It is relatively easy to find the shortest time needed to complete a projectif the project consists of only a few activities. But as the tasks increase innumber, the problem becomes more difficult to solve by inspection alone.

In the 1950s the U.S. government was faced with the need tocomplete very complex systems such as the U.S. Navy Polaris Submarineproject. In order to do this efficiently, a method was developed calledPERT (Program Evaluation and Review Technique). This techniquetargeted tasks that were critical to the earliest completion of the project.The path of targeted tasks from the start to the finish of the projectbecame known as the critical path.

Recall the graph in Lesson 4.1 that represented the Central Highyearbook project. How might you go about finding a systematic way toidentify the critical path for this project? To do this, an earliest-start time(EST) for each task must be found. The EST is the earliest that an activitycan begin if all the activities preceding it begin as early as possible.

To calculate the EST for each task, begin at the start. Then label eachvertex with the smallest possible time that is needed before the task canbegin. The label for C in Figure 4.3 is found by adding the EST of B to the1 day that it takes to complete task B(1 + 1 = 2). Task G cannot becompleted until both predecessors, D and E, have been completed. Hence,G cannot begin until 7 days have passed.

In the case of the yearbook staff, the earliest time in which theproject can be completed is 15 days. As paradoxical as it may seem, theleast amount of time that it takes to complete all of the tasks in the projectcorresponds to the time it takes to complete the longest path through thegraph from start to finish.

Lesson 4.2

0

3

A

Critical Paths

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176 Chapter 4 • Graphs and Their Applications

A path with this longest time is the desired critical path. In Figure4.3, the critical path is Start-ABCDGH-Finish.

Figure 4.3. Yearbook diagram showing the earliest-start time for each task.

Example1. Copy the graph and label the vertices with the EST for each task.

Then determine the earliest completion time for the project. Alltimes are in minutes.

2. Identify the critical path.

Solution:

1.

(1)

(2) (5)(7)

Start Finish0 1

1

1 1

1 3

2

22

5(0) (12)

(7)(2)

(15)

3

A B

EG

H

C DF

Start Finish0

3

3

3

3

1

7

6

6

B D

A

C

G

E

(3)

(0)

Start

(10)

(3)

Finish0

3

7

6

3

6

3

1

3

(15)(12)

(9)

A

C E

G

DB

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177Lesson 4.2 • Critical Paths

The earliest time in which the projectcan be completed is 15 minutes.

2. Since the critical path is the longestpath from the start to the finish, thecritical path is Start-ACEG-Finish.

If it is desirable to cut the completiontime of a project, it can be done by shorteningthe length of the critical path once it is found.In the preceding example, one way to shortenthe time it takes to complete the project is tocut the time it takes to complete task E. If taskE’s time is cut from 3 minutes to 2 minutes,the completion time for the project is cut to 14 minutes.

Exercises1.

Complete the following.

Vertex Earliest-Start Time

A 0B 7CDEFG

Minimum project time = Critical path(s) =

Start Finish0

7

7

4

3

3

1

5

5

7

3

A

B

C

D

E

F

G

The efficient management of largeprojects like the construction of abuilding requires the use of criticalpath analysis.

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178 Chapter 4 • Graphs and Their Applications

In Exercises 2 and 3, list the vertices of the graphs and give theirearliest-start time, as in Exercise 1. Determine the minimum project timeand all of the critical paths.

2.

3.

4. Using the information from the following table, construct a graphand label each of the vertices with its earliest-start time. Determinethe minimum project time and critical path.

Task Time Prerequisites

Start 0 ___

A 2 NoneB 4 NoneC 3 A, BD 1 A, BE 5 C, DF 6 C, DG 7 E, F

5 8

Start Finish

0

0

6

8

10

10 9

7

6

6

10

C EA

B D F

G

H

Start Finish

0

0

0

3

5

42

2

2

1

2

3

3

2

4

6

5

A

B

C

D

EI

F

G

H

J

L

K

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179Lesson 4.2 • Critical Paths

5.

a. Copy the graph and label each vertex with its earliest-start time.

b. How quickly can the project be completed?

c. Determine the critical path.

d. What happens to the minimum project time if task A’s time isreduced to 9 days? To 8 days?

e. Will the project time continue to be affected by reducing thetime of task A? Explain why or why not.

6. Construct a graph with three critical paths.

7. Determine the minimum project time and the critical path for thefollowing graph.

Start Finish

10

10

67

4

6

86

0

0

0

A

C

E

B

D

F

G

Start Finish

0

0

0

10

5

5

9

6

18

18

2

8

A

B

C

D

E

G

F

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180 Chapter 4 • Graphs and Their Applications

8. Task Time Prerequisites

Start 0 ___

A 13 NoneB 10 NoneC 4 AD 8 BE 6 BF 7 C, D, EG 5 FH 8 F

Finish

a. Draw a graph using the information in the table.

b. Label each vertex with its earliest-start time.

c. Determine the minimum project time.

d. Determine the critical path(s).

9. In the following graph, each vertex has been label with its EST, andthe critical path is marked.

a. Task E can begin as early as day 9. If it begins on day 9, whenwill it be completed? If it begins on day 10? On day 11? Whatwill happen if it begins on day 12?

b. What is the latest day on which task E can begin if task G is tobegin on day 18?

If an activity is not on the critical path, it is possible for it to start laterthan its earliest-start time and not delay the project. The latest a task canbegin without delaying the project’s minimum completion time is known asthe latest-start time (LST) for the task. For example, the LST for E is day 11.

(10)(4)

(0)

(4)

40

4

Start

(9)

(18)

6

5

5

8

7

2

(20)Finish

C

B

A

E

D

G

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181Lesson 4.2 • Critical Paths

c. In order to find the LST for vertex C, the times of the two verticesD and E need to be considered. Since vertex D is on the criticalpath, the latest it can start is day 10. For D to begin on time,what is the latest day on which C can begin? In part b, youfound that the latest E can start is day 11. In that case, what isthe latest C can begin? From this information, what is the latest(LST) that C can begin without delaying either task D or E?

10. To find the LST for each task, it is necessary to begin with the Finishand work through the graph in reverse order to the Start. Each ofthe vertices in the following graph is labeled with its EST. The LSTsfor several of the tasks have been calculated and are shown belowthe ESTs on the vertices. Find the LSTs for the remaining tasks.

11. Write an algorithm to find the LSTs for the tasks in a graph. Testyour algorithm on the graph in Exercise 1.

Project12. Interview the yearbook sponsors in your school to find out how

they organize the publication of your school’s yearbook. Create atask table that shows the approximate times and prerequisite tasksthat must be completed before your yearbook can go to thepublisher. Design a graph with the EST for each task, and identifythe critical path.

Modeling Project13. Use the Internet or other sources to research and report on

businesses or people who use PERT or similar evaluation techniquessuch as Gantt Charts to model project planning.

5 8 6

16(0) (6)10 9

( ) ( )

( )( )

(0)

0

0

7

10

Start(32)

(5) 1516

2122

2525

16

Finish

6

9

76

HB

A

D

C

F

E G

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