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

11
Example 2, contd
  1. 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
  1. For each source, follow the arcs until you reach
    a sink.
  2. 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
  1. 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
  1. Apply the decreasing-time algorithm to create a
    schedule and find the finishing time.
  2. Find the critical path and the critical time.
  3. 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
  1. Determine the critical-path priority list.
  2. 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
  • Two processors will complete the lasagna project.

145
Initial Problem Solution, contd
  • We will use the critical-path priority list.
  • The maximal paths are listed, with their weights.
  • The greatest weight is 83, so the first task in
    the list is T1.

146
Initial Problem Solution, contd
  • Remove T1.
  • The maximal paths are listed, with their weights.
  • The greatest weight is 73, so the next task in
    the list is T2.

147
Initial Problem Solution, contd
  • Remove T2.
  • The maximal paths are listed, with their weights.
  • The greatest weight is 68, so the next task in
    the list is T3.

148
Initial Problem Solution, contd
  • Remove T3.
  • The maximal paths are listed, with their weights.
  • The greatest weight is 58, so the next task in
    the list is T4.

149
Initial Problem Solution, contd
  • Remove T4.
  • The maximal paths are listed, with their weights.
  • The greatest weight is 48, so the next task in
    the list is T5.

150
Initial Problem Solution, contd
  • Remove T5.
  • The maximal paths are listed, with their weights.
  • The greatest weight is 46, so the next task in
    the list is T6.

151
Initial Problem Solution, contd
  • Remove T6.
  • The maximal paths are listed, with their weights.
  • The greatest weight is 45, so the next task in
    the list is T8.

152
Initial Problem Solution, contd
  • Remove T8.
  • The maximal paths are listed, with their weights.
  • The task are assigned in the order T7, T9, T10.
  • The critical-path priority list is T1, T2, T3,
    T4, T5, T6, T8, T7, T9 , T10 .

153
Initial Problem Solution, contd
  • The Gantt chart is shown above.
  • The finishing time is equal to the critical time
    of 83 minutes, so we know the schedule is
    optimal.
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