Question

Problem 5-05

Leisure Air, a regional airline, provides service for Pittsburgh, Newark, Charlotte, Myrtle Beach, and Orlando. Leisure Air has two Boeing 737-400 airplanes, one based in Pittsburgh and the other in Newark. Both airplanes have a coach section with a 132-seat capacity. Each morning the Pittsburgh-based plane flies to Orlando with a stopover in Charlotte, and the Newark-based plane flies to Myrtle Beach, also with a stopover in Charlotte. At the end of the day, both planes return to their home bases. We restrict our attention to the Pittsburgh-Charlotte, Charlotte-Orlando, Newark-Charlotte, and Charlotte-Myrtle Beach flight legs for the morning flights.

Leisure Air uses two fare classes: a discount-fare Q class and a full-fare Y class. Reservations using the discount-fare Q class must be made 14 days in advance and must include a Saturday night stay in the destination city. Reservations using the full-fare Y class may be made any time, with no penalty for changing the reservation at a later date. Leisure Air established fares and developed forecasts of customer demand for each of 16 ODIFs. These data are shown in the table below.

FARE AND DEMAND DATA FOR 16 LEISURE AIR ORIGIN-DESTINATION-ITINERARY FARES (ODIFs)
ODIF	Origin	Destination	Fare Class	ODIF Code	Fare($)	Forecasted Demand
1	Pittsburgh	Charlotte	Q	PCQ	178	33
2	Pittsburgh	Myrtle Beach	Q	PMQ	268	44
3	Pittsburgh	Orlando	Q	POQ	228	45
4	Pittsburgh	Charlotte	Y	PCY	380	16
5	Pittsburgh	Myrtle Beach	Y	PMY	456	6
6	Pittsburgh	Orlando	Y	POY	560	11
7	Newark	Charlotte	Q	NCQ	199	26
8	Newark	Myrtle Beach	Q	NMQ	249	56
9	Newark	Orlando	Q	NOQ	349	39
10	Newark	Charlotte	Y	NCY	385	15
11	Newark	Myrtle Beach	Y	NMY	444	7
12	Newark	Orlando	Y	NOY	580	9
13	Charlotte	Myrtle Beach	Q	CMQ	179	64
14	Charlotte	Myrtle Beach	Y	CMY	380	8
15	Charlotte	Orlando	Q	COQ	224	46
16	Charlotte	Orlando	Y	COY	582	10

But because demand cannot be forecasted perfectly, the number of seats actually sold for each origin-destinationitinerary fare (ODIF) may turn out to be smaller or larger than forecasted. Suppose that Leisure Air believes that economic conditions have improved and that its original forecast may be too low. To account for this possibility, Leisure Air is considering switching the Boeing 737-400 airplanes that are based in Pittsburgh and Newark with Boeing 757-200 airplanes that Leisure Air has available in other markets. The Boeing 757-200 airplane has a seating capacity of 158 in the coach section.

A. Because of scheduling conflicts in other markets, suppose that Leisure Air is only able to obtain one Boeing 757-200. Should the larger plane be based in Pittsburgh or in Newark? Answer: Newark

The total revenue of basing the larger plane in Newark is bigger than basing the larger plane in Pittsburgh.

B. Based upon your answer in part (a), determine a new allocation for the ODIFs.

Original allocation:

THE SOLUTION FOR THE LEISURE AIR REVENUE MANAGEMENT PROBLEM
Optimal Objective Value = 103103.0000
Variable		Value		Reduced Cost
PCQ		33.00000		0.00000
PMQ		44.00000		0.00000
POQ		22.00000		0.00000
PCY		16.00000		0.00000
PMY		6.00000		0.00000
POY		11.00000		0.00000
NCQ		26.00000		0.00000
NMQ		36.00000		0.00000
NOQ		39.00000		0.00000
NCY		15.00000		0.00000
NMY		7.00000		0.00000
NOY		9.00000		0.00000
CMQ		31.00000		0.00000
CMY		8.00000		0.00000
COQ		41.00000		0.00000
COY		10.00000		0.00000
Constraint		Slack/Surplus		Dual Value
1		0.00000		4.00000
2		0.00000		70.00000
3		0.00000		179.00000
4		0.00000		224.00000
5		0.00000		174.00000
6		0.00000		85.00000
7		23.00000		0.00000
8		0.00000		376.00000
9		0.00000		273.00000
10		0.00000		332.00000
11		0.00000		129.00000
12		20.00000		0.00000
13		0.00000		55.00000
14		0.00000		315.00000
15		0.00000		195.00000
16		0.00000		286.00000
17		33.00000		0.00000
18		0.00000		201.00000
19		5.00000		0.00000
20		0.00000		358.00000

SOLVE: Using a larger plane based in Newark, the optimal allocations are:

PCQ	=	PMQ	=	POQ	=
PCY	=	PMY	=	POY	=
NCQ	=	NMQ	=	NOQ	=
NCY	=	NMY	=	NOY	=
CMQ	=	CMY	=
COQ	=	COY	=

Briefly summarize the major differences between the new allocation using one Boeing 757-200 and the original allocation summarized above.

SOLVE: The main differences between the original allocations and the new allocations are in the variables:



SOLVE: C. Suppose that two Boeing 757-200 airplanes are available. Determine a new allocation for the ODIF’s using the two larger airplanes. Using a larger plane based in Pittsburgh and a larger plane based in Newark, the optimal allocations are:

PCQ	=	PMQ	=	POQ	=
PCY	=	PMY	=	POY	=
NCQ	=	NMQ	=	NOQ	=
NCY	=	NMY	=	NOY	=
CMQ	=	CMY	=
COQ	=	COY	=

Briefly summarize the major differences between the new allocation using two Boeing 757-200 airplanes and the original allocation shown in part (b).

SOLVE: The main differences between the allocations in part b and the new allocations are in the variables:


SOLVE: This solution provides an in (increase or decrease) in revenue of $ ___________ .

SOLVE: D. Consider the new solution obtained in part (b). Which ODIF has the highest bid price?

SOLVE: What is the interpretation for this bid price?

SOLVE: The bid price for this solution is $_______ which means that if there was one more (Y or Q) class seat revenue would (increase or decrease) by $_________ .

Answer 1

To develop a linear programming model that can be used to determine how many seats Leisure Air should allocate to each fare class, we need to define 16 decision variables, one for each origin-destination-itinerary fare alternative. Using P for Pittsburgh, N for Newark, C for Charlotte, M for Myrtle Beach, and O for Orlando, the decision variables take the following form:

PCQ = number of seats allocated to Pittsburgh–Charlotte Q class
PMQ = number of seats allocated to Pittsburgh–Myrtle Beach Q class
POQ = number of seats allocated to Pittsburgh–Orlando Q class

PCY = number of seats allocated to Pittsburgh–Charlotte Y class

NCQ = number of seats allocated to Newark–Charlotte Q class

COY = number of seats allocated to Charlotte–Orlando Y class

The objective is to maximize total revenue. Using the fares shown in Table, we can write the objective function for the linear programming model as follows:

Max 178PCQ + 268PMQ + 228POQ + 380PCY + 456PMY + 560POY + 199NCQ + 249NMQ + 349NOQ + 385NCY + 444NMY + 580NOY + 179CMQ + 380CMY + 224COQ + 582COY

Next, we must write the constraints. We need two types of constraints: capacity and demand.
We begin with the capacity constraints.
Consider the Pittsburgh–Charlotte flight leg in Figure. The Boeing 737-400 airplane has a 132-seat capacity. Three possible final destinations for passengers on this flight (Charlotte, Myrtle Beach, or Orlando) and two fare classes (Q and Y) provide six ODIF alternatives:

(1) Pittsburgh–Charlotte Q class,

(2) Pittsburgh–Myrtle Beach Q class,
(3) Pittsburgh–Orlando Q class,

(4) Pittsburgh–Charlotte Y class,

(5) Pittsburgh–Myrtle Beach Y class, and

(6) Pittsburgh–Orlando Y class. Thus, the number of seats allocated to

the Pittsburgh–Charlotte flight leg is PCQ PMQ POQ PCY PMY POY. With the capacity of 132 seats, the capacity constraint is as follows:

PCQ + PMQ + POQ + PCY + PMY + POY … 132 Pittsburgh– Charlotte

The capacity constraints for the Newark–Charlotte, Charlotte–Myrtle Beach, and Charlotte– Orlando flight legs are developed in a similar manner. These three constraints are as follows:

NCQ +NMQ +NOQ +NCY +NMY + NOY <= 132


The demand constraints limit the number of seats for each ODIF based on the forecasted demand. Using the demand forecasts in Table 5.3, 16 demand constraints must be added to the model. The first four demand constraints are as follows:
PCQ <= 33
PMQ <= 44
POQ <= 45
PCY <= 16
Pittsburgh – Charlotte Q class
Pittsburgh– Myrtle Beach Q class
Pittsburgh – Orlando Q class
Pittsburgh– Charlotte Y class

The complete linear programming model with 16 decision variables, 4 capacity constraints, and 16 demand constraints is as follows:

Max 178PCQ + 268PMQ + 228POQ + 380PCY + 456PMY + 560POY+ 199NCQ + 249NMQ + 349NOQ + 385NCY + 444NMY + 580NOY + 179CMQ + 380CMY + 224COQ + 582COY

The optimal solution to the Leisure Air revenue management problem is shown. The value of the optimal solution is $103,103. The optimal solution shows that
PCQ 33, PMQ 44, POQ 22, PCY 16, and so on.

Thus, to maximize revenue,
Leisure Air should allocate 33 Q class seats to Pittsburgh–Charlotte, 44 Q class seats to
Pittsburgh–Myrtle Beach, 22 Q class seats to Pittsburgh–Orlando, 16 Y class seats to
Pittsburgh–Charlotte, and so on.