Problem $10(10$pts$)$ Three sellers offer three foods, which contain three different nutrients. We are given Table $2$ with the nutritional content of a unit of each food, the minimum requirement of each nutrient, and the selling price for a unit of each food.

Design a diet so as to satisfy the minimum requirement of the nutrients and minimize the total price to purchase the foods in the diet.

$($a$).$ Formulate the diet problem as a linear programming model.?

$($b$).$ Solve the LP by Excel and obtain the sensitivity analysis report $($please insert a screenshot of your sensitivity analysis figure here$).$

$($c$).$ The seller of pizza would like to increase his revenue by increasing the price of pizza quoted to us$.$ Is it possible for him to do so$?$ If it is possible, how much more revenue can he obtain? If not, what should the seller do to increase his revenue?

$($d$).$ How about the seller of burger? Can he increase his revenue by increasing the price of burger? If yes, how much more revenue can he obtain? If no$,$ what should the seller do to increase his revenue?

$($e$).$ Can we save any money if the minimum requirement of protein is decreased to $16?$ If yes, what is the amount of the savings?

$($f$).$ Does our optimal cost increase or decrease if the minimum requirement of fat is decreased to $5?$

$($g$).$ If one hot dog provides $2$ units of carbohydrates, $3$ units of fat and $3$ units of protein, and it is sold at a price of $3\mathrm{.}2.$ Do we want to include it in the diet?

$($h$).$ If the amount of protein contained in burger is increased to $5\mathrm{.}5,$ but its fat is decreased to $2.$ Do we want to have it in the diet?