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Critical Paths. Considering Critical Paths. When there are only a few tasks to complete in a project it is relatively easy to find the shortest time to complete the project. But as the number of tasks increases the problem becomes more difficult to solve by inspection alone.

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considering critical paths
Considering Critical Paths
  • When there are only a few tasks to complete in a project it is relatively easy to find the shortest time to complete the project.
  • But as the number of tasks increases the problem becomes more difficult to solve by inspection alone.
  • This is so important that in the 1950s the US government came up with PERT.
slide3
PERT
  • PERT is the Program Evaluation and Review Technique.
  • The goal of PERT is to identify the tasks that are most critical to the earliest completion of the project.
  • This path of targeted tasks from the start to the finish of a project became known as the critical path.
yearbook project
Yearbook Project
  • Remember the graph from doing the yearbook project.
  • How would we find 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 activity can begin if all the activities preceding it begin as early as possible.
calculating the est
Calculating the EST
  • To calculate the EST for each task, begin at the start and label each vertex with the smallest possible time that will be needed before the task can begin.
  • To find the time for consecutive steps add the times for the prerequisites.
slide6

Graph

(7)

(2)

(5)

3

2

2

1

2

0

1

C

D

F

3

1

1

5

Start

A

B

H

Finish

1

(0)

(1)

(15)

G

(12)

E

(7)

(2)

finding the est
Finding the EST
  • Notice that the label for C in the graph is found by adding the EST of B to the one day that it takes to complete task B (1+1= 2).
  • With G, however, G can not be completed until both predecessors, D and E, have been completed. Therefore, G can not begin until seven days have passed.
critical path
Critical Path
  • In the yearbook example, we can see that the earliest time in which the project can be completed is 15 days.
  • The time that it takes to complete all of the tasks in the project corresponds to the total time for the longest path from start to finish.
  • A path with this longest time is the desired critical path.
  • The critical path for our example would be Start-ABCDGH-Finish.
example
Example
  • Copy the graph and label the vertices with the EST for each task, and determine the earliest completion time for the project. The times are in minutes. Find the critical path.

B

D

7

3

1

A

G

0

3

6

3

Start

Finish

3

6

C

E

possible solutions
Possible Solutions
  • The solutions are as follows:

(3)

(10)

(0)

B

D

7

3

1

A

G

0

3

6

3

Start

Finish

3

(12)

(15)

6

C

E

(3)

(9)

possible solutions cont d
Possible Solutions (cont’d)
  • The earliest time that the project can be completed is 15 minutes.
  • Since the critical path is the longest path from the start to finish, the critical path is Start-ACEG-Finish.
shortening the project
Shortening the Project
  • If you would like to cut the completion time of a project, it can be done by shortening the length of the critical path, once you know what it is.
  • For example, in the example problem if we cut the time needed to complete task E to 2 minutes instead of 3 minutes, we reduce the EST from 15 minutes to 14 minutes.
practice problems
Practice Problems
  • Use the following graph to complete the table:

B

D

F

3

1

7

7

3

0

5

3

Start

G

Finish

A

5

7

4

C

E

practice problems cont d1
Practice Problems (cont’d)

In the next exercises (2 and 3), list the vertices of the graphs and give their earliest start time. Then determine the minimum project time and all of the critical paths.

practice problems cont d2
Practice Problems (cont’d)

2.

G

C

A

E

9

6

10

7

0

10

6

8

Start

0

Finish

10

5

6

8

B

F

D

H

practice problems cont d3
Practice Problems (cont’d)

3.

A

D

G

5

6

0

5

5

B

E

4

H

9

0

7

Finish

Start

8

0

C

F

I

8

8

10

practice problems cont d4
Practice Problems (cont’d)

4. From the table below, construct a graph to represent the information and label the vertices with their earliest-start time. Determine the minimum project time and the critical path.

practice problems cont d5
Practice Problems (cont’d)

5.

A

B

10

4

0

10

D

G

C

0

7

6

6

Start

Finish

E

F

0

8

6

practice problems cont d6
Practice Problems (cont’d)
  • Copy the graph, and label the vertices with the earliest-start time.
  • How quickly can the project be completed?
  • Determine the critical path.
  • What will happen to the minimum project time if task A’s time can be reduced to 9? To 8 days?
  • Will the project time continue to be affected by reducing the time of task A? Why or why not?
practice problems cont d7
Practice Problems (cont’d)
  • Construct a graph with three critical paths.
  • Determine the minimum project time and the critical path.

10

18

A

8

0

5

D

18

5

F

0

6

2

E

0

B

9

Finish

Start

G

C

practice problems cont d8
Practice Problems (cont’d)

8. In the graph below, the ESTs for the vertices are labeled and the critical path is marked.

B

D

6

8

4

(4)

(10)

A

2

0

5

G

(0)

Finish

Start

(4)

(9)

4

(18)

(20)

7

5

E

C

practice problems cont d9
Practice Problems (cont’d)
  • Task E can begin as early as day 9. If it begins on day 9, when will it be completed? If it begins on day 10? On day 11? What will happen if it begins on day 12?
  • To complete task E by day 18, the day on which task G is to begin, what is the latest day on which E can begin?
latest start time
Latest-Start Time
  • If an activity is not on the critical path, it is possible for it to start later than its earliest-start time.
  • The latest that a task can begin without delaying the project’s minimum completion time is known as the latest-start time (LST) for the task.
practice problems cont d10
Practice Problems (cont’d)
  • To find the LST for vertex C, the times of the two vertices (D and E) need to be considered. Since vertex D is on the critical path, 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, we found that the latest E can start is on day 11. In that case, what is the latest C can begin? From this information, what is the latest LST that can begin without delaying either task D or E?