CPM (Critical Path Method)

CPM Objective

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

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)

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)

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

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)

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

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

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.

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

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

Sample CPM problem

Draw the network diagram and find the critical path

A Design new premises - 14

B Obtain tenders from the contractors A 4

C Select the contractor B 2

D Arrange details with the selected contractor C 1

E Decide which equipment to be used A 2

F Arrange storage of equipment E 3

G Arrange disposal of other equipment E 2

H Order new equipment E 4

I Take delivery of new equipment H,L 3

J Renovations take place K 12

K Remove old equipment D,F,G 4

L Cleaning J 2

M Return old equipment H,L 2

Solution with critical path

1

7

2

9

3

4

5

6

10

8

11

A (14)

B(4)

C (2)

D(1)

E (2)F(3)G(2)

H (4)

M(2)

J(12)K (4)

12

I (3)

L(2)

E1=0L1=0

E2=14L2=14

E4=16L4=18

E3=18L3=18

E5=20L5=20

E6=18L6=21

E7=21L7=21

E8=25L8=25

E9=37L9=37

E10=39L10=39

E11=41L11=42

E12=42L12=42

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.

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

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)

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

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

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

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

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

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