# Reliability Analysis for Unrepairable Automotive Components

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## Abstract

**:**

## 1. Introduction

## 2. Subject of Research

## 3. Failure Data and Probability Distribution Fitting

- -
- H
_{0}: the distribution represents the data, - -
- H
_{1}: the distribution does not represent the data.

_{1},T

_{2},...,T

_{n}, m—number of suspended data points at S

_{1},S

_{2},...,S

_{m}, $k$—the number of estimated parameters, ${T}_{i}$—failure time of the i-th component, S

_{j}—suspension of the j-th component, ${\theta}_{1},{\theta}_{2},\dots ,{\theta}_{k}$—k unknown parameters which need to be estimated, $f\left({T}_{i};{\theta}_{1},{\theta}_{2},\dots ,{\theta}_{k}\right)$—probability density function pdf and $F\left({S}_{j};{\theta}_{1},{\theta}_{2},\dots ,{\theta}_{k}\right)$—cumulative density function cdf.

## 4. Results Analysis

#### 4.1. Data Interpretation

#### 4.2. Optimal Maintenance Strategy

#### 4.3. The Method of Solving the Problem

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Boot lid contactor; (

**a**) front site view, (

**b**) front site with an open cover-electronic switch system-button, (

**c**) the place where the contactor is located.

**Figure 2.**View of the gap between the chrome strip and the vehicle body, through the water entered, causing damage to the boot lid contactor.

**Figure 9.**Diagram showing the place of installation of the boot lid contactor (marked with a red line); (

**a**) vehicle before modification (has a chrome strip), (

**b**) vehicle after modification (without the strip).

**Table 1.**Failure/suspension date of contactor (F/S–failure, suspension, car body type: H–hatchback, E–Estate).

F/S | Mileage [km] | Car Body Type | F/S | Mileage [km] | Car Body Type | F/S | Mileage [km] | Car Body Type |
---|---|---|---|---|---|---|---|---|

F | 52,621 | H | F | 126,639 | E | F | 272,945 | H |

F | 57,801 | H | F | 128,676 | H | S | 21,147 | E |

F | 58,001 | H | F | 133,390 | H | S | 21,169 | E |

F | 61,628 | H | F | 143,159 | E | S | 21,772 | E |

F | 68,100 | H | F | 144,041 | H | S | 23,071 | E |

F | 79,051 | H | F | 151,230 | H | S | 27,165 | H |

F | 85,974 | H | F | 153,345 | E | S | 36,019 | E |

F | 87,451 | E | F | 158,420 | E | S | 40,208 | E |

F | 88,050 | H | F | 161,200 | H | S | 44,374 | H |

F | 92,003 | E | F | 163,141 | H | S | 45,895 | E |

F | 95,636 | H | F | 163,952 | E | S | 49,990 | E |

F | 102,178 | H | F | 169,230 | H | S | 63,422 | E |

F | 105,600 | H | F | 176,965 | E | S | 63,519 | H |

F | 106,639 | H | F | 193,500 | H | S | 86,527 | H |

F | 110,558 | H | F | 199,263 | E | S | 99,784 | E |

F | 113,359 | H | F | 204,898 | E | S | 120,111 | E |

F | 115,000 | H | F | 206,460 | E | S | 128,523 | E |

F | 115,210 | H | F | 214,521 | H | S | 142,302 | E |

F | 116,762 | E | F | 231,403 | H | S | 151,209 | E |

F | 125,489 | H | F | 253,241 | H | S | 164,287 | E |

S | 171,356 | H |

Distribution | (K-S) | (rho) | LKV |
---|---|---|---|

1P-Exponential | 98.6344 | 13.157 | −373.271 |

2P-Exponential | 46.7258 | 5.121 | −357.329 |

Normal | 36.3504 | 5.114 | −360.985 |

Lognormal | 0.00136 | 2.168 | −357.914 |

2P-Weibull | 13.3673 | 3.803 | −359.422 |

3P-Weibull | 0.0004 | 2.316 | −356.841 |

Gamma | 1.6526 | 2.682 | −358.212 |

G-Gamma | 0.0018 | 2.170 | −357.914 |

Logistic | 10.3364 | 3.544 | −361.249 |

Loglogistic | 0.00697 | 2.172 | −358.661 |

Gumbel | 58.8010 | 7.855 | −366.116 |

Distribution | (K-S) | (rho) | LKV |
---|---|---|---|

1P-Exponential | 91.569 | 17.371 | −160.982 |

2P-Exponential | 95.994 | 20.586 | −151.111 |

Normal | 0.594 | 4.003 | −146.373 |

Lognormal | 0.020 | 4.846 | −147.220 |

2P-Weibull | 2.379 | 3.807 | −146.021 |

3P-Weibull | 2.918 | 3.758 | −146.001 |

Gamma | 0.016 | 4.626 | −146.840 |

G-Gamma | 9.291 | 6.338 | −143.857 |

Logistic | 0.328 | 3.580 | −146.749 |

Loglogistic | 0.002 | 3.919 | −147.255 |

Gumbel | 5.040 | 4.615 | −146.121 |

Distribution | K-S | rho | LKV | WDV | Ranking |
---|---|---|---|---|---|

3P-Weibull | 1 | 4 | 1 | 130 | 1 |

Lognormal | 2 | 1 | 4 | 290 | 2 |

G-Gamma | 3 | 2 | 3 | 290 | 2 |

Loglogistic | 4 | 3 | 6 | 490 | 3 |

Gamma | 5 | 5 | 5 | 500 | 4 |

2P-Exponential | 9 | 9 | 2 | 550 | 5 |

2P-Weibull | 7 | 7 | 7 | 700 | 6 |

Logistic | 6 | 6 | 9 | 750 | 7 |

Normal | 8 | 8 | 8 | 800 | 8 |

Gumbel | 10 | 10 | 10 | 1000 | 9 |

1P-Exponential | 11 | 11 | 11 | 1100 | 10 |

Distribution | K-S | rho | LKV | WDV | Ranking |
---|---|---|---|---|---|

3P-Weibull | 7 | 2 | 2 | 400 | 1 |

2P-Weibull | 6 | 3 | 3 | 420 | 2 |

Logistic | 4 | 1 | 6 | 470 | 3 |

Normal | 5 | 5 | 5 | 500 | 4 |

Gamma | 2 | 7 | 7 | 500 | 4 |

G-Gamma | 9 | 9 | 1 | 500 | 4 |

Loglogistic | 1 | 4 | 9 | 530 | 5 |

Gumbel | 8 | 6 | 4 | 580 | 6 |

Lognormal | 3 | 8 | 8 | 600 | 7 |

1P-Exponential | 10 | 10 | 11 | 1050 | 8 |

2P-Exponential | 11 | 11 | 10 | 1050 | 8 |

Body Type | Year of Registration | Sum | ||||||
---|---|---|---|---|---|---|---|---|

2009 | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | ||

Hatchback | 2402 | 3876 | 2418 | 1705 | 1207 | 1766 | 1597 | 14,971 |

Estate | 538 | 1379 | 1338 | 1131 | 1084 | 1817 | 1049 | 8336 |

Type of Body Car | Minimal Replacement Cost [EUR] | Optimal Time Interval [km] | Cost/Per Car [EUR] | Cost for all Cars (2009–2015) [EUR] |
---|---|---|---|---|

Hatchback | 0.000223 | 157,047.98 | 35.02 | 349,540 |

Estate | 0.000139 | 201,946.70 | 28.07 | 155,998 |

Total cost | 505,538 |

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**MDPI and ACS Style**

Ulbrich, D.; Selech, J.; Kowalczyk, J.; Jóźwiak, J.; Durczak, K.; Gil, L.; Pieniak, D.; Paczkowska, M.; Przystupa, K.
Reliability Analysis for Unrepairable Automotive Components. *Materials* **2021**, *14*, 7014.
https://doi.org/10.3390/ma14227014

**AMA Style**

Ulbrich D, Selech J, Kowalczyk J, Jóźwiak J, Durczak K, Gil L, Pieniak D, Paczkowska M, Przystupa K.
Reliability Analysis for Unrepairable Automotive Components. *Materials*. 2021; 14(22):7014.
https://doi.org/10.3390/ma14227014

**Chicago/Turabian Style**

Ulbrich, Dariusz, Jaroslaw Selech, Jakub Kowalczyk, Jakub Jóźwiak, Karol Durczak, Leszek Gil, Daniel Pieniak, Marta Paczkowska, and Krzysztof Przystupa.
2021. "Reliability Analysis for Unrepairable Automotive Components" *Materials* 14, no. 22: 7014.
https://doi.org/10.3390/ma14227014