An Optimization Approach for Airport Slot Allocation under IATA Guidelines

Transcription

1 An Optmzaton Approach for Arport Slot Allocaton under IATA Gudelnes Abstract Ar traffc demand has grown to exceed avalable capacty durng extended parts of each day at many of the busest arports worldwde. Absent opportuntes for capacty expanson, ths may requre the use of demand management measures to restore the balance between scheduled traffc and avalable capacty. The man demand management mechansm n use today s the admnstratve schedule coordnaton process operated by the Internatonal Ar Transport Assocaton (IATA), whch s n place at the great majorty of busy arports outsde the Unted States. Ths paper proposes a novel mult-objectve Prorty-based Slot Allocaton Model (PSAM) that optmzes slot allocaton, whle complyng wth the complex set of prortes and requrements specfed by the IATA gudelnes, as well as wth the declared capacty constrants at the arports. It presents an effcent computatonal approach that provdes optmal slot allocaton solutons at arports sgnfcantly larger than has been possble to date. The model s appled to two Portuguese arports, a small one (Madera) and a md-sze one (Porto) usng hghly detaled data on arlne slot requests and arport capacty constrants. Results suggest that PSAM can mprove the effcency of current practce by provdng slot allocatons that match better the slot requests of arlnes. Equally mportant, PSAM can also quantfy the senstvty of slot allocaton decsons to the varous prortes and requrements specfed n the IATA gudelnes. Keywords: arport demand management; slot allocaton; IATA gudelnes; nteger programmng. 1

2 1. Introducton Ar traffc growth coupled wth lmtatons n avalable nfrastructure and ar traffc management operatons have created severe mbalances between demand and capacty at the world s busest arports. Lmted capacty at busy arports can result n congeston and schedule unrelablty. In 2015, 19% and 18% of commercal flghts experenced an arrval delay of 15 mnutes or more n the Unted States and n Europe, respectvely (FAA, 2016), wth the trend pontng upward n both cases. Moreover, these constrants can mpose longterm economc mpacts due to lost demand, hgher arfares, and lmtatons n arlnes route development. In the absence of supply-sde nterventons amed to ncrease system capacty through nfrastructure expanson and/or operatonal mprovements, arport congeston mtgaton may requre the use of demand management mechansms. Demand management conssts of nterventons that lmt the number of flghts scheduled at busy arports at peak hours. These nterventons fall conceptually nto two categores: () economc approaches, whch nvolve market-based mechansms such as congeston prcng and slot auctons, and () admnstratve approaches, whch nvolve non-monetary adjustments to arport flght schedules mposed by a desgnated schedule coordnaton entty. The former has receved sgnfcant attenton n the economcs and operatons research lterature (see, e.g., Ball et al., 2006, and Gllen et al., 2016, for revews). On the economcs sde, much research has amed to desgn optmal congeston prcng schemes (Brueckner, 2002; Pels and Verhoef, 2004; Czerny and Zhang, 2011; Czerny and Zhang, 2014) and to compare prce-based vs. quanttybased aucton mechansms (Brueckner, 2009; Czerny, 2010; Basso and Zhang, 2010; Verhoef, 2010). On the operatons research sde, Ball et al. (2006) and Harsha (2009) developed optmzaton models to support auctonng of arport slots. In practce, however, exstng demand management practces are almost exclusvely based on admnstratve approaches. The foremost demand management mechansm currently n use s the schedule coordnaton process developed by the Internatonal Ar Transport Assocaton (IATA). Wth mnor varatons dependng on geographc locaton and local or regonal regulatons (e.g., n Europe), ths process, wth essentally dentcal gudelnes and prorty rules, s currently 2

3 appled at 175 schedule coordnated ( Level 3 ) arports worldwde, ncludng the great majorty of the busest ones outsde the Unted States (IATA, 2017). In Europe, for nstance, the process s mandatory for coordnated arports and drven by the EU regulaton (EC, 2002). Despte there are some dfferences, between the IATA gudelnes and the EU regulaton, n general the rules and prortes are very smlar. Ths paper proposes a novel model, the Prorty-based Slot Allocaton Model (PSAM), to optmze slot allocaton decsons based on slot avalablty and arlne slot requests. The model mnmzes the costs of schedule coordnaton to the arlnes and other arport stakeholders, as measured by the dsplacement from arlne requests, whle accountng for the many prortes and requrements ncluded n the IATA gudelnes. It develops an effcent computatonal approach that makes t possble to apply the model at even medum-sze arports, wth up to 100,000 arcraft movements per year, for an entre season of operatons. The paper then presents detaled applcatons at the Crstano Ronaldo Internatonal Arport of Madera and the Francsco Sá Carnero Arport of Porto, both located n Portugal, usng fne-gran data on arlne slot requests. The computatonal results suggest that such applcatons may offer mportant benefts by acceptng all slot requests, whle sgnfcantly reducng the largest flght dsplacement, the total schedule dsplacement, and the number of flghts dsplaced that are necessary to accommodate all requests. Before summarzng the paper s contrbutons n more detal n Secton 1.2, we provde addtonal nformaton on current schedule coordnaton processes and procedures IATA Slot Allocaton Process Ths secton provdes some background on the slot allocaton process endorsed by IATA, ncludng: () an overvew of ts dfferent stages and the scope of ths paper; () some mportant defntons and concepts; () ts prortes and requrements; and (v) the man sources of complexty of the problem consdered. The IATA schedule coordnaton process s carred out b-annually to provde arlnes wth access to schedule coordnated arports. Ths access s granted n the form of a landng or takeoff slot, defned as the permsson to use the full range of an arport s nfrastructure to perform arcraft arrvals or departures on a specfc day and at a specfc tme. For each season ( Summer or Wnter ), the IATA slot allocaton process nvolves fve man steps: 3

4 1) Settng of Declared Capacty: Each arport provdes the values of ts declared capacty, whch specfes the number of slots made avalable n each tme nterval of a day. Declared capactes are commonly specfed as hourly lmts on the number of flght movements (landngs and takeoffs) that may be scheduled, but may also be specfed at a fner level of granularty for () dfferent elements of the arport (e.g., runway capacty, apron capacty and termnal capacty), () dfferent types of movements (e.g., arrvals, departures and total), and () dfferent block duratons (e.g., capactes per hour, per 15-mnute perod, per 5-mnute perod, as well as per day, per week, per month, or per year), etc. The declared capactes of each schedule coordnated arport are announced about one year before the start of each season. 2) Slot Requests: The arlnes submt ther desred schedule of flghts at each arport to the schedule coordnator for the upcomng season. Flght schedulng requests are submtted n one of two forms. If a flght s to take place at least fve tmes over a season on the same day of the week and at the same tme of the day, the correspondng request must be submtted n the form of a seres of slots (e.g. a flght that takes place every Monday n July and August at 10:15). If the flght does not satsfy these crtera, the request s provded as an ndvdual slot. Requests for seres of slots are submtted approxmately fve months n advance of each season. Indvdual slots may be requested up to the actual day of operatons and may be awarded dependng on avalablty of slots at the requested tme. 3) Intal Slot Allocaton: At each arport, the schedule coordnator s tasked to perform the ntal slot allocaton n an unbased, transparent and non-dscrmnatory way. No contact s allowed between the slot coordnator and the arlnes. The allocaton of slots s performed solely on the bass of the prortes and requrements specfed by the IATA gudelnes. The coordnator provdes the resultng ntal schedule to the arlnes about four months before the start of each season. Only seres of slots are allocated at ths stage. 4) Schedule Coordnaton Conference: Potental adjustments to the ntal slot allocaton are made n the sem-annual IATA Slot Conferences (SC), whch are attended by arlne representatves, slot coordnators, arport representatves and other nterested partes. 4

5 These adjustments prmarly nvolve the resoluton of conflcts stemmng from the tmng of slots allocated across multple arports, and, f relevant, dsputes among arlnes competng for these slots. 5) Slot Return: The arlnes may return slots to the coordnator untl two months before the start of each season, f they decde that they wll not use these slots. They can also request and perform other schedule adjustments, subject to approval by the schedule coordnator, untl the day of operaton. The objectve s to correct any neffcences (from the arlne's standpont) resultng from the schedule coordnaton process. Ths paper focuses on the ntal slot allocaton (Step 3), whch s the most crtcal step n the entre process. Consstently wth the scope of the ntal slot allocaton, we only consder the seres of slots, and not the ndvdual slots, whch are only allocated after the SC. The allocaton problem takes as nputs the arport s declared capactes (Step 1) and arlne requests for seres of slots for the upcomng season (Step 2). Based on these nputs, the schedule coordnator allocates slots to the arlnes, subject to slot avalablty and the prortes and requrements specfed by the IATA gudelnes. Frst, the schedule needs to exhbt some regularty: () t s requred that all flghts belongng to the same seres of slots (.e., slots for the same flght on the same day of the week, at least fve tmes over the season) be gven a slot at the same tme of the day, and () t s recommended that dfferent seres of slots belongng to the same slot request code (.e., dentcal seres of slots for dfferent days of the week submtted together Secton 2.1) be gven slots at the same tme of the day across multple days of the week. Second, the turnaround tmes between flghts need to be mantaned between pars of arrvng and departng flghts (or, at least, adjusted wth mnmal changes) to mantan the connectvty of arlnes networks of flghts. Thrd, slot allocaton must follow a set of prortes specfed as prmary crtera for allocaton, as well as, when necessary, some addtonal crtera. The prmary crtera for allocaton defne prortes across four groups of slots, and allocate the seres of slots sequentally across these groups. Hghest prorty s gven to hstorc slots, or grandfathered slots,.e., seres of slots already held by the arlne n the prevous equvalent season (Wnter or Summer) and operated at least 80% of the tme (known as the use-t-orlose-t rule). Second prorty s gven to change-to-hstorc slots,.e., flghts for whch an 5

6 arlne holds a hstorc slot, but requests a change (e.g., n tmng or n arcraft type). Some change-to-hstorc requests allow the flght to be scheduled at any tme between the requested slot tme and the hstorc slot tme, whle some others allow the flght to be scheduled only at the requested tme or, f the requested tme s unavalable, at the hstorc tme. Thrd prorty s gven to new entrant arlnes, whch, accordng to the gudelnes, must receve up to 50% of the remanng slots (f demand s suffcent). The defnton of new entrant s based on market penetraton (e.g., an arlne that holds fewer than fve slots n a certan day of the season qualfes as new entrant for that day) and, potentally, other polcy consderatons (e.g., flghts requested for on underserved routes). Fnally, any remanng slots are allocated to the other requests,.e., requests that do not qualfy under the frst three prorty classes. In addton to these prmary crtera, the IATA gudelnes also provde a set of secondary crtera to dfferentate slots belongng to the same prorty class. Foremost, slot requests that extend exstng year-round operatons are gven prorty over new slot requests. Moreover, slot allocaton decsons can also consder other factors, such as the type of route (exstng route vs. new route), the type of servce (scheduled, charter and cargo), the sze of the arcraft (narrow-body vs. wde-body), and the type of market (domestc, regonal and long haul). These crtera are mostly used for te-breakng purposes,.e., to determne whch flghts to schedule f several solutons acheve the same outcome per the prmary crtera. For ths reason, they are beyond the scope of ths paper: we focus on determnng the optmal outcome (or outcomes) of the ntal slot allocaton based on the prmary crtera. Note that the seres of slots ntroduce nterdependences between the dfferent days of operatons. The problem cannot be decomposed nto a seres of ndependent problems that nvolve makng slot allocatons separately for each ndvdual day. Ths mght result n flghts belongng to the same seres of slots (or to separate seres of slots belongng to the same slot request code see Secton 2.1) beng scheduled at dfferent tmes on dfferent days. Instead, slot allocaton has to be performed for the entre season all at once. From a modelng standpont, ths creates couplng constrants across the slot requests from one day to another. From a computatonal standpont, ths ncreases greatly the sze, and, n turn, the complexty of the underlyng models. 6

7 1.2. Pror Work and Contrbutons Current slot allocaton procedures are asssted by specalzed software (e.g., PDC SCORE). Slot requests are typcally treated sequentally n an ad hoc bass, whch provdes only lmted vsblty on the whole set of slot requests and ther nteractons. In recent years, optmzaton models have emerged n the lterature to support the schedule coordnaton process. Experence wth these models (lmted to date) has suggested that there s potental for mprovng sgnfcantly slot allocaton decsons. An extensve revew of the current slot allocaton models s provded n Zografos et al. (2017), whch dvdes slot allocaton models nto two dfferent categores: sngle-arport slot allocaton models (see below) and networkwde slot allocaton models (Castell et al. 2011; Castell et al. 2012; Coroll et al. 2014). Ths paper presents a new sngle-arport model, a focus consstent wth current practce for the ntal slot allocaton (Step 3 of the process). The sngle-arport focus makes t possble, from a computatonal standpont, to consder seres of slots, all at once, across the entre season, whch s crtcal to ensurng complance wth the requrements of the IATA gudelnes. Sngle-arport slot allocaton models have been the subject of sgnfcant recent research. Frst, some research has focused on developng optmzaton models to determne the approprate level of the declared capacty to mnmze delays, maxmze arlne proftablty and maxmze passenger welfare (Swaroop et al., 2012; Vaze and Barnhart, 2012). Then, Jacqullat and Odon (2015) and Pyrgots and Odon (2016) developed optmzaton models to nform and assess schedulng adjustments at US arports by quantfyng ther effects on arlne schedules of flghts and resultng arport on-tme performance. However, the US focus of ths research dd not motvate consderaton of seres of slots and of some of the IATA gudelnes. Fnally, Zografos et al. (2012) developed an optmzaton model of slot allocaton that captured, for the frst tme, some of the prmary crtera of the IATA gudelnes by consderng prortes for hstorc and new entrant slots (but wth no separate treatment of change-to-hstorc slots). Moreover, that paper was the frst to consder explctly the noton of a seres of slots, thus ncreasng the tme scale of decson-makng to the entre season. The aforementoned models share the objectve of mnmzng total dsplacement from the slot tmes requested by the arlnes, measured as the absolute total dfference between the allocated and requested slot tmes. Ths was recently extended to ncorporate farness objectves between the arlnes, both at schedule-coordnated arports and at US arports 7

8 (Zografos and Jang, 2016; Jacqullat and Vaze, 2017). From a computatonal standpont, these models have been appled for a sngle day of operatons at some of the busest arports (Pyrgots and Odon, 2016; Jacqullat and Odon 2015) or for an entre season at only moderately busy arports. To the best of our knowledge, the largest arport where slot allocaton decsons have been addressed usng exact optmzaton methods s the Heraklon Internatonal Arport n Greece, whch operates fewer than 50,000 arcraft movements per annum (Zografos et al., 2012). Our paper extends ths prevous work n three major ways. Frst, from a modelng standpont, the Prorty-based Slot Allocaton Model (PSAM) s the frst model that optmzes slot allocaton decsons at schedule-coordnated arports, whle fully complyng wth the prortes across all the slot classes specfed n IATA s prmary crtera. In addton, t adds two new slot allocaton objectves, namely mnmzng the number of slots rejected and mnmzng the number of slots dsplaced, and t explctly captures the trade-offs between all the objectves underlyng slot allocaton decsons. Second, from a computatonal standpont, the PSAM provdes an effcent model formulaton and soluton approach that ensures, for the frst tme, the tractablty of an exact optmzaton approach for slot allocaton problems for an entre season at md-sze arports. Ths has enabled the mplementaton of the model at the arport of Porto, Portugal, whch operates roughly 100,000 arcraft movements per annum. Ths volume of traffc s over twce as large as the one at the busest arports prevously consdered n the lterature (Zografos et al., 2012). Thrd, from a practcal standpont, we perform comparsons wth real-world slot allocaton outcomes at the arports of Madera and Porto, Portugal, by leveragng fne-gran data on arlne slot requests and the resultng arport slot allocaton decsons. Results suggest that the model mproves the decsons made by slot coordnators by reducng the largest dsplacement experenced by any flght by 10 to 25 mnutes, the total schedule dsplacement by 4% to 27%, and the number of flghts dsplaced by 1% to 7%, dependng on schedulng and capacty patterns. Extensve computatonal tests also provde detaled characterzatons of the optmal slot allocaton decsons, and show that the optmzaton model can provde benefts across all prorty classes of the IATA gudelnes. In summary, ths paper provdes a model-based tool that can yeld sgnfcant mprovements n slot allocaton processes at congested arports by supportng and optmzng schedule coordnaton decsons based on quanttatve objectves. 8

9 The remander of ths paper s organzed as follows. Secton 2 descrbes and syntheszes the slot allocaton data from the arports of Madera and Porto. Secton 3 formulates the model, ncludng the techncal aspects of capturng the IATA gudelnes and prortes n optmzng the allocaton of slots. Secton 4 strengthens the formulaton and quantfes the resultng mprovements n computatonal performance. Secton 5 presents the computatonal results and ther mplcatons for schedule coordnaton practce. Secton 6 summarzes our work and ndcates drectons for future research. 2. Case Study Data The case studes reported n ths paper are based on slot request and slot allocaton data for the Summer Season of 2014 (from March 30, 2014 to October 25, 2014) at the arports of Madera and Porto, Portugal. Slot allocaton n Portugal s performed by ANA Aeroportos de Portugal. Both arports have runway declared capactes for each 15-mnute perod and 60- mnute perod on a 5-mnute rollng horzon bass, reported n Table 1. In Madera, no more than 14 movements, 7 arrvals and 7 departures can be scheduled between 8:00 and 9:00, between 8:05 and 9:05, between 8:10 and 9:10, etc., and no more than 6 movements, 4 arrvals and 4 departures can be scheduled between 8:00 and 8:15, between 8:05 and 8:20, between 8:10 and 8:25, etc. In Porto, no more than 20 movements can be scheduled per hour, and no more than 7 movements can be scheduled per 15-mnute perod (note that there s no separate lmt on the number of arrvals and of departures n Porto). Table 1 Madera Arport Declared Capactes for the Summer Season of 2014 Madera Arport Porto Arport Total flght movements / 60 mn Lmt on number of arrvals / 60 mn 7 20 Lmt on number of departures/ 60 mn 7 20 Total flght movements / 15 mn 6 7 Lmt on number of arrvals / 15 mn 4 7 Lmt on number of departures/ 15 mn 4 7 In addton to the runway declared capactes shown n Table 1, the arports of Madera and Porto are also subject to termnal and apron capacty constrants and to nose restrctons. These constrants were not consdered n ths paper, as they are not lmtng at these two 9

10 arports. In fact, n the solutons provded by the slot coordnators, no slots were dsplaced due to these capacty constrants. 2.1 Format of Slot Requests The slot requests from the arlnes follow the standard code provded by Chapter 6 of the Standard Schedules Informaton Manual (IATA, 2014). Table 2 shows a sample of these slot request codes n Madera, as provded by the arlnes to the slot coordnators. Ths ncludes, for each slot request: () the prorty class (hstorc, change-to-hstorc, new entrant or other); () the arrval/departure flght ID; () the start and end date of operatons; (v) the days of the week the slots wll be operated; (v) the type of arcraft and expected number of seats; (v) the requested arrval and departure tmes; (v) (v) the days of the week the slots wll be operated; (v) the type of arcraft and expected number of seats; (v) the requested arrval and departure tmes, (v) the orgn and fnal stop ( destnaton ) of the arcraft s overall tnerary (v) the last arport that the arcraft wll vst before landng at the subject arport (n ths case Madera), as well as the next arport the arcraft wll vst after departng from the subject arport, and (x) the type of flght (e.g., J for scheduled passenger flght, C for chartered passenger flght, etc.). For nstance, the thrd request shown n Table 2 corresponds to an arcraft tnerary that starts n Pars Orly (ORY), fles to Madera from Porto (OPO), then fles from Madera to OPO, and, eventually, ends at ORY. Req. Code Table 2 Sample slot request codes at Madera Prorty Arr. ID Dep. ID Start Date End Date Days of week 1 F XY001 XY002 30MAR 25OCT CR XY003 XY004 30MAR 1JUN CL XY005 XY006 01APR 21OCT B XY JUL 23SEP N --- XY008 30MAR 25OCT Req. Code Orgn Prevous Stop Seats Arcraft Arr. Tme Dep. Tme Next Stop Destnaton Arr. Type Dep. Type 1 LIS LIS LIS LIS J J 2 OPO LIS LIS OPO J J 3 ORY OPO OPO ORY J J 4 OPO OPO C PDL PDL --- J 10

11 For the purpose of the model developed n ths paper, the relevant nformaton corresponds to Ponts (), (), (v) and (v) above that specfy the days and tmes of the slots requested and the prorty of each slot request. The remanng nformaton may be used n other stages of the slot allocaton process. In the remander of ths secton we dscuss further the fve codes shown n Table 2. Frst, note that they belong to dfferent prorty classes. Specfcally, Request Code 1 corresponds to a hstorc slot (Code F), Request Codes 2 and 3 correspond to change-to-hstorc slots (Codes CR and CL), Request Code 4 to a new entrant (Code B), and Request Code 5 to a slot that does not belong to any of the aforementoned prorty classes. The dfference between the two types of change-to-hstorc requests s that, when an arlne submts a CR code, t s wllng to accept any slot between the requested and hstorc slot tmes whenever the requested slot s not avalable whle, when an arlne submts a CL code, t s only wllng to accept the hstorc slot tme when the requested slot s not avalable. Note that addtonal codes may be used by the arlnes (e.g., to specfy the slots that operate on a year round bass), but ths paper focuses on these fve man types of requests. Arrvals and departures may be requested n the same slot request (e.g., Request Codes 1, 2 and 3), or solely arrvals (Request Code 4) or solely departures (Request Code 5). In most nstances, arlnes nclude both an arrval and a departure n the same slot request code to ensure that the slot coordnator can mantan approprate connecton tmes and control the number of arcraft n the apron at any tme. However, some slot requests are made specfcally for each type of movement (f ths s allowed by the coordnator). Such requests typcally come from the bgger arlnes, whch derve operatng flexblty from the large number of arcraft they may be operatng at the subject arport. Arlnes may request more than one seres of slots (for dfferent days of the week) n the same slot request code. For nstance, Request Code 1 ncludes only one seres of slots, to be operated on Sundays (ndcated by the code). In contrast, Request Code 2 ncludes seven seres of slots, one per day of the week (ndcated by the code). Request Code 3 ncludes four seres of slots, to be operated on Sundays, Tuesdays, Thursdays and Saturdays ( ). Note that these seres of slots dffer only wth respect to the day of the week all ther other parameters are dentcal. As mentoned n the ntroducton, the 11

12 IATA gudelnes recommend that seres of slots requested n the same slot request code be allocated to the same tme on the dfferent days of the week. Overall, each slot request code may nclude a large number of slots. For nstance, Request Code 1 apples to the entre season (the 30 weeks between March 30 and October 25), and conssts of one seres of arrval and departure slots. Therefore, ths request nvolves a total of 30x2x1=60 slots. Smlarly, Request Code 2 corresponds to seven seres of slots of arrvals and departures over 10 weeks (between March 30 and June 1), and therefore conssts of 10x2x7=140 slots. The structure of slot requests thus creates mportant combnatoral complextes n slot allocaton, whch motvate the optmzaton approach proposed n ths paper. 2.2 Summary of the Data A total of 13,196 slots were requested by the arlnes at Madera for the Summer of 2014, dstrbuted across 836 seres of slots and 332 slot request codes. About 50% of the slots were requested as hstorc slots, 35% as change-to-hstorc slots, 1.5% as new entrant slots and 13.5% as other slots. At Porto, the number of slots requested was equal to 40,597, dstrbuted across 1,920 seres of slots and 882 slot request codes. About 64% of the slots were requested as hstorc slots, 21% as change-to-hstorc slots, 1.6% as new entrant slots and 13.4% as other slots. Fgure 1 shows the demand for slots and the slot lmts durng the busest day of the Summer of 2014 (August 18 n Madera and August 1 n Porto). Fgures 1a and 1b (resp., Fgures 1c and 1d) show the number of slots requested n the most recent prevous 60 mnutes and 15 mnutes, respectvely, for every 5-mnute perod of the day n Madera (resp. Porto). Note that several perods of the day are subject to mbalances between demand and capacty, as the number of slots requested exceeds the declared capacty at the arport. For these specfc days, such mbalances occur durng 17 5-mnute perods n Madera and 14 n Porto. Ths s explaned by the fact that the mornng peak perod s slghtly longer n Madera than n Porto, and that the Madera arport also mposes separate lmts on arrvals and on departures. 12

13 a) Madera, 60 mnutes rollng capacty b) Madera, 15 mnutes rollng capacty c) Porto, 60 mnutes rollng capacty b) Porto, 15 mnutes rollng capacty Fgure 1 - Demand for slots at Madera and Porto arports for the busest day of the Summer of 2014 Imbalances between demand and capacty are also found durng other days of the season. Table 3 shows the number of 5-mnute perods wth mbalances between demand and capacty durng the entre season, for the Madera and Porto arports. Specfcally, t reports the number of days wth mbalances by day of the week and by month of the season. Note that, at both arports, most mbalances occur n July, August and September,.e., durng the peak of the Summer season. Turnng to the days of the week, the capacty restrctons are bndng only on Mondays and Thursdays n Madera, whle mbalances are found on any day of the week n Porto (the busest days beng Thursdays, Frdays, and Sundays). Note that, even though demand for slots may fall below declared capactes on the least busy days of the season, these days need to be consdered n the slot allocaton decsons nonetheless because of the nterdependences between slots over the days of the week. These nterdependences across the entre season underscore the complexty of the problem of fndng slot allocaton solutons that wll comply consstently wth the values of the arport's declared capactes, as well as wth the IATA prortes and requrements regardng slot seres 13

14 and slot requests. Ths agan motvates the development and use of large-scale optmzaton technques to help coordnators make slot allocaton decsons more effcently and faster. Table 3 Number of 5-mnute perods where demand exceeds capacty n Madera and Porto arports Madera Arport Imbalances per day of week, and per month Mon Tue Wed Thu Fr Sat Sun Mar/Apr May Jun Jul Aug Sep Oct Porto Arport Imbalances per day of week, and per month Mon Tue Wed Thu Fr Sat Sun Mar/Apr May Jun Jul Aug Sep Oct At the two arports consdered, the mbalances between demand and declared capactes, although sgnfcant, can be addressed by reschedulng slot requests to dfferent tmes of the day, wthout rejectng any slot request. Nonetheless, the model presented n the next secton consders the possblty of rejectng slots. Ths s motvated by two consderatons. Frst, t provdes a more general framework that can also be appled at arports where total demand s so hgh that some slot requests may have to be rejected to satsfy the declared capacty constrants. Second, even at arports where total demand falls below total declared capacty, t may be necessary to consder slot rejectons when the IATA slot prortes are consdered. For example, some new entrant requests may be rejected f they exceed 50% of the remanng capacty after slots have been allocated to hstorc and change-to-hstorc seres. 3. The Prorty-based Slot Allocaton Model (PSAM) In ths secton, we present PSAM. Ths optmzaton model takes as nputs the values of arport declared capactes (see Table 1) and the slot requests of the arlnes, as descrbed n Secton 2.1. It then produces a schedule that mnmzes the dsplacement from the slot requests, subject to the constrants resultng from the prortes and requrements specfed by the IATA gudelnes and from the capactes declared by the arports. We present sequentally 14

15 ts nputs, decson varables, baselne formulaton, and adjustments to account explctly for the IATA gudelnes. 3.1 Inputs a) Sets T {1,..., T} : set of tme perods, ndexed by t D {1,..., D} S {1,..., S} S arr S : set of arrvals S dep and j : set of days, ndexed by d : set of slot requests codes, ndexed by or j S: set of departures P S S : set of slot request pars, js S such that there s a connecton between C {1,..., C} : set of capacty tme scales, ndexed by c The set T conssts of the number of perods of the day plus a snk perod at the end of the tme horzon (perod T) used for slots rejected. Note that the set S processes the seres of slots provded n the same request code together. As descrbed n the ntroducton and n Secton 2.1, the IATA gudelnes requre that the flghts requested n the same seres of slots (.e., for a gven day of the week) be allocated at the same tme of day, and recommend that the seres of slots requested n the same request code (.e., same seres of slots for dfferent days of the week) be allocated at the same tme of the day. By processng together all the slot requests n a slot request code, the PSAM actually also requres the latter. Moreover, for the slot requests that nclude both an arrval and a departure (see Table 2), we nclude two dfferent requests n the set S, and track the types of movements and connectons wth the subsets Sarr and Sdep and the set of flght pars P, respectvely. Fnally, the set C ncludes all the dfferent tme scales that are subject to declared capacty constrants (e.g., n the case of Madera or Porto shown n Table 1, t ncludes a 60-mnute tme scale and a 15-mnute tme scale). 15

16 b) Parameters A t B d 1, 0, 1, 0, f slot s requested to operate no earler than perod t otherwse f slot s requested to operate on day d otherwse dep C tdc arr C tdc T C tdc L c = departure capacty at perod t, day d and tme scale c = arrval capacty at perod t, day d and tme scale c = total capacty at perod t, day d and tme scale c = length of tme scale c max T = maxmum allowable ncrease n the connecton tme of two slots n comparson to the requested connecton tme mn T = maxmum allowable decrease n the connecton tme of two slots n comparson to the requested connecton tme Note that the connecton parameters max T and mn T are not provded n the data, but are consdered n the model to ether force connecton tmes to be mantaned to ther requested values, or to explore the trade-off between changes n connecton tmes and resultng schedule dsplacement (see Secton 5.1.b for more detals). We also assume that the fnal snk perod (.e., perod T) has nfnte departure, arrval and total capactes (snce ths perod s only used for flghts rejected and s not capacty-constraned) Decson Varables Durng the slot allocaton process, each slot request may be subject to four possble outcomes: () a slot request may be allocated at the requested tme; () a slot request may be allocated at a later tme; () a slot request may be allocated at an earler tme; (v) a slot request may be rejected. The decson varables capture these four outcomes. Frst, the decson varables Yt ndcate the allocated tme of each slot requested. Then, the decson varables X and 16

17 X defne the dsplacement of each slot request, and the decson varables W and ndcate f a slot s dsplaced or not Last, the decson varables that ndcate whether a slot s rejected or not are denoted as Z. The logcal relatonshps between varables wll be defned as part of the model's constrants n Secton 3.4. W Y t 1, 0, f slot s rescheduled to arrve / depart no earler than perod t otherwse X = dsplacement of slot f rescheduled to a later tme perod X = dsplacement of slot f rescheduled to an earler tme perod W W 1, f slot s dsplaced to a later tme 0, otherwse 1, f slot s dsplaced to a earler tme 0, otherwse Z 1, 0, f slot s rejected otherwse Note that each row of the Y t varables s of the form (1,..,1,0,..,0), nstead of the (0,..,0,1,0,..,0) format used n Zografos et al. (2012) and Pyrgots and Odon (2016). Ths follows the formulaton n Jacqullat and Odon (2015), whch s nspred by some effcent ar traffc flow management optmzaton models (Bertsmas and Patterson, 1998). The other decson varables must satsfy one of the followng combnatons of values: () X X W W Z 0 f a slot request s scheduled at the requested tme; () X 0 ; X 0 ; W 1 ; W 0 ; Z 0 f a slot request s dsplaced to a later tme; () X 0 ; X 0 ; W 0 ; W 1 ; Z 0 f slot request s dsplaced to an earler tme; (v) X X W W 0 ; Z 1 f slot request s rejected. 17

18 3.3. Objectve We consder the followng objectve functon, where w1, w2 and w3 represent weghtng parameters: d max, d d mn w B Z w X X w B X X B W W S S dd S dd S dd (1) Ths objectve functon ncludes four terms. The frst corresponds to the total number of slots rejected. The second ndcates the maxmum dsplacement mposed on any slot. The thrd quantfes the total dsplacement across all the flghts throughout the season. The last term captures the total number of slots dsplaced. The parameters w1, w2 and w3 are used to set the relatve weght of each of these four objectves. In ths paper, we consder weghts such that w1 w2 w3 1. In other words, the prmary goal s to ensure that all the slot requests wll be scheduled and none wll be rejected. Then the man objectve s to allocate these slots as close as possble from the requested tmes. Ths s captured by our objectves of mnmzng the maxmum dsplacement, and then the total dsplacement. The order of these two objectves s manly motvated by equty reasons, as t ensures that no slots wll ncur a dsproportonately large dsplacement. Fnally, we add to ths model the novel objectve of mnmzng the number of slots dsplaced, to reduce the complexty of the process and ease the subsequent negotatons durng the slot conference. Ths order among the four objectves s motvated by current practce from the slot coordnators and the nterests of the arlnes, and s consstent wth the exstng lterature (Zografos et al., 2012; Jacqullat and Odon, 2015; Pyrgots and Odon, 2016). Note, fnally, that our modelng framework s flexble enough to capture other prortes among these dfferent objectves through varatons n the weght parameters w1, w2 and w3. We show examples of such varatons n Secton 5. 18

19 3.4. Constrants The constrants to nclude n the model are as follows: Y 1 1 S (2) Y Y S tt Y t,, t 1, T Z S (3) (4) 1 tt Y A X X A Z t t t tt S (5) W Y A Z S, t T t t W Y A S, t T t t sl c Sarr ts sl c Sdep ts sl c S ts arr Y 1,, t Y, t1 Bd Csdc st s T Lc d D c C dep Y 1,, t Y, t1 Bd Csdc st s T Lc d D cc Y Y B s s d D, c C T 1, t, t1 d Csdc T T Lc mn jt t jt t T j, tt Y Y A A T Z Z j P tt max Yjt Yt Ajt At T T Z Z j, tt j P tt (6) (7) (8) (9) (10) (11) (12) X, X 0 (13) Y, W, W, Z 1,0 t (14) Constrants (2) ensure that all the slots requested are allocated to some perod. Constrants (3) ensure that the varables Y are non-ncreasng n t, whch s consstent wth ther defnton. Constrants (4) to (7) defne the logcal relatonshps between the varables X, X, Y, W, W, Z t (see detals below and n Proposton 1). Constrants (8), (9) and (10) ensure that the arport capactes for arrvals, departures and total number of movements are 19

20 never exceeded over any day d. The formulaton of these three constrants s smlar to the one presented n Zografos et al. (2012) and enables the consderaton of capactes for dfferent tme scales c. Constrants (11) and (12) ensure that the tme between two connected flghts does not ncrease/decrease by more than the allowable lmts. The term ensures that the slots rejected are not constraned by the connectvty parameters mn T : Snce T s a large number, whch corresponds to the total number of perods n a day (.e. the maxmum number of perods by whch a tme connecton may ncrease or decrease), the constrants are necessarly not volated for rejected slots (.e., when Z=1 or Zj=1). Fnally, constrants (13) and (14) specfy the domans of the decson varables. We now descrbe how the logcal relatonshps between the dfferent varables are captured through constrants (4) to (7). At a hgh level, constrants (4) defne whether a slot s rejected or not, whch happens when the slot s dsplaced to the last tme perod T. Constrants (5) defne the dsplacement of each slot as the dfference between the requested tme (the parameters At) and the allocated tme (the decsons Yt). The term and forces the dsplacement of a rejected slot to be equal to zero to avod double-countng the penalty assocated wth flght rejectons. Constrants (6) and (7) defne the bnary varables W, whch ndcate whether a slot s dsplaced or not, by forcng each to be equal to 1 f there s any dscrepancy between slot s scheduled and requested tmes, and f slot request s not rejected. More specfcally, we show n Proposton 1 that the optmal soluton can only be of four types, and characterze these four cases. Proposton 1: Let us consder an optmal soluton to the model. Then, for each flght request n S, one of the followng four propertes s satsfed: t T 1 At Z T Z max T Z j () Z X X W W 0,.e., flght request s allocated to the requested tme. () Z 0 and X 0, X 0 and W 1, W 0,.e., flght request s rescheduled to a later tme. () Z 0 and X 0, X 0 and W 0, W 1,.e., flght request s rescheduled to an earler tme. 20

21 (v) Z 1 and X X W W 0,.e., flght request s rejected. Proof: Let us consder an optmal soluton Z / X / X / W / W / Y t. Let us also consder a gven slot request S. We frst show that ether X 0 or X 0. By contradcton, we assume that X 0 and X 0. Then, wthout loss of generalty, we assume that X X. We defne a new soluton X / X * * as follows: X X X, X 0, and X X, X X for all j. Ths * * * * * * soluton s a feasble soluton, as t satsfes Constrants (5) (because X X X X ), * * and all other constrants are unchanged. Moreover, we have max X, X max X, X, and X * X * X X. Ths contradcts the fact that X / X s an optmal S soluton. S We now nvestgate the case where three subcases: (a) X X 0 ; (b) X 0 and X 0 ; and (c) X 0 and X 0. We nvestgate these three cases sequentally, and show that they are equvalent to propertes (), (), and (), respectvely. j j j j S S Z 0. Per the result above, ths case s separated nto Frst, let us consder the case where Z 0 and X X 0 (case (a)). We have t T Y A 0 t t (constrant (5)) and, snce Y and A are both of the form t t (1,1,,1,0,,0), ths mples that Y A for all t T. From constrants (6) and (7), we obtan W 0 and W 0.e., W W 0 because the soluton s optmal. Ths proves (). t t Second, let us consder the case where Z 0 and X 0, X 0 (case (b)). We have Yt At X 0 t T t t t t (constrant (5)), so Y A. Snce Y and A are both of the form (1,1,,1,0,,0), ths mples that Y A for all t T and there exsts at least one t perod such that Y A. From constrants (6) and (7), we obtan W 1and W 0 st s s T T t t t 21

22 .e., W 1and W 0 because the soluton s optmal. Ths proves (). We proceed smlarly n the case where Z 0 and X 0, X 0 (case (c)) and prove (). Fnally, we nvestgate the case where Z 1. From constrants (3) and (4), we have Y 1 t for all t. From constrants (5), we have Yt 1 X X,.e.,. Snce the t T X X 0 soluton s optmal, ths mples that X X 0 (ths can be easly checked by contradcton as done n the frst part of ths proof). Snce Z 1, we have Y A Z 0 for all t T, so constrants (6) become W 0 and, snce the soluton s optmal, W 0. Moreover, snce Y 1 for all t T, Y A 0 for all t T, so constrants (7) become t t t W 0 and, snce the soluton s optmal, W 0. Ths proves (v) and concludes the proof. We now turn to the addtonal constrants that arse from the consderaton of the successve IATA prorty classes IATA Prorty Constrants The IATA gudelnes requre consderaton of the dfferent prortes assgned to the varous types of slot requests. Ths s acheved through a sequental approach that frst allocates hstorc seres of slots, followed by the change to hstorc seres of slots, followed by new entrant slots, and, fnally, by the remanng slots. From a modelng standpont, ths s formulated through a lexcographc approach that solves each of these four sub problems n sequence. Accordngly, we dvde the model nto four sub-models, one for each slot prorty. For each sub-model, we store the optmal value of the objectve functon (OV) for the slot prorty consdered (Equaton (1)). The reason why we fx ths optmal value rather than the decson varables s that there may be more than one optmal soluton for the prorty class consdered, so fxng the slot allocaton decsons mght be constranng the resultng slot allocaton for the followng prorty classes more than necessary. Specfcally, we partton the set of slot requests S nto subsets SH, SCH, SNE, SOS, whch nclude the hstorc slots, change-to-hstorc slots, new entrants, and other slots, respectvely. We frst solve the model for hstorc slots (see Secton 3.5.a), and store the optmal value of the objectve functon, denoted by OV H. Note that OVH s typcally equal to 22 t t

23 0, as hstorc slots are typcally not dsplaced (see below). Then, we add Constrants (15) to the sub-models of change-to-hstorc, new entrants and other slots, to ensure that the allocaton of slots remans optmal for the hstorc slots. We then turn to the sub-model for change-to-hstorc slots (Secton 3.5.b), store the optmal value of the objectve functon OV CH, add Constrants (16) to the sub-models of new entrants and other slots, solve the submodel of new entrants, store the optmal value of the objectve functon OV NE, add Constrants (17) to the sub-model of other slots, and solve the sub-model of other slots. max, w B Z w X X w B X X B W W OV 1 d 2 3 d d H S S H H dd SH dd SH dd max, w B Z w X X w B X X B W W OV 1 d 2 3 d d CH S d CL S S CR CL SCR D SCL SCR dd SCL SCR dd max, w B Z w X X w B X X B W W OV 1 d 2 3 d d NE S S d NE NE D S NE dd S NE dd (15) (16) (17) From a computatonal standpont, ths lexcographc approach mproves the tractablty of the model by decomposng t nto four smaller problems. On the negatve sde, t does not search for alternatve solutons that could potentally meet arlnes' requests to a greater extent through even modest adjustments n the IATA requrements. Ths can be addressed n future research by relaxng some of the constrants derved from the IATA gudelnes and quantfyng the resultng mpacts on slot allocaton effcency. Some addtonal constrants are now needed to capture the rules mandated by the IATA gudelnes for each of the prorty classes. These are formulated below a) Hstorc Slots Snce the hstorc slots have absolute prorty, they can be smply allocated ther requested tmes, wthout runnng an optmzaton model. We wll therefore have X X W W Z 0 for all slot requests n SH. As a result, the optmal value of the objectve functon OVH wll be equal to 0. Note that ths assumes that there s suffcent capacty to accommodate all the hstorc slots requests, whch wll be the case n practce, as long as the declared capacty does not decrease from year to year. 23

24 b) Change-to-hstorc Slots As descrbed n Secton 2.1, changes to hstorc slots may be requested n two dfferent ways: f the requested slots are not avalable, then CR code requests can be scheduled at any tme between the requested and hstorc slot tmes, whle CL code requests can only be scheduled at the requested or the hstorc slot tmes. We denote the subsets of SCH that nclude all CR code requests and all CL code requests by SCR and SCL, respectvely. We also ntroduce a new model parameter Ht, whch ndcates whether the hstorc slot tme of was no earler than perod t (ths parameter has the same (1,,1,0,,0) format as the parameter A and the decson varable Y). Constrants (16) and (17) ensure that CR slots are allocated between the hstorc and the requested slot tme. Constrants (18) and (19) ensure that CL slots are allocated ether at the requested slot tme or at the hstorc slot tme. X H A W S (16) t t CR tt X A H W S (17) t t CR tt X H A W S (18) t t CL tt X A H W S (19) t t CL tt In addton, we must make sure that the connecton tmes between two change-to-hstorc slots le between the requested connecton tmes and the hstorc connecton tmes. We ntroduce a set PC n P that ncludes all the pars of flghts n SCH. We defne, for each par (, j), the requested connecton tme and the hstorc connecton tme, denoted by A j and j t T, respectvely. Mathematcally, ths s expressed as A j A A jt t and j t T jt t A A A. We then ensure approprate connecton tmes wth Constrants (20) and (21): Yjt Yt mn A j, j T Z Z j tt H P (20) Yjt Yt mn A j, j T Z Z j tt H P (21) c c 24

25 Wth these new constrants, the model s solved wth respect to the objectve functon (1) to mnmze dsplacement across all the change-to-hstorc slots. Note that the constrants presented n ths secton wll be ncluded not only n the sub-model of change-to-hstorc slots, but also n the sub-models of new entrants and other slots, n order to ensure that the allocaton of change-to-hstorc slots s consstent wth the rules of IATA. c) New Entrant Slots Accordng to the IATA gudelnes, after allocatng hstorc slots and change-to-hstorc slots, 50% of the remanng slots must be allocated to new entrants, unless the number of requests from new entrants falls below that percentage. To capture ths requrement, we denote the remanng arrval capacty, departure capacty and total capacty for new entrants by arr, NE, dep, NE C c C c and T, NE respectvely. Expressons (22) defne T, NE as 50% of the remanng slots after the allocaton of slots to hstorc and change-to-hstorc requests. Ths value s computed through preprocessng before solvng the sub-model for new entrants. Analogous expressons C c C c are also used to defne arr, NE and dep, NE. C c C c T Lc1 slc, 1 T NE Cc Csdc Yt Y, t1 Bd c C 2 dd s1 SH SCR SCL ts (22) In order to ensure that the total capacty for new entrants, T, NE C c, s not exceeded, we add Constrants (23) to the model. Analogous constrants are specfed for arr, NE and dep, NE. C c C c T L 1 sl c c dd s1 SNE ts Y Y B C c C T, NE t, t1 d c (23) In cases where more slots than avalable are requested by new entrants, some of the requests wll be rejected. The slots rejected at ths stage wll be reconsdered at the next stage of the allocaton process wth the same prorty as all the other remanng slots. Note that, n practce, the number of slots rejected at ths stage wll almost always be equal to zero, as arports typcally have perods of the day wth very low demand. As a result, the total number of slots remanng after the changes-to-hstorc allocaton wll typcally be a 25

26 large number, lkely to exceed the number of slots requested by new entrants. Ths s why some slot coordnators tend to smplfy the new entrant rule by smply prortzng all new entrant slot requests over the remanng slot requests. In such cases, we can apply drectly the model presented n Sectons 3.1 to 3.4 (wthout Constrants (23)) to mnmze dsplacement of the new entrant slots. One can then check the resultng soluton, and make adjustments, f necessary. d) Remanng Slots Remanng slots are allocated accordng to the model formulaton presented n Sectons 3.1 to 3.5. In ths stage, we allocate slots wth no prorty, ncludng the new entrant slots rejected n the prevous stage Model Sze Table 4 shows the number of bnary and nteger varables and the number of constrants n the model presented n ths secton, as well as the resultng sze of the model for the cases analyzed at Madera and Porto. Note that the sze of the sets T and D s dentcal from arport to arport. Typcally, each slot perod corresponds to 5 mnutes, so each day s dvded nto T=288 perods. The length of the season s defned by IATA. For nstance, the Summer Season of 2014 lasted from March 30 to October 25, whch corresponds to D=210 days. In contrast, the set S vares from arport to arport. For that season, Madera receved 332 slot request codes, 275 of whch conssted of flght pars of movements. Therefore, S and P comprse ( ) 607 elements and 275 elements, respectvely. In the case of Porto, the arport receved 882 slot requests codes, 312 of whch conssted of flght pars of movements. Therefore, S and P comprse 1194 elements and 312 elements, respectvely. The sze of the arport therefore has a sgnfcant mpact on the sze of the model. Table 4 Sze of the Model Sze of the Model Madera Case Study Porto Case Study Number of bnary varables ST+3S Number of nteger varables 2S c C Number of constrants 2ST 3S P 3 T Lc 1 D

27 4. Model Enhancement In ths secton, we strengthen the formulaton of the PSAM by ntroducng new constrants that reduce the gap between the nteger formulaton of the model and ts lnear relaxaton, thus sgnfcantly mprovng ts computatonal performance. Ths enables, n turn, the consderaton of exact optmzaton methods to solve real-world nstances at md-sze arports, such as Porto, n reasonable computatonal tmes Formulaton Strengthenng As descrbed n Secton 4.2, the formulaton ntroduced n Secton 3 can lead to computatonal ntractablty even n cases nvolvng modest-sze arports. We therefore strengthen the formulaton of the model presented n Secton 3 to fnd better lnear relaxatons and, consequently, faster soluton tmes. To ths end, we ntroduce new constrants to remove portons of the feasble regon that contan fractonal solutons wthout elmnatng any feasble nteger solutons, thus ensurng that the optmal nteger soluton remans unchanged. In fact, the lnear relaxaton of the model proposed n Secton 3 yelds an optmal soluton equal to zero n all cases consdered, as long as all demands can be accommodated through temporal shfts and there s no need to reject slot requests (.e., total demand for slots falls below total capacty durng each day). Ths property stems from Constrants (5), whch make t possble to satsfy the capacty and connectvty constrants wth dsplacement varables equal to zero ( X X 0). To llustrate ths pont, Table 5 provdes a small example wth a sngle day dvded nto 5 perods, 3 slot requests (Slots 1 and 2 are requested n Perod 2, and Slot 3 n Perod 3), and a capacty of 1 slot per perod. The declared capacty constrants are only volated n Perod 2. An optmal soluton conssts of dsplacng Slot 1 from Perod 2 to Perod 1, and allocatng Slot 2 to Perod 2 and Slot 3 to Perod 3, as requested. The optmal value of the total dsplacement s equal to 1. 27

28 Table 5 Inputs, nteger soluton, and lnear relaxaton for a smple example Slot Requests Soluton of the Integer Model Soluton of the Lnear Model Slot Slots Slot Slots Slot Slots Capacty Capacty Requested Allocated Allocated Perod Capacty Dsplacement = 1 Dsplacement = 0 However, the lnear relaxaton of the model yelds a fractonal soluton, where half of Slots 1 and 2 are dsplaced to Perod 3, and half of Slot 3 s dsplaced to Perod 4. We then obtan from Constrants (5) a dsplacement of zero, shown n Equatons (24) and (25). Note that ths s a very smple example, more cases of fractonal solutons are found when we consder larger problems. X1 X1 X 2 X (24) 3 3 X X (25) Note that ths s a very smple example and more cases of fractonal solutons wth a dsplacement of zero are found when we consder larger problems. In order to elmnate such fractonal solutons, we replace constrants (5) wth constrants (26) and (27) defned below. The purpose of these constrants s to force one of the dsplacement varables ( and ) to be dfferent from zero whenever the dfference between Y and A s dfferent from zero. We refer to the model developed n Secton 3 (wth Constrants (5)) as the orgnal model and to the model obtaned by replacng Constrants (5) wth Constrants (26) and (27) as the modfed model. t t X X 1 1 tt tt A Y X A Z S t t t tt 1 A Y X S t t (26) (27) 28

29 Note that the dfference between Equatons (26) and (27) yelds exactly Equaton (5), so any feasble soluton of the modfed model s also a feasble soluton of the orgnal model. We then prove n Proposton 2 that both models yeld the same optmal nteger soluton. Proposton 2: Any optmal soluton of the orgnal model (wth Constrants (5)) s an optmal soluton of the modfed model (wth Constrants (26) and (27)). Proof: Let us consder an optmal soluton of the orgnal model, and show that t satsfes Constrants (26) and (27). We consder a gven slot request. From Proposton 1, we know that t satsfes one of four cases: () Z X X 0 ; () Z 0, X 0, X 0 ; () Z 0, X 0, X 0 ; or (v) Z 1, X X 0. Let us frst consder case (). As n the proof of Proposton 1, we have t T Y A 0 t t (Constrants (5)) and, snce Y and A are both of the form (1,1,,1,0,,0), ths mples that Yt At for all t T. Snce Yt and are both bnary, we then have for all t, so Y 1A 0 and A 1Y 0. Ths proves that Constrants (26) and (27) are satsfed. t t A Y 1A At 1 Yt 0 t t t T t t t T t t t We now consder case (). As n the proof of Proposton 1, we show that Y s A s, and therefore Y 1 for all perods t such that At 1. We then have AtYt At for all t t T (because f A 0, then AY 0, and f A 1, then Y 1 and AY 1). Ths gves the followng equalty: tt satsfed. Smlarly, because Y A X. Therefore, A 1 Y A A Y 0, provng that constrant (26) s satsfed. We can proceed smlarly for case () X 0 and X 0 (.e., where the flght s rescheduled to an earler tme). Fnally, we consder case (v) where slot s rejected. From constrants (3) and (4), we have t 1 for all. Therefore, 1 A Y 1 A, so constrant (26) s satsfed. Smlarly, t 1 1 A Y X A Z t t t t tt t t 1 At Y t t Y t t A t ty t T T T tt t tt t tt t t tt t tt t t Y t T t T A t tt 1Y 0, so constrant (27) s also satsfed. Ths concludes the proof. t t t Z 0. Ths proves that constrant (25) s t t tt t t 29

30 Note that constrants (26) and (27) render nfeasble the prevous fractonal soluton that led to an optmal dsplacement of zero. For the example shown n Table 5, equatons (28) to (31) X X provde the new values of and, based on constrants (26) and (27), for the soluton of the lnear relaxaton of the orgnal model. The total dsplacement s now equal to 3, whle the optmal nteger soluton s stll equal to 1. Therefore, the new soluton s not an optmal soluton of the lnear relaxaton of the modfed model. In turn, constrants (26) and (27) strengthened the nteger programmng formulaton by tghtenng the feasble regon of ts lnear relaxaton. X1 X2 (1 1) 1 (1 1) 0.5 (1 0) 0.5 (0 0) (28) X1 X2 1 (1 1) 1 (1 0.5) 0 (1 0.5) 0 (0 0) 0.5 (29) X 3 (1 1) 1 (1 1) 1 (1 1) 0.5 (1 0) (30) X 3 1 (1 1) 1 (1 1) 1 (1 0.5) 0 (1 0.5) 0.5 (31) Table 6 shows the nteger and lnear solutons for the modfed model. Even though the lnear relaxaton yelds a fractonal soluton, the optmal value of the objectve functon of the lnear relaxaton s now equal to 1, and s, n fact, dentcal to the optmal value for the nteger programmng model. For large nstances, the optmal values of the objectve functon may not be dentcal n all cases, but the modfed model presented n ths secton results n a much smaller gap between the nteger programmng model and ts lnear relaxaton. Whle we are stll not able to guarantee nteger solutons, the modfed model leads to much faster computatonal tmes than the orgnal model, as shown n the next secton. Table 6 Integer soluton and lnear relaxaton wth the modfed model Slot Requests Soluton of the Integer Model Soluton of the Lnear Model Slot Slots Slot Slots Slot Slots Capacty Capacty Requested Allocated Allocated Perod Capacty Dsplacement = 1 Dsplacement = 1 30

31 Fnally, we also added two vald nequaltes to the model proposed (Constrants (32) and (33) below), whch specfy that any slot s not dsplaced W W 0 varables are equal to zero X f the dsplacement X 0. We can prove formally that these constrants are satsfed by the optmal soluton of the problem consdered and restrct the feasble regon of ts lnear relaxaton, therefore mprovng the computatonal performance of the model. We omt ths proof to avod repeatng the same procedure as above. W X S (32) W X S (33) Note that we can now relax the ntegralty constrant for varables and, because t wll be automatcally satsfed based on constrants (26) and (27). Ths reduces the number of nteger varables and therefore further mproves the computatonal performance of the model Computatonal Performance We appled the model to the Madera and Porto arports usng CPLEX 12.5, mplemented usng GAMS as the modelng language. We looked for exact solutons (.e., wth a 0% optmalty gap). The model was run wth an GHz, 8 Gb RAM computer under a Wndows bt operatng system. Table 7 compares the computatonal performance of the orgnal model (wth constrants (5)) and the modfed model (wth constrants (26) and (27)), usng data from Madera for the frst three weeks of the season. For ths experment, the model was solved wth all the slot requests wthout prorty consderatons, wth the objectve of mnmzng the total dsplacement only. As expected, the soluton of the modfed model s obtaned n sgnfcantly lower computatonal tmes than that of the orgnal model. For the smallest nstances, that conssted of optmzng the slot allocaton for only the frst day of the season, the modfed model s more than 20 tmes faster. As the sze of the nstance ncreases, the computatonal tmes of the modfed model ncrease moderately, whle those of the orgnal model ncrease extremely fast. For the three-week nstance, the orgnal model does not termnate after 1 day 31 X X

32 (wth an optmalty gap of over 15%), whle the modfed model fnds the optmal soluton n only 86 seconds. Thus, the orgnal model cannot be scaled to provde even approxmate solutons for an entre season n reasonable tmes, whle the modfed model enables the consderaton of problems of sze realstc for larger arports. Note, moreover, that the optmal value of the objectve functon of the lnear relaxaton of the orgnal model s always zero, whle the modfed model yelds lnear relaxaton values that are much closer to the nteger soluton value. Table 7 Improvements n the model s performance wth constrants (26) and (27) 1 Day 1 Week 2 Weeks 3 Weeks Number of Perods (T) Number of Days (D) Number of Slot Requests (S) Solvng Tme wth Constrants (5) (sec) (49 mn) (15.5 h) >1day Solvng Tme wth Constrants (26) and (27) (sec) Optmal value of total dsplacement (mn) Lnear Relaxaton wth constrants (5) Lnear Relaxaton wth constrants (26) and (27) Overall, the modfed model was solved for the entre season (wthout consderng the IATA prortes) n about 15 mnutes for Madera and 45 mnutes for Porto. When consderng the IATA prortes, the PSAM (wth the objectve functon specfed n Secton 3.3) s solved n only 4 mnutes for Madera (1 to 2 mnutes for each prorty class) and n 8 mnutes for Porto (2 to 3 mnutes for each prorty class). As expected the computatonal tmes are lower than n the nstance where the IATA prortes are gnored, as the problem s decomposed nto four smaller problems. Fnally, note that the computatonal performance may be senstve to the values of the weghts w1, w2 and w3. For nstance, assgnng a hgh weght to the number of slots dsplaced can ncrease the model s solvng tmes by a factor of 2 to 4. Ths may be due to the fact that gvng top prorty to the mnmzaton of the number of slots dsplaced results n many more optmal, or close-to-optmal, solutons. The takeaways from these computatonal experments are threefold. Frst, the modelng and computatonal approach developed n ths paper provdes fast solutons for a full season of operatons at medum-sze arports and can thus be used n support of slot allocaton processes. Second, the PSAM can provde n reasonable computatonal tmes alternatve solutons reflectng dfferent weghts attrbuted to the dfferent objectves of slot allocaton. 32

33 Therefore, t permts exploraton of the set of Pareto-optmal solutons when more than one objectves are consdered. Thrd, the soluton of the model remans tractable even when the IATA prorty classes are not consdered. Ths makes t possble to perform senstvty analyses wth respect to the IATA rules, such as those presented n Secton 5.1 below. Thus, n addton to provdng a near-term decson-makng support tool for slot coordnators, the PSAM can also be used as a more strategc tool n support of polcy-makng to evaluate the mpact of exstng and alternatve rules on the slot allocaton process. 5. Model Results In ths secton, we present the results obtaned through the PSAM for the Madera and Porto arports. We frst nvestgate the senstvty of the slot allocaton outcomes to the varous constrants mposed by the IATA gudelnes, as well as to dfferent prortes among the PSAM objectves. We then compare the model s solutons at the two arports wth the allocaton that was made n practce to quantfy the potental benefts assocated wth the mplementaton of the model n support of slot coordnaton decsons. We do not consder the possblty of slots beng rejected, as a feasble soluton can be found at both arports consdered wthout rejectng any requests Senstvty Analyss to Slot Allocaton Constrants and Objectves of PSAM We frst quantfy the mpact of the varous requrements mposed by the IATA gudelnes on the optmal dsplacement from the arlne slot requests at Madera arport. We consder here two measures of dsplacement: the maxmum dsplacement dsplacement B d d X X, and the total S D. We compute the Pareto optmal fronter for these two objectves (.e., the set of solutons such that no other soluton can mprove one objectve wthout worsenng the other) by usng a goal programmng approach (Steuer, 1986; De Weck, 2004, Marler and Arora, 2004). In other words, we frst compute the optmal value of the maxmum dsplacement, denoted by δ, and the optmal value of the total dsplacement, denoted by Δ. We then mnmze the total dsplacement whle progressvely ncreasng the value of the maxmum dsplacement from δ by ncrements of 5 mnutes, untl the optmal value of the total dsplacement (.e. Δ ) s attaned. In Secton 5.1.e), we extend ths analyss through a senstvty analyss to the order prorty among the PSAM objectves. 33 max X, X S

34 a) Interdependences between Slots over the Season The complexty of the slot allocaton process largely stems from the nterdependences between slot requests across the season. To ensure consstent treatment of all the flghts n the same seres of slots or n the same slot request code, the allocaton of slots has to be performed all at once for the entre season, and not for each day ndvdually. In order to quantfy the mpact of these nterdependences on the slot allocaton decsons, we solved the model frst ndvdually for each sngle day, and then for the entre season. We dd not consder the IATA prortes of slot classes at ths stage. The Pareto-optmal solutons are shown n Table 8. The second column shows the total dsplacement obtaned for the entre season by optmzng slot allocaton decsons separately for each sngle day, whle the thrd column provdes the total dsplacement obtaned for the entre season when PSAM s solved at once for all the days of the season, consderng the nterdependences between slots on dfferent days. Table 8 - Pareto-optmal solutons for the Madera arport wthout and wth nterdependences Max Dsp (mn) Indvdual days Entre season (+83%) (+41%) (+37%) (+28%) (+26%) (+24%) (+24%) Note, frst, that the optmal value of the maxmum dsplacement s equal to 15 mnutes n both cases. For ths case, the total dsplacement for the soluton that consders the entre season s 83% larger than when solvng ndvdual days separately. Second, the soluton that mnmzes the total dsplacement yelds a total dsplacement 24% larger (8,675 mnutes vs. 6,990 mnutes) and a maxmum dsplacement 33% larger (40 mnutes vs. 30 mnutes) wth the nterdependences than wthout them. Slot nterdependences thus have a strong mpact on slot allocaton decsons by restrctng the soluton set, therefore leadng to sgnfcantly larger dsplacement values and renderng the computatonal requrements of the underlyng models much more complex. 34

35 b) Connectvty Parameters Accordng to the IATA gudelnes, the coordnator shall mantan the connecton tmes requested by the arlnes between two connected slots (arrval-departure par) or, f ths s not feasble, shall endeavor to mnmze the ncrease or decrease n connecton tmes. In fact, there exsts a trade-off between changes n connecton tmes and schedule dsplacement. We demonstrate ths trade-off n Table 9 and Fgure 2 by showng the Pareto-optmal values of the maxmal and total dsplacement for dfferent values of the connectvty parameters and mn T, whle mantanng T max mn max mn T. Values of force the connecton tmes to adhere to those requested by the arlnes, whle ncreasng them provdes some addtonal flexblty n the slot allocaton decsons. In all cases, the mnmum value of the maxmum dsplacement s equal to 15 mnutes. The connectvty constrants have an mportant mpact on the optmal total dsplacement, for any gven value of the maxmum dsplacement. If the maxmum dsplacement s mnmzed, the total dsplacement can vary by as much as 23% n response to varatons n the connectvty parameters. If the total dsplacement s mnmzed, the optmal value of the total dsplacement can vary by 15% and the optmal value of the maxmum dsplacement can vary from 35 mnutes to 45 mnutes (a 30% ncrease). T T 0 max T Table 9 - Pareto-optmal solutons for the Madera Arport wth dfferent connectvty parameters Connectvty Parameters Max Dsp (mn) Total Dsplacement (mn) 15 15,600 13,690 (-12.2%) 13,210 (-15.3%) 12,660 (-18.8%) 11,945 (-23,4%) 20 11,815 10,520 (-11%) 9,605 (-18.7%) 9,375 (-20.7%) 9,315 (-21%) 25 10,775 9,210 (-14.5%) 9,105 (-15.5%) 9,040 (-16.1%) 8,765 (-18,7%) 30 9,755 8,845 (-9.3%) 8,660 (-11.2%) 8,600 (-11.8) 8,480 (-13,1%) 35 9,750 8,695 (-10.8%) 8,565 (-12.2%) 8,500 (-12.8%) 8,310 (-14,8%) 40 8,435 (-12%) 8,365 (-12.7) 8,240 (-14,1%) 9,590 8,595 (-10.4%) 45 8,430 (-12.1%) 8,305 (13.4%) 8,180 (-14,7%) Note that the mpact of the connectvty parameters on the optmal dsplacement s non-lnear. In other words, lmted flexblty n the connectvty parameters (e.g., 5 mnutes) can lead to sgnfcant mprovements n the total dsplacement, rangng from 9.3% to 14.5%. Further ncreases n the connectvty parameters by the same amount (e.g., 5 mnutes) yeld 35

36 mprovements n the resultng dsplacement of a much smaller magntude. In fact, Fgure 2 shows that more sgnfcant reductons n the schedule dsplacement can be acheved by max mn ncreasng T and T from 0 to 5 mnutes than from 5 mnutes to nfnte values. These results ndcate that even small adjustments n the connecton tmes can have a postve mpact on overall dsplacement. Fgure 2 - Evoluton of the Pareto-optmal fronters for the Madera Arport consderng dfferent connectvty parameters c) IATA Prorty Constrants In the solutons obtaned thus far, all flghts were treated equally regardless of ther prorty class. For nstance, up to 20-30% of hstorc slots are dsplaced, contradctng the grandfather rghts accorded by the gudelnes. When we requre that hstorc slots cannot be dsplaced (constrant (12)), we obtan two Pareto-optmal solutons wth a maxmum dsplacement equal to 55 and 60 mnutes, respectvely, and a total dsplacement of 11,145 mnutes and 10,805 mnutes, respectvely. In other words, the hstorc slot constrants result n very large ncreases n the maxmum flght dsplacement (from 15 mnutes to 55 mnutes) and n sgnfcant ncreases n total schedule dsplacement, estmated of about 10%. In addton to hstorc slots, the slot coordnator must allocate slots herarchcally across the three remanng prorty classes: change-to-hstorc slots, new entrant slots, and other slots. For that purpose, we mplement the full lexcographc model presented n Secton 3.3, where 36

37 each prorty class s treated sequentally. In ths case, we obtan a sngle Pareto-optmal soluton,.e., the maxmum dsplacement and the total dsplacement can be jontly mnmzed and there s no trade-off between these two objectves. Ths soluton has a maxmum dsplacement equal to 70 mnutes (a 12.5% mprovement compared to 80 mnutes n the slot coordnator s soluton) and a total dsplacement of 11,620 mnutes (a 4.3% mprovement compared to 12,140 mnutes n the slot coordnator s soluton). d) Summary of the Senstvty Analyss to the IATA Constrants Fgure 3 shows the Pareto-optmal fronters between the maxmum dsplacement and the total dsplacement for the several nstances dealt wth n ths secton, assumng no ncrease/decrease n connecton tmes,.e., T 0. Note that the slot allocaton decsons are hghly constraned by the IATA gudelnes, as each one leads to sgnfcant ncreases n the maxmum and/or the total dsplacement. Frst, the nterdependences between slots lead to an ncrease n the total dsplacement by an estmated 20% to 30%, as compared to the case where the slot requests are treated for each day ndependently. Second, the consderaton of hstorc slots results n a smaller ncrease n the total dsplacement percent-wse, but n a very large ncrease n the maxmum dsplacement, from 15 mnutes to 55 mnutes. Last, the IATA prortes ncrease the maxmum dsplacement by another 25% and the total dsplacement by another 7%. Ths analyss hghlghts the mpact of these prortes on slot allocaton, and can then nform potental adjustments to the IATA gudelnes to enhance the outcomes of schedule coordnaton. Ultmately, the soluton obtaned wth the model can mprove current practce at schedule coordnated arports. Frst, the soluton s reasonably smlar to the slot coordnator s, confrmng the realsm of the PSAM model. But, t also results n a smaller maxmum flght dsplacement and a smaller total schedule dsplacement, by an estmated 12.5% and 4.3%, respectvely. Moreover, as dscussed n the next secton, ths soluton leaves all the connecton tmes unchanged, unlke the one mplemented n practce. Other consderatons may, of course, have affected the slot coordnator s decsons, such as arcraft sze or type of market served. Nonetheless, these consderatons are expected to have lmted mpact, as they are explctly ntended for te-breakng purposes. We further dscuss the benefts of our modelng and computatonal approach n the next secton. max T mn 37

38 Fgure 3 Pareto-optmal fronter at the Madera arport wth dfferent slot allocaton constrants e) Senstvty Analyss to the Objectves of PSAM As can be seen n Fgure 3, when all the capacty constrants and prorty rules are consdered, the PSAM soluton at the Madera arport conssts of a sngle pont that mnmzes smultaneously the maxmum dsplacement and the total dsplacement. In other words, there s no trade-off between maxmum and total dsplacement. Ths, of course, s not generally the case, as llustrated by Table 10, whch summarzes the results of a senstvty analyss wth respect to the objectve functon of PSAM at Madera and Porto. Note that the table also ncludes results for the thrd objectve mnmzng the number of slot dsplacements. (As already noted, no slot requests are rejected at ether arport.) Table 10 ponts to the fact that, n the case of Porto (and n contrast to Madera), there s ndeed a trade-off between maxmum dsplacement and total dsplacement captured by the two Pareto-optmal solutons, Sol. 1 and Sol 2, the frst of whch mnmzes the former and the second the latter. Note that, nonetheless, both solutons mprove both objectves, as 38

39 compared to the coordnator s soluton and, n fact, also mprove t wth respect to the number of slots dsplaced. More generally, Table 10 demonstrates that PSAM can support the selecton of the approprate slot allocaton soluton by assgnng dfferent prortes to the dfferent objectves and quantfyng the resultng trade-offs between these objectves. For example, Sol. 1 for Porto mnmzes lexcographcally frst the maxmum dsplacement, then the total dsplacement and, fnally, the number of slots dsplaced, whereas Sol. 2 follows the order total dsplacement, maxmum, dsplacement, number of slots dsplaced, Sol. 3 the order number of slots dsplaced, maxmum dsplacement, total dsplacement and Sol. 4 number of slots dsplaced, total dsplacement, maxmum dsplacement. Interestngly, all four solutons mprove all three objectves by sgnfcant margns, as compared to the coordnator s soluton. It s also noteworthy that after a certan number of slot dsplacements (Sol. 2), any further reductons n the number of dsplaced slots comes at a cost of ncreased total dsplacement. Table 10 Senstvty analyss to the PSAM objectves at the Porto and Madera arports Madera Arport Porto Arport Max Dsplacement (mn) Total Dsplacement (mn) Slots Dsplaced Slots Max Dsplacement (mn) Total Dsplacement (mn) Slots Dsplaced (slots ) CoordSol CoordSol Sol. 1 Sol % % % % % % Sol. 1 Sol. 2 Sol. 3 Sol % 60-25% 55-31% 60-25% % % % % % % % % 5.2. Results at the Madera Arport and the Porto Arport We now present n more detal the model s and the coordnator s solutons at the Madera and Porto arports, where we mantan the order of objectves (maxmum dsplacement, total dsplacement, number of slots dsplaced) ndcated n Secton 3.3, whch s the most consstent wth current practce and the exstng lterature. We dscuss below the schedule of flghts at each arport after the slot allocaton, the dsplacement per prorty class (hstorc slots, change-to-hstorc slots, new entrant slots, and remanng slots), the dstrbuton of 39

40 schedule dsplacement per day of week and per month, and the dstrbuton of flght dsplacement across slot requests. a) Flght schedule on the busest day of the season a) Madera mnute rollng capacty b) Madera mnute rollng capacty a) Porto 60-mnute rollng capacty b) Porto mnute rollng capacty Fgure 4 Number of slots allocated on the busest day at Madera and Porto arports Fgure 4 shows the number of slots allocated per rollng perod on the busest day of the Summer of 2014 at Madera and Porto arports. Specfcally, Fgures 4a and 4b (resp. Fgures 4c and 4d) plot for Madera (resp. Porto) the number of slots allocated over the precedng 60 mnutes and the precedng 15 mnutes, respectvely, for every 5-mnute perod of the day. These plots are the counterparts of Fgure 1, but ndcate that the declared capacty s never exceeded over the day after slot allocaton. Note, also, that the strct declared capactes lead to flat schedules at peak hours, especally at buser arports. 40

41 b) Dsplacement across slot prorty classes We now compare the PSAM solutons to the slot coordnator s solutons at both arports. Table 10 shows the maxmum dsplacement, total dsplacement, number of slots dsplaced, and changes n connectng tmes n the slot coordnator s solutons ( Coord_Sol ) and three model solutons: () the man soluton that strctly comples wth the IATA prmary crtera and the requested connecton tmes ( Mod_Sol ); () an alternatve soluton that uses mnor adjustments to the prmary crtera to allevate the dsplacement borne by new entrants ( Mod_Sol_NE ): and () an alternatve soluton that allows for small changes n connecton tmes to reduce the dsplacement ( Mod_Sol_Con ). We detal these three solutons and ther ratonale below. In general terms, they represent alternatve optons that can be used by decson-makers to select the most desrable solutons. Table 11 Coordnator and model solutons for slot allocaton at Madera and Porto arports Madera Arport Max Dsplacement (mn) Chg. Hst New Ent. Other Slots Total Total Dsplacement (mn) Chg. New Hst Ent. Other Slots Total Slots Dsplaced (slots) Chg. New Hst Ent. Other Slots Connectons (mn) Max ΔCon Sum ΔCon Coord Sol ,140 4, , Mod Sol % 11, % 4,780-3% 0 6, % % % % 0 0 Mod Sol Con % 10, % 4,780-3% 0 6,210-14% % % % 5-94% 630-6% Porto Arport Chg. Hst Max Dsplacement (mn) New Ent. Other Slots Total Total Dsplacement (mn) Chg. Hst New Ent. Other Slots Total Slots Dsplaced (slots) Chg. Hst New Ent. Other Slots Connectons (mn) Max ΔCon Sum Δcon Coord Sol ,140 9,600 4,515 39,025 2, , ,745 Mod Sol 25-44% % 55-31% 38,625-27% 3,560-67% 5, % 29,845-24% 2, % % 396-1,7% 1, % 0 0 Mod Sol NE 45-44% 15 0% 55-31% 37,840-29% 3,940-59% 4, % 29,780-24% 2, % % % % 0 0 Mod Sol Con 25-44% % 55-31% 36,105-32% 3,560-67% 5, % 27,325-30% 2, % % % 1, % 5-88% 2, % Frst, the man soluton of the model (Mod_Sol) mproves the outcomes of slot allocaton at both arports, as compared to the slot coordnator s. At Madera, we observe a reducton of 4.3% n the total dsplacement, 12.5% n the maxmum dsplacement, and 1.1% n the number 41

42 of slots dsplaced. At Porto, the mprovements are even more sgnfcant, wth a reducton of 27% n the total dsplacement, 44% n the maxmum dsplacement, and 7% n the number of slots dsplaced. Ths s not surprsng because Porto s a much buser arport than Madera and t s therefore much harder for the slot coordnators to fnd close-to-optmal solutons wthout the use of an advanced optmzaton model such as the one proposed n ths paper. Moreover, the benefts of the PSAM soluton (Mod_Sol) are even greater when the dfferent prorty classes are consdered. Foremost, the dsplacement of the change-to-hstorc slots (the second-hghest prorty class) greatly decreased at both arports. Ths effect s notceable n Madera (wth 3% and 1.1% reductons n the total dsplacement and n the number of slots dsplaced, respectvely), but much stronger n Porto (wth reductons as large as 44% for the maxmum dsplacement, 67% for the total dsplacement, and 42% for the number of slots dsplaced). No request from new entrants s dsplaced n Madera (as only 1.5% of all requests come from new entrants, and these are not concentrated at the busest tmes), and the dsplacement for the lowest-prorty class also declnes, even though the number of such slots that are dsplaced ncreases slghtly by 1.5%, or 4 slots. In Porto, the large mprovements for change-to-hstorc slots constrans the allocaton of slots to the lower prorty classes by lmtng the number of slots avalable at the busest hours. In consequence, the results for the low-prorty classes are mxed. New entrants are mpacted most negatvely, wth an ncrease n maxmum dsplacement of 10 mnutes and n total dsplacement of 16%. In the Mod_Sol_NE., we show another soluton n Porto that constrans the maxmum dsplacement from new entrants. Ths results n slghtly lower (albet stll very sgnfcant) mprovements for the change-to-hstorc slots, but t also provdes reductons n the total dsplacement of the new entrant slots and the remanng slots, as compared to the slot coordnator s soluton. Ths llustrates how ths model can be used to explore the tradeoffs between the dsplacements faced by the dfferent prorty classes, and determne the most desrable soluton accordngly. Fnally, we note that the slot coordnator s soluton nvolved some changes to arlne requested connecton tmes. Specfcally, the connecton tmes decreased by 5 mnutes for 6 slot pars and ncreased by 80 mnutes for 8 slot pars n Madera. In Porto, 76 slot pars faced changes n connecton tmes, rangng from 5 to 45 mnutes. Ths may be due to specal consderatons beyond our knowledge, or because allocatng the slots wth the requested 42

43 connecton tmes was nfeasble gven slot allocaton decsons for the hgher prorty classes. In order to compare our soluton to the schedule coordnator s, we allowed for slght ncreases or decreases n connecton tmes of 5 mnutes for each slot n the lowest prorty class. Ths s shown n Table 10 Mod_Sol_Con. Note that ths flexblty n connectng tmes results n a sgnfcant decrease n total dsplacement (by 5.4% n Madera and 6.5% n Porto, as compared to the man soluton). Ths confrms the observatons made n Secton 5.1.b by showng the mpact of small varatons n connecton tmes on the slot allocaton process. In addton, note that the sum of changes n connecton tmes s stll lower than n the slot coordnator s soluton for Madera. Ths does not hold for Porto, but could be mposed as an addtonal constrant n the model (n whch case the reducton n the total dsplacement from the slot coordnator s soluton would be between 27% and 32%). c) Dstrbuton of Average Daly Dsplacement Followng the results of the prevous secton, we calculate the average daly dsplacement obtaned by the PSAM and the slot coordnator (.e., the average number of slots dsplaced per day (.e., S d D S d D d B X X D d B W W D ), as well as ). The average dsplacement per day n Madera (resp. Porto) s equal to 55.2 mnutes (resp mnutes) per day, a reducton by 2.6 mnutes (resp mnutes) from the slot coordnator s soluton. The average number of slots dsplaced per day s 2.89 slots (resp slots), a reducton of 0.03 slots (resp slots) per day. The dstrbuton of these mpacts vares across the days. For nstance, out of the 210 days of the season, the dsplacement s mproved (resp. worsened) on 20 days (resp. 13 days) n Madera and 200 days (resp. 7 days) n Porto. Fgure 5 shows the average total dsplacement per day for each month of the season (Fgure 5.a n Madera, Fgure 5.c n Porto) and for each day of the week (Fgure 5.b n Madera, Fgure 5.d n Porto). Note, frst, that the months wth hgher average dsplacements are those wth most frequent mbalances between slot requests and declared capactes,.e., July, August and September (see Table 3). Moreover, the PSAM soluton mproves the average dsplacement for almost every month over the season wth the excepton of July n Madera, when the average dsplacement ncreases by 0.9 mnutes per day. Smlarly, the average dsplacement s hghest on the days of the week wth the hghest mbalances between slot requests and declared capactes n Madera,.e., Mondays and Thursdays. Note that almost 43

44 no flghts are dsplaced on the other days of the week, whch s consstent wth the fact that, on these days, demand for slots falls below the arport s declared capactes (Secton 2). Nonetheless, the total dsplacement on these days s stll postve, reflectng the nterdependences between slots over multple days, and the fact that some slots had to be dsplaced on the least busy days to satsfy capacty constrants on the busest days. In Porto, the largest dsplacement occurs on Thursdays, although Frdays exhbt, on average, more perods when slot demand exceeds declared capactes. Ths also stems from the nterdependences between slots over dfferent days of the week. Fnally, the model mproves the average dsplacement on Mondays n Madera (by 30 mnutes), but worsens t on Frdays (by 13 mnutes). In contrast, the average dsplacement s sgnfcantly mproved on all days of the week n Porto (by up to 121 mnutes for Saturdays). a) Madera average dsplacement per month b) Madera -- average dsplacement per day c) Porto average dsplacement per month d) Porto -- average dsplacement per day Fgure 5 Average daly dsplacement at Madera and Porto arports per month and day of week 44

45 d) Dstrbuton of dsplacement across slots In addton to reducng the total dsplacement, the model also reduces the number of slots dsplaced (.e., B W W ) wth respect to the slot coordnator s soluton. In total 607 slots were dsplaced n Madera and 2,379 n Porto, whch corresponds to 4.7% and 5.9% of the total number of slots, respectvely. The average dsplacement per dsplaced slot (.e., S d D d B X X B W W S dd d S dd d ) s equal to 19.1 mnutes n Madera and 16.2 mnutes n Porto both sgnfcantly lower than n the slot coordnator s soluton. However, the dstrbuton of the dsplacement across all seres of slots exhbts sgnfcant varablty. Fgure 6 shows the hstogram of the number of slots dsplaced per dsplacement value n Madera (Fgure 6.a) and Porto (Fgure 6.b). As seen n Table 10, the maxmum dsplacement s reduced by 80 to 70 mnutes n Madera, and by 80 to 55 mnutes n Porto as compared to the slot coordnator soluton. The fgure shows that ths reducton mpacts postvely a large number of flghts. Indeed, the coordnator mposes a dsplacement that s larger than the model s maxmum dsplacement for 14 slots (2.3% of the slots dsplaced) n Madera, and 128 slots (5% of the slots dsplaced) n Porto. The number of slots wth a dsplacement larger than 30 mnutes s reduced from 80 to 72 n Madera, and from 396 to 238 n Porto. Therefore, the model provdes benefts not only by reducng total dsplacement, and/or the number of flghts dsplaced, but also by reducng the tal of the dsplacement dstrbuton, thus allevatng the costs assocated wth the largest dsplacements. a) Madera 45

46 b) Porto Fgure 6 Hstogram of number of slots dsplaced per mnutes of dsplacement n Madera and Porto arports 6. Concluson In ths paper, we have developed a novel modelng and computatonal approach to optmze slot allocaton decsons at busy schedule-coordnated arports. We have proposed a new Prorty-based Slot Allocaton Model (PSAM) that mnmzes the dsplacement from the arlnes slot requests, whle fully complyng wth the prmary crtera of the IATA gudelnes and wth arport declared capactes. We have ntroduced a strong formulaton that provdes exact solutons n reasonable computatonal tmes for md-sze arports twce the sze of those prevously consdered n the lterature. The model has then been mplemented usng hghly-detaled data from the arports of Madera and Porto, Portugal. Comparsons wth the slot coordnator decsons have suggested that the model captures well the man decsons and trade-offs made n practce and also mproves the slot allocaton outcomes by reducng the dsplacement experenced by the arlnes by an estmated 4.5% and 27% at the two arports consdered. Computatonal experments also quantfed the mpact of the varous constrants mposed by the IATA gudelnes. The nsghts ganed can be used to nform potental future adjustments to the slot allocaton rules. The PSAM can thus provde sgnfcant benefts at major arports worldwde by enhancng the outcomes of slot allocaton processes and makng the eventual schedules of flghts more consstent wth the schedulng preferences of the arlnes and, mplctly, wth passenger demand. 46