Project Management and scheduling
• Objectives of project scheduling
• Network analysis
• Scheduling techniques
Objectives of project scheduling
• Produce an optimal project schedule in
terms of cost, time, or risk.
• Usually, it is difficult to optimize the three
variables at the same time. Thus,
• setting an acceptable limit for two of the
three varaibles and optimizing the project in
terms of the third variable.
Critical Path Method (CPM)
• Produce the earliest and lastest starting and
finishing times for each task or activity.
• Calculate the amount of slack associated
with each activity.
• Determine the critical tasks (Critical path).
• Forward pass and backward pass
computational procedures.
Network control
• Track the progress of a project on the basis
of the network schedule and taking
corrective actions when necessary.
• Evaluate the actual performance against
expected performance.
PERT/CPM
Node
3
4
Merge point Successor
Burst point
1
6
2
Arrow
5
Predecessor
7
8
Two models of PERT/CPM
• Activity-on-Arrow (AOA): Arrows are used
to represent activities or tasks. Nodes
represent starting and ending points of
activities.
• Activity-on-Node (AON): Nodes are used
to represent activities or tasks, while arrows
represent precedence relationships.
Recap - purpose of CPM
•
•
•
•
•
•
•
Critical path
Earliest starting time
ES
Earliest completion time
EC
Latest starting time
LS
Latest completion time
LC
Activity
Capital letter
Duration
t
Example
•
•
•
•
•
•
•
•
Activity
A
B
C
D
E
F
G
Predecessor
A
C
A
B, D, E
Duration
2
6
4
3
5
4
2
Activity on Node Network
F
4
A
2
End
D
3
G
2
B
6
Start
C
4
E
5
Forward pass analysis
2
2
0
A
2
0
2
F
4
6
11
5
D
3
0
0
Start
0
C
4
B
6
4
6
4
9
E
5
9
G
2
11
End
11
Backward pass analysis
2
2
0
A
4 2
0
6
2
0
0 B
3 6
0
0
4
C
0 4 4
6
9
4
11
11
5
D
6 3
Start
0
7
6
F
4
9
9
9
9
E
4 5 9
G
2
11
11
11
End
11
11
Slack Time in Triangles
5
2
0
4
2
7
A
4 2
0
6
0
0
3
B
6
0
0
6
9
11
11
5
4
D
6 3
4
Start
0
2
6
F
4
9
9
9
G
2
0
0
C
0 4
0
4
4
4
4
9
E
5 9
0
11
11
11
End
0
11
11
Critical path
F
4
A
2
End
D
3
B
6
Start
C
4
G
2
E
5
Computational analysis
of network
• Forward pass: each activity begins at its
earliest time. An activity can begin as soon
as the last of its predecessors is finished.
• Backward pass: begins at its latest
completion time and ends at the latest
starting time of the first activity in the
project network.
Rules for implementation forward pass
• The earliest start time (ES) for any node (j)
is equal to the maximum of the earliest
completion times (EC) of the immediate
predecessors of the node.
• The earliest completion time (EC) of any
activity is its earliest start time plus its
estimated time (its duration).
• The earliest completion time of the project
is equal to the earliest completion time the
very last activity.
Rules for implementation backward pass
• The latest completion time (LC) of any
activity is the smallest of the latest start
times of the activity’s immediate
successors.
• The latest start time for any activity is the
latest completion time minus the activity
time.
Calculate slack time for each
activity
• Slack time: the difference in time between
the two dates at the beginning of a job or
the two dates at the end of the job. Slack
time represents the flexiblity of the job.
• Thus, slack time = LS - ES or LC - EC
PERT
• PERT is an extension of CPM.
• In reality, activities are usually subjected to
uncertainty which determine the actual
durations of the activities.
• It incorporates variabilities in activity
duration into project entwork analysis.
• The poetntial uncertainties in activity are
accounted for by using three time estimates
for each activity
Variation of Task Completion Time
Average
Task A
2
4
6
4
Task B
3
4
5
4
4
4
PERT Estimates & Formulas
a+4m+b
te =
6
2
(b-a)
s2 = 36
a = optimistic time estimate
m = most likely time estimate
b = pessimistic time estimate (a < m < b)
te = expected time for the activity
s2=variance of the duration of the activity
PERT
• Calculate the expected time for each
activity
• Calculate the variance of the duration of
each activity
• Follow the same procedure as CPM does to
calculate the project duration, Te
• Calculate the variance of the project
duration by summing up the variances of
the activities on the critical path.
Sources of the Three Estimates
•
•
•
•
•
•
Furnished by an experienced person
Extracted from standard time data
Obtained from historical data
Obtained from regression/forecasting
Generated by simulation
Dictated by customer requirement
A PERT Example
• Activity Predecessor
a
m
b
te
s2
•
•
•
•
•
•
•
1
5
2
1
4
3
1
2
6
4
3
5
4
2
4
7
5
4
7
5
3
2.17
6.00
3.83
2.83
5.17
4.00
2.00
0.2500
0.1111
0.2500
0.2500
0.2500
0.1111
0.1111
A
B
C
D
E
F
G
A
C
A
B, D, E
What do Te &
2
S
tell us?
• How likely to finish the project in a
specified deadline.
• For example, suppose we would like to
know the probability of completing the
project on or before a deadline of 10 time
units (days)
Probability of finishing the
project in 10 days
S2 = V[C] + V[E] + V[G]
S= 0.7817
= 0.25 + 0.25 + 0.1111
= 0.6111
( 10-Te )
P( T<=Td ) = P(T<=10) = P(z<=
)
S
Te = 11
(10-11)
= P(z<=
) = P(z<= -1.2793)
0.7817
= 0.1003
About 10% probabilty fo finishing the project within 10 days
Probability of finishing the
project in 13 days
S2 = V[C] + V[E] + V[G]
S= 0.7817
= 0.25 + 0.25 + 0.1111
= 0.6111
( 13-Te )
P(T<=Td ) = P(T<=10) = P(z<=
)
S
Te = 11
(13-11)
= P(z<=
) = P(z<= 2.5585)
0.7817
= 0.9948
About 99% probabilty of finishing the project within 13 days
Gantt Chart
• Gantt chart is a matrix of rows and columns.
The time scale is indicated along the
horizontal axis. Activities are arranged
along the vertical axis.
• Gantt charts are usually used to represent
the project schedule. Gantt charts should be
updated periodically.
Gantt Chart
G
F
E
D
C
B
A
1
2
3
4
5
6
7
8
9 10
11
Gantt Chart Variations
•
•
•
•
•
•
•
Linked Bars
Progress - monitoring
Milestone
Task - combinations
Phase-Based
Multiple-Projects
Project-Slippage-tracking
Linked Bars Gantt Chart
G
F
E
D
C
B
A
1
2
3
4
5
6
7
8
9 10
11