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30S Applied MathMr. Knight – Killarney School
Slide 1
Unit: Linear ProgrammingLesson 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 MathMr. Knight – Killarney School
Slide 2
Unit: Linear ProgrammingLesson 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 MathMr. Knight – Killarney School
Slide 3
Unit: Linear ProgrammingLesson 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 MathMr. Knight – Killarney School
Slide 4
Unit: Linear ProgrammingLesson 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
Solution1. 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) = 11Point (2, 1): M = 2 + 3(1) = 5Point (5, 1): M = 5 + 3(1) = 8Point (4, 2): M = 4 + 3(2) = 10
Therefore, the value of M is maximized at (2, 3).
30S Applied MathMr. Knight – Killarney School
Slide 5
Unit: Linear ProgrammingLesson 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 0y 0
3x + 2y 62x + 3y 6
30S Applied MathMr. Knight – Killarney School
Slide 6
Unit: Linear ProgrammingLesson 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 0y 0
3x + 2y 62x + 3y 6
Solution1. 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) + 1.2 = 6.
The coordinates (1.2, 1.2) produce the maximum value of the expression 4x + y.
30S Applied MathMr. Knight – Killarney School
Slide 7
Unit: Linear ProgrammingLesson 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 4x + 5y 8
-x + 2y 6
30S Applied MathMr. Knight – Killarney School
Slide 8
Unit: Linear ProgrammingLesson 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 4x + 5y 8
-x + 2y 6
Solution1. 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) = 8.67.
The coordinates (-2, 2) produce the minimum value of the expression 3x + 2y.
30S Applied MathMr. Knight – Killarney School
Slide 9
Unit: Linear ProgrammingLesson 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 + 4x + 4y 7
-x + 2y 5
30S Applied MathMr. Knight – Killarney School
Slide 10
Unit: Linear ProgrammingLesson 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 + 4x + 4y 7
-x + 2y 5
Solution1. 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 MathMr. Knight – Killarney School
Slide 11
Unit: Linear ProgrammingLesson 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 MathMr. Knight – Killarney School
Slide 12
Unit: Linear ProgrammingLesson 5: Problem Solving
Example – Seven Steps
30S Applied MathMr. Knight – Killarney School
Slide 13
Unit: Linear ProgrammingLesson 5: Problem Solving
Example – Seven Steps cont’d
30S Applied MathMr. Knight – Killarney School
Slide 14
Unit: Linear ProgrammingLesson 5: Problem Solving
Example – Seven Steps cont’d
30S Applied MathMr. Knight – Killarney School
Slide 15
Unit: Linear ProgrammingLesson 5: Problem Solving
Example 2 – Seven Steps
30S Applied MathMr. Knight – Killarney School
Slide 16
Unit: Linear ProgrammingLesson 5: Problem Solving
Example 2 – Seven Steps cont’d
30S Applied MathMr. Knight – Killarney School
Slide 17
Unit: Linear ProgrammingLesson 5: Problem Solving
Example 2 – Seven Steps cont’d