1. Projects: Critical Paths
Dr. Ron Lembke
Operations Management

2. 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

3. 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?

4. PERT & CPM Steps Identify activities Determine sequence Create network
Determine activity times
Find critical path
Earliest & latest start times
Earliest & latest finish times
Slack

5. 3 1 2 Activity on Node (AoN) Project: Obtain a college degree (B.S.)
Attend class, study etc.
Receive diploma
Enroll 3 1 2
1 month
4? Years
1 day

6. 1 2 3 4 Activity on Arc (AoA) Project: Obtain a college degree (B.S.)
Attend class, study, etc.
Receive diploma
Enroll 1 2 3 4
1 month
4,5 ? Years
1 day

7. 1 2 3 4 AoA Nodes have meaning Project: Obtain a college degree (B.S.)
GraduatingSenior
Applicant
Student
Alum

8. Liberal Arts Sidebar
Alum = ?
Alumnus
Alumna
Alumni
Alumnae
Alumni

9. Network Example
You’re a project manager for Bechtel. Construct the network.
Activity Predecessors A -- B A C A D B E B F C G D H E, F

10. Network Example - AON D G B A E Z C H F

11. Network Example - AON F H C A E Z B G D

12. Network Example - AON E H C G A Z F B D

13. Network Example - AOA D G 3 6 8 B E A 1 2 5 H 7 9 C F 4

14. AOA Diagrams
A precedes B and C, B and C precede D B A 1 2 3 D 4 C 3 B A 1 2 C 4 D 5
Add a phantom arc for clarity.

15. 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

16. Critical Path Analysis Example

17. Network Solution B D E A G 2 6 3 1 1 C F 3 4

18. 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

19. Activity A Earliest Start SolutionD B C F G 1 6 2 3 4
For starting activities, ES = 0.

20. Earliest Start SolutionD B C F G 1 6 2 3 4

21. 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

22. Earliest Start SolutionD B C F G 1 6 2 3 4

23. Latest Finish SolutionD B C F G 1 6 2 3 4

24. Compute Slack

25. Critical Path A E D B C F G 1 6 2 3 4

26. New notation C 7 Compute ES, EF for each activity, Left to Right
LS LF
Compute ES, EF for each activity, Left to Right
Compute, LF, LS, Right to Left

27. Exhibit 6
C 7
F 8
A 21
G 2
B 5
D 2
E 5

28. Exhibit 6
21 28
28 36
C 7
F 8
0 21
36 38
A 21
G 2
28 33
21 26
26 28
B 5
D 2
E 5
F cannot start until C and D are done.
G cannot start until both E and F are done.

29. Exhibit 6
21 28
28 36
C 7
F 8
21 28
28 36
0 21
36 38
A 21
G 2
0 21
36 38
28 33
21 26
26 28
B 5
D 2
E 5
21 26
26 28
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.

30. Gantt Chart - ES A C F B D E G

31. Solved Problem 2
B 4
E 6
G 7
A 1
C 3
I 4
F 2
D 7
H 9

32. Solved Problem 2 B 4 E 6 G 7 C 3 A 1 I 4 F 2 D 7 H 9 1 5 5 11 11 18
1 5
5 11
B 4
E 6
11 18
1 5
5 11
G 7
11 18
0 1
1 4
18 22
A 1
C 3
I 4
0 1
6 9
8 10
18 22
F 2
1 8
9 11
8 17
D 7
H 9
2 9
9 18

33. 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)