Projects:
Critical Paths
Dr. Ron Lembke
Operations Management
PERT & CPM
• Network techniques
• Developed in 1950’s
• CPM by DuPont for chemical plants
• PERT by U.S. Navy for Polaris
missile
• Consider precedence relationships
& interdependencies
• Each uses a different estimate of
activity times
Questions Answered by
PERT & CPM
•
•
•
•
•
•
Completion date?
On schedule? Within budget?
Probability of completing by ...?
Critical activities?
Enough resources available?
How can the project be finished early at
the least cost?
PERT & CPM Steps
•
•
•
•
•
Identify activities
Determine sequence
Create network
Determine activity times
Find critical path
• Earliest & latest start times
• Earliest & latest finish times
• Slack
Activity on Node (AoN)
Project: Obtain a college degree (B.S.)
Enroll
1
1 month
Attend class,
study etc.
Receive
diploma
2
3
4? Years
1 day
Activity on Arc (AoA)
Project: Obtain a college degree (B.S.)
1
Attend
class,
study,
etc.
Enroll
2
1 month
Receive
diploma
3
4,5 ?
Years
4
1 day
AoA Nodes have
meaning
Project: Obtain a college degree (B.S.)
1
Applicant
2
Student
3
Graduating
Senior
4
Alum
Liberal Arts Sidebar
• Alum = ?
Alumnus
Alumni
Alumna
Alumnae
Alumni
Network Example
You’re a project manager for Bechtel.
Construct the network.
Activity
A
B
C
D
E
F
G
H
Predecessors
-A
A
B
B
C
D
E, F
Network Example AON
D
G
B
A
E
C
F
Z
H
Network Example AOA
B
1
A
3
D
6
G
8
E
2
5
C
F
4
H
7
9
AOA Diagrams
A precedes B and C, B and C precede D
1
A
B
2
3
D
4
C
3
B
1
A
2
C
Add a phantom arc for clarity.
4
D
5
Critical Path Analysis
• Provides activity information
• Earliest (ES) & latest (LS) start
• Earliest (EF) & latest (LF) finish
• Slack (S): Allowable delay
• Identifies critical path
• Longest path in network
• Shortest time project can be
completed
• Any delay on activities delays project
• Activities have 0 slack
Critical Path
Analysis Example
Event
ID
A
B
C
D
E
F
G
Pred.
None
A
B
B
D
C
E,F
Description
Time
(Wks)
Prepare Site
Pour fdn. & frame
Buy shrubs etc.
Roof
Do interior work
Landscape
Move In
1
6
3
2
3
4
1
Network Solution
A
1
B
D
E
6
2
3
G
1
C
F
3
4
Earliest Start & Finish
Steps
• Begin at starting event & work forward
• ES = 0 for starting activities
• ES is earliest start
• EF = ES + Activity time
• EF is earliest finish
• ES = Maximum EF of all predecessors for
non-starting activities
Activity A
Earliest Start Solution
Activity
A
B
C
D
E
F
ES
0
EF
1
LS
A
1
LF
Slack
B
D
E
6
2
3
C
F
3
4
For starting activities, ES = 0.
G
1
Earliest Start Solution
Activity
A
B
C
D
E
F
G
ES
0
1
1
7
9
4
12
EF
1
7
4
9
12
8
13
LS
A
1
LF
Slack
B
D
E
6
2
3
C
F
3
4
G
1
Latest Start & Finish
Steps
• Begin at ending event & work backward
• LF = Maximum EF for ending activities
• LF is latest finish; EF is earliest finish
• LS = LF - Activity time
• LS is latest start
• LF = Minimum LS of all successors for
non-ending activities
Earliest Start Solution
Activity
A
B
C
D
E
F
G
ES
0
1
1
7
9
4
12
EF
1
7
4
9
12
8
13
LS
LF
B
A
6
Slack
D E
2
1 C
3
3 G
F 1
4
13
Latest Finish Solution
Activity ES
EF
A
0
1
B
D E
B
1
7
34 G
C A 61 2
D 1 C7
F 9 1
E
9
12
3
4
F
4
8
G
12
13
LS
0
1
5
7
9
8
12
LF
1
7
8
9
12
12
13
Slack
Compute Slack
Activity
A
B
C
D
E
F
G
ES
0
1
1
7
9
4
12
EF
1
7
4
9
12
8
13
LS
0
1
5
7
9
8
12
LF
1
7
8
9
12
12
13
Slack
0
0
4
0
0
4
0
Critical Path
A
1
B
D
E
6
2
3
C
F
3
4
G
1
New notation
ES EF
C7
LS LF
• Compute ES, EF for each
activity, Left to Right
• Compute, LF, LS, Right to Left
Exhibit 6
F8
C7
A 21
G2
B5
D2
E5
Exhibit 6
21
28
28
F8
C7
0
36
21
36
A 21
38
G2
21
26
B5
26
28
D2
28
33
E5
F cannot start until C and D are done.
G cannot start until both E and F are done.
Exhibit 6
21
28
28
F8
C7
0
21
21
36
28
28
36
36
A 21
0
38
G2
21
36
21
26
B5
21
26
26
28
D2
26
28
28
38
33
E5
31
36
E just has to be done in time for G to start at 36, so it has slack.
D has to be done in time for F to go at 28, so it has no slack.
Gantt Chart - ES
A
C
F
B
D
E
G
0
5
10
15
20
25
30
35
40
Solved Problem 2
B4
E6
G7
A1
C3
I4
F2
D7
H9
Solved Problem 2
1
0
1
5
11
B4
E6
1
5
5
1
4
A1
C3
0
6
1
5
9
11
11
18
G7
11
18
8
D7
2
9
22
I4
8
10
18
F2
1
18
9
11
8
17
H9
9
18
22
Summary
• Activity on Node representation
• Calculated
– ES, EF for all activities
– LS, LF for all activities (working backwards)
– Slack for each activity
• Identified critical path(s)