Lot-size models with backlogging: Strong reformulations and cutting planes

Abstract

We examine mixed integer programming reformulations of the uncapacitated lot-sizing problem with backlogging. First we consider the effect of using a standard reformulation technique for fixed charge network flow problems which involves the introduction of new variables, leading to a known plant location reformulation and a shortest path reformulation. Each of these reformulations is strong in the sense that its linear programming relaxation solves the lot-sizing problem. Secondly we attempt to treat the problem in the space of the original variables. We give an implicit description of the convex hull of solutions, and show how the problem of finding a violated cutting plane can be solved as a linear program. We also describe a family of strong valid inequalities which can be generated rapidly by a heuristic and which have proved effective in a cut generation algorithm. The efficiency of both the shortest path formulation and the cutting plane algorithm have been tested on a series of multi-item capacitated lot-sizing problems with backlogging. Near optimal solutions have been found to problems with 8 periods and up to 100 times.