Quantitative Methods for Business chapter 14 pot

CHAPTER
Getting from A to B –
project planning using
networks
14
Chapter objectives
This chapter will help you to:
■ construct network diagrams
■ apply critical path analysis (CPA)
■ identify slack time available for project activities
■ use the Program Evaluation and Review Technique (PERT)
■ conduct cost analysis of projects using network diagrams
■ become acquainted with business uses of CPA/PERT
Many business operations involve planning and coordinating a project –
a series of interlinked tasks or activities all of which have to be performed
in the correct sequence and within the least amount of time in order to
achieve a successful conclusion to the venture. This is not only typical of
large-scale projects such as you would find in industries like construction
and shipbuilding, but also occurs on a more modest basis in administra-
tive processes such as organizing events like conferences and concerts.
To help plan and execute projects successfully managers can turn
to network analysis, a system of representing and linking the activities
involved in a project. Once they have designed a network they can use
436 Quantitative methods for business Chapter 14
critical path analysis (CPA) to establish the minimum duration of the
project by identifying those tasks whose completion on time is essential,
the critical activities. Beyond this they can bring into consideration the
probability distributions that reflect the chances of the activities being
completed by specific times using the program evaluation and review
technique (PERT). In this chapter we will look at these techniques.
14.1 Network analysis

We can apply network analysis to any project that consists of series
of distinct activities provided that we know which activities must be com-
pleted before other activities can commence. This is called the prece-
dence of the activities because it involves identifying the activities that
must precede each activity. These are crucial because the point of a network
diagram is to show how the activities in a project are linked.
The diagrams used for network analysis are built up using a single
arrow for each activity. The arrow begins at a circle representing the point
in time at which the activity begins and finishes at another circle that
represents the completion of the activity. These circles are known as event
nodes. We would represent a single activity as shown in Figure 14.1.
In Figure 14.1 the direction of the arrow tells us that the event
node on the right marks the completion of the activity. The activity is
labelled using a letter and the events nodes are numbered. Network
diagrams consist of many arrows and many event nodes, so logical
layout and labelling is important.
Networks should be set out so that the beginning of the project is rep-
resented by the event node on the extreme left of the diagram and the
completion by the event node on the extreme right. In compiling them
you should try to ensure that the event nodes are labelled sequentially
from left to right from event number 1, the start of the project.
All the arrows should point from left to right either directly or at
angle and certainly not from right to left. This is usually straightfor-
ward because the purpose of an arrow in a network is to represent the
position of the activity in relation to other activities and not its duration.
A network diagram is not intended to be to scale.
Figure 14.1
A single activity
1
2

A
Chapter 14 Getting from A to B – project planning using networks 437
Example 14.1
Avia Petitza is an independent airline flying short-haul routes in South America. The
new general manager is keen to improve efficiency and believes that one aspect of their
operations that can be improved is the time it takes to service planes between flights.
She has identified the activities involved and their preceding activities and compiled
the following table:
Produce a network diagram to portray this project.
Activity A has no preceding activities so we can start the network with this as shown
in Figure 14.2(a). Four subsequent activities, B, C, D and I depend on activity A
being completed so we can extend the network as shown in Figure 14.2(b). Activity G
must follow activity C and activity J follows activity I as depicted in Figure
14.2(c). Activity E follows activity B and activity F follows activity E as shown in Figure
14.2(d).
Before activity K can take place activity H must also have finished. This presents a
problem because we cannot have two or more activities starting at the same event node
and finishing at the same event node. If we did we would not be able to use the diagram
for the sort of scheduling analysis you will meet in the next section of this chapter.
To avoid activities F and H having the same event node marking their start and the
same event node marking their completion we can introduce a dummy activity, an activ-
ity that takes no time nor uses any resources. Its role is merely to help us distinguish
two activities that might otherwise be confused in later analysis. To distinguish between
a dummy activity and real activities it is portrayed as a dotted line. You can see a dummy
activity in Figure 14.2(e) used to ensure that the event node that marks the conclusion
of activity H is not the same as that marking the end of activity F.
Activity Description Precedence
A Drive service vehicles to plane None
B Attach stairway A
C Unload baggage A

D Refuel A
E Passengers disembark B
F Clean cabin E
G Load baggage C
H Load food and beverages E
I Stock up water tanks A
J Service toilets I
K Passengers embark F, H
L Detach stairway K
M Drive service vehicles from plane D, G, J, L
438 Quantitative methods for business Chapter 14
(d)
E
F
B
D
CG
I
J
A
(c)
D
B
CG
I
J
A
(b)
B
D

I
C
A
(a)
A
The dummy activity in Figure 14.2(e) helps us to identify two sep-
arate activities. A dummy activity used in this way is an identity dummy.
Dummy activities are also used to resolve logical difficulties that arise
in some networks. A dummy activity used to do this is a logical dummy.
Chapter 14 Getting from A to B – project planning using networks 439
(f)
7
8
6
5
4
3
2
1
9
10 11
HE
F
K
AD
B
CG
IJ
M
L

(e)
H
E
F
B
D
CG
I
J
A
The final diagram, Figure 14.2(f), incorporates activities K, L and M. Activities K and
L share the same finishing event as activities D and J. This event represents the begin-
ning of activity M, whose closing event marks the conclusion of the project. We can also
number the events now the diagram is complete.
Figure 14.2
Stages in the compilation of the network diagram for aircraft servicing in Example 14.1
Both types of dummy activity serve only to clarify the network and
avoid ambiguity. The important distinction between real and dummy
activities is that the latter do not take time or use resources.
Compiling network diagrams often brings an understanding of the way
in which activities that make up a project fit together. It enables you to
clarify uncertainties about the planning of the project. This is some-
thing usually achieved by constructing drafts of the network before
producing the final version.
Whilst an understanding of the sequence is important, there is much
more to successful project planning. Ascertaining the minimum time in
which the project can be completed and scheduling the activities over
440 Quantitative methods for business Chapter 14
Example 14.2
Two building workers, Bru and Chai, like a mug of black tea in their morning break at

work. Bru takes sugar, Chai does not. Making their tea involves four activities:
It is tempting to represent these activities in the form of the network in Figure 14.3(a),
but this implies that activity C, putting hot water in Chai’s mug, depends on activity B,
putting sugar in Bru’s mug, which is not the case. To get around this we can include a
dummy activity as shown in Figure 14.3 (b).
(a) (b)
AC
D
B
AC
DB
Figure 14.3
Incorrect (a) and correct (b) networks for Example 14.2
Activity Description Precedence
A Put tea bags in the mugs None
B Put sugar in Bru’s mug None
C Pour hot water in Chai’s mug A
D Pour hot water in Bru’s mug A, B
time are typically the central issues. The technique that enables us to do
this is critical path analysis.
14.2 Critical path analysis
If we know the duration of each activity we can use a network diagram
to find out the least amount of time it will take to finish the project.
A network diagram shows the way that activities are linked; in effect it
portrays the project as a series of paths of activities. Since every activity
must be finished for the project to be completed, the minimum dur-
ation of the project is the length of time required to carry out the most
time-consuming path of activities. This path is known as the critical path
since any delay in the completion of activities along it will prolong the
entire project. Finishing those activities on the critical path on time is

therefore critical for the project; they are known as critical activities.
Critical path analysis involves increasing the amount of information
in the network by enhancing the role of the event nodes. In drawing a
network diagram the event nodes are in effect the punctuation marks
in the diagram; they bring order to the sequence of arrows. In critical
path analysis they become distinct points in time. For each event node
we assign an earliest event time (EET) and a latest event time (LET). These
are written in the circle that represents the event node beneath the
event number, as illustrated in Figure 14.4.
The circle on the left in Figure 14.4 is the event node that marks the
point in time when activity X begins. The number in the upper part of
the circle is the event number, 20. The number below it to the left is the
earliest event time and the number below it on the right is the latest
event time. These numbers tell us that the earliest time that activity X
can start is time period 9 and the latest time it can start is time period
11. From the equivalent figures in the event node to the right of activity
X you can tell that the earliest time the activity can be completed is time
period 13 and the latest time it can be completed is time period 15.
To work out the earliest and latest times for the events in a network we
use the activity durations. Starting with the event node at the beginning
Chapter 14 Getting from A to B – project planning using networks 441
Figure 14.4
Earliest and latest
event times for a
single activity
X
21
13
15
20

11
9
of the network and working through the network from left to right we
write in the earliest time that each event node in the network can occur.
This is referred to as making a forward pass through the network.
When doing this you need to remember that where there are two or
442 Quantitative methods for business Chapter 14
Example 14.3
The durations of the activities undertaken during the ground servicing operation in
Example 14.1 are given in the following table:
Using these figures we can enter the earliest event times in the network. These are
included in Figure 14.5.
7
8
6
7
48
5
5
13
33
33
27
57
59
4
3
2
2
1

0
9
10
11
H
E
F
K
A
D
B
C
G
IJ
L
M
Figure 14.5
Network for ground servicing with earliest event times
Activity Description Duration (minutes)
A Drive service vehicles to plane 2
B Attach stairway 3
C Unload baggage 25
D Refuel 15
E Passengers disembark 8
F Clean cabin 20
G Load baggage 30
H Load food and beverages 10
I Stock up water tanks 5
J Service toilets 5
K Passengers embark 15

L Detach stairway 3
M Drive service vehicles from plane 2
more activities leading to the same event node it is the duration of the
longest of the activities that determines the earliest time for the event
since all the activities must be completed before the event is reached. We
write the earliest event times in the lower left-hand side of the event nodes.
If you look carefully at Figure 14.5 you will see that the earliest event
time for event 1 is 0 reflecting the fact that 0 minutes of project time
have elapsed at the beginning of the project. The earliest event time
for event 2 is 2, reflecting the 2 minutes needed for the completion of
the only activity between event node 1 and event node 2, activity B,
driving the service vehicles to the plane.
The earliest event time for event 8 is perhaps less obvious. The figure
entered, 33, has been worked out based on the longest route to it. The
activities leading to event node 8 are activity F, with its associated
dummy activity between events 7 and 8, and activity H. The earliest time
that activity F can start is the earliest event time on the event marking its
beginning; the 13 in event 4. If we add the 20 minutes that activity F,
cleaning the cabin, takes to the earliest event time for event 4 we get 33
minutes as the earliest time activity F can be completed. Event 4 also
marks the beginning of activity H, loading the food and beverages. If we
add the time this activity takes, 10 minutes, to the earliest event time of
activity 4 we get 23 minutes. This would be the earliest event time for
event 8 if we did not need to complete activity F, but since both activity
F and activity H have to be completed before event 8 can occur, the earli-
est time we can get there must allow for the longer activity, activity F, to
be concluded, hence the earliest event time for event 8 is 33 and not 23.
The event node indicating the completion of the project on the
extreme right of the network has an earliest event time of 59 minutes.
This is the minimum duration of the entire project; given the activity

durations it cannot be completed in a lesser amount of time. We now
need to turn our attention to the latest event times, as comparing these
with the earliest event times will enable us to identify the critical path
for the project.
Once we have established the minimum project duration, in the case
of Example 14.3, 59 minutes, we assume that the project manager will
want to complete it in that time. We now undertake the same sort of task
to find the latest event times as we used to find the earliest event times,
but this time we start on the right-hand side and work back through
the network ascertaining the latest time each event can occur if the
project is to be finished in the minimum time. This is referred to as
making a backward pass through the network.
The latest event times for Example 14.3 are included in Figure 14.6.
Each event now has a number entered in the lower right-hand side of
its node.
Chapter 14 Getting from A to B – project planning using networks 443
Looking at Figure 14.6 you can see that the latest event time for the
last event, event 11, is 59 minutes. If the project is to be completed in
59 minutes then the latest time we must reach this point is 59 minutes.
The latest event time of event 10 is 57 minutes, 59 minutes less the
2 minutes we must allow for activity M to be completed.
Several activities conclude at event 10, including activities G and L.
Activity G begins at event 5. The latest time event time of event 5 is 27 min-
utes, sufficient to allow the 30 minutes necessary for the completion of
activity G in time for event 10 to be reached in 57 minutes, which in turn
allows for the 2 minutes to complete activity M and thus conclude the
project in 59 minutes. Activity L begins at event 9. Since activity L, detach-
ing the stairway, takes 3 minutes, the latest event time for event 9 is 54
minutes, sufficient to allow the 3 minutes for the completion of activity L
so that event 10 can be reached in 57 minutes and thus leave 2 minutes

for activity M to finish in time for completing the project in 59 minutes.
If you study Figure 14.6 carefully you will see that some events, such
as event 5, have the same earliest and latest event times while other
events, such as event 6, have different ones. In the case of event 6 the
earliest event time is 7 minutes whereas the latest event time is 52 min-
utes. This implies that activity I can be completed by 7 minutes but it
doesn’t have to be finished until 52 minutes have elapsed. In other
words, there is time to spare for the completion of activities I and J;
they have what is called slack or float.
444 Quantitative methods for business Chapter 14
Figure 14.6
Network for ground servicing with latest event times
7
8
6
7
48
5
511 1319
33
33 39
39
27 27
57
52
57
54
4
3
2

22
1
00
9
10
H
E
F
K
A
D
B
CG
IJ
M
L
59 59
11
In contrast, the latest event time for event 5 is the same as its earliest
event time, 27 minutes. In this case the earliest time event 5 can be
reached is the same as the latest time it has to be reached if the project
is to be completed in 59 minutes. This implies there is no slack or float
for activities C and G; they must be undertaken at the earliest feasible
opportunity as their completion on time is critical for the conclusion
of the project, they are critical activities. If you look at the event nodes
that have the same earliest and latest event times you will see that they
connect activities A, C, G and M. These activities are driving service
vehicles to the plane, unloading and loading baggage, and driving
service vehicles from the plane. They form the longest or critical path
through the network; a delay in the execution of any of them will result

in the project being prolonged beyond 59 minutes. On the other hand,
any reduction in the duration of any of them will reduce the duration
of the project, but only up to a point; at some point another path will
become critical.
By definition the critical activities in a network have no slack or float,
but other activities will. A project manager is likely to be very interested
in the float available for these activities, as it indicates the degree of
latitude available in scheduling them.
To work out the total float for an activity we need to know the latest
event time for the event that marks its completion, its finishing event,
and the earliest event time for the event that represents its beginning,
its starting event. If we subtract the duration of the activity from the dif-
ference between these times what is left over is the total float for the
activity. We can summarize this procedure as:
Total float ϭ Latest event time (finishing event)
Ϫ earliest event time (starting event)
Ϫ activity duration
Using abbreviations:
TF ϭ LET (F) Ϫ EET (S) Ϫ AD
Look carefully at Table 14.1 in Example 14.4 and you will see that
some activities have a zero float. These are the activities on the critical
path – A, C, G and M. In contrast there is a float of 6 minutes in the path
B–E–F–K–L, which is the uppermost path in Figure 14.6. This is float
associated with the path rather than an individual activity; once used up
by, say, taking 9 minutes rather than 3 minutes for activity B, attaching
the stairway, it would not be available for the subsequent activities E, F,
K and L, which as a consequence would become as critical as the activi-
ties on the critical path.
Chapter 14 Getting from A to B – project planning using networks 445
Activity H, loading the food and beverages, has a total float of 16

minutes, 6 of which are shared with the other activities along its path;
B, E , K and L. The remaining 10 minutes are specific to activity H; it
must be completed by the same time as activity F, cleaning the cabin, if
the project is to be finished in the minimum time, yet it takes 10 min-
utes less to perform.
Activity D, refuelling, has a float of 40 minutes, which cannot be
shared with the other activities along its path, A and M, as they are
both critical. The relatively large amount of float for this activity allows
the project manager considerable flexibility in scheduling this activity.
Similar flexibility exists for activities I and J, the water operations,
which share a total float of 45 minutes.
At this point you may find it useful to try Review Questions 14.1 to
14.11 at the end of the chapter.
14.3 The Program Evaluation and
Review Technique (PERT)
So far we have assumed that the activities making up a project each
have fixed durations. In reality this is unlikely to be the case. In the
446 Quantitative methods for business Chapter 14
Example 14.4
Find the total floats for the ground servicing activities in Example 14.1.
Table 14.1
Total floats for activities in Example 14.1
Activity LET (F) (1) EET (S) (2) AD(3) TF(1)–(2)–(3)
A2020
B11236
C 27 2 25 0
D 57 2 15 40
E19586
F3913206
G5727300

H39131016
I522545
J577545
K5433156
L574836
M595720
ground servicing project introduced in Example 14.1 factors such as
the state of equipment, the time of the day or night, and weather con-
ditions will influence the time it takes to complete the activities. Rather
than being fixed, activity durations are more likely to be variable. PERT
allows us to take this into account.
In using PERT we assume that the duration of an activity is a continu-
ous variable that follows a continuous probability distribution. We
looked at continuous probability distributions like the normal distri-
bution in Chapter 13. The distribution used in PERT as the model for
activity durations is the beta distribution. Like the normal distribution it
is continuous, but unlike the normal distribution it can be skewed or
symmetrical and it has a minimum and a maximum value. The key
characteristics or parameters of the beta distribution are the minimum
or optimistic duration, a, the maximum or pessimistic duration, b, and the
most likely duration, m.
You can see two types of beta distribution in Figure 14.7. The upper
diagram portrays a symmetrical distribution with a minimum of 0 and a
Chapter 14 Getting from A to B – project planning using networks 447
Figure 14.7
Examples of the
beta distribution
0510
0510
Duration (minutes)

Duration (minutes)
maximum of 10. The most likely value of this distribution is 5 minutes,
midway between the extremes. In the asymmetrical beta distribution in
the lower diagram the minimum and maximum durations are the same
as in the upper diagram but the most likely value is rather lower.
If you know the optimistic (a), most likely (m) and pessimistic (b)
durations of an activity you can work out its mean duration using the
expression:
The standard deviation of the activity duration is:
The variance of the activity duration is the square of its standard deviation:
The mean duration of an entire project is the sum of the mean dur-
ations of the activities that are on the critical path, the critical activities.
The variance of the project duration is the sum of the variances of the
critical activities and the standard deviation of the project duration is
the square root of its variance. Note that you cannot get the standard
deviation of the project duration by adding together the standard devi-
ations of the durations of the critical activities; it is the variances that
sum, not the standard deviations.
␴
2
2
6
)
ϭ
Ϫ(ba
␴
)
ϭ
Ϫ(ba
6

␮
4
ϭ
ϩϩamb
6
448 Quantitative methods for business Chapter 14
Example 14.5
Using time sheets and airport records, the general manager in Example 14.1 has
established the optimistic, most likely and pessimistic durations of the activities
involved in the ground servicing of aircraft. They are set out in Table 14.2. Using appro-
priate figures from this table, work out the mean and standard deviation of the project
duration.
The mean project duration is the sum of the mean durations of the critical activities.
In this project the critical activities are A, C, G and M. The means of these activities are:
␮
C
22 (4 * 25) 31
25.5ϭ
ϩϩ
ϭ
6
␮
A
1 (4 * 2) 3
2ϭ
ϩϩ
ϭ
6
The mean duration of the critical path is the sum of these means:
␮

CP
ϭ ␮
A
ϩ ␮
C
ϩ ␮
G
ϩ ␮
M
ϭ 2 ϩ 25.5 ϩ 29 ϩ 2 ϭ 58.5 minutes
The standard deviation of the critical path is the square root of the variance of the
critical path, which is the sum of the variances of the critical activities, A, C, G and M.
The variances of these activities are:
␴
A
2
ϭ ((3 Ϫ 1)/6)
2
ϭ 4/36 ϭ 1/9
␴
C
2
ϭ ((31 Ϫ 22)/6)
2
ϭ 81/36 ϭ 9/4
␴
G
2
ϭ ((32 Ϫ 25)/6)
2

ϭ 49/36
␴
M
2
ϭ ((3 Ϫ 1)/6)
2
ϭ 4/36
The variance of the duration of the critical path is:
The standard deviation of the duration of the critical path is:
␴
CP
ϭ Ί␴
2
CP
ϭ Ί3.833 ϭ 1.958 minutes
␴␴␴␴␴
CP A C G M

4 81 49 4 138
36
3.833
22222
36
ϭϩϩϩϭ
ϩϩϩ
ϭϭ
␮
M
1 (4 * 2) 3
2ϭ

ϩϩ
ϭ
6
␮
G
25 (4 * 30) 32
29ϭ
ϩϩ
ϭ
6
Chapter 14 Getting from A to B – project planning using networks 449
Table 14.2
Optimistic, most likely and pessimistic durations of ground operations
Duration (minutes)
Optimistic Most likely Pessimistic
Activity Description (a)(m)(b)
A Drive vehicles to plane 1 2 3
B Attach stairway 2 3 5
C Unload baggage 22 25 30
D Refuel 14 15 17
E Passengers disembark 6 8 12
F Clean cabin 15 20 25
G Load baggage 25 30 32
H Load food and beverages 8 10 12
I Stock up water tanks 4 5 7
J Service toilets 3 5 10
K Passengers embark 8 15 20
L Detach stairway 2 3 4
M Drive vehicles from plane 1 2 3
We can also work out the timescale within which a certain proportion

of ground operations should be completed using the Z distribution.
450 Quantitative methods for business Chapter 14
Example 14.6
What is the probability that the ground servicing operation whose activity durations
were detailed in Example 14.5 can be completed in one hour?
If X is the project duration, we want:
If you check in Table 5 on pages 621–622 you will find that the probability that Z is
more than 0.77 is 0.2206. The probability that Z is more than 0.77 is 1 less 0.2206, 0.7794,
which is the probability that the ground servicing operation is completed within an hour.
PX P P() 60
60 58.5
1.958
( 0.766)Ͻϭ Ͻ
Ϫ
ϭϽZZ
⎛
⎝
⎜
⎞
⎠
⎟
Example 14.7
What is the timescale within which 90% of ground servicing operations should be
completed?
For this you will need the z value that cuts off a tail of 10%, z
0.10
. You can find it by look-
ing down the probabilities in Table 5 until you reach the figure closest to 10%, the 0.1003
in the row for 1.2 and the column headed 0.08, so z
0.10

is approximately 1.28. For a more
precise figure for z
0.10
use the bottom row of the related Table 6 on page 623. The figure
in this row and in the column headed 0.10 is 1.282 so 90% of the normal distribution is to
the left of, or less than, 1.282 standard deviations above the mean. We can apply this to
the distribution of the duration of the ground servicing operation; 90% of the operation
will be completed in a time less than 1.282 standard deviations above the mean:
␮
CP
ϩ 1.282␴
CP
ϭ 58.5 ϩ 1.282 * 1.958 ϭ 61.010 minutes
Unless there is clear evidence that the distributions of the durations
of the individual activities are consistently skewed, we can assume that
the distribution of the project duration is approximately normal. This
means we can use the Standard Normal Distribution to work out the
probability that the project is completed within a certain timescale or
beyond a certain timescale.
At this point you may find it useful to try Review Questions 14.12 to
14.16 at the end of the chapter.
Chapter 14 Getting from A to B – project planning using networks 451
Example 14.8
The general manager in Example 14.1 has found that three of the activities that make
up the ground servicing operation can be crashed; activities C, unloading the baggage,
F, cleaning the cabin, and G, loading the baggage. The crash durations and crash costs
of these activities are set out in Table 14.3.
14.4 Cost analysis: crashing the
project duration
PERT is designed to allow for random fluctuations in the duration of activ-

ities. These are by definition in large part difficult or impossible to antici-
pate, yet they can have a substantial impact on the duration of a project.
In contrast, it may be possible to reduce the project duration by cut-
ting activity durations by intent. Typically this involves incurring extra
cost, defined as the cost of crashing the duration of an activity below its
anticipated duration.
In most projects there are at least some activities that can be performed
more quickly, perhaps by providing more or better equipment, perhaps
by allocating more staff, perhaps by hiring a sub-contractor. These con-
tingencies will come at an additional cost, the crash cost, the cost of carry-
ing out the activity in its crash duration rather than its normal duration.
Although there may be several activities in a project that can be
crashed, it may not be worthwhile crashing all of them. The project
duration is the critical path, the path of critical activities, whose com-
pletion on schedule is crucial for the minimum duration of the project
to be achieved. If an activity is not on the critical path it is not worth
crashing, or at least not until it might have become critical following
the crashing of one or more critical activities. Crashing a non-critical
activity will not reduce the duration of the critical path, but crashing a
critical activity will always reduce the duration of the critical path.
Having said that, there is a limit. Once the critical path activities have
been crashed to the point that another path loses its float and becomes
critical it is not worth crashing critical activities any further. Any fur-
ther project reduction has to involve further crashing of the activities
on the original critical path as well as crashing activities that have
become critical after the initial crashing of the critical path activities.
There may be several critical activities than can be crashed. If this is
the case the activities to be crashed should be selected on the basis of
cost, with the cheapest options implemented first.
452 Quantitative methods for business Chapter 14

Although the cheapest activity to crash is activity F, cleaning the cabin, if you look back
at Figure 14.6 you will see that it is not on the critical path, indeed if you check Example
14.4 you will find that there are 6 minutes of float available for activity F. Cutting its dur-
ation will merely add to its float and not reduce the minimum duration of the project.
Activities C and G are both on the critical path, so it is worth considering crashing
them. The cheaper to crash is activity C, at $40 per minute compared to $50 per minute
for crashing activity G.
To what extent should activity C be crashed? The answer is up to the point when the
path with the least float becomes critical. The path B–E–F–K–L has 6 minutes of float,
so it would be worth reducing the duration of activity C by 6 minutes. Since the crash
duration of activity C is only 5 minutes, to achieve a reduction of 6 minutes in the pro-
ject duration we would have to reduce the duration of activity G by 1 minute as well.
A greater reduction in the project duration would involve further crashing of activity
G and crashing activity F. The possible stages in reducing the project duration and their
associated costs are:
Stage 1: Crash activity C by 5 minutes at a cost of $200 and reduce the project dur-
ation to 54 minutes.
Stage 2: Crash activity G by 1 minute at a cost of $50 and reduce the project duration
to 53 minutes. Total crashing cost $250.
Stage 3: Crash activity G by a further 3 minutes and crash activity F by 3 minutes at a
combined cost of $210 and reduce the project duration to 50 minutes. Total
crashing cost $460.
Would it be worth incurring the extra costs of crashing in a case like
Example 14.8? It depends on the circumstances. It may be that the
normal duration of the ground servicing operation is longer than
the allotted time slot at the airport and as a result the airline has to pay
penalty costs to the airport authority. Alternatively, reducing the
ground servicing time may enable the airline to schedule more flights
and thus increase revenue. In either case, deciding how far to go in
crashing the project will rest on balancing the costs of crashing and the

financial, and possibly other, benefits of doing so.
Table 14.3
Crash durations and crash costs of operations C, F and G
Crash cost Crash cost
Normal duration Crash duration
Activity (minutes) (minutes) ($) per minute ($)
C 25 20 200 40
F 15 8 140 20
G 30 26 200 50
Chapter 14 Getting from A to B – project planning using networks 453
At this point you may find it useful to try Review Questions 14.17 to
14.19 at the end of the chapter.
14.5 Road test: Do they really use
critical path analysis and PERT?
Although they are similar in nature, critical path analysis and PERT
were developed at about the same time, the middle of the twentieth
century, the contexts in which they originated were markedly different.
Critical path analysis was originally known as the Kelly–Walker method
after the two pioneers of the technique: J.E. Kelly of the Remington
Rand Company and M.R. Walker of the Du Pont Company, the giant
US chemicals corporation that developed Dynamite, Nylon, Teflon
and Lycra as well as many other technological innovations. At the time,
Remington Rand was the computer systems subsidiary of Du Pont.
The initial application of the technique was in the construction of a
major new chemical plant at Louisville, Kentucky.
While the origins of critical path analysis were civilian, the roots of
PERT lie in the military. In the early phase of the Cold War the US mili-
tary authorities were desperate to develop intercontinental ballistic mis-
siles. They were concerned about the projections for the completion time
of key projects: first the Atlas missile, then more prominently the Polaris

system. To expedite matters a central coordinator was appointed to
oversee the Polaris project. He assumed control of the entire project
and under him PERT evolved as the means by which the myriad of activ-
ities involving hundreds of subcontractors were planned so that the
Polaris programme was completed much sooner than the 10 years that
was initially anticipated.
The success of the Polaris project meant that PERT became widely
publicized. In telling the story of its development, Morris (1994, p. 31)
notes that by 1964 there were almost 1000 overwhelmingly enthusiastic
articles and books published about PERT. Whilst in retrospect this
amounted to overselling the technique, like its close relation critical
path analysis, it has become widely accepted as a key tool in successful
project management.
In his study of the use made of quantitative methods in US busi-
nesses, Kathawala (1988) found that 54% of companies in his survey
reported that they made moderate, frequent or extensive use of critical
path analysis and PERT. More recently such planning tools have been
used in projects like the expansion of the Kings Cross underground
station in London (Lane, 2003).
454 Quantitative methods for business Chapter 14
Morris (1994) provides an interesting history of the techniques
we have looked at in this chapter. For more on their capabilities try
Klein (2001).
Review questions
Answers to these questions, including fully worked solutions to the Key
questions marked with an asterisk (*), are on pages 663–665.
14.1* Bibb and Tukka own and operate a successful chain of clothing
stores. They plan to open a new outlet. The activities involved
are listed below with the activities that must precede them and
their durations.

Duration
Activity Description Precedence (days)
A Negotiate the lease — 10
B Install the fixtures A 8
C Install the furnishings B 3
D Appoint the staff A 2
E Train the staff D 10
F Arrange the opening D 7
ceremony
G Opening ceremony C, E, F 1
(a) Compile a network diagram to represent this venture.
(b) Find the earliest and latest event times for the events in your
network, and use them to identify the minimum duration of
the project and the activities that are on its critical path.
14.2 Marsh and Root Construction have won the contract to widen
and resurface a section of road. The site manager has identi-
fied the following tasks together with the tasks that must pre-
cede them and their anticipated durations:
Duration
Activity Description Precedence (days)
A Relocate bus stop — 2
B Install temporary — 1
traffic lights
C Install safety barriers A, B 1
D Plane road C 2
E Lift kerbstones C 3
(Continued)
Duration
Activity Description Precedence (days)
F Replace ironworks D 3

G Lay and roll tarmac F 1
H Replace kerbstones E 4
I Road painting G 2
J Remove safety barriers I 1
K Remove temporary J 1
traffic lights
L Restore bus stop H 2
(a) Construct a network to portray the project.
(b) Identify the critical path and hence ascertain the min-
imum duration for these roadworks.
14.3 The Raketa Racing Team are new entrants to motor racing.
They have been practising their pit stop procedures and their
performance is detailed below:
Duration
Activity Description Precedence (seconds)
A Guide driver to pit — 3
B Jack up the car A 2
C Remove the old wheels B 5
D Fit the new wheels C 5
E Refuel the car B 16
F Wipe driver’s visor B 2
G Release and clear jacks D, F 2
H Check all clear E, G 2
I Signal GO H 1
(a) Draw a network for the pit stop procedure.
(b) Using your diagram work out the earliest and latest time
for each event, find the minimum duration for the pit stop
and identify the critical path.
14.4 Slattkey Sweets proposes to re-launch one of its mature brands,
the Zubirot Bar. The brand manager has identified the tasks

involved and produced the following information about them:
Duration
Activity Description Precedence (weeks)
A Redesign the bar — 8
B Redesign the packaging — 4
(Continued)
Chapter 14 Getting from A to B – project planning using networks 455
Duration
Activity Description Precedence (weeks)
C Build pilot production A, B 13
line
D Trial production run C 1
E Consumer panel tests D 1
F Main pilot production E 6
G Design promotional E 4
material
H Test market redesigned F, G 10
product
I Produce report for the H 2
Board
Draw a network to portray this enterprise and use it to find the
minimum duration of the project and those activities that are
on the critical path.
14.5 The renowned rock band Kamien have just completed a highly
successful national tour. Tickets have sold so well that they would
like to put on an extra concert. The tasks involved in organizing
this, together with their preceding activities and expected dura-
tions, are:
Duration
Activity Description Precedence (days)

A Agree the leasing of — 10
the hall
B Engage the support acts A 6
C Hire security staff A 8
D Order and receive B 14
merchandising
E Organize ticket sales B 2
F Early promotional work E 4
G Book hotels and transport B 2
H Final promotional work F 5
I Stage rehearsal/sound G 1
checks
(a) Create a network for this project.
(b) Find the minimum duration of the project and indicate
the activities on the critical path.
456 Quantitative methods for business Chapter 14
14.6 Nat Chelnick has to go on an unexpected but urgent business trip
to a city in Central Asia. The arrangements he has to make are:
Duration
Activity Description Precedence (days)
A Renew his passport — 8
B Obtain a visa A 10
C Go for vaccination shots† — 6
D Order and receive — 4
currency
E Order and receive tickets B 1
F Book accommodation E 3
G Book airport parking E 2
† includes time to allow for side-effects of vaccine
Compile a network for this venture and using it, find how long

Nat will have to wait before he can make the trip, and state
which activities are critical.
14.7 Doris and Dennis Datcher, experts at DIY, want to build a sum-
merhouse in their garden. They will start by drawing up a proper
plan, which they can complete in 2 days. Once they have a plan
they can prepare the site, which will take 6 days, order the timber,
which will be delivered in 10 days from placing the order, and
purchase the ironmongery, which will take 2 days. Once they
have the timber and ironmongery they can construct the floor
panels, which will take 4 days, make the frames for the walls,
which will take 2 days, and construct the door, which will take 1
day. When they have made the wall frames they can board them
up, which will take 2 days. When they have prepared the site and
constructed the floor panels they can lay the floor, which will take
1 day. As soon as the floor is laid and the wall frames boarded they
can erect the walls, which will take 1 day. Once the walls are
erected the roof joists can be fixed, which will take 2 days. Having
fixed the roof joists the roof can be boarded and weatherproofed,
which will take 4 days. Once the door has been constructed and
the walls erected they can hang the door, which will take 1 day.
(a) List the activities involved in building the summerhouse
and identify the predecessor activities for each of them as
well as their durations.
(b) Draw a network diagram for the building of the summer-
house and from it identify the critical path and the min-
imum duration of the project.
Chapter 14 Getting from A to B – project planning using networks 457
14.8 The Dom Stila fashion house intend to stage a fashion show to
promote their autumn collection. Three designers, Mallover,
Millasha and Mockry, have been commissioned to produce

designs for the show. The tasks entailed in completing the pro-
ject, together with their durations and tasks that must precede
them, are:
Duration
Activity Description Precedence (days)
A Mallover prepares — 8
designs
B Millisha prepares designs — 16
C Mockry prepares designs — 14
D Select models A, B, C 1
E Make up Mallover D 5
designs
F Make up Millisha designs D 10
G Make up Mockry designs D 12
H Design show (music, A, B, C 6
choreography and
lighting etc.)
I Construct the set H 12
J Obtain props and I 1
accessories
K Design and print H 9
publicity
L Distribute publicity K 3
M Rehearse show E, F, G, J 2
Produce a network for this project, list the critical activities and
ascertain the minimum duration of the project.
14.9 The new Raz Develka TV programme invites participants to
design and re-decorate a room in a neighbour’s house with the
assistance of a designer and technical staff. One contestant has
drawn up the following schedule for working on her neigh-

bour’s bedroom:
Duration
Activity Description Precedence (hours)
A Clear the furniture — 1
B Remove the carpets A 1
(Continued)
458 Quantitative methods for business Chapter 14
Duration
Activity Description Precedence (hours)
C Sand the floor B 3
D Install multipoint B 2
lighting
E Paint the ceiling C, D 2
F Paint the walls E 4
G Paint motifs on the walls F 6
H Build fitted wardrobes F 4
I Refurbish the bed A 3
J Make fabric blinds — 5
K Hang blinds G, J 1
L Acquire decorative — 12
ornaments
M Lay out room K, L 2
(a) Construct a network to represent this project and, using it,
work out the minimum duration of the project and iden-
tify the activities on the critical path.
(b) The contestant ruins the bed after spending 2 hours refur-
bishing it. Building a new one will take 18 hours. Will this
delay completion?
14.10 The Easkritsy Car Valet Service defines the standard car clean it
offers as consisting of the following operations:

Duration
Activity Description Precedence (minutes)
A External pre-wash — 5
B External main wash A 8
C External rinse B 2
D External wax and polish C 12
E Polish external windows D 1
F Enhance tyre wall black D 4
G Remove floor mats — 2
H Vacuum seats G 4
I Vacuum floor H 6
J Polish fascia and door I 6
panels
K Apply air freshener J 1
Draw a network to portray this enterprise and use it to find the
minimum duration of the project and those activities that are
on the critical path.
Chapter 14 Getting from A to B – project planning using networks 459