Two-Stage Improved Group Plans for Burr Type XII Distributions

In this paper, we present two-stage group plan for Burr type XII distribution using lifetime percentile as quality parameter. The plan parameters such as acceptance number and group size are determined by considering producer’s risk and consumer’s risk at the same time. To find the plan parameters, we fixed the number of testers, experiment time and percentiles ratio. Many useful tables are generated for Burr type XII distribution as well as for log-logistic distribution. Two industrial examples are presented for the illustration of the proposed two-stage group sampling plan. Comparison of proposed plan with single-stage group plan is also discussed. Generally, the proposed two-stage sampling plan needs less number of test items than single-stage group plan does.


Introduction
The main aim of acceptance sampling plans is to reduce the inspection cost and provides the protection to producer and costumer. In this modern era, even some advance techniques including the quality assurance and quality control chats are available to improve the quality of the products but these techniques alone are incapable to achieve the product quality according to the customer specification. Especially, for electronic components testing, acceptance sampling schemes for life time experiments are widely used. Today, acceptance sampling schemes are widely used in industries for the testing purpose; for example, the application of an acceptance sampling scheme in food industry can be found in Bray and Lyon [1] and applications of acceptance sampling plans in testing of fiber optical can be found in Bhaumik and Bhargava [2]. Many acceptance sampling schemes including single and double acceptance sampling plans, have been widely used in industries. In the single acceptance sampling plan, decision about the lot is finalized on the basis of single sample selected from the production lot. A double acceptance sampling plan is used after the inspection of the first sample and an experimenter is not able to reach on the final decision about the submitted product. Then the second sample is selected from the lot to make decision about the submitted product. It is important to notice that acceptance sampling plans can mislead the experimenter. For example, using the acceptance sampling schemes, there is a chance to accept the bad lot or reject the good lot. So, the acceptance sampling plans are used to minimize these misclassification probabilities. Accepting the bad lot is called consumer's risk and rejecting a good lot is called the producer's risk. Acceptance sampling plans based on truncated life tests have been studied by many authors including Kantam et al. [3], Baklizi [4], Balakrishnan et al. [5] and Tsai and Wu [6].
Double acceptance sampling is more useful to minimize the producer's risk than the single acceptance sampling schemes. Double acceptance sampling is more efficient in reducing the number of items for inspection than the single acceptance sampling. Aslam and Jun [7] proposed the double acceptance sampling plan for generalized log-logistic distributions with known shape parameter.
Group acceptance sampling plans are the extension of the ordinary acceptance sampling plans and widely used in the laboratory which has facility to install more than one item in a single tester. Group acceptance sampling plans are useful to reduce inspection cost and provide more strict inspection than single acceptance sampling plans. The group plans are widely used in sudden death testing. Jun et al. [8] proposed the variable sampling plans for Weibull distribution using the sudden death testing scheme. Aslam and Jun [9] originally proposed the group plan for time truncated experiment for Weibull distribution using two points approach. Two stage group plans are extension of the single stage group plans. Recently, Aslam et al. [10] proposed the improved version of single stage group plan and two-stage group plan based on total failures from all the groups. They declared that the improved version of single stage group plan and two-stage group plan based on total failures from all the groups are more efficient than the ordinary acceptance sampling plans.
All the aforementioned works in the area of acceptance sampling plans were developed by using the mean life time of the distribution. But, recently, Lio et al. [11,12] developed the acceptance sampling plans for the lifetime percentiles. They argued that mean lifetime of the product may not satisfy the requirement of engineering design consideration. Acceptance sampling plans developed using mean life could pass a lot which has a low percentile below the required specification. Further, the Burr type XII distribution is skewed distribution and as stated Marshall and Olkin [13] that as a quality parameter the mean life does not perform well than the lifetime percentiles if the distribution under study is skewed. The works for developing acceptance sampling plans for the lifetime percentiles are an extended research works for developing the acceptance sampling plans for the mean life and for the median.
According to best of our knowledge, two-stage group sampling plans based on the Burr type XII distribution percentiles have not seen in literatures. The purpose of this paper is to develop the two-stage group plans for the Burr XII distribution percentiles.
The Burr type XII distribution has the cumulative distribution function (cdf) as follows: where η is the scale parameter, b and k are the two shape parameters. When k=1, the burr type XII distribution converts to the log-logistic distribution. The 100 q -th percentile of the burr type XII distribution is given as: The rest of the paper is organized as follows, the design of the proposed plans and some of tables used for discussions are given in Section 2. Two applications in industrial are presented in Section 3. Section 4 compares the proposed two-stage sampling plan with sign stage sampling plan and concluding remarks are given at the end.

Design of Two-Stage Group Plan Using Percentiles
In this section, the design of two-stage group sampling plan for Burr XII distribution percentiles is described by followed Aslam et al. g groups is smaller than or equal to 1a c . Truncate the test and reject the lot as soon as the total number of failures is larger than or equal to 1r c ( > 1a c ) during the 0 t units of time. Otherwise, go to the second stage.

(Second stage)
Draw the second random sample of size 2 n from the lot, allocate r items to each of 2 g groups so that 2 2 n rg = and put them on test for the duration of 0 t units of time again. Accept the lot if the total number of failures from 1 g and 2 g groups is smaller than or equal to 2a c (> 1a c ). Otherwise, truncate the test and reject the lot. The proposed double group sampling plan is characterized by five design parameters, namely, 1 g , 2 g , 1a c , 1r c and 2a c .It should be noticed that the proposed group sampling plan is different from the double acceptance sampling plans studied by Aslam et al. [14]. When 1 1 1 r a c c = + , the total number of failures from 1 g groups (denoted by 1 X ) follows a binomial distribution with parameters 1 n and p . Therefore, the lot acceptance probability at the first stage under the proposed double sampling plan is given by Aslam et al. [10]. 1 1 The lot rejection probability at the first stage is given by Now, the lot will be accepted from the second stage if the decision has not been made at the first stage and the total number of failures from 1 g and 2 g groups (denoted by 2 X ) is smaller than or equal to 2a c . Hence according to Aslam et al. [10], Therefore, the lot acceptance probability for the proposed double group sampling plan is given by (1) (2) ( ) .
where ( ) Particularly, the p in terms of the median life is represented as Where When the quality level based on the percentile ratio 0 / q q t t between the true percentile q t and targeted percentile 0 q t , the two-point approach of finding the design parameters is to determine the minimum number of groups, 1 g and 2 g , and acceptance numbers, 1a c , 1r c and 2a c , to satisfy the following two inequalities where, 1 δ is the percentile ratio at the consumer's risk and 2 δ is the percentile ratio at the producer's risk. Let 1 p and 2 p are the failure probabilities of corresponding to consumer'srisk and producer's risk, respectively. The ASN under LTRL, 2 p , is given by Then the optimization problem to be considered is as follows: Minimize (12c) Therefore, given α and β , the percentile ratio, 1 δ , at the consumer's risk and the percentile ratio, 2 δ , at the producer's risk, the design parameters of the proposed plan are determined such that the 2 ( ) ASN p is minimized and inequalities (12b) and (12c) are satisfied simultaneously for specified values of shape parameters, b and k , termination ratio 0 q δ and the number of testers, r .If the shape parameters, b and k , are unknown, then the previous data information could be used to estimate these parameters.          1. For given β and the group size r in each table, all the design parameters and ASN are non-increasing with respect to the percentile ratio 2 δ 2. When the group size increases from 5 to 10, the numbers of groups for both stages are generally decreased for a given β and the percentile ratio 2 δ 3. When the group size increases from 5 to 10, ASN generally non-decreases for a given β and the percentile ratio 2 δ . 4. When β increases, the numbers of groups for both stages are generally decreased for given group size r and the percentile ratio 2 δ .

Applications of Industry
In this section, two industrial examples are used to demonstrate the applications of the proposed two-stage acceptance sampling plans. Example-1 Lio et al. (2010) showed that the lifetime of small electric carts have a Burr type XII distribution with k =0.08 and b =5.47. Suppose that an experimenter would like to use the proposed two-stage sampling plan to establish the true unknown 10 th percentile lifetime for the product is at least 4 months and experiment will be stopped after 4 months. Further, suppose that in the laboratory the experimenter has facility to install five items on a tester. This information leads to 0

g =
The two-stage acceptance sampling plan is implemented as follows: randomly select 45 items and distribute five items into each tester and accept the product if no failure is recorded in 4 months and reject the product if more than 1 failure is recorded before 4 months; otherwise, randomly select another 35 items from the lot and distribute five items into each tester. If the total number of failure items from the two-stage testing within four months for each stage is less than 2 then the lot is accepted; otherwise, the lot is rejected. According to this two-stage sampling plan, the experiment must be repeated when the number of failure is 1 from the first-stage test for four months. The probability of acceptance for this plan is 99.99% and 47 ASN = .

Example-2
Lio et al. [12] showed that the lifetime of oil breakdown of an insulating fluid under hightest voltage follows the Burr type XII distribution with k =5.49 and b =0.85. Suppose that an experimenter would like to use the proposed two-stage sampling plan to establish the true unknown 10 th percentile lifetime for the product is at least 6 months and experiment will be stopped after 6 months. Further suppose that the in laboratory the experimenter has facility to install ten items on a single tester. This information leads to 0 q δ =1.0. Let β =0.05 and 0 / 4 q q t t = with α =0.05 for this experiment. Then from Table 5  g = The sampling plan is implemented as follows: randomly select 120 items and distribute into ten testers such that tenitems on a single tester. Accept the product lot if less than seven failure items are recorded in 6 months and reject the product if more than 7 failures is recorded before 6 months. Otherwise, randomly select another 50 items and distribute into five testers evenly and test for 6 months. If the total number of failures from both stages is less than 11, then the lot is accepted; otherwise, the lot is rejected. From Table 5, the probability of acceptance is 95.58% and ASN =122 for this plan.

Comparison of Plans
In this section, the proposed two-stage acceptance sampling plan is compared with a single stage acceptance sampling plan. It should be noticed that we cannot compare these two acceptance sampling plans by using the sample size, since the two-stage sampling plan could have a chance to use the first stage to make a decision. Therefore, the minimum sample size needed for a single sampling plan will be used to compared with the ASN for the two-stage sampling plan. To save space, only the comparison for the Burr type XII with parameters 5.49, 0.85 k b = = is displayed in this section. Table 6 shows the minimum number of groups and acceptance number for the total number of failures and 2 ( ) L p for single stage acceptance sampling plan. It could be noticed from Table 5 and Table 6, the proposed two-stage sampling plan generally has ASN less than the sample size needed for a single sampling plan under the given rest conditions in both Tables 5 and 6.

Conclusions
In this paper, a two-stage grouped acceptance sampling plan is developed for Burr type XII percentiles under the truncated life testing. Many useful tables for the proposed two-stage grouped acceptance sampling plans have been obtained for practical applications and two industrial examples have been presented to illustrate the applications of the proposed two-stage grouped sampling plan. The comparison between the proposed two-stage grouped acceptance sampling plan and single-stage acceptance sampling plan show that the proposed two-stage grouped acceptance sampling plan generally needs less test items than single-stage grouped acceptance sampling plan does.