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

*Corresponding author: Chih-Chiang Fang

Received  June 2019 Revised  June 2020 Published  August 2020

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.

Citation: Chih-Chiang Fang. Bayesian decision making in determining optimal leased term and preventive maintenance scheme for leased facilities. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020127
References:
[1]

N. ArasR. 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.  Google Scholar

[2]

A. Ben MabroukA. 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.  Google Scholar

[3]

S. BourjadeR. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

M. EbrahimiS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

H. GargM. 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.  Google Scholar

[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.  Google Scholar

[16]

B. HadjaissaK. AmeurS. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

H. JinL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

R. T. Kleiman, The characteristics of venture lease financing, Journal of Equipment Lease Financing, 19 (2001), 1-10.   Google Scholar

[28]

C. LiuY. FangC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

Y. SaitoT. 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.   Google Scholar

[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.  Google Scholar

[39]

M. SheikhalishahiH. Heidaryan-BaygyS. 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.   Google Scholar

[40]

J. ShinJ. 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.   Google Scholar

[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.  Google Scholar

[42]

J. Taheri-TolgariM. MohammadiB. NaderiA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45]

X. WangL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[49]

R. H. YehK. 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.  Google Scholar

[50]

R. H. YehK. 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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

N. ArasR. 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.  Google Scholar

[2]

A. Ben MabroukA. 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.  Google Scholar

[3]

S. BourjadeR. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

M. EbrahimiS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

H. GargM. 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.  Google Scholar

[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.  Google Scholar

[16]

B. HadjaissaK. AmeurS. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

H. JinL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

R. T. Kleiman, The characteristics of venture lease financing, Journal of Equipment Lease Financing, 19 (2001), 1-10.   Google Scholar

[28]

C. LiuY. FangC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

Y. SaitoT. 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.   Google Scholar

[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.  Google Scholar

[39]

M. SheikhalishahiH. Heidaryan-BaygyS. 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.   Google Scholar

[40]

J. ShinJ. 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.   Google Scholar

[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.  Google Scholar

[42]

J. Taheri-TolgariM. MohammadiB. NaderiA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45]

X. WangL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[49]

R. H. YehK. 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.  Google Scholar

[50]

R. H. YehK. 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.  Google Scholar

[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.  Google Scholar

Figure 1.  Timeline of the PM Model
Figure 2.  Maintenance Scheme under Imperfect Recovery
Figure 3.  The Flow Chart of the Heuristic Process for Obtaining $ N^* $ and $ q^* $
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 ${u_\alpha}$= 1.60, ${u_\beta}$= 2.10, ${\sigma_\alpha}$= 1.10, ${\sigma_\beta}$= 0.80
Age reduction factors ${\delta ^{M_P^1}}$ = 0.7 ${\delta ^{M_P^2}}$ = 0.8 ${\delta ^{M_P^3}}$ = 0.9
Base cost for a PM action $C_F^{M_P^1}$=600 $C_F^{M_P^2}$=750 $C_F^{M_P^3}$=900
Periodically increasing rates of PM cost ${\tau ^{M_P^1}}$=0.2 ${\tau ^{M_P^2}}$=0.225 data ${\tau ^{M_P^3}}$=0.25
Depreciation rate $\rho$=0.15
Interval of PM; Time segment $x$=0.5 years; $T_S$=0.5 year
The minimal and maximal planned lease terms $T_L^{Min}$=2 years; $T_L^{Max}$=12 years
Rental of per half-year $R_0$=9800
Time discount rate $\epsilon$=0.02
Production cost of an equipment $V$=9800
Penalty cost for repair time over the time limit $C_Penalty= 170$
Expectation of performing a minimal repair $E(t_r)$= 9 hours
Standard deviation of performing a minimal repair $\sigma(t_r)$= 5 hours
Tolerable waiting time limit for performing a minimal repair $\varphi$=4.5 hours
Expected cost of performing a minimal repair $C_mr$=350
Maintenance Plan 1 Maintenance Plan 2 Maintenance Plan 3
Parameters for the deterioration judged by experts ${u_\alpha}$= 1.60, ${u_\beta}$= 2.10, ${\sigma_\alpha}$= 1.10, ${\sigma_\beta}$= 0.80
Age reduction factors ${\delta ^{M_P^1}}$ = 0.7 ${\delta ^{M_P^2}}$ = 0.8 ${\delta ^{M_P^3}}$ = 0.9
Base cost for a PM action $C_F^{M_P^1}$=600 $C_F^{M_P^2}$=750 $C_F^{M_P^3}$=900
Periodically increasing rates of PM cost ${\tau ^{M_P^1}}$=0.2 ${\tau ^{M_P^2}}$=0.225 data ${\tau ^{M_P^3}}$=0.25
Depreciation rate $\rho$=0.15
Interval of PM; Time segment $x$=0.5 years; $T_S$=0.5 year
The minimal and maximal planned lease terms $T_L^{Min}$=2 years; $T_L^{Max}$=12 years
Rental of per half-year $R_0$=9800
Time discount rate $\epsilon$=0.02
Production cost of an equipment $V$=9800
Penalty cost for repair time over the time limit $C_Penalty= 170$
Expectation of performing a minimal repair $E(t_r)$= 9 hours
Standard deviation of performing a minimal repair $\sigma(t_r)$= 5 hours
Tolerable waiting time limit for performing a minimal repair $\varphi$=4.5 hours
Expected cost of performing a minimal repair $C_mr$=350
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
$\mathop E\limits_{{\text{Prior}}} [\Phi ({T_L},x,{\delta ^{M_P^q}},\alpha ,\beta )]$ PM Cost Repair Cost $V$ ${V_{residual}}$ $\mathop E\limits_{{\text{Prior}}} \left[ \pi \right]$
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
$\mathop E\limits_{{\text{Prior}}} [\Phi ({T_L},x,{\delta ^{M_P^q}},\alpha ,\beta )]$ PM Cost Repair Cost $V$ ${V_{residual}}$ $\mathop E\limits_{{\text{Prior}}} \left[ \pi \right]$
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
$E[\Phi ({T_L},x,{\delta ^{M_P^q}},\alpha ,\beta )]$ Repair Cost PM Cost $V$ ${V_{residual}}$ $E\left[ \pi \right]$
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
$E[\Phi ({T_L},x,{\delta ^{M_P^q}},\alpha ,\beta )]$ Repair Cost PM Cost $V$ ${V_{residual}}$ $E\left[ \pi \right]$
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
Table 4.  The impact of $E(\alpha)$ and $\sigma(\alpha)$ on expected failure times and repair cost
$E(\alpha)$
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
$\sigma(\alpha)$
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
$E(\alpha)$
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
$\sigma(\alpha)$
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
Table 5.  The impact of $E(\beta)$ and $\sigma(\beta)$ on expected failure times and repair cost
$E(\beta)$
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
$\sigma(\beta)$
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
$E(\beta)$
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
$\sigma(\beta)$
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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