A deterministic multi-period production scheduling model with backlogging

A deterministic multi-period production and inventory model that has concave production costs and piecewise concave [6] 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 [2], Wagner and Whitin [5], Wagner [4], and Veinott [3]. 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.