Problem $3(32$ points$)$

A city hotel is looking to optimize its revenue by allocating rooms among three different customer segments: Business travelers, Tourists, and Event attendees. The hotel has $150$ rooms available. Business travelers pay $$250$ per night, tourists pay $$150$ per night, and event attendees, who come in larger groups, pay a discounted rate of $$100$ per night. The hotel aims to maximize its total revenue while considering the following requirements:

Each customer segment has a minimum requirement: at least $20$ rooms should be allocated to business travelers, $30$ rooms to tourists, and $50$ rooms to event attendees to maintain relationships and ensure future bookings.

The hotel has agreements with event organizers that require it to reserve at least one$-$third of available rooms for event attendees when there are events in the city.

The hotel's market analysis suggests that having more than $60$ rooms for business travelers can negatively impact the hotel's reputation among leisure travelers due to the busy and crowded atmosphere.

The hotel has a loyalty program that guarantees room availability for members. The program dictates that $10\%$ of rooms must be available for loyalty program members.

a$.$ Formulate the algebraic LP model whose objective is to maximize revenue and help the Hotel Management decide how many of each type of room to allocate to Business travelers, Tourists, and Event attendees. Define the decision variables, objective function, and constraints $(18$ points$).$

b$.$ Formulate this same linear programming problem on a spreadsheet and solve using Excel Solver. Provide the corresponding "Excel Spreadsheet" and the "Answer Report". Include "managerial statements" that communicate the results of the analyses. $($i$.$e$.,$ describe verbally the results$).(14$ points$)$