Round Tree Manor is a hotel that provides two types of rooms with three rental classes: Super Saver, Deluxe, and Business. The profit per night for each type of room and rental class is as follows.

Rental Class

Super Saver Deluxe Business

Room Type I $$30$ $$35$

Type II $$20$ $$30$ $$40$

Round Tree's management makes a forecast of the demand by rental class for each night in the future. A linear programming model developed to maximize profit is used to determine how many reservations to accept for each rental class. The demand forecast for a particular night is $140$ rentals in the Super Saver class, $70$ rentals in the Deluxe class, and $60$ rentals in the Business class. Round Tree has $100$ Type I rooms and $120$ Type II rooms.

$($a$)$

Formulate and solve a linear program to determine how many reservations to accept in each rental class and how the reservations should be allocated to room types. $($Assume S$1$ is the number of Super Saver rentals allocated to room type I, S$2$ is the number of Super Saver rentals allocated to room type II$,$ D$1$ is the number of Deluxe rentals allocated to room type I, D$2$ is the number of Deluxe rentals allocated to room type II$,$ and B$2$ is the number of Business rentals allocated to room type II$.$ Write your answers expressing the amount allocated in dollars.$)$

Max

s$.$t$.$

Demand for Super Saver

Demand for Deluxe

Demand for Business

Type I Rooms

Type II Rooms

S$1,$ S$2,$ D$1,$ D$2,$ B$2>=0$

Report the number of reservations of each type management should accept for each rental class room type combination to maximize profit.

Rental Class

Super Saver Deluxe Business

Room Type I

$$

Type II

What is the profit $($in dollars$)$ obtained through this allocation of room reservations?

$

Is the demand for any rental class not satisfied?

If demand materializes as forecast, there will be

rooms not reserved in the

$---$Select$---$

class.

$($b$)$

For the solution in part $($a$),$ how many reservations can be accommodated in each rental class?

Super Saver

Deluxe

Business

$($c$)$

With a little work, an unused office area could be converted to a rental room. If the conversion cost is the same for both types of rooms, would you recommend converting the office to a Type I or a Type II room? Why?

$---$Select$---$

$,$ because this will increase profit by $

$.$

$($d$)$

Could the linear programming model be modified to plan for the allocation of rental demand for the next night? What information would be needed and how would the model change?

We would need a forecast of demand for each rental class on the next night to use as the right$-$hand sides of the first three constraints.

We would need to know if Type $1$ rooms can be used as Business class rooms the next night and add a variable to the objective function.

We would need to know whether the profit per night of each type of room and rental class will change and modify all the constraints appropriately.

We would need to know how many rooms of Type I and Type II there will be on the next night to use as the right$-$hand sides of the last two constraints.