The optimizing-simulator: An illustration using the military airlift problem

31 Pages·2009·0.55 MB·English

The Optimizing-Simulator: An Illustration
Using the Military Airlift Problem
TONGQIANGTONYWUandWARRENB.POWELL
PrincetonUniversity
and
ALANWHISMAN
AirMobilityCommandRetired
Therehavebeentwoprimarymodelingandalgorithmicstrategiesformodelingoperationalprob-
lemsintransportationandlogistics:simulation,offeringtremendousmodelingflexibility,andop-
timization,whichofferstheintelligenceofmathprogramming.Eachofferssignificanttheoretical
andpracticaladvantages.Inthisarticle,weshowthatyoucanmodelcomplexproblemsusinga
rangeofdecisionfunctions,includingbothrule-basedandcost-basedlogic,andspanningdifferent
classesofinformation.Weshowhowdifferenttypesofdecisionfunctionscanbedesignedusingup
tofourclassesofinformation.Thechoiceofwhichinformationclassestouseisamodelingchoice,
andrequiresmakingspecificchoicesintherepresentationoftheproblem.Weillustratetheseideas
inthecontextofmodelingmilitaryairlift,wheresimulationandoptimizationhavebeenviewedas
competingmethodologies.Ourgoalistoshowthatthesearesimplydifferentflavorsofaseriesof
integratedmodelingstrategies. 14
CategoriesandSubjectDescriptors:I.6.1[SimulationandModeling]:Simulationtheory;I.6.3
[SimulationandModeling]:Applications;I.6.5[SimulationandModeling]:ModelDevelop-
ment—Modelingmethodologies
GeneralTerms:Algorithms
AdditionalKeyWordsandPhrases:Approximatedynamicprogramming,militarylogistics,control
ofsimulation,modelinginformation,optimizing-simulator
ACMReferenceFormat:
Wu,T.T.,Powell,W.B.,andWhisman,A.2009.Theoptimizing-simulator:Anillustrationusing
themilitaryairliftproblem.ACMTrans.Model.Comput.Simul.,19,3,Article14(June2009),31
pages.DOI=10.1145/1540530.1540535 http://doi.acm.org/10.1145/1540530.1540535
ThisresearchwassupportedinpartbygrantAFOSRcontractFA9550-08-1-0195.
Authors’addresses:T.T.WuandW.B.Powell,Dept.ofOperationsResearchandFinancialEn-
gineering,PrincetonUniversity,Princeton,NJ08544;email:[email protected];A.Whisman,
AirMobilityCommand,ScottAirForceBase,IL62225.
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DOI10.1145/1540530.1540535 http://doi.acm.org/10.1145/1540530.1540535
ACMTransactionsonModelingandComputerSimulation,Vol.19,No.3,Article14,Publicationdate:June2009.
14:2 • T.T.Wuetal.
1. INTRODUCTION
Complexoperationalproblems,suchasthosethatariseintransportationand
logistics, have long been modeled using simulation or optimization. Typically,
theseareviewedascompetingapproaches,eachofferingbenefitsovertheother.
Simulationofferssignificantflexibilityinthemodelingofcomplexoperational
conditions,andinparticularisabletohandlevariousformsofuncertainty.Op-
timizationoffersintelligentdecisionsthatoftenallowmodelstoadaptquickly
tonewdatasets(simulationmodelsoftenrequirerecalibratingdecisionrules),
andofferadditionalbenefitssuchasdualvariables.Inthedesireforgoodsolu-
tions,optimizationseekstofindthebestsolution,buttypicallyrequiresmaking
anumberofsimplifications.
We encountered the competition between simulation and optimization in
thecontextofmodelingthemilitaryairliftproblemfacedbytheanalysisgroup
at the Airlift Mobility Command (AMC). The military airlift problem deals
with effectively routing a fleet of aircraft to deliver loads of people and goods
(troops, equipment, food, and other forms of freight) from different origins to
different destinations as quickly as possible under a variety of constraints.
In the parlance of military operations, loads (or demands) are referred to as
requirements. Cargo aircraft come in a variety of sizes, and it is not unusual
forasinglerequirementtoneedmultipleaircraft.Iftherequirementincludes
people, the aircraft has to be configured with passenger seats. Other issues
include maintenance, airbase capacity, weather, and the challenge of routing
aircraftthroughfriendlyairspaces.
There are two major classes of models that have been used to solve the
military airlift (and closely related sealift) problem: cost-based optimization
models[Mortonetal.1996;Rosenthaletal.1997;Bakeretal.2002],andrule-
based simulation models, such as MASS (Mobility Analysis Support System)
andAMOS(AirMobilityOperationsSimulator),whichareheavilyusedwithin
the AMC. The analysis group at AMC has preferred AMOS because it offers
tremendous flexibility, as well as the ability to handle uncertainty. However,
simulation models require that the user specify a series of rules to obtain re-
alistic behaviors. Optimization models, on the other hand, avoid the need to
specifyvariousdecisionrules,buttheyforcetheanalysttomanipulatethebe-
havior of the model (the decisions that are made) by changing the objective
function.Whileoptimalsolutionsareviewedasthegoldstandardinmodeling,
itisasimplefactthatformanyapplications,objectivefunctionsarelittlemore
thancoarseapproximationsofthegoalsofanoperation.
Ourstrategyshouldnotbeconfusedwithsimulation-optimization,whichis
well-known within the simulation community (see, for example, the excellent
reviews Swisher et al. [2003] and Fu [2002]). This strategy typically assumes
anoftenmyopicparameterizedpolicy,wherethegoalistofindthebestsettings
foroneormoreparameters.Forourclassofapplications,wearetryingtocon-
structasimulationmodelthatcandirectlycompetewithamathprogramming
model, while retaining many of the important features that classical simula-
tion methods bring to the table. For example, we need decisions that consider
theirimpactonthefuture.Atthesametime,weneedamodelthatwillhandle
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TheOptimizing-Simulator:AnIllustrationUsingtheMilitaryAirliftProblem • 14:3
uncertainty and a high level of detail, features that we take for granted in
simulationmodels.
Thisarticleproposestobringtogetherthesimulationandoptimizationcom-
munities who work on transportation and logistics. We do this by combining
mathprogramming,approximatedynamicprogramming,andsimulation,with
astrongdependenceonmachinelearning.Fromtheperspectiveofthesimula-
tioncommunity,itwilllookasifwearerunningasimulationiteratively,during
whichwecanestimatethevalueofbeinginastate(dynamicprogramming),and
we can also measure the degree to which we are matching exogenously speci-
fiedpatternsofbehavior(aformofsupervisorycontrolfromthereinforcement-
learningcommunity).Ateachpointintime,weusemathprogrammingtosolve
sequences of deterministic optimization problems. Math programming allows
us to optimize at a point in time, while dynamic programming allows us to
optimizeovertime.Thepatternmatchingallowsustobridgethegapbetween
cost-basedoptimizationandrule-basedsimulation.
Thisstrategyhastheeffectofallowingustobuildafamilyofdecisionfunc-
tionswithuptofourclassesofinformation:(1)thephysicalstate(whatweknow
aboutthesystemnow),(2)forecastsofexogenousinformationevents(newcus-
tomerdemands,equipmentfailures,weatherdelays),(3)forecastsoftheimpact
of decisions now on the future (giving rise to value functions used in dynamic
programming),and(4)forecastsofdecisions(whichwerepresentaspatterns).
The last three classes of information are all a form of forecast. If we just use
the first class, we get a classical simulation using a myopic policy, although
these come in two flavors: rule-based (popular in simulation) and cost-based
(popularinthetransportationcommunitywhensolvingdynamicproblems).If
weusethesecondinformationclass,weobtainarolling-horizonprocedure.The
thirdclassofinformationusesapproximatedynamicprogrammingtoestimate
thevalueofbeinginastate.Thefourthclassallowsustocombinecost-based
logic (required for any optimization model) with particular types of rules (for
example, “we prefer to use C-17s for loads originating in Europe”). This class
introduces the use of proximal point algorithms. We claim that any existing
modelingandalgorithmicstrategycanbeclassifiedintermsofitsuseofthese
fourclassesofinformation.
The central contribution of this article is to identify how simulation and
optimizationcanbecombinedtoaddresscomplexmodelingproblemsthatarise
intransportationandlogistics,illustratedusingthecontextofamilitaryairlift
application. This problem class has traditionally been solved using classical
deterministicoptimizationmethodsorsimulation,witheachapproachhaving
significantstrengthsandweaknesses.Weshowhowcost-basedandrule-based
logic can be combined within this broad framework. We illustrate these ideas
using an actual airlift simulation to show that we can vary the information
contentofdecisionstoproducedecisionswithincreasinglevelsofsophistication.
WebegininSection2withareviewofthefieldofmodelingmilitarymobility
problems(thisishowthiscommunityreferstothisproblemclass).Morethan
justaliteraturereview,thissectionallowsustocontrastthemodelingstylesof
differentcommunities,includingdeterministicmathprogramming,simulation,
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14:4 • T.T.Wuetal.
andstochasticprogramming.InSection3weprovideourownmodeloftheairlift
problem, providing only enough notation to illustrate the important modeling
principles. Section 4 shows how we can create different decision functions by
modeling the information available to make a decision. We also discuss rule-
based and cost-based functions, and show how these can be integrated into
asingle,generaldecisionfunctionthatusesallfourclassesofinformation.In
Section5wesimulateallthedifferentinformationclasses,andshowthataswe
increasetheinformation(thatis,useadditionalinformationclasses)weobtain
bettersolutions,measuredintermsofthroughput,acommonmeasureusedin
themilitary,andrealism—reflectedbyourabilitytomatchdesiredpatternsof
behavior.Section6concludesthearticle.
2. MODELINGOFMILITARYMOBILITYPROBLEMS
This section provides a summary of different modeling strategies for mili-
tary mobility problems: air and sea. After providing a review of the military
mobility literature in Section 2.1, we briefly summarize the three primary
modelingstrategiesthathavebeenusedinthisarea:deterministiclinearpro-
gramming(Section2.2),simulation(Section2.3),andstochasticprogramming
(Section2.4).Themodelingofmobilityproblemsisunusualinthatithasbeen
approachedindetailbyallthreecommunities.Wepresentthesemodelsinonly
enoughdetailtoallowustocontrastthedifferentmodelingstyles.
2.1 TheHistoryofMobilityModeling
Ferguson and Dantzig [1955] is one of the first to apply mathematical models
to air-based transportation. Subsequently, numerous studies were conducted
on the application of optimization modeling to the military airlift problem.
Severalmathematicalmodelingformulationsformilitaryairliftoperationsare
described by Mattock et al. [1995]. The RAND Corporation published a very
extensive analysis of airfield capacity in Stucker and Berg [1999]. According
toBakeretal.[2002],researchonairmobilityoptimizationattheNavalPost-
graduate School (NPS) started with the Mobility Optimization Model (MOM).
This model is described in Wing et al. [1991] and concentrates on both sealift
andairliftoperations.Therefore,theMOMmodelisnotdesignedtocapturethe
characteristicsspecifictoairliftoperations,butitisagoodmodelinthesense
that it is time-dependent. THRUPUT, a general airlift model developed by
Yost[1994],capturesthespecificsofairliftoperationsbutisstatic.TheUnited
States Air Force Studies and Analyses Agency in the Pentagon asked NPS to
combine the MOM and THRUPUT models into one model that would be time
dependentandwouldalsocapturethespecificsofairliftoperations[Bakeretal.
2002].TheresultingmodeliscalledTHRUPUTII,describedindetailinRosen-
thal et al. [1997]. During the development of THRUPUT II at NPS, a group
at RAND developed a similar model called CONOP (CONcept of OPerations),
describedinKillingsworthandMelody[1997].TheTHRUPUTIIandCONOP
models each possessed several features that the other lacked. Therefore, the
NavalPostgraduateSchoolandtheRANDCorporationmergedthetwomodels
intoNRMO(NPS/RANDMobilityOptimizer),describedinBakeretal.[2002].
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TheOptimizing-Simulator:AnIllustrationUsingtheMilitaryAirliftProblem • 14:5
Crinoetal.[2004]introducedthegroup-theoretictabusearchmethodtosolve
thetheaterdistributionvehicleroutingandschedulingproblem.Theirheuristic
methodology evaluates and determines the routing and scheduling of multi-
modaltheatertransportationassetsattheindividualassetoperationallevel.
Anumberofsimulationmodelshavealsobeenproposedforairliftproblems
(andrelatedproblemsinmilitarylogistics).Schanketal.[1991]reviewseveral
deterministic simulation models. Burke et al. [2004] at the Argonne National
Laboratory developed a model called TRANSCAP (Transportation System
Capability) to simulate the deployment of forces from Army bases. The heart
ofTRANSCAPisthediscrete-eventsimulationmoduledevelopedinMODSIM
III. Perhaps the most widely used model at the Air Mobility Command is
AMOS which is a discrete-event worldwide airlift simulation model used in
strategic and theater operations to deploy military and commercial airlift
assets.ItwasoncethestandardforallairliftproblemsinAMC,andallairlift
studieswerecomparedwiththeresultsproducedbytheAMOSmodel.AMOS
provides for a very high level of detail, allowing AMC to run analyses on a
widevarietyofissues.
Onefeatureofsimulationmodelsistheirabilitytohandleuncertainty,and
as a result there has been a steady level of academic attention toward incor-
porating uncertainty into optimization models. Dantzig and Ferguson [1956]
is one of the first to study uncertain customer demands in the airlift prob-
lem. Midler and Wollmer [1969] also takes into account stochastic cargo re-
quirements.Thisworkformulatestwotwo-stagestochasticlinearprogramming
models:amonthlyplanningmodel,andadailyoperationalmodelfortheflight
scheduling. Goggins [1995] extended Throughput II [Rosenthal et al. 1997] to
a two-stage stochastic linear program to address the uncertainty of aircraft
reliability. Niemi [2000] expands the NRMO model to incorporate stochastic
ground times through a two-stage stochastic programming model. To reduce
the number of scenarios for a tractable solution, the model assumes that the
set of scenarios is identical for each airfield and time period and a scenario is
determined by the outcomes of repair times of different types of aircraft. The
resulting stochastic programming model has an equivalent deterministic lin-
ear programming formulation. Granger et al. [2001] compared the simulation
model and the network approximation model for the impact of stochastic fly-
ing times and ground times on a simplified airlift network (one origin aerial
port of embarkation (APOE), three intermediate airfields and one destination
aerial port of debarkation (APOD)). Based on the study, they suspect that a
combinationofsimulationandnetworkoptimizationmodelsshouldyieldmuch
better performance than either one of these alone. Such an approach would
useanetworkmodeltoexplorethevariablespaceandidentifyparameterval-
ues that promise improvements in system performance, then validate these
by simulation. Morton et al. [2002] developed the Stochastic Sealift Deploy-
mentModel(SSDM),amulti-stagestochasticprogrammingmodeltoplanthe
wartime,sealiftdeploymentofmilitarycargosubjecttostochasticattack.They
usedscenariostorepresentthepossibilityofrandomattacks.
A related method in Yost and Washburn [2000] combines linear program-
mingwithpartiallyobservableMarkovdecisionprocessestoamilitaryattack
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14:6 • T.T.Wuetal.
probleminwhichaircraftattacktargetsinaseriesofstages.Theyassumethat
theexpectedvalueofthereward(destroyedtargets)andresource(weapon)con-
sumptionareknownforeachpolicywhereapolicyischoseninafinitefeasible
set.Theirmethodrequiresthatthenumberofpossiblestatesofeachobjectbe
smallandtheresourceconstraintsbesatisfiedontheaverage.Inthemilitary
airliftproblem,thenumberofpossiblestatesisverylarge,asarethenumberof
actionsandoutcomes.Asaresult,wecouldnothavetheexpectedvalueofthe
reward before we solve the problem. In general, we require that the resource
constraintsarestrictlysatisfiedinamilitaryairliftproblem.
2.2 TheNRMOModel
TheNRMOmodelhasbeenindevelopmentsince1996andhasbeenemployed
in several airlift studies. For a detailed review of the NRMO model, including
a mathematical description, see Baker et al. [2002]. The goal of the NRMO
model is to move equipment and personnel in a timely fashion from a set of
origin bases to a set of destination bases using a fixed fleet of aircraft with
differing characteristics. This deployment is driven by the movement and de-
liveryrequirementsspecifiedinalistcalledtheTime-PhasedForceDeployment
Document (or Dataset) (TPFDD). This list essentially contains the cargo and
troops,alongwiththeirattributes,thatmustbedeliveredtoeachofthebases
ofagivenmilitaryscenario.
Aircraft types for the NRMO runs reported by Baker et al. [2002] include
C-5, C-17, C-141, Boeing 747, KC-10, and KC-135. Different types of aircraft
havedifferentfeatures,suchaspassengerandcargo-carryingcapacity,airfield
capacity consumed, range-payload curve, and so on. The range-payload curve
specifiestheweightthatanaircraftcancarrygivenadistancetraveled.These
range-payloadcurvesarepiecewiselinearconcave.
The activities in NRMO are represented using three time-space networks:
the first one flows aircraft, the second the cargo (freight and passengers) and
thethirdflowscrews.Cargoandtroopsarecarriedfromtheonloadaerialport
ofembarkation(APOE)toeithertheoffloadaerialportofdebarkation(APOD)
ortheforwardoperatingbase(FOB).Certainrequirementsneedtobedelivered
totheFOBviasomeothermeansoftransportationafterbeingoffloadedatthe
APOD. Some aircraft, however, can bypass the APOD and deliver directly to
the FOB. Each requirement starts at a specific APOE dictated by the TPFDD
andtheniseitherdroppedoffatanAPODoraFOB.
NRMO makes decisions about which aircraft to assign to which require-
ment, which freight will move on an aircraft, which crew will handle a move,
as well as variables that manage the allocation of aircraft between roles such
as long range, intertheater operations and shorter, shuttle missions within a
theater.ThemodelcanevenhandletheassignmentofKC-10sbetweentherole
ofstrategiccargohaulerandmidairrefueler.
In the model, NRMO minimizes a rather complex objective function that
is based on several costs assigned to the decisions. There are a variety of
costs, including hard operating costs (fuel, maintenance) and soft costs to
encouragedesirablebehaviors.Forexample,NRMOassignsacosttopenalize
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TheOptimizing-Simulator:AnIllustrationUsingtheMilitaryAirliftProblem • 14:7
deliveries of cargo or troops that arrive after the required delivery date.
This penalty structure charges a heavier penalty the later the requirement
is delivered. Another term penalizes the cargo that is simply not delivered.
The objective also penalizes reassignments of cargo and deadheading crews.
Last,theobjectivefunctionoffersasmallrewardforplanesthatremainatan
APOE, as these bases are often in the continental US and are therefore close
to the home bases of most planes. The idea behind this reward is to account
foruncertaintyintheworldofwarandtokeepunusedplaneswellpositioned
incaseofunforeseencontingencies.
Theconstraintsofthemodelcanbegroupedintosevencategories:(1)demand
satisfaction;(2)flowbalanceofaircraft,cargoandcrews;(3)aircraftdeliveryca-
pacityforcargoandpassengers;(4)thenumberofshuttleandtankermissions
perperiod;(5)initialallocationofaircraftandcrews;(6)theusageofaircraftof
each type; and (7) aircraft handling capacity at airfields. A general statement
ofthemodel(seeBakeretal.[2002]foradetaileddescription)isgivenby:
(cid:2)T
min c x (1)
t t
(xt),t=0,...,T t=0
subjectto
(cid:2)t
Atxt − Bt−τ,txt−τ = Rt, t =0,1,...,T, (2)
τ=1
D x ≤ u , t =0,1,...,T, (3)
t t t
x ≥ 0, t =0,1,...,T, (4)
t
where,
t = thetimeatwhichanactivitybegins,
τ = thetimerequiredtocompleteanaction,
A = incidence matrix giving the elements of x that represent
t t
departuresfromtimet,
Bt−τ,t = incidence matrix giving the elements of xt that represent
flowsarrivingattimet,
D = incidence matrix capturing flows that are jointly con-
t
strained,
u = upperboundsonflows,
t
R = theresourcesavailableattimet,
t
x = thedecisionvectorattimet,whereanelementmightbex
t tij
telling us if aircraft i is assigned to requirement j at time
t,
c = thecostvectorattimet.
t
NRMOisimplementedwiththealgebraicmodelinglanguageGAMS[Brooke
etal.1992],whichfacilitatesthehandlingofstates.Eachstateisidentifiedby
an index combination of elements drawn from sets of aircraft attributes such
as time periods, requirements, cargo types, aircraft types, bases, and routes.
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14:8 • T.T.Wuetal.
Anexampleofanindexcombinationwouldbeanaircraftofacertaintypede-
livering a certain requirement on a certain route departing at a certain time.
Onlyfeasibleindexcombinationsareconsideredinthemodelsothatthecom-
putation becomes tractable. The information in the NRMO model is captured
intheTPFDD,whichisknownatthebeginningofthehorizon.
NRMOrequiresthatthebehaviorofthemodelbegovernedbyacostmodel
(whichsimulatorsdonotrequire).Theuseofacostmodelminimizestheneed
forextensivetablesofrulestoproducegoodbehaviors.Theoptimizationmodel
also responds in a natural way to changes in the input data (for example, an
increase in the capacity of an airbase will not produce a decrease in overall
throughput).Butlinearprogrammingformulationssufferfromweaknesses.A
significant limitation is the difficulty in modeling complex system dynamics.
For example, a simulation model can include logic such as, “if there are four
aircraftoccupyingalltheparkingspaces,afifthaircraftwillhavetobepulled
off the runway where it cannot be refueled or repaired.” In addition, linear
programmingmodelscannotdirectlymodeltheevolutionofinformation.This
limits their use in analyzing strategies that directly affect uncertainty (What
istheimpactoflast-minuterequestsonoperationalefficiency?Whatisthecost
of sending an aircraft through an airbase with limited maintenance facilities,
wheretheaircraftmightbreakdown?).
2.3 TheAMOSModel
AMOS is a rule-based simulation model. A rough approximation of the rules
proceeds as follows. AMOS starts with the first available aircraft, and then
triestoseeiftheaircraftcanmovethefirstrequirementthatneedstobemoved
(thereislogictocheckifthereisanaircraftattheoriginoftherequirement,but
otherwise the distance the aircraft has to travel is ignored). Given an aircraft
and requirement, the next problem is to evaluate the assignment. For simple
transportationmodels,thisstepistrivial(e.g.acostpermiletimesthenumber
of miles). For more complex transportation problems (managing drivers), it is
necessarytogothroughamorecomplexsetofcalculationsthatdependonthe
hoursofservice.Forthemilitaryairliftproblem,movinganaircraftfrom,say,
the East Coast to India, requires moving through a sequence of intermediate
airbases that have to be checked for capacity availability. This step is fairly
expensive,soitisdifficulttoevaluateallpossiblecombinationsofaircraftand
requirements.IfAMOSdeterminesthatanassignmentisinfeasible,itsimply
movestothenextaircraftinthelist.
Wenotethatitistraditionaltodescribesimulationmodelssuchasthiswith
virtually no notation. We can provide a very high level notational system by
defining:
S = the state vector of the system at time t, giving what we
t
knowaboutthesystemattimet,suchasthecurrentstatus
ofaircraftandrequirements;
Xπ(S ) = thefunctionthatreturnsadecision,x ,giveninformation
t t
S ,whenweareusingpolicyπ,whereπ issimplyanindex
t
identifyingthespecificfunctionbeingused;
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TheOptimizing-Simulator:AnIllustrationUsingtheMilitaryAirliftProblem • 14:9
t
1
Attack
t
2
e
m
Ti
t
3
t
4
Fig.1. ThesingleattackscenariotreeusedinSSDM.
x = the vector of decisions at time t (just as in the NRMO
t
model);
W = exogenousinformationarrivingbetweent−1andt;
t
SM(St,xt,Wt+1) = the transition function (sometimes called the system
model)thatgivesthestate St+1,attimet+1,giventhat
weareinstate S ,makedecision x ,andobservenewex-
t t
ogenousinformation Wt+1.
π
Givenapolicy(thatis,adecisionfunction X (S )),asimulationmodelcanbe
t
viewedasconsistingofnothingmorethanthetwoequations:
x ← Xπ(S ), (5)
t t
St+1 ← SM(St,xt,Wt+1(ω)). (6)
Amorecompletemodelwouldspecifythestatevariableandtransitionfunction
ingreaterdetail.Notethatwedonothaveanobjectivefunction.
2.4 TheSSDMModel
TheStochasticSealiftDeploymentModel(SSDM)[Mortonetal.2002]isamul-
tistage stochastic mixed-integer program designed to hedge against potential
enemy attacks on seaports of debarkation (SPODs) with probabilistic knowl-
edgeofthetime,locationandseverityoftheattacks.Theytestthemodelona
networkwithtwoSPODs(thenumberactuallyusedintheGulfWar)andother
locations such as a seaport of embarkation (SPOE) and locations near SPODs
whereshipswaittoenterberths.Tokeepthemodelcomputationallytractable,
they assumed only a single biological attack can occur. Thus, associated with
eachpossibleoutcome(scenariointhelanguageofstochasticprogramming)is
whetherornotanattackoccurs.Weassumethatthereisafinitesetofscenar-
ios (cid:5), and we let ω ∈ (cid:5) be a single scenario. If scenario ω occurs, t(ω) is the
timeatwhichitoccurs.TheythenletT(ω)={0,...,t(ω)−1}bethesetoftime
periodsprecedinganattackthatoccursattimet(ω).Thisstructureresultsina
scenariotreethatgrowslinearlywiththenumberoftimeperiods,asillustrated
inFigure1.ThestochasticprogrammingmodelforSSDMcanthenbewritten:
(cid:2)(cid:2)T
min p(ω)c (ω)x (ω), (7)
t t
xt(ω),t=0,...,T,ω∈(cid:5)ω∈(cid:5)t=0
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14:10 • T.T.Wuetal.
subjectto
(cid:2)t
Atxt(ω)− (cid:6)t−τ,txt−τ(ω) = Rˆt(ω) t =0,1,...,T, ω∈(cid:5),
τ=1
B x (ω) ≤ u (ω), t =0,1,...,T, ω∈(cid:5),
t t t
x (ω) = x (ω(cid:7)), forallt ∈T(ω)∩T(ω(cid:7)), ω∈(cid:5), (8)
t t
x (ω) ≥ 0 andinteger,forallt =0,1,...,T, ω∈(cid:5).
t
Here ω is the index for scenarios. The probability of scenario ω, is given by
p(ω).Thevariablest,τ, At,(cid:6)t−τ,t,Bt,and Rˆt arethesameasintroducedinthe
NRMO model. We let u (ω) be the upper bound on flows, where the capacities
t
of SPOD cargo handling are varied under different attack scenarios. The cost
vector for scenario ω is given by c (ω), and x (ω) is the decision vector. We let
t t
Rˆ (ω) be the exogenous supply of aircraft that first become known at time t.
t
The nonanticipativity constraints (8) say that if scenarios ω and ω(cid:7) share the
samebranchonthescenariotreeattimet,thedecisionattimet shouldbethe
sameforthosescenarios.
In general, the stochastic programming model is much larger than a deter-
ministiclinearprogrammingmodelandstillstruggleswiththesamedifficulty
inmodelingcomplexsystemdynamics.Multistagestochasticprogrammingno-
toriouslyexplodesinsizewhentherearemultiplestages,andmultiplescenar-
ios per stage. This particular problem exhibited enough structure to make it
possible to construct a model that was computationally tractable. As a result,
the SSDM model was able to make it possible to formulate the optimization
problemwhilecapturingtheuncertaintyinthepotentialattacks.
2.5 AnOverviewoftheOptimizing-Simulator
In the remainder of this article, we are going to describe a way to optimize
complex problems using the same framework as that described by Equations
(5)and(6).Theonlydifferencewillbeintheconstructionofthedecisionfunction
π
X (S ). In many simulation models, this function consists of a series of rules.
t
Typically, these rules are designed to mimic how decisions are actually made,
andthereisnoattempttofindthebestdecisions.
It is easy to envision a decision function that is actually a linear program,
whereweoptimizetheuseofresourcesusingwhatweknowatapointintime,
butignoringtheimpactofcurrentdecisionsonthefuture.Thisisanexampleof
acost-basedpolicy,andlikearule-basedpolicy,itwouldalsobecalledamyopic
policy, because it ignores the future. Alternatively, we could optimize over a
horizont,t+1,...,t+T,andthenimplementthedecisionwechooseattime
t. This is classically known as a rolling-horizon procedure. There are other
techniques to build ever more sophisticated decision functions that produce
bothmoreoptimalbehaviors(asmeasuredbyanobjectivefunction)andmore
realisticbehaviors(asmeasuredbythejudgmentofanexpertuser).
AsindicatedbyEquations(5)and(6),thedecisionfunction(whichcontains
theoptimizinglogic)andthetransitionfunction(whichcontainsthesimulation
logic)communicateprimarilythroughthestatevariable.Thedecisionfunction
ACMTransactionsonModelingandComputerSimulation,Vol.19,No.3,Article14,Publicationdate:June2009.