Optimal Design of Accelerated Life Tests with Multiple Stresses

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1 Optimal Desig of Accelerated Life Tests with Multiple Stresses Y. Zhu ad E. A. Elsayed Departmet of Idustrial ad Systems Egieerig Rutgers Uiversity 2009 Quality & Productivity Research Coferece IBM T. J. Watso Research Ctr., Yorktow Heights, NY Jue 3-5,

2 Outlie Motivatio for ALT with multiple stresses Review curret work o multiple stresses ALT Optimal Lati hypercube desigs for ALT Modified simulated aealig algorithm Numerical examples Summary 2

3 Motivatio for ALT with Multiple Stresses Defiitio: ALT is coducted to quickly obtai failure times i order to evaluate ad predict reliability at ormal operatig coditios Most of the previous work o desigig ALT plas uses sigle stress For highly reliable products, it is difficult to obtai failures i a short time with sigle stress Reliability may deped o several stresses operatig simultaeously. Multiple stresses represet realistic field coditios 3

4 Curret Work o Multiple Stresses ALT Two stresses ad two levels for each stress: Nelso (1990) - oe accelerated stress ad oe other factor Escobar ad Meeker (1995) - compromise pla Park ad Yum (1996) - factorial desig Elsayed ad Zhag (2009) - factorial desig Difficult to exted existig methods, because: Multiple stresses ad levels ca result a huge umber of stress-level combiatios (desigs). With limited umber of test uits ad time, how to choose the best or a small umber of optimal desigs? How to spread the desig poits across the desig regio to avoid extreme coditios? 4

5 Proposed Solutio Lati Hypercube Desig Lati hypercube desig (, k): A LHD with k factors ad levels (desig poits). Each factor has levels take values i { 12,, K, } Example, LHD(5, 3) : Volt Temp Humidity Advatages: { Z 1, L, Z 2, L, Z 3 L }: log time Three or more factors ad levels ca be ivestigated { Z 1, H, Z 2, H, Z 3 H } : Other failure modes Optimal LHD ca avoid extreme stress-level combiatios Sigificatly reduce the stress-level combiatios: k to. 5

6 Iter-site distace: Optimal LHD Criterio q=1,2,, correspod to rectagular, Euclidea, ad ifiite distace Example: LHD (, k ): k d( st, ) = sj t j = t ( ) LHD ( 5, 3 ): = ( 5) = Morris ad Mitchell (1995) proposed maximum miimum 1 p iter-site distace criterio: ( 2) Spread the desig poits across the desig regio s φ p j q 1 q 1 = p i = 1 di pairs of iter-site distace 6

7 D-optimality: Optimal ALT Criterio Esure the accuracy of parameter estimatio Maximize determiat of Fisher iformatio matrix of maximum likelihood estimate D-optimality may ot lead to optimal distace measure 7

8 Objective Fuctio A multi-objective formulatio: w det ( Fs ( X) ) ( w ) ( X) U pu, p, L [ ] p pl, Mi 1, 0,1 Det Φ Φ Φ Φ w w is the weight X is the desig matrix det F X is the determiat of the Fisher iformatio matrix Φ = 1 φ is the distace measure p ( s ( )) p Normalize the distace ad D-optimality criteria: Det U is the upper boud of the determiat Φ p, U ad Φ p, Lare the upper ad lower boud of the distace measure 8

9 Propositios Focus o three stresses Cocurretly applyig too may stresses is very demadig for test equipmet Extesio to more stresses is possible For a LHD,3, For a LHD, k, Φ ( F ) Det ( ) 0 det s U 3 3 ( ) ( ) π γ ( γ) 4 2 DetU = , > 1 p, U = ( ) ( ) 1 p k + 2 ( ) ( ) 1 6 Proofs are give i our workig paper [ ) Φ p, L Φp ΦpU,, q, p 1, Φ pl, ( i) 1 = p 2 p i = 1 k i ( 1 p) 9

10 Log-locatio Scale Model Log times to failure Y T is assumed to follow = l( ) ( βx) y μ( βx) 1 y μ fy = ( y; μ( βx), σ) = exp exp < y < σ σ σ The locatio parameter depeds o the stresses through ( βx) μ = β + β x + β x + L+ β x k k Trasformatio Y = μ ( βx) + σ Z Extreme value distributio fz ( z) = exp ( z exp ( z) ), < z < Log likelihood fuctio of failures observed i the m th ru ( β β σ) π ( σ) ( Nπ ) ( Nπ ) l = l L,...,, = N l + z exp z ( ) m m 0 k m m, j m, j j= 1 j= 1 m m 10

11 Fisher Iformatio Matrix Assume equal proportio of test uits for each desig poit F = Nπ F + Nπ F + K+ Nπ F 1 1, j 2 2, j, j j j j F = N 2 σ x1, m L L xk, m ( 1 γ ) m= 1 m= 1 2 x x x x L x x ( 1 γ ) x 1, m M M O M M 2 ( γ ) x x x x 1 km, x m= 1 m= 1 m= 1 m= 1 2 π x1, m xk, m m= 1 m= 1 6 1, m 1, m 2, m 1, m k, m 1, m m= 1 m= 1 m= 1 m= 1 m= 1 km, 1, m km, km, ( 1 γ) ( 1 γ) L L ( 1 γ) + ( γ 1) 2 where γ = is Euler s costat. 11

12 Geeratio of LHD φ p For the distace measure, Morris ad Mitchell (1995) used simulated aealig (SA) to geerate LHDs: Begi with a radom chose LHD Perturb the curret desig D by iterchagig two radomly chose elemets withi a radomly chose colum of the desig matrix If the perturbed matrix D try leads to a improvemet, it is adopted as the ew curret desig from which the ext perturbatio is geerated Otherwise, a replacemet is made with probability { ( D ) ( ) try D t} π = exp φ φ where t is the aealig temperature The algorithm stops either after give iteratios or a tolerace criterio is met 12

13 Modified SA to Geerate LHD for ALT Choose the elemets to iterchage based o probabilities LHD (,3) : 1 x2,1 x3,1 2 M M M x x 2, 3, Maximum determiat of the Fisher iformatio matrix is obtaied 2 2 whe a = x x = ad a = x x = ( + 1) 4 1 1, m 2, m m= 1 Choose colum based o probability ( ) p ( 1) 2 4 ( 1) 2 4, 1, ) a1 + ai + p i = 1,2 2 1, m 3, m m= 1 p p Choose row based o probability Φ (Joseph ad Hug, 2008 p, i Φ p, i i = 1 proposed p p ) φ φ p, i p, i i = 1 p 13

14 Numerical Example 1 LHD (5,3) Testig stress levels for each factor: LHD Volts/cm Temperature (ºC) Relative Humidity(RH) Set w =0.5 (p,q) (5,1) (5,2) LHD Volts/cm 2 ºC RH LHD Volts/cm 2 ºC RH Complete eumeratio & Modified SA

15 Numerical Example 1 Simulatio Give parameter values: β = [ ] σ = 0.94 Stress settigs: optimal desig from (p, q) =(5, 2) Geerate radom failure times: F = 1 exp exp( z l ) ( ti ) ( β0 + β1x1+ β2x2 + β3x3) z ( ) z = ( β β β β ) σ ( ) Maximum likelihood estimate (repeat 100 times): σ ( ) t exp i = 0 + 1x1+ 2x2 + 3x3 + l l 1 rad i βˆ = [ ] σ = 0.91 = [ ] = ˆ σ β σ σ 15

16 Numerical Example 2 LHD (25, 3) 500 iteratios 1500 iteratios Mi 3000 iteratios Util a tolerace criterio is met 16

17 Numerical Example Times Simulatio Cout smaller objective fuctio values from MSA tha those from SA Stop Criterio LHD (10, 3) LHD (25, 3) q = 1 q = 2 q = 1 q = iter. 85.1% 90.2% 98% 95% 1500 iter. 88.3% 91% 98.5% 96.7% 3000 iter. 92% 92.5% 99.33% 98% Tolerace 100% 100% 100% 100% Compare MSA ad SA i terms of objective fuctio values LHD (10,3) Mi Max Mea (p, q) = (15, 2) MSA SA MSA SA MSA SA 500 iter iter iter tolerace LHD (25,3) Mi Max Mea (p, q) = (15, 2) MSA SA MSA SA MSA SA 500 iter iter iter tolerace

18 Summary Multiple stresses ALT pla is developed usig Lati hypercube desig, such that the stress-level combiatios are dramatically reduced The D-optimality ad maximi distace measures are ormalized ad combied to geerate desired LHD for ALT Simulated aealig algorithm is modified to efficietly geerate optimal LHDs. Numerical examples validate the feasibility ad efficiecy of the proposed methodology 18

Mathematical Modeling of Optimum 3 Step Stress Accelerated Life Testing for Generalized Pareto Distribution

Mathematical Modeling of Optimum 3 Step Stress Accelerated Life Testing for Generalized Pareto Distribution America Joural of Theoretical ad Applied Statistics 05; 4(: 6-69 Published olie May 8, 05 (http://www.sciecepublishiggroup.com/j/ajtas doi: 0.648/j.ajtas.05040. ISSN: 6-8999 (Prit; ISSN: 6-9006 (Olie Mathematical