A 2-Phase Method for Solving Transportation Problems with Prohibited Routes

Abstract

The Transportation Problem (TP) is a mathematical optimization technique which regulates the flow of items along routes by adopting an optimum guiding principle to the total shipping cost. However, instances including road hazards, traffic regulations, road construction and unexpected floods sometimes arise in transportation to ban shipments via certain routes. In formulating the TPs, potential prohibited routes are assigned a large penalty cost, M, to prevent their presence in the model solution. The arbitrary usage of the big M as a remedy for this interdiction does not go well with a good solution. In this paper, a two-phase method is proposed to solve a TP with prohibited routes. The first phase is formulated as an All-Pairs Least Cost Problem (APLCP) which assigns respectively a non-discretionary penalty cost M*ij <= M to each of n prohibited routes present using the Floyd¢s method. At phase two, the new penalty values are substituted into the original problem respectively and the resulting model is solved using the transportation algorithm. The results show that, setting this modified penalty cost ( M*) logically presents a good solution. Therefore, the discretionary usage of the M <= ∞ is not a guarantee for good model solutions. The modified cost M*<= M so attained in the sample model, is relatively less than the Big M ( <= ∞) and gives a good solution which makes the method reliable.