Optimization for condition-based maintenance with semi-Markov decision process * Dongyan Chen a, * , Kishor S. Trivedi b a Department of Computer Sciences and Computer Engineering, Xavier University of Louisiana, New Orleans, LA 70125, USA b Center for Advanced Computing and Communications, Duke University Durham, NC 27708-0294, USA Received 1 July 2003; accepted 26 January 2004 Available online 25 December 2004 Abstract The semi-Markov decision model is a powerful tool in analyzing sequential decision processes with random decision epochs. In this paper, we have built the semi-Markov decision process (SMDP) for the maintenance policy optimization of condition-based preventive maintenance problems, and have presented the approach for joint optimization of inspection rate and maintenance policy. Through numerical examples, the improvement of this method is compared with the scheme, which optimizes only over the inspection rate. We also find that under a special case when the deterioration rate at each failure stage is the same, the optimal policy obtained by SMDP algorithm is a dynamic threshold-type scheme with threshold value depending on the inspection rate. q 2004 Elsevier Ltd. All rights reserved. Keywords: Preventive maintenance; Reliability; Availability; Markov decision process; Optimization 1. Introduction Preventive maintenance is defined as the activity undertaken regularly at pre-selected intervals while the device is satisfactorily operating, to reduce or eliminate the accumulated deterioration [5], while repair is the activity to bring the device to a non-failed state after it has experienced a failure. When the cost incurred by a device failure is larger than the cost of preventive maintenance (this cost could be cost of downtime, repair expenses, revenue lost, etc.), it is worthwhile to carry out preventive maintenance. Generally, there exist two types of preventive mainten- ance schemes, i.e. condition based and time based preventive maintenance [3]. For condition based preventive mainten- ance, the action taken after each inspection is dependent on the state of the system. It could be no action, or minimal maintenance to recover the system to the previous stage of degradation, or major maintenance to bring the system to as good as new state. For time based preventive maintenance, the preventive maintenance is carried out at pre-determined time intervals to bring the system to as good as new state [7]. In this paper we focus on the condition based preventive maintenance. Sim and Endrenyi introduced the multi-stage exponential device failure model in [5], in which the idea of minimal preventive maintenance was introduced. The minimal preventive maintenance is defined as the preventive maintenance activity with limited effort and effect [4]. For deterioration failures modeled as several stages of expo- nential distributions, minimal maintenance restores the system to the previous deterioration stage. Corresponding to the minimal preventive maintenance, the idea of major maintenance is defined as the maintenance operation by which the device is restored ‘as good as new’ status. With condition based preventive maintenance, the maintenance action taken after each inspection is dependent on the state of the system. There could be no action, or minimal maintenance to recover the system to the previous failure stage, or major maintenance to bring the system to as good as new state. Hosseini et al. [2] introduced 0951-8320/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ress.2004.11.001 Reliability Engineering and System Safety 90 (2005) 25–29 www.elsevier.com/locate/ress * This research was supported in part by the Air Force Office of Scientific Research under MURI Grant No. F49620-00-1-0327, and in part by DARPA and US Army Research Office under Award No. C-DAAD19 01-1- 0646. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author(s) and does not necessarily reflect the view of the sponsoring agencies. * Corresponding author. Fax: C1 504 5207908. E-mail address: cdongyan@xula.edu (D. Chen).