Southern Oil Company produces two grades of gasoline: regular and premium. The profit contributions are $$0\mathrm{.}40$ per gallon for regular gasoline and $$0\mathrm{.}60$ per gallon for premium gasoline. Each gallon of regular gasoline contains $0\mathrm{.}3$ gallons of grade A crude oil, and each gallon of premium gasoline contains $0\mathrm{.}6$ gallons of grade A crude oil. For the next production period, Southern has $16,800$ gallons of grade A crude oil available. The refinery used to produce the gasoline has a production capacity of $50,000$ gallons for the next production period. Southern Oil's distributors have indicated that demand for the premium gasoline for the next production period will be at most $20,000$ gallons.

$($a$)$

Formulate a linear programming model that can be used to determine the number of gallons of regular gasoline and the number of gallons of premium gasoline that should be produced in order to maximize total profit contribution. $($Let R be the gallons of regular gasoline and let P be the gallons of premium gasoline.$)$

Max

s$.$t$.$

Grade A crude oil available

Production capacity

Demand for premium

R$,$ P $>=0$

$($b$)$

What is the optimal solution?

R $=$

P $=$

profit $=$ $

$($c$)$

What is the value of the slack variable in the Grade A crude oil constraint?

Interpret this value.

After reaching the optimal solution, there is still this amount of grade A crude oil to be used.

After reaching the optimal solution, all available grade A crude oil has been used.

In order to reach the optimal solution, this amount of grade A crude oil is required.

What is the value of the slack variable in the Production capacity constraint?

Interpret this value.

After reaching the optimal solution, the refinery is still able to produce this amount of gasoline.

After reaching the optimal solution, the total production capacity is used.

In order to reach the optimal solution, the refinery must produce this amount of additional gasoline.

What is the value of the slack variable in the Demand for premium constraint?

Interpret this value.

After reaching the optimal solution, the produced amount of premium gasoline is this much less than the maximum demand.

After reaching the optimal solution, the maximum demand for premium gasoline has been reached.

In order to reach the optimal solution, this amount of additional premium gasoline needs to be produced in order to meet demand.

$($d$)$

What are the binding constraints? $($Select all that apply.$)$

production capacity

demand for premium

profit

grade A crude oil available