A Deterministic Multi-Period Production Scheduling Model with Backlogging

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Abstract
A deterministic multi-period production and inventory model that has concave production costs and piecewise concave [Zangwill, W. I. 1965. A dynamic multi-product, multi-facility production and inventory model. Technical Report 1, Program in Operations Research, Stanford University, Stanford, California, April 29.] inventory costs is analyzed. An essential feature of the model is that it permits backlogging of unsatisfied demand; otherwise the model is similar to the one considered by Manne [Manne, A. S. 1958. Programming of economic lot sizes. Management Sci. 4(2, Jan.) 115–135.], Wagner and Whitin [Wagner, H. M., T. M. Whitin. 1959. Dynamic version of the economic lot size model. Management Sci. 5(1, October) 89–96.], Wagner [Wagner, H. M. 1960. A postscript to “dynamic problems to the firm.” Naval Res. Logist. Quart. 7(March) 7–12.], and Veinott [Veinott, A. F., Jr. 1963. Unpublished notes. Stanford University, Stanford, California.]. By permitting backlogging, which mathematically means that inventories can be negative, the concavity assumptions of these authors are no longer appropriate. Instead we consider piecewise concave cost functions to find the form of the minimum cost production schedule. An efficient dynamic programming algorithm to calculate the minimum cost schedule is also presented.

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