Accelerated Life Test Sampling Plans under Progressive Type II Interval Censoring with Random Removals

This paper investigates the design of accelerated life test (ALT) sampling plans under progressive Type II interval censoring with random removals. For ALT sampling plans with two over-stress levels, the optimal stress levels and the allocation proportions to them are obtained by minimizing the asymptotic generalized variance of the maximum likelihood estimation of model parameters. The required sample size and the acceptability constant which satisfy given levels of producer’s risk and consumer’s risk are found. ALT sampling plans with three over-stress levels are also considered under some specific settings. The properties of the derived ALT sampling plans under different parameter values are investigated by a numerical study. Some interesting patterns, which can provide useful insight to practitioners in related areas, are found. The true acceptance probabilities are computed using a Monte Carlo simulation and the results show that the accuracy of the derived ALT sampling plans is satisfactory. A numerical example is also provided for illustrative purpose.

The number of removals at each failure was assumed to be pre-fixed in those works. However, in practice it might be infeasible to pre-determine the removal pattern and the decision of removing any units is based on the status of the experiment at that specific time, such as excessive heat or pressure, reduction of budget and facility, etc. Therefore, the number of removals should be a random outcome (Yuen & Tse, 1996). Tse and Yang (2003) discussed the design of reliability sampling plans for the Weibull distribution under progressive Type II censoring with random removals, where the number of units removed at each failure was assumed to follow a binomial distribution. In recent years the feature of random removal has been adopted by many researchers in designing various kinds of progressive censoring schemes, such as Ashour and Afify (2007), Wu, Chen, and Chang (2007), and S. Dey and T. Dey (2014).
Units are supposed to run at use condition in traditional reliability sampling plans. When it is desired to test the acceptance of highly reliable products, it is impractical to use such reliability sampling plans due to time constraint. Wallace (1985) stressed the need for introducing ALT into reliability sampling plans. Bai, Kim, and Chun (1993) studied the design of failure censored ALT sampling plans for lognormal and Weibull distributions. Hsieh (1994) investigated reliability sampling plans with ALT under Type II censoring for exponential distribution. The optimal design of ALT sampling plans with a non-constant shape parameter under both Type I and Type II censoring schemes was given by Seo, Jung, and Kim (2009).
Note that continuous inspections were assumed in the above works. Nevertheless, sometimes it is inconvenient to conduct a test with continuous inspections due to the high cost and/or possible danger in monitoring the test continuously. Under these circumstances, the interval inspection schemes, in which only the number of failures between two successive inspections is recorded, would be more favorable. Studies on life test and/or accelerated life test which employ interval censoring schemes are numerous. To number some of them, Tse, Ding, and Yang (2008) investigated the optimal design of accelerated life test under interval censoring with random removals for Weibull distribution; Chen and Lio (2010) compared the maximum likelihood estimation, moment estimation and probability plot estimation of parameters in the generalized exponential distribution under progressive Type I interval censoring; Ding, Yang, and Tse (2010) discussed the design of optimal ALT sampling plans under progressive Type I interval censoring with random removals. Most recently, Ding and Tse (2013) investigated the design of optimal ALT plans under progressive Type II interval censoring with binomial removals for the Weibull distribution. However, as far as our knowledge goes, there is no relevant study that investigates the design of ALT sampling plans under similar experimental settings with a Type II censoring scheme.
The optimal reliability sampling plans which combine ALT, interval inspection and progressive Type II censoring with random removals are developed in this paper. This study can be noted as an extension to the work of Ding and Tse (2013) along three directions: (i) the research topic is extended from the design of optimal ALT plans to the design of optimal ALT reliability sampling plans, in which both the consumer's risk and the producer's risk are satisfied. In this sense this paper resolves a more practical problem; (ii) instead of minimizing the asymptotic variance of an estimated quantile of units' lifetime distribution, this paper minimizes the asymptotic generalized variance of the maximum likelihood estimation of model parameters. It enables us to compare the outcomes derived using two different criteria in optimization; (iii) the true acceptance probabilities of the derived optimal ALT sampling plans are simulated, which provides us a way to evaluate the accuracy of the proposed method.
The rest of this paper is organized as follows: Section 2 describes the basic model of the proposed scheme. The design of optimal ALT sampling plans under progressive Type II interval censoring with random removals is discussed in Section 3. A numerical study is conducted in Section 4 to examine the properties of the derived sampling plans. In Section 5 the accuracy of the proposed ALT sampling plans is evaluated by a Monte Carlo simulation. Section 6 provides a numerical example. Conclusions are drawn in Section 7.

Model Description
Consider an ALT with the following settings: 1. A total of n identical and independent units are available at the beginning of the test.
2. There are m over-stress levels, i.e., 12  The process of this testing scheme is depicted in Figure 1. Suppose that the lifetime of a unit T follows a Weibull distribution with probability density function (pdf) (1) Further assume that the scale parameter  and stress level s are related as where 0  and 1  are unknown constants and the shape parameter  does not depend on s . Define Y has an extreme value distribution with cumulative distribution function (cdf) where  = ln = 0 The maximum likelihood estimates of 0 where 1 0 1 ( , , ) on each stress, the stress levels ( , 1, 2,..., 1) i s i m  and the allocation proportions ( , 1,2,..., 1) i im   to these stress levels are selected in such a way that the generalized asymptotic variance of 0  , 1  , and  , which is given by

Design of ALT Sampling Plans
Suppose that a sample of size n is randomly drawn from the lot and the test is conducted at the accelerated settings described in Section 2. Assume that the lifetime of a unit T follows a Weibull distribution ( , ) F  , where the relationship between the scale parameter  and the stress s is given by Eq.
(2) and the shape parameter  does not depend on s . Suppose that a unit with lifetime less than  is considered to be nonconforming. Define ln( ) YT  , then Y follows an extreme value distribution ( , ) G  and the lower specification limit for the log lifetime is given by  is the location parameter of (.) G at use condition and d is the acceptability constant. Since the stresses can be standardized such that 0 0 s  , 1 m s  and 0 1 ( 1,2,..., 1) By the invariance principle of the maximum likelihood method, the MLE of  is then given To judge whether a lot should be accepted or not,  is compared with the lower specification limit  . If   , the lot is accepted; otherwise, it is rejected.
Define the nonconforming fraction of the lot by f p , which is calculated as The sample size n and the acceptability constant d are determined such that lots with nonconforming fraction f pp   are accepted with a probability of at least 1 and lots with f pp   are rejected with a probability of at least 1   , where  and  are the given levels of producer's and consumer's risks, respectively.
is parameter-free and asymptotically standard normal, the operating characteristic (OC) curve is given by where (.)  is the cdf of the standard normal distribution.
The sample size n and the acceptability constant d are determined such that the OC curve goes through two points ( ,1 ) p   and ( , ) It follows that where 1 () z uz   . The acceptability constant d is calculated directly from the first part of Eq. (9), while the required sample size n can be obtained by a search method from the second part (the detailed algorithm is provided in Section 4.1).

ALT Sampling Plans with Two Over-stress Levels
The properties of the derived ALT sampling plans under different parameter values are evaluated by a numerical study in this section. The following settings are made: 1. Two over-stress levels 1 s , 2 s are employed, i.e., 2 m  . i  , which is proportional to the inspection length i l , is used in this numerical study since it is more convenient to use a relative value than an absolute one. The case of 12     is considered.    , when p =0 and 0.05, n increases as  increases; when p =0.1 and 0.3, n first decreases and then increases as  increases. This pattern can be interpreted in this way: Larger  means wider inspection intervals, from which the collected information on units' lifetime is less accurate and thus more units are required to judge whether to accept the lot or not. However, when 0 p  , a larger  also implies that units are less likely to be removed at the early stage of the test. Consequently, more information on the lifetime distribution is collected and the required sample size n is decreased. Taking these two kinds of effect into consideration, shorter inspection interval doesn't always yield smaller required sample size for ALT sampling plans under progressive Type II interval censoring with random removals.
b. For the cases of 0.5   and 1, n decreases as p increases for all values of  . For the cases of 2   , when  =0.1 and 0.3, n decreases as p increases; when  =0.02 and 0.05, n first decreases and then increases as p increases. This pattern is caused by the two-sided effects of the removal probability p . Generally speaking, a test is likely to be prolonged as p increases. Thus more information on the lifetime distribution can be observed and the required sample size n is decreased. Nevertheless, when the inspection intervals are too small, a non-zero removal probability p also causes more units being removed at the early stage of the test. In this case, less data can be collected and thus n is increased. In conclusion, except for several cases ( 2   and  =0.02/0.05), the removal probability p is helpful in reducing the required sample size n .

ALT Sampling Plans with Three Over-stress Levels
ALT plans with three over-stress levels are useful in practice since they can provide a way to check the assumed straight-line relationship between distribution parameter  and stress level s by adding a middle stress. The design of three over-stress levels ALT sampling plans under progressive Type II interval censoring with random removals is discussed in this section. They are developed under the following settings:  Note that these patterns are similar to those observed in the two over-stress levels case.

Accuracy of Large Sample Approximation
Since the proposed ALT sampling plans are derived based on asymptotic theory, there is a need to evaluate the finite sample behavior of them. The accuracy of the derived ALT sampling plans is assessed by a simulation study. The OC curve is set to go through two points, that is, ( ,1 ) p   =(0.00041, 0.95) and ( , ) p   =(0.01840, 0.10). For each combination of parameters, the nonconforming fraction of a lot under given acceptance probability (99% and 95%) on the pre-defined OC curve is computed, and then the true acceptance probability of a lot with that corresponding nonconforming fraction is calculated by a Monte Carlo simulation with 1000 runs. The results for ALT sampling plans with two over-stress levels and three over-stress levels are presented in Table 1 and Table 2, respectively. Actually, several different values of  (   0.5, 1, 2) are considered in this numerical study. Since they show similar patterns, only parts of the results of 1   are provided for simplicity.
We note from Table 1 and Table 2 that the simulated acceptance probabilities are close to their nominal values in most cases. This indicates that the optimal ALT sampling plans derived based on asymptotic approximation have satisfactory accuracy.

A Numerical Example
Suppose that there is an agreement between a consumer and a producer to determine the acceptability of a lot. In particular, if the nonconforming fraction of a lot is smaller than 0.00041, then the lot should be accepted with a probability of at least 0.95; while if the nonconforming fraction of a lot is larger than 0.01840, then it should be rejected with a probability of at least 0.90. Assume that an ALT reliability sampling plan with two over-stress levels is used to determine the acceptability of the lot. The probabilities for a unit to fail at use condition and high stress level are estimated to be 0.01 and 0.1, respectively. A progressive Type II interval censoring scheme is employed, and the censoring fractions on two stress levels are 0.8. The proportions of the inspection length to the corresponding distribution mean on both stresses are set to be 0.1. Besides, based on prior information, it is assumed that units' lifetimes are Weibull distributed with shape parameter 1   and a unit is likely to be removed at each inspection with probability 0.1. The problem is to determine the number of units used in this ALT sampling plan and to determine the low stress level and the allocation proportions to two stresses so that (1) both the consumer's risk and the producer's risk can be satisfied and (2) the maximum amount of information on units' lifetime distribution can be collected.
The optimal ALT sampling plan is obtained using the proposed method. The required sample size is 17, with 7 and 10 units allocated to the low and high stress levels, respectively. The low stress level should be settled at 0.02 multiplied by the actual high stress. Besides, the acceptability constant which is required to make the decision is 5.6560.

Conclusion
The design of ALT sampling plans under progressive Type II interval censoring with random removals was discussed in this paper. For ALT sampling plans with two over-stress levels, the optimal stress levels and the corresponding allocation proportions, which minimize the generalized asymptotic variance of the MLE of model parameters, were found. The sample size and the acceptability constant required to judge the acceptability of the lot were calculated.
The properties of the derived ALT sampling plans were examined by a numerical study. It is shown that generally the removal probability is helpful in reducing the required sample size. More importantly, when there exists random removal, short inspection interval doesn't always yield small required sample size, which is different from the case of no random removal. These interesting patterns would provide useful insights to experimenter in designing similar ALT sampling plans. The accuracy of the proposed sampling plans was evaluated by a Monte Carlo simulation. The results show the simulated acceptance probabilities are close to their nominal values in most cases, which indicates that the performance of the derived ALT sampling plans is satisfactory.