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DISSERTATION
Simulation-Based Analysis of
Forecast Performance Evaluations
for Airline Revenue Management
submitted to
Faculty of Business Administration and Economics
University of Paderborn
by
Dipl.-Wirt.Inf. Catherine Cleophas
Dean of the Faculty of Business Administration and Economics:
Prof. Dr. Peter F. E. Sloane
Referees:
1.) Prof. Dr. Natalia Kliewer
2.) Prof. Dr. Leena Suhl
Paderborn, July 2009
I
This thesis was created thanks to a cooperation between the International Graduate
School Dynamic Intelligent Systems at the University of Paderborn and the German
airline Deutsche Lufthansa AG. It includes the consideration of problems occurring in
applied revenue management under the aspect of academic research. The goal is to use
methodological approaches to airline revenue management, demand forecast and simula-
tion presented in the further text as well as expert knowledge and data available in the
industry.
The purpose of this text is the development of a new view of forecast performance, in
order to avoid some of the complications connected to evaluation of demand forecasts for
revenue management. To enable this, a theoretical concept of decomposing and evaluating
forecasts under the laboratory conditions provided by a simulation and using information
exclusive to simulation environments is developed. To demonstrate the potential of this
concept, the implementation of a simulation environment including a choice-based de-
mand model is documented. Finally, a number of statements about the implications of
forecast quality and forecast evaluation is expressed formally and tested using simulation
experiments to demonstrate the use of the proposed concept.
Subject classifications: Simulation, Forecasting, Revenue Management, Yield Manage-
ment, Inventory Control, Pricing, Price-Elasticity, Econometrics
Contents II
Contents
I. State of the Art and Research Opportunities 1
1. Introduction 3
1.1. Background and Terminology . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2. MotivationandGoals.............................. 6
1.3. Outline...................................... 9
2. Existing Research on Airline Revenue Management 12
2.1. Available Overview Literature . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2. State of the Art of Optimization . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3. Appraisals of Recent Challenges . . . . . . . . . . . . . . . . . . . . . . . . 15
3. Demand Forecasting for Revenue Management 21
3.1. DemandVolume ................................ 22
3.2. Unconstraining ................................. 25
3.3. DemandBehavior................................ 27
4. Demand Forecast Performance Measurements 33
4.1. Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2. Applied Forecast Performance Evaluation . . . . . . . . . . . . . . . . . . . 44
5. Research Gap and Opportunities 46
II. Solution Approach - Concept and Implementation 48
6. Simulation for Decomposition and Evaluation of RM Systems 50
6.1. OverallSystemView .............................. 50
Contents III
6.2. ForecastingComponent............................. 54
6.3. Demand Volume Aspect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6.4. UnconstrainingAspect............................. 59
6.5. Demand Behavior Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
7. Simulation Environment for Revenue Management 63
7.1. SimulationControl ............................... 65
7.1.1. DataManagement ........................... 65
7.1.2. Simulation Runs and Lists of Events . . . . . . . . . . . . . . . . . 68
7.1.3. Reporting ................................ 71
7.2. Supply and Demand Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
7.2.1. Supply Information . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
7.2.2. DemandModel ............................. 76
7.2.3. Exemplary Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7.3. Revenue Management Components . . . . . . . . . . . . . . . . . . . . . . 89
7.3.1. Forecast................................. 90
7.3.2. Optimization .............................. 96
7.3.3. Inventory ................................ 98
7.4. Market Implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.4.1. Demand Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.4.2. Supply Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
III. Experiments and Conclusions 110
8. Simulation Based Analysis of Forecast Performance 112
8.1. Observations on Long-Term Effects of Forecast Methods . . . . . . . . . . 112
8.2. Consequences of Possible Definitions of Psychic Forecasts . . . . . . . . . . 137
8.3. Evaluation of Standard Accuracy Indicators . . . . . . . . . . . . . . . . . 154
8.4. Definitions and Effects of Uncertainty of Demand . . . . . . . . . . . . . . 168
8.5. Evaluation Approaches for Price-Sensitive Forecasts . . . . . . . . . . . . . 182
8.6. Simulation-Based Findings Recaptured . . . . . . . . . . . . . . . . . . . . 187
Contents IV
9. Conclusion 191
9.1. Summary ....................................191
9.2. Outlook .....................................193
List of Figures V
List of Figures
6.1. Evaluating the RM System . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.2. Comparing Forecasts and Bookings . . . . . . . . . . . . . . . . . . . . . . 54
6.3. Evaluating the Forecast Component . . . . . . . . . . . . . . . . . . . . . . 55
6.4. Evaluating the Trend Component . . . . . . . . . . . . . . . . . . . . . . . 58
6.5. Evaluating the Unconstraining Component . . . . . . . . . . . . . . . . . . 60
6.6. Evaluating the Choice Component . . . . . . . . . . . . . . . . . . . . . . . 62
7.1. TheSimulationCycle.............................. 64
7.2. Revenue Management Simulation . . . . . . . . . . . . . . . . . . . . . . . 70
7.3. DefiningtheProduct.............................. 74
7.4. RequestGeneration............................... 77
7.5. Example–Product............................... 87
7.6. Example – Customer Types . . . . . . . . . . . . . . . . . . . . . . . . . . 88
7.7. Inventory: Protected and Available Seats . . . . . . . . . . . . . . . . . . . 100
7.8. Mix of Customer Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.9. SLF Average and Deviation depending on Error Term Deviation . . . . . . 105
7.10. Increase in Bookings by Additional Classes . . . . . . . . . . . . . . . . . . 109
8.1. Predicted Demand per Class with Exp050 . . . . . . . . . . . . . . . . . . 116
8.2. Decrease in Demand Predicted for Class “A” . . . . . . . . . . . . . . . . . 117
8.3. Protected Seats per Class with Exp050 . . . . . . . . . . . . . . . . . . . . 119
8.4. Decrease in Seats Protected for Class “A” . . . . . . . . . . . . . . . . . . 120
8.5. Observed Bookings per Class with Exp050 . . . . . . . . . . . . . . . . . . 122
8.6. Decrease in the Share of Bookings Observed for Class “A” . . . . . . . . . 123
8.7. Revenue in Percent of Revenue Earned in Run 1 . . . . . . . . . . . . . . . 125
8.8. Yield in Percent of Yield Earned in Run 1 . . . . . . . . . . . . . . . . . . 128
8.9. Yield in Percent of Yield Earned with First-Come-First-Serve . . . . . . . 129
List of Figures VI
8.10. Mean Absolute Deviation (MAD): Constrained FC from Observed BKD . . 131
8.11. Root Mean Squared Error (RMSE): : Constrained FC from Observed BKD 132
8.12. Mean Avg. Percentage Error (MAPE): Constrained FC from Observed BKD133
8.13. Theil’s U2 (U2): Constrained FC from Observed BKD . . . . . . . . . . . 134
8.14. Revenue Resulting from Exp050 and Exp050upd . . . . . . . . . . . . . . . 136
8.15. MAD during the Booking Horizon of Run 1 . . . . . . . . . . . . . . . . . 137
8.16. Uses of the Psychic Forecast in the Simulation . . . . . . . . . . . . . . . . 138
8.17. Average Revenue over 50 Runs in Percent of First-Come-First-Serve . . . . 145
8.18. Average Yield over 50 Runs in Percent of First-Come-First-Serve . . . . . . 147
8.19. Average SLF over 50 Runs in Percent of First-Come-First-Serve . . . . . . 149
8.20. Revenue in Percent of First-Come-First-Serve . . . . . . . . . . . . . . . . 152
8.21. Deviation of Revenue between Simulation Experimenets . . . . . . . . . . . 153
8.22. Possible Error Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 157
8.23. Rank of Methods according to MAD in Product-Based Scenario with “Vol.
=050,Dev.=00” ...............................158
8.24. Rank of Methods according to MAD in Product-Based Scenario with “Vol.
=100,Dev.=00” ...............................160
8.25. MAD: Constrained Psychic Forecasts vs. Actual Bookings in the Product-
BasedScenarios.................................162
8.26. MAD of Unconstrained Forecasts from Psychic Forecast in Product-Based
Scenario with “Vol. = 050, Dev. = 00” . . . . . . . . . . . . . . . . . . . . 163
8.27. Revenue in Percent of Revenue Earned by Psychic Forecast – Product-
BasedScenario .................................165
8.28. Ranks According to Different Error Measurements . . . . . . . . . . . . . . 167
8.29. Increase of Yield in Percent of FCFS from Vol. = 050 to Vol. 100 . . . . . 170
8.30. Revenue in Percent of FCFS on Price- and Product-Based Markets . . . . 171
8.31. Difference in MAD (Price-Based - Product-Based Market) in Percent . . . 173
8.32. MAPE Averaged over 50 Runs . . . . . . . . . . . . . . . . . . . . . . . . . 175
8.33. “Percent Better” Averaged over 50 Runs . . . . . . . . . . . . . . . . . . . 177
8.34. MAD for Naive Forecast Averaged over 50 Runs . . . . . . . . . . . . . . . 178
8.35. Runs Required for Confidence Level . . . . . . . . . . . . . . . . . . . . . . 179
8.36. Variance of MAD for Naive Forecast over 50 Runs . . . . . . . . . . . . . . 180
List of Figures VII
8.37. Revenue Robustness based on Rev. Averaged over 50 Runs . . . . . . . . . 181
8.38. Revenue in Percent of Revenue Earned in Run 1 . . . . . . . . . . . . . . . 184
8.39. MAD of Elasticity vs. Psychic Elasticity . . . . . . . . . . . . . . . . . . . 186
List of Tables VIII
List of Tables
7.1. Simulation Environment: Supply Lists . . . . . . . . . . . . . . . . . . . . 66
7.2. Simulation Environment: Demand Lists . . . . . . . . . . . . . . . . . . . . 67
7.3. Output of Simulation Experiments . . . . . . . . . . . . . . . . . . . . . . 72
7.4. Booking Classes Differentiated by Product-Feature . . . . . . . . . . . . . 106
7.5. Booking Classes Differentiated by Price . . . . . . . . . . . . . . . . . . . . 107
7.6. Booking Classes Differentiated by Product Characteristics and Price (Hy-
bridDifferentiation) ..............................107
8.1. Possible Variations of Choice in Psychic Forecasts . . . . . . . . . . . . . . 139
1
Part I.
State of the Art and Research Opportunities
2
In this part, an introduction to the topic of this thesis as well as a background in rev-
enue management and demand forecasting is provided. This includes existing research
on overall revenue management, optimization methods, and approaches to demand fore-
casting. In the course of recapturing the state of the art, current challenges for demand
forecasting and research opportunities with regard to forecast evaluation are pointed out.
Chapter 1: Introduction 3
1. Introduction
Two general approaches to pricing a product exist: Prices may be calculated in order
to ensure a break even (to cover the costs of production) or to maximize revenue. In
airline revenue management, the latter is achieved by calculating the optimal number of
products to sell at a set of prices, given products that create similar production costs.
For example, a ticket on a flight from Frankfurt to New York may cost 400 Euro three
months before departure, but the price for the same flight under the same conditions may
increase to 800 Euro when the ticket is bought three days before departure. The goal is
to sell all seats available on the flight at the highest price customers are willing to pay. In
the example, this is achieved by differentiating customer segments according to the time
of booking before departure. Revenue management is often cited to be the art of “selling
the right seats to the right customers at the right prices” (American Airlines (1987)).
In order to successfully apply revenue management, knowledge of customers is required.
It is impossible to sell the right seats at the right price if there is no information available
on the price customers are willing to pay and the conditions under which they are willing
to do so. This information is provided by a demand forecast. It aspires to predict the
amount of customers that will be willing to buy a ticket at a specific price and time.
Based on the forecasted demand, revenue can be maximized by optimizing inventory
controls. The result should be the maximum of revenue to be earned under the given
conditions. Depending on the forecast and the optimization method, these inventory
controls maximize revenue per flight or for a complete network of itineraries offered by
the airline.
In any case, the process described above indicates that the demand forecast has a
decisive influence on the outcome of revenue management. The forecast provides the
basis for any optimization and thereby influences the inventory controls. If a forecast is
underestimates valuable demand, too many tickets may be sold at a reduced fare, leading
Chapter 1: Introduction 4
to shortages in capacity when customers willing to pay high fares request to book. In
the case of overestimation, too many tickets may be reserved for customers expected to
pay high fares, leading to unsold tickets and empty seats at the day of departure. The
consequence of both errors is falling short of maximum revenue.
Due to their importance, it is necessary to evaluate the performance of demand forecast
methods. This task is aggravated by two complications. The definition of the term
“performance” is ambivalent: Most forecasts are rated by their accuracy, their ability to
correctly predict the future. Yet in revenue management, experience shows that sometimes
a forecast that lacks accuracy can still lead to high revenue - the major indicator of revenue
management success. The revenue resulting from the implementation of a forecast method
may therefore be considered another indicator of a method’s performance.
Even the evaluation of forecast accuracy alone is not trivial. The inventory controls that
are computed based on the forecast information limit the bookings to be observed. As the
historical data created by booking bookings often the the only information available on
actual demand, they are frequently used to evaluate the forecast. The forecast’s accuracy
is estimated based on an indicator it influenced – the risk of self-fulfilling prophecies arises.
In this thesis, a decomposition approach as a new concept for the analysis of the evalu-
ation of forecasts is presented. For this purpose, a decomposed view of revenue manage-
ment and demand forecasting is introduced. Based on this concept, systematic evaluation
processes are developed. A simulation environment is documented to demonstrate the
implementation of these processes. Finally, a number of hypotheses on the evaluation of
demand forecasts are formalized and tested using simulation experiments.
1.1. Background and Terminology
Revenue Management plays an important role in the business model of airlines all over
the world. Optimizing the seat allocation in order to maximize revenue has a history that
goes back to 1950. Before the deregulation of the airline industry introduced competition,
revenue management mostly meant overbooking settings optimized to avoid empty seats.
For this purpose, forecasts estimate the number of cancellations and no-shows. Can-
cellations are returns of booked tickets within the booking horizon; no-shows indicate
Chapter 1: Introduction 5
customers with booked tickets failing to show up at the day of departure. Without over-
booking, the number of seats sold is equal to the capacity of the aircraft assigned to a
flight. With overbooking, the number of seats sold exceeds capacity not every customer
that books a ticket is expected to require a seat. The risk included in the practice of over-
booking is the occurrence of denied boardings. Denied boardings happen when customers
that booked tickets are denied as not enough seats are available due to an overestimation
of cancellations and no-shows. They are connected to direct costs as alternative trans-
port or over-night-stays have to be made available by the airline and to indirect costs as
customer satisfaction decreases.
The idea of maximizing revenue by offering tickets at different prices gained in impor-
tance with the Airline Deregulation Act of 1978. With this act, the American government
opened airline markets by removing control from route planning, fares and market entry.
This lead to increasing competition and, as a consequence, to competitive pricing.
In order to gain advantages over competition, so-called early bird offers were introduced.
In this form of customer segmentation, as in the example used in the introduction, tickets
are offered at reduced fares in the beginning of the booking horizon. As time to departure
decreases, fares increase, and tickets become more expensive. The underlying assumption
is that those customers that are willing to pay high fares, such as business travelers,
request shortly before departure. Customers that book early and at reduced prices are
not expected to be willing to pay the regular fare. With the introduction of this concept,
demand forecasts predicting the number of customer requests to arrive for specific classes
at a given time in the booking horizon gained importance. The decision of how many
tickets to sell at reduced fares and how many tickets to reserve for valuable customers is
based on such predictions.
In addition to the timing of request within the booking horizon, customer segmentation
can also be based on features or restrictions attached to booking classes. Features may
include superior physical comfort in the form of seating in the business compartment
or positive conditions such as a flexible re-booking policies. Restrictions are aimed at
rendering tickets sold at reduced fares less attractive, for instance by imposing negative
conditions such as minimum stay or weekend stay.
Chapter 1: Introduction 6
With increased computing power, the focus shifted from a flight view to a network view
of revenue management. Also referred to as origin-destination revenue management,
this approach includes the idea that customers do not actually desire to book tickets on
isolated flights. Rather, they want to travel from an origin to a destination via a network
of itineraries offered by one or more airlines. Such itineraries connect two airports and
can consist of one or more legs. Legs are the edges of the network, described by two
vertices (airports) connected by direct flights. Instead of predicting demand to arrive for
single flights, a network-based demand forecast predicts requests for itineraries. Instead
of maximizing revenue per flight, a network-based optimization for revenue management
aims to maximizes revenue for the complete network.
In recent years, the Internet has improved market transparency. Additionally, no-frills
airlines now offer restriction-free classes. These are not differentiated by either features
or restrictions, but only by their fare. When customers are able to compare the offers of
different airlines easily and are offered classes only differentiated by price, the independent
or static model of demand collapses. Customers no longer request tickets in a specific class.
Instead, they may be willing to buy a valuable class but use opportunities for booking at
lower fares.
Models now need to include the idea of correlated and flexible demand. Customer re-
quests depend on which itineraries and classes are made available. With these challenges,
the importance of forecast performance (and its evaluation) increases.
1.2. Motivation and Goals
As pointed out for example in P¨olt (1998) and Weatherford & Belobaba (2002), the
accuracy of the demand forecast has a significant impact on the success of revenue man-
agement. The target function of most methods for optimizing revenue includes predicted
demand. Depending on the number of customers expected to request tickets at a higher
price in the future, current requests are accepted or denied. Examples of optimization
methods have been summarized in Weatherford & Bodily (1992), McGill & Ryzin (1999),
and Talluri & Van Ryzin (2004b).
Chapter 1: Introduction 7
Forecast accuracy can have indirect effects on long-term results of revenue management.
While the information drawn from the overall results of a revenue management system
offers conclusion toward the financial success, it does not provide insights toward the
accuracy of the forecast.
While it is important for the performance of an airline’s revenue management strat-
egy, measuring forecast accuracy is not trivial. Most analyses draw conclusions from
comparisons of demand forecasts to historical booking data – the result are error mea-
surements. This data has been shaped by the inventory controls in place at a given point
of time before departure. These controls the results of revenue optimization techniques.
The optimization uses the demand forecast as input. Therefore, evaluating a forecast by
comparing it to the bookings that resulted from its application is a biased approach.
Furthermore, while a forecast based on historical data offers information on expected
demand, bookings have been constrained by limitations in capacity. If demand exceeds
capacity, forecasts should exceed bookings. As a consequence, error measurements incor-
porate unconstraining of historical booking data or constraining of the forecast. However,
the transformation of bookings to demand is a component of the forecast. The transfor-
mation of a forecast to compare it to bookings is done using the inventory controls. Both
comparisons result in a bias as a part of the forecast method is used for its own evaluation.
Ideally, in order to make statements about the performance of a forecast, one would like
to know the actual demand it attempted to predict. As no detailed information about
each individual customer’s decision processes can be attained, this is not possible in
practice. The nearest approximation are customer surveys (stated and revealed preference
data) as documented in Algers & Beser (2001) or click-streams hinted at in Nason (2007).
Both are not a satisfying solution: While click-streams focus exclusively on web-based
points of sale, customer surveys carry the risk of bias – customers may lie consciously or
unconsciously.
Accepting that the actual demand may never be known, feasible alternatives have to be
explored. One is the comparison of the results gained from the implementation of different
forecast methods when all other factors remain the same (ceteris paribus). This would
mean using first one, then the other method while keeping the same optimization system
and in the exact same market environment. Again, this does not seem realistic: As the
Chapter 1: Introduction 8
same seat cannot be sold twice, the same flight event may not be optimized twice. Two
different flights either take place in geographically and therefore economically different
markets or at different points of time (under circumstances influenced by season and time
of day) or are bound to influence one another. Due to this, any implementation of this
method in practice is likely to be flawed.
Additionally, the question of whether accurate forecasts are good forecasts is still dis-
cussed in the revenue management research community. Simulation experiments have
shown that a forecast that is somewhat flawed, depending on the pricing and competition
situation given, can lead to higher revenue than what is earned when demand is estimated
correctly. For this reason, the further text will make a distinction between forecast per-
formance and accuracy. The latter is regarded as an influential part of the former, but
not as its synonym.
The main goal of this thesis is the development and demonstration of a decomposed
approach to evaluating forecast performance. To compare the performance of methods,
a clear separation between forecast, optimization, and further means of strategic manip-
ulation located in the inventory is necessary. With a decomposition concept, the conse-
quences of changes in any component can be analyzed separately. At the same time, the
interpretation of overall system performance is still feasible.
For the demonstration of the decomposed concept, a simulation environment offers
many advantages. It provides a stable and fully controllable framework. Two simula-
tion scenarios may confront a system with the exact same customer model. Under such
conditions, solely methodical factors can cause differences in bookings and revenue. The
applicability of the results of such a simulation can be secured by including a sufficiently
complex customer model as well as a realistic network of flights. The results offer insights
not just concerning the financial success and the accuracy of different forecasting meth-
ods. In addition, conclusions toward the connection of the two and their evaluation may
be drawn.
In order to achieve the defined goal and to demonstrate the advantages of a decomposed
concept of evaluation in a simulation environment, a number of tasks are defined:
•categorization of existing forecast methods;
•characterization of existing approaches to forecast evaluation;
Chapter 1: Introduction 9
•conceptualization of a decomposition of revenue management systems and demand
forecasts;
•formulation of processes to separately evaluate the components of revenue manage-
ment systems;
•implementation of a simulation environment to apply the concept;
•formalization of statements on forecast performance evaluation to base simulation
experiments on;
•analysis of the results of experiments conducted in the simulation environment.
The next section describes the form in which the approach to these tasks was docu-
mented in the following chapters of this thesis. It shortly summarizes the content of each
of the three parts.
1.3. Outline
This thesis is divided into three parts. In the first part, an introduction to revenue man-
agement and the state of the art regarding forecasts and their evaluation is provided.
A research gap is identified and the research opportunities derived from it are listed.
In the second part, the solution approach is outlined. For this purpose, the concept of
the decomposed evaluation of demand forecasts and the implementation of a simulation
environment are documented. In the third part, simulation experiments and the find-
ings resulting from them are summarized. Conclusions toward the evaluation of forecast
performance and future research are drawn.
Part One: As the motivation and the findings of this work are based on existing methods
of airline revenue management, first an overview of published research in this field is
presented in Chapter 2. This chapter starts with a general introduction to the history,
the motivation, and the terminology of revenue management. Approaches to maximizing
revenue according to linear optimization, dynamic programming, and heuristic methods
are presented next. Finally, recent challenges such as the increased overview of the market
place provided by Internet search engines and the advent of no-frills airlines are listed.
Chapter 1: Introduction 10
These developments as well as its general role in maximizing revenue are used to underline
the importance of accurate demand forecasting.
Approaches to demand forecasting are summarized in Chapter 3. They are categorized
by aspects of demand including overall volume, unconstraining of bookings and flexible
demand behavior. The multitude of mathematical methods available to predict the various
aspects demand to come becomes apparent in this chapter.
Approaches to quantifying forecast performance are presented in Chapter 4. Further-
more, examples of applied forecast performance measurements are provided.
In Chapter 5, research opportunities with regard to demand forecast performance mea-
surement are pointed out. These research opportunities serve as motivation and provide
aims for the further work.
Part Two: A new approach to decomposing and evaluating demand forecasts is pre-
sented in Chapter 6. This chapter describes how a simulation system can be used to iso-
late different components and aspects of the system. These parts can then be evaluated
separately. Documented are concepts for analyzing the complete revenue management
system, the isolated forecast component and the decomposed aspects of demand volume,
unconstraining and behavior.
To implement this concept of decomposition, a simulation environment is documented
in Chapter 7. General concepts of simulation control are introduced and the implementa-
tion of a supply and demand model is described in detail. In addition, the precise revenue
management methods included are explained. Finally, the market implementations pre-
pared for simulation experiments are presented.
Part Three: Based on the simulation environment introduced in Chapter 7, simulation
experiments have been conducted to provide examples for the simulation based analysis
of forecast performance. The hypotheses that were evaluated empirically are formally
listed in Chapter 8 together with the conclusions that can be drawn from the results of
the experiments. The aspects under which revenue management in general and demand
forecasting in particular were analyzed include the long-term effects of methods, the idea
Chapter 1: Introduction 11
of psychic forecasts used in the simulation, the key performance indicators traditionally
used, the consequences of demand uncertainty and first approaches to evaluating price-
sensitivity for revenue management.
Finally, Section 9 provides a summary of steps taken and the insights gained. In
addition, an outlook to potential further research is offered.
Chapter 2: Existing Research on Airline Revenue Management 12
2. Existing Research on Airline Revenue Management
A great body of published research has been devoted to airline revenue management.
Literature including overviews of existing research is listed in the first part of this section.
Next, some approaches to maximizing revenue given available knowledge of demand are
described. Finally, recent challenges that revenue management experts are confronted
with are documented.
2.1. Available Overview Literature
A first introduction to revenue management and its importance for business success is pro-
vided by Cross (1997). The most important research with regard to demand forecasting
for revenue management up to 1999 has been outlined in McGill & Ryzin (1999). More
information, especially on general theory, is provided by Pak & Piersma (2002) and Talluri
& Van Ryzin (2004b). One of the latest overview articles, Chiang et al. (2007), mostly
concentrates the application of revenue management. Weatherford & Bodily (1992), Bi-
tran & Caldentey (2003), and Boyd & Bilegan (2003) focus on other areas of revenue
management such as the development of a typology, pricing, and the implications of
e-commerce.
Some detail on how to model the restrictions and the objective of revenue management
is offered in Wang (1982). An account of the development of a revenue management
system at American Airlines is provided by Smith et al. (1992). Revenue management
in the broader context of operations research problems in the air transport industry is
presented in Barnhart et al. (2003). Methods for approaching the subject with differing
degrees of sophistication are presented in Vinod (2006).
Chapter 2: Existing Research on Airline Revenue Management 13
2.2. State of the Art of Optimization
The research listed here is concerned both with mathematically optimal solutions and
heuristic approaches. Furthermore, most models used in revenue management rely on
simplifying assumptions – these are described with the references.
Flight Optimization: In Littlewood (1972), a first approach to optimally allocating
availabilities based on demand predicted per flight and booking class is introduced. Its
optimality given the condition of static demand was proven by Mayer (1976). Similar
topics are considered in Bhatia & Parekh (1973), Richter (1982), Gerchak et al. (1985),
Alstrup et al. (1986), Kraft et al. (1986), Pratte (1986), Wollmer (1986a), Gerchak &
Parlar (1987), Pfeifer (1989), Wong (1990), Stone & Diamond (1992), Wollmer (1992),
M. Li (1997) and M. Z. F. Li & Oum (1998). The expected marginal seat revenue
(EMSR) approach to allocating seats, which is widely used in practice, was introduced
in P. Belobaba (1987a), P. Belobaba (1987b), and P. Belobaba (1989). The underlying
assumption of low-fare demand arriving before high-fare demand is discussed with regard
to optimality in Titze & Griesshaber (1983). The concept of nesting booking classes to
ensure the availability of valuable classes is outlined in W. Swan (1993c).
In M. Z. F. Li & Oum (2002), optimality conditions of models for flight-based revenue
maximization are discussed. One of the first considerations that demand for several
classes may be stochastically dependent is presented in Brumelle et al. (1990). Brumelle
& McGill (1993) maximizes revenue when demand is dependent on the fare; prices as
decision variables are also considered in Weatherford (1997a,b, 2001). The idea that
fares may not be monotonically increasing within a fare structure is included in Robinson
(1995). A consideration of how fare structures may influence demand and a concept of
encouraging sell-up between classes are described in Botimer & Belobaba (1999).
The task of maximizing revenue is modeled as a knapsack problem in Young & van
Slyke (1994) and Young & van Slyke (2000). Cancellations and no-shows are included
in the model used in Subramanian et al. (1999). Ryzin & McGill (2000) presents an
adaptive algorithm aimed at eliminating the strong reliance on demand forecasts. A
similar goal is pursued by Ball & Queyranne (2006) via the use of competitive analysis of
Chapter 2: Existing Research on Airline Revenue Management 14
online algorithms. A situation in which only limited demand information is available is
discussed in Lan & Gao (2007). All these papers consider leg-based optimization only.
Flight-based models that consider stochastic uncertainty with no strict assumptions
about the timing of demand arrival are often optimized using dynamic pricing. This
dates back to Kincaid & Darling (1963). For multiple classes and customers that do not
exhibit a strict arrival pattern, this is presented in T. C. Lee & Hersh (1993) and Zhao &
Zheng (2000). Brumelle & Walczak (1997) includes the possibility of customers arriving
in batches, Walczak & Brumelle (2007) builds up on this. A consideration of no-shows and
diversion between flights is given in Zhao & Zheng (1998). In Lautenbacher & Stidham
(1999), the maximization of revenue with dynamic pricing given dependent demand is
discussed. In Cooper & Mello (2002), the limitation of certain stochastic programs to
flight-based problems is pointed out.
Network Optimization: While flight-based optimization becomes more sophisticated
and relies less on simplifying assumptions, the maximization of revenue over complete
networks has come into focus. Models that include several segments being offered on
one flight may be regarded as the preparation of this. Ladany & Bedi (1977), Hersh &
Ladany (1978), Buhr (1982), Wang (1983), and Simpson (1985) discuss this possibility.
Smith & Penn (1988) and Vinod (1995) are also concerned with the optimization of
multiple segments.
The first approaches to modeling the passenger flow over a network have been docu-
mented in Glover et al. (1982), D’Sylva (1982), Dror et al. (1988), Curry (1990), Vinod &
Ratliff (1990); Vinod (1990), Phillips et al. (1991), Wong et al. (1993) and Talluri (1994).
As one of the first, Williamson (1992) uses a simulation system to demonstrate the impact
of different network and leg-based revenue optimization methods. A linear programming
approach to network optimization is outlined in Wollmer (1986b), while a non-linear pro-
gramming approach is considered in Vinod (1991). In Garcia-Diaz & Kuyumcu (1997),
a cutting plane approach is used to solve the network problem. While Williamson (1988)
and Vinod (1989) introduce the concept of virtual nesting, Williamson (1992) considers
the decision of whether to accept or deny a customer request as the comparison to a bid
price.
Chapter 2: Existing Research on Airline Revenue Management 15
In Talluri (2001), the idea that passengers may be routed in a way that creates balances
high- and low-fare demand over itineraries is introduced. The idea that different itineraries
might compete for demand is expanded in Coldren & Koppelman (2005).
Gallego & Ryzin (1997) expanded the dynamic pricing method in order to consider
networks rather than legs. In Gallego & Hu (2007), dynamic pricing is applied with
special regard to recent challenges such as restriction-free classes and diversion between
competing flights.
Cooper & Mello (2003) propose a combination of mathematical programming meth-
ods and heuristics to make network revenue management applicable to the practice. A
concept to avoid the distinction between forecast and optimization is offered in Chen
et al. (2003): Value functions for network revenue management are estimated via sta-
tistical learning. M¨oller et al. (2004) attempts to maximize network revenue via linear
programming using stochastic scenarios rather than straight-forward demand forecasts as
input. The introduction of a network-based revenue management system in practice is
documented in Swift (2002) and Cutshall & Weisbrodt (2006).
In de Boer (2003), among other concepts, the use of simulation to maximize network
revenue is proposed; this prepares the way for Bertsimas & de Boer (2005). Ryzin &
Vulcano (2006) proposes a simulation-based approach to optimization that extends the
work of Bertsimas & de Boer (2005) to include a continuous model of capacity and de-
mand. A concepts for the joint optimization of inventory controls and fleet assignment is
introduced in Frank et al. (2006) and evaluated with the help of a simulation system.
2.3. Appraisals of Recent Challenges
Some developments concerning the market place and distribution channels have compli-
cated revenue management for airlines during the last decades. As customers are able
to effortlessly compare fares via websites such as opodo.com, travelocity.com, or low-
fares.com, they have become more flexible and price-oriented. Considerations of the im-
pact of increased market transparency and the consequences for consumers can be found
in Nason (2007). The entry of so-called no-frills airlines into the market place helped
Chapter 2: Existing Research on Airline Revenue Management 16
to further stress the influences of prices and put traditional airlines under pressure. The
business model of such airlines is described in some detail in Calder (2006).
General introductions to this situation and the consequences for revenue management
can be found in Boyd & Kallesen (2004), Dunleavy & Westermann (2005) and Ratliff
& Vinod (2005). In Boyd (2004), special attention is paid to the impact of Internet
sales. The rest of this section is devoted to three major challenges: Customers basing
their booking decision on which classes are available (dependent demand), customers
considering competition offers, and strategic customers delaying their bookings as they
hope for reduced fare offers.
Dependent Demand: The consequences of ignoring flexible customers and optimizing
revenue under the assumption of static demand are analyzed in Cooper et al. (2006):
When customers purchase the lowest available fare, static forecasts become self-fulfilling
prophecies. Availabilities assigned to reduced-fare classes are used up by flexible cus-
tomers, creating bookings – the basis for future forecasts. As more and more demand is
predicted for tickets offered at a low fare and the optimization algorithm allocates more
availability to low fares, revenue suffers the spiral-down effect.
As the perils of not reacting to changes in customer behavior are established, much
research has been devoted to new approaches to revenue management. Special attention
was directed toward restriction-free environments. This term describes the offer of booking
classes differentiated only by price. Restrictions such as weekend-stays or minimum-
stays are not applied. This strategy is frequently implemented by no-frills airlines and
matched by traditional airlines striving for competitiveness as described in Dunleavy
& Westermann (2005). Consequences of restriction-free environments are demonstrated
using the Passenger Origin Destination Simulator (PODS, see also C. Hopperstad (2000),
Gorin (2000), Gorin (2004) and Reyes (2006) for introductions) in Cusano (2003) and
P. Belobaba & Gorin (2004).
In P. Belobaba & Hopperstad (2004), the idea of the spiral-down effect is summarized
and a definition of sell-up is given: The term describes the probability of a customer
buying his ticket for a specific fare, given he would have bought the lowest fare had it
Chapter 2: Existing Research on Airline Revenue Management 17
been available. The presentation introduces a modified EMSRb algorithm based on sell-up
estimates as well as a dynamic program including knowledge on sell-up.
A different view of price-sensitive customers, called buy-down, is introduced in Ozdaryal
& Saranathan (2004). The term describes the phenomenon of customers who would
previously have bought one specific class now buingy a cheaper class, if it is available. As
a counter agent, inventory fences limiting the availability of cheap classes are proposed.
More comparisons of low-cost environments versus the traditional network models of
revenue management can be found in Weber & Thiel (2004). As ways of dealing with
price-elasticities the authors suggest neural networks.
A summary of methods for restriction-free environments is provided in Cl´eaz-Savoyen
(2005). The author presents and evaluates two network-optimization methods based on
the knowledge of sell-up probabilities: (Q-Forecasting as introduced in B. P. Hopperstad
C.H. (2004) and Fare Adjustment as introduced in Fiig & Isler (2004)). Detailed results
for an evaluation of fare adjustment using PODS are also presented in Lua (2007a,b,c) as
well as in Kayser (2007).
The consequences of restriction-free environments and a more transparent market place
can also be phrased as a change in assumptions. Traditional revenue management re-
garded demand as static or independent: Customers buy tickets in one booking class,
if that class was not offered, they do not buy at all. The new view includes dependent
demand: Customers choices depend on the classes offered.
P. Belobaba (1987b) touches on this idea when describing possibilities for incorporation
of passenger shifts in the EMSR model. In Pfeifer (1989), decision rules are implemented
based on the probability that customers are “shoppers”, basing their booking decision on
fares. Among the first papers to consider a change in assumptions is also Brumelle et al.
(1990). The author expands Littlewood’s Rule to include dependent demand.
An approach that is not just limited to the airline industry is documented in Bodily
& Weatherford (1995). In this context, buy-down is termed diversion. Heuristic deci-
sion rules are suggested for multiple nested classes in order to avoid as much diversion
as possible. P. Belobaba & Weatherford (1996) also considers diversion and evaluates
decision rules to minimize it. In addition, the authors present a new heuristic. Zhao &
Chapter 2: Existing Research on Airline Revenue Management 18
Zheng (2001) introduces a dynamic threshold policy in order to limit diversion. The au-
thors differentiate between static and dynamic policies: Whereas traditional approaches
to maximizing revenue such as Littlewood’s rule provided static policies, dynamic rules
allow for a change customer behavior and according changes in the policy. Furthermore,
the model includes the restriction that fare classes, once they are closed, can not be
re-opened.
A different view of dependent customer behavior is taken in Gallego & Phillips (2004)
and Gallego et al. (2008). Used in order to develop flexible or callable products, dependent
demand is regarded as an opportunity rather than a risk.
Competition: As customers have access to websites summarizing the offers of several
airlines, they can comparison-shop much easier. This makes it necessary for airlines to
consider competition to a greater degree.
In Fischer & Kamerschen (2003), an analysis of demand aggregated to airport-pairs
with regard to the market situation in terms of competition is presented. The authors
employed the Rosse-Panzar test in order to measure the consequences of competition on
airline markets.
The conclusions of this examination are not so much revenue management recommen-
dations as they are general statements about economic implications: The more intense
competition is, the lower the average fare. It is stated that with regard to the data used,
airline competition is not as perfect as often supposed.
An approach that focuses more on game theory aspects of airline revenue management
under competition is documented in Netessine & Shumsky (2005). Considered are both
vertical (different airlines compete on different legs of a multi-leg itinerary) and horizontal
(different airlines compete on the same leg) competition. Conditions for a Nash equilib-
rium are provided as desirable characteristics of the demand distribution. However, no
method of estimating the real consequences on a given market is provided.
A game theoretical view of revenue management is also taken in Gallego et al. (2006).
Without special regard for the airline industry, revenue management under competition is
regarded as a sequential and as a repeated game. Again, conditions for a Nash equilibrium
Chapter 2: Existing Research on Airline Revenue Management 19
are outlined and the advantages of competitors cooperating are pointed out. That condi-
tion is very difficult to realize under the legal circumstances of airline revenue management
practice.
Another example of the inclusion of competitor information is provided by Walczak
(2005). However, while knowledge of the influence of competition on demand is included
in a dynamic program according to this presentation, this knowledge is regarded as given.
Its estimation is not part of the research.
In Coldren & Koppelman (2005), demand shares for travel along itineraries of competing
airlines are predicted. The model includes departure day, brand, and service as factors.
Both a multinomial logit model and variations of the nested model are considered in
order to analyze the influence of these factors. The price differences between the different
airlines’ offers are not considered.
In the course of research conducted with PODS, some concepts for including knowl-
edge of competition in the estimation of sell-up rates have been outlined. These and the
resulting findings have been presented at several PODS summits. Recent examples can
be found inCarrier (2003), Guo (2007), and C. Hopperstad (2007). In addition, research
concerned with fare adjustment as for instance Kayser (2007) underlines that the con-
sideration of customer reactions to prices becomes even more important in markets with
intense competition. Finally, matching the lowest competitor price is presented as an ap-
proach including competition in revenue management that avoids actually forecasting the
customer reaction in Lua (2007a,b,c). In all these publications, knowledge of competitor
prices is regarded as given.
It can be concluded that much research that considers the effects of competition does
so only with regard to its general consequences. Forecasting methods that do consider a
wider range of influence factors often offer the possibility of including competition prices
or the existence of competition in the model.
In Gallego & Hu (2007), demand models that consider more characteristics of the
product as decision factors for customers are examined with special regard to competi-
tion. Dynamic pricing is presented to include demand forecasts that incorporate such
a customer model in the optimization. With consideration of the single-leg model, the
authors extend both on Netessine & Shumsky (2005) and Gallego et al. (2006). With
Chapter 2: Existing Research on Airline Revenue Management 20
regard to the airline industry, cooperation is excluded. An open-loop Nash equilibrium is
presented. Changes in customer demand as caused by competition are ascribed to the
predictions of a customer choice model. A general introduction to dynamic pricing and
the opportunities it offers is provided in Westermann (2006).
Strategic Customers: Another phenomenon along with the idea of flexible, informed,
and price-sensitive customers is that of strategic as opposed to myopic customers as
described by Talluri & Van Ryzin (2004b). This idea implies that customers learn from
their previous booking experiences and use their knowledge for future decisions. For
instance, a customer observed an offer for a reduced-fare ticket coming up shortly before
departure might delay his next booking, hoping for cheap tickets to be offered late in the
booking horizon again. Such strategic behavior needs to be recognized and considered in
inventory policies in order to avoid revenue losses.
Such demand behavior is described in Anderson & Wilson (2003); the authors point
out the impact of customers implementing strategies to counteract airline revenue man-
agement strategies. While not offering a concept on how to quantify or react to strategic
customers, the article highlights their possible importance to future revenue management
research. Xu & Hopp (2005b) considers strategic behavior in the retail industry as well
as in the airline industry and compares empirical observations with regard to changes in
price-elasticity. In Cho et al. (2007), fare track systems designed to systematically mine
airline fare data to enable customers to minimize their expenses are analyzed. Potential
benefits of such systems both for customers and airlines are listed.
Levin et al. (2006) describes a dynamic pricing strategy based on the assumption that
the market is monopolistic and customers may delay their bookings strategically. Knowl-
edge on customers’ strategies is regarded as available. Once more the dire consequences
of ignoring strategic customers are pointed out. Another model of dynamic pricing under
similar conditions is presented in Su (2007). Wilson et al. (2006) considers a demand
model with customers reacting flexibly to offers by buying the lowest available fare and
strategically delaying their bookings. An approach to finding optimal booking limits
under these conditions is offered.
Chapter 3: Demand Forecasting for Revenue Management 21
3. Demand Forecasting for Revenue Management
Forecasting demand is one of the fundamental elements of Revenue Management. Knowl-
edge of the amount and the qualities of demand for seats on flights is crucial for a successful
optimization model. Research such as presented in P¨olt (1998) shows that forecast accu-
racy result has a positive impact on revenue. This has also been confirmed in Weatherford
& Belobaba (2002). An analysis of business risk presented in Lancaster (2003) shows that
much of this risk stems from faulty forecasts.
A succinct characterization of the demand model needed for revenue management is
offered by van Ryzin (2005):
[...] it is really the entire system for estimating demand and market response
– the data sources, the information technology for collecting and storing data,
the various statistical estimations models and algorithms used to process and
analyze these data and the infrastructure for deploying model outputs – in
short, everything that is required to turn raw data into actionable market
information. This is normally called the ’forecasting system’ in traditional
RM parlance, though forecasting is merely one of its many functions.
Developments such as described in Section 2.3 make forecasting for revenue manage-
ment even more complex. At the same time, a more automated approach to revenue
management, growing data management opportunities, and the resulting need for higher
quality forecasts increase the interest in forecast methods (see for example Zaki (2000),
Chiang et al. (2007), or Nason (2007)).
In order to provide an overview of forecasting for revenue maximization, a categorization
is introduced. Research is listed according to the characteristics of demand that it focuses
on. Three characteristics of demand are discussed:
Chapter 3: Demand Forecasting for Revenue Management 22
•Demand volume: The absolute amount of passengers that request tickets can be
split up into the arrival process of demand throughout the booking horizon and the
development of overall demand over several departures due to trends and seasonality.
•Unconstraining: The transformation of observed sales data to deduce actual de-
mand.
•Demand behavior: The reaction of customers requesting tickets to the alternatives
offered by the airline and its competitors.
More general information on mathematical methods mentioned as approaches to fore-
casting are described in the following sections can be found in Armstrong (2001).
3.1. Demand Volume
When within the booking horizon will customers request tickets? How many customers,
overall, will demand tickets for a certain flight departure? If information on the amount
of demand per product and the timing of its arrival is available, target functions can be
formulated to optimally allocate capacity. Research conducted in order to gain estimates
for both dimensions will be summarized in this section.
Much early research on forecasting for revenue management has been done to optimize
overbooking levels. This is not the key question considered in this thesis; further reading
related to overbooking can be found in Taylor (1962); Rothstein & Stone (1967); McGill
(1989).
Arrival Process: To determine the optimal availabilities within the booking horizon,
three aspects of demand have to be available: A fixed overall demand (regarded as the
state space according to a terminology derived from A. O. Lee (1990)), the rate of demand
arrival as well as the sequence of high- and low-fare demand.
First distributions that describe demand arrival for a single fare class and flight are
given in Beckmann & Bobkowski (1958). The resulting description of demand is referred
to as booking curve. In addition to the positive impact of accepted passenger requests,
cancellations have a negative impact on this curve.
Chapter 3: Demand Forecasting for Revenue Management 23
In order to reduce complexity, it is common to divide the booking horizon up and to then
observe bookings as they occurred in time slices. Method that solely considers bookings
that have already been accepted on the flight is referred to as advanced bookings method
according to A. O. Lee (1990). Those concepts that only refer to observed demand on
previous departures of the same flight numbers are regarded as historical booking methods.
It is noted that methods combined from both concepts tend to work best – accordingly,
current research uses information both from past flights and already accepted bookings.
Poisson processes as a special case of Markovian processes are a common way of mod-
eling demand arrival – one of the first examples is Rothstein (1968). As they assume
the state of the system they model to be influenced only by the latest event, the arrival
of demand is regarded to be independent of booking controls. Customers request tickets
exactly once, if their request is denied they do not return. Recent examples of the applica-
tion of Poisson processes can be found in Weatherford et al. (1993), Talluri & Van Ryzin
(2004a) and Walczak & Brumelle (2007). The arrival rate of the customers is regarded
as a parameter to be estimated either via ad-hoc or time-series methods or by analyzing
influence factors.
To optimally allocate seats at any time within the booking period, Chen et al. (2003)
propose a method of statistical learning that estimates a market value for tickets be-
ing purchased at a specific time. The paper extends a model based on a discrete-time
Markov decision problem in Lautenbacher & Stidham (1999) to the network level. Instead
of explicitly forecasting demand, value functions for remaining seats are estimated and
updated.
A similar topic, demand learning for dynamic pricing, is taken up in Xu & Hopp (2005a).
This paper includes the assumption of a piecewise deterministic and Markovian customer
arrival process, the distribution of which is regarded as known, homogeneous and price
independent. It introduces estimates key parameters of the distribution by observing
demand as it arrives.
Correlations in demand for different products at the same point of time as well as
correlations in demand for the same products at different points of time within the booking
horizon are considered in Stefanescu et al. (2004). A linear mixed effects model considers
a booking time component, weighting matrices for correlations of influences by external
Chapter 3: Demand Forecasting for Revenue Management 24
shocks, and a normally distributed error term. Data is unconstrained using the estimation
maximization algorithm. In order to estimate the parameters of the demand distribution,
maximum likelihood is applied.
Final Bookings: Overall demand volume can fluctuate over time due to economic trends,
the influence of seasons, trade fairs, and holidays, within weekly or daily patterns. As
described in Talluri & Van Ryzin (2004b) and Armstrong (2001), a variety of statistical
methods can be employed in order to pick up trends and patterns. Two views stand out:
Demand may be predicted by considering the data patterns and emulating them with ad-
hoc methods or it may be predicted by considering possible causal relationships between
the booking data and influence factors. The latter approach is also called associative.
Much current revenue management research focuses on demand behavior rather than
overall demand development. Research considering the macro level of airline demand as
needed for airline fleeting, strategy, and general economics stays important. Examples
can be found in Andersson (2001), Abed et al. (2001), Bhadra (2003), Battersby (2005)
and Cunningham & De Haan (2006). Brons et al. (2002) and Njegovan (2006) consider
the influence of prices on overall demand levels.
In Grosche et al. (2007), a rather macroeconomic view of demand estimation is taken,
too. Still, the authors do mention the possibility of using their estimates for the op-
timization of itineraries for which no historical data exists yet. In order to calculate
the connection between service-related and geo-economic forces and demand, two gravity
models are implemented. These assume the influencing factors to be independent.
Even if demand is considered stationary, some fluctuation is likely to occur from one
departure to the next. Although focusing on spill estimation for fleet assignment, W. Swan
(2002) indicates that a gamma distribution of average stationary demand makes sense for
revenue management. The paper follows up on research described in W. Swan (1993a,b,
1999).
The idea that overall demand volume may not be exclusively derived the time factor
is presented in Sa (1987). The author introduces a model of regression analysis that may
be applied to different markets to forecast final bookings. Apart from time and historical
bookings, this regression considers socio-demographic factors as well as level of service
Chapter 3: Demand Forecasting for Revenue Management 25
variables. An earlier overview of regression models for demand estimation can be found
in Taneja (1978).
In order to predict demand over a network, Neuling et al. (2004) proposes an analysis
of passenger name records. This data includes information on each passenger’s itinerary,
including all booked segments. In addition to a regular demand forecast, a no-show
forecast is offered.
A theory of time series influenced by a variety of factors is laid out in Armstrong et al.
(2004). The authors describe a concept of decomposing the series to represent the causal
forces that impact it.
Time series and their reaction to external factors are also the focus of hybrid methods
combining traditional statistics and neural networks. An example of such an effort is
presented in Aburto & Weber (2007), which connects neural networks with an ARIMA
model. However, the application considers supermarkets rather than airlines.
3.2. Unconstraining
Historical booking data as stored by airlines does not represent actual demand. Customer
requests are constrained by the amount of tickets offered. Without further information,
it is unknown whether more tickets could have been sold, unless offer exceeded demand.
A general introduction to this topic is also presented in P¨olt (2000). Unconstraining, also
called detruncation, refers to the transformation of bookings to demand.
Nahmias (1994) refers to the demand that is not accepted as bookings as lost sales. The
paper concentrates on retail rather than the airline industry and describes the application
of maximum likelihood estimation, a best linear unbias estimator and an additional new
estimator.
In Zeni (2001a) the author extensively presents and compares a number of unconstrain-
ing methods. The findings are also summarized in Zeni (2001b). Among the listed con-
cepts are: ignoring the censoring, discarding the censored data, mean imputation method,
the booking profile method,expectation maximization, and projection detruncation.
Chapter 3: Demand Forecasting for Revenue Management 26
The easiest alternative is to ignore the censoring or to discard the censored data. The
mean imputation method, a variation of pickup unconstraining, and the booking profile
method are both ways of estimating demand by extrapolating from the mean of book-
ings that were not truncated. Expectation maximization uses a normal distribution and
maximum likelihood in order to iteratively estimate the parameters influencing demand.
Finally, projection detruncation is a variation of expectation maximization that uses an
additional parameter to scale the amount of unconstraining applied to the data.
The conclusion of Zeni (2001b) is that anything preferable to ignoring the truncation.
While intricate and computationally intense, expectation maximization works best. Such
findings are also confirmed by Weatherford (2000) and Weatherford & P¨olt (2002).
McGill (1995) introduces a concept of censored regression in order to estimate customer
bookings. This model extends unconstraining by estimating demand depending on up to
nine factors. To that aim, the expectation maximization method is adapted to what the
authors call censored demand estimation maximization. Other examples of the use of
expectation maximization can be found in Talluri & Van Ryzin (2004b), Stefanescu et al.
(2004) and Ferguson et al. (2007).
While claiming not to use any forecasting techniques, Ryzin & McGill (2000) solves
the problem of unconstraining data by applying life tables. This method is taken from
survival analysis and is implemented to estimate parameters of a survival function. It
indicates how many additional requests arrived after a booking class was closed. Tests
indicate mixed performance, therefore the implementation is only recommended for small
or start-up airlines or when demand is difficult to predict.
A new method of unconstraining is proposed in Ferguson et al. (2007). It uses double
exponential smoothing, also called “Holt’s Method”. Two smoothing constants are intro-
duced to calculate the base demand and the trend. An application is described both
for monotonously closing booking classes and booking classes being re-opened. Based on
simulated customer requests, the new method is compared to an averaging method, esti-
mation maximization and projection detruncation taken from Weatherford & P¨olt (2002),
as well as the method of life tables described in Ryzin & McGill (2000). The results are
generally favorable but do vary with regard to the simulated customer behavior.
Chapter 3: Demand Forecasting for Revenue Management 27
In Lan & Gao (2007), another approach to dealing with limited demand information
is offered. The authors develop robust inventory controls when only upper and lower
boundaries for demand are known. The concept is shown to be effective for single-leg
problems. This method as well as its preceding research on competitive online algorithms
in Ball & Queyranne (2006) would have to be adapted for network models.
While other industries profit from available turn-down data (see for instance Zhu
(2006)), such data is not yet fully available for airlines. Turn-down data stores customer
requests that have been denied in addition to those that were accepted as bookings. If
such data was available, the transformation of sales data to get information on demand
would become superfluous.
The nearest thing to turn-down data available for airlines are so-called click-streams.
These record customer behavior as observed on travel websites. They are likely to be
biased as customers do not seriously consider all travel-itineraries clicked at. The possible
importance of this kind of data for future revenue management forecasting is hinted at in
Nason (2007). However, no published research on working with this kind of information
in the airline business could be found.
3.3. Demand Behavior
The last sections as well as traditional approaches consider the overall amount of demand
to arrive for a distinct product (itinerary and booking class). Changes in customer men-
tality make other considerations necessary. As laid out for example in van Ryzin (2005),
customers that are more informed and flexible require a shift of focus from products to
customers. It used to be feasible to merely ask how many units of one product (for in-
stance, business class tickets) would be requested. Now it makes more sense to ask how
many business class customers (that might also go for a bargain ticket if it is offered to
them) do consider leaving their origin at a certain time for a specific destination. Much
of the development that has lead to this kind of flexible customer behavior has been
described in Section 2.3
With a new view of demand comes a consideration of dynamic demand behavior. Cus-
tomers choose between alternatives that are offered by the airline and its competition. It
Chapter 3: Demand Forecasting for Revenue Management 28
is no longer the absolute number of customers that revenue management needs to con-
sider, but also their choices with regard to prices, competition, and other utilities such
as schedule time and transfers. As indicated in P. Belobaba (1987b), with such choice
behavior, it is possible to recapture rejected customer requests vertically (to a different
booking class) or horizontally (to a different itinerary).
Sell-Up and Buy-Down: Vertical choice behavior leads to buy-down, the possibility that
customers with a high willingness to pay are offered a low price and accept it. On the
other hand, it opens the way to sell-up, customers being forced by inventory controls to
buy a class that is more valuable than the cheapest class possible. Simulation results with
regard to the performance of traditional revenue management given buy-down and sell-up
are offered in Cusano (2003) and Ozdaryal & Saranathan (2004).
Under these conditions, the challenge of revenue management becomes to include in-
formation on customers’ price-elasticity in forecasting models. As will be listed in the
following paragraphs, different approaches formulate this using different concepts such as
as willingness to pay,Q-forecasting, or hybrid demand.
Willingness to Pay: An appraisal of the influence of prices on customer demand is
documented in Castelli et al. (2003). With an ordinary least squares regression as well as
amultilevel analysis-based methodology, the variance of price elasticity on different routes
in a network is analyzed. With its focus on overall demand as well as on specific routes for
specific airlines, this paper straddles the border of macro- and microeconomics. Weber &
Thiel (2004) presents an attempt to estimate customer demand curves based on price. An
artificial neural network is implemented in order to derive price elasticities from booking
data.
First ideas about demand behavior being based on price elasticities date back as early
as the 1970s. Notable examples are Lennon (1972) and Jung & Fuji (1976).
Q-Forecasting and Fare Modifiers: Some approaches to forecasting demand under the
assumption that customers are not segmented according to restrictions are presented in
Cl´eaz-Savoyen (2005). The findings are accompanied by simulation results as attained
Chapter 3: Demand Forecasting for Revenue Management 29
through PODS. Building up on research documented in Bohutinsky (1990), P. Belobaba
& Weatherford (1996), and Gorin (2000), the thesis concentrates on combinations of Q-
forecasting and fare modifiers.
Q-forecasting models the behavior of customers by calculating the amount of passengers
willing to buy the class with the lowest fare - “Q”. The amount of passengers willing to
buy the next higher priced class is predicted from sell-up rates. These are formulated as
Frat5-rates: the price-ratio of two classes at which fifty percent of the demand for the
lower priced class will be willing to buy the higher-priced class. Frat5-rates are estimated
dynamically over the booking horizon via a regression across time-frames.
Fare adjustment using fare modifiers as a concept developed in Fiig & Isler (2004) are
introduced to optimize revenue for flights that have both a restricted and an unrestricted
fare-structure, thereby catering both to price-sensitive customers and to independent de-
mand. Fare modifiers also rely on sell-up estimates; however, they do not necessarily
assume that all lower priced classes are closed (as this is unnecessary in a restricted fare
environment).
Hybrid Demand: Building up on Cl´eaz-Savoyen (2005), Reyes (2006) offers forecasting
methods for a combination of restricted and unrestricted classes referred to as hybrid fare
structures. He describes the challenge of separating price and product-oriented demand.
A similar statement can be found in Boyd & Kallesen (2004), where the two demand
segments are called priceable and yieldable. While a product-oriented customer is only
interested in purchasing a ticket with or without specific restrictions, a price-oriented
customer will be looking to buy at the lowest available price.
Two understandings of hybrid forecasting are introduced in Reyes (2006): The simulta-
neous deployment of two separate forecasting methods for the two customer segments or
the separate forecasting of two fare structures, restricted and unrestricted. The concept
is ascribed to P. Belobaba & Hopperstad (2004). Combined are two more methods: Fare
adjustment as outlined in Fiig & Isler (2004) and path categorization. A similar approach
to incorporate market-based demand forecasts in network optimization is also taken in
C. Hopperstad (1994).
Chapter 3: Demand Forecasting for Revenue Management 30
Path categorization assumes that willingness to pay is related to the amount of transfers
a passenger’s way over an O&D network includes, as well as to how dominant a market he
comes from. Tests conducted with the help of the PODS are cited to show that both fare
adjustment and path categorization can significantly improve network revenue if hybrid
forecasting is applied. A review of the performance of sell-up algorithms in PODS can be
found in Guo (2007).
Customer Choice: While the consideration of prices for customer decisions is gaining
importance with the rise of no-frills carriers, other factors contribute to customer choice
behavior, too. It is intuitively appealing that when buying tickets, customers should
consider the travel time as well as the amount of transfers and connecting times. An
example of the examination of travel choices with regard to risk can be found in Theis et
al. (2006). Reyes (2006) mentions a connection between willingness to pay and transfers.
The paper includes this connection by applying path categorization.
The idea that customer choice depends on the distribution channel and characteristics
of the itinerary is outlined in Walczak & Brumelle (2007). This article claims that the
decision to buy depends on the price and assumes that the customer’s demand function
with regard to the price is known. Arrival rates derived from the demand function are
included in a Poisson process. Customer profiles are included in the arrival process,
thereby making customers’ demand functions variable. Comprised is also the so-called
market state, the competitive situation. The article refers to Walczak (2005) at this point
(see also Section 2.3).
The quantification of the influence of product characteristics and market state on cus-
tomer demand is relegated to consumer choice behavior models. Among these, discrete
choice models are of special interest for revenue management - in these, customers have to
choose exactly one of several distinct alternatives. The theory of discrete choice models
with special regard to revenue management and parameter estimation via maximum like-
lihood has been described for the first time in Kanafani & Sadoulet (1977) and Ben-Akiva
& Lerman (1985). On this theoretical background, a multitude of research has evolved.
In Talluri & van Ryzin (2000), a multi-nomial logit model of demand is developed in
order to predict customer choice. This model implements logistic regression for more than
Chapter 3: Demand Forecasting for Revenue Management 31
two variables to be included in customers’ cost functiosn. The probability of a customer
buying a product is calculated as a natural logarithm defined by a linear function. The
model parameters (arrival rates as well as choice factors) are estimated via estimation
maximization. Given this model, revenue is optimized for the single-leg multiple-fare case
with the help of a dynamic program.
The complexity of such programs increases over the available seats as well as over the
number of itineraries and classes. This issue is also referred to as the curse of dimension-
ality – see for example Bertsimas & de Boer (2005) for a description as well as an attempt
of solution. Therefore, most research considering customer choice models and dynamic
programs only consider one leg at a time. A similar model is also applied in Talluri &
Van Ryzin (2004a); here, only the prices that are offered are considered as factors.
In Talluri & Van Ryzin (2004b), a customer choice model is implemented in order to
estimate the connection between prices and customer demand. In this case, customers’
cost functions only include fares. After the model parameters are estimated using max-
imum likelihood, the paper offers optimal policies for an independent view of demand,
amultinomial logit model, and a model in which customers always purchase the lowest
available fare.
Another introduction to choice-based revenue management is provided in Vulcano
(2006). Simulation based optimization is suggested to build up on forecasts that incor-
porate this model. Apart from the already mentioned multinomial logit model of choice,
alternatives such as finite-mixture logit,Markovian second choice, and general random
utility are mentioned. The author also distinguishes between the estimation of choice
parameters and that of volume parameters. Forecasting methods for the latter have been
presented in Section 3.1. In order to estimate choice parameters, both estimation maxi-
mization and maximum likelihood are proposed.
Finally, an examination of price sensitivity with special regard to customers that buy
airline tickets online is documented in Garrow et al. (2007). The authors use stated pref-
erence data in order to estimate a multinomial nested logit model of customer choice
behavior. As a consequence of using stated preference data, the article draws special
attention to matters of survey design and recruitment. Both ticket prices and sociodemo-
graphic factors are included in the analysis in an attempt to explain willingness to pay
Chapter 3: Demand Forecasting for Revenue Management 32
through other utilities. The use of knowledge of customer behavior is also outlined in
Fudenberg & Villas-Boas (2006).
Chapter 4: Demand Forecast Performance Measurements 33
4. Demand Forecast Performance Measurements
As can be seen from the overview presented in the previous sections, much development
has taken place with regard to airline revenue management in general and especially
demand forecasting. With a shift from product focus to customer focus and a trend toward
less restricted products, high quality forecasting has gained significance. The importance
of forecast accuracy has been underlined for instance in Weatherford & Belobaba (2002).
Considering the task of finding the best forecast for a given situation, it can also make
sense to consider the complete revenue management system. Benchmarks to analyze the
efficiency of a set-up of this kind are provided by so-called revenue opportunity models
(ROM ). Such models usually attempt to estimate an upper benchmark for the revenue
that may be earned in a market by analyzing historical booking data (an example of this
is described in Rannou & Melli (2003) with regard to the hotel industry).
Granger & Pesaran (2000) propose the use of decision theory in order to evaluate fore-
cast performance. This means evaluating the outcome of decisions based on the forecast
in order to make statements about its quality. In the case of predicting demand in order
to maximize revenue, the forecast that leads to the highest earnings accordingly must be
the best.
General statistics offer several possible methods to calculate forecast error measure-
ments. Without regarding the specific challenges of determining future sales by trans-
forming historical bookings, these measurements will be presented in Section 4.1. In the
literature presented, the consequences of their application to specific objects and aggrega-
tion levels of comparisons are only hinted at.
For example, the object of comparison for forecast evaluation needs to be agreed on
before computing error measurements. Armstrong (2001) suggests analogous data from
different geographical areas or control groups as well as “backcasting” (predicting the past
based on current data). Evaluation methods applied in revenue management tend to use
Chapter 4: Demand Forecast Performance Measurements 34
actual booking data (see for example P¨olt (1998)). However, they have to chose between
different transformations to make booking data comparable to demand forecasts. The
consequences of this choice require further examination.
Existing attempts to evaluate any share of the multitude of forecast methods offered
can be found in Section 4.2. This includes tests of forecasting concepts conducted in the
course of their introduction as well as overview articles comparing existing approaches.
Challenges that arise from the application of the theoretical indicators of forecast ac-
curacy presented in this Chapter to the reality of revenue management are pointed out
in Section 5. Regarding these difficulties, research opportunities are identified. Finally,
immediate steps that can be taken to create a new approach to forecast evaluation are
outlined.
4.1. Theoretical Background
A general introduction to the concept of evaluating forecasts by statistical means is pre-
sented in Armstrong (2001). Further literature considering the topic will be summarized
in the context of those explanations.
An introductory overview of the challenges and aspects of time series forecasting as well
as of the consequences of forecasting errors in economic settings is provided by Makridakis
(1986). General guidelines toward the evaluation of forecast for a specific problem is
presented in Fildes (1992). The author points out that rather than relying on prior
forecasting competitions, researchers should evaluate methods with regard to the data at
hand.
The underlying assumption of evaluating forecasts is that the considered alternatives
are all methodically valid. This means that the task of choosing the right method for a
certain situation has already been tackled. As was shown in the previous chapter, a range
of solutions tailored specifically to forecasting for airline revenue management does exist.
However, the question which approach is best remains. Note that forecast quality
depends on the priorities of the evaluator: It may indicate accuracy, robustness, or a
Chapter 4: Demand Forecast Performance Measurements 35
good fit with other revenue management methods. In this chapter, a forecast’s ability to
predict actual demand, its accuracy, is regarded as its main performance factor.
Armstrong (2001) differentiates between analyzing the input and the output of forecast-
ing methods. When rating two methods processing the same kind of data and reaching
comparable results, a statistical indicator can be computed.
In the case of forecasting for revenue management, similar inputs are accessed by most
methods: Historical bookings and in some cases survey data. Forecasting methods for
revenue management may be arranged by the input data they use as well as the infor-
mation they provide on demand. The assumptions used were already mentioned when
describing the available methods in Section 3.
Another aspect of error measurement mentioned in Armstrong (2001) may be relevant
when testing forecast methods for revenue management: Asymmetrically rated evaluation.
If for instance the underestimation of demand is thought to be more harmful to revenue
than their overestimation, negative errors in the forecast might be rated more severely.
Whether to consider such asymmetries in the initial measurement of forecast errors or to
include their analysis in a final interpretation of the evaluation is a matter of context.
In order to evaluate error measures, Armstrong (2001) suggests the comparison of
rankings derived from those measurements. However, what measurements should be
calculated and compared?
In order to define the forecast error, Fildes (1992) introduces a formal notation. It has
been modified to fit the model presented in the remainder of this text.
•Let sbe the index of departure days (in the real world) or simulation runs (in a
simulation model). In chronological order: s= 1, ..., Ns.
•Let tbe a point of time in the booking horizon, t= 0, ..., Ntwith t= 0 indicating
the start of the booking horizon and t=Ntindicating its end.
•Let fbe the index of flights with f∈F.
•Let cbe the index of classes with c∈C.
•Let func (f, c, t, s) be the forecast of unconstrained demand for flight fin class c
between the points of time t−1 and tof s.
Chapter 4: Demand Forecast Performance Measurements 36
•Let bunc (f, c, t, s) be the unconstrained bookings for flight fin class cbetween points
of time t−1 and tof s.
•Let eu-u
◦(f, c, t, s) be the forecast error computed according to method ◦for flight
f, class c, and time slice t−1 to tin the booking horizon of s. The inputs for the
computation of this error are func (f, c, t, s) and bunc (f, c, t, s).
Errors can be summarized according to two ways: Series Squared Errors (SSE) and
Series Absolute Percentage Errors (SAPE). Aggregation can happen across a time period
considered in one series or across several series given one point of time or the complete
period.
Ideally, indicators measuring forecast quality should allow for the criteria given by
Diebold & Lopez (1996) as well as for a minimization of error as described by Fildes
(1992). To that effect, a combination of error measures can be helpful, describing both
the behavior of errors as well as their overall volume.
A preference of unit-free measures not affected by the scale of the predictions avoids to
unfairly weight different markets with different demand volumes. Biased error measures
should also be avoided - an example is the use of the SAPE indicator mean absolute
percent error (MAPE) when considering positive numbers (such as bookings without
cancellations). MAPE is calculated by averaging the absolute percent errors (APE).
•Let ˆeu-u
◦(s) be the series error computed according to method ◦for s.
•Let eu-u
APE (f, c, t, s) be the APE computed as shown in Definition (4.1).
•Let ˆeu-u
MAPE (s) be the MAPE computed as shown in Definition (4.2).
eu-u
APE (f, c, t, s) := |func (f, c, t, s)−bunc (f, c, t, s)|
bunc (f, c, t, s)·100
∀f∈F;c∈C;t= 1, ..., Nt−1; s= 1, ..., Ns
(4.1)
ˆeu-u
MAPE (s) := Pf∈FPc∈CPNt
t=1 eu-u
APE (f, c, t, s)
|F|·|C|·Nt
∀s= 1, ..., Ns
(4.2)
As it calculates the difference between a positive forecast and another positive bench-
mark, the highest difference possible if demand is underestimated is 100%. Overestimation
Chapter 4: Demand Forecast Performance Measurements 37
may be infinitely high. With such measures, an adjustment as the unbiased absolute per-
centage error (UAPE) presented in Makridakis (1993) can be in helpful (see Definition
(4.3)). This error measurement is summarized in Definition (4.4). The result is also
referred to as symmetrical MAPE (SMAPE) by Tayman & Swanson (1999).
•Let eu-u
UAPE (f, c, t, s) be the UAPE computed as shown in Definition (4.3).
•Let ˆeu-u
SMAPE (s) be the SMAPE computed for sas shown in Definition (4.4).
eu-u
UAPE (f, c, t, s) := |func (f, c, t, s)−bunc (f, c, t, s)|
func(f,c,t,s)+bunc(f,c,t,s)
2·100
∀f∈F;c∈V;t= 1, ..., Nt−1; s= 1, ..., Ns
(4.3)
ˆeu-u
SMAPE (s) := Pf∈FPc∈CPNt
t=1 eu-u
UAPE (f, c, t, s)
|F|·|C|·Nt
∀s= 1, ..., Ns
(4.4)
Alternatively, in order to compensate for the measure’s vulnerability with regard to
underestimation, Fildes (1992) suggests the use of the Median Absolute Percentage Error
(MdAPE). The calculation of this measure is shown in Definition (4.5).
•Let ˆeu-u
MdAPE (s) be the MdAPE computed for sas shown in Definition (4.5).
•Let N=|F|·|C|·Ntbe the overall number of observations for flights, classes and
points of time before departure per s.
•Let Eu-u
APE (s) be the set of all observations of APE for s.
•Let arg (E, n) be a function returning the nth element of the ordered set E.
ˆeu-u
MdAPE (s) :=
arg(Eu-u
APE(s),N
2)+arg(Eu-u
APE(s),N
2+1)
2Nmod 2 = 0
arg Eu-u
APE (s),N
2+ 1Nmod 2 >0
∀s= 1, ..., Ns
(4.5)
Chapter 4: Demand Forecast Performance Measurements 38
Based on the claim that MAPE tends to overstate the error found in skewed forecasts,
Tayman & Swanson (1999) reason that a class of measures called Minimization Estimators
(M-estimators) are more valid. The authors list resistance and robustness as criteria for
a good forecast performance estimator. Resistance describes small subsamples having
only a limited effect on the results of the evaluation. Robustness implies insensitivity
to underlying assumptions e.g. about the distribution. The authors compare the use
of MUAPE and M-Estimators. The latter are based on maximum likelihood procedures
minimizing objective functions describing the relative deviation of observations
The performance of any forecast depends on the degree in which the data considered
can be predicted. If forecasts are evaluated between different markets, it makes sense to
consider their sensitivity to the degree of demand uncertainty. If for example departures
of the same flight rather than different markets are compared, each point of data includes
the same level of uncertainty. However, when comparing forecasts to actual data the pool
of data the researchers can analyze often is limited. In Sullivan et al. (2003), solutions to
this problem of shared data sets are presented.
A comparison to the naive forecast (random walk) provided by the so-called relative
absolute error (RAE) can be advisable to neutralize market effects. The random walk
predicts the next value of a series by assuming that it will be the same as the previous
value. It is introduced in Definition (4.6).
•Let ˆ
f(f, c, t, s) be the naive forecast for class con flight fbetween points of time
t−1 and tof sas shown in (4.6).
•Let eu-u
RAE (f, c, t, s) be the relative absolute error calculated for the forecast generated
for demand to arrive for flight f, class c, between points of time t−1 and tof run
sas shown in Definition (4.7).
•Let ˆeu-u
GMRAE (s) be the RAE summarized over its geometric mean (GMRAE) as
presented in Definition (4.8).
•Let ˆeu-u
MdRAE (s) be the RAE summarized over its median (MdRAE) as presented in
Definition (4.9).
Chapter 4: Demand Forecast Performance Measurements 39
ˆ
f(f, c, t, s) :=
0s= 1
bunc (f, c, t, s −1) s > 1
∀f∈F;c∈C;t= 1, ..., Nt
(4.6)
eu-u
RAE (f, c, t, s) := |func (f, c, t, s)−bunc (f, c, t, s)|
ˆ
func (f, c, t, s)−bunc (f, c, t, s)
∀f∈F;c∈C;t= 1, ..., Nt;s= 1, ..., Ns
(4.7)
ˆeu-u
GMRAE (s) := Y
f∈FY
c∈C
Nt
Y
t=1
eu-u
RAE (f, c, t, s)!1/(|F|+|C|+(Nt))
∀s= 1, ..., Ns
(4.8)
Note that the geometric mean is costly to calculate for large |F|+|C|+Nt.
ˆeu-u
MdRAE (s) :=
arg Eu-u
RAE (s),N+1
2Nmod 2 = 0
arg(Eu-u
RAE(s),N−1
2)+arg(Eu-u
RAE(s),N+1
2)
2Nmod 2 >0
∀s= 1, ...Ns
(4.9)
According to Armstrong & Collopy (1992), RAE provides a valid alternative to another
measure called Theil’s U2 as presented in Theil (1966) and discussed and recommended
in Bliemel (1973) (see Definition (4.12). In order to make RAE comparable to MAPE,
Armstrong & Collopy (1992) suggests the calculation of a Relative Absolute Percent Error
as shown in Definition (4.10).
•Let eu-u
RAPE (f, c, t, s) be the relative absolute percent error computed as shown in
Definition (4.10).
•Let ˆeu-u
MRAPE (s) be the mean relative absolute percent error computed for sas shown
in Definition (4.11).
Chapter 4: Demand Forecast Performance Measurements 40
•Let ˆeu-u
U2 (s) be Theil’s U2 as shown in Definition (4.12).
eu-u
RAPE (f, c, t, s) :=
|func(f,c,t,s)−b(f,c,t,s)|
b(f,c,t,s)
|ˆ
func(f,c,t,s)−b(f,c,t,s)|
b(f,c,t,s)
∀f∈F;c∈C;t= 1, ..., Nt;s= 1, ..., Ns
(4.10)
ˆeu-u
MRAPE (s) := Pf∈FPc∈CPNt
t=1 eu-u
RAPE (f, c, t, s)
|F|·|C|·Nt
∀s= 1, ..., Ns
(4.11)
ˆeu-u
U2 (s) := qPf∈FPc∈CPNt
t=1 (func (f, c, t, s)−b(f, c, t, s))2
qPf∈FPc∈CPNt
t=1 b(f, c, t, s)
∀s= 1, ..., Ns
(4.12)
While the relative absolute percent error is bounded between 0 and 1, RAE and U2 have
no finite upper boundary. Another estimate of forecast quality based on the comparison
to a random walk is Percent Better. As shown in Definition (4.15), this measure calculates
the percentage of forecasts for which a given method is more accurate than the random
walk or naive forecast.
•Let δPB
f,c,t,s as shown in Definition (4.13) be an indicator to define whether for one
combination of flight f, class c, and time slice t−1 to tof the booking horizon
of sthe difference of the forecast to the bookings is smaller than that of the naive
forecast to the bookings.
•Let nu-u
PB (s) as shown in Definition (4.14) be the number of cases in which the
difference between the unconstrained forecast computed the unconstrained bookings
is smaller than that between the naive forecast and unconstrained bookings for s.
•Let ˆeu-u
PB (s) as shown in Definition (4.15) be the percent better indicator.
Chapter 4: Demand Forecast Performance Measurements 41
δPB
f,c,t,s :=
1|func (f, c, t, s)−b(f, c, t, s)|<|ˆ
func (f, c, t, s)−b(f, c, t, s)|
0 else.
∀f∈F;c∈C;t= 1, ..., Nt;s= 1, ..., Ns
(4.13)
nu-u
PB (s) := X
f∈FX
c∈C
Nt
X
t=1
δPB
f,c,t,s
∀s= 1, ..., Ns
(4.14)
ˆeu-u
PB (s) := nu-u
PB (s)
|F|·|C|·(Nt)·100
∀s= 1, ..., Ns
(4.15)
Armstrong & Collopy (1992) emphasize the ease of explanation as an advantage of RAE
over U2. However, as described in Makridakis (1993), the use of RAE is not without
problems, either, as it can be blown out of proportion due to overly large random values.
On the other hand, the use of Theil’s U2 can be justified in a scientific context when
managerial understanding is not the first priority.
Armstrong (2001) argues against the use of both a coefficient of determination,R2, and
Root Mean Square Error (RMSE) when evaluating time series forecasting methods. R2,
as shown in Definition (4.16), is calculated from the ratio of the sum of squared errors
and the total sum of squares of a set of predicted and observed values. While an R2of 1
is regarded as an indication of a perfect fit of predicted values to the observed data, this
can be misleading. On the one hand, this figure does not consider a systematical bias. On
the other hand, with a variation of zero in the data, R2can turn out as zero, indicating
no correlation, even if the forecast is correct.
•Let b(s) be the average number of bookings per flight and class and slice of time of
the booking horizon observed in s.
•Let R2(s) as shown in Definition (4.16) be the coefficient of determination for fore-
casts computed in s.
Chapter 4: Demand Forecast Performance Measurements 42
b(s) := Pf∈FPc∈CPNt
t=1 b(f, c, t, s)
|F|·|C|·Nt
R2(s) := Pf∈FPc∈CPNt
t=1 (func (f, c, t, s)−b(f, c, t, s))2
b(f, c, t, s)−b(s)2
∀s= 1, ..., Ns
(4.16)
The RMSE as shown in Definition (4.17) is calculated as the square root of the mean
squared error (MSE) also referred to as mean square forecast error (MSFE) or mean
square deviation (MSD). It offers an alternative to the mean absolute error (MAE) or
mean absolute deviation (MAD). The RMSE is cautioned against in Armstrong & Collopy
(1992), Armstrong & Fildes (1995) and Armstrong (2001) as it overstates the impact of
large errors. Its interpretative pitfalls are also emphasized in Fildes (1992). While this
makes sense in so far as that larger errors can lead to larger economic losses, it mixes
the interpretation of results with their calculation. For the same reason, the use of MSE
is criticized in Chatfield (1988). It is recommended for use only when the predicted
quantities are of a comparable order. In the case of comparing for instance departures of
the same flight, this can hold true.
•Let ˆeu-u
RMSE (s) be the RMSE computed as shown in Definition (4.17).
eu-u
RMSE (s) := sPf∈FPc∈CPNt
t=1 (func (f, c, t, s)−b(f, c, t, s))2
|F|·|C|·Nt
∀s= 1, ..., Ns
(4.17)
In Fildes (1992), the calculation of a Geometric Root MSE Squared is suggested in order
to summarize the indicator. For reasons of computational effort, like U2, RMSE will be
averaged arithmetically in the further text.
Scale-dependent measures such as MSE cannot be used to compare forecast performance
over diverse series. Measures based on percentage errors such as MAPE are undefined if
zero values are predicted (division by zero), and measures based on relative errors such
as MRAE can overstate extreme values. Based on these problems, Hyndman & Koehler
(2006) introduce a new measure. The Mean Absolute Scaled Error (MASE) is calculated
Chapter 4: Demand Forecast Performance Measurements 43
as a mean of the RAE, scaling the forecast error to the error of the naive forecast (see
Definition (4.7)).
•Let ˆeu-u
MASE (s) be the MASE computed as shown in Definition (4.18).
ˆeu-u
MASE (s) := Pf∈FPc∈CPNt
t=1 |func (f, c, t, s)−b(f, c, t, s)|
|F|·|C|·Nt
∀s= 1, ..., Ns
(4.18)
Using the concept for computing MASE, the authors also define the Root Mean Squared
Scaled Error (RMSSE).
•Let ˆeu-u
RMSSE (s) be the RMSSE computed as shown in Definition (4.19).
ˆeu-u
RMSSE (s) = v
u
u
u
tX
f∈FX
c∈C
Nt
X
t=1
(func (f, c, t, s)−b(f, c, t, s))2
ˆ
func (f, c, t, s)−b(f, c, t, s)2
∀s= 1, ..., Ns
(4.19)
In order to directly compare the accuracy of two competing forecasts, Diebold & Mari-
ano (1995) suggests the use of an evaluation of a null hypothesis stating a lack of difference.
The authors claim that their approach allows for a more diverse range of forecast errors.
They formulate the hypothesis as the statement that the population mean of the loss-
differential series of the two forecasts is 0. A summary of such tests can be found in
Harvey (1997).
With regard to the variety of available forecast measures, some limitation is called for.
When considering the idea of forecast measures as such rather than comparing two speci-
fied methods, rankings appear to make more sense than straight comparisons. Therefore,
tests based on a hypothesis assuming equal forecast quality will be neglected from this
point on.
One flaw inherent to scaled quality indicators becomes clear on further consideration.
Whether forecasts are scaled to the performance of a naive approach or to the spread of
Chapter 4: Demand Forecast Performance Measurements 44
actual values, in the case of small numbers, division by zero is a constant issue. Especially
with regard to revenue management, however, this cannot be ignored: whether a forecast
predicts 5 or 100 bookings for a situation in which 0 bookings occur can make all the
difference.
The only indicators that avoid this trap are those that are not scaled. While for example
RMSE does not adjust for scale, it also considers every error as it occurs with the gravity
in which it occurs. Due to the lack of scaling, RMSE values cannot be compared over
data sets of different proportions. A simulation could allow for the artificial stabilizing
of market sizes: Three different methods can be evaluated based on the same customer
model and therefore within the same order of magnitude.
4.2. Applied Forecast Performance Evaluation
Much research is concerned with the theory of evaluating forecast accuracy. However,
there are also several reports of applied forecast evaluation in the area of revenue man-
agement. The topic is split between evaluations based on data from real-life systems and
evaluations using a simulation environment. While real-life data necessarily includes the
actual degree of complexity, it is also truncated and limited with regard to data collec-
tion methods. In contrast to this, simulation-based approaches offer more insight into
customer decisions but are limited by implicit model assumptions.
W. M. Swan (1990) represents one of the first reports of forecast evaluations based
on a simulation. In this case, the evaluation is focused on the spill (rejected customer
requests) caused by capacity allocation decisions based on demand forecasts. The results
of the analysis performed indicate that it is crucial – unconstraining is referred to as an
“estimate of spill” in this study.
Zeni (2001a) describes the evaluation of a range of unconstraining methods using a simu-
lation to constrain data and forecasting methods to unconstrain it again. The constrained
data is compared to actual airline data sets. The text has already been summarized in
3.2, the findings are also summarized in Zeni (2001b). The author judges any approach
preferrable to that of ignoring the truncation of data. He indicates that expectation max-
Chapter 4: Demand Forecast Performance Measurements 45
imization methods work best. Such findings are also confirmed by Weatherford (2000)
and Weatherford & P¨olt (2002).
Another research study concerned with evaluating unconstraining methods is presented
in Ferguson et al. (2007). This study actually uses hotel data but relates the methods and
tests considered to those applied to airline data in other studies. Estimation maximization
methods and a newly proposed double exponential smoothing approach are compared
under the assumption of strictly static demand.
Ratliffe (2008) considers the problem of unconstraining under the aspect of customers
flexibly choosing from a range of flights offered. The results presented are based on a
simulation study. Evaluation is performed by a combination of MAD and MAPE ranking,
comparing multi-flight and single-flight estimation maximization methods. The conclusion
is that overall demand volume can make a difference for the success of forecasting methods.
In Ryzin & McGill (2000), the success of a revenue management system without a sys-
tematic forecast is evaluated using a simulation system. A simulation framework for the
evaluation of revenue management strategies is also presented in Abdelghany & Abdel-
ghany (2007) and Abdelghany & Abdelghany (2008). With regard to the success of the
introduction of a network-based forecast method, a performance analysis is presented in
Rockmann & Alder (2009). In P¨olt (1998), some thoughts on forecast methods and their
evaluation are presented. P. P. Belobaba (1998) considers the same issue based on PODS
simulations. Forecasting approaches to estimating customer behavior, with special regard
to sell-up, are evaluated in C. Hopperstad (2007).
In Frank et al. (2008), a number of general principles for the development of simulations
to evaluate revenue management systems is provided. Thoughts regarding the calibration
of stochastic demand data for such simulations are offered in Kimms & M¨uller-Bungart
(2007).
Chapter 5: Research Gap and Opportunities 46
5. Research Gap and Opportunities
As has been shown, a body of research on the theory as well as some documentation on the
actual application of forecast evaluation for revenue management is available. Research
opportunities arise as a gap exists in the available literature. While simulation methods
have been applied to estimate the success of revenue management strategies, potential
of exclusive knowledge on the demand model included in a simulation has not been used
extensively. Additionally, few simulations include flexible demand as opposed to demand
streams conforming to the static assumptions included in many forecast methods.
Most theoretical evaluation methods focus on the calculation of forecast accuracy. This
is usually regarded to be the difference between actual observations and predictions. In
demand forecasting for revenue management, observations are indirectly influenced by
predictions – bookings only manifest if optimized availabilities allow them. The opti-
mization uses the forecast as input. New error measurements may be computed by not
comparing predictions to bookings but instead comparing them to a suitable transforma-
tion of demand knowledge as it is available in a simulation environment.
As a systematic connection between observed values and predictions rarely is considered
in theory, developments in error measurements over time are neglected. When predictions
may influence observations, phenomena such as self-fulfilling prophecies can arise. Signs
for such methodical flaws can be found in the indicator development. As a simulation
enables the quick modeling of long-term effects, the consequences of systematic develop-
ments can also be analyzed.
The revenue consequences of the use of a forecast method are often used as indicators
when the use of methods in the real world is examined. However, revenue may also be
impacted by the interaction of forecast and optimization. It does not in fact offer informa-
tion on how correct a forecast is but rather indicates how financially successful its use in a
certain environment has proven to be. A decomposition of the revenue management sys-
47
tem would offer new opportunities of comparing forecasts without necessarily considering
the effect they have on revenue.
Using a transparent demand model in a simulation, traditional evaluation methods
may be reconsidered. New approaches to forecasting that attempt to describe demand
behavior can be evaluated by comparing parameters drawn from the forecast and the
model.
From the research gap described, three further steps can be derived:
Decomposition: When the components of a system can be isolated, processes can be
designed to evaluate them separately. This way, the performance of individual parts of
the system can be analyzed, while the other parts are kept stable (ceteris paribus). Such
a decomposition may be applied to a revenue management system or to an approach to
forecasting. This way, the accuracy of a forecast and its consequences for revenue can be
considered separately.
Simulation Environment: A simulation environment offers the opportunity of imple-
menting a decomposed model of revenue management and of interchanging separate mod-
ules. In addition to parts of the revenue management system, the market as presented
by a demand model can be influenced in a simulation. The demand model is transparent
in such an environment and can be mined for analysis. Furthermore, the cycles of fore-
casting, optimization and customers booking tickets can be sped up to observe long-term
effects.
Concept Application: Based on a decomposed model and knowledge about forecast
and forecast evaluation methods, expectations toward the evaluation of forecasts can be
stated. Simulation experiments can be designed and conducted to test them. From the
results of these simulation experiments, insights toward forecast evaluation may be drawn.
48
Part II.
Solution Approach - Concept and
Implementation
49
A revenue management system can be described as three processes: Demand is pre-
dicted in the forecast component, availabilities to maximize revenue are computed in the
optimization component, and seats are allocated according to this optimization and pos-
sibly strategic goals in the inventory. Corresponding to the aspects described in Chapter
3, the process of forecasting may be divided up further.
Using a simulation system, the theoretical decomposition of revenue management can be
realized under laboratory conditions. The evaluation of methods of revenue management
in simulations has already been described in Section 4.2. The concept described in this
chapter goes one step further by systematically introducing knowledge of the demand
model (as is exclusive to the simulation environment) to the evaluation of forecasts.
As this thesis concentrates on the evaluation of demand forecasts, the decomposed view
of optimization and inventory is not described in further detail here. However, it may be
realized in a similar fashion.
Chapter 6: Simulation for Decomposition and Evaluation of RM Systems 50
6. Simulation for Decomposition and Evaluation of RM
Systems
First, a concept for the overall evaluation of a revenue management system in a simulation
environment is described in the chapter. Next, the forecast component as a whole as well
as the aspects of predicting demand volume, unconstraining, and behavior aspects are
considered.
6.1. Overall System View
The consideration of the performance of a complete revenue management system follows
the ideas of decision theory as outlined in Granger & Pesaran (2000). This approach does
not conclude whether any part of the system is a decisive factor in its success.
Traditionally, in order to evaluate a whole system, one looks at whether it does what
it is supposed to do: Maximize revenue. When considering mathematical methods, this
can be achieved by mathematically proving that given correct forecasts, under certain
assumptions, an optimization algorithm determines inventory controls that yield maximal
revenue. An example of this approach can be found in Mayer (1976).
However, the knowledge that given correct forecasts, under certain assumptions, a rev-
enue management system works optimally is not helpful in practice. The assumptions
used in mathematical proofs are often simplified – one example is the “low fare demand
before high fare demand” rule described in Section 2.2. Forecast quality cannot be as-
sumed to be perfect or even constant. Results based on those conditions do not necessarily
apply to real-world systems.
A more practical view of evaluating a complete revenue management system is to look
at its outcome. This may be done by implementing and testing it on a real-world market
or under laboratory conditions using artificial demand.
Chapter 6: Simulation for Decomposition and Evaluation of RM Systems 51
The advantage of a real-world implementation obviously lies in the confrontation of the
system with just the amount of complexity it is supposed to handle. At the same time,
results from the real-world may be tainted by events or economic trends and therefore
may not give clear information on the performance of a new method.
Artificial demand can be designed to include only a desired degree of volatility. Further-
more, the same demand may be used to test two or more methods. However, performance
when confronted with simplified model may not allow for conclusions on real-world per-
formance.
The results of a revenue management system are bookings. Beyond these bookings, ag-
gregated by departure, fare class, and time of booking, monetary indicators are yield (the
average fare paid) and revenue (the sum of fares paid by all booking customers). While
the first goal of revenue management is to maximize revenue, indicators on productivity
such as seat load factors can be helpful, too.
A simulation, in which detailed knowledge on demand behavior is available, offers the
opportunity of generating some additional results. These, as will be described subse-
quently, pertain information on rejected customers as well as on dependent demand be-
havior. As each customer can be observed requesting tickets and and being accepted or
turned down, the equivalents of turn-down or click-stream data as described in Section 3.2
can be derived. Therefore, in the simulation, one can easily find out how many requests
were rejected and for which reasons. This may happen for two reasons, termed spill and
spoilage.
If customer requests are rejected due to limited capacity, this is called spill. Ideally, the
phenomenon should apply to low-fare demand, which is rejected in favor of customers with
a higher willingness to pay. If customer requests are rejected in order to reserve seats for
more valuable demand that fails to materialize, this is called spoilage. Any rejection can
be categorized as one or the other. While spill can rarely be avoided, given that aircrafts
do have limited capacity, spoilage is the consequence of imperfect revenue management.
Vertically dependent demand may be observed within a simulation in terms of buy-
down and sell-up. As demand is artificially generated to include a cost function as well as
maximum constraints, each individual passenger’s maximum willingness to pay is known.
The demand model may include flexible choice behavior letting each customer minimize
Chapter 6: Simulation for Decomposition and Evaluation of RM Systems 52
costs by choosing the cheapest available booking class within the parameters of product
preference. This leads to an effect also observed in practice: Customers often pay less
than what they theoretically are willing to. As information on the acceptance of product
restrictions and the acceptance of price is available, comparisons between what was booked
and what might have been booked are feasible in a simulation. Each booking can be placed
in a continuum of up-sell (the passenger was forced to pay more than the lowest price for
the requested product) and buy-down (the passenger’s maximum willingness to pay was
not fully exploited).
Horizontally dependent demand, customers choosing between available itineraries, can
be documented as recapture. Recapture can be observed when a customer, on not being
able to book his first choice due to limited availabilities, books a seat on a different
itinerary rather than not booking at all. In a demand model with flexible customers,
measuring recapture is not trivial. When many factors play a role in customer choice, it
can be difficult to decide what the originally desired and what the alternatively accepted
itinerary was. One solution is to define “first choice” as the itinerary a customer would
have chosen if the all prices were the same.
Another feature of a simulation is the possibility of repeating processes while keeping
individual components (such as the demand model) stable. This way, long-term conse-
quences of the use of methods can be analyzed. In practice, economic trends as well as
changes in the market situation (i.e. the entry of competitors) may distort the impact of
a new method. In a simulation, artificial demand can be used repeatedly and methods
can be tested ceteris paribus. This way, for example, the consequences of using historical
data as generated using a forecast method can be observed by comparing the results over
time. If the bookings, revenue, spill, spoilage, booking behaviors or recapture change,
this will not be due to changes in customer behavior.
The scheme of how to compare such a system’s results is shown in Figure 6.1. Step (1)
is to generate artificial demand for as many periods as the simulation aims at modeling.
Some of this data will be used for history building in step (2). In the case depicted, this
includes periods 1 to t−1. Requests and knowledge about the demand model are used to
provide the basis for the forecasts to be evaluated. This is realized by creating inventory
controls using an optimization based on forecasts derived from the knowledge of artificial
Chapter 6: Simulation for Decomposition and Evaluation of RM Systems 53
generate
stochastic
requests
optimize
artificial
requests for t
AU-levels or
bid prices
reservation
system
simulation
3
2
1
create initial
forecast
requests
initially
forecasted
for t
artificial
requests for
1, …, t-1
artificial
(historical)
booking data
requests
forecasted
for t
simulation
RESULTS
forecast
variants
optimization
variants
inventory
controls
reservation
system
simulation
Figure 6.1.: Evaluating the RM System
requests. These inventory controls are then used to turn the requests into booking data,
constraining them.
The historical booking data created in step (2) presents the basis for further calculations
in step (3). Now different forecast methods come into play. They all share the same data
basis – historical bookings from step (2). As the simulation can be repeated using the
same demand while employing different methods, each method can be tested. The results
of the forecasts are each handed to the same optimization method to generate inventory
controls. In a reservation system, each of the different sets of inventory controls is used to
channel the requests generated for period t. The results of this process can be compared
over the variety of methods evaluated.
In the set-up illustrated, only forecast performance over the course of one period, t,
is actually measured. By generating historical booking data for periods 1 to t−nand
Chapter 6: Simulation for Decomposition and Evaluation of RM Systems 54
unconstraint forecast
vs.
actual bookings
constraint forecast
vs.
actual bookings
unconstraint forecast
vs.
unconstraint bookings
Figure 6.2.: Comparing Forecasts and Bookings
using the compared methods for optimizing availabilities during periods t−n+ 1 to t,
any number nof periods can be included. This way, long-term evaluations are feasible.
6.2. Forecasting Component
The quest for evaluating the performance of the forecast component in a revenue manage-
ment system is not new. Some research conducted with regard to it has been introduced
in Chapter 4. Most attempts at evaluating the quality of a forecast that do not consider
the complete system such as described in Section 6.1 use one of three available options
presented in Figure 6.2.
The straight-forward approach is to compare predicted demand to the actual bookings.
However, these can be quite different quantities as bookings are constrained. Therefore,
it seems sensible to transform either the forecast or the bookings in order to make a
comparison more meaningful. In order to do this, one either compares the results of
Chapter 6: Simulation for Decomposition and Evaluation of RM Systems 55
the forecast with unconstrained actual bookings, or constrained forecast results to actual
bookings.
These comparisons, however, do contain one major bias. The same method applied
to unconstrain historical booking data in order to build the forecast is applied to trans-
form actual bookings for the comparison. When constraining the forecast, the inventory
controls that were applied according to an optimization based on the forecasts are put
into place. Any forecast evaluation that is conducted like this will either include the
unconstraining method of the forecast or the inventory controls based on it.
generate
stochastic
requests
optimize
artificial
requests for t
AU-levels or
bid prices
reservation
system
2
1
transform
create
initial
forecasts
requests initially
forecasted for
1, ... , t-1
artificial
requests for
1, …, t-1
artificial
(historical)
booking data
compare
requests
forecasted
for t
system
simulation
RESULT
3
transform
without
constraining
forecast
variants
benchmark
forecast
Figure 6.3.: Evaluating the Forecast Component
An approach based on a simulation system as presented in Figure 6.3 may avoid these
catches. Once more, artificial demand is generated in step (1). Requests are generated
for a time-line from 1 to t−1, to ensure a test of the forecast’s ability to pick up trends
and seasonality. Requests for periods 1 to t−1 are used to build a history of bookings
for the forecast methods to be evaluated in step (2). In step (3), the different forecast
Chapter 6: Simulation for Decomposition and Evaluation of RM Systems 56
methods are tested by providing the same data basis in the form of historical bookings
and for the task of forecasting unconstrained bookings for period t.
Finally, the resulting forecasts can be evaluated by comparing them to the actual re-
quests as generated for period t. The forecast that did best in recognizing the underlying
patterns and unconstraining sales data should yield the result with the smallest deviation
to the actual requests. As different approaches to forecasting may offer different forms of
information, the requests need to be transformed in order to make a comparison feasible.
This means that a kind of perfect or psychic forecast needs to be created, employing the
same information and data format as the evaluated forecast method.
Of course, aspects of the process generating a psychic forecast may influence the test
results. However, a consistent approach in transforming requests, such as the general
principle of always exploiting maximum willingness to pay, can still enable a consistent
comparison of diverse methods.
The comparison can follow standard statistical procedures. This includes the calcula-
tion of key indicators as described in Chapter 4.
6.3. Demand Volume Aspect
Forecasting for revenue management consists of two linked tasks. Historical booking in-
formation needs to be transformed in order to learn about past demand (unconstraining).
Historical patterns need to be recognized and extrapolated in order to provide predictions
toward future demand (time series aspect). When evaluating forecasts, it can make sense
to separately consider a method’s ability to fulfill both tasks. This may provide insight
to build better approaches by combining existing concepts. While this section describes a
simulation-based evaluation of the time series aspect, the next section will consider how
the unconstraining aspect may be isolated and analyzed.
While unconstraining seems to be connected to the special nature of revenue manage-
ment, the consideration of time series is a problem common to all kinds of forecasting.
Once more, three aspects arise: Demand seasonality, long-term trends, and arrival timing.
Over long or short terms, developments depending on time can include recurring pat-
terns. The fluctuations based on these patterns are called seasonality. Apart from actual
Chapter 6: Simulation for Decomposition and Evaluation of RM Systems 57
seasons such as spring or summer, demand may also fluctuate periodically with regard to
the day of week or the time of day. By recognizing seasonality and accurately predicting
it, a forecast can use the knowledge of the time a flight departs to estimate expected
demand.
While economic cycles may be considered as the outcome of a seasonality spreading over
the course of several years, it is more common to view them as a separate aspect, called
the trend. Trends provide information on the general development of demand volume over
several seasons. Rather than assuming that the second week day of the tenth week of
every year will see the exact same demand, different overall demand level for the current
situation based on recent developments is predicted. Seasonality is helpful to predict
patterns within this development.
Last but not least, demand arrival within the sales period also follows certain patterns
that may depend on the customers requesting demand. With regard to the timing of
a departure, demand arrival for the different booking classes or fares offered needs to
be predicted in order to allow for a successful optimization. This, too, is time series
forecasting.
Existing approaches to measuring a method’s ability to predict time series tend to do
so by providing a set of data that is not constrained. In revenue management, that means
relying on historical bookings that occurred in classes that were always available. In
practice, such data can only be the result of consistently low demand. Basing a forecast
evaluation on it means basing it on a special demand situation – one cannot say how the
forecast would perform given high levels of demand or high fluctuations in demand.
A simulation approach to evaluating the time series aspect can provide booking data
that has not been constrained by inventory controls. In such a model, capacities can be
set to infinity. In order to not turn even one customer away, capacity may be neglected
entirely: By closing booking classes, sales constrained and no longer represent the com-
plete demand. Without fixed capacities, unconstraining is necessary no more. Demand,
no matter how high or low and no matter how volatile, directly translates into bookings
that can be used as a basis for forecasts.
Based on this idea, requests are generated for departures t= 1..T in step (1) of Figure
6.4. The requests for 1..T −1 are turned into historical bookings by assigning them to
Chapter 6: Simulation for Decomposition and Evaluation of RM Systems 58
generate
stochastic
requests
artificial
requests for t
2
1
transform
without
constraining
artificial
requests for
1, …, t-1
artificial
(historical)
booking data
compare
requests
forecasted
for t
RESULT
3
transform
without
constraining
forecast
variants
benchmark
forecast
Figure 6.4.: Evaluating the Trend Component
booking classes without constraining them in step (2). Forecast methods are provided
historical data and predict demand for the departure t=Tin step (3). The forecast is by
design unconstrained and can therefore be easily compared to the requests generated. Any
flaws in the prediction are due to demand fluctuations caused by seasonality or trends.
As with traditional methods, the three aspects of revenue management time series may
be heeded by different indicators used in the error measurements. A forecasts ability
to pick up trends can be observed by comparing the overall predicted demand to the
overall amount of requests. Seasonality patterns included in forecasts may be evaluated
by forecasting and evaluating requests for a number of departures t=T−n..T rather
than for just one departure. The correct identification of arrival patterns can be checked
by comparing the predicted demand arrival rates for a departure to the demand arrival
rates inherent to the original requests.
Chapter 6: Simulation for Decomposition and Evaluation of RM Systems 59
This section explained ways of evaluating the time series aspect alone based on the
simulation concept. Such is also the focus of much research on forecasts and forecast
evaluation as presented in Armstrong (2001). The further text will focus mainly on
evaluating the performance of unconstraining as well as conclusions toward dependent
demand behavior. This allows for an examination of problems particular to revenue
management forecasting.
6.4. Unconstraining Aspect
Traditionally, a forecast’s ability to deduct demand from sales data has been evaluated
using historical data with as little overall fluctuations as possible. For example, seasonality
may be filtered including only departures within one season, day of the week and time
slot in the analysis. Trend may be excluded by considering a very uniform market with
little fluctuations. By separately considering bookings for time slices before departure,
the necessity of anticipating arrival rates can be neglected. All this makes the evaluation
of a method possible only under very specific circumstances. The challenges a forecast
faces for example when faced with a market with volatile demand are not considered.
A new approach to evaluating the unconstraining component is illustrated in Figure 6.5.
In a simulation, requests can be generated repeatedly (including a normally distributed
distortion) for only one point of time, t, in step (1). They can be turned into bookings
by an optimization based on an initial forecast in step (2). With a history of bookings
for one departure, the forecast methods tested only have to unconstrain the sales data in
order to predict demand in step (3).
Unconstraining is only ever feasible within the range of available data. This means that
if a class was never open, no method will be able to guess whether there was demand in
a market for this class and if so, how much. A perfect forecast is likely to only yield one
kind of (restrictive) inventory controls. Therefore, depending on how the demand model is
calibrated, none of the applied methods are likely to correctly estimate all demand. They
can, however, be compared with one another given the same degree of information. In
order to further analyze methods, they might be compared based on different approaches
to initial optimization. For example, a first-come-first-served optimization without a
Chapter 6: Simulation for Decomposition and Evaluation of RM Systems 60
generate
stochastic
requests
optimize
artificial
requests for t
AU-levels or
bid prices
reservation
system
2
1
transform
create initial
forecast
requests initially
forecasted for t
artificial
(historical)
booking data
compare
requests
forecasted
for t
system
simulation
RESULT
3
transform
without
constraining
forecast
variants
benchmark
forecast
Figure 6.5.: Evaluating the Unconstraining Component
forecast may be offered in step (2). The forecast outcomes given the data basis may yield
information about its reliability.
Diverse long-term effects are also conceivable. If, for example, overall demand is un-
derestimated on the long run, this can lead to a spiral-down effect: Capacity could be
reserved for expected high-value demand is allocated and sold at low fares. Customers
with a high willingness to pay may still book cheap tickets if those are available. As a
result, even fewer bookings are recorded in the high-value classes and even lower forecasts
are calculated. Such effects may be analyzed by letting a simulation run repeatedly on the
same, flexible customer model. If the customer behavior is stable, no changes in revenue
outcomes and bookings should occur. Any trend then points toward systematic flaw in
the uncontraining component.
Chapter 6: Simulation for Decomposition and Evaluation of RM Systems 61
Systematic flaws may be inherent to a forecast method. For example, as already men-
tioned in Section 3, Internet usage has lead to a shift in customer behavior. Customers
book flexibly and bookings in one class or even on one flight or itinerary may be depen-
dent on current availabilities. To derive informations on customer behavior from historical
bookings and possibly additional information has become a new task of forecasting. The
next section offers some ideas of how to evaluate the success in this regard.
6.5. Demand Behavior Aspects
The idea of dependent customer behavior as opposed to a model in which customers
statically demand tickets in one booking class is still relatively new. Customer choice
may depend on the lowest price available, flight or itinerary alternatives, or even product
preferences as opposed to price sensitivity. In any case, availabilities have an influence
on what customers book, and no availability in one specific booking class and for one
specific flight does not automatically mean that a customer will not book anything else.
Literature describing approaches to incorporating this idea into revenue management has
been listed in Section 3.3.
Traditionally, models incorporating dependent demand have been evaluated by com-
paring them to concepts relying on static demand models. In practice, expected revenue
improvements are calculated based on expected up-sell. The accuracy of a method’s de-
scription of a customer model as derived from historical bookings, though, has rarely
been emphasized. For example, when considering classical customer choice models such
as introduced in Ben-Akiva & Lerman (1985), the assumption is that customer surveys
establish reasonable estimates of customer choice factors.
A simulation with a demand model including customer choice offers the possibility of
further analysis. By separately setting up the request generation including a cost function
as well as price and product restrictions, mixed situations can be generated and the results
of forecast methods applied can be compared. The actual factors of the cost function are
known in the simulation.
The comparisons of forecasts including dependent aspects within a simulation may be
conducted on any of the levels described in this section. Important is the consideration
Chapter 6: Simulation for Decomposition and Evaluation of RM Systems 62
that the inclusion of a dependency assumption all by itself may not automatically lead
to more accurate forecasts or even higher revenues. Once more, a simulation offers the
opportunity of testing the concept within a freely manageable environment and on a stable
data basis.
generate
stochastic
requests
optimize
artificial
requests for t
AU-levels or
bid prices
reservation
system
2
1
extract
create initial
forecast
requests initially
forecasted for t
artificial
(historical)
booking data
compare
forecasted
choice
factors
system
simulation
RESULT
3
extract
choice
factors
forecast
variants
choice
factors
Figure 6.6.: Evaluating the Choice Component
As Figure 6.6 shows, the evaluation of the choice component differs from the evaluation
of the unconstraining component only in so far as that not merely overall demand volumes
are compared. Instead, the choice factors predicted by the forecast are extracted from
the demand model and used for the evaluation.
Chapter 7: Simulation Environment for Revenue Management 63
7. Simulation Environment for Revenue Management
This section describes the revenue management simulation implemented and applied for
the consideration of demand forecast evaluations. The simulation system in place can
be divided into three major components. One is the simulation control, functioning as a
framework that controls the event-based timing of a simulation as well as the management
of information and the triggering of reporting functionalities. The choice-based demand
model is a complex component necessary to make the simulation realistic. Finally, the part
of the simulation that is modeled after an actual revenue management system: modules for
forecasting, optimization, and inventory control. The components interact as simulation
control triggers customer requests, forecasts and optimization, customer requests interact
with the inventory control to create bookings, the results of this interaction are saved and
processed, new forecasts and optimized seat allocations are calculated based on historical
bookings.
Figure 7.1 shows the simulation system in the context of set-up and analysis. Supply and
demand are prepared from schedule, fares and input parameters. The system transforms
this data into result indicators such as bookings and revenue according to parameters
such as required confidence intervals. The analysis component puts data into context,
generating information that conclusions may be drawn from.
In Section 7.1, the details of the simulation control as well as the data and parameters
involved are described. Subsequently, a description of the supply and demand model
used in the simulation is provided in Section 7.2. Finally, in Section 7.3, the revenue
management methods implemented are outlined.
Setting up a simulation system in order to evaluate forecasts, the evaluation approach
of replicating outputs as described in Armstrong (2001) is included to some extent. The
importance of a sufficiently large population of predictions and observations to measure
errors on is pointed out in Armstrong & Collopy (1992). In the case of a simulation
Chapter 7: Simulation Environment for Revenue Management 64
Simulation
Analysis
Set-Up
runs
parameters for
• itinerary generation
• request generation
schedule,
classes & fares
parameters for
simulation processing
parameters for
result analysis
supply &
demand
result
indicators
information
Figure 7.1.: The Simulation Cycle
system, this pertains the number of runs evaluated as well as the number of scenarios set
up. Both are within the influence of the researcher but constrained by limits in time and
effort. Evaluating a method by comparing its output to actual knowledge about what is to
be predicted may yield misleading results as stochastic elements can lead to lucky flukes.
However, repeating the stochastic process over the course of several simulation runs,
such outliers can be neutralized: Results that are averaged over a number of replications
differing only in stochastic error terms tend to be more significant.
The system presented here follows the theory of stochastic simulation as laid down
in Law & Kelton (1997). In addition, the guidelines offered in Frank et al. (2008) are
considered.
Chapter 7: Simulation Environment for Revenue Management 65
7.1. Simulation Control
Contemplating the design of a revenue management simulation system, several aspects can
be identified. Each simulation experiment needs to be prepared: Data has to be provided
and the nature of some processes as well as the reporting required need to be specified.
An experiment consists of several simulation runs; these have to be initialized and from
one run to the other, data needs to be updated or reset. Within each simulation run,
points of time within a booking horizon pass and events need to be handled as customers
request and book tickets. Finally, the simulation results need to be stored and processed.
7.1.1. Data Management
All data is stored as lists of records, with each record having a set of properties distin-
guishing it. The following overview outlines the information stored for each item; the
algorithms used to process or create the data will be described in more detail in the next
sections.
At the start of the simulation, supply and demand data is read into the system cache
for fast access. As the simulation proceeds, simulation control provides access to the
required data. After a simulation experiment has finished, reporting information stored
in the result records is processed and written out (see Section 7.1.3).
As illustrated by Figure 7.1, data from outside the system enters the simulation at
several moments. This data can be split into different sets as shown in the following list.
•Schedule, booking classes and fares: This information can be taken from the real
world and may be based on the actual schedule as well as a selection of the actual
fares
•Parameters for itinerary generation: This input describes how combinations of origin
and destination and itineraries connecting the two are generated from the schedule.
•Parameters for request generation: This input describes how demand is to be gen-
erated for the simulation.
•Parameters for simulation processing: These parameters describe how the simulation
is to be executed.
Chapter 7: Simulation Environment for Revenue Management 66
List Properties Description
Airports name, latitude, longitude, traffic area Airports connected by
flights included in the
schedule used in the sim-
ulation.
Flights carrier, departure airport, arrival airport,
flight number, departure time, duration,
days of operation, capacity
Flights from one airport
to another.
Legs origin, destination, traffic area, distance Connections; at least one
flight between the two
airports is needed to jus-
tify a leg, but one leg can
include several flights at
different times of day.
Booking
Classes
name, carrier, traffic area, one boolean
value describing whether each of the in-
cluded product characteristics applies
Classes offered in the
simulation.
Fares carrier, class name, pairing, price Prices of tickets in one
class on one flight.
Pairings origin, destination, traffic area, request
share, customer mix
Origin and destination
combination offered to
customers.
Itineraries origin, destination, included flights, travel
time
Travel itineraries linking
two airports in a pairing.
Table 7.1.: Simulation Environment: Supply Lists
Chapter 7: Simulation Environment for Revenue Management 67
List Properties Description
Customer
Types
name, request distribution, departure time
distributions, cost function, willingness to
pay, maximum accepted travel time, ac-
cepted deviation from the preferred de-
parture time, a boolean value describing
whether each of the included product char-
acteristics is accepted by this customer,
error term used for the individual distor-
tion of requests
Input for demand genera-
tion: Types of customers
that may request tick-
ets throughout the simu-
lation.
Requests run, time of request, preferred departure
time, pairing requested, actual cost func-
tion, actual price and product preferences
Output of demand gen-
eration: Customers that
request tickets during the
simulation.
Table 7.2.: Simulation Environment: Demand Lists
•Parameters for result analysis: These parameters describe how the basic result
indicators generated in the simulation can be refined and put into context.
The following parameters are required to prepare the generation of pairings of origin
and destinations and itineraries connecting them. They define which connections are
acceptable for travel and will become part of supply:
•minimum and maximum connection time,
•minimum and maximum travel distance,
•maximum alternative itineraries per pairing,
•maximum transfers.
The data input needed to set up a supply scenario and the results of the connection
builder process are listed in Table 7.1.
Table 7.2 describes the customer types needed for setting up demand generation and
the requests that are its result. To assign demand to markets, a customer mix needs to
be provided as input parameter, defining a distribution over pairings and customer types.
Chapter 7: Simulation Environment for Revenue Management 68
Parameters influencing the analysis may be based on the motivation of the simulation
experiment. Output of the simulation experiments will be described in Section 7.1.3.
7.1.2. Simulation Runs and Lists of Events
Apart from providing data access and reporting facilities, two tasks of simulation control
remain. On the one hand, between simulation runs, preparatory measures and the evalu-
ation of stochastic confidence need to be controlled. On the other hand, within each run,
a list of events needs to be processed.
Between Simulation Runs
The simulation includes a number of repetitions (runs) as defined during set-up. As
described in Law & Kelton (1997), a certain number of runs is necessary to make the
outcome statistically significant. Within the simulation, any number of modules commu-
nicate with each other and access the data, and any number of processes can be triggered.
Depending on the requirements as declared during set-up or hard-coded into the system,
information on the simulation process is written out. Before the first run, an initialization
needs to be performed. Before every successive run, data from previous runs needs to be
processed and made available.
The initial run of any simulation experiment lacks historical information from preceding
runs. If a forecast requiring such information is included in the model, a substitute needs
to be provided in the first run. The task of simulation control is to recognize to first run
of an experiment and to generate an initial forecast that may be methodically different
from the forecasts in subsequent runs. Alternatively, before the first run, historical data
might be imported from an external data set and provided for the forecast.
After the initial run, data created in the preceding run may be used in the upcoming
run. This can be the case if the updating of the forecast method applied in the simulation
experiment depends on historical bookings. The task of simulation control is then to store
the booking data, reset the inventory, and trigger a forecast update at the beginning of
the new run. After every run, some amount of information on the interaction of the
demand model and revenue management processes needs to be permanently stored – this
Chapter 7: Simulation Environment for Revenue Management 69
is the task of result reporting as documented in Section 7.1.3. Variables or data structures
intended to hold only the data created within one run need to be reset.
The overall number of runs Nsincluded in one simulation experiment may be set before
the simulation starts. Alternatively, it can be set to depend on a defined confidence
interval.
The convergence of the sum of bookings and the sum of revenue toward the true mean
is tested after every run based on a confidence interval. This interval is determined via the
student (t) distribution and parameters, αand δ. Inequality (7.1) is used to test whether
the probability that the expected value does not differ from the true mean by more than
a percentage δis equal to or lesser than α.
To describe the test for confidence, the following notation is needed.
•Let sbe number of runs already processed.
•Let xbe the average of the indicator (sum of bookings or revenue) over n.
•Let σbe the deviation of x.
•Let αbe the acceptable probability of error.
•Let δbe the slice of xthat is acceptable as a confidence interval.
•Let ˆ
Nsbe the number of runs recommended to reach the confidence interval.
tn−1,1−α
2·σ
√n≤ |δ·x|(7.1)
If Inequality (7.1) is not fulfilled within the set maximum number of overall runs Ns,
the number of runs recommended to reach the confidence interval, ˆ
Nsis calculated for
informational purposes as shown in Definition (7.2). If deemed reasonable, the experiment
can be repeated with the required number of runs.
ˆn:= tn−1,1−α
2·σ2
(δ·x)2(7.2)
When the indicators can be expected to develop systematically during a simulation
experiment, testing for a confidence interval is less useful. However, in this case, a number
Chapter 7: Simulation Environment for Revenue Management 70
Optimization
Market availabilities Inventory
bookings
inventory
controls
Forecaster
historical bookings,
passenger
name records
forecast
SIMULATION
Figure 7.2.: Revenue Management Simulation
of initial runs may be exempted from the test (and the calculation of x). The system can
then be tested for stability under an expected dynamic behavior.
Within Simulation Runs
A revenue management simulation can be set up to answer a multitude of questions.
Regardless of its purpose, certain modules seem to be compulsory and are depicted in
Figure 7.2. These modules interact and thereby create or update data to be analyzed
later.
Within each run, simulation control has to keep track of time and trigger various pro-
cesses. This happens based on events. An event may be the arrival of a customer request,
the end of the booking horizon and the departure of a flight, or the necessity for an update
of forecast and optimization within the booking horizon. Event-controlled simulation al-
lows for an efficient way of implementation: The alternative would be to account for each
minuscule time slice within the booking horizon to test whether any action is necessary.
As a run begins, simulation control first triggers the update or the initialization of the
forecast module. Based on this, inventory allocation is optimized for each flight included
Chapter 7: Simulation Environment for Revenue Management 71
in the experiment. The list of customer requests is processed in the order of their arrival.
Each request is scheduled for a specific time in the booking horizon. If an update was
scheduled for a point of time that lies between two subsequent request, this update is
triggered by the simulation control before the next request is processed.
Data is generated as the revenue management system starts working. This includes
forecasts, optimization results, availabilities, and bookings as well as possibly details on
customer choice behavior shown in the course of the booking process. Simulation control
keeps track of this data as it is updated within the run.
The simulation includes the basic parts needed for revenue management: A forecast
determines demand to come and thereby provides an objective function for revenue to
be maximized by the optimization. That, in turn, yields a set of inventory controls
intended to result in maximum revenue. The algorithms and the data used in these
processes depend on the methods included as documented in Section 7.3. The resulting
availabilities are managed by an inventory system. The inventory acts as the interface to
the market.
A simulation could also be set up to examine the effects of competition with minimal
effort. For this, several parallel revenue management systems would be implemented. This
means that several inventories based on separate forecast and optimization modules are
kept up. The required data can still be stored in the same component but is distributed
according to airline. All airline inventories interface with the same demand model, the
market. Keeping track of different methods and therefore modules assigned to different
airlines is also within the responsibility of simulation control.
7.1.3. Reporting
Once a simulation experiment has been concluded, a broad data basis can be analyzed.
This information may be used to evaluate the general performance of a system set-up,
test the functioning of an implemented simulation, or even calculate error measurements
considering the level of accuracy of a given forecast. The information needed depends on
the specification of the simulation experiment considered. The data that has to be col-
lected and processed in order to analyze the result of simulation experiments is presented
in Table 7.3.
Chapter 7: Simulation Environment for Revenue Management 72
Bookings: Per run, flight, class, point of time before departure.
Revenue: Per run, flight, class, point of time before departure.
Yield: Per run, flight, class, point of time before departure.
Available seats: Per run, flight, class, point of time before departure.
Forecast Error: Per run and point of time before departure, aggregated
over flight, class, time before departure.
Buy-Down and Sell-
Up:
Per customer request, aggregated over run and time be-
fore departure.
Denied and Accepted
Requests:
Per customer request, aggregated over run and time be-
fore departure.
Table 7.3.: Output of Simulation Experiments
7.2. Supply and Demand Data
The goal of revenue management is to sell a perishable product at the right price. In the
airline industry, the product is a seat on a flight differentiated by characteristics including
the price, restrictions and features such as comfort and flexibility. Customers choose what
to buy based on their preference. When the flight departs, the seats on it lose their value.
In order to implement a realistic revenue management simulation, both product (sup-
ply) and customer (demand) need to be defined in detail. Offering routes through a
network as well as a range of booking classes tied to diverse restrictions and features
confronts customers with alternative products. They choose from these according to a
rational choice model including a cost function and a set of maximum values and boolean
acceptance rules.
7.2.1. Supply Information
The supply of an airline consists of seats on flights, represented by tickets. The flights are
described in the airline’s schedule. The capacity of each flight is defined by the aircraft
assigned during the fleet assignment process. By transferring from one flight to another,
customers travel from origin to destination according to itineraries. These itineraries are
calculated by booking engines provided by the airline itself or third parties. They are an
Chapter 7: Simulation Environment for Revenue Management 73
integral part of a simulation that models flexible customers. Finally, tickets are sold for
different booking classes. Each class defines a set of restrictions, features and the price of
the ticket.
Flights and Itineraries
Flights are described by a departure airport, an arrival airport, and a departure date and
time. They are further distinguished by a flight number and a carrier code. A combination
of two airports connected by one or more flights is referred to as a leg. Based on airports
as vertices and legs as edges, networks can be defined.
As described in Section 2, state of the art research tends to consider network rather
than flight views. Especially with the advent of Internet booking portals, customers no
longer book single flights but rather decide for or against itineraries leading them on a
path through a network. While the decision of whether to include a network model in
forecasting and optimization is a methodological one, a realistic simulation should model
both customer choice behavior and the product range based on a network view.
In the revenue management simulation system presented, a network is created from
the legs included during simulation set-up. In order to do so, two data structures are
required. A list of pairings describes combinations of origin and destination linked by
one or more legs. Further information that can be provided for each pairing includes
the knowledge whether a direct flight from origin to destination is available, the traffic
area and the geographical distance involved. For each pairing, a set of itineraries can be
defined. These itineraries describe the legs that a customer might book tickets for in order
to travel from the origin of a pairing to its destination. The complete process is presented
in Figure 7.3; those parts of the model that enable the network view are marked bold.
The connection builder derives pairings and itineraries from a set of flights. For this
purpose, a shortest path algorithm needs to be implemented. As indicated in Figure 7.3,
a modified Dijkstra offering the n-shortest paths is used in this case. How many paths a
customer may chose from and what itineraries are regarded as valid depends on a range
of settings. For example, while it may make sense to travel from Munich to New York
via Frankfurt, the customer should not be offered a trip from Munich to Frankfurt via
Chapter 7: Simulation Environment for Revenue Management 74
Figure 7.3.: Defining the Product
New York. In order to avoid such invalid itineraries, constraints on the overall duration
of travel,the amount of transfers and connecting times are set by parameters.
The relationships of flights, pairings and itineraries are as presented in the following
formulas.
•Let f∈Fbe the flights included in the schedule.
•Let Θ represent the parameters defining maximum duration of travel, amount of
transfers and connecting times.
•Let q∈Qbe the pairings that are possible based on a given set of airports.
•Let ∆ (ˆq, F, Θ) be the Dijkstra function defining the itineraries connecting origin
and destination of pairing ˆqbased on a set of flights Fand a set of parameters Θ.
Chapter 7: Simulation Environment for Revenue Management 75
•Let i∈Ibe the itineraries derived from the schedule and the parameters.
•Let Iqbe the set of itineraries derived from the schedule and the parameters con-
necting origin and destination of pairing q.
•Let Fibe the set of flights included in itinerary i.
Function (7.3) shows the generation of itineraries from pairings. If no itineraries can be
computed according to the parameters, the pairing is removed from the set of pairings.
New itineraries that are found for a pairing are added to the set of itineraries.
∆ (ˆq, F, Θ) =
Iˆq→I:= I∪Iˆq
{∅} → Q:= Q\ˆq
∀ˆq∈Q
(7.3)
According to this process, Inequality (7.4) states that every pairing included in the set of
offered pairings needs to be associated to one or more itineraries.
|Iq| ≥ 1∀q∈Q(7.4)
Equation (7.5) shows that the set of offered itineraries is made up by subsets of itineraries
offered for each pairing.
I=[
q∈Q
Iq(7.5)
Classes and Fares
Tickets are categorized by booking classes characterized by a set of restrictions or features.
The price of the ticket (fare) is defined by a function over the booking class and the chosen
flight. In practice, fare classes present an additional differentiation, defining diverse tariffs
for the same booking class. The simplified model implemented in the simulation assumes
that every booking classes represents exactly one fare class.
Every booking class has distinguishing characteristics: A caption naming it, a set of
restrictions, and a set of product features. Examples of possible restrictions are a weekend
or a minimum stay. Example of features are special flexibility or the seating in the business
compartment. A lack of features may also be modelled as a restriction.
Chapter 7: Simulation Environment for Revenue Management 76
Each flight is assigned a set of booking classes. It seems intuitive that flights span-
ning greater distances are connected to a higher fare than short flights, even if the same
booking class is chosen. Furthermore, it common that the fare increases with increasing
features and decreasing restrictions as presented in Inequality (7.6). Therefore, the func-
tion defining the price of a ticket takes into account the booking class and the flight’s
traffic area.
•Let c∈Cbe the classes offered, ordered by decreasing features and increasing
restrictions.
•Let f∈Fbe the flights offered.
•Let p(f, c) define the price of booking class cfor flight f.
p(f, c)> p (f, c + 1) ∀f∈F;c∈C(7.6)
7.2.2. Demand Model
If there were no customers demanding tickets, revenue management would be senseless.
Therefore, any revenue management simulation needs to include a demand model. How-
ever, the degree of sophistication of this artificial demand depends on the simulation
requirements. Each part of the simulation component needs input and processes and
stores updated or additional data.
In order to provide challenges for current forecasting methods, the customer model
should include as few simplifying assumptions as possible. For example, customers should
aspire to travel via a network from origin to destination rather than statically booking
single flights. Furthermore, the customer model should allow for flexible decisions based
on individual availability situations as they are possible in the age of comparison shopping.
Finally, while the demand model implemented here strives to include new ideas on flex-
ible customer behavior, one caveat needs to be mentioned. As flexible customer choice
was excluded by the assumption of independence in early demand forecasts for revenue
management, other aspects of customer behavior may not be included in the model im-
plemented here. Such implicit assumptions may lead to results that deviate from real-life
observations, but cannot be avoided whenever a model of reality is designed based on
current knowledge.
Chapter 7: Simulation Environment for Revenue Management 77
Pairing?
Customer Type?
Week? Day? Hour?
Product Requirements
Factors? Maxima?
Request
by request
shares
by customer mix normally
distributed
error terms
DCP?
Poisson
Process
Figure 7.4.: Request Generation
Demand Volume and Arrival
The overall amount of requests is approximated within the set-up parameters. It is as-
signed as a percentage of the amount of seats offered based on the supply model. A
demand volume setting of 150% will therefore result in about 1.5 times as many requests
as there are seats. This is an approximation in two regards: Some of these requests can
end up as bookings on more than one flight, and the actual demand volume is based on
ainhomogenous Poisson process as described below. The demand volume is distorted by
an error term for each run and used as an input for the intensity of a Poisson process.
This avoids a too deterministic model of demand.
Definition (7.7) formalizes the distortion of overall demand volumes.
•Let Rbe the parameter defining the average amount of requests to be scheduled
per simulation run of the experiment.
Chapter 7: Simulation Environment for Revenue Management 78
•Let sbe the error term of demand for simulation run sdrawn from the normal
distribution with expected value 0 and a deviation defined by an input parameter.
•Let Rsbe the set of requests that is generated for simulation run s.
|Rs|:= R+s∀s= 1, ..., Ns(7.7)
Depending on the number of pairings included in the simulation, the share of requests
allocated to each can be fixed manually or automatically. In an aggregated, automated
approach, request shares depend on characteristics such as traffic area or the existence of
direct flights.
Equation (7.8) presents the underlying constraints.
•Let q∈Qbe the pairings offered.
•Let γ(q)∈[0,1] be the share of overall requests allocated to pairing q.
X
q∈Q
γ(q) = 1 (7.8)
In order to allow for sufficiently complex patterns in customer arrival and behavior, the
simulation is based on the concept of customer types. The customer type describes the
factors applied for a choice of itinerary given a combination of origin and destination, the
requirements of a booking class, a customer’s price-sensitivity and the arrival distribution.
The share of the requests assigned to one pairing that is connected to one customer type
is determined by the pairing’s customer mix. Again, this customer mix (a distribution
over the existing customer types) may be assigned manually per pairing or automatically
based on pairing characteristics.
Depending on the motivation for implementing a simulation system, the featured de-
mand model can include different degrees of complexity. In order to test the workings of
mathematical methods especially with regard to optimization, demand is frequently mod-
eled based on stochastic distributions. Demand arrival in the sense of the implementation
presented here is based on a Poisson process with separate parameters for each customer
type.
Depending on the number of pairings included in the simulation, the request share can
be fixed manually or automatically. In an aggregated, automated approach, pairings can
Chapter 7: Simulation Environment for Revenue Management 79
be assigned shares depending on characteristics such as traffic area or the existence of
direct flights.
Every pairing has a customer mix associated to it, which includes one or more customer
types. Equation (7.9) presents the underlying constraints.
•Let m∈Mbe the index of customer types.
•Let η(q, m)) ∈[0,1] be the share of requests allocated to pairing qthat is to be
based on customer type m.
X
m∈M
η(q, m)=1∀q∈Q(7.9)
The absolute amount of requests based on one customer type for one pairing can be
calculated from the request share and the customer mix as shown in Definition (7.10).
|Rq,m
s|:= |Rs|·γ(q)·η(q, m) (7.10)
Based on this, an inhomogeneous Poisson Process can be generated using the arrival
distribution defined for the customer type.
•Let λq,m be the overall intensity of the Poisson process for pairing qand customer
type mthroughout the booking horizon.
•Let P[N(q, m, t +τ)−N(q, m, t) = k] be the probability of krequests based on
customer type mto arrive for pairing qin the time slice tto t+τof the booking
horizon of simulation run s.
This means that the Poisson Process (Xi,q,m)i∈Nis defined by Xi,q,m is distributed accord-
ing to exp(λq,m). It defines P[N(q, m, t +τ)−N(q, m, t) = k].
When requests are generated from customer types, a normally distributed error term
is added to the cost function. A function that computes requests from customer types
therefore needs input variables as presented in Function (7.11).
•Let |Rq,m,Nt
s|be the number of requests based on customer type mplanned to arrive
for pairing qwithin the booking horizon of simulation run s.
•Let σbe the deviation of the normal distribution that error terms rare drawn
from.
Chapter 7: Simulation Environment for Revenue Management 80
•Let Ψ m, q, |Rq,m,Nt
s|, rbe the function that defines |Rq,m,Nt
s|individual requests
for pairing qbased on customer types mand a given error deviation r.
Ψm, q, |Rq,m,Nt
s|, σ=Rm,q,Nt
s
∀m∈M, q ∈Q, s = 1, ..., Ns(7.11)
As shown in Equation (7.12), the overall set of requests generated for one simulation run
can be divided into subsets of requests derived from specific customer types for specific
pairings.
Rs=[
m∈M[
q∈Q
Rq,m,Nt
s∀s= 1, ..., Ns(7.12)
Itinerary Choice
Given a departure day and a pairing, the choice of itinerary is based on the cost function
and product acceptance stored in each request. A discussion of the decision factors follows.
More factors are imaginable and could be implemented without much additional effort.
As this section formally describes the preference for one itinerary over the other given
by each request, some notation is required. First, there are some additional features of
itineraries and pairings:
•Let q∈Qbe the index of all pairings offered in the simulation.
•Let νdist (q) be the minimum distance between origin and destination airports of the
pairing q.
•Let νdur (q) be the minimum travel time required by pairing q
•Let i∈Ibe the index of all itineraries offered in the simulation.
•Let Iqbe the set of all itineraries connecting origin and destination of pairing q.
•Let Ii,q be a boolean matrix indicating whether itinerary iconnects pairing qas
shown in Definition (7.13).
•Let xdep (i) be the departure time of itinerary i.
•Let xdur (i) be the travel time attached to itinerary i.
Chapter 7: Simulation Environment for Revenue Management 81
•Let xtrans (i) be the number of transfers in itinerary i.
Ii,q :=
1i∈Iq
0i /∈Iq∀i∈I;q∈Q(7.13)
Next, the factors of the cost function and the product acceptance defined in the cus-
tomer type:
•Let βdep (m) be the weight of the deviation from the preferred departure time in the
cost function of the customer type m.
•Let βdur (m) be the weight of the difference between actual and minimum travel
time in the cost function of the customer type m.
•Let βtrans (m) be the weight of the number of transfers included in the chosen
itinerary in the cost function of the customer type m.
•Let βcar (m) be the cost factor attached to any itinerary that is not provided by the
preferred carrier of customer type m.
•Let δdep (m) be the factor for maximum acceptable deviation from wd(r), defined
by the customer type m.
•Let δdur (m) be the factor for maximum acceptable travel time, defined by the cus-
tomer type m.
•Let σbe the deviation of the normal distribution that the error terms are drawn
from.
Finally, some more characteristics of requests:
•Let r∈Rbe the index of requests included in the simulation.
•Let mrbe the customer type that request rwas generated from.
•Let qrbe the pairing that request rwas generated for.
•Let wdep (r) be the preferred departure time of request r.
•Let rbe the actual error term drawn from the normal distribution and attached to
the cost function of request r.
Chapter 7: Simulation Environment for Revenue Management 82
•Let ˆ
C(i, r) be the cost of itinerary iconsidered by request r, without regard for the
actual ticket price (given the assumption that all itineraries cost the same).
With regard to the set of itineraries acceptable according to one request, note the
relationship displayed in Definition (7.14). All sets of alternative itineraries are subsets of
the quantity of itineraries offered. In order to belong to the set of acceptable itineraries of
one request, the considered alternative needs to belong to the set of itineraries associated
with the pairing the request is targeted at.
Ir⊂Iqr⊂I∀r∈R(7.14)
Departure Deviation: The time of day that is the preferred departure time wdep (r) de-
pends on a daily distribution from which the preferred hour of departure is drawn when
the request is generated. The factor βdep (mr) weights the difference between wdep (r)
and the departure time of the itinerary considered, xdep (i), in the cost function. Only
itineraries that fulfill the constraint shown in Inequality (7.15) are considered. This in-
cludes a maximum acceptable deviation from the preferred departure time that depends
on a parameter δdep (mr) as well as the distance covered by the pairing qr,ν(qr).
wdep (r)−xdep (i)≤δdep (mr)·pν(qr)
∀r∈R;i∈Iqr
(7.15)
Travel Time: The factor βdur (mr) weights the difference between the minimum travel
time νdur (qr) connected to the pairing qrrequested and the travel time of the itinerary
considered, xdur (i), in the cost function. Only itineraries that fulfill the constraint shown
in Inequality (7.16) are considered. This constraint includes a maximum acceptable de-
viation from the minimum travel time that depends on a parameter δdur (mr) as well as
the distance covered by the pairing qr,ν(qr).
νdur (qr)−xdur (i)≤δdur (mr)·pν(qr)
∀r∈R;i∈Iqr
(7.16)
Transfers: The factor βtrans (mr) weights the number of transfers included in the con-
sidered itinerary, xtrans (i), in the cost function. Only itineraries that fulfill the constraint
Chapter 7: Simulation Environment for Revenue Management 83
shown in Inequality (7.17) are considered; this constraint includes a maximum acceptable
number of transfers that depends on a parameter δtrans (mr).
xtrans (i)≤δtrans (mr)
∀r∈R;i∈Iqr
(7.17)
Brand Preference: If a carrier is preferred by a customer type and several carriers are
included in the simulation experiment, a factor βcar (mr) is included in the cost function.
It adds a constant additional cost to any brand that is not the preferred carrier.
Cost Function Without Price: The cost function without regard for the price is shown
in Equation (7.18). Note that a normally distributed error depending on the customer
type, ris added to each request’s cost function to individualize it. By minimizing the
cost, the first choice itinerary for every request can be determined from the quantity
of acceptable itineraries according to the constraints shown above. However, when a
customer makes a booking decision, he can only really consider itineraries for which
tickets are available.
ˆ
C(i, r) = βdep (mr)νdep (qr)−xdep (i)+βdur (mr)νdur (qr)−xdur (i)
+βtrans (mr)·xtrans (i) + βcar (mr) + r
∀r∈R;i∈Iqr
(7.18)
Simplifying Model Assumptions: In the revenue management simulation presented in
this text, some transformations are based on pragmatic assumptions. One is the as-
sociation between maximum acceptable departure time deviations and travel times: A
connection to the overall distance traveled is known, but the precise functional form is
not established. For the data used in this simulation, the product of a factor and the
square-root of the distance worked well. However, different functional forms are con-
ceivable. The same transformation is used to make the customers’ willingness to pay
dependent on the distance traveled. Finally, the form of the cost function is assumed to
be linear. Much more complex functional forms are conceivable. Implementing them is
easily possible, but a comparison of cost functions was not the focus of this work. In
order to model flexible customer choice behavior, the condition for bookings depending
on availabilities, the linear cost function is sufficient.
Chapter 7: Simulation Environment for Revenue Management 84
Class Choice
In order to formally express the logic of class choice, additional variables have to be
defined:
•Let f∈Fbe the index of flights.
•Let Fibe the set of flights included in itinerary i.
•Let c∈Cbe the index of booking classes offered.
•Let p(f, c) be the fare associated to a ticket for flight fin booking class c.
•Let z∈Zbe the set of restrictions of booking classes - the absence of a feature,
such as comfort seating, is modeled as a restriction.
•Let Zcbe the restrictions included in booking class c.
•Let Zrbe the restrictions accepted by request r.
•Let δprice
rbe the factor defining maximum willingness to pay for customer type m.
•Let δprice
rbe the factor defining maximum willingness to pay for request r.
•Let βprice
mbe the weight of fare in the cost function of the customer type m.
•Let C(i, r) be the cost of itinerary ias defined by the cost function of request r
when a lowest available fare has been found.
•Let δproduct
r,c be a boolean matrix defining whether the product represented by booking
class cis acceptable according to the product requirements of request r.
•Let δwtp
r,c,i be a boolean matrix defining whether the price of class con itinerary iis
acceptable according to the willingness to pay of request r.
•Let cmin
rbe the cheapest acceptable class according to the product requirements of
request r.
Product Characteristics: Every customer type includes a list of acceptable restrictions
and required features for classes. These correspond to the restrictions and features that
are used to describe the booking classes in the supply model. Whether one of the classes
Chapter 7: Simulation Environment for Revenue Management 85
available for an itinerary is acceptable becomes a matter of aligning class characteristics
and customer requirements. Class cis acceptable for request rif it fulfills the condition
presented in Inequality (7.19):
δproduct
r,c :=
1Zc⊂Zr
0 else.
∀r∈R;c∈C
(7.19)
Based on product acceptance, the acceptable class with the lowest price can already
been defined. This is based on the model limitation that classes have the same descending
order of price and restrictions on all flights. The cheapest acceptable class of a request r
is determined according to Definition (7.20).
cmin
r:= c0| |p(◦, c0) = min p(◦, c)∀c∈C×δproduct
r,c 
∀r∈R(7.20)
Price Characteristics: Included in the definition of the customer type is the maximum
willingness to pay. It depends on the parameter δprice
mand the distance covered by the
pairing qr,ν(qr). δprice
ris calculated by distorting the underlying δprice
mrwith the normally
distributed error term ras shown in Definition (7.21). The choice of a combination of
class cand itinerary iis only acceptable if the overall fare fulfills the restriction shown in
Inequality (7.22).
δprice
r:= δprice
mr+r∀r∈R(7.21)
δwtp
r,i :=
1Pf∈Fipf, cmin
r≤δprice
r·pν(qr)
0 else.
∀r∈R;i∈Iqr
(7.22)
The demand model is based on the assumption that every customer will accept a class
that includes more features or less restrictions than required, if that class is offered at
a fare lower than the maximum willingness to pay. A customer will always chose the
cheapest of all acceptable classes of one itinerary. The cost of tickets in the cheapest
acceptable class of the flights included is added to each itinerary’s cost function. The
resulting costs for the considered alternatives are compared and the itinerary with the
Chapter 7: Simulation Environment for Revenue Management 86
lowest cost is chosen. The cheapest available and acceptable class on this itinerary is
booked.
Cost Function With Price: The cost function including the price is shown in Equation
(7.23). The minimum available and acceptable price is added to the function with a
weight factor βprice
mr.
C(i, r) = ˆ
C(i, r) + βprice
mr·X
f∈Fi
pf, cmin
r
∀r∈R;i∈Iqr
(7.23)
If no acceptable class is available, the itinerary is not considered. If no acceptable class
could be determined on any of the acceptable itineraries, no booking takes place.
7.2.3. Exemplary Scenario
An example may be useful in illustrating how a simulation scenario can be generated. This
section will describe the design of supply data and the generation of customer requests.
Picture a small network consisting of four airports (vertices): The domestic airports
FRA (Frankfurt) and HAM (Hamburg) as well as the intercontinental airports JFK (New
York) and BKK (Bangkok). The four airports are connected by three vice-versa flight
legs (edges): FRA-HAM/HAM-FRA, FRA-BKK/BKK-FRA, HAM-JFK/JFK-HAM.
As shown in Figure 7.5, part (a), such a network offers a maximum of six possible
pairings. If no other constraints are considered, a customer can travel from any of the
airports to any of the other airports. An example is traveling from HAM to BKK via
FRA or directly, routes described by the dotted line in part (b).
A connection builder including constraints such as a maximum number of acceptable
transfers and a minimum as well as a maximum connecting time, however, may enforce
a limitation of offered pairings. In the given example, traveling from BKK to JFK is
only possible by transfers at both FRA and HAM. With a restriction to a maximum of
one transfer per itinerary, the pairing BKK-JFK is no longer considered, with the result
shown in part (c).
Chapter 7: Simulation Environment for Revenue Management 87
HAM
FRA
JFK
BKK
HAM
FRA
JFK
BKK
HAM
FRA
JFK
BKK
HAM
FRA
JFK
BKK
a) b)
c) d)
Figure 7.5.: Example – Product
Other pairings, in the given example HAM-BKK, may be excluded due to connecting
times exceeding the maximum given by input parameters. The network presented in part
(d) of Figure 7.5 would remain.
In order to provide complete itineraries for customers to chose from, a connection
builder combines the existing flights to form paths through the network. In keeping
with the current example, four pairings and eight directed itineraries emerge: FRA-HAM
(direct), HAM-FRA (direct), FRA-BKK (direct), BKK-FRA (direct), HAM-JFK (direct),
JFK-HAM (direct), and FRA-JFK (via HAM) as well as JFK-FRA (via HAM). Other
theoretically possible paths such as FRA-JFK-FRA-HAM are excluded by restrictions of
the connection builder.
Before the three existing pairings can be assigned shares of customer types, these need
to be defined. This small example contains only two types of customers: Business travelers
and tourists.
The timing of requests for tickets needs to be fixed per customer type. Keeping in
line with classical assumptions, tourists plan their trips a long time before departure,
whereas business travelers spontaneously decide to travel. Accordingly, the bulk of tourist
Chapter 7: Simulation Environment for Revenue Management 88
Business Tourist
arrival distribution
seasonal distribution
weekly distribution
daily distribution
Figure 7.6.: Example – Customer Types
customers arrives during the first two thirds of the booking horizon, whereas most business
customers arrive during the last third. As presented in Section 7.2.2, this is modeled as
changes in the intensity of the Poisson process over time. The different distributions with
regard to arrival and desired departure time can be seen in Figure 7.6.
Tourists may include different factors in their cost functions and tend to display a lower
maximum willingness to pay than business travelers. In this example, the customer type
representing tourists is willing to accept a weekend stay as well as a minimum stay of
five days. The customer type representing business travelers requires extra flexibility and
seating in the business compartment.
Pairings are assigned a general share of requests as well as a customer mix. To keep a
simulation realistic, the amount of customer requests is calibrated to match productivity
indicators such as seat load factors. An orientation for the assignment of shares to cus-
tomer types may be taken from actual booking data for classes aimed at different product
segments.
Chapter 7: Simulation Environment for Revenue Management 89
For this example, HAM-JFK (vice-versa) is assigned a 40% share of pairings whereas
FRA-BKK (vice versa) gets a 20% share. Assuming an overall demand of a thousand
requests, this means 400 requests for HAM-JFK and 200 requests for FRA-BKK.
One of the pairings is a major business route while the other is set to be a predominantly
tourist market. The consequence is to assign HAM-JFK a customer mix that determines
an 80% share of classical business customers and a 20% share of tourist customers, whereas
FRA-BKK gets exactly the opposite, 20% business customers and 80% tourists.
When requests are generated, the request shares of the pairings are regarded as fixed.
For 200 requests, the fact that the customer will want to travel from FRA to BKK is
certain. For 40 of these, the customer type will be “business” whereas for the rest (160),
the customer type will be “tourist”. A random element enters the model when these
numbers are used as the intensity of the respective Poisson processes for tourist and
business customer types. The request arrivals generated from these Poisson processes can
still differ from the expected intensity.
The distributions underlying the customer types are presented in Figure 7.6. The de-
sired departure time is drawn from three distributions: First the week, then the week
day, finally the preferred hour of departure are drawn. The result might look something
like this: A business customer requests a flight from FRA to BKK, leaving in the tenth
week of the year, on Tuesday, at 9 a.m. – the request arrives ten days before departure.
The request will include the product restrictions defined in the type description, for ex-
ample an exclusive acceptance of seats in the business compartment combined with an
intolerance for weekend stays.
The underlying cost function is taken from the customer type. It is distorted by an
error term drawn from a normal distribution for each request.
7.3. Revenue Management Components
Apart from supply and demand data, a revenue management simulation needs to include
a model of the systems actually in use in airline revenue management. Required are a
forecast, an optimization, and an inventory control. The implementation details for these
modules are described in the following text.
Chapter 7: Simulation Environment for Revenue Management 90
7.3.1. Forecast
Following basic approaches to demand forecasting as described in Section 3, two forecast
methods have been implemented. The exponential smoothing forecast follows the tra-
ditional assumption of static demand by observing historical bookings for one class to
predict demand for the same class. The price sensitive forecast alternative considers pos-
sible buy-down and sell-up to occur between classes that are regarded as interchangeable
by the customers.
Initial Forecasts
Every forecast method needs past observations to predict the future. In the first run of
a simulation experiment, such information is not yet available. Several ways of handling
this are conceivable:
Zero Forecast: When this forecast is chosen as the initialization method, all demand to
come is set to zero. As actual bookings are observed, the forecast is expected to increase.
Random Forecast: When this forecast is chosen as the initialization method, a parame-
ter is provided to indicate the predicted seat load factor. From this, the absolute number
of demand is derived via the capacity. This demand is then predicted equally distributed
over the booking classes.
Real Forecast: Using the real-world forecast for the supply included in the simulation
could be a way of ensuring a realistic initial status. However, the condition for this
is a demand model that is accurately calibrated to match real-world demand. As full
information on real-world customer choice behavior is not available, this is condition
cannot be fulfilled.
Psychic Forecast: This is a forecast that is based on knowledge of demand as inherent
in the simulation. Different methods of computing it are conceivable and may lead to
different results.
Chapter 7: Simulation Environment for Revenue Management 91
Exponential Smoothing
The exponential smoothing forecast predicts demand volume based on historical obser-
vations. By predicting demand for time slices before departure, the forecast considers
demand arrival patterns. By predicting demand for each flight, departure time patterns
are included.
The inclusion of seasonal patterns in the forecast could be implemented by adding
methods taken from time series forecasting. However, as the focus of the simulation ex-
periments conducted is on the evaluation of the quality of unconstraining and recognition
of demand behavior, this has not been implemented.
In the exponential smoothing forecast the unconstraining aspect is included via additive
pick-up. Historical bookings are transformed unless the class in which they occurred was
available throughout the considered time slice. If it was available, the observation is
added to the history of bookings as shown in Definition (7.24). If it was not available,
the number of bookings observed is compared to that observed in previous runs while the
class was open. The higher value is used. This process is formally expressed in Definition
(7.25).
•Let s= 2, ..., Nsbe the runs included in a simulation for which a forecast update is
performed. s+ 1 occurs chronologically after sand can be based on historical data
derived from s. For s= 1, an initial forecast is supplied.
•Let t= 0, ..., Ntbe points of time in booking horizon, with t= 0 designating the
start of the booking horizon and t=Ntbeing the time of departure.
•Let c∈Cbe the booking classes ordered by descending price.
•Let f∈Fbe the flights included in the schedule.
•Let b(f, c, t, s) be the bookings observed for flight f, class c, between points of time
before departure t−1 and t, in simulation run s.
•Let ˆ
b(f, c, t, s) be the average of historical bookings during the runs 1 to sthat
occurred on flight fbetween points of time tand t−1 while booking class cwas
available.
Chapter 7: Simulation Environment for Revenue Management 92
•Let bunc (f, c, t, s) be the unconstrained bookings for flight f, class c, between points
of time t−1 and t, in simulation run s.
•Let a(f, c, t, s) be the seats available for sale for flight f, class c, at points of time
before departure tin simulation run s.
First, the unconstrained bookings observed while the class was available need to be
updated:
ˆ
b(f, c, t, s) :=
ˆ
b(f,c,t,s−1)·(s−1)+b(f,c,t,s)
sa(f, c, t −1, s)>0, a (f, c, t, s)>0
ˆ
b(f, c, t, s −1) else.
∀f∈F;c∈C;t= 1, ..., Nt;s= 2, ..., Ns
(7.24)
Next, the observed bookings need to be unconstrained if the class was not available
throughout the time slice:
bunc (f, c, t, s) :=
b(f, c, t, s)a(f, c, t −1, s)>0; a(f, c, t, s)>0
max ˆ
b(f, c, t, s), b (f, c, t, s)else.
∀f∈F;c∈C;t= 1, ..., Nt;s= 2, ..., Ns
(7.25)
The forecast is updated with the help of the unconstrained bookings. The emphasis
given to new bookings over the existent forecast is influenced by a parameter α. Definition
(7.26) shows how the forecast is updated after each run.
•Let func (f, c, t, s) be the predicted demand to arrive for flight fbetween points of
time t−1 and tper class cand simulation run s.
•Let func (t, c, t, 1) be the initial forecast.
•Let αexp be the weight of new bookings in the calculation of the updated uncon-
strained forecast.
func (f, c, t, s) := (1 −αexp)·func (f, c, t, s −1) + λexp ·bunc (f, c, t, s)
∀f∈F;c∈C;t= 1, ..., Nt;s= 2, ..., Ns;αexp ∈[0,1] (7.26)
A variation of the exponential smoothing method inlcudes updates of the forecast within
the booking horizon of one simulation run. A comparison of actual and predicted bookings
Chapter 7: Simulation Environment for Revenue Management 93
is used to determine the current difference in terms of a factor. This factor is applied to
future bookings, as shown in Definition (7.27):
func (f, c, t0, s) := func (f, c, t0, s)·1 + bunc (f, c, t, s)−func (f, c, t, s)
func (f, c, t, s)
∀t0=t+ 1, ..., Nt;f∈F;c∈C;s= 2, ..., Ns
if func (f, c, t, s)>0, a (f, c, t −1, s)>0, a (f, c, t, s)>0
(7.27)
Price-Sensitive Estimators
Alternative forecasting method uses price-sensitive estimators. The underlying assump-
tion is that every customer will buy the cheapest alternative if classes are only differenti-
ated by price. In the simulation implemented, this is true for booking classes that offer
the same set of characteristics. In that case, demand is influenced by two factors, the
price of the cheapest available booking class and the time before departure.
•Let c∈Cbe a set of booking classes sharing the same set of restrictions, ordered
by descending price – this means Ncis the booking class with the lowest price.
•Let b(f, c, t, s) be the bookings that were observed for flight fin class cbetween
points of time t−1 and tof simulation run s.
•Let o(f, c, t, s) be a boolean matrix indicating the lowest available class for flight f
between points of time t−1 and tof simulation run s.
•Let ωT
f,t,s be the vector of time elasticity for flight fdepending on the point of time
tbefore departure of run s.
•Let ωP
f,c,s be the vector of price elasticity for flight fdepending on the class cfor
run s.
•Let uT(f, c, t, s) be the time-based estimator before departure for flight f, class c,
and point of time tof run s.
•Let uP(f, c, t, s) be the price-based estimator for flight f, class c, and point of time
tof run s.
•Let αTbe the weight of the time-based estimator in the joint estimator.
Chapter 7: Simulation Environment for Revenue Management 94
•Let αPbe the weight of the estimator based on price in the joint estimator.
•Let uJ(f, c, t, s) be the joint estimator for flight f, class c, point of time tand run
s.
First of all, the cheapest class available needs to be determined as shown in Definition
(7.28). This information is stored as a boolean flag per time slice before departure, class,
and run.
o(f, c, t, s) :=
1a(f, c, t, s)>0; a(f, c0, t, s) = 0 ∀c0=c+ 1, ..., Nc
1c= 1; a(f, c0, t, s) = 0 ∀c0∈C
0 else.
∀f∈F;c∈C;t= 1, ..., Nt;s= 1, ..., Ns−1
(7.28)
According to the assumption that customers will book the cheapest available class,
bookings can only be expected for this class. Therefore, the observed bookings in the
class indicated by a(f, c, t, s) = 1 enter both the price and the time estimator matrices.
For those classes that were not the cheapest available during time slice t, values are derived
via price-elasticity and time-elasticity vectors. The rules according to which the price and
time estimator matrices are filled are formalized in Definitions (7.29) and (7.30).
uP(f, c, t, s) :=
b(f, c, t, s −1) o(f, c, t, s −1) = 1
uP(f, c −1, t, s)/ωP(f, c −1, s −1) Pc
c0=1 o(f, c0, t, s −1) = 1
uP(f, c + 1, t, s)·ωP(f, c, s −1) PNc
c0=c+1 o(f, c0, t, s −1) = 1
eP(f, c, t, s −1) else.
∀f∈F;c∈C;t= 1, ..., Nt;s= 2, ..., Ns
(7.29)
uT(f, c, t, s) :=
b(f, c, t, s −1) o(f, c, t, s −1) = 1
uT(f, c, t + 1, s −1) ·ωT(f,t,s−1)
ωT(f,t+1,s−1) PNt
t0=t+1 o(f, c, t0, s −1) = 1
uT(f, c, t −1, s −1) ·ωT(f,t,s−1)
ωT(f,t−1,s−1) Pt−1
t0=1 o(f, c, t0, s −1) = 1
uT(f, c, t, s −1) else.
∀f∈F;c∈C;t= 1, ..., Nt;s= 2, ..., Ns
(7.30)
Chapter 7: Simulation Environment for Revenue Management 95
The joint estimator matrix is computed as shown in (7.31) using the parameters αexp
(as introduced for exponential smoothing), αTand αP. Note that demand for one class c
is set to be higher or equal to that for the next, cheaper class c+ 1.
uJ(f, c, t, s) :=
(1 −αexp)·uJ(f, c, t, s −1)
+ min αexp ·αT·uT(f, c, t, s) + αP·uP(f, c, t, s), uJ(f, c + 1, t, s)
∀αexp, αT, αP∈[0,1] ; αT+αP= 1; f∈F;c∈C;t= 1, ..., Nt;s= 2, ..., Ns
(7.31)
The price-elasticity and time-elasticity vectors are updated based on the new joint
estimator matrix. Considered are those points of time up to which a class was the lowest
available and from which on it was no longer available. If no such change took place, the
elasticity vectors are not updated. This is formalized by Definitions (7.32) and (7.33):
ωP(f, c, s) :=
Nt
X
t=1
(o(f, c, t −1, s −1) −o(f, c, t, s −1)) ·uJ(t, c, s)
uJ(t−1, c + 1, s)
∀f∈F;c∈C;s= 2, ..., Ns
(7.32)
ωT(f, t, s) :=
X
c∈Co(f, c, t, s −1) ·uJ(f, c, t, s)
·Y
c∈C c
X
c0=1
o(f, c0, t, s −1)!·1
ωP
c,s Pc∈Co(f, c, t, s −1) >0
ωT(f, t, s −1) Pc∈Co(f, c, t, s −1) = 0
∀f∈F;t= 1, ..., Nt;s= 2, ..., Ns
(7.33)
In order to make this forecast a valid input for the implemented EMSR-b optimization
algorithm, it needs to be transformed into a pseudo-static version. For this purpose,
each customer is assumed to request the booking class that corresponds to his highest
willingness to pay. The transformation is formally described by Definition (7.34):
func (f, c, t, s) :=
uJ(f, c, t, s)−uJ(f, c + 1, t, s)c<Nc
uJ(f, c, t, s)c=Nc
∀f∈F;c∈C;t= 1, ..., Nt;s= 2, ..., Ns
(7.34)
Chapter 7: Simulation Environment for Revenue Management 96
7.3.2. Optimization
The optimization method implemented is EMSR-b as first presented in P. Belobaba
(1987b) (see also Section 2.2). As the focus of this work is foremost on forecast eval-
uation, EMSR-b was chosen for its adaptability to different forecast methods. This way,
the optimization method could be kept stable as forecasts vary and are transformed to
match its demands.
EMSR-b uses the accumulated forecast to allocate seats to booking classes. The decision
of whether or not to reserve capacity for more expensive classes is based on the expected
marginal seat revenue. This is calculated as a ratio of expected demand to arrive in one
class and the next cheaper class multiplied by their respective value.
When forecasts are updated within the booking horizon, the optimization can be up-
dated as well. In order to include this option, the point of time tbefore departure is
included in all calculations described here. If t= 0, the optimization takes place before
any bookings were observed. If availabilities are calculated again after a forecast update,
t > 0 and bookings may have already taken place. Whether or not to update the forecast
and the availabilities is a methodological decision.
•Let c∈Cbe a set of booking classes sharing the same set of restrictions, ordered
by descending price – this means Ncis the booking class with the lowest price.
•Let PNt
t=1 func (f, c, t, s) be the sum of the unconstrained demand to arrive for flight
fin class cuntil point of time tin the booking horizon of run s.
•Let σ(func (f, c, t, s)) be the standard deviation of the forecast of demand for flight
fin class cat point of time tin run s.
•Let b(f, c, t, s) be the bookings that arrived for flight fin class cin run sbetween
points of time t−1 and t.
•Let K(f, t, s) be the available capacity of the flight fthe point of time tbefore
departure of simulation run s.
•Let p(f, c) be the price of class cfor the flight f.
•Let p(f, c, t, s) be the expected marginal seat revenue for a seat in class c, flight f
at point of time tof simulation run s.
Chapter 7: Simulation Environment for Revenue Management 97
•Let U−1be the inverse of the normal distribution.
•Let S(f, c, t, s) be a “safety margin” of demand expected for class cof flight fduring
the time between t−1 and t, based on the standard deviation of predicted demand
and the inverse of the normal distribution.
•Let ˆa(f, c, t, s) be the protected seats assigned to flight fin class cin simulation
run sat the point of time t.
At the beginning of the booking horizon, revenue-maximizing availabilities are com-
puted based on the demand forecast. Protected seats are calculated as the part of the
capacity that is reserved for one particular booking class. In a nested structure, these
seats can be used for bookings in the class they are reserved for or any more expensive
class. Before the booking horizon has started, at t= 0, capacity K(f, 0, s) is equal to the
overall capacity of the aircraft assigned to the considered flight.
To err on the side of caution, the average price of the considered class cand all the
more expensive classes offered is calculated. This average is weighted by the predicted
demand as shown in Definition (7.35). The result is the expected marginal seat revenue:
p(f, c, t, s) := Pc
c0=1 PNt
t0=t+1 func (f, c0, t0, s)·p(f, c0)
Pc
c0=1 PNt
t0=t+1 func (t0, c0, s)
∀f∈F;c∈C;t= 0, ..., Nt−1; s= 1, ..., Ns
(7.35)
The standard deviation of the forecast up to class cis computed as shown in Definition
(7.36):
ˆσ(func (f, c, t, s)) := v
u
u
t
Nc
X
c0=c
σ(func (f, c, t, s))2
∀f∈F;c∈C;t= 1, ..., Nt;s= 1, ..., Ns
(7.36)
A safety margin S(f, c, t, s) is calculated using the inverse of the normal distribution,
U−1and the standard deviation of the forecast.
S(f, c, t, s) := U−11−p(f, c)
p(f, c, t, s)·ˆσ(func (f, c, t, s))
∀f∈F;c∈C;t= 0, ..., Nt−1; s= 1, ..., Ns
(7.37)
Chapter 7: Simulation Environment for Revenue Management 98
Finally, the protection levels are assigned as shown in Definition (7.38). Note that in
the case of availabilities being updated throughout the booking horizon, bookings that
already took place automatically become part of the protection levels. Protected seats are
allocated from top down: First, the most expensive class gets its share, left-over capacity
is assigned to the cheapest class.
K∗(f, c, t, s) := K(f, t, s)−
c−1
X
c0=1
(ˆa(f, c0, t, s))
∀f∈F;c∈C;t= 0, ..., Nt;s= 1, ..., Ns
F∗(f, t, c, s) :=
Nc
X
c0=c Nt
X
t0=t+1
func (f, c0, t0, s)!+S(f, c, t, s)!
− Nc
X
c0=c+1 Nt
X
t0=t+1
func (f, c0, t0, s)!+S(f, c + 1, t, s)!
∀f∈F;c∈C;t= 0, ..., Nt−1; s= 1, ..., Ns
ˆa(f, c, t, s) :=
min (K∗(f, c, t, s), F∗(f, t, c, s))
+
t
X
t0=0
(b(f, c, t0, s)) c<Nc
K∗(f, c, t, s) + Pt
t0=0 (b(f, c, t0, s)) c=Nc
∀f∈F;c∈C;t= 0, ..., Nt−1; s= 1, ..., Ns
(7.38)
7.3.3. Inventory
Areservation system (inventory) is needed to calculate up-to-date availabilities and han-
dle customer bookings. In the inventory, customer requests for tickets meet the class
providing the desired product characteristics at the lowest available fare. Availabilities
are either calculated based on authorization levels and a nesting structure or by comparing
the value of a booking class to the current bid price.
In the case of authorization levels, each booking class offered is assigned a certain
amount of protected seats. An example for an optimization method yielding results in this
Chapter 7: Simulation Environment for Revenue Management 99
form is EMSR-b as presented in Section 7.3.2. In order to make the system more robust
and allow for more than one booking class to be sold at any given time, the concept of
nesting as described for example in Talluri & Van Ryzin (2004b) is applied. As protected
seats are calculated using the EMSR-b method in the implemented simulation system,
this system uses a nested fare structure with availabilities that are equal to or exceed the
protected seats.
Any class is available for as long as the overall available seats exceed those seats pro-
tected for more expensive classes. The relationship is presented in Definition (7.39) ac-
cording to the notation introduced in Section 7.3.2.
a(f, c, t, s) :=
min K(f, t, s)− c−1
X
c0=1
(ˆa(f, t, c0, s)) −
c−1
X
c0=1 t
X
t0=1
b(f, t0, c0, s)!!,0!
∀f∈F;c∈C;t= 0, ..., Nt−1; s= 1, ..., Ns
(7.39)
According to this concept, once the protected seats reserved for one class are used up,
customers wanting to book a ticket in this class can access left-over protected seats in
any of the lower classes. Depending on the precise implementation, sold seats are either
subtracted from the lowest class’s protected seats or only from the protected seats of the
class directly nested under the desired class. Availabilities for every class are calculated
by adding up its protected seats as well as all the protected seats for the lower-nested
classes. Depending on whether the separation between the compartments is treated as
flexible or as fixed, a nesting structure can include all the classes offered or only the classes
offered within one compartment.
A graphic illustration of this concept is provided by Figure 7.7. Three classes are
included in the example shown. Class 1 is the most expensive one with the highest
nesting position. While 50 seats are protected for this class alone, all 100 seats making
up overall capacity may be sold in this class if none of the cheaper ones is booked. Class
2 represents an intermediate value. Based on unconstrained forecasts, 10 protected seats
have been calculated for this class. As 50 seats are protected for class 1, the remaining
50 seats may be sold in class 2 if the cheaper class is not booked. Class 3, the cheapest
option, is allocated the “left over” seats after protected seats for all more expensive classes
Chapter 7: Simulation Environment for Revenue Management 100
K(f, t, s)
100 seats
capacity
of flight f
at time t
of run s
a(f, 1, t, s)
protected
seats for
class 1 of
flight f at
time t of
run s
50 seats
â(f, 1, t, s)
available
seats for
class 1 of
flight f at
time t of
run s
a(f, 2, t, s)
protected
seats for
class 2 of
flight f at
time t of
run s
100 seats 10 seats 50 seats 40 seats 40 seats
â(f, 2, t, s)
a(f, 3, t, s) â(f, 3, t, s)
protected
seats for
class 3 of
flight f at
time t of
run s
available
seats for
class 2 of
flight f at
time t of
run s
available
seats for
class 3 of
flight f at
time t of
run s
Figure 7.7.: Inventory: Protected and Available Seats
have been reserved. These 40 seats may be sold in class 3. As the cheapest class, it cannot
access availabilities in any of the other classes.
If bid price controls are implemented, calculating availabilities becomes more straight
forward. Each booking class is assigned a value based on its fare and, possibly, the
buy-down it triggers according to a forecast including such information. This value,
also referred to as capacity allocation value, is compared to the bid price assigned by the
optimization component. If the class’s value is lower than the bid price, it is not available,
if it is equal or higher than the bid price, it is available. The bid price changes flexibly
with regard to the seats already sold and the customers expected to request seats in the
future.
Chapter 7: Simulation Environment for Revenue Management 101
When a customer request arrives, the lowest available classes complying to the cus-
tomer’s product preferences are returned. Now, the customer can chose whether to book
a ticket; he will automatically book the alternative minimizing his cost function if his
maximum willingness to pay has not been excelled by all offers. If a reservation is af-
firmed, the inventory updates the availabilities. In the case of a nesting structure, this
includes updating the protected seats of the lower-priced classes if necessary. In the case
of bid price controls, this includes updating the bid price according to the specifications
given by the bid price curve provided by the optimization.
7.4. Market Implementations
As the basis of the simulation experiments presented in this thesis, one particular instance
of the demand model is continually used. This means the creation of a number of specific
customer types, their assembly to a customer mix, and the confrontation of this demand
with some variations of supply.
To ensure consistency, demand is varied along two clearly defined parameters and con-
fronted three alternative supply structures. These combinations can be used to evaluate
the application of a number of forecast and forecast evaluation methods.
7.4.1. Demand Variations
A number of different customer types were implemented to allow for a variety of product-
oriented and price-oriented behaviors. Underlying is the assumption of rational choice
behavior: Customers will always try to minimize cost in terms of price and preference
factors. For the mode, this means that all customer types will buy the cheapest ticket
available given all other (product) conditions are equal. Furthermore, they will choose
the itinerary that offers the best conditions according to the factors of their cost function.
Only one basic demand model was realized for the simulation experiments conducted.
This is based on the theory that revenue management does not change basic market
characteristics but instead utilizes different market segments by developing a targeted set
Chapter 7: Simulation Environment for Revenue Management 102
of offers. According to this, the same demand model should lead to quite different result
indicators depending on both the fare structures and the inventory controls in place.
To allow for scenarios that are purely price-driven, purely product-driven, and hybrid,
requests based on customer choice behaviors including either priority need to be generated.
The implemented customer types are:
•No Frills 1: Main focus on price, will only accept “no-refund” restriction,
maximum willingness-to-pay is about 10% of the price span offered.
•No Frills 2: Main focus on price, will only accept “no-refund” restriction,
maximum willingness-to-pay is about 30% of the price span offered.
•Tourist Weekend: Main focus on price, will accept “no-refund” and “weekend-
stay” restriction, maximum willingness-to-pay is about 50 % of the price span of-
fered.
•Tourist Medium: Main focus on price, will accept all restrictions, maximum
willingness-to-pay is about 60% of the price span offered.
•Tourist Lux: Main focus is on travel time and departure time, will accept all
restrictions, maximum willingness-to-pay is over 100 % of the price span offered.
•Business Long: Main focus is on travel time and departure time, will accept
“no-refund” and “minimum stay” restrictions, maximum willingness-to-pay is about
90% of the price span offered.
•Business Flex: Main focus is on flexibility, travel time and departure time, will
not accept restrictions, maximum willingness-to-pay is about 95% of the price span
offered.
•Business Pure: Main focus is on flexibility, business compartment seats, travel
time and departure time, will not accept restrictions, maximum willingness-to-pay
is over 100% of the price span offered.
Figure 7.8 shows the distribution of these customer types in the mix applied to all
markets in the simulation. Depending on the error term added to the individual origin-
destination combinations, the customer mix presented may vary. This error term is based
Chapter 7: Simulation Environment for Revenue Management 103
No Frills 1
10%
No Frills 2
10%
Tourist Weekend
10%
Tourist Lux
5%
Business Long
5%
Business Flex
10%
Business Pure
10%
No Frills 1
10%
No Frills 2
10%
Tourist Weekend
10%
Tourist Medium
40%
Tourist Lux
5%
Business Long
5%
Business Flex
10%
Business Pure
10%
Figure 7.8.: Mix of Customer Types
on an input parameter defining the standard deviation of the normal distribution the error
term is drawn from.
Overall demand was varied according to two parameters, demand volume and the devi-
ation of the error term. While volume was controlled over the overall amount of requests
generated, the error term drawn from the normal distribution with a zero average and
the set standard deviation is included in multiple parts of the model. This way, the input
parameter deviation influences the uncertainty of demand.
The realized variations are listed below. The abbreviation “Vol.” describes the number
of requests in a percentage relation to the number of seats included in the simulation. For
example, “Vol.050” indicates that the number of generated requests equals 50% of the
number of seats. The abbreviation “Dev.” describes the standard deviation set for the
Chapter 7: Simulation Environment for Revenue Management 104
normal distribution the error term is drawn from. In the case of “Dev.00”, demand only
varies from one run to the other due to the effects of drawing from a Poisson distribution.
•Vol.050 Dev.00: 3.000 requests are scheduled to arrive. The standard deviation
of error terms is 0.
•Vol.050 Dev.01: 3.000 requests are scheduled to arrive. The standard deviation
of error terms is 1.
•Vol.050 Dev.05: 3.000 requests are scheduled to arrive. The standard deviation
of error terms is 5.
•Vol.050 Dev.10: 3.000 requests are scheduled to arrive. The standard deviation
of error terms is 10.
•Vol.050 Dev.20: 3.000 requests are scheduled to arrive. The standard deviation
of error terms is 20.
•Vol.100 Dev.00: 6.000 requests are scheduled to arrive. The standard deviation
of error terms is 0.
•Vol.100 Dev.01: 6.000 requests are scheduled to arrive. The standard deviation
of error terms is 1.
•Vol.100 Dev.05: 6.000 requests are scheduled to arrive. The standard deviation
of error terms is 5.
•Vol.100 Dev.10: 6.000 requests are scheduled to arrive. The standard deviation
of error terms is 10.
•Vol.100 Dev.20: 6.000 requests are scheduled to arrive. The standard deviation
of error terms is of 20.
The consequences of varied volume and error deviation are presented in Figure 7.9.
Shown are the results of first-come-first-serve inventory controls averaged over 50 simula-
tion runs. As expected, the average seat load factor is higher for “Vol.100”. In addition,
it decreases slightly with increasing error deviation. The resulting deviation of the indi-
cator seat load factor increases with increasing error deviation. Average revenue is lower
for “Vol.100”: With more overall requests, more requests for cheap booking classes are
Chapter 7: Simulation Environment for Revenue Management 105
70,82 71,07 71,14 70,74 69,83
64,36 60,01 56,75 55,67 54,54
0
10
20
30
40
50
60
70
80
0 1 5 10 20
SLF Average (%)
Deviation of Normally Distributed Error-Term
Vol.=100
Vol.=050
0,24
0,27
0,88
1,53
3,25
0,34 0,27
2,22
3,12
6,13
0
1
2
3
4
5
6
7
SLF Deviation (%)
Vol.=100
Vol.=050
70,82 71,07 71,14 70,74 69,83
64,36 60,01 56,75 55,67 54,54
0
10
20
30
40
50
60
70
80
0 1 5 10 20
SLF Average (%)
Deviation of Normally Distributed Error-Term
Vol.=100
Vol.=050
2.166 2.555 2.507
2.101 1.773
5.235
4.352
2.530
1.997 1.623
0
1.000
2.000
3.000
4.000
5.000
6.000
0 1 5 10 20
Rev. Average (tsd. €)
Deviation of Normally Distributed Error-Term
Vol.=100
Vol.=050
0,24 0,27
0,88
1,53
3,25
0,34 0,27
2,22
3,12
6,13
0
1
2
3
4
5
6
7
0 1 5 10 20
SLF Deviation (%)
Deviation of Normally Distributed Error-Term
Vol.=100
Vol.=050
100
68
53
89
239
82
147
50
100
150
200
250
300
Rev. Deviation (tsd.
€)
Vol.=100
Vol.=050
70,82 71,07 71,14 70,74 69,83
64,36 60,01 56,75 55,67 54,54
0
10
20
30
40
50
60
70
80
0 1 5 10 20
SLF Average (%)
Deviation of Normally Distributed Error-Term
Vol.=100
Vol.=050
2.166 2.555 2.507
2.101 1.773
5.235
4.352
2.530
1.997 1.623
0
1.000
2.000
3.000
4.000
5.000
6.000
0 1 5 10 20
Rev. Average (tsd. €)
Deviation of Normally Distributed Error-Term
Vol.=100
Vol.=050
0,24 0,27
0,88
1,53
3,25
0,34 0,27
2,22
3,12
6,13
0
1
2
3
4
5
6
7
0 1 5 10 20
SLF Deviation (%)
Deviation of Normally Distributed Error-Term
Vol.=100
Vol.=050
31
100
51 68 61
53
89
239
82
147
0
50
100
150
200
250
300
0 1 5 10 20
Rev. Deviation (tsd. €)
Deviation of Normally Distributed Error-Term
Vol.=100
Vol.=050
Figure 7.9.: SLF Average and Deviation depending on Error Term Deviation
Chapter 7: Simulation Environment for Revenue Management 106
Caption Description Fare Near Fare Far
C seat in business compartment 350 700
Y refundable ticket 250 500
G base fare 175 350
M minimum stay restriction 150 300
W weekend stay restriction 125 250
Table 7.4.: Booking Classes Differentiated by Product-Feature
accepted before valuable customers arrive. Average revenue also decreases with increasing
error deviation. The resulting deviation of revenue shows no clear trend over increasing
error deviation, however, low request volume react stronger to changes in error deviation.
This seems to hold true over all four observed indicators.
7.4.2. Supply Variations
As described in the previous section, the customer behavior is set to be stable throughout
the experiments. However, three representative supply variations are realized to model
three supply strategies found in applied airline revenue management: product-based dif-
ferentiation, price-based differentiation, and hybrid differentiation.
Table 7.4 lists the booking classes that present product-based differentiation. Table
7.5 lists the booking classes that present price-based differentiation. Table 7.6 lists the
booking classes that present hybrid differentiation. Classes are presented from top to
bottom in their nesting order. Two fares are assigned for flights spanning one (“Fare
Near”) or two traffic areas (“Fare Far”).
Parallel to the assumption of rational customer choice behavior described in Section
7.2.2, an assumption of rational supply planning is held with regard to the calibration of
the market scenarios realized. As it would not make sense in terms of economic rationality
for a customer to buy an expensive ticket if a cheap ticket that corresponds to his or her
product preferences was available, it seems unreasonable for an airline to offer a product
that does not trigger demand from a customer segment specific to this product.
Chapter 7: Simulation Environment for Revenue Management 107
Caption Description Fare Near Fare Far
G base fare 175 350
K reduced fare 100 200
L reduced fare 75 150
T reduced fare 50 60
E reduced fare 20 20
Table 7.5.: Booking Classes Differentiated by Price
Caption Description Fare Near Fare Far
C seat in business compartment 350 700
Y refundable ticket 250 500
G base fare 175 350
M minimum stay restriction 150 300
W weekend stay restriction 125 250
K reduced fare 100 200
L reduced fare 75 150
T reduced fare 50 60
E reduced fare 20 20
Table 7.6.: Booking Classes Differentiated by Product Characteristics and Price (Hybrid
Differentiation)
Chapter 7: Simulation Environment for Revenue Management 108
If the risk of buy-down is considered a cost, any new product differing from a base
fare and standard features is costly or neutral. Product characteristics such as a business
compartment differing in terms of physical comfort obviously gain added value through
higher production cost. Features such as the possibility of refund add uncertainty to the
airline’s plans. Tickets that are sold at rates lower than that of the highest rate offered
may cause buy-down and therefore include the risk of lost revenue.
Therefore, demand was calibrated according to the assumption that the airline would
not offer a new booking class if there was no reason to expect additional demand. Using
an expanding fare structure, a number of simulation experiments based on the variation
Vol.050 Dev.00 of the demand model described above were conducted. The fare structures
develop as follows:
•G: This is the version including a single, non-refundable base fare representing fare
structures in a time when neither product nor price differentiation was applied.
•G+Y: With the class Y, a more flexible class that is refundable is added at a price
that exceeds that of G.
•G Y+C: With the class C, an option of comfortable seating in a business class is
added at a price that exceeds that of Y.
•GYC+M: The class M is the first reduced-fare class that is added to the supply
– however, to avoid buy-down, a “minimum-stay” restriction is included.
•GYCM+W(product-based scenario): The class W is another reduced-fare
class that is added to the supply, with a fare that is lower than that of M – however,
to avoid buy-down, a “weekend-stay” restriction is included.
•GYCMW+K,L,T,E(hybrid scenario): In the tradition of no-frills airline,
more classes are introduced at low fares that are not differentiated by restrictions
but only differ in price. The underlying hope is that with good inventory controls,
the gain through increased bookings will exceed the loss due to buy-down behavior.
•GKLTE(price-based scenario): Finally, to include an option that represents
that of low-fare or no-frills airlines, all classes based on product-differentiation are
excluded from the scenario once more and only classes differentiated by price are
offered.
109
25,45 27,51
46,13
55,16
62,67
67,47
72,30
86,31 88,09
67,41
Vol.050 Dev.00: Average SLF over 10 runs
25,45 27,51
46,13
55,16
62,67
67,47
72,30
86,31 88,09
67,41
Vol.050 Dev.00: Average SLF over 10 runs
Figure 7.10.: Increase in Bookings by Additional Classes
The results of differentiation are illustrated in Figure 7.10. In this diagram, every
column presents seat load factors under first-come-first-serve seat allocation. The demand
model used was “Vol.050 Dev.00”. With every added class, seat load factors increase as
a new customer segment is addressed with a new product. The last column shows seat
load factors under a solely price-differentiated environment: Low prices activate customer
segments with a low willingness to pay, but a lack of product features such as business
compartment seats drives away a different customer segment.
In this section, market implementations for a revenue management simulation have been
presented. Using this data, the simulation including flexible demand, and the concept for
decomposition introduced earlier, simulation experiments can be conducted. These may
be used to analyze statements regarding forecast performance.
110
Part III.
Experiments and Conclusions
111
With the help of the simulation system described in Chapter 7 and the concept of
decomposition outlined in Chapter 6, experiments can be designed and executed to analyze
forecast performance and its evaluation. Common evaluation methods can be combined
with knowledge of the implemented demand model. The results of this are presented
in Chapter 8 as combinations of formal statements and simulation results visualized by
graphs.
In Chapter 9, a summary of the outcomes of this thesis is presented. This chapter
recaptures the goals as first introduced in Section 1.2 and detailed in Section 5. It ex-
plains the actions that were taken in order to fulfill the goals and their results. A list
of recommendations compiled from the thoughts documented Chapter 8 is included. Fi-
nally, more ideas on how to apply and extend the concept and the simulation environment
documented in Part II are offered.
Chapter 8: Simulation Based Analysis of Forecast Performance 112
8. Simulation Based Analysis of Forecast Performance
This chapter lists ideas on forecast performance and forecast evaluation methods. These
ideas are used to design simulation experiments that illustrate their ramification and
consequences. Considered are the long-term effect of revenue management methods, as-
pects of error measurements, the use of psychic forecasts, uncertainty of demand, and the
inclusion of price-sensitivity in forecasts.
8.1. Observations on Long-Term Effects of Forecast Methods
When forecasts use historical data generated under their own influence, the repeated
application of revenue management methods leads to an evolving dynamic. A common
example for this is the so-called spiral-down effect. Its theoretical background has been
described in Cooper et al. (2006). In this section, it is used as an example for long-term
effects of revenue management methods.
The spiral-down effect can be expected when forecasting methods based on the as-
sumption of independent demand meet flexible customers. Such customers tend to buy
the cheapest acceptable class available. As a consequence, the forecast method will sys-
tematically predict more demand for low-fare classes and less demand for valuable classes.
An optimization using this forecast reserves less seats for valuable classes and allows more
availability in low-fare classes. The forecast becomes a self-fulfilling prophecy as the in-
creased availability is used by flexible customers and more bookings are observed in cheap
classes.
This effect is the result of a combination of methods and demand model. Therefore, the
spiral-down effect can be observed best when the complete system is analyzed as proposed
in Granger & Pesaran (2000).
Chapter 8: Simulation Based Analysis of Forecast Performance 113
The simulation system offers ways of evaluating the development over time in fast-
forward mode. As the demand model can be kept stable, the effects of the cycle of
bookings-forecast-optimization-bookings-etc. can be observed over dozens of simulation
runs. The results of the complete system and the accuracy of the forecast component are
evaluated.
Revenue Management Configurations: To validate statements on long-term effects,
four forecast methods are applied to the price-based scenario with customers choosing be-
tween restriction-free classes. Implemented are three variations of an exponential smooth-
ing forecast in combination with EMSR-b. They differ in the weight that is attached to
new observations. While “Exp025” includes these new observation with a smoothing fac-
tor of α= 0.25, “Exp050” uses a smoothing factor of α= 0.5 and “Exp075” applies
a smoothing factor of α= 0.75. To provide a lower boundary, a first-come first-serve
strategy is provided and referred to as “FCFS”.
Given the combination of demand, supply, and methods, over the course of several
simulation runs, shifts in bookings and availability as well as plunging yield and revenue
should manifest. This does hold true as shown by the simulation results depicted in the
further text and figures. The spiral-down effect can be observed best during the first
simulation runs – for this reason, further examinations focus on s= 1..20. In consecutive
runs, the development slows down as it approaches a steady state.
Indicators: Several indicators can be used to describe the spiral-down effect. It has
consequences for:
•predicted demand (forecast),
•protected seats (availabilities),
•bookings,
•revenue,
•yield,
•forecast evaluations (error measurements).
Chapter 8: Simulation Based Analysis of Forecast Performance 114
With regard to their expected development, these indicators and their formulaic expression
are listed in the following paragraphs. The outcome of the respective experiment is used
to illustrate the ideas presented in this section.
•Let c∈Cbe the index of restriction-free booking classes ordered descending by
their price.
•Let f∈Fbe the index of flights.
•Let t= 0, ..., Ntbe points of time before departure; demand arrives after t= 0,
t=Ntis the time of departure.
•Let s= 1, ..., Nsbe the runs included in a simulation. s+ 1 occurs chronologically
after sand can be based on historical data derived from s.
•Let func (f, c, t, s) be the unconstrained demand forecast per class con flight ffor
the time between t−1 and tof simulation run s.
•Let ˆa(f, c, t, s) be the protected seats per class con flight fat point of time tin
simulation run s.
•Let a(f, c, t, s) be the available seats per class con flight fat point of time tin
simulation run s.
•Let b(f, c, t, s) be the bookings per class con flight fthat arrived between points
of time t−1 and tof simulation run s.
•Let Rsbe the set of requests assigned to run sin the simulation demand model.
For every simulation run, the demand model is equivalent, even though individual
requests differ due to the error terms included: Rs≡Rs+1
•Let r(s) be the overall revenue generated in simulation run s.
Forecast: When the forecast is based on historical bookings and customers book the
cheapest tickets available, the amount of predicted requests for valuable classes decreases
while that for cheap classes increases. In the real world, rather than being updated
for simulation runs s, the forecast is computed per departure. As no historical data is
available in the first run (s= 1), a psychic forecast is used to generate the first prediction.
Over consecutive runs, the forecast is updated using exponential smoothing.
Chapter 8: Simulation Based Analysis of Forecast Performance 115
•Let the psychic forecast be described by a function Fpsy.
•Let the exponential smoothing forecast be described by a function Fhist.
Definition (8.1) describes how forecasts are generated:
func (f, c, t, s) =
Fpsy (Rs, F, C, t, 1) s= 1
Fhist (b(f, c, t, s −1) , a (f, c, t, s −1) , func (f, c, t, s −1)) s > 1
∀f∈F;c∈C;t= 1, ..., Nt;s= 1, ..., Ns
(8.1)
If classes Care ordered in the descending order of their value, class 1 is the most
expensive class while class Ncis the cheapest. Given the conditions of the spiral-down
effect, the following development is expected: Forecasts for valuable classes decrease, while
forecasts for cheap classes increase. This can be expressed formally as in Hypothesis (8.2):
lim
s→∞ func (f, c, t, s)=0 c < Nc
∃ns∈N|func (f, Nc, t, s)≤func (f, Nc, t, s +ns)
∀f∈F;c∈C;t= 1, ..., Nt;s= 1, ..., Ns−1
(8.2)
An indicator independent of the overall amount of predicted demand can be derived
by computing the percentage of forecasted requests per class. This forecast-mix func % is
expected to show the behavior described in Hypothesis (8.2), normalized to 100%.
func % (f, c, t, s) = func (f, c, t, s)
Pc∈Cfunc (f, c, t, s)·100
∀f∈F;c∈C;t= 1, ..., Nt;s= 1, ..., Ns
(8.3)
lim
s→∞ func% (f, c, t, s) =
0c<Nc
100 c=Nc
∀f∈F;c∈C;t= 1, ..., Nt
(8.4)
Figure 8.1 shows the amount of demand predicted in the five classes as method “Exp050”
is applied. In order to make the development of the forecast-mix comparable over the
scenario variations, forecasts are expressed as percentages of overall predicted demand,
Chapter 8: Simulation Based Analysis of Forecast Performance 116
Vol. = 050, Dev. = 00
0%
20%
40%
60%
80%
100%
1 2
Simulation Run
E
D
C
B
A
Vol. = 050, Dev. = 01
0%
20%
40%
60%
80%
100%
1 2
Simulation Run
E
D
C
B
A
Vol. = 050, Dev. = 05
0%
20%
40%
60%
80%
100%
1 2
Simulation Run
E
D
C
B
A
Vol. = 050, Dev. = 10
0%
20%
40%
60%
80%
100%
1 2
Simulation Run
E
D
C
B
A
Vol. = 050, Dev. = 20
0%
20%
40%
60%
80%
100%
1 2
Simulation Run
E
D
C
B
A
Vol. = 100, Dev. = 00
0%
20%
40%
60%
80%
100%
1 2
Simulation Run
E
D
C
B
A
Vol. = 100, Dev. = 01
0%
20%
40%
60%
80%
100%
1 2
Simulation Run
E
D
C
B
A
Vol. = 100, Dev. = 05
0%
20%
40%
60%
80%
100%
1 2
Simulation Run
E
D
C
B
A
Vol. = 100, Dev. = 10
0%
20%
40%
60%
80%
100%
1 2
Simulation Run
E
D
C
B
A
Vol. = 100, Dev. = 20
0%
20%
40%
60%
80%
100%
1 2
Simulation Run
E
D
C
B
A
Share of Predicted Demand per Class
Figure 8.1.: Predicted Demand per Class with Exp050
Chapter 8: Simulation Based Analysis of Forecast Performance 117
func % as described in Definition (8.3). As can be seen, forecasts for the most valuable
class, “A”, decrease over the course of 50 runs as those for the cheaper classes offered,
“B”, “C”, “D”, and “E” increase.
27,3 27,7
31,4 32,5 31,5
15,2 15,8 16,1 16,3
20,0
0
5
10
15
20
25
30
35
Dev. = 00 Dev. = 01 Dev. = 05 Dev. = 10 Dev. = 20
Percent Points
Vol. = 050
Vol. = 100
Figure 8.2.: Decrease in Demand Predicted for Class “A”
The degree of this development differs over the different markets. When demand volume
is high as in “Vol. = 100”, it is not as steep as when observing markets with low demand
volume as in “Vol. = 050”. Additionally, the decrease grows stronger with increasing
deviation of the error term distribution. The decrease of the share of predicted demand
for the class “A” in percent points is illustrated by Figure 8.2. However, the trend
described is always the same and can be observed in all the variations of the price-based
scenario.
Availabilities: Availabilities are the result of the optimization, which is based on the
demand forecast. When EMSR-b is applied, the result is expressed as protections, seats
reserved for valuable classes. In a system of nested classes as described in Section 7.3.3,
protected seats are used to compute authorization levels, ensuring a minimum of avail-
ability for valuable classes.
As shown in Definition (8.5), the function Aemsrb uses the EMSR-b algorithm to generate
protections for class con flight fand run sfor the point of time before the booking horizon
Chapter 8: Simulation Based Analysis of Forecast Performance 118
starts, t= 0, ˆa(f, c, 0, s), from the forecast func (f, c, t, s) and the prices per class con
flight f,p(f, c):
ˆa(f, c, 0, s) := Aemsrb Nt
X
t=1
func (f, c, t, s), p (f, c)!
∀f∈F;c∈C;s= 1, ..., Ns
(8.5)
Given the conditions of the spiral-down effect, the following development is expected:
As forecasts for valuable classes decrease, so do the protected seats computed for these
classes. This can be expressed formally as presented in Hypothesis (8.6).
lim
s→∞ ˆa(f, c, 0, s)=0 c<Nc
∃ns∈N|ˆa(f, Nc,0, s)≤ˆa(f, Nc,0, s +ns)
∀f∈F;c∈C;s= 1, ..., Ns+ 1
(8.6)
An indicator that is independent of the overall capacity can be derived by computing
the percentage of seats allocated to each class. This availabilities-mix ˆa%is expected to
show the same behavior described in Hypothesis (8.6), normalized to 100%.
ˆa%(f, c, t, s) = ˆa(f, c, t, s)
Pc∈Cˆa(f, c, t, s)·100
∀f∈F;t= 1, ..., Nt;s= 1, ..., Ns
(8.7)
lim
s→∞ ˆa%(f, c, 0, s) =
0c < Nc
100 c=Nc
∀f∈F;c∈C
(8.8)
Figure 8.3 shows the amount of seats protected in the five classes as method “Exp050”
is applied. In order to make the development of the protection-mix comparable over the
scenario variations, protected seats are expressed as percentages of overall capacity, a%as
described in Definition (8.7). Protected seats for the most valuable class, “A”, decrease
over the course of 50 runs as those for the cheaper classes offered, “B”, “C”, “D”, and
“E” increase.
The degree of this development differs over the different markets. When demand volume
is high as in “Vol. = 100”, it is not as steep as when observing markets with low demand
Chapter 8: Simulation Based Analysis of Forecast Performance 119
Vol. = 050, Dev. = 00
0%
20%
40%
60%
80%
100%
1 2
Simulation Run
E
D
C
B
A
Vol. = 050, Dev. = 01
0%
20%
40%
60%
80%
100%
1 2
Simulation Run
E
D
C
B
A
Vol. = 050, Dev. = 05
0%
20%
40%
60%
80%
100%
1 2
Simulation Run
E
D
C
B
A
Vol. = 050, Dev. = 10
0%
20%
40%
60%
80%
100%
1 2
Simulation Run
E
D
C
B
A
Vol. = 050, Dev. = 20
0%
20%
40%
60%
80%
100%
1 2
Simulation Run
E
D
C
B
A
Vol. = 100, Dev. = 00
0%
20%
40%
60%
80%
100%
1 2
Simulation Run
E
D
C
B
A
Vol. = 100, Dev. = 01
0%
20%
40%
60%
80%
100%
1 2
Simulation Run
E
D
C
B
A
Vol. = 100, Dev. = 05
0%
20%
40%
60%
80%
100%
1 2
Simulation Run
E
D
C
B
A
Vol. = 100, Dev. = 10
0%
20%
40%
60%
80%
100%
1 2
Simulation Run
E
D
C
B
A
Vol. = 100, Dev. = 20
0%
20%
40%
60%
80%
100%
1 2
Simulation Run
E
D
C
B
A
Share of Protected Seats per Class
Figure 8.3.: Protected Seats per Class with Exp050
Chapter 8: Simulation Based Analysis of Forecast Performance 120
15,7 16,1
17,8 18,6 19
12 12,6 12,7 13,7 13,8
0
2
4
6
8
10
12
14
16
18
20
Dev. = 00 Dev. = 01 Dev. = 05 Dev. = 10 Dev. = 20
Percent Points
Vol. = 050
Vol. = 100
Figure 8.4.: Decrease in Seats Protected for Class “A”
volume as in “Vol. = 050”. Additionally, the decrease grows stronger with increasing
deviation of the error term distribution. The decrease of the share of protected seats for
the class “A” in percent points is illustrated by Figure 8.4. However, the trend described
is always the same and can be observed in all the variations of the price-based scenario.
Bookings: In the demand model implemented, given the same product, customers al-
ways book the cheapest class available. For a set of restriction-free classes c= 1, ..., Nc
with similar product characteristics and increasing value, if overall demand volume stays
constant, this means that the amount of bookings b(f, c, t, s) depends solely on the avail-
abilities a(f, c, t, s). As availabilities for cheap classes increase, so do the bookings in these
classes. Formally, this expectation can be expressed as follows:
lim
s→∞ b(f, c, t, s)=0 c<Nc
∃ns∈N|b(f, Nc, t, s)≤b(f, Nc, t, s +ns)
∀f∈F;c∈C;t= 1, ..., Nt;s= 1, ..., Ns−1
(8.9)
Chapter 8: Simulation Based Analysis of Forecast Performance 121
An indicator that is independent of the overall bookings can be derived by computing
the percentage of bookings for each class. This bookings-mix b%is expected to show the
same behavior described in Hypothesis (8.9), normalized to 100%.
b%(f, c, t, s) = b(f, c, t, s)
Pc∈Cb(f, c, t, s)·100
∀f∈F;c∈C;t= 1, ..., Nt;s= 1, ..., Ns
(8.10)
lim
s→∞ b%(f, c, t, s) =
0c<Nc
100 c=Nc
∀f∈F;c∈C;t= 1, ..., Nt
(8.11)
Figure 8.5 shows the amount of seats booked in the five classes as method “Exp050”
is applied. In order to make the development of the booking-mix comparable over the
scenario variations, bookings per class are expressed as percentages of overall bookings,
b%as described in Definition (8.10). As can be seen over all scenario variations, bookings
for the most valuable class, “A”, decrease over the course of 50 runs as those for the
cheaper classes offered, “B”, “C”, “D”, and “E” increase.
The degree of this development differs over the different markets. When the deviation
of the error term distribution is low, it is stronger for markets with high demand. When
the deviation of the error term is high, the opposite seems to be true. The decrease grows
stronger with increasing deviation of the error term distribution for “Vol. 050”, but not
for “Vol. 100”. The decrease of the share of bookings for the class “A” in percent points
is compared is illustrated by Figure 8.6. The trend described is always the same and can
be observed in all the variations of the price-based scenario.
Revenue: The major indicator in revenue management, overall revenue per run, r(s),
is computed as the sum over the product of bookings b(f, c, t, s) and the price of classes
p(f, c). Outside a simulation, this may indicate the revenue generated by all flights on
one departure day.
r(s) = X
f∈FX
c∈C p(f, c)·
Nt
X
t=1
b(f, c, t, s)!
∀s= 1, ..., Ns
(8.12)
Chapter 8: Simulation Based Analysis of Forecast Performance 122
Vol. = 050, Dev. = 00
0%
20%
40%
60%
80%
100%
1 2
Simulation Run
E
D
C
B
A
Vol. = 050, Dev. = 01
0%
20%
40%
60%
80%
100%
1 2
Simulation Run
E
D
C
B
A
Vol. = 050, Dev. = 05
0%
20%
40%
60%
80%
100%
1 2
Simulation Run
E
D
C
B
A
Vol. = 050, Dev. = 10
0%
20%
40%
60%
80%
100%
1 2
Simulation Run
E
D
C
B
A
Vol. = 050, Dev. = 20
0%
20%
40%
60%
80%
100%
1 2
Simulation Run
E
D
C
B
A
Vol. = 100, Dev. = 00
0%
20%
40%
60%
80%
100%
1 2
Simulation Run
E
D
C
B
A
Vol. = 100, Dev. = 01
0%
20%
40%
60%
80%
100%
1 2
Simulation Run
E
D
C
B
A
Vol. = 100, Dev. = 05
0%
20%
40%
60%
80%
100%
1 2
Simulation Run
E
D
C
B
A
Vol. = 100, Dev. = 10
0%
20%
40%
60%
80%
100%
1 2
Simulation Run
E
D
C
B
A
Vol. = 100, Dev. = 20
0%
20%
40%
60%
80%
100%
1 2
Simulation Run
E
D
C
B
A
Share of Bookings per Class
Figure 8.5.: Observed Bookings per Class with Exp050
Chapter 8: Simulation Based Analysis of Forecast Performance 123
9,4 9,1
10,3
13,4 13,4
12,4
11,4 10,7
12,84
10,6
0
2
4
6
8
10
12
14
16
Dev. = 00 Dev. = 01 Dev. = 05 Dev. = 10 Dev. = 20
Percent Points
Vol. = 050
Vol. = 100
Figure 8.6.: Decrease in the Share of Bookings Observed for Class “A”
If overall demand volume stays constant and the booking mix changes according to Hy-
pothesis (8.9), less bookings in valuable classes and more bookings in expensive classes
lead to decreasing overall revenue. Formally, this can be expressed as follows:
X
f∈FX
c∈C
Nt
X
t=1
b(f, c, t, s)≡X
f∈FXc∈C
Nt
X
t=1
b(f, c, t, s + 1)
→r(s)≤r(s+ 1)
∀s= 1, ..., Ns−1
(8.13)
However, the spiral-down effect may be beneficial for revenue if the gain in bookings due
to increased availabilities for cheap classes compensates for the loss in bookings in valuable
classes. There is actually a break-even point from which on the revenue lost to buy-down
is compensated by that gained through low-fare acquisition:
Nt
X
t=1
(b(f, Nc, t, s)−b(f, Nc, t, s + 1)) ·p(f, Nc)
≤
Nt
X
t=1
Nc−1
X
c=1
(b(f, c, t, s + 1) −b(f, c, t, s)·p(f, c))
∀f∈F;s= 1, ..., Ns−1
(8.14)
Chapter 8: Simulation Based Analysis of Forecast Performance 124
To observe the development of revenue over time, a percentage indicator may be cal-
culated. Given an initial simulation run s= 1, revenue for all future s= 2, ..., Nsmay be
converted to a percentage r%of the revenue earned during the initial run.
r%(s) = r(s)
r(1) ·100
∀s= 1, ..., Ns
(8.15)
X
f∈FX
c∈C
Nt
X
t=1
b%(f, c, t, s)≡X
f∈FX
c∈C
Nt
X
t=1
b%(f, c, t, s + 1)
→r%(s)≤r%(s+ 1)
∀s= 1, ..., Ns−1
(8.16)
Figure 8.7 shows the revenue earned as exponential smoothing methods “Exp025”,
“Exp050”, and “Exp075” are applied. In order to make the development of revenue
comparable over the scenario variations, it is expressed as percentages of the revenue
earned in run 1, r%as described in Definition (8.15). As can be seen over all scenario
variations, revenue decreases over the course of 50 runs.
The form of this development differs over the different markets. With high demand
volume “Vol. = 100”, as the conditions described by Hypothesis (8.14) do manifest
during the first runs, a small revenue increase can be observed in the beginning of the
simulation experiment. However, soon the loss of bookings in valuable classes stops being
compensated by the gain of bookings in low-fare classes and overall revenue decreases.
With low demand volume “Vol. = 050”, the decrease of revenue starts immediately after
the first run – its consequences are also more severe. When deviation is high as in “Dev.
= 20”, the development is not as straightforward as revenue shifts with volatile demand,
yet it is even steeper. The trend described is always the same and can be observed in all
the variations of the price-based scenario.
After a number of runs, revenue reaches a plateau that still exceeds what is earned
with first-come-first-serve controls. Inventory controls that were originally based on the
psychic forecast initialization keep some seats protected for valuable customers – and
as long as these customers request tickets before cheaper classes are available, bookings
will be observed in these classes. This leads to a halt in the spiral-down effect. Volatile
Chapter 8: Simulation Based Analysis of Forecast Performance 125
Vol. = 050, Dev. = 00
60%
80%
100%
120%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 01
60%
80%
100%
120%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 05
40%
60%
80%
100%
120%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 10
40%
60%
80%
100%
120%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 20
20%
40%
60%
80%
100%
120%
140%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 00
60%
80%
100%
120%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 01
60%
80%
100%
120%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 05
40%
60%
80%
100%
120%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 10
40%
60%
80%
100%
120%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 20
20%
40%
60%
80%
100%
120%
140%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Revenue in Percent of Run 1
Exp025 Exp050 Exp075
Figure 8.7.: Revenue in Percent of Revenue Earned in Run 1
Chapter 8: Simulation Based Analysis of Forecast Performance 126
customer arrival patterns, however, can still lead to a further shift toward cheap classes,
while no shift toward expensive classes can occur in a purely price-based market.
When several methods are compared, the percentage difference to a lowest benchmark
may be computed. A possible benchmark is the result of a first-come-first-serve seat
allocation, rfcfs (s). The indicator rfcfs % (s) is computed as the percentage by which the
revenue of the considered method exceeds that gained when first-come-first-serve was
applied.
rfcfs % (s) = r(s)−rfcfs (s)
rfcfs (s)·100
∀s= 1, ..., Ns
(8.17)
Based on forecasts becoming self-fulfilling prophecies, the spiral-down effect is even more
severe when the forecast method picks up new data quickly. In Figure 8.7 the development
of revenue given the application of the three exponential smoothing methods is shown.
The indicator used to depict the implications of spiral-down for revenue depending on the
method used is rfcfs % as described by Definition (8.17). Revenue decreases more quickly
when the αexp employed to weight new values in the forecast is higher.
Yield: Yield is the average revenue gained per booking. It can be computed as shown
by dividing the overall sum of revenue by the overall sum of bookings.
y(s) = Pf∈FPc∈Cp(f, c)·PNt
t=1 b(f, c, t, s)
Pf∈FPc∈CPNt
t=1 b(f, c, t, s)
∀s= 1, ..., Ns
(8.18)
The development predicted for revenue in case of spiral-down can be applied to yield as
well. However, while extreme gains in low-fare bookings may lead to an overall revenue
compensation, the yield decrease can be expected with certainty. As only the cheapest
class, c= 1, is booked any more, the yield ends up as the price of this class.
lim
s→∞ y(s) = Pf∈Fp(f, Nc)·PNt
t=1 b(f, Nc, t, s)
Pf∈FPNt
t=1 b(f, Nc, t, s)
∀s= 1, ..., Ns
(8.19)
Chapter 8: Simulation Based Analysis of Forecast Performance 127
To better observe the development of yield over time, a percentage indicator may be
calculated. Given an initial simulation run s= 1, yield for all future s= 2..Nsmay be
converted to a percentage y%of the yield observed during the initial run.
y%(s) = y(s)
y(1) ·100 (8.20)
X
f∈F
Nt
X
t=1
b(f, Nc, t, s)≤X
f∈F
Nt
X
t=1
b(f, Nc, t, s + 1)
and X
f∈F
Nt
X
t=1
b(f, 1, t, s)≥X
f∈F
Nt
X
t=1
b(f, 1, t, s + 1)
→y%(s)≤y%(s+ 1)
∀s= 1, ..., Ns−1
(8.21)
Figure 8.8 shows the amount yield earned as method “Exp050” is applied. In order to
make the development of yield comparable over the scenario variations, it is expressed as
percentages of the yield earned in run 1, y%as described in Definition (8.20). As can be
seen over all scenario variations, yield decreases over the course of 50 runs.
The form of this development differs over the different markets. As the yield is indepen-
dent of the amount of overall bookings, the observed decrease in yield is much smoother
than the decrease observed with regard to revenue. With low demand volume “Vol. =
050”, yield is lower even in the first run compared to high demand volume. In addition,
the decrease of yield is steeper. A high deviation of the error term distribution contributes
to this effect.
When several methods are compared, the percentage difference to a lowest benchmark
may be computed. A possible benchmark in the simulation is the yield resulting from a
first-come-first-serve seat allocation, yfcfs(s). The indicator yfcfs %(s) is computed as the
percentage by which the average yield of the considered method exceeds that observed
when first-come-first-serve was applied.
yfcfs % (s) = yfcfs (s)−y(s)
yfcfs (s)·100
∀s= 1, ..., Ns
(8.22)
Chapter 8: Simulation Based Analysis of Forecast Performance 128
Vol. = 050, Dev. = 00
40%
60%
80%
100%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 01
40%
60%
80%
100%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 05
40%
60%
80%
100%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 10
40%
60%
80%
100%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 20
40%
60%
80%
100%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 00
40%
60%
80%
100%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 01
40%
60%
80%
100%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 05
40%
60%
80%
100%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 10
40%
60%
80%
100%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 20
40%
60%
80%
100%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Yield in Percent of Run 1
Figure 8.8.: Yield in Percent of Yield Earned in Run 1
Chapter 8: Simulation Based Analysis of Forecast Performance 129
Vol. = 050, Dev. = 00
200%
300%
400%
500%
600%
700%
800%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 01
200%
300%
400%
500%
600%
700%
800%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 05
200%
300%
400%
500%
600%
700%
800%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 10
200%
300%
400%
500%
600%
700%
800%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 20
200%
300%
400%
500%
600%
700%
800%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 00
200%
300%
400%
500%
600%
700%
800%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 01
200%
300%
400%
500%
600%
700%
800%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 05
200%
300%
400%
500%
600%
700%
800%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 10
200%
300%
400%
500%
600%
700%
800%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 20
200%
300%
400%
500%
600%
700%
800%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Exp025 Exp050 Exp075
Yield in Percent of FCFS
Figure 8.9.: Yield in Percent of Yield Earned with First-Come-First-Serve
Chapter 8: Simulation Based Analysis of Forecast Performance 130
In Figure 8.9, presents the development of yield given the application of the three
exponential smoothing methods “Exp025”,“Exp050”, and “Exp075”. The indicator used
to depict the implications of spiral-down for revenue depending on the method used is
yfcfs % as described by Definition (8.22). As the diagrams show, yield decreases more
quickly when the αexp employed to weight new values in the forecast is higher.
Forecast Evaluations: The expected development of forecast quality under the con-
ditions of the spiral-down effect is the same for all error measurements based on the
comparison of forecasted demand and bookings as outlined in Chapter 4.
•Let ec−c
◦(s) be the average forecast error computed for simulation run sbased on
some to be defined method ◦comparing observed bookings b(f, c, t, s) and the
constrained demand forecast fconst (f, c, t, s).
As the forecast based on bookings predicts more demand to come for cheap classes and
availabilities based on the forecast allow for this demand to realize in more bookings,
the forecast becomes a self-fulfilling prophecy. This systematic flaw is interpreted as an
improvement of forecast quality:
ec−c
◦(s)≥ec−c
◦(s+ 1)
∀s= 1, ..., Ns−1(8.23)
When the conditions of the spiral-down effect are fulfilled and forecasts are evaluated
based on comparisons to actual bookings, their quality seems to improve. This can be
validated by observing the development of MAD, RMSE, MAPE and U2 applied to the
comparison of the constrained forecast fconst and observed bookings bover the course of
50 runs. All indicators show a decrease over time as forecast and bookings converge due
to the spiral-down effect.
Figure 8.10 presents the development of MAD as three exponential smoothing meth-
ods are applied. Figure 8.11 presents the development of RMSE as three exponential
smoothing methods are applied. Figure 8.12 presents the development of MAPE as three
exponential smoothing methods are applied. Figure 8.13 presents the development of U2
as three exponential smoothing methods are applied.
Chapter 8: Simulation Based Analysis of Forecast Performance 131
Vol. = 050, Dev. = 00
6
8
10
12
14
16
18
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 01
6
8
10
12
14
16
18
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 05
6
8
10
12
14
16
18
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 10
6
8
10
12
14
16
18
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 20
6
8
10
12
14
16
18
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 00
6
8
10
12
14
16
18
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 01
6
8
10
12
14
16
18
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 05
6
8
10
12
14
16
18
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 10
6
8
10
12
14
16
18
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 20
6
8
10
12
14
16
18
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Mean Absolute Deviation (MAD)
Exp025 Exp050 Exp075
Figure 8.10.: Mean Absolute Deviation (MAD): Constrained FC from Observed BKD
Chapter 8: Simulation Based Analysis of Forecast Performance 132
Vol. = 050, Dev. = 00
0
20
40
60
80
100
120
140
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 01
0
20
40
60
80
100
120
140
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 05
0
20
40
60
80
100
120
140
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 10
0
20
40
60
80
100
120
140
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 20
0
20
40
60
80
100
120
140
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 00
0
20
40
60
80
100
120
140
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 01
0
20
40
60
80
100
120
140
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 05
0
20
40
60
80
100
120
140
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 10
0
20
40
60
80
100
120
140
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 20
0
20
40
60
80
100
120
140
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Root Mean Squared Error (RMSE): FC const. from BKD
Exp025 Exp050 Exp075
Figure 8.11.: Root Mean Squared Error (RMSE): : Constrained FC from Observed BKD
Chapter 8: Simulation Based Analysis of Forecast Performance 133
Vol. = 050, Dev. = 00
0%
100%
200%
300%
400%
500%
600%
700%
800%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 01
0%
100%
200%
300%
400%
500%
600%
700%
800%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 05
0%
100%
200%
300%
400%
500%
600%
700%
800%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 10
0%
100%
200%
300%
400%
500%
600%
700%
800%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 20
0%
100%
200%
300%
400%
500%
600%
700%
800%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 00
0%
100%
200%
300%
400%
500%
600%
700%
800%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 01
0%
100%
200%
300%
400%
500%
600%
700%
800%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 05
0%
100%
200%
300%
400%
500%
600%
700%
800%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 10
0%
100%
200%
300%
400%
500%
600%
700%
800%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 20
0%
100%
200%
300%
400%
500%
600%
700%
800%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Mean Average Percentage Error (MAPE): FC const. from BKD
Exp025 Exp050 Exp075
Figure 8.12.: Mean Avg. Percentage Error (MAPE): Constrained FC from Observed BKD
Chapter 8: Simulation Based Analysis of Forecast Performance 134
Vol. = 050, Dev. = 00
0%
20%
40%
60%
80%
100%
120%
140%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 01
0%
20%
40%
60%
80%
100%
120%
140%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 05
0%
20%
40%
60%
80%
100%
120%
140%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 10
0%
20%
40%
60%
80%
100%
120%
140%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 20
0%
20%
40%
60%
80%
100%
120%
140%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 00
0%
20%
40%
60%
80%
100%
120%
140%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 01
0%
20%
40%
60%
80%
100%
120%
140%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 05
0%
20%
40%
60%
80%
100%
120%
140%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 10
0%
20%
40%
60%
80%
100%
120%
140%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 20
0%
20%
40%
60%
80%
100%
120%
140%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Theil's U2 (U2): FC const. from BKD
Exp025 Exp050 Exp075
Figure 8.13.: Theil’s U2 (U2): Constrained FC from Observed BKD
Chapter 8: Simulation Based Analysis of Forecast Performance 135
For all indicators, the trend of forecast quality over the course of a number of simulation
runs is more volatile if the deviation of the error term distribution is higher. As demand
volume shifts more strongly from one run to the next due to this, it becomes harder to
predict and a self-fulfilling prophecy takes longer to manifest. As for other indicators, the
development is steeper for markets with a low overall demand volume.
All indicators also show that just as the decrease of yield is steeper when new values are
weighted stronger, forecast quality seems to improve quicker. When the new observations
based on availabilities that are already influenced by the spiral-down effect are weighted
heavier, the forecast turns into a self-fulfilling prophecy even more quickly.
Effects of Updates within the Booking Horizon: Finally, the spiral-down effect can
even be observed within the booking horizon of a single run if forecasts are updated.
Figure 8.14 shows a comparison of revenue over 50 runs for all price-based scenarios when
exponential smoothing with a smoothing factor of 0.5 is applied. With “Exp050”, the
forecast is not updated throughout the booking horizon as bookings are observed. With
“Exp050upd”, the forecast is updated as described in Section 7.3.1. Revenue decreases
with every run when the forecast is updated. This is due to availabilities updated based
on decreasing shares of predicted demand for the most expensive classes as described
earlier in this section.
However, as presented in Figure 8.15, this updating of the forecast does not even lead
to better results concerning traditional forecast evaluation. Using MAD as an exemplary
error measurement, it becomes clear that the computed error of the forecast decreases as
the booking horizon progresses. This is due to the decreasing remaining time span for
which the forecast is valid. However, while this development can be observed both for
“Exp050” and “Exp050upd”, the overall level of mean absolute deviation is lower when
the forecast is not updated based on observed bookings.
The graph is shown only for those variations of demand where the deviation of the error
term is 0 (“Dev. = 00”). Not much variation can be observed as volume and deviation
change – this is due to both the fact that constrained values are compared and that the
comparison is limited to the first run, initialized using the psychic forecast.
Chapter 8: Simulation Based Analysis of Forecast Performance 136
Vol. = 050, Dev. = 00
0
1
2
3
4
5
6
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 01
0
1
1
2
2
3
3
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 05
0
1
1
2
2
3
3
4
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 10
0
1
1
2
2
3
3
4
4
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 20
0
1
1
2
2
3
3
4
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 00
0
1
2
3
4
5
6
7
8
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 01
0
1
2
3
4
5
6
7
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 05
0
1
2
3
4
5
6
7
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 10
0
1
2
3
4
5
6
7
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 20
0
1
2
3
4
5
6
7
0 10 20 30 40 50
Simulation Run
Revenue in Mio. €
Exp050 Exp050 updated
Figure 8.14.: Revenue Resulting from Exp050 and Exp050upd
Chapter 8: Simulation Based Analysis of Forecast Performance 137
Vol. = 050, Dev. = 00
0
2
4
6
8
10
12
14
16
18
20
050100150200250300350
Days before Departure
Vol. = 100, Dev. = 00
0
5
10
15
20
050100150200250300350
Days before Departure
MAD for Exp050 during the Booking Horizon of Run 1: FC const. from BKD
Exp050 Exp050 updated
Figure 8.15.: MAD during the Booking Horizon of Run 1
Conclusion: When applied over longer terms, flawed assumptions included in adaptive
forecast methods can cause systematic trends in inventory controls and revenue devel-
opment. In addition, when causing a spiral-down effect, forecast methods can become
self-fulfilling prophecies.
8.2. Consequences of Possible Definitions of Psychic Forecasts
In the previous section, the spiral-down effect was demonstrated by providing an initial
forecast and then applying an adaptive method (exponential smoothing). The initial
forecast was implicitly assumed to present a more accurate prediction of demand, with the
adaptive method based on a model of static demand leading to a spiral-down effect when
confronted with flexible customer behavior. The initial forecast was based on knowledge
of the demand model implemented in the simulation, which is not available in the real
world. This class of forecasts will be referred to as psychic forecasts in the further text.
The concept of psychic forecasts will be the focus of this section. When evaluating
approaches to demand forecasting based on a simulation system, there are two applications
Chapter 8: Simulation Based Analysis of Forecast Performance 138
for psychic forecasts: They may be used as initialization to observe the development of
indicators when adaptive methods are applied or as a benchmark to compare a method’s
ability of picking up the characteristics of the demand model.
generate
stochastic
requests
optimize
artificial
requests for t
AU-levels or
bid prices
reservation
system
2
1
transform
create
initial
forecast
requests initially
forecasted for t
artificial
(historical)
booking data
compare
requests
forecasted
for t
system
simulation
RESULT
3
transform
without
constraining
forecast
variants
benchmark
forecast
Figure 8.16.: Uses of the Psychic Forecast in the Simulation
Figure 8.16 emphasizes those parts of the system that have not been further examined
so far. For example when evaluating the unconstraing aspect as described in Section 6.4,
the previous text has regarded the psychic forecast as a constant method. It is employed
in two places, both marked in bold.
Several approaches are conceivable to calculate a psychic forecast. When used as an
initialization or as a benchmark for methods based on a static model of demand, the
psychic forecast can be varied along two parameters: Class choice and itinerary choice.
These choices relate to the ways in which customers decide which classes to book (class
choice) on which flights (itinerary choice) in order to reach their destination.
Chapter 8: Simulation Based Analysis of Forecast Performance 139
preferred itinerary alternative itineraries
maximum price c-max 1-i c-max 2-i
minimum price c-min 1-i c-min 2-i
acceptable prices c-frac 1-i c-frac 2-i
Table 8.1.: Possible Variations of Choice in Psychic Forecasts
Two variations of itinerary choice and three variations of class choice as well as their
combinations will be examined in this section. Together with their abbreviations, they
can be found in Table 8.1.
In order to express the options of psychic forecasting and their consequences formally,
additional notation is required. The following list recaptures some of the variables intro-
duced in Section 7.2.2.
•Let r∈Rbe the index of requests.
•Let Rsbe the set of requests included in the run s.
•Let trindicate the point of time tat which request rarrives.
•Let δtime
r,t be a boolean matrix indicating whether request rarrives at point of time
t:t=tr.
•Let srindicate the simulation run sfor which request rwas generated.
•Let i∈Ibe the index of itineraries offered.
•Let Irbe the set of itineraries acceptable according to the product requirements
specified for request r(i.e. origin, destination, travel time, departure day and time,
transfers).
•Let Fibe the set of flights included in itinerary i.
•Let Iit
f,i be a boolean matrix indicating whether flight f∈Fifor any fand i.
•Let δproduct
r,c be a boolean matrix defining whether the product represented by book-
ing class cis acceptable according to the product requirements of request r(see
Definition (7.19).
Chapter 8: Simulation Based Analysis of Forecast Performance 140
•Let δwtp
r,c,i be a boolean matrix defining whether the price of class con itinerary iis
acceptable according to the willingness to pay of request r(see Definition (7.21)).
•Let p(f, c) be the price of booking class con flight f.
•Let func(f, c, t, s) be the unconstrained demand forecast for flight f, class c, and
point of time tof simulation run s.
•Let ˆ
C(i, r) be the cost of itinerary iconsidered by request r, without regard for the
actual ticket price (given the assumption that all itineraries cost the same).
Preferred Itinerary (1-i): Psychic forecasts predicting demand to arrive only for the
preferred itinerary use knowledge of the customers’ cost function given by the simulation.
Demand is expected to arrive in a given booking class of the itinerary that would be chosen
if all itineraries were available at the same price. The hypothesis is that the demand is
most likely to manifest in the itinerary that presents the first choice according to the cost
function. The formal expression for the choice of itinerary in this psychic forecast is given
in Definitions (8.24).
•Let δi-choice
i,r be a boolean matrix indicating the chosen itinerary based on the cost
function of request r.
•Let nibe the number of itineraries that are chosen according to the method of
psychic forecast. For “1-i”, ni= 1.
δi-choice
i,r :=
1i=i0|maxi0∈Irˆ
C(i0, r)
0 else.
∀r∈R;i∈Ir
(8.24)
Alternative Itineraries (n-i): Psychic forecasts that predict demand for two or more
itineraries use knowledge of the customers’ cost function. Demand is expected to arrive
in a given booking class of those itineraries that make the “top list” if all itineraries were
available at the same price. The number of itineraries nithat demand is distributed over
and the weight αit given to each itinerary are parameters of the method. The hypothesis
is that demand is most likely to manifest in itineraries presenting good choices according
Chapter 8: Simulation Based Analysis of Forecast Performance 141
to the cost function, and that customers will divert from their first choice if availability
is lacking. Itineraries are defined as chosen in an iterative process, starting with the first
choice. The formal expression for the choice of itinerary in this psychic forecast is given
in Definitions (8.25).
δi-choice
i,r :=
1i=i0|maxi0∈Irδi-choice
i0,r −12·ˆ
C(i0, r)
0 else.
∀r∈R;i∈Ir
(8.25)
Maximum Price (c-max): Psychic forecasts predicting maximum prices use knowledge
of the customers’ maximum willingness to pay given in the simulation. They expect
demand to arrive in the most expensive acceptable booking class of a given itinerary.
The hypothesis is that when the forecast predicts customers to arrive according to their
highest willingness to pay, inventory controls will be restrictive enough to prevent much
of buy-down. The formal expression for this psychic forecast is given in Definitions (8.26)
and (8.27).
•Let δmax
r,c,i be a boolean matrix indicating the most expensive booking class available
on all flights of the given itinerary iand acceptable for r.
δmax
r,c,i :=
1c=c0|maxc0∈Cδproduct
r,c0·δprice
r,c0,i ·Pf∈Fip(f, c)
0 else.
∀r∈R;I∈Ir
(8.26)
func (f, c, t, s) := X
r∈Rs
δtime
r,t ·Pi∈IrIit
f,i ·δi-choice
i,r ·δmax
r,c,i
ni
∀f∈F;c∈C;t= 1, ..., Nt;s= 1...Ns
(8.27)
Minimum Price (c-min): Psychic forecasts predicting minimum prices use knowledge of
the customers’ acceptance of booking classes given by the simulation. Demand is expected
to arrive in the cheapest acceptable booking class of a given itinerary. The hypothesis
is that when the forecast predicts customers to arrive in the cheapest class according
to their acceptance of product characteristics given by booking classes, the worst case
Chapter 8: Simulation Based Analysis of Forecast Performance 142
of buy-down is already included in the forecast. The formal expression for this psychic
forecast is given in Definitions (8.28) and (8.29).
•Let δmin
r,c,i be a boolean matrix indicating the cheapest booking class available on all
flights of the given itinerary iand acceptable for r.
δmin
r,c,i :=
1c=c0|minc0∈Cδproduct
r,c0·δprice
r,c0,i ·Pf∈Fip(f, c)>0
0 else.
∀r∈R;I∈Ir
(8.28)
func (f, c, t, s) := X
r∈Rs
δtime
r,t ·Pi∈IrIit
f,i ·δi-choice
i,r ·δmin
r,c,i
ni
∀f∈F;c∈C;t= 1, ..., Nt;s= 1...Ns
(8.29)
Acceptable Prices (c-frac): Psychic forecasts predicting all acceptable prices use the
knowledge of customers’ maximum willingness to pay and acceptance of booking classes
given by the simulation. Fractional demand is expected to arrive in all acceptable book-
ing classes of a given itinerary. The hypothesis is that demand will manifest in one of
these classes respectively and the forecast strives to get a reasonable estimate including
the possibility of buy-down. The formal expression for this psychic forecast is given in
Definition (8.30).
func (f, c, t, s) := X
r∈Rs
δtime
r,t ·X
i∈Ir Iit
f,i ·δi-choice
i,r ·δproduct
r,c ·δprice
r,c,i
Pc∈Cδproduct
r,c ·δprice
r,c,i !/ni
∀f∈F;c∈C;t= 1, ..., Nt;s= 1...Ns
(8.30)
Psychic Forecasts as Benchmarks: To present the consequences of the use of the dif-
ferent psychic forecasts, a normalization to the effects of first-come-first-serve inventory
controls is performed. These effects can be calculated by applying the controls to the
same demand that is later used to evaluate psychic forecasts. First-come-first-serve con-
trols provide a benchmark as they are the simplest alternative to applying any forecast
at all.
•Let Pf∈FK(f, 0, s) be the overall available capacity at the beginning of run s.
Chapter 8: Simulation Based Analysis of Forecast Performance 143
•Let Pf∈FPc∈CPNt
t=1 b(f, c, t, s) be the overall bookings generated in simulation
run s.
•Let r(s) be the overall revenue generated in simulation run s.
•Let rfcfs (s) be the overall revenue generated in simulation run sgiven first-come-
first-serve inventory controls.
•Let r% fcfs
2-i c-max (s) be the overall revenue generated in simulation run susing (e.g.)
psychic forecast method “2-i c-max”, as a percentage of the revenue resulting from
the application of first-come-first-serve controls.
•Let yfcfs (s) be the average yield generated in simulation run sgiven first-come-first-
serve inventory controls.
•Let y% fcfs
2-i c-max (s) be the average yield generated in simulation run susing (e.g.) psy-
chic forecast method “2-i c-max”, as a percentage of the revenue resulting from the
application of first-come-first-serve controls.
•Let l(s) be the average seat load factor generated in simulation run s.
•Let lfcfs (s) be the average seat load factor generated in simulation run sgiven first-
come-first-serve inventory controls.
•Let l% fcfs
2-i c-max (s) be the average seat load factor generated in simulation run susing
(e.g.) psychic forecast method “2-i c-max”, as a percentage of lfcfs (s).
Revenue is computed as described by Definition (8.12). The normalization to the
benchmark generated by first-come-first-serve controls is shown in Definition (8.31). To
present the result of a complete simulation experiment in a single indicator, the normalized
revenue may be averaged over all runs as presented in Definition (8.32).
r% fcfs (s) := r(s)
rfcfs (s)·100
∀s= 1, ..., Ns
(8.31)
ˆr% fcfs := PNs
s=1 r% fcfs (s)
Ns(8.32)
Yield is computed as described by Definition (8.18). The normalization to the bench-
mark generated by first-come-first-serve controls is shown in Definition (8.33). To present
Chapter 8: Simulation Based Analysis of Forecast Performance 144
the result of a complete simulation experiment in a single indicator, the normalized rev-
enue may be averaged over all runs as presented in Definition (8.34).
y% fcfs (s) := y(s)
yfcfs (s)·100
∀s= 1, ..., Ns
(8.33)
ˆy% fcfs := PNs
s=1 y% fcfs (s)
Ns(8.34)
The seat load factor is computed as a function of capacity and bookings, as presented
in Definition (8.35). It can be normalized, as Definition (8.36) shows. To present the
result of a complete simulation experiment in a single indicator, the normalized seat load
factor may be averaged over all runs as presented in Definition (8.37).
l(s) := Pf∈FPc∈CPNt
t=1 b(f, c, t, s)
Pf∈FK(f, 0, s)·100
∀s= 1, ..., Ns
(8.35)
l% fcfs (s) := l(s)
lfcfs (s)·100
∀s= 1, ..., Ns
(8.36)
ˆ
l% fcfs := PNs
s=1 l% fcfs (s)
Ns(8.37)
The simulation experiments conducted to analyze the effect of psychic forecasts are
based on the “hybrid” market scenario described in Section 7.4. As in this scenario both
product- and price-oriented supply and demand are included, the broadest variety of
effects is expected. Variations of demand volume and deviation are included in the data
and will be pointed out whenever helpful.
Given the different approaches to translating willingness to pay and the acceptance of
booking classes into psychic forecasts, the assumption is that consequences of the choice
of class-forecast will become clear when considering revenue. The precise expectation is
formalized in Hypothesis (8.38)
ˆr% fcfs
c-max ≥ˆr% fcfs
c-frac ≥ˆr% fcfs
c-min ≥100 (8.38)
Chapter 8: Simulation Based Analysis of Forecast Performance 145
Vol. = 050, Dev. = 00
203% 205% 180% 178% 170%
100%
350%
c-max 1-i c-max 2-i c-frac 1-i c-frac 2-i c-min 1-i
Simulation Run
Vol. = 050, Dev. = 01
205% 209% 181% 179% 170%
100%
350%
c-max 1-i c-max 2-i c-frac 1-i c-frac 2-i c-min 1-i
Simulation Run
Vol. = 050, Dev. = 05
240% 250%
194% 191%
161%
100%
350%
c-max 1-i c-max 2-i c-frac 1-i c-frac 2-i c-min 1-i
Simulation Run
Vol. = 050, Dev. = 10
267% 281%
209% 205%
157%
100%
350%
c-max 1-i c-max 2-i c-frac 1-i c-frac 2-i c-min 1-i
Simulation Run
Vol. = 050, Dev. = 20
283% 301%
217% 213%
152%
100%
350%
c-max 1-i c-max 2-i c-frac 1-i c-frac 2-i c-min 1-i
Simulation Run
Vol. = 100, Dev. = 00
471% 503%
394% 401%
293%
100%
550%
c-max 1-i c-max 2-i c-frac 1-i c-frac 2-i c-min 1-i
Simulation Run
Vol. = 100, Dev. = 01
411% 426%
359% 363%
286%
100%
550%
c-max 1-i c-max 2-i c-frac 1-i c-frac 2-i c-min 1-i
Simulation Run
Vol. = 100, Dev. = 05
412% 418%
359% 363%
284%
100%
550%
c-max 1-i c-max 2-i c-frac 1-i c-frac 2-i c-min 1-i
Simulation Run
Vol. = 100, Dev. = 10
451% 454%
386% 388%
276%
100%
550%
c-max 1-i c-max 2-i c-frac 1-i c-frac 2-i c-min 1-i
Simulation Run
Vol. = 100, Dev. = 20
482% 487%
404% 405%
261%
100%
550%
c-max 1-i c-max 2-i c-frac 1-i c-frac 2-i c-min 1-i
Simulation Run
Average Revenue over 50 Runs in Percent of First-Come-First-Serve
Figure 8.17.: Average Revenue over 50 Runs in Percent of First-Come-First-Serve
Chapter 8: Simulation Based Analysis of Forecast Performance 146
Figure 8.17 shows the average revenue over 50 runs resulting from the application of
the listed variations of psychic forecasts over all runs. Revenue is expressed as ˆr% fcfs as
formalized in Definition (8.32). In the diagram, the deviation of the average revenue in
both directions is presented by a grey vertical line. As expected, the deviation of the
average revenue grows as the deviation of the error term distribution grows and demand
volume is more volatile between runs.
In all cases, as predicted by Hypothesis (8.38), the psychic forecasts based on “c-max”
result in higher revenue than those based on “c-frac”. At the same time, “c-frac” still leads
to higher revenues than “c-min”. As the simulation experiment was based on a hybrid
scenario including some customers that do not accept the cheapest booking class, even
“c-min” always resulted in revenues exceeding those observed when first-come-first-serve
controls were applied.
In addition, it is remarkable that revenue was always highest when the forecast was not
only based on “c-max” but also took into account two alternative preferred itineraries as
in the variant “2-i”. This effect is strongest in the case of “Vol. = 100, Dev. = 00”,
when demand volume is high and the deviation of the error term distribution is low, and
in the case of “Vol. = 050, Dev. = 20”, when demand volume is low and the deviation of
the error term is high. In both cases, the probability for a few itineraries being requested
significantly more often than others seems to be higher. In such cases, it is advantageous
to include demand predicted also for the second-choice itineraries in the optimization.
The effect can even be observed in markets with a high demand volume when “c-frac”
is applied. However, the revenue observed in markets with a high error term deviation
is not much higher for “2-i” than it is for “1-i”. The reason for this becomes clear when
considering yield and seat load factor.
As demand is distributed more equally among itineraries with alternative itinerary
forecasts, availability control may be expected to turn more restrictive. The consequence
of this would be rising yields but also declining bookings. The precise expectation is
formalized in Hypotheses (8.39) and (8.40).
ˆ
l% fcfs
2-i ≤ˆ
l% fcfs
1-i ≤100 (8.39)
ˆy% fcfs
2-i ≥ˆy% fcfs
1-i ≥100 (8.40)
Chapter 8: Simulation Based Analysis of Forecast Performance 147
Vol. = 050, Dev. = 00
217% 257%
183% 179% 173%
100%
400%
700%
c-max 1-i c-max 2-i c-frac 1-i c-frac 2-i c-min 1-i
Simulation Run
Vol. = 050, Dev. = 01
220% 263%
184% 180% 172%
100%
400%
700%
c-max 1-i c-max 2-i c-frac 1-i c-frac 2-i c-min 1-i
Simulation Run
Vol. = 050, Dev. = 05
266% 325%
195% 192% 161%
100%
400%
700%
c-max 1-i c-max 2-i c-frac 1-i c-frac 2-i c-min 1-i
Simulation Run
Vol. = 050, Dev. = 10
299%
369%
210% 206% 157%
100%
400%
700%
c-max 1-i c-max 2-i c-frac 1-i c-frac 2-i c-min 1-i
Simulation Run
Vol. = 050, Dev. = 20
320%
400%
218% 214% 152%
100%
400%
700%
c-max 1-i c-max 2-i c-frac 1-i c-frac 2-i c-min 1-i
Simulation Run
Vol. = 100, Dev. = 00
519%
644%
373% 376%
293%
100%
400%
700%
c-max 1-i c-max 2-i c-frac 1-i c-frac 2-i c-min 1-i
Simulation Run
Vol. = 100, Dev. = 01
447%
546%
342% 344% 289%
100%
400%
700%
c-max 1-i c-max 2-i c-frac 1-i c-frac 2-i c-min 1-i
Simulation Run
Vol. = 100, Dev. = 05
447%
542%
343% 346% 289%
100%
400%
700%
c-max 1-i c-max 2-i c-frac 1-i c-frac 2-i c-min 1-i
Simulation Run
Vol. = 100, Dev. = 10
496%
603%
369% 371%
279%
100%
400%
700%
c-max 1-i c-max 2-i c-frac 1-i c-frac 2-i c-min 1-i
Simulation Run
Vol. = 100, Dev. = 20
538%
659%
389% 390%
262%
100%
400%
700%
c-max 1-i c-max 2-i c-frac 1-i c-frac 2-i c-min 1-i
Simulation Run
Average Yield over 50 Runs in Percent of First-Come-First-Serve
Figure 8.18.: Average Yield over 50 Runs in Percent of First-Come-First-Serve
Chapter 8: Simulation Based Analysis of Forecast Performance 148
Figure 8.18 shows the average yield over 50 runs resulting from the application of the
listed variations of psychic forecasts over all runs. Yield is expressed as ˆy% fcfs as shown in
Definition (8.34). In the diagram, the deviation of the average yield in both directions is
presented by a grey vertical line. As can be expected, the deviation of the average yield
grows as the deviation of the error term distribution grows and demand volume is more
volatile between runs.
In all cases, the psychic forecasts based on “c-max” result in higher Yield than those
based on “c-frac”. At the same time, “c-frac” still leads to higher yield than “c-min”.
As the simulation experiment was based on a hybrid scenario including some customers
that do not accept the cheapest booking class, even “c-min” always resulted in yields
exceeding those observed when first-come-first-serve controls were applied.
As already remarked with regard to revenue and predicted by Hypothesis (8.40), yield is
always highest when the forecast is not only based on “c-max” but also takes into account
two alternative preferred itineraries as in the variant “2-i”. This effect is strongest in the
case of “Vol. = 100, Dev. = 20”, when demand volume is high and the deviation of
the error term distribution is also high. The consequence of the high deviation of the
error distribution is a less homogeneous distribution of demand volume over the existing
itineraries. This leads to more itineraries being especially desirable and more itineraries
being “second-choice”. Without a “2-i” forecast, the second-best alternatives are assigned
less restrictive inventory controls. However, in the case of “Vol. = 100, Dev. = 20”, the
consideration of second-best alternatives seems to have lead to losses in bookings: While
yield strongly exceeds the yield gained by “1-i”, the revenue as shown in Figure 8.17 was
not much higher.
Figure 8.19 shows the average seat load factor over 50 runs resulting from the appli-
cation of the listed variations of psychic forecasts over all runs. Seat load factor (SLF)
is expressed as ˆ
l% fcfs as shown in Definition (8.37). In the diagram, the deviation of the
average SLF in both directions is presented by a grey vertical line. However, in contrast
to what was observed for yield and revenue previously, the deviation of this indicator is
small across all the markets.
In all cases, the psychic forecasts based on “c-max” result in lower SLF than those based
on “c-frac”. At the same time, “c-frac” still leads to lower SLF than “c-min”. The reason
Chapter 8: Simulation Based Analysis of Forecast Performance 149
Vol. = 050, Dev. = 00
94% 80%
98% 99% 98%
0%
20%
40%
60%
80%
100%
120%
140%
c-max 1-i c-max 2-i c-frac 1-i c-frac 2-i c-min 1-i
Simulation Run
Vol. = 050, Dev. = 01
93% 79%
98% 99% 98%
0%
20%
40%
60%
80%
100%
120%
140%
c-max 1-i c-max 2-i c-frac 1-i c-frac 2-i c-min 1-i
Simulation Run
Vol. = 050, Dev. = 05
90% 77%
99% 100% 100%
0%
20%
40%
60%
80%
100%
120%
140%
c-max 1-i c-max 2-i c-frac 1-i c-frac 2-i c-min 1-i
Simulation Run
Vol. = 050, Dev. = 10
89% 76%
100% 100% 100%
0%
20%
40%
60%
80%
100%
120%
140%
c-max 1-i c-max 2-i c-frac 1-i c-frac 2-i c-min 1-i
Simulation Run
Vol. = 050, Dev. = 20
89% 76%
99% 99% 100%
0%
20%
40%
60%
80%
100%
120%
140%
c-max 1-i c-max 2-i c-frac 1-i c-frac 2-i c-min 1-i
Simulation Run
Vol. = 100, Dev. = 00
91% 78%
106% 107% 100%
0%
20%
40%
60%
80%
100%
120%
140%
c-max 1-i c-max 2-i c-frac 1-i c-frac 2-i c-min 1-i
Simulation Run
Vol. = 100, Dev. = 01
92% 78%
105% 105% 99%
0%
20%
40%
60%
80%
100%
120%
140%
c-max 1-i c-max 2-i c-frac 1-i c-frac 2-i c-min 1-i
Simulation Run
Vol. = 100, Dev. = 05
92% 77%
105% 105% 99%
0%
20%
40%
60%
80%
100%
120%
140%
c-max 1-i c-max 2-i c-frac 1-i c-frac 2-i c-min 1-i
Simulation Run
Vol. = 100, Dev. = 10
91% 75%
105% 105% 99%
0%
20%
40%
60%
80%
100%
120%
140%
c-max 1-i c-max 2-i c-frac 1-i c-frac 2-i c-min 1-i
Simulation Run
Vol. = 100, Dev. = 20
90% 74%
104% 104% 100%
0%
20%
40%
60%
80%
100%
120%
140%
c-max 1-i c-max 2-i c-frac 1-i c-frac 2-i c-min 1-i
Simulation Run
Average SLF over 50 Runs in Percent of First-Come-First-Serve
Figure 8.19.: Average SLF over 50 Runs in Percent of First-Come-First-Serve
Chapter 8: Simulation Based Analysis of Forecast Performance 150
for this are more restrictive inventory controls resulting from demand being predicted to
arrive in classes that are not the cheapest. In the case of high demand volume, “Vol. =
100”, the psychic forecasts of the variant “c-min” lead to SLF that are even higher than
those achieved with first-come-first-serve controls. This is due to more bookings being
accepted as availabilities are optimized.
As already remarked with regard to yield, the itinerary choice has a direct effect on
SLF. They are even lower when the forecast is not only based on “c-max” but also takes
into account two alternative preferred itineraries as in the variant “2-i”. This effect can
be observed regardless of demand volume. It is stronger when the deviation of error term
distribution is high. The consequence of the high deviation of the error distribution is a
less homogeneous distribution of demand volume over the existing itineraries. This leads
to more itineraries being especially desirable and more itineraries being “second-choice”.
With a “2-i” forecast, the second-best alternatives are assigned more restrictive inventory
controls.
The consideration of second-choice alternatives leads to losses in bookings as more
seats are reserved for valuable customers that book on their preferred itinerary. This is
the reason for the effect observed with regard to revenue: While yield strongly exceeds
the yield gained by “1-i”, the revenue as shown in Figure 8.17 was not much higher.
Psychic Forecasts as Initialization: Used as an initialization method, psychic forecasts
influence the first set of historical bookings that adaptive methods such as exponential
smoothing can be based on. The degree of their influence may be derived from a mea-
surement that describes in how far the results of the simulation experiments based on
different initial forecasts but adapted according to the same method diverge. Therefore,
new indicators need to be introduced.
•Let {c-max 1-i,c-frac 1-i,c-min 1-i,c-max 2-i,c-frac 2-i}be the set of available psy-
chic forecasts.
•Let r%fcfs (s) be the revenue in run sas a percentage of the revenue gained with
first-come-first-serve controls.
•Let σ(r(s)) be the deviation of revenues at run sover all simulation experiments
considered.
Chapter 8: Simulation Based Analysis of Forecast Performance 151
As the same adaptive method is applied in a range of simulation experiments start-
ing out with different initial forecasts, a development in the deviation of results can be
expected. The assumed relationship is formalized in Hypothesis (8.41).
lim
s→∞ σ(r(s)) = 0 (8.41)
Figure 8.20 shows r%fcfs (s) over the course of 50 runs for different initializations of
Exp050. When “zero FC” is used as initialization method, the forecast is set to be
zero for all flights and classes in the first run. It is then updated based on observed
unconstrained bookings using the exponential smoothing method “Exp050”. The other
options shown correspond to the psychic forecast variants presented previously.
The “zero FC” initialization leads to first-come-first-serve controls being applied in
the first run. As no demand is predicted to arrive in any class, no protected seats are
computed by the EMSR-b optimization. A slight spiral-up effect can be observed in
later runs: As the exponential smoothing method picks up on product-based demand,
protected seats are introduced and the inventory controls are no longer merely based on
first-come-first-serve.
When used as an initialization method, “c-max 2-i” seems superior to “c-max 1-i”. In
contrast to the constant use of the psychic forecast that predicts demand to arrive in
part for second-choice itineraries, the exponential smoothing avoids the effect of overly
restrictive inventory controls. Both methods that are based on predicting demand to arrive
according to customers’ highest willingness to pay result in constantly higher revenue than
the alternatives.
While leading to lower revenue than the “c-max” and even “c-frac” alternatives, “c-
min” is still more successful than the “zero FC” initialization. As the market that is being
analyzed in these simulation experiments includes both a class structure and demand that
is price- as well as product-oriented, a share of customers accord to the static demand
assumption. An accurate prediction of this demand even in the cheapest class customers
are willing to buy therefore still results in protected seats for more valuable classes. The
initialization method “c-min” may also be regarded to present the state a complete spiral-
down effect would finally lead to in a hybrid market: All customers are predicted to arrive
in the cheapest class they are willing to buy, which is not necessarily the cheapest class
available.
Chapter 8: Simulation Based Analysis of Forecast Performance 152
Vol. = 050, Dev. = 00
0%
50%
100%
150%
200%
250%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 01
0%
50%
100%
150%
200%
250%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 05
0%
50%
100%
150%
200%
250%
300%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 10
0%
50%
100%
150%
200%
250%
300%
350%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 20
0%
50%
100%
150%
200%
250%
300%
350%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 00
0%
100%
200%
300%
400%
500%
600%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 01
0%
100%
200%
300%
400%
500%
600%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 05
0%
100%
200%
300%
400%
500%
600%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 10
0%
100%
200%
300%
400%
500%
600%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 20
0%
100%
200%
300%
400%
500%
600%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
c-min 1-i
Revenue in Percent of FCFS
zero FC c-frac 2-i
c-frac 1-i
c-max 2-i
c-max 1-i
Figure 8.20.: Revenue in Percent of First-Come-First-Serve
Chapter 8: Simulation Based Analysis of Forecast Performance 153
Deviation of Overall Revenue
1.766,69 1.716,96
1.441,83 1.403,35 1.316,87
3.181,40
3.018,65
2.743,02 2.745,96
1.942,06
1.235,51 1.136,76
817,45 740,32
452,06
2.844,17
2.683,33
2.497,02
2.252,03
1.824,26
0
500
1.000
1.500
2.000
2.500
3.000
3.500
Vol. = 050,
Dev. = 00 Vol. = 050,
Dev. = 01 Vol. = 050,
Dev. = 05 Vol. = 050,
Dev. = 10 Vol. = 050,
Dev. = 20 Vol. = 100,
Dev. = 00 Vol. = 100,
Dev. = 01 Vol. = 100,
Dev. = 05 Vol. = 100,
Dev. = 10 Vol. = 100,
Dev. = 20
Tsd. €
Run 1
Run 50
Figure 8.21.: Deviation of Revenue between Simulation Experimenets
Figure 8.21 shows the development of σ(r(s)) from run 1 to run 50, given different
forecast initializations and the adaptive method Exp050 as already discussed with regard
to Figure 8.20. As can be seen, the prediction formalized in Hypothesis (8.41) holds
true: The deviation of revenue between simulation experiments decreases as the same
exponential smoothing method is applied to different initial forecasts.
The difference in deviation between run 1 and run 50 is greater for markets including
low demand volume (“Vol. = 050”) and high deviation (“Dev. = 20”). This is due to
the fact that the spiral-down effect that leads to similar (low) revenue results takes place
faster under these conditions. This has been demonstrated in Section 8.1.
In all market variations including high demand volume (“Vol. = 100”), the initial
deviation is higher than when demand volume is low. Due to the high number of requests,
inventory control can have a wider range of success as the maximum of revenue that may
be gained is greater. As the spiral-down effect does not immediately lead to lower revenues
in such cases (this also been explained in Section 8.1), revenues do not decrease from their
initial state as much as they do in markets with low demand volume. For this reason, the
deviation does not decrease in a similar way, either.
Chapter 8: Simulation Based Analysis of Forecast Performance 154
Conclusion: The concept of psychic forecasts can be used for two purposes in a sim-
ulation environment: To initialize adaptive forecast methods lacking historical bookings
and to serve as a benchmark for the evaluation of other forecast methods. However, the
choice of psychic forecast has a great influence on its success. The way in which the
psychic forecast is defined and computed from knowledge of the demand model has di-
rect consequences in the form of the resulting revenue. It influences both the success of
adaptive methods that use it as an initialization method and the outcome of evaluations
based on comparisons with it.
8.3. Evaluation of Standard Accuracy Indicators
The concept of simulation offers the opportunity of considering long-term developments
and comparisons with exclusive knowledge of the demand model. In the previous two
sections, these views have been applied to forecast methods. This section focuses on the
ways of evaluating forecasts as presented in Section 4.1 with the help of a simulation
system. It strives to analyze the advantages and fallacies of common approaches to
forecast evaluation.
In order to compare the effects of different methods, forecast evaluation needs to be
defined more clearly. Three dimensions can be identified that describe traditional forecast
error measurements: the objects of comparison, the level of comparison, and the method
of comparison.
Objects of Comparison: In general, the objects of comparison are always bookings and
forecasts. However, in the case of demand forecasting for revenue management, a differ-
entiation between the states “constrained” and “unconstrained” applies to both of these
indicators as already explained in Section 6.2. This view results in four indicators that
may be compared, two considering bookings and two considering forecasts. In addition,
the unconstrained or constrained psychic forecast available in the simulation may also be
used as a benchmark to compare the results of evaluation.
•Let b(c, f, t, s) be the bookings observed in class cof flight fbetween points of time
t−1 and tin the booking horizon of simulation run s.
Chapter 8: Simulation Based Analysis of Forecast Performance 155
•Let bunc (c, f, t, s) be the unconstrained bookings. The transformation from actual
to unconstrained bookings is implicit in the forecast method applied.
•Let func (c, f, t, s) be the demand predicted to arrive for class cof itinerary fbetween
points of time t−1 and tin the booking horizon of simulation run s. This is the
output of the forecast method and the input for the optimization method.
•Let fconst(c, f, t, s) be the constrained predicted demand. The transformation from
unconstrained to constrained is handled over the inventory controls that were applied
during the time period considered.
•Let func
1-i c-max(c, f, t, s) = func
psy (c, f, t, s) be the psychic forecast according to the
method using the preferred itinerary and maximum willingness to pay described
in Section 8.2. As it was shown to be a good upper benchmark in most scenarios,
this psychic forecast will be used representatively for all psychic forecasts in this
section.
•Let fconst
1-i c-max(c, f, t, s) = fconst
psy (c, f, t, s) be the constrained version of the psychic
forecast. The transformation from unconstrained to constrained is handled over the
inventory controls calculated on the basis of the psychic forecast.
Intuitively, it appears sensible to compare only indicators that are of the same trans-
formation. This would mean comparing only unconstrained bookings with the original
unconstrained forecast and only comparing actual bookings with a constrained forecast.
However, as bookings and forecasts appear originally in different states, an evaluation that
considers the difference between actual bookings and the original, unconstrained forecast
may also be of interest. In contrast to this, there seems to be no point in transform-
ing both indicators from their original state and comparing unconstrained bookings with
constrained forecasts.
The resulting options for comparison are:
•actual bookings vs. constrained forecast, ec−c;
•unconstrained bookings vs. unconstrained forecast, eu−u;
•actual bookings vs. unconstrained forecast, ec−u;
•unconstrained forecast (e.g. a psychic variant) vs. unconstrained forecast, e
eu−u.
Chapter 8: Simulation Based Analysis of Forecast Performance 156
Levels of Comparison: Different aggregation levels may be considered as well. Before
comparing them, indicators may be averaged over classes, itineraries or points of time be-
fore departure. Summing up over classes or itineraries does not seem useful with regard to
forecast evaluation: If this was done, errors that occur with regard to the distribution of
predicted demand over classes or itineraries may be compensated. However, the distribu-
tion of demand over classes and itineraries is a vital information for revenue management.
The question of whether or not to sum up forecasts over points of time before departure
to evaluate them is not that trivial. The EMSR-b heuristic implemented in the revenue
management system modeled does not consider the order of arrival of demand. Does this
mean the information is irrelevant? In the course of this section, two aggregation levels
will therefore be analyzed:
•Let ˆe(s) be the series error calculated for run scomparing bookings and forecast
per flight, class and point of time.
•Let ˆe(s) be the series error calculated for run scomparing bookings and forecast
summed up over points of time per flight and class.
Methods of Comparison: Finally, four ways of calculating error measurements have
been described in Section 4.1: The absolute measurements MAD (mean absolute devia-
tion) and RMSE (root mean squared error) as well as the percentage error measurements
MAPE (mean average percentage error) and U2 (Theil’s U2). To specify the error calcu-
lated, further notation is needed.
•Let the usage of MAD be indicated by eMAD.
•Let the usage of RMSE be indicated by eRMSE.
•Let the usage of MAPE be indicated by eMAPE.
•Let the usage of U2 be indicated by eU2.
From these options, a tree of possible forecast error measurements emerges. It is out-
lined in Figure 8.22. According to it, using the range presented so far, 32 different error
measurements may be calculated from the combination of different forecasts or bookings.
Chapter 8: Simulation Based Analysis of Forecast Performance 157
(
)
∑
=
−constconstcc fbEse ,)(
(
)
∑
=
−uncuncuu fbEse ,)(
(
)
∑
=
−uncconstcc
fbEse ,)(
(
)
∑
=
−uncuncuu
ffEse ,
~
)(
~
(
)
∑
=
−),,(),,,()( tciftcibEse uncuncuu
∑ ∑∑
===
−
tt
N
t
unc
N
t
uncuu tciftcibEse 11 ),,(,),,()(
(
)
∑
=
−
),,(),,,()( tciftcibMADse
uncuncuu
MAD
...)( =
−se uu
MAPE ...)(
2
=
−
se
uu
U
...)( =
−
se
uu
RMSE
object of comparison
level of comparison
method of comparison
… …
… …
……
……… …
e
Figure 8.22.: Possible Error Measurements
By applying the alternative error measurements described, the consequences of choos-
ing one over the other can be tested using the simulation system. Applying different
measurements to the same scenario and combining it with knowledge about the actual
demand and quality of forecast methods applied allows for a systematic evaluation of
three fields: the choice of indicators to compare, the choice of aggregation level, and the
choice of error to compute.
Consequences of the Choice of Object: As documented in Section 4.2, it is common
to use the comparisons eu−uand ec−cfor forecast evaluation. The condition for this is
the equivalence of both indicators. While the absolute numbers depend on the type of
Chapter 8: Simulation Based Analysis of Forecast Performance 158
comparison, their conclusion should be the same. This may be tested by comparing the
ranking of methods based on each error measurement as shown in Definition (8.42).
•Let R(◦, e, s) be the rank of forecasting method ◦resulting from its evaluation via
the error measurement ein simulation run s.
eu−u≡ec−c:= R◦, eu−u, s=R◦, ec−c, s(8.42)
MAD Constrained Forecast vs. Actual Bookings
0
1
2
3
4
5
6
1 3 5 7 9 11 13 15
Simulation Run
Rank
c-max 1-i
c-max 2-i
exp025
exp050
exp075
exp050upd
MAD Unconstrained Forecast vs. Unconstrained Bookings
0
1
2
3
4
5
6
1 3 5 7 9 11 13 15
Simulation Run
Rank
c-max 1-i
c-max 2-i
exp025
exp050
exp075
exp050upd
MAD Unconstrained Forecast vs. Actual Bookings
0
1
2
3
4
5
6
1 3 5 7 9 11 13 15
Simulation Run
Rank
c-max 1-i
c-max 2-i
exp025
exp050
exp075
exp050upd
Figure 8.23.: Rank of Methods according to MAD in Product-Based Scenario with “Vol.
= 050, Dev. = 00”
Chapter 8: Simulation Based Analysis of Forecast Performance 159
Figure 8.23 shows the resulting ranks of the computation of eu−u
MAD,ec−c
MAD, and ec−u
MAD
for the forecast methods “1-i c-max”, “2-i c-max”, “exp025”, “exp050”, “exp075” and
the version of “exp050” that is updated throughout the booking horizon, “exp050-upd”.
The methods are ranked based on observations made when applying them to a product-
sensitive market with low demand volume, “Vol. = 050”, and low deviation of the error
distribution “Dev. = 00”. This market shows the clearest results, however, similar trends
emerge when analyzing different error term deviations. As the ranks are constant after
the tenth run, only the first ten runs of the simulation experiment are shown in the graph.
As can be seen, after a number of runs, all error measurements included in the analysis
lead to the same result in terms of rank: While the Exp025 forecast is judged to be the
most accurate, the two psychic forecast options occupy the last places. The relationship
stated by Hypothesis (8.42) is confirmed.
Given a view of ranks rather than absolute errors, a comparison of actual bookings
and unconstrained forecasts becomes possible. While higher absolute quantities can be
expected due to the inherent quantitative difference of bookings and demand, intuitively
rankings may still be equivalent. A successful optimization should always exclude book-
ings in cheap classes if enough demand for more expensive classes is available, this may
not be true for scenarios including a high volume of demand. Here, a systematic (and de-
sirable) difference between demand and forecast may lead to a disadvantage for methods
that correctly predict the demand that never gets to manifest as bookings. The expec-
tation according to this logic is that possible equivalences disappear as absolute demand
increases. This is formally expressed in Hypothesis (8.43).
lim
|R|→∞ R◦, eu−u, s−R◦, ec−u, s≥1
lim
|R|→∞ R◦, ec−c, s−R◦, ec−u, s≥1
∀s= 1, ..., Ns
(8.43)
Figure 8.24 shows the resulting ranks of the computation of eu−u
MAD,ec−c
MAD, and ec−u
MAD
for the forecast methods “1-i c-max”, “2-i c-max”, “exp025”, “exp050”, “exp075” and
the version of “exp050” that is updated throughout the booking horizon, “exp050-upd”.
The methods are ranked based on observations made when applying them to a product-
sensitive market with high demand volume, “Vol. = 050”, and low deviation of the error
Chapter 8: Simulation Based Analysis of Forecast Performance 160
MAD Const. FC vs. Act. BKD - Vol. = 100, Dev. = 00
0
1
2
3
4
5
6
1 3 5 7 9 11 13 15
Simulation Run
Rank
c-max 1-i
c-max 2-i
exp025
exp050
exp075
exp050upd
MAD Unc. FC vs. Unc. BKD - Vol. = 100, Dev. = 00
0
1
2
3
4
5
6
1 3 5 7 9 11 13 15
Simulation Run
Rank
c-max 1-i
c-max 2-i
exp025
exp050
exp075
exp050upd
MAD Unc. FC vs. Act. BKD - Vol. = 100, Dev. = 00
0
1
2
3
4
5
6
1 3 5 7 9 11 13 15
Simulation Run
Rank
c-max 1-i
c-max 2-i
exp025
exp050
exp075
exp050upd
Figure 8.24.: Rank of Methods according to MAD in Product-Based Scenario with “Vol.
= 100, Dev. = 00”
Chapter 8: Simulation Based Analysis of Forecast Performance 161
distribution “Dev. = 00”. This market shows the clearest results, however, similar trends
emerge when analyzing different error term deviations. As the ranks are constant after
the tenth run, only the first ten runs of the simulation experiment are shown in the graph.
Contrary to what could be expected based on Hypothesis (8.43), there is no difference
to be found between the ranking of forecasts after 15 runs – and from then on, it stays
constant for “Dev. = 00”. However, in getting to this state, especially during the first
two or three runs, the ranks seem to change quicker when demand volume is high. This
may be due to the higher pressure caused by more valuable demand being in the market.
In the simulation system, a psychic forecast includes the most accurate information.
Accordingly, correct error measurements should rate it as the best of all options. As can
be seen in Figure 8.23 and Figure 8.24, this is not the case. With regard to eu−u, the
reason for this lies in deficiencies in the unconstraining methods applied by the adaptive
forecasts and used to transform bookings for the comparison. These do not only lead to a
spiral-down-effect but also make forecasts that predict a smaller overall demand volume
seemingly more attractive. With regard to ec−c, the buy-down that is still possible in
spite of restrictive inventory controls leads to a deviation from the predicted maximum
willingness to pay.
As it incorporates buy-down and does not leave much space for spiral-down, func
1−ic−min
can be expected to perform better according to ec−cthan func
1−ic−max does. This logic would
predict func
1−ic−frac to end up ranked between the two other alternatives. Hypothesis (8.44)
formalizes this expectation.
R1-i c-min, ec−c, s≤R1-i c-frac, ec−c, s≤R1-i c-max, ec−c, s
∀s= 1, ..., Ns(8.44)
Figure 8.25 shows that the prediction formalized in Hypothesis (8.44) does come true
when MAD is applied to variants of psychic forecasts in the product-sensitive scenario.
The graph illustrates the value of MAD averaged over 50 runs. The variant “1-i c-min”,
as predicted, always achieves the lowest deviation from actual bookings while “1-i c-max”
leads to the highest deviation. One exception to the rule is the case of high demand
volume “Vol. = 100” and low deviation “Dev. = 00” – while the order of “1-i c-max”
and “1-i c-min” stays the same, “1-i c-frac” is rated worst in this case.
Chapter 8: Simulation Based Analysis of Forecast Performance 162
MAD Const. FC vs. Act. BKD
(averaged over 50 runs)
10,8 10,8 10,9 10,9 10,3
14,1 13,9 13,7 13,6 13,3
10,1 10,2 10,0 9,7 9,0
14,3 13,8 13,6 13,5 13,0
9,2 9,2 9,1 8,8 8,2
13,5 13,3 13,0 12,9 12,4
0
2
4
6
8
10
12
14
16
Dev. =
00 Dev. =
01 Dev. =
05 Dev. =
10 Dev. =
20 Dev. =
00 Dev. =
01 Dev. =
05 Dev. =
10 Dev. =
20
Vol. = 050 Vol. = 100
c-max
c-frac
c-min
Figure 8.25.: MAD: Constrained Psychic Forecasts vs. Actual Bookings in the Product-
Based Scenarios
Finally, if nothing changes in customer behavior over time, the repeated application
of adaptive forecasts should lead to an improvement of forecast quality. In addition to
ec−cand eu−u, the error measurement alternative including psychic forecasts, e
eu−u, can
be used to test for this. The expected forecast behavior can be found Hypotheses (8.45).
lim
s→∞ e
eu−u(s) = 0
lim
s→∞ eu−u(s) = 0
lim
s→∞ ec−c(s) = 0
(8.45)
However, as shown in Figure 8.26, Hypothesis (8.45) does not hold for the adaptive
forecast methods based on exponential smoothing implemented here. In contrast to that,
the prediction of Hypothesis (8.45) can be observed to come true over the course of the fifty
simulation runs included in the experiment. The conclusion drawn from this observation
is that the seemingly adaptive effect that leads to an improvement of eu−uand ec−cis not
in fact due to an adaption to the real demand. Instead, it is the result of a spiral-down
effect as described in Section 8.1.
Chapter 8: Simulation Based Analysis of Forecast Performance 163
MAD Unc. FC vs. Psy. FC - Vol. = 50, Dev. = 00
0
20
40
60
80
100
120
140
160
180
200
0 5 10 15 20 25
Simulation Run
exp025
exp050
exp075
exp050upd
Figure 8.26.: MAD of Unconstrained Forecasts from Psychic Forecast in Product-Based
Scenario with “Vol. = 050, Dev. = 00”
Consequences of the Level of Comparison: Whether or not to aggregate bookings and
forecasts over time before departure before calculating error measurements can also be
considered on the basis of a simulation experiment. For this, the computation of a reverse
psychic forecast is introduced.
•Let func
1-i c-max (f, c, t, s) be the psychic forecast predicting demand to arrive for class c
and flight fbetween points of time t−1 and tin the booking horizon of simulation
run s.
•Let func
1-i c-max (f, c, t, s) be the mirror psychic forecast as presented in Definition
(8.46).
func
1-i c-max (f, c, t, s) := func
1-i c-max f, c, Nt−t+ 1, s
∀f∈F;c∈C;t= 1, ..., Nt;s= 1, ..., Ns(8.46)
Considering that EMSR-b does not take into account the order of demand arrival, func
and func may be expected to be equivalent with regard to their consequences. Using
Chapter 8: Simulation Based Analysis of Forecast Performance 164
revenue r(s) as an indicator of overall system performance, this expectation has been
formalized in Hypothesis (8.47).
•Let rbe the overall revenue resulting from the application of a forecast func to one
scenario in a simulation experiment.
•Let rbe the overall revenue resulting from the application of a mirrored forecast
func to the same scenario.
func ≡func →r=r(8.47)
However, many revenue management systems include re-optimization routines within
the booking horizon as described in Section 7.3. These are modifications of forecasts and
inventory controls based on the deviation of already observed bookings from the predicted
demand.
•Let f0unc be a forecast that is updated throughout the booking horizon.
•Let f0unc be a mirror forecast that is updated throughout the booking horizon.
•Let r0be the overall revenue resulting from the application of a mirror forecast that
is updated.
As updating is applied, a difference in revenue can be expected. As seen before, adaptation
to observed values does not always have a positive effect. For this reason, the expectation
phrased by Hypothesis (8.48) is neutral.
f0unc 6=f0unc →r06=r0(8.48)
Figure 8.27 shows the revenue earned when inventory controls based on the psychic
forecast are re-optimized, a mirrored psychic forecast is applied, or inventory controls
based on the mirrored psychic forecast are re-optimized. In order to render the difference
clearly visible without regard to variations of overall demand volume and error term
deviation, revenue is expressed as a percentage of the revenue earned under the same
conditions with the psychic forecast.
As can be seen, Hypothesis (8.47) holds true: The mirrored psychic forecast without
re-optimization leads to a result that is equivalent to that of the psychic forecast that was
not mirrored. The revenue percentage displayed is 100 %.
Chapter 8: Simulation Based Analysis of Forecast Performance 165
Revenue in Percent of Revenue Observed with Psychic FC
(averaged over 50 runs on the product-based market)
100% 100% 100% 100% 100% 100% 100% 100% 100% 100%
102% 102%
103% 104% 104%
105% 106% 106% 106% 107%
96% 96%
100%
101% 102% 102%
101% 101%
102%
103%
90%
92%
94%
96%
98%
100%
102%
104%
106%
108%
Dev. =
00 Dev. =
01 Dev. =
05 Dev. =
10 Dev. =
20 Dev. =
00 Dev. =
01 Dev. =
05 Dev. =
10 Dev. =
20
Vol. = 050 Vol. = 100
psyMirror
psyReOpt
psyMirrorReOpt
Figure 8.27.: Revenue in Percent of Revenue Earned by Psychic Forecast – Product-Based
Scenario
The consequences of updating as predicted by Hypothesis (8.48) can also be observed
in Figure 8.27. As shown, re-optimization combined with the psychic forecast leads to
higher revenue in all market variations included. This is due to the overly restrictive
inventory controls of the psychic forecast being corrected by re-optimization: While yield
decreases, the increase in the number of bookings is strong enough to compensate.
In situations with low demand volume and low error term deviation, combining the mir-
rored psychic forecast with a re-optimization leads to lower revenue. As valuable demand
is expected to arrive at incorrect times, the re-optimization does not compute inventory
controls that succeed at reserving seats. Instead, the number of additional bookings
gained when cheap tickets are available does not compensate for the loss in yield. How-
ever, in situations with high demand volume or a high error term deviation leading to a
non-homogeneous distribution of demand over itineraries, even the re-optimized mirrored
psychic forecast leads to higher revenue than the original psychic forecast. Again, this is
due to an increase in bookings that compensates for losses in yield.
Chapter 8: Simulation Based Analysis of Forecast Performance 166
Consequences of Choice of Method: So far, all simulation experiments have focused
on the use of mean absolute deviation, MAD, to compute errors. However, three other
methods of computing error have been mentioned earlier. Of these, two (MAPE and U2)
are based on percentage rather than absolute errors. RMSE is another absolute error
measurement.
Two assumptions may be held about these indicators. One is the equivalence of absolute
error measurements as well as that of percentage error measurements. The assumption
has been taken from the literature presented in Section 4.1. It states that errors based on
a percentage calculation are preferable as they do take into account the overall amount
of demand.
Equivalence of MAD to RMSE and MAPE to U2 can be tested for based on a set-up
similar to that applied when considering the equivalence of object choices. If MAD and
RMSE are equivalent, this is not necessarily a question of their absolute value but one
of the resulting rankings for different forecast methods. The same is true for MAPE and
U2. This statement is formalized in Hypothesis (8.49).
ec−c
MAD ≡ec−c
RMSE →RMAD, ec−c, s=RRMSE, ec−c, s
ec−c
MAPE ≡ec−c
U2 →RMAPE, ec−c, s=RU2, ec−c, s(8.49)
If error measurements based on percentage values were more accurate than those based
on absolute values, one consequence would be a general difference in ratings. This expec-
tation can be expressed in similar terms as the expected equivalence of methods described
by Hypothesis (8.49). It is presented in Hypothesis (8.50).
ec−c
MAD 6=ec−c
MAPE →RMAD, ec−c, s6=RMAPE, ec−c, s
ec−c
RMSE 6=ec−c
U2 →RRMSE, ec−c, s6=RU2, ec−c, s(8.50)
Figure 8.28 shows that the prediction does hold true: Different rankings result from
the application of absolute or percentage values. No statement about the quality of
these measurements can be made yet. One way of evaluating the quality of the error
measurements analyzed here has already been introduced – the comparison to a psychic
forecast. It may offer insights as the quality of the psychic forecast is known to be superior
to that of the adaptive alternatives.
Chapter 8: Simulation Based Analysis of Forecast Performance 167
U2
0
1
2
3
4
5
6
0 5 10 15 20 25 30
Simulation Run
Rank
U2
0
1
2
3
4
5
6
0 5 10 15 20 25 30
Simulation Run
Rank
Ranks according to Errors of Constraint FC vs. Act. Bookings
exp050-upd
exp050
c-max 2-i
c-max 1-i
exp075
exp025
MAPE
0
1
2
3
4
5
6
0 5 10 15 20 25
Simulation Run
Rank
MAPE
0
1
2
3
4
5
6
0 5 10 15 20 25
Simulation Run
Rank
RMSE
0
1
2
3
4
5
6
0 5 10 15 20 25 30
Simulation Run
Rank
RMSE
0
1
2
3
4
5
6
0 5 10 15 20 25 30
Simulation Run
Rank
MAD
0
1
2
3
4
5
6
0 5 10 15 20 25
Simulation Run
Rank
MAD
0
1
2
3
4
5
6
0 5 10 15 20 25
Simulation Run
Rank
Vol. = 050, Dev. = 00
Vol. = 100, Dev. = 20
Figure 8.28.: Ranks According to Different Error Measurements
Chapter 8: Simulation Based Analysis of Forecast Performance 168
Conclusion: Different error measurements may result in different evaluations of the ac-
curacy of forecasts. A consequence of the choice of objects of comparison, level of compar-
ison, and method of comparison may be quite different rankings of the method compared.
Additionally, some simulation experiments involving mirrored psychic forecasts have in-
dicated that an accurate forecast is not necessarily the most successful forecast in terms
of resulting revenue.
8.4. Definitions and Effects of Uncertainty of Demand
The previous sections have considered the effects of the application of certain forecast
methods over a long term, the consequences of psychic initializations and benchmarks
and the characteristics of error measurements. All statements about the analysis of fore-
cast performance under these aspects have been illustrated by the results of simulation
experiments over an array of market variations. Depending on the volume of demand
included and the deviation of the error term involved in distorting demand over several
simulation runs, different results could be observed. The goal of this section is to examine
in how far the uncertainty included in a market influences the forecast quality and how
this aspect can be included in forecast performance evaluation.
The concept that some markets may include more uncertainty than others and therefore
be more difficult to predict has already been mentioned in Section 4.1 with special regard
to Diebold & Lopez (1996). When the customer behavior is random and volatile, no
forecast can achieve good results. Uncertainty includes two dimensions: information on
customer behavior is not available (apparent randomness) and the uncertainty of demand
is high. Both dimensions are to be analyzed.
Uncertainty due to Lacking Information: A way of comparing forecasts given different
amounts of information about customer behavior is to change the decision parameters
included in the market. An example of this can be found in the comparison of the product-
oriented to the price-oriented scenario. Customers confronted with a class structure clearly
differentiated by restrictions and basing their decision mainly on their product acceptance
tend to book the same classes whenever they are available. Customers confronted with a
Chapter 8: Simulation Based Analysis of Forecast Performance 169
product structure that is only differentiated by price and basing their decision mainly on
the price tend to book the cheapest class available. A forecast that assumes static demand
includes the necessary information to predict product-oriented, but not price-oriented
demand. The consequence is a higher degree of uncertainty due to information (on buy-
down behavior, in this case) not being available. This lack of information becomes even
more influential to the success of the forecast in terms of revenue when overall demand
volume is low, as under such circumstances buy-down takes place more frequently.
•Let Sproduct indicate a combination of supply and demand that is based on a differ-
entiation by product characteristics (product-sensitive market).
•Let Sprice indicate a combination of supply and demand that is based on a differen-
tiation by price (price-sensitive market).
•Let Shybrid indicate a combination of supply and demand that is based on a differ-
entiation by price and product characteristics (hybrid market).
•Let S=Sproduct, Sprice, Shybridbe the set of market scenarios available.
•Let Ψ (S, σ, V, F◦) describe a simulation experiment based on a market structure
S, a deviation of the error term of demand σ(), overall demand volume Vand a
forecast F◦.
•Let yfcfs % (Ψ (S, σ, V, F◦)) be the average yield generated in a simulation experiment
as the percentage of the yield generated when inventory controls based first-come-
first-serve were applied.
•Let rpsy % (Ψ (S, σ, V, F◦)) be the average revenue generated in a simulation exper-
iment as the percentage of the revenue generated when inventory controls based
first-come-first-serve were applied.
•Let ypsy % (Ψ (S, σ, V, F◦)) be the average yield generated in a simulation experiment
as the percentage of the yield generated when inventory controls based on a “c-max
1-i” psychic forecast were applied.
•Let rpsy % (Ψ (S, σ, V, F◦)) be the average revenue generated in a simulation exper-
iment as the percentage of the revenue generated when “c-max 1-i” psychic under
first-come-first-serve inventory controls were applied.
Chapter 8: Simulation Based Analysis of Forecast Performance 170
•Let ec-c
MAD (Ψ (S, σ, V, F◦)) be the mean average deviation of the constrained forecast
compared to the bookings observed in simulation experiment averaged over all runs.
In Hypothesis (8.51), the following expectation for simulation experiments using a psy-
chic forecast “c-max, 1-i” is formalized: If the market is product-based, the decrease of
yield resulting from a decrease in demand volume will not be as steep as when it is price
based.
yfcfs % ΨSproduct, σ, V, Fc-max, 1-i
yfcfs % (Ψ (Sproduct, σ, V 0, Fc-max, 1-i)) ≤yfcfs % ΨSprice, σ, V, Fc-max, 1-i
yfcfs % (Ψ (Sprice, σ, V 0, Fc-max, 1-i))
∀V0≤V;σ∈R
(8.51)
Development of Yield in Percent of FCFS from Vol. = 050 to Vol. = 100
0%
10%
20%
30%
40%
50%
60%
70%
80%
Product Price
Dev. = 00
Dev. = 01
Dev. = 05
Dev. = 10
Dev. = 20
Figure 8.29.: Increase of Yield in Percent of FCFS from Vol. = 050 to Vol. 100
Figure 8.29 shows the increase of yield indicated by yfcfs% as demand volume grows
from “Vol. = 050” to “Vol. = 100” The increase is depicted in percent of yfcfs% given
“Vol. = 050”. While growing yield can also be observed in a product-based market, the
increase is higher on price-based markets. On a product-based market, yield increases in
yield as with absolute demand volume being higher, the absolute share of customers that
exclusively request more valuable classes is also higher. On a price-based market, yield
Chapter 8: Simulation Based Analysis of Forecast Performance 171
increases as the absolute number of customers with a high willingness to pay increases
and inventory controls are become more restrictive.
The effect of yield also seems to depend on the deviation given in the market (“Dev.
= 00” vs. “Dev. = 20”). In general, it can be stated that deviation leads to a weaker
growth in yield as demand volume increases.
Hypothesis (8.52) formalizes the expectation that in a price-based scenario, a static
forecast will be less successful at maximizing revenue than in a product-based scenario
even when it is based on “psychic” knowledge of maximum willingness to pay as described
by Section 8.2.
rfcfs % ΨSprice, σ, V, , Fc-max, 1-i≤rfcfs % ΨSproduct, σ, V, Fc-max, 1-i
∀σ∈R;V∈N
(8.52)
Revenue in Percent of First-Come-First-Serve
0%
100%
200%
300%
400%
500%
600%
700%
800%
900%
Vol. = 050 Vol. = 100 Vol. = 050 Vol. = 100
Product Price
Dev. = 00
Dev. = 01
Dev. = 05
Dev. = 10
Dev. = 20
Figure 8.30.: Revenue in Percent of FCFS on Price- and Product-Based Markets
Figure 8.30 shows a comparison of the revenue as a percentage of the average yield ob-
served under first-come-first-serve inventory controls over price- and product-based mar-
kets. Obviously, the expectation stated in Hypothesis (8.52) is confirmed.
Chapter 8: Simulation Based Analysis of Forecast Performance 172
In the product-based markets, the difference in revenue between first-come-first-serve
controls and the “c-max 1-i” psychic forecast is not large to start with: At most, the
psychic forecast exceeds first-come-first-serve by 37 % in cases with low deviation (“Dev. =
00”) and high demand volume. As customers request tickets for specific classes, and nested
inventory controls ensure availability in the most expensive classes, even well-computed
protection levels can not improve revenue much if enough capacity is available to accept
most requests. This is the reason for a difference of only 9 % for low demand volume and
high deviation (“Vol. = 050, Dev. = 20”). For product-based markets, inventory controls
are most useful for high demand, when bookings in cheap classes threaten to use capacity
that could be sold at higher prices.
The case is quite different when the price-based markets are considered. On these, the
difference between first-come-first-serve and the psychic forecast in terms of revenue is
at least 300%: Four times as much revenue can be gained when applying the inventory
controls based on “c-max 1-i” to low deviation and low demand (“Vol. = 050, Dev.
= 00”). As deviation rises, the distribution of demand becomes less homogenuous and
on some flights, demand exceeds capacity. In these cases, as already described for the
product-based scenarios, inventory controls based on the psychic forecast avoid using the
sparse capacity for customers requesting cheap tickets even on markets with low overall
demand. Revenue based on the psychic forecast exceeds that based on the first-come-first-
serve controls by up to 370 %. When demand volume is high, the effect of high deviation
does not add to the revenue. Instead, revenue is at its maximum when deviation is low,
exceeding what was earned with first-come-first-serve by 665 %.
Hypothesis (8.53) formalizes the expectation that in any scenario, a static forecast
based on exponential smoothing will be less successful at maximizing revenue when the
deviation of the distribution the error term is drawn from is high.
lim
σ→∞ rfcfs % ΨSprice, σ, V, F◦= 100
∀S∈Sproduct, Sprice, Shybrid;V∈N;Fexp25, Fexp50, Fexp75
(8.53)
As could be seen in Figure 8.30, this statement only holds true for high demand volume.
When overall demand volume is low, a high deviation of the error term distribution can
have positive effects that may lead to an increase in revenue: As the distribution of
Chapter 8: Simulation Based Analysis of Forecast Performance 173
demand over itineraries becomes less homogeneous, in spite of low demand volumes,
inventory controls become more effective at reserving seats for valuable customers.
Hypothesis (8.54) formalizes the expectation that in a price-based scenario, a static
forecast will be less successful at predicting demand accurately than in a product-based
scenario. The quality of demand forecast is measured as the mean absolute deviation
between the constrained forecast and the observed bookings.
ec-c
MAD ΨSprice, σ, V, , Fc-max, 1-i≥ec-c
MAD ΨSproduct, σ, V, Fc-max, 1-i
∀σ∈R;V∈N
(8.54)
Difference between MAD on Price- and Product-based Markets in Percent
0%
5%
10%
15%
20%
25%
Vol. = 050 Vol. = 100
Dev. = 00
Dev. = 01
Dev. = 05
Dev. = 10
Dev. = 20
Figure 8.31.: Difference in MAD (Price-Based - Product-Based Market) in Percent
Figure 8.31 shows the positive difference between the MAD observed on price- and
product-based markets. It indicates that the MAD computed from the constrained fore-
cast and actual bookings is always higher for price-based markets. As capacity and the
overall demand volume were kept constant for all simulation experiments, this fulfills the
prediction stated in Hypothesis (8.54).
Uncertainty due to Market Volatility: In the simulation, the uncertainty created by
the volatility of a market is modeled as the deviation of the distribution the error term is
Chapter 8: Simulation Based Analysis of Forecast Performance 174
drawn from. A potentially high error term can result in overall demand volume changing
strongly from one run to the other. Furthermore, the distribution of demand over different
routes becomes more skewed.
σ≤σ0
→ec-c
MAPE (Ψ (S, σ, V, F◦)) ≤ec-c
MAPE (Ψ (S, σ0
, V, F◦))
∀S∈Sproduct, Sprice, Shybrid;V∈N;{Fexp25, Fexp50, Fexp75}
(8.55)
In any scenario a static forecast will be less successful accurately forecasting demand
when the deviation of the distribution the error term is drawn from is high. Standard
error indicators will interpret this as a decrease in forecast performance, even though not
the forecast’s abilities but the potential for accurate forecasts has been diminished. For
the forecast error MAPE, this expectation is formalized by Hypothesis (8.55).
Figure 8.32 presents the value of eu-u
MAPE (Ψ (S, σ, V, F◦)) averaged over 50 runs. The
error indicator is displayed for simulation experiments applying the naive forecast Fnaive
as well as exponential smoothing methods initialized by the psychic forecast “c-max 1-
i”; Fexp025,Fexp050, and Fexp075; to markets Sproduct and Sprice. Overall demand volume
changes from “Vol. = 050” to “Vol. = 100” – the four possible combinations of market-
type and demand volume have been used to generate one graph each. The deviation
of the error term distribution changes from “Dev. = 00” via “Dev. = 01”, “Dev. =
05”, and “Dev. = 10” to “Dev. = 20”. The mean average percentage error (MAPE) is
illustrated by a group of bars for each method, with the color changing from dark blue
(low deviation) to light blue (high deviation).
As can be seen in Figure 8.32, the forecast error tends to increase as the deviation of the
error term distribution increases. Its level is higher for the exponential smoothing methods
than for the naive forecast. This is due to the initialization of the exponential smoothing
methods based on the psychic forecast: As explained in Section 8.3, when evaluated by
standard error indicators, the psychic forecast leads to high errors. Furthermore, the
spiral-down effect that takes place on price-based markets as explained in Section 8.1
leads to smaller errors on price-based markets.
Chapter 8: Simulation Based Analysis of Forecast Performance 175
MAPE: Product-Based Market, Vol. = 050
0%
200%
400%
600%
800%
1000%
1200%
1400%
naive Exp025 Exp050 Exp075
Dev. = 00
Dev. = 01
Dev. = 05
Dev. = 10
Dev. = 20
MAPE: Product-Based Market, Vol. = 100
0%
200%
400%
600%
800%
1000%
1200%
1400%
naive Exp025 Exp050 Exp075
Dev. = 00
Dev. = 01
Dev. = 05
Dev. = 10
Dev. = 20
MAPE: Price-Based Market, Vol. = 050
0%
200%
400%
600%
800%
1000%
1200%
1400%
naive Exp025 Exp050 Exp075
Dev. = 00
Dev. = 01
Dev. = 05
Dev. = 10
Dev. = 20
MAPE: Price-Based Market, Vol. = 100
0%
200%
400%
600%
800%
1000%
1200%
1400%
naive Exp025 Exp050 Exp075
Dev. = 00
Dev. = 01
Dev. = 05
Dev. = 10
Dev. = 20
Figure 8.32.: MAPE Averaged over 50 Runs
Chapter 8: Simulation Based Analysis of Forecast Performance 176
Indicators of Uncertainty: To normalize the development of the forecast error to the
degree of market uncertainty, a new indicator is introduced.
•Let eu-u
PB (Ψ (S, σ, V, , F◦)) be the percentage of cases in which deviation of uncon-
strained forecast Fused in simulation experiment Sfrom the observed bookings
was smaller than that of the naive forecast.
The performance of the naive forecast depends on the volatility of the market: If the
same amount of requests for the same classes arrives in every run, it is very successful.
While in such a situation, forecasting is also easier for other methods, the benchmark of
the naive forecast becomes harder to reach. If high market uncertainty leads to volatile
request volumes, the difference between the naive forecast and the actual observations
grows. While this makes forecasting harder for other methods, as well, the benchmark of
the naive forecast becomes easier to reach. This way, forecast performance is normalized
to the situation.
Figure 8.33 presents the value of eu-u
PB (Ψ (S, σ, V, F◦)) averaged over 50 runs. The
performance indicator “Percent Better” is displayed for simulation experiments applying
the naive forecast Fnaive as well as exponential smoothing methods initialized by the
psychic forecast “c-max 1-i”; Fexp025,Fexp050, and Fexp075; to markets Sproduct and Sprice.
Overall demand volume changes from “Vol. = 050” to “Vol. = 100” – the four possible
combinations of market-type and demand volume have been used to generate one graph
each. The deviation of the error term distribution changes from “Dev. = 00” via “Dev.
= 01”, “Dev. = 05”, and “Dev. = 10” to “Dev. = 20”. The percentage of cases for
which the respective method performed better than the naive method (PB) is illustrated
by a group of bars for each method applied, with the color changing from dark blue (low
deviation) to light blue (high deviation).
The introduction of a percent-better indicator does normalize the evaluation of forecast
accuracy according to the level of uncertainty included in a market. The number of cases
in which the evaluated method performed better than the naive forecast does not seem
to clearly depend on the deviation of the error distribution. While it grows with rising
deviation in the product-based market with high demand volume, it does not show a
trend for the other combinations of market and demand volume shown.
Chapter 8: Simulation Based Analysis of Forecast Performance 177
Percent Better: Product-Based Market, Vol. = 050
0%
200%
400%
600%
800%
1000%
1200%
1400%
Exp025 Exp050 Exp075
Dev. = 00
Dev. = 01
Dev. = 05
Dev. = 10
Dev. = 20
Percent Better: Product-Based Market, Vol. = 100
0%
200%
400%
600%
800%
1000%
1200%
1400%
Exp025 Exp050 Exp075
Dev. = 00
Dev. = 01
Dev. = 05
Dev. = 10
Dev. = 20
Percent Better: Product-Based Market, Vol. = 100
0%
200%
400%
600%
800%
1000%
1200%
1400%
Exp025 Exp050 Exp075
Dev. = 00
Dev. = 01
Dev. = 05
Dev. = 10
Dev. = 20
Percent Better: Product-Based Market, Vol. = 100
0%
200%
400%
600%
800%
1000%
1200%
1400%
1600%
Exp025 Exp050 Exp075
Dev. = 00
Dev. = 01
Dev. = 05
Dev. = 10
Dev. = 20
Figure 8.33.: “Percent Better” Averaged over 50 Runs
Chapter 8: Simulation Based Analysis of Forecast Performance 178
According to what was shown so far, a general indicator of the uncertainty included
in a market may be the quality of the naive forecast. In a very stable environment, a
prediction based on the values observed during the last run can be expected to be accurate
if no trend influences demand. With demand fluctuating from one run to the other as
the deviation of error terms increases, this changes. Hypothesis (8.56) formalizes this
expectation.
σ≤σ0
→ec-c
MAD ΨS, σ, V, Fnaive≤ec-c
MAD ΨS, σ0
, V, Fnaive
∀S∈Sproduct, Sprice, Shybrid;V∈N
(8.56)
Mean Absolute Deviation of Naive FC
(averaged over 50 runs)
6
7
8
9
10
11
12
Vol. = 050 Vol. = 100 Vol. = 050 Vol. = 100
Product Price
Dev. = 00
Dev. = 01
Dev. = 05
Dev. = 10
Dev. = 20
Figure 8.34.: MAD for Naive Forecast Averaged over 50 Runs
Figure 8.34 shows the development of ec-c
MAD ΨS, σ, V, Fnaiveover all markets as σ
increases. As was already observed to some extend with regard to the error indicator
MAPE, the mean absolute deviation of the constrained naive forecast from observed
bookings increases with an increase in the deviation of the error term distribution. The
exception is the product-based market with low overall demand volume: Even with the
resulting high error term, product-based customers apparently stay equally predictable
as long as demand does not exceed capacity.
Chapter 8: Simulation Based Analysis of Forecast Performance 179
Runs Required for the Achievement of Confidence Level
Delta (span): 0.01; Alpha (probability): 0.01
0
200
400
600
800
1000
1200
Dev. = 00
Dev. = 01
Dev. = 05
Dev. = 10
Dev. = 20
Dev. = 00
Dev. = 01
Dev. = 05
Dev. = 10
Dev. = 20
Dev. = 00
Dev. = 01
Dev. = 05
Dev. = 10
Dev. = 20
Dev. = 00
Dev. = 01
Dev. = 05
Dev. = 10
Dev. = 20
Vol. = 050 Vol. = 100 Vol. = 050 Vol. = 100
Product-Based Market Price-Based Market
FCFS
Naive
Figure 8.35.: Runs Required for Confidence Level
In a simulation environment, another indication of market uncertainty is the number of
runs needed to reach the desired confidence level. Based on the computation of confidence
described in Section 7.1, the number of runs a simulation experiment increases with the
deviation of the distribution error terms. This observation is illustrated by Figure 8.35.
The graph shows the development of the number of runs required for simulation experi-
ments including first-come-first-serve controls or a naive forecast and EMSR-b, based on
price- or product-based markets with low or high demand volume and a range of deviations
of the error term distribution.
As can be seen, both methods applied need roughly an equal amount of runs to reach
the required confidence level. Note that on price-based markets, a naive forecast that is
initialized with a zero forecast and first-come-first-serve lead to similar inventory controls
and to bookings only in the cheapest class included in the scenario. Adaptive methods
such as psychic forecasts and exponential smoothing need many more runs as they interact
with the volatility of the market. However, in both cases the number of runs required
increases with the deviation of the distribution that the demand error term is drawn from.
Chapter 8: Simulation Based Analysis of Forecast Performance 180
Without having to rely on information exclusive to simulation environments, the error
variance of the naive forecast can also indicate uncertainty. The more that the fluctuation
of demand is influenced by a high error term and thereby randomized, the more random
also the quality of a naive forecast. Hypothesis (8.57) formalizes this expectation.
•Let σ2ec-c
MAD ΨS, σ, V, Fnaivebe the variation of the mean absolute deviation
of the constrained naive forecast from observed bookings over the runs of simulation
experiment Ψ.
σ≤σ0
→σ2ec-c
MAD ΨS, σ, V, Fnaive≤σ2ec-c
MAD ΨS, σ0
, V, Fnaive
∀S∈Sproduct, Sprice, Shybrid;V∈N
(8.57)
Variance of MAD of Naive FC
(over 50 runs)
0
1
2
3
4
5
6
Vol. = 050 Vol. = 100 Vol. = 050 Vol. = 100
Product Price
Dev. = 00
Dev. = 01
Dev. = 05
Dev. = 10
Dev. = 20
Figure 8.36.: Variance of MAD for Naive Forecast over 50 Runs
Figure 8.36 shows the development of σ2ec-c
MAD ΨS, σ, V, Fnaiveas σincreases
over all markets. As predicted in Hypothesis (8.57), the variance of error increases rapidly
as the variance of the error term distribution increases.
Indicators of Robustness: Observing the development of revenue and forecast quality as
the deviation of the distribution of error terms increases can also give an indication of the
Chapter 8: Simulation Based Analysis of Forecast Performance 181
robustness of a forecast method. In contrast to an observation of forecast quality under
static conditions, robustness provides information on how well a forecast reacts to changes
in the market place. Hypothesis (8.58) formalizes the description of a measurement of
robustness based on the comparison of revenue as uncertainty represented by the deviation
of the distribution that error terms are drawn from increases.
•Let Orev (F◦, σ, σ0
) be a revenue-based indicator of the robustness of a forecast
method F◦given an increase in uncertainty from σto σ0
Orev (F◦, σ, σ0
) := r(Ψ (S, σ0
, V, F◦))
r(Ψ (S, σ, V, F◦))
∀S∈Sproduct, Sprice, Shybrid;V∈N
(8.58)
Revenue Robustness (Dev. 00 to Dev. 20)
63%
93%
99% 100%
59%
85%
109%
90%
58%
81%
94%
73%
0%
20%
40%
60%
80%
100%
120%
Vol. = 050 Vol. = 100 Vol. = 050 Vol. = 100
Product Price
FCFS
Psy
Exp50
Figure 8.37.: Revenue Robustness based on Rev. Averaged over 50 Runs
Figure 8.37 shows the revenue-based robustness of first-come-first-serve controls, the
psychic forecast based on “c-max 1-i”, and the exponential smoothing method Exp050 for
σ0
= 0 and σ= 20. As may be expected, the first-come-first-serve controls are the most
robust choice on most market variations: First-come-first-serve inventory controls are not
changed based on forecasts that may be mislead by demand with a high deviation of the
error term distribution. However, based on these controls, the lowest absolute revenue
Chapter 8: Simulation Based Analysis of Forecast Performance 182
is generated in most cases. The psychic forecast, which anticipates at least part of the
demand volatility caused by the error term, comes in on second place with regard to
robustness.
Conclusion: Some combination market characteristics, labeled uncertainty, has an effect
on the potential success of any forecast method. In the simulation environment used
for experiments, uncertainty is modeled by an error term influencing the volatility of
customer behavior. As the deviation of the normal distribution this error term is drawn
from increases and thereby uncertainty rises, forecasts perform worse both in terms of
accuracy and in terms of resulting revenue. Finally, some indicators based on first-come-
first-serve controls or the naive forecast, by which to estimate the level of uncertainty
included in a market, have been introduced.
8.5. Evaluation Approaches for Price-Sensitive Forecasts
As some markets include more uncertainty than others, some forecast methods include
more information than others. The previous section presented simulation experiments
designed to show how markets where customers make decisions based on price-sensitivity
pose a problem for static forecasts not taking into account information such as which class
is the lowest available.
This section intends to demonstrate the advantages of additional demand information
considered in both the forecast and its evaluation. As an example for additional infor-
mation, the price sensitivity of customers is considered and a forecast method including
predictions on customers’ buy-down behavior is applied to the price-sensitive scenario.
The new forecast is based on price-sensitive estimators depending on the cheapest class
available. Details of its implementation are described in Section 7.3.1. It is influenced by
a weight parameter αPbalancing the influence of price to that of time before departure.
In the further text, the results of the use of this forecast are marked as “price09” - the
parameter that determines the weight of the price-estimators was set to 0.9 for the ex-
periments presented here. The exponential smoothing parameter αexp can also be varied.
However, as the general consequences of this parameter have already been presented in
Chapter 8: Simulation Based Analysis of Forecast Performance 183
Section 8.1, it is kept at αexp = 0.5 for all experiments presented here. For this reason,
the exponential smoothing forecast used for comparison is also “Exp050”.
To maximize the observed effects, the price-oriented market scenario is used in all
simulation experiments with the price-sensitivity forecast. When booking classes are only
differentiated by price and customers base their decisions on the information which class
is the cheapest available, the new forecast should fully realize its potential.
Revenue: First, it needs to be established that the new forecast method actually is
advantageous compared to exponential smoothing approaches. Hypothesis (8.59) presents
the expectation that at least one parametrization of the price-sensitive forecast performs
better in term of revenue than the exponential smoothing alternative.
•Let Ψ Sprice, σ, V, F◦be a simulation experiment based on the price-sensitive
market scenario Sprice with error deviation σ, demand volume Vand the forecast
method F◦.
•Let rΨ(Sprice,σ,V,F◦)(s) be the revenue generated in run s= 1, ..., Nsof the above
described simulation experiment.
∃Fprice →rΨ(Sprice,σ,V,Fexp050)(s)≤rΨ(Sprice,σ,V,F price)(s)
∀σ∈R;V∈N
(8.59)
Figure 8.38 shows the results of the application of the psychic forecast “c-max, i-1”, the
exponential smoothing method “exp050” and the price-sensitive forecast “market09” to
the price sensitive market variations over volume and the deviation of the error term. To
provide a measure of comparison, revenue is presented as a percentage what was earned
when first-come-first-serve inventory controls are applied. All methods are initialized with
the Fc-max, 1-i psychic method.
Accuracy: With new parameters and variables included, new forecast evaluation meth-
ods become available. In addition to comparing observed bookings to the predicted de-
mand, observed customer behavior may now also be transformed to make it comparable
to the estimators used in the new forecast.
Chapter 8: Simulation Based Analysis of Forecast Performance 184
Vol. = 050, Dev. = 00
0%
20%
40%
60%
80%
100%
120%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 01
0%
20%
40%
60%
80%
100%
120%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 05
0%
20%
40%
60%
80%
100%
120%
140%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 10
0%
20%
40%
60%
80%
100%
120%
140%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 20
0%
20%
40%
60%
80%
100%
120%
140%
160%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 00
0%
20%
40%
60%
80%
100%
120%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 01
0%
20%
40%
60%
80%
100%
120%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 05
0%
20%
40%
60%
80%
100%
120%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 10
0%
20%
40%
60%
80%
100%
120%
140%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 20
0%
20%
40%
60%
80%
100%
120%
140%
160%
180%
200%
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Revenue in Percent of Run 1
c-max 1-i market05 exp050
market09
Figure 8.38.: Revenue in Percent of Revenue Earned in Run 1
Chapter 8: Simulation Based Analysis of Forecast Performance 185
•Let bunc (f, c, t, s) be the unconstrained bookings for flight fobserved in class c
during time slice t−1 to tbefore departure in simulation run s.
•Let ωPf,c,s
Fbe the vector of price elasticity for flight fdepending on the class cfor
run saccording to the forecast F.
•Let ωPf,c,s
bbe the vector of price elasticity observed for flight fdepending on the
class cfor run saccording to observed bookings.
•Let ˆeωMAD (s) be the mean absolute deviation of the forecasted price elasticity from
the observed price elasticity during run s.
Definition (8.60) shows the computation of the price elasticity from observed bookings.
As no consideration is given to time before departure and the assumption of the price-
sensitive forecast is that all customers always buy the lowest class available, elasticity can
be computed as the share of bookings in one class compared to the overall bookings.
ωPf,c,s
b=PNt
t=1 bunc (f, c, t, s)
PNc
c=1 PNt
t=1 bunc (f, c, t, s)
∀f∈F;c∈ {2, ..., Nc};s= 1, ..., Ns
(8.60)
Definition (8.61) shows the computation of the mean absolute deviation of forecasted
elasticity from observed elasticity. The only difference to the computation of the previ-
ously used MAD indicator is in the indicators used for comparison.
ˆeωMAD (s) := Pf∈FPNc
c=2 ωPf,c,s
F−ωPf,c,s
b
|F|·(|C|−1)
∀s= 1, ..., Ns
(8.61)
After spiral-down is completed, as presented in Section 8.1, exponential smoothing
methods assuming static demand will predict all demand to request the cheapest booking
class in a purely price-based scenario. According to this logic, the elasticity vector of
“Exp050” is filled with zeros as no sell-up is predicted.
In Figure 8.39, the development of eωMAD is shown for two experiments with price-
sensitive forecasts, “market05” and “market09”, over the course of 50 simulation runs.
Chapter 8: Simulation Based Analysis of Forecast Performance 186
Vol. = 050, Dev. = 00
0
0,001
0,002
0,003
0,004
0,005
0,006
0,007
0,008
0,009
0,01
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 01
0
0,001
0,002
0,003
0,004
0,005
0,006
0,007
0,008
0,009
0,01
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 05
0
0,001
0,002
0,003
0,004
0,005
0,006
0,007
0,008
0,009
0,01
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 10
0
0,001
0,002
0,003
0,004
0,005
0,006
0,007
0,008
0,009
0,01
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 050, Dev. = 20
0
0,001
0,002
0,003
0,004
0,005
0,006
0,007
0,008
0,009
0,01
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 00
0
0,001
0,002
0,003
0,004
0,005
0,006
0,007
0,008
0,009
0,01
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 01
0
0,001
0,002
0,003
0,004
0,005
0,006
0,007
0,008
0,009
0,01
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 05
0
0,001
0,002
0,003
0,004
0,005
0,006
0,007
0,008
0,009
0,01
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 10
0
0,001
0,002
0,003
0,004
0,005
0,006
0,007
0,008
0,009
0,01
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
Vol. = 100, Dev. = 20
0
0,001
0,002
0,003
0,004
0,005
0,006
0,007
0,008
0,009
0,01
0 5 10 15 20 25 30 35 40 45 50
Simulation Run
MAD of Price Elasticity vs. Psychic Price Elasticity
market09
market05
Figure 8.39.: MAD of Elasticity vs. Psychic Elasticity
Chapter 8: Simulation Based Analysis of Forecast Performance 187
As in the case of the price-sensitive forecast “market05”, the update includes more time-
sensitive information, the spiral-down effect occurs. The result of this could already been
seen in Figure 8.38, as revenue declined. In the case of the mean absolute deviation, this
results in higher errors compared to psychic knowledge of price elasticity.
Conclusion: The advantages of forecasts that include a model of demand that is not
static when confronting flexible demand structures have been pointed out. As a conclu-
sion, the need for new error measurements including objects of comparison that differ
from the traditional set of bookings and forecasted demand volume may be stressed.
8.6. Simulation-Based Findings Recaptured
A range of statements on the relationship between forecast performance, forecast accuracy,
and forecast evaluation methods have been formalized. Expectations concerning the long-
term effects of methods were drawn from existing research with regard to recent challenges
as described in Section 2.3. Possible consequences of definitions of psychic forecasts were
highlighted as the by-product of thoughts on customer-choice arising when designing the
implementation of the demand model as described in Section 7.2.2. The standard accuracy
indicators evaluated have been introduced previously in Section 4.1. The considerations
of the effect of uncertainty of demand were listed as the consequence of observations on
results depending on the deviation of the error term. Finally, some evaluation approaches
for price-sensitive forecasts were offered using the forecast method described in Section
7.3.1.
Theoretical statements were used to design experiments using the simulation system
described in Chapter 7. Based on the decomposition concept introduced in Chapter 6,
for the first time such theories were analyzed ceteris-paribus in a system that include
a volatile, flexible customer model and a sophisticated range of supply. Processing the
output of the simulation experiments lead to information concerning the conditions under
which the theoretical statements apply.
A number of findings has been verified using the concept for the decomposition and
evaluation of forecasts as documented in Chapter 6 based on the simulation environment
Chapter 8: Simulation Based Analysis of Forecast Performance 188
presented in Chapter 7. The following observations have been made with regard to the
long-term application of methods in revenue management as described in Section 8.1:
•When forecasts based on static demand assumptions are confronted with a flexible,
price-sensitive customer model and a restriction-free class structure, the so-called
spiral-down effect can be observed (see Cooper et al. (2006).
•The spiral-down effect leads to decreasing forecasts for more valuable classes, de-
creasing amounts of protected seats for more valuable classes, and decreasing num-
bers of bookings in the valuable classes. At the same time, forecasts, availability
and bookings for the cheapest class increase.
•The ultimate consequence of the spiral-down effect is a loss in revenue. However,
in markets with high demand volume, the increase in bookings can temporarily
compensate for a loss in yield, resulting in increasing revenue for a limited time.
•The spiral-down effect is more drastic in markets with low demand volume and
volatile customer behavior.
•For adaptive forecast methods based on static demand that include a higher adaptive
weight, the spiral-down effect is more drastic than for those including a low adaptive
weight.
The following observations have been made with regard to the consequences of possible
definitions of psychic forecasts as described in detail in Section 8.2:
•In a simulation system, psychic forecasts may be used to initialize or to evaluate
other forecast methods.
•When predicting demand to arrive per booking class and itinerary, psychic forecasts
may differ with regard to the interpretation of customers’ willingness to pay and
preferred itineraries.
•The method chosen for generating the psychic forecast has consequences with regard
to its revenue performance, ...
•... with regard to the revenue performance that adaptive methods based on it show,
...
Chapter 8: Simulation Based Analysis of Forecast Performance 189
•... and with regard to the evaluation of accuracy of forecast methods compared to
it.
The following observations have been made with regard to the evaluation of standard
accuracy indicators for demand forecasts in revenue management as described in Section
8.3:
•Forecast error measurements may lead to different results depending on the objects,
the level, and the method of comparison.
•Error measurements including different units or operate on different levels of aggre-
gation may be compared by comparing the resulting ranks of evaluated methods.
•With regard to the object of comparison, the decision whether to compare con-
strained forecasts to actual bookings, unconstrained forecasts to unconstrained
bookings, or unconstrained forecasts to actual bookings needs to be taken.
•With regard to the level of comparison, even when an optimization method does
not explicitly use information on the timing of demand arrival, it can be useful to
include timing in the evaluation of forecast accuracy. The information available on
this level may be used later in the process, for example when updating the forecast
within the booking horizon.
•The psychic forecast can be used as a benchmark for the self-improving effects of
forecasts that trigger a spiral-down effect.
The following observations have been made with regard to the definition and the effect of
uncertainty of demand as described in Section 8.4:
•The potential for forecast success in terms of accuracy and revenue depends on the
level of uncertainty included in a market.
•Uncertainty may be measured by the success of the naive forecast.
•In a simulation system, the robustness of forecast methods with regard to uncer-
tainty included in a market may be measured by comparing the results of the forecast
when normalized to the results of first-come-first-serve inventory controls at different
levels of uncertainty.
Chapter 8: Simulation Based Analysis of Forecast Performance 190
The following observations have been made with regard to the evaluation of price-sensitive
forecast methods as described in detail in Section 8.5:
•In price-sensitive markets including a restriction-free product structure and cus-
tomers that base their decisions on price, forecasts that include a non-static model
of demand allow only for a weak spiral-down-effect.
•Forecasts that predict price-sensitive behavior require new evaluation methods based
on other objects of comparison than overall bookings and predicted demand volume.
According to the method applied in this chapter, more theories may be formalized and
tested. After a summary of the ideas introduced in this thesis, the final chapter will
provide a number of suggestions concerning further fields of inquiry.
This chapter described analyses applying the simulation environment for revenue man-
agement to the decomposition and evaluation of demand forecasts. The documentation
of findings that have been made possible by the concept and the implementation of this
system represents the closing argument of this text.
Chapter 9: Conclusion 191
9. Conclusion
The following sections first summarizes the ideas discussed in this text. Finally, an outlook
of future possible research and open questions is provided.
9.1. Summary
In this thesis, a new concept to evaluate demand forecast methods for revenue management
based on a decomposed view of the system was introduced. With the help of such a
concept, the components of revenue management and demand forecasts can be isolated
and evaluated. Interferences stemming from the market environment and the interaction
of the parts of the system can be controlled and analyzed: When evaluated in the presented
framework rather than on an actual market, the success of a method can be isolated from
possible economic trends. When the output of individual components is analyzed, the
performance of specific methods can be separated from their fit with the system. These
merits were demonstrated using a simulation environment. This chapter summarizes the
steps taken.
Background: Having introduced revenue management in general and the problems of
forecast performance evaluation in particular, a list of tasks was formulated in the initial
chapter, 1. Consecutively, these tasks have been made seized and set as goals for research.
The results have been documented in the previous text.
Categorization of Forecast Methods: In order to provide a sound basis for a new
concept of forecast evaluation, existing approaches to demand forecasting for revenue
management were listed in Chapter 3. Research on demand forecasting was categorized
by three aspects: the prediction of volume, the unconstraining of bookings to compute
Chapter 9: Conclusion 192
historical demand figures, and customer behavior. With regard to all categories, the
influence of forecasting on the outcome of revenue management was emphasized. The
categorization serves as the basis for a decomposition approach.
Characterization of Forecast Evaluation Methods: In Chapter 4, demand forecast
performance measurements are listed and characterized by the object, the level, and the
method of evaluation. A theoretical background of existing indicators is provided and
instances of forecast performance measurement applied in existing publications are listed.
In addition, in Chapter 4, research opportunities related to the special difficulties of
forecast evaluation for revenue management were highlighted. Based on the established
knowledge of forecast methods and approaches to forecast evaluation, the need for further
investigation into the decomposed evaluation was further justified. The consequences of
different error measurements are analyzed in Chapter 8.
Conceptualization of the Decomposition of Revenue Management Systems: To in-
troduce such a decomposed evaluation, Chapter 6 describes revenue management systems
in the terms of separate forecast, optimization and inventory modules. The forecast mod-
ule is further decomposed along the lines of the categorization introduced in Chapter 3.
This results in separate components for the prediction of overall demand volume, the
unconstraining and the prediction of demand behavior. Such a framework enables the
isolated evaluation and comparison of forecast methods ceteris-paribus.
Development of Processes for the Evaluation of Demand Forecast Components: In
Chapter 6, a new concept for the decomposition and evaluation of revenue management
systems is presented. In the framework described, the special characteristics of demand
forecasting for revenue management can be confronted by isolating aspects and using
knowledge of the customer demand model to compute new benchmarks. Detailed de-
scriptions of the use of this system to analyze a complete revenue management system,
the whole forecasting component as well as the aspects of predicting volume, unconstrain-
ing bookings to compute historical demand and predicting customer behavior are offered.
Chapter 9: Conclusion 193
Implementation of a Simulation Environment: In Chapter 7, a simulation environ-
ment for revenue management that enables the decomposed view is documented. This
includes a description of simulation control as well as of the supply information and
the demand model included. Furthermore, the revenue management components imple-
mented are formally introduced. Market implementations available based on the supply
and demand model are also outlined. The simulation-based approach provides access to
information on customer behavior not available in the real world and allows for the ac-
celerated observation of long-term developments. Given this view, the problem caused by
forecasts’ influence on observed bookings that later serve as a quality benchmark can be
quantified and avoided.
Formalization of Statements on Forecast Performance Evaluation: A number of
statements on forecast performance are formally expressed in terms of the decomposed
concept in Chapter 8. This includes a long-term view of the effects of methods applied, a
discussion of alternative methods of computing forecasts from available knowledge of the
demand model, a comparison of existing key performance indicators for forecast evalua-
tion in a simulation, an analysis of the concept of uncertainty of markets and finally an
introduction to the advantages and additional features of price-sensitive forecasting.
Analysis of Simulation Experiments: Based on statements on forecast performance
evaluation, simulation experiments have been designed, conducted and documented in
Chapter 8. For example, the difficulties stemming from traditional forecast evaluation
with regard to the spiral-down effect are demonstrated. Open decisions concerning the
transformation of knowledge on the demand model are analyzed. The effects of differences
in the market place with special regard to the concept of uncertainty are demonstrated
and quantified.
9.2. Outlook
Some opportunities for future research in revenue management become obvious without
special regard to forecast evaluation. New forecasting methods taking into account flexible
Chapter 9: Conclusion 194
customer behavior based on an ever more transparent market are still required and so are
optimization algorithms to efficiently compute inventory controls that maximize revenue
based on this information.
With regard to topics of this thesis, three topics that further research may expand on
emerge. These are the decomposition concept presented in Chapter 6, the simulation
environment introduced in Chapter 7, and applications of the combination of concept and
environment to the analysis of forecast evaluation documented in Chapter 8.
First of all, as a theoretical approach, the idea of decomposing the revenue management
system to evaluate its components separately and ceteris-paribus may not only be applied
to forecasting. Optimization methods and inventory solutions can also be analyzed in this
way. The following list offers a number of problems that may be considered in a similar
way as forecasting has been analyzed so far:
•Given a fixed market and forecast, what revenue management optimization method
works best?
•Given a fixed market, how can pricing and product design be improved?
•Given certain conditions of demand, how can scheduling and flight planning be
evaluated?
•Given methods of revenue management, how can fleet assignment be designed to
use opportunities of synergy?
Secondly, on a larger scale, different parts of the airline planning problem could be modeled
as a process built from separate components that may be interlinked to different degrees.
•Given a choice behavior of crew, how can crew scheduling be processed to maximize
gain both for the employees and the airline?
•Given certain options of action and probabilities of disturbance, how can operations
handling be designed in a robust fashion?
Thirdly, apart from including more aspects of the airline planning problem, the simula-
tion environment presented as a method of realizing the concept of decomposition could
be expanded to present a more realistic model of revenue management. From the cus-
tomer cost function, which is strictly linear so far, to the inclusion of cancellations and
Chapter 9: Conclusion 195
no-shows, a whole array of improvements is conceivable. The following list offers a number
of features that may be included in future versions of such a simulation environment:
•More methods for demand forecasting (network-based forecasts, market-sensitivity,
estimation maximization, neural networks);
•more methods for revenue optimization (network-based optimization, linear and
dynamic programming, new heuristics);
•real-time dynamic planning to realize strategic goals by applying rule-based systems
in the inventory;
•expansions to the customer model: more decision factors, different functional forms
for the cost function, cancellations, group bookings;
•expansions to the time horizon: seasonal departures, updating of forecasting within
simulation runs for consecutive departures.
When the simulation environment is considered as an object of further research, the
existence of implicit assumptions within and their consequences for evaluations based
on it have to be kept in mind. For example, any aspect of customer choice that is
influential in the real world but has not been identified in the model so far may distort
the findings. Therefore, whenever additional layers are added to this system, the risk of
implicit assumptions needs to be noted.
Fourthly, there are still many open questions that may be approached empirically with
the help of a concept for decomposition based on a simulation environment. Some of these
problems could already be tackled by the existing system. The following list includes
topics that are closely related to those presented in the last chapter.
•In how far can a simulation system be calibrated to present realistic market behav-
ior?
•Can a calibrated simulation system be used to generate new boundaries for revenue
opportunity?
•Can more aspects of market uncertainty be isolated and quantified?
•Can forecasts be evaluated based on the number of dimensions of demand they
predict?
Chapter 9: Conclusion 196
Finally, a more abstract idea may be drawn from the concept and experiments pre-
sented here. Using a simulation system to evaluate the interaction of a clearly defined
set of rules with a number of entities basing their actions on a cost function is a concept
that can be applied to many other fields of research. As decribed with regard to customer
choice behavior in the context of airline revenue management, the same conditions can be
replicated and confronted with different strategies in any simulation system. The chal-
lenge is always the decomposition of the system, the efficient definition of the boundaries
of the model included in the simulation, and the formulation of effective analyses and
experiments.
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Appendix: Notation 216
Notation used in Formulas
αexp Weight of new bookings in the calculation of forecasts
based on exponential smoothing.
αPWeight of the estimator based on price in the joint esti-
mator.
αTWeight of the time-based estimator in the joint estima-
tor.
Aemsrb Function that computes protected seats based on the
EMSR-b algorithm.
a(f, c, t, s) Seats available for class con flight f, at point of time
before departure t, in simulation run s.
ˆa(f, c, t, s) Seats protected for class con flight f, at point of time
before departure t, in simulation run s.
βdep (m) Weight of the deviation from the preferred departure
time in the cost function of the customer type m.
βdur (m) Weight of the difference between actual and minimum
travel time in the cost function of the customer type m.
βcar (m) Weight factor attached to any itinerary that is not pro-
vided by the preferred carrier of customer type m.
βprice (m) Weight of the fare attached to the considered itinerary
in the cost function of the customer type m.
βtrans (m) Weight of the number of transfers included in the chosen
itinerary in the cost function of the customer type m.
b(f, c, t, s) Bookings observed for flight f, class c, between points
of time before departure t−1 and t, in simulation run
s.
ˆ
b(f, c, t, s) Average of historical bookings during the runs 1 to s
that occurred on flight fbetween points of time tand
t−1 while booking class cwas available.
bunc(f, c, t, s) Unconstrained bookings observed for flight f, class c,
between points of time before departure t−1 and t, in
simulation run s.
Appendix: Notation 217
γ(p)∈[0,1] Share of overall requests to be assigned to pairing p.
CSet of booking classes offered, ordered by descending
price.
C(i, r) Cost function of request rconsidering itinerary iwhen
a lowest available fare has been found.
ˆ
C(i, r) Cost of itinerary iconsidered by request r, without re-
gard for the actual ticket price (given the assumption
that all itineraries cost the same).
CrSet of classes acceptable according to the criteria of re-
quest r, ordered by descending price.
δdep (m) Factor for maximum acceptable deviation from wd(r),
defined by the customer type m.
δdur (m) Factor for maximum acceptable travel time, defined by
the customer type m.
δprice (m) Factor defining maximum willingness to pay for cus-
tomer type m.
δprice (r) Factor defining maximum willingness to pay for request
r.
rError term defining the distortion of request r.
sError term defining the distortion of demand in run s.
ec−c
◦(f, c, t, s);
ec−u
◦(f, c, t, s);
eu−u
◦(f, c, t, s)
Error comparing actual or unconstrained bookings to
constrained or unconstrained forecast based on method
◦per flight f, class cand run s.
ˆec−c
◦(s); ec−u
◦(s); eu−u
◦(s) Series error comparing actual or unconstrained book-
ings to constrained or unconstrained forecast based on
method ◦per run s.
FSet of flights included in the schedule.
FiSet of flights included in itinerary i.
fconst(f, c, t, s) Constrained demand predicted to arrive for flight f,
class c, between points of time before departure t−1
and t, in simulation run s.
Appendix: Notation 218
func(f, c, t, s) Unconstrained demand predicted to arrive for flight f,
class c, between points of time before departure t−1
and t, in simulation run s.
func % (f, c, t, s) Forecasted demand per class as percentage of forecasted
overall demand for the flight.
η(p, m)∈[0,1] Share of requests scheduled to arrive for pairing pbased
on customer type m.
IqSet of itineraries connecting the origin and destination
included in pairing q.
IrSet of itineraries acceptable according to the criteria of
request r
i∈IItineraries derived from the schedule.
K(f, t, s) Available capacity of the flight fthe point of time t
before departure of simulation run s.
λp,m,s Overall intensity of the Poisson process for pairing p
and customer type mthroughout the booking horizon
of simulation run s
λt,t+τ
p,m,s Intensity of the Poisson process defining the arrival pat-
tern of customer type mfor pairing pin the slice of
the booking horizon of simulation run sdefined by the
interval [t, t +τ]
lfcfs (s) Average seat load factor generated in simulation run s
given first-come-first-serve inventory controls.
l% fcfs
2-i c-max (s) Average seat load factor generated in simulation run s
as percentage of lfcfs (s).
l(s) Average seat load factor generated in simulation run s.
MSet of customer types included in the demand model.
m(r)→mFunction resulting in the customer type from which a
request rwas created.
νdist (p) Minimum distance between origin and destination air-
ports of the pairing p.
νdur (p) Minimum travel time required by pairing p
Appendix: Notation 219
o(f, c, t, s) Matrix of boolean values indicating the lowest available
class cfor flight fbetween points of time t−1 and tof
simulation run s.
Pp,m,s[N(p, m, t +τ)
−N(p, m, t) = k]
Poisson probability of krequests based on customer type
mto arrive for pairing pin time slice tto t+τof simu-
lation run s.
p(f, c) Price of a ticket for flight fin booking class c.
p(f, c, t, s) Expected marginal seat revenue for a seat in class c,
flight fat point of time tof simulation run s.
q∈ {1, ..., Np}Pairings derived from the schedule.
q(i)→pFunction resulting in the pairing for which an itinerary
iwas created.
RSet of customer requests.
RsCustomer requests created for simulation run s.
Rp,m,t
sRequests based on customer type mthat arrives for pair-
ing pup to point of time tin the booking horizon of
simulation run s.
RInput parameter defining the average number of requests
to be scheduled per run.
r(s) Overall revenue generated during simulation run s.
σDeviation of the normal distribution that the error terms
are drawn from.
σ(func (f, c, t, s)) Standard deviation of the forecast of demand for flight
fin class cat point of time tin run s.
Shybrid Hybrid market scenario.
Sproduct Product-sensitive market scenario.
Sprice Price-sensitive market scenario.
s= 1, ..., NsSimulation runs included in the simulation experiment
in chronological order with run s−1 to occur before run
s.
t= 0, ..., NtPoints of time before departure, demand arrives after
t= 0, t=Ntis the time of departure.
Appendix: Notation 220
uP(f, c, t, s) Price-based estimator for flight f, class c, and point of
time tof run s
uT(f, c, t, s) Time-based estimator for flight f, class c, and the point
of time tof run s.
uJ(f, c, t, s) Joint estimator for flight f, class c, point of time tand
run s.
VIndicator of the average demand volume per simulation
run.
wdep (r) Preferred departure time of request r.
xdep (i) Departure time of itinerary i.
xdur (i) Travel time attached to itinerary i.
xtrans (i) Number of transfers in itinerary i.
y(s) Average yield generated during simulation run s.
Ψ (M, σ, V, F) Simulation experiment based on a market structure M,
a deviation of the error term of demand σ, overall de-
mand volume V, and a forecast method F.
ωP
f,c,s Vector of price elasticity for flight fdepending on the
class cfor run s.
ωT
f,t,s Vector of time elasticity for flight fdepending on the
point of time tbefore departure of run s.
ZSet of possible restrictions of booking classes - the ab-
sence of a feature, such as comfort seating, is modeled
as a restriction.
Zclass (z, c)→ {0,1}Boolean function defining for every booking class c
whether or not it includes restriction z.
Zrequest (z, r)→ {0,1}Boolean function defining for every request rwhether or
not it accepts restriction z.
Appendix: Glossary 221
Glossary
arrival distribution Timing of requests over the booking horizon of a
flight.
authorization level Inventory control: A class is available if the book-
ings already accepted do not exceed the authoriza-
tion level.
bid price Inventory control: The fare of each class is compared
to a bid price, the class is available for sale if its fare
exceeds the bid price.
booking class Set of restrictions and features defining the condi-
tions under which a ticket is sold by a carrier; defined
by a caption.
booking horizon Period of time before the departure of a flight during
which tickets can be bought; also: reservation phase.
buy-down Phenomenon of customers buying a class that is
cheaper than the most expensive class acceptable ac-
cording to their willingness to pay.
cancellation Customers returning tickets before the departure day.
carrier Airline offering flights and tickets.
connecting time Time available between two connecting flights for
customers to transfer.
connection builder Function applied to generate pairings and itineraries
given flights and set-up parameters.
cost function Function weighting factors according to a customers
preferences, used to choose between itineraries.
Appendix: Glossary 222
cost function factor Weight of characteristics of itineraries and prices in
the customers’ cost function.
customer mix Distribution over customer types for a given pairing.
customer segmentation Concept used to enable revenue management; cus-
tomers’ are segmented according to their product re-
quirements and price acceptance.
customer type Template of customer characteristics including ar-
rival time, product requirements, willingness to pay,
and cost function.
denied boarding Customers not being able to get a seat on a booked
flight due to flaws in overbooking.
destination Airport at which a customer desires to end the jour-
ney.
EMSR-b Heuristic maximizing revenue in a flight-based rev-
enue management system given static forecasts per
flight and class.
error term Stochastic distortion of request information, nor-
mally distributed.
Exp025 Exponential smoothing method applying the weight
0.25 to new data.
Exp050 Exponential smoothing method applying the weight
0.50 to new data.
Exp075 Exponential smoothing method applying the weight
0.75 to new data.
exponential smoothing Forecast method extrapolating expected demand
from historical bookings based on a static view of
demand.
fare Price of a ticket in a booking class on a specific flight.
Appendix: Glossary 223
feature Positive condition attached to a ticket; examples can
be flexible refund or seat in the business compart-
ment.
flight Direct connection between two airports; defined by
a flight number, a carrier, a departure day and a
departure time.
flight view View of revenue management that strives to maxi-
mize revenue per flight.
itinerary Way of traveling from one origin to one destination
using one or more connecting flights.
leg Combination of two airports for which one or more
direct flights are offered.
MAD Error measurement: Mean absolute deviation.
MAPE Error measurement: Mean average percentage error.
naive forecast Forecast method extrapolating expected demand
from the bookings observed in the previous run based
on a static view of demand.
network view View of revenue management that strives to maxi-
mize revenue over a complete network.
no-frills airlines Airlines offering restriction-free classes at usually low
prices.
no-show Customers not using their ticket on the departure
day.
origin Airport from which a customer desires to start the
journey.
Appendix: Glossary 224
overbooking Technique to compensate for no-shows and cancella-
tions by selling more seats than available based on
capacity.
pairing Combination of origin and destination for which
itineraries are offered.
PODS Passenger Origin and Destination Simulator – MIT
simulation used to evaluate revenue management
strategies.
Poisson Process Stochastic process used to describe the arrival distri-
bution of customer types.
price acceptance Customers’ maximum willingness to pay.
product acceptance Customers’ requirements regarding restrictions and
features of a booking class.
protected seats Result of optimization defining the seats to be sold
exclusively in a specific class; used to calculate au-
thorization levels.
random walk See: naive forecast.
request Instance of a customer desiring to buy a ticket for a
combination of origin and destination given certain
requirements regarding product and price.
restriction Negative condition attached to a ticket; examples can
be minimum stay or a lack of features such as flexible
refund.
RMSE Error measurement: Root mean squared error.
sell-up Phenomenon of customers buying a class that is more
expensive than the cheapest class acceptable accord-
ing to their product requirements.
simulation ...
Appendix: Glossary 225
simulation experiment Simulation given specific data, methods, and set-up
parameters.
simulation scenario Data input for a simulation experiment.
traffic area Geographical area to categorize flights, legs,
itineraries, and pairings; example: continental vs. in-
tercontinental.
U2 Error measurement: also called Theil’s U2.
uncertainty Characteristic describing the volatility and thereby
the predictability of a market.