New Distributions in Designing of Double Acceptance Sampling Plan with Application

In this paper, acceptance sampling plans as, double for the lifetime tests is truncated at pre-fixed time to determine on acceptance or rejection of the submitted lots are provided. The probability distributions of the lifetime of the product are determined based on three distributions: generalized inverse Weibull, skewgeneralized inverse Weibull and compound inverse Rayleigh. The median lifetime of the test unit as the quality parameter is considered. The minimum sample sizes to assure that the actual median life is more than the specified life, OC values according to different quality levels and the minimum ratios of the actual median life to the specified life at the determined level of producer's risk for acceptance sampling plans are obtained. Numerical cases are introduced to illustrate the applications of acceptance sampling plans.


Introduction
The quality of any product is a random variable, even if we could control all factors of production (workers, raw materials, organization and administration and capital). Some products which are produced in the same factory, methods, materials and workers could be conforming or nonconforming. Deciding the quality of any product depends on some quality standards. Applying these standards could lead to considering the product unit conforming or nonconforming. If the produced unit is conforming, this means that all standards of quality are achieved. On the other hand, the produced unit is considered nonconforming if one or more of these standards is missing. In industry, the concept of quality does not mean to produce the best product but to produce a product conforming to the standards or to certain standards. For example, the product should be suitable for the purpose for which it has been designed, it should satisfy the desires and needs of customers, and its cost should be as low as possible to be acceptable to customers and compete in the market.

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If a final decision cannot be made accordingly to the inspection of the first sample, because the numbers of non-conforming units fall between the acceptance and rejection levels, a double sampling plan gives us the opportunity to draw a second sample. Despite the importance of the double sampling plan, yet the number of studies in this area as compared to the number of studies in the single sampling plan is fewer such as Aslam and Jun (2010), Aslam et. al. (2010a) and Muthulakshmi and Selvi (2013). The rest of the paper is organized as follows. In section 2, the definitions of some probability distributions based on life test are provided. In section 3, the double acceptance sampling plan is designed. In section 4, assessing acceptance sampling plans and numerical cases are introduced to illustrate the applications of acceptance sampling plans

Some New Distributions Based on Life Test
For the CIR distribution, the scale parameter ; this will lead to having the same result for both distributions.

Designing of Double Acceptance Sampling Plan
Let  be the actual median life and 0  is the specified median life of a product, under the hypothesis that the lifetime of the product follows GIW, SGIW or CIR distributions. If , the lot is accepted, otherwise the lot is rejected. In acceptance sampling schemes, this hypothesis is tested according to the number of failures in a sample with pre-determined time. To provide the operating procedure of the double acceptance sampling plan based on truncated life test was provided by (Aslam, et al. (2010a)).
Step 1: Draw the first sample of size 1 from a lot and put them on test for 0 units of time.
Step 2: Accept the lot, if there are 1 or few number of failures 1 that occurred before a pre-determined experiment time 0 . Reject the lot and terminate the test, if ( ) failures are recorded. If the number of failure 1 is between 1 then draw the second sample of size 2 from the same lot and put them on test for another 0 unit of time.
Step 3: Accept the lot, if the total number of failures from the first and second samples is less than or equal to 2 , i.e. ( ).
Otherwise, terminate the test and reject the lot.
The first term in equation (4.1) represents the acceptance probability on the first sample and the second term does the acceptance probability on the second sample (Aslam, et al. (2010a)). When 0 1 = c and 2 2 = c , i.e. (zero and two failure schemes), the probability that the lot is accepted, can be obtain as  (2010)) explains, consumers prefer an acceptance sampling plan with lower acceptance limits. They argue that when the lot is accepted with several failed items from a test, the consumers may not understand this although it may happen probabilistically. For the zero and one failure schemes, the lot acceptance probability is: 2) Here is failure probability before the termination time 0 .

The minimum sample sizes
The minimum sample sizes 1 and 2 for zero and one failure schemes ensuring ≥ 0 at the consumer's confidence level  or consumer's risk ( )  − 1 can be determined by solving the following inequality: Where  is the failure probability of an item before the termination time 0 and it is defined in equations (2.1), (2.9) and (2.10). There is more than one solution that may satisfying the inequality (3.3), so (Aslam and Jun (2010)) proposed to minimize the average sample number (ASN) to find 1 and 2 by putting the constraint 1 ≥ 2 .
For a double sampling plan, the average sampling number is given by: Here 1 is the probability of making a decision on the first sample. The probability 1 can be expressed as When 1 = 0 and 2 = 2 the ASN for double sampling plan can be obtaining as:

OC function based on double sampling
Once the minimum sample size 1 and 2 are obtained, one may be concerned to find the probability of acceptance the lot when the quality of an item is conforming enough. As mentioned previously, an item is considered to be conforming if the true median life to specified median life >1. The OC values according to

The minimum ratios
The producer may be interested in knowing the minimum product quality level that will ensure the producer's risk, say , at the specified level. For a given value , the minimum ratio to ensure the producer's risk is less than or equal to = 0.05 can be obtained by satisfying the following inequality:

Description of Tables and An Illustrative Case
In this subsection, the numerical cases study for ( ) Thus, for three lifetime distributions, 6, 19 and 16 units respectively have to be put on test for 750 hours. The lot is accepted if zero non-conforming item is recorded during the experiment and rejected if two or more non-conforming units are found. If exactly one non-conforming unit is found draw another sample for three distributions and put them on the same test. Accept the lot if a total of non-conforming units are one or fewer are recorded otherwise reject the lot. It is observed that the minimum sample sizes increase quickly as the shape parameter increases when the termination time ratio is short for GIW, SGIW and CIR distributions. This means, the lot is accepted with probabilities 0.661, 0.991 and 0.701 for GIW, SGIW and CIR distributions respectively if the true median life of the units in the lot is twice than the specified median life. For three distributions, the probability of accepting the lot increases up to "one" if the true median life is 6 times than the specified median life. Also, the producer's risk for GIW, SGIW and CIR distributions will be 0.339, 0.009 and 0.299 respectively. To know the ratio corresponding to the producer's risk of 0.05, can be found from the cumulative Table (3.7). For example, when the lifetime of product follows GIW, SGIW or CIR distributions, the minimum ratios 0   are 2.57, 1.802 and 2.818 respectively. In addition, the minimum ratios decrease as the shape parameters increases.

Concluding Remarks
In this paper, double acceptance sampling plans when the lifetime of product follows GIW, SGIW or CIR distributions in order to make the decision of accepting or rejecting 345 the lots was proposed. It was concluded that the SGIW is better distribution for fitting these plans compared to GIW and CIR distribution in spite of the sample size is larger for SGIW than the GIW and CIR distributions.