Interactive fuzzy programming for multi-level programming problems: a review

Abstract

Decision-making problems in decentralised organisations are often modelled as Stackelberg games, and they are formulated as two-level mathematical programming problems with two decision-makers. If they do not have any motivation to cooperate mutually and behave rationally, outcomes of the problems can be explained by Stackelberg equilibrium which is not always Pareto optimal. From computational aspects, it is known that solving two-level programming problems is NP-hard even if the objective functions and the constraint functions are linear. In contrast, if the decision-makers can select strategies cooperatively, the most important aspect is to derive a Pareto optimal solution favourable to the decision-makers. As a method of this line of approach interactive fuzzy programming has been developed, taking into account fuzziness of human judgements. In this paper, after reviewing the development of solution methods for two- and multi-level programming problems, we focus on cooperative decision-making in decentralised organisations and give interactive fuzzy programming for two-level linear programming problems, which provides satisfactory solutions in accordance with the preference of the decision-makers. Moreover, we present extensions of interactive fuzzy programming for two-level linear programming problems under multi-objective environments and under uncertainty.