Generating Future Flight Schedules for
Airports by Sequential Optimization
Master’s Thesis Operations Research
Geeske Veenstra (s2164825)
Master’s Thesis Operations Research
Generating Future Flight Schedules for
Airports by Sequential Optimization
Geeske Veenstra
Abstract
Acknowledgements
This thesis has been written as part of the completion of the requirements for the degree of Master of Science in Operations Research at the Faculty of Economics and Business of the University of Groningen.
To begin with, I would like to thank Maarten van der Vlerk for his important contribution during the first part of the process. Also later, he and his thoughts have been an inspiration and motivation. Furthermore, special gratitude goes to Bram de Jonge for taking over and providing me with valuable feedback. I also thank Ward Romeijnders for his constructive comments.
Lastly, I would like to thank the Royal Schiphol Group and specifically my supervisor Ramon van Schaik for setting-up this assignment and supporting me during my time at the depart-ment Traffic Analysis and Forecast.
Contents
1 Introduction 1
2 Literature Review 2
2.1 Timetabling studies . . . 3 2.2 Air transport studies . . . 4
3 Problem Description 5
3.1 Flight characteristics . . . 7 3.2 Capacity . . . 11
4 Solution Approach 11
4.1 Mixed-integer program . . . 13
5 Case Study - Amsterdam Airport Schiphol 22
5.1 Additional series of flights . . . 23 5.2 Schedules 2025 . . . 26 5.3 Practical insights . . . 36
6 Conclusions and Future Extensions 37
A Appendices 41
1
Introduction
The air transport industry has experienced substantial growth over the past few decades. Both the number of passengers and the number of air traffic movements showed significant increases and are expected to rise even further in the future. According to Airbus, one of the most important global aircraft manufacturers, the passenger traffic will increase by 4.5% per year in the period 2016-2035 [1]. This would lead to more than a doubling of the number of passengers in this period. As a consequence, the number of air traffic movements in 2035 is also expected to be more than twice as large as in 2016 [1]. Due to this growth, many airports face increasing loads for their capacities and the question is whether airport infras-tructures can sustain the high growth rates. Examples of vital considerations are whether the construction of an additional runway will be necessary in order to accommodate the future number of flights, and whether an additional pier will be essential to provide enough gates for boarding the airplanes. In order to answer these types of questions, it is necessary to be able to estimate accurately how many flights will use an airport at certain days in the future, typically ten or fifteen years ahead, and, for each day, how these flights are distributed over the day. Herein the difficulty is that these estimates are usually hard to determine and credibility is often lacking.
As an illustration, Amsterdam Airport Schiphol (abbreviated as AMS) is an airport which is becoming increasingly busy over time. It serves as an important European hub airport, since AMS has a lot of direct flights enabling passengers to transfer and reach their final destination [3]. AMS is the main airport in the Netherlands with 64 million passengers and 479 thousand flight movements in 2016 [2]. This results in AMS ranking third among European airports in terms of the number of passengers, and first in terms of the number of movements [18]. It is expected that the number of movements at AMS will continue to grow in the upcoming years. Therefore, it is of interest to construct future flight schedules that incorporate the expected growth. These schedules will be communicated to and used by many departments within Schiphol Airport. For example, the noise nuisance department is interested in the re-sults for their calculations considering several noise criteria. Additionally, departments that are concerned with long-term development of infrastructures such as gates, passenger filters and baggage transport would like to determine whether the current services are adequately regulated for processing the future flows of passengers and aircraft.
moments, we assume that additional flights will have the same preferences regarding flight moments as existing flights belonging to the same region and market segment. Therefore, we will perform data analysis on a recent historical flight schedule, resulting in so-called preferred flight characteristics belonging to a specific region and market segment.
The process of adding new flights consists of two parts. Firstly, additional series of flights per region and segment will be determined including weeks and days of operation on which both an arriving and a departing flight should be scheduled. Secondly, the process of scheduling the additional flights in a time period for each established day will be modeled as a mixed-integer program (MIP). Based on flight characteristics and the fact that airlines desire a consistent schedule, we derive desirabilities regarding flight times. The MIP schedules new flights in the periods which satisfy these desirabilities to the largest possible extent, which is measured by an objective that maximizes a weighted sum of the desirabilities. Scheduling all additional flights simultaneously will lead to an MIP that cannot be solved within reasonable time. Therefore, we will create subsets of the additional series of flights and schedule these sequentially by iteratively applying the MIP. Here, we will start with the subsets which are estimated to be the most difficult to schedule. At the same time, the number of time periods available for scheduling is limited due to operational and environmental restrictions at the airport. A future schedule including the additional flights cannot violate these restrictions. Thus, the MIP will assign additional flights to time periods by optimizing the flight desirabil-ities formulated as a weighted objective and by taking into account constraints concerning the capacity of an airport.
Nowadays, constructing future flight schedules for airports is often performed manually. This requires a lot of time, since flights have to be scheduled one by one. Each flight should obtain a flight time matching the preferences, while restrictions concerning the capacity of an airport cannot be exceeded. Generally, future schedules require the addition of thousands of flights. Therefore, the manual procedure is very time consuming. The method proposed in this study, which incorporates an optimization model, speeds up the process of composing future flight schedules considerably. Moreover, it facilitates the generation of multiple schedules. For ex-ample, an airport might be interested in schedules for a range of years or might desire future schedules for varying estimated numbers of air traffic movements. For such situations, pro-ducing manual solutions is practically impossible, because of time limitations. Instead, the method developed here is less time consuming and offers quick results.
The remainder of this paper is structured as follows. In Section 2 we discuss the relevant literature. Then, we describe the problem of determining future flight schedules in Section 3. The approach that we develop to solve the problem is presented in Section 4. In Section 5 we apply this approach to a case study of Amsterdam Airport Schiphol. Finally, in Section 6 we provide conclusions and suggestions for future research.
2
Literature Review
for airports by taking into account these demand expansions is of paramount importance for anticipating capacity restrictions and performing environment analyses. Scheduling flights at airports basically is a timetabling problem, which means that a number of events have to be assigned to a limited number of time periods [22]. Although the study of timetabling problems has attracted considerable attention from researchers, very few studies have addressed the specific question of scheduling flights at airports. First, we review some studies that discuss timetabling problems in general. Then, we present a review of the studies in air transport industry that most closely match our problem definition.
2.1 Timetabling studies
Here, we will review studies that concern timetabling problems. Burke and Petrovic [9] dis-cuss various approaches to timetabling problems by focusing on universities. These problems have the characteristic that many constraints are imposed, which are either hard constraints which cannot be violated or soft constraints which can be violated but are desirable. For example, each class can only attend one course at the same time, which is a requirement and thus a constraint of the first category. Moreover, a university might prefer to minimize the amount of free time between courses for students. Since this is not a requirement, it is a constraint of the second category. Another characteristic of timetabling problems is that many events need to be scheduled while a variety of constraints is imposed, resulting in large computation times. These computation times always need to be weighted against the solution qualities in order to determine the effectiveness of a solution approach. In the survey article, Burke and Petrovic [9] present several methods to solve difficult timetabling problems such as meta-heuristic methods and sequential methods.
Veenstra and Vis [21] apply a sequential method to school timetabling problems under distur-bances, which basically means that schedules need to be adjusted after certain events occur. The authors solve these problems either by applying an integer linear programming model (ILP) to all classes at the same time or by optimizing the classes iteratively, and determine the quality of a solution by using a weighted sum of several performance measures. The itera-tive approach that they use, has also been discussed by several others. For example, Petrovic et al. [17] extend the approach by scheduling the events in an order based on their estimated difficulty to schedule. Furthermore, Aubin and Ferland [4] first generate a timetable and then assign students to the scheduled courses. They formulate a mathematical model including an objective function that represents the quality of both subproblems.
2.2 Air transport studies
In this section, we review studies devoted to the scheduling of flights at airports. Barnhart et al. [5] wrote a survey article reviewing the applications of operations research in the air transport industry. They mention that detailed simulation models have been constructed that could answer vital questions such as at what point in time an additional runway is required at an airport. These models require future schedules of aircraft movements as input. However, the authors note that these schedules for ten to fifteen years ahead are often hard to obtain. Moreover, in case a schedule for a certain year in the future is composed, it is difficult to determine its quality because of the great amount of uncertainty involved.
An important component of the long-term airport planning could be the question how existing forecasts of passenger numbers should be translated to actual movements. K¨olker et al. [14] construct a model that determines the future aircraft movements and the deployed aircraft. Their frequency-capacity model incorporates the fact that airlines can deal with passenger growth on a particular route by either increasing the frequency of flights or by expanding capacity through using a larger aircraft. The main assumption they use is that the size of an aircraft and the flight frequencies depend on both the yearly passenger volume and the flight distance. Their model incorporates two types of inputs. Firstly, past schedules including information per flight on origin, destination, passenger volume, flight distance and equipment capacity are needed, allowing flights to be categorized in defined segments. Secondly, a prog-nosis of the yearly passenger growth rate applying to the years 2012 until 2030 is required. The authors note here that a forecasted growth rate specified per region instead of worldwide, could result in more appropriate outcomes.
While the model of K¨olker et al. [14] does not result in a schedule, Dennis [13] develops a methodology to construct the route traffic and flight schedule of a medium-sized European airport. Future passenger demand at an airport is obtained by applying forecasted growth rates disaggregated by country to the current demand. Then, this demand is redistributed into route traffic and appropriate frequencies and aircraft sizes are determined. Lastly, new flights are added to an existing schedule such that the frequencies match. This methodology of using a schedule as a basis for future schedules has also been applied in other fields of study such as in airline schedule planning. Lohatepanont and Barnhart [16] state that this type of planning usually begins with an existing schedule to which changes are introduced to reflect developments in demand and environment. Dennis [12] applies the approach developed by Dennis [13] to generate a schedule for the year 2025 at Aberdeen Airport. In this case study, they assign flight times to additional flights such that patterns in an existing schedule are replicated.
hub for important airlines [3]. These differences point out the contribution of this study to the existing literature within the industry. Additionally, it is innovative that we include the fact that the growth in air traffic could vary noticeably per market segment.
To the best of our knowledge, no methods are designed yet to construct future flight schedules for airports based on a timetabling approach. Moreover, the fact that we disaggregate figures both on region and on market segment is a new extension of the air transport literature.
3
Problem Description
In this section, we will describe the problem of generating future flight schedules for airports. Our focus will be on so-called fully coordinated airports, which are airports where the demand for air travel exceeds the capacity. These airports have to provide capacity declarations [19]. A capacity declaration is specified per season (summer or winter) and contains a maximum for the total number of (night) movements during the season. Moreover, days are subdivided into time brackets, and the declaration specifies a maximum for the number of arriving and departing flights per bracket. Because a declaration is valid for one season, scheduling at these fully coordinated airports is performed per season. Our study is motivated by the fully coordinated airport Amsterdam Airport Schiphol (AMS) and we will use this airport to pro-vide explanations where necessary.
Our approach will be to compose future schedules by adding new flights to already exist-ing, current schedules. This method, which is applied in practice, was also suggested by Dennis [13] and Lohatepanont and Barnhart [16], and is reasonable, since current services are expected to be carried out in later years as well. Thus, adding new flights to an already existing schedule results in a more realistic schedule than creating a completely new one. We note that two different methods are used to construct future flight schedules in practice. The first one focuses on short-term future schedules (less than five years ahead) and composes flight schedules from scratch by iteratively adding schedules provided by airlines. However, airlines either do not have or do not want to reveal their schedules for the long-term. There-fore, the approach of adding flights to an existing schedule is applied to construct future flight schedules for five or more years ahead. In this study we consider the second category. Hence, schedules will be generated for at least five years after the current year.
Additional numbers of flights will be calculated based on forecasted future numbers of air traffic movements (ATMs) disaggregated both on region and on market segment. The former disaggregation results from the suggestion by K¨olker et al. [14], the latter is incorporated since growth numbers vary widely among different market segments. We note that further disag-gregations are possible, for example by using exact destinations instead of regions. However, this is beyond the expected level of detail, since a further disaggregation is probable to result in more apparent differences between forecasts and reality. We assume that the estimates for the number of ATMs per region and market segment in the future period under consideration are available and accurate. The difference between these future numbers of flights and the existing numbers of flights, determines the numbers of flights that have to be added.
America and North America. The different market segments are hub, Europe, intercontinen-tal, leisure, low cost and cargo. The hub segment consists of all flights that are operated by so-called hub carriers. These constitute the SkyTeam airlines and KLM codeshare partners (e.g. KLM and Transavia). A flight in the Europe segment is defined in the following way. It is served by a non-hub carrier and has a European destination with more than 10,000 business passengers per year. The same yields for the intercontinental segment, but then for flights with an intercontinental destination with more than 10,000 business passengers per year. A flight is classified in the leisure segment in case it is executed by a non-hub carrier with a European or intercontinental destination with less than 10,000 business passengers per year. A cargo flight is full freighter traffic. The low cost segment does not possess a clear definition at Schiphol. However, it consists mainly of flights operated by airlines which have low fares in general. Examples of airlines in the low cost segment are EasyJet and Ryanair.
A portion of a schedule at AMS is shown in Table 1. This schedule corresponds to the summer season of 2016 (S16), which starts at March 27, 2016 and ends at October 29, 2016. The focus is on what type of movement (arrival or departure) occurs at which date and at what time. As already mentioned, all air traffic movements will also belong to a region and a market segment.
Table 1: Portion of a schedule. Arrival/
Departure Scheduled date Scheduled time Region Market segment
Arrival 27-03-2016 8:15 Europe Hub
Departure 27-03-2016 8:15 Europe Hub
..
. ... ... ... ...
Arrival 29-10-2016 16:05 Europe Low cost
Departure 29-10-2016 16:10 Africa Intercontinental
..
. ... ... ... ...
impossible. As a consequence, the flight should be scheduled at an earlier or a later time, depending on the restrictions. Some deviation from a consistent schedule is allowed. However, the larger the deviation, the higher the associated cost. Since existing flights belong to series of flights, we will schedule the additional flights as series as well, including days and weeks of operation, where we assume that series of flights can only be scheduled in consecutive weeks. Furthermore, we assume that multiple arrivals and departures on one day of the same series of flights are not possible. Another assumption is that each arriving flight should be followed by a departing flight on the same day. This means that so-called stopover flights are not allowed. Additionally, other airports are outside the scope of this research. This indicates that if a departing flight is scheduled to depart to a certain region, we assume that the aircraft can arrive at its destination.
Additional series of flights belonging to a region and a market segment need to be assigned to weeks, days and times. Because airlines do not share their preferences considering these flight moments, an existing schedule is the best option available to assess these preferences. Therefore, we assume that additional flights at an airport will have the same preferences regarding flight moments as existing flights at the same airport, provided that they belong to the same region and market segment. Thus, an existing schedule can be used to determine so-called preferred flight characteristics, which should be replicated as closely as possible by new scheduled flights. A comparable approach is applied by Dennis [12], who use an existing schedule at an airport to ensure that generated future schedules replicate the existing pat-terns. We continue to explain the flight characteristics that are specific for the regions and segments in the following sections.
3.1 Flight characteristics
The region and segment specific flight characteristics that we will consider include preferences concerning weeks, days, flight times and turnaround times. A flight time can be either an arrival or a departure time, and a turnaround time is defined as the time between an arrival and a departure of the same flight. Here, the discussion of these characteristics will be presented successively. We will use the S16 schedule at AMS to provide examples.
3.1.1 Weeks
Depending on the regions and segments, the demand for flights at an airport varies over the weeks of the season. Therefore, some flights operate more frequently during certain weeks than others. In order to investigate in which weeks additional flights should be scheduled, the number of flights in an existing schedule can be determined for each of the different weeks and per combination of a region and a segment.
(a) (b)
Figure 1: Number of flights per week for the Europe segment (a) and the leisure segment (b) from/to Europe.
be seen in Figure 1(b). The flights concentrate around the summer, which is typical for the leisure segment, because this segment mainly deals with the extra demand for flights resulting from school holidays. The histograms for the other regions in the leisure segment demonstrate that all numbers of flights at AMS in the leisure segment, independently from the region, have similar features to Figure 1(b). On the contrary, all others approach the constant frequency comparable to Figure 1(a). Additional flights per region and segment should be scheduled in weeks such that they match the pattern of the corresponding histogram of the number of flights per week.
3.1.2 Days
To develop a future schedule, additional flights should also be assigned to series of days. Sim-ilar to the evaluation of the frequencies per week above, we determine the number of flights per region and segment, and per day.
(a) (b)
Figure 2: Number of flights per day for the Europe segment (a) and the leisure segment (b) from/to Europe, where 0 corresponds to Sunday.
3.1.3 Flight times
Preferences for flight times should be taken into account when scheduling additional flights. These preferences vary widely per region, because of favorable flight times induced by time differences for example between North America and Europe. They also vary per segment, because of specific segment preferences. For instance, flights with many business passengers may have a preference to depart in the morning. Therefore, we will determine the number of flights per combination of a region and a segment, and per five minutes (accuracy of a flight schedule), while distinguishing between arrivals and departures.
To clarify the preferences for flight times, Figure 3 shows the histograms for arrivals and departures from/to North America in the hub segment at AMS. Here, we have grouped the five minutes together per hour to present an evident figure.
(a) (b)
Figure 3(a) shows an arrival peak for flights from North America in the hub segment during the morning. Figure 3(b) shows a departure peak which is after the arrival peak. These and other comparable histograms show departure waves as well as arrival waves specific for the market segments and regions. Additional flights should be scheduled such that these waves are taken into account.
3.1.4 Turnaround times
Each flight should have an acceptable turnaround time. Naturally, a departure should occur after an arrival. The turnaround time is a vital characteristic in flight scheduling for both airports and airlines. According to Wu and Caves [23], airlines need to balance schedule punctuality and aircraft utilization to minimize their cost. Furthermore, they state that air-ports benefit from correctly scheduled turnaround times because of associated aircraft ground operations such as gate planning. To investigate what the preferences of flights are regarding turnaround times, we will analyze the turnaround times in a current schedule per region and segment. We assume that current turnaround times are ideal. This means that additional flights prefer turnaround times which are close to the turnaround times currently present at the airport. Moreover, to ensure that aircraft have enough time between arrival and depar-ture, a minimum turnaround time will be imposed. Similarly, we will impose a maximum turnaround time, since a very large waiting time at an airport will result in undesirably big cost for the airlines and planning difficulties for the airport.
(a) (b)
Figure 4: Number of turnaround times per ten minutes for European flights in the hub segment (a) and in the low cost segment (b).
3.2 Capacity
Apart from preferences concerning the development of future flight schedules, capacity re-strictions have to be taken into account at the considered fully coordinated airports. These airports have to compose a capacity declaration which is valid for a whole season (summer or winter) and contains restrictions on the capacity. Firstly, a maximum number of arrivals and departures per time bracket is imposed. See Appendix A.1 for an example of a so-called bracket list at Amsterdam Airport Schiphol. Secondly, the number of night arrivals and departures during a season is restricted. We note that violation of any of these restric-tions results in an infeasible schedule. As an additional restriction, a maximum for the total number of arrivals and departures per season can be specified. However, this is not imposed here, since we assume that airports are able to accommodate the future growth in the long run. To summarize, we generate future flight schedules at capacity-constrained airports, where we assume that existing schedules can be used to deduce preferences for additional flights and that future schedules can be composed by adding flights to current schedules. At the same time, future schedules have to satisfy several capacity restrictions.
4
Solution Approach
Here, we will continue to define a solution approach to solve the flight scheduling problem as defined in the previous part. First, we will introduce the approach and clarify it by means of a flowchart. Thereafter, we will give a detailed description of the mixed-integer program (MIP) that will be used to schedule flights in periods.
First, we will define some sets. We will denote the set of regions and the set of market segments by R = {1, . . . , Rmax} and S = {1, . . . , Smax}, respectively. Here, Rmax denotes the last region and Smax denotes the last segment. Furthermore, we will use M = {ARR, DEP } to denote the set of movement types of flights, where ARR represents an arrival and DEP represents a departure. The set B = {1, . . . , Bmax} contains the time brackets in a day as established in the bracket list of a capacity declaration with Bmax representing the last time bracket. Moreover, we denote the set of weeks by W = {Wmin, . . . , Wmax}, where Wmin and Wmax are the first and the last week of the planning horizon. Lastly, D = {0, . . . , 6} denotes the set of days, where 0 corresponds to Sunday.
In order to construct a future flight schedule at a capacity-constrained airport, the following inputs are required:
1. A capacity declaration; 2. An existing flight schedule;
3. Accurate estimates of the number of flights (ATMs) per combination of a region and a segment during the future period under consideration.
m in week w ∈ W on day d ∈ D by Cbmwd. These are capacity restrictions for an airport. By taking an existing schedule into account, we can now determine how much capacity is left. Let N N be the number of air traffic movements in the existing schedule during the night, which implies that the remaining capacity for the night as denoted by RN equals RN = N − N N . Similarly, let N Cbmwd be the number of air traffic movements in the existing schedule in bracket b of type m in week w on day d. Then, the remaining capacity of bracket b of type m in week w on day d is given by RCbmwd = Cbmwd− N Cbmwd. Hence, RN and RCbmwd indicate how much capacity is left for scheduling additional flights during the night and in certain brackets, respectively.
The number of additional flights per region and segment can be determined based on an existing flight schedule and estimates of the number of flights in the future. The difference between the future amount of flights and the current amount of flights determines the amount of flights that we need to add per region and segment. All additional flights will belong to series of flights including weeks and days of operation, where the weeks should be consecutive. The total number of series of flights that needs to be added will be denoted by Imax. The set F = {1, . . . , Imax} contains the additional series of flights. We use binary parameters Uirs, i ∈ F , r ∈ R, s ∈ S to indicate to which region r and segment s each additional series of flights i belongs:
Uirs= (
1, if additional series of flights i belongs to region r and segment s, 0, if not.
We will develop a procedure that determines the weeks and days of operation for each ad-ditional series based on the flight characteristics concerning weeks and days as discussed in Section 3.1. The set of weeks and the set of days that additional series of flights i operates will be denoted by Wi ⊂ W and Di ⊂ D, respectively. The procedure to determine Wi and Di is very case specific, since the definitions of market segments and regions as well as the numbers of flights within these categories, vary widely among airports. In Section 5.1 we will explain how we determine these quantities for our case study that focuses on Amsterdam Airport Schiphol.
In the remainder of this section, we assume that the set of additional series of flights F and the attributes Uirs, Wi and Di have been established. All series of flights should be assigned to an arriving and a departing flight time for each determined combination of a day and a week.
It is well known that timetabling problems with a large number of variables can be very difficult to solve. Moreover, based on our own experience considering the complexity of this problem, formulating a single MIP for the entire problem will lead to a model that cannot be solved within reasonable time. Therefore, we will decompose the large problem of scheduling all additional series of flights at once into sub-problems. We compose subsets of the additional series of flights and schedule each of these subsets subsequently by iteratively using the MIP. A numerical analysis can be carried out to determine an appropriate size of these subsets such that the calculation time is acceptable. This sequential heuristic based on Petrovic et al. [17] involves sorting the items to schedule starting with the item that is estimated to be the most difficult to schedule. To achieve an ordering, several heuristics have been proposed which are discussed in Burke and Newall [8]. One of the common ordering heuristics used is the saturation degree heuristic, which first schedules the events with the least number of periods available, since no more options might be available if such an event is scheduled during a later iteration. For example, within course scheduling, we would first schedule courses taught by the lecturer with the least availability. Scheduling these courses later might be a problem if the options for this lecturer already have been exhausted through the scheduling of other courses.
We estimate the difficulty of scheduling per additional series of flights by applying the sat-uration degree heuristic. Therefore, we first select the series of flights which have the least number of periods available. Series of flights that consist of a large number of flights |Di|·|Wi| do not have many options for scheduling, since a more or less consistent schedule is required over the weeks and days. Moreover, series with a small difference between the maximum and minimum turnaround time are also prioritized, since this results in less possible combinations of arrival and departure periods. By first selecting the series with the largest number of flights and then ordering these series according to the smallest difference between maximum and minimum turnaround time, a sequence of series is achieved which starts with the item that is estimated to be the most difficult to schedule. We note that we could either apply the MIP to one series after the other, or use larger subsets of ordered additional series. It is of interest to investigate whether the use of a larger subset changes the resulting schedules substantially.
The heuristic approach iteratively optimizes the schedule for each subset of additional se-ries, starting with the subset that is classified as being the most difficult to schedule. After each application of the MIP, the current schedule is updated by adding the scheduled flights. This means that N Cbmwd and N N have to be determined again to reflect the new used night and bracket capacities. According to Burke and Newall [7], a considerably better timetable is expected to be obtained by first scheduling the most difficult additional series of flights compared to a timetable composed by adding additional series of flights in a random order. In Figure 5, we present a flowchart containing the explained steps. The output will be a forecasted flight schedule at an airport for a future period under consideration.
4.1 Mixed-integer program
Figure 5: Flowchart containing the steps required to generate a future schedule.
The goal of the MIP is to schedule all flights, belonging to a particular subset of additional series, in periods such that the resulting schedule is the most preferred one based on input data obtained from an existing schedule. We will denote a subset under consideration by I ⊂ F . Periods have length t, where t is chosen such that the lengths of all brackets b are a multiple of t. Moreover, the set of the number of periods on a day is denoted by P = {1, . . . , Pmax}, where Pmax is the last period.
4.1.1 Explanation
We use binary decision variables Ximwdp, i ∈ I, m ∈ M , w ∈ W , d ∈ D, p ∈ P to indicate whether a flight from additional series i of movement type m will be scheduled in week w on day d in period p: Ximwdp=
1, if a flight from additional series i of movement type m will be scheduled in week w on day d in period p,
We note that if series i is such that d ∈ Di and w ∈ Wi, both an arriving and a departing flight should be scheduled on day d in week w. This implies that for all m, Ximwdp will equal 1 for exactly one period p ∈ P , if w ∈ Wi and d ∈ Di.
Based on flight characteristics concerning flight times and turnaround times, and on the observation that airlines prefer a consistent schedule as discussed in Section 3, each addi-tional series of flights i has a set of desirable objectives. The following desirabilities will be considered in the formulation of the MIP:
1. The desirability that a flight from/to a particular region belonging to a particular seg-ment arrives/departs in a certain period;
2. The desirability for a flight belonging to a particular region and segment to have the optimal turnaround time for that combination of the region and the segment;
3. The desirability for a consistent schedule.
We note that the first two desirabilities depend on the region r and market segment s to which series i belongs, Uirs. The last one is a general desirability for all series. When scheduling the additional flights, the desirabilities should be taken into account as much as possible. In order to achieve this goal, we will formulate an objective that maximizes a weighted sum of five separate objectives. We will use the parameters α, β, γ, δ and to denote the nonnegative weights of the different terms. We will formulate the first desirability as the first term of the weighted objective by transforming histograms as Figure 3 into parameters. Additionally, we will formulate the other two desirabilities as soft constraints including violation variables. These soft constraints will contribute two times two additional terms to the weighted objec-tive function. In the remainder of this section, we will discuss the desirabilities and their formulations, successively.
Desirabilities
The number of existing flights per period, and per region and segment will be translated into scores for additional flights. Let gmprs be the score that a flight of type m in region r and segment s receives if it is scheduled in period p. We establish these gmprsin the following way. First, we determine the number of flights in an existing schedule of type m from/to region r belonging to segment s for all periods. Second, we rank these such that the period with the highest number of flights in the past, achieves a parameter value equal to Pmax. Periods with a lower number of flights in the past, achieve a lower value with a minimum equal to 1. By applying this ordering, periods which had more flights in the past, obtain higher associated values. We will model the desirability for flight times by incorporating all gmprs in the first term of the weighted objective, where α denotes its weight. This implies that we will try to optimize the arrival and departure periods of all arriving and departing flights per region and segment. Because gmprs resembles past observations, we ensure a preference for a replication of patterns that currently exist in schedules.
times in an existing schedule (rounded to the nearest integer) and denote this by OP Trs. The turnaround time is a vital element of the MIP and is defined as the time between an arriving and a departing flight. Because the turnaround time will appear frequently in the model, we decide to introduce a new variable Oiwd ∈ Z+ for the turnaround time in minutes for a flight from series i in week w on day d. It is defined as follows:
Oiwd = t · X p p · Xi,DEP,wdp− X p p · Xi,ARR,wdp ! .
We assume that each flight prefers to have a turnaround time equal to its optimal turnaround time OP Trs. However, formulating this as a constraint for all flights will probably result in an infeasible problem. Therefore, we will distinguish between hard constraints, which are essen-tial for a feasible solution and soft constraints, which are actually preferences and not essenessen-tial for a feasible solution. Cong et al. [11] introduce a so-called penalty method for handling soft constraints using linear programming. They note that violation of a soft constraint results in a cost and introduce violation variables denoting the amount of violation. Moreover, a penalty depending on these violation variables is defined. We will apply this method of Cong et al. [11] in order to formulate preference constraints concerning the turnaround times. Violation of the optimal turnaround time results in a penalty representing the associated cost. In case there is a deviation from the optimal turnaround time OP Trs, one of the following violation variables will have a nonzero value:
mmiwd = X
r X
s
(Oiwd− OP Trs) · Uirs ∈ Z+: number of minutes more than the optimal turnaround time for a flight from series i in week w on day d; mliwd = X r X s
(OP Trs− Oiwd) · Uirs ∈ Z+: number of minutes less than the optimal turnaround time for a flight from series i in week w on day d.
Nonzero values of mmiwd and mliwd will receive a penalty in the objective function to en-sure that turnaround times close to OP Trs are most beneficial. The nonnegative weights for scheduling a turnaround time of one minute more and one minute less than the optimal turnaround time are denoted by β and γ, respectively. We conclude that the second term of the weighted objective will represent the total penalty for all turnaround times exceeding the optimal turnaround times, weighted by β per minute exceedance. Similarly, the third term will represent the total penalty for all turnaround times smaller than the optimal turnaround times, weighted by γ per minute.
Desirability 3 concerns the preference for a consistent schedule. Violation of a consistent schedule does not result in infeasibility. However, airlines and airports prefer to have a sched-ule which is as consistent as possible. Based on an approach used by Lam et al. [15], we select the mean flight time as the preference. This preference is denoted by ¯Xim ∈ R+ and represents the mean arrival/departure time in minutes for series i of movement type m during the planning horizon. It is defined by:
Small day-to-day variations from this mean are allowed, but will be penalized in the objective, again by following the idea of Cong et al. [11]. In case there is a deviation from ¯Xim, one of the following violation variables will have a nonzero value:
amimwd= t X
p
p · Ximwdp− ¯Xim∈ R+: number of minutes more than the mean for a flight from series i of type m in week w on day d; alimwd= ¯Xim− t
X
p
p · Ximwdp∈ R+: number of minutes less than the mean for a flight from series i of type m in week w on day d. Nonzero values of amimwd and alimwd will obtain a penalty in the objective function. The nonnegative weights for scheduling a deviation of one minute more and one minute less than the mean arrival/departure time are denoted by δ and , respectively. The fourth part of the objective will capture the total penalty for all minutes of scheduled flight times that exceed the mean times, weighted by δ. Moreover, the fifth part will measure the total penalty for all minutes less than the mean times, weighted by .
Hard constraints
Two of the flight desirabilities above have been modeled as soft constraints. On the other hand, hard constraints will be imposed as well. A violation of one of these results in an infeasible schedule. Firstly, constraints concerning the capacity of an airport should be taken into account. A capacity for the number of arrivals and departures per time bracket is con-sidered and the remaining capacity can be determined by RCbmwd= Cbmwd− N Cbmwd. We assume that the lengths of all brackets b are a multiple of t. Let lb denote the set consisting of the periods that correspond to bracket b, lb ⊂ P . For instance, if we set the length of a time period t = 10 minutes, then l1 = {1, 2} in the bracket list in Appendix A.1, since the first bracket has a length of 20 minutes. Moreover, a maximum number for night arrivals and departures N is considered, where PARR⊂ P and PDEP ⊂ P denote the set of periods during which a flight is characterized as a night arrival and a night departure, respectively. Similar to the remaining bracket capacities, the remaining night capacity can be determined by RN = N − N N . Besides capacity constraints, hard constraints concerning the turnaround times exist. As explained, a maximum turnaround time cannot be exceeded. Additionally, a turnaround time cannot be shorter than a minimum. We set the maximum turnaround time M ATrs and the minimum turnaround time M ITrs equal to the 95% and 5% interval of the turnaround times per region r and segment s in an existing schedule, respectively.
In the next part, we will present the precise mathematical formulation of the explained MIP that will be used to schedule the additional flights in periods.
4.1.2 Mathematical formulation
Sets
The following sets will be used:
F := {1, . . . , Imax} set of additional series of flights;
W := {Wmin, . . . , Wmax} set of week numbers in the planning horizon, where Wmin and Wmax are the first and last week of the horizon; D := {0, . . . , 6} set of days, where d = 0 corresponds to Sunday;
P := {1, . . . , Pmax} set of time periods, where Pmax is the last time period of a day;
M := {ARR, DEP } set of movement types, where ARR and DEP represent arrival and departure, respectively;
R := {1, . . . , Rmax} set of regions, where Rmax is the last region;
S := {1, . . . , Smax} set of market segments, where Smax is the last segment; B := {1, . . . , Bmax} set of time brackets corresponding to the bracket list,
where Bmax is the last bracket. Furthermore, we define sets which are subsets of the sets above:
I ⊂ F set of additional series of flights under consideration in the MIP;
PARR⊂ P set of periods during which an arrival is characterized as a night arrival; PDEP ⊂ P set of periods during which a departure is characterized as a night departure;
lb ⊂ P set of periods corresponding to time bracket b in the bracket list. The indices can take the following values:
i ∈ I, w ∈ W, d ∈ D, p ∈ P, m ∈ M, r ∈ R, s ∈ S, b ∈ B.
To make the notation more concise, we will just write ∀d instead of ∀d ∈ D and d instead of d ∈ D in the remainder of this section. Similarly, for the other indices in case the bound consists of the set above.
Parameters
Analysis of an existing flight schedule will result in values for the following parameters: gmprs score that a flight of type m from/to region r belonging to segment s receives
if it arrives/departs in period p;
M ITrs minimum turnaround time in minutes for a flight belonging to region r and segment s;
M ATrs maximum turnaround time in minutes for a flight belonging to region r and segment s;
N Cbmwd number of air traffic movements in the existing schedule in bracket b of type m in week w on day d;
RCbmwd remaining capacity of bracket b for movement type m in week w on day d; N N number of air traffic movements during the night in the existing schedule;
RN remaining night arrival and departure capacity. Moreover, the following parameters should be established:
Cbmwd capacity of bracket b for movement type m in week w on day d; N arrival and departure capacity during the night for a season;
t length of a time period.
The nonnegative weights for the five different terms of the objective are defined as follows: α weight for scheduling a flight in a period which matches historical occurrences; β weight for scheduling a turnaround time of one minute more than the optimal
turnaround time;
γ weight for scheduling a turnaround time of one minute less than the optimal turnaround time;
δ weight for scheduling a deviation of one minute more than the mean arrival/ departure time;
weight for scheduling a deviation of one minute less than the mean arrival/ departure time.
The next parameters are dependent on the additional series of flights:
Uirs= (
1, if additional series of flights i belongs to region r and segment s, 0, if not;
Wi ⊂ W set of weeks that additional series of flights i operates; Di ⊂ D set of days that additional series of flights i operates. Decision variables
The following binary decision variables will be determined by the MIP:
Ximwdp=
1, if a flight from additional series i of movement type m will be scheduled in week w on day d in period p,
0, if not.
For ease of notation, the following auxiliary variables are specified:
Oiwd∈ Z+ turnaround time in minutes for a flight from series i in week w ∈ Wi on day d ∈ Di;
¯
In order to formulate the soft constraints as explained in Section 4.1.1, the next violation variables are defined:
mmiwd∈ Z+ number of minutes more than the optimal turnaround time for a flight from series i in week w ∈ Wi on day d ∈ Di;
mliwd∈ Z+ number of minutes less than the optimal turnaround time for a flight from series i in week w ∈ Wi on day d ∈ Di;
amimwd∈ R+ number of minutes more than the mean for a flight from series i of type m in week w ∈ Wi on day d ∈ Di;
alimwd∈ R+ number of minutes less than the mean for a flight from series i of type m in week w ∈ Wi on day d ∈ Di.
Model
Here, we will specify the objective and the constraints of the MIP. Thereafter, we will clarify all parts successively. Each iteration of the MIP, a subsequent subset of additional series of flights I from the sequence, which is ranked on estimated difficulty to schedule, is considered input for the MIP. This implies that all series dependent parameters Uirs, Wi and Di will be updated before starting a new iteration. After applying the MIP, the schedule is updated by including the scheduled flights. Then, the current used night and bracket capacities should be recalculated, while the rest of the parameters do not change. The next subset is selected and the procedure is repeated until all additional series of flights are scheduled.
X i X w∈Wi X d∈Di X p∈PARR Xi,m=ARR,wdp+ X i X w∈Wi X d∈Di X p∈PDEP Xi,m=DEP,wdp ≤ RN (11) Oiwd= t · X p p · Xi,DEP,wdp− X p p · Xi,ARR,wdp ! ∀i, w ∈ Wi, d ∈ Di (12) X r X s Oiwd· Uirs ≥ X r X s M ITrs· Uirs ∀i, w ∈ Wi, d ∈ Di (13) X r X s Oiwd· Uirs ≤ X r X s M ATrs· Uirs ∀i, w ∈ Wi, d ∈ Di (14) X r X s
(Oiwd− OP Trs) · Uirs− mmiwd+ mliwd = 0 ∀i, w ∈ Wi, d ∈ Di (15)
¯ Xim= tP w P d P p p · Ximwdp |Wi| · |Di| ∀i, m (16) tX p
p · Ximwdp− ¯Xim− amimwd+ alimwd= 0 ∀i, m, w ∈ Wi, d ∈ Di (17)
Ximwdp∈ {0, 1} ∀i, m, w, d, p (18)
Oiwd, mmiwd, mliwd ∈ Z+ ∀i, w ∈ Wi, d ∈ Di (19)
¯
Xim, amimwd, alimwd∈ R+ ∀i, m, w ∈ Wi, d ∈ Di (20) The objective function consists of a weighted sum of five terms, where α, β, γ, δ and are the weights. Firstly, (1) ensures that a flight is added in a period that is most beneficial considering its region and market segment. Periods which had a high number of flights in the past are more beneficial as measured by the parameters gmprs. We note that both arriving and departing flights contribute separately to this part of the objective. Parts (2) and (3) exist to lower the objective in case there is a deviation from the optimal turnaround time (penalty for the violation of a soft constraint). Similarly, due to parts (4) and (5), the ob-jective will receive a penalty if there is a deviation from the mean scheduled arrival and/or departure time. Simultaneously optimizing the arrival and departure periods and minimizing the deviations from the optimal turnaround times as well as minimizing the deviations from the mean times, can be conflicting. Therefore, we should be careful in selecting the weights such that the resulting schedules match the conceptions of airports.
during which an arrival is categorized as a night arrival. Similarly, PDEP consists of the periods corresponding to night departures. These sets should be specified by the airport. Constraints (12) determine the turnaround times for the series of flights in the subset for the established weeks and days per series. Constraints (13)-(15) consider the minimum, max-imum and optimal turnaround times. Because these times are specific for the regions and segments, Uirs is included in each of the constraints. Turnaround times that are shorter than a particular minimum are excluded by (13). At the same time, these constraints avoid that a departure occurs before an arrival. Undesirable long turnaround times are prohibited by (14). Constraints (15) are in place to determine the number of minutes more or less than the optimal turnaround times, such that these can be penalized in the objective ((2) and (3)). The purpose of constraints (16) is to determine the mean arrival and departure times for the series of flights in the subset. Based on these means, constraints (17) calculate the number of minutes more or less than the mean times (penalized in the objective by (4) and (5)). The binary decision variables are set binary as indicated by (18). Finally, constraints (19) and (20) define the domains of the auxiliary variables and the violation variables.
5
Case Study - Amsterdam Airport Schiphol
Amsterdam Airport Schiphol (AMS) is interested in future flight schedules that incorporate the expected growth at the airport. For ease of notation, we will number the regions and the segments as in Table 2. Thus, we have R = {1, . . . , 6} and S = {1, . . . , 6}. There will not be flights for all combinations of a region r ∈ R and a segment s ∈ S. At the airport, accurate estimates of the number of air traffic movements per combination of a region and a segment are available for future periods as far ahead as 2030. The most recent declared bracket list is shown in Appendix A.1. It can be seen that the length of each bracket is a multiple of 10 minutes, with a maximum of 30 minutes. At AMS current schedules have a precision of 5 minutes. Since we assume that the lengths of all brackets b should be a multiple of the length of a time period t, setting t = 5 would be an option. Instead we reduce the size of the model by setting t = 10 minutes. This implies that each day has Pmax = 24·6010 = 144 periods and thus that P = {1, . . . , 144}. Moreover, the maximum number of night arrivals and departures is N = 23, 219 for one season. Lastly, the set of periods during which an arrival is characterized as a night arrival is specified by PARR = {1, . . . , 43, 140, . . . , 144}. Similarly, night departures are specified by the set PDEP = {1, . . . , 41, 138, . . . , 144}.
Table 2: Numbering of the regions and the segments at AMS.
r Region
1 Africa
2 Europe
3 Far East 4 Middle East
AMS prefers to create schedules for summer seasons by using two different existing schedules. The short-term forecasted schedule for the summer of 2017 (S17) will be used to determine the current number of flights per region and segment, and the used bracket and night capac-ities. The input data gmprs, M ITrs, M ATrs and OP Trs that are used in the MIP, as well as the data that will be used for the determination of the additional series of flights, will be calculated based on the past flight schedule for the summer of 2016 (S16). To briefly repeat here, gmprsrepresents the score that a flight of movement type m from/to region r belonging to segment s receives if it arrives/departs in period p. Moreover, M ITrs, M ATrs and OP Trs denote the minimum, maximum and optimal turnaround time in minutes for a flight belong-ing to region r and segment s, respectively. Lasly, the data used for the determination of the additional series of flights will be explained in Section 5.1.
Slight differences exist between the number of flights in S17 compared to S16. However, this disadvantage does not outweigh the advantage that the historical schedule for S16 is more reliable than the forecast for S17. The S17 schedule is reliable for the aggregated num-bers such as the number of flights in a particular region and segment, but less reliable for the exact flight schedule. Therefore, S16 will be used for calculations regarding exact flight moments consisting of weeks, days and periods.
By looking at the flowchart in Figure 5 that contains the steps of our developed method, we note that the first step in the generation of future flight schedules is the determination of additional series of flights per region and segment with weeks and days of operation. There-fore, in Section 5.1, we will first explain a procedure to establish additional series of flights F = {1, . . . , Imax} including the series dependent parameters Uirs, Wi and Di, denoting the binary parameters that indicate the region and segment, the set of operative weeks and the set of operative days per series of flights i, respectively. Thereafter, in Section 5.2, we will present examples for the development of future schedules by following the subsequent steps in Figure 5. These steps include determining the additional series of flights and ordering them after that such that they can be scheduled iteratively by the MIP. Lastly, in Section 5.3, we will derive practical insights for our case study of AMS.
5.1 Additional series of flights
that the procedures are independent from the regions r. Furthermore, we note that sets of operative weeks Wi should consist of consecutive weeks for all additional series of flights. 5.1.1 Europe, hub, intercontinental, low cost and cargo
The numbers of flights per week in the Europe, hub, intercontinental, low cost and cargo segments are, independently from the regions, quite constant throughout the season. Figure 1(a) demonstrates this for flights belonging to the European region and the Europe segment. Since we prefer to replicate existing weekly patterns as closely as possible, the number of additional flights for all combinations of regions and segments except the leisure segment, should approximately be equal for all weeks. The number of weeks in S16 is 31. Therefore, we multiply the total number of flights to add per region and segment for one season by 311 and the result then indicates how many flights we should approximately add each week. We start with determining additional series of flights which operate all days of the week, thus 7 · 2 = 14 flights per week during the entire season (one arrival and one departure per day). We continue with series of 14 flights per week until the number of additional flights left (for a combination of a region and a segment) is less than 14 per week.
Next we establish additional series of flights which operate 6, 5, 4, 3, 2 and 1 day(s) per week, successively. In these cases, the days of operation are determined randomly by taking a sample without replacement from the set of days D. Thereafter, the maximum number of additional flights left per week is less than 2 and the total for the season is less than 31·2 = 62. At this point, it is no longer possible to determine additional series of flights which operate the entire season. If the total remainder exceeds 2, we constitute a final series. A random day is selected and we start assigning flights of this final series to this day and to weeks starting from the peak of the season. At AMS, this corresponds to the middle of the summer weeks, which is week number 31 (July 31-August 6). This approach is used, because the summer is the busiest period for an airport, and because the demand for (additional) flights is highest in this period. After selecting week 31, we continue establishing weeks around week 31, while starting from the left. Whenever the total remainder is smaller than 2, the assignment stops and the number of flights in additional series is now maximally 2 flights smaller than the number calculated beforehand. However, this is not a problem as the difference is consid-erably small in percentage terms. We note that we decide to select the days randomly and therefore do not apply the results from histograms as Figure 2. If we did use such figures, the days assigned would be too much dependent on the existing schedule and particularities in the data would be exaggerated. Nevertheless, if an airport prefers to replicate the patterns of frequencies over the days and thus assigns a higher probability of occurrence to days which had a high number of flights in the past, this can easily be adjusted.
For instance, suppose that the number of flights to add equals 1,001 for a particular region and segment (other than the leisure segment). We start with the determination of seasonal series of flights. We multiply 1,001 by 311 to see that 311 · 1, 001 = 32.29 flights should be added to all weeks w in W = {13, . . . , 43}. First, we determine whether (and how often) a series of flights can be added which operates all days of the week. Since 32.29
and 4.29
2·2 = 1. Hence, one series of flights can be determined and two days are selected by random sampling from D. This results for example in i = 3, W3= {13, . . . , 43}, D3= {1, 3}. Now, 4.29 − 2 · 2 = 0.29 flights are left per week, implying that 0.29 · 31 = 9 flights are left for the season of 31 weeks. Since, 9 < 62 we have a total remainder of 9 and we cannot determine more series of flights operating the whole season with a minimum of one day per week. For the final series, we commence at week 31 and assign two flights (ARR and DEP ) to Wednesday (d = 3), which is selected randomly. Then, 7 flights are left and we allocate these to weeks around week 31. Therefore, two flights should be scheduled on Wednesday in week 30, two flights on Wednesday in week 31 and two flights on Wednesday in week 29. Hence, i = 4, W4 = {29, . . . , 32} and D4 = {3}. Note that the last single flight cannot be scheduled, because an arrival and a departure always occur together. We conclude that we have determined four additional series of flights for a combination of a region and a segment, including weeks and days of operation. A similar approach applies to the other combinations of regions and segments for which the segment is the Europe, hub, intercontinental, low cost or cargo segment.
5.1.2 Leisure
As can be seen in Figure 1(b), European flights in the leisure segment show a completely different pattern compared to European flights in the Europe segment. The frequencies concentrate around the summer, which is typical for the leisure segment. The leisure segment for other regions shows a similar pattern. Therefore, we confirm that spreading the additional flights evenly over the season would not represent the typical pattern of the leisure segment. Because additional series of flights need to be scheduled in consecutive weeks, we choose to smooth the pattern by distinguishing four different options for the sets of operative weeks, Wi, for each additional series of flights i in the leisure segment:
1. Wi= {13, . . . , 43}; 2. Wi= {17, . . . , 43}; 3. Wi= {28, . . . , 34};
4. Wi consisting of the remaining weeks around week 31.
Figure 6: Smoothed number of flights per week for European flights in the leisure segment.
After executing this procedure, additional series of flights are determined which have the following attributes: a region and a market segment (Uirs), a set of operative weeks (Wi) and a set of operative days (Di), where the latter is subject to randomness if the number of operative days is smaller than 7.
5.2 Schedules 2025
To provide examples for the development of future schedules, we create schedules for the summer of 2025 (S25) by applying the solution approach that we discussed in Section 4. Based on expert opinions, we use the summer of 2017 for the determination of the current numbers of flights and for the calculations of the used night and bracket capacities. As already explained, the summer of 2016 will be used for the calculations regarding the procedures developed in Section 5.1. By subtracting the number of flights per region and segment in S17 from accurate estimates for the number of flights per region and segment in S25, we know how many flights we should add. These results are depicted in Table 3.
Table 3: Number of additional flights per region and segment for 2025.
Regions 1 2 3 4 5 6 Segments 1 896 24,572 1,662 584 1,066 1,699 2 0 4,363 0 0 0 0 3 0 0 0 44 18 0 4 387 0 0 34 112 0 5 83 5,766 0 141 0 0 6 166 611 643 392 131 139
objective weights to zero such that the first feasible solution instead of the optimal solution is selected each time the MIP is applied. In Section 5.2.3, we will perform sensitivity anal-ysis with respect to the weights. Then, in Section 5.2.4, we will repeat the procedures to determine the additional series of flights several times. In Section 5.2.5, we will examine different ordering methods. Lastly, in Section 5.2.6, we will investigate the subset sizes of the additional series of flights.
5.2.1 Base settings
Here, we will specify the base settings and generate a corresponding schedule for 2025. By using the numbers of additional flights from Table 3, we execute the first step in Figure 5 and determine additional series of flights per region and segment including weeks and days of operation. After applying the procedures explained in Section 5.1, we establish 131 additional series of flights and thus the set of additional series of flights F = {1, . . . , 131}. Table 4 shows a portion of these additional series.
Table 4: Portion of additional series of flights for 2025.
i Uirs= 1 Wi Di 1 U1,1,1 W1 = {13, . . . , 43} D1 = {0, . . . , 6} 2 U2,1,1 W2 = {13, . . . , 43} D2 = {0, . . . , 6} .. . ... ... ... 130 U130,5,4 W130= {28, . . . , 34} D130 = {3} 131 U131,5,4 W131= {21, . . . , 41} D131 = {0}
Of the 131 series, 38 series are such that the number of operative days is smaller than 7 (i.e. |Di| < 7). Therefore, these series have sets of operative day(s) which are not uniquely defined. In Section 5.2.4, we will execute the procedures to determine the additional series of flights repeatedly. Thereafter, we will generate corresponding schedules and compare them to investigate whether the results are affected significantly. In the remainder of this section, we will continue with the series already established and execute the second step in Figure 5, namely to create a sequence of subsets of additional series of flights based on their estimated difficulty to schedule. We assume that an additional series of flights is difficult to schedule if:
1. The series consists of a large number of flights (i.e. |Wi| · |Di| is high);
2. The series has a small difference between the maximum and minimum turnaround time (i.e. M ATrs− M ITrs is small).
the ordering criteria in later sections.
All calculations are performed with an Intel Core i7 processor at 2.9 GHz and 16.0 GB RAM. By using Figure 5 as a basis, the sequential heuristic is implemented using the program-ming language Python. Moreover, the MIP is iteratively solved by using the CPLEX solver. Based on expert opinions at AMS, we set the penalty β for scheduling a turnaround time of one minute more than the optimal turnaround time equal to the penalty γ for scheduling a turnaround time of one minute less. Similarly, we set the penalties concerning the deviations from the mean arrival/departure times equal, δ = . Besides this, the experts point out that a violation of a consistent schedule by one minute is twice as undesirable as a violation of an optimal turnaround time by one minute. By taking the opinions into account, we will set β = γ = 1 and δ = = 2 initially.
More uncertainty exists regarding the selection of the weight α for scheduling a flight in a period which matches historical occurrences, because the value for α should ensure a cor-rect balance between the different terms of the objective. Therefore, we generate schedules for different values of α and compare them. It appears that schedules for the whole season cannot be obtained due to extremely high computation times. For this reason, we will consider a one week schedule in these numerical experiments, which is still of utmost importance for AMS. Capacity planning at AMS is always performed based on schedules for only a few busy weeks such as a schedule for the May holiday and a schedule for an arrival or a departure peak week during the summer holiday. Moreover, the consistency of a weekly schedule can still be investigated, since the vast majority of series operates on several days (83% in this setting). We select week number 31, since this is expected to be the most busy week in 2025 and thus the most challenging to schedule. Moreover, it is important to note that all additional series of flights operate in this week. We determine schedules for a range of α between 0.25 and 4.00 and present the schedules to experts. The higher the value of α, the more a future schedule is directed at achieving arrival and departure periods that match historical observations. On the other hand, lowering α shifts the focus to obtaining a schedule which is more consistent and more in line with turnaround times observed in the past. The experts establish that a choice of α = 0.5 matches their conceptions most closely.
All 131 additional series of flights operate in week 31, but not all operate on all days. In total, 1,430 flights need to be assigned to an arrival or departure period. We set each Wi to {31} and adjust constraint (11) by assuming that if a flight is scheduled during the night, it will also be scheduled during the night in the other weeks. Therefore, we replace the re-maining night capacity RN on the right-hand side of constraint (11) by RN · |Wi|, where Wi corresponds to the original set of operative weeks. Similarly, we adjust the determination of the used night capacity after each iteration of the heuristic. For instance, if a series is such that Wi = {13, . . . , 43} and a daily arrival is scheduled during the night in week 31, the used night capacity increases with 31 · 7. Otherwise too many flights will obviously be allowed to be assigned to night periods in later iterations.
the MIP per additional series of flights, we observe that these times increase if more series with the same region and segment are scheduled. This can be explained by the fact that early considered series can be assigned to their preferred time periods, leading to a trivial solution. On the other hand, series which are considered later in the heuristic might not have this triviality such that computation times increase.
The total objective value as expressed in Section 4.1.2 equals 56,577. Part (1) of the ob-jective has a value of 74,371. Because we will use β = γ throughout this case study, we combine part (2) and part (3) when we present results. We will denote the sum of both parts as part (2)+(3), which equals -7,270 in this case. Similarly, we will use δ = in the numerical experiments and thus write part (4)+(5) to denote the sum of part (4) and part (5) of the objective. In the base case discussed here, part (4)+(5) equals -10,524. We observe that the selection of arrival and departure periods according to historical observations (i.e. part (1)) has a large contribution to the total objective value. Moreover, we deduce that the total number of minutes deviated from the optimal turnaround times equals 7,270, since β = γ = 1. This implies an average of 7,270131 = 55 minutes per series. Additionally, the total number of minutes deviated from the mean arrival and departure times is 10,5242 = 5, 262, which implies an average of 5,262131·2 ≈ 20 minutes per arrival/departure series per week.
Table 5: Portion of additional series scheduled with base settings.
i m w d p 94 ARR 31 0 83 94 DEP 31 0 86 94 ARR 31 1 83 94 DEP 31 1 86 94 ARR 31 2 83 94 DEP 31 2 86 94 ARR 31 3 83 94 DEP 31 3 86 94 ARR 31 4 83 94 DEP 31 4 86 94 ARR 31 5 83 94 DEP 31 5 86 94 ARR 31 6 83 94 DEP 31 6 86 i m w d p 26 ARR 31 0 79 26 DEP 31 0 84 26 ARR 31 1 79 26 DEP 31 1 85 26 ARR 31 2 79 26 DEP 31 2 85 26 ARR 31 3 79 26 DEP 31 3 85 26 ARR 31 4 79 26 DEP 31 4 86 26 ARR 31 5 79 26 DEP 31 5 85 26 ARR 31 6 79 26 DEP 31 6 85 i m w d p 86 ARR 31 0 60 86 DEP 31 0 77 86 ARR 31 1 62 86 DEP 31 1 77 86 ARR 31 2 62 86 DEP 31 2 76 86 ARR 31 3 62 86 DEP 31 3 79 86 ARR 31 4 62 86 DEP 31 4 76 86 ARR 31 5 64 86 DEP 31 5 77 86 ARR 31 6 62 86 DEP 31 6 77
and the low cost segment. Series 26 belongs to the European region and the hub segment. Series 86 belongs to the North American region and the hub segment. Additionally, all three series operate on all days.
By investigating Table 5, we observe that series 94 obtained a completely consistent schedule with a small turnaround time of (86 − 83) · 10 = 30 minutes. This is one of the best possible approximations of the optimal turnaround time for European flights in the low cost segment, which equals 35 minutes. On the other hand, series 26 and 86 are not scheduled completely consistently. However, deviations from mean arrival and departure times are small. For ex-ample, series 86, which is a hub flight from and to North America has a mean arrival period of 62 and a mean departure period of 77, corresponding to 10:10 and 12:40, respectively. This matches the patterns we observed for North American hub arrivals and departures as presented in Figure 3.
In the following sections, we will adjust parameters and input data of our model, keeping all else unchanged, and apply our solution approach to generate schedules. Thereafter, we will compare the obtained results with the results from the base case that we presented here and interpret the possible differences.
5.2.2 Zero objective
To investigate the effect of the weighted objective that we introduce, we set all weights of the objective to zero, α = β = γ = δ = = 0. In this way, each solution of the MIP corresponds to the first feasible solution found by the CPLEX solver. The heuristic approach schedules all additional series of flights in 0.23 hours, which is logically much faster than in the base case. Nevertheless, it proves that finding feasible solutions is already challenging, since otherwise the computation time would have been smaller. The objective value obviously equals zero for all the different parts. By examining the resulting schedule, we observe that extremely inconsistently scheduled series of flights result that do no replicate the historically observed patterns. The total number of minutes deviated from the optimal turnaround times equals 658,263. Moreover, the schedule shows a total deviation of 1,342,822 minutes from the mean arrival and departure times. Both numbers are significantly higher than in the base case. Obviously, this is not a schedule that is expected to be seen at any airport. Table 6 shows the results for series 94, 26 and 86.
