Optimal Facility Location Model Based on Genetic Simulated Annealing Algorithm for Siting Urban ...

14 Pages·2015·2.33 MB·English

Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2015, Article ID 981370, 13 pages
http://dx.doi.org/10.1155/2015/981370
Research Article
Optimal Facility Location Model Based on Genetic Simulated
Annealing Algorithm for Siting Urban Refueling Stations
DaweiChen,1ShunZhou,1YuanchangXie,2andXuhongLi1
1SchoolofTransportation,SoutheastUniversity,Nanjing210096,China
2DepartmentofCivilandEnvironmentalEngineering,UniversityofMassachusettsLowell,Lowell,MA01854,USA
CorrespondenceshouldbeaddressedtoYuanchangXie;yuanchang [email protected]
Received2October2014;Accepted18February2015
AcademicEditor:MarekLefik
Copyright©2015DaweiChenetal.ThisisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense,
whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperlycited.
Thispaperanalyzestheimpactfactorsandprinciplesofsitingurbanrefuelingstationsandproposesathree-stagemethod.The
mainobjectiveofthemethodistominimizerefuelingvehicles’detourtime.Thefirststageaimsatidentifyingthemostfrequently
traveledroadsegmentsforsitingrefuelingstations.Thesecondstagefocusesonaddingadditionalrefuelingstationstoservevehicles
whosedemandsarenotdirectlysatisfiedbytherefuelingstationsidentifiedinthefirststage.Thelaststagefurtheradjustsand
optimizestherefuelingstationplangeneratedbythefirsttwostages.Ageneticsimulatedannealingalgorithmisproposedtosolve
theoptimizationprobleminthesecondstageandtheresultsarecomparedtothosefromthegeneticalgorithm.Acasestudyis
alsoconductedtodemonstratetheeffectivenessoftheproposedmethodandalgorithm.Theresultsindicatetheproposedmethod
canprovidepracticalandeffectivesolutionsthathelpplannersandgovernmentagenciesmakeinformedrefuelingstationlocation
decisions.
1.Introduction Someexistingrefuelingstationlocationmodelsarebased
onqualitativeanalysis.Thesemethodscangenerallybecate-
WiththerapideconomicdevelopmentinChina,automobile gorizedintotwogroups:maximumcoverageandmaximum
ownershiphasincreaseddrasticallyinthepastdecade.Due profit methods [1]. The maximum coverage methods are
tothis,thereisanurgentneedforupgradingtransportation mainlyusedforhighpopulationdensityandcongestedareas.
infrastructure to accommodate the newly generated traffic. Their objective is to maximize the coverage of refueling
Refuelingstations,eithergasolinestationsorelectricvehicle stationsandreducetheunnecessarytraveldistancecausedby
chargingstations,areaveryimportanttypeofservicefacility vehiclessearchingforrefuelingstations.Themaximumprofit
and are essential for the appropriate operations of urban methodsaredesignedforlowpopulationdensityareas.Their
transportationsystems.Thelocationsofrefuelingstationscan primary goal is to ensure that refueling stations at selected
directlyaffecttheefficacyofurbanroadtrafficsystem.Inap- locations can have adequate demand to maintain certain
propriatelocationdesignscansubstantiallyincreasevehicle levelofprofit[2].Bothmethodscannotprovidequantitative
travel distances for finding refueling stations and cause results to demonstrate the economic and traffic impacts of
unnecessary congestion and emissions. Therefore, investi- differentrefuelingstationlocationplans.
gatingtheoptimalrefuelingstationlocationshasimportant Several related quantitative researches have also been
practicalsignificancefordevelopingcountriessuchasChina. conducted. Current et al. [3] investigated a network design
Thisresearchisalsoimportantforsomedevelopedcountries problem.TheyformulateditasaMaximumCovering/Short-
thatareinvestingheavilyinalternative-fueledvehicles(e.g., estPath(MCSP)modelwithtwoobjectives.Thefirstobjec-
electricvehicles),whereneedsforoptimallylocatingelectric tive was to find the shortest path between any pair of start
vehiclechargingstationsexist. and end points, and the second objective was to maximize
2 MathematicalProblemsinEngineering
thenumberofdemandnodescoveredbytheshortestpath. presented. Based upon this analysis, a three-stage method
Intheirresearch,ifanodeisontheshortestpathorwithin isproposedtooptimallylocaterefuelingstations.Insteadof
certaindistanceofanynodesontheshortestpath,thenitis focusing on the regional level and considering each city as
considered as being covered. Bapna et al. [4] extended the a single demand point [3–5], this study centers on urban
MCSPformulationtoaMaximumCovering/ShortestSpan- areasandtheentirecityisdividedintosmalltrafficanalysis
ning Subgraph Problem (MC3SP) that considers multiple zones(TAZs).Inaddition,themodelintroducedinthisstudy
pairsofstartandendpoints.Thefocusoftheirresearchwasto explicitlyconsidersthediverseroutechoicebehaviorbetween
optimallylocateunleadedgasolinestationstofacilitatetrips origins and destinations and can better reflect real world
amonglargeandpopulatedcities.BuildingupontheMCSP trafficscenarios.
and MC3SP, Y.-W. Wang and C.-R. Wang [5] formulated a
flow-basedsetcoveringlocationmodel.Similarto[3,4],they 2.Principlesfor SitingUrban
assumedtherefuelingdemandofacitytobeconcentratedon
RefuelingStations
thecenterofthecity.In otherwords,acitywasconsidered
as a single demand point. All these models [3–5] tried to
2.1. Principles of Location Design. Siting urban refueling
cover as many nodes or population along the shortest path
stationsshouldconsideracomprehensivesetoffactors.Gov-
aspossiblewhilekeepingthetotalcostlow.However,amore
ernment agencies are mostly concerned about congestion,
important issue in refueling station location modeling is to
safety, environment, and land use factors, while refueling
covermoreOD(originanddestination)flowswithrefueling
station operators are more concerned about demand and
demand. Also different from these studies, the focus of our
profit. Besides, fuel price difference at various refueling
researchistoinvestigaterefuelingstationlocationproblems
stationsisalsoanimportantfactortoconsider.However,in
withinurbanareasratherthanforintercitytrips.
China gasoline prices are decided by the government and
GoodchildandNoronha[6]proposedacombinedmodel
thereisalmostnodifferenceintermsofpriceamongdifferent
tositeafinitenumberofrefuelingstationstomaximizethe
gasstations.Therefore,thepricecompetitionamongdifferent
refuelingmarketshareofafirm.AspointedoutbyKubyand
refuelingstations/companiesisnotconsideredinthisstudy.
Lim [7], a major problem of this combined model is that it
In general, siting refueling stations should (a) minimize
used link flow data instead of OD flow data as the input.
their impacts on urban traffic, (b) be on main roads to
Therefore, OD flows may be double-counted as they pass
ensureprofitability,(c)keepanappropriatedistancebetween
through several road links. Hodgson [8] conducted one of
refueling stations, (d) be far from residential and other
thepioneerstudiesonsitingrefuelingstationsanddeveloped
sensitiveareas,and(e)bebasedonurbanlanduseplanning.
a Flow CapturingLocationModel (FCLM). The goal of the
Principle (a) is from the perspective of minimizing the
FCLMwastocapturemoreODflows(insteadofmorelink
influence of refueling stations on urban road traffic. For
flowsasin[6])ontheshortestpathsbetweenODpairs.This
example, a vehicle with refueling need should not have to
can avoid double-counting OD flows captured by multiple
detourtoofarforrefueling.Intuitively,thisrequiresrefueling
refueling stations along their shortest paths. A number of
stationstobelocatedonornearthepathsofmostvehicles,so
improvementshavebeendonebasedontheFCLM.Hodgson
thattheydonotneedtomakeadetourorjustneedtomake
and Rosing [9] extended the FCLM and proposed a hybrid
averyshortdetour.
flow capturing plus p-median model with two conflicting
Principle(b)considerstheeconomicbenefitofrefueling
objectives,whichweremaximizingthecoverageofODflows
stations. The number of refueling vehicles that may poten-
andminimizingdemand-weightedtraveldistance.Lateron,
tially be served by a refueling station should be taken into
KubyandLim[7]proposedaFlow-RefuelingLocationModel
considerationduringthelocationdesign.Ideally,thisnumber
(FRLM) based on the FCLM. A major difference between
shouldbewithinareasonablerangetoensurethatarefueling
FCLM and FRLM is that a facility combination concept
station’sprofitisabovecertainlevelandtherefuelingdemand
was proposed. OD flows were covered by a combination of
doesnotexceeditscapacity.
facilitiesinsteadofindividualfacilitiesintheFRLMmodel.
Recently,LimandKubyproposedanumberofheuristicalgo- Principle (c) takes the service ranges of refueling sta-
rithmsforsolvingthedevelopedFRLM[10,11].Thesestudies tions into consideration. It is to make sure that refueling
[7–10]allconsideredasingleshortestpathconnectingeach stationsarenottooconcentratedincertainareas.Keepingan
ODpair.Theircommonobjectivewastooptimallylocatea appropriatedistancebetweenrefuelingstationssometimesis
prespecifiedamountofrefuelingstationssuchthatthetotal necessaryduetoreasonssuchassafety.
numberofshortestpathsbeingcoveredcouldbemaximized. Principle(d)ismainlyaboutthesafetyandenvironmen-
Intherealworld,trafficflowbetweenanODpairrarelytakes talconcerns.Basedonthisprinciple,sitingrefuelingstations
asingleshortestpath.Travelerstypicallyfollowtheuserequi- should consider the safety of refueling station facilities and
librium(UE)principleandtrytominimizetheirindividual equipment,refuelingvehicles,travelers,andnearbyresidents.
travel times. This usually results in many different routes Also, refueling stations should be far away from some
betweenanODpair.Unfortunately,thisdiverseroutechoice environmentallyandsociallysensitiveareas.
behaviorisnotconsideredintheserefuelingstationlocation Lastbutcertainlynotleast,principle(e)optimizesrefuel-
studies[7–10]. ingstationlocationsfromtheurbanplanningpointofview.
In the following subsection, a comprehensive analysis Specifically,urbanrefuelingstationlocationsshouldtakeinto
of factors affecting refueling station location planning is accountlandusedevelopmentandfuturelanduseplanning.
MathematicalProblemsinEngineering 3
Land use directly affects trip productions and attractions. simplyremoved/unselected.Forthelackofcapacityscenario,
Consequently, it has major impacts on the distribution of additional refueling stations need to be added. To add new
futurerefuelingdemandandrefuelingstationlocations. refuelingstations,thefirststepistofindrefuelingvehicleODs
Fromthestandpointofmodeldevelopmentforoptimally being assigned to the overloaded refueling stations. Next,
siting refueling stations, principle (a) can be considered in road segments on these OD pairs’ paths and the refueling
theobjectivefunction,andprinciples(b)through(e)canbe trafficvolumeoneachofthesesegmentsareidentified.Based
eitherincludedasconstraintsorusedtodeterminecandidate on the principles discussed in the previous section, these
locationsforfuturerefuelingstations. segments’refuelingvolumesarethensortedinadescending
orderandusedtoselectnewrefuelingstations.Duringthis
2.2.Three-StageModel. Basedontheanalysisintheprevious adjustmentprocess,atleastonenewstationthathelpswith
section,ifprinciple(a)isadoptedastheobjective,therewill diverting refueling demand should be selected. The new
be two cases in the refueling station location planning. The locationplanmayneedtobefedintothesecondstagetobe
firstcaseisthatrefuelingstationsarelocatedonthepathsof furtherimproved.Thisthree-stage modelissummarizedin
somerefuelingvehicles.Hence,thesevehiclesdonotneedto Figure1.
detourforrefueling.Theothercaseisthatrefuelingstations
are not the paths of the remaining refueling vehicles and
3.OptimizationModelfor SitingUrban
these vehicles have to detour for refueling. In light of these
RefuelingStations
two cases, siting refueling stations can be implemented in
three stages. In the first stage, candidate refueling stations
3.1.VariablesandRefuelingDemand
on road segments traveled by most refueling vehicles are
selected.Inthesecondstage,thefocusisonrefuelingvehicles
3.1.1. Definitions of Variables and Parameters. Given a road
that are not covered in the first stage. The objective is network𝐺(𝑁,𝐴),𝑁isthesetofallnodesin𝐺(𝑁,𝐴),and𝐴is
to minimize the total detour distance of these vehicles by thesetofallroadsectionsin𝐺(𝑁,𝐴),𝑎𝑚 (𝑚=1,2,...,𝑛)∈
properly selecting additional refueling stations. The third 𝐴.Supposeallmajorandminorarterialsegments,𝑐𝑚 (𝑚 =
stageisbasedontheresultsofthefirstandsecondstages.It 1,2,...,𝑛) ∈ 𝐶, are considered as candidate locations for
considerstheredistributionofthepathsforrefuelingvehicles
refueling stations. Before the models are introduced, the
duetotrafficcongestionandrefuelingstationcapacities.The
following variables and parameters used in the models are
redistribution result is used to fine-tune the location plan
definedbelow.
generatedinthefirstandsecondstages.
Specifically,thefirststagestartswithassigningallvehicles (1)𝑥𝑎 isthetrafficflowofroadsegment𝑎𝑖;
𝑖
(including refueling vehicles) to the road network. Based (2)𝑡𝑎 isthetraveltimeofroadsegment𝑎𝑖;
on the traffic assignment results, the paths of refueling 𝑖
vehiclesareidentified.Fromthesepaths,roadsegmentsthat (3)𝜁 is a binary variable. 1 stands for constructing a
coverthehighestrefuelingvehicleODdemandareselected stationand0fornot;
for constructing refueling stations. The refueling ODs cov- (4)𝑀istheminimumrefuelingdemandforconstructing
ered/servedbytheseroadsegmentsarethenremovedfrom
arefuelingstation;
theoriginalrefuelingvehicleOD.TheregularvehicleODand
theunsatisfiedrefuelingvehicleODareassignedagaintothe (5)𝑆istheserviceareaoftheroadnetwork;
roadnetworktofindadditionalcandidatelocations. (6)𝑝 (𝑝=1,2,...,𝑛)isthesetofcandidatelocationsfor
Attheendofthefirststage,thedemandsofsomerefueling constructingrefuelingstations;
ODs may still be unmet. Therefore, additional refueling
stationsneedtobeconstructedandsomerefuelingvehicles (7)𝑏𝑝 is the cost of constructing a refueling station
havetodetour.Whenselectinglocationsfortheseadditional at candidate location 𝑝, and 𝐵max stands for the
maximumaveragecost;
refueling stations, it is natural to consider the objective of
minimizingthetotaldetourdistance. (8)𝑡𝑝𝑝󸀠isthetraveltimebetweenrefuelingstations𝑝and
In the first and second stages, refueling vehicles are 𝑝󸀠andshouldbeintherangeof[𝑡𝑑min,𝑡𝑑max];
assignedtotheroadnetworkinanincrementalmannerand
(9)𝜌 stands for the density of refueling stations and its
refueling station locations are added gradually. In the third
stage,allvehicles(includingrefuelingvehicles)areassigned rangeshouldbe[𝜌min,𝜌max];
to the network simultaneously given the locations selected (10)𝑄𝑖𝑗isthetraveldemandbetweenorigin𝑖anddestina-
inthefirstandsecondstages.Thismostlikelywillresultin tion𝑗and𝑞𝑖𝑗isthecorrespondingrefuelingdemand;
a traffic flow pattern (e.g., choice of stations for refueling
(11)𝑓𝑖𝑗isthetrafficvolumeonpath𝑘betweenorigin𝑖and
vehicles) that is different from those in the previous two 𝑘
destination𝑗.
stages. For instance, some previously overloaded refueling
stations(basedontheresultofthesecondstage)mayendup
withnothavingenoughdemand.Forthosewithoutenough 3.1.2.VehicleRefuelingDemand. Exceptforthevehiclerefuel-
demand in the second stage, they may not have adequate ingdemand𝑞𝑖𝑗,mostofthevariablesandparametersdefined
capacity to satisfy the refueling demand. For the lack of in the previous section either are given or can be readily
demandscenario,thecorrespondingrefuelingstationscanbe calculated.Thevehiclerefuelingdemandhoweverisclosely
4 MathematicalProblemsinEngineering
Layout model of first stage based on FCLM
Urban road network, vehicle travel OD in
future and refueling vehicle OD
Searching maximum section
of refueling vehicles
Setting station location and
finishing the redistribution
Whether reaching of traffic volume
the minimum Yes
demand for building
refueling station
No
End of layout model of first stage
Layout model of second stage for arranging public
refueling station
Establishing layout model
Model solution algorithm
Public refueling station location
End of layout model of second stage
Layout model of third stage
Updating road network and traffic assignment
Judging level of service for each refueling
station
Whether the No
refueling station location
is reasonable
Yes
Final location result
End of layout model of third stage
Figure1:Flowchartforthethreestages.
MathematicalProblemsinEngineering 5
relatedtothevehicletraveldemand(𝑄𝑖𝑗)andshouldbedeter-
Layout model of first stage based on
minedcarefully.
FCLM
For different OD pairs, even the same vehicle’s average
refuelingfrequenciescanbedifferent.Thisismainlydueto Urban road network, vehicle travel OD in
two factors. First, different OD pairs have different average future and refueling vehicle OD
traveldistances,whichnaturallyresultindifferentfuelcon- SUE
sumptions. Second, vehicles from different OD pairs may
Assigning vehicle travel OD
experiencedifferentlevelsofcongestion,whichwillalsoaffect
theirfuelconsumptions.Onaparticularday,traveldemand
ofanODpairisseparatedintodemandwithrefuelingneeds Searching maximum section
anddemandwithoutrefuelingneeds.Thedemandbetween for refueling vehicles
origin 𝑖 and destination 𝑗 with refueling needs is estimated
Updating road network
using (1). This study assumes that travelers do not have a
and distributing traffic volume
special preference for certain refueling stations. This is a
reasonableassumptionforplanningpurpose,sinceinChina SUE
gasolinepricesarethesameatdifferentstationsanditishard Removing the refueling vehicle
ODs which have met refueling
totellwhichrefuelingstation(s)willbemoreattractiveinthe
demand
future.Consider
𝐴(V)⋅𝑢 Adopting this section as the
𝑖𝑗
𝑞𝑖𝑗 =𝐷𝑖𝑗(𝑢𝑖𝑗)=𝑄𝑖𝑗⋅ 𝐶 , ∀𝑖,𝑗. (1) alternative losctaattiioonn of refueling
gas
Whether reaching the
Yes
In(1),𝐷𝑖𝑗 isthedemandfunctionbetween𝑖and𝑗.This minimum demand for building
refueling station
monotonicfunctiondependsonthetraveldistancebetween
𝑖and𝑗andhasanupperlimit;𝑢𝑖𝑗istheaveragetraveldistance
between 𝑖 and 𝑗 in kilometers; 𝐴(V) is the average vehicle No
fuelconsumptionmeasuredinliter/100km.Itisafunctionof
theactualdrivingspeedVinkm/hr,whichcanbeestimated End of layout model of first stage
usingthemethodin[11];and𝐶 istheaveragefueltanksize
gas
measuredinliters. Figure2:First-stagelocationmodelbasedonFCLM.
3.2.First-StageModelBasedonFCLM. Thefirst-stagemodel
is shown in (2) below and illustrated in Figure2. It is
built upon the Flow-Capturing Location-Allocation Model
(FCLM)proposedbyHodgson[8]andtheideaforthefirst Thisassignmentisbasedontheresultantroadnetworkofthe
stagediscussedpreviously.Onehas first stage. Thisbilevelsecond-stagemodelisalsoshownin
Figure3.Thethirdstagedoesnothaveaspecificmodel.Itis
Maximum ∑𝑞𝑎𝜁𝑎 simplytofine-tunetheresultofthesecond-stagemodel.
𝑖 𝑖
𝑎∈𝐴
𝑖
Step1(upper-levellocationmodel). Consider
s.t. 𝑞𝑎 ≥𝑀 (𝑎𝑖 ∈𝑃)
𝑖 (2)
𝜁 ={0 𝑎𝑖 ∉𝑃 min 𝑧(𝑥)=∑∑𝑞𝑖𝑗(𝑡𝑖󸀠𝑗−𝑡𝑖𝑗) (3)
𝑎 { 𝑖 𝑗
𝑖 1 𝑎 ∈𝑃.
{ 𝑖
s.t. 𝑡𝑑min <𝑡𝑝𝑝󸀠 <𝑡𝑑max 𝑝,𝑝󸀠 ∈𝑃 (4)
3.3. Location Models for the Second Stage. After completing
the first stage, the refueling demands of some vehicles are ∑ 𝜁 𝑏
𝑝∈𝑃 𝑝 𝑝
stillunmet.Whenaddingadditionalrefuelingstations(also ∑ 𝜁 ≤𝐵max (5)
calledpublicrefuelingstationsinthisstudy)toaddressthis 𝑝∈𝑃 𝑝
issue,ourobjectiveistominimizethetotaldetourdistance
∑ 𝜁
forthoseODswithunmetrefuelingdemands.Specifically,a 𝑝∈𝑃 𝑝
𝜌 ≤ ≤𝜌 . (6)
bilevellocationmodelisproposedthatreflectstheconcerns min 𝑆 max
ofbothgovernmentagenciesandtravelers.Intheupper-level
model,thetotaldetourtimeisusedastheobjectivefunction.
Constraintsconsideredincludetraveltimebetweenrefueling In (3), 𝑡𝑖󸀠𝑗 is the average travel time of refueling vehicles for
stations, average construction cost, and refueling station ODpair(𝑖,𝑗),𝑡𝑖𝑗standsfortheaveragetraveltimeofvehicles
density. The lower-level model assigns the unmet refueling thatdonotneedrefueling,andtheremainingvariableshave
ODstothenetworkusingStochasticUserEquilibrium(SUE). allbeendefinedpreviously.
6 MathematicalProblemsinEngineering
𝑓𝑖𝑗,𝑓𝑖𝑝,𝑓𝑝𝑗 ≥0, ∀𝑘,𝑖,𝑗,𝑝
Layout model of second stage for arranging public 𝑘 𝑘 𝑘
refueling station 𝑞󸀠,𝑞 ,𝑞 ≥0, ∀𝑖,𝑗,𝑝,
𝑖𝑗 𝑖𝑝 𝑝𝑗
𝑝∈𝑃,
(7)
Establishing upper-level layout model
𝑡 +𝑡 <𝑡 +𝑡 , 𝑝opt ∈𝑃,
𝑖𝑝opt 𝑝opt𝑗 𝑖𝑃󸀠 𝑃󸀠𝑠
(8)
Controlling reasonable interval 𝑃󸀠 ={𝑝|𝑝=𝑃−𝑝opt}.
between refueling stations
Shortest In constraint (8), 𝑝opt represents the refueling station
Controlling average cost of
refueling stations time for closesttoanODpairamongallcandidaterefuelingstations
detouring 𝑃and𝑃󸀠isthesetofrefuelingstationsexcluding𝑝opt.
Controlling arrangement
density of refueling stations
4.Algorithms
4.1.Overview. Thisstudyreliesontwocoremodels:thefirst-
stagemodelbasedontheFCLMandthesecond-stagemodel
Establishing lower-level layout model based on
foraddingandfine-tuningrefuelingstations.Thefirst-stage
Wardrop theory
modelisrelativelysimple.Thus,thispaperwillbrieflydiscuss
the algorithm for the first-stage model and will focus on
Genetic developingthealgorithmforsolvingthesecond-stagemodel.
simulated Comparison Genetic Thetrafficassignmentinthefirst-stagemodelisbasedon
annealing algorithm SUEassignment.Fromtheresultofeachassignment,aroad
algorithm +SUE
segmentwiththehighestrefuelingtrafficvolumeisselected.
+SUE
Thissegmentisselectedforconstructingarefuelingstationif
itsrefuelingvolumeisgreaterthanaprespecifiedvalue.The
correspondingODwillbeexcludedfromfurtherconsidera-
Public refueling station location
tions.Thisprocessrepeatsuntilnoqualifiedsegmentscanbe
foundforconstructingrefuelingstations.
Theupper-levelofthesecond-stagemodelisanonlinear
program and the lower-level is a SUE problem. Heuristic
End of layout model of second stage
algorithmsareverypopularforsolvingnonlinearprogram-
mingproblems.Forinstance,geneticalgorithm(GA)isoften
used for finding global optimal solutions, and simulated
Figure3:Locationmodelsofthesecondstage. annealing (SA) has good local search capability. This paper
proposes an integrated genetic simulated annealing (GSA)
algorithmtosolvetheupper-levelmodel,generatingfeasible
solutions for the lower-level model. The lower-level model
Step2(lower-levellocationmodel). Consider
basicallyperformsSUEtrafficassignmentsandprovidesthe
min 𝑍(𝑥,𝑞)= ∑ ∫𝑥𝑎𝑖 𝑡𝑎 (𝜔)𝑑𝜔 uanpdpeGr-SleAvelalgmooridtehlmwsitahre{𝑥co𝑎𝑖n,s𝑡𝑎id𝑖}e.reInd athnids tshtuediry,rebsoutlhtsGarAe
𝑎∈𝐴 0 𝑖 compared.
𝑖
𝑞
−∑∫ 𝑖𝑝𝐷−1(𝜔)𝑑(𝜔) 4.2.GeneticSimulatedAnnealing(GSA)Algorithm. GAwas
𝑖𝑝
𝑖𝑝 0 motivated by Darwin’s theory of natural selection, while
SA was inspired by the process of heating and cooling of
−∑∫𝑞𝑝𝑗𝐷−1(𝜔)𝑑(𝜔) materials in a controlled setting. By integrating these two
𝑝𝑗 algorithms,wehopetofurtherimproveGAbyenhancingits
𝑝𝑗 0
localsearchwithSA.
The proposed algorithm consists of an inner loop and
𝑞󸀠
−∑∫ 𝑖𝑗𝐷−1(𝜔)𝑑(𝜔) anouterloop,whicharebasedonGAandSA,respectively.
𝑖𝑗
𝑖𝑗 0 Theinnerloopgeneratesbinaryvectors(e.g.,chromosomes)
representingvariousrefuelingstationlocationsolutionsand
s.t. ∑𝑎 𝑓𝑘𝑖𝑗 =𝑞𝑖󸀠𝑗, ∑𝑎 𝑓𝑘𝑖𝑝 =𝑞𝑖𝑝, ∑𝑎 𝑓𝑘𝑝𝑗 =𝑞𝑝𝑗, eovuateluratloesopthaenirdfiatnppeslise.sThSAe atlogofruitrhthmerthimenprsowvietcthhees fitotntehses
of these chromosomes locally. Upon finishing the local
∀𝑖,𝑗,𝑝 search, the algorithm returns to the inner loop and applies
MathematicalProblemsinEngineering 7
crossoverandmutationoperatorstoavoidbeingstuckatlocal of𝐴1and𝐴2are𝑓1and𝑓2,respectively;thecrossoverprocess
minimumsolutions.Thecrossoverandmutationoperators, iscarriedoutasfollows.
initialtemperatureofannealing,andtheannealingrateareall
(a)Calculate the probabilities for genes to be selected
importantfactorsthatmayaffectthealgorithm’sperformance
and should be carefully considered. The following steps from𝐴1and𝐴2
describetheproposedalgorithminmoredetail. 𝑓
For 𝐴1 : 𝑃1 = 𝑓 +1𝑓 ,
Step1. Sincethisisessentiallyafacilitylocationproblem,a 1 2
binaryvectoroflength𝑛isusedtorepresenteachrefueling 𝑓 (9)
stationlocationsolution.Eachvectorelement(oreachgene) For 𝐴2 : 𝑃2 = 𝑓 +2𝑓 .
represents a candidate location. 1 means the corresponding 1 2
location is selected to construct a refueling station, and 0
means it is not selected. Pseudorandom number generator (b)Generate a sequence 𝛼𝑖 (𝑖 = 1,2,...,𝑘) with each
(PRNG)isusedtogenerateinitialrefuelingstationlocation elementbeingarandomnumberwithin[0,1].Com-
solution set Pop(𝑘), 𝑘 = 0. The solution set size is Popsize, pare each element in this sequence with 𝑃1 or 𝑃2 to
meaningitincludesPopsizesolutions.Also,theinitialtem- determinewhetheragenefrom𝐴1 or𝐴2 shouldbe
panerdattuhreemisasxeitmtou𝑇m0;nfiunmalbteermopfeirteartuatrieoinss𝑇f𝑒o;rcothoelinogutrearteloiso𝑇p𝑅is; gseelneectferdo.mFo𝐴r1exwaimllpblee,siefle𝑃c1te<d𝑃fo2r,ftohre𝛼𝑖t𝑖h<g𝑃en1,etihnet𝑖hthe
settoOitermax. newsolution;otherwise,the𝑖thgenefrom𝐴2willbe
selected.
Step 2. This step calculates the fitness values for individual
solutions/chromosomes. Based on the results, the fittest
(ii)MutationOperator.Assumeaparentsolutionselectedfor
chromosome𝑖anditsfitnessvalue𝐹(𝑖)aredetermined.This mutation is 𝑋 = 𝑥1𝑥2𝑥3⋅⋅⋅𝑥𝑛. 𝑥𝑘 (𝑘 = 1,2,3...,𝑛) is the
chromosomeistheonethathas thebest objectivefunction mutationpointanditisdefinedonthedomainof[𝐴𝑘,𝐵𝑘].A
value (i.e., fitness value) among all chromosomes in the newsolutionisgeneratedbasedonmutationasfollows:
current population. Therefore, we set 𝑆best = 𝑖 and 𝐹min =
𝐹(𝑖).Also,𝑓𝑖(𝑇𝑘)isusedtodenotethetargetvalueforchro- {𝑥𝑘+𝑓(𝑡,𝐵𝑘−𝑥𝑘)rand(0,1)=0
mosome 𝑖 under temperature 𝑇𝑘, where 𝑖 ∈ Pop(𝑘) is a 𝑥𝑘󸀠 ={
refuelingstationlocationsolutionincurrentsolutionset. {𝑥𝑘+𝑓(𝑡,𝑥𝑘−𝐴𝑘)rand(0,1)=1 (10)
Step 3. The entire algorithm will stop upon reaching the 𝑓(𝑡,𝑦)=𝑦(1−𝑟𝛽(1−(𝑡/𝑇))),
finaltemperature𝑇𝑒anditwilloutputthebestchromosome
𝑆best and the optimal fitness value 𝐹min. Otherwise it will where 𝑟 is a random real number uniformly distributed
randomlyselectastate𝑗 ∈ 𝑁(𝑖)fromtheneighborhoodof between0and1,𝑡isthecurrentnumberofiterations,𝑇sets
each chromosome 𝑖 ∈ Pop(𝑘) in the current solution set. the maximum number of iterations, and 𝛽 is a prespecified
Thenewstate𝑗willbeeitheracceptedorrejectedaccording parameterandcanbeadjustedbasedon𝑡.Itissettoalarge
to the acceptance probability 𝑝𝑖𝑗 = min{1,exp[−((𝑓𝑗(𝑇𝑘) − valueatthebeginningandreducesgraduallyas𝑡increases.
𝑓𝑖(𝑇𝑘))/𝑇𝑘)]}.𝑓𝑖(𝑇𝑘)and𝑓𝑗(𝑇𝑘)arethetargetvaluesofstates rand(0,1)randomlyselects0or1astheoutput.Finally,the
𝑖 and 𝑗, respectively. This step takes Popsize iterations and mutatedchromosomesandtheirparentchromosomes𝑋are
generatesanewsolutionsetNewpop1forrefuelingstations. bothincludedinthesolutionset.
The mutation operator is applied to chromosomes with
Step 4. This step calculates the fitness values 𝑓𝑖(𝑇𝑘) = thehighestandlowestfitnessvalues.Applyingittothefittest
exp{−((𝑓𝑖(𝑇𝑘)−𝑓min)/𝑇𝑘)}ofallsolutionsinNewpop1,where chromosome is for accelerating the speed for local search,
𝑓min is the minimum fitness value among all solutions. At while applying it to the least-fit chromosome is to increase
theend,Popsizesolutions/chromosomesfromNewpop1are thediversityofthepopulationandtoavoidprematurecon-
selected with probabilities inversely proportional to their vergences.
fitnessvaluestoformanewsolutionsetNewpop2.
Step6. SetPop(𝑘+1)=mutpopandfindchromosome𝑖that
Step 5. Crossover and mutation are applied to Newpop2, minimizes𝑓𝑖(𝑇𝑘).If𝑓𝑖(𝑇𝑘) ≤ 𝑓min,set𝑆best = 𝑖and𝐹min =
whichgeneratecrosspopandmutpop,respectively.Thecross- 𝑓𝑖(𝑇𝑘).Finally,set𝑇𝑘 = 𝑇0exp(−𝐶⋅𝑘1/𝑁)and𝑘 = 𝑘+1and
overandmutationprocessesaredescribedbelow. gotoStep3.
5.CaseStudy
(i) Crossover Operator. For crossover, two parent chromo-
somes(representingtworefuelingstationlocationsolutions)
5.1.DescriptionofCaseStudyProblem
𝐴1and𝐴2areselectedtoformanewchromosome.Eachgene
ofthenewchromosomeisselectedfromthecorresponding 5.1.1.RoadNetworkandParameters. Tovalidatetheproposed
gene in either 𝐴1 or 𝐴2 based on probabilities inversely model and algorithm, a case study was conducted using
proportionaltotheirfitnessvalues.Supposethefitnessvalues the urban road network in Figures 4 and 5. This network
8 MathematicalProblemsinEngineering
159 3 48 2 59 27 69 47 79 66 89 85 99 100 108 117125 149 126 151 143 183 144
179160 6 491 5 6260 29 7460 49 8650 68 9840 99 611 741 150 181 184
1 87 182 151
11899691 200 5740 8 683201 31 748501 51 896671 7720 98998812 8991 111010010 110042118111009 112119 111452243 114486 111522278 115553 111794421 180 119145
22303 11 591 10 6322 33 7522 53 8712 90 301 120 123 114322 144 115429 157 177140 178 145816
3 14 52 13 63 35 73 55 83 74 93 93 102 106 111
124532034 193 5213 16 6344 37 7544 57 8734 76 9924 95 110503 108 112212 125114121 142 165130 159 117539 176 146817
119254 19 5154 18 6563 39 7565 59 8755 94 107 124 139 851 173 187
19 55 3 78 180 203 104 110113 127 120 140 131 161 138 174 148
1121205882100483822 2220004629215258711125915202260042721612245883316124911424624700648862144738561121922767076586642166380818877981679837 828391278 119969290258645 91128912620111123211901111109007556 111211522429111132211261280311916465 111233322184111122513153173311919778 211123366463 111116016224176293338324 111267626258 111176122913711693935796 111726903800 121123159820111880740189
Figure4:Roadnetworkwithnodeandsegmentnumbers.
consistedof128nodesand234segments.InFigure5,green demandwasestimatedtobe6,775pcu/h.Duetopagelimit,
lines represent major arterials, yellow lines are for minor detailedstepsfortheabovecalculationsarenotincludedin
arterials,andbluelinesdenotesecondaryroads.Thenumbers this paper. Also, based on the refueling station density and
nexttoeachsegmentareitscapacityandfree-flowtraveltime. capacityconstraints,itwasassumedthatbetween14and23
Theentirestudyareawasabout90km2.Inordertomitigate refuelingstationsshouldbeconstructed.
traffic congestion in downtown, no refueling stations were
allowedwithintheredcircleinFigure5.Also,itwasassumed
5.2.ModelingResults
thatnorefuelingstationscanbebuiltonsecondaryroads.
Additionally, the following assumptions were made: the 5.2.1.ResultsoftheFirst-StageModel. TransCAD5.0wasused
maximum average construction cost for a refueling station tosolvethefirst-stagemodel.AsshowninTable1,thefirst-
is 3 million Chinese yuan; the density of refueling stations stagemodelfinishedafter8iterations.Duringthisstage,the
2
is between 0.1 and 0.3 stations/km ; the minimum distance capacity of refueling stations was not considered. In other
between two refueling stations is 4km; and each station words, we assumed that a refueling station can satisfy all
shouldserveatleast300vehiclesinordertobeprofitableand passingvehicles’refuelingdemand.
nomorethan500vehiclesduetocapacityconstraint. Table1showsthattherefuelingdemandofSegment194
is 278pcu/h, which is less than 300pcu/h and does not
5.1.2. Vehicle Refueling Demand. It was assumed that each meet the minimum demand requirement for constructing
a refueling station. Therefore, the first-stage model stopped
node is the centroid of a Traffic Analysis Zone (TAZ). The
travel demand between two TAZs was estimated by 𝑓𝑖𝑗 = afterthe8thiteration.TheresultinTable1suggeststhatthere
𝑃𝑖𝑃𝑗/𝑑𝑖1𝑗.5, where 𝑃𝑖 and 𝑃𝑗 stand for the trip production of irsefnuoeloinbgvioduemscaonrdre.lThatieonrebseutlwtoeefnthseegfimrsetn-sttflagoewmanoddeslegismaelsnot
nodes 𝑖 and the trip attraction of node 𝑗, respectively. Both
showninFigure6.
𝑃𝑖and𝑃𝑗wereassumedtobe5000trips/dayforallnodes.𝑑𝑖𝑗 The stations selected during the first stage can directly
isthedistancebetweentwonodes. satisfyarefuelingdemandof3,430pcu/h,whichisabouthalf
Consideringpeak-hourfactorandmodesplit,thepeak- ofthetotaldemand,meaningthatapproximately50%ofthe
hourtraveldemandbetweentwoTAZswasestimatedtobe refueling vehicles do not need to detour. In the following
38,537pcu/h. By applying (1), the corresponding refueling section,theunmetrefuelingdemandwillbediscussed.
MathematicalProblemsinEngineering 9
42000402200400249384511211 22231240330102320230800300007000 1200002112001200351163170131131142221018231000303403000030301003000333000000000120112222117 3121421110120320383000060016408000005066505060056008910091080134212112110208041034000901304100000020202000002000125110211172132241112100102323000900089018050000012200112001200017631108218 113411216202390100000441000093152189006500650065341112102132107003601090010016323033201200021002113214120019123004091080060500831115710000020200021242131120236026000210330000 9314100105600006500653241110102266202030160080400 175122431003000303000 4311122201582007094000000500 411118611002442004200
420078 5126620910 1200171 51610600003411 150670000659851600800002521 51600901200170 51600806900655161010 15812005161020 2510002 5162070 1110065 5163040 7410003 5163000 6110024
424200004200420042001279563759441212327010282320205000032900020071200012102081142201039242000211207030016121300591380002333100205704001131110083210020090100812530000003063000302000111800952127701204056012000320105680700930009586080056005600944110829650212120268301030110020380004902000000202000002020078011795416510114121111213280224900709070240000010200120000211200210010392842702961 42113111211022038105090007113006000878111708189006500006006056550654131112201101232007980303000180800024208116516844100210120001200211200134112111120221039390000709441000003117621119112410000200000220000020231112401222620120633010030064400030 61110478189900655600000066550065 41211311222202602602020790050809000 31112022501171100000303000000333000 1421213322037022000001680190000509012048231980951100042020442000024
Figure5:Segmentcapacityandtraveltime.
Table1:Resultofthefirst-stagemodel.
Numberof Refuelingdemand
Segmentnumber Segmentdirection Segmentflow(pcu/h)
iterations (pcu/h)
1 125 112→121 4637 788
2 162 133→132 2902 610
3 111 105→106 3966 552
4 103 101→102 2341 421
5 56 75→74 2612 418
6 154 128→129 1983 333
7 98 98→107 1853 308
8 194 154→202 2141 278
5.2.2.ResultoftheSecond-StageModel mutationratewas0.0001∼0.1,crossoverprobabilitywas0.9,
mutationprobabilitywas0.04,andthemaximumnumberof
(i)ModelInitialization.Asdescribedpreviously,thesecond- iterationsOitermax=2000.
stage model first generated Popsize = 100 chromosomes The constraints in the second-stage model were con-
and each chromosome consisted of 151 genes. Each gene is sidered by introducing a penalty. All solutions not meeting
a binary variable, representing whether the corresponding constraints(4)∼(6)werepunishedbyamultiplicationof10.
segmentisselectedforconstructingarefuelingstation. Theelitiststrategywasadoptedtoselectchromosomestobe
includedinthenextgeneration,meaningchromosomeswere
(ii)ParameterSetting.Topreventprematureconvergence,the selected with probabilities inversely proportional to their
following parameters were used: initial temperature in the fitnessvalues.
annealingprocess𝑇0 = 1000,terminatingtemperature𝑇𝑒 =
0,coolingspeed𝑇𝑅 = 0.95,maximumnumberofiterations (iii)ModelingResult.VBandGISDKinTransCADwereused
foreachtemperaturewas100,crossoverratewas0.40∼0.99, toimplementthesecond-stagealgorithm.Amongthem,VB
10 MathematicalProblemsinEngineering
Table2:Resultofthesecond-stagemodel.
Numberof Shortesttotal
Detourtimesofothersolutions(min)
iterations detourtime(min)
1 26752 27078 28244 27927 28470 ⋅⋅⋅ 28154 28637
2 26066 27129 28141 28923 28228 ⋅⋅⋅ 27130 26639
3 25045 28991 26479 27130 27910 ⋅⋅⋅ 28192 28087
4 25384 28136 28467 26486 27130 ⋅⋅⋅ 25631 26177
5 25218 25596 25922 26140 26551 ⋅⋅⋅ 26841 25676
6 23806 24469 25921 26685 24534 ⋅⋅⋅ 25170 25191
7 22362 22796 22344 22867 22845 ⋅⋅⋅ 25917 22948
8 22269 23082 23505 23600 22796 ⋅⋅⋅ 22796 25692
... ... ... ... ... ... ... ... ...
872 6991 8729 9455 7086 7138 ⋅⋅⋅ 7283 7136
873 6782 8007 8795 6926 6792 ⋅⋅⋅ 8869 6936
874 6688 6688 6688 8360 8024 ⋅⋅⋅ 6688 6688
875 6688 6688 6688 8125 7727 ⋅⋅⋅ 6688 6688
876 6688 6688 6688 6688 6688 ⋅⋅⋅ 6688 6688
×107
80
70
60
6
4 50
40
1 30
5
20
10
3 2 0
00000000000000000000000000
0628406284062840628406284
09988877666554443322211
1
Average fitness
7
Optimal fitness
Urban refueling station
Figure7:Iterativeprocessofthesecond-stagemodel.
Figure 6: Locations selected for refueling station during the first
stage.
Table3:Algorithmperformancecomparison.
Averagenumberof
was used for coding the GSA algorithm, and GISDK was Algorithm AverageCPUtime
iterations
usedforimplementingtheSUEassignmentalgorithm.After
Geneticsimulated
876coolingiterations,thetemperaturewas3.06E-17andan 876 24min46s
annealingalgorithm
optimalsolutionwasobtained.Theaverageruntimewas24
Geneticalgorithm 1226 34min39s
minutesand46seconds.Theobtainedshortesttotaldetour
timeforrefuelingvehicleswas6,688minutes.
Table2liststheshortesttotaldetourtimeandthedetour
times for different chromosomes at each iteration step. The The newly added stations in the second stage are shown
iterative process is also illustrated in Figure7, in which the inFigure8.Thesestationsprovideservicetotheremaining
horizontal axis stands for temperature and the vertical axis 49.37%refuelingvehicles.
standsforfitnessvalue.Figure7showsthattheoptimalfitness
curve goes up and down at the beginning and tends to (iv) Comparison of Algorithms. In this section, GA and the
stabilizeandleveloutafter876iterations.Thisoptimalfitness GSA algorithms are compared. The result is reported in
curveisnotmonotonicallydecreasingbecausethesimulated Table3. As can be seen, on average it takes the GA more
annealing module sometimes accepts chromosomes with iterations(1226)thantheGSAalgorithmtoobtainthesame
suboptimal(i.e.,worse)fitnessvalueswithasmallprobability. solution.Also,theGArequireslongerCPUtime.Foralarger