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Lecture 7.2 bt

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Lecture 7.2 bt

  1. 1. Today’s Agenda  Attendance / Announcements  Questions from Yesterday  Sections 7.2  Quiz Today
  2. 2. Exam Schedule  Exam 4 (Ch 6,7) Fri 11/15  Exam 5 (Ch 10) Thur 12/5  Final Exam (All) Thur 12/12
  3. 3. 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.
  4. 4. Linear Programming The Objective Function is what we need to maximize or minimize. For us, this will be a function of 2 variables, f(x, y)
  5. 5. Linear Programming The Constraints are the inequalities that provide us with the Feasible Region.
  6. 6. Linear Programming (pg 400)
  7. 7. The general idea… (pg 398) Find max/min values of the objective function, subject to the constraints. Objective Function f ( x, y) 2x 5 y Constraints 3x 2 y 6 2x 4 y 8 x y 1 x 0, y 0
  8. 8. The general idea… (pg 398) Graph the Feasible Region
  9. 9. The general idea… (pg 398) The Feasible Region makes up the possible inputs to the Objective Function f ( x, y) 2x 5 y
  10. 10. Corner Point Thm (pg 400) If a feasible region is bounded, then the objective function has both a maximum and minimum value, with each occurring at one or more corner points.
  11. 11. Find the minimum and maximum value of the function f(x, y) = 3x - 2y. We are given the constraints: • y≥2 • 1 ≤ x ≤5 • y≤x+3
  12. 12. 1 ≤ x ≤5 8 7 6 5 4 3 y≤x+3 y≥2 2 1 1 2 3 4 5
  13. 13. Need to find corner points (vertices) 1 ≤ x ≤5 8 7 6 5 4 3 y≤x+3 y≥2 2 1 1 2 3 4 5
  14. 14. • The vertices (corners) of the feasible region are: (1, 2) (1, 4) (5, 2) (5, 8) • Plug these points into the function f(x, y) = 3x - 2y Note: plug in BOTH x, and y values.
  15. 15. Evaluate the function at each vertex to find min/max values 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
  16. 16. So, the optimized solution is: • f(1, 4) = -5 minimum • f(5, 2) = 11 maximum
  17. 17. Find the minimum and maximum value of the function f(x, y) = 4x + 3y With the constraints: y y y x 2 1 x 2 4 2x 5
  18. 18. y ≥ 2x -5 Need to find corner points (vertices) 6 5 y ≤ 1/4x + 2 4 y ≥ -x + 2 3 2 1 1 2 3 4 5
  19. 19. Evaluate the function at each vertex to find min/max values f(x, y) = 4x + 3y • f(0, 2) = 4(0) + 3(2) = 6 • f(4, 3) = 4(4) + 3(3) = 25 7 1 1 7 • f( 3 , - 3 ) = 4( 3 ) + 3(- 3 ) = 28 3 -1 = 25 3
  20. 20. So, the optimized solution is: • f(0, 2) = 6 minimum • f(4, 3) = 25 maximum
  21. 21. Classwork / Homework • Page 403 • 1, 3, 7, 9, 11

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