Networks
Project Networks and PERT/CPM
Introduction
For our purposes, a project is a collection of related activities undertaken to achieve some goal. The activities use time and other
resources and there are precedence relations among them (certain activities must be completed before other activities can be
started.)
In the simplest case, we are concerned with the duration of the project – how long will it take to complete. We also are concerned to
know which activities are critical (delaying these activities will delay the project as a whole). the project network models can also
be used to manage a project (keeping track of schedule, use of resources, budgeting, making decisions about putting more resources
into activities to speed them up, etc.).
some background:
The Critical Path Method (CPM) was developed in 1957 by UNIVAC of Sperry Rand for DuPont – the purpose was planning and
management of construction projects – large chemical plants. Generally the activities were things that had been done before, so there
were good estimates of the time and resources required for each activity.
The Program Evaluation and Review Technique (PERT) was developed in the same period by the Navy with Booz Allen Hamilton
and Lockheed for planning and overseeing research and development projects – the first big use was in the development of the
Polaris missile-launching submarine [The method has been credited with bringing the program in earlier than anticipated and with
budget savings]. PERT differs from CPM in focusing more on probability – based estimates.
Most people don’t worry too much about the distinction any more and refer to PERT/CPM .
All major construction projects, all government –funded development projects require PERT/CPM analysis as part of the bid/contract
process. A typical large project will involve thousands of activities to be coordinated and managed.
We will first look at the setup of such models and at construction of a spreadsheet model for finding critical path and early and late
schedules, then move to refinements and elaborations:
These are descriptive and analytic models – unlike the optimization models we have been considering.
The first step in developing a project network model is identification of the activities together with the immediate predecessors and
the duration of each. We will leave that to the management team, and focus on the development and use of the network itself.
An example:
Angela has just finished her undergraduate degree. After exploring the job market for over a year she has decided to go to school to
work for an MBA degree. She has not applied to any school yet, but wishes to anticipate how long the admission process will take.
She has identified the following activities:
Activity
A. Select universities
B. Obtain application
C. Take GMAT-send score
D. Complete application
E. Wait for answer
F. Make final decision
Predecessor
A
A
B
C,D
E
Duration(days)
14
21
50
14
30
10
With this information, we draw the project network [much of this information can be put on the Gantt chart – but we want to look at
the network anyway]
[Note: We will discuss the “Activity on Node” form of the network – there is also an “Activity on Arc” form – it contains the same
information and is the older form. The Activity on Node form seems to be taking over – and it’s the form described in your text]
Each activity is a node in the network.
In working by hand, we often add a “start” node and a “finish” node to tie the whole network together [like the single “source” and
single “sink” node in a flow network] .
We put an arc to a node from each of its immediate predecessors (we don’t mark all predecessors – just the immediate ones – the
network takes care of the rest) – A is an immediate predecessor of B if B must wait for A to finish, but does not have to wait
for anything else that depends on A.
The network is always drawn so that the arcs go from left to right or so that the arcs go from top to bottom [often easier on a vertical
sheet of paper
Each activity is marked with some label (typically letters A, B, C etc.) and with its duration.
To find the shortest length of time in which the project can be completed:
We work forward through the network and mark each activity with its early start time (ES) and early finish time . The early start
time for the first activity is the starting date of the project (present time, or time 0 if there is no fixed date). The early finish time is
then the early start time plus the duration. For other activities, the early start time is the time that the last of the immediate
predecessors finishes (latest of the EF times on the immediate predecessors). The early finish time is early start time plus duration.
The latest EF time is the shortest time in which th eproject can be compelted [since all activities must be completed for the project to
be complete]
To find which activities must be rigidly scheduled [critical activities] and which have slack:
We then work through the network in reverse and label each activity with its latest finish time (LF) and latest start time (the latest
times that the activity can start and finish without delaying the project as a whole). The late finish time for the project (for the
“finish” activity, or for all activities with no successors) is the time when the project must be completed or (if there is no such
time/date) the latest of the “early finish” times (the earliest possible completion date for the project). The late start time is the late
finish time minus duration. For earlier activities, the late finish time is the earliest of the late start times on the successors (the time
the activity must be completed to avoid delaying the project) and the late start time is late finish minus late finish.
The difference between early start and late start [or between early finish and late finish) is the slack or float on an activity –
the amount by which it could be delayed without delaying the project. An activity with slack 0 is a critical activity . Any delay in a
critical activity will delay the project. There is a path through the network (there may be more than one) made up of critical activities
– this is the critical path - the sequence (or sequences) of activities that must be managed and watched most closely.
With this information, we can get more detailed information about th eeffect of delays or problems in individual activities. We will
further develop anlaysis of projects based on the critical path and on slack in the non-critical activities.