_{1}

^{*}

Whenever purchasing durable goods, customers expect it to perform properly at least for a reasonable period of time. Over the last 30 years there has been a heightened interest in improving quality, productivity and reliability of manufactured products. As discussed by Murthy, Xie and Jiang, [1] analyzed “Aircraft windshield failure data” and fitted 2-fold Weibull mixture model for this data set. They estimated the model parameters from WPP plot. In this study, a set of competitive 2-fold mixture models (including Weibull, Exponential, Normal, Lognormal, Smallest extreme value distribution) are applied to find out the suitable statistical models. The data consist of both failure and censored lifetimes of the windshield. In the existing literature, there are many uses of mixture models for the complete data, but very limited literature available about it uses for the censored data case. Maximum likelihood estimation method is used to estimate the model parameters and the Akaike Information Criterion (AIC), Anderson-Darling (AD) and Adjusted Anderson Darling (AD*) test statistics are applied to select the suitable models among a set of competitive models. Various characteristics of the mixture models, such as the cumulative distribution function, reliability function, mean time to failure, B10 life, etc. are estimated to assess the reliability of the component.

Customers expect purchased products to be reliable and safe. They wish that system, vehicle, machine, device, and so on should, with high probability, be able to perform their intended function under usual operating conditions within a desired time period. So, it is the responsibility of the manufacturers to inform their customers about the average life span of their products. Again for costly items, customers expect that the product will perform properly for a minimum life span without any disturbance. If this happens, the producer or his agent should give free service to repair or replace the item. Improving reliability of product is an important part of improving product quality. In recent years many manufacturers have collected and analyzed field failure data to improve the quality and reliability of their products and to develop customer satisfaction.

In this article we have considered both the nonparametric and parametric estimation procedures. The Akaike Information Criterion (AIC), Anderson-Darling (AD), Adjusted Anderson Darling (AD*), Root Mean Square Error (RMSE) & Kolmogrov-Smirnov test statistics (KSts) are applied to select the best models for the data sets. This article deals with the analysis of product failure data to estimate a variety of quantities of interest used in investigating product reliability.

The article is organized as follows: Section 2 describes the product failure data sets which will be analyzed in this paper. Section 3 derives the lifetime models. Section 4 explains the parameter estimation procedures of the lifetime models of the components. Section 5 discusses the results obtained from the analysis. Finally, Section 6 concludes the article with additional implementation issues for further research.

Field failure data is superior to laboratory test data, because, it contains valuable information on the performance of any goods in actual usage conditions. There are many sources of collecting product reliability data. Warranty claim data is used as an essential source of field failure data which can be collected economically and efficiently through repair service networks and therefore, different procedures have been developed for collecting and analyzing warranty claim data refer to the literatures [

Failures of the Aircraft Windshield involve damage or delamination of the nonstructural outer ply or failure of the heating system. These failures do not result in damage to the aircraft but do result in replacement of the windshield.

Data on failure and service times for a particular model windshield are given in

In the real world, problems arise in many different contexts. Always models play an important role in solving the problem. A variety class of statistical models have been developed and studied extensively in analyzing the product failure data [

Mixture Models

A general k-fold mixture model involves k subpopulations and is given by

where

The density function is given by:

where _{i}(t).

The hazard function h(t) is given by

Failure Times | Service Times | |||||
---|---|---|---|---|---|---|

0.040 | 1.866 | 2.385 | 3.443 | 0.046 | 1.436 | 2.592 |

0.301 | 1.876 | 2.481 | 3.467 | 0.140 | 1.492 | 2.600 |

0.309 | 1.899 | 2.610 | 3.478 | 0.150 | 1.580 | 2.670 |

0.557 | 1.911 | 2.625 | 3.578 | 0.248 | 1.719 | 2.717 |

0.943 | 1.912 | 2.632 | 3.595 | 0.280 | 1.794 | 2.819 |

1.070 | 1.914 | 2.646 | 3.699 | 0.313 | 1.915 | 2.820 |

1.124 | 1.981 | 2.661 | 3.779 | 0.389 | 1.920 | 2.878 |

1.248 | 2.01 | 2.688 | 3.924 | 0.487 | 1.963 | 2.950 |

1.281 | 2.038 | 2.823 | 4.035 | 0.622 | 1.978 | 3.003 |

1.281 | 2.085 | 2.89 | 4.121 | 0.900 | 2.053 | 3.102 |

1.303 | 2.089 | 2.902 | 4.167 | 0.952 | 2.065 | 3.304 |

1.432 | 2.097 | 2.934 | 4.240 | 0.996 | 2.117 | 3.483 |

1.480 | 2.135 | 2.962 | 4.255 | 1.003 | 2.137 | 3.500 |

1.505 | 2.154 | 2.964 | 4.278 | 1.010 | 2.141 | 3.622 |

1.506 | 2.190 | 3.000 | 4.305 | 1.085 | 2.163 | 3.665 |

1.568 | 2.194 | 3.103 | 4.376 | 1.092 | 2.183 | 3.695 |

1.615 | 2.223 | 3.114 | 4.449 | 1.152 | 2.240 | 4.015 |

1.619 | 2.224 | 3.117 | 4.485 | 1.183 | 2.341 | 4.628 |

1.652 | 2.229 | 3.166 | 4.570 | 1.244 | 2.435 | 4.806 |

1.652 | 2.300 | 3.344 | 4.602 | 1.249 | 2.464 | 4.881 |

1.757 | 2.324 | 3.376 | 4.663 | 1.262 | 2.543 | 5.140 |

1.795 | 2.349 | 3.385 | 4.694 | 1.360 | 2.560 |

where h_{i}(t) is the hazard function associated with subpopulation i, and

with

From (4), we see that the failure rate for the model is a weighted mean of the failure rate for the subpopulations with the weights varying with t.

Special Case: Two-Fold Mixture Model (k = 2)

The CDF of the two-fold mixture model is given by

For example, suppose,

The probability density function

Other two-fold mixture models can be derived by using different CDFs in (6) from different lifetime distributions, similarly.

In this article we have applied both the nonparametric (Kaplan-Meier estimate) and parametric (Weibull Probability paper plot and maximum likelihood method) estimation procedures. The Akaike Information Criterion (AIC), Anderson-Darling (AD), adjusted Anderson Darling, Root Mean Square Error (RMSE) & Kolmogrov- Smirnov test statistics (KSts) are applied to select the best fitted models for the data sets.

In the early 1970s a special paper was developed for plotting the data under this transformation and was referred to as the Weibull probability paper (WPP) and the plot called the WPP plot. Weibull Probability Paper plot (WPP plot) is a special case of the probability paper plot. It is based on the Weibull transformations:

A plot of y versus x is called the Weibull probability plot.

For censored data the likelihood function is given by

where δ_{i} is the failure-censoring indicator for t_{i} (taking on the value 1 for failed items and 0 for censored). Taking log on both sides we get,

In the case of Weibull-Exponential mixture model putting the value of CDF and pdf of the model in Equation (9), we obtain the log-likelihood function of Weibull-Exponential mixture model, which is:

The maximum likelihood estimates of the parameters are obtained by solving the partial derivative equations of (10) with respect to

The value of −2log-likelihood, Akaike Information Criterion (AIC), AD, Adjusted AD, KSts & RMSE of the different seven mixture models are estimated for Windshield failure data. The results are displayed in

Here, the Weibull-Exponential mixture model contains the smallest AIC value 362.9783, the Normal-Log- normal mixture model contain the smallest value of AD and Adjusted AD test statistics and the Weibull-Normal mixture model contains the smallest value of KS.ts and RMSE among all of the seven mixture models. Hence, we can say that, among these mixture models, Weibull-Exponential, Weibull-Normal and Normal-Lognormal mixture models can be selected as the best three models according to the value of AIC, AD, Adj AD, RMSE and Kolmogrov-Smirnov test statistic, respectively for Windshield failure data.

Now, the parameters of Weibull-Exponential, Normal-Lognormal and Weibull-Normal mixture models, estimated by applying ML method are displayed in

Murthy et al. [

Here, we observe that, the reliability function obtained from the MLE is closer to the Kaplan-Meier estimate than that of the reliability function obtained from the WPP plot. So, we may say that, the maximum likelihood estimate procedure is much better than Weibull probability paper (WPP) plot procedure.

The CDF of Weibull-Exponential, Normal-Lognormal and Weibull-Normal mixture models, using K-M and ML estimating methods are estimated and displayed the results in

Mixture Model | −2 logL | AIC | AD | Adj AD | KS.ts | RMSE |
---|---|---|---|---|---|---|

1. Weibull-Weibull | 354.9248 | 364.9248 | 5.3241 | 5.4377 | 0.0656 | 0.0259 |

2. Weibull-Exponential | 354.9783 | 362.9783 | 5.2892 | 5.4019 | 0.0654 | 0.0257 |

3. Weibull-Normal | 353.326 | 363.326 | 4.9906 | 5.0970 | 0.0374 | 0.0145 |

4. Weibull-Lognormal | 355.1653 | 365.1653 | 4.5138 | 4.6100 | 0.0780 | 0.0276 |

5. Normal-Exponential | 359.5963 | 367.5963 | 5.7191 | 5.8411 | 0.0777 | 0.0328 |

6. Normal-Lognormal | 355.4454 | 365.4454 | 4.5031 | 4.5991 | 0.0782 | 0.0276 |

7. Lognormal-Exponential | 355.3063 | 363.3063 | 4.5211 | 4.6175 | 0.0772 | 0.0275 |

Mixture Models | Parameters | Estimated Values |
---|---|---|

Weibull-Exponential | 2.6584 | |

3.4276 | ||

4.6373 | ||

0.9855 | ||

Normal-Lognormal | 0.2984 | |

0.1810 | ||

1.0514 | ||

0.4540 | ||

0.0256 | ||

Weibull-Normal | 6.6674 | |

4.4085 | ||

2.1679 | ||

0.8130 | ||

0.4236 |

From the previous figure, we see among the three mixture models the CDF based on Weibull-Normal mixture model belong very closely to the CDF based on the K-M estimate. Hence we may consider the Weibull-Normal model as the best fitted model for the data set.

Now, the value of Mean Time to Failure (MTTF) of Weibull-Exponential, Normal-Lognormal & Weibull- Normal Mixture models are given below in

The MTTF of the best fitted Weibull-Normal mixture model is 2.9919. And the MTTF value obtained from Weibull-Exponential & Nomal-Lognormal mixture model are 3.0054 & 3.0987, respectively. which are very close to the MTTF of the best fited Weibull-Normal mixture model.

In this article we have also estimated the B10 life, median and B90 life of the CDF based on Kaplan-Meier estimate, ML method and WPP plot for Windshield failure data and displayed the result in

From the above table, we may conclude that, 10% of the total components fail at time 1.432 for K-M procedure, at time 1.432 for MLE method and at time 1.262 for WPP plot method. 50% of the total components fail at time 2.934 for K-M procedure, at time 2.964 for MLE and at time 2.878 for WPP plot. 90% of the total components

Mixture Model | MTTF |
---|---|

Weibull-Exponential | 3.0054 |

Normal-Lognormal | 3.0987 |

Weibull-Normal | 2.9919 |

Estimates of | KM | ML | WPP |
---|---|---|---|

B10 life | 1.432 | 1.432 | 1.262 |

B50 life or median | 2.934 | 2.964 | 2.878 |

B90 life | 4.570 | 4.663 | 4.694 |

fail at time 4.570 for K-M procedure, at time 4.663 for MLE and at time 4.694 for WPP plot method. Hence we may say the WPP plot under estimate the B10 life and the median life and over estimate the B90 life compared with K-M and ML methods.

Maximum likelihood estimate procedure gives much better fit than Weibull probability paper (WPP) plot procedure. The Weibull-Exponential, Normal-Lognormal & Weibull-Normal mixture models fit well for the Windshield failure data, based on the value of AIC, AD, RMSE & Kolmogrov-Smirnov test statistic, respectively. While comparing the CDF values graphically, we found that the Weibull-Normal mixture model belongs very closely to the CDF based on the K-M estimate. The MTTF of the best fitted Weibull-Normal mixture model is 2.9919, which is very close to the other two mixture models. The WPP plot under estimate the B10 life, median life and over estimate the B90 life compared with K-M and ML methods for Windshield failure data.

This article analyzed the product failure data. However, the proposed methods and models are also applicable to analyze lifetime data available in the fields, such as, biostatistics, medical science, bio-informatics, etc. The article considered the first failure data of the product. If, there are repeat failures for any product, application of an approach of modeling repeated failures based on renewal function would be relevant. Finally, further investigation on the properties of the methods and models by simulation study would be useful.

Sabba Ruhi, (2015) Application of Mixture Models for Analyzing Reliability Data: A Case Study. Open Access Library Journal,02,1-8. doi: 10.4236/oalib.1101815