American Institute of Mathematical Sciences

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  January 2021 Early access  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 and 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. [2] F. Beichelt, A unifying treatment of replacement policies with minimal repair, Naval Research Logistics, 40 (1993), 51-67. [3] M. Berg and R. Cléroux, A marginal cost analysis for an age replacement policy with minimal repair, INFOR, 20 (1982), 258-263. [4] H. W. Block, W. S. Broges and T. H. Savits, Age-dependent minimal repair, J. Appl. Probab., 22 (1985), 370-385.  doi: 10.2307/3213780. [5] M. Brown and F. Proschan, Imperfect repair, J. Appl. Probab., 20 (1983), 851-859.  doi: 10.2307/3213596. [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. [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. [8] C.-C. Chang, S.-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. [9] M. Chen, S. Mizutani and T. Nakagawa, Random and age replacement policies, International Journal of Reliability, Quality and Safety Engineering, 17 (2010), 27-39. [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. [11] M. Kijima, Some results for repairable systems with general repair, J. Appl. Probab., 26 (1989), 89-102.  doi: 10.2307/3214319. [12] M. Kijima, H. 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. [13] T. Nakagawa, Periodic and sequential preventive maintenance policies, J. Appl. Probab., 23 (1986), 536-542.  doi: 10.1017/S0021900200029843. [14] T. Nakagawa, Sequential imperfect preventive maintenance policies, IEEE Transactions on Reliability, 37 (1988), 295-298. [15] T. Nakagawa, Maintenance Theory of Reliability, Springer, London, 2005. [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. [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. [18] H. Pham and H. Wang, Imperfect maintenance, European Journal of Operational Research, 94 (1996), 425-438. [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. [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. [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. [22] T. Sugiura, S. Mizutani and T. Nakagawa, Optimal random replacement policies, Tenth ISSAT International Conference on Reliability and Quality in Design, (2004), 99-103. [23] H. Wang and H. Pham, Optimal Imperfect Maintenance Models, Pham, H. (ed), Handbook of Reliability Engineering, London: Spring, (2003), 397–414. [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. [25] X. Zhao, T. 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. [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. [27] X. Zhao, C. Qian and T. Nakagawa, Optimal policies for cumulative damage models with maintenance last and first, Reliability Engineering and System Safety, 110 (2013), 50-59.

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References:
 [1] R. Barlow and L. Hunter, Optimum preventive maintenance policies, Operations Res., 8 (1960), 90-100.  doi: 10.1287/opre.8.1.90. [2] F. Beichelt, A unifying treatment of replacement policies with minimal repair, Naval Research Logistics, 40 (1993), 51-67. [3] M. Berg and R. Cléroux, A marginal cost analysis for an age replacement policy with minimal repair, INFOR, 20 (1982), 258-263. [4] H. W. Block, W. S. Broges and T. H. Savits, Age-dependent minimal repair, J. Appl. Probab., 22 (1985), 370-385.  doi: 10.2307/3213780. [5] M. Brown and F. Proschan, Imperfect repair, J. Appl. Probab., 20 (1983), 851-859.  doi: 10.2307/3213596. [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. [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. [8] C.-C. Chang, S.-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. [9] M. Chen, S. Mizutani and T. Nakagawa, Random and age replacement policies, International Journal of Reliability, Quality and Safety Engineering, 17 (2010), 27-39. [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. [11] M. Kijima, Some results for repairable systems with general repair, J. Appl. Probab., 26 (1989), 89-102.  doi: 10.2307/3214319. [12] M. Kijima, H. 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. [13] T. Nakagawa, Periodic and sequential preventive maintenance policies, J. Appl. Probab., 23 (1986), 536-542.  doi: 10.1017/S0021900200029843. [14] T. Nakagawa, Sequential imperfect preventive maintenance policies, IEEE Transactions on Reliability, 37 (1988), 295-298. [15] T. Nakagawa, Maintenance Theory of Reliability, Springer, London, 2005. [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. [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. [18] H. Pham and H. Wang, Imperfect maintenance, European Journal of Operational Research, 94 (1996), 425-438. [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. [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. [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. [22] T. Sugiura, S. Mizutani and T. Nakagawa, Optimal random replacement policies, Tenth ISSAT International Conference on Reliability and Quality in Design, (2004), 99-103. [23] H. Wang and H. Pham, Optimal Imperfect Maintenance Models, Pham, H. (ed), Handbook of Reliability Engineering, London: Spring, (2003), 397–414. [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. [25] X. Zhao, T. 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. [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. [27] X. Zhao, C. Qian and T. Nakagawa, Optimal policies for cumulative damage models with maintenance last and first, Reliability Engineering and System Safety, 110 (2013), 50-59.
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
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
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