Bayes Theorem in Failure Mode and Effects Analysis
Peter G Blaine, Professor PJ Vlok, Professor AH Basson, and Mr RT Dobson
Energy Postgraduate Conference 2013
Bayes Theorem in Failure Mode and Effects Analysis
Peter G Blaine, Professor PJ Vlok, Professor AH Basson, and Mr RT Dobson
Introduction
Risk and Safety
• It is important to identify Threats and Risks
• Where there is a lot of failure data
Threats and Risks are unacceptably high
• The highest Threats and Risks must be identified
• Their probability of happening must be calculated
• Classic statistical analysis requires much data to
be able to assign a probability to an event
Bayes Theorem in Failure Mode and Effects Analysis
Peter G Blaine, Professor PJ Vlok, Professor AH Basson, and Mr RT Dobson
Natural Convection
Reactor Cavity Cooling Loop
Sittmann, I. (2010). Characterisation of boiling and condensation heat transfer coefficients for different flow patterns. Private Comm
Bayes Theorem in Failure Mode and Effects Analysis
Peter G Blaine, Professor PJ Vlok, Professor AH Basson, and Mr RT Dobson
Failure Mode & Effects Analysis
Present Application
Crowe, D. and Feinberg, A. (2001). Design for Reliability. CRC Press.
ASQC/AIAG Task Force (2001). Potential failure mode and effects analysis (FMEA) Reference Manual. 3rd edn. ASQC/AIAG Task Force,.
Bayes Theorem in Failure Mode and Effects Analysis
Peter G Blaine, Professor PJ Vlok, Professor AH Basson, and Mr RT Dobson
Failure Mode & Effects Analysis
Proposed Application
Bayes Theorem in Failure Mode and Effects Analysis
Peter G Blaine, Professor PJ Vlok, Professor AH Basson, and Mr RT Dobson
Bayes Theorem (1764)
Bayes, T. (1764). An essay toward solving a problem in the doctrine of chances.
Philosophical Transactions of the Royal Society of London, vol. 53, pp. 370-418.
P
(B

A
)
P
(A
|B
)
P
(B
)
Conditional Probability

P
(B

A
) P
(B
|A
)P
(A
)a
n
d
s
oP
(B

A
) P
(B
|A
)P
(A
)
n
o
w
P
(B

A
) P
(AB
)
b
u
tP
(B
)
P
(B

A
) P
(B

A
)
s
o
P
(B
)
P
(B
|A
)P
(A
)
P
(B
|A
)P
(A
)
Total Probability
where
P(A) is the prior probability
P(A|B) is the posterior probability
P(B|A) is the likelihood
Given a prior state of knowledge or belief, Bayes' Theorem tells
us how to update beliefs based upon observations (current data)
Bayes Theorem in Failure Mode and Effects Analysis
Peter G Blaine, Professor PJ Vlok, Professor AH Basson, and Mr RT Dobson
0
20
What is the
probability that
10
`
the speed will
reach 9Mb/s?
30
Bayes Theorem in Failure Mode and Effects Analysis
Peter G Blaine, Professor PJ Vlok, Professor AH Basson, and Mr RT Dobson
40
50
60
68
Bayes Theorem in Failure Mode and Effects Analysis
Peter G Blaine, Professor PJ Vlok, Professor AH Basson, and Mr RT Dobson
Data of the failure of the
Condensate Air Removal
System (CARS) pumps over
the period from June 1999
to October 2004 at the
South Texas Project Nuclear
Power Plant.
The plant used two Westinghouse
pressurised water reactors.
Sun, A., Kee, E., Yu, W., Popova, E., Grantom, R. and Richards, D. (2005).
Application of Crow-AMSAA analysis to nuclear power plant equipment performance.
13th International Conference on Nuclear Engineeing, ICONE13- 50049.
Bayes Theorem in Failure Mode and Effects Analysis
Peter G Blaine, Professor PJ Vlok, Professor AH Basson, and Mr RT Dobson
Model
Prior
Likelihood
Prior*Like
Posterior
successes
1
Failures
14
0.0000
0.0909
0.0000
0.00E+00
0.0000
0.0000
0.0020
0.0909
0.0272
2.48E-03
0.0215
0.0000
0.0040
0.0909
0.0530
4.81E-03
0.0417
0.0002
0.0060
0.0909
0.0772
7.02E-03
0.0609
0.0004
0.0080
0.0909
0.1001
9.10E-03
0.0789
0.0006
0.0100
0.0909
0.1217
1.11E-02
0.0959
0.0010
0.0120
0.0909
0.1420
1.29E-02
0.1119
0.0013
0.0140
0.0909
0.1611
1.46E-02
0.1270
0.0018
0.0160
0.0909
0.1790
1.63E-02
0.1411
0.0023
0.0180
0.0909
0.1959
1.78E-02
0.1544
0.0028
0.0200
0.0909
0.2116
1.92E-02
0.1668
1.15E-01
1.0000
1.0000
Model*Post
Likelihood
0.1668
0.0033
0.0137
1 failure in 14 days
gives 0.0714 failure
per day
Bayes Theorem in Failure Mode and Effects Analysis
Peter G Blaine, Professor PJ Vlok, Professor AH Basson, and Mr RT Dobson
successes
1
Failures
6
0.0000
0.0000
0.0000
0.00E+00
0.0000
0.0000
0.0020
0.0215
0.0119
2.54E-04
0.0034
0.0000
0.0040
0.0417
0.0234
9.78E-04
0.0131
0.0001
0.0060
0.0609
0.0347
2.11E-03
0.0283
0.0002
0.0080
0.0789
0.0458
3.61E-03
0.0483
0.0004
0.0100
0.0959
0.0565
5.42E-03
0.0726
0.0007
0.0120
0.1119
0.0670
7.50E-03
0.1004
0.0012
0.0140
0.1270
0.0772
9.81E-03
0.1313
0.0018
0.0160
0.1411
0.0872
1.23E-02
0.1647
0.0026
0.0180
0.1544
0.0969
1.50E-02
0.2003
0.0036
0.0200
0.1668
0.1064
1.77E-02
0.2376
7.47E-02
1.0000
1.0000
0.2376
2 failures over 20 days gives 0.1000 failures/day
0.0048
0.0154
Bayes Theorem in Failure Mode and Effects Analysis
Peter G Blaine, Professor PJ Vlok, Professor AH Basson, and Mr RT Dobson
MLE Maximum Likelihood Estimate
30 failures over 1949 days gives 0.0154 failures/day
Bayes Theorem in Failure Mode and Effects Analysis
Peter G Blaine, Professor PJ Vlok, Professor AH Basson, and Mr RT Dobson
Risk
Analysis
Flow
Chart
MechRell® Jones, J. (2010). Handbook of Reliability Prediction Procedures for Mechanical Equipment.
US Naval Surface Warfare Center.
WinNUPRAl® Canavan, K. (2009). Safety risk technology and application. Tech. Rep.,
Electric Power Research Institute
Bayes Theorem in Failure Mode and Effects Analysis
Peter G Blaine, Professor PJ Vlok, Professor AH Basson, and Mr RT Dobson
Discussion
• FMEAs provide a useful method of identifying and catagorising risk
• Assigning and updating probabilities in FMEAs makes them an
integral and continual part of risk analysis
• When there is few data, Bayes' Theorem provides updated
probabilities as new data is obtained.
• Bayes' Theorem accommodates expert judgement provided
probabilities
• These probabilities can be used in Fault and Event Trees
• A continuous risk analysis feedback system is set up resulting in
continual improvement in and understanding of threats and risk
and their consequences.
Bayes Theorem in Failure Mode and Effects Analysis
Peter G Blaine, Professor PJ Vlok, Professor AH Basson, and Mr RT Dobson
Conclusions
1. Bayes' Theorem allows probabilities to be calculated
for very low incidence events even using expert judgement
2. The methodology allows FMEAs to become an integral
part of a system of continually assessing risk and resultant
corrective actions.
3. When a previously defined high risk component has its
failure probability reduced as a consequence, then the
original FMEA can be used to identify a new higher risk
component for analysis using Bayes Theorem.