Interactive compensatory fuzzy programming for decentralized multi-level linear programming (DMLLP) problems

This paper presents interactive compensatory fuzzy programming for decentralized multi-level linear programming (DMLLP) problems. By adjusting the cooperative decision making process between the different levels and also between the decision makers of the same level; our aim is to obtain a preferred compensatory compromise Pareto-optimal solution for DMLLP. For this, the weights of objectives at each level are assigned by the next upper level decision maker (DM) by using analytic hierarchy process (AHP) or any other weighting methods. The weight of any objective for whole system is equal to the product of the weights on the path tying it to the top decision maker DM0. Using these weights, equivalence is established such that the satisfactory levels of all objectives are proportional to their own weights. Werners’ compensatory “fuzzy and’’ operator is offered to solve DMLLP problem. The most important idea to be emphasized is that equivalence is established such that the satisfactory levels of all objectives are proportional to their own weights. Thanks to this equivalence, DMLLP problem has been transformed to the multi-objective linear programming (MOLP) problem at level 0, the equivalence is reflected to the compensatory model within the constraints, and the equivalence also enables all DMs to obtain proportional satisfactions with their weights as much as possible. So, in our compensatory model, a reduction on equivalent satisfactory level of one DM can be compensated for by an increase in the equivalent satisfactory level of another DM. Furthermore, being developed a finite interactive iterative procedure with maximum interaction step, a set of compensatory solutions which are also Pareto-optimal is obtained, depending on compensation parameter γ. Giving a theorem, we will show that the solutions generated by Werners’ compensatory “fuzzy and’’ operator do guarantee Pareto-optimality for our DMLLP problem. And comparing it with some other computational efficient compensatory fuzzy aggregation operators we will conclude that this operator is more appropriate for DMLLP. Illustrative numerical example is provided to demonstrate the feasibility and efficiency of the proposed interactive fuzzy compensatory method for DMLLP.