Quarter 1, 2002:  Volume 3, Issue 1

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Predicting Warranty Returns Based on Customer Usage Data

In recent years, there has been a lot of research in understanding how the customer is using the product. Many products are designed with built-in devices (such as e-prom chips) that collect data as to how the product is used, the stresses it experiences, the environmental conditions under which it operates, etc. In addition, many companies conduct surveys in order to obtain customer usage information. Despite the significant investment into these efforts, much of this customer usage information is being underutilized, or is not being used at all, because the proper analysis method is unknown. 

Clearly, the stress conditions under which a product operates will have a direct effect on its life and reliability. At the same time, the stress conditions depend on the way the product is used and not every customer uses the product in the same way. Certain customers operate the product at higher stress levels than others. For example, every user does not accumulate 12,000 miles a year on a vehicle and every user does not print the same number of pages per week on a printer. In addition, products operate at different geographical locations where the environmental conditions are different. 

This article presents a methodology for utilizing customer usage data in order to perform reliability predictions. The approach presented here is based on a combination of quantitative accelerated life testing (QALT) data analysis methodologies and stress-strength interference analysis, a method used in probabilistic design. An example that takes into account customer usage profiles for washing machines is provided. Two software packages, ReliaSoft’s ALTA 6 and Weibull++ 6, were used together to perform the analysis.

Background Theory
Given a distribution that describes the stress levels a unit will experience during its use and a distribution that describes the strength of this unit (e.g. material strength, material properties, design properties, etc.), the analyst can determine the probability of failure for this unit based on the probability of the stress exceeding strength, or:

The relationship of stress, strength and region of failure is shown graphically in Figure 1. This plot displays the probability density functions (pdfs) for the stress distribution and the strength distribution. A failure occurs when the stress exceeds the strength. 

Figure 1: Stress distribution, strength distribution and region of failure

Using this idea, the customer usage information represents the data set required to obtain the stress distribution in the above equation. This distribution describes the percentage of users who operate the product at each stress level. Therefore, if we can obtain the strength distribution, we will be able to use Eqn (1) to calculate the overall probability of failure of the product, while taking into account the variation in usage patterns among customers. In order to determine the strength distribution, we will need a way to relate life with stress and to estimate the percentage of units failing at each stress level. Utilizing quantitative accelerated life tests and QALT analysis methods, we can obtain this life-stress relationship and then obtain the probability of failure at different stress levels for a given time. 

The idea is that if we know the fraction of users operating the product at a given stress level and the percentage of units failing at that stress level (for a given time), we can determine the percentage of the population failing at that stress level. For example, consider a population of 200 units. From the stress distribution (usage distribution), we have determined that 10% of these units operate at a stress level S. Now we would like to know the probability of failure at S. This probability has to be associated with a time as well. If the probability of failure is 30% at time t and stress level S, then we know that 30% of the 10% of the units that operate at stress level S will fail by time t, or 6 out of the 200 total units will fail at stress level S. This will need to be repeated for all stress levels that are experienced in the field in order to estimate the overall percentage of units failing by time t. Given the stress distribution and the strength distribution, this metric can be obtained from Eqn (1). 

The following example will be used to demonstrate the procedure to perform this type of analysis. 

Example for Washing Machine Usage
Consider an electric motor that has been designed for a washing machine. The actual load sizes for the motors in the field will vary depending on the way that the individual user employs each machine. The manufacturer provides a warranty of 1000 cycles for this motor. In order to make adequate preparations to support the warranty, the manufacturer wants to estimate the percentage of returns that can be expected during the warranty period. The life of the motor is clearly dependent on the applied load and the applied load varies based on customer usage patterns. So the question to be answered is which load size should be used in predicting the percentage of returns during warranty. 

As a first step, the manufacturer decided to obtain information on the life of the motor at different loads, and the data set in Table 1 was collected. This data set contains cycles-to-failure information at three different loads (or stress levels) of 6 lbs, 8 lbs and 12 lbs. Five motors were suspended at 6500 cycles at the 6 lb load. One motor was suspended at 6500 cycles at the 8 lb load. 

Table 1: Test data for various load sizes.

This data set can be analyzed using accelerated life models. In this case, the Weibull-Inverse Power model was fitted to the data set, using the ALTA 6 software. Figure 2 displays the Weibull probability plot obtained from the analysis. The plot contains a separate line to represent the probability over time for each load size (or stress level). 

Figure 2: Weibull probability plot for three load sizes

Traditionally, the analyst would use an average usage level in order to predict the probability of failure for the product within the warranty period. For this motor, it was assumed that 7 lbs is the average use level stress and that 1.39% of the motors will fail within the 1,000 cycle warranty period. Figure 3 displays the probability of failure result in ALTA’s QCP, calculated based on the analysis for an average use stress level of 7 lbs. 

Figure 3: Probability of failure in ALTA's QCP, calculated for an average use stress level of 7 lbs

However, because customer usage information is also available for this motor, the analyst can make more realistic estimates for the probability of failure for the units in the field. The customer usage information was obtained by conducting a survey on a representative sample of customers and recording the sizes of the loads that they placed into their washing machines. From this data set, the distribution that gives the percentage of units at different loads can be determined. Using the Weibull++ software, the three parameter Weibull distribution was fitted to the data set and the following parameters were obtained: beta = 1.78714, eta = 7.14339, gamma = 1.36493. Although the data set is too large to be included in this article, Figure 4 displays the pdf for this distribution, based on the customer usage information.

Figure 4: Pdf from the customer usage data

The question now is how to relate the load usage distribution to the life of the motor at different loads. For this, we will go back to the life-stress model and obtain the probability of failure at 1000 cycles for different load sizes. Using the analysis from ALTA 6, the percentages failing at 1000 cycles for five different load sizes were obtained and the results are shown in Table 2.

Table 2: Percentage failing at 1000 cycles for various load sizes.

Using this data set, we can obtain the distribution of the percentage of units failing during the warranty period of 1000 cycles at each load size. This can be performed using the “Free Form” data type in Weibull++. The Weibull distribution was fitted and the parameters were obtained: beta = 3.57032 and eta = 23.12145. This is our strength distribution. 

Now we have two distributions, one giving the percentage of units operating at each load size and the second giving the percentage of units that fail at each load size during the warranty period of 1000 cycles. The stress-strength interference analysis can now be used to obtain the percentage failing during warranty from the whole range of load sizes applied in the field. Figure 5 displays the stress distribution compared with the strength distribution and Figure 6 displays the results in the Weibull++ Stress-Strength Wizard. 

Figure 5: Comparison of stress and strength distributions

Figure 6: Weibull++ stress-strength comparison results

Based on this analysis, it is estimated that 4.16% of the units will fail under warranty. This value is significantly higher than the 1.39% obtained by assuming an average load size of 7 lbs. 

Conclusion
This article describes a method for incorporating a type of customer usage profile in reliability predictions. This allows the organization to maximize the effectiveness of the customer usage information that it collects and to make better decisions about warranty limits and services. As the example illustrates, when the customer usage profile is taken into account, the reliability analysis results may be different than when an “average usage” estimate is used. In some cases, this can result in more realistic estimates as to the behavior that can be expected from the products in the field because the analysis more accurately reflects the conditions in the field. Using a combination of QALT and stress-strength interference analysis methods, such predictions are possible. Analysts may wish to investigate this or other options for incorporating a customer usage profile into their reliability analyses. The appropriate analysis method will depend on the data available and on the particular application.

[Editorial Note: In the printed edition of Volume 3, Issue 1, the x-axis labels in Figures 4 and 5 were incorrect. The correct label -- "Load (lbs)" -- is displayed in this on-line version. We apologize for any inconvenience.]