Chapter 3, Project Management
Problem 1: For the following project, draw a AON network and find all the paths and
duration of the paths.
Activity
A
B
C
D
E
F
G
H
Predecessors
Start
A
A
B
B, C
C
D, E
E,F
Paths
Critical path:
Project completion time:
t
Successors
3
9
6
3
8
5
6
4
Duration
Problem 2: For the project from problem #1, determine the earliest and latest schedules
and the slacks of all the activities.
B
9
D
3
G
6
END
A
3
START
E
8
H
4
C
6
F
5
Project start:
Activity
t
A
3
B
9
C
6
D
3
E
8
F
5
G
6
H
4
Earliest
Start
Project completion time:
Earliest
Finish
Latest
Start
Latest
Finish
Slack
Critical
Problem 3: The following represent activities in a major construction project.
Activity
A
B
C
D
E
F
G
H
a
2
4
1
1
2
6
7
2
Activity
a
m
b
A
2
5
14
B
4
7
10
C
1
2
9
D
1
2
3
E
2
4
6
F
6
6
6
G
7
9
17
H
2
8
8
m
5
7
2
2
4
6
9
8
b
14
10
9
3
6
6
17
8
Expected time (t)
Predecessors
Start
Start
A
A
B
B
C, E
D, F
Variance 2)
Draw the network to represent this project. Find the Critical Path, expected project
completion and project variance.
Problem 4:
a.
What is the probability of finishing the project in 24 weeks or less?
b.
What is the probability of finishing the project in 20 weeks or less?
c.
The due date for the project is set so that there is a 90% chance that the project will
be finished by this date. What is the date?
Problem 5:
Development of a new software product is being considered. The activities are listed in
the table below along with their costs and completion times in weeks.
Activity
A
B
C
D
E
F
G
Immediate
Predecessor
A
A
B, D
C
E
Normal
Time
4
2
3
8
6
3
4
Crash
Time
3
1
3
4
3
2
2
Normal
Cost
2,000
2,200
500
2,300
900
3,000
1,400
Crash
Cost
2,600
2,800
500
2,600
1,200
4,200
2,000
a.
What is the project expected completion date?
b.
What is the total cost required for completing this project on normal time?
c.
Activity
A
If you wish to reduce the time required to complete this project by 1 week, which
activity should be crashed, and how much will this increase the total cost?
MCA
Cost difference =
(Normal time – Crash time)
Crash cost – Normal cost
B
C
D
E
F
G
MCA = Maximum crashing allowed
Crashing cost/week
d.
1
Solver Model
C
A
B
Class practice problem #5
2
Activity
D
E
Normal
Time
(NT)
Crash
time
(CT)
Normal
Cost
(NC)
Crash
cost
(CC)
F
MCA = NT - CT
G
H
CC - NC
Crashing
cost/week =
(CC-NC)/MCA
I
Predecessors
3
A
4
3
2000
2600
Start
4
B
2
1
2200
2800
Start
5
C
3
3
500
500
A
6
D
8
4
2300
2600
A
7
E
6
3
900
1200
B, D
8
F
3
2
3000
4200
C
9
G
4
2
1400
2000
E
10
Normal cost =
J
Successors
A
11
12
B
C
D
E
F
G
H
Weeks
crashed
Actual duration
ES
EF
LS
LF
Slack
Project start
Activity
13
A
14
B
15
C
16
D
17
E
18
F
19
G
20
Project duration
21
Normal project cost =
22
Cost of crashing =
23
Total cost =
Solver Parameters:
Step 1: Minimize project completion time
Set Target cell
□ Maximize □ Minimize
(Set objective)
Changing cells
Constraints
Options
Step 2: Minimize project cost for the minimum project completion time
Set Target cell
Changing cells
Constraints
Options
□ Maximize □ Minimize