Leisure Air, a regional airline, provides service for Pittsburgh, Newark, Charlo
ID: 3071934 • Letter: L
Question
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?
Newark
Explain.
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
c.
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
=
d.
Briefly summarize the major differences between the new allocation using one Boeing 757-200 and the original allocation summarized above.
The main differences between the original allocations and the new allocations are in the variables:
CMQ, COQ, PMQ, NMQ, and POQ
e. 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
=
f.
Briefly summarize the major differences between the new allocation using two Boeing 757-200 airplanes and the original allocation shown in part (b).
The main differences between the allocations in part b and the new allocations are in the variables:
CMQ, COQ, NMQ, and POQ
This solution provides an increase in revenue of $ .
g. Consider the new solution obtained in part (b). Which ODIF has the highest bid price?
COY
What is the interpretation for this bid price?
The bid price for this solution is $ which means that if there was one more Y class seat revenue would increase by $ .
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
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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