Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 325274, 9 pages http://dx.doi.org/10.1155/2013/325274 Research Article LQ Optimal Sliding Mode Control of Periodic Review Perishable Inventories with Transportation Losses Piotr LeVniewski and Andrzej Bartoszewicz Institute of Automatic Control, Technical University of Lodz, 18/22 Bohdana Stefanowskiego Street, 90-924 Lodz, Poland Correspondence should be addressed to Andrzej Bartoszewicz; andrzej.bartoszewicz@p.lodz.pl Received 30 July 2013; Accepted 16 September 2013 Academic Editor: Xudong Zhao Copyright © 2013 P. Le´ sniewski and A. Bartoszewicz. his is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this work we apply the control-theoretic approach to design a new replenishment strategy for inventory systems with perishable stock. Such systems are supposed to efectively satisfy an unknown and permanently time-varying consumers’ demand. he main obstacle of achieving this goal is the need of obtaining supplies from a distant source. During the supply process goods are inevitably lost due to various causes. Furthermore, those goods which successfully arrive at the distribution center still deteriorate while stored in its warehouse. We explicitly take into account both of these factors in designing our control strategy. We propose a sliding mode strategy and choose its parameters to minimize a quadratic quality criterion. his approach allows us to ameliorate the bullwhip efect (the ampliication of the demand variations when going up in the supply chain). he control strategy proposed in this work ensures bounded orders, guarantees full consumers’ demand satisfaction, and eliminates the risk of exceeding the warehouse capacity. hese properties are stated in three theorems and proved in the paper. 1. Introduction In a competitive economy, the problem of eicient manage- ment of supply chains has become increasingly important. herefore, recently many attempts to solve the problem have been proposed. hese attempts difer in various aspects including supply chain modeling, speciic performance mea- sures, primary objectives, and methods applied to accomplish the objectives [1]. A good overview of the results obtained in this area can be found in [2–7] and numerous references cited in those papers. he earliest application of the control theory techniques to the management of logistic processes was reported about sixty years ago when Simon (H. A. Simon for his contribution to the ield received the Nobel prize in economics in 1978) in paper [8] applied servomechanism control algorithm to get a feasible strategy of goods replen- ishment for continuous-time, single product inventory. Soon ater that a discrete time servomechanism control algorithm for managing of goods replenishment process has also been proposed [9]. hen, block diagram representation of con- ventional inventories and production management systems was introduced by Towill [10]. One of the most signiicant developments in this area was the result of Forrester [11, 12], who analyzed the ampliication of demand luctuations when moving upstream in the supply chain. he phenomenon has later been termed bullwhip efect. Important contribution to the study of this efect has been presented in [13–19]. he authors of those papers were able to smooth ordering policies and stock levels and have demonstrated that appli- cation of control theory methods helps efectively prevent the occurrence of undesirable bullwhip efect. Over the last twenty years, numerous valuable solutions in this ield have been presented. herefore, in this section we are only able to mention a few of them. In [16, 20], an autoregressive moving average (ARMA) system structure has been applied in order to model uncertain demand. Furthermore, model predictive control of supply chains has been widely used (see, e.g., papers [7, 21–24]), and in [25] a robust controller for the continuous-time system with uncertain processing time and delay has been designed by minimizing H-ininity norm. Also estimation techniques have been used for the purpose of inventory management. he recursive least squares method and Kalman ilter were applied in [21, 26] for delay identii- cation and in [16, 20] for demand prediction, respectively. In a number of papers [27–30] the classical and modern tools of time-delay systems theory were applied to the problem of