Fuzzy approach for multi-level programming problems

Multi-level programming techniques are developed to solve decentralized planning problems with multiple decision makers in a hierarchical organization. These become more important for contemporary decentralized organizations where each unit or department seeks its own interests. Traditional approaches include vertex enumeration and transformation approaches. The former is in search of a compromise vertex based on adjusting the control variable(s) of the higher level and thus is rather inefficient. The latter transfers the lower-level programming problem to be the constraints of the higher level by its Karush-Kuhn-Tucker conditions or penalty function; the corresponding auxiliary problem become non-linear and the decision information is also implicit. In this study, the authors use the concepts of tolerance membership functions and multiple objective optimization to develop a fuzzy approach for solving the above problems. The upper-level decision maker defines his or her objective and decisions with possible tolerances which are described by membership functions of fuzzy set theory. This information then constrains the lower-level decision maker’s feasible space. A solution search relies on the change of membership functions instead of vertex enumeration and no higher order constraints are generated. Thus, the proposed approach will not increase the complexities of original problems and will usually solve a multi-level programming problem in a single iteration. To demonstrate our concept, we have solved numerical examples and compared their solutions with classical solutions.