Weibull Point Process Applied To Repairable Systems
(an application to the Cuban sugar industry)
Ing. Segismundo Mojicar Caballero, Universidad de
In spite of any proactive kind of maintenance, we must face
failures that happen in our production assets. These failures
have to be treated by a minimal repair, that is, an action
that just restore the operability of the failed item to the
same condition as before fail. Usually these failures take
by surprise production and maintenance people and as repair
support is not ready, availability can be seriously affected.
We may ask ourselves: Why do these failures happen even when
a preventive (or predictive) maintenance program has been implemented
in our industry? Three basic reasons let us discuss about that:
- Generally, all the items of all the systems are not into
the preventive program. Failures that can not be forecasted
can not be prevented. So random failures are beyond the scope
of any preventive action and in complex systems we may find
many items that run to failure.
- Maintenance economics limits the maintenance inherent reliability,
so we must deal with a probability of failure. If a preventive
program of maintenance ensures an item reliability of 95%,
we are accepting the risk of 5% of failure probability.
- The preventive maintenance program fails. Sometimes maintenance
does not complain possible changes in production rates or
the incoming of new production technologies. As the number
of failures that should be avoided by the preventive maintenance
increases, the effectiveness of the preventive program becomes
worse and it has to be improved.
If we want save availability (for example, in a JIT production
process) minimal repairs must be avoided, that is, surprising
failure events must be reduced to the minimum.
Cuban sugar industry operates continuously during few month
per year and the highest technical availability is required
as in any other JIT process. Between successive seasons (time
that the industry operates) there is a general repair, that
is an action that restores the systems to a mean level of reliability
similar at the beginning of each season. Under these conditions
we can implement the following method for failure analysis:
Let's suppose that we have recorded the time when each failure
of a subsystem has happened during a season. The arrangement
of these failures (black points in Figure 1) along the time
axis is called a stochastical realization of failure events.
Figure 1. Stochastical realization of failures during
The position of each failure along the time axis is purely
random. The preventive (routine) maintenance takes place in
fixed intervals (yellow points in Figure 1). In order to achieve
such a representation we have assumed that non-operative time
in maintenance and minimal repairs (due to failures) is negligible
related to the time in operation.
Estimating reliability from a sample of stochastical process
Just a realization is not enough to make statistical inferences
about the failures that happen in a season. We need a sample
composed of several of such realizations.
A sample of realizations may be describe by three parameters:
Z: number of realizations in the sample.
R: indicate the type of subsystem (repairable or non-repairable
T: time when observation of all the realizations is stopped.
Figure 2. Sample of Z=5 realizations (seasons) observed
until T=140 days in a repairable subsystem.
As a general repair between seasons lets us assume that realizations
are independent one to each other, we can estimate an average
number of failure function:
where mz(t) is the total number of failure events that can
be counted in all the realizations till time t.
Then, N*(t) is a cumulative function. For t = 0, N*(t) =
0 and always it will show an increasingly trend (positive slope).
Table 1 shows the values that take the N*(t) function for
the sample shown in Figure 2.
Fitting a mathematical model.
The N*(t) function can be fitted to a mathematical model
using a non-linear regression procedure. In this case we use
a Weibull model, where the expected number of failures is given
where b is the shape parameter and q is the average time
per season when the first failure happens.
Table 2 shows the values of the fitted function N(t) for
the example we are carrying out. The estimated parameters are:
b =1.39 and q =88.5 days. Figure 3 shows both functions N*(t)
and N(t). The goodness of the fit is measured by R2 = 0.98.
Figure 3. Functions N*(t) and N(t).
The function N(t) may show different outcomes. We are going
to delve in the possible values of the shape parameter.
If b = 1, the N(t) function is a linear function with a constant
slope . It means that failures, in spite of the preventive
maintenance program, happen randomly.
If b < 1, the N(t) function slope becomes less abrupt as
time runs. In this case, failures happen mostly at the beginning
of the observation period and they are due to repair errors
(if the start of each realization is defined by the fulfillment
of a general repair as in our study in the Sugar Industry).
If b > 1, the N(t) function slope becomes more abrupt as
time runs. In this case failures are time dependent and it
points that preventive actions are not able to avoid the evolution
of these failures.
From the N(t) function other important reliability features
can be derived, as it is shown in Table 3.
The procedure, that for simplicity we called Weibull Point
Process (WPP), leads to important conclusions about the Maintenance
Inherent Reliability (MIR) and lets us implement new mathematical
approaches in order to control maintenance effectiveness and
optimize its variables. In next articles we will discuss these