Planning of ALT Test

Even the analyzing methods are powerful in obtaining an estimate, the estimates might be significantly different from the true lifetime at use stress. Planning of ALT mitigates this pressure. Usually, the sample size, the test duration and the expected proportion of failures are known due to the availability of samples and testing time. Hereby, decision variables in test planning involves the stress levels, number of units

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allocated to each stress level in CSALT or holding time assigned at each stress in SSALT. Nelson (1990) summarized the normally used optimization criteria:

1. minimizing the variance of the LSE of reliability interest at use stress (for test plans with complete data); Or

2. minimizing the variance of the MLE of reliability interest at use stress (for tests with censored data); Or

3. minimizing the variance of an estimate over a range of stress; Or 4. minimizing the variance of the estimate of a particular coefficient; Or 5. minimizing the variance of the estimate of the scale parameter; Or 6. be most sensitive to detect non-linearity of the stress-life relationship.

Tian (2002) thoroughly reviewed the models, plans and applications of SSALT. Miller & Nelson (1983) designed a simple two stress SSALT with completed data. Bai et al (1993), Tang et al (1996) and Khamis & Higgins (1996) presented ideas on three- stress SSALT designs. Tang (1999, 2003) not only optimized stress levels and the holding time, but also achieved a target Acceleration Factor (AF) to satisfy the test time constraint with a desirable fraction of failure in SSALT planning, see also Yeo & Tang (1999). They concluded that the statistically optimal way to increase AF in an ALT was to increase the lower stress levels and shorten their holding time. In most of the above-mentioned papers, a certain Weibull distribution has been assumed to model the failure time distribution and the cumulative exposure (CE) model has been employed to model the stress-life relationship.

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In additions, Xiong & Milliken (1999) investigated the statistical models in SSALT and planned the optimal tests when the stress changing time was an order statistic from an exponential lifetime family. Park & Yum (1998) presented a modified SSALT and CSALT based on an exponential life distribution and a linear acceleration relationship by minimizing the asymptotic variance of the MLE of the mean lifetime at use stress. When the sample size was small, SSALT performs well to capture reliability information (Khamis, 1997).

Statistically optimal CSALT plan (Nelson et al, 1978) consists of two stress levels. It yields the most accurate estimate but is unable to validate the assumed linearity of the stress-life relationship. It is not robust to mis-specifications of some pre-estimates either. Three-stress designs have eliminated this disadvantage. The three-stress best standard plan (Nelson & Kielpinski, 1976), which provides poor estimates, sets three equally spaced stresses and equally allocated test units (with allocation ratio 1/3:1/3:1/3) to those stress levels. Three-stress compromised CSALT plan (Meeker 1975, Meeker & Hahn 1985) uses three stresses in which the middle one is the average of the other two, and the allocation ratio for the three stresses is 4/7:2/7:1/7. It is more robust to avoid deviation from the linear stress-life assumption than the statistically optimal plan and more efficient to produce an accurate estimate than the best standard plan. But in this situation, no optimization has been addressed on the middle stress level. It is proved that the statistically optimal plan produces the smallest asymptotic variance. In the above-mentioned plans, the underlying life distribution is assumed to be a Weibull distribution, in which the scale parameter keeps constant for all stresses, and the stress-life relationship is simply linear. Optimality is achieved by minimizing the asymptotic variance of the MLE of a certain percentile of the time–to-failure

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distribution at the design stress. Under the same assumptions, Yang & Jin (1994) addressed the sample allocations in three-stress CSALT by minimizing the asymptotic variance of the MLE of the mean life at design stress and the total running time. Tan (1999) solved the same problem by constraining the expected failures at design stress not less than a specific amount.

By assuming that the failure time follows a Weibull distribution with non-constant scale parameter at each stress and the allocations at each stress is 4:2:1, Meeter & Meeker (1994) planned the three-stress CSALT by minimizing the asymptotic variance of MLE of the mean life at use stress. While, Tang (1999) gave the optimal stress levels and their allocations by minimizing the variability of scale parameter at each stress, see also Tang et al (2002). Wu et al (2001) addressed the limited failure censored life test plans. Sun (1999) proposed the failure free test plans.

Test plans with other lifetime distributions such as the normal and the exponential distribution are also studied in Yang (1994) and Park & Yum (1996).

Optimality in most of the above-mentioned literature is defined as achieving a minimum asymptotic variance for a certain percentile of reliability interest, that is, to obtain the “best design” that corresponds to “optimization”. In practice, stress levels of the best design may be too harsh to implement and also there may be constraints due to limitation of experiment budget and availability of products and/or test duration. Thus pursuing the best design, sometimes, enhances the difficulty of conducting experiments and increases the experimental expenses. To solve these problems, we consider relaxing the usual optimization criteria to obtain alternate plans. This idea is first initiated by the goal-softening approach.

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Ho et al (1992) and Ho (1996) introduced the goal-softening approach as to “settle for the good enough solutions with high probability”. Ho et al (1992) argued ordinal rather than cardinal optimization concentrating on finding good, better or best designs rather than estimating accurately the performance value of these designs. The interest of goal softening approach is on whether solution A is better than solution B (say, solution A > solution B), not on how much solution A is better than solution B (say, solution A – solution B). Barnhart et al (1994), Ho & Deng (1997) showed examples of this approach for comparison with traditionally cardinal optimization. Lee et al (1999) also explained the role of goal softening in ordinal optimization: other than using accurate performance that presumably takes a long time to obtain, one could use the relative order of performance estimate as a basis for comparing and choosing design. Other than picking one single design that is exactly the true optimum in the design space, which is important in the presence of large estimate errors, one could pick a subset in which some good enough designs are contained with high probability. Using the order statistics formulation, they examined the feasibility of this approach to discrete event dynamic systems. It was stated that goal-softening approach was exponentially efficient in terms of matching good designs in a selected group.

We intend to apply goal-softening approach to ALT planning in chapter 2. Instead of finding the unique design solution that corresponds to the smallest asymptotic variance, we are going to solve for the design space in which the asymptotic variances are within a tolerable bound.

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