A Polynomial Algorithm for the Multi-Stage Production-Capacitated Lot-Sizing Problem

The multi-stage lot-sizing problem with production capacities (MLSP-PC) deals with a supply chain that consists of a manufacturer with stationary production capacity and intermediaries (distribution centers or wholesalers) and a retailer to face deterministic demand. An optimal supply chain plan for the MLSP-PC specifies when and how many units each organization of the supply chain has to produce or transport to ultimately fulfill the demand at the retailer with the objective of minimizing total supply chain cost. All the production, transportation and inventory holding costs in each organization are assumed to be concave. The single-stage uncapacitated lot-sizing problem for a manufacturer was introduced by Wagner and Whitin [6] and the multi-stage version of the uncapacitated problem was solved by Zangwill [5]. To address the manufacturer’s production capacitated situation, Florian and Klein [1] solved the single-stage capacitated lot-sizing problem. Optimal algorithms for the multi-stage problem with production capacity were first presented by Kaminsky and Simchi-Levi for the two-stage case (2LSPPC) [2]. Van Hoesel et al. [3] generalized the 2LSP-PC to the multi-stage lot-sizing problem MLSP-PC. For the multi-stage dynamic lot-sizing problem with production capacities, Van Hoesel et al. [3] provide an O(LT 4 + T ) algorithm when no speculative motive exits in transportation (we call it the MLSP-PC with non-speculative cost structure) where L is the number of stages in the supply chain and T is the length of the planning horizon. For the most general MLSP-PC problem with concave cost structure,