Automotive reliability inference based on past data and technical knowledge

Reliability Engineering and System Safety 76 (2002) 129±137

www.elsevier.com/locate/ress

Automotive reliability inference based on past data and technical knowledge Maurizio Guida a, Gianpaolo Pulcini b,* a

b

Department of Information and Electrical Engineering, University of Salerno, 84084 Fisciano, Italy Department of Statistics and Reliability, Istituto MotoriÐCNR, Via G. Marconi 8, 80125 Napoli, Italy Received 30 June 2001; accepted 8 October 2001

Abstract The constantly increasing market requirements of high quality vehicles ask for the automotive manufacturers to carry outÐbefore starting mass productionÐreliability demonstration tests on new products. However, due to cost and time limitation, a small number of copies of the new product are available for testing, so that, when the classical approach is used, a very low level of con®dence in reliability estimation results in. In this paper, a Bayes procedure is proposed for making inference on the reliability of a new upgraded version of a mechanical component, by using both failure data relative to a previous version of the component and prior information on the effectiveness of design modi®cations introduced in the new version. The proposed procedure is then applied to a case study and its feasibility in supporting reliability estimation is illustrated. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: Automotive reliability; Bayes procedure

1. Introduction The constantly increasing market requirements of high quality vehicles ask for the automotive manufacturers (or their suppliers) to verifyÐbefore starting mass productionÐif new components or parts attain a ®eld reliability target Rp at a speci®ed time period t0 (e.g. warranty period). To this end, reliability demonstration tests are usually carried out by running a number of copies of the new product for a pre®xed time interval either under use or under more severe conditions (accelerated testing). The test time is sometimes chosen to simulate the average mileage during t0 ; say x0 : However, when the reliability target is high and/or each unit is expensive, the test time is usually prolonged in order to simulate a mileage greater than x0 so that the sample size can be reduced. Most of the estimation methods currently used in the industry use only data observed during the demonstration test and ignore any prior knowledge about the product or its predecessors that could be available. However, in the automotive industry, many new products are evolutionary and not revolutionary, thus allowing prior information about their predecessors (past products) to be easily available. Such prior information can be used both in conjunction * Corresponding author. Fax: 139-81-239-6097. E-mail addresses: [email protected] (G. Pulcini), [email protected] (G. Pulcini).

with the results of demonstration tests in order to obtain a posterior estimation of the reliability, and by itself in order to predict the product reliability before performing experimental tests and/or to plan more carefully demonstration tests. In the light of these considerations, the Bayes method appears to be a valid alternative to classical methods [1±3]. In this paper a Bayes procedure is proposed for analyzing failure data of mechanical components in a reliability demonstration test. The use of an informative prior probability density function (pdf) on the unreliability of the upgraded product is suggested on the basis of past data and technical knowledge on the product and/or its predecessor. In particular, we suppose that information on the failure probability of the past product is available through observed failure data possibly coming from heterogeneous contexts (e.g. different type of cars, different environmental and/or operating conditions, different production plants,¼). Moreover, we suppose that information is available on the effectiveness of design or process modi®cations introduced into the new product in order to remove the most critical failure modes of the past product. Both point and interval reliability estimates are then obtained. Finally, a numerical example is developed in order to illustrate the proposed inference procedure. It is shown that a correct use of past data and suf®ciently accurate prior information on the effectiveness of improvement actions allow one to make inference on the reliability of the new product even before performing the demonstration tests.

0951-8320/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0951-832 0(01)00132-6

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M. Guida, G. Pulcini / Reliability Engineering and System Safety 76 (2002) 129±137

2. Prior inference on the reliability of the new product Many new products in the automotive industry are evolutionary and not revolutionary, so that failure data relative to a previous version of a new upgraded product are generally available. These data can be used, in conjunction with the designer belief on the effectiveness of design modi®cations, in order to provide information on the reliability of the new product. 2.1. Past data analysis Information on the failure probability of the past product typically is in the form of the number of failures observed in the population of the past product in use during a given time period t0 ; usually warranty period. In fact, from the manufacturer's viewpoint the fraction failing in the calendar time t0 ; say p…t0 †; conveys information on the warranty costs and on the extent of customers' satisfaction. For most components, however, life is measured rather by the number x of kilometers run than by the calendar time of use. Hence, strictly speaking, the fraction failing in t0 is not a measure of the component unreliability, since different items run a different number of kilometers in t0 : The fraction failing in t0 ; however, can be viewed as an `average unreliability' [4] in t0 when averaging the reliability function with respect to the distribution f …xut0 † of the kilometers run in t0 over the population of the owners. In many cases the past product operates under heterogeneous conditions, for example, it is mounted on cars that differ by weight, engine power or body style. Hence, the product population has to be partitioned in order to obtain sets characterized by homogeneous conditions. Let l denote the number of sets (heterogeneous among themselves) of the whole past population. Each set contains Mi …i ˆ 1; ¼; l† items that have operated, under the same environmental conditions, up to a common time value ti and data consist of the number ri of items failed before ti : Time ti is generally equal to t0 but, sometimes, it can be smaller than t0 when products of set i have not completed the warranty period. Thus, the likelihood function relative to each set is given by ! Mi r L…ri upi † ˆ pi i …1 2 pi †Mi 2ri i ˆ 1; ¼; l …1† ri where pi ˆ pi …ti † denote the unknown average unreliability (up to ti ) of set i: Since the truncation times ti differ among the l sets and generally differ from the time period t0 ; a functional form has to be assumed which describes the behavior of the average unreliability function with the operating time. Past data provided by repair service during warranty period include both early failures and deterioration failures. Hence, a complex model for the product average reliability should be assumed. Unfortunately, these complex models, such as a mixture of Weibull distributions [5], involve a

high number of unknown parameters (at least ®ve) which makes the estimation problem almost intractable. Hence, a simpler model, based on single failure mechanism, has to be assumed, taking in mind that a simple model provides an approximation to the real reliability behavior. This approximation works well when the fraction of early failures is not large with respect to the fraction of failures observed in average until t0 (as a consequence of application of burnin techniques and high quality production process) and when the truncation times ti are not very smaller than t0 (for example, t0 =2 # ti # t0 ). Preliminary analyses, based on actual failure data of a number of mechanical components, have shown that the Weibull function "   # t b p…t† ˆ 1 2 exp 2 a; b . 0; t $ 0 …2† a adequately describes the fraction of failures observed until t, and hence the Weibull model (2) is assumed to describe the behavior of the average unreliability with time. Thus, the likelihood function (1) can be reparameterized in terms of the shape parameter b and of the average unreliability up to b t0 ; namely p0;i ˆ pi …t0 † ˆ 1 2 …1 2 pi †…t0 =ti † : L…ri up0;i ; b†

! Mi h

ˆ

ri

1 2 …1 2 p0;i †…ti =t0 †

b

iri

…1 2 p0;i †…Mi 2ri †…ti =t0 †

b

i ˆ 1; ¼; l

(3)

In order to combine all data sets of past product, we assume that use conditions do not modify the failure mechanism, so that the Weibull function relative to each set is indexed by the same shape parameter b; whereas the average unreliability p0 varies among sets according to a Beta pdf f …p0 † ˆ

pA21 …1 2 p0 †B21 0 Be…A; B†

…4†

whose parameters A and B are unknown and have to be estimated by past data. To this end the parametric empirical Bayes estimation method (see, for example, Ref. [6]) is used. As suggested by a referee, an alternative approach could be the two-stage Bayesian approach [7,8], which however appears to be notably more computationally onerous in the present context. The likelihood function relative to l sets is the product of l likelihood function (3) averaged with respect to the Beta pdf (4) L…past datauA; B; b† Z1 h

/

l Y iˆ1

0

b

1 2 …1 2 p0;i †…ti =t0 †

i ri

b

…1 2 p0;i †…ti =t0 †

…Mi 2ri †1B21 A21 p0;i

dp0;i

Be…A; B†

…5†

M. Guida, G. Pulcini / Reliability Engineering and System Safety 76 (2002) 129±137

An estimate of the hyperparameters A and B can be obtained by maximizing the log-likelihood ` ˆ ln‰L…past datauA; B; b†Š /

l X iˆ1

1n

Z 1 h 0

b

1 2 …1 2 p0;i †…ti =t0 †

iri

b

…1 2 p0;i †…ti =t0 †

…Mi 2ri †1B21 A21 p0;i

 dp0;i

…6† Since closed form solutions for the estimates A^ and B^ are not available, a numerical maximization algorithm is needed. Some guidelines on the selection of appropriate maximization algorithms and on initial guesses are provided in Appendix A. Note that some criticisms on the use of averaged likelihood functions have been raised in Refs. [9,10]. When the size Mi …i ˆ 1; ¼; l† of the sets is not very large, namely a few hundreds, it is convenient to rewrite the likelihood (3) as ! r i Mi X di; j …1 2 p0;i †si; j …7† L…ri up0;i ; b† ˆ ri jˆ0 where di; j ˆ …21†

ri

!

j

and

si; j ˆ …ti =t0 †b …Mi 2 ri 1 j†

Then, the integration with respect to p0;i of the product of Eqs. (4) and (7) can be analytically performed and the averaged likelihood (5) becomes L…past datauA; B; b† /

ri l X Y iˆ1 jˆ0

di;j

Be…A; B 1 si;j † Be…A; B†

…8†

When the set sizes are large, the evaluation of Eq. (8) can be very onerous and can be affected by numerical errors. 2.2. Prior belief on the effectiveness of design modi®cations In the development phase of the new product, on the basis of the observed failure data and failure modes analyses, the designer attempts to remove, through design or process modi®cations, the critical failure modes. When effective, these modi®cations will produce an improvement of the reliability of the new product with respect to the past one. The effectiveness of design modi®cation is generally expressed by an improvement factor d , 1 that measures the ratio of the average unreliability F0 ˆ F…t0 † of the new product (at given time t0 ) to the average unreliability p0 ˆ p…t0 † of the past product, given an observed value of p0 :

d ˆ F0 =p0

p0 used as a basis for the failure modes analyses, so that in principle d cannot be considered a priori independent of p0 : However, since the ¯uctuation of p0 ; expressed by the Beta pdf (4), is generally small, we can assume that, in the range of variation of the past product unreliability, the prior information on d is independent of the value of p0 : Under such an assumption, we have that F0 ˆ d p 0

2l ln‰Be…A; B†Š

j

131

…9†

The uncertainty on the effectiveness of design modi®cation cannot be ignored and thus the improvement factor has to be treated as a random variable. Of course, any information the designer possesses on d depends on the estimated value of

is the product of two a priori independent random variables. Once the prior information on d; say g…d†; is formalized, the prior information on the average unreliability F0 of the new product can be obtained. In order to derive workable prior pdf on d; we observe that the designer is often able to anticipate an interval …d1 ; d2 † that contains, with high probability, the unknown value of the improvement factor. In this framework, two alternatives for g…d† are suggested, which represent different levels of detail of the prior knowledge on the improvement factor. 1. The prior belief of the designer regards the overall effectiveness of design modi®cations, so that, by assuming that each value of d in the interval …d1 ; d2 † is equally likely, a uniform pdf is used for d : gU … d† ˆ

1 d2 2 d1

d1 # d # d2

…10†

2. The failure modes analyses supply the designer with the relative frequency fj …j ˆ 1; ¼; r† of each failure mode, namely the fraction of observed failures caused by the failure mode j to the total number of observed failures P ( rjˆ1 fj ˆ 1). This allows the designer to express a prior belief on the effectiveness of design modi®cations with respect to each failure mode j; say dj …j ˆ 1; ¼; r†; so that the effectiveness of all design modi®cations is given by the weighed sum of dj s, where the weighs are the relative frequencies fj :

dˆ

r X jˆ1

fj dj

Assuming that the prior belief on the improvement factor dj consists of an interval …d1;j ; d2;j † of equally likely values for dj : g…dj † ˆ

1 d2; j 2 d1; j

d1; j # dj # d2; j

the subjective pdf of d can be approximated by a Beta pdf gB …d† ˆ

da21 …1 2 d†b21 Be…a; b†

…11†

with mean and variance equal to the exact mean and

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M. Guida, G. Pulcini / Reliability Engineering and System Safety 76 (2002) 129±137

variance of d : r X md ˆ fj …d1; j 1 d2; j †=2

with mean and variance and

jˆ1

r X

s d2 ˆ

jˆ1

fj2 …d2; j 2 d1; j †2 =12

Of course, if no modi®cation is introduced to reduce the failure mode i, then: d1;i ˆ d2;i ˆ 1: The Beta parameters a and b are then easily evaluated as a ˆ …1 2 md †…md =s d †2 2 md

and

b ˆ a…1 2 md †=md By combining the Beta pdf (4) with the subjective pdf (10) or (11), the subjective pdf on the average unreliability F0 of the new product at time t0 is obtained. Of course, these ^ the subjective pdfs contain, through the estimates A^ and B; information provided by the past data. 1. By using the uniform pdf (10) we have that

A^ a ; a 1 b A^ 1 B^

VB {F0 } ˆ

^ A^ 1 1† A… a…a 1 1† ^ A^ 1 B^ 1 1† …a 1 b†…a 1 b 1 1† …A^ 1 B†… 2 ‰EB {F0 }Š2

Eq. (12) or (13) provides the formalization of the prior information on the average unreliability F0 of the new product up to the speci®ed time t0 ; based on failure data of past products and prior belief on the effectiveness of design modi®cations. On the basis of Eq. (12) or (13), a predictive estimate of the average reliability of the upgraded product can be obtained. For example, the prior subjective probability that the new product attains the ®eld reliability target Rp is given by integrating Eq. (12) or (13) over the interval …0; 1 2 Rp † : Pr{R0 $ Rp } ˆ

gU …F0 upast data† ˆ

Zpmax ^ 1 1 ^ pA22 …1 2 p0 †B21 dp0 ^ B† ^ d2 2 d1 pmin 0 Be…A;

ˆ

A^ 1 B^ 2 1 1 ^ A 2 1 d2 2 d1

Z 1 2 Rp 0

gX …F0 upast data†dF0

…14†

and a point estimate of the reliability of the new product is given by EX {R0 } ˆ 1 2 EX {F0 }

^ 2 BI…pmin ; A^ 2 1; B†Š ^  ‰BI…pmax ; A^ 2 1; B† 0 # F0 # d 2

(12)

where

In the case that the predicted value of R0 is far from the reliability target, then such predictive estimates can provide evidence of the need of introducing further improvements into the new design, before performing demonstration tests. 3. Demonstration test and likelihood function

pmax ˆ min…1; F0 =d1 †; and

BI…x; r; s† ˆ

pmin ˆ min…F0 =d2 ; pmax †

Zx 0

z

r21

…1 2 z†

s21

dz=Be…r; s†

is the incomplete Beta function. The mean and variance of Eq. (12) are, respectively d 1 d1 A^ ; EU {F0 } ˆ 2 2 A^ 1 B^ VU {F0 } ˆ

^ A^ 1 1† A… d32 2 d31 2 ‰EU {F0 }Š2 ^ A^ 1 B^ 1 1† 3…d2 2 d1 † …A^ 1 B†…

2. By using the Beta pdf (11) we obtain gB …F0 upast data† ˆ £

EB {F0 } ˆ

Z1 F0

^

F0a21 ^ B† ^ Be…a; b†Be…A; ^

p0A2a21 …1 2 p0 †B21 …1 2 F0 =p0 †b21 dp0

0 # F0 # 1

(13)

In the automobile context, demonstration tests cannot simulate the different mileage values run during t0 by the population of owners, and hence tests are generally carried out by simulating the mileage x0 ˆ E{xut0 } that is run, in average, during t0 : In such a way, tests allow one to make inference rather on the reliability (evaluated at the average usage x0 ) than on the average reliability R0 : However, using results of reliability theory, the average reliability can be approximated by the reliability R…x† evaluated at x0 [11] Z1 R…x†f …xut0 †dx ù R…x0 † R0 ; E{R…xut0 †} ˆ 0

Thus, the same functional law can be used to describe the component unreliability with mileage and the fraction failing with calendar time (at least in the range of t values where the above approximation works well) and test results allow inference on the average reliability to be made. It can be easily shown that the above approximation works well when the product reliability is high and the standard deviation of f …xut0 † is quite smaller than the mean value x0 : Furthermore, the reliability target of the new product is

M. Guida, G. Pulcini / Reliability Engineering and System Safety 76 (2002) 129±137

often high, each test is very expensive, and only few copies of the new product are available for testing. Thus, tests are often prolonged by simulating a mileage x greater than x0 (prolonged test). Let Ni …i ˆ 1; ¼; m† denote the number of exchangeable copies of the new product tested for a mileage xi . x0 …i ˆ 1; ¼; m†: Suppose that ki …i ˆ 1; ¼; m† items fail before xi : Hence, the likelihood function relative to the above data is ! m Y Ni Fiki …1 2 Fi †Ni 2ki …15† L…new datauF1 ; ¼; Fm † ˆ k iˆ1 i where Fi ˆ F…xi † …i ˆ 1; ¼; m† is the unreliability at xi of the new product. In order to make inference about the unreliability F0 at x0 (or, approximately, about average unreliability at t0 ), we assume that the application of burn-in techniques and a high quality production process assure a very small fraction of early failures. Thus, deterioration failures constitute the predominant part of all failures observed both during the whole test and until x0 : The Weibull function R…x† ˆ exp‰2…x=u†b Šbis then assumed so that, substituting Fi ˆ 1 2 …1 2 F0 †…xi =x0 † in Eq. (15), the likelihood function is rewritten in terms of the shape parameter b and the unreliability F0 at x0 : m h i Y b ki 1 2 …1 2 F0 †…xi =x0 † L…new dataub; F0 † / iˆ1

b

£ …1 2 F0 †…Ni 2ki †…xi =x0 † ˆ where qi; j ˆ …21† j

ki j

ki m X Y iˆ1 jˆ0

…16† qi;j …1 2 F0 †hi;j

! and

hi; j ˆ …xi =x0 †b …Ni 2 ki 1 j†

4. Posterior inference on the reliability of the new product The likelihood function (16) relative to prolonged tests is parameterized in terms of the product unreliability F0 and of the shape parameter b: Thus, a joint prior pdf on F0 and b has to be formalized. Under the assumption that F0 and b are prior independent random variables, the joint pdf is the product of a subjective pdf on F0 and a subjective pdf on b: The prior information on F0 is provided by Eq. (12) or (13), according to the level of detail of prior information on the improvement factor. However, the use of each one of the above exact subjective pdfs in conjunction with the likelihood function (16) does not provide mathematical tractability. Thus, it is convenient to approximate the exact subjective pdf on F0 with a pdf function that assures a good mathematical tractability of the posterior pdf. At this aim, we approximate the subjective pdf on F0 through a

133

Beta pdf g…F0 upast data† ˆ

F0c21 …1 2 F0 †d21 Be…c; d†

…17†

which has the same mean and variance of the exact pdf. Thus, the Beta parameters c and d are given by c ˆ ‰1 2 E{F0 }Š dˆc

‰E{F0 }Š2 2 E{F0 } V{F0 }

and

1 2 E{F0 } E{F0 }

Several applications have shown that the above pdf is a good approximation of the exact pdf of F0 ; especially when the prior information on the improvement factor is more detailed and is formalized through the Beta pdf (11). In order to formalize a subjective pdf g…b†; we suppose that the uncertainty on the true value of b can be represented by an interval …b1 ; b2 † including the unknown b value. Since the assumption of the Weibull model arises from the hypothesis that deterioration is the predominant mechanism of the observed failures, then b1 $ 1: If b can assume with equal probability any value in the range …b1 ; b2 †; then a uniform pdf is chosen g…b† ˆ

1 b2 2 b1

b1 # b # b2

…18†

By combining the likelihood function (16) with the subjective pdfs (17) and (18), the joint posterior pdf on b and F0 follows. For mathematical tractability, the likelihood is rewritten as L…new dataub; F0 † / where q ˆ …21†

j1 1¼1jm

k1 j1

k1 X

¼

j1 ˆ0

! ¼

km X jm ˆ0

km

q…1 2 F0 †h

…19†

! ;

jm

h ˆ …x1 =x0 †b …N1 2 k1 1 j1 † 1 ¼ 1 …xm =x0 †b …Nm 2 km 1 jm † Expression (19) does not suffer from numerical problems since the number ki …i ˆ 1; ¼; m† of failures observed during the demonstration test is generally small (a few units). Thus, the joint posterior subjective pdf results in

p…b; F0 uall data† ˆ

k1 km X 1 X ¼ q…1 2 F0 †h1d21 F0c21 …20† D j ˆ0 j ˆ0 1

m

where Dˆ

k1 Zb2 X

b1 j ˆ0 1

¼

km X

qBe…c; h 1 d†db

jm ˆ0

The marginal posterior subjective pdf on the average unreliability of the new product at t0 is

134

M. Guida, G. Pulcini / Reliability Engineering and System Safety 76 (2002) 129±137

Table 1 Failure data of past component C Model i

Volume sold, Mi

Observation time, ti (years)

No. of failures, ki

1 2 3 4 5 6 7 8 9 10

20,000 25,000 30,000 40,000 50,000 30,000 30,000 6000 7000 5000

1.00 0.75 0.50 1.00 1.00 1.00 0.75 1.00 1.00 0.50

416 195 75 74 184 160 156 53 60 10

p…F0 uall data† ˆ

k1 km X F0c21 Zb2 X ¼ q…1 2 F0 †h1d21 db D b1 j ˆ0 j ˆ0 1

m

…21† and the posterior subjective probability that the new product attains the reliability target Rp is given by Pr{…R0 uall data† $ Rp } k1 km X 1 Zb2 X ¼ qBI…1 2 Rp ; c; h 1 d†Be…c; h 1 d†db ˆ D b1 j ˆ0 j ˆ0 1

m

…22† A point estimate of the average reliability of the new product is given by E{R0 uall data} ˆ 1 2

k1 km X 1 Zb2 X ¼ qBe…c 1 1; h 1 d†db D b1 j ˆ0 j ˆ0 1

m

In the case that no failure occurs during demonstration tests (success run hypothesis), the likelihood function (16) becomes …23† L…no failureub; F0 † / …1 2 F0 †h0 Pm b where h0 ˆ iˆ1 Ni …xi =x0 † is often denoted as the equivalent number of copies put on test. The posterior subjective pdf on F0 is then

p…F0 uno failure† ˆ D0 ˆ

Zb2 b1

F0c21 Zb2 …1 2 F0 †h0 1d21 db D 0 b1

Be…c; h0 1 d†db

and the posterior subjective probability results in Pr{…R0 uno =failure† $ Rp } ˆ

1 Zb2 BI…1 2 Rp ; c; h0 1 d†Be…c; h0 1 d†db D0 b1

…24†

The subjective probability (24) can represent a useful tool for sizing the demonstration test. In fact, by iterative procedures, the analyst can determine a set of values

…N1 ; ¼; Nm ; x1 ; ¼; xm † which should allow one to verify, in case no failure occurs during the test, that the new product attains the reliability target Rp with a pre®xed subjective probability level. 5. Numerical application The proposed methodology has been applied to a real case study. Data have been slightly modi®ed from actual ones for security reasons. The manufacturer intends to verify if the upgraded version of the component C attains the ®eld reliability target Rp ˆ 0:993 (at the warranty period t0 ˆ 1 year) at a subjective probability level g ˆ 0:9: The original version of the component is mounted on 10 models that differ for years and body style. Past data were collected by repair service during warranty period, but some models were not 1 year old at the end of the observational study, so that their failure data refer to a smaller time period ti , t0 : Table 1 shows the data of the past component C by model years and body styles. By using the DBCONG routine of the IMSL Math/ Library w [12] with the initial guesses b ˆ 1:5 and A ˆ 1:98 B ˆ 277 (solutions of Eq. (A1)), the maximum likelihood estimates of the likelihood (5) are A^ ˆ 2:90; B^ ˆ 335 and b^ ˆ 1:75: Thus, the average unreliability p0 of the past component C, at t0 ˆ 1 year; varies according to the Beta pdf (4) with parameters A^ ˆ 2:90 and B^ ˆ 335: The mean value and the standard deviation of p0 are equal to 0.00858 and 0.00501, respectively. Failure modes analysis, based on the total number of observed failures, supplies the designer with the relative frequency fj of each failure mode of the component C, and some modi®cations are introduced into the original design in order to reduce the failure occurrences relative to three failure modes. No modi®cation is introduced with respect to the remaining failure modes, whose overall relative frequency is equal to 0.2. Then, the designer is able to express his prior belief on the effectiveness of these modi®cations in terms of the interval …d1;j ; d2;j † …j ˆ 1; 2; 3† for each improvement factor dj (see Table 2). On the basis of these information, the mean and the

M. Guida, G. Pulcini / Reliability Engineering and System Safety 76 (2002) 129±137 Table 2 Prior information on improvement factor Failure mode j

Relative frequency, fj

Prior interval, …d1;j ; d2;j †

1 2 3 Others

0.1 0.5 0.2 0.2

(0.2,0.8) (0.4,1.0) (0.6,0.9) No action

it is necessary to introduce further improvements. The attainment of the reliability target can be veri®ed after performing demonstration tests, whose results can reduce the uncertainty about the reliability of the new product and thus increase the subjective probability that R0 . Rp : The demonstration test consists in prolonged tests which simulate the component use during x1 ˆ 3:3x0 and x2 ˆ 2:0x0 km; where the average mileage in t0 ˆ 1 year is x0 ˆ 15; 000 km: The prolongation factors xi =x0 …i ˆ 1; 2† depend on the time available for testing and on the procedures used to prolong the tests. The posterior subjective probability (24) under the success run hypothesis is then used for sizing the demonstration test, by determining the numbers N1 and N2 of copies of the new component to be tested at x1 and x2 ; respectively, in order to assure (in case that no failure occurs) the pre®xed subjective probability level. A pair of acceptable values is N1 ˆ 26 and N2 ˆ 51: The demonstration test is then performed by testing N1 ˆ 26 copies of the component C at 49,500 km, and N2 ˆ 51 copies at 30,000 km. Test results consist of k1 ˆ 2 failures observed in the ®rst sample and no failure in the second sample. Since no strong information on the parameter b is available (except that b . 1), the uniform pdf (18) over the interval …1; 3† is used as subjective pdf for b: The exact prior subjective pdf (13) on F0 is approximated by the Beta pdf (17) with parameters c ˆ 2:75 and d ˆ 425: Then, the posterior subjective pdf on F0 results in

δ

standard deviation of the effectiveness of all design modi®cations are equal to 0.75 and 0.0904, respectively, so that the prior belief on d is formalized by the Beta pdf (11) with parameters a ˆ 16:47 and b ˆ 5:49 (see Fig. 1). By combining the Beta pdf on p0 with the Beta pdf on d; the prior subjective pdf (13) on the average unreliability F0 of the upgraded component C (at t0 ˆ 1 year) is obtained. The prior mean and the standard deviation are EB {F0 } ˆ 0:00643 and ‰VB {F0 }Š1=2 ˆ 0:00386; respectively. In Fig. 2, the subjective pdf on F0 is compared to the subjective pdf on p0 and the reliability growth that should be obtained through design modi®cations is depicted. The prior subjective probability (14) that the upgraded component C attains the reliability target Rp ˆ 0:993 is equal to 0.636. Since the prior mean EB {R0 } ˆ 0:99357 is greater than Rp and the prior subjective probability is not very far from g ˆ 0:9; then the management does not think

p…F0 uall data† ˆ δ

where

δ

135

q ˆ …21†

j

2 j

2 F0c21 Z3 X q…1 2 F0 †h1d21 db D 1 jˆ0

…25†

! h ˆ …3:3†b …26 2 2 1 j† 1 …2:0†b 51

The posterior mean of the average reliability R0 at t0 ˆ 1 year is E{R0 uall data} ˆ 0:99469; and the posterior subjective probability that the new component C attains the reliability target is: Pr{…R0 uall data† $ Rp } ˆ 0:768: Test results produced a posterior point estimate of R0 a little greater than the prior estimate, and an appreciable increase of the subjective probability that the reliability target has been attained, as a consequence of the reduction of the uncertainty on the R0 value. The reduction is clearly shown in Fig. 3, where the prior and the posterior subjective pdfs of F0 are shown, as well as the posterior subjective probability that R0 is greater then or equal to 0.993. Note that the exact subjective pdf of F0 and the Beta approximation are practically indistinguishable.

δ

δ

δ

δ

δ

Fig. 1. Prior information on the improvement factor d:

Acknowledgements The authors would like to thank the referee for his/her

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M. Guida, G. Pulcini / Reliability Engineering and System Safety 76 (2002) 129±137

contrary, an error in the sign of the derivatives. Thus, very high precision has to be used in the numerical evaluation of Eq. (6) and caution has to be used in assessing the accuracy (neither low nor too high) of the estimate of derivatives. Hence, a maximization algorithm that uses a user-supplied gradient function is recommended. A second problem is the value of the initial guess in the maximization routine (starting point). We suggest obtaining a starting point as it follows. For each set of past data (Mi ; ri ) …i ˆ 1; ¼; l†; we ®rst estimate the average unreliability at the time ti as

F0

p0

p^ i ˆ p^i …ti † ˆ ri =Mi Fig. 2. Prior pdfs of average unreliability p0 of the past component (dotted line) and of average unreliability F0 of the upgraded component (solid line).

Now, we estimate the average unreliability at t0 by using a plausible value, say b0 ; for the unknown Weibull parameter b

useful comments. This research has been partially supported by Centro Ricerche FIAT and Elasis contracts.

p^ i …t0 † ˆ 1 2 …1 2 p^ i †…t0 =ti †

and use the weighted sample moment estimators given by Martz and Waller [13] for estimating the hyperparameters A and B of the Beta pdf (4)   MT p w 2 Mw2 A~ 1 B~ ˆ and MT Mw2 2 lp w 2 …MT 2 l†p2w …A1†

Appendix A The main problem in the search of the maximum likelihood estimates is due to the fact that the log-likelihood (6) has to be evaluated via numerical integration and its derivatives have to be estimated through a ®nite-difference method that involves, for each derivative, a double numerical evaluation of Eq. (6). Hence, the attempt to obtain very accurate estimates of the derivatives can produce, on the

~ A~ ˆ pw …A~ 1 B† where l X

MT ˆ p w ˆ

b0

Mi

is the total size of past population

Mi p^i …t0 †=MT

is the weighted mean of p^ i …t0 †

iˆ1 l X iˆ1

Mw2 ˆ

l X iˆ1

Mi ‰p^ i …t0 †Š2 =MT

is the weighted second moment of p^ i …t0 †

These values are then used as initial guess in the maximization routine. References

1 -R *

Fig. 3. Prior (dotted line) and posterior (solid line) subjective pdfs of the average unreliability F0 of the upgraded component.

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