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Mathematical Problems in Engineering
Volume 2013, Article ID 325274, 9 pages
Research Article

LQ Optimal Sliding Mode Control of Periodic Review Perishable Inventories with Transportation Losses

Institute of Automatic Control, Technical University of Lodz, 18/22 Bohdana Stefanowskiego Street, 90-924 Lodz, Poland

Received 30 July 2013; Accepted 16 September 2013

Academic Editor: Xudong Zhao

Copyright © 2013 Piotr Leśniewski and Andrzej Bartoszewicz. This 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 effectively satisfy an unknown and permanently time-varying consumers’ demand. The 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. This approach allows us to ameliorate the bullwhip effect (the amplification of the demand variations when going up in the supply chain). The control strategy proposed in this work ensures bounded orders, guarantees full consumers’ demand satisfaction, and eliminates the risk of exceeding the warehouse capacity. These properties are stated in three theorems and proved in the paper.