Abstract
Abstract This article attempts to study cost minimizing multi-objective fractional solid transportation problem with fuzzy cost coefficients c ˜ i j k r {\tilde{c}}_{ijk}^{r} , fuzzy supply quantities a ˜ i {\tilde{a}}_{i} , fuzzy demands b ˜ j {\tilde{b}}_{j} , and/or fuzzy conveyances e ˜ k {\tilde{e}}_{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.
Highlights
Solid transportation problem (STP) is a generalization of the well-known classical transportation problem (TP), where three item properties are taken into account in the constraint set of the STP instead of two constraints
Vejda [5] developed an algorithm for a multi-index TP, which is the extension of the distribution modification method
The zero-point method for finding the optimal solution of TP was introduced by Pandian and Natarajan [6]
Summary
Solid transportation problem (STP) is a generalization of the well-known classical transportation problem (TP), where three item properties are taken into account in the constraint set of the STP (namely, supply, demand, and mode of transportation or conveyance) instead of two constraints (source and destination). Bit et al [4] applied fuzzy programming approach to solve the multi-objective STP with real-life applications. Singh et al [20] formulated a general model of the multiobjective STP with some random parameters and they proposed a solution method by using the chanceconstraint programming technique to solve the model of multi-objective STP. Kumar et al [21] proposed a new computing procedure for solving fuzzy Pythagorean TP, where they extended the interval basic feasible solution, existing optimality method to obtain the cost of transportation. Stanojevic and Stanojevic [26] applied the efficiency test introduced by Lotfi et al (2010) to the proposed two procedures for deriving weakly and strongly efficient solutions in multi-objective LFP problems.
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