A linear programming computer package is needed.

Hanson Inn is a $96-$room hotel located near the airport and convention center in Louisville, Kentucky. When a convention or a special event is in town, Hanson increases its normal room rates and takes reservations based on a revenue management system. A large profesional organization has scheduled its annual convention in Louisville for the first weekend in June. Hanson Inn agreed to make at least $50\%$ of its rooms available for convention attendees at a special convention rate in order to be listed as a recommended hotel for the convention. Although the majority of attendees at the annual meeting typically request a Friday and Saturday two$-$night package, some attendees may select a Friday night only or a Saturday night only reservation. Customers not attending the convention may also request a Friday and Saturday two$-$night package, or make a Friday night only or Saturday night only reservation. Thus, six types of reservations are possible: Convention customers$/$two$-$night package; convention customers$/$Friday night only; convention customers$/$Saturday night only; regular customers$/$two$-$night package; regular customers$/$Friday night only; and regular customers$/$Saturday night only. The cost for each type of reservation is shown below.

Two$-$Night

Package Friday Night

Only Saturday Night

Only

Convention $$225$ $$123$ $$130$

Regular $$295$ $$146$ $$152$

The anticipated demand for each type of reservation is as follows.

Two$-$Night

Package Friday Night

Only Saturday Night

Only

Convention $402015$

Regular $203025$

Hanson Inn would like to determine how many rooms to make available for each type of reservation in order to maximize total revenue.

$($a$)$

Formulate a linear programming model for this revenue management application. $($Let CT $=$ number of convention two$-$night rooms, CF $=$ number of convention Friday only rooms, CS $=$ number of convention Saturday only rooms, RT $=$ number of regular two$-$night rooms, RF $=$ number of regular Friday only rooms, RS $=$ number of regular Saturday only rooms.$)$

Max

s$.$t$.$

anticipated demand for convention two$-$night rooms

anticipated demand for convention Friday night only rooms

anticipated demand for convention Saturday night only rooms

anticipated demand for regular two$-$night rooms

anticipated demand for regular Friday night only rooms

anticipated demand for regular Saturday night only rooms

Friday night rooms available for convention attendees only

Saturday night rooms available for convention attendees only

total rooms available for Friday night

total rooms available for Saturday night

CT$,$ CF$,$ CS$,$ RT$,$ RF$,$ RS $>=0$

$($b$)$

What is the optimal allocation?

CT $=$

$40$

CF $=$

$20$

CS $=$

$15$

RT $=$

$20$

RF $=$

$30$

RS $=$

$25$

From this allocation, what is the anticipated total revenue $($in dollars$)?$

$

$27590$

$($c$)$

Suppose that one week before the convention the number of regular customers$/$Saturday night only rooms that were made available sell out. If another nonconvention customer calls and requests a Saturday night only room, what is the value $($in dollars$)$ of accepting this additional reservation?

If the hotel accepts this additional reservation, then the total profit would increase by $