IMM Trykt af IMM, DTU Crew Scheduling Airline Tracking During Jesper Holm 2002 LYNGBY THESIS MASTER NR.??/02

Transcription

1 IMM Trykt f IMM, DTU Crew Scheduling Airline Trcking During Jesper Holm 2002 LYNGBY THESIS MASTER NR.??/02

2 Prefce M.Sc. sis is finl requirement for obtining degree: Mster This Science in Engineering. work hs been crried out in period from of of September 2001 to 28th of Februry 2002 t Opertions Reserch 1st t Informtics nd Mmticl Modeling, Technicl University of section Denmrk. work hs been supervised by Professor Jens Clusen. sis hs been crried out in collbortion with Scndinvin IT where Kim Milvng-Jensen nd Steven Lursen hve been mu col- Group lbortors. Lyngby, Februry 28th, 2002 Holm Jesper c iii i

3 Abstrct sis investigtes irline crew scheduling, which covers problem This ssigning crew to plnned deprtures. problem is often subdivided of severl phses: piring phse, ssignment phse, trcking into nd dy-to-dy phse. In this sis, trcking phse hs been phse min focus. An existing system developed by Scndinvin IT Group been used s strting point for developed softwre. hs properties of trcking phse mke it desirble to construct individul work shifts for ech crew. This pproch is in contrst to usul tken during ssignment nd piring phses. Hence, set of pproches work shifts is constructed for ech crew to be considered. From individul set, work shift for ech crew to mn is chosen using Set Covering ech formultion. construction of work shifts is done by enumerting subset of ll work shifts by using heuristics or by using n existing genertor possible by Scndinvin IT Group. resulting Set Covering Problem developed solved using Simulted Anneling. is totl of 16 rel-life problems re considered. smllest contins A more thn 200 crew nd 150 open flights nd lrgest contins slightly bove 600 crew nd 900 open flights. Using developed heuristics just 90% of plnned deprtures were covered. In comprison, round existing heuristic resulted in coverge round 60%. Airline crew scheduling, Simulted Anneling, Combintoril Keywords Heuristics. Optimiztion, v Contents 1 Introduction 1.1 Terminology Phses of irline crew scheduling Report orgniztion Problem description 2.1 Project bckground Literture review 3.1 Piring nd ssignment Trcking Dy-to-dy Trcking phse t ss tp-i Initil piring Reordering Min piring pros nd cons of tp-i v

4 CONTENTS vii 5 Solution Approch Piring genertion Resources Pirings Depth first serch Depth-best first serch Piring cost Serching from middle nd out Preprocessing of open flights Piring selection Solving set covering problem Implementtion Progrm design Memory consumption Results Dt Model prmeters Prmeters in Simulted Anneling Ghostcrew cost Piring cost weights Bounding depth first serch Serchfront length for depth-best first serch Performnce tests Serching from middle nd out dfs versus dbfs dbfs versus sig Preprocessing CONTENTS vi 8 Conclusion Outlook Bibliogrphy 9

5 Chpter 1 Introduction crew scheduling is problem of ssigning personnel (crew) to Airline deprtures. Becuse crew cost mke up one of lrgest direct plnned when operting n irline compny, optimizing crew utiliztion expenses result in huge gins. Thus re of crew scheduling is importnt. my obtining good solutions to irline crew scheduling problem Unfortuntely, is hrd. First of ll becuse problem is combintoril in nture; re huge number of wys one cn mn hundreds of deprtures re irline compny serve per dy. However, problem is not restricted n single dy nd often everything between couple of dys nd up to to month hs to be considered. On top of this lrge set of rules given by vition uthorities nd union greements hs to be respected. Just leglity of given solution is difficult nd computtionlly checking expensive. project hs been crried out in collbortion with Scndinvin Airline This (ss). For furr detils see section 2.1. Systems n introduction to terminology used in field of irline crew Below will be presented. Followed by more in-depth introduction to scheduling crew scheduling. Finlly n overview of rest of report is irline given Terminology 1.1 Terminology order to be ble to give more specific problem formultion n introduc In to some of specilized terminology used in irline crew schedulin tion will be presented below. A crew is single person. re re two min types of crew: Cbi Crew pssengers) nd cockpit (steering ircrft). In thi (servicing only cbin crew re considered nd hence crew will be use project n synonym for cbin crew. However ides nd method use s be esy to pply to cockpit crew s well. should A bse is n irport. A homebse is bse where crew belongs Bse ss this is one of following: Copenhgen (CPH), Stockholm For nd Oslo (OSL). (ARN), time period of time between crew rrives t bs Connection one flight nd deprt with nor. with A piring is work shift for single crew. On figure 1.1 th Piring of piring is shown. A piring strts nd ends on th structure bse (on figure CPH). A piring is constructed from one o sme duty periods which re series of flights (lso known s legs) wit more smll connection time. period of time between two legs in period is clled sit. Ech duty period strts with briefing n duty with debriefing. When connection time between tw termintes is lrge it is clled stop, which seprtes duty periods. Piring legs lso be referred to s rosters or slings ( ltter only used b might ss). flight Open flight is used to denote severl slightly different things Open of ll n open flight isflight which lcks one or more crew First one open flight is lso used to denote tht one crew is missin But flight. This mens tht if flight is lcking two crew this fligh on two open flights; one for ech crew. represents trnsfer is lso known s dedheding. A crew is sid to be on Pssive trnsfer when she is ssigned to flight tht is not in lck o pssive This is done to trnsport her from one bse to nor, ei crew. she is needed t rriving bse or to return her to he becuse homebse. lso known s reserve crew is crew tht cn be clled on wor Stndby reltive short time of notice (often crew hs to be t he with homebse within n hour).

6 A piring is sid to be legl if it complies with set of rules Leglity by vition uthorities s well s union greements. This given to difficulties of irline crew scheduling solution is trditionlly Due into number of phses. On figure 1.2 se phses re sketched. divided first phse consist of longterm plnning where destintions to serve type of service (number of flights per dy) re determined. This nd im of next phse, piring, is to generte pirings from th flights from plnning phse. objective is to include ec plnned pirings hve been constructed individul crews re ssigned to th When pirings in ssignment phse. This step ends with schedul generted relesed schedule is not complete, s not ll flights re ssigned th of crews which re needed. In or words relesed schedul number likely contins open flights. Some of open flights in schedul most inherited from previous phse. Ors emerges during trckin re due to new pssenger forecsts 1 nd illness. objective of th phse phse is to ssign crew to se open flights. trcking on dy of opertion, schedule is executed in dy-to-d Finlly, Remining open flights, new which rise, delys etc. re delt wit phse. description This chpter gives more precise description o Problem problem trgeted in project long with description of th problems of irline crew scheduling during trckin solving used by ss. phse, pproch This chpter describes pproch tken in thi Solution for solving problem. project comprison with results obtined with progrm currently use ss. by 1 number of crew needed on flight is determined by type of flight nd th 1.3 Report orgniztion 1.2 Phses of irline crew scheduling 3 Duty period Duty period set hs to be stisfied by ll pirings. plnned flight exctly once in generted pirings. Brief Leg Sits Leg Debrief Stop Brief Leg Debrief gives reltion between crews nd flights; showing flights n which crews tht re to mn m. At this point schedule is relesed, s CPH ARN ARN OSL OSL CPH ech crew gets to see flights she is to mn. Piring/roster/sling 1.1: building blocks of piring. possibly by cncelltion. 1.2 Phses of irline crew scheduling 1.3 Report orgniztion Schedule relese bckground. project phse t ss This chpter describes n existing pproch fo Trcking results in set of ll flights to fly. Plnning Piring Assignment Trcking Dy to dy Here n overview of implementtion is given. Implementtion This chpter presents numericl result obtined long wit Results dys estimted number of pssengers. 1.2: Phses of irline crew scheduling.

7 Open flight Blnk dys Spretime Usble tim Chpter 2 Problem description Stndby Duty No problem considered in this project is crew scheduling during trcking phse. As described in section 1.2 schedule hs lredy been con- when entering trcking phse. However, open flights re still structed during trcking phse nd problem is to ssign crew to se round To be ble to solve this problem re hs to be crew vilble tht flights. be ssigned to open flights. refore, different kinds of stndby cn re llocted in schedule during ssignment phse, which cn time be used during trcking nd dy-to-dy phses. n grphicl presenttion of problem is given on figure 2.1. On figure A n open flight oriented view of schedule is given simply showing 2.1() open flights. Notice tht open flights C nd D origintes from sme flight between Copenhgen nd London which lcks two crew physicl members. This shows up in schedule s two different open flights. figure 2.1(b) schedule is shown from crew point of view which for On crew gives different ctivities tht she hs been ssigned. Duty ech is time where crew hs lredy been ssigned to flights; she is working. time Stndby is time where crew cn be clled on work with reltively notice (often within n hour). Or types of stndby re blnk-dys short usble-time. differences lie in specific rules for how nd when nd cn be used to close open flights. Spretime is time where crew is y work nd refore cnnot work. Finlly re re holes in schedule off where crew is not ssigned to specific ctivity. Crew 3 hs hole in her Project bckground () Open flight oriented view of schedule Now Time (b) Crew oriented view of schedu 2.1: A smple schedule showing open flights (left) nd crew schedul (right). fter returning to Copenhgen from Oslo. Holes might be use schedule closing open flights. when in one, stndby, blnk-dys, usble-time nd holes in schedule re re All to s resource periods; time in schedule where crew might be use ferred to close open flights. This gives following problem sttement: s mny open flights s possible using resource periods Close in schedule s cheply s possible, with respect to llocted some cost mesure. top of this crew oriented pproch should be tested when solving th On This mens tht informtion bout crew tht is to mn problem. should be sought used when piring is generted. This is i piring to pproch described bove where pirings re generted i contrst piring phse nd ssigned in ssignment phse. 2.1 Project bckground project hs been mde in collbortion between Informtics n Modelling (imm), Technicl University of Denmrk nd Scn Mmticl IT Group (sig) ltter being 100% owned by Scndinvi dinvin System (ss). Airline hs developed system clled tp-i tht prior to 1998 ws successfull sig to solve scheduling problem during trcking phse. However used OSL CPH CPH CPH ARN LHR OSL OSL ARN OSL CPH CPH Crew CPH LHR OSL ARN CPH LHR LHR OSL CPH CPH E D C B A

8 2.1 Project bckground 7 to introduction of new union rules heuristics deployed by tpi due hs become invlid. From 1998 nd onwrds trcking phse hs been delt with mnully. However, combintoril nture of refore problem indictes tht n OR-pproch might led to notble svings. hs piring genertion prt tht hs been sought used to generte tp-i of open flights which n could be used by stff. However, pirings no informtion bout vilble resources in schedule is used it since not gurnteed tht generted pirings cn be ssigned to crew. is shows tht slings generted by tp-i often do not fit with Prctices resources in schedule nd tp-i is not currently used by stff. Chpter 3 Literture review re hs been much work crried out trgeting piring Currently, nd dy-to-dy phses. But lmost no (published) work h ssignment done directly trgeting trcking phse. It hs only been possible t been one pper [13] which dels directly with this phse. But technique find ides used in piring, ssignment nd dy-to-dy phses re nd when considering scheduling during trcking. usble literture cn roughly be divided into 3 clsses: Those tht del wit piring nd ssignment phses, one tht dels with trcking n those tht try to solve dy-to-dy problems. 3.1 Piring nd ssignment problems hve been trgeted in two min wys; s two seprte se nd s one. In both cses vrition of Set Prtitionin problems Problem is used to perform some kind of selection mong pirings:

9 cjxj (3.1) ijxj = 1; for i = 1;:::;m (3.2) xj integer; for j = 1;:::;n (3.3) 3.1 Piring nd ssignment 9 Minimize Z = Subject to nx j=1 nx j=1 rises from mtrix A where ech row corresponds to flight nd ij column to piring. xj denotes if piring (column) j is chosen, ech cost is cj. (3.2) ensures tht ech flight is included exctly corresponding in set of chosen pirings. Alterntively, set covering model might once chosen where equl sign in (3.2) is replced by greter-thn-orequl be sign. n ech flight is covered t lest once by chosen pirings. models re known to be NP-hrd. However, fesible solution to Both Covering Problem cn esily be found given it exist: Just include ll Set in solution. This pproch cn obviously not be used when columns with Set Prtitioning Problem becuse of over coverge of rows deling not llowed. This mkes Set Covering Problem much nicer to work is However, since over coverge of rows is llowed one might get multiple with. coverge of flights in Set Covering Problem formultion. common element is grph G which represents possible pirings. Anor pth in grph represents (legl) piring which n gin corre- Ech to column in A. Some kind of cost or restriction is often introduced sponds G. in [8] genetic or evolutionry lgorithm[12] is developed tht solves In Prtitioning Problem rising from crew scheduling s described bove. Set piring construction nor ssignment of pirings to crew re Neir considered. [14] insted of using individul flight legs s building blocks, set of duty In (see figure 1.1) is constructed. se duty periods re n selected periods Set Covering Problem so tht set of duty periods covering flight in is obtined. selected duty periods re orgnized in grph where legs correspond to pirings. problem is solved by column genertion pths shortest pth lgorithm is used on grph to generte columns where 3.2 Trcking 1 mster problem. This gin is Set Prtitioning Problem with row to selected duty periods nd columns representing pirings representing very simple form of leglity is sought enforced on generted piring A grph representtion. initil construction of duty period through is not considered, nor is ssignment to crew. [4, 6] grph is constructed directly from flight legs nd used t In possible pirings. In [6] pirings re constructed once n enumerte Set Prtitioning Problem is solved. In [4] grph G k is constructe n ech crew k nd pth in G k now represents legl piring for crew k for shortest pth problem on se grphs re used s subproblem in A genertion scheme. mster problem gin becomes Se column Problem. Becuse piring lwys hs crew ssocited th Prtitioning problem is lso solved. G k is constructed by enumerting ssignment possible nd legl pirings for crew k. formultes mmticl model tht given set of piring P k fo [3] crew k selects exctly one piring from ech set P k so tht ll flight ech covered exctly once. This model is solved using brnch-nd-boun re technique. [9] Simulted Anneling pproch is tken to solve ssignmen In (it is ssumed tht set of pirings is given). Firstly, n initi problem of crew to pirings is mde using some heuristic. neigh ssignment is defined by eir moving one piring from one crew to no bourhood or by swpping two pirings between two crew. 3.2 Trcking mentioned bove, only pper found deling directly with trckin As is [13]. Firstly, it seprtes introduced phses of irline crew phse into two met phses; plnning phse nd n opertion scheduling This is outlined on figure 3.1. phse.

10 3.1: division of phses of crew scheduling into to phses: plnning phse nd opertionl phse. met defines opertionl irline crew scheduling problem s tht of modifying [13] individul monthly work schedules for irline crew members during pproch tken is column genertion. mster problem is modelled Set Prtitioning Problem over open flights. subproblem gen- s columns corresponding to pirings. subproblem is formulted ertes shortest pth problem on duty grph G k for ech crew k. Ech pth s grph corresponds to legl piring. cost ssocited with ech in in G k corresponds to mrginl cost of new piring in pth dy-to-dy problem is lso referred to s irline irregulr opertions or crew recovery [7], becuse it consists of trgeting those problem tht [15] due to disruption from mintennce problems, bd wer conditions rise etc. [15] depth first serch in brnch-nd-bound tree is used to ssign crew In flights tht hve become open due to disruption. brnch is done on to formultion which ensures tht ech open flight iscovered t les covering once. 3.3 Dy-to-dy Dy-to-dy 11 Plnning Opertion Plnning Piring Assignment Trcking Dy to dy dys opertionl phse of plnned schedule. mster problem. specific problem considered in this pper ws tht of cockpit crew. 3.3 Dy-to-dy ssignment of crew to n open flight. [7] grph G k In representing pirings is build for ech crew k s n extension of lredy flown pirings by k. This is n used in set

11 Trcking phse t ss chpter describes solution pproch used currently by ss when This with trcking phse. deling uses progrm clled tp-i, bsed on self developed heuristics (which ss be described below) to close open flights during trcking phse. will 1998 tp-i ws ble to efficiently close open flights. This ws done Before reordering (from Dnish omdisponering ), which bsiclly swps using in nd out of schedule, possibly chnging lredy plnned duty pirings more detiled description follows). But in 1998 new union greement ( consists of two min prts: One tht does reordering, nd one, tp-i genertes pirings from initil set of open flights. To overcome tht of new union greement sig took pproch of letting tpi difficulties generte pirings which could n be ssigned to crew using llocted Thus mirroring piring nd ssignment phses from figure reordering. ssignment is done mnully tp-i piring genertion prt pproch oftp-i cn be divided into tw n initil piring genertion, which ws genertor used befor steps; nd (min) piring genertion. ltter which hs been buil 1998, top of initil piring genertion fter on piring genertion nd reordering prts of tp-i uses th Both piring genertion to build n initil set of pirings from give initil closer look will be tken t reordering pproch used prior t Secondly, nd thirdly, min piring genertion used to dy will be described 1998, initil piring uses two lists, which re sketched on figure 4.1 All ope re kept in list L1, which is sorted ccordingly to incresing de flights time. L1 is trversed severl times with different heuristics tryin prture construct pirings. All heuristics re greedy in sense tht to t hed of L1 nd dd first flight to piring currentl strt construction, which fulfill some requirements given by heuristic under time piring is successfully constructed it is stored in L2 nd th Ech which mke up piring re mrked s used in L1 (illustrte flights crosses on figure). As severl heuristics will be tried in turn onl with not mrked s used will be considered when trying to construct flights 4.1: 2 lists used when tp-i does initil piring open flights. on 4.1 tp-i 1 13 Chpter 4 set of open flights. Firstly, description of this initil piring is given Initil piring limited possibilities of chnging plnned duty time during hevily phse. Hence, usefulness of this pproch ws limited. trcking L1: Open flights A B C D E F G H piring. in schedule (such s stndby, blnk-dys, usble-time nd holes ) resources of chnging lredy plnned duty which ws cse when using insted L2: Pirings of open flights C F H B G In following more in-depth description of tp-i will be given.

12 described, initil piring consist of number of heuristics. y cn As divided into two min ctegories: A preprocessing (enforcing some rules) be number of heuristics which constructs pirings. two ctegories nd be described next. will smll mount of preprocessing is pplied. This forces flights to be First, nd locked 1 if 50 minute rule 2 pply. Similr trivsel connected rule 3 is checked, nd flights tht re required to be connected will be stop nd locked. connected following heuristics, which tries to build pirings from flights Next L1 re pplied. heuristics re tried successively in order y in first heuristic tries to construct piring strting from homebse X, one or more stops with good connection time (less thn 3 hours) t with (here nd b) nd returning to X. This is illustrted on non-homebses 4.2. figure 4.2: Piring with good connection time ending nd strting t homebse X. second heuristic relxes requirement on connection time; now times up to 30 hours for leg returning to homebse connection connection time to next leg, y should be cptin nd pilot on tht next minutes leg. 3 Trivsel stop rule forces flights toger in pirings tht will keep crew hppy. ccepted. Hence, requirement becomes < 30h t figure 4.2 for th re time t bse b. connection third heuristic tries to build piring consisting of exctly two flights good connection time nd strt nd end t sme homebse X with 4.3: Piring with exctly 2 legs, good connection time strt nd end t homebse X. nd fourth heuristic relxes requirement of good connection time, henc connection time of up to 26.6 hours is ccepted (see figure 4.4()). How re re two exceptions illustrted in figure 4.4(b) nd figure 4.4(c) ever, flight creting more thn 3 hours connection time on homebse Y is no A if previous bse c or next bse e re homebses (illus ccepted, t figure 4.4(b)). A long stop (more thn 3 hours) is not ccepte trted bse d (see figure 4.4(c)) if it is possible to insert two pssive trnsfer on lines) forming two new blnced pirings. two pssive trns (dotted my only hve 5 hours of connection time nd should not overlp wit fers thn 2 hours. more 4.1 tp-i tp-i 15 This is illustrted t figure 4.3. X X Preprocessing <3h Heuristics re listed. X b b X <3h <3h Pirings cn be locked so y will not be chnged t ny point in future. 1 rule sttes tht if cptin or pilot, for given leg, hve less thn 50 2

13 Connection time > (b) hours on homebse Y 3 Inserting 2 pssive flights, forming 2 new (c) with t most 5 hours connection time pirings, 4.4: Relxtion on connection time (), with 2 exceptions (b) nd (c). fifth heuristic relxes requirement from figure 4.4 where piring strt nd end t sme homebse. Hence, piring should strt should sixth heuristic is vrint of figure 4.2. It tries to connect flight, strts on homebse X, with one or more flights with good connection which But now n ending on bse (even non-homebse) different from time. stting bse X is ccepted. seventh heuristic tries to build pirings strting from ny bse, with connection time, ending on homebse (see figure 4.5). A possible good connection time of up to 24 hours is ccepted on leg rriving on bd homebse, with exception illustrted on figure 4.4(c). 4.6: Piring strting nd ending t bse with good time t b. connection finl processing ll remining unmrked legs in L1 from figure 4.1 r As into pirings consisting of one flight, nd re inserted into L2. converted reordering pproch ws developed bsed on interviews with st prior to tp-i solved scheduling problems during trckin who description of reordering is given in [11]. Below rough outline o depth ides used in reordering will be given. is process where pirings re swpped in nd out of crew Reordering using some heuristics. process cn be described s loc schedule 4.1 tp-i tp-i 17 X b b X c Y Y e b b c c d d X <26.7h <26.7h >3h <3h <3h <24h <2h 4.5: Piring strting from ny bse with good connection nd ending on homebse with possible bd connection time Piring with t most 26.6 hours connection () time with strt nd end t homebse X. nd c or e lso homebse. time. c d d c <5h <5h eighth heuristic tries to construct piring strting on homebse three or more flights with good connection time (< 3 hours) nd endin with on ny bse. e d d e ninth heuristic tries to construct pirings consisting of two flights strt t ny bse, with good connection time, nd ending t bse (se ing b b figure 4.6). nd no more thn 2 hours of overlp. <3h nd end on homebse, but not necessrily sme Reordering phse mnully. gol ws to develop system which mirrored th heuristics tht stff consciously or unconsciously would use. An in

14 heuristics tht tries to trnsform current schedule by swpping serch (of open flights) into schedule (possible) replcing or pir- pirings which n become open flights. hope is, tht if pirings tht ings swpped into schedule, re lwys hrder to cover thn ones re re replcing, set of open pirings cn, t some point in time, be y into schedule without replcing ny or pirings. inserted be ble to use bove sketched locl serch definition of hrd To cover nd esy to cover pirings/flights hs to be introduced. to Severl short pirings re considered esier to cover thn one long. Length hope is tht smll pirings cn be more esily fitted into thn long ones. schedule Flights between three homebses re considered esy to Destintion This is nturl since huge number of personl re trnsported cover. pssive flights between homebses nd y cn be used to with flights. cover time Pirings which re closer to opertion re considered Deprture it is not possible to get ll open pirings closed, set of remining If pirings (fter reordering process is terminted) should hopefully open esier to cover thn initil set. Due to lst preference listed be time) some importnt time hs been chieved becuse open (deprture in crew schedules. pirings produced by initil piring re llocted towrds reordering of crew schedules, becuse it ws originlly optimized to produce initil set of pirings from set of open flights used build reordering prt of tp-i (s described bove). Pirings used in in step re (in generl) shorter thn vilble resources in crew reordering refore some furr piring is introduced to optimize use schedules. piring process cn run in one of two modes. In mode 1 one open flight only included in one piring (similr to wy initil piring works, is mrking open flights s used). This wy one cn be sure, tht flight i by overcovered, becuse multiply instnces of sme open flight isno not 2 does not mrk flight s used reby producing severl piring Mode contining sme open flight. This is delt with by greed possible tht postprocesses set of generted pirings nd selects heuristics tht do not contin ny duplicte use of flights. subset two different modes reflects two different (nd greedy) wys of delin set prtitioning problem tht lies beneth; generte pirings th with following descriptions of heuristics used in two modes r In given. to initil piring number of heuristics re tried in turn. Similr function on list of pirings sorted by deprture time (see figure 4.7) ll heuristics strt t hed of list, nd try to build new pirin from pirings from list. When new piring X is built it i constructed to list nd piring used (B, E nd G) re mrked s use ppended ignored fterwrds. nd first heuristic trverses list of pirings looking for pirings, whic or ends on homebse. When such piring is found it is checked strts 4.1 tp-i tp-i 19 present in severl pirings. following chrcteristics re considered: covers ech open flight exctly once. Mode 1 more importnt to cover thn pirings which re fr from opertion. A B C D E F G X flights re moved forwrd in time. B Min piring im of min piring (in following just referred to s piring) is to produce pirings which cn be covered with resources process E G 4.7: List of pirings. of resources. it strts on homebse. If it does not strt on homebse pssive trnsfe

15 dded (if possible) to mke it strt on homebse (mking sure is is still legl). If this succeeds result is piring like Piring 1 piring figure 4.8, where X is homebse, leg X-b is possibly pssive nd on is homebse. Next list of pirings is trversed looking for nor Y Piring 2 with connection time less thn 5 hours if X=Y or less piring 15 hours if X6=Y. Piring 2 is ppended to Piring 1 if new thn new piring is blnced. ffl If it strts nd ends on homebse nd covers t most two dys. ffl If it covers t most two dys nd cn be blnced by dding pssive ffl trnsfer. second heuristic is just like first with exception tht pssive re considered (when trying to connect two pirings) if re is flights most 24 hours between m nd if it does not crete night stop on t bse where piring strts. third heuristic tries to blnced pirings, which eir strts or ends homebse, by dding pssive trnsfers. Optionlly it is lso ttempted on ll pirings tht contin night stop on strting bse re broken Finlly, smller pirings on those bses. into 2 is very similr to mode 1 except tht it does not mrk piring Mode used. This is illustrted on figure 4.9, which corresponds to figure 4.7 s ll possible combintions of pirings strting with B, tht fulfills th Hence of given heuristic, re built. requirements 2 refore produces pirings, where severl instnces of sm Mode flight my be present in, severl different pirings. As describe open problem now is to solve set prtitioning problem. This is don bove, greedy heuristic which sorts pirings by density. densit using piring is defined s totl number of flights in piring minu of number of pssive trnsfers. Hence, density is piring qulit heuristic chooses piring with highest density firs mesure. rule out ll or pirings tht contins flights lso present in th nd finl result from piring process (despite mode) is set o covering ech flight exctly once. pirings mode 2 is not considered fully developed nd hs refore no However, put into production. been 4.1 tp-i tp-i 21 Mode 2 A B C D E F G X Y piring is legl, nd if one of following holds: B B pirings re ppended for s long s possible, under rules described New long with 5/15 hours rule, forming one long piring. bove Piring 1 Piring 2 E G F G X b c Y Y d e f X=Y: <5h X!=Y: <15h 4.9: List of piring. 4.8: Piring 2 re ppended t Piring 1 by 5/15 hour rule. chosen piring. This wy best first principle is used. to remove first or lst leg if this would blnce piring. fourth heuristic tries to connect unblnced pirings tht strts nd on homebse (possible with pssive trnsfer) to mke m bl- ends nced.

16 4.2 pros nd cons of tp-i 4.2 pros nd cons of tp-i 23 lredy mentioned, usefulness of reordering prt of tp-i hs As hevily limited by new union greements. been of two modes tht does piring genertion, only mode 1 hs mde Out into production. problem with mode 2 is tht it uses huge mount it memory nd is slow compred with mode 1. Mode 1 is ble to quickly of set of pirings which covers open flights. However, since no produce bout vilble resources in schedule re used, re is no informtion tht generted pirings cn be ssigned to crew. In prctice gurntee hs turned out, tht pirings generted by mode 1 often do not fit it vilble resources in schedule. Thus mode 1 is rrely used by stff t ss. drwbck re heuristics used in initil piring. As one Anor hve noticed y re redundnt nd still reside in Prolog, where might rest of tp-i is implemented in C/C++. Hence, mintennce of piring is hrder nd mking tp-i more complex s whole more initil complex. Chpter 5 Solution Approch lredy described new union greements hve mde tp-i effec As useless. ttempt mde to overcome se new restrictions ws t tively tp-i pure genertor which could be used by stff in trck mke deprtment. However, s lredy described it hs not been successfu ing to qulity of generted pirings. refore sig hs looked for due to improve generted pirings nd, if possible, wy of ssignin wy crew to generted pirings s well. usul pproch for piring construction does not include informtio crew tht (t some point in time) is going to fly piring bout mkes sense in piring phse (see figure 1.2) becuse vilbl This in ssignment phse re quite uniform mong crew; no o resources work hs been ssigned to crew t this point. However, in trckin little resources re spred more non-uniformly mong crew becuse lrg phse of pirings lredy hve been ssigned to crew. refore th number tht re going to be ssigned in trcking phse hs to b pirings to crew tht hs to cover m. Firstly, crew hs to b tilored in period of time piring covers. Hence she is going to b vilble some form of stndby (stndby, blnk-dy, usble-time etc.). Secondly on strt nd end bses of piring hve to fit with bse t which th is stndby or it hs to be possible to use pssive trnsfers to trnspor crew crew to/from strt/end of piring. Thirdly, pirings hv to be legl. 2

17 min ide which sig hd considered ws this tiloring of pirings. Generting tilored pirings for ech crew lso lie in line of reviewed (see chpter 3). Here, grph G k which for ech crew k represents literture set of pirings tilored for k ws widely used. Especilly, in trcking nd dy-to-dy phses for resons described bove. given by G k ws used s columns in Set Covering Problem or pirings Prtitioning Problem. Below, solution pproch using se elements Set be presented. Firstly, genertion of pirings (G k ) will be covered will gol of piring genertion is for ech crew to generte set of tht she might fly. This is conceptully done by serching through pirings grph G k for ech crew k which represents pirings constructed from open flights which re tilored for k. more precise definition of how resources re identified will be Firstly, followed by presenttion of number of heuristics for serching for given, resource is crew tht hs one or more stndby llocted in her schedule. A resource period is one or more successive stndby periods in crew A holes (described in chpter 2). However, mjority of resource nd llocted in schedule is stndby nd extension might not be time forwrd due to differences in rules concerning use of stright resources. Thus only stndby is considered. different figure 5.1 two resource periods re shown in which crew might be ble On cover some open flights. first period (between t 1 nd t 2 ) is simple; to which cn be ssigned during this period is t 2 t 1. next period time t 3 nd t 8 ) is formed by three successive stndby. Here t 8 t 3 (between 5.1: A smple schedule for crew. Solid lines represents nd dotted lines stndby production t 3 nd t 8 leglity of production is bounded by sum o between individul stndby. lredy described piring is work shift for crew. estimte As time which piring tkes up is sum of length of dut duty term prtil piring will be used to denote piring which is no (Strting nd ending nd sme bse). blnced construction of G k iswyofenumerting ll legl pirings for crew in given resource period. As first ttempt depth first serch throug k open flights which lie within given period (with respect to time) i ll used. figure 5.2 smple serch tree is shown. Here resource is vilbl On t 1 nd t 2 t bse. Ech node in tree corresponds to le between ech edge to connection time. Ech pth in tree, from root t nd lef, corresponds to legl piring to which crew my be ssigned. 5.1 Piring genertion Piring genertion 25 Period Period t3 t4 t5 t6 t7 t8 Time t2 t1 followed by selection of pirings ech crew is to mn. 5.1 Piring genertion Pirings pirings in G k. periods tht mkes up piring (see figure 1.1) Resources Depth first serch This definition might be extended to blnk-dys, usble-time schedule. it consists only of 1 stndby. An upper bound on totl production is n upper bound but not prticulrly tight one. A better bound is (t 4 t 3 )+(t 6 t 5 )+(t 8 t 7 ), becuse llthough one my ssign production

18 closeup on serch tree is shown in figure 5.3. Here four open flights A follow p i. Two to which it is directly connected, nmely, p j nd p k nd cn p l nd p m, to which it is connected through pssive flights k l nd k m two, respectively. tble 5.1 pseudocode for recursive depth first serch which genertes On pirings is given. As rguments, DepthFirstSerch tkes pir- p corresponding to pth to current node in serch tree, ing initilly is empty piring. Next it tkes list of prtil pirings which 1 ;:::;l n ], which initilly consists of one prtil piring for ech open flight [l mtches given resource period with respect to time. [l 1 ;:::;l n ] which sorted by incresing deprture time. Finlly, P r is set of generted is for resource period r. pirings 3-9 check possible direct connections connecting p with n open Line This corresponds to connection between p i 7! p j nd p i 7! p k on flight check for possible connections between p nd n open flight Line pssive connection, which corresponds to p i 7! k l 7! p l nd through check for pssive flights tht would blnce current piring. Line corresponds to node k n on figure 5.3. Which of 3 prts (line intervls) described bove hve more or less Ech structure. Firstly, y check if given connection should be tried sme pk 5.3: Anode p i in serchtree with successors p j, p k p l p m. p l nd p m re connected by two pssive flights k l nd nd okto predictes in line 3, 10 nd 19. If connection shoul with tried y check if legl piring hs been generted (lines 5, 12 n be in which cse it is sved (line 6, 13 nd 22). If connection does no 21) in legl piring y check if subtree beneth current nod result is predicte tht decides if l should be ppended t oktoappend(p,l,r) when trying to build piring for resource r. p figure 5.4 sitution is sketched. Firstly, it is checked whtever l On bse c mtches p's rrivl bse b nd if connection tim deprture p nd l, t 3 t 2, lies within predefined intervl. Secondly, it i between whtever l lies within current resource periods time interv checked 4» t 5 nd t 1» t 3 ). Finlly, itischecked whtever estimted duty o (t bigger thn estimted duty for given resource period. p[lis kl pl 5.1 Piring genertion Piring genertion 27 pi Time km t1 t2 pj pm kn 5.2: A smple serch tree for depth first serch. Nodes corresponds to legs nd edges to connection time. k m. should be explored (lines 7 nd 14). A more in-depth description of different predictes follows. oktoappend(p,l,r) figure 5.3. p i 7! k m 7! p m on figure 5.3.

19 DepthFirstSerch(p,[l 1 ;:::;l n ],P r,r) 1 for l i 2 [l 1 ;:::;l n ] 2 if oktoappend(p,l i,r) 3 if isvlid(p [ l i,r) 5 P r = P r [ [p [ l i ] 6 else if oktocontinue(p [ l i,r) 7 DepthFirstSerch(p [ l i,[l i+1 ;:::;l n ],P r,r) 8 end 9 else if oktoappendwithpssive(p,l i,r) 10 k = GetPssive(p,l i,r) 11 if isvlid(p [ [k] [ l i,r) 12 P r = P r [ [p [ [k] [ [l i ]] 13 else if oktocontinue(p [ [k] [ l i,r) 14 DepthFirstSerch(p [ [k] [ l i,[l 1 ;:::;l n ],P r,r) 15 end 16 end 17 end 18 if oktoblncewithpssive(p,r) 19 k = GetPssive(p,r) 20 if isvlid(p [ [k],r) 21 P r = P r [ [p [ [k]] 22 end : Pseudocode for depth first serch procedure tht Tble pirings. genertes checks whtever piring p is vlid ccording to union nd isvlid(p,r) rules. This is done by system clled rve, which ss governmentl uses to ensure leglity ofschedule. check is performed currently inserting piring p into crew r's schedule nd n vlidte by 5.4: Prtilly piring l sought ppended toprtilly piring p. corresponding resource period strt nd end t bse checks if subtree rooted t p should be explore oktocontinue(p,r) resource period r. Here prtil leglity check is used to ensure th for illegl node is explored ny furr. However, since piring is onl no only prtil leglity check cn be performed. piring p i prtil, into crew r's schedule, however ll ctivities which lie fter p r inserted from schedule before leglity check is performed. This i removed since piring p is not blnced nd refore do not (yet) fit wit done prt of schedule which lies fter it. checks whtever p nd l cn be connecte oktoappendwithpssive(p,l,r) pssive flight. requirement is tht (see figure 5.4) p rriv through different bse thn l (b 6= d), nd tht time intervl between on l is so big enough for pssive flight to be inserted. nd returns pssive connection to connect p nd l if pos GetPssive(p,l,r) This is done by querying sig utility for pssive connection. sible. 5.1 Piring genertion Piring genertion 29 b c d Bses p l Time t1 t2 t3 t4 t5 from time t 1 to t 5. oktocontinue(p,r) 24 end oktoappendwithpssive(p,l,r) isvlid(p,r) GetPssive(p,l,r) schedule using rve.

20 checks whtever it is relistic to blnce oktoblncewithpssive(p,r) p (see figure 5.5) with pssive flight. requirement is tht piring between rrivl t b (t 2 ) nd end of given resource period time 3 ) is big enough for t pssive flight nd tht p is not blnced lredy (t 5.5: Prtilly piring l sought ppended toprtilly piring p. corresponding resource period strt nd end t bse bove description of depth first serch n implicit priority is used In serch becuse [l 1 ;:::;l n ]is sorted by incresing deprture time. during wy flights with smll connection times re sought connected before This with long connection times. This priority might not be optiml. flights cost could be ssigned to piring n t ech node ll possible If could be generted nd sorted ccording to this cost. At ech successors this would form serchfront with given length L SF. nodes is node serchfront could n be explored in turn by incresing cost. cost of piring is not strightforwrd. Mny fctors influence Estimting on qulity of piring. Currently only mesurement used sig is density of piring (introduced in section 4.1.3) which is by number of flights in piring minus number of pssive flights. totl however, is very corse estimte. refore more flexible estimt This, introduced. is with sig, four spects of piring hve been identified whic Toger qulity of piring nd refore should be reflected in th influence mount of open flight duty which piring covers is importnt. Th duty of n open flight hs lredy been introduced in sectio (estimted) duty of piring Estimted duty of resource period Estimted or words, this specify wht you get ( duty covered by piring In to wht you py ( duty llocted in crew schedule). Since th reltive grows, s mount ofduty tht is covered grows rtio ctull rtio wht you gin. refore 1 minus rtio is used s cost. A mesures ssumption is mde here tht remining time in resourc implicit is not usble. Even though this might not be cse ll time period sems resonble during trcking. Here gol is to use resourc it llocted in schedule nd not to build up schedule for from periods pssive duty time in piring is lso importnt since it bsiclly i where crew gets pyed but do not work. Some pssive legs in time might bechnged to ordinry legs, before crew sees piring piring this is fr from lwys cse nd refore pssive flights shoul However, pssive duty of piring Estimted duty of resource period Estimted 5.1 Piring genertion Piring genertion 31 oktoblncewithpssive(p,r) cost. se spects will be introduced below. (b 6= ). p b Bses Duty Time s sum of duty periods tht mke up piring. To mk comprble between different resource periods cost for duty is see this t2 reltively to estimted duty of period piring is intended for: t1 t3 Duty cost = 1 from time t 1 to t Depth-best first serch scrtch. Pssive flights Piring cost be voided. Once gin rtio is used to mke mesure comprble: Pssive cost =

21 notion of problem flight rises becuse some open flights re more to cover n or, or put in or words some open flights desirble flight from Copenhgen to Alborg deprting t is esy to cover A with flight from Stockholm to Munich t refore it compred more desirble to cover second flight thn first t this point in is solution process. This ide is somewht similr to tht of reordering Time of deprture (Erly deprtures re problems). ffl Time of rrivl (Lte rrivls re problems). ffl Length of flight (Long flights re problems). ffl If flight cretes night stop (Night stops re problems). ffl rtio is used to mke cost comprble between different resource Agin periods. of problem legs Duty duty of resource period Estimted cover is verge rtio of crew which hve resource periods which Crew (with respect to time) legs in given piring. This gives mtches estimte of how likely it is tht legs in piring cn be covered rough nor crew. by serch strtegy forces given prtil piring to be prt of pir tht re generted. This is illustrted on figure 5.6. Here prti ings from b to c is used s strting point for 2 depth first serches piring serches bckwrds in time towrds strt of given period n One modified version of depth first serch described in section i A lrgest modifiction is tht between t 1 nd t 2 serch hs t used. 5.6: A smple serch tree for depth first serching bcknd forwrd from given piring. constructing pirings for different resource periods (different crew When open flights will be sought connected over nd over gin. refor severl preprocessing of open flights where ll possible successors re foun ech flight might improve performnce. This ide forms clone betwee for clssicl pproch, where ll pirings re generted first nd n crew ssigned, nd ide tht is tested in this project where crew r re ide is to construct two mps which for ech leg gives possibl successors nd possible successors through pssive leg. Thought o direct 5.1 Piring genertion Piring genertion 33 Problem flight Serching from middle nd out re more difficult to cover thn or. nor forwrd in time towrds end of period. section 4.1.2). Here flights re lso rnked ccording to how big (see y re. And flights tht re bigger problems re soughtswpped problem into schedule insted of less problemtic ones. with sig following 4 spects of n open flight hve been identified Toger which chrcterize problem flight: be bckwrds in time form b towrds. b c Time t2 t3 t1 t4 Problem Cost = 1 Crew cover Preprocessing of open flights cost of piring is weighted sum of introduced costs: considered in genertion process. Cost =Duty Weight Duty Cost+ Problem Weight Problem Cost+ Pssive Weight Pssive Cost+ in terms of grphs two mps represents met grph G tht contin Crew Cover Weight Crew Cover Cost

22 cixi (5.1 ijxi 1; for j =1;:::;n (5.2 X i2ip xi» 1; for ll crew p (5.3 xi non negtive integer ( Piring selection 35 pirings tht re going to be considered. pths represented by ll specific grph G k re n subset of pths in G. Pseudocode crew for generting mps is given in tble 5.2. ConstructSuccessorMps([l1;:::;ln],DS,PS,P) 1 for li 2 [l1;:::;ln] 2 for lj 2 [li+1;:::;ln] 3 if directsuccessor(li,lj) 4 DS[li] =DS[li] [ lj 5 else if pssivesuccessor(li,lj) 6 PS[li]=PS[li][lj 7 P[(li;lj)] = GetPssive(li,lj) 8 end 9 end end Tble 5.2: Pseudocode for construction of successor mps for ech leg. tkes 4 rguments: list of open flights ConstructSuccessorMps nd 3 mps. DS is mp between leg li nd list of pos- [l1;:::;ln] direct successors of li. PS mps between leg li nd possiblsible successors through pssive leg. P mps pir of open legs (li;lj) corresponding pssive leg tht cn be used to connect li nd lj. to is predicte tht indictes if lj is possible di- directsuccessor(li,lj) successor of li. pssivesuccessor(li,lj) is predicte tht indictes rect li nd lj cn be connected through pssive leg. if slightly modified version of depth-first serch from tble 5.1 cn Now used to serch open flights given by mps. be 5.2 Piring selection previous section set of pirings hs been constructed for ech resource In period. In this section we focus on selection of subset of those which will be ones ctully flown. problem consists of pirings t most one piring for ech resource period tht correspond- choosing crew hs to cover. problem is modelled s Set Covering Problem ing some extr constrints. se constrints ensure tht crew is not plus 5.2 Piring selection 3 two different flights t sme time. This is done by groupin ssigned pirings ccording to crew y where generted for (see tble 5.3 nd n llow t most one piring to be selected within ech group. choice of Set Covering Problem in fvour of Set Prtitioning Prob rest in nture of two problems. Since overcoverge is llowed i lem Set Covering Problem, it is lwys esy to construct fesible solutio tht ll rows re covered by t lest one column ech); just includ (given columns. On or hnd Set Prtitioning Problem does not ll for overcoverge, nd just finding fesible solution might be hrd. Th low of using covering model is tht some flights might be covere drwbck times. severl A mmticl formultion of problem follows. Mximize Z = Subject to mx i=1 mx i=1 is binry decision vrible tht decides wher piring i is used in th xi or not nd ci is corresponding cost. ij indictes if open flight solution covered by piring i. Ip is set of indices for columns corresponding t is p. first set of constrints ensures tht ll open flights re covere crew lest once (5.2). second set of constrints ensures tht ech crew t with t most one piring to solution (5.3). contribute ensure tht fesible solution lwys exists, so clled ghostcrew r To A ghostcrew hs property tht it hs exctly one pir introduced. ssocited nd this piring covers exctly one open flight. Hence on ing is introduced for ech open flight. Anor use of ghostcrew ghostcrew to control serch process, since cost plced on ghostcrew co is will ffect how desirble it is to include non-ghostcrew columns th umn covers sme row s given ghostcrew column.