1. 3.1 Earliest Start Times

2. 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.

3. 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.

4. EST How would we find the critical path for a 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.

5. 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.

6. Graph (7) (2) (5) 3 2 2 1 2 1 C D F 3 1 1 5 Start A B H Finish 1 (0)1 C D F 3 1 1 5
Start A B H
Finish 1
(0)
(1)
(15) G
(12) E
(7)
(2)

7. Finding the EST
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.

8. Critical Path
In the 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.

9. 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 3 6 3
Start
Finish 3 6 C E

10. Possible Solutions The solutions are as follows: (3) (10) (0) B D 7 3G 3 6 3
Start
Finish 3
(12)
(15) 6 C E
(3)
(9)

11. 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.

12. Practice Problems Use the following graph to complete the table: B D F3 1 7 7 3 5 3
Start G
Finish A 5 7 4 C E

13. Practice Problems (cont’d)
Vertex
Earliest-Start Time A B C D E F G
Minimum project time =
Critical Path (s) =

14. Practice Problems (cont’d)
Vertex
Earliest-Start Time A B 7 C D E F G
Minimum project time =
Critical Path (s) =

15. Practice Problems (cont’d)
Vertex
Earliest-Start Time A B 7 C D E F G
Minimum project time =
Critical Path (s) =

16. Practice Problems (cont’d)
Vertex
Earliest-Start Time A B 7 C D 10 E F G
Minimum project time =
Critical Path (s) =

17. Practice Problems (cont’d)
Vertex
Earliest-Start Time A B 7 C D 10 E 11 F G
Minimum project time =
Critical Path (s) =

18. Practice Problems (cont’d)
Vertex
Earliest-Start Time A B 7 C D 10 E 11 F 16 G
Minimum project time =
Critical Path (s) =

19. Practice Problems (cont’d)
Vertex
Earliest-Start Time A B 7 C D 10 E 11 F 16 G 23
Minimum project time =
Critical Path (s) =

20. Practice Problems (cont’d)
Vertex
Earliest-Start Time A B 7 C D 10 E 11 F 16 G 23
Minimum project time = 26
Critical Path (s) =

21. Practice Problems (cont’d)
Vertex
Earliest-Start Time A B 7 C D 10 E 11 F 16 G 23
Minimum project time = 26
Critical Path (s) = ACEFG

22. 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 a critical path.

23. Practice Problems (cont’d)2. G C A E 9 6 10 7 10 6 8
Start
Finish 10 5 6 8 B F D H

24. Practice Problems (cont’d)3. A D G 5 6 5 5 B E 4 H 9 7
Finish
Start 8 C F I 8 8 10

25. 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.
Task
Time
Prerequisites
Start A 2
None B 4 C 3
A, B D 1 E 5
C, D F 6 G 7
E, F

26. Practice Problems (cont’d)5. A B 10 4 10 D G C 7 6 6
Start
Finish E F 8 6

27. Practice Problems (cont’d)
How quickly can the project be completed?
Determine the critical path.

28. Practice Problems (cont’d)
Determine the minimum project time and the critical path. 10 18 A 8 5 D 18 5 F 6 2 E B 9
Finish
Start G C

29. 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 5 G
(0)
Finish
Start
(4)
(9) 4
(18)
(20) 7 5 E C