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Linear Programming?!?! Sec. 7.5

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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.

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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)

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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!)

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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)

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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)

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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)

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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

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