Research Article Acceptance Sampling Plans from …downloads.hindawi.com/archive/2013/302469.pdfAcceptance Sampling Plans from Life Tests Based on Percentiles of Half Normal Distribution

Hindawi Publishing CorporationJournal of Quality and Reliability EngineeringVolume 2013 Article ID 302469 7 pageshttpdxdoiorg1011552013302469

Research ArticleAcceptance Sampling Plans from Life Tests Based onPercentiles of Half Normal Distribution

B Srinivasa Rao1 Ch Srinivasa Kumar2 and K Rosaiah3

1 Department of Mathematics amp Humanities RVR amp JC College of Engineering Guntur Aandhra Pardesh 522 019 India2Department of Mathematics KLUniversity Guntur Aandhra Pardesh 522 502 India3 Department of Statistics Acharya Nagarjuna University Guntur Aandhra Pardesh 522 010 India

Correspondence should be addressed to B Srinivasa Rao boyapatisrinuyahoocom

Received 15 April 2013 Revised 9 October 2013 Accepted 11 October 2013

Academic Editor Christian Kirchsteiger

Copyright copy 2013 B Srinivasa Rao et alThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The design of acceptance sampling plans is developed under truncated life testing based on the percentiles of half normaldistribution The minimum sample size necessary to ensure the specified life percentile is obtained under a given consumerrsquos riskThe operating characteristic values of the sampling plans as well as the producerrsquos risk are presented The results are illustrated byexamples

1 Introduction

Acceptance sampling is concerned with inspection and deci-sion making regarding lots of products and constitutes oneof the oldest techniques in quality control If the lifetime ofthe product represents the quality characteristics of interestthe acceptance sampling is as follows a company receivesa shipment of product from a vendor This product isoften a component or raw material used in the companyrsquosmanufacturing process A sample is taken from the lot andthe relevant quality characteristic of the units in the sampleis inspected On the basis of the information in the samplea decision is made regarding lot disposition Traditionallywhen the life test indicates that the mean life of productsexceeds the specified one the lot of products is acceptedotherwise it is rejected Accepted lots are put into productionwhile rejected lots may be returned to the vendor or maybe subjected to some other lot disposition actions For thepurpose of reducing the test time and cost a truncated lifetest may be conducted to determine the sample size to ensurea certainmean life of products when the life test is terminatedat a time 119905

0 and the number of failures does not exceed a given

acceptance number 119888A common practice in life testing is to terminate the

life test by a predetermined time 1199050and note the number of

failures One of the objectives of these experiments is to set alower confidence limit on the mean life It is then to establish

a specified mean life with a given probability of at least 119901lowastwhich provides protection to consumers The test may beterminated before the time is reached or when the numberof failures exceeds the acceptance number 119888 in which case thedecision is to reject the lot

Studies regarding truncated life tests can be found inEpstein [1] Sobel and Tischendrof [2] Goode and Kao[3] Gupta and Groll [4] Gupta [5] Fertig and Mann [6]Kantam andRosaiah [7] Baklizi [8]Wu andTsai [9] Rosaiahand Kantam [10] Rosaiah et al [11] Tsai and Wu [12]Balakrishnan et al [13] Srinivasa Rao et al [14] SrinivasaRao et al [15] Aslam et al [16] and Srinivasa Rao et al[17] All these authors designed acceptance sampling plansbased on the mean life time under a truncated life testIn contrast Lio et al [18] considered acceptance samplingplans for percentiles using truncated life tests and assumingBirnbaum-Saunders distribution Srinivasa Rao and Kantam[19] developed similar plans for the percentiles of log-logisticdistribution

Normal distribution is the most preferred distribution instatistical studies But for life test models it is not suitabledistribution because of its domain [minusinfin +infin] Half normaldistribution is an increasing failure rate (IFR) model whichis most useful in reliability studies Because of this IFRnature with domain [0 +infin] we are motivated to study thisdistribution

2 Journal of Quality and Reliability Engineering

In this paper acceptance sampling plans are developed forpercentiles of half normal distribution life test and are givenin Section 2 Operating characteristic (OC) and producerrsquosrisk are given in Section 3 Examples based on real fatiguelife data sets are provided for an illustration in Section 4 Thepaper is closed with summary and conclusions in Section 5

2 Acceptance Sampling Plans

The probability density function (pdf) of a half normaldistribution is given by

119901 (119909) =2120579

120587119890minus11990921205792120587 119909 ge 0 (1)

Its cumulative distribution function (cdf) is

119875 (119909) = erf ( 120579119909radic120587) 119909 ge 0 (2)

where erf is an error function and 120579 is a parameter Assumethat the life time of a product follows half normal distributionwith 120590 as scale parameter Its cumulative distribution func-tion 119865(sdot) is given by

119865 (119905) = erf (120579 (119905120590)radic120587

) 119905 ge 0 120590 gt 0 (3)

Given 0 lt 119902 lt 1 the 100th percentile is given by

119905119902= 120590radic120587

120579erfminus1 (119902) (4)

Substituting 120590 in (3) in the scaled form we get

119865 (119905) = erf (120579 (119905119905119902) (radic120587120579) erfminus1 (119902)radic120587

)

997904rArr erf (1205750erfminus1 (119902))

(5)

where 1205750= 119905119905119902 Here erfminus1 is simulated by

erfminus1 (119911) = 12

radic120587[119911 +120587

121199113+71205872

4801199115

+1271205873

403201199117+4369120587

4

58060801199119sdot sdot sdot ]

(6)

A common practice in life testing is to terminate the lifetest by a predetermined time 119905 to require the probability ofrejecting a bad lot to be at least 119901lowast and to have the maximumpermissible number of bad items to accept the lot equalto 119888 The acceptance sampling plan for percentiles under atruncated life test is to set up the minimum sample size 119899 fora given acceptance number 119888 such that the consumerrsquos riskthe probability of accepting a bad lot does not exceed 1 minus119901lowastA bad lot means that the true 100th percentile 119905

119902 is below a

specified percentile 1199050119902Thus the probability119901lowast is a confidence

level in the sense that the chance of rejecting a bad lot with

119905119902lt 1199050

119902is at least equal to 119901lowast Therefore for given 119901lowast the

proposed acceptance sampling plan can be characterized bythe triplet (119899 119888 120575

0) = (119899 119888 119905119905

0

119902) where 120575 = 1199051199050

119902

We consider large sized lots so that the binomial distri-bution can be applied The problem is to determine for givenvalues of 119901lowast 1199050

119902 and 119888 the smallest positive integer 119899 required

to assert that 119905119902gt 1199050

119902must satisfy the relation

119888

sum

119894=0

119901119894

0(1 minus 119901

0)119899minus119894le 1 minus 119901

lowast (7)

where 1199010= 119865119905(1205750) is the probability of a failure during the

time 119905 = 1205751199050119902for the specified 100qth percentile of lifetime 1199050

119902

The value 1199010depends only on 120575

0= 1199051199050

119902 Since 120597119865

119905(120575)120597120575 gt 0

it is an increasing function of 120575 Accordingly we have

119865119905(120575) le 119865

119905(1205750) lArrrArr 120575 le 120575

0 (8)

or equivalently

119865 (119905 119905119902) lt 119865 (119905 119905

0

119902) lArrrArr 119905

119902ge 1199050

119902 (9)

The smallest sample size 119899 satisfying (9) can be obtained forany given 119902 1199051199050

119902 or 119901lowast In contrast only the input values

119905120590 119901lowast are needed to calculate the smallest sample size 119899 Inparticular if we take 119902 = 050 the value of 119899 would becomeldquothe smallest sample size required to test that the populationmedian life exceeds a given specified valuerdquo Half normaldistribution is a skewed distribution for our present skewedpopulation the median is a more appropriate average fordecision making about the quality of the life than populationmean Thus we may conclude that population median basedsampling plans of half normal distribution model are moreeconomical than those based on population mean withrespect to sample size To save space only the results of smallsample sizes for 119902 = 050 119901lowast = 075 090 095 099 119888 =0(1)10 120575 = 01 02 04 06 08 10 15 20 25 30 are dis-played in Table 1

3 Operating Characteristic of the SamplingPlan and Producerrsquos Risk

The operating characteristic (OC) function 119871 of the samplingplan (119899 119888 1199051199050

119902) is the probability of accepting a lot as a

function of 119901 = 119865119905(120575) with 120575

0= 119905119905119902 It is given as

119871 (119901) =

119888

sum

119894=0

(119899

119894) 119901119894(1 minus 119901)

119899minus119894 (10)

Therefore we have

119901 = 119865119905(120575) = 119865(

119905

1199050

119902

1

119889119902

) (11)

where 119889119902= 1199051199021199050

119902 Using (10) the OC values can be obtained

for any sampling plan (119899 119888 119905119905005) (Table 5) To save space we

Journal of Quality and Reliability Engineering 3

Table 1 Minimum sample sizes necessary to assert the median lifeof a product

119901lowast119888

1199051199050

05

01 02 04 06 08 10 15 20 25 30075 0 26 13 6 4 3 3 2 2 2 1075 1 50 25 12 8 6 5 3 3 2 2075 2 72 36 18 12 9 7 5 4 4 3075 3 94 47 23 16 12 10 7 5 5 4075 4 116 58 29 19 15 12 8 7 6 5075 5 137 68 34 23 17 14 10 8 7 6075 6 158 79 39 26 20 16 11 9 8 8075 7 179 89 45 30 23 18 13 11 9 9075 8 200 100 50 33 25 21 14 12 10 10075 9 220 110 55 37 28 23 16 13 12 11075 10 241 120 60 40 31 25 18 14 13 12090 0 42 21 10 7 5 4 2 2 2 2090 1 71 35 17 11 8 7 4 3 3 2090 2 98 48 24 16 12 9 6 5 4 4090 3 123 61 30 20 15 12 8 6 5 5090 4 147 73 36 24 18 14 10 8 6 6090 5 171 85 42 28 21 17 11 9 8 7090 6 194 96 48 32 24 19 13 10 9 8090 7 217 108 53 35 27 21 15 12 10 9090 8 240 119 59 39 29 24 16 13 11 10090 9 262 130 65 43 32 26 18 14 12 11090 10 284 141 70 47 35 28 20 16 14 12095 0 55 27 13 8 6 5 3 2 2 2095 1 87 43 21 14 10 8 5 4 3 3095 2 115 57 28 18 13 11 7 5 4 4095 3 142 70 34 23 17 13 9 7 6 5095 4 168 83 41 27 20 16 11 8 7 6095 5 193 96 47 31 23 18 12 10 8 7095 6 218 108 53 35 26 21 14 11 9 8095 7 242 120 59 39 29 23 16 12 11 10095 8 266 132 65 43 32 26 18 14 12 11095 9 289 143 71 47 35 28 19 15 13 12095 10 312 155 77 51 38 30 21 16 14 13099 0 84 41 20 13 9 7 4 3 2 2099 1 121 59 29 19 14 11 7 5 4 3099 2 154 76 37 24 17 14 9 7 5 5099 3 184 91 44 29 21 17 11 8 7 6099 4 212 105 51 33 25 19 13 10 8 7099 5 240 119 58 38 28 22 15 11 9 8099 6 267 132 65 42 31 25 16 13 10 9099 7 293 145 71 47 35 27 18 14 12 10099 8 319 158 77 51 38 30 20 15 13 12099 9 345 171 84 55 41 33 22 17 14 13099 10 370 183 90 59 44 35 24 18 15 14

present the OC values for sampling plans with 119902 = 050 119901lowast =075 090 095 099 120575 = 01020406081015202530119905051199050

05= 10 15 20 25 30 35 40 45 50 for 119888 = 1 and

119888 = 5 which are given in Table 2 and Table 3 respectively

The producerrsquos risk is defined as the probability ofrejecting the lot when 119905

119902gt 1199050

119902 For a given value of the

producerrsquos risk say 120574 we are interested in knowing the valueof 119889119902to ensure that the producerrsquos risk is less than or equal

to 120574 if a sampling plan (119899 119888 119905119905119902) is developed at a specified

confidence level119901lowastThus one needs to find the smallest value119889119902according to (10) as

119888

sum

119894=0

(119899

119894) 119901119894(1 minus 119901)

119899minus119894 (12)

where

119901 = 119865(119905

1199050

119902

1

119889119902

) (13)

To save space based on the sampling plans (119899 119888 1199051199050119902) given

in Table 1 the minimum ratios of 11988905

for the acceptability of alot at the producerrsquos risk of 120574 = 005 are presented in Table 4

4 Illustrative Examples

In this section we consider two examples with real data setsto illustrate the proposed acceptance sampling plans

41 Example 1 The first data refers to software reliabilitypresented by Wood [20] The data set was reported in hoursas 519 968 1430 1893 2490 3058 3625 4422 and 5218 Theconfidence level is assumed for this acceptance sampling planonly if the lifetimes are from half normal distribution withshape parameter 120590 So in order to apply this example for ourtables we have to confirm the goodness of fit of half normaldistribution to the data in the example This is done throughthe well-known QQ-plot method and the value we get is 119877 =09287 Hence we conclude that the half normal distributionprovides a reasonable goodness of fit for the data

Suppose that the experimenter would like to establish theunknown 50th percentile life time for the softwarementionedabove to be at least 300 h and the life test would be ended at600 h which should have led to the ratio 1199051199050

05= 20 Thus

with 119888 = 1 and 119901lowast = 075 the experimenter may take fromTable 1 the sample size 119899 which must be at least 3 thus thesampling plan to the data is given by

(119899 119888119905

1199050

05

) = (3 1 20) (14)

Since there is one itemwith a failure time less than or equal to600 h in the given sample of 9 observations the lot is acceptedas the result indicates that the 50th percentile life time 119905

05is

at least 300 h with a confidence level of 119901lowast = 075The OC values for acceptance sampling plan in (14) and

confidence 119901lowast = 075 for a half normal distribution may betaken from Table 2

The producerrsquos risk is almost equal to 03683 (= 1 minus06317) when the true percentile is greater than or equal to20 times the specified 50th percentile


Table 2 OC values of sampling plans

119888 = 1 1199051199050

05

119901lowast

119899 120575001

10 15 20 25 30 35 40 45 50

075

50 01 02420 04605 06093 07075 07741 08207 08545 08796 0898925 02 02345 04576 06090 07084 07754 08221 08559 08810 0900112 04 02408 04764 06301 07281 07928 08373 08690 08924 091018 06 02283 04759 06347 07342 07989 08430 08742 08970 091426 08 02173 04788 06425 07427 08070 08502 08805 09026 091915 10 01875 04609 06328 07373 08038 08482 08793 09018 091853 15 02309 05616 07262 08143 08662 08992 09214 09370 094843 20 00832 04276 06317 07464 08157 08602 08905 09120 092722 25 01752 06334 07938 08680 09084 09327 09484 09593 096702 30 00843 05920 07711 08535 08983 09253 09428 09548 09634

090

71 01 00993 02723 04274 05466 06354 07019 07524 07915 0822135 02 00980 02752 04332 05534 06424 07085 07586 07971 0827217 04 00961 02833 04473 05695 06583 07235 07723 08095 0838411 06 00953 02950 04654 05891 06772 07409 07879 08235 085098 08 00958 03109 04880 06124 06989 07604 08053 08388 086457 10 00625 02634 04449 05767 06698 07365 07854 08221 085034 15 00928 03766 05734 06949 07725 08243 08605 08866 090613 20 00832 04276 06317 07464 08157 08602 08905 09120 092773 25 00237 03441 05687 07000 07805 08329 08688 08942 091302 30 00843 05930 07711 08535 08983 09253 09428 09548 09364

095

87 01 00484 01766 03176 04390 05362 06127 06730 07209 0759443 02 00468 01773 03210 04440 05420 06186 06788 07264 0764521 04 00441 01804 03302 04566 05558 06325 06920 07388 0776014 06 00377 01753 03298 04595 05605 06380 06978 07445 0781510 08 00404 01948 03602 04932 05935 06688 07259 07699 080438 10 00352 01951 03671 05033 06047 06798 07363 07795 081315 15 00354 02431 04397 05795 06761 07441 07933 08297 085754 20 00194 02436 04558 06002 06969 07633 08105 08450 087103 25 00237 03441 05687 07000 07805 08329 08688 08942 091303 30 00054 02983 05323 06726 07597 08166 08557 08836 09041

099

121 01 00098 00662 01603 02633 03593 04433 05148 05749 0625659 02 00100 00695 01679 02740 03718 04565 05280 05878 0637829 04 00086 00688 01709 02807 03811 04672 05392 05991 0648919 06 00075 00694 01766 02908 03938 04812 05536 06133 0662614 08 00066 00714 01851 03044 04103 04987 05712 06303 0678811 10 00059 00751 01971 03221 04307 05199 05919 06501 069747 15 00047 00942 02439 03841 04975 05859 06446 07083 075095 20 00042 01328 03177 04694 05819 06644 07257 07721 080794 25 00029 01730 03843 05395 06469 07222 07764 08164 084673 30 00054 02983 05323 06726 07597 08166 08557 08836 09041

From Table 4 the experimenter could get the values of11988905

for different choices of 119888 and 119905119905005

in order to assert thatthe producerrsquos risk is less than 005 In this example the valueof11988905

should be 79120 for 119888 = 1 119905051199050

05= 20 and119901lowast = 075

This means that the product can have a life of 79120 timesthe required 50th percentile life time in order that under theabove acceptance sampling plan the product is accepted withprobability at least 075

42 Example 2 The second data refers to the data obtainedfrom Aarset [21] It represents the lifetimes of 50 devices inhours The data set was reported in hours as 01 02 1 1 1 11 2 3 6 7 11 12 18 18 18 18 18 21 32 36 40 45 46 47 5055 60 63 63 67 67 67 67 72 75 79 82 82 83 84 84 84 8585 85 85 85 86 and 86 On the similar lines of Example 1for this data the goodness of fit for 50 observations showedthat 119877 = 09175



119888 = 5 1199051199050

05

119901lowast

119899 120575001

10 15 20 25 30 35 40 45 50

075

137 01 02484 06314 08348 09234 09623 09802 09891 09936 0996168 02 02493 06407 08427 09284 09652 09820 09901 09943 0996634 04 02408 06506 08533 09354 09694 09845 09916 09952 0997223 06 02228 06527 08596 09400 09723 09862 09926 09959 0997617 08 02359 06888 08833 09527 09790 09898 09947 09971 0998314 10 02120 06898 08883 09561 09809 09909 09953 09974 0998510 15 01711 07193 09118 09685 09871 09942 09971 09985 099918 20 01556 07826 09428 09816 09930 09970 09986 09993 099967 25 01297 08341 09625 09889 09960 09983 09992 09996 099986 30 02323 09326 09880 09969 09989 09996 09998 09999 10000

090

171 01 00008 00143 00541 01154 01868 02595 03288 03925 0449785 02 00007 00137 00536 01156 01879 02615 03314 03955 0453142 04 00005 00129 00534 01173 01918 02673 03387 04038 0461928 06 00004 00117 00519 01169 01930 02701 03428 04087 0467521 08 00002 00108 00516 01185 01970 02760 03501 04170 0476317 10 00001 00096 00501 01182 01983 02789 03542 04219 0481811 15 00001 00120 00654 01499 02431 03321 04117 04810 054049 20 00000 00093 00619 01492 02459 03377 04193 04898 054998 25 00000 00078 00604 01503 02499 03439 04268 04980 055847 30 00000 00109 00779 01823 02907 03885 04722 05422 06006

095

193 01 00003 00071 00328 00787 01376 02019 02661 03273 0384196 02 00002 00068 00324 00787 01382 02031 02680 03298 0386947 04 00002 00067 00335 00824 01449 02124 02793 03425 0400631 06 00001 00063 00339 00849 01499 02197 02884 03529 0411723 08 00001 00062 00352 00892 01575 02301 03008 03666 0426218 10 00001 00068 00395 00991 01728 02495 03228 03901 0450312 15 00000 00070 00462 01166 02004 02845 03624 04320 0493010 20 00000 00046 00399 01092 01938 02794 03590 04299 049208 25 00000 00078 00604 01503 02499 03439 04268 04980 055847 30 00000 00109 00779 01823 02907 03885 04722 05422 06006

099

240 01 00000 00015 00110 00339 00700 01153 01657 02177 02691119 02 00000 00015 00109 00343 00710 01172 01683 02211 0273158 04 00000 00015 00117 00369 00764 01254 01792 02341 0287738 06 00000 00015 00122 00393 00813 01331 00189 00246 0301228 08 00000 00015 00132 00428 00883 01435 02026 02617 0318322 10 00000 00016 00149 00480 00978 01571 02195 02810 0339215 15 00000 00014 00159 00535 01096 01750 02424 03075 0368111 20 00000 00023 00256 00793 01518 02299 03057 03756 043839 25 00000 00034 00365 01055 01912 02781 03588 04307 049378 30 00000 00045 00457 01256 02197 03115 03945 04668 05289

Suppose that wewould like to establish the unknown 50thpercentile life time for the devices to be at least 10 hours andthe life test would be ended at 15 hrs which should have ledto the ratio of 119905

051199050

05= 15 Thus with 119888 = 5 119901lowast = 099

the experimenter may take from Table 1 the sample size 119899whichmust be at least 15Thus the sampling plan to the givendata is given by (119899 119888 1199051199050

05) = (15 5 15) Since there are 13

items with a failure time less than or equal to 15 hours inthe given sample of 50 observations the lot is accepted as the

result indicates that the 50th percentile lifetime 11990505

is at least10 hours with a confidence level of 119901lowast = 099

5 Summary and Conclusions

This paper provides the minimum sample size required todecide upon acceptingrejecting a lot based on its specified50th percentile (median) when the data follows half normaldistribution Assuming that the size of the given sample


Table 4 Minimum ratio for the acceptability of a lot with producerrsquos risk 005

119901lowast

119888 01 02 04 06 08 10 15 20 25 30075 0 272995 273264 252820 253354 253887 317358 318683 424911 531138 322612075 1 75211 74707 70638 69551 68415 70279 59340 79120 59356 71227075 2 46982 46579 45740 44857 43925 41606 42240 42572 53215 42191075 3 36690 36354 34860 35693 34898 35650 35335 30410 38013 31989075 4 31398 31106 30490 29280 30224 29465 27276 30537 30610 26281075 5 27959 27492 26938 26752 25684 25816 25929 25693 26240 23656075 6 25649 25411 24575 24019 24074 23411 22305 22542 23369 28043075 7 23985 23630 23429 22910 22887 21705 21968 23400 21320 25584075 8 22725 22520 22076 21353 21038 21610 19808 21336 19782 23739075 9 21635 21442 21022 20753 20432 20464 19775 19748 21740 22298075 10 20847 20577 20176 19725 19931 19550 19712 18485 20464 21137090 0 440826 441095 420653 442167 421723 422257 318683 424911 531138 637366090 1 107007 104994 100945 96860 92735 100728 82474 79120 98900 71227090 2 64092 62376 61553 60698 59809 54907 52368 56320 53215 63857090 3 48111 47384 45906 45183 44422 43623 41426 38877 38013 45615090 4 39865 39303 38153 37515 36839 35015 35805 36368 30610 36731090 5 34961 34497 33544 32966 32348 32105 29167 30187 32117 31488090 6 31546 30983 30488 29954 29380 28428 27494 26193 28177 28043090 7 29122 28770 27767 26995 27270 25857 26252 26377 25401 25584090 8 27309 26877 26215 25512 24762 25130 23481 23904 23338 23739090 9 25801 25413 25003 24353 23657 23516 22950 22008 21740 22298090 10 24598 24243 23678 23422 22768 22240 22503 22469 23107 21137095 0 577180 566969 546527 505105 505640 527154 476037 424911 531138 637366095 1 131233 129222 125182 124144 117015 115919 105419 109966 98900 118680095 2 75278 74222 72090 68610 65094 68151 62408 56320 53215 63857095 3 55593 54474 52215 52290 50756 47596 47466 47113 48596 45615095 4 45600 44766 43623 42449 41238 40540 40012 36368 38171 36731095 5 39491 39029 37669 36688 35670 34192 32373 34572 32117 31488095 6 35477 34916 33770 32917 32025 31754 30050 29739 28177 28043095 7 32501 32015 31018 30256 29455 28609 28366 26377 29250 30482095 8 30289 29859 28972 28279 27544 27464 27084 26410 26670 28005095 9 28479 27993 27389 26749 26067 25540 24517 24208 24685 26088095 10 27041 26687 26127 25529 24889 24024 23882 22469 23107 24557099 0 881383 860673 840232 819790 757389 736945 633385 634716 531138 637366099 1 182712 177675 173645 169596 165526 161433 151091 140558 137457 118680099 2 100941 99229 95790 92329 86206 87969 82360 83211 70400 84480099 3 72133 71017 67979 66493 63403 63445 59462 55234 58892 58316099 4 57617 56786 54559 52305 52214 48799 48365 47740 45460 45805099 5 49170 48504 46741 45363 43954 42512 41877 38890 37734 38540099 6 43502 42779 416472 39820 38621 38379 35116 36659 32741 33812099 7 39395 38776 37517 36770 35993 34087 32557 32162 32971 30482099 8 36363 35820 34483 33804 33092 32113 30646 28874 29880 32004099 9 34034 33549 32557 31533 30872 30576 29161 28494 27510 29622099 10 32100 31574 30673 29738 29118 28460 27972 26279 25624 27713

Table 5 OC values of sampling plans for 119899 = 3 119888 = 1 and 1205750= 20

119905051199050

0510 15 20 25 30 35 40 45 50

OC 00832 04276 06317 07464 08157 08602 08905 09120 09272


is minimum we can use the tables in this paper to pickup acceptance number 119888 and the upper bound in order toacceptreject the lot for a given probability of acceptance say119901lowast This paper also provides the sensitivity of the sampling

plans in terms of theOCWe advise the industrial practitionerand the experimenter to adopt this plan in order to save thecost and time of the experiment This plan can be furtherstudied for many other distributions and various quality andreliability characteristics as future research

Acknowledgments

The authors thank the editor and the reviewers for theirhelpful suggestions comments and encouragement whichhelped in improving the final version of the paper

References

[1] B Epstein ldquoTruncated life tests in the exponential caserdquo Annalsof Mathematical Statistics vol 25 pp 555ndash564 1954

[2] M Sobel and J A Tischendrof ldquoAcceptance samplingwith skewlife test objectiverdquo inProceedings of the 5thNational SyposiumonReliability and Qualilty Control pp 108ndash118 Philadelphia PaUSA 1959

[3] H P Goode and J H K Kao ldquoSampling plans based on theWeibull distributionrdquo in Proceedings of the 7th National Sypo-sium on Reliability and Qualilty Control pp 24ndash40 Philadel-phia Pa USA 1961

[4] S S Gupta and P A Groll ldquoGamma distribution in acceptancesampling based on life testsrdquo Journal of the American StatisticalAssociation vol 56 pp 942ndash970 1961

[5] S S Gupta ldquoLife test sampling plans for normal and lognormaldistributionrdquo Technometrics vol 4 pp 151ndash175 1962

[6] F W Fertig and N R Mann ldquoLife-test sampling plans for two-parameter Weibull populationsrdquo Technometrics vol 22 pp165ndash177 1980

[7] R R L Kantam and K Rosaiah ldquoHalf Logistic distribution inacceptance sampling based on life testsrdquo IAPQR Transactionsvol 23 no 2 pp 117ndash125 1998

[8] 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

[9] C-J Wu and T-R Tsai ldquoAcceptance sampling plans forbirnbaum-saunders distribution under truncated life testsrdquoInternational Journal of Reliability Quality and Safety Engineer-ing vol 12 no 6 pp 507ndash519 2005

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

[11] K Rosaiah R R L Kantam and Ch Santosh Kumar ldquoReli-ability test plans for exponnetiated log-logistic distributionrdquoEconomic Quality Control vol 21 pp 165ndash175 2006

[12] T-R Tsai and S-J Wu ldquoAcceptance sampling based on trun-cated life tests for generalized Rayleigh distributionrdquo Journal ofApplied Statistics vol 33 no 6 pp 595ndash600 2006

[13] N Balakrishnan V Leiva and J Lopez ldquoAcceptance sam-pling plans from truncated life tests based on the generalizedBirnbaum-Saunders distributionrdquo Communications in Statis-tics Simulation and Computation vol 36 no 3 pp 643ndash6562007

[14] G Srinivasa Rao M E Ghitany and R R L KantamldquoReliability test plans for Marshall-Olkin extended exponentialdistributionrdquo Applied Mathematical Sciences vol 3 no 53-56pp 2745ndash2755 2009

[15] G Srinivasa Rao M E Ghitany and R R L Kantam ldquoAneconomic reliability test plan for Marshall-Olkin extendedexponential distributionrdquoAppliedMathematical Sciences vol 5no 1ndash4 pp 103ndash112 2011

[16] M Aslam C-H Jun and M Ahmad ldquoA group samplingplan based on truncated life test for gamma distributed itemsrdquoPakistan Journal of Statistics vol 25 no 3 pp 333ndash340 2009


[18] Y L Lio T-R Tsai and S-J Wu ldquoAcceptance sampling plansfrom truncated life tests based on the birnbaum-saunders distri-bution for percentilesrdquoCommunications in Statistics Simulationand Computation vol 39 no 1 pp 119ndash136 2010

[19] G Srinivasa Rao and R R L Kantam ldquoAcceptance samplingplans from truncated life tests based on log-logistic distributionfor percentilesrdquoEconomicQuality Control vol 25 no 2 pp 153ndash167 2010

[20] AWood ldquoPredicting software reliabilityrdquo IEEE Transactions onSoftware Engineering vol 22 pp 69ndash77 1996

[21] M V Aarset ldquoHow to identify a bathtub hazard raterdquo IEEETransactions on Reliability vol 36 no 1 pp 106ndash108 1987

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


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Navigation and Observation



DistributedSensor Networks



In this paper acceptance sampling plans are developed forpercentiles of half normal distribution life test and are givenin Section 2 Operating characteristic (OC) and producerrsquosrisk are given in Section 3 Examples based on real fatiguelife data sets are provided for an illustration in Section 4 Thepaper is closed with summary and conclusions in Section 5

2 Acceptance Sampling Plans

The probability density function (pdf) of a half normaldistribution is given by

119901 (119909) =2120579

120587119890minus11990921205792120587 119909 ge 0 (1)

Its cumulative distribution function (cdf) is

119875 (119909) = erf ( 120579119909radic120587) 119909 ge 0 (2)

where erf is an error function and 120579 is a parameter Assumethat the life time of a product follows half normal distributionwith 120590 as scale parameter Its cumulative distribution func-tion 119865(sdot) is given by

119865 (119905) = erf (120579 (119905120590)radic120587

) 119905 ge 0 120590 gt 0 (3)

Given 0 lt 119902 lt 1 the 100th percentile is given by

119905119902= 120590radic120587

120579erfminus1 (119902) (4)

Substituting 120590 in (3) in the scaled form we get

119865 (119905) = erf (120579 (119905119905119902) (radic120587120579) erfminus1 (119902)radic120587

)

997904rArr erf (1205750erfminus1 (119902))

(5)

where 1205750= 119905119905119902 Here erfminus1 is simulated by

erfminus1 (119911) = 12

radic120587[119911 +120587

121199113+71205872

4801199115

+1271205873

403201199117+4369120587

4

58060801199119sdot sdot sdot ]

(6)

A common practice in life testing is to terminate the lifetest by a predetermined time 119905 to require the probability ofrejecting a bad lot to be at least 119901lowast and to have the maximumpermissible number of bad items to accept the lot equalto 119888 The acceptance sampling plan for percentiles under atruncated life test is to set up the minimum sample size 119899 fora given acceptance number 119888 such that the consumerrsquos riskthe probability of accepting a bad lot does not exceed 1 minus119901lowastA bad lot means that the true 100th percentile 119905

119902 is below a

specified percentile 1199050119902Thus the probability119901lowast is a confidence

level in the sense that the chance of rejecting a bad lot with

119905119902lt 1199050

119902is at least equal to 119901lowast Therefore for given 119901lowast the

proposed acceptance sampling plan can be characterized bythe triplet (119899 119888 120575

0) = (119899 119888 119905119905

0

119902) where 120575 = 1199051199050

119902

We consider large sized lots so that the binomial distri-bution can be applied The problem is to determine for givenvalues of 119901lowast 1199050

119902 and 119888 the smallest positive integer 119899 required

to assert that 119905119902gt 1199050

119902must satisfy the relation

119888

sum

119894=0

119901119894

0(1 minus 119901

0)119899minus119894le 1 minus 119901

lowast (7)

where 1199010= 119865119905(1205750) is the probability of a failure during the

time 119905 = 1205751199050119902for the specified 100qth percentile of lifetime 1199050

119902

The value 1199010depends only on 120575

0= 1199051199050

119902 Since 120597119865

119905(120575)120597120575 gt 0

it is an increasing function of 120575 Accordingly we have

119865119905(120575) le 119865

119905(1205750) lArrrArr 120575 le 120575

0 (8)

or equivalently

119865 (119905 119905119902) lt 119865 (119905 119905

0

119902) lArrrArr 119905

119902ge 1199050

119902 (9)

The smallest sample size 119899 satisfying (9) can be obtained forany given 119902 1199051199050

119902 or 119901lowast In contrast only the input values

119905120590 119901lowast are needed to calculate the smallest sample size 119899 Inparticular if we take 119902 = 050 the value of 119899 would becomeldquothe smallest sample size required to test that the populationmedian life exceeds a given specified valuerdquo Half normaldistribution is a skewed distribution for our present skewedpopulation the median is a more appropriate average fordecision making about the quality of the life than populationmean Thus we may conclude that population median basedsampling plans of half normal distribution model are moreeconomical than those based on population mean withrespect to sample size To save space only the results of smallsample sizes for 119902 = 050 119901lowast = 075 090 095 099 119888 =0(1)10 120575 = 01 02 04 06 08 10 15 20 25 30 are dis-played in Table 1

3 Operating Characteristic of the SamplingPlan and Producerrsquos Risk

The operating characteristic (OC) function 119871 of the samplingplan (119899 119888 1199051199050

119902) is the probability of accepting a lot as a

function of 119901 = 119865119905(120575) with 120575

0= 119905119905119902 It is given as

119871 (119901) =

119888

sum

119894=0

(119899

119894) 119901119894(1 minus 119901)

119899minus119894 (10)

Therefore we have

119901 = 119865119905(120575) = 119865(

119905

1199050

119902

1

119889119902

) (11)

where 119889119902= 1199051199021199050

119902 Using (10) the OC values can be obtained

for any sampling plan (119899 119888 119905119905005) (Table 5) To save space we



119901lowast119888

1199051199050

05

01 02 04 06 08 10 15 20 25 30075 0 26 13 6 4 3 3 2 2 2 1075 1 50 25 12 8 6 5 3 3 2 2075 2 72 36 18 12 9 7 5 4 4 3075 3 94 47 23 16 12 10 7 5 5 4075 4 116 58 29 19 15 12 8 7 6 5075 5 137 68 34 23 17 14 10 8 7 6075 6 158 79 39 26 20 16 11 9 8 8075 7 179 89 45 30 23 18 13 11 9 9075 8 200 100 50 33 25 21 14 12 10 10075 9 220 110 55 37 28 23 16 13 12 11075 10 241 120 60 40 31 25 18 14 13 12090 0 42 21 10 7 5 4 2 2 2 2090 1 71 35 17 11 8 7 4 3 3 2090 2 98 48 24 16 12 9 6 5 4 4090 3 123 61 30 20 15 12 8 6 5 5090 4 147 73 36 24 18 14 10 8 6 6090 5 171 85 42 28 21 17 11 9 8 7090 6 194 96 48 32 24 19 13 10 9 8090 7 217 108 53 35 27 21 15 12 10 9090 8 240 119 59 39 29 24 16 13 11 10090 9 262 130 65 43 32 26 18 14 12 11090 10 284 141 70 47 35 28 20 16 14 12095 0 55 27 13 8 6 5 3 2 2 2095 1 87 43 21 14 10 8 5 4 3 3095 2 115 57 28 18 13 11 7 5 4 4095 3 142 70 34 23 17 13 9 7 6 5095 4 168 83 41 27 20 16 11 8 7 6095 5 193 96 47 31 23 18 12 10 8 7095 6 218 108 53 35 26 21 14 11 9 8095 7 242 120 59 39 29 23 16 12 11 10095 8 266 132 65 43 32 26 18 14 12 11095 9 289 143 71 47 35 28 19 15 13 12095 10 312 155 77 51 38 30 21 16 14 13099 0 84 41 20 13 9 7 4 3 2 2099 1 121 59 29 19 14 11 7 5 4 3099 2 154 76 37 24 17 14 9 7 5 5099 3 184 91 44 29 21 17 11 8 7 6099 4 212 105 51 33 25 19 13 10 8 7099 5 240 119 58 38 28 22 15 11 9 8099 6 267 132 65 42 31 25 16 13 10 9099 7 293 145 71 47 35 27 18 14 12 10099 8 319 158 77 51 38 30 20 15 13 12099 9 345 171 84 55 41 33 22 17 14 13099 10 370 183 90 59 44 35 24 18 15 14


05= 10 15 20 25 30 35 40 45 50 for 119888 = 1 and



119902gt 1199050





119888

sum

119894=0

(119899

119894) 119901119894(1 minus 119901)

119899minus119894 (12)

where

119901 = 119865(119905

1199050

119902

1

119889119902

) (13)








05= 20 Thus


(119899 119888119905

1199050

05

) = (3 1 20) (14)


05is






119888 = 1 1199051199050

05

119901lowast

119899 120575001

10 15 20 25 30 35 40 45 50

075

50 01 02420 04605 06093 07075 07741 08207 08545 08796 0898925 02 02345 04576 06090 07084 07754 08221 08559 08810 0900112 04 02408 04764 06301 07281 07928 08373 08690 08924 091018 06 02283 04759 06347 07342 07989 08430 08742 08970 091426 08 02173 04788 06425 07427 08070 08502 08805 09026 091915 10 01875 04609 06328 07373 08038 08482 08793 09018 091853 15 02309 05616 07262 08143 08662 08992 09214 09370 094843 20 00832 04276 06317 07464 08157 08602 08905 09120 092722 25 01752 06334 07938 08680 09084 09327 09484 09593 096702 30 00843 05920 07711 08535 08983 09253 09428 09548 09634

090

71 01 00993 02723 04274 05466 06354 07019 07524 07915 0822135 02 00980 02752 04332 05534 06424 07085 07586 07971 0827217 04 00961 02833 04473 05695 06583 07235 07723 08095 0838411 06 00953 02950 04654 05891 06772 07409 07879 08235 085098 08 00958 03109 04880 06124 06989 07604 08053 08388 086457 10 00625 02634 04449 05767 06698 07365 07854 08221 085034 15 00928 03766 05734 06949 07725 08243 08605 08866 090613 20 00832 04276 06317 07464 08157 08602 08905 09120 092773 25 00237 03441 05687 07000 07805 08329 08688 08942 091302 30 00843 05930 07711 08535 08983 09253 09428 09548 09364

095

87 01 00484 01766 03176 04390 05362 06127 06730 07209 0759443 02 00468 01773 03210 04440 05420 06186 06788 07264 0764521 04 00441 01804 03302 04566 05558 06325 06920 07388 0776014 06 00377 01753 03298 04595 05605 06380 06978 07445 0781510 08 00404 01948 03602 04932 05935 06688 07259 07699 080438 10 00352 01951 03671 05033 06047 06798 07363 07795 081315 15 00354 02431 04397 05795 06761 07441 07933 08297 085754 20 00194 02436 04558 06002 06969 07633 08105 08450 087103 25 00237 03441 05687 07000 07805 08329 08688 08942 091303 30 00054 02983 05323 06726 07597 08166 08557 08836 09041

099

121 01 00098 00662 01603 02633 03593 04433 05148 05749 0625659 02 00100 00695 01679 02740 03718 04565 05280 05878 0637829 04 00086 00688 01709 02807 03811 04672 05392 05991 0648919 06 00075 00694 01766 02908 03938 04812 05536 06133 0662614 08 00066 00714 01851 03044 04103 04987 05712 06303 0678811 10 00059 00751 01971 03221 04307 05199 05919 06501 069747 15 00047 00942 02439 03841 04975 05859 06446 07083 075095 20 00042 01328 03177 04694 05819 06644 07257 07721 080794 25 00029 01730 03843 05395 06469 07222 07764 08164 084673 30 00054 02983 05323 06726 07597 08166 08557 08836 09041




should be 79120 for 119888 = 1 119905051199050

05= 20 and119901lowast = 075





119888 = 5 1199051199050

05

119901lowast

119899 120575001

10 15 20 25 30 35 40 45 50

075

137 01 02484 06314 08348 09234 09623 09802 09891 09936 0996168 02 02493 06407 08427 09284 09652 09820 09901 09943 0996634 04 02408 06506 08533 09354 09694 09845 09916 09952 0997223 06 02228 06527 08596 09400 09723 09862 09926 09959 0997617 08 02359 06888 08833 09527 09790 09898 09947 09971 0998314 10 02120 06898 08883 09561 09809 09909 09953 09974 0998510 15 01711 07193 09118 09685 09871 09942 09971 09985 099918 20 01556 07826 09428 09816 09930 09970 09986 09993 099967 25 01297 08341 09625 09889 09960 09983 09992 09996 099986 30 02323 09326 09880 09969 09989 09996 09998 09999 10000

090

171 01 00008 00143 00541 01154 01868 02595 03288 03925 0449785 02 00007 00137 00536 01156 01879 02615 03314 03955 0453142 04 00005 00129 00534 01173 01918 02673 03387 04038 0461928 06 00004 00117 00519 01169 01930 02701 03428 04087 0467521 08 00002 00108 00516 01185 01970 02760 03501 04170 0476317 10 00001 00096 00501 01182 01983 02789 03542 04219 0481811 15 00001 00120 00654 01499 02431 03321 04117 04810 054049 20 00000 00093 00619 01492 02459 03377 04193 04898 054998 25 00000 00078 00604 01503 02499 03439 04268 04980 055847 30 00000 00109 00779 01823 02907 03885 04722 05422 06006

095

193 01 00003 00071 00328 00787 01376 02019 02661 03273 0384196 02 00002 00068 00324 00787 01382 02031 02680 03298 0386947 04 00002 00067 00335 00824 01449 02124 02793 03425 0400631 06 00001 00063 00339 00849 01499 02197 02884 03529 0411723 08 00001 00062 00352 00892 01575 02301 03008 03666 0426218 10 00001 00068 00395 00991 01728 02495 03228 03901 0450312 15 00000 00070 00462 01166 02004 02845 03624 04320 0493010 20 00000 00046 00399 01092 01938 02794 03590 04299 049208 25 00000 00078 00604 01503 02499 03439 04268 04980 055847 30 00000 00109 00779 01823 02907 03885 04722 05422 06006

099

240 01 00000 00015 00110 00339 00700 01153 01657 02177 02691119 02 00000 00015 00109 00343 00710 01172 01683 02211 0273158 04 00000 00015 00117 00369 00764 01254 01792 02341 0287738 06 00000 00015 00122 00393 00813 01331 00189 00246 0301228 08 00000 00015 00132 00428 00883 01435 02026 02617 0318322 10 00000 00016 00149 00480 00978 01571 02195 02810 0339215 15 00000 00014 00159 00535 01096 01750 02424 03075 0368111 20 00000 00023 00256 00793 01518 02299 03057 03756 043839 25 00000 00034 00365 01055 01912 02781 03588 04307 049378 30 00000 00045 00457 01256 02197 03115 03945 04668 05289


051199050

05= 15 Thus with 119888 = 5 119901lowast = 099


05) = (15 5 15) Since there are 13








119901lowast

119888 01 02 04 06 08 10 15 20 25 30075 0 272995 273264 252820 253354 253887 317358 318683 424911 531138 322612075 1 75211 74707 70638 69551 68415 70279 59340 79120 59356 71227075 2 46982 46579 45740 44857 43925 41606 42240 42572 53215 42191075 3 36690 36354 34860 35693 34898 35650 35335 30410 38013 31989075 4 31398 31106 30490 29280 30224 29465 27276 30537 30610 26281075 5 27959 27492 26938 26752 25684 25816 25929 25693 26240 23656075 6 25649 25411 24575 24019 24074 23411 22305 22542 23369 28043075 7 23985 23630 23429 22910 22887 21705 21968 23400 21320 25584075 8 22725 22520 22076 21353 21038 21610 19808 21336 19782 23739075 9 21635 21442 21022 20753 20432 20464 19775 19748 21740 22298075 10 20847 20577 20176 19725 19931 19550 19712 18485 20464 21137090 0 440826 441095 420653 442167 421723 422257 318683 424911 531138 637366090 1 107007 104994 100945 96860 92735 100728 82474 79120 98900 71227090 2 64092 62376 61553 60698 59809 54907 52368 56320 53215 63857090 3 48111 47384 45906 45183 44422 43623 41426 38877 38013 45615090 4 39865 39303 38153 37515 36839 35015 35805 36368 30610 36731090 5 34961 34497 33544 32966 32348 32105 29167 30187 32117 31488090 6 31546 30983 30488 29954 29380 28428 27494 26193 28177 28043090 7 29122 28770 27767 26995 27270 25857 26252 26377 25401 25584090 8 27309 26877 26215 25512 24762 25130 23481 23904 23338 23739090 9 25801 25413 25003 24353 23657 23516 22950 22008 21740 22298090 10 24598 24243 23678 23422 22768 22240 22503 22469 23107 21137095 0 577180 566969 546527 505105 505640 527154 476037 424911 531138 637366095 1 131233 129222 125182 124144 117015 115919 105419 109966 98900 118680095 2 75278 74222 72090 68610 65094 68151 62408 56320 53215 63857095 3 55593 54474 52215 52290 50756 47596 47466 47113 48596 45615095 4 45600 44766 43623 42449 41238 40540 40012 36368 38171 36731095 5 39491 39029 37669 36688 35670 34192 32373 34572 32117 31488095 6 35477 34916 33770 32917 32025 31754 30050 29739 28177 28043095 7 32501 32015 31018 30256 29455 28609 28366 26377 29250 30482095 8 30289 29859 28972 28279 27544 27464 27084 26410 26670 28005095 9 28479 27993 27389 26749 26067 25540 24517 24208 24685 26088095 10 27041 26687 26127 25529 24889 24024 23882 22469 23107 24557099 0 881383 860673 840232 819790 757389 736945 633385 634716 531138 637366099 1 182712 177675 173645 169596 165526 161433 151091 140558 137457 118680099 2 100941 99229 95790 92329 86206 87969 82360 83211 70400 84480099 3 72133 71017 67979 66493 63403 63445 59462 55234 58892 58316099 4 57617 56786 54559 52305 52214 48799 48365 47740 45460 45805099 5 49170 48504 46741 45363 43954 42512 41877 38890 37734 38540099 6 43502 42779 416472 39820 38621 38379 35116 36659 32741 33812099 7 39395 38776 37517 36770 35993 34087 32557 32162 32971 30482099 8 36363 35820 34483 33804 33092 32113 30646 28874 29880 32004099 9 34034 33549 32557 31533 30872 30576 29161 28494 27510 29622099 10 32100 31574 30673 29738 29118 28460 27972 26279 25624 27713


119905051199050

0510 15 20 25 30 35 40 45 50

OC 00832 04276 06317 07464 08157 08602 08905 09120 09272




Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119901lowast119888

1199051199050

05

01 02 04 06 08 10 15 20 25 30075 0 26 13 6 4 3 3 2 2 2 1075 1 50 25 12 8 6 5 3 3 2 2075 2 72 36 18 12 9 7 5 4 4 3075 3 94 47 23 16 12 10 7 5 5 4075 4 116 58 29 19 15 12 8 7 6 5075 5 137 68 34 23 17 14 10 8 7 6075 6 158 79 39 26 20 16 11 9 8 8075 7 179 89 45 30 23 18 13 11 9 9075 8 200 100 50 33 25 21 14 12 10 10075 9 220 110 55 37 28 23 16 13 12 11075 10 241 120 60 40 31 25 18 14 13 12090 0 42 21 10 7 5 4 2 2 2 2090 1 71 35 17 11 8 7 4 3 3 2090 2 98 48 24 16 12 9 6 5 4 4090 3 123 61 30 20 15 12 8 6 5 5090 4 147 73 36 24 18 14 10 8 6 6090 5 171 85 42 28 21 17 11 9 8 7090 6 194 96 48 32 24 19 13 10 9 8090 7 217 108 53 35 27 21 15 12 10 9090 8 240 119 59 39 29 24 16 13 11 10090 9 262 130 65 43 32 26 18 14 12 11090 10 284 141 70 47 35 28 20 16 14 12095 0 55 27 13 8 6 5 3 2 2 2095 1 87 43 21 14 10 8 5 4 3 3095 2 115 57 28 18 13 11 7 5 4 4095 3 142 70 34 23 17 13 9 7 6 5095 4 168 83 41 27 20 16 11 8 7 6095 5 193 96 47 31 23 18 12 10 8 7095 6 218 108 53 35 26 21 14 11 9 8095 7 242 120 59 39 29 23 16 12 11 10095 8 266 132 65 43 32 26 18 14 12 11095 9 289 143 71 47 35 28 19 15 13 12095 10 312 155 77 51 38 30 21 16 14 13099 0 84 41 20 13 9 7 4 3 2 2099 1 121 59 29 19 14 11 7 5 4 3099 2 154 76 37 24 17 14 9 7 5 5099 3 184 91 44 29 21 17 11 8 7 6099 4 212 105 51 33 25 19 13 10 8 7099 5 240 119 58 38 28 22 15 11 9 8099 6 267 132 65 42 31 25 16 13 10 9099 7 293 145 71 47 35 27 18 14 12 10099 8 319 158 77 51 38 30 20 15 13 12099 9 345 171 84 55 41 33 22 17 14 13099 10 370 183 90 59 44 35 24 18 15 14


05= 10 15 20 25 30 35 40 45 50 for 119888 = 1 and



119902gt 1199050





119888

sum

119894=0

(119899

119894) 119901119894(1 minus 119901)

119899minus119894 (12)

where

119901 = 119865(119905

1199050

119902

1

119889119902

) (13)








05= 20 Thus


(119899 119888119905

1199050

05

) = (3 1 20) (14)


05is






119888 = 1 1199051199050

05

119901lowast

119899 120575001

10 15 20 25 30 35 40 45 50

075

50 01 02420 04605 06093 07075 07741 08207 08545 08796 0898925 02 02345 04576 06090 07084 07754 08221 08559 08810 0900112 04 02408 04764 06301 07281 07928 08373 08690 08924 091018 06 02283 04759 06347 07342 07989 08430 08742 08970 091426 08 02173 04788 06425 07427 08070 08502 08805 09026 091915 10 01875 04609 06328 07373 08038 08482 08793 09018 091853 15 02309 05616 07262 08143 08662 08992 09214 09370 094843 20 00832 04276 06317 07464 08157 08602 08905 09120 092722 25 01752 06334 07938 08680 09084 09327 09484 09593 096702 30 00843 05920 07711 08535 08983 09253 09428 09548 09634

090

71 01 00993 02723 04274 05466 06354 07019 07524 07915 0822135 02 00980 02752 04332 05534 06424 07085 07586 07971 0827217 04 00961 02833 04473 05695 06583 07235 07723 08095 0838411 06 00953 02950 04654 05891 06772 07409 07879 08235 085098 08 00958 03109 04880 06124 06989 07604 08053 08388 086457 10 00625 02634 04449 05767 06698 07365 07854 08221 085034 15 00928 03766 05734 06949 07725 08243 08605 08866 090613 20 00832 04276 06317 07464 08157 08602 08905 09120 092773 25 00237 03441 05687 07000 07805 08329 08688 08942 091302 30 00843 05930 07711 08535 08983 09253 09428 09548 09364

095

87 01 00484 01766 03176 04390 05362 06127 06730 07209 0759443 02 00468 01773 03210 04440 05420 06186 06788 07264 0764521 04 00441 01804 03302 04566 05558 06325 06920 07388 0776014 06 00377 01753 03298 04595 05605 06380 06978 07445 0781510 08 00404 01948 03602 04932 05935 06688 07259 07699 080438 10 00352 01951 03671 05033 06047 06798 07363 07795 081315 15 00354 02431 04397 05795 06761 07441 07933 08297 085754 20 00194 02436 04558 06002 06969 07633 08105 08450 087103 25 00237 03441 05687 07000 07805 08329 08688 08942 091303 30 00054 02983 05323 06726 07597 08166 08557 08836 09041

099

121 01 00098 00662 01603 02633 03593 04433 05148 05749 0625659 02 00100 00695 01679 02740 03718 04565 05280 05878 0637829 04 00086 00688 01709 02807 03811 04672 05392 05991 0648919 06 00075 00694 01766 02908 03938 04812 05536 06133 0662614 08 00066 00714 01851 03044 04103 04987 05712 06303 0678811 10 00059 00751 01971 03221 04307 05199 05919 06501 069747 15 00047 00942 02439 03841 04975 05859 06446 07083 075095 20 00042 01328 03177 04694 05819 06644 07257 07721 080794 25 00029 01730 03843 05395 06469 07222 07764 08164 084673 30 00054 02983 05323 06726 07597 08166 08557 08836 09041




should be 79120 for 119888 = 1 119905051199050

05= 20 and119901lowast = 075





119888 = 5 1199051199050

05

119901lowast

119899 120575001

10 15 20 25 30 35 40 45 50

075

137 01 02484 06314 08348 09234 09623 09802 09891 09936 0996168 02 02493 06407 08427 09284 09652 09820 09901 09943 0996634 04 02408 06506 08533 09354 09694 09845 09916 09952 0997223 06 02228 06527 08596 09400 09723 09862 09926 09959 0997617 08 02359 06888 08833 09527 09790 09898 09947 09971 0998314 10 02120 06898 08883 09561 09809 09909 09953 09974 0998510 15 01711 07193 09118 09685 09871 09942 09971 09985 099918 20 01556 07826 09428 09816 09930 09970 09986 09993 099967 25 01297 08341 09625 09889 09960 09983 09992 09996 099986 30 02323 09326 09880 09969 09989 09996 09998 09999 10000

090

171 01 00008 00143 00541 01154 01868 02595 03288 03925 0449785 02 00007 00137 00536 01156 01879 02615 03314 03955 0453142 04 00005 00129 00534 01173 01918 02673 03387 04038 0461928 06 00004 00117 00519 01169 01930 02701 03428 04087 0467521 08 00002 00108 00516 01185 01970 02760 03501 04170 0476317 10 00001 00096 00501 01182 01983 02789 03542 04219 0481811 15 00001 00120 00654 01499 02431 03321 04117 04810 054049 20 00000 00093 00619 01492 02459 03377 04193 04898 054998 25 00000 00078 00604 01503 02499 03439 04268 04980 055847 30 00000 00109 00779 01823 02907 03885 04722 05422 06006

095

193 01 00003 00071 00328 00787 01376 02019 02661 03273 0384196 02 00002 00068 00324 00787 01382 02031 02680 03298 0386947 04 00002 00067 00335 00824 01449 02124 02793 03425 0400631 06 00001 00063 00339 00849 01499 02197 02884 03529 0411723 08 00001 00062 00352 00892 01575 02301 03008 03666 0426218 10 00001 00068 00395 00991 01728 02495 03228 03901 0450312 15 00000 00070 00462 01166 02004 02845 03624 04320 0493010 20 00000 00046 00399 01092 01938 02794 03590 04299 049208 25 00000 00078 00604 01503 02499 03439 04268 04980 055847 30 00000 00109 00779 01823 02907 03885 04722 05422 06006

099

240 01 00000 00015 00110 00339 00700 01153 01657 02177 02691119 02 00000 00015 00109 00343 00710 01172 01683 02211 0273158 04 00000 00015 00117 00369 00764 01254 01792 02341 0287738 06 00000 00015 00122 00393 00813 01331 00189 00246 0301228 08 00000 00015 00132 00428 00883 01435 02026 02617 0318322 10 00000 00016 00149 00480 00978 01571 02195 02810 0339215 15 00000 00014 00159 00535 01096 01750 02424 03075 0368111 20 00000 00023 00256 00793 01518 02299 03057 03756 043839 25 00000 00034 00365 01055 01912 02781 03588 04307 049378 30 00000 00045 00457 01256 02197 03115 03945 04668 05289


051199050

05= 15 Thus with 119888 = 5 119901lowast = 099


05) = (15 5 15) Since there are 13








119901lowast

119888 01 02 04 06 08 10 15 20 25 30075 0 272995 273264 252820 253354 253887 317358 318683 424911 531138 322612075 1 75211 74707 70638 69551 68415 70279 59340 79120 59356 71227075 2 46982 46579 45740 44857 43925 41606 42240 42572 53215 42191075 3 36690 36354 34860 35693 34898 35650 35335 30410 38013 31989075 4 31398 31106 30490 29280 30224 29465 27276 30537 30610 26281075 5 27959 27492 26938 26752 25684 25816 25929 25693 26240 23656075 6 25649 25411 24575 24019 24074 23411 22305 22542 23369 28043075 7 23985 23630 23429 22910 22887 21705 21968 23400 21320 25584075 8 22725 22520 22076 21353 21038 21610 19808 21336 19782 23739075 9 21635 21442 21022 20753 20432 20464 19775 19748 21740 22298075 10 20847 20577 20176 19725 19931 19550 19712 18485 20464 21137090 0 440826 441095 420653 442167 421723 422257 318683 424911 531138 637366090 1 107007 104994 100945 96860 92735 100728 82474 79120 98900 71227090 2 64092 62376 61553 60698 59809 54907 52368 56320 53215 63857090 3 48111 47384 45906 45183 44422 43623 41426 38877 38013 45615090 4 39865 39303 38153 37515 36839 35015 35805 36368 30610 36731090 5 34961 34497 33544 32966 32348 32105 29167 30187 32117 31488090 6 31546 30983 30488 29954 29380 28428 27494 26193 28177 28043090 7 29122 28770 27767 26995 27270 25857 26252 26377 25401 25584090 8 27309 26877 26215 25512 24762 25130 23481 23904 23338 23739090 9 25801 25413 25003 24353 23657 23516 22950 22008 21740 22298090 10 24598 24243 23678 23422 22768 22240 22503 22469 23107 21137095 0 577180 566969 546527 505105 505640 527154 476037 424911 531138 637366095 1 131233 129222 125182 124144 117015 115919 105419 109966 98900 118680095 2 75278 74222 72090 68610 65094 68151 62408 56320 53215 63857095 3 55593 54474 52215 52290 50756 47596 47466 47113 48596 45615095 4 45600 44766 43623 42449 41238 40540 40012 36368 38171 36731095 5 39491 39029 37669 36688 35670 34192 32373 34572 32117 31488095 6 35477 34916 33770 32917 32025 31754 30050 29739 28177 28043095 7 32501 32015 31018 30256 29455 28609 28366 26377 29250 30482095 8 30289 29859 28972 28279 27544 27464 27084 26410 26670 28005095 9 28479 27993 27389 26749 26067 25540 24517 24208 24685 26088095 10 27041 26687 26127 25529 24889 24024 23882 22469 23107 24557099 0 881383 860673 840232 819790 757389 736945 633385 634716 531138 637366099 1 182712 177675 173645 169596 165526 161433 151091 140558 137457 118680099 2 100941 99229 95790 92329 86206 87969 82360 83211 70400 84480099 3 72133 71017 67979 66493 63403 63445 59462 55234 58892 58316099 4 57617 56786 54559 52305 52214 48799 48365 47740 45460 45805099 5 49170 48504 46741 45363 43954 42512 41877 38890 37734 38540099 6 43502 42779 416472 39820 38621 38379 35116 36659 32741 33812099 7 39395 38776 37517 36770 35993 34087 32557 32162 32971 30482099 8 36363 35820 34483 33804 33092 32113 30646 28874 29880 32004099 9 34034 33549 32557 31533 30872 30576 29161 28494 27510 29622099 10 32100 31574 30673 29738 29118 28460 27972 26279 25624 27713


119905051199050

0510 15 20 25 30 35 40 45 50

OC 00832 04276 06317 07464 08157 08602 08905 09120 09272




Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119888 = 1 1199051199050

05

119901lowast

119899 120575001

10 15 20 25 30 35 40 45 50

075

50 01 02420 04605 06093 07075 07741 08207 08545 08796 0898925 02 02345 04576 06090 07084 07754 08221 08559 08810 0900112 04 02408 04764 06301 07281 07928 08373 08690 08924 091018 06 02283 04759 06347 07342 07989 08430 08742 08970 091426 08 02173 04788 06425 07427 08070 08502 08805 09026 091915 10 01875 04609 06328 07373 08038 08482 08793 09018 091853 15 02309 05616 07262 08143 08662 08992 09214 09370 094843 20 00832 04276 06317 07464 08157 08602 08905 09120 092722 25 01752 06334 07938 08680 09084 09327 09484 09593 096702 30 00843 05920 07711 08535 08983 09253 09428 09548 09634

090

71 01 00993 02723 04274 05466 06354 07019 07524 07915 0822135 02 00980 02752 04332 05534 06424 07085 07586 07971 0827217 04 00961 02833 04473 05695 06583 07235 07723 08095 0838411 06 00953 02950 04654 05891 06772 07409 07879 08235 085098 08 00958 03109 04880 06124 06989 07604 08053 08388 086457 10 00625 02634 04449 05767 06698 07365 07854 08221 085034 15 00928 03766 05734 06949 07725 08243 08605 08866 090613 20 00832 04276 06317 07464 08157 08602 08905 09120 092773 25 00237 03441 05687 07000 07805 08329 08688 08942 091302 30 00843 05930 07711 08535 08983 09253 09428 09548 09364

095

87 01 00484 01766 03176 04390 05362 06127 06730 07209 0759443 02 00468 01773 03210 04440 05420 06186 06788 07264 0764521 04 00441 01804 03302 04566 05558 06325 06920 07388 0776014 06 00377 01753 03298 04595 05605 06380 06978 07445 0781510 08 00404 01948 03602 04932 05935 06688 07259 07699 080438 10 00352 01951 03671 05033 06047 06798 07363 07795 081315 15 00354 02431 04397 05795 06761 07441 07933 08297 085754 20 00194 02436 04558 06002 06969 07633 08105 08450 087103 25 00237 03441 05687 07000 07805 08329 08688 08942 091303 30 00054 02983 05323 06726 07597 08166 08557 08836 09041

099

121 01 00098 00662 01603 02633 03593 04433 05148 05749 0625659 02 00100 00695 01679 02740 03718 04565 05280 05878 0637829 04 00086 00688 01709 02807 03811 04672 05392 05991 0648919 06 00075 00694 01766 02908 03938 04812 05536 06133 0662614 08 00066 00714 01851 03044 04103 04987 05712 06303 0678811 10 00059 00751 01971 03221 04307 05199 05919 06501 069747 15 00047 00942 02439 03841 04975 05859 06446 07083 075095 20 00042 01328 03177 04694 05819 06644 07257 07721 080794 25 00029 01730 03843 05395 06469 07222 07764 08164 084673 30 00054 02983 05323 06726 07597 08166 08557 08836 09041




should be 79120 for 119888 = 1 119905051199050

05= 20 and119901lowast = 075





119888 = 5 1199051199050

05

119901lowast

119899 120575001

10 15 20 25 30 35 40 45 50

075

137 01 02484 06314 08348 09234 09623 09802 09891 09936 0996168 02 02493 06407 08427 09284 09652 09820 09901 09943 0996634 04 02408 06506 08533 09354 09694 09845 09916 09952 0997223 06 02228 06527 08596 09400 09723 09862 09926 09959 0997617 08 02359 06888 08833 09527 09790 09898 09947 09971 0998314 10 02120 06898 08883 09561 09809 09909 09953 09974 0998510 15 01711 07193 09118 09685 09871 09942 09971 09985 099918 20 01556 07826 09428 09816 09930 09970 09986 09993 099967 25 01297 08341 09625 09889 09960 09983 09992 09996 099986 30 02323 09326 09880 09969 09989 09996 09998 09999 10000

090

171 01 00008 00143 00541 01154 01868 02595 03288 03925 0449785 02 00007 00137 00536 01156 01879 02615 03314 03955 0453142 04 00005 00129 00534 01173 01918 02673 03387 04038 0461928 06 00004 00117 00519 01169 01930 02701 03428 04087 0467521 08 00002 00108 00516 01185 01970 02760 03501 04170 0476317 10 00001 00096 00501 01182 01983 02789 03542 04219 0481811 15 00001 00120 00654 01499 02431 03321 04117 04810 054049 20 00000 00093 00619 01492 02459 03377 04193 04898 054998 25 00000 00078 00604 01503 02499 03439 04268 04980 055847 30 00000 00109 00779 01823 02907 03885 04722 05422 06006

095

193 01 00003 00071 00328 00787 01376 02019 02661 03273 0384196 02 00002 00068 00324 00787 01382 02031 02680 03298 0386947 04 00002 00067 00335 00824 01449 02124 02793 03425 0400631 06 00001 00063 00339 00849 01499 02197 02884 03529 0411723 08 00001 00062 00352 00892 01575 02301 03008 03666 0426218 10 00001 00068 00395 00991 01728 02495 03228 03901 0450312 15 00000 00070 00462 01166 02004 02845 03624 04320 0493010 20 00000 00046 00399 01092 01938 02794 03590 04299 049208 25 00000 00078 00604 01503 02499 03439 04268 04980 055847 30 00000 00109 00779 01823 02907 03885 04722 05422 06006

099

240 01 00000 00015 00110 00339 00700 01153 01657 02177 02691119 02 00000 00015 00109 00343 00710 01172 01683 02211 0273158 04 00000 00015 00117 00369 00764 01254 01792 02341 0287738 06 00000 00015 00122 00393 00813 01331 00189 00246 0301228 08 00000 00015 00132 00428 00883 01435 02026 02617 0318322 10 00000 00016 00149 00480 00978 01571 02195 02810 0339215 15 00000 00014 00159 00535 01096 01750 02424 03075 0368111 20 00000 00023 00256 00793 01518 02299 03057 03756 043839 25 00000 00034 00365 01055 01912 02781 03588 04307 049378 30 00000 00045 00457 01256 02197 03115 03945 04668 05289


051199050

05= 15 Thus with 119888 = 5 119901lowast = 099


05) = (15 5 15) Since there are 13








119901lowast

119888 01 02 04 06 08 10 15 20 25 30075 0 272995 273264 252820 253354 253887 317358 318683 424911 531138 322612075 1 75211 74707 70638 69551 68415 70279 59340 79120 59356 71227075 2 46982 46579 45740 44857 43925 41606 42240 42572 53215 42191075 3 36690 36354 34860 35693 34898 35650 35335 30410 38013 31989075 4 31398 31106 30490 29280 30224 29465 27276 30537 30610 26281075 5 27959 27492 26938 26752 25684 25816 25929 25693 26240 23656075 6 25649 25411 24575 24019 24074 23411 22305 22542 23369 28043075 7 23985 23630 23429 22910 22887 21705 21968 23400 21320 25584075 8 22725 22520 22076 21353 21038 21610 19808 21336 19782 23739075 9 21635 21442 21022 20753 20432 20464 19775 19748 21740 22298075 10 20847 20577 20176 19725 19931 19550 19712 18485 20464 21137090 0 440826 441095 420653 442167 421723 422257 318683 424911 531138 637366090 1 107007 104994 100945 96860 92735 100728 82474 79120 98900 71227090 2 64092 62376 61553 60698 59809 54907 52368 56320 53215 63857090 3 48111 47384 45906 45183 44422 43623 41426 38877 38013 45615090 4 39865 39303 38153 37515 36839 35015 35805 36368 30610 36731090 5 34961 34497 33544 32966 32348 32105 29167 30187 32117 31488090 6 31546 30983 30488 29954 29380 28428 27494 26193 28177 28043090 7 29122 28770 27767 26995 27270 25857 26252 26377 25401 25584090 8 27309 26877 26215 25512 24762 25130 23481 23904 23338 23739090 9 25801 25413 25003 24353 23657 23516 22950 22008 21740 22298090 10 24598 24243 23678 23422 22768 22240 22503 22469 23107 21137095 0 577180 566969 546527 505105 505640 527154 476037 424911 531138 637366095 1 131233 129222 125182 124144 117015 115919 105419 109966 98900 118680095 2 75278 74222 72090 68610 65094 68151 62408 56320 53215 63857095 3 55593 54474 52215 52290 50756 47596 47466 47113 48596 45615095 4 45600 44766 43623 42449 41238 40540 40012 36368 38171 36731095 5 39491 39029 37669 36688 35670 34192 32373 34572 32117 31488095 6 35477 34916 33770 32917 32025 31754 30050 29739 28177 28043095 7 32501 32015 31018 30256 29455 28609 28366 26377 29250 30482095 8 30289 29859 28972 28279 27544 27464 27084 26410 26670 28005095 9 28479 27993 27389 26749 26067 25540 24517 24208 24685 26088095 10 27041 26687 26127 25529 24889 24024 23882 22469 23107 24557099 0 881383 860673 840232 819790 757389 736945 633385 634716 531138 637366099 1 182712 177675 173645 169596 165526 161433 151091 140558 137457 118680099 2 100941 99229 95790 92329 86206 87969 82360 83211 70400 84480099 3 72133 71017 67979 66493 63403 63445 59462 55234 58892 58316099 4 57617 56786 54559 52305 52214 48799 48365 47740 45460 45805099 5 49170 48504 46741 45363 43954 42512 41877 38890 37734 38540099 6 43502 42779 416472 39820 38621 38379 35116 36659 32741 33812099 7 39395 38776 37517 36770 35993 34087 32557 32162 32971 30482099 8 36363 35820 34483 33804 33092 32113 30646 28874 29880 32004099 9 34034 33549 32557 31533 30872 30576 29161 28494 27510 29622099 10 32100 31574 30673 29738 29118 28460 27972 26279 25624 27713


119905051199050

0510 15 20 25 30 35 40 45 50

OC 00832 04276 06317 07464 08157 08602 08905 09120 09272




Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119888 = 5 1199051199050

05

119901lowast

119899 120575001

10 15 20 25 30 35 40 45 50

075

137 01 02484 06314 08348 09234 09623 09802 09891 09936 0996168 02 02493 06407 08427 09284 09652 09820 09901 09943 0996634 04 02408 06506 08533 09354 09694 09845 09916 09952 0997223 06 02228 06527 08596 09400 09723 09862 09926 09959 0997617 08 02359 06888 08833 09527 09790 09898 09947 09971 0998314 10 02120 06898 08883 09561 09809 09909 09953 09974 0998510 15 01711 07193 09118 09685 09871 09942 09971 09985 099918 20 01556 07826 09428 09816 09930 09970 09986 09993 099967 25 01297 08341 09625 09889 09960 09983 09992 09996 099986 30 02323 09326 09880 09969 09989 09996 09998 09999 10000

090

171 01 00008 00143 00541 01154 01868 02595 03288 03925 0449785 02 00007 00137 00536 01156 01879 02615 03314 03955 0453142 04 00005 00129 00534 01173 01918 02673 03387 04038 0461928 06 00004 00117 00519 01169 01930 02701 03428 04087 0467521 08 00002 00108 00516 01185 01970 02760 03501 04170 0476317 10 00001 00096 00501 01182 01983 02789 03542 04219 0481811 15 00001 00120 00654 01499 02431 03321 04117 04810 054049 20 00000 00093 00619 01492 02459 03377 04193 04898 054998 25 00000 00078 00604 01503 02499 03439 04268 04980 055847 30 00000 00109 00779 01823 02907 03885 04722 05422 06006

095

193 01 00003 00071 00328 00787 01376 02019 02661 03273 0384196 02 00002 00068 00324 00787 01382 02031 02680 03298 0386947 04 00002 00067 00335 00824 01449 02124 02793 03425 0400631 06 00001 00063 00339 00849 01499 02197 02884 03529 0411723 08 00001 00062 00352 00892 01575 02301 03008 03666 0426218 10 00001 00068 00395 00991 01728 02495 03228 03901 0450312 15 00000 00070 00462 01166 02004 02845 03624 04320 0493010 20 00000 00046 00399 01092 01938 02794 03590 04299 049208 25 00000 00078 00604 01503 02499 03439 04268 04980 055847 30 00000 00109 00779 01823 02907 03885 04722 05422 06006

099

240 01 00000 00015 00110 00339 00700 01153 01657 02177 02691119 02 00000 00015 00109 00343 00710 01172 01683 02211 0273158 04 00000 00015 00117 00369 00764 01254 01792 02341 0287738 06 00000 00015 00122 00393 00813 01331 00189 00246 0301228 08 00000 00015 00132 00428 00883 01435 02026 02617 0318322 10 00000 00016 00149 00480 00978 01571 02195 02810 0339215 15 00000 00014 00159 00535 01096 01750 02424 03075 0368111 20 00000 00023 00256 00793 01518 02299 03057 03756 043839 25 00000 00034 00365 01055 01912 02781 03588 04307 049378 30 00000 00045 00457 01256 02197 03115 03945 04668 05289


051199050

05= 15 Thus with 119888 = 5 119901lowast = 099


05) = (15 5 15) Since there are 13








119901lowast

119888 01 02 04 06 08 10 15 20 25 30075 0 272995 273264 252820 253354 253887 317358 318683 424911 531138 322612075 1 75211 74707 70638 69551 68415 70279 59340 79120 59356 71227075 2 46982 46579 45740 44857 43925 41606 42240 42572 53215 42191075 3 36690 36354 34860 35693 34898 35650 35335 30410 38013 31989075 4 31398 31106 30490 29280 30224 29465 27276 30537 30610 26281075 5 27959 27492 26938 26752 25684 25816 25929 25693 26240 23656075 6 25649 25411 24575 24019 24074 23411 22305 22542 23369 28043075 7 23985 23630 23429 22910 22887 21705 21968 23400 21320 25584075 8 22725 22520 22076 21353 21038 21610 19808 21336 19782 23739075 9 21635 21442 21022 20753 20432 20464 19775 19748 21740 22298075 10 20847 20577 20176 19725 19931 19550 19712 18485 20464 21137090 0 440826 441095 420653 442167 421723 422257 318683 424911 531138 637366090 1 107007 104994 100945 96860 92735 100728 82474 79120 98900 71227090 2 64092 62376 61553 60698 59809 54907 52368 56320 53215 63857090 3 48111 47384 45906 45183 44422 43623 41426 38877 38013 45615090 4 39865 39303 38153 37515 36839 35015 35805 36368 30610 36731090 5 34961 34497 33544 32966 32348 32105 29167 30187 32117 31488090 6 31546 30983 30488 29954 29380 28428 27494 26193 28177 28043090 7 29122 28770 27767 26995 27270 25857 26252 26377 25401 25584090 8 27309 26877 26215 25512 24762 25130 23481 23904 23338 23739090 9 25801 25413 25003 24353 23657 23516 22950 22008 21740 22298090 10 24598 24243 23678 23422 22768 22240 22503 22469 23107 21137095 0 577180 566969 546527 505105 505640 527154 476037 424911 531138 637366095 1 131233 129222 125182 124144 117015 115919 105419 109966 98900 118680095 2 75278 74222 72090 68610 65094 68151 62408 56320 53215 63857095 3 55593 54474 52215 52290 50756 47596 47466 47113 48596 45615095 4 45600 44766 43623 42449 41238 40540 40012 36368 38171 36731095 5 39491 39029 37669 36688 35670 34192 32373 34572 32117 31488095 6 35477 34916 33770 32917 32025 31754 30050 29739 28177 28043095 7 32501 32015 31018 30256 29455 28609 28366 26377 29250 30482095 8 30289 29859 28972 28279 27544 27464 27084 26410 26670 28005095 9 28479 27993 27389 26749 26067 25540 24517 24208 24685 26088095 10 27041 26687 26127 25529 24889 24024 23882 22469 23107 24557099 0 881383 860673 840232 819790 757389 736945 633385 634716 531138 637366099 1 182712 177675 173645 169596 165526 161433 151091 140558 137457 118680099 2 100941 99229 95790 92329 86206 87969 82360 83211 70400 84480099 3 72133 71017 67979 66493 63403 63445 59462 55234 58892 58316099 4 57617 56786 54559 52305 52214 48799 48365 47740 45460 45805099 5 49170 48504 46741 45363 43954 42512 41877 38890 37734 38540099 6 43502 42779 416472 39820 38621 38379 35116 36659 32741 33812099 7 39395 38776 37517 36770 35993 34087 32557 32162 32971 30482099 8 36363 35820 34483 33804 33092 32113 30646 28874 29880 32004099 9 34034 33549 32557 31533 30872 30576 29161 28494 27510 29622099 10 32100 31574 30673 29738 29118 28460 27972 26279 25624 27713


119905051199050

0510 15 20 25 30 35 40 45 50

OC 00832 04276 06317 07464 08157 08602 08905 09120 09272




Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119901lowast

119888 01 02 04 06 08 10 15 20 25 30075 0 272995 273264 252820 253354 253887 317358 318683 424911 531138 322612075 1 75211 74707 70638 69551 68415 70279 59340 79120 59356 71227075 2 46982 46579 45740 44857 43925 41606 42240 42572 53215 42191075 3 36690 36354 34860 35693 34898 35650 35335 30410 38013 31989075 4 31398 31106 30490 29280 30224 29465 27276 30537 30610 26281075 5 27959 27492 26938 26752 25684 25816 25929 25693 26240 23656075 6 25649 25411 24575 24019 24074 23411 22305 22542 23369 28043075 7 23985 23630 23429 22910 22887 21705 21968 23400 21320 25584075 8 22725 22520 22076 21353 21038 21610 19808 21336 19782 23739075 9 21635 21442 21022 20753 20432 20464 19775 19748 21740 22298075 10 20847 20577 20176 19725 19931 19550 19712 18485 20464 21137090 0 440826 441095 420653 442167 421723 422257 318683 424911 531138 637366090 1 107007 104994 100945 96860 92735 100728 82474 79120 98900 71227090 2 64092 62376 61553 60698 59809 54907 52368 56320 53215 63857090 3 48111 47384 45906 45183 44422 43623 41426 38877 38013 45615090 4 39865 39303 38153 37515 36839 35015 35805 36368 30610 36731090 5 34961 34497 33544 32966 32348 32105 29167 30187 32117 31488090 6 31546 30983 30488 29954 29380 28428 27494 26193 28177 28043090 7 29122 28770 27767 26995 27270 25857 26252 26377 25401 25584090 8 27309 26877 26215 25512 24762 25130 23481 23904 23338 23739090 9 25801 25413 25003 24353 23657 23516 22950 22008 21740 22298090 10 24598 24243 23678 23422 22768 22240 22503 22469 23107 21137095 0 577180 566969 546527 505105 505640 527154 476037 424911 531138 637366095 1 131233 129222 125182 124144 117015 115919 105419 109966 98900 118680095 2 75278 74222 72090 68610 65094 68151 62408 56320 53215 63857095 3 55593 54474 52215 52290 50756 47596 47466 47113 48596 45615095 4 45600 44766 43623 42449 41238 40540 40012 36368 38171 36731095 5 39491 39029 37669 36688 35670 34192 32373 34572 32117 31488095 6 35477 34916 33770 32917 32025 31754 30050 29739 28177 28043095 7 32501 32015 31018 30256 29455 28609 28366 26377 29250 30482095 8 30289 29859 28972 28279 27544 27464 27084 26410 26670 28005095 9 28479 27993 27389 26749 26067 25540 24517 24208 24685 26088095 10 27041 26687 26127 25529 24889 24024 23882 22469 23107 24557099 0 881383 860673 840232 819790 757389 736945 633385 634716 531138 637366099 1 182712 177675 173645 169596 165526 161433 151091 140558 137457 118680099 2 100941 99229 95790 92329 86206 87969 82360 83211 70400 84480099 3 72133 71017 67979 66493 63403 63445 59462 55234 58892 58316099 4 57617 56786 54559 52305 52214 48799 48365 47740 45460 45805099 5 49170 48504 46741 45363 43954 42512 41877 38890 37734 38540099 6 43502 42779 416472 39820 38621 38379 35116 36659 32741 33812099 7 39395 38776 37517 36770 35993 34087 32557 32162 32971 30482099 8 36363 35820 34483 33804 33092 32113 30646 28874 29880 32004099 9 34034 33549 32557 31533 30872 30576 29161 28494 27510 29622099 10 32100 31574 30673 29738 29118 28460 27972 26279 25624 27713


119905051199050

0510 15 20 25 30 35 40 45 50

OC 00832 04276 06317 07464 08157 08602 08905 09120 09272




Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Acknowledgments


References
























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Acceptance Sampling Plans from …downloads.hindawi.com/archive/2013/302469.pdfAcceptance Sampling Plans from Life Tests Based on Percentiles of Half Normal Distribution