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3-5: Linear Programming

3-5: Linear Programming. Learning Target. I can solve linear programing problem. Terms. Optimization – is finding the minimum or maximum value of some quantity.

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3-5: Linear Programming

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  1. 3-5: Linear Programming

  2. Learning Target • I can solve linear programing problem.

  3. Terms • Optimization – is finding the minimum or maximum value of some quantity. • Linear programming is a form of optimization where you optimize an objective function with a system of linear inequalities called constraints. • The overlapped shaded region is called the feasible region.

  4. Solving a linear programming problem 1. Graph the constraints. 2. Locate the ordered pairs of the vertices of the feasible region. 3. If the feasible region is bounded (or closed), it will have a minimum & a maximum. If the region is unbounded (or open), it will have only one (a minimum OR a maximum). 4. Plug the vertices into the linear equation (C=) to find the min. and/or max.

  5. A note about: Unbounded Feasible Regions • If the region is unbounded, but has a top on it, there will be a maximum only. • If the region is unbounded, but has a bottom, there will be a minimum only.

  6. 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.

  7. 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

  8. 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.

  9. 1 ≤ x ≤5 8 7 6 5 4 y ≥ 2 3 2 y ≤ x + 3 1 3 5 4 1 2

  10. 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

  11. 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

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

  13. 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

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

  15. 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 =

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

  17. Pair-share Work on page 166 to 167 on #9 to 17 Homework Homework on page 167 #18 to #20

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