تعاملی برنامه ریزی خطی فازی تصادفی دوسطحی با مراجعه ساده

In this paper, assuming cooperative behavior of the decision makers, two-level linear programming problems involving random variables in constraints are considered. Using the concept of simple recourse, the formulated stochastic two-level simple recourse problems are transformed into deterministic two-level programming ones. Taking into account vagueness of judgments of the decision makers, interactive fuzzy programming is presented. In the proposed interactive method, after determining the fuzzy goals of the decision makers at both levels, a satisfactory solution is derived efficiently by updating the satisfactory degree of the decision maker at the upper level with considerations of overall satisfactory balance between both levels. An illustrative numerical example is provided to demonstrate 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 and fuzzy-theoretic one. Stochastic programming, as an optimization method based on the probability theory, have been developed in various ways [47], [48], [6] and [27], including two-stage programming [2], [3], [9], [10], [12], [20], [49] and [50] and chance constrained programming [7], [8], [27] and [47]. Fuzzy mathematical programming representing the vagueness in decision making situations by fuzzy concepts have been studied by many researchers [22] and [23]. Fuzzy multiobjective linear programming, first proposed by Zimmermann [51], have been also developed by numerous researchers, and an increasing number of successful applications has been appearing [14], [18], [23], [24], [25], [40], [45], [46] and [52]. In particular, after reformulating stochastic multiobjective linear programming problems using several models for chance constrained programming, Sakawa et al. [26], [28] and [29] presented an interactive fuzzy satisficing method to derive a satisficing solution for the decision maker (DM) as a generalization of their previous results [23], [37], [38], [39] and [40]. However, decision making problems in decentralized organizations are often formulated as two-level programming problems with a DM at the upper level (DM1) and another DM at the lower level (DM2) [32]. Under the assumption that these DMs do not have motivation to cooperate mutually, the Stackelberg solution [5], [19], [42] and [43] is adopted as a reasonable solution for the situation. On the other hand, in the case of a project selection problem in the administrative office of a company and its autonomous divisions, the situation that these DMs can cooperate with each other seems to be natural rather than the noncooperative situation. Assuming that the DMs essentially cooperate with each other, Lai [13] and Shih et al. [41] proposed solution concepts for two-level linear programming problems. In their methods, the DMs identify membership functions of the fuzzy goals for their objective functions, and in particular, the DM at the upper level also specifies those of the fuzzy goals for the decision variables. The DM at the lower level solves a fuzzy programming problem with a constraint with respect to a satisfactory degree of the DM 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 the DMs [33] and [34]. The subsequent works on two-level or multi-level programming under fuzziness have been developed [1], [15], [21], [30], [31], [35], [36] and [44]. Realizing the importance of considering not only the fuzziness but also the randomness of coefficients of objective functions or constraints in mathematical programming, some researchers developed two-stage or multi-stage fuzzy stochastic programming [11], [16] and [17]. However, there is no study which focuses on the simultaneous consideration of two-level decision making situations and fuzzy stochastic programming approaches. Under these circumstances, we propose a novel fuzzy stochastic two-level programming model which incorporates interactive two-level fuzzy programming into two-stage stochastic programming. In two-level programming under a cooperative relationship between the two DMs, the upper-level DM needs to select a solution that takes a balance between his/her own objective function value and the lower-level DM’s objective function value. In addition, it is significant to properly represent the imprecision of the satisfaction of DMs with respect to the goals of objective function values. From these viewpoints, in the proposed interactive method, after determining the fuzzy goals of the DMs at both levels, a satisfactory solution is derived efficiently by updating the satisfactory degree of the DM at the upper level with considerations of overall satisfactory balance between the both level DMs. The proposed method has an advantage that the problem for deriving a satisfactory solution can be strictly solved by some convex programming techniques like the sequential quadratic programming method. A numerical example of two-level production planning problems is provided to illustrate the feasibility and efficiency of the proposed method.

In this paper, we focused on two-level linear programming problems with random variables in constraints. Through the use of the simple recourse model, the formulated two-level simple recourse problems are transformed into deterministic two-level programming ones. Taking into account vagueness of judgments of the DMs, interactive fuzzy programming has been proposed. In the proposed interactive method, after determining the fuzzy goals of the DMs at both levels, a satisfactory solution is derived efficiently by updating the satisfactory degree of the DM at the upper level with considerations of overall satisfactory balance between both levels. It is significant to note here that the transformed deterministic problems to derive a satisfactory solution can be easily solved through some convex programming technique like the sequential quadratic programming method. An illustrative numerical example was provided to demonstrate the feasibility and efficiency of the proposed method. As a future work, the proposed method may be extended by incorporating the sensitivity analysis. Extensions to other stochastic programming models will be considered elsewhere. Also extensions to two-level integer programming problems or multi-level linear programming problems involving random variables will be required in the near future.