Double-AcceptanceSamplingPlanforExponentiatedFre´chet ...

Research ArticleDouble-Acceptance Sampling Plan for Exponentiated FrechetDistribution with Known Shape Parameters

M Sridhar Babu1 G Srinivasa Rao 2 and K Rosaiah3

1Department of Sciences St Maryrsquos College Yousufguda Hyderabad 500045 India2Department of Mathematics and Statistics e University of Dodoma Dodoma POBox 259 Tanzania3Department of Statistics Acharya Nagarjuna University Guntur 522007 India

Correspondence should be addressed to G Srinivasa Rao gaddesraoyahoocom

Received 24 July 2020 Revised 1 October 2020 Accepted 18 February 2021 Published 28 February 2021

Academic Editor Hussein Abulkasim

Copyright copy 2021 M Sridhar Babu et al +is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We suppose that a productrsquos lifetime follow the exponentiated Frechet distribution of defined shape parameters Based on thisassumption a double-acceptance sampling plan is constructed +e zero and one failure framework is essentially thought of if noerrors 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 ifone failure is observed then the second sample is taken and decided for the same length as the first one +e cumulative samplesizes of the first and second samples are determined on the basis of the stated confidence level of the consumer to ensure that theactual median is longer than the given life As indicated by the various ratios of the actual median life to specified median lifetimethe operating characteristics are calculated and placed in presented tables To decrease the risk of the producer at the predefinedlevel the minimum ratios of this sort are additionally obtained Lastly examples are provided for representation reasons for theproposed model

1 Introduction

Goods or products have lifetime differences despite the factthat they are manufactured by the same producer andmachine and under similar sates of assembling Producersare exceptionally wary about the quality of their productswith the goal that when the customer shows up to buy themthey do not confront any challenges in getting them It is notgenerally conceivable to test the full life of all goods from anenormous size assortment because of the expense and timeof the test +e choice on the acceptance or rejection of themerchandise involving the manufacturer and the customeris therefore filled with vulnerabilities +e risk of the con-sumer is characterized as the probability of choosing a fewitems based on a sample from the lot that has the mean ormedian lifespan less than the predefined lifetime +e risk ofthe producer is the chance that a good quality product wouldbe rejected as a defective product by the customer Anapproach to limit the producerrsquos risk and consumerrsquos risk is

to raise the size of the sample however this change will leadthe producer to lose To lessen such kind of risks we shouldutilize a viable acceptance sampling plan To lessen the trialperiod truncated life tests are frequently implemented

We use the distribution of statistical probabilities tocalculate the variation in product lifetime +e underlyingstatistical distribution could be used to understand thesampling design parameters Various studies have been doneto develop single-sample plans under various statisticaldistributions dependent on life of the truncated tests In asingle sampling scheme a lot is rejected and we stop theexperiment based on the truncated life test if beyond cfailures happen during the pre-fixed experiment time In theevent that the experiment time initiates c or fewer failuresthe lot will be accepted

Because of the simplicity of usage of the plan muchexertion has been made in the course of recent decades toexplore the acceptance of single sampling plans for thetesting and review of items for various sampling

HindawiMathematical Problems in EngineeringVolume 2021 Article ID 7308454 9 pageshttpsdoiorg10115520217308454

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] Kantamet al [5] Baklizi [6] Baklizi and Masri [7] Tsai and Wu [8]Balakrishnan et al [9] Rosaiah and Kantam [10] Rosaiahet al [11] Rao et al [12 13] and Rosaiah et al [14] +eabove-listed plan has also recently been designed for variouslifetime distributions

In cases where normal distribution is often adopted thedouble-sampling schemes were known to decrease the size ofthe sample or the risk of the producer in the area of qualitycontrol eg Duncan [15] +e choice of double samplingdecisions is based on the data collected from earlier decisionson the process Aslam [16] developed Rayleigh distributionbased on double-acceptance sampling depend on truncatedlife-tests Aslam et al [17 18] proposed double-acceptancesampling plans based on truncated life tests for the Weibulldistribution and general life distributions

A double-acceptance sampling plan for generalized log-logistic distribution with known shape parameters havebeen developed by Aslam and Jun [19] Rao [20 21]considered double-acceptance sampling plans for theMarshallndashOlkin extended exponential and MarshallndashOlkinextended Lomax distributions depend on average life timeof the truncated life testing data Aslam et al [22] discusseddouble-acceptance sampling plans for Burr type-XII dis-tribution percentiles under the truncated life tests Ingeneralized exponential distribution Ramaswamy andAnburajan [23] presented double acceptance samplingbased on truncated life-tests Gui [24] has developed adouble-acceptance sampling plan for time-truncated life-tests based on Maxwell distribution Malathi andMuthulakshmi [25] developed a zero-one double-accep-tance sampling plan based on MarshallndashOlkin extendedexponential distribution for truncated life tests Tripathiet al [26] developed an application of time-truncatedsingle-acceptance sampling plan based on generalized half-normal distribution Tripathi et al [27] studied double- andgroup-acceptance sampling plan for truncated life testbased on inverse log-logistic distribution

In the field of quality control the normal distribution isoften implemented double sampling schemes have been re-ported tominimize the size of the sample or the producerrsquos risksee eg Duncan [15] in the area of reliability Nevertheless aplan for double sampling has also not been developed Based onthe assumption that a productrsquos lifetime follows the expo-nentiated Frechet distribution the main objective of this articleis to recommend the double-acceptance sampling plans fortruncated lifetime tests In survival research we especially usethis distribution because of the monotonous nature of thehazard function We essentially consider zero and one failuresystem where we accept a lot for no failures from the firstsample and we reject if at least two failures are found Whenthere is only one defect we will select the second sample andcheck with the same duration as the first sample At the definedconsumerrsquos level of confidence both the initial and secondsample minimum sizes are calculated +e operational prop-erties are calculated according to the ratio of truemedian life tothe defined lifetime+e minimum ratios are often achieved in

order to reduce the risk of the producer to the degree specifiedSection 2 describes the exponential Frechet distribution Sec-tion 3 accounts for the proposed double sampling scheme andSection 4 assesses its operating characteristics Examples of thesampling technique are given in Section 5 and finally con-clusions are given in Section 6

2 The Exponentiated FrechetDistribution (EFD)

Consider the productrsquos lifetime following the exponentiatedFrechet distribution which was introduced and studied byNadarjah and Kotz [28] the probability density function andcumulative distribution function of EFD respectively aregiven by

f(t σ λ θ) σλλθ 1 minus eminus (σt)λ

1113876 1113877θminus 1

tminus (1+λ)

eminus (σt)λ

F(t σ λ θ) 1 minus 1 minus eminus (σt)λ

1113876 1113877θ tgt 0 σ λ θgt 0

(1)

Here the scale parameter is σ and the shape parameters areλ and θ Remember that this CDF represents the chance offailure of a parallel system with θ items having a Frechetdistributed lifetime +e exponentiated Frechet distributionwill then be used to check the reliability of the system Inconstructing single-sampling plans this exponentiated Frechetdistribution was considered in Rao et al [13] When θ 1 it iscalled the Frechet distribution For this analysis the shapeparameters λ and θ are assumed to be a priori known

+e parameters λ and θ can be predicted when data onfailure are available It is understood that for θ 1 the failurerate (hazard function) declines in the case of λle 1 while itrises and then declines in the case of λgt 1 +is distributionis used as a model for lifetime in this study as its failure ratepattern is very flexible

Generally the mean cannot necessarily be defined in aclosed form while exponential Frechet distribution ispossible in the closed form However the median life is the50th percentile of the exponentiated Frechet distributionderived by

m σ minus ln 1 minus (05)1θ

1113872 11138731113960 1113961minus (1λ)

(2)

When the parameters λ and θ are fixed the median isdirectly proportional to the scale parameter Note that themedian reduces to m σ independently of λ for the Frechetdistribution (θ1) Harlow [29] developed the Frechetdistribution function for applications Nadarajah andGupta [30] studied the beta Frechet distribution Abd-Elfattah and Omima [31] addressed the estimation of theunknown parameters of the generalized Frechet distribu-tion Abd-Elfattah et al [32] introduced the goodness of fittests for generalized Frechet distribution Al-Nassar andAl-Omari [33] studied the acceptance sampling plan basedon truncated life tests for exponentiated Frechet distri-bution Kotz and Nadarajah [34] studied theory and ap-plications of extreme value distributions

2 Mathematical Problems in Engineering

3 Design of Suggested Sampling Plan

Assume that the median (m) lifetime is used to determinethe productrsquos quality We presumed that the lot of goodquality is submitted if the statistics enables the below nullhypothesis H0 mgem0 towards the alternative hypothesisH1 mltm0 +e level of significance for the test the risk ofthe consumer is used through 1 minus plowast where plowast is theconsumerrsquos confidence level

Based on the truncated life test the following double-acceptance sampling plan is developed

(i) Choose a sample of size n1 at random and examine itIf C1 or fewer failures occurred prior to t0 thepredefined experiment time the lot is approved +eexperiment has been truncated before time t0 if(C2 + 1) failures are found when the lot is rejected if(C1 ltC2)

(ii) If the failures in number lie between C1 + 1 and C2by time t0 then select the second sample of size n2and test them during time t0 From both the samplesa maximum of C2 failures are found then the lot isapproved Failing that the lot would be rejected

For a given factor multiplier a it is always feasible to fixthe terminating time as a multiple of the defined lifetimem0in which case t0 am0 +e suggested sampling method isthen described by the following five parameters(n1 n2 C1 C2 a) if C1 ltC2

+e procedure for a double-acceptance sampling plan(DASP) for life test flow chart is given below

Flow chart for DASP

Yes

No

Yes

No

Yes

No

Draw the secondsample of n2 units

If F1 le C1

If F1 gt C2

If F1 + F2 gt C2

Accept the lot

Reject the lot

Draw a sample ofsize n1

Reject the lot

Accept the lot

+e lot size is known as sufficiently large to calculate theacceptance probability of the lot in order to use the binomialdistribution see for instance Stephens [35] for furtherexplanation of the application of binomial distributionUnder the suggested double-acceptance sampling plan thelot acceptance probability is obtained by

Pa 1113944

n1

i0

n1

i1113888 1113889p

i(1 minus P)

n1minus i+ 1113944

C2

xC1+1

n1

x1113888 1113889p

x

middot (1 minus P)n1minus x

1113944

C2minus x

i0

n2

i1113888 1113889p

i(1 minus P)

n2minus i⎡⎢⎣ ⎤⎥⎦

(3)

In the equation above the probability that an item willfail before time t0 is p which is given by

p 1 minus 1 minus eminus σt0( )

λ

1113876 1113877θ1 minus 1 minus e

minus mm0( )ηqa( 1113857λ

1113890 1113891

θ

(4)

where ηq [minus ln(1 minus (1 minus q)1θ)]minus (1λ)In general we are focused in C1 0 and C2 1 and for

the suggested double sampling plan which would beregarded as zero and one failure scheme as consumersprefer a sampling plan for acceptance with lower acceptancenumbers If the lot is accepted with several failed items froma test this may not be understood by consumers though itmay happen based on probabilities In the zero and onefailure plans the probability of lot acceptance equation (3)reduces to

Pa (1 minus p)n1 1 + n1p(1 minus p)

n2minus 11113960 1113961 (5)

+e minimum sizes of sample n1 and n2 guaranteemgem0 at the level of confidence for consumers Plowast can beachieved as a solution to the following inequality

1 minus p0( 1113857n1 1 + n1P0 1 minus p0( 1113857

n2minus 11113960 1113961le 1 minus P

lowast (6)

where the probability p0 in equation (3) obtained at m m0 as

p0 1 minus 1 minus eminus 1ηqa( 1113857

λ

1113890 1113891

θ

ASN n1p1 + n1 + n2( 1113857 1 minus p1( 1113857

(7)

+e chance of rejection or acceptance is P1 on the basisof first sample and is given by

P1 1 minus 1113944

c2

ic1+1

n1

i1113888 1113889p

i(1 minus p)

n1minus i (8)

For C1 0 and C2 1 we have

ASN n1 + n1n2p(1 minus p)n1minus 1

(9)

+e following optimization problem is then determinedin terms of the minimum sample sizes for zero and onefailure scheme in our double-acceptance sampling plan

ASN n1 + n1n2p0 1 minus p0( 1113857n1minus 1

(10a)

Subject to

Mathematical Problems in Engineering 3

1 minus P0( 1113857n1 1 + n1p0 1 minus P0( 1113857

n2minus 11113960 1113961le 1 minus P

lowast (10b)

1le n2 le n1 (10c)

n1 n2 integers (10d)

A quick search can solve this problem by modifying thevalues of n1 and n2+e size of the sample relating to the single-sampling method that estimates the initial values of n1 and n2

1 minus P0( 1113857n le 1 minus P

lowast (11)

For example Table 1 displays the lowest possible samplesizes for the first and second samples under the EFDaccording to various values of Plowast ( 075 090 095 099)

and a (05 07 09 11 13 15 17 19) three combina-tions of (θ 05 10 20 and λ 2) were taken into con-sideration It has been found that the sizes of sample rapidlyincrease as (λ or θ) increases when the experiment time iscomparatively less but they remain about the same irre-spective of (λ or θ) when the experiment duration is longer

4 Operating Characteristics (OC)

+erefore we have to know the operating characteristics ofthe planned proposal on the basis of the ratio of actualmedian life to defined life tqt0q ie (mm0) clearly a schemewill become more appropriate if its OC increases very closeto one Tables 2ndash4 display the EFD OC values with of Plowast

(075 090 095 099) and a (05 07 09 11 13 15

17 19) three combinations of (θ 05 10 20 and λ 2)Table 5 display the OC values of EFD for the estimatedparameters of1113954λ 07130 and 1113954θ 16684 When the actualmedian life increases more than the defined life the chance ofacceptance will increase +erefore we have to know the OCrsquosfor the suggested plan according to the ratio of the actualmedianlife to the defined life ie (mm0) Obviously a method will bemore suitable if its OC increases very close to one

We may compare the DASP with the existing singlesampling plan (when c 0 and 1) in terms of OC valuesFrom Table 4 the design parameters of DASP with c1

0 c2 1 when θ 05 λ 2 β 025 at δq 05 aren1 9 and n2 9 So sample size for single sampling plan is9 +erefore we compare DASP with single sampling planwith n 9 c 0 and also for n 9 c 1 It is clear visible ofthese 3 plans in Figure 1 +e OC values for DASP passthrough middle of the two single sampling plans It ulti-mately proves DASP provides better results than single stagegroup sampling plan

5 Use of Tables and Example

+e data obtained from the cleanup gradient test wells onvinyl chloride are considered Vinyl chloride is an un-stable chemical compound +is factor is of great interestas it is both anthropogenic and carcinogenic in envi-ronmental investigations In addition this component isfound to be weak in a number of well-tracking historyLower level identifications of this product are attributed towater or air cross-contamination or to the testing methoditself in safe historical wells Bhaumik and Gibbons [36]and Krishnamoorthy et al [37] discussed this principle inthe development of predictive and tolerance intervals forgamma variables

Data on vinyl chloride from safe groundwater obser-vation wells (μgL)

+e validity of our model by plotting the superimposeddata indicates that the EFD is a reasonable fit and also thegoodness of fit is shown by the Q-Q plot seen in Figure 2+emaximum likelihood estimates of the EFD two-parametersfor breaking carbon fiber stress are1113954λ 07130 and 1113954θ 16684 and using the Kolmogorov-Smirnov test we found that the maximum distance betweenthe data and the fitted of the EFD is 011804 with p value is07306 +e EFD therefore suits the vinyl chloride dataperfectly

Table 1 ASN and minimum sample sizes under exponentiated Frechet distribution

aθ λ Plowast 05 07 09 11 13 15 17 19

(20 20)

075 119 93 (14698) 11 7 (1285) 4 3 (465) 2 2 (239) 2 1 (210) 2 1 (205) 1 1 (111) 1 1 (108)090 176 137 (20226) 15 14 (1756) 6 3 (636) 3 3 (336) 2 2 (221) 2 1 (205) 2 1 (203) 2 1 (201)095 216 183 (23984) 19 16 (2087) 7 5 (743) 4 3 (419) 3 2 (308) 2 2 (210) 2 1 (203) 2 1 (201)099 315 309 (32860) 28 20 (2875) 10 6 (1016) 6 3 (605) 4 2 (403) 3 2 (303) 3 1 (300) 2 2 (203)

(10 20)

075 28 20 (3374) 7 4 (797) 4 2 (437) 3 1 (314) 2 2 (230) 2 1 (210) 2 1 (207) 2 1 (205)090 40 33 (4624) 10 6 (1090) 5 4 (553) 4 2 (416) 3 2 (315) 2 2 (221) 2 2 (214) 2 1 (205)095 49 45 (5483) 12 9 (1293) 6 5 (646) 4 4 (433) 3 3 (323) 3 2 (308) 3 1 (302) 2 2 (210)099 73 59 (7542) 17 14 (1751) 9 5 (913) 6 4 (609) 5 2 (503) 4 2 (403) 4 1 (401) 3 2 (303)

(05 20)

075 9 9 (1153) 5 3 (566) 3 3 (367) 3 2 (332) 2 2 (238) 2 2 (231) 2 1 (213) 2 1 (211)090 14 11 (1586) 7 4 (755) 5 3 (533) 4 2 (419) 3 3 (334) 3 2 (316) 3 1 (306) 3 1 (305)095 17 14 (1863) 8 7 (873) 6 3 (622) 5 2 (511) 4 2 (412) 4 2 (408) 3 2 (312) 3 2 (309)099 25 19 (2571) 12 8 (1225) 8 6 (817) 7 3 (705) 6 2 (603) 5 3 (505) 4 4 (410) 4 3 (405)

(16684 07130)

075 6 5 (724) 4 3 (468) 3 3 (364) 3 2 (333) 3 1 (313) 2 2 (235) 2 2 (230) 2 1 (213)090 9 6 (987) 6 4 (652) 5 2 (520) 4 2 (421) 3 3 (338) 3 2 (320) 3 2 (315) 3 1 (306)095 11 7 (1167) 7 5 (747) 6 3 (620) 5 3 (518) 4 3 (421) 4 2 (410) 3 3 (323) 3 2 (312)099 15 14 (1552) 10 8 (1026) 8 5 (813) 7 3 (706) 6 3 (605) 5 4 (509) 5 2 (503) 4 4 (410)


Table 2 OC values for exponentiated Frechet distribution with θ 2 and λ 2

Ratio m (m0)Plowast n1 n2 a 2 4 6 8 10

075

119 93 05 09984 09989 09998 10000 1000011 7 07 09976 09992 09998 09999 100004 3 09 09927 09982 09991 09995 100002 2 11 09988 09991 09998 09999 100002 1 13 09887 09956 09991 09989 099992 1 15 09548 09789 09943 09991 099991 1 17 08897 09823 09912 09992 099991 1 19 08000 09899 09956 09990 09999

090

176 137 05 09900 09956 09976 09999 1000015 14 07 09891 09956 09991 09998 100006 3 09 09919 09978 09991 09999 100003 3 11 09966 09989 09994 09996 100002 2 13 09685 09879 09923 09998 099992 1 15 08835 09567 09964 09995 099992 1 17 07424 09145 09989 09991 099982 1 19 05789 09698 09987 09990 09999

095

216 183 05 09849 09878 09987 09997 1000019 16 07 09719 09919 09984 09996 099997 5 09 09691 09899 09923 09988 099994 3 11 09466 09879 09923 09998 099993 2 13 09385 09567 09954 09985 099992 2 15 08835 09145 09989 09995 100002 1 17 07424 09098 09927 09939 099992 1 19 05789 09198 09897 09919 09999

099

315 309 05 09987 09991 09991 09989 1000028 20 07 09912 09986 09991 09989 0999910 6 09 09799 09879 09923 09998 099996 3 11 09433 09567 09974 09995 099994 2 13 09114 09245 09969 09991 099983 2 15 07994 09145 09929 09989 099983 1 17 05948 09098 09917 09979 099992 2 19 03955 09196 09789 09945 09987



075

28 20 05 09914 09979 09992 10000 100007 4 07 09789 09982 09996 09999 100004 2 09 09969 09982 09991 09995 100003 1 11 09714 09991 09998 09999 100002 2 13 09018 09856 09971 09988 099992 1 15 07945 09689 09983 09996 099992 1 17 06721 09823 09902 09982 099992 1 19 05540 09899 09916 09929 09999

090

40 33 05 09678 09951 09978 09999 1000010 6 07 09599 09956 09991 09998 100005 4 09 09449 09968 09991 09999 100004 2 11 09349 09979 09991 09996 100003 2 13 08530 09879 09923 09998 099992 2 15 07091 09577 09955 09991 099992 2 17 05604 09145 09989 09991 099982 1 19 04303 09691 09937 09929 09999


Table 3 Continued


095

49 45 05 09788 09878 09982 09996 1000012 9 07 09697 09908 09974 09995 099996 5 09 09519 09868 09905 09989 099994 4 11 09466 09879 09925 09988 099993 3 13 08285 09567 09954 09986 099993 2 15 06664 09145 09986 09993 100003 1 17 05045 09091 09923 09935 099992 2 19 03685 09198 09895 09917 09999

099

73 59 05 09916 09981 09997 09999 1000017 14 07 09899 09984 09992 09989 099999 5 09 09769 09876 09917 09998 099996 4 11 09169 09566 09973 09995 099995 2 13 07498 09245 09969 09991 099984 2 15 05455 09145 09935 09988 099994 1 17 03667 09098 09917 09969 099993 2 19 02358 09193 09789 09935 09977



075

9 9 05 09899 09965 09979 09992 099995 3 07 09813 09959 09979 09992 099983 3 09 09543 09913 09986 09994 099973 2 11 08795 09976 09993 09997 099992 2 13 07874 09967 09956 09991 099992 2 15 06944 09675 09739 09945 099962 1 17 06089 09679 09833 09914 099952 1 19 05336 09384 09799 09944 09989

09

14 11 05 09898 09978 09996 09998 099997 4 07 09865 09923 09946 09989 099995 3 09 09164 09887 09967 09977 099994 2 11 07931 09903 09981 09996 099963 3 13 06565 09836 09879 09923 099983 2 15 05324 09655 09767 09961 099963 1 17 04293 09402 09215 09978 099943 1 19 03469 08869 09668 09977 09998

095

17 14 05 09968 09991 09996 09998 099998 7 07 09704 09945 09969 09994 099996 3 09 08833 09776 09896 09933 099985 2 11 07254 09689 09874 09921 099974 2 13 05645 09506 09567 09954 099954 2 15 04299 09046 09147 09987 099963 2 17 03264 08956 09098 09924 099373 2 19 02493 08471 09198 09896 09991

099

25 19 05 09895 09989 09993 09993 0999912 8 07 09559 09878 09985 09992 099998 6 09 07715 09999 09879 09923 099987 3 11 05334 09873 09542 09974 099956 2 13 03445 09783 09845 09968 099985 3 15 02199 09229 09545 09941 099994 4 17 01426 08286 09398 09908 099894 3 19 00948 07112 09296 09789 09925


Suppose the productrsquos lifetime follows exponentiatedFrechet distribution with parameters θ 2 and λ 2 It canalso be known that the manufacturer would like to learnwhether themedian life of the product is above or equivalent to1000 hours at a level of confidence 075 +e researcher wantsto end an experiment at 500 hours under the zero and onefailure plan of the double sampling plan It refers to the a 07terminator of the experiment From Table 1 the requiredminimum sizes of sample are n1 11 and n2 7 +e doublesampling plan shall be described as follows+e very first thingto do here is to monitor eleven items for 500 hours and acceptthe lot if no failure occurs during the experiment When theexperiment produces at least two errors the lot is rejected +esecond sample of seven items is drawn and tested for 500 hourswhere only one failure was reported+e lot would be acceptedif no failure exists in the second sample When quality im-proves the product can be correlated with the probability ofacceptance and they want to reduce the risk of the producerSuppose the supplier knows what degree of quality leads to arisk of less than 005 from the product

Table 5 OC values for exponentiated Frechet distribution with 1113954θ 16684 and 1113954λ 07130


075

6 5 05 09636 09990 10000 10000 100004 3 07 08898 09926 09994 09999 100003 3 09 07951 09761 09968 09995 099993 2 11 06978 09484 09906 09981 099963 1 13 06076 09111 09797 09950 099872 2 15 05280 08672 09636 09896 099682 2 17 04593 08196 09427 09813 099362 1 19 04008 07705 09178 09702 09889

090

9 6 05 09385 09982 09999 10000 100006 4 07 08235 09871 09989 09999 100005 2 09 06884 09592 09944 09991 099984 2 11 05616 09139 09837 09967 099933 3 13 04539 08557 09651 09912 099773 2 15 03664 07901 09385 09818 099443 2 17 02968 07221 09049 09679 098893 1 19 02418 06553 08660 09494 09806

095

11 7 05 09218 09976 09999 10000 100007 5 07 07839 09831 09985 09998 100006 3 09 06316 09476 09926 09988 099985 3 11 04969 08915 09787 09956 099904 3 13 03884 08216 09550 09885 099694 2 15 03045 07454 09218 09763 099263 3 17 02405 06687 08806 09585 098543 2 19 01918 05956 08339 09353 09748

099

15 14 05 08792 09961 09998 10000 1000010 8 07 06865 09727 09975 09997 100008 5 09 04965 09178 09880 09981 099977 3 11 03473 08351 09659 09928 099846 3 13 02406 07372 09291 09813 099495 4 15 01671 06364 08792 09621 098805 2 17 01171 05408 08194 09345 097649 6 05 09385 09982 09999 10000 10000

10

08

06

04

02

00

OC

000 005 010 015 020 025 030p

DASPSSP when C = 0SSP when C = 1

Figure 1 OC curve of double- and single-sampling plans


6 Conclusion

A double-sampling procedure for the decision to approve orreject the lot submitted was built on the basis of a truncatedlife test +e lifespan of the product is expected to followexponentiated Frechet distribution which is useful in systemreliability analysis because the failure rate is very flexible Itwas observed that the necessary sample sizes declinedsteadily as the time of the experiment grew and that the sizeof the sample for the reasonable duration of the experimentwas not very sensitive to the confidence level or the shapeparameter It has been revealed by examples that the doublesampling plan would be more appropriate than a singlesampling plan in terms of the OC values A variable sam-pling plan will be preferred because it utilizes all the detailsavailable As a result the potential research will establish adouble acceptation sampling plan on variables

Data Availability

+e used data sets are given in the manuscript

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

Methodology and computation were carried out by GSR andwriting and data collection done by MS and KR Both au-thors read and approved the final manuscript

Acknowledgments

Corresponding author from low income country got a 100discount on article processing charge (APC) for the acceptedarticles

References

[1] B Epstein ldquoTruncated life tests in the exponential caserdquo eAnnals of Mathematical Statistics vol 25 no 3 pp 555ndash5641954

[2] H P Goode and J H K Kao ldquoSampling plans based on theWeibull distributionrdquo in Proceeding of the Seventh NationalSymposium on Reliability and Quality Control pp 24ndash40Philadelphia PA USA 1961

[3] S S Gupta and S S Gupta ldquoGamma distribution in ac-ceptance sampling based on life testsrdquo Journal of the AmericanStatistical Association vol 56 no 296 pp 942ndash970 1961

[4] S S Gupta ldquoLife test sampling plans for normal and log-normal distributionsrdquo Technometrics vol 4 no 2pp 151ndash175 1962

[5] R R L Kantam K Rosaiah and G S Rao ldquoAcceptancesampling based on life tests log-logistic modelrdquo Journal ofApplied Statistics vol 28 no 1 pp 121ndash128 2001

[6] A Baklizi ldquoAcceptance sampling based on truncated life testsin the Pareto distribution of the second kindrdquo Advances andApplications in Statistics vol 3 pp 33ndash48 2003

[7] A Baklizi and A E Q El Masri ldquoAcceptance sampling basedon truncated life tests in the Birnbaum Saunders modelrdquo RiskAnalysis vol 24 no 6 pp 1453ndash1457 2004

[8] T-R Tsai and S-J Wu ldquoAcceptance sampling based ontruncated life tests for generalized Rayleigh distributionrdquoJournal of Applied Statistics vol 33 no 6 pp 595ndash600 2006

[9] N Balakrishnan V Leiva and J Lopez ldquoAcceptance samplingplans from truncated life tests based on the generalizedBirnbaum-Saunders distributionrdquo Communications in Sta-tistics-Simulation and Computation vol 36 no 3 pp 643ndash656 2007

[10] K Rosaiah and R R L Kantam ldquoAcceptance sampling basedon the inverse Rayleigh distributionrdquo Economic QualityControl vol 20 no 2 pp 277ndash286 2005

[11] K Rosaiah R R L Kantam and C Santosh Kumar ldquoReli-ability of test plans for exponentiated log-logistic distribu-tionrdquo Economic Quality Control vol 21 no 2 pp 279ndash2892006

08

06

04

02

00

0 2 4 6 8x

(a)

8

6

4

2

0

0 2 4 6 8Fitted quantiles

(b)

Figure 2 +e empirical and theoretical cdfs and Q-Q plots for the vinyl chloride data (a) Empirical and fitted PDFs (b) Q-Q plot


[12] G S Rao K Rosaiah K Kalyani and D C U Sivakumar ldquoAnew acceptance sampling plans based on percentiles for oddsexponential log logistic distributionrdquo e Open Statistics ampProbability Journal vol 7 no 1 pp 45ndash52 2016

[13] G S Rao K Rosaiah M S Babu and D C U SivaKumar ldquoAnew acceptance sampling plans based on percentiles forexponentiated Frechet distributionrdquo Economic Quality Con-trol vol 31 no 1 pp 37ndash44 2016

[14] K Rosaiah G S Rao D C U Sivakumar and K Kalyanildquo+e odd generalized exponential log logistic distribution anew acceptance sampling plans based on percentilesrdquo In-ternational Journal of Advances in Applied Sciences vol 8no 3 pp 176ndash183 2019

[15] A J Duncan Quality Control and Industrial Statistics IrwinEd Richard D Irvin Inc Homewood IL USA 5th edition1986

[16] M Aslam ldquoDouble acceptance sampling based on truncatedlife tests in Rayleigh distributionrdquo European Journal of Sci-entific Research vol 17 no 4 pp 605ndash610 2005

[17] M Aslam C H Jun and M Ahmad ldquoA double acceptancesampling plan based on the truncated life tests in the Weibullmodelrdquo Journal of Statistical eory and Applications vol 8no 2 pp 191ndash206 2009

[18] M Aslam C-H Jun and M Ahmad ldquoDesign of a time-truncated double sampling plan for a general life distribu-tionrdquo Journal of Applied Statistics vol 37 no 8pp 1369ndash1379 2010

[19] M Aslam and C-H Jun ldquoA double acceptance sampling planfor generalized log-logistic distributions with known shapeparametersrdquo Journal of Applied Statistics vol 37 no 3pp 405ndash414 2010

[20] G S Rao ldquoDouble acceptance sampling plans based ontruncated life tests for the Marshall-Olkin extended expo-nential distributionrdquo Austrian Journal of Statistics vol 40no 3 pp 169ndash176 2011

[21] G S Rao ldquoDouble acceptance sampling plans based ontruncated life tests for Marshall-Olkin extended Lomax dis-tributionrdquo Journal of Modern Applied Statistical Methodsvol 10 no 1 p 12 2011

[22] M Aslam Y Mahmood Y L Lio T-R Tsai andM A KhanldquoDouble acceptance sampling plans for Burr type XII dis-tribution percentiles under the truncated life testrdquo Journal ofthe Operational Research Society vol 63 no 7 pp 1010ndash10172012

[23] A S Ramaswamy and P Anburajan ldquoDouble acceptancesampling based on truncated life tests in generalized expo-nential distributionrdquo Applied Mathematical Sciences vol 6no 64 pp 3199ndash3207 2012

[24] W Gui ldquoDouble acceptance sampling plan for time truncatedlife tests based on Maxwell distributionrdquo American Journal ofMathematical and Management Sciences vol 33 no 2pp 98ndash109 2014

[25] D Malathi and S Muthulakshmi ldquoSpecial double samplingplan for truncated life tests based on the Marshall-Olkinextended exponential distributionrdquo International Journal ofComputational Engineering Research vol 5 no 1 pp 56ndash622015

[26] H Tripathi M Saha and V Alha ldquoAn application of timetruncated single acceptance sampling inspection plan basedon generalized half-normal distributionrdquo Annals of DataScience 2020

[27] H Tripathi S Dey and M Saha ldquoDouble and group ac-ceptance sampling plan for truncated life test based on inverse

log-logistic distributionrdquo Journal of Applied Statistics vol 12020

[28] S Nadarajah and S Kotz ldquo+e exponentiated Frechet dis-tribution Inter Statrdquo Electronics Journal 2006

[29] D G Harlow ldquoApplications of the Frechet distributionfunctionrdquo International Journal of Materials and ProductTechnology vol 17 no 56 pp 482ndash495 2002

[30] S Nadarajah and A K Gupta ldquo+e beta Frechet distributionrdquoFar East Journal-eory and Statistics vol 14 no 1 pp 15ndash242004

[31] A M Abd-Elfattah and A M Omima ldquoEstimation of theunknown parameters of the generalized Frechet distributionrdquoJournal of Applied Sciences Research vol 5 no 10pp 1398ndash1408 2009

[32] A M Abd-Elfattah A F Hala and A M Omima ldquoGoodnessof fit tests for Generalized Frechet distributionrdquo AustralianJournal of Applied Sciences vol 4 no 2 pp 286ndash301 2010

[33] A D Al-Nassar and A I Al-Omari ldquoAcceptance samplingplan based on truncated life tests for exponentiated Frechetdistributionrdquo Journal of Statistics vol 25 no 2 pp 107ndash1192013

[34] S Kotz and S Nadarajah Extreme Value Distributions eoryand Applications Imperial College Press London UK 2000

[35] K S StephenseHandbook of Applied Acceptance SamplingPlans Principles and Procedures ASQ Quality Press Mil-waukee WI USA 2001

[36] D K Bhaumik and R D Gibbons ldquoOne-sided approximateprediction Intervals for at LeastpofmObservations from agamma population at each ofrLocationsrdquo Technometricsvol 48 no 1 pp 112ndash119 2006

[37] K Krishnamoorthy T Mathew and S Mukherjee ldquoNormal-based methods for a gamma distributionrdquo Technometricsvol 50 no 1 pp 69ndash78 2008


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] Kantamet al [5] Baklizi [6] Baklizi and Masri [7] Tsai and Wu [8]Balakrishnan et al [9] Rosaiah and Kantam [10] Rosaiahet al [11] Rao et al [12 13] and Rosaiah et al [14] +eabove-listed plan has also recently been designed for variouslifetime distributions

In cases where normal distribution is often adopted thedouble-sampling schemes were known to decrease the size ofthe sample or the risk of the producer in the area of qualitycontrol eg Duncan [15] +e choice of double samplingdecisions is based on the data collected from earlier decisionson the process Aslam [16] developed Rayleigh distributionbased on double-acceptance sampling depend on truncatedlife-tests Aslam et al [17 18] proposed double-acceptancesampling plans based on truncated life tests for the Weibulldistribution and general life distributions

A double-acceptance sampling plan for generalized log-logistic distribution with known shape parameters havebeen developed by Aslam and Jun [19] Rao [20 21]considered double-acceptance sampling plans for theMarshallndashOlkin extended exponential and MarshallndashOlkinextended Lomax distributions depend on average life timeof the truncated life testing data Aslam et al [22] discusseddouble-acceptance sampling plans for Burr type-XII dis-tribution percentiles under the truncated life tests Ingeneralized exponential distribution Ramaswamy andAnburajan [23] presented double acceptance samplingbased on truncated life-tests Gui [24] has developed adouble-acceptance sampling plan for time-truncated life-tests based on Maxwell distribution Malathi andMuthulakshmi [25] developed a zero-one double-accep-tance sampling plan based on MarshallndashOlkin extendedexponential distribution for truncated life tests Tripathiet al [26] developed an application of time-truncatedsingle-acceptance sampling plan based on generalized half-normal distribution Tripathi et al [27] studied double- andgroup-acceptance sampling plan for truncated life testbased on inverse log-logistic distribution

In the field of quality control the normal distribution isoften implemented double sampling schemes have been re-ported tominimize the size of the sample or the producerrsquos risksee eg Duncan [15] in the area of reliability Nevertheless aplan for double sampling has also not been developed Based onthe assumption that a productrsquos lifetime follows the expo-nentiated Frechet distribution the main objective of this articleis to recommend the double-acceptance sampling plans fortruncated lifetime tests In survival research we especially usethis distribution because of the monotonous nature of thehazard function We essentially consider zero and one failuresystem where we accept a lot for no failures from the firstsample and we reject if at least two failures are found Whenthere is only one defect we will select the second sample andcheck with the same duration as the first sample At the definedconsumerrsquos level of confidence both the initial and secondsample minimum sizes are calculated +e operational prop-erties are calculated according to the ratio of truemedian life tothe defined lifetime+e minimum ratios are often achieved in

order to reduce the risk of the producer to the degree specifiedSection 2 describes the exponential Frechet distribution Sec-tion 3 accounts for the proposed double sampling scheme andSection 4 assesses its operating characteristics Examples of thesampling technique are given in Section 5 and finally con-clusions are given in Section 6

2 The Exponentiated FrechetDistribution (EFD)

Consider the productrsquos lifetime following the exponentiatedFrechet distribution which was introduced and studied byNadarjah and Kotz [28] the probability density function andcumulative distribution function of EFD respectively aregiven by

f(t σ λ θ) σλλθ 1 minus eminus (σt)λ

1113876 1113877θminus 1

tminus (1+λ)

eminus (σt)λ

F(t σ λ θ) 1 minus 1 minus eminus (σt)λ

1113876 1113877θ tgt 0 σ λ θgt 0

(1)

Here the scale parameter is σ and the shape parameters areλ and θ Remember that this CDF represents the chance offailure of a parallel system with θ items having a Frechetdistributed lifetime +e exponentiated Frechet distributionwill then be used to check the reliability of the system Inconstructing single-sampling plans this exponentiated Frechetdistribution was considered in Rao et al [13] When θ 1 it iscalled the Frechet distribution For this analysis the shapeparameters λ and θ are assumed to be a priori known

+e parameters λ and θ can be predicted when data onfailure are available It is understood that for θ 1 the failurerate (hazard function) declines in the case of λle 1 while itrises and then declines in the case of λgt 1 +is distributionis used as a model for lifetime in this study as its failure ratepattern is very flexible

Generally the mean cannot necessarily be defined in aclosed form while exponential Frechet distribution ispossible in the closed form However the median life is the50th percentile of the exponentiated Frechet distributionderived by

m σ minus ln 1 minus (05)1θ

1113872 11138731113960 1113961minus (1λ)

(2)

When the parameters λ and θ are fixed the median isdirectly proportional to the scale parameter Note that themedian reduces to m σ independently of λ for the Frechetdistribution (θ1) Harlow [29] developed the Frechetdistribution function for applications Nadarajah andGupta [30] studied the beta Frechet distribution Abd-Elfattah and Omima [31] addressed the estimation of theunknown parameters of the generalized Frechet distribu-tion Abd-Elfattah et al [32] introduced the goodness of fittests for generalized Frechet distribution Al-Nassar andAl-Omari [33] studied the acceptance sampling plan basedon truncated life tests for exponentiated Frechet distri-bution Kotz and Nadarajah [34] studied theory and ap-plications of extreme value distributions









Flow chart for DASP

Yes

No

Yes

No

Yes

No


If F1 le C1

If F1 gt C2

If F1 + F2 gt C2

Accept the lot

Reject the lot


Reject the lot

Accept the lot


Pa 1113944

n1

i0

n1

i1113888 1113889p

i(1 minus P)

n1minus i+ 1113944

C2

xC1+1

n1

x1113888 1113889p

x


1113944

C2minus x

i0

n2

i1113888 1113889p

i(1 minus P)


(3)



λ

1113876 1113877θ1 minus 1 minus e


1113890 1113891

θ

(4)




n2minus 11113960 1113961 (5)



n2minus 11113960 1113961le 1 minus P

lowast (6)



λ

1113890 1113891

θ

ASN n1p1 + n1 + n2( 1113857 1 minus p1( 1113857

(7)


P1 1 minus 1113944

c2

ic1+1

n1

i1113888 1113889p

i(1 minus p)

n1minus i (8)



(9)



(10a)

Subject to



n2minus 11113960 1113961le 1 minus P

lowast (10b)

1le n2 le n1 (10c)




lowast (11)





(075 090 095 099) and a (05 07 09 11 13 15









aθ λ Plowast 05 07 09 11 13 15 17 19

(20 20)

075 119 93 (14698) 11 7 (1285) 4 3 (465) 2 2 (239) 2 1 (210) 2 1 (205) 1 1 (111) 1 1 (108)090 176 137 (20226) 15 14 (1756) 6 3 (636) 3 3 (336) 2 2 (221) 2 1 (205) 2 1 (203) 2 1 (201)095 216 183 (23984) 19 16 (2087) 7 5 (743) 4 3 (419) 3 2 (308) 2 2 (210) 2 1 (203) 2 1 (201)099 315 309 (32860) 28 20 (2875) 10 6 (1016) 6 3 (605) 4 2 (403) 3 2 (303) 3 1 (300) 2 2 (203)

(10 20)

075 28 20 (3374) 7 4 (797) 4 2 (437) 3 1 (314) 2 2 (230) 2 1 (210) 2 1 (207) 2 1 (205)090 40 33 (4624) 10 6 (1090) 5 4 (553) 4 2 (416) 3 2 (315) 2 2 (221) 2 2 (214) 2 1 (205)095 49 45 (5483) 12 9 (1293) 6 5 (646) 4 4 (433) 3 3 (323) 3 2 (308) 3 1 (302) 2 2 (210)099 73 59 (7542) 17 14 (1751) 9 5 (913) 6 4 (609) 5 2 (503) 4 2 (403) 4 1 (401) 3 2 (303)

(05 20)

075 9 9 (1153) 5 3 (566) 3 3 (367) 3 2 (332) 2 2 (238) 2 2 (231) 2 1 (213) 2 1 (211)090 14 11 (1586) 7 4 (755) 5 3 (533) 4 2 (419) 3 3 (334) 3 2 (316) 3 1 (306) 3 1 (305)095 17 14 (1863) 8 7 (873) 6 3 (622) 5 2 (511) 4 2 (412) 4 2 (408) 3 2 (312) 3 2 (309)099 25 19 (2571) 12 8 (1225) 8 6 (817) 7 3 (705) 6 2 (603) 5 3 (505) 4 4 (410) 4 3 (405)

(16684 07130)

075 6 5 (724) 4 3 (468) 3 3 (364) 3 2 (333) 3 1 (313) 2 2 (235) 2 2 (230) 2 1 (213)090 9 6 (987) 6 4 (652) 5 2 (520) 4 2 (421) 3 3 (338) 3 2 (320) 3 2 (315) 3 1 (306)095 11 7 (1167) 7 5 (747) 6 3 (620) 5 3 (518) 4 3 (421) 4 2 (410) 3 3 (323) 3 2 (312)099 15 14 (1552) 10 8 (1026) 8 5 (813) 7 3 (706) 6 3 (605) 5 4 (509) 5 2 (503) 4 4 (410)




075

119 93 05 09984 09989 09998 10000 1000011 7 07 09976 09992 09998 09999 100004 3 09 09927 09982 09991 09995 100002 2 11 09988 09991 09998 09999 100002 1 13 09887 09956 09991 09989 099992 1 15 09548 09789 09943 09991 099991 1 17 08897 09823 09912 09992 099991 1 19 08000 09899 09956 09990 09999

090

176 137 05 09900 09956 09976 09999 1000015 14 07 09891 09956 09991 09998 100006 3 09 09919 09978 09991 09999 100003 3 11 09966 09989 09994 09996 100002 2 13 09685 09879 09923 09998 099992 1 15 08835 09567 09964 09995 099992 1 17 07424 09145 09989 09991 099982 1 19 05789 09698 09987 09990 09999

095

216 183 05 09849 09878 09987 09997 1000019 16 07 09719 09919 09984 09996 099997 5 09 09691 09899 09923 09988 099994 3 11 09466 09879 09923 09998 099993 2 13 09385 09567 09954 09985 099992 2 15 08835 09145 09989 09995 100002 1 17 07424 09098 09927 09939 099992 1 19 05789 09198 09897 09919 09999

099

315 309 05 09987 09991 09991 09989 1000028 20 07 09912 09986 09991 09989 0999910 6 09 09799 09879 09923 09998 099996 3 11 09433 09567 09974 09995 099994 2 13 09114 09245 09969 09991 099983 2 15 07994 09145 09929 09989 099983 1 17 05948 09098 09917 09979 099992 2 19 03955 09196 09789 09945 09987



075

28 20 05 09914 09979 09992 10000 100007 4 07 09789 09982 09996 09999 100004 2 09 09969 09982 09991 09995 100003 1 11 09714 09991 09998 09999 100002 2 13 09018 09856 09971 09988 099992 1 15 07945 09689 09983 09996 099992 1 17 06721 09823 09902 09982 099992 1 19 05540 09899 09916 09929 09999

090

40 33 05 09678 09951 09978 09999 1000010 6 07 09599 09956 09991 09998 100005 4 09 09449 09968 09991 09999 100004 2 11 09349 09979 09991 09996 100003 2 13 08530 09879 09923 09998 099992 2 15 07091 09577 09955 09991 099992 2 17 05604 09145 09989 09991 099982 1 19 04303 09691 09937 09929 09999


Table 3 Continued


095

49 45 05 09788 09878 09982 09996 1000012 9 07 09697 09908 09974 09995 099996 5 09 09519 09868 09905 09989 099994 4 11 09466 09879 09925 09988 099993 3 13 08285 09567 09954 09986 099993 2 15 06664 09145 09986 09993 100003 1 17 05045 09091 09923 09935 099992 2 19 03685 09198 09895 09917 09999

099

73 59 05 09916 09981 09997 09999 1000017 14 07 09899 09984 09992 09989 099999 5 09 09769 09876 09917 09998 099996 4 11 09169 09566 09973 09995 099995 2 13 07498 09245 09969 09991 099984 2 15 05455 09145 09935 09988 099994 1 17 03667 09098 09917 09969 099993 2 19 02358 09193 09789 09935 09977



075

9 9 05 09899 09965 09979 09992 099995 3 07 09813 09959 09979 09992 099983 3 09 09543 09913 09986 09994 099973 2 11 08795 09976 09993 09997 099992 2 13 07874 09967 09956 09991 099992 2 15 06944 09675 09739 09945 099962 1 17 06089 09679 09833 09914 099952 1 19 05336 09384 09799 09944 09989

09

14 11 05 09898 09978 09996 09998 099997 4 07 09865 09923 09946 09989 099995 3 09 09164 09887 09967 09977 099994 2 11 07931 09903 09981 09996 099963 3 13 06565 09836 09879 09923 099983 2 15 05324 09655 09767 09961 099963 1 17 04293 09402 09215 09978 099943 1 19 03469 08869 09668 09977 09998

095

17 14 05 09968 09991 09996 09998 099998 7 07 09704 09945 09969 09994 099996 3 09 08833 09776 09896 09933 099985 2 11 07254 09689 09874 09921 099974 2 13 05645 09506 09567 09954 099954 2 15 04299 09046 09147 09987 099963 2 17 03264 08956 09098 09924 099373 2 19 02493 08471 09198 09896 09991

099

25 19 05 09895 09989 09993 09993 0999912 8 07 09559 09878 09985 09992 099998 6 09 07715 09999 09879 09923 099987 3 11 05334 09873 09542 09974 099956 2 13 03445 09783 09845 09968 099985 3 15 02199 09229 09545 09941 099994 4 17 01426 08286 09398 09908 099894 3 19 00948 07112 09296 09789 09925





075

6 5 05 09636 09990 10000 10000 100004 3 07 08898 09926 09994 09999 100003 3 09 07951 09761 09968 09995 099993 2 11 06978 09484 09906 09981 099963 1 13 06076 09111 09797 09950 099872 2 15 05280 08672 09636 09896 099682 2 17 04593 08196 09427 09813 099362 1 19 04008 07705 09178 09702 09889

090

9 6 05 09385 09982 09999 10000 100006 4 07 08235 09871 09989 09999 100005 2 09 06884 09592 09944 09991 099984 2 11 05616 09139 09837 09967 099933 3 13 04539 08557 09651 09912 099773 2 15 03664 07901 09385 09818 099443 2 17 02968 07221 09049 09679 098893 1 19 02418 06553 08660 09494 09806

095

11 7 05 09218 09976 09999 10000 100007 5 07 07839 09831 09985 09998 100006 3 09 06316 09476 09926 09988 099985 3 11 04969 08915 09787 09956 099904 3 13 03884 08216 09550 09885 099694 2 15 03045 07454 09218 09763 099263 3 17 02405 06687 08806 09585 098543 2 19 01918 05956 08339 09353 09748

099

15 14 05 08792 09961 09998 10000 1000010 8 07 06865 09727 09975 09997 100008 5 09 04965 09178 09880 09981 099977 3 11 03473 08351 09659 09928 099846 3 13 02406 07372 09291 09813 099495 4 15 01671 06364 08792 09621 098805 2 17 01171 05408 08194 09345 097649 6 05 09385 09982 09999 10000 10000

10

08

06

04

02

00

OC

000 005 010 015 020 025 030p




6 Conclusion


Data Availability






Acknowledgments


References












08

06

04

02

00

0 2 4 6 8x

(a)

8

6

4

2

0


(b)






































Flow chart for DASP

Yes

No

Yes

No

Yes

No


If F1 le C1

If F1 gt C2

If F1 + F2 gt C2

Accept the lot

Reject the lot


Reject the lot

Accept the lot


Pa 1113944

n1

i0

n1

i1113888 1113889p

i(1 minus P)

n1minus i+ 1113944

C2

xC1+1

n1

x1113888 1113889p

x


1113944

C2minus x

i0

n2

i1113888 1113889p

i(1 minus P)


(3)



λ

1113876 1113877θ1 minus 1 minus e


1113890 1113891

θ

(4)




n2minus 11113960 1113961 (5)



n2minus 11113960 1113961le 1 minus P

lowast (6)



λ

1113890 1113891

θ

ASN n1p1 + n1 + n2( 1113857 1 minus p1( 1113857

(7)


P1 1 minus 1113944

c2

ic1+1

n1

i1113888 1113889p

i(1 minus p)

n1minus i (8)



(9)



(10a)

Subject to



n2minus 11113960 1113961le 1 minus P

lowast (10b)

1le n2 le n1 (10c)




lowast (11)





(075 090 095 099) and a (05 07 09 11 13 15









aθ λ Plowast 05 07 09 11 13 15 17 19

(20 20)

075 119 93 (14698) 11 7 (1285) 4 3 (465) 2 2 (239) 2 1 (210) 2 1 (205) 1 1 (111) 1 1 (108)090 176 137 (20226) 15 14 (1756) 6 3 (636) 3 3 (336) 2 2 (221) 2 1 (205) 2 1 (203) 2 1 (201)095 216 183 (23984) 19 16 (2087) 7 5 (743) 4 3 (419) 3 2 (308) 2 2 (210) 2 1 (203) 2 1 (201)099 315 309 (32860) 28 20 (2875) 10 6 (1016) 6 3 (605) 4 2 (403) 3 2 (303) 3 1 (300) 2 2 (203)

(10 20)

075 28 20 (3374) 7 4 (797) 4 2 (437) 3 1 (314) 2 2 (230) 2 1 (210) 2 1 (207) 2 1 (205)090 40 33 (4624) 10 6 (1090) 5 4 (553) 4 2 (416) 3 2 (315) 2 2 (221) 2 2 (214) 2 1 (205)095 49 45 (5483) 12 9 (1293) 6 5 (646) 4 4 (433) 3 3 (323) 3 2 (308) 3 1 (302) 2 2 (210)099 73 59 (7542) 17 14 (1751) 9 5 (913) 6 4 (609) 5 2 (503) 4 2 (403) 4 1 (401) 3 2 (303)

(05 20)

075 9 9 (1153) 5 3 (566) 3 3 (367) 3 2 (332) 2 2 (238) 2 2 (231) 2 1 (213) 2 1 (211)090 14 11 (1586) 7 4 (755) 5 3 (533) 4 2 (419) 3 3 (334) 3 2 (316) 3 1 (306) 3 1 (305)095 17 14 (1863) 8 7 (873) 6 3 (622) 5 2 (511) 4 2 (412) 4 2 (408) 3 2 (312) 3 2 (309)099 25 19 (2571) 12 8 (1225) 8 6 (817) 7 3 (705) 6 2 (603) 5 3 (505) 4 4 (410) 4 3 (405)

(16684 07130)

075 6 5 (724) 4 3 (468) 3 3 (364) 3 2 (333) 3 1 (313) 2 2 (235) 2 2 (230) 2 1 (213)090 9 6 (987) 6 4 (652) 5 2 (520) 4 2 (421) 3 3 (338) 3 2 (320) 3 2 (315) 3 1 (306)095 11 7 (1167) 7 5 (747) 6 3 (620) 5 3 (518) 4 3 (421) 4 2 (410) 3 3 (323) 3 2 (312)099 15 14 (1552) 10 8 (1026) 8 5 (813) 7 3 (706) 6 3 (605) 5 4 (509) 5 2 (503) 4 4 (410)




075

119 93 05 09984 09989 09998 10000 1000011 7 07 09976 09992 09998 09999 100004 3 09 09927 09982 09991 09995 100002 2 11 09988 09991 09998 09999 100002 1 13 09887 09956 09991 09989 099992 1 15 09548 09789 09943 09991 099991 1 17 08897 09823 09912 09992 099991 1 19 08000 09899 09956 09990 09999

090

176 137 05 09900 09956 09976 09999 1000015 14 07 09891 09956 09991 09998 100006 3 09 09919 09978 09991 09999 100003 3 11 09966 09989 09994 09996 100002 2 13 09685 09879 09923 09998 099992 1 15 08835 09567 09964 09995 099992 1 17 07424 09145 09989 09991 099982 1 19 05789 09698 09987 09990 09999

095

216 183 05 09849 09878 09987 09997 1000019 16 07 09719 09919 09984 09996 099997 5 09 09691 09899 09923 09988 099994 3 11 09466 09879 09923 09998 099993 2 13 09385 09567 09954 09985 099992 2 15 08835 09145 09989 09995 100002 1 17 07424 09098 09927 09939 099992 1 19 05789 09198 09897 09919 09999

099

315 309 05 09987 09991 09991 09989 1000028 20 07 09912 09986 09991 09989 0999910 6 09 09799 09879 09923 09998 099996 3 11 09433 09567 09974 09995 099994 2 13 09114 09245 09969 09991 099983 2 15 07994 09145 09929 09989 099983 1 17 05948 09098 09917 09979 099992 2 19 03955 09196 09789 09945 09987



075

28 20 05 09914 09979 09992 10000 100007 4 07 09789 09982 09996 09999 100004 2 09 09969 09982 09991 09995 100003 1 11 09714 09991 09998 09999 100002 2 13 09018 09856 09971 09988 099992 1 15 07945 09689 09983 09996 099992 1 17 06721 09823 09902 09982 099992 1 19 05540 09899 09916 09929 09999

090

40 33 05 09678 09951 09978 09999 1000010 6 07 09599 09956 09991 09998 100005 4 09 09449 09968 09991 09999 100004 2 11 09349 09979 09991 09996 100003 2 13 08530 09879 09923 09998 099992 2 15 07091 09577 09955 09991 099992 2 17 05604 09145 09989 09991 099982 1 19 04303 09691 09937 09929 09999


Table 3 Continued


095

49 45 05 09788 09878 09982 09996 1000012 9 07 09697 09908 09974 09995 099996 5 09 09519 09868 09905 09989 099994 4 11 09466 09879 09925 09988 099993 3 13 08285 09567 09954 09986 099993 2 15 06664 09145 09986 09993 100003 1 17 05045 09091 09923 09935 099992 2 19 03685 09198 09895 09917 09999

099

73 59 05 09916 09981 09997 09999 1000017 14 07 09899 09984 09992 09989 099999 5 09 09769 09876 09917 09998 099996 4 11 09169 09566 09973 09995 099995 2 13 07498 09245 09969 09991 099984 2 15 05455 09145 09935 09988 099994 1 17 03667 09098 09917 09969 099993 2 19 02358 09193 09789 09935 09977



075

9 9 05 09899 09965 09979 09992 099995 3 07 09813 09959 09979 09992 099983 3 09 09543 09913 09986 09994 099973 2 11 08795 09976 09993 09997 099992 2 13 07874 09967 09956 09991 099992 2 15 06944 09675 09739 09945 099962 1 17 06089 09679 09833 09914 099952 1 19 05336 09384 09799 09944 09989

09

14 11 05 09898 09978 09996 09998 099997 4 07 09865 09923 09946 09989 099995 3 09 09164 09887 09967 09977 099994 2 11 07931 09903 09981 09996 099963 3 13 06565 09836 09879 09923 099983 2 15 05324 09655 09767 09961 099963 1 17 04293 09402 09215 09978 099943 1 19 03469 08869 09668 09977 09998

095

17 14 05 09968 09991 09996 09998 099998 7 07 09704 09945 09969 09994 099996 3 09 08833 09776 09896 09933 099985 2 11 07254 09689 09874 09921 099974 2 13 05645 09506 09567 09954 099954 2 15 04299 09046 09147 09987 099963 2 17 03264 08956 09098 09924 099373 2 19 02493 08471 09198 09896 09991

099

25 19 05 09895 09989 09993 09993 0999912 8 07 09559 09878 09985 09992 099998 6 09 07715 09999 09879 09923 099987 3 11 05334 09873 09542 09974 099956 2 13 03445 09783 09845 09968 099985 3 15 02199 09229 09545 09941 099994 4 17 01426 08286 09398 09908 099894 3 19 00948 07112 09296 09789 09925





075

6 5 05 09636 09990 10000 10000 100004 3 07 08898 09926 09994 09999 100003 3 09 07951 09761 09968 09995 099993 2 11 06978 09484 09906 09981 099963 1 13 06076 09111 09797 09950 099872 2 15 05280 08672 09636 09896 099682 2 17 04593 08196 09427 09813 099362 1 19 04008 07705 09178 09702 09889

090

9 6 05 09385 09982 09999 10000 100006 4 07 08235 09871 09989 09999 100005 2 09 06884 09592 09944 09991 099984 2 11 05616 09139 09837 09967 099933 3 13 04539 08557 09651 09912 099773 2 15 03664 07901 09385 09818 099443 2 17 02968 07221 09049 09679 098893 1 19 02418 06553 08660 09494 09806

095

11 7 05 09218 09976 09999 10000 100007 5 07 07839 09831 09985 09998 100006 3 09 06316 09476 09926 09988 099985 3 11 04969 08915 09787 09956 099904 3 13 03884 08216 09550 09885 099694 2 15 03045 07454 09218 09763 099263 3 17 02405 06687 08806 09585 098543 2 19 01918 05956 08339 09353 09748

099

15 14 05 08792 09961 09998 10000 1000010 8 07 06865 09727 09975 09997 100008 5 09 04965 09178 09880 09981 099977 3 11 03473 08351 09659 09928 099846 3 13 02406 07372 09291 09813 099495 4 15 01671 06364 08792 09621 098805 2 17 01171 05408 08194 09345 097649 6 05 09385 09982 09999 10000 10000

10

08

06

04

02

00

OC

000 005 010 015 020 025 030p




6 Conclusion


Data Availability






Acknowledgments


References












08

06

04

02

00

0 2 4 6 8x

(a)

8

6

4

2

0


(b)
































n2minus 11113960 1113961le 1 minus P

lowast (10b)

1le n2 le n1 (10c)




lowast (11)





(075 090 095 099) and a (05 07 09 11 13 15









aθ λ Plowast 05 07 09 11 13 15 17 19

(20 20)

075 119 93 (14698) 11 7 (1285) 4 3 (465) 2 2 (239) 2 1 (210) 2 1 (205) 1 1 (111) 1 1 (108)090 176 137 (20226) 15 14 (1756) 6 3 (636) 3 3 (336) 2 2 (221) 2 1 (205) 2 1 (203) 2 1 (201)095 216 183 (23984) 19 16 (2087) 7 5 (743) 4 3 (419) 3 2 (308) 2 2 (210) 2 1 (203) 2 1 (201)099 315 309 (32860) 28 20 (2875) 10 6 (1016) 6 3 (605) 4 2 (403) 3 2 (303) 3 1 (300) 2 2 (203)

(10 20)

075 28 20 (3374) 7 4 (797) 4 2 (437) 3 1 (314) 2 2 (230) 2 1 (210) 2 1 (207) 2 1 (205)090 40 33 (4624) 10 6 (1090) 5 4 (553) 4 2 (416) 3 2 (315) 2 2 (221) 2 2 (214) 2 1 (205)095 49 45 (5483) 12 9 (1293) 6 5 (646) 4 4 (433) 3 3 (323) 3 2 (308) 3 1 (302) 2 2 (210)099 73 59 (7542) 17 14 (1751) 9 5 (913) 6 4 (609) 5 2 (503) 4 2 (403) 4 1 (401) 3 2 (303)

(05 20)

075 9 9 (1153) 5 3 (566) 3 3 (367) 3 2 (332) 2 2 (238) 2 2 (231) 2 1 (213) 2 1 (211)090 14 11 (1586) 7 4 (755) 5 3 (533) 4 2 (419) 3 3 (334) 3 2 (316) 3 1 (306) 3 1 (305)095 17 14 (1863) 8 7 (873) 6 3 (622) 5 2 (511) 4 2 (412) 4 2 (408) 3 2 (312) 3 2 (309)099 25 19 (2571) 12 8 (1225) 8 6 (817) 7 3 (705) 6 2 (603) 5 3 (505) 4 4 (410) 4 3 (405)

(16684 07130)

075 6 5 (724) 4 3 (468) 3 3 (364) 3 2 (333) 3 1 (313) 2 2 (235) 2 2 (230) 2 1 (213)090 9 6 (987) 6 4 (652) 5 2 (520) 4 2 (421) 3 3 (338) 3 2 (320) 3 2 (315) 3 1 (306)095 11 7 (1167) 7 5 (747) 6 3 (620) 5 3 (518) 4 3 (421) 4 2 (410) 3 3 (323) 3 2 (312)099 15 14 (1552) 10 8 (1026) 8 5 (813) 7 3 (706) 6 3 (605) 5 4 (509) 5 2 (503) 4 4 (410)




075

119 93 05 09984 09989 09998 10000 1000011 7 07 09976 09992 09998 09999 100004 3 09 09927 09982 09991 09995 100002 2 11 09988 09991 09998 09999 100002 1 13 09887 09956 09991 09989 099992 1 15 09548 09789 09943 09991 099991 1 17 08897 09823 09912 09992 099991 1 19 08000 09899 09956 09990 09999

090

176 137 05 09900 09956 09976 09999 1000015 14 07 09891 09956 09991 09998 100006 3 09 09919 09978 09991 09999 100003 3 11 09966 09989 09994 09996 100002 2 13 09685 09879 09923 09998 099992 1 15 08835 09567 09964 09995 099992 1 17 07424 09145 09989 09991 099982 1 19 05789 09698 09987 09990 09999

095

216 183 05 09849 09878 09987 09997 1000019 16 07 09719 09919 09984 09996 099997 5 09 09691 09899 09923 09988 099994 3 11 09466 09879 09923 09998 099993 2 13 09385 09567 09954 09985 099992 2 15 08835 09145 09989 09995 100002 1 17 07424 09098 09927 09939 099992 1 19 05789 09198 09897 09919 09999

099

315 309 05 09987 09991 09991 09989 1000028 20 07 09912 09986 09991 09989 0999910 6 09 09799 09879 09923 09998 099996 3 11 09433 09567 09974 09995 099994 2 13 09114 09245 09969 09991 099983 2 15 07994 09145 09929 09989 099983 1 17 05948 09098 09917 09979 099992 2 19 03955 09196 09789 09945 09987



075

28 20 05 09914 09979 09992 10000 100007 4 07 09789 09982 09996 09999 100004 2 09 09969 09982 09991 09995 100003 1 11 09714 09991 09998 09999 100002 2 13 09018 09856 09971 09988 099992 1 15 07945 09689 09983 09996 099992 1 17 06721 09823 09902 09982 099992 1 19 05540 09899 09916 09929 09999

090

40 33 05 09678 09951 09978 09999 1000010 6 07 09599 09956 09991 09998 100005 4 09 09449 09968 09991 09999 100004 2 11 09349 09979 09991 09996 100003 2 13 08530 09879 09923 09998 099992 2 15 07091 09577 09955 09991 099992 2 17 05604 09145 09989 09991 099982 1 19 04303 09691 09937 09929 09999


Table 3 Continued


095

49 45 05 09788 09878 09982 09996 1000012 9 07 09697 09908 09974 09995 099996 5 09 09519 09868 09905 09989 099994 4 11 09466 09879 09925 09988 099993 3 13 08285 09567 09954 09986 099993 2 15 06664 09145 09986 09993 100003 1 17 05045 09091 09923 09935 099992 2 19 03685 09198 09895 09917 09999

099

73 59 05 09916 09981 09997 09999 1000017 14 07 09899 09984 09992 09989 099999 5 09 09769 09876 09917 09998 099996 4 11 09169 09566 09973 09995 099995 2 13 07498 09245 09969 09991 099984 2 15 05455 09145 09935 09988 099994 1 17 03667 09098 09917 09969 099993 2 19 02358 09193 09789 09935 09977



075

9 9 05 09899 09965 09979 09992 099995 3 07 09813 09959 09979 09992 099983 3 09 09543 09913 09986 09994 099973 2 11 08795 09976 09993 09997 099992 2 13 07874 09967 09956 09991 099992 2 15 06944 09675 09739 09945 099962 1 17 06089 09679 09833 09914 099952 1 19 05336 09384 09799 09944 09989

09

14 11 05 09898 09978 09996 09998 099997 4 07 09865 09923 09946 09989 099995 3 09 09164 09887 09967 09977 099994 2 11 07931 09903 09981 09996 099963 3 13 06565 09836 09879 09923 099983 2 15 05324 09655 09767 09961 099963 1 17 04293 09402 09215 09978 099943 1 19 03469 08869 09668 09977 09998

095

17 14 05 09968 09991 09996 09998 099998 7 07 09704 09945 09969 09994 099996 3 09 08833 09776 09896 09933 099985 2 11 07254 09689 09874 09921 099974 2 13 05645 09506 09567 09954 099954 2 15 04299 09046 09147 09987 099963 2 17 03264 08956 09098 09924 099373 2 19 02493 08471 09198 09896 09991

099

25 19 05 09895 09989 09993 09993 0999912 8 07 09559 09878 09985 09992 099998 6 09 07715 09999 09879 09923 099987 3 11 05334 09873 09542 09974 099956 2 13 03445 09783 09845 09968 099985 3 15 02199 09229 09545 09941 099994 4 17 01426 08286 09398 09908 099894 3 19 00948 07112 09296 09789 09925





075

6 5 05 09636 09990 10000 10000 100004 3 07 08898 09926 09994 09999 100003 3 09 07951 09761 09968 09995 099993 2 11 06978 09484 09906 09981 099963 1 13 06076 09111 09797 09950 099872 2 15 05280 08672 09636 09896 099682 2 17 04593 08196 09427 09813 099362 1 19 04008 07705 09178 09702 09889

090

9 6 05 09385 09982 09999 10000 100006 4 07 08235 09871 09989 09999 100005 2 09 06884 09592 09944 09991 099984 2 11 05616 09139 09837 09967 099933 3 13 04539 08557 09651 09912 099773 2 15 03664 07901 09385 09818 099443 2 17 02968 07221 09049 09679 098893 1 19 02418 06553 08660 09494 09806

095

11 7 05 09218 09976 09999 10000 100007 5 07 07839 09831 09985 09998 100006 3 09 06316 09476 09926 09988 099985 3 11 04969 08915 09787 09956 099904 3 13 03884 08216 09550 09885 099694 2 15 03045 07454 09218 09763 099263 3 17 02405 06687 08806 09585 098543 2 19 01918 05956 08339 09353 09748

099

15 14 05 08792 09961 09998 10000 1000010 8 07 06865 09727 09975 09997 100008 5 09 04965 09178 09880 09981 099977 3 11 03473 08351 09659 09928 099846 3 13 02406 07372 09291 09813 099495 4 15 01671 06364 08792 09621 098805 2 17 01171 05408 08194 09345 097649 6 05 09385 09982 09999 10000 10000

10

08

06

04

02

00

OC

000 005 010 015 020 025 030p




6 Conclusion


Data Availability






Acknowledgments


References












08

06

04

02

00

0 2 4 6 8x

(a)

8

6

4

2

0


(b)

































075

119 93 05 09984 09989 09998 10000 1000011 7 07 09976 09992 09998 09999 100004 3 09 09927 09982 09991 09995 100002 2 11 09988 09991 09998 09999 100002 1 13 09887 09956 09991 09989 099992 1 15 09548 09789 09943 09991 099991 1 17 08897 09823 09912 09992 099991 1 19 08000 09899 09956 09990 09999

090

176 137 05 09900 09956 09976 09999 1000015 14 07 09891 09956 09991 09998 100006 3 09 09919 09978 09991 09999 100003 3 11 09966 09989 09994 09996 100002 2 13 09685 09879 09923 09998 099992 1 15 08835 09567 09964 09995 099992 1 17 07424 09145 09989 09991 099982 1 19 05789 09698 09987 09990 09999

095

216 183 05 09849 09878 09987 09997 1000019 16 07 09719 09919 09984 09996 099997 5 09 09691 09899 09923 09988 099994 3 11 09466 09879 09923 09998 099993 2 13 09385 09567 09954 09985 099992 2 15 08835 09145 09989 09995 100002 1 17 07424 09098 09927 09939 099992 1 19 05789 09198 09897 09919 09999

099

315 309 05 09987 09991 09991 09989 1000028 20 07 09912 09986 09991 09989 0999910 6 09 09799 09879 09923 09998 099996 3 11 09433 09567 09974 09995 099994 2 13 09114 09245 09969 09991 099983 2 15 07994 09145 09929 09989 099983 1 17 05948 09098 09917 09979 099992 2 19 03955 09196 09789 09945 09987



075

28 20 05 09914 09979 09992 10000 100007 4 07 09789 09982 09996 09999 100004 2 09 09969 09982 09991 09995 100003 1 11 09714 09991 09998 09999 100002 2 13 09018 09856 09971 09988 099992 1 15 07945 09689 09983 09996 099992 1 17 06721 09823 09902 09982 099992 1 19 05540 09899 09916 09929 09999

090

40 33 05 09678 09951 09978 09999 1000010 6 07 09599 09956 09991 09998 100005 4 09 09449 09968 09991 09999 100004 2 11 09349 09979 09991 09996 100003 2 13 08530 09879 09923 09998 099992 2 15 07091 09577 09955 09991 099992 2 17 05604 09145 09989 09991 099982 1 19 04303 09691 09937 09929 09999


Table 3 Continued


095

49 45 05 09788 09878 09982 09996 1000012 9 07 09697 09908 09974 09995 099996 5 09 09519 09868 09905 09989 099994 4 11 09466 09879 09925 09988 099993 3 13 08285 09567 09954 09986 099993 2 15 06664 09145 09986 09993 100003 1 17 05045 09091 09923 09935 099992 2 19 03685 09198 09895 09917 09999

099

73 59 05 09916 09981 09997 09999 1000017 14 07 09899 09984 09992 09989 099999 5 09 09769 09876 09917 09998 099996 4 11 09169 09566 09973 09995 099995 2 13 07498 09245 09969 09991 099984 2 15 05455 09145 09935 09988 099994 1 17 03667 09098 09917 09969 099993 2 19 02358 09193 09789 09935 09977



075

9 9 05 09899 09965 09979 09992 099995 3 07 09813 09959 09979 09992 099983 3 09 09543 09913 09986 09994 099973 2 11 08795 09976 09993 09997 099992 2 13 07874 09967 09956 09991 099992 2 15 06944 09675 09739 09945 099962 1 17 06089 09679 09833 09914 099952 1 19 05336 09384 09799 09944 09989

09

14 11 05 09898 09978 09996 09998 099997 4 07 09865 09923 09946 09989 099995 3 09 09164 09887 09967 09977 099994 2 11 07931 09903 09981 09996 099963 3 13 06565 09836 09879 09923 099983 2 15 05324 09655 09767 09961 099963 1 17 04293 09402 09215 09978 099943 1 19 03469 08869 09668 09977 09998

095

17 14 05 09968 09991 09996 09998 099998 7 07 09704 09945 09969 09994 099996 3 09 08833 09776 09896 09933 099985 2 11 07254 09689 09874 09921 099974 2 13 05645 09506 09567 09954 099954 2 15 04299 09046 09147 09987 099963 2 17 03264 08956 09098 09924 099373 2 19 02493 08471 09198 09896 09991

099

25 19 05 09895 09989 09993 09993 0999912 8 07 09559 09878 09985 09992 099998 6 09 07715 09999 09879 09923 099987 3 11 05334 09873 09542 09974 099956 2 13 03445 09783 09845 09968 099985 3 15 02199 09229 09545 09941 099994 4 17 01426 08286 09398 09908 099894 3 19 00948 07112 09296 09789 09925





075

6 5 05 09636 09990 10000 10000 100004 3 07 08898 09926 09994 09999 100003 3 09 07951 09761 09968 09995 099993 2 11 06978 09484 09906 09981 099963 1 13 06076 09111 09797 09950 099872 2 15 05280 08672 09636 09896 099682 2 17 04593 08196 09427 09813 099362 1 19 04008 07705 09178 09702 09889

090

9 6 05 09385 09982 09999 10000 100006 4 07 08235 09871 09989 09999 100005 2 09 06884 09592 09944 09991 099984 2 11 05616 09139 09837 09967 099933 3 13 04539 08557 09651 09912 099773 2 15 03664 07901 09385 09818 099443 2 17 02968 07221 09049 09679 098893 1 19 02418 06553 08660 09494 09806

095

11 7 05 09218 09976 09999 10000 100007 5 07 07839 09831 09985 09998 100006 3 09 06316 09476 09926 09988 099985 3 11 04969 08915 09787 09956 099904 3 13 03884 08216 09550 09885 099694 2 15 03045 07454 09218 09763 099263 3 17 02405 06687 08806 09585 098543 2 19 01918 05956 08339 09353 09748

099

15 14 05 08792 09961 09998 10000 1000010 8 07 06865 09727 09975 09997 100008 5 09 04965 09178 09880 09981 099977 3 11 03473 08351 09659 09928 099846 3 13 02406 07372 09291 09813 099495 4 15 01671 06364 08792 09621 098805 2 17 01171 05408 08194 09345 097649 6 05 09385 09982 09999 10000 10000

10

08

06

04

02

00

OC

000 005 010 015 020 025 030p




6 Conclusion


Data Availability






Acknowledgments


References












08

06

04

02

00

0 2 4 6 8x

(a)

8

6

4

2

0


(b)































Table 3 Continued


095

49 45 05 09788 09878 09982 09996 1000012 9 07 09697 09908 09974 09995 099996 5 09 09519 09868 09905 09989 099994 4 11 09466 09879 09925 09988 099993 3 13 08285 09567 09954 09986 099993 2 15 06664 09145 09986 09993 100003 1 17 05045 09091 09923 09935 099992 2 19 03685 09198 09895 09917 09999

099

73 59 05 09916 09981 09997 09999 1000017 14 07 09899 09984 09992 09989 099999 5 09 09769 09876 09917 09998 099996 4 11 09169 09566 09973 09995 099995 2 13 07498 09245 09969 09991 099984 2 15 05455 09145 09935 09988 099994 1 17 03667 09098 09917 09969 099993 2 19 02358 09193 09789 09935 09977



075

9 9 05 09899 09965 09979 09992 099995 3 07 09813 09959 09979 09992 099983 3 09 09543 09913 09986 09994 099973 2 11 08795 09976 09993 09997 099992 2 13 07874 09967 09956 09991 099992 2 15 06944 09675 09739 09945 099962 1 17 06089 09679 09833 09914 099952 1 19 05336 09384 09799 09944 09989

09

14 11 05 09898 09978 09996 09998 099997 4 07 09865 09923 09946 09989 099995 3 09 09164 09887 09967 09977 099994 2 11 07931 09903 09981 09996 099963 3 13 06565 09836 09879 09923 099983 2 15 05324 09655 09767 09961 099963 1 17 04293 09402 09215 09978 099943 1 19 03469 08869 09668 09977 09998

095

17 14 05 09968 09991 09996 09998 099998 7 07 09704 09945 09969 09994 099996 3 09 08833 09776 09896 09933 099985 2 11 07254 09689 09874 09921 099974 2 13 05645 09506 09567 09954 099954 2 15 04299 09046 09147 09987 099963 2 17 03264 08956 09098 09924 099373 2 19 02493 08471 09198 09896 09991

099

25 19 05 09895 09989 09993 09993 0999912 8 07 09559 09878 09985 09992 099998 6 09 07715 09999 09879 09923 099987 3 11 05334 09873 09542 09974 099956 2 13 03445 09783 09845 09968 099985 3 15 02199 09229 09545 09941 099994 4 17 01426 08286 09398 09908 099894 3 19 00948 07112 09296 09789 09925





075

6 5 05 09636 09990 10000 10000 100004 3 07 08898 09926 09994 09999 100003 3 09 07951 09761 09968 09995 099993 2 11 06978 09484 09906 09981 099963 1 13 06076 09111 09797 09950 099872 2 15 05280 08672 09636 09896 099682 2 17 04593 08196 09427 09813 099362 1 19 04008 07705 09178 09702 09889

090

9 6 05 09385 09982 09999 10000 100006 4 07 08235 09871 09989 09999 100005 2 09 06884 09592 09944 09991 099984 2 11 05616 09139 09837 09967 099933 3 13 04539 08557 09651 09912 099773 2 15 03664 07901 09385 09818 099443 2 17 02968 07221 09049 09679 098893 1 19 02418 06553 08660 09494 09806

095

11 7 05 09218 09976 09999 10000 100007 5 07 07839 09831 09985 09998 100006 3 09 06316 09476 09926 09988 099985 3 11 04969 08915 09787 09956 099904 3 13 03884 08216 09550 09885 099694 2 15 03045 07454 09218 09763 099263 3 17 02405 06687 08806 09585 098543 2 19 01918 05956 08339 09353 09748

099

15 14 05 08792 09961 09998 10000 1000010 8 07 06865 09727 09975 09997 100008 5 09 04965 09178 09880 09981 099977 3 11 03473 08351 09659 09928 099846 3 13 02406 07372 09291 09813 099495 4 15 01671 06364 08792 09621 098805 2 17 01171 05408 08194 09345 097649 6 05 09385 09982 09999 10000 10000

10

08

06

04

02

00

OC

000 005 010 015 020 025 030p




6 Conclusion


Data Availability






Acknowledgments


References












08

06

04

02

00

0 2 4 6 8x

(a)

8

6

4

2

0


(b)


































075

6 5 05 09636 09990 10000 10000 100004 3 07 08898 09926 09994 09999 100003 3 09 07951 09761 09968 09995 099993 2 11 06978 09484 09906 09981 099963 1 13 06076 09111 09797 09950 099872 2 15 05280 08672 09636 09896 099682 2 17 04593 08196 09427 09813 099362 1 19 04008 07705 09178 09702 09889

090

9 6 05 09385 09982 09999 10000 100006 4 07 08235 09871 09989 09999 100005 2 09 06884 09592 09944 09991 099984 2 11 05616 09139 09837 09967 099933 3 13 04539 08557 09651 09912 099773 2 15 03664 07901 09385 09818 099443 2 17 02968 07221 09049 09679 098893 1 19 02418 06553 08660 09494 09806

095

11 7 05 09218 09976 09999 10000 100007 5 07 07839 09831 09985 09998 100006 3 09 06316 09476 09926 09988 099985 3 11 04969 08915 09787 09956 099904 3 13 03884 08216 09550 09885 099694 2 15 03045 07454 09218 09763 099263 3 17 02405 06687 08806 09585 098543 2 19 01918 05956 08339 09353 09748

099

15 14 05 08792 09961 09998 10000 1000010 8 07 06865 09727 09975 09997 100008 5 09 04965 09178 09880 09981 099977 3 11 03473 08351 09659 09928 099846 3 13 02406 07372 09291 09813 099495 4 15 01671 06364 08792 09621 098805 2 17 01171 05408 08194 09345 097649 6 05 09385 09982 09999 10000 10000

10

08

06

04

02

00

OC

000 005 010 015 020 025 030p




6 Conclusion


Data Availability






Acknowledgments


References












08

06

04

02

00

0 2 4 6 8x

(a)

8

6

4

2

0


(b)































6 Conclusion


Data Availability






Acknowledgments


References












08

06

04

02

00

0 2 4 6 8x

(a)

8

6

4

2

0


(b)































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