$1.($Feedbag Food Store$)$ The Feedbag Food Store sells a feed mix made of alfalfa, barley, and corn meal. The table below gives the percentage of fat, fiber, and protein in each ingredient. Thus, we can see that alfalfa is not only the richest in protein, but also highest in fiber. Each $50$ kilograms of mix must contain no more than $8$ kilograms of fat, $5$ kilograms of fiber, and at least $11$ kilograms of protein.

Nutrient Alfalfa Barley Corn Meal Min kg Max kg

Fat $2\mathrm{.}5\%1\mathrm{.}8\%3\mathrm{.}8\%8$

Fiber $18\mathrm{.}0\%8\mathrm{.}0\%1\mathrm{.}5\%5$

Protein $33\mathrm{.}0\%21\mathrm{.}5\%18\mathrm{.}6\%11$

Cost$/$kg $$0\mathrm{.}817$ $$0\mathrm{.}844$ $$0\mathrm{.}723$

How many kilograms of each ingredient should be used in the mix so that the cost of each $50-$kilogram bag is minimized?

a$)$ Formulate the problem as an LP problem.

b$)$ Set this model in Excel. Solve the LP problem using Solver. Attach the Excel output with the LP model, answer report and sensitive report on one page.

c$)$ According to your optimal solution, how many kilograms of each ingredient should be used in the mix? What is the minimum cost per $50-$kilogram bag?