Fairbrother, Jamie

Zografos, Konstantinos G

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ON THE DEVELOPMENT OF A FAIR AND EFFICIENT SLOT SCHEDULINGMECHANISM AT CONGESTED AIRPORTSTRB Paper number: 18-05366Jamie FairbrotherCentre for Transport and Logistics, Management SchoolLancaster University, Lancaster, LA1 4YZ, United Kingdom.Tel: +44 1524 592364 Email: j.fairbrother@lancaster.ac.ukKonstantinos G. ZografosCentre for Transport and Logistics, Management SchoolLancaster University, Lancaster, LA1 4YW, United KingdomTel: +44 1524 592384 Email: k.zografos@lancaster.ac.ukKeywords: airport slot scheduling, airport capacity management, slot scheduling mechanism,multiobjective optimization, slot scheduling fairnessAcknowledgements: The work in this paper has been supported by the Engineering and PhysicalSciences Research Council (EPSRC) through the Programme Grant “Mathematical Models andAlgorithms for Allocating Scarce Airport Resources (OR-MASTER)” EP/M020258/1.Fairbrother and Zografos 2INTRODUCTIONIn many airports around the world, demand to use the airport infrastructure exceeds the availablecapacity, leading to congestion-related delays or infeasible slot scheduling. Outside of the US,capacity is managed according to an administrative scheme detailed in the IATA Worldwide slotguidelines (1). To use infrastructure at a congested airport, an airline must be assigned a slot. Aslot is a time period during which an aircraft can use the airport for landing or take-off.A number of optimization models have been developed to solve the problem of allocatingslots to airlines. The problem was first explicitly modelled in the paper (2) which formulatedthe problem as an integer linear program. More recently, fairness has been incorporated into slotallocation models (3, 4). Fairness is a key component of a schedule which is acceptable to theparticipating airlines.Another key component in constructing a schedule which is acceptable to airlines is theincorporation of airline’s preferences as to which flights are displaced from their preferred slots.Although this has been addressed in the context of air traffic flow management (5, 6), as far as weare aware, no work has been done in this direction for the slot allocation problem.The aim of the present work is to propose a mechanism for allocating slots which is bothfair and takes into account airlines’ preferences. The scheme consists of first constructing a fairbaseline reference schedule, followed by using airline preferences modify this schedule in order toimprove its acceptability.METHODOLOGYOverall SchemeThe overall scheme we propose is a two-stage approach. In this first stage, we construct a baselinewhich is fair in the sense that airlines are assigned total displacement costs in proportion to theircontribution to congestion. In order to achieve this, we develop a new measure of fairness, andcalculate the efficient frontier between the total schedule displacement and the level of fairness. Afair baseline schedule is then selected from this efficient frontier.In the second stage, the airlines suggest improvements to the baseline schedule. In par-ticular, for each of its requests, the airline indicates how much schedule displacement it would bewilling to accept subject to its total preferred schedule displacements being at least equal to its totalassigned schedule displacement in the baseline schedule. The proposed mechanism then attemptsto meet as many requests for improvement as possible. The total schedule displacement assignedto the airline is referred to its budget and hence we call this the budget displacement mechanism.The notation required to present our new mechanism is shown in Table 1. For sake ofbrevity, we do not present the full slot allocation model here, and refer the reader to (2) for furtherdetails.Congestion-based fairnessBy congestion, we mean excess of demand for slots in a given time period with respect to the air-port capacity constraints. Note that it does not make sense to displace requests for movements inuncongested periods. However, using the previous fairness approaches based on assigning sched-ule displacement to airlines in proportion to the number of movement requests, airlines with lotsof flights in off-peak periods may have their peak flights disproportionately displaced.We therefore propose to modify the fairness index of (7) to take into account the numberof requests an airline makes during congested periods. By vdt we denote a congestion indicatorFairbrother and Zografos 3SetsT set of time coordination intervals in a dayA set of airlinesM (Ma) set of movement requests (by airline a)D (Dm) set of dates in scheduling season (for which movement m is requested)Parametersf tm displacement cost of assigning movement request m a slot at time tvtd indicates whether time period t is congested at time t on date dσm schedule displacement cost assigned to movement request m in fair baselinescheduleδm airline’s preferred schedule displacement cost for movement request mΣa schedule displacement cost assigned to airline a in fair baseline scheduleDecisions and functionsxtm indicating whether movement request m is assigned slot at time tym indicates whether request for improved displacement by airline m can be sat-isfied (in budget displacement mechanism)sa = ∑m∈Ma∑t∈T ftmxtm total schedule displacement cost assigned to airline aS= ∑a∈A sa total schedule displacementµa fairness index for airline a (see (1))TABLE 1 Notationwhich takes the value 1 if the time interval t on date d is congested, and 0 otherwise. Congestionfor a given time period does not just occur because of an excess of demand for that time interval inparticular, but also by excess demand for adjacent time intervals whose requests may be displacedto other time periods in order to construct a feasible schedule. Hence, we construct our congestionindicator from a schedule which is optimal with respect to total displacement. Specifically, if anextra request for a slot would cause a capacity constraint to be broken in the optimal schedule, thenwe deem the time interval of the request to be congested.Given the congestion indicator we propose the following fairness index for each airlinea ∈A :µa :=saS∑m∈Ma ∑d∈Dm vdtm∑m∈M ∑d∈Dm vdtm(1)That is, µa is the proportion of schedule displacement allocated to airline a divided by the propor-tion of movement requests made during congested periods by airline a. We refer to this denom-inator as the contribution to congestion of an airline. Based on the principle that airlines shouldbe allocated schedule displacement in proportion to contribution to congestion, it is desirable thatthis value should be as close as possible to 1 for all airlines. This can be ensured by adding thefollowing fairness objective to our slot allocation optimization model, which we refer to as themaximum deviation from absolute (MDA) fairness:maxa∈A|µa−1|,A similar objective was proposed in (7).Fairbrother and Zografos 4Displacement Budget MechanismThe incorporation of displacement preferences can be achieved through the modification of the slotallocation model of (2). These modifications depend on the following new sets of parameters:σm = baseline displacement cost for movement mΣa = ∑m∈Maσa = total baseline displacement cost for airline aδm = preferred displacement cost for movement mThe baseline displacement cost for a movement m is calculated with respect to a referenceschedule. The parameters δm ≥ 0 are displacement costs chosen by the operating airlines whichthey would prefer a movement did not exceed. To ensure each airline is assigned a fair amountof delay, we require the total displacement cost each airline uses to be at least equal to the totaldisplacement cost for the baseline schedule, that is:∑m∈Maδm ≥ Σa for a ∈AGiven these preferences, we add the following constraints to the base model:∑t∈Tf tmxtm ≤ σm− (δm−σm)ym for m ∈M if δm < σmwhere ym is a new decision variable that can be interpreted as follows:ym ={1 if m assigned slot such that schedule delay is less than δm,0 otherwise.For all other requests, we add the following constraints:∑t∈Tf tmxtm ≤ δm for m ∈M if δm ≥ σm.These constraints ensure that all requests are assigned a delay at most max{δm,σm}. Theobjective of this model is then lexicographical optimize the number of satisfied the number ofrequests for improved schedule displacement and the total displacement:lexmin(− ∑m∈M :δm<σmym, ∑m∈M∑t∈Tf tmxtm)FINDINGSWe test our overall slot allocation scheme using real data for a medium-sized airport. For sim-plicity, we solve the problem without priority classes. This problem has 898 requests. The integerlinear programs are solved using Gurobi(8).Reference schedule and simulation of preferencesThe reference schedule is found by solving the total displacement-fairness problem for congestion-based fairness with ε = 0.9. This yields a total displacement cost of 8123, a relatively smallincrease as compared to the minimum total displacement cost of 8003 found by solving the problemwithout fairness constraints. Displacement preferences for the airlines are simulated assuming eachFairbrother and Zografos 5airline has a cost function for the displacement of each movement of the form:ctm = rm|t− tm|and then for each airline selecting preferred delays which minimize their displacement costs assum-ing all requests are satisfied. The base costs rm are randomly generated from a uniform distributionon the interval (0,1).ResultsA summary of the results from the displacement budget mechanism over all airlines and for theairline with the largest displacement costs are given in Table 2. These show that about a fifth ofrequests for improved displacements were satisfied. However, this has led to the total displacementcosts increasing by around a third. Averaged over all airlines, the additional displacement cost persatisfied improvement request is 11946−812343 = 88.9, Similar observations also hold for the airlinewith the largest displacement costs.All airlinesInitial displacement costs 8123Preferred displacement costs 8472New displacement costs 11946Number improvement requests 214Satisfied improvement requests 43Airline with largest displacement costsInitial displacement costs 1463Preferred displacement costs 1463New displacement costs 1839Number of improvement requests 50Satisfied improvement requests 9TABLE 2 Summary of prioritization resultsIn Figure 1 are plotted the baseline delay costs with preferred delay costs and the newdelay costs after the displacement budget mechanism. Note that all requests which had no initialdisplacement and for which the airline requested no displacement have been omitted from this plot.The plot demonstrates that when an airline allows a larger displacement cost than the baseline, thisflexibility is often not used, or only partially used. This is because the mechanism will not assignrequests a greater displacement unless it will help another request for improved displacement to befulfilled.CONCLUSIONIn this paper we have proposed a slot allocation mechanism which incorporates efficiency, fairness,and takes into account the airlines’ priorities. The scheme consists of first constructing a fairreference schedule, and then using airlines preferences to make adjustments to this.The method used to construct a fair reference schedule improves over previous schemesby using the principle that the schedule delay an airline receives should be proportional to itscontribution to congestion rather than simply the number of requests it makes.Fairbrother and Zografos 60 50 100 150 200 250 300 350 400Displacement Cost010203040506070Request IDPreferred delaysNew delaysBaseline delaysFIGURE 1 Displacement costs for an airline before and after displacement budget mecha-nismThe mechanism for taking into account the airlines’ priorities consisted of allowing theairlines specify how the displacement costs assigned to them in the reference schedule should bedistributed among their flights, and then meeting as many prioritized requests as possible. In anumerical experiment, we demonstrated that a significant number of these requests could be met,but at the expense of some increase in the total schedule displacement.Fairbrother and Zografos 7REFERENCES[1] International Air Transport Association, Worldwide Slot Guidelines, 2017, http://www.iata.org/policy/infrastructure/slots/Pages/slot-guidelines.aspx, ac-cessed 25 April 2017.[2] Zografos, K. G., Y. Salouras, and M. A. Madas, Dealing with the efficient allocation ofscarce resources at congested airports. Transportation Research Part C: Emerging Technolo-gies, Vol. 21, No. 1, 2012, pp. 244–256.[3] Jacquillat, A. and V. Vaze, Inter-airline equity in airport scheduling interventions, 2015, work-ing paper.[4] Zografos, K. and Y. Jiang, Modelling fairness in slot scheduling decisions at capacity-constrained airports, 2017, presented at Transportation Research Board Annual Conference2017.[5] Pilon, N., A. Cook, S. Ruiz, A. Bujor, and L. Castelli, Improved Flexibility and Equity forAirspace Users During Demand-capacity Imbalance. In 6th SESAR Innovation Days, 2016.[6] Bertsimas, D. and S. Gupta, Fairness and collaboration in network air traffic flow management:an optimization approach. Transportation Science, Vol. 50, No. 1, 2015, pp. 57–76.[7] Zografos, K. and Y. Jiang, Modelling and solving the airport slot scheduling problem withefficiency, fairness, and accessibility considerations. In TRISTAN Symposium, 2016.[8] Gurobi Optimization, I., Gurobi Optimizer Reference Manual, 2017.

On the Development of a Fair and Efficient Slot Scheduling Mechanism at Congested Airports

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On the Development of a Fair and Efficient Slot Scheduling Mechanism at Congested Airports

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