1. 1 What you will learn  Lots of vocabulary!  How to find the maximum and minimum value of a function given a set of “rules”

2. Objective: 3.4 Linear Programming 2 Linear Programming Optimization is the process of 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. The graph of the system of constraints is called the feasible region.

3. Objective: 3.4 Linear Programming 3 An Example  Find the minimum value and the maximum value of C = 3x + 4y subject to the following constraints:

4. Objective: 3.4 Linear Programming 4 Steps If an objective function has a maximum or a minimum value then it must occur at a vertex of the feasible region. 1. Graph the constraints and shade. 2. Find the vertices and “plug” those values of x and y into the objective function. 3. Determine the minimum and maximum values.

5. Objective: 3.4 Linear Programming 5 You Try  Find the minimum value and the maximum value of C = -x + 3y subject to the following constraints:

6. Objective: 3.4 Linear Programming 6 Another Example  Find the minimum value and the maximum value of C = 5x + 6y subject to the following constraints:

7. Objective: 3.4 Linear Programming 7 What If the Feasible Region is Unbounded? If the feasible region has an unbounded portion (no end), you can still get a maximum or minimum. Example: (provided by Mrs. C.)

8. Objective: 3.4 Linear Programming 8 A Real-World Example  Two manufacturing plants make the same kind of bicycle. For the two plants combined, the manufacturer can afford to use up to 4000 hours of general labor, up to 1500 hours of machine time, and up to 2300 hours of technical labor per week. Plant A earns a profit of $60 per bicycle and Plant B earns a profit of $50 per bicycle. How many bicycles per week should the manufacturer make in each plant to maximize profits.

9. Objective: 3.4 Linear Programming 9 The Facts Ma’am  Step 1: Write an objective function. Let a and b represent the number of bicycles made in Plant A and Plant B. Therefore, the maximum profit will be: P = 60a + 50b

10. Objective: 3.4 Linear Programming 10 More Facts The Constraints: ResourceHours per bicycle in Plant A Hours per bicycle in Plant B Gen Labor101 Machine Time13 Technical Labor52

11. Objective: 3.4 Linear Programming 11 The Graph Get the vertices:

12. Objective: 3.4 Linear Programming 12 Plug It in Plug It In Plug the vertices back into the original objective function: P = 60a +50b(0, 500) (300, 400) (380, 200) (400, 0) (0, 0)

13. Objective: 3.4 Linear Programming 13 Homework Page 166, 10, 14, 18, 22