Formulating a linear programming model for this problem of United Aluminum Company of Cincinnati:

The objective is to minimize the cost of operating the mills while meeting the contract requirements.

Lex X$1$ be the number of days to operate the mill $1,$ and x$2$ be the number of days to operate mill $2.$

Decision Variables:

X$1$: Number of days to operate Mill $1$

X$2$: Number of days to operate Mill $2$

Objective Function:

Z$=6000$x$1+7000$x$2$

Constraints:

Production Requirements: Aluminum grade:

High: $6$x$1+2$x$2>=12$

Medium: $2$x$1+2$x$2>=8$

Low: $4$x$1+10$x$2>=5$

Non$-$negativity:

X$1>=0,$ X$2>=0$

The objective function minimizes the mills' overall operating costs.

Each constraint guarantees that the overall output of each aluminum grade equals or exceeds the stipulated quantity.

The non$-$negativity condition assures that the number of days of operation is not negative.

This linear programming model will assist the United Aluminum Company of Cincinnati in determining the ideal number of operating days for each mill while satisfying contract criteria at the lowest possible cost.

Graphically solve the linear programming model formulated for United Aluminum Company.

a$.$ How much extra $($i$.$e$.,$ 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 $$6,000$ to $$7,500$ per day?

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