1. Slide 7 - 1 Copyright © 2009 Pearson Education, Inc. 7.6 Linear Programming

2. Slide 7 - 2 Copyright © 2009 Pearson Education, Inc. Linear Programming In a linear programming problem, there are restrictions called constraints. EEach is represented as a linear inequality. TThe list of constraints forms a system of linear inequalities. When the system of inequalities is graphed, often a feasible region is obtained. TThe points where two or more boundaries intersect are called the vertices of the feasible region.

3. Slide 7 - 3 Copyright © 2009 Pearson Education, Inc. Fundamental Principle of Linear Programming If the objective function, K = Ax + By, is evaluated at each point in a feasible region, the maximum and minimum values of the equation occur at vertices of the region.

4. Slide 7 - 4 Copyright © 2009 Pearson Education, Inc. Solving a Linear Programming Problem 1.Determine all necessary constraints. 2.Determine the objective function. 3.Graph the constraints and determine the feasible region. 4.Determine the vertices of the feasible region. 5.Determine the value of the objective function at each vertex. The solution is determined by the vertex that yields the maximum or minimum value of the objective function.

5. Slide 7 - 5 Copyright © 2009 Pearson Education, Inc. Example Planer Carpentry makes rocking horses and rocking airplanes. Each rocking horse requires 5 hours of woodworking and 4 hours of finishing. Each airplane requires 10 hours of woodworking and 3 hours of finishing. Each month Planer Carpentry has 600 hours of labor for woodworking and 240 hours for finishing. The profit on each rocking horse is $40 and on each airplane is $75. How many of each product should be made in order to maximize profit?

6. Slide 7 - 6 Copyright © 2009 Pearson Education, Inc. Example continued Constraints x = number of rocking horses y = number of rocking airplanes P = 40x + 75y (profit function) 5x + 10y  600 (woodworking hours) 4x + 3y  240 (finishing hours) x  0 and y  0

7. Slide 7 - 7 Copyright © 2009 Pearson Education, Inc. Example continued P = 40(24) + 75(48) = 4560(24, 48) P = 40(0) + 75(60) = 4500(0, 60) P = 40(60) + 75(0) = 2400(60, 0) P = 40(0) + 75(0) = 0(0, 0) P = 40x + 75yVertices The maximum profit would be from making 24 rocking horses and 75 rocking airplanes.

8. Slide 7 - 8 Copyright © 2009 Pearson Education, Inc. Homework: p. 462 # 8 – 18 even Ch. 7.5- 7.6 quiz next class