1. RELIABILITY Dr. Ron Lembke SCM 352

2. Uncertainty? What is it? How do you prepare for it? Known unknowns? Unknown unknowns?

3. Future States of Nature Demand hi, medium, low Traffic: normal, terrible

4. Black Swans Cognitive Bias: Framing: compare to known things A game-changing, totally unexpected event Housing bubble? Internet? Cell phones?

5. Risks Preventable risk Strategic risk – some benefits from the exposure External risk – how to protect? Limited by ability to predict

6. Reliability Ability to perform its intended function under a prescribed set of conditions Probability product will function when activated Probability will function for a given length of time

7. Measuring Probability Depends on whether components are in series or in parallel Series – one fails, everything fails

8. Measuring Probability Parallel: one fails, everything else keeps going

9. Reliability Light bulbs have 90% chance of working for 2 days. System operates if at least one bulb is working What is the probability system works?

10. Reliability Light bulbs have 90% chance of working for 2 days. System operates if at least one bulb is working What is the probability system works? Pr = 0.9 * 0.9 * 0.9 = 0.729 72.9% chance system works

11. Parallel 90% 80%75%

12. Parallel 0.9 prob. first bulb works 0.1 * 0.8 First fails & 2 operates 0.1 * 0.2 * 0.75 1&2 fail, 3 operates =0.9 + 0.08 + 0.015 = 0.995 99.5% chance system works 90% 80%75%

13. Parallel – Different Order 80% 75%90%

14. Parallel 0.8 prob. first bulb works 0.2 * 0.75 First fails & 2 operates 0.2 * 0.25 * 0.90 1&2 fail, 3 operates =0.8 + 0.15 + 0.045 = 0.995 99.5% chance system works Same thing! 80% 75%90%

15. Parallel – All 3 90% 0.9 prob. first bulb works 0.1 * 0.9 First fails & 2 operates 0.1 * 0.1 * 0.9 1&2 fail, 3 operates =0.9 + 0.09 + 0.009 = 0.999 99.9% chance system works

16. Practice.95.9.75.80.95.9.95

17. 1: 0.95 * 0.95 2: Simplify 0.8 * 0.75 = 0.6 and 0.9 * 0.95 * 0.9 = 0.7695 Then 0.6 + 0.4 * 0.7695 = 0.6 + 0.3078 = 0.9078 Solutions

18. Practice.9.95

19. Solution 2 Simplify: 0.9 * 0.95 = 0.855 0.95 * 0.95 = 0.9025 Then 0.855 + 0.145 * 0.9025 = = 0.855 + 0.130863 = 0.985863

20. Practice.90.95.9.75.80.9.95

21. Simplify 0.9 * 0.95 = 0.855 0.8 * 0.75 = 0.6 0.9 * 0.95 * 0.9 = 0.7695.855.60.7695

22. Simplify 0.6 + 0.4 * 0.7695 = 0.6 + 0.3078 =0.9078.8550.9078 0.855 + 0.145 * 0.9078 = 0.855+0.13631 =0.986631

23. These 3 are in parallel 0.855 + 0.145 * 0.6 + 0.145*0.4*.7695 =0.855 + 0.087 + 0.044631 = 0.986631 (1-0.855)*(1-0.6)*(1-0.7695) =0.013369 = Prob. of failure =0.986631 success.855.60.7695

24. Lifetime Failure Rate 3 Distinct phases: Infant Mortality StabilityWear-out Failure rate time, T

25. Exponential Distribution MTBF = mean time between failures Probability no failure before time T Probability does not survive until time T = 1- f(T) e = 2.718281828459045235360287471352662497757

26. Example Product fails, on average, after 100 hours. What is the probability it survives at least 250 hours? T/MTBF = 250 / 100 = 2.5 e^-T/MTBF = 0.0821 Probability surviving 250 hrs = 0.0821 =8.21%

27. Normally Distributed Lifetimes Product failure due to wear-out may follow Normal Distribution