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8.6 Linear Programming

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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)

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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 restricts to 1 st quadrant (10, 0), (0, 8) (5, 0), (0, 12) lines meet at … 8x + 10y = 80 12x + 5y = 60 –2( ) (0, 8) (5, 0) (0, 0) (2.5, 6) 8x + 10y = 80 –24x – 10y = –120 –16x = –40 x = 2.5 20 + 10y = 80 10y = 60 y = 6 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

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

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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 1x + 4y ≤ 16 2x + 2y ≤ 20 x ≥ 0 y ≥ 0 dye hours want (+) scarfs P = 3x + 4y (0, 4): 3(0) + 4(4) (8, 2): 3(8) + 4(2) (10, 0): 3(10) + 4(0) = $16 = $32 = $30 (16, 0), (0, 4) (10, 0), (0, 10) –2x – 8y = –32 2x + 2y = 20 –6y = –12 y = 2 MAX Sky should make 8 square scarves & 2 rectangle scarves x + 8 = 16 x = 8 (0, 4) (10, 0) (8, 2)

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

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1 Chapter 14 Making the most of things: Linear Programming.

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