Production Smoothing Under Piecewise Concave Costs, Capacity Constraints and Nondecreasing Requirements

The production smoothing problem with known demands that are assumed to increase with time is treated. The production and inventory costs are concave. The cost of increasing production from one period to the next is a concave function of the increase; similarly, the cost of decreasing production is a concave function of the decrease. Backlogging is not permitted. In each period, a fixed production capacity that does not vary with time is available. The problem it similar to that discussed by Zangwill, except for the important difference of capacity constraint. Feasible production plans are partitioned into sets on the basis of production differences from period to period---those with production increases in all N periods, those with a decrease only in the final period, etc. A minimum cost plan is an extreme point of one of these sets. We show that an extreme point plan is such that in between periods with zero inventory there is at most one sequence of periods when production is neither zero nor capacity and within these periods, production does not change. An algorithm is proposed for generating extreme point production plans. It is shown that finding the minimum cost production plan is equivalent to finding the shortest route through an acyclic network of extreme point production plans. The approach put forth enables a complete solution to the problem discussed.