Statistical analysis of Accelerated life testing under Weibull distribution based on fuzzy theory

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1 Statstcal analyss of Accelerated lfe testng under Webull dstrbuton based on fuzzy theory Han Xu, Scence & Technology on Relablty & Envronental Engneerng Laboratory, School of Relablty and Syste Engneerng, Behang Unversty Xaoyang L, Scence & Technology on Relablty & Envronental Engneerng Laboratory, School of Relablty and Syste Engneerng, Behang Unversty Le Lu, Scence & Technology on Relablty & Envronental Engneerng Laboratory, School of Relablty and Syste Engneerng, Behang Unversty Key Words: accelerated lfe testng; fuzzy theory; lfete evaluaton; Partcle Swar Optzaton; relablty SUMMARY & COCLUSIOS Soetes the accelerated lfe testng (ALT) of soe products stll tae te and t s hard to pleent real-te ontorng on saples to chec whether the saples fal or not. Under ths condton, perodc nspecton s always conducted to ontor the saple perforance, obtanng the falure te data n ALT. But perodc nspecton ay cause probles n soe cases. When the falure s detected durng the nspecton, ts tng pont s recorded by estaton. So the falure te s not accurate snce t ay happen at any te n the adjacent nspecton nterval and cannot be detected edately after the falure happen. The extree case s that the saple fals edately when the nspecton s accoplshed. Thus, there exsts bg uncertanty n falure te data f the nterval s qute long whch wll affect the accuracy of evaluaton results. However, there are soe ethods whch ay help to detect the falure happen ndrectly. For exaple, the fluctuaton or excurson of the output dgtal sgnal of saples ay ndcate the falure to soe degree. Even so, the falure te cannot be recognzed drectly because the judgng crtera of falure of dfferent nspectors ay slghtly vary. Ths leads to the estaton of the exact falure te s not precse. Therefore, the falure te data of ALT s not precse and should be fuzzy data. The lfe estaton of ALT should be a fuzzy nterval value rather than just a sngle nuber. Hence, n ths paper, a statstcal ethod for constant-stress ALT wth Type II censored saples based on fuzzy theory s proposed whch assues that the lfe te of the product follows Webull dstrbuton. 1 ITRODUCTIO Accelerated lfe testng has been nternatonally researched as t saves te and oney than tradtonal lfe testng. Many statstcal ethods of ALT have been proposed by researchers around the world. Maxu lelhood estaton (MLE) s one of coonly used ethod to analyze the falure te data. Fan [1] dscusses a paraeter estaton ethod of the generalzed gaa lfete dstrbuton under the constant ALT based on MLE. Chandra [2] provdes an optu plan for step-stress ALT whle axu lelhood functon s derved. As dscussed above, the falure data of ALT s not precse and nd of epstec uncertanty. Fuzzy theory s one of the ost portant ethods dealng wth uncertanty. Huang [3] proposes a statstcal odel on copettve falure process when the degradaton data s fuzzy. Jahaneh [4] analyzes the syste relablty wth fuzzy Webull dstrbuton. Wu [5] provdes a Bayesan approach to estate syste fuzzy relablty. Vertl [6] partcularly descrbes soe statstcal odels wth fuzzy data. Therefore, fuzzy theory s chosen together wth MLE ethod to establsh a statstcal odel of constant ALT wth Webull dstrbuton. 2 ACCELERATED LIFE TESTIG AF FUZZY THEORY 2.1 The statstcal odel of accelerated lfe testng The statstcal nference of ALT bases on two assuptons. Frstly, the lfete dstrbuton of the product follows Webull dstrbuton under the noral stress level and accelerated stress levels. It can be expressed as t (1) F t 1exp And ts probablty densty functon s 1 t t (2) f t exp In Eq.(1) and Eq.(2), s scale paraeter; s shape paraeter. Secondly, the relatonshp between the scale paraeter, also /15/$ IEEE

2 nown as the characterstc lfe, and the stress s below: ln a bs (3) In Eq.(3),(S ) s a functon defned by the stress nflcted on the saples. In CSALT, assung there are n saples worng at constant but dfferent stress levels. Presung an ALT has accelerated stress levels, denoted as S 1 <S 2 < <S, noral stress level s S 0. At each accelerated stress level S, n specens are run to falure untl r specens faled. The falure te data are called Type II censored data. Mar falure tes of jth saple under ts accelerated stress as t j, =1,2,,, j=1,2,, n. Statstcal odel of constant ALT based on MLE wth Type II censored data has been deeply researched. The jont densty functon s r n! t 1 j La, b, tj exp 1 j1 n r! (4) nr t r exp Eq.(5) s Eq.(4) n logs. ln La, b, (5) r n! ln rln rln 1 ln j 1 t n r! j1 r tj tr n r 1 j1 After further dervaton, Eq.(6)-(8) can be obtaned. Cobnng Eq.(3) and usng Newton Raphson teraton ethod, the paraeter a,b and can be calculated. 0 tj tr n r r0 r tj tr n r r 1 j1 r 1 j1 r r r t 1 j t r (8) ln t ln t n r ln t 0 j j r 1 j1 j1 2.2 Fuzzy theory A fuzzy nuber s descrbed by a characterzng functon, also called ebershp functon (.) whch denotes the degree of ebershp of eleent of the unverse X. x : X 0:1 (9) x X [0,1] An -cut of, wrtten as, s defned as x x x X (10) 0 1 As the value of s settled, a fuzzy nuber can be turned nto a fnte closed nterval as [7] (6) (7) x x, x (11) Consderng that the recorded falure te s not precse and the falure happens between the adjacent nspecton ntervals, such a ebershp functon s chosen n ths paper. t t j j 1 l tj t j gl 1 tj tj t t j j tj tj 1 t t j j r g u 0 else (12) whle g l, g r l tj u tj In Eq.(12), t j represents the exact falure te of saples. represents the recorded falure te of saples by researchers. l and r represent the boundary value of the th nspecton nterval when the t j falure s detected. g l or g u represents the gap between the recorded te and the edge of the nspecton nterval. As the Eq.(12) shows, f s precse, the degree of ebershp of the exact falure te t j s 1. If the recorded te far fro the exact falure te t j, the degree of ebershp approaches to zero. 3 THE OPTIMIZAITO MODEL OF ALT As dscussed n the frst part, the recorded falure te data s fuzzy and has a ebershp functon le Eq.(11). Therefore, all the t j n the Eq.(6)-(8) should be changed nto. 3.1 Paraeter optzaton of ALT based on PSO Partcle Swar Optzaton(PSO) s a nd of optzaton algorth. By usng tes of teratons PSO fnds the optal value when gven an ntal value. In ths paper, the falure te data becoes a fnte closed fuzzy nterval value. When t j changes n ts nterval, the value of the paraeter a,b and wll be change as a result. The extenson theore of fuzzy theory s often used when dealng wth fuzzy nubers and ther ebershp functon. However, accordng to the for of Eq.(6)-(8), the operaton of fuzzy nuber wll be too coplcated and ts ebershp functon s hard to express. Therefore, a nuercal ethod s chosen to obtan proper value of paraeters. Slar ethods have been proposed by Huang [8] and Jahaneh [9]. The process s shown n Fgure 1 as below.

3 Basc setup of PSO PSO begn Input rando ntal value Newton teraton begn Return the results of Newton teraton to PSO Newton teraton end Calculate the ftness of partcles Satsfy the ternal condton No Update the speed and locaton of partcles based on local optu and global optu Output the optzed value of paraeters PSO end Fgure 1-The process of optzaton of ALT Based on the PSO, the process provdes the range of the three paraeters and when gven a -cut level of fuzzy falure data. The -cut level restrans the range of whch represents the uncertanty of falure data. equals to 1 eans the falure data s precse. When the exact falure te s totally unnown, s zero. The core equatons of optzaton are below. Eq.(13) and Eq.(14) just show the way to calculate paraeter. Paraeter and are calculated n the sae way. Results wll be obtaned after tes of teratons cobnng these equatons and PSO., 6 (8) a a a solve Eq t j C T (13) where L a nsolveeq6 (8) t j C T (14) U a axsolveeq6 (8) t j C T C T T, T Fuzzy nterval of characterstc lfe under noral stress level S 0 then wll be gven through Eq.(15) by nvong PSO a second te. % exp a% b % ( S0) (15) Fuzzy nterval of relablty wll also obtaned by PSO. t R (16) t exp 4 CASE STUDY Soe detals wll be dscussed cobnng wth a case study. The data s quoted fro reference [10]. It s a constantstress accelerated lfe testng (CSALT) of soe tantalu electrolytc capactor wth four stress level. Teperature s the senstve stress of saples. In order to estate the lfete and relablty of the product under ts noral worng Yes condton(50), such a CSALT s conducted. Test condtons and results have been presented n Table 1. Test condton and results S 1 =85,n 1 =60, r 1 =26 S 2 =125,n 2 =60, r 2 =33 S 3 =150,n 3 =20, r 3 =12 S 4 =175,n 4 =20, r 4 =14 Table 1 Test condtons and results Falure te of saples(unt: hour) ,445.05,500,600,836,850,867.20,1250,1250, 1750,1750,1750,1750,1750,1750,1993,2050.5,2500, 2500,2500,2608.1,2608.1,2608.1,2609.1,3500,3500 2,11.3,20,22,50,50,52,52,121.45,126.25,127.4, ,146.0,152,152,162,196.3,199.3,200,205.3, ,285.3,315,315,315,317,400,450.45,493,493, 493, ,26.06,32,36.42,42.06,60.10,83.18,92.1,92.1, 100,120, ,18.18,23.48,23.54,24.12,24.24,40.18,48.36, 48.42,48.48,50,50.12,74.24, The reference [10] does not provde the nspecton nterval. As all now, the hgher stress-level s, the shorter s the nspecton nterval. So ths paper assues the value of dfferent ntervals as below to llustrate the ethod. Besdes, the taen te of nspecton tself can be gnored. The nspecton nterval of S 1 s 48 hours and expressed as h 1 =48. And h2=10, h3=2, h4=0.5. The unt s hour. Based on fuzzy theory and the ebershp functon provded, the data can be transferred nto fuzzy nterval data. For exaple, the frst data under S1 s Gven the assuptons above, the nspecton nterval of S1 s 48 hours. So the frst falure s detected n the 6 th nspecton. The exact tng pont s between 240(48*5) and 288. Therefore ts ebershp functon can be expressed as t j t j tj tj tj tj tj else (17) The fuzzy nterval value of paraeter and can be estated by atlab as below. when =0.4 (18) a a, a 14.65, 14,89 b b, b , , 0.952, Accordng to Eq.(15) and PSO, fuzzy characterstc lfe of saples under noral stress can be obtaned as, , (19) If becoes bgger, whch shows that the data s ore precse, the fuzzy nterval wll be saller as below. when =0.7 (20) a a, a 14.7, 14,8 b b, b , , 0.987, Accordng to Eq.(15) and PSO, fuzzy characterstc lfe of saples under noral stress can be obtaned as

4 , , (21) The length of nterval s shorter whch shows a ore defnte range s obtaned. Assung the data s precse whch eans =1, the result of tradtonal statstcal ethod based on MLE s as below. The value s ncluded n the fuzzy ntervals above. when =1 (22) a ; b ; The value of s settled by consderng the accuracy of the falure te data. More accurate the data s, s closer to 1. Test condtons and the dfferences between nspectors also should be easured. Soetes expert coents ay be requred to ae a decson. Therefore, based on the analyss above, the fuzzy relablty should be plotted as curve surface gven a par of value of the characterstc lfe and the shape paraeter ore than a sngle lne. Tae the data when =0.7 as an exaple. Two peces of surface restran the range of the value of fuzzy relablty as te goes. R(relablty) Surface plots of nterval value of R worng hours Fgure 2- Curve surface of fuzzy relablty 5 COCLUSIOS AD FUTURE WORK In ths paper, fuzzy theory s frst used n ALT cobng wth MLE n order to solve a nd of uncertanty proble. As the recorded falure data s not precse, the output of ALT should be nterval estaton whch s ore reasonable and credble. Therefore, based on fuzzy theory and axu lelhood estaton, a nuercal ethod s proposed. Ths ethod can also be used n other statstcal odel of accelerated tests when test data possesses nd of uncertanty. However, when the statstcal odel s coplcated, the output of Newton Raphson teraton ethod s very strct wth ntal value and ay be not accurate enough. Besdes, the teratons ay tae too uch te. More optzaton ethods need to be researched to solve the equatons of MLE n further study. REFERECES 1. Fan Tsa-Hung, Yu Cha-Hsang. Statstcal Inference on Constant Stress Accelerated Lfe Tests under Generalzed Gaa Lfete Dstrbutons. Qualty and Relablty Engneerng Internatonal, 2013, 29(5): Chandra N, Khan M A. Optu Plan for Step-Stress eanlfe of saples/hour Accelerated Lfe Testng Model under Type-I Censored Saples. Journal of Modern Matheatcs and Statstcs, 2013, 7(5-6): Wang Zhong-la, Huang Hong-zhong, Du L. Relablty analyss on copettve falure processes under fuzzy degradaton data. Appled Soft Coputng, 2011, 11(3): Jahaneh E B. Analyzng Syste Relablty Usng Fuzzy Webull Lfete Dstrbuton. Internatonal Journal of Appled, 2014, 4(1): Wu Hsen-Chung. Fuzzy relablty estaton usng Bayesan approach. Coputers & Industral Engneerng, 2004, 46(3): Vertl R. Statstcal ethods for fuzzy data. John Wley & Sons, Zadeh L A. Fuzzy sets. Inforaton and control, 1965, 8(3): Huang Hong-zhong, Sun Zhan-quan. Bayesan relablty analyss for fuzzy lfete data. Fuzzy Sets and Systes, 2006, 157(12): Jahaneh E B. An Evaluaton of the Systes Relablty Usng Fuzzy Lfete Dstrbuton. Journal of Appled Matheatcs, Islac Azad Unversty of Lahjan, 2011, 7(28): Mao Sh-song, Huang Lng-lng. Accelerated Lfe Testng. Scence Press, 1997: BIOGRAPHIES Han XU Scence & Technology on Relablty & Envronental Engneerng Laboratory, School of Relablty and Systes Engneerng, Behang Unversty Bejng , P.R. Chna e-al: xuhan1214@163.co Han Xu receved hs B.E. n Qualty and Relablty Engneerng fro Behang Unversty n Hs research nterests n accelerated test odelng ncludng ADT and ALT, uncertanty analyss. Xaoyang L Scence & Technology on Relablty & Envronental Engneerng Laboratory, School of Relablty and Systes Engneerng, Behang Unversty Bejng , P.R. Chna Cty, State or Provnce, Postal Code, Country e-al: leexy@buaa.edu.cn Xaoyang L receved the Ph. D. degree n syste engneerng of aeronautcs and astronautcs fro Behang Unversty, Bejng, n She joned the departent of syste engneerng, Behang Unversty, as lecturer n Her research nterests nclude relablty testng, accelerated test odelng, lfe predcton and plannng experents. Le Lu Scence & Technology on Relablty & Envronental

5 Engneerng Laboratory, School of Relablty and Systes Engneerng, Behang Unversty Bejng , P.R. Chna Cty, State or Provnce, Postal Code, Country e-al: Le Lu receved hs B.E. n Qualty and Relablty Engneerng fro Behang Unversty n He s currently a Ph.D. student at the School of Relablty and Systes Engneerng, Behang Unversty. Hs research nterests nclude accelerated testng, relablty odelng, uncertanty analyss.