A novel reduction method for type-2 uncertain normal critical values and its applications on 4D profit transportation problem involving damageable and substitute items

In this paper, we developed and investigated some four-dimensional profit maximization transportation problems considering damageablity and substitutability, where the parameters are of a type-2 normal uncertain variable in nature. Here, unit transportation cost, distances of the paths, fixed charges, rate of breakability, availability, demands, capacities of conveyances, the unit selling price, unit purchasing cost, unit procurement cost are regarded as type-2 normal uncertain variables. Two models are formulated based on two separate criteria of the substitution items. We use a critical value-based reduction method to reduce type-2 normal uncertain variables into type-1 normal uncertain variables and then apply some properties of uncertainty theory to convert the profit transportation parameters into deterministic form. The two deterministic models are solved using the optimization software named LINGO -18.0. A real-life numerical example and optimal results are presented here to show the application of the proposed model and the proposed reduction method.