# Bayesian Statistical Method Enhance the Decision-Making for Imperfect Preventive Maintenance with a Hybrid Competing Failure Mode

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## Abstract

**:**

## 1. Introduction

## 2. Preventive Maintenance for Deteriorating Systems with a Hybrid Deterioration

#### 2.1. Hybrid Competing Failure Mode

- (1)
- The system’s deterioration behaves as a non-homogeneous Poisson process (NHPP).
- (2)
- The system’s deterioration is composed of maintainable and non-maintainable failure modes.
- (3)
- A PM cannot restore the whole system to a brand-new state; instead, it can restore the whole system to some state as better-than-now.
- (4)
- Any breakdowns occurring within the interval between two PM actions cause a minimal repair.

#### 2.2. Estimation of a System’s Failures under Preventive Maintenances

#### 2.3. Evaluation of Repair and Maintenance Costs of a Facility

#### 2.4. Optimal Preventive Maintenance Schedule with Consideration of Multiple PM Alternatives

**Proposition 1.**

**Proof.**

## 3. Bayesian Decision Process by Using Domain Experts’ Judgment and Collected Information

#### 3.1. Analysis by the Natural Conjugate Probability Distribution

#### 3.2. The Bayesian Decision Process

#### 3.3. Computerized Information System Design

## 4. Application and Sensitivity Analyses

#### 4.1. Application of Prior and Posterior Analyses

#### 4.2. Sensitivity Analyses

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Park, D.H.; Jung, G.M.; Yum, J.K. Cost minimization for periodic maintenance policy of a system subject to slow degradation. Reliab. Eng. Syst. Saf.
**2000**, 68, 105–112. [Google Scholar] [CrossRef] - Yeh, R.H.; Lo, H.C. Optimal preventive-maintenance warranty policy for repairable products. Eur. J. Oper. Res.
**2001**, 134, 59–69. [Google Scholar] [CrossRef] - Jung, G.M.; Park, D.H. Optimal maintenance policies during the post-warranty period. Reliab. Eng. Syst. Saf.
**2003**, 82, 173–185. [Google Scholar] [CrossRef] - Seo, J.H.; Bai, D.S. An optimal maintenance policy for a system under periodic overhaul. Math. Comput. Model.
**2004**, 39, 373–380. [Google Scholar] [CrossRef] - Yeh, R.H.; Chang, W.L. Optimal threshold value of failure-rate for Leased products with preventive maintenance actions. Math. Comput. Model.
**2007**, 46, 730–737. [Google Scholar] [CrossRef] - Das, A.N.; Sarmah, A.N. Preventive replacement models: An overview and their application in process industries. Eur. J. Ind. Eng.
**2010**, 4, 280–307. [Google Scholar] [CrossRef] - Yeh, R.H.; Kao, K.C.; Chang, W.L. Preventive-maintenance policy for leased products under various maintenance costs. Expert Syst. Appl.
**2011**, 38, 3558–3562. [Google Scholar] [CrossRef] - Bouguera, S.; Chelbi, A.; Rezg, N. A decision model for adopting an extended warranty under different maintenance policies. Int. J. Prod. Econ.
**2012**, 135, 840–849. [Google Scholar] [CrossRef] - Chang, W.L.; Lin, J.H. Optimal maintenance policy and length of extended warranty within the life cycle of products. Comput. Math. Appl.
**2012**, 63, 144–150. [Google Scholar] [CrossRef] [Green Version] - Beaurepaire, P.; Valdebenito, M.A.; Schuëller, G.I.; Jensen, H.A. Reliability-based optimization of maintenance scheduling of mechanical components under fatigue. Comput. Methods Appl. Mech. Eng.
**2012**, 221–222, 24–40. [Google Scholar] [CrossRef] - Schutz, J.; Rezg, N. Maintenance strategy for leased equipment. Comput. Ind. Eng.
**2013**, 66, 593–600. [Google Scholar] [CrossRef] - Kim, B.S.; Ozturkoglu, Y. Scheduling a single machine with multiple preventive maintenance activities and position-based deteriorations using genetic algorithms. J. Adv. Manuf. Technol.
**2013**, 67, 1127–1137. [Google Scholar] [CrossRef] - Khojandi, A.; Maillart, L.M.; Prokopyev, O.A. Optimal planning of life-depleting maintenance activities. IIE Trans.
**2014**, 46, 636–652. [Google Scholar] [CrossRef] - Yuan, X.; Lu, Z. Efficient approach for reliability-based optimization based on weighted importance sampling approach. Reliab. Eng. Syst. Saf.
**2014**, 132, 107–114. [Google Scholar] [CrossRef] - Lu, B.; Zhou, X.; Li, Y. Joint modeling of preventive maintenance and quality improvement for deteriorating single-machine manufacturing systems. Comput. Ind. Eng.
**2016**, 91, 188–196. [Google Scholar] [CrossRef] - Wang, K.; Djurdjanovic, D. Joint Optimization of Preventive Maintenance, Spare Parts Inventory and Transportation Options for Systems of Geographically Distributed Assets. Machines
**2018**, 6, 55. [Google Scholar] [CrossRef] [Green Version] - Zhou, Y.; Kou, G.; Xiao, H.; Peng, Y.; Alsaadi, F.E. Sequential imperfect preventive maintenance model with failure intensity reduction with an application to urban buses. Reliab. Eng. Syst. Saf.
**2020**, 198, 106871. [Google Scholar] [CrossRef] - García, F.J.Á.; Salgado, D.R. Analysis of the Influence of Component Type and Operating Condition on the Selection of Preventive Maintenance Strategy in Multistage Industrial Machines: A Case Study. Machines
**2022**, 10, 385. [Google Scholar] [CrossRef] - Diatte, K.; O’Halloran, B.; Van Bossuyt, D.L. The Integration of Reliability, Availability, and Maintainability into Model-Based Systems Engineering. Systems
**2022**, 10, 101. [Google Scholar] [CrossRef] - Paulsen, J.; Cooke, R.; Nyman, R. Comparative evaluation of maintenance performance using subsurvival functions. Reliab. Eng. Syst. Saf.
**1997**, 58, 157–163. [Google Scholar] [CrossRef] - Cooke, R.; Paulsen, J. Concepts for measuring maintenance performance and methods for analysing competing failure models. Reliab. Eng. Syst. Saf.
**1997**, 55, 135–141. [Google Scholar] [CrossRef] - Fang, C.C.; Hsu, C.C.; Liu, J.H. The Decision-Making for the Optimization of Finance Lease with Facilities’ Two-Dimensional Deterioration. Systems
**2022**, 10, 210. [Google Scholar] [CrossRef] - Salinas-Torres, V.H.; Pereira, C.A.B.; Tiwari, R.C. Bayesian nonparametric estimation in a series system or a competing risk model. J. Nonparametr. Stat.
**2002**, 14, 449–458. [Google Scholar] [CrossRef] - Yousef, M.M.; Hassan, A.S.; Alshanbari, H.M.; El-Bagoury, A.H.; Almetwally, E.M. Bayesian and Non-Bayesian Analysis of Exponentiated Exponential Stress–Strength Model Based on Generalized Progressive Hybrid Censoring Process. Axioms
**2022**, 11, 455. [Google Scholar] [CrossRef] - Wang, J.; Miao, Y. Optimal preventive maintenance policy of the balanced system under the semi-Markov model. Reliab. Eng. Syst. Saf.
**2021**, 213, 107690. [Google Scholar] [CrossRef] - Alotaibi, R.; Nassar, M.; Ghosh, I.; Rezk, H. Elshahhat, A. Inferences of a Mixture Bivariate Alpha Power Exponential Model with Engineering Application. Axioms
**2022**, 11, 459. [Google Scholar] [CrossRef] - Liu, T.; Zhang, L.; Jin, G.; Pan, Z. Reliability Assessment of Heavily Censored Data Based on E-Bayesian Estimation. Mathematics
**2022**, 10, 4216. [Google Scholar] [CrossRef] - Yousef, M.M.; Hassan, A.S.; Al-Nefaie, A.H.; Almetwally, E.M. Bayesian Estimation Using MCMC Method of System Reliability for Inverted Topp–Leone Distribution Based on Ranked Set Sampling. Mathematics
**2022**, 10, 3122. [Google Scholar] [CrossRef] - Zequeira, R.I.; Berenguer, C. Periodic imperfect preventive maintenance with two categories of competing failure modes. Reliab. Eng. Syst. Saf.
**2006**, 91, 460–468. [Google Scholar] [CrossRef] - El-Ferik, S.; Ben-Daya, M. Age-based hybrid model for imperfect preventive maintenance. IIE Trans.
**2007**, 38, 365–385. [Google Scholar] [CrossRef] - Kahrobaee, S.; Asgarpoor, S. A hybrid analytical-simulation approach for maintenance optimization of deteriorating equipment: Case study of wind turbines. Electr. Power Syst. Res.
**2013**, 104, 80–86. [Google Scholar] [CrossRef] - Rafiee, K.; Feng, Q.; Coit, D.W. Condition-based maintenance for repairable deteriorating systems subject to a generalized mixed shock model. IEEE Trans. Reliab.
**2015**, 64, 1164–1174. [Google Scholar] [CrossRef] - Zhou, X.; Wu, C.; Li, Y.; Xi, L. A preventive maintenance model for leased equipment subject to internal degradation and external shock damage. Reliab. Eng. Syst. Saf.
**2016**, 154, 1–7. [Google Scholar] [CrossRef] - Yang, L.; Zhao, Y.; Peng, R.; Ma, X. Hybrid preventive maintenance of competing failures under random environment. Reliab. Eng. Syst. Saf.
**2018**, 174, 130–140. [Google Scholar] [CrossRef] - Cao, Y. Modeling the effects of dependence between competing failure processes on the condition-based preventive maintenance policy. Appl. Math. Model.
**2021**, 99, 400–417. [Google Scholar] [CrossRef] - Liu, J.; Zhuang, X.; Pang, H. Reliability and hybrid maintenance modeling for competing failure systems with multistage periods. Probabilistic Eng. Mech.
**2022**, 68, 103254. [Google Scholar] [CrossRef] - Basílio, M.P.; Pereira, V.; Costa, H.G.; Santos, M.; Ghosh, A. A Systematic Review of the Applications of Multi-Criteria Decision Aid Methods (1977–2022). Electronics
**2022**, 11, 1720. [Google Scholar] [CrossRef] - Huang, Y.-S.; Bier, V.M. A Natural Conjugate Prior for the Nonhomogeneous Poisson Process with a Power Law Intensity Function. Commun. Stat.-Simul. Comput.
**1998**, 27, 525–551. [Google Scholar] [CrossRef]

**Figure 2.**Preventive maintenances between perfect recovery and imperfect recovery under a hybrid deterioration.

**Figure 3.**Heuristic algorithm for obtaining the minimal cost by setting ${N}^{*}$ and ${M}_{P}^{q*}$.

**Figure 6.**The overall costs per unit and year for all PM alternatives estimated by the prior analysis.

**Figure 7.**The overall costs per unit and year for all PM alternatives estimated by the prior and posterior analyses.

**Figure 8.**The impacts of ${\mu}_{{\alpha}_{o}},{\mu}_{{\alpha}_{p}}$, ${\mu}_{{\beta}_{o}}$, and ${\mu}_{{\beta}_{p}}$ on the overall cost per unit and year.

**Figure 9.**The Impacts of ${C}_{F}^{q}$, ${\tau}_{q}$, ${C}_{mr}$, and ${C}_{pl}$ on the overall cost per unit and year.

${T}_{L}$: the lifetime of a equipment or facility. |

$t$: the age of a equipment or facility. |

$x$: the time interval between two PMs. |

${t}_{k}^{-}$: the effective age of a equipment or facility before the time point of the kth PM. |

${t}_{k}^{+}$: the effective age of a equipment or facility after the time point of the kth PM. |

${\alpha}_{o}$: the scale factor of the intensity function of non-maintainable failure mode. |

${\beta}_{o}$: the shape factor of the intensity function of non-maintainable failure mode. |

${\alpha}_{p}$: the scale factor of the intensity function of maintainable failure mode |

${\beta}_{p}$: the shape factor of the intensity function of maintainable failure mode. |

$f\left(\alpha ,\beta \right)$: the prior probability distribution of the power-law intensity function. |

$g\left(\alpha ,\beta \right)$: the posterior probability distribution of the power-law intensity function. |

$\delta $: the age reduction factor, where $\delta \in \left[0,1\right]$. |

${\lambda}_{o}(t|{\alpha}_{o},{\beta}_{o})$: the intensity function of non-maintainable failure mode of the system deterioration. |

${\lambda}_{p}(t|{\alpha}_{p},{\beta}_{p})$: the intensity function of maintainable failure mode of the system deterioration. |

${\lambda}_{h}\left(t|{\alpha}_{o},{\beta}_{o},{\alpha}_{p},{\beta}_{p}\right)$: the intensity function of the hybrid mode of the system deterioration. |

$N$: the number of PM action during the whole system lifetime. |

${N}_{mr}(\cdot )$: the expected number of performing minimal repairs of the system. |

${C}_{mr}$: the average cost to perform a minimal repair. |

${C}_{p{m}_{k}}$: the cost to perform the kth PM. |

${C}_{rp}$: the cost of the overall replacement of a equipment or facility. |

$\mathsf{\Phi}\left({t}_{r}\right)$: the probability density function of the time for performing a minimal repair. |

${C}_{pl}$: the penalty cost if the actual repair time over the time threshold $\phi $ |

$\phi $: the time threshold for performing a minimal repair. |

${C}_{F}$: the base cost for a PM action, which is influenced by the degree of PM. |

$\tau $: the increasing rate of PM base cost |

Parameters for the two categories deterioration, which were judged by experts | ${\mu}_{{\alpha}_{o}}=0.6$$,{\mu}_{{\beta}_{o}}=0.15$$,{\sigma}_{{\alpha}_{o}}=1.25$$,{\sigma}_{{\beta}_{o}}=0.3125$$;{\mu}_{{\alpha}_{p}}=0.85$$,{\mu}_{{\beta}_{p}}=0.2125$$,{\sigma}_{{\alpha}_{p}}=1.75$$,{\sigma}_{{\beta}_{p}}=0.4375$ |

Interval between two PM actions | $x=0.5\mathrm{years}$ |

PM’s Base cost of the five candidate PM alternatives | ${C}_{F}^{q}=\left\{\$780,\$790,\$800,\$880,\$890\right\}$ |

Age reduction factors of the five candidate PM alternatives | ${\delta}_{pm}^{q}=\left\{0.8,0.85,0.9,0.95,1.0\right\}$ |

Periodically increasing rates of PM cost of the five candidate PM alternatives | ${\tau}_{q}=\left\{0.19,0.195,0.2,0.235,0.24\right\}$ |

Replacement cost | ${C}_{rp}=\$20,000$ |

Expected cost of performinga minimal repair | ${C}_{mr}$= $250 |

Penalty cost if the repair time exceed the time limit $\phi $ | ${C}_{pl}=\$90$ |

Expected value and standard deviation of performing a minimal repair | $E\left({t}_{r}\right)=5\mathrm{h}$$,\sigma \left({t}_{r}\right)=3\mathrm{h}$ |

The limit of tolerable waiting time for performing a minimal repair | $\phi =6.5\mathrm{h}$ |

**Table 3.**The expected preventive cost, repair cost, penalty cost, replacement cost and overall cost per unit and year for PM Alternatives 2, 3, 4 estimated by the prior analysis.

${\mathit{T}}_{\mathit{L}}$ | $\frac{{\mathit{C}}_{\mathit{p}\mathit{m}}^{\mathit{q}}\left({\mathit{C}}_{\mathit{F}}^{\mathit{q}},{\mathit{\tau}}_{\mathit{q}},\mathit{x},{\mathit{T}}_{\mathit{L}}\right)}{{\mathit{T}}_{\mathit{L}}}$ | $\frac{{\mathit{C}}_{\mathit{m}\mathit{r}}\underset{\mathit{P}\mathit{r}\mathit{i}}{\mathit{E}}\left[{\mathit{N}}_{\mathit{m}\mathit{r}}\left(\mathit{\xb7}\right)\right]}{{\mathit{T}}_{\mathit{L}}}$ | $\frac{{\mathit{C}}_{\mathit{p}\mathit{l}}\mathit{E}\left[{\mathit{t}}_{\mathit{r}}|\mathit{\omega},\mathit{\eta}\right]\underset{\mathit{P}\mathit{r}\mathit{i}}{\mathit{E}}\left[{\mathit{N}}_{\mathit{m}\mathit{r}}\left(\mathit{\xb7}\right)\right]}{{\mathit{T}}_{\mathit{L}}}$ | $\frac{{\mathit{C}}_{\mathit{r}\mathit{p}}}{{\mathit{T}}_{\mathit{L}}}$ | $\underset{\mathit{P}\mathit{r}\mathit{i}}{\mathit{E}}\left[\mathit{C}(\mathit{N}|{\mathit{M}}_{\mathit{P}}^{\mathit{q}})\right]$ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{M}}_{\mathit{P}}^{2}$ | ${\mathit{M}}_{\mathit{P}}^{3}$ | ${\mathit{M}}_{\mathit{P}}^{4}$ | ${\mathit{M}}_{\mathit{P}}^{2}$ | ${\mathit{M}}_{\mathit{P}}^{3}$ | ${\mathit{M}}_{\mathit{P}}^{4}$ | ${\mathit{M}}_{\mathit{P}}^{2}$ | ${\mathit{M}}_{\mathit{P}}^{3}$ | ${\mathit{M}}_{\mathit{P}}^{4}$ | ${\mathit{M}}_{\mathit{P}}^{2}$ | ${\mathit{M}}_{\mathit{P}}^{3}$ | ${\mathit{M}}_{\mathit{P}}^{4}$ | ||

0.5 | 0 | 0 | 0 | 261 | 261 | 261 | 219 | 219 | 219 | 40,000 | 40,480 | 40,480 | 40,480 |

1 | 867 | 880 | 983 | 295 | 291 | 287 | 247 | 244 | 240 | 20,000 | 21,409 | 21,415 | 21,510 |

1.5 | 1207 | 1227 | 1380 | 325 | 317 | 308 | 272 | 266 | 258 | 13,333 | 15,138 | 15,142 | 15,280 |

2 | 1416 | 1440 | 1630 | 353 | 340 | 328 | 296 | 285 | 275 | 10,000 | 12,064 | 12,066 | 12,233 |

2.5 | 1572 | 1600 | 1822 | 379 | 363 | 346 | 318 | 304 | 290 | 8000 | 10,269 | 10,267 | 10,458 |

3 | 1702 | 1733 | 1984 | 404 | 384 | 364 | 339 | 322 | 305 | 6667 | 9111 | 9106 | 9319 |

3.5 | 1816 | 1851 | 2129 | 429 | 405 | 380 | 359 | 339 | 319 | 5714 | 8318 | 8310 | 8543 |

4 | 1922 | 1960 | 2264 | 452 | 425 | 397 | 379 | 356 | 332 | 5000 | 7753 | 7741 | 7993 |

4.5 | 2021 | 2062 | 2392 | 476 | 444 | 412 | 399 | 372 | 346 | 4444 | 7340 | 7323 | 7594 |

5 | 2115 | 2160 | 2515 | 499 | 463 | 428 | 418 | 389 | 358 | 4000 | 7032 | 7012 | 7301 |

5.5 | 2207 | 2255 | 2634 | 522 | 482 | 443 | 437 | 404 | 371 | 3636 | 6802 | 6777 | 7084 |

6 | 2296 | 2347 | 2751 | 544 | 501 | 457 | 456 | 420 | 383 | 3333 | 6630 | 6601 | 6925 |

6.5 | 2383 | 2437 | 2865 | 567 | 519 | 472 | 475 | 435 | 395 | 3077 | 6502 | 6468 | 6809 |

7 | 2468 | 2526 | 2978 | 589 | 537 | 486 | 494 | 450 | 407 | 2857 | 6408 | 6371 | 6729 |

7.5 | 2553 | 2613 | 3090 | 611 | 555 | 500 | 512 | 466 | 419 | 2667 | 6343 | 6301 | 6676 |

8 | 2637 | 2700 | 3201 | 633 | 573 | 514 | 531 | 480 | 431 | 2500 | 6300 | 6254 | 6646 |

8.5 | 2719 | 2786 | 3311 | 655 | 591 | 528 | 549 | 495 | 442 | 2353 | 6276 | 6225 | 6634 |

9 | 2802 | 2871 | 3420 | 677 | 608 | 541 | 567 | 512 | 454 | 2222 | 6268 | 6214 | 6637 |

9.5 * | 2883 | 2956 | 3529 | 699 | 626 | 555 | 586 | 525 | 465 | 2105 | 6273 | 6212 * | 6654 |

10 | 2964 | 3040 | 3637 | 720 | 643 | 568 | 604 | 539 | 476 | 2000 | 6289 | 6223 | 6681 |

10.5 | 3045 | 3124 | 3744 | 742 | 661 | 581 | 622 | 554 | 487 | 1905 | 6314 | 6243 | 6718 |

11 | 3126 | 3207 | 3851 | 764 | 678 | 595 | 640 | 568 | 499 | 1818 | 6348 | 6272 | 6763 |

11.5 | 3206 | 3290 | 3958 | 786 | 695 | 608 | 659 | 583 | 510 | 1739 | 6389 | 6308 | 6815 |

12 | 3286 | 3373 | 4065 | 807 | 712 | 621 | 677 | 597 | 520 | 1667 | 6436 | 6350 | 6873 |

12.5 | 3365 | 3456 | 4171 | 829 | 730 | 634 | 695 | 612 | 531 | 1600 | 6489 | 6397 | 6936 |

13 | 3445 | 3538 | 4277 | 851 | 747 | 647 | 713 | 626 | 542 | 1538 | 6547 | 6449 | 7005 |

13.5 | 3524 | 3621 | 4383 | 872 | 764 | 660 | 731 | 640 | 553 | 1481 | 6609 | 6506 | 7077 |

14 | 3603 | 3703 | 4489 | 894 | 781 | 672 | 750 | 655 | 564 | 1429 | 6676 | 6567 | 7154 |

14.5 | 3682 | 3785 | 4595 | 916 | 798 | 685 | 768 | 669 | 574 | 1379 | 6745 | 6631 | 7233 |

15 | 3761 | 3867 | 4700 | 938 | 815 | 698 | 786 | 683 | 585 | 1333 | 6818 | 6698 | 7316 |

**Table 4.**The expected preventive cost, repair cost, penalty cost, replacement cost and overall cost per unit and year for PM Alternative for the prior and posterior analyses.

${\mathit{T}}_{\mathit{L}}$ | $\frac{{\mathit{C}}_{\mathit{p}\mathit{m}}^{\mathit{q}}\left({\mathit{C}}_{\mathit{F}}^{\mathit{q}},{\mathit{\tau}}_{\mathit{q}},\mathit{x},{\mathit{T}}_{\mathit{L}}\right)}{{\mathit{T}}_{\mathit{L}}}$ | $\frac{{\mathit{C}}_{\mathit{m}\mathit{r}}\mathit{E}\left[{\mathit{N}}_{\mathit{m}\mathit{r}}\left(\mathit{\xb7}\right)\right]}{{\mathit{T}}_{\mathit{L}}}$ | $\frac{{\mathit{C}}_{\mathit{p}\mathit{l}}\mathit{E}\left[{\mathit{t}}_{\mathit{r}}|\mathit{\omega},\mathit{\eta}\right]\mathit{E}\left[{\mathit{N}}_{\mathit{m}\mathit{r}}\left(\mathit{\xb7}\right)\right]}{{\mathit{T}}_{\mathit{L}}}$ | $\frac{{\mathit{C}}_{\mathit{r}\mathit{p}}}{{\mathit{T}}_{\mathit{L}}}$ | $\underset{\mathit{P}\mathit{r}\mathit{i}}{\mathit{E}}\left[\mathit{C}(\mathit{N}|{\mathit{M}}_{\mathit{P}}^{\mathit{q}})\right]$ | $\underset{\mathit{P}\mathit{o}\mathit{s}}{\mathit{E}}\left[\mathit{C}(\mathit{N}|{\mathit{M}}_{\mathit{P}}^{\mathit{q}})\right]$ | ||
---|---|---|---|---|---|---|---|---|

Prior | Posterior | Prior | Posterior | |||||

0.5 | 0 | 261 | 252 | 219 | 212 | 40,000 | 40,480 | 40,464 |

1 | 880 | 291 | 287 | 244 | 240 | 20,000 | 21,415 | 21,407 |

1.5 | 1227 | 317 | 313 | 266 | 262 | 13,333 | 15,142 | 15,135 |

2 | 1440 | 340 | 334 | 285 | 280 | 10,000 | 12,066 | 12,055 |

2.5 | 1600 | 363 | 354 | 304 | 297 | 8000 | 10,267 | 10,251 |

3 | 1733 | 384 | 372 | 322 | 312 | 6667 | 9106 | 9083 |

3.5 | 1851 | 405 | 388 | 339 | 325 | 5714 | 8310 | 8279 |

4 | 1960 | 425 | 404 | 356 | 339 | 5000 | 7741 | 7702 |

4.5 | 2062 | 444 | 419 | 372 | 351 | 4444 | 7323 | 7276 |

5 | 2160 | 463 | 433 | 389 | 363 | 4000 | 7012 | 6956 |

5.5 | 2255 | 482 | 446 | 404 | 374 | 3636 | 6777 | 6712 |

6 | 2347 | 501 | 460 | 420 | 385 | 3333 | 6601 | 6525 |

6.5 | 2437 | 519 | 472 | 435 | 396 | 3077 | 6468 | 6382 |

7 | 2526 | 537 | 485 | 450 | 406 | 2857 | 6371 | 6274 |

7.5 | 2613 | 555 | 497 | 466 | 416 | 2667 | 6301 | 6193 |

8 | 2700 | 573 | 508 | 480 | 426 | 2500 | 6254 | 6134 |

8.5 | 2786 | 591 | 520 | 495 | 436 | 2353 | 6225 | 6094 |

9 | 2871 | 608 | 531 | 512 | 445 | 2222 | 6214 | 6069 |

9.5 * | 2956 | 626 | 542 | 525 | 454 | 2105 | 6212 * | 6057 |

10 ** | 3040 | 643 | 553 | 539 | 463 | 2000 | 6223 | 6056 ** |

10.5 | 3124 | 661 | 563 | 554 | 472 | 1905 | 6243 | 6064 |

11 | 3207 | 678 | 574 | 568 | 481 | 1818 | 6272 | 6080 |

11.5 | 3290 | 695 | 584 | 583 | 489 | 1739 | 6308 | 6103 |

12 | 3373 | 712 | 594 | 597 | 498 | 1667 | 6350 | 6132 |

12.5 | 3456 | 730 | 604 | 612 | 506 | 1600 | 6397 | 6166 |

13 | 3538 | 747 | 614 | 626 | 514 | 1538 | 6449 | 6205 |

13.5 | 3621 | 764 | 623 | 640 | 522 | 1481 | 6506 | 6248 |

14 | 3703 | 781 | 633 | 655 | 530 | 1429 | 6567 | 6295 |

14.5 | 3785 | 798 | 642 | 669 | 538 | 1379 | 6631 | 6345 |

15 | 3867 | 815 | 651 | 683 | 546 | 1333 | 6698 | 6398 |

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**MDPI and ACS Style**

Fang, C.-C.; Hsu, C.-C.; Liu, J.-H.
Bayesian Statistical Method Enhance the Decision-Making for Imperfect Preventive Maintenance with a Hybrid Competing Failure Mode. *Axioms* **2022**, *11*, 734.
https://doi.org/10.3390/axioms11120734

**AMA Style**

Fang C-C, Hsu C-C, Liu J-H.
Bayesian Statistical Method Enhance the Decision-Making for Imperfect Preventive Maintenance with a Hybrid Competing Failure Mode. *Axioms*. 2022; 11(12):734.
https://doi.org/10.3390/axioms11120734

**Chicago/Turabian Style**

Fang, Chih-Chiang, Chin-Chia Hsu, and Je-Hung Liu.
2022. "Bayesian Statistical Method Enhance the Decision-Making for Imperfect Preventive Maintenance with a Hybrid Competing Failure Mode" *Axioms* 11, no. 12: 734.
https://doi.org/10.3390/axioms11120734