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Extended age maintenance models and its optimization for series and parallel systems

  • S.I.: Statistical Reliability Modeling and Optimization
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Abstract

In this paper, extended preventive replacement models for series and parallel system with n independent non-identical components are proposed. It is assumed that the system suffers from two types of failure. One is repairable (type-I) failure, at that time the system can be rectified by minimal repair. Another is non-repairable (type-II) failure, then the whole system is replaced. In the proposed models, the system is replaced at the planned time, at random working time, or at the time when type-II failure occurs, with options whichever occurs first or whichever occurs last. The average cost rate (ACR) function and the failure rate function (FRF) of the series and parallel system under the different cases are obtained respectively. Moreover, the optimal preventive replacement time of models based on minimization of the ACR function is obtained theoretically. Numerical examples are presented to evaluate the cost of the system and verify the performance of our results.

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Acknowledgements

This work was supported by National Natural Science Foundation of China [Grant Numbers 61573014, 11501433], the Fundamental Research Funds for the Central Universities [Grant Number JB180702] and the China Postdoctoral Science Foundation [Grant number 2019M650260]. The authors would like to thank sincerely the editor and the anonymous referees for furnishing components and valuable suggestions that improved the quality of this paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Xidian University, Xi’an, 710071, P.R. China

    Junyuan Wang, Jimin Ye & Liang Wang

  2. School of Mathematics, Yunnan Normal University, Kunming, 650500, P.R. China

    Liang Wang

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  1. Junyuan Wang
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  2. Jimin Ye
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  3. Liang Wang
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Correspondence to Junyuan Wang.

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This work was supported by National Natural Science Foundation of China [Grant Numbers 61573014, 11501433], and the Fundamental Research Funds for the Central Universities [Grant Number JB180702] and the China Postdoctoral Science Foundation [Grant number 2019M650260].

Appendices

Appendix A

Derivation of Eq. (8)

$$\begin{aligned} E(U)&= T {\overline{F}}_p(T) \prod _{i=1}^n {\overline{G}}_i(T) + \int _0^T t {\overline{F}}_p(t) d \Big (1 - \prod _{i=1}^n {\overline{G}}_i(t) \Big ) + \int _0^T t \prod _{i=1}^n {\overline{G}}_i(t) d {F}_p(t) \\&= \left( - T {\overline{F}}_p(T) \prod _{i=1}^n {\overline{G}}_i(T) + \int _0^T {\overline{F}}_p(t) \prod _{i=1}^n {\overline{G}}_i(t) dt + \int _0^T t \prod _{i=1}^n {\overline{G}}_i(t) d{\overline{F}}_p(t) \right) \\&\quad + T {\overline{F}}_p(T) \prod _{i=1}^n {\overline{G}}_i(T) + \int _0^T t \prod _{i=1}^n {\overline{G}}_i(t) d F_p(t) \\&= \int _0^T {\overline{F}}_p(t) \prod _{i=1}^n {\overline{G}}_i(t) dt. \end{aligned}$$

Derivation of Eq. (9)

$$\begin{aligned}&\varLambda _q(T) {\overline{F}}_p(T) \prod _{i=1}^n {\overline{G}}_i(T) + \int _0^T \varLambda _q(t) {\overline{F}}_p(t) d \Big (1-\prod _{i=1}^n {\overline{G}}_i(t)\Big ) + \int _0^T \varLambda _q(t)\prod _{i=1}^n {\overline{G_i}} {(t)} d F_p(t) \\&\quad = \varLambda _q(T) {\overline{F}}_p(T) \prod _{i=1}^n {\overline{G}}_i(T) - \int _0^T \varLambda _q(t) {\overline{F}}_p(t) d \prod _{i=1}^n {\overline{G}}_i(t) + \int _0^T \varLambda _q(t) \prod _{i=1}^n {\overline{G}}_i(t) d F_p(t) \\&\quad = \varLambda _q(T) {\overline{F}}_p(T) \prod _{i=1}^n {\overline{G}}_i(T) - \varLambda _q(T) {\overline{F}}_p(T) \prod _{i=1}^n {\overline{G}}_i(T) + \int _0^T \varLambda _q(t) \prod _{i=1}^n {\overline{G}}_i(t) d {\overline{F}}_p(t) \\&\qquad + \int _0^T {\overline{F}}_p(t) \prod _{i=1}^n {\overline{G}}_i(t) q r_s(t) dt + \int _0^T \varLambda _q(t) \prod _{i=1}^n {\overline{G}}_i(t) d F_p(t) \\&\quad = \int _0^T {\overline{F}}_p(t) \prod _{i=1}^n {\overline{G}}_i(t) q r_s(t) dt. \end{aligned}$$

Derivation of Eq. (10)

$$\begin{aligned} E(V)&= C_T {\overline{F}}_p(T) \prod _{i=1}^n {\overline{G}}_i(T) + C_Y \int _0^T {\overline{F}}_p(t) d \Big ( 1 - \prod _{i=1}^n {\overline{G}}_i(t) \Big ) \\&\quad + C_{II} \int _0^T \prod _{i=1}^n {\overline{G}}_i(t) d F_p(t) + C_m \int _0^T {\overline{F}}_p(t) \prod _{i=1}^n {\overline{G}}_i(t) q r(t) dt \\&= -C_T \int _0^T {\overline{F}}_p(t) d\Big (1-\prod _{i=1}^n {\overline{G}}_i(t)\Big )-C_T\int _0^T \prod _{i=1}^n {\overline{G}}_i(t) dF_p(t) + C_T \\&\quad + C_Y \int _0^T {\overline{F}}_p(t) d \Big ( 1 - \prod _{i=1}^n {\overline{G}}_i(t) \Big ) + C_{II} \int _0^T \prod _{i=1}^n {\overline{G}}_i(t) d F_p(t) \\&\quad + C_m \int _0^T {\overline{F}}_p(t) \prod _{i=1}^n {\overline{G}}_i(t) q r_s(t) dt \\&= C_T + (C_Y-C_T)\int _0^T {\overline{F}}_p(t) d \Big ( 1 - \prod _{i=1}^n {\overline{G}}_i(t) \Big ) \\&\quad + (C_{II}-C_T)\int _0^T \prod _{i=1}^n {\overline{G}}_i(t) d F_p(t) + C_m \int _0^T {\overline{F}}_p(t) \prod _{i=1}^n {\overline{G}}_i(t) q r_s(t) dt. \end{aligned}$$

Appendix B

Derivation of Eq. (22)

$$\begin{aligned} E(U)&= T{\overline{F}}_p(T) \Big ( 1 - \prod _{i=1}^n {\overline{G}}_i(T) \Big ) + \int _T^\infty t {\overline{F}}_p(t) d \Big ( 1 - \prod _{i=1}^n {\overline{G}}_i(t) \Big ) \\&\quad + \int _0^T t d F_p(t) + \int _T^\infty t \prod _{i=1}^n {\overline{G}}_i(t) d {F}_p(t) \\&= T{\overline{F}}_p(T) \Big ( 1 - \prod _{i=1}^n {\overline{G}}_i(T) \Big ) + T{\overline{F}}_p(T) \prod _{i=1}^n {\overline{G}}_i(T) \\&\quad + \int _T^\infty {\overline{F}}_p(t) \prod _{i=1}^n {\overline{G}}_i(t)d t + \int _T^\infty t \prod _{i=1}^n {\overline{G}}_i(t)d {\overline{F}}_p(t) \\&\quad - T{\overline{F}}_p(T) + \int _0^T {{\overline{F}}}_p(t) d t + \int _T^\infty t \prod _{i=1}^n {\overline{G}}_i(t) d {F}_p(t) \\&= \int _0^T {\overline{F}}_p(t)d t + \int _T^\infty {\overline{F}}_p(t) \prod _{i=1}^n {\overline{G}}_i(t) d t. \end{aligned}$$

Derivation of Eq. (23)

$$\begin{aligned}&\varLambda _q(T) {{\overline{F}}}_p(T) \Big ( 1 - \prod _{i=1}^n {\overline{G}}_i(T) \Big ) + \int _T^\infty \varLambda _q(t) {\overline{F}}_p(t) d \Big ( 1 - \prod _{i=1}^n {\overline{G}}_i(t) \Big ) \\&\qquad + \int _0^T \varLambda _q(t)d F_p(t) + \int _T^\infty \varLambda _q(t) \prod _{i=1}^n {\overline{G}}_i(t) d {F}_p(t) \\&\quad = \varLambda _q(T) {{\overline{F}}}_p(T) \Big ( 1 - \prod _{i=1}^n {\overline{G}}_i(T) \Big ) + \varLambda _q(T) {{\overline{F}}}_p(T) \prod _{i=1}^n {\overline{G}}_i(T) \\&\qquad + \int _T^\infty {\overline{F}}_p(t)\prod _{i=1}^n {\overline{G}}_i(t)q r_s(t)d t + \int _T^\infty \prod _{i=1}^n {\overline{G}}_i(t)\varLambda _q(t)d {\overline{F}}_p(t) \\&\qquad + \int _0^T \varLambda _q(t)d F_p(t) + \int _T^\infty \prod _{i=1}^n {\overline{G}}_i(t)\varLambda _q(t)d {F}_p(t) \\&\quad = \varLambda _q(T) {{\overline{F}}}_p(T) + \int _T^\infty {\overline{F}}_p(t)\prod _{i=1}^n {\overline{G}}_i(t)q r_s(t)d t + \int _0^T \varLambda _q(t)d F_p(t) \\&\quad = \int _0^T {\overline{F}}_p(t) q r_s(t)dt + \int _T^\infty {\overline{F}}_p(t)\prod _{i=1}^n {\overline{G}}_i(t)q r_s(t)d t. \end{aligned}$$

Derivation of Eq. (24)

$$\begin{aligned} E(V)&= C_T {\overline{F}}_p(T) \left( 1 - \prod _{i=1}^n {\overline{G}}_i(T) \right) + C_Y \int _T^\infty {\overline{F}}_p(t) d \left( 1 - \prod _{i=1}^n {\overline{G}}_i(t) \right) \\&\quad + C_{II} \left( F_p(T) + \int _T^\infty \prod _{i=1}^n {\overline{G}}_i(t) d {F}_p(t) \right) \\&\quad + C_m \left( \int _0^T {\overline{F}}_p(t) q r_s(t)dt + \int _T^\infty {\overline{F}}_p(t)\prod _{i=1}^n {\overline{G}}_i(t)q r_s(t)d t \right) \\&= - C_T \int _T^\infty {\overline{F}}_p(t)d \Big ( 1-\prod _{i=1}^n {\overline{G}}_i(t) \Big ) - C_T \int _T^\infty \Big ( 1-\prod _{i=1}^n {\overline{G}}_i(t) \Big ) d {\overline{F}}_p(t) \\&\quad + C_Y \int _T^\infty {\overline{F}}_p(t) d \Big ( 1 - \prod _{i=1}^n {\overline{G}}_i(t) \Big )+C_{II} \left( F_p(T)+\int _T^\infty \prod _{i=1}^n {\overline{G}}_i(t) d {F}_p(t) \right) \\&\quad + C_m \left( \int _0^T {\overline{F}}_p(t) q r_s(t)dt + \int _T^\infty {\overline{F}}_p(t)\prod _{i=1}^n {\overline{G}}_i(t)q r_s(t)d t \right) \\&= (C_Y -C_T) \int _T^\infty {\overline{F}}_p(t) d \Big ( 1 - \prod _{i=1}^n {\overline{G}}_i(t) \Big )-C_T \int _T^\infty \prod _{i=1}^n {\overline{G}}_i(t) d {\overline{F}}_p(t) \\&\quad + C_{II} \left( F_p(T) + \int _T^\infty \prod _{i=1}^n {\overline{G}}_i(t) d {F}_p(t) \right) \\&\quad + C_m \left( \int _0^T {\overline{F}}_p(t) q r_s(t)dt + \int _T^\infty {\overline{F}}_p(t)\prod _{i=1}^n {\overline{G}}_i(t)q r_s(t)d t \right) \\&= C_T + (C_Y -C_T) \int _T^\infty {\overline{F}}_p(t) d \Big ( 1 - \prod _{i=1}^n {\overline{G}}_i(t) \Big ) \\&\quad + ( C_{II}-C_T)\left( F_p(T) + \int _T^\infty \prod _{i=1}^n {\overline{G}}_i(t) d {F}_p(t) \right) \\&\quad + C_m \left( \int _0^T {\overline{F}}_p(t) q r_s(t)dt + \int _T^\infty {\overline{F}}_p(t)\prod _{i=1}^n {\overline{G}}_i(t)q r_s(t)d t \right) . \end{aligned}$$

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Wang, J., Ye, J. & Wang, L. Extended age maintenance models and its optimization for series and parallel systems. Ann Oper Res 312, 495–517 (2022). https://doi.org/10.1007/s10479-019-03355-3

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