POMS $5010$ Extra$-$Credit Assignment $(10$ percent course bonus$)$

$1.$ Food glorious food $(100$ pts$)$

You are the head planner for the local school system and your task is to design a subsidized lunch program. Speci$$cally$,$ you are to formulate the least costly lunch menu that will satisfy the following state and federal nutritional guidelines:

$($i$)$ A meal must be between $500$ and $800$ calories

$($ii$)$ It must contain at least $200$ calories of protein, at least $200$ calories of carbohydrates, but no more than $400$ calories of fat

$($iii$)$ It must provide at least $200$ calories from a food that is classi$$ed as a fruit or a vegetable

The table below provides information on the complete list of foods that could feature on your menu.

FOOD Cost per serving $($$$)$ Calories per serving $\%$ Protein $\%$ Carbs $\%$ Fat Fruit$/$Veg$.$

Applesauce $0\mathrm{.}3010001000$ Yes

Brussels sprouts $0\mathrm{.}4015020800$ Yes

Chicken $0\mathrm{.}9025055540$ No

French fries $0\mathrm{.}2040053560$ No

Garden salad $0\mathrm{.}90100154045$ Yes

Mac & cheese $0\mathrm{.}50430203050$ No

Turkey breast $1\mathrm{.}5030067033$ No

$($a$)$ Write down an optimization problem to minimize the cost of your lunch o$$ering $(50$ points$)$

Hint: Use the following decision variables:

A: Number of servings of Applesauce

B: Number of servings of Brussels sprouts C: Number of servings of Chicken

F: Number of servings of French fries G: Number of servings of Garden salad M: Number of servings of Mac & cheese T: Number of servings of Turkey breast

$($b$)$ Build an Excel model to solve your optimization problem. What is your optimal lunch o$$ering and how much does it cost$?(50$ points$)$