Past lessons have focused on cash flows. We have
considered methods for estimating costs and revenues that occur
throughout a project. We have used these estimates to decide
whether or not to proceed with a project, and which of several
alternative solutions to choose. Our decisions were based on
economic criteria. Time has played an important role because
of the time value of money. The economic evaluation considers
time through the measures of NPW, NAW and IRR.
In most of the previous lessons the timing of events is given.
Thus we prescribe when investments occur, how long the project
will last, and when costs and revenues are realized.
In this and the following lessons, the
timing of events is the major question. We consider primarily
the construction or installation phase of a project. It might
be the construction of a building, the design of a product or
the programming of new software, or any of a variety of tasks
that have a fixed goal and a collection of activities required
to meet the goal. The problem addressed in this section is to
schedule the activities of the project over time.
As an example, a builder has the task of constructing
a building to meet the specifications of a design provided by
an architect. The builder identifies a set of activities that
must be performed to accomplish this goal such as clear the
land, lay access roads, pour the foundation, erect the walls,
and so on. Each of the activities requires time, money and resources.
Often there are precedence relations that restrict the start of
some activity until others are completed. For
example, the erection of the walls cannot begin before the foundation
Before the project begins, the builder needs to
schedule the activities involved. The primary purposes of a schedule
- predict when the project will be completed
- predict when a subcontract should be let to perform an
- determine when limited resources should be obtained
- predict the cash flow over time for project expenditures
After the project begins, it is necessary to track
the performance of the various activities to assure that the
project is on schedule and on budget. Some activities may require
more time than originally estimated, while others may require
less. Resources thought to be available may be delayed. Any change
from projected values might require a new schedule to be prepared
or corrective actions to be taken.
This lesson describes the model we use for project
scheduling. The model includes a list of activities that comprise
the project, a list of the immediate predecessors for each activity,
and estimates of the time, money and resources required for each
activity. Of course an actual problem
may not exactly fit this model, but it is the one we use for
description and analysis. Most introductory and educational discussions
make the same simplifications as the ones described here, but
commercial project scheduling software has features
that describe more complex situations.
Because of the limited time available for the class,
not all the topics alluded to below are required for the course.
See the syllabus for a list of the required topics.
Given a word description of a project,
identify the activities and the immediate
predecessors of each activity. Also identify resources
used by the project and cash flows associated with
Construct an activity-on-node network
describing the project.
For a given project, use the Project
Management add-in to
find the critical path and describe the associated
schedule with activity start and end times.
Use the Project Management add-in to
find schedules that satisfy restrictions on resources.
If it is impossible to satisfy restrictions, find a
solution that minimizes some measure of infeasibility.
For a given project and schedule, use
the Project Management
add-in to find the
cash flow associated with the schedule.
Chapter 9 supports several lessons
in this part of the course. Section 9.1 introduces the project
scheduling problem, provides a historical perspective and relates
the subject of the chapter to the work breakdown schedule (WBS).
Read Section 9.2 about estimating activity durations. We have
used some of these techniques earlier for estimating costs. For
this and the next few lessons we assume durations are fixed,
but we later consider durations modeled as random variables.
Section 9.3 is about the learning effect. You should able to
apply these methods (used earlier for costs) to time estimation
for WBS activities. Precedence relations, discussed in Section 9.4 are
of primary importance for project scheduling.
The discussion below mentions Chapters 10, 11 and
12 from the text. These readings are not required, but the references
provide a more complete discussion of topics relevant to Project
Management than the lessons on this site.
For homework problems we will use
the Project Management add-in to simplify computations
and address problems of meaningful size. Students should become familiar with all the aspects of the add-in as discussed
in the documents reached by clicking the link below. Many of
the graphics in this section are screen shots of Excel spreadsheets
constructed by the add-in.
The model for project scheduling must include
all the physical tasks necessary to complete the project. Earlier
we described the WBS that breaks down a project into work packages
(WP). Necessary characteristics of the WBS is that WPs cannot
overlap in function and that they must collectively describe
everything that must be done to complete the project. When discussing
project scheduling we usually use the term activities to
describe the work contents of a project; however, sometimes we
use the word tasks. In our discussions we use the terms
WP, activities or tasks interchangeably. The important thing
is that the activities do not overlap in function and they collectively
describe all the work that must be done.
To illustrate we use an example
from earlier in this course called the Oil
The first step in the analysis is to create a list
of activities that comprise the project. We list them
below and assign an alphabetic designation to each.
The list will always include two artificial activities: a Start activity
that precedes all other activities, and an End activity
that follows all other activities.
The set of activities must
describe all of the steps necessary to complete a
project. Activities cannot overlap in function.
The oil collection narrative describes various ordering restrictions
in terms of which activity must precede another or which activity
should follow another. An obvious example is that the "Pour
activity that must precede the "Cure Slab" activity.
These are called precedence relations.
of an activity must be completed before the activity
The predecessors for the
example are given below. Some of the activities have more than one
It is only necessary
to identify immediate predecessors. Although we see that
D (Cure Slab) and E (Deliver Pump) precede H (Mount Pump), it
is also true that the predecessors of D and E, activities A (Pour
Slab) and B (Order Pump), also precede H. We do not list A
and B as predecessors of H because these relations are implied
by the other entries. Although the analysis does not fail if
other than immediate predecessors are listed, it is inefficient
to do so.
Indirectly, the table also shows successors
of the activities. Whenever an activity appears in the list of
predecessors, the rows in which the name appears identify the
successors of the activity. For example, we see C (Cut Path) listed
as a predecessor of F (Run Flow Line) and G (Run Electric Line).
This implies that the immediate successors of C are activities
F and G.
The successors of an activity
cannot begin until the activity
Although not shown, this table implies relations
between the start and end activities as well as the
remaining activities. An activity that has no listed predecessors
(A, B and C) is an immediate successor of the start activity.
Similarly, an activity that has no listed successors (M and N)
is an immediate predecessor of the end activity.
The list of activities and precedence relations
describes the structure of the problem. An equivalent
description of the project is the project network. Click
the icon below to see the project network for the example problem.
The circles, called nodes,
represent the activities. The lines connecting the nodes represent
the precedence relations and are called arcs.
The graphical aid of the project network is helpful when building
the project description and for checking the validity of precedence
To better understand the network we often use
a simplified version that shows the names rather than the descriptions.
In the next figure, the entries below the names
are the activity times, discussed in the next paragraph, and
the red nodes and arcs show the critical
path, also described later. This is the format of the network
constructed by the add-in.
This form of the network is called the activity-on-node network.
Another form is called the activity-on-arc network.
We choose the former because it is easier to construct and is
used by the Project Management add-in. The
process of constructing this network is called networking,
and it is an important part of project management.
Time is the primary consideration
of project scheduling.
The activity time or duration is the time required
from the moment the activity begins to the moment when it ends.
Since the analysis takes place before the project begins, the
time required for an activity must be an estimate. Of course
when the activity is actually performed the time may turn out
to be different than the estimate. The estimate times for our
example are in the column labeled Time in the chart
below. The time measurement in this case is days.
A schedule is a prescription of start times for
each activity. A feasible schedule is one where each activity
begins after all its predecessors are complete. A feasible schedule
is shown in the last two columns of the above table. The scheduled finish
time for an activity is the scheduled start time plus the activity
duration. For the example, we see that A, B and C all start at
time 0. Activity D starts at time 2, immediately after its predecessor
A is complete. Activity E starts at time 1 after its precedence
B is complete. Activity H, with predecessors D and E, starts
at time 5 when both D and E are complete. Although D finishes
at time 3, activity H must wait to begin until both D and E are
finished at time 5.
An alternative representation of the schedule
is shown in the Gantt chart below. Each activity is depicted as
a colored bar drawn from the start to the finish time. The particular
schedule shown results in the project completion time of 12.
This is the earliest possible time.
The goal of project scheduling is to find the schedule
that results in the smallest possible completion time. The activities
shown in bright red on the Gantt chart are the critical activities.
We say these activities are on the critical path. They
must be started exactly when specified or the project will be
delayed beyond the minimum of 12 days. The activities colored in maroon
may be delayed without increasing the project completion time.
We address the problem of finding the critical path in the
Lesson 21. Horizontal lines in the chart separate the critical and noncritical activities.
Most of the analyses in this section assume activity
times are fixed constants. An alternative is to describe an activity
time as a random variable with a specified probability distribution
function. Lesson 22 considers the problem of estimating the critical
activities when the duration times are uncertain.
Limited resources complicate the
scheduling problem. In the example, a crew of fixed size is assigned to each
activity as shown in the Crew column below. A limited
number of workers could make a schedule infeasible.
Using the earliest time schedule given in the Gantt chart,
the pattern of worker usage over time is shown in the figure
below. For illustration we set the number of workers available
to 8. We see that at times 1, 2, 4 and 5, the number of workers
used is greater than the number available indicating a
shortage in workers. The schedule is infeasible. Assigning
a cost of 1 for each unit of shortage, we find
that this solution has an infeasibility cost of 7.
Although the schedule is infeasible with respect
to the labor resource, it may still be possible to implement it.
The shortage in workers may be remedied by assigning overtime to workers or by hiring temporary workers. The shortage cost
could be set to reflect overtime charges or the extra costs of
temporary workers. Our goal will be to minimize shortages,
but it may be impossible to eliminate them entirely. The add-in
allows the project to be delayed beyond the minimum time at a
prescribed cost per unit delayed. This added flexibility
allows schedules that reduce shortages. Although the chart above includes
14 days of data, the example was limited to a completion
time of 12 days so the last two days are irrelevant.
When resources are limited, finding a schedule that is
feasible or optimal is theoretically a very difficult
problem. The Project
Management add-in has
a heuristic search procedure that attempts to find better solutions.
For the example, we obtained a better schedule which is
shown below. The activities Cut Path and Run Electric
Line have been delayed as indicated by the black bands.
The schedule still has the same finish time, but
with a reduced infeasibility cost of 3.
When resources are part of the model, the question
must be answered: how should the project be scheduled so that
available resources are not exceeded? Because of time limitations,
we do not answer this question in this class but the add-in
provides tools for estimating resource usage as a function of the
schedule. Automatic search methods look for schedules that minimize
resource constraint violations.
The add-in also offers an alternative
to the objective of minimizing shortages subject to fixed resource limits.
The alternative objective is to level the amount of resources
used and is achieved by minimizes the sum of the squares of the difference between the resources available and the amount used. This is similar to minimizing the variance of
resource utilization over the project duration. The link below
opens the instructions for the add-in with regard to resources.
The textbook provides a chapter on the management of resources
in project scheduling.
For some projects, the cash flow
is important. The table below shows the oil collection problem expanded
to include cash flow data. The earliest time schedule is also
shown. For a given activity,
entry in the column labeled Initial
C.F. is the amount expended at the start time of the
The entry in the column labeled Final
the amount expended at the end time of the activity.
The entry in the column labeled Uniform
the amount expended in each period from the start time to the
end time. For the start and end times, the uniform amount
is added to the initial and final amounts,
The term buckets refers to the time intervals
for the cash flow analysis. For the example, each bucket is one
day in length. For other problems buckets may span several
time periods or even factional time periods.
To illustrate the cash flow data consider the Pour
Slab activity. The initial cash flow, 1000, is
the cost of obtaining materials and setting up the forms to
hold the slab concrete. The uniform cost, 80, is
the cost of hiring the four person crew per day. This is expended
during each day of pouring, 2 days for the example. The final
cost, 200, is the cost of removing the forms and cleaning up
the area. Because the pour slab operation requires two days, the
cost of the first day is the initial cost plus the uniform cost
for one day, 1080. The cost for the second day is the final cost
plus the uniform cost for one day, 280. Similar justification could be provided for each
of the cost entries.
The table below shows the cash flow expenditures
in each period for the example problem using the earliest time
schedule. This might be important to the person in charge of the project's
finances. The first
three rows of the table describe the cumulative initial, uniform,
and final payments. The fourth row is the total cumulative payments
in each time bucket. The last row shows the cash amounts in
the individual buckets. The example has activities A, B and
C scheduled to begin at time 1. Thus we see that the total initial
cost is 1050, (1000 + 50 + 0). The uniform cost in the first period
is 220, (80 + 20 + 120). The final cost at time 1 is 0. Although
activity B ends in time 1 since it only lasts for one period,
its final cost is 0. The table reflects a time range of 14 days,
but the schedule shown has a 12-day duration. The maximum time
in the table is 15 days to capture the cumulative values for the 14-day time horizon.
The cash flow depends on the schedule. In the example,
which uses the earliest time schedule, most of the costs are
at the beginning of the project. Costs (and revenues) are delayed
as activities are delayed. Because of the time value of money,
there may be benefits in delaying activities that involve costs
and moving forward activities that involve revenue. The add-in
allows an objective to be defined that reflects the net present
value (NPW) of the cash flow. By minimizing that criterion,
the scheduling activity will reflect the financial measures that
could be important. The particular example only involves costs
and a small time period, but there are projects that last many
years and involve large economic impacts where the financial
measures are important.
In the graph
below, two cumulative cash flows are shown. The cash flow for
the current schedule is shown in green (it is also the early schedule which is normally shown in orange). The cash flow shown
in red is for a schedule that allowed the project to be completed
in 14 days. To meet this deadline, it was necessary to schedule the activities as late as possible. This allowed the activity expenditures
to be delayed. In general, if you plot a schedule with delays, it will be different than the early schedule.
The Project Management add-in
provides tools for estimating and evaluating cash flows.
The textbook provides a chapter on the management of budgets for
Critical path analysis is a planning
technique for scheduling future activities. After the project
actually begins, activities start and become "in progress."
Eventually activities finish. At any given time, some activities
are finished, others are in progress and others have yet to begin.
Finished activities may have required different durations than originally
estimated, and estimates for in-progress activities may also
require adjustment. The process of tracking the progress of an
ongoing project and making adjustments to estimates and schedules
is called Project Control. This is an extremely important
in practice because projects rarely proceed as planned. We cannot
provide an extensive discussion in this brief introduction, but
the interested student can read further on the topic in the references
The Project Management add-in has a control procedure called Update.
It can be used to monitor the project schedule and update time
and resource estimates.
The textbook provides a chapter on the control of projects.
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