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8.6 Linear Programming

8.6 Linear Programming. Linear Program : a mathematical model representing restrictions on resources using linear inequalities combined with a function to be optimized Objective Function : function to be minimized or maximized Constraints : the linear inequalities

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8.6 Linear Programming

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  1. 8.6 Linear Programming

  2. Linear Program: a mathematical model representing restrictions on resources using linear inequalities combined with a function to be optimized • Objective Function: function to be minimized or maximized • Constraints: the linear inequalities • Feasible region: the solution to the system of linear inequalities • If a linear program has a solution it will occur at a vertex • To Solve a Linear Program: • 1) Solve the system of inequalities • 2) Determine the vertices of the feasible region • 3) Evaluate the objective function at each vertex • 4) Identify the max/min of the objective function (graph them)

  3. Ex 1) Determine the values of x & y that will maximize the function P = 300x + 150y under the constraints • 8x + 10y ≤ 80 • 12x + 5y ≤ 60 • x ≥ 0 • y ≥ 0  (10, 0), (0, 8)  (5, 0), (0, 12) (0, 8) (2.5, 6) • restricts to • 1st quadrant • lines meet at … • 8x + 10y = 80 • 12x + 5y = 60 (0, 0) (5, 0) • 8x + 10y = 80 • –24x – 10y = –120 –2( ) • 20 + 10y = 80 • 10y = 60 • y = 6 –16x = –40 x = 2.5 • P = 300x + 150y • (0, 0): 300(0) + 150(0) • (0, 8): 300(0) + 150(8) • (2.5, 6): 300(2.5) + 150(6) • (5, 0): 300(5) + 150(0) = 0 = 1200 = 1650 = 1500 want MAX P(2.5, 6) = 1650 maximum

  4. Ex 2) Determine the values of x & y that will minimize the function C = 7x + 3y under the constraints 30 • 4x + 3y ≤ 75 • x + 3y ≥ 30 • x ≥ 0 • y ≥ 0  (18.75, 0), (0, 25) (0, 25)  (30, 0), (0, 10) • restricts to • 1st quadrant (0, 10) (15, 5) 5 • lines meet at … • 4x + 3y = 75 • –x – 3y = –30 5 30 • 15 + 3y = 30 • 3y = 15 • y = 5 3x = 45 x = 15 want MIN • C = 7x + 3y • (0, 10): 7(0) + 3(10) • (0, 25): 7(0) + 3(25) • (15, 5): 7(15) + 3(5) = 30 = 75 = 120 C(0, 10) = 30 minimum

  5. Ex 3) Sky tie-dyes silk scarves. She makes square & rectangular scarves. A square scarf requires 1 oz of dye and a rectangular scarf requires 4 oz of dye. It takes 2 hours to make either type of scarf. She makes $3 profit on each square scarf and $4 on each rectangular scarf. This week she has 16 oz of dye on hand and can work for 20 hours. Assuming she can sell all of the scarves she makes, determine how many of each type she should make to maximize the profit. Let x = # of square y = # of rectangle  (16, 0), (0, 4) • 1x + 4y ≤ 16 • 2x + 2y ≤ 20 • x ≥ 0 • y ≥ 0 dye  hours   (10, 0), (0, 10) want (+) scarfs (0, 4) (8, 2) • –2x – 8y = –32 • 2x + 2y = 20 (10, 0) • P = 3x + 4y • (0, 4): 3(0) + 4(4) • (8, 2): 3(8) + 4(2) • (10, 0): 3(10) + 4(0) = $16 = $32 = $30 –6y = –12 y = 2 • x + 8 = 16 • x = 8 MAX Sky should make 8 square scarves & 2 rectangle scarves

  6. Homework #806 Pg 422 #1–7 odd, 11, 14, 17, 22, 24, 25

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