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Optimizing Profit and Costs with Linear Programming

Linear programming is a powerful method used by businesses to maximize profit or minimize costs while adhering to various constraints. This guide illustrates how to find the minimum and maximum values of functions like f(x, y) = 0.3x - 0.2y under given constraints. By graphing inequalities and identifying the vertices of the polygon formed, you can evaluate the function at these points. For instance, the vertices (1, 2), (1, 4), (5, 2), and (5, 8) show that the minimum is f(1, 4) = -5 and the maximum is f(5, 2) = 0.11.

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Optimizing Profit and Costs with Linear Programming

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  1. 3-5: Linear Programming

  2. Linear Programming Businesses use linear programming to find out how to maximize profit or minimize costs. Most have constraints on what they can use or buy.

  3. Find the minimum and maximumvalue of the function f(x, y) = 3x - 2y. We are given the constraints: • y ≥ 2 • 1 ≤ x ≤5 • y ≤ x + 3

  4. Linear Programming • Find the minimum and maximum values by graphing the inequalities and finding the vertices of the polygon formed. • Substitute the vertices into the function and find the largest and smallest values.

  5. 1 ≤ x ≤5 8 7 6 5 4 y ≥ 2 3 2 y ≤ x + 3 1 3 5 4 1 2

  6. Linear Programming • The vertices of the quadrilateral formed are: (1, 2) (1, 4) (5, 2) (5, 8) • Plug these points into the function f(x, y) = 3x - 2y

  7. Linear Programming f(x, y) = 3x - 2y • f(1, 2) = 3(1) - 2(2) = 3 - 4 = -1 • f(1, 4) = 3(1) - 2(4) = 3 - 8 = -5 • f(5, 2) = 3(5) - 2(2) = 15 - 4 = 11 • f(5, 8) = 3(5) - 2(8) = 15 - 16 = -1

  8. Linear Programming • f(1, 4) = -5 minimum • f(5, 2) = 11 maximum

  9. Find the minimum and maximum value of the function f(x, y) = 4x + 3y We are given the constraints: • y ≥ -x + 2 • y ≤ x + 2 • y ≥ 2x -5

  10. y ≥ 2x -5 6 5 4 3 y ≥ -x + 2 2 1 1 2 3 4 5

  11. Vertices f(x, y) = 4x + 3y • f(0, 2) = 4(0) + 3(2) = 6 • f(4, 3) = 4(4) + 3(3) = 25 • f( , - ) = 4( ) + 3(- ) = -1 =

  12. Linear Programming • f(0, 2) = 6 minimum • f(4, 3) = 25 maximum

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