# Customizing Random Replacement Model and Flexible Warranty Model from the Perspective of Screening Reliability

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## Abstract

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## 1. Introduction

## 2. Random Warranty Design

#### 2.1. Random Warranty Definition

- ◆
- If no failure occurs until the time threshold $\tau $, then such a product will continue to undergo minimal repair at all failures until the extended time span $w$;
- ◆
- If the first failure occurs before the time threshold $\tau $, then such a failure is removed with minimal repair and the related product will continue to undergo minimal repair at all failures until the $m\mathrm{th}$ task cycle completes or at the extended time span $\omega $, whichever occurs first.

#### 2.2. The Measures of the FRFRW

#### 2.3. Derivative Models of the FRFRW

## 3. Customization of a Random Maintenance after the FRFRW Expiration

#### 3.1. The Customization of the Random Maintenance Model

- ◆
- For the product through the FRFRW when the $m\mathrm{th}$ task cycle completes or at the extended time span $\omega $, it will be replaced at the first failure, the operating time $T$ or when the $N\mathrm{th}$ task cycle completes, whichever occurs earliest;
- ◆
- For the product through the FRFRW at the extended time span $w$, it will be replaced at the first failure, the operating time $T$ or when the $(N+M)\mathrm{th}$ task cycle completes, whichever occurs earliest.

#### 3.2. The Objective Function of the CRR Model

#### 3.2.1. The Expected Length of the Renewing Cycle

#### 3.2.2. The Expected Total Cost during the Renewing Cycle

#### 3.2.3. The Expected Cost Rate Model

#### 3.3. Derivative Models of the Expected Cost Rate

## 4. Numerical Experiment

#### 4.1. Illustration of the FRFRW Properties

#### 4.2. Illustration of the CRR Properties

#### 4.3. Illustration of the CRR Performance

- ①
- Computing ${L}_{M=0}=\underset{\begin{array}{l}m\to \infty \\ M\to 0\end{array}}{\mathrm{lim}}EL(1,{T}^{*})\times \underset{m\to \infty}{\mathrm{lim}}TSC(1,{T}^{*})$ and ${L}_{M\ne 0}=\underset{m\to \infty}{\mathrm{lim}}EL(1,{T}^{*})\times \underset{\begin{array}{l}m\to \infty \\ M\to 0\end{array}}{\mathrm{lim}}TSC(1,{T}^{*})$; where $\underset{\begin{array}{l}m\to \infty \\ M\to 0\end{array}}{\mathrm{lim}}EL(1,{T}^{*})$ and $\underset{m\to \infty}{\mathrm{lim}}EL(1,{T}^{*})$ are the denominators of (32, 32), and $\underset{\begin{array}{l}m\to \infty \\ M\to 0\end{array}}{\mathrm{lim}}TSC(1,{T}^{*})$ as well as $\underset{m\to \infty}{\mathrm{lim}}TSC(1,{T}^{*})$ are the numerators of (33, 32);
- ②
- CRR should be selected if ${L}_{M\ne 0}>{L}_{M=0}$; any of both can be selected if ${L}_{M\ne 0}={L}_{M=0}$; RARF should be selected if ${L}_{M=0}>{L}_{M\ne 0}$.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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Optimal CRR | Parameter Value | |||
---|---|---|---|---|

$\mathit{\tau}=0.5$ | $\mathit{\tau}=0.6$ | $\mathit{\tau}=0.7$ | $\mathit{\tau}=0.8$ | |

${T}^{*}$ | 1.4103 | 1.2267 | 1.0480 | 0.8743 |

$\underset{m\to \infty}{\mathrm{lim}}ECT(1,{T}^{*})$ | 5.8180 | 5.6481 | 5.4855 | 5.3292 |

Optimal CRR | Parameter Value | |||
---|---|---|---|---|

$\mathit{\omega}=0.5$ | $\mathit{\omega}=0.6$ | $\mathit{\omega}=0.7$ | $\mathit{\omega}=0.8$ | |

${T}^{*}$ | 1.4427 | 1.4361 | 1.4295 | 1.4230 |

$\underset{m\to \infty}{\mathrm{lim}}ECT(1,{T}^{*})$ | 5.8729 | 5.8619 | 5.8510 | 5.8400 |

Optimal CRR | Parameter Values | |||
---|---|---|---|---|

$\mathit{w}=0.5$ | $\mathit{w}=0.7$ | $\mathit{w}=0.9$ | $\mathit{w}=1.1$ | |

${T}^{*}$ | 2.4465 | 2.0295 | 1.6338 | 1.2557 |

$\underset{m\to \infty}{\mathrm{lim}}ECT(1,{T}^{*})$ | 6.8917 | 6.4550 | 6.0586 | 5.6939 |

${\mathit{C}}_{\mathit{R}}$ | ${\mathit{C}}_{\mathit{P}}=9$ | ${\mathit{C}}_{\mathit{P}}=10$ | ${\mathit{C}}_{\mathit{P}}=11$ | ${\mathit{C}}_{\mathit{P}}=12$ | ||||
---|---|---|---|---|---|---|---|---|

${\mathit{T}}^{*}$ | $\underset{\mathit{m}\to \mathit{\infty}}{\mathrm{lim}}\mathit{E}\mathit{C}\mathit{T}(1,{\mathit{T}}^{*})$ | ${\mathit{T}}^{*}$ | $\underset{\mathit{m}\to \mathit{\infty}}{\mathrm{lim}}\mathit{E}\mathit{C}\mathit{T}(1,{\mathit{T}}^{*})$ | ${\mathit{T}}^{*}$ | $\underset{\mathit{m}\to \mathit{\infty}}{\mathrm{lim}}\mathit{E}\mathit{C}\mathit{T}(1,{\mathit{T}}^{*})$ | ${\mathit{T}}^{*}$ | ||

15 | 0.8304 | 5.5875 | 1.4103 | 5.8180 | 2.2466 | 5.9934 | 3.6288 | 6.1541 |

16 | 0.5647 | 5.7721 | 1.0280 | 6.0629 | 1.6351 | 6.2681 | 2.5233 | 6.4363 |

17 | 0.3514 | 5.9103 | 0.7436 | 6.2750 | 1.2207 | 6.5261 | 1.8582 | 6.7146 |

18 | 0.1722 | 5.9982 | 0.5186 | 6.4485 | 0.9159 | 6.7586 | 1.4103 | 6.9816 |

Comparison Type | $\mathbf{Optimal}\mathbf{Values}\mathbf{with}\mathit{M}=0$ | $\mathbf{Optimal}\mathbf{Values}\mathbf{with}\mathit{M}\ne 0$ | Cycle Length | Relationship | ||||
---|---|---|---|---|---|---|---|---|

$\underset{\begin{array}{l}\mathit{m}\to \mathit{\infty}\\ \mathit{M}\to 0\end{array}}{\mathrm{lim}}\mathit{E}\mathit{L}(1,{\mathit{T}}^{*})$ | $\underset{\begin{array}{l}\mathit{m}\to \mathit{\infty}\\ \mathit{M}\to 0\end{array}}{\mathrm{lim}}\mathrm{lim}(1,{\mathit{T}}^{*})$ | $\underset{\mathit{m}\to \mathit{\infty}}{\mathrm{lim}}\mathit{E}\mathit{L}(1,{\mathit{T}}^{*})$ | $\underset{\mathit{m}\to \mathit{\infty}}{\mathrm{lim}}\mathit{T}\mathit{S}\mathit{C}(1,{\mathit{T}}^{*})$ | ${\mathit{L}}_{\mathit{M}=0}$ | ${\mathit{L}}_{\mathit{M}\ne 0}$ | |||

$M=0$ and $M=1$ | 1.8541 | 13.5891 | 2.0797 | 14.7201 | 27.2911 | 28.2613 | ${L}_{M=0}<{L}_{M\ne 0}$ | |

$M=0$ and $M=2$ | 1.8541 | 13.5891 | 2.2096 | 15.4265 | 28.6023 | 30.0265 | ${L}_{M=0}<{L}_{M\ne 0}$ | |

$M=0$ and $M=3$ | 1.8541 | 13.5891 | 2.2753 | 15.8038 | 29.3018 | 30.9193 | ${L}_{M=0}<{L}_{M\ne 0}$ | |

$M=0$ and $M=4$ | 1.8541 | 13.5891 | 2.3047 | 15.9789 | 29.6265 | 31.3188 | ${L}_{M=0}<{L}_{M\ne 0}$ | |

$M=0$ and $M=5$ | 1.8541 | 13.5891 | 2.3165 | 16.0515 | 29.7611 | 31.4792 | ${L}_{M=0}<{L}_{M\ne 0}$ |

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## Share and Cite

**MDPI and ACS Style**

Shang, L.; Liu, B.; Yang, L.; Gao, K.
Customizing Random Replacement Model and Flexible Warranty Model from the Perspective of Screening Reliability. *Axioms* **2023**, *12*, 358.
https://doi.org/10.3390/axioms12040358

**AMA Style**

Shang L, Liu B, Yang L, Gao K.
Customizing Random Replacement Model and Flexible Warranty Model from the Perspective of Screening Reliability. *Axioms*. 2023; 12(4):358.
https://doi.org/10.3390/axioms12040358

**Chicago/Turabian Style**

Shang, Lijun, Baoliang Liu, Li Yang, and Kaiye Gao.
2023. "Customizing Random Replacement Model and Flexible Warranty Model from the Perspective of Screening Reliability" *Axioms* 12, no. 4: 358.
https://doi.org/10.3390/axioms12040358