Comparative study on credibility measures of type-2 and type-1 fuzzy variables and their application to a multi-objective profit transportation problem via goal programming

In real world applications supply, demand and transportation costs per unit of the quantities in multi-objective transportation problems may be hardly specified accurately because of the changing economic and environmental conditions. It is also significant that the time required for transportation should be minimized. In this paper, we have presented three reduction methods for a type-2 triangular fuzzy variable (T2TrFV) by adopting the critical value (CV). Three generalized expected values (optimistic, CV and pessimistic) are derived for T2TrFVs with some special cases. Then a multi-objective profit transportation problem (MOPTP) with fixed charge (FC) cost has been formulated and solved in type-2 fuzzy environment. Unit transportation costs, FC, selling prices, unit transport times, loading and unloading times, total supply capacities and demands are all considered as triangular Type-2 fuzzy numbers. The MOPTP has been converted into a single objective by using the goal programming technique and the weighted sum method. The deterministic model is then solved using the Generalized Reduced Gradient method Lingo 14.0. Numerical experiments with some sensitivity analysis are illustrated the application and effectiveness of the proposed approaches.