Automatica 46 (2010) 1982–1993 Contents lists available at ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica Linear–quadratic optimal control strategy for periodic-review inventory systems ✩ Przemysław Ignaciuk ∗ , Andrzej Bartoszewicz Institute of Automatic Control, Technical University of Łódź, 18/22 Stefanowskiego St., 90-924 Łódź, Poland article info Article history: Received 7 December 2008 Received in revised form 28 July 2010 Accepted 2 August 2010 Available online 23 October 2010 Keywords: Optimal control Inventory control Riccati equations Discrete-time systems abstract The paper addresses the problem of efficient inventory management in production–inventory systems focusing on the dynamical nature of goods flow process. In the considered systems, the stock used to satisfy an unknown, time-varying demand is replenished either from a single or from multiple supply sources. The replenishment orders issued in each review period are realized with a delay, which differs among the suppliers and transport alternatives. For the analyzed setting, modeled as a discrete-time nth-order deterministic system, a new inventory policy is developed using a strict control-theoretic methodology. In contrast to the classical, stochastic approaches, the proposed control law is obtained by minimizing a quadratic cost functional, which guarantees the optimal dynamical performance of production–inventory systems with (possibly) different lead-time delays in the supply path. The designed policy ensures that the demand is always entirely satisfied from the on-hand stock (yielding zero lost-sales cost) and the warehouse capacity is not exceeded (which eliminates the risk of high-cost emergency storage). The closed-form solution of the linear–quadratic (LQ) optimization problem allows for a straightforward implementation of the developed control strategy in real systems. © 2010 Elsevier Ltd. All rights reserved. 1. Introduction A properly designed and implemented inventory policy has long been identified as a decisive factor behind the success of pro- duction and goods distribution systems (Zipkin, 2000). It is vital both for establishing sound foundations for a new company, or an- other stage in the production process, as well as for maintaining high effectiveness of the existing business entities and the overall supply chain performance (Silver, Pyke, & Peterson, 1998). The tra- ditional approaches to inventory control primarily concentrate on the statistical analysis of the long-term variables and (static) opti- mization performed on the averaged values of various cost compo- nents. However, the technological advances used to facilitate the flow of information in production–inventory systems (e.g. Inter- net ordering (Doukidis, Pramatari, & Lekakos, 2008; Teich, Wal- lenius, Wallenius, & Koopius, 2004), automatic warehouse review ✩ This work was financed by the Polish State budget in the years 2010–2012 as a research project N N514 108638 ‘‘Application of regulation theory methods to the control of logistic processes’’. P. Ignaciuk gratefully acknowledges financial support provided by the Foundation for Polish Science (FNP). The material in this paper was partially presented at 48th IEEE Conference on Decision and Control, December 16–18, 2009, Shanghai, China. This paper was recommended for publication in revised form by Associate Editor Suresh P. Sethi under the direction of Editor Berç Rüstem. ∗ Corresponding author. Tel.: +48 42 6312556; fax: +48 42 6312551. E-mail addresses: pignaciuk@poczta.onet.pl, przemyslaw.ignaciuk@p.lodz.pl (P. Ignaciuk), andrzej.bartoszewicz@p.lodz.pl (A. Bartoszewicz). (Giordano, Zhang, Naso, & Lewis, 2008), RFID enhancements in sup- ply chain (Tajima, 2007)) call for new solutions, which will not only provide good steady-state performance, but will also guaran- tee prompt response to the changing market conditions. Therefore, in order to answer the challenges of the highly competitive and rapidly varying environment, and in this way safeguard the com- pany prosperity in the 21st century, the traditional approaches to the design of inventory management schemes must be augmented by a solid analysis of the dynamical features of the system. The recognized need for studying the dynamical aspects of production–inventory systems has resulted in a number of research efforts employing the tools of control theory. A good, comprehensive summary of the initial proposals is given in Axsäter (1985), whereas more recent approaches are discussed in the review paper by Ortega and Lin (2004), in the special issue of Automatica (Sethi & Zhang, 2006), and in the extensive survey provided by Sarimveis, Patrinos, Tarantilis, and Kiranoudis (2008). One of the first attempts to apply the control theory to production–inventory systems were the works of Simon (1952) and Vassian (1955). Simon (1952) noticed an analogy between the servomechanism systems and the techniques involved in production planning and inventory ordering decisions, and used this observation to elaborate a continuous-time production rate controller for a single-commodity manufacturing system. Vassian (1955), in turn, extended the servomechanism concept to discrete- time inventory control. The early applications of the deterministic maximum principle in production, finance and marketing, were 0005-1098/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2010.09.010