Optimum Time-Censored Constant-Stress PALTSP for the Burr Type XII Distribution Using Tampered Failure Rate Model

This paper presents optimum design of time-censored constant-stress partially accelerated life test sampling plan (PALTSP) in which each item runs either at use or at accelerated conditions and product life follows Burr type XII. The optimal plan consists in finding out sample proportions allocated to both use and accelerated conditions by minimizing the asymptotic variance of test statistic for deciding on acceptance/rejection of the lot such that producer’s and consumer’s interests are safeguarded. The method developed has been illustrated using an example. Sensitivity analysis has also been carried out.


Introduction
Quality control methods are commonly used to determine the acceptability of a product with regard to its usefulness at the time it is put into the service.Reliability is related to the concept of quality of performance.Achieving desirable standards of reliability requires careful analysis in the product design phase.Analysis of data, obtained on a timely basis during product performance, keeps the design and production parameters updated and ensures that the product continues to perform in an acceptable manner.Reliability is built in through quality of design.
Acceptance sampling plans (ASPs) are commonly used to determine the acceptability of a product.If the sample data are related to the reliability of items (e.g., lifetimes or number of failures), then the corresponding ASP is called life test sampling plan or reliability acceptance sampling plan.
Many modern high reliability products are designed to operate without failure for a very long time.Life testing for these products under normal operating conditions is time consuming as it takes a lot of time to obtain reasonable failure information.Introducing ALT in life test sampling plan can be a good way to overcome such a difficulty.This has necessitated the formulation of ALTSPs.ALTSPs differ in the assumed life distribution, censoring schemes, failure monitoring method, test condition, and so forth.
Life test under accelerated environmental conditions may be fully accelerated or partially accelerated.In fully accelerated life testing all the test units are run at accelerated condition, while in partially accelerated life testing they are run at both normal and accelerated conditions.Commonly used stress loadings are constant-stress and step-stress (see Nelson [1]).Under constant-stress PALT each item is tested either at normal operating condition or at accelerated condition only.However, in step-stress PALT, a test item is first run at normal operating condition and, if it does not fail then, it is run at accelerated condition until failure occurs or the observation is censored.Accelerated test condition includes stresses in the form of temperature, voltage, pressure, vibration, cycling rate, humidity, and so forth.
Several authors have studied the problem of designing PALT under constant-stress loading.Yang [2] has indicated that constant-stress accelerated life tests are widely used to save time and money.DeGroot and Goel [3] have considered a PALT and estimated the parameters of the exponential distribution and the acceleration factor using the Bayesian approach.Bai et al. [4] have used the maximum likelihood method to estimate the scale parameter and the acceleration factor for the log normally distributed lifetime, using type I censoring data.Ismail [5] has used maximum likelihood and Bayesian methods for estimating the acceleration factor 2 Journal of Quality and Reliability Engineering and the parameters of Pareto distribution of the second kind.Also, Bhattacharyya and Soejoeti [6] have estimated the parameters of the Weibull distribution and acceleration factor using maximum likelihood method.Bai and Chung [7] have considered optimal designs for both constant and step PALTs under type I censoring.Abdel-Hamid [8] has used the constant-stress PALT and estimated the parameters of the Burr type XII distribution and the acceleration factor under progressive type II censoring However, no work seems to exist in the literature with PALT incorporated in acceptance sampling plan to facilitate formulation of decision rules for accepting or rejecting a lot of high reliability items satisfying producer's and consumer's requirements.In this paper optimum timecensored constant-stress PALT sampling plans have been designed using hazard acceleration and the Burr type XII life distribution.Single sample acceptance sampling plans by variables are used, and acceptance/rejection decision of a lot is based on Schneider's [9] approach to -method wherein sample mean and standard deviation are replaced by MLEs of location parameter and scale parameter of log life distribution, respectively.The asymptotic variance of the test statistic so obtained is used for deciding on lot disposition.The optimal plan consists in finding out sample proportions allocated to both use and accelerated conditions for the constant PALT by minimizing the asymptotic variance of test statistic for deciding on acceptance/rejection of the lot such that producer's and consumer's interests are safeguarded.

Burr Type XII Life Distribution
The Burr type XII distribution has a nonmonotone hazard function, which can accommodate many shapes of hazard function.
Unlike fully accelerated life testing wherein a regression structure on the stress variable (e.g., temperature and voltage) is specified, the acceleration factor   ,  = 1, 2, . . ., , is assumed to be a parameter of the model in PALT.

Lot Acceptance Sampling Procedure
Assume that one-sided lower specification limit  exists for the lifetime of a product.Let for convenience  = ln , instead of using the actual life time .The lower specification limit on  is   = ln .Then, pdf and cdf of an item tested at normal operating condition and at accelerated conditions using   (),    (),   (), and    () in Section 4.1 are given by under normal condition The reliability function of an item tested at normal and accelerated conditions  * 1 and  * 2 are, respectively, given by under used condition where  = ln() is location parameter and  = () −1 .
The following lot acceptance sampling procedure is considered.
(i) Samples of  items are randomly selected from the lot and are tested according to the above test procedure.
(ii) Maximum likelihood estimators (MLEs) μ and σ, respectively, of location parameter  and scale parameter  at normal operating condition, respectively, of Burr type XII distribution are obtained from the test data.
(iii) Schneider's [9] approach to -method is used wherein sample mean and standard deviation are replaced by MLEs of location parameter and scale parameter of log life distribution, respectively.In this paper  1 instead of  has been used as acceptability constant.
The value of test statistic   = μ −  1 σ is compared with   ; the lot is accepted if   ≥   and rejected otherwise.The sample size  and acceptability constant  1 determined using variance optimality criterion subject to the constraints that the lots with fraction nonconforming  ≤  α are accepted with probability of at least 1 − α and lots with  >  β are rejected with probability of at least 1 − β.

Model Formulation
Maximum likelihood method has been used to estimate the model parameters , , and acceleration factors,  1 and  2 , from the test data.
6.1.Likelihood Function.The likelihood function based on "" observation is where ) ) ) ) ) The first-order partial derivatives of log-likelihood function of the th unit with respect to  1 ,  2 , , and  are given by On summing these partial derivatives and equating them to zero, likelihood equations are obtained.Since the closed form solutions of above likelihood equations are very hard to obtain, so further numerical treatment is required to obtain the MLEs of  1 ,  2 , , and .

Fisher Information
where the values of these elements are given in Appendix A.

Asymptotic Variance of Test Statistic
.For any plan, the asymptotic variance-covariance matrix of the MLEs Â1 , Â2 , α, ĉ is the inverse of the corresponding Fisher information matrix; that is, The asymptotic variance of the test statistic   is given by where  is obtained in Appendix B.

OC Curve
The operating characteristic (OC) curve measures the performance of a sampling plan and plots the probability of accepting the lot versus the proportion nonconforming of the lot.Based on the asymptotic distribution theory, The standardized variate using ( 14) is The OC curve is obtained by plotting the probability of accepting the lot,   , given as against the fraction nonconforming, , where   = (  − )/ is the quantile of the standard logistic distribution corresponding to the fraction nonconforming , and Φ(⋅) is the standard normal distribution function.Therefore, the following equations have to be solved for  * 1 and  * in order to obtain an optimal sampling plan for two given points ( α, 1 − α) and ( β, β) on the OC curve, where   α and   β denote the quantiles of the standard normal distribution and  α and  1− β are the quantiles of the standard log-life distribution.Thus, on solving (18), where and  is given in (14).See Schneider [9] for reference.

Formulation of an Optimization Problem
in ( 14) is a function of  1 ,  2 , , , , and .Thus, the optimal design problem can be formulated as a nonlinear optimization problem: Since the sample size is unknown while minimizing , therefore instead of giving minimum mean number of failures (MMNF) a prespecified value in advance, 0.1, is assigned to the ratio of MMNF and the sample size at each stress level.Another suitable number for MMNF can be specified according to cost, time, and precision implications (see Yang [2]).The optimum values of   ,  = 1, 2, 3, are found by minimizing  under type I censoring, using NMinimize option of Mathematica 8.

A Numerical Example
In this section, a hypothetical constant-stress PALTSP experiment is considered to illustrate the methods described in this paper.

Optimal Plan. Optimal 𝜙 *
,  = 1, 2, 3, and  * are using (22).These are given in Table 1 using the following hypothetical data set: Table 1 shows that optimal sample size decreases as  1 increases and optimal sample proportions allocated to accelerated condition  2 increase as  1 increases.

Optimal Acceptance Sampling Plan.
Let us consider the lower specification limit of  as   (ln ) indicating that items with lifetime shorter than ln  are nonconforming.For given lower specification limit   , lot is accepted if   = μ0 −  1 σ0 >   , lot is accepted and otherwise rejected.Figure 1 depicts the OC curve obtained from   , ,  * and the two points ( α, 1 − α) and ( β, β).

Sensitivity Analysis
To formulate an optimum sampling plan, the information about acceleration factors  1 and parameters of the model  and  is needed as incorrect choice of these results in poor estimates of the parameters.The effects of incorrect preestimates of  1 ,  2 , , and  in terms of the relative increase in asymptotic variance of test statistic have been studied.The percentage deviations (PD) of the resulting optimal settings are measured by PD = (| * * −  * |/ * ) × 100, where  * is the setting obtained with the given design parameters, and  * * is the one obtained when the parameter is misspecified.Tables 4 and 5 show that irrespective of whether the incorrect variance is smaller or larger than the true variance, the proposed optimum plan is robust since the percentage deviation in variance is small.

Conclusion
In this paper we have formulated an optimum time-censored constant-stress PALTSP for the Burr type XII life distribution using hazard acceleration.Single sampling plan by variables has been used.The acceptance/rejection decision of the lot involves -method in which Schneider's [9] approach has been used with sample mean replaced by MLE of location parameter and standard deviation replaced by MLE of scale parameter of the log-lifetime distribution with lower specification limit specified.The optimum plan consists in finding optimum allocation at normal operating condition and accelerated conditions by minimizing the variance of the test statistic meant for deciding on lot acceptability.The sensitivity analysis has been carried out and it has been found that if the misspecified values of the parameters are not too far from the true values, the optimum plan is robust.

2. 1
. Assumptions (a) The lifetime of an item tested either at normal operating condition or at accelerated conditions follows Burr type XII distribution.(b) The lifetimes of test units are independent and identically distributed random variables.(c) Tampered failure rate (TFR) model is assumed (Bhattacharyya and Soejoeti [6]).(d) The log-lifetimes   ,  = 1, 2, . . .,
Matrix.The Fisher information matrix ( 1 ,  2 , , ) ≡  is the 4 × 4 symmetric matrix of expectation of negative second-order partial derivatives of the log likelihood function with respect to  1 ,  2 , , and .