A Dual Algorithm for the Economic Lot-Sizing Problem

Abstract

In the economic lot-sizing problem we have to decide when and how much to produce of a given item so as to satisfy a known demand in each of n time periods and so as to minimize total cost consisting of production and inventory costs. The inventory costs are linear in the number of items in stock at the end of each time period. The production costs decompose into two parts; a fixed set-up cost is incurred whenever a non-zero production occurs in a period in addition to a cost linear in the number of items produced. This problem is well-know to be solved in O(n2) by dynamic programming. A linear programming formulation for the economic lot-sizing problem was given by Barany, Van Roy and Wolsey. This formulation contains a class of so called (S,m)-inequalities for each m;m=1,..,n and for each S, S ⊆ {1,2,..,m}. We present an O(n2) dual algorithm for this linear programming formulation. This dual algorithm provides an alternative proof of the fact that the linear programming formulation is a complete linear description for the economic lot-sizing problem. The dual algorithm was constructed to be able to perform sensitivity analysis. The result of which will be presented in another paper.