Reliability HotWire

Reliability HotWire

Issue 127, September 2011

Reliability Basics

Using RENO to Examine Interval Width

Sometimes when making a prediction of item life, the amount of usage of the item, such as the number of miles a car was driven or the number of prints a copier produced, is the variable of interest (rather than the time the product was in service). Typically, it is known when each item was put into service, so the time in the field for each unit can be calculated. If a failure occurs during the warranty period, the item is returned for repair or replacement and the usage of that item can be recorded. However, if the item does not fail, the amount of usage it has accumulated during its time in service is often unknown. The usage format warranty folio in Weibull++ has a built-in method of estimating the usage of non-failed items based on the time the items spent in the field and a user-specified usage distribution. With this feature, the user is asked to define an interval width that is used when calculating the usage values assigned to the suspended units. This article presents a simulation in RENO that illustrates the effect of changing the interval width on the way usage values are assigned to the suspended units in the Weibull++ usage format warranty folio.

Consider the following simple scenario. 100 cars were put into service on January 1, 2010. Of those 100 cars, one failed after 4,000 miles, one failed after 6,700 miles, and three failed after 12,500 miles. The remaining 95 cars did not fail as of the end of the observation period on January 1, 2011. We created a usage format warranty folio in Weibull++. We chose to use a Weibull distribution and the MLE analysis method, and entered 1/1/2011 in the Calculations End Date field on the Analysis tab. (Click here to download the Weibull++ file.) Figures 1 and 2 show the data we entered in the Sales and Returns sheets of the folio.

Sales Data
Figure 1 - Sales Data

Returns Data
Figure 2 - Returns Data

Next we entered the usage information into the Usage tab of the control panel. We chose to use a distribution and set a Usage Distribution Period of 1 year. We know that customer surveys have shown that the 50th percentile user drives about 10,000 miles per year, while the 95th percentile user drives about 25,000 miles per year. As shown in Figure 3, we used the Parameter Experimenter to calculate the parameters of the 2-parameter Weibull distribution corresponding to this information. Figure 4 shows the calculated parameters that we pasted into the appropriate fields on the Usage tab of the warranty folio.

Usage Distribution Parameters
Figure 3 - The Parameters of the Usage Distribution

Usage Tab of the Warranty Folio
Figure 4 - The Usage Tab of the Warranty Folio

The last field on the Usage tab is Interval Width. The number in this field corresponds to the granularity of the usage values assigned to the suspended units. In other words, an interval width of 1,000 used in conjunction with a usage distribution period of 1 year will allocate suspended units into bins of 1,000 miles per year. It is important to set an appropriate value for this field so that a histogram of the usage values of the suspended units replicates the shape of the user-specified usage distribution shown in Figure 5. If the chosen interval width is too small, such as 2 miles per year, the histogram of suspensions will approach a horizontal line, and therefore it will be very flat compared to the specified usage distribution. If the chosen interval width is too large, such as 20,000 miles per year, the histogram of suspensions will approach a delta function, and therefore it will be tall and narrow compared to the specified usage distribution.

Usage Distribution pdf plot
Figure 5 - Usage Distribution Defined by Weibull Parameters β = 1.5974 and η = 12579

We created a simulation in RENO in order to easily generate histograms of the usage values assigned to the suspended units for a particular interval width value, and compare them to the user-specified usage distribution. (Click here to download the RENO file.) Note that this is a simplified version of the algorithm used in the Weibull++ software, so it may not match the values in Weibull++ exactly for every combination of usage distribution and interval width. Figure 6 shows the RENO flowchart.

RENO flowchart
Figure 6 - RENO Flowchart for Creating Histogram of Suspended Units and pdf of Usage Distribution

In the present scenario, usage values must be assigned to the 95 units that are still operating without failure after 1 year in the field. The RENO flowchart accomplishes this in two steps. The first section of the flowchart (Loop A) builds a table containing the number of suspended units assigned to each bin of a histogram. The size of the bins of the histogram is defined by the constant "Interval_Width." This constant is changed for each simulation in order to investigate the effect of the interval width on how well the histogram matches the user-specified usage distribution. The second section of the flowchart (Loop B) both scales the histogram such that the total area of the histogram is equal to 1 and calculates the value of the pdf of the user-specified usage distribution at the upper edge of each bin of the histogram.

For example, consider the case of an Interval_Width of 1,000 miles per year. There are 95 units, so they are assigned to evenly spaced percentiles of the usage distribution. As shown in Table 1, the usage distribution is used to convert these percentiles to mileages, and the interval width is used to assign the mileages to bins. Table 2 show the Histogram_Data table at the end of Loop A.

Table 1 - Assignment of each unit to a Histogram Bin

Unit Number Usage Distribution Percentile Mileage from Usage Distribution Mileage from Interval Width Histogram Bin Number
1 0.005263157895 471.8902026 1,000 1
2 0.01578947368 941.817729 1,000 1
3 0.02631578947 1301.076204 2,000 2
4 0.03684210526 1611.572729 2,000 2
5 0.04736842105 1892.598368 2,000 2
... ... ... ... ...
91 0.9526315789 25281.51118 26,000 26
92 0.9631578947 26566.29567 27,000 27
93 0.9736842105 28230.52984 29,000 29
94 0.9842105263 30650.98823 31,000 31
95 0.9947368421 35507.24307 36,000 36

Table 2 - Histogram_Data Table at completion of Loop A

Row Number Number of Units in Bin
1 2
2 3
3 4
4 5
5 5
... ...
31 1
32 0
33 0
34 0
35 0
36 1

In order to compare the histogram of the suspended units and the pdf of the user-specified usage distribution, the histogram must be scaled appropriately. Each value in the histogram is normalized in order to make the total area of the histogram equal to the area under the pdf, (i.e., equal to 1). This is done by dividing each value in the histogram data table by the product of the interval width and the number of units. An array of these normalized values is stored in the "Histogram_Data" storage variable. In addition, the flowchart calculates the values of the pdf at the upper edge of each bin. An array of these values is stored in the "pdf_Data" storage variable. (Note that at the end of a simulation, the values in a table are reset to the original data, so the histogram data must be passed to a storage variable for plotting.) As shown in Figure 7, at the end of Loop B, the storage variables contain all of the data necessary to build a plot to compare the histogram and the pdf. The blue points represent the values in the Histogram_Data storage variable and the thick pink line represents the pdf of the user-specified usage distribution.

Plot of Interval Width of 1,000
Figure 7 - Comparison of Histogram and pdf for an Interval Width of 1,000

Figure 8 show the results of changing the Interval_Width constant to 1,500 and simulating the flowchart again. Figure 9 shows the plot for an Interval_Width of 2,000 and Figure 10 shows the plot for an Interval_Width of 2,500. (Note: These plots, along with plots for interval widths of 500 and 3,000, are in the "Interval Width Comparisons" spreadsheet in the RENO file.)

Plot of Interval Width of 1,500
Figure 8 - Comparison of Histogram and pdf for an Interval Width of 1,500

Plot of Interval Width of 2,000
Figure 9 - Comparison of Histogram and pdf for an Interval Width of 2,000

Plot of Interval Width of 2,500
Figure 10 - Comparison of Histogram and pdf for an Interval Width of 2,500

Increasing the interval width causes the histogram to track the pdf more closely, but at widths of 2,500 and greater, the beginnings of both the histogram and the pdf are truncated significantly. Therefore we choose to use an interval width of 2,000 in the Weibull++ warranty folio. Figure 11 shows the complete list of failure and suspension mileages, along with the calculated parameters for the warranty data set.

Mileages Extracted from Usage Format Warranty Folio
Figure 11 - Mileages Extracted from the Usage Format Warranty Folio

Conclusions

In this article, we used RENO to examine the Interval Width field of the usage format warranty folio in Weibull++. We saw that the choice of an interval width will have a large effect on whether the mileages assigned to suspended units will track the user-specified usage distribution well. Therefore it is important to carefully consider the choice of interval width when using the usage format warranty folio in Weibull++ to ensure that non-failed units are properly accounted for.