1. 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

2. Introduction Risk and Safety Where there is a lot of failure data Threats and Risks are unacceptably high It is important to identify Threats and Risks 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

3. 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

4. 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

5. 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

6. Conditional Probability Bayes Theorem (1764) 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) Total Probability 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. Bayes Theorem in Failure Mode and Effects Analysis Peter G Blaine, Professor PJ Vlok, Professor AH Basson, and Mr RT Dobson

7. 0 What is the probability that the speed will reach 9Mb/s? ` 10 20 30 Bayes Theorem in Failure Mode and Effects Analysis Peter G Blaine, Professor PJ Vlok, Professor AH Basson, and Mr RT Dobson

8. 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

9. 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

10. ModelPriorLikelihoodPrior*LikePosteriorModel*Post successes1Failures14 0.00000.09090.00000.00E+000.0000 0.00200.09090.02722.48E-030.0215 0.0000 0.00400.09090.05304.81E-030.0417 0.0002 0.00600.09090.07727.02E-030.0609 0.0004 0.00800.09090.10019.10E-030.0789 0.0006 0.01000.09090.12171.11E-020.0959 0.0010 0.01200.09090.14201.29E-020.1119 0.0013 0.01400.09090.16111.46E-020.1270 0.0018 0.01600.09090.17901.63E-020.1411 0.0023 0.01800.09090.19591.78E-020.1544 0.0028 0.02000.09090.21161.92E-020.1668 0.0033 1.00001.15E-011.00000.0137 1 failure in 14 days gives 0.0714 failure per day Likelihood Bayes Theorem in Failure Mode and Effects Analysis Peter G Blaine, Professor PJ Vlok, Professor AH Basson, and Mr RT Dobson

11. successes1Failures6 0.0000 0.00E+000.0000 0.00200.02150.01192.54E-040.0034 0.0000 0.00400.04170.02349.78E-040.0131 0.0001 0.00600.06090.03472.11E-030.0283 0.0002 0.00800.07890.04583.61E-030.0483 0.0004 0.01000.09590.05655.42E-030.0726 0.0007 0.01200.11190.06707.50E-030.1004 0.0012 0.01400.12700.07729.81E-030.1313 0.0018 0.01600.14110.08721.23E-020.1647 0.0026 0.01800.15440.09691.50E-020.2003 0.0036 0.02000.16680.10641.77E-020.2376 0.0048 1.00007.47E-021.00000.0154 2 failures over 20 days gives 0.1000 failures/day Bayes Theorem in Failure Mode and Effects Analysis Peter G Blaine, Professor PJ Vlok, Professor AH Basson, and Mr RT Dobson

12. MLE Maximum Likelihood Estimate 30 failures over 1949 days gives 0.0154 failures/day

13. Risk Analysis Flow Chart Bayes Theorem in Failure Mode and Effects Analysis Peter G Blaine, Professor PJ Vlok, Professor AH Basson, and Mr RT Dobson MechRel l® Jones, J. (2010). Handbook of Reliability Prediction Procedures for Mechanical Equipment. US Naval Surface Warfare Center. WinNUPRA l® Canavan, K. (2009). Safety risk technology and application. Tech. Rep., Electric Power Research Institute

14. Discussion FMEAs provide a useful method of identifying and catagorising risk When there is few data, Bayes' Theorem provides updated probabilities as new data is obtained. Assigning and updating probabilities in FMEAs makes them an integral and continual part of risk analysis These probabilities can be used in Fault and Event Trees Bayes' Theorem accommodates expert judgement provided probabilities 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

15. Conclusions 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. 1. Bayes' Theorem allows probabilities to be calculated for very low incidence events even using expert judgement Bayes Theorem in Failure Mode and Effects Analysis Peter G Blaine, Professor PJ Vlok, Professor AH Basson, and Mr RT Dobson