A generalized time-cost trade-off transportation problem

In the classical transportation probem if the unit costs and transportation durations are considered, the time-cost trade-off solutions can be determined by the well-known threshold approach assuming that all the transportations are permitted to be simultaneous in time. If all the unit costs are linear functions of time over a specified interval of time, a parametric technique can be applied for identifying all the time-cost trade-off solutions pertaining to this interval. In this paper, the unit costs considered are piecewise linear non-increasing functions of time and transportations are allowed to be simultaneous. It is shown that a parametric method involving a finite sequence of parametric transportation problems reveals all the time-cost trade-off solutions of this generalized trade-off problem. Computational experience is included. If the transportation problem has considerable degeneracy, the parametric approach may pose some computational difficulty. This difficulty can be reduced by using an alternative method involving the bicriteria optimization approach of Aneja and Nair. Also, a direct method is outlined for the case where a finite set of discrete alternatives of unit cost-time pairs is available.