Production, Manufacturing and Logistics A reduced variable neighborhood search algorithm for uncapacitated multilevel lot-sizing problems Yiyong Xiao a , Ikou Kaku b , Qiuhong Zhao c,⇑ , Renqian Zhang c a School of Reliability and System Engineering, Beihang University, Beijing 100191, China b Department of Management Science and Engineering, Akita Prefectural University, Yulihonjo, Akita 015-0055, Japan c School of Economics and Management, Beihang University, Beijing 100191, China article info Article history: Received 29 July 2010 Accepted 19 April 2011 Available online 27 April 2011 Keywords: Meta-heuristics Uncapacitated multilevel lot-sizing (MLLS) problem Material requirement planning (MRP) Reduced variable neighborhood search (RVNS) algorithm Production planning abstract Multilevel lot-sizing (MLLS) problems, which involve complicated product structures with interdepen- dence among the items, play an important role in the material requirement planning (MRP) system of modern manufacturing/assembling lines. In this paper, we present a reduced variable neighborhood search (RVNS) algorithm and several implemental techniques for solving uncapacitated MLLS problems. Computational experiments are carried out on three classes of benchmark instances under different scales (small, medium, and large). Compared with the existing literature, RVNS shows good performance and robustness on a total of 176 tested instances. For the 96 small-sized instances, the RVNS algorithm can find 100% of the optimal solutions in less computational time; for the 40 medium-sized and the 40 large-sized instances, the RVNS algorithm is competitive against other methods, enjoying good effective- ness as well as high computational efficiency. In the calculations, RVNS updated 7 (17.5%) best known solutions for the medium-sized instances and 16 (40%) best known solutions for the large-sized instances. Ó 2011 Elsevier B.V. All rights reserved. 1. Introduction Material requirement planning (MRP) systems coordinate replenishment decisions for materials/components with complex product structures along with their bill of material. As a key oper- ating strategy in MRP systems, multilevel lot-sizing (MLLS) prob- lems concern how to determine lot sizes for producing/procuring multiple items at different levels with quantitative interdepen- dences so as to minimize the total costs of production/procure- ment setup and inventory holding occurring over a infinite planning horizon. The problem is of great importance for the effi- cient operation of modern manufacturing and assembly processes and has been widely studied both in practice and in academic fields over past half century. Optimal algorithms exist for the problem, including dynamic programming formulations (Zangwill, 1968, 1969), an assembly structure-based method (Crowston and Wagner, 1973), branch and bound algorithms (Afentakis et al., 1984; Afentakis and Gavish, 1986), etc. However, those methods can only solve small instances due to the NP-hard feature of the problem (Steinberg and Napier, 1980). Consequently, quite a few heuristic approaches have been developed. Early work consisted first of the sequential application of single-level lot-sizing models to each component of the product structure (Yelle, 1979; Veral and LaForge, 1985), and later, of the approximate application of multilevel lot-sizing models (Blackburn and Millen, 1982, 1985; Coleman and McKnew, 1991). Due to the rise in customer demand and fierce competition, we are faced nowadays with a challenging environment that involves both increasing product complexity and decreasing marginal prof- it. MLLS problems become more challenging with larger product size, greater product complexity, and growing requirements for accuracy of solution. In the last decade, meta-heuristic algorithms, including genetic algorithms (Dellaert and Jeunet, 2000; Dellaert et al., 2000; Homberger, 2008), simulated annealing (Tang, 2004; Jeu- net and Jonard, 2005; Raza and Akgunduz, 2008; Homberger, 2010), particle swarm optimization (Han et al., 2009), the soft opti- mization approach based on segmentation (Kaku and Xu, 2006; Kaku et al., 2010), ant colony optimization systems (Pitakaso et al., 2007; Homberger and Gehring, 2009), and recently, a vari- able neighborhood based algorithm (Xiao et al., 2011), have been developed to solve MLLS problems. Although satisfying solutions having been obtained by those methods in reasonable computing time, it is still a challenge to improve calculating efficiency and solution quality when solving modern MLLS problems with increasing complexity, especially in the area of developing a simple but robust algorithms to cover the variety of problem attributes. In this paper, we present a succinct approach — reduced vari- able neighborhood search (RVNS), to solve uncapacitated MLLS 0377-2217/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2011.04.015 ⇑ Corresponding author. Tel.: +86 10 82316181; fax: +86 10 82328037. E-mail addresses: xiaoyiyong@buaa.edu.cn (Y. Xiao), ikou_kaku@akita-pu.ac.jp (I. Kaku), qhzhao@buaa.edu.cn (Q. Zhao), zhangrenqian@buaa.edu.cn (R. Zhang). European Journal of Operational Research 214 (2011) 223–231 Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor