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November
2021, 17(6): 3445-3473.
doi: 10.3934/jimo.2020127
Bayesian decision making in determining optimal leased term and preventive maintenance scheme for leased facilities
School of Computer Science and Software, Zhaoqing University, Guangdong 526061, China |
Under a business competitive environment, quite a few enterprises choose capital leasing to reduce tax payment and investment risk instead of buying facilities. Since the durability and service life of leased facilities will be longer, the breakdowns and deterioration of leased facilities are inevitable during lease period. Accordingly, in order to reduce the related costs and keep the facility's health during lease period, preventive maintenances are required to perform to reduce the cost of free-repair warranty and maintain customers' satisfaction. However, performing preventive maintenance is not easy to scheme due to the scarcity of historical failure data. Accordingly, the study integrates lease and maintenance decisions into a synthetic strategy, and it can be applied under the situation of only expert's evaluation and/or scare historical failure data by employing Bayesian analyses. In this study, the mathematical models and corresponding algorithms are developed to determine the best preventive maintenance scheme and the optimal term of contract for leased facilities to maximize the expected profit. Moreover, the computerized architecture is also proposed, and it can help the lessor to solve the issue in practice. Finally, numerical examples and the sensitive analyses are provided to illustrate the managerial strategies under different leased period and the preventive maintenance policies.
Mathematics Subject Classification:
Primary: 62C10, 62C25; Secondary: 90B25.
Citation:
Chih-Chiang Fang. Bayesian decision making in determining optimal leased term and preventive maintenance scheme for leased facilities. Journal of Industrial and Management Optimization,
2021, 17
(6)
: 3445-3473.
doi: 10.3934/jimo.2020127
![]()
References:
| [1] |
N. Aras, R. Güllü and and S. Yürülemz,
Optimal inventory and pricing policies for remanufacturable leased products, International Journal of Production Economics, 133 (2011), 262-271.
doi: 10.1016/j.ijpe.2010.01.024. |
| [2] |
A. Ben Mabrouk, A. Chelbi and M. Radhoui,
Optimal imperfect maintenance strategy for leased equipment, International Journal of Production Economics, 178 (2016), 57-64.
doi: 10.1016/j.ijpe.2016.04.024. |
| [3] |
S. Bourjade, R. Huc and C. Muller-Vibes,
Leasing and profitability: Empirical evidence from the airline industry, Transportation Research Part A, 97 (2017), 30-47.
doi: 10.1016/j.tra.2017.01.001. |
| [4] |
J. Cao and W. Xie,
Optimization of a condition-based duration-varying preventive maintenance policy for the stockless production system based on queueing model, Journal of Industrial and Management Optimization, 15 (2019), 1049-1083.
doi: 10.3934/jimo.2018085. |
| [5] |
Y. H. Chun,
Optimal number of periodic preventive maintenance operations under warranty, Reliability Engineering and System Safety, 37 (1992), 223-225.
doi: 10.1016/0951-8320(92)90127-7. |
| [6] |
J. S. Dagpunar and N. Jack,
Preventive maintenance strategy for equipment under warranty, Microelectronics Reliability, 34 (1994), 1089-1093.
doi: 10.1016/0026-2714(94)90073-6. |
| [7] |
P. Desai and D. Purohit,
Leasing and selling: Optimal marketing strategies for a durable goods firm, Management Science, 44 (1998), 19-34.
doi: 10.1287/mnsc.44.11.S19. |
| [8] |
A. N. Das and A. N. Sarmah,
Preventive replacement models: An overview and their application in process industries, European Journal of Industrial Engineering, 4 (2010), 280-307.
doi: 10.1504/EJIE.2010.033332. |
| [9] |
S. H. Ding and S. Kamaruddin,
Maintenance policy optimization-literature review and directions, The International Journal of Advanced Manufacturing Technology, 76 (2015), 1263-1283.
doi: 10.1007/s00170-014-6341-2. |
| [10] |
M. Ebrahimi, S. M. T. F. Ghomi and B. Karimi,
Application of the preventive maintenance scheduling to increase the equipment reliability: Case study - bag filters in cement factory, Journal of Industrial and Management Optimization, 16 (2020), 189-205.
doi: 10.3934/jimo.2018146. |
| [11] |
C. C. Fang and Y. S. Huang,
A study on decisions of warranty, pricing, and production with insufficient information, Computers and Industrial Engineering, 59 (2010), 241-250.
doi: 10.1016/j.cie.2010.04.005. |
| [12] |
H. Garg,
Reliability, availability and maintainability analysis of industrial systems using PSO and fuzzy methodology, MAPAN-Journal of Metrology Society of India, 29 (2014), 115-129.
doi: 10.1007/s12647-013-0081-x. |
| [13] |
H. Garg, Bi-criteria optimization for finding the optimal replacement interval for maintaining the performance of the process industries, Modern Optimization Algorithms and Application in Engineering and Economics, 25 (2016), 33 pp.
doi: 10.4018/978-1-4666-9644-0.ch025. |
| [14] |
H. Garg, M. Rani and S. P. Sharma,
Preventive maintenance scheduling of the pulping unit in a paper plant, Japan Journal of Industrial and Applied Mathematics, 30 (2013), 397-414.
doi: 10.1007/s13160-012-0099-4. |
| [15] |
H. Garg and S. P. Sharma, A two-phase approach for reliability and maintainability analysis of an industrial system, International Journal of Reliability, Quality and Safety Engineering, 19 (2012), 1250013.
doi: 10.1142/S0218539312500131. |
| [16] |
B. Hadjaissa, K. Ameur, S. M. Ait Cheikh and N. Essounbouli,
Bi-objective optimization of maintenance scheduling for power systems, The International Journal of Advanced Manufacturing Technology, 85 (2016), 1361-1372.
doi: 10.1007/s00170-015-8053-7. |
| [17] |
F. Hu and Q. Zong,
Optimal periodic preventive maintenance policy and lease period for leased equipment, Journal of Tianjin University Science and Technology, 41 (2008), 248-253.
|
| [18] |
Y. S. Huang,
A structural design of decision support systems for deteriorating repairable systems, Computers and Operations Research, 31 (2004), 1135-1145.
doi: 10.1016/S0305-0548(03)00069-8. |
| [19] |
Y. S. Huang and V. M. Bier,
A natural conjugate prior for the nonhomogeneous poisson process with a power law intensity function, Communications in Statistics-Simulation and Computation, 27 (1998), 525-551.
doi: 10.1080/03610919808813493. |
| [20] |
Y. S. Huang and V. M. Bier,
A natural conjugate prior for the nonhomogeneous Poisson process with an exponential intensity function, Communications in Statistics-Simulations and Computation, 28 (1999), 1479-1509.
doi: 10.1080/03610929908832368. |
| [21] |
B. P. Iskandar and H. Husniah,
Optimal preventive maintenance for a two dimensional lease contract, Computers and Industrial Engineering, 113 (2017), 693-703.
doi: 10.1016/j.cie.2017.09.028. |
| [22] |
R. Jamshidi and M. M. Seyyed Esfahani,
Maintenance policy determination for a complex system consisting of series and cold standby system with multiple levels of maintenance action, The International Journal of Advanced Manufacturing Technology, 78 (2015), 1137-1346.
doi: 10.1007/s00170-014-6727-1. |
| [23] |
V. Jayabalan and D. Chaudhuri,
Cost optimization of maintenance scheduling for a system with assured reliability, IEEE Transactions on Reliability, 41 (1992), 21-25.
doi: 10.1109/24.126665. |
| [24] |
H. Jin, L. Hai and X. Tang,
An optimal maintenance strategy for multi-state systems based on a system linear integral equation and dynamic programming, Journal of Industrial and Management Optimization, 16 (2020), 965-990.
doi: 10.3934/jimo.2018188. |
| [25] |
B. S. Kim and Y. Ozturkoglu,
Scheduling a single machine with multiple preventive maintenance activities and position-based deteriorations using genetic algorithms, The International Journal of Advanced Manufacturing Technology, 67 (2013), 1127-1137.
doi: 10.1007/s00170-012-4553-x. |
| [26] |
C. S. Kim, I. Djamaludin, I. and D. N. P. Murthy, Warranty and discrete preventive maintenance, Reliability Engineering and System Safety, 84 (2004), 301-309.
doi: 10.1016/j.ress.2003.12.001. |
| [27] |
R. T. Kleiman,
The characteristics of venture lease financing, Journal of Equipment Lease Financing, 19 (2001), 1-10.
|
| [28] |
C. Liu, Y. Fang, C. Zhao and J. Wang,
Multiple common due-dates assignment and optimal maintenance activity scheduling with linear deteriorating jobs, Journal of Industrial and Management Optimization, 13 (2017), 713-720.
doi: 10.3934/jimo.2016042. |
| [29] |
S. Martorell, A. Sanchez, A. and V. Serradell, Age-dependent reliability model considering effects of maintenance and working conditions, Reliability Engineering and System Safety, 64 (1999), 19-31.
doi: 10.1016/S0951-8320(98)00050-7. |
| [30] |
W. T. Moore and S. N. Chen,
The decision to lease or purchase under uncertainty: A Bayesian approach, The Engineering Economist, 29 (1984), 195-206.
doi: 10.1080/00137918408967711. |
| [31] |
P. Müller and G. Parmigiani,
Optimal design via curve fitting of Monte Carlo experiments, Journal of the American Statistical Association - Theory and Methods, 90 (1995), 1322-1330.
doi: 10.2307/2291522. |
| [32] |
A. Nisbet and A. Ward,
Radiotherapy equipment-purchase or lease?, The British Journal of Radiology, 74 (2000), 735-744.
doi: 10.1259/bjr.74.884.740735. |
| [33] |
R. Niwas and H. Garg, An approach for analyzing the reliability and profit of an industrial system based on the cost free warranty policy, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 40 (2018), Art. 265.
doi: 10.1007/s40430-018-1167-8. |
| [34] |
I. A. Papazoglou,
Bayesian decision analysis and reliability certification, Reliability Engineering and System Safety, 66 (1999), 177-198.
doi: 10.1016/S0951-8320(99)00035-6. |
| [35] |
D. F. Percy,
Bayesian enhanced strategic decision making for reliability, European Journal of Operational Research, 139 (2002), 133-145.
doi: 10.1016/S0377-2217(01)00177-1. |
| [36] |
J. Pongpech and D. N. P. Murthy,
Optimal periodic preventive maintenance policy for leased equipment, Reliability Engineering and System Safety, 91 (2006), 772-777.
doi: 10.1016/j.ress.2005.07.005. |
| [37] |
Y. Saito, T. Dohi and W. Y. Yun,
Uncertainty analysis for a periodic replacement problem with minimal repair: Parametric bootstrapping approach, International Journal of Industrial Engineering: Theory, Applications and Practice, 21 (2014), 337-347.
|
| [38] |
J. Schutz and N. Rezg,
Maintenance strategy for leased equipment, Computers and Industrial Engineering, 66 (2013), 593-600.
doi: 10.1016/j.cie.2013.05.004. |
| [39] |
M. Sheikhalishahi, H. Heidaryan-Baygy, S. Abdolhossein Zadeh and A. Azadeh,
Comparison between condition-based, age-based and failure-based maintenance policies in parallel and series configurations: A simulation analysis, International Journal of Industrial Engineering: Theory, Applications and Practice, 24 (2017), 295-305.
|
| [40] |
J. Shin, J. R. Morrison and A. Kalir,
Optimization of preventive maintenance plans in G/G/M queueing networks and numerical study with models based on semiconductor wafer fabs, International Journal of Industrial Engineering: Theory, Applications and Practice, 23(5) (2016), 302-317.
|
| [41] |
D. W. Steeneck and S. C. Sarin,
Product design for leased products under remanufacturing, International Journal of Production Economics, 202 (2018), 132-144.
doi: 10.1016/j.ijpe.2018.04.025. |
| [42] |
J. Taheri-Tolgari, M. Mohammadi, B. Naderi, A. Arshadi-Khamseh and A. Mirzazadeh,
An inventory model with imperfect item, inspection errors, preventive maintenance and partial backlogging in uncertainty environment, Journal of Industrial and Management Optimization, 15 (2019), 1317-1344.
doi: 10.3934/jimo.2018097. |
| [43] |
G. Walter and S. D. Flapper,
Condition-based maintenance for complex systems based on current component status and Bayesian updating of component reliability, Reliability Engineering and System Safety, 168 (2017), 227-239.
doi: 10.1016/j.ress.2017.06.015. |
| [44] |
H. Wang and H. Pham,
Some maintenance models and availability with imperfect maintenance in production systems, Annals of Operations Research, 91 (1999), 305-318.
doi: 10.1023/A:1018910109348. |
| [45] |
X. Wang, L. Li and M. Xie,
Optimal preventive maintenance strategy for leased equipment under successive usage-based contracts, International Journal of Production Research, 57 (2019), 5705-5724.
doi: 10.1080/00207543.2018.1542181. |
| [46] |
C. W. Yeh and C. C. Fang,
Optimal pro-rata warranty decision with consideration of the marketing strategy under insufficient historical reliability data, International Journal of Advanced Manufacturing Technology, 71 (2014), 1757-1772.
doi: 10.1007/s00170-013-5596-3. |
| [47] |
R. H. Yeh and W. L. Chang,
Optimal threshold value of failure-rate for Leased products with preventive maintenance actions, Mathematical and Computer Modelling, 46 (2007), 730-737.
doi: 10.1016/j.mcm.2006.12.001. |
| [48] |
R. H. Yeh and H. C. Lo,
Optimal preventive-maintenance warranty policy for repairable products, European Journal of Operational Research, 134 (2001), 59-69.
doi: 10.1016/S0377-2217(00)00238-1. |
| [49] |
R. H. Yeh, K. C. Kao and W. L. Chang,
Optimal preventive maintenance policy for leased equipment using failure rate reduction, Computers and Industrial Engineering, 57(1) (2009), 304-309.
doi: 10.1016/j.cie.2008.11.025. |
| [50] |
R. H. Yeh, K. C. Kao and W. L. Chang,
Preventive-maintenance policy for leased products under various maintenance costs, Expert Systems with Applications, 38 (2011), 3558-3562.
doi: 10.1016/j.eswa.2010.08.144. |
| [51] |
Y. Zhang, X. Zhang, J. Zeng, J. Wang and S. Xue, Lessees satisfaction and optimal condition-based maintenance policy for leased system, Reliability Engineering and System Safety, 191 (2019), Art. 106532.
doi: 10.1016/j.ress.2019.106532. |
show all references
References:
| [1] |
N. Aras, R. Güllü and and S. Yürülemz,
Optimal inventory and pricing policies for remanufacturable leased products, International Journal of Production Economics, 133 (2011), 262-271.
doi: 10.1016/j.ijpe.2010.01.024. |
| [2] |
A. Ben Mabrouk, A. Chelbi and M. Radhoui,
Optimal imperfect maintenance strategy for leased equipment, International Journal of Production Economics, 178 (2016), 57-64.
doi: 10.1016/j.ijpe.2016.04.024. |
| [3] |
S. Bourjade, R. Huc and C. Muller-Vibes,
Leasing and profitability: Empirical evidence from the airline industry, Transportation Research Part A, 97 (2017), 30-47.
doi: 10.1016/j.tra.2017.01.001. |
| [4] |
J. Cao and W. Xie,
Optimization of a condition-based duration-varying preventive maintenance policy for the stockless production system based on queueing model, Journal of Industrial and Management Optimization, 15 (2019), 1049-1083.
doi: 10.3934/jimo.2018085. |
| [5] |
Y. H. Chun,
Optimal number of periodic preventive maintenance operations under warranty, Reliability Engineering and System Safety, 37 (1992), 223-225.
doi: 10.1016/0951-8320(92)90127-7. |
| [6] |
J. S. Dagpunar and N. Jack,
Preventive maintenance strategy for equipment under warranty, Microelectronics Reliability, 34 (1994), 1089-1093.
doi: 10.1016/0026-2714(94)90073-6. |
| [7] |
P. Desai and D. Purohit,
Leasing and selling: Optimal marketing strategies for a durable goods firm, Management Science, 44 (1998), 19-34.
doi: 10.1287/mnsc.44.11.S19. |
| [8] |
A. N. Das and A. N. Sarmah,
Preventive replacement models: An overview and their application in process industries, European Journal of Industrial Engineering, 4 (2010), 280-307.
doi: 10.1504/EJIE.2010.033332. |
| [9] |
S. H. Ding and S. Kamaruddin,
Maintenance policy optimization-literature review and directions, The International Journal of Advanced Manufacturing Technology, 76 (2015), 1263-1283.
doi: 10.1007/s00170-014-6341-2. |
| [10] |
M. Ebrahimi, S. M. T. F. Ghomi and B. Karimi,
Application of the preventive maintenance scheduling to increase the equipment reliability: Case study - bag filters in cement factory, Journal of Industrial and Management Optimization, 16 (2020), 189-205.
doi: 10.3934/jimo.2018146. |
| [11] |
C. C. Fang and Y. S. Huang,
A study on decisions of warranty, pricing, and production with insufficient information, Computers and Industrial Engineering, 59 (2010), 241-250.
doi: 10.1016/j.cie.2010.04.005. |
| [12] |
H. Garg,
Reliability, availability and maintainability analysis of industrial systems using PSO and fuzzy methodology, MAPAN-Journal of Metrology Society of India, 29 (2014), 115-129.
doi: 10.1007/s12647-013-0081-x. |
| [13] |
H. Garg, Bi-criteria optimization for finding the optimal replacement interval for maintaining the performance of the process industries, Modern Optimization Algorithms and Application in Engineering and Economics, 25 (2016), 33 pp.
doi: 10.4018/978-1-4666-9644-0.ch025. |
| [14] |
H. Garg, M. Rani and S. P. Sharma,
Preventive maintenance scheduling of the pulping unit in a paper plant, Japan Journal of Industrial and Applied Mathematics, 30 (2013), 397-414.
doi: 10.1007/s13160-012-0099-4. |
| [15] |
H. Garg and S. P. Sharma, A two-phase approach for reliability and maintainability analysis of an industrial system, International Journal of Reliability, Quality and Safety Engineering, 19 (2012), 1250013.
doi: 10.1142/S0218539312500131. |
| [16] |
B. Hadjaissa, K. Ameur, S. M. Ait Cheikh and N. Essounbouli,
Bi-objective optimization of maintenance scheduling for power systems, The International Journal of Advanced Manufacturing Technology, 85 (2016), 1361-1372.
doi: 10.1007/s00170-015-8053-7. |
| [17] |
F. Hu and Q. Zong,
Optimal periodic preventive maintenance policy and lease period for leased equipment, Journal of Tianjin University Science and Technology, 41 (2008), 248-253.
|
| [18] |
Y. S. Huang,
A structural design of decision support systems for deteriorating repairable systems, Computers and Operations Research, 31 (2004), 1135-1145.
doi: 10.1016/S0305-0548(03)00069-8. |
| [19] |
Y. S. Huang and V. M. Bier,
A natural conjugate prior for the nonhomogeneous poisson process with a power law intensity function, Communications in Statistics-Simulation and Computation, 27 (1998), 525-551.
doi: 10.1080/03610919808813493. |
| [20] |
Y. S. Huang and V. M. Bier,
A natural conjugate prior for the nonhomogeneous Poisson process with an exponential intensity function, Communications in Statistics-Simulations and Computation, 28 (1999), 1479-1509.
doi: 10.1080/03610929908832368. |
| [21] |
B. P. Iskandar and H. Husniah,
Optimal preventive maintenance for a two dimensional lease contract, Computers and Industrial Engineering, 113 (2017), 693-703.
doi: 10.1016/j.cie.2017.09.028. |
| [22] |
R. Jamshidi and M. M. Seyyed Esfahani,
Maintenance policy determination for a complex system consisting of series and cold standby system with multiple levels of maintenance action, The International Journal of Advanced Manufacturing Technology, 78 (2015), 1137-1346.
doi: 10.1007/s00170-014-6727-1. |
| [23] |
V. Jayabalan and D. Chaudhuri,
Cost optimization of maintenance scheduling for a system with assured reliability, IEEE Transactions on Reliability, 41 (1992), 21-25.
doi: 10.1109/24.126665. |
| [24] |
H. Jin, L. Hai and X. Tang,
An optimal maintenance strategy for multi-state systems based on a system linear integral equation and dynamic programming, Journal of Industrial and Management Optimization, 16 (2020), 965-990.
doi: 10.3934/jimo.2018188. |
| [25] |
B. S. Kim and Y. Ozturkoglu,
Scheduling a single machine with multiple preventive maintenance activities and position-based deteriorations using genetic algorithms, The International Journal of Advanced Manufacturing Technology, 67 (2013), 1127-1137.
doi: 10.1007/s00170-012-4553-x. |
| [26] |
C. S. Kim, I. Djamaludin, I. and D. N. P. Murthy, Warranty and discrete preventive maintenance, Reliability Engineering and System Safety, 84 (2004), 301-309.
doi: 10.1016/j.ress.2003.12.001. |
| [27] |
R. T. Kleiman,
The characteristics of venture lease financing, Journal of Equipment Lease Financing, 19 (2001), 1-10.
|
| [28] |
C. Liu, Y. Fang, C. Zhao and J. Wang,
Multiple common due-dates assignment and optimal maintenance activity scheduling with linear deteriorating jobs, Journal of Industrial and Management Optimization, 13 (2017), 713-720.
doi: 10.3934/jimo.2016042. |
| [29] |
S. Martorell, A. Sanchez, A. and V. Serradell, Age-dependent reliability model considering effects of maintenance and working conditions, Reliability Engineering and System Safety, 64 (1999), 19-31.
doi: 10.1016/S0951-8320(98)00050-7. |
| [30] |
W. T. Moore and S. N. Chen,
The decision to lease or purchase under uncertainty: A Bayesian approach, The Engineering Economist, 29 (1984), 195-206.
doi: 10.1080/00137918408967711. |
| [31] |
P. Müller and G. Parmigiani,
Optimal design via curve fitting of Monte Carlo experiments, Journal of the American Statistical Association - Theory and Methods, 90 (1995), 1322-1330.
doi: 10.2307/2291522. |
| [32] |
A. Nisbet and A. Ward,
Radiotherapy equipment-purchase or lease?, The British Journal of Radiology, 74 (2000), 735-744.
doi: 10.1259/bjr.74.884.740735. |
| [33] |
R. Niwas and H. Garg, An approach for analyzing the reliability and profit of an industrial system based on the cost free warranty policy, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 40 (2018), Art. 265.
doi: 10.1007/s40430-018-1167-8. |
| [34] |
I. A. Papazoglou,
Bayesian decision analysis and reliability certification, Reliability Engineering and System Safety, 66 (1999), 177-198.
doi: 10.1016/S0951-8320(99)00035-6. |
| [35] |
D. F. Percy,
Bayesian enhanced strategic decision making for reliability, European Journal of Operational Research, 139 (2002), 133-145.
doi: 10.1016/S0377-2217(01)00177-1. |
| [36] |
J. Pongpech and D. N. P. Murthy,
Optimal periodic preventive maintenance policy for leased equipment, Reliability Engineering and System Safety, 91 (2006), 772-777.
doi: 10.1016/j.ress.2005.07.005. |
| [37] |
Y. Saito, T. Dohi and W. Y. Yun,
Uncertainty analysis for a periodic replacement problem with minimal repair: Parametric bootstrapping approach, International Journal of Industrial Engineering: Theory, Applications and Practice, 21 (2014), 337-347.
|
| [38] |
J. Schutz and N. Rezg,
Maintenance strategy for leased equipment, Computers and Industrial Engineering, 66 (2013), 593-600.
doi: 10.1016/j.cie.2013.05.004. |
| [39] |
M. Sheikhalishahi, H. Heidaryan-Baygy, S. Abdolhossein Zadeh and A. Azadeh,
Comparison between condition-based, age-based and failure-based maintenance policies in parallel and series configurations: A simulation analysis, International Journal of Industrial Engineering: Theory, Applications and Practice, 24 (2017), 295-305.
|
| [40] |
J. Shin, J. R. Morrison and A. Kalir,
Optimization of preventive maintenance plans in G/G/M queueing networks and numerical study with models based on semiconductor wafer fabs, International Journal of Industrial Engineering: Theory, Applications and Practice, 23(5) (2016), 302-317.
|
| [41] |
D. W. Steeneck and S. C. Sarin,
Product design for leased products under remanufacturing, International Journal of Production Economics, 202 (2018), 132-144.
doi: 10.1016/j.ijpe.2018.04.025. |
| [42] |
J. Taheri-Tolgari, M. Mohammadi, B. Naderi, A. Arshadi-Khamseh and A. Mirzazadeh,
An inventory model with imperfect item, inspection errors, preventive maintenance and partial backlogging in uncertainty environment, Journal of Industrial and Management Optimization, 15 (2019), 1317-1344.
doi: 10.3934/jimo.2018097. |
| [43] |
G. Walter and S. D. Flapper,
Condition-based maintenance for complex systems based on current component status and Bayesian updating of component reliability, Reliability Engineering and System Safety, 168 (2017), 227-239.
doi: 10.1016/j.ress.2017.06.015. |
| [44] |
H. Wang and H. Pham,
Some maintenance models and availability with imperfect maintenance in production systems, Annals of Operations Research, 91 (1999), 305-318.
doi: 10.1023/A:1018910109348. |
| [45] |
X. Wang, L. Li and M. Xie,
Optimal preventive maintenance strategy for leased equipment under successive usage-based contracts, International Journal of Production Research, 57 (2019), 5705-5724.
doi: 10.1080/00207543.2018.1542181. |
| [46] |
C. W. Yeh and C. C. Fang,
Optimal pro-rata warranty decision with consideration of the marketing strategy under insufficient historical reliability data, International Journal of Advanced Manufacturing Technology, 71 (2014), 1757-1772.
doi: 10.1007/s00170-013-5596-3. |
| [47] |
R. H. Yeh and W. L. Chang,
Optimal threshold value of failure-rate for Leased products with preventive maintenance actions, Mathematical and Computer Modelling, 46 (2007), 730-737.
doi: 10.1016/j.mcm.2006.12.001. |
| [48] |
R. H. Yeh and H. C. Lo,
Optimal preventive-maintenance warranty policy for repairable products, European Journal of Operational Research, 134 (2001), 59-69.
doi: 10.1016/S0377-2217(00)00238-1. |
| [49] |
R. H. Yeh, K. C. Kao and W. L. Chang,
Optimal preventive maintenance policy for leased equipment using failure rate reduction, Computers and Industrial Engineering, 57(1) (2009), 304-309.
doi: 10.1016/j.cie.2008.11.025. |
| [50] |
R. H. Yeh, K. C. Kao and W. L. Chang,
Preventive-maintenance policy for leased products under various maintenance costs, Expert Systems with Applications, 38 (2011), 3558-3562.
doi: 10.1016/j.eswa.2010.08.144. |
| [51] |
Y. Zhang, X. Zhang, J. Zeng, J. Wang and S. Xue, Lessees satisfaction and optimal condition-based maintenance policy for leased system, Reliability Engineering and System Safety, 191 (2019), Art. 106532.
doi: 10.1016/j.ress.2019.106532. |
Figure 2.
Maintenance Scheme under Imperfect Recovery
Figure 4.
Flowchart for the Bayesian Solution Algorithm
Figure 5.
Computerized Implementation Architecture
Figure 6.
Average Profits per Unit and Year for Maintenance Plans 1, 2, 3 Estimated by Prior Analysis
Figure 7.
Average Profits per Unit and Year for Maintenance Plan 1 Estimated by Prior and Posterior Analyses
Figure 8.
The Impact of $ E $ ($ \alpha $ ), $ E $ ($ \beta $ ), $ \sigma(\alpha) $ and $ \sigma(\beta) $ on Average Profit
Figure 9.
The Impact of Minimal Repair Cost on Average Profit
Figure 10.
The Impact of Base Cost for a PM action on Average Profit
Figure 11.
The Impact of Increasing Rate of PM Cost on Average Profit
Figure 12.
The Impact of Time Discount Rate on Average Profit
Figure 13.
The Impact of Depreciation Rate on Average Profit
Table 1.
The detailed information of three maintenance plans
| Maintenance Plan 1 | Maintenance Plan 2 | Maintenance Plan 3 | |
| Parameters for the deterioration judged by experts | |||
| Age reduction factors | |||
| Base cost for a PM action | |||
| Periodically increasing rates of PM cost | data |
||
| Depreciation rate | |||
| Interval of PM; Time segment | |||
| The minimal and maximal planned lease terms | |||
| Rental of per half-year | |||
| Time discount rate | |||
| Production cost of an equipment | |||
| Penalty cost for repair time over the time limit | |||
| Expectation of performing a minimal repair | |||
| Standard deviation of performing a minimal repair | |||
| Tolerable waiting time limit for performing a minimal repair | |||
| Expected cost of performing a minimal repair | |||
| Maintenance Plan 1 | Maintenance Plan 2 | Maintenance Plan 3 | |
| Parameters for the deterioration judged by experts | |||
| Age reduction factors | |||
| Base cost for a PM action | |||
| Periodically increasing rates of PM cost | data |
||
| Depreciation rate | |||
| Interval of PM; Time segment | |||
| The minimal and maximal planned lease terms | |||
| Rental of per half-year | |||
| Time discount rate | |||
| Production cost of an equipment | |||
| Penalty cost for repair time over the time limit | |||
| Expectation of performing a minimal repair | |||
| Standard deviation of performing a minimal repair | |||
| Tolerable waiting time limit for performing a minimal repair | |||
| Expected cost of performing a minimal repair | |||
Table 2.
Expected failures, repair costs, preventive costs, production cost, residual value and average profits per unit and year for maintenance plan 1, 2, 3 estimated by prior analysis
| PM Cost | Repair Cost | |||||||||||||
| Time | Plan1 | Plan2 | Plan3 | Plan1 | Plan2 | Plan3 | Plan1 | Plan2 | Plan3 | Plan1 | Plan2 | Plan3 | ||
| 2 | 3.06 | 2.74 | 2.44 | 2760 | 3495 | 4194 | 1500 | 1343 | 1194 | 98000 | 70805 | 3292 | 3003 | 2729 |
| 2.5 | 4.24 | 3.69 | 3.17 | 3600 | 4575 | 5490 | 2077 | 1807 | 1553 | 98000 | 65279 | 3472 | 3190 | 2926 |
| 3 | 5.61 | 4.75 | 3.96 | 4500 | 5738 | 6885 | 2746 | 2327 | 1938 | 98000 | 60184 | 3625 | 3352 | 3099 |
| 3.5 | 7.17 | 5.93 | 4.79 | 5460 | 6983 | 8379 | 3514 | 2904 | 2349 | 98000 | 55487 | 3752 | 3491 | 3251 |
| 4 | 8.95 | 7.23 | 5.69 | 6480 | 8310 | 9972 | 4387 | 3542 | 2786 | 98000 | 51157 | 3854 | 3608 | 3381 |
| 4.5 | 10.97 | 8.66 | 6.63 | 7560 | 9720 | 11664 | 5373 | 4243 | 3250 | 98000 | 47164 | 3932 | 3703 | 3492 |
| 5 | 13.22 | 10.23 | 7.64 | 8700 | 11213 | 13455 | 6480 | 5011 | 3741 | 98000 | 43483 | 3987 | 3779 | 3584 |
| 5.5 | 15.75 | 11.94 | 8.70 | 9900 | 12788 | 15345 | 7716 | 5848 | 4261 | 98000 | 40089 | 4021 | 3835 | 3659 |
| *6 | 18.55 | 13.79 | 9.81 | 11160 | 14445 | 17334 | 9090 | 6758 | 4809 | 98000 | 36961 | *4033 | 3874 | 3718 |
| 6.5 | 21.65 | 15.80 | 10.99 | 12480 | 16185 | 19422 | 10611 | 7744 | 5386 | 98000 | 34076 | 4025 | 3896 | 3761 |
| *7 | 25.08 | 17.98 | 12.23 | 13860 | 18008 | 21609 | 12289 | 8810 | 5993 | 98000 | 31417 | 3998 | *3902 | 3790 |
| 7.5 | 28.85 | 20.32 | 13.53 | 15300 | 19913 | 23895 | 14135 | 9959 | 6631 | 98000 | 28965 | 3951 | 3893 | 3805 |
| *8 | 32.98 | 22.85 | 14.90 | 16800 | 21900 | 26280 | 16160 | 11194 | 7301 | 98000 | 26704 | 3885 | 3869 | *3808 |
| 8.5 | 37.50 | 25.55 | 16.33 | 18360 | 23970 | 28764 | 18375 | 12521 | 8002 | 98000 | 24620 | 3802 | 3831 | 3798 |
| 9 | 42.43 | 28.46 | 17.83 | 19980 | 26123 | 31347 | 20793 | 13943 | 8736 | 98000 | 22698 | 3701 | 3780 | 3778 |
| 9.5 | 47.81 | 31.56 | 19.40 | 21660 | 28358 | 34029 | 23426 | 15464 | 9504 | 98000 | 20927 | 3583 | 3716 | 3746 |
| 10 | 53.65 | 34.88 | 21.03 | 23400 | 30675 | 36810 | 26289 | 17089 | 10306 | 98000 | 19294 | 3448 | 3640 | 3705 |
| 10.5 | 59.99 | 38.41 | 22.74 | 25200 | 33075 | 39690 | 29394 | 18822 | 11143 | 98000 | 17788 | 3296 | 3553 | 3654 |
| 11 | 66.85 | 42.18 | 24.52 | 27060 | 35558 | 42669 | 32758 | 20668 | 12016 | 98000 | 16400 | 3128 | 3455 | 3595 |
| 11.5 | 74.28 | 46.19 | 26.38 | 28980 | 38123 | 45747 | 36396 | 22632 | 12926 | 98000 | 15120 | 2944 | 3346 | 3527 |
| 12 | 82.29 | 50.45 | 28.31 | 30960 | 40770 | 48924 | 40324 | 24719 | 13874 | 98000 | 13940 | 2744 | 3227 | 3451 |
| PM Cost | Repair Cost | |||||||||||||
| Time | Plan1 | Plan2 | Plan3 | Plan1 | Plan2 | Plan3 | Plan1 | Plan2 | Plan3 | Plan1 | Plan2 | Plan3 | ||
| 2 | 3.06 | 2.74 | 2.44 | 2760 | 3495 | 4194 | 1500 | 1343 | 1194 | 98000 | 70805 | 3292 | 3003 | 2729 |
| 2.5 | 4.24 | 3.69 | 3.17 | 3600 | 4575 | 5490 | 2077 | 1807 | 1553 | 98000 | 65279 | 3472 | 3190 | 2926 |
| 3 | 5.61 | 4.75 | 3.96 | 4500 | 5738 | 6885 | 2746 | 2327 | 1938 | 98000 | 60184 | 3625 | 3352 | 3099 |
| 3.5 | 7.17 | 5.93 | 4.79 | 5460 | 6983 | 8379 | 3514 | 2904 | 2349 | 98000 | 55487 | 3752 | 3491 | 3251 |
| 4 | 8.95 | 7.23 | 5.69 | 6480 | 8310 | 9972 | 4387 | 3542 | 2786 | 98000 | 51157 | 3854 | 3608 | 3381 |
| 4.5 | 10.97 | 8.66 | 6.63 | 7560 | 9720 | 11664 | 5373 | 4243 | 3250 | 98000 | 47164 | 3932 | 3703 | 3492 |
| 5 | 13.22 | 10.23 | 7.64 | 8700 | 11213 | 13455 | 6480 | 5011 | 3741 | 98000 | 43483 | 3987 | 3779 | 3584 |
| 5.5 | 15.75 | 11.94 | 8.70 | 9900 | 12788 | 15345 | 7716 | 5848 | 4261 | 98000 | 40089 | 4021 | 3835 | 3659 |
| *6 | 18.55 | 13.79 | 9.81 | 11160 | 14445 | 17334 | 9090 | 6758 | 4809 | 98000 | 36961 | *4033 | 3874 | 3718 |
| 6.5 | 21.65 | 15.80 | 10.99 | 12480 | 16185 | 19422 | 10611 | 7744 | 5386 | 98000 | 34076 | 4025 | 3896 | 3761 |
| *7 | 25.08 | 17.98 | 12.23 | 13860 | 18008 | 21609 | 12289 | 8810 | 5993 | 98000 | 31417 | 3998 | *3902 | 3790 |
| 7.5 | 28.85 | 20.32 | 13.53 | 15300 | 19913 | 23895 | 14135 | 9959 | 6631 | 98000 | 28965 | 3951 | 3893 | 3805 |
| *8 | 32.98 | 22.85 | 14.90 | 16800 | 21900 | 26280 | 16160 | 11194 | 7301 | 98000 | 26704 | 3885 | 3869 | *3808 |
| 8.5 | 37.50 | 25.55 | 16.33 | 18360 | 23970 | 28764 | 18375 | 12521 | 8002 | 98000 | 24620 | 3802 | 3831 | 3798 |
| 9 | 42.43 | 28.46 | 17.83 | 19980 | 26123 | 31347 | 20793 | 13943 | 8736 | 98000 | 22698 | 3701 | 3780 | 3778 |
| 9.5 | 47.81 | 31.56 | 19.40 | 21660 | 28358 | 34029 | 23426 | 15464 | 9504 | 98000 | 20927 | 3583 | 3716 | 3746 |
| 10 | 53.65 | 34.88 | 21.03 | 23400 | 30675 | 36810 | 26289 | 17089 | 10306 | 98000 | 19294 | 3448 | 3640 | 3705 |
| 10.5 | 59.99 | 38.41 | 22.74 | 25200 | 33075 | 39690 | 29394 | 18822 | 11143 | 98000 | 17788 | 3296 | 3553 | 3654 |
| 11 | 66.85 | 42.18 | 24.52 | 27060 | 35558 | 42669 | 32758 | 20668 | 12016 | 98000 | 16400 | 3128 | 3455 | 3595 |
| 11.5 | 74.28 | 46.19 | 26.38 | 28980 | 38123 | 45747 | 36396 | 22632 | 12926 | 98000 | 15120 | 2944 | 3346 | 3527 |
| 12 | 82.29 | 50.45 | 28.31 | 30960 | 40770 | 48924 | 40324 | 24719 | 13874 | 98000 | 13940 | 2744 | 3227 | 3451 |
Table 3.
Expected failures, repair costs, preventive cost, production cost, residual value and average profits per unit and year for prior and posterior analyses
| Repair Cost | PM Cost | ||||||||
| Time | Prior | Posterior | Prior | Posterior | Prior | Posterior | |||
| 2 | 3.06 | 3.90 | 1500 | 1911 | 1500 | 98000 | 70805 | 3292 | 3087 |
| 2.5 | 4.24 | 5.82 | 2077 | 2856 | 2077 | 98000 | 65279 | 3472 | 3161 |
| 3 | 5.61 | 8.09 | 2746 | 3965 | 2746 | 98000 | 60184 | 3625 | 3219 |
| 3.5 | 7.17 | 10.68 | 3514 | 5233 | 3514 | 98000 | 55487 | 3752 | 3261 |
| 4 | 8.95 | 13.58 | 4387 | 6655 | 4387 | 98000 | 51157 | 3854 | 3287 |
| *4.5 | 10.97 | 16.79 | 5373 | 8226 | 5373 | 98000 | 47164 | 3932 | *3298 |
| 5 | 13.22 | 20.29 | 6480 | 9944 | 6480 | 98000 | 43483 | 3987 | 3295 |
| 5.5 | 15.75 | 24.09 | 7716 | 11805 | 7716 | 98000 | 40089 | 4021 | 3277 |
| *6 | 18.55 | 28.18 | 9090 | 13807 | 9090 | 98000 | 36961 | *4033 | 3247 |
| 6.5 | 21.65 | 32.54 | 10611 | 15946 | 10611 | 98000 | 34076 | 4025 | 3204 |
| 7 | 25.08 | 37.19 | 12289 | 18222 | 12289 | 98000 | 31417 | 3998 | 3150 |
| 7.5 | 28.85 | 42.10 | 14135 | 20631 | 14135 | 98000 | 28965 | 3951 | 3085 |
| 8 | 32.98 | 47.29 | 16160 | 23173 | 16160 | 98000 | 26704 | 3885 | 3009 |
| 8.5 | 37.50 | 52.74 | 18375 | 25845 | 18375 | 98000 | 24620 | 3802 | 2923 |
| 9 | 42.43 | 58.46 | 20793 | 28645 | 20793 | 98000 | 22698 | 3701 | 2829 |
| 9.5 | 47.81 | 64.43 | 23426 | 31573 | 23426 | 98000 | 20927 | 3583 | 2725 |
| 10 | 53.65 | 70.67 | 26289 | 34627 | 26289 | 98000 | 19294 | 3448 | 2614 |
| 10.5 | 59.99 | 77.15 | 29394 | 37805 | 29394 | 98000 | 17788 | 3296 | 2495 |
| 11 | 66.85 | 83.89 | 32758 | 41107 | 32758 | 98000 | 16400 | 3128 | 2369 |
| 11.5 | 74.28 | 90.88 | 36396 | 44532 | 36396 | 98000 | 15120 | 2944 | 2236 |
| 12 | 82.29 | 98.12 | 40324 | 48077 | 40324 | 98000 | 13940 | 2744 | 2097 |
| Repair Cost | PM Cost | ||||||||
| Time | Prior | Posterior | Prior | Posterior | Prior | Posterior | |||
| 2 | 3.06 | 3.90 | 1500 | 1911 | 1500 | 98000 | 70805 | 3292 | 3087 |
| 2.5 | 4.24 | 5.82 | 2077 | 2856 | 2077 | 98000 | 65279 | 3472 | 3161 |
| 3 | 5.61 | 8.09 | 2746 | 3965 | 2746 | 98000 | 60184 | 3625 | 3219 |
| 3.5 | 7.17 | 10.68 | 3514 | 5233 | 3514 | 98000 | 55487 | 3752 | 3261 |
| 4 | 8.95 | 13.58 | 4387 | 6655 | 4387 | 98000 | 51157 | 3854 | 3287 |
| *4.5 | 10.97 | 16.79 | 5373 | 8226 | 5373 | 98000 | 47164 | 3932 | *3298 |
| 5 | 13.22 | 20.29 | 6480 | 9944 | 6480 | 98000 | 43483 | 3987 | 3295 |
| 5.5 | 15.75 | 24.09 | 7716 | 11805 | 7716 | 98000 | 40089 | 4021 | 3277 |
| *6 | 18.55 | 28.18 | 9090 | 13807 | 9090 | 98000 | 36961 | *4033 | 3247 |
| 6.5 | 21.65 | 32.54 | 10611 | 15946 | 10611 | 98000 | 34076 | 4025 | 3204 |
| 7 | 25.08 | 37.19 | 12289 | 18222 | 12289 | 98000 | 31417 | 3998 | 3150 |
| 7.5 | 28.85 | 42.10 | 14135 | 20631 | 14135 | 98000 | 28965 | 3951 | 3085 |
| 8 | 32.98 | 47.29 | 16160 | 23173 | 16160 | 98000 | 26704 | 3885 | 3009 |
| 8.5 | 37.50 | 52.74 | 18375 | 25845 | 18375 | 98000 | 24620 | 3802 | 2923 |
| 9 | 42.43 | 58.46 | 20793 | 28645 | 20793 | 98000 | 22698 | 3701 | 2829 |
| 9.5 | 47.81 | 64.43 | 23426 | 31573 | 23426 | 98000 | 20927 | 3583 | 2725 |
| 10 | 53.65 | 70.67 | 26289 | 34627 | 26289 | 98000 | 19294 | 3448 | 2614 |
| 10.5 | 59.99 | 77.15 | 29394 | 37805 | 29394 | 98000 | 17788 | 3296 | 2495 |
| 11 | 66.85 | 83.89 | 32758 | 41107 | 32758 | 98000 | 16400 | 3128 | 2369 |
| 11.5 | 74.28 | 90.88 | 36396 | 44532 | 36396 | 98000 | 15120 | 2944 | 2236 |
| 12 | 82.29 | 98.12 | 40324 | 48077 | 40324 | 98000 | 13940 | 2744 | 2097 |
| 1.2 | 1.4 | 1.6 | 1.8 | 2.0 | 2.2 | 1.2 | 1.4 | 1.6 | 1.8 | 2.0 | |
| Time | Expected failure times | Expected repair cost | |||||||||
| 2 | 2.32 | 2.69 | 3.06 | 3.43 | 3.79 | 4.15 | 1136 | 1319 | 1500 | 1680 | 1858 |
| 3 | 4.08 | 4.84 | 5.61 | 6.38 | 7.16 | 7.95 | 2000 | 2371 | 2746 | 3125 | 3507 |
| 4 | 6.30 | 7.61 | 8.95 | 10.34 | 11.75 | 13.21 | 3085 | 3727 | 4387 | 5065 | 5759 |
| 5 | 9.00 | 11.07 | 13.22 | 15.47 | 17.81 | 20.25 | 4412 | 5423 | 6480 | 7581 | 8726 |
| 6 | 12.26 | 15.31 | 18.55 | 21.98 | 25.59 | 29.42 | 6005 | 7500 | 9090 | 10769 | 12541 |
| 7 | 16.10 | 20.42 | 25.08 | 30.07 | 35.41 | 41.13 | 7890 | 10005 | 12289 | 14736 | 17352 |
| 8 | 20.60 | 26.51 | 32.98 | 40.01 | 47.61 | 55.88 | 10096 | 12989 | 16160 | 19603 | 23331 |
| 9 | 25.82 | 33.69 | 42.43 | 52.06 | 62.60 | 74.20 | 12653 | 16508 | 20793 | 25508 | 30675 |
| 10 | 31.83 | 42.09 | 53.65 | 66.54 | 80.83 | 96.74 | 15596 | 20622 | 26289 | 32602 | 39606 |
| 11 | 38.69 | 51.83 | 66.85 | 83.80 | 102.81 | 124.23 | 18959 | 25398 | 32758 | 41060 | 50378 |
| 12 | 46.49 | 63.08 | 82.29 | 104.23 | 129.14 | 157.52 | 22781 | 30910 | 40324 | 51075 | 63280 |
| 0.9 | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 | 0.9 | 1.0 | 1.1 | 1.2 | 1.3 | |
| Time | Expected failure times | Expected repair cost | |||||||||
| 2 | 3.04 | 3.05 | 3.06 | 3.07 | 3.08 | 3.09 | 1488 | 1494 | 1500 | 1505 | 1509 |
| 3 | 5.71 | 5.66 | 5.61 | 5.56 | 5.51 | 5.47 | 2800 | 2772 | 2746 | 2723 | 2701 |
| 4 | 9.36 | 9.14 | 8.95 | 8.78 | 8.63 | 8.48 | 4584 | 4480 | 4387 | 4304 | 4228 |
| 5 | 14.14 | 13.65 | 13.22 | 12.85 | 12.51 | 12.20 | 6928 | 6688 | 6480 | 6295 | 6128 |
| 6 | 20.27 | 19.34 | 18.55 | 17.85 | 17.24 | 16.68 | 9932 | 9478 | 9090 | 8749 | 8446 |
| 7 | 27.98 | 26.41 | 25.08 | 23.93 | 22.92 | 22.02 | 13709 | 12940 | 12289 | 11725 | 11230 |
| 8 | 37.53 | 35.05 | 32.98 | 31.20 | 29.66 | 28.30 | 18390 | 17175 | 16160 | 15290 | 14532 |
| 9 | 49.23 | 45.51 | 42.43 | 39.83 | 37.58 | 35.61 | 24124 | 22300 | 20793 | 19515 | 18412 |
| 10 | 63.43 | 58.05 | 53.65 | 49.96 | 46.80 | 44.07 | 31080 | 28442 | 26289 | 24480 | 22933 |
| 11 | 80.51 | 72.95 | 66.85 | 61.78 | 57.48 | 53.78 | 39447 | 35747 | 32758 | 30271 | 28163 |
| 12 | 100.91 | 90.56 | 82.29 | 75.48 | 69.75 | 64.87 | 49444 | 44376 | 40324 | 36986 | 34179 |
| 1.2 | 1.4 | 1.6 | 1.8 | 2.0 | 2.2 | 1.2 | 1.4 | 1.6 | 1.8 | 2.0 | |
| Time | Expected failure times | Expected repair cost | |||||||||
| 2 | 2.32 | 2.69 | 3.06 | 3.43 | 3.79 | 4.15 | 1136 | 1319 | 1500 | 1680 | 1858 |
| 3 | 4.08 | 4.84 | 5.61 | 6.38 | 7.16 | 7.95 | 2000 | 2371 | 2746 | 3125 | 3507 |
| 4 | 6.30 | 7.61 | 8.95 | 10.34 | 11.75 | 13.21 | 3085 | 3727 | 4387 | 5065 | 5759 |
| 5 | 9.00 | 11.07 | 13.22 | 15.47 | 17.81 | 20.25 | 4412 | 5423 | 6480 | 7581 | 8726 |
| 6 | 12.26 | 15.31 | 18.55 | 21.98 | 25.59 | 29.42 | 6005 | 7500 | 9090 | 10769 | 12541 |
| 7 | 16.10 | 20.42 | 25.08 | 30.07 | 35.41 | 41.13 | 7890 | 10005 | 12289 | 14736 | 17352 |
| 8 | 20.60 | 26.51 | 32.98 | 40.01 | 47.61 | 55.88 | 10096 | 12989 | 16160 | 19603 | 23331 |
| 9 | 25.82 | 33.69 | 42.43 | 52.06 | 62.60 | 74.20 | 12653 | 16508 | 20793 | 25508 | 30675 |
| 10 | 31.83 | 42.09 | 53.65 | 66.54 | 80.83 | 96.74 | 15596 | 20622 | 26289 | 32602 | 39606 |
| 11 | 38.69 | 51.83 | 66.85 | 83.80 | 102.81 | 124.23 | 18959 | 25398 | 32758 | 41060 | 50378 |
| 12 | 46.49 | 63.08 | 82.29 | 104.23 | 129.14 | 157.52 | 22781 | 30910 | 40324 | 51075 | 63280 |
| 0.9 | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 | 0.9 | 1.0 | 1.1 | 1.2 | 1.3 | |
| Time | Expected failure times | Expected repair cost | |||||||||
| 2 | 3.04 | 3.05 | 3.06 | 3.07 | 3.08 | 3.09 | 1488 | 1494 | 1500 | 1505 | 1509 |
| 3 | 5.71 | 5.66 | 5.61 | 5.56 | 5.51 | 5.47 | 2800 | 2772 | 2746 | 2723 | 2701 |
| 4 | 9.36 | 9.14 | 8.95 | 8.78 | 8.63 | 8.48 | 4584 | 4480 | 4387 | 4304 | 4228 |
| 5 | 14.14 | 13.65 | 13.22 | 12.85 | 12.51 | 12.20 | 6928 | 6688 | 6480 | 6295 | 6128 |
| 6 | 20.27 | 19.34 | 18.55 | 17.85 | 17.24 | 16.68 | 9932 | 9478 | 9090 | 8749 | 8446 |
| 7 | 27.98 | 26.41 | 25.08 | 23.93 | 22.92 | 22.02 | 13709 | 12940 | 12289 | 11725 | 11230 |
| 8 | 37.53 | 35.05 | 32.98 | 31.20 | 29.66 | 28.30 | 18390 | 17175 | 16160 | 15290 | 14532 |
| 9 | 49.23 | 45.51 | 42.43 | 39.83 | 37.58 | 35.61 | 24124 | 22300 | 20793 | 19515 | 18412 |
| 10 | 63.43 | 58.05 | 53.65 | 49.96 | 46.80 | 44.07 | 31080 | 28442 | 26289 | 24480 | 22933 |
| 11 | 80.51 | 72.95 | 66.85 | 61.78 | 57.48 | 53.78 | 39447 | 35747 | 32758 | 30271 | 28163 |
| 12 | 100.91 | 90.56 | 82.29 | 75.48 | 69.75 | 64.87 | 49444 | 44376 | 40324 | 36986 | 34179 |
| 1.5 | 1.7 | 1.9 | 2.1 | 2.3 | 2.5 | 2.7 | 1.5 | 1.7 | 1.9 | 2.1 | 2.3 | 2.5 | 2.7 | |
| Time | Expected failure times | Expected repair cost | ||||||||||||
| 2 | 3.02 | 3.07 | 3.08 | 3.06 | 3.03 | 2.98 | 2.92 | 1481 | 1502 | 1507 | 1500 | 1482 | 1458 | 1428 |
| 3 | 4.88 | 5.14 | 5.38 | 5.61 | 5.81 | 6.01 | 6.20 | 2391 | 2517 | 2636 | 2746 | 2849 | 2946 | 3038 |
| 4 | 7.07 | 7.65 | 8.28 | 8.95 | 9.66 | 10.40 | 11.17 | 3463 | 3749 | 4059 | 4387 | 4732 | 5094 | 5476 |
| 5 | 9.64 | 10.66 | 11.87 | 13.22 | 14.73 | 16.40 | 18.26 | 4723 | 5226 | 5816 | 6480 | 7219 | 8038 | 8946 |
| 6 | 12.65 | 14.25 | 16.22 | 18.55 | 21.25 | 24.35 | 27.92 | 6199 | 6981 | 7950 | 9090 | 10410 | 11932 | 13681 |
| 7 | 16.17 | 18.47 | 21.45 | 25.08 | 29.42 | 34.59 | 40.70 | 7923 | 9050 | 10510 | 12289 | 14418 | 16947 | 19943 |
| 8 | 20.27 | 23.42 | 27.66 | 32.98 | 39.52 | 47.50 | 57.20 | 9932 | 11475 | 13552 | 16160 | 19366 | 23276 | 28026 |
| 9 | 25.04 | 29.19 | 34.97 | 42.43 | 51.82 | 63.53 | 78.08 | 12269 | 14301 | 17138 | 20793 | 25394 | 31132 | 38261 |
| 10 | 30.58 | 35.88 | 43.54 | 53.65 | 66.64 | 83.17 | 104.12 | 14982 | 17582 | 21336 | 26289 | 32653 | 40752 | 51016 |
| 11 | 37.00 | 43.63 | 53.52 | 66.85 | 84.31 | 106.94 | 136.13 | 18129 | 21377 | 26225 | 32758 | 41313 | 52400 | 66704 |
| 12 | 44.44 | 52.56 | 65.08 | 82.29 | 105.23 | 135.44 | 175.07 | 21777 | 25754 | 31890 | 40324 | 51561 | 66367 | 85783 |
| 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 | 1.2 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 | 1.2 | |
| Time | Expected failure times | Expected repair cost | ||||||||||||
| 2 | 3.08 | 3.08 | 3.07 | 3.06 | 3.04 | 3.02 | 2.99 | 1507 | 1508 | 1505 | 1500 | 1491 | 1480 | 1466 |
| 3 | 5.78 | 5.72 | 5.66 | 5.61 | 5.56 | 5.52 | 5.50 | 2832 | 2802 | 2773 | 2746 | 2724 | 2706 | 2696 |
| 4 | 9.29 | 9.15 | 9.03 | 8.95 | 8.92 | 8.94 | 9.03 | 4551 | 4482 | 4426 | 4387 | 4369 | 4378 | 4424 |
| 5 | 13.63 | 13.42 | 13.27 | 13.22 | 13.29 | 13.51 | 13.94 | 6680 | 6574 | 6505 | 6480 | 6511 | 6618 | 6833 |
| 6 | 18.85 | 18.59 | 18.47 | 18.55 | 18.88 | 19.55 | 20.76 | 9239 | 9107 | 9052 | 9090 | 9249 | 9579 | 10172 |
| 7 | 24.99 | 24.71 | 24.72 | 25.08 | 25.91 | 27.46 | 30.19 | 12245 | 12110 | 12114 | 12289 | 12698 | 13455 | 14792 |
| 8 | 32.07 | 31.87 | 32.12 | 32.98 | 34.69 | 37.75 | 43.25 | 15716 | 15615 | 15740 | 16160 | 16997 | 18496 | 21192 |
| 9 | 40.15 | 40.11 | 40.79 | 42.43 | 45.53 | 51.06 | 61.40 | 19673 | 19652 | 19985 | 20793 | 22309 | 25019 | 30088 |
| 10 | 49.25 | 49.50 | 50.83 | 53.65 | 58.82 | 68.23 | 86.78 | 24134 | 24256 | 24905 | 26289 | 28824 | 33432 | 42524 |
| 11 | 59.42 | 60.12 | 62.36 | 66.85 | 75.04 | 90.32 | 122.54 | 29118 | 29458 | 30558 | 32758 | 36768 | 44259 | 60044 |
| 12 | 70.70 | 72.03 | 75.53 | 82.29 | 94.71 | 118.73 | 173.42 | 34644 | 35295 | 37010 | 40324 | 46407 | 58178 | 84977 |
| 1.5 | 1.7 | 1.9 | 2.1 | 2.3 | 2.5 | 2.7 | 1.5 | 1.7 | 1.9 | 2.1 | 2.3 | 2.5 | 2.7 | |
| Time | Expected failure times | Expected repair cost | ||||||||||||
| 2 | 3.02 | 3.07 | 3.08 | 3.06 | 3.03 | 2.98 | 2.92 | 1481 | 1502 | 1507 | 1500 | 1482 | 1458 | 1428 |
| 3 | 4.88 | 5.14 | 5.38 | 5.61 | 5.81 | 6.01 | 6.20 | 2391 | 2517 | 2636 | 2746 | 2849 | 2946 | 3038 |
| 4 | 7.07 | 7.65 | 8.28 | 8.95 | 9.66 | 10.40 | 11.17 | 3463 | 3749 | 4059 | 4387 | 4732 | 5094 | 5476 |
| 5 | 9.64 | 10.66 | 11.87 | 13.22 | 14.73 | 16.40 | 18.26 | 4723 | 5226 | 5816 | 6480 | 7219 | 8038 | 8946 |
| 6 | 12.65 | 14.25 | 16.22 | 18.55 | 21.25 | 24.35 | 27.92 | 6199 | 6981 | 7950 | 9090 | 10410 | 11932 | 13681 |
| 7 | 16.17 | 18.47 | 21.45 | 25.08 | 29.42 | 34.59 | 40.70 | 7923 | 9050 | 10510 | 12289 | 14418 | 16947 | 19943 |
| 8 | 20.27 | 23.42 | 27.66 | 32.98 | 39.52 | 47.50 | 57.20 | 9932 | 11475 | 13552 | 16160 | 19366 | 23276 | 28026 |
| 9 | 25.04 | 29.19 | 34.97 | 42.43 | 51.82 | 63.53 | 78.08 | 12269 | 14301 | 17138 | 20793 | 25394 | 31132 | 38261 |
| 10 | 30.58 | 35.88 | 43.54 | 53.65 | 66.64 | 83.17 | 104.12 | 14982 | 17582 | 21336 | 26289 | 32653 | 40752 | 51016 |
| 11 | 37.00 | 43.63 | 53.52 | 66.85 | 84.31 | 106.94 | 136.13 | 18129 | 21377 | 26225 | 32758 | 41313 | 52400 | 66704 |
| 12 | 44.44 | 52.56 | 65.08 | 82.29 | 105.23 | 135.44 | 175.07 | 21777 | 25754 | 31890 | 40324 | 51561 | 66367 | 85783 |
| 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 | 1.2 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 | 1.2 | |
| Time | Expected failure times | Expected repair cost | ||||||||||||
| 2 | 3.08 | 3.08 | 3.07 | 3.06 | 3.04 | 3.02 | 2.99 | 1507 | 1508 | 1505 | 1500 | 1491 | 1480 | 1466 |
| 3 | 5.78 | 5.72 | 5.66 | 5.61 | 5.56 | 5.52 | 5.50 | 2832 | 2802 | 2773 | 2746 | 2724 | 2706 | 2696 |
| 4 | 9.29 | 9.15 | 9.03 | 8.95 | 8.92 | 8.94 | 9.03 | 4551 | 4482 | 4426 | 4387 | 4369 | 4378 | 4424 |
| 5 | 13.63 | 13.42 | 13.27 | 13.22 | 13.29 | 13.51 | 13.94 | 6680 | 6574 | 6505 | 6480 | 6511 | 6618 | 6833 |
| 6 | 18.85 | 18.59 | 18.47 | 18.55 | 18.88 | 19.55 | 20.76 | 9239 | 9107 | 9052 | 9090 | 9249 | 9579 | 10172 |
| 7 | 24.99 | 24.71 | 24.72 | 25.08 | 25.91 | 27.46 | 30.19 | 12245 | 12110 | 12114 | 12289 | 12698 | 13455 | 14792 |
| 8 | 32.07 | 31.87 | 32.12 | 32.98 | 34.69 | 37.75 | 43.25 | 15716 | 15615 | 15740 | 16160 | 16997 | 18496 | 21192 |
| 9 | 40.15 | 40.11 | 40.79 | 42.43 | 45.53 | 51.06 | 61.40 | 19673 | 19652 | 19985 | 20793 | 22309 | 25019 | 30088 |
| 10 | 49.25 | 49.50 | 50.83 | 53.65 | 58.82 | 68.23 | 86.78 | 24134 | 24256 | 24905 | 26289 | 28824 | 33432 | 42524 |
| 11 | 59.42 | 60.12 | 62.36 | 66.85 | 75.04 | 90.32 | 122.54 | 29118 | 29458 | 30558 | 32758 | 36768 | 44259 | 60044 |
| 12 | 70.70 | 72.03 | 75.53 | 82.29 | 94.71 | 118.73 | 173.42 | 34644 | 35295 | 37010 | 40324 | 46407 | 58178 | 84977 |
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