Unit – graph theory

Concept Characteristics What is a Graph……Really?

Graph Theory Learning Objectives— Part 1 You will understand how to model a project using graphs. You will be able to identify how many edges and vertices a graph has. You will be able to find the “critical path” of a project graph from start to finish and find Earliest Start Times for each task. You will be able to determine how much time it will take for a project to be complete.

Graphs:Vertices and Edges How many Edges does this graph have? How many vertices does this graph have? The vertices are “tasks” and the “ “edges” show the time the task takes.

Brushing your teeth If the “project” is brushing your teeth, what are the “tasks” that occur from start to finish? Tasks? Let’s make sure they are in order. Let’s draw a directed graph of these tasks…..

Imagine Name the things you do each morning before you get to school. Lets make a list The things you do to before you get to school are TASKS, and the preparation/tasks you do before arriving at school “are” the PROJECT. Is there a way to show your “project” using a graph? Is this graph linear?

Let’s take our list of things (tasks) you do before you arrive at school. Are there things that have to happen before other things? What if you can’t do task B before task A is complete? What would you have to do to show this in a graph? What if tasks A and B can be done simultaneously? What if you add tasks C & D that must be done after A is complete? What if you add task E that is dependent on task B & D?

Directed Graph A directed graph is a graphic way using vertices and edges to show the “path” or “paths” of a project leading from task to task. A “directed graph” from start to finish can be thought of as a “time line” from beginning to end…..

Example of a directed graph What can you guess from the graph below?

Project:Mowing the lawn WARM-UP:PUT THESE TASKS IN ORDER AND IDENTIFY ANY OF THE TASKS THAT CAN TAKE PLACE SIMULTANEOUSLY. TASKSTIME A.Picking up/ Clearing the lawn15 minutes B.Mowing the lawn45 minutes C.Gassing up the mower5 minutes D.Start mower3 minutes E.Getting mower out5 minutes F.Cleaning the mower off10 minutes G.Checking oil5 minutes H.Putting the mower away5 minutes

Graphing the Project Where do we start? What has to be done first? Can more than one thing be done first? What task(s) come next? From where? Next? Where to we end?

Practical Example of a project 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 left to finish. The remaining tasks and time that it takes to complete each task are listed in the following table.

Yearbook Tasks Is it possible to complete the project if the tasks have to be done in order? TaskTime (in days) Start0 ABuy Film1 BLoad Camera1 CTake Club Photos3 DTake Sports Photos2 ETake Teacher Photos1 FDevelop Film2 GDesign the Layout5 HPrint and Mail Pages3

Yearbook Tasks What if some of the jobs can be done simultaneously? Draw a graph using arrows as edges representing a task being done and vertices as tasks. TaskTime (in days)Prerequisite Task Start0None ABuy Film1None BLoad Camera1A CTake Club Photos3B DTake Sports Photos2C ETake Teacher Photos 1B FDevelop Film2D, E GDesign the Layout5D, E HPrint and Mail Pages 3G, F

Graph the Tasks

You Practice TASKTIMEPREREQUISITES Start0--- A5NONE B6A C4A D4B E8B, C F4C G10D, E, F FINISH

One More TASKTIMEPREREQUISITES START0---- A4NONE B3A C1A D6A E2B F3C, D G3E H1E, F FINISH

Critical Path Critical path is defined as the shortest path that will take you to the completion of a project AND ensure that ALL tasks are completed as well. There’s an entire profession devoted to this called Project Management….

Critical Path Identify the Earliest Start Time for each task! Begin at the start Label each vertex with the smallest possible time needed for that task to begin based on the prerequisites. The critical path is actually the LONGEST path…..Why??????

Warm-up – Graph the Project Management Chart TASKTIMEPREREQUISITE START0---- A5NONE B8A, D C9B, I D7NONE E8B F12I G4C, E, F H9NONE I5D, H FINISH

Make a Chart of Tasks, Times and Prerequisites--- Homework Back

Make a Chart of Tasks, Times and Prerequisites TaskTimePrerequisites

Propulsion System of a Nuclear Submarine -- - Project Management Graph

Game Room Project--Partners Pretend you are planning to build a game room in your home. Sketch out what you want it to look like. List the tasks that need to be done to complete your project. Estimate the time it will take to do each task in your project. Determine if any task in your project is “dependent” on another part. From start to finish, draw a “directed graph” from start to finish…think of this as a “time line” from beginning to end….. Are there parts of your project that can be done simultaneously? If so, how do you show that graphically?

Let’s Find the Earliest Start Time

Graph and Find the Earliest start times for each task, Critical Path AND Minimum Time to completion for the project TaskTimePrerequisites Start0--- A13NONE B10NONE C4A D8B E6B F7C, D, E G5F H8F FINISH

Create the project chart and find the earlies start times and critical path

Warm-up Problem Create the Project Management Chart/Table. Find the Earliest Start Times for Each Task….. Calculate the minimum time needed to complete this project. What is the critical path?

Draw a Graph to Represent the Project Find the Critical Path AND Minimum Time TaskTimePerequisites Start0---- A2NONE B4 C3A, B D1 E5C, D F6 G7E, F

Partner Practice Create the Project Table. Find the Earliest Starting times for each task…

Game Room Project What Should Be Finished TODAY….By Thursday!! Sketch of your Game Room Finalize your Task List and Time Estimates Determine the prerequisite tasks for each task Create a Project Management Chart/Table with four columns Vertices Labels (alphabetical order) Task description Time for task Prerequisite(s)

Warm-Up

Review Problem TaskTimePrerequisites A5None B7A C3A D2B, C E8D Finish Draw the Graph based on the Chart Below….. Find the Earliest Start Times for Each Task….. Calculate the minimum time needed to complete this project. What is the critical path?

Worksheet #1

Worksheet #2

Worksheet #3

TASKTIMEPREREQUISITESE. S. T. START A B C D E F G H I J

Create the Project Management Chart/Table. Find the Earliest Start Times for Each Task….. Calculate the minimum time needed to complete this project. What is the critical path?

Warm-up Problem Create the Project Management Chart/Table. Find the Earliest Start Times for Each Task….. Calculate the minimum time needed to complete this project. What is the critical path?

Game Room Project What Should Be Finished TODAY…. Finalize your Task List and Time Estimates Determine the prerequisite tasks for each task Sketch of your Game Room Create a Project Management Chart/Table with four columns Vertices Labels (alphabetical order) Task description Time for task Prerequisite(s) Due in 30 minutes