The transportation problem with objective of simultaneous minimization of total transportation cost and transportation time to each destination

For the Vector Minimization Problem (VMP) of simultaneous minimization of total transportation cost and transportation time to each destination two algorithms are proposed, which give a sequence of strictly non-dominated solutions (SNDS) by constructing suitable cost function involving penalty weights in the framework of a sequence of standard transportation problems and solving them sequentially. Whenever multiple optima are confronted in this procedure to the standard transportation problem, a second algorithm branching off from the main algorithm, tests the optimal feasible solution for dominance to the VMP, and if it is dominated, yields a non-dominated solution, which is a member of the sequence of SNDS. Apart from their reduced number as compared to the number of non-dominated solutions, the SNDS introduce a well-defined preference order in the solution as one goes up in the sequence. Thus, SNDS constitute a superior class of dominated solutions and render the task of choosing a preferred solution by the decision-maker less difficult. Several special cases in which the proposed algorithms can be applied are briefly described to highlight their usefulness to real-life problems.