Linear Programming. ICM Unit 3 Day 1 Part 1

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1 Linear Programming ICM Unit 3 Day 1 Part 1

Arrival: Tools of the Trade Pencils Grab

Get out a NEW sheet of notebook paper for

2 Arrival: Tools of the Trade Pencils Grab a couple to SHARE with your partner. Graph Paper Two pieces per student for today. Ruler One per student. Get out a NEW sheet of notebook paper for Test Corrections! Put Phones in Red Pockets!! (Turned off)

3 Homework Discussion What questions do you have last night s review? How are we doing with graphing linear inequalities and solving systems of equations? These skills are CRITICAL for success in this unit!!

4 Foundational Skills for this unit Last night s review HW included skills that are CRITICAL for success in this unit!! Having trouble with this or ANYTHING from today s lesson come to tutorial ASAP!! Tutorials: Most mornings ~7 AM Monday & Wednesday 1 st Half of Lunch We re nearly ½ way through the Quarter!! Do you have 2 Tutorials yet?

5 Homework Tonight Finish the Back of Unit Outline) Complete Packet p. 2

7 Here is a scenario... Mrs. Walden is deciding what to order at Chick-Fil-A. An order of fries costs $2 and a spicy chicken sandwich costs $3. She has a total of $12 available for her purchase. Volunteer to answer the following: What would it cost to buy two orders of fries? $2 * 2 = $4 What would it cost to buy three orders of fries? $2 * 3 = $6 What would it cost to buy x orders of fries? $2 * x = 2x What would it cost to buy y spicy chicken sandwiches? $3 * y = 3y

8 Inequality for our scenario... Mrs. Walden is deciding what to order at Chick-Fil-A. An order of fries costs $2 and a spicy chicken sandwich costs $3. She has a total of $12 available for her purchase. Let x: represent # of orders of fries y: represent # of spicy chicken sandwiches The inequality that represents the combined amount spent on fries and spicy chicken sandwiches would be: 2x + 3y 12

9 Today s 1 st Topic: Review of Graphing Linear Inequalities Basic Steps 1. Graph boundary line using x and y intercept method 2. Test the point (0, 0) to determine the shading 3. Determine and label the feasible region Our First Example Problem: Graph 2x 3y 12 Continued on Next Slide...

10 STEP 1: Graph Boundary Line using x- and y-intercept method To graph the boundary line we can temporarily ignore the inequality symbol. For our example, 2x 3y 12 will become 2x 3y 12 Continued on Next Slide...

11 Step 1 continued: Graph boundary line using x and y intercept method X-intercept: (some number, 0) (, 0) Y-intercept: (0, some number) (0, ) 2x 3y 12 SHOW YOUR Work!! Plot these two points on your graph paper. Connect the points with a straight line using your RULER!! NEATNESS COUNTS!!

12 2x 3y 12 2x + 3(0) = 12 2x = 12 SHOW YOUR Work x = 6 X-intercept: (some number, 0) ( 6, 0) 2(0) + 3y = 12 3y = 12 y = 4 Y-intercept: (0, some number) (0, 4 ) Plot these two points on your graph paper. Connect the points with a straight line

13 STEP 2: Test the point (0,0) to determine shading This requires going back to the inequality statement 2x 3y 12 Substitute the point (0,0) into the inequality: If the statement is TRUE, then the Test Point should be included in the shaded region. If the statement is FALSE, then shade away from the Test Point (so that this point is NOT in the shaded region). SHOW YOUR Work!!

14 STEP 2: Test the point (0,0) to determine shading 2x 3y 12 Substitute the point (0,0) into the inequality: 2(0) + 3(0) If the statement is TRUE, then the Test Point should be included in the shaded region. Our statement was TRUE, so we shade to include (0, 0) If the statement is FALSE, then shade away from the Test Point (so that this point is NOT in the shaded region). SHOW YOUR Work!!

Let s revisit the scenario... Mrs. Walden is deciding what to order at Chick-Fil-A. An order of fries costs $2 and a spicy chicken sandwich costs $3. She has a total of $12 available for her purchase.

15 Let s revisit the scenario... Mrs. Walden is deciding what to order at Chick-Fil-A. An order of fries costs $2 and a spicy chicken sandwich costs $3. She has a total of $12 available for her purchase. Let x: represent # of orders of fries y: represent # of spicy chicken sandwiches 2x 3y 12 The region we shaded includes negative values for x and y. Is this reasonable? No! You can t have a negative amount of food!

Defining the Feasible Region Mrs. Walden is deciding what to order at Chick-Fil-A. An order of fries costs $2 and a spicy chicken sandwich costs $3. She has a total of $12 available for her purchase.

16 Defining the Feasible Region Mrs. Walden is deciding what to order at Chick-Fil-A. An order of fries costs $2 and a spicy chicken sandwich costs $3. She has a total of $12 available for her purchase. Let x: represent # of orders of fries y: represent # of spicy chicken sandwiches We agree that it is NOT reasonable, in the context of this problem, to have a negative number of x and y values. So we want to restrict our shading to just the first quadrant. This region will be defined and labeled as our feasible region. 2x 3y 12 x 0 y 0

17 NOTATION specific to this course We will use equations that have x 1 and x 2 for x for y The reason for this will become apparent when we get into later problems. For now, just go with it! Example: 4x 3y 24 4x 3x 24 would become

18 Important question: What quadrant would always contain points that would meet the criteria of x 0 and x 0? Quadrant I These equations are called non-negativity constraints. They will be VERY important to us! We do A LOT of application problems! Quadrant II Quadrant III x 2 Quadrant I Quadrant IV x 1

19 Graphing Constant Inequalities x x Overall Feasible Region

20 Taking this topic one step further System of Inequalities: Two or more inequalities graphed on the same grid Example: the graph we just did is a system of inequalities Example: Graph x x x -9x -9 4x +2x 16 We ll graph this system on the next slide

21 Day 1 Practice Graph the following system of inequalities 1) 3x -9x -9 4x +2x 16 *Find x 1 & x 2 intercepts of the 1 st inequality. x 1 intercept: (number, 0) x 2 intercept: (0, number) 2) Graph x 2 *Connect intercepts with a line *Test point (0,0) x 1 *Shade NOW REPEAT those steps for the 2 nd inequality!

22 Sharing Student Presentation Selected student shares graphing practice problem Document Camera

23 Please put in your notes Check-In State the 1 and intercepts for the following: 4x 5x 20 x x intercept: (, ) 3x 9 x x x 2 intercept: (, ) intercept: (, ) intercept: (, ) THEN Graph and Shade them on your graph paper. Be sure to use your RULER & Colored Pencils! x x

24 Check-In ANSWERS to first steps State the and x intercepts for the following: 4x 5x 20 x x intercept: ( 5, 0) x intercept: ( 0, 4) 3x 9 x x1 2 intercept: ( 9, 0) x intercept: ( 0, -3)

25 Graphs Check-in Selected student shares graphing practice problem 4x 5x 20 x 3x 9

26 Sharing Student Presentation Selected student shares graphing practice problem Document Camera

27 Day 1 Notes Part 2 Linear Programming Vocabulary Formulating a Linear Programming Problem

28 Agenda for the Rest of Today... Questions about Part 1 of the lecture? History behind Linear Programming Linear Programming Terminology Maximizing Profit Lego Furniture Implementing the use of technology in linear programming Example problem

29 History Behind Linear Programming Mathematical Advisor for the US Air Force Required to come up with a plan for the military for training, supplying and deploying combat units Military referred to a plan as a program Developed an algorithm for finding the maximum or minimum of an objective given a set of constraints

30 Currently... Shipping companies How many planes and where to deploy delivery trucks Oil industry How much of raw product should become gasoline and how much should be used for petroleum-based products Revenue management Advertising Telecommunications LOTS more Sources: MINDSET Chapter 2

31 Linear Programming Terminology WRITE THESE DOWN!!! Decision Variables (x 1,x 2, x 3 etc): represent the quantities we are solving for; look for How Many part of problem statement Objective Function: statement that represents the goal to be maximized or minimized Optimal solution: solution that results in the maximized or minimized objective function Constraints: restrictions on the decision variables; stated as inequalities Feasible Region: graphical representation of the set of solutions that satisfies the constraints Ex : P =2x +7x Ex : 3x -9x 18 4x +2x 16 Corner Points: the intersection of one or more constraints; the maximum or minimum will occur at one of these locations

32 Exploring a Contextual Problem LEGO Activity Packet p. 1 GOAL: If a furniture company obtains 6 large and 8 small pieces every day, what production rate of tables and chairs generates the most profit? For #2, you may want to sketch diagrams of the tables and chairs. Be sure to stay within your restrictions! You will be working with your desk partner to work through the packet page. Available Resources 2008 North Carolina State University Adapted with permission form MINDSET Chapter 2 materials

33 Lego Furniture (continued) Obtaining Maximum Profit If a furniture company obtains 6 large and 8 small pieces every day, what production rate of tables and chairs generates the most profit? Formulating the problem on paper--notes

34 Notes: How to Formulate a Linear Programming Problem Using the Lego Problem Step 1: Define the decision variables x 1 : # of tables x 2 : # of chairs production rate Important to put # of Step 2: Write the objective function (profit statement) 16x x 2 P(x 1, x 2 ):

35 Formulating the Lego Problem Step 3: Write the constraints TABLES CHAIRS Resources 2x 1 + 1x 2 2x 1 2x Large Pieces 8 Small Pieces 2x 1x 6 2x 2x 8 **Remember don t be a dummy. Be sure to write the dummy constraints. These are the non-negativity constraints** x 0 x 0 1 2

36 Formulating the Lego Problem Step 4: Graph the constraints and plot intersection points of the constraints. Document Camera Volunteer to Guide Us? Note: More about why the corner points are important during Tomorrow s lesson.

37 Student Presenters Using the document camera. Bring your Work Presentation: LEGO Furniture Company Your presenters will be: &

38 Homework Tonight Finish the CW Handout (back of unit outline) Complete Packet p. 2 **********Now Writing an EXIT TICKET*********** On 2 sticky notes, write down: 1 st Sticky Note) One thing you don t understand from today s lesson (put this sticky on RED or YELLOW) AND 2 nd Sticky Note) One thing you are comfortable with (put this sticky on GREEN)

Concept: Solving Inequalities Name:

Concept: Solving Inequalities Name: You should have completed Equations Section 7 Part A: Solving Inequalities before beginning this handout. COMPUTER COMPONENT Instructions: In follow the Content Menu