Reliability

A frequently encountered problem in many areas of applied mathematics, including both reliability and
simulation is that of modelling a task duration when that time is subject to random variation.
Probability plotting allows us to
assess how well a specified probability density function model fits a sample
data set.
An obvious difficulty with
this
in practice is that it can be difficult to know how well our techniques
work. For example, it may be th
at the life of a
battery
would best be described by say a normal distribution but
because of the quirks of randomness in the particular sample analysed
,
it may be deemed that the best model
might be lognormal.
And we can never know if and when that might h
appen in a real world scenario.
Simulation affords us an opportunity
to
learn
something about the frequency with which some anomalies occur.
We can randomly generate data
from a known density with known parameters, then apply our probability
plotting tech
nique and see how good the technique is at identifying the
true
nature of the data.
Example
Suppose we have a device
, say a disposa
b
l
e
battery
and
we
know that the
true
distribution for its
mean life is
normal with mean
50 hours and SD 3 hours.
If
the design life is 47
hours (i.e. we require / desire that the battery
should survive for at least this time)
that we can readily calculate the reliability to be 84.13%
Now suppose we were to
randomly generate 10 sample lifespans consistent with the true
distribution. Such
samples are
called deviates
and are easily obtained
using
a
mathematical algorithm, which is incorporated in to
Minitab. Such a sample is shown below.
50.6
45.5
44.5
43.6
49.8
51.8
47.5
47.6
45.2
50.1
What model and what parameters are most consistent with this data? (We will restrict
the model form to be one
of the normal, the lognormal, the exponential and the Weibull
but of
course there may be another model form
which might fit our data well)
Soluti
on
A sample of size 10 is quite small but nevertheless, a reasonable starting point is to construct a histogram and
(using Minitab) superimpose each of our four candidate model forms on this histogram. This is shown below.
5
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We can discount the exponential model, but the other three seem to fit the data about equally well.
Rather than
use density plots (histograms) we will use probability plots. The closer a probability plot for a given model is to
a straight line, the better than model fits the data. A statistic r
2
called the coefficient of determination takes a
value in t
he interval [0, 1].
Values close
to 1
suggest
the probability plot is
a
straight line.
Unfortunately, whilst Minitab will readily generate
probability
plots, it does not report the r
2
value so we will
have to re
–
do the calculations in Excel. The plots from Minitab are shown below (the generation of these in
Excel
will be demonstrated in class). Probability plots for the normal, lognormal and Weibull are shown below.
6
0
5
5
5
0
4
5
4
0
2
1
0
–
1
–
2
C
2
S
c
o
r
e
M
e
a
n
4
7
.
6
3
S
t
D
e
v
2
.
8
5
8
N
1
0
A
D
0
.
3
1
1
P
–
V
a
The extent to which the data falls on a straight line
indicates how well each model fits. Visually, both the
normal and lognormal models appear to fit about
equally well.
When we construct the same plots in
Excel we will be able to calculate the r
2
value and
choose the
model with the straightest line.
(
I
n fact the lognormal model is actually a marginally
better fit based on the r
2
value).
Note the sample mean and
standard deviation are 47.63 and 2.858 respectively
.
R
emember the population
‘s
mean and standard deviation
are 50 and 3 respectively.
Thus our simulated sample of 10 batteries tended to fail
earlier and i
f we calculate the reliability
using the sample dat
a parameters we get 57.8% for the normal and the
56.9% for the lognormal. These reliability estimates are reasonably close so the fact that we would have
erroneously chosen the lognormal rather than the normal would not matter so much.
But the true reliability measure of about 84% is quite a distance from either of these estimates. (84% suggests
that about 1 device in 6 will fail before its design life, 56
–
57% suggests a figure of
close to
1 in 2)
.
So our sample of size gave rise to ra
ther misleading findings. But perh
aps if we repeated this simulation
a
number of times we might find that that such a misleading result was unusually and happens
only
occasionally.
Further analysis might show that such misleading findings arise less freque
ntly with larger sample sizes.
The a
ssignment
The first assignment requires you to do a more extensive analysis in the same vein as above.
You are free to
select
most of the details and write your report as you see fit subject to the following.
?
Produce
at least 20 sample data sets. Make sure you document each one (PDF & parameters)
?
Use the spreadsheet
calculator
available on Moodle to help you choose parameters and design life. As a
guide, opt for a (true) reliability measure in the range [60%, 90%]
?
Inc
lude samples from all four distributions, Weibull, normal, lognormal and exponential
?
Include a mixture of small (8
–
15) and large samples (>30) but you may find it convenient to have a
single small sample size and a single large sample size.
?
Document your f
indings
(s
ee sample narrative fragments
for guide). All numerical calculations should
be computed automatically in Excel and should be clearly referenced from your report (use a separate
workbook sheet for every distinct data set). The text of your report
should extend to about 500 words.
?
Your report should include histogram plots with superimpose PDF models (as above) for one small
sample and one large sample.
The Excel sheet containing the data/calculations should also be submitted.
Sample report fragmen
t
(fictitious!)
Eight samples, all of size 12 were generated from a Weibull distribution with
?
= 1.9 and
?
?
= 11. For a design
life of 5 years the reliability
would be 80%. Three of the eight returned a “best” model other than the Weibull
(twice the norma
l was selected and once the lognormal). Reliabilities calculated for the eight samples (using the
best model in each case) ranged from 49% up to 91%.
See table 1.
So while 4 out of 5 such devices would
survive for at least 5 years, a real world analyses mi
ght well have suggested that this figure could be as low as 1
in 2, or as high as about 9 in 10 in a relatively low number of trails (eight). This would suggest that samples of
size 12 are simply too small
Sample
1
2
3
4
5
6
7
8
“Best” model
Weibull
Weibull
Normal
Normal
Weibull
Weibull
Lognormal
Weibull
Reliability
89%
64%
49%
60%
91%
88%
63%
79%
Table 1. Sample deviates from a Weibull distribution with
?
= 1.9 and
??
= 11
. Design life is taken to be 5 years,
which yields a reliability measure of
80%
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