برنامه ریزی خطی تصادفی تعاملی دوسطحی فازی از طریق بهینه سازی معیار فراکتال

In this paper, assuming cooperative behavior of the decision makers, solution methods for decision making problems in hierarchical organizations under fuzzy random environments are considered. To deal with the formulated two-level linear programming problems involving fuzzy random variables, αα-level sets of fuzzy random variables are introduced and an αα-stochastic two-level linear programming problem is defined for guaranteeing the degree of realization of the problem. Taking into account the vagueness of judgments of decision makers, fuzzy goals are introduced and the αα-stochastic two-level linear programming problem is transformed into the problem to maximize the satisfaction degree for each fuzzy goal. Through the use of the fractile criterion optimization 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, the sequential quadratic programming or the combined use of the bisection method and the sequential quadratic programming. An illustrative numerical example is provided to demonstrate the feasibility and efficiency of the proposed method.

Fuzzy random variables, first introduced by Kwakernaak [1], have been developing in various ways [2], [3] and [4]. An overview of the developments of fuzzy random variables was found in the article of Gil et al. [5]. Studies on linear programming problems with fuzzy random variable coefficients, called fuzzy random linear programming problems, were initiated by Wang and Qiao [6], Qaio et al. [7] 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. [8] and [9] and further developed by Liu [10] and [11] and Rommelfanger [12]. A brief survey of major fuzzy stochastic programming models was found in the paper by Luhandjula [13]. As we look at recent developments in the fields of fuzzy random programming, we can see continuing advances [14], [15], [16], [17], [18], [19], [12], [20], [21] and [22]. However, decision making problems in hierarchical managerial or public organizations are often formulated as two-level mathematical programming problems [23]. 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 [24], [25], [26], [27], [28], [29], [30] and [31]. Compared with this, for decision making problems, for example, in decentralized large firms with divisional independence, it is quite natural to suppose that there exist communication and some cooperative relationship among the decision makers [23]. Lai [32] and Shih et al. [33] 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. 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 [34] and [35]. Subsequent works on two-level or multi-level programming have appeared [36], [37], [38], [39], [40], [41], [42], [43], [44] and [23]. 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, αα-level sets of fuzzy random variables are introduced and an αα-stochastic two-level linear programming problem is defined for guaranteeing the degree of realization of the problem. Taking into account the vagueness of judgments of decision makers, fuzzy goals are introduced [45] and the αα-stochastic two-level linear programming problem is transformed into the problem to maximize the satisfaction degree for each fuzzy goal. Following the fractile criterion optimization model [46], 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, the sequential quadratic programming or the combined use of the bisection method and the sequential quadratic programming. 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, αα-level sets of fuzzy random variables were introduced and an αα-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 αα-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, the sequential quadratic programming or the combined use of the bisection method and the sequential quadratic programming. An illustrative numerical example demonstrated the feasibility and efficiency of the proposed method. As future works, the proposed method may be extended by incorporating the sensitivity analysis, which helps a decision maker to find easily the proper values of parameters such as αα, θ1θ1 and θ2θ2. It would be interesting to consider extensions to three-level [54] and [55] or multi-level programming [56] under cooperative environments. Extensions to other stochastic programming models such as expectation optimization and variance minimization, as well as comparison of the proposed models with those models, will be discussed elsewhere. Also extensions to fuzzy random two-level linear programming problems with two decision makers under noncooperative environments will be required in the near future.