# Study of a Random Warranty Model Maintaining Fairness and a Random Replacement Next Model Sustaining Post-Warranty Reliability

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## Abstract

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## 1. Introduction

## 2. Random Warranty Model to Maintain Fairness

#### 2.1. Warranty Definition

- The warranty service including the former stage warranty and the latter stage warranty sustains the reliability of the product, under which each failure is minimally repaired;
- The former stage warranty is confined to a coverage range formed by the warranty period $w$ or the $n\mathrm{th}$ random mission cycle completion, whichever occurs first;
- If the first stage warranty expires at $w$, then the reliability of the related product will be sustained by the second stage warranty whose coverage range is confined to a region formed by the warranty period $w$ or the $n\mathrm{th}$ random mission cycle completion, whichever occurs first;
- If the former stage warranty expires at the $n\mathrm{th}$ random mission cycle completion, then the reliability of the related product will still be sustained by the latter stage warranty, whose coverage range is confined to a region formed by the warranty period $w$ or the $n\mathrm{th}$ random mission cycle completion, whichever occurs last.

#### 2.2. The Cost Measure Modeling for the Two-Stage 2DFRW

#### 2.2.1. The Cost Measure of the Former Stage Warranty

#### 2.2.2. The Cost Measure of the Latter Stage Warranty

#### 2.2.3. The Cost Measure of the Two-Stage 2DFRW

#### 2.2.4. Derivative Models of the Two-Stage 2DFRW

## 3. Random Replacement Next Model Sustaining the Post-Warranty Reliability

#### 3.1. The Design of the Random Replacement Next Model

- The product through the two-stage 2DFRW is minimally repaired at each failure before replacement.
- If the $N\mathrm{th}$ random mission cycle is completed before the working time $T$ is reached, then the product through the two-stage 2DFRW will be replaced at next random mission cycle completion, i.e., the $(N+1)\mathrm{th}$ random mission cycle completion; otherwise, it will be replaced at the working time $T$.

#### 3.2. The Expected Cost Rate

#### 3.2.1. The Length of Renewable Cycle

#### 3.2.2. The Total Cost during the Renewable Cycle

#### 3.2.3. The Expected Cost Rate

#### 3.2.4. Other Expected Cost Rates

## 4. Numerical Examples

#### 4.1. Exploration of the Characteristics of the Designed Warranty

#### 4.2. Exploration of the Characteristics of RNNs

## 5. Conclusions

- ◆
- Flexible warranty models under the case of the multi-failure mode;
- ◆
- Customized maintenance models to sustain the different post-warranty reliabilities.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Liu, B.; Wu, J.; Xie, M. Cost analysis for multi-component system with failure interaction under renewing free-replacement warranty. Eur. J. Oper. Res.
**2015**, 243, 874–882. [Google Scholar] [CrossRef] - Qiao, P.; Shen, J.; Zhang, F.; Ma, Y. Optimal warranty policy for repairable products with a three-dimensional renewable combination warranty. Comput. Ind. Eng.
**2022**, 168, 108056. [Google Scholar] [CrossRef] - Chen, C.-K.; Lo, C.-C.; Weng, T.-C. Optimal production run length and warranty period for an imperfect production system under selling price dependent on warranty period. Eur. J. Oper. Res.
**2017**, 259, 401–412. [Google Scholar] [CrossRef] - Wang, L.; Pei, Z.; Zhu, H.; Liu, B. Optimising extended warranty policies following the two-dimensional warranty with repair time threshold. Eksploat. Niezawodn. Maint. Reliab.
**2018**, 20, 523–530. [Google Scholar] [CrossRef] - Wang, X.; Ye, Z.-S. Design of customized two-dimensional extended warranties considering use rate and heterogeneity. IISE Trans.
**2020**, 53, 341–351. [Google Scholar] [CrossRef] - Wang, X.-L. Design and pricing of usage-driven customized two-dimensional extended warranty menus. IISE Trans.
**2022**, 1–33. [Google Scholar] [CrossRef] - Ye, Z.; Murthy, D.N.P.; Xie, M.; Tang, L. Optimal burn-in for repairable systems sold with a two-dimensional warranty. IIE Trans.
**2013**, 45, 164–176. [Google Scholar] [CrossRef] - Gavish, B.; Sobol, M. Warranty Policy Impact On Net Revenues Due To Optional Purchases. Int. J. Inf. Technol. Decis. Mak.
**2010**, 9, 507–523. [Google Scholar] [CrossRef] - Wu, S.; Longhurst, P. Optimising age-replacement and extended non-renewing warranty policies in lifecycle costing. Int. J. Prod. Econ.
**2011**, 130, 262–267. [Google Scholar] [CrossRef] [Green Version] - Su, C.; Wang, X. A two-stage preventive maintenance optimization model incorporating two-dimensional extended warranty. Reliab. Eng. Syst. Saf.
**2016**, 155, 169–178. [Google Scholar] [CrossRef] - Wang, X.; Li, L.; Xie, M. An unpunctual preventive maintenance policy under two-dimensional warranty. Eur. J. Oper. Res.
**2019**, 282, 304–318. [Google Scholar] [CrossRef] - Peng, S.; Jiang, W.; Wei, L.; Wang, X.-L. A new cost-sharing preventive maintenance program under two-dimensional warranty. Int. J. Prod. Econ.
**2022**, 254, 108580. [Google Scholar] [CrossRef] - Liu, B.; Pandey, M.D.; Wang, X.; Zhao, X. A finite-horizon condition-based maintenance policy for a two-unit system with dependent degradation processes. Eur. J. Oper. Res.
**2021**, 295, 705–717. [Google Scholar] [CrossRef] - Li, H.; Zhu, W.; Dieulle, L.; Deloux, E. Condition-based maintenance strategies for stochastically dependent systems using Nested Lévy copulas. Reliab. Eng. Syst. Saf.
**2021**, 217, 108038. [Google Scholar] [CrossRef] - Wang, J.; Qiu, Q.; Wang, H. Joint optimization of condition-based and age-based replacement policy and inventory policy for a two-unit series system. Reliab. Eng. Syst. Saf.
**2020**, 205, 107251. [Google Scholar] [CrossRef] - Zhu, W.; Fouladirad, M.; Berenguer, C. Condition-based maintenance policies for a combined wear and shock deterioration model with covariates. Comput. Ind. Eng.
**2015**, 85, 268–283. [Google Scholar] [CrossRef] - Qiu, Q.; Maillart, L.M.; Prokopyev, O.A.; Cui, L. Optimal Condition-Based Mission Abort Decisions. IEEE Trans. Reliab.
**2022**, 1–18. [Google Scholar] [CrossRef] - Wang, J.; Qiu, Q.; Wang, H.; Lin, C. Optimal condition-based preventive maintenance policy for balanced systems. Reliab. Eng. Syst. Saf.
**2021**, 211, 107606. [Google Scholar] [CrossRef] - Zhao, X.; Sun, J.; Qiu, Q.; Chen, K. Optimal inspection and mission abort policies for systems subject to degradation. Eur. J. Oper. Res.
**2020**, 292, 610–621. [Google Scholar] [CrossRef] - Zhang, N.; Tian, S.; Cai, K.; Zhang, J. Condition-based maintenance assessment for a deteriorating system considering stochastic failure dependence. IISE Trans.
**2022**, 1–11. [Google Scholar] [CrossRef] - Zhang, N.; Fouladirad, M.; Barros, A.; Zhang, J. Condition-based maintenance for a K-out-of-N deteriorating system under periodic inspection with failure dependence. Eur. J. Oper. Res.
**2020**, 287, 159–167. [Google Scholar] [CrossRef] - Chen, Y.; Qiu, Q.; Zhao, X. Condition-based opportunistic maintenance policies with two-phase inspections for continuous-state systems. Reliab. Eng. Syst. Saf.
**2022**, 228, 108767. [Google Scholar] [CrossRef] - Shang, L.; Si, S.; Sun, S.; Jin, T. Optimal warranty design and post-warranty maintenance for products subject to stochastic degradation. IISE Trans.
**2018**, 50, 913–927. [Google Scholar] [CrossRef] - Shang, L.; Qiu, Q.; Wang, X. Random periodic replacement models after the expiry of 2D-warranty. Comput. Ind. Eng.
**2021**, 164, 107885. [Google Scholar] [CrossRef] - Afsahi, M.; Kashan, A.H.; Ostadi, B. A Bi-Objective Simulation-Based Optimization Approach for Optimizing Price, Warranty, and Spare Part Production Decisions Under Imperfect Repair. Int. J. Inf. Technol. Decis. Mak.
**2021**, 20, 903–932. [Google Scholar] [CrossRef] - Park, M.; Jung, K.M.; Park, D.H. A Generalized Age Replacement Policy for Systems Under Renewing Repair-Replacement Warranty. IEEE Trans. Reliab.
**2015**, 65, 604–612. [Google Scholar] [CrossRef] - Park, M.; Pham, H. Cost models for age replacement policies and block replacement policies under warranty. Appl. Math. Model.
**2016**, 40, 5689–5702. [Google Scholar] [CrossRef] - Zhao, X.; Fan, Y.; Qiu, Q.; Chen, K. Multi-criteria mission abort policy for systems subject to two-stage degradation process. Eur. J. Oper. Res.
**2021**, 295, 233–245. [Google Scholar] [CrossRef] - Ye, S.Z.; Xie, M. Stochastic modelling and analysis of degradation for highly reliable products. Appl. Stoch. Model. Bus. Ind.
**2015**, 31, 16–32. [Google Scholar] [CrossRef] - Qiu, Q.; Cui, L. Gamma process based optimal mission abort policy. Reliab. Eng. Syst. Saf.
**2019**, 190, 106496. [Google Scholar] [CrossRef] - Yang, L.; Chen, Y.; Qiu, Q.; Wang, J. Risk Control of Mission-Critical Systems: Abort Decision-Makings Integrating Health and Age Conditions. IEEE Trans. Ind. Inform.
**2022**, 18, 6887–6894. [Google Scholar] [CrossRef] - Zhao, X.; Chai, X.; Sun, J.; Qiu, Q. Joint optimization of mission abort and protective device selection policies for multistate systems. Risk Anal.
**2022**, 42, 2823–2834. [Google Scholar] [CrossRef] [PubMed] - Qiu, Q.; Cui, L.; Wu, B. Dynamic mission abort policy for systems operating in a controllable environment with self-healing mechanism. Reliab. Eng. Syst. Saf.
**2020**, 203, 107069. [Google Scholar] [CrossRef] - Qiu, Q.; Kou, M.; Chen, K.; Deng, Q.; Kang, F.; Lin, C. Optimal stopping problems for mission oriented systems considering time redundancy. Reliab. Eng. Syst. Saf.
**2020**, 205, 107226. [Google Scholar] [CrossRef] - Qiu, Q.; Cui, L. Optimal mission abort policy for systems subject to random shocks based on virtual age process. Reliab. Eng. Syst. Saf.
**2019**, 189, 11–20. [Google Scholar] [CrossRef] - Qiu, Q.; Cui, L.; Dong, Q. Preventive maintenance policy of single-unit systems based on shot-noise process. Qual. Reliab. Eng. Int.
**2019**, 35, 550–560. [Google Scholar] [CrossRef] - Shang, L.; Qiu, Q.; Wu, C.; Du, Y. Random replacement policies to sustain the post-warranty reliability. J. Qual. Maint. Eng. 2022; ahead-of-print. [Google Scholar] [CrossRef]
- Shang, L.; Yu, X.; Wang, X.; Qiu, Q. Study of A Two-stage Random Warranty to Maintain Fairness. Procedia Comput. Sci.
**2022**, 214, 437–440. [Google Scholar] [CrossRef] - Shang, L.; Liu, B.; Cai, Z.; Wu, C. Random maintenance policies for sustaining the reliability of the product through 2D-warranty. Appl. Math. Model.
**2022**, 111, 363–383. [Google Scholar] [CrossRef] - Nakagawa, T. Maintenance Theory of Reliability; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2005. [Google Scholar]
- Zhao, X.; Qian, C.; Nakagawa, T. Comparisons of replacement policies with periodic times and repair numbers. Reliab. Eng. Syst. Saf.
**2017**, 168, 161–170. [Google Scholar] [CrossRef] - Qiu, Q.; Cui, L.; Gao, H. Availability and maintenance modelling for systems subject to multiple failure modes. Comput. & Ind. Eng.
**2017**, 108, 192–198. [Google Scholar] - Barlow, R.E.; Proschan, F. Mathematical Theory of Reliability; John Wiley & Sons: New York, NY, USA, 1965. [Google Scholar]
- Sheu, S.-H.; Liu, T.-H.; Zhang, Z.-G. Extended optimal preventive replacement policies with random working cycle. Reliab. Eng. Syst. Saf.
**2019**, 188, 398–415. [Google Scholar] [CrossRef] - Zhang, Q.; Yao, W.; Xu, P.; Fang, Z. Optimal age replacement policies of mission-oriented systems with discounting. Comput. Ind. Eng.
**2023**, 177, 109027. [Google Scholar] [CrossRef]

$\mathit{n}$ | $\mathit{w}=2$ | $\mathit{w}=3$ | $\mathit{w}=4$ | ||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{N}}^{*}$ | ${\mathit{T}}^{*}$ | $\mathit{E}\mathit{C}{\mathit{R}}_{\mathit{a}}({\mathit{N}}^{*},{\mathit{T}}^{*})$ | ${\mathit{N}}^{*}$ | ${\mathit{T}}^{*}$ | $\mathit{E}\mathit{C}{\mathit{R}}_{\mathit{a}}({\mathit{N}}^{*},{\mathit{T}}^{*})$ | ${\mathit{N}}^{*}$ | ${\mathit{T}}^{*}$ | $\mathit{E}\mathit{C}{\mathit{R}}_{\mathit{a}}({\mathit{N}}^{*},{\mathit{T}}^{*})$ | |

2 | 26 | 3.1342 | 3.0287 | 25 | 3.1279 | 3.0495 | 24 | 3.1270 | 3.0513 |

3 | 24 | 2.8049 | 2.9686 | 24 | 2.7775 | 3.0329 | 24 | 2.7748 | 3.0411 |

4 | 23 | 2.5050 | 2.8612 | 23 | 2.4250 | 2.9931 | 22 | 2.4142 | 3.0174 |

$\mathit{k}$ | $\mathit{\lambda}=2$ | $\mathit{\lambda}=3$ | $\mathit{\lambda}=4$ | ||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{N}}^{*}$ | ${\mathit{T}}^{*}$ | $\mathit{E}\mathit{C}{\mathit{R}}_{\mathit{a}}({\mathit{N}}^{*},{\mathit{T}}^{*})$ | ${\mathit{N}}^{*}$ | ${\mathit{T}}^{*}$ | $\mathit{E}\mathit{C}{\mathit{R}}_{\mathit{a}}({\mathit{N}}^{*},{\mathit{T}}^{*})$ | ${\mathit{N}}^{*}$ | ${\mathit{T}}^{*}$ | $\mathit{E}\mathit{C}{\mathit{R}}_{\mathit{a}}({\mathit{N}}^{*},{\mathit{T}}^{*})$ | |

2 | 16 | 1.9893 | 3.0398 | 24 | 2.7775 | 3.0329 | 30 | 3.1705 | 2.9773 |

3 | 15 | 1.7025 | 3.2991 | 23 | 2.6090 | 3.2694 | 30 | 3.1575 | 3.0477 |

4 | 14 | 1.4184 | 3.5938 | 22 | 2.4474 | 3.5239 | 30 | 2.9298 | 3.3493 |

$\mathit{\lambda}$ | The Optimal RNN | The Optimal Random Periodic Replacement | ||
---|---|---|---|---|

$\mathit{R}\mathit{C}{\mathit{L}}_{\mathit{a}}({\mathit{N}}^{*},{\mathit{T}}^{*})$ | $\mathit{E}\mathit{C}{\mathit{R}}_{\mathit{a}}({\mathit{N}}^{*},{\mathit{T}}^{*})$ | $\underset{\overline{\mathit{G}}(\mathit{y})\to 1}{\mathbf{lim}}\mathit{R}\mathit{C}{\mathit{L}}_{\mathit{a}}({\mathit{N}}^{*},{\mathit{T}}^{*})$ | $\underset{\overline{\mathit{G}}(\mathit{y})\to 1}{\mathbf{lim}}({\mathit{N}}^{*},{\mathit{T}}^{*})$ | |

4 | 4.1707 | 2.9773 | 4.0592 | 3.1639 |

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## Share and Cite

**MDPI and ACS Style**

Shang, L.; Zhang, N.; Yang, L.; Shang, L.
Study of a Random Warranty Model Maintaining Fairness and a Random Replacement Next Model Sustaining Post-Warranty Reliability. *Axioms* **2023**, *12*, 258.
https://doi.org/10.3390/axioms12030258

**AMA Style**

Shang L, Zhang N, Yang L, Shang L.
Study of a Random Warranty Model Maintaining Fairness and a Random Replacement Next Model Sustaining Post-Warranty Reliability. *Axioms*. 2023; 12(3):258.
https://doi.org/10.3390/axioms12030258

**Chicago/Turabian Style**

Shang, Lifeng, Nan Zhang, Li Yang, and Lijun Shang.
2023. "Study of a Random Warranty Model Maintaining Fairness and a Random Replacement Next Model Sustaining Post-Warranty Reliability" *Axioms* 12, no. 3: 258.
https://doi.org/10.3390/axioms12030258