Multicommodity Production Planning: Qualitative Analysis and Applications

We develop a qualitative analysis theory for the convex-cost dynamic multicommodity production planning problem, which can be used, without performing any computational work, to provide invaluable insight to managers when faced with the task of deciding how to respond to changes in problem environment. We first formulate the problem as a multicommodity flow problem with parameters associated with each arc–commodity pair. We then reduce the problem to an equivalent single-commodity flow problem and develop a complete characterization of conformality among production, sales, and inventory activities in various instances of the problem. By combining the conformality characterizations with the monotonicity theory of Granot and Veinott [Granot F, Veinott AF Jr (1985) Substitutes, complements and ripples in network flow. Math. Oper. Res. 10:471–497] for single-commodity problems, we study the effects of changes in problem environment on optimal production, sales, and inventory schedules in the multicommodity problem. Numerous applications are presented and analyzed.