Innovative Applications of O.R. Optimal maintenance of a production-inventory system with idle periods C.C. Karamatsoukis, E.G. Kyriakidis * Department of Financial and Management Engineering, University of the Aegean, 31 Fostini Str., 82100 Chios, Greece article info Article history: Received 20 September 2007 Accepted 4 April 2008 Available online 15 April 2008 Keywords: Maintenance Dynamic programming Renewal-reward process Control-limit policies abstract In this paper we consider a production-inventory system in which an input generating installation sup- plies a buffer with a raw material and a production unit pulls the raw material from the buffer with con- stant rate. The installation deteriorates in time and the problem of its optimal preventive maintenance is considered. It is assumed that the installation after the completion of its maintenance remains idle until the buffer is evacuated. Under a suitable cost structure it is shown that the average-cost optimal policy for fixed buffer content is of control-limit type, i.e. it prescribes a preventive maintenance of the instal- lation if and only if its degree of deterioration is greater than or equal to a critical level. Using the usual regenerative argument, the average cost of a control-limit policy is computed exactly and then, the opti- mal control-limit policy is determined. Furthermore, the stationary probabilities of the system under the optimal policy are computed. Ó 2008 Elsevier B.V. All rights reserved. 1. Introduction In modern industry the maintenance of a production system is an important issue. The optimal strategy for the maintenance can result in substantial saving in operation and, also, in increased availability of the system. In the last 50 years a great number of maintenance prob- lems have been studied and various mathematical models have been proposed for their solution. A survey of different kinds of maintenance policies of deteriorating systems was given by Wang [19]. The Markov decision model (see e.g. Chapter 3 in Tijms [17]) has been proved to be a powerful and flexible tool for the description and solution of many problems, which are related to the optimal corrective or preventive maintenance of a system. For example, in the papers of Federgruen and So [7,8], Ozekici and Gunluk [13], Douer and Yechiali [6], Stadje and Zuckerman [15], Chen and Feldman [2], Benyamini and Yechiali [1], Su et al. [16], Chen et al. [3], Moustafa et al. [12] and Grosfeld-Nir [9] suitable Markov decision models were constructed for various maintenance models. In the present paper we modify a model (see Kyriakidis and Dimitrakos [11]), in which the problem of the optimal preventive mainte- nance of a production-inventory system was considered. In that model it was assumed that a deteriorating machine supplies a buffer with some raw material and a production unit pulls the raw material from the buffer at a constant rate. If the machine is found to be at failed condition, a corrective maintenance must be commenced, while, if it found to be at an operative condition, a preventive maintenance may be initiated. It was assumed that the times required for a preventive or a corrective maintenance are geometrically distributed. A suitable cost structure was introduced and the problem of finding the policy that minimises the long-run expected average cost per unit time was considered. In the present paper we modify the above model by assuming that, after the completion of a preventive or a corrective maintenance, the machine remains idle until the buffer is evacuated. It is possible to prove that the average-cost optimal policy for fixed buffer content is of control-limit type, i.e. it initiates a preventive maintenance of the machine if and only if the degree of its deterioration is greater than or equal to a critical level. The average cost of a control-limit policy can be computed exactly by applying the usual regenerative argument since the expected time and the expected cost of a cycle under a control-limit policy can be computed through recursive equations. The control-limit policy, which has the smallest average cost among all control-limit policies, is the overall optimal policy. It is also possible to compute exactly the stationary probabilities when the system is controlled by the optimal policy. The rest of the paper is organised as follows. The description of the model is given in the next section and in Section 3 the optimality of a control-limit policy is proved. In Section 4 we show how the average cost of a control-limit policy can be computed exactly. We also present two numerical examples, in which the optimal policy is computed. In Section 5 we show how the stationary probabilities of the system under the optimal policy can be computed. The conclusions of the paper are given in the last section and in the Appendix we explain in detail how a formula is derived. 0377-2217/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2008.04.012 * Corresponding author. Tel.: +30 2271035464. E-mail addresses: k.karamatsoukis@fme.aegean.gr (C.C. Karamatsoukis), kyriak@fme.aegean.gr, kyriak@aegean.gr (E.G. Kyriakidis). European Journal of Operational Research 196 (2009) 744–751 Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor