Reliability HotWire

Reliability HotWire

Issue 125, July 2011

Reliability Basics

Truncation of the Integral Used in the Comparison Wizard in Weibull++

In the August 2010 issue of Hotwire, the Reliability Basics article discussed the integral used in the Weibull++ Comparison Wizard to calculate the probability that Design A will outlast Design B. This comparison is performed over the entire time domain, therefore the limits of integration are time = 0 and time = infinity. This article uses simulations in RENO to explain how changing the limits of integration affects the output of the Comparison Wizard in Weibull++.

One day when Joe the reliability engineer was using the Comparison Wizard in Weibull++ to help him decide between two designs for his upcoming project, he noticed that there is a place in the wizard to change the limits of the integration. "Hmm," Joe thought. "I wonder if changing the upper limit could help me to compare two designs whose lives have a maximum, say, in the case of a warranty."

Joe used his data sets for Design A and Design B from his previous work, as shown in Table 1.

Table 1 - Failure times for Designs A and B

Design A Design B
1059 4869
2245 5348
2781 5951
3130 6257
3535 6561
4512 6763
5632 6796
6218 7774
6819 7992
6891 8133

He recalled that the probability that Design B would outlast Design A was 80.7452% using the default limits of integration from 0 to infinity. Then he reviewed the MultiPlot of the reliability functions of the designs and noted that most of the components from both Design A and Design B have failed by 10,000 hours, as shown in Figure 1. (Click here to access Joe’s Weibull++ file.)

Figure 1
Figure 1 - Input data and calculated parameters for Design B

Based on this information, Joe decided to create a table of outputs of the Comparison Wizard using values of 1,000 to 10,000 for the warranty period.

Next, Joe opened the Comparison Wizard in Weibull++. On the Setup tab, he selected the Override auto-calculated limits check box to allow him to enter new limits of integration. He decided to set the number of quadratures (i.e., the number of intervals that the integral is divided into) to 1,000. Then he set the lower limit to 0 and the upper limit (i.e., warranty time) to 1,000. Finally, he chose to compare Design A to Design B (the data for these are in data sheets named "A" and "B" of the same folio). These settings are displayed next.

Figure 2
Figure 2 - Comparison Wizard settings

Joe returned to the Common Probability tab of the Comparison Wizard and clicked Compare to view the result, as shown in Figure 3. He copied the probability that A would outlast B (0.0002%) and pasted it into a table he created in a general spreadsheet.

Figure 3
Figure 3 - Comparison of Designs A and B from 0 to 1,000 hours

Next, he calculated the probability that Design B will outlast Design A. Then he entered the calculated probability (4.6180%) into the general spreadsheet. He repeated this procedure for multiples of 1,000 hours up to 10,000 hours. The results are shown in Table 2.

Table 2 - Comparison Wizard results for warranty times from 1,000 to 10,000 hours
Table 2

Joe was surprised to see that, for a given warranty time, the probability that Design A outlasts Design B and the probability that Design B outlasts Design A did not add up to 100%. "For a warranty time of 1,000 hours, what happens in the remaining 95% of the time that is not represented in the table?" he wondered. He did notice that as the warranty time gets closer to infinity, the sum of the probabilities in each row approaches 100%. In addition, the probability that Design B outlasts Design A for a warranty time of 10,000 hours was identical to the value he calculated using the default Comparison Wizard limits of integration. Joe looked at the equation that is used by the Comparison Wizard to compute the probability that Design B outlasts Design A, as shown below, and decided to come up with an hypothesis.

For a given value of time, t, the expression inside the integral represents the probability that Design A lasts exactly t and Design B lasts longer than t. The integral is the sum of those probabilities for all possible outcome values of life. When a finite upper limit is used, outcomes in which the life of Design A is greater than the upper limit are excluded from the calculation. Similarly, when computing the probability that Design A outlasts Design B, the outcomes in which the life of Design B is greater than the upper limit are excluded from the calculation. This observation prompted Joe to create a table of the possible combinations of the outcomes of the lives of Designs A and B, as shown in Table 3. (Note that there is also a very small probability that the lives of both designs will be less than the upper limit and equal to each other. This case is excluded from Table 3.) Based on his analysis of the above equation, Joe realizes that only the first column of Table 3 can contribute to the Comparison Wizard result for the probability that Design B outlasts Design A, and only the first row of the table can contribute to the Comparison Wizard result for the probability that Design A outlasts Design B. So the result of comparing Design A to Design B using the Comparison Wizard in Weibull++ is the sum of the italicized values in Table 3, and the result of comparing Design B to Design A using the Comparison Wizard is the sum of the underlined values. He hypothesized that the remaining outcomes—i.e., those in the bottom right cell of Table 3—account for the fact that the rows of Table 2 do not add up to 100%. He decided to test this hypothesis by building a simulation in RENO to examine the effect of an upper limit. (Click here to access his RENO file.)

Table 3 - Combinations of lives of Designs A and B using an upper limit
Table 3

The flowchart that Joe created for his RENO simulation follows these steps, as illustrated in Figure 4:

  1. Choose one random failure time using the Weibull parameters for each design. For any failure time that is greater than the warranty time, set that failure time equal to the warranty time.
  2. Subtract the failure time for Design B from Design A. Use this difference to determine which design lasts longer.
    1. If the difference is positive, Design A outlasts Design B. Add 1 to the value of the "A Outlasts B" storage block.
    2. If the difference is negative, Design B outlasts Design A. Add 1 to the value of the "B Outlasts A" storage block.
    3. If the difference is zero, one cannot tell the difference in the lives of Designs A and B.
  3. Subtract the original failure time for Design A (i.e., the failure time before adjusting for warranty time) from the warranty time. Use this difference to determine if both designs fail before or after the warranty time.
    1. If the difference is positive, Designs A and B fail at the same time (less than the warranty time). Add 1 to the value of the storage block labeled "A Equal to B and Less Than Warranty Time."
    2. If the difference is negative or zero, both Designs A and B last as long as or longer than the warranty time. Add 1 to the value of the storage block labeled "A and B Greater Than or Equal to Warranty Time."

Figure 4
Figure 4 - Comparison Wizard upper limit simulation flowchart

At the end of each simulation, the results in the storage blocks are divided by the number of simulations to obtain the probability of each outcome occurring. Joe used 100,000 simulations per run, and he used one run for each warranty time in Table 2. The results of his simulations are shown in Table 4, along with the results of Table 2 for comparison. Note that in all simulations for all warranty times considered, the life of Design A was never both equal to the life of Design B and less than the warranty time.

Table 4 - Simulation results for warranty times from 1,000 to 10,000 hours (shown with Comparison Wizard results)
Table 4

Joe was pleased to confirm that his hunch was correct, as seen by comparing the simulation and Comparison Wizard results in Table 4. In addition, he realized that the probability that both Designs A and B outlast the warranty time is the product of the probabilities that each design lasts longer than the warranty time. Since the probability that an item lasts longer than a specific time is the reliability, R, of that item at the specified time, the probability, P, that both A and B outlast the warranty time is given by:

Joe computed these values for each warranty time using the Function Wizard in the general spreadsheet that he had created in Weibull++ and added them to the information above as shown in Table 5. He confirmed that the sum of the results of the two Comparison Wizard calculations and the probability that both designs outlast the warranty time computed from the above equation is 100% for each row of the table.

Table 5 - Simulation and Comparison Wizard results for warranty times from 1,000 to 10,000 hours, including probabilities that both designs last longer than the warranty time
Table 5

Joe was excited to see his results confirmed his intuition about the use of the upper limit in the Comparison Wizard. He decided to repeat the process for the lower limit. Joe imagined that one might use the lower limit to examine the effect of burn-in on the lives of two designs.

First, Joe made a table of possible combinations of the outcomes of the lives of Designs A and B when using a lower limit, as shown in Table 6. Again, he thought about the equation that the Comparison Wizard employs. Using a lower limit excludes the outcomes in which the life of Design A is less than the lower limit from the calculation of the probability that Design B outlasts Design A, and it excludes the outcomes in which the life of Design B is less than the lower limit from the calculation of the probability that Design A outlasts Design B. So the first column of the table is excluded from the calculation of the probability that Design B outlasts Design A, and the first row of the table is excluded from the calculation of the probability that Design A outlasts Design B. Therefore, only the outcomes in the lower right cell of Table 6 are used to compute the probability that Design A outlasts Design B (italicized value) or that Design B outlasts Design A (underlined value). Any outcomes in which at least one design fails before the burn-in time are excluded from the results of the Comparison Wizard calculations.

Joe knew that the probability that at least one of the designs would fail before the burn-in time is the complement of the probability that both designs last at least as long as the burn-in time. This probability is given by:

Joe computed these values for each burn-in time using the Function Wizard in the general spreadsheet that he had created in Weibull++.

Table 6 - Combinations of the lives of Designs A and B using a lower limit
Table 6

Next, Joe created his simulation in RENO. The flowchart he created to examine the effect of a lower limit follows these steps, as illustrated in Figure 5:

  1. Choose one random failure time using the Weibull parameters for each design. For each design, compare the failure time to the burn-in time and, if either failure time is less than the burn-in time, add 1 to the value of the "A and/or B Does Not Survive Burn In" storage block.
  2. Subtract the failure time for Design B from Design A. Use this difference to determine which design lasts longer.
    1. If the difference is positive, Design A outlasts Design B. Add 1 to the value of the "A Outlasts B" storage block.
    2. If the difference is negative, Design B outlasts Design A. Add 1 to the value of the "B Outlasts A" storage block.
    3. If the difference is zero, Designs A and B fail at the same time. Add 1 to the value of the storage block labeled "A Equal to B and Greater Than Burn In Time."

Table 5
Figure 5 - Comparison Wizard lower limit simulation flowchart

Finally, Joe constructed a table of results from the simulation and the Comparison Wizard, as shown in Table 7. The agreement between the results using the two methods gives Joe confidence that now he has a firm understanding of the use of the upper and lower limits in the Comparison Wizard tool in Weibull++.

Table 7 - Simulation and Comparison Wizard results for burn-in times from 1,000 to 10,000 hours, including probabilities that at least one design fails before the burn in time
Table 7

Conclusion

In this article, we discussed the use of the upper and lower limits in the Comparison Wizard in Weibull++ and illustrated the use of the limits by performing simulations in RENO. We saw that using limits causes the Comparison Wizard to exclude from the results some of the possible outcomes for whether Design A will outlast Design B or Design B will outlast Design A. Specifically, when using an upper limit, the percentage of time where both Design A and Design B are greater than the upper limit is excluded from the calculations. When using a lower limit, the percentage of time that either Design A or Design B is less than the lower limit is excluded from the calculations.