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Group maintenance policies for an R-out-of-N system with phase-type distribution

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Abstract

This paper presents an extension of our earlier paper on the 1-out-of-N repairable cold standby system (i.e., Barron IIE Trans 47:1139–1151, 2015). Specifically, we consider an R-out-of-N repairable system where the lifetimes of the units follow phase-type distribution. The system is functioning if at least R out of its N components work. Each working component is subject to failure. There are fixed, unit repair, and replacement costs associated with the maintenance facility, which is carried out after a fixed lead time \(\tau \). A penalty cost is incurred when the number of good components decreases to \(R-1\). We assume that the repair takes no time and repaired units are as good as new. By applying renewal theory and matrix-geometric methods, we derive the expected discounted costs under three classes of group maintenance policies: m-failure, T-age, and (\(m,T,\tau \)), which is a refinement of the classical (m, T) policy. Illustrative examples, a comparative study and insights are provided.

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References

  • Ahmadi, R. (2014). Maintenance scheduling for a complex manufacturing system subject to deterioration. Annals of Operation Research, 217, 1–29.

    Article  Google Scholar 

  • Asmussen, S. (2000). Matrix-analytic models and their analysis. Scandinavian Journal of Statistics, 27, 193–226.

    Article  Google Scholar 

  • Assaf, D., & Levikson, B. (1982). Closure of phase type distributions under operations arising in reliability theory. The Annals of Probability, 10, 265–269.

    Article  Google Scholar 

  • Assaf, D., & Shanthikumar, J. G. (1987). Optimal group maintenance policies with continuous and periodic inspection. Management Science, 33, 1440–1452.

    Article  Google Scholar 

  • Barron, Y., Frostig, E., & Levikson, B. (2004). Analysis of \(R\)-out-of-\(N\) repairable systems: The case of phase-type distribution. Advances in Applied Probability, 36, 116–138.

    Google Scholar 

  • Barron, Y., Frostig, E., & Levikson, B. (2006). Analysis of \(R\)-out-of-\(N\) systems with several repairmen, exponential life times and phase type repair times: An algorithmic approach. European Journal of Operational Research, 169, 202–225.

    Article  Google Scholar 

  • Barron, Y. (2015). Group replacement policies for a repairable cold standby system with fixed lead times. IIE Transactions, 47, 1139–1151.

    Article  Google Scholar 

  • Bevilacqua, M., & Braglia, M. (2000). The analytic hierarchy process applied to maintenance strategy selection. Reliability Engineering & Systems Safety, 70, 71–83.

    Article  Google Scholar 

  • Bohlin, M., & Wärja, M. (2015). Maintenance optimization with duration-dependent costs. Annals of Operations Research, 224, 1–23.

    Article  Google Scholar 

  • Cao, J. H., & Cheng, K. (2006). Introduction to reliability mathematics. Beijing: Higher Education Press.

    Google Scholar 

  • Chang, M. F., Shi, W., & Fuchs, W. K. (1990). Optimal diagnosis procedures for \({k}\)-out-of-\({n}\) structures. IEEE Transactions on Computers, 39(4), 559–564.

    Article  Google Scholar 

  • Chien, Y. H., Chang, C. C., & Sheu, S. H. (2010). Optimal age-replacement model with age-dependent type of failure and random lead time based on a cumulative repair-cost limit policy. Annals of Operation Research, 181, 723–744.

    Article  Google Scholar 

  • Coetzee, J. (2004). Maintenance. Victoria: Trafford Publishing.

    Google Scholar 

  • Ji-Wen, S., Li-feng, X., Shi-Chang, D., & Er-Shun, P. (2010). Tool maintenance optimization for multi-station machining systems with economic consideration of quality loss and obsolescence. Robotics and Computer-Integrated Manufacturing, 26, 145–155.

    Article  Google Scholar 

  • Kang, H. G., & Kim, H. E. (2012). Unavailability and spurious operation probability of \(k\)-out-of-\(n\) reactor protection systems in consideration of CCF. Annals of Nuclear Energy, 49, 102–108.

    Article  Google Scholar 

  • Kuo, W., & Zuo, M. J. (2003). Optimal reliability modeling: Principles and applications. New York: Wiley.

    Google Scholar 

  • Li, H., & Xu, S. H. (2004). On the coordinated random group replacement policy in multivariate repairable systems. Operations Research, 52(3), 464–477.

    Article  Google Scholar 

  • Liu, G. S. (2012). Three \(m\)-failure group maintenance models for \(M/M/N\) unreliable queueing service systems. Computers & Industrial Engineering, 62, 1011–1024.

    Article  Google Scholar 

  • Lorén, S., & de Maré, J. (2015). Maintenance for reliability—A case study. Annals of Operations Research, 224, 111–119.

    Article  Google Scholar 

  • Misra, K. B. (2008). Handbook of performability engineering. London: Springer.

    Book  Google Scholar 

  • Neuts, M. F. (1981). Matrix-geometric solutions in stochastic models: An algorithmic approach. Baltimore: Johns Hopkins University Press.

    Google Scholar 

  • Neuts, M. F., Pérez-Ocón, R., & Torres-Castro, I. (2000). Repairable models with operating and repair times governed by phase type distributions. Advances in Applied Probability, 32, 468–479.

    Article  Google Scholar 

  • Neuts, M. F., & Takahashi, Y. (1981). Asymptotic behavior of the stationary distributions in the GI/PH/c Queue with heterogenous servers. Zeitschrift für. Wahrscheinlichkeitstheorie und verwandte. Gebiete, 57, 441–452.

    Article  Google Scholar 

  • Niu, C., Liang, X., Ge, B., Tian, X., & Chen, Y. (2016). Optimal replacement policy for a repairable system with deterioration based on a renewal-geometric process. Annals of Operations Research, 244(1), 49–66.

  • Park, K. S., & Yoo, Y. K. (2004). Comparison of group replacement policies under minimal repair. International Journal of Systems Science, 35, 179–184.

    Article  Google Scholar 

  • Pham, H., & Wang, H. (2000). Optimal \((\tau; T)\) opportunistic maintenance of a \(k\)-out-of-\(n\):\(G\) system with imperfect PM and partial failure. Naval Research Logistics, 47, 223–239.

    Article  Google Scholar 

  • Popova, E., & Wilson, J. G. (1999). Group replacement policies for parallel systems whose components have phase distributed failure times. Annals of Operations Research, 91, 163–198.

    Article  Google Scholar 

  • Ritchken, P., & Wilson, J. G. (1990). \((m, T)\) group maintenance policies. Management Science, 36, 632–639.

    Article  Google Scholar 

  • Ruiz-Castro, J. E. (2016). Complex multi-state systems modelled through marked Markovian arrival processes. European Journal of Operational Research, 252, 852–865.

    Article  Google Scholar 

  • Safaei, N., Banjevic, D., & Jardine, A. K. (2011). Workforce-constrained maintenance scheduling for military aircraft fleet: A case study. Annals of Operations Research, 186, 295–316.

    Article  Google Scholar 

  • Sheu, S. H., Chang, C. C., & Chien, Y. H. (2011). Optimal age-replacement time with minimal repair based on cumulative repair-cost limit for a system subject to shocks. Annals of Operation Research, 186, 317–329.

    Article  Google Scholar 

  • Smith, M. A. J., & Dekker, R. (1997). Preventive maintenance in a \(1\)-out-of-\(n\) system: The uptime, downtime and costs. European Journal of Operational Research, 99, 565–583.

    Article  Google Scholar 

  • Wells, C. E. (2014). Reliability analysis of a single warm-standby system subject to repairable and nonrepairable failures. European Journal of Operational Research, 235, 180–186.

    Article  Google Scholar 

  • Wang, Y., Li, L., Huang, S., & Chang, Q. (2012). Reliability and covariance estimation of weighted k-out-of-n multi-state systems. European Journal of Operational Research, 221, 138–147.

    Article  Google Scholar 

  • Wu, W., Tang, Y., Yu, M., & Jiang, Y. (2014). Reliability analysis of a \(k\)-out-of-\(n\):\(G\) repairable system with single vacation. Applied Mathematical Modelling, 38, 6075–6097.

    Article  Google Scholar 

  • Yeh, R. H., Chang, W. L., & Lo, H. C. (2010). Optimal threshold values of age and two-phase maintenance policy for leased equipments using age reduction method. Annals of Operations Research, 181, 171–183.

    Article  Google Scholar 

  • Yoo, Y. K. (2011). Operating characteristics of a failure counting group replacement policy. International Journal of Systems Science, 42, 499–506.

    Article  Google Scholar 

  • Zhong, C. Q., & Jin, H. (2014). A novel optimal preventive maintenance policy for a cold standby system based on semi-Markov theory. European Journal of Operational Research, 232, 405–411.

    Article  Google Scholar 

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  1. Department of Industrial Engineering and Management, Ariel University, Ariel, 40700, Israel

    Yonit Barron

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  1. Yonit Barron
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Barron, Y. Group maintenance policies for an R-out-of-N system with phase-type distribution. Ann Oper Res 261, 79–105 (2018). https://doi.org/10.1007/s10479-017-2617-x

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  • DOI: https://doi.org/10.1007/s10479-017-2617-x

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