Group Acceptance Sampling Plans for Lifetimes Following an Odds Exponential LogLogistic Distribution

A group acceptance sampling plan is developed for odds exponential log logistic distribution (OELLD) based on truncated lifetime with a known shape parameter. At a specified level of consumer risk and at the predefined test termination time, we derived the minimum number of groups required for a given group size and the acceptance number using the two-point approach by satisfying the producer’s and the consumer’s risks at the specified quality levels, while the termination time and the number of testers are specified. The values of operating characteristic (OC) function according to various quality levels are derived and the minimum ratios of true average life to the specified life for given producer’s risk are obtained. The results are illustrated with real lifetime data set.


Introduction 1.1 Introduction to the Research Problem
The reputation of companies depends upon the high reliability of their products.These companies compete with each other on the basis of quality and reliability.Thus quality control techniques became one of the most important tools to differentiate between the competitive ISSN-2424-6271 IASSL enterprises in a global business market.In order to control the quality of the purchased goods, two major alternatives are open to a buyer.First type is the complete inspection, in which each and every item in the lot is to be inspected/ tested.This is often impractical and uneconomical and also involves cost and time for inspection.Secondly, the partial inspection in which a sample of items are taken which are inspected and the whole lot is accepted or rejected depending on whether few or more defective items are found in the sample.
Acceptance sampling plan is an essential tool in the statistical quality control.In most of the statistical quality control experiment, it is not possible to have a100% inspection due to various reasons.In usual sampling plan, the decision of accepting or rejecting a lot is on the basis of a sample of items.
Acceptance sampling plans based on truncated life tests having single item as group were discussed for various distributions by Epstein (1954), Sobel and Tischendrof (1959), Goode and Kao (1961), Gupta (1962), Gupta and Groll (1961), Fertig and Mann (1980), Kantam  The ordinary acceptance sampling scheme is used when the experimenter has the facility to test only one item at a time.In single sampling plans the experimenter needs more time to inspect the products and also samples to be tested.On the other hand, group acceptance sampling plans (GASPs) are used when the tester has the facility to install more than one item at the same time in a single tester.By doing so, substantial testing time and cost can be reduced if the tester to accommodate a multi number of items are utilized for testing purpose.The other importance of GASPs is that it provides strict inspection before the product is sent for the consumer's use.As such GASPs perform better than ordinary plans in terms of reduction of the time, strict inspection, cost, energy, minimize the risks and labor.

Research Problem
The main purpose of this article is to develop the group acceptance sampling plans for the odds exponential log-logistic distribution (OELLD) introduced by Rosaiah et al. (2016).Gupta (1962) suggested that for a skewed distribution, the median represents a better quality parameter than the mean.On the other hand, for a symmetric distribution, mean is preferable to use as a quality parameter.Since OELLD is a skewed distribution, we prefer to use the percentile lifetime as the quality parameter and it will be denoted by   .The rest of this article is organized as follows.In Section 2, we describe concisely the OELLD distribution and group acceptance sampling for lifetime percentiles under a truncated life test is discussed.The description of the proposed methodology with real lifetime data is presented in Section3.Comparison of distributions in Section 4 and finally, conclusions are made in Section 5.
This OELLD distribution has increasing failure rate (IFR) for the shape parameter 1   , decreasing failure rate (DFR) for 1   .This shows that this distribution can be used for industrial applications and reliability studies.The 100q-th percentile of the OELLD is given by  

1
, where ln(1 ) Hence, for the fixed values of 0 0 and      , the quantile q t given in Equation ( 3) is the function of scale parameter 0   , that is Note that 0  depends on 00 and  , to build up acceptance sampling plans for the OELLD, ascertain 0 qq tt  , equivalently that  exceeds 0  .

The Group Acceptance Sampling Plan (GASP)
Here, we discussed group acceptance sampling plans (GASPs) when lifetime of the product follows OELLD.We considered the GASP under the truncated life test, which is based on the total number of failures from all groups.The procedure for the proposed plan is as follows [Aslam et al. (2011)]:  Randomly draw a sample of size n from a production lot, allocate r items to each of g groups (or testers) so that n = rg and put them on test until the pre-determined test time 0 t . Accept the lot when the number of failures from each g groups is smaller than or equal to c. Truncate the test and reject the lot as soon as the number of failures from all g groups is larger than c before 0 t .The probability of accepting a lot for the group acceptance sampling plan P( ) p based on the number of failures from all groups under a truncated life test at the test time schedule 0 t is Where g is the number of testers, c is the acceptance number, r is the group size and p is the probability of getting a failure within the life test time schedule, 0 t .If the product lifetime follows OELLD, then 0 0 0 0 ( ; , , ) Usually, it would be convenient to determine the experiment termination time, 0 t , as 00 0 qq tt   for a constant 0 q  and the targeted 100q-th lifetime percentile, 0 q t .Let q t be the true 100q-th lifetime percentile.Then, p can be rewritten as The design parameters of the proposed GASP are obtain using two points approach on the OC curve by considering the producer's and consumer's risks.In this method the quality level is studied with the ratio of its quantile to the lifetime, 0 qq tt .The producer's point of view is the probability of lot acceptance should be at least 1   at the acceptable reliability level (ARL), So the producer demands that a lot should be accepted at various levels, say 0 qq tt =2, 4, 6, 8 in Equation ( 6).Whereas, the consumer's viewpoint is the lot rejection probability should be at most  at the lot tolerance reliability level (LTRL), p 2 .In this way, the consumer considers that a lot should be rejected when 0 / qq tt =1, in Equation ( 6).Now, let us consider For a specified producer's risk 0.05  4. The operating characteristic values are also calculated for these values.From Tables 1 to 4, we observe that the percentile ratio increases when the number of groups g reduces.Similarly, as r increases from 5 to 10, the number of groups reduces.

Description of the Proposed Methodology with Real Lifetime Data
The proposed methodology is illustrated with following real lifetime data on breaking stress of carbon fibers (in Gba) and it is taken from Nichols and Padgett (2006).0.39,0.81,0.85,0.98,1.08,1.12,1.17,1.3.11,3.11,3.15, 3.15,3.19,3.19,3.22,3.22,3.27,3.28,3.31,3.31,3.33,3.39,3.39,3.51,3.56 .The goodness of fit is tested using the Kolmogorov-Smirnov test, we found that the distance between the data and the fitted OELLD is 0.0604 with p-value is 0.8582.Hence, the two-parameter OELLD provides reasonable fit for the breaking stress of carbon fibers and Figure 1 also emphasized the same.
Since the life times of a product given in the above data follows OELLD with MLEs ˆ20.4032 and 2.7932

 
. Suppose that it is desired to develop the group acceptance sampling plan to satisfy that the 50 th percentile lifetime is greater than breaking stress of carbon fibers0.40 through the experiment to be completed breaking stress of carbon fibers by 0.80 to protect the producer's risk at 5%.For ˆ20.4032 and 2.7932

 
, the consumer's risk is 25%, r=5, 0 1.0 q   and 0 / qq tt =2, the minimum number of groups and acceptance number given by g=2 and c=3 from Table 4. Thus the design can be implemented as follows: select a total of 10 products, and allocate five items to each of the two groups.We can accept the lot when no more than three failures occur before breaking stress of carbon fibers0.80 from each of two groups.According to this plan, the breaking stress of carbon fibers could have been accepted because there is only one failure before the termination time 0.80.

Comparison of Distributions
In the Table 5 we  The acceptance number for the OELLD is smaller as compared to generalized log-logistic distribution by using to 50 th percentile.

Conclusions
In this article, we studied a group acceptance sampling plans when lifetime of the follows OELLD.The plan parametric quantities like the number of groups, g, and acceptance number, care determined by considering the consumer's risk and producer's risk simultaneously.
Our proposed plan noticed that if the percentile ratio increases, the number of groups g reduces and as r increases the number of groups reduces for all the parametric combinations considered in this article.
The plan also illustrated with real lifetime data for industrial purpose.

Figure 1 .
Figure 1.Estimated density and Q-Q plot for OELLD.
compare the plan parameters for the generalized log logistic distribution (GLLD) studied by Aslam et al. (2011), Type-II generalized log logistic distribution (TGLLD) developed by Rosaiah et al. (2016) and OELLD when 2 To save time and cost in the life test, it is very often to put a number of items in a tester.In this life test, a tester is called a group(g) and the number of items (r) in each tester is called the group size.The acceptance sampling via the group life test is called the group acceptance sampling plan (GASP) which is also often implemented under a truncated life test.For such a type of test, the determination of the sample size n=rg is equivalent to determine the number of groups.This type of testers is frequently used in the case of so-called sudden death testing that is discussed byPascualand Meeker (1998) and Vlceket al. (2003).Recently, Jun et al. (2006) introduced this group concept into acceptance sampling plan and developed variable sampling plans for sudden death testing for the Weibull distribution.