A Restricted Multi-Objective Solid Transportation Problem with Budget Constraint Involving Stochastic Variable and Interval Type-2 Fuzzy Number

This paper proposes a new concept in transportation problem in which if the negligible amount quantity is transported through a distribution center, then the decision maker (DM) cannot deliver that negligible amount quantity through the particular distribution center and the DM puts some restriction on transportation problem. Here, we develop six transportation models with budget at each destination, of which three models are without restriction and another three models are with restriction. Also, apart from source, demand and capacity constraints, an extra constraint on the total budget at each destination is also imposed. Here, all the parameters in Models 1 and 2 are crisp in nature and are solved in crisp environment, whereas all the parameters in Models 3–6 are interval type-2 fuzzy and random in nature, respectively, and are solved in uncertain environment. To reduce Models 3–6 into its crisp equivalent, we use the expected value of fuzzy number and chance constraint programming technique, respectively. Weighted sum method is also used to give the preference of the objective function and a gradient-based optimization technique-generalized reduced gradient (GRG) method are applied and using LINGO-13 software to get the optimal solutions. A numerical example is provided to illustrate the models and programming. Finally, a sensitivity analysis is presented for Models 1 and 2 with respect to the weight function.