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 follow Room Max Type I Type II s.t. Demand for Super Saver Demand for Deluxe Demand for Business Type I Rooms Type II Rooms Super Saver $30 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 160 rentals in the Super Saver class, 70 rentals in the Deluxe class, and 60 rentals in the Business class. Since these are the forecasted demands, Round Tree will take no more than these amounts of each reservation for each rental class. Round Tree has a limited number of each type of room. There are 100 Type I rooms and 120 Type II rooms. D.D. 8. 20 Rental Class $20 (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 S₁ is the number of Super Saver rentals allocated to room type 1, S₂ Is the number of Super Saver rentals allocated to room type II, D, is the number of Deluxe rentals allocated to room type 1, D₂ Is the number of Deluxe rentals allocated to room type II, B, is the number of Business rentals allocated to room type II. Write your answers expressing the amount allocated in dollars.) Deluxe $35 $30 Business $40
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 follow Room Max Type I Type II s.t. Demand for Super Saver Demand for Deluxe Demand for Business Type I Rooms Type II Rooms Super Saver $30 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 160 rentals in the Super Saver class, 70 rentals in the Deluxe class, and 60 rentals in the Business class. Since these are the forecasted demands, Round Tree will take no more than these amounts of each reservation for each rental class. Round Tree has a limited number of each type of room. There are 100 Type I rooms and 120 Type II rooms. D.D. 8. 20 Rental Class $20 (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 S₁ is the number of Super Saver rentals allocated to room type 1, S₂ Is the number of Super Saver rentals allocated to room type II, D, is the number of Deluxe rentals allocated to room type 1, D₂ Is the number of Deluxe rentals allocated to room type II, B, is the number of Business rentals allocated to room type II. Write your answers expressing the amount allocated in dollars.) Deluxe $35 $30 Business $40
MATLAB: An Introduction with Applications
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Author:Amos Gilat
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![## Room Reservation and Profit Optimization
### Optimization Problem
The problem is to determine the number of reservations management should accept for each rental class and room type combination to maximize profit.
#### Table of Room Reservations
| Rental Class | Super Saver | Deluxe | Business |
|---------------|-------------|--------|----------|
| **Room Type** | | | |
| Type I | | | — |
| Type II | | | |
- **Profit Calculation:**
- What is the profit (in dollars) obtained through this allocation of room reservations?
- **Profit $**: [Input box]
### Questions
**(b) Reservation Accommodation**
- Determine how many reservations can be accommodated in each rental class under the optimal solution:
- **Super Saver**: [Input box]
- **Deluxe**: [Input box]
- **Business**: [Input box]
- **Unfulfilled Demand**
- Is the demand for any rental class not satisfied?
- If unmet demand occurs, there will be [Input box] rooms not reserved in the [Dropdown: Select] class.
**(c) Conversion of Office Space**
- Consider converting an unused office area to a rental room.
- If conversion costs are identical for both types of rooms, would converting to a Type I or Type II room increase profitability?
- **Recommendation**: [Dropdown: Select], as it will increase profit by $ [Input box].
**(d) Linear Programming Model Modification**
- Modifying the linear programming model for next night's demand:
- Would you need a forecast of demand for each rental class on the next night?
- ○ [Option] Forecast of demand for each rental class to use as constraints.
- ○ [Option] Number of rooms of Type I and II for use in constraints.
- ○ [Option] Know if Type I can be used as Business class and modify objective function.
- ○ [Option] Know the profit per night and modify all constraints.
This setup is aimed at maximizing the revenue from the rooms by carefully selecting the rental class and type of room allocations. Adapting the model based on forecasts and understanding conversion potential can play pivotal roles in achieving optimal profitability.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15c8605-c3c4-4922-bcea-8a06f4b6dc06%2F0d1c078a-8aea-4dd3-a6a9-c667f8d816f2%2F99dtk4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Room Reservation and Profit Optimization
### Optimization Problem
The problem is to determine the number of reservations management should accept for each rental class and room type combination to maximize profit.
#### Table of Room Reservations
| Rental Class | Super Saver | Deluxe | Business |
|---------------|-------------|--------|----------|
| **Room Type** | | | |
| Type I | | | — |
| Type II | | | |
- **Profit Calculation:**
- What is the profit (in dollars) obtained through this allocation of room reservations?
- **Profit $**: [Input box]
### Questions
**(b) Reservation Accommodation**
- Determine how many reservations can be accommodated in each rental class under the optimal solution:
- **Super Saver**: [Input box]
- **Deluxe**: [Input box]
- **Business**: [Input box]
- **Unfulfilled Demand**
- Is the demand for any rental class not satisfied?
- If unmet demand occurs, there will be [Input box] rooms not reserved in the [Dropdown: Select] class.
**(c) Conversion of Office Space**
- Consider converting an unused office area to a rental room.
- If conversion costs are identical for both types of rooms, would converting to a Type I or Type II room increase profitability?
- **Recommendation**: [Dropdown: Select], as it will increase profit by $ [Input box].
**(d) Linear Programming Model Modification**
- Modifying the linear programming model for next night's demand:
- Would you need a forecast of demand for each rental class on the next night?
- ○ [Option] Forecast of demand for each rental class to use as constraints.
- ○ [Option] Number of rooms of Type I and II for use in constraints.
- ○ [Option] Know if Type I can be used as Business class and modify objective function.
- ○ [Option] Know the profit per night and modify all constraints.
This setup is aimed at maximizing the revenue from the rooms by carefully selecting the rental class and type of room allocations. Adapting the model based on forecasts and understanding conversion potential can play pivotal roles in achieving optimal profitability.
Transcribed Image Text:**Round Tree Manor Room Rental Optimization**
Round Tree Manor is a hotel offering 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:
| Room Type | Super Saver | Deluxe | Business |
|-----------|-------------|--------|-----------|
| Type I | $30 | $35 | - |
| Type II | $20 | $30 | $40 |
**Forecasted Demand and Room Availability**
Round Tree's management forecasts demand by rental class per night to determine reservation acceptance, aiming to maximize profit using a linear programming model. Forecasts for a particular night are:
- Super Saver: 160 rentals
- Deluxe: 70 rentals
- Business: 60 rentals
Round Tree restricts the number of reservations to these forecasts due to room availability, with 100 Type I and 120 Type II rooms available.
**Linear Programming Problem**
Formulate and solve a linear program to determine the optimal number of reservations to accept for each rental class and room type allocation. The variables are defined as follows:
- \( S_1 \): Super Saver rentals for Type I rooms
- \( S_2 \): Super Saver rentals for Type II rooms
- \( D_1 \): Deluxe rentals for Type I rooms
- \( D_2 \): Deluxe rentals for Type II rooms
- \( B_2 \): Business rentals for Type II rooms
**Objective**
Maximize profit subject to constraints.
**Constraints**
- Demand maximum for each class
- Type I and Type II room availability
- Non-negativity: \( S_1, S_2, D_1, D_2, B_2 \geq 0 \)
**Solution Structure**
Complete the solution by filling in the expected rental numbers for each equation, ensuring alignment with forecasts and room availability. Express allocations in dollars.
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