In: Statistics and Probability
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 $_________ .
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.