only solve problem $14$ by using excel solver$($must show all the steps and screenshots of excel $)12.$ United Aluminum Company of Cincinnati produces three grades $($high$,$ medium, and low$)$ of aluminum at two mills. Each mill has a different production capacity $($in tons per day$)$ for each grade, as follows:

The company has contracted with a manufacturing firm to supply at least $12$ tons of high$-$grade aluminum, $8$ tons of medium$-$grade aluminum, and $5$ tons of low$-$grade aluminum. It costs United $\backslash (\backslash$$ $6,000\backslash )$ per day to operate mill $1$ and $\backslash (\backslash$$ $7,000\backslash )$ per day to operate mill $2.$ The company wants to know the number of days to operate each mill to meet the contract at the minimum cost.

Formulate a linear programming model for this problem.

$13.$ Solve the linear programming model formulated in Problem $12$ for United Aluminum Company graphically.

a$.$ How much extra $($i$.$c$.,$ surplus$)$ high$-,$ medium$-,$ and low$-$grade aluminum does the company produce at the optimal solution?

b$.$ What would be the effect on the optimal solution if the cost of operating mill $1$ increased from $\backslash (\backslash$$ $6,000\backslash )$ to $\backslash (\backslash$$ $7,500\backslash )$ per day?

c$.$ What would be the effect on the optimal solution if the company could supply only $10$ tons of high$-$grade aluminum?

$14.$ Solve the linear programming model formulated in Problem $12$ for United Aluminum Company by using the computer.

a$.$ Identify and explain the shadow prices for each of the aluminum grade contract requirements,

b$.$ Identify the sensitivity ranges for the objective function coefficients and the constraint quantity values.

c$.$ Would the solution values change if the contract requirements for high$-$grade aluminum were increased from $12$ tons to $20$ tons? If yes, what would the new solution values be$?$