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Some properties of nonconvex oriented distance function and applications to vector optimization problems
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 |
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.
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. 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. 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. 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. 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. 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. |
[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. 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. |
[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. 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. |
[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. Sugiura, S. 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. |
[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. Google Scholar |
[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. 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. |
[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. 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. 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. 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. 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. |
[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. 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. |
[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. 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. |
[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. Sugiura, S. 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. |
[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. Google Scholar |
[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. Google Scholar |
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 |
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 |
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 | 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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