Revenue Management for a Tour Operator A Stochastic Programming Approach

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Revenue Management for a Tour Operator

A Stochastic Programming Approach

Jelmer Lichthart

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Master’s Thesis Econometrics, Operations Research and Actuarial Studies Supervisor: Prof. Dr. M.H. van der Vlerk

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Revenue Management for a Tour Operator

A Stochastic Programming Approach

Jelmer Lichthart

Abstract

In this thesis we discuss a generic problem faced by a tour operator (TO). After presenting a brief theoretical framework on revenue management, we provide a detailed description of the particular TO problem. The objective of this research is to develop a mathematical model that matches the characteristics of this underlying problem. It turns out that a mixed-integer recourse model is an appropriate candidate for this purpose. We motivate why this type of model is appropriate. Subsequently, we perform a case study which serves as a numerical illustration of the mathematical model. We demonstrate that the model is correct in the sense that it indeed determines an optimal price such that total expected future profits are maximized. However, some issues we met during this research will need more research attention.

keywords: revenue management; tour operator; stochastic programming; mixed-integer recourse model.

Groningen, May 26th 2010

Master’s Thesis Econometrics, Operations Research and Actuarial Studies Specialization: Operations Research

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Preface

By having written this master’s thesis, I complete my master’s program Econometrics, Op-erations Research and Actuarial Studies at the University of Groningen. As this thesis also marks the end of my study period in Groningen, I would like to express my sincere gratitude toward several people.

This thesis is written based on an internship at Bizztravel, a tour operator in Groningen. Therefore I would first and foremost like to thank Bizztravel for providing me the opportu-nity and inspiration for conducting this research, in particular Jacco Tas.

Secondly, I would like to thank prof. dr. Maarten van der Vlerk for his excellent guidance during my graduation project. We had quite some consultations together and his useful com-ments, interesting ideas and critical feedback substantially contributed in achieving this final result. I can say that I have learned a lot from him, and I deeply appreciate his willingness all along to respond to my queries. Furthermore, I would like to thank prof. dr. Ruud Teunter for being my co-assessor.

During my studies, I spent quite some time with Tijn Schulting and Thijs van Dongen. We studied together for exams and worked together on several assignments. I would like to thank both of you for all your cooperation.

Finally, I would like to thank my girlfriend, parents, family and friends for all their uncondi-tional support and encouragements in times of the progress of this project was not according to plan.

I hope you enjoy reading my thesis, Sincerely yours,

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1 Introduction

This Master’s Thesis is a research into Revenue Management (RM) for tour operators. Nowa-days, RM is an active and broad research area. However, in the world of tour operators, the potential of RM is only partly recognized. This inspired us to perform this research. The specific tour operator for which we performed this research is called Bizztravel.

A tour operator (TO) sells holiday packages that consist of several items. In many cases these items are air (or ground) travel, accommodation facilities and optionally, rental cars. In order to be able to offer these packages, a TO reserves capacity at several other companies. They need capacity for transportation (flight or bus seats), capacity for accommodation (hotels or apartments) and they need to reserve a number of rental cars. A TO’s product is a mix of these different types of resources.

TOs negotiate and operate as intermediary between so-called suppliers—airlines, accommo-dations and rental car companies—and customers. They recognize the common interests of both parties. The interest of a customer is that he can search for a suitable vacation package almost effortless. Therefore, packages are published in catalogs and on the Internet. On the other hand, a TO generates the possibility for suppliers to reach a wide range of customers. In general, these are all different types of leisure travelers.

Generally, resources need to be purchased in advance and cannot be resold to the suppliers later on. This implies that a TO carries a substantial financial risk. Therefore it is important to determine how much capacity is needed. This is an interesting problem on itself, but falls outside the scope of this thesis. The problem discussed in this thesis starts after capacity is purchased. Namely, it is the challenge to maximize revenue (or profit) that can be gained from a collection of resources. This can be accomplished by customer segmentation. A TO is able to serve different types of customers by offering packages that fit specific market segment. Every package has its own characteristics, adjusted to the type of customers in the market segment. Ideally, all capacity is sold to the customers that have the highest willingness to pay for their holidays. Unfortunately, for a TO, the market does not only consist of these types of customers. Lower market segments should be targeted too. This means that resources need to be allocated to different types of customers. For a TO it becomes the challenge to sell the right resource to the right customer at the right time for the right price such that total revenue is maximized. This RM problem is the focal point of this thesis.

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which reflect the decision problems of an airline. It turned out that these models were ap-propriate tools and that they substantially contributed to increases in revenue.

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2 Theoretical Framework

In this chapter we introduce Revenue Management (RM) on a conceptual level and we give a review of the literature on this concept. We begin by explaining the principle of RM in Section2.1. We describe in which circumstances RM can have potential. RM is not without difficulties. We describe some of them in Section2.2.

In Section 2.3 we discuss a number of RM related papers on a more detailed level. These papers contain some ideas that are useful for the sequel of this thesis.

2.1 Overview of Revenue Management

What is Revenue Management?

Companies selling products in order to generate profit face a number of fundamental deci-sions. Examples of such decisions are when, to whom and for which price to sell the product? If products are not sold, which decisions should a company take? RM tries to support in these decisions. But what is RM exactly? According toChen(2006), RM involves the application of quantitative techniques to improve profits by controlling the prices and availabilities of various products that are produced with scarce resources, while e.g. Williamson(1992) describes RM as the process of determining how many seats to sell at what prices and, thus, is comprised of both pricing and reservations control.

It is difficult to give a universal and comprehensive formulation or definition of RM. On the one hand, some definitions are related to airline RM while on the other hand, there exist definitions which are not necessarily airline related. However, it is beyond dispute that RM concentrates on activities involved with the sales side of a firm, as Talluri and Van Ryzin

(2004) write. They state that RM is concerned with sales decisions and the methodology and systems required to make them. It involves managing the firm’s ‘interface with the market’ as it were—with the objective to increase revenues. In this description of RM, the word market shows up. A market consists of all different types of customers and sellers. Every customer has a different willingness to pay for a product. This willingness to pay can vary over time. RM is a tool that exploits customers’ differences in preferences and their varying purchase behavior. This can be done by means of customer segmentation. It becomes the challenge to sell the right product (or resource) to the right customer (segment) at the right time for the right price.

The main mechanism that is used to segment customers is the time of purchase, see Weath-erford and Bodily(1992). Price-sensitive customers are inclined to reserve early. They are willing to trade away some flexibility for a reduced price. Less price-sensitive customers wait until the last minute and pay extra for their inflexibility.

This already implies that time is an important aspect of RM. In this perspective, a slightly different and probably more plausible definition of RM is the one given by Perishable-Asset Revenue Management (PARM), proposed by Weatherford and Bodily (1992). They define this by the optimal revenue management of perishable assets through price segmentation. PARM tries to answer two key questions:

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This alternative definition should allow for finding the optimal trade-off between average price paid and capacity utilization, which can be meaningful in many applicable industries. This definition was proposed, since the initial definition of RM was misleading. Not only yield is important, the aspect of capacity utilization is relevant too.

The classical definition of RM originates from airline industries. Below we discuss how airline RM and its definition came into existence.

Background of Revenue Management

RM originates from airline industries and started about thirty years ago. Deregulations in the United States during the late seventies of the previous century made it possible for low-cost airlines to enter the market. Up to then, prices had been strictly regulated by the Civil Avi-ation Board (CAB). From then on, the established airlines had to deal with two conflicting goals simultaneously. On the one hand they had to compete with the new entrants, which implied cheaper tickets, while on the other hand they still had to maximize revenues. A controlled and effective pricing strategy became crucial for these airlines in order to survive. Introducing low fare tickets was a way to respond on the fresh competition. However, selling only low fare tickets did not yield enough revenue. Therefore, airlines charged higher fares when the passengers would like to make use of extra facilities and services. Typically, such ad-ditional features consisted out of the type of seat, cancellation options or the time of booking. Passengers were differentiated into classes for which different prices were charged. However, as the seat capacity is fixed, the airline industries came into touch with quantity decisions as well: they had to determine the availability of the different classes, since a plane filled with only low fare passengers is not desired from a revenue point of view. The problem of control-ling the availability of the different price classes is known as the classical (airline) RM problem. The big successes that followed attracted many other industries’ attention. However, not all companies can benefit from RM. Generally, three conditions are viewed as necessary. We discuss them below.

Conditions for Revenue Management

Companies that apply RM are most likely benefited if they (see Weatherford and Bodily

(1992)):

• sell products with a perishable character; • have a fixed capacity;

• have the possibility of segmenting price-sensitive customers.

The first two mean that a fixed capacity of resources has to be sold before a certain deadline. After this deadline the product is aged or, even worse, completely worthless. Equivalently we can say that the products can only be sold during a finite horizon. Consider e.g. a seat for a theater show. At the moment the show begins, the seat cannot be sold anymore and thus is worthless. The final point in time of the (finite) horizon is basically the moment the show starts.

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2.2 Difficulties in Applying Revenue Management

When a firm is interested in RM it needs to realize that several difficulties arise. Determining how many resources to reserve for each customer segment, it would be convenient for firms to know in advance at least something about the level of demand to be anticipated. However, several problems are concerned with modeling demand. First, as suggested, the level demand is not known in advance: it is a stochastic process, depending on many factors. Second, even if this process of demand would be modeled and subsequently estimated accurately, it will be non-trivial how to deal with it.

Second, the aspect of time is also of great importance in RM. Reservations can occur all through a so-called booking horizon. This is the period of time in which bookings can occur. When a booking comes in, it is not evident whether a firm should accept or reject this booking. There exists the possibility that bookings with higher revenue will be made closer to the end date of the horizon. Of course, companies prefer the latter, but they realize at the same time that it is uncertain whether they will observe such a booking. However, it would also be naive when such possible future reservations would not be taken into account. Especially regarding that customers with a high willingness to pay are inclined to book later than customers who have a low willingness to pay. The former do prefer some level of flexibility, while the latter get a discount in exchange for a loss in flexibility.

Many papers have been published for some decades in order to research how to support in making RM decisions. Initially, the focus was on airline RM, starting with small and simple models in the early seventies of the previous century. Since then, many contributions have been made in order to make more comprehensive models. The research area was not restricted to airline companies anymore, as many other companies recognized the potential of applying RM too. Indeed, more and more industries became interested in applying RM or related activities. RM has been applied in railways or parking lot industries, see for example Rojas

(2006) orBharilla and Rangaraj(2008). Moreover, in his Ph.D. Thesis, Pak(2005) discusses applications of RM to hotel and cargo industries. In our literature review, which we present in Section2.3, we will restrict the discussion to RM in airlines, because this problem received a lot of research attention. Furthermore, the problem of a TO is quite similar to the problem of airline companies. Indeed, this research area provides some major modeling ideas that serve as inspiration and building blocks for this thesis.

2.3 Literature on Airline Revenue Management

Single Leg Revenue Management

The problem of accepting and rejecting low fare bookings such that one obtains an optimal ratio between low and high fare passengers became known as the seat inventory control prob-lem. The challenge is to find an effective control method or decision rule to reject or accept passengers’ reservations in a fare class at a specific point in time. Littlewood(1972) presents one of the first decision rules in order to control the mix between low and high fare passen-gers.

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high and the low fare level respectively. The idea is to accept low fare bookings as long as the expected revenue of selling a high fare ticket does not exceed the revenue of a low fare booking. In other words, we accept bookings in fare class l if fhP (Dh > nh) < fl holds, where

Dh denotes demand in the high fare class and P (Dh > nh) is the probability that demand is

greater than nh seats. The smallest number nh for which fhP (Dh> nh) < fl holds is known

as the protection level of the high fare class. Let C be the total number of seats on a flight. Protecting nh seats for high fare passengers is equivalent to introducing a booking limit bl

equal to C − nh for the low fare class. The booking limit bl for class l is the maximal number

of low fare bookings to be accepted.

The idea of Littlewood was the start of what became known as the Expected Marginal Seat Revenue (EMSR) problem. By comparing the revenue of a booking in a given fare class with the expected revenue of bookings in higher fare classes, it can be decided whether to accept or reject a booking.

Of course, it still should be possible to fill up the plane with high fare passengers only. If bl

and bh denote the booking limits of the low and high fare class respectively, it should hold

that C = bh> bl. This is the principle of nesting. Below, we will elaborate on this principle.

Using only two fare classes is not enough to segment all different types of customers. In order to differentiate better among customers, more fare classes are needed. Belobaba (1987) was the first to extend the idea to multiple fare classes. Belobaba assumes the existence of m fare classes for which f1 > f2 > . . . > fm holds. His goal was to determine nested booking

limits b1, . . . , bm for all m fare classes. With a nested booking limit, the capacity available

to different classes overlaps in a hierarchical manner, seeTalluri and Van Ryzin (2004). The booking limit bj is the maximum number of bookings that can be accepted in classes j and

all lower classes j + 1, . . . , m. Hence, bookings in class j will be accepted as long as the total capacity available for classes j, . . . , m is not fully occupied yet. For nested booking limits it should hold that C = b1 > b2 > . . . > bm. When there are only high fare bookings, they all

should be accepted. Indeed, the booking limit for the highest fare class is equal to the total capacity.

Belobaba suggests to use the decision rule of Littlewood recursively. He assumes that for all classes k = 2, . . . , m passengers in fare class k book before passengers in class k − 1. Now we will explain the principle of this decision rule. Assume that fare class j + 1 is open, meaning that bookings in the fare classes j + 1, j, . . . , 1 are accepted. The goal is to determine nested booking limits for the remaining fare classes j, . . . , 1. For this purpose, Belobaba determines the protection levels for all classes j, . . . , 1. First, he considers some fare class k out of the fare classes j, . . . , 1 and he compares class k with j + 1 separately. He applies the decision rule of Littlewood onto these two classes, that is, he determines the smallest nj+1_k such that

fkP (Dk> nj+1_k ) ≤ fj+1,

which is equivalent to

nj+1_k = F−1(1 − fj+1 fk

).

Here, Dkis demand in class k and P (Dk> nj+1_k ) is the probability that more than nj+1_k seats

will be sold in class k and F−1(nk) is the inverse of the cumulative distribution function of

de-mand F (nk) = P (Dk≤ nj+1k ). By definition, F −1

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Then the protection level of class j is given by nj =Pj_k=1nj+1_k . The nested booking limits

are defined by bj = C − nj, j = 1, . . . , m.

This method is known as the EMSR-a method. This method differs slightly from the EMSR-b method. In the latter, nested booking limits are derived by first determining combined pro-tection levels. This is done by applying Littlewood’s rule to aggregated demand and weighted averages of fare classes. Neither of the two methods do give theoretical optimal nested book-ing limits. We refer toTalluri and Van Ryzin(2004) for a proof.

In his paper, Belobaba assumes that • low fare passengers book first, • no cancellations or no-shows occur, • no batch bookings occur and

• demand is independent between the booking classes.

In the decades after these initial airline RM models, many papers have been published in which one or more of these assumptions are relaxed. For an extensive overview on this sub-ject, we refer the interested reader toPak and Piersma (2002).

Network Revenue Management

Rather than optimizing revenues over all flight legs separately, airlines are more benefited by optimizing revenues over a network of flights that are offered by the airline simultaneously. Many papers that focus on this so-called network seat inventory control problem have been published. The paper by De Boer, Freling, and Piersma (2002) is of special interest to us. In this paper a Stochastic Linear Program (SLP) is formulated. The idea is to formulate an objective function that consists out of two separate parts. The first part defined as the revenue that would be gained if all planes would depart completely loaded. Since there exists the possibility that demand is lower than capacity, one has to correct for this in the objective function. They model that empty seats can be observed with positive probabilities. The reductions in revenues are modeled by means of so-called recourse variables.

2.4 Literature on Stochastic Programming

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In stochastic programming (SP), recourse models are the most important class of models as

Klein Haneveld and Van der Vlerk (2007) point out. A lot of theory and algorithms have been developed relevant to this field during the last few decades. Moreover, recourse models are frequently used in other application as well. Examples of applications can be found in

Wallace and Ziemba(2005).

For an introduction into SP theory, we refer the interested reader to text books like Kall and Wallace (1994),Birge and Louveaux(2000) or the lecture notes by Klein Haneveld and Van der Vlerk (2007) or Shapiro and Ruszczynski (2009). Furthermore, we would like to mention that in the bibliographyVan der Vlerk(2007) one can find a comprehensive overview of publications related to SP. For a comprehensive overview of SP and related information we refer to the web page of COSP (2009). On this web page, one can find lectures and introductory texts on SP. Information on SP events such as conferences is published here too.

2.5 Demand

Estimating and modeling the stochastic process of demand is one of the main issues in RM. RM models require an estimate of future demand, prior to price or sales related decisions can be made. Obviously, demand is uncertain. No firm has perfect insights into the future. Fur-thermore, demand varies over time. This is due to e.g. (uncertain) seasonal trends, trends and uncertainties in economics, holidays, external shocks, competition and so on. This so-called multi-dimensional character of demand is a necessary condition to apply RM, according to

Talluri and Van Ryzin(2004).

All kinds of difficulties are involved in estimating the distribution of demand. Generally, one has to make assumptions about the demand behavior. According toPak (2005), when concentrating on airline RM it is common in this procedure to assume that

• no cancellations, no-shows or overbooking occur, • no group bookings occur and

• demand is independent of the booking control policy used.

The first step in order to predict demand is to investigate historical data. By means of sta-tistical tools one should estimate demand. Then by looking at for example seasonal trends, economy forecasts and other predicted future events, one has to forecast demand. Some fu-ture events cannot be predicted based on historical data.

Analyzing historical data should not be confused with forecasting. Of course, historical book-ing patterns are not the same as future demand. However, since it is very hard to get a grip on future demand, historical data is used in practice to support in constructing estimates of future demand.

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3 Problem Description and Approach

This chapter is organized as follows. In Section3.1 we state our research question based on an initial description of the generic problem discussed in this research. Because the problem is rather complex, we should determine an appropriate scope of our research. Section 3.2 discusses certain restrictions and simplifying assumptions in order to accomplish this. Finally, we describe our approach in Section3.3. In this description we motivate why a recourse model is an appropriate model. Furthermore, we explain how we model future uncertainty to be incorporated in our mathematical model.

3.1 Formulating Research Question

As mentioned, the inspiration for this thesis comes from Bizztravel, a TO in Groningen. The objective of this research is to develop a generic model related to a generic problem faced by a TO. Before we develop a model, we first need to understand the main issues of the problem. We describe them in this section. Based on an initial description, we formulate our research question. Subsequently, we elaborate on the problem by discussing some additional features which are important for our research as well.

TOs sell all types of holiday packages. These packages consist of transportation and an ac-commodation. For this purpose, a TO reserves a number of seats on two flights, in order to transport passengers, and a number of rooms at several accommodations, in order to lodge the passengers. A TO carries the financial risks of the purchased resources, as it is uncertain how much it will earn on its packages. Holiday packages are worthless when the flight has departed to its destination. Since a TO purchases capacity for many destinations with mul-tiple departure dates in the season, the total financial risk is substantial.

Once the total capacity is determined, a TO has the objective to maximize revenue that can be generated from it. Revenue is defined by price multiplied by volume, where volume depends on price. Generally, a lower price implies a higher volume and vice versa. Therefore, the central question becomes how to adjust price over time to generate a maximal amount of revenue. Below we elaborate on this fundamental principle by describing in more detail the decisions to be taken by a TO.

At the start of the bookings period, a TO determines all its prices. As the season evolves, a TO observes mutations in its inventories. When a TO conjectures that they run out of inventories too fast or too slow, it needs to take measures. The first option is to adjust prices. If a TO runs out of inventory too slow, it is plausible that it decreases its prices in order to stimulate demand. Indeed, a TO tries to skim the market in general. First, it asks high prices to sell a part of its resources to customers who have the highest willingness to pay. Subsequently, it tries to sell its resources to customers who have a second highest willingness to pay and so on. Therefore, the prices tend to decrease over time. If a TO conjectures that it runs out of inventories too fast, a second option is to buy additional capacity. Of course, the airline and accommodation owners should be willing to cooperate in that case.

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evaluates its inventories again. If they are not satisfied with the size of the remaining capacity, it decides what to do next. An interplay between decisions (price adjustments) and demand, dependent on price, arises.

As becomes clear, the problem is dynamic, since a TO takes decisions at multiple points in time. Successive decisions are dependent on each other in general, as an iterative process arises of adjusting prices and observing demand. This mechanism continues until the final date of the booking period.

As mentioned, determining or adjusting a price does not yield revenue immediately. Bookings as a result from this new price level do contribute to revenue. Indeed, a TO has to make decisions ‘now’ such that future revenues, which result from future bookings, are maximized. Based on this initial problem description, we state our research question. The question this research tries to answer is:

How to model initial price related decisions such that total expected revenues resulting from all future bookings are maximized?

A TO needs to take initial price related decisions, decisions that need to be taken ‘now’, fac-ing an uncertain future. Moreover, the future depends on initial decisions. Hence, we should somehow model the dependence between initial price related decisions and their future con-sequences.

In attempting to come up with an answer, we address several sub-questions: • How are price related decisions taken currently?

• Which type of mathematical model is appropriate for modeling the problem discussed in this thesis?

• How should future uncertainty be modeled?

• How should the dependence between price (decision) and demand (observation) be mod-eled?

In Section3.3we answer these sub-questions in the above-mentioned order. First, we proceed by elaborating on the problem. We discuss several other characteristics of the problem that are relevant to our research (question) as well.

The main objective of a TO is to maximize profit. Maximizing revenue can support in ac-complishing this, since price is an important tool in this perspective. Furthermore, contracts with airlines and accommodations need to be paid. A TO runs into liquidity problems and can go bankrupt when it does not generate enough revenue in time. If a TO is not able to pay the contracts with the suppliers, there is the possibility that the suppliers are not willing to cooperate anymore during next seasons. This is disastrous for a TO, since it needs suppliers in order to be able to offer products.

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• the destination;

• the (type of) accommodation; • the date of departure;

• the remaining time to departure;

• the demand to be anticipated in the remainder of the booking horizon; • the length of stay;

• the type of the room; • the capacity of the room;

• the occupation rate of the room;

• and the composition of the group of travelers.

It is extremely difficult to determine the right price for all different products such that total revenues are maximized.

Indeed, the demand for these products quite likely is correlated to each other. If an accom-modation is expensive, customers might be inclined to search for a cheaper alternative. Or, if a customer is looking for a certain destination and he is disappointed by the corresponding accommodations, he or she might search for a completely different destination.

A TO has to deal with customer behavior. One can imagine that customers are inclined to wait for lower prices. Running the risk that their product is not available anymore, they might be rewarded for this behavior, as the prices of TOs tend to decrease over time. This illustrates that demand depends on time.

Rationally, a customer will book at a TO which offers him or her the desired package for the lowest price. It is obvious that the competition is crucial too. Demand depends also on the prices of their competitors, who often offer the same or equivalent products. A customer is able to compare these packages on the Internet at low search costs.

Demand is likely to be low for departure dates in the off-season. In order to stimulate de-mand for these departure dates, a TO is inclined to drop its prices drastically. As a possible consequence, the selling price can become lower than the purchase price, i.e., the profit can become negative. The reason to do so is that a TO has already paid for its resources. Empty seats and rooms are missed opportunities to generate revenue. Selling under cost price is necessary then, since generating some revenue is preferable over generating no revenue at all. One can argue that a TO should not sell packages for off-season departure dates, since this has negative impacts on profit. Unfortunately, this is impossible. Owners of accommodations are not interested in letting out their rooms only during the high season. They are willing to cooperate, only if a TO purchases capacity for off-season dates as well. It is thus the challenge to maximize revenue taking into account that demand for peak season dates is high and demand for off-season dates is low. The profits of the high season should compensate the off-season dates’ losses.

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instrument in this perspective. The profit of a package is defined by the selling price minus the purchase price of a package. A booking in a very expensive hotel is attractive from a revenue point of view. However, a booking in a cheaper hotel does not imply that the profit is lower. From a profits point of view, these bookings could be different, e.g. the cheaper one may actually be more lucrative.

In determining its prices, a TO has to take into account so-called mismatch, which is called scheefstand in Dutch. For the first departure dates of the high season, many flight seats from the Netherlands to the destinations are occupied, while many seats on the flights back to the Netherlands will be empty for this period. This is due to the fact that there are almost no passengers that travel back at the start of the peak season. On the other hand, generally there are almost no passengers that travel from the Netherlands to the destinations at the final dates of the high season. Then all flights back are fully occupied.

As becomes clear, the world of a TO is rather complex. In this thesis we cannot take into account all aspects that are involved. We should restrict the scope of our research. Therefore, we have to make simplifying assumptions and impose several restrictions while keeping a relevant level of detail in our research. This is the topic in Section3.2.

3.2 Limiting the Scope and the Level of Detail of the Research

We should limit the scope and level of detail of our research. First, because some issues which are important for a TO, are related to different types of problems. They fall outside the scope of this research. Second, some assumptions are needed in order to simplify the eventual model. In this section we describe which restrictions and assumptions we impose on this research.

Separability Assumptions

Because of the dependence of demand among different packages, we should actually consider all of them in one model. However, the model will become too large then. Therefore, we have to make strong assumptions that enable the model to be separated into multiple smaller models. To accomplish this, we assume that demand:

• for different departure dates is independent; • for different destinations is independent; • for different accommodations is independent.

Indeed, these are very strong assumptions. However, we have good reasons to make them. The first reason is that this research does not focus on conducting comprehensive analyses of demand. This potentially is something for future research. We are mainly interested in developing a generic model in which (initial) price related decisions, to be taken under future uncertainty, are modeled.

Second, the level of model detail should be consistent with the type of data available, asLaw

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collection, if possible, may be quite expensive and does not guarantee substantial better in-sights or decisions.

Finally, it will turn out in Chapter 6 that, even with these strong assumptions, the time required to solve a model instance to optimality is still rather large. An optimal solution is generally not found within several hours.

These so-called separability assumptions enable us, in developing a mathematical model, to focus on one specific departure date, one destination and one specific accommodation and to subsequently perform numerical experiments. A necessary condition for separability though is that a package always consists of only two flights: one flight to the destination and a flight back. A package cannot be a combination of two different destinations. Then the demand between destinations is not independent anymore.

Using Benchmark Prices

The prices within an accommodation can vary a lot, depending on for example the type of room, the size of the room, the occupation rate of the room and so on. Additional booking options are more expensive by some fixed amount compared to the so-called benchmark price, which is the price of e.g. a room for two persons (fully occupied), based on a holiday of eight days. We assume that, if this price is adjusted, all other prices are adjusted accordingly. Buying Additional Flight Capacity

A TO has the possibility to buy additional capacity during the booking horizon. We do not consider these type of decisions in designing the model. Capacity decisions are decisions related to supply management, which can be interpreted as the complement of applying RM according to Talluri and Van Ryzin(2004). This is an interesting problem on itself, but falls outside the scope of this thesis.

Modeling Demand

Modeling the stochastic process of demand is an important issue in RM. As the objective of this thesis is to develop a decision model, rather than developing a descriptive or predictive model of demand, this will not be our focal point. This is a research area on itself. We assume that demand distributions are known in advance. Still, some distributions have been constructed in order to perform numerical experiments. We elaborate on this in Chapter 5.

3.3 Approach

In this section we try to answer the sub-questions that were stated in Section3.1.

3.3.1 Current Situation

Conjecturing that current practice is more or less the same for all TOs, we describe the situ-ation at Bizztravel as observed during an internship.

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will result in substantial extra bookings.’. Then, after a small period of time this decision is evaluated and a new decision might be proposed. This procedure iterates until the last day before the date of departure, i.e., the final day of the booking horizon.

Stock mutations are displayed in a so-called flight utilization list. By examining this overview, Bizztravel tries to adjust the relevant prices in an attempt to stimulate demand. Historical booking patterns, which are stored in a central registration system called Dynatravel, or demand forecasts are left out of consideration during this process. Hopefully, this research contributes in improving on this myopic view.

3.3.2 Model Selection

We completed the problem description of this research in the previous section. One important characteristic of the problem is that decisions are taken under future uncertainty. Further-more, the problem is dynamic: (price related) decisions have to be taken at multiple points in time. Between successive decisions we may observe random demand that depends not only on time, but also on price. Therefore, we should model the dependence between price related decisions and demand carefully.

Not only price related decisions are important. Volume is important too, as revenue is de-fined by price multiplied by its corresponding volume. Moreover, a TO would like to track its performance as the season evolves.

In this research we use a multistage recourse model. Below we motivate why this type of model is appropriate for our study.

Uncertainty concerning certain model parameters is an important feature of the problem. This uncertainty can be incorporated in a recourse model by assuming that these parameters can be modeled as random variables with known distributions. Indeed, the idea is to model price related decisions that need to be taken ‘now’, having only probabilistic information on uncertain parameters. A recourse model is appropriate to model these so-called here-and-now decision problems, which is a class of a decision problems that model optimal decisions to be taken prior to a realization of some random variable such that expected total future ‘costs’ are maximized.

Recourse models are mathematical models which are pre-eminently appropriate to model the impact of future observations and decisions carefully. We realize that initial decisions, deci-sions to be taken ‘now’, should be modeled. However, by modeling future observations and future decisions too, initial decisions can be evaluated by calculating their associated expected future ‘costs’.

An essential limitation of standard recourse models is the assumption that decisions do not influence the distributions of the uncertain parameters. This is however in contradiction with the theory that demand (observation) is dependent on price (decision): the lower the price, the higher demand will be. Obviously, this characteristic should be modeled. For this purpose we made up a modeling trick to deal with it: the idea is to ‘manipulate’ the observed distribu-tion of demand by means of binary variables. We will explain this in more detail in Chapter4.

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the mathematical model. A so-called two-stage recourse model is not capable to capture the time-dynamics of demand. We overcome this by introducing multiple stages.

By definition, revenue is price multiplied by its corresponding volume. Therefore, we should take quantity decisions into account as well. These are sales corresponding to a certain price level at a certain point in time.

Rather than maximizing sales at any particular moment in time, a TO would like to max-imize revenue over the entire booking horizon. Capacity that eventually remains unsold in this process comes at a cost. Therefore, instead of waiting until the final date of the horizon, when it is too late to take corrective actions, it may be sensible to set targets at certain points in time and to steer on them.

Recourse variables are appropriate modeling tools for this purpose. In applications, they are frequently used to model deviations from targets to be steered on. Considering the nature of the problem in this research, we e.g. can specify targets in terms of sales. Deviations from targets are then penalized according to a piecewise linear convex penalty function modeled in terms of recourse variables v− and v+, representing respectively sales shortages and sales surpluses with respect to certain targets.

Shortages and surpluses can also be penalized according to a convex piecewise linear penalty function. This is called multiple simple recourse (MSR). If for example the targets are defined as ranges, a MSR structure is needed. We refer the interested reader toKlein Haneveld(1986) andVan der Vlerk (2005) for a discussion on MSR.

A multistage recourse model is not the only candidate to deal with the problem. One can think of e.g. a simulation study, which could be appropriate too. Below we explain why we do not select this type of study.

Recall that we aim at modeling decisions that optimize a certain objective function. Fol-lowing Law (2007), if it is possible to develop an analytical model, then this will generally be preferable to a simulation model. Namely, a (stochastic) simulation model only produces estimates of a model’s true characteristics for one particular set of input parameters. Mul-tiple independent runs are necessary for each set of input parameters in order to determine optimality. Generally, this is rather difficult too. Furthermore, one has to reckon with the stochastic nature of the output produced by the different simulation runs.

The advantage of using a multistage recourse model, is that we can determine an optimal (or approximate) solution for a particular model instance. We can calculate the model instance’s characteristics exactly. We do not have to analyze stochastic output by means of statistics. Moreover, by using a recourse model we only need a single run to determine an optimal so-lution for a given set of input parameters.

As mentioned, a multistage recourse model is a mathematical model in which future decisions are modeled dependent on future observations. These future decisions correspond to certain points in time. These points in time are referred to as stages. Between two successive decisions we may observe a realization of some random parameter. Decisions at stage t are taken dependent on the decisions at xt−1 and on the random observation(s) at stage t. In

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t ∈ {2, . . . , T } we take decisions xt(xt−1; ξt), depending on the (t − 1)th-stage decisions xt−1

and the observation of ξt.

This implies that every (future) decision xt is made dependent on the information that is

available up to time t having probabilistic knowledge about the remainder of the horizon. First-stage decisions x1are taken having only probabilistic information on future observations.

Figure 1: The underlying dynamics of multistage recourse models In Section3.3.3we will explicate how we model future uncertainty in this thesis.

3.3.3 Modeling Future Uncertainty

In this thesis we model uncertain parameters as random variables. We assume in addition that the distributions of these random variables are known. This is an assumption common to all stochastic programming models.

Following Van der Vlerk(2003), we discretize the booking horizon into stages , so that the model has a finite number of time periods. These time periods do not necessarily have equal length. Every period t reflects the time span [t − 1, t). At the beginning of each period, i.e. at stage t − 1, (optimal) decisions xt−1 are to be taken. During the time span [t − 1, t) that

elapses, a realization of some random parameter ξt is revealed. Subsequently, decisions at

stage t are to be taken, knowing the realization of the uncertain parameter corresponding to the time span [t − 1, t) and only having probabilistic information on the future observations to come in the remainder of the horizon. In such a way, the interplay between decisions and observations, as depicted in Figure1, arises.

Let us consider a random (demand) process ξ2, . . . , ξT, where the subscript refers to stage.

When all random variables ξt, t ∈ {2, . . . , T } are modeled as discrete random variables with a

finite number of realizations, the total number of possible sequences of realizations ξ₂s, . . . , ξ_Ts of the random process is finite too. Every possible sequence constitutes a so-called future scenario. These scenarios can be depicted combined in a so-called scenario tree. It consists of nodes, denoted by (t, n), which correspond to the stage t and the realizations of our random (demand) process. Conventionally, at every stage t, the index n of node (t, n) is ordered top-down. In Figure 2 we depict a scenario tree with four stages, twenty-two nodes and eleven scenarios.

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realization. Subsequently, at the third stage, the scenario tree has as many nodes as the number of realizations of ξn₃, where n refers to the index of node (2, n) and so on. As suggested, the number of realizations of ξn₃ is dependent on the node (2, n).

Figure 2: Scenario tree with four stages, twenty-two nodes and eleven scenarios A scenario can be seen as a path from the root node to an end node, a so-called leaf of the scenario tree. Every scenario tree consists of S unique future scenarios. Let S := {1, . . . , S} be the collection of scenarios. To every scenario s ∈ S there is associated some positive probability of occurrence, denoted by ˜ps. Obviously, P

s∈Sp˜s= 1.

Rather than considering a scenario tree, the scenarios can also be considered separately, as displayed in Figure3. Then, for all s ∈ S optimal decisions xs _{= (x}s

1, . . . , xsT) are to be taken.

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4 Mathematical Model

In this chapter we translate the problem discussed in Chapter3into a mathematical model. First, in Section4.1, we give a verbal description of the model. In this section the reader can find which modeling choices are made and why we made these choices. Second, in Section4.2 we give a mathematical formulation of the model.

Subsequently, we briefly explain which recourse structure the model has in Section4.3. Fi-nally, we know from SP literature that there exist mathematical models that are closely related to recourse models. We discuss them in the final section of this chapter.

The model should be used by means of a rolling horizon approach. Actually, the only decision that is relevant for a TO is the first-stage decision z0 indicating which price related decision

should be taken. This decision is to be implemented. Subsequently, at time t = 1, a decision can be reevaluated and a new initial decision will be determined. Of course, the final date of the booking horizon should be kept fixed in this approach. This is however not the case in a rolling horizon approach in general.

4.1 Verbal Description

This section is devoted to explaining the model on a conceptual level. In Section 4.1.1 we discuss how we model the objective function of our model. In Section 4.1.2 we explicate how we model the relevant issues described in Chapter3. Subsequently, in Section 4.1.3we describe which model parameters are uncertain. Furthermore, we define some of the model variables. Finally, in Section4.1.4, we elaborate on how we model certain constraints.

4.1.1 Objective Function

The mathematical objective function of a TO, to be formulated based on the objective of a TO that it has in practice, consists of several parts. First, a TO has the objective to maximize profits. An additional objective of a TO is that it aims at tracking its performances as the season evolves and take corrective actions in time. Therefore, it might be sensible to model targets corresponding to certain points in time and take them into account in the objective. Third, a TO can possibly be interested in maximizing revenues rather than maximizing profits. A TO faces the problem to maximize total profits over a finite booking horizon. The revenue should be generated from a fixed (and perishable) capacity. Since a TO does not know in advance how many profit it will generate, it is reasonable to model decisions that maximize in terms of expected total future profits. Indeed, determining a price ‘now’ does not generate profit immediately. Only future bookings that will be accepted do this.

The most important tool for a TO in order to accomplish this objective, is that it can vary its prices over time. Indeed, a TO periodically adjusts its price in order to maximize profits. Of course, prices are not the same as profits, but please do note that there is a close relation between the two. We elaborate on this in the upcoming sections.

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to steer on them.

We model this additional aspect of the objective function by introducing recourse variables v_t− and v+_t measuring respectively sales shortages and sales surpluses with respect to targets at certain points in time. Sales shortages are not desired and therefore come at a cost. Sales surpluses are desired—if at a desired price level—and are rewarded. The larger the shortage or surplus, the bigger the penalty or reward should be. This can be modeled by using a simple recourse structure, in which sales shortages (or surpluses) are penalized (or rewarded) according to a piecewise linear convex penalty function.

Rather than maximizing profits, a TO might be more interested in maximizing revenues. By definition, the profit π of a package is defined by its selling price p minus its purchase price cp, i.e., π := p − cp. The objective function can easily be reformulated in that case.

When customers are obliged to pay immediately after they booked, a TO has the opportu-nity to deposit the money on a bank account. The interest that accumulates is an additional source of ‘revenue’ and it might be sensible to model this.

One could argue that maximizing total revenue is equivalent to maximizing total profits. Af-ter all, the profit is defined by total revenue minus total purchasing costs. The latAf-ter is a fixed amount, determined prior to the start of the booking horizon, and thus the difference between the two objective values consists of a constant only.

Indeed, this holds true if we consider all accommodations separately. However, in our numer-ical experiment we aggregate accommodations and assign aggregated demand among classes with the same profit parameters, which are not necessarily the same price parameters. Be-cause every package has its own purchase price, total revenue will not be the same as total profit anymore.

We will be talking about price related decisions in the sequel of this thesis, as this agrees most with the intuition of the reader.

4.1.2 Decisions Pricing Decisions

In order to accomplish in the above-mentioned objectives, a TO has to make a vast number of decisions. One important, if not most important, decision is related to price. At the start of the booking period, initial prices have to be determined. Subsequently, the prices have to be adjusted as the season evolves. These price adjustments have to be significant, because generally their purpose is to stimulate demand. Customers will not react on say a one percent price decrease. However, a five percent price decrease is probably enough to persuade customers to book.

We model price related decisions as follows. We introduce classes i ∈ I which can be opened and closed over time by means of binary variables: if xit= 1, it implies that class i is opened

at time t; if xit = 0, it means that class i is closed at time t. There can be opened at most

one price class at every t. To every class i there is associated some price parameter pit. If for

example xit= 1, it means that the price will be equal to the corresponding parameter pit.

Indeed, price parameters are pre-specified on a so-called price grid denoted by pit, i ∈ I, t ∈

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our multistage recourse model can be described by: • open a price class at some point in time;

• observing random demand, which should be dependent on the price parameter of the opened class, during a subsequent period;

• depending on this observation, react by opening a new class if necessary.

This is an iterative procedure that continues until the final date of the booking horizon. Mak-ing use of classes, rather than continuous price variables, prevents that the objective function becomes bilinear and thus non-convex. Fixing the prices on a grid implies that price itself is not a decision variable anymore.

However, binary variables come at a large cost. They cause solvability issues if we would like to do calculations. Hence, we should have a rather good reason to make use of them and we have. The big advantage is that they can be used to influence (actually manipulate) the observed distribution of demand. Indeed, it is essential that the model we develop allows for this, as demand depends on price.

Volume Decisions

Opening a price class, i.e. determining a price, does not contribute to revenues directly. Future sales, to be captured in opened classes, do generate revenues. Hence, sales should be modeled as well. Without loss of generality, we model sales by continuous variables (rather than integer variables) since this yields very reasonable approximations for the level of sales. After all, our focal point is on modeling price related decisions. It is obvious that sales are non-negative variables.

4.1.3 Stochastic Parameters and Definitions Demand

Of course, demand is a stochastic parameter. We model demand as discrete random variables with finitely many mass points.

An essential characteristic of the problem discussed in this research is that demand is depen-dent on price. Standard recourse models do not allow for dependence between observations and their preceding decisions. We made up a modeling trick to model that demand is depen-dent on price. The idea of the trick can be best illustrated by means of an example, which we present below.

Consider Table 1. Let p denote the possible prices of a package. Let η(p) be defined as the number of customers who are willing to pay at most ep. In this example, there are 6 cus-tomers that pay at moste700, 20 customers that pay at most e650 and 14 customers that pay at moste600.

Customers who are willing to pay at moste700 are also willing to pay any price lower than e700, i.e., e650 or e600 in this example. Analogous, customers who are willing to pay at most e650 are also willing to pay e600. Therefore, cumulative demand is defined by D(p) :=P

ˆ

p≥pη(ˆp). The third column in Table1 displays the values of demand D(p) for this

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revenue R(600) is equal to p × D(p) =e24,000.

p (e) η(p) D(p) R(p) (e)

600 14 40 24,000

650 20 26 16,900

700 6 6 4,200

Table 1: Values of p, η(p), D(p) and R(p) for numerical example

Actually, the example above assumes a deterministic setting. However, it perfectly illustrates the underlying principle to be used in the mathematical model. In the sections that follow we will elaborate on this modeling trick. Hopefully, it is clear by now that binary variables are pre-eminently appropriate to model that demand D(p) is dependent on price p. Considering this numerical example, we need to define three classes: each of them corresponding to one of the values of price parameter p with a corresponding level of cumulative demand equal to D(p). Cancellations

Next to sales, cancellations also imply capacity mutations. If cancellations or no-shows occur often, it might be sensible to take them into account. Generally, cancellations occur close to the date of departure and the profit of the package is relatively low then. When a TO has to resell these packages in a short period of time, it in fact loses revenue.

We model cancellations as random variables, since they are uncertain parameters. Price of Competitors

As far as we know, there are few RM decision models in which the competition is explicitly modeled. This research distinguishes itself by modeling prices of competitors. Because a TOs’ market is rather transparent and competitive, we think it is reasonable to model decisions that reflect this aspect.

As we explained in Section 3.1, there are multiple TOs that offer the same or comparable packages. A rational customer will always buy his or her desired package at the TO which charges the lowest price for its products. Indeed, an important determinant for customers is the level of the price. Therefore, the price of a package should be such that it does exceed the lowest price for the equivalent package of the competition by at most some small fraction αt.

We denote the lowest price of the competition at time t by pct. Since all parameters pct are

uncertain, they are modeled as random variables. Model Variables

Some variables are defined in terms of parameters and decision variables. These are so-called model variables. At every stage t, given demand Dit, we can keep track of for instance sales

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4.1.4 Constraints

In this section we explain how we model the restrictions of the model. Capacity

Capacity can be modeled in two ways. There is the possibility to model it as a so-called hard constraint, or it can be modeled as a soft constraint. Below we discuss both alternatives in the mentioned order.

The total number of passengers that eventually book for a departure date and an accommo-dation cannot exceed the corresponding initial capacity. The initial capacity is defined by C0 := min{Cseats, Cbeds}, where Cseats and Cbeds are the initial capacities of flight seats and

beds respectively.

Obviously, capacity varies over time. Define Bt := P_i∈Iyit− ant, the actual sales at stage

t. The available capacity at time t defined by Ct := C0 −Pτ <tBτ. Conceptually, capacity

at time t is defined by initial capacity minus the total level of actual sales up to time t. At the first and the second stage, capacity is equal to initial capacity, since no sales have been observed then.

In practice, multiple accommodations are used to fill a single flight. Therefore, rather than a hard constraint, we can model capacity as a soft constraint. This constraint should then be interpreted as follows. Suppose a TO reserves a number of seats for customers within a certain collection of accommodations, anticipating that the remaining flight capacity will be sold to packages which contain different accommodations. If demand for the specific collec-tion of accommodacollec-tions exceeds this number, a TO can decide to allocate some more seats to these accommodations in order to be sure that at least these flight seats are sold. However, then there are less resources for other packages left of course.

By means of a multiple simple recourse (MSR) structure, with appropriate cost parameters, we can model capacity as a soft constraint: sales shortages with respect to the lower bound of a capacity range are penalized and sales surpluses with respect to the upper bound of a range are penalized as well, meaning that selling some more than the reserved capacity cor-responding to a collection of accommodations can be accepted to a certain extent.

Sales

Sales are constrained by several factors. Sales at stage t cannot exceed capacity and demand at stage t, since a TO cannot sell more than demand or available resources. Furthermore, sales at stage t can only be captured in the price class that is opened at stage t − 1.

Competition

The model we develop is able to incorporate anticipated future prices of competitors. By taking into account these prices in the decisions to be taken, the model distinguishes itself from other RM decision models as far as we know. Below, we will explain how decisions are affected by this additional feature.

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a TO observes that its package contains some additional features that are not offered by the competition, it can charge an additional amount for this product differentiation modeled by 1t. Combining, for all stages t at which a price class needs to be opened, it should hold that

pitxit≤ (1 + αt)pct+ 1t. Obviously, if xit= 0 for some class i and some stage t (meaning that

the price level is not equal to the corresponding price pit), this constraint holds. Furthermore,

at every stage t, only the classes i for which pit≤ (1+αt)pct+1tholds, are allowed to be open.

Of course, we should prevent the situation that no class can be opened at all. Otherwise, there will exists no feasible solution(s) then. At the risk of laboring the obvious, let us therefore mention that the mass points of the distribution of pct should be greater or equal than the

value of the price parameter pit corresponding to the lowest available class at stage t, if we at

least assume that αt= 1t= 0.

The reason to incorporate anticipated future prices of competitors is that we would like to model that demand is dependent on other prices in the market. Indeed, it is plausible that the level of a TO’s demand decreases if it does not react on price decreases of its rivals. After all, customers will in general book at the TO which charges the lowest price.

Furthermore, if a TO has decreased its price, it is possible that there are customers that actually have a higher willingness to pay. Because it is plausible that at least a part of them will book at other TOs, only a fraction of demand (which is captured in higher price classes) will be observed.

Let us consider the parameter 2t, necessary to model the sensitivity of demand upon the

prices in the market of other TOs. We define the fraction witof demand we observe at stage t

in class i by wit:= min{pct_p+_it2t, 1}. Please note that 2t< 0 is allowed to model that demand

is rather sensitive for price decreases of competitors.

4.2 Mathematical Formulation

Now we are ready to present a mathematical formulation of the model. Sets

L := {1, . . . , L} set of departure dates A := {1, . . . , A} set of accommodations

I := {1, . . . , I} set of price classes in ascending order S := {1, . . . , S} set of scenarios

T := {1, . . . , T } set of stages

T1 := {1, . . . , T − 1} subset of stages at which classes need to be opened

T2 := {2, . . . , T } subset of stages at which sales and deviations

with respect to targets are modeled Indices

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Please note that there is no set that corresponds to destination. This set actually is redun-dant, since we already make a distinction in accommodations: every accommodation a ∈ A is associated with a single destination.

For convenience, we drop the indices of the sets L, A and S, which correspond to the departure date,the accommodations and the scenarios respectively. Recall that we assume independent demand between different departure dates and between different accommodations, so that we may indeed restrict the model to a single accommodation and date. For all other departure dates and accommodations, the mathematical model representation is equivalent.

Decision Variables

xit binary variable indicating whether a price class is open (xit= 1) or closed (xit= 0)

in period t + 1; xit∈ {0, 1}, i ∈ I, t ∈ T1

yit continuous variable modeling sales in class i in period t; i ∈ I, t ∈ T2

v_t+ continuous variable modeling sales surpluses in period t; t ∈ T2

v_t− continuous variable modeling sales shortages in period t; t ∈ T2

Deterministic Parameters C0 initial capacity

pit price in class i in period t, pit< pi+1,t;i ∈ I, t ∈ T2

λt interest to be accumulated over the period T − t; t ∈ T2

cp cost of the package

πit profit in class i in period t, πit:= pit− cp, πit < πi+1,t; i ∈ I, t ∈ T2

1t tolerance parameter modeling additional amount for packages in period t

compared to competitors; t ∈ T2

2t tolerance parameter for modeling the dependence between demand in period t

and (the price of) competitors; t ∈ T2

αt fraction indicating the maximal deviation of the price of a TO

compared to the competition’s prices in period t; t ∈ T2

gt goal on level of sales in period t; t ∈ T2

Gt cumulative goal on level of sales up to and including period t, Gt=P_{τ ≤t}gτ; t ∈ T2

q+_t reward parameter for sales surplus v+_t in period t; t ∈ T2

q−_t penalty parameter for sales shortage v_t− in period t;t ∈ T2

M sufficiently large number for modeling purposes Stochastic Parameters

dit demand in class i in period t; i ∈ I, t ∈ T2

wit fraction of observed demand in class i in period t, 0 ≤ wit≤ 1; i ∈ I, t ∈ T2

Dit cumulative demand in class i in period t, Dit:=P_j≥iwjtdjt; i ∈ I, t ∈ T2

pct lowest price of competitors in period t; pct∈ [p1t, pIt]; t ∈ T2

ant cancellations in period t; t ∈ T2

Model Variables

C_ts value of available capacity in period t in scenario s; t ∈ T2, s ∈ S

Rt expected value of cumulative revenue up to and including period t; t ∈ T2

Ut expected value of cumulative generated profits up to and including period t; t ∈ T2

υt expected sales in period t; t ∈ T2

Υt expected value of cumulative sales up to and including period t; t ∈ T2

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4.2.1 Decisions and Constraints

We model price related decisions by making use of binary variables xit, i ∈ I, t ∈ T2. Opening

a profit class i at stage t, i.e. xit = 1, means that the price of the package is equal to the

corresponding parameter pitduring period t + 1. Obviously, at most one class can be opened

at any time.

The price is allowed to vary over time. However, the course of the prices might be determined partly by a TO’s competition. The future prices of a TO in period t are allowed to be higher than the prices of competitors by at most the parameter (1 + αt). Furthermore, an additional

tolerance parameter 1t, which indicates the maximal deviation that can be charged for extra

facilities belonging to the package in period t, has to be incorporated. Below, we present the restrictions that model the above-mentioned aspects.

X i∈I xit = 1, t ∈ T1, xit ≤ (1 + αt)pct+ 1t pit , i ∈ I, t ∈ T1.

Sales, necessary in order to define total profit, are restricted by several factors. First, sales in period t can only be captured in the class that is opened during that period. Second, sales cannot exceed demand or available resources. We refer to Section4.2.2for the definitions of capacity Ct and cumulative demand Dit.

We claimed that we modeled that demand (observation) is dependent on price (the decision). Actually, the realizations of the random parameters dit are not affected by their preceding

decision xi,t−1, neither are the realizations of the stochastic parameters Dit. However, by

combining the definition of cumulative demand and the constraint that sales can only be captured in one specific price class, we can influence the observed distribution of demand. To show how, consider the second and third constraints below. The parameter Dit decreases

with the index i: opening a higher class results in a lower value of cumulative demand Dit.

The price parameter pit is higher in that case. Hence, in maximizing total revenues (loosely

speaking defined byP

i,tpityit), an optimal compromise between sales yit and price pit should

be found.

yit ≤ M xi,t−1, i ∈ I, t ∈ T2,

yit ≤ Dit, i ∈ I, t ∈ T2,

yit ≤ Ct, i ∈ I, t ∈ T2.

Define Bt:=P_i∈Iyit−ant, the actual sales at stage t. Then deviations from targets (Bt−gt)+

and (Bt− gt)− both are penalized according to a piecewise linear convex penalty function,

modeled using nonnegative surplus and shortage variables v_t+and v_t−, t ∈ T2 respectively. By

definition (b)+:= max{0, b} and (b)−:= max{0, −b}, for b ∈ R.

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4.2.2 Defining Parameters

Some of these relations and definitions were discussed above. We repeat them here for the sake of completeness.

TOs are usually interested in maximizing expected total future profits. The profit πit of a

package is equal to the selling price pit of a package minus the cost cp (or purchase price) of

a package. Assuming that the purchase price does not vary over time, we define: πit := pit− cp, i ∈ I, t ∈ T2.

If a TO has specified goal parameters gt corresponding to stage t, the parameters Gt,

indi-cating the cumulative goal on sales up to and including stage t, are defined by: Gt :=

X

τ ≤t

gτ, t ∈ T2.

Demand is dependent on the price: a price cut should have a non-negative effect on the level of demand. In our research we model price related decisions by making use of price classes which can be opened and closed over time. Every price class i corresponds to a specific price level pi, with pi< pi+1. If the highest price class is opened, only the demand corresponding to

this price class will be observed. If the second but highest price class is opened, the demand in this price class will be observed. However, as we may only open one price class simultaneously, the demand in the highest price class should be observed in the second but highest price class as well. This is generalized to all price classes by defining cumulative demand:

Dit := X j∈I|j≥i wjtdjt, i ∈ I, t ∈ T2, wit := min{ pct+ 2t pit , 1}, i ∈ I, t ∈ T2.

The parameters wit refines the definition of cumulative demand. We define witby the fraction

of observed demand in class i at stage t. For example, if wit = 1 it means that all demand

in class i at stage t will be observed, while wit< 1 means that only a fraction of demand is

observed in class i at stage t. 4.2.3 Model Variables

We give definitions of the model variables in this section.

Capacity varies over time. Sales υtand cancellations ant imply a stock mutation in capacity.

We should have that initial capacity is available at the second stage too, because capacity mutates after sales or cancellations have occurred. For every stage t ∈ T2 we define capacity

by:

Ct := C0, t ∈ {1, 2},

C_ts := C_t−1s − Bs

t, t ∈ T2\{1, 2}.

In order to evaluate the performance of a TO, we would like to track certain model variables. For this purpose, we define sales υt at stage t by:

Revenue Management for a Tour Operator A Stochastic Programming Approach

Revenue Management for a Tour Operator

A Stochastic Programming Approach

Revenue Management for a Tour Operator

A Stochastic Programming Approach

Preface

Contents

1

Introduction

2

Theoretical Framework

3

Problem Description and Approach

4

Mathematical Model