Multi-dimensional transportation problems in multiple environments: a simulation based heuristic approach

Here, a general methodology is proposed to formulate and solve any multidimensional balanced/unbalanced, constrained/unconstrained transportation problems(TP) in different environments(crisp/fuzzy/rough). To understand the general model easily, here, at first, a multi-item 5-dimensional fixed charge profit maximization TP under budget and time constraint is presented. A potential solution of the problem is coded as a permutation of the different cells of the allocation matrix. A general decoding rule is proposed to determine the actual allocation from this coded solution. A heuristic approach is applied on a set of randomly generated coded solution of the target problem to determine the marketing decision. Applying swap operations on the coded solutions, the perturbation rules of the heuristic Particle Swarm Optimization(PSO) are modified to solve the problem. In a particular case, the problem is analysed as a bi-criteria decision making problem with the maximization of the total profit as well as the minimization of the total shipment time under a budget constraint. The bi-criteria TP is formulated as a single objective optimisation problem using a proposed rule and the same heuristic is run for a finite number of times to determine the pareto optimal front. To formulate the problem in the fuzzy(rough) environment an approach is proposed using credibility(trust) measure of fuzzy(rough) events. Proper fuzzy(rough) simulation algorithms are also proposed to solve the problem for any type of fuzzy(rough) estimation. Using this approach no crisp equivalent of any imprecise parameters is used for the marketing decision. Due the unavailability of the test data in the literature, different hypothetical data sets are used for the illustration of the models.