Integrated Airline Fleet and Crew Robust Planning

Transcription

1 Integrated Arlne Fleet and Crew Robust Plannng Chunhua Gao * Ells Johnson * Barry Smth # * # H. Mlton Stewart Shool o Industral and Systems Engneerng Georga Insttute o Tehnology 765 Ferst Drve NW Atlanta GA gao@sye.gateh.edu eohnson@sye.gateh.edu Sabre Holdngs 3150 Sabre Drve Southlake TX Barry.Smth@sabre-holdngs.om Abstrat The arlne leet assgnment problem nvolves assgnng arrats to lghts to mamze prot. Derent leet assgnment solutons ause dramatally derent perormane n subsequent rew plannng and operatonal proesses. We have developed an ntegrated leet and rew robust plannng method to provde leet assgnment solutons that are both rendly to rew plannng and robust to real tme operatons. The three hallenges o ths work are 1 to understand the nluene o leet assgnment on rew shedulng; 2 to address rew shedulng n a tratable way n the ntegrated model; and 3 to aheve robustness. We address these hallenges by developng a new approah that ntegrates rew onnetons wthn the leet assgnment model and mposes staton purty by lmtng the number o leet types and rew bases allowed to serve eah arport. Computatonal results demonstrate that the proposed approah an redue rew plannng ost mprove robustness and solve ndustral sze problems wth good omputatonal eeny. 1

2 1 Introduton The arlne plannng proess has evolved tsel nto a sequene o deson-makng phases wth the deson rom one phase used as nput or the subsequent phases. Speally the arlne plannng proess starts wth shedule onstruton and leet plannng that are sueeded by arrat mantenane routng and rew shedulng. Arrat mantenane routng and rew shedulng are two onurrent proedures both workng on deomposed shedules o separate leets. By makng desons sequentally there s lmted normaton shared and lttle nteraton between the phases. Derent leet assgnment solutons ould ause dramatally derent perormane n the subsequent rew shedulng phase. Durng the day o operaton the shedule s eeuted as planned no dsruptons our. However n realty delays and dsruptons aused by adverse weather mehanal alures rew skness and ar tra ontrol are unavodable. Arlne plannng solutons that al to aount or reovery lebltes would result n hgh operatonal ost. Thereore t has been a maor hallenge to ntegrate both plannng untons and operatonal onsderatons n the plannng proess. To aheve the goal o ntegraton over the plannng untons and the tmelne between plannng and operatons researhers have started to work on arlne ntegrated plannng and robust plannng. Integrated plannng s ntended to ntegrate the untonal phases at the plannng stage and robust plannng s ntended to make desons at the plannng stage that are beneal to the operatons. Sne a omprehensve model ntegratng all untons or ntegratng over both untons and the tmelne s stll not omputatonally attanable alternatve ntegratng strateges are nvestgated to aheve partal ntegraton. These ntegratng strateges nlude: 1 ntegratng shedule desgn and leet assgnment Reng et al Lohatepanont and Barnhart 2004; 2 ntegratng leet assgnment and arrat mantenane routng Clarke et al Barnhart et al. 1998b; 3 ntegratng arrat routng and rew parng Klaban et al Cohn and Barnhart 2003 Cordeau et al ; 4 ntegratng leet assgnment and rew shedulng Clarke et al Barnhart et al. 1998a 2

3 Sandhu and Klaban In parallel researhes on robust plannng nlude: 1 robust leet assgnment Ageeva 2000 Rosenberger 2004 Smth and Johnson 2006; 2 robust arrat routng Kang and Clarke 2003 Lan et al. 2003; 3 robust rew shedulng Yen and Brge 2006 Shaeer et al Ehrgott and Ryan 2002 Chebalov and Klaban The obetve o ths paper s to ntegrate leet assgnment and rew shedulng and to also provde robust solutons to real-tme operatons. There are three hallenges n ths work: 1 To understand the nluene o leet assgnment on the perormane o rew shedulng. Ths nludes what an lead to poor rew solutons or eample rew double overnght rests. 2 To address the rew shedulng problem n the ntegrated model. The rew problem beomes muh harder n the ntegrated model sne t s neessary to work on the omplete shedule nstead o the sub-shedules or derent leet types. Thus omputatonal tratablty s an mportant ssue to be onsdered. 3 To aheve robustness n an ntegrated ramework so that the resultng plans are robust or both arrat and rew reovery n operatons. In regards to ntegratng leet and rew plannng Clarke et al was the rst attempt to address rew shedulng ssues n the leet assgnment model FAM. In ther work a ost s added on eah lonely double overnght and the optmzaton model s then used to balane the osts between lonely double overnghts and leetng. Barnhart et al. 1998a propose an ntegrated appromate model or leet assgnment and rew parng optmzaton whh ombnes the bas FAM and a duty-based model or the rew parng problem. Sandhu and Klaban 2004 propose an ntegrated plannng model whh ntegrates leetng arrat routng and rew parng smultaneously. Crew parngs are modeled epltly and the arrat rotaton problem s aptured by the plane ount onstrants. The ntegrated plannng problem s truly omputatonally ntensve. A omputng envronment onsstng o a luster o 27 dual 900 MHz Itanum 2 proessors s adopted to ondut omputatonal eperments n ther work. Prevous researh on robust arlne rew shedulng nludes stohast and determnst methods. Yen and Brge 2006 propose a stohast rew shedulng model and devse a 3

4 soluton methodology or ntegratng dsruptons n the evaluaton o rew shedules. Shaeer et al 2005 propose a stohast etenson to the determnst rew shedulng problem. They mody the oeent vetor o the obetve unton to relet the epeted ost o eah deson varable rather than determnst ost. Monte Carlo smulaton s used to estmate the operatonal ost. Ehrgott and Ryan 2002 propose a b-rtera optmzaton model. The measure alled non-robustness s evaluated or eah parng based on the eet o potental delays wthn the parng. The non-robustness measure s then treated as a seond obetve. Chebalov and Klaban 2006 present a determnst method that addresses robustness by onsderng rews that an be swapped n operatons. They add a seond obetve or mamzng the number o move-up rew to the tradtonal rew shedulng model. Smth and Johnson 2006 propose a robust leet assgnment model mposng staton purty whh an also be alled leet purty. Fleet purty ensures that the number o leet types servng a gven staton does not eeed a speed lmt. Addng leet purty an greatly redue planned rew osts mantenane osts and mprove robustness. However t has sgnant negatve mpat on the omputatonal eeny o FAM. To mprove the omputatonal eeny a olumn generaton soluton approah alled staton deomposton was proposed. In ths paper an ntegrated leet and rew robust plannng approah s developed. In ths approah to avod the urse o dmensonalty rew onnetons rather than eplt rew parngs or dutes are modeled to represent the rew shedulng problem wthn the ntegrated model. To aheve robustness we etend the staton purty dea whh orgnally reers to leet purty only to both leet purty and rew base purty so that the numbers o both leet types and rew bases allowed to serve eah arport are lmted. A move-up rew s a rew that s ready to ly a derent lght whh means that t s qualed to operate the spe arrat and t s rom the same rew base and t has the same number o days untl the end o the parng Chebalov and Klaban Imposng rew base purty an nrease the 4

5 opportuntes o ndng a move-up rew n rew reovery. Furthermore t has the benet o mprovng omputatonal eeny and qualty o the FAM solutons. The ontrbutons o ths paper nlude: developng an ntegrated leet and rew robust plannng approah whh provdes leet assgnment solutons both rendly to rew plannng and robust to real tme operatons. Eperments are onduted usng data rom a maor U.S. arlne. studyng staton purty shemes nludng both leet purty and rew base purty and nvestgatng the mpat o staton purty on FAM prot omputatonal eeny robustness and perormane o the rew soluton. In addton to the savngs n mantenane and operatonal osts we demonstrate that or the ndustral sze testng shedule 1 addng rew base purty an avod loked rotatons n FAM soluton and redue CPU tme by 10~100 tmes wthout sgnantly redung the FAM prot; 2 addng leet purty an mprove the rew soluton by 2~3% n pay-and-redt whh amounts to rew shedulng ost reduton o 5~8 mllon dollars per year. The ollowng s an overvew o the remander o ths paper. In Seton 2 we nvestgate the mpat o leetng solutons on the subsequent plannng and operatonal proesses and eplore opportuntes or mprovng the leetng soluton. Etended staton purty shemes nludng both rew base purty and leet purty are proposed n Seton 3. We study the mpat o staton purty on FAM prot omputatonal eeny robustness and rew plannng n Seton 4. Imposng staton purty an ntegrated leet and rew robust plannng model s proposed n Seton 5. Soluton algorthm approah and omputatonal results are presented n Seton 6. Fnally we summarze our work n Seton 7. 2 Arlne Fleet Assgnment Modelng Gven a predened lght shedule the leet assgnment problem determnes whh arrat leet type s assgned to a gven lght segment to mamze prot. The alulaton o 5

6 prot s based on estmates o operatng ost and revenue or eah possble leet/lght assgnment. In ths seton we ntrodue the bas FAM ormulaton Seton 2.1 and nvestgate the mpat o leetng solutons on the subsequent plannng and operatonal proesses Seton 2.2 or the sake o eplorng opportuntes to mprove the leetng model and solutons. In Seton 2.3 the robust FAM model and the onept o leet purty proposed n Smth and Johnson 2006 are brely ntrodued whh provde a bass or the work n ths paper. 2.1 Bas FAM model The bas FAM ormulaton mamzes operatng prot: revenue mnus operatng ost. The leet assgnment model an be lassed as a large multommodty low problem wth sde onstrants. A tme lne network s dened or eah staton and leet type ombnaton. Ground ars are used to trak the number o planes on the ground. There are two sets o varables: lght-leet assgnment varables and ground ar varables. The three sets o onstrants are: Cover onstrants every lght must be assgned to one and only one leet type; Balane onstrants ensure low onservaton n the tme lne network; Plane ount onstrants the total number o planes n the ar and on the ground an not eeed the avalable leet sze. Thus the bas FAM s ormulated as: Mamze: F L R C 1 Subet to: F = 1 L y + y 0 s t s t + s t = I s t L s S O s t L y s t + N F m CL

7 {01} F L 5 y 0 s t 6 s t The dentons o the sets appeared n the ormulaton are: S: Set o statons ndeed by s. F: Set o leet types ndeed by. L: Set o lght legs ndeed by. CL: Set o lght legs rossng the ountng lne lown by leet. Ist: Ost: Set o lght legs nbound to {st}. Set o lght legs outbound rom {st}. The deson varables are: 1 leg L s assgned leet. = 0 otherwse. y : s t y + : s t The number o arrat on the ground or leet type at staton s on the ground ar ust pror to tme t. The number o arrat on the ground or leet type at staton s on the ground ar ust ollowng tme t. The parameters dened n the model nlude: R : Revenue or lght leg t s assgned leet type. C : Cost or lght leg t s assgned leet type. N : The number o arrat avalable o leet type. Ths bas FAM model s desrbed thoroughly n Hane et al

8 2.2 Impat o leetng solutons on subsequent proesses FAM solutons gnorng the subsequent proesses ould easly lose ts savngs n leet assgnment to eessve rew ost and hgh operatonal ost. Moreover FAM solutons that do not satsy mantenane onstrants or FAM solutons that ause arrat loked rotatons are unaeptable to the arlne. The goal o our work s to provde modelng deves that norporate aspets o arrat routng mantenane rew shedulng and operatonal ssues nto the leet assgnment model whle retanng ts omputatonal eeny. Thus t s neessary to analyze the mpat o leet solutons on the subsequent proesses Arrat routng Arlnes must meet mantenane requrements n makng arrat rotatons. In order to ensure ar travel saety the Federal Avaton Admnstraton FAA requres arlnes to perorm our types o arrat mantenane alled A- B- C- and D-heks. Arlnes have ther sel-mposed nspeton and mantenane requrements whh are more strngent than FAA requrements. The our types o mantenane vary n sope duraton o tme and requeny. Among them C- and D-hek requrements do not aet daly shedulng and routng sne the arrat s taken out o the shedule. A-heks nvolve a routne vsual nspeton o all maor systems whh requre 4 hours. It s ommon or arlnes to perorm A-heks n every 35 hours o lyng. B-heks nlude a thorough vsual nspeton plus lubraton o all movng parts and are perormed every 300 to 600 lyng hours. The B-heks requre a 10 to 15 hour long stay at the mantenane hangar. Clarke et al norporates B-hek requrements nto the leet assgnment problem and Clarke et al norporates A-hek requrements nto the arrat rotaton problem. Arrat rotaton s the routng o ndvdual arplanes whh s determned ater leet assgnment. Typally the arlnes requre that eah arrat n a leet ly an dental route onsstng o all the legs assgned to that leet whh mples that the route must be a yle. When the lghts overed by arrat are separate rom the lghts overed by another arrat 8

9 n the leet t s alled a broken rotaton. A loked rotaton s a broken rotaton where the separate sequenes do not have any loatons n ommon Clarke et al Loked rotatons are unaeptable to almost all arlnes. To a loked rotaton requres a new soluton rom leet assgnment. In Seton 4.2 o ths paper we wll show addng rew base purty an help avod loked rotatons aused by FAM soluton. Fgure 1 llustrates a lght network and ts adaeny graph lown by a ertan leet type. The nodes n the adaeny graph represent statons the ars represent that there s at least one lght between the onnetng two statons. It s apparent that the estene o unonneted sub-graphs n the adaeny graph results n loked rotatons. The rotatons that over the lghts between statons A-B A-C and A-D an not have any loatons n ommon wth the rotatons that over the lghts between statons E and F. The loked rotatons an also ause rew shedulng problem neasble unless rews are deadheaded to the separated network whh mples etra rew lyng tme and ost. B A C D D A E F E F a tme-spae lght network B C b adaeny graph Fgure 1: Loked rotatons aused by unonneted adaeny graph Crew shedulng The rew shedulng problem s to nd a mnmum ost assgnment o lght rews to a gven lght shedule. The lght shedule onsdered nludes all lght legs assgned to a 9

10 rew ompatble leet. The rew shedulng problem s typally broken nto two sequentally solved subproblems the rew parng problem and the rew rosterng problem. The rew parng problem generates mnmum ost parngs that over the olleton o lght legs n the shedule whle the rew rosterng problem ombnes the parngs nto month-long rew shedules and assgns the month-long plans to ndvdual rew members. A rew parng s a sequene o lghts that starts at a rew base and ends at that same rew base. It an span several days n whh rew members wll rest usually overnght at some loaton other than where they resde. The perods between rests n a parng are alled dutes. Throughout ths paper we all the two subsequent lghts wthn a duty as rew day onneton and the two subsequent lghts onnetng one duty wth another va an overnght rest as rew nght onneton. The ost struture o dutes and parngs are dened n Equatons 7 and 8 respetvely. Usually the overall qualty o a rew shedulng soluton s evaluated by pay and redt o the rew parng soluton whh s the eess ost o rew beyond the requred lyng hours see Equaton 9. duty ost = ma{ bloktme oe1*elapsed tme mn duty guarantee}. 7 parng ost = ma{ duty os t oe2*tafb #dutes*average duty guarantee}. 8 pay&redt = parng ost total bloktme total bloktme 9 Anbl et al dsuss three man auses o pay and redt or parngs: 1 long or requent sts wthn a duty; 2 long overnght rests between dutes and 3 deadheadng. Long overnght rests and long sts result n hgh TAFB Tme-Away-From-Base penalty whh s the seond tem n the parng ost. Fgure 2 llustrates an eample o long overnght aused by leet assgnment. A lonely double overnght ours when a rew arrves late at nght at a staton that s not ts base and the arrat t arrved on leaves beore the rew has had suent rest and there s no other departure o that leet that day Clarke et al In the ase shown n Fgure 2 the rew arrvng va leg L 1 at 11pm annot leave va leg L 2 untl 10

11 ater a 31 hour layover. However a par o mdday lghts L 3 and L 4 are also assgned to the same leet the rew arrvng va leg L 1 an leave n the mddle o the net day va L 4 and the rew arrvng va leg L 3 an leave the net mornng va L 2. Ths opton s alled mdday breakouts. L 2 L 4 6:00 23:00 L 3 L 1 Fgure 2: Lonely double overnghts aused by leet assgnment In Gao and Johnson 2006 we propose shedule analyss methods or rew rendlness o a gven shedule when the thrd tems domnate the parng ost and demonstrate that an unbalaned leet assgnment ould ause more dutes requred n the parng soluton or ertan leets and result n hgh pay and redt beause o the ost nurred by average duty guarantee Operatonal reovery Arlne operatons are requently dsrupted by nlement weather mehanal problems and ar tra ontrol and ground delay programs o FAA. FAM soluton an sgnantly aet the tme and ost requred to return to the planned operatons sne arrat and rew reovery are all predetermned by leet assgnment. Ageeva 2000 mody the leet assgnment problem to reward opportuntes or swappng planes. Suh opportuntes an help to loalze the mpat o a dsrupton aused by an unavalable arrat. Rosenberger et al develop a FAM ormulaton to nrease operatonal robustness by redung hub onnetvty. They show through smulaton that dereased hub onnetvty results n ewer 11

12 anellatons and delays. Smth and Johnson 2006 develop leet assgnment solutons that nrease plannng leblty and redue ost by mposng leet purty thereore lmtng the number o leet types allowed to serve eah arport n the shedule. Imposng leet purty on the leet assgnment model an lmt arrat dsperson n the network and make solutons more robust relatve to rew plannng mantenane plannng and operatons. In ths paper we urther etend the noton o staton purty to nlude both rew base purty and leet purty. In the ollowng subseton the robust FAM model and the onept o leet purty n Smth and Johnson 2006 wll be brely ntrodued as a bass o our work. 2.3 Robust leet assgnment mposng staton purty Smth and Johnson 2006 address the rew mantenane and operatonal ssues smultaneously through staton purty. Staton purty ensures that the number o leet types servng a gven staton does not eeed a speed lmt. Ths lmt on the number o leets servng a staton s the staton s purty level. The purty level an be dened on rew-ompatble leet amles whh ensures opportuntes to swap arrat or rews or ether operatonal or protablty reasons. To mplement leet purty an aulary varable w s s dened to ndate whether leet or leet amly serves staton s n the FAM soluton. The ollowng new onstrants are added to the bas FAM ormulaton: w s I s L F s S 10 F w s w s SP s S s 11 {01} F s S 12 1 lght serves staton s. I s = 0 otherwse. 13 The parameter SP s denes the leet/amly purty level. Smth and Johnson 2006 demonstrate that mposng staton purty on the leet assgnment model an lmt arrat dsperson n the network and make solutons more robust 12

13 relatve to rew plannng mantenane plannng and operatons. It s also ound that staton purty an sgnantly degrade omputatonal eeny. A olumn generaton soluton approah staton deomposton and a prmal-dual method are developed to mprove soluton qualty and omputatonal eeny. It s estmated that the annual net benet o staton purty beause o redued mantenane and rew shedulng osts s greater than $100 mllon or a maor U.S. domest arlne. 3 Etended Staton Purty A move-up rew s not only requred to operate a ompatble leet amly but also requred to ome rom the same rew base. To urther mprove robustness n rew shedulng also to altate ondutng ntegrated plannng we etend the noton o staton purty to rew base purty. Crew base purty mples that the number o rew bases servng eah staton s lmted. The nature o the hub-and-spoke network ensures that rew base purty s ratonal and easble. In at most spoke statons are only onneted to a ew rew bases nearby. It s not reommended to send rews rom rew bases ar away to vst the spoke statons. The benet s quk reovery rom dsruptons by ndng a move-up rew or deadheadng rew home ast. In ths seton we rst dene rew base purty and dsuss how to mplement rew base purty. Then we show va an eample how to dene etended staton purty senaros by ombnng leet purty and rew base purty. 3.1 Crew base purty Crew base purty s determned by the adaeny graph o the lght network. In a hub-and- spoke network most spoke statons are only onneted to a ew rew bases nearby. We dene naturally pure spoke and med spoke dependng on a spoke s onneted to only one rew base or more. For a naturally pure spoke ts rew base purty level s restrted as 1 and the spe rew base whh t onnets to s assgned to ths spoke. For a med spoke we usually need to assgn all the onneted rew bases to t. Otherwse we may 13

14 possbly lose easble rew shedulng solutons. For a rew base staton we allow other rew bases wthn dstanes o 2 n the adaeny graph.e. they are onneted through non-stop or 1-stop lghts to have day vsts at ths staton. In addton to the rew base purty level dened on statons we an urther restrt the rew base assgnment to lghts at med spokes. Leg pure s one o the strateges whh means a lght segment onnets a spoke wth a rew base we assgn ths ed rew base to ths lght. I a shedule does not have rewbase-to-rewbase or spoke-to-spoke lghts a omplete mplementaton o leg pure plan mples that the shedule s dvded nto separated subsets and derent rew bases wll be assgned to the orrespondng subsets o the shedule. Leg pure plan provdes a way to smply the rew shedulng problem. 3.2 Epermental shedule A lght shedule wth an ndustral problem sze s used to test the mpat o staton purty as well as the ntegrated leet and rew plannng model. Ths lght shedule s based on a daly shedule o a maor U.S. domest arrer. The senaro lght network s onstruted by deletng parts o the shedule that are qute solated rom the rest o the network. Consequently there are 1388 daly lghts and s rew bases n ths shedule. The leet onssts o 3 leet amles JET-1 JET-2 and TURBO. Table 1 shows the sze o the leets. JET-1 s the largest leet and nludes three rew-ompatble sub-leets. One o the man haratersts o ths lght network s that the leets are dvded up among rew bases resultng n 15 derent leet types n the problem. It s noted that qute a ew regonal arlnes or ommuter arlnes operate wth ths haraterst.e. leets are dvded up among hubs or rew bases and rews serve the arrats rom ther own base. 14

15 Table 1: Fleet amly and sze Fleet amly Sze JET JET-2 24 TURBO 12 In our tests large spokes are dened as havng more than 40 daly operatons medum spokes as havng rom 21 to 40 daly operatons and small spokes as havng less than 20 daly operatons. Based on these rtera 98 out o the total 100 spokes are small and medum spokes. The number o naturally pure spokes s 66 and most o the med spokes only onnet to two rew bases. Table 2 lsts the statsts o the epermental lght network and Fgure 3 shows the adaeny graph o the lght network rom the epermental shedule. Table 2: Statsts o the epermental lght network Ctes 106 Flghts 1388 Fleet types 15 Fleet amles 3 Crew bases 6 Large spokes 2 Medum spokes 13 Small spokes 85 Naturally pure spokes 66 Med Connet to 2 rew bases 27 spokes Connet to 3 or more rew bases 7 15

16 CB2 CB3 CB4 CB5 CB6 CB1 Fgure 3: Adaeny graph o the epermental lght network 3.3 Staton purty senaros The etended staton purty proposed n ths paper nludes both leet purty and rew base purty. Nne staton purty senaros are desgned to study the mpat o staton purtes. Fleet purty s related to both the sze o the staton and the rew base purty. For leet purty three ases are onsdered. In the rst ase no purty s mposed on leet type or leet amly as n the bas FAM ormulaton. The seond ase s alled Famly purty n whh 1 or rew bases or large spokes nstead o mposng hard onstrants on staton leet/amly purty level penaltes on leet amly and staton ombnatons are dened n the obetve unton; 2 or small or medum szed spokes hard onstrants on leet amly purty at these statons are dened to be 1. The thrd ase s alled Fleet purty n whh 1 or rew bases or large spokes penaltes on leet type nstead o leet amly and staton ombnatons are dened n 16

17 the obetve unton; 2 or a small or medum szed spoke dependng on whether t s a naturally pure spoke or med spoke hard onstrant on leet type purty or leet amly purty s dened to be 1 respetvely at ths staton. Smlarly or rew base purty three ases are onsdered. In the rst ase no rew base purty s mposed. The seond ase s alled CB purty 1 n whh rew base purty level as ntrodued n Seton 3.1 s mposed on all naturally pure spokes med spokes and rew bases. Besdes leg pure plan s mposed on those med spokes onnetng to more than two rew bases and a bonus s dened n the obetve unton to enourage leg pure plan at other statons. The thrd ase s reerred to CB purty 2 whh s based on the seond ase. The only derene s that n the thrd ase leg pure plan s also mposed on the med spokes between rew bases CB1 and CB2. Consequently the ombnaton o derent settngs o leet purty and rew base purty denes nne staton purty senaros. In the net seton we eamne the mpat o the etended staton purty on FAM prot omputatonal eeny robustness and rew shedulng usng the epermental shedule. 4 Impat o the Etended Staton Purty The mpat o the etended staton purty s tested usng the robust FAM model ntrodued n Seton 2.3. The rew base purty s mplemented by restrtng the legal lght-leet assgnments. In the epermental shedule leets are dstrbuted among rew bases and rews are enouraged to ly ther own leets. Thereore rew base purty mples that some lght-leet assgnments are dsabled. For nstane the lghts onnetng to a naturally pure spoke o one rew base should not be assgned leet types dstrbuted at other rew bases. Our proposed method an also be appled to general problems wthout the haraterst that leets are dvded up among the rew bases as long as the plane ount s restrted on the real leet types nstead o the leets dstrbuted at the rew bases. To mplement leet purty the two onstrants dened n Equatons 10 and 11 are nluded n the robust FAM model. In 17

18 ths seton the mpat o the etended staton purty on FAM prot omputatonal eeny robustness and rew shedulng s nvestgated usng the epermental shedule. 4.1 Impat on FAM prot and omputatonal tme Table 3 gves the problem szes o the robust FAM model under derent staton purty senaros. It s shown that addng rew base purty an redue the sze o the problem and addng Fleet purty has a larger problem sze than addng Famly purty. Table 4 shows the robust FAM results or derent staton purty senaros. By addng Famly purty FAM prot dereases by 2.9% on average whle CPU tme nreases by 300 tmes on average. By addng Fleet purty FAM prot dereases by 3.6% on average whle CPU tme nreases by 900 tmes on average. However by addng rew base purty FAM prot only dereases by 0.6% on average. The most notable benets o addng rew base purty s the mprovement o omputatonal eeny. By addng CB purty 1 the CPU tme s redued by 16 tmes on average. By addng CB purty 2 the CPU tme s redued by 10 ~ 100 tmes. Fgure 4 and Fgure 5 learly llustrate the mpat o staton purty on CPU tme as well as on FAM prot. Table 3: Problem szes No CB purty CB purty 1 CB purty 2 No Combo* Rows Cols No Combo Rows Cols No Combo Rows Cols No leet purty Famly purty Fleet purty *Combo reers to legal lght-leet assgnments 18

19 Table 4: Robust FAM results No CB purty CB purty 1 CB purty 2 No. Tme* Prot** No. Tme Prot No. Tme Prot No leet purty Famly purty Fleet purty * n CPU seonds ** n mllon dollars Tme No leet purty Famly purty Fleet purty No CB purty CB purty 1 CB purty 2 Fgure 4: Impat o staton purty on runtme FAM Prot No leet purty Famly purty Fleet purty No CB purty CB purty 1 CB purty 2 Fgure 5: Impat o staton purty on FAM prot 19

20 In ths problem there s one large leet amly and two small leet amles. The reason or FAM prot loss s that by settng the leet/amly purty level as 1 only a small number o statons an be served by the two small leet amles. The leet/amly purty onstrants an be relaed by settng the leet/amly purty level as 2. Further omputatonal eperments show that FAM prot only dereases by 0.09% wth amly purty level set as 2 or the ases o Famly purty ; and FAM prot dereases by 0.13% wth amly and leet type purty level set as 2 or the ases o Fleet purty. In prate ompromse would be neessary or balanng the benets o robustness and the loss o FAM prot by adustng the leet purty level. 4.2 Impat on rew shedulng and robustness Takng the FAM soluton rom the robust FAM model rew shedulng problems are solved or eah leet. It s ound that wthout mposng rew base purty the FAM soluton tends to produe loked rotatons whh means that the lghts overed by some arrats are separate rom the lghts overed by other arrats n the leet. Loked rotatons are unaeptable to most arlnes. The rew shedulng problem or a FAM soluton wth loked rotatons s not easble. Deadheads are requred to lead rews to those separate routes. Addng rew base purty s an eetve way to avod the loked rotatons sne t enourages rew and leet not to ly ar rom ts base staton. The denton o rew base purty on naturally pure spokes and med spokes as well as the hard or sot restrtons on leg pure plan an altogether lmt the statons that a leet an vst wthn dstanes o 2 to ts base staton n the adaeny graph. Ths means the statons that a leet an vst are onneted by a non-stop or 1-stop tnerary to ts base staton. The resultng sub-network or eah leet tends to be onneted. The rew shedulng problems are solved or all three senaros o CB purty 1. Table 5 shows rew soluton perormane under derent leet purty n terms o atual parng ost n mnutes eess ost n Pay & redt total number o dutes n the parng soluton and the 20

21 mamum st tme needed to produe a easble rew soluton. In order to eamne the perormane o rew shedulng we dd not add a lmt to the mamum st tme when solvng the rew parng problem. The purpose s to make the rew parng problem easble wthout usng deadhead. The obetve unton o the rew parng model ould enore the soluton lookng or small st tme rew onnetons. The mamum st tme ested n the parng soluton n turn demonstrates the soluton qualty and omputatonal omplety o the rew parng problem sne long st tme mples hgh Pay & redt and larger sze o the rew parng model. Table 5 shows that Soluton 6 Fleet purty has the best rew results. Soluton 5 Famly purty has muh better rew results than Soluton 4 but s not as good as Soluton 6. Table 5: Crew soluton results No leet purty Famly purty Fleet purty * n mnutes Soluton No. Parng ost* Pay & redt Number o dutes Mamum St tme* % % % Table 6 ompares the number o rew double overnghts nluened by derent leet purtes. By addng leet purty the total number o leetamly-staton ombnatons and amly-staton sngletons the number o amly-staton ombnatons wth only one arrval and departure or a amly are redued. Thereore the rew lonely double overnghts an also be redued whh has a sgnant postve eet on the rew perormane. In at rew lonely double overnght s nluened by both amly-staton sngletons and rewbase-staton sngletons the number o rewbase-staton ombnatons wth only one arrval and departure served a rew base. Analyss s onduted on the staton atvtes o the orgnal shedule and t s ound that the orgnal shedule nludes 3 double overnght rests ntrnsally. Table 21

22 6 shows that Soluton 6 Fleet purty has the least number o double overnghts. Soluton 5 Famly purty has three more double overnghts than Soluton 6. Addng leet purty Famly purty or Fleet purty an greatly redue the estene o rew double overnghts whh s one o the maor reasons that ause hgh rew ost. Note that Soluton 6 has better rew perormane than Soluton 5. Ths s beause n the ase o Famly purty there are more hanes or lonely rew base vsts than n the ase o Fleet purty whh may ause more rew double overnght rests. The lonely rew base vsts at statons are not preerable n the operatonal proess. Table 6: Impat o leet purty on rew lonely double overnghts No leet purty Famly purty Fleet purty Sol No. Famly staton pars Fleet Staton pars Famlystaton FS sngletons Crew lonely double ovn aused by FS sngletons Crewbasestaton sngletons Double ovn n rew soluton Robustness s mproved when eah leet and rew base servng a staton operates at relatvely hgh requeny nto and out o that staton. The number o Famly-staton pars Fleet-staton pars Famly-staton sngletons and Crewbase-staton sngletons n Table 6 provde a measure o robustness. Table 6 shows that mposng staton purty an greatly redue the lonely leets and lonely rew bases servng a staton and also reate more opportuntes or arrat and rew swappng to over operatonal dsruptons. To summarze the mpat and value o staton purty are as ollows. 1 Addng rew base purty an avod loked rotatons n FAM soluton. It wll not degrade the FAM prot sgnantly. Most mportantly the CPU tme needed to solve the robust FAM model an be redued by 10~100 tmes. 22

23 2 By addng leet/amly purty the rew soluton s mproved by 2~3% n pay-and-redt whh gves a savng o rew shedulng ost by up to 5~8 mllon dollars per year. The value o purty also nludes the savngs o $ per year or eah amly-staton par Smth and Johnson 2006 and the savngs n arrat mantenane ost and operatonal osts. 5 Integrated Fleet and Crew Robust Plannng Model Imposng staton purty to FAM ormulaton an mprove rew shedulng beause FAM solutons wth leets servng a smaller number o statons wth greater requeny provde more leblty or rew assgnment to redue rew ost. In the omputatonal eperments onduted n Seton 4 the best rew soluton Soluton 6 obtaned or the epermental shedule has s double overnghts and a mamum st tme o 250 mnutes. In addton to the three ntrns double overnghts o the shedule three more double overnghts were aused by leet assgnment. There were also some long st tmes n the rew soluton. The mamum st tme s preerably less than 180 mnutes. These ndngs suggest that urther researh on ntegrated leet and rew plannng s neessary. Addressng the rew shedulng problem n the ntegrated model s perhaps the most hallengng task n ntegrated plannng. Crew shedulng or separate leets s already omputatonally epensve. In an ntegrated model the rew problem beomes muh harder to solve sne t s neessary to work on the omplete shedule. In the epermental shedule the number o dutes enumerated on the omplete shedule eeeds two mllons. It s etremely omputatonally epensve to model parngs or dutes epltly n the ntegrated model. Thereore we ntegrate leet assgnment wth rew onnetons n our ntegrated plannng model. Crew onnetons nstead o eplt parngs or dutes are modeled to represent the routngs o rew or the sake o retanng a omputatonal tratable ntegrated model. Crew 23

24 lonely double overnghts and day onnetons wth long st tme are the maor reasons that ause poor rew soluton. Addng rew onneton ators n the leet assgnment model an help produe rew-rendly FAM solutons. Day and nght onnetons are enumerated a pror based on the mnmum and mamum st or rest tme whh onsttute the onneton varables. At spokes a lght has ether a day onneton or a nght onneton whle at rew bases a lght an ether be the begnnng/endng o a parng or have day onnetons. The soluton o ths model provdes leet assgnment soluton and a pseudo rew parng soluton. Based on rew base and leet dvson provded by the soluton legal parngs an be obtaned by solvng the deomposed sub-problems. Ths model mantans the haratersts and benets o staton purty. Derent rom the bas and robust FAM models n ths ntegrated model eah lght s assgned both a leet type and a rew base. As to onneton varables the two lghts that onsttute a onneton should be assgned wth the same rew base and rew-ompatble leet types sne they represent how rew rotatons are ormed. Net we ntrodue the mathematal ormulaton o ths ntegrated leet and rew robust plannng model whh wll be alled n bre the ntegrated model hereater n ths paper. 5.1 Sets L: Set o legs n the shedule ndeed by. S: Set o statons ndeed by s. CB: Set o statons that are rew bases. F: Set o leet types ndeed by. CF: Set o ompatble rew base and leet pars ndeed by. CL: Set o lght legs rossng the ountng lne lown by leet. Ist: Ost: Set o lght legs nbound to {st}. Set o lght legs outbound rom {st}. 24

25 5.2 Deson varables 1 leg L s assgned leet and rew base. = 0 otherwse. y s t : The number o arrat on the ground or leet type at staton s on the ground ar ust pror to tme t. y + s t : The number o arrat on the ground or leet type at staton s on the ground ar ust ollowng tme t. 1 leet F serves staton s S n the soluton. w s = 0 otherwse. s : Total number o leet-staton ombnatons n the soluton. 1 day onneton leg leg s assgned rew base leet type. day = 0 otherwse. 1 nght onneton leg leg s assgned rew base leet type. nte = 0 otherwse. 1 departure leg at rew base s the begnng o a parng. start = 0 otherwse. 1 arrval leg at rew base s the end o a parng. end = 0 otherwse. 5.3 Parameters and data R : Revenue or lght leg t s assgned leet type. C : Cost or lght leg t s assgned leet type. B : Bonus or lght leg t s assgned leet type. Ctm day : Conneton tme or day onneton day n mnutes. Ctm nte : Conneton tme or nght onneton nte n mnutes.. 25

26 26 : N The number o arrat avalable o leet type. s I = otherwse. 0 serves staton s. lght 1 s SP : Fleet purty level at staton s. 5.4 Formulaton Mamze: s P Ctm Ctm B C R nte nte nte day day day L * + 14 Subet to: L = 1 15 t s y y L t s O t s L t s I t s 0 = L CL S s t s F N y m 17 L I w s s 18 S s SP w s F s 19 = F S s s w s 20 + = nte day nomng L at staton CB s 21 + = nte day outgong L at staton CB s 22 + = end day nomng L at rewbase 23 + = start day outgong L at rewbase 24 = day nomng L at staton CB s but s 25 = day outgong L at staton CB s but s 26

27 {01} L 27 w s day y 0 s t 28 s t {01} F s S 29 s 0 30 {01 } 31 nte start end The obetve unton nludes our tems: FAM prot revenue mnus operatng ost bonus dened on leg pure plan osts nurred by rew onneton tme and penalty dened on leetamly-staton pars. The ost tems on rew onneton tme n the obetve unton wll make the solutons o ths model avod lonely double overnghts and long st tme. The last tem n the obetve unton s penalty dened on leetamly-staton pars. At rew base statons or large spokes t s napproprate to mpose a small leet purty level; however a penalty on leetamly-staton pars n the obetve unton an enourage leet purty as well. Constrants orrespondng to those n the bas FAM ormulaton are over onstrants staton leet balane onstrants and plane ount onstrants respetvely. The only derene s that or eah lght not only a leet type but also a rew base s assgned here. Constrants are related to leet purty whh are same as those dened n the robust FAM model Smth and Johnson Equaton 19 denes hard onstrants on leet/amly purty. Varable s ndates the total number o leet/amly staton ombnatons n the soluton. Constrants are related to rew onnetons. Equatons are appled to statons that are not rew bases. Equaton 21 mples that eah nomng lght leg to suh a staton has ether a day onneton or a nght onneton. Equaton 22 mples that eah outgong lght leg rom suh a staton has ether a day onneton or a nght onneton. The onstrants 23 to 26 are appled to statons that are rew bases. Ths means a departure lght at rew base s ether the begnnng o a parng or t has day onnetons. 27

28 Aordngly an arrval lght at rew base s ether the end o a parng or t has day onnetons. I rews rom one rew base vst another rew base they an only have day onnetons there. 6 Computatonal Results 6.1 Algorthm and mplementaton In preproessng legal leet and rew base assgnments are generated or lghts and onnetons based on the senaros o staton purtes. For eah staton a set o rew base statons s dened to denty whh rew bases are allowed to serve ths staton. Eah rew an only ly the leets rom ther own base. One the rew base purty or eah staton s determned there are ompatble pars o leets and rew bases to be assgned to lghts. As or onnetons by hekng the arrval staton and the departure staton o a onneton only the rew bases n the nterseton set o the two orrespondng rew base sets are allowed to serve ths onneton. Ths sheme allows rew rom one rew base to vst another rew base but then return to ts own onneted statons. The omputatonal eperments are onduted on a Pentum 4 proessor 1.83GHz 1.5G RAM usng ILOG CPLEX 9.0. The models are ormulated n ILOG Conert 2.0. The lnear programmng LP relaaton problem s solved by dual steepest-edge smple. In order to get nteger soluton lght varables and onneton varables wth value n the LP soluton are ed to 1. As a result related onneton varables wth derent leet or rew base assgnment an be ed to 0. The resultng med nteger programmng MIP problem s then solved by CPLEX s branhng and bound proess. 6.2 Results The ntegrated leet assgnment and rew onneton model s appled to solve the epermental problem desrbed n Seton 3.2. The sze o the MIP model and the perormane o ths method are summarzed n Table 7. Four purty senaros are tested. The 28

29 ndes o these senaros are n subsequent order wth the earler senaros ntrodued n Seton 4.1. Senaro 10 has Fleet purty same meanng as dened n Seton 3.3 and bas rew base purty nludng: 1 rew base purty mposed on all naturally pure spokes med spokes and rew bases; 2 leg pure plan on those med spokes onnetng to more than two rew bases eept the statons that wll generate double overnghts. Senaro 11 has Fleet purty rew base purtes as n senaro 10 and bonus dened n the obetve unton to enourage leg pure plan at other statons. Senaro 12 has Fleet purty rew base purtes as n senaro 11 and leg pure plan on parts o the med spokes between rew bases CB1 and CB2. Derent rom senaro 12 senaro 13 has Famly purty nstead o Fleet purty. Table 7 shows that the CPU tme to solve the ntegrated model s redued dramatally as the purty senaros mproved. Senaro 13 an be solved n thrteen mnutes whh s about 100 tmes aster than solvng senaro 10. Moreover ompared to the results n Table 4 the FAM prot ddn t degrade beause o addng rew onneton varables and onstrants. On the ontrary the CPLEX MIP gap demonstrates that the qualty o the nteger soluton s mproved. Table 7: Results o the ntegrated model No. Fleet purty type Crew base purty Rows Cols Tme** LP IP Total Prot* Gap*** 10 Fleet base Fleet Bonus on leg pure 12 Fleet Fored leg pure Famly Fored leg pure * FAM prot n mllon dollars ** n CPU seonds ***MIP gap n perent returned by CPLEX. 29

30 The deomposed rew shedulng problems are solved or senaro 12 and senaro 13 respetvely. Table 8 shows rew soluton perormanes and Table 9 ompares the number o rew double overnghts nluened by derent leet purtes. For both senaros there are no more double overnghts besdes the ones ntrns to the orgnal shedule. Moreover the mamum st tme s 3 hours as requred. Fleet purty as n senaro 12 gves better rew soluton. In senaro 13 the Famly purty ase there are more hanes or lonely rew base vsts. Although these lonely rew base vsts are not lonely double overnght rests t s not as leble or rew shedulng or rew reovery as the Fleet purty ase. However the Famly purty gves better FAM prot and has better omputatonal eeny. Table 8: Crew soluton o the ntegrated model Soluton No. Parng ost Pay & redt Number o dutes Mamum St tme Fleet purty % 380 [20180] Famly purty % 392 [20180] Table 9: Statsts o the ntegrated model Fleet purty Famly purty Sol No. Famly staton pars Fleet Staton pars Famlystaton FS sngletons Crew lonely double ovn aused by FS sngletons Crewbasestaton sngletons Double ovn n rew soluton Dsusson Besdes the mprovement o rew shedulng nurred by addng leet purty the ntegrated model urther mproves rew soluton perormane through modelng the rew 30

31 onnetons. It mantans the robustness and omputatonal eeny by mposng approprate leet purty and rew base purty. From the results n Seton 4.2 and Seton 6.2 we an see that leg pure plan s eetve n mprovng omputatonal eeny. Ths s beause the strt onstrants or bonuses on leg plan help breakng the symmetry wthn leet or rew base assgnment. However t must be noted that sometmes a omplete leg pure plan s too restrtve to produe any easble soluton. Ths s espeally true when rew base balane onstrants are requred that s the number o hours that rews loated at some base spent away rom ther rew base must be wthn speed lmts Homan and Padberg Another stuaton s that at some statons leg pure plan reates lonely double overnght. As shown n Fgure 6 the alphabets n rles represent the arrval or departure statons whh are rew bases n ths ase o the lghts leg pure plan mples rew onnetons o L 1 -L 2 and L 3 -L 4 n whh L 1 -L 2 leads to double overnght. However the rew onnetons o L 1 -L 4 and L 3 -L 2 wthout mposng leg pure plan generate two legal short overnght rests whh s preerred. Ths eample shows that leg pure plan should be appled approprately n order to avod generatng rew lonely double overnghts. A B L 2 6:30 L 4 8:30 L 3 21:00 23:00 L 1 B A Fgure 6: leg pure plan and rew lonely double overnght Crew base purty and rew onneton plan at spokes altogether make the subsequent rew shedulng problem muh easer to solve. Shaw 2003 nvestgates the advantages o 31

32 hub-to-hub strngs n onstrutng parngs whh nlude reduton n the amount o storage spae and n the tme requred to do prng out. Crew onnetons at spokes are mostly hub-to-hub strngs. Furthermore they provde a easble spoke plan or rew. Crew base purty urther restrts the rew base assgnment on rew onnetons. Hene rew base purty and rew onneton plan remove unavorable optons n rew shedulng so as to mprove omputatonal eeny. When leet purty s ombned wth rew base purty the omputatonal eeny o the robust FAM model an also be mproved. The prmary obetve o ths paper s to obtan ntegrated and robust plannng approah whle retanng ts omputatonal eeny through desgnng eetve modelng mehansms. Future researh dreton nludes ntegratng arlne mantenane routng wth the urrent ntegrated model. The nteraton between arrat routng and rew shedulng s due to rew short onnet Cohn and Barnhart 2003 Cordeau et al That s the mnmum st tme or rew s longer than the mnmum turn tme or arrat but the onneton tme or rew an be less than ts mnmum st tme as long as rew ollows the route o the same arrat. In our model ssues o rew short onnet an be modeled. It would be onvenent or the urrent ntegrated model to represent rew short onnets and denty rew short onnets n the soluton where same arrat onstrants should be mposed. 7 Summary The ntegrated leet and rew robust plannng approah provdes modelng deves that norporate aspets o arrat routng mantenane rew shedulng and operatonal ssues nto the leet assgnment model whle retanng ts omputatonal eeny. We study staton purty shemes nludng both leet purty and rew base purty and nvestgate the mpat o staton purty on FAM prot omputatonal eeny robustness and perormane o the rew soluton. We ondut omputatonal eperments and demonstrate that or the ndustral sze testng shedule n addton to the savngs n mantenane and operatonal osts: 1 addng rew base purty an avod loked rotatons n FAM soluton and redue CPU tme by 32

33 10~100 tmes wthout sgnant reduton o the FAM prot; 2 addng leet purty an mprove the rew soluton by 2~3% n pay-and-redt whh amounts to a reduton n rew shedulng ost o 5~8 mllon dollars per year. We developed an ntegrated leet and rew robust plannng approah whh provde leet assgnment solutons both rendly to rew plannng and robust to real tme operatons. Computatonal results show that the ntegrated model urther mproves rew soluton perormane and mantans the robustness and omputatonal eeny by mposng approprate leet purty and rew base purty. 8 Aknowledgments We would lke to thank the employees o Ameran Arlnes or ther assstane. We also thank Dr. Mhael Clarke rom Sabre Holdngs or valuable omments and dsussons. 9 Reerenes Abara J Applyng nteger lnear programmng to the leet assgnment problem. Interaes Ageeva Y Approahes to norporatng robustness nto arlne shedulng. PhD Dssertaton Massahusetts Insttute o Tehnology. Anbl R. Tanga R. Johnson E. L A global approah to rew-parng optmzaton. IBM Systems Journal Ball M. Barnhart C. Nemhauser G.L. and Odon A Ar transportaton: rregular operatons and ontrol. In: Transportaton Vol. 14 Handbooks o Operatons Researh and Management Sene Barnhart C. and Laporte G. Eds.. Barnhart C. Johnson E. L. Anbl R. Hatay L A olumn-generaton tehnque or the long-haul rew-assgnment problem. In: Optmzaton n ndustry 2: Mathematal programmng and modelng tehnques n prate Cran T. A. Leahman R. C. Eds. John Wley & Sons Ltd pp Barnhart C. Lu F. Sheno R. 1998a. Integrated arlne shedule plannng. In: Operatons Researh n the Arlne Industry G. Yu Eds. Kruwer Aadem Publshers pp Barnhart C. Boland N. L. Clarke L. W. Johnson E. L. Nemhauser G. L. Sheno R. G. 1998b. Flght strng models or arrat leetng and routng. Transportaton Sene

34 Barnhart C. Cohn A.M. Johnson E.L. Klaban D. Nemhauser G. and Vane P.H. 2002a. Crew shedulng. In: Handbook o Transportaton Sene 2nd Edton R.W. Hall Eds. pp Barnhart C. Belobaba P. Odon A. R Applatons o operatons researh n the ar transport ndustry. Transportaton Sene Barnhart C. Cohn A Arlne shedule plannng: aomplshments and opportuntes. Manuaturng and Serve Operatons Management Buthers E. R. Day P. R. Golde A. P. Mller S. Meyer J. A. Ryan D. M. Sott A. C. Wallae C. A Optmzed rew shedulng at Ar New Zealand. Interaes Chebalov S. Klaban D Robust arlne rew shedulng: move-up rews. Transportaton Sene Chu H. D. Gelman E. Johnson E. L Solvng large sale rew shedulng problems. European Journal o Operatonal Researh Clarke L. Johnson E. Nemhauser G. Zhu Z The arrat rotaton problem. Annals o Operatons Researh Clarke L. W. Hane C. A. Johnson E. L. Nemhauser G. L Mantenane and rew onsderatons n leet assgnment. Transportaton Sene Clarke M. Smth B Impat o Operatons Researh on the Evoluton o the Arlne Industry. Journal o Arrat Cohn A. M. Barnhart C Improvng rew shedulng by norporatng key mantenane routng desons. Operatons Researh Cordeau J. F. Stokov G. Soums F. Desrosers J Benders deomposton or smultaneous arrat routng and rew shedulng. Transportaton Sene Cordeau J. F. J. Merer A. Soums F A omputatonal study o Benders deomposton or the ntegrated arrat routng and rew shedulng problem. Computers & Operatons Researh Ehrgott M. Ryan D Construtng robust rew shedules wth brtera optmzaton. Journal o Mult-Crtera Deson Analyss Gao C. Johnson E Rethnkng the arlne rew shedulng proess. Submtted to AGIFORS Anna Velek Award Competton Gao C Arlne ntegrated plannng and operatons. Ph.D. Dssertaton Georga Insttute o Tehnology. Gershko I Optmzng lght rew shedules. Interaes