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 setup 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(n2) 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.
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