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1. CPM (Critical Path Method)

2. CPM Objective
• To estimate the total project duration and to
assign starting and finishing times to all
activities involved in the project

3. What factors should we consider
for project scheduling?
•
•
•
•
Total completion time of the project
Earlier and latest start time of each activity
Critical activities and critical path
Float for each activity (I.e. the amount of
time by which the completion of a non
critical activity can be delayed without
delaying the total project completion time)

4. Important Notations
• Ei = Earliest occurrence time of an event (I.e.
The earliest time for an event to occur
immediately after al the preceding activities
have been completed without delaying the
entire project)
• Li = Latest allowable time for an event (I.e.
The latest time at which an event can occur
without causing a delay in already determined
project’s completion time)

5. Important Notations (contd…)
• ESij = Early starting time for an activity (i,j). It
is the earliest time an activity can possibly
start without affecting the project completion
• LSij = Late starting time for an activity (i,j) . It
is the latest possible time an activity can start
without delaying the project completion time

6. Important Notations (contd…)
• EFij = Early finishing time for an activity
(i,j). It is the earliest time an activity can
possibly finish without affecting the project
completion
• LFij = Late finishing time for an activity (i,j)
. It is the latest possible time an activity can
start without delaying the project
completion time
• Tij = Duration of an activity (i.j)

7. How to calculate earliest
occurrence and latest available
times of events?
• Forward pass method for calculating
earliest event time
• Backward pass method for latest available
event time

8. Forward pass method
• Start with initial event numbered 1. Earliest
occurrence time for event 1 is always zero.
I.e E1 = 0 where i = 1
• Calculate earliest start time for each activity
that begins at event ( i = 1). This is equal to
the earliest start time of event I (tail event)
(I.e) ESij = Ei for all activities (i.j) starting at
event i

9. Forward pass method (Contd..)
• Calculate earliest finish time of each
activity that begins at event (i) . This is
equal to the earliest start time of the activity
plus the duration of the activity
(I.e) EFij = ESij + tij = Ei + tij, for all activities
(i,j) beginning at event i.

10. Forward pass method (Contd..)
• Proceed to the next event j ; j > I
• Calculate the earliest occurrence time for the event
j. This is the maximum of the earliest finish times
of all activities ending into that event
Ej = Max { EFij} = Max { Ei + tij }, for all immediate
predecessor activities
 If N is the final event then earliest finish time for
the project (or) the earliest occurrence time EN for
the final event is given by
EN = Max {EFij} = Max {EN-1+ tij}, for all terminal
activities

11. Backward pass method (For
latest allowable event time)
• Set the latest occurrence event N equal to
its earliest occurrence time (known from
forward pass method) LN = EN , j = N
• Latest finish time of each activity which
ends at event j, LFij = Lj
• Latest start time of each activity ending at j,
LSij= LFij-tij
• Latest occurrence time for event ( i) where
i<j , L = Min {LSij}= Min {Lj-tij}
• L1=0

12. Sample CPM problem
Draw the network diagram and find
the critical path

13. A
B
C
D
E
F
G
H
I
J
K
L
M
Design new premises
Obtain tenders from the contractors
Select the contractor
Arrange details with the selected contractor
Decide which equipment to be used
Arrange storage of equipment
Arrange disposal of other equipment
Order new equipment
Take delivery of new equipment
Renovations take place
Remove old equipment
Cleaning
Return old equipment
A
B
C
A
E
E
E
H,L
K
D,F,G
J
H,L
14
4
2
1
2
3
2
4
3
12
4
2
2

14. Solution with critical path
E3=18
L3=18
C (2)
5
3
D(1)
E6=18
L6=21
B(4)
A (14)
E1=0
L1=0
2
E (2)
E2=14
L2=14
E7=21
L7=21
K (4)
8
J(12)
7
6
G(2)
1
E8=25
L8=25
E5=20
L5=20
F(3)
4
9
E9=37
L9=37
L(2)
M(2)
12
10
I (3)
H (4)
E4=16
L4=18
11
E11=41
L11=42
E10=39
L10=39
E12=42
L12=42

15. From the earlier example what is
the critical path?
• The critical path is the sequence of critical
activities that form a continuous path
between the start of a project & its
completion
• It is the longest path
• Critical events : where E = L
• Critical activities float is 0.

16. Floats
• Calculated for non critical activity or non
critical event
• The length of time to which a non critical
activity and/or an event can be delayed or
extended without delaying the total project
completion time
• Float can be called as slack or free time also

17. Float on event
• Float slack of an event is the difference
between its latest occurrence time and its
earliest occurrence time
• Event float = Li-Ei
• Event float is 0 for critical events (I.e. L = E)

18. Float on activities
• The length of free time available within the
estimated times of non-critical activities
• 3 types of float for non-critical activities
– Total float
– Free float
– Independent float

19. Total float
• Length of time by which an activity can be
delayed if all its predecessor activities are
completed at its earliest possible time and
all successor activities can be delayed until
their latest permissible time
• Total float TFij= (Lj-tij)-Ei

20. Free float
• Time by which the completion of an
activity can be delayed without causing any
delay in its immediate succeeding activities
• Free float FFij= (Ej-Ei)-tij

21. Independent float
• Amount of time available when preceding
activities are completed at their latest
permissible times and the following
activities can still be completed at their
earliest possible times
• Independent float IFij=(Ej-Li)-tij
• Negative value of independent float is
considered as 0

22. Further on floats
• Latest occurrence time of an event is always
greater than or equal to its earliest occurrence
time. This implies
Independent float < Free float < Total float
• Floats help in identifying underutilized
resources,flexibility in the total schedule and
possibilities of redeployment of resources
• Once a float of an activity is disturbed, float for
all other activities of the project is changed and
should be recalculated

23. What does a float value mean?
• If float value is negative, it means project
completion is behind the schedule data.
Resources are not adequate and activity may
not finish in time
• Float = 0. Just enough resources, no
possibility for delay
• Float is positive means resources are surplus
and can be deployed elsewhere as required