GraphsGraphs

Their ApplicationsTheir Applications

NC Standard Course of NC Standard Course of StudyStudy

Competency Goal 1Competency Goal 1: The learner : The learner will use matrices and graphs to will use matrices and graphs to model relationships and solve model relationships and solve problems. problems.

Objective 1.02:Objective 1.02: Use graph theory to Use graph theory to model relationships and solve model relationships and solve problems. problems.

PlanningPlanning

How does a contractor organize all of the How does a contractor organize all of the jobs needed to complete a project?jobs needed to complete a project?

How do your parents manage to get all How do your parents manage to get all parts of a Thanksgiving dinner done at the parts of a Thanksgiving dinner done at the same time?same time?

Although day-to-day planning may be a Although day-to-day planning may be a simple activity, the planning of extensive simple activity, the planning of extensive projects may require very detailed projects may require very detailed preparation.preparation.

Organized PlanningOrganized Planning

Business owners Business owners usually can not usually can not function properly function properly when haphazard when haphazard planning is used.planning is used.

A more scientific, A more scientific, organized method organized method must be used.must be used.

ExampleExample

The Central High yearbook staff has The Central High yearbook staff has only 16 days left before the deadline only 16 days left before the deadline for completing their yearbook.for completing their yearbook.

They are running behind schedule They are running behind schedule and still have several tasks to finish.and still have several tasks to finish.

The tasks remaining and the times it The tasks remaining and the times it takes to do each task are listed in takes to do each task are listed in the following table.the following table.

Example (cont’d)Example (cont’d)

TaskTask TimeTime

A. Buy Film A. Buy Film 00

B. Load CamerasB. Load Cameras 1 day1 day

C. Take Photos of ClubsC. Take Photos of Clubs 33

D. Take Sports PhotosD. Take Sports Photos 22

E. Take photos of E. Take photos of TeachersTeachers

11

F. Develop FilmF. Develop Film 22

G. Design LayoutG. Design Layout 55

H. Print and Mail PagesH. Print and Mail Pages 33

Example (cont’d)Example (cont’d)

Will it be possible for the yearbook to be Will it be possible for the yearbook to be completed on time if the tasks have to be completed on time if the tasks have to be done one after the other?done one after the other?

If some tasks can be done at the same If some tasks can be done at the same time, could the deadline be met?time, could the deadline be met?

As you can see, some of the yearbook staff As you can see, some of the yearbook staff jobs can be done at the same time and yet jobs can be done at the same time and yet many of them can not be started until many of them can not be started until others have been completed.others have been completed.

Example (cont’d)Example (cont’d) Assuming the following prerequisites, Assuming the following prerequisites,

how soon can the project be how soon can the project be completed?completed?TaskTask TimeTime Prerequisite Prerequisite

TaskTask

StartStart 00

A. Buy filmA. Buy film 11 NONENONE

B. Load camerasB. Load cameras 11 AA

C. Take photos of clubsC. Take photos of clubs 33 BB

D. Take sports photosD. Take sports photos 22 CC

E. Take photos of E. Take photos of Teachers Teachers

11 BB

F. Develop filmF. Develop film 22 D, ED, E

G. Design layoutG. Design layout 55 D, ED, E

H. Print and mail pagesH. Print and mail pages 33 G, FG, F

Making a GraphMaking a Graph

Drawing a diagram or a Drawing a diagram or a graphgraph of this of this information makes it easier to see information makes it easier to see the relationships among the tasks.the relationships among the tasks.

In the following graph, tasks are In the following graph, tasks are represented by points, or represented by points, or vertices,vertices, and the arrows, or and the arrows, or edges, edges, indicate indicate which tasks must be finished before which tasks must be finished before a new task can begin. a new task can begin.

Graphing (cont’d)Graphing (cont’d)

Each edge also shows the number of Each edge also shows the number of days it takes to complete the days it takes to complete the preceding task.preceding task.

Note that tasks with the same Note that tasks with the same prerequisites are aligned vertically. prerequisites are aligned vertically. Although this is not necessary, it Although this is not necessary, it makes the graph easier to read.makes the graph easier to read.

GraphGraph

StartA B

C

E

DF

G

H Finish

0 11

3 2

22

3

1

1

51

Practice ProblemsPractice Problems

In Exercises 1 In Exercises 1 through 4, draw a through 4, draw a graph and label the graph and label the vertices for each vertices for each task table.task table.

1. 1. TaskTask

TimTimee

PrerequisitePrerequisitess

StartStart 00

AA 55 NONENONE

B B 66 AA

CC 44 AA

DD 44 BB

EE 88 B, CB, C

FF 44 CC

GG 1010 D, E, FD, E, F

FinishFinish

Practice Problems (cont’d)Practice Problems (cont’d)

2.2. TasTaskk

TimeTime Prereq.Prereq.

StartStart 00

AA 11 NoneNone

BB 22 NoneNone

CC 33 A, BA, B

DD 55 BB

EE 55 CC

FF 55 C, DC, D

GG 44 D, ED, E

HH 44 E, FE, F

FinishFinish

3.3. TasTaskk

TimeTime Prereq.Prereq.

StartStart 00

AA 55 NoneNone

BB 77 AA

CC 44 AA

DD 33 BB

EE 77 B, CB, C

FF 55 CC

GG 88 D, E, FD, E, F

FinishFinish

Practice Problems (cont’d)Practice Problems (cont’d)4.4. TaskTask TimeTime PrerequisitPrerequisit

eses

StartStart 00

AA 55 NoneNone

BB 88 A, DA, D

CC 99 B, IB, I

DD 77 NoneNone

EE 88 BB

FF 1212 II

GG 44 C, E, FC, E, F

HH 99 NoneNone

II 55 D, HD, H

FinishFinish

Practice Problems (cont’d)Practice Problems (cont’d)5.5. In order to get a family dinner completed in the In order to get a family dinner completed in the

shortest amount of time, Mrs. Shu listed the shortest amount of time, Mrs. Shu listed the following tasks:following tasks:TaskTask TimeTime Prerequisite Prerequisite

TaskTask

StartStart 00

A. Wash HandsA. Wash Hands

B. Defrost hamburgerB. Defrost hamburger

C. Shape meat into C. Shape meat into pattiespatties

D. Cook hamburgersD. Cook hamburgers

E. Peel and slice E. Peel and slice potatoespotatoes

F. Fry potatoesF. Fry potatoes

G. Make saladG. Make salad

H. Set tableH. Set table

I. Serve foodI. Serve food

Practice Problems (cont’d)Practice Problems (cont’d)

a.a. Construct reasonable time Construct reasonable time estimates in minutes for each of estimates in minutes for each of these tasks and label the these tasks and label the prerequisites.prerequisites.

b.b. Construct a graph of the table.Construct a graph of the table.

c.c. What is the least amount of time What is the least amount of time needed to prepare dinner?needed to prepare dinner?

Practice Problems (cont’d)Practice Problems (cont’d)

6.6. Your best friend, Matt, has always been Your best friend, Matt, has always been very disorganized. He is now trying to very disorganized. He is now trying to get ready to leave for college and get ready to leave for college and desperately needs your help.desperately needs your help.

a. Develop a table of at least six a. Develop a table of at least six activities that will need to be activities that will need to be completed to get Matt on his way. completed to get Matt on his way. Give the times and prerequisites of Give the times and prerequisites of these activities.these activities.

Practice Problems (cont’d)Practice Problems (cont’d)

b.b. Construct a corresponding graph.Construct a corresponding graph.

c.c. What is the least amount of time it What is the least amount of time it will take to get Matt off to school?will take to get Matt off to school?

Practice Problems (cont’d)Practice Problems (cont’d)

7.7. Consider the following graph:Consider the following graph:

Start

0

0

A

B

2

4

C

3 D

E4

2F

1

3

G

4

Finish

Practice Problems (cont’d)Practice Problems (cont’d)Complete the task table below for this Complete the task table below for this

graph.graph.TaskTask TimeTime Prerequisite Prerequisite

TaskTaskStartStart 00

AA

BB

CC

DD

EE

FF

GG

FinishFinish