Performance measurement in airline revenue management: a simulation-based assessment of the network-based revenue opportunity model [original]

Dissertation

Performance Measurement in

Airline Revenue Management - A

Simulation-based Assessment of the

Network-based Revenue

Opportunity Model

Dipl. Wirt.-Inform. Christian Temath

Schriftliche Arbeit zur Erlangung des akademischen Grades

doctor rerum politicarum (dr. rer. pol.)

im Fach Wirtschaftsinformatik

eingereicht an der

Fakult¨at f¨ur Wirtschaftswissenschaften der

Universit¨at Paderborn

Paderborn, im September 2010

Datum der m¨undlichen Pr¨ufung: 11.04.2011

Gutachter:

1. Prof. Dr. Leena Suhl

2. Prof. Dr. Alf Kimms

Acknowledgements

This thesis is a result of a research cooperation between the Decision Support and

Operations Research Lab (DSOR) at the University of Paderborn and Lufthansa

German Airlines. It would not have been possible to complete this thesis without

the valuable support of many people.

First and foremost, I want to express my gratitude to Prof. Dr. Leena Suhl for

her continuous and always caring support while supervising my thesis. Moreover,

I have enjoyed the open and friendly atmosphere at her research department and

the help of the entire working group. In addition, I want to thank Prof. Dr. Alf

Kimms for being second referee of my thesis.

I also owe a big thank you to Dr. Stefan P¨olt and Dr. Michael Frank -

my supervisors at Lufthansa German Airlines - for providing me with so many

practical advice and introducing me into the secrets of revenue management.

Furthermore, I would like to thank Martin Friedemann for our joint sessions to

help me implement reasonable parts of the simulation environment.

Finally, I would like to thank my family for their encouragement and support

in the course of writing this thesis. I would like to express my gratitude to my

parents without their continuous encouragement this thesis would not have been

possible at all. I also thank my sister Bettina, who gave me precious input in

making my thoughts understandable. Lastly, I thank my wife Julia, who has

always been there for me and always had trust in me.

Christian Temath

Cologne, September 2010

Contents

1. Introduction 1

1.1. Airline Revenue Management . . . . . . . . . . . . . . . . . . . . 2

1.2. Performance Measurement of Revenue Management . . . . . . . . 7

1.3. Measuring Performance with the ROM . . . . . . . . . . . . . . . 10

1.3.1. Model Definition and Terminology . . . . . . . . . . . . . 10

1.3.2. Main Properties of the ROM . . . . . . . . . . . . . . . . . 11

1.3.3. Major Developments in Airline Revenue Management Af-

fecttheROM......................... 14

1.3.4. Consideration of Practical Aspects in the ROM . . . . . . 15

1.4. Scope and Purpose of the Thesis . . . . . . . . . . . . . . . . . . 16

2. Airline Revenue Management and Performance Measurement: State-

of-the-art 19

2.1. Airline Revenue Management . . . . . . . . . . . . . . . . . . . . 19

2.1.1. Optimization Models with Independent Demand . . . . . . 20

2.1.2. Modeling, Unconstraining and Forecasting Customer De-

mand.............................. 22

2.1.3. Optimization Models with Dependent Demand . . . . . . . 26

2.2. Performance Measurement of Revenue Management . . . . . . . . 27

2.3. TheROM ............................... 30

2.4. Research Opportunities and Goals of the Thesis . . . . . . . . . . 32

3. A Novel Simulation-based Approach to Investigate ROM Properties 35

3.1. The Simulation Environment . . . . . . . . . . . . . . . . . . . . . 35

3.1.1. Modeling Customer Demand and Request Generation . . . 37

3.1.2. Unconstraining and Demand Forecasting . . . . . . . . . . 42

3.1.3. Optimization Models and Seat Inventory Control . . . . . 46

3.2. Measuring ROM Robustness . . . . . . . . . . . . . . . . . . . . . 48

3.2.1. ErrorMeasures ........................ 49

3.2.2. Similarity Measures . . . . . . . . . . . . . . . . . . . . . . 50

3.3. The Simulation Scenarios . . . . . . . . . . . . . . . . . . . . . . . 52

3.3.1. TheBaseCase ........................ 52

vii

3.3.2. Adjusting the Unconstraining Error . . . . . . . . . . . . . 53

3.3.3. Further Scenarios . . . . . . . . . . . . . . . . . . . . . . . 54

3.4. Summary ............................... 56

4. The Network-based ROM with Independent Demand 57

4.1. ModelDefinition ........................... 57

4.2. Main Properties of Network-based ROM with Independent Demand 59

4.3. Computational Results . . . . . . . . . . . . . . . . . . . . . . . . 62

4.3.1. Comparing Model- vs. Data-related Errors . . . . . . . . . 62

4.3.2. Analyzing the Effect of Unconstraining Errors . . . . . . . 63

4.3.3. Analyzing the Effect of Further Scenarios . . . . . . . . . . 68

4.4. Summary ............................... 72

5. The Network-based ROM with Dependent Demand 75

5.1. Extensions to the Network-based ROM with Independent Demand 75

5.2. Properties of the Network-based ROM with Dependent Demand . 79

5.3. Computational Results . . . . . . . . . . . . . . . . . . . . . . . . 81

5.3.1. Base Case and Unconstraining Error Scenarios . . . . . . . 82

5.3.2. Analyzing the Effect of Further Scenarios . . . . . . . . . . 89

5.3.3. Analyzing the Effect of Different Sell-up Rates . . . . . . . 93

5.4. Summary ............................... 94

6. Disaggregation of ROM Measures to Single Legs 97

6.1. Relation between Network and Leg Level . . . . . . . . . . . . . . 98

6.2. Prorating Fares to Single Legs . . . . . . . . . . . . . . . . . . . . 99

6.2.1. Mileage ............................ 100

6.2.2. BidPrices........................... 101

6.3. Model Definition on a Leg Base . . . . . . . . . . . . . . . . . . . 102

6.4. Computational Results . . . . . . . . . . . . . . . . . . . . . . . . 104

6.4.1. No-connecting-traffic Flight Network: Network Level . . . 104

6.4.2. No-connecting-traffic Flight Network: Leg Level . . . . . . 106

6.4.3. Realistic Flight Network: Leg Level . . . . . . . . . . . . . 109

6.5. Summary ............................... 115

7. Disaggregation of ROM Measures to Single Components 117

7.1. Extending the Network-based ROM to Overbooking and Upgrading117

7.1.1. Potential Revenue with Upgrading . . . . . . . . . . . . . 118

7.1.2. Potential Revenue with Overbooking . . . . . . . . . . . . 119

7.1.3. Actual Revenue after No-shows and Cancelations . . . . . 121

7.1.4. No RM Revenue after No-shows and Cancelations . . . . . 122

viii

7.2. Measuring Overbooking and Upgrading Success . . . . . . . . . . 125

7.2.1. Incremental Revenue due to Overbooking and Upgrading . 126

7.2.2. ROM with Upgrading . . . . . . . . . . . . . . . . . . . . 127

7.2.3. ROM with Overbooking . . . . . . . . . . . . . . . . . . . 128

7.2.4. ROM with Overbooking and Upgrading . . . . . . . . . . 130

7.3. Computational Results . . . . . . . . . . . . . . . . . . . . . . . . 131

7.4. Summary ............................... 133

8. Summary and Concluding Remarks 137

A. Detailed Test Results 141

A.1. The Network-based ROM with Independent Demand . . . . . . . 141

A.2. The Network-based ROM with Dependent Demand . . . . . . . . 146

A.2.1. The Base Case: Sell-up Rate 30% . . . . . . . . . . . . . . 146

A.2.2. Sell-up Rate 10% . . . . . . . . . . . . . . . . . . . . . . . 155

A.2.3. Sell-up Rate 50% . . . . . . . . . . . . . . . . . . . . . . . 165

A.3. Disaggregation of ROM Measures to Single Legs . . . . . . . . . . 175

A.4. Disaggregation of ROM Measures to Single Components . . . . . 178

List of Figures 189

List of Tables 192

List of Algorithms 193

Notations 195

Acronyms 209

Bibliography 211

1. Introduction

Airlines all around the world repeatedly face difficult environments and develop-

ments in the airline market. Be it the oil crisis in the seventies or most recently

the global depression in the course of the financial crisis, many external effects

put tremendous pressure on airlines’ profitability. Additionally, airlines nowa-

days face an increasing competition by the entrance of low cost carriers into the

market and declining revenues caused by customers using the new opportunities

to search for cheap tickets in the internet.

For many decades airlines have been applying methods of revenue management

(RM) to maximize revenues. They basically aim at ”selling the right seats to

the right customers at the right prices” (see Smith et al., 1992) and thus to

increase their revenues significantly. Lieberman (1991) and Skugge (2004) refer

to additional earnings by 3 to 7 percent that are possible for companies with

successful RM.

However, the question ”Am I making as much money as I should be?” raised

by Rannou and Melli (2003) is posed in many RM departments all over the world.

Even though RM uses optimization techniques for the inventory control to max-

imize revenues, its success strongly depends on forecast accuracy. The stochas-

ticity of the demand and the necessary manual adjustments in the course of a

booking period influence the results and the quality of the revenue optimization.

At the end of each booking period, the question of RM performance remains. In

the past, many techniques to measure RM performance have been developed and

proposed. Basic measures like seat load factor (SLF) or revenue per available seat

kilometer (RASK) can easily be calculated by using data from the inventory sys-

tems. More sophisticated concepts for performance measurement (PM) include

comparing two different time periods to analyze the performance of the revenue

management system (RMS). Moreover simulation plays an important role in in-

vestigating the performance of new RM methods, before implementing them into

the operational RMS. Drawbacks related to the above methods are either the

inability for continuous measurement or the inability to isolate RM contribution

from the overall success.

A widely used technique that allows to continuously measure but also isolate

the contribution of the RMS is the revenue opportunity model (ROM). While

1. Introduction

adapting the ROM to major developments in RM science - i.e. the advancement

from leg-based to network-based RM controls and the recent transition from

independent to dependent demand structures - the question of applicability and

in particular the effect of errors in the input data on the quality of the ROM

became increasingly important. These new developments in RM science pose new

questions and challenges on the ROM. In this thesis we model both independent

and dependent demand structures in a network-based ROM and investigate main

properties. Furthermore we consider different practical aspects of airline RM to

enable the application of the ROM in practice.

We start with an introduction into airline RM and briefly describe some of

the major developments in airline RM science in Section 1.1. In Section 1.2

we describe basic approaches to measure the performance of RM. Section 1.3

formally introduces the ROM. In addition, we describe some main properties of

the ROM, elaborate the effect of major developments in airline RM science on the

ROM and discuss important aspects of the application of the ROM in practice.

Finally we conclude the chapter in Section 1.4 with a summary of the scope and

purpose of this thesis.

1.1. Airline Revenue Management

As already described in the previous section, the main goal of RM is ”selling the

right seats to the right customers at the right prices”. This definition is also ap-

plicable to many other industries, for example the sale of empty rooms in a hotel.

According to Weatherford and Bodily (1992) products or services, for whose sale

the application of RM methods is useful, share three common characteristics.

First, these products are ”perishable”. This means that the product on sale is

no longer available after a certain point of time and cannot be sold any more.

An example is a plane leaving with empty seats. These empty seats cannot be

sold any more to a customer after the departure of the plane. Second, these

products have a ”fixed capacity”. It is hardly possible to increase the number

of products available for sale within a short time period and it mostly incurs

high cost to extend the amount of products that can be offered. An example

for limited quantity is again a plane, which has a fixed number of seats that can

be sold to customers. The third characteristic is that the potential customers

can be grouped into different ”segments”. A classical segmentation at airlines

is the differentiation between business travelers, who normally book their tickets

shortly before departure and are not too price-sensitive, and private customers,

who normally book early before the departure date, but look thoroughly at the

1.1. Airline Revenue Management

prices. The segmentation of products is normally done with restrictions that

apply to a given product. These so called fencing rules could be for example a

purchase with a minimum of 21 days in advance or a mandatory Saturday night

stay for return tickets. These two rules are typically applied to discount tickets

offered to leisure customers. A business customer usually does not buy a ticket

more than 21 days in advance and also does not want to stay over the weekend.

Thus, these rules prevent the business customer to opt for a discount ticket. An

example of how price discrimination helps to improve revenues is illustrated in

Figure 1.1. If an airline would only offer a product for price p, the demand would

Price

Quantity

Additional revenue with 2

additional fare classes

p2 One-price

Revenue

(OPR)

qq1 q2

Figure 1.1.: Effect of Introducing Additional Customer Segments

be at quantity q. The resulting revenues would be p∗q. To simplify our example

we do not consider any capacity restrictions. If it is possible to create customer

segments at different price levels additional revenue can be earned. For example

two additional price points p1and p2would lead to additional revenues (p1−p)∗q1

and p2∗(q2−q). Weatherford and Bodily (1992) introduce the term perishable-

asset revenue management to consider for the three previously mentioned main

characteristics. Applications of perishable-asset revenue management or RM can

be found in many industries apart from airlines, such as hotels, car rentals, cruise

lines, the steel industry or in the broadcasting business.

The development of airline RM has a long history. At the beginning of the

seventies airlines in Europe started offering reduced fares for seats on their flights.

With the start of the Super Saver Fares of American Airlines in 1977 and the

deregulation of the national and international air traffic in the US with the Airline

Deregulation Act in 1978, reduced fares were introduced in the US on a large scale.

As a consequence airlines had to decide if a booking for a discount ticket should

1. Introduction

be accepted or not. A major contribution to support this decision was made by

Littlewood’s rule (see Littlewood, 1972). It proposes that a booking for a reduced

fare should be accepted as long as the value of the booking exceeds the expected

value of a future booking for a normal fare. Starting with Littlewood’s rule

techniques and methods of RM have made intensive progress in the last decades.

In this thesis we focus on two major developments in RM science. We illustrate

them in Figure 1.2.

Leg-based

control

Dependent

demand

Independent

demand

Network-based

control

Figure 1.2.: Major Developments in RM Science

In the beginning RM methods were focussed on leg-based controls. Leg-based

controls assume that a customer only demands a single resource, which in the air-

line case would be a single flight leg and that no interdependencies exist between

the different flights offered to the customers. Thus, it was possible to manage

each flight leg independently. The first major development we focus on in this

thesis is the advancement from leg-based controls to network-based controls.

Since the 1980s more and more airlines established hub and spoke networks

with an increasing number of passengers buying itineraries which included more

than one flight leg. These so called origin & destination (O&D) itineraries de-

manded an availability decision on all flight legs contained in the itinerary at the

time of a booking request. Network-based controls take the interdependencies

between the flight legs into account, decide on the availability on a network level

and thus help to maximize the revenues for the total flight network.

The second major development we focus on in this thesis is the change from

independent to dependent demand structures. For a long time the assumption

of independence of the demand between booking classes prevailed. Fencing fare

rules for separating booking classes helped the airlines to uphold the assumption

of the independence of the demand between the booking classes for a long time.

1.1. Airline Revenue Management

However, this situation has changed. Nowadays the internet offers more and

more transparency about available fares. Additionally, low-cost-carriers entered

the market removing fencing rules and applying restriction-free pricing. As a

result, customers became increasingly price-sensitive and began to search for the

lowest available fare. A lot of research has been accomplished in this field and

many airlines have started upgrading their RMS to consider dependent demand

structures in the last years.

The support of the availability decisions on flight legs or O&D itineraries re-

quires a set of quite complex and sophisticated models and methods. The com-

ponents of a complete RMS and the interaction with other systems such as the

reservation system is presented in Talluri and van Ryzin (2004b, see Chapter 1,

page 19) and in Klein and Steinhardt (2008, see Chapter 1, page 27). We show

their illustration in Figure 1.3. It describes the main steps of RM for an airline:

Data consolidation, forecasting and optimization. These steps will be accom-

plished many times during a booking period. Usually airlines recollect the actual

(booking) data at a data collection point (DCP) and readjust the optimization

settings according to the new information. At all times a manual intervention

by revenue managers or analysts is possible, to react on special events or short

notice changes.

In the step of data consolidation all relevant input data for RM is collected and

adjusted for the upcoming RM steps. One data source contains the capacities

and fares. The fares are set by the pricing department in a separate process

and are considered to be fixed in classical, quantitative RM. Of course, fares can

vary over time, but in the optimization models fixed prices are assumed. The

actual capacities are the result of the fleet assignment process, which normally

takes place around a year before the departure of a plane and which makes use

of the results of the network planning. Usually airlines readjust the capacities to

the current booking situation some weeks before the departure in a second fleet

assignment process. Additionally equipment changes are possible on short notice

to react on last-minute developments or events. Other than that the capacities of

the planes will normally be considered as fixed in the optimization steps of RM.

The other data sources contain information about historical bookings, historical

cancelations and no-shows. A cancelation occurs, if a customer cancels his book-

ing during the booking period before the departure of the plane. In contrast, a

no-show occurs if a customer has a valid ticket for a flight and does not show-up

at the airport when the plane departs. Usually this applies to customers with

flexible tickets, who can change their reservation to a different flight without any

additional cost. Additionally, the databases contain the information about the

actual booking situation.

1. Introduction

Historical

booking data

Actual

booking data

Capacities,

products, fares

system

Reservation

system

Global

distribution

system

Travel agencies

Call-center Webserver,

e-commerce

Data consolidation

Forecasting

Optimization

Proprietary

distribution

systems

Monitoring and controlling

Figure 1.3.: A RMS and its Interaction with Other Systems (Adapted Illustration

- See Talluri and van Ryzin (2004b) and Klein and Steinhardt (2008))

All of this data is elementary input for the forecasting methods that are per-

formed in a second step. Typically airlines forecast the customer demand, the

cancelations and the no-shows. The process of forecasting is essential in a RMS,

because the forecasted data is the key input for the optimization models. Lee

(1990, see page 2) for example refers to significant revenue improvements, if de-

mand forecast accuracy increases. His analyses show that a 10% increase in

forecast accuracy results in 0.5 to 3% more revenue. A main part of forecasting

is unconstraining. Unconstraining is necessary, because an airline is not able to

observe the total demand in a booking period, but only the number of bookings

and the availability of booking classes. With unconstraining an airline estimates

the total demand that existed for a given flight leg or O&D itinerary that has

already departed.

1.2. Performance Measurement of Revenue Management

The actual optimization takes place in the third step. During the optimization

it is decided how many seats shall be reserved for which customer segment for a

given product. This optimization is based on the demand forecasts, the actual

booking data and the fares and capacities. To account for cancelations and no-

shows and to prevent a high demand flight from leaving with empty seats, airlines

apply techniques of overbooking during the optimization. Overbooking virtually

increases the capacity of a plane to consider no-shows and cancelations. If can-

celation and no-show forecasts are accurate and the overbooking optimization

performs well, the real capacity will be sufficient to accommodate all passengers.

However, sometimes overbooking leads to a situation in which more passengers

are booked on a flight than seats are available. In this case not every customer

can be boarded, which is called denied boarding. By civil aviation law airlines are

obligated to pay a compensation for each passenger who is denied boarding. One

method to prevent a denied boarding is upgrading: A passenger is offered a seat

in a higher valued compartment than he has purchased a ticket for. Upgrading

can also be applied during the booking period, if for example there is forecasted

excess demand in the economy compartment and forecasted free seats in the busi-

ness compartment. In this case an airline might want to virtually increase the

capacity of the economy compartment to sell the surplus seats in the business

compartment that might otherwise stay unsold. The results of the optimization

process are reported to the reservation system, in which the seat inventory is

controlled. Different parties are able to access the reservation system. A main

part of the customer requests for traditional airlines is handled by a global dis-

tribution system (GDS). Airlines increasingly offer their products through own

sales channels, in particular their own websites, but also call centers.

1.2. Performance Measurement of Revenue

Management

Since the application of RM methods in airline operations different methods to

measure the performance of RM are prevalent. Generally, performance measures

aim at describing how well the RMS in conjunction with the revenue managers

were able to achieve the goal of RM, which is usually to maximize overall revenues

(see McGill and van Ryzin, 1999). Talluri and van Ryzin (2004b) distinguish

between ”revenue-opportunity assessment” and ”revenue-benefits measurement”.

While the first one is usually performed before the introduction and implementa-

tion of (parts of) a RMS, the second one is usually conducted continuously after

the implementation. The main motivation for revenue opportunity assessment

1. Introduction

is to estimate the potential that the introduction of a RMS is able to generate.

After the implementation of the RMS or update of single components of the RMS

the management and the RM department aim at justifying the investment into

RM techniques. They want to know if the investment was successful and how

much of the revenue potential is actually gained. Most importantly this assess-

ment has to be performed continuously to track performance over time and to

identify and eliminate weaknesses in the RMS (see also P¨olt, 2001).

One main method for revenue opportunity assessment is according to Talluri

and van Ryzin (2004b) to estimate an upper bound for the achievable revenue

with RM techniques using perfect hindsight information and compare it with

the actual revenue. This approach reveals the potential revenue gains that are

achievable with the introduction of a RMS. Although usually only a fraction of

the revenue potential will be captured, even a fraction of the revenue potential

surely justifies the investment into RM. A better approach according to Tal-

luri and van Ryzin (2004b) is to assess the potential of RM methods by using

techniques of simulation. With simulation the performance of a complete (or

parts of a) RMS can be evaluated. In a simulation environment it is furthermore

possible to thoroughly model the customer behavior and investigate the likely

performance of the RM methods before implementation and operational service.

In addition sensitivity and what-if analysis help to examine and to understand

basic relations between the RM components and to reveal critical parts. A well

known example in this field is the passenger origin-destination simulator (PODS)

at the MIT, which is funded by a consortium consisting of seven airlines. PODS

was originally developed at Boeing Company by Craig A. Hopperstad and allows

to simulate a complete RM environment (see Gorin, 2000). PODS has been used

by many authors to analyze the impact and performance of new RM methods

and, according to Barnhart et al. (2003), is able to ”realistically simulate large

networks”. The main drawback of simulation is the clinical environment of the

modules and the data, but also the fact, that user controls and influences of the

revenue managers are currently not considered. Furthermore it cannot be used

to continuously measure RM performance once the system and the modules are

in place and operational.

To assess the revenue benefits of an operational RMS Talluri and van Ryzin

(2004b) and other authors propose different methods. As a main prerequisite the

revenue benefits measurement should be based on actual data from the opera-

tional RMS. As a simple classification there are usually three major categories:

•Comparison of pre and post RMS implementation performance

•Use of classical performance measures

1.2. Performance Measurement of Revenue Management

•Assessment of the achieved revenue potential

To compare the pre and post RMS implementation performance Talluri and

van Ryzin (2004b) distinguish between the comparison of two time periods - one

before, one after the implementation of the RMS - and a parallel test of markets or

flights - some controlled with and some controlled without the RMS. The first one

is a suitable method to justify the implementation of a RMS ex-post. One major

challenge for this method is to choose two time periods which are comparable

in terms of overall market structure. Although the contribution of the RMS to

the overall success can be isolated in that approach, it is not very well suitable

to continuously measure RM performance. Another approach is the parallel test

approach. It allows to examine the performance implications in the introduction

phase of the new RMS. Some flights or markets will be controlled with the new

RMS and others without. Positive differences in the resulting performance can

be attributed to the RMS and the revenue managers. For this approach it is

very important to choose comparable market situations to retrieve meaningful

results from this test phase. Furthermore this approach is usually not suitable

for continuous measurement as eventually the new RMS will be used to control

all markets or flights.

The second category includes the continuous use of classical performance mea-

sures. Classical performance measures include SLF or RASK and are often used

in the annual financial reports of an airline. Performance measures range from

indicators for the overall success of the RMS to indicators of single parts of the

RMS, such as forecast accuracy and quality. However, it is hard to isolate the

contribution of RM from the overall success for all of these classical measures. For

example, it is possible that the RM control is still equally good, but the RASK

and the SLF decrease dramatically because of the entrance of a new competitor.

A widely used technique at airlines for allowing continuous measurement but

also the isolation of the contribution of the RMS is the ROM. Smith et al. (1992)

describe the basic idea of applying different RM control strategies to a past time

period to estimate the potential revenue gains by the RMS and to investigate

the achieved revenue gains accordingly. The focus of their approach in contrast

to the previously mentioned revenue opportunity assessment discussed by Talluri

and van Ryzin (2004b) is on the continuous examination of the revenue benefits

of the RMS after the implementation. We introduce the ROM in more detail in

the next section.

1. Introduction

1.3. Measuring Performance with the ROM

This section introduces the general approach of the ROM and the necessary

terminology. We focus on the main concept, neglecting the difference between

leg- and network-based ROM.

1.3.1. Model Definition and Terminology

The idea of the ROM is to compare the revenue that has been actually achieved

during a booking period with two reference points which are estimated at hind-

sight (see e.g. Smith et al., 1992). Figure 1.4 shows the three main values that

are the starting point for the ROM: the potential revenue, the actual revenue and

the no RM revenue.

Potential

Revenue

Actual

Revenue

No RM

Revenue

Revenue Opportunity

Achieved

Revenue

Opportunity

Lost

Revenue

Opportunity

Figure 1.4.: Concept of the ROM

The first reference point is the potential revenue and is used as an indicator

of how much revenue would have been potentially achievable during the past

booking period. The no RM revenue as the second reference point indicates

the amount of revenue the airline would have earned by not applying any RM

controls and simply accepting all booking requests if capacity allows. Usually

the no RM revenue is estimated using a ’first come, first served’ (FCFS) strategy

assuming that no RMS and no other (manual) RM controls are in place. Both

estimations use the knowledge of the estimated unconstrained demand and are

deterministic as they are performed after the end of the booking period. In

contrast to this, the actual revenue is the result of the joint control decisions made

by the revenue managers and the supporting RMS. The main ROM measures for

isolated RM performance are defined based on the potential, the actual and the

no RM revenue:

•Revenue opportunity (RO) = Potential revenue - No RM revenue

•Achieved revenue opportunity (ARO) = Actual revenue - No RM revenue

1.3. Measuring Performance with the ROM

•Percentage achieved revenue opportunity (PARO) = Achieved revenue op-

portunity / Revenue opportunity

The RO indicates the possible revenue gains achievable with RM techniques

and the ARO shows how much of this revenue potential was actually earned. The

absolute measure ARO might for example be used to compare the costs of the

RMS and the revenue managers controlling the system with the gain the airlines

get out of it. In contrast to the absolute measures, the PARO indicates the

relative success of the RM control in comparison to its theoretical potential.

1.3.2. Main Properties of the ROM

One main property of the ROM is that it isolates the RM performance from all

other revenue influencing factors, such as the overall demand during the booking

period. Variations of the overall demand correlate highly with the revenue earned

in a given booking period. This behavior is reflected in common performance

measures like SLF and RASK. These measures usually increase in high demand

situations and decrease in low demand situations. The ROM, however, takes

the variations in customer demand into account as the potential and the no RM

revenue change with the demand level. An example of this characteristic is shown

in Figure 1.5, in which we compare the actual revenues and the PAROs of 20

42,2

42,4

42,6

42,8

43,2

1 2 3 4 5 6 7 8 9 1011121314151617181920

Flight departure

Actual Revenue

(Million EUR)

81%

82%

83%

84%

85%

PARO (Percent)

Actual Revenue PARO

Figure 1.5.: Comparison of Actual Revenue and PARO

different flight departures. Although at some flight departures we observe higher

total revenues the relative level of revenue gained by the RM controls applied

decreases. P¨olt (2001) presents a similar analysis and characterizes the ability to

1. Introduction

isolate the RM contribution from the overall success as one key property of the

ROM.

For the ROM measures obtained in a specific situation some special cases can

be observed. For example it is possible that the actual revenues fall below the

estimates for the no RM revenue. This is due to the fact that a very restrictive

RM control leads to the rejection of too many low-fare customers. Table 1.1

illustrates this case. In this example, we assume a plane with one compartment

Bkg Act. No RM Act. No RM

Class Fare Forecast Demand limit bkgs. bkgs. rev. rev.

1 200 20 5 20 5 5 1,000 1,000

2 100 50 45 30 30 45 3,000 4,500

Sum 70 50 50 35 50 4,000 5,500

Table 1.1.: Actual Revenue Gets Less Than No RM Revenue due to Restrictive

Controls

and a capacity of 50 seats. Due to the high forecasted demand for booking class

one, the optimization model reserves 20 seats for booking class one. However, the

real demand is only five. This leads to the rejection of many low fare customers

and to very bad revenue results in comparison to the no RM revenue. As a

consequence the ARO is negative. Another special case with the application

of the ROM is that the RO is zero. This is the case if the potential and no

RM revenue are equal. Basically this happens in low demand situations. If the

estimated unconstrained demand is less than the capacity of the flight leg all

bookings are accepted in the revenue estimations both for the potential and the

no RM revenue. The previously mentioned special cases also have an effect on

the PARO. If for example the ARO is negative, the PARO also is negative. In

all situations, in which the RO is zero, the PARO cannot be determined, because

of a division by zero. These special cases may occur in practical applications and

should then be interpreted considering the given definition of the ROM and the

current RM context.

One very important aspect of the application of the ROM is the validity of

its measures. The validity of the ROM is influenced by two sources of error.

We distinguish model-related errors from data-related errors. The first source of

errors describes all errors that occur, because the ROM does not reflect reality

correctly in the estimation of the potential and the no RM revenue. These errors

are mainly due to the practical limitations of the RMS in place. The main

model-related error is caused by the fact that the demand data at an airline

is usually aggregated at DCP-level. Deriving the correct booking order from

1.3. Measuring Performance with the ROM

aggregated data is not possible and thus within the ROM definition, a decision

has to be taken, which demand order is assumed in between two subsequent

DCPs. If for example a FCFS strategy is used to estimate the no RM revenue

the accuracy strongly depends on the real booking order. In the examples in this

section we assumed a strict low-before-high (LBH) booking order. This means

that customers willing to purchase low fare tickets are showing up first and the

customers opting for high fare tickets are coming afterwards. However, in reality

this will rarely be the case potentially leading to less accurate estimates for the

no RM revenue.

The second source of errors are data-related errors. If we assume that the

ROM reflects reality accurately, it still relies on estimated unconstrained demand,

which does not match real demand due to unconstraining errors. These errors

in the ROM induced by incorrect input data also lead to wrong estimations of

the potential and the no RM revenue. An example of how unconstraining errors

might influence the validity of the ROM measures is presented in Table 1.2. In

Class Fare Actual Real demand

Actual Actual No RM No RM Potential Potential

bookings revenue Demand bookings revenue bookings revenue

1 200 10 2,000 15 5 1,000 15 3,000

2 100 40 4,000 45 45 4,500 35 3,500

Sum 50 6,000 60 50 5,500 50 6,500

Resulting RO=1,000, ARO=500 and PARO=50%

Actual Estimated unconstrained demand

1 200 10 2,000 17 0 0 17 3,400

2 100 40 4,000 50 50 5,000 33 3,300

Sum 50 6,000 67 50 5,000 50 6,700

Resulting RO=1,700, ARO=1,000 and PARO=59%

Table 1.2.: Errors in Unconstrained Demand Lead to Wrong ROM Measures

this example, we again assume a plane with one compartment and a capacity of

50 seats. A total of 50 bookings have been accepted by the actual RM controls

with a total revenue of 6,000. The real demand is 15 for class one and 45 for

class two. This leads to a no RM revenue of 5,500 if the booking requests arrive

in LBH order and to an potential revenue of 6,500. According to those values,

the RO is 1,000, the ARO is 500 and the PARO is 50%. If we estimate the

unconstrained demand to be 17 for class one and 50 for class two, we derive

different ROM values. The no RM revenue would be 5,000 and the potential

1. Introduction

revenue 6,700. This would result in a RO of 1,700, an ARO of 1,000 and a PARO

of 59%. This difference between the PAROs for real and estimated unconstrained

demand could lead to a misinterpretation of the results and the evaluation of the

RM controls. Authors like P¨olt (2001) have already considered the problem of

invalid results because of errors in the estimated unconstrained demand.

1.3.3. Major Developments in Airline Revenue Management

Affect the ROM

As described in Section 1.1 science in RM has made significant progress in the

last decades of airline history. The two major developments not only have a

significant impact on how airlines model and setup their RMS, but they also have

a significant impact on the validity of the ROM. These two major developments

and their impact on the ROM are a key topic in the remainder of this thesis.

The first major development - the advancement from leg-based controls to

network-based controls - had a tremendous effect on the validity and applicability

of the ROM. Network-based controls take the interdependencies of connecting

itineraries into account and evaluate a connecting booking request as one O&D

itinerary. With network-based controls, it might be revenue optimal for the

network to accept a connecting passenger at a feeder flight in a low booking

class. This is particularly true if the passenger connects to a long-haul flight with

moderate demand. A leg-based ROM examines every flight separately and might

evaluate the actual booking in the low booking class at the feeder flight as poor

control. Let us assume it would have been revenue optimal for the entire flight

network to accept a booking from Hamburg via Frankfurt to New York for 1,000

and the associated fare value for the flight leg Hamburg to Frankfurt is 50. In this

case a leg-based ROM evaluates all rejected booking requests against bookings

on flight leg Hamburg to Frankfurt with an associated fare value higher than

50 as poor RM control, because on a leg base it would for example have been

better to accept a local flight from Hamburg to Frankfurt with fare value of 200.

These misleading and invalid results caused many airlines with network-based

seat inventory controls to stop using their leg-based ROMs.

The second main development with a major impact on the validity of the

ROM is the transition from independent to dependent demand structures. In

Table 1.3 a simple example is presented on how dependent demand structures

influence the estimation of the no RM revenue. If we assume that demand is

independent and demand for booking class one is 20 and for booking class two

the demand is 30, than the no RM revenue for a flight with capacity larger than

50 is 7,000. However, out of the 20 customers demanding booking class one

1.3. Measuring Performance with the ROM

there are 10 customers who would buy-down into booking class two, if it is also

available. This leads to a correct no RM revenue of 6,000. Similar examples

can be presented for the effect of dependent demand structures on the potential

revenue. Subsequently this also has a tremendous effect on the derived ROM

measures like the PARO.

Ind. No RM No RM Dep No RM No RM

Class Fare dmd. bkgs. rev. dmd bkgs. rev.

1 200 20 20 4,000 10 10 2,000

2 100 30 30 3,000 40 40 4,000

Sum 50 50 7,000 50 50 6,000

Table 1.3.: Effect of Dependent Demand Structures on No RM Revenue

1.3.4. Consideration of Practical Aspects in the ROM

Considering practical aspects in the ROM is also very important. We start with

introducing the basic process of the application of the ROM in practice. It

involves four main process steps. These steps are described in detail by Chandler

and Ja (2007). We illustrate these steps in Figure 1.6. The first step is to gather

Data

input

Uncon-

straining

ROM

calculation

Output and

evaluation

Figure 1.6.: Process of ROM Application

and input all relevant data of the booking period to be assessed. This includes

actual bookings and availability information. In the second step, the main input

for the ROM calculation is generated, the estimated unconstrained demand. As

we laid out earlier in this thesis, this is a very important task that is discussed in

detail in the remainder of this thesis. The first two steps may also be merged, if

the estimated unconstrained demand can directly be taken from the forecasting

module. In a third step, the ROM measures are calculated based on the estimated

unconstrained demand. Lastly, the measures are analyzed and potentially split

further.

In many cases, the ROM is not only used to measure the overall performance

of the RMS, but also of different components or parts of it. As described earlier

1. Introduction

overbooking and upgrading play an important role in airline RM. P¨olt (2001) for

example proposes a split between fare-mix, overbooking and upgrading success.

The choice of a specific way of considering components of the RMS strongly

depends on the airline’s context. We describe other potential ways to split the

ROM measures in Chapter 2.

When applying the ROM to network-based controls new challenges arise. The

main proposals calculate one single measure for the total network. However, for

many airlines it might be very interesting to disaggregate the ROM measures

for the total network to subparts of the network. Chandler and Ja (2007) for

example propose a disaggregation to market level or even a single flight leg. A

main challenge of this disaggregation is that usually the fares of the itineraries

have to be distributed to subparts or even single flight legs of the itinerary. The

distribution of the fares can be accomplished by prorating the fares. Methods

of prorating distribute the fares of an itinerary to subparts according to a given

allocation formula.

We focus on questions concerning the consideration of the practical aspects

mentioned above in the ROM in Chapters 6 and 7 of this thesis.

1.4. Scope and Purpose of the Thesis

In the previous section we introduced the ROM as an important method to

measure RM performance. We also described two major developments in RM

science that pose new questions and challenges on the ROM. In particular the

increasing importance of modeling dependent demand structures is at the center

of attention in the RM departments of many airlines.

Until now, the effect of the two major developments in RM science on the

ROM have not been reflected in detail. Since leg-based ROMs are not showing

valid results in a network-based RM environment, the question of validity and

applicability of the network-based ROM is crucial. Moreover the advancement to

dependent demand structures has not been discussed in detail in the context of

the ROM. In addition, we put further attention into considering practical aspects

in the network-based ROM with independent and dependent demand. This being

said, our main areas of interest in this thesis are:

•To assess the validity and applicability of a network-based ROM, in partic-

ular we aim at measuring the robustness of the network-based ROM

•To model dependent demand structures in a network-based ROM and to

analyze the validity and robustness of the extended ROM

1.4. Scope and Purpose of the Thesis

•To discuss and apply enhancements of the ROM to consider practical as-

pects

This thesis consists of eight chapters and is structured as follows. In this chap-

ter we introduced airline RM and main methods to measure RM performance.

We described the ROM with its main facets and motivated our research in this

field. In Chapter 2 we give an overview of state-of-the-art methods in the field of

airline RM, methods to measure RM performance, and in particular the ROM.

For the state-of-the-art of airline RM science we focus on the two major devel-

opments in demand modeling and optimization. We conclude this chapter with

an appraisal of research opportunities in the context of the ROM and the goals

of this thesis. The concept of a novel simulation-based approach to investigate

ROM properties is introduced in Chapter 3. We describe the basic approach

and the components of the simulation environment. In addition, we introduce a

novel method to measure the robustness of a ROM against errors in the input

data. We conclude this chapter with a detailed description of the scenarios used

to analyze the robustness and further properties of the ROM. The network-based

ROM with independent demand will be discussed in Chapter 4. We describe

main properties of the estimated potential and no RM revenue and analyze the

robustness of the ROM under various scenarios. An enhancement of the ROM to

dependent demand structures is introduced in Chapter 5. We describe in detail

how dependent demand can be modeled in the ROM formulation and discuss

main characteristics. In addition, an investigation of the ROM properties, in

particular of its robustness, is conducted using our simulation environment. The

consideration of practical aspects in the ROM is the focus of Chapters 6 and 7.

We evaluate the possibility to disaggregate the ROM measures to single flight

legs in Chapter 6. The consideration of no-shows, cancelations and subsequently

overbooking and upgrading is described in Chapter 7. Furthermore we introduce

an extension of the ROM to derive sub-measures for the success of single RM

components, like overbooking and upgrading. We will summarize and conclude

this thesis in Chapter 8 and give an outlook to further research.

1. Introduction

2. Airline Revenue Management

and Performance Measurement:

State-of-the-art

In this chapter we give an overview of state-of-the-art methods for both airline

revenue management and performance measurement of airline revenue manage-

ment. For the airline RM part in Section 2.1 we give a brief summary about main

overview literature and focus on the transition from leg-based to network-based

RM controls and the advancement from independent to dependent demand struc-

tures, in particular in unconstraining and forecasting techniques. In Section 2.2

we give an overview about methods and techniques to measure the performance

in RM. A thorough overview of the state-of-the-art of the ROM is presented in

Section 2.3. We conclude this chapter with an appraisal of recent challenges and

goals of this thesis in Section 2.4.

2.1. Airline Revenue Management

In the last decades numerous works have been published in the field of airline

RM. A detailed introduction into the topic of RM is given by Cross (1995).

An overview of the development in RM science up to the end of the 1990s is

described by McGill and van Ryzin (1999). Chiang et al. (2007) present a more

recent overview of the advances and recent developments in RM. Weatherford

and Bodily (1992) thoroughly describe the characteristics of problems for which

RM is applicable and introduce a taxonomy to classify different kinds of problems

in this area. Talluri and van Ryzin (2004b) not only introduce the art of RM, but

also give a detailed overview about the different aspects that RM deals with from

both a theoretical and a practical perspective. Other publications that introduce

RM in a detailed and structured way are e.g. Cross (1997), Klein and Steinhardt

(2008) and Phillips (2005). The broad range of application areas of RM is for

example discussed by Yeoman and McMahon-Beattie (2004), Kimms and Klein

(2005), Chiang et al. (2007) and Talluri and van Ryzin (2004b). Kimms and

2. Airline Revenue Management and Performance Measurement: State-of-the-art

Klein (2005) describe the application of RM methods in the airline sector, the

tourism industry and discuss an application in a manufacturing environment.

Some very special areas of application are presented by Yeoman and McMahon-

Beattie (2004). They present for example an application of RM with saunas.

The structure of this section follows the major line of developments in RM

science that we presented in Section 1.1. First, we focus on the transition of

optimization models from leg-based controls to network-based controls under

the general assumption of the independence of the demand between booking

classes in Section 2.1.1. The progression from independent to dependent demand

structures is considered both in Section 2.1.2 and 2.1.3. In Section 2.1.2 we discuss

the advancement in modeling customer demand to consider dependencies. In

Section 2.1.3 we present optimization models that incorporate dependent demand

structures. Please note that we do not provide a complete literature review in

this section, but highlight major contributions and ideas in the areas that are

relevant for our thesis.

2.1.1. Optimization Models with Independent Demand

Leg-based Controls

A thorough overview on publications on optimization models on a single flight leg

with independent demand structures is provided by McGill and van Ryzin (1999)

and Talluri and van Ryzin (2004b). The authors introduce and discuss a vari-

ety of contributions for the leg-based seat inventory control. A milestone in the

development of optimization models for leg-based controls was the introduction

of Littlewood’s rule for the two-fare-class problem in Littlewood (1972). Be-

lobaba (1987, 1989) extended Littlewood’s rule to multiple booking classes and

introduced the expected marginal seat revenue (EMSR) heuristic to determine

booking limits for the seat inventory control. Methods to obtain optimal booking

limits have been introduced for example by Curry (1990), Brumelle and McGill

(1993) and Wollmer (1992). The EMSR heuristics, and in particular the EMSRb

heuristic, are still widely in use, because they are easy to implement and deliver

very satisfactory results compared to the optimal booking limits. All of the meth-

ods mentioned above share certain underlying assumptions. One assumption -

next to the assumption about the independence of the demand between book-

ing classes - is sequential booking classes. Many RM controls take advantage of

a sequential order of booking classes applying a principle called nesting. With

nesting all protected seats for a given booking class are also available for any

higher booking class. This means that if a booking class is available for sale,

2.1. Airline Revenue Management

all higher booking classes are available, too. Another main assumption is the

LBH booking order, which we already introduced in the previous chapter. With

the introduction of optimization methods based on dynamic programming (DP)

the LBH booking order assumption could be neglected. Lee and Hersh (1993)

for example introduce a discrete-time DP model formulation to generate optimal

booking limits with batch bookings. A detailed description about applying DP

in RM can be found in Talluri and van Ryzin (2004b).

Network-based Controls

Concise overviews about optimization models considering network effects are pro-

vided by Barnhart et al. (2003), McGill and van Ryzin (1999) and Talluri and

van Ryzin (2004b). The main challenge with network-based controls is to ac-

count for the dependencies between the flight legs in the network, because an

accept/deny decision on a booking request for an itinerary potentially involves

deciding on the availability of multiple flight legs. Methods to account for net-

work effects using adjusted leg-based optimization methods have been discussed

by Williamson (1992, 1988), Smith et al. (1992), Vinod (2005) and Talluri and

van Ryzin (2004b). Williamson (1992, 1988) presents the prorated EMSR heuris-

tic that splits the fares of an itinerary onto the contained flight legs according to a

given prorating scheme. With the prorated fares booking limits for each flight leg

can be calculated. Another main approach that allows using leg-based inventory

controls called virtual nesting is described by Smith et al. (1992), Vinod (2005),

Talluri and van Ryzin (2004b) and also Williamson (1992). Virtual nesting de-

fines virtual classes on each leg and assigns sets of itineraries to these classes.

This assignment process is also known as indexing. If an itinerary is requested

by a customer, the booking request will be accepted if all the virtual classes that

the itinerary is assigned to are available.

A very simple approach to obtain booking limits on a network level using a

deterministic linear program (DLP) is described for example in Talluri and van

Ryzin (2004b) and Williamson (1988). The objective function of the DLP aims at

optimizing the total network revenue, while considering the capacity constraints

of the flight legs and the forecasted demand of the itineraries. The primal solution

of a DLP can be used as booking limits for the respective itineraries. Among

others, Talluri and van Ryzin (2004b) describe the use of bid prices as another

approach for network-based RM controls. In this approach to control the seat

inventory a bid price for each leg is calculated. If the fare for an itinerary exceeds

the sum of the bid prices of the legs contained in the itinerary the booking

request is accepted, otherwise it is rejected. A simple way to obtain bid prices

2. Airline Revenue Management and Performance Measurement: State-of-the-art

is by solving a DLP and using the shadow prices on the leg capacity constraints.

Different variants and characteristics of DLPs are described in detail by Talluri

and van Ryzin (2004b). One drawback of the DLP is that it only considers mean

demand. Talluri and van Ryzin (1999) propose an extension of the DLP called

the randomized linear program (RLP), which accounts for variance in demand

and results in better bid prices.

However, bid prices have to be updated regularly within the booking period

as every accepted booking and change in forecasted demand has the potential to

change the adequate bid price. Bertsimas and Popescu (2003) basically propose

to solve the DLP for each incoming booking request and call their approach ”cer-

tainty equivalent control”. Other methods to obtain better bid prices include the

application of DP. Due to the fact that the state space of a DP even for small

networks grows enormously, these methods usually apply a decomposition of the

network problem to multiple leg problems such as the virtual nesting control.

Talluri and van Ryzin (2004b) describe two widely used approaches. The first

approach is used to improve the indexing process in a method called displacement

adjustment virtual nesting (DAVN). The other approach calculates bid price vec-

tors, that contain an appropriate bid price for each number of remaining seats.

Methods that do not make use of decomposition but of simulation-based ap-

proaches to improve bid prices are presented by Klein (2007), Bertsimas and

de Boer (2005) as well as van Ryzin and Vulcano (2008b). Klein (2007) intro-

duces a heuristic for self-adjusting bid prices considering the current booking

situation. Bertsimas and de Boer (2005) and van Ryzin and Vulcano (2008b)

propose simulation-based approaches to improve bid prices that do not make use

of a decomposition approach either and show reasonably good results with ad-

equate computing time. M¨oller et al. (2004) propose a stochastic programming

formulation for network-based RM controls. The method shows good revenue

results on small network instances, but is currently computationally infeasible

for flight networks that are used in practice.

2.1.2. Modeling, Unconstraining and Forecasting Customer

Demand

The problem of modeling and forecasting customer demand is one of the most

important areas in RM research. The assumptions about customer behavior, for

example the LBH booking order or the independence of demand between booking

classes are integral decisions for the optimization models applied. Furthermore

the question of forecast accuracy for a given demand model significantly corre-

sponds to the RM success. A concise overview about the different aspects of

2.1. Airline Revenue Management

handling and modeling customer demand is provided by Ratliff et al. (2008),

Cleophas (2009) and Cleophas et al. (2009a). In the following we use a cate-

gorization of demand models introduced by Ratliff et al. (2008). The authors

mainly distinguish between three major model types. They refer to single-class,

multiple-class and multiple-flight models. In this section we focus in particular

on major demand models and approaches to unconstrain and forecast demand

for these three types.

Independent Demand Models

One classic assumption in modeling customer demand is to assume independence

of the demand between booking classes. Among others Talluri and van Ryzin

(2004b) describe the independent demand model. Basically this simplifying as-

sumption was justified with the application of fencing rules as described in Section

1.1. Although this assumption was never completely appropriate, it was and is

still widely applied by both researchers and practitioners. Ratliff et al. (2008)

refer to the independent demand model as the single class model.

For all forecasting methods, the handling of censored booking data is an ele-

mentary part. The actual bookings for a booking period usually do not reflect

the overall demand that existed in the market. This is due to the fact that some

booking classes are closed during the booking period according to the results of

the optimization. A booking request for these booking classes cannot be seen

in the actual bookings. Thus airlines strive to unconstrain the demand in those

time periods, in which the booking classes have not been available. The process

of unconstraining plays an important role in forecasting.

One of the seminal works on unconstraining for independent demand structures

is provided by Zeni (2001). The author describes and compares major methods

like booking profile method,mean imputation method,projection detruncation and

expectation maximization. Another detailed introduction into unconstraining is

given by P¨olt (2000). Crystal et al. (2007) introduce another unconstraining

method called double exponential smoothing. Other important publications that

deal with unconstraining are provided by Zeni (2003), Zeni and Lawrence (2004),

Talluri and van Ryzin (2004a) and Weatherford and P¨olt (2002). Weatherford

and P¨olt (2002) describe and quantify the positive revenue effect, that occurs

when the unconstraining quality increases.

Among others Lee (1990) and Talluri and van Ryzin (2004b) discuss different

approaches for demand forecasting. A widely used method is exponential smooth-

ing, because it is simple, robust and obtains a good forecast accuracy. The idea

of exponential smoothing is to calculate the new forecast based on the historical

2. Airline Revenue Management and Performance Measurement: State-of-the-art

forecast and actual observations weighted with a smoothing factor α. The higher

the smoothing factor αis, the higher the share of the actual observations in the

new forecast gets. Talluri and van Ryzin (2004b) discuss the effect of different

levels of α. The actual observations used in this approach is usually the estimated

unconstrained demand for the given booking period.

The forecasting of demand on itineraries in a flight network is more difficult

than the forecasting of the demand on single flight legs. Williamson (1992) points

out that due to the high number of itineraries offered by an airline, a significant

portion of them has a probability to be traveled near or equal to zero. This

characteristic makes forecasting for these itineraries very difficult. According

to McGill and van Ryzin (1999) airlines tackle this problem by grouping these

itineraries. In addition to just using the booking numbers from the inventory

system Neuling et al. (2004) for example propose an analysis of the passenger

name records to improve the quality of forecasts on itineraries. Analyses on the

positive effect of better forecast accuracy were for example accomplished by Lee

(1990) and Weatherford and Belobaba (2002).

However, the assumption of the independence of demand between booking

classes is increasingly inadequate. Moreover Cooper et al. (2006) describe the so

called spiral down effect. This effect in particular occurs when classical RM envi-

ronments that assume independent demand are confronted with a price-sensitive

customer behavior and fare structures with no restrictions. In such a situation

a price-sensitive customer looks for the lowest available fare of a ticket. This

behavior leads to increased bookings in the lower booking classes and decreased

bookings in the upper booking classes. The forecaster incorporates this infor-

mation into the new forecasts and the optimization model reserves less seats

for higher booking classes. This feedback loop repeats and the revenue results

decrease further. Eventually, the forecasts only predict demand for the lowest

available fare. As a remedy against this behavior demand models that incorpo-

rate dependencies have been proposed.

Dependent Demand Models

To react to the effects described by Cooper et al. (2006) models for dependent

demand structures have been introduced. A thorough overview about available

literature is provided by Ratliff et al. (2008), Cleophas et al. (2009a) and Cleophas

(2009). In this section we focus on selected dependent demand models and rel-

evant methods for unconstraining and forecasting dependent demand. We both

discuss multiple class and multiple flight models. Single-class models have already

been described in the previous section.

2.1. Airline Revenue Management

Main demand models for dependent demand include sell-up models,hybrid

demand and customer-choice models. Sell-up models are based on independent

demand and incorporate a sell-up probability describing a customer purchasing

a ticket in a higher booking class, if his first choice is not available. Talluri and

van Ryzin (2004b) give a comprehensive introduction into sell-up models. A

combination of product-oriented and price-oriented customers is modeled with

hybrid demand. The product-oriented or yieldable demand is expected to have

no dependencies to other booking classes, i.e. a customer will not sell-up or buy-

down into another booking class. The yieldable demand matches the independent

demand described in the previous section. The other part is the price-oriented

or priceable demand. This demand is a consequence of restriction-free-pricing

and models customers that have a certain willingness-to-pay. According to their

willingness-to-pay they will opt for the cheapest booking class available. Hybrid

demand models are described intensively by Fiig and Isler (2004), Boyd and

Kallesen (2004), Walczak et al. (2010) and Fiig et al. (2010). A general extension

of the hybrid demand model is the model described by Winter (2010). The

demand is modeled using a directed acyclic buy-down graph. For each booking

class the buy-down into a lower booking class is estimated. The buy-down occurs,

if the lower booking class is available. In the graph the buy-down is modeled

with buy-down arcs. Additionally, for each booking class the total demand is

estimated, if all buy-down can be prevented. It is possible to show, that the

demand models described by Fiig and Isler (2004), Boyd and Kallesen (2004),

Walczak et al. (2010) and Fiig et al. (2010) are a special case of the model

introduced by Winter (2010). A general description of customer demand that

takes into account multiple criteria for selecting a ticket is modeled using so-

called customer-choice models. These models define the choice behavior according

to preferences in categories like price, travel time, strategic behavior, offers by

competitors and so forth. Kimms and M¨uller-Bungart (2006) for example give a

comprehensive introduction into customer-choice behavior. A detailed overview

on available literature is given by Cleophas (2009). Customer-choice behavior is

modeled formally using binary probit,binary logit or multinomial logit models

(see e.g. Talluri and van Ryzin, 2004b). We refer the interested reader to the

publications mentioned above to learn more about customer-choice models.

Ratliff et al. (2008) present a concise overview about methods to unconstrain

and forecast demand for multiple classes. Mishra (2003) for example introduces

a method called cumulative expected bookings. It is primarily used to estimate

dependent demand in restriction-free fare environments. An application of the

expectation maximization algorithm is introduced by McGill (1995). Skwarek

(1996) investigates unconstraining and forecasting with sell-up behavior. An-

2. Airline Revenue Management and Performance Measurement: State-of-the-art

other method is Q-forecasting. Authors like Cleaz-Savoyen (2005), Belobaba

and Hopperstad (2004), Gorin and Belobaba (2004), Kambour et al. (2001) and

Reyes (2006) describe the method in detail. Q-forecasting is mainly applied in

the PODS context and primarily used to estimate the priceable demand part in

hybrid demand models.

The multiple-flight models are also covered by several publications. Ratliff

et al. (2008) introduce a recapture heuristic to estimate the unconstrained de-

mand based on actual bookings for multiple flights. Stefanescu (2009) and Ste-

fanescu et al. (2004) describe a multivariate demand model and present a method

to unconstrain and forecast correlated demand based on censored sales data. The

expectation maximization approach to unconstrain dependent demand for mul-

tiple flights is described and applied by Talluri and van Ryzin (2004a), Vulcano

et al. (2010) and Vulcano et al. (2009). Ja et al. (2001) apply a regression-based

demand and recapture estimation to unconstrain and forecast demand for con-

nected flights.

2.1.3. Optimization Models with Dependent Demand

In this section we introduce some major optimization models that consider hybrid

demand or customer-choice based demand models. We primarily focus on some

major models that are relevant for our analyses. As a starting point we refer to

Weatherford and Ratliff (2010). The authors discuss existing approaches to deal

with dependent demand structures in optimization.

Optimization models using hybrid demand are getting increasingly common.

Authors like Fiig and Isler (2004), Boyd and Kallesen (2004), Cleaz-Savoyen

(2005), Belobaba and Hopperstad (2004), Reyes (2006), Walczak et al. (2010)

and Fiig et al. (2010) discuss these kind of optimization models. The main idea

behind these approaches is the application of fare adjustment or fare transfor-

mation and demand transformation. The fare transformation incorporates the

opportunity cost of potential buy-down into the fares. With demand transforma-

tion the hybrid demand consisting of yieldable and priceable demand is changed

into an equivalent yieldable demand. Walczak et al. (2010) lay out that this

transformation from a dependent demand model into a transformed independent

demand model leads to the same optimization results. The main advantage is

that the existing optimization methods, i.e. the operational optimization systems

at an airline, can still be used. For details on the transformation and the charac-

teristics of this optimization approach we refer to Walczak et al. (2010) and Fiig

et al. (2010).

Optimization models for general customer-choice demand models are discussed

2.2. Performance Measurement of Revenue Management

by Brumelle et al. (1990), Gallego et al. (2009), Bront et al. (2009), van Ryzin

and Vulcano (2008a) and Talluri and van Ryzin (2004a). Brumelle et al. (1990)

propose a method to allocate seats between stochastically dependent demands.

Gallego et al. (2009) extend the EMSR heuristic to consider choice-based cus-

tomer behavior for single-leg RM with demand dependencies. A column genera-

tion algorithm for choice-based network RM is presented by Bront et al. (2009). A

customer-choice demand model to compute virtual nesting controls in a network-

based environment is considered by van Ryzin and Vulcano (2008a). Talluri and

van Ryzin (2004a) introduce an optimization approach for customers with general

choice behavior.

2.2. Performance Measurement of Revenue

Management

Until now, a variety of sophisticated methods to measure the performance of RM

has been introduced. In this section we introduce the most common approaches.

We also refer to the motivation of PM, highlight some main approaches and con-

tributions, exemplify the broad variety of methods available and take a broader

view on RM performance by discussing some organizational challenges for suc-

cessful RM. A summary of the state-of-the-art of the ROM use is presented in

the next section. As a good starting point for literature that covers PM of RM

we refer to Chiang et al. (2007). The authors give a detailed overview about

existing approaches.

The motivation to measure performance in RM is explained by numerous au-

thors. Talluri and van Ryzin (2004b) mention the assessment of the revenue

potential of a RMS and the continuous measurement of captured benefits. After

stressing the importance to justify the investment into the RMS they also high-

light the importance of continuous improvement of the RM process and methods.

Vinod (2006), among others, names the validation of the performance of a re-

cently introduced RMS and ”getting the most out of revenue management in a

steady-state operating environment”. One argument in favor of PM according

to Curry (1992) is the fact that continuously measuring performance is able to

prevent costly mistakes. He adds that in the course of a booking period revenue

managers or the RMS in use can make wrong decisions and methods of PM are

able to detect them. Additionally it can help companies to fine-tune their RMS,

as PM tools allow to display in which areas of the system further revenue can be

generated. As a basic property PM methods for RM should therefore be able to

isolate the contribution of RM from the overall success. P¨olt (2001) adds that

2. Airline Revenue Management and Performance Measurement: State-of-the-art

PM also helps to track the RM performance over time, which is key for continu-

ous improvement of the RM controls. This not only facilitates the identification

of weaknesses in the RMS but it also allows to quantify and objectify the impact

of RM decisions.

Applications of simulation to analyze the performance of RM methods are

widely spread and used for different purposes. A major contribution in this field

is PODS. Many authors used the PODS environment to assess the performance of

different RM components in a realistic environment. In the following we will just

name some of these assessments and do not account for completeness. Among

others Skwarek (1996), Reyes (2006), Carrier (2003), Cleaz-Savoyen (2005), Gorin

(2000), Zickus (1998) and Gorin and Belobaba (2004) used the PODS environ-

ment to accomplish research on forecasting models, hybrid forecasting, fare ad-

justment and other topics.

More applications of simulation can be found in Weatherford (2004b, 2002,

2004a), Belobaba and Weatherford (1996), Weatherford and Belobaba (2002) and

Weatherford and P¨olt (2002). Similar to PODS, these authors also investigate

different aspects of RMS components. The topics range from evaluating the

impact of different optimization and forecasting models on the RM success to

comparing the performance of different methods of unconstraining.

Variants of simulation studies focussing on single regional markets are for ex-

ample provided by Oliveira (2003) and Eguchi and Belobaba (2004). In his study

Oliveira (2003) assesses the consequences of RM application in the Brazilian

airline market. Eguchi and Belobaba (2004) analyze the impact of RM method-

ologies on the domestic airline market in Japan.

A novel approach to apply a simulation environment to investigate forecast

performance is presented by Cleophas (2009) and Cleophas et al. (2009b). The

authors apply a simulation-based approach based on a decomposition of the sin-

gle components of a RMS to evaluate the performance of forecasts and classical

measures of forecast accuracy considering customers with a choice-based demand

model. Furthermore simulation environments are being used to examine strate-

gic decisions or to train the revenue managers. Basumallick and Singh (2009),

for example, propose a simulation environment that is fed and calibrated with

data from the operational RMS to analyze the impact of strategic RM decisions.

Gerlach and Frank (2010) introduce the revenue management training for ex-

perts (ReMaTE) simulator. In this simulation setup the revenue managers are

able to replay real life situations to better understand the influencing factors for

RM success. The simulator basically reflects the operational RMS with the same

underlying RM methods and control screens used. ReMaTE furthermore allows

to simulate competition against other airlines in selected markets.

2.2. Performance Measurement of Revenue Management

Core principles in the implementation of a RM simulator can be found for

example in Talluri and van Ryzin (2004b), Frank et al. (2008) or Vinod (2006).

Vinod (2006) describes a ”passenger simulation model” and states that using this

model can help airlines to point out the revenue gains through the application

of RM. Talluri and van Ryzin (2004b) and Frank et al. (2008) describe basic

principles about setting up a RM simulation environment.

Various methods of PM using actual data from the operational RMS are pro-

posed in the RM community. The comparison of two time periods is described

by Williams (1995), Jain and Bowman (2005) as well as Lieberman and Raskin

(2005). Williams (1995) uses a multi-regression analysis to evaluate the positive

effect of RM on the overall performance. Jain and Bowman (2005) introduce a

method to measure the performance of a length-of-stay control for the hotel in-

dustry. The authors conclude that this model provides accurate results by remov-

ing the influence of internal and external factors. Lieberman and Raskin (2005)

introduce a method named ”comparable challenges” which normalizes market

conditions and provides an indicator of the efficiency of RM decisions. However,

although the contribution of the RMS to the overall success can be isolated in

these approaches, they are not well suitable for continuous measurement. One

application of parallel testing of old and new RM methods using actual data is in-

troduced by Talluri et al. (2010). The authors propose a method called ”sandbox

testing” to evaluate the revenue potential of a new RM methodology.

Another method to assess the success of RM called ”performance monitor”

is introduced by Anderson and Blair (2002, 2004). The first article deals with

assessing the relative performance of a location to benchmarks gathered from

different locations, markets and also time periods. In their second article they

describe a disaggregation of the lost revenue opportunities to single components.

Vinod (2006), P¨olt (2001) and Talluri and van Ryzin (2004b) present a com-

prehensive overview of classical or traditional performance measures, which can

easily be calculated using data from the operational RMS. Vinod (2006) and P¨olt

(2001) focus on the airline industry, while Talluri and van Ryzin (2004b) name

common performance measures used in other industries. Vinod (2006) also pro-

poses to distinguish between pre- and post-departure measures. Pre-departure

measures give an indication of how well the RM is performing within the booking

period. A major pre-departure measure is the booked seat load factor. Post-

departure measures are calculated after the departure of the plane and describe

retrospectively the overall success or the isolated contribution of the RMS. Widely

used classical measures are RASK or SLF. Phillips (2005) names the RASK as

the key classical measure as it not only incorporates the revenue gained, but also

considers the supply that was offered - namely the seat kilometers that have been

2. Airline Revenue Management and Performance Measurement: State-of-the-art

offered to customers. Beyond focussing on monetary performance or the utiliza-

tion of the plane several measures are known to measure forecast accuracy. In

a seminal work about forecasting, Armstrong (2001) provides a comprehensive

overview of common measures to examine forecast accuracy and performance.

According to authors like Skugge (2004) and Lieberman (1991) meeting the

organizational requirements is key for a successful implementation of RM (see

e.g. Lieberman, 1991). Based on an empirical test, Crystal (2007) identifies the

”technical capability” and the ”social support capability” as key drivers of RM

success. Lieberman (1991) proposes ten guidelines for a successful application

of RM including the importance of training the employees. Skugge (2004) also

emphasizes the relevance of training as a main driver for RM performance par-

ticularly the use of interactive case studies and simulation tools. One previously

mentioned example in this area is the ReMaTE simulator described by Gerlach

and Frank (2010). This simulation tool intends to train the revenue managers to

obtain the full possibilities of RM. In his article Lieberman (2003) concludes that

six key criteria exist for successful RM: ”measuring performance”,”developing

supporting business policies and processes”,”ensuring decision-making authority

and accountability”,”integrating RM with other departments”,”knowing the lim-

its of the RMS” and ”providing career path support and progression”. In addition,

Wishlinski (2006) discusses the organizational requirements for successful RM in

detail.

2.3. The ROM

Leg-based ROMs have been applied at many airlines and some publications can

be found covering the topic. First ideas to apply perfect RM controls in hind-

sight were presented by Kempka (1991) and Smith et al. (1992). Kempka (1991)

proposes a model for calculating the optimal revenue on a single leg. Smith

et al. (1992) discuss different hindsight control strategies in detail and provide a

comprehensive introduction to the ROM in a leg-based airline RM environment.

Daudel and Vialle (1992, p. 110) also propose to compare the actual revenues

with estimates for perfect hindsight and no RM revenues. P¨olt (2001) provides

a thorough summary of the leg-based ROM. Rannou and Melli (2003) use a pro-

cedure very similar to the ROM to evaluate the performance of a RMS in the

Western European hotel-industry. Similar to Smith et al. (1992) they define and

discuss various control strategies to estimate the potential and no RM revenue

in hindsight. For the no RM revenue estimate they propose not only to apply a

FCFS strategy, but to assume certain (rule-based) user interactions that lead to

2.3. The ROM

higher revenues.

Adler (1993) describes several major issues that arise when a leg-based ROM is

applied in a network-based RM environment and suggests to introduce adjusted

variants of the ROM. Proposals for network-based ROMs are introduced by Tal-

luri and van Ryzin (2004b), Vinod (2006), Chandler and Ja (2007) and Temath

et al. (2009). Talluri and van Ryzin (2004b) and Vinod (2006) propose to use

a LP-formulation to determine the optimal revenue for independent demand at

hindsight. Chandler and Ja (2007) describe the whole process of ROM applica-

tion at an airline and introduce the approaches used to estimate the potential

and no RM revenue for a network-based ROM in detail. Temath et al. (2010)

present computational results on ROM robustness for the network-based ROM

with independent demand.

Adler (1993) points out that the ROM is only able to measure ”within the

current infrastructure”. Chandler and Ja (2007) examine this characteristic for

the assumption of the independence of the demand between booking classes and

describe the negative implications on the potential revenue estimations if depen-

dent demand structures are not considered. Temath et al. (2009) introduce an

extension of the network-based ROM to dependent demand structures to account

for this problem.

Some authors comment on main properties of the ROM. Curry (1992) intro-

duces the notion of ”achievable opportunity” to consider forecast errors during

the booking period and its implications on the ROM results. Adler (1993) names

the ability to isolate the RM contribution of the overall success as one main

property. The importance of accurate estimates for the unconstrained demand

for the ROM is emphasized by Chandler and Ja (2007), Adler (1993), P¨olt (2001)

and Zeni (2001, 2003). P¨olt (2001) shows an analysis in a leg-based airline RM

context in which he investigated the effect of an independent and unbiased un-

constraining error on the validity of the ROM. He points out that the effect in

that context is minor and can be neglected since the ROM errors balance out

at an aggregated level. Adler (1993) states that the ROM is ”is only as good

as the unconstrained demand forecasts”. Zeni (2001, 2003) also emphasizes this

problem in his detailed work on unconstraining.

The practical applications of the ROM plays an important role in ROM dis-

cussions as well. Cross (1995) characterizes the ROM as a very useful method to

keep the revenue managers ”focussed” and to search for continuous improvements.

Bach (1999) presents an analysis which shows a positive correlation between the

performance of revenue managers and their ROM results. Adler (1993) and P¨olt

(2001) consider the relation between ROM measures and classical performance

measures. Adler (1993) proposes the ”key performance measures basket concept”,

2. Airline Revenue Management and Performance Measurement: State-of-the-art

to achieve the best results using performance measures. P¨olt (2001) suggests to

use ROM measures in combination with other performance measures to increase

reliability.

Ideas to customize the ROM to investigate single components of RM control are

presented by Smith et al. (1992), P¨olt (2001) and Chandler and Ja (2007). Smith

et al. (1992) describe two variants of the ROM, which focus on specific parts

of RM. On the one hand they describe the overbooking ROM, which estimates

the contribution of overbooking to the overall success. On the other hand they

present the discount allocation ROM that describes the performance of reserving

seats for different price categories. One key challenge for the customized use of

the ROM measures according to Smith et al. (1992) is to ”avoid double counting

benefits”. P¨olt (2001) adds to that and identifies upgrading, overbooking and

fare-mix as potential areas of investigation. Chandler and Ja (2007) propose

to split the ROM measures into ”dilution” and ”spoilage”. With the analysis

of dilution they aim at assessing the revenue mix of the passengers accepted.

The analysis of spoilage in contrast aims at investigating the revenue loss caused

by empty seats that could have been sold to customers. For the network-based

ROM the authors propose to disaggregate the ROM measures for the total flight

network to market or leg level.

Beyond the application of the ROM to continuously measure the performance

of an operational RMS, some authors use the ROM to describe the performance

of new RM methods in simulation studies. In the following we present some ex-

amples. Mak (1992), for example, uses the ROM to evaluate the performance of

different optimization techniques in a simulation study. Dar (2006) uses a no RM

control strategy to measure performance improvements achieved by different RM

methods in a PODS study. An examination of customer lifetime value consider-

ations in RM is presented by von Martens and Hilbert (2010). They utilize the

ROM to determine upper and lower bounds for the achievable revenue. Imhof

et al. (2010) use revenues based on a simulated FCFS strategy and ex-post opti-

mal revenues to classify the performance of different approaches to optimize the

availability of rental cars considering upgrades.

2.4. Research Opportunities and Goals of the

Thesis

In the previous sections we gave an overview about current literature in airline

RM, performance measurement of RM and the ROM. While revisiting the existing

literature we identified some research opportunities.

2.4. Research Opportunities and Goals of the Thesis

Performance measurement is an important facet of the application of RM at

airlines. Among other approaches discussed in literature the ROM allows to con-

tinuously measure and to isolate the contribution of RM from the overall success.

The ROM is well documented in leg-based environments. Authors like Smith

et al. (1992) and P¨olt (2001) discussed the ROM and its properties intensively.

In particular the effect of errors in the input data on the ROM is considered the

main driver for validity of the ROM. However, a systematic evaluation of the

implications of unconstraining errors on the robustness of the ROM has not yet

been presented. In addition the airline market and subsequently the airline RM

has significantly changed. The advancement from leg-based to network-based

RM controls and the consideration of dependent demand models instead of in-

dependent demand models are becoming increasingly important. As Barnhart

et al. (2003) point out, applications of network-based controls with independent

demand in airline RM are nowadays common with airlines using hub and spoke

networks. On the contrary the necessity of applying forecasting and optimization

models for dependent demand strongly increased in the last decade caused by

low-cost-carries removing fare-restrictions and the ability to search the internet

for cheap fares. Research in this field has made tremendous progress and as

Weatherford and Ratliff (2010) point out, a lot of work exists in this field which

prove applicable in practice. Only a limited number of publications exist covering

the necessary modifications and adjustments of the ROM to consider these new

developments. Proposals to apply the ROM in a network-based RM environment

with independent demand have been described for example by Chandler and Ja

(2007). However, an enhancement of the ROM to dependent demand structures

and a concise analysis of the effect of dependent demand structures is missing.

Proposals to consider practical aspects, for example to integrate overbooking and

upgrading into the ROM, have been made so far, but no detailed analysis has

been presented.

Building on the research work accomplished so far and the developments air-

line RM is heading at, we derive some research opportunities and goals for this

thesis. First, we want to thoroughly assess the effect of errors in the uncon-

strained demand on the robustness of the network-based ROM. Therefore we

aim at implementing a simulation environment that reflects reality in the best

possible way and to develop a novel approach to measure the robustness of the

ROM. In addition we plan to assess the properties of the ROM with the help of

different scenarios and sensitivity analysis. The simulation environment should

be capable to allow those kinds of analyses. In a second step, we want to use

the novel simulation-based approach to investigate the properties of the network-

based ROM with independent demand. A special focus will be on the comparison

2. Airline Revenue Management and Performance Measurement: State-of-the-art

of data- and model-related errors and on the robustness of the ROM against er-

rors in the unconstrained demand. Another main goal of this thesis is to consider

the latest developments in RM science and to enhance the network-based ROM

with independent demand to dependent demand structures. Therefore we want

to make use of a state-of-the-art dependent demand model and an optimization

method which is able to handle dependent demand. We aspire to enhance the

given formulation of the ROM to dependent demand structures. Similar to the

ROM with independent demand questions about the robustness of the ROM with

dependent demand are to be analyzed. As a last goal we strive to consider prac-

tical aspects in the ROM. The disaggregation of the ROM measures to subparts

of the flight network seems to be useful in practice. We will investigate if this

can be done and how reliable the results are. At last the assessment of single RM

components needs to be discussed. In particular we aim to integrate common RM

components into the ROM and to explore ways of splitting the overall success to

single parts of the RMS.

3. A Novel Simulation-based

Approach to Investigate ROM

Properties

In this chapter we present our novel simulation-based approach to investigate

ROM properties. The simulation environment introduced in this chapter is an

essential part of our assessment of the ROM. As written earlier, we aim at test-

ing the robustness, but also at investigating further properties of the ROM in an

environment which reflects the real life applications and network structures of a

large network carrier as realistically as possible. The potential to use the ROM

to measure RM performance strongly depends on the evaluation of the revenue

managers that it delivers valid results. To assess if the network-based ROM with

independent demand and dependent demand is able to deliver valid results, a spe-

cial simulation environment has to be set up. Most notably a systematic way to

analyze the robustness of the ROM against errors in the estimated unconstrained

demand has to be implemented.

In this chapter we describe the structure of our simulation environment and

the interaction of the core modules applied in Section 3.1. In the same section we

also provide a detailed description of the components of the simulated RMS. In

Section 3.2 we introduce our novel simulation-based approach to measure ROM

robustness. Details on the simulation scenarios in which we investigate the prop-

erties of the ROM are provided in Section 3.3. We conclude this chapter with

Section 3.4.

3.1. The Simulation Environment

For our investigation of the properties of the network-based ROM we use the sim-

ulation environment presented in Figure 3.1. Basically we simulate a complete

RMS and add an additional module to calculate and evaluate the ROM. The

setup of the simulated RMS follows the principles of a RM simulator that are

presented in Frank et al. (2008) and is based on an existing simulation environ-

3. A Novel Simulation-based Approach to Investigate ROM Properties

Request

generator

Forecaster Optimization Inventory

Unconstraining

Capacities/

Fares

Booking

requests

Real

demand

Actual

bookings

Unconstrained

demand

Forecasts Bid prices

Actual

bookings

Unconstrained

demand

ROM calculation and evaluation

Figure 3.1.: The Simulation Environment to Investigate ROM Properties

ment available at Lufthansa German Airlines. For our simulation environment

we apply a decomposition approach very similar to the approach introduced by

Cleophas (2009). This allows us to perform various scenarios and analyses to

investigate the effect on the ROM. Therefore, the existing simulation environ-

ment is extended and revised in most components, for example to implement the

different simulation scenarios. In addition, the RMS used in this thesis reflects all

main components and used RM methods of a large network airline, in particular

regarding state-of-the-art demand modeling and optimization models. In addi-

tion we calibrate the input data to be as realistic as possible to achieve results

that allow a transfer of the findings to the operational RM controls.

In order to generate a sufficient number of observations numerous simulation

runs are applied, i.e. simulated consecutive booking periods. Before the first sim-

ulation run some data structures are initialized with default values, for example

the forecaster with an initial forecast. At the beginning of each simulation run

all booking requests for the simulation run are generated and stored. After the

request generation the flow of the simulation is very similar to an operational

RMS. At the beginning of each simulation run the current forecast is used in

the optimization module to calculate bid prices for the inventory control. The

optimization also incorporates the fares for the itineraries and capacities of the

compartments of the flight legs. Within a simulation run the booking requests are

handled by the inventory and either accepted or rejected. At predefined DCPs

the bid prices are reoptimized to react to the current booking situation. After

3.1. The Simulation Environment

the simulation run the actual bookings and availability information are used in

the unconstraining module to estimate the unconstrained demand. The uncon-

strained demand together with the old forecast is the base for the forecast of the

next run.

After each simulation run four data streams are provided to the module that

calculates and evaluates the ROM. In the first data stream the capacities of the

flight legs and the fares of the itineraries are provided. The second data stream

are the booking requests from the request generator, which serve as the real

demand in the following calculations. The actual bookings from the inventory

not only serve as a key input to the unconstraining module, but they also define

the third data stream used in the ROM calculation. The fourth input stream

is the estimated unconstrained demand from the unconstraining module. We

provide a detailed description of the different modules and data streams in the

following sections.

3.1.1. Modeling Customer Demand and Request Generation

In this section we start with a formal definition of the applied models of inde-

pendent and dependent demand in our simulation. Afterwards we describe how

the customer requests are generated in the request generator and how they are

handled in the simulation.

Basic Notation and Independent Demand Model

Let us start this section with some general definitions. Let Idenote the set of

all itineraries ioffered. These O&D itineraries contain the flight legs that are

traversed to get from an origin to a destination. The available flight legs lare

stored in set L, whereas Licontains only those legs lthat are part of an itinerary

i. In analogy to the last definition Jicontains the set of all available booking

classes jon itinerary i.Mdenotes the set of all compartments mand the set Ml

the compartments that are available in leg l. The respective compartment for a

booking on leg lin booking class jis labeled with ml,j. The physical capacity of

such a compartment min leg lis named capl,m. In addition, Tdenotes the set

of all time periods tand Sthe set of all available simulation runs s. The sets

Ji,T,Mland Sare sequentially ordered. For example t= 1 defines the first

time period, whereas t=|T|defines the last time period in T. The successor of

a given time period tis marked with t+ 1. The sequential order of the booking

classes is based on their respective fares. The successor of a booking class jis

defined as j+ 1. The highest available booking class in a compartment mfor

3. A Novel Simulation-based Approach to Investigate ROM Properties

itinerary iis denoted with j+

i,m and the lowest available booking class with j−

i,m.

The same order applies to compartments. The highest compartment of a leg lis

defined as m+

l. The next simulation run from a given run sis denoted with s+1.

As described in Section 2.1.2 an independent demand model describes the cus-

tomer demand without considering any dependencies to other available products.

The assumption of the independence of the demand increasingly loses importance,

since more and more models to handle dependencies have been introduced and

proven in practical applications. However, many airlines still use independent

demand models and some state-of-the-art optimization methods transform and

reduce dependent demand models back to equivalent independent demand mod-

els. To model the independent demand in our simulation we refer to a classic

definition, which is widely used in the airline world. In the following di,j,t,s rep-

resents the mean demand for itinerary ifor booking class jin time period tin

simulation run s.sis only appended in the subscript if we have to differentiate

between two simulation runs s. If sis omitted, di,j,t describes the independent

demand in the current simulation run. This also applies to all other definitions

for which we have values for each simulation run.

Dependent Demand Model

We introduced some state-of-the-art dependent demand models in Section 2.1.2.

For our simulation of the effect of dependent demand structures on the ROM

we refer to the model definition by Winter (2010). Like the dependent demand

models described by Walczak et al. (2010) and Fiig et al. (2010) and in accordance

to the classification by Ratliff et al. (2008), it models the dependencies of the

demand between the booking classes for a single itinerary. A wide range of

practical applications is available for these kinds of models as well as numerous

optimization models that are based on these model definitions. One advantage

of the model definition by Winter (2010) is the fact that it allows more degrees

of freedom to model dependencies of the demand.

The basic idea of the definition introduced by Winter (2010) is to model the

demand using a buy-down graph. The graph models the customer-choice options

in an acyclic directed graph. This means that a logical ordering of buy-down

behavior is given. Buy-down can occur from one booking class to another, but

not in reverse direction. In Figure 3.2 we provide an example of such an acyclic

directed buy-down graph. The graph illustrates the dependencies between the

booking classes. As a basic assumption the availability of booking classes is often

sequentially ordered. The sequential opening order is defined by feasible actions.

In our example there are five feasible actions. It is only possible to make the

3.1. The Simulation Environment

Buy-down graph xtotal demand

buy-downy

Figure 3.2.: Dependent Demand Modeled in an Acyclic Directed Buy-down

Graph

next booking class available in the given sequential order. It is for example not

possible to have booking class one and three available, but booking classes two,

four and five unavailable. The realized demand we observe in a booking class is

the total demand minus the buy-down into other booking classes. In our example

the total demand of booking class one is ten. If only booking class one is available

this is also the realized demand. However, a buy-down will materialize if booking

class two or additionally three are also available. The buy-down from booking

class one to booking class two is five. Given the feasible action of making booking

classes one and two available the realized demand for booking class one is five

and for booking class two it is 16. We list the realized demands in Table 3.1.

Realized demand in BC

BCs open 1 2 3 4 5 P

1 10 - - - - 10

1,2 5 16 - - - 21

1,2,3 2 6 15 - - 23

1,2,3,4 2 6 15 12 - 35

1,2,3,4,5 2 6 5 4 30 47

Table 3.1.: Realized Demand According to Opened Booking Classes

We formalize this model with the following definitions. The total demand for

itinerary ifor booking class jin time period tis denoted with dtd

i,j,t and the buy-

3. A Novel Simulation-based Approach to Investigate ROM Properties

down for itinerary ifor booking class jinto a lower booking class j0in time period

twith dbd

i,j,j0,t. With yieldable demand we describe the demand for a booking class,

if all buy-down occurred. A formal definition is given in Equation 3.1.

dyd

i,j,t =dtd

i,j,t −X

j0∈Ji,j

dbd

i,j,j0,t ∀i∈I, ∀j∈Ji,∀t∈T(3.1)

In the given equation Ji,j describes the set of all booking classes that are lower

than booking class jand for which a buy-down relation exists. A cross com-

partment buy-down is not considered in this thesis. Please note that the total

demand contains the buy-down into lower booking classes and thus cannot be

used to estimate the total number of customers in the market.

In this thesis we differentiate between different types of demand. The esti-

mated unconstrained demand is indicated by di,j,t. It refers to the estimated

unconstrained demand that is generated in the unconstraining module. ri,j,t de-

notes the real demand. The real demand is taken from the request generator and

is described in detail in the next section. The forecasted demand as the basic

input for the optimization module is named fi,j,t.

Request Generation

A main part of each simulation environment is the generation of customer re-

quests. The booking requests in our simulation environment are generated ac-

cording to the proposal of Frank et al. (2008). Customer requests are assumed to

be Poisson distributed and generated using a non homogeneous Poisson process.

For a detailed description about the application of a non homogeneous Poisson

process we refer the interested reader to Kimms and M¨uller-Bungart (2007). The

intensity of the Poisson distribution may vary over time and is based on histori-

cal demand profiles for each itinerary and booking class used in the simulation.

In the case of independent demand we generate customer requests for a given

itinerary iin a given booking class jwithin a given time period t. The number

of all customer requests for itinerary iin booking class jand time period tis de-

noted with ci,j,t. Although not used in the notation, we assume, that a customer

request can occur at any given point in time and we are able to use this infor-

mation for a special analysis. No-show behavior is applied to a customer request

using a no-show probability and a Bernoulli process to determine whether a cus-

tomer request ends up being a no-show or not. The information about no-show

probability are also taken from historical observations in the operational RMS.

Cancelations could be modeled in the same manner, but are not considered in this

thesis. It is also possible to consider seasonal demand deviations in the historical

3.1. The Simulation Environment

demand profiles. This allows us to simulate realistic demand profiles known from

the operational RMS.

In the case of dependent demand structures an additional information to each

customer request is applied. Given a probability that a customer would also be

willing to purchase the next higher booking class, a Bernoulli process is applied to

determine the number of booking classes the customer is willing to sell-up. The

sell-up order of our customer requests is sequentially along the booking classes.

In analogy to the definition with independent demand the number of customer

requests for a given itinerary iwithin a given time period tis denoted with ci,j,j0,t.

Booking class jrefers to the highest booking class the customer would purchase

and booking class j0refers to the lowest booking class the customer would opt

for.

The booking requests from the request generator module are not only handled

by the inventory availability decision, but they also reflect the real demand. This

is one key input stream for our analysis setup. For our standard analysis we will

use aggregated real demand information. For some analysis however, we make

use of the single customer requests because they contain the exact information,

at which time point the requests occur. With independent demand the definition

of ri,j,t is simple. It equals ci,j,t in all cases, as presented in Equation 3.2.

ri,j,t =ci,j,t ∀i∈I, ∀j∈Ji,∀t∈T(3.2)

With dependent demand the transformation from customer requests to the

aggregated real total demand, yieldable demand and buy-down is a bit more

complicated. Using the definitions made before, Algorithm 3.1 describes, how

the values of rtd

i,j,t,ryd

i,j,t and rbd

i,j,j0,t are generated.

First, all values of ryd

i,j,t and rbd

i,j,j0,t are set to zero (Lines 3 to 6). Afterwards

for all potential combinations of customer requests in which the customers are

willing to purchase a ticket within the range from booking class jto j0the values

of the real yieldable demand and buy-down are determined. The values of the

yieldable demand are increased for the lowest booking class the customers are

looking for (Line 10). From this booking class no buy-down is intended. For

all booking classes in between (jto j0−1) buy-down into a lower booking class

can occur and thus the values of rbd

i,j,j0,t are increased (Line 12). At the end the

total real demand rtd

i,j,t is determined as the sum of the real yieldable demand and

buy-down (Line 15).

3. A Novel Simulation-based Approach to Investigate ROM Properties

Algorithm 3.1: Generating Real Demand out of Single Booking Requests

1foreach t∈Tdo

2foreach i∈Ido

3set values to zero

4foreach j∈Jido

5ryd

i,j,t = 0

6rbd

i,j,j+1,t = 0

7determine real aggregated yieldable demand and buy-down

8foreach j∈Jido

9for j0=jto |Ji|do

10 ryd

i,j0,t =ryd

i,j0,t +ci,j,j0,t

11 for j00 =jto j0−1do

12 rbd

i,j00,j00 +1,t =rbd

i,j00,j00 +1,t +ci,j,j0,t

13 determine real aggregated total demand

14 foreach j∈Jido

15 rtd

i,j,t =ryd

i,j,t +rbd

i,j,j+1,t

3.1.2. Unconstraining and Demand Forecasting

All common unconstraining methods make use of actual bookings and information

about the availability of booking classes as a basic input. bi,j,t denotes the number

of actual bookings and ai,j,t the availability information of itinerary ifor booking

class jin time period t. Many unconstraining methods also use average historical

bookings, which are labeled with hi,j,t.

Independent Demand

In Section 2.1.2 we listed several commonly used unconstraining methods such

as additive pick-up, expectation maximization or projection detruncation. As

we investigate the effect of different levels of unconstraining errors on the ROM

robustness later in this thesis, the choice of a specific unconstraining method

is not important. We apply the additive pick-up unconstraining method and

estimate the unconstrained demand using Equation 3.3.

di,j,t =(bi,j,t ai,j,t = 1

max(hi,j,t;bi,j,t)ai,j,t = 0 ∀i∈I, ∀j∈Ji,∀t∈T(3.3)

3.1. The Simulation Environment

This definition also adheres to one basic proposition, which is that the lower

bound of the demand estimates for a given time period are the actual bookings

bi,j,t. We formalize this assumption in Proposition 3.1.1. Keeping this proposition

in mind, the basic idea of our unconstraining approach is to use all booking

observations in those time periods in which the booking classes were available all

the time. For all other time periods we use the maximum of the average historical

bookings hi,j,t and the actual bookings bi,j,t.

Proposition 3.1.1 di,j,t ≥bi,j,t ∀i∈I, ∀j∈Ji,∀t∈T

In addition to the unconstrained demand we also recalculate the average his-

torical bookings and the forecast for the next simulation run. For both updates

we use exponential smoothing, which is described in detail in Talluri and van

Ryzin (2004b, Chapter 9.3.1.2 Exponential Smoothing, page 436). Equation 3.4

shows, how the average historical bookings are updated.

hi,j,t,(s+1) =(α∗bi,j,t,s + (1 −α)∗hi,j,t,s ai,j,t,s = 1

hi,j,t,s ai,j,t,s = 0 (3.4)

∀i∈I, ∀j∈Ji,∀t∈T

fi,j,t,(s+1) =α∗di,j,t,s + (1 −α)∗fi,j,t,s (3.5)

∀i∈I, ∀j∈Ji,∀t∈T

To update the average historical bookings we use all time periods in which the

booking class was available the whole time. The bookings of these time periods

are learned using exponential smoothing with smoothing factor α. If the booking

class was not available for sale in a time period, the old value is kept for the next

simulation run. The forecast for the next simulation run s+ 1 simply integrates

the estimated unconstrained demand with the same smoothing factor αas shown

in Equation 3.5.

Dependent Demand

The unconstraining and forecasting of dependent demand is a very challenging

task compared to the independent demand case. The estimation of independent

unconstrained demand is able to directly make use of actual observations or of

historical observations. Thus, the accuracy of the estimated unconstrained de-

mand is usually quite good. For dependent demand structures it is not possible to

estimate the unconstrained demand based on direct observations with the same

accuracy. In particular the buy-down is hard to estimate. As presented in Sec-

tion 2.1.2 some methods have been proposed to solve this challenging task. The

3. A Novel Simulation-based Approach to Investigate ROM Properties

approach we apply in this thesis uses some simplifying assumptions. One sim-

plifying assumption is that we only estimate buy-down from one booking class j

to the next booking class j+ 1 which corresponds to the customer demand we

generated in the request generator. The unconstraining method is first and fore-

most intended to support us in our research objective at assessing the impact of

different levels of unconstraining errors on the ROM. For the dependent demand

case we are also able to define a basic proposition. It is depicted in Proposition

3.1.2.

Proposition 3.1.2 dtd

i,j,t ≥bi,j,t ∀i∈I, ∀j∈Ji, t ∈T

It states, that the total demand for a booking class is greater than or equal to

the actual bookings. In addition, we consider the average historical bookings

to be the average historical yieldable demand in the case of dependent demand

structures. According to our model these values are easy to observe in comparison

to the total demand and the buy-down.

The unconstraining process is described in detail in Algorithm 3.2. It basically

consists of three steps. First, the yieldable demand is estimated. This is done

in Lines 4 to 16. According to our demand model we assume that all buy-down

for a booking class jmaterialized, if booking class jwas open and also booking

class j+ 1. If this was not the case we use historical average yieldable demand

as an estimator. The buy-down and total demand are estimated by using the

estimations for the total demand of a booking class j+ 1 and the help of an

estimated sell-up rate θi,j,t in a second step. These steps are described in Lines

17 to 24. To adhere to Proposition 3.1.2 the total demand in a booking class

is always estimated to be greater than or equal to the bookings we observed. If

this is not the case in the first place the difference between the two estimators

is attributed to the yieldable demand and buy-down using coefficient ω. The

third step is enforcing consistency. It is possible that after the first two steps the

estimations of the buy-down into a lower booking class are greater than the total

demand in the lower booking class. This is not possible by definition and will be

corrected in a similar way we described before (Lines 25 - 34). At the end of the

algorithm we have estimations for dtd

i,j,t,dyd

i,j,t and dbd

i,j,j0,t.

The historical average yieldable demand is updated in a very similar way to the

average historical bookings for the independent demand model. If for a booking

class jit holds true that it was a) the last booking class j−

i,m in a compartment

mand it was available or b) it was not the last booking class in a compartment

mand both booking classes jand j+ 1 were available, then we assume that

we observed the yieldable demand. This observation is then learned by using

3.1. The Simulation Environment

Algorithm 3.2: Process to Unconstrain Dependent Demand

1foreach t∈Tdo

2foreach i∈Ido

3foreach m∈Mido

4Estimate yieldable demand

5for j←j+

i,m to j−

i,m do

6if (j=j−

i,m)then

7if ((ai,j,t = 1) ∨(bi,j,t > hi,j,t)) then

8dyd

i,j,t =bi,j,t

9else

10 dyd

i,j,t =hi,j,t

11 dtd

i,j,t =dyd

i,j,t

12 else

13 if ((ai,j,t = 1) ∧(ai,(j+1),t = 1)) then

14 dyd

i,j,t =bi,j,t

15 else

16 dyd

i,j,t =hi,j,t

17 Estimate buy-down and total demand

18 for j←j−

i,m −1downto j+

i,m do

19 dbd

i,j,(j+1),t =dtd

i,(j+1),t ∗θi,j,t

20 dtd

i,j,t =dbd

i,j,(j+1),t +dyd

i,j,t

21 if dtd

i,j,t < bi,j,t then

22 dbd

i,j,(j+1),t =dbd

i,j,(j+1),t +ω∗(bi,j,t −dtd

i,j,t)

23 dtd

i,j,t =bi,j,t

24 dyd

i,j,t =dtd

i,j,t −dbd

i,j,(j+1),t

25 Enforce consistency to observed bookings

26 for j←j+

i,m to j−

i,m −1do

27 if dtd

i,(j+1),t < dbd

i,j,(j+1),t then

28 if (j+ 1) < j−

i,m then

29 dbd

i,(j+1),(j+2),t,s =

dbd

i,(j+1),(j+2),t,s +ω∗(dbd

i,j,(j+1),t −dtd

i,(j+1),t)

30 dtd

i,(j+1),t =dbd

i,j,(j+1),t

31 dyd

i,(j+1),t =dtd

i,(j+1),t −dbd

i,(j+1),(j+2),t,s

32 else

33 dtd

i,(j+1),t =dbd

i,j,(j+1),t

34 dyd

i,(j+1),t =dtd

i,(j+1),t

3. A Novel Simulation-based Approach to Investigate ROM Properties

exponential smoothing. A formal definition is given in Equation 3.6.

hi,j,t,(s+1) =









α∗bi,j,t,s + (1 −α)∗hi,j,t,s (j < j−

i,m ∧ai,j,t,s = 1

∧ai,(j+1),t,s = 1)∨

(j=j−

i,m ∧ai,j,t,s = 1)

hi,j,t,s Otherwise

(3.6)

∀i∈I, ∀j∈Ji,∀t∈T

The process of forecasting is the same as with independent demand. Again we

use exponential smoothing as depicted in Equations 3.7 - 3.9.

ftd

i,j,t,(s+1) =α∗dtd

i,j,t,s + (1 −α)∗ftd

i,j,t,s (3.7)

∀i∈I, ∀j∈Ji,∀t∈T

fyd

i,j,t,(s+1) =α∗dyd

i,j,t,s + (1 −α)∗fyd

i,j,t,s (3.8)

∀i∈I, ∀j∈Ji,∀t∈T

fbd

i,j,j0,t,(s+1) =α∗dbd

i,j,j0,t,s + (1 −α)∗fbd

i,j,j0,t,s (3.9)

∀i∈I, ∀j∈Ji,∀t∈T

3.1.3. Optimization Models and Seat Inventory Control

Optimization with Independent Demand

In the optimization module, we use a bid price control, which uses shadow prices

from a DLP and a decomposition approach to multiple leg problems in conjunc-

tion with DP to generate bid prices as described in Talluri and van Ryzin (see

2004b, Chapter 3.4.4). Based on the given forecasts bid prices πl,m are calculated

for each compartment mon leg l.πl,m denotes the current valid bid price for

one seat in compartment mand leg l. Usually airlines store bid price vectors

containing values for each number of seats left. As in practice, the bid prices are

recalculated at selected DCPs. In the inventory for each booking request the fare

is evaluated against the sum of the current bid prices. If the fare is greater than

the sum of the bid prices, the booking request is accepted, otherwise rejected.

We have chosen this optimization approach because it is common in practical

applications of network-based RM at airlines and thus supports us in assessing

the ROM in a realistic environment.

Optimization with Dependent Demand

In case of dependent demand structures we will use a hybrid optimization ap-

proach as described by Fiig et al. (2010) and Walczak et al. (2010) with fare

3.1. The Simulation Environment

and demand transformation which is common in practical RM applications and

nowadays becoming increasingly important for airlines. One advantageous fea-

ture of this approach is that it allows to further use the previously mentioned

optimization model with independent demand because the dependent demand is

transformed to an equivalent independent demand.

For the inventory two adjustments have to be performed. First, instead of

comparing the bid prices against the original fare of a booking request, their sum

is evaluated against the transformed fare. If the transformed fare is greater than

the sum of the bid prices the booking request is accepted, otherwise rejected.

And secondly, the booking request contains a range of booking classes in which

the customer is willing to buy a ticket as described with the request generation.

Thus, starting with the the lowest booking class j0all booking classes between j0

and jare evaluated until the booking request is accepted or eventually rejected.

Applying Upgrading and Overbooking

In our simulation environment we also able to apply upgrading and overbooking.

These two RM components play a crucial role at an airline to consider no-shows

and cancelations. For these two components of RM many sophisticated methods

have been described in literature and used in practice. Because we also apply dif-

ferent scenarios of the overbooking and upgrading controls, we have implemented

simple methods.

If upgrading is applied in our simulation environment, we follow a basic ap-

proach presented by P¨olt (2002). We determine an adjusted virtual capacity

capU

l,m for each compartment mon a leg l. The capacity capU

l,m of a compartment

mis increased if there is excess demand for compartment mand there is fore-

casted spare capacity in the next higher compartment m−1. If the capacity is

adjusted according to this principle, capU

l,(m−1) is decreased by the same number

of seats. This ensures that the RM control does not offer more seats for a flight

leg than actually available.

If overbooking is applied, the capacity of the compartments is virtually in-

creased. A simple way in calculating the adjusted virtual capacity of a compart-

ment is depicted in Equation 3.10 (see Talluri and van Ryzin, 2004b, Chapter

4.2.2 for details)

capO

l,m =round(capl,m

ql,m

)∀l∈L, ∀m∈Ml(3.10)

The capacity of each compartment mon a given leg lis divided by the estimated

show-up rate ql,m for this compartment. The adjusted capacity capO

l,m is rounded

to an integer value, because the capacity of a compartment cannot take fractional

3. A Novel Simulation-based Approach to Investigate ROM Properties

values by definition. If both upgrading and overbooking are applied, the upgrad-

ing is performed first and afterwards the overbooking. Equation 3.11 changes the

definition of capO,U

l,m accordingly.

capO,U

l,m =round(capU

l,m

ql,m

)∀l∈L, ∀m∈Ml(3.11)

Virtually increasing the capacity of a compartment can lead to a situation, in

which more bookings than the total capacity of the plane are accepted. This

might in particular result in denied boardings. We discuss this matter in more

detail in Chapter 7.

3.2. Measuring ROM Robustness

A key prerequisite for a ROM to present valid performance measures is that it

reflects the general method of the RMS in place, e.g. network-based controls.

Furthermore the question of robustness against errors in the input data is a

main determinant for its validity. For this reason we present in this section a

simulation-based approach to measure the robustness of a ROM. Although we

focus on the application of this approach to airline RM, the approach can easily

be applied to different industries and scenarios.

An operational RMS observes the actual bookings and estimates the uncon-

strained demand based on these bookings. The estimated unconstrained demand

serves as the input to calculate ROM measures. Since the estimated uncon-

strained demand contains errors, there will be errors in the ROM measures as

well. However, we do not know how severely the error in the estimated un-

constrained demand affects the quality and validity of the ROM measures. To

analyze this effect we take advantage of the previously defined simulation envi-

ronment. Figure 3.3 illustrates the principle of our approach. In our simulation

environment, we do not only have the actual bookings and thus the estimated

unconstrained demand at hand, but also the real demand. This allows us to

quantify the degree of error between estimated unconstrained and real demand,

as well as the degree of similarity between the ROM measures that are calculated

based upon this underlying estimated unconstrained and real demand. In such

an environment, we are also able to incorporate further scenarios - for example

different forecast error levels - and to simulate the implications they have on

the similarity of the ROM measures. To decide whether or not to consider the

ROM robust against input errors, we define two thresholds. One threshold is

aminimum level of similarity, which defines the minimum degree of similarity

3.2. Measuring ROM Robustness

Est. unc.

Demand

Real

Demand

ROM

measure

ROM

measure

Data available in

operational RM-

Systems, but also

in simulation

Data only

known in

simulation

Error

measure

Similarity

measure

Figure 3.3.: Simulation-based Approach to Measure ROM Robustness

between the ROM measures that we consider sufficient to apply the ROM in real

life applications. We also define a maximum error level in the estimated uncon-

strained demand based upon our worst case expectations in reality. By the use

of sensitivity analysis we examine whether the similarity measures are above our

threshold for given error levels up to the defined maximum error level and even

beyond. If for all error levels applied up to the defined worst case expectations

the similarity measures are above the required level, we consider the ROM robust

against errors in the input data. Please note that the decision whether or not to

consider the ROM robust heavily depends on the scenarios and the assumptions

on the real world used in the simulation. In the next sections, we will give formal

definitions of error and similarity measures.

3.2.1. Error Measures

Many methods and measures to quantify the error level between correct and

estimated figures are available. Armstrong (2001) discusses several error measures

in the area of forecasting in detail. In our approach, we use the mean absolute

error (MAE) and the percentage mean absolute error (PMAE) as error measures.

These measures are defined in Equations 3.12 and 3.13. The MAEDdefined in

Equation 3.12 measures the average absolute error between the real demand and

the estimated unconstrained demand. The definition makes use of the cumulated

estimated unconstrained demand Di,j and real demand Ri,j for itinerary ifor

booking class jup to the end of the booking period. In comparison to the MAED

the PMAEDdefined in Equation 3.13 determines the relative error level between

the sum of absolute errors and the total real demand: The higher the error

3. A Novel Simulation-based Approach to Investigate ROM Properties

measures, the higher the error level for a certain scenario. These two measures

are common in practical applications and thus we are able to define maximum

error levels for different error scenarios by considering the worst case. We will

come back to these error levels in Section 3.3.

MAED=Pi∈IPj∈Ji|Di,j −Ri,j|

Pi∈I|Ji|(3.12)

PMAED=Pi∈IPj∈Ji|Di,j −Ri,j |

Pi∈IPj∈JiRi,j

(3.13)

With dependent demand we do not only have to consider yieldable demand,

but also buy-down and the resulting total demand. In our simulation environ-

ment we are able to measure the unconstraining or forecast error for all demand

components, because we know them from the demand generation. This is of

course not possible in reality, but in our simulation environment it helps us, to

perform different kinds of sensitivity analysis. The formulas to determine the

error measures on total demand, yieldable demand and buy-down are the same

as presented in Equations 3.12 and 3.13. The only change is using the respective

real and estimated unconstrained demand figures (i.e. Dtd

i,j,Rtd

i,j,Dyd

i,j,Ryd

i,j,Dbd

i,j

and Rbd

i,j).

3.2.2. Similarity Measures

The proposed simulation environment generates pairs of ROM measures for each

run, one ROM measure calculated based on real demand and one calculated based

on the estimated unconstrained demand. These ROM measures include values

not only for the potential and no RM revenue, but also for the RO, the ARO

and the PARO. In our definition of similarity measures, we use the PARO as

one example. However, the similarity measures are easily applicable to the other

ROM measures. In the following, PARORdenotes the PARO calculated with

the real demand and PARODdenotes the PARO calculated with the estimated

unconstrained demand. We illustrate our definitions with an example in Figures

3.4 and 3.5, in which we compare PARORand PARODout of 20 simulated flight

departures per run and in a scatter plot.

In case of perfect similarity, we would observe PAROR=PARODfor all runs.

However, this will rarely be the case in reality and is not the case in our exam-

ple. To measure the degree of similarity, we propose a combination of quantitative

measures and a visual inspection of the scatter plot. The first measure we propose

is the mean absolute error between PARORand PAROD. A low MAEP ARO im-

plies a high similarity between PARORand PAROD. For example, if MAEP ARO

3.2. Measuring ROM Robustness

78%

80%

82%

84%

86%

88%

1 3 5 7 9 11 13 15 17 19

Runs

PARO

Real Demand Est. Unc. Dmd

Figure 3.4.: Comparing PAROs per

Run

75%

80%

85%

90%

95%

75% 80% 85% 90% 95%

Real Demand

Est. Unc. Demand

Figure 3.5.: Comparing PAROs in a

Scatter Plot

is zero then all pairs are identical and we observe perfect similarity. In practice

all values for MAEP ARO below 5% indicate a very high similarity. In cases in

which MAEP ARO does not indicate a high degree of similarity we could still be

able to observe a linear relation between PARORand PAROD, for example be-

cause of a biased over- or underestimation of PAROD. In this case, there could

be a high degree of similarity, which cannot be measured with the MAEP ARO as

the MAEP ARO will by definition increase with the level of bias in this case. By

a visual inspection of the scatter plot, we would be able to observe this linear

relation. If there still exists a linear relation values of MAEP ARO up to 15%

are considered sufficient to indicate a high degree of similarity. To quantify the

linear relation between PARORand PARODand to introduce the second mea-

sure for similarity we propose the Pearson’s correlation coefficient and denote it

with rP ARO. Pearson’s correlation coefficient is a basic measure of linear rela-

tions between two paired sets of values. Values of rP ARO range between -1 and 1

and values greater than 0.5 indicate a good linear correlation. In those cases in

which the MAEP ARO does not indicate a good similarity but in which there is

a good linear relation between PARORand PAROD, Pearson’s rP ARO helps us

to quantify the quality of this linear relation. As a conclusion, we propose to use

the MAEP ARO, the Pearson’s correlation coefficient rP ARO in combination with

a visual inspection of the scatter plot to determine the level of similarity. For

the MAEP ARO we define 5% - 15% as the minimum level of similarity depending

on the values for rP ARO. We consider 0.5 as the lower bound for rP ARO. In our

example, MAEP ARO is 0.9% and rP ARO is 0.83. These values indicate a high

3. A Novel Simulation-based Approach to Investigate ROM Properties

level of similarity.

3.3. The Simulation Scenarios

To measure the robustness of the ROM as described in the previous section and

to investigate further properties of the ROM we apply different scenarios. Besides

simulating various unconstraining errors, the scenarios include the possibility to

adjust the forecast errors, to apply different kinds of seasonality to the customer

demand, to adjust the bid prices to simulate open or restrictive RM controls and

to adjust the overbooking and upgrading controls. All scenarios are performed

ceteris paribus. Besides the module, in which an adjustment is applied, all other

modules work under normal conditions. A detailed description of the scenarios

is given in the following sections.

3.3.1. The Base Case

The starting point of all of our investigations is a base case that reflects the re-

ality of a large network carrier in the most realistic way possible. We consider

nine booking classes (two business and seven economy classes). The flight net-

work consists of 728 continental and intercontinental flights and includes 1,605

different itineraries which are taken from a realistic flight network. The demand

level applied leads to an average SLF of around 75%, which varies according to

an observed seasonality in reality by around 10% over time. The share of con-

necting passengers is around 30%. On continental flights, the capacity can be

flexibly distributed between business and economy class bookings. The fares and

capacities are kept constant within a simulated booking period. We simulate one

network day for each scenario. The number of total runs for each scenario is 180,

out of which 150 runs are used for the analyses. 30 preliminary runs in a start-up

phase are not considered in the final evaluation. If dependent demand is applied

we assume 30% as an average sell-up rate. If no-shows are applied, an average

no-show rate of around 6% is assumed as observed in reality. The smoothing

factor αis set to 15%.

For some analyses we change the structure of our realistic flight network and

set the share of connecting passengers to 0%. We keep the demand level on

the flight legs constant compared to the base case defined on the realistic flight

network. We refer to this network as the no-connecting-traffic flight network.

3.3. The Simulation Scenarios

3.3.2. Adjusting the Unconstraining Error

One main focus of this thesis is to assess the robustness of the ROM against errors

in the estimated unconstrained demand. Therefore we apply various unconstrain-

ing error scenarios and measure the effect on the ROM measures. In this section

we describe how we adjust the unconstraining error based on the base case. All

of the following approaches have in common, that they influence the forecast of

the next simulation run and thus the results of the optimization module. These

feedback loops are well-known in reality; a special case was described with the

spiral-down effect.

Independent Demand

For the unconstraining error we assume that we only apply errors in those time

periods in which the booking classes were closed. Furthermore Proposition 3.1.1

still holds. Equation 3.14 depicts the adjusted principle.

di,j,t =(bi,j,t ai,j,t = 1

max((1 ±i,j,t)hi,j,t;bi,j,t)ai,j,t = 0 (3.14)

∀i∈I, ∀j∈Ji,∀t∈T

i,j,t describes a random error factor from a uniform distributed interval [l−

d..l+d]. ldescribes the average error level and dthe error deviation. In our

simulation setup, we are able to apply a biased overestimation, a biased under-

estimation and an unbiased error for the unconstrained demand in the case of a

closed booking class. If we apply a biased overestimation the value of i,j,t always

increases the estimated unconstrained demand. The contrary is true for a biased

underestimation. When using the unbiased error the estimated unconstrained

demand is randomly increased or decreased with the same probability. In our

scenarios we set dto 10% and apply three different error levels l: 30%, 60% and

90%. Based on observations in practice1we define the 60% error level to be the

worst case in reality. Furthermore we expect the unbiased unconstraining error

to be more common because a strong bias into one direction is usually prevented

by the forecasting modules in an operational RMS.

Dependent Demand

The application of additional error to the estimated unconstrained demand and

the forecast is done in a similar way with dependent demand structures. As a first

1Based on information discussed in personal communication with Dr. P¨olt - Lufthansa German

Airlines

3. A Novel Simulation-based Approach to Investigate ROM Properties

step we estimate the unconstrained demand using Algorithm 3.2. Afterwards we

apply an error to the yieldable demand or the buy-down in a two-stepped ap-

proach. First, the demand estimates are adjusted. The yieldable demand is

adjusted according to Equation 3.15 and the buy-down is adjusted in correspon-

dence to Equation 3.16.

dyd

i,j,t =









dyd

i,j,t (j < j−

i,m ∧ai,j,t,s = 1

∧ai,(j+1),t,s = 1)∨

(j=j−

i,m ∧ai,j,t,s = 1)

(1 ±i,j,t)∗dyd

i,j,t Otherwise

(3.15)

∀i∈I, ∀j∈Ji,∀t∈T

dbd

i,j,j0,t = (1 ±i,j,t)∗dbd

i,j,j0,t (3.16)

∀i∈I, ∀j∈Ji,∀t∈T

The definition of i,j,t,land dis the same as in the independent demand

case. The main difference between the error for the yieldable demand and for the

buy-down is, that in the buy-down case the error is applied for every booking

class, whereas for the yieldable demand it is only applied, if for a booking class j

it holds true that it was a) the last booking class j−

i,m in a compartment mand it

was available or b) it was not the last booking class in a compartment mand both

booking classes jand j+ 1 were available. After adjusting the yieldable demand

and buy-down in the described way, the total demand is preliminary set to dtd

i,j,t =

dyd

i,j,t +dbd

i,j,j0,t. Afterwards the consistency check of Algorithm 3.2 in Lines 25 -

34 is performed to enforce Proposition 3.1.2. The applied scenarios are like with

independent demand a biased overestimation, a biased underestimation and an

unbiased error for either the unconstrained yieldable demand or the unconstrained

buy-down. The error levels lremain the same with 30%, 60% and 90% and the

same worst case assumption.

3.3.3. Further Scenarios

Besides analyzing the robustness of the ROM with the help of sensitivity analyses

with different error scenarios in the estimated unconstrained demand, we apply

further scenarios to assess the properties of the ROM.

Adjusting the Forecast Error

The application of an additional forecast error increases the effect of a simulated

unconstraining error. By increasing the forecast error we expect that the quality

3.3. The Simulation Scenarios

of the RM control will decrease further and in particular we expect that it will

decrease stronger than in the unconstraining error scenario. The adjustment of

the forecast error with independent demand follows the same principle as the

modification of the unconstrained demand. The only difference is that we do

not consider whether the booking classes were available or not, but apply the

additional error in all cases. With dependent demand the only change is that the

error for the yieldable demand is applied to every booking class, no matter if the

booking class was available or not. For the forecast error we also apply a biased

overestimation, a biased underestimation and an unbiased error with error levels

lof 30%, 60% and 90%.

Adjusting the Seasonality of Customer Demand

As described in Section 3.1.1 we are able to apply seasonality to the demand

generation. In Section 3.3.1 we explained which kind of seasonality we assume for

the base case scenario. Beyond that we apply a seasonality with less or stronger

deviation in customer demand. Another kind of seasonality that we apply is a

demand deviation that follows a saw tooth curve. Within five simulation runs we

decrease the demand level from 130% to 70% and jump back to 130%. Another

saw tooth curve scenario describes a decrease from 120% to 80% demand level

within five simulation runs. The adjustment of the seasonality aims at assessing

the effect of different kinds of seasonality on the robustness of the ROM. Moreover

we want to assess how the overall RM control is affected by different kinds of

seasonality.

Adjusting the Bid Prices

In addition, it is possible to influence the RM control to be either more open or

more restrictive. In the scenario for the less restrictive RM control, we decrease

the bid prices πl,m from the optimization module by a certain percentage β. For

more restrictive control, we increase the bid prices respectively. Equation 3.17

provides a formal definition of the modification of the bid prices.

˜πl,m = (1 ±β)∗πl,m ∀l∈L, m ∈Ml(3.17)

As we are not applying any additional error to the estimated unconstrained de-

mand or the forecasted demand we expect to observe effects in the overall RM

success, but no significant effects on the robustness of the ROM.

3. A Novel Simulation-based Approach to Investigate ROM Properties

Adjusting the Overbooking and Upgrading Levels

As part of the scenarios in which we applied overbooking we are also able to

adjust the overbooking level. Equation 3.18 shows the formal definition of the

modification. Basically we apply an adjustment level βto the overbooking level

obtained from the overbooking control. We are able to increase and to decrease

capO

l,m.

capO

l,m =round((1 ±β)∗capO

l,m)∀l∈L, m ∈Ml(3.18)

In the scenario presented before we only considered overbooking in the RM

simulation. If the RM control considers only upgrading, the virtual capacity of

the compartments capU

l,m is adjusted according to Equation 3.19. In case both

upgrading and overbooking are applied Equation 3.20 is to be used to modify

capO,U

l,m .

capU

l,m =round((1 ±β)∗capU

l,m)∀l∈L, m ∈Ml(3.19)

capO,U

l,m =round((1 ±β)∗capO,U

l,m )∀l∈L, m ∈Ml(3.20)

3.4. Summary

In this chapter we described a novel simulation-based approach to investigate

ROM properties. The simulation environment makes use of state-of-the-art de-

mand modeling and optimization methods for both independent and dependent

demand structures in a network-based RM context. They are similar to those

methods applied at large network carriers. Common RM components in prac-

tice such as overbooking and upgrading are also incorporated. In addition we

calibrated the input data to reflect the reality of a network carrier as well as pos-

sible. The simulation environment furthermore allows us to perform sensitivity

analyses on the robustness of the ROM by adjusting the unconstraining error

in different ways. With our simulation environment we are also able to assess

many other additional scenarios and the subsequent effect of these scenarios on

the ROM. This enables us to apply a holistic simulation-based assessment of the

network-based ROM.

4. The Network-based ROM with

Independent Demand

In this chapter we focus on the network-based ROM with independent demand.

We describe the estimation of the potential, actual and no RM revenue in detail.

Furthermore we highlight some main properties of the network-based ROM with

independent demand with a special focus on model- and data-related errors.

We put a main emphasis on investigating the robustness of the ROM against

errors in the estimated unconstrained demand. Therefore we present and analyze

computational results on the effect of different (error) scenarios on the validity

and robustness of the network-based ROM.

4.1. Model Definition

The basic idea of a network-based ROM was described in detail by Chandler

and Ja (2007). We follow their approach and define the potential, the actual

and the no RM revenue used in the ROM according to their proposal. The

potential revenue is calculated by solving a DLP. As described in Section 1.3.2

this linear program (LP) does not take into account any stochasticity and simply

maximizes the potential revenue for the past booking period under the given

constraints of the model. In this chapter we do not consider any no-shows or

cancelations of passengers. As a consequence overbooking or upgrading are also

not applied. However, we extend the described model to no-shows, cancelations

and the application of overbooking and upgrading by some modifications to the

demand inputs and the LP formulation in Chapter 7.

Max X

i∈IX

j∈JiX

t∈T

pi,j,t ∗x+

i,j,t (4.1)

i∈IlX

j∈Ji,l,m X

t∈T

i,j,t ≤capl,m ∀l∈L, ∀m∈Ml(4.2)

0≤x+

i,j,t ≤di,j,t ∀i∈I, ∀j∈Ji,∀t∈T(4.3)

The objective function 4.1 maximizes the revenue as the sum of fare pi,j,t times

accepted bookings x+

i,j,t over all itineraries i∈Iwith booking class j∈Jiin time

4. The Network-based ROM with Independent Demand

period t∈T. Constraint 4.2 ensures that the capacities of the compartments are

not exceeded. Ji,l,m denotes the set of all booking classes jthat are booked into

compartment mof flight leg l. Finally, Constraint 4.3 ensures that the number

of accepted bookings is bound by the estimated unconstrained demand. The

potential revenue Rev+corresponds to the solution of the objective function.

Rev+can also be deducted by taking the solution of the x+

i,j,t variables applied

to Equation 4.4.

Rev+=X

i∈IX

j∈JiX

t∈T

pi,j,t ∗x+

i,j,t (4.4)

To estimate the no RM revenue, we simulate a FCFS strategy over the booking

period for each itinerary. The estimated unconstrained demand di,j,t is available

as curves over |T|time periods within the booking period, which are defined by

|T|+ 1 DCPs. As we do not have any information on the booking order between

two DCPs, we assume that booking requests arrive in a LBH order within each

single time period tdefined by two subsequent DCPs. Thus, we first sort all

booking requests for an itinerary iin booking class jby their fare ascending

within each single time period tdefined by two DCPs and store them in Pt. Then,

starting with the first time period in the booking period, all booking requests

that still fit into the given capacity of the associated planes are accommodated.

Algorithm 4.1 describes the process to estimate the no RM revenue in detail.

First, the free capacity capf

l,m of all compartments is set to the capacity of the

compartments capl,m. Then for each booking request the number of seats left sl

are determined and the remaining capacity is adjusted. After the algorithm has

been applied the estimations for the bookings are used to determine Rev−, which

is formally defined in Equation 4.5.

Rev−=X

i∈IX

j∈JiX

t∈T

pi,j,t ∗x−

i,j,t (4.5)

As we do not assume any no-shows or cancelations, the actual revenue is calcu-

lated as the sum of all accepted bookings bi,j,t times their fares pi,j,t. This infor-

mation is available in the inventory system. Rev is formally defined in Equation

4.6.

Rev =X

i∈IX

j∈JiX

t∈T

pi,j,t ∗bi,j,t (4.6)

Based on the calculated values for the potential, the actual and the no RM

revenue, the other ROM measures are deducted. According to the definition in

4.2. Main Properties of Network-based ROM with Independent Demand

Algorithm 4.1: Estimation of No RM Revenue

Input:Pt∀t∈T

1foreach l∈Ldo

2foreach m∈Mldo

3capf

l,m =capl,m

4for t= 1 to |T|do

5foreach (i, j)∈Ptdo

6Determine seats left

7sl =∞

8foreach l∈Lido

9sl =min(sl, capf

l,ml,j )

10 Take seats

11 x−

i,j,t =x−

i,j,t +min(sl, di,j,t)

12 foreach l∈Lido

13 capf

l,ml,j =capf

l,ml,j −min(sl, di,j,t)

Section 1.3.1 we define RO,ARO and PARO formally in Equations 4.7, 4.8 and

4.9.

RO =Rev+−Rev−(4.7)

ARO =Rev −Rev−(4.8)

PARO =ARO

RO (4.9)

4.2. Main Properties of Network-based ROM with

Independent Demand

By definition the network-based ROM with independent demand isolates the RM

contribution from the overall success. The estimated unconstrained demand is

the key input data to determine the potential and the no RM revenue and thus

the ROM measures consider demand deviations.

Moreover, without the application of any enhancements or further analyses the

network-based ROM with independent demand generates only one aggregated set

of measures for the entire flight network for each booking period considered. This

means that we obtain one estimate of the potential and the no RM revenue and

derive from that the RO, the ARO and the PARO. It is unlikely, that we observe

4. The Network-based ROM with Independent Demand

the special cases mentioned in Section 1.3.2, i.e. that the RO is zero or the ARO

is negative.

A main focus in this chapter is on the validity of the ROM measures. The

validity is strongly driven by data-related errors, i.e. the accuracy of the esti-

mated unconstrained demand used in the revenue estimations. We analyze the

robustness against errors in the input data in detail in the following section.

Our definition of the network-based ROM with independent demand also incurs

model-related errors. As already described in Section 1.3.2 model-related errors

denote errors in the ROM measures caused by incorrect modeling of the reality

in the revenue estimations. One source of errors is the LP-relaxation. Bookings

in reality are integer. This means that the number of bookings never takes frac-

tional values. However, we have defined the potential revenue estimation as an

LP, which relaxes the integer constraint. Besides reasons of solvability, this is

mainly due to the fact that demand estimations are representing mean demand

values, which in most cases are fractional. These values cannot simply be rounded

or transformed into integer demand. An example of how the LP-relaxation leads

to different results is depicted in Figure 4.1. Let us assume we have a flight net-

A B

D E

Figure 4.1.: Effect of LP-relaxation on Potential Revenue Estimate

work which consists of 5 flight legs AB, BC, BD, CD, and DE each with a capacity

of one. The itineraries offered to the customers are ABCD, ABDE, and BCDE.

If we assume a demand for each itinerary of one and a fare of 500, the optimal

integer solution is 500, because only one itinerary can be sold to customers. The

solution of the LP is 750 (each itinerary is taken 0.5 times). This is of course an

extreme case to illustrate that there might be deviations between the IP and LP

solution. In our simulations however, we observed no differences between the IP

and LP solutions and expect the effect of the LP-relaxation to be very minor on

a large network.

Another main source of model-related errors is the assumed booking order.

One main assumption that is usually made for the no RM revenue is the LBH

booking order. This assumption was also one key assumption at the beginning of

RM research. In a leg-based ROM this would automatically lead to a decreased

4.2. Main Properties of Network-based ROM with Independent Demand

revenue estimate. For a network-based ROM however, this is not always the case.

In Table 4.1 we describe an example with three different booking orders and the

effect on the no RM revenue estimates. We assume that we have a very simple

Booking order

Itinerary Fare LBH Revenue Real 1 Revenue Real 2 Revenue

AB-1 100 AB-2 50 AC-2 505 BC-1 1,000

AB-2 50 AB-1 rejected AC-1 rejected AB-1 100

BC-1 1000 BC-2 500 BC-2 rejected BC-2 rejected

BC-2 500 AC-2 rejected AB-1 rejected AC-1 rejected

AC-1 1010 BC-1 rejected AB-2 rejected AC-2 rejected

AC-2 505 AC-1 rejected BC-1 rejected AB-2 rejected

Sum 550 Sum 505 Sum 1,100

Table 4.1.: No RM Revenue Depending on Booking Order

network with destinations A, B and C. The available itineraries are AB, BC and

AC, which means there are two local itineraries and one connecting itinerary.

If we now suppose that the capacity on each flight (AB and BC) is limited to

one, we could observe very different estimations for the no RM revenue. If we

apply a LBH booking order - which is indicated in column ’LBH’ - the request

for ’AB-2’ comes first, followed by ’AB-1’ and so forth. The request for ’AB-2’

will be accepted, the request for ’AB-1’ rejected and finally the request for ’BC-1’

accepted. All other remaining booking requests have to be rejected due to the

capacity constraints. Both accepted requests lead to a total revenue of 550. If the

real booking order is as presented in column ’Real 1’, then the no RM revenue is

different. The request for ’AC-2’ will be accepted and all other requests rejected.

This only leads to a total revenue of 505. If we apply the booking order ’Real 2’,

the total demand goes up to 1,100. As we can see, the total revenue according

to a given order can be higher or lower than the total revenue estimated with

a LBH booking order. However, this is an extreme example. Most likely the

correct total revenue will be higher than the total revenue estimated with the

LBH assumption. This effect decreases significantly if the whole booking period

is split into multiple time periods. Usually airlines divide their booking periods in

20-25 time periods. Theoretically this model-related effect completely disappears

if we divide the booking period in so many time periods that in each time period

only one booking occurs. In the following section we compare the influence of

the main model-related errors with the data-related errors in the ROM.

4. The Network-based ROM with Independent Demand

4.3. Computational Results

In this section, we investigate the properties and in particular the robustness of

the network-based ROM with independent demand. We start with a comparison

of the effect of model- and data-related errors, followed by a detailed inspection of

the main data-related error, i.e. errors in the estimated unconstrained demand.

We also assess the effect of other relevant scenarios on the validity and the results

of the ROM. For each scenario we apply 180 simulation runs out of which we

discard the first 30 runs. The base case as defined in Section 3.3.1 serves as

the starting point of our analysis. Based on the base case we derive all further

scenarios.

4.3.1. Comparing Model- vs. Data-related Errors

In Section 4.2 we discussed one main model-related source of errors in the ROM.

We described the effect of a wrong assumption on the booking order on the

no RM revenue. In this section, we present a detailed analysis to compare the

magnitude of this effect against the effect of the data-related errors, i.e. the

errors in the estimated unconstrained demand. We make use of the simulation-

based environment to investigate ROM properties. In particular we examine the

base case introduced in Section 3.3.1 and the scenarios to simulate unconstraining

errors introduced in Section 3.3.2. In our simulation environment we are not only

able to adjust and measure unconstraining errors, but we are also able to utilize

the single booking requests generated in the request generator (see also Section

3.1.1 for further details). This allows us to determine the no RM revenue using

the correct booking order. In our flight network the correct no RM revenue based

on a real FCFS booking order is on average 39.3 million. If we use the aggregated

real demand in conjunction with Algorithm 4.1 presented in Section 4.1, we obtain

a slightly smaller average no RM revenue of 39.0 million. We observe that the gap

between the two no RM revenues with real demand is marginal with 0.3 million

or 0.8%.

In a second analysis, we investigate the effect of errors in the unconstrained

demand on the no RM revenue estimation derived with Algorithm 4.1 compared

to the average no RM revenue obtained with real demand (Rev−,R). Detailed

results can be found in Table 4.2. In the first two rows we describe the different

error scenarios and the error level. We start with the base case in the second

column and continue with the three main error scenarios: An error with a biased

underestimation, an error with a biased overestimation and an unbiased error

of the estimated unconstrained demand. For all error scenarios, we apply error

4.3. Computational Results

Base Biased Biased

Case underestimation overestimation Unbiased error

Error level - 30% 60% 90% 30% 60% 90% 30% 60% 90%

Deviation to

Rev

−,R (million) 0.0 0.8 1.8 2.7 -0.8 -1.4 -2.0 -0.1 -0.1 -0.1

Deviation to

Rev

−,R (%) -0.1 2.1 4.5 7.0 -2.1 -3.7 -5.1 -0.2 -0.2 -0.3

Table 4.2.: Deviations in No RM Revenue Estimates Caused by Errors in Uncon-

strained Demand

levels of 30%, 60% and 90%, out of which we assume 60% to be the worst case

as already described in Chapter 3. In the third row the absolute deviations

between the average no RM revenue obtained with the corresponding estimated

unconstrained demand using the simulated FCFS-strategy are listed. The relative

deviations are presented in the subsequent row. We observe that with the base

case - which already contains unconstraining errors as defined before - and the

unbiased unconstraining errors the deviations are lower than the deviation caused

by the model-related error. However, the deviations are by far higher with a

biased over- or underestimation of the estimated unconstrained demand.

We conclude that the model-related error induced by an incorrect assumption

on the booking order has a minor effect on the validity of the no RM revenue

estimate. On average the errors induced by incorrect estimations of the uncon-

strained demand are significantly higher. In the remainder of this thesis, we will

focus on the analysis of the effect of errors in the estimated unconstrained de-

mand. Thus, we use the no RM revenue estimate for real demand obtained with

the simulated FCFS algorithm to avoid overlapping error effects. Furthermore,

in practice the correct booking order is unknown and an assumption in the ROM

has to be made that leads to reasonable results.

4.3.2. Analyzing the Effect of Unconstraining Errors

In this section, we analyze the main data-related error in the ROM. We investi-

gate the effect of different unconstraining errors on the validity of the ROM and

determine its robustness. We analyze the general average effect of the different

unconstraining error scenarios on the potential and no RM revenue estimates

and furthermore the resulting effect on the two derived absolute ROM measures

RO and ARO. Afterwards we focus on the PARO and in particular assess its

robustness against errors in the input data.

4. The Network-based ROM with Independent Demand

The first analysis we conduct is comparing the estimations of the potential

and the no RM revenue between the different scenarios already introduced in the

previous section. Figure 4.2 compares the potential and no RM revenue estimates

for the base case and the nine unconstraining error scenarios. The error scenarios

36,0

38,0

40,0

42,0

44,0

46,0

48,0

50,0

52,0

Base

Case

-30% -60% -90% +30% +60% +90% ±30% ±60% ±90%

Opt Rev No RM Rev Act Rev

Figure 4.2.: Effect of Unconstraining Errors on the Potential and No RM Revenue

of a biased underestimation are marked with a minus (e.g. −30%), the error

scenarios of a biased overestimation with a plus (e.g. +30%) and the unbiased

unconstraining error scenarios are marked with a plus/minus sign (e.g. ±30%).

We observe that for an unbiased error the effect on the revenue estimates is minor.

Both the estimates of the potential and the no RM revenue remain more or less

constant. However, if we overestimate the unconstrained demand the potential

revenue increases and the no RM revenue decreases. An underestimation of the

unconstrained demand leads to contrary results. The potential revenue estimate

decreases and the no RM revenue estimate increases. We also observe that the RO

as the difference between potential and no RM revenue deviates to a significant

degree from the base case scenario if a biased under- or overestimation of the

unconstrained demand is given. The effect on the potential revenue, the no

RM revenue and subsequently on the RO and the ARO is examined in detail

in Table 4.3. In the first three data rows the table shows the average potential

revenue for both real (Rev+,R) and estimated unconstrained demand (Rev+,D)

and the difference between them. The average actual revenue Rev is listed in the

fourth data row. The subsequent data rows present the average no RM revenue,

the average ARO and the average RO calculated with the real demand and the

estimated unconstrained demand (Rev−,R,ARORand ROR) and with estimated

4.3. Computational Results

Base Biased Biased

Case underestimation overestimation Unbiased error

Error level - 30% 60% 90% 30% 60% 90% 30% 60% 90%

Rev+,R (million) 46.5 46.5 46.5 46.5 46.5 46.5 46.5 46.5 46.5 46.5

Rev+,D (million) 46.6 46.2 45.3 43.4 46.8 47.0 47.1 46.6 46.6 46.6

Diff. (million) -0.1 0.3 1.2 3.1 -0.3 -0.5 -0.6 -0.1 -0.1 -0.1

Rev (million) 44.6 44.4 43.8 42.8 44.8 44.9 44.9 44.6 44.6 44.6

Rev

−,R (million) 39.0 39.0 39.0 39.0 39.0 39.0 39.0 39.0 39.0 39.0

Rev

−,D (million) 39.0 39.8 40.8 41.8 38.2 37.6 37.1 39.0 39.0 38.9

Diff. (million) 0.0 -0.8 -1.8 -2.8 0.8 1.4 1.9 0.0 0.0 0.1

AROR(million) 5.6 5.3 4.8 3.7 5.8 5.8 5.9 5.6 5.6 5.6

AROD(million) 5.6 4.5 3.0 1.0 6.6 7.3 7.9 5.7 5.7 5.7

Diff. (million) 0.0 0.8 1.8 2.7 -0.8 -1.5 -2.0 -0.1 -0.1 -0.1

ROR(million) 7.5 7.5 7.5 7.5 7.5 7.5 7.5 7.5 7.5 7.5

ROD(million) 7.5 6.3 4.5 1.6 8.6 9.4 10.1 7.6 7.6 7.7

Diff. (million) 0.0 1.2 3.0 5.9 -1.1 -1.9 -2.6 -0.1 -0.1 -0.2

Table 4.3.: Effect of Errors in the Unconstrained Demand on RO and ARO

unconstrained demand (Rev−,D,ARODand ROD) and the respective differences.

The error scenarios are still the same. Besides assessing the base case we again

investigate the three main unconstraining error scenarios. As already indicated

in Figure 4.2 the average RO remains constant with an unbiased unconstraining

error. It increases with a biased overestimation and strongly decreases with a

biased underestimation. This is in particular due to the effect that a strong biased

underestimation leads to an estimated total demand which is only slightly above

the total number of bookings. Additionally, we observe that the ARO mainly

shows the same characteristics as the RO. However, we furthermore conclude

that the ARO can be used to justify the application of a RMS. For all error

scenarios up to the expected worst case of 60%, the average ARO is pretty stable.

In particular the values for the unbiased unconstraining error are very stable,

which is the most likely scenario of an unconstraining error in reality. Thus, an

indication of the absolute revenue contribution of the RM controls applied at an

airline can be given by the ARO.

After having assessed the basic average effects of the different unconstraining

errors on the revenue estimates and the absolute ROM measures, in the following

we investigate the robustness of the PARO in detail. We analyze the base case

scenario and additionally the unconstraining error scenarios as already introduced

before. In the base case, we compare the PARORwith the PAROD, which is

derived from the unconstraining module without applying any additional error

4. The Network-based ROM with Independent Demand

in this case. In the scatter plot presented in Figure 4.3, each dot represents a

pair of PARO values generated in a single simulation run. For the x-value, we

30%

50%

70%

90%

30% 50% 70% 90%

Real Demand

Est. Unc. Demand

Figure 4.3.: Base Case with Independent Demand

take the PARORvalues and for the y-value, we take the PARODvalues. As

the PAROs range from 30% to 90% for most of the scenarios analyzed, we only

use this scale in the graphs to improve visibility and comparability. In case that

the PARO values are not within this graphs we adjust the range accordingly

and point to the adjustment. In Table 4.4, we list the key metrics of the base

case and the error scenarios on the unconstrained demand. The columns are the

Base Biased Biased

Case underestimation overestimation Unbiased error

Error level - 30% 60% 90% 30% 60% 90% 30% 60% 90%

PAROR(%) 74.7 71.0 64.1 49.6 76.8 78.0 78.7 74.8 74.6 74.5

PAROD(%) 74.7 71.6 66.9 59.4 76.3 77.4 78.2 74.4 74.0 73.9

MAEP ARO (%) 0.3 0.6 2.8 9.8 0.5 0.6 0.6 0.4 0.6 0.7

rP ARO 0.94 0.87 0.75 0.64 0.97 0.97 0.96 0.94 0.90 0.86

R(thousand) 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6

D(thousand) 87.2 79.8 72.6 65.4 95.1 102.9 110.7 87.5 87.8 88.2

MAED0.56 0.73 1.11 1.55 0.76 1.17 1.66 0.75 1.15 1.62

PMAED(%) 9.3 12.0 18.2 25.6 12.6 19.4 27.5 12.3 18.9 26.8

Table 4.4.: Effect of Unconstraining Errors on PARO

same as listed in the previous tables. The results for the base case are presented

in column two. In the first two data rows we present the average values for

4.3. Computational Results

the PAROs calculated with real and estimated unconstrained demand in the

following denoted with PARORand PAROD. According to these underlying

PARO values, the two subsequent data rows show the values for the derived

similarity measures MAEP ARO and rP ARO. The average total real demand R

and the average total estimated unconstrained demand Dare shown in data rows

five and six. The values of the error measures MAEDand PMAED, which were

derived by comparing the total real and the estimated unconstrained demand

for each itinerary are presented in the last two data rows. For the following

unconstraining error scenarios, we present the results using similar scatter plots

and tables. For the base case, the MAEP ARO is very low with 0.3%. By inspecting

the scatter plot, we are also able to observe a very strong linear relation between

PARORand PAROD. This visual observation is supported by a high value of

the correlation coefficient rP ARO = 0.94. The levels of total demand are very

similar, and, for the error measures, we observe MAED= 0.56 and PMAED=

9.3%. The values for the error measures seem to be low compared to error levels

we normally observe for forecasts in real-life applications. This is because of the

fact that for unconstraining we only have to estimate the demand for those time

periods in which the booking classes were closed. In this and also in the following

scenarios, this was the case in about 20% of the time periods, which is comparable

to what we observe in reality. This circumstance reduces the error potential of the

unconstraining significantly. Overall, we observe similarity measures indicating

a very high similarity combined with moderate error levels for the base case.

Besides the base case we also investigated the error scenarios on the uncon-

strained demand. We again analyzed an error with a biased underestimation,

an error with a biased overestimation and an unbiased error of the estimated

unconstrained demand. The results are listed in Table 4.4 and the scatter plots

for the respective scenarios can be found in Figures 4.4, 4.5 and 4.6. Overall,

because of the average error levels applied, the error measures approximately

tripled compared to the base case from 0.56 to around 1.60 for the MAEDand

from 9.3% to around 27% for the PMAED. However, for all error scenarios the

similarity measures indicate a very high resemblance. For the biased underesti-

mation of the unconstrained demand, we observe a significant effect on the ROM

measures. With increasing error level, the MAEP ARO increases from 0.3% for

the base case to 9.8% for the highest error level. We also observe a strong de-

crease in overall RM success, which is indicated by a decrease of the values for

PAROR. Please note that for the highest error level, the average total demand

Ddecreased to around 65.4 thousand. This is only slightly above the average

number of actual bookings, which is, by definition, the absolute lower bound for

the estimation of the unconstrained demand (see Proposition 3.1.1). However,

4. The Network-based ROM with Independent Demand

30%

50%

70%

90%

30% 50% 70% 90%

Real Demand

Est. Unc. Demand

Error 0.30 Error 0.60 Error 0.90

Figure 4.4.: Effect of a Biased Un-

derestimation of Un-

constrained Demand on

PARO

30%

50%

70%

90%

30% 50% 70% 90%

Real Demand

Est. Unc. Demand

Error 0.30 Error 0.60 Error 0.90

Figure 4.5.: Effect of a Biased Overes-

timation of Unconstrained

Demand on PARO

we still observe a linear relation in the scatter plot shown in Figure 4.4 and the

correlation coefficient remains above our minimum level of similarity with rP ARO

at 0.64, even at an error level of 90%. In our worst-case scenario at 60% error

level, the similarity measures indicate a very high resemblance. The application

of a biased overestimation and an unbiased error on the unconstrained demand

do not have a significant effect on the similarity measures. Although the error

measures strongly increase, the similarity measures indicate a high similarity.

The MAEP ARO stays below 1% and the correlation coefficient rP ARO is larger

than 0.86 for all cases. The results of our scenarios also validate the analysis of

the effect of an unbiased unconstraining error on the ROM accomplished by P¨olt

(2001) in a leg-based airline RM context.

4.3.3. Analyzing the Effect of Further Scenarios

Besides analyzing the robustness of the ROM - in particular the PARO - against

errors in the unconstrained demand, we apply additional scenarios to investigate

the ROM properties. In the following we analyze the effect of forecast errors,

open/restrictive RM control and adjusted seasonality on the ROM and on its

robustness.

4.3. Computational Results

30%

50%

70%

90%

30% 50% 70% 90%

Real Demand

Est. Unc. Demand

Error 0.30 Error 0.60 Error 0.90

Figure 4.6.: Effect of an Unbiased Unconstraining Error on PARO

Effect of Forecast Errors on ROM

We start by analyzing the effect of forecast errors on the ROM. The applied

scenarios are described in detail in Section 3.3.2. Compared to the unconstrained

demand scenarios we expect the error and similarity measures to be quite similar,

but we also expect a decreased overall RM success, which is indicated by the

values for PAROR. The detailed results are presented in Table 4.5. Figures

4.7 and 4.8 show the scatter plot for the biased over- and underestimation of the

forecasted demand. Please note, that the 90% error scenarios are not in the range

of the graphs. The scatter plots showing these error scenarios and the scatter

plot of the unbiased forecast error can be found in the appendix. In the table we

added a data row with the average forecasted demand Fand two data rows with

the measures for the forecast error MAEFand PMAEF. A first observation

is that the similarity measures of the PARO are basically the same as those

with the error on the unconstrained demand scenario. For example the values of

rP ARO develop in the same direction throughout the error scenarios, whereas they

decrease a bit stronger than with the unconstraining error scenarios. However,

the values of rP ARO are above 0.76, except for the 90% underestimation of the

forecast. Moreover, the values of MAEP ARO remain moderate in most cases.

Another main observation is that the increase in the forecast error leads to worse

RM results as it was expected during the scenario setup. The values for PAROR

decreased for each forecast error scenario. By looking at the forecast error it

becomes obvious that applying an error on all booking classes leads on average

4. The Network-based ROM with Independent Demand

Base Biased Biased

Case underestimation overestimation Unbiased error

Error level - 30% 60% 90% 30% 60% 90% 30% 60% 90%

PAROR(%) 74.7 65.0 43.2 15.1 69.0 53.8 35.5 74.7 74.4 73.9

PAROD(%) 74.7 67.4 52.1 33.4 69.3 57.3 44.0 74.3 73.8 73.2

MAEP ARO (%) 0.3 2.4 8.9 18.3 0.4 3.5 8.5 0.4 0.6 0.8

rP ARO 0.94 0.81 0.76 0.49 0.99 0.99 0.98 0.93 0.89 0.84

R(thousand) 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6

D(thousand) 87.2 80.0 72.7 65.2 95.1 104.6 115.6 87.5 87.8 88.2

F(thousand) 87.1 61.2 35.2 8.9 112.9 138.5 163.8 87.1 87.1 87.1

MAED0.56 0.75 1.13 1.57 0.78 1.31 2.02 0.75 1.15 1.63

PMAED(%) 9.3 12.3 18.6 25.9 12.9 21.7 33.4 12.3 19.0 26.9

MAEF1.83 2.37 3.76 5.47 2.50 3.85 5.44 1.89 2.05 2.29

PMAEF(%) 30.3 39.0 62.0 90.2 41.4 63.7 90.0 31.2 33.9 37.7

Table 4.5.: Effect of Forecast Errors on PARO

to the applied forecast error defined in the error scenario both for a biased over-

and underestimation. The unbiased forecast error scenario shows smaller forecast

errors, due to the fact that the forecast is updated using exponential smoothing

and the error method applied randomly overestimates or underestimates it. This

behavior is expected to be more realistic than a constant overestimation of one

part of the itineraries and a constant underestimation of the other part.

Effect of Adjusted RM Control and Seasonality on ROM

Apart from analyzing the robustness of the ROM against unconstraining and

forecast errors in various scenarios and biases, we studied the effects of poor -

i.e., very open or very restrictive - RM controls on the PAROs. Therefore, we

increased and decreased the bid prices by a certain percentage. The adjustment

levels applied were 25% and 50%. In contrast to the forecast error scenario we

expect the overall RM success to decrease, but the errors in the unconstrained

demand should remain basically constant. Detailed results can be found in Table

4.6. We also show the scatter plot for the open RM control scenario in Figure 4.9.

The scatter plot for the restrictive RM control case can be found in the appendix.

As we adjusted the final bid prices, not the forecasts in the simulation runs, we

observe similar error levels in the estimated unconstrained demand compared to

the base case. Furthermore, the similarity measures indicate a very high similarity

in all cases. We not only observe high similarity of the PAROs for both the open

RM control and the restrictive RM control scenario. Moreover the results of

4.3. Computational Results

30%

50%

70%

90%

30% 50% 70% 90%

Real Demand

Est. Unc. Demand

Error 0.30 Error 0.60

Figure 4.7.: Effect of Biased Under-

estimation of Forecasted

Demand on PARO

30%

50%

70%

90%

30% 50% 70% 90%

Real Demand

Est. Unc. Demand

Error 0.30 Error 0.60

Figure 4.8.: Effect of Biased Overesti-

mation of Forecasted De-

mand on PARO

the average PAROs confirm that the quality of the RM control decreased with

increasing adjustment level. This was the assumption underlying the scenarios.

The last scenarios we conducted were adjusting the underlying seasonality of

the booking requests. On the one hand we increased and decreased the amplitude

of the underlying seasonality and on the other hand we applied a saw tooth curve

to the request generator. Within five runs we change the demand level from

130% down to 70% or from 120% down to 80% and after these five runs we

jump back to the starting value. We expect the error measures to decrease if we

decrease the seasonality and to increase in the opposite case. The results of the

scenarios are also presented in Table 4.6 and we show the scatter plot of the saw

tooth curve scenarios in Figure 4.10. As expected, the overall unconstraining

errors slightly decreases from 0.56 to 0.54 for the MAEDand from 9.3% to

8.8% for the PMAEDif we decrease the amplitude of the seasonality. If we

increase the amplitude, the error measures increase to 0.60 and to 9.9%. The

similarity measures remain the same, with a slightly increased rP ARO of 0.96 for

the higher amplitude. The saw tooth curve only leads to a minor increase in the

unconstraining error. However, the correlation coefficient increases significantly

to nearly one, because the dispersion of the PARO values is significantly higher.

We conclude that demand deviations help to increase the value of rP ARO, while

MAEP ARO remains moderate.

4. The Network-based ROM with Independent Demand

Base Bid price Bid price Adjust Apply

Case decrease increase seasonality saw tooth curve

Adj. level - 25% 50% 25% 50% -50% +50% 130%-70% 120%-80%

PAROR(%) 74.7 62.5 36.7 66.3 59.6 75.1 74.2 69.5 72.7

PAROD(%) 74.7 62.9 36.7 66.4 59.5 75.1 74.3 70.8 73.2

MAEP ARO (%) 0.3 0.5 0.5 0.3 0.4 0.2 0.3 1.2 0.6

rP ARO 0.94 0.90 0.94 0.98 0.98 0.94 0.96 0.99 0.99

R(thousand) 87.6 87.6 87.6 87.6 87.6 87.4 87.8 87.2 87.2

D(thousand) 87.2 87.5 88.0 86.8 86.6 87.3 86.9 83.8 85.6

MAED0.56 0.61 0.66 0.58 0.60 0.53 0.60 0.67 0.59

PMAED(%) 9.3 10.1 11.0 9.6 10.0 8.8 9.9 10.7 9.6

Table 4.6.: Effect of Adjusted RM Control and Seasonality on PARO

4.4. Summary

In this chapter we described the network-based ROM with independent demand

in detail. We highlighted some of the main properties of the ROM in this RM

context and investigated the magnitude of model- and data-related errors on

the validity of the ROM. After having analyzed the effect of a wrong booking

order assumption in comparison to various unconstraining errors, we conclude

that model-related errors do not play a major role for the validity of the ROM.

As a consequence, we focus on analyzing the effect of data-related errors on the

ROM in the remainder of the thesis. In this chapter we therefore also analyzed

the robustness of the ROM in detail not only considering different kinds of un-

constraining errors, but also further scenarios including forecast errors, applying

adjusted RM controls and different sorts of seasonality. In all scenarios applied

the values of the similarity measures showed results above our minimum level of

similarity defined in Section 3.2. As we tested all scenarios with error levels up to

the expected worst case and even beyond, we consider the network-based ROM

with independent demand robust against errors in the input data for all error

rates we would expect in real life. In addition, the effect of the other scenarios

on the ROM was as expected, which also supports our conclusion to consider

the ROM robust to deliver valid information about the RM success on a network

level.

4.4. Summary

30%

50%

70%

90%

30% 50% 70% 90%

Real Demand

Est. Unc. Demand

Base Case Adj. -25% Adj. -50%

Figure 4.9.: Effect of Open RM Con-

trols on PARO

30%

50%

70%

90%

30% 50% 70% 90%

Real Demand

Est. Unc. Demand

Base Case 130% to 70% 120% to 80%

Figure 4.10.: Effect of High Deviation

in Customer Demand on

PARO

4. The Network-based ROM with Independent Demand

5. The Network-based ROM with

Dependent Demand

Considering dependent demand structures in practical RM applications becomes

increasingly important and common (see e.g. Weatherford and Ratliff, 2010). In

this chapter we therefore introduce an enhancement of the network-based ROM

with independent demand to a ROM, which considers dependent demand struc-

tures. We describe in detail how the estimations of the potential and no RM

revenue are adjusted. Furthermore we discuss main properties of the network-

based ROM with dependent demand. In the remainder of the chapter, we present

computational results on the properties of the previously defined ROM with a

special focus on the robustness on unconstraining errors. Additionally we inves-

tigate the effect of further scenarios on the ROM.

5.1. Extensions to the Network-based ROM with

Independent Demand

In this section we explain the enhancement of the network-based ROM with

independent demand to dependent demand structures in detail. Modifications

have to be made to the estimations of the potential and the no RM revenue,

because the demand model changes. The actual revenue as the result of the

actual RM control is derived as explained in Chapter 4.

First, we introduce the enhancement of the potential revenue estimation in-

troduced for independent demand. Therefore we start with the definition of the

LP used for the network-based ROM with independent demand, which is shown

again in Equations 5.1 to 5.3.

Max X

i∈IX

j∈JiX

t∈T

pi,j,t ∗x+

i,j,t (5.1)

i∈IlX

j∈Ji,l,m X

t∈T

i,j,t ≤capl,m ∀l∈L, ∀m∈Ml,∀t∈T(5.2)

0≤x+

i,j,t ≤di,j,t ∀i∈I, ∀j∈Ji,∀t∈T(5.3)

5. The Network-based ROM with Dependent Demand

To estimate the potential revenue with dependent demand we still want to max-

imize the bookings multiplied with the given fare as shown in the objective func-

tion 5.1. All booking decisions made by the LP are also still bound to Constraint

5.2 considering the capacity of the compartments in the legs. However, we assume

that the demand structure has changed from the independent demand model to

a dependent demand model. In contrast to independent demand the realized

demand depends on the availability of booking classes in a dependent demand

context. Accordingly, Constraint 5.3 has to be adjusted to consider dependent

demand. We will use the definition of dependent demand introduced in Chap-

ter 3 in Section 3.1.1. The main idea of the enhancement is to let the LP for

the potential revenue estimation also optimize the availability of booking classes.

Therefore we introduce a variable yi,j,t to indicate, whether a booking class jfor

itinerary iin a given time period tis open or not. If the booking class is open yi,j,t

takes the value 1, otherwise yi,j,t takes the value 0. Using this new yi,j,t variable

Constraint 5.3 changes to 5.4. In addition we add Constraint 5.5 to ensure that

yi,j,t ∈ {0,1}.

0≤x+

i,j,t ≤yi,j,t ∗dtd

i,j,t −X

j0∈Ji,j

yi,j0,t ∗dbd

i,j,j0,t ∀i∈I, ∀j∈Ji,(5.4)

∀t∈T

yi,j,t ∈ {0,1} ∀i∈I, ∀j∈Ji,(5.5)

∀t∈T

We illustrate the adjusted demand constraint in the following examples. Using

the buy-down graph introduced in Section 3.1.1 the demand constraint translates

for a given itinerary ifor booking classes one, two, three, four, and five in a given

time period tto the following equations.

0≤xi,1,t ≤yi,1,t ∗10–yi,2,t ∗5–yi,3,t ∗3

0≤xi,2,t ≤yi,2,t ∗16–yi,4,t ∗10

0≤xi,3,t ≤yi,3,t ∗15–yi,5,t ∗10

0≤xi,4,t ≤yi,4,t ∗12–yi,5,t ∗8

0≤xi,5,t ≤yi,5,t ∗30

yi,j,t ∈ {0,1},∀j∈ {1,2,3,4,5}

The values of each x-variable are limited to the total demand of the respec-

tive booking class minus the buy-down, which is realized according to the given

availability constellation. xi,1,t for example has an upper bound of ten minus

five if booking class two is open and additionally minus three if booking class

5.1. Extensions to the Network-based ROM with Independent Demand

three is open, which leads to an upper bound of two in this case. The following

example illustrates the resulting bounds of the demand constraints, if the first

three booking classes are open and the other two booking classes are closed (i.e.

yi,1,t, yi,2,t, yi,3,t = 1 and yi,4,t, yi,5,t = 0).

0≤xi,1,t ≤1∗10–1 ∗5–1 ∗3 = 2

0≤xi,2,t ≤1∗16–0 ∗10 = 16

0≤xi,3,t ≤1∗15–0 ∗10 = 15

0≤xi,4,t ≤0∗12–0 ∗8 = 0

0≤xi,5,t ≤0∗30 = 0

Besides adjusting the demand constraint, an additional constraint to model

the feasible actions has to be included in the model. According to the definition

in Section 3.1.1 the opening order of the booking classes is sequentially ordered.

In the previous examples we have implicitly considered this requirement, but not

enforced it formally. The feasible actions can easily be modeled using the newly

introduced yi,j,t variables.

yi,(j+1),t ≤yi,j,t ∀i∈I, ∀m∈Mi,(5.6)

∀j∈Ji,m \ {j−

i,m},∀t∈T

For each booking class jConstraint 5.6 ensures that the next lower booking

class j+ 1 is closed, if booking class jis closed. The feasible action constraint

is applied to all booking classes jin a compartment m. The booking classes

are taken from the set of all booking classes jin itinerary ithat are related

to compartment m Ji,m except the lowest booking class j−

i,m. We illustrate the

constraint using a simple example. If booking classes one and two belong to the

same compartment, then the feasibility constraint yi,2,t ≤yi,1,t has to be fulfilled.

If for example booking class one is closed (i.e. yi,1,t = 0), it follows that also

yi,2,t = 0.

Up until now we only discussed the case in which yi,j,t was either one or zero.

This means that a given booking class was either opened or closed during the

entire time period. However, for the ROM with dependent demand we assume

that a booking class can be closed in the course of a given time period t, as it is the

case in reality. As a result yi,j,t can take all (fractional) values ∈[0,1]. In Figure

5.1 we illustrate the basic assumption of closing a booking class. At the beginning

of the time period all booking classes are open. After the completion of 30% of

the time period booking class five is closed. All other booking classes remain open

at that point in time. After 50% of the time period has been completed booking

5. The Network-based ROM with Dependent Demand

Status of booking class

Start of

time period

End of

time period

open

closed

Figure 5.1.: Linear Opening Constraint During a Time Period

class four is also closed. Until the end of the time period the first three booking

classes remain open. This closing assumption is reflected in the following values

of y.yi,1,t, yi,2,t, yi,3,t = 1, yi,4,t = 0.5 and yi,5,t = 0.3. If we include these fractional

values of yinto the demand constraint we implicitly assume that the demand is

uniformly distributed in the DCP intervals. This assumption is also made by

Boyd and Kallesen (2004), Walczak et al. (2010) and Fiig et al. (2010) during

the demand transformation process. We illustrate the example from Figure 5.1

by calculating the upper bounds for the demand constraint.

0≤xi,1,t ≤1∗10–1 ∗5–1 ∗3=2

0≤xi,2,t ≤1∗16–0.5∗10 = 11

0≤xi,3,t ≤1∗15–0.3∗10 = 12

0≤xi,4,t ≤0.5∗12–0.3∗8 = 3.6

0≤xi,5,t ≤0.3∗30 = 9

In this example the upper bound of xi,2,t is set to eleven. The total demand of

16 is only decreased by five, because booking class four was only open half of

the time period. The realized demand for booking class four consists of 50% of

the total demand and is decreased by 30% of the buy-down into booking class

five. It is to be noted that it generally holds that the upper bound for the

adjusted demand constraint is always greater or equal to zero. This directly

follows from dtd

i,j,t ≥Pj0∈Ji,j dbd

i,j,j0,t and the fact that the respective yi,j,t values

adhere to Constraint 5.6.

The complete LP formulation of the network-based ROM with dependent de-

5.2. Properties of the Network-based ROM with Dependent Demand

mand is listed in Equations 5.7 to 5.11.

Max X

i∈IX

j∈JiX

t∈T

pi,j,t ∗x+

i,j,t (5.7)

i∈IlX

j∈Ji,l,m X

t∈T

i,j,t ≤capl,m ∀l∈L, ∀m∈Ml,(5.8)

∀t∈T

0≤x+

i,j,t ≤yi,j,t ∗dtd

i,j,t −X

j0∈Ji,j

yi,j0,t ∗dbd

i,j,j0,t ∀i∈I, ∀j∈Ji,(5.9)

∀t∈T

yi,(j+1),t ≤yi,j,t ∀i∈I, ∀m∈Mi,(5.10)

∀j∈Ji,m \ {j−

i,m},∀t∈T

yi,j,t ∈[0,1] ∀i∈I, ∀j∈Ji,(5.11)

∀t∈T

For the no RM revenue estimation we apply a simulation of a FCFS strategy

again. Algorithm 4.1 introduced in Section 4.1 can be applied again. During

the simulation of a FCFS strategy we assume that all booking classes are open.

The effect on the demand used in the algorithm is that all buy-down into lower

booking classes is realized. The demand that correlates to this is the yieldable

demand dyd

i,j,t.

The result of the estimations of the potential and the no RM revenue x+

i,j,t and

x−

i,j,t and the actual results of the booking period bi,j,t are applied to Equations 4.4

to 4.5. The derived ROM measures are determined in the same way as presented

in Chapter 4 in Equations 4.7, 4.8 and 4.9.

5.2. Properties of the Network-based ROM with

Dependent Demand

In Section 4.2 we discussed main properties of the network-based ROM with

independent demand. The properties regarding the isolation of the RM contri-

bution from the overall success and the probability of special cases for the RO, the

ARO and the PARO are the same for the network-based ROM with dependent

demand.

As a main model-related error we described the assumption on the booking

order in the FCFS simulation to determine the no RM revenue for the independent

demand case. With dependent demand the booking order not only plays a role

in the no RM revenue estimation, but also in the estimation of the potential

5. The Network-based ROM with Dependent Demand

revenue. As described in the previous section we assume that booking classes

can be closed at some point within a given time period and the realized demand

in the demand constraint is distributed uniformly according to the availability of

the booking classes defined by the y-variables. The following example shows that

even with perfect aggregated data input, the calculation of the potential revenue

with the LP defined in Equations 5.7 to 5.11 might lead to results that are below

the actual revenue (for reasons of readability we omit the indices for the itinerary

and the time period).

max 300 ∗x1+ 150 ∗x2+ 100 ∗x3

0≤x1≤y1∗4−y2∗2

0≤x2≤y2∗4−y3∗2

0≤x3≤y3∗4

0≤x1+x2+x3≤6

0≤y3≤y2≤y1≤1

We consider one flight leg with a capacity of six seats in this example. Three

booking classes one, two and three are available to the customers. On an aggre-

gated level the demand is described by the three demand constraints. The fourth

constraint ensures the feasible actions and the value range of the y-variables.

If we solve this example using an LP-solver, the optimal solution is 1,250 with

x1= 3, x2= 1 and x3= 2. The yvalues are y1= 1, y2= 0.5 and y3= 0.5. Given

our current definition of the potential revenue, this would be the respective esti-

mate. However it is possible, that the RMS and the revenue managers were able

to anticipate the customer requests in a better way leading to an actual revenue

that is larger than the estimate for the potential revenue. The example in Table

5.1 gives an illustration. The column with heading ’Req.’ describes the single

requests. The listed booking classes describe the set of booking classes the given

passenger is willing to purchase. Request ’2,1’ for example describes a passenger

who starts looking for a ticket in booking class two and then looks for a ticket in

booking class one if the former is not available. In column ’Avail.’ all booking

classes that are available at that point in time are listed. The availability ’1,2’

for example represents the situation in which booking classes one and two are

open. The column ’Dec.’ lists the results of the accept/deny-decision, which is

based on the given request and the current availability of the booking classes.

The result could either be the booking class the customer is booked into or ’rej.’,

if the request was rejected. In column ’Rev.’ the resulting revenue is presented.

In the given example the revenue manager made booking classes one and two

available at the beginning. This leads to the rejection of the first two booking

5.3. Computational Results

Req. Avail. Dec. Rev.

3 1,2 rej. -

3,2 1,2 2 150

2,1 1 1 300

1 1 1 300

Sum 1,500

Table 5.1.: Actual Revenue with Restrictive Control and Low-before-high Book-

ing Order

requests. The next two requests are accepted in booking class two leading two a

revenue of 150 for each booking. After these two bookings the revenue manager

decides to close booking class two and to only leave booking class one open. As a

result four bookings in booking class one will be made based on the requests. In

total this leads to a revenue of 1,500. In this case the actual revenue eventually

was larger than the potential revenue estimated with the LP. However, the effect

that the actual revenue might become larger than the potential revenue decreases

strongly with increasing number of DCPs considered in the ROM. We expect the

effect of data-related errors again to be larger and of higher importance than

the model-related errors. Moreover, the assumption made within the ROM -

particularly for the potential revenue - reflects the modeling decisions that had

to be taken for the RM methodology applied in practice and we expect this

definition in general to be a very good estimation of the potential revenue.

5.3. Computational Results

In this section we investigate the properties and in particular the robustness of the

network-based ROM with dependent demand. We start with a detailed inspection

of the main data-related error, i.e. errors in the estimated unconstrained demand.

With dependent demand, we have to consider errors in the yieldable demand and

errors in the buy-down as well. As in Chapter 4 the results of each scenario

are based on the evaluation of 150 simulation runs. We conclude this section by

assessing the effect of other relevant scenarios on the validity and the results of

the ROM.

5. The Network-based ROM with Dependent Demand

5.3.1. Base Case and Unconstraining Error Scenarios

In this section we investigate the effect of different unconstraining errors on the

validity of the ROM. According to Section 4.3.2 in Chapter 4 we start with

analyzing the effect on the absolute ROM measures. Afterwards we examine the

PARO in detail and in particular assess its robustness against errors in the input

data.

The first analysis we conduct is comparing the estimations of the potential

and the no RM revenue of the different error scenarios for the estimated uncon-

strained demand. As already described in Section 3.3.2 we are able to apply

errors for the yieldable demand, but also for the buy-down. Figures 5.2 and 5.3

compare the potential and no RM revenue estimates for the base case and the

nine unconstraining error scenarios. The scenarios are the same as in the pre-

36,0

38,0

40,0

42,0

44,0

46,0

48,0

50,0

52,0

Base

Case

-30% -60% -90% +30% +60% +90% ±30% ±60% ±90%

Opt Rev No RM Rev Act Rev

Figure 5.2.: Effect of Errors in the Unconstrained Yieldable Demand on the Po-

tential and No RM Revenue

vious chapter. Again the error scenarios of a biased underestimation are marked

with a minus (e.g. −30%), the error scenarios of a biased overestimation with

a plus (e.g. +30%) and the unbiased unconstraining error scenarios are marked

with a plus/minus sign (e.g. ±30%). If we apply an unconstraining error on

the yieldable demand we observe results similar to the ROM with independent

demand. We again observe that for an unbiased error the effect on the revenue

estimates is minor. Both the estimates of the potential and the no RM revenue

remain more or less constant. An overestimation of the unconstrained yieldable

demand leads to an increase of the potential revenue and a decrease of the no

RM revenue. An underestimation of the unconstrained yieldable demand leads

5.3. Computational Results

36,0

38,0

40,0

42,0

44,0

46,0

48,0

50,0

52,0

Base

Case

-30% -60% -90% +30% +60% +90% ±30% ±60% ±90%

Opt Rev No RM Rev Act Rev

Figure 5.3.: Effect of Errors in the Unconstrained Buy-down on the Potential and

No RM Revenue

to contrary results. The potential revenue estimate decreases and the no RM

revenue estimate increases. If an error in the estimated unconstrained buy-down

is applied, the effects on the potential and no RM revenue are lower. The poten-

tial revenue slightly increases if the buy-down is underestimated. The estimates

for the no RM revenue remain very stable for all error scenarios. These observa-

tions are supported by Tables 5.2 and 5.3. The subsequent effects on the RO

and ARO are very similar to the ROM with independent demand if we apply

errors to the unconstrained yieldable demand. However, the absolute amount of

the RO is higher on average. As already indicated in Figure 5.2, the average

RO remains constant with an unbiased unconstraining error. It increases with

a biased overestimation and strongly decreases with a biased underestimation.

In addition, we observe that the ARO mainly shows the same characteristics as

the RO. Applying errors to the unconstrained buy-down leads to significantly

lower effects. In particular the ARO is very stable if errors are applied to the

unconstrained buy-down, because the no RM revenue is very stable in this case.

We again conclude that the ARO can be applied to quantify the contribution of

the RMS in use.

In the following, we focus on the effects of the described unconstraining error

scenarios on the PARO. We start with a comparison of the base case simulated

with independent demand and the base case simulated with dependent demand.

The scatter plots are presented in Figures 5.4 and 5.5. They show that the ROM

with dependent demand is also robust for the base case. The detailed results are

5. The Network-based ROM with Dependent Demand

Base Biased Biased

Case underestimation overestimation Unbiased error

Error level - 30% 60% 90% 30% 60% 90% 30% 60% 90%

Rev+,R (million) 48.6 48.6 48.6 48.6 48.6 48.6 48.6 48.6 48.6 48.6

Rev+,D (million) 47.9 47.1 46.0 44.2 49.2 50.1 50.9 48.3 48.7 49.1

Diff. (million) 0.7 1.5 2.6 4.4 -0.6 -1.5 -2.3 0.3 -0.1 -0.5

Rev (million) 44.8 44.6 44.0 43.0 44.8 44.7 44.6 44.8 44.7 44.6

Rev

−,R (million) 39.0 39.0 39.0 39.0 39.0 39.0 39.0 39.0 39.0 39.0

Rev

−,D (million) 39.2 39.9 40.8 41.0 38.6 38.1 37.8 39.2 39.2 39.0

Diff. (million) -0.2 -0.9 -1.8 -2.0 0.4 0.9 1.2 -0.2 -0.2 0.0

AROR(million) 5.8 5.5 5.0 4.0 5.8 5.7 5.5 5.8 5.7 5.6

AROD(million) 5.6 4.6 3.2 2.0 6.2 6.6 6.8 5.6 5.5 5.6

Diff. (million) 0.2 0.9 1.8 2.0 -0.4 -0.9 -1.3 0.2 0.2 0.0

ROR(million) 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6

ROD(million) 8.7 7.2 5.2 3.2 10.6 12.0 13.1 9.1 9.5 10.1

Diff. (million) 0.9 2.4 4.4 6.4 -1.0 -2.4 -3.5 0.5 0.1 -0.5

Table 5.2.: Effect of Errors in the Unconstrained Yieldable Demand on ROM

Measures

Base Biased Biased

Case underestimation overestimation Unbiased error

Error level - 30% 60% 90% 30% 60% 90% 30% 60% 90%

Rev+,R (million) 48.6 48.6 48.6 48.6 48.6 48.6 48.6 48.6 48.6 48.6

Rev+,D (million) 47.9 47.8 47.6 47.6 48.3 48.8 49.4 48.1 48.4 49.1

Diff. (million) 0.7 0.8 1.0 1.0 0.3 -0.2 -0.8 0.5 0.2 -0.5

Rev (million) 44.8 44.8 44.8 44.8 44.8 44.7 44.7 44.8 44.8 44.8

Rev

−,R (million) 39.0 39.0 39.0 39.0 39.0 39.0 39.0 39.0 39.0 39.0

Rev

−,D (million) 39.2 39.2 39.2 39.1 39.2 39.2 39.2 39.2 39.2 39.4

Diff. (million) -0.2 -0.2 -0.2 -0.1 -0.2 -0.2 -0.2 -0.2 -0.2 -0.4

AROR(million) 5.8 5.8 5.8 5.8 5.8 5.7 5.7 5.8 5.8 5.8

AROD(million) 5.6 5.7 5.7 5.7 5.6 5.5 5.4 5.6 5.6 5.3

Diff. (million) 0.2 0.1 0.1 0.1 0.2 0.2 0.3 0.2 0.2 0.5

ROR(million) 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6

ROD(million) 8.7 8.6 8.5 8.4 9.1 9.6 10.2 8.9 9.1 9.6

Diff. (million) 0.9 1.0 1.1 1.2 0.5 0.0 -0.6 0.7 0.5 0.0

Table 5.3.: Effect of Errors in the Unconstrained Buy-down on ROM Measures

5.3. Computational Results

30%

50%

70%

90%

30% 50% 70% 90%

Real Demand

Est. Unc. Demand

Figure 5.4.: Base Case with Indepen-

dent Demand

30%

50%

70%

90%

30% 50% 70% 90%

Real Demand

Est. Unc. Demand

Figure 5.5.: Base Case with Depen-

dent Demand

presented in Table 5.4. They include another column listing the values of the base

case obtained from the independent demand scenarios. The values belonging to

this scenario are listed in the rows of the total demand (Dtd etc.). The tables are

also enhanced by data rows to list the average real yieldable demand (Ryd) and the

average real buy-down (Rbd). The values for the estimated unconstrained demand

are also added to the table (Dyd and Dbd). The error measures on the estimated

unconstrained demand are complemented accordingly with the respective values

(MAEDyd ,MAEDbd ,PMAEDyd and PMAEDbd ). The similarity measures for

the ROM with dependent demand indicate a high similarity with rP ARO = 0.91.

The MAEP ARO increases to 3.8%, which is mainly due to an underestimation

of the potential revenue for all scenarios with dependent demand. However, the

ROM proves itself robust against errors in the unconstrained demand also with

dependent demand for the base case scenario.

In the following we analyze main results obtained by the error scenarios in the

unconstrained demand for both yieldable demand and buy-down. The numeri-

cal results are listed completely, whereas we focus on some main scatter plots.

The scatter plots for all error scenarios are included in the appendix. As it has

been observed before, adjusting the estimated unconstrained yieldable demand

leads to larger effects. In Figures 5.6 and 5.7 we present the effect of a biased

underestimation and a biased overestimation of the unconstrained yieldable de-

mand. A biased underestimation of the unconstrained yieldable demand has a

significant effect on the robustness of the ROM measures. The values for rP ARO

decrease from 0.91 to 0.42 for the case with 90% error. We also observe a strong

5. The Network-based ROM with Dependent Demand

Base Base

case case Biased Biased

ind. dep. underestimation overestimation Unbiased error

Error level - - 30% 60% 90% 30% 60% 90% 30% 60% 90%

PAROR(%) 74.7 60.8 57.9 51.9 42.1 60.7 59.6 57.9 60.2 59.5 58.9

PAROD(%) 74.7 64.6 64.1 61.9 63.0 58.9 54.9 51.7 61.1 58.0 55.7

MAEP ARO (%) 0.3 3.8 6.2 10.0 20.9 1.8 4.7 6.2 1.0 1.5 3.1

rP ARO 0.94 0.91 0.84 0.69 0.42 0.97 0.98 0.98 0.92 0.91 0.92

Rtd (thousand) 87.6 117.5 117.5 117.5 117.5 117.5 117.5 117.5 117.5 117.5 117.5

Ryd (thousand) - 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6

Rbd (thousand) - 29.9 29.9 29.9 29.9 29.9 29.9 29.9 29.9 29.9 29.9

Dtd (thousand) 87.2 120.4 111.6 103.6 97.9 134.2 147.8 161.3 122.8 125.4 129.3

Dyd (thousand) - 88.7 79.5 71.1 65.3 102.3 115.9 129.4 90.8 93.1 96.6

Dbd (thousand) - 31.7 32.1 32.5 32.6 31.8 31.9 31.9 32.0 32.3 32.7

MAEDtd

0.56 1.37 1.47 1.77 2.02 1.87 2.65 3.51 1.67 2.22 2.80

MAEDyd

- 0.76 0.91 1.30 1.64 1.26 2.06 2.95 1.08 1.69 2.32

MAEDbd

- 0.87 0.88 0.89 0.90 0.87 0.87 0.87 0.87 0.88 0.90

PMAEDtd

(%) 9.3 16.9 18.1 21.8 24.8 23.0 32.6 43.3 20.6 27.3 34.4

PMAEDyd

(%) - 12.5 14.9 21.4 26.9 20.8 34.0 48.6 17.8 27.8 38.3

PMAEDbd

(%) - 42.0 42.5 43.2 43.3 42.0 42.0 42.0 42.3 42.8 43.4

Table 5.4.: Effect of Errors in the Unconstrained Yieldable Demand on PARO

5.3. Computational Results

30%

50%

70%

90%

30% 50% 70% 90%

Real Demand

Est. Unc. Demand

Error 0.30 Error 0.60 Error 0.90

Figure 5.6.: Effect of Biased Underes-

timation of Unconstrained

Yieldable Demand on

PARO

30%

50%

70%

90%

30% 50% 70% 90%

Real Demand

Est. Unc. Demand

Error 0.30 Error 0.60 Error 0.90

Figure 5.7.: Effect of Biased Overesti-

mation of Unconstrained

Yieldable Demand on

PARO

increase in MAEP ARO from 3.8% to 20.9%. However, up to error level 60%,

the similarity measures for the ROM still indicate a high similarity. Because an

error level of 90% is unlikely in practice, we consider the ROM robust even for a

biased underestimation of the unconstrained demand. In comparison to a biased

underestimation, the ROM proves itself to be much more robust against a biased

overestimation of unconstrained yieldable demand for all error levels. The values

for rP ARO actually increase from 0.94 for the base case to 0.98 for the 90% error

scenario. The values for MAEP ARO increase to 6.2%, which is still a very low

value for the 90% error case. An unbiased unconstraining error leads to similar

results.

Errors in the estimated unconstrained buy-down have a lower effect on the

ROM. The numerical results for the buy-down scenarios are listed in Table 5.5.

We focus on presenting the scatter plot for the unbiased error scenario in this

section. It is presented in Figure 5.8. The effect of an unbiased error in the

estimated unconstrained buy-down is minor. It is also minor for a biased over-

and underestimation of the unconstrained buy-down. The values for rP ARO are

above 0.90 in all cases. The values for MAEP ARO are also very promising. The

maximum value measured is 6.8% given a biased underestimation with 90% er-

ror. We conclude that the ROM is also robust with unconstraining errors in the

5. The Network-based ROM with Dependent Demand

Base Base

Case Case Biased Biased

ind. dep. underestimation overestimation Unbiased error

Error level - - 30% 60% 90% 30% 60% 90% 30% 60% 90%

PAROR(%) 74.7 60.8 60.9 60.9 60.8 60.5 60.0 59.1 60.7 60.5 60.2

PAROD(%) 74.7 64.6 65.9 67.0 67.6 61.2 57.6 53.4 63.4 61.0 55.3

MAEP ARO (%) 0.3 3.8 5.1 6.1 6.8 0.8 2.4 5.8 2.7 0.6 4.9

rP ARO 0.94 0.91 0.91 0.90 0.90 0.94 0.95 0.96 0.93 0.93 0.94

Rtd (thousand) 87.6 117.5 117.5 117.5 117.5 117.5 117.5 117.5 117.5 117.5 117.5

Ryd (thousand) - 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6

Rbd (thousand) - 29.9 29.9 29.9 29.9 29.9 29.9 29.9 29.9 29.9 29.9

Dtd (thousand) 87.2 120.4 111.6 102.9 94.2 130.1 140.0 150.1 120.9 121.7 123.6

Dyd (thousand) - 88.7 89.3 89.8 90.5 88.9 89.1 89.6 89.1 89.7 91.2

Dbd (thousand) - 31.7 22.4 13.1 3.8 41.3 50.8 60.5 31.8 32.0 32.5

MAEDtd

0.56 1.37 1.41 1.67 2.08 1.62 2.07 2.64 1.51 1.85 2.26

MAEDyd

- 0.76 0.77 0.78 0.80 0.77 0.77 0.79 0.77 0.79 0.86

MAEDbd

- 0.87 0.95 1.32 1.85 1.13 1.62 2.21 1.04 1.46 1.99

PMAEDtd

(%) 9.3 16.9 17.3 20.6 25.5 20.0 25.5 32.5 18.6 22.8 27.8

PMAEDyd

(%) - 12.5 12.7 12.9 13.2 12.6 12.8 13.0 12.7 13.0 14.2

PMAEDbd

(%) - 42.0 45.8 63.9 89.3 54.8 78.2 106.9 50.2 70.7 96.5

Table 5.5.: Effect of Errors in the Unconstrained Buy-down on PARO

5.3. Computational Results

30%

50%

70%

90%

30% 50% 70% 90%

Real Demand

Est. Unc. Demand

Error 0.30 Error 0.60 Error 0.90

Figure 5.8.: Effect of an Unbiased Error in the Unconstrained Buy-down on

PARO

estimated unconstrained buy-down.

5.3.2. Analyzing the Effect of Further Scenarios

In this section we investigate the effect of further scenarios on the ROM with

dependent demand. We make use of the set of standard scenarios which have

already been applied to the ROM with independent demand. We assess the

effect of forecast errors, adjusted RM control and adjusted seasonality.

Effect of Forecast Errors on ROM

We start by analyzing the effect of an additional forecast error on the ROM. The

results for an error in the forecasted yieldable demand are listed in Table 5.6.

We also show the scatter plots for the biased under- and overestimation of the

forecasted yieldable demand in Figures 5.9 and 5.10. Please note that we left out

the 90% error scenario because the values are not within the 30% to 90% range.

Scatter plots with a range from 0% to 100% showing the 90% error scenarios can

be found in the appendix.

If an additional error in the forecast is applied, the PARORdecreases signifi-

cantly in comparison to the scenarios with an error in the estimated unconstrained

demand. The similarity measures of the PARO, however, show similar tenden-

cies. Except for a biased underestimation with 90% error they are above our

5. The Network-based ROM with Dependent Demand

Base Biased Biased

Case underestimation overestimation Unbiased error

Error level - 30% 60% 90% 30% 60% 90% 30% 60% 90%

PAROR(%) 60.8 56.9 48.6 16.8 56.0 44.9 31.0 59.4 56.7 52.8

PAROD(%) 64.6 62.7 56.0 47.5 53.4 39.4 25.7 59.0 53.1 46.9

MAEP ARO (%) 3.8 5.8 7.4 30.7 2.6 5.4 5.3 0.6 3.6 5.8

rP ARO 0.91 0.70 0.52 0.86 0.98 0.98 0.96 0.94 0.95 0.95

Rtd (thousand) 117.5 117.5 117.5 117.5 117.5 117.5 117.5 117.5 117.5 117.5

Ryd (thousand) 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6

Rbd (thousand) 29.9 29.9 29.9 29.9 29.9 29.9 29.9 29.9 29.9 29.9

Dtd (thousand) 120.4 111.7 104.7 93.2 134.6 149.7 166.5 123.0 125.7 129.6

Dyd (thousand) 88.7 79.6 70.6 54.7 102.8 117.8 134.5 91.0 93.3 96.8

Dbd (thousand) 31.7 32.1 34.1 38.6 31.9 31.9 32.0 32.1 32.4 32.8

Ftd (thousand) 120.4 107.5 100.1 90.0 147.7 174.7 201.6 127.5 136.8 147.8

Fyd (thousand) 88.7 73.2 61.0 45.8 115.8 142.8 169.7 94.4 101.7 110.6

Fbd (thousand) 31.7 34.3 39.1 44.2 31.8 31.9 32.0 33.1 35.1 37.2

MAEDtd

1.37 1.47 1.86 2.50 1.89 2.75 3.79 1.68 2.26 2.88

MAEDyd

0.76 0.91 1.41 2.47 1.29 2.20 3.31 1.09 1.72 2.41

MAEDbd

0.87 0.88 0.93 1.08 0.87 0.87 0.88 0.88 0.89 0.90

PMAEDtd

(%) 16.9 18.1 22.9 30.7 23.3 33.8 46.7 20.7 27.8 35.4

PMAEDyd

(%) 12.5 15.0 23.2 40.7 21.3 36.4 54.6 18.0 28.4 39.7

PMAEDbd

(%) 42.0 42.6 45.0 52.1 42.1 42.2 42.5 42.4 42.9 43.6

MAEFtd

2.27 2.41 2.75 3.17 3.01 4.41 6.08 2.41 2.73 3.21

MAEFyd

1.86 2.09 2.59 3.32 2.63 4.11 5.82 1.98 2.25 2.65

MAEFbd

0.99 1.03 1.19 1.40 0.99 0.99 0.99 1.01 1.05 1.12

PMAEFtd

(%) 28.0 29.7 33.8 39.0 37.2 54.5 75.1 29.7 33.7 39.6

PMAEFyd

(%) 30.8 34.5 42.7 54.7 43.6 68.0 96.3 32.7 37.2 43.8

PMAEFbd

(%) 47.7 50.2 57.5 67.9 47.8 47.8 47.8 48.9 51.1 54.4

Table 5.6.: Effect of Errors in the Forecasted Yieldable Demand on PARO

5.3. Computational Results

30%

50%

70%

90%

30% 50% 70% 90%

Real Demand

Est. Unc. Demand

Error 0.30 Error 0.60

Figure 5.9.: Effect of Biased Under-

estimation of Forecasted

Yieldable Demand on

PARO

30%

50%

70%

90%

30% 50% 70% 90%

Real Demand

Est. Unc. Demand

Error 0.30 Error 0.60

Figure 5.10.: Effect of Biased Over-

estimation of Forecasted

Yieldable Demand on

PARO

minimum level of similarity we defined before. We conclude that the ROM re-

mains robust even if we apply a strong forecast error. As expected the quality of

the RM control decreases with increasing error level. A biased underestimation of

the forecasted yieldable demand, for example, leads to a decrease in RM success

from 60.8% to 16.8%.

Applying an additional forecast error is comparable to the scenarios with an er-

ror in the estimated unconstrained buy-down. This is in particular due to the fact

that the unconstraining error on the buy-down is applied for all booking classes

no matter if they are open or not - consequently the results are approximately

the same. Details are presented in the appendix.

Effect of Adjusted RM Control and Seasonality on ROM

We also applied scenarios in which we investigated the effect of an adjusted RM

control and seasonality on the ROM. The results are principally identical to those

obtained from the same scenarios applied to the ROM with independent demand.

Detailed results are listed in Table 5.7. If the bid prices are adjusted, we observe

lower average values for PAROR. The similarity measures still indicate a high

similarity. MAEP ARO remains moderate with a maximum value of 5.8%. The

correlation coefficient rP ARO is also very high ranging from 0.69 to 0.96. The error

5. The Network-based ROM with Dependent Demand

Base Bid price Bid price Adjust Apply

Case decrease increase seasonality saw tooth curve

Adj. level - 25% 50% 25% 50% -50% +50% 130% - 70% 120% - 80%

PAROR(%) 60.8 55.3 33.2 55.0 49.9 61.0 60.5 55.3 58.4

PAROD(%) 64.6 60.9 39.1 58.2 52.4 65.1 64.3 59.8 62.6

MAEP ARO (%) 3.8 5.6 5.8 3.3 2.5 4.0 3.8 4.7 4.2

rP ARO 0.91 0.69 0.87 0.95 0.96 0.82 0.94 1.00 1.00

Rtd (thousand) 117.5 117.5 117.5 117.5 117.5 117.2 117.7 117.0 117.0

Ryd (thousand) 87.6 87.6 87.6 87.6 87.6 87.4 87.8 87.3 87.3

Rbd (thousand) 29.9 29.9 29.9 29.9 29.9 29.8 29.9 29.8 29.8

Dtd (thousand) 120.4 120.1 118.7 119.2 118.4 120.6 119.9 115.6 118.3

Dyd (thousand) 88.7 88.6 87.8 87.8 87.3 88.8 88.3 85.2 87.2

Dbd (thousand) 31.7 31.6 30.9 31.4 31.1 31.8 31.6 30.5 31.2

MAEDtd

1.37 1.34 1.30 1.34 1.32 1.33 1.41 1.49 1.40

MAEDyd

0.76 0.75 0.73 0.75 0.75 0.72 0.80 0.87 0.78

MAEDbd

0.87 0.86 0.85 0.86 0.85 0.86 0.87 0.87 0.86

PMAEDtd

(%) 16.9 16.5 16.0 16.5 16.3 16.4 17.4 18.6 17.4

PMAEDyd

(%) 12.5 12.4 12.1 12.4 12.4 12.0 13.2 14.1 12.9

PMAEDbd

(%) 42.0 41.6 41.1 41.5 41.3 41.8 42.2 43.3 42.4

Table 5.7.: Effect of Adjusted RM Control and Seasonality on PARO

5.3. Computational Results

measures are influenced on a minor level by the bid price adjustment. Compared

to the base case the average values for the MAE and PMAE remain constant.

The scatter plot of the restrictive RM control case is presented in Figure 5.11.

The scatter plot open of the RM control case can be found in the appendix.

If we adjust the amplitude of the seasonality we observe the expected effects.

If the amplitude of the seasonality is decreased the error measures decrease and

MAEP ARO decreases. The contrary result occurs, if we increase the magnitude

of the amplitude of seasonality. For both scenario the values of rP ARO remain

above 0.84. The saw tooth curve applied to the overall demand level leads to

slightly increased error measures. The MAEP ARO also increases a bit, whereas

the correlation coefficient rP ARO again goes up to 1. This is in particular due to

the wide range of realized PARO results along the 150 simulation runs. In Figure

5.12 the saw tooth curve scenarios are shown.

30%

50%

70%

90%

30% 50% 70% 90%

Real Demand

Est. Unc. Demand

Base Case Adj. +25% Adj. +50%

Figure 5.11.: Effect of Restrictive RM

Controls on PARO

30%

50%

70%

90%

30% 50% 70% 90%

Real Demand

Est. Unc. Demand

Base Case 130% to 70% 120% to 80%

Figure 5.12.: Effect of High Deviation

in Customer Demand on

PARO

5.3.3. Analyzing the Effect of Different Sell-up Rates

In Chapter 3 we defined an average sell-up rate of 30% to be our base case.

The real sell-up rate however might be higher or lower. Thus, we also assessed

scenarios with sell-up rates of 10% and 50%. The complete set of result tables and

figures can be found in the appendix. We focus on the unconstraining errors on

5. The Network-based ROM with Dependent Demand

the estimated yieldable demand to compare the behavior of the ROM according

to the sell-up rate in the flight network. Table 5.8 shows the detailed results. One

main result of our investigation is that no matter which sell-up rate is applied, the

ROM proves itself to be robust against the basic error scenarios. The values of

rP ARO again show high values, with a decrease for the biased underestimation of

the yieldable demand. In accordance to this observation the MAEP ARO obtains

values below our defined threshold except for the biased underestimation of the

estimated unconstrained yieldable demand. For the base case it we observe that

the lower the sell-up rate is, the better the values of MAEP ARO get. It increases

from 1.3% with 10% sell-up rate to 8.6% with 50% sell-up rate. However, this is

not a general trend. For the error scenarios the values of MAEP ARO are always

higher with a sell-up rate of 10%. We illustrate the unbiased unconstraining error

on the yieldable demand in Figures 5.13 and 5.14. Detailed results can be found

in the appendix.

30%

50%

70%

90%

30% 50% 70% 90%

Real Demand

Est. Unc. Demand

Error 0.30 Error 0.60 Error 0.90

Figure 5.13.: Sell-up Rate 10%: Effect

of an Unbiased Error in

Unconstrained Yieldable

Demand on PARO

30%

50%

70%

90%

30% 50% 70% 90%

Real Demand

Est. Unc. Demand

Error 0.30 Error 0.60 Error 0.90

Figure 5.14.: Sell-up Rate 50%: Effect

of an Unbiased Error in

Unconstrained Yieldable

Demand on PARO

5.4. Summary

In this chapter we introduced the network-based ROM with dependent demand in

detail. We made use of a common way of modeling dependent demand structures

5.4. Summary

Sell-up rate 10%

Base Biased Biased

Case underestimation overestimation Unbiased error

Error level - 30% 60% 90% 30% 60% 90% 30% 60% 90%

PAROR(%) 67.9 64.8 57.8 43.5 67.6 66.0 64.0 67.2 66.4 65.5

PAROD(%) 66.5 66.2 64.6 68.2 61.2 57.2 53.9 63.4 60.6 58.1

MAEP ARO (%) 1.3 1.5 6.8 24.7 6.4 8.8 10.1 3.9 5.8 7.3

rP ARO 0.89 0.75 0.57 0.39 0.96 0.97 0.97 0.90 0.91 0.89

Rtd (thousand) 95.6 95.6 95.6 95.6 95.6 95.6 95.6 95.6 95.6 95.6

Ryd (thousand) 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6

Rbd (thousand) 8.0 8.0 8.0 8.0 8.0 8.0 8.0 8.0 8.0 8.0

Dtd (thousand) 97.6 89.4 81.7 74.8 110.7 123.6 136.7 100.0 102.5 105.4

Dyd (thousand) 88.4 79.6 71.5 64.4 101.4 114.4 127.4 90.5 92.7 95.3

Dbd (thousand) 9.2 9.7 10.2 10.3 9.3 9.3 9.3 9.5 9.8 10.2

MAEDtd

1.00 1.10 1.41 1.77 1.50 2.27 3.12 1.30 1.85 2.46

MAEDyd

0.70 0.84 1.23 1.66 1.18 1.95 2.80 1.01 1.59 2.25

MAEDbd

0.47 0.49 0.51 0.52 0.47 0.47 0.47 0.48 0.49 0.51

PMAEDtd

(%) 15.1 16.7 21.3 26.7 22.7 34.4 47.2 19.7 27.9 37.2

PMAEDyd

(%) 11.5 13.9 20.3 27.3 19.4 32.2 46.3 16.6 26.2 37.1

PMAEDbd

(%) 85.1 88.8 92.7 93.7 85.1 84.9 84.8 87.0 89.5 92.5

Sell-up rate 50%

Base Biased Biased

Case underestimation overestimation Unbiased error

Error level - 30% 60% 90% 30% 60% 90% 30% 60% 90%

PAROR(%) 53.1 50.3 45.1 38.2 53.2 52.7 51.7 52.6 52.1 51.6

PAROD(%) 61.7 60.3 57.6 56.1 55.8 52.0 49.0 58.0 54.9 52.8

MAEP ARO (%) 8.6 10.1 12.5 17.9 2.7 0.7 2.7 5.4 2.7 1.3

rP ARO 0.92 0.86 0.74 0.64 0.95 0.96 0.96 0.92 0.92 0.87

Rtd (thousand) 153.0 153.0 153.0 153.0 153.0 153.0 153.0 153.0 153.0 153.0

Ryd (thousand) 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6

Rbd (thousand) 65.3 65.3 65.3 65.3 65.3 65.3 65.3 65.3 65.3 65.3

Dtd (thousand) 157.1 147.8 139.7 134.6 171.4 185.8 200.1 159.5 162.7 167.4

Dyd (thousand) 89.1 79.4 71.3 66.4 103.4 117.6 131.9 91.3 94.2 98.4

Dbd (thousand) 67.9 68.4 68.4 68.2 68.0 68.2 68.3 68.2 68.5 69.0

MAEDtd

1.77 1.89 2.13 2.30 2.25 3.03 3.91 2.07 2.60 3.14

MAEDyd

0.85 1.01 1.36 1.64 1.36 2.20 3.12 1.19 1.79 2.41

MAEDbd

1.26 1.27 1.27 1.26 1.26 1.26 1.26 1.26 1.27 1.28

PMAEDtd

(%) 16.8 17.8 20.1 21.7 21.3 28.6 37.0 19.6 24.5 29.7

PMAEDyd

(%) 14.1 16.6 22.4 27.0 22.5 36.2 51.5 19.6 29.5 39.8

PMAEDbd

(%) 27.8 28.0 28.0 27.8 27.8 27.9 27.9 27.9 28.1 28.3

Table 5.8.: Applying Sell-up Rates of 10% and 50% to Flight Network

5. The Network-based ROM with Dependent Demand

in reality to enhance the network-based ROM with independent demand. After

discussing main properties of the ROM, we analyzed the robustness of the ROM,

particularly against unconstraining errors in the yieldable demand and the buy-

down. In all scenarios applied, the values of the similarity measures showed results

above our minimum level of similarity defined in Section 3.2. Because we tested

all scenarios with error levels up to the expected worst case and even beyond, we

consider the network-based ROM with dependent demand robust against errors

in the input data for all error levels we would expect in real life. Moreover, the

effects of applying different sell-up rates in the flight network basically do not

change the results. The ROM proved itself robust for all scenarios applied.

However, in comparison to the independent demand case, the magnitude of

unconstraining errors is significantly higher with dependent demand. This applies

in particular to the estimated unconstrained buy-down, but also for the estimated

unconstrained yieldable demand. The higher average unconstraining error leads

to a decrease of the similarity measures.

6. Disaggregation of ROM

Measures to Single Legs

In this chapter, we build on the previously defined ROMs with independent and

dependent demand and assess their potential to be used in practical applications.

One important dimension is the disaggregation of the aggregated ROM measures

to subparts of the total flight network and in particular to single legs.

Although the overall RM control is network-based, there are several reasons

why a leg-based perspective in RM departments is important. One reason is that

RM control is historically leg-based. Revenue managers were historically respon-

sible to control several legs because network structures did not exist. This situ-

ation has significantly changed: Many airlines operate complex flight networks.

However, even today many RM departments all over the world are organized ac-

cording to legs or markets. Furthermore many RM controls are on a leg base, e.g.

overbooking or upgrading. These decisions are leg-based because they require di-

rect adjustments to the available capacity of a single leg. A very popular way of

controlling bookings in a network-based RM environment makes use of bid-price

models. These models define bid prices for using single resources, which are again

the legs in the flight network. Adjustments to a more open or restrictive RM con-

trol are usually done by increasing the bid price on a specific leg. Because many

RM decisions have to be taken on a leg level, there is huge interest in obtaining

performance measures for a single leg. One important example is the SLF, which

not necessarily indicates the RM performance, but primarily the utilization of

the resource. Another reason to disaggregate the aggregated ROM measure to

subparts of the network is simply having not only one aggregated measure for

the entire network, but several measures suitable to the organizational structure

of the revenue managers.

However, a disaggregation to a single leg usually incurs errors. The revenue

optimization is performed on a network level. As with all network problems,

a local optimum not necessarily corresponds to the network optimum. Thus,

performance measures on a leg level can always only be a supplemental indicator.

Simply maximizing the performance on one leg must not necessarily lead to the

desired results for the entire flight network.

6. Disaggregation of ROM Measures to Single Legs

In the first section of this chapter we investigate the relationship between leg-

based and network-based ROMs. In a second step we introduce several prorating

methods to distribute the fares of an itinerary to single legs. After formally

defining the leg-based ROM measures obtained in a network-based RM context

we assess their potential applicability on a leg level. We present computational

results on the robustness of the ROM disaggregated to leg level and investigate

further properties.

6.1. Relation between Network and Leg Level

In this section we discuss the basic relation between the network and leg level

and introduce some new definitions. In a network-based RM context passengers

book itineraries containing one or more resources, i.e. legs. As a result a booking

on an itinerary might lead to the use of multiple legs. This holds particularly

true for airlines using hub and spoke network structures. Often these airlines try

to increase the number of bookings on intercontinental flights departing from a

hub with passengers from other spoke locations using feeder flights. As already

described in Chapter 1 network airlines significantly use network-based controls

to handle these overlapping network traffic flows.

One typical differentiation for network airlines is to differentiate between con-

necting and local traffic. Local traffic describes bookings on itineraries that only

contain one leg, whereas connecting traffic describes itineraries in which the pas-

senger takes at least two flights. Airlines usually measure the degree of connecting

traffic within their flight network. This can be accomplished by simply calculat-

ing the share of bookings on connecting itineraries in comparison to the total

number of bookings. For a common network carrier the share of bookings on

connecting itineraries out of their total bookings is around 30% - 50%1.

The share of connecting traffic can also be determined on a leg level. To

quantify the degree of connecting traffic on a leg level, we start with some formal

definitions. We recall the fact that Lidenotes the set of all legs lthat are

contained by itinerary i. In contrast to this, Ildenotes all itineraries that use leg

land Iγ

ldenotes the set of all itineraries that contain more than one leg l(i.e.

|Li|>1) and which are considered connecting traffic. Given these notations we

define the number of cumulated bookings on a given leg las Bl. The number

of cumulated connecting traffic bookings for a given leg lis accordingly denoted

1Based on information discussed in personal communication with Dr. P¨olt - Lufthansa German

Airlines

6.2. Prorating Fares to Single Legs

with Bγ

l. The definitions are shown in Equations 6.1 and 6.2.

Bl=X

i∈IlX

j∈JiX

t∈T

bi,j,t ∀l∈L(6.1)

Bγ

l=X

i∈Iγ

j∈JiX

t∈T

bi,j,t ∀l∈L(6.2)

Using these definitions we introduce γlto measure the share of connecting

traffic on a leg l. A definition of γlcan be found in Equation 6.3.

γl=Bγ

∀l∈L(6.3)

γldescribes the ratio of all cumulated connecting traffic bookings Bγ

lto all cumu-

lated bookings Blon a leg l. The average share of connecting traffic passengers

for a network airline is usually between 45% and 65%. This rate is higher than

the rate based on itineraries because each connecting itinerary is counted multiple

times. For each leg in an itinerary we count a booking on the respective leg.

In case we do not have any connecting traffic in the flight network, i.e. |Li|=

1,∀i∈Iwe observe a special case. The estimation of the bookings for each

itinerary iderived from the LP-model and the FCFS simulation algorithm do not

contain any network effects. This leads to the observation that the solution for

the network-based ROM is equivalent to the solution of multiple independent leg-

based ROMs. The network-based ROMs defined in Chapters 4 and 5 presents a

complicated way to maximize revenues on one leg in comparison to the approaches

usually used for leg-based ROMs. But this special case also has some advantages

for our analyses. The ROM measures on a leg level do not contain any errors due

to network effects, which allows us to examine the influence of network effects on

the measures. In the remainder of this chapter we use this property to describe

and quantify the network effects on the ROM, if disaggregated to leg level.

6.2. Prorating Fares to Single Legs

For the calculation of the different revenue estimates (i.e. potential and no RM

revenue) on a leg base we utilize the bookings on the itineraries and combine

them to the respective legs. For all itineraries ithat only contain one leg the

assignment of the fare to the respective leg is simple. The fares of the leg l

pi,j,l,t correspond to the fares of the itinerary pi,j,t. If an itinerary icontains more

than one leg, this assignment does not work. The fare of the itinerary has to be

distributed to the respective legs. This procedure is called prorating and several

6. Disaggregation of ROM Measures to Single Legs

ways to perform this procedure are described in detail in this section. The basic

idea is to split the fares pi,j,t for each itinerary ito the flight legs it contains

using prorate factors ρi,j,l obtained for each leg lin the respective itinerary iand

booking class j. A formal definition is given in Equations 6.4 - 6.6.

pi,j,l,t =pi,j,t ∗ρi,j,l ∀i∈I, ∀j∈Ji,∀l∈Li,∀t∈T(6.4)

l∈Li

pi,j,l,t =pi,j,t ∀i∈I, ∀j∈Ji,∀t∈T(6.5)

0≤pi,j,l,t ≤pi,j,t ∀i∈I, ∀j∈Ji,∀l∈Li,∀t∈T(6.6)

The prorated fare pi,j,l,t for a leg lis defined in Equation 6.4. It is a share of

the total fare pi,j,t of an itinerary i. The share is defined by ρi,j,l. Equations 6.5

and 6.6 ensure that the fare of an itinerary iis fully distributed to the legs and

that no leg obtains a higher prorated fare than the fare of the itinerary.

A very important question is how to obtain the prorating factors ρi,j,l. In

literature some ways to determine ρi,j,l have been proposed. For example Talluri

and van Ryzin (2004b) and Williamson (1992) describe some common ways to

prorate fares to legs and comment on the applicability of the different methods.

Among others the following methods are commonly used in practice:

•Prorating by mileage

•Prorating by fares

•Prorating by bid prices

The prorating by mileage is basically static because it uses the length of the

flight legs as an input which is known beforehand and does not change over time.

The prorating based on fares is a semi-static approach. Fares might change over

the course of a booking period. In contrast to this the prorating by bid prices is

dynamic and basically changes with every single flight departure. We concentrate

on the prorating by mileage and the prorating by bid prices for the remainder of

this thesis in order to analyze effects both for dynamic and static approaches.

6.2.1. Mileage

One simple proposal to prorate the fares of an itinerary between its flight legs is

to use the distance of the flight legs. Equation 6.7 gives a formal definition:

ρi,j,l =υl

Pl0∈Liυl0

∀i∈I, ∀j∈Ji,∀l∈Li(6.7)

100

6.2. Prorating Fares to Single Legs

With υlwe denote the distance of a flight leg l. The prorate factor ρi,j,l is derived

as the share of the distance of a leg to the total flight distance of all legs of the

itinerary. Because the distance of a leg is always greater than zero, a division by

zero cannot occur. This method usually privileges legs with a long distance, in

particular if the other legs in the itinerary are very short. This is often the case

for combinations of continental feeder flights to intercontinental flights departing

from hubs.

6.2.2. Bid Prices

Another way of prorating fares is using bid prices. This method incorporates

information on the importance of a single leg into the distribution of fares. The

basic idea is to increase the fare ratio for those legs that have a high bid price

and thus are a very important and constrained resource in the flight network. In

Equation 6.8 we define the method formally.

ρi,j,l =max(πl,ml,j , pmin

l,ml,j )

Pl0∈Limax(πl0,ml0,j , pmin

l0,ml0,j )∀i∈I, ∀j∈Ji,(6.8)

∀l∈Li

First, the bid prices πl,m for each compartment mon leg lhave to be determined

(please recall that ml,j corresponds to the compartment on leg lwhich is related to

booking class j). They can e.g. be the shadow prices from the capacity constraint

of the LP model solved for the potential revenue estimation (see e.g. Equations

4.2 and 5.8). It is also possible to use the bid prices of the respective booking

period for each compartment. Vinod (2006), for example, proposes to store the

bid prices after the departure of a plane to use them for ex-post PM. Bid prices

have the disadvantage that in some cases πl,m might be zero and could potentially

lead to a division by zero. This usually happens in low demand situations. To

prevent this, we propose to use the maximum of the given bid price πl,m and the

minimum fare for the compartment pmin

l,m . For example the minimum fare for the

local itinerary containing that leg can be used. This procedure ensures that a

value greater than zero is used for each leg in the calculation of the prorating

factors. However, this procedure leads to the effect that the distribution of the

fares is stronger aligned to the distribution of local fares, because low bid prices

are increased to the minimum local fare pmin

l,m .

If we do not want to discard any bid prices with a value of zero, we propose a

101

6. Disaggregation of ROM Measures to Single Legs

slightly more aggressive distribution of the fares. It is defined in Equation 6.9

ρi,j,l =









πl,ml,j

Pl0∈Liπl0,ml0,j Pl0∈Liπl0,ml0,j >0

pmin

l,ml,j

Pl0∈Lipmin

l0,ml0,j

otherwise (6.9)

∀i∈I, ∀j∈Ji,∀l∈Li

In this case the bid prices of a compartment are used unless all bid prices of the

itinerary are zero. Only in this case we make use of the minimum revenue of the

compartments. This leads to a much more aggressive split of the fares, because

all legs with a zero bid price do not get any revenue share. This only changes if

all bid prices for the itinerary are zero.

6.3. Model Definition on a Leg Base

The ROM measures on a leg base are calculated with the results of the estimations

described in Chapters 4 and 5. Basic input is the number of estimated bookings

for the potential and no RM revenue x+

i,j,t and x−

i,j,t, but also the number of actual

bookings bi,j,t. As described in the previous sections the bookings on an itinerary

are applied to all related legs together with the prorated fares pi,j,l,t. Based on

this information the potential revenue Rev+

l, the actual revenue Revland the no

RM revenue Rev−

lare calculated as described in Equations 6.10, 6.11 and 6.12.

Rev+

l=X

i∈IlX

j∈JiX

t∈T

pi,j,l,t ∗x+

i,j,t ∀l∈L(6.10)

Revl=X

i∈IlX

j∈JiX

t∈T

pi,j,l,t ∗bi,j,t ∀l∈L(6.11)

Rev−

l=X

i∈IlX

j∈JiX

t∈T

pi,j,l,t ∗x−

i,j,t ∀l∈L(6.12)

The values for Rev+

l,Revland Rev−

lare used as the input for calculating the

ROM measures on a leg base. A formal definition is given in Equations 6.13, 6.14

and 6.15. The ROM measures on a leg base enable to compare the performance of

single legs. As mentioned earlier these measures incur errors because of network

effects, if there is connecting traffic in the flight network. In the remainder of

this thesis we analyze the extent of these effect.

102

6.3. Model Definition on a Leg Base

ROl=Rev+

l−Rev−

l∀l∈L(6.13)

AROl=Revl−Rev−

l∀l∈L(6.14)

PAROl=AROl

ROl

∀l∈L(6.15)

The definitions for ROl,AROland PAROlare basically the same as for the

network level, but in contrast to the network level, we obtain ROM measures for

each leg l. However, this also leads to a potential problem, that was not very

likely on the aggregated network level. For some legs it might be the case that

the RO is zero. This happens if the potential and the no RM revenue are the

same. Mainly this constellation occurs, if there is low demand on a flight and

the RM control is not able to increase revenues. To handle these situations we

extend the definition for the PARO in Equation 6.16.

PAROl=(AROl

ROlROl>0

1otherwise ∀l∈L(6.16)

We define the PARO to be 100%, if the RO is zero or below zero. Values below

zero may occur in particular in network-based environments, if for example the

simulated FCFS strategy accepts many bookings of itineraries using a leg, but

the LP, which estimated the potential revenue bookings, only allocates a few

bookings to this leg.

We have not yet discussed the case in which the PARO becomes greater than

100% or smaller than 0%, i.e. turns negative. This happens when the ARO gets

larger than the RO or the ARO turns negative. Especially the case of a negative

ARO is quite common. It usually describes situations in which the RM control

was very restrictive, most likely due to a demand forecast that was too high.

In the following we propose to use only PARO values between 0% and 100%.

Very poor RM control leading to PARO values below zero will be set to 0% and

PARO values above 100% indicating very good RM control will be set to 100%.

We base this proposal on the fact, that values out of this range sometimes take

arbitrary high or low values. We show some examples in the next section. In the

remainder of this thesis we call this adjustment capping. Equation 6.17 defines

capping formally.

PAROl=









PAROl0≤PAROl≤1

0PAROl<0

1PAROl>1

∀l∈L(6.17)

(6.18)

103

6. Disaggregation of ROM Measures to Single Legs

After we have obtained ROM measures for each leg and adjusted them in

the described manner, we suggest to filter out some flight departures for further

analysis. As it has been defined before, all cases in which the RO is less or equal

to zero the PARO has an arbitrary value. We defined the value to be 100%.

However, these 100% are not comparable to the 100% obtained in case of perfect

RM controls. It is difficult to make an interpretation of those cases possible in

which the RO for a flight departure is zero or below zero. We propose to filter out

these flight departures and examine them separately. We refer to this approach

as filtering in the following.

All the definitions made in this section are based on the consideration of one

flight departure. In RM practice it is common to observe performance measures

for a longer time period. Forecast errors, for example, are measured on a monthly

base. Thus, we propose to examine ROM measures that are based on multiple

flight departures. As the input for the ROM measures we therefore use average

values of the actual and the estimated potential and no RM revenue over a time

period of two weeks (14 days) or one month (30 days). With the use of averaging

we aspire to decrease the negative effect of unconstraining errors and network

effects on the leg-based measures. This concept is denoted with averaging.

6.4. Computational Results

In this section we present computational results of the disaggregation of the

ROM measures to single legs. We focus on the PARO and in particular on the

influences of network effects and errors in the estimated unconstrained demand

on the robustness of the PARO.

6.4.1. No-connecting-traffic Flight Network: Network Level

We start off with an investigation of a flight network with no connecting traffic.

The flight network consists of the same 728 flights as the realistic flight network.

As described earlier, there are no negative network effects on the validity of the

ROM measures with flight networks that do not contain any connecting traffic.

First, we analyze the aggregated PAROs over all flight legs as conducted in

Chapters 4 and 5. For this analysis and the rest of this section we focus on the

base case and the unbiased unconstraining error scenarios with error level of 30%,

60% and 90% for both independent and dependent demand. The scatter plots of

the base cases are presented in Figures 6.1 and 6.2. Please note that the scatter

plots range from 40% to 100% (instead of from 30% to 90%). Detailed results of

all scenarios are presented in Table 6.1. The structure of the table is similar to the

104

6.4. Computational Results

40%

60%

80%

100%

40% 60% 80% 100%

Real Demand

Est. Unc. Demand

Figure 6.1.: No-connecting-traffic

Flight Network with

Independent Demand

Aggregated to Network

Level

40%

60%

80%

100%

40% 60% 80% 100%

Real Demand

Est. Unc. Demand

Figure 6.2.: No-connecting-traffic

Flight Network with

Dependent Demand

Aggregated to Network

Level

one used in the previous chapters. The scenarios presented in the columns are the

base case and the three unbiased unconstraining error scenarios for independent

demand. The base case and the three unbiased unconstraining error scenarios

of the yieldable demand and the three unbiased unconstraining error scenarios

of the buy-down for dependent demand are also included. For both independent

and dependent demand the PAROs show good results. The values of MAEP ARO

are comparable to our realistic flight network scenarios. The scatter plot also

shows a high linear relation for both base cases. The correlation coefficient rP ARO

decreases slightly compared to the realistic flight network. However, with values

Independent demand Dependent demand

Base Error level Base Error level YD Error level BD

Case 30% 60% 90% Case 30% 60% 90% 30% 60% 90%

PAROR(%) 89.2 89.3 89.3 89.3 81.2 81.7 82.1 82.3 81.4 81.6 81.9

PAROD(%) 90.3 89.8 89.2 88.7 88.9 84.1 80.2 77.1 87.8 85.6 79.8

MAEP ARO (%) 1.2 0.8 0.9 1.2 7.7 2.4 2.1 5.2 6.4 4.0 2.2

rP ARO 0.88 0.88 0.88 0.84 0.62 0.73 0.76 0.83 0.68 0.75 0.80

Table 6.1.: Results for No-connecting-traffic Flight Network Aggregated over All

Flight Legs

105

6. Disaggregation of ROM Measures to Single Legs

over 0.62 for all error scenarios the aggregated PAROs are also considered robust

against unconstraining errors.

6.4.2. No-connecting-traffic Flight Network: Leg Level

Using the same flight network with no connecting traffic we now analyze the

ROM measures on a leg level. We again performed 150 simulation runs and

make use of five (averaged) flight departures. In our analysis we obtain PARO

values with real and estimated unconstrained demand for each simulation run

and flight leg and present them in Figure 6.3. The scatter plot shows some

extreme cases, in which the PARO values are significantly below 0% or above

100%. These extreme cases were already predicted in the previous section. They

are usually not observed on an aggregated network level, which is illustrated

in the scatter plots in Chapters 4 and 5. We do not measure the correlation

coefficient and the MAE in this case, because we expect the outliers to disturb

the similarity measures significantly. However, we analyze the number of cases

in which the PARO values are below 0% or above 100% and also the cases in

which the RO is equal to or below zero. The results are shown in Table 6.2.

In the first three result rows we list the share of cases in which the condition

Independent demand Dependent demand

Base Error level Base Error level YD Error level BD

Case 30% 60% 90% Case 30% 60% 90% 30% 60% 90%

Cases for est. unc. dmd, in which condition holds (%)

PAROl<0 (%) 0.7 1.0 1.2 1.3 1.0 2.0 3.5 3.8 1.1 1.5 2.3

PAROl>1 (%) 0.0 0.0 0.0 0.0 1.0 1.1 3.2 4.4 0.6 0.3 0.1

ROl≤0 (%) 45.4 45.1 45.3 45.9 44.4 43.2 40.9 39.6 44.3 44.1 43.8

Cases, in which fulfillment of condition matches between real and est. unc. dmd. (%)

PAROl<0 (%) 99.3 99.1 98.9 98.3 98.1 97.2 95.9 96.5 98.2 97.8 97.0

PAROl>1 (%) 100.0 100.0 100.0 100.0 99.0 98.9 96.8 95.6 99.4 99.7 99.9

ROl≤0 (%) 99.1 99.1 98.6 97.7 56.1 57.3 59.5 60.8 56.2 56.4 56.7

Table 6.2.: Analyzing Special Cases for No-connecting-traffic Flight Network

denoted in the first column holds for the ROM measures that were obtained

with estimated unconstrained demand. In the last three rows we analyze the

share of cases in which the fulfillment of the given condition matches between

real and estimated unconstrained demand. It can be observed that there is only

a small percentage of cases in which the PARO actually becomes less than 0%

or greater than 100%. The share of matches of the fulfillment of the condition

between the results obtained with real demand are very high with above 95% for

106

6.4. Computational Results

both independent and dependent demand. Using the estimated unconstrained

demand the RO is estimated to be smaller or equal to zero in about 40% to 45%

of the cases for both independent and dependent demand. However, the number

of matches between real and estimated unconstrained demand is very high for

independent demand at around 97% and significantly smaller at around 55% for

dependent demand. We conclude that if we cap the PAROs calculated with real

-10000%

-5000%

5000%

10000%

15000%

-6000% -4000% -2000% 0% 2000%

Real Demand

Est. Unc. Demand

Figure 6.3.: No-connecting-traffic

Flight Network with

Dependent Demand - No

Capping and Filtering

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Real Demand

Est. Unc. Demand

Figure 6.4.: No-connecting-traffic

Flight Network with

Dependent Demand

demand to 0% or to 100% the PAROs calculated with estimated unconstrained

demand would be capped the same way. The same is true for filtering out the

flight departures with the independent demand scenario. For dependent demand

the congruence of the filter is smaller. This means that filtering out a flight

departure due to the RO obtained with the estimated unconstrained demand is

not necessary, because the RO for real demand is greater than zero. Thus, the

results of the similarity measures might incur an error due to filtering. This effect

has to be kept in mind while analyzing scenarios with a filter being applied.

Detailed results, in which we capped and filtered the PAROs are presented

in Table 6.3. The first results describe the scenario in which we capped the

PAROs. For independent demand this already leads to good values for rP ARO.

For dependent demand there is still no linear relation observable. If we also apply

the filter mentioned above the values for rP ARO are high for both independent and

dependent demand. rP ARO is above 0.56 even for a very high unconstraining error.

The MAEP ARO also indicates strong similarity with better values for independent

107

6. Disaggregation of ROM Measures to Single Legs

Independent demand Dependent demand

Base Error level Base Error level YD Error level BD

Case 30% 60% 90% Case 30% 60% 90% 30% 60% 90%

Capping

MAEP ARO (%) 2.9 3.4 4.5 6.8 48.6 48.3 48.1 48.1 48.1 47.4 47.0

rP ARO 0.83 0.82 0.76 0.62 -0.15 -0.15 -0.10 -0.10 -0.16 -0.18 -0.20

Filtering

flight dep. incl. (%) 54.6 54.9 54.7 54.1 55.6 56.8 59.1 60.4 55.7 55.9 56.2

MAEP ARO (%) 5.1 6.1 8.0 11.6 8.5 10.0 13.0 14.8 7.8 6.8 6.6

rP ARO 0.81 0.77 0.71 0.56 0.85 0.75 0.69 0.64 0.85 0.87 0.89

Averaging over two weeks

flight dep. incl. (%) 67.3 67.7 67.6 67.7 67.4 68.9 71.6 73.6 67.4 67.7 68.2

MAEP ARO (%) 3.4 4.0 4.9 5.9 11.9 7.3 6.5 6.9 10.6 5.6 5.7

rP ARO 0.89 0.85 0.82 0.80 0.85 0.93 0.93 0.93 0.87 0.97 0.96

Averaging over one month

flight dep. incl. (%) 69.9 70.2 70.2 70.1 70.2 71.5 74.2 76.3 70.1 70.3 70.8

MAEP ARO (%) 3.1 3.5 4.2 5.0 12.9 6.8 5.7 5.9 11.6 5.5 5.6

rP ARO 0.91 0.88 0.86 0.84 0.85 0.96 0.96 0.95 0.87 0.97 0.97

Table 6.3.: Similarity Measures for the No-connecting-traffic Flight Network

demand. This result can also be observed in the scatter plot depicted in Figure

6.4. In our analysis we also observe that the decrease in similarity observed on a

network level for dependent demand is also strongly reflected on a leg level. The

results for independent demand are significantly better.

In practical applications it is common to use average values of certain figures

over a given time period. In the following we investigate two weeks and one month

as common time periods for averaging. In Figures 6.5 and 6.6 we present the

corresponding results. The detailed results can also be found in Table 6.3. The

results show that the correlation coefficient rP ARO significantly increases for both

independent and dependent demand and also MAEP ARO decreases. Averaging

over two weeks leads to values of rP ARO above 0.80 and they further increase to

0.84 if we use averaging over one month. The values of MAEP ARO on average

decrease over all error scenarios.

We conclude that the similarity measures obtained for the main error scenarios

indicate that the PAROs can be used to track performance on a leg level if no

connecting traffic is applied in the flight network. Capping the PAROs to 0%

and 100% is a very useful concept to allow the application of the ROM. Filtering

out flight departures is another powerful method to increase the robustness of

the ROM. We also conclude that using an average over a certain time period

further increases robustness. We propose to apply a monthly averaging, because

108

6.4. Computational Results

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Real Demand

Est. Unc. Demand

Figure 6.5.: No-connecting-traffic

Flight Network with

Dependent Demand -

Averaged over 2 Weeks

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Real Demand

Est. Unc. Demand

Figure 6.6.: No-connecting-traffic

Flight Network with

Dependent Demand

- Averaged over One

Month

it shows good results and this time period is already known from tracking forecast

errors in operational RMS. However, even if we do not have any errors induced

by network effects, the use of PAROs on a leg level needs specific treatment.

In Figures 6.5 and 6.6 we observe some outliers, in particular for lower PARO

values. If the ROM is intended to be applied in a real life system, it might be

worth analyzing which common characteristic these outliers share to filter them

out later on.

6.4.3. Realistic Flight Network: Leg Level

The assumption of a flight network without connecting traffic is not applicable

for a network carrier in reality. In this section we analyze the potential to disag-

gregate the ROM measures to leg level if connecting traffic is applied. We base

our analyses on the realistic flight network that we already used in the previous

chapters. The fares are prorated by mileage to the legs per default. Although it

is likely that some model-related errors occur while disaggregating the network

results to leg level, we still consider the values obtained with the real demand

to be the best estimates for the correct values. Thus, we continue to compare

the ROM measures calculated with real demand to those measures that were

calculated with the estimated unconstrained demand. We begin by analyzing the

number of special cases. The results are presented in Table 6.4. The analysis of

109

6. Disaggregation of ROM Measures to Single Legs

Independent demand Dependent demand

Base Error level Base Error level YD Error level BD

Case 30% 60% 90% Case 30% 60% 90% 30% 60% 90%

Cases for est. unc. Dmd, in which condition holds (%)

PAROl<0 (%) 5.7 5.3 4.8 5.9 9.1 10.8 14.1 14.8 9.5 10.1 14.1

PAROl>1 (%) 3.7 3.9 4.2 4.5 2.1 2.0 3.3 3.6 2.4 2.0 1.0

ROl≤0 (%) 39.9 39.9 39.9 40.5 37.8 37.5 35.4 34.2 37.7 38.0 37.5

Cases, in which fulfillment of condition matches between real and est. unc. dmd. (%)

PAROl<0 (%) 95.6 95.5 94.9 94.0 91.3 89.9 87.6 85.6 91.0 90.6 88.0

PAROl>1 (%) 97.5 97.0 95.7 95.0 97.4 97.4 96.1 95.9 97.2 97.6 98.5

ROl≤0 (%) 96.3 95.6 95.1 93.8 64.0 64.2 66.1 67.4 64.3 63.8 64.1

Table 6.4.: Analyzing Special Cases for Realistic Flight Network

special cases of the realistic flight network shows an increased number of PAROs

to be capped. Nevertheless, the share of matches between real and estimated

unconstrained demand remains high. The number of cases in which the RO is

smaller than or equal to zero is very similar to the no-connecting-traffic flight

network. However, for dependent demand the number of matches between real

and estimated unconstrained demand increases to approximately 65%.

If we cap the PAROs, filter out the flight departures with RO being smaller

than or equal to zero and take the average over one month we obtain the results

presented in Table 6.5. If we only cap the PAROs, the results of the similarity

Independent demand Dependent demand

Base Error level Base Error level YD Error level BD

Case 30% 60% 90% Case 30% 60% 90% 30% 60% 90%

Capping

flight dep. incl. (%) 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0

MAEP ARO (%) 7.2 7.8 9.4 11.3 43.8 44.1 44.0 44.6 43.2 43.0 43.1

rP ARO 0.76 0.74 0.68 0.62 -0.01 -0.02 -0.01 -0.01 0.00 -0.02 -0.06

Filtering

flight dep. incl. (%) 60.1 60.1 60.1 59.5 62.2 62.5 64.6 65.8 62.3 62.0 62.5

MAEP ARO (%) 10.4 11.1 13.6 16.4 13.5 14.4 17.2 19.5 12.9 12.1 12.6

rP ARO 0.74 0.72 0.64 0.57 0.76 0.75 0.67 0.61 0.78 0.79 0.80

Averaging over one month

flight dep. incl. (%) 70.0 70.0 70.2 70.2 71.8 72.5 74.2 75.5 71.6 72.0 72.5

MAEP ARO (%) 4.3 4.2 5.2 6.0 11.7 8.8 9.9 11.2 11.1 7.7 9.3

rP ARO 0.91 0.92 0.89 0.87 0.83 0.90 0.87 0.84 0.84 0.92 0.92

Table 6.5.: Similarity Measures for the Realistic Flight Network

110

6.4. Computational Results

measures are similar to the no-connecting-traffic flight network. With indepen-

dent demand the values of rP ARO indicate a high similarity. With dependent de-

mand we again measure no correlation. The MAEP ARO with dependent demand

for any given error scenario is also very high with around 45%. This changes if we

apply the filter to the RO, too. The scatter plots in Figures 6.7 and 6.8 confirm

the results shown in the result table. The MAEP ARO decreases significantly to

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Real Demand

Est. Unc. Demand

Figure 6.7.: Realistic Flight Net-

work with Independent

Demand

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Real Demand

Est. Unc. Demand

Figure 6.8.: Realistic Flight Network

with Dependent Demand

values between 10% and 20% and the correlation coefficient is above 0.61 for all

error scenarios. However, in particular with dependent demand the scatter plot

reveals some outliers that disturb the robustness. The detailed results and the

scatter plots lead to the conclusion that the similarity and thus the robustness

for independent demand is significantly higher than for dependent demand.

The robustness of the ROM can further be increased by using an average over

one month. The scatter plots of the base cases both for independent and de-

pendent demand support these findings. They are presented in Figures 6.9 and

6.10. The correlation coefficients rP ARO are above 0.84 for all scenarios and the

values for MAEP ARO also show a significant decrease. In addition, the number

of flight departures included in the evaluation increased on average by 10% to

values of around 70%. Moreover it can be observed that the application of a

realistic flight network - and that means including network effects into the ROM

calculation - leads to worse results compared to the no-connecting-traffic flight

network scenario. However, in particular the correlation coefficient rP ARO indi-

cates a high similarity and shows comparable results. The values of MAEP ARO

111

6. Disaggregation of ROM Measures to Single Legs

are much higher with the realistic flight network.

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Real Demand

Est. Unc. Demand

Figure 6.9.: Realistic Flight Network

with Independent De-

mand - Averaged over

One Month

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Real Demand

Est. Unc. Demand

Figure 6.10.: Realistic Flight Network

with Dependent Demand

- Averaged over One

Month

In Figure 6.11 we present a scatter plot of an unbiased error in the yieldable

demand with an error level of 30%. As we already observed in Chapter 5 the

unbiased unconstraining error in the yieldable demand increases robustness in

our simulation environment. This also holds true for the results on a leg level.

The correlation coefficient rP ARO is higher for error levels 30% and 60% than for

the base case and also the values of MAEP ARO are smaller. These findings are

supported by the scatter plot in which we see less outliers and a better linear

relation.

Prorating of fares was not necessary with a no-connecting-traffic flight network.

All itineraries contain only one flight leg and the fares of the itineraries can be

applied to the legs without splitting them. In the analyses performed so far,

we prorated the fares based on the distance of the contained flight legs. In

the following we investigate the effect of using prorating methods based on bid

prices. We used the moderate bid-price prorating and the aggressive bid-price

prorating introduced in Section 6.2. The results are listed in Table 6.6 and shown

in Figures 6.12 and 6.13. They show that the robustness of the ROM increases if

the moderate bid-price method is applied. For all error scenarios the correlation

coefficient rP ARO increased. The minimum value of rP ARO is 0.87, which is already

a very high value. The values of MAEP ARO also decrease in all assessed cases.

The scatter plot shows a decreased number of outliers and a stronger linear

112

6.4. Computational Results

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Real Demand

Est. Unc. Demand

Figure 6.11.: Realistic Flight Network with Dependent Demand with 30% Un-

biased Error on Unconstrained Yieldable Demand - Averaged over

One Month

relation. The increase in similarity especially relates to the dependent demand

scenarios. In contrast, the aggressive bid-price prorating method does not lead to

better results. For some scenarios an improvement in similarity can be observed

and in some cases the similarity decreases compared to the prorating by mileage.

We conclude that it might be worth to further explore the moderate bid-price

prorating method. One main challenge in practice will be to convince the revenue

managers to accept this way of prorating, because usually a static mileage- or

semi static fare-based prorating approach is applied.

Another main differentiation is usually taken between continental and intercon-

tinental flights. We also analyze the robustness of the ROM for both continental

and intercontinental flights in our flight network. We use a miles-based prorat-

ing method and apply capping, filtering and averaging. The detailed results are

presented in Table 6.7 and Figures 6.14 and 6.15. The results for continental

flights are comparable to the results of all flight legs. The values of rP ARO and

MAEP ARO do not differ much. The scatter plot supports these findings. It looks

similar to the scatter plot obtained for the entire flight network. In contrast to

this, we observe very good results for intercontinental flights, in particular with

independent demand. Values of rP ARO and MAEP ARO are very good. The very

high linear relation can be observed in the respective scatter plot. Addition-

ally, the number of flight departures included is almost 100% for intercontinental

flights. From the given results we conclude that intercontinental flights are more

robust against errors in the estimated unconstrained demand than continental

113

6. Disaggregation of ROM Measures to Single Legs

Independent demand Dependent demand

Base Error level Base Error level YD Error level BD

Case 30% 60% 90% Case 30% 60% 90% 30% 60% 90%

Moderate bid-price prorating

flight dep. incl. (%) 70.6 70.6 70.9 70.9 72.3 73.1 74.8 76.0 72.2 72.6 73.0

MAEP ARO (%) 4.0 4.0 4.8 5.5 10.8 7.5 8.4 9.5 10.1 6.4 7.8

rP ARO 0.91 0.92 0.90 0.89 0.87 0.93 0.90 0.87 0.88 0.95 0.94

Aggressive bid-price prorating

flight dep. incl. (%) 56.3 56.5 57.1 57.7 60.6 61.7 69.2 71.4 60.1 60.5 61.1

MAEP ARO (%) 4.4 4.7 5.1 5.8 13.2 7.9 7.3 7.8 12.1 6.8 7.8

rP ARO 0.88 0.85 0.85 0.83 0.79 0.92 0.93 0.91 0.81 0.94 0.94

Table 6.6.: Similarity Measures for the Realistic Flight Network - Focus on Bid-

price Prorating

Independent demand Dependent demand

Base Error level Base Error level YD Error level BD

Case 30% 60% 90% Case 30% 60% 90% 30% 60% 90%

Continental flights

flight dep. incl. (%) 57.2 57.2 57.4 57.4 59.0 59.8 61.4 62.7 58.9 59.3 59.7

MAEP ARO (%) 4.9 4.8 5.9 6.8 13.1 9.4 10.3 11.5 12.4 8.4 9.9

rP ARO 0.91 0.92 0.88 0.86 0.83 0.90 0.87 0.84 0.84 0.92 0.92

Intercontinental flights

flight dep. incl. (%) 97.9 97.9 97.9 97.9 97.9 97.9 97.9 97.9 97.9 97.9 97.7

MAEP ARO (%) 1.4 1.5 1.9 2.6 5.3 5.8 8.0 9.3 5.1 4.6 6.3

rP ARO 0.98 0.98 0.97 0.95 0.94 0.90 0.82 0.77 0.94 0.94 0.94

Table 6.7.: Similarity Measures for the Realistic Flight Network - Separating Con-

tinental and Intercontinental Flights

114

6.5. Summary

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Real Demand

Est. Unc. Demand

Figure 6.12.: Realistic Flight Network

with Dependent Demand

- Bid Price Moderate

and Averaged over One

Month

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Real Demand

Est. Unc. Demand

Figure 6.13.: Realistic Flight Network

with Dependent Demand

- Bid Price Aggressive

and Averaged over One

Month

flights. We observe less outliers that have to be examined further. The results

for continental flights also indicate sufficient similarity. The scatter plot indi-

cates that the outliers for the entire flight network can mainly be found within

continental flights.

6.5. Summary

As a general conclusion of the investigation of the ROM measures in a realistic

flight network with connecting traffic we consider the ROM on a leg level robust

against errors in the estimated unconstrained demand for the base cases and the

most important unconstraining error scenarios. We consider both the ROM with

independent and dependent demand to be robust. We observed a decrease in

the quality of the ROM measures due to network effects, however the decrease

is minor compared to the effect of the transition from independent to dependent

demand. The capping of the PAROs to values between 0% and 100%, the filtering

out of flight departures with a RO of less or equal zero and the averaging of

the flight departures over one month are important, in particular with regard to

dependent demand. The robustness of the ROM in our scenarios was significantly

higher with independent demand. For dependent demand the errors already

observed on an aggregated network level lead to subsequent errors on a leg level.

115

6. Disaggregation of ROM Measures to Single Legs

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Real Demand

Est. Unc. Demand

Figure 6.14.: Realistic Flight Net-

work with Dependent

Demand - Continental

Flights and Averaged

over One Month

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Real Demand

Est. Unc. Demand

Figure 6.15.: Realistic Flight Network

with Dependent Demand

- Intercontinental Flights

and Averaged over One

Month

The analysis of the outliers will be a main task in making the ROM applicable

in a real life situation. As the analysis between intercontinental and continental

flights proved, most of the outliers might be found within the continental flights.

It might also be worth looking at other common characteristics of these outliers

to filter out these flight departures. This could lead to less flight departures that

are part of the performance evaluation done by the ROM. However, it would lead

to an increased acceptance by the revenue managers, due to its higher robustness

and validity.

116

7. Disaggregation of ROM

Measures to Single Components

So far we have neither considered no-shows and cancelations nor overbooking

and upgrading in the ROM. In practice overbooking and upgrading are com-

monly used to further improve revenues. For practical applicability of the ROM

it is crucial that it reflects these main approaches to increase revenues with RM.

Therefore we first extend the network-based ROM in general to consider over-

booking and upgrading and introduce the necessary enhancements in this section.

In the second part we introduce ways to isolate the contribution of these RM com-

ponents from the overall success and provide computational results.

7.1. Extending the Network-based ROM to

Overbooking and Upgrading

As described in Chapter 1, airlines often face the challenge that passengers with

a valid ticket do not show up at the time of flight departure. The reasons are

flexible tickets or simply delays in connecting flights. With qi,j we denote the

show-up rate of passengers booked on itinerary iin booking class j, i.e. the share

of passengers showing up at the departure of a plane. The show-up rate is usually

determined by the ratio of cumulated bookings for an itinerary after no-shows

and the total number of cumulated bookings. Usually the show-up rates can

be derived from the operational systems of an airline. In the remainder we use

the terms no-shows and show-ups simultaneously. Cancelations might already

occur during the booking period. A cancelation occurs, if a passenger cancels

and returns his ticket before the departure of the plane. After a cancelation the

seat is available for sale again. In the following we denote the cancelation rate

ki,j,t to describe the share of bookings on itinerary iand booking class jthat are

canceled in this time period. ki,j,t applies to all bookings that have been booked

until the end of time period t.

In order to incorporate overbooking and upgrading into the ROM the calcu-

lations of the potential, the actual and the no RM revenue have to be adjusted.

117

7. Disaggregation of ROM Measures to Single Components

The necessary adjustments are described in the following subsections.

7.1.1. Potential Revenue with Upgrading

The first enhancement we apply is the consideration of upgrading in the potential

revenue estimation. If upgrading is applied in the RM control, empty seats in

a higher valued compartment are made available for passengers who are willing

to book into a lower valued compartment. This is usually done by virtually

increasing the capacity of the lower valued compartment if excess demand for

this compartment is forecasted and for the higher valued compartment on that

flight the forecast indicates empty seats at the end of the booking period. The

LP presented in Chapter 5 can simply be enhanced to allow upgrading. We show

the model again in Equations 7.1 to 7.4.

Max X

i∈IX

j∈JiX

t∈T

pi,j,t ∗x+

i,j,t (7.1)

i∈IlX

j∈Ji,l,m X

t∈T

i,j,t ≤capl,m ∀l∈L, ∀m∈Ml,(7.2)

∀t∈T

0≤x+

i,j,t ≤yi,j,t ∗dtd

i,j,t −X

j0∈Ji,j

yi,j0,t ∗dbd

i,j,j0,t ∀i∈I, ∀j∈Ji,(7.3)

∀t∈T

yi,j0,t ≤yi,j,t ∀i∈I, ∀j∈Ji,(7.4)

∀j0∈Ji,j,∀t∈T

For example, a plane with an economy compartment with 100 seats and a

business compartment with 30 seats could have the virtual extended capacity of

130 seats for the economy compartment. If we exchange the capacity constraint

as presented in Equation 7.5 we allow the LP to increase the potential number

of bookings into a given compartment by the capacity of all higher valued com-

partments. In our example the potential number of economy bookings would

be increased to 130. The capacity of the highest valued compartment - in this

case the business compartment - would not be changed, because upgrading is not

possible for this compartment. This leads to the necessity to introduce another

capacity constraint shown in Equation 7.6 to ensure that the LP does not dis-

tribute more bookings on a leg than seats are available. In the given constraint

we introduced the set Ji,l which contains all booking classes that are booked on

118

7.1. Extending the Network-based ROM to Overbooking and Upgrading

leg lin itinerary i.

i∈IlX

j∈Ji,l,m X

t∈T

x+,U

i,j,t ≤X

m0∈Ml:m0≤m

capl,m0∀l∈L, ∀m∈Ml,∀t∈T(7.5)

i∈IlX

j∈Ji,l X

t∈T

x+,U

i,j,t ≤capl∀l∈L, ∀t∈T(7.6)

With the use of x+,U

i,j,t the potential revenue with consideration of upgrading

Rev+,U can be calculated using the basic formula 4.4 known from Chapter 4.

7.1.2. Potential Revenue with Overbooking

The other important enhancement is calculating the potential revenue with over-

booking: We assume that we know the no-show and cancelation rates and are

able to apply perfect overbooking controls and to compensate for cancelations in

the LP model. In order to integrate this assumption the estimated unconstrained

demand has to be adjusted to the show-up rates and the cancelation rates.

In the independent demand case, the adjustment is simple as illustrated in

Equation 7.7. We apply the cancelation rate to the demand for each time period

in the future t0starting with the current time period t. This set of time periods

is denoted with Tt. Afterwards we apply the show-up rate to the demand that

was already adjusted to the expected cancelations. We denote the resulting value

with ˆ

di,j,t.

di,j,t =di,j,t ∗Y

t0∈Tt

(1 −ki,j,t0)∗qi,j ∀i∈I, ∀j∈Ji,∀t∈T(7.7)

If we assume dependent demand structures, the adjustment is more compli-

cated, because we have to adjust the estimates for total demand and buy-down

simultaneously to ensure consistency. The necessary steps are presented in Al-

gorithm 7.1. The basic idea is again to apply the cancelation rates ki,j,t and

show-up rate qi,j for a booking class jto the demand of this booking class.

However, the buy-down dbd

i,j,j0,t is by definition part of the total demand dtd

i,j,t in

booking class j, but also of total demand dtd

i,j0,t in booking class j0. Thus, it is not

possible to simply apply the cancelation and show-up rate of a booking class to

both total demand and buy-down. It is solely possible to apply the cancelation

and show-up rates to the total demand of the lowest booking class because there

is no buy-down into lower booking classes (Line 5). To ensure consistency in all

other booking classes, the buy-down dbd

i,j,j0,t from a booking class jinto another

booking class j0is changed using the average resulting cancelation and show-up

rate of the total demand of booking class j0(Line 10). Afterwards the cancelation

119

7. Disaggregation of ROM Measures to Single Components

Algorithm 7.1: Estimation of Dependent Demand after No-shows and

Cancelations

1foreach t∈Tdo

2foreach i∈Ido

3for j= 1 to |Ji|do

4if j=|Ji|then

5ˆ

dtd

i,j,t =dtd

i,j,t ∗Qt0∈Tt(1 −ki,j,t0)∗qi,j

6foreach j0∈Ji,j do

7ˆ

dbd

i,j,j0,t = 0

8else

9foreach j0∈Ji,j do

10 ˆ

dbd

i,j,j0,t =dbd

i,j,j0,t ∗ˆ

dtd

i,(j+1),t

dtd

i,(j+1),t

11 ˆ

dtd

i,j,t =

(dtd

i,j,t −Pj0∈Ji,j dbd

i,j,j0,t)∗Qt0∈Tt(1−ki,j,t0)∗qi,j +Pj0∈Ji,j

dbd

i,j,j0,t

12 ˆ

dyd

i,j,t =ˆ

dtd

i,j,t −Pj0∈Ji,j

dbd

i,j,j0,t

and show-up rates are applied to the yieldable demand, which is the difference

between total demand and all buy-downs into lower booking classes (Line 11) to

determine the adjusted total demand ˆ

dtd

i,j,t. The application of the cancelation

and no-show rates is performed in the same way as introduced in Equation 7.7.

Lastly, the adjusted yieldable demand ˆ

dyd

i,j,t is derived based on the adjusted total

demand and buy-down.

In a second step we modify Constraint 7.3 in the LP to determine the potential

revenue under consideration of no-shows and cancelations. Constraint 7.8 shows

the adjustment. The estimates for the unconstrained demand dtd

i,j,t and dbd

i,j,j0,t are

replaced by the estimates considering no-shows and cancelations ˆ

dtd

i,j,t and ˆ

dbd

i,j,j0,t.

The Equation 4.3 is modified to use ˆ

di,j,t instead of di,j,t in the independent

demand case in a similar way.

0≤x+,O

i,j,t ≤yi,j,t ∗ˆ

dtd

i,j,t −X

j0∈Ji,j

yi,j0,t ∗ˆ

dbd

i,j,j0,t ∀i∈I, ∀j∈Ji,(7.8)

∀t∈T

The y-constraint defined in Equation 7.4 ensuring the feasible actions remains

unchanged. The estimated bookings for the potential revenue with overbooking

120

7.1. Extending the Network-based ROM to Overbooking and Upgrading

are denoted with x+,O

i,j,t in the remainder of the thesis. The potential revenue with

overbooking Rev+,O is calculated like the other potential revenues.

The described enhancement can be applied no matter if upgrading is also con-

sidered or not. However, in practical applications usually both overbooking and

upgrading play an important role and are applied in the RM controls. In this

case we extend the potential revenue with overbooking to also allow upgrading

and denote the estimated bookings with x+,O,U

i,j,t . The potential revenue with over-

booking and upgrading Rev+,O,U is defined according to the other definitions.

7.1.3. Actual Revenue after No-shows and Cancelations

The actual revenue as one important input of the ROM has to also be adjusted

for no-shows and cancelations. We denote the bookings after consideration of no-

shows and cancelations with ˆ

bi,j,t and define them formally in Equation 7.9. This

formula draws back on the same way of deducting cancelations as in Equation

7.7.

bi,j,t =bi,j,t ∗Y

t0∈Tt

(1 −ki,j,t0)∗qi,j ∀i∈I, ∀j∈Ji,∀t∈T(7.9)

If overbooking is applied it is possible that even after consideration of no-shows

there are more passengers for a flight than seats available in the plane. Before

explaining how airlines usually deal with this situation, we define the cumulated

bookings after no-shows and cancelations ˆ

Bl,m, the excess bookings Bex

l,m and the

free capacity of a compartment min Equations 7.10, 7.11 and 7.12.

Bl,m =X

i∈IlX

j∈Ji,l,m X

t∈T

bi,j,t ∀l∈L, ∀m∈Ml(7.10)

Bex

l,m =max(ˆ

Bl,m −capl,m,0) ∀l∈L, ∀m∈Ml(7.11)

capf

l,m =max(capl,m −ˆ

Bl,m,0) ∀l∈L, ∀m∈Ml(7.12)

For all compartments with excess bookings we decide whether these oversold

seats lead to upgrading, downgrading or denied boarding. If there are excess

bookings on a compartment an airline’s priority is to prevent denied boardings.

This can be done via upgrading or downgrading. An airline would of course prefer

upgrading instead of downgrading because a customer would always be willing

to take a seat in a higher valued compartment but would expect a compensatory

payment if placed into a lower valued compartment. An airline would start with

the highest valued compartment trying to upgrade excess passengers. If this is

121

7. Disaggregation of ROM Measures to Single Components

not possible, it would try to downgrade them. If not all passengers could be up-

graded or downgraded, the residual passengers are denied boarding, which also

incurs another compensatory payment. In Algorithm 7.2 we describe the general

case for an unspecified number of compartments per leg to determine the num-

ber of upgraded, downgraded and denied boarded passengers per compartment.

Starting with the highest valued compartment the algorithm tries to upgrade

excess passengers first, then tries to downgrade them and finally marks them as

denied boardings.

After passengers that are upgraded, downgraded and denied boarded have been

determined the actual revenue after no-shows and cancelations RevNis adjusted

according to Equation 7.13. In this formula we observe one interaction between

network and leg level in network-based RM. The general revenue figures are a

result of bookings on network-based itineraries. The compensation payments, i.e.

the denied boarding costs pdb

l,m and downgrading costs pdg

l,m, are determined on a

leg level, because they depend on the availability and capacity of the resources

of the flight network, the single flight legs.

RevN=X

i∈IX

j∈JiX

t∈T

pi,j,t ∗ˆ

bi,j,t −X

l∈L,m∈Ml

Bdb

l,m ∗pdb

l,m −X

l∈L,m∈Ml

Bdg

l,m ∗pdg

l,m(7.13)

7.1.4. No RM Revenue after No-shows and Cancelations

The adjustment of the no RM revenue to consider no-shows and cancelations re-

quires only minor modifications, too. Because we assume a no RM situation and

thus simulate the FCFS strategy, we do not consider any upgrading or overbook-

ing in the estimation of the no RM revenue. We estimate values for x−

i,j,t using an

extended version of Algorithm 4.1 introduced in Chapter 4. Cancelations already

occur during the course of a booking period and thus the handling of cancelations

has to be included into our algorithm to simulate a FCFS revenue estimation.

The algorithm remains the same except for the fact that at the end of each time

period the cancelations are deducted from the number of bookings and the ca-

pacity is increased accordingly. The enhancement is shown in Algorithm 7.3 in

Lines 12 to 18 in detail. In the algorithm we use the unadjusted unconstrained

demand di,j,t in the independent demand case or the unadjusted unconstrained

yieldable demand dyd

i,j,t in the dependent demand case. After having accomplished

Algorithm 7.3 we have to apply the show-up rate to the estimates to obtain the

estimated bookings after no-shows x−,N

i,j,t as described in Equation 7.14.

x−,N

i,j,t =x−

i,j,t ∗qi,j ∀i∈I, ∀j∈Ji,∀t∈T(7.14)

122

7.1. Extending the Network-based ROM to Overbooking and Upgrading

Algorithm 7.2: Determine Number of Upgraded, Downgraded and Denied

Boarded Passengers

1foreach l∈Ldo

2Initialize values

3foreach m∈Mldo

4Bex

l,m =max(ˆ

Bl,m −capl,m,0)

5capf

l,m =max(capl,m −ˆ

Bl,m,0)

6Bdg

l,m = 0

7Bup

l,m = 0

8for m← |Ml|downto 1do

9ex =Bex

l,m

10 Determine upgrades

11 m0=m+ 1

12 while (ex > 0∧m0≤ |Ml|)do

13 up =min(ex, capf

l,m0)

14 Bup

l,m =Bup

l,m +up

15 capf

l,m0=capf

l,m0−up

16 ex =ex −up

17 m0=m0+ 1

18 Determine downgrades

19 m0=m−1

20 while (ex > 0∧m0≥1) do

21 dg =min(ex, capf

l,m0)

22 Bdg

l,m =Bdg

l,m +dg

23 capf

l,m0=capf

l,m0−dg

24 ex =ex −dg

25 m0=m0−1

26 Determine denied boardings

27 Bdb

l,m =Bex

l,m −Bdg

l,m −Bup

l,m

123

7. Disaggregation of ROM Measures to Single Components

Algorithm 7.3: Estimation of No RM Revenue after No-shows and Can-

celations

Input:Pt,∀t∈T

1foreach l∈Ldo

2foreach m∈Mldo

3capf

l,m =capl,m

4for t= 1 to |T|do

5foreach (i, j)∈Ptdo

6sl =∞

7foreach l∈Lido

8sl =min(sl, capf

l,ml,j )

9x−

i,j,t =x−

i,j,t +min(sl, di,j,t)

10 foreach l∈Lido

11 capf

l,ml,j =capf

l,ml,j −min(sl, di,j,t)

12 foreach (i, j)∈Ptdo

13 canc = 0

14 for t0= 1 to tdo

15 canc =canc +x−

i,j,t0∗(1 −ki,j,t)

16 x−

i,j,t0=x−

i,j,t0∗(1 −ki,j,t)

17 foreach l∈Lido

18 capf

l,ml,j =capf

l,ml,j +canc

124

7.2. Measuring Overbooking and Upgrading Success

After the show-up rates have been applied, the actual revenue after no-shows

and cancelations RevNhas to be calculated using the basic formula 4.5 introduced

in Chapter 4.

The calculation of the derived ROM measures has to be adjusted to the new

estimates for the potential, the actual and the no RM revenue. Depending on the

scenario and the applied RM methodology one single adjusted revenue estimate

has to be chosen. In practical applications usually both upgrading and over-

booking are applied. Thus, we only present the normal case in which Rev+,O,U

serves as the potential revenue, RevNas the actual revenue and Rev−,N as the

no RM revenue. This changes the definitions of RO, ARO and PARO according

to Equations 7.15 to 7.17.

RO =Rev+,O,U −Rev−,N (7.15)

ARO =RevN−Rev−,N (7.16)

PARO =ARO

RO (7.17)

7.2. Measuring Overbooking and Upgrading

Success

In the previous section we enhanced the ROM to incorporate cancelations, no-

shows, overbooking and upgrading. However, we did not introduce new measures

that are able to indicate the performance of the RM methods mentioned above.

In Chapter 2 we discussed several ways to measure the success of RM components.

Because overbooking and upgrading are important components of common RM

systems we follow the suggestion by P¨olt (2001) and introduce methods to split

the success into overbooking, upgrading and fare-mix success.

The basic idea is to split the overall RO and ARO into three subparts, namely

one for overbooking, one for upgrading and one residual part for the fare-mix.

However, the success of a component cannot be exactly isolated. All RM compo-

nents are interdependent and influence each other. Smith et al. (1992) and P¨olt

(2001) describe the interdependence of different sub-measures as an important

challenge. Thus, for all proposed methods of disaggregation we assume a certain

interdependency.

In the following we assume that overbooking and upgrading increased the num-

ber of bookings relative to the case in which none of the above components were

applied. The main idea is to estimate the number of potential additional bookings

and to compare them to the number that has been estimated as a contribution

125

7. Disaggregation of ROM Measures to Single Components

by overbooking and upgrading. The ARO and the RO depend on the average

incremental revenue we apply to the additional bookings.

If both overbooking and upgrading are applied, the separation of the number

of additional bookings into overbooking and upgrading gets more complicated.

In the following sections we therefore first introduce the estimation of additional

bookings for overbooking and upgrading separately and describe the common

case with both overbooking and upgrading applied afterwards. We start with

defining the incremental average revenue.

7.2.1. Incremental Revenue due to Overbooking and

Upgrading

A crucial point is the definition of the incremental revenue that is applied to the

number of bookings that are considered to be the outcome of overbooking and

upgrading. One way is taking the yield or the average revenue of the respective

compartment. We define the average revenue pavg

l,m for a compartment mon leg l

in Equation 7.18.

pavg

l,m =Pi∈IlPj∈Ji,l,m Pt∈Tpi,j,l,t ∗bi,j,t

Bl,m

∀l∈L, ∀m∈Ml(7.18)

The formula sums up all bookings multiplied with the respective prorated fare

and divides it by the total number of bookings for the compartment Bl,m. If

the incremental revenue applied to the additional bookings is calculated through

this formula, then it can be assumed that the fare-mix remains constant for

all additional bookings due to overbooking and upgrading. A more realistic

assumption is that the bookings with the highest fares will be accepted no matter

if overbooking was applied or not. This is due to the following assumptions: First,

that the RM control has protected seats for the higher fare classes and second,

that overbooking and upgrading only have allowed additional bookings with low

fares.

We present our approach to obtain the incremental revenue due to low fare

bookings in Algorithm 7.4. The algorithm calculates the average incremental

revenue for a given number of additional bookings Badd

l,m . Another input of the

algorithm is an ordered set Pl,m. It stores all valid booking combinations (i, j, t)

∈Il×Ji,l,m ×Tand these are sorted by their fare pi,j,l,t in an ascending order.

Starting with the booking combination (i,j,t) with the lowest fare associated to

it, the algorithm determines the total incremental revenue rev that is achieved

with additional bookings. bl denotes the number of additional bookings left that

still have to be included into the total revenue. Finally the average incremental

126

7.2. Measuring Overbooking and Upgrading Success

Algorithm 7.4: Determination of Incremental Revenue per Compartment

Input:Badd

l,m and Pl,m ,∀l∈L, ∀m∈Ml

1foreach l∈Ldo

2foreach m∈Mldo

3if Badd

l,m >0then

4bl =Badd

l,m

5while bl > 0do

6get next (i, j, t)∈Pl,m

7rev =rev + (pi,j,l,t ∗min(bl, bi,j,t))

8bl =bl −min(bl, bi,j,t)

9pinc

l,m =rev/Badd

l,m

10 else

11 pinc

l,m = 0

revenue pinc

l,m is calculated (see Line 9). In the following we primarily focus on the

incremental revenue pinc

l,m due to its more realistic assumptions.

7.2.2. ROM with Upgrading

We first separate the ROM measures into upgrading and fare-mix success. In this

section we therefore assume that the RM control does not apply overbooking, each

customer is showing up at the departure of the flight and no customer cancels a

booking. The main number to be estimated is the number of additional bookings

that we consider a result of upgrading. In our current scenario the estimation is

simple. Because the RM control has not applied overbooking we neither observe

denied boardings nor downgrades. Thus, the number of excess bookings in a

compartment can be clearly related to upgrading. The number of additional

bookings BU

l,m is simply the number of all passengers exceeding the capacity (see

Equation 7.19).

l,m =(0m=m+

Bex

l,m otherwise ∀l∈L, ∀m∈Ml(7.19)

Figure 7.1 illustrates this. In the highest valued compartment m+

lthe RM con-

trol has not increased the capacity. Thus there are no passengers exceeding the

capacity. The value for BU

l,m is zero in this compartment.

The ARO for upgrading AROU

lis the number of additional bookings BU

l,m

multiplied by the incremental revenue pinc

l,m (see Equation 7.20). As described

127

7. Disaggregation of ROM Measures to Single Components

Capacity

Additional

bookings -

upgrading

Maximum bookings

without upgrading

Figure 7.1.: Additional Bookings Related to Upgrading

earlier the ARO for the fare-mix AROF

lis the residual part of AROl(see Equation

7.21).

AROU

l=X

m∈Ml

l,m ∗pinc

l,m ∀l∈L(7.20)

AROF

l=AROl−AROU

l∀l∈L(7.21)

The definition of the RO for upgrading ROU

lis similar to the definition of the

ARO. First, a theoretical number of additional bookings that can be related to

upgrading XU

l,m is determined and then multiplied by the incremental revenue

pinc

l,m. In analogy to BU

l,m,XU

l,m is the difference between the cumulated potential

bookings on the compartment X+,U

l,m and the capacity of the compartment capl,m.

l,m =max(X+,U

l,m −capl,m,0) ∀l∈L, ∀m∈Ml(7.22)

ROU

lis determined in analogy to AROU

land the RO of the fare-mix is defined

as the residual RO after deducting ROU

l(see Equations 7.23 and 7.24).

ROU

l=X

m∈Ml

l,m ∗pinc

l,m ∀l∈L(7.23)

ROF

l=ROl−ROU

l∀l∈L(7.24)

7.2.3. ROM with Overbooking

In this section we focus on the separation of the ROM measures into overbooking

and fare-mix-success. To estimate the number of additional bookings that are

applicable to overbooking we make another important assumption. We assume

that without applying overbooking RM controls the maximum number of book-

ings Bl,m is the capacity of the compartment capl,m. At the end of the booking

period passengers showed up with the average show-up rate ql,m. We furthermore

assume that as a result the maximum number of bookings for the given compart-

ment after no-shows is capl,m ∗ql,m. The overbooking control virtually increases

the capacity to prevent too many empty seats because of no-shows. Taking all of

128

7.2. Measuring Overbooking and Upgrading Success

these assumptions together we define the difference between the bookings after

no-shows and cancelations ˆ

Bl,m and the adjusted capacity capl,m ∗ql,m as those

additional bookings that can be applied to overbooking BO

l,m (see Equation 7.25).

l,m =max(ˆ

Bl,m −capl,m ∗ql,m,0) ∀l∈L, ∀m∈Ml(7.25)

Figure 7.2 illustrates the definition.

Capacity

Additional

bookings -

overbooking

Capacity less

no-shows

Maximum bookings

without overbooking

Figure 7.2.: Additional Bookings Related to Overbooking

However, it is possible that passenger bookings exceed the capacity of the

compartments. In this case we have to apply Algorithm 7.2 to find out how

many passengers can be upgraded and which ones have to be downgraded or

denied boarding. We assume that the costs for denied boarding and downgrading

decrease the ARO of overbooking AROO

l(see Equations 7.26 and 7.27).

AROO

l=X

m∈Ml

(BO

l,m ∗pinc

l,m −Bdb

l,m ∗pdb

l,m −Bdg

l,m ∗pdg

l,m)∀l∈L(7.26)

AROF

l=AROl−AROO

l∀l∈L(7.27)

In general, denied boardings do not increase the ARO of overbooking. Surely

excess bookings increase the number of additional bookings applied to overbook-

ing. However, as pinc

l,m is usually smaller than pdb

l,m, the denied boarding costs are

higher than the additional achieved revenue by the excess bookings.

The theoretical amount of additional bookings that can be obtained with over-

booking XO

l,m are derived in accordance to BO

l,m. We use the difference between

the estimates for the potential revenue with overbooking X+,O

l,m and the adjusted

capacity of the compartment malready used before.

l,m =max(X+,O

l,m −capl,m ∗ql,m,0) ∀l∈L, ∀m∈Ml(7.28)

The theoretical RO that is achievable with overbooking ROO

lis derived in

analogy to ROU

l. Like the RO of upgrading the RO of overbooking is always ≥

0. The RO for the fare-mix ROF

lis again the residual RO (see Equations 7.29

129

7. Disaggregation of ROM Measures to Single Components

and 7.30).

ROO

l=X

m∈Ml

l,m ∗pinc

l,m ∀l∈L(7.29)

ROF

l=ROl−ROO

l∀l∈L(7.30)

7.2.4. ROM with Overbooking and Upgrading

In practice, usually both overbooking and upgrading are part of the RMS. The

concepts described in the two previous sections can further be used. However,

adjustments have to be made because it has to be decided, whether an additional

booking belongs to overbooking or upgrading. Figure 7.3 shows how we separate

the additional bookings into overbooking and upgrading success.

Capacity

Additional

bookings -

overbooking

Maximum bookings without

overbooking and upgrading

Capacity less

no-shows

Additional

bookings -

upgrading

Figure 7.3.: Additional Bookings Related to Overbooking and Upgrading

We assume that all bookings exceeding the capacity belong to upgrading and

the bookings exceeding the adjusted capacity belong to overbooking. The def-

inition of BU

l,m stays the same. It still represents all excess bookings in a com-

partment, except in the highest valued compartment. By definition, upgrading

is not possible here. The definition of the number of additional bookings related

to overbooking BO

l,m is adjusted by deducting BU

l,m. The denied boardings and

downgrades are calculated using Algorithm 7.2. We assume that they are a result

of overbooking and thus solely decrease AROO

l. The adjusted determination of

l,m is depicted in Equation 7.31.

l,m =max(Bl,m −capl,m ∗ql,m −BU

l,m,0) ∀l∈L, ∀m∈Ml(7.31)

The definition of the AROs of overbooking and upgrading remains the same.

Only the ARO of fare-mix AROF

lhas to be redefined in Equation 7.32

AROF

l=AROl−AROU

l−AROO

l∀l∈L(7.32)

The separation of XU

l,m and XO

l,m equals the split between the estimates of

the achieved additional bookings related to overbooking and upgrading BO

l,m and

130

7.3. Computational Results

l,m. The main input is the estimated number of potential bookings X+,O,U

l,m

on a compartment munder consideration of overbooking and upgrading (see

Equations 7.33 and 7.34).

l,m =max(X+,O,U

l,m −capl,m,0) ∀l∈L, ∀m∈Ml(7.33)

l,m =max(X+,O,U

l,m −capl,m ∗ql,m −XU

l,m,0) ∀l∈L, ∀m∈Ml(7.34)

The definition of ROF

lhas also to be adjusted:

ROF

l=ROl−ROU

l−ROO

l∀l∈L(7.35)

7.3. Computational Results

In this section we present computational results on the isolation of upgrading,

overbooking and fare-mix success from the overall ROM measures. In contrast

to the analyses performed so far, the RMS applies overbooking and upgrading if

needed in the scenario. As a prerequisite the request generator creates requests

with no-show behavior. Although we modeled cancelations in the formal defini-

tions, we did not apply them in our RM simulations. In our simulations, we focus

on the base case for dependent demand and on the realistic flight network. The

results for the no-connecting-traffic flight network and for independent demand

are very similar and can be found in the appendix.

In Chapters 4 and 5 we did not consider upgrading and overbooking in the

analysis on the robustness of the ROM. In Table 7.1 we analyze the similarity

measures and error measures, if overbooking and upgrading are applied. The

Overbooking Reg. Reg. -50% None None Reg. -50%

Upgrading Reg. -50% Reg. Reg. -50% None None

PAROR(%) 63.3 63.0 59.1 60.9 60.5 63.4 59.0

PAROD(%) 67.5 67.2 62.6 64.7 64.3 67.6 62.6

MAEP ARO (%) 4.2 4.2 3.5 3.8 3.7 4.3 3.6

rP ARO 0.90 0.90 0.90 0.92 0.91 0.89 0.90

Table 7.1.: PAROs on an Aggregated Network Level with Upgrading and Over-

booking Applied

assessed scenarios can be divided into three groups: 1) both overbooking and

upgrading have been applied, 2) only overbooking has been applied and 3) only

upgrading has been applied. ’Reg.’ in a column header denotes that overbooking

or upgrading was applied with regular settings and no adjustments. If the entry

131

7. Disaggregation of ROM Measures to Single Components

in the column header is ’-50%’ the overbooking or upgrading levels were reduced

by 50%.

Figure 7.4 shows the scatter plot for the base case, in which both overbooking

and upgrading are applied with unadjusted levels. We observe that the error

40%

60%

80%

100%

40% 60% 80% 100%

Real Demand

Est. Unc. Demand

Figure 7.4.: Realistic Flight Network with Dependent Demand with Overbooking

and Upgrading Aggregated to Network Level

measures remain constant. The similarity measures indicate a very high resem-

blance. We conclude that enhancing the ROM to overbooking and upgrading on

a network level does not affect the robustness.

In a first analysis to examine the potential isolation of single RM components

we focus on the ARO and assess it for both upgrading and overbooking. In

practice, particularly the values of AROO

lare of interest because they describe

the absolute success of overbooking. This is important because overbooking also

incurs a risk of denied boardings and a loss in customer goodwill.

In Table 7.2 we compare the AROs calculated with the average revenue pavg

l,m

and the incremental revenue pinc

l,m. We observe that the AROs in all cases roughly

double if we assume that the average revenue on a leg is the correct revenue to

be applied on additional bookings. We also observe that the values of the ARO

are quite stable and reflect the underlying overbooking and upgrading controls.

If we decrease overbooking levels by 50%, the AROOalso decreases by nearly

50%. In the second part of the table we compare the AROs for overbooking and

upgrading to the actual revenue. Depending on the incremental revenue assumed,

overbooking is responsible for 2.7% or 1.3% of total revenue. Upgrading is not

as important as overbooking and only contributes with 0.4% or 0.2%.

Beyond using the ARO to justify the application of upgrading and overbooking

132

7.4. Summary

Overbooking Reg. Reg. -50% None None Reg. -50%

Upgrading Reg. -50% Reg. Reg. -50% None None

AROObased on pavg

l,m (million) 1.17 1.16 0.68 - - 1.15 0.57

AROObased on pinc

l,m (million) 0.56 0.56 0.35 - - 0.55 0.29

Diff. (million) 0.60 0.60 0.33 - - 0.60 0.28

AROUbased on pavg

l,m (million) 0.16 0.11 0.06 0.16 0.10 - -

AROUbased on pinc

l,m (million) 0.09 0.06 0.03 0.08 0.05 - -

Diff. (million) 0.08 0.05 0.03 0.08 0.05 - -

Rev (million) 43.3 43.2 42.8 44.9 44.9 43.2 42.7

AROD(million) 6.5 6.4 6.0 5.8 5.7 6.3 5.9

AROO/Rev based on pavg

l,m (%) 2.7 2.7 1.6 - - 2.7 1.3

AROO/Rev based on pinc

l,m (%) 1.3 1.3 0.8 - - 1.3 0.7

AROU/Rev based on pavg

l,m (%) 0.4 0.3 0.1 0.4 0.2 - -

AROU/Rev based on pinc

l,m (%) 0.2 0.1 0.1 0.2 0.1 - -

Table 7.2.: Comparison of RO and ARO between Incremental and Average Rev-

enues

the question of robustness is also relevant for the PAROs calculated for overbook-

ing, upgrading and fare-mix. The analyzed scenarios remain the same. In Table

7.3 we present the similarity measures for PAROO,PAROUand PAROF. In

figures 7.5, 7.6 and 7.7 we show the corresponding scatter plots for the base case

with dependent demand with regular overbooking and upgrading settings. We

focus on the realistic flight network scenarios and refer the reader for the no-

connecting-traffic flight network to the appendix. We observe that the PAROs

calculated for overbooking and upgrading are very robust. The scatter plots con-

tain only a few outliers. The scatter plot and the similarity measures for the

fare-mix are comparable to the overall PARO measure. Averaging over multiple

flight departures is not necessary for upgrading and overbooking and does not

improve the similarity measures. The share of flight departures included grows

as well as the MAEP AROOand MAEP AROU. The results for fare-mix however

increase with the application of averaging.

7.4. Summary

In this chapter we assessed the potential to disaggregate the ROM measures to

isolate the contribution of overbooking, upgrading and fare-mix from the overall

success. We conclude that an isolation of overbooking, upgrading and fare-mix

133

7. Disaggregation of ROM Measures to Single Components

Overbooking Reg. Reg. -50% None None Reg. -50%

Upgrading Reg. -50% Reg. Reg. -50% None None

flight dep. incl. (%) 34.3 30.9 34.3 - - 34.2 30.0

MAEP AROO

(%) 1.7 2.2 1.6 - - 1.6 2.2

rP AROO0.96 0.92 0.97 - - 0.97 0.91

flight dep. incl. (%) 27.6 27.4 25.9 18.1 16.0 - -

MAEP AROU

(%) 2.5 2.6 2.5 3.1 1.5 - -

rP AROU0.95 0.93 0.90 0.89 0.93 - -

flight dep. incl. (%) 62.7 62.7 62.8 62.2 62.3 62.7 62.8

MAEP AROF

(%) 13.9 13.7 13.6 13.8 13.6 14.0 13.7

rP AROF0.78 0.79 0.77 0.75 0.76 0.77 0.77

Table 7.3.: Comparing PAROs for Overbooking, Upgrading and Fare-mix on Re-

alistic Flight Network

Overbooking Reg. Reg. -50% None None Reg. -50%

Upgrading Reg. -50% Reg. Reg. -50% None None

flight dep. incl. (%) 53.0 53.0 52.4 - - 52.9 52.5

MAEP AROO

(%) 2.8 2.9 3.8 - - 2.7 3.7

rP ARO 0.97 0.96 0.86 - - 0.97 0.81

flight dep. incl. (%) 38.3 38.5 37.9 33.5 32.8 - -

MAEP AROU

(%) 5.9 5.8 2.4 3.8 2.2 - -

rP ARO 0.85 0.79 0.79 0.91 0.95 - -

flight dep. incl. (%) 72.3 72.4 72.4 71.9 72.0 72.3 72.3

MAEP AROF

(%) 14.2 14.0 13.0 12.0 11.9 14.1 12.9

rP ARO 0.76 0.77 0.78 0.82 0.83 0.76 0.78

Table 7.4.: Comparing PAROs for Overbooking, Upgrading and Fare-mix on Re-

alistic Flight Network Using Averaging

134

7.4. Summary

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Real Demand

Est. Unc. Demand

Figure 7.5.: Realistic Flight Network

with Dependent Demand -

Overbooking

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Real Demand

Est. Unc. Demand

Figure 7.6.: Realistic Flight Network

with Dependent Demand -

Upgrading

success is reasonable and leads to good results. In particular the isolation of

overbooking and upgrading success leads to promising results. The values ob-

tained from one simulation run are robust both for independent and dependent

demand. Furthermore we propose to use the incremental revenue and not the av-

erage revenue on a flight leg to value the overbooking and upgrading success: A

realistic assumption of the additional bookings attributable to overbooking and

upgrading suggests that these are the ones which correspond to the lowest fare.

This of course leads to a significant decrease in total revenue that is attributed

to overbooking and upgrading.

135

7. Disaggregation of ROM Measures to Single Components

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Real Demand

Est. Unc. Demand

Figure 7.7.: Realistic Flight Network

with Dependent Demand -

Fare-mix

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Real Demand

Est. Unc. Demand

Figure 7.8.: Realistic Flight Network

with Dependent Demand

and Averaged over One

Month - Overbooking

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Real Demand

Est. Unc. Demand

Figure 7.9.: Realistic Flight Network

with Dependent Demand

and Averaged over One

Month - Upgrading

20%

40%

60%

80%

100%

0% 20% 40% 60% 80% 100%

Real Demand

Est. Unc. Demand

Figure 7.10.: Realistic Flight Network

with Dependent Demand

and Averaged over One

Month - Fare-mix

136

8. Summary and Concluding

Remarks

In this thesis we addressed the topic of performance measurement in airline RM

and in particular the network-based ROM with independent and dependent de-

mand. Performance measurement is an important part of the application of RM

controls at an airline. It is employed to continuously assess the contribution of

the RM techniques in use, to give the revenue managers feedback on their ac-

tions and to fine-tune the RMS. However, the major developments in airline RM

science - advancing from leg-based to network-based RM controls and from inde-

pendent to dependent demand structures - pose new questions and challenges to

the ROM.

In the first chapter we introduced the main concepts and terminology of airline

RM and in addition discussed methods to measure its performance. In particular

we described the ROM as one way to measure RM performance in detail. In

Chapter 2 we provided a literature review of airline RM with a specific focus

on the major developments in demand modeling and optimization techniques

in the last decades. We gave a thorough overview about the transition from

leg-based to network-based RM controls and the advancement from independent

demand models to dependent demand structures. We described state-of-the-

art methods for both demand modeling and optimization techniques. After a

detailed overview about current developments in performance measurement and

specifically the ROM we motivated our research on network-based ROMs with

independent and dependent demand.

In Chapter 3 we introduced a novel simulation-based approach to investigate

ROM properties. First, it comprises a simulation environment that reflects state-

of-the-art models and methods of operational RMS of large network airlines and

uses input data that corresponds to reality as well as possible. Second, we

presented a novel approach to measure the robustness of the ROM using the

previously described simulation environment and furthermore designed several

scenarios to thoroughly investigate the properties of the ROM.

A detailed assessment of the ROM with independent demand was presented in

Chapter 4. After describing the network-based ROM with independent demand

137

8. Summary and Concluding Remarks

in detail, we discussed some of its main properties and in particular the effect

of model- and data-related errors on the ROM. Computational results show that

data-related errors, i.e. errors in the input data, have a higher effect on the

validity of the ROM results than model-related errors. In addition, the ROM

proved itself to be robust against errors in the unconstrained demand up to our

previously defined worst case scenarios and even beyond. Thus, we consider the

ROM applicable in practical RMS.

In Chapter 5 we enhanced the network-based ROM with independent demand

to dependent demand structures. We have chosen a modeling approach for the

dependent demand which is common in practical RMS that consider demand de-

pendencies. We described the modification of the ROM in detail and discussed

the main properties of the adjusted ROM. As with independent demand the

network-based ROM with dependent demand proved to be robust and is con-

sidered applicable in real life applications. However, in comparison to the ROM

with independent demand we observed a decrease in robustness which is due to

a higher overall error level in the estimated unconstrained demand.

After having assessed the ROM for both independent and dependent demand

on a network level, we investigated ways to consider practical aspects in the

ROM. In Chapters 4 and 5 we focussed our research on the main properties and

correlation between the ROM and different error scenarios. We did neither incor-

porate common RM components such as overbooking or upgrading nor intended

to disaggregate the measures to sub-measures that are useful in a practical RM

context. However, these are important questions in reality.

Therefore we discussed a potential disaggregation of the ROM measures to

subparts of the flight network in Chapter 6. We reviewed several methods to

prorate fares to single legs and techniques to increase the quality of the leg-based

ROM measures such as capping, filtering and averaging. Our analysis shows

that the ROM proved itself to be robust even if used on a leg level. However, the

effect of unconstraining errors increases significantly in particular with dependent

demand. Thus, we suggest to thoroughly analyze which flight departures should

be included in the ROM evaluation and which should be evaluated differently.

The integration of single RM components into the ROM and a potential dis-

aggregation of the ROM measures to those components is discussed in Chapter

7. First, we integrated no-shows and cancelations into the network-based ROM

with independent and dependent demand. In a second step we proposed a dis-

aggregation into upgrading, overbooking and fare-mix success. The ROM that

considers no-shows and cancelations also proved to be robust and applicable in

real life. The disaggregation to single RM components delivered very promising

results. In particular the ROM measures for overbooking and upgrading showed

138

a high robustness.

Overall, we assessed the ROM for a large network airline with its main facets

in a very detailed manner. However, the results might be different for airlines

with other network characteristics such as low cost carriers or regional airlines.

Moreover, competitors and airline alliances have not explicitly been considered

in our investigations, but play a very important role in today´s revenue man-

agement environments. The analysis of different network characteristics and the

consideration of competitors and alliances could be one field for further research.

In addition, we did not focus on customer lifetime value models, but on pure

transaction-based RM models. Also the growing importance of dynamic pricing

was not taken into account in this thesis. Incorporating these developments into

the ROM could be another stream of future research. In this thesis we defined the

estimations of the potential and the no RM revenue based on some main assump-

tions and practical consideration such as availability of demand data. However, it

might be worth thinking about different ways of defining the reference points to

compare the actual revenue with. Moreover, the research on dependent demand

structures in airline RM still shows significant developments and an increasing

number of airlines is going to implement methods to support these methodolo-

gies. For example a lot of research is done with general customer-choice models.

We suggest to integrate new findings in this area into the ROM continuously.

Moreover, we propose to investigate the interaction and combination of ROM

measures with other performance measures. In the future, not only an integra-

tion of the ROM into training tools for revenue managers could be investigated,

but also the potential to use the ROM as a pre-departure performance measure.

139

8. Summary and Concluding Remarks

140

A. Detailed Test Results

In the next sections we list the detailed results of the analyses we conducted

in the course of this thesis. We show additional scatter plots and tables that

supplement the findings and conclusions of our thesis.

A.1. The Network-based ROM with Independent

Demand

In this section we list supplemental scatter plots of our analyses of the network-

based ROM with independent demand. In addition, we present extended result

tables that include the similarity measures for all ROM measures.

25%

50%

75%

100%

0% 25% 50% 75% 100%

Real Demand

Est. Unc. Demand

Error 0.30 Error 0.60 Error 0.90

Figure A.1.: Effect of a Biased Un-

derestimation of the Fore-

casted Demand on PARO

25%

50%

75%

100%

0% 25% 50% 75% 100%

Real Demand

Est. Unc. Demand

Error 0.30 Error 0.60 Error 0.90

Figure A.2.: Effect of a Biased Over-

estimation of the Fore-

casted Demand on PARO

141

A. Detailed Test Results

30%

50%

70%

90%

30% 50% 70% 90%

Real Demand

Est. Unc. Demand

Error 0.30 Error 0.60 Error 0.90

Figure A.3.: Effect of an Unbiased Er-

ror of the Forecasted De-

mand on PARO

30%

50%

70%

90%

30% 50% 70% 90%

Real Demand

Est. Unc. Demand

Base Case Adj. +25% Adj. +50%

Figure A.4.: Effect of Restrictive RM

Controls on PARO

30%

50%

70%

90%

30% 50% 70% 90%

Real Demand

Est. Unc. Demand

Base Case Adj. -50% Adj. +50%

Figure A.5.: Effect of Adjusted Sea-

sonality on PARO

142

A.1. The Network-based ROM with Independent Demand

Base Biased Biased

Case underestimation overestimation Unbiased error

Error level - 30% 60% 90% 30% 60% 90% 30% 60% 90%

P AROR(%) 74.7 71.0 64.1 49.6 76.8 78.0 78.7 74.8 74.6 74.5

P AROD(%) 74.7 71.6 66.9 59.4 76.3 77.4 78.2 74.4 74.0 73.9

MAEP ARO (%) 0.3 0.6 2.8 9.8 0.5 0.6 0.6 0.4 0.6 0.7

rP ARO 0.94 0.87 0.75 0.64 0.97 0.97 0.96 0.94 0.90 0.86

AROR(million) 5.6 5.3 4.8 3.7 5.8 5.8 5.9 5.6 5.6 5.6

AROD(million) 5.6 4.5 3.0 1.0 6.6 7.3 7.9 5.7 5.7 5.7

MAEARO (million) 0.1 0.8 1.8 2.7 0.8 1.4 2.0 0.1 0.1 0.1

rARO 0.98 0.97 0.94 0.73 0.98 0.98 0.98 0.97 0.95 0.90

ROR(million) 7.5 7.5 7.5 7.5 7.5 7.5 7.5 7.5 7.5 7.5

ROD(million) 7.5 6.3 4.5 1.6 8.6 9.4 10.1 7.6 7.6 7.7

MAERO (million) 0.1 1.2 2.9 5.8 1.1 1.9 2.6 0.1 0.2 0.2

rRO 0.98 0.98 0.95 0.81 0.98 0.98 0.97 0.97 0.96 0.91

Rev+,R (million) 46.5 46.5 46.5 46.5 46.5 46.5 46.5 46.5 46.5 46.5

Rev+,D (million) 46.6 46.2 45.3 43.4 46.8 47.0 47.1 46.6 46.6 46.6

MAERev+

(million) 0.0 0.4 1.2 3.1 0.3 0.5 0.6 0.1 0.1 0.1

rRev+1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

Rev (million) 44.6 44.4 43.8 42.8 44.8 44.9 44.9 44.6 44.6 44.6

Rev−,R (million) 39.0 39.0 39.0 39.0 39.0 39.0 39.0 39.0 39.0 39.0

Rev−,D (million) 39.0 39.8 40.8 41.8 38.2 37.6 37.1 39.0 39.0 38.9

MAERev−

(million) 0.1 0.8 1.8 2.7 0.8 1.4 2.0 0.1 0.1 0.1

rRev−

0.99 0.99 0.99 0.98 0.99 0.99 0.99 0.99 0.99 0.98

R(thousand) 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6

D(thousand) 87.2 79.8 72.6 65.4 95.1 102.9 110.7 87.5 87.8 88.2

F(thousand) 87.1 79.8 72.6 65.4 95.0 102.9 110.7 87.4 87.7 88.1

MAED0.56 0.73 1.11 1.55 0.76 1.17 1.66 0.75 1.15 1.62

P MAED(%) 9.3 12.0 18.2 25.6 12.6 19.4 27.5 12.3 18.9 26.8

MAEF1.83 1.97 2.28 2.63 2.02 2.40 2.86 1.85 1.89 1.95

P MAEF(%) 30.3 32.4 37.6 43.4 33.4 39.7 47.3 30.5 31.2 32.2

Table A.1.: Effect of Unconstraining Errors on ROM Measures

143

A. Detailed Test Results

Base Biased Biased

Case underestimation overestimation Unbiased error

Error level - 30% 60% 90% 30% 60% 90% 30% 60% 90%

P AROR(%) 74.7 65.0 43.2 15.1 69.0 53.8 35.5 74.7 74.4 73.9

P AROD(%) 74.7 67.4 52.1 33.4 69.3 57.3 44.0 74.3 73.8 73.2

MAEP ARO (%) 0.3 2.4 8.9 18.3 0.4 3.5 8.5 0.4 0.6 0.8

rP ARO 0.94 0.81 0.76 0.49 0.99 0.99 0.98 0.93 0.89 0.84

AROR(million) 5.6 4.9 3.2 1.1 5.2 4.0 2.7 5.6 5.6 5.5

AROD(million) 5.6 4.3 2.4 0.5 6.0 5.5 4.7 5.7 5.6 5.6

MAEARO (million) 0.1 0.5 0.8 0.6 0.8 1.5 2.1 0.1 0.1 0.1

rARO 0.98 0.95 0.91 0.72 0.99 0.99 0.98 0.97 0.94 0.89

ROR(million) 7.5 7.5 7.5 7.5 7.5 7.5 7.5 7.5 7.5 7.5

ROD(million) 7.5 6.4 4.7 1.6 8.6 9.7 10.7 7.6 7.6 7.7

MAERO (million) 0.1 1.1 2.8 5.9 1.1 2.2 3.3 0.1 0.2 0.2

rRO 0.98 0.98 0.96 0.87 0.98 0.97 0.96 0.97 0.94 0.91

Rev+,R (million) 46.5 46.5 46.5 46.5 46.5 46.5 46.5 46.5 46.5 46.5

Rev+,D (million) 46.6 46.0 44.5 41.2 46.9 47.2 47.7 46.6 46.6 46.6

MAERev+

(million) 0.0 0.5 2.0 5.3 0.3 0.7 1.2 0.1 0.1 0.1

rRev+1.00 1.00 1.00 0.99 1.00 1.00 1.00 1.00 1.00 1.00

Rev (million) 44.6 43.9 42.3 40.2 44.2 43.1 41.7 44.6 44.6 44.6

Rev−,R (million) 39.0 39.0 39.0 39.0 39.0 39.0 39.0 39.0 39.0 39.0

Rev−,D (million) 39.0 39.6 39.9 39.7 38.2 37.5 37.0 39.0 39.0 39.0

MAERev−

(million) 0.1 0.5 0.8 0.6 0.8 1.5 2.1 0.1 0.1 0.1

rRev−

0.99 1.00 1.00 1.00 0.99 0.99 0.99 0.99 0.99 0.98

R(thousand) 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6

D(thousand) 87.2 80.0 72.7 65.2 95.1 104.6 115.6 87.5 87.8 88.2

F(thousand) 87.1 61.2 35.2 8.9 112.9 138.5 163.8 87.1 87.1 87.1

MAED0.56 0.75 1.13 1.57 0.78 1.31 2.02 0.75 1.15 1.63

P MAED(%) 9.3 12.3 18.6 25.9 12.9 21.7 33.4 12.3 19.0 26.9

MAEF1.83 2.37 3.76 5.47 2.50 3.85 5.44 1.89 2.05 2.29

P MAEF(%) 30.3 39.0 62.0 90.2 41.4 63.7 90.0 31.2 33.9 37.7

Table A.2.: Effect of Forecast Errors on ROM Measures

144

A.1. The Network-based ROM with Independent Demand

Base Bid price Bid price Adjust Apply

Case decrease increase seasonality saw tooth curve

Adj. level - 25% 50% 25% 50% -50% +50% 130% - 70% 120% - 80%

P AROR(%) 74.7 62.5 36.7 66.3 59.6 75.1 74.2 69.5 72.7

P AROD(%) 74.7 62.9 36.7 66.4 59.5 75.1 74.3 70.8 73.2

MAEP ARO (%) 0.3 0.5 0.5 0.3 0.4 0.2 0.3 1.2 0.6

rP ARO 0.94 0.90 0.94 0.98 0.98 0.94 0.96 0.99 0.99

AROR(million) 5.6 4.7 2.7 5.0 4.5 5.6 5.6 5.3 5.4

AROD(million) 5.6 4.8 2.9 5.0 4.4 5.6 5.6 5.1 5.4

MAEARO (million) 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.5 0.3

rARO 0.98 0.96 0.96 0.98 0.98 0.95 0.99 1.00 1.00

ROR(million) 7.5 7.5 7.5 7.5 7.5 7.4 7.6 7.4 7.4

ROD(million) 7.5 7.6 7.8 7.5 7.4 7.5 7.6 7.1 7.3

MAERO (million) 0.1 0.2 0.3 0.1 0.1 0.1 0.1 0.8 0.5

rRO 0.98 0.97 0.95 0.98 0.98 0.95 0.99 1.00 1.00

Rev+,R (million) 46.5 46.5 46.5 46.5 46.5 46.6 46.5 46.0 46.3

Rev+,D (million) 46.6 46.6 46.7 46.5 46.5 46.6 46.5 45.8 46.2

MAERev+

(million) 0.0 0.0 0.2 0.0 0.0 0.0 0.0 0.3 0.2

rRev+1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

Rev (million) 44.6 43.7 41.8 44.0 43.5 44.7 44.5 43.8 44.3

Rev−,R (million) 39.0 39.0 39.0 39.0 39.0 39.1 38.9 38.5 38.9

Rev−,D (million) 39.0 38.9 38.9 39.1 39.1 39.1 38.9 38.7 38.9

MAERev−

(million) 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.5 0.3

rRev−

0.99 1.00 1.00 0.99 0.99 0.99 1.00 1.00 1.00

R(thousand) 87.6 87.6 87.6 87.6 87.6 87.4 87.8 87.2 87.2

D(thousand) 87.2 87.5 88.0 86.8 86.6 87.3 86.9 83.8 85.6

F(thousand) 87.1 87.4 87.9 86.8 86.6 87.3 86.8 83.8 85.6

MAED0.56 0.61 0.66 0.58 0.60 0.53 0.60 0.67 0.59

P MAED(%) 9.3 10.1 11.0 9.6 10.0 8.8 9.9 10.7 9.6

MAEF1.83 1.83 1.83 1.83 1.83 1.77 1.92 2.11 1.91

P MAEF(%) 30.3 30.2 30.3 30.3 30.2 29.2 31.8 36.0 32.1

Table A.3.: Effect of Adjusted RM Controls and Seasonality on ROM Measures

145

A. Detailed Test Results

A.2. The Network-based ROM with Dependent

Demand

In this section we list supplemental scatter plots and tables of our analyses of the

network-based ROM with dependent demand. Most of the results for the base

case scenario with a sell-up rate of 30% are already presented in Chapter 5. In

the contrary, most of the results for the scenarios with sell-up rates of 10% and

50% are listed in the following sections. As with the previous section we present

extended result tables that include the similarity measures for all ROM measures.

A.2.1. The Base Case: Sell-up Rate 30%

30%

50%

70%

90%

30% 50% 70% 90%

Real Demand

Est. Unc. Demand

Error 0.30 Error 0.60 Error 0.90

Figure A.6.: Effect of an Unbiased Er-

ror of the Unconstrained

Yieldable Demand on

PARO

146

A.2. The Network-based ROM with Dependent Demand

30%

50%

70%

90%

30% 50% 70% 90%

Real Demand

Est. Unc. Demand

Error 0.30 Error 0.60 Error 0.90

Figure A.7.: Effect of a Biased Un-

derestimation of the Un-

constrained Buy-down on

PARO

30%

50%

70%

90%

30% 50% 70% 90%

Real Demand

Est. Unc. Demand

Error 0.30 Error 0.60 Error 0.90

Figure A.8.: Effect of a Biased Over-

estimation of the Un-

constrained Buy-down on

PARO

25%

50%

75%

100%

0% 25% 50% 75% 100%

Real Demand

Est. Unc. Demand

Error 0.30 Error 0.60 Error 0.90

Figure A.9.: Effect of a Biased Un-

derestimation of the Fore-

casted Yieldable Demand

on PARO

25%

50%

75%

100%

0% 25% 50% 75% 100%

Real Demand

Est. Unc. Demand

Error 0.30 Error 0.60 Error 0.90

Figure A.10.: Effect of Biased Over-

estimation of the

Forecasted Yieldable

Demand on PARO

147

A. Detailed Test Results

30%

50%

70%

90%

30% 50% 70% 90%

Real Demand

Est. Unc. Demand

Error 0.30 Error 0.60 Error 0.90

Figure A.11.: Effect of an Unbiased

Error of the Forecasted

Yieldable Demand on

PARO

30%

50%

70%

90%

30% 50% 70% 90%

Real Demand

Est. Unc. Demand

Error 0.30 Error 0.60 Error 0.90

Figure A.12.: Effect of a Biased Un-

derestimation of the

Forecasted Buy-down

on PARO

30%

50%

70%

90%

30% 50% 70% 90%

Real Demand

Est. Unc. Demand

Error 0.30 Error 0.60 Error 0.90

Figure A.13.: Effect of a Biased

Overestimation of the

Forecasted Buy-down

on PARO

30%

50%

70%

90%

30% 50% 70% 90%

Real Demand

Est. Unc. Demand

Error 0.30 Error 0.60 Error 0.90

Figure A.14.: Effect of an Unbiased

Error of the Forecasted

Buy-down on PARO

148

A.2. The Network-based ROM with Dependent Demand

30%

50%

70%

90%

30% 50% 70% 90%

Real Demand

Est. Unc. Demand

Base Case Adj. -25% Adj. -50%

Figure A.15.: Effect of Open RM Con-

trols on PARO

30%

50%

70%

90%

30% 50% 70% 90%

Real Demand

Est. Unc. Demand

Base Case Adj. -50% Adj. +50%

Figure A.16.: Effect of Adjusted Sea-

sonality on PARO

149

A. Detailed Test Results

Base Biased Biased

Case underestimation overestimation Unbiased error

Error level - 30% 60% 90% 30% 60% 90% 30% 60% 90%

P AROR(%) 60.8 57.9 51.9 42.1 60.7 59.6 57.9 60.2 59.5 58.9

P AROD(%) 64.6 64.1 61.9 63.0 58.9 54.9 51.7 61.1 58.0 55.7

MAEP ARO (%) 3.8 6.2 10.0 20.9 1.8 4.7 6.2 1.0 1.5 3.1

rP ARO 0.91 0.84 0.69 0.42 0.97 0.98 0.98 0.92 0.91 0.92

AROR(million) 5.8 5.5 5.0 4.0 5.8 5.7 5.5 5.8 5.7 5.6

AROD(million) 5.6 4.6 3.2 2.0 6.2 6.6 6.8 5.6 5.5 5.6

MAEARO (million) 0.2 0.9 1.7 2.0 0.4 0.9 1.2 0.2 0.2 0.1

rARO 0.97 0.95 0.88 0.77 0.98 0.98 0.98 0.97 0.95 0.93

ROR(million) 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6

ROD(million) 8.7 7.2 5.2 3.2 10.6 12.0 13.1 9.1 9.5 10.1

MAERO (million) 0.8 2.3 4.4 6.3 1.0 2.4 3.5 0.4 0.1 0.5

rRO 0.96 0.94 0.87 0.77 0.96 0.96 0.95 0.95 0.93 0.90

Rev+,R (million) 48.6 48.6 48.6 48.6 48.6 48.6 48.6 48.6 48.6 48.6

Rev+,D (million) 47.9 47.1 46.0 44.2 49.2 50.1 50.9 48.3 48.7 49.1

MAERev+

(million) 0.7 1.4 2.6 4.3 0.6 1.5 2.3 0.3 0.1 0.5

rRev+1.00 1.00 1.00 0.99 1.00 1.00 0.99 1.00 1.00 0.99

Rev (million) 44.8 44.6 44.0 43.0 44.8 44.7 44.6 44.8 44.7 44.6

Rev−,R (million) 39.0 39.0 39.0 39.0 39.0 39.0 39.0 39.0 39.0 39.0

Rev−,D (million) 39.2 39.9 40.8 41.0 38.6 38.1 37.8 39.2 39.2 39.0

MAERev−

(million) 0.2 0.9 1.7 2.0 0.4 0.9 1.2 0.2 0.2 0.1

rRev−

0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.98 0.97

Rtd (thousand) 117.5 117.5 117.5 117.5 117.5 117.5 117.5 117.5 117.5 117.5

Ryd (thousand) 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6

Rbd (thousand) 29.9 29.9 29.9 29.9 29.9 29.9 29.9 29.9 29.9 29.9

Dtd (thousand) 120.4 111.6 103.6 97.9 134.2 147.8 161.3 122.8 125.4 129.3

Dyd (thousand) 88.7 79.5 71.1 65.3 102.3 115.9 129.4 90.8 93.1 96.6

Dbd (thousand) 31.7 32.1 32.5 32.6 31.8 31.9 31.9 32.0 32.3 32.7

Ftd (thousand) 120.4 111.6 103.5 97.8 134.1 147.7 161.3 122.7 125.3 129.2

Fyd (thousand) 88.7 79.5 71.0 65.2 102.3 115.9 129.4 90.8 93.0 96.5

Fbd (thousand) 31.7 32.1 32.5 32.6 31.8 31.9 31.9 32.0 32.3 32.7

MAEDtd

1.37 1.47 1.77 2.02 1.87 2.65 3.51 1.67 2.22 2.80

MAEDyd

0.76 0.91 1.30 1.64 1.26 2.06 2.95 1.08 1.69 2.32

MAEDbd

0.87 0.88 0.89 0.90 0.87 0.87 0.87 0.87 0.88 0.90

P MAEDtd

(%) 16.9 18.1 21.8 24.8 23.0 32.6 43.3 20.6 27.3 34.4

P MAEDyd

(%) 12.5 14.9 21.4 26.9 20.8 34.0 48.6 17.8 27.8 38.3

P MAEDbd

(%) 42.0 42.5 43.2 43.3 42.0 42.0 42.0 42.3 42.8 43.4

MAEFtd

2.27 2.37 2.65 2.86 2.66 3.35 4.17 2.34 2.45 2.63

MAEFyd

1.86 1.99 2.33 2.61 2.25 2.97 3.80 1.92 2.03 2.20

MAEFbd

0.99 1.00 1.01 1.01 0.99 0.99 0.99 0.99 1.00 1.01

P MAEFtd

(%) 28.0 29.2 32.6 35.2 32.8 41.4 51.4 28.8 30.3 32.4

P MAEFyd

(%) 30.8 32.9 38.5 43.1 37.2 49.1 62.8 31.7 33.6 36.3

P MAEFbd

(%) 47.7 48.3 49.0 49.1 47.8 47.8 47.8 48.1 48.5 49.0

Table A.4.: Effect of Unconstraining Errors of the Yieldable Demand on ROM

Measures

150

A.2. The Network-based ROM with Dependent Demand

Base Biased Biased

Case underestimation overestimation Unbiased error

Error level - 30% 60% 90% 30% 60% 90% 30% 60% 90%

P AROR(%) 60.8 60.9 60.9 60.8 60.5 60.0 59.1 60.7 60.5 60.2

P AROD(%) 64.6 65.9 67.0 67.6 61.2 57.6 53.4 63.4 61.0 55.3

MAEP ARO (%) 3.8 5.1 6.1 6.8 0.8 2.4 5.8 2.7 0.6 4.9

rP ARO 0.91 0.91 0.90 0.90 0.94 0.95 0.96 0.93 0.93 0.94

AROR(million) 5.8 5.8 5.8 5.8 5.8 5.7 5.7 5.8 5.8 5.8

AROD(million) 5.6 5.7 5.7 5.7 5.6 5.5 5.4 5.6 5.6 5.3

MAEARO (million) 0.2 0.2 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.4

rARO 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97

ROR(million) 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6

ROD(million) 8.7 8.6 8.5 8.4 9.1 9.6 10.2 8.9 9.1 9.6

MAERO (million) 0.8 1.0 1.1 1.1 0.4 0.1 0.6 0.7 0.4 0.1

rRO 0.96 0.96 0.97 0.97 0.96 0.96 0.96 0.96 0.96 0.96

Rev+,R (million) 48.6 48.6 48.6 48.6 48.6 48.6 48.6 48.6 48.6 48.6

Rev+,D (million) 47.9 47.8 47.6 47.6 48.3 48.8 49.4 48.1 48.4 49.1

MAERev+

(million) 0.7 0.8 0.9 1.0 0.2 0.2 0.9 0.5 0.2 0.5

rRev+1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

Rev (million) 44.8 44.8 44.8 44.8 44.8 44.7 44.7 44.8 44.8 44.8

Rev−,R (million) 39.0 39.0 39.0 39.0 39.0 39.0 39.0 39.0 39.0 39.0

Rev−,D (million) 39.2 39.2 39.2 39.1 39.2 39.2 39.2 39.2 39.2 39.4

MAERev−

(million) 0.2 0.2 0.1 0.1 0.2 0.2 0.2 0.2 0.2 0.4

rRev−

0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99

Rtd (thousand) 117.5 117.5 117.5 117.5 117.5 117.5 117.5 117.5 117.5 117.5

Ryd (thousand) 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6 87.6

Rbd (thousand) 29.9 29.9 29.9 29.9 29.9 29.9 29.9 29.9 29.9 29.9

Dtd (thousand) 120.4 111.6 102.9 94.2 130.1 140.0 150.1 120.9 121.7 123.6

Dyd (thousand) 88.7 89.3 89.8 90.5 88.9 89.1 89.6 89.1 89.7 91.2

Dbd (thousand) 31.7 22.4 13.1 3.8 41.3 50.8 60.5 31.8 32.0 32.5

Ftd (thousand) 120.4 111.6 102.9 94.2 130.1 139.9 150.0 120.9 121.7 123.6

Fyd (thousand) 88.7 89.2 89.8 90.5 88.9 89.1 89.5 89.1 89.7 91.1

Fbd (thousand) 31.7 22.4 13.1 3.8 41.2 50.8 60.5 31.8 32.0 32.5

MAEDtd

1.37 1.41 1.67 2.08 1.62 2.07 2.64 1.51 1.85 2.26

MAEDyd

0.76 0.77 0.78 0.80 0.77 0.77 0.79 0.77 0.79 0.86

MAEDbd

0.87 0.95 1.32 1.85 1.13 1.62 2.21 1.04 1.46 1.99

P MAEDtd

(%) 16.9 17.3 20.6 25.5 20.0 25.5 32.5 18.6 22.8 27.8

P MAEDyd

(%) 12.5 12.7 12.9 13.2 12.6 12.8 13.0 12.7 13.0 14.2

P MAEDbd

(%) 42.0 45.8 63.9 89.3 54.8 78.2 106.9 50.2 70.7 96.5

MAEFtd

2.27 2.28 2.43 2.71 2.45 2.78 3.22 2.29 2.32 2.38

MAEFyd

1.86 1.88 1.89 1.91 1.87 1.87 1.88 1.87 1.88 1.90

MAEFbd

0.99 1.03 1.36 1.85 1.24 1.70 2.26 1.00 1.04 1.10

P MAEFtd

(%) 28.0 28.0 29.9 33.3 30.2 34.3 39.7 28.2 28.6 29.4

P MAEFyd

(%) 30.8 31.0 31.2 31.5 30.8 30.9 31.0 30.9 31.1 31.5

P MAEFbd

(%) 47.7 49.8 65.6 89.6 60.2 82.3 109.6 48.5 50.4 53.4

Table A.5.: Effect of Unconstraining Errors of the Buy-down on ROM Measures

151

A. Detailed Test Results

Base Biased Biased

Case underestimation overestimation Unbiased error

Error level - 30% 60% 90% 30% 60% 90% 30% 60% 90%

P AROR(%) 60.8 56.9 48.6 16.8 56.0 44.9 31.0 59.4 56.7 52.8

P AROD(%) 64.6 62.7 56.0 47.5 53.4 39.4 25.7 59.0 53.1 46.9

MAEP ARO (%) 3.8 5.8 7.4 30.7 2.6 5.4 5.3 0.6 3.6 5.8

rP ARO 0.91 0.70 0.52 0.86 0.98 0.98 0.96 0.94 0.95 0.95

AROR(million) 5.8 5.4 4.6 1.6 5.4 4.3 3.0 5.7 5.4 5.1

AROD(million) 5.6 4.6 3.1 2.1 5.7 4.8 3.6 5.4 5.1 4.8

MAEARO (million) 0.2 0.9 1.6 0.5 0.3 0.5 0.6 0.3 0.3 0.3

rARO 0.97 0.95 0.88 0.86 0.98 0.98 0.97 0.97 0.96 0.95

ROR(million) 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6

ROD(million) 8.7 7.3 5.5 4.5 10.6 12.3 13.8 9.2 9.6 10.1

MAERO (million) 0.8 2.3 4.1 5.1 1.1 2.7 4.3 0.4 0.1 0.6

rRO 0.96 0.94 0.88 0.76 0.96 0.96 0.95 0.95 0.93 0.89

Rev+,R (million) 48.6 48.6 48.6 48.6 48.6 48.6 48.6 48.6 48.6 48.6

Rev+,D (million) 47.9 47.2 46.1 43.0 49.3 50.7 52.3 48.4 48.9 49.4

MAERev+

(million) 0.7 1.4 2.5 5.6 0.8 2.2 3.7 0.1 0.4 0.9

rRev+1.00 1.00 0.99 0.99 1.00 1.00 0.99 1.00 0.99 0.99

Rev (million) 44.8 44.5 43.7 40.6 44.4 43.3 42.0 44.7 44.4 44.1

Rev−,R (million) 39.0 39.0 39.0 39.0 39.0 39.0 39.0 39.0 39.0 39.0

Rev−,D (million) 39.2 39.9 40.6 38.5 38.7 38.5 38.4 39.3 39.3 39.3

MAERev−

(million) 0.2 0.9 1.6 0.5 0.3 0.5 0.6 0.3 0.3 0.3

rRev−

0.99 0.99 0.99 0.97 0.99 0.99 0.98 0.99 0.99 0.98