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Optimum design and replacement policies for k-out-of-n systems with deviation time and cost

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Abstract

In this paper, a new repairable k-out-of-n system model is proposed, in which the deviation cost of the system is taken into account. Firstly, the standard k-out-of-n system is replaced at planned time or at the failure time. Next, replacement policies are studied at planned time or periodic time when k is a constant. Thirdly, replacement policies are studied at planned time or periodic time when k is a random number, respectively. Finally, replacement policies are planned at the completion of random working time. The expected cost rate functions and its optimum replacement policies are derived analytically for each model. Meanwhile, optimum number of units is designed. The analytical solution of replacement policies for each model is discussed. Numerical examples are given to demonstrate our results.

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Abbreviations

\(r_{n,k}(t)\) :

Failure rate function of k-out-of-n \((1 \le k \le n)\) system

\(F_{n, k}(t)\) :

Failure distribution function of k-out-of-n system

T :

Constant time for preventive replacement

X :

Failure time

\(c_d\) :

The deviation cost per unit of time between T and X

\(c_1\) :

Acquisition cost of each component for the k-out-of-n system

\(c_f\) :

Corrective cost for a failed system

C(T; n):

Expected cost rate

\(\widehat{C}_{1}(T;n)\) :

Expected cost contain the deviation cost

\(C_1{(T; n)}\) :

Expected cost rate contain the deviation cost

\(R_{r: n, k}(t)\) :

The reliability of the k-out-of-n system when k is a random variable

\(\widehat{C}_{1}(K;n)\) :

Expected cost contain the deviation cost for periodic times \(K \tau \)

\(C_1{(K; n)}\) :

Expected cost rate contain the deviation cost for periodic times \(K \tau \)

C(N; n):

Expected cost rate when it has working times N

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Acknowledgements

This work is supported by the Fundamental Research Funds for the Central Universities of China (No. JB180702), National Natural Science Foundation (No. 71801126) and Natural Science Foundation of Jiangsu Province (No. BK20180412).

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Authors and Affiliations

  1. College of Sciences, China Jiliang University, Hangzhou, 310018, China

    Junyuan Wang

  2. College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, China

    Xufeng Zhao

  3. College of Mechanical and Electrical Engineering, Wenzhou University, Wenzhou, 325035, China

    Xufeng Zhao & Jiawei Xiang

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  1. Junyuan Wang
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Correspondence to Junyuan Wang.

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Appendix

Appendix

In Section 2.1, when \(X \le T\) and \(X \ge T\), the deviation cost is also different. Therefore, \(c_d\) should be divided into two parts. One part is failure time X less than the planned time T, i.e., the replacement time is postponed. The deviation cost per unit of time between X and T is \(c_{d_1}\). \(c_{d_1}\) is called the downtime cost per unit of time after failure. And another is planned time T less than failure time X, i.e., the replacement time is advanced. The deviation cost per unit of time between X and T is \(c_{d_2}\). \(c_{d_2}\) is called the uptime cost per unit of time before failure. In this case, the expected cost becomes

$$\begin{aligned} \widehat{C}_{1}(T;n ) = c_{d_1} \int _0^{T} F_{n, k}(t) {{\,\mathrm{d\!}\,}}t + c_{d_2} \int _T^{\infty } \left[ 1 - F_{n, k}(t) \right] {{\,\mathrm{d\!}\,}}t + {nc_1 + c_f F_{n, k}(T) }, \end{aligned}$$
(57)

and the expected cost rate becomes

$$\begin{aligned} C_{1}(T;n) = \frac{ c_{d_1} \int _0^{T} F_{n, k}(t) {{\,\mathrm{d\!}\,}}t + c_{d_2} \int _T^{\infty } \left[ 1 - F_{n, k}(t) \right] {{\,\mathrm{d\!}\,}}t + {nc_1 + c_f F_{n, k}(T) } }{\int _0^T \left[ 1 - F_{n, k}(t) \right] {{\,\mathrm{d\!}\,}}t }. \end{aligned}$$
(58)

When \(c_{d_1}=0\),

$$\begin{aligned} C_{1}(\infty ; n) = \frac{ nc_1 + c_f }{ \mu _n } = C(\infty ; n), \end{aligned}$$

which denotes the system is just corrective replacement and the cost rate agree with Eq. (3).

The optimum number of units and replacement policy is also obtained by adopting the similar method.

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Wang, J., Zhao, X. & Xiang, J. Optimum design and replacement policies for k-out-of-n systems with deviation time and cost. Ann Oper Res (2022). https://doi.org/10.1007/s10479-022-05043-1

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