Research Article Received 7 September 2014, Revised 31 August 2015, Accepted 1 September 2015 Published online in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/asmb.2143 Optimal replacement and allocation of multi-state elements in k-within-m-from-r/n sliding window systems Hui Xiao a * † , Rui Peng b and Gregory Levitin c,d This paper proposes a new model that generalizes the linear sliding window system to the case of multiple failures. The considered k-within-m-from-r/n sliding window system consists of n linearly ordered multi-state elements and fails if at least k groups out of m consecutive groups of r consecutive multi-state elements have cumulative performance lower than the demand W. A reliability evaluation algorithm is suggested for the proposed system. In order to increase the system availability, maintenance actions can be performed, and the elements can be optimally allocated. A joint element allocation and maintenance optimization model is formu- lated with the objective of minimizing the total maintenance cost subjected to the pre-speciied system availability requirement. Basic procedures of genetic algorithms are adapted to solve the optimization problem. Numerical experiments are presented to illustrate the applications. Copyright © 2015 John Wiley & Sons, Ltd. Keywords: multi-state reliability; sliding window system; preventive replacement; universal generating function; genetic algorithms 1. Introduction A sequence of n elements is linearly ordered if they are arranged on a line. The linear consecutive k-out-of-r-from-n system consists of n linearly ordered binary elements. It fails if at least k elements fail within any group of r consec- utive elements. This system was formally introduced by Grifith [1]. It was also mentioned in some earlier research [2–6]. Examples of the linear consecutive k-out-of-r-from-n system can be found in quality control, inspection proce- dures, service systems, and radar detection problems. In the literature, various models have been proposed to evaluate the reliability of such systems. For example, Bollinger [7] and Sfakianakis et al. [8] considered the reliability evaluation of such systems with identical units. Papastavridis and Sfakianakis [9] and Cai [10] derived some reliability bounds of such systems. In the literature, the binary k-out-of-n system and the binary consecutive-k system have been generalized to multi-state k-out-of-n systems [11] and multi-state consecutive-k system [12], respectively. Likewise, the linear consecutive k-out- of-r-from-n system has been also generalized to the linear sliding window system (SWS) when the system elements are multi-state. That is, each element can have more than two performance states from perfect functioning to complete failure. The SWS consists of n linearly ordered multi-state elements (MEs). Each ME e j can have H j different performance states. The SWS fails if the sum of the performance of any r consecutive MEs is lower than the pre-speciied demand W. The algorithm for evaluating the reliability of the SWS based on universal generating function (UGF) is proposed by Levitin [13], and the optimal allocation of the MEs is studied in [14]. This paper proposes a new model that generalizes the SWS to the case of multiple failures. The considered k-within- m-from-r/n SWS consists of n linearly ordered MEs and fails if at least k groups out of m consecutive groups of r a School of Statistics, Southwestern University of Finance and Economics, Chengdu, China b Dongling School of Economics and Management, University of Science and Technology Beijing, Beijing, China c The Israel Electric Corporation Ltd., Haifa, Israel d Collaborative Autonomic Computing Laboratory, School of Computer Science, University of Electronic Science and Technology of China, Chengdu, China *Correspondence to: Hui Xiao, School of Statistics, Southwestern University of Finance and Economics, Chengdu, China. † E-mail: msxh@swufe.edu.cn Copyright © 2015 John Wiley & Sons, Ltd. Appl. Stochastic Models Bus. Ind. 2015