Graphs Their Applications

NC Standard Course of Study Competency Goal 1: The learner will use matrices and graphs to model relationships and solve problems. Competency Goal 1: The learner will use matrices and graphs to model relationships and solve problems. Objective 1.02: Use graph theory to model relationships and solve problems. Objective 1.02: Use graph theory to model relationships and solve problems.

Planning How does a contractor organize all of the jobs needed to complete a project? How do your parents manage to get all parts of a Thanksgiving dinner done at the same time? Although day-to-day planning may be a simple activity, the planning of extensive projects may require very detailed preparation.

Organized Planning Business owners usually can not function properly when haphazard planning is used. Business owners usually can not function properly when haphazard planning is used. A more scientific, organized method must be used. A more scientific, organized method must be used.

Example The Central High yearbook staff has only 16 days left before the deadline for completing their yearbook. The Central High yearbook staff has only 16 days left before the deadline for completing their yearbook. They are running behind schedule and still have several tasks to finish. They are running behind schedule and still have several tasks to finish. The tasks remaining and the times it takes to do each task are listed in the following table. The tasks remaining and the times it takes to do each task are listed in the following table.

Example (cont’d) Task Time A. Buy Film 0 B. Load Cameras 1 day C. Take Photos of Clubs 3 D. Take Sports Photos 2 E. Take photos of Teachers 1 F. Develop Film 2 G. Design Layout 5 H. Print and Mail Pages 3

Example (cont’d) Will it be possible for the yearbook to be completed on time if the tasks have to be done one after the other? Will it be possible for the yearbook to be completed on time if the tasks have to be done one after the other? If some tasks can be done at the same time, could the deadline be met? If some tasks can be done at the same time, could the deadline be met? As you can see, some of the yearbook staff jobs can be done at the same time and yet many of them can not be started until others have been completed. As you can see, some of the yearbook staff jobs can be done at the same time and yet many of them can not be started until others have been completed.

Example (cont’d) Assuming the following prerequisites, how soon can the project be completed? Assuming the following prerequisites, how soon can the project be completed? TaskTime Prerequisite Task Start0 A. Buy film 1NONE B. Load cameras 1A C. Take photos of clubs 3B D. Take sports photos 2C E. Take photos of Teachers 1B F. Develop film 2 D, E G. Design layout 5 D, E H. Print and mail pages 3 G, F

Making a Graph Drawing a diagram or a graph of this information makes it easier to see the relationships among the tasks. Drawing a diagram or a graph of this information makes it easier to see the relationships among the tasks. In the following graph, tasks are represented by points, or vertices, and the arrows, or edges, indicate which tasks must be finished before a new task can begin. In the following graph, tasks are represented by points, or vertices, and the arrows, or edges, indicate which tasks must be finished before a new task can begin.

Graphing (cont’d) Each edge also shows the number of days it takes to complete the preceding task. Each edge also shows the number of days it takes to complete the preceding task. Note that tasks with the same prerequisites are aligned vertically. Although this is not necessary, it makes the graph easier to read. Note that tasks with the same prerequisites are aligned vertically. Although this is not necessary, it makes the graph easier to read.

Graph Start AB C E D F G H Finish

Practice Problems In Exercises 1 through 4, draw a graph and label the vertices for each task table. 1. Task TimePrerequisites Start0 A5NONE B6A C4A D4B E8 B, C F4C G10 D, E, F Finish

Practice Problems (cont’d) 2. Task TimePrereq. Start0 A1None B2None C3 A, B D5B E5C F5 C, D G4 D, E H4 E, F Finish 3. Task TimePrereq.Start0 A5None B7A C4A D3B E7 B, C F5C G8 D, E, F Finish

Practice Problems (cont’d) 4. Task TimePrerequisites Start0 A5None B8 A, D C9 B, I D7None E8B F12I G4 C, E, F H9None I5 D, H Finish

Practice Problems (cont’d) 5. In order to get a family dinner completed in the shortest amount of time, Mrs. Shu listed the following tasks: TaskTime Prerequisite Task Start0 A. Wash Hands B. Defrost hamburger C. Shape meat into patties D. Cook hamburgers E. Peel and slice potatoes F. Fry potatoes G. Make salad H. Set table I. Serve food

Practice Problems (cont’d) a. Construct reasonable time estimates in minutes for each of these tasks and label the prerequisites. b. Construct a graph of the table. c. What is the least amount of time needed to prepare dinner?

Practice Problems (cont’d) 6. Your best friend, Matt, has always been very disorganized. He is now trying to get ready to leave for college and desperately needs your help. a. Develop a table of at least six activities that will need to be completed to get Matt on his way. Give the times and prerequisites of these activities.

Practice Problems (cont’d) b. Construct a corresponding graph. c. What is the least amount of time it will take to get Matt off to school?

Practice Problems (cont’d) 7. Consider the following graph: Start 0 0 A B 2 4 C 3 D E 4 2 F 1 3 G 4 Finish

Practice Problems (cont’d) Complete the task table below for this graph. TaskTime Prerequisite Task Start0 A B C D E F G Finish