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Predicting Equipment Failures
Using Weibu11 Analysis and SAS* Software
Bruce Kay, Federal Express Corporation
Douglas E. Price, American Airlines
ABSTRACT
Predicting engine failures is a two-step
process. First, the parameters must be
estimated. Then the estimates are used to
Ball bearings, aircraft engines and missile
guidance systems have one thing in common:
they are all subject to unexpected mechanical
failure. Practically every industry is
concerned with increased cost~ and loss of
productivity and customer satisfaction
associated with equipment failure. Although
breakdowns are inevitable, it is possible to
predict when a failure is likely to occur
using statistical techniques.'
predict subsequent failures. We shall begin
with a brief discussion of the theory behind
Weibull analysis. Next, we shall describe a
step-by-step procedure for estimating the
Weibull parameters. Then we shall consider
the unique circumstances surrounding
predicting engine failures and some typical
problems that might arise during an
analysis. We shall conclude with a
description of how the analysis might be used
This paper will investigate a method for
estimating failure parameters and how to use
these parameters to predict equipment
failure.
to forecast engine failures.
BACKGROUND
In particular, it will examine
Weibull analysis, a statistical technique of
wide applicability. Using this technique in
Although there are a variety of techniques
for analyzing failure data, three
conjunction with historical data, we can
predict failures more accurately and minimize
their impact. Our discussion will center on
predicting aircraft engine failures; however,
requirements are common to each: an
unambiguously defined time origin, a scale
for measuring the passage of time, and a
clear meaning for the term failure.
this technique applies to a wide range of
applications.
Provided
these three ingredients are present, the
technique to be described applies equally
well to failures of ball bearings, aircraft
HITRODUCTION
engines or missile guidance systems.
Although the Weibull distribution contains
three parameters in its most general form, it
Federal Express Corporation specializes in
the door-to-door express delivery of packages
and documents throughout the United States
is commonly expressed as a two-parameter
distribution. The frequently omitted third
parameter is called the location parameter
and to many foreign countries. The company
operates an extensive fleet of aircraft that
serves sorting facilities in the United
because it serves as a time offset for
products where failure cannot occur until at
States and Europe. Reliability of its
aircraft is a key ingredient in the Federal
or after the value of the location
Express commitment to service.
Unfortunately, aircraft or equipment of any
parameter. Since most products can normally
fail at any time ~fter time zero, the
kind is subject to unexpected breakdown.
Since Federal Express offers a money-back
two-parameter Weibull distribution is the one
which we shall discuss here.
failures Can be especially costly.
If a random variable t has a Weibull
distribution, then the Weibull probability
density function (pdf) is:
guarantee for service failures, equipment
Because
equipment failures cannot be eliminated
entirely, our objective is'to reduce the
uncertainty as much as possible.
f(t)
One method used to predict equipment failures
is based on the Weibull distribution.
Waloddi Weibull (1951) introduced "A
Statistical Distribution Function of Wide
Applicability" to model a variety of product
life and reliability problems. The
widespread use of the Weibull distribution is
due to the great range of shapes of the
distribution which enables it to fit many
types of data. Our interest centers on using
the Weibull distribution to predict aircraft
engine failures.
= (y/ay)ty-1exp[-(t/a)y],
where t > O. The parameter y is the shape or
Weibull slope. The parameter a is the scale
parameter; it is also called the
"characteristic life since it is always the
ll
value of t where the cumulative probability
of failure is 63.2 percent (i.e., it is the
63.2 percentile of the cumulative
distribution function). Both parameters must
be positive. "( is unitless while
expressed in the same units as t.
0:
is
However, the technique that
will be described is equally applicable to
The Weibull cumulative distribution function
(CDF) describes the cumulative probability of
numerous other problems.
219
3.
4.
failure of an item at a given point in time.
The reliability function, which is a term
more familiar to some, is related to the COF
as the difference between unity and the CDF.
The cumulative distribution function is:
F(t)
=
The first step in the estimation of Weibull
parameters is to define precisely the time
origin for the analysis and the meaning of
the tenns "timet! and IIfailure.
For aircraft
engines the time origin is marked by the
installation of the engine on the aircraft.
The passage of time is measured in flight
hours of operation, and failure is the
inability of an engine to perform to
standards.
1 - exp[-(t/a)Y],
1I
and the reliability function is:
R(t)
=
exp[-(t/a)Y].
Additional information on theWeibull
distribution is available in Kalbfleisch and
Prentice (1980) and Nelson (1982).
After defining the scope, the next step is to
obtain data on each engine in the analysis.
Each observation is identified as either a
failure or censored, where the latter term
refers to engines which had not failed prior
to the conclusion of the sampling period.
This type of censoring is often called Type I
or time censoring (Kalbfleisch and Prentice,
1980) •
One feature which explains the Weibull
distribution's wide applicability is that
several other statistical distributions are
either related to, or can be approximated by,
the Weibull distribution. For example, when
r equals 1, the distribution is exponential;
when Y equals 2, it is the Rayleigh
distribution. For values of y near 3.4, the
shape of the Weibull distribution is
approximately that of the normal
distribution; for r greater than 10, the
shape of the Weibull is nearly that of the
smallest extreme value distribution.
The third step is to plot the failure data to
verify that they conform to a Weibull
distribution. Computing the plotting
positions for the failures involves
approximating the true values of the Weibull
cumulative distribution function. Nelson
(1982) presents a variety of methods for
computing both probability and hazard
plotting positions. Regardless of the
technique employed, the data must be
plotted. Only the failures are plotted
although the censored data define the
relative plotting position. The LIFETEST
procedure eliminates most of the drudgery of
transforming and plotting the data. In
addition to providing graphical
representations of the data, PROC LIFETEST
can be used to compute nonparametric
estimates of the survival distribution. A
probability plot of data on our Pratt and
Whitney JTBD-7 engine appears in Figure 1.
The plotted points should lie in a relatively
straight line if the data conform to a
Weibull distribution. Once this requirement
has been satisfied, the Weibull parameters
may be estimated.
The flexibility of the Weibull distribution
explains why it applies to so many different
types of data. Nelson (1982) provides
numerous examples of Weibull analysis applied
to batteries, alarm clocks, generator field
windings, circuit breakers and pistons. Mann
(1968) mentions applications of the Weibull
distribution to the incidence of droughts,
the analysis of water droplet size and thrust
levels of rocket engine systems. Of
particular importance here is the
applicability of the Weibull distribution to
the analysis of aircraft engine failures.
Three articles provide the foundation for the
present study. Pratt and Whitney Aircraft
(1967) presents an empirical discussion of
estimating Weibull parameters. Pratt and
Whitney (1980) describes the results of an
extensive study to improve engine reliability
and reduce operating costs. The third
article, Cavoli and Rokicki (1981),
summarizes how to use the estimates to
predict engine failures. Much of this paper
is based on improvements to ideas presented
in those articles and advantages provided by
the SAS* system for such an analysis.
There are two generally accepted methods for
estimating the Weibull parameters: ordinary
least squares regression and maximum
likelihood estimation. Fortunately, the SAS
system can accommodate either technique. We
have compared the results from PROC REG with
those from PROC LIFEREG and have found that
the latter procedure provides more consistent
results. The LIFEREG procedure fits
parametric models to failure-time data that
may be time censored. The parameters are
estimated by maximum likelihood using a
ridge-stabilized Newton-Raphson algorithm.
The procedure iteratively solves for the
parameters until they reach a specified
converge,nce criteri on. For further detai 1s,
consult ·Nel son (1982).
METHODOLOGY
Now that we have discussed the theoretical
underpinnings of a Weibull analysis, let us
turn our attention to a step-by-step
description of the methodology. There is a
four-step procedure to the estimation of the
Weibull parameters.
1.
2.
Plot the data.
Estimate the parameters.
Define the scope.
Collect the data.
220
ANALYSIS
the planner and are not true failures.
Instead, convenience is another example of
time censoring.
Now that the reader has a general
understanding of the technique, we would like
to take a closer look at the analysis of our
JT8D-7 engines. Figure 2 shows the data as
they were originally collected and plotted.
It represents three years of failure data:
there are 104 failures and 126 time censored
values. The data as they are plotted
obviously do not fit a Weibull distribution
We have chosen to focus on premature and
foreign object damage failures because they
represent over three-fourths of our engine
removals and are very costly in terms of our
service commitment. Since life-limited
failures and convenience are both scheduled
events, they have considerably less impact on
our system. We have computed the results for
premature and foreign object damage failures'
separately. The results are shown in Figures
3 and 4.
because one observation has severely
distorted the linearity of the graph. The
SAS statistics guide cautions that extreme
observations can unduly influence the fitted
model. The results of our empirical studies
for both ordinary least squares and maximum
Before turning to the prediction of engine
failures, one other topic deserves a
likelihood indicate that if the number of
outliers is small, the best strategy is to
eliminate them. Considerable improvement is
obtained by removing only one extreme
observation (See Figure 1).
comment. Since the lIFEREG procedure uses
the maximum likelihood technique, it is
possible that the model will not converge.
It has been our experience that
nonconvergence can he a problem when sample
sizes are small. In those rare instances, an
If there is still a problem with the fit,
there are a variety of goodness-of-fit tests
for the Weibull distribution (Stephens, 1974,
1977l. In addition, SAS Communications*
(1987) provides details and appropriate code
to accompl ish this task using the
Anderson-Darling statistic. Since our model
is intended to be used by someone with little
statistical
expertise~
estimate of the Weibull parameters can be
obtained using the Benard and Bos-Levenbach
formula for median plotting positions
(Nelson, 1982) and the REG procedure.
However, the results should be interpreted
with caution.
PREDICTION
we have chosen not to
include a goodness-of-fit test. Although
plots do have limitations, they are more
easily understood by the 1ayman, and they are
extremely useful in detecting outliers.
To model future engine failures, we use the
Weibull parameters in a simulation. The
six-step procedure is diagrammed in Fi gure 5.
The analysis of failure data for a mechanism
as complex as a jet aircraft. engine is
considerably more complicated than we have
presented thus far. Aircraft engines
actually fit a mixed distribution model. For
the sake of simplicity, we shall develop our
1. Increment the time variable.
2. Check for a life-limited failure.
3. Calculate the conditional probability.
own taxonomy for engine failure.
6. Repeat the process.
4. Generate a random number.
S. Compare the conditional probability and
random number.
Engine Failure
Premature
life1imited
Foreign
Object
Damage
The procedure begins with four inputs for
each engine type: the Weibull parameters,
the current engine times, the time until the
next scheduled life-limited removal, and an
estimate of the average utilization. We have
chosen to increment the time variable on a
monthly basis to reduce the amount of
Conveni eoce
computer time required.
Our interest will center on predicting
premature failures which result from a part
or parts wearing out. We shall also consider
fails, it
time, the
continues
fail, the
foreign object damage (FOD); this occurs when
a bird or other object is drawn into the
engine causing sudden or catastrophic
failure. Life-limited failures are a
category of parts or modules of an engine
which have a mandated replacement time.
life-limited failures are easy to forecast
since it is simply a matter of tracking the
life-limited parts. The final category is
convenience.
Each engine is
treated separately during the course of the
simulation. After the monthly utilization is
added to the current engine time, it is
checked for a life-limited failure. If it
is replaced by an engine with zero
event is recorded, and a new engine
the simulation. If it does not
engine is checked for a premature
or foreign object damage failure.
The first step in identifying a random
failure is to calculate the conditional
probability of the event. Given that an
engine has reached time t1, the probability
it will reach t2 can be calculated using
These are at the discretion of
221
the following relationship:
REFERENCES
Cavo1i, N.C. and G.J. Rokicki, (1981),
IICommercial Engine Logistics Cost
Projections - A Dynamic and Flexible
Approach, II
We can use this equation in conjunction with
the cumulative distribution function to
calculate a probability of failure. This
probability is then compared to a random
number generated from a uniform
distribution. If the probability is less
than the random number, the engine begins a
new month. If the probability is greater
than or equal to the random number, the
engine fails and is replaced by a zero-time
engine.
Kalbfleisch, J.D. and R.L. Prentice,
(1980), The Statistical Analysis of
Failure Tlme Data, New York: John Wiley
and Sons.
Mann, N.R., (1968), "Point and Interval
Estimation Procedures for the
We have chosen one year as the
Two-Parameter Weibu11 and Extreme-Value
forecast horizon and typically run the model
100 times to improve its accuracy.
Oistributions, II Technometrics, 10,
The
231-256.
results are reported in tabular form by
engine serial number and reason for failure.
Nelson, W., (1982), Applied Life Data
Analysis, New York: John Wlley and Sons.
They are also reported in a histogram (Figure
6) so the planner can judge the range of
failures for each type of engine. Of course,
there is no guarantee that a particular
engine will fail since the process is based
on a random number generator.
Paper presented at the Gas
Turbine Conference and Products Show,
Houston, TX.
Pratt and Whitney Aircraft, (1967),
Introduction to Weibul1 Analysis, PWA
3001, East Hartford, CT.
On the other
hand, this method can provide considerably
more insight than applying a flat rate. It
also takes the age of the engine into account
since engines with more running time are more
likely to fail.
Pratt and Whitney Aircraft, (1980), JT8D
Maintenance Technology Study, PWA 5738,
East Hartford, CT.
SAS Communications*, (1987), "Distribution
Testing Using Base SAS* Software," SAS
Institute Inc., XIII, 30-31.
CONCLUSION
This paper has considered the application of
Weibu11 analysis to the prediction of
aircraft engine failures. There are distinct
advantages and disadvantages to this
approach. On the negative side, Weibu11
analysis has more startup time and requires
more technical expertise. It is also more
difficult for the layman to understand and
apply the technique. On the positive side,
SAS software eliminates much of the mundane
work with predefined procedures. That means
that more time can be spent analyzing the
data and less time coding. In addition, by
Stephens, M.A., (1974), "EOF Statistics for
Goodness of Fit and Some Comparisons,1I
Journal of the American Statistical
Association, 69, 730-736.
Stephens fl.A., (1977), "Goodness of Fit
for the Extreme Value Distribution,lI
Biometrika, 64, 583-588.
Weibull, W., (1951), "A Statistical
Distribution Function of Wide
Applicability," Journal of Applied
Mechanics, 18, 293-297.
taking the equipment age into consideration,
Weibul1 analysis provides a more
comprehensive approach to equipment failure
than rate analysis.
* SAS and SAS Communications are the
registered trademarks of SAS Institute
Inc., Cary, NC, USA.
Although this project is still relatively
young, we believe the advantages of this
approach will far outweigh the increased
For further information please contact:
startup costs by improving our planning,
budgeting and inventory control.
Furthermore, the versatility of the approach
means that it can be applied in other areas
of the company as well.
Bruce Kay
Federal Express Corp.
2831 Airways Blvd.
Memphis, TN 38132
222
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FIGURE.
223
STARTUP
AND LlFELIMITED TIMES
WEIBULL
PARAMETERS
UTILIZATION
RATE
CALCULATE
CONDITIONAL
PROBABILI'IY
GENERATE
RANDOM
NUMBER
y
N
FIGURE 5
JT80-1 FAILURE DISTf'llBUTlON
fREQUENCY BAR CHART
fREQUENCY
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flGURE 8
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