A cereal manufacturer wishes to produce 1000 pounds of a cereal that contains ex

Question

A cereal manufacturer wishes to produce 1000 pounds of a cereal that contains exactly 10% fiber, 2% fat, and 5% sugar (by weight). The cereal is to be produced by combining four items of raw food material in appropriate proportions. These four items have certain combinations of fiber, fat, and sugar content, and are available at various prices per pound:

The manufacturer wishes to find the amounds of each iterm to be used to produce the cereal in the least expensive way. Formulate the problem as a linear programming problem. What can you say about the existence of a solution to this problem?

Item 1 2 3 4 %fiber 3 8 16 4 %fat 6 46 9 9 %sugar 20 5 4 0 Price / pound 2 4 1 2

Explanation / Answer

Let the amount (in pounds) of item 1 used be x1.

Let the amount (in pounds) of item 2 used be x2.

Let the amount (in pounds) of item 3 used be x3.

Let the amount (in pounds) of item 4 used be x4.

The amout of cereal produced = 1000 lbs

=> x1 + x2 + x3 + x4 = 1000 ....(i)

Amount of fibre in cereal = 10% of 1000 = 100 lbs

=> (3(x1) + 8(x2) + 16(x3) + 4(x4))/100 = 100

=> 3(x1) + 8(x2) + 16(x3) + 4(x4) = 10000 ....(ii)

Amount of fat in cereal = 2% of 1000 = 20 lbs

=> (6(x1) + 46(x2) + 9(x3) + 9(x4))/100 = 20

=> 6(x1) + 46(x2) + 9(x3) + 9(x4) = 2000 ....(iii)

Amount of sugar in cereal = 5% of 1000 = 50 lbs

=> (20(x1) + 5(x2) + 4(x3) + 0(x4))/100 = 50

=> 20(x1) + 5(x2) + 4(x3) = 5000 ....(iv)

Cost of ingredients, C = 2(x1) + 4(x2) + 1(x3) + 2(x4)

problem:

Minimize C = 2(x1) + 4(x2) + 1(x3) + 2(x4)

subject to (i), (ii), (iii), (iv), and x1>=0, x2>=0, x3>=0, x4>=0

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