Volume 5, Issue 1

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Optimal Allocations of Stress Levels and Test Units in Accelerated Testing

[Editor's Note: In the online version of this article, we have corrected one equation used in the printed edition of Volume 5, Issue 1. The equation for the asymptotic variance of the ML estimate of Yp is given here in its entirety. In addition, we have modified the article to reflect the fact that the accelerated test design utility will be available in the upcoming release of ALTA 7, rather than ALTA 6 PRO.]

Before launching a new product, the manufacturer is always faced with decisions regarding the optimum method to estimate the reliability of the product or service. Accelerated testing (with accelerated time or accelerated stress) might be the recommended or required approach. Conducting a quantitative accelerated life test (QALT) requires the determination or development of an appropriate life-stress relationship model. Moreover, a test plan needs to be developed to obtain appropriate and sufficient information in order to accurately estimate reliability performance at operating conditions, significantly reduce test times and costs and achieve other objectives. Nelson (1990), Meeker and Escobar (1998) and Nelson (2003) provide a substantial review of the literature on how to develop optimum QALT plans. Such plans are becoming very popular and are starting to be used in engineering, materials science and manufacturing industries. However, the rate of increase of this popularity has been slow due to the limited tools available at this time for designing optimum accelerated life testing plans.

With the upcoming release of Version 7, ReliaSoft's ALTA will be the only software package capable of analyzing, modeling, planning and evaluating a quantitative accelerated life test. In this article, we will use ALTA's new accelerated life test planning module to investigate the procedure for designing QALT plans and apply the techniques to develop an example test plan for MOS capacitors.

Development of Accelerated Life Testing Plans
A detailed test plan is usually designed before conducting an accelerated life test. The plan requires the determination of the type of stress, method of applying stress, stress levels, number of units to be tested at each stress level and an applicable accelerated life testing model that relates the failure times at accelerated conditions to those at normal conditions.

General Assumptions
Most accelerated life testing plans use the following model and testing assumptions that correspond to many practical QALT problems.

1. The log-time-to-failure for each unit follows a location-scale distribution such that:

Equation

where μ and σ are the location and scale parameters respectively and Φ(·) is the standard form of the location-scale distribution.

2. Failure times for all test units, at all stress levels, are statistically independent. Without loss of generality, we follow the Nelson (1990) and Meeker and Escobar (1998) standardization of stress S by defining x = (S - SU)/(SH - SU), where SU is the design stress and SH is the highest test stress. S, SU and SH are the transformed stresses according to different life-stress relationships (linear, inverse power law and Arrhenius).

3. The location parameter μ is a linear function of stresses. Specifically, we assume that:

Equation

4. The scale parameter σ does not depend on the stress levels. All units are tested until η, a pre-specified test time.

5. Two of the most common models used in QALT are the linear Weibull and lognormal models. The Weibull model is given by:

Equation

where SEV denotes the smallest extreme value distribution. The lognormal model is given by:

Equation

That is, log life Y is assumed to have either an SEV or a normal distribution with location parameter μ(z), expressed as a linear function of z and constant scale parameter σ.

Problem Formulation
The appropriate criteria for choosing a test plan depend on the purpose of the experiment. For censored data, most references minimize the asymptotic variance of the maximum likelihood (ML) estimate of a (log) percentile of the life distribution at the design stress. In ALTA, we use this optimization criterion. Under the constraints of available test units, test time and failure distribution at each stress level, the problem is to optimally allocate stress levels and test units so that asymptotic variance of the ML estimate of a (log) percentile of the life distribution at the design stress is minimized. The optimal decision variables (x1*,...,x2*,p1*,...,p3*) are chosen by solving the following optimization problem with a nonlinear objective function and both linear and nonlinear constraints.

Min:

Equation

Subject to:

Equations

The ML estimate of the p quantile Yp at the normal stress Xo is:

Equation

where zp is the p percentile of the underlying standardized distribution. For SEV (Weibull), we have [zp = log[-log(1-p)] and for Normal (lognormal), zp is equal to the standard normal p percentile, Φnor-1(p). Thus, the asymptotic variance of the ML estimate of Yp is defined by:

Where:

Equations

ALTA Test Planning Module
Figures 1 and 2 show ALTA's new accelerated life test planning module (note that early BETA development screen designs are subject to change before release). The software will provide five types of test plans that can be determined for a single accelerating stress:

  • The 2 Level Statistically Optimum Plan minimizes the variance of the estimate of the percentile.
  • The 3 Level Best Standard Plan has three equally spaced stress levels with equal allocations. Subject to this restriction, the lowest stress value is chosen to minimize the variance of the estimate of the percentile.
  • The 3 Level Best Compromise Plan puts a specified proportion of the test units at the middle stress level, which is chosen as halfway between the low and high stress levels. Subject to these restrictions, the low and high stress levels are chosen to minimize the variance of the estimate of the percentile.
  • The 3 Level Best Equal Expected Number Failing Plan is like a best compromise plan except that the allocations are chosen so that the expected number of units failing is the same at each stress level.
  • The 3 Level 4:2:1 Allocation Plan has three stress levels with a 4:2:1 allocation of test units to the low, middle and high stress levels, respectively. The low and middle test stresses are chosen to minimize the variance of the estimate of the percentile, with the middle stress as close as possible to halfway between the low and high stresses subject to the restriction that the probability of failure at the middle stress is at least twice the percentile that is to be estimated.

Figure 1: Test Plans for a single accelerating stress

Figure 1: Test Plans for a single accelerating stress

ALTA also provides two types of test plans for multiple accelerating stresses:

  • The 3 Level Optimum Plan is obtained by first finding a degenerate optimum plan and splitting this degenerate plan into an appropriate two-factor plan with the same variance.
  • The 5 Level Best Compromise Plan is obtained by first finding a degenerate compromise plan and splitting this degenerate plan into an appropriate two-factor plan with the same variance.

Figure 2: Test Plans for multiple accelerating stresses

Figure 2: Test Plans for multiple accelerating stresses

Example: Application to MOS Capacitors
An accelerated life test is to be conducted at different temperature levels for MOS capacitors in order to estimate the 10th percentile of the life distribution at a design temperature of 50° C (323.16 K) after ten years of operation. The test needs to be completed in 300 hours. The total number of items placed under test is 200 units. To avoid the introduction of failure mechanisms other than those expected at the design temperature, it has been decided through engineering judgment that the testing temperature cannot exceed 250�� C (523.16 K). Assume that a reasonable guess for the probability of failure in the 300 hour test at 50° C (323.16 K) is 0.05% and 80% of the test units would fail within 300 hours at 250° C (523.16 K). Additionally, a Weibull distribution and an Arrhenius life-stress relationship are assumed. How should this accelerated life test plan be designed?

According to the specifications, the design stress level for this particular application is 323.16 K and the highest stress level is 523.16 K. After these input values have been entered in the ALTA test planning module, and assuming the failure data follow the Weibull distribution, it is possible to use the utility to develop a suitable QALT plan. We began with the 2 level statistically optimum plan. As shown in Figure 3, the optimized standardized low stress condition is x′L = 0.708. This translates into an actual stress of TL = 169.91° C (443.08 K). Therefore, 138 of the 200 test units would be assigned to 169.91° C (443.08 K) with the remaining 62 units tested at 250° C. 14.12%, or 19, of the 138 test units at 169.91° C (443.08 K) would be expected to fail during the 300 hour test.

Figure 3: 2 level statistically optimum test plan results

Figure 3: 2 level statistically optimum test plan results

After reviewing these results, it was determined that the statistically optimum plan was not intuitively satisfying because it limits the test program to merely two temperatures and because of the relatively high value for the lowest stress condition, 169.91° C (443.08 K). Instead, it seemed appropriate to trade off some of the 19 expected failures at the lower stress for the sake of reducing the temperature and permitting testing at a middle stress condition. This led to the optimized 4:2:1 plan, which is shown in Figure 4. With this plan, the ratio of the asymptotic variance of the estimator of the 100pth percentile of the time-to-failure distribution, Ratio(p), is shown in Figure 4 to be 1.22 relative to that for the statistically optimum plan. Therefore, this approach increases the variance by 22%. This is the price that one may be paying for using the more robust and intuitively appealing optimized 4:2:1 plan.

Figure 4: 3 level 4:2:1 allocation plan results

Figure 4: 3 level 4:2:1 allocation plan results

After reviewing the results for the second plan, the low test temperature, 156° C (429 K), was thought to involve too much stress extrapolation relative to the design temperature of 50° C and hence a lower temperature seemed desirable for the standardized low stress condition. Thus, we adjusted the optimized 4:2:1 plan by reducing the low stress value to some fraction of the low stress value in the optimum plan. We can use different fractions to adjust the low stress value as long as the selected plan results in at least 3.33% ( (100p/3)% ) failures and at least five expected failures at the low stress (Ref. 1). As shown in Figure 5, a plan with 0.9 fraction low stress was selected. The probability of failure at the low stress level is 0.0533 (which satisfies the minimum requirement of 3.33%) and the expected number of failures is six.

Figure 5: Adjusted 3 level 4:2:1 allocation plan results

Figure 5: Adjusted 3 level 4:2:1 allocation plan results

ALTA also provides a "Sensitivity Analysis" option, which allows the analyst to evaluate the test plans under consideration. This includes necessary sample size determination, robust analysis to misspecified models and sensitivity analysis to the guess value.

Up to this point, we have assumed that the number of available test specimens was predetermined by economic or other practical constraints. When this is not strictly so, it may be possible to choose a sample size that is large enough to provide a specified degree of precision. In ALTA, the number of test units that is required in the entire test program to estimate the 100pth percentile at the design stress to within of (1 + γ ) (i.e., an error of less than 100 γ %) with probability (1 + α) is approximately:

Equation

where z(1 - α/2) is the percentage point of the standard normal distribution, σ is a guess value for the scale parameter and V is the asymptotic variance of the estimate of the 100pth percentile of the time to failure distribution at the design stress multiplied by n2.

Accelerated life testing plans developed under an assumed model are suitable only if the model is correct. Although these plans perform well under some models, they may or may not perform equally well under other models. For robust analysis to a misspecified model, ALTA uses R(WL/LL) and R(LW/WW) to analyze the possibility of bias due to model misspecification. If Weibull is the assumed distribution, R(WL/LL) is calculated. R(WL/LL) denotes the ratio of the variance for the plan obtained with an assumed Weibull distribution to that for the actual lognormal distribution, both being evaluated under the actual lognormal. R(LW/WW) is defined similarly for an assumed lognormal distribution and an actual Weibull distribution.

On the other hand, the calculated accuracy or sample size for the optimum plan also depends on the assumed values of the model parameters (the guess values of the failure probabilities). Sensitivity analysis will be investigated with respect to the guess values of the failure probabilities to test the robustness of the theoretical models. Traditionally, the model parameters are either estimates from the preliminary experiments or based on the experience of the experimenters and these assumed values differ from the true ones. Thus the calculated accuracy or sample size differs from the correct one. It is useful to re-evaluate a plan using other assumed values, changing one parameter at a time. If the plan or accuracy is sensitive to a parameter value, then one must consider changing the plan. Such an analysis can also be carried out on other characteristics of the plan, such as the probability of no failures at the lower test stress level.

ALTA uses the ratio of the variance of a plan generated under new modified guess values and the original guess value to analyze the sensitivity of plans to the guess value of the failure probabilities. Figure 6 shows these results for the MOS capacitor test plan, which include sample size calculation and some other sensitivity analysis.

Figure 6: Sensitivity Analysis in ALTA 6 PRO

Figure 6: Sensitivity Analysis in ALTA

Conclusion
This article introduced ALTA new functions for developing a QALT plan. The objective is to optimally allocate stress levels and test units so that asymptotic variance of the ML estimate of a (log) percentile of the life distribution at the design stress is minimized under the constraints of available test units, test time and failure distribution at each stress level. More information on ALTA is available on the Web at http://ALTA.ReliaSoft.com. The new utility will be available in Version 7.

References
1. Meeker, W. Q. & Hahn, G. J. (1985). How to Plan an Accelerated Life Test - Some
     Practical Guidelines. Statistical Techniques, 10.

2. Meeker, W. Q. & Escobar, L. A. (1995). Planning Accelerated Life Tests with Two or More
     Experimental Factors. Technometrics 37(4), 411-427.

3. Meeker, W. Q. & Escobar, L. A. (1998). Statistical Methods for Reliability Data. New York:
     John Wiley and Sons, Inc.

4. Nelson, W. (1990). Accelerated Testing - Statistical Models, Test Plans, and Data
     Analyses
. New York: John Wiley and Sons, Inc.

5. Nelson, W. (2003). “Bibliography of Accelerated Test Plans,” available from the author at
     739 Huntington Dr. Schenectady, NY 12309-2917.

 

End Article

 

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