A Long-Term Fleet Composition Study for a European Network

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A Long-Term Fleet Composition Study for

a European Network

A Study for KLM, the Royal Dutch Airlines

Nicole Horsman

May 2006

Master Thesis

Econometrics and Operations Research Supervisors:

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Foreword

Hereby I present my thesis paper which concludes my study of Econometrics and Operational Re-search at the University of Groningen. The study for my thesis paper was carried out for the Network Planning Department at KLM where I spent seven months working as an intern.

KLM was the company that first came to mind when I began applying for internships. The organiza-tion and complexity of such a large logistical operaorganiza-tion as that of an airline intrigued me. Fortunately, I was given the opportunity to experience and work for a prestigious company like KLM.

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1 Introduction

The Royal Dutch Airlines, commonly known as KLM, is dealing with present and future decisions on the expansion of fleet and its composition for the long-term period of 10 years. The reason why KLM is interested in making a study on its European fleet is because the last ’long-term study’ was done in 2000, and the aviation industry has changed dramatically since then. There have been (economic) crises that affected demand in air travel, many new competitors have entered the market, and fuel costs are continuously rising. Consequently, KLM has been focusing on improving cost efficiency in its European fleet by increasing in number of seats and reducing cost levels. Given these changes, KLM has reached a point where it is concerned with the long-term review of fleet expansion, which is not yet an annual embedded process.

The company has come to a point where they need to replace some of their old aircraft. Given that high growth rates are expected in the airline industry, KLM wants to expand by aligning their capacity growth with expected growth in demand. Therefore, it is important for KLM to make decisions on what kind of aircraft to buy, and in order to be prepared for the future, these decisions need to be made as soon as possible. However, it is an ever-changing uncertain world and there are outside factors that can affect the expected growth in demand for KLM, such as competition. Uncertainty plays an important role in long term planning; uncertainty of what will actually happen in the future. KLM is interested to know how uncertainty can be taken into account when making future decisions on their fleet. What effects can uncertainty in demand have on KLM’s fleet expansion plans and how can they be dealt with? The central question asked by the Network Planning Department at KLM and to be answered in this thesis paper is:

How can the number of aircraft and its distribution over capacity segments for KLM’s European fleet be determined for 2010 and 2015 such that a growth path can be determined and that the fleet

compositions for these years satisfy future demand at a minimum cost?

In order to answer the central question, other questions need to be answered as well. First of all, how should uncertainty in demand be handled? What are the main components of uncertainty and how do they affect future demand for KLM? KLM further posed the question: provided the fleet composi-tions, what would be an optimal growth path for the European fleet? How can these compositions be integrated in order to determine this growth path?

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2 European Airline Environment: Predicting Demand

This chapter is concerned with the uncertainty part of the long-term fleet composition planning prob-lem at hand. The uncertainty in this probprob-lem is future demand since the decisions on aircraft for the fleet will depend on how many people are predicted to choose to fly with KLM. Though there are several factors that can influence future demand, the analysis on demand was restricted to two fac-tors which are most important according to KLM. The chapter begins with an introduction on how dynamic the airline industry can be and what changes have happened in the last 5 years. In 2.2, it is explained why Gross Demand Product can help predict demand. In 2.3, it is justified why competition is an influential factor when predicting demand. Please take note that this study commenced in 2005 so all data and information gathered is from that year.

2.1 Industry under pressure

The aviation industry is a dynamic environment that has changed dramatically in the last 5 years. Crises such as 9/11, SARS, and the war in Iraq have led to significant disruptions in travel patterns and developments, as can be seen in Figure 1. These crises caused airline companies to lose significant amounts of money due to the high costs maintained and the large decrease in numbers of passengers. Fuel prices are continuously increasing, hurting the aviation industry by constantly causing airline’s cost margins to rise. What we see further in Europe is that competition has grown immensely since 2000, with many new companies offering very low prices to customers. Also playing a big role in the industry, as to contradict competition, is the forming of alliances, which are maturing and taking steps towards further consolidation in Europe. Air travel growth has been unstable, and these events have led to the following generic trends among airlines: an increase in costs, leading to a decrease in yields. Nevertheless, economic growth seems to be returning since 2005, which gives a positive out-look for the airline industry. The following section will discuss the relationship between GDP growth and growth in demand for air transport.

40,000 42,000 44,000 46,000 48,000 50,000 52,000 54,000 56,000 Se p-0 0 Nov -00 Ja n-01 Ma r-0 1 Ma y-0 1 Ju l-01 Sep -01 No v-0 1 Ja n-02 Ma r-0 2 Ma y-0 2 Ju l-02 Se p-0 2 Nov -02 Ja n-03 Ma r-03 Ma y-0 3 Ju l-03 Sep -03 No v-0 3

AEA Total Scheduled Passenger Traffic (in million RPK)

(corrected for seasonal variations)

9/11

War in Iraq economic downturn

SARS

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2.2 Gross Demand Product and its relationship with demand for air travel

The relationship to be discussed is that between the growth rate of GDP and the growth rate of de-mand for air travel. In the airline industry, one of the terms used for measuring dede-mand for air travel is Revenue-passengers-kilometres (RPK) (see Appendix A for definition). Studies made by Air France have shown that the relationship can be specified asRP K ≈ k × GDP , where k is a so-called growth

rate multiplier, and which is equivalent to 2 in their case (see [9]). By definition, the growth rate in air travel is equal to about twice the growth rate of GDP.

The McKinsey Company, however, claims that the annual passenger growth rate is slower than twice the annual GDP growth rate, meaning that the growth rate multiplier is lower than 2, as shown in Figure 2. They say that ”significant steps must be taken to improve convenience and lower the cost to the passenger if long-term growth rates for traffic (passenger demand) are not to decline toward GDP levels.” For example, in Boeing’s Current Market Outlook document [5], the growth rate of World GDP was forecasted to be 3.0% per year and world air traffic was forecasted at a growth rate of 5.2%, which is 1.7 times as much. This could be an indication that the multiplier should be lower than 2 on average. Since this study is concerned with KLM’s European fleet, we will now look at what holds for the Netherlands, KLM’s origin, and the rest of Europe.

Figure 2: McKinsey Company, from [6] Figure 3: Boeing Company, from [5]

KLM tested this relationship in 2003 at their hub airport, Schiphol Airport (SPL) in Amsterdam. (See Appendix A, under definition of hub and spoke). Figure 4, on the following page, compares the growth rates of GDP in the Netherlands and the air traffic growth rates at Schiphol from the years 1960 to 2002. In order to define the GDP growth rate for a flight, the average of the GDP growth rate of the country of origin and the country of destination is calculated. That way, the GDP growth rate can be compared to the growth rate in air traffic for a particular flight. In the graph, the term origin-destination (O&D) represents the traffic of passengers at Schiphol. The values of the GDP growth rate percentages are found on the vertical axis on the left-hand side and the values of the O&D growth rate percentages are found on the vertical axis on the right-hand side of the graph. The graph shows that the two growth rates are highly correlated, indicating that the relationship exists and can be used for calculating demand at Schiphol.

The graph in Figure 5 shows what the GDP growth rate multiplier was from 1960 to 2002. It also gives a prediction of what the multiplier could be from 2002 to 2010. The multiplier is equal to the ratio

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0.0 1.0 2.0 3.0 4.0 5.0 6.0 19 60 19 62 19 64 19 66 19 68 19 70 19 72 19 74 19 76 19 78 19 80 19 82 19 84 19 86 19 88 19 90 19 92 19 94 19 96 19 98 20 00 20 02 G D P g r o w t h % -2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00 O & D S P L g r o w t h % O&D SPL growth % (10 year rolling average)

GDP NL growth % (10 year rolling average)

Figure 4: GDP vs. O&D Schiphol growth rates, from [8]

between Pax and GDP growth rates, where Pax is another term for passengers. The box represents the range of this predicted GDP growth rate multiplier. What falls above it is considered to represent a high growth scenario for KLM and what falls under, a low growth rate scenario. One can see that the GDP multiplier was predicted to be about 2 from 2005 to 2010, which means the relationship

2×Passenger demand (growth rate) ≈ GDP (growth rate) can hold for Schiphol.

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 19601962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 19841986 1988 1990 1992 1994 1996 1998 20002002 20042006 2008 2010 Multiplier Pax/GDP (10 Year rolling average)

High growth scenario

Low growth scenario

Figure 5: Expected multiplier for KLM, from [8]

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de-tailed definitions. In general, advanced economies are considered to have had a stable and low GDP growth rate, while emerging economies are those countries that have had a volatile growth rate. We conclude that GDP developments can help forecast demand for air travel. In order to make the fleet composition results as realistic as possible it is important to look at each country’s economic situation. This information will help define an annual GDP growth rate of a country and help calculate an expected annual passenger growth rate. A GDP multiplier will be used to calculate expected demand for each of KLM’s flights and is provided in Chapter 4. By request of KLM, a sub study was made on the economic developments of emerging countries. The information on these countries can be found in detail in Appendix B. The following subsection will explain the influences of competition on demand, our second factor for calculating demand.

2.3 Competition and its effects on demand for air travel

KLM’s biggest threat at the moment is that of growing competition, which could have a large impact on their passenger demand. There are two types of competitors, Low Cost Carriers (LCCs) and Network Carriers (NCs). KLM is considered to be a NC. More information on the differences between the two can be found in Appendix A. LCCs are a relatively new type of competitor and are entering the airline industry with ease because they offer a cheap product in comparison to NCs. NCs are less of a threat due to the many new alliances and merges among themselves. This is because it has become more competitive between alliances rather than between NC airlines. Therefore we will focus on the threats of LCCs. Section 2.3.1 looks at what effects LCCs have had on the market and its competitors. The second section, 2.3.2, gives details on the future growth plans of LCCs and discusses the competition expected in the future.

2.3.1 The impact of Low Cost Carriers

The LCCs largest impact on the airline industry has been the generation of new demand for air travel. LCCs caused people to change their choices in types of travel since in many cases it became cheaper to travel by air than by land. Due to their much lower prices, on average from 40 to 60 % lower fares [13], flying was no longer a luxury product. It was the case for many destinations that when LCCs entered the market, only part of the demand they captured was from that of NCs. Figure 6 shows examples of the impact of LCCs on different routes. The term incumbents in each graph refers to all NCs flying that specific route.

Though the effects of LCCs’ entrance to market seem small towards demand for NCs in the graphs of Figure 6, in reality it did have a large impact on NCs like KLM. In order not to lose too much market share and to compete with the new LCCs, NC’s had to lower their fares, causing their yield levels to drop and forcing them to lower costs. The graphs in Figure 7 give examples. The first graph shows that demand grew immensely after Ryanair started up London-Dublin. The second figure shows the entrance of LCC Southwestern in the United States forced other companies to drop their yields. Thus it is expected that the entrance of new LCCs on routes served by KLM will have a substantial impact on passenger demand for KLM.

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Figure 6: Examples of LCCs effect on routes from Amsterdam (from [8]

Figure 7: Impact of LCC presence in market on volume and yield level (from [11] 2.3.2 Present and future developments of competition

LCCs expect to continue growing in the airline industry, and this is proven by their fleet expansion plans. Figure 8 shows the fleet growth plans for the 3 largest European LCCs, Ryanair, easyJet, and Air Berlin. As can be seen in the figure, the total number of planes planned for the future is 253, which is a 98% increase in only 5 years, from the year 2005 to 2010. Ryanair alone has announced its order for 149 Boeing 737-800 to be delivered by 2010 [22]. 1 Figure 9 shows the size of each European carrier fleet and their publicized orders. One can see that by 2010 Ryanair and easyJet will each have a fleet that is twice the size of KLM’s European fleet. This means that KLM and other NCs can expect strong competition.

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108 134 149 176 197 96 121 147 167 186 53 60 69 74 79 218 91 204 77 88 47 2004 2005 2006 2007 2008 2009 2010 Air Berlin/Niki Ryanair easyJet

Figure 8: Largest LCCs and their planned fleet developments ([10]

0 50 100 150 200 250 300 Wiz zair Sky Eur ope Ger man win gs Hlx Tran savi a Airb erlin /Nik i Rya nair Eas yjet _KLM Alit alia Iber ia _BA Lufth ansa A F firm orders Total European Fleet

Figure 9: Airline future fleet development plans for 2009 (from [15]

Another development we are seeing in the airline industry is the expansion of LCCs in countries with emerging economies. For example, Eastern Europe is seen as a potential market, where the economy is growing fast, businesses are expanding and tourism is on the rise. New LCCs originating from this area, such as SkyEurope and Wizz Air, are taking advantage of this economic growth and expanding rapidly. The second part of the sub study for KLM focuses on the LCC developments. More detailed information on the most potential LCC’s and their expansion plans can be found in Appendix C. Given competition is expected to increase in the years to come, it was decided that a competition parameter would also be used in order to predict demand. Since the competition expected is mainly from LCCs, the parameter is given the name LCC multiplier. Based on the information in Appendix C, the LCC multiplier will be applied to each destination in KLM’s network where LCCs are present.

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2.4 Summary

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3 Plan of Approach

This chapter focuses on describing the problem presented by the Network Planning Department at KLM and the objectives for its fleet composition study. My approach to answering their questions is through means of a mathematical model. Section 3.1 focuses on restating the problem presented by KLM in terms of concrete sub-questions. In Section 3.2 some different possible models are compared and the reasons for selecting a certain type of model are given. In Section 3.3, the multistage recourse model is introduced and it is explained why this approach is suitable for answering the central question and the sub-questions.

3.1 Formulation of problem

The objective of the long term European fleet study is to determine an optimal fleet composition for 2010 and 2015, as well as a growth path between now and 2015 (recall that ’now’ refers to the situa-tion in 2005). The Network Planning Department at KLM wants to know how many aircraft they need and in which capacity segments they will need to reduce/expand, in order to satisfy future demand. For such long term planning, demand is uncertain and can be very variable. Different demand expec-tations can have very different impacts on the fleet composition. The most realistic approach would be to consider the influence of variable demand scenarios. Which fleet compositions are robust enough to envelop such variation? Furthermore, how can the compositions for 2010 and 2015 be related in order for KLM to be able to define a realistic growth path? These are some of the questions to be answered.

Furthermore, the reason why it is important for KLM to make these decisions now for 2010 and 2015 is to be able to determine a growth path between the fleet compositions such that investment plans can be made. Since investing in new aircraft and/or lease contracts is very expensive and depends on available capital, these plans are often spread out over a long-term period of 3-5 years.

KLM’s present fleet is another reason why KLM is concerned with long-term decisions for their fleet. There are several aircraft in KLM’s present fleet that will have to be replaced or removed within 10 years either because of lease contracts that are due to expire or because of ’old’ age. Furthermore, these are also types that have gone out of production. The chart on the following page in Figure 10, shows which capacity segments are being eliminated and which segments are popular among the Eu-ropean single-aisle2fleets. The seating bands, on the horizontal axis, refer to the capacity segments, and the vertical axis represents the total number of aircraft in Europe for all capacity segments. In addition, the types in KLM’s current European fleet that are out of production are pointed out in the graph.

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KLM Mainline KLM Cityhopper 0 50 100 150 200 250 300 350 400 450 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 1 1 0 1 2 0 1 3 0 1 4 0 1 5 0 1 6 0 1 7 0 1 8 0 1 9 0 2 0 0 2 1 0 2 2 0 2 3 0 Seating band F le et Production types Out-of-production types 737-300737-400 737-800 737-900 F100 F70

European single-aisle fleet

Figure 10: Overview of types and capacity segments (from [17])

Given some of these types need to be replaced within the next 10 years, and given uncertain future demand, KLM’s Network Planning department, for whom the study is being done, is concerned with the long-term decisions on its European fleet. Which capacity segments should expand or decrease in size? Which capacity segments will provide the best results? Fleet compositions do not only depend on demand, but also on the costs of the types considered for each capacity segment. The study presented in this paper looks into what kind of mathematical model is needed, and how it can be built in order to answer these questions.

3.2 Approach: Selection of model

The fleet mix optimisation model available at KLM is based on a short-term model developed by Berge and Hopperstad [2], which is called the D3 model: Demand Driven Dispatch. This model is used for short-term situations where aircraft are assigned to flights given expected demand yet are allowed to switch flight assignments in order to compensate for changes between expected demand and actual demand. It is a short-term model because expected demand is based on the latest estimates of passenger demand and the flight assignments are brought up to date shortly before the actual flight operations. It is a deterministic model because it only takes one possibility of demand, for each flight, into account. However, a long-term model is needed for this study, which means that uncertainty in future demand plays an important role in the problem and thus needs to be incorporated in the model. While researching on suitable mathematical models, I came across a robust fleet composition model. The stochastic model was developed by Listes and Dekker [19]. This model was a good basis since it took uncertainty in future demand into account. It determined a fleet composition that was dependent

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on several expected passenger demand scenarios, but which also had to be feasible for a given schedule and maximize profits. These components, such as passenger demand and schedules, will be explained in Chapter 4. However, the model needs to be expanded in order to address some modelling issues, such as:

1. How can different passenger demand scenarios, representing different passenger demand growth rates, be taken into account when making decisions on future fleet compositions? 2. The model needs to determine a fleet composition for both 2010 and 2015 simultaneously. 3. How can the dependencies between the fleet compositions of today (2005), 2010 and 2015 be

modelled?

The following section discusses how the model of Listes & Dekker can be adjusted in order to address KLM’s long-term planning problem including the modelling issues above.

3.3 Adjustment: Multistage Recourse

There are certain characteristics in KLM’s problem and questions which can be captured by a mul-tistage recourse model. First of all, the problem is dynamic since we are dealing with decisions for different points in time, including the decisions that need to be made now (which is represented by 2005 in the model). Secondly, the decisions on the fleet compositions need to take uncertainty in fu-ture demand into account. Thirdly, in order to determine a growth path between the fleet compositions at different moments in time, the fleet compositions need to be related and therefore the decisions on the fleet compositions need to be integrated.

To begin with the dynamics of the problem, there are three different realizations of demand growth possible between 2005 and 2010. Depending on the demand realization in 2010, there are again three different realizations of demand growth possible between 2010 and 2015. As a result, there are 9 sce-narios in total which model the uncertain growth in demand between 2005 and 2015. These scesce-narios are depicted in a scenario tree in Figure 11.

In 2005 (t = 0), the decision of the fleet composition for 2010 needs to be made, when only

prob-abilistic information about the future is available. The decisions on the fleet composition for 2015, however, only need to be taken in 2010 (t = 1) so that the decision is dependent on the demand

growth that appeared for 2005 to 2010 and on the fleet composition for 2010. Thus a multistage re-course model allows for these decisions to be conditional on the demand growth up until 2010. Nodes 2-4 in the scenario tree represent when these decisions are made.

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On the other hand, the decision on the 2010 fleet taken in 2005 does take into account the conse-quences for future decisions, which is measured by expected future costs. That way the dependency between the the current fleet and the decisions on the different fleet compositions is modelled.

5 2 6 7 8 3 1 9 10 11 4 12 13 t = 0 t = 1 t = 2 Figure 11: 3-stage scenario tree A multistage recourse model also allows for the long-term decisions to depend on expected short-term decisions in the future. These short-term decisions refer to the fleet assignments for the 2010 fleet and the 2015 fleet. They are short-term decisions because they are decided at the moment when the fleet compositions are implemented, given a schedule. These can be explained in a time-line, such as that in Figure 12. In 2005 (t = 0), the long-term decision for the 2010 fleet composition is made. In 2010 (t = 1), the short-term decisions on flight assignments for fleet 2010 are carried out, and the long-term decision for the 2015 fleet composition is made depending on what kind of demand growth actually happened in 2010. In 2015 (t = 2), we again have the short-term decisions on the flight assignments for the fleet 2015 (depending on what resulted in 2010). D 1 D2 z 0 z 1, x1 x2 t = 0 t = 1

t = 2 (2005) (2010) (2015) time

Figure 12: Time-line for Decisions

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z0 fleet composition for 2010 is decided in 2005

D1 demand for 2010 is revealed

z1 = z1(z0, D1) fleet composition for 2015 is decided in 2010

x1 = x1(z0, D1) flight assignments for schedule 2010

D2 demand for 2015 is revealed

x2 = x2(z1, D1, D2) flight assignments for schedule 2015

What is essential in the multistage recourse model is that uncertainty can now be explicitly modelled and the decisions for the fleet compositions in the second stage are conditional. This means that the decisions on fleet compositions depend on revealed information, past fleet compositions decisions, and future probabilistic information.

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4 Modelling Passenger Demand and Network Schedules

The decisions for the fleet compositions of 2010 and 2015 are evaluated by looking at how well the solution works with different scenarios, which is determined by low expected total costs. In order to do this, the input for the problem, demand and schedules, needs to be modelled. Demand is in this case the random variable in our stochastic problem. In Section 4.1, the derivation of demand scenarios is explained, where the demand variable is dependent on GDP growth and LCC competition, as explained in Chapter 2. Section 4.2 describes the flight schedules, which are fixed yet different for 2010 and 2015. The mathematical formulation will follow in Chapter 5.

4.1 Demand data

Demand in our model is a random variable because of the uncertainty of what it will look like in the future. Before explaining our modelling of random demand, we will first look at how demand is defined at KLM. Passenger demand for a route is divided into 4 categories, the main reason being that each category provides KLM with a different yield. Furthermore, given their differences, each can have a different growth rate. This is because a number of passengers are transfer passengers, who use the route to get to their primary destination, inside or outside of Europe. Since passenger demand is related to economic growth, economic growth rates of countries outside of Europe should be consid-ered as well. This is because it is predicted that air travel between Europe and the rest of the world will grow faster than air travel within Europe. However, in order to simplify the model and reduce its size, it was decided that the four categories be aggregated into a single input of passenger demand per flight. Thus the different growth rates per category were calculated first and then added together. We will no explain the four different categories, which are the following:

3rd/4th EUR 6th EUR 6th ICA OC Feeder

The terms 3rd, 4th, 6th represent types of so-called freedoms of the air. The freedoms of the air are agreements and laws between countries giving one another permission to land in each other’s country. A 3rd/4th EUR represents flights between 2 European destinations. A 6th freedom of air represents a flight from a nation A to a nation C, with a stop/transfer in a nation B. A 6th EUR is a flight within Europe and 6th ICA is a route between Europe and a country outside of Europe, where ICA stands for Intercontinental. The OC Feeder represents a flight between two intercontinental destinations with a stop in Europe, this stop being Schiphol for KLM.

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de-creases by 25%. This means that if the presence of LCC competition is already present and expected to be high, then the demand growth rate for KLM will be low, and vice versa. As for the growth in demand of the two remaining categories, 6th ICA and OC Feeder, these were calculated by KLM.

Table 1: Growth multipliers used for calculating demand GDP multiplier:GDPs Low(s = L) 1.5 Medium(s = M ) 2.0 High(s = H) 2.5 LCC multiplier No LCC present: 1 LCC present: 0.75

Even though it has been clearly illustrated in the above section that demand is treated as a single input for each flight, we return to the fact that it remains a random variable. Demand is modelled such that there are three possible realizations for 2010 and three possible realizations for 2015, where the realizations in 2015 are conditional on which scenario will realize in 2010. Thus there are 9 scenarios in total.

Before we explain how these 9 scenarios were modelled, we will explain what scenarios were pro-vided by KLM. Three demand scenarios were given: a low, medium, and high. Each scenario has two moments when demand is revealed, one moment in 2010 and the other in 2015. Figure 13 depicts these scenarios in the form of a scenario tree. The following paragraphs explain how the demand scenarios provided by KLM were modelled in order to get 9 scenarios.

H ~ HH H M M ~ MM L L ~ LL t = 2005 t = 2010 t = 2015

Figure 13: The scenarios given by KLM presented in a scenario tree

We will now explain how the demand scenarios provided by KLM were modelled in order to get 9 scenarios. Since we are dealing with two moments in the future, we will define two random vari-ables, one for the demand in 2010 and the other for the demand in 2015. Uncertainty in the future is represented by different possible realizations of demand at each point in time, 2010 and 2015. The random variable for 2010, or att = 1, is ω1, which can result in either a low, medium or high demand

scenario. The random variable for 2015 is ωs

2, which can result in a low, medium or high demand

scenario, but is conditional on realizations of ω1, as explained in Chapter 3 (see Figure 11).

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In order to model demand for 2010, we define the following:

δs

i1: expected demand for flighti in realization s at t = 1

δi1: average demand for flighti at t = 1

gi1: growth in demand for flighti at t = 1

ei1: percentage GDP growth rate for flighti at t = 1

Pi0: previous demand for flighti at t = 0

The GDP growth rate eit is dependent on the country (see Appendix B). Growthgi1 in demand,

de-pends on average demand δi1, which is the average between the three demand scenarios. Average

demand and growth in demand are calculated in the following way:

δs i1 = (GDP1s× (1 + ei1))5× LCC × Pi0 δi1 = pLi1δLi1+ pi1Mδi1M + pHi1δi1H gi1 = (δ H i1−δi1)+(δi1−δLi1) 2 = (δH i1−δ L i1) 2 whereGDPs

1 is the GDP multiplier dependent on scenarios ∈ S1= {L, M, H} for t = 1 and psi1is

the probability of expected demand corresponding to scenarios ∈ {L, M, H}.

Assuming that each scenario has an equal probability of happening, we defineω1 to have a uniform

distribution with realizations{−1, 0, 1} and thus ps

i1 = 1/3. Hence with ω1, demandDi1 for flighti

is modelled by:

Di1:= δi1+ ωi1gi1so thatDi1is uniformly distributed with realizations{δi1− gi1, δi1, δi1+ gi1}

Demand for 2015 is modelled in a similar way, but the difference is that it is now conditional of re-alizations in 2010. Expected demand with realization s is dependent on demand Di1 in the previous

stage. Again we have average demandδs_i2, which is the average of the three demand scenarios of 2015 provided by KLM, and growthgs

i2 in demand. We define the following:

δs

i2: expected demand for flighti in realization s at t = 2

δi2: average demand for flighti at t = 2

gi2: growth in demand for flighti at t = 2

ei2: percentage GDP growth rate for flighti at t = 2

These are all calculated in the same way as with t = 1, but I would like to emphasize that we are

looking at two moments in time, and separating these allows for different expected GDP growth rate percentageseitor even different growth rate multipliers GDP₂2 than int = 1, though this is not the

case in this model. We have:

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whereps

i2is the probability of expected demand corresponding to scenarios ∈ {LL, M M, HH} and

GDPs

2 is dependent on 9 possible realizationss ∈ S2 = {LL, LM, LH, M L, M M, M H, HL, HM, HH}

fort = 2, with s1 = L for s ∈ {LL, LM, LH}, and analogously for s1 = M, H. Three GDP

mul-tipliers were provided for 3 different growth rates in demand, thusGDPs_{holds for each scenario}_s

in 2015, independent of what demand scenario realized in 2010. For example, the low growth rate multiplierGDPLis applied to each low growth rate scenario for 2015, thusLL, M L, and HL.

We defineωs

2 wheres ∈ S1 = {L, M, H} so that it is conditional on the scenario result of 2010.

Each of the three random variablesω₂shas three possible realizations which are uniformly distributed: a low, medium, and high demand scenario. Thus it holds that ps

i2 = 1/3. The distributions of ω2sare

as follows:

ωL

2 has a uniform distribution with realizations{−1, −0.7, −0.3}

ωM

2 has a uniform distribution with realizations{−0.5, 0, 0.5}

ωH

2 has a uniform distribution with realizations{0.4, 0.7, 1}

Hence, withωs

2, demandDsi2for flighti at t = 2 is uniformly distributed and is modelled by:

DL i2:= δi2+ ωLi2gi2with realizations{δi2− gi2, δi2− 0.7gi2, δi2− 0.3gi2} DM i2 := δi2+ ωi2Mgi2with realizations{δi2− 0.5gi2, δi2, δi2+ 0.5gi2} DH i2 := δi2+ ωHi2gi2with realizations{δi2+ 0.4gi2, δi2+ 0.7gi2, δi2+ gi2}

Hence, we now have 9 possible demand scenarios for 2015. For each of the 3 realizations in 2010, the corresponding scenarios can be distinguished only in 2015. Figure 14 is a visual representation of the scenario tree that has been built. Thus the 9 demand scenarios each represent a different outcome in the growth of demand. Furthermore, one can see that there are three realizations possible for 2010 and depending on the outcome in 2010, there are three possible realizations for 2015.

4.2 Network schedule

This section is divided into two subsections. The first focuses on the conditions and assumptions that were taken into account while defining the schedule input. The second describes the elements that make up a network schedule, and how the schedule is defined for the mathematical model.

4.2.1 Conditions and assumptions

The main assumption for the schedule is that it is fixed. This could have consequences since it is not likely that the schedule made for the year 2010 will indeed be used in 2010, which holds even stronger for the 2015 schedule. Nevertheless, the schedules were provided by KLM and are reasonable predic-tions of the network in the future. In practice, the schedule is constantly being evaluated and changed in order to improve performance. Destinations and frequencies may be added or eliminated, depar-ture and arrival times may be shifted, etc. This can depend on several things, such as the financial performance of flights, the time slots offered by airports, the performance on connectivity with other European flights and Intercontinental flights, and accommodating partner airlines in alliances.

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Figure 14: The scenario tree

A more realistic approach would be to take different schedules into consideration for 2015, given a low, medium or high scenario was revealed in 2010. For example, if a high scenario were to happen, it is most likely that the schedule will consist of more frequencies and destinations, than if a medium scenario resulted. Analogously, there would be larger than if a low scenario resulted. This could have an effect on the fleet compositions for 2015 since they would not only be dependent on the information revealed in 2010 (low/medium/high demand), but also on a low/medium/high schedule provided for 2015. This type of model is known as random recourse since there is more than one random variable in the model. Given a high demand scenario were to occur in 2010, a high scenario schedule should be used for 2015, and analogously for the medium and low scenario cases. An increase/decrease in the number of destinations could mean an increase/decrease in the number of planes. More frequencies could mean that the demand is more spread out, and that smaller planes are needed rather than larger ones. Variable schedule input could not be implemented into the model since there was no solver available able to solve this random recourse problem. It is recommended to look into this for further research.

The regular schedule also includes maintenance, yet it is not taken into account in the modelling of the schedule. The number of planes used in a schedule without maintenance is thus a nett result, which means only the planes used to fly the schedule are accounted for. In the case of KLM, their present total European fleet consists of 95 planes while their nett fleet consists of 86 planes. Furthermore, cargo flights are excluded.

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that only fly 3 times a week and on other days of the week than Monday, yet they were included in Monday’s schedule so as to take every possible destination/flight, and therefore demand, into account. The European schedule for the model has an exception of 6 intercontinental destinations which are included in the schedule since they are operated with aircraft from the European fleet. The day of the schedule was taken from the 1st week of June, which is during the summer period. A summer sched-ule is usually busier than a winter schedsched-ule in Europe, thus the maximum aircraft capacity needed is considered.

When maximizing results for fleet assignments, a split between business passengers and economy passengers is often included. This is not only because business passengers bring in a higher rev-enue than economy passengers, but also because it divides an aircraft into two seating sections, which means that there is a maximum number of seats for both passengers. However, the fraction of business class passengers has decreased immensely in recent years for European flights, as shown in Figure 15. Thus, the split in demand is just omitted. Nevertheless, the yield per flight and per passenger, provided by KLM, consists of a weighted average between the yields of an economy passenger and a business passenger. That way the presence of business passengers is still taken into account.

0 10 20 30 40 50 60 70 80 90 100 1997 1998 1999 2000 2001 2002 2004 S p li t o f p a s s e n g e r s ( % )

First or Business Premium Economy or Economy

Figure 15: Split of passengers between classes for KLM [17]

As mentioned in Section 2.2, demand is determined for each one-way flight i.e. a flight leg. At one point during the process of modelling the problem, we considered adjusting the schedule to round trips. This was considered because the schedules at KLM consist of return flights, with only a few exceptions. In terms of modelling, such an assumption is an advantage simply because it halves the size of the model. Instead of 576 flight legs, the schedule would consist of 288 round trips. However, the out-bound plane of almost half of KLM’s European flights does not return until the next morning, otherwise known as a night stop. By using single flight legs, it is possible to model a plane staying overnight, which is functional since the schedule represents a single day. Furthermore, the exceptions were 4 flights where the plane did not return to where it originated. For example, there was a 50-seater aircraft, a Fokker 50, which first flew the route Rotterdam to London City and then London City to Amsterdam. The schedule needed to be in terms of flight legs in order to model such trajectories. However, towards the end of the project, it was decided to eliminate all flights that do not return directly to Amsterdam since it was assumed that the types assigned would stay the same. Though the modelling of round-trips was not implemented, it is suggested for further research.

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4.2.2 Description of schedule

In this section we go into detail on how the schedule is modelled. In order to create a schedule, a time network is constructed (for more specific detail on KLM’s network see [3]). This space-time network, which allows the fleet composition problem to be formulated as a flow problem, is taken from Listes and Dekker [19]. The network is built with nodes and directed arcs, such that the activity at each airport can be portrayed given a period of time. This space-time network is copied for each type of aircraft.

To begin with, a node represents the activity at the airport of a destination for a period of time. For Amsterdam, which is the hub airport in this network, the day is divided up into periods of one hour, except for the first and last period. The first period is from 0:00-5:00 and the last period is 21:00-24:00. These periods are longer because of the fact that there are very little arrivals and departures during these hours. Combining these time periods also reduces the size of the network. For the other airports in the network, the total number of flights from Amsterdam to each airport represents the total number of nodes for this airport. These nodes have been numbered in order of arrival time, for each airport, yet they do not represent a specific time in the network. For example, the first node at the airport London Heathrow, LHR1, is in a different time zone than the first node at Budapest airport, BUD1. However, what prevents one from assuming that they are in the same time period is the flight’s departure/arrival time in Amsterdam. Taking the previous example into account, the flight departing from LHR1 arrives at node AMS4 while the flight departing from BUD1 arrives at node AMS12. The reason why it is useful for other airports not to have a specified time-line is because, once again, this allows the network we are dealing with to be much smaller, making the model easier to solve. It would only be needed in the case that KLM would offer routes between destinations other than Amsterdam. For example, if KLM would offer a straight flight between London and Budapest, the times at both destinations would have to be specified. However, in the case of KLM’s network, all flights are either from or to the hub airport, Amsterdam.

An arc in the network can represent either a flight arc or a ground arc. A flight arc corresponds to a flight leg. Each flight leg in a schedule is completely specified by a departure and an arrival destina-tion, and a departure and arrival time. Thus a flight arc connects two nodes from two different airports, the first node denoting the airport of departure and the second, the airport of arrival. A ground arc connects two consecutive nodes of an airport, indicating how many planes are on the ground at that airport during a period of time. Most of the ground arcs at all airports except Amsterdam, will have a value 0 because of the fact that each aircraft flying in from Amsterdam returns before the follow-ing flight to the same destination. This is not the case for night-stop flights, where the aircraft stays overnight at the airport. This may suggest that a large number of ground arcs can be eliminated from the network. It is recommendable since this helps reduce the size of the model.

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The time it takes to execute each flight can be divided into a block time and a turn around time. The block time represents the time it takes for a plane to leave the gate of departure and arrive at the gate of arrival of its flight destination, and it is measured in hours. The block time includes the runway time before take off, the take off itself, the time it is flying, the landing, and the runway time after landing. The Turn-around time is the time it takes for an aircraft to prepare itself for the next de-parture after its arrival. Turn-around times include the time it takes to unload and load passengers, luggage, catering food, to get fuel and to clean the aircraft.

As for the turn-around time, it needs to be included in the schedule because an aircraft cannot depart until after its turn-around time. Thus the time of arrival of each flight in this network is the actual arrival time plus the turn-around time. That way the network is constructed such that an aircraft de-parts immediately after arrival. However, we must realize that in reality, these times are dependent on the type of aircraft. Turn-around times for larger aircraft are longer on average than those for smaller aircraft. However, because the schedule is fixed in this model, they are assumed to be the same for each type. The minimum turnaround time assumed for Amsterdam is 20 minutes [1]. schedule can be adjusted such that the flights that would most likely use a large aircraft have a longer turn-around time than flights that will most likely use a small aircraft.

There are three other important constraints in order for the schedule to work. One constraint states that each flight is only allowed to be operated by one plane. Another constraint states that after the arrival of a plane at an airport, it must either leave (flight arc), or stay at the airport (ground arc). A third constraint states that the number of planes needed to fly the schedule is equal to the sum of the number of planes at each airport at the start of the day. Figure 16 is a small example of what a network looks like. The different types of aircraft are represented by the different lines, which shows that the network is copied for each capacity segment. Furthermore, the diagonal lines represent the flight arcs and the horizontal lines represent the ground arcs.

KLM Model: 3 destinations and 2 types of aircraft

Timeline Source Sink MAD1 MAD AMS2 AMS AMS1 AMS3 CPH CPH1 CPH2 1 0:00 – 5:00 6:00 – 7:00 7:00 - 8:00 50-seater 100-seater MAD = Madrid AMS = Amsterdam CPH = Copenhagen

Figure 16: A small example of a network

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In order to define which aircraft are allowed to operate each flight, the maximum range of an aircraft should be compared to the distance of the destination. However, instead of distance, the block time of a flight was used. This was because the data for operational costs was only provided per round trip or per block hour. Therefore all flight distances were translated into block times. The disadvantage of using block hour dependent costs is that only one block time is being considered for all types. It would be realistic to have the block times depend on type. It is obvious that a B737-400 (see Appendix ??, which travels at an average speed of 803 km/hr, will arrive much earlier than a F50, which travels on average at 505 km/hr. Thus a B737-400 would have a much shorter block time than a F50. However, since the schedule is fixed, it is not possible to include this variability. Nevertheless, it was assumed that the block times provided were more or less related to the type that would be used for the distance. For example, the route Amsterdam to Athens is usually flown by a 170-seater, a B737-400, which takes about 3 hours and 35 minutes. If a 50-seater aeroplane like the F50 were used, not only would this type of plane take about 5 hours to reach the destination given its speed, it is not able to fly this far due to its limited range. Therefore an aircraft is allowed to operate a flight only if its maximum range exceeds the block time of the destination. Though the schedule would be more accurate if block times were dependent on type, these approximations should have a minimum effect on the long-term planning of fleet compositions.

4.2.3 Summary and conclusions

Demand is random, and for each flight in the model it is:

• Provided in the form of aggregated demand over KLM’s 4 passenger categories • Depending on GDP growth rates and the presence of LCC competition

The following assumptions on the network schedule have been made:

• Schedule of a single day • Schedule input is fixed

– Block time per type fixed – Turn-around times per type fixed

• Schedule consists of flight legs and not round trips • Two schedules: one for 2010 and one for 2015 • Maintenance is excluded

• Cargo is excluded

• Distinction between business and economy passengers was not modelled

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5 Fleet Composition Model

The main focus of this chapter is to describe the mathematical model built in order to address KLM’s questions. The fleet composition model is a multistage integer recourse model since the fleet compo-sition variables are integer. This type of model is used for modelling the dynamic decisions, which are the long-term decisions for the fleet compositions of 2010 and 2015. Moreover, it allows for decisions to be taken right now which depend on the uncertainty of what future demand will look. Uncertainty in future demand plays an important role, and therefore the goal of our model is to satisfy demand at a minimal expected cost. Such dynamics in decisions and uncertainty imply that all the decisions to be made after the first stage, represented by 2005, are conditional on new information revealed and on remaining uncertainty. The model is first described verbally, followed by the section where the problem is translated into mathematical terms. Section 5.2.2 specifies the data and parameters used for the model.

5.1 Verbal description

Decisions

In this fleet composition model there are different types of decisions to be made. The main decisions are the ones on the fleet compositions. There is one decision for 2010 fleet composition that needs to be made now, in order to be able to set up an investment plan for the upcoming years. The decision for 2010 is made under the uncertainty of future demand, which is in this case dependent on future economic growth and presence of in competition. Another decision is that for the 2015 fleet composi-tion, where different possibilities can be looked at now, yet this decision should be made in 2010. The decision for 2015 is also made under the uncertainty of the growth in demand between 2010 and 2015, however it can be based on the information revealed at that time. That is why the decision should be taken only in 2010 so that it can be based on the information made available. This information refers to the decision taken in 2005 for the 2010 fleet composition and the demand growth scenario that occurred between 2005 and 2010.

The decisions can be defined as being inherently dynamic, since they depend on previous decisions and/or expected future decisions. The different moments in time, 2005, 2010 and 2015, represent 3 stages in time when decisions are taken. Though no decision is made for the 2015 fleet composition until 2010, the 2010 fleet composition is influenced by the possible 2015 fleet compositions, which are conditional on the demand realization of 2010. The decisions in the second and third stage are called the recourse decisions, which are taken when new information on demand is available such that corrective actions can be taken in the case the goal of satisfying demand is not reached.

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The outcome of the fleet composition model is that there is a single decision for the 2010 fleet in the first stage, and there are several possible decisions for the 2015 fleet in the second stage. The reason why there are several decisions for 2015 is because there is a separate decision for each possible demand scenario to result in 2010. Hence, instead of a single growth path from 2010 to 2015, there are a number of such paths provided, which are 3 in this case. These paths and decisions are illustrated in Figure 17 using a scenario tree. As we can see, they each have the same starting point, which is the fleet composition for 2010. Yet the growth paths have different finishing points since they depend on the scenario that appeared for 2010.

Decision on 2010 Fleet Composition made in 2005, without knowing what demand scenario will realize in 2010 &

2015 Decision on 2015 Fleet Composition made in 2010,

given the 2010 fleet composition and HIGH demand

scenario realized in 2010, and without knowing what demand scenario will realize in 2015

Analogous Decision in the case a MEDIUM demand scenario realized in 2010 Analogous Decision in the case a LOW demand scenario realized in 2010 2005 2010 2015

Figure 17: Decisions pointed out in scenario tree

Note that only the first-stage decision, i.e. the fleet composition for 2010, is actually implemented. All recourse decisions included in the model serve to evaluate the expected future costs resulting from first-stage decisions. Thus, the fleet composition for 2015 will be reevaluated in 2010.

Objective

The fleet composition is determined by the types selected for the flights and these decisions are based on minimizing the costs for using the planes as well as minimizing the penalty costs for the loss in potential revenue when the demand of passengers is not satisfied. Thus our objective is to determine a fleet composition for 2010 that will minimize the sum of current and expected future costs, i.e. as the expected total costs.

The model management system used for solving this recourse model is called SLP-IOR [16], designed for stochastic linear programming. A second program called GAMS [4] was used to specify the model and create mps files which could be imported by SLP-IOR. The solver used was HOPDM version 2.3 (Higher order primal-dual method) by J. Gondzio, which solved the model in about 30 seconds. Due to the dimensions of our model, no specific recourse solvers could be used.

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5.2 Mathematical model

In this section the mathematical formulation of the fleet composition problem is presented. As ex-plained in Chapter 4, the network schedule is based on the model by Listes and Dekker [19]. In Section 5.2.2 the data for the model will be presented. The last section goes into detail of how the model can be refined with a modelling adjustment that allows for our goals for demand to be depen-dent on different load factors.

5.2.1 Formulation of model Sets

Time is discrete, with stagest = 0 (now), representing the 1st stage, until the model horizon t = T ,

where T represents the last stage.

Kt set of aircraft capacity segments, at timet

Nt set of flight legs in a schedule, at timet

Mk

t set of flight legs not allowed to be flown by a plane from capacity segmentk, in time t

Lt set of airports serviced by schedule, at timet

Gt set of ground arcs, at timet

Vt set of all nodes excluding sources and sinks, at timet

Ωt set of all demand realizations, at timet

Indices

i index of flight legs

k index of capacity segments

l index of airports being served

g index of ground arcs

v index of all nodes (except sources and sinks)

t index of timet which is equivalent to stage t + 1, t = 0, ..., T − 1

st index of demand scenario at timet, where s1 ∈ S1 = {H, M, L} represents the realizations

att = 1

ands2∈ S2 = {HH, HM, HL, M H, M M, M L, LH, LM, LL} the realizations at t = 2

Variables: Stage 1 (t = 0) to Stage T − 1

zkt number of aircraft in capacity segmentk in the fleet at time t + 1,

k = 1, ..., Kt in integers; the fleet composition is represented by the K-vector zt :=

(z1t...zKt)

f_kt+ planned addition of planes in capacity segmentk between the fleet for t

and the fleet fort + 1, in number of aircraft

f_kt− planned removal of planes in capacity segmentk between the fleet for t

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Variables: Stage 2 (t = 1) to Stage T xk

it a binary variable, where value 1 indicates flighti is flown by

a plane from capacity segmentk, and 0 otherwise, i = 1, ..., Nt,k = 1, ..., Kt

ykgt number of aircraft from capacity segmentk on ground arc g, k = 1, ..., Kt

d+_it surplus in number of passengers for flighti at time t, i = 1, ..., Nt

d−_it shortage in number of passengers for flighti at time t, i = 1, ..., Nt

Stochastic parameters

Dit random demand in number of passengers for each flighti, at stage t with realization Dits in

scenarios

Deterministic parameters

zkc initial fleet i.e. current fleet, aK0-vector of all capacity segmentsk at t = 0

γk capacity in plane segmentk in number of seats

bi block time per flight legi in hours

rk limit block time range per capacity segmentk in hours

pkt fixed daily costs in euros of using an aircraft from capacity segment k at time t, t =

0, ..., T − 1

h+_kt fleet deviation unit penalty cost in euros for adding an aircraft to capacity segmentk at time t, t = 0, ..., T − 1

h−_kt fleet deviation unit penalty cost in euros for eliminating an aircraft from capacity segment

k at time t, t = 0, ..., T − 1 ck

it daily costs in euros of assigning an aircraft from capacity segmentk to flight leg i at time t,

t = 1, ..., T

q+_it passenger surplus unit penalty cost in euros for flighti at time t, t = 1, ..., T q−_it passenger shortage unit penalty cost in euros for flighti at time t, t = 1, ..., T Zmin

1 minimum total number of aircraft needed to fly schedule for 2010

Zmin

2 minimum total number of aircraft needed to fly schedule for 2015

Output

z0 aK0-vector representing fleet composition for stage 2 (t = 1) decided in stage 1 (t = 0),

Zt:= (z1t...zKt)

zs

1 aK1-vector representing fleet composition for stage 3 (t = 2), where s ∈ S1 = {L, M, H}

Constraints

First stage constraints (t = 0):

X

k∈K0

zk0 ≥ Z1min (1)

zk0− f_k0+ + f_k0− = zkc, k ∈ K0 (2)

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Constraint (1) represents the total number of all planes over each capacity segment needed to fly the schedule for 2010. When solving the model, the constraint is defined as an equality constraint, mean-ing that the total sum of aircraft is equal to a given minimum. Constraint (2) allows to count the number of aircraft removed or added to the initial fleet of 2005 and sets this equal to the fleet com-position for 2010. This constraint penalizes large deviations between the fleet comcom-positions of 2005 when determining that of 2010. This determines the first part of our growth path.

Second and Third Stage Constraints (t = 1, 2):

X k∈K zkt≥ Ztmin (3) zkt− fkt++ fkt−= zkt−1 (4) X k∈Kt xk_it= 1, i ∈ Nt (5) y_gkt|in v− y k gt|out v+ X i∈arr(v) xkit− X i∈dep(v) xkit= 0, k ∈ Kt, v ∈ Vt (6) zkt−1− X l∈Lt y_gkt|in l = 0, k ∈ Kt (7) X k∈Kt X i∈Nt:bi≥rk xk_it= 0 (8) X k∈Kt X i∈Mk t xk_it= 0 (9) d+_it+ d−_it = Ds_it− X k∈Kt γkxk_it, ∀i ∈ Nt∀s ∈ St (10)

The second and third stage consist of the same constraints, except for constraint (3) which is used only in the stages where a fleet composition is determined. Thus this constraint is used until stageT − 1,

which is the year 2010 in our case, the 2nd stage in our multistage model. Constraint (3) and (4) have the same functions as the constraints in the first stage. In constraint (3), the parameterZmin

2 represents

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the nodes. It accounts for every movement of an aircraft from where it starts until where it finishes in the network. This means that each aircraft coming into an airport, must either leave, or stay at the airport. Thus, the number of aircraft on all incoming arcs is equal to the number of aircraft on all outgoing arcs. The third constraint states that the sum of the number of planes at airportl at the start

of the dayP

l∈Lty

k

gt|in l is equal to the number of planes of typek needed to fly the schedule zkt−1.

Constraint (8) states that if the distance of a flight exceeds the maximum range of a segment type k

plane, both measured in block hours, then such a plane is not allowed to operate flighti. Constraint (9)

applies to the regulations of some European airports, where it is prohibited for certain types of planes to land, for reasons such as noise pollution. Thus, flighti cannot be assigned to capacity segment k if i ∈ Mtk.

The last constraint listed is the recourse constraint for demand. Our goal constraint is that each flight assignment with corresponding capacity segmentk satisfies demand Difor flighti. The variables d+it

andd−_itrepresent the deviations from this goal for the schedule at timet, thus respectively the surplus

of passengers and shortage of passengers with respect to the assigned capacity for flighti.

Objective function min X k∈K0 pk0zk0+ X k∈K0 h+_k0f_k0+ + X k∈K0 h−_k0f_k0− + Q1(z0) with, fort = 1, ..., T Qt(zt−1) =ED t[vt(zt−1, Dt)] vt(zt−1, Dt) = min{ X i∈Nt   X k∈Kt ck_itxk_it+ q_it+d+_it− q_it−d−_it   + X k∈Kt pktzkt+ X k∈Kt h+_ktf_kt++ X k∈Kt h−_ktf_kt− + Qt+1(zt) : constraints}

where in the last stage,t = T , it holds that Qt+1(zt) = 0

The objective is to minimize expected total costs. Expected total costs is the sum of costs given current decisions and expected future costs. The first stage consists of the fixed costs for fleet composition in 2010 , the penalty costs for removing aircraft from the initial fleet in capacity segment k, and the

penalty costs for adding aircraft to the initial fleet in capacity segmentk. The second stage consists

of the recourse penalty cost functionv1(z0, D1) which are the costs for the fleet assignments for the

2010 fleet, the fixed costs and the fleet deviation costs for 2015 fleet compositions. The third, and last stage in the model for KLM, consists of only the recourse penalty costs ofv2(z1, D2) for the fleet

assignments of the 2015 fleet.

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5.2.2 Data Parameters

The parameters can be divided into two groups, the ones related to the schedule and the ones related to the aircraft representing each capacity segment. Recall that the schedules are fixed in the model and the schedules provided for 2010 and 2015 are each just one of the many possibilities. The ca-pacity segment related parameters, such as fixed costs, operational costs, and range, were provided in KLM’s Quick Reference Guide (QRG) [7], a guide that provides information on many different types of aircraft. Furthermore, the time value of money is reflected in the costs that were discounted. All costs having to do with the 2015 fleet composition and the 2015 fleet assignments were discounted with 82%, which corresponds to about 3.3% per year.

The table below lists the parameters for the 2010 and 2015 schedules provided by KLM. As explained under constraints in Section 5.2.1, a minimum total of aircraft is assigned for each schedule. This minimum was determined by running the model with fixed costs equal to 1 and all other costs equal to 0.

Table 2: Parameters for 2010 and 2015 schedules

Parameter Schedule 2010 Schedule 2015

Number of flights Nt 576 630

Number of airports Lt 77 82

Total number of planes Zmin

t 92 103

Next, the parameters related to the capacity segments can be found in Table 3. Each capacity segment is represented by a particular type, listed in the second column. The actual names of the types can be found in Appendix D.

The Range of an aircraft is defined by block times (in hours) instead of distance in kilometres. This is because the costs were only available per block hour and therefore the length of each flight in the schedule is given in number of block hours.

The operational costs for a flight depends on the type of aircraft and the length of the flight, which is measured in block time. These operating costs consist of costs for maintenance, cockpit crew, cabin crew, fuel, flight operations, handling, and catering.

The fixed costs in the model were originally listed under operational costs in the QRG as ownership costs. In order to determine a fixed cost per day per capacity segment, these ownership costs per block hour were converted into costs per day using the average block time per year for each type.

The Capacity of planek refers to the capacity segment it is in. Take note that the capacities of these

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Table 3: Capacity segment parameters

Capacity Reference Range Fixed cost per day Operational cost per Capacity

Segment Type (hours) (Euros) block hour (Euros) (seats)

k rk pk ck γk 50 F50 1hr 15min (1.25) 1299 2674 50 70 J70 2hrs 45min (2.75) 4175 3405 70 80 F70 2hrs 30min (2.5) 1578 3628 80 100 J100 2hrs 45min (2.75) 5121 4274 100 110 J110 2hrs 45min (2.75) 5696 4429 110 130 B737 5hrs 30min (5.5) 6022 5489 130 150 B734 3hrs 30min (3.5) 3442 5765 150 170 B738 5hrs 30min (5.5) 7412 6356 170 180 B739 5hrs 30min (5.5) 7873 6545 180

The initial fleet represents KLM’s nett European fleet in 2005. However, since the capacity segment parameters are type-dependent, not all types of the initial fleet were included. Only the types that will be considered for the fleet compositions 2010 and 2015 appear in the initial fleet. For example, we take the F100 which is part of KLM’s 2005 fleet and belongs to the capacity segment of 100 seats. It is assumed to be eliminated by 2010, which means that only a new type will be purchased for this capacity segment. Therefore the initial fleet has 0 aircraft of the new type for 100 capacity segment rather than the number of F100. The same reasoning applies to KLM’s B737-300, which belongs to capacity segment 130. Table 4 provides the data for the initial fleet.

Table 4: Initial fleet: 2005 Capacity Segmentk Initial fleet: 2005

k zc 50 7 70 0 80 20 100 0 110 0 130 0 150 12 170 13 180 5

The fleet deviation penalty costs are the penalty costs and are assigned to each capacity segment in the fleet composition, for removing or adding a plane to the previous fleet. These costs are ‘artificial’ costs and were determined through means of a sensitivity analysis. Since the 2005 (initial fleet) is related to the 2010 fleet, the fleet deviation penalty costs assigned to the 2010 fleet control the changes

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from the initial fleet. The same holds for the relation between the 2010 and 2015 fleet compositions: the fleet deviation penalty costs assigned to the 2015 fleet control the changes from the 2010 fleet. Given these relations, the 2015 fleet is also related to the 2005 fleet. A high penalty cost for the addition of a plane was assigned to the types that KLM were considering to eliminate in 2010 and 2015. A penalty cost of 0 was assigned to new types. A high penalty cost for the removal of a plane was assigned to new types in the 2015 fleet composition, such that if these new types were included in the fleet composition of 2010, then they will not be removed in 2015. This is because if new types are purchased, then these are intended to be used for the fleet composition of 2015 as well. In order to enforce a maximum number of planes for a capacity segment, upper bounds on zk0and zk1 were

used. This assumption is was used for the two cases, which are presented further on in this chapter. Table 5: Deviation penalty cost parameters:

Schedule 2010 Schedule 2015 Capacity Segment Addition Removal Addition Removal

k h+_k0 h−_k0 h+_k1 h−_k1 50 10,000 0 8,200 0 70 0 0 0 8,200 80 10,000 0 8,200 0 100 0 0 0 8,200 110 0 0 0 8,200 130 0 0 0 8,200 150 10,000 0 8,200 0 170 0 1,000 0 8,200 180 10,000 1,000 8,200 8,200

Passenger surplus is demand that exceeds the available capacity of an aircraft that has been assigned to a flight. In other words, it is the number of passengers that are not taken on the flight, which is known as spillage in aviation terminology. The penalty cost for passenger surplus is represented by yield, which is different for each flight. Thus the costs for leaving passengers behind represents the potential loss in yields. The higher the surplus unit penalty cost per flight means the higher the loss in yield per passenger. This way the yields of flights also play a role in the model.

A Long-Term Fleet Composition Study for a European Network