Optimizing Intratheater Military Airlift in Iraq and Afghanistan - index

18 Pages·2013·1.15 MB·English

ABSTRACT withintheirpowerstoreducethenumberof
ground convoys and thus reduce exposure
TheAirTaskingandEfficiencyModel ofpersonneltotheever-increasingnumber
(ATEM)hasbeenusedsince2006.Its ofimprovisedexplosivedevices.Thedirec- Optimizing
development was motivated by an torofCDDOCpressedhisstaff,‘‘Wecannot
urgentneedtoplanandevaluateintratheater fly everything, but we need to fly every- Intratheater
airlift of passengers and palletized freight thingwecan.’’
for Operation Iraqi Freedom in Iraq and The United States Transportation Com- Military
Operation Enduring Freedom in Afghani- mand divides intratheater air routes into
stan.ATEMplansroutesandaircraftconfig- two broad categories, frequency channel
Airlift in
urations (capacity of passenger seats and routes and requirements channel routes
pallet positions)for aheterogeneous fleetof (United States Transportation Command,
Iraq and
aircraft flying between multiple airfields. 2005). Frequency channel routes are pub-
ATEMrespectslimitsoncrewdutyperiods, lished in a schedule, much like an airline,
timesandabilities ofeachairfield to handle based on anticipated demands for a given Afghanistan
andfueleachaircrafttype,andaircraftspeed time period. Requirements channel routes
and carrying capacity. Initially, ATEM ad- are decided each day, and move emergent
vised improving daily and weekly route demand as necessary. Author Brau was
Dr. Gerald G. Brown,
ensembles,conveyingmorepassengersand stationed at Camp Arifjan in October 2005,
pallets and using fewer aircraft than prior and at that time he found that air planners Dr. W. Matthew Carlyle,
manually generated solutions. This early were manually scheduling CENTCOM and Dr. Robert F. Dell
usereducedtherequirednumberofground intratheater airlift using basic tools such as
convoys and thereby exposure to impro- whiteboards and simple Microsoft Excel
NavalPostgraduateSchool
visedexplosivedevices.Later,ATEMadvised spreadsheetstokeeptrackofassetsandma-
[email protected],
where to advantageously move aircraft to teriel. He immediately started working on
[email protected],
new home airfields, how to shift aircraft an optimization model to assist planners in
[email protected]
betweentheaters,andwhentobringaircraft creating the requirements channel routes
homefromwar. and prescribing air and ground movement
John W. Brau, Jr.
of passengers (PAX) and air freight pallets
ThatIhavehoistedsailtoallthewinds,
(PALS), hereafter referred to collectively as
which should transport me farthest
cargo. He contacted the other authors soon USTRANSCOM/TCAC-D
fromyoursight.
afterhestartedthiseffort,andbyDecember [email protected]
Shakespeare,Sonnet
2005 our team had begun development of
adecisionsupporttoolcalledtheAirTrans-
portation and Efficiency Model (ATEM) for
INTRODUCTION quicklycreatingrequirementschannelroutes,
tohelpclearbackloggedcargo,andtodesign
The logistics of transporting, arming, high-qualityweeklyfrequencychannelroutes
feeding,clothing,sheltering,andfuelinghun- for future demands. The solutions provided
dreds of thousands of personnel involved by ATEM maximize the flow of PAX and
with our military operations in Iraq and PALSonintratheaterairplanes:byanalyzing
Afghanistan has been a daunting challenge daily operations data, we realized this pre-
for US Central Command (CENTCOM), sented the greatest opportunity to make a
theunifiedcombatantcommandwhosearea significantnear-termimprovement.
of responsibility includes those countries. ATEMwouldbeoflittleutilitywithout
The CENTCOM Deployment and Distribu- an interface making it easy for nonanalyst
tion Operations Center (CDDOC) at Camp plannerstounderstandanduse.So,wede-
Arifjan, Kuwait, receives specific demand veloped a portable, laptop-based graphical
signalsandmanageswhatisessentiallythe user interface using Microsoft Excel and APPLICATIONAREAS:
last echelon in a world-wide supply chain. Visual Basic (Microsoft, 2012a,b), and a Airlift,VehicleRouting,
This echelon conducts intratheater move- mathematical modeling suite including ConvoyMitigation,
ment of materiel, using convoys of trucks the General Algebraic Modeling System IEDMitigation,
and,whenpossible,airlift,tomoveperson- (http://www.gams.com) and several com- DecisionSupport
nelandtheirequipmenttowheretheyare mercial optimization packages. A trip to
ORMETHODS:
neededinsupportofmilitaryoperations. Kuwaitbytwoofourco-authorsconfirmed
Optimization,
In2005theCENTCOMcommanderdi- that such a decision support tool would be
Spreadsheets,Heuristics
rectedUSmilitaryleaderstodoeverything asignificantimprovementinbothfrequency
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OPTIMIZINGINTRATHEATERMILITARYAIRLIFTINIRAQANDAFGHANISTAN
andrequirementschannelplanning,andhelped ofStaff2000).Strategicdistribution,themove-
solidifythestructureofthemodelbasedonour ment of forces from Continental US (CONUS)
direct experience with air logistics in Iraq and bases into a theater of operations, flows along
onourdiscussionswithplannersintheatre. intertheaterdistributionchannels.Operational
distribution, the movement of materiel within
aCOCOMtheater,flowsalongintratheaterlogis-
MILITARYAIRLIFTPLANNINGTOOLS tics channels. VRPs are inherent in both inter-
theater and intratheater distribution, and the
Our intratheater distribution problem is Department of Defense employs many tools to
a variant of the vehicle routing problem (VRP) solveboththeseproblems.
(Dantzig and Ramser, 1959). The objective of The major strategic distribution models in
VRPistodesignasetofroutesforagivenfleet use when we started ATEM and still in use to-
of vehicles that satisfies customer demands at day are the Global Deployment Analysis Sys-
dispersed locations from one or more depots, tem (GDAS), Joint Flow and Analysis System
atlowestcost.Sideconstraintsaddedtotheba- for Transportation (JFAST), Model for Inter-
sic VRP create variants of the problem. Typical theater Deployment by Air and Sea (MIDAS),
variantsincludethecapacitatedvehiclerouting the Mobility Simulation Model (MobSim), and
problem, wherevehicles have limited capacity, theConsolidatedAirMobilityPlanningSystem
the vehicle routing problem with pickups and (CAMPS). Barnes and McKinzie (2004) survey
deliveries, where customers can both receive thesesystems.Inadditiontothese,othersimu-
material from and return material to a depot, lation models in use today include the Air
the vehicle routing problem with time win- Mobility Operations Simulation (AMOS) and
dows,wherevisits tocustomers mustbemade theAnalysisofMobilityPlatform(AMP).
withinaspecifiedtimeepoch,andtheshuttling GDAS uses alternating greedy cargo and
problem,whereeachcustomerisbothanorigin greedy vehicle heuristics combined with route
and a destination for shipments from and to insertiontoassignpassengerstomodesoftrans-
other customers, and the dial-a-ride problem, portation (air, ground, and sea) for transport
where customer pickups and drop-offs are between points of CONUS embarkation to
merged into a set of vehicle routes. See Toth pointsofdebarkationinsomeareaofoperations.
and Vigo (2002) for a survey of VRP problems JFAST employs local search to plan strategic
andsolutiontechniques. lift from CONUS to some deployment area of
We find intratheater airlift as required in operations(seeKoprowski[2005],andhisrefer-
Operation Iraqi Freedom (OIF) and Operation ences for a summary). MIDAS is a simulation
Enduring Freedom (OEF) to be a unique VRP. thatrepresentsstrategicdeploymentofunitper-
Feillet, Dejax, and Gendreau (2005) present the sonnel and equipment by air and sealift into
profitable arc tour problem that shares much theater ports of debarkation (Military Surface
incommon,butinadditiontothefeaturesthey Deployment and Distribution Transportation
describewealsomustplanformultipledepots, EngineeringAgencyCommand,2005).MobSim
multiple (here, two) types of cargo capacity on is a network-based discrete event stochastic
eachvehicle,andalimitedheterogeneousvehi- model.CAMPS,thereplacementforbothADANS
clefleet. (Air Mobility Command Deployment Analysis
JointPub3-0(JointChiefsofStaff2001)de- System) and CMARPS (Combined Mating and
fines three levels of war: strategic, operational, Ranging Planning System), is a heuristic plan-
andtactical.Militarylogisticsmustenableoper- ning tool (Becker et al., 2004). AMOS is a rule-
ations at all three levels. The Joint Staff and based, discrete-event simulation of worldwide
service staffs concentrate on strategic logistics airlift with the ability to model detailed air to
matters. Combatant Commanders (COCOM) airrefuelingandairfieldcongestion(seeMason
link strategic- and operational-level logistics. [2009]andWuandPowell[2009]).AMPisasim-
Finally, subordinate commanders blend oper- ulation that includes MIDAS and other legacy
ational and tactical logistics to accomplish simulation models (Raytheon, 2013). Noel and
missionsassignedbytheCOCOM(JointChiefs Stratton (2012) catalog the portfolio of more
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OPTIMIZINGINTRATHEATERMILITARYAIRLIFTINIRAQANDAFGHANISTAN
than 40 models (including those referenced be served by each route and what capacity, in
above)thathavehadsomeuseatAirMobility termsofPAXandPALS,itcanliftoneachflight
Command. in that route: any cargo from a demand line
Intratheaterdistributiontoolsarelesscom- whose origin precedes its destination on that
mon. Barnes et al. (2004) provide approximate routecouldconceivablybecarriedbyanaircraft
solutions to an intratheater VRP with a tabu flyingthatroute,althoughitmightrequiresev-
searchmetaheuristic,butweknowofnoinstal- eral flight legs to get there. ATEM allows such
lation of their product. They claim that, at the loading of cargo and carrying it through one
timeofpublicationoftheirpaper,allotherthe- ormoreintermediateairfieldsbeforeunloading
ater distribution tools were simulation-based. it at its destination (planners call this through-
These include ELIST (Enhanced Logistics Intra- put); the number of legs per load is controlled
theater Support Tool), TRANS (Transportation by a single parameter, maxloadhops, (where the
Resource Assessment Network Simulator), and numberofhopsaparticularpieceofcargomakes
MASS (Mobility Analysis Support System). isjustthenumberofconsecutiveflightlegsbe-
Transway(Pohl2006)usesatabusearchmeta- tween its origin and destination), and setting
heuristic to help plan intertheater (and intra- thisparametertoonepreventsanythroughput
theater) distribution. Burks et al. (2010) also (i.e., every passenger or pallet can travel on at
tackleintratheaterdistributionwithtabusearch. mostoneleginaroute).
We have no evidence that either of these Finally, for each airport ATEM needs to
products were ever used by the Department knowthemaximumnumberoflandingsallowed
of Defense. eachday,andthenumberofaircraftofeachtype
thatarebasedatthatairport.Givenanenumera-
tion of every admissible route for each aircraft
ATEM INTEGER LINEAR PROGRAM type, configuration, and home airport, ATEM
FORMULATION seeks an optimal assignment of a single route
toeachaircraftandanallocationofcargotothose
Wepresentanintegerlinearprogram(ILP) routes that maximizes (prioritized) PALS and
formulationofATEM.ATEMtakesasinputalist PAX conveyed. It may not always be possible
of airport pairs and the number of PAX and to fill each flight on a route with cargo. For in-
PALS traveling between them; each of these stance,anemptyflight(adeadhead)maybeneces-
pairs and associated demands is known as sarytorepositionanairplane,ortobringitback
ademandline,becauseitisrepresentedbyasin- homeattheendofitsroute.
gle line on a demand spreadsheet used by the
planners. Indices [;cardinality]
ATEM also takes as input a (typically very
large) list of routes, each of which is specific to a 2A setofairfields(aliasa,a)½;20(cid:2)
i j
anairplanetypeandtoaparticularconfiguration c 2C set of cargo types (passengers
of that airplane. A route for one airplane on andcargopallets)½;2(cid:2)
one day consists of a sequence of flight legs, p 2P setofairplanetypes½;5(cid:2)
where each leg is a single nonstop flight be- p 2P setofairplanesoftypepbased
ai
tween an origin and a destination airfield. The atairfielda
i
firstlegdepartsfromtheairplane’shomeairfield, g 2G set of configurations for air-
p
and the last leg returns to this same home air- planetypep½;5(cid:2)
(cid:2) (cid:3)
field. Each route consists of a feasible sequence a;a 2L set of possible flight legs or
i j
offlightlegs,inthattheassociatedaircraftcan, segments(airportofembarka-
in one day, make all of the flights on the route tion(APOE),airportofdebar-
inthesequencegivenwithoutviolatinganyop- kation(APOD)pairs)½;100(cid:2)
erational restriction or resource requirement r 2R setofroutesforairplanetypep
pai
(suchasfuelandtotalflighttime). startingatairfielda½#100;000(cid:2)
This route and configuration data allows s 2f1;2;.;Sg ordinalofstoponaroute(alias
ATEM to determine which demand lines can s#,so)[;7]
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OPTIMIZINGINTRATHEATERMILITARYAIRLIFTINIRAQANDAFGHANISTAN
Data X
LOADrss#c#caprcSELECTr
len numberofstopsonrouter ðs;s#Þ2Hr:
r s#so,s#
aðr;sÞ airfieldatstopnumbersonroute
r,where1#s#len "r;1#so,lenr;c (4)
r
carry amountofcargoconleg(cid:2)a;a(cid:3)that X
aiajcr canbeairliftedbyrouter i j SELECTr#landingaj "aj 2A (5)
(cid:2) (cid:3) r;s#:
cdpemaiaajicajc dperimoraintydoufncaitrsgooncolengle(cid:2)gai;aaij;(cid:3)ajfor 1a,ðrs;#s##Þ5lenajr;
cargoc
caprc capacityonrouterofcargotypec The objective function (1) calculates the
landing maximumnumberoflandings in priority-weighted value of all cargo conveyed
a
airfielda on all routes. For each airfield and each air-
planespa numberofairplanetypepstarting plane type based there, constraint (2) limits
fromairfielda the number of routes selected to the number
maxloadhops the maximum number of consecu- of airplanes available. For each demand line
tive legs the cargo from a single and cargo type, constraint (3) prevents air-
demandlinecantravelononeroute planes from lifting more that the demand
availableandaccountsforanyextraliftcapac-
ityavailable.Foreachroute,andeachstopon
Calculated Sets
thatroutethatisnotthelaststop,andeachcargo
ðs;s#Þ2H stop s’ comes after, and within type, constraint (4) limits loading on the flight
r
departing that stop by the airplane capacity for
maxloadhops of, stop s on route r:
1#s,s##len ,s#2s#maxloadhops each cargo type, for the configuration used on
r
thatroute.Foreachairfield,constraint(5)limits
totaldailylandingsbyallroutesselected;note
Variables that each route may land at an airfield more
thanonce,andmaycarryportionsofsomede-
SELECTr integer number of times airplane mandlineontwoormorelegs.
typep(r)isflownonrouter Inthespecialcasewheremaxloadhops¼1,
LOADrss#c integer cargo units airlifted by air- thecapacityavailableoneachflightlegonaroute
plane type p(r) flying route r from isjustthecapacityoftheaircraftassignedtothe
APOE airfield s to APOD airfield route,becausetheaircraftiscompletelyemptied
s#ofcargotypec at each stop on the route. We can simplify the
model and eliminate many constraints and all
oftheLOAD variablesby defining anewset of
ATEM-ILP Formulation
variables, called EXTRA, that measure the un-
usedcapacityoneachlegofeachrouteselected.
X
Maximize cpaðr;sÞaðr;s#ÞcLOADrss#c (1) The ‘‘Solving ATEM’’ section of this paper
r;ðs;s#Þ2Hr;c more fully discusses the benefits of this
Subjectto: simplification.
X
SELECTr#planespai "ai 2A;p2Pai (2) New Variables
r2Rpai
X EXTRA unusedcapacityofselectedroutes
LOADrss#c#demaiajc "ðai;ajÞ2L;c forcargoc,oaniajcleg(cid:2)a;a(cid:3)
r;s;s#: i j
ðs;s#Þ2Hr; With these variables accounting for the overall
aðr;sÞ5ai; loadingofeachlegontheroutewecanremove
aðr;s#Þ5aj theLOADvariablesandreplaceconstraints(3)
(3) and(4)with:
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OPTIMIZINGINTRATHEATERMILITARYAIRLIFTINIRAQANDAFGHANISTAN
Figure1. MapofIraqindicatingasampleofairfieldlocationsbydots.AlUdeid(located350nauticalmilestothesouth-
east)inQatar,andIncirlik(located325nauticalmilestothenorthwest)inTurkey,arenotshown.(AfterNationalIm-
agingandMappingAgency,2003;http://www.lib.utexas.edu/maps/middle_east_and_asia/iraq_planning_2003.jpg)
Xcarry SELECT # dem which calculates the total carrying capacity of
aiajcr r aiajc eachroute,andthensubtractsanyunusedcapac-
r
1EXTRAaiajc "ðai;ajÞ2L;c; (6) itytoarriveatthetotalamountofcargolifted.
which explicitly accounts for unused space on ATEMalsoallows,foreachaircraft,theselec-
each flight in a route, and replace the objective tionofadifferentrouteoneachdayofamultiday
functionwith: planninghorizon(e.g.,oneweek).Forsimplicity
X ofexposition,wehavenotincludedadayindex,
Maximize cp carry SELECT
aiajc aiajcr r althoughoneexistsinATEM.However,because
Xðai;ajÞ2L;c;r airplanesreturntotheirhomeairbaseaftereach
2 cp EXTRA ; (7)
aiajc aiajc day offlying, the days in a plan areonly linked
ðai;ajÞ2L;c bythecargoremainingtobelifted.
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OPTIMIZINGINTRATHEATERMILITARYAIRLIFTINIRAQANDAFGHANISTAN
CENTCOM INTRATHEATER AIRLIFT Table1. Exampleofdemandlines,eachwithanor-
igin and destination airfield, a number of PAX and
Weconsiderthefrequencyandrequirement PALStocarry,andapriority.Thereare126passengers
channel route selection problems faced by the andonepallettoflyfromAlSalemtoAlasad,withpri-
CENTCOM Combined Air Operations Center orityone.Bycontrast,the44passengersandonepallet
(CAOC), located at Al Udeid Air Force Base, toflyfromAlSalemtoTallilhavepriority100,signi-
fyingeachofthesePAXandPALSis100timesmore
nearDoha,Qatarin2005and2006.
importanttocarrythanapriority1demand.Priority
isasimplewaytohighlightfrustratedcargo.Thisde-
Airfield Locations and Demand Data
mandmaybeforaday,aweek,orsomelongerplan-
ningepoch.
Weusedabout20airfieldstosupportintra-
theaterairdistributioninIraq,includingseveral APOE AOPD PAX PALS PRIORITY
outsidethecountry;Figure1showslocationsof
AlSalem Alasad 126 1 1
someofthesefordistributioninIraq.Table1pro-
AlSalem AlTaq 159 1 1
videsanexcerptfromalongersheetofdemand
AlSalem Balad 108 5 1
lines. Each demand line expresses the number
AlSalem Baghdad 273 3 1
of PALS and PAX that need to be transported AlSalem Mosul 47 4 1
between an APOE and an APOD. A standard AlSalem Kirkuk 33 2 1
‘‘463L’’ pallet is 88 inches long by 108 inches AlSalem QWest 5 3 1
wide, and each carries a maximum of 10,000 AlSalem AlSahra 51 1 1
pounds(DepartmentoftheArmy,1993).Apas- AlSalem Tallafar 52 2 1
sengerrepresentsasinglepersonandpersonal AlSalem Tallil 44 1 100
gear. AlSalem AlUdeid 113 8 1
Alasad AlSalem 117 1 1
Ouroverallobjectiveistomaximizethepri-
Alasad AlTaq 17 3 1
oritizednumberofPAXandPALSlifted,where
Alasad Balad 7 2 1
some PAX and PALS have different priority
Alasad AlUdeid 5 2 1
than others (e.g., delayed, or frustrated, cargo
AlTaq AlSalem 135 6 1
might be given a higher priority to encourage AlTaq Kuwait 1 7 1
clearing out the backlog). Each cargo type and AlTaq Alasad 14 17 1
eachindividualdemandlineisassignedapri- AlTaq Balad 2 2 1
ority.ATEMmultiplieseachdemandbyitscargo AlTaq Baghdad 2 1 1
type priority and by its line priority, and maxi- AlTaq AlUdeid 3 1 1
mizes the total resulting prioritized demand Balad AlSalem 288 3 1
thatismoved.Table1isanexampleofsuchde- Balad Kuwait 4 8 1
Balad Alasad 6 2 1
mand data, including demand line priorities.
Balad AlTaq 7 5 1
Datapresentedinthistableandtheremaining
Balad Baghdad 13 7 1
tables are a sample of possible ATEM input
Balad Mosul 10 6 1
valuesprovidedbyCENTCOMplannersforun-
classifieduse.Plannerscanchangethevaluesof
theseATEMinputparametersatwillasthesitu- C-130J1,C-130J2,C-17(see Figure 4), and IL-76.
ation dictates. Figure 2 presents a summary of Each airplane type, with the exception of the
oneweekofdemandforpalletsandpassengers IL-76, can beflowninany ofanumberof alter-
inIraq. nateconfigurations,witheachconfigurationpro-
viding the passenger seat and pallet position
Airplanes and Load Configurations capacities.Table2listsconfigurationsbyairplane
type (with the exception of C-130J2) along with
Thenumberandtypesofairplanesavailable associatedpassengerandpalletcapacities.
fordistributionalsovaries,butusuallyconsisted Airplane and aircrew endurance limit the
whenATEMwasfirstusedofabout35airplanes lengthofaroute,asshowninTable3,whichalso
of five different types for CENTCOM intra- shows a sample of possible fuel and crew con-
theaterairlift,includingC-130E(seeFigure3), straints that limit routes. ‘‘Max #landings in
Page40 MilitaryOperationsResearch,V18N32013
OPTIMIZINGINTRATHEATERMILITARYAIRLIFTINIRAQANDAFGHANISTAN
Figure 2. PALS and PAX demand for a seven-day Iraq scenario. Pallet demands are depicted in the upper,
darkerbarsegments.Thereareabout20x20¼400airfieldpairs,butonly85ofthesehavedemandlineshere.De-
mandlinesvarywidelyinmagnitude,andincludesomepure-passengerandsomepure-palletdemands.Airfield
pairsaresuppressedforbrevity.
box before recrew’’ represents the number of airplanehomeairfieldsandthenumbersofair-
landings into a combat area, where approach planesbasedateach.Flyingtimesbetweenair-
anddepartureinvolveevasivetacticalmaneu- fields are given for each airplane type. Other
vers, and may be conducted at night without inputincludesthenumberoflandingspermit-
lighting.Inadditiontotheseconstraints,plan- tedateachairfieldperday,aswellastheability
nersalsolimitthemaximumnumberofstops of each airfield to accommodate each airplane
anairplanecanmakeduringaroute.Forexam- type.
ple, a C-17, which is nominally based outside
Iraq, can make no more than three landings in
Iraq before it needs to return to its home base. ROUTE GENERATION
However,itcanrecrew,andreturntolandinIraq
atmost threemore times before returningtoits Givendemanddata,afleetofairplaneswith
homeairfield. theirhomeairfields,and a timehorizon,ATEM
prescribes a configuration and a route for each
Airfield Capabilities airplaneoneachdayintheplanninghorizonthat
movesasmuchcargoaspossible.
Table 4 is a sample of ground times at air- Acompleteflightensemblespecifiesaroute
fieldsbyairplanetype.Similarly,Table5samples for each airplane on each day in the planning
airfieldrefueltimesbyairplanetype.Table6lists horizon. We have discovered, by hand, that
Figure3. C-130EatAlUdeid.Thetailrampispositionedtotransfercargopalletsbyforkliftorroll-on,roll-off
k-loadervehicle.TheC-130hasasingleroller-railtrackformanuallypushingcargopalletsforward,oraft,oroff.
Forscale,authorDellisstandingundertheinboardstarboardengine.Intheimageontheright,anaircrewmem-
berfoldsuphammockseatsinaconfigurationchangetomakeroomforanothercargopallet.
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OPTIMIZINGINTRATHEATERMILITARYAIRLIFTINIRAQANDAFGHANISTAN
Figure4. C-17atAlUdeid.Theaftk-loadervehiclesarereceivingrowsofpassengerseatspushedoff(andaban-
doneduntilalaterevolutionhere)inaconfigurationchangetoaccommodatemorecargopallets.TheC-17has
two,parallelroller-railtracksformanuallypushingcargopalletsforward,oraft.Forscale,ontheright,authors
BrauandDellappearstandingonthetailramp,workingthroughthedetailsofloadconfigurationchangeswith
theairplane’sloadmaster.
conjuring these routes while synchronously Givenaflighttimet . 0foreacharc(i,j)in
ij
determiningtheirjointfeasibilityandeffective- A,andathresholdvalueT[MAXROUTEHRS,
nessisthecruxofsolvingthisproblem.Manual the pseudocode in Figure 5 enumerates the fi-
planning experience suggests thumb rules that nitenumberofs-tpathsoflengthboundedbyT.
canbeagoodguidehere,butnomanualplanner Forsimplicityofexposition,wedonotclut-
can be expected to solve this complex problem terthisalgorithmwiththeadmissibilitytestsfor
everydaywithanythingapproachingoptimality. refuelingrequirementsalongtheroute,themin-
Thereareanumberofwell-knownwaystoauto- imumandmaximumroutelengths,orthemax-
mate route construction. In this case, there are imum number of flights (hops) per route (see
few enough airfields, and few enough demand Table 3 for examples of these data). However,
lines, and each route can include at most few thesetestsareeasilyincludedatthesamepoint
enoughhops,thatwecanenumerateeveryfeasi- inthealgorithm asthetestfor totalroutetime.
bleroute,andthusnotworryaboutmissingany Although in general the number of paths in a
good one. For each airplane and its origin air- single graph can be exponential in the number
field,ATEMsuccessivelyconstructseverypossi- ofnodesinthegraph,fortheOIFandOEFcases
ble permutation of airfield-hops that can be wehaveexaminedwecanaccommodateexhaus-
visitedinonedutyperiod,endingwithareturn tive enumeration. With data similarto thatpre-
totheoriginairfield.Simplerules,suchasnever sented here, we generate on the order of a few
flyingtwodeadheadlegsinarow,helplimitthe hundredthousandfeasibleroutes.
number of routes. Additional rules eliminate Filteringroutestolimittheminimumnum-
routeslackingarequiredrefuelorcrewreplace- berof hops,the maximum numberof landings
mentevent. ‘‘in the box,’’ and so forth, is a simple matter
WedefineadirectedgraphG¼(N,A)with ofkeepingtrackofthesenumbers,suppressing
node set N defined by a start node s, an end thecompleteroutesthatdonotmeetthelower
node t (here, s ¼ t), and an intermediate eche- thresholdsandfathomingenumerationofpar-
lon of nodes for every stop on a route up to tial paths as soon as they exceed the upper
maxroutehops-1.Forspecificity,eachofthesein- thresholds.
termediatenodesisnamed(airfield,hopnum- Routes are generated separately for each
ber).WespecifyadirectedarcinAconnecting airplanetypeandeachorigin homeairfield for
eachnodewith‘‘hopnumber’’witheveryother that airplane type. Any route longer than the
nodewith‘‘hopnumber11’’towhichwemight given maximum time before refueling for that
flydirectly. airplanetyperequiresa(single)stopatarefueling
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OPTIMIZINGINTRATHEATERMILITARYAIRLIFTINIRAQANDAFGHANISTAN
Table2. Loadconfigurationsbyairplanetypewith ensemblecouldbeusedasaninitialincumbent
passengerandpalletcapacities.AC-130Ehaspassen- for a local search heuristic. However, we seek
gerhammockseatsthatcanbefoldedupormovedout a certificate of optimality for our solutions, so
ofthewaytomakeroomforcargopalletstobepushed weusetheintegerlinearprograminthe‘‘ATEM
aboardonarollerrailandcancarrybetweenzeroand
integer linear program formulation’’ section to
fivecargopalletswitheachadditionalpalletconfigura-
chooseanoptimalsetofroutes(outofallfeasible
tionreducingpassengercapacityindiscretestepsfrom
routes)andtoestablishanupperboundonhow
72 down to nine. Configuration changes don’t take
muchadditionalcargocouldhavebeenmoved.
long,andsomecanbecompletedin-flightinanticipa-
tionofthenextload.Bycontrast,theC-17hastwopar-
allel cargo pallet roller rails, and some configuration
changesrequirerowsofpassengerseatsmountedon SOLVING ATEM
theserailstoberolledoffandleftonairfield,tobere-
coveredlater.Theseconfigurationchangesare better The ATEM integer linear programming
assisted by special ground vehicles called k-loaders. model has hundreds of thousands of variables
The IL-76 is chartered to carry pallets only. (Not andconstraintsfortheoptimizationmodelwith
listed:C-130J2.)
throughput,andtensofthousandsofconstraints
Airplane Passenger Pallet for the reformulation without. If throughput is
Configuration type capacity capacity notallowedwecansolveallofourproblemin-
stances using, e.g., GAMS/CPLEX [2012]. If we
Acp2 C-130E 9 5 allow throughput (i.e., maxloadhops . 1), then
Acp3 C-130E 25 4
theintegerprogramsbecomesignificantlylarger
Acp4 C-130E 42 3
duetothequadraticdependenceonmaxloadhops
Acp5 C-130E 56 2
of the number of LOAD variables. Some cases
ap1 C-130E 72 0
canstillbesolvedwithCPLEX,butwefindthat
jac2 C-130J1 0 8
Jacp2 C-130J1 18 7 theexecutiontimesaremuchlongerthaninthe
Jacp3 C-130J1 34 6 case of no throughput. Schedulers do not have
Jacp4 C-130J1 50 5 immediate access to commercial solvers, so we
Jacp5 C-130J1 60 4 developed a companion heuristic algorithm to
Jacp6 C-130J1 74 3 provide good solutions to ATEM in a very
Jacp7 C-130J1 94 2 short runtime. Our basic heuristic consists of
jap1 C-130J1 115 1
data preprocessing followed by greedy selec-
C17c C-17 0 18
tion of routes. The preprocessing reduces de-
C17cp C-17 54 11
mand lines that appear on routes that may
C17p C-17 189 4
have been fixed by the planner. Aftergenerat-
Ilc IL-76 0 9
ing all of the routes a greedy heuristic makes
a single pass through the list of aircraft (and
their home cities) and for each of those and
airfieldbetweenminimumtimebeforerefueland foreachdayintheplanninghorizonitchooses
maximumtimebeforerefuel.Similarly,foranair- theroutethatmaximizesliftofremainingcargo
plane type with a crew replacement restriction (withoutthroughput)onthatroute,assignsthat
(i.e.,maximumnumberoflandingsintheboxbe- route to that aircraft on that day, and then
forecrewreplacementisasmallinteger,suchas removes that lifted cargo from those demand
three)anyroutemustreturntoitsoriginairfield lines.IncontrastwiththeILP,theheuristicdoes
beforethenumberoflandingsatairfields‘‘inthe not consider throughput. Bridges [2006] de-
box’’followingthelastlandingattheoriginair- scribes two additional iterative improvement
fieldexceedsthislimit.Routesviolatingeitherre- stepstoimproveagreedysolution.Healsocon-
quirementaresuppressed. siders generalizing relaxations including air-
Wefindthatselectingafeasibleensembleof plane configuration changes during a route,
routes that moves a large amount of cargo is andthroughput.
relatively straightforward once we have enu- The results worksheet (Figure 6) serves as
merated all the feasible routes. This feasible bothanoutputdisplayandanimportantinput.
MMiilliittaarryyOOppeerraattiioonnssRReesseeaarrcchh,,VV1188NN3322001133 PPaaggee4433
OPTIMIZINGINTRATHEATERMILITARYAIRLIFTINIRAQANDAFGHANISTAN
Table3. Routelimitationsbyairplanetype.Limitationsincludethemaximumtimeanairplanecanoperate,time
betweenrefueling,numberoflandingsinthebox,andnumberoftimesanairplanecanreplaceaircrews.Each
routeforeachairplanetypeisalsolimitedbysomeMaxRouteHopsnumberoflandings.
Airplanetype C-17 C-130E C-130J1 C-130J2 IL-76
Maxroutetime(hrs) 12.00 9.50 9.50 9.50 12.00
Maxtimebeforerefuel(hrs) 12.00 5.00 5.00 5.00 12.00
Mintimebeforerefuel(hrs) 12.00 3.00 3.00 3.00 12.00
Max#landingsinboxbeforerecrew 3 20 20 20 20
Maxnumberrecrews 1 0 0 0 0
TheplannercanuseATEMrepeatedlytorefine EXAMPLES
a plan. The planner can designate the first col-
umnofafavoredroutewith‘‘fix’’andthenum- Wepresentfourrepresentativescenariosof
ber of days the route will be repeated by a intratheater airlift for Iraq and Afghanistan in
specificairplane.SubsequentATEMrouterevi- 2005 and 2006. The first and smallest of these
sions will honor this guidance. These distin- isasingle-dayrequirementforlogisticsdistri-
guished routes have either been automatically butiontoandwithinAfghanistan.Oursecond
generated by ATEM, or manually entered by scenario is for a seven-day planning horizon
theplanner.BecauseATEMconsidersalladmis- thatsuggestsfrequencychannelsinIraq.Next,
sibleroutes,theonlyreasonforaplannertokey wetestalarge,single-daydistributionrequire-
inarouteistoviolatesomebusinessruleATEM ment within Iraq. Finally, we show how to
wouldhonor. parametrically remove airplanes from a chan-
The routes selected worksheet in Figure 7 nel design. The ATEM-ILP integer linear pro-
will indicate a planned refueling by a prefix, gram is generated by GAMS (2012) and solved
for example, ‘‘F_Al_Salem.’’ A planned crew- withGAMS/CPLEX12.3(2012)ona3-GHzlap-
replacementeventwouldappear‘‘C_Al_Salem.’’ top,usingavalueof0.01fortherelativeoptimal-
Ifanypassengerorcargoiscarriedformorethan itycriterion (CPLEXterminates afterfindinga
onehop,thisexceptionwouldappearasaninter- solution that is guaranteed to be within 1% of
mediate line tracing the throughput in addition optimal).
tothecustomaryone-hopresultsshown.
Table5. Sampleofairfieldrefuelingtimes.Refuel-
ingisconcurrentwithgroundtime.Forexample,if
Table 4. Sample of airfield ground times. Ground aC-17requiresrefuelinginBalad,itstotalground
timevariesbyairfieldandairplanetype.Inthisexam- time is 90 minutes. All other airplanes require 75
ple,aC-17requires45minutesonthegroundatBalad, minutesonthegroundtorefuelatBalad.An‘‘n’’in-
while all other airplane types require 30 dicates that the airplane type cannot refuel at
minutesthere. thatairfield.
Groundtime(mins) Refueltime(mins)
Airfield C-17 C-130E C-130J1 C-130J2 IL-76 Airfield C-17 C-130E C-130J1 C-130J2 IL-76
Alasad 45 60 60 60 60 Alasad 90 60 60 60 60
AlTaq 45 30 30 30 30 AlTaq 90 60 60 60 60
Balad 45 30 30 30 30 Balad 90 75 75 75 75
Baghdad 45 60 60 60 60 Baghdad 90 90 90 90 90
Mosul 45 30 30 30 30 Mosul n n n n n
Kirkuk 45 45 45 45 45 Kirkuk 90 75 76 77 77
Basrah 45 45 45 45 45 Basrah n n n n n
QWest 45 45 45 45 45 QWest n n n n n
AlSahra 45 30 30 30 30 AlSahra n n n n n
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