CHAPTER 10
CRITICAL PATH METHOD
1.The critical path method (CPM) can be employed to aid
in the scheduling and coordination of the many
interrelated activities that make up a large, complex
project.
2.This technique was first developed in 1957 to aid in the
planning of the construction and maintenance of
chemical process plants.
3.Since then it has become one of the best known and
most widely used techniques within operations
research.
It has been successfully applied in many diverse areas,
including scheduling a highway construction project,
developing a countdown and "hold" procedure for the
launching of space flights, installing a computer system
and building a ship.
For our purposes, a project will be defined by:
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1. a list of activities
2. each with an estimated completion time,
3. and a set of precedence relationships between
these activities.
It is these relationships which are of primary importance,
as they will allow us to determine which of the individual
activities has the greatest potential for disrupting the
overall project schedule if their durations are for some
reason delayed beyond the estimated time.
It is the goal of a CPM analysis to not just determine the
minimum completion time for the project (we already
know how to do this from the previous chapter); but also
to highlight those activities to which special effort should
be applied to guarantee their successful on-time
completion.
We proceed by first laying the foundation for our analysis
by demonstrating how to represent any project in the
form of a network. We next develop the central ideas of
CPM and conclude by investigating one of many possible
extensions to the basic methodology.
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10.1 DEVELOPING THE PROJECT NETWORK
In applying a CPM analysis it is helpful, space
permitting, to first represent the project schematically in
the form of a network. To do this we need the following
three items of data:



List of jobs to be performed
Best estimate of the duration of each job
Predecessors of each job (ie. for each job, a list of
jobs that must be completed before it can begin).
It will often be helpful to organize this data in the form of
a table, as illustrated in Figure 10.1.
Activity
Foundation
Walls
Roof
Interior
Duration
(days)
27
10
5
33
10.3
Immediate
Predecessors
--Foundation
Walls
Roof
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Figure 10.1
Once the data is organized in this form, a project
network is then designed to represent the precedence
relationships between the activities. We point out
immediately that any project network must be acyclic
(think about why this is the case). Now, there are two
basic ways of assigning a network to a project and they
differ in their representations of the activities.

In the activity-on-node representation, or simply
activity network, each activity is represented by a
node and a directed arc is drawn from node i to node j
if the job corresponding to node i immediately precedes
the job corresponding to node j. This type of network
representation is probably the most natural, and it is a
very straightforward process to construct the (unique!)
activity network corresponding to any project.
However, for reasons that are not quite clear, the
activity network has not received much attention
throughout the literature. For this reason, we will
concentrate our attention on the second approach for
representing projects with networks.

In the activity-on-arc representation, or simply event
network, the activities are represented by directed
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arcs with values equal to the duration of the activities.
The precedence relationship among the activities is
established through the nodes, which correspond to
events representing the completion or initiation (or
both) of some set of activities.
We state immediately that it is not always a simple
matter to construct an event network for a given
project. Furthermore, it is often the case that the
representation is not unique, so that several different
networks can be legitimately developed for the same
project. In general, there will always exist an event
network representation for any given project containing
less nodes and arcs than the corresponding activity
network; however, there is no known efficient algorithm
for constructing this minimal-sized event network. We
will outline a process here for constructing event
networks that, while not guaranteeing the minimum
number of nodes and arcs, has proven to be fairly
efficient and will be quite adequate for our purposes. For
further information on this topic sees Syslo, Section 4.2.1
To illustrate a simple example of how precedence
relationships for a project are modelled, the case where
activities A and B must be completed before activity C
begins. Therefore, A and B are predecessors of C and this
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relationship can be represented schematically as shown in
Figure 10.2 where nodes 1 and 2 represent the beginning
of jobs A and B respectively; node 3 represents the
completion of both these tasks and the simultaneous
initiation of job C; and node 4 represents the completion
of job C.
A
1
C
3
4
B
2
Figure 10.2
As another example, if activity A must be completed
before both activities B and C begin, the network
representation would appear as follows:
1
A
B
3
C
4
2
Figure 10.3
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Here node 1 represents the initiation of job A; node 2
represents the completion of A and beginning of both
jobs B and C; and nodes 3 and 4 represent the completion
of jobs B and C respectively.
We can now formalize our description of this process. In
general, when constructing an event network for all of the
jobs in a project, there are a few rules, which must be
followed.
 First, the start of the project is represented by an origin
labelled as node 1 - all arcs incident from this node
correspond to jobs which have no predecessors (ie. jobs
which can begin as soon as the project begins).
 Second, there is a single node, termed the terminal
node, corresponding to the completion of the entire
project.
 Lastly, the nodes must be numbered in such a way that
the node representing the completion of any activity
must have a larger label than the node representing the
beginning of that activity - in other words, arcs must
always be directed to higher labelled nodes. We know
that this can generally be accomplished because the
fact that all project networks are acyclic allows us to
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appeal to Lemma 9.1.1, which states that any acyclic
network must admit such a labelling scheme.
Notice that these rules tell us how the project network
must ultimately look; but really does not provide much
information on the appropriate steps for developing such
a network. This is a process, which would probably be
more difficult to explain than to actually perform so we
will instead provide an illustration of our approach
through a simple example.
10.1.1 Example
If we were to decide to build a house, then one way to
schedule the appropriate activities for this project might
be as follows: First, lay the foundation. When this is
done, rough the walls. Once this is complete, we can
begin roughing the floors and also putting up exterior
siding. We begin painting when the exterior siding is
complete. When the floor is laid, we can begin the
electrical work and the plumbing. When the electrical
work is complete, we begin the interior painting. The
interior fixtures are placed once the plumbing and interior
painting is complete. Final cleaning begins when all other
tasks are have been completed.
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We can organize this information (and add our best
estimate of duration times for each task) in the form
given in Figure 10.1. Observe that it is sufficient to
specify the immediate predecessors, rather than
predecessor for each activity. The data is summarised in
the following table.
Activity
Duratio
n (days)
3
5
13
6
4
8
7
3
2
1
A = foundation
B = walls
C = floor
D = siding
E = exterior
F = electrical
G = plumbing
H = interior paint
I = fixtures
J = cleaning
Immediate
Predecessors
---A
B
B
D
C
C
F
G,H
E,I
Observe that, while we only list job B as a predecessor of
job C, it is obviously true that job A must also be
completed before job C can begin. However, because we
have already accounted for the fact that B can not begin
until A is completed (ie. A is a predecessor of B), it
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would be redundant to now declare A as a predecessor of
C also. Therefore, in general, when describing
precedence relationships in the form of the above table,
we need to specify only the immediate predecessors of
each activity. Organizing the data in this form will make
it a bit easier to develop the project network directly from
the table.
On the other hand, we point out that it is often very
difficult for large problems to work form a verbal
description and distinguish only the immediate
predecessors from other "non-immediate" predecessors
for each job. Therefore, since nothing has been lost if
some redundancies do exist in the table, we need not
insist that the table appear in precisely this way.
The network for this project is displayed in Figure 10.4.
Notice that there is one origin and one terminal node
(nodes 1 and 9 respectively) and that each activity ends at
a higher labelled node than from which it began. Thus,
our network representation in Figure 10.4 of the given
project is a valid project network. Finally, along with
each arc we place the associated task name (or symbol)
and the estimated duration time. 
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F,8
6
H
3
G,7
4
7
I
2
C
13
3 3
B,5
2
CHAPTER 10: CRITICAL PATH METHOD
D
6
A
13
9
J,1
8
E
4
5
1
Figure 10. 4
This was a relatively simple example, so it is important to
point out here that there are two types of complexities
that will often arise throughout the formulation of a
project network. In particular, in order that a network
appears in an appropriate form for application of the
critical path method, we must now impose certain
restrictions on the structure of this network.
First, consider a project where jobs A and B begin at
exactly the same time and are both predecessors of job
C. From the rules we have thus far established, it would
be legitimate to represent this relationship as follows:
A
10.11
1
B
2
C
3
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Figure 10.5
However, we will from here on require that, for any
project network, two nodes can be connected by at most
one arc. In order to satisfy this requirement for the
network in Figure 10.5, we need to introduce to the
project a dummy activity with duration of zero. When
we add this dummy activity to the network representation
of the project in Figure 10.5, we indicate the arc
corresponding to the dummy activity by a dotted arrow.
Furthermore, we must also create an additional node to
represent the event corresponding to the initiation of the
dummy activity. We can now put aside the network in
Figure 10.5 and represent the same relationship as
follows:
1
3
A
B
C
2
Dummy activity
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Figure 10.6
Notice that the networks of Figures 10.5 and 10.6 are
equivalent in terms of the precedence relationships they
define; however, as explained above, the latter will be
more amenable to analysis with the critical path method.
To illustrate the second instance where we will need to
use a dummy activity, consider a project where jobs A
and B both begin at the same time, job A must be
completed before job C can begin, B must be completed
before job D can begin, and both A and D must be
completed before job E begins. One network that
captures these relationships is as follows:
4
C
2
E
A
1
B
3
D
5
Figure 10.7
10.13
E
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However, we will from here on want to impose the
constraint that an activity will not be represented by
more than one arc in the project network. To satisfy this
requirement, we need to add a dummy activity to the
given project and then develop the following network,
equivalent to the one above in Figure 10.7
4
C
Dummy activity
2
A
6
Figure
10.85
D
E
In summary then, dummy activities are introduced into a
project to avoid the occurrence of either two nodes being
connected by more than one arc or some activity being
represented by more than one arc in the associated project
network. Each dummy activity is represented by a dotted
arc, does not correspond to any actual activity and must
always have a duration time equal to zero.
1
B
3
10.1.2 Example
Consider the project described by the following table.
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Acti Duratio Immediate
vity
n
Predecessors
(days)
A
4
----B
2
----C
5
A
D
11
A,B
E
8
----F
3
C,E
G
2
C,E
We can develop a project network in accordance with all
our rules and requirements as shown in Figure 10.9.
You should be aware of how to utilize dummy activities
when developing project networks; however, from here
on, to avoid any unnecessary complications, we will
apply the critical path method only to networks that do
not contain these sorts of activities.
E,8
1
A,4
2
C,5
4
0
B,2
3
10.15
F,3
G
D,11
6
2
5
0
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Figure 10.9
10.2 DETERMINING THE CRITICAL PATH
In describing the critical path method, we will assume
throughout this section to be working on a project with a
network representation as follows:
5
D,4
A,3
1
B,2
2
C,2
3
E,1
G
7
4
F, 3
6
Figure 10.10
Once again, notice that we display an activity name and
duration with each arc of the network. For our purposes
here, the relevant notation is as follows:
tm,n := duration of the job represented by arc (m,n).
T := earliest completion time of the entire project.
Notice that the earliest completion time, T, corresponds
to the earliest time that the terminal node of the network
(in this case node 6) is reached. In general, we assume
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that the network is comprised of N nodes with the origin
corresponding to node 1 and the terminal node to node N.
The fundamental steps for a CPM analysis involve the
computation of two descriptive statistics for each node
(ie. event) of the network.
10.2.1 Definition
The early event time for node n, denoted ET(n), is the
earliest time that the event corresponding to node n can
occur, namely all jobs preceding node n have been
completed. 
10.2.2 Definition
The late event time for node n, denoted LT(n), is the
latest time that the event corresponding to node n can
occur without delaying the completion time T of the
entire project. 
So we see that the early event times describe how the
project can proceed if each activity is begun at the
earliest possible time and there is no delay in its
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performance. The late event times provide information on
how much room for delay there is at each event of the
network.
We now describe how these times are
computed. The early event time of a node is specified by
the length of the longest path from the origin to that node.
This may not seem quite right - the first thought might be
that the earliest time for a node should be determined by
the shortest path to that node. This is perhaps more
intuitive, but also incorrect. Remember that, in
performing a shortest path calculation we consider all of
the paths leading to the node under consideration and
then select the one with the shortest length. On the other
hand, in order to reach a node of a project network we
need to move through all of the paths leading to this node
(notice that this is equivalent to stating that we must
complete all jobs preceding the event corresponding to
the node). Therefore, the earliest time that the node can
be reached is dependent on the path that takes the longest
to traverse.
As a further explanation, imagine that you are trying to
determine when is the earliest time you can leave your
house for a vacation. Before leaving you must wait for
three friends, all of whom leave their initial destination at
2:00, to arrive at your house. The first friend estimates
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his travel time as two hours, the second as three hours,
and the third as six hours. The earliest time that you
could expect to leave your house is given by their
departure time plus the time required for the longest trip ie. 2:00 + 6 hours = 8:00.
Therefore, to specify the early event time for each node
of a project network, we can appeal to our dynamic
programming equations of (9.12) for specifying the
length of a longest path to from the origin to any node of
a network. This yields the following (remember that, as
defined in Chapter 9, P (n) represents the set of
immediate predecessors of node n):
ET(n)  max ET(m)  t m,n  , n  2, , N
(10.1a)
mP( n)
ET(1)  0
(10.1b)
These equations are solved exactly as before, namely
recursively, for n=1,...,N (in this order).
10.2.3 Example
To compute the early event times for all nodes in the
project network of Figure 10.10 we simply solve the
equations of (10.1) as follows:
ET(1)  0
ET(2)  max ET(m)  tm,2  max {0  3}  3
m{1}
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ET(3)  max ET(m)  tm,3  max {0  2}  2
m{1}
ET(4)  max ET(m)  t m,4  max {3  2,2  1}  5
m{2,3}
ET(5)  max ET (m)  t m,5  max {3  4}  7
m{2}
ET(6)  max ET(m)  t m,6  max {5  3, 7  7}  14
m{4,5}
As mentioned earlier, the earliest completion time for the
project is given by the earliest event time for the terminal
node, so that T = ET(6) = 14. 
We now compute the latest event times for each node.
Remember that our definition of this statistic is the latest
time that an event can be initiated so as not to delay the
completion time for the entire project. We therefore know
that the event corresponding to the completion of the
project, represented by node N, must have a latest time
equal to T. So we have the following initial condition:
LT(N) = T = ET(N)
(10.2)
To compute the latest event times for all other nodes we
will work recursively backward through the network. The
reasoning behind working in this direction is that we are
now interested in determining the latest possible starting
time for an event such that the remaining schedule is not
disrupted. We therefore need to know what are the latest
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possible start times for all events following the event
under consideration. We then determine latest times for
each event so that all succeeding events can be initiated
within their latest times.
For example, imagine that you will begin and perform
two jobs simultaneously, the first taking six hours and the
second taking four. There is one group waiting on you to
complete the first job and another waiting on you to
finish the second; and both say that the latest they can
begin their activities is 10:00. It is easy to see that you
must begin your work at 4:00 to satisfy the first group
and at 6:00 so as not to disrupt the second. In order to
not disrupt any part of the overall schedule you must
begin at the earlier of these times, namely 4:00.
Combining this reasoning with the initial condition in
(10.2), we can now specify the dynamic programming
equations for computing late event times for each node of
a project network as follows (remember that S(m) was
defined in Chapter 9 as the set of immediate successors of
node m):
LT(n)  min LT(m)  t  , n  1, , N 1 (10.3a)
n,m
mS(n)
LT( N)  T  ET (N)
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(10.3b)
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We solve these equations
=N1,N2,...,1 (in this order).
recursively,
for
m
10.2.4 Example
For our sample network in Figure 10.10, the
computations of late event times proceed as follows:
LT(6)  ET(6)  T  14
LT(5)  min T(m)  t5,m  min{14  7}  7
m{6}
LT(4)  min T(m)  t4,m  min{14  3}  11
m{6}
LT(3)  min T(m)  t3,m  min{11 1}  10
m{4}
LT(2)  min T(m)  t2,m  min{7  4,11  2}  3
m{4,5}
LT(1)  min T(m)  t1,m  min{3  3,10  2}  0
m{2,3}
Notice that LT(1) must always equal zero (think about
why this is the case) - this fact is useful for checking the
calculations.
After calculating both the early and late event times for
each node, it is useful to highlight this information in the
project network by redrawing the network with the nodes
displayed in the following format:
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n
ET(n)
LT(n)
Figure 10.11
Our sample problem, originally represented with the
network in Figure 10.10, will now appear as shown in
Figure 10.12.
5
D,4
A,3
2
7 1414
7 11
141411
G
14
7
C,2
3 3 11
11
1
0000
0 0
B,2
3
4
E,1
F, 3
5 11 11
11
2 10 11
11
6
14 1414
14 11
141411
14
Figure 10.12
Now, it is important to remember that the arc values tm,n in
any project network are merely estimates of the duration
of each activity collected before the project actually
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begins. In other words, the actual duration of any job is
unknown when this sort of analysis would take place it
could turn out to be less than our estimate, but, more
critically, it also could be more. We therefore need to
develop a statistic to measure the amount of time that can
be employed for the performance of each activity, beyond
our initial duration estimate, before the completion of the
overall project is delayed.
We already have all the
information we need.
10.2.5 Definition
The total float for the activity corresponding to arc (m,n),
denoted TF(m,n), is the amount by which the duration of
the activity corresponding to that arc can be increased
without delaying the completion of the entire project,
assuming no other activities are delayed. 
To compute this statistic, notice that activity (m,n) begins
at node m. Assuming that no earlier activities are
delayed, then this activity can be initiated at the earliest
possible time, ET(m). The activity is completed at node
n. Assuming that no later activities will be delayed, then
we can allow this activity to be completed at the latest
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possible time, LT(n). The maximum amount of time
available for the performance of this job without delaying
the overall project completion time (assuming no other
jobs are delayed) is therefore given by LT(n)ET(m).
Since our initial estimate for the duration of this task is
given by tm,n, the total float associated with this job is
given by
TF(m,n) = LT(n)  ET(m)  tm,n
(10.4)
10.2.6 Example
Using the equation of (10.4), the total float for each
activity for our sample network is calculated as follows:
TF(A) = TF(1,2) = LT(2)  ET(1) t1,2 = 3  0  3 =
0
TF(B) = TF(1,3) = LT(3) ET(1)  t1,3 = 10  0 2
= 8
TF(C) = TF(2,4) = LT(4) ET(2) t2,4 = 11 3 2 =
6
TF(D) = TF(2,5) = LT(5) ET(2) t2,5 = 7 3  4 =
0
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TF(E) = TF(3,4) = LT(4) ET(3) t3,4 = 11 2 1 =
8
TF(F) = TF(4,6) = LT(6) ET(4) t4,6 = 14 5 3 =
6
TF(G) = TF(5,6) = LT(6) ET(5) t5,6 = 14 7  7
=0
The concept of total float is intended to provide a
quantitative measure of the importance of keeping each
activity's actual duration from exceeding our initial
estimate. The smaller the value of the total float for any
particular job, the less leeway we have for overshooting
our performance estimate for this job without adversely
affecting the entire project.
10.2.7 Definition
A critical activity is a job whose total float is equal to
zero. 
It will be particularly
important to monitor the
performance of these critical activities, since any delay in
their durations will necessarily result in a corresponding
delay in the completion of the entire project. Notice also
that, if one were concerned with decreasing the estimated
overall project dur ation, then these same critical
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activities are the ones whose durations must be shortened.
In general, project completion time is sensitive to
changes in the durations of these activities in either
direction (increasing/decreasing). However, we will
continue to take the pessimistic view and therefore speak
exclusively in terms of avoiding delays. There will be
several critical activities throughout the project.
10.2.8 Definition
A critical path is a path in the project network from the
origin (node 1) to the terminal node (node N) that
consists entirely of critical activities. 
Any project network will always have at least one
critical path. This is the one corresponding to the
longest path from the origin to the terminal node.
Remember that in order to complete the project we must
cover all paths from the origin to terminal node, so the
earliest time for completing the entire project is
determined by the longest path. Furthermore, there could
be more than one critical path - depending on the
placement of the critical activities throughout the
network.
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10.2.9 Example
We see from our computations of total float in Example
10.2.6 that the critical activities for our sample network
in Figure 10.10 are jobs A, D and G, so that the (in this
case the only) critical path is given by 1, 2, 5, 6. 
Because any delays in critical activities will delay the
completion of the entire project, it is important to closely
monitor the performance of all activities along the critical
paths and take all appropriate actions.
Now, notice that the total float statistic indicates the
maximum amount of time that an activity can be allowed
to go beyond its estimated duration before the overall
project is delayed. There is a related measurement that
indicates the amount of additional time available for an
activity before the earliest start times of any proceeding
jobs are affected.
10.2.10 Definition
The free float for the activity corresponding to arc (m,n),
denoted FF(m,n), is the amount by which the duration of
this activity can be increased without delaying the start of
any later activity beyond its earliest possible starting
time. 
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To compute this statistic notice that, for any activity
(m,n), the earliest possible start time is ET(m). If no
succeeding jobs are to be delayed then this activity must
finish no later than ET(n), so that the total available time
is ET(n)ET(m).
Since the estimated activity duration is given by tm,n, we
have (assuming no dummy activities):
FF(m,n) = ET(n)  ET(m) tm,n (10.5)
10.2.11 Example
Using the recipe given in (10.5), the free float for each
activity in our sample network is computed as follows:
FF(A) = FF(1,2) = ET(2) ET(1) t1,2 = 3 0 3 = 0
FF(B) = FF(1,3) = ET(3) ET(1)  t1,3 = 2  0 2 =
0
FF(C) = FF(2,4) = ET(4)  ET(2) t2,4 = 5 3 2 =
0
FF(D) = FF(2,5) = ET(5)  ET(2)  t2,5 = 7 3 4 =
0
FF(E) = FF(3,4) = ET(4) ET(3) t3,4 = 5 2 1 = 2
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FF(F) = FF(4,6) = ET(6) ET(4)  t4,6 = 14 5 3 =
6
FF(G) = FF(5,6) = LT(6) ET(5) t5,6 = 14 7 7 =
0
Notice that the total float for activities B and C from
Example 10.2.6 are positive valued while the free float
for these same activities from Example 10.2.11 are zero.
In general, free float figures must always be less than or
equal to total float because it is defined so as to allow less
deviation from the overall schedule. Remember that total
float tells us the amount of time available before the
completion of the project is delayed while free float
indicates the amount of time available before any part of
the schedule is delayed. The relative importance of these
two statistics for helping to monitor the schedule and
allocate resources among activities will depend to a large
extent on the overall objectives of the project and perhaps
even the personal style of the project manager.
The final step of any CPM analysis is to display the
results in the form of a graph, which highlights the
important information. Simply redraw the project
network on a horizontal axis representing time and
indicate the performance of each critical activity with a
solid arrow and each non-critical activity with a dotted
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arrow. The graph for our sample network in Figure 10.10
is as shown in Figure 10.13.
Event Chart
1
2
3
5
6
4
0 1 2 3 4 5 6 Time
7 8 9 10 11 12 13 14
e
Figure 10.13
Alternatively, we can draw the famous Gantt Chart for
the project. This type of chart, shown in Figure 10.14,
displays the activities on a time scale and indicates their
respective total floats.
Gantt Chart
A
B
C
D
E
F
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Figure 10.14
This sort of graphical presentation will prove very useful
to project management for tracking actual versus planned
performance and also for making decisions on what steps
should be taken to keep the overall project running on (or
at least close to) schedule.
Finally, it is important to note that the output of a CPM
study should not be considered to provide the ultimate
"answer" to the scheduling and underlying resource
allocation problem. Remember that CPM is applied
given a set of estimates for activity durations and that
these estimates are based on a particular allocation of
resources among the various activities of the project.
Furthermore, it is usually the case that the project
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manager will have a good deal of flexibility for altering
these estimates before initiation of the project by
changing his current allocation strategy (for example,
manpower and machinery can often be transferred from
one activity to another). So we see that determining the
optimal allocation of these resources is a very important
problem since it will effectively specify the input data for
the CPM analysis (and, obviously, the output of the
analysis will be very sensitive to this data).
We will not have any more to say about this problem.
However, we will point out that once a commitment has
been established to an initial resource allocation and
activity durations have been estimated and CPM has been
performed, there is a form of sensitivity analysis that can
be applied in order to consider the relative costs and
benefits of making slight changes and additions to the
present allocation. For more information on this topic see
Markland, Section 13.3.
10.3 CPM WITH LINEAR PROGRAMMING
As mentioned above, there will always be a critical path
corresponding to the longest path in any project network.
We can develop a linear program to determine the length
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of this path and then employ the output from the simplex
method to specify the critical activities.
Given a project, we associate one decision variable with
each node of the corresponding project network. These
variables are defined as
xj = time that the event corresponding to node j is
scheduled to occur.
There will be one constraint associated with each activity
(ie. arc of the network). For any arc (i,j), we know that
the corresponding activity is scheduled to begin at node i
at time xi and end at node j at time xj. Given that the
activity has duration of ti,j, in order to satisfy the
precedence constraints on schedule, the following
condition must be satisfied:
(10.6)
x j  xi  ti, j , j, i P( j)
Given these restrictions, we want to specify values for the
decision variables that minimize the overall duration of
the project. Remembering that the first node is labelled 1,
so that x1 is the scheduled start time for the project, and
the last node is labelled N, so that x is the scheduled
completion time for the project, the objective function
and optimality criteria are given by
N
min Z = xN  x1
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Combining (10.6) and (10.7) with non-negativity
restrictions on the decision variables establishes a linear
program whose optimal objective function value specifies
the minimum duration of the project, or equivalently, the
length of longest path from the origin (node 1) to the
terminal node (node N) in the corresponding project
network. In general, this linear program will appear as
follows:
min x N  x1
x
(10.8)
s.t.
x j  xi  ti, j
xj  0 ,
, j, i P( j)
j  1, , N
10.3.1 Example
We can determine the longest path in our sample network
of Figure 10.10 with the following linear program:
min Z  x6  x1
x
s.t.
x2  x1  3
x3  x1  2
x 4  x2  2
x5  x2  4
x 4  x3  1
(10.9)
(Activity A)
(Activity B)
(Activity C)
(Activity D)
(Activity E)
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x6  x 4  3 (Activity F)
x6  x5  7 (Activity G)
x1, , x6  0
One optimal solution to this program is given by x =
(0,3,2,5,7,14) which yields Z= 14.
Notice that these xj values for the particular solution
correspond to the early event times, ET(j), for nodes
j=1,...,N calculated in Example 10.2.3. However, this LP
problem also has many other optimal solutions. In
general, the value of xj in an optimal solution may assume
any value between the early and late event times for the
corresponding node in the network (think back to our
definitions of these statistics to understand why this is the
case).
To specify the critical activities, remember what it means
for an activity to be critical. The total float is equal to
zero, so that any delay in the duration of that activity will
result in a corresponding delay in the completion time for
the project. Notice that the duration time for each
activity is located on the right side of the constraint
associated with that activity; and that the duration time
for the project is measured by the objective function
value, Z. The shadow price associated with a constraint
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measures the increase in the objective function value
resulting from an increase in the value of the right-hand
side of that constraint. Therefore, since we know that
increasing the duration of a critical activity by one time
unit (ie. increasing the right-hand side of the associated
constraint by one) will delay the project completion by
one time unit, all critical activities will be characterized
by a shadow price of +1.
Furthermore, the shadow price for a constraint is always
specified by the optimal value of the dual variable
associated with that constraint. We have shown in
Chapter 7 that the values of these dual variable are
located in the Z-row of the final primal tableau after an
application of the simplex method.
So we see that the critical activities for the project are
easily recognized as those whose corresponding
constraints in the linear programming formulation of
(10.8) have associated dual variable values of 1.
Finally, we note here that, in applying a CPM analysis to
a project network, this linear programming formulation is
almost always abandoned in favor of the dynamic
programming approach developed in Section 10.2. Given
that this is the case, it is reasonable to assume
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(surprisingly, there is little discussion of this issue in the
usual references) that there are certain structural
characteristics of these types of networks which generally
allow for a more efficient solution with dynamic
programming than with linear programming. There are
several possible explanations for this; however, we will
not go through these now, but rather will leave this issue
as a consideration for the reader.
10.4 PROJECT EVALUATION AND REVIEW
TECHNIQUE (PERT)
We have stressed throughout our development of the
critical path method that the duration times for each task
used in the project network are only estimates, the actual
times being unknown until the project is actually
completed. With CPM we have thus far relied on point
estimates, so that we in some sense ignore the
uncertainty associated with these task times and simply
insert our best guess into the analysis.
The Project Evaluation and Review Technique (PERT)
explicitly accounts for the uncertainty associated with
task completion times by treating each estimate as a
random variable having some probability distribution.
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Note: Don't panic if your background in statistics is
not too strong - we will only assume a basic
knowledge of random variables and probability
distributions.
The advantage of adding this additional level of
complexity is that, after making certain simplifying
assumptions concerning the nature of these probability
distributions, we will be able to extend our analysis by
determining the probability that the project will be
completed on time.
We first need to characterize the probability distributions
associated with each task completion time. To this end,
we replace the simple point estimates for each activity
duration with three values representing different points of
the distribution:
m = most likely estimate (intended to provide the most
realistic estimate of the time under "normal"
conditions)
a = optimistic estimate (represents the best possible
time under the most favorable conditions)
b = pessimistic estimate (represents the worst possible
time under the least favorable conditions)
Statistically speaking,
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 m is the estimate of the mode, of the underlying
probability distribution;
 a represents the minimum possible duration of the
activity and
 b represents the maximum possible duration of the
activity.
It is worthwhile to put aside technical matters for a
moment to say a word about how these estimates are
obtained. Remember that the person performing the
analysis is presumably competent in operations research
and the mechanics of CPM and PERT, but most likely
has very little knowledge of the project and associated
activities under consideration. For this reason it is not at
all the analyst's responsibility to provide the values for
these estimates - on the contrary, it would be wrong to
attempt to do so. It is, however, a very important
responsibility of the analyst to determine what is the best
source for providing this information and to develop an
effective style for communicating her requirements and
collecting the appropriate data.
In determining the best source of information, we need to
decide whose opinion would be most "expert" for
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assessing task completion times. Perhaps it is the project
manager, or perhaps the person working on a particular
task, or maybe even a source not directly related to this
project but with a wealth of experience working on
similar projects. Once the appropriate information source
is discovered, we must make certain that we obtain
exactly the information we need by communicating in
language familiar to our source. It is unlikely that the
person with which we deal will understand and be able to
respond to a request for the mode or lower bound of a
probability distribution. We must therefore be able to
express these concepts in laymen terms and develop some
method for verifying that our source is relaying the
information we want. The ability to extract information in
this way is a very subtle skill whose importance within an
analysis (everybody is familiar with the expression
"garbage in, garbage out") is too often overlooked in
favor of other, more technical, issues. There is certainly a
good deal more to say on this topic, but for the time being
it will suffice to simply be aware of the challenges
associated with the seemingly simple task of assembling
the data requirements for a study.
Now, we summarize all of our information on each
probability distribution into two statistics,
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 the expected value or mean (ti,j) and
 the variance ( i,2 j ).
To do this, we assume that each job completion time
follows a beta distribution. This will allow us to make
the following approximations:
1
2m  12 (a  b)

3
1
2
  (b  a)
6

ti, j 
(10.10)
 i,2 j
(10.11)
Now that we have estimates for the mean and variance
for the duration time for each individual task, we want to
develop comparable statistics for the entire project. To
do this we first perform our usual CPM analysis using the
values ti,j for estimates of activity durations; then, after
specifying a critical path for the given project, we need to
make two simplifying assumptions.
1.First, we assume that, no matter what the duration times
of each task actually turn out to be once the project is
completed, the path we have specified as critical from
our analysis will in fact have turned out to be the
critical one. In other words, we know that actual task
times can vary significantly from their expected values .
In particular, if jobs along the critical path that we have
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specified turn out to be completed in times well below
their means, and a different path has its jobs completed
in times longer than their means, then this second path
could very well have turned out to be the critical one.
Our assumption here asserts that this will never happen.
2. Secondly, we assume that duration times for all
activities are statistically independent, so that the mean
and variance for overall project time is determined as
simply the sum of means and variances of times for the
activities along a critical path.
Given these assumptions, we denote CP as the set of arcs
along the critical path and specify the mean and variance
for overall project completion time as follows:
(10.12)
E(T)  Expected value of T =

ti, j
Var(T)  Variance of T =
(i, j)CP

 i,2 j
(i, j)CP
(10.13)
Finally, we make a third assumption that our critical path
contains a sufficiently large number of activities so that
we can appeal to the central limit theorem. Remembering
that the random variable T is determined as the sum of
many independent random variables ti,j, this theorem
states that the probability distribution of such a sum is
normal under a broad range of conditions.
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Now, given estimates for the mean and variance of T in
(10.12) and (10.13) and the fact that it is normally
distributed, it is straightforward to reference a table for
normal variables found in any probability or statistics
book to estimate the probability that the actual project
time T will attain a value no greater than the scheduled
completion time.
10.4.1 Example
Consider again our sample project network from Figure
10.10 and assume that, instead of the point estimates for
durations displayed with each arc, we rely on the
following data for each activity:
Acti m a
b
vity
A 2
1
9
B 2
1
3
C 2
1
3
D 4 1.5
6.5
E 1 0.5
1.5
F 3
1
5
G 6
4
14
We can compute the mean and variance for each activity
duration from (10.10) and (10.11):
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8 1  9
tA 
3
6
8 1  3
tB 
2
6
8  1 3
tC 
2
6
16 1.5  6.5
tD 
4
6
4  0.5  1.5
tE 
1
6
12 1  5
tF 
3
6
24  4 14
tG 
7
6
CHAPTER 10: CRITICAL PATH METHOD
(9 1)2

 1.78
36
2
(3
1)
 B2 
 0.11
36
(3 1)2
2
C 
 0.11
36
2
(6.5
1.5)
 D2 
 0.69
36
2
(1.5

0.5)
 E2 
 0.03
36
2
(5
1)
 F2 
 0.44
36
(14  4)2
2
G 
 2.78
36
 2A
We now determine a critical path by an application of
CPM with duration estimates for each activity given by
the means ti,j calculated above.
Since these values are the same as our initial estimates in
Figure 10.10, we can appeal to our results from Section
10.2 - in particular, we have from Example 10.2.7 that the
critical path is made up of jobs A, D and G.
Allowing for the appropriate assumptions, we have from
(10.12) and (10.13) that the mean and variance for
project duration is as follows:
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E(T)  t A  t D  tG  3  4  7  14
Var(T)   2A   2D   G2  1.78  0.69  2.78  5.25
We see that the expected project completion time is 14
days.
If we were hoping to finish the project in just 12 days,
then, assuming that we can appeal to the central limit
theorem and thereby assert that the T is normally
distributed, we can compute the probability of this event
occurring.
First standardize the random variable:
 :
T  E(T)
Var(T)
We now compute the probability of the event as follows:
Pr(T  12)  Pr( 
12 E( T)
Var(T ) ) 
Pr(  1214
5.25 )  Pr(   0.8729 )
To calculate this probability, we simply consult a table
for the standard normal distribution in any statistics
textbook to obtain
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Pr(  0.8729)  0.19
So we see from our PERT analysis that there is
approximately a 19% chance that the project will be
completed at least 2 days ahead of schedule. 
Finally, our development of this approach has relied at
several different stages on various assumptions, so that
the usefulness of this method will obviously depend to a
large extent on the validity of these assumptions. As we
introduced each assumption throughout the discussion
above, we explained why each was necessary but gave no
indication of the likelihood that the required conditions
would be satisfied in a "real-world" situation. This is
something that always needs to be assessed for each
individual study.
For example, in order that the mean and variance of total
project time be equal to the sum of the corresponding
statistics for individual jobs along any path, we needed to
assume that task times for these individual activities are
statistically independent.
The implication of this
assertion is that any factor influencing the duration time
of one job will not influence the duration time of any
other. How likely is it that this assumption will hold?
Probably not very likely, in fact. For example, if the
project were being performed outdoors, then weather
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conditions will almost certainly impact all completion
times. Similarly, if the same person were to work on
several different jobs within a project, then she may very
well significantly affect the times for each. On the other
hand, situations will certainly exist where project times
are completely independent or the level of dependence is
small enough to be considered negligible so that
independence can be assumed. The important point is
that this must always be checked.
Descriptions of algorithms and solution methods often
casually state assumptions which in real life are very
difficult to guarantee. Before applying any method,
always make sure to guarantee that the assumptions
supporting the approach are indeed realized in the system
under study.
10.48