b$.$ What is the optimal solution? Problem $7-41$

Southern Oil Company produces two grades of gasoline: regular and premium. The profit contributions are $$0\mathrm{.}30$ per gallon for regular

gasoline and $$0\mathrm{.}50$ 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 $18,000$ gallons of grade

A crude oil available. The refinery used to produce the gasolines 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. If required, round your answers to

two decimal places.

Let $R=$ number of gallons of regular gasoline produced

$P=$ number of gallons of premium gasoline produced

Gallons of regular gasoline

Gallons of premium gasoline

Total profit contribution

$

c$.$ What are the values and interpretations of the slack variables?

d$.$ What are the binding constraints?

Grade A crude oil available

Production capacity

Binding

Demand for premium