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.
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
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
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.
Graph
(2) 3
(5)
2
1
0
Start
1
C
1
A
B
(0)
(1)
(7)
2
2
D
F
1
5
H
1
E
(2)
3
G
(7)
(12)
Finish
(15)
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
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 StartABCDGH-Finish.
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
B
D
7
path. 3
1
0
A
G
3
6
3
Start
Finish
3
6
C
E
Possible Solutions
The solutions are as follows:
(3)
(0)
B
3
0
(10)
7
D
1
A
G
3
6
3
Start
3
(12)
Finish
(15)
6
C
E
(3)
(9)
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
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
1. Use the following graph to complete the
table:
B
D
F
3
1
7
7
3
0
5
3
Start
G
A
5
7
C
4
E
Finish
Practice Problems (cont’d)
Vertex
Earliest-Start Time
A
B
C
D
0
7
E
F
G
Minimum project time =
Critical Path (s) =
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’d)
2.
A
0
G
C
6
10
E
9
7
10
6
8
Start
0
5
B
8
D
10
6
F
H
Finish
Practice Problems (cont’d)
3.
A
D
G
5
6
0
5
B
5
E 4
H
9
0
7
Finish
Start
0
8
C
F
I
8
8
10
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
0
A
2
None
B
4
None
C
3
A, B
D
1
A, B
E
5
C, D
F
6
C, D
G
7
E, F
Practice Problems (cont’d)
A
5.
B
10
4
0
10
0
C
D
7
G
6
6
Start
Finish
0
E
F
8
6
Practice Problems (cont’d)
Copy the graph, and label the vertices with the
earliest-start time.
b. How quickly can the project be completed?
c. Determine the critical path.
d. What will happen to the minimum project time
if task A’s time can be reduced to 9? To 8
days?
e. Will the project time continue to be affected by
reducing the time of task A? Why or why not?
a.
Practice Problems (cont’d)
6. Construct a graph with three critical
paths.
7. Determine the minimum project time and
the critical path.
10
A
0
5
8
D
18
5
0
0
18
F
6
B
2
E
9
Finish
Start
G
C
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
A
4
(4)
2
0
Start
(10)
5
(0)
4
(4)
G
(18)
(9)
7
C
5
E
Finish
(20)
Practice Problems (cont’d)
a. 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?
b. 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
If an activity is not on the critical path, it is
possible for it to start later than its earlieststart 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’d)
c. 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?