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TOPIC : CRITICAL PATH ANALYSIS 950 / 3
Compiled by: Goh PC 1
LEARNING OUTCOMES:
When you have completed this module, you should be able to
(Networks)
Identify activities on arcs and activities on nodes; construct
an activity network for a project;
(Critical paths)
calculate the earliest and latest start and finish times for
activities; calculate and interpret the total float for an
activity; identify critical activities and critical paths;
determine the minimum completion time of a project;
(Scheduling and crashing)
construct a Gantt chart and resource histogram for a project;
determine the minimum number of workers to complete a project in a
given time; determine the minimum time to complete a project for a
given number of workers; determine the effect of adjusting the
duration of an activity on the critical path and
completion time;
determine the effect of adjusting the number of workers on the
critical path and completion time.
INTRODUCTION
Critical path analysis (CPA) is a technique used in project
management for planning, scheduling,
and controlling a complex project. CPA involves the logical
sequencing of the activities,
managing the time required for each activity, and determining
the most efficient plan for
carrying out the various activities to ensure the whole scheme
is completed in the minimum time.
NETWORK
A project defines a combination of interrelated activities that
must be executed in a certain order
before the entire task can be completed. The sequence in which
the activities in a project must be
carried out is summarized in a table known as the dependence
table or precedence table.
An example of a precedence table is as follow.
Activity Preceding Activity
Designing Purchasing Designing
Cutting Purchasing
Sewing Cutting
A project network is a diagram that shows the sequence in which
the activities of a project will
take place and their interdependencies. It is also known as a
network diagram.
Two popular forms of project network are:
activity on arc (AOA) network diagram, and activity on node
(AON) network diagram.
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TOPIC : CRITICAL PATH ANALYSIS 950 / 3
Compiled by: Goh PC 2
Activity on arc (AOA) network diagram
The basic elements in an AOA network diagram are activities and
events.
An activity in a project is a job or task requiring both time
and resources for its completion. For example, training of workers,
laying tiles etc.
In an AOA network diagram, an activity is represented by an
arrow (or a directed arc).
An event represents a point in time that signifies the
completion of some activities and the beginning of new ones.
In an AOA diagram, an event is represented by a node
(circle).
Rules in AOA network construction:
Network typically flows from left to right.
A complete network must have a single starting event and a
single ending event.
Each activity must have a preceding event (tail event) and a
succeeding event (head event).
The tail event has a smaller number than the head event.
There can be more than one activity having the same tail event
or head event.
Each event has at least a preceding activity and succeeding
activity except the start event and the end event.
tail head
(denotes the start
of an activity)
(denotes the completion
of an activity)
A
1
A
B C
D 1
2
3 4 Start event End event
(No preceding
activity) (No succeeding
activity)
Tail event
1 2
Tail event Head event
A
(Denotes the beginning of
Activity A)
(Denotes the end of
Activity A)
Head event
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TOPIC : CRITICAL PATH ANALYSIS 950 / 3
Compiled by: Goh PC 3
An activity cannot start until the tail event has occurred.
An event cannot occur until all the preceding activities are
completed.
Arrows (or arcs) cannot cross each other.
Two activities CANNOT have the same tail event and head
event.
Loop (a sequence of activities that starts and ends at the same
event) is NOT allowed.
Dummy Activity
A dummy activity is an activity which does not consume time or
resources. Dummy activities
are represented by broken arrows .
A dummy activity is used in the following situations.
When two activities are having the same tail event and head
event.
To illustrate logical dependencies of activities.
For example:
Activities A and B precede activity C.
Activity B precedes activity D.
1 2
A
B
1 2
3
A
B C
1 2
A
B
1 3
A
B
2
Dummy (Not allowed)
(Use dummy
activity)
Dummy
A
B
C
D
(Use dummy
activity)
A
B
C
D
(Incorrect)
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TOPIC : CRITICAL PATH ANALYSIS 950 / 3
Compiled by: Goh PC 4
Activity on node (AON) network diagram
An AON network diagram uses boxes (or nodes) to denote
activities. These boxes are connected
from beginning to end with arrows to depict a logical sequence
and the interdependencies
between the schedule activities. The following diagram is an
example of an AON network
diagram.
In the above AON network diagram, activity A precedes activities
B and E, whereas activity E
succeeds activities A and D.
Rules in AON network construction:
Network typically flows from left to right.
An activity cannot begin until all the preceding activities are
completed.
Arrows indicate precedence and flow and can cross over each
other.
Looping is not allowed.
The network has a unique start node and a unique end node.
Each activity has at least one entering arrow and leaving arrow
except the start and end
nodes.
The precedence table for a project is given as follow.
Activity Preceding Activity
A B A
C A
D B, C
E C
AOA network diagram:
AON network diagram:
Dummy
A
C
B 1 2
3
4 5 D
E
Start Finish
B
A
C E
D
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TOPIC : CRITICAL PATH ANALYSIS 950 / 3
Compiled by: Goh PC 5
Exercise 1
1. Base on the network diagram, construct a precedence
table.
2. Base on the precedence table, draw an AOA network
diagram.
Activity Preceding Activity
A B C A
D B
E C
F D, E
3. Draw an AOA network and an AON network for each of the
following projects.
(a)
Activity Preceding Activity
A B A
C A
D A
E B, C
F D
(b)
Activity Preceding Activity
A B C D A
E B
F C
G D, E
H F, G
(c)
Activity Preceding Activity
A B C A
D A, B
E A, B
F C
G C, D
A
B
C
D 1 3
2
4 5 E
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TOPIC : CRITICAL PATH ANALYSIS 950 / 3
Compiled by: Goh PC 6
CRITICAL PATHS
A path in a network diagram is a continuous line leading from
the first event (or start) and
connecting adjacent activities until the last event (or finish).
A critical path is the path with the
longest duration in the network. It gives the shortest time in
which the entire project can be
completed.
In this diagram, the paths are A-C-G, A-E-F and B-D-F.
Path Duration (weeks)
A-C-G 2 + 4 + 3 = 9
A-E-F 2 + 3 + 5 = 10
B-D-F 1 + 2 + 5 = 8
The critical path is A-E-F and the minimum time required to
complete the project is 10 weeks.
The critical activities are A, E, and F.
In this diagram, the paths are A-C-E-F, B-E-F and B-D-F.
Path Duration (days)
A-C-E-F 2 + 1 + 5 + 3 = 11
B-E-F 3 + 5 + 3 = 11
B-D-F 3 + 7 + 3 = 13
The critical path is B-D-F and the minimum time required to
complete the project is 13 days.
The critical activities are B, D and F.
Note:
A network may have more than one critical path. All the
activities on the critical path are known as the critical
activities. A delay in any of the critical activities will increase
the project duration.
A
B
C
D 1
3
2
4
5
E 2
1
4
2
3
F
5
G 3
6
Act t
A 2
B 3
Start
C 1
E 5
D 7
Finish F 3
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TOPIC : CRITICAL PATH ANALYSIS 950 / 3
Compiled by: Goh PC 7
Exercise 2
1. Base on the following network diagram, list all the possible
paths of the project and their
corresponding total duration (days). Hence, determine the
critical path and the minimum time
required to complete the project.
2. Draw an AOA network diagram for each of the following
projects, list all the possible
paths of the project and their corresponding total duration.
Hence, determine the critical path and
the minimum time required to complete the project.
(a)
Activity Preceding Activity Duration (days)
A 2
B - 3
C A, B 4
D B 7
E C 5
(b)
Activity Preceding Activity Duration (weeks)
A 3
B A 1
C A 2
D B 6
E B, C 5
F E 2
3. Draw an AON network diagram for the following project, list
all the possible paths of the
project and their corresponding total duration. Hence, determine
the critical activities and the
minimum time required to complete the project.
Activity Preceding Activity Duration (weeks)
A 3
B 1
C B 2
D A 3
E A 4
F C, D 7
G E 5
H F, G 2
A
B
C
D 1
3
2 4 5
E
2
3
4
6
F
4
3 Dummy
5
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TOPIC : CRITICAL PATH ANALYSIS 950 / 3
Compiled by: Goh PC 8
Calculation using AOA network diagram
Earliest and Latest Event Time
ie = the earliest time event i can occur
il = the latest time event i can occur
Calculation of the earliest event time (forward pass):
Begins from the start event and move to the end event.
The earliest time for the start event is 0.
One tail event:
22012 Atee
53223 Btee
86224 Ctee
More than one tail events:
1138
1055
4
3
E
D
te
te
11
11 ,10max.5
e
The earliest time of the end event is the shortest time taken to
complete the entire project.
1541156 Ftee
i A B
il ie
B
1
0
A
2
Start
event
D
5
C
3 E
4
End
event F 3
6
3
4
5
6
2
2e
3e
4e
5e 6e
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TOPIC : CRITICAL PATH ANALYSIS 950 / 3
Compiled by: Goh PC 9
Calculation of the latest event time (backward pass):
Begins from the end event and move to the start event.
For the end event, the latest event time equals the earliest
event time.
1566 el
One head event:
1141565 Ftll
831154 Etll
651153 Dtll
More than one head events:
336
268
3
4
B
C
tl
tl
2
3 ,2min.2
l
The latest time for the start event is 0.
The AOA network diagram showing the earliest and latest event
times:
Earliest Start Time (EST), Earliest Finish Time (EFT), Latest
Start Time (LST) and Latest Finish
Time (LFT) of An Activity
EST of activity A = ie
EFT of activity A = Atei
LST of activity A = Atl j
LFT of activity A = jl
B
1
0
A
2
Start
event
D
5
C
3 E
4
End
event F 3
6
3
4
5
6
2
2l
3l
4l
5l 6l 15 11 2
5
8
B
1
0
A
2
Start
event
D
5
C
3 E
4
End
event F 3
6
3
4
5
6
2
15 11 2
5
8
0 15 11
8
6
2
i A
il ie
j
je jl At
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TOPIC : CRITICAL PATH ANALYSIS 950 / 3
Compiled by: Goh PC 10
Calculation using AON network diagram
An AON network diagram can be constructed using the following
notations.
Calculation of the earliest start time of an activity (forward
pass):
Begins from the Start activity and move to the Finish
activity.
The earliest start time for the start activity is 0 (the
duration of start activity is 0).
One preceding activity:
EST of A = 0
EST of B = 0
EST of D = EST of B + Bt = 0 + 5 = 5
More than one preceding activities:
EST of A + At = 0 + 2 = 2 EST of C = max. (2, 5)
EST of B + Bt = 0 + 5 = 5 = 5
EST of C + Ct = 5 + 4 = 9 EST of E = max. (8, 9)
EST of D + Dt = 5 + 3 = 8 = 9
The earliest start time of the Finish activity is the shortest
time taken to complete the
entire project.
EST of Finish = EST of E + Et = 9 + 3 = 12
Act t
EST LST
C 4
Finish
A 2
B 5
Start
0
D 3
E 3
Act
t
EST
LST
EFT
LFT Act
EST EFT t
LST T.F. LFT
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TOPIC : CRITICAL PATH ANALYSIS 950 / 3
Compiled by: Goh PC 11
Calculation of the latest start time of an activity (backward
pass):
Begins from the Finish activity and move to the Start
activity.
For the Finish activity, the latest start time equals the
earliest start time.
LST of Finish = EST of Finish = 12
One succeeding activity:
LST of E = LST of Finish Et = 12 3 = 9
LST of D = LST of E Dt = 9 3 = 6
LST of C = LST of E Ct = 9 4 = 5
LST of A = LST of C At = 5 2 = 3
More than one succeeding activities:
LST of D Bt = 6 5 = 1 LST of B = min. (0, 1)
LST of C Bt = 5 5 = 0 = 0
The latest start time for the Start activity is 0.
Conditions for a critical activity:
In an AOA network diagram
A is a critical activity if and only if
ie = il
je = jl
Aijij tllee
In an AON network diagram
A is a critical activity if and only if
EST of A = LST of A
EST of B = LST of B
EST of B EST of A = duration of A
A 3
2 2
B 4
5 5
A i
ie
j
il je jl At
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TOPIC : CRITICAL PATH ANALYSIS 950 / 3
Compiled by: Goh PC 12
Float
Float is the spare time or slack by which an activity can be
delayed without increasing the overall
project completion time.
An activity with a slack of zero is critical since it must be
completed on time to avoid increasing
the completion time of the project.
Three varieties of float are distinguished:
Total float the delay possible for an activity if all preceding
activities start as early as possible whilst all subsequent
activities start as late as possible.
Free float the delay possible for an activity if all preceding
activities start as early as possible whilst all subsequent
activities start at their earliest time.
Independent float the delay possible for an activity if all
preceding activities start as late as possible whilst all
subsequent activities start at their earliest time.
All non-critical activities have positive total float. These
activities can be delayed by the amount
of time equals to the total float without changing the minimum
completion time of the project.
However, it affects the latest finish times of the preceding
activities and the earliest start times of
the subsequent activities.
Calculations of the floats:
Total float = tel ij = 19 8 3 = 8 days
Free float = tee ij = 15 8 3 = 4 days
Independent float = tle ij = 15 10 3 = 2 days (if the value is
negative, then the
independent float is considered zero).
i
8 10
j
15 19
A
3 days(t)
8 10 13 15 19
maximum time available
3 days
A Total float
A
3 days
Free float
A
3 days
Independent
float
ei ej li lj
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TOPIC : CRITICAL PATH ANALYSIS 950 / 3
Compiled by: Goh PC 13
Calculations of the total floats in an AOA network diagram
Activity Duration
(days)
Earliest Latest Total Float
Start Finish Start Finish
A 2 0 2 0 2 0
B 3 2 5 3 6 1
C 6 2 8 2 8 0
D 5 5 10 6 11 1
E 3 8 11 8 11 0
F 4 11 15 11 15 0
Only activities B and D have positive total float. So activities
B and D are non-critical activities.
Both activities B and D can be delayed by one day without
delaying the minimum completion
time of the project.
The critical activities are A, C, E and F and the minimum
completion time of the project is 15
days.
Calculations of the total floats in an AON network diagram
Activity Duration
(days)
Earliest Latest Total Float
Start Finish Start Finish
A 2 0 2 3 5 3
B 5 0 5 0 5 0
C 4 5 9 5 9 0
D 3 5 8 6 9 1
E 3 9 12 9 12 0
Only activities A and D have positive total float. So activities
A and D are non-critical activities.
Activity A can be delayed by three days while activity D can be
delayed by 1 day without
delaying the minimum completion time of the project.
The critical activities are B, C and E and the minimum
completion time of the project is 12 days.
B
1
0
A
2
Start
event
D
5
C
3 E
4
End
event F 3
6
3
4
5
6
2
15 11 2
5
8
0 15 11
8
6
2
C 4
5 5
Finish
12 12
A 2
0 3
B 5
0 0
Start
0 0
D 3
5 6
E 3
9 9
Act t
EST LST
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TOPIC : CRITICAL PATH ANALYSIS 950 / 3
Compiled by: Goh PC 14
The following activity network shows the earliest start time,
earliest finish time, latest start time,
latest finish time and the total float for each activity in the
project.
Activity Preceding Activities Duration (days)
A - 2
B - 3
C A, B 1
D A 5
E C, D 2
Exercise 3
1. Based on the following network diagram, construct a table
showing the earliest start time
(EST), earliest finish time (EFT), latest start time (LST) and
latest finish time (LFT) for each
activity.
2. Determine the earliest and latest time for each event in the
following network diagram.
Hence, construct a table showing the earliest start time (EST),
earliest finish time (EFT), latest
start time (LST) and latest finish time (LFT) and the total
float for each activity.
0 2 2
0 0 2
A
0 3 3
3 3 6
B
2 5 7
2 0 7
D
3 1 4
6 3 7
C
7 2 9
7 0 9
E
9
9
FINISH
3
1 1
5
7 8
E 2 2
2 3
A
3
1
0 0
6
10 10
4
6 6 B
C
D
F
G
5
2
5
4
1
Act
EST EFT t
LST T.F. LFT
A
B
C
D
E
1
1
0
3
2
5
6
5
10
3
4
4
F
2 Dummy
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TOPIC : CRITICAL PATH ANALYSIS 950 / 3
Compiled by: Goh PC 15
3. The following table shows the activities for a project.
Activity Preceding Activity Duration (weeks)
A - 2
B - 1
C A 3
D B 5
E C, D 2
F B 4
G E, F 3
H B 7
(a) Draw an activity network for the project showing the
earliest and latest start time for each activity.
(b) Determine the critical path and the minimum time required to
complete the project.
4. The following table shows the activities for a project.
Activity Preceding Activities Duration (days)
A - 1
B - 1
C - 3
D A 2
E B 3
F B 5
G D, E 5
H C, F 4
(a) Draw an AOA network for the project. (b) Construct a table
showing the earliest start time, earliest finish time, latest
start
time, latest finish time and the total float for each
activity.
(c) Hence, determine the critical activities and the minimum
completion time for the project.
5. The following table shows the activities for a project.
Activity Preceding Activities Duration (weeks)
A - 3
B A 5
C A 6
D B 4
E C 8
F B 2
G D, E 11
(a) Draw an AOA activity network for the project. (b) Construct
a table showing the duration, earliest start time, earliest finish
time,
latest start time, latest finish time and total float for each
activity.
(c) Hence, state the critical path and the minimum time for the
project to be completed.
(d) If the duration of activity B has to be extended for 3
weeks, determine whether the project will be delayed.
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TOPIC : CRITICAL PATH ANALYSIS 950 / 3
Compiled by: Goh PC 16
6. The following table shows the activities involved in a
project.
Activity Preceding Activities Duration (Hours)
A - 6
B A 4
C - 5
D B, C 1
E D 1
F A 7
G E, F 4
(a) Draw an activity network for the project showing the
earliest start time and the latest start time for each
activity.
(b) Determine the total float for each activity. (c) Determine
the critical path and the minimum time required to complete the
project.
(d) If activity B is extended to 7 hours, determine the number
of hours the project will be delayed.
7. The following table shows the activities, their durations and
their preceding activities for
a project.
Activity Duration (weeks) Preceding activities
A 6 -
B 4 -
C 2 A
D 3 A
E 7 D
F 10 B, C
G 5 C
H 1 E, G
(a) Draw an AOA network for the project.
(b) Construct a table showing the earliest start time, earliest
finish time, latest start
time and latest finish time for each activity. Hence, determine
the critical activities and find the
minimum time needed to complete the project.
(c) Determine the minimum time needed to complete the
project,
(i) if the duration for activity E is reduced by a week,
(ii) if the duration for activity F is reduced by a week.
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TOPIC : CRITICAL PATH ANALYSIS 950 / 3
Compiled by: Goh PC 17
SCHEDULING AND CRASHING
Scheduling
The objective of critical path analysis is to obtain a schedule
or time chart that determines the
start and finish dates of each activity in the project.
Gantt chart
A Gantt chart is a horizontal-bar chart, which provides a
graphical illustration of a schedule
showing activity start, duration and completion. It is
constructed with a horizontal axis
representing the total time span of the project, broken down
into increments (days, weeks or
months) and a vertical axis representing the activities (or
tasks) that make up the project.
A Gantt chart can be created from an activity network:
Start by scheduling critical activities. Since a critical
activity doesnt have spare time or float, its start date is fixed
at the earliest start time (EST) and finish date is fixed at
the
latest finish time (LFT).
Next, scheduling the non-critical activities. For each
non-critical activity, the duration between the earliest start time
and the latest finish time is determined. Non-critical
activities can be scheduled anywhere within this duration as
long as it doesnt affect the precedence relationships.
o If the total float equals the free float, the activity can be
scheduled anywhere within this duration without affecting the
precedence relationships.
o If the free float is less than the total float, the start time
of the activity can be delayed from the earliest start time by not
more than the free float, without
affecting the precedence relationships.
Two commonly used schedules are: o Earliest schedule schedule
all non-critical activities as earliest as possible. o Latest
schedule schedule all non-critical activities as late as
possible.
Scheduling must take into consideration: o Parallelism tasks can
be undertaken simultaneously. o Dependency task has an effect on
subsequent tasks.
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TOPIC : CRITICAL PATH ANALYSIS 950 / 3
Compiled by: Goh PC 18
A project involves 10 activities. The following network diagram
shows the dependency
relationships between activities and the duration (days) for
each activity.
The table below shows the number of workers needed for each of
the activities.
Activity Number of workers Activity Number of workers
A 3 F 2
B 2 G 4
C 4 H 6
D 3 I 3
E 5 J 4
The Gantt chart below shows the schedule (the earliest and the
latest) for each activity in the
project.
Time
1 2 3 4 5 6 7 8 9 10 11 12 13 14
A(3)
C(4)
H(6)
J(4)
B(2)
D(3)
E(5)
F(2)
G(4)
I(3)
5 5 11 12 12 12 4 15 15 13 8 8 8 4
3 5 5 12 12 12 12 6 12 12 8 11 11 11
Row (i) shows the number of workers needed each day if each
activity starts as early as possible.
Row (ii) shows the number of workers needed each day if each
activity starts as late as possible.
3
2 3
4
7 7
E
3
2
3 3
A
3
1
0 0
7
14 14
5
7 7
B
C
D
F G
4
2 6
4
2 4
4
6
10 10
3 H
I
J Dummy
(i)
(ii)
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TOPIC : CRITICAL PATH ANALYSIS 950 / 3
Compiled by: Goh PC 19
Resource histogram
The number of workers needed in a project for each time period
may be illustrated in a resource
histogram as follow. The non critical activities, activities
which are not on the critical path, do
not have fixed starting and finishing times but are constrained
by the earliest and latest starting
and finishing times. This situation offers the planner chance
for adjusting the demand for
resources.
(i) Resource histogram showing the number of workers needed when
activities are scheduled on their earliest start time.
No. of workers Time
1 2 3 4 5 6 7 8 9 10 11 12 13 14
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
5 5 11 12 12 12 4 15 15 13 8 8 8 4
(ii) Resource histogram showing the number of workers needed
when activities are scheduled on their latest start time.
No. of workers Time
1 2 3 4 5 6 7 8 9 10 11 12 13 14
12
11
10
9
8
7
6
5
4
3
2
1
3 5 5 12 12 12 12 6 12 12 8 11 11 11
The peaks and valleys in the above resource histograms indicate
high day-to-day variation in the
resource demand. Resource leveling shifts non-critical
activities within their float times so as to
move resources from the peak periods (high usage) to the valley
periods (low usage), without
delaying the project.
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TOPIC : CRITICAL PATH ANALYSIS 950 / 3
Compiled by: Goh PC 20
Scheduling limited resource
A resource conflict occurs when at any point in the schedule
several activities are in parallel and
the total amount of required resource(s) exceeds the
availability limit. The solution to this
situation is to give the resource to higher priority activities
and delay the others until the earliest
time the resource become available again. Resource is assigned
to the activities having:
the least total float; and the earliest latest start time.
The following network diagram shows a simple project involving
three activities A, B and C.
Assuming activity A needs 2 workers, B needs 4 workers and C
needs 2 workers and the
resource available is 6 workers per day.
No. of workers Time (day)
1 2 3 4 5
8 A
7
6 C
5
4
B 3
2
1
8 8 6 4 4
The above resource histogram shows resource usage if all
activities are scheduled on their
earliest start time. Note that activities A, B & C require
more than 6 workers at time period 1 & 2.
No. of workers Time (day)
1 2 3 4 5
6 A C
5
4
B 3
2
1
6 6 6 6 6
The above resource histogram shows resource usage if activity C
is delayed by 2 days, its total
slack.
A
B
C
5
1
0 0
3
2 5
2
3 5
3
4
5 5
2
Resource
available 6
workers/day
-
TOPIC : CRITICAL PATH ANALYSIS 950 / 3
Compiled by: Goh PC 21
The activities of a project along with their durations,
predecessors and resource used are given in
the following table.
Activity Preceding Activities Duration (days) Resource
(men/day)
A - 6 3
B - 2 6
C A 10 4
D A, B 16 4
E B 6 2
F C 8 3
G D, E 10 5
H F 6 2
The AON network is drawn and the project completion time is 32
days without considering the
resource limits.
Time (day) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
22 23 24 25 26 27 28 29 30 31 32
A(3) D(4) G(5)
B(6)
C(4)
E(2)
F(3)
H(2)
9 9 5 5 5 5 10 10 8 8 8 8 8 8 8 8 7 7 7 7 7 7 8 8 7 7 7 7 7 7 5
5
The Gantt chart above shows the activities scheduled at their
earliest time and the resource
histogram shows the resource usage per day. The project can be
completed in 32 days with a
requirement of at least 10 workers per day.
Act t
EST LST
C 10
6 8
Finish
32 32
A 6
0 0
B 2
0 4
Start
0
D 16
6 6
G 10
22 22
E 6
2 16
F 8
16 18
H 6
24 26
10
9
8
7
6
5
4
3
2
1
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TOPIC : CRITICAL PATH ANALYSIS 950 / 3
Compiled by: Goh PC 22
If the resource is limited to 8 men per day, determine the
activities schedule start and finish times
so that the daily resource usage does not exceed the resource
limits.
Current
time
Eligible
activities
Resource Duration ELS Decision Finish time
0 A
B
3
6
6
2
0
4
Start
Delay
6
-
6 B
C
6
4
2
10
4
8
Start
Delay
8
-
8 D
C
E
4
4
2
16
10
6
6
8
16
Start
Start
Delay
24
18
-
18 D
E
F
4
2
3
16
6
8
-
16
18
Continue
Start
Delay
24
24
-
24 F
G
3
5
8
10
18
22
Start
Start
32
34
32 G
H
5
2
10
6
-
26
Continue
Start
34
38
Therefore, the project completion time is 38 days.
Time (day) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
A(3) B(6) D(4) G(5)
C(4) E(2) F(3) H(2)
3 3 3 3 3 3 6 6 8 8 8 8 8 8 8 8 8 8 6 6 6 6 6 6 8 8 8 8 8 8 8 8
7 7 2 2 2 2
Resource (men/day)
8
7
6
5
4
3
2
1
-
TOPIC : CRITICAL PATH ANALYSIS 950 / 3
Compiled by: Goh PC 23
Crashing
Crashing an activity refers to the speeding up or shortening of
the duration of an activity by
using additional resources. These include overtime, hiring
temporary staff, renting more efficient
equipment, and other measures. Project crashing refers to the
process of shortening the duration
of the project by crashing the duration of a number of
activities. Since it generally results in an
increase of the overall project costs, the challenge faced by
the project manager is to identify the
activities to crash and the duration reduction for each activity
such that the project crashing is
done in the least expensive manner possible.
The key to project crashing is to attain maximum reduction in
schedule time with minimum cost.
The time to stop crashing is when it no longer becomes cost
effective.
A simple guideline for schedule crashing is:
Crash only activities that are critical. Crash from the least
expensive to the most expensive. Crash an activity only until
o It reaches its maximum time reduction. o It causes another
path to also become critical. o It becomes more expensive to crash
than not to crash.
The AOA network diagram of a project along with the durations
(days) of the activities is given
as follow. The minimum completion time of the project is 21
days.
The critical activities are A, F and H and the minimum
completion time for the project is 21 days.
The duration of the project can be reduced by crashing one of
the activities A, F or H.
First, calculate the free floats for all the non-critical
activities and identify those activities with
free float more than zero. These activities are C, D and G.
Activity Duration (days) Free float
A 7 0
B 4 0
C 5 2
D 10 1
E 5 0
F 8 0
G 7 2
H 6 0
3
7 7
7
21 21 6 4
2
4 5
A
5
1
0 0
4
7 7
B
C
D
E
G
10
7
5
8
7
5
12 14
6
15 15
F H
X
-
TOPIC : CRITICAL PATH ANALYSIS 950 / 3
Compiled by: Goh PC 24
To crash activity A: o Reduce the duration of activity A by 1
day and recalculate the free floats for
activities C, D and G.
o The free floats for activities C and D have reduced by 1 day
each. So activity A can only be reduced by 1 day (minimum of the
original FF of C and D).
o In fact, by reducing the duration of activity A by 1 day has
caused the path B-D-H to become critical path. So, activity A can
be crashed by at most 1 day.
o The project completion time has been reduced by 1 day to 20
days.
To crash activity F: o Reduce the duration of activity F by 1
day and recalculate the free floats for
activities C, D and G.
o The free floats for activities D and G have reduced by 1 day
each. So activity A can only be reduced by 1 day (minimum of the
original FF of D and G).
o In fact, by reducing the duration of activity A by 1 day has
caused the path B-D-H to become critical path. So, activity A can
be crashed by at most 1 day.
o The project completion time has been reduced by 1 day to 20
days.
3
6 6
7
20 20 6 4
2
4 4
A
5
1
0 0
4
6 6
B
C
D
E
G
10
7
5
8
6
5
11 13
6
14 14
F H
X
3
7 7
7
20 20 6 4
2
4 4
A
5
1
0 0
4
7 7
B
C
D
E
G
10
7
5
7
7
5
12 13
6
14 14
F H
X
-
TOPIC : CRITICAL PATH ANALYSIS 950 / 3
Compiled by: Goh PC 25
To crash activity H: o Reduce the duration of activity H by 1
day and recalculate the free floats for
activities C, D and G.
o The free floats for activity G has reduced by 1 day. So
activity A can be reduced by 2 day (original FF of G).
o By reducing the duration of activity H by 2 days will cause
the path A-E-G to become critical path. So, activity A can be
crashed by 2 days.
o The project completion time has been reduced by 2 days to 19
days.
3
7 7
7
20 20 5 4
2
4 5
A
5
1
0 0
4
7 7
B
C
D
E
G
10
7
5
8
7
5
12 13
6
15 15
F H
X
3
7 7
7
19 19 4 4
2
4 5
A
5
1
0 0
4
7 7
B
C
D
E
G
10
7
5
8
7
5
12 12
6
15 15
F H
X
-
TOPIC : CRITICAL PATH ANALYSIS 950 / 3
Compiled by: Goh PC 26
Exercise 4
1. The following table shows the activities for a project, their
preceding activities, durations
and resource usage for each activity.
Activity Preceding Activities Duration (days) Resource
(men/day)
A - 5 3
B - 3 2
C B 6 4
D A 2 1
E A 3 6
F C, D 5 2
G E 7 5
H F 4 3
(a) Draw an AOA network for the project. (b) Determine the
critical path and the minimum time to complete the project. (c)
Draw a Gantt chart if all activities are scheduled on their
earliest start time. (d) Based on your Gantt chart in (c), draw a
resource histogram showing the day-to-
day resource usage. State the minimum number of workers required
at any given
time.
(e) If the resource is limited to 8 men per day, draw a Gantt
chart and a resource histogram to show the schedule start and
finish times and resource allocation for
all the activities so that the daily resource usage does not
exceed the resource
limits. State the minimum project completion time.
2. The following table shows the activities for a project, their
preceding activities, durations
and resource used.
Activity Preceding Activities Duration (days) Resource
(men/day)
A - 4 3
B - 6 6
C - 2 4
D A 8 1
E D 4 4
F B 10 1
G B 16 4
H F 8 2
I E, H 6 4
J C 6 5
K G, J 10 2
(a) Draw an AON network for the project. (b) Determine the
critical path and the minimum time to complete the project. (c)
Draw a Gantt chart and a resource histogram if all activities are
scheduled on their
earliest start time.
(d) Determine the minimum number of workers required at any
given time by shifting the non-critical activities within their
total float times without delaying
the project. Illustrate the resource allocation using a resource
histogram.
(e) Determine the maximum number of days activity K can be
crashed without affecting the dependency relationships between the
activities. State the project
completion time after crashing activity K.
-
TOPIC : CRITICAL PATH ANALYSIS 950 / 3
Compiled by: Goh PC 27
STPM 2012
A company is involved in construction projects. One of the
projects awarded to the company
contains seven activities. The activities, the preceding
activities and the duration required for
each activity are shown in table below.
Activity Preceding Activity Duration (weeks)
A
B
C
D
E
F
G
-
-
A
A
B
B
C, D, E
3
9
4
10
2
15
8
(a) Draw an activity network for the project. [3]
(b) Determine the critical activities of the project, and find
the minimum number of weeks required to complete the project.
[4]
(c) Shortly before the company starts to implement the project,
a technical assistant points out that the duration required to
undertake activity F could be shortened to 11 weeks
with a new innovative approach. Determine whether the new
approach adopted by the
company for activity F would affect your answer in (b). [3]
STPM 2011
The network of the activities on nodes of a project is shown
below.
(d) Determine the values of r and s. [4]
(e) State the critical path, and determine the time required to
complete the project. [3]
(f) Calculate the total floats for activities A and J. [2]
(g) If the duration of activity J is extended to four weeks,
determine whether the project will be delayed. [2]
C 6
3 6
A 0
6 0
B 0
4 5
D 6
5 15
E 6
1 s Start
F 9
4 9
G r
2 16
I 13
5 13
H 13
6 14
J 19
3 20
Finish
K 18
5 18
Activity
Duration (weeks)
Earliest start time
Latest start time
-
TOPIC : CRITICAL PATH ANALYSIS 950 / 3
Compiled by: Goh PC 28
STPM 2010
A group of students are involved in an orientation programme for
new form six students. Nine
activities are required in order to organise the programme. The
activities and the duration for
each activity are shown in the table below.
Activity Preceding Activities Duration (days)
A
B
C
D
E
F
G
H
I
-
-
A
A, B
A
C
C
E, F
D, G
3
3
4
2
5
2
6
4
2
(h) Draw an activity network for the programme. [3]
(i) Construct a table which shows the earliest start time,
latest finish time and the total float for each activity. Hence,
determine the critical path and the minimum number of
days needed to complete the programme. [8]
(j) If the duration of activity B has to be extended to four
days, determine whether the programme will be extended or not. Give
a reason for your answer. [2]
STPM 2009
A company wishes to develop a theme park on a 120-acre land. The
major activities of the
project are listed in the table below.
Activity Duration
(month) Preceding activities
A Project application and approval 10 -
B Project design 6 A
C Project design approval 3 B
D Land clearing 2 A
E Machinery and equipment purchase 6 C
F Building construction 8 C, D
G Landscaping 6 C, D
H Park construction 10 C, D
I Testing 3 E, F, G, H
J Opening ceremony 1 I
(a) Draw an activity network for the project. [4]
(b) (i) List all the possible paths of the project and their
corresponding total duration. [4]
(ii) Determine the critical path. [1]
(iii) Find the minimum time required to complete the project.
[1]
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TOPIC : CRITICAL PATH ANALYSIS 950 / 3
Compiled by: Goh PC 29
STPM 2008
The following table shows the activities for a project and their
preceding activities and duration.
Activity Preceding Activities Duration (weeks)
A - 3
B - 4
C A 2
D B, C 5
E B 2
F D, E 3
(a) Draw an activity network for the project showing the
earliest start time and the latest
start time for each activity. [7]
(b) State the critical activities of the project and the minimum
time required to complete
the project. [2]
(c) If the duration of activity E has to be extended for a week,
determine whether the
project will be delayed. [3]
STPM 2007
The following table shows the activities for a project and their
preceding activities and duration.
Activity Preceding Activities Duration (weeks)
A - 11
B - 5
C A 9
D B 6
E B 8
F E 6
G C, D, F 8
H E, G 7
I H 9
(a) Draw an activity network for the project. [3]
(b) Construct a table showing the duration, earliest start time,
latest finish time and total
float for each activity. Hence, determine the critical path and
the minimum duration of the
project. [9]
STPM 2006
The following table shows the activities, their durations and
their preceding activities for a
project.
Activity Duration (weeks) Preceding activities
A 2 -
B 1 -
C 3 A
D 2 B
E 3 C, D
F 2 E
G 1 F
(a) Draw an activity network for the project. [2]
(b) Construct a table showing the earliest start time, earliest
finish time, latest start time
and latest finish time for each activity. Hence, determine the
critical activities and find the
minimum time needed to complete the project. [8]
(c) If the durations for activities C and D are each reduced by
a week, determine whether
the project can be completed within 10 weeks. [2]
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TOPIC : CRITICAL PATH ANALYSIS 950 / 3
Compiled by: Goh PC 30
STPM 2005
The following table shows the activities, their preceding
activities and their durations for a
project.
Activity Preceding Activities Duration (weeks)
A - 7
B A 3
C A 3
D B, C 4
E B 5
F A 3
G D, E, F 6
(a) Draw an activity network for the project. [3]
(b) Construct a table which shows the earliest start time,
earliest finish time, latest start
time, latest finish time, total float, free float and
independent float for each activity. [7]
(c) Determine the critical path and the minimum time required to
complete the project. [2]
(d) If the duration of activity D has to be extended to 8 weeks,
determine the number of
weeks the project will be delayed. [3]
STPM 2004
The following table shows the activities involved in a
particular project.
Activity Preceding
activities
Duration
(days)
Earliest start Latest finish
A - 5 0 5
B - 1 0 5
C B 2 1 7
D A, C 4 5 11
E A 6 5 11
F D, E 3 11 14
(a) Draw an activity network for the project. [3]
(b) Calculate the total float and free float of each activity.
Hence, determine the critical
path of the project. [7]
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TOPIC : CRITICAL PATH ANALYSIS 950 / 3
Compiled by: Goh PC 31
STPM 2003
A training programme for young man managers involves seven
activities. The activities and the
duration for each activity are shown in the table below.
Activity Preceding Activities Duration (weeks)
A - 2
B - 5
C - 1
D B 10
E A, D 3
F C 6
G E, F 8
(a) Draw the network diagram for the training programme. [3]
(b) Determine the critical activities, and find the minimum time
needed to complete the
training programme. [8]
STPM 2002
A project on setting up a student-registration system of a
college involves seven activities. The
activities and their duration times (in days) are listed as
follows:
Activity Preceding Activities Duration (weeks)
A - 4
B - 2
C - 3
D A 8
E B 6
F C 3
G D, E 4
(a) Draw a network diagram for the project. [3]
(b) Determine the minimum duration for the project to be
completed. [5]
(c) Calculate the total float for each activity and state the
critical path of the project. [3]