We have been solving linear inequalities. i.e. Word Problem from homework. I see that there are many solutions, but what is the BEST one?
Optimization: how to find the maximum or minimum of a quantity. Linear Programming: method of finding a maximum or minimum value (optimization) of a function given the constraints. Constraint : an INEQUALITY that restricts one of the variables Feasible Region : POSSIBLE SOLUTION
Have students graph on whiteboards and hold up. Intercepts: (8, 0) (0, 4) (4, 0), (6, 0) ** Must be careful with shading
Have students switch who has board and graph. DON’T FORGET TO COPY DOWN ALL IMPORTANT INFO FROM PROBLEM. Write inequalities on board.
Objective Function: The objective function may have a minimum, a maximum, neither, or both depending on the feasible region. **Typically a linear equation ** What is being maximized or minimized
Objective Function: The objective function may have a minimum, a maximum, neither, or both depending on the feasible region. **Typically a linear equation Why do you need the feasible region from Example 1 to solve the problem? Why do you think the vertices as well as points inside the feasible region can be used to evaluate the function? ** What is being maximized or minimized?
Recall: What does the Vertex Principle tell us? Set up chart
Call up students to fill in answers 0, 1350, 1650, 1440 WORKSHEET
<ul><li>Bell Ringer: </li></ul><ul><li>Graph the system of inequalities from problem # 36 on your homework last night. </li></ul><ul><li>2. Is (-1,1) a solution of the system? </li></ul><ul><li>Why or why not? </li></ul>
optimization linear programming constraint feasible region New Vocabulary
Check It Out! Example 1 Graph the feasible region for the following constraints. (Hint: Find the vertices) x ≥ 0 y ≥ 1.5 2.5 x + 5 y ≤ 20 3 x + 2 y ≤ 12
Yum’s Bakery bakes two breads, A and B . One batch of A uses 5 pounds of oats and 3 pounds of flour. One batch of B uses 2 pounds of oats and 3 pounds of flour. The company has 180 pounds of oats and 135 pounds of flour available. Write the constraints for the problem and graph the feasible region. Example 1: Graphing a Feasible Region
The feasible region is a quadrilateral with vertices at (0, 0), (36, 0), (30, 15), and (0, 45). Check A point in the feasible region, such as (10, 10), satisfies all of the constraints.
Why would we want to find a feasible region? objective function:
Yum’s Bakery wants to maximize its profits from bread sales. One batch of A yields a profit of $40. One batch of B yields a profit of $30. Use the profit information and the data from Example 1 to find how many batches of each bread the bakery should bake. Example 2: Solving Linear Programming Problems
Example 2 Continued Step 1 Let P = the profit from the bread. Write the objective function: P = 40 x + 30 y Step 2 Recall the constraints and the graph from Example 1. x ≥ 0 y ≥ 0 5 x + 2 y ≤ 180 3 x + 3 y ≤ 135
Example 2 Continued Step 3 Evaluate the objective function at the vertices of the feasible region. Yum’s Bakery should make 30 batches of bread A and 15 batches of bread B to maximize the amount of profit. ( x , y ) 40 x + 30 y P($) (0, 0) 40(0) + 30(0) (0, 45) (30, 15) (36, 0)
Ticket out the Door Use your own words to define Vertex Principle of Linear Programming.