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

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