January  2021, 17(1): 501-516. doi: 10.3934/jimo.2020149

Optimal preventive "maintenance-first or -last" policies with generalized imperfect maintenance models

1. 

Department of Marketing Management, Takming University of Science and Technology, Taipei 114, Taiwan

2. 

Department of Chains and Franchising Management, Takming University of Science and Technology, Taipei 114, Taiwan

3. 

Department of Decision Sciences, Western Washington University, Bellingham, WA 98225-9077, USA

* Corresponding author: Chin-Chih Chang

Received  April 2016 Revised  March 2018 Published  September 2020

This paper presents modified preventive maintenance policies for an operating system that works at random processing times and is imperfectly maintained. The system may suffer from one of the two types of failures based on a time-dependent imperfect maintenance mechanism: type-Ⅰ (minor) failure, which can be rectified by minimal repair, and type-Ⅱ (catastrophic) failure, which can be removed by corrective maintenance. When the system needs to be maintained, two policies "preventive maintenance-first (PMF) and preventive maintenance-last (PML)" may be applied. In each maintenance interval, before any type-Ⅱ failure occurs, the system is maintained at a planned time $ T $ or at the completion of a working time, whichever occurs first and last, which are called PMF and PML, respectively. After any maintenance activity, the system improves but its failure characteristic is also altered. At the $ N $-th maintenance, the system is replaced rather than maintained. For each policy, the optimal preventive maintenance schedule ($ T $, $ N $)$ ^{*} $ that minimizes the mean cost rate function is derived analytically and determined numerically in terms of its existence and uniqueness. The proposed models provide a general framework for analyzing the maintenance policies in reliability theory.

Citation: Yen-Luan Chen, Chin-Chih Chang, Zhe George Zhang, Xiaofeng Chen. Optimal preventive "maintenance-first or -last" policies with generalized imperfect maintenance models. Journal of Industrial & Management Optimization, 2021, 17 (1) : 501-516. doi: 10.3934/jimo.2020149
References:
[1]

R. Barlow and L. Hunter, Optimum preventive maintenance policies, Operations Res., 8 (1960), 90-100.  doi: 10.1287/opre.8.1.90.  Google Scholar

[2]

F. Beichelt, A unifying treatment of replacement policies with minimal repair, Naval Research Logistics, 40 (1993), 51-67.   Google Scholar

[3]

M. Berg and R. Cléroux, A marginal cost analysis for an age replacement policy with minimal repair, INFOR, 20 (1982), 258-263.   Google Scholar

[4]

H. W. BlockW. S. Broges and T. H. Savits, Age-dependent minimal repair, J. Appl. Probab., 22 (1985), 370-385.  doi: 10.2307/3213780.  Google Scholar

[5]

M. Brown and F. Proschan, Imperfect repair, J. Appl. Probab., 20 (1983), 851-859.  doi: 10.2307/3213596.  Google Scholar

[6]

C. C. Chang, Optimum preventive maintenance policies for systems subject to random working time, replacement, and minimal repair, Computers & Industrial Engineering, 67 (2014), 185–194. Google Scholar

[7]

C. C. Chang, S. H. Sheu and Y. L. Chen, Optimal number of minimal repairs before replacement based on a cumulative repair-cost limit policy, Computers & Industrial Engineering, 59 (2010), 603–610. Google Scholar

[8]

C.-C. ChangS.-H. Sheu and Y.-L. Chen, A bivariate optimal replacement policy for a system with age-dependent minimal repair and cumulative repair-cost limit, Comm. Statist. Theory Methods, 42 (2013), 4108-4126.  doi: 10.1080/03610926.2011.648789.  Google Scholar

[9]

M. ChenS. Mizutani and T. Nakagawa, Random and age replacement policies, International Journal of Reliability, Quality and Safety Engineering, 17 (2010), 27-39.   Google Scholar

[10]

Y. L. Chen, A bivariate optimal imperfect preventive maintenance policy for a used system with two-type shocks, Computers & Industrial Engineering, 63 (2012), 1227-1234.   Google Scholar

[11]

M. Kijima, Some results for repairable systems with general repair, J. Appl. Probab., 26 (1989), 89-102.  doi: 10.2307/3214319.  Google Scholar

[12]

M. KijimaH. Morimura and Y Sujuki, Periodical replacement problem without assuming minimal repair, European J. Oper. Res., 37 (1988), 194-203.  doi: 10.1016/0377-2217(88)90329-3.  Google Scholar

[13]

T. Nakagawa, Periodic and sequential preventive maintenance policies, J. Appl. Probab., 23 (1986), 536-542.  doi: 10.1017/S0021900200029843.  Google Scholar

[14]

T. Nakagawa, Sequential imperfect preventive maintenance policies, IEEE Transactions on Reliability, 37 (1988), 295-298.   Google Scholar

[15]

T. Nakagawa, Maintenance Theory of Reliability, Springer, London, 2005. Google Scholar

[16]

T. Nakagawa and X. Zhao, Comparisons of replacement policies with constant and random times, J. Oper. Res. Soc. Japan, 56 (2013), 1-14.  doi: 10.15807/jorsj.56.1.  Google Scholar

[17]

D. G. Nguyen and D. N. P. Murthy, Optimal preventive maintenance policies for repairable systems, Oper. Res., 29 (1981), 1181-1194.  doi: 10.1287/opre.29.6.1181.  Google Scholar

[18]

H. Pham and H. Wang, Imperfect maintenance, European Journal of Operational Research, 94 (1996), 425-438.   Google Scholar

[19]

P. S. Puri and H. Singh, Optimum replacement of a system subject to shocks: A mathematical lemma, Oper. Res., 34 (1986), 782-789.  doi: 10.1287/opre.34.5.782.  Google Scholar

[20]

S. H. Sheu, A generalized age and block replacement of a system subject to shocks, European Journal of Operational Research, 108 (1998), 345-362.   Google Scholar

[21]

S. H. Sheu and C. T. Liou, A generalized sequential preventive maintenance policy for repairable system with general random repair costs, International Journal of Systems Science, 26 (1995), 681-690.   Google Scholar

[22]

T. SugiuraS. Mizutani and T. Nakagawa, Optimal random replacement policies, Tenth ISSAT International Conference on Reliability and Quality in Design, (2004), 99-103.   Google Scholar

[23]

H. Wang and H. Pham, Optimal Imperfect Maintenance Models, Pham, H. (ed), Handbook of Reliability Engineering, London: Spring, (2003), 397–414. Google Scholar

[24]

X. Zhao and T. Nakagawa, Optimization problem of replacement first or last in reliability theory, European J. Oper. Res., 223 (2012), 141-149.  doi: 10.1016/j.ejor.2012.05.035.  Google Scholar

[25]

X. ZhaoT. Nakagawa and C. Qian, Optimal imperfect preventive maintenance policies for a used system, Internat. J. Systems Sci., 43 (2012), 1632-1641.  doi: 10.1080/00207721.2010.549583.  Google Scholar

[26]

X. Zhao, M. Chen and T. Nakagawa, Optimal time and random inspection policies for computer systems, Applied Mathematics & Information Sciences, 8, (2014) 413–417. Google Scholar

[27]

X. ZhaoC. Qian and T. Nakagawa, Optimal policies for cumulative damage models with maintenance last and first, Reliability Engineering and System Safety, 110 (2013), 50-59.   Google Scholar

show all references

References:
[1]

R. Barlow and L. Hunter, Optimum preventive maintenance policies, Operations Res., 8 (1960), 90-100.  doi: 10.1287/opre.8.1.90.  Google Scholar

[2]

F. Beichelt, A unifying treatment of replacement policies with minimal repair, Naval Research Logistics, 40 (1993), 51-67.   Google Scholar

[3]

M. Berg and R. Cléroux, A marginal cost analysis for an age replacement policy with minimal repair, INFOR, 20 (1982), 258-263.   Google Scholar

[4]

H. W. BlockW. S. Broges and T. H. Savits, Age-dependent minimal repair, J. Appl. Probab., 22 (1985), 370-385.  doi: 10.2307/3213780.  Google Scholar

[5]

M. Brown and F. Proschan, Imperfect repair, J. Appl. Probab., 20 (1983), 851-859.  doi: 10.2307/3213596.  Google Scholar

[6]

C. C. Chang, Optimum preventive maintenance policies for systems subject to random working time, replacement, and minimal repair, Computers & Industrial Engineering, 67 (2014), 185–194. Google Scholar

[7]

C. C. Chang, S. H. Sheu and Y. L. Chen, Optimal number of minimal repairs before replacement based on a cumulative repair-cost limit policy, Computers & Industrial Engineering, 59 (2010), 603–610. Google Scholar

[8]

C.-C. ChangS.-H. Sheu and Y.-L. Chen, A bivariate optimal replacement policy for a system with age-dependent minimal repair and cumulative repair-cost limit, Comm. Statist. Theory Methods, 42 (2013), 4108-4126.  doi: 10.1080/03610926.2011.648789.  Google Scholar

[9]

M. ChenS. Mizutani and T. Nakagawa, Random and age replacement policies, International Journal of Reliability, Quality and Safety Engineering, 17 (2010), 27-39.   Google Scholar

[10]

Y. L. Chen, A bivariate optimal imperfect preventive maintenance policy for a used system with two-type shocks, Computers & Industrial Engineering, 63 (2012), 1227-1234.   Google Scholar

[11]

M. Kijima, Some results for repairable systems with general repair, J. Appl. Probab., 26 (1989), 89-102.  doi: 10.2307/3214319.  Google Scholar

[12]

M. KijimaH. Morimura and Y Sujuki, Periodical replacement problem without assuming minimal repair, European J. Oper. Res., 37 (1988), 194-203.  doi: 10.1016/0377-2217(88)90329-3.  Google Scholar

[13]

T. Nakagawa, Periodic and sequential preventive maintenance policies, J. Appl. Probab., 23 (1986), 536-542.  doi: 10.1017/S0021900200029843.  Google Scholar

[14]

T. Nakagawa, Sequential imperfect preventive maintenance policies, IEEE Transactions on Reliability, 37 (1988), 295-298.   Google Scholar

[15]

T. Nakagawa, Maintenance Theory of Reliability, Springer, London, 2005. Google Scholar

[16]

T. Nakagawa and X. Zhao, Comparisons of replacement policies with constant and random times, J. Oper. Res. Soc. Japan, 56 (2013), 1-14.  doi: 10.15807/jorsj.56.1.  Google Scholar

[17]

D. G. Nguyen and D. N. P. Murthy, Optimal preventive maintenance policies for repairable systems, Oper. Res., 29 (1981), 1181-1194.  doi: 10.1287/opre.29.6.1181.  Google Scholar

[18]

H. Pham and H. Wang, Imperfect maintenance, European Journal of Operational Research, 94 (1996), 425-438.   Google Scholar

[19]

P. S. Puri and H. Singh, Optimum replacement of a system subject to shocks: A mathematical lemma, Oper. Res., 34 (1986), 782-789.  doi: 10.1287/opre.34.5.782.  Google Scholar

[20]

S. H. Sheu, A generalized age and block replacement of a system subject to shocks, European Journal of Operational Research, 108 (1998), 345-362.   Google Scholar

[21]

S. H. Sheu and C. T. Liou, A generalized sequential preventive maintenance policy for repairable system with general random repair costs, International Journal of Systems Science, 26 (1995), 681-690.   Google Scholar

[22]

T. SugiuraS. Mizutani and T. Nakagawa, Optimal random replacement policies, Tenth ISSAT International Conference on Reliability and Quality in Design, (2004), 99-103.   Google Scholar

[23]

H. Wang and H. Pham, Optimal Imperfect Maintenance Models, Pham, H. (ed), Handbook of Reliability Engineering, London: Spring, (2003), 397–414. Google Scholar

[24]

X. Zhao and T. Nakagawa, Optimization problem of replacement first or last in reliability theory, European J. Oper. Res., 223 (2012), 141-149.  doi: 10.1016/j.ejor.2012.05.035.  Google Scholar

[25]

X. ZhaoT. Nakagawa and C. Qian, Optimal imperfect preventive maintenance policies for a used system, Internat. J. Systems Sci., 43 (2012), 1632-1641.  doi: 10.1080/00207721.2010.549583.  Google Scholar

[26]

X. Zhao, M. Chen and T. Nakagawa, Optimal time and random inspection policies for computer systems, Applied Mathematics & Information Sciences, 8, (2014) 413–417. Google Scholar

[27]

X. ZhaoC. Qian and T. Nakagawa, Optimal policies for cumulative damage models with maintenance last and first, Reliability Engineering and System Safety, 110 (2013), 50-59.   Google Scholar

Table 1.  Optimal PMF policies and minimum cost rates of the imperfect maintenance models for varied failure rates. $ C_{O} = 500, C_{R} = 1500, C_{B} = 500, c_{\infty} = 1000, C\sim N (100, 25^{2}), G (t) = 1-e^{-t}, \beta = 2 $
$ \alpha_{i} =(1.05)^{{(i-1)}/2} $ $ \alpha_{i} =(1.15)^{{(i-1)}/2} $
$ \delta $ $ q $ $ N_{F}^{*} $ $ T_{F}^{*} $ $ J_{F} (T_{F}^{*}, N_{F}^{*}) $ $ N_{F}^{*} $ $ T_{F}^{*} $ $ J_{F} (T_{F}^{*}, N_{F}^{*}) $
1 0.9 10 3.6330 823.6007 6 3.6504 895.7270
9/11 0.8 10 3.1563 887.5352 6 3.2182 967.3210
7/11 0.7 9 2.7194 974.0102 5 2.8597 1062.4599
5/11 0.6 9 2.2434 1088.4253 5 2.3676 1188.1660
4/11 0.5 9 2.0239 1158.3947 5 2.1378 1264.9318
3/11 0.4 9 1.8197 1238.3947 5 1.9228 1352.3516
2/11 0.3 8 1.6632 1328.3542 5 1.7239 1451.5804
1/11 0.1 8 1.4887 1430.3649 5 1.5420 1563.8858
$ \alpha_{i} =(1.05)^{{(i-1)}/2} $ $ \alpha_{i} =(1.15)^{{(i-1)}/2} $
$ \delta $ $ q $ $ N_{F}^{*} $ $ T_{F}^{*} $ $ J_{F} (T_{F}^{*}, N_{F}^{*}) $ $ N_{F}^{*} $ $ T_{F}^{*} $ $ J_{F} (T_{F}^{*}, N_{F}^{*}) $
1 0.9 10 3.6330 823.6007 6 3.6504 895.7270
9/11 0.8 10 3.1563 887.5352 6 3.2182 967.3210
7/11 0.7 9 2.7194 974.0102 5 2.8597 1062.4599
5/11 0.6 9 2.2434 1088.4253 5 2.3676 1188.1660
4/11 0.5 9 2.0239 1158.3947 5 2.1378 1264.9318
3/11 0.4 9 1.8197 1238.3947 5 1.9228 1352.3516
2/11 0.3 8 1.6632 1328.3542 5 1.7239 1451.5804
1/11 0.1 8 1.4887 1430.3649 5 1.5420 1563.8858
Table 2.  Optimal PML policies and minimum cost rates of the imperfect maintenance models for varied failure rates. $ C_{O} = 500, C_{R} = 1500, C_{B} = 500, c_{\infty} = 1000, C\sim N (100, 25^{2}), G (t) = 1-e^{-t}, \beta = 2 $
$ \alpha_{i} =(1.05)^{{(i-1)}/2} $ $ \alpha_{i} =(1.15)^{{(i-1)}/2} $
$ \delta $ $ q $ $ N_{L}^{*} $ $ T_{L}^{*} $ $ J_{L} (T_{L}^{*}, N_{L}^{*}) $ $ N_{L}^{*} $ $ T_{L}^{*} $ $ J_{L} (T_{L}^{*}, N_{L}^{*}) $
1 0.9 6 2.4933 522.8141 4 2.4939 568.2837
9/11 0.8 6 2.2631 590.6646 4 2.2725 642.1931
7/11 0.7 6 2.0081 682.0260 4 2.0243 741.7077
5/11 0.6 6 1.7438 802.4905 4 1.7642 872.8862
4/11 0.5 6 1.6133 875.6308 4 1.6347 952.5069
3/11 0.4 6 1.4863 958.5882 4 1.5081 1042.7916
2/11 0.3 6 1.3641 1052.3871 4 1.3857 1144.8479
1/11 0.1 6 1.2477 1158.1623 4 1.2689 1259.9027
$ \alpha_{i} =(1.05)^{{(i-1)}/2} $ $ \alpha_{i} =(1.15)^{{(i-1)}/2} $
$ \delta $ $ q $ $ N_{L}^{*} $ $ T_{L}^{*} $ $ J_{L} (T_{L}^{*}, N_{L}^{*}) $ $ N_{L}^{*} $ $ T_{L}^{*} $ $ J_{L} (T_{L}^{*}, N_{L}^{*}) $
1 0.9 6 2.4933 522.8141 4 2.4939 568.2837
9/11 0.8 6 2.2631 590.6646 4 2.2725 642.1931
7/11 0.7 6 2.0081 682.0260 4 2.0243 741.7077
5/11 0.6 6 1.7438 802.4905 4 1.7642 872.8862
4/11 0.5 6 1.6133 875.6308 4 1.6347 952.5069
3/11 0.4 6 1.4863 958.5882 4 1.5081 1042.7916
2/11 0.3 6 1.3641 1052.3871 4 1.3857 1144.8479
1/11 0.1 6 1.2477 1158.1623 4 1.2689 1259.9027
[1]

Hanyu Gu, Hue Chi Lam, Yakov Zinder. Planning rolling stock maintenance: Optimization of train arrival dates at a maintenance center. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020177

[2]

Pablo D. Carrasco, Túlio Vales. A symmetric Random Walk defined by the time-one map of a geodesic flow. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020390

[3]

Mohammed Abdulrazaq Kahya, Suhaib Abduljabbar Altamir, Zakariya Yahya Algamal. Improving whale optimization algorithm for feature selection with a time-varying transfer function. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 87-98. doi: 10.3934/naco.2020017

[4]

Hedy Attouch, Aïcha Balhag, Zaki Chbani, Hassan Riahi. Fast convex optimization via inertial dynamics combining viscous and Hessian-driven damping with time rescaling. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021010

[5]

Vivina Barutello, Gian Marco Canneori, Susanna Terracini. Minimal collision arcs asymptotic to central configurations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 61-86. doi: 10.3934/dcds.2020218

[6]

Timothy Chumley, Renato Feres. Entropy production in random billiards. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1319-1346. doi: 10.3934/dcds.2020319

[7]

Jong Yoon Hyun, Boran Kim, Minwon Na. Construction of minimal linear codes from multi-variable functions. Advances in Mathematics of Communications, 2021, 15 (2) : 227-240. doi: 10.3934/amc.2020055

[8]

Patrick W. Dondl, Martin Jesenko. Threshold phenomenon for homogenized fronts in random elastic media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 353-372. doi: 10.3934/dcdss.2020329

[9]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, 2021, 15 (1) : 159-183. doi: 10.3934/ipi.2020076

[10]

Min Xi, Wenyu Sun, Jun Chen. Survey of derivative-free optimization. Numerical Algebra, Control & Optimization, 2020, 10 (4) : 537-555. doi: 10.3934/naco.2020050

[11]

Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121

[12]

Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324

[13]

Josselin Garnier, Knut Sølna. Enhanced Backscattering of a partially coherent field from an anisotropic random lossy medium. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1171-1195. doi: 10.3934/dcdsb.2020158

[14]

Luca Battaglia, Francesca Gladiali, Massimo Grossi. Asymptotic behavior of minimal solutions of $ -\Delta u = \lambda f(u) $ as $ \lambda\to-\infty $. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 681-700. doi: 10.3934/dcds.2020293

[15]

Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115

[16]

Xinpeng Wang, Bingo Wing-Kuen Ling, Wei-Chao Kuang, Zhijing Yang. Orthogonal intrinsic mode functions via optimization approach. Journal of Industrial & Management Optimization, 2021, 17 (1) : 51-66. doi: 10.3934/jimo.2019098

[17]

Wolfgang Riedl, Robert Baier, Matthias Gerdts. Optimization-based subdivision algorithm for reachable sets. Journal of Computational Dynamics, 2021, 8 (1) : 99-130. doi: 10.3934/jcd.2021005

[18]

Manxue You, Shengjie Li. Perturbation of Image and conjugate duality for vector optimization. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020176

[19]

Marek Macák, Róbert Čunderlík, Karol Mikula, Zuzana Minarechová. Computational optimization in solving the geodetic boundary value problems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 987-999. doi: 10.3934/dcdss.2020381

[20]

Sören Bartels, Jakob Keck. Adaptive time stepping in elastoplasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 71-88. doi: 10.3934/dcdss.2020323

2019 Impact Factor: 1.366

Article outline

Figures and Tables

[Back to Top]