An optimal maintenance strategy for multi-state systems based on a system linear integral equation and dynamic programming

Haibo Jin; Long Hai; Xiaoliang Tang

doi:10.3934/jimo.2018188

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doi: 10.3934/jimo.2018188

An optimal maintenance strategy for multi-state systems based on a system linear integral equation and dynamic programming

Haibo Jin ^1,, , Long Hai ^2, and Xiaoliang Tang ^3,

1.	School of Software, Liaoning Technique University, Huludao, Liaoning, 125105, China
2.	School of information engineering, Shenzhen University, Shenzhen, Guangdong, 518060, China
3.	Quanzhou institute of equipment manufacturing haixi institutes, Chinese Academy of Sciences, Quanzhou, Fujian, 362124, China

^* Corresponding author: Haibo Jin

Received May 2018 Revised August 2018 Published December 2018

Fund Project: The first author is supported by the Education Foundation of Liaoning Province in China (grant no. 161129). The second author is supported by the National Natural Science Foundation of China (grant no. 61701314). The third author is supported by the National Natural Science Foundation of China (grant no. 61401185)

An optimal preventive maintenance strategy for multi-state systems based on an integral equation and dynamic programming is described herein. Unlike traditional preventive maintenance strategies, this maintenance strategy is formulated using an integral equation, which can capture the system dynamics and avoid the curse of dimensionality arising from complex semi-Markov processes. The linear integral equation of the system is constructed based on the system kernel. A numerical technique is applied to solve this integral equation and obtain all of the mean elapsed times from each reliable state to each unreliable state. An analytical approach to the optimal preventive maintenance strategy is proposed that maximizes the expected operational time of the system subject to the total maintenance budget based on dynamic programming in which both backward and forward search techniques are used to search for the local optimal solution. Finally, numerical examples concerning two different scales of systems are presented to demonstrate the performance of the strategy in terms of accuracy and efficiency. Moreover a sensitivity analysis is provided to evaluate the robustness of the proposed strategy.

Keywords: Optimal maintenance strategy, multi-state systems, dynamic programming, linear integral equation.

Mathematics Subject Classification: Primary: 60K20, 65K05; Secondary: 90C31.

Citation: Haibo Jin, Long Hai, Xiaoliang Tang. An optimal maintenance strategy for multi-state systems based on a system linear integral equation and dynamic programming. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018188

References:

[1]	Z. Cheng, Z. Yang and B. Guo, Optimal opportunistic maintenance model of multi-unit systems, Journal of Systems Engineering and Electronics, 24 (2013), 811-817. Google Scholar
[2]	A. H. Christer and N. Jack, An integral-equation approach for replacement modelling over finite time horizons, IMA Journal of Mathematics Applied in Business and Industry, 3 (1991), 31-44. Google Scholar
[3]	M. Compare, F. Martini and E. Zio, Genetic algorithms for condition-based maintenance optimization under uncertainty, European Journal of Operational Research, 244 (2015), 611-623. Google Scholar
[4]	M. Compare and E. Zio, Genetic algorithms in the framework of dempster-shafer theory of evidence for maintenance optimization problems, IEEE Transactions on Reliability, 64 (2015), 645-660. Google Scholar
[5]	A. Csenki, An integral equation approach to the interval reliability of systems modelled by finite semi-Markov processes, Reliability Engineering and System Safety, 47 (1995), 37-45. Google Scholar
[6]	L. Cui, H. Li and J. Li, Markov repairable systems with history-dependent up and down states, Stochastic Models, 23 (2007), 665-681. doi: 10.1080/15326340701645983. Google Scholar
[7]	V. Dominguez, High-order collocation and quadrature methods for some logarithmic kernel integral equations on open arcs, Journal of Computational and Applied Mathematics, 161 (2003), 145-159. doi: 10.1016/S0377-0427(03)00583-1. Google Scholar
[8]	J. Driessen, H. Peng and G. van Houtum, Maintenance optimization under non-constant probabilities of imperfect inspections, Reliability Engineering and System Safety, 165 (2017), 115-123. Google Scholar
[9]	E. El-Neweihi and F. Proschan, Degradable systems:a survey of multistate system theory, Communications in Statistics, 13 (1984), 405-432. doi: 10.1080/03610928408828694. Google Scholar
[10]	S. Eryilmaz, Modeling dependence between two multi-state components via copulas, IEEE Transactions on Reliability, 63 (2014), 715-720. Google Scholar
[11]	M. Gu, X. Lu, J. Gu and Y. Zhang, Single-machine scheduling problems with machine aging effect and an optional maintenance activity, Applied Mathematical Modelling, 40 (2016), 8862-8871. doi: 10.1016/j.apm.2016.01.038. Google Scholar
[12]	S. V. Gurov and L. V. Utkin, The time-dependent availability of repairable m-out-of-n cold standby systems by arbitrary distributions and repair facilities, Microelectronics and Reliability, 35 (1995), 1377-1393. Google Scholar
[13]	A. Horenbeek, L. Pintelon and P. Muchiri, Maintenance optimization models and criteria, International Journal of System Assurance Engineering and Management, 35 (2010), 189-200. Google Scholar
[14]	N. Jack, Repair replacement modeling over finite-time horizons, Journal of The Operational Research Society, 42 (1991), 759-766. Google Scholar
[15]	F. Kayedpour, M. Amiri, M. Rafizadeh and A. S. Nia, Multi-objective redundancy allocation problem for a system with repairable components considering instantaneous availability and strategy selection, Reliability Engineering and System Safety, 160 (2017), 11-20. Google Scholar
[16]	V. P. Koutras, S. Malefaki and A. N. Platis, Optimization of the dependability and performance measures of a generic model for multi-state deteriorating systems under maintenance, Reliability Engineering and System Safety, 166 (2017), 73-86. Google Scholar
[17]	I. N. Kovalenko, N. Y. U. Kuznetsov and P. A. Pegg, Wiley Series in Probability and Statistics Mathematical Theory of Reliability of Time Dependent Systems with Practical Applications, Wiley, New York, 1997.Google Scholar
[18]	G. Levitin and A. Lisnianski, A new approach to solving problems of multi-state system reliability optimization, Quality and Reliability Engineering International, 17 (2001), 93-104. Google Scholar
[19]	N. Limnios and G. Oprisan, A unified approach for reliability and performability evaluation of semi-Markov systems, Applied Stochastic Models in Business and Industry, 15 (1999), 353-368. doi: 10.1002/(SICI)1526-4025(199910/12)15:4<353::AID-ASMB399>3.0.CO;2-2. Google Scholar
[20]	A. Lisnianski, Extended block diagram method for a multi-state system reliability assessment, Reliability Engineering and System Safety, 92 (2007), 1601-1607. Google Scholar
[21]	A. Lisnianski, I. Frenkel and L. Khvatskin, On sensitivity analysis of ageing multi-state system by using LZ-transform, Reliability Engineering and System Safety, 166 (2017), 99-108. Google Scholar
[22]	E. López-Santana, R. Akhavan-Tabatabaei, L. Dieulle, N. Labadie and A. L. Medaglia, On the combined maintenance and routing optimization problem, Reliability Engineering and System Safety, 145 (2016), 199-214. Google Scholar
[23]	E. Y. A. Maksoud and M. S. Moustafa, A semi-markov decision algorithm for the optimal maintenance of a multi-stage deteriorating two-unit standby system, Operational Research, 9 (2009), 167-182. Google Scholar
[24]	D. Montoro-Cazorla and R. Pérez-Ocón, A redundant n-system under shocks and repairs following Markovian arrival processes, Reliability Engineering and System Safety, 130 (2014), 69-75. Google Scholar
[25]	M. L. Neves, L. P. Santiago and C. A. Maia, A condition-based maintenance policy and input parameters estimation for deteriorating systems under periodic inspection, Computers and Industry Engineering, 61 (2011), 503-511. Google Scholar
[26]	M. Nourelfath, E. Châtelet and N. Nahas, Joint redundancy and imperfect preventive maintenance optimization for series-parallel multi-state degraded systems, Reliability Engineering and System Safety, 103 (2012), 51-60. Google Scholar
[27]	H. Pham and H. Wang, Imperfect maintenance, European Journal of Operational Research, 94 (1996), 425-438. Google Scholar
[28]	K. Prem and P. Pratap, Computational methods for linear integral equations, Birkhäuser Boston, c/o Sprintger-Verlag, New York, Inc., 175 Fifth Avenue, New York, USA, 2002.Google Scholar
[29]	G. Rubino and B. Sericola, Interval availability analysis using denumerable Markov-processes application to multiprocessor subject to breakdowns and repair, IEEE Transactions on Computers, 44 (1995), 286-291. Google Scholar
[30]	J. E. Ruiz-Castro, Markov counting and reward processes for analysing the performance of a complex system subject to random inspections, Reliability Engineering and System Safety, 145 (2016), 155-168. Google Scholar
[31]	S. H. Sheu, C. Chang, Y. Chen and Z. George, Optimal preventive maintenance and repair policies for multi-state systems, Reliability Engineering and System Safety, 140 (2015), 78-87. Google Scholar
[32]	A. Sharma, G. S. Yadava and S. G. Deshmukh, A literature review and future perspectives on maintenance optimization, Journal of Quality in Maintenance Engineering, 17 (2011), 5-25. Google Scholar
[33]	I. W. Soro, M. Nourelfath and D. Aït-Kadi, Performance evaluation of multi-state degraded systems with minimal repairs and imperfect preventive maintenance, Reliability Engineering and System Safety, 95 (2010), 65-69. Google Scholar
[34]	R. Srinivasan and A. K. Parlikad, Semi-Markov Decision Process With Partial Information for Maintenance Decisions, IEEE Transactions on Reliability, 63 (2014), 891-898. Google Scholar
[35]	D. Tang, V. Makis, L. Jafari and J. Yu, Optimal maintenance policy and residual life estimation for a slowly degrading system subject to condition monitoring, Reliability Engineering and System Safety, 134 (2015), 198-207. Google Scholar
[36]	S. Wu and P. Longhurst, Optimising age-replacement and extended non-renewing warranty policies, International Journal of Production Economics, 130 (2011), 262-267. Google Scholar
[37]	T. Xia, L. Xi, X. Zhou and J. Lee, Condition-based maintenance for intelligent monitored series system with independent machine failure modes, International Journal of Production Research, 51 (2013), 4585-4596. Google Scholar
[38]	M. Zhang, O. Gaudoin and M. Xie, Degradation-based maintenance decision using stochastic filtering for systems under imperfect maintenance, European Journal of Operational Research, 245 (2015), 531-541. doi: 10.1016/j.ejor.2015.02.050. Google Scholar
[39]	X. Zhao, T. Nakagawa and M. J. Zuo, Optimal replacement last with continuous and discrete policies, IEEE Transactions on Reliability, 63 (2014), 868-880. Google Scholar
[40]	Z. Zheng, L. R. Cui and H. Li, Availability of semi-Markov repairable systems with history-dependent up and down states, In: Proceedings of the Third Asian international workshop, 2008,186--193.Google Scholar
[41]	X. Zhou, L. Xi and J. Lee, Reliability-centered predictive maintenance scheduling for a continuously monitored system subject to degradation, Reliability Engineering and System Safety, 92 (2007), 530-534. Google Scholar
[42]	X. Zhou, L. Xi and J. Lee, Opportunistic preventive maintenance scheduling for a multi-unit series system based on dynamic programming, International Journal of Production Economics, 118 (2009), 361-366. Google Scholar

show all references

References:

[1]	Z. Cheng, Z. Yang and B. Guo, Optimal opportunistic maintenance model of multi-unit systems, Journal of Systems Engineering and Electronics, 24 (2013), 811-817. Google Scholar
[2]	A. H. Christer and N. Jack, An integral-equation approach for replacement modelling over finite time horizons, IMA Journal of Mathematics Applied in Business and Industry, 3 (1991), 31-44. Google Scholar
[3]	M. Compare, F. Martini and E. Zio, Genetic algorithms for condition-based maintenance optimization under uncertainty, European Journal of Operational Research, 244 (2015), 611-623. Google Scholar
[4]	M. Compare and E. Zio, Genetic algorithms in the framework of dempster-shafer theory of evidence for maintenance optimization problems, IEEE Transactions on Reliability, 64 (2015), 645-660. Google Scholar
[5]	A. Csenki, An integral equation approach to the interval reliability of systems modelled by finite semi-Markov processes, Reliability Engineering and System Safety, 47 (1995), 37-45. Google Scholar
[6]	L. Cui, H. Li and J. Li, Markov repairable systems with history-dependent up and down states, Stochastic Models, 23 (2007), 665-681. doi: 10.1080/15326340701645983. Google Scholar
[7]	V. Dominguez, High-order collocation and quadrature methods for some logarithmic kernel integral equations on open arcs, Journal of Computational and Applied Mathematics, 161 (2003), 145-159. doi: 10.1016/S0377-0427(03)00583-1. Google Scholar
[8]	J. Driessen, H. Peng and G. van Houtum, Maintenance optimization under non-constant probabilities of imperfect inspections, Reliability Engineering and System Safety, 165 (2017), 115-123. Google Scholar
[9]	E. El-Neweihi and F. Proschan, Degradable systems:a survey of multistate system theory, Communications in Statistics, 13 (1984), 405-432. doi: 10.1080/03610928408828694. Google Scholar
[10]	S. Eryilmaz, Modeling dependence between two multi-state components via copulas, IEEE Transactions on Reliability, 63 (2014), 715-720. Google Scholar
[11]	M. Gu, X. Lu, J. Gu and Y. Zhang, Single-machine scheduling problems with machine aging effect and an optional maintenance activity, Applied Mathematical Modelling, 40 (2016), 8862-8871. doi: 10.1016/j.apm.2016.01.038. Google Scholar
[12]	S. V. Gurov and L. V. Utkin, The time-dependent availability of repairable m-out-of-n cold standby systems by arbitrary distributions and repair facilities, Microelectronics and Reliability, 35 (1995), 1377-1393. Google Scholar
[13]	A. Horenbeek, L. Pintelon and P. Muchiri, Maintenance optimization models and criteria, International Journal of System Assurance Engineering and Management, 35 (2010), 189-200. Google Scholar
[14]	N. Jack, Repair replacement modeling over finite-time horizons, Journal of The Operational Research Society, 42 (1991), 759-766. Google Scholar
[15]	F. Kayedpour, M. Amiri, M. Rafizadeh and A. S. Nia, Multi-objective redundancy allocation problem for a system with repairable components considering instantaneous availability and strategy selection, Reliability Engineering and System Safety, 160 (2017), 11-20. Google Scholar
[16]	V. P. Koutras, S. Malefaki and A. N. Platis, Optimization of the dependability and performance measures of a generic model for multi-state deteriorating systems under maintenance, Reliability Engineering and System Safety, 166 (2017), 73-86. Google Scholar
[17]	I. N. Kovalenko, N. Y. U. Kuznetsov and P. A. Pegg, Wiley Series in Probability and Statistics Mathematical Theory of Reliability of Time Dependent Systems with Practical Applications, Wiley, New York, 1997.Google Scholar
[18]	G. Levitin and A. Lisnianski, A new approach to solving problems of multi-state system reliability optimization, Quality and Reliability Engineering International, 17 (2001), 93-104. Google Scholar
[19]	N. Limnios and G. Oprisan, A unified approach for reliability and performability evaluation of semi-Markov systems, Applied Stochastic Models in Business and Industry, 15 (1999), 353-368. doi: 10.1002/(SICI)1526-4025(199910/12)15:4<353::AID-ASMB399>3.0.CO;2-2. Google Scholar
[20]	A. Lisnianski, Extended block diagram method for a multi-state system reliability assessment, Reliability Engineering and System Safety, 92 (2007), 1601-1607. Google Scholar
[21]	A. Lisnianski, I. Frenkel and L. Khvatskin, On sensitivity analysis of ageing multi-state system by using LZ-transform, Reliability Engineering and System Safety, 166 (2017), 99-108. Google Scholar
[22]	E. López-Santana, R. Akhavan-Tabatabaei, L. Dieulle, N. Labadie and A. L. Medaglia, On the combined maintenance and routing optimization problem, Reliability Engineering and System Safety, 145 (2016), 199-214. Google Scholar
[23]	E. Y. A. Maksoud and M. S. Moustafa, A semi-markov decision algorithm for the optimal maintenance of a multi-stage deteriorating two-unit standby system, Operational Research, 9 (2009), 167-182. Google Scholar
[24]	D. Montoro-Cazorla and R. Pérez-Ocón, A redundant n-system under shocks and repairs following Markovian arrival processes, Reliability Engineering and System Safety, 130 (2014), 69-75. Google Scholar
[25]	M. L. Neves, L. P. Santiago and C. A. Maia, A condition-based maintenance policy and input parameters estimation for deteriorating systems under periodic inspection, Computers and Industry Engineering, 61 (2011), 503-511. Google Scholar
[26]	M. Nourelfath, E. Châtelet and N. Nahas, Joint redundancy and imperfect preventive maintenance optimization for series-parallel multi-state degraded systems, Reliability Engineering and System Safety, 103 (2012), 51-60. Google Scholar
[27]	H. Pham and H. Wang, Imperfect maintenance, European Journal of Operational Research, 94 (1996), 425-438. Google Scholar
[28]	K. Prem and P. Pratap, Computational methods for linear integral equations, Birkhäuser Boston, c/o Sprintger-Verlag, New York, Inc., 175 Fifth Avenue, New York, USA, 2002.Google Scholar
[29]	G. Rubino and B. Sericola, Interval availability analysis using denumerable Markov-processes application to multiprocessor subject to breakdowns and repair, IEEE Transactions on Computers, 44 (1995), 286-291. Google Scholar
[30]	J. E. Ruiz-Castro, Markov counting and reward processes for analysing the performance of a complex system subject to random inspections, Reliability Engineering and System Safety, 145 (2016), 155-168. Google Scholar
[31]	S. H. Sheu, C. Chang, Y. Chen and Z. George, Optimal preventive maintenance and repair policies for multi-state systems, Reliability Engineering and System Safety, 140 (2015), 78-87. Google Scholar
[32]	A. Sharma, G. S. Yadava and S. G. Deshmukh, A literature review and future perspectives on maintenance optimization, Journal of Quality in Maintenance Engineering, 17 (2011), 5-25. Google Scholar
[33]	I. W. Soro, M. Nourelfath and D. Aït-Kadi, Performance evaluation of multi-state degraded systems with minimal repairs and imperfect preventive maintenance, Reliability Engineering and System Safety, 95 (2010), 65-69. Google Scholar
[34]	R. Srinivasan and A. K. Parlikad, Semi-Markov Decision Process With Partial Information for Maintenance Decisions, IEEE Transactions on Reliability, 63 (2014), 891-898. Google Scholar
[35]	D. Tang, V. Makis, L. Jafari and J. Yu, Optimal maintenance policy and residual life estimation for a slowly degrading system subject to condition monitoring, Reliability Engineering and System Safety, 134 (2015), 198-207. Google Scholar
[36]	S. Wu and P. Longhurst, Optimising age-replacement and extended non-renewing warranty policies, International Journal of Production Economics, 130 (2011), 262-267. Google Scholar
[37]	T. Xia, L. Xi, X. Zhou and J. Lee, Condition-based maintenance for intelligent monitored series system with independent machine failure modes, International Journal of Production Research, 51 (2013), 4585-4596. Google Scholar
[38]	M. Zhang, O. Gaudoin and M. Xie, Degradation-based maintenance decision using stochastic filtering for systems under imperfect maintenance, European Journal of Operational Research, 245 (2015), 531-541. doi: 10.1016/j.ejor.2015.02.050. Google Scholar
[39]	X. Zhao, T. Nakagawa and M. J. Zuo, Optimal replacement last with continuous and discrete policies, IEEE Transactions on Reliability, 63 (2014), 868-880. Google Scholar
[40]	Z. Zheng, L. R. Cui and H. Li, Availability of semi-Markov repairable systems with history-dependent up and down states, In: Proceedings of the Third Asian international workshop, 2008,186--193.Google Scholar
[41]	X. Zhou, L. Xi and J. Lee, Reliability-centered predictive maintenance scheduling for a continuously monitored system subject to degradation, Reliability Engineering and System Safety, 92 (2007), 530-534. Google Scholar
[42]	X. Zhou, L. Xi and J. Lee, Opportunistic preventive maintenance scheduling for a multi-unit series system based on dynamic programming, International Journal of Production Economics, 118 (2009), 361-366. Google Scholar

Figure 1. The whole operation process of the system

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Figure 2. An operation cycle consisting of the maintenance and deterioration stages

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Figure 3. Backward search technique for every state $ X_{t_{q+1}} \in \mathbf{U} $

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Figure 4. Backward search technique for every state $ X_{t_{q+1}} \in \mathbf{U} $

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Figure 5. Forward search technique for every remaining state $ X_{t_{q}} \in \mathbf{U} $

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Figure 6. Time-spans from state 1 to state X_t₁ ∈ U and from X_{t_last} ∈ U to failure state n

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Figure 7. Discrete values of $ \phi_{13}(t_{m}) $, $ \phi_{26}(t_{m}) $, $ \phi_{38}(t_{m}) $, $ \phi_{47}(t_{m}) $ under the Weibull distribution with parameters $ \lambda = 0.4 $ and $ \alpha = 2 $

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Figure 8. The running time of Algorithm 1 for the small-scale system with the Weibull distribution corresponding to Case 1

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Figure 9. The running time of Algorithm 1 for the small-scale system with the general distribution corresponding to Case 2

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Figure 10. The running time of Algorithm 1 for the large-scale system with the Weibull distribution corresponding to Case 3

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Figure 11. The running time of Algorithm 1 for the large-scale system with the general distribution corresponding to Case 4

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Figure 12. Error of $ T^{*}(t, a_{t_q}) $ vs Errors of $ \lambda $ and $ \alpha $

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Algorithm 1. Optimal maintenance strategy based on dynamic programming

1: Initialization: $ q = 1, \mathit{\boldsymbol{A}}^{*} = \left(0\right)_{1\times(n-k-1)}, $ $ \mathit{\boldsymbol{B}}^{*} = \left(0\right)_{1\times(n-k-1)} $, $ \mathit{\boldsymbol{C}}^{*} = \left(0\right)_{1\times(n-k-1)} $ and $ D = \{\underbrace{\{\}, \cdots, \{\}}_{n-k-1}\} $

2: while

$ \min\{c_{1}, \cdots, c_{n-k-1}\} < C $

3: Construct matrices $ \mathit{\boldsymbol{\Theta_{t_q}}} $, $ {\bf{E}}\left[\mathit{\boldsymbol{\Psi}}_{t_q}\right] $ and $ \mathit{\boldsymbol{C}}_{t_q} $:

$\mathit{\boldsymbol{\Theta_{t_q}}} = \left[ \theta_{X_{t_q}, Z(X_{t}, a_{t_q})} \right]_{(n-k-1)\times k} \ {\bf{E}}\left[\mathit{\boldsymbol{\Psi}}_{t_q}\right] = \left[ {\bf{E}}\left[\psi_{Z(X_{t}, a_{t_q}), X_{t_{q+1}}} \right] \right]_{k \times (n-k-1)} \\\ \mathbf{C}_{t_q} = \left[ c_{X_{t_q}, Z(X_{t}, a_{t_q})} \right]_{(n-k-1)\times k} $

4: Calculate the optimal vectors $ \zeta_{X_{t_{q+1}}}^* $, $ {(\theta_{q}+{\bf{E}}[\psi_{q}])}_{X_{t_{q+1}}}^{*} $ and $ c^* $:

$ \begin{aligned} \zeta_{X_{t_{q+1}}}^* = \max \Vert ( \mathit{\boldsymbol{\theta}}_{X_{t_q}, \bullet} +{\bf{E}}[\mathit{\boldsymbol{\psi}}_{\bullet, X_{t_{q+1}}}] )\cdot \mathit{\boldsymbol{c}}_{X_{t_q}, \bullet}^{-1} \Vert_{\infty} \text{for} X_{t_q} = k+1, \cdots, n-1 \end{aligned} $

$\begin{aligned} {(\theta_{q}+{\bf{E}}[\psi_{q}])}_{X_{t_{q+1}}}^{*} = \arg \max \limits_{\mathit{\boldsymbol{\odot}}} \Vert ( \mathit{\boldsymbol{\theta}}_{X_{t_q}, \bullet} +{\bf{E}}[\mathit{\boldsymbol{\psi}}_{\bullet, X_{t_{q+1}}}] )\cdot \mathit{\boldsymbol{c}}_{X_{t_q}, \bullet}^{-1} \Vert_{\infty} \\\ \text{for} X_{t_q} = k+1, \cdots, n-1 \end{aligned} $

$c^* = \arg \max \limits_{\mathit{\boldsymbol{\oslash}}} \Vert ( \mathit{\boldsymbol{\theta}}_{X_{t_q}, \bullet} +{\bf{E}}[\mathit{\boldsymbol{\psi}}_{\bullet, X_{t_{q+1}}}] )\cdot \mathit{\boldsymbol{c}}_{X_{t_q}, \bullet}^{-1} \Vert_{\infty} \text{for} X_{t_q} = k+1, \cdots, n-1 $

5: Assign $ \zeta_{X_{t_{q+1}}}^* $, $ {(\theta_{q}+<bold>E</bold>[\psi_{q}])}_{X_{t_{q+1}}}^{*} $ }and $ c^* $ for all $ X_{t_{q+1}} \in \mathbf{U} $ to $ \mathit{\boldsymbol{A}}^{*}_{q} $, $ \mathit{\boldsymbol{B}}^{*}_{q} $ and $ \mathit{\boldsymbol{C}}^{*}_{q} $, respectively, i.e.,

$\mathit{\boldsymbol{A}}^{*}_{q} = \left(\zeta_{k+1}^*, \zeta_{k+2}^*, \cdots, \zeta_{n-1}^*\right) $

$\mathit{\boldsymbol{B}}^{*}_{q} = \left({(\theta_{q}+{\bf{E}}[\psi_{q}])}_{k+1}^{*}, {(\theta_{q}+{\bf{E}}[\psi_{q}])}_{k+2}^{*}, \cdots, {(\theta_{q}+{\bf{E}}[\psi_{q}])}_{n-1}^{*} \right) $

$\mathit{\boldsymbol{C}}^{*}_{q} = \left(c_{k+1}^*, c_{k+2}^*, \cdots, c_{n-1}^*\right) $

6: Identify the optimal paths corresponding to $ \zeta_{X_{t_{q+1}}}^* $ for all $ X_{t_{q+1}} $, i.e.,

$\begin{aligned} (X_{t_q}, Z(X_{t}, a_{t_q}), X_{t_{q+1}})^* \arg \max \limits_{\mathit{\boldsymbol{\oplus}}} & \Vert ( \mathit{\boldsymbol{\theta}}_{X_{t_q}, \bullet} +{\bf{E}}[\mathit{\boldsymbol{\psi}}_{\bullet, X_{t_{q+1}}}] )\cdot \mathit{\boldsymbol{c}}_{X_{t_q}, \bullet}^{-1} \Vert_{\infty} \\\ & \text{for} X_{t_{q+1}} = k+1, \cdots, n-1 \end{aligned} $

7: Store each $ (X_{t_q}, Z(X_{t}, a_{t_q}), X_{t_{q+1}})^* $ in List D.

8: Judge whether or not there exists the case of not one-to-one correspondence. If yes, the forward search technique is used for all of the remaining states $ X_{t_q} \in \overline{\mathbf{R}} $ to identify $ \zeta_{X_{t_{q}}}^* $ and the corresponding optimal path, i.e.,

$ \begin{aligned} \zeta_{X_{t_{q}}}^* = \max &\Vert ( \mathit{\boldsymbol{\theta}}_{X_{t_q}, \bullet} +{\bf{E}}[\mathit{\boldsymbol{\psi}}_{\bullet, X_{t_{q+1}}}] )\cdot \mathit{\boldsymbol{c}}_{X_{t_q}, \bullet}^{-1} \Vert_{\infty} \\\ & \text{for} X_{t_{q+1}} = k+1, \cdots, n-1; X_{t_q} \in \overline{\mathbf{R}} \end{aligned}$

${({X_{{t_q}}}, Z({X_t}, {a_{{t_q}}}), {X_{{t_{q + 1}}}})^*} = {\rm{ }}\arg \mathop {\max }\limits_ \oplus ({\rm{ }}{\mathit{\boldsymbol{\theta }}_{{X_{{t_q}}}, \bullet }} + {\bf{E}}[{\mathit{\boldsymbol{\psi }}_{ \bullet , {X_{{t_{q + 1}}}}}}]) \cdot {\rm{ }}\mathit{\boldsymbol{c}}_{{X_{{t_q}}}, \bullet }^{ - 1}{_\infty }\\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{ for}}{X_{{t_{q + 1}}}} = k + 1, \cdots , n - 1;{X_{{t_q}}} \in \overline {\bf{R}} $

and append them into the tail of corresponding sub-lists of $ D $. Else, continue.

9: Update the vectors $ \mathit{\boldsymbol{A}}^{*} $, $ \mathit{\boldsymbol{B}}^{*} $ and $ \mathit{\boldsymbol{C}}^{*} $, i.e., $ \mathit{\boldsymbol{A}}^* \leftarrow \mathit{\boldsymbol{A}}^*+\mathit{\boldsymbol{A}}^*_{q} $, $ \mathit{\boldsymbol{B}}^* \leftarrow \mathit{\boldsymbol{B}}^*+\mathit{\boldsymbol{B}}^*_{q} $ and $ \mathit{\boldsymbol{C}}^* \leftarrow \mathit{\boldsymbol{C}}^*+\mathit{\boldsymbol{C}}^*_{q} $.

10: Judge whether each element of $ \mathit{\boldsymbol{C}}^* $ is greater than or equal to the total maintenance budget $ C $. If yes, the optimal search on this optimal path is over. Otherwise, the optimal search is continued.

11: Update List $ D $ by

$D \leftarrow \textrm{Append} \left(D_{X_{t_{q}}}, (X_{t_q}, Z(X_{t_q}, a_{t_q}), X_{t_{q+1}})^*\right) $

12: Set $ q = q+1 $

13: end while

14: Determine all time-spans in cycle $ t_0 $ and in cycle $ t_{\textrm{last}} $, i.e.,

${\bf{E}}\left[\mathit{\boldsymbol{\Psi}}_{t_1}\right] = \left( {\bf{E}}\left[\psi_{1, k+1}\right], {\bf{E}}\left[\psi_{1, k+2}\right], \cdots {\bf{E}}\left[\psi_{1, n-1}\right]\right) $

${\bf{E}}\left[\mathit{\boldsymbol{\Psi}}_{t_{\textrm{last}}}\right] = \left( {\bf{E}}\left[\psi_{k+1, n}\right], {\bf{E}}\left[\psi_{k+2, n}\right], \cdots {\bf{E}}\left[\psi_{n-1, n}\right]\right) $

15: Update $ \mathit{\boldsymbol{B}}^* $ by $ \mathit{\boldsymbol{B}}^* \leftarrow \mathit{\boldsymbol{B}}^* + {\bf{E}}\left[\mathit{\boldsymbol{\Psi}}_{t_0}\right] + {\bf{E}}\left[\mathit{\boldsymbol{\Psi}}_{t_{\textrm{last}}}\right] $ and store all paths$ \left(1, X_{t_1}\right) $ and $ \left(X_{t_{\textrm{last}}}, n\right) $ in the header and tail, respectively, of the corresponding sub-list.

16: Determine $ T^{*}\left(t, a_t\right) $ and the corresponding $ D^* $, i.e., $ T^{*}(t, a_t) = \Vert \mathit{\boldsymbol{B}}^* \Vert_{\infty} \ D^{*} = \arg \Vert \mathit{\boldsymbol{B}}^* \Vert_{\infty} $

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