# Double-Acceptance Sampling Plan for Exponentiated Fréchet Distribution with Known Shape Parameters

Authors: M. Sridhar Babu; G. Srinivasa Rao; K. Rosaiah
Journal: Mathematical Problems in Engineering
 (2021)

---

## Abstract

We suppose that a product’s lifetime follow the exponentiated Fréchet distribution of defined shape parameters. Based on this assumption, a double-acceptance sampling plan is constructed. The zero and one failure framework is essentially thought of: if no errors are found from the first sample, then the lot is approved; also, if at least two failures occur, it is rejected. In the first sample, if one failure is observed, then the second sample is taken and decided for the same length as the first one. The cumulative sample sizes of the first and second samples are determined on the basis of the stated confidence level of the consumer to ensure that the actual median is longer than the given life. As indicated by the various ratios of the actual median life to specified median lifetime, the operating characteristics are calculated and placed in presented tables. To decrease the risk of the producer at the predefined level, the minimum ratios of this sort are additionally obtained. Lastly, examples are provided for representation reasons for the proposed model.


---

## Body

## 1. Introduction

Goods or products have lifetime differences, despite the fact that they are manufactured by the same producer and machine and under similar sates of assembling. Producers are exceptionally wary about the quality of their products, with the goal that when the customer shows up to buy them, they do not confront any challenges in getting them. It is not generally conceivable to test the full life of all goods from an enormous size assortment because of the expense and time of the test. The choice on the acceptance or rejection of the merchandise involving the manufacturer and the customer is therefore filled with vulnerabilities. The risk of the consumer is characterized as the probability of choosing a few items based on a sample from the lot that has the mean or median lifespan less than the predefined lifetime. The risk of the producer is the chance that a good quality product would be rejected as a defective product by the customer. An approach to limit the producer’s risk and consumer’s risk is to raise the size of the sample; however, this change will lead the producer to lose. To lessen such kind of risks, we should utilize a viable acceptance sampling plan. To lessen the trial period, truncated life tests are frequently implemented.We use the distribution of statistical probabilities to calculate the variation in product lifetime. The underlying statistical distribution could be used to understand the sampling design parameters. Various studies have been done to develop single-sample plans under various statistical distributions dependent on life of the truncated tests. In a single sampling scheme, a lot is rejected and we stop the experiment based on the truncated life test if beyondc failures happen during the pre-fixed experiment time. In the event that the experiment time initiates c or fewer failures, the lot will be accepted.Because of the simplicity of usage of the plan, much exertion has been made in the course of recent decades to explore the acceptance of single sampling plans for the testing and review of items for various sampling circumstances. A number of authors have developed single-sampling plans for different distributions: Epstein [1]; Goode and Kao [2]; Gupta and Groll [3]; Gupta [4]; Kantam et al. [5]; Baklizi [6]; Baklizi and Masri [7]; Tsai and Wu [8]; Balakrishnan et al. [9]; Rosaiah and Kantam [10]; Rosaiah et al. [11]; Rao et al. [12, 13]; and Rosaiah et al. [14]. The above-listed plan has also recently been designed for various lifetime distributions.In cases where normal distribution is often adopted, the double-sampling schemes were known to decrease the size of the sample or the risk of the producer in the area of quality control; e.g. Duncan [15]. The choice of double sampling decisions is based on the data collected from earlier decisions on the process. Aslam [16] developed Rayleigh distribution based on double-acceptance sampling depend on truncated life-tests. Aslam et al. [17, 18] proposed double-acceptance sampling plans based on truncated life tests for the Weibull distribution and general life distributions.A double-acceptance sampling plan for generalized log-logistic distribution with known shape parameters have been developed by Aslam and Jun [19]. Rao [20, 21] considered double-acceptance sampling plans for the Marshall–Olkin extended exponential and Marshall–Olkin extended Lomax distributions depend on average life time of the truncated life testing data. Aslam et al. [22] discussed double-acceptance sampling plans for Burr type-XII distribution percentiles under the truncated life tests. In generalized exponential distribution, Ramaswamy and Anburajan [23] presented double acceptance sampling based on truncated life-tests. Gui [24] has developed a double-acceptance sampling plan for time-truncated life-tests based on Maxwell distribution. Malathi and Muthulakshmi [25] developed a zero-one double-acceptance sampling plan based on Marshall–Olkin extended exponential distribution for truncated life tests. Tripathi et al. [26] developed an application of time-truncated single-acceptance sampling plan based on generalized half-normal distribution. Tripathi et al. [27] studied double- and group-acceptance sampling plan for truncated life test based on inverse log-logistic distribution.In the field of quality control, the normal distribution is often implemented; double sampling schemes have been reported to minimize the size of the sample or the producer’s risk; see, e.g., Duncan [15] in the area of reliability. Nevertheless, a plan for double sampling has also not been developed. Based on the assumption that a product’s lifetime follows the exponentiated Fréchet distribution, the main objective of this article is to recommend the double-acceptance sampling plans for truncated lifetime tests. In survival research, we especially use this distribution because of the monotonous nature of the hazard function. We essentially consider zero and one failure system where we accept a lot for no failures from the first sample and we reject if at least two failures are found. When there is only one defect, we will select the second sample and check with the same duration as the first sample. At the defined consumer’s level of confidence, both the initial and second sample minimum sizes are calculated. The operational properties are calculated according to the ratio of true median life to the defined lifetime. The minimum ratios are often achieved in order to reduce the risk of the producer to the degree specified. Section 2 describes the exponential Fréchet distribution. Section 3 accounts for the proposed double sampling scheme, and Section 4 assesses its operating characteristics. Examples of the sampling technique are given in Section 5 and finally conclusions are given in Section 6.


## 2. The Exponentiated Fréchet Distribution (EFD)

Consider the product’s lifetime following the exponentiated Fréchet distribution which was introduced and studied by Nadarjah and Kotz [28]; the probability density function and cumulative distribution function of EFD, respectively, are given by(1)ft,σ,λ,θ=σλλθ1−e−σ/tλθ−1t−1+λe−σ/tλ,Ft,σ,λ,θ=1−1−e−σ/tλθ,t>0,σ,λ,θ>0.Here, the scale parameter isσ and the shape parameters are λ and θ. Remember that this CDF represents the chance of failure of a parallel system with θ items having a Fréchet distributed lifetime. The exponentiated Fréchet distribution will then be used to check the reliability of the system. In constructing single-sampling plans, this exponentiated Fréchet distribution was considered in Rao et al. [13]. When θ = 1, it is called the Fréchet distribution. For this analysis, the shape parameters λ and θ are assumed to be a priori known.The parametersλ and θ can be predicted when data on failure are available. It is understood that for θ = 1, the failure rate (hazard function) declines in the case of λ≤1 while it rises and then declines in the case of λ>1. This distribution is used as a model for lifetime in this study, as its failure rate pattern is very flexible.Generally, the mean cannot necessarily be defined in a closed form, while exponential Fréchet distribution is possible in the closed form. However, the median life is the50th percentile of the exponentiated Fréchet distribution derived by(2)m=σ−ln1−0.51/θ−1/λ.When the parameters λ and θ are fixed, the median is directly proportional to the scale parameter. Note that the median reduces to m=σ independently of λ for the Fréchet distribution (θ = 1). Harlow [29] developed the Fréchet distribution function for applications. Nadarajah and Gupta [30] studied the beta Fréchet distribution. Abd-Elfattah and Omima [31] addressed the estimation of the unknown parameters of the generalized Fréchet distribution. Abd-Elfattah et al. [32] introduced the goodness of fit tests for generalized Fréchet distribution. Al-Nassar and Al-Omari [33] studied the acceptance sampling plan based on truncated life tests for exponentiated Fréchet distribution. Kotz and Nadarajah [34] studied theory and applications of extreme value distributions.


## 3. Design of Suggested Sampling Plan

Assume that the median (m) lifetime is used to determine the product’s quality. We presumed that the lot of good quality is submitted if the statistics enables the below null hypothesis, H0:m≥m0 towards the alternative hypothesis, H1:m<m0. The level of significance for the test, the risk of the consumer is used through 1−p∗, where p∗ is the consumer’s confidence level.Based on the truncated life test, the following double-acceptance sampling plan is developed.(i)
Choose a sample of sizen1 at random and examine it. If C1 or fewer failures occurred prior to t0 the predefined experiment time, the lot is approved. The experiment has been truncated before time t0 if (C2+1) failures are found, when the lot is rejected if (C1<C2).(ii)
If the failures in number lie betweenC1+1 and C2 by time t0, then select the second sample of size n2 and test them during time t0. From both the samples, a maximum of C2 failures are found; then the lot is approved. Failing that, the lot would be rejected.For a given factor multipliera, it is always feasible to fix the terminating time as a multiple of the defined lifetime m0, in which case t0=am0. The suggested sampling method is then described by the following five parameters n1,n2,C1,C2,a, if C1<C2.The procedure for a double-acceptance sampling plan (DASP) for life test flow chart is given below.The lot size is known as sufficiently large to calculate the acceptance probability of the lot in order to use the binomial distribution; see, for instance, Stephens [35], for further explanation of the application of binomial distribution. Under the suggested double-acceptance sampling plan, the lot acceptance probability is obtained by(3)Pa=∑i=0n1n1ipi1−Pn1−i+∑x=C1+1C2n1xpx1−Pn1−x∑i=0C2−xn2ipi1−Pn2−i.In the equation above, the probability that an item will fail before timet0 is p, which is given by(4)p=1−1−e−σ/t0λθ1−1−e−m/m0/ηqaλθ,where ηq=−ln1−1−q1/θ−1/λ.In general, we are focused inC1=0 and C2=1 and for the suggested double sampling plan, which would be regarded as zero and one failure scheme, as consumers prefer a sampling plan for acceptance with lower acceptance numbers. If the lot is accepted with several failed items from a test, this may not be understood by consumers though it may happen based on probabilities. In the zero and one failure plans, the probability of lot acceptance equation (3) reduces to(5)Pa=1−pn11+n1p1−pn2−1.The minimum sizes of samplen1 and n2 guarantee m≥m0 at the level of confidence for consumers P∗ can be achieved as a solution to the following inequality:(6)1−p0n11+n1P01−p0n2−1≤1−P∗,where the probability p0 in equation (3) obtained at m=m0 as(7)p0=1−1−e−1/ηqaλθ,ASN=n1p1+n1+n21−p1.The chance of rejection or acceptance isP1 on the basis of first sample and is given by(8)P1=1−∑i=c1+1c2n1ipi1−pn1−i.ForC1=0 and C2=1, we have(9)ASN=n1+n1n2p1−pn1−1.The following optimization problem is then determined in terms of the minimum sample sizes for zero and one failure scheme in our double-acceptance sampling plan:(10a)ASN=n1+n1n2p01−p0n1−1.Subject to(10b)1−P0n11+n1p01−P0n2−1≤1−P∗,(10c)1≤n2≤n1,(10d)n1,n2:integers.A quick search can solve this problem by modifying the values ofn1 and n2. The size of the sample relating to the single-sampling method that estimates the initial values of n1 and n2:(11)1−P0n≤1−P∗.For example, Table1 displays the lowest possible sample sizes for the first and second samples under the EFD according to various values of P∗=0.75,0.90,0.95,0.99 and a=0.5,0.7,0.9,1.1,1.3,1.5,1.7,1.9 three combinations of θ=0.5,1.0,2.0andλ=2 were taken into consideration. It has been found that the sizes of sample rapidly increase as λorθ increases when the experiment time is comparatively less, but they remain about the same irrespective of λorθ when the experiment duration is longer.Table 1
ASN and minimum sample sizes under exponentiated Fréchet distribution.
aθ, λP∗0.50.70.91.11.31.51.71.9(2.0, 2.0)0.75119, 93 (146.98)11, 7 (12.85)4, 3 (4.65)2, 2 (2.39)2, 1 (2.10)2, 1 (2.05)1, 1 (1.11)1, 1 (1.08)0.90176, 137 (202.26)15, 14 (17.56)6, 3 (6.36)3, 3 (3.36)2, 2 (2.21)2, 1 (2.05)2, 1 (2.03)2, 1 (2.01)0.95216, 183 (239.84)19, 16 (20.87)7, 5 (7.43)4, 3 (4.19)3, 2 (3.08)2, 2 (2.10)2, 1 (2.03)2, 1 (2.01)0.99315, 309 (328.60)28, 20 (28.75)10, 6 (10.16)6, 3 (6.05)4, 2 (4.03)3, 2 (3.03)3, 1 (3.00)2, 2 (2.03)(1.0, 2.0)0.7528, 20 (33.74)7, 4 (7.97)4, 2 (4.37)3, 1 (3.14)2, 2 (2.30)2, 1 (2.10)2, 1 (2.07)2, 1 (2.05)0.9040, 33 (46.24)10, 6 (10.90)5, 4 (5.53)4, 2 (4.16)3, 2 (3.15)2, 2 (2.21)2, 2 (2.14)2, 1 (2.05)0.9549, 45 (54.83)12, 9 (12.93)6, 5 (6.46)4, 4 (4.33)3, 3 (3.23)3, 2 (3.08)3, 1 (3.02)2, 2 (2.10)0.9973, 59 (75.42)17, 14 (17.51)9, 5 (9.13)6, 4 (6.09)5, 2 (5.03)4, 2 (4.03)4, 1 (4.01)3, 2 (3.03)(0.5, 2.0)0.759, 9 (11.53)5, 3 (5.66)3, 3 (3.67)3, 2 (3.32)2, 2 (2.38)2, 2 (2.31)2, 1 (2.13)2, 1 (2.11)0.9014, 11 (15.86)7, 4 (7.55)5, 3 (5.33)4, 2 (4.19)3, 3 (3.34)3, 2 (3.16)3, 1 (3.06)3, 1 (3.05)0.9517, 14 (18.63)8, 7 (8.73)6, 3 (6.22)5, 2 (5.11)4, 2 (4.12)4, 2 (4.08)3, 2 (3.12)3, 2 (3.09)0.9925, 19 (25.71)12, 8 (12.25)8, 6 (8.17)7, 3 (7.05)6, 2 (6.03)5, 3 (5.05)4, 4 (4.10)4, 3 (4.05)(1.6684, 0.7130)0.756, 5 (7.24)4, 3 (4.68)3, 3 (3.64)3, 2 (3.33)3, 1 (3.13)2, 2 (2.35)2, 2 (2.30)2, 1 (2.13)0.909, 6 (9.87)6, 4 (6.52)5, 2 (5.20)4, 2 (4.21)3, 3 (3.38)3, 2 (3.20)3, 2 (3.15)3, 1 (3.06)0.9511, 7 (11.67)7, 5 (7.47)6, 3 (6.20)5, 3 (5.18)4, 3 (4.21)4, 2 (4.10)3, 3 (3.23)3, 2 (3.12)0.9915, 14 (15.52)10, 8 (10.26)8, 5 (8.13)7, 3 (7.06)6, 3 (6.05)5, 4 (5.09)5, 2 (5.03)4, 4 (4.10)


## 4. Operating Characteristics (OC)

Therefore, we have to know the operating characteristics of the planned proposal on the basis of the ratio of actual median life to defined lifetq/tq0, i.e., m/m0 clearly, a scheme will become more appropriate if its OC increases very close to one. Tables 2–4 display the EFD OC values with of P∗=0.75,0.90,0.95,0.99 and a=0.5,0.7,0.9,1.1,1.3,1.5,1.7,1.9 three combinations of θ=0.5,1.0,2.0andλ=2. Table 5 display the OC values of EFD for the estimated parameters ofλ^=0.7130andθ^=1.6684. When the actual median life increases more than the defined life, the chance of acceptance will increase. Therefore, we have to know the OC’s for the suggested plan according to the ratio of the actual median life to the defined life, i.e. m/m0. Obviously, a method will be more suitable if its OC increases very close to one.Table 2
OC values for exponentiated Fréchet distribution withθ = 2 and λ = 2.
Ratiom (m0)P∗n1n2a2468100.75119930.50.99840.99890.99981.00001.00001170.70.99760.99920.99980.99991.0000430.90.99270.99820.99910.99951.0000221.10.99880.99910.99980.99991.0000211.30.98870.99560.99910.99890.9999211.50.95480.97890.99430.99910.9999111.70.88970.98230.99120.99920.9999111.90.80000.98990.99560.99900.99990.901761370.50.99000.99560.99760.99991.000015140.70.98910.99560.99910.99981.0000630.90.99190.99780.99910.99991.0000331.10.99660.99890.99940.99961.0000221.30.96850.98790.99230.99980.9999211.50.88350.95670.99640.99950.9999211.70.74240.91450.99890.99910.9998211.90.57890.96980.99870.99900.99990.952161830.50.98490.98780.99870.99971.000019160.70.97190.99190.99840.99960.9999750.90.96910.98990.99230.99880.9999431.10.94660.98790.99230.99980.9999321.30.93850.95670.99540.99850.9999221.50.88350.91450.99890.99951.0000211.70.74240.90980.99270.99390.9999211.90.57890.91980.98970.99190.99990.993153090.50.99870.99910.99910.99891.000028200.70.99120.99860.99910.99890.99991060.90.97990.98790.99230.99980.9999631.10.94330.95670.99740.99950.9999421.30.91140.92450.99690.99910.9998321.50.79940.91450.99290.99890.9998311.70.59480.90980.99170.99790.9999221.90.39550.91960.97890.99450.9987Table 3
OC values for exponentiated Fréchet distribution withθ = 1 and λ = 2.
Ratiom (m0)P∗n1n2a2468100.7528200.50.99140.99790.99921.00001.0000740.70.97890.99820.99960.99991.0000420.90.99690.99820.99910.99951.0000311.10.97140.99910.99980.99991.0000221.30.90180.98560.99710.99880.9999211.50.79450.96890.99830.99960.9999211.70.67210.98230.99020.99820.9999211.90.55400.98990.99160.99290.99990.9040330.50.96780.99510.99780.99991.00001060.70.95990.99560.99910.99981.0000540.90.94490.99680.99910.99991.0000421.10.93490.99790.99910.99961.0000321.30.85300.98790.99230.99980.9999221.50.70910.95770.99550.99910.9999221.70.56040.91450.99890.99910.9998211.90.43030.96910.99370.99290.99990.9549450.50.97880.98780.99820.99961.00001290.70.96970.99080.99740.99950.9999650.90.95190.98680.99050.99890.9999441.10.94660.98790.99250.99880.9999331.30.82850.95670.99540.99860.9999321.50.66640.91450.99860.99931.0000311.70.50450.90910.99230.99350.9999221.90.36850.91980.98950.99170.99990.9973590.50.99160.99810.99970.99991.000017140.70.98990.99840.99920.99890.9999950.90.97690.98760.99170.99980.9999641.10.91690.95660.99730.99950.9999521.30.74980.92450.99690.99910.9998421.50.54550.91450.99350.99880.9999411.70.36670.90980.99170.99690.9999321.90.23580.91930.97890.99350.9977Table 4
OC values for exponentiated Fréchet distribution withθ = 0.5 and λ = 2.
Ratiom (m0)P∗n1n2a2468100.75990.50.98990.99650.99790.99920.9999530.70.98130.99590.99790.99920.9998330.90.95430.99130.99860.99940.9997321.10.87950.99760.99930.99970.9999221.30.78740.99670.99560.99910.9999221.50.69440.96750.97390.99450.9996211.70.60890.96790.98330.99140.9995211.90.53360.93840.97990.99440.99890.914110.50.98980.99780.99960.99980.9999740.70.98650.99230.99460.99890.9999530.90.91640.98870.99670.99770.9999421.10.79310.99030.99810.99960.9996331.30.65650.98360.98790.99230.9998321.50.53240.96550.97670.99610.9996311.70.42930.94020.92150.99780.9994311.90.34690.88690.96680.99770.99980.9517140.50.99680.99910.99960.99980.9999870.70.97040.99450.99690.99940.9999630.90.88330.97760.98960.99330.9998521.10.72540.96890.98740.99210.9997421.30.56450.95060.95670.99540.9995421.50.42990.90460.91470.99870.9996321.70.32640.89560.90980.99240.9937321.90.24930.84710.91980.98960.99910.9925190.50.98950.99890.99930.99930.99991280.70.95590.98780.99850.99920.9999860.90.77150.99990.98790.99230.9998731.10.53340.98730.95420.99740.9995621.30.34450.97830.98450.99680.9998531.50.21990.92290.95450.99410.9999441.70.14260.82860.93980.99080.9989431.90.09480.71120.92960.97890.9925Table 5
OC values for exponentiated Fréchet distribution withθ^ = 1.6684 and λ^ = 0.7130.
Ratiom (m0)P∗n1n2a2468100.75650.50.96360.99901.00001.00001.0000430.70.88980.99260.99940.99991.0000330.90.79510.97610.99680.99950.9999321.10.69780.94840.99060.99810.9996311.30.60760.91110.97970.99500.9987221.50.52800.86720.96360.98960.9968221.70.45930.81960.94270.98130.9936211.90.40080.77050.91780.97020.98890.90960.50.93850.99820.99991.00001.0000640.70.82350.98710.99890.99991.0000520.90.68840.95920.99440.99910.9998421.10.56160.91390.98370.99670.9993331.30.45390.85570.96510.99120.9977321.50.36640.79010.93850.98180.9944321.70.29680.72210.90490.96790.9889311.90.24180.65530.86600.94940.98060.951170.50.92180.99760.99991.00001.0000750.70.78390.98310.99850.99981.0000630.90.63160.94760.99260.99880.9998531.10.49690.89150.97870.99560.9990431.30.38840.82160.95500.98850.9969421.50.30450.74540.92180.97630.9926331.70.24050.66870.88060.95850.9854321.90.19180.59560.83390.93530.97480.9915140.50.87920.99610.99981.00001.00001080.70.68650.97270.99750.99971.0000850.90.49650.91780.98800.99810.9997731.10.34730.83510.96590.99280.9984631.30.24060.73720.92910.98130.9949541.50.16710.63640.87920.96210.9880521.70.11710.54080.81940.93450.9764960.50.93850.99820.99991.00001.0000We may compare the DASP with the existing single sampling plan (whenc=0and1) in terms of OC values. From Table 4, the design parameters of DASP with c1=0,c2=1 when θ=0.5,λ=2,β=0.25atδq=0.5 are n1=9andn2=9. So, sample size for single sampling plan is 9. Therefore, we compare DASP with single sampling plan with n=9,c=0 and also for n=9,c=1. It is clear visible of these 3 plans in Figure 1. The OC values for DASP pass through middle of the two single sampling plans. It ultimately proves DASP provides better results than single stage group sampling plan.Figure 1
OC curve of double- and single-sampling plans.


## 5. Use of Tables and Example

The data obtained from the cleanup gradient test wells on vinyl chloride are considered. Vinyl chloride is an unstable chemical compound. This factor is of great interest as it is both anthropogenic and carcinogenic in environmental investigations. In addition, this component is found to be weak in a number of well-tracking history. Lower level identifications of this product are attributed to water or air cross-contamination, or to the testing method itself in safe historical wells. Bhaumik and Gibbons [36] and Krishnamoorthy et al. [37] discussed this principle in the development of predictive and tolerance intervals for gamma variables.Data on vinyl chloride from safe groundwater observation wells (μg/L).The validity of our model by plotting the superimposed data indicates that the EFD is a reasonable fit and also the goodness of fit is shown by the Q-Q plot seen in Figure2. The maximum likelihood estimates of the EFD two-parameters for breaking carbon fiber stress are λ^=0.7130andθ^=1.6684, and using the Kolmogorov-Smirnov test, we found that the maximum distance between the data and the fitted of the EFD is 0.11804 with p value is 0.7306. The EFD therefore suits the vinyl chloride data perfectly.Figure 2
The empirical and theoretical cdfs and Q-Q plots for the vinyl chloride data. (a) Empirical and fitted PDFs, (b) Q-Q plot.
(a)(b)Suppose the product’s lifetime follows exponentiated Fréchet distribution with parametersθ=2 and λ=2. It can also be known that the manufacturer would like to learn whether the median life of the product is above or equivalent to 1000 hours at a level of conﬁdence 0.75. The researcher wants to end an experiment at 500 hours under the zero and one failure plan of the double sampling plan. It refers to the a = 0.7 terminator of the experiment. From Table 1, the required minimum sizes of sample are n1 = 11 and n2 = 7. The double sampling plan shall be described as follows. The very first thing to do here is to monitor eleven items for 500 hours and accept the lot if no failure occurs during the experiment. When the experiment produces at least two errors, the lot is rejected. The second sample of seven items is drawn and tested for 500 hours where only one failure was reported. The lot would be accepted if no failure exists in the second sample. When quality improves, the product can be correlated with the probability of acceptance, and they want to reduce the risk of the producer. Suppose the supplier knows what degree of quality leads to a risk of less than 0.05 from the product.


## 6. Conclusion

A double-sampling procedure for the decision to approve or reject the lot submitted was built on the basis of a truncated life test. The lifespan of the product is expected to follow exponentiated Fréchet distribution, which is useful in system reliability analysis because the failure rate is very flexible. It was observed that the necessary sample sizes declined steadily as the time of the experiment grew and that the size of the sample for the reasonable duration of the experiment was not very sensitive to the confidence level or the shape parameter. It has been revealed by examples that the double sampling plan would be more appropriate than a single sampling plan in terms of the OC values. A variable sampling plan will be preferred, because it utilizes all the details available. As a result, the potential research will establish a double acceptation sampling plan on variables.



---

*Source: 7308454-2021-02-28.xml*