1. SOME THOUGHTS ON THE TCI REPORT Weibull Analysis

2. WEIBULL ANALYSIS OF WHEELS - SOME THOUGHTS ON THE TCI REPORT The Weibull random variable: 1939 Swedish physicist Ernest Hjalmar Wallodi Weibull. The Weibull random variable provides a way to evaluate the probability that a certain equipment (or system) will fail before a certain instant “t”.

3. WEIBULL CUMULATIVE PROBABILITY FUNCTION

4. WEIBULL PARAMETERS Equation: “t” is the argument, “t 0 ”, “  ” and “  ” are the “parameters” of the Weibull random variable. For different equipment, the values of these parameters will vary.

5. FINDING VALUES FOR WEIBULL PARAMETERS: SAMPLING Question: for a specific equipment (such as a wheel manufactured by a certain company), what is the set of numeric values to be assigned to the parameters that best represents the probability that one wants to evaluate? The only way to answer is sampling: –how many of them failed –how old (in hours or in miles) they were when they failed.

6. WEIBULL PARAMETERS “t 0 ” is often called “Minimum Life” or “Intrinsic Reliability”: P(T<t 0 )=0. “t 0 ” = zero?  is the “Weibull slope”.  > 1 => “increasing failure rate function” (the older the equipment gets, the more likely to fail it becomes).  decreasing failure rate.

7. WEIBULL PARAMETERS  is called the “Characteristic Life” of the equipment. One can show that approximately 2/3 of the equipment will fail before instant , and only 1/3 will live longer than . http://www.weibull.com

8. Wheels (a) – Table I Manufacture r and Year B1 Life (1% Fail) Mean Life (1000) Characterist ic Life (1000) Weibull Slope Total Wheels Suspensions Failures TOP: All Modes of Removal for Wheel Causes A – 1995 403,9441,64.6738383281557 B – 1995 415,7456,44.35692256351287 M – 1995 474,2526,73.5544443820624 C – 1995 430,9476,93.7927222165557

9. Wheels (a) – Table II – Catastrophic Case Manufacture r and Year B1 Life (1% Fail) (1000) Mean Life (1000) Characterist ic Life (1000) Weibull Slope Total Wheels Suspensions Failures BOTTOM: For Broken Wheel Causes Only A – 1995 104811296.12383838371 B – 1995 502556243.04692269211 M – 1995 9119855.60444444386 C – 1995 Est 43433.002722 0 523 1,230 436

10. Wheels (a) – Table II – Catastrophic Case Manufacture r and Year B1 Life (1% Fail) (1000) Mean Life (1000) Characterist ic Life (1000) Weibull Slope Total Wheels Suspensions Failures BOTTOM: For Broken Wheel Causes Only A – 1995 104811296.12383838371 B – 1995 502556243.04692269211 M – 1995 9119855.60444444386 C – 1995 Est 43433.002722 0 523 1,230 436

11. Wheels (b) – Table III – Different Wheels Manufacture r and Year B1 Life (1% Fail) Mean Life (1000) Characterist ic Life (1000) Weibull Slope Total Wheels Suspensions Failures TOP: All Modes of Removal for Wheel Causes A – 1995 2.6x10 8 1.2x10 8 0,47680667997 F – 1995 131914281,304081402457

12. Wheels (a) – Table II Data – Bernoulli Model

14. Wheels (a) – Table II Data – Bernoulli Model Sensitivity Analysis

16. WEIBULL ANALYSIS: CONCLUSION V.1 - Top: Sufficient statistical evidence that manufacturers A and M have a smaller proportion of failures than manufacturers B and C. Not enough data to reach a consistent conclusion concerning the Weibull random variable. However, experience in dealing with similar experiments plus the sample sizes (minimum of 2,722) and the number of failures (minimum of 557) all together strongly suggest that the mean life for wheels manufactured by manufacturer M is significantly greater than all other manufacturers (A, B and C).

17. WEIBULL ANALYSIS: CONCLUSION V.1 – Bottom: Table II presents results based on single digit observations. Consequence: point estimates become meaningless. Sensitivity analysis using Binomial random variable shows solution instability. No statistically significant conclusions can be taken from the results presented on the bottom of Table II.