Preventive Maintenance and the
Cost Per Unit Time Equation
Components that wear out are
candidates for preventive maintenance. When the cost to replace
a component before it fails is less than the cost to replace the
component after it fails, it makes sense to maintain the
component preventively. The optimum preventive maintenance time
can be found using the
Weibull++,
BlockSim
and RCM++
software packages from ReliaSoft. These methods are based on the
cost per unit time to maintain the component, and they are
presented in this article.
Consider a nonrepairable
component with a failure rate behavior that is independent of
the age of the system in which it is installed. The component
has a life described by a Weibull distribution with
β
= 3 and η
= 150 days. It will cost $8 each time the component is replaced
after it fails (corrective maintenance), while it will cost $2
to replace the component before it fails (preventive
maintenance). It is the job of the reliability engineer to
determine the optimum preventive maintenance time for the
component.
The optimum solution to this
problem will be to choose the preventive maintenance time that
minimizes the cost per unit time. The equation describing cost
per unit time is as follows:
where CPUT is the cost
per unit time, C_{P} is the cost of a planned
(preventive) replacement, C_{U} is the cost of an
unplanned (corrective) replacement, R(t) is the
reliability function for the component and t is the
preventive maintenance time. Note that the costs used for this
model can be due to a variety of causes, which include things
such as monetary cost to replace the component, cost of
diminished company reputation and cost of lawsuits associated
with failures. The numerator of this expression represents the
average cost for a single replacement. It is the costs of
preventive and corrective maintenance actions weighted by the
probabilities that the component will survive or not survive the
preventive maintenance interval. The denominator of this
expression represents the average time until a single component
is replaced. The expression in the denominator is not easy to
explain as it is written. However, if the denominator is
integrated by parts, the cost per unit time equation becomes:
where f(t) is the
probability density function (pdf) for the component. The
first term is the replacement time multiplied by the probability
that the component will survive until the scheduled maintenance.
The second term is the expected value of the pdf on the
interval from 0 to the replacement time. In other words, it
represents the expected failure time of the components that fail
before the scheduled maintenance time, weighted by the
percentage of the components that do not survive until the
scheduled preventive maintenance action.
In order to determine the
lowest cost per unit time for this component, the time, t,
must be chosen to minimize the CPUT equation. This can be
accomplished by taking the derivative of the CPUT
equation, setting this derivative to zero, and solving for t.
The mathematical details are beyond the scope of this article.
One approach to solving this
problem is to use the Optimum Replacement report template in
Weibull++. First, the components failure rate behavior must
be defined in a Standard Folio. The analyst could use a Folio
containing component failure times and calculate the parameters
of the pdf for the component, but for the purpose of this
example, we will assume that the parameters describing the
failure distribution of the component are known from a previous
analysis. The analyst creates a new timestofailure Folio but
does not enter any data in the Data Sheet. When she clicks
Calculate, the Parameter Input window appears, allowing her
to enter the parameters of the distribution, as shown in Figure
1.
Figure 1: The Parameters of the
pdf (β
= 3 and η
= 150 hours) Are Prescribed
When the analyst clicks OK,
the parameters entered in the dialog box appear in the Folio's
Control Panel, as shown in Figure 2.
Figure 2: The Weibull++ Folio
Containing the Parameters for the Component
Next, the analyst chooses
Project > Add Report and the Report Wizard opens. In order
to link the report to the Data Sheet with the components
parameters, the analyst clicks Select and selects the
Data Sheet in the Select Folio/Data Sheet window, as shown in
Figure 3.
Figure 3: The Data Sheet Is
Linked to the Report
The analyst then selects the
Based on an existing Template check box and chooses the
Optimum Replacement template, as shown in Figure 4.
Figure 4: Choosing the Optimum
Replacement Report Template
When the analyst clicks OK,
the report shown in Figure 5 is created.
Figure 5: The Default Optimum
Replacement Report
The cost per unit time values
in column B are computed using the formula for CPUT given
above. Note that clicking a cell in column B (for example,
consider cell B9) displays an equation using the builtin
function COSTPERUNITTIME in the data entry box at the top of the
report. The arguments for this function are:
 Default, which
contains either a link to or the name of the data sheet
containing the components parameters.
 A9, which contains
the time at which the cost per unit time is computed.
 H5 and H6,
which contain the costs for preventive and corrective
maintenance actions, respectively.
The report displays cost per
unit time values for times ranging from 100 to 2100 days using
the default settings. However, since the characteristic life, or
the time by which 63.2% of the components are expected to fail,
is 150 days, the analyst must reduce the time values to capture
the optimum replacement time. In addition, the default costs in
the report are not the actual costs for the component. After the
analyst alters the top rows of the report to reflect these
changes, the report appears as shown in Figure 6.
Figure 6: Using the Optimum
Replacement Report in Weibull++ to Determine Optimum Preventive
Maintenance Time
The optimum preventive
maintenance time and corresponding cost per unit time are shown
in both the table and graphically. The optimum preventive
maintenance time for this scenario is around 80 days and the
corresponding cost per unit time is 3.69 per day.
A second way to determine the
optimum maintenance time for a nonrepairable component is to
use the Optimum Replacement tool in BlockSim. The analyst
creates a single block to represent the component, then
doubleclicks the block to open the Block Properties window. On
the Reliability tab, she enters the parameters of the
distribution, as shown in Figure 7, and then clicks OK.
Figure 7: Assigning the
Parameters of the
pdf to the Block
The analyst then rightclicks
the block and chooses Optimum Replacement in the shortcut
menu that appears. She enters the planned and unplanned
replacement costs and clicks Calculate to obtain the
results shown in Figure 8.
Figure 8: Using the Optimum
Replacement Tool in BlockSim to Determine Preventive Maintenance
Time
A third method to determine the
optimum preventive maintenance time is to use the Optimum
Maintenance tool in RCM++. The optimum preventive
maintenance interval in RCM++ is calculated at the task
level, using information specified for the task and for the
cause.
The analyst doubleclicks the
cause of interest to open the Cause Properties window. On the
Probablility page, she enters the parameters of the Weibull
distribution, as shown in Figure 9.
Figure 9: Entering the
Probability Parameters for the Cause
She enters the costs associated
with corrective maintenance on the Corrective Maintenance page,
as shown in Figure 10.
Figure 10: Entering the
Corrective Maintenance Costs
Next, the analyst clicks the
Tasks button to open the Task Manager, where she clicks
the Add Task button. In the Task Properties window that
appears, she will set up the replacements. She enters a Task
Name, chooses Repair/Replace in the Type field, chooses
Day in the Units field for the Assigned Interval and
verifies that the Assigned Interval is based on Item Age,
as shown in Figure 11.
Figure 11: Defining a New Task
To assign costs associated with
the preventive maintenance action, the analyst goes to the PM
Resources page of the Task Properties window. She then enters
the costs, as shown in Figure 12.
Figure 12: Entering the
Preventive Maintenance Costs
The analyst returns to the Task
page and clicks the Calculate Optimum button. The
Calculated Optimum Interval window shown in Figure 13 appears
and prompts her to choose how the calculated optimum interval
will be used. This gives the analyst the option to use this
calculated value as the assigned interval for the task or, if
desired, to store it as a proposed interval and to round up or
down, according to her needs, for the actual assigned interval.
In this case, she selects Set as Assigned Interval
and clicks OK.
Figure 13: Specifying How the
Calculated Optimum Interval Will Be Used
This optimum interval for
preventive maintenance, 83.123 days, corresponds to the value
obtained using BlockSim's Optimum Replacement tool, and
both confirm the approximate value determined using the Optimum
Replacement report template in
Weibull++.
