Leisure Air is a regional airline that provides service from Pittsburgh, Newark,
ID: 447060 • Letter: L
Question
Leisure Air is a regional airline that provides service from 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.
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 anytime, with no penalty for changing the reservation at a later date. The itinerary and fare alternatives that Leisure Air can offer to its customers based on both the origin-destination of each flight and the fare classes are synthetize in the following table.
ODIF
Origin
Destination
Fare class
ODIF code
Fare
Demand
1
Pittsburgh
Charlotte
Q
PCQ
$178
33
2
Pittsburgh
Myrtle Beach
Q
PCQ
$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
Determine how many discount-fare and full-fare seats Leisure Air should make available in their reservation system to maximize its daily total expected revenue from running the two planes.
ODIF
Origin
Destination
Fare class
ODIF code
Fare
Demand
1
Pittsburgh
Charlotte
Q
PCQ
$178
33
2
Pittsburgh
Myrtle Beach
Q
PCQ
$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
Explanation / Answer
Answer:
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.
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