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March  2020, 16(2): 965-990. doi: 10.3934/jimo.2018188

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

 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  March 2020 Early access  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.

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 and Management Optimization, 2020, 16 (2) : 965-990. doi: 10.3934/jimo.2018188
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##### References:
The whole operation process of the system
An operation cycle consisting of the maintenance and deterioration stages
Backward search technique for every state $X_{t_{q+1}} \in \mathbf{U}$
Backward search technique for every state $X_{t_{q+1}} \in \mathbf{U}$
Forward search technique for every remaining state $X_{t_{q}} \in \mathbf{U}$
Time-spans from state 1 to state Xt1U and from XtlastU to failure state n
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 $\alpha = 2$
The running time of Algorithm 1 for the small-scale system with the Weibull distribution corresponding to Case 1
The running time of Algorithm 1 for the small-scale system with the general distribution corresponding to Case 2
The running time of Algorithm 1 for the large-scale system with the Weibull distribution corresponding to Case 3
The running time of Algorithm 1 for the large-scale system with the general distribution corresponding to Case 4
Error of $T^{*}(t, a_{t_q})$ vs Errors of $\lambda$ and $\alpha$
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  $\min\{c_{1}, \cdots, c_{n-k-1}\} < C$  do 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 $\zeta_{X_{t_{q+1}}}^*$, ${(\theta_{q}+E[\psi_{q}])}_{X_{t_{q+1}}}^{*}$ }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., $T^{*}(t, a_t) = \Vert \mathit{\boldsymbol{B}}^* \Vert_{\infty} \ D^{*} = \arg \Vert \mathit{\boldsymbol{B}}^* \Vert_{\infty}$
 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  $\min\{c_{1}, \cdots, c_{n-k-1}\} < C$  do 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 $\zeta_{X_{t_{q+1}}}^*$, ${(\theta_{q}+E[\psi_{q}])}_{X_{t_{q+1}}}^{*}$ }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., $T^{*}(t, a_t) = \Vert \mathit{\boldsymbol{B}}^* \Vert_{\infty} \ D^{*} = \arg \Vert \mathit{\boldsymbol{B}}^* \Vert_{\infty}$
Comparison of the absolute and relative asymptotic errors
 $L$ 30 31 32 33 34 35 36 ${\bf{E}}\left[\mathit{\boldsymbol\psi}_{1, 3} \right]_{L}$ 9.6186 9.6059 9.5939 9.5827 9.5721 9.5621 9.5527 absolute error 0.0127 0.0119 0.0112 0.0105 0.0099 0.0094 relative error 0.13% 0.12% 0.12% 0.11% 0.10% 0.098% 37 38 39 40 41 42 43 44 9.5438 9.5354 9.5274 9.5198 9.5126 9.5057 9.4992 9.4929 0.0088 0.0084 0.0079 0.0075 0.0072 0.0068 0.0065 0.0062 0.092% 0.088% 0.082% 0.078% 0.075% 0.071% 0.068% 0.065% 45 46 47 48 49 50 9.4869 9.4812 9.4758 9.4705 9.4655 9.4607 0.0059 0.0057 0.0054 0.0052 0.0050 0.0048 0.062% 0.060% 0.056% 0.054% 0.052% 0.050%
 $L$ 30 31 32 33 34 35 36 ${\bf{E}}\left[\mathit{\boldsymbol\psi}_{1, 3} \right]_{L}$ 9.6186 9.6059 9.5939 9.5827 9.5721 9.5621 9.5527 absolute error 0.0127 0.0119 0.0112 0.0105 0.0099 0.0094 relative error 0.13% 0.12% 0.12% 0.11% 0.10% 0.098% 37 38 39 40 41 42 43 44 9.5438 9.5354 9.5274 9.5198 9.5126 9.5057 9.4992 9.4929 0.0088 0.0084 0.0079 0.0075 0.0072 0.0068 0.0065 0.0062 0.092% 0.088% 0.082% 0.078% 0.075% 0.071% 0.068% 0.065% 45 46 47 48 49 50 9.4869 9.4812 9.4758 9.4705 9.4655 9.4607 0.0059 0.0057 0.0054 0.0052 0.0050 0.0048 0.062% 0.060% 0.056% 0.054% 0.052% 0.050%
Comparison of running times of Algorithm 1 for cases 1 to 4
 Running time The small-scale system The large-scale system Weibull general Weibull general maximum time 0.0057 0.0051 0.089 0.093 minimum time 0.0043 0.0038 0.058 0.062 mean time 0.0052 0.0048 0.0751 0.0747
 Running time The small-scale system The large-scale system Weibull general Weibull general maximum time 0.0057 0.0051 0.089 0.093 minimum time 0.0043 0.0038 0.058 0.062 mean time 0.0052 0.0048 0.0751 0.0747
Modified parameters of the two distributions for both the small-scale system and the large-scale system
 Errors Weibull (Case 1 & Case 3) General (Case 2 & Case 4) $\lambda$ $\alpha$ $\lambda$ -10% 0.36 1.8 1.08 -5% 0.38 1.9 1.14 0% 0.4 2 1.2 5% 0.42 2.1 1.26 10% 0.44 2.2 1.32
 Errors Weibull (Case 1 & Case 3) General (Case 2 & Case 4) $\lambda$ $\alpha$ $\lambda$ -10% 0.36 1.8 1.08 -5% 0.38 1.9 1.14 0% 0.4 2 1.2 5% 0.42 2.1 1.26 10% 0.44 2.2 1.32
Sensitivity analysis of $\lambda$ and $\alpha$ on $T^{*}(t, a_{t_q})$ and$D^*$ for the small-cale system with the Weibull distribution (Case 1)
 errors of $\lambda$ and $\alpha$ $\lambda$ $\alpha$ $T^{*}(t, a_{t_q})$ errors of $T^{*}(t, a_{t_q})$ $D^*$ -10% 0.36 1.8 400.80 -0.17% {1 7 3 8 2 8 3 5 4 8 2 6 3 6 3 9} -5% 0.38 1.9 401.16 -0.08% {1 7 3 8 2 8 3 5 4 8 2 6 3 6 3 9} 0%s 0.4 2 401.48 0% {1 7 3 8 2 8 3 5 4 8 2 6 3 6 3 9} 5% 0.42 2.1 401.80 0.08% {1 7 3 8 2 8 3 5 4 8 2 6 3 6 3 9} 10% 0.44 2.2 402.04 0.14% {1 7 3 8 2 8 3 5 4 8 2 6 3 6 3 9}
 errors of $\lambda$ and $\alpha$ $\lambda$ $\alpha$ $T^{*}(t, a_{t_q})$ errors of $T^{*}(t, a_{t_q})$ $D^*$ -10% 0.36 1.8 400.80 -0.17% {1 7 3 8 2 8 3 5 4 8 2 6 3 6 3 9} -5% 0.38 1.9 401.16 -0.08% {1 7 3 8 2 8 3 5 4 8 2 6 3 6 3 9} 0%s 0.4 2 401.48 0% {1 7 3 8 2 8 3 5 4 8 2 6 3 6 3 9} 5% 0.42 2.1 401.80 0.08% {1 7 3 8 2 8 3 5 4 8 2 6 3 6 3 9} 10% 0.44 2.2 402.04 0.14% {1 7 3 8 2 8 3 5 4 8 2 6 3 6 3 9}
Sensitivity analysis of $\lambda$ on $T^{*}(t, a_{t_q})$ and $D^*$ for the small-scale system with the general distribution (Case 2)
 errors of $\lambda$ $\lambda$ $T^{*}(t, a_{t_q})$ errors of $T^{*}(t, a_{t_q})$ $D^*$ -10% 1.08 352.60 -0.13% {1 8 4 7 3 6 3 7 2 5 4 6 4 9} -5% 1.14 352.84 -0.06% {1 8 4 7 3 6 3 7 2 5 4 6 4 9} 0% 1.2 353.05 0% {1 8 4 7 3 6 3 7 2 5 4 6 4 9} 5% 1.26 353.26 0.06% {1 8 4 7 3 6 3 7 2 5 4 6 4 9} 10% 1.32 353.44 0.11% {1 8 4 7 3 6 3 7 2 5 4 6 4 9}
 errors of $\lambda$ $\lambda$ $T^{*}(t, a_{t_q})$ errors of $T^{*}(t, a_{t_q})$ $D^*$ -10% 1.08 352.60 -0.13% {1 8 4 7 3 6 3 7 2 5 4 6 4 9} -5% 1.14 352.84 -0.06% {1 8 4 7 3 6 3 7 2 5 4 6 4 9} 0% 1.2 353.05 0% {1 8 4 7 3 6 3 7 2 5 4 6 4 9} 5% 1.26 353.26 0.06% {1 8 4 7 3 6 3 7 2 5 4 6 4 9} 10% 1.32 353.44 0.11% {1 8 4 7 3 6 3 7 2 5 4 6 4 9}
Sensitivity analysis of $\lambda$ and $\alpha$ on $T^{*}(t, a_{t_q})$ and$D^*$ for the large-cale system with the Weibull distribution (Case 3)
 errors of $\lambda$ and $\alpha$ $\lambda$ $\alpha$ $T^{*}(t, a_{t_q})$ errors of $T^{*}(t, a_{t_q})$ $D^*$ -10% 0.36 1.8 710.29 -0.05% {1 42 10 34 26 32 14 40 27 45 2 35 14 46 11 40 26 48 19 46 2 44 29 45 5 50} -5% 0.38 1.9 710.43 -0.03% {1 42 10 34 26 32 14 40 27 45 2 35 14 46 11 40 26 48 19 46 2 44 29 45 5 50} 0% 0.4 2 710.65 0% {1 42 10 34 26 32 14 40 27 45 2 35 14 46 11 40 26 48 19 46 2 44 29 45 5 50} 5% 0.42 2.1 710.80 0.02% {1 42 10 34 26 32 14 40 27 45 2 35 14 46 11 40 26 48 19 46 2 44 29 45 5 50} 10% 0.44 2.2 710.93 0.04% {1 42 10 34 26 32 14 40 27 45 2 35 14 46 11 40 26 48 19 46 2 44 29 45 5 50}
 errors of $\lambda$ and $\alpha$ $\lambda$ $\alpha$ $T^{*}(t, a_{t_q})$ errors of $T^{*}(t, a_{t_q})$ $D^*$ -10% 0.36 1.8 710.29 -0.05% {1 42 10 34 26 32 14 40 27 45 2 35 14 46 11 40 26 48 19 46 2 44 29 45 5 50} -5% 0.38 1.9 710.43 -0.03% {1 42 10 34 26 32 14 40 27 45 2 35 14 46 11 40 26 48 19 46 2 44 29 45 5 50} 0% 0.4 2 710.65 0% {1 42 10 34 26 32 14 40 27 45 2 35 14 46 11 40 26 48 19 46 2 44 29 45 5 50} 5% 0.42 2.1 710.80 0.02% {1 42 10 34 26 32 14 40 27 45 2 35 14 46 11 40 26 48 19 46 2 44 29 45 5 50} 10% 0.44 2.2 710.93 0.04% {1 42 10 34 26 32 14 40 27 45 2 35 14 46 11 40 26 48 19 46 2 44 29 45 5 50}
Sensitivity analysis of $\lambda$ on $T^{*}(t, a_{t_q})$ and $D^*$ for the large-scale system with the general distribution (Case 4)
 errors of $\lambda$ $\lambda$ $T^{*}(t, a_{t_q})$ errors of $T^{*}(t, a_{t_q})$ $D^*$ -10% 1.08 652.28 -0.05% {1 41 5 34 20 35 2 31 7 33 15 32 26 34 16 46 26 34 29 44 14 49 8 50} -5% 1.14 652.48 -0.02% {1 41 5 34 20 35 2 31 7 33 15 32 26 34 16 46 26 34 29 44 14 49 8 50} 0% 1.2 652.61 0% {1 41 5 34 20 35 2 31 7 33 15 32 26 34 16 46 26 34 29 44 14 49 8 50} 5% 1.26 652.81 0.03% {1 41 5 34 20 35 2 31 7 33 15 32 26 34 16 46 26 34 29 44 14 49 8 50} 10% 1.32 652.94 0.05% {1 41 5 34 20 35 2 31 7 33 15 32 26 34 16 46 26 34 29 44 14 49 8 50}
 errors of $\lambda$ $\lambda$ $T^{*}(t, a_{t_q})$ errors of $T^{*}(t, a_{t_q})$ $D^*$ -10% 1.08 652.28 -0.05% {1 41 5 34 20 35 2 31 7 33 15 32 26 34 16 46 26 34 29 44 14 49 8 50} -5% 1.14 652.48 -0.02% {1 41 5 34 20 35 2 31 7 33 15 32 26 34 16 46 26 34 29 44 14 49 8 50} 0% 1.2 652.61 0% {1 41 5 34 20 35 2 31 7 33 15 32 26 34 16 46 26 34 29 44 14 49 8 50} 5% 1.26 652.81 0.03% {1 41 5 34 20 35 2 31 7 33 15 32 26 34 16 46 26 34 29 44 14 49 8 50} 10% 1.32 652.94 0.05% {1 41 5 34 20 35 2 31 7 33 15 32 26 34 16 46 26 34 29 44 14 49 8 50}
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