1. Linear Programming?!?! Sec. 7.5

2. Linear Programming In management science, it is often required to maximize or minimize a linear function called an objective function. This is a linear programming problem. In two dimensions, the objective function takes the form f = ax + by, and is used with a system of inequalities, called constraints. The solution to a linear programming problem occurs at one of the vertex points, or corner points, along the boundary of the region.

3. Linear Programming Practice Problems Find the maximum and minimum values of the objective function f = 5x + 3y, subject to the constraints given by the system of inequalities. Start with a graph! (0,8) (0,3) (4,0) (9,0)

4. Linear Programming Practice Problems Find the maximum and minimum values of the objective function f = 5x + 3y, subject to the constraints given by the system of inequalities. Start with a graph! Next, find the corner points! Then evaluate f at the corner points! (9,0), (0,8), (3,2) (x, y) f (9, 0) 45 (0, 8) 24 (3, 2) 21 at (3, 2)none! (unbounded region!)

5. Linear Programming Practice Problems Find the maximum and minimum values of the objective function f = 5x + 8y, subject to the constraints given by the system of inequalities. Where’s the graph? (0,10) (0,14/3) (5,0) (7,0)

6. Linear Programming Practice Problems Find the maximum and minimum values of the objective function f = 5x + 8y, subject to the constraints given by the system of inequalities. Where’s the graph? Corner points? Evaluation of f ? (0,0), (0,14/3), (5,0), (4,2) (x, y) f (0, 0) 0 (0, 14/3) 112/3 (5, 0) 25 (4, 2) 36 at (0, 0) at (0, 14/3)

7. Linear Programming Practice Problems Find the maximum and minimum values of the objective function f = 3x – 2y, subject to the constraints given by the system of inequalities. Where’s the graph? (0,8) (0,1) (0,10) (2,0)(3,0) (10,0)

8. Linear Programming Practice Problems Find the maximum and minimum values of the objective function f = 3x – 2y, subject to the constraints given by the system of inequalities. Where’s the graph? Corner points? Evaluation of f ? (2/5,8), (7,8), (90/49,40/49), (10/3,2/3) (x, y) f (2/5, 8) –14.8 (7, 8) 5 (90/49, 40/49)(10/3, 2/3) at (2/5,8) at (10/3,2/3) 190/4926/3