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The Munchies Cereal Company makes a cereal from several ingredients. Two of the ingredients, oats and...

The Munchies Cereal Company makes a cereal from several ingredients. Two of the ingredients, oats and rice, provide vitamins A and B. The company wants to know how many ounces of oats and rice it should include in each box of cereal to meet the minimum requirements of 48 milligrams of vitamin A and 12 milligrams of vitamin B while minimizing cost. An ounce of oats contributes 8 milligrams of vitamin A and 1 milligram of vitamin B, whereas an ounce of rice contributes 6 milligrams of A and 2 milligrams of B. An ounce of oats costs $0.05, and an ounce of rice costs $0.03.

a. Formulate a linear programming model for this problem.

b. Solve this model by using graphical analysis.

math   Statistics-And-Probability   
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Answer #1

a) Linear programming model:

Let x1 = oats

x2 = rice

Minimize: Z = 0.052.1 + 0.032.2

Subject to constraints:

81 6r2 48 OT

1r1 2r212

1,2

b) Graphical solution:

8 7 6 5 3 2 8x1+6x2=348 x1 +2x2-12 02 1 2 345678 9 10 11 12

The value of the objective function at each of these extreme points is as follows Extreme Point Coordinates (x12) Objective function value z0.05x,+0.03x Lines through Extreme Point 1 -Sr1 + 6x2 2 48 3→X120 4(0, 8) 0.05(0) + 0.03(8) 0.24 12 24 +0.03 = 0.26 2x1+2x2 2 12 2x12x2 2 12 C(12, 0) 0.05(12) + 0.03(0)0.6 The minimum value of the objective function z 0.24 occurs at the extreme point (0, 8) Hence, the optimal solution to the given LP problem is : x,-0, x2-8 and min z 0.24

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