Decision making problems in decentralized organizations are often formulated as two-level programming problems with a decision maker (DM) at the upper level (DM1) and another DM at the lower level (DM2). Under the assumption that these DMs do not have motivation to cooperate mutually, the Stackelberg solution [1- 4] is adopted as a reasonable solution for the situation. However, the above assumption is not always reasonable when we deal with decision making problems in a decentralized firm where top management is DM1 and an operation division of the firm is DM2 because it is supposed that there exists cooperative relationship between them.
Namely, top management or an executive board is interested in overall management policy such as long-term corporate growth or market share. In contrast, operation divisions of the firm are concerned with coordination of daily activities. After the top management chooses a strategy in accordance with the overall management policy, each division determines goals which are relevant to the strategy chosen by the top management, and it tries to achieve them. In this way, decision making problems in a decentralized firm are often formulated as two-level programming problems where there is essentially cooperative relationship between DM1 and DM2.
Lai  and Shih et al.  proposed solution concepts for two-level linear programming problems or multi-level ones such that decisions of DMs in all levels are sequential and all of the DMs essentially cooperate with each other. 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 DMs [7, 8]. The subsequent works on two-level or multi-level programming have been developing [9-18].
In actual decision making situations, however, 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 developing in various ways [19, 20], including two stage problems considered by Dantzig  and chance constrained programming proposed by Charnes et al. . Especially, for multiobjective stochastic linear programming problems, Stancu-Minasian  considered the minimum risk approach, while Leclercq  and Teghem Jr. et al.  proposed interactive methods.
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(Author: Masatoshi Sakawa, Takeshi Matsui