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Optimal allocation of relevations in coherent systems

Part of: Special processes Operations research and management science Distribution theory - Probability

Published online by Cambridge University Press:  22 November 2021

Rongfang Yan Open the ORCID record for Rongfang Yan [Opens in a new window] ,
Jiandong Zhang Open the ORCID record for Jiandong Zhang [Opens in a new window]  and
Yiying Zhang Open the ORCID record for Yiying Zhang [Opens in a new window]
Rongfang Yan*
Affiliation:
Northwest Normal University
Jiandong Zhang*
Affiliation:
Northwest Normal University
Yiying Zhang*
Affiliation:
Southern University of Science and Technology
*
*Postal address: College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China.
*Postal address: College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China.
**Postal address: Department of Mathematics, Southern University of Science and Technology, Shenzhen 518055, Guangdong, China. Email address: zhangyiying@outlook.com
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Abstract

In this paper we study the allocation problem of relevations in coherent systems. The optimal allocation strategies are obtained by implementing stochastic comparisons of different policies according to the usual stochastic order and the hazard rate order. As special cases of relevations, the load-sharing and minimal repair policies are further investigated. Sufficient (and necessary) conditions are established for various stochastic orderings. Numerical examples are also presented as illustrations.

Keywords

Relevationcoherent systemload-sharing modelminimal repairstochastic orders

MSC classification

Primary: 90B25: Reliability, availability, maintenance, inspection
Secondary: 60E15: Inequalities; stochastic orderings 60K10: Applications (reliability, demand theory, etc.)
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 4 , December 2021 , pp. 1152 - 1169
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Arriaza, A., Navarro, J. and Suárez-Llorens, A. (2018). Stochastic comparisons of replacement policies in coherent systems under minimal repair. Naval Res. Logistics 65, 550–565.10.1002/nav.21820CrossRefGoogle Scholar
Barlow, E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing: Probability Models (International Series in Decision Processes). Holt, Rinehart and Winston.Google Scholar
Barlow, R. and Hunter, L. (1960). Optimum preventive maintenance policies. Operat. Res. 8, 90–100.10.1287/opre.8.1.90CrossRefGoogle Scholar
Baxter, L. A. (1982). Reliability applications of the relevation transform. Naval Res. Logistics Quart. 29, 323–330.10.1002/nav.3800290212CrossRefGoogle Scholar
Belzunce, F., Lillo, R. E., Ruiz, J.-M. and Shaked, M. (2001). Stochastic comparisons of nonhomogeneous processes. Prob. Eng. Inf. Sci. 15, 199–224.10.1017/S0269964801152058CrossRefGoogle Scholar
Belzunce, F., Martnez-Riquelme, C., Mercader, J. A. and Ruiz, J. M. (2021). Comparisons of policies based on relevation and replacement by a new one unit in reliability. TEST 30, 211–227.10.1007/s11749-020-00710-6CrossRefGoogle Scholar
Belzunce, F., Martnez-Riquelme, C. and Ruiz, J. M. (2019). Allocation of a relevation in redundancy problems. Appl. Stoch. Models Business Industry 35, 492–503.10.1002/asmb.2328CrossRefGoogle Scholar
Belzunce, F., Riquelme, C. M. and Mulero, J. (2016). An Introduction to Stochastic Orders. Academic Press.Google Scholar
Birnbaum, Z. and Saunders, S. C. (1958). A statistical model for life-length of materials. J. Amer. Statist. Assoc. 53, 151–160.10.1080/01621459.1958.10501433CrossRefGoogle Scholar
Boland, P. J., El-Neweihi, E. and Proschan, F. (1992). Stochastic order for redundancy allocations in series and parallel systems. Adv. Appl. Prob. 24, 161–171.10.2307/1427734CrossRefGoogle Scholar
Chen, J., Zhang, Y., Zhao, P. and Zhou, S. (2017). Allocation strategies of standby redundancies in series/parallel system. Commun. Statist. Theory Meth. 47, 708–724.10.1080/03610926.2017.1313984CrossRefGoogle Scholar
Cui, L., Kuo, W., Loh, H. and Xie, M. (2004). Optimal allocation of minimal & perfect repairs under resource constraints. IEEE Trans. Reliab. 53, 193–199.10.1109/TR.2004.829143CrossRefGoogle Scholar
Deshpande, J., Dewan, I. and Naik-Nimbalkar, U. (2010). A family of distributions to model load sharing systems. J. Statist. Planning Infer. 140, 1441–1451.10.1016/j.jspi.2009.12.005CrossRefGoogle Scholar
Di Crescenzo, A., Kayal, S. and Toomaj, A. (2019). A past inaccuracy measure based on the reversed relevation transform. Metrika 82, 607–631.10.1007/s00184-018-0696-6CrossRefGoogle Scholar
Fang, R. and Li, X. (2016). On allocating one active redundancy to coherent systems with dependent and heterogeneous components’ lifetimes. Naval Res. Logistics 63, 335–345.10.1002/nav.21692CrossRefGoogle Scholar
Fang, R. and Li, X. (2017). On matched active redundancy allocation for coherent systems with statistically dependent component lifetimes. Naval Res. Logistics 64, 580–598.10.1002/nav.21774CrossRefGoogle Scholar
Joag-dev, K., Kochar, S. and Proschan, F. (1995). A general composition theorem and its applications to certain partial orderings of distributions. Statist. Prob. Lett. 22, 111–119.10.1016/0167-7152(94)00056-ECrossRefGoogle Scholar
Kapodistria, S. and Psarrakos, G. (2012). Some extensions of the residual lifetime and its connection to the cumulative residual entropy. Prob. Eng. Inf. Sci. 26, 129–146.10.1017/S0269964811000271CrossRefGoogle Scholar
Karrlin, S. (1968). Total Positivity, vol. 1. Stanford University Press.Google Scholar
Krakowski, M. (1973). The relevation transform and a generalization of the gamma distribution function. Revue Française d’Automatique, Informatique et de Recherche Opérationnelle: Recherche Opérationnelle 7, 107–120.Google Scholar
Liao, G.-L. (2016). Production and maintenance policies for an EPQ model with perfect repair, rework, free-repair warranty, and preventive maintenance. IEEE Trans. Syst. Man Cybernet. Systems 46, 1129–1139.10.1109/TSMC.2015.2465961CrossRefGoogle Scholar
Misra, N., Misra, A. K. and Dhariyal, I. D. (2011). Standby redundancy allocations in series and parallel systems. J. Appl. Prob. 48, 43–55.10.1239/jap/1300198135CrossRefGoogle Scholar
Navarro, J., Arriaza, A. and Suárez-Llorens, A. (2019). Minimal repair of failed components in coherent systems. Europ. J. Operat. Res. 279, 951–964.10.1016/j.ejor.2019.06.013CrossRefGoogle Scholar
Psarrakos, G. and Di Crescenzo, A. (2018). A residual inaccuracy measure based on the relevation transform. Metrika 81, 37–59.10.1007/s00184-017-0633-0CrossRefGoogle Scholar
Romera, R., Valdes, J. and Zequeira, R. (2004). Active redundancy allocation in systems. IEEE Trans. Reliab. 53, 313–318.10.1109/TR.2004.833309CrossRefGoogle Scholar
Sankaran, P. G. and Kumar, M. D. (2018). Reliability properties of proportional hazards relevation transform. Metrika 82, 441–456.10.1007/s00184-018-0681-0CrossRefGoogle Scholar
Sarkar, J. and Sarkar, S. (2000). Availability of a periodically inspected system under perfect repair. J. Statist. Planning Infer. 91, 77–90.10.1016/S0378-3758(00)00128-2CrossRefGoogle Scholar
Schechner, Z. (1984). A load-sharing model: the linear breakdown rule. Naval Res. Logistics 31, 137–144.10.1002/nav.3800310114CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, G. (2007). Stochastic Orders. Springer.10.1007/978-0-387-34675-5CrossRefGoogle Scholar
Shanthikumar, J. G. and Baxter, L. A. (1985). Closure properties of the relevation transform. Naval Res. Logistics Quart. 32, 185–189.10.1002/nav.3800320121CrossRefGoogle Scholar
Wang, Y. T. and Morris (1985). Load sharing in distributed systems. IEEE Trans. Comput. 100, 204–217.10.1109/TC.1985.1676564CrossRefGoogle Scholar
Yan, R., Lu, B. and Li, X. (2018). On redundancy allocation to series and parallel systems of two components. Commun. Statist. Theory Meth. 48, 4690–4701.10.1080/03610926.2018.1500603CrossRefGoogle Scholar
Yan, R. and Luo, T. (2018). On the optimal allocation of active redundancies in series system. Commun. Statist. Theory Meth. 47, 2379–2388.10.1080/03610926.2015.1054942CrossRefGoogle Scholar
Yun, W. Y. and Cha, J. H. (2010). A stochastic model for a general load-sharing system under overload condition. Appl. Stoch. Models Business Industry 26, 624–638.10.1002/asmb.824CrossRefGoogle Scholar
Zhang, X., Zhang, Y. and Fang, R. (2020). Allocations of cold standbys to series and parallel systems with dependent components. Appl. Stoch. Models Business Industry 36, 432–451.10.1002/asmb.2497CrossRefGoogle Scholar
Zhang, Y. (2018). Optimal allocation of active redundancies in weighted k-out-of-n systems. Statist. Prob. Lett. 135, 110–117.10.1016/j.spl.2017.12.002CrossRefGoogle Scholar
Zhang, Y., Amini-Seresht, E. and Ding, W. (2017). Component and system active redundancies for coherent systems with dependent components. Appl. Stoch. Models Business Industry 33, 409–421.Google Scholar
Zhang, Y. and Zhao, P. (2019). Optimal allocation of minimal repairs in parallel and series systems. Naval Res. Logistics 66, 517–526.10.1002/nav.21865CrossRefGoogle Scholar
Zhang, Z. and Balakrishnan, N. (2016). Stochastic properties and parameter estimation for a general load-sharing system. Commun. Statist. Theory Meth. 46, 747–760.10.1080/03610926.2015.1004095CrossRefGoogle Scholar