Section 3.4 - Linear Programming ALGEBRA TWO CHAPTER THREE: SYSTEMS OF LINEAR EQUATIONS AND INEQUALITIES Section 3.4 - Linear Programming

LEARNING GOALS Goal One - Solve linear programming problems. Goal Two - Use linear programming to solve real-life problems.

VOCABULARY Optimization means finding the maximum or minimum value of some quantity. Linear programming is the process of optimizing a linear objective function subject to a system of linear inequalities called constraints. 1

VOCABULARY The graph of the system of constraints is called the feasible region. If an objective function has a maximum or a minimum value, then it must occur at a vertex of the feasible region. Moreover, the objective function will have both a maximum and a minimum value if the feasible region is bounded. 1

Solving a Linear Programming Problem Find the minimum and maximum values of the objective function C = 3x + 2y subject to the following constraints 3x + 4y < 20 3x - y < 5 x > 0 y > 0 √ STEP 1: Graph all the inequalities. 3x + 4y < 20 4y < -3x + 20 y < (-3/4)x + 5 1

Solving a Linear Programming Problem Find the minimum and maximum values of the objective function C = 3x + 2y subject to the following constraints 3x + 4y < 20 3x - y < 5 x > 0 y > 0 √ √ STEP 1: Graph all the inequalities. 3x - y < 5 -y < -3x + 5 y > 3x - 5 1

Solving a Linear Programming Problem Find the minimum and maximum values of the objective function C = 3x + 2y subject to the following constraints 3x + 4y < 20 3x - y < 5 x > 0 y > 0 √ √ √ STEP 1: Graph all the inequalities. x > 0 1

Solving a Linear Programming Problem Find the minimum and maximum values of the objective function C = 3x + 2y subject to the following constraints 3x + 4y < 20 3x - y < 5 x > 0 y > 0 √ √ √ √ STEP 1: Graph all the inequalities. y > 0 1

Solving a Linear Programming Problem Find the minimum and maximum values of the objective function C = 3x + 2y subject to the following constraints 3x + 4y < 20 3x - y < 5 x > 0 y > 0 √ √ √ √ STEP 1: Graph all the inequalities. Now you must color in the area where the shading overlaps. 1

Solving a Linear Programming Problem Find the minimum and maximum values of the objective function C = 3x + 2y subject to the following constraints 3x + 4y < 20 3x - y < 5 x > 0 y > 0 √ √ √ √ STEP 1: Graph all the inequalities. Identify the ordered pairs at each vertex. (0, 5) (8/3, 3) (0, 0) (5/3, 0) 1

Solving a Linear Programming Problem Find the minimum and maximum values of the objective function C = 3x + 2y subject to the following constraints 3x + 4y < 20 3x - y < 5 x > 0 y > 0 √ √ √ √ The feasible region determined by the constraints is shown. The three vertices (0, 5), (0, 0), and (5/3, 0) are intercepts. The fourth vertex (8/3, 3) is found by solving the intersecting lines. 1

Solving a Linear Programming Problem Find the minimum and maximum values of the objective function C = 3x + 2y subject to the following constraints 3x + 4y < 20 3x - y < 5 x > 0 y > 0 √ √ √ √ To find the minimum and maximum values of C, evaluate C = 3x + 2y at each of the four vertices. 1

Solving a Linear Programming Problem Find the minimum and maximum values of the objective function C = 3x + 2y subject to the following constraints 3x + 4y < 20 3x - y < 5 x > 0 y > 0 √ √ √ √ C = 3x + 2y C = 3(0) + 2(5) = 10 C = 3(0) + 2(0) = 0 Minimum C = 3(5/3) + 2(0) = 5 C = 3(8/3) + 2(3) = 14 Maximum Minimum = 0; Maximum = 14 1

ASSIGNMENT READ & STUDY: pg. 156-158. WRITE: pg. 159-162. #13, #15, #17, #19, #21, #37, #39