Implement an uncertain vector approach to solve entropy-based four-dimensional transportation problems with discounted costs

In this research paper, using uncertainty theory we introduced and developed entropy-based uncertain four-dimensional transportation problem with fixed charges, discounted costs, and vehicle costs. In this transportation system, we considered a discount policy on the transportation cost which depends on the basis of the transported amount. Here, the discounted costs are in the form of all unit discounts (AlUD), incremental quantity discounts (InQD), and the combination of these two. The main objective is to minimize the total transportation cost via maximum entropy which ensures the number of items to be transported from some source to some destinations by some conveyances through some routes. For optimizing the proposed model, using uncertain programming techniques, we have developed two different models such as expected value programming model and expected constrained programming model. Then, Using minimizing distance method and linear weighted method we formulated and solved the equivalent deterministic transformation of these two constructed models. Finally, to show the application of the proposed models and methods we presented a numerical example with optimal results.