Hacettepe Journal of Mathematics and Statistics 2651-477X 2651-477X Hacettepe University 10.15672/hujms.1798557 Probability Theory Stochastic Analysis and Modelling Statistics (Other) Olasılık Teorisi Olasılıksal Analiz ve Modelleme İstatistik (Diğer) Optimal condition-based maintenance strategy for a three-state system under dependent competing failures https://orcid.org/0000-0002-5043-6967 Guo Zhouxia Northwest Normal University https://orcid.org/0000-0002-2418-8859 Yan Rongfang Northwest Normal University 03 09 2026 55 2 752 782 10 07 2025 02 16 2026 Copyright © 2002, Hacettepe Journal of Mathematics and Statistics 2002 Hacettepe Journal of Mathematics and Statistics

Industrial systems function in dynamic settings characterized by changing conditions. This manuscript develops a reliability model and a condition-based maintenance approach for a single-component system with three states (normal, defective, and failed). The system is prone to failures resulting from both internal degradation and external shocks, and these shocks can cause abrupt rises in degradation. Internal degradation is simulated through a Wiener process that features elevated drift and diffusion coefficients when the system is in the defective state. External shocks follow a non-homogeneous Poisson process and are classified as safe, damaging, and fatal shocks. A preventive degradation threshold is defined according to the state of the system, and particle swarm optimization is utilized to determine the inspection interval and the optimal preventive threshold. Finally, the proposed maintenance model is verified using a numerical example.

 Competing failure degradation expected cost per unit time reliability shocks National Natural Science Foundation of China (No. 12361060); Industrial Support Program of Gansu Provincial Department of Education (No. 2025CYZC-016) [1] J. M. Graham, Soft failures, Cochlear Implants Int. 5(4), v–vi, 2004. [2] S. Taghipour, D. Banjevic and A. K. S. Jardine, Periodic inspection optimization model for a complex repairable system, Reliab. Eng. Syst. Saf. 95 (9), 944–952, 2010. [3] K. Rafiee, Q. Feng and D. W. Coit, Reliability analysis and condition-based maintenance for failure processes with degradation-dependent hard failure threshold, Qual. Reliab. Eng. Int. 33 (7), 1351–1366, 2016. [4] G. Wei, X. Zhao, S. He and Z. He, Reliability modeling with condition-based maintenance for binary-state deteriorating systems considering zoned shock effects, Comput. Ind. Eng. 130, 282–297, 2019. [5] M. N. Mustafa, Classification of maintenance techniques and diagnosing failures methods, J. Phys.: Conf. Ser. 2060 (1), 012014, 2021. [6] J. Liu, X. Zhuang and H. Pang, Reliability and hybrid maintenance modeling for competing failure systems with multistage periods, Probab. Eng. Mech. 68, 103254, 2022. [7] A. Shangguan, G. Xie, L. Mu, R. Fei and X. Hei, Reliability modeling: Combining selfhealing characteristics and dynamic failure thresholds, Qual. Technol. Quant. Manag. 21 (3), 363–385, 2024. [8] Y. Zhang, S. Zhao and B. Wu, Reliability and maintenance analysis of two-component system subject to zoned shocks and degradation processes, Reliab. Eng. Syst. Saf. 257, 110812, 2025. [9] F. Dufresne, H. U. Gerber and E. S. W. Shiu, Risk theory with the gamma process, ASTIN Bull. 21 (2), 177–192, 1991. [10] A. Csenki, Mission availability for repairable semi-Markov systems: Analytical results and computational implementation, Statistics 26 (1), 75–87, 1995. [11] A. G. Malliaris, Wiener process, In Time Series and Statistics, pp. 316–318, 1990. [12] J. Lawless and M. Crowder, Covariates and random effects in a gamma process model with application to degradation and failure, Lifetime Data Anal. 10, 213–227, 2004. [13] X. Wang and D. Xu, An inverse gaussian process model for degradation data, Technometrics 52 (2), 188–197, 2010. [14] Z.-S. Ye and N. Chen, The inverse gaussian process as a degradation model, Technometrics 56 (3), 302–311, 2014. [15] Z. Huang, Z. Xu, W. Wang and Y. Sun, Remaining useful life prediction for a nonlinear heterogeneous wiener process model with an adaptive drift, IEEE Trans. Reliab. 64 (2), 687–700, 2015. [16] R. Kumar, D. B. H. Cline and P. Gardoni, A stochastic framework to model deterioration in engineering systems, Struct. Saf. 53, 36–43, 2015. [17] H. P. Suryawan, Existence and regularization of the local times of a gaussian process, Hacet. J. Math. Stat. 47 (5), 1206–1215, 2018. [18] L. A. Rodríguez-Picón, A. P. Rodríguez-Picón and A. Alvarado-Iniesta, Degradation modeling of 2 fatigue-crack growth characteristics based on inverse gaussian processes: A case study, Appl. Stoch. Model. Bus. Ind. 35 (3), 504–521, 2019. [19] Z. Zhang, X. Si, C. Hu and Y. Lei, Degradation data analysis and remaining useful life estimation: A review on wiener-process-based methods, Eur. J. Oper. Res. 271 (3), 775–796, 2018. [20] M. Leroy, C. Bérenguer, L. Doyen and O. Gaudoin, Statistical inference for a wienerbased degradation model with imperfect maintenance actions under different observation schemes, Appl. Stoch. Model. Bus. Ind. 39 (3), 352–371, 2023. [21] M. Tanwar and N. Raghavan, Continuous degradation modeling under random shocks and imperfect maintenance using generalized renewal process, J. Qual. Maint. Eng. 31 (1), 1–16, 2024. [22] W. Chen and S. Hao, Condition-based operation and maintenance strategy for loadsharing systems based on wiener process, IEEE Trans. Reliab., 74 (3), 4402-4416, 2025. [23] J. A. M. van der Weide and M. D. Pandey, Stochastic analysis of shock process and modeling of condition-based maintenance, Reliab. Eng. Syst. Saf. 96 (6), 619–626, 2011. [24] A. H. S. Garmabaki, A. Ahmadi, Y. A. Mahmood and A. Barabadi, Reliability modelling of multiple repairable units, Qual. Reliab. Eng. Int. 32 (7), 2329–2343, 2015. [25] N. C. Caballé and I. T. Castro, Analysis of the reliability and the maintenance cost for finite life cycle systems subject to degradation and shocks, Appl. Math. Model. 52, 731–746, 2017. [26] C. Bai, P. Li and Y. Liu, Reliability assessment for dependent competing failure process with shifting-threshold and imperfect shock model, In 2019 Int. Conf. Qual. Reliab. Risk Maint. Saf. Eng. (QR2MSE), pp. 736–741, 2019. [27] S. Gan, Z. Song and L. Zhang, A maintenance strategy based on system reliability considering imperfect corrective maintenance and shocks, Comput. Ind. Eng. 164, 107886, 2022. [28] Q. Liang, S. Liu and C. Peng, Reliability analysis of multi-component systems subjected to dependent degradation processes and random shocks in dynamic environments, Process Saf. Environ. Prot. 190, 1546–1561, 2024. [29] W. Dong, Y. Cao and J. Zhang, Reliability modeling and optimal maintenance strategies for stochastically deteriorating systems with random degradation latency, Ann. Oper. Res. 345 (1), 105–124, 2025. [30] J. Niu, R. Yan and J. Zhang, Preventive replacement policies of parallel/series systems with dependent components under deviation costs, Reliab. Eng. Syst. Saf. 260, 111033, 2025. [31] M. Park and S.-I. Sung, Reliability analysis of the dependent degradation-thresholdshock model for a system, Qual. Technol. Quant. Manag., 23 (2), 169–187, 2026. [32] J. H. Cha, Generalizations of the Poisson process and their reliability applications, In Reliab. Anal. Maint. Optim. Complex Syst., pp. 85–95, 2025. [33] A. Lisnianski and G. Levitin, Multi-state system reliability: assessment, optimization and applications, World Scientific, 2003. [34] H. Pham, A. Suprasad and R. B. Misra, Availability and mean life time prediction of multistage degraded system with partial repairs, Reliab. Eng. Syst. Saf. 56 (2), 169–173, 1997. [35] G. Levitin, Block diagram method for analyzing multi-state systems with uncovered failures, Reliab. Eng. Syst. Saf. 92(6):727–734, 2007. [36] M. Ram, Reliability measures of a three-state complex system: a copula approach, Appl. Appl. Math. 5 (2), 10, 2010. [37] S. Eryilmaz, Dynamic reliability and performance evaluation of multi-state systems with two components, Hacet. J. Math. Stat. 40 (1), 125–133, 2011. [38] K. Rafiee, Q. Feng and D. W. Coit, Reliability modeling for dependent competing failure processes with changing degradation rate, IIE Trans. 46 (5), 483–496, 2014. [39] R. Peng, H. Liu and M. Xie, A study of reliability of multi-state systems with two performance sharing groups, Qual. Reliab. Eng. Int. 32 (7), 2623–2632, 2016. [40] J. Yu, S. Zheng, H. Pham and T. Chen, Reliability modeling of multi-state degraded repairable systems and its applications to automotive systems, Qual. Reliab. Eng. Int. 34 (3), 459–474, 2018. [41] H. Hao and C. Li, Multi-state reliability analysis based on general wiener degradation process and random shock, Shock Vib. 2022, 1–13, 2022. [42] H. Dui, Y. Lu and S. Wu, Competing risks-based resilience approach for multi-state systems under multiple shocks, Reliab. Eng. Syst. Saf. 242, 109773, 2024. [43] A. Erem Halilsoy and F. Iscioglu, Reliability analysis of (r, s)-out-of-n multi-state systems using copulas, Eng. Comput. 42 (1), 234–254, 2025. [44] G. Levitin and A. Lisnianski, Optimization of imperfect preventive maintenance for multi-state systems, Reliab. Eng. Syst. Saf. 67 (2), 193–203, 2000. [45] W. Li and H. Pham, An inspection-maintenance model for systems with multiple competing processes, IEEE Trans. Reliab. 54 (2), 318–327, 2005. [46] M.-C. Fitouhi and M. Nourelfath, Integrating noncyclical preventive maintenance scheduling and production planning for multi-state systems, Reliab. Eng. Syst. Saf. 121, 175–186, 2014. [47] K. Atashgar and H. Abdollahzadeh, A joint reliability and imperfect opportunistic maintenance optimization for a multi-state weighted k-out-of-n system considering economic dependence and periodic inspection, Qual. Reliab. Eng. Int. 33 (8), 1685– 1707, 2017. [48] L. Yang, X. Ma and Y. Zhao, A condition-based maintenance model for a three-state system subject to degradation and environmental shocks, Comput. Ind. Eng. 105, 210–222, 2017. [49] L. KnopiK and K. MigAwA, Multi-state model of maintenance policy, Eksploat. Niezawodn. 20 (1), 125–130, 2018. [50] P. Do Van and C. Bérenguer, Condition-based maintenance with imperfect preventive repairs for a deteriorating production system, Qual. Reliab. Eng. Int. 28 (6), 624–633, 2012. [51] Y. Wang, Y. Liu, J. Chen and X. Li, Reliability and condition-based maintenance modeling for systems operating under performance-based contracting, Comput. Ind. Eng. 142, 106344, 2020. [52] Z. Dahia, A. Bellaouar and J.-P. Dron, A dynamic approach for maintenance evaluation and optimization of multistate system, J. Ind. Eng. Int. 17 (1), 1, 2021. [53] G. Satoshi and J. Lu, Threshold policy of condition-based maintenance for a multistate system and deteriorating sensor, Total Qual. Sci. 7 (2), 89–101, 2022. [54] M. Han, Z. Ma, S. Ma, W. Li and J. Yang, Joint optimization of condition-based maintenance and spare ordering strategy for a multistate competing failure system, IEEE Access 11, 37006–37020, 2023. [55] Y. Lu, S. Wang, C. Zhang, R. Chen, H. Dui, D. Mazurkiewicz and Y. Zhang, A dynamic imperfect inspection-based maintenance optimization considering dependent competing failure, Measurement 253, 117470, 2025. [56] J. Wang, S. Zhou, R. Peng, Q. Qiu and L. Yang, An inspection-based replacement planning in consideration of state-driven imperfect inspections, Reliab. Eng. Syst. Saf. 232, 109064, 2023. [57] H. Che, S. Zeng, J. Guo and Y. Wang, Reliability modeling for dependent competing failure processes with mutually dependent degradation process and shock process, Reliab. Eng. Syst. Saf. 180, 168–178, 2018. [58] W. Wang, An inspection model based on a three-stage failure process, Reliab. Eng. Syst. Saf. 96 (7), 838–848, 2011. [59] A. Jiang, L. Huang and Y. Xiang, Reliability analysis for competitive failure processes with multi-state degradation, In 2020 Asia-Pacific Int. Symp. Adv. Reliab. Maint. Model. (APARM), pp. 1–7, IEEE, 2020. [60] S. M. Ross, Introduction to probability models, Academic Press, 2014. [61] S. M. Ross, Stochastic processes, John Wiley & Sons, 1995. [62] J. Kennedy and R. Eberhart, Particle swarm optimization, In Proc. ICNN’95, 4, 1942–1948, 1995. [63] N. Chen, Z.-S. Ye, Y. Xiang and L. Zhang, Condition-based maintenance using the inverse gaussian degradation model, Eur. J. Oper. Res. 243 (1), 190–199, 2015.