Variable Activities do not have a fixed time, rather
the durations of these activities are determined by the start
and finish times of other activities. They have variable activity
times. To illustrate, consider again the pump installation example.
In addition to the situation described on the Example
and Cash Flow pages, we add some
new features to the project. We realize that a complicated cost
and revenue structure is probably not reasonable for this project
with such a short duration, but the example illustrates this
new modeling capability.
- The company must rent temporary buildings and other equipment
that are to be used for the entire duration of the project.
There is a deposit of $500 and the rent is $10 per hour.
One employee is hired to work in the office at a cost of
$20 per hour. At the end of the project the deposit will
- During the process of pouring and curing the slab, special
concrete equipment must be rented. The cost of this equipment
is $10 per hour. These payments begin with activity A and
end with activity D.
- The company must hire a Registered Professional Engineer
for all activities regarding electricity. The requirement
begins with the running of the electrical line (G) and ends
when the motor is tested (K). The cost of the engineer is
$50 per hour. There is an additional paperwork charge of $200
when the engineering is hired.
- The company will outsource the ordering and installation
of the pump unit. The requirement begins with the ordering
of the pump unit (B) and ends when the unit is installed (H).
There is an initial payment of $200 for this service and a
final payment of $300. The crew requirements are reduced to
zero for activities B and H, but one person is allocated to
this variable activity for supervision. The cost of this person
is $20 per hour.
- In order to finance the project the company will take out
a construction loan of $10,000. The money will be received
at the start of the project. The company must return to the
lender $10,500 at the end of the project. We also include
in this activity the management costs for the project of $80
These activities are different than those formerly described
in that their starting times and ending times depend on the
start and end times of other activities and are not fixed.
For a variable activity we specify two fixed time activities,
From Activity and a To Activity. The activity
described by paragraph 2 above starts at the same time as A
(the From Activity) and ends at the same time as
D (the To Activity) . When the From activity
is left blank, the variable activity starts at the beginning
of the project. When the To
is left blank the variable activity ends when the project is
For an additional variation, the company receives
income for the project. The company will be paid $27,000,
to be received in three payments. The first payment of $9,000
will be received at the completion of the slab curing step
(activity D). The second payment of $9,000 will be received
when the electric motor is installed (activity I). The last
payment of $9,000 will be received when the project is completed
activity). Three variable activities are added to represent
the payments. The variable activities are defined in the table
The project is defined
using the dialog below. We enter 8 as the number of Variable
button on the project definition page allows this number to
be increased or decreased.
The data form for the complete problem is shown below. We
have adjusted the cash flow data for some of the activities
to reflect the new information. The crew resources for G and
H have been reduced to 0 because these activities have been
outsourced. The variable activities require some additional
crew resources. Costs are represented as positive
cash flows and revenues are negative. We have associated the payments
with the final cash flows for the activities that determine
the timing of the payments.
Since the fixed times
have not changed for this example, the critical path and other
time-related results are the same as when variable activities
were not considered. The variable activities do not affect the
early and late schedule, however, they do simplify the modeling
of costs and resources when activities span a series of fixed-time
activities. These will affect the optimum schedule when costs
and resources are considered.
Of course it is important that a variable activity not end
before it begins. This illogical result is possible if one chooses
a From activity that is not restricted by the precedence
relations from starting after the end of the To activity.
The program does not check for this, but in most reasonable
cases the normal relationships between fixed-time activities
will prevent it.
Clicking the Solve button builds the equations that determine
the scheduled start and finish times. The example below shows
some nonzero delays. They are determined during the scheduling
process. The variable activities start and stop based on the From and To activities.
Clicking the Cash Flow button at the top of the schedule
worksheet computes the cash flow for three schedules: the schedule
with the activities beginning at their Earliest start
times, the schedule currently specified on the worksheet, and
the schedule with the activities beginning at their Latest start
times. The three results are shown in the green field above
the graph. Only the first few buckets are shown and the table
continues to the right.
The cumulative cash flows are shown in a chart as below. The
sudden changes in the cash flow indicates large receipts and
expenditures. The last observations on the right shows the
cumulative cash flows at the end of the project. These are
different for the three cases because the variable activities
have different durations for the cases and hence impose different
cash flows. The current schedule uses delays to reduce the
cost of shortages. The cumulative cost at the end of the time
horizon indicate whether the project makes a profit. The cumulative
cost for the early time schedule is -1340, indicating a profit
of 1340. The profit for the late schedule is 2170, while the
profit for the current schedule is 2580.