A bin packing approach to solve the aircraft maintenance task allocation problem

Delft University of Technology

A bin packing approach to solve the aircraft maintenance task allocation problem

Witteman, Max; Deng, Qichen; Santos, Bruno F.

DOI

10.1016/j.ejor.2021.01.027

Publication date

2021

Document Version

Final published version

Published in

European Journal of Operational Research

Citation (APA)

Witteman, M., Deng, Q., & Santos, B. F. (2021). A bin packing approach to solve the aircraft maintenance

task allocation problem. European Journal of Operational Research, 294(1), 365-376.

https://doi.org/10.1016/j.ejor.2021.01.027

Important note

To cite this publication, please use the final published version (if applicable).

Please check the document version above.

Copyright

Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons. Takedown policy

Please contact us and provide details if you believe this document breaches copyrights. We will remove access to the work immediately and investigate your claim.

This work is downloaded from Delft University of Technology.

ContentslistsavailableatScienceDirect

European

Journal

of

Operational

Research

journalhomepage:www.elsevier.com/locate/ejor

Innovative

Applications

of

O.R.

A

bin

packing

approach

to

solve

the

aircraft

maintenance

task

allocation

problem

Max

Witteman,

Qichen

Deng

∗

,

Bruno F.

Santos

Air Transport and Operations, Faculty of Aerospace Engineering, Delft University of Technology, The Netherlands

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 17 June 2020 Accepted 16 January 2021 Available online 22 January 2021 Keywords:

Scheduling Bin packing problem Worst-fit decreasing Task allocation Aircraft maintenance

a

b

s

t

r

a

c

t

Thispaperaddressestheschedulingofaircraftmaintenancetasksthatmustbecarriedoutinmultiple maintenancecheckstokeepafleetofaircraftairworthy.Theallocationofmaintenancetasksto mainte-nanceopportunities,alsoknownasthetaskallocationproblem(TAP),isacomplexcombinatorial prob-lemthatneedstobesolveddailybymaintenanceoperators.Weproposeanovelapproachcapableof efficientlysolvingthemulti-yeartaskallocationproblemforafleetofaircraftinafewminutes.We for-mulatethisproblemas atime-constrainedvariable-sized binpackingproblem (TC-VS-BPP),extending thewell-knownvariable-sizedbinpackingproblem(VS-BPP)byaddingdeadlines,intervals,andarrivals fortherepetitionoftasks.Inparticular,wedividetheplanninghorizonintovariablesizebinstowhich multidimensionaltasksareallocated, subjecttoavailablelaborpowerandtaskdeadlines.Tosolve this problem,weproposeaconstructiveheuristicbasedontheworst-fitdecreasing(WFD)algorithmfor TC-VS-BPP.Theheuristicistestedandvalidatedusingthemaintenancedataof45aircraftfromaEuropean airline. Comparedwiththe solution obtainedwith anapproachusing anexact method,the proposed heuristicismorethan30%fasterforallthetestcasesdiscussedwiththeairline.Mostofthecaseshave optimalitygapsbelow3%.Evenfortheextremecase,theoptimalitygapisstillsmallerthan5%.

© 2021TheAuthors.PublishedbyElsevierB.V. ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/)

1. Introduction

Modern airliners have thousands of parts, systems, and com-ponents that need to be recurrentlymaintained afterundergoing certain flight hours(FH),flight cycles(FC),calendar days(DY), or months(MO).TheFH,FC,DY,andMOareknownasusage param-eters,andtheirmaximumsallowedinoperationaredefinedas in-spectionintervals.Theoptimalallocationofthemaintenancetasks to the best maintenance opportunities is a challenging problem solved dailybymaintenanceplanners.Thecommonapproach fol-lowedbytheseplannersistogrouptasksintomaintenancechecks (e.g.,A-,B-1_{,C-andD-check)toensurea consistentmaintenance}

program inwhich all tasksare performedbeforetheir associated duedates.AtypicalA-checkincludesinspectionoftheinterioror exterior oftheairplanewithselectedareasopened,e.g., checking and servicing the oil, filter replacement, and lubrication (Ackert, 2010 ).C-checkrequiresthoroughinspectionsofindividualsystems

∗_{Corresponding author.}

E-mail address: q.deng@tudelft.nl (Q. Deng).

1 B-checks are rarely mentioned in practice. The tasks that could be included in B-checks are commonly incorporated into successive A-checks.

andcomponents for serviceability andfunction. D-check2

uncov-erstheairframe,supportingstructure andwingsforinspectionof moststructurallysignificantitems.

To determine the optimal start date of the tasks, it is com-monin practicetoadopta sequentialprocess:first, schedulethe A-,C-andD-checks andthen allocatemaintenance tasksto each check. Although some tasks can quickly be packaged into these letter checks,a large number of other tasks (more than 70% for an AirbusA320aircraft) aredephasedfromtheintervalsofthese checks.It means that they eitherhaveto be allocated toa more frequentlettercheckormanuallyallocatedbymaintenance opera-torstodifferentmaintenanceeventsbasedonthesuitabilityofthe tasktothatcheck andtheurgencyofperformingthe taskindue time.Inpractice,bothapproachesareconductedaccordingtothe experienceofmaintenanceplanners,leadingtoinefficiencies.

Thetaskallocationproblem(TAP)inaircraftmaintenancerefers to the process of optimally allocating tasks in predefined main-tenance checks. It determines the optimal start dates of aircraft maintenance tasks so that all preventive tasks are performedas closetotheirduedatesaspossible.TAPiscomplicatedbecauseof

2 Many airlines merge D-check into C-check and label it as a heavy C-check or structural check.

https://doi.org/10.1016/j.ejor.2021.01.027

Fig. 1. Snapshot of maintenance overlap situation between aircraft.

itscombinatorialnature,andithastobesolvedfortheentirefleet atthesametime.Inreal-lifeapplications,multipleaircraftchecks can be scheduled in parallel, andtasks allocatedto these checks will share the maintenance resources. For example, Fig. 1 illus-tratesacaseforfiveC-checksoverlappingintime.Maintenance re-sourcesincludematerial,equipment,andasetoflaborhoursfrom differentskills.Furthermore,theallocationprocessisintricatealso because themaintenance tasks involvedin thesechecksare usu-allyassociatedwithdifferentintervalsandelapsedtime.

In this paper, we propose a novel approach to efficiently ad-dresstheTAP,whichcansolvetheproblemveryfastwithout com-promising thequality of thesolution. Maintenanceplansare fre-quentlybeingaffectedbyflightschedulesdisruptionsortheneed for unscheduled maintenance tasks, andthey constantly need to be revised or evenre-planed (Steiner, 2006 ). Inspired by the bin packing problem (BPP), we consider pre-scheduled maintenance checks to be bins of different (time) dimensions and sharing a multi-dimensionalcapacity,referringtothemultipletypesoflabor skillsinvolvedintheexecutionofthetasks.Theitemsarethetasks that needtobe packedinthebins,andtheyalsosubjecttotime constraints that limit thebin options.We formulate the problem asanextension ofthevariablesizebinpackingproblem(VS-BPP) (Friesen & Langston, 1986 )inwhichitemsarerepeatedwithintime intervals, andbinshaveavariabletimedimension.Thisextension oftheVS-BPPisnamedtime-constrainedVS-BPP(TC-VS-BPP).We presenta constructivealgorithm tosolve thisproblemefficiently. Wetestthisheuristicinacasestudyusingdatafromamajor Eu-ropeanairlineandcomparetheresultswiththeonesobtained us-ing an exactmethod. The main contributionof thisresearch can besummarizedinthefollowing:

• This work is the first to formulate the TAP as a bin packing problemandsolveitwithanefficientconstructivealgorithm.

• Forthefirsttime,tothebestknowledgeofauthors,theclassic VS-BPP formulationis extended toconsider time intervals for theallocation ofrepeateditemsandvariabletime dimensions forthebins.

• Weadapttheworst-fitdecreasingalgorithmfortheclassic BPP toefficientlysolvetheTC-VS-BPP.Theresultingconstructive al-gorithm is validatedwith areal casestudy andbenchmarked against the solution obtained using a commercial linear pro-grammingsolver.

The outline of this paper is as follows: Section 2 gives an overview of the relevant literature on maintenance related TAPs and bin packing problems (BPPs). The formulation of the TC-VS-BPP for aircraft maintenance is described in Section 3 .

Section 4 presentsataskallocation framework andanassociated heuristic algorithm. Section 5 shows a case study from a Euro-peanairlineandthealgorithmperformanceanalysis.Thelast sec-tion summarizesthe research withconcludingremarks andgives anoutlookonfuturework.

2. Relatedwork

In this section, we briefly discuss previous works. We divide thisliterature overviewintotwo subsections.Thefirst subsection reviewstheresearchworksdealingwiththeTAPforaircraft main-tenance, withdifferent perspectivesand methodologies. The sec-ond subsection discusses the literature on the bin packing prob-lem.

2.1. Maintenancetaskallocation

In one of the initial studies on TAP of aircraft maintenance,

Van Buskirk et al. (2002) combinedthe maintenance task alloca-tion with aircraft operation to one single problem. The authors presentedatwo-stagesystemthatsupportsmaintenancechiefsin planningboth aircraft operationsandmaintenance activities. The firststage assignstheplanesto flightoperationsusinga custom-built, multi-level greedy search algorithm. The second stage schedulesall maintenanceactivities accordingtoaconstraint sat-isfactionproblem.Theauthorstestedthesystemwith17jets,and results indicate that the system can schedule 3750 maintenance activities fora3-monthplanninghorizonwithin20minutes. The authorsalsostatethatthegoalwastoplantheactivitiesgiven var-iousconstraints:calendar-based actionshavetobe donewithin a specifictime window;usage-basedactionshavetobedone when theusageclockonapartorsubsystemreachesaparticularvalue; personnelhastobeavailabletodothejob(mechanicscanonlydo jobsthattheyarequalified for),andmaintenancejobshavetobe inspectedbyaquality/safetyinspectorandsoforth.However,this initial work does not optimize the maintenance schedule given thatsupportfortheflightoperationwasthetoppriority.

In contrast to Van Buskirk et al. (2002) , Steiner (2006) pre-sented a heuristic for aircraft maintenance planning, aiming at minimizing the overall number of maintenance actions and uni-formlydistributingthecapacityandflyinghoursoveragiventime horizon. The main idea wasto split the whole process into sub-processesthatcould behandledcomputationallyfastatthesame time.Determiningtheoptimalpositionofthemaintenanceactions was the least difficult one, whereas the balancing step was the mostchallengingone.Evenundervarioussettingsandconstraints, the proposed algorithms have shown to work reliably, fast, and withgoodoptimizationresults.According tothe casestudyfora 5-yeartime horizon,thenumberoftasksscheduledperfleetwas around50–500.Thetimetocomputeanewmaintenanceplanwas about15minutes.

In practice,many large airlines adoptthe top-down approach by appropriately grouping maintenance tasks into large packages andfitting theminto letterchecks. Muchiri and Smit (2009) fol-lowedthisapproachanddevelopedamaintenanceitemallocation model(MIAM)toclusteraircraftmaintenancetasksintopackages. The MIAMfirst simulates theaircraft utilization, calculates when

a maintenance item turns due, and then fits each maintenance item intoa package.The authorsusethe conceptofde-escalation to assessthe quality oftheir MIAM, whichcan be interpreted as thelossassociatedwithmaintenanceitemsbeingperformedmore frequently than necessary. The authors proposed a translation of thede-escalationintoadditionallabor costsessentialinthe long-termtoperform extramaintenanceactivities. Accordingtoa case studyofaBoeing737-NGaircraft,theauthorsclaimedthat intro-ducinganinitialde-escalation,i.e.,performingthefirstbase main-tenance before its duedate, leadsto a lower de-escalation labor cost overtime. Theauthorsobtainedthe bestresultforan initial de-escalation of30days, leadingtoa savings of 248labor hours (or €13,902) for a singleaircraft. The importance of Muchiri and Smit (2009) is that it provides an alternative of assessing main-tenance costs using thecausal relationship betweenexpense and laborhours.

Maintenanceoperation costs, inmore detail,includethe costs of maintenance tools, labor hours, and aircraft spareparts. Each maintenance task associates a cost. Since there are 1000–3000 tasks involved in aircraft maintenance, and many tasks can be performedin parallel,one of thebiggest challengesis toexecute the right maintenance task at the right time. Assigning priori-ties to maintenance tasks, such as the rule of “the most urgent task first”, can significantly reduce problem complexity. Hölzel, Schröder, Schilling, and Gollnick (2012) consideredthisaspectand presentedanoptimizationmethodforaircraftmaintenancetask al-location integratingsimulationsofaircraftlife-cycles.Inareal-life application,theauthorsobtainedthebestresultswhensortingthe tasks by cost (laborhours) in descending order. In this way,the optimizer allocatedthemostexpensivetasks to maintenance op-portunitiesclosertotheendofthelivesofthecomponents.

From an efficiency perspective, finding the best maintenance opportunities andallocatingmaintenance tasks one afteranother is exceptionally time-consuming. Since each taskhas some basic propertiesto indicatesimilarities, suchasATAcode,maintenance interval, zone, and check type, it ismore convenient to combine several similar tasks into a work package and reduce the total number of tasks. Li, Zuo, Lei, Liang, and Lu (2015) followed this ideaandgavedifferentweightsonpropertiestoindicatetask sim-ilarities. Based on engineeringexperience,weighting factors 0.05, 0.8, 0.05, and0.1are assignedtoATAcode,maintenanceinterval, zone,andcheck,respectively.TheauthorssolvedtheTAPofan air-lineusingafuzzyC-meansclusteringalgorithm.Although conver-gence and improvementswere both achieved, the authors stated there are still some pitfalls that need tobe investigated, such as theinfluenceofmodelparametersonsolutionqualityand conver-gencerate.

Ingeneral,theliteratureonTAP,especiallyforalongterm plan-ning horizon, is very limited. Some of them address TAP on air-craft level(Li et al., 2015; Muchiri & Smit, 2009 ), whileotherson fleet level (Hölzel et al., 2012; Steiner, 2006; Van Buskirk et al., 2002 ).EvenintheresearchworkofTAPinfleetlevel,theauthors tackledtaskallocationofeach aircraftindependently,and eventu-ally looped over the entire fleet. Furthermore, noneof those re-latedworkshasassessedtheoptimalityoftheproposedmodelsor heuristics. Thereisnocomparisonofhowclosethesolutionfrom proposedmodelsorheuristicstolocal/globaloptimum.

2.2. Thebinpackingproblem

Despite the various task allocation models and methods dis-cussed before, the TAP ofaircraft maintenance is very analogous withbin packingproblem(BPP), whereforTAP,the maintenance opportunities are equivalent to bins, and maintenance tasks are considered as items. The keys to solving BPP are the bin se-lection and item allocation. For bin selection strategies, Johnson

(1974) listsfourfundamentalandwidelyusedalgorithms, next-fit (NF),first-fit(FF),best-fit(BF),andworst-fit(WF):

• Next-Fit(NF):If theitem fits inthe samebin astheprevious item, put it there. Otherwise, open a new bin and put it in there.

• First-Fit (FF): Put each item asyou come toit into theoldest (earliestopened)binintowhichitfits.Onlyopenanewbinif anitemdoesnotfitintoanypreviousbin.

• Best-Fit(BF):Put itemsinbins inawaythat itmaximizesthe utilizationofthebinsthatalreadyhavebeenopened.

• Worst-Fit (WF): Put each item into the emptiest bin among thosewithsomethinginthem.Onlystartanewbiniftheitem doesnotfitintoanybinthathasalreadybeenstarted. Ifthere aretwoormorebinsalreadystartedwhicharetiedfor empti-est,usethebinopenedearliestfromamongthosetied. Ifallitemsarethesamesize,thereisnodifferenceinthefour algorithms.Since itemsare verylikelyto havedifferentsizes, the allocationofitemstobinsbecomesintricateandtime-consuming. Andthismayinvolveshifting bincontents continuouslyuntil the item list is empty. Thus, some researchers proposed prioritizing the items before putting them in bins. Johnson (1972) has sug-gestedsome alternatives tothe FFandBF. Theauthorstates that ifthe itemsare sorted indescending order (i.e.,the largest item goes first), the worst-case behavior of bin packing problems can be significantly improved. Therefore, it is now a common step to prioritize items before allocation when solving the BPP. The resulting algorithms are the equivalent first-fit decreasing (FFD) and best-fit decreasing (BFD) algorithms. Similarly, there are also next-fitdecreasing(NFD)andworstfitdecreasing(WFD)algorithms. Inpractice,notonlyitemscanhavevarioussizes,butalsobins canhave differentcapacities,andthisleadsto variable-sizedBPP (VS-BPP).VS-BPPisan extensionoftheclassic BPP,inwhichbins nolongerhavethesamesize,andthecostofabinisproportional to its size (Friesen & Langston, 1986 ). VS-BPP is more challeng-ing since putting itemsin bins affects the selections of opening newbins later on anditem allocations andvice versa.VS-BPP is NP-hard(Correia, Gouveia, & da Gama, 2008 ).Researcherstendto solveitusingapproximationalgorithmsinsteadoffindingthe ex-actglobaloptimum.Friesen and Langston (1986) listedsome algo-rithmsforVS-BPP,suchasnext-fitusinglargestbinsonly(NFL),and firstfitdecreasingusinglargest binsandattheendrepackto small-est possible bins (FFDLR). The authors also showed that allowing repackingsmallbinsandshiftingbincontentsimprovesalgorithm efficiency.AndtheFFDLRhasbetterworst-caseperformance than NFLbecausethereisnorepackingintheNFL.Friesen and Langston (1986) further developed a new algorithm first fit decreasing us-ingthelargestbins,butshiftingasnecessary(FFDLS)todynamically shiftingbincontentsduringtheconstructionofpacking.Case stud-ies provethat withdynamicallyshifting bincontents, FFDLS out-performsbothNFLandFFDLRintheworstcases.

WhileFriesen and Langston (1986) isoneofthefirstworksin VS-BPP, research in this topic continues and flourishes in many otherstudies (Correia et al., 2008; Csirik, 1989; Haouari & Serairi, 2009; Kang & Park, 2003 ).The main focusof thesestudiesis on the development of algorithms. There is no deadline for putting each item in bins in all of those studies. VS-BPP in scheduling, especiallymaintenanceplanning,isverydistinctfromotherfields duetotimeconstraints.Forexample,eachmaintenancetask asso-ciatesaduedate.InVS-BPP,itisequivalenttoimposingadeadline foreach item (eachitemhas tobe putin abin beforea specific time).Besides, and some tasks have to be performedrepeatedly. Oncethetaskisexecuted,wecananticipatethenextarrivaltime ofthesametask.

Thearrival timesofitemsanditem allocation deadlinesmake the maintenance scheduling related VS-BPP unique and more

complex. Some researcherscategorize the VS-BPP,in which each item has an associated arrival time and allocation deadline, as time-constrainedVS-BPP(TC-VS-BPP).Inoneoftheveryfew avail-able references, Fazi, van Woensel, and Fransoo (2012) presents a Markov Chain Monte Carlo (MCMC) heuristic to address the TC-VS-BPP in a working paper. The main difference between TC-VS-BPP andVS-BPP is that in TC-VS-BPP, the arrival times of the items have specific patterns, e.g., a probability distribution in Fazi et al. (2012) , and each item has to be allocated before a particulardeadline. TheMCMCheuristicisa combinationoflocal search andMonte Carlo sampling. It starts witha simple greedy approach to obtain an initial feasible solution. In this step, the authors createtwo non-ordered lists forbins anditems, respec-tively,andapply theFFalgorithmtoputitemsinbins.Afterthat, the authors use MCMC to improve the initial feasible solutions iteratively. One interesting finding from Fazi et al. (2012) is that when timeconstraintsare introduced,smallerandfasterbinsare preferredtomeetthedeadlines.ButintheclassicalVS-BPP,items are often concentrated in few high capacitated bins. Two main features in TC-VS-BPP, arrival times of the items and deadlines of theitems (Fazi et al., 2012 ), are alsocommon inmaintenance scheduling. Since most of the maintenance tasks have deadlines and follow periodic patterns, once a task is performed, we can alreadyanticipateitsnextexecution.

The review of the literature on TAP, BPP, VS-BPP, TC-VS-BPP, and corresponding solution techniques indicates that an aircraft maintenance TAP is similar to TC-VS-BPP in the model formula-tion in terms of maintenance capacity constraints, availability of each maintenance hangar, the different costs in task execution, workloadsofperformingtasks,taskexecutionintervals, and dead-lines ofthe maintenance tasks,meaning that the solution strate-gies, suchasNFD/FFD/BFD/WFD,toBPP/VS-BPP/TC-VS-BPP,canbe usedtoaddressTAP.Basedonthefindingsfromtheliterature,we propose a constructive heuristic based on the WFD algorithm to solve thelong-termaircraftmaintenance TAP.Themain reasonis that morethan55% ofthetasksbelongtoheavy maintenance (C-/D-check), andwe wantto let theavailable workforceaddress as many heavy maintenance tasks (largetask blocks) as possible in aircraftC-checks.Inourproblem,wearenot tryingtoreducethe number of bins being used – these were already pre-defined in the maintenance schedule and asa consequence ofthe overlap-ping ofmultiple checksintime. Furthermore,we wantto spread the tasksover themultiplebins insuch awaythat we avoid re-source limitations at anypoint. So the idea is always to allocate the item tothe binwiththe minimumload(or higherresources available).Sinceourworkfocusesonpracticalapplication,instead of worst-caseperformance analysis, we comparetheresults from theheuristictoasolutionfromexactmethods.

3. Problemformulation

In this section we define the TC-VS-BPP for aircraft mainte-nance task allcoation. We start the section with specifying the problem andits scope (Section 3.1 ), followed by a description of theassumptionsfollowed(Section 3.2 ).InSection 3.3 weintroduce some modelconsiderations,includingtheconceptoftimesegment and thegeneration ofthe taskitems inour TC-VS-BPP.Finally, in

Section 3.4 wepresenttheoptimizationmodelformulation.

3.1. Problemdefinitionandscope 3.1.1. TaskClasses

Intheaircraftmaintenancecontext,taskscanrepresentregular maintenance jobsneededforthecontinuousairworthinessofthe aircraft or repairing works that need to be performed to correct

malfunctionsordamage.Accordingly,thetaskscanbedividedinto twomainclasses(Ackert, 2018 ):

• RoutineTasks:thesearetheregulartasksoutlinedina Main-tenance Planning Document (MPD) provided by the aircraft manufacture or defined by the airline in their Operator Ap-proved MaintenanceProgram(OAMP). Thesetasks have tobe scheduledwithin certain fixed intervals, specifiedin terms of usageparameterssuch asFH,FC,andcalendardays.Aroutine taskhasto be performedbefore one ofthe usage parameters reachesthespecifiedinterval.

• Non-RoutineTasks:thesearenon-scheduledtasksthatcan re-sultfromdefaultsordamageidentified whenexecutinga rou-tinetask,pilot reports,orabnormaleventssuch ashard land-ings or ground damages. They can also represent abnormal maintenanceinterventionssuggestedby,e.g.,theaircraft man-ufacturer(service bulletins)orthe regulatorybody (airworthi-ness directives). When generated, these tasks are also associ-ated with a time window for their execution. And this time windowcan vary fromhaving toperform the taskbefore the nextflighttoacoupleofweeksaftertheyweregenerated.

3.1.2. Taskintervals

The aircraft maintenance tasks, regardless ofbeing routine or non-routine, have to be allocated to a maintenance event. These eventsincludelinemaintenanceinspections(i.e.,performedatthe ramporremotestandsduringtheturn-aroundtimeoftheaircraft) andhangarinspections.Inthisarticle,we onlyconsiderthelatter andignorethesmalltasksusually performedduringline mainte-nanceinspections.

Workforce

Theavailableworkforceconstrainsthetaskallocationto main-tenancecheck;eachmaintenancetaskisassociatedwiththe work-force requirements to perform the task. The maintenance work-force is divided per skill types (e.g., engines and flight control systems,avionics, aircraftmetallic structure, andpainting techni-cians). Itis limitedper dayorshift,accordingto thedaily work-force schedule.In thisstudy, theavailabilityofthe workforceper skill isan input to the model.The number of hours needed per skill type is a characteristic ofthe task, which can only be allo-catedto a maintenanceopportunity ifthere isenough workforce forallskilltypesinvolvedintaskexecution.

3.1.3. Timehorizon

Giventhat routinetasks havetobe scheduled basedon inter-valsandthattheseintervalsarere-startedeverytimethetasksare performed, the TAP should consider a time horizonthat is large enoughtocoveratleasttwofollowingtaskexecutions.Thereason beingthat,otherwise,apossibleactioncouldbe todelaythefirst taskasmuch a possible,disregardingthe possibilityofexecuting the tasks the next time. And this can resultin a poor or unfea-sible solution inthe long-term. Forthis reason, given that some taskshavingverylargeintervals(i.e.,some arenotperformed ev-eryyear),amulti-yearplanninghorizonisadopted.

3.1.4. Sequentialapproach

To plan hangar inspection tasks, we follow a sequential ap-proach,consistentwiththepracticeofmostairlines.Thatis,we as-sumethat theaircraftmaintenancecheck scheduling(AMCS)was solved beforehand and that an optimal letter check schedule is provided.According tothis schedule,eachcheck is consideredas amaintenanceopportunitytoperformamaintenancetask. Conse-quently,thegoaloftheTAPistoallocatethemaintenancetasksto theopportunitiesthatareascloseaspossibletotheirduedates.

Fig. 2. Overlapped maintenance checks are divided into several time segments.

3.2. Assumptions

Thisresearchissubjecttothefollowingassumptions:

A. There are sufficient aircraft spareparts andavailable mainte-nance tools andequipment, without constraining the optimal allocationoftasks.

B. Theoptimalallocationoftasksisconstrainedbytheworkforce available.The optimaldistributionoftaskspershiftorworker isnotconsideredintheTAP.

C. A-checktaskscanbeperformedinaC-check,butnottheother wayaround.

D. Non-routine tasks generated while executing other tasks can also be performedduringthesame check, andthisis consid-eredbyaugmentingthetaskdurationandlaborpowerneeded accordingto“non-routinerates” estimatedfromhistoricaldata. The firsttwo assumptionsare reasonable,considering thatthe TAP is a long-term problem and spare parts, maintenance tools andequipment,andlaborforceareplannedfollowingthe mainte-nanceschedule.Assumptions A3andA4arecommon inpractice. Thefirst,becausetheresources,skills,andtimeneededtoperform mostC-checktasksarenotcompatiblewiththeplanningofan A-check.Thesecond,becausethedifferingtasksfromahangarcheck can result in pressure toperform these tasks another day, even-tually causingdisruptionsinoperations.Therefore,airlinesusually prefertopre-allocateatimeandworkforcebufferineach mainte-nancechecktoexecutethesenon-routinetasks.

3.3. Modelconsiderations 3.3.1. Timesegments

Inpractice,maintenanceoperatorsaretypicallyconfrontedwith situationsofoverlappedmaintenancechecks,inwhichseveral air-craftundergothesametypeofmaintenancecheckatthetimeand thereforecompetingforthelimitedmaintenance resources.Fig. 2

depictsanexampleofsuchaschedule,andfiveaircraftare sched-uledtoperformC-checksmaintenancebetweenApr21st andMay 30th . During these overlap periods, resources have to be shared, constrainingtheoptimalallocationoftasks.

WedividetheplanninghorizondepictedinFig. 2 intotime seg-ments. Atime segment is created every time the overlap condi-tions change.In Fig. 2 , theoverlapof checkschangeon Apr24th and 30th , May12th , 15th ,18th , 21st and30th .Therefore, we cre-ateseventime segments:Apr21st_–24th_,Apr24th_–30th_,Apr30th_–

May12th , May12th –15th ,May15th –18th , May18th –21st andMay 21st –30th .Each time segmentof anaircraft isconsidered tobe a bin,witha givenduration(in days)andconstrainedby thelabor available onthesedaysforeachgivenskilltype.Forexample, AC-1hasfourbins,T4–T8;AC-5hasonlyonebin,T3;AC-16hasfour bins, T1–T4; AC-17 hasfive bins, T2–T6; AC-21 has two bins, T6 andT7. It isworth mentioningthat all thebins andtheir associ-ated sizes are definedbased onthe maintenance check schedule

andkept open.Unlikethe classicBPP, we donot needto open a binwhenweallocateitems(tasks).Fortherestofthepaper,when we refertoTC-VS-BPP,we alsoimplythat allthebins are prede-termined.

3.3.2. Taskitemsandmaintenanceopportunities

Most routine tasks have to be scheduled more than once for thesame aircraftover thetime horizonconsidered. Forexample, ataskthat hastobe performedineveryA-check(aboutevery7– 8weeks),mayhavetobe executed38 timesin a5-year horizon. Inourapproach,weconsidereachoccurrenceofthesetaskstobe an item inour TC-VS-BPP. That is,a routine taskthat has to be executed atmost N timesin the planninghorizonwill be trans-latedintoN tasksitemsin ouroptimizationmodel.Todoso, we havetoestimatethemaximumnumberofrepetitionsinthe plan-ninghorizon. Table 1 illustrates our approach fora giventaskof aspecificaircraft.Inthisexample,themaintenancetaskhastobe performed every ten weeks,while the aircraftA-checks are per-formedeverysevenweeks.Therearefivemaintenanceevents dur-ingthetime horizonfortheexecutionofthetask(fouraircraft A-checksandoneaircraftC-check,presentedinchronologicalorder). Thistaskcan beexecuted fromtwo times(only inA2andA3) to fivetimes(in every maintenance check),whichcan betranslated asfivetaskitemsinthetaskallocation.Theprocedureforcreating taskitemsanddefiningthe respectivemaintenance opportunities canbesummarizedasfollows:

- Step1: The maintenance opportunities forthe first execution ofthetaskaredetermined,accordingtothestateofthetaskat thestartoftheplanninghorizonanditsinspectioninterval. - Step2:Iftheearliestmaintenanceopportunityfortheprevious

taskitemisthelastmaintenanceeventintheplanninghorizon, westop.Otherwise,wecreateanewtaskitem(nextexecution). - Step3:Forthenewtaskitem(newexecution),

◦ Step 3.1: the first maintenance opportunity is the mainte-nanceeventrightaftertheearliestmaintenanceopportunity fromtheprevioustaskitem;

◦ Step3.2:thelastmaintenanceopportunityisthelast main-tenanceevent,withintheplanninghorizon,thatcanbe con-sideredbeforetheendofitsfixedinterval.

◦ Step 3.3: all maintenance events between the first and last maintenanceopportunitiesare consideredinthesetof maintenanceopportunities.

- Step4:GobacktoStep2.

Forthe taskitemswhich can potentially be allocatedto a main-tenance checkafterthe endoftheplanninghorizon, we createa fictitiousmaintenanceopportunity(bin).Thefictitiousbinisneeded because,eventually,notall taskitemshavetobe allocatedwithin theplanninghorizontokeep theaircraftairworthy.Thefictitious binisaddedonthedayrightaftertheendoftheplanninghorizon, associated withinfinite resources and no costs, andit is consid-eredasabinforalltaskitemsthatcanbescheduledaftertheend

Table 1

Illustration of the maintenance opportunities for repeated items of one maintenance task with an inspection interval of 10 weeks (Task 1 1 –1 5 represent the 1 st –5 th exe- cution of the same task). The value of “1” indicates that the associated maintenance check (column) is a possible maintenance opportunity for the execution (row).

A1 A2 C1 A3 A4 Fictitious

week 1 week 8 week 12 week 15 week 22 opportunity

Task 1 1 1 1 0 0 0 0

Task 1 2 0 1 1 1 0 0

Task 1 3 0 0 1 1 1 1

Task 1 4 0 0 0 1 1 1

Task 1 5 0 0 0 0 1 1

of the planninghorizon. Thisstep-wise approach,repeatedto all maintenancetasks,willresultinalistoftaskitemsN_kperaircraft k andtherespectivesetofmaintenance opportunitiesRi,k

associ-atedwitheachtaskiinthelist.

There are several task execution plans for the example pre-sentedinTable 1 .Eachplanisassociatedwithade-escalationcost, dependingon theletterchecksthat thetaskisexecuted. Wecan choose a high-cost plan in which the task is executed in every maintenance check (i.e.,five times inthe planning horizon) ora low-costplaninwhichthetaskisexecutedonlytwice,duringthe A1 and the A3 checks. Even fortwo planswith the same num-beroftotalexecutionsofatask,thede-escalationisdifferent.For instance, performing the task in A2, C1 andA3 results in a de-escalationcostof:

(

10− 8

)

+[10−

(

12− 8

)

]+[10−

(

15− 12

)

]=15weeks (1)

executingthetaskinA2,A3andA4resultsinade-escalationcost of:

(

10− 8

)

+[10−

(

15− 8

)

]+[10−

(

22− 15

)

]=8weeks (2)

Wecanobservefrom(1) and(2) thatthelatterexecutionplanhas alowerde-escalationcost,andthegoaloftaskallocationistofind the taskexecutionplanwiththe lowestcost, giventheresources availableandtheurgencyofothertasks“competing” forthesame maintenanceopportunities. 3.4. Problemformulation 3.4.1. Nomenclature • Sets • i:taskindicator • K:setofaircraft

• N_k:setoftaskitemsforaircraftk(k_∈K)

• Tk:setoftimesegmentsforaircraftk(k∈K)

• Ri,k:setoftimesegments fortaskitemi (i∈Nk)ofaircraft

k(k_∈K)

• J:setofskills

• Oi,kunitsetwiththetaskitemthatfollowstaskitemi (i∈

N_k)ofaircraftk(k_∈K)

• Parameters • ct

i,k:costofallocatingtaskitemi(i∈Nk)fromaircraftk(k∈

K)tomaintenanceopportunitybelongingtotimesegmentt (t∈Tk)

• GRtj:amountofavailablelaborhoursofskilltypej(j∈J)at

timesegmentt

• GR_i,kj :amountoflaborhoursofskilltypejprescribedto per-formtaskitemiofaircraftk

•

σ

j,l_{:“non-routinerate” indicatingtheamountoflaborhours}

needed from skill type l for every labor-hour prescribed fromskilltypej(note:

σ

j, j≥ 1.0

∀

j∈J)

• d_i,k: maximumnumber ofdays betweenrescheduling task itemi(t∈Tk)foraircraftk(k∈K)

• dt_{:number ofdaysfromthe startof theplanninghorizon}

tillmaintenanceopportunitybelongingtotimesegmentt

• f h_i,k:maximumnumberofflight-hoursbetween reschedul-ingtaskitemiforaircraftk

• f ht_{:number ofaccumulated flight-hoursfromthe start of}

the planning horizontill maintenance opportunity belong-ingtotimesegmentt

• f ci,k:maximumnumberofflight-cyclesbetween

reschedul-ingtaskitemiforaircraftk

• f ct_{: number ofaccumulated flight-cyclesfromthe start of}

the planning horizontill maintenance opportunity belong-ingtotimesegmentt

• O_dayi:totaldaysofaircraftoperationsfromthestartofthe

planninghorizontotheduedateofperformingtaskitemi, followingthetaskfixintervalandifnoresourceconstraints areconsidered

• inter

v

ali: average fix interval for task item i measured in

days

• labor_ratej:laborrate,perhour,ofskilltypej(j∈J) • other_costsi,k: non-labor costs associated with task item i

(i∈Nk)ofaircraftk(k∈K),suchascostsofsparepartsand

tooling

• Decisionvariables • xt

i,k:1iftaskitemi isassignedtomaintenance opportunity

belongingtotimesegmenttforaircraftk,and0otherwise MixedIntegerLinearProgramming(MILP)Formulation

Givenalong-termaircraftA-andC-checkschedule,we formu-latedtheTAPasa0–1MILPmodel.

min k∈ K  i∈ Nk  t∈ Ri,k ct i,k× xti,k (3) subjectto:  t∈ Ri,k xt i,k=1

∀

i∈Nk

∀

k∈K (4)  k∈ K  i∈ Nk  j∈ J GR_ij_,k× xt i,k×

σ

j,l≤ GR l t

∀

t∈Tk

∀

l∈J (5)  m∈ R_p,k dm_{× x}m p,k−  t∈ R_i,k dt_{× x}t i,k≤ di,k

∀

i∈Nk

∀

p∈Oi,k

∀

k∈K (6)  m∈Rp,k f hm_{× x}m p,k−  t∈Ri,k f ht_{× x}t

i,k≤ f hi,k

∀

i∈Nk

∀

p∈Oi,k

∀

k∈K

(7)  m∈Rp,k f cm_{× x}m p,k−  t∈Ri,k f ct_{× x}t

i,k≤ f ci,k

∀

i∈Nk

∀

p∈Oi,k

∀

k∈K

xt

i,k∈

{

0,1

}

∀

k∈K

∀

i∈Nk

∀

t∈Tk (9)

Theobjectivefunction(3) aimsatminimizingthetotal mainte-nance costs,which reflectthede-escalationcosts associated with scheduling the task earlier than its due date and, consequently, havingtoperformthetaskmorefrequentlyinthefuture.To com-pute these costs, we estimate the due date to allocate the task item beforehand.Forexample,ifamaintenance taskistoreplace an aircraftcomponent, based on its previous execution date and theassociatedmaintenanceinterval,wesimulatetheutilizationof thecomponentusingtheaverageaircraft’sdailyutilization.Inthis way,we canestimate thenext duedateofreplacingthis compo-nent andits idealmaximum utilizationO_dayi.The de-escalation

costs can then be calculated by comparing how earlierthe task itemisallocatedwhencomparedwithitsdesiredday(Hölzel et al., 2012 ): ct i,k= O_dayi− dt inter

v

ali ×



 j∈J



 l∈J GRl i,k×

σ

l, j



× labor_ratej+other_costsi



(10)

Thede-escalationcostsindicatedby(10) isareferencecostusedas aproxy ofthegoalofschedulingthetasksaslateraspossible,or aslessfrequentaspossible.In(10) ,thecostofallocatingtaskitem i of aircraft k to maintenance opportunity t is a function ofthe wastedintervalofthetask(firstterm),thelaborhoursrequiredto performthetask(secondterm),thelaborhourscostperlaborskill (thirdterm)andadditionalcostsassociatedwithmaintenancetask i such as the cost for materials or expensive tooling (last term). And thisformulationaims atallocatingtasks tothe maintenance opportunityclosertoitsduedatewhilegivingahigherpriorityto labor-intensivetasksandtasksinvolvingmanylaborskillsorhigh additionalcosts.

Constraints (4) guaranteethat each task item is allocated ex-actlyonce,eithertoamaintenanceeventortothefictitious main-tenance event after the planning horizon. Constraints (5) make sure that the available labor hours foreach skill type is not ex-ceeded ineach ofthe maintenance time segments.The left-hand side oftheseconstraintssumsthelaborhours neededtoperform eachtaskitem,includingtheworkforceneededtoperformthetask and, eventually,associated “non-routine” tasks. Thesetwo setsof constraints arethe onesthatdefine the classicVS-BPP.The other three setofconstraints(6) –(8) arethe featuresof TC-VS-BPPand alsoonesthat representthemaintenancetime-intervals.They im-ply the arrivals and deadline of tasks. Constraints (6) guarantee that a subsequent task item is scheduled within the number of daysdefinedinthefixintervalfortherespectivetask,while con-straints (7) and(8) reflectthefixintervalintermsofflight-hours andflight-cycles,respectively.

4. Taskallocationframework

The sameasBPP,TC-VS-BPPisalsoNP-hard(Garey & Johnson, 1990 ). OptimalsolutionstosmallTC-VS-BPPs canbeobtained us-ing exactmethods.Still,unfortunately,whenthesizeofthe prob-lemgrows,therunningtimesoftheseexactmethodsbecome pro-hibitive, especially forpractical implementations. Forthis reason, weproposeaconstructiveheuristictosolvetheTAPefficiently.The proposed approach is an iterative process based on the WFD al-gorithm. Tothe TAPforaircraftmaintenance, we start by sorting thetasksfromthemultipleaircraftintodecreasingorderof prior-ityandthen allocatethosetasksoneafteranothertothesuitable bin that has a lower load. In this section, we provide details on theproposed constructiveheuristic,explainingthegeneral frame-work, including the input data (Section 4.1 ), the necessary pre-computation(Section 4.2 )andthealgorithmitself(Section 4.3 ).

4.1. Inputdata

Foursets ofinputdata areneededto formulateandsolve the TAP.Thefirstsetconsistsofmaintenancetaskinformationpresent in the OAMP for the considered aircraft fleet. This information is not necessarily limited to maintenance tasks described in the MPD. It could include additional maintenance tasks as required by theairline, servicebulletins, airworthinessdirectives, deferred defects,or modifications(Ackert, 2018 ). Furthermore,information aboutthelastexecuted dateoftheroutinetasksisusedto calcu-latethefirstdue-dateofthemaintenancetask.Thesecond set in-cludestheestimateddailyaircraftutilization,inDY,FH,andFC,of eachaircraftfortheentiretimehorizon.Fortheshortterm,these valuescould beobtainedusingaircraftroutesorflight schedules, whileinthelongrun,themostcommonapproachistouse aver-ageaircraftutilizationperdayoftheweek,permonth,orseason. It is convenient, however, to use the same input values used to producethemaintenancecheck schedule.Thethirdsetofinputis theavailableworkforceperskilltype,perday,fortheentiretime horizon.Again,detaileddailyschedulescould beprovidedforthe shortterm, whilethe averageworkforceper daycan be usedfor the longerterm. The last setof data usedis the A-and C-check schedule,defining thestartingdatesanddurationofallchecksin theplanninghorizonforeachaircraftinthefleet.

4.2. Pre-computation

Aset of pre-computationsteps are necessary beforeinitiating theconstructive taskallocationalgorithm. Thesesteps canbe di-vided into task items and bins related pre-computations. Start-ingwiththetaskitemsrelatedsteps,maintenancetasksfromthe sameaircraftthat haveidenticalintervals,intermsofFH,FC,and DY,areclusteredtogethertoreducethenumberoftaskstobe con-sidered.Fortheresultingtasks,asetoftaskitemsarecreated, fol-lowingtheprocedureexplainedinSection 3.3 .Thefollowingstep istocomputethedue-datesforthefirst itemofthemaintenance tasks.Andthisisdonebyconsideringtheinitialstateofeachtask (i.e.,number ofFH, FC,andDY sinceits previous execution), the taskintervalsasdefinedbytheOEMorairline,andthesimulation ofthe aircraftutilizationovertime. Some tasks,such asdeferred defectsormodifications,canbeinputalreadywithfixedduedates insteadoftaskintervals.

For the bin related steps, the checks schedule is used to di-vide the maintenance opportunities into bins, as explained in

Section 3.3.1 .The binsare variableinsizeanddiscrete,composed byasetofdays.Afterthat,wecontinuetoconvertthelaborpower obtainedperdayintolaborpoweravailableperbin.

4.3. Constructiveheuristic

A constructive heuristic based on WFD is proposed for task allocation. The pseudo-code of the heuristic is presented in

Algorithm 1 , while the main procedures of the heuristic are ex-plainednext.

SortTaskList

After uploading the input data, the first procedure is to sort thetaskitemslistaccordingtotheprioritiesoftheitemsincluded inthe list.Theprioritization is doneaccordingto a prioritization function p

(

i

)

that classifies each task item i. This prioritization functiondividestaskitemsintothreeclasses:

• High Priority – these are items from maintenance tasks that haveanintervalequaltotheintervaloftheaircraftchecks.The allocation process for these items is trivial since those tasks haveto be allocated to all equivalent checks inthe schedule.

Algorithm1 TaskAllocationAlgorithm.

1: N_← setoftaskitemsfromallaircraft,N_=∪N_k

2: GR_tj← availablelaborhoursfromskill jinbint

3: GR_ij_,k← amountoflaborhoursofskill jprescribedtoperform taskitemiofaircraftk

4:

σ

j,m← “non-routinerate” fromskillmfromeveryhourofskill j

5: procedureSortTaskItemsList

6: SortandreindexNsothatp

(

i₁

)

≥ p

(

i₂

)

≥ … ≥ p

(

in

)



Prioritizationoftaskitems

7: endprocedure

8: procedureTaskItemsLoop

9: whileN_=∅do

10: SelectifromN Selectthefirsttaskinthelist

11: Ri←Ri,k∪t0 Addt₀ asafictitiousopportunity

12: SortandreindexRisothat

 j∈ J GRtj_i,1≥  j∈ J GRtj_i,2≥ … ≥  j∈ J GR_tj i,n

13: procedureAllocatetoBin

14: n←0 15: whilen<

|

Ri

|

do 16: n←n+1 17: ifGR_tj_≥ j∈ J GR_ij ,k×

σ

j,m

∀

m∈Jthen 18: Allocateitoti,n 19: SetGRtj_i,n=GR j t_i,n−  j∈ J GR_ij_,k×

σ

j,m

∀

m∈J

20: Computenextdue-datefortaskitemi

21: ifNextdue-datenotwithintimehorizonthen

22: N←N

\

{

i

}

Removethemaintenancetask

23: else

24: SortTaskItemsList

25: endif

26: break

27: endif

28: ifn=

|

Ri

|

then Incaseofnoallocationpossible

29: Allocateitot₀ 30: ReportAlert 31: endif 32: endwhile 33: endprocedure 34: endwhile 35: endprocedure

Thisstrategyofstartingtheallocationprocesswiththesetasks followsthe schedulingpracticeobserved inpractice,assigning theworkforcenecessarytothesetasksbeforestartingthe allo-cationofmaintenancetaskswithmoreflexibility.

• Medium Priority – these are the maintenance tasks dephased from the aircraftchecks intervals. Each ofthese tasks has an interval length larger than the A-check interval (e.g., the task inTable 1 )andhencetheywillnotnecessarilybe allocatedto everymaintenancecheck.

• Low Priority – these are the maintenance tasks with a low frequency of occurrence.They are dephased from the aircraft checksby,atleast,beingabletoskipatleastoneA-checkfrom any day within the planning horizon. These tasks have some flexibility,andtheycanbeallocatedatlast.

Thetasks withineachoftheseclassesaresortedbythe main-tenancecosts,asexpressedinthesecondandthirdtermsin(10) .

4.3.1. Taskitemsloop

Taskitemloop(TIL)isthemainprocedureofthealgorithm.The goalistochoosethebestmaintenanceopportunitythatminimizes themaintenance costs, asdefinedin (10) ,andto selectfromthe bins theone that less compromisesthe best allocationof subse-quenttaskitems.Aftersortingallthetaskitemsaccordingtotheir costs,thefirst taskitem hasthehighestpriority;thesecondtask itemhasthesecond-highestpriority,andsoforth.Wedefinealist ofbinsthatwouldallowafeasibleallocationoftaskitemibefore theassociatetaskintervalisexpiredaccordingtothemaintenance checkschedule.Afictitiousbin(t₀ )isadded tothislistofbinsin casenone of the available bins has enough resources to allocate thetaskitem.Otherthanthat,wewillnotcreateanynewbin dur-ingthetaskallocation.

After that, we sort the available bins for task item i accord-ing to the maintenance resources within bins in descending or-der. Namely, the bin withthe most resources is always the first toassignthetaskinit.After that,theallocationofeachtaskitem followinga“worstbin” selectionprocessinthefourthstep. There-fore,theTILproceduregivesahigherpreferencetothebinscloser totheduedateofthetaskitemand,amongthesebins,totheones thathavemoreavailablemaintenanceresources.

4.3.2. Allocationoftaskstobins

The next procedureis to allocatethe taskitemsto a bin, fol-lowingthe sorted listof bins.If thebin underconsideration has enough available labor hours for the necessary skills to perform the respective maintenance task,we allocate an item to the bin. Inthiscase,wesubtractthelaborhours consumedtoexecutethe taskfromthe total available labor hours fromthat bin. The next stepis tocheck the needto remove thetaskthat has been allo-cated.Fora routine task,wesimulatethe evolution ofusage pa-rameters after allocation and estimate its new date according to aircraftdailyutilization.Ifthenextduedateisbeyondtheendof thetimehorizon,wejustremove thetaskfromthetaskitemlist. Fora non-routine task,since they arenot recurrentlyperformed, we generatea new duedate after theend ofthe planning hori-zon.Forthecaseofrunningoutofbinstowhichthetaskcanbe allocated,we generateanalert andputthetaskintofictitiousbin t0.Thisfictitiousbinincludes allthetasksthat havenotbeen

al-locatedto anyavailable bin,andwewillinformthemaintenance controllerandletthemaddressthosetasks.

5. Casestudy

Inthis section, we presenta casestudyon a major European airlineandillustratetheapplicabilityoftheTAPapproach.The in-putdataincludesaircraftutilization,a4-year maintenance sched-ule generated by the dynamic programming based methodology describedinthepaperofDeng, Santos, and Curran (2020) ,task in-formationfromaheterogeneousfleetof45aircraft,andan associ-atedestimationofavailableworkforceperday.Ourairlinepartner currentlyfollowsamanualprocesstoallocatetheaircraft mainte-nancetasks tochecks, supported by adigital solutionthat keeps track of the open tasks and suggests a prioritization of mainte-nanceactivities.Therearetwomaintenanceplannersintheairline doingthisjobfortheentirefleet.

Weconsidereightskilltypesandthattheproductivityfactorof each worker is equivalent to 4.8productive labor hours per day, following the airline practice. The remaining hours of the labor shiftarededicatedtotransitioningmeetingsbetweenworkshifts, collectionofmaterials orequipment, obtaininginformationabout themaintenancetask,reporting,andancillaryactivities.

The results from thiscase study are discussed in Section 5.1 , followedby an analysisofthecurrentairlinepractice ofnot per-forminganyaircraftC-check tasks inan A-check(Section 5.2 ). In

Fig. 3. Labor hours distribution per aircraft and skill type.

Fig. 4. The amount of labor hours used for each skill type during the time segments within the overlap situation. Each time segment has eight different bars and each bar represent a particular skill type (Group 1, Group 2,..., Group 8).

Section 5.3 ,we validate the results obtained using the proposed taskallocation heuristic. Wesuggest assessing thealgorithm per-formance bycomparingit withthesolution obtainedwhen using anexactmethodforsolvingtheMILPpresentedinSection 3.4 . Fur-thermore,allresultsobtainedbytheproposedheuristicwere vali-datedbythemaintenanceplannersoftheairlinepartner.

5.1. Optimizationresults

In this subsection, we apply the proposed task allocation al-gorithm to the casestudy, followingthe airline current policy of notallowingtoallocateC-checktaskstoA-checkmaintenance op-portunities. The problemwas solved in lessthan 14 minutes by

thealgorithm.The outcomeis a4-year,fleet-widetaskallocation plan that satisfies labor-hour constraints and tasks fix intervals. Theplanincludesaround85thousandtaskitems,fromwhich24% ofthemareC-checktasks,and76%areA-checktasks.Despitethis, the C-check tasks consume about65.5% of the labor hours allo-catedtoperformthetasks.Thealgorithmachievesanaverage de-escalationof205daysforC-checktasksand19.3daysforA-check tasks.

Fig. 3 shows the distribution of labor hours per skill for the maintenanceofallaircraftinthefleetforthefullplanninghorizon. Thereissignificantdiversityintherequiredlaborhoursamongthe aircraft.Andthedifferenceinaircraftage,thenumberofC-check eventsinthemaintenance schedule,andthedifferencesinterms

Fig. 5. Average wasted interval in days for increased C-check task labor hours thresholds.

of aircraft utilization cause this diversity. For instance, aircraft AC-41 is phased-out a few days after the start of the planning horizon, while AC-24 is phased out one and half years afterthe beginningoftheplanninghorizon,followingaminorC-checkand 10A-checks.Inthesameway,itispossibletoidentifytheaircraft that perform a C-check early in the planning horizonand hence have to undergo three C-check before the end of the planning horizon.AndthisappliestoaircraftAC-25,AC-26,andAC-29.

Toanalyzethemaintenanceplaninmoredetail,wedecidedto focus on the overlapsituation presented inFig. 2 .The allocation oflabor hours per timesegment isdepictedinFig. 4 .In this fig-ure,thereareeightbarsper time-segment,representingtheeight different skilltypes. We observethat thefirst six time-segments consume all the available labor hours of the Group 2 skill type. Andthisrestrictstheallocationoftasksrequiringlaborhoursfrom Group 2 skillforAC-5, AC-16, andAC-17 since theseaircraftwill haveafullyconstrainedoverlapsituation.Andthisforcessome of theA-checktasksfromtheseaircrafttobeallocatedtoaprevious A-check. Similarly, there are also C-check tasks beinganticipated atan earlierC-check. Inthelattercase,it meansthatsome com-ponentsareinspectedorreplacedabouttwoyearsearlierthan in-tended.

5.2. Flexibletaskallocationpolicy

Inthissubsection,wequestionthecurrentairlinepolicyofnot allocatinganyC-checktasktoA-checkmaintenance opportunities, even thoughwe observethat thereis asurplusoflabor hours in the A-checks scheduled.Several smallC-check tasks would fit in an A-check, interms oftime andresources needed. Forthis rea-son, weperformedasimulationinwhich theseC-check tasksare allowedtobeallocatedtoA-checkopportunities.Wecarryoutthe analysisconsideringdifferentthresholdsforthesizeofthesetasks. After discussing with maintenance planners from the airline, we agree on usingthe labor hours needed forthe taskasthe refer-encemetricfortasksize,andtoconsiderathresholdvaryingfrom zeroto2.5laborhours.

Thesimulationresults(presentedinFig. 5 )indicatethatthe de-escalationofC-checktaskscanbereducedfrommore205daysto 132 days when allowing C-check tasks within 2.5 labor hours to beexecutedonA-checkopportunities.Fromtheresults,itcanalso beconcludedthatthemarginalgainofextendingthethreshold

re-ducesasthethresholdincreasesinvalue.Infact,itcanbeinferred fromFig. 5 that, forthis airline,after a labor hours thresholdof 2.0or2.5, thereare barelyanybenefitsofextending this thresh-old.The reasonbeingthatvery fewC-check tasksconsumemore than2.5laborhoursandcanstillbeallocatedinanA-check with-outcompromisingtheallocationoftheA-checktaskstotheirbest A-checkopportunities.

5.3. Algorithmperformanceanalysis

Toanalyzetheperformanceofthetaskallocationalgorithm,we compare it withthe performance ofan approach usingan exact methodto solve the TAPformulated inSection 3.4 . Toprovide a moredetailedcomparison,wedecidedtovarytheproductivity fac-toroftheworkforce,fromtheinitialconsidered4.8laborhoursper daytoarestrictedcaseof3.2laborhoursperday.

To compute solutions with the exact method in a reasonable time for the more restricted cases, we follow an iterative pro-cessforthecreationoftaskitemsandmaintenanceopportunities foreachmaintenancetask(Section 3.3.1 ). Thatis,weinitially run theMILP, thenadd newitemsandmaintenance opportunitiesfor those taskitems that hadthe constraints violated and rerun the MILPuntilthe problembecomesfeasible.Forthe 4.8laborhours case, the MILPformulation resultedin 1.15million decision vari-ablesand373thousandconstraints.The taskallocation algorithm iscodedinPython3.7,whiletheexactmethodisaddressedusing the commercial solver Gurobi. The results from both approaches arecomputedonanIntelCorei72.6GHzlaptopwith8GBram.

WesummarizetheresultsinTable 2 .EachlineofTable 2 com-paresthecomputationtimesandpresentstheoptimalitygapfora givenproductivityfactorbetweentwodifferentapproaches,where theresultsobtainedfromthesolverisused asareference. While thecomputationtimeoftheexactmethod(MILPsolver)explodes with the decrease of the productivity factor, the same does not happentotheproposedheuristicalgorithm.Theproposed heuris-ticismorethan30%fasterthanthe exactmethodforthedefault productivity factor of 4.8 labor hours, and the optimality gap is only0.03%. Eventhoughthe productivitylabor hoursdecrease to 3.2,theoptimalitygapisstillwithin5%.

Itisworthmentioningthat forthemostconstrainedtest case, theexactmethodrequiresabout4.9hourstocomputetheoptimal solution,whiletheproposed heuristicneedslessthan15minutes.

Table 2

Simulation results of performance analysis.

Productivity Computational time (s) Solution Gap Labor hours MILP solver Heuristic Heuristic vs. Solver

4.80 1,135 773 0.03% 4.72 1,148 775 0.03% 4.64 1,154 776 0.03% 4.56 1,159 778 0.04% 4.48 1,168 779 0.05% 4.40 1,172 787 0.11% 4.32 1,181 792 0.23% 4.24 1,186 798 0.36% 4.16 1,187 803 0.54% 4.08 1,195 811 0.81% 4.00 1,639 818 1.17% 3.92 1,821 822 1.44% 3.84 1,903 828 1.61% 3.76 2,570 835 1.43% 3.68 3,097 839 1.90% 3.60 3,857 846 2.45% 3.52 4,702 851 2.88% 3.44 5,679 861 3.38% 3.36 8,243 866 3.89% 3.28 13,828 871 4.47% 3.20 17,636 879 4.95%

In summary,from a perspective of solution quality, the solution gap between the heuristic algorithm and the optimalsolution is within5%foralltestcases.Forcaseswheretheproductivityfactor washigherthan4.0laborhours,thesolutiongapisbelow1%,and thisconfirmsthat theheuristicalgorithm iscapableofproducing goodsolutionsinminutesforarealisticTAPwithafleetof45 air-craft.

6. Conclusion

The task allocation problem (TAP) of aircraft maintenance is definedasassigningtasks totheir optimalmaintenance opportu-nities. In this research, we formulate TAP as a time-constrained variable-sizedbinpackingproblem(TC-VS-BPP),inwhichwetreat maintenance opportunities as bins and the tasks as items, and there are time constraints on both bins and items. TC-VS-BPP is NP-hard and, therefore, challenging to solve for large case in-stances. For thisreason, we proposed a constructive heuristic to solve the TAP(TC-VS-BPP).The proposed approach isan efficient iterative process based on the worst-fit decreasing (WFD) algo-rithm.Accordingtoareal-lifecasestudyonaheterogeneousfleet of 45 aircraft, the heuristic is more than 30% faster than an ex-act method,while the solution gapis smaller than 0.1%. Forthe most restricted test case, the solution from the heuristic is only 5%worsethanthesolutionobtainedfromtheexactmethod,while beingmuchfaster.ThecomputationtimeofTAPisessentialinthe aircraftmaintenancedomainsincechangestothepriority/urgency of existing tasks or new (non-routine) tasks can requirerunning theproposedconstructiveheuristicmanytimesperday.Therefore, an algorithm that runsin a reasonableand stablecomputational time, regardless of how restrictive is the problem, is something veryuseful.

Duringthecasestudy,wearetoldthatsomeairlinetechnicians workjustpart-timeatthehangar,andweoverestimatedthe main-tenance capacityif we set the productivitylabor hours to 8 (all techniciansareworkingfulltime,8hoursaday).Themaintenance capacityconstraint(5) isnot themain restrictionduringthetask allocation process. Since there isno other data to support sensi-tivityanalysis, we changetheproductivitylaborhours totestthe proposedheuristicinamoreconstrainedcontext.

The research presented in this paper is also one of the re-quirementsfromtheairline,continuingtheworkofaircraft

main-tenance check scheduling optimization described in Deng et al. (2020) . The dynamic programming based methodology in Deng et al. (2020) first determines theoptimalstart dates ofall main-tenance check for the entire fleet, and the optimalmaintenance checkscheduleindicates inwhichchecksamaintenance taskcan beallocated.Otherwise,withoutamaintenancecheckschedule,it is very time-consumingto know when,which aircraft,and what maintenancetasks shouldbe performed.The resultsofthe main-tenancetaskallocationarethetaskexecutionplansforall mainte-nancechecks,whichhelpthetechnicianstoexecutetherighttask, ontherightaircraft,attherighttime.

We structure the task allocation problem of aircraft mainte-nance as a bin packing problem (BPP) so that it can be solved quickly using the worst-fit decreasing algorithm. Whenever un-scheduledmaintenancetasks occur,we canusethedynamic pro-grammingbased methodologypresented inDeng et al. (2020) to obtainanewmaintenancecheckschedule,andthenapplythetask allocationframeworktoupdatethetasksaccordingly.Thetask al-locationframeworkissuitableforreal-lifeapplications.Itcan pro-vide near-optimalsolutions to the TAP,significantly reducing the workloadcurrentlyrequiredinpracticeforthecreationof mainte-nanceplans. Besides,given that itrunsin minutes,it can poten-tiallybeusedtodynamicallyadjustthetaskallocationplansgiven flightscheduledisruptionsduringoperationsoremergencyof un-scheduledtasksduringthe executionof maintenanceinspections. Furthermore, the task allocation framework can be used to test oranalyze differentmaintenance concepts orpolicies,as demon-stratedinSection 5.2 .

Futureresearchonthisworkmayconsiderthestochasticity as-sociatedwiththeTAPproblem,orexplore theuncertaintyrelated to, e.g., theemerge of “non-routinetasks” or the aircraft utiliza-tionovertheplanninghorizon.Andthiscouldenhancethe robust-nessoftheoutcomingtaskexecutionplan.Furthermore,a stochas-tic approach could extend the current work to consider health prognosticsanddiagnostics,investigating thepossibility of incor-poratingcondition-basedmaintenanceintheproposedframework. An alternativeinteresting futureresearch directionis tointegrate themaintenancecheckscheduleoptimizerwiththetaskallocation frameworkproposed.Andthiscouldimprovetheoverallqualityof themaintenance plan,including checksscheduleandtask alloca-tionpercheck.

Acknowledgments

This research work is part of AIRMES project, which received fundingfrom the CleanSky 2 JointUndertaking under the Euro-pean Union’s Horizon 2020 research and innovation programme undergrant agreement No 681858.We would liketo show grat-itude to the associated AIRMES partners for providing their air-craftmaintenance dataandvalidationof theproposed methodol-ogyandits practical relevance. Formore informationpleasevisit

www.airmes-project.eu .

References

Ackert, S. P. (2010). Basics of aircraft maintenance programs for financiers. http://aircraftmonitor.com/uploads/1/5/9/9/15993320/basics _ of _ aircraft _ maintenance _ programs _ for _ financiers _ v1.pdf . (Accessed on June 6, 2020) http://aircraftmonitor.com/uploads/1/5/9/9/15993320/basics _ of _ aircraft _ maintenance _ programs _ for _ financiers _ v1.pdf .

Ackert, S. P. (2018). Aircraft Maintenance Handbook for Financiers. http: //www.aircraftmonitor.com/uploads/1/5/9/9/15993320/aircraft _ mx _ handbook _ for _ financiers _ v1.pdf . (Accessed on June 6, 2020) http://www.aircraftmonitor. com/uploads/1/5/9/9/15993320/aircraft _ mx _ handbook _ for _ financiers _ v1.pdf . Correia, I., Gouveia, L., & da Gama, F. S. (2008). Solving the variable size bin pack-

ing problem with discretized formulations. Computers & Operations Research, 35 , 2103–2113. https://doi.org/10.1016/j.cor.2006.10.014 .

Csirik, J. (1989). An on-line algorithm for variable-sized bin packing. Acta Informat- ica, 26 , 697–709. https://doi.org/10.10 07/BF0 0289157 .

Deng, Q., Santos, B. F., & Curran, R. (2020). A practical dynamic programming based methodology for aircraft maintenance check scheduling optimization. European Journal of Operational Research, 281 , 256–273. https://doi.org/10.1016/j.ejor.2019. 08.025 .

Fazi, S., van Woensel, T., & Fransoo, J. C. (2012). A Stochastic Variable Size Bin Packing Problem with Time Constraints. https://pure.tue.nl/ws/files/3678899/ 565767590859814.pdf . (Accessed on June 6, 2020) https://pure.tue.nl/ws/files/ 3678899/565767590859814.pdf .

Friesen, D. K., & Langston, M. A. (1986). Variable sized bin packing. SIAM Journal on Computing, 15 (1), 222–230. https://doi.org/10.1137/0215016 .

Garey, M. R. , & Johnson, D. S. (1990). Computers and intractability; a guide to the

theory of NP-Completeness . USA: W. H. Freeman & Co. .

Haouari, M., & Serairi, M. (2009). Heuristics for the variable sized bin-packing prob- lem. Computers & Operations Research, 36 (10), 2877–2884. https://doi.org/10. 1016/j.cor.2008.12.016 .

Hölzel, N., Schröder, C., Schilling, T., & Gollnick, V. (2012). A Maintenance Packaging and Scheduling Optimization Method for Future Aircraft. In Air transport and operations symposium . https://doi.org/10.3233/978- 1- 61499- 119- 9- 343 .

Johnson, D. S. (1972). Fast allocation algorithms. In Proceedings of the 13th annual symposium on switching and automata theory . https://doi.org/10.1109/SWAT.1972. 4 .

Johnson, D. S. (1974). Fast algorithms for bin packing. Journal of Computer and System Science, 8 , 272–314. https://doi.org/10.1016/S0 022-0 0 0 0(74)80 026-7 .

Kang, J., & Park, S. (2003). Algorithms for the variable sized bin packing problem. European Journal of Operational Research, 147 , 365–372. https://doi.org/10.1016/ S0377-2217(02)00247-3 .

Li, H., Zuo, H., Lei, D., Liang, K., & Lu, T. (2015). Optimal combination of aircraft maintenance tasks by a novel simplex optimization method. Mathematical Prob- lems in Engineering . https://doi.org/10.1155/2015/169310 .

Muchiri, A. K., & Smit, K. (2009). Application of maintenance interval de-escalation in base maintenance planning optimization. Enterprise Risk Management, 1 (2). https://doi.org/10.5296/erm.v1i2.179 .

Steiner, A. (2006). A heuristic method for aircraft maintenance scheduling under various constraints. In Proceedings of the 6th swiss transport research conference . Van Buskirk, C. , Dawant, B. , Karsai, G. , Sprinkle, J. , Szokoli, G. , Suwanmongkol, K. , & Currer, R. (2002). Computer-Aided Aircraft Maintenance Scheduling. Technical