Design of a reliable logistics network with hub disruption under uncertainty

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Design of a reliable logistics network with hub

disruption under uncertainty

Mehrdad Mohammadi, Reza Tavakkoli-Moghaddam, Ali Siadat, Jean-Yves

Dantan

To cite this version:

Mehrdad Mohammadi, Reza Tavakkoli-Moghaddam, Ali Siadat, Jean-Yves Dantan. Design of a

re-liable logistics network with hub disruption under uncertainty. Applied Mathematical Modelling,

Elsevier, 2016, 40 (9-10), pp.1-22. �10.1016/j.apm.2016.01.011�. �hal-01794858�

Design

of

a

reliable

logistics

network

with

hub

disruption

under

uncertainty

Mehrdad

Mohammadi

a,b,c,∗

_,

_Reza

_{Tavakkoli-Moghaddam}

a

_,

_Ali

_Siadat

b

_,

Jean-Yves

Dantan

b

a School of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran b LCFC Laboratory, Arts et Métiers ParisTech, 4 rue Augustin Fresnel, 57078 Metz, France

c Ecole des Mines de Saint-Etienne, Department of Manufacturing Sciences and Logistics, CMP, CNRS UMR 6158 LIMOS, 880 avenue de Mimet, 13541 Gardanne, France

a

b

s

t

r

a

c

t

Inthis study, wedesign a reliablelogistics networkbased ona hublocation problem, whichisless sensitiveto disruptionand it performsefficiently whendisruption occurs. Anewmixed-integerprogrammingmodel isproposedtominimizethetotalsum ofthe nominalandexpectedfailurecosts.Thismodelconsiderscompleteandpartialdisruption athubs.Inaddition,weproposeanewhybridmeta-heuristicalgorithmbasedongenetic andimperialistcompetitivealgorithms.Wecomparetheperformanceoftheproposed al-gorithmwithanewlowerboundmethodintermsoftheCPUtimeandsolutionquality. Furthermore,weconcludethataconsiderableimprovement inthereliabilityofthe net-workcanbeachievedwithonlyaslightincreaseinthetotalcost.Finally,wedemonstrate thatthenetworksdesignedusingourmodelarelessconservativeandmorerobustto dis-ruptioncomparedwiththosedesignedbasedonotherrobustnessmeasures.

1. Introduction

Strategicdecisionsaboutthelocationsofnewfacilities,boostingdistributionnetworks,andcooperatingwithnewparties are requiredinmost industries.Industrial companies mustlocate manufacturing facilities aswell asdistribution centers, which need to be positioned near retailoutlets. Thus, the success of theseindustries depends greatly on the quality of thelocation-relateddecisions.Similarly, governmentsmustmake decisionsaboutthelocationsofhospitals, policeoffices, schools,firestations,andotherfacilities.Ineachcase,thelocationofafacilityhasasignificanteffectonthequalityofthe logisticsnetwork.

Ingeneral,logisticsnetworksareresponsiblefortransferringanddistributingflowsbetweenasetoforiginand destina-tionnodesthroughanumberofintermediatefacilities.Inmostpreviousstudiesofthisproblem,ithasbeenassumedthat thefacilitiesarealwaysavailable.However,logisticsnetworksdesignedaccordingtothisassumptionmayencounterserious problemsiftheyfailtoconsiderdisruptioninanycomponentofthenetwork,andthustheywouldbecomeineffectivewhen thefacilitiesaresubjecttofailure.Recently,logisticsnetworkshavebecomeincreasinglycomplicatedduetotheirsize,the servicesrequired,thenatureofcustomerassignments,theutilizationofseveralcomponents,andassociatednetworkflows.

Thus,asmallfailureinacomponentmaypropagatethroughoutthesystemandcauseahugeinconsistencyinthenetwork, whichfurtherimposesatremendouscost,withirrecoverableordifficulttorecovereffectsonthenetwork[1].

Facilities mightfail due toseveral intentional/unintentionalreasons such asnaturaldisasters, equipment breakdowns, suppliercutouts,explosionsduetoprocesshazards,laborstrikes,andterroristattacks.Pengetal.[2]highlightedthe short-termand long-termeffects ofdisruption in logisticsnetworks such asincreasesin costs and thetime of transportation, deliverydelays,andinventoryshortages,aswellasnegativefinancialeffects.Thus,enterprisesmustaimtodesignareliable logisticsnetworkinordertocopewiththeseshort-andlong-terminefficiencies.

Inthisstudy,weconsidertheproblemofdesigningalogisticsnetworkasahublocationproblem(HLP),whichcomprises origin, hub, and destination nodes.In a hub network, a set of intermediate nodes are located as hubs, and origin and destinationnodes(spokes)areallocatedtothesehubs.Hubsareresponsibleforconsolidating,transferring,anddistributing flows through the network to obtain economic profits. After a hub network has been designed, it can be very difficult andcostlytomodifythedesign.Therefore,itisnecessarytodesignareliablelogisticsnetworkthatremainsavailable and efficientinthepresenceofmanytypesofdisruptions.

InclassicalHLPs,itisusuallyconsideredthat thetopologyofthehubnetworkisnotchangedafterthehubshavebeen locatedandspokesareallocatedtothehubs. Itisalsoassumedthatthehubnodesarealwaysavailable andnoeventcan disrupttheirperformance.However,thesearenotvalidassumptionsifthelocatedhubsaresubjecttofailureandtheymay becomeunavailableduetodisruptionscausedbyintentionalorunintentionalreasons.Disruptionathubscanbelongtotwo types:completeandpartialdisruptions.

Whenahubnodeiscompletelydisrupted,thehubbecomesunavailableandthespokesoriginallyallocatedtoitmustbe reallocatedtoother (operational)hub nodes,which usuallyincurshigherconnection costs(i.e.,re-allocationcost). During partialdisruption,althoughthehubnodemaybestillavailable,theservicerateorthecapacityofthehubisdegradedtoa lowerlevel.Ifserviceratedegradationoccurs,thehubsbecomecongestedandtheincomingflowmustwaittobeprocessed. During capacitydegradation, the capacityof the hub is degraded to a lower level andthe hub is unable to process the entirearrivalflow(i.e.,thecapacityofthehubisdecreased).Therefore,disruptionsinahubnetworkcangreatlyaffectthe performanceofthelogisticsnetwork.

Themain aimofthisstudyistointroduce anewmathematicalmodelfordesigninga reliablelogisticsnetworkbased onaHLPunderdisruption,wheretheflowsshouldbetransferredbetweeneachpairoforigin-destinationnodesviaagiven setoflinks. Abriefsurveyofprevious research(Section2) showsthatfew previousstudies haveaddressedthedesign of a reliablehub network,wherehubsaresubjecttocomplete andpartialdisruption,andthusthey mayfailto processthe flows.Inaddition,inrealsettings,afterdesigningahubnetwork,someparameters(e.g.,demand,distance,time,andcost) intheproblemmaychangeduetouncertainty,butmostpreviousstudieshavenotadequatelyaddressedhowuncertain pa-rametersmightaffectthenetworkdesign.Theuncertaintyinparametersmaycomefromavarietyofsources,butcollecting historicaldataisasdifficultasestimatingtheprobabilitiesoftheparameters.Hence,weutilizetherobustoptimization ap-proachproposedbyPengetal.[2],whichprovidesanalternativemethodforcopingwithuncertaintybutwithoutrequiring probabilisticinformationabouttheinputparameters.

Themaincontributionsofthisstudyareasfollows.

• Wedesignandmodela newreliablelogisticsnetworkasaHLPbyconsideringthepossibilityofdisruptioninthehub facilities.

• Weconsidertheeffectsoftwodifferenttypesofdisruptions:(1)hubavailability,i.e.,completedisruption;and(2)hub capacity,i.e.,partialdisruption.

• Wedevelopaservicelevelconstraintthatconsidersdisruptionofthecapacity.

• Anew hybridsolution approachis proposed by combininga numberofefficient solutionapproaches fromrecent re-search,i.e.,stochasticprogrammingandrobustoptimization.

• Anefficientlowerboundapproachisproposedtofindanear-optimalsolutioninareasonableamountofcomputational timewithfewgaps.

• Weproposeanewhybridself-adaptivemeta-heuristicalgorithmbasedongeneticalgorithmsandimperialistcompetitive algorithms(ICAs).

Theremainder ofthispaperisorganizedasfollows.Section 2describesprevious researchinthearea ofhub networks byconsideringdisruptionsinthehubfacilities.Section3explainstheproblemandthemathematicalformulation,andthe robustoptimizationapproachispresentedinSection4.Thenewlower boundandmeta-heuristicmethods areintroduced inSection5.ComputationalresultsareprovidedinSection6.Finally,wegiveourconclusionsinSection7.

2. Literaturereview

Mostprevious studiesofdisruptioninnetworksarebasedontheclassicalp-median[3]anduncapacitatedfixed-charge location problems [4]. Bothof theseproblems locate facilities andassign customersto the located facilities to minimize thetotaltransportation(and/orconstruction)cost,whereallfacilitiesareassumedtobetotallyavailableandreliable.Some recentstudieshaveconsideredfacilitylocationinthepresenceofrandomdisruptionsandinterestedreadersmayrefertoa comprehensivereviewbySnyder[5].Mostpreviousstudiesinthisareaconsidereddisruptionsinafacilitylocationproblem, somostofthissectiondealswithstudiesrelatedtothefacilitylocationprobleminthepresenceofdisruption.

Oneofthefirst mathematicalmodels usedtostudya reliablefacilitylocation problemwaspresentedby Dreznerand Wesolowsky[6],whereanunreliablep-medianand(p,q)-centerlocationproblemswereconsideredandfacilitieshadagiven probabilityofbecominginactive.SnyderandDaskin[7]presentedtworeliabilitymodels(i.e.,reliablep-medianandreliable uncapacitatedfixed-chargelocation)forfacilitylocation,whereeachcustomerisassignedtoaprimaryfacilityandanumber ofbackupfacilities,andatleastonefacilitymustbeavailable.Ifthecurrentfacilityfails,thecustomerisservedbythenext availablebackup facility.Theyalsoconsidered thatthe probabilitiesoffailure are equalandmutuallyindependentforall facilities.Cuietal.[1],Bermanetal.[8],LiandOuyang[9],Limetal.[10,11],Shenetal.[12],SnyderandDaskin[13],and Zhanetal.[14]alldescribedmodelssimilartoSnyderandDaskin[7],buttheyrelaxedtheuniformdisruptionprobability assumptionusingdifferentmodelingapproaches.

LiandOuyang[9]studied facilitylocationusinga three-echelonsupplychainproblemby consideringrandom disrup-tionsforbothsuppliersandretailers.Theirmodeldeterminestheoptimallocationsofretailers,customerassignments,and inventorypolicy.Pengetal.[2]introducedthep-robustnesscriterion,whichallowsadesignednetworktoperformwellin bothdisruptedandnormalconditions.Theyproposedahybridmeta-heuristicalgorithmbasedonageneticalgorithm,local improvement,andtheshortestaugmentingpathmethod.Accordingtodifferentnumericalexperiments,theydemonstrated thesuperiorperformance ofthe proposedheuristic comparedwithCPLEXintermsof thecomputationaltime, whilestill obtaininganexcellentqualitysolution.Theyalsoshowedthatthereliabilityofthenetworkcouldbeimprovedsubstantially withonly a slight increase in cost. Finally,they found that their model produced lessconservative solutions than those generatedbycommonrobustnessmeasures.

ItshouldbenotedthatPengetal.[2]definedsomespecificdisruptionscenariosandconsideredthathubsareabsolutely disruptedin each scenario. In the presentstudy, we alsoconsider a set of disruption scenarios andthe probability that eachhub isdisruptedineach scenario.We believethat ourassumptionis moreapplicablewhen decisionmakerscannot determinewhetherthereisanydisruptionineachscenario.Inaddition,weemploytheirproposedrobustmethodtocope withtheuncertaintiesoftheinputparameters(e.g.,costs,disruptionprobabilities,andcapacitydegradationfactors).

Lietal.[15]presentedtwo relatedmodels (i.e.,reliablep-medianandreliable uncapacitatedfixed-chargelocation)for designing reliable distribution networks. Both models were formulated by nonlinear integer programmingand heteroge-neousfacilityfailureprobabilitieswerealsoconsidered,withonelayerofsupplierbackupandfacilityfortificationwithina finitebudget.TheNP-hardnessofthesemodelswasalsoproved.

Tothebestofourknowledge,onlythreepreviousstudieshaveaddressedthehubnetworkdesignproblembyconsidering hub disruptions. Parvaresh et al.[16] formulated a bi-level multiple allocation p-hub median problemunder intentional disruptionsby a bi-level model withbi-objective functionsat an upper level anda singleobjective at a lower level. In theirmodel,theleaderaimstoidentifythelocationsofhubssuch thatthenormalandworst-casetransportationcosts are minimizedwhileconsideringthenormalandfailureconditions.Finally,theworst-casescenarioismodeledatalowerlevel, wherethefollower’sobjectiveis toidentifythehubsthat wouldreduce theserviceefficiencymostifthey arelost.They alsodevelopedtwomulti-objectivemeta-heuristicsbasedonsimulatedannealingandtabusearchtosolve theirproposed model.In a similar study,Parvaresh etal.[17] developed a multiple allocationp-hub median problemunderintentional disruptionsusingdifferentdefinitionsofthefailureprobabilityforahubcomparedwiththeirpreviousstudy.

Morerecently,Mohammadietal.[18]proposedabi-objective stochasticp-hubcenterproblemtoinvestigatetheeffect ofdelivery servicerequirements ontheconfigurationof ahub-and-spokenetwork withfuzzyparameters, wherethehub operationalnodesmaybedisruptedandfailtooperateshipments.Theyconsideredthatthedeliverytimeoftheshipments isthesumofthetransittimeoverthelinksandthetimespentatthehubs.Theycalculatedthetimespentwaitingforthe fuzzyarrivalofshipmentsbyutilizinga fuzzyM/M/1queuingsystem. Theyalsoassumedthat thequeuingsysteminthe hubsisdegradedstochasticallyandthenrestoredtoitsperformancelevel.

3. Problemdescriptionandformulation

Inthis study, we design a logistics network withfacility disruptionsbased on a hub location network. This problem generalizestheclassicalHLPby makingdecisions regardingthelocationsofhubs,the allocationoforiginanddestination nodesto the located hub,and routing the flows through thenetwork. Thus, we propose a new mixed-integer nonlinear programmingmodelfordesigningareliablehubnetwork.Hubsareaffectedbytwo typesofdisruption,i.e.,disruptionsin hubfacilitiesanddisruptioninthedesignedcapacityofhubs.Anewassumptionthatincludestheprobabilityoffailurefor thehubs isconsidered forhub disruption. Whenhub disruption occurs, the hub becomesunavailableandno originand destinationnodesareserved.Inthissituation,backuphubsareneededtoroutetheflows.Therefore,asetofpotentialhubs are located andthe originanddestination nodes are allocatedto the hubslevel by level (i.e.,backup hubs).Thus, for a non-hubnode,after thecorresponding hub inlevelone isdisrupted,the non-hubnode isreallocatedto thebackuphub inleveltwo.Thisreallocationpolicyisrepeatedforthenon-hubnodeaftera disruptionoccursatahigherlevelanditis continuedafterthenon-hubnodehasbeenallocatedtoadummynon-failablehub.

By contrast, duringcapacity disruption at a hub, the hub is available and only the size of the flow that enters the hubsmustbe decreased. Inaddition,the capacityofa hub issubjectto stochasticdegradation, so weconsider acertain probabilitythat atraffic flowenteringa hubwillexceed theactualcapacityanddisrupttheflow process.Toaddressthis issue,itisdesirabletoensurethattheprobabilityofsuchanoccurrenceislowerthanaspecifiedorsatisfactorylevel.

r=1 r=2 r=1 r=2 r=1 r=1 r=2 r=3 Non-failable Hub Failable Hub Non-hub Node Connection Link

a) Allocation Structure

b Tree Structure

1 2 3 4 1 2 3 4 5 1 2 3 4

Fig. 1. Logistics of the proposed RHLP.

Intermsofthestructure ofthehubnetwork, moststudies ofHLPshaveconsideredthat ahub networkisacomplete graphwhereallofthehubsarefullyinterconnected.ThisassumptionmakesiteasiertomodeltheHLPs,butthereissome justificationfordesigningincompletehubnetworksforvariousapplications.Forexample,indistributionnetworks,sending separatetrucksfromadistributioncenter(i.e.,hub)toallotherdistributioncentersisexpensiveintermsofthefixedcost ofusingahighnumberoftrucks.Instead,usingthesametruckswithsufficientcapacitytovisitmorethanonedistribution centermaydecreasethetotalinvestmentcostconsiderably.Similarly,inairlinepassengertransportation,anairlinecompany mayrouteandforceitsflights topassthroughalarge numberofdestinationsbecausetherequirementformoreseparate aircraftandairstaff ateach destinationimposes highercongestionandinvestmentcosts inairports andairnetworksfor thecompany[19].AsuitablestructureforincompleteHLPsisatreestructure,whichplacesasingleallocationp-HLPwhere

phubsmustbeconnectedusinga(non-directed)tree[20].Thus,weconsideratreestructureforthehubnetwork.Inorder tounderstandtheHLPmodeleasily,Fig.1illustratesthelogisticsnetworkwithareallocationpolicyandthetreestructure of thehub network. As shownin Fig. 1a,non-hub node 1is allocated tofailable hub 4 atthe first level (i.e.,r=1) and afterthehub4isdisrupted,thenon-hubnodeisallocatedtodummynon-failable hub2atlevel2(i.e.,r=2),andthe re-allocationpolicyisthenstopped.ThetreestructureofthehubnetworkisshowninFig.1b,wherethehubsareconnected byatree-shapegraph.

Inthefollowingsubsections,weformulatethesingle-objectivereliableHLP(SRHLP)tominimizethesumofthenominal transportationandexpectedfailurecosts.First,wedevelopaservicelevelthatconsidersthehubcapacityreliabilityandwe thenintroducetheassumptionsoftheproblem,beforedescribingtheinitialnonlinearmodeloftheSRHLP.

3.1. Hubcapacityreliability

As mentioned inSection 1,the capacityof ahub may be subjecttostochastic degradation, wherethe flows entering thehubwillexceedthedesignedcapacitybyacertainprobability.Itisdesirabletoensurethattheprobabilityofsuchan occurrenceislowerthanaspecifiedorsatisfactorylevel.Weintroducethehubcapacityreliabilityastheprobabilityofthe flowenteringahubexceedingitscapacity,whichisreferred toasthecapacityexceedanceprobability

η

.Thehubcapacity reliabilitycanbeformulatedmathematicallyas:

P

{

Γ

≤ EF

}

≤

η

, (1)

where EF and



are the flows entering the hub andthe capacityof the hub, respectively. It should be noted that the hubcapacityreliabilityrequirementcandifferfordifferenthubs.Ininequality(1),



isavariablespecifiedbyaparticular probability density function.The left-hand side of inequality (1) can be considered asa cumulative distribution function (CDF)of



,whichiswrittenasfollows.

F_Γ

(

EF

)

= P

{

Γ

≤ EF

}

(2)

Usingthisequation,inequality(1)canberewrittenas:

F_Γ

(

EF

)

≤

η

. (3)

TheCDFsaremonotonicone-to-onefunctions,sowecantaketheinverseofEq.(3)andwritethefollowingequation. EF≤ F −1

Γ

(

η

)

. (4)

By specifying the CDF of the hub capacity



andthe acceptable capacityexceedance probability

η

, Eq. (4) becomes a deterministic constraint. For simplicity,we assume that the hub capacityfollows a uniformdistribution defined by an upperbound(i.e.,designcapacity)andalowerbound(i.e.,worst-degradedcapacity).Generalizingtheconsiderationtoother probabilitydistributions(e.g.,GammaandtruncatedGamma)canbeaccomplishedwiththeMellinTransformtechnique,as

discussedin[21].Itshouldbe notedthat severalstudies[21–24]suggestthatauniformdistributionismoreapplicableto thetransportationdomainwhenaffectedbydisruption.

Itshouldbementionedthatotherdistributionfunctions(e.g.,GammaandtruncatedGamma)canbeusedtomodelthe capacitydegradation.Furthermore,weconsiderthat thelowerbound isafraction

θ

ofthedesigncapacity.Forauniform distribution,theinverseCDFof



canbewrittenas:

F_Γ−1

(

η

)

=

θ

_Γ

¯ ₊

_η

_Γ

¯

₍

_{1 −}

_θ

₎

₌

_Γ

¯_[

_θ

₊

_η

₍

_{1 −}

_θ

₎

_{] , (5)}

where

Γ

¯ isthedesigncapacityofthehub,whichhasadeterministicvalue.ByputtingEq.(5)intoinequality(4),weobtain thefollowinghubcapacityreliability.

EF ≤ ¯

Γ

[

θ

+

η

(

1 −

θ

)

] . (6)

3.2.Assumptions

TheSRHLPisformulatedundersomespecificassumptionsasfollows. • Thenumberofhubsthatshouldbelocatedisgiven.

• Eachnon-hubnodecanbeallocatedtoexactlyonehubateachlevel(i.e.,singleallocation). • Thegraphofhubsisformedasanindirecttreestructure.

• Locatingthehubsincursafixedcost.

• Afixedfailureprobabilityisassociatedwitheachhub.

3.3.MathematicalformulationoftheSRHLP

Inthissection,wefirstpresentthenotationsfortheproblemandthemathematicalformulationisthenproposed.

Sets

i,j=

{

1,...,I

}

setofnodes

k_,l_,m_,n₌

{

1_,...,K₊1

}

setofhubswherethe

(

K₊1

)

thhubisadummynon-failablehub

r=

{

1,...,R

}

setofallocationlevels

Parameters

H numberofhubsthatshouldbelocatedinthehubnetwork

R maximumnumberofhubsthatcanserveanode(R≤ H)

W_ij flowtobesentfromnodeitonodej

Cij transportationcostofaunitofflowfromnodeitonodej

Fk fixedcostoflocatingahubatnodek ¯

Γ

k designedcapacityofthecollectingflowathubk(i.e.,flowintohubk)

η

k capacityexceedanceprobabilityathubk

θ

k capacitydisruptionfactorathubk

Oi= j∈I

Wi j totalfloworiginatingatnodei

Di= j∈I

Wji totalflowdestinedfornodei

qk failureprobabilityathubk

α

∈ [0, 1] costdiscountfactorbetweentwohubs

δ

∈[0,1] costdiscountfactorbetweennon-hubnodesandhubs

M abignumber

Variables Xr

ik 1,ifnodeiisallocatedtohubkatlevelr;0,otherwise

Zk 1,ifahubislocatedatnodek

Pr

ik probabilitythathubkservesnodeiatlevelr

γ

k variablecapacityofhubkaffectedbydisruptionfactor

θ

k,where

γ

k∼ U

(

θ

k

Γ

¯k,

Γ

¯k

)

Lkl 1,ifthereisahublinkbetweenhubkandhubl(k<l);0,otherwise

Ukl

mn 1,ifthehublink{m,n}isusedonthepathfromhubktohublinthedirectionfrommton;0,otherwise Remark. A ‘‘level-r’’assignment is one for whichr failable hubsthat are open. Ifr = 0,this is a primary assignment; otherwise,itisabackupassignment.Eachnon-hubnodejhasalevel_{−r assignment}foreachr₌0_, 1_,...,R_{− 1},unlessjis assignedto alevel−l hubthat isdummynon-failable,wherel<r.Inother words,non-hubnode jisallocatedtoone hub atlevel0,anotherhubatlevel1,andsoonuntiljisallocatedtoalloftheopenhubsatsome level,orjisallocatedtoa dummynon-failablehub[7].

Byemployingthespecifiednotations,theproposedSRHLPispresentedasfollows. min K  k=1 FkZk+ I  i=1 K+1  k=1 R  r=1

δ

Cik

(

Oi + Di

)

PikrXikr + K_ +1 m=1 K_ +1 n=1

α

Cmn

⎡

⎢

⎢

⎣

I  i=1 I  j=1 j=i I  k=1 K+1  l=1 l=k R  r=1 R  s=1 Wi jUmnklPimrXimrPjnsXsjn

⎤

⎥

⎥

⎦

, (7) s.t. K  k=1 Xr ik+ r  s=1 Xs i(K+1)= 1

∀

i ∈ I,r ∈

{

1 ,...,R

}

, (8) R  r=1 X_ir_(K+1)= 1

∀

i ∈ I, (9) P1 ik= 1 − q k

∀

i ∈ I,k ∈

{

1 ,...,K + 1

}

, (10) Pr il=

(

1 − q l

)

K  k=1 qk 1 − q k Pr−1 ik X r−1 ik

∀

i∈ I,l ∈

{

1 ,...,K + 1

}

,r ∈

{

2 ,...,R

}

, (11) K  k=1 Zk = H, (12) ZK+1 = 1 , (13) R  r=1 X_ikr ≤ Z k

∀

i∈ I,k∈

{

1 ,...,K

}

, (14) K+1  m=1 I  i=1 I  j=1 j=i I  k=1 K_ +1 l=1 l=k R  r=1 R  s=1 Wi jUmnklPimr X r imP s jnX s jn≤ ¯

Γ

n[

θ

n

(

1 −

η

n

)

+

η

n]

∀

n ∈

{

1 ,...,K

}

, (15) K+1  k=1 K+1  l=1 k<l Lkl= H− 1 , (16) Lkl≤ 1 2

(

Zk+ Zl

)

∀

k,l ∈

{

1 ,...,K + 1

}

, k<l, (17) Ukl mn+ Unmkl ≤ 1 3

(

Lmn + Zk+ Zl

)

∀

m,n,k,l ∈

{

1 ,...,K + 1

}

,m<n,k  = l, (18) K_ +1 n=1 n=m Uml mn≥ Z m + Zl− 1

∀

m,l ∈

{

1 ,...,K + 1

}

,m  = l, (19) K_ +1 n=1 n=m Unmkl − K+1  n=1 n=m Umnkl = 0

∀

m,k,l ∈

{

1 ,...,K + 1

}

,m  = k,m = l,k = l, (20) K_ +1 n=1 n=m Ukm nm≥ Z m + Zk− 1

∀

m,k∈

{

1 ,...,K + 1

}

,m  = k, (21)

Xr

ik,Zk,Umnkl,Lkl∈

{

0 , 1

}

∀

i ∈ I; k, m,n∈

{

1 ,...,K + 1

}

; r ∈ R, (22) Pr

ik≥ 0

∀

i∈ I; k, l ∈

{

1 ,...,K + 1

}

; r ∈ R. (23) Objectivefunction (7)isthe sumofthefixed andexpectedtransportationcosts. Eq.(8)ensures thatforeachnon-hub nodeiandeachlevelr,eitheriisallocatedtoaregularhubatlevelroritisassignedtothereliablehubK+1atcertainlevel

s≤ r(takingr_s=1Xs

i,K+1=0ifr=1).Constraint(9)ensuresthateachnon-hubnodeisallocatedtothedummynon-failable hubatacertainlevel.Constraint(10) showsthattheprobabilitythathub kservesnon-hubnodei atlevel1issimplythe probabilitythatkisnotdisrupted.Constraint(11)isthe“transitionalprobability” equation,whichshowsthattheprobability ofhublservingnon-hubnodeiatlevelr(i.e.,Pr

il)iscalculatedwhenhublremainsopen(i.e.,withprobability

(

1− ql

)

)and otherhubstowhichnodeihasbeenpreviouslyallocatedinlowerlevelsthanrhavebeendisrupted.Constraint(12)ensures thatthenumberofhubsshould beequalto thegivenvalue H.Eq.(13)requiresthat thedummynon-failablehub K+1 is open.Constraint(14)forcesnon-hubnodestobeallocatedtovalidhubs.Constraint(15)limitstheamountofflowsentering thehubs. Accordingto constraint(16), thevariablesLdefine a treesinceH−1edges are selected.Inorderto establisha hublink{k,l},bothendnodes(i.e.,nodeskandl)ofthatlinkneedtobehubnodes(i.e.,constraint(17)).

Constraint(18)alsoensuresthatatmostoneofUkl

mnandUnmkl variablescanbe1becausetheyneedtoprovideasimple directed path.Constraints (19)–(21) are the flow balance constraints inthe hub network. According to theseconstraints, everyhubnode sendsandreceives oneunitofflow,andtheconnectivityinthehub networkisestablished.Byconstraint (19), ifboth nodes mand l arehubs, thenthe originhub m sends one unit of flow tothe destination hub l inthe hub network.By constraint(21), ifnodesm andk are bothhub nodes,then the destinationhub m receives one unit offlow fromtheoriginhubkinthehubnetwork.Ifhubnodemisnottheoriginorthedestination,thentheincomingflowmust equaltheoutgoingflowbyconstraint(20).Constraints(22)and(23)aredomainconstraints.

4. RobustSRHLP

Inrealsettings,afterdesigningahubnetwork,some parameters(e.g.,demand,distance,time,andcost)intheproblem maychangeduetouncertainty.Forexample,dueto variabilityinthetransportation time froman originto adestination, there isa possibilitythat the flow might not be delivered on time. A failure inon-time delivery may havea huge and non-measureable cost (e.g., lost-opportunity cost andlost-sale cost dueto customer churning)due to the compensation ofanunsatisfied customer.Nevertheless,mostproposed HLPmodelstreatdata asknownanddeterministic. Hence,many researchershaveattemptedtomodeltheuncertaintyinthedesignofoptimizationproblemsinrecentstudies[25–34].

Onewayofhandlinguncertaintyisto estimatethe probabilitydistribution oftheparameters, butvarious sources can leadtoparameteruncertainty,andthuscollecting historicaldatacanbedifficultandtheprobabilitiesaredifficultto esti-mate.Similarto Pengetal.[2],we employrobust optimizationsothereis noneedto estimateprobabilistic information. Theproposedapproachisaformofscenario-basedrobustoptimization,whichdeliverssolutionswithreasonableobjective valuesunderanyscenariospecifiedbythedecisionmaker.Manypreviousstudiesinthisarea,particularlyfacilitylocation problems,haveconsidered various robustnesscriteria(see Snyder[5]for furtherdetails),where themostcommonly ap-pliedrobustnesscriteriaaretheminimaxcost andminimaxregret,whichminimizethemaximumcostorregretamongall scenarios.Oneofthemostefficientmethodsforrobustoptimizationisthep-robustnessmeasure,wheretherelativeregret ineachscenariomustbenomorethanaconstantp[35].Thismethodhasalsobeenappliedtotwowell-knownproblems tominimizetheexpectedtotalcost,i.e.,thep-robustk-medianandp-robustuncapacitatedfixed-chargelocationproblems bySnyderandDaskin[13].

Inthefollowing,weapplythep-robustnessmeasureintheproposedSRHLPandwerefertothismethodasthe p-robustnessSRHLP(p-SRHLP).Theobjectivefunctionofthep-SRHLPmodelistominimizethetotalexpectedcostwhile restrictingtherelativeregretineachscenariotonomorethanpforaconstantp>0.Ascenario-basedmodelingapproach isused,wherespecificvaluesfortheparametersareconsideredineachscenario.Tothebestofourknowledge,thisisthe firstuseofthep-robustnessmeasuretodesignareliablehublocationmodelinordertoensurehigh-levelperformance duringdisruption.First,weintroducetheuncertainparametersanddecisionvariables,whichhavedifferentvaluesineach scenario.Anindex

ω

torepresentascenarioisaddedtothepreviousvariablesgiveninSection3.3.Inaddition,random parametersarerewrittenbyaddingtheindex

ω

toeachofthem.

Set

ω

=

{

1,...,

}

setofscenarios

Parameters

W_{i j}ω flowtobesentfromnodeitonodejinscenario

ω

.

C_ikω transportationcostofaunitofflowfromnodeitonodejinscenario

ω

. F_kω fixedcostofopeningahubatnodekinscenario

ω

.

η

ω

k capacityexceedanceprobabilityathubkinscenario

ω

.

θ

ω

k capacitydisruptionfactorathubkinscenario

ω

.



ω probabilityofscenario

ω

.

ζ

∗

p desiredrobustnesslevel,p_>0;

Oω_i = j∈I

W_{i j}ω totalfloworiginatingatnodeiinscenario

ω

.

Dω_i = j∈I

Wω_ji totalflowdestinedfornodeiinscenario

ω

.

qω_k probabilityoffailureforhubkinscenario

ω

.

Variables Xrω

ik is1ifnodeiisallocatedtohubkatlevelrinscenario

ω

;0,otherwise

Prω

ik probabilitythathubkservesnodeiatlevelrinscenario

ω

.

OFVp objectivefunctionvalueforap-robustmodel

4.1. Definitionofp-robustness

Inthissection,thedefinitionof‘‘p-robust’’givenbySnyderandDaskin[13]isstatedasfollows.

Definition. Givena set of scenarios (

), consider Dω as the deterministic minimization problem for scenario

ω

(i.e., a deterministic HLP) andits optimalobjectivefunction value is shownas

ζ

_ω∗. Inaddition, let ZandX be feasible vectors ofthe location andallocation variables,respectively, UandL are feasiblevectors ofthetree structurevariables, P isthe feasiblevectoroftheprobabilityvariables,and

ζ

_ω(X,Z,U,L,P)arethevaluesoftheobjectivefunctionforthevariableset (X,Z,U,L,P)inscenario

ω

.Then,(X,Z,U,L,P)iscalledp-robustifforall

ω

∈

,

ζ

ω

(

X,Z,U,L,P

)

−

ζ

ω∗

ζ

ω∗ ≤ p

∀

ω

∈

. (24)

Equivalently,constraint(24)canbestatedby:

ζ

ω

(

X,Y,Z,U,L,P

)

≤

(

1 + p

)

ζ

_ω∗

∀

ω

∈

, (25)

wherep≥ 0isagivenconstant, whichindicatesthedesiredrobustnesslevel.Theleft-handside ofconstraint(25)isthe relativeregretoftheobjectivefunctioninscenario

ω

.Thep-robustmeasuresetsupperboundsonthemaximumallowable relative regret forthe objective function in each scenario. Finally, the p-robustness measure can be combined with the originalproblemtoobtainamodelinthefollowingform.

min  ω∈



ω

ζ

ω, (26) s.t.

ζ

ω

(

X,Z,U,L,P

)

≤

(

1 + p

)

ζ

_ω∗

∀

ω

∈

, (27) g

(

X,Z,U,L,P

)

≤

(

= or ≥

)

b. (28) 4.2. Formulationofthep-SRHLP

Weproposethefollowingmixed-integerprogrammingmodelforthep-SRHLP.

min OF V p =  ω=1



ω

K  k=1 F_kωZk+ I  i=1 K+1  k=1 R  r=1

δ

Cω_ik



Oω_i + Dω_i



Prω ik Xikrω + K+1  m=1 K+1  n=1

α

C_mnω

⎡

⎢

⎢

⎣

I  i=1 I  j=1 j=i I  k=1 K+1  l=1 l=k R  r=1 R  s=1 W_{i j}ωUkl mnPimrωXimrωPsjnωXsjnω

⎤

⎥

⎥

⎦

⎫

⎪

⎪

⎬

⎪

⎪

⎭

, (29) s.t. K  k=1 F_kωZk+ I  i=1 K+1  k=1 R  r=1

δ

C_ikω



Oω_i + Dω_i



P_ikrωX_ikrω + K_ +1 m=1 K_ +1 n=1

α

C_mnω

⎡

⎢

⎢

⎣

I  i=1 I  j=1 j=i I  k=1 K+1  l=1 l=k R  r=1 R  s=1 W_{i j}ωUkl mnPimrωXimrωPsjnωXsjnω

⎤

⎥

⎥

⎦

≤

(

1 + p

)

ζ

_ω ∗

∀

ω

∈

, (30)

K  k=1 Xrω ik + r  s=1 Xsω i(K+1)= 1

∀

i ∈ I,r ∈

{

1 ,...,R

}

,

ω

∈

, (31) R  r=1 Xrω i(K+1)= 1

∀

i∈ I,

ω

∈

, (32) P1ω ik = 1 − q ωk

∀

i ∈ I,k ∈

{

1 ,...,K + 1

}

,

ω

∈

, (33) Prω il =



1 − q ωl



_K k=1 qω_k 1 − q ω_kP r−1,ω ik X r−1,ω ik

∀

i ∈ I,l ∈

{

1 ,...,K + 1

}

,r∈

{

2 ,...,R

}

,

ω

∈

, (34) R  r=1 Xrω ik ≤ Z k

∀

i∈ I,k∈

{

1 ,...,K

}

,

ω

∈

, (35) K_ +1 m=1 I  i=1 I  j=1 j=i I  k=1 K_ +1 l=1 l=k R  r=1 R  s=1 W_{i j}ωUkl mnPimrωX rω imP sω jnX sω jn ≤ ¯

Γ

n[

θ

nω

(

1 −

η

ωn

)

+

η

nω]

∀

n ∈

{

1 ,...,K

}

,

ω

∈

, (36) (12),(13),(16)-(21) Xrω ik ,Zk,Umnkl,Lkl∈

{

0 , 1

}

∀

i∈ I; k, m,n ∈

{

1 ,...,K + 1

}

; r ∈ R,

ω

∈

, (37) Prω ik ≥ 0

∀

i ∈ I; k, l ∈

{

1 ,...,K + 1

}

; r ∈ R,

ω

∈

. (38)

4.3.Identifyingdisruptionscenarios

An importantissue ishow to determine thedisruption scenarios. For a givenset of Spossible scenarios, there are a totalof2S _{possibledisruptionscenarios,whichleadstoexponentialgrowthoftheproblemsizeandahugecomputational} timeisrequiredtosolvetheproblem.Inaddition,iftherearealargenumberofscenarios,carefullydeterminingpishighly importantformaintainingthefeasibility.

The numberof scenarios ismuch smaller inreal settings, so decisionmakers maydetermine scenarios based on the preferencesofexperts.Onemethodtodetermineasetofscenariosmayinvolveasituationwhereexactlyonehubnodeis disrupted(i.e.,complete orpartialdisruptions).Thissetofscenariosisdetermined byconsideringthehubsthat aremore likelyto be disrupted according tohistorical data orthe a priori beliefs ofdecision makers [2]. In realworld cases, the probability of morethan two unrelated hubs beingdisrupted atthe same time is very small,so thisset of scenarios is neglectedinthisstudy.Therefore,inthecaseofmultiplehubdisruption,thehubslocatedinthesameareahaveahigher probabilityofbeingdisruptedduetocertainevents(e.g.,earthquakesandelectricityshortages)[2].

Inthecaseofa largenumberofscenarios, differentsetsofscenarios maybe determinedapproximatelyusingsample averageapproximation[36],whereeachscenariocanbegeneratedusingtheMonteCarlosimulationmethod.

5. Lowerboundandmeta-heuristicalgorithm

Themodelpresentedin Section3.3isa mixed-integer nonlinearprogrammingmodel.Severaloptimizationtechniques havebeen ever developedto solve this model (e.g.,branch-and-bound, branch-and-cut, andbranch-and-price). However, thesetechniquescanbecomputationallydemanding,especiallywhenthenumberofscenariosislarge,butdecisionmakers areoftenmoreinterestedinobtainingnear-optimalsolutionsinarelativelyshortcomputationaltime.

TheproposedmodelwascodedinGAMS22.9butsolvinglargesizeinstances(upto15nodes)requiredacomputational time ofseveralweeks.Due tothislimitation, we developed an efficientmeta-heuristicalgorithm calledthe self-adaptive imperialistcompetitive algorithm (SAICA) to finda sufficiently good (i.e.,near optimal) solution ina reasonable amount of computational time, especially for real sized problems. Furthermore, we compared the performance of the proposed SAICAwithanefficientlowerboundmethodbasedon[27].Inthefollowing,theproposedSAICAisexplainedforthegiven problem.First,atailoredsolutionrepresentationdevisedfortheproblemisillustratedandthedetailsoftheSAICAarethen elaborated.Finally,inordertoevaluatetheperformanceofthemeta-heuristicalgorithm,alowerboundisdevelopedbased ontheapproachofMohammadietal.[27].

Potential Nodes

1 2 3 4 5 6

→ H=3 →

0.94 0.45 0.02 0.67 0.11 0.59 1 0 0 1 0 1

Fig. 2. Hub location scheme.

Potential Hub Nodes

1

2

3

4

5

6

Non-failable hub

Location of Hubs

1

0

0

1

0

1

1

Hub

1

0.51

0

0

0.79

0

0.60

0.67

Non-hub

2

0.40

0

0

0.85

0

0.03

0.75

Non-hub

3

0.72

0

0

0.15

0

0.27

0.74

Hub

4

0.91

0

0

0.03

0

0.04

0.39

Non-hub

5

0.63

0

0

0.84

0

0.99

0.65

Hub

6

0.09

0

0

0.93

0

0.12

0.17

Hub

1

-

-

-

-

-

-

-Non-hub

2

-

-

-

1

-

-

2

Non-hub

3

-

-

-

-

-

-

1

Hub

4

-

-

-

-

-

-

-Non-hub

5

-

-

-

2

-

1

3

Hub

6

-

-

-

-

-

-

-Fig. 3. Allocation scheme.

5.1. Solutionrepresentation

AnencodedHLPsolutionshould showthelocationsofthehub nodesandtheallocationsoftheoriginanddestination nodesforthelocatedhubs.Inthisstudy,acontinuoussolutionrepresentation(CSR)isdevelopedfortheHLP,whichavoids generatinginfeasiblesolutionsduringthesearch processandthesolutionprocess ismademucheasier.TheproposedCSR comprisesthreecomponents:(1)locationsofthehubs,(2)allocationsofnon-hubnodestothelocatedhub,and(3)structure ofthehubnetwork,whichareexplainedindetailasfollows.

5.1.1. CSRforthelocationofhubs

The first matrix corresponds to thelocation decisionrepresented by a (1 × K)matrix (i.e.,location matrix), where K denotesthenumberofpotentialhubnodes.Thismatrixisfilledwithrandomnumbersfrom[0,1].Inthismatrix,thefirst maximumHnumbersareconsideredtobelocatedhubs,asshowninFig.2.Inthisfigure,threehubsmustbelocatedamong sixnodes;therefore,thefirst,fourth,andsixthnodesarelocatedashubs.

5.1.2. CSRfortheallocationofnon-hubnodestothelocatedhub

This section representsthe allocation of non-hubnodesto the located hubs. The non-hubnodes are allocatedto the failable hubslevelby level afterthey havebeen allocatedto thedummynon-failable hub ata certain level.Thispart of thesolutioncontainsthematrix

(

I× K+1

)

,whichisfilledwithrandomnumbersfrom[0,1].First,eachrowofthematrix is multipliedby the location matrix.Next,each non-hubnode is allocatedto its biggest corresponding non-zero column as the first level ofallocation andthis value is replaced by 1.Moreover, the non-hob node is allocated to the next big valueforthenextlevelofallocationandthisvalueisreplacedby2.ThisprocedureisrepeatedRtimes(i.e.,themaximum numberofallocationlevels)afterthenon-hubnodehasbeenallocatedtothedummynon-failablehubinthelastcolumn. Ifthenon-hubhasnotbeenallocatedtothedummynon-failablehubat(R−1)thlevel,thenthatnon-hubnodeisallocated intentionallytothedummynon-failablehubatlevelR.Thus,thelastcolumnofthatnon-hubnodeisreplacedbyR.Fig.3 showsthenon-huballocationforthepreviousexamplewithR=3.

Forexample,inFig. 3,non-hubnumber2is allocatedtofailable hub number4atthe firstlevel andallocatedtothe dummynon-failablehubatthesecond level.Furthermore,non-hubnumber3isonlyallocatedtothedummynon-failable hubatlevelone.Moreover,non-hubnumber5isallocatedtofailablehubsnumbers6and4andthedummynon-failable hubforthefirst,second,andthirdlevelsofallocation,respectively.

5.1.3. CSRforthestructureofthehubnetwork

Thispartofthesolutionrepresentationdepictsthestructureofthehubnetwork.Thenetworkofhubsisatreestructure, soaPrüfersequence methodisused[37].ThePrüfersequence ofalabeled treeisa uniquesequenceassociatedwiththe tree.Thesequenceforatreeonnverticeshasalengthofn−2anditcanbegeneratedbyasimpleiterativealgorithm.

Fig. 4. Pseudo code of the SAICA’s Initialization phase.

5.2.ProposedSAICA

Similartomostother evolutionaryalgorithms suchasgeneticalgorithms [38,39]andparticleswarm optimization,the ICAisapopulation-basedalgorithm[40,41]anditwasintroducedbyAtashpaz-GargariandLucas[42].Thus,theICAbegins withaninitialpopulationofrandomsolutions,whereeachmemberofthepopulationiscalledacountry(seeMohammadi etal.[26]forfurtherdetails).Inthisstudy,weproposeaself-adaptivevariantoftheICAcalledSAICA,whichobtainshigher qualitysolutionscomparedwithpureICA.IntheproposedSAICA,aself-adaptivecrossoveroperatorisappliedinadditionto theoriginalassimilationandrevolutionoperators inICA.Variouscrossoveroperators (e.g.,one point,twopoints,uniform, andcyclecrossover)havebeenintroducedpreviously,butemployingallofthemsimultaneouslyinasinglealgorithmwould increasethecomputationaltimetremendously.Therefore,manystudieshaveusedonlyonetosearchthesolutionspace.By contrastintheproposedSAICA,severalcrossoveroperatorsareappliedsimultaneouslywithoutincreasingthecomputational time.

TheSAICAcontainsinitializationandmainphases.Intheinitializationphase,eachcrossoveroperatorisrewardedwitha scoreaftercomparingitwithotheroperatorsifitcangenerateabettersolutionateachiteration.Afterendingthe initializa-tionphasefollowingapredefinednumberofiterations(i.e.,MaxIteration),theselectionprobability(SP)metriciscalculated foreachoperatorbydividingthescoreobtainedbythenumberofiterations,whereSP=1.Aftertheinitializationphase, themainphase isstarted, whichincludessearchingthesolution spaceusingtheassimilation,self-adapted crossover,and revolutionoperators.Inthisstudy,thecrossoveroperatorscompriseonepoint,twopoints,threepoints,uniform,andthree parentcrossoveroperators[43].Thepseudo-codesforthetwophasesoftheproposedSAICAareshownseparatelyinFigs.4 and5.InordertounderstandtheproposedSAICAmoreeasily,flowchartsoftheinitializationandmainphasesaredepicted inFigs.6and7.

5.3.Lowerboundprocedure

Inorderto evaluatethe effectivenessofthe proposed meta-heuristicalgorithm,a newlower bound procedureis pre-sentedin this section. As shownin the proposed model, non-hubnodes are reallocated to other hubs ata higher level whenanylowerhubfailstoprovideservices.Ifallofthehubsfailforanon-hubnode,thereisadummyhubdenotedby

K+1,whichhasnofixedcostwithafailureprobabilityqK+1=0andatransportationcostc1i,K+1=

φ

i,fornon-hubnodei. Letqω_[1]≤ qω[2]≤ ...≤ qω[l−1]≤ q[ωl] beanordering ofthefailure probabilitiesinKforscenario

ω

.Anoptimistic versionof

Prω

Fig. 5. Pseudo code of the SAICA’s main phase. asprovedpreviouslyin[44]. Lωkr=

⎧

⎪

⎨

⎪

⎩



_r  l=1 qω_[l]



(

1 − q k

)

k∈ K, r_ −1 l=1 q[l] k= K + 1 . (39)

Calculate OFV for each

Solution

j

 1

Choose Parents using Binary

Tournament Selection

i

 Max Iter

Yes

Calculate Selection Probability

for each XO Operator SPi

No

Apply all Crossover Operators

- XO

i

Calculate OFV for New

Obtained Solutions

mini OFVXOi

XOi

SXO(i)=SXO(i)+1

Finish

Generating Initial Population

Set

SXO(i) 0

Set the Parameters

Start

Fig. 6. Flowchart of the initialization phase.

Approximatedmodel

P

: min  ω=1



ω

_K  k=1 F_kωZk+ I  i=1 K+1  k=1 R  r=1

δ

C_ikω



Oω_i + Dω_i



Lω_krXrω ik + K+1  m=1 K+1  n=1

α

Cωmn

⎡

⎢

⎢

⎣

I  i=1 I  j=1 j=i I  l=1 K+1  k=1 k=l R  r=1 R  s=1 W_{i j}ωUmnkl LωmrXimrωLωnsXsjnω

⎤

⎥

⎥

⎦

⎫

⎪

⎪

⎬

⎪

⎪

⎭

, (40) s.t. K  k=1 F_kωZk + I  i=1 K+1  k=1 R  r=1

δ

Cω_ik



Oω_i + Dω_i



LωkrX rω ik + K+1  m=1 K+1  n=1

α

Cmnω

⎡

⎢

⎢

⎣

I  i=1 I  j=1 j=i I  l=1 K+1  k=1 k=l R  r=1 R  s=1 W_{i j}ωUkl mnLωmrXimrωLωnsXjnsω

⎤

⎥

⎥

⎦

≤

(

1 + p

)

ζ

_ω ∗

∀

ω

∈

, (41)

Start

Set the

Parameters

Generating

Initial Countries

Create Initial

Imperialisms

i

 1

Assimilate Colonies

Towards Their Relevant

Empire

NFC

is

satisied?

Finish

Apply roulette wheel

selection to choose

crossover operator based

on SPi metric

i



i



1

iN-imp

Yes

Calculate the Total Power

of imperialist

No

Imperialist Competition

Is there an Empire with no Colony

Eliminate Empire

Yes

No

No

Best Solution  Archive

Yes

Set NFC 0

Apply crossover operator

based on IInitialization

phase

Perform

Revolution

among colonies

Evaluate itness of

new created

solutions

Archive the Best Solution

among the all Countries

Self-Adaptive Crossover

Fig. 7. Flowchart of the main phase.

K+1  m=1 I  i=1 I  j=1 j=i I  l=1 K_ +1 k=1 k=l R  r=1 R  s=1 W_{i j}ωUkl mnLωmrXimrωLωnsXjnsω≤ ¯

Γ

n[

θ

nω

(

1 −

η

ωn

)

+

η

nω]

∀

n∈

{

1 ,...,K

}

,

ω

∈

, (42) (12),(13),(16)-(21),(31),(32),(35),(37)

Inaddition to theabove approximationmethod,according toMohammadi etal.[27], weuse apartial relaxationofa sub-problem of theapproximated problem

P

. Inthis procedure, a sub-problem of

P

can be createdby dividing the set ofhubs, whereanytwo sets ofhubs, SgandSq,are distinctifthere isatleastone hub k∈Sg andk∈Sq.Thus, there are

C_H|K|=K!/

(

(

K− H

)

!H!

)

distinct hub sets in a network. For a selected set of hubs Sg, where

|

Sg

|

=H, the corresponding location decisionvariablesZkarefixed at1and

P

isthenreducedtoa mixed-integernon-linearassignmentsub-problem (SPg) overthebinarydecisionvariablesX,U,L.Now,let RP

(

SPg

)

bethe partialrelaxationofSPg,wherethe integralityof theotherbinaryvariablesisrelaxed(i.e.,0<Xrω

ik , Lkl, Umnkl <1).Moreover,letOFVRP(SPg)be theoptimalobjectivefunction

valueofR_P

(

S_Pg

)

.ThisvaluecanbedeclaredasalowerboundfortheoptimalvalueofSPg,sotheminimumvalue among thesevaluesfromthepossibleC_H|K| setsdeterminesa lowerboundfortheoptimalvalue oftheoriginalproblem

P

,where OFV_LB(P)=min

g OFVRP(SPg).

6. Computationalresults

Weperformedanumberofnumericalexperimentstoevaluatetheperformanceoftheproposedalgorithm.Weencoded thealgorithm inC++ andexecutedit ona Pentium4 computerwitha 3.0GHzCPU and16GBRAM,whichoperated

Mi-Table 1

Randomly generated parameters. Parameters

W C F η θ 

Values ∼U (20,500) ∼U (50,1500) ∼U (5e3,2e4) ∼U (0,0.3) ∼U (0.5,1) ∼U (1e3,5e3)

Table 2

SAICA parameters settings.

Parameters Settings Parameters Settings

Population size 200 ξ 0.05

Imperialist no. 8 β 1.8

Assimilation rate 0.80 NFC 50,0 0 0 Revolution rate 0.20

crosoftWindows7Ultimate.WebenchmarkedourresultsusingthenonlinearbranchandreducedtheapproachinBARON solver.Thecomputationaltimesarereportedinseconds.

6.1. Experimentaldesign

Wegenerated45random testproblemswithdifferentsizesintermsofthenumberofnodes(I), numberofhubs(H), numberofassignment levels(R),andnumberof scenarios(

), whichranged from10to 200,4to 25,2to 5,and10to 50,respectively.Thetestproblemswerelabeledas‘‘I_{− H− R−}

’’.Forexample,instance‘‘40_{− 8− 4− 15}’’had40non-hub nodes,eighthubnodes,fourassignmentlevels,and15scenarios.Forallofthetestproblems,itwasconsideredthatall non-hubnodescould beoriginanddestinationnodesatthesametime. Disruptionscenarios were generatedrandomlywhere each hub couldbe disrupted witha probability ofqω_{k =}0_.15.This probability wasintentionally selectedas highinorder todetermine the impactof disruptionsandto assessthe performance ofthe modelwhen disruptionswere asignificant factor.For duplicate scenarios, a newscenario was generatedand theprocedure was repeateduntil

unique scenarios weregenerated.

Thefeasibilityofthe modeldependsstrongly onthevalue ofp.Forinstance,p=1will usuallyguaranteethe feasibil-ityofthe model,butit isless realistic becauseit allows a largeincrease in thecost of disruptedscenarios. Bycontrast, smallvaluesofpleadtoinfeasibility.Heuristicalgorithmsareavailable tofindtheminimumvalueofpinordertoobtain feasiblesolutions anda method forfinding theminimum feasiblep inthe proposed SRHLPis introduced inSection 6.4. Weperformedalloftheexperimentswiththesamevalueofp=0.15forconsistency.Theotherparametersweregenerated uniformlyasshowninTable1.

6.2.PerformanceofSAICA

Inthissection,wepresentcomparisonsbetweentheperformanceoftheproposedSAICAandthelowerboundprocedure (LB).First, the requiredparameters in theproposed SAICAalgorithm were tuned using thewell-known response surface methodology (RSM) andthey are tabulated in Table 2. Due to space limitations, the details of the RSM results are not presentedbutthey can be provided onrequest. The parametersettings were pre-testedover randomlyselected problem instances beforethey were used in thecomparative study. Allevolutionary algorithms are stochastic innature andtheir performanceoftenfluctuates,soforall45testproblems,eachinstancewasgenerated20timesandtheaveragevalueofthe resultsispresented.The resultsofthe performancecomparisonare showninTable 3.The‘‘TestProblem’’columnshows theinstancename.Thecolumns“Time” and“OFV” presenttheruntimeandtheobjectivevalueforboththeLBandSAICA. First,inordertodeterminetheaccuracyoftheproposedLB,thedifferencebetweenOFVLB(P) andtheoptimalsolution

wascalculatedfor15smalltestproblemsas:[100×

(

OFVp− OFVLB(P)

)

/OFVLB(P)].AccordingtoFig.8,themeandifference

betweenthe LB andoptimalsolution was0.48%. Therefore,the accuracy ofthe LB wasabout0.48% in comparisonwith optimalsolutions.

TheLBmethodandSAICAarecomparedinthe“%GAP” column,wherethe“Time” columnshowsthepercentageofCPU timeunder theLB requiredbythe proposed SAICAalgorithmand“OFV” indicatesthe difference(as apercentage) inthe objectivefunctionvalues.A valuelessthan 100%inthe“Time” columnindicates thebetter performance oftheproposed SAICAintermsoftheCPUtime,whichwasthecaseforallofthetestproblems.Accordingtothe‘‘OFV’’columnas“%GAP,” themeandifference betweentheproposed SAICAandLB procedurewas0.89%.The meandifference betweenthe LBand optimalsolutionwas0.48%,sothemeandifferencebetweentheproposedSAICAandtheoptimalsolutionwas0.41%(i.e., 0.89–0.48%).

Asshown inTable3, comparedwiththe LB,our proposed algorithmfound near-optimalsolutions forall ofdata sets withameandifferenceof0.89%,butitrequiredonlyasmallfractionofthetimetakenbyLB(18.76%onaverage).Itshould alsobenotedthattheCPUtimegenerallyincreasedwiththenumberofscenariosinmostcases.

Table 3

SAICA performance – vs. LB.

Test problem LB SAICA % GAP

Time OFV Time OFV Time (%) OFV (%) 10-3-2-10 18 286,572 13 288,119 72 .22 100 .54 10-4-2-10 19 291,881 14 293,486 73 .68 100 .55 15-3-2-12 26 315,627 19 317,299 73 .07 100 .53 15-4-2-16 32 321,438 20 323,238 62 .50 100 .56 15-5-3-14 30 305,772 21 307,576 70 .00 100 .59 25-3-2-16 93 410,319 31 412,822 33 .33 100 .61 25-4-2-16 96 416,278 36 418,775 37 .50 100 .6 25-5-3-18 112 407,551 31 410,119 27 .67 100 .63 25-6-3-18 117 412,715 33 415,563 28 .21 100 .69 25-7-3-10 101 396,406 34 398,943 33 .66 100 .64 40-5-2-10 257 513,284 46 516,826 17 .90 100 .69 40-6-3-10 263 507,426 49 511,080 18 .63 100 .72 40-7-3-12 289 519,320 48 523,007 16 .61 100 .71 40-8-3-12 293 511,815 51 515,654 17 .41 100 .75 70-6-2-12 615 731,492 79 736,832 12 .84 100 .73 70-7-3-10 609 728,391 77 734,145 12 .64 100 .79 70-8-3-30 623 724,555 81 730,134 13 .00 100 .77 70-9-4-40 691 731,581 82 737,068 11 .86 100 .75 70-10-4-45 759 729,290 84 734,832 11 .07 100 .76 100-8-4-16 1831 1,422,073 138 1,433,165 7 .54 100 .78 100-10-3-16 1842 1,410,332 144 1,421,474 7 .82 100 .79 100-10-4-18 1948 1,417,610 147 1,428,951 7 .54 100 .8 100-12-4-20 1944 1,421,672 149 1,433,188 7 .66 100 .81 100-14-4-30 2132 1,428,100 153 1,439,668 7 .18 100 .81 100-16-4-45 2159 1,431,209 158 1,443,088 7 .32 100 .83 120-12-3-20 2895 1,620,319 203 1,634,416 7 .01 100 .87 120-12-4-30 3214 1,689,372 210 1,704,407 6 .53 100 .89 120-14-4-35 3231 1,722,119 208 1,738,479 6 .43 100 .95 120-14-5-40 3244 1,673,291 215 1,688,853 6 .62 100 .93 120-16-3-45 2958 1,734,880 235 1,751,882 7 .94 100 .98 120-18-4-50 3031 1,702,315 229 1,718,487 7 .55 100 .95 150-14-5-25 4752 1,844,351 378 1,863,717 7 .95 101 .05 150-16-5-25 4763 1,851,618 383 1,871,615 8 .04 101 .08 150-18-4-30 4979 1,923,511 392 1,944,862 7 .87 101 .11 150-18-5-35 5022 2,021,632 389 2,042,657 7 .75 101 .04 150-20-4-40 5104 2,104,371 392 2,129,203 7 .68 101 .18 150-20-5-45 5149 1,978,439 391 2,0 0 0,795 7 .59 101 .13 150-20-6-50 5348 2,112,412 403 2,136,916 7 .53 101 .16 200-16-3-35 7482 2,648,528 638 2,6 83,4 89 8 .52 101 .32 200-16-4-35 7495 2,521,339 629 2,553,864 8 .39 101 .29 200-18-4-40 7745 2,768,341 641 2,804,329 8 .28 101 .3 200-18-5-40 7753 2,782,669 653 2,818,287 8 .42 101 .28 200-20-5-40 7784 2,828,102 659 2,865,716 8 .46 101 .33 200-22-5-50 8647 2,791,420 712 2,830,779 8 .23 101 .41 200-25-6-50 8679 2,879,922 724 2,921,393 8 .34 101 .44

0.34

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Ga

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Sample Test Problem

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1

150000

CPU Time

25-6-3-18 40-6-3-10 70-8-3-30 100-8-4-16 150-14-5-25 200-16-4-35

Fig. 9. Algorithm performance vs. different p values.

0 1000 2000 3000 4000 5000 6000 7000 8000 0.15 0.2 0.4 0.6 0.8 1 150000

CPU Time

25-6-3-18 40-6-3-10 70-8-3-30 100-8-4-16 150-14-5-25 200-16-4-35

Fig. 10. LB performance vs. different p values.

WealsotestedtheproposedSAICAusinginstanceswithsmallerdisruptionprobabilities(3%,5%,and10%),buttherewas nosignificantdifferenceinperformance.Therefore,thedetailedresultsareomittedduetospacelimitations.

WealsoinvestigatedtheeffectofchangingthevalueofpontheperformanceoftheproposedSAICA.Thus,weperformed severalnumericalexperiments withdifferentvaluesofp usingvarious datasets. Asshown inFig.9, theperformance of SAICAwasaffectedlittlebythevalueofpforallofthetestdatasets.However,theCPUtimerequiredwashigherwhenthe valueofpwassmaller.IncontrasttoSAICA,theperformanceofLBwasaffectedgreatlybydifferentvaluesofp,asshown inFig.10.Therefore,theproposedSAICAmethodperformedefficientlywithsmallerandlargervaluesofp.

6.3.Minimaxcostandminimaxregretcriteria

Twoother minimax cost andminimaxregretrobustnessmeasures are oftencriticizedasbeingtooconservative when they are measured against a worst-case scenario that occurswith a small probability [2].Thus, the relative regrets are increasedgreatly inotherscenarios.Inreality,managersareeagertomaximize profitsandminimize costs,andthusthey arenotpreparedonlyfortheworstsituation.Theproposedrobustapproachismoresuitablefororganizationsthatplanfor theminimumcostormaximumprofitoveralong-termhorizon,whilelimitingthescenariocostsattemptstodecreasethe short-termeffectofdisruption.In ordertodemonstratethat ourproposedrobust approachislessconservative compared withtheminimax cost andminimax regret measures,the resultsof severalexperimentsare presented inTable 4.These minimaxcost andminimax regretmeasures were calculatedaccordingto models(43)and(44),andsolved to optimality usingtheBARONsolver.

Minimax Cost min max

ω∈

ζ

ω

(

X,Z,U,L,P

)

s .t .

(

12

)

,

(

13

)

,

(

16

)

−

(

21

)

,

(

31

)

,

(

32

)

,

(

35

)

,

(

37

)

,

(

41

)

,

(

42

)

,

Table 4

Comparison with the minimax cost and minimax regret criteria.

ω ζω ∗ p -Robust Minimax cost Minimax regret

OFV Regret OFV % GAP Regret OFV % GAP Regret 0 311 ,317 312 ,935 1 .0052 335,195 7 .11 1 .0767 315,768 0 .91 1 .0143 1 291 ,469 295 ,724 1 .0146 338,686 14 .86 1 .1620 299,018 1 .11 1 .0259 2 304 ,118 307 ,919 1 .0125 329,329 6 .95 1 .0829 317,286 3 .04 1 .0433 3 321 ,555 324 ,545 1 .0093 337,504 3 .99 1 .0496 328,307 1 .16 1 .0210 4 305 ,852 315 ,578 1 .0318 314,935 −0 .20 1 .0297 315,241 −0 .11 1 .0307 5 298 ,270 300 ,208 1 .0065 332,839 10 .86 1 .1159 309,216 3 .00 1 .0367 6 300 ,729 303 ,826 1 .0103 321,449 5 .80 1 .0689 313,389 3 .15 1 .0421 7 307 ,100 320 ,489 1 .0436 319,414 −0 .33 1 .0401 323,130 0 .82 1 .0522 8 325 ,628 327 ,223 1 .0049 354,218 14 .22 1 .0878 346,370 5 .85 1 .0637 9 348 ,628 370 ,731 1 .0634 360,516 −2 .75 1 .0341 365,292 −1 .46 1 .0478 10 298 ,331 307 ,937 1 .0322 324,912 5 .51 1 .0891 306,326 −0 .52 1 .0268 11 301 ,4 4 4 306 ,749 1 .0176 333,155 8 .61 1 .1052 319,018 3 .99 1 .0583 12 331 ,529 334 ,114 1 .0078 349,564 4 .62 1 .0544 344,723 3 .17 1 .0398 13 294 ,720 297 ,991 1 .0111 317,118 6 .42 1 .0760 307,127 3 .06 1 .0421 14 299 ,138 306 ,048 1 .0231 311,911 1 .92 1 .0427 309,099 0 .99 1 .0333

10

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OFV

x 10

4

Scenario

p-SRHLP

Minimax Cost

Minimax Regret

Fig. 11. Comparison between three robustness measures: OFV.

Minimax Regret min max

ω∈ ζω(X,Z,U,L,P)−ζω∗ ζω∗ s .t .

(

12

)

,

(

13

)

,

(

16

)

−

(

21

)

,

(

31

)

,

(

32

)

,

(

35

)

,

(

37

)

,

(

41

)

,

(

42

)

. (44)

Thenumericalexperimentswereconductedusingasetofrandomlygenerateddatawithasizeof15-5-3-14andp=0.15. Table4 showsthe optimalscenariocosts

ζ

_ω∗,aswell asthescenariocost (“OFV”)andrelativeregret(“Regret”) foreach model(p-SRHLP,minimaxcost,andminimaxregret)ineachscenario.Thepercentagedifferencesbetweenthescenariocosts forp-SRHLParealsocompared withthe minimaxcost andminimax regretmodelsinthe“%GAP” column. Thecosts and regretsofeachmodelforeachscenarioaredepictedinFigs.11and12,respectively.Table4andFigs.11and12showthat theminimaxcostandregretmodelsmostlyconsideredtheworst-casescenario(i.e.,scenario9)inthisdataset.Incontrast totheminimaxmodels,theproposedp-SRHLPmodelhadahighercostinscenario9,butlowercostsinallbutthreeofthe otherscenarios.

Inaddition, therelativeregrets were verylarge intheminimax costmodel (≥ 14%) comparedwiththep-SRHLP and minimaxregretmodels.Moreover,thep-SRHLPmodelobtainedsmallerrelativeregretscomparedwiththeminimaxregret modelinmostofthescenarios,whileittriedtominimizethecostofthenominalscenarioanditwasaffectedlessbythe disruption scenarios.Thus, theproposed p-SRHLP waslessconservativethanthe othertwo models.Furthermore,another advantageofthep-SRHLPmodelisitsflexibilityintermsofadjustingthevalue ofptoobtaindifferentsolutions,whereas theothertwomodelsprovideonlyonesolution.

0.4

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9

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12

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14

Regret

Scenario

p-SRHLP

Minimax Cost

Minimax Regret

Fig. 12. Comparison between three robustness measures: Regret.

Table 5

OFV vs. the maximum regret.

Test problem p OFV OFV inc (%) Max regret Max regret dec (%)

25-6-3-18 ∞ 348,110 0 0 .8535 0 0 .8535 352 ,537 1 .27 0 .7245 17 .80 0 .7245 353 ,371 0 .23 0 .6475 11 .89 0 .6475 354 ,112 0 .21 0 .6015 7 .64 0 .6015 354 ,714 0 .17 0 .5725 5 .06 0 .5725 355 ,224 0 .14 0 .5485 4 .37 0 .5485 357 ,890 0 .75 0 .5265 4 .18 0 .5265 359 ,474 0 .44 0 .5045 4 .36 0 .5045 360 ,331 0 .23 0 .4985 1 .20 0 .4985 361 ,668 0 .37 0 .4945 0 .81 0 .4945 362 ,777 0 .31 0 .4915 0 .61 0 .4915 364 ,482 0 .47 0 .4905 0 .20 70-8-3-30 ∞ 629 ,311 0 0 .7135 0 0 .7135 635 ,629 1 .0 0 .6355 12 .27 0 .6355 636 ,003 0 .06 0 .5585 13 .78 0 .5585 636 ,667 0 .11 0 .4845 15 .27 0 .4845 637 ,101 0 .07 0 .4315 12 .28 0 .4315 638 ,193 0 .17 0 .4045 6 .67 0 .4045 639 ,722 0 .24 0 .3725 8 .59 0 .3725 640 ,110 0 .06 0 .3515 5 .97 0 .3515 641 ,403 0 .20 0 .3485 0 .86 0 .3485 643 ,220 0 .28 0 .3445 1 .16 0 .3445 644 ,339 0 .17 0 .3425 0 .58 0 .3425 646 ,208 0 .29 0 .3405 0 .58 6.4.Priceofrobustness

One of the most importantissues in the proposed p-SRHLP is the “priceof robustness,” i.e., the extra cost required todesign a morerobust andreliable hub networkwithmuch lower scenariocosts when disruptionsoccur. According to Snyderand Daskin[13],more robust solutions can be found witha smallincrease inthe cost. Inthis section,we show thatourproposedmodelingapproachalsoobtainssimilartrends.Thus,weinvestigatedthetrade-off betweenthemaximal relativeregretandthenominalcostasfollows.First, weconsidered p₌1andsolved theproblem, andwe calculatedthe maximumrelativeregretoverallthescenarios.Next,wesetptothemaximumrelativeregretminus0.0005andre-solved theproblem.Thisprocedurewasrepeatedbyreducing pateach iterationandwere-solved theproblemuntil nofeasible solutioncouldbefound.Withfurtheriterations,theobjectivevalueincreasedwhilethemaximumregretdecreased when thep-robustnessconstraintsweretightened.

Weappliedthisprocedure totwotest problems,“25-6-3-18” and“70-8-3-30,” andtheresultsare reportedinTable5. Thecolumn“p” showsdifferentvaluesofp.Thethirdcolumn(“OFV”)showstheobjectivefunctionvalue obtainedby the