1 An Application of Weibull Analysis to Determine Failure Rates in Automotive Components Jingshu Wu, PhD, PE, Stephen McHenry, Jeffrey Quandt National Highway Traffic Safety Administration (NHTSA) U.S. Department of Transportation Paper No Abstract This paper focuses on the automotive component failure rate detection using Weibull analysis and other survival analysis techniques. Detailed attention is paid to three areas: 1) overall failure rates are described statistically first, and data cleaning and definitions of failed and censored data within the research time or warranty period are made; 2) Kaplan-Meier life curves and Log-rank tests are used to compare the component reliability over time and explore risks factor related to the component failure; 3) Weibull regressions, with two and three parameters, are applied to fit real-world reliability data from different test conditions, and to predict the automotive component failure trend over future time. The analysis results agree well with real-world test data, and provide reasonable prediction of future failure trends. 2. Introduction Automotive components fail over time, especially beyond the warranty time period. It is important to study the reasons why automotive components fail, and to predict the component reliability trends, associated with various manufacturing, environmental and testing conditions. For effective use of a predictive failure analysis of automotive components, all of the failures within a specific driving time period need to be captured. Because any repairs to failed components are free to the owner and paid by the manufacturer during the manufacturer s basic warranty coverage period (usually 36 s or 36,000 miles), most repair records are kept well. These are available to investigators and provide information on what components have been repaired or replaced at specific mileages and time in service. However, as soon as the basic warranty (free repair) period is over, and owners are required to pay for the repairs, there is no central repository to collect comprehensive information on failures due to data availability, which may lead to limited or incomplete analysis beyond the relatively short basic warranty window. In order to overcome this obstacle and to capture more insightful failure information within a relatively longer period, a data analysis technique was developed to utilize the manufacturer s extended warranty data to create a richer and more representative sample, from which the data analysis could be efficiently utilized, although it is not easy to completely avoid data bias when the extended warranty programs are slected. Most manufacturers will offer an extended service plan to an owner that provides coverage for a specified period beyond the basic warranty. For example the extended warranty plan may be for 5 years from purchase and 60,000 miles, or 5 years/75,000 miles, or 5 years/100,000 miles, or for 4 years/75,000 miles, etc. and that there are typically extended service plans offering different levels of coverage on different components (e.g., Gold Plan, Platinum Plan, Powertrain Plan). The driver may choose a plan that best fits his/her need. The data technique developed here requires that 1) all vehicles sold with all extended service plans be identified, 2) the plan(s) that cover repairs on the subject component be identified, 3) that a calculation is made to determine average miles per driven for the subject vehicles using records of warranty repairs, which provide time in service and mileage at the time of repair. All detailed records of repairs associated with time and mileages are then used for data analysis. Wu 1
2 For example, one powertrain component under consideration has a population sample size of over 200,000, the component failures starts around 30,000 miles, with extended service plans for 60 s, or 100,000 miles (warranty period or agreements), whichever comes first, the data set is termed as Old K60 data of type-1 in this study. Second type of test data of this same component may have a different failure mode or with a mix of failure modes. For this type-2 test, there are three new data sets of interest first new data set has the capture information of 60 s and 100,000 miles, i.e., the maximum to failure is 60 s, and maximum mileage of 100,000 miles ( New K60 data); Similarly, 2nd new data set has the information of 48 /100,000 mile capture, i.e., the maximum to failure is 48, and with maximum mileage of 100,000 ( New K48 ); The third contains the info of 72 /100,000 mile capture ( New K72 ). In total, there are four sets of data for type 1 and type 2 tests of this same component, which are summarized in Table 1. Most reliability test data are closely time-related, and failures happened within the warranty time window or beyond, a technique of time-to-event or survival analysis is very suitable for such reliability data, especially Weibull model is a well established tool to fit the test data and to predict the future failure trend beyond the available test duration. The main goals of this project are to apply a Weibull model to the automotive component reliability analysis, and then explore the failure rate over service time or mileage, and to provide some statistically-based insights into the component failures by the following process: Provide descriptive summary of available failure data of a component; Verify the reliability difference over time of the component under two different conditions; Explore the component failure probability over test time, and compare the failure rates of the same component from a few different data sets; Improve the data fitting and prediction by using a three-parameter Weibull model, and compare with real-world test results. 3. Descriptive Summary of Data Sample The focused automotive component test data, with service plan for 60 s, or 100,000 miles ( Old K60 of Type-1), can be divided into two sub-groups: The first group is failed event data (234 failures as following Table 1); The second group, Suspension or Censored group, has 4483 observations and has no failures within 60 s, further, their future failure behavior beyond 60 s are unknown. The service time of the censored group, at least, all passed the Cut-Off line of 60 s. Figure 1 on next page is helpful to explain the failures within and beyond a warranty time, or research time window. The similar descriptive summaries of three new data sets (New K60, New K48, and New K72) are also listed in following Table 1, with the failure data descriptive summary. Table1: Descriptive Statistics for Failed Components mean Std Min Max failure/ all % Dev. Type-1: Old K60 Data (234 failures, 4483 censors) Month % Type-2: New K60 Data (280 failures & 4474 censors) Month % Mile 70,597 17,326 27, ,288 Type-2: New K72 Data (164 failures & 2282 censors) Month % Mile 60,097 17,513 21,290 98,958 Type-2: New K48 Data (54 failures and 871 censors) Month % Mile 69,798 20,370 28,888 97,687 The ratio of failures /all % (all=failures + censors) in the last column in Table 1 is an important parameter that gives the overall failure rate at the end of service time, and will be used later to compare with the failure rate predicted by a Weibull model at the same time. For example, the failure rate at the time of 60 s is 5%, for data Old K60, or 5.89% for New K60 data. 4. Methods of Modeling Survival and Failure Rates It is of great interest to observe the component failure rate, F(t), or from an opposite point of view, the survival probability varying over a test time, S(t), while there is a simply relationship between the two: S(t)= 1 F(t), i.e., 20% failures means 80% survival rate among a fixed sample. One of the most useful tools to compare the survival probability over time is a method proposed by Wu 2
3 Kaplan and Meier 1. The Kaplan-Meier survival curve is described by the following formula: d i s i S ˆ ( t ) = (1 ) = ( ) t i <= t n i t i <= t n i (1) Where d i is deceased subject or failed automotive component, and s i is the survivor or alive component, and n i is the total (both failed and suspension components) in the study at any moment beyond time zero. Or, turning the problem around, the failure probability over test time, F(t), equal to 1-S(t), it can be further expressed by following Eq. (2) in Weibull model 2 : β ( t / η ) F ( t ) = 1 e (2) Or, equivalently it can be visualized by the following linear transformation, as Eq. (3): 2 log( log( S ( t)) = β log( t) β log( η ) (3) 5. Case Studies of Modeling Survival and Failure Rates Case 1: Compare Survival Rates Over Time of Two different Conditions It is often required to investigate automotive component reliability or survival rates over time under two different conditions, either manufactured during different time periods, or, by two different designs, or being used under different environmental conditions. Here is one example of discussing failed event and censor within the research time window and beyond there are 234 failures from data set of Old K60 within the study time of 60 s, however, the Suspension or Censored 4483 observations have service time beyond the Cut-Off line of 60 s, and their future statuses beyond 60 s are unknown (see Figure 1). In the above Eq.(3), S(t) is survival function, which can be estimated from the Kaplan-Meier curve discussed earlier, and F(t) of Eq. (2) is the accumulation of failure probability as time increases, here β is regarded as the Slope of the linear plot, or Shape parameter, and η is a Scale parameter and is related to the intercept of the linear plot. When a plot of test data is not visualized as a linear plot as Eq. (3), especially at the earlier time stage, a Weibuull with three parameters, as Eq. (4), provides a better data fitting, where a time shift, or threshold, t 0, is included as Eq. (4) - β (( t t0 ) /η ) F ( t) = 1 e (4) Three cases are studies in details using above Eq. (1) to (4), from some investigation data and examples, as shown as following Sections 5, where Case1 compares the reliability curves over time of two different conditions; and Cases 2-3 applies Weibull models, with two or three parameters, for data fitting and future trend predictions. Computing procedures by SAS Institute, LifeTest, LifeReg, and Reliability, are used for calculations 3. Some extra attentions should be paid to - how to clean the time data using SAS program; to determine the test interval from start to end; to convert the time data from calendar time (with different starting time each) to study time where all subjects have the same starting time; and to determine the relative status of censor over time. 1,3 Figure 1: Research Time Window, censor & failures In this project, the research time frame is 0-60 s (warranty duration or other agreement), some components have test time less than 60 s without failures, and they are treated as censored data; all suspension data are still alive without failures beyond 60 s ( Censors, or 0 ), and their service time are all assumed to be at least 60 s. On the other hand, all failed data (234 failures, coded as 1 ) have specific time of failures for each, prior to 60 s (Figure 1). Wu 3
4 The following Kaplan-Meier life curves (Figure 2), using Old K60 failed data only, are studied in detail to verify if the reliability of products built during Period A might be different from the reliability of products built at other time. Figure 2 indicates that no significant difference of reliability is observed between the products built during Period-A (orange color) and other dates ( blue or 0 ). The log-rank test, which compares two survival rate curves over time, provides a p- value of 21% for the two Kaplan-Meier plots of Figure 2, where X-axis has a unit of and Y-axis displays the probability from zero to 1.0. Condition_A Condition_B miles Figure 3: Survival Plots of Two Groups with two Using Conditions, p-value< Period_A One useful feature of the Kaplan-Meier life curve is to permit comparison of the survival rates over time between two different conditions, and evaluate the effect of one single factor (such as manufacturing time as Figure 2, or using condition as Figure 3) on reliability, if more risk factors are considered simultaneously, the Cox proportional hazard model is a better tool to evaluate multiple risk factors 3. Case 2: Weibull Modeling with Linear Plot using Different Data Sets Figure 2: Survival Plots of Component built during Period_A (Orange Color) vs. Other Dates. On the other hand, a significant difference (p-value <0.0001) of reliability is observed (Figure 3), from the failed data of another automotive component whose mileage (X-axis) information for each is available, under two different using conditions (1673 failures under Condition-A, and 531failures under Condition-B). One specific automotive component can be tested under different conditions, and the different tests (such as load or temperature) may lead to different failure modes. A linear Weibull model, based on Eq. (2) (3), is used first to plot the failure rates over time. For various tests of this same component, the similar approach of treating failed events ( 1 ) and censor data ( 0 ), as Figure 1 of Case 1, is applied to all data sets: Old K60, New K60, New K72 and New K48. Figure 4 shows the linear plots of Weibull modeling of failure probability from data set of Old K60, where X-axis has a unit of and Y-axis displays the probability from zero to 1.0. Wu 4
5 Table 3: Weibull Parameters and Failure Prediction (One Old & Three New Data Samples) Old New New New parameter 60K 60K 72K 48K β (shape) η (scale) % fail. 60 mo 56 Mo 59 Mo 45 mo 10% fail 76 mo 73 mo 88 mo 62 mo Figure 4: Weibull Plot with Data of Old K60, where 234 are failures and 4483 are suspension Figure 5 provides the Weibull modeling results using two data sets of Old K60 and New K60 (with the same warranty program) as one example. The predictions from the two different data sets agree well, and indicates that the same component fails at a similar rate although under two slightly different test modes and conditions In Figure 4, the failure rate will reach approximately 5 percent afters 60 s, and this prediction agrees very well with the test result as Table 1 (the last column and 3 rd row). On the other hand, the linear Weibull model with only two parameters of β ( Slope or Shape ) and η ( Scale ) does not fit the earlier time data very well (under 30 s), more discussions will be given later in Case 3, where the non-linear plot under 30 s will be described further Old_60K_Data Mean Failure_60K The data fitting results from using all data ( failed and censored data) and from failed data only are listed in Table 2. Table 2: Weibull Parameters and Failure Prediction (Two Samples out of Old K60 Data) Parameter Failed, (1) and Failed data Suspension, (0) ( 1 ) Only β (slope/shape) η (scale) % failure 60 s (*) 23 s When a few test data sets of the same component are available, it is of interest to compare the failure rates from using old data (Old K60), together with the results using new data (New K60, New 72K, and New 48K), as shown in Table 3. The failure rate from the shortest warranty program (48 s) is slightly faster, which reflects the reality of components in use Figure 5: Compare Mean Failures (Y-axis) vs. Month (X-axis) from Old Data (Old K60) vs. New Data (New K60). Case 3: Improving Data Fitting and Prediction by Considering the Time Shift The Weibull linear plot with only two parameters, as Figure 4, is not fitting data well enough, especially at the earlier time stage (time <30 s). A Weibull model with three parameters, shown as Eq. (4), provides a better fit. The data fitting parameters displayed in Figure 6 are for the Old K60 data. Wu 5
6 6. Conclusion Modeling of automotive component reliability is a data mining and learning process at NHTSA. From the simple statistical description, to estimation of a reliability curve over time, to a proper mathematical model to fit the test data and to predict the future component failure trend. Figure 6: Failure (Y-axis) vs. Month (X-axis) using Eq (4), Old 60K data Furthermore, the three parameters of Figure 6 (Slope β=1.728, Scale η=232.82, and threshold or time-shift t 0 =18.529) are used in Eq. (4), and provides the future trend prediction as Figure Failure Rate Prediction Month Rate % % % % % % Month Figure 7: Mean Failure Prediction vs. Month using Results of Figure 6, by Eq(4), with Slope (1.728), Scale ( ), and time-shift (18.529), Old 60K data. Employing the Kaplan-Meier life curve permits us to compare the component reliability over time between two different conditions, and to evaluate the effect of one single factor with statistical reliability. A Weibull model with two parameters (slope, β, and scale, η) can reasonably predict the mean failure with a linear model, while a Weibull model with three parameters can treat some nonlinearity at earlier time stage much better. For example, about 5 percent of products fail when service time has reached 60 s, and this prediction agrees well with the known test. The modeling results from using a few different test data sets from a same component provide meaningful comparisons, and such comparisons permit insights into component failure modes under different manufacturing regimes. 7. References 1. Kaplan, E. L. and Meier, P. (1958), Nonparametric Estimation from Incomplete Observations, J. of the American Statistical Association, 261, Abernethy, Robert, The New Weibull Handbook, (3 rd Edition), Allison, P. (1999), Survival Analysis using SAS: A Practical Guide, SAS Publishing. A similar Weibull curve as Figure 6 but using mileage for the X-axis is also plotted from a data set New K60, and the plot of failure against mileage can be more intuitive and realistic sometimes than the similar curve of failure against time (a car can be kept in a garage for a long time). Wu 6
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