DTIC ADA549418: Optimizing Assignment of Tomahawk Cruise Missile Missions to Firing Units

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by  Defense Technical Information Center

Optimizing Assignment of Tomahawk Cruise Missile Missions to Firing Units
AlexandraM.Newman,1RichardE.Rosenthal,2JavierSalmerón,2GeraldG.Brown,2*WilsonPrice,2
AntonRowe,2CharlesF.Fennemore,3RobertL.Taft3
1DivisionofEconomicsandBusiness,ColoradoSchoolofMines,Golden,Colorado80401
2OperationsResearchDepartment,NavalPostgraduateSchool,Monterey,California93943
3NavalSurfaceWarfareCenter,Dahlgren,Virginia22448
Received12September2008;revised6July2009;accepted4August2009
DOI10.1002/nav.20377
Publishedonline6October2009inWileyOnlineLibrary(wileyonlinelibrary.com).
Abstract: The Tomahawk land-attack cruise missile can be launched from a ship or submarine, and can deliver its warhead
preciselytoatargetatlongrange.ThereareseveralvariantsoftheTomahawkmissile,eachwithspecializedcapabilities.Foreach
TomahawkMissileSequenceNumber(MSN)task(i.e.,mission),theTomahawkmissilevariantscanberankedwithrespectto
theirabilityandcosteffectivenesstoperformthattask.AgivenlandattackstrikeordermayinvolvealargenumberofTomahawk
missilesandnumerousTomahawklaunchplatforms.Operationalplannersselect,inrealtime,feasiblelaunchplatformstoexecute
Tomahawktaskings.TheTomahawktaskinginastrikeorderincludesnotonlyprimaryassignmentsbutalsobackupassignments.On
boardeachlaunchplatform,thepreciseallocationofspecificTomahawkmissilestotheTomahawkMSNtaskassignedisoptimized
withamodeldescribedhere.Wehelpnavaloperationalplannersselect,inrealtime,appropriatefeasiblelaunchplatformstofulfill
theTomahawktaskinginastrikeorder. ©2009WileyPeriodicals,Inc.NavalResearchLogistics58:281–295,2011
Keywords: weapon-target assignment; Tomahawk; multi-objective optimization; integer programming; operations research;
militaryoperationsresearch
“Beswiftlikelightningintheexecution; operationalplannersselect,inrealtime,feasiblelaunchplat-
Andletthyblows,doublyredoubled, formstofulfillthetaskinginastrikeorder.Predesignationis
Falllikeamazingthunderonthecasque theprocessbywhichasetofoneormorespecificTomahawk
Ofthyadverseperniciousenemy” missiles,ononeormorespecificfiringplatformsismatched
RichardII,Shakespeare to the mission requirements in a “strike plan.” For a given
target(thatis,asetofaimpointstobeattackedinasingle,
coordinated action), a strike plan specifies how many and
1. INTRODUCTION whatkindofTomahawkmissilesarerequested.Inaddition,
we address specific aspects of a goal hierarchy involved in
The Tomahawk (Fig. 1) is a long-range, subsonic, land-
the coordination of a large number of Tomahawk missiles
attack missile manufactured by Raytheon. It is a tactical,
andnavalplatforms.
conventional weapon that can be launched from U.S. Navy
There is a rich and long-standing literature on missile
surface ships and submarines, and some Royal Navy sub-
assignment. For example, the early work of Bracken and
marines.Itfliesaplannedroute,deliveringitswarheadpre-
McCormick [4] presents a nonlinear approximation to a
ciselytoatargetatlongrange,ataspecifiedtime.Agiven
weapon allocation problem where the goal is to maximize
landattackstrikeordermayinvolvealargenumberofToma-
expectedinflicteddamage.Matlin[26]reviewsmuchofthe
hawkmissilesofdifferenttypes(calledTomahawkvariants)
earlywork,whichisalignedwiththeTomahawk-to-taskallo-
andnumerousTomahawklaunchplatforms.
cationcarriedoutbytheTacticalTomahawkWeaponControl
Inthisarticle,wedescribea“predesignation”optimization
System(TTWCS)Urioste[34].
modelandafastheuristicalgorithmtohelptheU.S.Navy’s
Weapon assignment is part of a large class of opti-
mization models known as resource allocation problems.
*Correspondenceto:G.G.Brown([email protected]) Jaiswal[19, pp.132–137,andreferencestherein]dividesthe
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1. REPORT DATE 3. DATES COVERED
JUL 2009 2. REPORT TYPE 00-00-2009 to 00-00-2009
4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER
Optimizing Assignment of Tomahawk Cruise Missile Missions to Firing
5b. GRANT NUMBER
Units
5c. PROGRAM ELEMENT NUMBER
6. AUTHOR(S) 5d. PROJECT NUMBER
5e. TASK NUMBER
5f. WORK UNIT NUMBER
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION
Naval Postgraduate School,Operations Research REPORT NUMBER
Department,Monterey,CA,93943
9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S)
11. SPONSOR/MONITOR’S REPORT
NUMBER(S)
12. DISTRIBUTION/AVAILABILITY STATEMENT
Approved for public release; distribution unlimited
13. SUPPLEMENTARY NOTES
14. ABSTRACT
The Tomahawk land-attack cruise missile can be launched from a ship or submarine, and can deliver its
warhead precisely to a target at long range. There are several variants of the Tomahawk missile, each with
specialized capabilities. For each Tomahawk Missile Sequence Number (MSN) task (i.e., mission), the
Tomahawk missile variants can be ranked with respect to their ability and cost effectiveness to perform
that task. A given land attack strike order may involve a large number of Tomahawk missiles and
numerous Tomahawk launch platforms. Operational planners select, in real time, feasible launch
platforms to execute Tomahawk taskings. The Tomahawk tasking in a strike order includes not only
primary assignments but also backup assignments. On board each launch platform, the precise allocation
of specific Tomahawk missiles to the Tomahawk MSN task assigned is optimized with a model described
here.We help naval operational planners select, in real time, appropriate feasible launch platforms to fulfill
the Tomahawk tasking in a strike order.
15. SUBJECT TERMS
16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF
ABSTRACT OF PAGES RESPONSIBLE PERSON
a. REPORT b. ABSTRACT c. THIS PAGE Same as 15
unclassified unclassified unclassified Report (SAR)
Standard Form 298 (Rev. 8-98)
Prescribed by ANSI Std Z39-18
282 NavalResearchLogistics,Vol. 58(2011)
Figure1. TacticalTomahawkMissile.[Source:NavalSurfaceWarfareCenter,DahlgrenDivision,2003].
complex decision process of planning these missions into missile’s position. A digital optical system that compares a
threehierarchicallystructuredclasses:(a)weaponmixprob- storedimageofthetargetwiththeobservedtargetprovides
lems,whichdeterminetheacquisitionofaweaponsmixthat guidanceintheterminaltargetarea.
inflictsmaximumdamagetoasetoftargets,subjecttocon- The Tomahawk has proven to be a remarkably flexible
straintsincludingcost,manpower,andweaponavailability; weapon. In 1991, hundreds of Tomahawks were launched
(b) weapon deployment problems, which assume a given against land targets in Iraq during Operation Desert Storm,
weapon mix and candidate locations where these weapons principally to attack the Iraqi air defense network. In
can be deployed, and which seek a weapon deployment to September 1995, USS Normandy (CG 60) fired 13
maximize damage on a given set of targets; and (c) the Tomahawks against Bosnian Serb targets, contributing
weapon-to-target allocation problem, which is “Given the to the subsequent cease-fire in Bosnia [3]. Tomahawks
mix of weapons deployed at specified locations, determine were used again in 1996 during Operation Desert Strike
the allocation of weapons to each target to inflict desired (Fig.2).
damage.”Theresourceallocationconceptsfromourliterature
reviewhavebeenmodified,abstracted,andextendedinorder
toconformtothetask-to-firingunitallocationaddressedin
thisarticle.
2. THETOMAHAWKMISSILEANDITS
TACTICALUSE
TheTomahawkenteredservicein1983,andfirstsawcom-
batinthe1990–1991GulfWar.Tomahawksarenowavailable
inanumberofvariants,differinginthetypeofmunitionscar-
ried, guidance system, and other features. Newer versions
have improved satellite communications that allow repro-
grammingthemissilewhileinflight,fasterlaunchtimes,and
theabilitytoloiterinanareaofemergingtargets.Tomahawks
usebothinertialandGPSnavigationsystems,aswellaster-
Figure 2. Flying her battle flag, USS Shiloh (CG-67) fires a
rain contour matching radar guidance [13] that compares a Tomahawk land attack missile in Operation Desert Strike,
storedmapreferencewiththeactualterraintodeterminethe 3September1996.
NavalResearchLogisticsDOI10.1002/nav
Newmanetal.: OptimizingAssignmentofTomahawkCruiseMissiles 283
Figure3. ImpactofaTomahawkmissilesequencenumbertask-to-firingunitallocationonsubsequentstrikeplancapability.
EquippingsubmarineswiththehighlyaccurateTomahawk considermanypotentiallyconflictingobjectivesandprefer-
addsanewconventionallandattackstrikewarfarecapabil- ences,suchas[7]:
ity.TheemploymentofTomahawksenablessurfaceshipsand
submarinestohitamyriadofkeytargetssuchascommand • Allocatingtaskstofiringunitssothateachfiringunit
centers,fortifiedpositions,andairfields. hasaTomahawkvariantthatcanperformthetaskallo-
Beforeleavingport,navalshipsandsubmarines(referred catedinacost-effectivemanner.Forexample,ifthe
toasthe“firingunits”inthisarticle)areloadedwithTom- targetcanbestruckbyaTomahawkvariantwithter-
ahawk missiles. The fleet command determines the num- raincontourmatchingradarguidancecapability,we
ber of Tomahawks and the specific variants loaded on a prefernottolaunchan“overqualified”variantwhich
firing unit. The weapon station’s (port facility) and firing alsohasGPSnavigationcapability.
unit’scommanderdecidewhereeachspecificTomahawkis • Leavingeachfiringunitwithasufficientlylargeresid-
loadedinthelauncherortorpedoroom.ForAEGIScruisers ualstrikecapacity,bothindividuallyandaspartofa
and destroyers [20], the Naval Surface Warfare Center Port battle group, so that it continues to be effective as
HuenemeDivisiondesignedandmaintainsacomputerpro- a naval force after expending the assigned missiles.
gram that suggests the exact Tomahawk loadout positions. Firing unit assignments allocate missiles to existing
The program takes into account the capabilities and limi- targetswhileplatformresidualcapacitypreservesthe
tations of the vertical launch system (VLS) on the AEGIS abilitytoaddresspotentialtargets(asexemplifiedin
ships. theexamplefollowingthisbulletedlist).
A Tomahawk Missile Sequence Number (MSN) task-to- • Expending missiles first from firing units that will
firing unit allocation is decided at the fleet command level. soon depart the theater of operations, and avoiding
This allocates each Tomahawk MSN task in a strike plan assignments that interfere with firing units engaged
toaparticularfiringunit;amissile-to-TomahawkMSNtask inotherduties.
allocationonafiringunitisassignedbyTacticalTomahawk
WeaponsControlSystemoneachfiringunit.Thisallocates The TSC has a very difficult job that must often be per-
a specific Tomahawk loaded on the firing unit to each par- formed under great time pressure, and in conjunction with
ticularTomahawkMSNtaskassignedtothatfiringunit;this otherduties.Asanillustrationofthecomplexityoftheprob-
capabilityhasbeendocumentedinanumberofNavalSurface lem,considerasimpleexampleinwhichthereareonlytwo
WarfareCenterDahlgrenDivision(NSWCDD)[27]techni- firingunitsinanavalstrikeforce:anAEGISdestroyerand
calreports(see,e.g.,Refs.1,14and15)andisnotdiscussed an AEGIS cruiser. On both AEGIS platforms, missiles are
furtherinthisarticle. carriedinVLSmodulesthatcancontaineightcells,eachof
ForagivensetofTomahawkMSNtasksinastrikeplan, which can hold a Tomahawk. Each module is divided into
aTomahawkStrikeCoordinator(TSC),usuallyaLieutenant twohalf-modulesoffourcells.AtmostoneTomahawkcan
Commander or Commander, develops one, or more, Tom- bepoweredupatatimeinaVLShalf-module.Forthisexam-
ahawk MSN task-to-firing unit allocation alternatives and ple,Fig.3showstheimpactofaTomahawkmissilesequence
presents these to the fleet command for approval. The TSC number task-to-firing unit allocation on subsequent strike
mustconsiderandbalancemanyfactorsinbuildingastrike plan capability. The upper part of the figure shows hypo-
plan. These include geographical (e.g., distance to targets theticalinitialloadoutsinaVLShalf-moduleontheAEGIS
fromavailablefiringunits,aimpoints,flightroutecoordina- destroyer and an AEGIS cruiser. The AEGIS destroyer is
tion),logistical(e.g.,inventoryofmissilesavailable,techni- loadedwithfourTomahawksinasinglehalf-moduleofthe
cal conflicts with other missiles, preparation time, required firing unit’s VLS (and no other Tomahawks). Specifically,
time on target), and operational (e.g., need for redundancy therearetwovariantT1Tomahawks,onevariantT2Tom-
in case of failure) factors. Simultaneously, the TSC must ahawk,andoneT3inthehalf-module.TheAEGIScruiser
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284 NavalResearchLogistics,Vol. 58(2011)
is loaded with four Tomahawks in a single half-module of Naval Surface Warfare Center, Port Hueneme Division, as
the firing unit’s VLS (and no other Tomahawks). There is a stand-alone capability, (b) subsequently implemented by
one variant T1 Tomahawk, two T2s, and one T3 in the NSWCDDwithinTTWCS,and(c)laterformulatedindepen-
half-module. dentlybyKuykendall[25],allreflectthatthemissile-to-task
The TSC creates a strike plan that has two MSN tasks. allocation on board a firing unit has to be performed in a
ThefirstMSNtask(MSN1)requiresthelaunchofasingle mannerthatmaximizesfuturelaunchcapabilityofthefiring
missile,andcanbeperformedbyeitheraTomahawkvariant unit.
T1oraTomahawkvariantT2;but,itismorecosteffective
touseaT2becausetheT1hasallthecapabilitiesofaT2as
wellasadditionalcapabilitiesthatarenotneededtoperform 3. PROBLEMDEFINITION
MSN1.Thus,usinganoverqualifiedT1wouldwastecapa-
Wefirstprovidesomedefinitionsandrulesneededtodefine
bilityandmightdegradethereadinessoftheforcetolaunch
theproblemandformulateamodel.
futurestrikes.
TheTomahawkselectedtoperformMSN1mustbepow-
ered up and launched at the same time as the Tomahawk 3.1. ConflictingTasks
launch time for the other MSN task (MSN 2). This second
TwoTomahawkMSNtasksaresaidtobeinconflictifthe
task requires the launch of a single Tomahawk variant T1.
preparationandlaunchtimeepochsforeachmissileallocated
BecausebothMSNtaskshavethesamemissilelaunchtime,
toperformtheTomahawkMSNtasksoverlap.Timeconflict
theyconflict,andmaynotbelaunchedfromthesameplat-
tasksetsareconstructedbasedonthepreparationandlaunch
form. Therefore, one task must be allocated to the AEGIS
timeepochsoftheTomahawkMSNtasks,andthevariousfir-
destroyer,andonetasktotheAEGIScruiser.
ingunitcapabilitiesandrestrictions.ATomahawkMSNtask
Suppose we assign MSN 1 to the AEGIS destroyer and
may be in more than one time conflict task set. Constraints
MSN 2 to the AEGIS cruiser. Then, onboard the AEGIS
aregeneratedtoreflecteachfiringunit’scapabilitytohandle
destroyer, the Tactical Tomahawk Weapon Control System
taskswithconflictingtimes.
(TTWCS)missile-to-TomahawkMSNtaskallocationwould
assigntheTomahawkvariantT2toMSN1inordertomax-
3.2. FiringUnitCharacteristics
imize the future launch capability of the AEGIS destroyer.
OnboardtheAEGIScruiser,theTomahawkvariantT1would Thereareseveraltypesofsurfaceshipandsubmarinefiring
be assigned to MSN 2 because it is the only Tomahawk units.Eachtypeoffiringunithasparticularweaponsstorage
missile that can perform the task. Notice that as a result of andlaunchcharacteristicsthatimposedifferentconstraints.
this allocation it would not be possible to perform a sub- ThesurfaceshipsstoretheirTomahawksinVLSs.Eachship
sequent strike plan requiring two T1s to be launched at mayholdadifferentnumberofmissiles,andthecell-location
the same time, nor a strike plan that might be performed ofthemissilesmaydifferfromshiptoship.Dependingonthe
by launching two, more cost-effective, T2s at the same classofsubmarine,Tomahawkscanbestoredinthetorpedo
time. room or in a vertical launch canister, with capacity varying
However,ifweallocateMSN1totheAEGIScruiser,and bysubmarineclass.
MSN 2 to the AEGIS destroyer, then onboard the AEGIS Allocating a Tomahawk MSN task to a firing unit that
destroyertheTTWCSmissile-to-TomahawkMSNtaskallo- isalreadyengagedinsomeotheractivityincursaso-called
cationwouldassignaT1toMSN2;and,onboardtheAEGIS employmentpenalty;thus,anotherwiseengagedfiringunit
cruiseraT2wouldbeassignedtoMSN1becausetheTom- isusedtoperformsuchataskonlyifnecessary.
ahawk variant T2 is more cost-effective to use than the Each firing unit is characterized as either an expend or a
TomahawkvariantT1.Noticethattheresultofthisalloca- non-expendunit.Afiringunitmightbedesignatedasexpend
tionallowsustoexecuteasubsequentstrikeplanthatrequires ifitisscheduledtoleavethetheaterofoperationssoon,and
twoTomahawkvariantT1stobelaunchedatthesametime, thusitisadvantageoustotheremainingforceintheaterifthe
orasubsequentstrikeplaninwhichtwomorecost-effective departing firing unit launches its weapons. For non-expend
missilescanbelaunchedatthesametime. firingunits,weallocateTomahawkMSNtaskstomaximize
While it might be easy to find the preferred Tomahawk theabilitytoperformfollow-onstrikes.
MSNtask-to-firingunitallocationinthissimpleexampleby
inspection, this is too complex to do when multiple Toma-
3.3. TasksandTask-Parts
hawkMSNtaskswithconflictingtimesneedtobeallocated
to multiple firing units in a naval strike force. As indicated A Tomahawk MSN “task” is made up of one or more
in the example above, the missile-to-Tomahawk MSN task missiles that are to be prepared for launch to fly a partic-
allocationcapabilitiesthatwere(a)initiallyimplementedby ularover-landrouteandattackaspecifiedaimpoint.Atask
NavalResearchLogisticsDOI10.1002/nav
Newmanetal.: OptimizingAssignmentofTomahawkCruiseMissiles 285
can have one or more task-parts (defined below), each of A number of constraints guide the assignment of task-
which requires at least one missile. “Strike packages” are parts:
sets of tasks or missions created by the TSC. For exam-
ple,astrikepackagemaycontainthefollowinginformation: • Tasksareclassifiedasseparableornonseparable:if
“Task1,requiresoneprimarypart,oneready-sparepart,and ataskisseparable(nonseparable)thenallthemissile
onebackuppart,TomahawkvariantT1,withearliest/latest requirements of a task-part associated with the task
time to launch 00:35/00:40; Task 2, requires one primary can(cannot)beallocatedtodifferentfiringunits.
part and one backup part, Tomahawk variant T2, with ear- • Somefiringunitsaredesignatedasspread,and,ifpos-
liest/latesttimetolaunch00:37/00:42;etc.”Afterthestrike sible,theprimaryandback-uptask-partassignments
packagesarecreated,theTSCsendsthem(viathedesignated should be allocated among as many of these firing
communication system) to the firing units. The firing units units as possible. Note: A related concept is known
returnexceptionreportsdetailingmissionstheycannotcom- as“task-partspreading.”Forexample,aprimarytask-
pleteasspecified.Notethatinoursystem,theallocationof partthatisdesignatedasspreadshouldbeallocated
successivestrikepackagesisdynamicallyincorporatedinto amongdifferentfiringunits.Thisarticleincorporates
theoptimization.Thatis,theexecutionsequencefreezesthe spreadfiringunits,butnotspreadtask-parts.
assets selected for previous allocations and proceeds with • Aready-sparetask-partmustbeassignedtoafiring
subsequentpackagesfromtheremainingassets. unitthatisassignedanassociatedprimarytask-part.
Therearefivetypesoftask-parts: Also, in the allocation of ready-spare and primary
task-parts for a specific separable task, for each fir-
ingunit,therequirednumberofready-sparemissiles
• A primary task-part requires one or more primary
needed for the ready-spare task-part should be no
missiles that are to be launched to fly a particular
morethanthenumberofprimarymissilesneededfor
over-landrouteandaretoattackaspecifiedaimpoint.
theprimarytask-part.
• Aready-sparetask-partisassociatedwithaspecific
• A backup task-part must not be assigned to a firing
primary task-part, and requires one or more ready-
unitthatisassignedanassociatedprimarytask-part.
spare missiles (located on the same firing units as
Some firing units, especially submarines, may be
the task’s primary missiles). The ready-spare mis-
excluded from being assigned backup task-parts by
siles provide redundancy for the primary missiles,
theTSC.
in case some of the primary missiles do not launch
• A task is considered “allocated” only when all the
successfully.
missilesrequiredforeachofitstask-partshavebeen
• Abackuptask-partisassociatedwithaspecificpri-
assigned. Partial allocation of a task is not allowed
mary task-part, and requires one or more backup
inpracticeandourformulationexplicitlyavoidsthat
missileslocatedonfiringunitsotherthanthosethat
possibility.
havethetask’sprimarymissiles.Thebackupmissiles
provide a second-level redundancy for the primary
missiles,incasetherequiredtotalnumberofprimary 3.4. LaunchAreas
andready-sparemissilesisnotlaunchedsuccessfully
Each firing unit is located in a specific launch area, and
fromprimaryfiringunits.
canonlybeassignedtostriketargetsthataregeographically
feasiblefromthatarea.Relocationofthefiringunittoadif-
For completeness, we describe two other types of task- ferentlaunchareaisbeyondourplanningscope,andisnot
parts that are set aside at the outset, that is, they are not considered.
availableintheallocationproblemmodeledinthisarticle.
3.5. Missile-to-MissionMatching
• Acall-for-firetask-partrequiresoneormoremissiles There are multiple Tomahawk missile variants, and each
bereservedforlaunchinginsupportoffriendlyforce one is designated by its Expanded Missile Identification
shoreoperationsduringaspecifiedtimeinterval. (XMID). For the missile-to-task allocation that takes place
• A pooled task-part designates one or more missiles ontheindividualfiringunits,theTomahawksthatarecapa-
tobereadytobeusedinsupportoflast-minutetasks ble of performing each task are identified, prioritized, and
duringaspecifiedtimeinterval.Ifalast-minuteneed allocated based on a number of factors. Although the TSC
arisesforafiringunitwithpooledmissilestoperform does not have information concerning all of the factors, he
aspecificprimarytask,thentheappropriatenumber hasknowledgeconcerningtheTomahawkmissileloadouton
ofpooledmissilescanbeselectedforlaunchfromthe eachfiringunit,andthecosteffectivenessofusingaparticu-
firingunittoattackthespecifiedaimpoint. larTomahawkvariant(i.e.,XMID)toperformatask.Forany
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286 NavalResearchLogistics,Vol. 58(2011)
task,aMissileMissionMatching(orM3)listofXMIDsthat ofataskisnotallowed,thisobjectivecouldbemodifiedto
areabletoperformthetaskisconstructedandprioritizedfrom reflecttheimportanceandrelativepriorityofthevarioustasks
themosttotheleastcost-effectiveTomahawkvarianttoper- andtask-parts.
formthetask.Thetask-to-firingunitallocationshouldassign (cid:1)
task-parts to firing units in order to maximize the potential Objectivek =1: min z1 ≡ Utpm,
follow-onlaunchcapabilityoftheforcebyallowingthefir- t∈T,p∈P,
ing units to perform Tomahawk-to-task allocations that use m∈Mtp
themostcosteffectiveTomahawks.
where Utpm equals 1 if the m-th missile requirement for
task-partpintasktisunmet,and0otherwise.
3.6. TomahawkStrikeCoordinatorOverrides
While our intent here is to develop an automated deci- OBJECTIVE 2(Use of firing units currently engaged in
sionaidtoallocatetask-partstofiringunits,therearecases other tasking; to be minimized): This objective penalizes
inwhichthesuggestedallocationshouldnotbeappliedand distractionoffiringunitsengagedinothermissions.
othercasesinwhichtheTSCmaywishtooverridethetask
(cid:1)
parttofiringunitallocation,e.g.: Objectivek =2: min z2 ≡ psVs,
• Taskscanbedividedintotwogroups:executetasks s∈S
and launch-plan tasks. “Execute” tasks are tasks
whose parts have already been allocated to firing whereVsequals1ifshipsisused,and0otherwiseandwhere
units, and the firing units’ commanders have con- ps isthepenaltyforusingships.
firmed that Tomahawks can be allocated to the task
OBJECTIVE 3(Number of primary task-parts allocated
parts. Within the automated decision aid to allocate
todesignated“expend”firingunits;tobemaximized):This
task-partstofiringunits,anypriorexecutetaskallo-
objective rewards firing units regarded as “expend” if they
cations to firing units should be accounted for, but
areassignedprimarytask-parts(notredundancies).
not changed. “Launch-plan” tasks are those whose
taskpartshavenotyetbeenallocatedtofiringunits. (cid:1)
The job of the automated decision aid is to allocate Objectivek =3: max z3 ≡ Ystpm
launch-plan task-parts to firing units, while taking t∈T,s∈St∩SE,
priorexecutetaskingintoaccount. p∈P,m∈Mtp
• TheTSCcanallocateaparticulartask-part,orapor-
whereYstpmequals1ifshipsisassignedtoperformthem-th
tion of a task-part, to a specific firing unit, or even
missilerequirementfortask-partpintaskt,and0otherwise.
designatelaunchofaspecificmissile.Suchgeneral,
adhocconstraintsofferingtheTSCcompletecontrol
OBJECTIVE 4(Deviations in residual missile inventory
overpredesignationareakeytogainingandkeeping
acrossfiringunits;tobeminimized):Thisobjectiveseeksto
the confidence and trust of the TSC. In our experi-
level residual missile inventories across firing units (except
ence,plannersdistrust“blackbox”toolsoverwhich
forexpendships)afterprimarytask-partsareassigned.The
they have no control, and find ways to avoid using
rationaleforthisobjectiveistoavoidweakeningnon-expend
suchtools.
shipsforpotentialfutureengagements.
4. MATHEMATICALMODELING (cid:1)
Objectivek =4: min z ≡ RD ,
4 s
4.1. ObjectiveFunctions
s∈S\SE
We have developed objective functions by interviewing
TSCs,theircommandingofficersandseniornavalofficials. whereRDs isthedifferencebetweentheresidualnumberof
missilesonshipsandtheaverageresidualnumberofmissiles
The objective functions listed below are for a streamlined
acrossall“non-expend”ships.
modelformulationpresentedinthesectionsthatfollowthat
dealsonlywiththeallocationofTomahawktaskstoAEGIS
ships.ThenotationisdefinedindetailintheAppendix,but OBJECTIVE 5(Number of “spread” firing units used
thedefinitionsofvarioustermsarerepeatedhere. for primary task-part assignments; to be maximized): This
objective counts the number of firing units included in our
OBJECTIVE 1(Number of unassigned task-parts; to be predesignation.Maximizingthisnumberspreadsthealloca-
minimized): This objective expresses the number of task- tion of primary task-parts, minimizes the impact of losing
parts not allocated to firing units. While partial allocation one or several ships, and encourages participation by many
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Newmanetal.: OptimizingAssignmentofTomahawkCruiseMissiles 287
firingunits.Someshipscanbemarkedtobe“preferred”for Also, if based on the information held by the TSC, one
primarytask-partsbytheTSC. ormoreweaponsoftypewwouldremainonshipsinhalf-
(cid:1) modulehafterfiringallprimarytask-partmissiles,atmost1
Objectivek =5: max z5 ≡ PRs, Dshw forthat(s,h)isallowedtobeequalto1,andmustbe0
s∈SPR otherwise.
where PRs equals 1 if any primary task-part is allocated to
ships(onlyfors ∈SPR),and0otherwise.SPR ⊆Sisasub- 4.2. ModelFormulation
setofAEGISshipfiringunitstowhichwewishtoallocate
We present a simplified model formulation that covers
primarytask-parts.
most problem features. In this streamlined formulation, we
have suppressed geographic proximity, substituting instead
OBJECTIVE 6(Number of “spread” firing units used
lists of geographically feasible assignments. Here, we con-
for backup task-part assignments; to be maximized): This
sideronlyAEGISshipsasfiringunits.Fordetailsonmod-
objective is the same as Objective 5, but applies to backup
elingwithSSNsubmarines,seeRef.9.FordetailsonSSGN
task-parts.
modeling,seeRef.8.Theinclusionofsubmarinesdoesnot
(cid:1)
affect the solvability of the problem any more than adding
Objectivek =6: max z6 ≡ BUs, newsurfaceships.Zumwalt-classships[33]havenotyetbeen
s∈SBU
inserted in the formulation. (For specifications of various
whereBUs equals1ifanyback-uptask-partisallocatedto combatant ship classes, see Ref. 21.) Only primary, ready-
ships(onlyfors ∈SBU),and0otherwise.SBU ⊆Sisasub- spare, and backup task-parts are discussed in the following
setofAEGISshipfiringunitstowhichwewishtoallocate formulation.Astructuredlistofvariableandparameterdefi-
backuptask-parts. nitionsandaformaldefinitionofconstraintsareprovidedin
theAppendix.
OBJECTIVE 7(Useofnon-cost-effectivemissiles;tobe
minimized): This objective assesses the cost effectiveness
4.3. SolvingtheMultiobjectiveOptimizationProblem
of the missiles that may be allocated to tasks on the firing
units,andminimizingtheuseofnon-cost-effectivemissiles Themodelthatwehavedescribedisamixed-integerlin-
preservespotentialfollow-onlaunchcapability. earprogramwithmultipleobjectivefunctions.Ingeneral,a
(cid:1) multiobjectiveoptimizationproblemmaybewrittenas(e.g.,
Objectivek =7: min z7 ≡ m3twpXshwtpm, Ref.11,p.16):
t∈T,s∈St,h∈Hs,
w∈Wsht,p∈P,m∈Mtp “minimize”F[z1(x),z2(x),..., z|K|(x)]
wherem3 ,isthepenaltyforallocatingweaponwtopartp subjecttox ∈θ
twp
intaskt(basedonM3positionofweaponwfortaskt,andthe
iwfesihgihptsoifspaasrstipgntoedmtaotcphertfhoerMm3thliestm).-Athlsmo,isXsislhewrtpeqmueiqreumalesn1t wanhderze1x,.=..,{zx|K1,|xa2,re..o.,bxjenc}tiivseafvuencctotiroonfsdoefcitshioesnevadreicaibslieosn,
variables, θ is the set of feasible decisions allowed by the
fortask-partpintaskt(tentativelyusingaweaponoftypew
constraints, and F is some vector or scalar function of the
inhalf-moduleh),and0otherwise.
objectivefunctionvalues.Bynecessity,theterm“minimize”
mayhaveseveralmeaningsandthechoiceamongthemissub-
OBJECTIVE 8(Relative value of residual salvo; to be
jective.Ifthereisavalueofxthatsimultaneouslyoptimizes
maximized): This objective assesses the potential missile
eachzk(x),thenthemultiobjectiveproblemwouldbeclear
loadoutonthefiringunitsafterTomahawksarelaunchedto
andsimple,butsuchanidealsolutioncannotbeguaranteed,
performprimarytaskparts,inordertomaximizethepotential
andisrareinpractice.
follow-onsalvo,thatis,launchcapability.(Theformulation
For convenience, we assume that any objective that is to
presentedbelowonlyconsidersthe“mostvaluable”missile
bemaximizedintheoriginalformulationisredefinedasits
typeinalauncherhalfmodule.Theobjectivecanbemodified
negation,sothatallobjectivesaretobeminimized.
toreflectallthemissiletypesinalauncherhalfmodule.)
(cid:1)
Objectivek =8: max z8 ≡ vwDshw, 4.3.1. TheValueFunction
s∈S,h∈Hs,w∈Wsh
IfFisawell-defined,scalar-valuedfunctionthatallinter-
wherevwistherelativevalueofweaponwforresidualsalvo ested parties agree is an accurate representation of their
computation. preferences and priorities, then the multiobjective problem
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288 NavalResearchLogistics,Vol. 58(2011)
reducestosingle-objectiveoptimization,forwhichstandard 4.3.4. HierarchicalOptimization
methodscanbeemployed.SuchFfunctionsarecalledvalue
Anotherintuitivelysimpleideaisprioritizingthemultiple
functionsinadeterministicsetting(whichisthecasehere)or
objectivefunctionsandthenoptimizingthem,oneatatime,
theyarecalledutilityfunctionsinastochasticsetting. (See
inpriorityorder.Attainmentofoptimal(ornear-optimal)val-
Keeney and Raiffa [22, pp. 82–116 and 131–218, respec-
uesofhigherpriorityobjectivesbecomesaconstraintwhen
tively]fordevelopmentofthesefunctionsandHarrisonand
optimizing the lower priority ones. This method requires
Rosenthal[16]fortheiruseinoptimizationmodels.)
solving the following sequence of increasingly restricted
single-objectiveproblems(Hk),fork =1,..., |K|:
4.3.2. LinearValueFunction
The most commonly used value function is a weighted (Hk): zk∗ =min zk(x)
s.t. x ∈θ
sum of the objective functions that has the advantage of
being intuitively appealing to many people. This also has zk(cid:6)(x)=zk∗(cid:6), ∀k(cid:6) <k
well-documented drawbacks. The problem becomes
intractableifanonlinearweightingisused.Despiteitspoten- Oneweaknessofthishierarchicaloptimization(alsoknown
tialweakness,alinearvaluefunction(thatis,weightedsums) aspre-emptivepriorities)isthattheoptimizationateachstep
iswidelyusedandisoftenanadvisableapproach,especially dependsupontheexistenceofmultipleoptimaintheprevious
when under time pressure. There is no way to avoid some step. Without multiple optima in step k, the method con-
subjective judgment on empirical observations of prelimi- vergesbeforeconsideringobjectivesk+1,..., |K|.Another
nary results when setting the weights. We look at the need valid criticism of hierarchical optimization is that it may
forsubjectivejudgmentinmultiobjectiveoptimization(and bar consideration of some sensible opportunities to trade
other areas of OR) as an important distinguishing feature off a small decrement in a high-priority objective in return
ofORversus,say,physics,because“weareengagedinthe foralargeimprovementinalow-priorityone.Theseissues
modelingofsystemsthatinvolvepeople,notparticles.”1 canbeaddressedbyrelaxingthehigher-priorityobjective’s
Theformulationofthelinear-value-functionorweighted- aspirations, replacing each equality constraint in Hk by an
sumis: inequality bounded by some given fraction of the optimal
(cid:1) valuez∗.
k
(W): zo∗ =minzo = ωkzk(x) Iftheweightsarechosensuchthatω1 >>ω2 >>···>>
k∈K ω|K|,thentheweighted-sumobjectiveyieldsthesamesolu-
s.t.x ∈θ tionashierarchicaloptimization.Inpractice,alargedisparity
in the magnitude of the weights and objective coefficients
whereωk >0istheweightgiventoobjectivek. canbecomputationallyproblematic.Thisisbecauseinorder
for the low-priority objectives’ weights to have any influ-
ence on the solution, the optimality tolerance needs to be
4.3.3. ParetoOptimality
set very close to zero, which in turn may cause the lower-
For a multiobjective problem, it is always preferred to priorityobjectives’contributionstothevaluefunctiontobe
chooseaPareto-optimalsolution(e.g.,Ref.[11],p.24),that ignored by the optimization numerics. Normalization and
is,onefromwhichitisimpossibletofindanotherpointthat scalingacrossallobjectivesshouldbeimplementedtohelp
improvestheattainmentofsomeobjective(s)withoutsimul- reducethisrisk.
taneouslydiminishingtheattainmentofsomeotherobjective. The(W)and(H)formulationsabovecanbesolvedaspre-
There are usually many Pareto-optimal points, perhaps an sented,orapproximatedbyaproblemcascadeapproach,as
infinite number. Where |K| =2, the Pareto-optimal set is in Brown et al. [6, p. 341]. This strategy is also known as
acurve,theefficientfrontier,whichiseasytogenerateand fix-and-relax[10,12].Thistechniquerelaxestheintegrality
extremelyusefulforpresentingalternativestodecisionmak- of most of the decision variables to make the problem eas-
ers.Thecaseof|K|>2ismoredifficultandhasbeentreated ier.Forexample,onlythosevariablesrelatedtoasubsetof
byZeleny[37],Rosenthal[28],Steuer[31],andWhite[35]. tasks are forced to assume integer values. Then, these vari-
One fortunate result in the theory of multiobjective opti- able values are fixed and a new subset of variable values is
mization is Soland’s Theorem [30], from which it follows forcedtobeintegral,untilallofthemarefixed.Algorithmic
that solving problem (W) yields a Pareto-optimal solution, detailsareprovidedbyDemir[9].Wefindthisapproachmay
althoughnotallefficientpointsmaybederivedasthesolution helpreducecomputationaltimeforcasesinwhich(W)or(H)
toaproblemlike(W). take a long time to converge, without much degradation in
theachievedobjectives.However,italsoincreasesthecom-
1ThisobservationisclassicRickRosenthal. putationaltimeineasierinstancesduetothemultiplemodels
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Newmanetal.: OptimizingAssignmentofTomahawkCruiseMissiles 289
thatneedtobesolved.Asourcomputationalexperimentsare Table1. Predesignationoptimizationproblemcharacteristicsfor
labandfleetexercisetestcases.
restrictedbyashorttimelimit,wedonotdiscussthecascade
approach. No.of No.of No.of
Testcases firingunits tasks tasks-parts
4.3.5. HeuristicOptimization 22Labproblems[2] 4–8 13–26 104–232
13Exerciseproblems[36] 4–8 18–100 28–128
Inadditiontotheconventionalmultiobjectiveoptimization
methods of solving (W) and (Hk), we have also developed
a specialized, custom heuristic for finding good solutions
includescomparisonsofwhatwouldhavehappenedhadour
quickly. The heuristic uses sequential insertion: Queues of
modelbeenused,versustheactualexerciseresults.
task-parts, firing units, and missiles are maintained, with
Our runs were made on a 2 GHz laptop with 2 GB of
prioritiesinthequeuesdeterminedbyabatteryofproblem-
RAM. Exact models (H) and (W) were implemented in
specificcriteria.Asassignmentsaremadefromthequeues,
GAMS [5] and solved with CPLEX 9.0 [18]. The heuristic
thequeuesarereplenishedandreordered.Occasionally,we
wasimplementedinJava[32].
invokearestarttogivetheheuristicanopportunitytoimprove
thesolutionwithlargechanges,ratherthanwithsmallincre-
5.2. ResultsforLabProblems
ments.DetailsofthisheuristicappearinSalmeron[29]and
Arnold [2], but initial steps in exploring heuristics for this Table1showsproblemcharacteristicsforbothlab[2]and
problemareduetoKirk[23]andHodge[17]. ThirdFleetexercises[36]testcases.Notethat“x-y”indicates
Overtheperiodinwhichtheyhavebeeninvolvedinmod- arangefromaminimumofxtoamaximumofy.Tables2
elingTomahawkmissileassignment,theNavalPostgraduate and3showdetailedresultsforthe22labproblemsandthe
Schoolteamofstudentsandfacultyhasworkedcloselywith 13exerciseproblems,respectively.
experiencedoperationsofficers.Thisclosecontactwithoper- Bycolumn,Table2showsthecasename,theoptimalsolu-
ationalpracticehasallowedtheteamtogaindomain-specific tiontoeachoftheeightgoals,andthecomputationaltimefor
knowledge that allows the resolution of various modeling each of the three methods: Heuristic, (W) and (H) models,
issues. For example, Kubu [24] systematically studies situ- respectively.Forallofthesetestcases,theexactoptimization
ations in which the heuristic is unable to assign all tasks to solutions(W)and(H)areidentical.Insomecases,theheuris-
firingunits.Heintroducesadiagnosticmodulethatprovides ticsolution,whichisdisplayedbyanunderlinedalternative
theTSCwithlikelyreasonsastowhyataskcannotbecov- value, is of inferior quality. For example, all three methods
ered.Healsoprovidesaprescriptivemoduleforsuggesting achieve the same solutions for cases “1” and “1A,” but for
changestotheinputssothatalltaskscanbeallocated.Kubu case “1B” the heuristic solution is 3 for goal k = 6, when
collects domain-specific knowledge about our problem and the optimal objective should be 4. Of course, the heuristic
assemblesacatalogofsuggestedadjustmentstodiagnoseand solution might be superior for goals k+1,...,8 (see, e.g.,
eliminateinfeasibilities.Themosteffectivetoolisanadjust- case“2”)butthatcanonlyhappenifitisofinferiorquality
ment of time windows performed by sliding the earliest or for a hierarchically superior goal k. Overall, heuristic solu-
latestlaunchtimesinordertoavoidtaskconflicts. tions are of acceptable quality. In the 22 lab problems, the
heuristic never fails to optimize the first goal, only fails to
optimizethesecondgoalinoneoftheproblems(case“3”),
andthethirdgoalinanotherproblem(case“6”).Goalk =4,
5. TESTPROBLEMSANDCOMPUTATIONAL
“levelingmissilesleftonfiringunitsbyminimizingabsolute
EXPERIENCE
deviations” is apparently the most difficult goal, since the
heuristic misses the optimal solution in 10 of the remain-
5.1. TestProblems
inglabcases.Indeed,mostofthetimeconsumedbythe(H)
Wehavedevelopedadatasetof35testproblems.Werefer methodinsolvingtheeightproblems(Hk),k = 1,...,8,is
tothefirst22testproblemsaslabproblems,andtheother13 spentinsolving(H ).
4
asexerciseproblems.Thelabproblemsarecompleteinven-
tions[2],baseduponwar-fighterandengineeringjudgment,
5.3. ResultsforExerciseProblems
but without empirical basis. In contrast, the exercise prob-
lemsarebasedentirelyoncarefullyrecordedobservationsof In Table 3, “R-Sn” denotes the n-th strike scenario for
naval exercises at sea attended by Wingeart [36]. Wingeart the Rimpac exercise, and “T-Sn-m” denotes the m-th salvo
collectedbothinputsandoutputsofallthetaskstofiringunit withinthen-thstrikescenariofortheTexasexercise.(Rim-
allocations completed in two major U.S. Navy Third Fleet pacstrikeshaveoneuniquesalvoeach.)Strikesareplanned
exercisesknownasRimpacandTexas-Thunder.Hisanalysis independently in that they may involve different groups of
NavalResearchLogisticsDOI10.1002/nav