Projects:Critical Paths

Dr. Ron Lembke

Operations Management

PERT & CPMPERT & 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 & CPMQuestions Answered by PERT & CPM

PERT & CPM StepsPERT & 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)Activity on Node (AoN)

2

4? Years

EnrollReceive diploma

Project: Obtain a college degree (B.S.)

1 month

Attend class, study etc.

1

1 day

3

Activity on Arc (AoA)Activity on Arc (AoA)

4,5 ? Years

Enroll

Receive diploma

Project: Obtain a college degree (B.S.)

1 month

Attend class, study,

etc.1

1 day

2 3 4

AoA Nodes have meaningAoA Nodes have meaning

GraduatingSenior

Applicant

Project: Obtain a college degree (B.S.)

1

Alum

2 3 4

Student

Liberal Arts Sidebar

• Alum = ? Alumnus

Alumna

Alumni

Alumnae

Alumni

Network ExampleNetwork Example

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

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

Network Example - AONNetwork Example - AON

A

C

E

F

BD

G

H

Z

Network Example - AOANetwork 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 AnalysisCritical 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 ExampleCritical 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 SolutionNetwork Solution

AA

EEDDBB

CC FF

GG

1

6 2 3

1

43

Earliest Start & Finish StepsEarliest 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

Activity AEarliest Start Solution

For starting activities, ES = 0.For starting activities, ES = 0.

AAEEDDBB

CC FF

GG

1

6 2 3

1

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 SolutionEarliest Start Solution

AAEEDDBB

CC FF

GG

1

6 2 3

1

43

Latest Start & Finish StepsLatest 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 SolutionEarliest Start Solution

AAEEDDBB

CC FFGG

1

6 2 31

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 SolutionLatest Finish Solution

AAEEDDBB

CC FF

GG

1

6 2 3

1

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 SlackCompute Slack

Critical PathCritical Path

AA

EEDDBB

CC FF

GG

1

6 2 3

1

43

New notation

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

• Compute, LF, LS, Right to Left

C 7C 7LS LF

ES EF

Exhibit 6

A 21A 21

E 5E 5D 2D 2B 5B 5

C 7C 7 F 8F 8

G 2G 2

Exhibit 6

A 21A 21

E 5E 5D 2D 2B 5B 5

C 7C 7 F 8F 8

G 2G 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 21A 21

E 5E 5D 2D 2B 5B 5

C 7C 7 F 8F 8

G 2G 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 1A 1

B 4B 4

C 3C 3

D 7D 7

E 6E 6

F 2F 2

H 9H 9

I 4I 4

G 7G 7

Solved Problem 2

A 1A 10 1

0 1

B 4B 41 5

1 5

C 3C 36 9

1 4

D 7D 72 9

1 8

E 6E 65 11

5 11

F 2F 29 11

8 10

H 9H 99 18

8 17

I 4I 418 22

18 22

G 7G 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)