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TWO-STAGE GROUP ACCEPTANCE SAMPLING PLAN FOR HALF-NORMAL DISTRIBUTION
C. Geetha1, S. Jayabharathi2, Mohammed Ahmar Uddin3, and Pachiyappan D4*
1Associate Professor, Department of Statistics, Government Arts college, Salem-7, Tamil Nadu, India. geethani sha1 @gmail.com 2Associate Professor, Department of Mathematics, Sona college of technology,
Salem-5, Tamil Nadu, India. [email protected] 3Department of Finance and Economics, College of Commerce and Business Administration,
Dhofar University, Salalah, Dhofar, Oman. [email protected]
4Research Assistant, Department of Finance and Economics, College of Commerce and Business Administration, Dhofar University, Salalah, Dhofar, Oman * Correspondence Email: [email protected]
Abstract
This paper proposes a time-truncated life test based on a two-stage group acceptance sampling plan for the percentile lifetime following a half-normal distribution. The optimal parameters for this plan are determined to simultaneously satisfy both producer's and consumer's risks for a given experimentation time and sample size. The efficiency of the proposed sampling plan is evaluated by comparing the average sample number with that of existing sampling plans. Industrial examples are provided to illustrate the application of the proposed sampling plan.
Keywords: Two-stage group sample, producer and consumer risk, normal distribution.
I. Introduction
In today's modern industrial environment, producing high-quality products using modern statistical quality control techniques has become essential. Product quality is a crucial factor contributing to business success, growth, and competitive advantage [1]. SQC techniques are now vital for any manufacturing process, as their application helps improve product quality by reducing process and product variability. SQC plays a significant role in the success of any industry. SQC involves a set of operational activities that an enterprise implements to ensure that its products meet the required quality levels set by consumers. According to [2], product quality can be evaluated based on various dimensions, including durability, serviceability, performance, aesthetics, features, reliability, and conformance to standards. These dimensions collectively determine the overall quality of a product. Therefore, quality has become the most significant factor in consumer satisfaction when selecting among competing products and services.
SQC involves a set of operating activities that an enterprise implements in order to get certified that the quality of its products is at required levels of the consumers. According to [3] one can evaluate the quality of the product in terms of its durability, serviceability, performance, aesthetics,
C. Geetha, S. Jayabharathi, Mohammed Ahmar Uddin, and Pachiyappan D RT&A, No 4(80) TWO-STAGE GROUP ACCEPTANCE SAMPLING PLAN_Volume 19 December, 2024
features, reliability, quality and its conformance to standards collectively and they are termed as the
dimensions of quality. Hence, we can say that in selecting among competing products and services,
the quality has become the most significant factor for consumer's satisfaction.
The quality of any finished product can be judged by inspecting a few items taken randomly
from a lot of products and the process of taking such samples is called the sampling. In quality
management, the acceptance sampling plans are vital tools in making a decision about the product
whether to accept or reject based on the inspection of sampled items and the sampling plans
prescribe the experimenter how many items in the sample should be selected from the submitted lot
for inspection and how many defectives can be allowed in this sample in order to satisfy both the
producer and the consumer. The probability of rejecting a good lot is called the producer's risk and
the chance of accepting bad lot is called the consumer's risk. The cost of any life test experiment is
directly proportional to the sample size. Therefore, sampling plans that provide smaller sample size
for inspection and minimize two risks are considered as efficient sampling plans.
In general, the acceptance sampling plans can be classified into two types namely attribute
sampling plans and variables sampling plans. The attribute sampling plans are implemented for
quality characteristics which are expressed on a ''go, no go'' basis whereas the variables sampling
plans can be applied where the quality characteristics of interest are measured on a numerical scale
[4]. Various types of sampling plans such as single sampling plan, double sampling plan, multiple
sampling plan, and sequential sampling plan are available in the literature [5]. Group acceptance
sampling plan is one of the types of sampling plans which involves a number of testers to be used
for testing so that cost and time can be reduced. The inspection of multiple items simultaneously can
be made easy to the experimenter for testing. Two stage group acceptance sampling plan is the
extension of GASP which involves two groups. The GASP is more advantageous than the
conventional sampling plans in terms of minimum inspection so that the considerable testing time
and cost can be reduced [6]. The advantage of two stage group sampling plan is that it reduces the
average sample number as compared to the GASP.
Several authors have investigated the sampling plans under various life time distributions,
which are available in the literature of acceptance sampling [7,8,9,10,11,12] proposed the SSP based
on half normal distribution. By exploring the literature on two stage group sampling plans, it can be
seen that no work is available based on the half normal distribution. In this paper, we will present
the designing of two stage group sampling plan when the life time of an item follows the half normal
distribution. The structure of proposed plan is presented and efficiency is compared with the
existing sampling plan. The application of the proposed sampling plan is explained with the help of
industrial illustrative examples.
2. Half-normal distribution
As far as the variables sampling plans are concerned, the normal distribution is the most preferred statistical distribution. But for life testing problems, normal distribution is not preferred because of its range [-ro, to] . However, one of the normal family distributions called the half-normal distribution is the widely used probability distribution for nonnegative data modeling, particularly, in life time testing. [13] investigated the properties of half normal distribution. [14] investigated the maintenance performance of the system under half-normal failure lifetime model as well as a repair-time model. The probability density function of a half normal distribution with 0 mean and its parameter 9 with domain y £ [0, to] is given by
2e y202
f(x) = ~,y > 0,0 > 0 (1)
Its cumulative distribution function is given by
F(y) = erf(-g),y>0,d>0 (2)
Here erf is the "Error Function" defined by
errf (y)=^i^e-t2dt (3)
Consider that life time of product follows a half-normal distribution with a as a scale parameter. Its cdf is given by
F(y) = erf ^^J, t>0,9>0,a>0 (4)
The 100th percentile of the half normal distribution with 0 < p < 1, is defined as
t_ = ^erf-1 (p), for 0 < p < 1 (5)
" 0
Here erf-1 is inverse function of error function. The Maclaurin series of erf-1 (.) is given by
erf-1(v) = *Jn(-y + —ny3 + —n2y5 + 127 n3y7 + ••• ) (6)
J \J J 24 960 80640 J ) V '
According to Pewsey, "If Z is a standard normal random variable, Z~N(0,1), then Y=IZI follows a standard positive half normal distribution and -Y=-1ZI. follows a standard negative halfnormal distribution. The half-normal distribution is a central chi-square distribution with one degree of freedom and a special case of truncated and folded normal distributions". The half-normal distribution is also a limiting case of skewed normal distribution [15]. The applications of halfnormal in reliability analysis can be seen in Ayman and Kristen [15]. As half-normal distribution has positively skewed shape, there is a need to model monotone hazard rates. The half-normal distribution is very widespread model to describe the lifetime process of any device under fatigue (see [16].
3. Two Stage Group Acceptance Sampling Plan
Naveed et al. [17] proposed the two-stage group sampling plan. The operating procedure of this plan is explained below.
3.1 Stage one
• Extract the first random sample of size n1 from a lot submitted for inspection.
• Randomly assign r items to each of g1 groups or testers so that n1 = rg1 and set them on test for the duration of t0 units of time.
• Accept the lot if the total number, Y1, of failures from g1 groups is smaller than or equal to c1a.
• Truncate the test and reject the lot as soon as the number of failures Y1 reaches c1r (> c1a) during the test. Otherwise, go to stage two.
3.2 Stage two
• Extract a second random sample of size n2 from the same lot.
• Randomly assign r items to each of g2 groups so that n2 = rg2 and set them on test for the duration of t0 units of time again.
• Let the total number of failures from the second sample be Y2.
• Accept the lot if the total number, Y1 + Y2, of failures from g1 and g2 groups is smaller than or equal to c2a(> c1a). Otherwise, truncate the test and reject the lot.
The design parameters of above two stage group sampling plans are g1, g2, c1a, c1r, and c2a. The acceptance number, c2a, in the second stage is larger than the acceptance number, c1a, in the first stage, since in two stage sampling total number of failures from both stages is used in decision making.
4. Designing of the proposed sampling plan Based on the operating procedure of two-stage group sampling plan, the lot acceptance probability at the first stage is given as
P(1 =P[Y1< cla} = (rf) vj (1 - p)rgi-j (7)
The lot rejection probability at the stage one is as follows (see Aslam et al. (2012)).
Pr(1)=Zrji1Clr (raj1)pi(1-p)rai-i = 1-rj^1 (rf)pj(1-p)rai-j (8)
To accept the lot based on stage two, the total number, Y1 + Y2, of failures from both groups g1 and g2 must be smaller than or equal to c2a. So, the lot acceptance probability at this stage under the proposed two stage sampling plan is as follows
P(2) = P{C1a + 1<Y1<C1r-1,Y1+Y2< C2a} (9)
Thus, under the proposed two stage group acceptance sampling plan, the lot acceptance probability is as follows.
L(p) = P™ + P(2) (10)
Stated that in order to implement an acceptance sampling plan to assure the percentile lifetime in a truncated life test, it is convenient to determine the experiment time in terms of the specified percentile lifetime as t0 = 5ptp, where 5p is called the termination time ratio. The probability of failure of an item before time t0 is given as
p = F(t0) = F(8pt°) (11)
or
p = F(t0) = erf (Sperf-1(pd)/(tp/tp0)) (12)
Where tp = true unknown population pth percentile. The erf (.) and erf-1 (.) functions have been defined in Eqs. (3) and (6) respectively. The above expression is represented in terms of tp/ and is also called failure probability. Let a be the producer's risk and / be the consumer's risk. The producer wishes that the lot acceptance probability should be larger than 1 - a at various values of percentile ratio tp/and the consumer wishes that it should be smaller than / at tp/tp = 1. Therefore, the plan parameters of the proposed plan will be determined by minimizing ASN at consumer's risk using the following non-linear optimization problem [16]. Minimize ASN (p2) = rg! + rg2 (1 - Pa(1) - Pr(1)), Subject to L(p2) >1-a
L(pi) < 3 (13)
gi > 1,32 > 1,r > 1,Cir > Cla > 0,C2a > cXa > 0 (14)
5. Description of tables and industrial examples
Tables for the selection of optimal parameters of the proposed sampling plan are given for various specified requirements such as consumer's risk ( / = 0.25,0.10,0.05,0.01 ), producer's risk (a = 0.05), percentile ratio (d2 = 2,4,6,8), (r = 5 or 10), 5p = 0.5 or 1.0 and d1 = 1. The tables are presented for average life (p = 0.5 and p = 0.25). The tables can be made for any other specified parameters. The R codes are available with the authors upon request.
The optimal parameters of the proposed sampling plan along with the ASN and the lot acceptance probability are presented in Tables 1-4. Tables 1 and 2 provide the optimal parameters of the two-stage group acceptance sampling plan to ensure median life time of the products and Tables 3 and 4 show the optimal parameters of the two-stage group acceptance sampling plan to
C. Geetha, S. Jayabharathi, Mohammed Ahmar Uddin, and Pachiyappan D RT&A, N° 4(80) TWO-STAGE GROUP ACCEPTANCE SAMPLING PLAN_V°lume 19, December, 2024
ensure lower percentile life. From Tables 1-4, we observe that when the ratio d2 increases the number
of groups for the experiment and the acceptance numbers decrease. Tables 2-4 will be made available
on request.
6. Case study
For the implementation of proposed sampling plan, we will consider the data from a leading ballbearing manufacturing company in Korea. The failure data are well fitted to the half normal distribution. Suppose that the company is interested to test the product using the proposed sampling plan. Let a = 5%,p = 5%, and p = 0.5. The number of testers of the product is limited to r = 5 From Table 1, we have c1r = 3, c1a = 0, c2a = 2,gI = 3 and g2 =2. A sample of size 15 items are selected and distributed into three groups. The failure times for each of three groups are given as follows
Table 1: Optimal parameters of the two-stage group sampling plan under half-normal distribution with 5q=0.5, q=0.5
p d2 r = 5 r = 10
c1r c1a c2a fa R2. ASN L(P2.) c1r c1a c2a R2. ASN L(P2.)
0.25 2 9 6 13 7 5 41.34 0.953 10 7 13 4 2 44.54 0.956
4 4 2 3 3 1 16.06 0.9671 4 2 5 2 1 21.14 0.9558
6 3 0 2 2 1 12.19 0.9732 4 2 3 2 1 21.14 0.9717
8 2 0 3 2 1 10.84 0.9574 3 1 9 2 1 20.53 0.9717
0.1 2 13 0 17 11 6 63.20 0.9554 16 10 17 6 3 72.45 0.9552
4 4 2 5 4 3 21.72 0.9505 5 1 4 2 1 23.42 0.9534
6 3 1 3 3 2 16.36 0.9644 4 4 2 1 21.14 0.9857
8 3 1 2 3 1 15.68 0.9753 3 1 7 2 1 20.53 0.9717
0.05 2 14 4 20 12 9 71.26 0.9502 15 10 26 7 6 77.64 0.9529
4 5 2 6 5 4 27.96 0.9614 5 8 3 2 30.92 0.9514
6 4 1 3 4 2 21.67 0.9598 4 1 3 2 1 21.67 0.9598
8 3 0 2 3 2 16.90 0.9519 3 1 4 2 1 20.53 0.9713
0.01 2 19 8 30 18 15 97.62 0.9521 19 2 31 9 8 98.13 0.9540
4 6 1 8 7 5 36.74 0.9634 7 3 7 4 2 41.28 0.9528
6 4 0 4 5 3 26.08 0.9580 4 0 6 3 2 30.51 0.9547
8 3 0 3 4 3 21.03 0.9569 4 2 3 3 1 30.19 0.9648
Group-1 Group-2 Group-3
0.6825 1.5650 0.9232
1.8024 0.8981 0.0607
0.0509 0.7322 0.4541
1.2080 2.1866 1.0035
0.4275 0.4223 0.6611
Let 5p = 0.5 and tp = 1.5, these leads t0 = 0.075. We note one failure from group 1, no failure from group 2 and 1 failure from group 3 before time t0 = 0.075. So, the total number of failures from three groups is 2. As the total number of failures lies between c1a = 0 and cIr = 3, a decision about the disposition of lot will be made on the basis of second sample. A second sample of size 10 is selected from the lot and distributed into two groups. The number of failures from two groups is given in the following table
Group-1 Group-2
0.8472 0.0701
0.7845 0.4341
0.5452 0.1104
0.1316 0.7054
0.2624 0.8239
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From group 1, we note that no failure occurs before experiment time and from group 2, we note that only one failure occurs before the experiment time. Since, the total number of failures from both samples is larger than c2a = 2, the lot of products will be rejected.
7. Comparison
In this section, a comparison is made between the GASP and the proposed two-stage group sampling plan. This comparison is made based on the ASN needed for both sampling plans. It is to be pointed out that the two-stage group sampling plan could have a chance to use the sample from the first stage or combined samples from both stages to make a decision. For example, if the product lifetime has a half-normal distribution, the next step would be to decide whether to use a GASP or to use the proposed two stage group sampling plan which will have a minimum ASN. Here we compare both plans for experiment termination time (5 = 0.50 and 1.0) with p = 0.5 median life time quality level.
Tables 5 and 6 provide the ASN for both acceptance sampling plans, where presumed producer's risk was set as a = 0.05. It has been witnessed from tables that the ASN for the proposed two-stage group sampling plan is much smaller than the GASP at lower ARL mean ratios with termination time 5p = 0.5 and 1.0. So, it is concluded that the two-stage group acceptance sampling plan is better than the GASP as it provides lesser ASN at lower percentile ratios for accepting or rejecting a lot in case of median life time quality.
As in Table 5, when 5p = 0.5 with p = 0.5 and consumer's risk is set as 0.05 with ratio 2, ASN for GASP when = 5 is 110 and the ASN of the proposed two stage group acceptance sampling plan reduces to 71.26. Likewise, from Table 6, when 5p = 1.0 and p = 0.5 with same consumer's risk and ratio, the ASN for GASP is 50 but for proposed plan, it reduces to 38.50. Table 6 will be made available on request.
8. Concluding remarks
In this paper, two stage sampling plan has been proposed for the inspection of products whose life time follows a half-normal distribution. Tables have been presented for industrial applications of the proposed sampling plan. The efficiency of the proposed sampling has been compared with the existing single stage group sampling plan. It is concluded that the proposed sampling plan is more efficient in reducing the ASN for the life test experiment. The real time applications of the proposed sampling plan are given using the industrial data. The proposed sampling plan can also be used in testing of software
Table 5: ASN for GASP and two-stage group sampling plan under half-normal distribution 5p = 0.5 when p = 0.5.
B d2 GASP with r = 5 GASP with r = 10 Proposed two-stage pl an
r = 5 r = 10
0.25 2 65 70 41.34 44.54
4 20 * 16.06 *
6 15 * 12.19 *
0.1 2 95 100 63.20 72.45
4 30 30 21.72 23.42
6 25 30 16.36 21.14
8 20 15.68
0.05 2 110 40 71.26 77.64
4 40 30 27.96 30.92
6 30 160 21.67 21.67
8 30 60 16.90 20.53
0.01 2 155 40 37.62 98.13
4 60 40 26.74 41.28
6 40 21.03 30.51
8 35
Optimal plan does not exist. reliability, testing/lot sizing of electronic product, automobile industry and mobile manufacturing industry. The efficiency of the proposed sampling plan using a cost model can be considered as future research.
Discloser statement
The authors declare no potential conflict of interest.
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