1. INTRODUCTION
With the advancement in manufacturing technology, the amount of production has increased its pace in manifold in last two decades. But products or items have variations even though they are produced by the same machine and under same manufacturing conditions. So, to check the lifetime and quality of products, the concept of acceptance sampling is gaining popularity. Acceptance sampling is a scrutiny method used to determine, whether to accept or reject the lot of the products based on the results of the inspection of sample items. The main problem in acceptance sampling plan is the total time lost during the experiment of the life test of the sample items. So, our main focus is to find the smallest sample size to ensure a certain quality level. The lifetime of a product is expected to be high, so the life testing experiments are usually terminated before a prefixed time denoted by t_{o}. This type of test for evaluating the lifetime of a product is called the truncated life test. In truncated life tests, experiments carried out to obtain the lifetime of an item. We are especially interested in determining the probability that a unit which has the satisfactory operation during experiment is classified as a nondefective unit. When acceptance sampling plans are implemented, the decision of the lot is immediately made by the inspection of the sample items so that it saves time and reduces cost of experiment. There may be a chance to classify a nondefective lot as a bad lot or classify defective lot as a good one. When the nondefective lot is rejected the producer will be affected and when a defective lot is accepted, the consumer will be affected. Therefore, the probability of rejecting a good lot or accepting a bad lot is usually referred to as producer’s risk (α) and consumer’s risk (β) respectively.
In the field of acceptance sampling, there are many sampling schemes including single acceptance sampling plan, double acceptance sampling plan, repetitive sampling plan. Due to its simplicity, the single acceptance sampling plan is most widely used. In this plan, the examined lot is rejected if the number of failures is larger than a preassigned acceptance number. One may refer to, AlOmari and AlHadhrami(2018) described acceptance sampling plan for extended exponential distribution and reveals by an application of a real data set that the single acceptance sampling plan can be exercised in the industry. Chowdhury (2016) discussed to obtain the minimum sample size required to guarantee a certain median life of the investigational units in acceptance sampling plan for generalized Weibull distribution. AlOmari (2018) concluded that the acceptance sampling plans for the Garima distribution gives smaller sample sizes than several other acceptance sampling plans under the identical conditions. Mahdy et al. (2018) suggested, the Skew generalized inverse Weibull distribution is considered to be best than generalized inverse Weibull distribution using single acceptance sampling plan as it yields more operating characteristic values. AlNasser et al. (2018) introduced single acceptance sampling plan in which they discussed that both consumers and producers are directed to assume this plan in order to save time and minimize production cost. In double acceptance sampling plan, two samples are taken, and the total number of failures are observed to decide about the acceptance or rejection. For more information about double acceptance sampling plan refer to Singh et al. (2015) in which they revealed that the double acceptance sampling plan require smaller sample size than the single acceptance sampling plan. Aslam et al. (2010) discussed that the least sample size of the both first and second samples are determined to certify that the true median life is longer than the given life at the specified consumer’s risk. The preference of any sampling scheme mainly depends on the degree of accuracy desired by the organization as well as resources available for inspection. It is evident that acceptance sampling provides economy in inspection with a good chance of maintaining the desired quality standard.
Many researchers have developed new plans which give the best results under the constraint of two risks and time termination ratio. Few of such plans are Sequential Sampling (SS) plan and repetitive acceptance sampling plan (RASP) and multiple deferred state sampling plan. Recently, Balamurali et al. (2017) studied multiple deferred state sampling plan for Weibull distributed life time assuring mean life and revealed that average sample number of the multiple deferred state sampling plan is small when compared with repetitive group sampling plan and single sampling plan for both consumer’s risk and producer’s risk. RASP is an attribute sampling plan first used for normal distribution. Singh et al. (2018a), Singh et al. (2018b) studied RASP for generalized inverted exponential distribution and Inverse Weibull distribution respectively and they compared the results of RASP with a single acceptance sampling plan in terms of average sample number. Balamurali and Jun (2006) studied a variable RASP for the normal distribution and compared their proposed RASP with the single acceptance sampling plan for normal distribution. In the end, they concluded that the variable RASP was more efficient than the variable single acceptance sampling plan in terms of smallest sample size required. Aslam et al. (2016) developed multiple dependent state repetitive group sampling plan for Burr XII distribution in which they discussed the advantages of the proposed plan over single sampling plan. Aslam et al. (2017) discussed the efficiency of the control chart for detection of the nonrandom change in the manufacturing process using repetitive sampling plan. Aldosari et al. (2017) introduced a new attributed control chart which is more sensitive than the existing control charts in terms of small change in the production process. Yen et al. (2018) discussed the multiple dependent state repetitive sampling plan which provide the better results than the existing plan in terms of operating characteristic curve and average sample numbers. Aslam et al. (2018) explained that the multiple dependent state repetitive sampling plan is more economical than the single sampling plan and repetitive sampling plan. SS plan is different from single, double or multiple sampling plans. In SS plan the sequence of samples are taken from a lot of finished items. In the past, sequential sampling plan have been used in truncated life test for Weibull distribution by Rasay et al. (2018) and they reveal that the SS plan can considerably reduce the ASN and has an enhanced performance with respect to the zero one double sampling plan and repetitive group sampling plan. The main objective of this article is to minimize the average sample number subject to some constraints and give the significant comparison between the sequential sampling plan and some existing sampling plan in terms of average sample number. Also furthermore, we present a significant comparative study of proposed plan with Rasay et al. (2018). To our best knowledge, the SS plan for assuring mean life under the generalized Pareto distribution has not shown in a literature yet.
2. GENERALIZED PARETO DISTRIBUTION
The Pareto distribution is named after Italian economist Vilfredo Pareto, is a powerlaw probability distribution. The generalized Pareto distribution (GPD) was introduced by Pickands (1975). The GPD is a family of continuous probability distributions. It is specified by three parameters namely location, scale, and shape. The GPD contains uniform, exponential and Pareto distributions as special cases. It has application in the various fields, including reliability studies, acceptance sampling theory and the analysis of environmental extreme events. The probability distribution function and cumulative distribution function of GPD are given as below:
where $\gamma <t<\infty ,\hspace{0.17em}\hspace{0.17em}\xi >0,\text{}\sigma 0,\text{}\delta 0,\hspace{0.17em}\gamma $ is location parameter, ξ is scale parameter and (σ, δ) are shape parameter, Γ(.) is gamma function.
The mean and variance of GPD are,
In the field of probability and quality control of the product, the product quality is usually expressed in terms of the percentage of the defectives or the fraction of nonconforming. The quality levels corresponding to the producer’s risk and consumer’s risk are respectively termed as acceptance quality level (AQL) and limiting quality level (LQL). A producer wants to use an acceptance sampling plan which helps him to guarantee a high acceptance probability for a submitted lot when the defective level is greater than or equal to AQL. From the consumer's perspective, he uses that acceptance sampling plan which ensures a low acceptance probability for a submitted lot when the defective level is smaller than or equal to LQL. In this paper, we consider a product whose lifetime follow GPD.
3. DESIGN OF SEQUENTIAL SAMPLING ACCEPTANCE PLAN FOR TRUNCATED LIFE TEST
Sequential Sampling (SS) plan is simply an extension of double acceptance sampling plan. At each stage of sampling, the cumulated results are analyzed to take a decision of accepting or rejecting a lot. If at any stage no final decision can be taken, then another sample is drawn to take decision. How many total samples examined, is a function of the results of the sampling process. We will take one sample at a time and it is called item by item sequential sampling. The operation of this plan is illustrated below.
There are two boundary lines which are parallel to each other known as the acceptance line and rejection line. At each stage, the cumulative observed number of inspected items and cumulative observed number of defective items are plotted on the chart as one point. If the point lies below the acceptance line then the lot is accepted. If this point falls above the rejection line, then the lot is rejected and if it falls between these two lines, another sample must be taken. Hence, this sampling procedure will be terminated as soon as the point does not fall between these two boundary lines for the given values of α, P_{1}, β, P_{2}. The boundary lines can be calculated using the belowmentioned equations.
where h_{1}, h_{2} and s are given by
and k is solved by the equation
4. COMPUTING THE OPERATING CHARACTERISTIC CURVE AND AVERAGE SAMPLE NUMBER
We are interested to check the mean lifetime of the product which is denoted by μ_{0} . The lot is regarded as good if there is enough evidence that μ≥μ_{0} where μ denotes the actual life of the product and regarded as bad if μ<μ_{0}. The test termination ratio t_{0} is a multiple of specified mean lifetime μ_{0} and is given by ${t}_{0}=a{\mu}_{0}$ where a is constant coefficient for the test termination time. The quality level of each item is denoted by r and it is a ratio of mean time to specified mean time i.e. $r=\frac{\mu}{{\mu}_{0}}$.
An operating characteristic curve displays the probability of lot acceptance in each quality level. The quality level r_{1} is corresponding to consumer’s risk and r_{2} is the quality level corresponding to producer’s risk. Corresponding to these quality levels we calculate the probability that the test item fails during the test termination time point t_{0}. For the GPD it is given by
where P is independent of location parameter γ and scale parameter ξ.
We apply the following model to derive the parameter of the proposed sampling plan.
Minimize Average Sample Number (ASN) Subject to
P(P_{1}) is acceptance probability at consumer’s risk and P(P_{2}) is acceptance probability at producer’s risk where P_{1} and P_{2} are calculated corresponding to r_{1} and r_{2} using equation (11) and ASN is calculated from the formula
P and P_{a} are nonconforming proportion and the probability of lot acceptance respectively and can be obtained based on these equations.
Procedure 1

1. From equation (11), compute the value of P for the specified value of r and it is denoted by P_{r}.

2. By substituting the value of P in (15) compute the corresponding value of θ.

3. By putting the value of θ in equation (16) compute the value of P_{a}.

4. Insert the values of P and P_{a} in equation (14) and calculate the corresponding value of ASN.
The value of ASN and the probability of lot acceptance for SS plan for shape parameter δ = σ = 2 are reported in Table 1. We mentioned that two different values of a such as 0.7 and 1.0, four different levels of consumer’s risk β, namely 0.25, 0.10, 0.05, 0.01 and producer’s risk α as 0.05 and 0.01 are taken. The quality level at consumer’s risk r_{1} is considered as 1 and producer’s risk r_{2} is considered as 2, while four different levels of product quality r = 2, 4, 6, 8 are taken into the consideration. With the increase in product quality level the ASN decreases, this pattern is clearly visible for all levels of the consumer’s risk, producer’s risk and shape parameters δ = σ = 2, 3 in Table 1, 2, 3 and 4. Whereas the probability of acceptance increases up to 1.000 for all levels of consumer’s risk and producer’s risk. In Table 3, the quality level at producer’s risk is considered as 4 and remaining all the values for quality levels are same as that of Table (1). The ASN values for shape parameter δ = σ = 2 and producer’s quality level r_{2} = 2 in Table 1 are higher than the value of ASN in Table 3 for the shape parameter δ = σ = 2 , but at different level of producer’s quality r_{2} = 4. Same pattern is observed for the shape parameters δ = σ = 3 in Table 2 and 4.
5. DESIGN OF REPETITIVE ACCEPTANCE SAMPLING PLAN
The Repetitive Acceptance Sampling plan (RASP) under a truncated life test is described as follows:

Step 1: From the submitted lot take a random sample of size n units and put them on life test, separately until a predecided experiment time t_{0}.

Step 2: After the termination of time t_{0} , if the number of failures D is smaller than or equal to c_{1}, then the lot is accepted. Reject the lot and stop the experiment as soon as the number of failures exceeds c_{2}.

Step 3: If c_{1} < D ≤ c_{2} then repeat the experiment starting from step I.
If c_{1} = c_{2} then the above plan matches with a single acceptance sampling plan. RASP is different from the double acceptance sampling plan because it uses an individual sample to a decision until the decision is reached whereas the double acceptance sampling plan uses the combined sample for making the second decision if the first is inconclusive. The OC function of the RASP, according to Sherman (1965) is given by:
where P is the probability that a product under test fails before t_{0}, P_{a} is the probability that lot is accepted based on the single sample and P_{R} is the probability that a lot is rejected based on the single sample. These probabilities can be calculated as follows:
and
where D denotes the number of failures by time t_{0} . The operating characteristic function of is given by
Procedure 2

1. Specify the values of ${r}_{1},\hspace{0.17em}{r}_{2},\hspace{0.17em}r,\hspace{0.17em}\alpha ,\hspace{0.17em}\beta $ and shape parameters.

2. From equation (11), compute the values of ${P}_{1},\hspace{0.17em}{P}_{2}$ associated with ${r}_{1},\hspace{0.17em}{r}_{2}$ respectively.
Balamurali and Jun (2006) suggested that the parameters be determined to minimize average sample number ASN given by equation subject to inequality (21), (22) simultaneously.
where P_{a} and P_{R} in ASN are calculated corresponding to an incoming quality level r of an item.
6. AN ILLUSTRATIVE EXAMPLE
Suppose that a lifetime of a product follows GPD with shape parameter δ = 2, σ = 2 . Now we are interested to find out the design parameters of the SS plan to assure that the mean lifetime of this product becomes more than 1000 hours, while the experiment must be terminated after 700 hours. The consumer wants that risk of accepting the lot with the mean value of 1000 hours to become less than 0.01 and the producer wants the risk of rejecting the lot with the mean value of 4000 becomes less than 0.05. Based on the given data, we obtain a = 0.7, r_{1} = 1, β = 0.01, r_{2} = 4, α = 0.05. Now firstly we obtain values of P_{1} and P_{2}. Based on the equation (11) by the SS plan P_{1} = 0.4101 and P_{2} = 0.0367. Based on the equations (7, 8, 9, and 10) the values of k, s, h_{1}, and h_{2} are equal to 2.903, 0.1689, 1.5682, and 1.0282 respectively. Hence, the acceptance line (X_{A}) and rejection line (X_{R}) are
The result of acceptance and rejection numbers for value of sample size n from 1 to 20 are presented in Table 5. For example, consider the case of computing the rejection and acceptance number for n =12 .
Inserting n=12, we get
Acceptance and rejection number are always the integer, hence the value of X_{A} is rounded downward and values of X_{R} are rounded upward. Hence the rejection and acceptance number for n =12 are 4 and 0 respectively. Concluding the above results, for n =12 , if the collective number of observed defects, until this stage is 1, 2, 3 then the sampling process must be continued. If the collective number of defects until this stage is 0 then the lot is accepted and if the collective number of defects until this stage, is more than or equal to 4 then the lot is rejected. In the same way the values of acceptance and rejection numbers can be calculated for given values of shape parameter, producer’s risk, consumer’s risk and test termination constant.
Where n denotes the number of items checked; AC: acceptance number, RC: rejection number a: means that it is not possible to accept the lot; b: means that it is not possible to reject the lot.
7. APPLICATION IN INDUSTRY
In this section, the industrial application of SS and RASP is illustrated in the truncated life test based on the real example. Suppose a bulb manufacturer would like to know whether the mean life of the bulb is longer than the specified mean life μ_{o} = 2000 hours. Further now the manufacturer wants to run the experiment for 1400 hours. If the lifetime of bulbs follows a generalized pareto distribution with shape parameter δ = σ = 2 . Suppose that the consumer’s risk is 5% when the true mean life is 2000 hours and the producer’s risk is 5% when the true mean is 8000 hours. From the above given problem, we get $a=0.7,\hspace{0.17em}{r}_{1}=1,\hspace{0.17em}{r}_{2}=4,\hspace{0.17em}\alpha =0.05,\hspace{0.17em}\beta =0.05,\hspace{0.17em}r=6$
From the plan of sequential sampling, the equations of the boundary lines are given by:
Based on the procedure1, we find the values of ASN from Table 3 is 6.45. For example, if n = 7 then the lot is accepted if the cumulative number of failures is 0 and rejected if the cumulative number of failures is more than or equal to 3. If the cumulative number of failures are 1 or 2 then the procedure is repeated.
Based on the procedure 2, the following results are obtained for RASP in this example:
This result means that a sample with size n = 7 is drawn from the submitted lot and put them on the test for t_{o}=1400 hours. If the number of failures is 0 then accept the lot and if the number of failures is 2 or more then reject the lot. Otherwise, it is necessary to take another sample from the lot. The procedure is repeated until the lot is accepted or rejected. From above results, we observed that the sequential sampling plan give smaller ASN than RASP.
8. COMPARATIVE STUDY
The comparison between SS plan with RASP and also the comparison of ASN for SS plan under the generalized pareto distribution with Rasay et al. (2018) are discussed in this section.
8.1 Comparison of SS plan and RASP
In this subsection, we are comparing the above proposed plans with each other in terms of ASN. The results are presented in a Table 6. The values of the parameters n, c_{1}, c_{2} for RASP are calculated according to the procedure 2 given in section 5 at α = 0.05, a = 0.7, 1.0, r_{1} = 1, r_{2} = 2, β = 0.01, δ = σ = 2. The values of ASN of RASP are calculated at different levels of the product quality (r = 2, 4, 6, 8) at the values of the parameters $n,\hspace{0.17em}{c}_{1},\hspace{0.17em}{c}_{2}$ and the values of ASN for SS plan are taken from Table 1 at different levels of the product quality (r = 2, 4, 6, 8). Based on the results the inference is drawn that the SS plan is more beneficial as this plan gives smaller ASN than the RASP. For example, when r_{2} = 2, β = 0.01 = r = 2, α = 0.05, δ = σ = 2 and a = 0.7, then ASN under the proposed plan for generalized Pareto distribution is 23.15 and for repetitive acceptance sampling plan is 31.13, but when we change only the test termination ratio a = 0.7 to a = 1.0 and all other values of producer’s risk, consumer’s risk, product quality level and shape parameters remains same, the ASN under the proposed plan for generalized Pareto distribution is reduced to 14.92 and for repetitive acceptance sampling plan is reduced to 21.05 as given in the Table 6.
8.2 Comparison of Proposed Plan with Rasay et al. (2018) using ASN
In this subsection, we observe that the proposed plan under GPD provides a smaller amount of ASN than the Rasay et al. (2018). For example, when r_{2} = 2, β = 0.10, r = 8, α = 0.01 and a = 1.0, then ASN under the proposed plan for GPD is 2.76 and for Rasay et al. (2018) is 3.69 as given in the Table 7, which is smaller than the Rasay et al. (2018) and this type of difference observed for all the levels of the product quality (r = 2, 4, 6, 8). So the proposed plan under the GPD is more efficient than the Rasay et al. (2018).
9. CONCLUDING REMARKS
In this paper, SS plan and RASP have been developed for truncated life test on products whose lifetime follows the generalized Pareto distribution. A procedure is given for calculating operation characteristic function and average sample number in the SS plan. The procedure of RASP is also presented for calculating the sampling plan parameters. The ASN was determined for the SS plan and RASP by satisfying both consumer’s and producer’s risks simultaneously. The comparative study is done with the help of an application of industry which reveals that the SS plan gives smaller ASN than RASP. This indicates that the proposed plan is better than RASP in the view of guarding both the consumer and producer with minimum inspection of the products during the life test experiment. Using ASN, we performed a comparative study of the proposed plan along with repetitive acceptance sampling plan and proposed plan with Rasay et al. (2018) in the comparison section which indicates that the proposed plan gives better outcomes than Rasay et al. (2018). The current study can also be extended to some other distributions, for example, generalized exponential distribution.