PROBLEM SOLVING # $1($TOPIC: PRODUCTIVITY OPTIONS$)$

Frida is the sales director of Hilton Hotel and she wants to determine the number of rooms that the hotel should overbook per night to maximize room revenue. Currently the hotel does not have an overbooking policy but based on past experience, some hotel guest who made advance reservations do not show up on their reserved dates. At any given day the probability that all rooms reserved in advance are occupied by the listed guest is only $18\%.$ The probability that the number of rooms which do not get occupied by a listed guest is $23\%$ for one room, $34\%$ for two rooms, $17\%$ for three rooms, and $8\%$ for four rooms. The hotel charges $$136\mathrm{.}00$ per room per night and the cost of maintaining a room is $$49\mathrm{.}00$ per night. If the number of overbooked rooms is more than the number of actual vacant rooms due to no$-$shows, the hotel will have to shoulder the cost of booking the bumped off guest to a nearby hotel. This overflow cost of $$64\mathrm{.}00$ per month includes the communication fee, transportation expense, and other charges made in moving the guest to the nearby hotel so that the guest will not feel too inconvenienced. How many rooms should the hotel overbook to minimize lost revenue due to no$-$shows? How much will be the savings with the new overbooking policy?