Three-phase time minimization transportation problem

The present article discusses a special variant of the multiphase transportation problem dealing with the transportation of a commodity from various origins to terminals when the set of origin-terminal links is divided into three disjoint partitions, namely Phase-1 links, Phase-2 links and Phase-3 links. Transportation of the commodity via Phase-2 links starts only after the completion of transportation in Phase-1, and similarly transportation in Phase-3 commences only after transportation in Phase-2 has been completed. The aim is to find a solution that minimizes the total time of transportation of the commodity subject to the constraints pertaining to origin availabilities and terminal requirements. The overall availability of the commodity at various origins is equal to the total demand at various terminals. An iterative polynomially bounded algorithm is proposed for obtaining an optimal feasible solution of the three-phase problem. Theoretical justification and numerical illustrations are provided to support the algorithm.