$(03)$ A diet problem: Determine the optimal selection of proportions of foods from the list of foods provided in the table below, by answering the following questions. Note that $\backslash ($ g $\backslash )$ stands for grams.

Let $\backslash ($ x$\_\{1\},$ x$\_\{2\},$ x$\_\{3\},$ x$\_\{4\},$ x$\_\{5\}\backslash )$ and $\backslash ($ x$\_\{6\}\backslash )$ be the optimal proportions $($units$)$ of foods from Bread, Milk, Cheese, Potato, Fish and Yogurt, respectively.

$($a$)$ Protein intake: the total protein intake should be less than or equal to $10$ g $.$ Write down a constraint mathematically for protein intake.

$($b$)$ Carbohydrate intake: required minimum is $10$ g $.$ Write down a constraint mathematically for carbohydrate intake.

$($c$)$ Fat intake: required minimum is $8$ g $.$ Write down a constraint mathematically for fat intake.

$($d$)$ Write down all the nonnegativity constraints.

$($e$)$ Additional Constraint $-$ Fish: the diet should contain at least $0\mathrm{.}5$ units of fish. Write down a constraint mathematically to explain the requirement.

$($f$)$ Additional Constraint $-$ Milk: the diet should contain less than or equal to $1$ unit of milk. Write down a constraint mathematically to explain the requirement. $($g$)$ Total calorie intake: required minimum is $300$ calories. Write down a constraint mathematically for total calorie requirement.

$($h$)$ Objective function is the Total Cost: write down the objective function mathematically and indicate if it should be maximized or minimized.

$($i$)$ Solve the problem using the Simplex Method in MS Excel. Completely explain the solution obtained.

$(04)$ The cost of transportation per truck load from each of the six factories to each of the $9$ warehouses is given in the matrix below. Determine the optimal transportation plan using the Simplex Method in MS Excel. The Excel file with the dataset is available on Canvas for your convenience. $[$You do not need to upload Excel output as a part of your answer. Clearly explain the answer obtained$].$