A New Acceptance Sampling Plans Based on Percentiles For Type II Generalized Half-logistic Distribution

A New Acceptance Sampling Plans Based on Percentiles For Type II Generalized Half-logistic Distribution

G Srinivasa Rao ¹ , Sd. Jilani* ² , A Vasudeva Rao ³ and S. Bhanu Prakash ⁴

1 Department of Statistics,

The University of Dodoma, Dodoma, PO. Box: 259, TANZANIA.

² Research Scholar (RFSMS), Dept. of Statistics, Acharya Nagarjuna University, A.P., INDIA.

3 Professor & Head, Dept. of Statistics, Acharya Nagarjuna University, A.P., INDIA.

4 Research Scholar, Dept. of Statistics,

Acharya Nagarjuna University, Guntur-522510, A.P., INDIA.

email:[email protected], [email protected], [email protected]

* Corresponding author: [email protected].

(Received on: May 28, 2019) ABSTRACT

Time to failure owing to fatigue is one among the common quality characteristics in material engineering applications. In this article, acceptance sampling plans are developed for the Type II Generalized Half-logistic distribution using percentiles when the life test is truncated at a pre-specified time. The minimum sample size necessary to ensure the required lifetime percentile is obtained under a given customer’s risk and producer’s risk at the same time. The operating characteristics value of the sampling plans is given. The result is illustrated by an example.

Keywords: Acceptance sampling, Operating Characteristic Function, Type II Generalized Half-logistic distribution, Truncated life tests, Producer’s risk, Consumer’s risk.

1. INTRODUCTION

In statistical quality control, acceptance sampling for products is one aspect of quality

assurance. If the quality characteristic is regarding the lifetime of the product, the acceptance

sampling problem becomes a life test. Quality personnel would like to know whether the

lifetimes of products reach the consumer’s minimum standard or not. Traditionally, when the

life test indicates that the mean life of products exceeds the specified one, the lot of products

is accepted, otherwise it is rejected. For the aim of reducing the test time and cost, a truncated life test could also be conducted to determine the tiniest sample size to ensure a explicit mean life of products once the life test is terminated at a pre-assigned time t 0 , and also number of failures determined doesn’t exceed a given acceptance number c. The decision is to accept the lot if a pre-determined mean life can be reached with a pre-determined high probability P ^* that provides protection to consumers. Therefore, the life test is over all at the time the failure is observed or at the pre-assigned time t 0 , whichever is earlier. For such a truncated life test and the associated decision rule, we are interested in obtaining the smallest sample size to arrive at a decision.

Statistical inference is one of the core topics of mathematical statistical. Reliability study plays a vital role in the quality control, and it can save time and money by realizing early conclusions. If a genuine product is rejected on the bias of sample information it is called Type I error. On other hand if an in-genuine product is accepted, then it is called Type II error.

According to Duncan (1986), the decision to accept or reject lot is subjected to the risks associated with these two errors. These procedures are formed as “reliability test plan “or acceptance sampling based on life test

In the literature of acceptance sampling, there are many methods of testing. Epstein (1954) was the first who considered truncated life tests in the exponential distribution. Some other studies regarding truncated life tests can be found in Sobel and Tischendrof (1959), Goode & Kao(1961), Gupta and Groll (1961a), Fertig and Mann(1980), Kantam and Rosaiah (1998), Kantam et al.(2001), Baklizi and El Masri (2004), Tsai and Wu (2006). Balakrishnan et al. (2007), Rao et al. (2008).

All these authors considered the design of acceptance sampling plans based on the population mean under a truncated life test. 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. When the quality of a specified low percentile is concerned, the acceptance sampling plans based on the population mean could pass a lot which has the low percentile below the required standard of consumers. Actually, percentiles provide more information regarding a life distribution than the mean life does. When the life distribution is symmetric, the 50th percentile or the median is equivalent to the mean life. Hence, developing acceptance sampling plans based on percentiles of a life distribution can be treated as a generalization of developing acceptance sampling plans based on the mean life of items. Kantam and Rosaiah (1998) and Kantam et al.

(2001) developed acceptance sampling plans based on half logistic and log-logistic

distributions. Baklizi (2003) developed acceptance sampling based on truncated life tests in

the pareto distribution of the second kind. Balakrishnan et al. (2007) proposed acceptance

sampling plans can be used for the quantiles and derived the formule. Lio et al. (2009,

2010)consider acceptance sampling plans based on truncated life tests to Birnbaum-Saunders

(BS) and Burr Type XII for percentiles and they proposed that the acceptance sampling plans

based on mean may not satisfy the requirement of engineering on the specific percentile

strength or breaking stress. Rao and Kantam (2010) developed acceptance sampling plans the

percentiles from truncated life tests based on the log-logistic. Rao et al. (2012) developed an

acceptance sampling procedure for the inverse Rayleigh distribution percentile under a truncated life test. Rao et al. (2013) developed acceptance sampling plans for percentiles assuming linear failure rate distribution. Rao et al. (2013a) studied acceptance sampling plans for percentiles based on the Marshall-Olkin extended Lomax distribution. Balamurali et al.

(2013) developed acceptance sampling plans based on median life for Fréchet distribution. Al- Omari (2014) constructed acceptance sampling plans based on truncated life tests for three parameter Kappa distribution. Gui and Aslam (2015) discussed acceptance sampling plans based on truncated life tests for weighted exponential distribution. Rao et al. (2016, 2016a) developed some new acceptance sampling plans for odds exponential log-logistic and exponentiated Frechet distributions based on median (50 ^th percentile) life time and in these plans, the minimum sample size is determined under a given both consumer’s risk and producer’s risk jointly.These arguments motivated us to develop acceptance sampling plans based on percentiles. Since Type II Generalized Half-logistic distributionis a skewed distribution we prefer to use the percentile point as the quality parameter, denoted by 𝑡 _𝑞 .

The article is organized as follows. In Section 2, we describe briefly the Type II Generalized Half-logistic distribution. The design of proposed acceptance sampling plan for lifetime percentiles under a truncated life test is mentioned in Section 3. In Section 4, we present the outline of the proposed plan. A real data set is considered for an illustration. Finally, conclusions are drawn in Section 5.

2. THE TYPE II GENERALIZED HALF-LOGISTIC DISTRIBUTION

In this section, we provide a brief summary about a new lifetime model named the Type II Generalized Half-logistic distribution (GHLD).Consider a series system of components with individually and identically distributed (iid) individual lifetimes, for example, F t . The reliability function of such a system is given by _{1 F t} ; hence, the distribution function of the lifetime random variable of a series system is _{1 1 F t} . Taking F t as the standard half logistic model, Kantam et al. (2014) proposed a new model called the standard Type II generalized half logistic distribution (GHLD), whose probability density function (pdf) and cumulative distribution function (cdf) are respectively given by

1 ; 0 0 .

( ; ) 2 1

t

t

t and

f t e

e

(1)

2 ; 0 0

1 ( ; ) 1

t

t and

e

F t (2) If we introduce a scale parameter in the above standard distribution, then the resultant distribution may be called as two-parameter Type II GHLD, whose pdf and cdf are respectively given by

1 ; 0, , 0 . ( ; , ) 2

1

t

t

e t f t

e

(3)

2 ; 0, , 0 . 1

( ; , ) 1

t t

e

F t (4)

where is a scale parameter and is the shape parameter. The Type II Generalized Half- logistic distribution is special case of (1) for 1 .

The 100-th quantile of the Type II GHLD is given as:

, where ln 2(1 ) 1 1

q q q

t q

(5) From the above equation, we may note that for the fixed values of ₀ , the quantile t _q is a function of the scale parameter and at a pre-specified value of t say _q t , we may obtain the _q ⁰ value of , say ₀ , as

0

0 1

ln 2(1 ) 1

t q

q

. (6)

Obviously ⁰

0

q q

t t and we may also note that ₀ depend on ₀ . And to build up acceptance sampling plans for the Type II GHLD it is ascertained that

t q exceeds t _q ⁰ equivalently exceeds ₀ .

3. THE ACCEPTANCE SAMPLING PLAN

The problem considered is that of finding the minimum sample size necessary to ensure a percentile life when the life test is terminated at a pre-assigned time t _q ⁰ and when the confirm number of failures does not exceed a given acceptance number c. The decision procedure is to accept a lot only if the specified percentile of the lifetime is established with a pre-assigned high probability α, that provides protection to the consumer. The life test experiment gets terminated at the time at which (c+1) ^th failure is observed or at quantile time

t q , whichever is earlier.

The probability of accepting a lot based on the number of failures from a sample of n items under a truncated life test at the test time schedule t ₀ is given by

0

( ) 1

c i n i

i

L p n p p

i (7) Where n is the sample size, c is the acceptance number, and p is the probability of getting a failure within the life test schedule, t ₀ . If the product lifetime follows a Type II GHLD, then p=F( t ₀ ). Usually, it would be convenient to determine the experiment termination time, t ₀ , as t _{0 q} t _q ⁰ for a constant _q and the targeted 100q-th lifetime percentile, t . Suppose _q ⁰ t q is the true 100q-th lifetime percentile. Then, p can be rewritten as

0 0 0

2 2

1 exp 1 exp

1 1

q q q q

t t p

t

. (8)

In order to find the design parameters of the proposed acceptance sampling plan, we prefer the approach based on two points on the OC curve by considering the producer’s and consumer’s risks. In our approach, the quality level is measured through the ratio of its percentile lifetime to the lifetime; t _q t _q ⁰ .These percentile ratios are very helpful for the producer to enhance the quality of products. From the producer’s perspective, the probability of lot acceptance should be at least 1 at the acceptable reliability level (ARL), p

1 . So the producer demands that a lot should be accepted at various levels, say t _q t _q ⁰ =2, 4, 6, 8, 10 in Equation (7). On the other hand, from the consumer’s viewpoint, the lot rejection probability should be at most at the lot tolerance reliability level (LTRL), p

2 . Therefore, the consumer considers that a lot should be rejected when t _q / t _q ⁰ =1. From Equation (7), we have

1 1 1

0

( ) (1 ) 1

c

i n i

i

L p n

p p

i

(9)

2 2 2

0

( ) (1 )

c

i n i

i

L p n

p p

i

(10)

where p ₁ and p ₂ are given by

1 2

0 0 0

2 and 2

1 exp

1 exp

1 1

q q q q

q q

p p

t t

(11) The parametric quantities of the new acceptance sampling plan for different values of the parameter θ are constructed. Given the producer's risk α=0.05 and termination time schedule

0 0

0 q q

t t with _q ⁰ 1.0, 1.5, 2.0, 2.5 & 3.0 the smallest sample sizes (n) those guarantee the median (50 ^th percentile) life time are determined at different levels of the consumer’s confidence levels 0.25, 0.10, 0.05 & 0.25; and are presented through Tables 1-3 for θ=1.5, θ=2.0 and θ=2.5 respectively. The OC values of the sampling plans are also presented. From the tables, in all the cases, we may observe that the sample size (n) decreases as the percentile ratio ( t _q / t _q ⁰ ) increases.

4. EXPLANATION OF THE PROPOSED METHODOLOGY AND REAL DATA EXAMPLE 4.1Explanation of the Proposed Plan

Assume that the producer wish to implement a single sampling plan for assuring the median (50 ^th percentile) life of the products under inspection is at least 1000 hours when =0.10 and

=0.05 at the percentile ratio t _q t _q ⁰ =2. He wants to run this experiment 1000 hours. From the past data, it is observed that the lifetime of the product follows Type II Generalized Half- logistic distribution with 2.5. Hence, from Table 3, for given =0.05, =0.10, t _q t _q ⁰ =2 and

q 1.0 the optimal plan is obtained as n=44 and c=17 with 0.9562 probability of acceptance.

4.2 Real Data example

In this section, we present the analysis of real data, partially considered in Ghitany et al.

(2008), for illustrative purposes. The data represent the waiting times (in minutes) before customer service in a bank. Waiting time (in minutes) before customer service in a Bank: 0.1 0.2 0.3 0.70.9 1.1 1.2 1.8 1.9 2.0 2.2 2.3 2.3 2.32.5 2.6 2.7 2.7 2.9 3.1 3.1 3.2 3.4 3.4 3.5 3.9 4.0 4.24.5 4.7 5.3 5.6 5.6 6.2 6.3 6.6 6.8 7.3 7.5 7.7 7.7 8.0 8.0 8.5 8.5 8.7 9.5 10.710.9 11.0 12.1 12.3 12.8 12.9 13.2 13.714.5 16.0 16.5 28.0

The MLEs of the parameters of Type II GHLD based on the waiting times data are obtained as ˆ =3.6291 and ˆ =0.7594. To test the goodness of fit we apply the Kolmogorov-Smirnov test, we found that the maximum distance between the observed and fitted waiting times is 0.0739 with p-value 0.8985. Thus Type II GHLD provides a reasonable fit for the waiting times before customer service in a Bank. To emphasize the goodness of fit we also plot histogram and superimposed empirical density and theoretical density. Further, goodness of fit is stressed with Q-Q plot, both plots are displayed in the Figure 1.

Suppose that it is desired to develop the single acceptance sampling plan to satisfy that the 50 ^th percentile lifetime is greater than waiting time of 0.5 minutes through the experiment to be completed by completed by waiting time of 0.5 minutes. Let us fix that the consumer's risk is at 25% when the true 50 ^th percentile is waiting time of 0.5 minutes and the producer's risk is 5% when the true 50 ^th percentile is waiting time 1.00. Since ˆ =0.7594, the consumer's risk is 25%, _q ⁰ 1.0 and t _q / t _q ⁰ =2, the minimum sample size and acceptance number given by n =25 and c =10 from Table 4. Thus the design can be implemented as follows. Select a sample of 25 waiting times, we will accept the lot when no more than 10 customer before waiting time 1.00 minutes. According to this plan, the Waiting time before customer service in a Bank have been accepted because there are only five customers before the termination time of waiting time 1.00.

Figure 1: Histogram with superimposed theoretical density Plotand Q-Q Plot of the fitted Type II GHLD

for the waiting times.

Table-1: The smallest sample size n required to assert the 50 ^th percentile life time and the corresponding OC values of TGHLD for θ =1.5at given producer’s risk α =0.05.

0 q q

t t ^{q 0 1.0 q 0 1.5 q 0 2 q 0 2.5 q 0 3}

C n

( 1 )

L p ^{c n} L p ( ₁ ) ^{c n} L p ( ₁ ) ^{c n} L p ( ₁ ) ^{c n} L p ( ₁ )

0.25 2 11 27 0.9579 11 20 0.9502 11 16 0.9616 13 17 0.9590 14 17 0.9513

4 3 10 0.9585 3 7 0.9613 4 7 0.9808 3 5 0.9539 4 6 0.9620

6 2 7 0.9774 2 5 0.9773 3 4 0.9776 2 4 0.9594 2 3 0.9795

8 1 5 0.9554 2 5 0.9896 2 4 0.9776 1 2 0.9692 1 2 0.9563

10 1 5 0.9704 1 4 0.9605 1 3 0.9640 1 2 0.9799 1 2 0.9715

0.10 2 15 39 0.9550 15 28 0.9572 15 23 0.9534 15 20 0.9553 18 22 1.0000

4 4 14 0.9624 4 10 0.9609 4 8 0.9603 4 7 0.9538 4 6 0.9620

6 2 9 0.9530 2 6 0.9593 2 5 0.9517 2 4 0.9594 3 5 0.9782

8 2 9 0.9775 2 6 0.9809 2 5 0.9773 2 4 0.9812 2 4 0.9692

10 2 9 0.9876 2 6 0.9896 2 5 0.9876 2 4 0.9898 2 4 0.9832

0.05 2 19 51 0.9560 19 37 0.9506 18 28 0.9564 19 26 0.9518 - - - 4 5 18 0.9674 5 13 0.9633 5 10 0.9695 4 7 0.9538 5 8 0.9324

6 3 13 0.9707 3 9 0.9720 3 7 0.9737 3 6 0.9717 3 5 0.9782

8 2 11 0.9604 2 7 0.9692 2 6 0.9593 2 5 0.9591 2 4 0.9692

10 2 11 0.9777 2 7 0.9829 2 6 0.9773 2 5 0.9773 2 4 0.9832

0.01 2 - - - - - - - - - - - - - - -

4 7 27 0.9707 6 17 0.9533 6 13 0.9610 6 11 0.9603 6 10 0.9500 6 4 19 0.9706 4 13 0.9727 4 10 0.9752 4 9 0.9646 4 8 0.9603

8 3 17 0.9702 3 11 0.9767 3 9 0.9720 3 7 0.9888 3 6 0.9801

10 2 14 0.9856 2 9 0.9640 2 7 0.9636 2 6 0.9593 2 5 0.9632

Table-2: The smallest sample size n required to assert the 50 ^th percentile life time and the corresponding OC values of TGHLD for θ =2.0at given producer’s risk α =0.05.

0 q q

t t ^{q 0 1.0 q 0 1.5 q 0 2 q 0 2.5 q 0 3}

C n

( ) 1

L p ^{c n} L p ( ) ₁ ^{c n} L p ( ) ₁ ^{c n} L p ( ) ₁ ^{c n} L p ( ) ₁

0.25 2 11 27 0.9538 12 21 0.9680 11 16 0.9616 13 17 0.9605 14 17 0.9570

4 3 10 0.9553 3 7 0.9589 4 7 0.9797 3 5 0.9524 4 6 0.9604

6 2 7 0.9757 2 5 0.9758 2 4 0.9763 2 4 0.9575 2 3 0.9782

8 1 5 0.9529 2 5 0.9888 2 4 0.9763 1 2 0.9679 1 2 0.9537

10 1 5 0.9687 2 5 0.9940 1 3 0.9622 1 2 0.9790 1 2 0.9694

0.10 2 17 44 0.9596 16 30 0.9568 15 23 0.9534 17 23 0.9551 17 21 0.9524

4 4 14 0.9590 4 10 0.9581 4 8 0.9581 4 7 0.9520 4 6 0.9611

6 3 12 0.9761 2 6 0.9568 3 7 0.9719 2 4 0.9575 3 5 0.9771

8 2 9 0.9758 2 6 0.9795 2 5 0.9758 2 4 0.9801 2 4 0.9677

10 2 9 0.9866 2 6 0.9887 2 5 0.9867 2 4 0.9892 2 4 0.9822

0.05 2 19 51 0.9502 - - - 19 30 0.9540 19 26 0.9540 - - -

4 5 18 0.9639 5 13 0.9603 5 10 0.9675 5 9 0.9546 5 8 0.9529

6 3 13 0.9680 3 9 0.9698 3 7 0.9719 3 6 0.9700 4 7 0.9797

8 2 11 0.9574 2 7 0.9670 2 6 0.9568 2 5 0.9568 2 4 0.9677

10 2 11 0.9759 2 6 0.9816 2 6 0.9757 2 5 0.9758 2 4 0.9822

0.01 2 - - - - - - - - - - - - - - -

4 7 27 0.9669 7 19 0.9671 6 13 0.9583 6 11 0.9584 7 11 0.9728 6 4 19 0.9674 4 13 0.9702 4 10 0.9731 4 9 0.9622 4 8 0.9581

8 3 17 0.9673 3 11 0.9746 3 9 0.9698 3 7 0.9772 3 6 0.9788

10 2 14 0.9534 2 9 0.9613 2 7 0.9611 2 6 0.9568 2 5 0.9611

Table-3: The smallest sample size n required to assert the 50 ^th percentile life time and the corresponding OC values of TGHLD θ =2.5 at given producer’s risk α =0.05.

0 q q

t t q ^{0 1.0}

0 1.5

q

0 2

q

0 2.5

q

0 3

q

C n

( ) 1

L p ^{c n} L p ( ₁ ) ^{c n} L p ( ) ₁ ^{c n} L p ( ) ₁ ^{c n} L p ( ) ₁

0.25 2 11 27 0.9508 12 21 0.9669 12 18 0.9519 13 17 0.9616 14 17 0.9570

4 3 10 0.9530 3 7 0.9572 4 7 0.9789 3 5 0.9513 4 6 0.9604

6 2 7 0.9745 2 5 0.9748 2 4 0.9754 2 4 0.9562 2 3 0.9782

8 1 5 0.9512 2 5 0.9883 2 4 0.9887 2 4 0.9793 1 2 0.9537

10 1 5 0.9675 1 4 0.9570 1 3 0.9610 2 4 0.9887 1 2 0.9694

0.10 2 17 44 0.9562 16 30 0.9552 17 26 0.9622 17 23 0.9565 17 21 0.9551

4 4 14 0.9565 4 10 0.9561 4 8 0.9565 4 7 0.9507 4 6 0.9604

6 3 12 0.9746 3 8 0.9800 3 7 0.9706 2 4 0.9562 3 5 0.9763

8 2 9 0.9745 2 7 0.9655 2 5 0.9748 2 4 0.9793 2 4 0.9666

10 2 9 0.9858 2 7 0.9806 2 5 0.9860 2 4 0.9887 2 4 0.9814

0.05 2 - - - - - - 19 30 0.9506 - - - - - -

4 5 18 0.9641 5 13 0.9581 5 10 0.9661 5 9 0.9533 5 8 0.9520

6 5 13 0.9661 3 9 0.9682 3 7 0.9706 3 6 0.9688 4 7 0.9789

8 2 11 0.9553 2 7 0.9655 2 6 0.9550 2 5 0.9552 2 4 0.9666

10 2 11 0.9746 2 7 0.9806 2 6 0.9746 2 5 0.9748 2 4 0.9814

0.01 2 - - - - - - - - - - - - - - -

4 7 27 0.9640 7 19 0.9649 7 15 0.9665 7 13 0.9609 7 11 0.9721 6 4 19 0.9652 4 13 0.9683 4 10 0.9717 4 9 0.9605 4 8 0.9565

8 3 17 0.9653 3 11 0.9731 3 9 0.9682 3 7 0.9761 3 6 0.9779

10 2 14 0.9510 2 9 0.9595 2 7 0.9594 2 6 0.9550 2 5 0.9596

Table-4: The smallest sample size n required to assert the 50 ^th percentile life time and the Corresponding OC values TGHLD for ˆ =3.6291 and ˆ =0.7594 at given producer’s risk α =0.05.

0 q q

t t _q ⁰ 1.0 _q ⁰ 1.5 _q ⁰ 2 _q ⁰ 2.5 _q ⁰ 3

c n

( ) 1

L p ^{c n} L p ( ) ₁ ^{c n} L p ( ) ₁ ^{c n} L p ( ₁ ) ^{c n} L p ( ) ₁

0.25 2 10 25 0.9566 9 16 0.9525 9 13 0.9539 11 14 0.9619 11 13 0.9531

4 3 10 0.9664 3 7 0.9672 3 6 0.9544 3 5 0.9576 4 6 0.9642

6 2 7 0.9816 1 3 0.9518 2 4 0.9807 1 2 0.9512 2 3 0.9816

8 1 5 0.9618 1 3 0.9723 1 3 0.9518 1 2 0.9724 1 2 0.9604

10 1 5 0.9749 1 3 0.9821 1 3 0.9686 1 2 0.9824 1 2 0.9746

0.10 2 14 37 0.9568 14 26 0.9604 15 23 0.9534 15 20 0.9515 18 22 0.9544

4 4 14 0.9708 4 10 0.9678 3 6 0.9544 4 7 0.9580 4 6 0.9642

6 2 9 0.9614 2 6 0.9657 2 5 0.9582 2 4 0.9640 3 5 0.9808

8 2 9 0.9820 2 6 0.9844 2 5 0.9810 2 4 0.9840 2 4 0.9732

10 1 7 0.9509 2 6 0.9917 2 5 0.9899 1 3 0.9518 2 4 0.9857

0.05 2 18 49 0.9604 17 33 0.9516 18 28 0.9564 - - - - - -

4 5 18 0.9756 4 11 0.9505 5 10 0.9741 4 7 0.9580 5 8 0.9572

6 3 13 0.9773 3 9 0.9775 3 7 0.9782 3 6 0.9757 3 5 0.9808

8 2 11 0.9681 2 7 0.9747 2 6 0.9657 2 5 0.9649 2 4 0.9732

10 2 11 0.9823 2 7 0.9863 2 6 0.9813 1 3 0.9518 2 4 0.9857

0.01 2 - - - - - - - - - - - - - - -

4 6 25 0.9603 6 17 0.9632 6 13 0.9674 6 11 0.9649 6 10 0.9536

6 4 19 0.9782 3 11 0.9527 3 8 0.9624 3 7 0.9532 3 6 0.9544

8 3 17 0.9773 3 11 0.9818 3 8 0.9861 3 7 0.9826 3 6 0.9833

10 2 14 0.9654 2 9 0.9707 2 7 0.9699 2 6 0.9657 2 5 0.9686

5. CONCLUSIONS

In this article, we developed the single acceptance sampling plan based on the Type II GHLD percentiles, when the life test is truncated at a pre-fixed time; and these plans are named new acceptance sampling plans. For these plans, the smallest sample size, which guarantee the median life time, is determined under a given both consumer’s risk and producer’s risk. In other words, we have designed sampling plans based on two points on the OC curve approach for assuring percentile life time of the products. Some tables are provided for practical purpose in industry. The proposed plan is illustrated with a real data set, which is well fitted by the proposed Type II GHLD.

REFERENCES

1. Al-Omari, A. I. Acceptance sampling plan based on truncated life tests for three parameter Kappa distribution, Economic Quality Control, 29 (1), 53–62 (2014).

2. Baklizi, A. Acceptance sampling based on truncated life tests in the Pareto distribution of the second kind, Advances and Applications in Statistics, 3(1), 33-48 (2003).

3. Baklizi, A. and EI Masri, A.E.K. Acceptance sampling based on truncated life tests in the Birnbaum-Saunders model, Risk Analysis, 24(6), 1453-1457 (2004).

4. Balakrishnan, N., Leiva, V. and Lopez, J. Acceptance sampling plans from truncated life tests based on the generalized Birnbaum-Saunders distribution, Communication in Statistics-Simulation and Computation, 36, 643-656 (2007).

5. Balamurali, S., Aslam, M. and Fallah Nezhad, M.S. An acceptance sampling plan under Frechet distribution assuring median life, Research Journal of Applied Sciences, Engineering and Technology, 6(24), 4519-4523 (2013).

6. Epstein, B. Truncated life tests in the exponential case, Annals of Mathematical Statistics, 25, 555-564 (1954).

7. Fertig F.W., Mann N.R. Life-test sampling plans for two-parameter Weibull populations, Technometrics, 22(2),165-177 (1980).

8. Ghitany, M. E., Atieh, B., Nadarajah, S. Lindley distribution and its application.

Mathematics and Computers in Simulation, 78, 493-506 (2008).

9. Goode H.P., Kao J.H.K. Sampling plans based on the Weibull distribution. Proceedings of Seventh National Symposium on Reliability and Quality Control, Philadelphia, 1961, 24-40 (1961).

10. Gupta, S.S., and Groll, P.A. Gamma distribution in acceptance sampling based on life tests, J. Amer. Statist. Assoc., 56, 942-970 (1961a).

11. Gupta, S.S. Life test sampling plans for normal and log-normal distribution, Technometrics, 4, 151- 175 (1962).

12. Gui, W., and Aslam, M. Acceptance sampling plans based on truncated life tests for weighted exponential distribution, Communications in Statistics - Simulation and Computation, 46(3), 2138-2151 (2015). DOI: 10.1080/03610918.2015.1037593

13. Kantam, R.R.L., and Rosaiah, K. Half logistic distribution in acceptance sampling based

on life tests, IAPQR Transactions, 23(2), 117-125 (1998).

14. Kantam, R.R.L., Rosaiah, K. and Rao, G.S. Acceptance sampling based on life tests: Log- logistic models, Journal of Applied Statistics, 28, 121-128 (2001).

15. Kantam R.R.L., Ramakrishna V., Ravikumar M.S. Estimation and testing in type 1generalized half logistic distribution. Modern applied statistical methods, 12(1), 198-206 (2014).

16. Lio, Y.L., Tsai, T.R., & Wu, S.J. Acceptance sampling plan based on the truncated life test in the Birnbaum Saunders distribution for percentiles, Communications in Statistics- Simulation and Computation, 39, 119-136 (2009).

17. Lio, Y. L., Tsai, T.R., & Wu, S.J. Acceptance sampling plans from truncated life tests based on the Burr type XII percentiles, Journal of the Chinese Institute of Industrial Engineers, 27(4), 270-280 (2010).

18. Rao, G.S., Ghitany, M.E., and Kantam, R.R.L. Acceptance sampling plans for Marshall- Olkin extended Lomax distribution, International Journal of Applied Mathematics, 21, 315-325 (2008).

19. Rao, G.S. and Kantam, R.R.L. Acceptance sampling plans from truncated life tests based on the log-logistic distributions for percentiles, Economic Quality Control, 25(2), 153–

167 (2010).

20. Rao, G.S., Kantam, R.R.L., Rosaiah, K. Pratapa Reddy, J. Acceptance sampling plans for percentiles based on the inverse Rayleigh distribution, Electron. J. App. Stat. Anal., 5(2), 164 -177 (2012).

21. Rao B., Priya M. Ch. and Kantam R.R.L.Acceptance Sampling Plans for Percentiles Assuming the Linear Failure Rate Distribution, Economic Quality Control, 29(1), 1–9 (2013).

22. Rao, G.S. Acceptance Sampling Plans for percentiles based on the Marshall-Olkin extended Lomax distribution, International Journal of Statistics and Economics, 11(2), 83-96 (2013a).

23. Rao, G S, Rosaiah K, Kalyani K and Sivakumar D. C. U. Odds exponential log logistic distribution: A new acceptance sampling plans based on percentiles, the open statistics &

probability journal, 7,45-52 (2016).

24. Rao, G.S., Rosaiah K, Sridhar babu M., and Sivakumar D. C. U. A New Acceptance Sampling Plans Based on Percentiles For Exponentiated Fréchet Distribution, Econ.Qual.Control, 31(1), 37-44 (2016a).

25. Sobel, M., and Tischendrof, J. A. Acceptance sampling with new life test Objectives, Proceedings of Fifth National Symposium on Reliability and Quality Control, Philadelphia, Pennsylvania, 108-118 (1959).

26. Tsai, T.R., and Wu, S.J. Acceptance sampling based on truncated life tests for generalized

Rayleigh distribution, Journal of Applied Statistics, 33, 595-600 (2006).