A Mathematical View of Our World

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A Mathematical View of Our World

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Title: A Mathematical View of Our World

1
A Mathematical View of Our World

1st ed.
Parks, Musser, Trimpe, Maurer, and Maurer

2
Chapter 7

Scheduling

3
Section 7.1Basic Concepts of Scheduling

Goals
Study project scheduling
Study tasks
Find finishing times
Study weighted digraphs
Study maximal paths
Find critical times

4
7.1 Initial Problem

A model railroad club is preparing for an open
house.
Estimate the minimum preparation time required.
The solution will be given at the end of the
section.

5
Tasks

A project consists of 2 or more smaller
activities called tasks.
A task cannot be broken into smaller jobs.
A task is done by 1 machine or 1 person.
The amount of time needed to complete a project
depends on
How many people or machines are available to do
the work.
How the tasks are assigned.

6
Example 1

A group of college students are making lasagna.
Divide the project into tasks.

7
Example 1, contd

Solution A possible list of tasks is given.

8
Tasks, contd

The people or machines performing the tasks are
called processors.
The amount of time it takes to perform a task is
called its completion time.

9
Tasks, contd

If one task must be completed before another task
can be started, we say the tasks have a
precedence relation or order requirement between
them.
Tasks that may be done in any order are called
independent.
One way to represent a precedence relation
between 2 tasks is to use points to denote the
tasks and an arrow to indicate the relationship.

10
Example 2

The tasks of putting on socks and putting on
shoes have a preference relation.

11
Example 2, contd

The tasks of putting on a hat and putting on a
coat do not have a preference relation. These
are independent tasks.

12
Question

Choose the list of tasks that has no preference
relations.
a. bake the pizza, grate some cheese, make pizza
sauce
b. fold some clothes, wash some clothes, hang
clothes on the line to dry
c. bake the cake, toss the green salad, set the
table
d. frost the cake, bake the cake, set the table

13
Digraphs

A collection of points (tasks) and straight or
curved arrows (precedence relations) connecting
them is called a directed graph or digraph.
A point in a digraph is called a vertex.
An arrow in a digraph is called an arc.

14
Example 3

Use a digraph to represent the precedence
relations for a man dressing to go for a run in
the summer.
The tasks consist of putting on
A Shirt
Shoes
Shorts
Socks

15
Example 3, contd

Solution Determine if there are any order
requirements among the tasks.
Socks must be put on before shoes.
It is much easier to put on shorts before shoes.
The shirt can be put on at any point in the
process.

16
Example 3, contd

Solution, contd A possible digraph is shown
below.

17
Digraphs, contd

A digraph that contains at least one precedence
relation is called an order-requirement digraph.
A vertex in a digraph with no arcs attached to it
is said to be isolated.

18
Digraphs, contd

Example
The digraph from the previous example problem is
an order-requirement digraph.
Vertex put on shirt is isolated.

19
Example 4

Construct an order-requirement digraph for the
tasks of
Making blackberry jam
Making blackberry cobbler
Picking blackberries

20
Example 4, contd

Solution
The berries must be picked first.
The jam and cobbler can be made in any order.

21
Example 5

Construct an order-requirement digraph for the
tasks of
Putting on socks
Putting on a sweatshirt
Putting on snow pants
Putting on a hat
Putting on boots
Strapping on a snowboard

22
Example 5, contd

Solution
The socks probably go on before the pants.
The pants go on before the boots.
The boots go on before the snowboard.
The sweatshirt goes on before the coat.
The hat can be put on at any point.

23
Example 5, contd

Solution, contd The digraph is shown below.

24
Paths

Any list of 2 or more vertices connected by
arrows such that the list goes in the same
direction as the arcs in the corresponding
digraph is called a directed path.
Because we consider only directed paths, we
simply call the list a path.

25
Example 6

Identify all possible paths in the
order-requirement digraph representing a simple
version of the project of replacing old brake
shoes.

26
Example 6, contd

Solution There are 6 possible paths.

27
Paths, contd

A path that cannot be extended by adding a vertex
at either end is called a maximal path.
In the previous example, the maximal paths were

28
Sources and Sinks

A vertex in a digraph that is at the end of no
arc is called a source.
A vertex in a digraph that is at the start of no
arc is called a sink.
An isolated vertex is both a source and a sink.
A maximal path starts at a source and ends at a
sink.

29
Sources and Sinks, contd

In this digraph from the previous example,
Vertex T1 is a source.
Vertex T4 is a sink.

30
Question

Identify all sources and sinks.
a. Sources put on socks Sinks put on hat
b. Sources put on hat, put on coat, strap on
snowboard Sinks put on socks, put on sweatshirt
c. Sources put on hat Sinks put on socks
d. Sources put on socks, put on sweatshirt
Sinks put on hat, put on coat, strap on
snowboard

31
Finding all Maximal Paths

To find all maximal paths in a digraph
Locate all the sinks that are not isolated
vertices.
Locate all the sources that are not isolated
vertices.

32
Finding Maximal Paths, contd

For each source, follow the arcs until you reach
a sink.
If there is more than one arc at any vertex along
the path formed in Step 3, start again at the
same source and choose a different route.

33
Question

Which path listed below is not a maximal path?
a. put on hat
b. put on pants, put on boots
c. put on sweatshirt, put on coat
d. put on socks, put on pants, put on boots,
strap on snowboard

34
Example 7

Identify all maximal paths in the digraph.

35
Example 7, contd

Solution
There is only one sink T6
There is only one source T1

36
Example 7, contd

Solution, contd
A maximal path from T1 to T6 is
Another maximal path from T1 to T6 is

37
Example 8

Identify all maximal paths in the digraph.

38
Example 8, contd

Solution
There is only one sink T10

39
Example 8, contd

Solution, contd
There are 4 sources T1, T4 , T6, and T8.

40
Example 8, contd

Solution, contd
There are 4 maximal paths

41
Weighted Digraphs

The completion time associated with a vertex
(task) is the weight of the vertex.
A digraph with weights at each vertex is called a
weighted digraph.
For example This weighted digraph shows that it
takes 20 seconds to put on sock and 25 seconds to
put on shoes.

42
Example 9

Construct a weighted digraph.

43
Example 9, contd

Solution

44
Project Finishing Time

The time from the beginning of a project until
the end of the project is called the finishing
time for the project.
The finishing time depends on
The task completion times.
The number of processors available.
How the processors are scheduled.
Whether or not some tasks can be done at the same
time.

45
Example 10

If only one processor works on the lasagna
project, what is the finishing time?

46
Example 10, contd

Solution We assume the processor can only
perform one task at a time.
Add up all the task completion times 10 5
30 10 10 2 6 7 8 30 118
minutes.
It will take one processor almost 2 hours to
finish the project.

47
Weighted Digraphs, contd

The weight of a path in a weighted
order-requirement digraph is the sum of the
weights at the vertices on the path.
A critical path in an order-requirement digraph
is a path having the largest possible weight.
The weight of a critical path is called the
critical time for the project.

48
Weighted Digraphs, contd

Any critical path must be a maximal path.
To find the critical paths
Find all the maximal paths.
Select the maximal path(s) with the largest
weight.

49
Example 11

Find a critical path in the lasagna project
digraph.

50
Example 11, contd

Solution Previously we found 4 maximal paths.

51
Example 11, contd

Solution, contd

52
Example 11, contd

Solution, contd The largest weight for any
maximal path is 83.
The one critical path is
The critical time for the project is 83 minutes.
It will take a minimum of 83 minutes, with enough
processors, to make the lasagna.
This example illustrates the fact that the
finishing time of a project is always greater
than or equal to its critical time.

53
7.1 Initial Problem Solution

A model railroad club is preparing for an open
house.
What is the minimum completion time?

54
Initial Problem Solution, contd

The first 6 tasks need to be done first and in
order.
Task 7 must be done before task 8.
Everything else must be done before task 10.

55
Initial Problem Solution, contd

A weighted digraph is created.

56
Initial Problem Solution, contd

Find all maximal paths, along with their weights.

57
Initial Problem Solution, contd

The critical time of the project is 52 hours.
The finishing time is greater than or equal to
the critical time, so it will take at least 52
hours to prepare for the open house.

58
Section 7.2List-Scheduling Algorithm

Goals
Study project scheduling
Study priority lists
Use the list-scheduling algorithm
Create and read Gantt charts
Study optimal schedules

59
7.2 Initial Problem

Suppose 2 students will make the lasagna in the
project examined in Section 7.1.
What is a good way for the 2 students to divide
the work and how long will it take?
The solution will be given at the end of the
section.

60
Priority Lists

A priority list is an ordered list of all the
tasks in a project.
When tasks are performed, precedence relations
override a priority list.
A priority list tells you which task to do first
among those you can do at a certain point in time.

61
Example 1

A woodworking project consists of 7 tasks.

62
Example 1, contd

The weighted digraph above represents the
project, with completion times in minutes.
Determine the critical time.
Create a priority list based on completion time,
from shortest to longest.

63
Example 1, contd

Solution The 3 maximal paths, and their weights,
are listed.

64
Example 1, contd

Solution, contd The largest weight of a maximal
path in this case is 90 minutes.
The critical time is 90 minutes.

65
Example 1, contd

Solution, contd The tasks are placed in a
priority list according to their completion
times.
The priority list is T1, T5, T2, T3, T6, T7, and
T4.

66
Priority Lists, contd

A priority list in which tasks are arranged from
shortest completion time to longest completion
time is called an increasing-time priority list.
A priority list in which tasks are arranged from
longest completion time to shortest completion
time is called a decreasing-time priority list

67
Question

Create the decreasing-time priority list for the
project whose digraph is shown below.
a.
b.
c.
d.

68
Example 2

How would one processor finish the woodworking
project from the previous example using the
increasing-time priority list T1, T5, T2, T3, T6,
T7, and T4?

69
Example 2, contd

Solution Use the precedence relations and the
priority list to determine the order of the
tasks.
Task T1 is a source and is first on the list, so
it will be done first.
Next, either T2 or T3 could be done. Choose T2
because it is first in the list.
At this point, the tasks that can be done are T3
or T5. According to the priority list, choose T5.

70
Example 2, contd

Solution, contd The order of the rest of the
tasks is determined in the same manner.
The priority list is T1, T5, T2, T3, T6, T7, T4.
The order in which the tasks will be completed is
T1, T2, T5, T3, T6, T4, T7 .

71
Gantt Charts

We can keep track of the order in which tasks are
done and how long it takes using a diagram called
a Gantt chart.
Numbers at the bottom of the chart show the
elapsed time.
The boxes represent the tasks.

72
Gantt Charts, contd

A Gantt chart for the woodworking project in the
previous example is shown below.

73
Definitions

A task T will be called ready if all tasks
required to begin T have been completed.
A processor that has not yet been assigned a task
and is not working on a task will be called idle.
At the beginning of a project we assume all
processors are idle and at least 1 task is ready.

74
List-Processing Algorithm

The lowest-numbered idle processor is assigned to
the highest-priority ready task until either all
processors are assigned or all ready tasks have
been assigned.

75
List-Processing Algorithm, contd

When a processor completes a task, that processor
becomes idle. One of 3 cases will occur.
If there are ready tasks, repeat Step 1.
If there are no ready tasks but not every task
has been done, the processor remains idle until
more tasks are completed.
If all tasks have been completed, the project is
finished.

76
Example 3

Apply the algorithm to the wood-working project
using 2 processors and the increasing-time
priority list T1, T5, T2, T3, T6, T7, T4.

77
Example 3, contd

Solution Step 1
P1 is assigned to T1.
P2 remains idle.

78
Example 3, contd

Solution, contd At 5 minutes, Step 2, Case A.
Ready T2, T3.
Not completed T2, T3, T4, T5, T6, T7.
Priority list T1, T5, T2, T3, T6, T7, T4.

79
Example 3, contd

Solution, contd Step 1
P1 and P2 are idle.
P1 is assigned to T2.
P2 is assigned to T3.

80
Example 3, contd

Solution, contd At 20 minutes, Step 2, Case A.
Ready T5.
Not completed T3, T4, T5, T6, T7.
Priority list T1, T5, T2, T3, T6, T7, T4.

81
Example 3, contd

Solution, contd Step 1
P1 is idle.
P1 is assigned to T5.

82
Example 3, contd

Solution, contd At 25 minutes, Step 2, Case A.
Ready T4, T6.
Not completed T4, T5, T6, T7.
Priority list T1, T5, T2, T3, T6, T7, T4.

83
Example 3, contd

Solution, contd Step 1
P2 is idle.
P2 is assigned to T6.

84
Example 3, contd

Solution, contd At 30 minutes, Step 2, Case A.
Ready T4.
Not completed T4, T6, T7.
Priority list T1, T5, T2, T3, T6, T7, T4.

85
Example 3, contd

Solution, contd Step 1
P1 is idle.
P1 is assigned to T4.

86
Example 3, contd

Solution, contd At 50 minutes, Step 2, Case B.
Ready none.
Not completed T4, T7.
Priority list T1, T5, T2, T3, T6, T7, T4.

87
Example 3, contd

Solution, contd Step 1
P2 remains idle.

88
Example 3, contd

Solution, contd At 65 minutes, Step 2, Case A.
Ready T7.
Not completed T7.
Priority list T1, T5, T2, T3, T6, T7, T4.

89
Example 3, contd

Solution, contd Step 1
P1 and P2 are idle.
P1 is assigned to T7.

90
Example 3, contd

Solution, contd At 95 minutes, Step 2, Case C.
Ready none.
Not completed none.
The finishing time with 2 processors is 95
minutes.
This is less than 140 minutes for 1 processor,
but more than the critical time of 90 minutes.

91
Example 4

Apply the algorithm to the wood-working project
using 3 processors and the increasing-time
priority list T1, T5, T2, T3, T6, T7, T4.

92
Example 4, contd

Solution Step 1
P1 is assigned to T1.
P2 and P3 remain idle.

93
Example 4, contd

Solution, contd At 5 minutes Step 2 and Step 1
P1, P2 and P3 are idle.
T2 and T3 are ready.
P1 is assigned to T2.
P2 is assigned to T3.
P3 remains idle.

94
Example 4, contd

Solution, contd At 20 minutes Step 2 and Step
1
P1 and P3 are idle.
T5 is ready.
P1 is assigned to T5.
P3 remains idle.

95
Example 4, contd

Solution, contd At 25 minutes Step 2 and Step
1
P2 and P3 are idle.
T6 and T4 are ready.
P2 is assigned to T6.
P3 is assigned to T4.

96
Example 4, contd

Solution, contd At 30 minutes Step 2 and Step
1
P1 is idle.
No tasks are ready.
P1 remains idle.

97
Example 4, contd

Solution, contd At 50 minutes Step 2 and Step
1
P1 and P2 are idle.
No tasks are ready.
P1 and P2 remain idle.

98
Example 4, contd

Solution, contd At 60 minutes Step 2 and Step
1
P1, P2 and P3 are idle.
T7 is ready.
P1 is assigned to T7.

99
Example 4, contd

Solution, contd At 90 minutes Step 2 and Step
1
P1, P2 and P3 are idle.
No tasks are ready.
All tasks have been completed.

100
Example 4, contd

The completion time with 3 processors is 90
minutes, the critical time.
The project completion is summarized in the Gantt
chart.

101
Example 5

Apply the algorithm to the wood-working project
using 2 processors and the decreasing-time
priority list T4, T7, T6, T3, T2, T5, T1.

102
Example 5, contd

Solution Step 1
P1 is assigned to T1.
P2 remains idle.

103
Example 5, contd

Solution, contd At 5 minutes Step 2 and Step 1
P1 and P2 are idle.
T2 and T3 are ready.
P1 is assigned to T3.
P2 is assigned to T2.

104
Example 5, contd

Solution, contd At 20 minutes Step 2 and Step
1
P2 is idle.
T5 is ready.
P2 is assigned to T5.

105
Example 5, contd

Solution, contd At 25 minutes Step 2 and Step
1
P1 is idle.
T4 and T6 are ready.
P1 is assigned to T4.

106
Example 5, contd

Solution, contd At 30 minutes Step 2 and Step
1
P2 is idle.
T6 is ready.
P2 is assigned to T6.

107
Example 5, contd

Solution, contd At 55 minutes Step 2 and Step
1
P2 is idle.
No tasks are ready.
P2 remains idle.

108
Example 5, contd

Solution, contd At 60 minutes Step 2 and Step
1
P1 and P2 are idle.
T7 is ready.
P1 is assigned to T7.
P2 remains idle.

109
Example 5, contd

Solution, contd At 90 minutes Step 2 and Step
1
P1 and P2 are idle.
No tasks are ready.
All tasks are completed.
The finishing time with the decreasing-time
priority list and 2 processors is 90 minutes.

110
Example 5, contd

Solution, contd

111
Question

Given the priority list
and the digraph for the project, which tasks
will be assigned first if there are 3 processors?

112
Question contd

a. Processor 1 Task 4 Processor 2 Task 1
Processor 3 Task 8
b. Processor 1 Task 1 Processor 2 Task 4
Processor 3 Task 6
c. Processor 4 Task 1 Processor 2 Task 1
Processor 3 Task 6
d. Processor 1 Task 1 Processor 2 Task 4
Processor 3 Task 8

113
Optimal Schedules

An optimal schedule is a schedule assigning
processors to tasks in such a way that it results
in the shortest possible finishing time for that
project with that number of processors.

114
Optimal Schedules, contd

If the finishing time is equal to the critical
time, we know the schedule is optimal.
If the finishing time is more than the critical
time, the schedule may or may not be optimal.
Typically, the decreasing-time priority list
gives a better finishing time than does the
increasing-time priority list.

115
Decreasing-Time Algorithm

Form a decreasing-time priority list.
Apply the list-processing algorithm.
If 2 or more tasks have the same completion time,
they may be chosen in any order.

116
7.2 Initial Problem Solution

Determine a schedule and the finishing time for
the lasagna project with 2 processors.

117
Initial Problem Solution, contd

Use a decreasing-time priority list T3, T10, T1,
T4, T5, T9, T8, T7, T2, T6.
The Gantt chart that results from applying the
algorithm is shown next.

118
Initial Problem Solution, contd

The finishing time for this project with 2
processors and a decreasing-time priority list is
90 minutes.
The critical time was 83 minutes.
This may or may not be an optimal schedule.

119
Section 7.3The Critical-Path Algorithm

Goals
Study the critical-path priority list algorithm
Create the critical-path priority list
Schedule tasks according to the critical-path
priority list

120
7.3 Initial Problem

Find an optimal schedule for the lasagna project
using 2 processors.
The solution will be given at the end of the
section.

121
Example 1

A digraph for the project of making sandwiches is
shown below.
Two processors will be assigned.

122
Example 1, contd

Apply the decreasing-time algorithm to create a
schedule and find the finishing time.
Find the critical path and the critical time.
Find the finishing time when one processor is
assigned to the tasks in the critical path and
one to the other tasks.

123
Example 1, contd

Solution The decreasing-time priority list is
T1, T2, T3, T4, T5, T6, T7.
The Gantt chart shows a finishing time of 39
minutes.

124
Example 1, contd

Solution The maximal paths and their weights are
shown below.
The critical time is 30 minutes.

125
Example 1, contd

Solution Instead of using a priority list, all
tasks in the critical path are assigned to one
processor.
The Gantt chart shows a finishing time of 30
minutes.

126
Critical-Path Priority List

List all the maximal paths and isolated vertices.
Find the greatest of all the weights of the
maximal paths and isolated vertices.
If the weight belongs to a maximal path, the task
at the head of the path goes next in the priority
list.
If the weight belongs to an isolated vertex, that
task goes next in the priority list.

127
Critical-Path Priority List, contd

Remove the task selected in Step 2 and all
attached edges from the digraph.
Using the new digraph, repeat Step 1 and Step 2.

128
Example 2

Find the critical-path priority list.

129
Example 2, contd

Solution The maximal paths are listed. There
are no isolated vertices.
The greatest weight is 42, so the first task in
the list is T2.

130
Example 2, contd

Solution, contd The task and all attached edges
are removed from the digraph.

131
Example 2, contd

Solution, contd The maximal paths are listed.
There are no isolated vertices.
The greatest weight is 38, so the next task in
the list is T1.

132
Example 2, contd

Solution, contd Remove T1 and create a new
digraph and table.
The greatest weight is 30, so the next task in
the list is T5.

133
Example 2, contd

Solution, contd Remove T5 and create a new
digraph and table.
The greatest weight is 22, so the next task in
the list is T6.

134
Example 2, contd

Solution, contd Remove T6 and create a new
digraph and table.
The greatest weight is 20, so the next task in
the list is T3.

135
Example 2, contd

Solution, contd Remove T3 and create a new
digraph and table.
The greatest weight is 15, so the next task in
the list is T7.

136
Example 2, contd

Solution, contd Remove T7 and create a new
digraph and table.
The greatest weight is 11, so the next task in
the list is T4.

137
Example 2, contd

Solution, contd Remove T4 and create a new
digraph and table.
The next task in the list is T8, followed by T9.
The priority list is T2, T1, T5, T6, T3, T7, T4,
T8, T9.

138
Question

Find the critical-path priority list for the
project.
a.
b.
c.
d.

139
Critical-Path Scheduling Algorithm

Determine the critical-path priority list.
Apply the list-processing algorithm.

140
Question

Suppose the critical-path algorithm is being
used to schedule a project using 3 processors.
The priority list is . At 2 minutes, the
status is that Processors 1 and 2 are both
working on task, while Processor 3 is idle. What
should happen at this point?
a. Processor 3 is assigned
Task 8.
b. Processor 3 is assigned
Task 2.
c. Processor 3 is assigned
Task 5.
d. Processor 3 remains idle.

141
Example 3

Apply the critical-path scheduling algorithm
using 2 processors.

142
Example 3, contd

Solution The critical-path priority list was
found in the last example T2, T1, T5, T6, T3,
T7, T4, T8, T9.
The Gantt chart is shown below.

143
Example 3, contd

Solution, contd The finishing time with 2
processors using the critical-path priority list
is 42 minutes.
There were 2 periods of time during which
Processor 2 had to remain idle.

144
7.3 Initial Problem Solution