IJCRR

IJCRR - 3(11), November, 2011

Pages: 23-35

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TIME TRUNCATED GROUP ACCEPTANCE SAMPLING PLANS FOR LIFETIME PERCENTILES UNDER
GENERALIZED LOG-LOGISTIC DISTRIBUTIONS

Author: Muhammad Aslam, Muhammad Shoaib, Chi-Hyuck Jun, Nadia Saeed

Category: General Sciences

Abstract:A group acceptance sampling plan is considered for a truncated life test when a multiple
number of items as a group can be tested simultaneously in a tester. Group acceptance sampling
plans under the truncated life test are designed for lifetime percentiles when the lifetime of a
product follows the generalized log-logistic distribution or the Burr type XII distribution. The
design parameters such as the number of groups and the acceptance number required are
determined by satisfying the consumer's risk and producer's risk at the specified quality levels,
while the number of testers and the termination time are specified. The comparison between the
distributions is given using the percentiles life of the products. The results are discussed with
real life industrial examples. The extensive tables and graphs are given to explain the procedure
developed under the generalized log-logistic distribution or the Burr type XII distribution.

Keywords: Generalized log-logistic distribution, Burr type XII distribution, Percentile, Consumer‘s risk, producer‘s risk, truncated life test

Full Text:

1. INTRODUCTION

In practice, it is difficult to test the complete life time of every item from a large lot due to the cost and the time required for the inspection. So, the decision about the acceptance or rejection of submitted lots should be based on sampled items selected from the lot. The single acceptance sampling plan is often adopted in laboratory for the life test purpose due to its simplicity. In this sampling scheme, the capacity to install items on a tester is limited to one. Therefore, the experimenter needs the number of testers equal to the number of items selected from the lot. Obviously, installing a single item to a single tester requires lots of efforts, time, and cost. Saving these resources including cost and time is an important issue in life testing. The cost and the time are the factor which is directly related to the number of items selected from a lot. The larger the sample size the larger the producer‘s loss. Therefore, researchers have been trying to propose or improve the sampling plan to require smaller sample size in a life testing. Nowadays testers accommodating a multiple number of items at a time are used in practice because testing time and cost can be saved by testing these items simultaneously. For more detail, reader can refer to Aslam and Jun (2009). Items in a tester can be regarded as a group and the number of items in a group is called the group size. The acceptance sampling planused to determine these groups of items will be called a group acceptance sampling plan (GASP). The sudden death testing scheme is always implemented in groups. Many authors including Pascual and Meeker (1998), Vlcek et al. (2003) and Jun et al. (2006) discussed the sudden death testing in groups. More recently, Aslam and Jun (2009) developed a group acceptance sampling plan based on truncated life test when the lifetime of a product is best fitted to the inverse Rayleigh or log-logistic distribution and Srinivasa Rao (2010) developed a group acceptance sampling plan based on truncated life test for the Marshall-Olkin extended Lomax distribution. The ordinary acceptance sampling plans and the group acceptance sampling plans based on time truncated life in the literature are proposed using the mean or median life of the product for assuring the quality and reliability of the product. As stated by Lio et al. (2010a) and Lio et al. (2010b), the existing acceptance sapling plans may not assure the engineering consideration on the specific percentile of item life time. When the quality of a major focus is a low percentile, the acceptance sampling plans based on the mean life could not pass a lot which has the low percentile below the required customer standard. Furthermore, a small decrease in the average lifetime with a simultaneous small increase in the variance can result in a significant downward shift in small percentile of interest. This means that a lot of products could be accepted due to a small decrease in the mean life after inspection. The accepted lot may not meet the consumer‘s expectation if the low percentile is used for the lifetime of products. The mean life may not be applicable to a skewed distribution but the percentile is more suitable to apply on the distribution for making the required results. The median lifetime is suitable when the distribution is skewed. See, for example, Marshall and Olkin (2007) and Aslam et al. (2010). It is important to note that the items produced under the same environment have some random variation in their lifetimes. This variation in failure time can be modeled by a probability distribution. The life time distribution also plays a vital rule to design an acceptance sampling plan. The ordinary acceptance sampling plans based on the truncated life test using various distributions have been discussed by many authors in the literature including Epstein (1954), Goode and Kao (1961), Kantam and Rosaiah (1998), Kantam et al. (2001), Baklizi (2003), Rosaiah et al. (2006), Rosaiah and Kantam (2005), Tsai and Wu (2006), Rosaiah et al. (2007), Aslam and Kantam (2008), and Balakrishnan et al. (2007).

Two risks are always associated with any type of sampling scheme. The probability of accepting a bad lot is called the consumer‘s risks and the chance of rejecting a good lot is called the producer‘s risk. The acceptance sampling schemes including the variable sampling, attribute sampling, skip-lot sampling and the normal to tightened sampling are used to reduce the producer‘s risk and the consumer‘s risk. So, the determination of the design parameters such as the sample size and the acceptance number satisfying both risks is preferable to the single-point approach. Further, as stated by Aslam and Jun (2009), a sampling plan obtained by satisfying only the consumer‘ risk may not always satisfy the producer‘s risk. The two-point approach on the OC curve for designing the variable acceptance sampling plan has been developed and implemented by Fertig and Mann (1998), Jun et al. (2006).

The main purpose of this paper is to propose a GASP based on truncated life tests when the lifetime of an item follows the generalized log-logistic distribution or the Burr type XII distribution. As the best of author‘s knowledge, no attention has been paid to use these distributions to develop the group plans for the lifetime percentiles of the product using the two points on OC curve approach.

2. Generalized Log-Logistic and Burr Type XII Distributions

The generalized log-logistic distribution and the Burr type XII distribution are the life time distributions, which have been widely used in the area of reliability and the acceptance sampling plan for the testing purpose. These two distributions are not symmetric and different generalized forms of the log-logistic distribution. The generalized log-logistic distribution was applied to a breast cancer survival data by Singh et al. (1994). The application of the generalized log-logistic distribution in a double acceptance sampling plan is discussed by Aslam and Jun (2010). Kantam et al. (2001) used the log-logistic distribution in acceptance sampling plans. Recently, Lio et al. (2010a) and Lio et al. (2010b) used the Burr type XII distribution and the generalized Birnbaum-Saunders distribution to develop the ordinary acceptance sampling plan using the percentiles as life time. They showed that both distributions are well fitted to real data. To develop the group plan, we will assume that the lifetime of a product either follows the generalized log-logistic distribution or the Burr type XII distribution. The brief introduction of these two distributions is given as follows: The probability density function (pdf) and the cumulative distribution function (cdf) of the generalized log-logistic distributions are respectively given as

5. CONCLUSION

]We develop the group acceptance sampling plan based on a truncated life test under the assumption that the lifetime of a product follows the generalized log-logistic distribution and burr type XII distribution with known and unknown shape parameter. The two point approach was used for determining the design parameters such as the number of groups and the acceptance number. Our proposed plan indicate that the generalized log-logistic distribution provide the larger number of groups as compared to burr type XII distribution by using the 10th percentile but in 50th percentile the two distribution groups are not quite different. The log-logistic distribution is better than the Weibull distribution. As in acceptance sampling schemes, there is still a capacity available to reduce sample size to save the time and the cost of the experiment. Therefore, there is need to modify the proposed plan using the percentiles of the distributions as a future research.

ACKNOWLEDGEMENTS

The authors are deeply thankful to reviewers and the editor for several valuable comments.

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