ON-LINE PROCEDURE FOR TERMINATING AN ACCELERATED DEGRADATION TEST

Transcription

1 Statistica Sinica 8(1998), ON-LINE PROCEDURE FOR TERMINATING AN ACCELERATED DEGRADATION TEST Hng-Fwu Yu and Sheng-Tsaing Tseng Natinal Taiwan University f Science and Technlgy and Natinal Tsing-Hua University Abstract: Accelerated degradatin testing (ADT) is a useful technique t extraplate the lifetime f highly reliable prducts under nrmal use cnditins if there exists a quality characteristic f the prduct whse degradatin ver time can be related t reliability. One practical prblem arising frm designing a degradatin experiment is hw lng shuld an accelerated degradatin experiment last fr cllecting enugh data t allw ne t make inference abut the prduct lifetime under the nrmal use cnditin? In this paper, we prpse an intuitively appealing prcedure t determine an apprpriate terminatin time fr an ADT. Finally, we use sme light-emitting dide (LED) data t demnstrate the prpsed prcedure. Key wrds and phrases: Accelerated degradatin test (ADT), degradatin path, highly reliable prduct, terminatin time. 1. Intrductin Traditinally, reliability assessment f new prducts has been based n accelerated life tests (ALTs) that recrd failure and censring times f prducts subjected t elevated stress. Hwever, this apprach may ffer little help fr highly reliable prducts which are nt likely t fail during an experiment f reasnable length. An alternative apprach is t assess the reliability frm the changes in perfrmance (degradatin) bserved during the experiment, if there exists a quality characteristic f the prduct whse degradatin ver time can be related t reliability. Usually, in rder t facilitate bserving the degradatin phenmenn r shrten the degradatin experiment under a nrmal use cnditin, it is practical t cllect the degradatin data at higher levels f stress and, then, carry ut extraplatin in stress t estimate the reliability under nrmal use cnditins. Such an experiment is called an accelerated degradatin test (ADT). Nelsn (1990), chapter 11 and Meeker and Escbar (1993) survey the scant literature n the subject. Carey and Kenig (1991) describe a data-analysis strategy and a mdel-fitting methd t extract reliability infrmatin frm bservatins n the degradatin f integrated lgic devices that are cmpnents in a new generatin f submarine cables.

2 208 HONG-FWU YU AND SHENG-TSAING TSENG In rder t cnduct an ADT efficiently, there are several factrs (fr example, number f stresses, the stress levels, the sample size fr each stress level and the terminatin time, etc.) that need t be cnsidered carefully. Bulanger and Escbar (1994) address the prblem f determining bth the selectin f stress levels and sample size fr each stress level under a pre-determined terminatin (life-testing) time. The results are interesting. Hwever, the terminatin time nt nly affects the cst f perfrming an experiment, but als affects the precisin f estimating a prduct s mean lifetime (MTTF). We use an example (in Sectin 2) t explain why it is mre apprpriate nt t fix the terminatin time in advance. Thus, determining an apprpriate terminatin time fr an ADT is a real challenge fr reliability engineers. Tseng and Yu (1997) prpse a simple rule t determine the terminatin time fr a nn-accelerated degradatin mdel. Hwever, fr highly-reliable prducts, the result can be applied nly t estimate the prduct s MTTF (under the nrmal use cnditin) when the acceleratin factr (AF) is knwn. When the AF is unknwn, we need t cnduct an efficient ADT t estimate the prduct s MTTF. In this paper, by cmbining the apprach f Tseng and Yu (1997) with an ALT mdel, we prpse a prcedure t achieve the abve gal. Finally, we als use sme LED (light emitting dide) data t demnstrate this prcedure. The rest f the paper is rganized as fllws: Sectin 2 gives an explanatin why the terminatin time is s imprtant. Sectin 3 prpses a stpping rule t determine an apprpriate terminatin time fr an ADT. Sectin 4 applies the prpsed prcedure t a numerical example. Sectin 5 cnducts a simulatin study f the prpsed stpping rule. Finally, Sectin 6 addresses sme cncluding remarks. 2. Why the Terminatin Time is Imprtant? Suppse that an ADT f a prduct is cnducted at m higher stress levels: S u S 1 S 2 S m, (1) where S u dentes the nrmal use cnditin. Fr the ith stress level S i,thereare n i devices (items) which are randmly selected fr perfrming a degradatin test. Let G(t, Θ ij ) dente the quality characteristic f the jth item under the stress level S i, which degrades ver time t and Θ ij is a vectr f parameters. Assume that D is a critical value fr the degradatin path. Then the failure time τ ij is defined as the time when the degradatin path crsses the critical degradatin level D. Thus, if Θ ij is knwn, the lifetime f the jth item under S i can be expressed by τ ij = τ(d; Θ ij ). (2)

3 ON-LINE PROCEDURE FOR TERMINATING 209 Fr example, if G(t; Θ ij )=e α ijt β ij,then ( ln D ) 1 β τ ij = ij. (3) α ij Applying an accelerated life test (ALT) mdel, the lifetime distributin under a nrmal use cnditin (say S u ) can then be easily btained. In practical situatins, hwever, Θ ij is unknwn. In additin, due t the measurement errrs, the bserved degradatin path at time t, LP ij (t), can nly be expressed as fllws: LP ij (t) =G(t; Θ ij )+ɛ ij (t), (4) where ɛ ij (t) is the measurement errr term which is assumed t fllw a distributinwithmean0andvarianceσ 2 ɛ. T btain a precise estimate f a prduct s MTTF, the ascertainment f the terminatin time is an imprtant issue t the experimenter. We use the fllwing example fr illustratin. the standardized light intensity time(hur) Figure 1. A typical degradatin path f an LED prduct Example 1. Figure 1 shws a typical degradatin path f an LED prduct. Frm the plt, it is seen that G(t, Θ) =e αtβ is an apprpriate mdel fr the degradatin path. Nw, if the experiment is terminated at 3000 hurs, then

4 210 HONG-FWU YU AND SHENG-TSAING TSENG the MLEs fr α and β are ˆα= and ˆβ= Hwever, if the experiment is terminated at 8000 hurs, then ˆα= , and ˆβ= Assume that D=0.50. Then the crrespnding estimated lifetimes are and hurs, respectively. It is clear that the terminatin time has a significant impact n the precisin f estimating a prduct s lifetime. Fr the ADT case, we nw prvide a three-dimensinal plt fr illustratin. In Figure 2, suppse that the experiment is cnducted up t the time t l. Then, based n the bserved data {(t k,lp ij (t k ))} l k=1, the least squares estimatr (LSE) f Θ ij and the crrespnding jth prduct s lifetime (under S i ) can be btained. Then, by using a statistical life-stress ALT mdel, we can extraplate t btain the MTTF under the nrmal use cnditin S u. LetMTTF(l) ˆ dente the estimated MTTF when the ADT is cnducted up t the time t l. Frm the plts f { MTTF(l)} ˆ l 1, it is seen that the curve (path) will scillate drastically at the beginning; hwever, as the terminatin time t l increases, mre data are cllected and the path f MTTF(l) ˆ appraches an asymptte. Life Time Testing Time Stress Level Figure 2. A typical trend f the estimatrs f MTTF under nrmal use cnditin fr an ADT. Frm Figure 2, it is bvius that the experiment can be terminated nly if the sequence MTTF(l) ˆ is cnvergent. Hwever, ne usually needs t cnduct a very lng life-testing time t achieve a cnvergent value. This is impractical fr experimenters. In the fllwing sectin, we prpse an intuitive prcedure t determine an apprpriate terminatin time fr an ADT.

5 ON-LINE PROCEDURE FOR TERMINATING Determining the Terminatin Time fr an ADT The prcedure fr determining an apprpriate terminatin time fr an ADT cnsists f three majr steps labelled (A) t (C) as fllws: (A) Use the degradatin paths t estimate the lifetimes f devices under each testing stress. Suppse that an ADT is cnducted up t the time t l. Based n the degradatin data {(t k,lp ij (t k ))} l k=1, the least squares estimatr (LSE) ˆΘ ij (l) fθ ij can be btained by minimizing SSE(Θ ij )= l {LP ij (t k ) G(t k ; Θ ij )} 2 (5) k=1 and the crrespnding lifetime τ ij can be estimated by [ ˆτ ij (l) =τ D; ˆΘ ] ij (l). (6) (B) Find a suitable life-stress mdel and use an ML prcedure t estimate the MTTF f the device under S u. Applying an ALT mdel t extraplate the lifetime distributin under nrmal use cnditins requires the fllwing steps: 1. use prbability plts t assess the lifetime distributin f {ˆτ ij (l)} n i j=1, fr all 1 i m; 2. use scatter plts f {ˆτ ij (l)} n i j=1, 1 i m, t determine a suitable life-stress relatinship; and 3. use an ML prcedure t estimate the unknwn parameters in a suitable lifestress mdel and then the MLE f the prduct s MTTF at nrmal use cnditin S u can be btained. (C) Investigate the limiting prperty f MTTF(l) ˆ and prpse an apprpriate terminatin time. Intuitively, the grwth trend f MTTF(l) ˆ may scillate drastically at the beginning. As t l increases, the grwth trend will cnverge. Assume that l 0 is a starting pint at which { MTTF(k)} ˆ l k=l 0 has a cnvergent pattern. A cnvergent pattern is indicated by ne f the fllwing three cases: (1) mntnically increasing t a target; (2) mntnically decreasing t a target; and (3) slightly scillating arund a target value. Due t the asympttic prperty, there exists a sigmidal grwth curve f l (t) which fits { MTTF(k)} ˆ l k=l 0 (Seber and Wild (1989), Chapter 7). T btain a mre precise estimatr f MTTF, we can define an asympttic MTTF as f l ( )(= lim t f l (t)). The physical meaning f f l ( ) is that the predicted prduct s MTTF will cnverge asympttically t this value

6 212 HONG-FWU YU AND SHENG-TSAING TSENG when the experiment is cnducted up t the time t l. Obviusly, f l ( ) prvides a better estimatr than MTTF(l). ˆ T measure the relative rate f change f the asympttic mean lifetime, we cnsider the fllwing h-perid mving-average: ρ(l) = 1 { l h 1 f k( ) } f k=l h+1 k 1 ( ). (7) Obviusly, when h =1,ρ(l) reduces t a ne-perid change rate f the asympttic mean lifetime. T avid the irregular pattern f the relative change rate, we chse h = 3 in this study. Thus, a rule fr terminating the experiment can be stated as fllws: t l is an apprpriate terminatin time if ρ(m) ε, m l, where ε is an allwable tlerance which is cmmnly specified by the experimenters. Nw, we state an algrithm t summarize the abve prcedure. Algrithm fr determining an apprpriate terminatin time Step 0. At the beginning, arbitrarily chse l = 4 as a starting pint. Step 1. Use Equatins (5) and (6) t cmpute the estimated lifetime ˆτ ij (l) f the jth item under the stress level S i,1 j n i,1 i m. Step 2. Use scatter plts t assess the life-stress relatinship and cmpute the MLE fr MTTF. Step 3. Plt the grwth trend f { MTTF(k)} ˆ l k=2. If there exists a cnvergent pattern g t Step 4. Otherwise, let l = l +1andgtStep1. Step 4. Chse a suitable starting pint l 0 such that the plt f { MTTF(k)} ˆ l k=l 0 has a cnvergent trend. Then, find a suitable functin f l (t) t fit { MTTF(k)} ˆ l k=l 0 and cmpute f l ( ). Step 5. Cmpute ρ(l). If ρ(m) ε, m l, thent l is an apprpriate terminatin time. Otherwise, let l = l +1andgtStep1. In the next sectin, we use a numerical example t illustrate the prcedure. 4. A Numerical Example Light emitting dides (LEDs) have becme widely used in a variety f fields. The fields f applicatin range frm cnsumer electrnics t ptical fiber transmissin systems. Very-high-reliability is especially required in ptical fiber transmissins. Thus, designing an efficient experiment t estimate its lifetime is a challenge t the prducers. Frm engineering knwledge, electric current is a suitable accelerated variable fr LED prducts (see Ralstn and Mann (1979)); s, three higher stress

7 ON-LINE PROCEDURE FOR TERMINATING 213 levels, S 1 =10mA,S 2 =20mA,andS 3 = 30 ma, are carefully chsen t perfrm an ADT. The gal is t estimate the prduct s MTTF under nrmal use cnditins (say, 5 ma). There are n 1 = 16, n 2 = 14, and n 3 =18itemswhich are randmly selected fr perfrming an ADT under 10 ma, 20 ma, and 30 ma, respectively. A key quality characteristic f LED is its light intensity. It degrades ver time. Let LP ij (t) dente the bserved standardized light intensity f the jth LED under S i. Figure 3 shws the degradatin paths f the standardized light intensity f LEDs fr these three stress levels LP 2j (t) LP 1j (t) time(hur) (a) time(hur) (b) LP 3j (t) time(hur) (c) Figure 3. (a), (b), and (c) are the sample degradatin paths under 10 ma, 20 ma, and 30 ma, respectively. The experiment was cnducted up t 9998 hurs fr each stress. A practical decisin that the experimenter faces is: Is 9998 hurs lng enugh t prvide a precise estimatin fr the prduct s MTTF? If the testing time is lng enugh, what is the mst apprpriate terminatin time? Next, we apply the prpsed methd t address this prblem.

8 214 HONG-FWU YU AND SHENG-TSAING TSENG (A) Estimate the lifetimes f devices under each testing stress Figure 4 is a plt f lg( lg LP ij (t)) vs lg t. Frm the linear patterns, it is seen that G(t; Θ ij )=G(t; α ij,β ij )=e α ij t β ij is an apprpriate mdel t describe the LED data ln( ln LP 1j ) ln t (a) ln( ln LP 2j ) ln t (b) ln( ln LP 3j ) ln t (c) Figure 4. (a), (b) and (c) are the plts f ln( ln LP ij )vslnt fr10ma, 20 ma, and 30 ma, respectively. Basednthebservatins{(t k,lp ij (t k ))} l k=1 and Equatin (5), the LSEs (ˆα ij (l), ˆβ ij (l)) f (α ij,β ij ) can be cmputed. Then the lifetimes {ˆτ ij (l)} n i j=1 can als be btained by the fllwing equatin: ˆτ ij (l) = [ ] 1 ln D ˆβ ij (l). (8) ˆα ij (l) (B) Find a suitable life-stress relatin and use an ML prcedure t estimate prduct s MTTF Figure 5 shws tw typical lgnrmal prbability plts f {ˆτ 1j (l)} 16 j=1, {ˆτ 2j (l)} 14 j=1,and{ˆτ 3j(l)} 18 j=1 fr l = 46 (7984 hurs) and l = 58 (9998 hurs). It is seen that the lgnrmal distributin is an apprpriate mdel t fit the

9 ON-LINE PROCEDURE FOR TERMINATING 215 lifetime data. Besides, the patterns f three apprximately parallel lines in these prbability plts imply that the scale parameters are equal Cumulative Percent ma ma ma hurs (a) Cumulative Percent ma ma ma hurs (b) Figure 5. (a) and (b) are the lgnrmal prbability plts f {ˆτ ij (l)} ni j=1, i =1, 2, 3, fr l =46andl = 58, respectively. ln ˆτ ij (l) ln ˆτ ij (l) ln ma (a) ln ma (b) Figure 6. (a) and (b) are the scatter plts f ln ˆτ ij (l) vslnmafrl =46and l = 58, respectively. Furthermre, frm the lg-lg scale scatter plts shwn in Figure 6, it is seen that the inverse-pwer relatinship is an apprpriate mdel t describe the life and current relatin. Hence, the lgnrmal-inverse pwer is a suitable life-stress mdel. Let ˆµ l and ˆσ l dente the MLEs f the lcatin and scale parameters f lg lifetime under the nrmal use cnditin 5 ma. The ˆµ l,ˆσ l,andmttf(l) ˆ fr 4 l 58 are listed in Table 1. Figure 7 shws the grwth trends f { MTTF(l)} ˆ 58 l=4.

10 216 HONG-FWU YU AND SHENG-TSAING TSENG Table 1. The estimates ˆµ l,ˆσ l, ˆ MTTF(l), ˆf l ( ), ρ(l), and ρ (l) l time t l ˆµ l ˆσ l ˆ MTTF(l) ˆfl ( ) ρ(l) ρ (l) (hurs)

11 ON-LINE PROCEDURE FOR TERMINATING 217 MTTF(l) ˆ ρ(l) time(hur) l= time(hur) Figure 7. The trend f { MTTF(l)} ˆ 58. Figure 8. The trend f ρ(l). (C) Investigate the limiting prperty f MTTF(l) ˆ and determine an apprpriate terminatin time. Observing Figure 7, it is seen that the MTTF(l) ˆ curve changes drastically befre t 34 = 5976 hurs. After t 34, there appears an expnentially decreasing pattern and the curve f MTTF(k) ˆ levels ff after t 42 = 7312 hurs. Hence, we use the fllwing grwth curve t describe { MTTF(k)} ˆ l k=34 fr l 42: f l (t) =a l + e (b l+c l t). (9) Obviusly, f l ( ) = lim f l (t) =a l. (10) t Using the methd f nn-linear least squares, we btain the asympttic mean lifetime â l = ˆf l ( ) and the value ρ(l). The results are shwn in Clumns 6 and 7 f Table 1. Figure 8 als shws the plt f ρ(l). Frm Table 1, it is seen that the estimated asympttic mean lifetime is near hurs if the experiment is cnducted up t 9998 hurs. Besides, frm Figure 8, we can btain a reasnable estimate f MTTF within 1% errr if the experiment time is cnducted at least 7480 hurs (which is abut 11% f the prduct s MTTF). 5. A Simulatin Study f the Prpsed Rule The prpsed stpping rule is very intuitive. Due t the cmplexity f the mdel, it is nt easy t prvide analytical supprt fr this rule. Instead, we

12 218 HONG-FWU YU AND SHENG-TSAING TSENG cnducted a simulatin study t investigate the perfrmance f this rule. Assume that the degradatin path LP ij (t) satisfies equatin (4), where G ij (t) =e α ijt β ij and ɛ ij (t) fllws N(0,σɛ 2 ). In rder t cnduct a simulatin study, we specify the jint distributin f (α ij,β ij ), 1 j n i, 1 i 3. Then, we use the terminatin time f 9998 hurs as a benchmark t estimate these values. The LSEs ( αˆ ij, β ˆ ij )f(α ij,β ij ) have the fllwing apprximate relatinships: where and ln β ˆ ij = p i1 + p i2 αˆ ij, αˆ ij (α il,α ir ), ( , ), fr i=1, (p i1,p i2 )= ( , ), fr i=2, ( , ), fr i=3, (0.3127, ), fr i=1, (α il,α ir )= (0.4636, ), fr i=2, (0.2927, ), fr i=3. In additin, the R 2 values fr these three mdels are , and , respectively. Thus, the fllwing mdel is apprpriate fr describing the relatinship between α ij and β ij : ln β ij = p i1 + p i2 α ij + η ij, α ij (α il,α ir ), (11) where η ij is N(0,ση). 2 Frm Sectin 4, we btain σ ɛ 0.01 and σ η Thus, we chse varius cmbinatins f σ η =(1+δ 1 ) and σ ɛ =(1+ δ 2 ) 0.01 (where 5% δ 1 5% and 20% δ 2 20%) fr the simulatin study. Set n 1 = 16, n 2 = 14, and n 3 = 18, the sample sizes used in the example f Sectin 4. Nw, the simulatin prcedure is summarized as fllws: Fr 1 j n i,1 i 3, 1. Generate (α ij,β ij ) frm Equatin (11). 2. Generate a degradatin path {LP ij (t k )} 58 k=1 frm Equatin (4). 3. Use the prcedure given in Sectin 3 t estimate {τ ij } and the crrespnding MTTF under nrmal use cnditins. 4. Determine the terminatin time t l and the crrespnding asympttic mean lifetime ˆf l ( ) with a tlerance errr ε =0.01. Fr each cell f (δ 1,δ 2 ), we cnduct 100 trials and the fllwing quantities are cmputed: M f : the sample mean f asympttic mean lifetime { ˆf l ( )}; S f : the standard errr f asympttic mean lifetime { ˆf l ( )}; φ tl : the sample mean f terminatin time {t l }. These values are given in Table 2. Frm the results, it is seen that:

13 ON-LINE PROCEDURE FOR TERMINATING The value f M f in each cell is very clse t hurs (the asympttic mean lifetime which was btained in Sectin 4). The largest abslute errr is less than 3.5%. It shws the prpsed stpping rule is quite rbust t variatin f δ 1 and δ The values f S f are mderately affected by the values f δ 1 and δ 2.Thus,the values f σ ɛ and σ η have a mderate impact n the precisin f the asympttic mean lifetime. 3. The values f φ tl are less than 7480 hurs (the terminatin time which was btained in Sectin 4). It means the terminatin time f the simulatin data is shrter than that f the real LED data. This may be due t the reasn that the real LED data in Sectin 4 fluctuate mre irregularly than ur simulatin data. Table 2. The values f M f, S f,andφ tl under varius cmbinatins f (1 + δ 1 ) and (1 + δ 2 ) 0.01 δ 1-5% 0% +5% M f = M f = M f = % S f = S f = S f = φ tl = φ tl = φ tl = M f = M f = M f = % S f = S f = S f = φ tl = φ tl = φ tl = M f = M f = M f = % S f = S f = S f = φ tl = φ tl = φ tl = Cncluding Remarks Determining an apprpriate terminatin time fr cnducting an ADT is an imprtant decisin prblem fr experimenters. By mdifying Tseng and Yu (1997), we prpse an intuitive methd t achieve the abve gal. The methd cnsists f using the traditinal ALT and ML prcedures t estimate the unknwn parameters and MTTF f the device under a nrmal use cnditin. Finally, an apprpriate terminatin time is determined by using the limiting prperty f the estimatr f MTTF. Finally, sme cncluding remarks abut the methd are as fllws: (1) The prpsed methd prvides the decisin maker an n-line real-time infrmatin abut the prduct lifetime. It assesses the lifetime distributin f the prduct at each testing time. Thus, sme imprtant reliability measures, such as MTTF, hazard functin and pth percentile under the nrmal use cnditins can be easily btained. Taking the LED data mentined abve, fr example,

14 220 HONG-FWU YU AND SHENG-TSAING TSENG if the experiment is terminated at t 46 = 7984 hurs and the decisin-maker wishes t estimate the 5th percentile f the prduct s lifetime, then, frm Table 1, we have ˆµ 46 = and ˆσ 46 = Thus, the 5th percentile f the prduct s lifetime is hurs. (2) This methd als prvides the decisin-maker with a simple criterin t measure the difference between the estimated MTTF and the asympttic mean lifetime. It can be expressed as fllws: ρ (l) = 1 MTTF(l) ˆ ˆf l ( ). (12) Clumn 8 f Table 1 lists the values f ρ (l). It shws that the differences are nt significant (less than 3%) if the experiment is cnducted ver 7816 hurs. (3) Althugh there is n analytical supprt fr the prpsed stpping rule, we cnducted a simulatin study t assess its perfrmance. The results in Table 2 indicate that the prpsed rule is quite rbust in estimating the asympttic mean lifetime. Acknwledgement We deeply appreciate the valuable cmments by the referees. Als, the helpful cmments by the Chair Editr have made the paper mre readable. Besides, we thank Mr. Tseng, M. and LITEON crpratin fr kindly prviding the LED data set. References Bulanger, M. and Escbar, L. A. (1994). Experimental design fr a class f accelerated degradatin tests. Technmetrics 36, Carey, M. B. and Kenig, R. H. (1991). Reliability assessment based n accelerated degradatin: A case study. IEEE Trans. Reliability 40, Lu, C. J. and Meeker, W. Q. (1993). Using degradatin measures t estimate a time-t-failure distributin. Technmetrics 35, Meeker, W. Q. and Escbar, L. A. (1993). A review f recent research and current issues in accelerated testing. Internat. Statist. Rev. 61, Nelsn, W. (1990). Accelerated Testing: Statistical Mdels, Test Plans, and Data Analysis. Jhn Wiely, New Yrk. Ralstn, J. M. and Mann, J. W. (1979). Temperature and current dependence f degradatin in red-emitting GaP LED s. J. Appl. Phys. 50, Seber, G. A. F. and Wild, C. J. (1989). Nnlinear Regressin. Jhn Wiley, New Yrk. Tseng, S. T. and Yu, H. F. (1997). A rule fr terminating degradatin experiments. IEEE Trans. Reliability 46, Department f Industrial Management, Natinal Taiwan University f Science and Technlgy, Taipei, Taiwan. Institute f Statistics, Natinal Tsing Hua University, Hsinchu 30043, Taiwan. sttseng@stat.nthu.edu.tw (Received Octber 1995; accepted May 1997)