HW Unit $7$: Multiple Objective Optimization $(60$ points$)$ Submit all answers via Canvas.

$1.$ 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:

The only requirement is that the blended meat consists of at least $\backslash (75\backslash \%\backslash )$ protein and at most $\backslash (10\backslash \%\backslash )$ 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 $\backslash$#$1\backslash$& $2$

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, $\backslash (5\backslash$mathbf$\{$p t s$\}\backslash ))$

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$)$