ORIGINAL ARTICLE Fixed-quantity dynamic lot sizing using simulated annealing Lotfi K. Gaafar & Ashraf O. Nassef & Ahmed I. Aly Received: 6 May 2007 / Accepted: 18 February 2008 / Published online: 31 May 2008 # Springer-Verlag London Limited 2008 Abstract In this paper, simulated annealing (SA) is applied to the deterministic dynamic lot-sizing problem with batch ordering and backorders. Batch ordering requires orders that are integer multiples of a fixed quantity that is larger than 1. The performance of the developed SA heuristic is compared to that of a genetic algorithm (GA) and a modified silver-meal (MSM) heuristic developed in the literature, based on the frequency of obtaining the optimum solution and the percentage average deviation from the optimum solution. In addition, the effects of three factors on the performance of the SA, GA, and the MSM are investigated in a 2 3 factorial experiment. The investigated factors are the demand pattern, the batch size, and the length of the planning horizon. Results indicate that the SA heuristic has the best performance, followed by GA, in terms of the frequency of obtaining the optimum solution and the average deviation from the optimum solution. SA is also the most robust of the investigated heuristics as its performance is only affected by the length of the planning horizon. Keywords Simulated annealing . Lot sizing . Batch ordering . Silver-meal . Genetic algorithm 1 Introduction This paper develops a simulated annealing (SA) heuristic for solving the dynamic lot-sizing problem under a batch ordering constraint that requires order quantities to be an integer multiple of a fixed quantity (Q, Q >1). In general, the dynamic lot-sizing problem seeks to determine the economic lot sizes that satisfy a series of deterministic time-varying demands over a discrete plan- ning horizon. The economic lot sizes are the ones that minimize the total cost of ordering and holding. A dynamic programming algorithm to obtain the optimum solution to this general dynamic lot-sizing problem was developed by Wagner and Whitin [1]. This paper adds the constraint that lot sizes must be multiples of a fixed quantity (Q). In the investigated problem, backorders are allowed in order to accommodate any discrepancies between the ordered quantities and the actual demand. Vander Eecken [2] extended the Wagner- Whitin (WW) algorithm to the case where orders must be integer multiples of a batch size (Q >1), with the restriction of no backorders. Elmaghraby and Bawle [3] extended Vander Eecken’ s work to the case where backorders are allowed, but with the restriction that the ordering cost is constant over time. Several authors provided extensions to Elmaghraby and Bawle’ s work [4], but the most general extension was developed by Li et al. [5] to the case where all cost elements are time varying with allowance for backordering. Even though the WW algorithm and its extensions provide the optimum solution for various versions of the lot-sizing problem, they are not widely used because of their perceived complexity by practitioners [6], who prefer other suboptimal heuristics that are simpler and that are more easily tailored for various extensions of the lot-sizing problem for which no optimization algorithm has yet been developed [7]. In addition, in a rolling-horizon situation, an optimization algorithm becomes a heuristic too, and is often surpassed by simple heuristics [8]. One of the most Int J Adv Manuf Technol (2009) 41:122–131 DOI 10.1007/s00170-008-1447-z L. K. Gaafar (*) : A. O. Nassef : A. I. Aly Mechanical Engineering, The American University in Cairo, 113 Kasr Elaini St., Cairo 11511, Egypt e-mail: gaafar@aucegypt.edu