Abstract — Inventory replenishment system design and optimisation has gained increasing levels of attention in mathematical and engineering literature in recent times. In response to many organisations expanding and globalising across geographic and political boundaries, any inventory held poses significant cost implications and, hence, also presents immense scope for optimization. In this paper, the inventory efficient management problem is investigated from a control theory viewpoint, also the factors which are often considered in purely mathematical models of operational research literature are considered in this work, but formulated differently, in the control theory domain. The goods are replenished with delay from a remote supplier. The main objective is to obtain the inventory keeping cost-benefit trade-off by reduction of the undesirable influences of system uncertainties, such as unpredictable demand and lead time, on inventory level stability. Furthermore, attention is given to developing as simple as possible a mathematical representation of the system, as is practicable. Being straightforward in application, the developed tool attempts to bridge the gap between the precision afforded by methods of control theory and the expectations of supply chain supervisors. I. INTRODUCTION Inventory replenishment optimisation becomes a more and more extensive research area in the operational research field which reflects the increasing importance of obtaining inventory holding benefit-expenses balanced for excellent stock management. Although the holding inventory within the distribution arms of supply chains ensures satisfaction and rapid response to customer requirements, it generates an expense, which has already been recognized as one of the highest logistics costs [3]. Obtaining optimal order quantities, for balancing the inventory levels on daily basis, in dynamically changing conditions, such as unpredictable varying demand, lead time delay and deterioration of products, becomes extremely complex process, if the engineering approach is not adopted. Uncontrolled inventory level fluctuations, however, can cause several other problems such as unnecessary deterioration of products, varying storage costs, difficulty in required employees planning, extension of storage capacity and/or backorders. As a consequence, the need for designing an engineering replenishment decision framework has been dealt with in * J. E. Orzechowska, Control Theory and Applications Centre, Coventry University, Control Theory and Application Centre; Priory Street, Coventry CV1 5FD Coventry, UK, e-mail: orzechoj@uni.coventry.ac.uk A. Bartoszewicz, Politechnika Łódzka, Instytut Automatyki; 90-924 Łódź, ul. Bohdana Stefanowskiego 18/22. e-mail: andrzej.bartoszewicz@p.lodz.pl K J. Burnham, Control Theory and Applications Centre, Coventry University, Control Theory and Application Centre; Priory Street, Coventry CV1 5FD Coventry, UK, e-mail: k.burnham@coventry.ac.uk D. Petrovic, Control Theory and Applications Centre, Coventry University, Control Theory and Application Centre; Priory Street, Coventry CV1 5FD Coventry, UK, e-mail: d.petrovic@coventry.ac.uk many earlier publications. Techniques such as convex programming [4], genetic algorithms [14] or heuristic methods [6] have been commonly applied to improve warehouse operations and efficiencies. Nevertheless, in most of these cases involving techniques for the optimisation of replenishment systems, the demand was not considered as being dynamic, as its mathematical consideration is at least not straightforward, and probably very difficult to formulate. This implies that the above models are applicable only in cases where the demand is already known. However, in most real-world replenishment systems, the demand is at best only partially predictable or at worst highly unpredictable. In the paper the control theory approach to inventory modelling and control has been applied, which allows for straightforward consideration of system dynamics in a simple yet elegant way, and allows for on-line optimisation of orders. The initial state space representation of the inventory balance, considering the lead time delay, unknown demand and deterioration rate of products, is presented in Section II A. Then, as a starting point, a model predictive control (MPC) approach, is briefly discussed in Section II B. Although application of control techniques to inventory decision making is not a new field of research, relatively only a handful of researchers have dedicated their work to this problem. The model predictive control (MPC) approach to developing an optimisation tool for inventory management has been used by several researchers [1,2,5,13,15]. A more detailed literature review on the application of MPC and other control theory approaches can be found in [9]. Although control theory and mathematical optimisation techniques, significantly facilitate inventory management, its application in real-world case studies requires an in-depth knowledge of mathematics and control theory, and, as a result, might not be appreciated and easily adopted by the non-engineering warehouse managers. Therefore, in this paper a simplified implementation of the model predictive based and cost reduction oriented replenishment decision framework is proposed. The control strategy finally presented in this paper was found by mathematical reformulation of the initial MPC for the initially defined inventory model. The reduced mathematical formulation was based on noticing patterns in the MPC formulation and followed a series of propositions and their demonstrations leading to the final approach. The mathematical demonstration of the equivalence of the initial and final formulation is not shown in the paper due to the length limitations. Nevertheless, for the non-perishable products case, it can be found in [7,10,11], if needed. The final formulation is presented in the form of a proposition in Section II C, together with an elaboration of its simplicity in A Simple Approach to MPC of Perishable Inventory Systems Joanna E. Orzechowska, Andrzej Bartoszewicz, Keith J. Burnham and Dobrila Petrovic*