A study on uncertain fixed charge bi-objective 4-dimensional transportation problems under budget constraints

Generally, in traditional transportation problems, single items are transported from a set of origins to a set of destinations in such a way that the transportation cost is minimized. To face the challenges that arise in modelling transportation problem various parameters are to be considered. In this work to provide an optimal transportation policy to the decision maker, we have studied a fixed-charge bi-objective multi-product  4-dimensional transportation problem with vehicle speed under budget constraints under uncertain environment. The objective of the proposed model is to have the maximum profit with the minimum time taken of transporting the goods. An equivalent deterministic model of the proposed model is obtained to deal the uncertain variable using expected value-chance constraint properties on uncertainty theory and its compromise solution is obtained by using the goal programming technique. A numerical example is discussed to provide more clarity on the proposed model.