An Imperfect Repair Model with Delayed Repair under Replacement and Repair Thresholds
Abstract
:1. Introduction
- A novel model for imperfect delayed repair is built by using extended geometric processes.
- Replacement and repair thresholds are involved.
- Two kinds of replacement policies and are considered.
- The explicit expressions of the long-run average cost rate are obtained.
- The existence of optimal policies is proved, and numerical examples are presented to demonstrate the application of the results obtained in the paper.
2. Problem Definition
3. Optimization Model Development
3.1. Long-Run Average Cost Rate under Replacement Policy
3.2. Special Cases
4. Numerical Example
4.1. Long-Run Average Cost Rate under Policy
Algorithm 1 Long-Run Average Cost Rate under Policy |
Input , , , , , , , , , . |
Step 1. Compute as defined by Equation (17). |
Step 2. Find the optimal threshold to minimize ; output and . |
Step 3. For to , compute as defined by Equation (17). |
Step 4. Plot against threshold . |
Stop. |
4.2. Long-Run Average Cost Rate under Policy
Algorithm 2 Long-Run Average Cost Rate under Policy |
Input , , , , , , , , , . |
Step 1. Compute as defined by Equation (16). |
Step 2. Find the optimal and to minimize . |
Step 3. Input ; for to , to , compute as defined by Equation (16). |
Step 4. Plot against and . |
Stop. |
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
- Lam, Y. A note on the optimal replacement problem. Adv. Appl. Probab. 1988, 20, 479–482. [Google Scholar] [CrossRef] [Green Version]
- Lam, Y. Geometric processes and replacement problem. Acta Math. Appl. Sin. 1988, 4, 366–377. [Google Scholar] [CrossRef]
- Wu, D.; Peng, R.; Wu, S.M. A review of the extensions of the geometric process, applications, and challenges. Qual. Reliab. Eng. Int. 2020, 36, 436–446. [Google Scholar] [CrossRef]
- Zhang, Y.L.; Wang, G.J. An extended geometric process repair model with delayed repair and slight failure type. Commun. Stat.-Theory Methods 2017, 46, 427–437. [Google Scholar] [CrossRef]
- Chan, J.S.K.; Yu, P.L.H.; Lam, Y.; Ho, A.P.K. Modelling SARS data using threshold geometric process. Stat. Med. 2006, 25, 1826–1839. [Google Scholar] [CrossRef]
- Wu, S.M. Doubly geometric processes and applications. J. Oper. Res. Soc. 2018, 69, 66–77. [Google Scholar] [CrossRef] [Green Version]
- Sarada, Y.; Shenbagam, R. On a random lead time and threshold shock model using phase-type geometric processes. Appl. Stoch. Models Bus. Ind. 2018, 34, 407–422. [Google Scholar] [CrossRef]
- Braun, W.J.; Li, W.; Zhao, Y.Q. Properties of the geometric and related processes. Nav. Res. Logist. 2005, 52, 607–616. [Google Scholar] [CrossRef] [Green Version]
- Sun, Q.Z.; Ye, Z.S.; Zhu, X. Managing component degradation in series systems for balancing degradation through reallocation and maintenance. IISE Trans. 2020, 52, 797–810. [Google Scholar] [CrossRef]
- Zhang, Y.L.; Wang, G.J. An extended geometric process repair model for a cold standby repairable system with imperfect delayed repair. Int. J. Syst. Sci. Oper. Logist. 2016, 3, 163–175. [Google Scholar] [CrossRef]
- Zhang, Y.L.; Wang, G.J. An extended geometric process repair model with imperfect delayed repair under different objective functions. Commun. Stat.-Theory Methods 2018, 47, 3204–3219. [Google Scholar] [CrossRef]
- Wang, J.Y.; Ye, J.M.; Ma, Q.R.; Xie, P.F. An extended geometric process repairable model with its repairman having vacation. Ann. Oper. Res. 2022, 311, 401–415. [Google Scholar] [CrossRef]
- Zhang, Y.L. A bivariate optimal replacement policy for a repairable system. J. Appl. Probab. 1994, 31, 1123–1127. [Google Scholar] [CrossRef]
- Wang, G.J.; Zhang, Y.L. Optimal periodic preventive repair and replacement policy assuming geometric process repair. IEEE Trans. Reliab. 2006, 55, 118–122. [Google Scholar] [CrossRef]
- Wang, G.J.; Zhang, Y.L. A bivariate mixed policy for a simple repairable system based on preventive repair and failure repair. Appl. Math. Model. 2009, 33, 3354–3359. [Google Scholar] [CrossRef] [Green Version]
- Chang, C.C.; Sheu, S.H.; Chen, Y.L. A bivariate optimal replacement policy for a system with age-dependent minimal repair and cumulative repair-cost limit. Commun. Stat.-Theory Methods 2013, 42, 4108–4126. [Google Scholar] [CrossRef]
- Sheu, S.H.; Chien, Y.H.; Chang, C.C.; Chiu, C.H. Optimal trivariate replacement policies for a deteriorating system. Qual. Technol. Quant. Manag. 2014, 11, 307–320. [Google Scholar] [CrossRef]
- Dong, Q.L.; Cui, L.R.; Gao, H.D. A bivariate replacement policy for an imperfect repair system based on geometric processes. Proc. Inst. Mech. Eng. Part O J. Risk Reliab. 2019, 233, 670–681. [Google Scholar] [CrossRef]
- Sun, Q.Z.; Ye, Z.S.; Chen, N. Optimal inspection and replacement policies for multi-unit systems subject to degradation. IEEE Trans. Reliab. 2018, 47, 401–413. [Google Scholar] [CrossRef]
- Qiu, Q.A.; Maillart, L.; Prokopyev, O.; Cui, L.R. Optimal condition-based mission abort decisions. IEEE Trans. Reliab. 2022, 1–18. [Google Scholar] [CrossRef]
- Mendes, A.A.; Ribeiro, J.L.D.; Coit, D.W. Optimal time interval between periodic inspections for a two-component cold standby multistate system. IEEE Trans. Reliab. 2017, 66, 559–574. [Google Scholar] [CrossRef]
- Wang, J.J.; Qiu, Q.A.; Wang, H.H. Joint optimization of condition-based and age-based replacement policy and inventory policy for a two-unit series system. Reliab. Eng. Syst. Saf. 2021, 205, 107251. [Google Scholar] [CrossRef]
- Yang, L.; Chen, Y.; Qiu, Q.A.; Wang, J. Risk control of mission-critical systems: Abort decision-makings integrating health and age conditions. IEEE T. Ind. Inform. 2022. [Google Scholar] [CrossRef]
- Dong, Q.L.; Cui, L.R.; Si, S.B. Reliability and availability analysis of stochastic degradation systems based on bivariate Wiener processes. Appl. Math. Model. 2020, 79, 414–433. [Google Scholar] [CrossRef]
- Dong, Q.L.; Cui, L.R. Reliability analysis of a system with two-stage degradation using Wiener processes with piecewise linear drift. IMA J. Manag. Math. 2021, 32, 3–19. [Google Scholar] [CrossRef]
- Yu, M.M.; Tang, Y.H.; Wu, W.Q.; Zhou, J. Optimal order-replacement policy for a phase-type geometric process model with extreme shocks. Appl. Math. Model. 2014, 38, 4323–4332. [Google Scholar] [CrossRef]
- Zhang, Y.L. A geometrical process repair model for a repairable system with delayed repair. Comput. Math. Appl. 2008, 55, 1629–1643. [Google Scholar] [CrossRef] [Green Version]
- Zhang, M.M.; Ye, Z.S.; Xie, M. A condition-based maintenance strategy for heterogeneous populations. Comput. Ind. Eng. 2014, 77, 103–114. [Google Scholar] [CrossRef]
- Xie, W.; Liao, H.T.; Jin, T.D. Maximizing system availability through joint decision on component redundancy and spares inventory. Eur. J. Oper. Res. 2014, 237, 164–176. [Google Scholar] [CrossRef]
- Zhao, X.F.; Mizutani, S.; Nakagawa, T. Which is better for replacement policies with continuous or discrete scheduled times? Eur. J. Oper. Res. 2015, 242, 477–486. [Google Scholar] [CrossRef]
- Liu, B.; Xie, M.; Xu, Z.G.; Kuo, W. An imperfect maintenance policy for mission-oriented systems subject to degradation and external shocks. Comput. Ind. Eng. 2016, 102, 21–32. [Google Scholar] [CrossRef] [Green Version]
- Tsai, H.N.; Sheu, S.H.; Zhang, Z.G. A trivariate optimal replacement policy for a deteriorating system based on cumulative damage and inspections. Reliab. Eng. Syst. Saf. 2017, 160, 74–88. [Google Scholar] [CrossRef]
- Zhao, X.; Guo, X.X.; Wang, X.Y. Reliability and maintenance policies for a two-stage shock model with self-healing mechanism. Reliab. Eng. Syst. Saf. 2017, 172, 185–194. [Google Scholar] [CrossRef]
- Zhao, X.J.; Gaudoin, O.; Doyen, L.; Xie, M. Optimal inspection and replacement policy based on experimental degradation data with covariates. IISE Trans. 2019, 51, 322–336. [Google Scholar] [CrossRef]
- Gao, K.Y.; Peng, R.; Qu, L.; Wu, S.M. Jointly optimizing lot sizing and maintenance policy for a production system with two failure modes. Reliab. Eng. Syst. Saf. 2020, 202, 106996. [Google Scholar] [CrossRef]
- Chen, K.; Zhao, X.; Qiu, Q.A. Optimal task abort and maintenance policies considering time redundancy. Mathematics 2022, 10, 1360. [Google Scholar] [CrossRef]
Literature | Models | Delayed Repair | Policy | Other Factors |
---|---|---|---|---|
Zhang [27] | GPRM | Independent of the working time; No cost | N | Average cost rate |
Zhang & Wang [10] | EGPRM | Independent of the working time; No cost | N | Cold standby and average cost rate |
Zhang & Wang [4] | EGPRM | Independent of the working time; No cost | N | Average reward rate |
Zhang & Wang [11] | EGPRM | Independent of the working time; No cost | N | Average cost rate and average availability rate |
Dong et al. [18] | GPRM | Dependent of the working time; a penalty proportional to the delayed repair time | System availability and average cost rate | |
Wang et al. [12] | EGPRM | Independent of the working time; No cost | N | The repairman has multiple vacation |
This paper | EGPRM | Dependent of the working time; a penalty proportional to the delayed repair time | , | Average cost rate |
3 | 3.0000 | 4.5377 |
5 | 5.0000 | 4.2703 |
8 | 8.0000 | 3.9904 |
10 | 9.8398 | 3.9459 |
15 | 10.8330 | 4.0076 |
20 | 12.0509 | 4.1864 |
25 | 13.4093 | 4.4771 |
30 | 14.8871 | 4.8706 |
N | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|
0 | 0 | 0 | 3.5386 | 6.4304 | 8.5769 | 10.3333 | 11.8186 | |
9.9598 | 7.5453 | 6.5230 | 6.0469 | 5.8372 | 5.7723 | 5.7868 | 5.8386 | |
N | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 |
13.0631 | 14.0577 | 14.7908 | 15.2778 | 15.5656 | 15.7148 | 15.7812 | 15.8058 | |
5.8999 | 5.9534 | 5.9918 | 6.0153 | 6.0277 | 6.0333 | 6.0354 | 6.0361 |
N | 7 | N | |
---|---|---|---|
8.5769 | 13.5929 | ||
5.7723 | 7.0073 |
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Sun, M.; Dong, Q.; Gao, Z. An Imperfect Repair Model with Delayed Repair under Replacement and Repair Thresholds. Mathematics 2022, 10, 2263. https://doi.org/10.3390/math10132263
Sun M, Dong Q, Gao Z. An Imperfect Repair Model with Delayed Repair under Replacement and Repair Thresholds. Mathematics. 2022; 10(13):2263. https://doi.org/10.3390/math10132263
Chicago/Turabian StyleSun, Mingjuan, Qinglai Dong, and Zihan Gao. 2022. "An Imperfect Repair Model with Delayed Repair under Replacement and Repair Thresholds" Mathematics 10, no. 13: 2263. https://doi.org/10.3390/math10132263
APA StyleSun, M., Dong, Q., & Gao, Z. (2022). An Imperfect Repair Model with Delayed Repair under Replacement and Repair Thresholds. Mathematics, 10(13), 2263. https://doi.org/10.3390/math10132263