A polynomial algorithm for a two-stage time minimizing transportation problem

The present paper studies a two-stage minimizing transportation problem in which the total availability of a homogeneous product at the sources is more than the minimum requirement of the same at the destinations. It is different from the conventional imbalanced time minimizing transportation problem in the sense that just enough of the product is sent from the sources to the destinations in the Stage-I so as to satisfy the minimum requirement of each destination and on completion of the Stage-I, the surplus quantity (if any) at the sources is transported to some of the destinations in Stage-II. In both the stages, transportation of the product from the sources is done in parallel. Aim is to minimize the sum of the transportation times for Stage-I and Stage-II. A polynomial bound algorithm, involving scanning of lexicographic optimal solutions of a standard time minimizing transportation problem and its restricted versions, is developed to obtain optimal schedules for Stage-I and Stage-II such that the sum of the corresponding transportation times is the least.