A Deterministic Multi-Period Production Planning Model with Piecewise Concave Production and Holding-Backorder Costs

A single product, finite horizon production planning model with known requirements is considered. Production and holding-backorder cost functions are assumed to be piecewise concave, thereby allowing an arbitrarily close approximation to a wide range of cost functions which one might encounter in practice. In each period production, inventories and backlogged orders may not exceed prescribed levels. Production (inventory) breakpoints are the endpoints of the intervals over which the production (holding-backorder) cost functions are concave. It is shown that there is an optimal production schedule which has the property that between successive periods in which ending inventories are at inventory breakpoint levels there is at most one period in which production is not at a production breakpoint level. This property, which is an extension of recent results obtained by Florian and Klein [Florian, Michael, Morton Klein. 1971. Deterministic production planning with concave costs and capacity constraints. Management Sci. 18 (1, September) 12-20.] and Love [Love, Steven F. 1973. Bounded production and inventory models with piecewise concave costs. Management Sci. 20 (3, November) 313-318.], suggests a straight-forward dynamic programming algorithm for obtaining an optimal solution.