Computers & Operations Research 34 (2007) 934 – 937 www.elsevier.com/locate/cor A polynomial algorithm for the production/ordering planning problem with limited storage J. Gutiérrez ∗ , A. Sedeño-Noda, M. Colebrook, J. Sicilia Dpto. de Estadística, I.O. y Computación, Universidad de La Laguna, Tenerife, Islas Canarias, Spain Available online 11 July 2005 Abstract This paper concerns the dynamic lot size problem where the storage capacity is limited and shortages are allowed. The planning horizon is divided into T periods and, for each period, concave functions to define the holding/stockout and production costs are considered. It is proved that the results derived in a previous work for the dynamic lot size problem assuming time-varying storage capacities remain valid for the case with backlogging.  2005 Elsevier Ltd. All rights reserved. Keywords: Inventory-production: Policies; Capacity; Dynamic programming 1. Introduction We address the production planning problem assuming time-varying capacities and allowing stock- outs. The holding/stockout and production costs are defined by concave functions. The planning horizon is divided into T periods and the demand for each period is known in advance. The goal consists of determining a production plan which satisfies the demand for each period at minimum cost. This problem was first studied by Love [1], who developed an O(T 3 ) algorithm based on the dynamic programming approach. Recently, Gutiérrez et al. [2] proposed a new characterization of the optimal plans when stockouts are not allowed. Such a characterization yields a O(T 3 ) dynamic programming algorithm that runs on average 27% faster than Love’s procedure. Furthermore, the procedure computes an optimal solution in O(T ) expected time when the demand in each period varies between zero and the storage capacity for such period. ∗ Corresponding author. Tel.: +34 922 31 91 89; fax: +34 922 31 92 02. E-mail address: jmgrrez@ull.es (J. Gutiérrez). 0305-0548/$ - see front matter  2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.cor.2005.05.029