Healthy Choice $(35$ Points$)$

A school dietitian wants to determine the best blend of meats to use in an order of $500$ pounds of ground meat for hamburgers to be used for lunch at a local school. Three types of meat can be used in the blend with each having the following characteristics:

Meat $1$ Meat $2$ Meat $3$

Cost per pound $$0\mathrm{.}74$ $$0\mathrm{.}95$ $$1\mathrm{.}02$

$\%$ Fat $15\%10\%5\%$

$\%$ Protein $68\%75\%85\%$

$\%$ Water $13\%9\%8\%$

$\%$ Filler $2\%5\%6\%$

The only requirement is that the blended meat consists of at least $75\%$ protein and at most $10\%$ each of water and filler. Although the dietitian wants to minimize his cost of filling the order, she would also like to provide a healthy lunch for the school. As a result, the owner also wants to minimize the fat content of the blended meat as well as the cost.

Hint: See Unit $7$ Practice Problem #$1$ & $2$

a$.$ State the decisions to be made, the objective and the constraints in words using $<=,>=$ or $=$ in the constraints. $($Submit copy of Handwritten answer, $5$ pts$)$

b$.$ Formulate this as a linear two objective algebraic optimization model. Carefully define your variables. $($Submit copy of Handwritten answer, $5$ pts$)$

c$.$ Using Fat as objective $1$ and Cost as objective $2,$ find the fat range of the Trade$-$off Problem. Note: Temporarily ignore cost and solve for Min and Max possible fat to get a range of fat values for a Trade$-$off curve. $($Submit copy of Handwritten answer, $5$ pts$)$

d$.$ Using your trade$-$off problem fat range in $($c$),$ generate the Trade$-$off curve of Cost versus Fat using Excel. Note: Use the fat range calculated above, solve for cost a couple times, and use the sensitivity report to create the graph. $($Submit answer in an Excel file, $5$ pts$)$

e$.$ Identify the Efficient Frontier. $($Submit answer in an Excel file, $5$ pts$)$

f$.$ What point on the Efficient Frontier Minimizes Fat? What is its corresponding Efficient Solution? $($Submit answer in an Excel file, $5$ pts$)$

g$.$ What point on the Efficient Frontier minimizes Cost? What is its

corresponding Efficient Solution? $($Submit answer in an Excel file, $5$ pts$)$