Question

A hospital dietitian prepares breakfast menus every morning for the hospital patients. Part of the dietitian’s responsibility is
to make sure that minimum daily requirements for vitamins A and B are met. At the same time, the cost of the menus must
be kept as low as possible. The main breakfast staples providing vitamins A and B are eggs, bacon, and cereal. The vitamin
requirements and vitamin contributions for each staple follow:
Vitamin Contributions
Vitaminmg/Egg mg/Bacon Stripmg/Cereal CupMinimum Daily Requirements
A 2 4 1 16
B 3 2 1 12
An egg costs $0.04, a bacon strip costs $0.03, and a cup of cereal costs $0.02. The dietitian wants to know how much of
each staple to serve per order to meet the minimum daily vitamin requirements while minimizing total cost.

a. Formulate a linear programming model for this problem.
b. Solve the model by using the computer.

Answer 1

Answer:

2 eggs, 3 bacon strips, no cereal for a minimum cost of $0.17

Explanation:

Optimization

We are required to find the optimum values for each staple to serve per order of breakfast to reduce the cost at a minimum while meeting the minimum daily vitamin requirements.

Each staple contributes in different amounts to the daily vitamin requirements. One egg costs $0.04 and provides 2 units of vitamin A and 3 of vitamin B

One bacon costs $0.03 strip provides 4 units of vitamin A and 2 of vitamin B

One cup of cereal costs $0.02 provides 1 unit of vitamin A and 1 of vitamin B

Let's assume we serve x eggs, y bacon strips and z cups of cereal per breakfast unit. The total contribution to vitamin A is 2x+4y+z and it must be not less than 16, thus

$2x+4y+z \geq 16$

The total contribution to vitamin B is 3x+2y+z and it must be not less than 12, thus

$3x+2y+z \geq 12$

The total cost is

$C=0.04x+0.03y+0.02z$

a. The linear programming model for this problem is

Minimize

$C=0.04x+0.03y+0.02z$

Subject to

$2x+4y+z \geq 16$

$3x+2y+z \geq 12$

$x \geq0,\ y\geq 0,\ z\geq 0$

b. We used Excel's solver to find the optimum solution for x, y, and z as follows

x=2 eggs, y=3 bacon strips and z=0 cups of cereal, which produce the minimum cost

C=$0.17

Answer 2

Linear programming model is given by $2x+4y+z\geq 16$ and $3x+2y+z\geq 12$ and the value of (x = 2), (y = 3), and (z = 0). The total cost is $0.17 and this can be determined by using the given data.

Given :

• A hospital dietitian prepares breakfast menus every morning for the hospital patients.
• Part of the dietitian’s responsibility is  to make sure that minimum daily requirements for vitamins A and B are met.
• The main breakfast staples providing vitamins A and B are eggs, bacon, and cereal.
• An egg costs $0.04, a bacon strip costs $0.03, and a cup of cereal costs $0.02.

Let the total number of eggs be 'a', the total number of bacon strips be 'b', and the total number of cereal be 'c'.

The total contribution to vitamin A is given by:

$2x+4y+z\geq 16$

The total contribution to vitamin B is given by:

$3x+2y+z\geq 12$

The total cost is given by:

C = 0.04x + 0.03x + 0.02z

b) by using the excel solver in order to find the values of x, y, and z.

x = 2 eggs

y = 3 bacon strip

z = 0 cups of cereals

So, the value of C is:

C = $0.17

For more information, refer to the link given below:

https://brainly.com/question/21835898