision, and then the other who knows the decision of the opponent makes a decision. 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 [4], [28], [54] and [55]. 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 [46] and [47]. In order to deal with such cooperative two-level programming problems, Lai [17] and Shih et al. [53] proposed solution concepts 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. 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. To overcome such a 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 [49] and [50]. The subsequent works on two-level or multi-level programming have been appearing [1], [18], [29], [32], [39], [44], [45], [46], [47], [51], [52] and [56]. 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 [34], [35], [36], [37], [38], [60], [61], [64], [66] and [67] and randomness [5], [11], [42], [43], [57] and [58] would be required. Fuzzy random variables, first introduced by Kwakernaak [16], have been developing [15], [24] and [30], and an overview of the developments of fuzzy random variables was found in [9]. Studies on linear programming problems with fuzzy random variable coefficients, called fuzzy random linear programming problems, were initiated by Wang and Qiao [59], Qiao, Zhang and Wang [31] as seeking the probability distribution of the optimal solution and optimal value. Optimization models for fuzzy random linear programming problems were first considered by Luhandjula et al. [25] and [27] and further developed by Liu [22] and [23] and Rommelfanger [33]. A brief survey of major fuzzy stochastic programming models was found in the paper by Luhandjula [26]. As we look at recent developments in the fields of fuzzy random programming, we can see continuing advances [2], [3], [8], [10], [13], [14], [19], [20], [21], [33], [40], [41], [48], [62], [63] and [65]. Under these circumstances, in this paper, assuming cooperative behavior of the decision makers, we consider solution methods for two-level linear programming problems involving fuzzy random variables. By introducing α-level sets of fuzzy random variables, an α-stochastic two-level linear programming problem for guaranteeing the degree of realization of the problem is defined. Considering vague natures of the decision makers’ judgment, it is assumed that decision makers may have fuzzy goals for each of the objective functions in the α-stochastic two-level linear programming problem. Having elicited the membership functions which well represent the fuzzy goals of the decision makers at both levels, we propose a new decision making problem through the fractile criterion optimization model [12] together with the equivalent transformation of the α-stochastic two-level linear programming problem into the problem to maximize the satisfaction degree for each fuzzy goal. Following the proposed model, 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. An illustrative numerical example is provided to demonstrate the feasibility and efficiency of the proposed 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 the 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 fractile criterion optimization 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 and the simplex method. An illustrative numerical example demonstrated the feasibility and efficiency of the proposed method. Applications of the proposed method to real-world decision making problems in decentralized organizations together with extensions to other stochastic programming models will be required in the near future. Also extensions to fuzzy random two-level linear programming problems with two decision makers under noncooperative environments will be considered elsewhere.