Experimental Designs for Planning Efficient Accelerated Life Tests

Transcription

1 Experimental Designs for Planning Efficient Accelerated Life Tests Kangwon Seo and Rong Pan School of and Decision Systems Engineering Arizona State University ASTR 2015, Sep 9-11, Cambridge, MA 1

2 Outline Part I Mo@va@on Accelerated Life Tests Planning ALT and Op@mality criteria Part II Modeling Weibull model and Likelihood func@on Objec@ve func@on deriva@on Part III ALTopt : R package for op@mal ALT design Crea@ng and evalua@ng op@mal ALT designs Conclusions ASTR 2015, Sep 9-11, Cambridge, MA 2

3 Part I. MOTIVATION ASTR 2015, Sep 9 11, Cambridge, MA 3

4 Accelerated Tests [1] AT tests units at of higher- than- usual stress levels of variables like temperature. AT (QualAT) product weaknesses / study the failure s root cause. Improve the product design or manufacturing process. Quan@ta@ve AT (QuanAT) Obtain informa@on about the distribu@on. Conduct sta@s@cal inference and predic@on of at the use level through failure data. Failure mechanism at the higher stress levels should be the same as one of field failure. In literature, accelerated life tests (ALTs) o`en refer to QuanAT. ASTR 2015, Sep 9-11, Cambridge, MA 4

5 of ALT Non- normal failure - O`en skewed - Exponen@al / Weibull - Lognormal Use- condi@on X X X X Test- condi@on Censored data data Stress Extrapola@on - Use condi@on is outside of test region - Physical / chemical models are needed Censored data - Limited : Right censoring - Periodic inspec@ons : Interval censoring ASTR 2015, Sep 9-11, Cambridge, MA 5

6 Planning ALT To make more accurate and reliable inference on well- planned ALTs are needed for acquiring useful failure experimental designs may not be very helpful due to the of ALT. Many components to be decided [2] Total number of test units and the levels of stress factors and censoring strategy Factor level and the of test units Much more ASTR 2015, Sep 9 11, Cambridge, MA 6

7 Design of ALT A well- designed ALT test plan seeks to achieve some op@mal criteria in the sense of Minimizing the uncertainty (variance) in parameter es@ma@on in model (D- op@mal). Minimizing the uncertainty (variance) in predic@on under the use condi@on. A single point of use condi@on (U- op@mal) Some region of use condi@on (I- op@mal) Given other informa@on, these criteria depend on the design (factor levels and alloca@on). ASTR 2015, Sep 9-11, Cambridge, MA 7

8 Part II. MODELING ASTR 2015, Sep 9 11, Cambridge, MA 8

9 Physical Model Physical models provide valuable clues to the life- stress that is necessary for the Arrhenius model, Inverse power model, Eyring model Generally most physical models propose the log linear between life and natural stress variables. log t 50 = β 0 + β 1 s β k s k e.g., Stress Temperature Humidity Natural stress variable k= ev/k is the Boltzmann s constant. T=temp is the temperature in degrees Kelvin. RH is rela@ve humidity. ASTR 2015, Sep 9 11, Cambridge, MA 9

10 Weibull Model Weibull is commonly used for model because of its flexibility. Let T i be the of i th test unit with stress level x i. T i ~ ind. Weibull( λ i, α), i=1, 2,, n - Scale parameter (Intrinsic failure rate) - Depends on stress level by log linear rela@onship log λ i = x i β - Shape parameter - Related to failure mode - Common to all stress levels The probability density func@on is given by f( t i )= λ i α t i α 1 e λ i t i α h( t i ) R( t i ) ASTR 2015, Sep 9-11, Cambridge, MA 10

11 Likelihood of Data with Right Censoring with right censoring ( t i, c i ), i=1, 2,, n Failure or c i ={ 1, failed &0, censored The likelihood of observa@ons L= i=1 n f( t i ) c i R( t i ) 1 c i = i=1 n h( t i ) c i R( t i ) = i=1 n ( λ i Let λ i t i α = μ i, then L= i=1 n α c i t i c i μ i c i e μ i The kernel of the likelihood func@on of Poisson random variable c i with mean μ i ASTR 2015, Sep 9-11, Cambridge, MA 11

12 Generalized Linear Model Maximizing the likelihood of s w.r.t. is the same as maximizing the likelihood func@on of s which can be formulated as following GLM model. (log link func@on) We are interested in the variance of es@mated parameters. Asympto@c variance- covariance matrix of is given by is design matrix (test loca@on of each test unit). is weight which is dependent on planning values [3] (shape parameter, regression coefficients and ASTR 2015, Sep 9-11, Cambridge, MA 12

13 D- Expression [4],[5] max ξ X (ξ) WX(ξ)/n U- op@mality min ξ x use (X (ξ) WX(ξ)) 1 x use I- op@mality X(ξ) :design matrix W=diag{ σ i 2 } x use :use condition Ω :use region S Ω :area of use region min ξ Ω x use (X (ξ) WX(ξ)) 1 x use d x use / S Ω ASTR 2015, Sep 9-11, Cambridge, MA 13

14 More about Weight matrix Replace by its expected value Define. ASTR 2015, Sep 9-11, Cambridge, MA 14

15 Interval censoring case Model with interval censoring can be constructed in a similar fashion using Binomial GLM model. Observa@ons with interval censoring j=1 j=2 j=k+1 (0, t 1, r i1 ),( t 1, t 2, r i2 ),,( t k,, r i,k+1 ) i=1, 2,, n Beginning and ending of j th interval r ij ={ 1, fail within j th interval &0, not fail With an assump@on that interval have the same length, t, the weight matrix is given by W=diag{ ((j 1) α j α ) 2 t 2α exp {2 x i Dimension: β n(k+1) n(k+1) j α t α e x i β }/1 exp { ((j 1) α j α ) t α e ASTR 2015, Sep 9-11, Cambridge, MA 15 x i β } }

16 Part III. ALTopt : R PACKAGE FOR OPTIMAL ALT DESIGN ASTR 2015, Sep 9-11, Cambridge, MA 16

17 ALTopt Package R is an open- source so`ware for sta@s@cal analysis. ALTopt [6] is an R package for crea@ng and evalua@ng op@mal ALT designs. (hlp://cran.r- project.org/package=altopt) Crea@ng op@mal designs for ALT Graphical display of a design Evaluate the objec@ve func@on value for a given design PV (Predic@on Variance) contour plot FUS (Frac@on of Use Space) plot VDUS (Variance Dispersion of Use Space) plot ASTR 2015, Sep 9 11, Cambridge, MA 17

18 Seong Up Experimental Region Suppose an ALT is conducted with 2 stress factors. Stress variable Use condi@on Test stress region Low High Temperature () Humidity () s 1 = 1/kT =11605/( ) s 2 = ln RH Natural stress variable Temperature () Humidity () Use condi@on Low Test stress region High 18

19 Seong Up Experimental Region Transform the original natural stress variables to the coded variables. s 2 High x 2 Use (1.758, 3.159) Low Use x 1 = s 1 s 1 H / s 1 L s 1 H Low (1, 1) x 2 = s 2 s 2 H / s 2 L s 2 H Design region s 1 High (0, 0) x 1 convert.stress.level(lowstlv, highstlv, actual = NULL, stand = NULL) ASTR 2015, Sep 9 11, Cambridge, MA 19

20 Create ALT Design Suppose we have 100 test units and is units with the right censoring plan. We assume that are distributed (α=1) and the following linear predictor is postulated from previous experiments of similar products. η i = log λ i = x x x 1 x 2 > set.seed(10) > DR <- altopt.rc("d", N = 100, tc = 30, nf = 2, alpha = 1, + formula = ~x1 + x2 + x1:x2, coef = c(0, , , 0.01)) Star@ng with randomly selected 100 ini@al points in the design region, op@miza@on is performed to find the D- op@mal design. ASTR 2015, Sep 9-11, Cambridge, MA 20

21 September 3, 2015 ASTR 2015, Sep 9-11, St. Cambridge, MA 21

22 Create ALT Design > DR $call altopt.rc(opttype = "D", N = 100, tc = 30, nf = 2, alpha = 1, formula = ~x1 + x2 + x1:x2, coef = c(0, , , 0.01)) $opt.design.rounded x1 x2 allocation grouped by rounding $opt.value.rounded [1] $opt.design.kmeans x1 x2 allocation grouped by k- means clustering $opt.value.kmeans [1] ASTR 2015, Sep 9-11, Cambridge, MA 22

23 Create ALT Design > UR <- altopt.rc("u", N = 100, tc = 30, nf = 2, alpha = 1, + formula = ~x1 + x2 + x1:x2, coef = c(0, , , 0.01), + usecond = c(1.758, 3.159)) > IR <- altopt.rc("i", N = 100, tc = 30, nf = 2, alpha = 1, + formula = ~x1 + x2 + x1:x2, coef = c(0, , , 0.01), + uselower = c(1.489, 2.710), useupper = c(2.046, 3.710)) > design.plot(dr$opt.design.rounded, xaxis = x1, yaxis = x2) > design.plot(ur$opt.design.rounded, xaxis = x1, yaxis = x2) > design.plot(ir$opt.design.rounded, xaxis = x1, yaxis = x2) D- op@mal U- op@mal I- op@mal ASTR 2015, Sep 9-11, Cambridge, MA 23

24 Evaluate ALT Designs Compare the values of a factorial design with the D-, U- or I- op@mal design. ## create factorial design > x1 <- c(0, 1, 0, 1) > x2 <- c(0, 0, 1, 1) > allocation <- c(25, 25, 25, 25) > fact <- data.frame(x1, x2, allocation) ## evaluate obj. function value > alteval.rc(fact, "D", tc = 30, nf = 2, alpha = 1, formula = ~x1 + x2 + x1:x2, coef = c(0, , , 0.01)) Op@mality Criteria Objec@ve func@on value Factorial D- op@mal U- op@mal I- op@mal D (max) 16,978 27, U (min) I (min) ASTR 2015, Sep 9-11, Cambridge, MA 24

25 Compare ALT Designs Compare D- and U- designs using the Variance) contour plot. > pv.contour.rc(dr$opt.design.rounded, xaxis = x1, yaxis = x2, tc = 30, nf = 2, alpha = 1, formula = ~x1 + x2 + x1:x2, coef = c(0, , , 0.01), usecond = c(1.758, 3.159)) > pv.contour.rc(ur$opt.design.rounded, xaxis = x1, yaxis = x2, tc = 30, nf = 2, alpha = 1, formula = ~x1 + x2 + x1:x2, coef = c(0, , , 0.01), usecond = c(1.758, 3.159)) PV contour of D- op@mal PV contour of U- op@mal ASTR 2015, Sep 9-11, Cambridge, MA 25

26 Compare ALT Designs Compare a D- op@mal test plan and a U- op@mal test plan using the FUS(Frac@on of Use Space) plot [7]. > fusdr <- pv.fus.rc(dr$opt.design.rounded, tc = 30, nf = 2, alpha = 1, formula = ~x1 + x2 + x1:x2, coef = c(0, , , 0.01), uselower = c(1.489, 2.710), useupper = c(2.046, 3.710)) > fusur <- pv.fus.rc(ur$opt.design.rounded, tc = 30, nf = 2, alpha = 1, formula = ~x1 + x2 + x1:x2, coef = c(0, , , 0.01), uselower = c(1.489, 2.710), useupper = c(2.046, 3.710)) > compare.fus(fusdr, fusur) ASTR 2015, Sep 9 11, Cambridge, MA 26

27 Compare ALT Designs Compare D- and U- designs using the VDUS(Variance Dispersion of Use Space) plot [8]. > vdusdr <- pv.vdus.rc(dr$opt.design.rounded, tc = 30, nf = 2, alpha = 1, formula = ~x1 + x2 + x1:x2, coef = c(0, , , 0.01), uselower = c(1.489, 2.710), useupper = c(2.046, 3.710)) > vdusur <- pv.vdus.rc(ur$opt.design.rounded, tc = 30, nf = 2, alpha = 1, formula = ~x1 + x2 + x1:x2, coef = c(0, , , 0.01), uselower = c(1.489, 2.710), useupper = c(2.046, 3.710)) > compare.vdus(vdusdr, vdusur) ASTR 2015, Sep 9-11, Cambridge, MA 27

28 Conclusions We are able to construct test plans of ALTs with right censoring or interval censoring. criteria are used for increasing the efficiency of a test plan. Engineers can easily create and evaluate ALT op@mal designs using ALTopt package. With uncertain planning values, sensi@vity analysis can be performed [9]. ASTR 2015, Sep 9-11, Cambridge, MA 28

29 References [1] L. A. Escobar and W. Q. Meeker. A review of accelerated test models. Sta@s@cal Sciences, 21(4): , [2] W. B. Nelson. An Updated bibliography of Accelerated Test Plans. Proceedings of the Reliability and Maintainability Symp., , 2015 Request a searchable Word file of the updated bibliography from WNconsult@aol.com. [3] W. Q. Meeker and L. A. Escobar. Sta@s@cal Methods for Reliability Data, volume 314. John Wiley & Sons, [4] E. M. Monroe, R. Pan, C. M. Anderson- Cook, D. C. Montgomery, and C. M. Borror. A generalized linear model approach to designing accelerated life test experiments. Quality and Reliability Engineering Interna@onal, 27(4): , [5] T. Yang and R. Pan. A novel approach to op@mal accelerated life test planning with interval censoring. Reliability, IEEE Transac@ons on, 62(2): , [6] K. Seo and R. Pan. ALTopt: Op@mal Experimental Designs for Accelerated Life Tes@ng, URL hlp://cran.r- project.org/package=altopt. R package version [7] A. Zahran, C. M. Anderson- Cook, and R. H. Myers. Frac@on of design space to assess predic@on capability of response surface designs. Journal of Quality Technology, 35(4): , [8] A. Giovannio- Jensen and R. H. Myers. Graphical assessment of the predic@on capability of response surface designs. Technometrics, 31(2): , [9] E. M. Monroe, R. Pan, C. M. Anderson- Cook, D. C. Montgomery, and C. M. Borror. Sensi@vity analysis of op@mal designs for accelerated life tes@ng. Journal of quality technology, 42(2): , ASTR 2015, Sep 9-11, Cambridge, MA 29

30 Biography Kanwon Seo is currently a Ph.D. student in the School of Compu@ng, Informa@cs, and Decision Systems Engineering at Arizona State University. He received the M.S. degree in Industrial Engineering from Arizona State University. His research interests include data analysis, design of experiments and sta@s@cal learning. Rong Pan is an Associate Professor in the School of Compu@ng, Informa@cs, and Decision Systems Engineering at Arizona State University. He received his Ph.D. in Industrial Engineering from Penn State University in His research interests include data analysis, design of experiments, mul@variate sta@s@cal quality series analysis, and control. He is a senior member of ASQ and IIE, and a member of SRE and IEEE. ASTR 2015, Sep 9-11, Cambridge, MA 30