Round Tree Manor is a hotel that provides two types of rooms with three rental c
ID: 432269 • Letter: R
Question
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 $35 $40 - Type II $25 $35 $45 Type I rooms do not have high-speed wireless Internet access and are not available for the Business rental class. 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 150 rentals in the Super Saver class, 55 in the Deluxe class, and 40 in the Business class. Round Tree has 100 Type I rooms and 110 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. Rental Class with room type No of Reservations Super Saver rentals allocated to room type I Super Saver rentals allocated to room type II Deluxe rentals allocated to room type I Deluxe rentals allocated to room type II Business rentals allocated to room type II (b) For the solution in part (a), how many reservations can be accommodated in each rental class? Rental Class No of Reservations Super Saver I Deluxe Business Demand for rental class was not satisfied. (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? Type I Type II Shadow Price $ $ Convert an unused office area to room. Explain. The input in the box below will not be graded, but may be reviewed and considered by your instructor. (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? Explain. The input in the box below will not be graded, but may be reviewed and considered by your instructor.
Explanation / Answer
There are two types of rooms with three rental classes:
The given information is tabulated as follows:
Profit per unit $
TypesClasses Super Saver Deluxe Business Availability
Type-I 30 35 NA 90
Type-II 15 25 35 110
Demand 110 55 40
From above the problem is similiar to that of (unbalanced) Transportation Problem (specil type of linear programming problem).
Decision variables are defined as the number of rooms of each type assigned to each rental class as Xij where i represent types taking values 1, and 2 and j represents the rental class taking the values 1, 2, and 3 and these variables are non-negative (being numbers).
Objective is to have Maximum Total Profit given as Sigma Cij*Xij where Cij is the per unit profit of assigning ith type to jth rental class.
Constraints are in the form of availability of types as Sigma(j) X1j <= 90, and Sigma(j) X2j <= 110
Constraints are also in the form of demand for rental classes as Sigma(i) Xi1 <= 110, Sigma(i) Xi2 <= 55, and X23 <= 40 (note normally demand constraints are >= types but here total demand (205) is more than total availability.
Excel format formulation and solution using Excel Solver is as follows:
Decision Variables
TypesClasses Super Saver Deluxe Business Total Availability
Type-I 90 0 0 90 <= 90
Type-II 15 55 40 110 <= 110
Total 105 55 40
<= <= <=
Demand 110 55 40
Maximum Profit SumProduct -Sigma CijXij 5700
Total Profit is $5,700
Demand of Business class demand (40) from Type-II, Demand of Deluxe (55) from Type-II, All type-I (90) to Super saver class with remaining (15) of Type-II to Super saver class.
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