Interactive fuzzy programming for decentralized two-level linear fractional programming (DTLLFP) problems

This paper presents two new interactive fuzzy programming approaches for a decentralized two-level linear fractional programming (DTLLFP) problem with a single decision maker (DM0) at the upper level and multiple DMs (DMi,i=1,…,k) at the lower level. In the first approach, DM0 specifies the minimal satisfactory level for own objective without considering the satisfactory levels of own decision variables and decreases it in favour of objectives at the lower level. Whereas, in the second approach, DM0 does not specify the minimal satisfactory level for own objective, but instead DM0 transfers the degree of satisfaction for not only own objective but also the own decision variables to the lower level. In both our approaches, with the help of analytic hierarchy process (AHP) method [Saaty TL. The analytical hierarchy process. New York: McGraw-Hill, 1980], DM0 assigns weights w1,w2,…,wk to objectives at the lower level. 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. To obtain an overall satisfactory balance between both levels, by updating the satisfactory degree of the DM0 which is in the first approach or the tolerances of the DM0's decision variables which is in the second approach, transformed main problems are constructed corresponding to DTLLFP. Maximizing the least degree of equivalent satisfaction among all DMs, they efficiently find a satisfactory or compromise solution from a Pareto optimal set for DTLLFP problem. If the DM0 is not satisfied with this solution, a strongly efficient satisfactory solution can be reached by interacting with him or her. An illustrative numerical example is provided to demonstrate the feasibility and efficiency of the proposed methods.