Optimal Accelerated Life Test Designs for Burr Type XI1 Distributions under Periodic Inspection and Type I Censoring

Nesar Ahmad and A. Islam Department of Statistics and Operations Research, Aligarh Muslim University,

Aligarh 202 002, India

This article develops optimal accelerated life test designs for Burr Type XI1 distributions under periodic inspection and Type I censoring. It is assumed that the mean lifetime (the Burr XI1 scale parameter) is a log-linear function of stress and that the shape parameters are independent of stress. For given shape parameters, design stress and high test stress, the test design is optimized with respect to the low test stress and the proportion of test units allo- cated to the low stress. The optimality criterion is the asymptotic variance ofthe maximum- likelihood estimator of log mean life at the design stress with the use of equally spaced inspection times. Computational results for various values of the shape parameters show that this criterion is insensitive to the number of inspection times and to misspecification of imputed failure probabilities at the design and high test stresses. Procedures for planning an accelerated life test, including selection of sample size, are also discussed. It is shown that optimal designs previously obtained for exponential and Weibull distributions are similar to those obtained here for the appropriate special cases of the Burr XI1 distribution. Thus the Burr XI1 distribution is a useful and widely applicable family of reliability models for ALT design. 0 1996 John Wiley & Sons, Inc.

1. INTRODUCTION

Accelerated life testing (ALT) is used to obtain information about the life distribution of a material or product with a long expected life at the design stress (use conditions). Test units are induced to early failure by applying higher-than-usual stress levels. The test data obtained at accelerated conditions are then extrapolated to estimate the life distribution at the design stress.

Two inspection modes are prevalent in ALT. One is continuous inspection results in exact failure times [ 3,7]. An alternative is intermittent or periodic inspection of test units at predetermined points in time. Many authors [ 4,6, 13, 171 have used periodic inspection because it requires less testing effort and is easier to administrate.

Most previous work on optimal design for ALT assumes that the life distribution is either Exponential or Weibull [2-4, 6, 7, 9, 13, 171. Normal and log-normal models have also been used [ 7,8 1.

This article considers optimal ALT designs based on the Burr Type XI1 distribution [ 13. This family includes exponential, Weibull and log-logistic as special cases (see Figure 1 ) . It also covers the curve shape characteristics of normal, log-normal, gamma, logistic, and Pearson type X distributions, as well as a significant portion of the curve shape character- istics for Pearson Type I (Beta), 11, V, VII, IX, and XI1 families (see Rodriguez [ 1 I , 121 and Wingo [ 15, 161, and also Figure 2). Another advantage is that Burr XI1 has simple

NavulReseurch Logistics, Vol. 43, pp. 1049- 1077 (1996) Copyright 0 1996 by John Wiley & Sons, Inc. CCC 0894-069)</96/08 1049-29

1050 Naval Research Logistics, Vol. 43 ( 1996)

t

Figure 1. Some typical Burr Type XI1 density shapes.

t

algebraic forms for the reliability and hazard rate functions (see Figure 3); this is particu- larly important in handling censored data. Thus Burr XI1 provides a wide variety of density shapes along with functional simplicity (see Figure 1 ).

A useful technique for visualizing the richness ofthe Burr XI1 family is the moment ratio diagram developed by Rodriguez [ 1 1, 121 (see Figure 2). We are using Burr XI1 because it has simple functional forms, even though Burr 111 is wider.

Ahmad and Islam: Accelerated Life Test Designs

I I I

- 1.5 -1.0 -0.5

I05 I

I I 1

0.5 I .n 1 15

In this article we develop asymptotically optimal ALT plans for the Burr XI1 distribution under Type I censoring and periodic inspection at two test stress levels. We assume that the mean lifetime (the Burr XI1 scale parameter) is a log-linear function of stress and that the shape parameters are independent of stress. The unknown parameters in the log-linear

1052 Naval Research Logistics, Vol. 43 ( 1996)

__ 3.5 I I

0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 t

Figure 3. Burr Type XI1 hazard curves.

relationship are estimated by maximum likelihood. For given levels of the shape parame- ters, design stress, and high test stress, the low test stress and associated proportion of test units are optimally determined. The optimality criterion is the asymptotic variance of the maximum-likelihood estimator of log mean life at the design stress. This may be viewed as an extension of Wingo [ 151 to ALT with censoring and periodic inspection and as an extension of Wingo [ 161 to ALT with periodic inspection. It also extends Islam and Ahmad [ 61, Yum and Choi [ 171 and Seo and Yum [ 131. The basic approach is similar to Yum

Ahmad and Islam: Accelerated Life Test Designs 1053

and Choi [ 171 but the computational techniques are new. A software program to carry out the computations is available from the authors.

Computational studies are conducted to examine how optimal plans vary with respect to various combinations of the parameters, and to assess their sensitivity to misspecifica- tion of imputed failure probabilities at the design and high test stresses. It is shown that optimal designs previously obtained for exponential and Weibull distributions are similar to those obtained here for the appropriate special cases of the Burr XI1 distribution. Pro- cedures for planning an accelerated life test, including selection of sample size, are also discussed.

2. THE MODEL

Basic Assumptions

1. Three stress levels so < sI < s2 are considered, where so is the design stress representing use conditions and s, and s2 are elevated stresses representing accelerated conditions, for example, higher-than-usual temperature, voltage, et cetera.

2. At any level of stress s, the lifetimes ( T) of test items follow the Burr Type XI1 distribution with shape parameter m, 6 and scale parameter 0. That is,

For m = 1, the distributionf( t ) becomes a log-logistic distribution with shape parameter 6 and scale parameter 19. That is,

f ( t ) = ( s / ~ ) ( t / e ) ~ - ' ( 1 + ( t / e ) * ) - 2 , t 2 0, 6 , e > 0. (2)

For m = 2, the life distribution becomes

f ( t ) = 2(6/e) ( t /e )*- l ( 1 + ( t / ~ ) * ) - ~ , t 2 0, 6 , e =- 0, ( 3 )

and so on. 3. The mean lifetime ( d ) and the stress level s are exponentially related as

where Po and PI are unknown parameters depending on the nature of the prod- uct and the test method (see [ 6, 7 ] ).

4. 6 and m are known parameters independent of stress. 5. The lifetimes of test units at stress level si are independent and identically

distributed.

1054 Naval Research Logistics, Vol. 43 ( 1996)

Test Procedure

1. The use test stress (so) and high test stress (sz) are prespecified, whereas the

2. Test units ( ni ) allocated to s; out of total N test units, is test stress ( sI ) is to be optimally determined.

2 ni = L Y ~ N , 2 (Y; = 1, ai > 0, i = 1, 2,

i= 1

where aI is to be optimally determined. 3. The test items (n , ) are initially placed on life test at stress level s, and run until

a prespecified time t,, (i.e., Type I censoring is assumed). 4. At specified points in time t, I , f r 2 , . . . , t , , K ( , ) , the inspections are carried out,

where t r , h ( , ) = f,,, f r o = 0, and & K c , ) + , = 00. Let the number oftest items failed during t,,) at stress level s, be x,, and the probability of failure during (fiJ-,,f,,)beP,,,j= 1,2 ,..., K(i)+ 1.

The grouped data {xo, i = 1,2,; j = 1,2, . . . , K( i) + 11 are used to estimate Po and PI . The estimated relationship is then extrapolated to estimate mean lifetime at the use con- dition. The logarithm of mean lifetime 6 at the use condition is defined by

po = In B0 = Po + ,Olso. ( 5 )

Let t, be the qth quantile of the life distribution at use condition. We want to estimate

Let b0 and 6, be the ML estimates of Po and P I , respectively. Then y, is estimated as

where

3. MAXIMUM-LIKELIHOOD ESTIMATION

The grouped data (xu, j = 1, 2, . . . , K( i) + l} is multinomially distributed with pa- rameters ni and { PG, j = l , 2, . . . , K( i ) + l ) at stress level si . The likelihood function is

Ahmad and Islam: Accelerated Life Test Designs 1055

2 2 L' = J-J L: = nj!

i= 1 i= 1 j = I

The logarithm of the above equation gives

2 2 K ( I ) + l

L = In L' = 2 In L: = C + C x, ln(P,), I= I j = I j=I

where Cis constant and Pi, is defined as

fo r i= l , 2 a n d j = 1,2,. . . , K ( i ) + 1. The ML estimates of Po and PI can be obtained by solving the following equations:

where

f o r i = 1 , 2 , a n d j = 0 , 1 , 2 ,..., K ( i ) + l .

Fisher Information Matrix

The Fisher information matrix F ( see Escobar and Meeker [ 51 and Rao [lo]) is obtained by taking expectations of negative second partial derivatives of L with respect to Po and PI ; that is

F =

where

(7)

1056 Naval Research Logistics, Vol. 43 ( 1996)

Table 1. Optimal ALT plans when 6 = 0.5 and m = 1.

pu ph P O PI K s: (Y: NAsVar(&) Ratio

0.0001 0.9

0.5

0. I

0.01

0.001 0.9

0.5

0.1

0.01 0.9

0.5

18.4201 -22.8146

18.420 -18.4201

18.420 - 14.0257

18.4201 -9.2299

13.8135 -18.2080

13.8135 - 13.8 135

13.8135 -9.4191

9.1902 -13.5847

9.1902 -9.1902

1 5

10

1 5

10

1 5

10

1 5

10

1 5

10

1 5

10

1 5

10

1 5

10

I 5

10

co

co

co

CI)

co

co

co

co

m

0.596 0.616 0.622 0.632 0.664 0.668 0.668 0.670 0.624 0.624 0.624 0.624 0.444 0.444 0.444 0.444

0.494 0.520 0.528 0.538 0.554 0.558 0.558 0.560 0.440 0.442 0.442 0.442 0.322 0.356 0.366 0.380 0.328 0.336 0.336 0.338

0.647 0.684 0.697 0.7 15 0.181 0.793 0.795 0.794 0.845 0.845 0.845 0.845 0.889 0.889 0.889 0.889 0.688 0.718 0.729 0.747 0.8 15 0.82 1 0.822 0.822 0.885 0.884 0.884 0.884 0.772 0.788 0.796 0.807 0.822 0.884 0.885 0.884

775.4652 635.7888 596.9892 547.6198

1379.3949 1344.6050 1339.428 1 1335.6889 5066.4662 5063.8 17 1 5063.6003 5063.4864

21 100.5332 21 100.4593 21 100.4539 21 100.4512

435.90 18 365.5392 345.6581 320.12 18 72 1.0074 705.2172 702.8597 701.1697

2080.3997 2079.5 80 1 2079.5 13 1 2079.4779

192.7697 169.0355 162.0368 152.84 1 1 273.0988 269.17 I6 268.5830 268.16 10

1.4161 1.1610 .0902

.0327

.0067

.0028

.0006

.ooo 1 1 .oooo 1 1 .oooo 1 .oooo 1 .oooo 1

1.3617 1.1419 1.0798 1 1.0283 1.0058 1.0024 1 1.0004 I .oooo 1 .oooo 1

1.2612 1.1060 I .0602 1 1.0184 1.0038 1.0016 1

~

After some algebraic manipulation, we obtain

2

f 00 = C aiQi7 i= 1

Ahmad and Islam: Accelerated Life Test Designs

Table 2. Optimal ALT plans when 6 = 1 and m = 1.

1057

P U ph P O PI K s: a: NAsVar&) Ratio 0.0001 0.9

0.5

0.1

0.0 1

0.001 0.9

0.5

0. I

0.0 1 0.9

0.5

9.2101

9.2101

9.2101

9.2101

6.9068

6.9068

6.9068

4.595 1

4.595 1

- 1 1.4073

-9.2 101

-7.0128

-4.6 150

-9.1040

- 6.906 8

-4.7095

-6.7923

-4.595 1

1 5

10

1 5

10

1 5

10

1 5

10

a3

00

a3

m

1 5

10

1 5

10

1 5

10

1 5

10

I 5

10

co

m

m

00

a2

0.596 0.628 0.634 0.636 0.664 0.670 0.670 0.670 0.624 0.624 0.624 0.624 0.444 0.444 0.444 0.444

0.494 0.534 0.542 0.544 0.554 0.558 0.560 0.560 0.440 0.442 0.442 0.442

0.322 0.376 0.386 0.390 0.328 0.336 0.338 0.338

0.647 0.71 1 0.724 0.732 0.787 0.793 0.794 0.794 0.845 0.845 0.845 0.845 0.889 0.889 0.889 0.889 0.688 0.743 0.753 0.76 1 0.8 15 0.823 0.822 0.822 0.885 0.884 0.884 0.884 0.772 0.804 0.81 1 0.816 0.882 0.885 0.884 0.885

193.8663 140.3335 132.8058 128.91 17 344.8487 334.3 100 3 3 3.8 599 333.7 163

1266.6 165 1265.8973 1265.8747 1265.8676 5275.1333 5275. I 135 5275. I129 5275.1 127

108.9754 81.8131 77.897 1 75.8599

180.25 19 175.4684 175.2642 175.1988 520.0999 5 19.8774 5 19.8704 5 19.8682 48.1924 38.8593 37.4289 36.675 1 68.2747 67.0847 67.0333 67.0169

1 SO39 1.0886 1.0302 1 1.0334 1.0018 1.0004 1 1.0006 1 .oooo 1 .oow 1 1 .oooo I .oooo 1 .oooo 1

1.4365 1.0785 1.0269 1 1.0288 1.0015 1 .OW4 1 1.0004 I .oooo 1 .oooo 1

1.3140 1.0596 1.0206 1 1.0188 1.0010 I .0002 1

Note that

where K ( i ) + l

Qi = C (Aij-1 - Aij)*/Pjj, for i = 1,2. j = I

Asymptotic Variance of ML Estimates of the Parameters The asymptotic covariance matrix Vof &, and BI is the inverse of the Fisher information

matrix F:

1058 Naval Research Logislics, Vol. 43 ( 1996)

Table 3. Optimal ALT plans when 6 = 2 and m = 1.

pu ph P O PI K s: a: NAsVar(io) Ratio

0.0001 0.9

0.5

0.1

0.0 1

0.001 0.9

0.5

0.1

0.0 1 0.9

0.5

4.6050

4.6050

4.6050

4.6050

3.4534

3.4534

3.4534

2.2976

2.2976

-5.7036

-4.6050

- 3.5064

-2.3075

-4.5520

-3.4534

-2.3548

-3.3962

-2.2976

1 5

10

1 5

10

1 5

10

I 5

10

1 5

10

1 5

10

1 5

10

1 5

10

I 5

10

03

03

03

03

03

03

03

03

03

0.596 0.636 0.636 0.636 0.664 0.670 0.670 0.670 0.624 0.624 0.624 0.624 0.444 0.444 0.444 0.444

0.494 0.544 0.544 0.546 0.554 0.558 0.560 0.560 0.440 0.442 0.442 0.442

0.322 0.388 0.390 0.390 0.328 0.338 0.338 0.338

0.647 0.727 0.731 0.733 0.787 0.794 0.794 0.794 0.845 0.845 0.845 0.845 0.889 0.889 0.889 0.889

0.688 0.757 0.76 1 0.760 0.8 15 0.823 0.822 0.822 0.885 0.884 0.884 0.884

0.772 0.814 0.8 16 0.817 0.882 0.884 0.884 0.885

48.4666 32.6862 32.2960 32.1529 86.2 122 83.5562 83.4594 83.4287

316.6541 316.4801 316.4701 3 16.4670

1318.7833 13 18.7785 1318.7782 1318.7782

27.2439 19.2058 19.0008 18.9256 45.0630 43.8577 43.8 135 43.7995

130.0250 129.9712 129.968 I 129.967 1

12.048 I 9.2588 9.1824 9.1541

17.0687 16.7690 16.7577 16.7542

1 SO74 1.0166 1.0045 1 1.0334 1.0015 1.0004 1 1.0006 1 .oooo 1 .oooo 1 1 .oooo 1 .oooo 1 .oooo 1

1.4395 1.0148 1.0040 1 1.0288 1.0013 1.0003 1 1.0004 1 .oooo 1 .oooo 1

1.3161 1.01 14 1.003 1 1 1.0188 1.0009 1.0002 1

Then the asymptotic variance of jLo is obtained as

AsVar(io) = [ l , S~]F-~[I , SO]' = N-l(f,f,~ - f & - l ( h ~ +s&& - ~ J O ~ I ) . (9)

This is also the asymptotic variance of go for any q .

Ahmad and Islam: Accelerated Life Test Designs 1059

Table 4. Optimal ALT plans when 6 = 0.5 and m = 2. PId ph P O PI K 3: a: NAsVar(h) Ratio

0.0001 0.9

0.5

0.1

0.0 1

0.001 0.9

0.5

0.1

0.0 1 0.9

0.5

19.8065

19.8065

19.8065

19.8065

15.2002

15.2002

1 5.2002

10.5816

10.5816

-21.3489

- 18.0438

- 13.9724

-9.2250

- 16.7426

- 13.4375

-9.366 I

- 12.1239

-8.8 188

1 5

10

I 5

10

1 5

10

1 5

10

1 5

10

1 5

10

1 5

10

1 5

10

1 5

10

00

co

00

co

00

co

co

co

a2

0.634 0.660 0.664 0.668 0.686 0.688 0.688 0.688 0.630 0.630 0.630 0.630 0.446 0.446 0.446 0.446 0.534 0.566 0.572 0.578 0.578 0.580 0.580 0.580 0.448 0.448 0.448 0.448

0.356 0.400 0.408 0.416 0.358 0.360 0.362 0.362

0.678 0.723 0.734 0.744 0.814 0.817 0.8 18 0.8 I8 0.850 0.850 0.850 0.850 0.889 0.889 0.889 0.889 0.7 14 0.753 0.762 0.769 0.839 0.842 0.842 0.842 0.889 0.889 0.889 0.889 0.789 0.812 0.8 18 0.823 0.893 0.895 0.895 0.895

831.6177 662.4729 634.3262 609.5900

1333.7367 13 18.2556 13 16.4945 I3 15.4163 5005.2878 5003.9674 5003.8650 5003.8125

21076.4578 21076.4165 21076.4135 21076.4120

460.468 I 376.0482 36 1.7372 349.0932 697.1685 690.0983 689.2978 688.8079

2058.7 185 2058.3090 2058.2772 2058.2610

197.6643 169.7398 164.7724 160.3237 264.4008 262.5929 262.3880 262.2624

I .3642 1.0868 1.0406 1 1.0139 1.0022 1.0008 1 1.0003 1 .oOOo 1 .oooo 1 I .oooo 1 .oooo I .oooo 1

1.3190 1.0772 1.0362 1 1.0121 1.0019 1.0007 1 1.0002 1 .oooo I .moo 1

1.2329 1.0587 1.0277 1 1.0082 1.0013 1.0005 1

Design Problem and Optimality Criterion

Optimal design problem of an ALT under periodic inspection can now be stated as fol- lows: given N, so, s2, 6, m, { tci, i = 1,2 1 and { K( i), i = 1,2 } , determine al, sl, and { ti i , i = 1,2;j = 1,2, . . . , K( i ) - I } such that the AsVar(io) is minimized.

We assume that the design stress level (so) is adjusted to be zero. Then, Eq. (9) becomes

After some algebraic manipulation, we obtain

I060 NavalResearch Logistics, VoI. 43 (1996)

Table 5. Optimal ALT plans when 6 = 1 and m = 2. P,, PIl Bo PI K 3: a: N AsVar &) Ratio

~~~ ~

0.0001 0.9 9.9033 -10.6744 1 0.634 0.678 207.9044 1.3823

0.5

0.1

0.0 1

0.001 0.9

0.5

0.1

0.0 1 0.9

0.5

9.9033

9.9033

9.9033

7.6001

7.600 1

7.6001

5.2908

5.2908

-9.02 19

-6.9862

-4.6 125

-8.3713

-6.7 187

-4.683 1

-6.06 19

-4.4094

5 10

1 5

10

1 5

10

1 5

10

I 5

10

1 5

10

I 5

10

1 5

10

I 5

10

03

a!

03

03

co

co

03

co

00

0.668 0.670 0.670 0.686 0.688 0.688 0.688 0.630 0.630 0.630 0.630 0.446 0.446 0.446 0.446 0.534 0.576 0.578 0.578 0.578 0.580 0.580 0.580 0.448 0.448 0.448 0.448

0.356 0.4 14 0.418 0.4 18 0.358 0.362 0.362 0.362

0.739 0.744 0.746 0.814 0.818 0.818 0.8 18 0.850 0.850 0.850 0.850 0.889 0.889 0.889 0.889

0.714 0.767 0.772 0.774 0.839 0.842 0.842 0.842 0.889 0.889 0.889 0.889 0.789 0.82 1 0.824 0.825 0.893 0.895 0.895 0.895

154.1652 15 I .3659 150.4098 333.4342 329.0084 328.8546 328.806 3

I25 1.32 19 1250.9659 1250.9548 1250.95 13 5269.1 145 5269.1034 5269.1030 5269.1029

115.1 170 88.1774 86.7423 86.2537

174.292 I 172.2722 172.2023 172.1803 5 14.6796 5 14.5692 5 14.5658 5 14.5647 49.416 1 40.40 1 3 39.8935 39.7 195 66.1002 65.5838 65.5657 65.5601

1.0250 1.0064 1 1.0141 1.0006 1 .ooo 1 1 1.0003 1 .oooo 1 .oooo 1 1 .oooo I .oooo 1 .oooo I 1.3346 I .0223 1.0057 1 1.0123 1.0005 1 .ooo 1 1 1.0002 1 .oooo 1 .oooo 1

1.244 I 1.0172 1.0044 1 1.0082 1.0004 1 .ooo 1 1

Minimizing the asymptotic variance of io of the life distribution at design stress is used as the optimality criterion, but note that it may not be a good optimization criterion for heavy censor- ing and/or small sample sizes, because in these cases there may be a significant bias in the MLE.

The present method of an ALT under periodic inspection may be extended to general 1 levels and can now be stated as follows: given N, so, sI, 6, rn, { t,,, i = 1,2, . . . , 1} and { K( i), i = 1 , 2 , . . . ,l},determine {a,, i = 1 , 2 , . . . , I - l } , {si, i = 1 ,2 , . ..,I- 1) and {ti,, i = 1,2, . . . , 1 } ; j = 1,2, . . . , K( i) - 1 } such that the AsVar( F o ) is minimized.

4. OPTIMAL PLANS Assumptions

1. Censoring times at sI and s2 are the same. That is, t c l = tc2 = t,.

Ahmad and Islam: Accelerated Life Test Designs

Table 6. Optimal ALT plans when 6 = 2 and m = 2.

1061

P U ph P O P I K s: a: N AsVar (Go) Ratio

0.0001 0.9

0.5

0.1

0.0 I

0.001 0.9

0.5

0.1

0.0 1 0.9

0.5

4.95 16

4.95 16

4.95 16

4.95 16

3.800 1

3.8001

3.8001

2.6454

2.6454

-5.3372

-4.5 109

-3.4931

-2.3062

-4.1856

-3.3594

-2.3415

-3.0310

- 2.2047

1 5

10

1 5

10

I 5

10

1 5

10

1 5

10

I 5

10

1 5

10

1 5

10

1 5

10

a,

cg

co

a,

a,

a,

a,

a,

a,

0.634 0.668 0.670 0.670 0.686 0.688 0.688 0.688 0.630 0.630 0.630 0.630 0.446 0.446 0.446 0.446 0.534 0.578 0.578 0.578 0.578 0.580 0.580 0.580 0.448 0.448 0.448 0.448

0.356 0.416 0.4 18 0.4 18 0.358 0.362 0.362 0.362

0.678 0.745 0.745 0.746 0.8 14 0.818 0.8 18 0.8 18 0.850 0.850 0.850 0.850 0.889 0.889 0.889 0.889 0.7 14 0.77 I 0.773 0.774 0.839 0.842 0.842 0.842 0.889 0.889 0.889 0.889

0.789 0.824 0.825 0.825 0.893 0.895 0.895 0.895

5 1.976 I 37.9634 37.6825 37.5922 8 3.3 5 8 5 82.2652 82.2170 82.201 8

3 12.8305 3 12.7446 3 12.7395 3 12.7379

13 17.2786 13 17.2759 13 17.2758 1317.2757

28.7793 2 1.7487 2 1.6044 2 1.5582 43.5730 43.074 1 43.052 1 43.0452

128.6699 128.6433 128.641 7 128.64 12 12.3540 9.996 I 9.9446 9.9280

16.5250 16.3976 16.39 18 16.3900

1.3826 1.0099 1.0024 1 1.0141 I .0008 I .0002 1 1.0003 1 .oooo 1 .oooo 1 1 .oooo 1 .oooo 1 .oooo 1

1.3350 1.0088 I .002 1 1 1.0123 1.0007 I .0002 1 1.0002 1 .oooo 1 .oooo 1

1.2444 I .0069 1.0017 1 1.0082 1.0005 1 .ooo 1 1

2. The number of inspections at each stress level is the same. That is, K( 1 ) =

K( 2) = K. Further, we standardize the parameters such that s2 = 1 as well as t , = 1. In Section 3, the design stress level was already adjusted to be 0. Such standardization does not alter the nature of our problem (see the Appendix).

Optimization Method

The optimal plan is developed by determining optimal values of sI and a I for given N, K, 6, and m such that AsVar(io) is minimized. A two-step procedure for minimizing AsVar( io) with respect to s1 and aI is given in the Appendix. The optimization procedure

1062 Naval Research Logistics, Vol. 43 ( 1996 )

Table 7. Sensitivities of AsVar (h) when 6 = 0.5, m = I , and K = 2.

0.000 I 0.9 F u / F h 0.7000 0.8000 0.9000 0.9500 0.9900

0.0000 1 0.00003 0.00005 0.000 10 0.00020 0.00030 0.00050

0.1 F d F h

1.6610 1.4088 1.41 13 1.2217 I .3032 1.1482 1.1875 I .0748 1.0996 1.0294 I .066 1 1.0185 1.0457 1.0274

0.0600 0.0800

1.1244 1.0388 1.0128 1 1.0178 1.0463 1.1077

0.1000

I .0285 1.3936 1.0196 1 S988 1.0341 1.7314 1.0774 1.9797 1.1606 2.3078 1.2388 2.5703 1.3650 2.9885

0.1200 0. I400

0.0000 I 0.00003 0.00005 0.00010 0.00020 0.00030 0.00050

0.01 F u / F h

I .0929 1.0171 1.0019 1.0077 I .0622 1.1274 1.2654

0.0060

1.1140 1.0302 I .0083 1.0012 I .0326 1.0798 1.1827

0.0080

1.1311 1.0412 1.0142 1 1.0186 1.0542 1.1337 0.0 100

1.1432 1.0505 1.0216 1.0010 1.01 10 1.036 1 1.1004

0.0 120

1.1554 1.0573 1.0260 1.0026 I .0064 1.0273 1.0787 0.0 140

0.0000 1 1.1703 1.2219 1.2625 1.2933 1.3255 0.00003 1.0319 1.0587 1.0852 1.1069 1.1310 0.00005 1.0025 1.0151 I .030 1 1.0464 1.0638 0.000 10 1.0228 1.004 1 1 1.0025 1.007 8 0.00020 1.1738 1.0887 1.046 1 1.0229 1.0108 0.00030 1.3936 1.2212 1.1383 1.0888 I .0576 0.00050 1.887 1 1.5964 1.3937 1.2744 1.2024

0.00 1 0.9 F u / F h 0.7000 0.8000 0.9000 0.9500 0.9900

0.00010 1.7569 1.4933 1.1955 1.0640 1.2459 0.00030 1.4097 1.2383 1.0606 1.0170 1.3860 0.00050 1.2768 1.1490 1.0222 1.0209 I .4902 0.00 100 1.1392 1.0573 I 1.0589 1.690 1 0.00200 1.0556 1.0174 1.0282 1.1570 1.9839 0.00300 I .0427 1.0266 I .0798 1.2502 2.2 196 0.00500 1.0834 1.092 1 1.1942 1.4232 2.6 135

0.7000 0.3000 0.4000 0.5000 0.6000 0.5 F u / F h

0.000 I0 1.1944 1.1932 0.00030 1.0547 I .0596 0.00050 1.0184 I .0204 0.00 100 1.0052 1.0010 0.00200 1.0594 1.0345 0.00300 1.1445 1.0929 0.00500 1.3477 1.2370

.1712 1.1442 1.1 126

.0550 1.0457 1.0343

.O 193 1.0166 1.0138 1.0016 1.0098

.0264 1.0280 1.0419

.0729 1.0708 1.085 1

.1870 1.1669 1.1789

Ahmad and Islam: Accelerated Life Test Designs

Table 7. Continued.

1063

0.1 Fuf& 0.0600 0.0800 0.1000 0.1200 0.1400

0.000 10 1.1825 1.229 1 I .2622 0.00030 I .0336 I .0607 1.084 I 0.00050 1.0033 1.0156 1.0306 0.00 100 1.0223 1.0037 1 0.00200 1.1742 I .0877 1.0466 0.00300 1.3958 1.2233 1.1388 0.00500 1.9161 1 S985 1.3914

.2969 1.3194

.I078 1.1234

.0465 1.0608

.0023 1.0066

.0245 1.0120

.0887 1.0593

.2748 1.2019

0.0 1 0.9 kf4 0.7000 0.8000 0.9000 0.9500 0.9900

0.00 100 0 .OO 300 0.00500 0.0 1000 0.02000 0.03000 0.05000

0.5 FJFh

1.9539 1.4129 1.2274 1.066 1 1.0582 1.1812 I .6290

0.3000

I .6934 1.3586 1.2833 1.1 123 1.1433 1.0407 1.0302 1 1.039 1 1.0582 1.1420 1.1675 1.4722 1.4498 0.4000 0.5000

1.1774 1.0413 1.0169 1.0342 1.1393 I .2684 1.5508

0.6000

1.1528 1.1869 1.2366 1.3536 I S478 1.7150 I .0076 0.7000

0.00 100 1.3331 1.3692 1.3733 1.3525 1.3185 0.00300 1.0723 1.1045 1.1210 1.1238 1.1 179 0.00500 1.0121 I .0298 1.0437 1.0526 I .0536 0.01000 1.0316 1.0048 1 1.0023 1.0080 0.02000 1.2895 1.1325 I .0675 1.0392 1.0314 0.03000 1.4859 1.3633 1.2064 1.1334 1.0976 0.05000 1.4862 1.486 1 1.4860 I .4027 1.2836

is initiated by first providing guesstimates of PI, and Ph, which are used instead of Po and PI :

P,, = Probability that test unit fails in (0, t,) at the use condition.

Ph = Probability that test unit fails in (0, t,) at the high stress.

It is clear that P , and Ph are more familiar and easier to estimate than Po and P I . The corresponding Po and PI are obtained as follows:

The equally spaced inspection times are calculated at stress level si as

1064 Naval Research Logistics, Vol. 43 ( 1996)

Table 8. Sensitivities ofAsVar (Lo) when 6 = 1, m = 1, and K = 2.

P” p h

0.000 1 0.9 Fu/Fh 0.7000 0.8000 0.9000 0.9500 0.9900

0.0000 I 1.5829 1.3674 1.1246 1.028 1 1.3626 0.00003 1.3439 1.1930 1.0388 1.0174 1 S456 0 .OOOO 5 I .2546 1. I305 1.0128 I .0292 1.6664 0.000 10 1.1510 1.0625 1 1.0687 1.884 1 0.00020 1.0750 1.0219 1.0155 1.1470 2.1971 0.00030 I .0484 1.0152 1.0425 1.2147 2.4405 0.00050 1.0374 1.0285 1.1072 1.3332 2.8060

0.1 F u / P h 0.0600 0.0800 0.1000 0.1200 0.1400

0.0000 I 0.0000 3 0.00005 0.000 10 0.00020 0.00030 0.00050

0.01 FJFh

1.0928 1.0170 1.0019 1.0077 1.0622 1.1275 1.2655 0.0060

1.1 139 1.030 1 1.0083 1.0013 1.0326 1.0798 1.1827

0.0080

1.131 1 1.041 1 1.0142 1 1.0186 1.0542 1.1337

0.0100

1.1432 1.0505 1.0216 1.0009 1.01 10 1.036 1 1.1004 0.0 120

1.1554 1.0573 1.0259 1.0026 1.0064 1.0273 1.0787 0.0 140

0.0000 1 0.00003 0.00005 0.000 10 0.00020 0.00030 0.00050

0.00 1 0.9 F u / F h

0.000 10 0.00030 0.000 50 0.00 100 0.00200 0.00300 0.00500

0.5 F u / F h

,1703 1.2219 1.2625 1.2933 1.3255 .03 19 1.0587 1.0852 1.1069 1.1310 .0025 1.0151 1.030 1 1.0464 1.0638 .0228 1.004 1 1 1.0025 1.0078 .I738 1.0887 1.046 1 1.0229 1.0108 .3936 1.2212 1.1383 1.0888 1.0576 387 1 1.5964 I .3937 1.2744 1.2024

0.7000

1.6611 1.3459 1.2268 1.1068 1.0405 1.0380 1.0929 0.3000

0.8000

1.449 I 1.2100 1.1276 1.0452 1.0140 1.027 1 1.098 1

0.4000

0.9000

1.1882 1.0604 1.020 1 1 1.028 1 1.0754 1.1869

0.5000

0.9500

1.0648 1.0168 1.0181 1.052 1 1.1409 1.2288 1.3938 0.6000

0.9900

1.2309 1.3562 1.4497 1.6294 1.9017 2.1220 2.477 1

0.7000

0.00010 1.1868 1.1869 1.1728 1.1472 1.1205 0.00030 1.0510 1.0565 1.0558 1.047 I I .037 1 0.00050 1.0164 1.0202 1.0198 1.0172 1.0143 0.00100 1.0054 1.0010 1 1.0013 1.0073 0.00200 1.0618 1.0363 1.0258 1.0265 1.0362 0.00300 1.1485 1.0959 1.0720 1.0660 1.0774 0.00500 1.3467 1.236 1 1.1855 1.1639 1.1683

Ahmad and Islam: Accelerated Life Test Designs 1065

Table 8. Continued.

pu ph

0.1 j u / j h 0.0600 0.0800 0.1000 0.1200 0.1400

0.000 10 1.1824 1.2290 1.262 1 0.00030 1.0336 1.0606 1.0840 0.00050 1.0033 1.01 56 1.0306 0.00 100 1.0223 1.0038 1 0.00200 1.1742 1.0877 1.0467 0.00300 1.3958 1.2234 1.1388 0.00500 1.9162 1 S986 1.3915

0.0 1 0.9 j u / j h

0.00 100 0.00300 0.00500 0.0 1000 0.02000 0.03000 0.05000

0.5 j u / j h

0.7000 0.8000 0.9000

.2968 1.3194

.lo78 1.1234

.0464 1.0608

.0023 1.0066

.0245 1.0120

.0887 1.0593

.2749 1.2020

0.9500 0.9900

1.8226 1.3409 1.1791 1.0465 1.0608 1.1954 1.6607 0.3000

1.6205 1.2499 1.1215 1.0225 1.042 1 1.1459 1.4799 0.4000

1.3420 1.1086 1.0387 1 1.0544 1.1607 1.4300

0.5000

1.1750 1.0417 1.0158 1.0290 1.1268 1.246 I 1.5114

0.6000

1.1527 1.1761 1.2212 1.3289 1.5160 1.6759 1.9632 0.7000

0.00 100 1.322 1 1.3591 1.3724 1.3614 1.3285 0.00300 1.0682 1.1002 1.1206 1.1281 1.1221 0.00500 1.01 12 1.029 1 1.0435 1.0528 1.0556 0.0 1000 1.0317 1.0049 1 1.0025 1.0073 0.02000 1.2945 1.1329 1.0676 1.0373 1.0262 0.03000 1.4924 I .3638 1.2066 1.1273 1.0890 0.05000 1.4926 1.4926 1.4925 1.3926 1.2703

for i = 1,2, a n d j = 1,2, . . . , K . For given values of K, P,, P h , 6, and rn, optimal values of sI and aI are determined by a

two-step procedure such that AsVar(Go) is minimized. For K = co , the optimal values of sl, al, and AsVar(bo) are determined with the method described by Nelson and Meeker [ 93. It can also be determined by the two-step procedure. Further, ratio of AsVar(bo(K)) to AsVar( Go( 03 )) is obtained. A FORTRAN implementation of the two-step procedure has been written and was run on a VAX- 1 1 1780.

Sensitivity Analysis

For optimal ALT plans, we require knowledge of P,, and P h (or equivalently Po and PI ). Chernoff [ 21 termed this situation locally optimal design and suggested sensitivity analysis. The sensitivity analysis is conducted for some selected plausible values of P,, Ph, 6, and m, to see how AsVar( G o ) varies around these plausible values. Let F,, and p h be the estimated values of P,, and p h , respectively. s ? and a: are determined with these estimated values with K = 2 as $ and i i y , respectively. Then, the sensitivity is defined as the ratio of

1066 Naval Research Logistics, Vol. 43 ( 1996)

Table 9. Sensitivities of AsVar when 6 = 2. m = 1. and K = 2.

P U ph

0.000 1 0.9 F u / F h 0.7000 0.8000 0.9000 0.9500 0.9900 ~~ ~

0.0000 1 I .439 1 1.2942 1.1 159 1.0310 1.2764 0 .OOOO 3 1.2397 1.1423 1.0374 1.01 12 1.4214 0.00005 1.1677 1.0899 1.01 18 1.0179 1.5139 0.000 10 1.0874 1.0360 1 1.0486 1.6926 0.00020 1.0353 1.0102 1.0145 1.1 116 1.9409 0.00030 1.0225 1.01 10 1.0407 1.1705 2.1273 0.00050 1.0305 I .0344 I .0987 1.2758 2.4277

0.1 F u / F h 0.0600 0.0800 0.1000 0.1200 0.1400

0.00001 1.0929 1.1 140 1.1311 1.1432 1.1554 0.00003 1.0171 1.0302 1.0412 1.0505 1.0573 0.00005 1.0019 1.0083 1.0142 1.0216 1.0260 0.00010 1.0077 1.0012 1 1.0010 1.0026 0.00020 1.0622 1.0326 1.0186 1.01 10 1.0064 0.00030 1.1274 1.0798 1.0542 1.036 1 1.0273 0.00050 1.2654 1.1827 1.1337 1.1004 1.0787

0.0 1 F u / F h 0.0060 0.0080 0.0100 0.0 120 0.0140

0.0000 1 0.00003 0.00005 0.000 10 0.00020 0.00030 0.00050

1.1703 1.0319 1.0025 1.0228 1.1738 1.3936 1.887 I

0.7000

1.2219 I .0587 1.0151 1.004 1 1.0887 1.2212 1.5964

0.8000

1.2625 1.0852 1.030 1 1 I .046 1 1.1383 1.3937

0.9000

I .2933 1.1069 1.0464 1.0025 1.0229 1.0888 1.2744

0.9500

1.3255 1.1310 1.0638 I .0078 1.0108 1.0576 I .2024

0.9900 ~

0.000 10 1.4976 1.3586 1.1832 1.0709 1.1785 0.00030 1.2379 1.1584 1.0574 1.0156 1.2718 0.00050 1.1435 1.0888 1.0203 1.01 12 1.3463 0.00 100 1.0550 1.0255 1 1.0354 1.4953 0.00200 1.0200 1.0105 I .0255 1.1081 1.71 18 0.00300 1.0353 1.0350 1.0710 1.1804 1.8886 0.00500 1.1 151 1.1 118 1.1732 1.3273 2.1907

0.5 F u I F h 0.3000 0.4000 0.5000 0.6000 0.7000

0.000 10 1.1850 1.1854 1.1722 1.1537 1.1271 0.00030 I .050 1 I .0557 1.0555 1.0473 1.0392 0.00050 1.0158 1.0197 1.0196 1.0172 1.0145 0.00 100 1.0054 1.0010 1 1.0010 I .0049 0.00200 1.0623 1.0367 I .026 1 1,0244 1.0297 0.00300 1.1494 1.0966 1.0723 1.0653 1.0682 0.00500 1.3482 1.2372 1.1861 1.1586 1.1552

Ahmad and Islam: Accelerated Life Test Designs 1067

Table 9. Continued.

PU ph

0.1 F”lA 0.0600 0.0800 0.1000 0.1200 0.1400 ... ..

0.00010 0.00030 0.00050 0.00 100 0.00200 0.00300 0.00500

0.0 1 0.9 F u f F h

1.1825 1.0336 1.0033 I .0223 1.1742 1.3958 1.9161

0.7000

1.229 1 1.0607 1.0156 I .0037 1.0877 1.2233 1.5985

0.8000

1.2622 1.084 I 1.0306 1 1.0466 1.1388 1.3914

0.9000

1.2969 1.1078 1.0465 1.0023 1,0245 1.0887 1.2748

0.9500

1.3195 1.1234 1.0608 1.0066 1.0120 1.0593 1.2019

0.9900

0.00 I00 1.6184 1.5138 0.00300 1.2347 1.1859 0.00500 1.1085 1.0864 0.0 1000 1.0205 1.0103 0.02000 1.0773 I .0496 0.03000 I .2404 1.1649 0.05000 1.743 1 1.5077

0.5 Fulk 0.3000 0.4000

0.00 100 1.3196 1.3569 0.00300 1.0672 1.0992 0.00500 1.0108 1.0286 0.0 1000 1.0320 1.0050 0.02000 1.2955 1.1335 0.03000 1 A936 1.3649 0.05000 1.4939 1.4938

1.3236 1.1850 1.1032 1.0480 I .0380 1.0152 1 1.0184 1.0522 1.1007 1.1522 1.2056 1.408 1 1.4468

0.5000 0.6000

1.1332 1.1399 1.1719 1.2603 1.4209 I S636 1.8237

0.7000

1.3714 1.3627 1.1201 1.1284 I .0432 1.055 1 1 1.0023 I .0680 1.0356 1.2073 1.1246 1.4937 1.3882

I .3393 1.1305 1.060 1 1.008 1 1.0223 1.0829 1.2575

AsVar(L,($, 6 : ) ) to minimum AsVar(io(s:, a:)) for various cases with K = 2. Sensi- tivity analysis may be conducted for various values of K.

Sample Size

For an optimal ALT plan, the desired precision of the MLE of the log mean life or qth quantile with known shape parameters is determined by the sample size N*. To specify the desired precision of i, (or Lo), one requires that with high probability 4, iq must fall between tq/ h and ht, for a specified h > 1 ; that is,

P,{ t q / h 5 tq I ht,} 2 4,

where h( > 1 ) and 4 are given constants. Then, we can rewrite the above equation as

P,(y , - In h I yq I y , + In h } 2 4.

Then it can be shown that the approximate sample size ( N ) that achieves the above equation is given by

I068 Naval Research Logistics, Vol. 43 ( 1996)

Thouundr 60

40 3 I =30

20

10

0

\ \

I .01 .l A .9 'h

a

0 i .1 .6 .9

C

P* - 0.001 I

. \ -_

.01 .1 .6

. - '.

b

.,., '". 0.1

- .? .O

'h

d

Figure 4. (a) Optimal ALT plans for Burr Type XI1 distribution when m = 1. ( b ) Optimal ALT plans for Burr Type XI1 distribution when m = 1. (c) Optimal ALT plans for Burr Type XI1 distri- bution when m = 1. (d) Optimal ALT plans for Burr Type XI1 distribution when m = 1. Dots, 6 =

0.5; tic marks, 6 = 1.0; asterisks, 6 = 2.0; squares; 6 = 3.0.

where Wis the ( 1 + 4)/2 quantile ofthe standard normal distribution.

5. COMPUTATIONAL RESULTS AND COMPARATIVE STUDY The optimum plans are summarized in Tables 1-6 which present the optimum s 7 , a T,

and AsVar(io) for various combinations of P,, Ph, K, 6, and rn. Tables 7-9 illustrate

Ahmad and Islam: Accelerated Life Test Designs 1069

--+---- lool 0 -4----

.l .I .S %

C

.01 .1 .6 'n

b

- - I 1

9 30 .

. __ 2/li 10 10

0 .r P, -8

d

Figure 5. ( a ) Optimal ALT plans for Burr Type XI1 distribution when m = 2. (b ) Optimal ALT plans for Burr Type XI1 distribution when m = 2. (c) Optimal ALT plans for Burr Type XI1 distri- bution when m = 2. (d ) Optimal ALT plans for Burr Type XI1 distribution when m = 2. Dots, 6 = 0.5; tic marks, 6 = 1 .O; asterisks, b = 2.0; squares, d = 3.0.

computational results of sensitivity analysis for various values of P,,, Ph, and 6 with m = 1 and K = 2. The tables exhibit the following trends:

1. As P,, increases and/or Ph decreases, AsVar( io) is insensitive to K for given

2. For selected values of P,, Ph, 6, and m, s ? and a: are fairly stable over K. 3. For given 6 and m, AsVar( b 0 ) decreases as P,, increases and it increases when

values of shape parameters 6 and m .

Ph decreases [see Figures 4 ( a ) -4 (d ) and 5 (a) -5 ( d ) 1.

1070 Naval Research Logistics, Vol. 43 ( 1996)

a

0.8 I

I “ - 1

0.6

I

I 0.4

0.2

i

,001 .01 1 .6 Ph - .9

b

0 1

e.4

.01 .6 Ph -

C d

Figure 6. ( a ) Optimal low test stress (s T) for Burr XI1 when d = 1 and m = 1. (b) Optimal propor- tion of test units (a: ) for Burr XI1 when d = 1 and m = 1. (c) Optimal low test stress (s:) for exponential given by Yum and Choi. (d ) Optimal proportion of test units (a:) for exponential given by Yum and Choi. Dots, P, = 0.0001; tic marks, P, = 0.001; asterisks, P, = 0.0 1.

4. For all the cases considered, the values of ratio indicate that the number of inspections (K) need not be too large, which is an encouraging result in terms of testing efforts.

5. For given 6 and rn, as P,, increases and/or P,, decreases the values of s and a 7 tend to zero (the design stress) and 1, respectively [see Figures 6 (a), 6( b), 7( a), and 7 (b)] . This implies almost no need for an ALT. Similar trends are observed when PI, values are small for P,, < 0.1.

6. For given values of rn, as 6 increases, asymptotic variance of io decreases [see Figures 4 (a)-4( d ) and 5 (a)-5 (d)] .

Ahmad and Islam: Accelerated Life Test Designs 107 1

4 0 8

0.7 o.8--I t 8

+ 4 1

C

0 .001 .01 1 .5 ph - 9

b

.001 .O 1 6 Ph-- 9

d

Figure 7. ( a ) Optimal low test stress (s:) for Burr XI1 when 6 = 0.5 and m = 1. (b ) Optimal proportion of test units (a:) for Burr XI1 when 6 = 0.5 and m = 1. ( c ) Optimal low test stress (s:) for Weibull given by Islam and Ahmad when 6 = 0.5. (d ) Optimal proportion of test units (a:) for Weibull given by Islam and Ahmad when 6 = 0.5. Dots, P,, = 0.0001; tic marks, P,, = 0.00 1; asterisks, P,, = 0.01.

7. For given 6, if the shape parameter m increases and P,, increases, then

8. For given 6 and P,,, when shape parameter rn increases, AsVar( Go) decreases

9. For given values of 6 and m, sensitivity values are fairly stable over K.

AsVar( Go) increases except for P,, = 0.1.

as Ph decreases [see Figures 4 (a) -4 ( d ) and 5 ( a ) -5 ( d )] .

10. The sensitivity values are very close to 1, implying that the AsVar(Go) is robust against the true P, and Ph from their guessed values, implying that it may not be necessary to adjust the sample size.

1072 Naval Research Logistics, Vol. 43 ( 1996)

.OOI .01 .1 .6 Fh .@

a

Thousands

l 2 I

b

Figure 8. (a) Optimal ALT plans for Burr XI1 when 6 = 1 and rn = 1 . (b ) Optimal ALT plans for exponential given by Yum and Choi. Dots, P, = 0.000 1; tic marks, P, = 0.001; asterisks, P,, = 0.01.

1 1. It has been found that the results of Burr Type XI1 for specific values, when it approximates the exponential distribution, are strikingly similar to the result obtained by Yum and Choi [17] [see Figures 6(a)-6(d), 8(a), and 8(b)]. The results of Seo and Yum [ 131, who have used the extreme value distribu- tion instead of the main Weibull distribution, are somewhat different from our results for Burr Type XI1 distribution when it approximates the Weibull distribution. However, the results of Islam and Ahmad [ 61, who have em- ployed Weibull distribution, are strikingly similar to those of Burr Type XI1 when it approximates the Weibull distribution [see Figures 7 (a)-7 (d) and

Ahmad and Islam: Accelerated Life Test Designs 1073

.001 .01 .l .6 ,, .a

a b

Thouundr 1200 1

.001 .Ol .1

C d Figure 9. (a) Optimal ALT plans for Burr XI1 when 6 = 0.5 and m = 1 . (b ) Optimal ALT plans for Burr XI1 when 6 = 3 and m = 1. (c) Optimal ALT plans for Weibull given by Islam and Ahmad when 6 = 0.5. ( d ) Optimal ALT plans for WeibuII given by Islam and Ahmad when 6 = 3. Dots, P,, = 0.000 1 ; tic marks, P,, = 0.00 1 ; asterisks, P,, = 0.0 1.

9(a)-9( d)]. Specific curves have been drawn for the asymptotic variances, optimal low test stress, and optimal proportion of test units of the Weibull distribution and Burr Type XI1 distribution, when it approximates the Wei- bull distribution [see Figures 7(a)-7(d) and 9(a)-9(d)]. The similarity be- tween the two is quite obvious. The curves drawn for asymptotic variance, optimal low test stress, and optimal proportion of test units of the exponen- tial distribution and the Burr Type XI1 distribution when it approximates the exponential distribution is also similar [see Figures 6 (a)-6 ( d) , 8 (a) and

1074 Naval Research Logistics, VoI. 43 ( 1996)

8 (b)] . Therefore it can be safely concluded that Burr Type XI1 distribution as a reliability model in ALT designs is more useful and has wider applica- tions.

6. PROCEDURES FOR PLANNING AN ALT

Based upon the above discussion, we suggest the following guidelines for planning an ALT.

1. Provide guesstimates of P,, and Ph and the ranges of their plausible values. 2. Determine the censoring time. 3. For various values ofK, determine optimal plans with given 6 and m. 4. For given d and rn, conduct a sensitivity analysis with respect to the plausible

values of P, and Ph, to see how AsVar(co) varies around these plausible val- ues.

5. Check the necessity of an ALT based upon s ? and a 7 . 6. Determine the sample size. 7. Adjust N* if necessary based upon the results of the sensitivity analysis.

7. CONCLUDING REMARKS

Asymptotically optimal ALT plans are developed for the Burr Type XI1 distribution assuming periodic inspection and Type I censoring at two test stress levels.

We conclude that the proposed procedure for Burr XI1 distribution yields better optimal ALT designs and is a widely applicable family of reliability models for ALT designs. Com- putational results for various values of the shape parameters indicate that the number of inspections need not be too large and the proposed plan is insensitive to misspecification of imputed failure probabilities at the design and high test stresses. It has also been observed that schemes with equally spaced inspection times at each stress level are administratively convenient and statistically optimal.

APPENDIX

Standardization of Parameters

Let us consider design stress so = 0 and high stress s2 = 1. Under this reparameterzation, let with and without the prime represent the original and standardized scale, respectively. It has the following transformation:

s = (s'- sb ) / ( s ; - sb).

It can be written as

s' = s(s; - sb) + sb.

Next, consider the standardization oftime. Let t , , = tc2 = t:. in the original time scale. In the standardized scale, every parameter whose unit is time must bedivided by I ; . That is,

Ahmad and Islam: Accelerated Life Test Designs

0 = & ‘ / I : ,

and

I = t r / t : .

Therefore, we have the following relationship:

0 = e t ~ b + ~ ; i ’ ) / r , = e ( ~ b + ~ ; ~ s ( s ; ~ ” b ) + s b l ) e - ~ n ‘ ; = ,1(0b+a;lb-1”1C)+0;5(5;--6b)l.

Because & = e(@0+~1’), we have

or, equivalently,

and

1075

& = ~ o + I n t : - ~ ’ l s ~ = ~ o + I n ~ : - P l s ~ / ( s ~ - s b ) .

From Eqs. (6) and ( I I ), one can see

Also.

Let = & + @‘, sb. Then it can be shown that

AsVar(j&) = AsVar(io),

or, equivalently,

AsVar(&) = AsVar(&).

Therefore, no generality is lost under the above standardization.

Two-step Procedure The two-step procedure was adopted to optimize AsVar( Go) with respect to sI and a]. This technique is applied

to our problem according to the following steps:

I . Weoptimizea, (says:). That is, from Eq. ( l o ) ,

1076 Naval Research Logistics, Vol. 43 ( 1996)

The optimal value of al, for 0 < aI < I , is given by

Here aI is the proportion oftest item allocated to s, . Hence it is always positive. Becausedenomi- nator s:QI - Qz > 0 [see Eq. (8 ) ] , \rs:Q,Qz should be positive, as -Qz is negative.

2. We employ the grid search with respect to sI to optimize AsVar( G o ) . That is, a: and the minimum of AsVar( G o ) are determined on the grid s, = d , 2d, 3 d , . . . , where d i s the grid size. Finally, the minimum value of AsVar(co) is determined by optimal sI (i.e., s:) and the corresponding a: among all the grid points considered.

In actual computational experiments. the above method was tried for a given set ofP,,, Ph (and corresponding Po, PI), 6, m, and Kwith 500 different grid points. Besides, the grid size for sI is set to 0.002. Finally, N is set to I , the optimal solution consists ofs: (and the corresponding a:) for which AsVar( Go) attains the minimum among all the grid points considered.

ACKNOWLEDGMENTS

The research of Nesar Ahmad was supported by the Council of Scientific and Industrial Research, India, under Grant No. 9/ 1 12(278)/95-EMR-I. The authors thank the editor, the associate editor, and the referees for their helpful comments and suggestions, which led to improvements in the article. We are also pleased to thank Professor Zaheerudddin for assistance in the preparation of this revision.

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Manuscript received April 1994 Revised manuscript received March 1996 Accepted May 24, 1996