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 (Mountain View)	$35	$40	-
	Type II (Street View)	$25	$35	$45

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 120 rentals in the Super Saver class, 65 in the Deluxe class, and 50 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 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. If an amount is zero, enter “0”.
	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 - Select your answer -DeluxeSuper SaverBusinessItem 9 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 - Select your answer -Type IType IIItem 12 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?
	- Select your answer -YesNoItem 14
	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.

Answer 1

a)

Rental Class Super Saver Deluxe Business Type II 35 25 35 45 4 0 Decision Variables Number of reservations to be accepted for each rental class Rental Class Super Saver Deluxe Business Type I Type II 10 Obiective function 12 13 14:SUMPRODUCT(B9:C11,B3:C5) Maximise the profit Constraints 16 B9+C9 17 B10+C10 18-B11+C11 19 SUM(B9:B11 20:SUM(C9:C11) 21 100 110

2 Rental ClassTypeType lI 3 Super Saver 4 5 Business 35 Deluxe 35 45 Decision Variables Number of reservations to be accepted for each rental class 8 RenalClass Type Type I 9Super Saver 10 11 BusinesS 12 Objective function 13 Maximise the profit Deluxe 0 Constraints 16 17 18 19 0 0 0 0 100 110 21

Solver Parameters Set Objective SAS14 To: ⓔMax OMin。Value Of: By Changing aiable Cells: SBS9:SBS10,scs9:SCS11 Subject to the Constraints: SAS16:SAS20SCS16:SCS20 SBS9:SCS110 Add Change Delete Reset All Load/Save Make Unconstrained Variables Non-Negative Select a Solving Method: Simplex LP Options Solving Method Select the GRG Nonlinear engine for Solver Problems that are smooth nonlinear. Select the LP Simplex engine for linear Solver Problems, and select the Evolutionary engine for Solver problems that are non-smooth. Help Solve Close

Decision Variables Number of reservations to be accepted for each rental class 8 Rental ClassTypeType I 9Super Saver 95 0 60 50 Deluxe 11 BusinesS 12 Objective function 13 Maximise the profit 0 7875 Constraints 16 I 95 120 65 50 100 110 18 19 20 21 50 100 110

b)

Super Saver - 95

Deluxe - 65

Business - 50

c)

6 Variable Cells Final Reduced Objective Allowable Allowable Value Cost Coefficient Increase Decrease Cell Name 9SB$9 Super Saver Type I 10 SB$10 Deluxe Type I 11 $C$9 Super Saver Type II 12 $C$10 Deluxe Type II 13 ŞC$11 Business Type II 95 0 0 0 0 35 25 35 45 0 1E+30 15 15 1E+30 15 Constraints 16 17 Cell 18 ŞA$16 Constraints 19 SA$17 Constraints 20 ŞA$18 Constraints 21 ŞA$19 Constraints 22 SA$20 Constraints 23 ŞB$9 Super Saver Type I 24 ŞC$9 Super Saver Type II 25 ŞB$10 Deluxe Type I 26 ŞC$10 Deluxe Type II 27 ŞB$11 Business Type I 28 ŞC$11 Business Type II Final Shadow Constraint Allowable Allowable Value Price R.H. Side Increase Decrease 25 Name 95 0 120 1E+30 65 95 65 15 100 110 95 35 100 110 0 25 95 0 95 1E+30 0 1E+30 1E+30 1E+30 1E+30 0

Shadow price of type I = $35

Shadow price of type II = $30

Shadow price of type I is higher. Therefore, unused office space must be converted into type I room.

d)

Linear programming can be modified to include the demand for next night too. Additional information that is required is , average number of days for which a booking takes place is each type of room.