30S Applied Math Mr. Knight – Killarney School Slide 1 Unit: Linear Programming Lesson 5: Problem Solving Problem Solving with Linear Programming Learning Outcome B-1 LP-L5 Objectives: To solve complex problems using Linear Programming techniques.

30S Applied Math Mr. Knight – Killarney School Slide 2 Unit: Linear Programming Lesson 5: Problem Solving The process of finding a feasible region and locating the points that give the minimum or maximum value to a specific expression is called linear programming. It is frequently used to determine maximum profits, minimum costs, minimum distances, and so on. Theory – Intro

30S Applied Math Mr. Knight – Killarney School Slide 3 Unit: Linear Programming Lesson 5: Problem Solving Graph the following system of inequalities and identify the corner points of the feasible region. Then find the values of x and y that maximize the expression M = x + 3y. Example - Maximize the Value of a Specific Expression x + y  6 x + 2y  8 x  2 y  1

30S Applied Math Mr. Knight – Killarney School Slide 4 Unit: Linear Programming Lesson 5: Problem Solving Graph the following system of inequalities and identify the corner points of the feasible region. Then find the values of x and y that maximize the expression M = x + 3y. Example - Maximize the Value of a Specific Expression x + y  6 x + 2y  8 x  2 y  1 Solution 1. Graph the system: The feasible region is the green shaded area shown 2. Find the vertices of the feasible region: The coordinates of the corner points are (2, 3), (2, 1), (5, 1), and (4, 2). 3. Substitute each vertice into the equation to find maximum: The value of M for each point is Point (2, 3): M = 2 + 3(3) = 11 Point (2, 1): M = 2 + 3(1) = 5 Point (5, 1): M = 5 + 3(1) = 8 Point (4, 2): M = 4 + 3(2) = 10 Therefore, the value of M is maximized at (2, 3).

30S Applied Math Mr. Knight – Killarney School Slide 5 Unit: Linear Programming Lesson 5: Problem Solving Graph the following system of inequalities and identify the corner points of the feasible region. Then find the values of x and y that maximize the expression M = 4x + y. Test Yourself - Maximize the Value of a Specific Expression x  0 y  0 3x + 2y  6 2x + 3y  6

30S Applied Math Mr. Knight – Killarney School Slide 6 Unit: Linear Programming Lesson 5: Problem Solving Graph the following system of inequalities and identify the corner points of the feasible region. Then find the values of x and y that maximize the expression M = 4x + y. Test Yourself - Maximize the Value of a Specific Expression x  0 y  0 3x + 2y  6 2x + 3y  6 Solution 1. Graph the system: The feasible region is the green shaded area shown 2. Find the vertices of the feasible region: The coordinates of the corner points are (0, 3), (0, 2), and (1.2, 1.2). 3. Substitute each vertice into the equation to find maximum: Using (0, 3), M = 4(0) + 3 = 3. Using (0, 2), M = 4(0) + 2 = 2. Using (1.2, 1.2), M = 4(1.2) = 6. The coordinates (1.2, 1.2) produce the maximum value of the expression 4x + y.

30S Applied Math Mr. Knight – Killarney School Slide 7 Unit: Linear Programming Lesson 5: Problem Solving Graph the following system of inequalities and identify the corner points of the feasible region. Then find the values of x and y that minimize the expression M = 3x + 2y. Test Yourself – Minimize the Value of a Specific Expression x + y  4 x + 5y  8 -x + 2y  6

30S Applied Math Mr. Knight – Killarney School Slide 8 Unit: Linear Programming Lesson 5: Problem Solving Graph the following system of inequalities and identify the corner points of the feasible region. Then find the values of x and y that minimize the expression M = 3x + 2y. Test Yourself – Minimize the Value of a Specific Expression x + y  4 x + 5y  8 -x + 2y  6 Solution 1. Graph the system: The feasible region is the green shaded area shown 2. Find the vertices of the feasible region: The coordinates of the corner points are (-2, 2), (3, 1), and (0.67, 3.33). 3. Substitute each vertice into the equation to find minimum: Using (-2, 2), M = 3(-2) + 2(2) = -2. Using (3, 1), M = 3(3) + 2(1) = 11. Using (0.67, 3.33), M = 3(0.67) + 2(3.33) = The coordinates (-2, 2) produce the minimum value of the expression 3x + 2y.

30S Applied Math Mr. Knight – Killarney School Slide 9 Unit: Linear Programming Lesson 5: Problem Solving The constraints for manufacturing two types of hockey skates are given by the following system of inequalities. Find the maximum value of Q over the feasible region if Q = 3x + 5y. Test Yourself – Maximize the Value of a Specific Expression y  -1x + 4 x + 4y  7 -x + 2y  5

30S Applied Math Mr. Knight – Killarney School Slide 10 Unit: Linear Programming Lesson 5: Problem Solving The constraints for manufacturing two types of hockey skates are given by the following system of inequalities. Find the maximum value of Q over the feasible region if Q = 3x + 5y. Test Yourself – Maximize the Value of a Specific Expression y  -1x + 4 x + 4y  7 -x + 2y  5 Solution 1. Graph the system: The feasible region is the green shaded area shown 2. Find the vertices of the feasible region: The coordinates of the corner points are (1, 3), (-1, 2), and (3, 1). 3. Substitute each vertice into the equation to find maximum: Using (1, 3), Q = 3(1) + 5(3) = 18. Using (-1, 2), Q = 3(-1) + 5(2) = 7. Using (3, 1), Q = 3(3) + 5(1) = 14. The coordinates (1, 3) produce a maximum value for Q over the feasible region where Q = 3x + 5y.

30S Applied Math Mr. Knight – Killarney School Slide 11 Unit: Linear Programming Lesson 5: Problem Solving Here is a plan of the steps used to solve word problems using linear programming: 1.After reading the question, make a chart to see the information more clearly. 2.Assign variables to the unknowns. 3.Form expressions to represent the restrictions. 4.Graph the inequalities. 5.Find the coordinates of the corner points of the feasible region. 6.Find the vertex point that maximizes or minimizes what we are looking for. 7.State the solution in a sentence. Theory – Solving Problems Using Linear Programming

30S Applied Math Mr. Knight – Killarney School Slide 12 Unit: Linear Programming Lesson 5: Problem Solving Example – Seven Steps

30S Applied Math Mr. Knight – Killarney School Slide 13 Unit: Linear Programming Lesson 5: Problem Solving Example – Seven Steps cont’d

30S Applied Math Mr. Knight – Killarney School Slide 14 Unit: Linear Programming Lesson 5: Problem Solving Example – Seven Steps cont’d

30S Applied Math Mr. Knight – Killarney School Slide 15 Unit: Linear Programming Lesson 5: Problem Solving Example 2 – Seven Steps

30S Applied Math Mr. Knight – Killarney School Slide 16 Unit: Linear Programming Lesson 5: Problem Solving Example 2 – Seven Steps cont’d

30S Applied Math Mr. Knight – Killarney School Slide 17 Unit: Linear Programming Lesson 5: Problem Solving Example 2 – Seven Steps cont’d