Reliability Assessment Based on D-S Evidence Theory

Yongquan SUN, Jianying GUO Department of Measurement Technology and Instrument

Harbin University of Science and Technology Harbin, China

E-mail: sunyongquan@sina.com.cn; guojy@hrbust.edu.cn

Abstract—For many high-reliability products, reliability assessment relying on the tests has become unrealistic, but expert experience shows its advantage to the above problem. In order to resolve the problem that how to fuse the expert experience, we convert the expert experience into the reliability probability density distribution, determine the knowledge framework and basic probability assignment function based on the definition of consistent degree, fuse the basic probability distribution based on D-S fusion rule, and get the fusion distribution weights, finally obtain the reliability of the product. This method has been validated by an example, and shows perfect.

Keywords- D-S evidence theory; reliability assessment; expert experience; consistent degree

I. INTRODUCTION In the process of assessing the reliability of the complex

and high-reliability products, the traditional method based on mathematical statistic theories is no longer applicable to this situation, because of the limitation of the cost, time and so on. Reliability engineers, system designers and domain experts accumulate a large number of knowledge and experience that reflect the actual situation of the product, so expert experience should be given attention. Different experts have different decisions, so it is necessary to fuse different decisions in order to get higher precise and credible result [1,2].

Evidence theory is a new important reasoning method under uncertainty, which is suitable for making decisions when there are uncertain factors, there is advantage to deal with subjective judgments and synthesize the uncertainty knowledge using this method [3]. Booker discussed expert information extraction and the application of fuzzy theory [4,5]; Dey described how to change the expert information to be the probability density distribution[6]; Szwed, Hora and Bram studied the methods to fuse expert experience with different forms[7-9].

Based on the definition of consistent degree of expert experience and fusion rule, we convert the reliability confidence intervals into the probability density distribution, determine the weights, and finally get the product life distribution and product reliability.

II. EXPERT EXPERIENCE FUSION BASED ON D-S EVIDENCE THEORY

A. Description of Expert Experience Experts give vary reliability data according to experience

and field situation, such as failure rate, reliability, failure distribution and fuzzy data [10]. This research fuses the reliability confidence intervals.

B. D-S Fusion rule Definition 1: U is a knowledge framework, the function

m:2U-> [0,1] (2U is the set constituted by all the subsets of U) meets the following conditions

( )( )

0

1

m

m⊂

=⎧⎪⎨ =⎪⎩∑A U

Φ

A (1)

Then we consider m(A) is the basic probability assignment function for A. m(A) expresses confidence in the proposition A, that is, the support of the A.

Definition 2: U is a knowledge framework, function m:2U-> [0,1] is the basic probability assignment function on U, the definition of function BEL: 2U -> [0,1] is

( ) ( )( )BEL m⊂

= ∀ ⊂∑B A

A B A U (2)

Then the function BEL(A) can be considered to be the belief function of A on U. It expresses the summation of the possibility of all the subsets of A, which means the total trust of the A.

Definition 3: BEL1 and BEL2 are two belief functions on the same knowledge framework Θ, m1 and m2 are the corresponding basic probability assignment functions, focal elements respectively are A1,A2,…Ak, and BB1,B2B ,…BBr, then D-S fusion rule for two evidence is as follows [11,12]:

978-1-4244-4905-7/09/$25.00©2009 IEEE 411

( ) ( )1 2

1

,( ) 10

i j

i jm m

m K∩ =

⎧⎪⎪ ∀ ⊂ ≠= ⎨ −⎪

≠⎪⎩

∑A B C

A BC U C ΦC

C Φ

(3)

where, ( ) ( )1 1 2,

1

i j

i ji j

K m m

∩ =

= <∑A B Φ

A B .

C. The Expert experience conversion The reliability is determined through design, manufacture,

management, so the reliability of a special product is fixed. Reliability engineers, product designers and domain experts can give the reliability estimation based on experience when there are no reliability tests, and then the estimations are random variable. Assume that every expert is reliable, then the reliability estimation will swing around the true value, and the nearer away from the true value the greater the likelihood here, so we consider the random variables are under normal distribution N(μ,δ2) according to Central Limit Theorem. In the field, experts give the reliability confidence intervals (RL,RU) with confidence level 1-α, that is, P(RL<R<RU)=1-α, shown in Fig.1.

Reliability confidence intervals given by experts can be considered the reliability is under normal distribution N(μ,δ2), as well as the probability density function (pdf) is f(R), and the probability of the reliability falls in (RL,RU) is 1-α, which is the area surrounded by the probability density curve and reliability in (RL,RU). Then the expert experience can be changed to be the probability density distribution by this method.

There are two parameters in the normal distribution which is symmetrical, and the parameter μ takes the midpoint of the valuation interval. So the parameter μ can be shown in (4),

2L UR Rμ +

= (4)

The variance δ2 can be estimated through the integral in the range (RL, RU),

( ) 1U

L

R

Rf R dR α= −∫ (5)

D. Consistent Degree of Expert Experience The ith, jth experts give reliability confidence intervals,

which are under distributions N(μi,δ2i), N(μj,δ2

j) correspondingly, the pdf are fi®and fj®which have superposition, shown in Fig.2.

Figure 1. Converted probability density distribution

Figure 2. Expert experience degree of consistency

Definition 4: fi(R), fj(R) are pdf converted from expert experience, and then the consistent degree of expert experience is the ratio of the area S1 and S2,

( ) 1

2

( ), ( )i jSCM f R f RS

= (6)

Where, S1 — the superposition area of fi(R), fj(R), S2 — the incorporative area of fi(R), fj(R).

( )( ), ( )i jCM f R f R is recorded as , which shows

the degree one expert experience agrees with another. The greater the value, the more reliable the expert experience.

ijCM

It is easy to find that the consistent degree of expert experience has characters from the definition, shown as follows,

ij jiCM CM= (7)

E. Information Fusion Based on D-S Evidence Theory n experts assess the reliability of the product, each expert’s

assessment Ei(i=1,2,…,n) can be changed to be pdf fi(R). Every two expert assessments Ei and Ej have consistent degree according to the definition, so there is n × n matrix based on all expert assessments.

412

12 1

21 2

1 2

1 ...

1 ...

... ... ...

... 1

n

n

n n

CM CM

CM CM

CM CM

⎡ ⎤⎢ ⎥⎢

= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

CM⎥

(8)

The consistent degree matrix is symmetric matrix, and all the main diagonal elements are ‘1’ according to the consistent degree character.

To normalize the elements in the same row in the matrix, and then we get a new matrix . 'CM

('

1

, 1, 2,...,ijij n

ijj

CMCM i j n

CM=

= =

∑) (9)

All the expert experience are converted into pdf, and then we obtain set {Sj(R)|j=1,2…,n} which is recognized as knowledge framework Θ, is the probability the proposition is considered to be true which the i

'ijCM

th expert considers the reliability’s pdf is fj(R). To construct the basic pdf m:2Θ->[0,1],

{ }{ }

'( ) ( ) | 1...

( ) 0 ( ) | 1... ,

i ij j

i j

m A CM A f R j n

m A A f R j n A

⎧ = = =⎪⎨

= ≠ =⎪⎩ Θ⊆ (10)

First of all, we get the basic probability assignment of the distribution through fusing the first and second expert information. Combined method is shown formula (11)

{ }

{ }

1 2,

1 2,

( ) ( )

( ) | 1... ; 1...1 ( ) ( )( )

0 ( ) | 1... ,

i j

i j

i ji j

A B Aj

i ji j

A B

j

m A m B

A f R j n i nm A m Bm A

A f R j n A

∩ =

∩ =

⎧⎪⎪ = = =⎪ −=⎨⎪⎪⎪ ≠ = ⊆⎩

∑

∑Φ

Θ

(11)

To fuse n evidences can be carried out by n-1 times fusion of two evidences.

Finally we consider (j=1, 2…n) as the weights of the pdf after fusion.

( )(j jw m f R= )

F. Product Reliability The probability density distribution is transformed from

expert experience, so the weight of probability density distribution is the weight of expert experience. The product reliability is

1 1 2 2 ...s nR w w w nμ μ μ= ⋅ + ⋅ + + ⋅ (12)

The fused reliability distribution is

1 1 2 2 ...s n nf w f w f w f= ⋅ + ⋅ + + ⋅ (13)

III. EXAMPLE ANALYSIS During reliability assessment, four experts give reliability

confidence intervals, as follows.

P1(0.6<R<0.8)=0.85, P2(0.7<R<0.85)=0.9

P3(0.65<R<0.8)=0.8, P4(0.7<R<0.8)=0.8.

To convert the expert judgments into normal distribution respectively and the distribution parameters (μ, δ2) are as follows,

(0.7000, 0.06942), (0.7750, 0.04572)

(0.7250, 0.05862), (0.7500, 0.03912)

The consistency degree matrix is shown

4 4

1 0.3286 0.7512 0.34160.3286 1 0.4461 0.61240.7512 0.4461 1 0.60280.3416 0.6124 0.6028 1

×

⎡ ⎤⎢ ⎥

= ⎢ ⎥⎢ ⎥⎣ ⎦

CM

We get the new matrix after normalization

4 4

0.4130 0.1357 0.3102 0.14110.1377 0.4189 0.1869 0.25650.2683 0.1593 0.3571 0.21530.1336 0.2395 0.2358 0.3911

×

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

CM＇

To fuse the first and second expert experience through the D-S fusion rule, the result is shown in Table 1.

TABLE I. TWO EXPERT EXPERIENCE FUSION

m1(•) m1(f1) m1(f2) m1(f3) m1(f4)

m2(f1) (0.05687)f1 (0.01869)Φ (0.04271)Φ (0.01943)Φm2(f2) (0.17301)Φ (0.05684)f2 (0.12994)Φ (0.05911)Φm2(f3) (0.07719)Φ (0.02536)Φ (0.05798)f3 (0.02637)Φm2(f4) (0.10593)Φ (0.03481)Φ (0.07957)Φ (0.03619)f4

Conflictive coefficient K1 can be estimated.

K1=0.17301+0.07719+0.10593+0.01869+0.02536+0.03481+0.04271+0.12994+0.07957+0.01943+0.05911+0.02637 = 0.79212

413

Then we get the basic probability assignment of the expert experience by fusion.

12 11

0.05687( )= 0.27361-K

m f ≈

12 21

0.05684( )= 0.27341-K

m f ≈

12 31

0.05798( )= 0.27891-K

m f ≈

12 41

0.03619( )= 0.17411-K

m f ≈

To fuse n-1 times, then the final fusion results are shown in Table 2.

TABLE II. INFORMATION FUSION RESULT

f1 f2 F3 f4

m1⊕2 0.2736 0.2734 0.2789 0.1741 m1⊕2⊕3 0.2912 0.1728 0.3952 0.1487

m1⊕2⊕3⊕4 0.1739 0.1850 0.4165 0.2600

Figure 3. The probability density distribution before and after fused

To obtain the distribution weights after D-S fusion.

w1= 0.1739, w2= 0.1850

w3= 0.4165, w4=0.2600

Finally, the product reliability and the fused distribution parameters are as follows.

Rs= w1μ1 + w2μ2 + w3μ3 + w4μ4 =0.7621

μs=0.7621, σ2s=0.03032

The probability density curves before and after fused are shown in Figure 3.

IV. CONCLUSIONS In order to resolve the problem it is not realistic to carry out

tests on high-reliability product, we introduce the D-S evidence theory to fuse expert experience, and obtain good results. Meanwhile, there are some other questions during fusing expert experience, such as conflicts, expressions of experience, relevancy, credibility, and so on. How to deal with these questions is the important content in reliability assessment.

ACKNOWLEDGMENT The authors would like to thank all the authors of the

referees and the editors for their elicitation and suggestions which led to a substantially improved version of the article.

REFERENCES [1] G.H. Fang, H. Zhao, “The Application of the D-S Evidence Theory in

the Reliability Assessment”. Quality and Reliability. 2006, No.6, pp.27-29.

[2] G.H. Fang, “Research on the Multi-source Information Fusion Techniques in the Process of Reliability Assessment”. Hefei Industry university, 2006.

[3] J. Zhang, G.P. Tu, “A New Method to Deal With the Conflicts in the D-S Evidence Theory”. Statistics and Decision, 2004, No.7, pp.21-22.

[4] J. M. Booker, M. A. Meyer, “Elicitation and Analysis of Expert Judgment”. Encyclopedia of Statistical Sciences, (Samuel Kotz,editor),2000.

[5] J. M. Booker, M. A. Meyer, “A Uncertainty Quantification: Method and Examples From Probability and Fuzzy Theories”. World Automation Congress, 2002, Vol.13, pp.135-140.

[6] D. K. DEY, J. LIU, “Prior Elicitation from Expert Opinion: An Interactive Approach”. University of ConnecticutDivision of Biostatistics,2004.

[7] P. S. Szwed, J. R. Drop, “A Bayesian Model for Rare Event Risk Assessment Using Expert Judgment About Paired Scenario Comparisons”. ASEM National Conference Proceedings, 2002, pp.444-453.

[8] S. Hora, M. Jensen, “Expert judgement elicitation”. SSI rapport,2002. [9] W. Bram, B. Tim and Q. John, “Expert Judgment Combination Using

Moment Methods”. Reliability Engineering and System Safety, 2007, pp.1-25.

[10] J. Yang, X.Y. Wu and S.H. Ma, “Aggregation of Expert Judgments in Reliability Test and Estimation”. Aeronautical Computing Technique, 2007, No.5, pp.14-17.

[11] Y. He, G.H. Wang, “Multisensor Information Fusion With Applications (Second Edition)”. Publishing House of Electronics Industry, 2007.

[12] K. Sentz, S. Ferson, “Combination of evidence in dempster-shafer theory”. Sandia National Laboratories: Technical report: SAND2002- 0835, 2002.

[13] R. Sun, H.Z. Huang, Q. Miao, “Improved information fusion approach based on D-S evidence theory,” Journal of Mechanical Science and Technology, 2008, Vol. 22, No. 12, pp.2417-2425.

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