some thoughts on the tci report weibull analysis

SOME THOUGHTS ON THE TCI REPORT

Weibull Analysis

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”.

WEIBULL CUMULATIVE PROBABILITY FUNCTION

00 _,1)( ttparatt

ExptTP

WEIBULL PARAMETERS

• Equation:• “t” is the argument,

• “t0”, “” and “” are the “parameters” of the

Weibull random variable. • For different equipment, the values of these

parameters will vary.

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.

WEIBULL PARAMETERS

• “t0” is often called “Minimum Life” or “Intrinsic

Reliability”: P(T<t0)=0. “t0” = zero?

is the “Weibull slope”. > 1 => “increasing failure rate function” (the

older the equipment gets, the more likely to fail it becomes).

< 1 => decreasing failure rate.

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

Wheels (a) – Table I

Manufacturer and Year

B1 Life (1% Fail)

Mean Life

(1000)

Characteristic Life (1000)

Weibull Slope

Total Wheels

Suspensions Failures

TOP: All Modes of Removal for Wheel Causes

A – 1995   403,9 441,6 4.67 3838 3281 557

B – 1995   415,7 456,4 4.35 6922 5635 1287

M – 1995   474,2 526,7 3.55 4444 3820 624

C – 1995   430,9 476,9 3.79 2722 2165 557

Wheels (a) – Table II – Catastrophic Case

Manufacturer and Year

B1 Life (1% Fail)

(1000)

Mean Life

(1000)

Characteristic Life (1000)

Weibull Slope

Total Wheels

Suspensions Failures

BOTTOM: For Broken Wheel Causes Only

A – 1995   1048 1129 6.12 3838 3837 1

B – 1995   5025 5624 3.04 6922 6921 1

M – 1995   911 985 5.60 4444 4438 6

C – 1995 Est   4343 3.00 2722 2722 0

523

1,230

436

Wheels (a) – Table II – Catastrophic Case

Manufacturer and Year

B1 Life (1% Fail)

(1000)

Mean Life

(1000)

Characteristic Life (1000)

Weibull Slope

Total Wheels

Suspensions Failures

BOTTOM: For Broken Wheel Causes Only

A – 1995   1048 1129 6.12 3838 3837 1

B – 1995   5025 5624 3.04 6922 6921 1

M – 1995   911 985 5.60 4444 4438 6

C – 1995 Est   4343 3.00 2722 2722 0

523

1,230

436

Wheels (b) – Table III – Different Wheels

Manufacturer and Year

B1 Life (1% Fail)

Mean Life

(1000)

Characteristic Life (1000)

Weibull Slope

Total Wheels

Suspensions Failures

TOP: All Modes of Removal for Wheel Causes

A – 1995   2.6x108 1.2x108 0,47 6806 6799 7

F – 1995   1319 1428 1,30 4081 4024 57

Wheels (a) – Table II Data – Bernoulli Model

MANUF Total Wheels

Proportion “p” of failures: Point

estimate

95% Confidence Interval: Lower Limit for “p”

95% Confidence Interval: Upper Limit for “p”

Length of the Confidence

Interval

A 3,838 0.0003 -0.0003 0.0008 0.0011

B 6,922 0.0001 -0.0001 0.0004 0.0005

M 4,444 0.0014 0.0003 0.0024 0.0021

C 2,722 0.0000 - - -

Wheels (a) – Table II Data – Bernoulli Model

Wheels (a) – Table II Data – Bernoulli ModelSensitivity Analysis

MANUF Total Wheels

Proportion “p” of failures:

Point estimate

95% Confidence Interval: Lower

Limit for “p”

95% Confidence Interval: Upper

Limit for “p”

Length of the Confidence

Interval

A 3,838 0.0005 -0.0002 0.0012 0.0014

B 6,922 0.0003 -0.0001 0.0007 0.0008

M 4,444 0.0011 0.0001 0.0021 0.0020

C 2,722 0.0004 -0.0004 0.0011 0.0015

Wheels (a) – Table II Data – Bernoulli ModelSensitivity Analysis

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).

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.