Projects:Critical Paths

Dr. Ron LembkeOperations 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

• 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?

Questions Answered by PERT & CPM

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)

24? Years

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Activity on Arc (AoA)

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AoA Nodes have meaning

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Network Example

You’re a project manager for Bechtel. Construct the network.

Activity PredecessorsA --B AC AD BE BF CG DH E, F

Network Example - AON

A

C

E

F

BD

G

H

Z

Network Example - AOA

2

4

51

3 6 8

7 9A

C F

EBD

H

G

AOA Diagrams

2 31A

C

BD

A precedes B and C, B and C precede D

2 41A C

B

D

3

5

4

Add a phantom arc for clarity.

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 Pred. Description Time

(Wks) A None Prepare Site 1 B A Pour fdn. & frame 6 C B Buy shrubs etc. 3 D B Roof 2 E D Do interior work 3 F C Landscape 4 G E,F Move In 1

Network Solution

A

EDB

C F

G

1

6 2 3

1

43

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 ES EF LS LF SlackA 0 1BCDEF

Activity AEarliest Start Solution

For starting activities, ES = 0.

AEDB

C F

G1

6 2 31

43

Activity ES EF LS LF Slack A 0 1 B 1 7 C 1 4 D 7 9 E 9 12 F 4 8 G 12 13

Earliest Start Solution

AEDB

C F

G1

6 2 31

43

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

Activity ES EF LS LF SlackA 0 1B 1 7C 1 4D 7 9E 9 12F 4 8G 12 13 13

Earliest Start Solution

AEDB

C FG

16 2 3

1

43

Activity ES EF LS LF Slack A 0 1 0 1 B 1 7 1 7 C 1 4 5 8 D 7 9 7 9 E 9 12 9 12 F 4 8 8 12 G 12 13 12 13

Latest Finish Solution

AEDB

C F

G

1

6 2 31

43

Activity ES EF LS LF Slack A 0 1 0 1 0 B 1 7 1 7 0 C 1 4 5 8 4 D 7 9 7 9 0 E 9 12 9 12 0 F 4 8 8 12 4 G 12 13 12 13 0

Compute Slack

Critical Path

A

EDB

C F

G

1

6 2 3

1

43

New notation

• Compute ES, EF for each activity, Left to Right

• Compute, LF, LS, Right to Left

C 7LS LF

ES EF

Exhibit 6

A 21

E 5D 2B 5

C 7 F 8

G 2

Exhibit 6

A 21

E 5D 2B 5

C 7 F 8

G 2

21 28 28 36

36 38

28 3326 2821 26

0 21

F cannot start until C and D are done.G cannot start until both E and F are done.

Exhibit 6

A 21

E 5D 2B 5

C 7 F 8

G 2

21 26

0 21

26 28 31 36

36 38

21 28 28 36

21 28 28 36

36 38

28 3326 2821 26

0 21

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

0 5 10 15 20 25 30 35 40

A

B

C

D

E

F

G

Solved Problem 2

A 1

B 4

C 3

D 7

E 6

F 2

H 9

I 4

G 7

Solved Problem 2

A 10 1

0 1

B 41 5

1 5

C 36 9

1 4

D 72 9

1 8

E 65 11

5 11

F 29 11

8 10

H 99 18

8 17

I 418 22

18 22

G 711 18

11 18

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)