Nationaal Lucht- en Ruimtevaartlaboratorium National Aerospace Laboratory NLR

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1 Nationaal Lucht- en Ruimtevaartlaboratorium National Aerospace Laboratory NLR NLR- Algorithms or CDM on slot allocation: slot swapping and slot shiting By Menkes van den Briel Distribution: IW LL J.M. v/d Akker (IW) (3X) P.R. de Waal (IW) M.H.L. v/d Briel (2X) J.P.M. van Hoesel (Univ. Maastricht) H.H. Hesselink (ID) H. de Jonge (LL) W. Post (LL) No part o this document may be reproduced and/or disclosed, in any orm or by any means, without the prior written permission o NLR. Division: Order-/codenumber: Inormation and Communication Technology Prepared: Issued: M.H.L.vdB./ May 2000 Approved: Classiication title: J.M.vdA./ P.R.dW./ R.J.P.G./ Unclassiied

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3 -3- Acknowledgements This document is the Master s thesis o M.H.L. van den Briel. It describes the project that was carried out at the Department o Mathematical Models and Methods (IW) o the Dutch National Aerospace Laboratory (NLR) in Amsterdam. The project was guided by Dr. Ir. J.M. van den Akker and Dr. Ir. P.R. de Waal at NLR, and by Dr. Ir. S. van Hoesel at the Faculty o Economics and Business Administration o Maastricht University.

4 -4- Summary Collaborative Decision Making (CDM) has been identiied as one o the key elements o uture Air Traic Management (ATM) strategies. CDM is about improving the way air traic management, airlines and airports work together. It is a process in which airspace participants improve the exchange o inormation and delegate decisions to the participant that is at the best position to make them. Slot swapping and slot shiting are two potential CDM applications that give airlines greater control over the operation o their lights. Slot swapping and slot shiting are proposed as an enhancement or the problem o slot allocation. Slot allocation is about planning lights beore departure in order to avoid congestion at runways and airspace sectors. Slot swapping is a more collaborative approach to slot allocation in a way that airlines can prioritise their lights. With prioritisation the sequence o air traic can be adjusted rom the basic irst-come-irst-served sequence in order to increase throughput. Prioritisation is seen as an important actor in maximising the eiciency o the uture ATM system. Slot shiting is a more collaborative approach to slot allocation in the case o delayed departures o lights. Slot shiting avoids very large delays or lights missing their slot and minimises the number o slots that go unused, and hence minimises wasted capacity. In the problem o slot swapping, three algorithms are developed. Two o the algorithms swap slots through exchanging times, the other algorithm swaps a light s presence in a certain sector at a certain time period. In the problem o slot shiting one algorithm is developed that sequentially builds a single chain o lights to be shited. Computational results or both applications are encouraging, because they show a possibility or a more lexible interpretation o slot allocation.

5 -5- Contents 1 Introduction 7 2 Slot allocation Background Slot allocation models Optislot model 9 3 Collaborative decision making Background Potential applications 13 4 Slot swapping Overview Slot swapping process Swapping method Swap at departure time Example Algorithm Conclusion Swap at sector arrival time Example Algorithm Conclusion Swap at load contribution Example Algorithm Conclusion Swap costs and beneits Implementation aspects Swaplist Load matrix Swap bounds Computational results Test problems 37

6 Algorithm perormance Swap characteristics Conclusions 43 5 Slot shiting Overview Slot shiting process Shiting method Example Algorithm Remark Shit costs and beneits Implementational aspects Shitlist Recursive procedure Computational results Algorithm perormance Shit characteristics Conclusions 59 6 Concluding remarks 60 7 Reerences 61 (62 pages in total)

7 -7-1 Introduction World-wide the concept o Collaborative Decision Making (CDM) is gaining credits to deal with the ongoing growth in air traic. Currently, CDM is the cornerstone o uture Air Traic Management (ATM) strategies in Europe and already a number o CDM applications have been implemented successully in the USA. Although CDM in Europe is dierent rom CDM in the USA, the concept o CDM remains the same: sharing inormation between airspace users and taking decisions collaboratively to achieve maximum eiciency and saety within the airspace system. This document is devoted to slot swapping and slot shiting which are two CDM applications that give airlines greater control over the operation o their lights. Slot swapping and slot shiting are proposed as an enhancement or the problem o slot allocation. Slot allocation is about planning lights beore departure in order to avoid congestion at runways and airspace sectors. Slot swapping is a more collaborative approach to slot allocation in a way that airlines can prioritise their lights. With prioritisation the sequence o air traic can be adjusted rom the basic irst-come-irst-served sequence in order to increase throughput. Prioritisation is seen as an important actor in maximising the eiciency o the uture ATM system. Slot shiting is a more collaborative approach to slot allocation in the case o delayed departures o lights. Slot shiting avoids very large delays or lights missing their slot and minimises the number o slots that go unused, and hence minimises wasted capacity. This document is organised as ollows. Chapter 2 and Chapter 3 are two introductory chapters that give an overview o slot allocation and CDM. In Chapter 4 slot swapping is discussed. The problem is described and three algorithms are presented. The results o the algorithms are given at the end o the chapter. Chapter 5 deals with slot shiting. It has the same structure as Chapter 4 however only one algorithm is presented. Some concluding remarks and recommendations or urther research are given in the last chapter.

8 -8-2 Slot allocation Slot allocation is about planning lights beore departure in order to avoid congestion at runways and airspace sectors. In the ollowing sections we give a summary o air traic low management and we present a model or the slot allocation problem. 2.1 Background Most aircrat passengers have no idea about what takes place to get their light rom one airport to another. From the moment an aircrat is ready or take-o, until it has landed saely, it will be under constant supervision o some controller. Ground controllers take care o taxiing aircrat on the ground, tower controllers take care o landing and departing (taking o) aircrat, and air traic controllers take care o aircrat lying in higher parts o airspace. European airspace is divided into 240 sectors, all permanently monitored by air traic controllers, at least one controller per sector. When an aircrat enters a sector the air traic controller will guide the aircrat across the sector and pass it to the controller o the next sector until a ground controller takes over again. Air traic controllers maintain a certain distance between two lights in order to ensure saety. Since air traic controllers can only monitor a limited amount o aircrat at the same time, all sectors have a certain limited capacity. The capacity o a sector is deined as the number o lights that are allowed to enter the sector during a given time period. Limited capacity can lead to overloads when demand is high. Sectors that tend to be overloaded, because demand exceeds capacity, are called regulated sectors. On a typical day about 100 sectors are regulated. In order to avoid congestion and overload, Air Traic Flow Management (ATFM) is applied. ATFM concerns planning lights beore departure. One o the measures taken by ATFM is slot allocation. Slot allocation is the assignment o departure time intervals to lights in such a way that none o the airspace sectors or airports becomes overloaded. Such an interval (usually about 15 minutes) is called a slot. Note that slot allocation imposes ground delays to selected lights in order to avoid congestion at airspace sectors. Such congestion would lead to airborne delays, i.e. delays during the light. In practice, the Central Flow Management Unit (CFMU) in Brussels perorms slot allocation. This proceeds as ollows: All aircrat operators must notiy the CFMU and Air Traic Control (ATC) o their intention to operate a light, by iling a light plan (FPL). A FPL is a record o light data, which include place o departure and destination, scheduled departure time, and light times. FPLs must be iled at least one hour beore departure and at least one hour beore being processed by the CFMU. The CFMU calculates or each aircrat a departure slot such that during its light the aircrat will not encounter any overloaded sector. On a normal day up to 24,000 FPLs are iled and processed.

9 -9- For allocating a slot to each light in Western Europe the CFMU uses the Computer Assisted Slot Allocation (CASA) system. The CASA heuristic uses a irst-come-irst-serve algorithm, which is applied to the lights scheduled arrival time at regulated sectors. The results are obtained very ast and satisy equity rules i.e. rules that ensure that dierent airlines are treated equally. However, recent studies show that delays can be signiicantly decreased by using optimisation methods, or example see: Maugis [1] and Vranas and Psaratis [2]. 2.2 Slot allocation models The problem o slot allocation can be ormulated as ollows: determine a departure slot or each light such that overall delay costs are minimal, and all capacity constraints o both sectors and airports are satisied. I a easible solution exists, then each light will have an allocated departure slot. I the allocated slot or a light alls later than its scheduled departure time, then the light will have a ground delay. Thereore slot allocation is also known as ground holding. Ground delays are preerred to airborne delays, since the latter are much costlier and less sae. Costs include uel, maintenance, depreciation, and saety costs. In other words, it might be advantageous to deliberately delay a light at departure. Slot allocation can be viewed as an optimisation problem in which the delay costs are minimised. Dierent slot allocation models are available. Static and dynamic versions o slot allocation models can be distinguished. In the static versions, the ground holds are decided once at the beginning o a given time period, whereas in the dynamic versions they are updated all the time. Note that slot allocation is a dynamic problem: FPLs are iled at any time o the day and capacity changes when weather conditions change. However, static models could be used dynamically by solving the problem repeatedly as better capacity estimates and new FPLs become available. Deterministic and probabilistic versions o slot allocation models can also be distinguished, according to whether airport and sector capacities are considered deterministic or probabilistic. 2.3 Optislot model The Deutsche Centrum Fur Lut- und Raumahrt (DLR) and the Nationaal Lucht- en Ruimtevaartlaboratorium (NLR) have developed a slot allocation algorithm called Optislot. Optislot is based on an integer linear programming (ILP) model. Various ILP slot allocation models have been described in the literature. For a detailed review see, or instance: Maugis [1], van den Akker [3], Andreatta and Brunetta [4], Andreatta, Brunetta and Guastella [5] and Bertsimas and Stock [6]. The ILP model used by Optislot is described below. The ollowing notation is used:

10 -10- Term F S s T Description Set o lights. Flight. Set o regulated sectors. Regulated sector. Time period in minutes (or sector capacity). F(s) Set o lights crossing sector s. S ( ) Set o sectors crossed by light. Γ (s) Set o time periods o sector s. * D ( ) Set o all possible departure slots variables or light. ω d The cost or light i it departs in slot with slot number d. δ τ G d 0 s in Length o a departure slot in minutes. Time period length or sector capacities in minutes. Maximum delay allowed or lights in minutes. Scheduled departure slot or light, relative to 00:00 AM. d, Time light needs to ly to enter sector s. C st Total capacity o sector s in time period T. Table 1: Optislot notation Using the notations above, the slot allocation problem can be ormulated as an ILP model as ollows: Minimise Subject to: (a) ω d x d F * d D ( ) d * d D ( ) x = 1 F (b) d F ( s) * d D ( ): d + d s, in T (c) { 0,1} d x C s S, T Γ( s) st x F, d D ( ) * Basically, the model is the ollowing: allocate a departure slot to each light such that total delay costs are minimised. A departure slot is a time interval [t 1, t 2 > o length δ, where t 1 and t 2 are expressed in minutes, relative to a reerence time t 0. Each slot has a slot number d. The slot number o slot [t 1, t 2 > with reerence time t 0 equals (t 1 t 0 )/(t 2 t 1 ). x d are variables indicating

11 -11- a decision on the departure slot d or a light. The earliest slot in which a light can depart is its scheduled departure slot d 0. The actual departure slot d is calculated such that total delay costs are minimised. The cost or a light to depart in slot with slot number d is given by ω d. Three types o constraints can be identiied. Constraints (a) are assignment constraints. These constraints take care that exactly one departure slot must be selected or every given light. Constraints (b) are the capacity constraints, limiting the number o lights that may enter the sector during a certain time period. To deine sector capacities, time periods are used. A time period T is a time interval [t 1, t 2 > o length τ, where τ = k δ or some integer k. The capacity o sector s at time period T is given by C st. The load o a sector in a certain time period can be calculated by adding every x d variable that causes light to enter sector s in time period T. Constraints (c) are the integrality constraints and orm the basis o the ILP model. Optislot uses the ollowing algorithm. First, a easible solution is computed by letting the lights depart in chronological order. Second, the (c) constraints o the ILP model are relaxed to a ractional orm, ater which the optimal LP-relaxation solution is computed using column and row generation. Finally, the Integer Programming problem restricted to the variables generated by the column generation algorithm is solved to optimality. The new ILP is an integer optimisation problem but it has much ewer decision variables. To test the CDM applications slot swapping and slot shiting, slot allocation solutions rom Optislot are used. A more detailed description o Optislot can be ound in: van den Akker [3] and Zwaal [7].

12 -12-3 Collaborative decision making Collaborative Decision Making (CDM) has been identiied as one o the key elements o uture Air Traic Management (ATM) strategies. CDM is about improving the way air traic management, airlines and airports work together. It is a process in which airspace participants improve the exchange o inormation and delegate decisions to the participant that is at the best position to make them Background Between the early 1970s and the late 1980s, air traic in Europe doubled in volume causing serious congestion and delays. Thereore, in 1990 the European Air Traic Control and Integration Program (EATCHIP) was launched. EATCHIP is a co-operative program o the member states o the European Civil Aviation Centre (ECAC). The program strongly emphasises integrating European ATM, such that operation o the ATM system unctions as i it were a single unit. It has already delivered increased airspace and control capacity through the harmonisation o existing air traic systems. However, the present ATM organisation and inrastructure will be unable to keep up with the expected air traic demand, which is expected to double again by the year 2015 when compared to the air traic volume in Thereore, a new approach has been developed which is described in the ATM Strategy or [9]. The ATM Strategy or involves much more interaction between airspace participants than beore. CDM has been identiied as one o the cornerstones o the ATM Strategy or the years The ATM Strategy or describes CDM as ollows: Both the collective requirements o all airspace users and the individual aircrat operator s preerences will be taken into account in determining solutions to events. The open systems environment and better inormation management will allow a permanent dialogue between the various parties (ATM, Aircrat Operators Operations Centres, Pilots and Airport Operations) beore departure, and as the light progresses through the ATM system. This exchange o inormation will enable the various organisations to continuously update each other on relevant events in real-time and provide the basis or more eicient decision making. Aircrat operators will have up-to-date and accurate inormation on which to base decisions about their lights, and will be able to apply actors which are not known to ATM, such as leet management priorities, uel consumption igures and other aircrat operating parameters, when determining solutions.

13 -13- In the past, although inormation was available at several sites it was very oten not shared. With the use o CDM, inormation sharing will be improved by better managing and distributing inormation. In order or a number o airspace participants to plan collaboratively, they must have access to consistent sets o inormation. This does not necessarily implies that all participants must have access to the same set o inormation, each participant only needs the pieces o inormation that are relevant to him. The concept o CDM has always existed in the ATM system, but until today it has been a very low-level process. In the uture, the exchange o inormation should improve light operations and increase capacity, eiciency and saety. 3.2 Potential applications Recently, EUROCONTROL produced a report identiying 22 potential applications o collaborative decision making in Europe. These applications include our dierent levels o CDM. The irst level o CDM is the distribution o existing inormation. To some amount, inormation is already distributed, however in some cases it can not be used eectively since the available inormation is not complete. The second level o CDM can be seen as co-operation between participants to improve planning estimates. The third level o CDM is changing the planning process so that additional participants priorities are taken into account. The ourth level o CDM that is included in some o the applications is redistribution o decision making, so that the best placed participant is responsible or decision making. A complete description o the applications can be ound in Potential applications o collaborative planning and decision making inal report [9]. Slot swapping and slot shiting are considered among the 22 applications o CDM in Europe. They can be seen as the most important applications related to slot allocation. Slot swapping is concerned with light prioritisation and is described in Chapter 4. Slot shiting is about delayed departures and minimising the number o unused slots. Slot shiting is described in Chapter 5.

14 -14-4 Slot swapping Slot swapping is identiied as an application o CDM on slot allocation. With slot swapping airlines have the opportunity to prioritise their lights. They can exchange delays between two lights provided that these lights are subject to the same regulation and provided that other lights are not aected by the swap. 4.1 Overview The concept o delay exchange is introduced by Carr, Erzberger and Neuman [10]. In the same way slot swapping can be deined as a air method o accommodating an airline s request or an earlier departure by advancing the departure time o one aircrat, while simultaneously delaying the departure time o another aircrat rom the same airline. Any time advance gained by the time-advanced aircrat must be compensated by delay or another aircrat. In this manner slot swapping will not cause additional delays or aircrat rom other airlines. The objective o slot swapping is to give airlines the possibility to prioritise lights that are subject to the same regulations. Prioritisation would be achieved by changing the order o departure o two lights. Through swapping, airlines will be able to prioritise lights over others within their own set o lights. Flights that have high priorities will be reerred to as time critical lights. Flights can be time critical or several reasons, or example i they carry many passengers who have to change to another light or have crew hour s restrictions. A swap is requested to advance the slot time o a time critical light to take a less critical slot rom another light. A light s departure slot is determined by slot allocation. Slot allocation, perormed by the CFMU with the CASA algorithm, is based on a irst scheduled at the regulation, irst served principle. This principle deines the Most Penalising Regulation (MPR) or a light. The MPR is the regulation that gives the light the greatest delay i it would be the only regulation. All lights in Europe are scheduled with CASA. Flights allocated through FCFS at their MPR can only be swapped with lights that have the same MPR (see [11]). However, in this study we use Optislot. Optislot does not allocate slots using a light s MPR. Thereore, lights to be swapped do not necessarily need the same MPR, but must have at least one common regulation through which they pass. At present slot swapping is perormed manually. This means one must check by hand that changing the order o two lights does not negatively aect other lights. This is very timeconsuming and can only be considered in very rare occasions. A prototype o an automated process is implemented and tested in this study. Three dierent algorithms are distinguished and evaluated. Feasibility o a slot swap is automatically checked by the algorithms, reducing the time needed to approve or reject swap requests, or to ind swap possibilities.

15 Slot swapping process In the report by Eurocontrol [8] a proposed slot swapping process is given which is also shown in Figure 1. Two actors are involved in the slot swap decision making: the Airline Operations Centre (AOC) and the CFMU. Several other actors are involved in the slot swapping process due to the changes in the light plan o the swapped aircrat. These actors include all ATC units concerned (ATC at ADEP, ATC at ADES, ACCs and TMAs) as well as both departure and destination airport authorities (AA at ADEP and AA at ADES). The AOC, in co-ordination with their commercial department, decide which lights are time critical and hence have the highest priority to be time advanced. Thereore, the AOC will be responsible or deciding which lights it wishes to swap, and requesting this swap through a Slot Revision Request (SRR) (step 1). Currently, the CFMU can approve or reject swaps by sending a Slot Revision Message (SRM) because it is the only actor that has a ull picture o the available capacity and expected loading at the regulations. Thereore the CFMU will be responsible or checking the easibility o a requested swap (step 2). However, i in the uture AOCs get access to the CFMU inormation, this may change. Furthermore, in order to help the AOCs making swap requests that have every change o being approved, the CFMU must provide the AOCs with updated inormation about possible swap opportunities. When a swap request is approved and accepted (step 3) the CFMU must send the updated FPLs to all other actors involved (step 4). AOC CFMU ATC at ADEP ATC at ADES ACCs TMAs AA at ADEP AA at ADES 1: AOC sends slot revision request to the CFMU SRR 2: CFMU sends AOC slot revision message SRM 3: AOC sends acceptance Accept FPL 4: CFMU sends up-dated FPL to ATS and airport authorities Figure 1: Proposed process or slot swapping [8]

16 Swapping method Three dierent slot swapping algorithms are examined in this study: 1 Slot swapping at departure times. 2 Slot swapping at sector arrival times. 3 Slot swapping at load contribution. All three algorithms have been implemented and tested on light schedule solutions computed by Optislot. Beore we explain the algorithms, we introduce some terminology in Table 2 (some terminology is already deined in Table 1). Term Description a light. s a sector. T a time period. st pair o time period T and sector s, also reerred to as sector time period. L st load o sector s at time period T. C st capacity o sector s at time period T. d swap departure slot o light ater it has been swapped. number o swapped minutes or light. 1 2 light to be time advanced/ light to be swapped. light to be delayed/ light to be swapped with. Table 2: description o terminology We use the notation st to represent a sector time period pair. Henceorth, this will be reerred to as sector time period or st combination. The capacity o a sector time period is deined in a capacity matrix C st. In the capacity matrix C st, rows correspond to sectors and columns correspond to time periods. In this manner all st combinations can be speciied. For those st combinations having no capacity speciication, ininite capacity is assumed. In the Optislot model the load o a st combination is deined as the number o aircrat that enter sector s in time period T. The load o all st combinations is calculated in a load matrix. The load matrix, represented by L st, has the same structure and size as the capacity matrix C st. Thus, rows correspond to sectors and columns to time periods. Each slot allocation solution computed by Optislot satisies the assignment, capacity, and integrality constraints. With slot swapping the departure times o two lights are changed. This will not aect the assignment and integrality constraints. However dierent departure times, give dierent sector arrival times and as a result, the load at one or more st combinations might

17 -17- change. Thereore, one must check whether the capacity constraints, given by L st C st, still satisy. Since slot swapping is perormed on a easible slot allocation, it can be assumed that the capacity constraints or the st combinations that are not aected by the swap keep satisied. Only the st combinations aected by the swap have to be checked on overload. Slot swapping always involves two lights that can be distinguished as: 1 A time critical light to be time advanced. 2 A light that compensates the time advance o the time critical light by taking up an additional delay. In the three algorithms described in the ollowing sections, lights stated in 1 and 2 are reerred to as 1 and 2 respectively. The light 1 corresponds to the light to be swapped to an earlier slot and the light 2 corresponds to the light to be swapped with 1. A swap results in new departure times or both lights involved. The actual departure slot o a light ater it has been successully swapped is reerred to as d light is swapped is given by. swap. The number o minutes a

18 Swap at departure time The literal translation o slot swapping is exchanging departure time intervals between two lights. This algorithm does exactly that: it perorms slot swapping based on the actual departure times o lights. In Optislot d 0 is the scheduled departure time o a light and d is the actual departure time o a light. The scheduled departure time is also reerred to as the earliest departure time o a light. The departure time computed by Optislot is the actual departure time. Thus when a light is not delayed d is equal to d Example Suppose an airline has two lights 1 and 2. Assume both lights depart rom the same airport, have the same cruising speed, go through the same single regulated sector, but have dierent destinations. Regulation Airport departure Sector arrival (capacity = 1 light / st) d 0 08:00 08:30 d 1 08:30 09:00 2 d 0 08:00 08:30 d 2 08:00 08: Figure 2: Situation beore swapping where T is 60 minutes relative to 00:00 This scenario is illustrated in Figure 2. Optislot notation is used to indicate the scheduled and actual departure times o the two lights. The times printed near the regulation are the scheduled and actual sector arrival times o light 1 and 2. The light routes are indicated by a dotted line in the igure. The time critical light 1 is delayed by 30 minutes. This delay ollows rom the lack o capacity at the regulation. The regulation has a capacity o controlling at most one light per sector time period. The time period T or this sector is 60 minutes reerenced to midnight. This means one light can enter the sector in the st combination [00:00, 01:00> and another one in [01:00, 02:00> and so on. A delay or the time critical light 1 is very expensive. Thereore, the AOC may request to swap the two lights and thereby have the time critical light 1 depart with the less penalising slot o 2. Since both lights are subject to only one and the same regulation, the slot swap will not aect any other regulations and should be possible.

19 -19- Figure 3 illustrates the same two lights as rom Figure 2, however now their actual departure times have been exchanged. The new departure time o light 1 is now equal to the original departure time o light 2, which is 08:00. The reverse holds or light 2, its new departure time is now equal to the original departure time o light 1, which is 08:30. The algorithm checks that changing the order o these lights does not cause capacity problems in any part o airspace. Airport departure Sector arrival d 0 08:00 08:30 d 1 swap 08:00 (was 08:30) 08:30 2 d 0 08:00 08:30 d 2 swap 08:30 (was 08:00) 09:00 Regulation (capacity = 1 light / st) 1 2 Figure 3: Flights swapped at departure time Algorithm To understand how the algorithm works it is broken down into the ollowing steps. Step 0 Step 1 Step 2 Step 3 Step 4 Step 5 Step 6 Step 7 Step 8 Initialisation. Choose a time critical light 1 to be swapped to an earlier slot. Choose a light 2 to be swapped with. Determine i 2 satisies slot interval conditions. Update (decrement) load. Swap slots. Update (increment) load. Check easibility by checking load (approve or reject swap). Output. Now the steps o the algorithm will be explained in more detail. Step 0: Initialisation. The initialisation computes the load or all st combinations rom a given slot allocation solution. The load o a st combination is equal to the number o lights that have an arrival time at sector s in time period T. The load must be known in order to check i a swap is easible.

20 -20- Swapping will change the order o two lights and it is very likely that, as a result, the load will change at one or more st combinations. The advantage o computing the load matrix L st at the beginning is that we have to do it only once and changes in the load can be made through relatively small load updates. When the load is computed, the algorithm can be perormed as many times, without computing the total load again. Step 1 & step 2 In practice, the airline understands its operational needs better than anyone else. Thereore, the AOC will be responsible or requesting a swap. The algorithm supports three dierent swap requests. First, a request can be made where both 1 and 2 will be given by the AOC. In this situation the algorithm checks i 1 and 2 can be swapped by going through all steps exactly once. The second type o request that can be made is that the AOC announces that it wants to swap 1 without giving a light 2 to be swapped with. In this situation the algorithm basically puts a loop around step 2 to ind all possible aircrat that can be swapped with 1. In this case, step 2 to 7 will be perormed several times until the loop terminates. The output o the algorithm now exists o a list o lights that can be swapped with 1. We will call this list a swaplist. The third type o request that can be made is that neither 1 nor 2 are given. In this situation, the algorithm will try to ind all possible swaps that can be made rom a given set o lights. Again the output will be stored in a swaplist. Step 3: Determine i 2 satisies slot interval conditions. The slot interval conditions or 2 to be swapped with 1 are: 2 d < d 1 (1) 2 d 0 2 d (2) d d 0 G (3) The irst slot interval condition restricts 2 to have an earlier departure slot than 1. Hence, we do not want the time critical light 1 to be swapped to a slot later than its actual departure time. The second condition restricts the actual departure time o 2 to be later or equal to the scheduled departure time o light 1. Hence, ater swapping, 1 can not depart beore its earliest departure time. The third condition states that the new delay o 2 may not exceed its maximum delay. The delay can be calculated by subtracting the scheduled departure time o a light rom its actual departure time. Since d will be the new departure slot o light 2 the delay will be equal to 2 0 d d. We will call a light 2 satisying the swap interval conditions a swap candidate.

21 -21- Step 4: Update (decrement) load. I 2 is a swap candidate it has the potential to be swapped with 1. Beore assigning the new departure slots to the two lights, the load has to be updated. The two lights are deleted rom the load matrix L st. This is done by decreasing the L st by one at all st combinations that are crossed by the two lights. Ater step 4 is inished the L st matrix represents the total load without including the two lights that are being swapped. The lights involved will be added to the load again when their new departure slots are assigned. Step 5: Swap slots. Ater the total load is decreased in step 4, the actual swap takes place. The departure times are exchanged between the lights and new arrival times at sectors are calculated. swap : 2 d = d (4) 2 swap : d = d (5) The two equations assign the new departure times to the two lights. The new departure slot o light 1 will be the original departure time o light 2 and vice versa. Step 6: Update (increment) load. By comparing the L st load matrix with the C st capacity matrix we can determine i the slot swap is easible. However, comparing load against capacity must be done with an updated L st matrix. The L st must represent the actual load including the two lights that have been swapped. In step 5 the new departure slots have been assigned and the new sector arrival times have been calculated, thereore the L st can be updated. The L st matrix is increased by one at all st combinations that are crossed by the two swapped lights. Step 7: Check easibility by checking load (approve or reject swap). The swap is easible when no overload occurs in any o the regulated sectors. In other words, at all st combinations the ollowing condition must hold: L st C st (6) It can be assumed that the condition above still holds or the st combinations not aected by the swap. Only the st combinations that are crossed by 1 and 2 have to be checked on overload. I one o the relevant st combinations is overloaded the swap will be ineasible and rejected. In that case 1 and 2 must be given back their original slot. This can be done by redoing steps 4, 5

22 -22- and 6: step 4 takes 1 and 2 out rom the load matrix, step 5 gives back the original slots, and step 6 updates the load matrix back to its original state. When no overload occurs the swap will be easible and approved. We will call 2 a swap partner when it can be easibly swapped with 1. When the swap is accepted by the airline, the new data is stored and send to all actors involved. However, i the swap is not accepted by the airline, steps 4, 5 and 6 will be taken again to give 1 and 2 back their original slots. In case o inding all possible swaps, such that the algorithm is perormed repeatedly, we must bring the situation back to its original state each time when we have checked two lights. Output The algorithm output will be a swaplist. A swaplist is a record o data o all approved swaps. It includes inormation on the two lights, the number o times a light can be swapped, as well as the amount o time advance or additional delay the lights obtain through swapping Conclusion The swap example shown in Section is kept as simple as possible. Since both lights departed rom the same airport it is called a single airport swap. However, i in the example the lights departed rom dierent airports the swap would still be easible. I a swap can be made such that the lights do not depart rom the same airport it is called a multi airport swap. The swap at departure time algorithm can be used or single as well as multi airport swaps. However, it does not seem natural to swap lights at their departure times when they do not depart rom the same airport. Note that the load o a sector time period is determined by the arrival time o lights. Thereore, another possibility is to swap lights at their sector arrival times. The algorithm that swaps lights at their sector arrival times is described in the ollowing section.

23 Swap at sector arrival time The second algorithm perorms slot swapping based on light arrival times at sectors. In Optislot d 0 + d s, in represents the scheduled arrival time o a light crossing sector s. It is the sum o the scheduled departure time, d 0, and the time light needs to ly to enter sector s, d s, in. In the same way the actual arrival time o a light can be represented by d + d s, in. When a light has no delay d 0 + d s, in equals d + d s, in Example Figure 4 shows two lights lying through one and the same regulation. The lights do not depart rom the same point, which means they do not necessarily depart rom the same airport. The scheduled ( d 0 + d s, in ) and actual ( d + d s, in ) sector arrival times o the lights 1 and 2 are printed near the regulation. Airport departure 08:10 08:50 08:00 0 s, in d s, in d + d 08:20 swap d + 09: Sector arrival 2 s, in d + d 08:20 Regulation (capacity = 1 light / st) :00 d swap + d s, in 08:20 Figure 4: Situation beore swapping, where T is 60 minutes reerenced to 00: The time critical light 1 is delayed by 40 minutes and can not be swapped with light 2 using the swap at departure time algorithm. Flight 2 does not satisy the slot interval condition stated 2 in (2). The actual departure slot o 2 ( d = 08:00) lies beore the scheduled departure slot o 1 ( d 0 = 08:10). The lights can be swapped at their sector arrival time. The actual arrival time o 1 is 09:00 and the actual arrival time o 2 is 08:20. Swapping exchanges the sector arrival times o the lights. Thus ater swapping 1 arrives at the regulation at 08:20 and 2 at 09:00. The updated times are given in Figure 5. This swap will be easible because there is no light penalised by the swap.

24 -24- Airport departure 08:10 08:10 08:00 08:40 0 s, in d + d 08:20 swap s, in d + d 08:20 (was 09:00) s, in d + d 08:20 2 swap Sector arrival 2 s, in d + d 09:00 (was 08:20) Regulation (capacity = 1 light / st) 1 2 Figure 5: Flights swapped at sector arrival time Algorithm The second algorithm looks in many ways just like the irst algorithm. The steps identiied in section can also be deined or this algorithm. However, the steps 2, 3 and 5 are signiicantly dierent and thereore these steps will be evaluated. The other steps will not be evaluated, as they are already thoroughly discussed. Step 2 Choose a light 2 to be swapped with The search process or choosing a light 2 is somewhat dierent rom the irst algorithm. The irst algorithm selects every light exactly once by going through the set o lights. Here some lights can be selected several times while other lights will never be selected at all. For selecting lights, all sectors that are crossed by light 1 are used as a search space. A light 2 is selected when it has a coinciding sector with 1. As a result, i a light 2 has two coinciding sectors with 1, it will be selected twice, or three coinciding sectors it will be selected three times and so on. Flights can be selected several times because the dierence in arrival times might be dierent at another sector. Thus, it might be possible that a light is selected and rejected in step 3 at one sector, but selected and accepted in step 3 at another sector. Step 3 Determine i 2 satisies slot interval conditions The slot interval conditions or swapping at sector arrival time are: 2 2 s, in 2 s, in s s, in s, in s d + d < d + d (7) 2 0 d + d d + d (8) d 2 + G where ( ) ( ) 2 2 = d + d d + d (9) s, in s, in The irst condition restricts that the actual arrival time o light 2 at sector s to be earlier than the actual arrival time o light 1. The second condition restricts that the actual arrival time o

25 -25- light 2 to be no earlier than the scheduled arrival time o light 1. Hence, we want the time critical light 1 to be advanced in time, but not more than it is delayed. The third condition states that the new delay or light 2 may not to exceed its maximum delay. s is the number o swapped units or light at sector s. s will be equal to the dierence in the arrival times o the lights at sector s. Step 5 Swap slots The new departure slots will be given by: d d swap = d 2 swap = d s s 1 (10) 2 + (11) The time advance or light 1 will be the same as the additional delay obtained by light 2. The number o swapped minutes s will be equal to dierence in arrival times at the sector o the two lights. Thereore, the new departure slot o 1 will be equal to its original departure slot minus the number o swapped minutes. For 2 the new departure slot will be its original slot plus the number o swapped minutes Conclusion Some swaps are identiied by both the arrival and the departure swap algorithms. However, some swaps are only identiied by the swap at arrival time algorithm (see example section ) or only by the swap at departure time algorithm. But i airports are implemented as sectors, then the swap at arrival time algorithm will ind the same swaps as the swap at departure time algorithm plus swaps uniquely deined or swapping at arrival times.

26 Swap at load contribution Unlike the irst two algorithms this algorithm does not swap slots through exchanging times, but by changing a light s presence in a sector during a certain time period. Each aircrat that passes a sector contributes to the load o that sector at a certain time period. Thus lights passing a sector s during time period T contribute to the load o that st combination. Thereore, we can say that this algorithm swaps at load contribution Example Figure 6 shows two lights lying through the same regulation. The lights do not necessarily depart rom the same airport or have similar destinations. The scheduled and actual departure times, as well as their sector arrival times are given. Note that the two previous algorithms ail to swap the lights. The lights can not be swapped at their departure time (irst algorithm) because light 2 departs beore the scheduled departure time o light 1, i.e. equation (2) is violated. Neither can the lights be swapped at their sector arrival time (second algorithm). Using the second algorithm would obtain a 50-minute swap. However, light 1 is only delayed by 40 minutes. Hence, equation (8) is violated. Airport departure Sector arrival Scheduled 08:10 09:20 Actual 08:50 10:00 Scheduled 08:00 09:10 Actual 08:00 09:10 Regulation (capacity = 1 light / st) 1 2 Figure 6: Situation beore swapping where T is 60 minutes relative to 00:00 Table 3 is a load contribution table and is a tabular representation o Figure 6. In a load contribution table the regulation is represented by a number o columns equal to the number o time periods that are deined. In a load contribution table, lights are represented by row vectors. Ones in a light vector represent the time periods the light crosses. Zeros represent the time periods the light could cross according to its scheduled time. The load o a sector time period is the sum all ones in the corresponding st column. In our example the lights 1 and 2 ly through only one regulation. From Table 3 we can immediately see that the lights can be swapped. Just exchange the presence o the lights in the st combinations to which they are allocated. The arrows indicate how the swap is achieved.

27 -27- Regulation st :00-09:00 09:00-10:00 10:00-11:00 11:00-12:00 12:00-13: LsT CsT Table 3: Load contribution table o Figure 6 (arrows indicate a possible swap) Ater swapping the load does not exceed capacity so the swap will be easible. Now we know the lights can be swapped we must compute how many minutes they can be swapped. I light 2 enters the regulation at 10:00 it will be allocated to the time period [10:00, 11:00>. For light 1 to enter the regulation in the time period o [09:00, 10:00>, it can enter the sector as early as 09:20, based on its scheduled departure time o 08:10. Airport departure Sector arrival Scheduled 08:10 09:20 Actual 08:10 (was 08:50) 10:00 Scheduled 08:00 09:10 Actual 08:50 (was 08:00) 09:10 Regulation (capacity = 1 light / st) 1 2 Figure 7: Flights swapped at load contribution The number o swapped minutes is 50 minutes additional delay or light 2 and 40 minutes time advance or light 1. The new swapped times are given in Figure 7. Swapping the lights results in an extra 10 minutes o total delay. I this is still acceptable or the airline the swap will be approved.

28 Algorithm The algorithm to swap slots at load contribution is also broken down into a step by step algorithm. Since steps 0, 1, 2, and 5 are equal to steps 0, 1, 2, and 8 in the slot swapping at arrival time algorithm (see section ) they will not be evaluated here. Step 0 Initialisation. Step 1 Choose a time critical light 1 to be swapped to an earlier slot. Step 2 For all sectors crossed by 1, choose a light 2 to be swapped with that crosses sector s. Step 3 Determine i 2 satisies slot interval conditions. Step 4 Swap. Step 5 Output. We need the help o the ollowing deinitions and propositions to explain step 3 and 4. Deinition 1: We will call a slot allocation solution light-by-light optimal i or every light we have that can not be time advanced without delaying another light. Actually, deinition 1 deines a local optimum or a slot allocation solution. A local optimum can be seen as an optimum where the total delay can not decrease by time advances o single lights. Using this deinition we can prove the ollowing proposition. Proposition 1: Suppose that in a light-by-light optimal slot allocation solution 1 and 2 can be swapped. Then, 1 and 2 must have at least one common st combination that 1 crosses ater the swap and 2 crosses beore the swap. Proo: From the contrary, when 1 and 2 can be swapped such that 1 ater the swap, and 2 beore the swap, do not have one common st combination. Then 2 could never have made capacity available at a st combination or 1. In that case, 1 could have been time advanced without delaying 2, but then our slot allocation solution was not light-by-light optimal, which yields to a contradiction rom our assumption. Thus, when 1 and 2 can be swapped in a light-by-light optimal slot allocation solution there must be at least one st combination that is crossed by 1 ater the swap and by 2 beore the swap.

29 -29- The st combination mentioned in the proo o proposition 1 deines the time period that 2 crosses beore swapping and (through exchange) 1 crosses ater swapping. This st combination will be called a common swap st combination. Thus: Deinition 2: We say that st is a common swap st combination or 1 and 2 or a given sector s, i 2 crosses the st combination beore swapping and 1 crosses that st combination ater swapping. Proposition 2: Suppose that in a light-by-light optimal slot allocation solution 1 and 2 can be swapped. Then, or a common sector s, there is at least one common swap st combination that lies strictly beore the time period T that 1 crossed beore swapping. Proo: Swapping gives 1 a time advance, thus or each sector that 1 crosses the ollowing condition holds: ater swapping 1 it can never cross a st combination later than the st combination it originally passed or that sector. Hence, no st combination that 1 crosses ater swapping can be ater the corresponding st combination that 1 passed beore swapping. Hence, all st combinations that 1 crosses ater swapping are either beore or at the same st combinations that 1 passed beore swapping. Assume that ater swapping, at all common regulations with 2, the st combinations that 1 crosses ater swapping are at the same place as the st combinations that 1 crossed beore swapping. In that case 1 could have been time advanced without delaying 2, i.e. the slot allocation solution was not light-by-light optimal, a contradiction with our assumptions. Hence, there must be at least one common swap st combination that lies strictly beore the time period T that 1 originally crossed beore swapping or a common sector s. Step 3 Determine i 2 satisies slot interval conditions. Note that at this point 1 is ixed (chosen in step 1). For all sectors s crossed by light 1 crosses, all possible swap candidates 2 that cross that same sector have been ound in step 2. In step 3 we will check i the lights 2 satisy the slot interval conditions. Using proposition 2 we deine the slot interval conditions or 2 as: d d 2 2 ( d + d s, in ) ( d d s, in ) modτ s ( d + d s, in ) ( d d s, in ) modτ s 2 s, in < + 2 s, in d (12) + d (13)

30 -30- In detail, the irst condition states that the time period o light 2 must be strictly beore the time period that 1 crosses beore the swap. Using Optislot notation, the time that 1 passes the sector s, in 1 1 is given by d + d minus the surplus o time in the sector time period given by ( d d s, in ) modτ s +. An example: i the sector time period is deined by [11:00, 12:00> and the sector arrival time o light 1 at the sector is somewhere in the interval [11:00, 12:00>, then all swap candidates must have a sector arrival time earlier than 11:00. This restriction is actually the result o proposition 2. The second condition is a much more clear condition. It states that the time period o light 2 must not be earlier than the earliest time period 1 can cross. In Table 4 1 is the light to be time advanced at sector s. The lights 2A, 2B, and 2C ly through the same sector and have been selected in step 2 as lights to be tested on the interval conditions. These lights will become a swap candidate i their arrival time is in a time period strictly beore the arrival time period o 1 (irst condition), but not beore the earliest arrival time period o 1 (second condition). From the lights in Table 4 only 2C will become a swap candidate. Regulation st 1 2A 2B 2C... 08:00-08:59 09:00-09:59 10:00-10:59 11:00-11:59 12:00-12: ( d d ) ( d + d ) modτ 0 + s, in 0 s, in s d + d ( ) ( ) s, in < d + d s, in d + d s, in mod τ s Table 4: Slot interval conditions or a light to be swapped with 1 using load contribution Step 4 Swap The swapping itsel is broken down into even smaller steps: step 4.1 Remove 2. step 4.2 Find new (earlier) slot or 1. step 4.3 I ound, ind new (later) slot or 2. Step 4.1 Remove 2 First, the presence o the light 2 is removed rom the load matrix L st. In this way we make ree space at all st combinations crossed by 2. This ree space may be used by light 1 when it is advanced in time. Step 4.2 Find new (earlier) slot or 1.