(11C) Weibull Distribution Parameter Calculation

[Gamo]

For detail information:
https://en.wikipedia.org/wiki/Weibull_distribution

This program was adapted from HP-55 Statistic Programs Book. P.62
Using the Median Rank Regression method (MRR)

As stated in the book:
A common application is to use Weibull analysis for failure data
where samples are tested to failure. To use the program, list the
items in order increasing time to failure.

The reliability test and longevity of the shelf life of a product or survival of affected patients.
One reason it is widely used in reliability and life data analysis is due to its flexibility.
It can mimic various distributions like the normal.

Example use from this book P.63

Procedure: "each new problem must clear Statistic Register"

f [∑]
[A] input n (Data Count)
[B] input Data (Data count equal to n)
[C] Calculate β [R/S] ϴ

Example: f [USER] f [∑]
Xi: 34, 60, 75, 95, 119, 158 (hours to failure)
(Xi's must be entered in incresing order.)
n=6
β=1.95
ϴ=104.09

β value type of failure meaning
β < 1 infant mortality high probability of failing at early stages
β =1 random failures failures are independent of time
1 < β < 4 early wear out can be due to generic failure modes, such as corrosion
β > 4 rapid wear out steep curve with fast wear out at some point


f [∑]
6 [A] display 6.00
34 [B] display 1.00
60 [B] display 2.00
75 [B] display 3.00
95 [B] display 4.00
119 [B] display 5.00
158[B] display 6.00
[C] display 1.95 [R/S] 104.09

Program:

Code:


LBL A // Data Count
STO7 1 STO6 X<>Y RTN
----------------------------
LBL B  // Enter Data
LN RCL6 .3 - RCL7 .4 + ÷ 1 STO+6 X<>Y
- 1/x LN LN X<>Y ∑+ RTN
-----------------------------
LBL C  // Answer
L.R. X<>Y R/S ÷ CHS e^x RTN

Remark:
The Weibull hazard function or failure rate depends upon the value of β.
Because the value indicates whether newer or older parts are more likely to fail,
the Weibull hazard function can represent different parts of the bathtub curve:

• Infant Mortality. In electronics and manufacturing, infant mortality refers to a
higher probability of failure at the start of the service life. When the value is
less than 1.0, the Weibull probability plot indicates that newer parts are more
likely to fail during normal usage, which is known as a decreasing instantaneous failure rate.
To end infant mortality in electronic and mechanical systems
with high failure rates, manufacturers provide production acceptance tests,
“burn-in” and environmental stress screenings prior to delivering such systems to
customers. Providing that the part survives infant mortality, its failure rate should
decrease, and its reliability should increase. In this case, because such parts tend
to fail early in life, old parts are considered better than new parts. Overhaul of
parts experiencing high infant mortality is generally not appropriate.

• Random Failures. Assuming that the Weibull probability plot is based on a
single failure mode, a value of 1.0 indicates that the failure rate is constant or
independent of time. This means that of those parts that survive to time t, a
constant percentage will fail in the next unit of time, which is known as a
constant hazard rate or instantaneous failure rate. This makes the Weibull
probability plot identical to the exponential distribution. Because old parts are
assumed to be as good as new parts, overhaul is generally not appropriate. The
only way to increase reliability for components or systems that experience
random failures is by redesigning them.

• Early Wear-out. Unexpected failures during the design life are often due to
mechanical problems. When the value is greater than 1.0 but less than 4.0,
overhauls or part replacements at low B-lives may be cost effective.
B-lives indicate the ages at which given percentages of the population are expected to fail.
For example, the B-1 life is the age at which 1 percent of the population is
expected to fail, and the B-10 life is the age at which 10 percent of the population
is expected to fail. Reliability and cost performance for parts experiencing early
wear-out may be improved by optimizing the preventative maintenance schedule.

• Rapid Wear-out. Although a value greater than 4.0 within the design life of a
part is a major concern, most Weibull probability plots with steep slopes have a
safe period within which the probability of failure is negligible, and the onset of
failure occurs beyond the design life. The steeper the slope, the smaller variation
in the times to failure and the more predictable the results. For parts that have
significant failures, overhauls and inspections may be cost effective.
Because scheduled maintenance can be costly, it is usually only considered when older
parts are more likely to wear out and fail, which is known as an increasing
instantaneous failure rate.

Gamo 2/2020