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Basis Paths and a Polynomial Algorithm for the Multistage Production-Capacitated Lot-Sizing Problem

We consider the multilevel lot-sizing problem with production capacities (MLSP-PC), in which production and transportation decisions are made for a serial supply chain with capacitated production and concave cost functions. Existing approaches to the multistage version of this problem are limited to nonspeculative cost functions---up to now, no algorithm for the multistage version of this model with general concave cost functions has been developed. In this paper, we develop the first polynomial algorithm for the MLSP-PC with general concave costs at all of the stages, and we introduce a novel approach to overcome the limitations of previous approaches. In contrast to traditional approaches to lot-sizing problems, in which the problem is decomposed by time periods and is analyzed unidirectionally in time, we solve the problem by introducing the concept of a basis path, which is characterized by time and stage. Our dynamic programming algorithm proceeds both forward and backward in time along this basis path, enabling us to solve the problem in polynomial time.