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1 Copyrght IEEE. Reprnted, wth permsson, from Huaru Guo, Pengyng Nu, Adamantos Mettas and Doug Ogden, On Plannng Accelerated Lfe Tests for Comparng Two Product Desgns, Relablty and Mantanablty Symposum, January,. Ths materal s posted here wth permsson of the IEEE. Such permsson of the IEEE does not n any way mply IEEE endorsement of any of RelaSoft Corporaton's products or servces. Internal or personal use of ths materal s permtted. However, permsson to reprnt/republsh ths materal for advertsng or promotonal purposes or for creatng new collectve works for resale or redstrbuton must be obtaned from the IEEE by wrtng to y choosng to vew ths document, you agree to all provsons of the copyrght laws protectng t.
2 On Plannng Accelerated Lfe Tests for Comparng Two Product Desgns Huaru Guo, Ph.D, RelaSoft Corporaton Pengyng Nu, RelaSoft Corporaton Adamantos Mettas, RelaSoft Corporaton Doug Ogden, RelaSoft Corporaton Key Words: accelerated lfe test, power and sample sze, test desgn SMMARY & CONCLSIONS Accelerated lfe test (ALT) plannng s one of the most mportant and challengng tasks for relablty engneers. Snce the late 97s, methods for effcent ALT plannng have been studed extensvely and over 5 research papers have been publshed []. Most of the exstng methods focus on desgnng tests to mnmze the estmaton precson of model parameters or ther functons. Popularly used test desgns such as the level statstcally optmum plan, 3level best compromse plan and 3level best standard plan are all based on ths theory. However, although these desgns are very useful for estmatng dstrbuton parameters or gven relablty metrcs, they are not effcent for plannng tests that compare dfferent products. In ths paper, we wll present two methods for desgnng ALT to compare two dfferent desgns n terms of ther lfe. The probablty of detectng a gven amount of dfference of the lves s the focus of the proposed methods. Ths probablty usually s called detecton power. Comparng the estmated lves of two desgns s the same as comparng two random varables snce each lfe estmated through the ALT data s a random varable. Accordng to the requred detecton probablty, the sample sze of a comparson test can be determned by ether the analytcal or the smulaton method gven n ths paper. An example s used n the paper to llustrate the theory and the applcatons of the proposed methods. The presented methods are general methods and can be extended to other stuatons and appled beyond the example used n ths paper. PROLEM STATEMENT Assume the lfe of a product s affected by voltage. Hgh voltage wll shorten the lfe and the usage level voltage s 7 volts. Accelerated lfe tests are conducted for one desgn opton of the product and the test results are gven n Table. For convenence, we ll refer to ths desgn as desgn A. F/S Tme Voltage F/S Tme Voltage F 55 F F 7 F F 856 F F 968 F F 995 F 95 F 5 F 3 95 F 7 F 95 F 68 F 8 95 F 8 F F 3 F F 436 F F 445 F F 588 F 7 95 F 654 F F 786 F F 8 F 8 95 F 863 F F 937 F F 68 F F 334 F F F F F 6 95 F F F F F S 3 95 F S 3 95 F S 3 95 Table  Test Data for Desgn A at v and 95v From Table, we can see that samples were tested at v and 34 were tested at 95v. sng these data, we ft a lfestress model and predct the lfe at the usage level. The model and the predcton are gven n the followng secton //$6. IEEE
3 . Accelerated Lfe Test Data Analyss The nverse power lawwebull (IPLWebull) model was used here. The cumulatve dstrbuton functon (cdf) of t s: n ( ) FtV (, ) e KV t = () where t s lfe tme; V s voltage; K and n are model parameters n the lfestress relaton functon; and s the shape parameter of the Webull dstrbuton. For data n Table, the model parameters calculated usng maxmum lkelhood estmaton (MLE) are: K=.686, n=.56 and =.34. Applyng ths model, the predcted lfe at the usage level of 7v s,63 hours. Its onesded upper bound at a confdence level of 9% s,478. (For detals on accelerated lfe data modelng and parameter estmaton, please refer to [, 3].) However, the requred lfe at the use stress level should be at least,5 hours. Clearly, the current desgn couldn t meet ths requrement. Therefore, ths desgn s modfed. The modfed desgn s called desgn. Due to cost and tme constrants, only samples of desgn were manufactured and tested for 3, hours at v n order to quckly show the mprovement. The test data at v s gven n Table. F/S Tme Voltage F/S Tme Voltage F 876 F 59 F 3 F 7 F 345 F 4 F 473 F 693 F 57 F 7 F 58 F 75 F 638 F 86 F 796 F 96 F 94 S 3 F 934 S 3 Table  Test Data for Desgn at v After applyng the Webull dstrbuton to data n Table, the calculated lfe for desgn at v s,94. If we also ft a Webull dstrbuton to the falures for desgn A at v n Table, the predcted lfe s 78. Therefore, the new desgn shows sgnfcant mprovement at a voltage level of v. The rato of these two lves s about.55. Wll ths rato stll hold for the lfe at the usage voltage level? If so, the new desgn wll meet the relablty requrement of a lfe of,5 hours. In order to answer ths queston, test data at another stress level of desgn are needed. Test engneers decded to test more samples at a voltage level of 95v. The queston s how many samples are needed? The engneers only have 5, hours for testng, and manufacturng test samples s costly. Due to these constrants, the engneers want to buld and test as few samples as possble. The engneers decded to determne the sample sze based on the requrement of the detecton probablty. For desgn A, we know that the predcted lfe at the usage stress level s,63 wth a onesded 9% upper bound of,478. For desgn, the lfe wll be predcted usng data at a voltage level of v (Table ) and data at a voltage level of 95v (to be tested). The requrement for the detecton probablty s to have at least an 8% chance to detect ths dfference, f the rato of the two lves s at least.5. In other words, hopefully through choosng the rght sample sze to be tested at 95v for desgn, we wll have an 8% chance of detectng the dfference of the two lves. Statstcally speakng, the requrement of plannng the test s that the detecton probablty must be at least 8% at a confdence level of 9% when the rato of the lves of the two desgns s.5. In secton, we wll llustrate the theory of determnng the mnmal sample sze based on the requred detecton probablty. AN ANALYTICAL SOLTION. Calculate the Varance of the Lfe The lfe s the 9% percentle of the falure tme dstrbuton. If we defne x as the log transform of lfe, t can be calculated by: ln ( ln(.9) ) x = ln( K) nln( V) ().54 = ln( K ) n ln( V ) sng the delta method [4, 6], the varance of x s calculated by: x x x Var( x) = Var( θ) + Cov( θ, θ ) = θ = =, θ θ (3) where θ s model parameter, K and n. Var( θ ) and Cov( θ, θ ) are obtaned from the Fsher nformaton matrx of the falure data [4, 6]. Once the predcted value of x and ts varance are obtaned usng Eq. () and (3), the confdence bounds of x can be calculated by assumng that x s normally dstrbuted wth the predcted mean and varance.. Calculate the Detecton Probablty For the desgn A data n Table, the predcted lfe at 7v s,63. Therefore ts log transform s x A =ln()=ln(,63)= The calculated varance of x A A s.66, so the standard devaton std( x ) s.57 by takng the square root of the varance. x A usually s assumed to be normally dstrbuted. Ths assumpton has been proven by smulaton studes and s used by [4, 5]. The onesded upper bound of x A s: A A A x = x + Z.9 ( ) 7.984; CL= std x = where Z Cl =.9 s the 9% percentle of the standard normal dstrbuton. If the lfe of desgn s.5 tmes the lfe of desgn A, the log of the lfe wll be x = x A +ln(.5)= = In other words, the predcted x at 7v of desgn wll be a random number
4 followng a normal dstrbuton wth mean of Its standard devaton s determned by the test data at v (Table ) and 95v (to be tested). If the calculated x s larger A than x the upper bound of x A at a confdence level of CL, we conclude that the lfe of desgn s larger than desgn A at a confdence level of CL. Therefore, we have detected the dfference of the lves. The probablty that the calculated x A s larger than x s: A Detecton Probablty = Pr( x < x ) = Pr(7.984 < x ) (4) = Pr( < z) = Φnorm.8 std( x ) std( x ) where z s the standard normal random varable. s the Φnorm cdf of the standard normal dstrbuton. The calculated detecton probablty should be larger than the requred value of.8. In order to meet ths requrement, from Eq. (4) we calculate that std( x ) must be less than.93. Ths standard devaton s used to determne the sample sze at 95v for desgn..3 Determne the Sample Sze.3. Theory of Fsher Informaton Matrx From the theory of Fsher nformaton matrx, we know the standard devaton of the estmated dstrbuton parameters and the functon of them are affected by the test sample sze. Therefore, we can determne the sample sze of the test at 9v for desgn based on the solved standard devaton n secton.. Frst, let s dscuss how the Fsher nformaton matrx s developed when maxmum lkelhood estmaton s used. It s known that f falure tme t has a Webull dstrbuton wth parameters of and η, the natural logarthm y=ln(t) of t has a smallest extreme value (SEV) dstrbuton. The cdf for SEV s ( y μ )/ e F( y) = e (5) where μ = ln( η); = /. For convenence, the IPL lfestress functon used n Eq. () s reparameterzed as: ηv = ln( ηv) = ln( K) nln( V) n KV μ = α + α S V where μv = ln( ηv), ln( K) = α, α = n, S = ln( V). Let z = ( y μv ) / for falure tmes, and for the suspenson tme, we defne ln( T) α αs Φ ( w) =Φ where T s the suspenson tme. Φ s the cdf of the standard SEV dstrbuton. The log lkelhood functon s: l f N s l N z, ( ln( ) e z, ) ln ( ( w, ) ) Λ= + + Φ = = = = (6) (7) where z, s from the th falure tme at the th stress; l s the f total number of stresses; N s the number of falures at the th s stress; N s the number of suspensons at the th stress; w, s for the th suspenson tme at the th stress. From Eq. (7), the Fsher nformaton matrx for the test data n terms of, α and α s: Λ Λ Λ E E E α α Λ Λ Λ F = E E E (8) α α α α Λ Λ Λ E E E α α α α For the calculaton of each element n Eq. (7), please refer to Appendx and [6]. The asymptotc covarance matrx of the estmated dstrbuton parameters s the nverse of the Fsher nformaton matrx n Eq. (8). That s, Var( ) Cov(, α) Cov(, α) Σ= Cov( α, ) Var( α) Cov( α, α) = F (9) Cov( α, ) Cov( α, α) Var( α).3. Calculate Fsher Informaton Matrx for Desgn For desgn, snce there are no test data at 95v yet, we cannot estmate the model parameters and calculate the local Fsher nformaton based on falure data and the estmated parameters. However, we can use the nformaton from desgn A to calculate the plannng value of the model parameters for desgn. From the current test results, t s known the falure mechansm under study s the same for both desgns. Therefore, we assume the IPLWebull model s also applcable to desgn, and and n n the model are the same for both desgns. The dfference of the lfe s only reflected by parameter K. Snce we are nterested n the stuaton where the lfe of desgn s.5 tmes the lfe of desgn A, ths leads us to set K for desgn to be.5 tmes smaller than the K for desgn A. Therefore, usng the estmated parameters of desgn A n secton., the plannng values for desgn are: =.34 ; K = KA /.5 =.75E 6 ; n =.56, or n terms of, α and α : =.433 ; α = ; α =.56 sng the above plannng values n Eq. (7) we can calculate the Fsher nformaton matrx for desgn for the data n Table. It s, F = () F wll be combned wth F 95, the one we wll dscuss shortly, to get the total Fsher nformaton matrx. Snce no data were avalable yet for the test at 95v, we have to use the expected Fsher nformaton. For any test sample at 95v, t
5 would ether fal or survve by the end of a test tme of T=5,. Let f a test unt fals I = f a test unt survves Then the log lkelhood Λ for the th test unt at stress S s, ( ln( ) z ) ( )ln ( ( )) Λ = I e + z + I Φ w () For all the m test unts, the log lkelhood functon s m. Snce we don t yet have observatons at 95v, F 95 = Λ= Λ s calculated usng the plannng model parameters and the expected value of the dervatves n Eq. (8). For detals of the calculaton, please see Eq. (A.) n Appendx and [6]. The expected Fsher nformaton matrx for the m test unts at 95v s: F95 = m () The total Fsher nformaton matrx for desgn s F = F + F95 (3) m s the only unknown n Eq. (3) and t can be solved by settng the calculated standard devaton std( x ) to be equal to.93, the standard devaton value obtaned based on the requred detecton probablty n secton.. The calculated value of m s 4. When m=4, the total F s: F = The varance and covarance matrx s Σ= F = From Eq. (), we know x s calculated by.54 x = ln( K ) n ln(95) =.54 + α + α Defne L = (.54,, ), the varance of x and the standard devaton std( x ) are calculated by Var( x ) = L Σ L ' =.8; std( x ) = Var( x ) =.9 It confrms that the calculated std(x ) s less than the requred value of.93 when 4 samples are tested. If we use 3 samples, the predcted standard devaton std(x ) s.937 whch s slghtly larger than the requred value. Therefore at least 4 unts should be tested at 95v for desgn n order to have an 8% chance of detectng the dfference of the lves. 3 A SIMLATION APPROACH In secton, an analytcal soluton was found for determnng the test sample sze n order to meet the detecton probablty requrement. In ths secton we wll dscuss a novel smulaton soluton by generatng the expected falure tmes at 95v. Smlar to the assumptons used n secton, we assume the falure tmes at 95v follow a Webull dstrbuton wth the followng plannng values as gven n secton : =.34 ; K = KA /.5 =.75E 6 ; n =.56 From the above plannng values, we can solve the scale n parameterη = / ( KV ) = Let s assume m samples are tested. When the th falure occurs, the estmated probablty of falure usng the approxmated medan rank method s [6]: F =.3 / ( m+.4) (4) ( ) From Eq. (4) the expected falure tme t for the th falure s solved from: F ( t η ) = e (5) For example, when samples are tested, the estmated medan rank for the st falure s: F = (.3) / ( +.4) =.673 Settng Eq. (5) equal to.673, the expected falure tme for the st falure of the samples s calculated to be usng the plannng values for and η. Wth the calculated falure tme for each falure from Eq. (5) and the plannng values for model parameters, we can get the Fsher nformaton matrx, varance/covarance matrx and the expected standard devaton for the lfe. If the calculated standard devaton s bgger than the requred value, we need to repeat the above procedure by ncreasng the sample sze untl the one that meets the requrement s found. From Secton, we know the analytcal soluton for the sample sze s 4. sng the above smulaton procedure wth a sample sze of 4, we can fnd the expected falure tme for each sample usng Eq.(5). Snce the avalable test tme s 5, hours, any falures occur beyond 5, are treated as suspenson. The expected falure tmes of the 4 samples are gven n Table 3. Expected Falure Sample Medan Rank Falure Tme /Suspenson F F F F F F F F , S , S Table 3 Expected Results of 4 Test Samples for Desgn at 95v sng the data n Table 3 and the plannng parameters gven n secton.3. n terms of, α and α we got the expected Fsher nformaton matrx
6 F95 = (6) Comparng Eq. (6) wth the analytcal soluton of Eq. () by settng m=4, the relatve dfferences for all the elements n these two matrxes can be calculated as:.38%.89%.89% D Fsher _95 =.89%.63%.63% (7).89%.63%.63% From Eq. (7) t can be seen that the smulaton approach for obtanng expected falure tmes provdes a result that s very close to the analytcal soluton gven n secton. Smlar to the analytcal soluton, we can get the expected value of the total Fsher nformaton matrx by combnng Eq. () and Eq. (6). From here the expected varance/covarance matrx can be calculated and the expected value of std(x ) s calculated to be Ths result s almost dentcal to the analytcal soluton of.9. Therefore, both the analytcal and the smulaton methods suggest that at least 4 samples should be tested at 95v for desgn n order to have an 8% chance of detectng the dfference of the lves of desgn A and desgn. 4 CONCLSIONS In ths paper, we presented two methods for determnng sample sze accordng to the requred detecton probablty of an accelerated lfe test for comparng two desgns. An example s used to llustrate the theory of these two methods. Although part of the test data (test at v) of desgn and all the test data of desgn A have been gven n ths example, the proposed method can be appled to cases where no data are avalable. As long as the plannng values and the detecton probablty are provded and the mproved desgn s expected to have the same falure mechansms as the old desgn, the methods presented n ths paper can be appled. Any unknown varable n the test plan such as the stress level, sample sze at each stress level, and the test tme can be solved gven that other varables are provded. The analytcal soluton provded n ths paper requres numercally solvng the expected Fsher nformaton matrx. For engneers wth a strong math background, ths method may be faster than the smulaton method snce no tral and error s requred to fnd the sutable sample sze. Wth the smulaton method, once the falure tmes are smulated, the Fsher nformaton matrx can be easly calculated usng equatons gven n the Appendx. ecause ths s a very straghtforward method, t wll often be easer than the analytcal method for engneers to understand and use. 5 APPENDIX From Eq. (), the sx second partal dervatves n the Fsher nformaton matrx are: Λ gg k z w = Ie + ( I) e ;, k =, α αk Λ g z w = Ize ( I ) we ;, α + = (A.) Λ z w = I ( + z e ) + ( I) w e where g =, g = S = ln( V), and ln( T) α αs w = For an observaton, ether t s a falure or a suspenson, Eq. (A.) can be appled drectly by settng I= or. It s used to get the Fsher nformaton matrx for data at v and the smulated falure data at 95v n the smulaton method for desgn. If there are no observatons yet, the expected Fsher nformaton matrx can be calculated by takng the expectaton of the terms n Eq. (A.). They are: Λ gg k E = [ Φ ( w) ];, k =, α αk w Λ g e x w E = ln( x) xe dx + ( Φ ( )) ; w we α =, w Λ e x w E ( ) = ( ) ln ( ) ( ) w x xe dx Φ + + Φ w w e (A.) Eq. (A.) s used to calculate the expected Fsher nformaton matrx for the planned samples at 95v for desgn n the analytcal method. 6 REFERENCES. Nelson, W. (), A blography of Accelerated Test Plans, Part IIReferences, IEEE Trans. on Relablty 54, Sep. 5, Nelson, W. Accelerated Testng: Statstcal Models, Test Plans, and Data Analyss, John Wley & Sons, Inc., New York, RelaSoft Corporaton (7), Accelerated Lfe Testng Reference, RelaSoft, Tucson, AZ, Nelson, W. and Meeker, W. (978), Theory for Optmum Accelerated Censored Lfe Tests for Webull and Extreme Value Dstrbutons, Technometrcs,, Meeker, W. and Nelson W. (977), Webull Varances and Confdence Lmts by Maxmum Lkelhood for Sngly Censored Data, Technometrcs, 9, Meeker, W. Q., and Escobar, L. A., Statstcal Methods for Relablty Data, John Wley & Sons, Inc., New York, 998. IOGRAPHIES Huaru Guo RelaSoft Corporaton
7 45 S. Eastsde Loop Tucson, AZ, 857 emal: Dr. Huaru (Harry) Guo s the Drector of the Theoretcal Development Department at RelaSoft Corporaton. He receved hs Ph.D. n Systems & Industral Engneerng and M.S. n Relablty & Qualty engneerng; both from the nversty of Arzona. Hs research and publcatons cover relablty areas, such as lfe data analyss, reparable system modelng and relablty test plannng, and qualty areas, such as process montorng, analyss of varance and desgn of experments. In addton to research and product development, he s also part of the tranng and consultng arm and has been nvolved n varous proects from automoble, medcal devce, ol and gas, and aerospace ndustry. He s a certfed relablty professonal (C.R.P), ASQ certfed CQE. He s a member of IIE, SRE and ASQ. Pengyng Nu RelaSoft Corporaton 45 S. Eastsde Loop Tucson, AZ, 857 emal: Pengyng Nu s a research scentst at RelaSoft Corporaton. She s currently playng a key role n the development of Lambda Predct. efore onng RelaSoft, she worked at Texas Instruments where she was nvolved n IC desgn and testng. She receved her Masters degree from the Natonal nversty of Sngapore and M. E. n Electrcal and Computer Engneerng from the nversty of Arzona. She has done extensve work on AC/DC and DC/AC converters. Her current research nterests nclude relablty predcton and physcs of falure for electronc components such as MOSFET, IGT and electronc systems. She s also an ASQ Certfed Relablty Engneer(CRE) and a Certfed Qualty Engneer(CQE). Adamantos Mettas RelaSoft Corporaton 45 S. Eastsde LP Tucson, Arzona 857, SA emal: Adamantos Mettas s the VP of Product Development at RelaSoft Corporaton, spearheadng RelaSoft s nnovatve product development group. In hs prevous poston as RelaSoft s Senor Scentst, he played a key role n the development of RelaSoft's software ncludng Webull++, ALTA, RGA and locksm by dervng statstcal formulatons and models and developng numercal methods for solvng them. Mr. Mettas has traned more than, engneers throughout the world on the subects of Lfe Data Analyss, FMEA, Warranty Analyss, System Relablty, RCM, DOE and Desgn for Relablty, as well as advanced topcs such as Accelerated Testng, Reparable Systems and Relablty Growth. He has publshed numerous papers on these topcs. In addton to tranng, Mr. Mettas s part of RelaSoft s consultng arm and has been nvolved n varous proects across dfferent ndustres, ncludng Ol & Gas, Power Generaton, Automotve, Semconductor, Defense and Aerospace. Mr. Mettas holds an M.S. n Relablty Engneerng from the nversty of Arzona and he s a Certfed Relablty Professonal (CRP). Doug Ogden RelaSoft Corporaton 45 S. Eastsde LP Tucson, Arzona 857, SA emal: Mr. Ogden oned RelaSoft n 997 and has served n a varety of executve management roles. In these capactes he has been nstrumental n the growth and evoluton of the sales organzaton through the creaton of sales plans, ppelne development, sales segmentaton and growth strateges. Mr. Ogden attended the nversty of Mnnesota and s a member of the Socety of Relablty Engneers.