The dietician at a camp is planning breakfast for the first

The dietician at a camp is planning breakfast for the first day of camp. The dietician has the responsibility of providing a menu that satisfies the minimum nutrient requirements at the lowest cost. Two types of foods are being considered for the breakfast: toast and sausage.A piece of toast contains 2 milligrams of vitamin A, 3 milligrams of vitamin B, and 2 milligram of iron. A sausage contains 4 milligrams of vitamin A, 1.5 milligrams of vitamin B, and 2 milligram of iron. The minimum breakfast requirements of these nutrient elements are estimated to be:

Nutrient         Requirement (mg)

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Vitamin A                   20

Vitamin B                   15

Iron                             16

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The American Medical Association has published an article which reported that having more than four sausages for breakfast is not recommended for young people. The dietitian considers this one of the most important constraints. The unit costs of the food are: toast, 4 cents; sausage, 8 cents. Formulate this problem as a linear programming problem, solve it by the software and answer the following questions.

1.

How many variables are there in your LP model?

A) one
B) two
C) three
D) four
E) more than four

2.

How many constraints does your LP model have (excluding the nonnegative constraints)?

A) one
B) two
C) three
D) four
E) more than four

3.

What is the cost of this breakfast that the dietitian would provide for everyone in the camp?

A) $0.20
B) $0.40
C) $1.20
D) $1.50
E) none of the above

4.

How many toasts should be given in the breakfast?

A) one
B) two
C) three
D) four
E) more than four

5.

How many milligrams of Vitamin B does this breakfast menu provide?

A) 15 milligrams
B) 18 milligrams
C) 21 milligrams
D) 24 milligrams
E) None of the above

6.

How many milligrams of iron does this breakfast menu provide?

A) 12 milligrams
B) 16 milligrams
C) 20 milligrams
D) 24 milligrams
E) none of the above

Solution

Let x and y be the pieces of toast and sausage required.

Then the objective function is

Min Z=0.04x+0.08y

Constraints:

Vitamin A, 2x+4y20

Vitamin B, 3x+1.5y15

Iron, 2x+2y16

Sausage, y4

The non-negative constraints are x0, y0

Thus, the linear programming model is

Min Z=0.04x+0.08y

Subject to,

2x+4y20

3x+1.5y15

2x+2y16

y4

x0, y0

1) There are two variables in the LP model. So, option (B) is correct.

2) The LP model contains four constraints (excluding the non negative constraints). So, option (D) is correct.

3) Solving the LP model using a software, we get the solution as

z=0.4, at x=6 and y=2

So, The cost of the breakfast is $ 0.4. Option (B) is correct.

4) The breakfast should contains 6 toasts.

5) The amount of Vitamin B in the breakfast = 3(6)+1.5(2)=18+3=21 milligrams. So, option (C) is correct.

6) The amount of Iron in the breakfast= 2(6)+2(2)=12+4=16 milligrams. So, option (B) is correct.