reaches 60,000 people in the first group and 30,000 in the second. We require that at least 6 units of Tv advertising be used and that no more than 12 units of magazine advertising be used for policy reasons. The advertising budget is 360,000 Formulate this problem as a linear programming problem, defining all variables used.(10 points) Solution: we set advertise xI TV unit and x2 magazine unit, the linear programming mo maxZ=2(20000x1+60000x2)+(80000x+30000x2) 40000x1+24000x2≤360000 x1≥6 s t x,≤12 x1,x2≥0 3.A manufacturing firm has discontinued the production of a certain unprofitable product line. This act created considerable excess production capacity. Management is considering devoting this excess capacity to one or more of three products; call them products 1, 2, and 3. The available capacity on the machines that might limit output is summarized in the following table Machine Ty Available Time(in Machine Hours per W Milling machine 500 Lathe 350 The number of machine hours required for each unit of the respective products is as Machine type Product 1 Product 2 Product 3 Lathe 953 4 Grinde 2 The sales department indicates that the sales potential for products I and 2 exceeds the maximum production rate and that the sales potential for product 3 is 20 units per week. The unit profit would be $50, $20, and $25, respectively, on products 1, 2 and 3 the objective is to determine how much of each product the firm should produce to maximize profit Formulate a linear programming model for this problem. (10 points) Solution: we set the production of production 1, 2, 3 are x1, X2, X3. So the linear programming model is maxZ=50x1+20x,+25 9x1+3x2+5x3≤500 5x,+4x,+0x2≤350 s3x1+0x,+2x2≤150 x,x2 reaches 60,000 people in the first group and 30,000 in the second. We require that at least 6 units of TV advertising be used and that no more than 12 units of magazine advertising be used for policy reasons. The advertising budget is 360,000. Formulate this problem as a linear programming problem, defining all variables used. (10 points) Solution: we set advertise x1 TV unit and x2 magazine unit, the linear programming model is ⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ ≥ ≤ ≥ + ≤ = + + + , 0 12 6 40000 24000 360000 . . max 2(20000 60000 ) (80000 30000 ) 1 2 2 1 1 2 1 2 1 2 x x x x x x st Z x x x x 3.A manufacturing firm has discontinued the production of a certain unprofitable product line. This act created considerable excess production capacity. Management is considering devoting this excess capacity to one or more of three products; call them products 1,2, and 3. The available capacity on the machines that might limit output is summarized in the following table: Machine Type Available Time (in Machine Hours per Week) Milling machine Lathe Grinder 500 350 150 The number of machine hours required for each unit of the respective products is as follows: Machine Type Product 1 Product 2 Product 3 Milling machine Lathe Grinder 9 5 3 3 4 0 5 0 2 The sales department indicates that the sales potential for products 1 and 2 exceeds the maximum production rate and that the sales potential for product 3 is 20 units per week. The unit profit would be $50, $20, and $25, respectively, on products 1,2 and 3. the objective is to determine how much of each product the firm should produce to maximize profit. Formulate a linear programming model for this problem. (10 points) Solution: we set the production of production 1, 2, 3 are x1, x2, x3. So the linear programming model is ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ ≥ ≤ + + ≤ + + ≤ + + ≤ = + + , , 0 20 3 0 2 150 5 4 0 350 9 3 5 500 . . max 50 20 25 1 2 3 3 1 2 3 1 2 3 1 2 3 1 2 3 x x x x x x x x x x x x x st Z x x x