Reliability HotWire: eMagazine for the Reliability Professional
Reliability HotWire

Issue 73, March 2007

Hot Topics

Usage-Based Warranty Analysis

Warranty data analysis is a central activity in reliability analysis for manufacturing companies. It is one of the most important ways for companies to keep track of their products' behavior in the hands of customers and to perform reliability analysis and forecasts that are in line with the realities seen in the field. Warranty data analysis relies on the estimation of a failure distribution based on data including the age and number of returns and the age and number of surviving units in the field. When working in the time domain, this is relatively simple, as one has knowledge of the time a part failed or survived as of the analysis date. When the driving factor of reliability is usage rather than time, however, the analysis becomes more involved.

There are many applications in which failures are dependent upon usage, not time. For example, in the automotive industry, the failure behavior in the majority of the products is mileage-dependent rather than time-dependent. These kinds of products present a challenge for data analysis. For surviving units still working in the field, how do we know their usage (life) and how do we incorporate it into the estimation of the failure distribution?


Analysis Procedure

Suppose that you have been collecting sales (units in service) and returns data. For the returns data, you can determine the number of failures and their usage (by reading the odometer value, for example). Determining the number of surviving units (suspensions) and their ages is a straightforward step. By taking the difference between the analysis date and the date when a unit was put in service, you can determine the age of the surviving units.

 

What is unknown, however, is the exact usage accumulated by each surviving unit. The key part of the usage-based warranty analysis is the determination of the usage of the surviving units based on their age. Therefore, the analyst needs to have an idea about the usage of the product. This can be obtained, for example, from customer surveys or by designing the products to collect usage data. For example, in automotive applications, engineers often use 12,000 miles/year as an average usage. Based on this average, the usage of an item that has been in the field for 6 months and has not yet failed would be 6,000 miles. So to obtain the usage of a suspension based on an average usage, one could take the time of each suspension and multiply it by this average usage. As you see, in this situation the analysis becomes straightforward. With the usage values and the quantities of the returned units, a failure distribution can be constructed and subsequent warranty analysis (such as warranty return forecasts) becomes possible.

 

Alternatively, and more realistically, instead of using an average usage, an actual distribution that reflects the variation in usage and customer behavior can be used. This distribution describes the usage of a unit over a certain time period (e.g. 1 year, 1 month, etc). This probabilistic model can be used to estimate the usage for all surviving components in service and the percentage of users running the product at different usage rates. In the automotive example, for instance, such a distribution can be used to calculate the percentage of customers that drive 0-200 miles/month, 200-400 miles/month, etc. We can take these percentages and multiply them by the number of suspensions to find the number of items that have been accumulating usage values in these ranges.

 

To proceed with applying a usage distribution, the usage distribution is divided into increments based on a specified interval width denoted as Z. The usage distribution, Q, is divided into intervals of 0+Z, Z + Z, 2Z + Z, etc., or xi = xi-1 + Z, as shown in the next figure.

 

 

The interval width should be selected such that it creates segments that are large enough to contain adequate numbers of suspensions within the intervals.

 

The percentage of suspensions that belong to each usage interval is calculated as follows:

 

F(xi) = Q(xi)-Q(xi-1)

 

Where:

  • Q() is the usage distribution Cumulative Density Function, cdf.

  • x represents the intervals used in apportioning the suspended population.

We also define a suspension group as a collection of suspensions that have the same age.

 

The above percentage of suspensions can be translated to numbers of suspensions within each interval, xi. This is done by taking each group of suspensions and multiplying it by each F(xi), or:

 

N1,j = F(x1) NSj
N2,j = F(x2) NSj
.
.
.
Nn,j = F(xn) NSj

 

Where:

  • Nn,j is the number of suspensions that belong to each interval.

  • NSj is the jth group of suspensions from the data set.

This is repeated for all the groups of suspensions.

 

The age of the suspensions is calculated by subtracting the Date In-Service (DIS), the date at which the unit started operation, from the calculation End Date (ED). This is the Time In-Service (TIS) value that describes the age of the surviving unit.

 

TIS = ED DIS

 

Note: TIS is in the same time units as the period in which the usage distribution is defined.

 

For each Nk,j, the usage is calculated as:

 

Uk,j = xi TISj

 

After this step, the usage of each suspension group is estimated. The data can be combined with the failures and a failure distribution can be fitted.

Example Using Weibull++

Weibull++ (Version 7.1.4 and above) now offers a usage-based warranty feature in addition to its previously existing selection of warranty analysis data formats. This feature automates the analysis procedure described above, thereby facilitating a process that would be quite tedious otherwise, particularly when dealing with large warranty data sets.

 

Suppose that an automotive manufacturer collects warranty returns and sales data for a particular part. The sales data for one region, as entered in Weibull++, are shown in the next figure.

 

 

The returns data (accumulated mileage at time of failure and the date each failed unit was put in service, DIS) are shown in the next figure.

 

 

The above figure also shows the calculation End Date (ED), which can be set on the Analysis page of the Control Panel. In this example, the warranty data were collected until 12/1/2006.

 

The user has the option to input either a constant value that describes an average of usage or a distribution with intervals to be considered in the analysis. In this example, the manufacturer has been documenting the mileage accumulation per year for this type of product across the customer base in comparable regions for many years. The yearly usage has been determined to follow a lognormal distribution with μT' = 9.38, σT' = 0.085. Graphically, the distribution looks as follows.

 

 

 The usage distribution is specified on the Usage page of the Control Panel, as shown next.

 

 

Note that the Interval Width was defined to be 1000 miles (as seen in the above figure). This interval facilitates the calculations of mileage usage for the suspensions in the data set.

 

In addition to 14 failures, the data set contains 208 suspensions spread according to the defined usage distribution. This data set contains 12 groups of suspensions that have the same age. To illustrate the analysis procedure explained above, let us use one suspension group, the first group. The first group contains 9 units that went in service on a given date; 1 unit failed from that batch. Therefore, 8 suspensions have survived from December 2005 until the beginning of December 2006, a total of 12 months. The calculations are summarized as follows.

The two columns on the right constitute the calculated suspension data (number of suspensions and their usage) for the first group. The calculation in the above table is then repeated for each of the remaining 11 suspension groups in the data set. We then add the resulting data and the failure data and proceed with estimating a failure distribution model.

 

The data set that is used to perform the estimation of the failure distribution model can be extracted and displayed by selecting Transfer Life Data to New Folio from the Data menu. The failures and suspensions data set, as presented in the Standard Folio, is partially shown next. (Note that the Folio contains 66 rows.) 

 

 

The plot that follows shows the failures and suspensions contained in the above data set.

 

 

The entered sales and return data set, the usage distribution and the interval width in the Warranty Folio can be used to derive a failure distribution. The manufacturer assumes that the observed failures are typically modeled by a lognormal distribution. The parameters of the distribution are estimated to be:

 

μT' = 10.4747

σT' = 1.1159

 

The reliability plot (with mileage being the random variable driving reliability), including the 90% confidence bounds on reliability, is shown next.

 

 

The obtained model serves as the basis for all subsequent calculations. For example, the warranty returns forecasts for the next 10 months, starting from the analysis end date of December 2006, can be calculated by selecting Generate Forecasts from the Data menu. The next plot shows the estimated returns forecast.

 

 

Conclusion

This article presented a methodology for analyzing warranty data of products for which usage is the driving factor. With the help of Weibull++, the data analysis can be performed and useful information can be made available for important business and engineering decision-making.

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