Yet Another Note on “An Efficient Zero-One Formulation of the Multilevel Lot-Sizing Problem”

In “An Efficient Zero-One Formulation of the Multilevel Lot-Sizing Problem” [8] MCKNEW, SAYDAM, and COLEMAN claim the polynomial solvability of this particular production planning problem. Both, the proof given by MCKNEW et al. and a statement of its incorrectness by RAJAGOPALAN [10] contain errors, and the question remains whether the approach leads to a polyomial time algorithm. We show by means of an example that solving the linear program is not sufficient. Subject Areas: Materials Requirement Planning, Assembly Systems, Integer/Binary Program, and Computational Complexity INTRODUCTION In “An Efficient Zero-One Formulation of the Multilevel Lot-Sizing Problem” [8] MCKNEW, SAYDAM, and COLEMAN present a zero-one formulation for the solution of the following production planning problem: Given an assembly system of n items, instantaneous and uncapacitated production, and lead times of zero from one stage to the next, determine a production plan that minimizes the total setup and inventory holding cost over a planning horizon of T periods. The formulation presented in [8] is not flawless, however, has been corrected in [5]—for details check also the electronic source we provide at the end of this note. MCKNEW, SAYDAM, and COLEMAN assert that the polyhedron P defined by the given system of O n T 2 linear inequalities in O n T 2 variables is integral. RAJAGOPALAN [10] pointed out that the proof’s fundamental argument of total unimodularity of the corresponding coefficient matrix does not hold. Although his claim is true, since the manipulations of the coefficient matrix performed in [8] do not preserve total unimodularity, the given counterexample of RAJAGOPALAN contains an error. Numerical investigations of P using PORTA [3] helped us to identify valid counterexamples and reveal e.g. for a tiny example with n 2 and T 4 that about 10% of P’s vertices are fractional. Still, the possibility remained that—with respect to the objective function given in [8]—at