Fuzzy stochastic solid transportation problem using fuzzy goal programming approach

• A fixed charge fuzzy stochastic solid transportation problem (FCFSSTP) is considered.
• ‘⩾̃’ and ‘⩽̃’ fuzzy inequalities are used in constraints and appropriately defuzzified.
• New derandomization method is used for solid transportation problem with random goals.
• Fuzzy goal programming (FGP) approaches are proposed for FCFSSTP model’s solution.
• A real-life transportation problem with raw data is formulated as FCFSSTP and solved.

A fixed charge fuzzy stochastic solid transportation problem (FCFSSTP) is formulated with random budget and time constraints, random sources, demands and capacities of conveyances. Fuzzy stochastic constraints involving the symbols ‘⩾̃’ (approximately or fuzzily greater than or equal to) and ‘⩽̃’ (approximately or fuzzily less than or equal to) are used and appropriately transformed to deterministic ones. Fuzzy goal programming (FGP) approach is applied to solve the said FCFSSTP under several constraints. This paper also presents additive FGP models for the FCFSSTP. This method aggregates the membership functions of the stochastic constraints with the help of crisp and fuzzy weights based on importance of the objectives to construct the relevant decision function. From this general formulation, different particular models can be derived. As an example, one particular model with two fuzzy-stochastic constraints has been formulated. Moreover, as a particular case, three dimensional representation of an existing model is also presented. Transformed deterministic models are derived and solved by a gradient based non-linear optimization method-Generalized Reduced Gradient (GRG) technique. Two dimensional (with single conveyance) representation of a proposed FCFSSTP is derived and solved numerically. The optimum results of this model are compared with the solid transportation model. The suggested models and approaches are illustrated by a real-life practical problem.