A Transportation Problem in Which Costs Depend on the Order of Arrival

This paper describes a problem in which the “cost” of satisfying the demand at a particular location is a weighted average of the travel times for the items that are supplied. The greatest weight is given to the first-arriving item, with decreasing weights given to each succeeding item. In the case of known demand, the problem is transformed into an equivalent transportation problem. In the case of stochastic demand, the problem is transformed into a transportation problem whose objective is to minimize the sum of a linear function and a convex function of the sum of the flows on selected arcs. This problem is linearized by substituting for the convex function the product of a parameter and a linear term. The parameterized problem is solved by parametric linear programming and by “updating the slope.”