NP-hard and polynomial cases for the single-item lot sizing problem with batch ordering under capacity reservation contract

• Published in: European journal of operational research 2017-03, Vol.257 (2), p.483-493
• Main Authors: Akbalik, Ayse, Hadj-Alouane, Atidel B., Sauer, Nathalie, Ghribi, Houcem
• Format: Article
• Language: English
• Subjects: Algorithms; Capacity reservation contract; Complexity; Computer Science; Dynamic programming; Inventory; Lot sizing; Operations Research; Polynomial time algorithm; Polynomials; Studies
• ISSN: 0377-2217; 1872-6860
• DOI: 10.1016/j.ejor.2016.07.028

•Single-item lot sizing problem is studied under a capacity reservation contract.•Batch deliveries are allowed and the overall replenishment cost is stepwise.•Four NP-hard cases are identified and an efficient FPTAS is proposed.•Pseudo-polynomial time dynamic programming algorithm is given for the general case.•Polynomial time algorithms are proposed under restricted parameters. In this paper, we study the single-item lot sizing problem under a capacity reservation contract. A manufacturer is replenished by an external supplier with batch deliveries and a certain capacity is reserved at the supplier level with an advantageous cost. In addition to the classical ordering and inventory holding costs, for each batch ordered under the reserved capacity a fixed cost per batch is incurred; and for batches exceeding this capacity a higher fixed cost per batch is paid, typically through the purchase from the spot market. We identify various NP-hard cases, propose a pseudo-polynomial time dynamic programming algorithm under arbitrary parameters, show that the problem admits an FPTAS and give polynomial time algorithms for special cases. We finally state a list of open problems for further research.