Design of Optimum Reliability Test Plans under Multiple ... - CiteSeerX

Design of Optimum Reliability Test Plans under Multiple Stresses Elsayed A. Elsayed and Hao Zhang Department of Industrial and System Engineering, Rutgers University 96 Felinghuysen Road, Piscataway, NJ 08854-8018 Phone: 732-445-3859, Fax: 732-445-5467 [email protected], [email protected] Abstract: Accelerated life testing (ALT) is used to obtain failure time data quickly under high stress levels in order to predict product life and performance under the design stress. Most of the previous work on designing ALT plans is focused on the application of a single stress. However, as components or products become more reliable due to technological advances, it becomes more difficult to obtain a significant amount of failure data within a reasonable amount of time using single stress only. Multiple-stresses ALTs have been employed as a means of overcoming such difficulties. In this paper, we design optimum multiple stresses ALT plans based on the proportional hazards model. We do not make assumptions about the failure distribution of the test units. We develop efficient multiple stress plans using the proportional hazards model and the proportional odds model. We then validate the reliability estimates predictions of the two approaches. Keywords: Accelerated Life Testing, Test Plans, Multiple Stresses, and Proportional Hazards Model 1 – Introduction Accelerated life testing (ALT) is used to obtain failure time data quickly under high stress levels in order to predict product reliability performance under the design stress. ALT plans are used to design the ALT to increase the reliability prediction accuracy by determining the optimum stress levels, the number of units at every stress level, and the duration of the test. Most of the previous work on designing ALT plan is focused on the application of a single stress. However, as components or products become more reliable due technological advances, it becomes more difficult to obtain a significant amount of failure data within a reasonable amount of time using single stress only. Multiple-stresses ALTs have been employed as a means of overcoming such difficulties. For instance, Kobayashi et al. (1978), Minford (1982), Mogilevsky and Shirn (1988), and Munikoti and Dhar (1988) use two stresses to test capacitors, and Weis et al. (1988) employ two stresses to estimate the lifetime of silicon photodetectors. Unlike the case of the single stress ALT, little work has been done on designing multiplestresses accelerated life testing plans. Escobar and Meeker (1995) develop statistically optimal and practical plans with two stresses with no interaction between them. However, if prior information does not support the nonexistence of interaction or if the so-called sliding level technique cannot be employed to avoid the potential interaction, then the analysis based on the main effects only could lead to serious bias in estimation.

Park and Yum (1996) develop ALT plans in which two stresses are employed with possible interaction between them. The lifetime distribution of test units is assumed to be exponential. In this paper, we design two-stresses ALT plans based on the proportional hazards (PH) model. We also propose multi-stresses ALT plans based on the proportional odds model. We do not make assumptions about the failure distribution of the test units. The plans are optimized such that the error in reliability prediction at normal operating condition is minimized. 2 – Two Stresses ALT Plans Based on PH Model 2.1 –Assumptions Figure 1 shows the candidate test combinations for the two-stress ALT plans. The stresses z1D and z2 D are the design stresses corresponding to stresses z1 and z2 respectively. Stress level zlm ( l = 1, 2 and m = 0,1 ) represents the type l stress at the m level. (z1Upper, z2Upper)

z2 High (z21)

Low (z20)

(z1D, z2D)

p01

p11

p00

p10

Low (z10)

High (z11)

z1

Figure 1: Candidate test combinations in two-stress ALT plans We assume the following model for the accelerated life test with two stresses. 1. Two stresses z1 and z2 are used in the test. 2. The proportional hazards model is employed to relate the reliability performance under different stress levels, it is expressed as:

λ (t ; z1 , z2 ) = λ0 (t ) exp( β1 z1 + β 2 z2 )

3. The baseline hazard function λ0 (t ) is quadratic

λ0 (t ) = γ 0 + γ 1t + γ 2t 2 γ 0 , γ 1 and γ 2 are unknown parameters. 4. Four candidate test combinations are considered, as shown in Figure 1. The upper bounds of each stress are pre-specified, whereas the two levels of each stress are to be optimally determined. 5. The total number of test units n is given. The proportion of units allocated to the ith level of the first stress z1 and jth level of the second stress z2 is denoted by pij ( i = 0,1; j = 0,1 ) and will be optimally determined. 6. The lifetimes of test units are s-independent. 7. The test is terminated at a pre-specified censoring time τ . The proposed factorial arrangement of test units shown in Figure 1 may not be statistically optimal; however, this arrangement is motivated by the actual practice of reliability engineers. It enables one to conduct the entire test simultaneously by utilizing the available equipment in an efficient manner, which can lead to significant savings in completion time. In addition, it allows for testing of interactions after the data are collected. The hazard function λ (t ; z1 , z2 ) is obtained by substituting λ0 (t ) into the PH model as

λ (t ; z1 , z2 ) = (γ 0 + γ 1t + γ 2t 2 ) exp( β1 z1 + β 2 z2 )

(1)

We obtain the corresponding cumulative hazard function Λ (t ; z1 , z2 ) , reliability function R (t ; z1 , z2 ) and density function f (t ; z1 , z2 ) respectively as follows: t

t

Λ (t; z1 , z2 ) = ∫ λ (u )du = ∫ (γ 0 + γ 1u + γ 2u 2 ) exp( β1 z1 +β 2 z2 )du 0

= (γ 0t +

0

γ 1t 2

2

+

γ 2t 3 3

)e

(2)

β1z1 + β 2 z2

R(t ; z1 , z2 ) = exp(−Λ (t ; z1 , z2 )) = exp(−(γ 0t +

γ 1t 2 2

+

γ 2t 3 3

)e β1z1 + β2 z2 )

(3)

f (t ; z1 , z2 ) = λ (t ; z1 , z2 ) R (t ; z1 , z2 ) = (γ 0 + γ 1t + γ 2t 2 )e β1z1 + β 2 z2 exp(−(γ 0t +

γ 1t 2 2

+

γ 2t 3 3

)e β1z1 + β 2 z2 )

(4)

2.2 – The Log likelihood Function The log likelihood of an observation t (time to failure) under two stress levels z1 and z2 with Type I censoring is given below. We define the indicator function I = I (t ≤ τ ) in terms of the censoring time τ as: 1 I = I (t ≤ τ ) =  0

if t ≤ τ , failure observed before time τ , if t > τ , censored at time τ .

The log likelihood of a Type I censored observation at stresses z1 and z2 is: ln L(t ; z ) = I ln(λ (t ; z )) − Λ (t ; z ) = I ln(γ 0 + γ 1t + γ 2t 2 ) + I β1 z1 + I β 2 z2 − (γ 0t +

γ 1t 2 2

+

γ 2t 3 3

)e( β1z1 + β 2 z2 )

(5)

Suppose that the ith observation ti corresponds to a value zi and the corresponding log likelihood is li. Then the sample likelihood l0 for n independent observations is: l0 = l1 + l2 + " + ln 2.3 – Maximum Likelihood Estimation For a single observation, the five first partial derivatives with respect to the model parameters are:

∂ ln L(t ; z ) I = − te( β1z1 + β 2 z2 ) 2 ∂γ 0 γ 0 + γ 1t + γ 2t

(6)

∂ ln L(t ; z ) It t 2 ( β1z1 + β 2 z2 ) = − e γ 0 + γ 1t + γ 2t 2 2 ∂γ 1

(7)

∂ ln L(t ; z ) It 2 t 3 ( β1z1 + β 2 z2 ) = − e ∂γ 2 γ 0 + γ 1t + γ 2t 2 3

(8)

∂ ln L(t ; z ) γ γ = Iz1 − z1 (γ 0t + 1 t 2 + 1 t 3 )e( β1z1 + β 2 z2 ) ∂β1 2 3

(9)

∂ ln L(t ; z ) γ γ = Iz2 − z2 (γ 0t + 1 t 2 + 1 t 3 )e( β1z1 + β2 z2 ) ∂β 2 2 3

(10)

Summing Eq. (6) through (10) over all test units and setting them equal to zero will provide the maximum likelihood estimates for the model parameters.

Suppose that we only consider the correlation among γ 0 , γ 1 and γ 2 , the second-partial derivatives are as shown below:

∂ 2 ln L(t ; z ) I =− 2 (γ 0 + γ 1t + γ 2t 2 ) 2 ∂γ 0

(11)

∂ 2 ln L(t ; z ) It 2 = − 2 (γ 0 + γ 1t + γ 2t 2 )2 ∂γ 1

(12)

∂ 2 ln L(t ; z ) It 4 =− 2 (γ 0 + γ 1t + γ 2t 2 )2 ∂γ 2

(13)

∂ 2 ln L(t ; z ) It =− ∂γ 0 ∂γ 1 (γ 0 + γ 1t + γ 2t 2 ) 2

(14)

∂ 2 ln L(t ; z ) It 2 =− ∂γ 0 ∂γ 2 (γ 0 + γ 1t + γ 2t 2 ) 2

(15)

∂ 2 ln L(t ; z ) It 3 =− ∂γ 1∂γ 2 (γ 0 + γ 1t + γ 2t 2 ) 2

(16)

∂ 2 ln L(t ; z ) γ γ 2 = − z1 (γ 0t + 1 t 2 + 2 t 3 )e( β1z1 + β2 z2 ) 2 2 3 ∂β1

(17)

∂ 2 ln L(t ; z ) γ γ 2 = − z2 (γ 0t + 1 t 2 + 2 t 3 )e( β1z1 + β 2 z2 ) 2 2 3 ∂β 2

(18)

Eqs. (11) through (18) are given in terms of the random quantities I and z and the model parameters. The elements of the Fisher information matrix for an observation are the negative expectations of the above equations: τ  ∂ 2 ln L(t ; z )  1 γ γ E − e β1z1 + β 2 z2 exp(−(γ 0t + 1 t 2 + 2 t 3 )e β1z1 + β 2 z2 )dt =  2 ∫ 0 γ + γ t + γ t2 2 3 ∂γ 0   0 1 2

(19)

τ  ∂ 2 ln L(t ; z )  t2 γ γ = E − e β1z1 + β 2 z2 exp(−(γ 0t + 1 t 2 + 2 t 3 )e β1z1 + β 2 z2 )dt  2 ∫ 0 γ + γ t + γ t2 2 3 ∂γ 1 0 1 2  

(20)

τ  ∂ 2 ln L(t ; z )  t4 γ γ = E − e β1z1 + β 2 z2 exp(−(γ 0t + 1 t 2 + 2 t 3 )e β1z1 + β 2 z2 )dt  2 2 ∫ 0 γ +γ t +γ t 2 3 ∂γ 2 0 1 2  

(21)

τ  ∂ 2 ln L(t ; z )  t γ γ E − e β1z1 + β 2 z2 exp(−(γ 0t + 1 t 2 + 2 t 3 )e β1z1 + β 2 z2 )dt  = ∫0 2 ∂γ 0γ 1  γ 0 + γ 1t + γ 2t 2 3 

(22)

τ  ∂ 2 ln L(t ; z )  t2 γ γ E − e β1z1 + β 2 z2 exp(−(γ 0t + 1 t 2 + 2 t 3 )e β1z1 + β 2 z2 )dt  = ∫0 2 ∂γ 0γ 2  γ 0 + γ 1t + γ 2t 2 3 

(23)

τ  ∂ 2 ln L(t ; z )  t3 γ γ E − e β1z1 + β 2 z2 exp(−(γ 0t + 1 t 2 + 2 t 3 )e β1z1 + β 2 z2 )dt  = ∫0 2 γ 0 + γ 1t + γ 2t 2 3 ∂γ 1γ 2  

(24)

 ∂ 2 ln L(t ; z )  γ γ 2 ∞ (γ 0t + 1 t 2 + 2 t 3 )e β1z1 + β 2 z2 f (t ; z1 , z2 )dt = E − z  1 2 ∫ 0 2 3 ∂β1  

(25)

 ∂ 2 ln L(t ; z )  γ 1 2 γ 2 3 β z +β z 2 ∞ E −  = z2 ∫ 0 (γ 0t + t + t )e 1 1 2 2 f (t ; z1 , z2 )dt 2 2 3 ∂β1  

(26)

The Fisher information matrix F for all units under test is expressed as F = np00 F00 + np01 F01 + np10 F10 + np11 F11 where n p00 p01 p10 p11 F00 F01 F10 F11

total number of test units placed on test proportion of test units allocated to test combination (z10, z20) proportion of test units allocated to test combination (z10, z21) proportion of test units allocated to test combination (z11, z20) proportion of test units allocated to test combination (z11, z21) Fisher’s information matrix for a unit at test combination (z10, z20) Fisher’s information matrix for a unit at test combination (z10, z21) Fisher’s information matrix for a unit at test combination (z11, z20) Fisher’s information matrix for a unit at test combination (z11, z21)

which is a function of γ 0 , γ 1 , γ 2 , β , zlm (l = 1, 2; m = 0,1) and pij (i = 0,1; j = 0,1) . The asymptotic variance-covariance matrix Σ of the ML estimates γˆ , γˆ , γˆ , βˆ and βˆ is 0

the inverse of the Fisher information matrix F

1

2

1

2

 Var (γˆ0 )   Cov(γˆ0 , γˆ1 )  Σ =  Cov(γˆ0 , γˆ2 )   ˆ  Cov(γˆ0 , β1 ) Cov(γˆ , βˆ ) 0 2 

Cov(γˆ0 , βˆ2 )   Var (γˆ1 ) Cov(γˆ1 , γˆ2 ) Cov(γˆ1 , βˆ2 )  1 1  −1 Cov(γˆ1 , γˆ2 ) Var (γˆ2 ) Cov(γˆ2 , βˆ1 ) Cov(γˆ2 , βˆ2 )  = F  Var ( βˆ1 ) Cov( βˆ1 , βˆ2 )  Cov(γˆ1 , βˆ1 ) Cov(γˆ2 , βˆ1 ) Cov(γˆ1 , βˆ2 ) Cov(γˆ2 , βˆ2 ) Cov( βˆ1 , βˆ2 ) Var ( βˆ2 )  Cov(γˆ0 , γˆ1 )

Cov(γˆ0 , γˆ2 )

Cov(γˆ0 , βˆ1 ) Cov(γˆ , βˆ )

Each ALT plan is characterized by the level of each stress, the proportion of test units allocated to each test combination, the total number of test items available, and the censoring time. We assume that the design and highest levels of each stress, the total number of test units, and the censoring time are pre-specified. The objective is to determine the low and high levels of each stress and the proportion of test units allocated to each test condition such that the asymptotic variance of the MLE of the reliability estimate at the normal operating conditions is minimized. The nonlinear optimization problem can be formulated as follows: 2.4 – Problem Formulation Objective function T

Min

∫ Var[(γˆ

0

ˆ

ˆ

+ γˆ1t + γˆ2t 2 )e( β1z1 D + β2 z2 D ) ]dt

0

Subject to 0 < pij < 1, i = 0,1 j = 0,1

∑p

ij

=1

i, j

npij Pr[t ≤ τ | z1i , z2 j ] ≥ MNF , i = 0,1

j = 0,1

Where, MNF is the minimum number of failures. The last constraint ensures a desired minimum number of failures at any of the test combinations. ˆ ˆ The term Var[(γˆ0 + γˆ1t + γˆ2t 2 )e( β1z1 D + β2 z2 D ) ] is calculated as follows: ˆ ˆ Var[(γˆ0 + γˆ1t + γˆ2t 2 )e( β1z1 D + β2 z2 D ) ] = Var[λˆ(γˆ0 , γˆ1 , γˆ2 , βˆ1 , βˆ2 )] T

 ∂λˆ ∂λˆ ∂λˆ ∂λˆ ∂λˆ   ∂λˆ ∂λˆ ∂λˆ ∂λˆ ∂λˆ  = , , , , , , , , Σ   ∂γˆ0 ∂γˆ1 ∂γˆ2 ∂β1 ∂β 2   ∂γˆ0 ∂γˆ1 ∂γˆ2 ∂β1 ∂β 2  .

3 – Numerical Example

Suppose we develop an accelerated life test plan for a certain type of capacitor using two stresses: temperature and voltage. The reliability estimate at the design condition over a 10-year period of time is of interest. The design condition is characterized by 50oC and 5V. The high levels of temperature and voltage are pre-specified as 250oC and 10V, respectively. The allowed test duration is 200 hours, and the total number of test units placed under test is 200. The minimum number of failures at any test combination is specified as 10. The test plan is determined through the following steps: 1. According to the Arrehenius model, we use 1/(Absolute Temperature) as the first covariate z1 and 1/(Voltage) as the second covariate z2 in the ALT model. 2. The PH model is used in conducting reliability data analysis and designing the optimal ALT plan. The model is given by:

λ (t ; z , E ) = λ0 ( t ) exp ( β1 z1 + β 2 z2 )

λ0 (t ) = γ 0 + γ 1t + γ 2t 2

where

3. A baseline experiment is conducted to obtain initial estimates for the model parameters. These values are:

γˆ0

0.0001

γˆ1

0.5

γˆ2

0

βˆ1

-3800

βˆ2

-10

4. The optimization problem is formulated as, Objective function T

Min

f (x) = ∫ Var[(γˆ0 + γˆ1t + γˆ2t 2 )e( β z1D + β z2 D ) ]dt ˆ

ˆ

0

Subject to 0 < pij < 1, i = 0,1 j = 0,1

∑p

ij

=1

i, j

npij Pr[t ≤ τ | z1i , z2 j ] ≥ MNF , i = 0,1 j = 0,1 where, MNF is the minimum number of failures

and, x = [ z10 , z11 , z20 , z21 ,

p00 ,

p01 ,

p10 ]

5. We use numerical methods to solve the optimization problem: Input the initial baseline values for the model parameters γ 0 , γ 1 , γ 2 , β1 and β 2 . Input the design stress level z1D, z2D and highest stress level z1H, z2H as well as total test units n, test duration τ and minimum number of failures for each stress level MNF. 6. The optimum plan that optimizes the objective function and meets the constraints is shown as follows: T1* = 140oC, V1* = 5.04V, T2* = 209oC, V2* = 7.48V, p00* = 0.37, p01* = 0.17, p10* = 0.32, p11* = 0.14 4 – PO Model and ALT Plan

The assumptions of proportional hazards model are not always valid for all failure data. Brass (1971) observes that the ratio of the death rates, or hazard rates, of two populations under different stress levels (for example, one population for smokers and the other for non-smokers) is not constant with age, or time, but follows a more complicated course, in particular moving closer to unity for older people. So the PH model is not suitable for this case. Brass (1971) proposes a more realistic model as F (t ) F0 (t ) = eβ z 1 − F (t ) 1 − F0 (t )

(27)

This model is referred to Proportional Odds (PO) model since the odds ratios under different stress levels are proportional to each other. Assuming two stresses exist, after mathematical manipulation, Eq. (27) could be expressed as

λ (t ) =

e β1z1 + β2 z2 λ0 (t ) 1 − (1 − e β1z1 + β2 z2 ) F0 (t )

(28)

The baseline hazard function λ0 (t ) is assumed to be quadratic: λ0 (t ) = γ 0 + γ 1t + γ 2t 2 . We rewrite Eq. (28) as

λ (t; z ) =

e β1z1 + β2 z2 (γ 0 + γ 1t + γ 2t 2 ) e β1z1 + β 2 z2 + (1 − e β1z1 + β 2 z2 )e − (γ 0t +γ 1t

We also have t

λ 0 ( u ) du 2 3 F0 (t ) = 1 − e ∫0 = 1 − e − (γ 0t +γ 1t / 2+γ 2t / 2) −

2

/ 2 +γ 2 t 3 / 2)

.

(29)

R 0 (t ) = exp[−Λ 0 (t )] = 1 − F0 (t ) = e − (γ 0t +γ 1t

2

/ 2+γ 2t 3 / 2)

t

Λ 0 (t ) = ∫ λ 0 (u )du = γ 0t + γ 1t 2 / 2 + γ 2t 3 / 3 0

t

t

e β z λ 0 (u )

0

0

e β z + (1 − e β z ) R 0 (u )

Λ (t; z ) = ∫ λ (u; z )du = ∫ =∫

(30)

e β1z1 + β2 z2 (γ 0 + γ 1t + γ 2t 2 )

t

0

du

e β1z1 + β2 z2 + (1 − e β1z1 + β2 z2 )e − (γ 0t +γ 1t

2

/ 2 +γ 2t 3 / 2)

du

With Eq. (29) and Eq. (30), we can construct the log likelihood function for a single failure unit based on the PO model by ln L(t; z ) = I ln(λ (t; z )) − Λ (t; z ) = I ln(

e β1z1 + β2 z2 (γ 0 + γ 1t + γ 2t 2 ) e β1z1 + β2 z2 + (1 − e β1z1 + β2 z2 )e

)−∫ − ( γ t +γ t 2 / 2 +γ t 3 / 2) 0

1

2

e β1z1 + β2 z2 (γ 0 + γ 1t + γ 2t 2 )

t

0

e β1z1 + β2 z2 + (1 − e β1z1 + β 2 z2 )e − (γ 0t +γ 1t

2

/ 2 +γ 2t 3 / 2)

du

Suppose that the ith observation ti corresponds to a value zi and the corresponding log likelihood is li. Then the sample log likelihood l for n independent observations is: l = l1 + l2 + " + ln The values of ( γ 0 , γ 0 , γ 0 , β1 , β 2 ) can be estimated by maximizing the above log likelihood function. The maximum likelihood estimates ( γˆ , γˆ , γˆ , βˆ , βˆ ) are obtained 0

0

0

1

2

by numerical method. After construction of the log likelihood function based on the PO model, we then follow the same procedures mentioned in Section 2 to design the optimal ALT plans. 5 – Conclusions

We develop the first analytically based accelerated life testing plans that utilize the Proportional Hazards model with multiple stresses for reliability prediction. The plans determine the levels for each stress type and the number of test units allocated to each level in order to minimize the variance of reliability estimate at normal operating conditions. Similar plans based on Proportional Odds model with multiple-stresses are also developed following the same procedures.

Reference:

Brass, W., “On the scale of mortality”, In: Brass, W., editor. Biological aspects of Mortality, Symposia of the society for the study of human biology. Volume X. London: Taylor & Francis Ltd.: 69-110, 1971 Chernoff, H., “Optimal accelerated life design for estimation”, Technometrics, 4, 381408, 1962 Cox, D. R., “Regression models and life tables (with discussion)”, Roy. Stat. Soc. B34, 187-208, 1972 Elsayed, E. A., Reliability engineering, MA: Addison-Wesley Longman, 1996 Elsayed, E. A. and Zhang, H., “design of optimum simple step-stress accelerated life testing plans”, Proceedings of the Second International Industrial Engineering Conference. Riyadh, Kingdom of Saudi Arabia, 2004 Elsayed, E. A. and Jiao, L., “optimal design of proportional hazards based accelerated life testing plans”, International J. of Materials & Product Technology, 17 411-424, 2002 Escobar, L. A. and Meeker, W. Q., “Planning accelerated life tests with two or more experimental factors”, Technometrics 37(4), 411-427, 1995 Fletcher, R., Practical methods of optimization, John Wiley and Sons, 1987. Gill, P.E., Murray, W. and Wright, M. H., Practical optimization, London, Academic Press, 1981. Hannerz, H., “An extension of relational methods in mortality estimation”, Demographic Research, Vol. 4, p. 337-368, 2001 Hock, W. and Schittkowski, K., “A Comparative Performance Evaluation of 27 Nonlinear Programming Codes”, Computing, Vol. 30, p. 335, 1983 Jiao. L., “Optimal allocations of stress levels and test units in accelerated life tests”, Ph.D. Dissertation, Department of Industrial and Systems Engineering, Rutgers University, 2001 Kalbfleisch, J. D. and Prentice, R. L., The statistical analysis of failure time data, New Jersey: John Wiley & sons, Inc, 2002 Kobayashi, T., Ariyoshi, H. and Masua, A., “Reliability evaluation and failure analysis for multilayer ceramic chip capacitors”, IEEE Transactions on Components, Hybrids, and Manufacturing Technology, CHMT-1, 316-324, 1978 Minford, W. J., “Accelerated life testing and reliability of high K multilayer ceramic capacitors.” IEEE Transactions on Components, Hybrids, and Manufacturing Technology, CHMT-5, 297-300, 1982 Mogilevsky, B. M. and Shirn, G. A., “Accelerated life tests of ceramic capacitors.” IEEE Transactions on Components, Hybrids, and Manufacturing Technology, CHMT-11, 351357, 1988

Munikoti, R. and Dhar, P., “Highly accelerated life testing (HALT) for multilayer ceramic capacitor qualification”, IEEE Transactions on Components, Hybrids, and Manufacturing Technology, CHMT-11, 342-345, 1988 Nelson, W. and Meeker, W., “Theory for optimum censored accelerated life tests for weibull and extreme value distributions”, Technometrics, 20, 171-177, 1978 Nelson, W., Accelerated testing: statistical models, test plans, and data analyses, New York: John Wiley & sons, Inc, 1990 Park, J. W. and Yum, B. J., “Optimum design of accelerated life tests with two stresses”, Naval Research Logistic, 43, 863-884, 1996 Powell, M.J.D., "Variable metric methods for constrained optimization", Mathematical Programming: The State of the Art, (A. Bachem, M. Grotschel and B. Korte, eds.) Springer Verlag, pp 288-311, 1983. Tang, L. C., “Planning for accelerated life tests”, International J. of Reliability, Quality, and Safety Engineering, 6, 265-275, 1999 Yang, G. B., “Optimum constant-stress accelerated life test plans”, IEEE Trans. on Reliability, 43, 575-581, 1994