A fuzzy goal programming approach for solving the decentralized bi-level optimization problem with imprecise cooperation relations

This paper studies the decentralized bi-level multi-objective programming (DBLMOP) problem with one decision-maker (DM) at the higher level and more than one DM at the lower level. The number of objective functions to be optimized by each DM may be different. The fuzzy relation technique is first incorporated into fuzzy goal programming (FGP) to solve the hierarchical optimization problem. After characterizing the fuzzy goals of the objective functions and the higher-level DM´s decision vector by the corresponding membership functions, the concept of fuzzy binary relation is introduced to define the imprecise cooperation relations between the two DMs located at the different levels. An FGP model is formulated to reach the satisfactory values of all the fuzzy goals as close to the optimums as possible by minimizing their deviation variables and thereby finding a candidate solution. The proposed solution algorithm involving an interactive procedure with two orientations enables the higher-level DM to adjust the decision powers of the DMs through changing the cooperation relations, if the solution needs to be re-evaluated. A numerical example is presented to illustrate and demonstrate the proposed method.