Technical Note - Solving Linear Cost Dynamic Lot-Sizing Problems in O(n log n) Time

Ravindra K. Ahuja and Dorit S. Hochbaum Abstract In this paper, we study capacitated dynamic lot sizing problems with or without backorders, under the assumption that production costs are linear, that is, there are no set up costs. These two dynamic lot sizing problems (with or without backorders) are minimum cost flow problems on an underlying network that possess a special structure. We show how the well-known successive shortest path algorithm for the minimum cost flow problem can be used to solve these problems in O(n) time, where n is the number of periods in the dynamic lot sizing problems, and how with the use of dynamic trees we can solve these problems in O(n log n) time. Our algorithm also extends to the dynamic lot sizing problem with integer variables and convex production costs with running time O(n log n log U), where U is the largest demand value.