Contents lists available at ScienceDirect Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie Reliability modeling and optimal random preventive maintenance policy for parallel systems with damage self-healing Wenjie Dong a, ⁎ , Sifeng Liu a , Yingsai Cao a , Saad Ahmed Javed b , Yangyang Du a a College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, Jiangsu, PR China b School of Business, Nanjing University of Information Science and Technology, Nanjing 210044, Jiangsu, PR China ARTICLE INFO Keywords: Damage self-healing Cumulative shock damage Reliability analysis Preventive replacement Micro-electro-mechanical systems (MEMS) ABSTRACT Materials with intrinsic self-healing phenomenon possess the ability to heal in response to external random shocks. Introducing a recovery factor to quantitatively measure the damage self-recovery efciency, this paper designs a self-healing mechanism corresponding to both damage load and shock arrival numbers for a parallel redundant system consisting of multiple non-identical components. From the actual engineering perspective, each shock arriving on the system selectively afects one component or more but not necessarily all units in parallel, and consequently, random shocks are categorized according to their sizes, attributes and afected components. This study investigates novel reliability models and schedules optimal preventive maintenance policies, in which the closed-form reliability quantities are derived analytically and the optimum preventive replacement interval is demonstrated theoretically. In addition, a Nelder-Mead downhill simplex method is introduced to seek the optimal replacement age in minimizing the long-run average maintenance cost rate for the condition system failure distribution is rather complex. A micro-electro-mechanical system (MEMS) whose constitutional materials are integrated by microcrystalline silicon, where polymer binders with self-healing capability are always synthesized, is designed to verify the results we obtained numerically, illustrating the signifcance of considering damage self-healing phenomena. 1. Introduction Most industrial systems fail to work due to two competing failure processes incorporating a soft failure process and a hard failure process. The soft failure occurs owing to internal performance degradation whereas the hard failure happens as a result of external random shocks. In general, these two failure processes are dependent as well as com- peting with each other, i.e., random shocks infuence the increments on degradation amount or accelerate the degradation rate or both. That is, systems are sufering from dependent and competing failure processes (DCFP). Degradation modeling and reliability analysis for systems subject to DCFP have sought a lot of attention from researchers in the literature (Caballe & Castro, 2019; Hao & Yang, 2018; Qi, Zhou, Niu, Wang, & Wu, 2018; Wang, Li, Bai, & Zuo, 2020). For the soft failure process, degradation models such as a general degradation path, a gamma process or a Wiener process are commonly developed in previous works (Cha, Finkelstein, & Levitin, 2018). Peng, Feng, and Coit (2010) considered a linear degradation path for the wear volume due to internal continuous degradation. Diferent from Peng et al.’s work (2010), Shen, Elwany, and Cui (2018) regarded that the degradation measurement between two adjacent shocks was regulated by a gamma process, and random shocks caused a jump in degradation level and accelerated the degradation rate simultaneously. Wei, Zhao, He, and He (2019) modeled system internal degradation as a two-phase Wiener process, which has a larger shift and difusion parameter when the system transfers from the normal state to the weakened state. On the other aspect, random shocks resulting in catastrophic failures are considered as seven diferent types (Rafee, Feng, & Coit, 2017): (a) extreme shock model, in which a system fails as soon as the frst shock exceeds a specifed threshold value; (b) standardly cumulative shock model, when hard failure occurs until the cumulative shock damage is beyond a critical level; (c) mixed shock model, where a system fails as soon as an extreme shock failure occurs or a cumulative shock failure occurs, whichever takes place frst; (d) m shock model, where a system experiences failure after shocks whose magnitudes are all larger than a threshold level; (e) shock model, where failure happens when the time lag between two successive shocks is less than a threshold ; (f) run shock model, in which a system fails if there is a run of n https://doi.org/10.1016/j.cie.2020.106359 ⁎ Corresponding author at: College of Economics and Management, Nanjing University of Aeronautics and Astronautics, 29 General Avenue, Jiangning District, Nanjing City 211106, Jiangsu Province, China. E-mail address: dongwenjie@nuaa.edu.cn (W. Dong). Computers & Industrial Engineering 142 (2020) 106359 Available online 12 February 2020 0360-8352/ © 2020 Elsevier Ltd. All rights reserved. T