Fuzzy goal programming: Complementary slackness conditions and computational schemes

Goal programming (GP) is an important category in linear programming and multiobjective versions, which minimize the deviation among value of each objective function and its goal with satisfying other constraints of problem. In this paper, we deal with fuzzy goal programming (FGP), which is a generalized problem in vague nature, where parameters, relations and goals are given as fuzzy quantities with respect to imprecise data. Recently, Inuiguchi et al. [M. Inuiguchi, J. Ramik, T. Tanio, M. Vlach, Satisficing solutions and duality in interval and fuzzy linear programming, Fuzzy Sets and Systems, 135 (2003) 151-177.] have proved some important dual theorems on FGP, which are very close to the same crisp versions. In continuation to their works, based on fuzzy extensions of relations, we get complementary slackness conditions for FGP, which have an essential role in many optimization techniques. Moreover, we obtain the most optimistic and pessimistic satisficing solutions, employing these conditions and present a convex combination of them which covers the satisficing solutions for a fixed level of certainty. Our practical schemes implement an efficient interior point algorithm as sub-procedure, so they are polynomial-time algorithms.