Optimal maintenance policy for a k-out-of-n system with replacement first and last

Abstract

In this paper, an optimal maintenance policy is developed for a k-out-of-n system that works in a random cycle. A k-out-of-n system, which consists of $n$ independent components and functions if at least $k$ of the $n$ components function, is one of the most important systems in reliability engineering. Consider that a component suffers from either minor or catastrophic failure, which is minimally repaired or lying idle, respectively. When $n - k + 1$ idle components occur, the system completely fails and undergoes corrective replacement. The notations of preventive replacement first and last are adopted; in other words, the system undergoes preventive replacement before complete failure at a planned time or at a random working time, whichever occurs first or last. The present paper aims to develop a two-stage maintenance methodology for a k-out-of-n system, and determine the optimal number of components and preventive replacement time that minimize the mean cost rate functions, respectively. For each model, the mean cost rate function is formulated analytically and the optimization problem is solved theoretically. Finally, a numerical example is given to demonstrate the applicability of the methodology derived from this paper. The proposed models provide a more general framework to analyze the maintenance policy for the k-out-of-n system.

Appendices

Appendix 1 Proof of Theorem 1

The inequalities $C(n + 1) \ge C(n)$ and $C(n) < C(n - 1)$ imply inequalities (9). If $\zeta (n)$ is increasing in $n$, then $L(n)$ is also increasing in $n$ since

$$ L(n + 1) - L(n) = \left[ {\zeta (n + 1) - \zeta (n)} \right]\int_{0}^{\infty } {\overline{H}_{n - k + 2} (t){\text{d}}t} > 0 $$

Furthermore, $L(0) = 0$ and $\mathop {\lim }\limits_{n \to \infty } L(n) \to \infty$ since $\mathop {\lim }\limits_{n \to \infty } \zeta (n) \to \infty$. There exists an unique $n^{*}$ satisfies inequalities (7), which minimizes $C(n)$ in Expression (6). □

Appendix 2 Proof of Theorem 2

Taking the first order derivative of $C_{f} (T)$ with respect to $T$ and setting it to zero imply (17). If $\varphi_{f} (T)$ is increasing in $T$, it causes that $Q_{f} (T)$ is also increasing in $T$. Further, $Q_{f} (0) = 0$ and

$$ \begin{gathered} Q_{f} (\infty ) \equiv \varphi_{f} (\infty )\int_{0}^{\infty } {\overline{H}_{{n^{*} - k + 1}} (t)\overline{G}(t){\text{d}}t} \hfill \\ \, - c_{2} \int_{0}^{\infty } {\overline{G}(t){{d}}{H}_{{n^{*} - k + 1}} (t)} - c_{m} \int_{0}^{\infty } {\left( {\sum\limits_{j = 1}^{{n^{*} - k}} {\overline{H}_{j} (t)} + k\overline{H}_{{n^{*} - k + 1}} (t)} \right)\overline{G}(t)qr(t){\text{d}}t} , \hfill \\ \end{gathered} $$

If $\varphi_{f} (\infty ) > \theta$, then $Q_{f} (\infty ) > n^{*} c_{1}$. Thus, form the increasing property of $Q_{f} (T)$, there exists a finite and unique $T_{f}^{*}$ $(0 < T_{f}^{*} < \infty )$ satisfies (17), which minimizes $C_{f} (T)$. If $T_{f}^{*}$ is the solution, from Eqs. (17) and (18), $C_{f} \left( {T_{f}^{*} } \right) = \varphi_{f} \left( {T_{f}^{*} } \right)$. □

Appendix 3 Proof of Theorem 3

Taking the first order derivative of $C_{l} (T)$ with respect to $T$ and setting it to zero imply (28). If $\varphi_{l} (T)$ is increasing in $T$, it causes that $Q_{l} (T)$ is also increasing in $T$. Further,

$$ Q_{l} (\infty ) \equiv \varphi_{l} (\infty )\int_{0}^{\infty } {\overline{H}_{{n^{*} - k + 1}} (t){\text{d}}t} - c_{2} - c_{m} \int_{0}^{\infty } {\left( {\sum\limits_{j = 1}^{{n^{*} - k}} {\overline{H}_{j} (t)} + k\overline{H}_{{n^{*} - k + 1}} (t)} \right)qr(t){\text{d}}t} , $$

If $\varphi_{l} (\infty ) > \delta$, then $Q_{l} (\infty ) > n^{*} c_{1}$. Thus, form the increasing property of $Q_{l} (T)$, there exists a finite and unique $T_{l}^{*}$ satisfies (28), which minimizes $C_{l} (T)$. If $T_{l}^{*}$ is the solution, from Eqs. (28) and (29), $C_{l} \left( {T_{l}^{*} } \right) = \varphi_{l} \left( {T_{l}^{*} } \right)$. □