Application of Bayes Statistics to Reduce Sample-Size, Considering a Lifetime-Ratio

Anna Krolo Institute of Machine Components University of Stuttgart 0 Stuttgart Bettina Rzepka 0 Institute of Machine Components University of Stuttgart 0 Stuttgart Bemd Bertsche 0 Institute of Machine Components University of Stuttgart Stuttgart

Key Words: Bayes statistics, Lifetime-ratio, Sample-size, Prior knowledge, Weibull distribution, Knowledge factor

SUMMARY & CONCLUSIONS

Due to short development schedules, the amount of time available for testing is often unequal to the required lifetime of the product. Either a reduction or increase of the test time affects the confidence level of the reliability test.

In this paper the so-called "lifetime-ratio" was introduced for the Kleyner et al. method and for the Bayes approach, which uses a uniform distribution as prior knowledge. With the lifetime-ratio, it is possible to consider a case where the test time is unequal to the projected lifetime. Apart from the simple case of the success run, we also took into account that failures may occur during the test. In addition, we showed that the lifetime-ratio is dependent on the number of items that must be tested for a required reliability and on the desired confidence level.

The knowledge factor was introduced to the BeyedLauster method. The methods of Beyer/Lauster and Kleyner et al. were compared and discussed using a practical example where the target reliability of an automotive component must be en- sured by a success run. Furthermore, we discussed the condi- tions under which the methods yield the same posterior distri- butions and under which the results may vary. The methods yield the same results if the knowledge factor is zero or one. If the results from the preceding test can be completely trans- ferred, use of the BeyedLauster method is recommended due to its simplicity. As far as a knowledge factor O<p< 1 is concemed, the BeyedLauster method leads to results that are more conservative compared with the results by Kleyner et al.. Thus, the tests would need to be more extensive.

1. INTRODUCTION

As technology advances, customers continually make high demands on a product performance and quality. Therefore, the companies are exposed to an increasing competition among manufacturers. An ever decreasing development time and in- creasing complexity of technical products make it increasingly difficult to maintain market position.

Shorter development times contrasted with customer de- mands for higher reliability and product life require a care-

fully thought-out and optimized product development process (ref: I ) . This also pertains to the product test program (ref: 2).

The higher the product reliability requirements, the more ex- tensive the test must be to demonstrate the required reliability. If no failures occur during the test, the number of test items can be derived from the well-known equation for the "success run", which is a special case of binomial sampling (ref: 2).

There exist different Bayesian methods for reducing the level of testing. These methods consider previous knowledge obtained from previous tests or calculations and include it in the planning of subsequent reliability tests. Two interesting methods were published by BeyerILauster (ref: 3) and Kleyner et al. (ref: 4).

Notation &Acronyms parameters of the beta disbibution parameters of the beta distribution resulting from the first test stage confidence level event lifetime-ratio conditional probability of R given E reliability reliability at C = 63.2% as prior knowledge from BeyedLauster scale parameter (characteristic lifetime) of the Weibull distribution shape parameter of the Weibull distribution probability density function (pdf) prior pdf of the reliability posterior pdf of the reliability

sample-size sample-size of the first test stage sample-size of the second test stage time location parameter of the Weibull distribution specified product lifetime

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4 test time X number of failures B(.) Beta function P knowledge factor pdf probability density function

2. LIFETIME-RATIO

Given short development times, the amount of time avail- able for testing is often unequal to the required product life- time (ref: 5). Either a reduction or an increase in the test time affects the confidence level of the reliability test. Therefore, the so-called lifetime-ratio (ref: 2, 6) is used to consider a case where the test time tt is not equal to the required lifetime t, . The lifetime-ratio L, is defined as

2.1 Weibull Distribution

The usual description of the failure behavior of automotive systems or components is provided by the three-parameter Weibull distribution (e.g. ref: I ) , with the pdf

For the two-parameter Weibull distribution where the loca- tion parameter to = 0, the reliability at the required product lifetime t, in the field is given by

-($Y R( t s ) = e ( 3 ) If the test time tl is unequal to the product's lifetime, the re-

liability is

-(q R(tl) = e .

The logarithm of eq. (3) and (4) yields b

In R(t,) = -( 5) and

hR(t()=-($) b .

Thus, the quotient of eq. ( 5 ) and (6) results in the following conditions

As obtained from eq. (7), the reliability for a given test time tt depends on the reliability at a specified lifetime t, and the lifetime-ratio L, ( reJ2)

b R(t,) = R(t,)Lr . (8)

Besides the two-parameter Weibull distribution, the location parameter in the three-parametric case has to be considered. As a result, one obtains the following equation for the life- time-ratio (ref: 2)

t -to

t, -to (9) L, =t.

2.2 Success Run

If no failures occur during the test, the number of test items can be derived from the well-known equation for the "success run", which is a special case of binomial sampling (ref: 2, 5). Thus, the confidence level C for reaching the required reliabil- ity within the test time tt can be determined as follows

C = 1 - R(t t )n . (10)

C=l-R(t,)L' n . (1 1)

With eq. (8) the confidence level can be calculated from b

As can be seen from eq. (1 l), the shape parameter of the Weibull distribution has to be known if the time available for testing is not equal to the required lifetime of the product.

3. METHODS BY MEANS OF BAYES THEOREM

The methods described in the following chapters are all based on the Buyes theorem (ref: 7). The posterior pdf is de- fined as

(12) P ( E I R ) f ( R )

J P ( E I R ) f ( R ) U

f ( R I E ) =

-m

for the continuous case (ref: 5) where f ( R ) is the prior pdf

of the reliability R and P(E1R) is equivalent to the condi-

tional probability of E , given R . Hence, the confidence level can be determined by integrating eq. (12).

In this paper the Buyes theorem will be used for determining the amount of testing necessary to ensure the required reliabil- ity of an automotive component. Moreover, the lifetime-ratio will be added.

3.1 Uniform Prior pdf

If no prior information is available, the uniform prior pdf within the Bayes theorem can be used (ref: 8). Considering the lifetime-ratio (see eq. (8)) and integrating the posterior pdf, yields the confidence level for the simple case of success run with n test items

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(13) C = R - - 1 - RL,bn+' I /R.L'b"dR.

0

If one has to take into account that a number x of failures may occur during the test, the confidence level for the re- quired reliability may be found from

considering binomial sampling. Eq. (14) can only be solved numerically.

3.2 Beyer/Lauster Method

The BeyedLauster method (re$ 3) does not require the prior distribution as a whole, but considers only one value as prior information - namely the reliability Ro at the confidence level C = 63.2%. This results from the derivation where the Weibull distribution is transformed into an exponential one (see ref 3 for more details). The confidence level for the suc- cess run can be determined by the following formula, consid- ering the lifetime-ratio

c = 1 - RL,bn+l/ln(l/Ro) (15) For a number x of failed items, the confidence level may be

found from

(16) The reduction of the sample-size can be calculated for the

success run, dependent on the prior reliability Ro

If one considers the lifetime-ratio for a success run, the con- fidence level becomes

and is only solvable with numerical methods. Eq. (1 8) leads to eq. (1 3) for a knowledge factor p = 0 . The beta distribution with the parameters A and B is obtained by the evaluation of failure data of an older product or from preceding tests.

With regard to the occurrence of x failures during the test, the confidence level can be determined by

(19) A numerical solution of eq. (19) is unavoidable. Eq. (19)

corresponds to eq. (14) for a knowledge factor p = 0.

4. CASE STUDY: TEST PLAN FOR AN AUTOMOTIVE COMPONENT

In this case study the required reliability targets for an auto- motive component have to be assured. First, the component is tested on the test bed. Afterwards, it is tested under real oper- ating conditions. The results of the test bed are to serve as previous knowledge for further tests that are necessary to prove the reliability requirements.

4.1 Reliability Targets and Test Conditions

The component's reliability is required to be 90% at a speci- fied lifetime, considering a confidence level of 90%. For the test on the test bed, 15 test articles are available. The test time is equal to the service life, which means that the lifetime-ratio is equal to one. Further tests under real use conditions have to be planned in order to assure the reliability targets, consider- ing the test results of the test bed,

3.3 Kleyner et al. Method 4.2 Results of the Test Bed

Kleyner et al. (reJ 4 ) propose to use a mixture of a uniform distribution and a beta distribution as prior information. The beta distribution results from an older product or preceding tests for example. The two distributions are combined accord- ing to weights of the so-called "knowledge factor". The knowledge factor p represents how similar the new product is to the old one. For a knowledge factor p = 0 , the new product is not similar to the old one. For a knowledge factor p = 1, the information about the older product is totally transferable to the new product. The knowledge factor cannot be determined quantitatively but has to be estimated.

On the test bed nl = 15 items were tested successfully. Con- sequently, the test was a success run. Figure 1 shows the con- fidence level C as a hnction of the cumulative test time E t , . The diagram is valid for a reliability of 90% which is the tar- get for the component. The shape parameter has been esti- mated with b = 2 . Because of internal data the abscissa is standardized by the service life to conceal the real values of the component's lifetime at the reliability of 90%. In this case the test time for one component becomes exactly tt = 1 . Thus, the abscissa ends at the cumulative test time that is equal to the number of items available for testing - Le., in our case 15.

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A uniform distribution has been considered as prior knowl- edge before starting the tests on the test bed. Hence, the con- fidence level starts with C=O.1 at a test time tt = O . The effect is that a confidence level of 10% has already been reached before performing any test.

1

0.8 0

4 0.6

- e,

e, u 8

0" 0.4

0.2

0

c

0 5 10 15 Cumulative test time Et,

Figure 1: Confidence level C dependent on the cumulative test time Zft (Reliability R(t,) = 90% , shape parameter b = 2 )

As shown in Figure 1, the rate of increase in the confidence level is higher in the beginning when starting the test at a low confidence. While testing the first 7 items, the confidence level increases from 10% to 57% - a difference of 47%. Fur- ther tests with the next 8 items yield a confidence level of 8 1.5% in the end. Therefore, the increase is only 24.5%.

The requirement for the component is to reach a confidence level of 90% for the reliability of 90%. The tests on the test bed yield a confidence level of 8 1.5%. Consequently, further tests are necessary to receive the additional 8.5%.

4.3 Results of the Tests Under Use Conditions

In the following, the methods of BeyerILnuster and Kleyner et al. are used for the determination of the posterior distribu- tion that results from the tests under use conditions. First, one must derive the prior information from the results of the tests performed on the test bed. In the second test stage, i.e. the test under real use conditions, n2 = 3 items are available for test- ing. Now, the test time must be determined in order to attain a confidence level of 90% for a reliability of 90% at the end of the tests.

The Kleyner- et al. method requires the beta pdf of the reli- ability. With the number of items from the first test and the consideration of a uniform prior pdf, the posterior pdf that results from the first test stage is a beta distribution with the parameters A l = 16 and B , = 1 .

The similarity of the results from the preceding tests to the following tests under normal operating conditions are consid- ered by the knowledge factor. Thus, the prior distribution for the second test stage is given by

-- - RI5 +(l-p) = 16pRI5 +(l-p) , P(16,1)

After successfully performing the tests under normal use conditions, the confidence level may be found from eq. (18)

The knowledge factor cannot be calculated but has to be es- timated. Therefore, the lifetime-ratio was determined for dif- ferent knowledge factors. Figure 2 shows the confidence level C depending on the lifetime-ratio L,. and different knowledge factors p for a success run test with 3 items. As obtained from the posterior distribution, the knowledge factor p = 1 yields a lifetime-ratio 1.4. Consequently, the 3 items have to be tested without failure at least 1.4 times longer than the re- quired lifetime of the component.

If it is not valid to use the results of the preceding tests for the determination of the component's reliability, one has to choose the knowledge factor p=O. In this case, the confi- dence level can be calculated directly from eq. (13). Thus, the results from the test bed have not been taken into account within the calculation. As a result, the tests have to be started fiom the beginning, where the confidence level was only 10%.

According to Figure 2, the knowledge factor influences the confidence level that has been reached by the preceding test.

1

0.8 u 4 0.6

c1 e,

e,

8 0.4

0.2

0

8 u

0 0.5 1.0 1.5 2.0 2.5 3 Lifetime-ratio L,

Figure 2: Confidence level C dependent on the lifetime-ratio L, for various knowledge factors p by means of Kleyner et al. (sample- size m2 = 3 for success run, reliability R ( t , ) = 90% , shape parame- ter b = 2 )

The higher the knowledge factor has been estimated, the higher the confidence level becomes in the beginning. For p = 1 the confidence level starts with the value that has been

verified by the preceding tests ( C(Lr = 0) = 81.5% in Figure

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2). As shown by Figure 2, the variation of the lifetime-ratio is smaller for a knowledge factor 0.5 2 p S 1 than for a knowl- edge factor 0 I p 1 0 . 5 . The values of the lifetime-ratio are summarized in Table 1.

1

The confidence level using Beyer/Lauster is calculated with eq. (1 5 ) if no failures occur during the test. The prior reliabil- ity & = 93.94% is given from eq. (22). Solving eq. (15) for the lifetime-ratio L,. yields

I .40

Table 1: Some values of the lifetime-ratio L, dependent on the knowledge factor p (according to Figure 2) /I &owledge Lifetime-ratio L,. factor p

0.9 1.48 I

I b = 2 II I

Figure 3 shows the lifetime-ratio as a function of the knowl- edge factor for the success run with 3 test articles in the sec- ond test stage. The reliability is 90% as well as the confidence level, according to the reliability target. As shown in Figure 3, the lifetime-ratio is greater than one for every value of the knowledge factor. Thus, the test time is in any case greater than the required lifetime, provided that the reliability targets are maintained.

3 R(t,) = 90%

1 0 0.2 0.4 0.6 0.8 1

Knowledge factor p

Figure 3: Lifetime-ratio L, dependent on the knowledge factor p by means of Kleyner et al.

For the Beyer/Lauster method the value of the reliability at a confidence level of 63.2% is needed as prior information. As obtained from eq. (1 3) with a lifetime-ratio L,. = 1 , the reli- ability becomes

1 1 Ro = ( l - C ) Z =(1-0.632)= =93.94%,

after the 15 items performed the test successfully.

= {m = 1.4.

Hence, the Beyer/Lauster method yields the same result as the Kleyner et al. method when using a knowledge factor

In the following, the knowledge factor is introduced to the Beyer/Lauster method. Accordingly, the reliability at the con- fidence level of 63.2% that has been reached by the preceding test is now derived from eq. (20). Figure 4 shows the curves for the confidence level as dependent on the lifetime-ratio for various knowledge factors. Additionally, the graphs deter- mined by means of Kleyner et al. are included for comparison. The diagram shows that the methods yield different results although the value of the knowledge factor is equal, except for the special cases when p = 0 and p = 1.

p = l .

1

0.8 u 0 ..$ 0.6 e, V

8

8 0.4

0.2 u

p = 0.5

- Kleyner i

I B eyer/Lauster 0 , , I I I I I

0 0.5 1.0 1.5 2.0 2.5 3 Lifetime-ratio L,

Figure 4: Confidence level C dependent on the lifetime-ratio L, for various knowledge factors p by means of Beyer/Lauster, compared with the results by means of Kleyner et al. (sample-size n2 = 3 for success run, reliability R(t,) = 90% , shape parameter b = 2 )

The differences in the determination of the lifetime-ratio are shown in Figure 5 for a reliability and confidence level of 90%. According to Figure 5, the Beyer/Lauster method leads to results that are more conservative compared with the results using Kleyner et al.. Therefore, it is obvious that the test would have to be more extensive to prove the reliability re- quirements.

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2.5 4 1 -

0 0.95 - - 0 0 3

% 0.90 - a G F: 6 0.85

0 m .d c

L : 2

!$ .e 3 c

1.5

i-

1, - ,,,, BeyertLauster I

n2 = 3 (Success Run)

n z = l O \ n, = 8 \

0.80 1 I I I I I 1

Figure 5: Comparison of the lifetime-ratio L, dependent on the 0 0.5 1.0 1.5 2.0 2.5 3 knowledge factor p by means of Beyer/Lauster and Kleyner et al. Lifetime-ratio L,

4.4 Interpretation of the Results Figure 6: Confidence level C dependent on the lifetime-ratio L, for various sample-sizes n2 by means of BeyedLauster and Kleyner et at.

The methods of Beyer/Lauster and Kleyner et al. yield the same results, if the knowledge factor is zero or one. Figure 6 shows the confidence level for different sample-sizes as de- pendent on the lifetime-ratio determined by the two different methods with a knowledge factor p = 1. The preceding test

for a knowledge factor P = 1 (Reliability Wt,) = 90% , shape pa- rameter b = 2 )

REFER E K E S corresponds to the success run with 15 test articles where the lifetime-ratio has been equal to one. As can be seen from Fig- ure 6, the curves for the methods of BeyedLauster and Kley- ner et al. are one on top of the other. Hence, if the results from the preceding test can be completely transferred, use of the Beyer/Lauster method is recommended due to its simplicity. If components are modified during the test or tested under dif- ferent operating conditions, it is advisable to introduce the knowledge factor. Otherwise, there exists the risk of failing to meet the product requirements. One should keep in mind that the BeyedLauster method leads to results that are more con- servative compared with the results using Kleyner et al. as far as a knowledge factor 0 < p < 1 is concerned. Therefore, the tests would need to be more extensive. Anyway, there is the disadvantage of having to estimate a suitable value of the knowledge factor.

1. Bertsche, B.; Lechner, G.: "Zuverlassigkeit im Maschinenbau".

2. VDA: "Qualitatsmanagement in der Automobilindustrie - Zuverlassig- Springer-Verlag, 1999

keitssicherung bei Automobilherstellem und Lieferanten". Verband der Automobilindustrie e.V. (VDA), 3. uberarbeitete und erweiterte Auflage, Frankfurt, 2000

sichtigung von Vorkenntnissen". QZ 35 (1990), Heft 2, pp 93-98

"Bayesian techniques to reduce the sample size in automotive electronics attribute testing". Microelectron. Rehab., Vol. 37, No. 6, pp 879-883, 1997

5. Meeker, W.Q.; Escobar, L.A.: "Statistical Methods for Reliability Data". Jon Wiley & Sons, Inc., 1998

6. Rach, E.: "Terms and Procedures of Reliability Engineering". Quality Assurance in the Bosch Group: Technical Statistics, Vol 13, 1996

7. Comfield, J.: "Bayes theorem". Review of the intemational statistical institute, Volume 35:1, 1967

8. Martz, H.F.; Waller, R.A.: "Bayesian Reliability Analysis". Jon Wiley & Sons, Inc., 1982

3. Beyer, R.; buster, E.: "Statistische Lebensdauerprufplane bei Beruck-

4. Kleyner, A,: Bhagath, S.; Gasparini, M.; Robinson, J.; Bender, M.:

BIOGRAPHIES

Anna Krolo, Dipl.-Ing. Institute of Machine Components (IMA), University of Stuttgart Pfaffenwaldring 9 70569 Stuttgart, Germany

Internet (e-mail): krolo@ima.uni-stuttgart.de

Anna Krolo studied mechanical engineering at the University of Stuttgart in Germany and got the academic degree Dipl.-Ing. in 1999. She is a research assistant in reliability engineering at the Institute of Machine Components.

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Bettina Rzepka, Dipl.-lng. Dipl.-Kffr. Institute of Machine Components (IMA), University of Stuttgart Pfaffenwaldring 9 70569 Stuttgart, Germany

Internet (e-mail): rzepka@ima.uni-stuttgart.de

Bettina Rzepka studied business administration at the University of Augs- burg and graduated with the academic degree Dip1.-Kffr. in 1995. After studying mechanical engineering from 1995 - 2000 she got the academic degree DipLIng.. She is a research assistant in reliability engineering at the Institute of Machine Components.

Bemd Bertsche, Prof. Dr.-Ing Institute of Machine Components (IMA), University of Stuttgart Pfaffenwaldring 9 70569 Stuttgart, Germany

Internet (e-mail): bertsche@ima.uni-stuttgart.de

Bemd Bertsche studied mechanical engineering at the University of Stuttgart i n Germany and got the academic degree Dipl.-Ing. in 1984. From 1984 - 1989 he was a research assistant in reliability engineering at the Institute of Machine Components. He was conferred the academic degree Dr.-Ing. from the University of Stuttgart in 1989. From 1989 - 1992 he was a development engineer at DaimlerChrysler AG in Stuttgart, Germany. His main projects were: development of automotive transmissions and four-wheel drive cars. From 1993 - 1996 he was a professor of mechanical engineering at the Uni- versity of Applied Science in Albstadt, Germany. Since 1996 he has been a professor of mechanical engineering at the Institute of Machine components, University of Stuttgart, and head of the reliability engineering department. Since April 2000 he has also been CEO of the Technology Transfer Initiative (TTI GmbH) of the University of Stuttgart. Since May 2001 he has been head of the Institute of Machine Components.

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