critical paths
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
Critical PathsCritical Paths
Considering Critical PathsConsidering Critical Paths
When there are only a few tasks to When there are only a few tasks to complete in a project it is relatively easy to complete in a project it is relatively easy to find the shortest time to complete the find the shortest time to complete the project.project.But as the number of tasks increases the But as the number of tasks increases the problem becomes more difficult to solve by problem becomes more difficult to solve by inspection alone.inspection alone.This is so important that in the 1950s the This is so important that in the 1950s the US government came up with PERT.US government came up with PERT.
PERTPERT
PERT is the Program Evaluation and PERT is the Program Evaluation and Review Technique.Review Technique.
The goal of PERT is to identify the tasks The goal of PERT is to identify the tasks that are most critical to the earliest that are most critical to the earliest completion of the project.completion of the project.
This path of targeted tasks from the start This path of targeted tasks from the start to the finish of a project became known as to the finish of a project became known as the the critical path.critical path.
Yearbook ProjectYearbook Project
Remember the graph from doing the Remember the graph from doing the yearbook project. yearbook project. How would we find the critical path for this How would we find the critical path for this project?project?To do this, an To do this, an earliest-start time (EST)earliest-start time (EST) for each task must be found.for each task must be found.The EST is the earliest that an activity can The EST is the earliest that an activity can begin if all the activities preceding it begin begin if all the activities preceding it begin as early as possible.as early as possible.
Calculating the ESTCalculating the EST
To calculate the EST for each task, begin To calculate the EST for each task, begin at the start and label each vertex with the at the start and label each vertex with the smallest possible time that will be needed smallest possible time that will be needed before the task can begin. before the task can begin.
To find the time for consecutive steps add To find the time for consecutive steps add the times for the prerequisites.the times for the prerequisites.
GraphGraph
StartA B
C
E
DF
G
H Finish
0 11
3 2
22
3
1
1
51
(0) (1)
(2)(7)
(2) (5) (7)
(12)(15)
Finding the ESTFinding the EST
Notice that the label for C in the graph is Notice that the label for C in the graph is found by adding the EST of B to the one found by adding the EST of B to the one day that it takes to complete task B (1+1= day that it takes to complete task B (1+1= 2).2).
With G, however, G can not be completed With G, however, G can not be completed until both predecessors, D and E, have until both predecessors, D and E, have been completed. Therefore, G can not been completed. Therefore, G can not begin until seven days have passed.begin until seven days have passed.
Critical PathCritical Path
In the yearbook example, we can see that the In the yearbook example, we can see that the earliest time in which the project can be earliest time in which the project can be completed is 15 days. completed is 15 days. The time that it takes to complete all of the tasks The time that it takes to complete all of the tasks in the project corresponds to the total time for in the project corresponds to the total time for the longest path from start to finish.the longest path from start to finish.A path with this longest time is the desired A path with this longest time is the desired critical path.critical path.The critical path for our example would be Start-The critical path for our example would be Start-ABCDGH-Finish.ABCDGH-Finish.
ExampleExample
Copy the graph and label the vertices with Copy the graph and label the vertices with the EST for each task, and determine the the EST for each task, and determine the earliest completion time for the project. earliest completion time for the project. The times are in minutes. Find the critical The times are in minutes. Find the critical path.path.
Start
A
B
C
D
E
G
Finish
03
3
7
6
6
3
13
Possible SolutionsPossible Solutions
The solutions are as follows:The solutions are as follows:
Start
A
B
C
D
E
G
Finish
03
3
7
6
6
3
13
(0)
(3) (10)
(3) (9)
(12)
(15)
Possible Solutions (cont’d)Possible Solutions (cont’d)
The earliest time that the project can be The earliest time that the project can be completed is 15 minutes.completed is 15 minutes.
Since the critical path is the longest path Since the critical path is the longest path from the start to finish, the critical path is from the start to finish, the critical path is Start-ACEG-Finish.Start-ACEG-Finish.
Shortening the ProjectShortening the Project
If you would like to cut the completion time If you would like to cut the completion time of a project, it can be done by shortening of a project, it can be done by shortening the length of the critical path, once you the length of the critical path, once you know what it is.know what it is.
For example, in the example problem if we For example, in the example problem if we cut the time needed to complete task E to cut the time needed to complete task E to 2 minutes instead of 3 minutes, we reduce 2 minutes instead of 3 minutes, we reduce the EST from 15 minutes to 14 minutes.the EST from 15 minutes to 14 minutes.
Practice ProblemsPractice Problems
1.1. Use the following graph to complete the Use the following graph to complete the table:table:
Start
A
C E
B D F
G Finish
0
7
7
4
3
3 1
5
5
7
3
Practice Problems (cont’d)Practice Problems (cont’d)
VertexVertex Earliest-Start TimeEarliest-Start Time
AA 00
BB 77
CC
DD
EE
FF
GG
Minimum project time = Minimum project time =
Critical Path (s) =Critical Path (s) =
Practice Problems (cont’d)Practice Problems (cont’d)
In the next exercises (2 and 3), list the In the next exercises (2 and 3), list the vertices of the graphs and give their vertices of the graphs and give their earliest start time. Then determine the earliest start time. Then determine the minimum project time and all of the critical minimum project time and all of the critical paths.paths.
Practice Problems (cont’d)Practice Problems (cont’d)
2. 2.
Start
0A
6C
10E 9
G
7
Finish0
B5
D
8
10
F8
6
6H
10
Practice Problems (cont’d)Practice Problems (cont’d)
3. 3.
Start Finish
0
0
0
6
9
8
8
5
5
4
10
5
7
8
A
B
C
D
E
F
G
H
I
Practice Problems (cont’d)Practice Problems (cont’d)
4. From the table below, construct a graph to represent the 4. From the table below, construct a graph to represent the information and label the vertices with their earliest-start information and label the vertices with their earliest-start time. Determine the minimum project time and the time. Determine the minimum project time and the critical path.critical path. TaskTask TimeTime PrerequisitesPrerequisites
StartStart 00
AA 22 NoneNone
BB 44 NoneNone
CC 33 A, BA, B
DD 11 A, BA, B
EE 55 C, DC, D
FF 66 C, DC, D
GG 77 E, FE, F
Practice Problems (cont’d)Practice Problems (cont’d)
5. 5.
Start Finish
6
8
6
4
6
7
10
10
0
0
0
AB
GC D
E F
Practice Problems (cont’d)Practice Problems (cont’d)
a.a. Copy the graph, and label the vertices with the Copy the graph, and label the vertices with the earliest-start time.earliest-start time.
b.b. How quickly can the project be completed?How quickly can the project be completed?
c.c. Determine the critical path.Determine the critical path.
d.d. What will happen to the minimum project time What will happen to the minimum project time if task A’s time can be reduced to 9? To 8 if task A’s time can be reduced to 9? To 8 days?days?
e.e. Will the project time continue to be affected by Will the project time continue to be affected by reducing the time of task A? Why or why not?reducing the time of task A? Why or why not?
Practice Problems (cont’d)Practice Problems (cont’d)
6.6. Construct a graph with three critical Construct a graph with three critical paths.paths.
7.7. Determine the minimum project time and Determine the minimum project time and the critical path.the critical path.
StartFinish
0
0
0
10
5
5
9
18
18
6
8
2
A
B
C
D
E
F
G
Practice Problems (cont’d)Practice Problems (cont’d)
8. In the graph below, the ESTs for the 8. In the graph below, the ESTs for the vertices are labeled and the critical path is vertices are labeled and the critical path is marked.marked.
Start Finish
A
B D
G
C E
0
4
6
8
2
7
5
4
5
(0)
(4) (10)
(18) (20)(4) (9)
Practice Problems (cont’d)Practice Problems (cont’d)
a.a. Task E can begin as early as day 9. If it Task E can begin as early as day 9. If it begins on day 9, when will it be begins on day 9, when will it be completed? If it begins on day 10? On completed? If it begins on day 10? On day 11? What will happen if it begins on day 11? What will happen if it begins on day 12?day 12?
b.b. To complete task E by day 18, the day To complete task E by day 18, the day on which task G is to begin, what is the on which task G is to begin, what is the latest day on which E can begin?latest day on which E can begin?
Latest-Start TimeLatest-Start Time
If an activity is not on the critical path, it is If an activity is not on the critical path, it is possible for it to start later than its earliest-possible for it to start later than its earliest-start time. start time.
The latest that a task can begin without The latest that a task can begin without delaying the project’s minimum completion delaying the project’s minimum completion time is known as thetime is known as the latest-start time latest-start time (LST)(LST) for the task. for the task.
Practice Problems (cont’d)Practice Problems (cont’d)
c.c. To find the LST for vertex C, the times of the To find the LST for vertex C, the times of the two vertices (D and E) need to be considered. two vertices (D and E) need to be considered. Since vertex D is on the critical path, the latest Since vertex D is on the critical path, the latest it can start is day 10. For D to begin on time, it can start is day 10. For D to begin on time, what is the latest day on which C can begin? what is the latest day on which C can begin? In part b, we found that the latest E can start is In part b, we found that the latest E can start is on day 11. In that case, what is the latest C on day 11. In that case, what is the latest C can begin? From this information, what is the can begin? From this information, what is the latest LST that can begin without delaying latest LST that can begin without delaying either task D or E?either task D or E?