A new characterization for the dynamic lot size problem with bounded inventory

Abstract

In this paper, we address the dynamic lot size problem with storage capacity. As in the unconstrained dynamic lot size problem, this problem admits a reduction of the state space. New properties to obtain optimal policies are introduced. Based on these properties a new dynamic programming algorithm is devised. Superiority of the new algorithm to the existing procedure is demonstrated. Furthermore, the new algorithm runs in O( T) expected time when demands vary between zero and the storage capacity. Computational results are reported for randomly generated problems. Scope and purpose Dynamic lot size problems describe a relevant class of production/inventory systems which are often met in practice. The goal consists of finding the production/order plan satisfying the demands over a given number of periods at minimum cost. When the lot sizes to be produced are restricted by bounds, the problem is called capacitated. On the other hand, when the inventory levels are bounded variables, this problem is usually known as bounded/ limited inventory model. Although these two latter problems are mathematically related, the principles which characterize the optimal plans in both cases are different. The capacitated version of the dynamic lot size problem has been studied in detail by many authors. These authors have considered distinct assumptions on the cost functions and the boundaries of the production quantities. In contrast, the bounded inventory model can be found in few references in the literature. In this paper, we introduce new properties to determine optimal plans of the dynamic lot size problem with storage capacity. As we will show, this new approach is conceptually more understandable than the one proposed previously by Love (Management Sci. 20(3) (1973) 313). Moreover, the computational results indicate that the algorithm introduced in this paper is almost 30 times faster than Love's procedure. Besides, it can be proved that the algorithm runs in linear expected time when demands take values in the interval [0, S].