1 / 17

Problem Solving with Linear Programming

Problem Solving with Linear Programming. LP-L5 Objectives: To solve complex problems using Linear Programming techniques. Learning Outcome B-1.

alden-hinton
Download Presentation

Problem Solving with Linear Programming

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Problem Solving with Linear Programming LP-L5 Objectives:To solve complex problems using Linear Programming techniques. Learning Outcome B-1

  2. 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

  3. x + y £ 6 x + 2y £ 8 x ³ 2 y ³ 1 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

  4. x + y £ 6 x + 2y £ 8 x ³ 2 y ³ 1 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. • Solution • Graph the system:The feasible region is the green shaded area shown • Find the vertices of the feasible region: The coordinates of the corner points are (2, 3), (2, 1), (5, 1), and (4, 2). • Substitute each vertice into the equation to find maximum:The value of M for each point isPoint (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) = 10Therefore, the value of M is maximized at (2, 3). Example - Maximize the Value of a Specific Expression

  5. x ³ 0y ³ 03x + 2y £ 62x + 3y ³ 6 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

  6. x ³ 0y ³ 03x + 2y £ 62x + 3y ³ 6 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. • Solution • Graph the system:The feasible region is the green shaded area shown • Find the vertices of the feasible region: The coordinates of the corner points are (0, 3), (0, 2), and (1.2, 1.2). • 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) + 1.2 = 6.The coordinates (1.2, 1.2) produce the maximum value of the expression 4x + y. Test Yourself - Maximize the Value of a Specific Expression

  7. x + y £ 4x + 5y ³ 8-x + 2y £ 6 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

  8. x + y £ 4x + 5y ³ 8-x + 2y £ 6 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. • Solution • Graph the system:The feasible region is the green shaded area shown • Find the vertices of the feasible region: The coordinates of the corner points are (-2, 2), (3, 1), and (0.67, 3.33). • 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) = 8.67.The coordinates (-2, 2) produce the minimum value of the expression 3x + 2y. Test Yourself – Minimize the Value of a Specific Expression

  9. 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. y £ -1x + 4x + 4y ³ 7-x + 2y £ 5 Test Yourself – Maximize the Value of a Specific Expression

  10. 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. y £ -1x + 4x + 4y ³ 7-x + 2y £ 5 • Solution • Graph the system:The feasible region is the green shaded area shown • Find the vertices of the feasible region: The coordinates of the corner points are (1, 3), (-1, 2), and (3, 1). • 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. Test Yourself – Maximize the Value of a Specific Expression

  11. Here is a plan of the steps used to solve word problems using linear programming: • After reading the question, make a chart to see the information more clearly. • Assign variables to the unknowns. • Form expressions to represent the restrictions. • Graph the inequalities. • Find the coordinates of the corner points of the feasible region. • Find the vertex point that maximizes or minimizes what we are looking for. • State the solution in a sentence. Theory – Solving Problems Using Linear Programming

  12. Example – Seven Steps

  13. Example – Seven Steps cont’d

  14. Example – Seven Steps cont’d

  15. Example 2 – Seven Steps

  16. Example 2 – Seven Steps cont’d

  17. Example 2 – Seven Steps cont’d

More Related