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Reliability Acceptance Sampling Plan for One Parameter Polynomial Exponential Distribution
Anumita Mondal and Sudhansu S. Maiti
•
Department of Statistics, Visva-Bharati University Santiniketan-731 235, West Bengal, India anumitamndl35@gmail.com, dssm1@rediffmail.com
Abstract
In this study, we construct a reliability acceptance sampling inspection plan to decide whether to accept or reject a lot of products where the One Parameter Polynomial Exponential (OPPE) family of distributions governs the lifetimes. The OPPE distribution has infinite support. To utilise finite support, it has transformed into its unit form, i.e. having the support (0, 1). The design of the plan, Operating characteristic curve, and Sampling procedure are discussed. Determination of the plan parameters using an algorithm is stated. The optimal sample size is determined to protect the consumer's confidence level. Two simplest particular choices of the OPPE family - the exponential and the Lindley are chosen as examples, and optimal plan parameters are tabulated and compared. The plan is executed with three real-life data sets.
Keywords: Consumer's risk, Operating characteristic function, Scale-invariant family of distributions, Truncated life test, unit-Lindley distribution.
1. Introduction
Quality control takes centre stage if one wants multiple copies of a product. The first question that arises throughout the repetitive process is what should the quality features of the product be for it to be satisfactory? The same prototype of products can be seen with the naked eye in the early days. As a result, making items identical is one of the most crucial aspects. Variation in the products is inevitable. In 1924, Walter A. Shewhart introduced many statistical approaches to assess product quality variation.
The sampling inspection plan aims to sentence a lot of products and make decisions to reject or accept that lot. In practice, two types of sampling plans are attribute sampling and variable sampling. In attribute sampling, products of a lot are judged as defective or non-defective, whereas the variable sampling plan measures the product's actual quantitative information. The main advantage of a variable sampling plan is that it requires fewer samples than an attribute plan with the same protection.
Acceptance sampling is traditionally used to decide on acceptance or rejection of the lot, not to determine the quality of the product by using estimation methods. As a result, most acceptance sampling plans are not adequately designed and worrying truth because buying a lot without knowing its quality seems risky. Therefore, to make an appropriate decision, it is better to develop a procedure for evaluating the value of the fraction defective of the lot.
In a sampling inspection plan, if sample observations represent the lifetime of products in the test, one may be interested in the hypothesis that the population average surpasses a specified
average. Suppose the population average represents the average lifetime, defined as If is the specified minimum value, then one would like to test the hypothesis which means the
population average surpasses the specified average. In the test, n samples are placed for testing over a period of time t. The lot cannot be accepted if the observed failures exceed acceptance number (c). This sampling inspection plan is called Reliability Acceptance Sampling Plan (RASP) and it is characterised by the triplet (n, c, t).
Many authors chalked out the RASP for quality characteristics following different parametric distributions, like exponential [22] , Weibull [9] , Gamma [11], Normal and Log-Normal [10], half Logistics [12], Log-Logistic [17], Birnbaum-Saunders [14] , exponential Frechet [1], three-parameter kappa [2] , generalised inverse exponential [21], generalised Weibull[8], Ishita [3] , transmuted generalised inverse Weibull [4] , generalised Pareto [20] , Quasi Shanker [5], etc.
The popularity of exponential distribution is well known in the context of life testing because of its simplicity in analysis. The constancy properties of failure rate and residual mean life limit the distribution in the present industrial scenario. The Lindley distribution is an excellent alternative to an exponential distribution, but it is overlooked for life-testing purposes. It is a mixture of two parametric distributions, exponential and gamma. The Lindley distribution is more flexible than the exponential distribution because of its increasing, decreasing and upside-down bathtub failure rate for the parameter at different values. [7] has proposed a new and unified approach in generalizing the Lindley's distribution. They investigated some structural properties like moments, skewness, kurtosis, median, mean deviations, Lorenz curve, entropies and limiting distribution of extreme order statistics; reliability properties like reliability function, hazard rate, stress-strength reliability, stochastic ordering; and estimation methods like the method of moment and maximum likelihood. We call the distribution as the one-parameter polynomial exponential (OPPE) family of distributions. The proposed RASP will be constructed assuming only OPPE distribution considering the rejectable quality level. The RASP for the Lindley distribution will be discussed in detail as particular example and compare it with that of the exponential distribution.
Few works have been done on acceptance sampling inspection plans assuming the Lindley distribution (see [15]; [?] ); [6] and [23] worked on an acceptance sampling plan under a truncated life test assuming two-parameter Lindley distribution. Plan parameters are estimated based on two-point approaches on Operating characteristic (OC) curve-acceptable and rejectable quality levels.
Almost all works on RASP are done for scale-invariant distributions. Minimum sample size n and acceptance number (c) are determined for different times per mean (^). However, the OPPE distribution does not belong to a scale-invariant family of distributions. Therefore, the utilisation of time per mean is beyond our scope. We may directly chalk the plan with plan parameters (n, c, t). Since the OPPE distribution has support (0,ro), we may utilise it with the finite support by transforming into its unit form, i.e. having the support (0,1) with the transformation V = e-T. We make tables using the unit-Lindley form by choosing V and the mean in the interval (0,1). Utilising this benefit, we choose optimal (n, c) and then revert to plan parameter t from the relation of the transformation. So, in a nutshell, our objective is to develop a RASP for the OPPE distributed quality characteristic. Based on the time-truncated life test, the plan has the advantage of saving the organisation's time and cost while also being very helpful in determining whether to accept or reject a lot. The OC is derived for choosing the optimal plan based on the consumer's confidence level. Tables of minimum sample sizes are examples for easy understanding and execution of the proposed plan. It is put into practice for real-life experimental data, and the OC surface is depicted to provide a clear picture of the plan.
The following is the arrangement of the rest of the paper. The OPPE and unit-OPPE distributions are described in section 2. In section 3, we describe the sampling design, operating
characteristics function, and operating procedure. In section 4, an algorithm for calculating the minimal sample size of the proposed RASP is stated for the OPPE distribution, and examples for the Lindley and exponential distributions, in particular, are presented in tabular form. In section 5, we use the said sampling plan to work on real-world data. Section 6 concludes.
2. The One Parameter Polynomial Exponential Distribution and its unit
version
The probability density function (PDF) of a random variable T of the OPPE distribution can be written as
fT(t,9) = h(9)p(t)e-m, t, 9 > 0, (1)
where, h(9) = —-—1 r(k+1) , p(t) = Yk=0 aktk, a-k's are known non-negative constants and r is ^k=0 ak (k+1 known non-negative integer.
The distribution can also be written as
fr (t, 9) = h(9) £ aktke
k„-9t
k=0
= £k=0 ak ^ fGA (t; k + 1,9)
Tr a, i(k+i) , (2
yk=0 ak 9k+l
where fGA (t; k + 1,9) is the PDF of a gamma distribution with shape parameter (k + 1) and scale parameter 9. The distribution is a finite mixture of (r + 1) gamma distributions. The cumulative density function (CDF) is given by
' yr akr(k+1)r(k+1,9t) \
Fr(t, 9) = 1 - | U=° r 9k+1- | , t, 9 > 0, (3)
v-r „ _kj_
yk=0 ak 9k+1 )
where T (m, t) = j^my ftœ e uum 1 du.
The s-th order raw moment of OPPE is given by
Hi = E(TS)
yr a r(k+s+1) yk=0 ak 9k+s+1
yr a r(k+1~ yk=0 ak 9k+1
(4)
Now, if we take a transformation V = e r, then the OPPE turns into unit-OPPE in range of (0,1). The PDF and CDF of unit-OPPE is given by ,
fv(v, 9) = h(9) y ak( - ln v)kv
k„ ,9-1
k=0
£k=0 ak ^ fuGA (v; k + 19)
= -£r aT(k+1)-, 0 <v < 1, (5)
yk=0 ak 9k+1
where fUGA (v; k + 1,9) = Y(^~1)( — ln v)k+1v9-1 is the PDF of a unit-gamma distribution with shape parameter (k + 1 ) and scale parameter 9, and
akr(k+1)r(k+1,-9lnv) '
_9k+1
rr a _fcL yk=0 ak 9k+1
Fv(v, 9) = 1 - ( ^^^-^^- | ,0 < v < 1, 9 > 0, (6)
respectively.
The s-th order raw moment of unit-OPPE is given by
PS = E(Vs)
yr a r(fc+l)
= yk=o(s+e)k+1 (7)
yr a IM1) . (7)
yk=0 0k+1
The Lindley distribution (for r = 1, fl0 = «1 = 1), introduced by Lindley (1958) to analyse failure time data has the PDF, CDF and hazard rate function (HRF) as
fT(t; 0) = 0+1 (1 + t)e-0t t > 0, 0 > 0, (8)
1 + 0 + 0t,_ ef
Ft (t; 0) = 1 - 0 + 1 e (9)
and
02(1 +t)
hT(t; 0)= 1+0+07 t > 0,0 > 0, (10)
respectively.
The mean of the random variable T is
0+2
P = 0(1+0) • (11)
The unit-Lindley distribution with parameter 0 has the PDF , CDF and HRF respectively , as follows:
f (v; 0) = TT0 (1 - log(v)) (v0-1) 0 < v < 1,0 > 0, (12)
0) = v0(1 + 0((1 - 1og(v))) 0 < v < 1,0 > 0, (13)
1 + 0
h(v; 0)= /.2(1 - -^ 0 < v < 1,0 > 0. (14)
v ' v(01og(v) - (1 + 0)(1 - v-0))
The shapes of the PDF, CDF and HRF of unit-Lindley distribution for different 0 are shown in Figure 1. Notably, the HRF is an increasing function. So, the distribution is capable of modeling life time data. Furthermore, the first moment about the origin of unit-Lindley distribution can be
obtained as m1 = 0(1(++)3) = Pv.
For comparison purpose, we will choose the exponential distribution (r = 0, fl0 = 1) and its corresponding unit version. The unit version of the exponential distribution with parameter 0 has the PDF, CDF and HRF as
f (v, 0) = 0v0-1, 0 < v < 1,0 > 0, (15)
F(v, 0) = v0, 0 < v < 1,0 > 0, (16)
0v0-1
h(v, 0) = --tt, 0 < v < 1,0 > 0, (17)
1 v0
respectively. In this case, = 1+:^ implies 0 = ^
Pv
3. Reliability Acceptance Sampling Plan for OPPE distribution
According to the product's mean life, a product lot is labelled as good or bad in this sampling plan. The RASP has the plan parameters n, c, and t. For proper implementation of the plan, engineers and practitioners use the tabulated value or algorithm. Tables are presented for fixed t and c, the optimal value of n. Since the value of t € (0, to), fixing t is tedious, whereas choosing v € (0, 1) is easy and comprehensive.
Figure 1: PDF,CDF and HRF o/wn/t-L/nd/ey d/str/but/on/or different 0
3.1. Design of the sampling plan
A product is defective if it fails before truncation time v. The fraction defective i.e. the probability that a product is defective is
/yr akr(k+1)r(k+1,v0) \
p(v) = F(v; 0) = 1 - I -- I ,0 < v < 1, 0 > 0. (18)
y yk=0 ak 0k+1 J
In particular, for the Lindley distribution,
p(v) = F(v;0)= v0(1 + 0((1 -/og(v))) 0 < v < 1,0 > 0. (19)
1 + 0
In this equation, we replace the shape parameter 0 by product's mean life (p). We can say that 0 = g(p), and we get the value of 0 by solving the equation by numerical method and hence we have p(v) = F(v, pv).
The Operating Characteristic (OC) function plays a vital role in product control techniques. It gives the probability of acceptance of an individual lot from finite production. For some fixed p, our sampling plan characterized by (n, c, t) or equivalently by (n, c, v), where v = e-t. For sufficient large lots, the binomial distribution can be applied. The OC function can be formulated as
n(p) = yc=0 (n)(1 - p)n-i = 1 - Bp(c + 1, n - c)
with p = F(v; pv), and Bp(c + 1, n - c) is the incomplete beta function. For determining small positive integer n for given c, v and p^ , we use
n(p0) = E(T)p0(1 - p0)n-i < 1 - p*, (20)
i=0 V'/
where, P* is the level of confidence. If p is very small and n is very large, and ft = np is finite, then the binomial distribution can be approximated by the Poisson distribution. Then, the OC
function becomes
n(p) = yc=0 = 1 - r(c + 1, ft), with r(k, w) = /0Wxk-1 e-xdx, the incomplete gamma function.
3.2. Sampling Procedure
The RASP is conducted as follows.
1. Put n items on test.
2. Choose the acceptance number c and specify the maximum test time t.
3. Perform the experiment and count the number of failures.
4. Accept the lot if the number of failures is at most c by the experiment time t.
5. Terminate the experiment as soon as (c+1)th failure occurs and reject the lot.
4. Estimation of the plan parameters
A sampling scheme for unit-OPPE distribution with parameter (n, c, v) satisfies the consumer's risk given in the previous section. To determine the smallest integer of sample size n for given
( c, v) is
Min(n\c^,P*)n subject to
cn
i=0
E i - Po(1 - Po)n-t < 1 - P*, (21)
( Er akr(k+1)r((k+1,vg(^'V)
where Po = 1 -
ELo - I ,0 < v < 1, g(v°v) > 0.
Ek=0 ak
\
g(.F0° )k+1
Algorithm for determination of the plan parameters is as follows.
1. Specify the confidence level P*.
2. Choose t and ^o.
Er ak r(k+s+1) Er ak r(k+s+1)
3. Calculate v = e-t and $ = ~E==°—r++i} , is obtained by solving = —E=°—^++1}
numerically.
4. Choose a value of c.
5. For given (c,t,^V), choose minimize n such that (21) satisfies.
The minimum values of n and c satisfying the inequality are obtained and shown in Table 1- 2 for the Lindley and that are in Tables 3-4 for the exponential distribution for P* = 0.95,0.99. The representative tables are shown for an easy and comprehensive understanding of the proposed plan. The contents of each table are described as follows. The first row is reserved for specifying confidence level (P*). The first column represents the pre-specified mean value, ^ of unit-Lindley/unit-exponential distribution and the corresponding mean value, ^0 of Lind-ley/exponential distribution in the parenthesis. The second row represents the truncation time, v, of unit-Lindley/unit-exponential distribution and the corresponding time, t, of Lind-ley/exponential distribution in the parenthesis. For a combination of (^, v) or (^0, t), in the cell, the optimal choices of n, the sample size and c, the acceptance number are presented.
Table 2: Determination of optimal sample size for Lindley set up
P* = 0.99
PV (P0 )v(t) 6.74 x 10-3(5) 4.54 x 10-5(10) 3.05 x 10-8(15)
c n c n c n
0.1162(5) 0 3 0 5 0 8
1 5 1 8 1 12
2 6 2 10 2 16
0.0447(10) 0 2 0 2 0 3
1 3 1 4 1 5
2 4 2 5 2 6
0.0236(15) 0 1 0 2 0 2
1 3 1 3 1 3
2 4 2 4 2 5
Table 1: Determination of optimal sample size for Lindley set up
p* = 0.95
pV (P0 )v(t) 6.74 x 10-3(5) 4.54 x 10-5(10) 3.05 x 10-8(15)
c n c n c n
0.1162(5) 0 2 0 3 0 6
1 4 1 6 1 9
2 5 2 8 2 12
0.0447(10) 0 1 0 2 0 2
1 3 1 3 1 4
2 4 2 4 2 5
0.0236(15) 0 1 0 1 0 2
1 2 1 3 1 3
2 3 2 4 2 4
Table 3: Determination of optimal sample size for Exponential set up
P* = 0.95
pV (P0 )v(t) 6.74 x 10-3(5) 4.54 x 10-5(10) 3.05 x 10-8(15)
c n c n c n
0.1662(5) 0 7 0 21 0 94
1 11 1 34 1 150
2 15 2 45 2 197
0.0909(10) 0 4 0 7 0 16
1 6 1 11 1 25
2 8 2 15 2 34
0.0625(15) 0 3 0 5 0 8
1 5 1 8 1 14
2 7 2 10 2 18
A few observations from the Tables are noted below.
(i) With the confidence level (P*) increase, the minimum sample size increases.
(ii) The optimal sample size for the Lindley distribution is smaller than that for the exponential distribution.
(iii) The sample size increases with the increase of truncation time (t) or mean (p) or both.
Table 4: Determination of optimal sample size for Exponential set up
P* = 0.99
vV (F0 )v(t) 6.74 x 10-3(5) 4.54 x 10-5(10) 3.05 x 10-8(15)
c n c n c n
0.1662(5) 0 10 0 32 0 72
1 15 1 46 1 122
2 20 2 60 2 166
0.0909(10) 0 5 0 11 0 12
1 8 1 16 1 21
2 11 2 20 2 27
0.0625(15) 0 4 0 7 0 7
1 6 1 10 1 11
2 8 2 13 2 16
5. Real life examples
Data Set 1: The data is from [16] and arose in test on the cycle at which the yarn failed. The data are the number of cycles until failure of the yarn(100 units):
15, 20, 20, 38, 38, 40, 40, 42, 55, 55, 61, 61, 65, 71, 76, 81, 86, 88, 90, 93, 98, 105, 121,124,124,131, 135, 135, 137,143,146, 149, 151,157, 166, 169,175,176, 180,180,180, 182, 185,185,186, 188, 188, 193, 194, 195,196,198, 198, 203, 203, 211, 220, 224, 229, 229, 236, 239, 244, 246, 246, 249, 250, 251, 262, 264, 264, 264, 277, 279, 282, 284, 286, 290, 292, 315, 321, 325, 337, 338, 341, 350, 353, 364, 393, 396, 398, 400, 400, 423, 497, 568, 571, 597, 653, 829.
First, we have checked whether the considered data set is well fitted with the exponential or Lindley distribution by goodness-of-fit test. For this purpose, we have used the Akaike Information Criterion [AIC=-2log(likelihood)+2k, k is the parameter number] to verify which data fit better. The model that best fits the data could be the one with the lowest AIC value.
Table 5: Comparison of Exponential and Lindley Distribution for Data Set 1
Distribution Estimate of 9 Negative Log-likelihood AIC
Exponential Lindley 0.00450 0.0089687 640.2587 625.6708 1282.517 1253.3410
Table 5 shows that the Lindley distribution gives a better fit as the AIC value is less than the exponential distribution. The histogram with fitted distributions and P-P plots is shown in Figure 2.
Suppose the truncation time of the testing number of cycles until failure of the yarn is 150, 200, 250 and 300. We construct the decision table for specified mean life as 150, 200, 150, and 300. The estimated average life of the number of cycles is 221.98. The sample size n(=100) is fixed, so Tables 6 and 7 are constructed for (c,v) or (c,t) values, and the decision regarding acceptance or rejection of the lot is made accordingly for the Lindley and the exponential distributions. Based on the observations, we must make a sentence for the lot, whether it will be accepted or rejected. In this example, let us assume that truncation time, t is 200, that after transformation of t (v = e-t), v is 0.0497, and that we take the specified mean of the Lindley distribution to be 150, with the corresponding unit-Lindley mean to be 0.3001. We accept the lot only if the number of failures before the specified mean,150, is less than or equal to acceptance number 60 (see Table 6). In this case, the decision is to accept the lot. We arrive at the same decision for exponential assumption, but the acceptance number, c(=94), is larger than that for the Lindley distribution. So, there may be a chance to come to a wrong decision if Lindley is a better fit, which will be
O
CO
o o o
c
<D
Q
o
o _ o
O
o o
o J
o
o r 0
Figure 2: Histogram withfitted distributions and P-P plot for Data Set 2
noticed in the case of Data Set 2. The OC surface of the plan for n = 100 and c = 72 has been shown in Figure 3.
Table 6: RASP for Lindley distribution for Data Set 2
p* = 0.95
mV (Mo)v(t) 0.1353 (150) 0.0497 (200) 0.0183 (250) 0.0067 (300)
c n decision c n decision c n decision c n decision
0.3001(150) 72 100 Accept 60 100 Accept 49 100 Accept 39 100 Accept
0.2065(200) 82 100 Accept 74 100 Accept 65 100 Accept 57 100 Accept
0.1515(250) 88 100 Accept 82 100 Accept 76 100 Accept 70 100 Accept
0.1162(300) 91 100 Accept 87 100 Accept 83 100 Accept 78 100 Accept
Table 7: RASP for Exponential distribution for Data set 2
p* = 0.95
mV (M0)v(t) 0.1353 (150) 0.0497 (200) 0.0183 (250) 0.0067 (300)
c n decision c n decision c n decision c n decision
0.0067(150) 96 100 Accept 94 100 Accept 93 100 Accept 93 100 Accept
0.0049(200) 96 100 Accept 95 100 Accept 95 100 Accept 94 100 Accept
0.0039(250) 97 100 Accept 96 100 Accept 95 100 Accept 95 100 Accept
0.0033(300) 97 100 Accept 96 100 Accept 96 100 Accept 95 100 Accept
Data Set 2:
This data represents the failure times in minutes for a sample of 15 electronic component in
accelerated life test (see, [13]), also used by [19] and [18] :
1.4, 5.1, 6.3, 10.8, 12.1, 18.5, 19.7, 22.2, 23, 30.6, 37.3, 46.3, 53.9, 59.8, 66.2.
OC surface for the plan with n=100, c=72
Figure 3: OC Surface of the proposed plan for Data Set 1
Histogram and fitted data
P-P plot
I-1
I-1
0 10 20 30 40 50 60 70
0.0 0.2 0.4 0.6 0.8 1.0
Theoritical distribution
Figure 4: Histogram with fitted distributions and P-P plot for Data Set 2
Table 8: Comparison of Exponential and Lindley Distributions for Data Set 2:
Distribution Estimate of в Negative Log-likelihood AIC
Exponential Lindley 0.03630 0.07025 64.7386 64.40554 131.4764 130.8110
Table 8 shows that the Lindley distribution gives a better fit as the AIC value is less than that for the exponential distribution. The histogram with fitted distributions and the P-P plot shown in Figure 4 substantiates the claim.
The estimated mean life is 27.54 for the fitted Lindley distribution. We have selected the testing time for failure times as 20, 30, 35 and 40. The corresponding hypothesis for testing is v > V0 against v < v0, where v0 is the specified mean, and v is the average lifetime. We construct Tables 9 and 10 with sample size n =15 for the Lindley and exponential distributions, respectively. For example, let the truncation time is 30, and the specified mean is 35. The decision regarding the lot is to reject under the Lindley assumption and accept under the exponential assumption. Hence the exponential assumption leads to a wrong conclusion as the data fits better with the Lindley distribution. Figure 5 shows the OC surface of the plan for n = 15 and c = 9.
Table 9: RASP for the Lindley distribution for Data set 2
p* = 0.95
vV (№)v(t) 0.3679 (20) 0.1353 (30) 0.0820 (35) 0.0498 (40)
c n decision c n decision c n decision c n decision
0.4853(20) 9 15 Accept 5 15 Accept 4 15 Reject 3 15 Reject
0.3002(30) 11 15 Accept 8 15 Reject 7 15 Reject 6 15 Reject
0.2465(35) 11 15 Accept 9 15 Reject 8 15 Reject 8 15 Reject
0.2065(40) 12 15 Accept 10 15 Reject 9 15 Reject 9 15 Reject
Table 10: RASP for Exponential distribution for Data set 2
p* = 0.95
VV (V0)v(t) 0.3679 (20) 0.1353 (30) 0.0820 (35) 0.0498 (40)
c n decision c n decision c n decision c n decision
0.0476(20) 12 15 Accept 11 15 Accept 10 15 Accept 10 15 Accept
0.0323(30) 12 15 Accept 11 15 Accept 11 15 Accept 11 15 Accept
0.0278(35) 12 15 Accept 12 15 Accept 11 15 Accept 11 15 Accept
0.0244(40) 12 15 Accept 12 15 Accept 11 15 Accept 11 15 Accept
OC surface for the plan with n=15, c=9
Figure 5: OC Surface of the proposed plan for Data Set 2
Data set 3: This data set shows that the cycle-to-failure numbers for 25 (100-cm) specimens of yarn tested at a particular strain level (see, [13]) are:
15, 20, 38, 42, 61, 76, 86, 98,121,146, 149, 157, 175, 176,180,180,198, 220, 224, 251, 264, 282, 321, 325, 653.
Table 11 shows that the OPPE with a0 = 9, ai = 4, a2 = 0.005 distribution is a better fit as the AIC value is less than that for the exponential distribution. The histogram with fitted distributions and the P-P plot in Figure 6 justifies the claim. Table 12 shows different plans and their decisions for the fitted OPPE model. Table 13 shows that for the exponential model. The OPPE assumption shows its superiority. The OC surface of the plan for n = 25 and c = 18 with the fitted OPPE is at Figure 7.
Table 11: Comparison of Exponential and OPPE(9,4,0.005)/or Data Set 3
Distribution Estimate of 0 Negative Log-likelihood AIC
Exponential OPPE(9,4,0.005) 0.0056 0.01115 154.5078 152.5078 311.179 307.0156
Histogram and fitted density
P-P plot
~ CO
(fl o f= o
Exponential oppe(9,4,0.005)
-1-1-1-1-
100 200 300 400
—I-1-1
500 600 700
Theoritical distribution
Figure 6: Histogram withfitted distributions and P-P plotfor Data Set 3
Table 12: RASP for the OPPE(9,4,0.005) distribution for Data set 3
0
p* = 0.95
MV (M0)v(t) 0.4065 (100) 0.2466 (150) 0.1469 (200) 0.0907 (250)
c n decision c n decision c n decision c n decision
0.5208(100) 18 25 Accept 11 25 Accept 8 25 Accept 6 25 Reject
0.4054(150) 16 25 Accept 14 25 Accept 11 25 Accept 9 25 Reject
0.3282(200) 18 25 Accept 16 25 Reject 13 25 Reject 12 25 Reject
0.2730(250) 19 25 Accept 17 25 Reject 15 25 Reject 14 25 Reject
Table 13: RASP for the Exponential distribution for Data set 3
p* = 0.95
MV (M0)v(t) 0.4065 (100) 0.2466 (150) 0.1469 (200) 0.0907 (250)
c n decision c n decision c n decision c n decision
0.0099(100) 23 25 Accept 23 25 Accept 22 25 Accept 22 25 Accept
0.0066(150) 23 25 Accept 23 25 Accept 23 25 Accept 22 25 Accept
0.0049(200) 23 25 Accept 23 25 Reject 23 25 Accept 23 25 Accept
0.0039(250) 23 25 Accept 23 25 Accept 23 25 Accept 23 25 Accept
OC surface for the plan n=25,c=18
8°
0.
Figure 7: OC Surface of the proposed plan for Data Set 3
6. Concluding remarks
A reliability acceptance sampling plan is formulated for the OPPE distributed quality characteristic. The OPPE family of distributions does not belong to the scale-invariant family, whereas most of the RASP chalked out for the scale-invariant family in the literature. The optimal plan parameters are estimated by transforming the OPPE distribution into its unit form to utilize the advantage of finite range (in this case, (0,1)). A few examples are presented for finding optimal sample sizes for the proposed plan for the Lindley distribution, a particular choice of the OPPE family, which will be helpful to scientists and quality practitioners for implementation. Three data sets are analyzed for implementing the proposed plan. The approach may be adopted to construct RASP for other lifetime quality characteristic distributions that do not belong to the scale-invariant family. Acknowledgement
The authors thank the anonymous referee for insightful comments that help improve the paper. Declarations
Disclosure of Conflicts of interest/competing interests: The authors declare that they have no conflict of interest.
Authors contributions: Each author has equal contribution. All authors jointly write, review and edit the manuscript.
Funding: The authors received no specific funding for this study.
Data Availability Statements: All cited data analysed in the article are included in References. Data sets are also provided in the article.
Ethical Approval: This article does not contain any studies with human participants performed by any of the authors.
Code availability: Codes are available on request.
[1] Al-Nasser, A. D. and Al-Omari, A. I. (2013): Acceptance sampling plan based on truncated life tests for exponentiated Frechet distribution, Journal of Statistics and Management, 16(1),
[2] Al-Omari, A. I. (2014): Acceptance Sampling Plan Based on Truncated Life Tests for Three Parameter Kappa Distribution, Economic Quality Control, 29, 53-62.
[3] Al-Nasser, A. D., Al-Omari, A. I., Bani-Mustafa, A. and Jaber, K. (2018): Developing Single-Acceptance Sampling Plans Based On A Truncated Lifetime Test For An Ishita Distribution, Statistics in Transition New Series, 19, 393-406.
References
13-24.
Al-Omari, A. I. (2018): The transmuted generalized inverse Weibull distribution in acceptance sampling plans based on life tests, Transactions of the Institute of Measurement and Control, 40(16), 4432-4443.
Al-Omari, A. I., Almanjahie, I. M., and Dar, J. G. (2021): Acceptance sampling plans under two-parameter Quasi Shanker distribution assuming mean life with an application to manufacturing data, Science Progress 104(2), 1-17.
Al-Omari, A. I. and Al-Nasser, A. D. (2019): A Two-Parameter Quasi Lindley Distribution in Acceptance Sampling Plans from Truncated Life Tests, Pakistan Journal of Statistics and Operation Research, 15(1), 39-47
Bouchahed, L. and Zeghdoudi, H. (2018): A new and unified approach in generalizing the Lindley's distribution with applications, Statistics in Transition, 19(1), 61-74. Chowdhury, S. (2016): Acceptance sampling plans based on truncated life test for the generalized Weibull model, 2016 IEEE International Conference on Industrial Engineering and Engineering Management (IEEM), 886-889.
Goode, H. P. and Kao, J. H. K. (1961): Sampling plans based on the Weibull distribution,
Proceedings of Seventh National Symposium on Reliability and Quality Control, Philadelphia, Pennsylvania, 24-40.
Gupta, S. (1962): Life Test sampling plans for Normal and Log-Normal Distribution,
Technometrics, 4,151-175.
Gupta, S. S. and Groll, P. A. (1961): Gamma distribution in acceptance sampling based on life tests, Journal of the American Statistical Association, 56, 942-970.
Kantam, R.R. L. and Rosaiah, K. (1998): Half logistic distribution in acceptance sampling based on life tests, IAPQR Transactions, 23, 117-125.
LawlessJ. F., Statistical Models and Methods for Lifetime Data, John Wiley Sons, New York, 2003.
Lio, Y. L., Tsai, T. R., and Wu, S. J. (2010): Acceptance sampling plans from truncated life tests based on the Birnbaum-Saunders distribution for percentiles, Communications in Statistics-Simulation and Computation, 39,119-136.
Mukherjee, S. and Maiti, S. S. (2014): Sampling Inspection Plan by variable for Lindley distributed quality characteristic, Proceedings ofIMBIC - MSAST, 3, 213-223. Picciotto R. Tensile fatigue characteristics of a sized polyester viscose yarn and their effect on weaving performance. North Carolina State, University at Raleigh, USA, 1970. Rosaiah, K., Kantam, R.R. L. and Kumar,S. Ch. (2006): Reliability Test plans for Exponentiated Log-Logistic Distribution, Economic Quality Control, 21(2), 279-289.
Saha, M., Tripathi, H., Dey, S., and Maiti, S. S. (2021): Acceptance sampling inspection plan for the Lindley and power Lindley distributed quality characteristics, International Journal of System Assurance Engineering and Management, 12,1410-1419.
Shanker, R. and Shukla, K. K (2016): On modelling of lifetime data using three-parameter generalized Lindley and generalized gamma distributions, Biometrics & Biostatistics International Journal, 4, 283-288.
Singh, N., Sood, A., and Buttar, G. S. (2020): Acceptance Sampling Plan for Truncated Life Tests Based on Generalized Pareto Distribution using Mean Life, textitIndustrial Engineering and Management Systems, 19, 694-703.
Singh, S.,Tripathi,Y. M, and Jun, C. H. (2015): Sampling Plans Based on Truncated Life Test for a Generalized Inverted Exponential Distribution, Industrial Engineering & Management Systems, 14(2), 183-195.
Sobel, M and TischendorfJ. A. (1959): Acceptance sampling with new life test objectives, Proceedings of fifth National Symposium on Reliability and Quality Control, Philadelphia Pennsylvania, 108-118.
Wu, C. W., Shu, M. H., and Wu, N. Y. (2020): Acceptance sampling schemes for two-parameter Lindley lifetime products under a truncated life test, Quality Technology and Quantitative Management, 18, 382-395.
