A GROUP ACCEPTANCE SAMPLING PLAN FOR TRUNCATED LIFE TEST HAVING THE (P-A-L) EXTENDED WEIBULL DISTRIBUTION

The main point of this paper is the group acceptance sampling plans under the time truncated life test is proposed when the product lifetime


Introduction
Accepting sampling plans is concerned with inspection and decision making regarding product, one of the oldest aspects of quality assurance. The acceptance sampling plans are concerned with acceptance or rejection of a submitted lot of size of products on the basis of the quality of products inspected in a sample taken from the lot. An acceptance sampling plan is specified plan that establishes the minimum sample size to be used for testing. In most acceptance sampling plans for a truncated life test, the major issue is to determine the sample size from a lot under consideration. It is implicitly assumed in the ordinary acceptance sampling plan under a truncated life test is the simple sampling plan since it is customary to install one item to a single tester during the experiment. The ordinary acceptance sampling plans for different distributions have been developed by many researchers including Kantam et al. [11], Baklizi [1], Balakrishnan et al. [10] and Lio et al. [12] and [13]. However, it requires more cost, time and supervision to gather the sample for making a decision of either accepting or rejecting the lot of products. Therefore, the ordinary acceptance sampling plan is usually expensive to be implemented. On the other hands, in group sampling case, the experimenter can reduce the experiment expenditure by installing more than on item on a single tester. The other advantage of the group plan is that it provides more strict inspection of the product than the ordinary acceptance sampling because sample is distributed over than one group. The condition of acceptance is applied to each group for a better product inspection. Therefore, the group acceptance sampling plan (GASP) has attracted researchers due to its advantage over the ordinary acceptance sampling plan under a truncated life test. The sudden death life testing is always performed in GASP. Jun et al. [2] proposed variables sampling plans for the Weibull distribution under the sudden death testing. Various GASPs under a time truncated life test are available in literature. For example, Aslam and Jun [7] studied GASP for the inverse Rayleigh and the log-logistic distributions. Aslam and Jun [6] proposed the GASPs for the Weibull distribution and Aslam et al. [8] proposed the plans for the gamma distribution. More recently, Aslam et al. [9] proposed a modified group plan for the Weibull distribution.
The purpose of this paper is to propose a GASP based on truncated life tests when the lifetime of a product follows the (P-A-L) extended Weibull distribution. Further, we obtain the number of groups and the acceptance number simultaneously for given values of both risks using the two point approach. The rest of the paper is organized as follows: we will introduce the glance about the (P-A-L) extended Weibull distribution in Section 2. The proposed GASP along with the operating characteristics is described in Section 3. The results are explained with some examples in section 4. Then, we conclude with some remarks in Section 5.

The (P-A-L) Extended Weibull Distribution
The (P-A-L) extended Weibull distribution can be used for the life modelling in reliability analysis, life testing problems and acceptance sampling plan. The probability density function (cdf) and the cumulative distribution function (cdf) of the (P-A-L) extended Weibull distribution, respectively, are given by: It is clear that the (P-A-L) extended Weibull distribution is very flexible. This is so since several other distributions follow as special cases from the (P-A-L) extended Weibull distribution, by selecting the appropriate values of the parameters. These special cases include eleven distributions.
The mean of the (P-A-L) extended Weibull distribution is given by: The median of the (P-A-L) extended Weibull distribution is given by:

Design of GASP for the (P-A-L) extended Weibull Distribution
Let μ represent the true value of the median of lifetime distribution of a product and 0 μ denote the specified median, under the assumption that lifetime of an item follows a (P-A-L) extended Weibull distribution. We interested in designing a sampling plan in order to assure that the median life of items in a lot ( ) μ is greater than the specified life ( ). Let us propose the following group acceptance sampling plan based on the truncated life test: (1) Select the number of groups g and allocate predefined r items to each group so that the sample size for a lot will be . g r n × = A Group Acceptance Sampling Plan for Truncated Life Test … 205 (2) Select the acceptance number (or action limit) c for a group and the experiment time . (4) Truncate the experiment if more than c failures occur in any group and reject the lot.
Note that the proposed group sampling plan reduces to the ordinary sampling plan if , 1 = r when the sample size n is equal to g. We are interested in determining the number of groups g and the action limit c which satisfies both the risks at the same time, whereas the group size r and the termination time 0 t are assumed to be specified.
Suppose that the life time of a product follows the (P-A-L) extended Weibull distribution which has the cdf given by equation (2). As pointed out by Grant and Leavenworth [3] and Stephen [5], binomial distribution can be used to express the OC curve for a sampling plan when the lot size is large enough and the experimenter focuses only on two options either to accept or reject the lot. The lot of products is accepted only if there are c or less failures in each of g groups, so the lot acceptance probability will be ( ) ( ) where P is the probability that an item in any group fails before the termination time .  The probability of rejection a good lot is called the producer's risk, whereas the probability of accepting a bad lot is known as the consumer's risk. The consumer demands that the lot acceptance probability should be smaller than the specified consumer's risk β at a lower quality level (usual at ratio 1), whereas the producer requires that the lot rejection probability should be smaller than the specified producer's α at higher quality level. When the quality level is expressed by the ratio mentioned earlier, the proposed two-point approach of finding the design parameters is to determine the number of groups and the acceptance number that satisfy the following two inequalities simultaneously: where 1 r is the mean ratio at the consumer's risk and 2 r is the mean ratio at the producer's risk. Let 1 p be the failure probability corresponding to the consumer's risk and 2 p be the failure probability corresponding to be producer's risk, the minimum number of groups and action number required can be determined by considering the consumer's risk and producer's risk at the same time through the following inequalities: A Group Acceptance Sampling Plan for Truncated Life Test … 207 The design parameter in terms of integers can be found by using a search, which can be implemented by MATLAB program.    Note: The cells with hyphens (-) indicate that g and c cannot satisfy the conditions. The cells with hyphens (*) indicates that g and c are found to be large.
In these tables, note that as the ratio 2 r increases, the number of groups and the acceptance numbers decrease at the same time. We need a smaller number of groups if the termination ratio increases at fixed group size. For example, from

Description of Tables and Examples
Suppose, for example that the lifetime of a product follows a (P-A-L) extended Weibull distribution with the shape parameter 8 = δ and other parameters are Suppose that it is desired to design a group sampling plan to assure that the median life is greater than 1000 h through the experiment to complete by 1000 h using testers equipped with three products each. It is assumed that the consumer's risk is 1% when the true median is 1000 h and the producer's risk is 5% when the true median is  Table 1. This means that a total of 30 products are needed and five items will be allocated to each of the five testers. We will accept the lot if no more than three failures occurs before 1000 h in each of the five groups. For this proposed sampling plan under the (P-A-L) extended Weibull distribution, the number of groups decreases and the OC values increase as follows when the true median increases, which summarized from Table 1.

Concluding Remarks
The group sampling plans attribute was proposed and designed using the two-point approach under the assumption that the lifetime of a product follows the (P-A-L) extended Weibull distribution. The two-point approach determines the plan parameters such as the number of groups and the acceptance number. This group sampling plan would be beneficial in terms of test time and cost because a group of items can be tested simultaneously. This paper only deals with the (P-A-L) extended Weibull distribution. Further study is required to propose more distributions as special cases of the (P-A-L) extended Weibull distribution.