Optimal Test Plan for Two Stresses
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This example compares the results from the statistically optimum test plan in the ALTA test plan tool for two stresses.
Reference Case
The data set is from Example 20.8 on page 549 in the book Statistical Methods for Reliability Data by Dr. Meeker and Dr. Escobar, John Wiley & Sons, 1998.
Data
A Weibull distribution with the following life stress relationship is used.
- [math]\displaystyle{ ln(\eta)=\beta_{0}+\beta_{1} \times log(vpm) + \beta_{2} \frac{11605}{temp}\,\! }[/math]
where temp is the temperature in K and vpm is the voltage stress in volts/mm. The planning values for the model parameters used in the test plan are:
- [math]\displaystyle{ \beta = \frac{1}{\sigma} = 1.485\,\! }[/math] , [math]\displaystyle{ \beta_{1} = 12.28\,\! }[/math] and [math]\displaystyle{ \beta_{2} = 0.3878\,\! }[/math]
These planning values are given on page 535.
The use condition stress values are vpm = 80 volts/mm and temp = 120°C (393.15°K).
The highest stress values that can be used in the test are vpm = 200 volts/mm and temp = 260°C (533.15°K).
A total of 170 units are available for testing for 1,000 hours. Three combinations of temperature and voltage levels can be used in the test. The objective of the test plan is to:
- Determine the three stress combinations used in the test.
- Determine the number of test units at each stress combination.
Tests will be conducted using the created test plan. The failure data from the test will be used to estimate the model parameters. The estimated parameters are then used to predict the B10 life at vpm = 80 volts/mm and temp = 120°C.
We need to find a test plan that will minimize the estimation variance of the B10 life at the given usage stress level.
Result
- The three stress combinations and the number of test units at each of them are:
- 62 units will be tested at voltage of 124 and temperature of 260°C (533.15°K)
- 42 units will be tested at voltage of 159 and temperature of 120°C (393.15°K)
- 66 units will be tested at voltage of 200 and temperature of 260°C (533.15°K)
- The estimated standard error for the B10 life at vpm = 80 volts/mm and temp = 120°C from this test plan is [math]\displaystyle{ Ase \left[log(\hat{t}_{0.1}(50))\right]\,\! }[/math] = 0.3670 .
The standard error is calculated from the result in Table 20.6. In Table 20.6, the value for [math]\displaystyle{ \frac{n}{\sigma^{2}}Var\left[log(\hat{t}_{0.1}(50)) \right]\,\! }[/math] = 50.5. Since n = 170 and [math]\displaystyle{ \frac{1}{\sigma^{2}} = \beta^{2}\,\! }[/math] = 2.205229, so [math]\displaystyle{ Var\left [log(\hat{t}_{0.1}(50)) \right]\,\! }[/math] = 0.1347. Therefore, [math]\displaystyle{ Ase \left[log(\hat{t}_{0.1}(50)) \right]\,\! }[/math] = 0.3670.
Results in ALTA
The planning information is entered in ALTA as shown below.