On characterizing solution for multi-objective fractional two-stage solid transportation problem under fuzzy environment

This article attempts to study cost minimizing multi-objective fractional solid transportation problem with fuzzy cost coefficients (C) over tilde (r)(ijk), fuzzy supply quantities ( a) over tilde (i), fuzzy demands (b) over tilde (j), and/or fuzzy conveyances (e) over tilde (k). The fuzzy efficient concept is introduced in which the crisp efficient solution is extended. A necessary and sufficient condition for the solution is established. Fuzzy geometric programming approach is applied to solve the crisp problem by defining membership function so as to obtain the optimal compromise solution of a multi-objective two-stage problem. A linear membership function for the objective function is defined. The stability set of the first kind is defined and determined. A numerical example is given for illustration and to check the validity of the proposed approach.