Fixed charge transportation problem with type-2 fuzzy variables

• Formulation of two fixed charge transportation problems with type-2 fuzzy parameters.
• Transportation Costs, supplies, demands, conveyance capacities are taken as type-2 fuzzy variables.
• CV-based reduction methods and centroid method are applied for defuzzification.
• Chance-constrained programming model is formulated using generalized credibility measure for the second problem.
• Sensitivity analysis is done for different generalized credibility levels.

This paper considers two fixed charge transportation problems with type-2 fuzzy parameters. Unit transportation costs, fixed costs in the first problem and unit transportation costs, fixed costs, supplies and demands in the second problem are type-2 fuzzy variables. For the first problem, to get corresponding defuzzified values of the type-2 fuzzy cost parameters, first critical value (CV)-based reduction methods are applied to reduce type-2 fuzzy variables into type-1 fuzzy variables and then centroid method is used for complete defuzzification. Besides this, we also apply geometric defuzzification method to the type-2 fuzzy cost parameters in the first problem to provide a comparison of the results. Coming to the second problem, a chance-constrained programming model is formulated using generalized credibility measure for the objective function as well as the constraints with the CV-based reductions of corresponding type-2 fuzzy parameters. Next, the reduced model is turned into equivalent parametric programming problem. The deterministic problems so obtained are then solved by using the standard optimization solver – LINGO. We have provided numerical examples illustrating the proposed models and techniques. Some sensitivity analyzes for the second model are also presented.