{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,4]],"date-time":"2026-08-04T01:50:12Z","timestamp":1785808212657,"version":"3.56.0"},"reference-count":26,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2021,5,4]],"date-time":"2021-05-04T00:00:00Z","timestamp":1620086400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,5,4]],"date-time":"2021-05-04T00:00:00Z","timestamp":1620086400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Department of Mathematics, Imperial College London","award":["Research Impulse Award (Reference: DRI122PB)"],"award-info":[{"award-number":["Research Impulse Award (Reference: DRI122PB)"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Ann Oper Res"],"published-print":{"date-parts":[[2022,8]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In many real-life scenarios, system failure depends on dynamic stress-strength interference, where strength degrades and stress accumulates concurrently over time. In this paper, we consider the problem of finding an optimal replacement strategy that balances the cost of replacement with the cost of failure and results in the minimum expected cost per unit time under cumulative damage model with strength degradation. In the most general setting, we propose to find optimal choices of three thresholds on operation time, number of arriving shocks and amount of cumulative damage such that replacement of the system due to failure or reaching any of the three thresholds, whichever occurs first, results in the minimum expected cost per unit time. The existing recommendations are applicable only under the assumption of Exponential damage distribution including Poisson arrival of shocks and\/or with fixed strength. As theoretical evaluation of the expected cost per unit time turns out to be very complicated, a simulation-based algorithm is proposed to evaluate the expected cost rate and find the optimal replacement strategy. The proposed method is easy to implement having wider domain of application including non-Poisson arrival of shocks and non-Exponential damage distributions. For illustration, the proposed method is applied to real case studies on mailbox and cell-phone battery experiments.<\/jats:p>","DOI":"10.1007\/s10479-021-04080-6","type":"journal-article","created":{"date-parts":[[2021,5,4]],"date-time":"2021-05-04T12:03:56Z","timestamp":1620129836000},"page":"1345-1371","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Optimal replacement policy under cumulative damage model and strength degradation with applications"],"prefix":"10.1007","volume":"315","author":[{"given":"Phalguni","family":"Nanda","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0026-2193","authenticated-orcid":false,"given":"Prajamitra","family":"Bhuyan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Anup","family":"Dewanji","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2021,5,4]]},"reference":[{"issue":"4","key":"4080_CR1","first-page":"766","volume":"51","author":"P Bhuyan","year":"2017","unstructured":"Bhuyan, P., & Dewanji, A. (2017a). Estimation of reliability with cumulative stress and strength degradation. Statistics\u2014A Journal of Theoretical and Applied Statistics, 51(4), 766\u2013781.","journal-title":"Statistics\u2014A Journal of Theoretical and Applied Statistics"},{"issue":"4","key":"4080_CR2","doi-asserted-by":"publisher","first-page":"2701","DOI":"10.1080\/03610918.2015.1057288","volume":"46","author":"P Bhuyan","year":"2017","unstructured":"Bhuyan, P., & Dewanji, A. (2017b). Reliability computation under dynamic stress-strength modeling with cumulative stress and strength degradation. Communications in Statistics - Simulation and Computation, 46(4), 2701\u20132713.","journal-title":"Communications in Statistics - Simulation and Computation"},{"issue":"1","key":"4080_CR3","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1002\/nav.3800280103","volume":"28","author":"S Chikte","year":"1981","unstructured":"Chikte, S., & Deshmukh, S. (1981). Preventive maintenance and replacement under additive damage. Naval Research Logistics Quarterly, 28(1), 33\u201346.","journal-title":"Naval Research Logistics Quarterly"},{"key":"4080_CR4","volume-title":"Simulated annealing in modern heuristic techniques for combinatorial problems","author":"K Dowsland","year":"1995","unstructured":"Dowsland, K. (1995). Simulated annealing in modern heuristic techniques for combinatorial problems. New York: McGraw-Hill."},{"key":"4080_CR5","doi-asserted-by":"publisher","first-page":"9","DOI":"10.1007\/BF00773665","volume":"45","author":"N Ebrahimi","year":"1993","unstructured":"Ebrahimi, N., & Ramallingam, T. (1993). Estimation of system reliability in brownian stress-strength models based on sample paths. Annals of the Institute of Statistical Mathematics, 45, 9\u201319.","journal-title":"Annals of the Institute of Statistical Mathematics"},{"issue":"4","key":"4080_CR6","doi-asserted-by":"publisher","first-page":"1450021","DOI":"10.1142\/S0218539314500211","volume":"21","author":"A Endharta","year":"2014","unstructured":"Endharta, A., & Yun, W. (2014). A comparison study of replacement policies for a cumulative damage model. International Journal of Reliability Quality and Safety Engineering, 21(4), 1450021.","journal-title":"International Journal of Reliability Quality and Safety Engineering"},{"key":"4080_CR7","doi-asserted-by":"publisher","first-page":"40","DOI":"10.1016\/j.cie.2016.11.017","volume":"103","author":"S Eryilmaz","year":"2017","unstructured":"Eryilmaz, S. (2017). Computing optimal replacement time and mean residual life in reliability shock models. Computers & Industrial Engineering, 103, 40\u201345.","journal-title":"Computers & Industrial Engineering"},{"issue":"4598","key":"4080_CR8","doi-asserted-by":"publisher","first-page":"671","DOI":"10.1126\/science.220.4598.671","volume":"220","author":"S Kirkpatrick","year":"1983","unstructured":"Kirkpatrick, S., Gelatt, C., & Vecchi, M. (1983). Optimization by simulated annealing. Science, 220(4598), 671\u2013680.","journal-title":"Science"},{"key":"4080_CR9","doi-asserted-by":"publisher","DOI":"10.1002\/0471722065","volume-title":"Continuous multivariate distributions : Models and applications","author":"S Kotz","year":"2000","unstructured":"Kotz, S., Balakrishnan, N., & Johnson, N. L. (2000). Continuous multivariate distributions\u202f: Models and applications. Hoboken: John Wiey and Sons Inc."},{"key":"4080_CR10","first-page":"3","volume":"35","author":"H Lee","year":"2018","unstructured":"Lee, H., & Cha, J. H. (2018). A bivariate optimal replacement policy for a system subject to a generalized failure and repair process. Applied Stochastic Models in Business and Industry, 35, 3.","journal-title":"Applied Stochastic Models in Business and Industry"},{"issue":"4","key":"4080_CR11","doi-asserted-by":"publisher","first-page":"895","DOI":"10.1057\/jors.1976.178","volume":"27","author":"T Nakagawa","year":"1976","unstructured":"Nakagawa, T. (1976). On a replacement problem of a cumulative damage model. Operations Research Quaterly, 27(4), 895\u2013900.","journal-title":"Operations Research Quaterly"},{"key":"4080_CR12","volume-title":"Maintenance theory of reliability","author":"T Nakagawa","year":"2005","unstructured":"Nakagawa, T. (2005). Maintenance theory of reliability. London: Springer-Verlag Ltd."},{"key":"4080_CR13","volume-title":"Shock and damage models in reliability theory","author":"T Nakagawa","year":"2007","unstructured":"Nakagawa, T. (2007). Shock and damage models in reliability theory. London: Springer-Verlag London Ltd."},{"key":"4080_CR14","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61786-7","volume-title":"Quadpack: A subroutine package for automatic integration","author":"R Piessens","year":"1983","unstructured":"Piessens, R., Doncker-Kapenga, E. D., Uberhuber, C., & Kahaner, D. (1983). Quadpack: A subroutine package for automatic integration. Berlin: Springer-Verlag."},{"issue":"4","key":"4080_CR15","doi-asserted-by":"publisher","first-page":"1164","DOI":"10.1109\/TR.2015.2461217","volume":"64","author":"K Rafiee","year":"2015","unstructured":"Rafiee, K., Feng, Q., & Coit, D. W. (2015). Condition-based maintenance for repairable deteriorating systems subject to a generalized mixed shock model. IEEE Transactions on Reliability, 64(4), 1164\u20131174.","journal-title":"IEEE Transactions on Reliability"},{"key":"4080_CR16","doi-asserted-by":"publisher","first-page":"313","DOI":"10.1016\/S0895-7177(00)00101-1","volume":"31","author":"T Satow","year":"2000","unstructured":"Satow, T., Teramoto, K., & Nakagawa, T. (2000). Optimal replacement policy for a cumulative damage model with time deterioration. Mathematical and Computer Modelling, 31, 313\u2013319.","journal-title":"Mathematical and Computer Modelling"},{"key":"4080_CR17","doi-asserted-by":"publisher","first-page":"331","DOI":"10.1016\/0951-8320(95)00056-9","volume":"51","author":"P Scarf","year":"1996","unstructured":"Scarf, P., Wang, W., & Laycock, P. (1996). A stochastic model of crack growth under periodic inspection. Reliability Engineering & System Safety, 51, 331\u2013339.","journal-title":"Reliability Engineering & System Safety"},{"key":"4080_CR18","doi-asserted-by":"publisher","first-page":"563","DOI":"10.1016\/j.apm.2019.04.015","volume":"73","author":"SH Shue","year":"2019","unstructured":"Shue, S. H., Liu, T. H., Tsai, H. N., & Zhang, Z. G. (2019). Optimization issues in k-out-of-n systems. Applied Mathematical Modelling, 73, 563\u2013580.","journal-title":"Applied Mathematical Modelling"},{"key":"4080_CR19","doi-asserted-by":"publisher","first-page":"652","DOI":"10.2307\/1427411","volume":"19","author":"K Sobczyk","year":"1987","unstructured":"Sobczyk, K. (1987). Stochastic models for fatigue damage of materials. Advances in Applied Probability, 19, 652\u2013673.","journal-title":"Advances in Applied Probability"},{"key":"4080_CR20","volume-title":"Random fatigue: From Data to Theory","author":"K Sobczyk","year":"1992","unstructured":"Sobczyk, K., & Spencer, B. (1992). Random fatigue: From Data to Theory. New York: Academic Press."},{"issue":"1","key":"4080_CR21","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1002\/nav.3800220102","volume":"22","author":"H Taylor","year":"1975","unstructured":"Taylor, H. (1975). Optimal replacement under additive damage and other failure models. Naval Research Logistics Quarterly, 22(1), 1\u201318.","journal-title":"Naval Research Logistics Quarterly"},{"key":"4080_CR22","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-70456-2","volume-title":"Advanced maintenance policies for shock and damage models","author":"X Zhao","year":"2018","unstructured":"Zhao, X., & Nakagawa, T. (2018). Advanced maintenance policies for shock and damage models. Berlin: Springer International Publishing AG."},{"key":"4080_CR23","doi-asserted-by":"publisher","first-page":"50","DOI":"10.1016\/j.ress.2012.09.004","volume":"110","author":"X Zhao","year":"2013","unstructured":"Zhao, X., Qian, C., & Nakagawa, T. (2013). Optimal policies for cumulative damage models with maintenance last and first. Reliability Engineering and System Safety, 110, 50.","journal-title":"Reliability Engineering and System Safety"},{"key":"4080_CR24","doi-asserted-by":"publisher","first-page":"383","DOI":"10.1016\/j.cie.2018.03.005","volume":"118","author":"X Zhao","year":"2018","unstructured":"Zhao, X., Cai, K., Wang, X., & Song, Y. (2018). Optimal replacement policies for a shock model with a change point. Computers & Industrial Engineering, 118, 383\u2013393.","journal-title":"Computers & Industrial Engineering"},{"key":"4080_CR25","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.ress.2016.05.005","volume":"154","author":"X Zhou","year":"2016","unstructured":"Zhou, X., Wu, C., Li, Y., & Xi, L. (2016). A preventive maintenance model for leased equipment subject to internal degradation and external shock damage. Reliability Engineering and System Safety, 154, 1.","journal-title":"Reliability Engineering and System Safety"},{"issue":"4","key":"4080_CR26","doi-asserted-by":"publisher","first-page":"549","DOI":"10.1002\/nav.3800240404","volume":"24","author":"D Zuckerman","year":"1977","unstructured":"Zuckerman, D. (1977). Replacement models under additive damage. Naval Research Logistics Quarterly, 24(4), 549\u2013558.","journal-title":"Naval Research Logistics Quarterly"}],"container-title":["Annals of Operations Research"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10479-021-04080-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10479-021-04080-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10479-021-04080-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,9,20]],"date-time":"2022-09-20T04:23:50Z","timestamp":1663647830000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10479-021-04080-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,4]]},"references-count":26,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2022,8]]}},"alternative-id":["4080"],"URL":"https:\/\/doi.org\/10.1007\/s10479-021-04080-6","relation":{},"ISSN":["0254-5330","1572-9338"],"issn-type":[{"value":"0254-5330","type":"print"},{"value":"1572-9338","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,5,4]]},"assertion":[{"value":"9 April 2021","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 May 2021","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}