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Algorithms and Reformulation Techniques for Three-Stage Interdiction and Fortification Problems

Permanent Link: http://ufdc.ufl.edu/UFE0045177/00001

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Title: Algorithms and Reformulation Techniques for Three-Stage Interdiction and Fortification Problems
Physical Description: 1 online resource (130 p.)
Language: english
Creator: Prince, Michael D
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: facility -- fortification -- interdiction -- mixed-integer -- network -- optimization -- procurement
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In this paper we present several optimization problems in supply chains and logistics. The first group of problems we analyze involve a procurement optimization problem in which a firm seeks to minimize its procurement costs. It has a demand for some item that can be satisfied through purchases from a set of capacitated suppliers. We consider the scenario in which the capacities are are uncertain as well as scenarios in which competing buyers exist. The firm may have the opportunity to pay a fixed cost amount to some/all of the suppliers to ensure that their capacities are available. The second group of problems we consider are closely relate to the classic p-median and k-center combinatorial optimization problems. We consider extensions of these problems in which the facilities’ locations are fixed, but before the demand node-facility assignments are made, interdiction and fortification stages occur. In the median version of the problem, the network owner seeks to minimize its total assignment cost when assigning each demand node to a facility and in the center version of the problem, the network owner seeks to minimize the maximum of all demand node-facility assignments costs. However, before the network owner makes these assignments, an adversary selects a subset of facilities to interdict (or remove), so that the network owner’s assignment objective is maximized. Prior to the interdiction stage, the network owner 9 may fortify a subset of facilities so they cannot be interdicted. Additionally, we consider problems in which there are limitations on the number of demand nodes that can be assigned to each facility. Each of these problems contain distinct sets of decisions made at different points in time as well as certain aspects of uncertainty. The modeling approach we use is a three-stage interdiction and fortification framework resulting in min-max-min games between two opposing entities, the leader and the follower. The leader represents the protagonistic decision maker while the follower may represent an intelligent adversary or merely some type of worst-case scenario. The leader’s goal is to minimize its objective function while the follower’s goal is to maximize this objective function. Through the use of dualization and reformulation techniques, we transform the three-stage optimization models into single- or two-stage models. Further, we use linearization and penalty methods to increase tractability of the models and solve the them using specially-designed and enhanced cutting-plane techniques with CPLEX as a mixed-integer programming solver. We present computational results for all models and, when applicable, compare to the results for existing algorithms.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Michael D Prince.
Thesis: Thesis (Ph.D.)--University of Florida, 2013.
Local: Adviser: Smith, Jonathan Cole.
Local: Co-adviser: Geunes, Joseph Patrick.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045177:00001

Permanent Link: http://ufdc.ufl.edu/UFE0045177/00001

Material Information

Title: Algorithms and Reformulation Techniques for Three-Stage Interdiction and Fortification Problems
Physical Description: 1 online resource (130 p.)
Language: english
Creator: Prince, Michael D
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: facility -- fortification -- interdiction -- mixed-integer -- network -- optimization -- procurement
Industrial and Systems Engineering -- Dissertations, Academic -- UF
Genre: Industrial and Systems Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In this paper we present several optimization problems in supply chains and logistics. The first group of problems we analyze involve a procurement optimization problem in which a firm seeks to minimize its procurement costs. It has a demand for some item that can be satisfied through purchases from a set of capacitated suppliers. We consider the scenario in which the capacities are are uncertain as well as scenarios in which competing buyers exist. The firm may have the opportunity to pay a fixed cost amount to some/all of the suppliers to ensure that their capacities are available. The second group of problems we consider are closely relate to the classic p-median and k-center combinatorial optimization problems. We consider extensions of these problems in which the facilities’ locations are fixed, but before the demand node-facility assignments are made, interdiction and fortification stages occur. In the median version of the problem, the network owner seeks to minimize its total assignment cost when assigning each demand node to a facility and in the center version of the problem, the network owner seeks to minimize the maximum of all demand node-facility assignments costs. However, before the network owner makes these assignments, an adversary selects a subset of facilities to interdict (or remove), so that the network owner’s assignment objective is maximized. Prior to the interdiction stage, the network owner 9 may fortify a subset of facilities so they cannot be interdicted. Additionally, we consider problems in which there are limitations on the number of demand nodes that can be assigned to each facility. Each of these problems contain distinct sets of decisions made at different points in time as well as certain aspects of uncertainty. The modeling approach we use is a three-stage interdiction and fortification framework resulting in min-max-min games between two opposing entities, the leader and the follower. The leader represents the protagonistic decision maker while the follower may represent an intelligent adversary or merely some type of worst-case scenario. The leader’s goal is to minimize its objective function while the follower’s goal is to maximize this objective function. Through the use of dualization and reformulation techniques, we transform the three-stage optimization models into single- or two-stage models. Further, we use linearization and penalty methods to increase tractability of the models and solve the them using specially-designed and enhanced cutting-plane techniques with CPLEX as a mixed-integer programming solver. We present computational results for all models and, when applicable, compare to the results for existing algorithms.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Michael D Prince.
Thesis: Thesis (Ph.D.)--University of Florida, 2013.
Local: Adviser: Smith, Jonathan Cole.
Local: Co-adviser: Geunes, Joseph Patrick.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045177:00001


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ALGORITHMSANDREFORMULATIONTECHNIQUESFORTHREE-STAGEINTERDICTIONANDFORTIFICATIONPROBLEMSByMIKEPRINCEADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013MikePrince 2

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Idedicatethisworktomyfamily 3

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ACKNOWLEDGMENTS Irstwouldliketothankmyadvisors,Dr.J.ColeSmithandDr.JosephGeunesfortheirsupportandguidance.Iambeyondgratefulforhavingtheopportunitytoworkwiththemandcouldnothaveaskedforbetteradvisors.Theircombinedknowledge,experience,attentiontodetail,andenthusiasmhelpedmoldmeintotheresearcherIamtoday.Ourmanyconversations,consistingofbothacademicandothertopics,wereinvaluableinkeepingmeontrackandfocused,whilenevertoostressed.Withouttheircontinuedhelpandadvice,thisdissertationwouldnotbepossible.IwouldliketothankDr.Jean-PhilippeRichardandDr.AnandPaulforservingonmysupervisorycommitteeandfortheirinsightfulcommentsandsuggestions.Also,aspecialthankstoDr.RichardforthefoundationalknowledgehepassedontomeinhisFundamentalsofMathematicalProgrammingandIntegerProgrammingcourses.Ihadthepleasureofworking,studying,andspendingtimewithmanyamazinggraduatestudentcolleagueswhomademyexperienceattheUniversityofFloridatrulyremarkable.TheuncountablehoursspentpreparingforthegeneralexamwithSoheilHemmati,ClayKoschnick,AndrewRomich,CinthiaPerez,andJohannaAmayawasanindispensableexperience.Inaddition,IwouldliketothankBehnamBehdani,KellySullivan,SiqianShen,ShantihSpanton,DeonBurchett,BitaTadayon,JoseWalteros,JorgeSefair,ChrysasVogiatzis,AyseArslan,RuiweiJiang,YiqiangSu,JingMa,andRezaSkandariforthemanywonderfulmemorieswehavetogether.Finally,IwouldliketothankmyfamilyfortheirconstantsupportthroughoutmytimeattheUniversityofFloridaandlifeingeneral.Withoutthem,thisachievementwouldneverhavebeenpossible. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 11 2ATHREE-STAGEPROCUREMENTOPTIMIZATIONPROBLEMUNDER A UNCERTAINTY .................................... 14 2.1MotivationandLiteratureSurvey ....................... 14 2.2ModelFormulation ............................... 17 2.2.1ProblemFormulationandMethodOverview ............. 18 2.2.2ShortestPathReformulation ...................... 21 2.2.3Two-StageFormulation ......................... 24 2.2.4SolutionMethod ............................ 25 2.3Enhancements ................................. 27 2.3.1M-valuefor( 2b ) ........................... 27 2.3.2InequalitiesfromReduced-SizeSubproblems ............ 30 2.4IllustrationandComputationalResults .................... 34 3PROCUREMENTALLOCATIONPLANNINGWITHMULTIPLESUPPLIERSUNDERCOMPETITION ............................... 44 3.1MotivationandLiteratureSurvey ....................... 44 3.2NotationandProcurementModel ....................... 47 3.3OptimisticProblem ............................... 51 3.4GeneralizedBilevelProblem .......................... 55 3.4.1GeneralizedBilevelModel ....................... 56 3.4.2InvariantFeasibleRegionModel ................... 57 3.4.3Cutting-PlaneAlgorithm:Big-MModel ................ 59 3.4.4Cutting-PlaneAlgorithm:M-FreeModel ............... 64 3.4.5AlgorithmEnhancements ....................... 66 3.5ComputationalResults ............................. 67 4CAPACITATEDFACILITYINTERDICTIONPROBLEMSWITHFORTIFICATIONUNDERMEDIANANDCENTEROBJECTIVEFUNCTIONS ........... 71 4.1MotivationandLiteratureSurvey ....................... 71 4.2ProblemDescriptions,Notation,andModels ................. 74 5

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4.3BilevelModels ................................. 77 4.3.1IMF-CBilevelModel .......................... 77 4.3.2IMFBilevelModels ........................... 79 4.3.3ICF-CBilevelModel .......................... 81 4.3.4ICFBilevelModels ........................... 88 4.4SolutionMethod ................................ 92 4.4.1Cutting-PlaneApproach ........................ 92 4.4.2RobustModelforThird-StageInfeasibility .............. 97 4.5ComputationalExperiments .......................... 99 5CONCLUSIONSANDFUTURERESEARCH ................... 106 APPENDIX APROOFS ....................................... 109 BDETAILEDCOMPUTATIONALRESULTS ..................... 116 REFERENCES ....................................... 125 BIOGRAPHICALSKETCH ................................ 130 6

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LISTOFTABLES Table page 2-1Exampleproblem ................................... 34 2-2Suppliercapacitysets ................................ 37 2-3Buyerdemandasapercentageoftotalsuppliercapacity ............ 37 2-4ResultsforN2f11,...,25g ............................. 42 3-1Suppliercapacitysets ................................ 68 3-2Buyerdemandasapercentageoftotalsuppliercapacity ............ 68 3-3AverageincreaseinCPUtimeandnumberofiterationswhen>0 ....... 70 4-1AveragecomputationtimesandnumberofiterationsforIMF .......... 101 4-2AveragecomputationtimesandnumberofiterationsforICF ........... 102 4-3SummaryofCPUtimesforICF ........................... 103 4-4AveragecomputationtimesandnumberofiterationsforIMF-C ......... 105 4-5AveragecomputationtimesandnumberofiterationsforICF-C ......... 105 B-1ComputationaltimeforN=5,6,7withvaryingbuyerdemands ......... 116 B-2ComputationaltimeforN=8,9,10withvaryingprotectioncosts ........ 117 B-3AverageCPUtimesandnumberofiterationsforN=5,6,7withvaryingbuyerdemands ....................................... 118 B-4AverageCPUtimesandnumberofiterationsforN=8,9,10withvaryingprotectioncosts .................................... 119 B-5AverageCPUtimesandnumberofiterationsforN=5,6,7withvaryingbuyerdemands ....................................... 120 B-6ComputationtimesandnumberofiterationsforIMF ............... 121 B-7ComputationtimesandnumberofiterationsforICF ............... 122 B-8ComputationtimesandnumberofiterationsforIMF-C .............. 123 B-9ComputationtimesandnumberofiterationsforICF-C .............. 124 7

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LISTOFFIGURES Figure page 2-1Twosuppliercostcurvesbeforeandafterinterdiction ............... 20 2-2Shortestpathnetwork ................................ 22 2-3Computationtimesfordifferentdemandlevelsandstrategies .......... 38 2-4Computationtimesfordifferentcapacitysets ................... 39 2-5Computationtimesfordifferentprotectioncosts .................. 40 2-6Computationtimesfordifferentstrategies ..................... 40 4-1IllustrationofaugmentationnetworkGkintheproofofProposition 4.4 ,wheredij-valuesarelistedalongsidetheedgesandm1=m2=m3=2. ........ 84 4-2IllustrationofauxiliarynetworkintheproofofProposition 4.4 .......... 88 4-3Numberofsubproblemssolved ........................... 102 8

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyALGORITHMSANDREFORMULATIONTECHNIQUESFORTHREE-STAGEINTERDICTIONANDFORTIFICATIONPROBLEMSByMikePrinceMay2013Chair:J.ColeSmithCochair:JosephGeunesMajor:IndustrialandSystemsEngineeringInthispaperwepresentseveraloptimizationproblemsinsupplychainsandlogistics.Therstgroupofproblemsweanalyzeinvolveaprocurementoptimizationprobleminwhicharmseekstominimizeitsprocurementcosts.Ithasademandforsomeitemthatcanbesatisedthroughpurchasesfromasetofcapacitatedsuppliers.Weconsiderthescenarioinwhichthecapacitiesareareuncertainaswellasscenariosinwhichcompetingbuyersexist.Thermmayhavetheopportunitytopayaxedcostamounttosome/allofthesupplierstoensurethattheircapacitiesareavailable.Thesecondgroupofproblemsweconsiderarecloselyrelatetotheclassicp-medianandk-centercombinatorialoptimizationproblems.Weconsiderextensionsoftheseproblemsinwhichthefacilities'locationsarexed,butbeforethedemandnode-facilityassignmentsaremade,interdictionandforticationstagesoccur.Inthemedianversionoftheproblem,thenetworkownerseekstominimizeitstotalassignmentcostwhenassigningeachdemandnodetoafacilityandinthecenterversionoftheproblem,thenetworkownerseekstominimizethemaximumofalldemandnode-facilityassignmentscosts.However,beforethenetworkownermakestheseassignments,anadversaryselectsasubsetoffacilitiestointerdict(orremove),sothatthenetworkowner'sassignmentobjectiveismaximized.Priortotheinterdiction 9

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stage,thenetworkownermayfortifyasubsetoffacilitiessotheycannotbeinterdicted.Additionally,weconsiderproblemsinwhichtherearelimitationsonthenumberofdemandnodesthatcanbeassignedtoeachfacility.Eachoftheseproblemscontaindistinctsetsofdecisionsmadeatdifferentpointsintimeaswellascertainaspectsofuncertainty.Themodelingapproachweuseisathree-stageinterdictionandforticationframeworkresultinginmin-max-mingamesbetweentwoopposingentities,theleaderandthefollower.Theleaderrepresentstheprotagonisticdecisionmakerwhilethefollowermayrepresentanintelligentadversaryormerelyuncertaintyinwhichcasewewishtoexaminetheworst-casescenario.Theleader'sgoalistominimizeitsobjectivefunctionwhilethefollower'sgoalistomaximizethisobjectivefunction.Throughtheuseofdualizationandreformulationtechniques,wetransformthethree-stageoptimizationmodelsintosingle-ortwo-stagemodels.Further,weuselinearizationandpenaltymethodstoincreasetractabilityofthemodelsandthensolvethemusingspecially-designedandenhancedcutting-planetechniqueswithCPLEXasamixed-integerprogrammingsolver.Wepresentcomputationalresultsforallmodelsand,whenapplicable,comparetotheresultsforexistingalgorithms. 10

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CHAPTER1INTRODUCTIONEvenwiththecontinuedimprovementsincomputerperformanceandoptimizationsoftware,mathprogrammersstilloftenfaceverychallengingoptimizationproblems.Manycharacteristicsmaycontributetomakingtheseproblemshardtosolveandvarioustechniqueshavebeendevelopedtocountertheseissues.Inourresearch,weapplytwowidely-usedtechniques,reformulationanddecomposition,insolvingsupply-chain/logisticsrelatedoptimizationproblems.Eachoftheproblemsencounteredinourresearchcontainsomeorallofthefollowingcharacteristics: Nonlinearity Uncertainty Integerandbinaryvariables Constraintswhichcontainoptimizationproblemsthemselves LargenumbersofvariablesandconstraintsInourrstlineofresearch,weexamineaproblemfacedbyarmprocuringamaterialinputorgoodfromasetofsuppliers.Thecosttoprocurethematerialfromanygivensupplierisconcaveintheamountorderedfromthesupplier,uptoasupplier-speciccapacitylimit.Thisproblem,introducedin[ 18 ],isNP-hard,butcanbesolvedviaapseudo-polynomiallysizeddynamicprogram.InChapter 2 weextendthisresarchundertheobservationthatcapacitiesareoftenuncertaininpracticedue,forinstance,toproductionshortagesatthesuppliersorcompetitionfromotherrms.Weaccommodatethisuncertaintyinaworst-case(robust)fashionbymodelinganadversarialentity(whichwecallthefollower)withalimitedprocurementbudget.Thefollowerreducessuppliercapacityinordertomaximizetheminimumcostrequiredforthermtoprocureitsrequiredgoods.Toguardagainstuncertainty,thermcanprotectanysupplieratacost(e.g.,bysigningacontractwiththesupplierthatguaranteessupplyavailabilityorbyinvestinginmachineupgradesthatguaranteethesupplier'sabilitytoproducegoodsatadesiredlevel)ensuringthattheanticipatedcapacityofthatsupplierwillindeedbeavailable.Theproblemweconsideristhusathree-stagegameinwhichtherm 11

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rstchooseswhichsuppliers'capacitiestoprotect,thefolloweractsnexttoreducecapacityfromunprotectedsuppliers,andthermthensatisesitsdemandusingtheremainingcapacity.Weformulateathree-stagemixed-integerprogramthatiswellsuitedtodecompositiontechniquesanddevelopaneffectivecutting-planealgorithmforitssolution.Thecorrespondingalgorithmicapproachsolvesasequenceofscaledandrelaxedprobleminstances,whichenablessolvingproblemshavingmuchlargerdatavalueswhencomparedwithstandardtechniques.InChapter 3 wefurtheranalyzethisprocurementproblemandconsiderthetwofollowingextensionsinwhichthefollowerrepresentsacompetitororsetofcompetitors: (a) thefollower'sgoalissimplytominimizeitsownprocurementcost,or (b) thefollowerseekstomaximizetheleadersprocurementcostssubjecttoarequirementthattheirownprocurementcostiswithinaspeciedtoleranceoftheirminimumpossiblecost.Wepresentabilevelmixedintegerprogramming(MIP)formulationfor(a)andathree-stageMIPmodelfor(b).Onceagain,reformulationanddecompositiontechniquesareusedtoproducemodelswhichcanbesolvedusingcutting-planemethodsandaMIPsolver.InChapter 4 ,weconsiderinterdictionandforticationextensionsoftheclassicp-medianandk-centerproblems.Variousresearchershaveexaminedther-interdictionmedianproblemwithfortication(IMF)[ 3 4 23 36 48 49 ],inwhichanetworkownerfortiesfacilitiessothataftersomeadversarialentity(realorhypothetical)hasinterdicteduptoroftheunfortiedfacilities,asetofdemandpointsisassignedtofacilitieswiththegoalofminimizingthetotalassigneddistance.Wedevelopasolutionapproachforthisproblemthatsolvesinstancesmuchquickerthanexistingalgorithms.Further,wedevelopamodelforthek-centerversionofthisproblem,ICF,inwhichtheobjectiveistominimizethemaximumassigneddistance.Inaddition,ourmethods 12

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enableustosolvetheseproblemswhenthereareconstraintsrestrictingthenumberofdemandpointsassignedtoeachfacility. Remark1.Chapters 2 4 areeachindividuallyself-contained.WhiletheworkinChapters 2 and 3 iscloselyrelated,alotofthenotationsharedbetweenthechaptersisre-introducedatthebeginningofChapter 3 .ThisenablesthereadertoskipChapter 2 ifdesired.Further,somenotationsuchasthecharacterx(usedasaparameterinChapters 2 and 3 andavariableinChapter 4 ),hasdifferentusesindifferentchapters.Thusthereaderisadvisedtorecalltheself-containmentofthechapterswhenencounteringsuchinstances.2 13

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CHAPTER2ATHREE-STAGEPROCUREMENTOPTIMIZATIONPROBLEMUNDERUNCERTAINTY 2.1MotivationandLiteratureSurveyConsiderarmthatseekstoprocureagivenquantityofagoodorproductioninputfromasetofcapacitatedsuppliers.Thecostofprocuringaquantityfromanygivensupplierisaconcavefunctionofthequantityorderedfromthesupplier.Concaveprocurementcoststructuresarisefrequentlyinpractice,e.g.,whenaxedcostexistsfororderingapositiveamountofgoodsfromasupplier,orwhenthesupplieroffersbulkdiscountsforlargeorders.Weconsidertheproblemofminimizingtheprocurementcostsincurredbytherm,whichisknowntobeNP-hardevenwhenalldataisdeterministic.Insuchsettingsitisquitepossiblethatsomeofthesupplierswillnotbeabletoprovidethermtheirinitiallystatedcapacityoroutputlevels.Forinstance,asuppliermayactuallycorrespondtooneoftherm'sownplants,whosecapacitiesaregovernedbyanominalproductionrate.Ifamachinemalfunctionsorifotherfactorsatthisplantaffectitsnominalproductioncapacity,thentherm'sprocurementstrategymaybecomeinvalid(andexpensive,ifthepossibilityofthesefailuresisnottakenintoaccountaheadoftime).Inanothersetting,asuppliermaycorrespondtoanexternalrmwhoseavailablecapacitycanbereducedbyprocurementordersfromcompetitors,orbyothersourcesofuncertainty,asintheinternalcapacitycase.Thischapterconsidersamulti-phasestrategythatthermcantaketominimizeitsworst-casecostundersupplycapacityuncertainty.Initially,thermmaypro-tect(orguarantee)someofthesuppliers'capacitiesatacost.Thiscostmaybeassociatedwithreplacingequipmentwhosefailurecouldresultinproductionshortages,providingbackupproductioncapacity,or(inthecaseofexternalsuppliers)contractuallyensuringsomeminimumamountofsupplyavailability.Whileeachprotectedsupplierisguaranteedtoofferthermitsnominalcapacitylevel,theother(unprotected)suppliers 14

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mayhaveshortagesintheircapacities.Giventheactualrealizedsupplycapacities,thermcanthenmakeitsprocurementdecisionsatminimumcost.Themethodofrepresentinguncertaintyinunprotectedsupplycapacitiesisthusanimportantmodelingdecision.Wetaketherobustoptimizationviewpointinthischapterinordertominimizetheworst-caseprocurementcostsincurredbytherm.Anextremelyconservativemodelwouldsetthenominalcapacitiesofunprotectedsupplierstotheirsmallestpossiblevalues.Ourapproachisakintothebudgeteduncertaintyof[ 13 ],inwhichthetotalamountofreductioninthesuppliercapacitiesisbounded,butoccursinsuchawaythatmaximizestherm'soptimalprocurementcosts.RobustoptimizationwasintroducedasamainstreammodelingtoolbySoyster[ 56 ]andreceivedrenewedattentioninthelate1990s(seeforinstance[ 7 9 26 27 ]).ThebudgeteduncertaintystrategyofBertsimasandSim[ 13 ]thatweuseinthischapterisrelatedtotheworkofBen-TalandNemirovski[ 10 ]fornon-polyhedraluncertaintysets(e.g.,setscontainingallpossibleuncertaindatarealizations).WerefertothetutorialchapterbyBertsimasandThiele[ 14 ]andthebookofBen-Taletal.[ 6 ]forathoroughdiscussionoftherobustoptimizationliterature.Itisusefultoconceiveofthisprocurementplanningproblemwithuncertaincapacityasathree-stagegameplayedbetweenaleaderrmandanadversarial(andpossiblyhypothetical)follower,asfollows.Theleaderplaysrsttoprotectsuppliers'capacitylevelsforsomesubsetofsuppliers.Thefollowerplayssecondtoreduceunprotectedsupplycapacities,withthegoalofmaximizingtheleader'sthird-stageobjective:tominimizeprocurementcostsfromtheremainingcapacitiesateachsupplier.Notethatalthoughthisproblemisenvisionedtoaidagainstworst-caseaccidentaldisruptionstosupplies,itequivalentlymodelsthecaseinwhicharealpredatorycompetitor(thefollower)seekstostrategicallyeliminatecapacityinamannerthatdrivesuptheleaderrm'scosttothegreatestextentpossible. 15

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Inviewingthefollowerasapredatorycompetitor,itisworthnotingthatpredatorybusinesspracticesaregenerallyillegal.However,itisnotuncommonforacompanytomakepurchasingdecisionsthatcanputcompetitorsatadisadvantage,e.g.,withtheintentofmaximizingtheoptimalpurchasingcostsincurredbycompetitors.Evidenceoftheexistenceofpredationcanbeseenin[ 5 41 43 ],alongwithrationaleastowhysuchstrategiesmightbebenecial[ 31 ].Furthermore,evenifthecompetitor'smotivationsarenotpredatory,itmaybeimpossibletoanticipatewhichprocurementdecisionswillbeoptimalforthem,orwhethertheywillactuallyidentifyandfollowanoptimalstrategy.Thedifcultyinpredictingacompetitor'sactionbecomesevenmorecomplexwhenasinglecompetitoractuallyrepresentsagroupofseveralindependentbuyers.Alargebodyofliterature(e.g.,[ 25 35 57 ])existsontheusageandbenetsofquantitydiscountsinsupplychains.Furthermore,variousresearchershaveexaminedproblemsinwhichasinglesuppliersellstocompetingbuyers.Forexample,BernsteinandFedergruen[ 12 ]andIngeneandParry[ 33 ]discusscoordinationgamesinwhichpricingandorderingdecisionsaremadeinordertomaximizeprots.Incontrast,relativelylittleliteratureexiststhatconsiderscompetitivebuyingfrommultiplecapacitatedsuppliers.Minner[ 39 ]discussesvariousmodelsforasinglebuyerwhomustallocateitsrequirementstomultiplesuppliers.In[ 17 18 20 ],asinglebuyerseekstominimizeitsprocurementcostsfromasetofcapacitatedsuppliersofferingquantitydiscounts.Bycontrast,ourcontributionconsiderspurchasingdecisionsforasinglebuyeroperatingunderuncertainty,orequivalentlyfortwormswhocompetefortheprocurementofaproductamongmultiplesuppliers.Oursolutionapproachtransformsthethree-stagegamewedescribedintoatwo-stagemathematicalprogrammingformulation,whichwesolveusingacutting-planealgorithm.(Seealso[ 52 ]forarelatedapproachinthecontextofatwo-playerproductintroductiongame.)Here,wegeneratecuttingplanesbasedonthesolutiontoaspecially-structurednonlinearknapsackproblem.Overviewsofnetworkinterdiction 16

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problemsandnetworkinterdictionwithforticationproblemscanbefoundin[ 50 51 53 58 ];three-stagegamesinvolvingnetworkdesign,nationaldefense,facilitylocation,andsupplychainsarediscussedin[ 54 ],[ 15 16 ],[ 23 44 49 ],and[ 55 ],respectively.Seealso[ 19 ],whichdevelopsanimplicitenumerationalgorithmforfortifyingashortest-pathnetworkagainst(worst-case)interdiction.Ouralgorithmicstrategyconstructsashortest-pathproblemonalargenetwork,whichrepresentstheleader'sprocurementproblem.Thearcsofthisnetworkrepresentpossiblequantitiestopurchasefromeachsupplier.Hence,intherststage,theleadermaychoosetofortifygroupsofarcs(byprotectingasupplier)sothattheycannotbedestroyed.Inthesecondstage,thefollowerdestroyssomeoftheunfortiedarcsinordertomaximizethelengthoftheshortestpaththattheleadersolvesinthethirdstage.Thischapterthuscontributesanetworkinterdictionmodelingframeworkfortheanalysisofcapacitymanagementinsupplychains.Wealsoprovidevariousalgorithmicandmodelingcontributionsforsolvingproblemswithinthisclass.Inparticular,wedemonstratehowtosolvelargerproblemsthanstandardtechniquescanhandle,bysolvinganinterestingsequenceofscaledproblemsandproblemrelaxations.Theremainderofthischapterisorganizedasfollows.InSection 2.2 weformulateathree-stagenonlinearmixed-integerprogram(MIP)basedonthescenariodescribedabove.WeprovidemodelandalgorithmenhancementsdevisedtoincreasethescopeofproblemsthatcanbesolvedwithinreasonablecomputationallimitsinSection 2.3 .Finally,wepresentcomputationalresultsthatdemonstratetheefcacyofourapproachinSection 2.4 2.2ModelFormulationInSection 2.2.1 ,weintroducetheformulationforthethree-stagegame,alongwithanoverviewoftheapproachthatwetakeforsolvingitinthischapter.Wethenreformulatethethird-stageproblemasashortest-pathprobleminSection 2.2.2 .Bytakingthedualofthethirdstage,wecombinethesecondandthirdstagesintoasingle 17

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optimizationproblem,whichresultsinthetwo-stageproblempresentedinSection 2.2.3 .TheoverallsolutionmethodisthenpresentedinSection 2.2.4 2.2.1ProblemFormulationandMethodOverviewIntheproblemweconsider,twormspurchaseacommonproductfromsomesetofsuppliers(wherethesecondrmmaymerelyrepresentuncertaintyinsuppliers'capacities).Theleaderhastheoptionofprotectingsomeor(potentially)allofthesuppliers.Thefollowerthenmakesitspurchasesfromtheunprotectedsuppliers.Finally,theleadersatisesitsdemandwithanyoftheremainingcapacityfromallsuppliers.Weassumethatthecostofprocuringfromeachsupplierisaconcavefunctionoftheamountprocuredfromthesupplier;thatis,thecostofprocuringqiunitsfromsupplieri2S=f1,...,Ngequalspi(qi),wherepi(qi)isanonnegative,nondecreasing,concavefunction.Also,associatedwitheachsupplieri2Sisaprotectioncost,ri,andproductioncapacity,ui.TheleaderisrequiredtopurchaseQLunits,whilethefollowercanpurchaseuptoQFunits.Letzidenoteabinaryvariablethatdetermineswhethertheleaderchoosestoprotect(zi=1)ornotprotect(zi=0)supplieri2S.Letyidenoteanintegervariablethatrepresentsthenumberofunitsthefollowerpurchasesfromsupplieri2S.Thisleadstothefollowingthree-stageformulation,P: minz2f0,1gNrTz+maxy2YminXi2Spi(qi) (2a) s.t.qiui)]TJ /F3 11.955 Tf 11.96 0 Td[(yi,8i2S, (2b) Xi2SqiQL, (2c) q0andinteger, (2d) whereY=y0jPi2SyiQF;yiui(1)]TJ /F3 11.955 Tf 11.95 0 Td[(zi),8i2S;yiinteger,8i2S.First,notethat( 2c )canequivalentlybewrittenasanequality;theinequalityherebecomesusefulinoursubsequentanalysis.Next,whileitisadvantageousforthefollowertopurchaseitsmaximumallowablenumberofunits,wecannotreplacethefollower'sdemandconstraintwithPi2Syi=QF,orelsethefollower'sfeasibleregionmaybe 18

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empty.Forinstance,assumingthatQF1,thefollower'sfeasibleregionisemptywhenallzi=1. Remark2.WeassumethatQL+QFPi2Sui.Ifthiswerenotthecase,thenthefollowercouldmakethethirdstageinfeasibleunlesstheleaderprotectedsupplierswithenoughcombinedcapacitytosatisfyitsdemand.Inthiscase,wewouldneedtoaddthefollowinginequalitytotherststage: Xi2SuiziQL.(2)Theremaininganalysisandmethodspresentedinthischaptercouldthencontinuetobeemployedwithoutmodicationtohandlethiscase.2Figure 2-1 illustratescostcurvesfortwosuppliersfromtheleader'sperspective,beforeandafterthefollowerpurchasesitsunits.(Thedottedpartofthecurvecorrespondstoaninfeasibleprocurementamountfortheleader.)Notethatafterinterdictionoccurs,thetoppartsofthecostcurvesaretruncatedbecausethefollowerhaspurchasedunitsfromeachofthemthatpreventstheleaderfrompurchasingthefullcapacityfromeithersupplier.Whenthisoccurs,iteffectivelyforcestheleadertopayalargerper-unitcostfromasupplierthanhadthefollowernotconsumedsomeofthesupplier'scapacity.Denetheinnerminimizationproblem(IMP)asthethird-stageproblemfacedbytheleader,i.e.,theproblem( 2 )inwhichzandyhavebeenxed.WenotethatIMPisNP-hardasaresultof[ 29 ].Thisobservationhasthefollowingimplicationsforourmethodologicalapproach. Inmaximinoptimizationproblems,suchasthefollower's(second-stage)optimizationproblem,acommonapproachistodualizetheinnerproblem(whichisIMPinthiscontext).TheminimizationdualtoIMPcouldthenbecombinedwiththefollower'sproblemtoformasingle,possiblynonlinear,mathematicalprogram.However,becauseIMPisnotaconvexoptimizationproblem(evenwhenintegralityrestrictionsonqarerelaxed),wecannotdirectlyformulateitsdual.Toovercomethisdifculty,wederiveanequivalent(pseudo-polynomial)linearprogrammingformulationforIMP,alongwithitsdual,inSection 2.2.2 19

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Figure2-1. Twosuppliercostcurvesbeforeandafterinterdiction GiventhisdualIMPformulation,wethencombinethefollower'sproblemandIMPintoasingleMIP,asdemonstratedinSection 2.2.3 .Atthispoint,theoverallthree-stageproblemweconsiderinthischaptercannowbeviewedasthefollowingtwo-stageminimaxproblem:Theleaderrstminimizesthefollower'smaximumobjective,andthenthefollowersolvesthecombinedmaximinMIPformulationasthesecond-stageproblem. Next,inSection 2.2.4 ,weaddressthesolutionoftheforegoingtwo-stageminimaxproblem.Onceagain,thesecond-stage(follower's)problemcannotbedualized,becauseitisaMIP.Inthiscase,weadoptacutting-planestrategytosolvetheproblem.Thevalidityofthesecuttingplanesexploitsthefactthattherst-stagevariables(z)arebinary-valued,andthatthesecond-stagefeasibleregionobtainedinSection 2.2.3 isindependentofz. 20

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2.2.2ShortestPathReformulationFollowingclassicalapproachesintheliterature,weemployapseudo-polynomial-timedynamicprogrammingapproachforsolvingIMP,whichusesashortest-pathrepresentationonanappropriatenetwork.Weconstructanetworkwithsourcenodev00,nodesvijforj=0,...,QLandi2SnfNg,wherenodevijcorrespondstoasolutioninwhichjunitsarepurchasedfromthersti2SnfNgsuppliers,andsinknodevNQL.WedenethesetsFi=fjjmaxf0,QL)]TJ /F14 11.955 Tf 12.16 8.97 Td[(PNs=i+1usgjminfQL,Pis=1usgg,8i2S(whereF0=f0gandPNs=N+1us0,sothatFN=fQLg),thesetsIik=fj2Fi)]TJ /F7 7.97 Tf 6.59 0 Td[(1jk)]TJ /F3 11.955 Tf 12.41 0 Td[(uijkg,8i2S,k2Fi,andthesetsOik=fj2Fi+1jkjk+ui+1g,8i2SnfNg,k2Fi.ThesetFicontainsthefeasibleprocurementamountsfromtherstisuppliers(basedonthetotalcapacityoftherstisuppliersandtheremainingN)]TJ /F3 11.955 Tf 12.05 0 Td[(isuppliers).Giventhatkunitsareprocuredfromtherstisuppliers,thesetIikcontainsthefeasibleprocurementamountsforthersti)]TJ /F4 11.955 Tf 12.43 0 Td[(1suppliers,andthesetOikcontainsthefeasibleprocurementamountsfromthersti+1suppliers.Thenetworkconsistsofallarcsoftheform(vi)]TJ /F7 7.97 Tf 6.59 0 Td[(1j,vik),81iN,k2Fi,j2Iik.Thebinaryvariableyi,qtakesavalueof1ifthefollowerpurchasesatleastqunitsfromsupplieri,and0otherwise,8i2S,0qiminfui,QFg.Arc(vi)]TJ /F7 7.97 Tf 6.58 0 Td[(1j,vik)hascapacity1)]TJ /F4 11.955 Tf 12.42 0 Td[(yi,ui)]TJ /F6 7.97 Tf 6.58 0 Td[(k+j+1andcostpi(k)]TJ /F3 11.955 Tf 12.13 0 Td[(j)fori2S,k2Fi,andj2Iik.Letfijkbetheowonarc(vi)]TJ /F7 7.97 Tf 6.59 0 Td[(1j,vik).AnillustrationoftheresultingnetworkisshowninFigure 2-2 .Givenavectory,weformulatethethird-stageshortest-pathproblem(representingIMP)usingthefollowinglinearprogram(LP): PL:minXi2SXk2FiXj2Iikpi(k)]TJ /F3 11.955 Tf 11.95 0 Td[(j)fijk (2a) s.t.Xk2F1f10k=1, (2b) Xj2Iikfijk=Xj2Oikfi+1kj,8i2SnfNg,k2Fi, (2c) )]TJ /F14 11.955 Tf 17.26 11.35 Td[(Xj2FN)]TJ /F16 5.978 Tf 5.76 0 Td[(1fNjQL=)]TJ /F4 11.955 Tf 9.3 0 Td[(1, (2d) 21

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Figure2-2. Shortestpathnetwork fijk1)]TJ /F4 11.955 Tf 12.25 0 Td[(yi,ui)]TJ /F6 7.97 Tf 6.58 0 Td[(k+j+1,8i2S,k2Fi,j2Iik, (2e) fijk0,8i2S,k2Fi,j2Iik. (2f) TheObjective( 2a )minimizestotalprocurementcosts,whileConstraints( 2b )( 2d )serveasow-balanceconstraints.Constraints( 2e )enforcetheconditionthatsuppliercapacityconstraintsarenotviolatedafterthefollowerhaspurchaseditssupply,andConstraints( 2f )enforcenonnegativityoftheowvariables.Werefertothefeasibleregionof( 2 )asF(PL).Let00,ik,NQL,andijkdenotedualvariablesassociatedwithConstraints( 2b ),( 2c ),( 2d ),and( 2e ),respectively.Thedualofproblem( 2 )isgivenas: max00)]TJ /F5 11.955 Tf 11.96 0 Td[(NQL)]TJ /F14 11.955 Tf 11.96 11.36 Td[(Xi2SXk2FiXj2Iikijk(1)]TJ /F4 11.955 Tf 12.24 0 Td[(yi,ui)]TJ /F6 7.97 Tf 6.59 0 Td[(k+j+1) (2a) s.t.i)]TJ /F7 7.97 Tf 6.59 0 Td[(1j)]TJ /F5 11.955 Tf 11.95 0 Td[(ik)]TJ /F5 11.955 Tf 11.96 0 Td[(ijkpi(k)]TJ /F3 11.955 Tf 11.95 0 Td[(j),8i2S,k2Fi,j2Iik, (2b) 0. (2c) 22

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Notethatbecause( 2d )islinearlydependentupontheotherowbalanceconstraints,wefurtherrestrictNQL=0,andsoijrepresentstheshortest-pathcostfromnodevijtonodevNQLinthenetwork(ifsuchapathexists),giveny(see[ 2 ]).Nextwesimplify( 2 )bymakingasubstitutionforthe-values.Wedenethefollowingnotationtoaidinouranalysis.LetLP(vik,v^^k)denotethelongestpathcostfromnodeviktonodev^^kintheuninterdictednetwork,letSP(vik,v^^k)denotetheshortest-pathcostfromnodeviktonodev^^kintheuninterdictednetwork,andletSPI(vik,v^^k)denotetheshortest-pathcostfromnodeviktonodev^^kintheinterdictednetwork.(Notethatthesevaluesareobtainableinpseudo-polynomialtimebecausetheshortest-pathnetworkisacyclic.)Anarc(vi)]TJ /F7 7.97 Tf 6.59 0 Td[(1j,vik)issaidtobeavailableifitscapacityequals1(i.e.,yi,ui)]TJ /F6 7.97 Tf 6.59 0 Td[(k+j+1=0).Wesaythatnodev^^kisreachablefromnodevikifthereisapathofavailablearcsfromviktov^^kintheinterdictedgraph(where^>iand^kk). Proposition2.1. Thereexistsanoptimalsolutionto( 2 )inwhichallijk=Bijkyi,ui)]TJ /F6 7.97 Tf 6.58 0 Td[(k+j+1,whereBijkisdenedaccordingtothefollowingtwocases: Case1:Ifnodevi)]TJ /F7 7.97 Tf 6.58 0 Td[(1jcanneverbedisconnectedfromnodevNQLbythefollower(i.e.,vNQLisalwaysreachablefromvi)]TJ /F7 7.97 Tf 6.58 0 Td[(1j),thenBijk=LP(vi)]TJ /F7 7.97 Tf 6.59 0 Td[(1j,vNQL))]TJ /F4 11.955 Tf 11.06 0 Td[([pi(k)]TJ /F3 11.955 Tf 11.06 0 Td[(j)+SP(vik,vNQL)].Case2:Otherwise,Bijk=LP(v00,vNQL))]TJ /F4 11.955 Tf 10.88 0 Td[([SP(v00,vi)]TJ /F7 7.97 Tf 6.58 0 Td[(1j)+pi(k)]TJ /F3 11.955 Tf 10.87 0 Td[(j)+SP(vik,vNQL)]. Proof. SeeAppendixA. Makingthesesubstitutions(notingthatyisbinary)andrestrictingNQL=0,wereformulate( 2 )as: DL:max00 (2a) s.t.i)]TJ /F7 7.97 Tf 6.59 0 Td[(1j)]TJ /F5 11.955 Tf 11.95 0 Td[(ikpi(k)]TJ /F3 11.955 Tf 11.95 0 Td[(j)+Bijkyi,ui)]TJ /F6 7.97 Tf 6.59 0 Td[(k+j+1,8i2S,k2Fi,j2Iik. (2b) WerefertothefeasibleregionofDL,inwhichbothandyarevariables,asF(DL). 23

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2.2.3Two-StageFormulationSincethesecondstageofPisamaximizationproblem,wecombinethesecond-stageproblemwithDLintooneMIP.First,usingthenewdouble-subscriptedy-variables,werewritethefollower'sfeasibleregionas:Y=8<:yi,j2f0,1gjXi2Sminfui,QFgXj=1yi,jQF;minfui,QFgXj=1yi,jui(1)]TJ /F3 11.955 Tf 11.96 0 Td[(zi);yi,j+1yi,j,8i2S,j=0,...,minfui,QFg)]TJ /F4 11.955 Tf 20.59 0 Td[(1).ThecombinedMIPisthengivenas: max00 (2a) s.t.(y,)2F(DL), (2b) Xi2Sminfui,QFgXj=1yi,jQF, (2c) minfui,QFgXj=1yi,jui(1)]TJ /F3 11.955 Tf 11.95 0 Td[(zi),8i2S, (2d) yi,j+1yi,j,8i2S,j=1,...,minfui,QFg)]TJ /F4 11.955 Tf 20.59 0 Td[(1, (2e) yi,j2f0,1g,8i2S,j=1,...,minfui,QFg, (2f) whereNQLisxedto0byeliminatingtheredundantConstraint( 2d ).Inordertoderiveacutting-planealgorithmforsolvingtheentirethree-stageproblem,wewishtoobtainafeasibleregionto( 2 )thatisindependentofz.Thisisaccomplishedbyremovingthe(1)]TJ /F3 11.955 Tf 12.63 0 Td[(zi)terminConstraint( 2d )andpenalizingtheobjectivefunctionifthefollowerpurchasesanyunitsfromaprotectedsupplier.Dene^Yasthefeasibleregiondescribedby( 2c ),( 2e ),( 2f ),andPminfui,QFgj=1yi,jui,8i2S.Anequivalentformulationof( 2 )whosefeasibleregionisindependentofzisgivenby: max00)]TJ /F3 11.955 Tf 11.96 0 Td[(MXi2Sziyi,1 (2a) s.t.(y,)2F(DL), (2b) 24

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y2^Y, (2c) whereMisasufcientlylargepenaltyvalue.(Fornow,wetreatMasanarbitrarilylargenumber,butinSection 2.3 wereturntotheissueofndingthesmallestpermissiblepenaltyvaluethatisvalidinthisformulation.)Usingthisreformulation,werewritePasfollows: minz2f0,1gNrTz+max00)]TJ /F3 11.955 Tf 11.95 0 Td[(MXi2Sziyi,1 (2a) s.t.(y,)2F(DL), (2b) y2^Y. (2c) Weshowinthenextsubsectionthatremovingzfromthelowerlevelproblem'sfeasibleregionenablesustosolvetheproblemusingacutting-planealgorithm. 2.2.4SolutionMethodWenowpresentacutting-planealgorithmtosolve( 2 ).Lettingdenotethesetofallsolutions,(y0,0),thatsatisfytheconstraintsof( 2 ),werewrite( 2 )as: minrTz+w (2a) s.t.w000)]TJ /F3 11.955 Tf 11.96 0 Td[(MXi2Sziy0i,1,8(y0,0)2, (2b) z2f0,1gN. (2c) Werefertothisformulationasthemasterproblem(MP).Notethat( 2b )doesnotneedtobeincludedforeverysolutionin.Insteadweonlyneedtoincludethevectorsinthatareoptimalsolutionstothesecondstageof( 2 )foragivenz.Wedenotethissetofsolutionsby .WhenonlyasubsetofInequalities( 2b )ispresent(i.e.,forsomeset^ ),thisrelaxation,alongwithasimplenonnegativityboundw0,isreferredtoastherelaxedmasterproblem(RMP).Werefertoproblem( 2 )asthesubproblem(SP).ObservethatInequalities( 2b )arevalidbecauseeverypossiblesolution(y0,0)2remainsfeasibletotheSP,regardlessofz. 25

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Algorithm1Cutting-planealgorithm 1: Set^=; 2: SetLB=0,UB=1,andz?i=0,8i2S 3: whileLB0). Remark3.DuringStep 5 ,insteadofsolving( 2 )directly,wesolve( 2 )becauseitexplicitlyrestrictsziyi,1=0,8i2Sintheconstraintsratherthanviaobjectivepenalty.Theresultingoptimalsolutionto( 2 )isthereforeoptimalto( 2 ),andacuttingplaneoftheform( 2b )isderivedfromitssolution.2 Remark4.Wenextpresentaformulationtobeusedwhenasupplierrequiresaminimumprocurementquantityifitistobeprotected.Thisscenarioarises,forinstance, 26

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whenasupplierisprotectedbysigninganexclusivityrightsagreementwiththeleaderrm.Inthissituation,thesuppliermayclearlywishtoinsistthattheleaderwillindeedpurchaseaminimumquantityinordertoensurethatitsownsalesareguaranteed.Letlidenotetheminimumnumberofunitsthatmustbepurchasedfromsupplieriifitisprotected.Thethird-stageshortest-pathnetworkisalteredbyrevisingarccapacitiesto(1)]TJ /F4 11.955 Tf 12.26 0 Td[(yi,ui)]TJ /F6 7.97 Tf 6.59 0 Td[(k+j+1)(1)]TJ /F3 11.955 Tf 11.98 0 Td[(zi)foralli2S,k2Fi,andj2Iiksuchthatk)]TJ /F3 11.955 Tf 11.98 0 Td[(j
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LetV(y)betheleader'soptimalshortest-pathcostforagiveny,andlety(z)bethefollower'soptimalsolutiongivenleader'schoiceofprotection,z.Notethatgiveny(z),0canbecomputedbysolving( 2 ).Wenowcomparetheleader'sshortest-pathcostwhenemployingsomepotentialprotectionstrategy,^z,totheshortest-pathcostwhenapreviousprotectionstrategy,z0,wasused.Givenz0andcorrespondingsolution(y0,0)toSP,letBimax=maxk2Fi,j2IikfBijky0i,ui)]TJ /F6 7.97 Tf 6.58 0 Td[(k+j+1g,8i2S.LetTSbesomesubsetofsuppliers,let^zi=18i2T,^zi=08i=2T,andletT=fi2Tjy0i,1=1g(i.e.,Tisthesetofsuppliersfromwhichthefollowerpreviouslypurchasedunits,butwhicharenowprotectedinstrategy^z).WeshowthatthesumofthevaluesforBijky0i,ui)]TJ /F6 7.97 Tf 6.58 0 Td[(k+j+1onanyoptimalpathcorrespondingtoanewprotectionstrategy,^z,cannotexceedthesumoftheBimaxvaluesforalli2T.Givenapath,P=fv00,v1k1,v2k2,...,vN)]TJ /F7 7.97 Tf 6.59 0 Td[(1kN)]TJ /F16 5.978 Tf 5.76 0 Td[(1,vNQLg,deneA(P)=f(i,j,k)jvi)]TJ /F7 7.97 Tf 6.59 0 Td[(1j,vik2PgasthesetofarcsinpathP. Lemma1. Consideravectorz0,andafeasiblesolutiontothecorrespondingsubprob-lem,(y0,0).LetTSbesomesetofsuppliers,andlet^zandTbeasdenedabove.Further,lety00ij=y0ij8i=2T,j=1,...,ui,andlety00ij=08i2T,j=1,...,ui.Giveny00,letPbetheleader'soptimalpathfromsourcetosink.ThenX(i,j,k)2A(P)Bijky0i,ui)]TJ /F6 7.97 Tf 6.58 0 Td[(k+j+1Xi2TBimax. Proof. First,notethat^zissomeprotectionstrategyandy00issomepotentialstrategyforthefollower,given^z.WepartitionA(P)intosetsA+=f(i,j,k)2A(P)jy0i,1>0gandA0=f(i,j,k)2A(P)jy0i,1=0g,whereA+(A0)isthesetofarcsinPcorrespondingtosuppliersfromwhichthefollowerpurchased(didnotpurchase)unitsinthesolution(y0,0).Further,wecanpartitionA+intothesets^A+=f(i,j,k)2A+jy00i,1>0gand^A0=f(i,j,k)2A+jy00i,1=0g,where^A+isthesetofarcsinPcorrespondingtosuppliers 28

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fromwhichthefollowerpurchasedunitsinthesolution(y0,0)andstilldoeswithstrategy^z(i.e.,^zi=0),and^A0isthesetofarcsinPcorrespondingtosuppliersfromwhichthefollowerpurchasedunitsinthesolution(y0,0)butcannotwithprotectionstrategy^z(i.e.,^zi=1).Weknowthat: X(i,j,k)2A(P)Bijky0i,ui)]TJ /F6 7.97 Tf 6.58 0 Td[(k+j+1=X(i,j,k)2A0Bijky0i,ui)]TJ /F6 7.97 Tf 6.58 0 Td[(k+j+1+X(i,j,k)2^A0Bijky0i,ui)]TJ /F6 7.97 Tf 6.59 0 Td[(k+j+1+X(i,j,k)2^A+Bijky0i,ui)]TJ /F6 7.97 Tf 6.59 0 Td[(k+j+1=X(i,j,k)2^A0Bijky0i,ui)]TJ /F6 7.97 Tf 6.58 0 Td[(k+j+1Xi2TBimax.Thesecondequalityholdsbecausey0i,ui)]TJ /F6 7.97 Tf 6.58 0 Td[(k+j+1=0forall(i,j,k)2A0[^A+.(ArcsfromthesetA0correspondtosuppliersfromwhichthefollowerdidnotpurchaseanyunitsandarcsfromtheset^A+A+musthavebeenavailableintheprevioussolution.)Theinequalityholdsbecausefi2Sj(i,j,k)2^A0g=TTandbecauseforeachi,onlyoneBijky0i,ui)]TJ /F6 7.97 Tf 6.59 0 Td[(k+j+1(whichisnomorethanBimax)canappearinthesummation. Giventhisresult,wenowshowthattheimprovementintheleader'sshortest-pathcostwithprotectionstrategy^zwhencomparedtostrategyz0isnomorethanPi2TBimax. Proposition2.2. Givenavectorz0andsolution(y0,0)tothecorrespondingsubprob-lem,letTSbesomesetofsuppliers,andlet^zandTbeasdenedabove.Thenthefollowinginequalityholds: V(y(^z))V(y0))]TJ /F14 11.955 Tf 11.96 11.35 Td[(Xi2TBimax.(2) Proof. SinceV(y0)=000andbecause(y0,0)isfeasibletothesubproblemwhenz=z0,thefollowinginequalitieshold: Bijky0i,ui)]TJ /F6 7.97 Tf 6.59 0 Td[(k+j+1i)]TJ /F7 7.97 Tf 6.59 0 Td[(10j)]TJ /F5 11.955 Tf 11.96 0 Td[(i0k)]TJ /F3 11.955 Tf 11.96 0 Td[(pi(k)]TJ /F3 11.955 Tf 11.96 0 Td[(j),8i2S,8k2Fi,j2Iik. 29

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Deney00asinLemma 1 ,wherey00ij=y0ij8i=2T,j=1,...,ui,andy00ij=08i2T,j=1,...,uiandletP=fv00,v1k1,v2k2,...,vN)]TJ /F7 7.97 Tf 6.59 0 Td[(1kN)]TJ /F16 5.978 Tf 5.76 0 Td[(1,vNQLgbetheleader'soptimalpathfromsourcetosink.SummingtheBijky0i,ui)]TJ /F6 7.97 Tf 6.58 0 Td[(k+j+1valuescorrespondingtopathP,wehavethefollowing: X(i,j,k)2A(P)Bijky0i,ui)]TJ /F6 7.97 Tf 6.59 0 Td[(k+j+1000)]TJ /F5 11.955 Tf 11.95 0 Td[(10k1)]TJ /F3 11.955 Tf 11.96 0 Td[(p1(k1)+10k1)]TJ /F5 11.955 Tf 11.95 0 Td[(20k2)]TJ /F3 11.955 Tf 11.96 0 Td[(p2(k2)]TJ /F3 11.955 Tf 11.95 0 Td[(k1)++N)]TJ /F7 7.97 Tf 6.58 0 Td[(10kN)]TJ /F16 5.978 Tf 5.76 0 Td[(1)]TJ /F5 11.955 Tf 11.95 0 Td[(N0QL)]TJ /F3 11.955 Tf 11.95 0 Td[(pN(QL)]TJ /F3 11.955 Tf 11.96 0 Td[(kN)]TJ /F7 7.97 Tf 6.59 0 Td[(1)=000)]TJ /F5 11.955 Tf 11.95 0 Td[(N0QL)]TJ /F14 11.955 Tf 11.95 13.27 Td[(p1(k1)+p2(k2)]TJ /F3 11.955 Tf 11.96 0 Td[(k1)++pN(QL)]TJ /F3 11.955 Tf 11.95 0 Td[(kN)]TJ /F7 7.97 Tf 6.58 0 Td[(1)=V(y0))]TJ /F3 11.955 Tf 11.96 0 Td[(V(y00),becauseN0QL=0.Therefore,combiningthisresultwiththeresultfromLemma 1 wehave: V(y00)V(y0))]TJ /F14 11.955 Tf 11.96 11.36 Td[(Xi2TBimax.(2)Now,notingthaty(^z)isanoptimalactionforthefollowergiven^z,weknowV(y(^z))V(y00),whichalongwith( 2 )implies( 2 ).Thiscompletestheproof. Giventheseresults,wereplace( 2b )with: w000)]TJ /F14 11.955 Tf 11.95 11.35 Td[(Xi2SBimaxziy0i,1,8(y0,0)2.(2) 2.3.2InequalitiesfromReduced-SizeSubproblemsRecallthatthemagnitudeofthedataimpactsthesizeofSP,whichhasapseudo-polynomialnumberofdualvariablessincetheshortest-pathgraphcorrespondingtoIMPhasO(NQL)nodes.Wenextpresentamethodforobtainingvalidinequalitiesoftheform( 2 )bysolvingsubproblemsthataresubstantiallysmallerthanSP.Weapplytwodifferenttechniquesinordertoformulatethesmallersubproblemsfromwhichthesevalidinequalitiesareobtained.Therstisanadjustmenttechnique,whichmodiesdataelements(QL,QT,ui,8i)ofPtoobtainrevisedproblemswhose 30

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subproblemscanbesolvedtoobtainvalidinequalitiesforP.Intheadjustedproblemthatwedene,morecapacityiseffectivelyavailabletotheleader,assuppliercapacitiesareadjustedtovaluesatleastaslargeasthoseintheoriginalproblem,andthequantitythattheleadermustprocureisnogreaterthanthequantityintheoriginalproblem.Moreover,themaximumquantitythatthefollowermayprocureisnogreaterthanthatintheoriginalproblem.Asaresult,thesolutionoftheadjustedproblemprovidesalowerboundonthesolutionofbothIMPandoftheoriginalproblemP.Theadjustedproblemsarecreatedinsuchawaytoensureproblemdatathatcontainscommonfactors.Givenanadjustedproblem,wethenapplyacoarsediscretization(CD)techniquethatreducesthesizeofthesubproblem.Inthediscretizedversion,weonlyallowthefollowertopurchaseunitsinmultiplesofsomecommonfactorofthedata.AdiscretizedproblemisgeneratedbydividingQL,QF,andui,8i2S,bythiscommonfactor,andadjustingtheprocurementcostsaccordingly.Aswewillshow,wecanthensolvethesmallerdiscretizedsubproblemstoobtainvalidinequalitiesforP.Althoughtheinequalitiesgeneratedfromthesediscretizedproblemswillnotingeneralbesufcienttoforcethecutting-planealgorithmtoconverge,thegoalistoreducethenumberoftimesthefullSPmustbesolvedtondanoptimalsolutiontoP.WerstintroducesomeadditionalnotationtodescribetheadjustmentandCDtechniques.Let(N,QL,QF,u1,...,uN)describeaninstanceofP,andletAbetheintegeradjustmentfactorthatwillbeappliedtoP.GivenA,wesetQ0L=QL AA,Q0F=QF AA,andu0i=ui AA,8i2S.LetPA,SPA,RMPA,andIMPArepresenttheadjustedversionsforP,SP,RMP,andIMP.WhenA=1,thedataisnotadjustedandthesuperscriptwillbeomitted.FortheCDtechnique,letGbeacommonfactorofQ0L,Q0F,u01,...,u0NandletCD-PA(G),CD-SPA(G),CD-RMPA(G),andCD-IMPA(G)representthediscretizedversionsforPA,SPA,RMPA,andIMPA,inwhichweaddtherestrictionthatthefollowercanonlypurchaseunitsinmultiplesofG.Burkeetal.[ 18 ]showthatthereexistsanoptimalsolutiontotheleader'sprocurementprobleminwhichtheleaderpurchasesall 31

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ornoneofasupplier'scapacityforallsuppliersexceptforatmostone.SinceQ0Landu0i,8i,aredivisiblebyG,thismeansthatoncethefollowerpurchasesinmultiplesofG,thereexistsanoptimalsolutiontotheleader'sprocurementprobleminwhichtheleader'sprocurementamountfromeachsupplierisalsodivisiblebyG.Therefore,wesolveareduced(butequivalent)versionofthesubprobleminwhicheachdataelementisdividedbyG,therebydecreasingtheproblemsize. Example:Supposethat(N,QL,QF,u1,u2,u3)=(3,24,26,13,21,4)andA=10.ThentheadjusteddataforP10is(N,Q0L,Q0F,u01,u02,u03)=(3,20,20,20,30,10).WeknowthatG=5isacommonfactorofthedata(excludingN)forproblemP10.Therefore,thediscretizedsubproblemdataforCD-P10(5)is(N,Q00L,Q00F,u001,u002,u003)=(3,4,4,4,6,2).Notethatinthediscretizedproblem,p00i(k)=pi(5k),8i,k. Remark5.Asillustratedintheforegoingexample,wemayhaveu0i>uiforsomei.Inthiscase,forallunitspurchasedinexcessofthecapacityui,weapplyaprocurementcostequaltothecheapestunitcost(correspondingtotheuithunit).Thismaintainstheconcavecostcurveproperty.Forexample,supposeui=10andtherst5unitseachcost10,whilethenext5unitseachcost7.Ifu0i=15,thenunits1115wouldeachcost7.2WenowpresentfourlemmasthatdemonstratehowtheadjustmentandCDtechniquesresultinproblemsthatprovidevalidinequalitiesforP. Lemma2. LetAbesomepositiveinteger,let(N,QL,QF,u1,...,uN)beaninstanceofP,andletPArepresentproblemPwithadjustmentfactorA.Also,let(z)betheoptimalobjectivefunctionvaluetoSPgivenz,usingdataforP,anddeneA(z)analogouslyforPA.ThenA(^z)(^z)foranyfeasiblevector^ztoP. Proof. Givenabinaryvector^z,consideranyfeasiblesolutionyAtoSPA,whichreducestheprocurementcapacitiesinIMPAtou0i)]TJ /F4 11.955 Tf 13.22 0 Td[(yAi.Notingthatuiu0iandQ0FQF,therealsoexistsasolutiony0suchthatui)]TJ /F4 11.955 Tf 13.19 0 Td[(y0iu0i)]TJ /F4 11.955 Tf 13.19 0 Td[(yAi,8i2S(e.g.,bysetting 32

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y0i=minfui,yAig).ThenbecauseQ0LQL,thischoiceofy0leadstoanIMPinstancethatisarestrictionofIMPAforthegivenyA.ThisimpliesthattheoptimalobjectivefunctionvaluetoIMPisatleastaslargeasthatforIMPA.BecausethisistrueforanyyAthatisfeasibletoSPAforthegiven^z,wehavethatA(^z)(^z). Theorem2.1. ConsideraninstanceP,apositivefactorA,andanadjustedproblemPA.Anyinequalityoftheform( 2 )generatedforproblemPAisvalidforP. Proof. Theresultfollowsbecause( 2 )isvalidforproblemRMPAgivenanypotentialvector^z.DuetoLemma 2 ,( 2 )mustalsobevalidforRMPgivenany^zaswell. Nextwewillshowthatanycutsobtainedfromadiscretizedversionofaproblemarevalidforthenon-discretizedandnon-adjustedversions. Theorem2.2. Aninequalityoftheform( 2 )obtainedforCD-RMPA(G)isvalidforRMPA,andisthusvalidforRMPaswell. Proof. AvalidinequalityforCD-RMPA(G)providesalowerboundontheoptimalobjectivefunctionvalueforCD-SPA(G).SinceCD-SPA(G)isarestrictionofSPA,thislowerboundisalsovalidfortheoptimalobjectivefunctionvalueofSPA.Therefore,theinequalityisvalidforRMPA,andTheorem 2.1 thusguaranteesthattheinequalityisvalidforRMP. WenowpresentAlgorithm 2 basedontheadjustmentandCDtechniques.Ineachiteration,az-vectorischosenbysolvingRMPwiththevalidinequalitiesthathavebeenaddedsofar,andthenasubproblemissolvedtoseeifanewvalidinequalitymustbeadded.Thealgorithmstartsoutbysolvingsmallsubproblems(largevaluesofAandG),beforeeventuallysolvingatleastoneinstanceofCD-SPA(G)whenA=G=1.WeinitializethesetofRMPvalidinequalitiestobetheemptysetinStep 1 ,andchoosethesetofvaluesforAinStep 2 .ForeachvalueofA,wethenselectasetofdiscretizationfactorsinStep 4 .GivenAandG,inStep 6 ,wethensolveRMP(containingallpreviouslygeneratedvalidinequalities),usingCD-SPAi(Gj)asits 33

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Algorithm2Adjustmentanddiscretizationalgorithm 1: Set=; 2: Determineasetofadjustmentfactors:Aset=fA1,...AjAsetjgwhereA1>>AjAsetj=1 3: fori=1,...,jAsetjdo 4: GivenAi,ndasetofcommonfactors:Gset=fG1,...,GjGsetjgwhereG1>>GjGsetj=1 5: forj=1,...,jGsetjdo 6: Applycutting-planealgorithmtoRMPusingCD-PAi(Gj)astheSP 7: endfor 8: endfor subproblem,untilnomorevalidinequalitiescanbeaddedtoRMP.ThealgorithmterminatesaftercompletingStep 6 withAi=Gj=1,resultinginanoptimalsolution. 2.4IllustrationandComputationalResultsWerstuseathree-supplierexampletoillustrateAlgorithm 1 .Foreaseofexposition,wepresentthesolutiontoSPusingthesingle-subscriptedy-variables,omitthevaluesofthecontinuousvariables,anduseagenericbig-Mvalue. Example:N=3,QL=50,QF=11,u1=25,u2=25,u3=25,r1=1,r2=1,r3=1,p1(q1)=log2(q1+1),p2(q2)=p q2,p3(q3)=1.53p q3,andM=100. Table2-1. Exampleproblem RMPSolution SPSolution Iteration Value z1 z2 z3 w y1 y2 y3 Cut UB 1 0 0 0 0 0 5 0 6 w11.71)]TJ /F3 11.955 Tf 11.95 0 Td[(M(z1+z3) 11.71 2 1 1 0 0 0 0 2 9 w11.39)]TJ /F3 11.955 Tf 11.95 0 Td[(M(z2+z3) 11.71 3 1 0 0 1 0 7 4 0 w11.28)]TJ /F3 11.955 Tf 11.95 0 Td[(M(z1+z2) 11.71 4 2 0 1 1 0 11 0 0 w9.39)]TJ /F3 11.955 Tf 11.95 0 Td[(Mz1 11.39 5 2 1 1 0 0 0 0 11 w9.70)]TJ /F3 11.955 Tf 11.95 0 Td[(Mz3 11.39 6 2 1 0 1 0 0 0 0 w9.09 11.09 7 11.09 1 0 1 9.09 Intherstiteration,sincenocutshaveyetbeengenerated,anoptimalsolutiontoRMPisobtainedbysettingallrst-stagevariablesto0.Thefollower'soptimalchoiceistopurchase5unitsfromsupplier1and6unitsfromsupplier3.Thisresultsinatotalprocurementcostof11.71,which 34

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becomestheupperbound,sincethisistheincumbentsolution.Thecutgeneratedforceswtobenolessthan11.71unlesssupplier1orsupplier3isprotected.Notethatz2doesnotshowupinthecut,becausecurrently,therewouldbenoadvantagegainedbyprotectingonlysupplier2.Notethatthecutobtainedbyprotectingonlysupplier2isexactlythesameastheoneobtainedhere.Hence,ouralgorithmwillneverexaminethissolution.Initerations2and3,procurementcostsof11.39and11.28areobtained,butwhencombinedwiththecostofprotectingthesupplier,thevalueisgreaterthanthecurrentupperbound.Initeration4,anewupperboundisobtainedbyprotectingsuppliers2and3andpayingaprocurementcostof9.39.Initeration5,abettersolutionisnotfound,sotheupperboundremainsthesame.Aninterestingsituationarisesiniteration6.Supplier2isnotprotectedbytheleader,butthefollowerdoesnotpurchaseanyunitsfromthissupplier.Whatleadstothissituationistheexistenceofalternativeoptimalsolutions:Thefollowercouldhavepurchasedanywherebetween0and11unitsfromsupplier2withoutchangingtheobjectivefunctionvalue.Inthisalgorithm,whentherearealternativeoptimalsolutions,itisdesirabletoobtainasolutioninwhichthefollowerpurchasesasfewunitsaspossiblebecausethisleadstostrongercuts.Inthisexample,becauseofthecutobtainediniteration6,thereisnoneedtoexaminethechoiceofprotectingallsuppliers,sincethereisnoneedtoprotectsupplier2whentheothersuppliersarealreadyprotected.Initeration7,anoptimalsolutionisobtainedbyprotectingsuppliers1and3,withanoverallcostof11.09.SeeTable 2-1 forasummaryofthestepstakeninthisexample.2Totestouralgorithmmorerigorously,weperformedanumberofcomputationaltestswithN=5,...,25,focusingmorecloselyoncasesinwhichonN10.Basedonananalysisofrelatedliteratureandpubliclyavailableinformationonrms 35

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thatassemblecomponentsintoendproducts,itappearsthatrelativelyfewcasesariseinwhichrmshavegreaterthan25suppliersforagivencomponent,andinmanycases,10suppliersaresufcient.Forexample,Apple( http://images.apple.com/supplierresponsibility/pdf/Apple_Supplier_List_2011.pdf ),Dell( http://content.dell.com/us/en/gen/d/corp-comm/cr-ca-list-suppliers ),andHewlett-Packard( http://www.hp.com/hpinfo/globalcitizenship/environment/supplychain/supplier_list.pdf )eachuseatmosttensuppliersforcentralprocessingunits.AgrawalandNahmias[ 1 ]developamodeltodeterminetheoptimalnumberofsuppliersandsuggestthatinpractice,thesetofsupplierswillbemodest(usingN=10intheirexample).InBergeretal.[ 11 ],theauthorsmodelthedecision-makingprocesswhendeterminingtheoptimalnumberofsupplierstouseinthepresenceofpotentialsupplychaininterruptions.Intheiranalysis,theoptimalnumberofsuppliersisnevergreaterthansix.InKauffmanandPopokowskiLeszczyc[ 34 ],theauthorsciteasurveyinwhich58purchasersweresurveyedand74%reportedthattheaveragenumbersuppliersfromwhichtheypurchaseanygivenitemisnomorethanthree.Theyestimatetheirmodelusingempiricaldataforprocurementauctionsofsteelpipeand,intheircomputationalexperiments,themaximumoptimalnumberofsuppliersacrossalltestinstancesis12.Ruiz-TorresandMahmoodi[ 45 ]optimizethenumberofsupplierswhenasubsetofsuppliersmayfailtodeliver.Intheirexperiments,theyneverndanoptimalsolutionwithgreaterthan13suppliers.Whiletheoptimalnumberofsupplierswillvaryfromindustrytoindustryandcompanytocompany,consideringamaximumvalueofN25andfocusingmoreintenselyoninstancesinwhichN10appearstothuscoveraverybroadcrosssectionofindustrialapplications.ForN=5,6,7weexaminedvedifferentsetsofsuppliercapacitiesandfourdifferentsetsofleaderandfollowerdemands,foratotalof354=60problemproles.Table 2-2 showsthevesetsofsuppliercapacitiesthatwereused.Table 2-3 showstheleaderandfollowerdemandsasapercentageoftotalsuppliercapacity 36

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(fractionalcapacitieswereroundeddown).Thesumofthepercentagesislessthan100%tomaketheleader'sprocurementdecisionnon-trivial.WerefertoarowofTable 2-3 asademandlevel. Table2-2. Suppliercapacitysets CapacitySet# u1 u2 u3 u4 u5 u6 u7 u8 u9 u10 1 25 25 25 25 25 25 25 25 25 25 2 10 25 15 20 40 45 35 20 50 20 3 25 24 22 23 26 26 27 21 49 25 4 12 17 20 21 25 23 25 24 16 19 5 30 47 50 45 36 46 27 32 43 36 Table2-3. Buyerdemandasapercentageoftotalsuppliercapacity Leader Follower 20% 20% 40% 40% 20% 70% 70% 20% Fourdifferentadjustmentanddiscretizationstrategieswereused.Forstrategy0weletAset=f1g,i.e.,noadjustmentsweremade.Strategy1hadAset=f20,10,5,2,1gandstrategy2hadAset=f23,11,5,2,1g.Forstrategy3,Asetcontainedthesetofuptovefactorsthatthemostdataelementshadincommon.Forexample,ifQL=QF=30,u1=36,andu2=40,thenAset=f10,6,5,2,1g(becausef2,1garecommonfactorsandf10,6,5garecommonfactorsforthreeofthefourdataelements).Forstrategy0,Gset=f1gandforstrategies1,2,and3,weletGsetcontainallcommonfactorstotheadjusteddata.Foreachprole,wegeneratedtensetsofsuppliercostfunctions.Eachcostfunctionisapiecewiselinearconcavefunctionwithvesegments,whereeachsegmenthasaslopeintheinterval[1,10].Foreachsupplieri,weletri=3(pi(ui))]TJ /F3 11.955 Tf 12.37 0 Td[(pi(ui)]TJ /F4 11.955 Tf 12.37 0 Td[(1))(i.e.,threetimesthesupplier'scheapestunit).Usingsupplierprotectioncostsofthismagnitudeseemedtoresultinsolutionsinwhichsome,butnotall,ofthesuppliers 37

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wereprotectedbytheleader.Wethenusedthefourstrategiestosolveeachofthe10instancesforeachprole.TheresultsarepresentedinTable B-1 ofAppendixB. Figure2-3. Computationtimesfordifferentdemandlevelsandstrategies InTable B-1 ,asuperscriptonanentryindicatesthenumberofproblems(outoften)thatweresolvedwithintheone-hourtimelimit,whiletheentryitselfprovidestheaveragesolutiontimeforthoseproblemssolvedwithinanhour(a-indicatesthatnoinstancesweresolvedwithinanhourandthelackofasuperscriptindicatesall10instancesweresolved).Foreachprole,onlyoneinstancewasrunusingstrategy0becausepreliminarycomputationtestsclearlydemonstratedthatitwastheleasteffectivestrategy. 38

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Figure2-4. Computationtimesfordifferentcapacitysets TheresultsinTable B-1 aresummarizedbyfourgraphsinFigure 2-3 .Foreachofthefourdemandlevels,wepresenttheaveragecomputationtime(atimeof3600wasusedforinstancesthatdidnotsolve)usingeachofstrategies1,2,and3withN=5,6,7.Thepairofnumbersaboveeachgraphgivestheleader'sandfollower'sdemandsasapercentageoftotalsuppliercapacity,respectively.Asseeninthegraphs,strategy1performsbestinthreeoutofthefourdemandlevelsandstrategy2performsworstinthreeoutofthefourdemandlevels.Also,inthreeoutofthefourdemandlevels,strategy3performsalmostaswellas,orbetterthanstrategy1.Thiswouldsuggestthatstrategies1and3shouldbeusedinfuturecomputationalexperiments(atleastwhenN=5,6,7).Whencomparingresultsforthedifferentbuyerdemandlevels,itbecomesclearthatincreasingQLrelativetothetotalsuppliercapacitymakestheproblemhardertosolve,whileanincreaseinQFrelativetothetotalsuppliercapacityhaslessofaneffectonsolutiontime.Figure 2-4 showstheaveragecomputationtimeoverstrategies1forN=5,6,7foreachofthecapacitysets.Clearly,prolesusingcapacityset1aretheeasiesttosolve.Thisismostlikelyduetothefactthatlessadjustmentanddiscretization 39

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isneededsincethecapacitiesareallthesame.Prolesusingcapacityset5arehardesttosolve,mostlikelybecausethecapacitiesarelargeandhavefewcommonfactors.Anothernoteworthyobservationisthatthenumberofcutsrequiredtosolvetheproblemsisrelativelyconstantacrossthedifferentproles.Therefore,thedifferencesinsolutiontimesareafunctionofsubproblemsolutiontime. Figure2-5. Computationtimesfordifferentprotectioncosts Figure2-6. Computationtimesfordifferentstrategies 40

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ForN=8,9,10weagainusedthevedifferentsetsofsuppliercapacities,butthistimexedtheleaderandfollowerdemandstoeachbe20%ofthetotalcapacity.Wevariedtheprotectioncosts,ri=xi(pi(ui))]TJ /F3 11.955 Tf 11.25 0 Td[(pi(ui)]TJ /F4 11.955 Tf 11.26 0 Td[(1))8i,byvaryingthevalueofxi,wherexiisascalarusedtocalculatetheprotectioncostforsupplieri2S.TheresultsareshowninTable B-2 ofAppendixB.Figure 2-5 showsasummaryofcomputationtimeasafunctionofN,forthecasesofxi=1,8i;xi=38i;xi=58i2S;andwhenxiisrandomlyselectedfromthesetf1,...,5g.Interestingly,thecomputationtimesdecreaseasxiincreases.Thisisadirectresultoftheincreaseinnumberofiterationsrequiredtosolvetheproblem.Whentheprotectioncostsaregreater,fewerprotectionstrategiesarebenecial,whichresultsinfewersubproblemsneedingtobesolved.Ifthexi-valuesarelargeenough,theproblembecomestrivialbecausenosupplierswouldbeprotectedandonlyonesubproblemwouldneedtobesolved.Ontheotherhand,ifxi=0,8i,theproblemisalsotrivialsincetheleaderwouldchoosetoprotectallsuppliers,andonlyasinglethird-stageproblem(anLP)wouldneedtobesolved.Figure 2-6 showstheaveragecomputationtimeforeachofthestrategiesforN=5,...,10whenxi=38iandtheseconddemandlevel,(20,20),isused.Thegraphshowsthatstrategy1ismostefcientforsolvingproblemswhenQLissmallrelativetotheoverallsuppliercapacity.ItisalsoimportanttonotethattheincreaseincomputationtimeasNincreasesissubstantial,asisexpectedforanyNP-hardproblem.Tofurthertesttheefciencyofouralgorithm,wegeneratedprolesinwhichN2f11,...,25g.ForeachvalueofN,wecreated6prolesbyrandomlygeneratingeachsupplier'scapacityfromthesetsf5,...,15g,f10,...,20g,f15,...,25g,f20,...,30g,f25,...,35g,andf30,...,40g.Withtheseproles,themaximumdifferenceincapacitybetweenanytwosuppliersis10.Wealsogenerated5moreprolesbyrandomlygeneratingcapacitiesfromthesetsf5,...,20g,f5,...,25g,f5,...,30g,f5,...,35g,andf5,...,40g.Withtheseproles,thesuppliers'capacitiesaremorevariablebuteach 41

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hasaminimumof5units.Foreachofthe11proles,wexedtheleaderandfollowerdemandstoeachbe20%ofthetotalcapacityandtheprotectioncosttobethreetimesthesuppliers'mostexpensiveunit(i.e.,ri=3(pi(ui))]TJ /F3 11.955 Tf 12.37 0 Td[(pi(ui)]TJ /F4 11.955 Tf 12.36 0 Td[(1)),8i2S).Strategy1wasusedasthesolutionmethodandforeachprolewerandomlygeneratedthreesetsofsuppliercostfunctions.Asbefore,eachwasapiecewiselinearconcavefunctionwithvesegments,eachhavingaslopeintheinterval[1,10]. Table2-4. ResultsforN2f11,...,25g ui N f5,...,15g f10,...,20g f15,...,25g f20,...,30g f25,...,35g f30,...,40g 11 3.3 7.3 13.7 48.3 58.3 203.3 12 3.0 16.3 50.3 71.3 80.7 151.0 13 10.7 11.3 53.0 162.7 118.3 592.7 14 6.7 20.3 44.3 1116.3 487.7 298.3 15 8.3 21.3 197.3 933.3 720.0 508.3 16 52.3 112.7 204.0 219.7 729.3 799.7 17 77.3 83.7 382.3 788.7 2097.01 1715.3 18 64.0 163.3 588.3 639.3 676.01 2225.52 19 743.3 230.7 875.7 629.0 871.02 1680.02 20 319.7 1015.7 844.01 577.02 2275.52 2255.01 21 629.3 1744.7 323.52 2313.52 22 1385.7 440.01 1626.01 2600.01 23 1062.3 1577.52 796.01 2613.01 24 802.02 3479.01 25 2549.3 2213.01 N f5,...,15g f5,...,20g f5,...,25g f5,...,30g f5,...,35g f5,...,40g 11 3.3 4.0 13.7 11.7 6.7 11.3 12 3.0 13.7 9.0 25.3 73.0 96.3 13 10.7 11.3 17.7 11.3 147.0 65.3 14 6.7 17.3 33.3 46.7 115.7 132.3 15 8.3 16.0 72.0 157.7 141.0 340.7 16 52.3 23.7 115.7 192.0 700.3 356.0 17 77.3 34.0 628.3 171.0 266.0 176.3 18 64.0 85.0 355.7 820.0 107.0 450.3 19 743.3 166.3 243.3 472.7 914.01 1055.01 20 319.7 581.5 275.3 990.3 631.01 847.3 21 629.3 857.7 204.3 2717.01 2905.52 266.01 22 1385.7 1146.0 1273.3 587.52 2310.01 1255.01 23 1062.3 365.52 3089.01 469.01 3504.01 24 802.02 2020.01 1558.02 1151.01 434.01 25 2549.3 1763.01 1096.01 1085.0 42

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InTable 2-4 ,weprovidetheaveragesolutiontimesforeachoftheproles.(Asuperscriptonanentryindicatesthenumberofproblems(outofthree)thatweresolvedwithintheone-hourtimelimit.)ThetophalfofTable 2-4 containsthecomputationaltimesfortherst6oftheseprolesandthebottomhalfcontainsthecomputationaltimesfortheother5proles(thesecondcolumnisthesameinboththetopandbottomhalves).Whenui2f5,...,15g8i2S,allinstancesexceptforone(whenN=24)weresolvedwithintheonehourtimelimit.Asseeninbothhalvesofthetable,asthecapacitiesincrease,thecomputationaltimesincreaseandfewerinstancesaresolvedwithintheonehourtimelimit.Whencomparingthetophalftothebottomhalf,itappearsasifthemagnitudeofthecapacitieshasagreaterimpactonthecomputationaltimesthandoesthevarianceofthecapacities.Forexample,whenui2f30,...,40g8i2S,notasingleinstanceinwhichN>20wassolved.Alternatively,whenui2f5,...,40g8i2S,theonlyvalueofNforwhichnoinstancesweresolvedwasN=25.Weseethissametrendwhencomparingthetopandbottomhalvesofthetableforcolumns3. 43

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CHAPTER3PROCUREMENTALLOCATIONPLANNINGWITHMULTIPLESUPPLIERSUNDERCOMPETITION 3.1MotivationandLiteratureSurveyManufacturersandre-sellersofconsumergoodsmustoftendrawonmultiplesourcesforsupplyofinputcomponentsandcommodities.Utilizingmultiplesourcesistypicallynecessaryduetooutputcapacitylimitsatanygivensupplysource.InBurkeetal.[ 18 ],forexample,arm(whichwewilllaterrefertoastheleader)wishestosatisfyitsdemandforsomegoodorproductioninputfromasetofcapacitatedsuppliers.Becauseofeconomiesofscaleinproductionanddistribution,thecosttoprocurefromagivensupplierisaconcavefunctionofthequantitypurchased.Quantitydiscountingisquitecommoninpracticeandappliestovarioussituations,e.g.,whenthesupplierchargesaxedcosttoplaceanorderorwhenthesupplieroffersdiscountsfororderinginbulk.Evenwhensuchdiscountsarenotexplicitlyoffered,theconcavecoststructureoftenarisesduetotransportationpricesthatreecteconomiesofscaleintransportation.Chapter 2 extendstheworkin[ 18 ]byintroducinguncertaintyinsuppliercapacities,which,forexample,maybeduetoproductionshortagesasaresultofuncontrollableorunpredictablefactors,orfromcompetitionfromotherrms.Unpredictabilityinsuppliercapacityduetocompetitionishandledinthefollowingmanner.First,allcompetitorsaremodeledasasinglecompetingrm(thefollower)withaknowndemandfortheitem.Second,theleaderassumesaworst-casescenarioinwhichthefollowermakespurchasingdecisionswithagoalofmaximizingtheleader'sprocurementcostsfromtheremainingsuppliercapacities.Inpractice,however,armrarelypursuesasingulargoalofmaximizingacompetitor'scost.Instead,competitivermstypicallyseektomaximizetheirownprotability,whilesimultaneouslytakingstepsthatmayputacompetitoratadisadvantage.Theworkwepresentinthischapterthereforecomplementstheprevious 44

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worst-caseapproachandfurtherbroadensitsapplicabilitytomoregeneralcompetitivesettingsbyaddressingscenariosinwhich: (a) thefollower'sgoalissimplytominimizeitsownprocurementcost,or (b) thefollowerseekstomaximizetheleader'sprocurementcostssubjecttoarequirementthattheirownprocurementcostiswithinaspeciedtoleranceoftheirminimumpossiblecost.Inparticular,onemotivationforthesecondscenarioabovecomesfromsituationsinwhichsomeuncertaintyexistsregardingthefollower'sdecisioninthesecondstage.Forexample,theleadermaynothavecondenceinthefollower'sabilitytosatisfyitsdemandatminimumcost,ortheleadermaynotpreciselyknowallofthefollower'sdata.Additionally,thefollowermayalsosimplybewillingtosuboptimizeitsobjectivebyasmallamountinordertoincreaseitscompetitor'scosts.Inthesecases,adecision-makermaywishtoconsidertheworst-case,assumingthatthefollower'sdecisionissuboptimalbynomorethansomevalue,withlargervaluesleadingtomoreconservativedecisions(notingthatthefollower'ssetofpotentialactionsbecomeslargerasgrows).Weconsideramulti-phaseprocurementstrategythattheleadertakestominimizeitsprocurementcosts.Intherstphase,theleaderisallowedtoprotect(orguarantee)somesubsetofthesuppliers'capacitiesbypayingsomeupfrontxedcost.Foragivensupplier,thiscostmayrepresentsomexedchargetheyrequiretoensurethatsomelevelofcapacityisavailable,oritmayrepresentthecostofsometypeofexclusivityagreementinwhichthesupplieragreestoonlyselltotheleader.Inscenario(b)above,thecompetitoractsinapredatorymannerwhen=1,i.e.,insteadofpurchasingitemswiththegoalofminimizingtheirownprocurementcosts,theywishtonegativelyimpacttheleader.Whilepredatorybusinesspracticesare,forthemostpart,illegal,theyarenotentirelyuncommon.Existenceofpredatorybusinesspracticesaswellaswhysuchstrategiesmightbeemployedbycompaniescanbefoundin[ 5 31 41 43 ]. 45

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Quantitydiscountsarewidelyappliedinallindustries,andagreatdealofliteratureregardingthispricingstrategyexists,forexamplein[ 25 35 57 ].However,fewerstudiesconsidercompetitorsbuyingfrommultiplecapacitatedsuppliers.Minner[ 39 ]discussesmodelsforasinglebuyerwithmultiplesuppliersand[ 17 18 20 ]discussmodelsforasinglebuyerpurchasingitemsfromasetofcapacitatedsupplierswithconcavecostcurves.Adifferent,yetrelated,problemhasbeenexaminedbyBernsteinandFedergruen[ 12 ]andIngeneandParry[ 33 ],inwhichasupplierseekstomaximizeprotsviapricingandorderingdecisionswhendealingwithmultiplecompetingbuyers.Thesolutionapproachdevisedinthischapterinvolvesreformulationtechniquesthatenableustotransformourthree-stagegamesintosingle-stageandtwo-stagemathematicalprogrammingformulations.Thesingle-stageformulationisthensolveddirectlyusingCPLEX,whileacutting-planealgorithmisdevelopedforthesolutionofthetwo-stageformulations.(See[ 52 ]forarelatedsolutionmethod.)Forapplicationsofthree-stagegamesinotherareassuchasfacilitylocation,networkdesign,supplychains,andnationaldefense,see[ 23 49 ],[ 54 ],[ 55 ],and[ 15 16 ],respectively.Ourreformulationtechniquedrawsonthedevelopmentofanetworkoptimizationformulationoftheleader'scost-minimizationproblem(providedinChapter 2 )thatoccursinthethirdstage.Thisnetworkformulationforthethird-stageproblemallowsustoviewthethree-stagegameasanetworkinterdictionandforticationproblem.Foroverviewsofnetworkinterdictionandforticationproblems,see[ 50 51 53 58 ].Inthischapter,wedeveloptwomodels:anoptimisticmodelandageneralizedbilevelmodel.Intheoptimisticmodel,thefollower'sgoalistominimizeitsownprocurementcostsandintheexistenceofalternativeoptimalsolutions,thefollowerchoosesasolutionthatisbestfortheleader(intermsoftheleader'sprocurementcosts).Inthegeneralizedbilevelmodel,thefollower'sgoalistomaximizetheleader'sprocurementcosts,subjecttotherestrictionthatthecostoftheirprocurementdecisionmustbewithinsomespeciedlimit,,oftheirminimum-costprocurementdecision. 46

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When=0,thefollower'sgoalistominimizeitsownprocurementcostsasintotheoptimisticmodel;however,intheexistenceofalternativeoptimalsolutions,thefollowerchoosesasolutionthatisworstfortheleader.(Werefertothecasewhen=0asthepessimisticmodel.)When=1,themodelfromChapter 2 results.Thischapterthereforecontributestotheliteraturebyprovidingaclassofnewprocurementplanningmodels(andcorrespondingsolutionapproaches)thatismoregenerallyapplicabletoawiderangeofcompetitiveprocurementscenarioswithmultiplesuppliers.Wealsointroduceaneffectiverelaxationandpenalizationmethodthatenablesacutting-planealgorithmforthesolutionofourmodels.Ourcomputationalresultsalsoprovidesomeinterestinginsightsonthevalueoftheworst-caseanalysisforthisproblemclass.Inparticular,inthevastmajorityofcases,theworst-caseoutcomecoincidedwiththecaseinwhichthefollowerminimizesitscost,indicatingthataself-interestedfollower'sactionsoftenleadtotheworst-caseoutcomefortheleadersimplyasaby-productofcompetition.Theremainderofthischapterisorganizedasfollows.InSection 3.2 weprovidenotationthatmoreformallydenesourproblems,alongwithashortest-pathformulationusedtosolvethethird-stageproblem(inwhichtheleaderprocuresitsrequireddemand).InSection 3.3 weintroducetheoptimisticproblemandformulateitasasingle-stagelinearmixed-integerprogram(MIP).Section 3.4 discussesthegeneralizedbilevelmodelandprovidesacutting-planealgorithmforitssolution.Section 3.5 containstheresultsofourcomputationalexperiments. 3.2NotationandProcurementModelThemodelsinthischapterconsideratwo-playerprocurementgameinwhichtwormspurchaseanitemfromthesamesetofsuppliers.Theleaderhastheoptionofprotectingsomesubsetofthesuppliers.Thefollowerthenprocuresfromtheremainingsuppliers.Afterthefollowerhasacted,theleadersatisesitsdemandatminimumcost,usingtheremainingcapacityfromthesuppliers.Aswenotedearlier,thefollowermay 47

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correspondtoacollectionofcompetitors.Thissequenceofeventsisintendedtomodelthefactthatsomedegreeofcapacitywillhavebeenreservedbythesecompetitorsatthetimetheleaderplacesitsorderswithsuppliers.Thecosttopurchaseqiunitsfromsupplieri2S=f1,...,Ngispi(qi),wherepi(qi)isanonnegative,nondecreasing,concavefunction.Eachsupplieri2Shasaprotectioncost,ri,andcapacity,ui.Theleader'sdemandisQLandthefollower'sdemandisQF.Fori2S,letzibeabinaryvariablethatequals1iftheleaderprotectssupplieri,and0otherwise.Thesecond-stageintegervariables,yi,8i2S,representthenumberofunitsthefollowerpurchasesfromsupplieri.Thethird-stageintegervariables,qi,8i2S,representthenumberofunitspurchasedfromsupplieribytheleader.Next,inordertomakethischapterasself-containedaspossible,theremainderofthissectionpresentsresultsfromChapter 2 ,wherethethird-stageoptimizationproblem(theleader'sprocurementcostminimizationproblem)isreformulatedasashortest-pathproblem.Thisthird-stageproblemisemployedineachofthenewmodelsweintroduceinSections 3.3 and 3.4 .Givenasetofsecond-stagedecisions,y,the(leader's)third-stageproblemisasfollows: minXi2Spi(qi) (3a) s.t.qiui)]TJ /F3 11.955 Tf 11.96 0 Td[(yi,8i2S, (3b) Xi2SqiQL, (3c) q2ZN+, (3d) whereConstraints( 3b )representthecapacityrestrictionsafterthefollowerhasacted,Constraints( 3c )ensurethattheleadersatisesitsdemand(andcanequivalentlybewrittenasanequality),andConstraints( 3d )arethenonnegativityandintegerrestrictions. 48

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ThereformulationstrategyinChapter 2 transforms( 3 )toanequivalentshortest-pathproblem.Theshortest-pathnetworkcontainssourcenodev00,nodesvijforj=0,...,QLandi2SnfNg,andsinknodevNQL.Asolutioninwhichthepathvisitsnodevijfori=0,...,Nandj=0,...,QLindicatesthatjunitsarepurchasedfromtherstisuppliers.WedenethesetFiasthefeasibleprocurementamountsfromtherstisuppliers.Also,fori=1,...,Nandk2Fi,thesetIikcontainsthefeasibleprocurementamountsforthersti)]TJ /F4 11.955 Tf 12.52 0 Td[(1suppliersgiventhatkunitsareprocuredfromtherstisuppliers,andthesetOikcontainsthefeasibleprocurementamountsfromthersti+1suppliersgiventhatkunitsareprocuredfromtherstisuppliers.Formally,wehaveFi=njjmaxf0,QL)]TJ /F14 11.955 Tf 11.96 8.97 Td[(PNs=i+1usgjminfQL,Pis=1usgo,8i2S,whereF0=f0gandPNs=N+1us0,sothatFN=fQLg,Iik=fj2Fi)]TJ /F7 7.97 Tf 6.59 0 Td[(1jk)]TJ /F3 11.955 Tf 12.21 0 Td[(uijkg,8i2S,k2Fi,andOik=fj2Fi+1jkjk+ui+1g,8i2SnfNg,k2Fi.Thenetworkconsistsofallarcs(vi)]TJ /F7 7.97 Tf 6.59 0 Td[(1j,vik),8i2S,k2Fi,j2Iik.Wenextreplacethesecond-stageintegerdecisionvariableswithanequivalentsetofbinaryvariables.Thatis,thebinaryvariableyi,qequals1ifthefollowerpurchasesatleastqunitsfromsupplieri,and0otherwise,8i2S,1qminfui,QFg.Thecapacityandcostforarc(vi)]TJ /F7 7.97 Tf 6.59 0 Td[(1j,vik)are1)]TJ /F4 11.955 Tf 12.4 0 Td[(yi,ui)]TJ /F6 7.97 Tf 6.58 0 Td[(k+j+1andpi(k)]TJ /F3 11.955 Tf 12.11 0 Td[(j),respectively,fori2S,k2Fi,andj2Iik.Letfijkbetheowonarc(vi)]TJ /F7 7.97 Tf 6.59 0 Td[(1j,vik).Givenabinaryvectorofyi,qvalues,y,weformulate( 3 )asashortest-pathproblemusingthefollowinglinearprogramming(LP)formulation: PL:minXi2SXk2FiXj2Iikpi(k)]TJ /F3 11.955 Tf 11.96 0 Td[(j)fijk (3a) s.t.Xk2F1f10k=1, (3b) Xj2Iikfijk=Xj2Oikfi+1kj,8i2SnfNg,k2Fi, (3c) )]TJ /F14 11.955 Tf 17.26 11.36 Td[(Xj2FN)]TJ /F16 5.978 Tf 5.76 0 Td[(1fNjQL=)]TJ /F4 11.955 Tf 9.29 0 Td[(1, (3d) fijk1)]TJ /F4 11.955 Tf 12.25 0 Td[(yi,ui)]TJ /F6 7.97 Tf 6.58 0 Td[(k+j+1,8i2S,k2Fi,j2Iik, (3e) 49

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fijk0,8i2S,k2Fi,j2Iik. (3f) TheObjective( 3a )minimizesthetotalprocurementcostsfortheleader,whileConstraints( 3b )( 3d )serveasow-conservationconstraints.Constraints( 3e )enforcethesuppliercapacityconstraintsafterthefollowerhasacted,andConstraints( 3f )enforcenonnegativity.WerefertothefeasibleregionofPLasF(PL)anditsobjectivefunctionasgPL(f).Let00,ik,NQL,andijkdenotethedualvariablesassociatedwithConstraints( 3b ),( 3c ),( 3d ),and( 3e ),respectively.ThedualofproblemPLiswrittenas: max00)]TJ /F5 11.955 Tf 11.96 0 Td[(NQL)]TJ /F14 11.955 Tf 11.96 11.36 Td[(Xi2SXk2FiXj2Iikijk(1)]TJ /F4 11.955 Tf 12.24 0 Td[(yi,ui)]TJ /F6 7.97 Tf 6.59 0 Td[(k+j+1) (3a) s.t.i)]TJ /F7 7.97 Tf 6.59 0 Td[(1j)]TJ /F5 11.955 Tf 11.95 0 Td[(ik)]TJ /F5 11.955 Tf 11.96 0 Td[(ijkpi(k)]TJ /F3 11.955 Tf 11.95 0 Td[(j),8i2S,k2Fi,j2Iik, (3b) 0. (3c) Notethatbecause( 3d )islinearlydependentupontheotherowbalanceconstraints,wemayrestrictNQL=0.Thisallowsijtorepresenttheshortest-pathcostfromnodevijtonodevNQLinthenetwork,assumingthatsuchapathexists(see[ 2 ]).Nextwesimplify( 3 )bymakingasubstitutionforthe-values.Denethefollowingnotation.LetLP(vik,v^^k)denotethelongest-pathcostfromnodeviktonodev^^kintheshortestpathnetworkwhennopurchasesaremadebythefollowerandletSP(vik,v^^k)denotetheshortest-pathcostfromnodeviktonodev^^kintheshortestpathnetworkwhennopurchasesaremadebythefollower.Anarc(vi)]TJ /F7 7.97 Tf 6.58 0 Td[(1j,vik)isavailableifitscapacityequals1(i.e.,yi,ui)]TJ /F6 7.97 Tf 6.59 0 Td[(k+j+1=0).Anodev^^kisreachablefromnodevikifthereisapathofavailablearcsfromviktov^^kintheshortest-pathnetworkafterthefollowerhaspurchaseditsitems(where^>iand^kk).InChapter 2 weshowthevalidityofthesubstitutionijk=Bijkyi,ui)]TJ /F6 7.97 Tf 6.59 0 Td[(k+j+1,8i2S,k2Fi,j2Iik,whereBijkisdenedasfollows: 50

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Case1:Ifnodevi)]TJ /F7 7.97 Tf 6.58 0 Td[(1jcanneverbedisconnectedfromnodevNQLbythefollower(i.e.,vNQLisalwaysreachablefromvi)]TJ /F7 7.97 Tf 6.58 0 Td[(1j),thenBijk=LP(vi)]TJ /F7 7.97 Tf 6.59 0 Td[(1j,vNQL))]TJ /F4 11.955 Tf 11.06 0 Td[([pi(k)]TJ /F3 11.955 Tf 11.06 0 Td[(j)+SP(vik,vNQL)].Case2:Else,Bijk=LP(v00,vNQL))]TJ /F4 11.955 Tf 11.96 0 Td[([SP(v00,vi)]TJ /F7 7.97 Tf 6.59 0 Td[(1j)+pi(k)]TJ /F3 11.955 Tf 11.96 0 Td[(j)+SP(vik,vNQL)].Makingthesesubstitutions,restrictingNQL=0,andnotingthatyisbinary,wereformulate( 3 )asfollows: DL:max00 (3a) s.t.i)]TJ /F7 7.97 Tf 6.59 0 Td[(1j)]TJ /F5 11.955 Tf 11.95 0 Td[(ikpi(k)]TJ /F3 11.955 Tf 11.95 0 Td[(j)+Bijkyi,ui)]TJ /F6 7.97 Tf 6.59 0 Td[(k+j+1,8i2S,k2Fi,j2Iik. (3b) WerefertothefeasibleregionofDLasF(DL)anditsobjectivefunctionasgDL().(NotethatwhilePLandDLaredependentony,wehavesuppressedthisdependencefornotationalclarity.) 3.3OptimisticProblemRecallthatintheoptimisticproblem,thefollower'sgoalistominimizeitsownprocurementcosts,andthefollowerchoosesfromitssetofoptimalsolutionsonethatisbestfortheleader.Theformulationfortheoptimisticmodelisasfollows: minrTz+Xi2Spi(qi) (3a) s.t.qiui)]TJ /F3 11.955 Tf 11.95 0 Td[(yi,8i2S, (3b) Xi2SqiQL, (3c) q2ZN+, (3d) z2f0,1gN, (3e) y2argmin(Xi2Spi(yi)jyiui(1)]TJ /F3 11.955 Tf 11.95 0 Td[(zi),8i2S,Xi2SyiQF,y2ZN+). (3f) Theobjectivefunction,( 3a ),minimizestheleader'stotalcosts.Constraints( 3b )restricttheleader'sprocurementamountfromsupplieritobenomorethantheremainingsupplyafterthefollowerhasacted.Constraint( 3c )ensuresthatthe 51

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leader'sdemandissatised.Constraints( 3d )and( 3e )arethenonnegativityandbinaryrestrictions.Constraint( 3f )ensuresthatyisanoptimalsolutiontothefollower'sprocurementproblem.Notably,however,themodelensuresthattheleaderchoosesanyythatoptimizesthefollower'sproblem,ensuringthatayischosenthatismostbenecialtotheleader. Remark6.Toensurethatallleaderandfollowerproblemsarefeasible,weaddadummysupplierwithcapacityQL+QFandarbitrarilylargepurchasecosts.Additionally,thez-variablecorrespondingtothisdummysupplierisxedtozero.Withoutlossofgenerality,fortheremainderofthechapterweassumethatthisdummysupplierisincludedinSasthelastsupplier,N.NotethatbecausethelastsupplierisalwaysavailabletotheleaderandhasasupplyofQL+QF,wehaveFi)]TJ /F7 7.97 Tf 6.59 .01 Td[(1Fi.2Tosolve( 3 ),weformulatethefollower'scost-minimizationproblemasashortest-pathproblem(inthesamemannerasinSection 3.2 fortheleader'scost-minimizationproblem).Wethenreplace( 3f )withtheshortest-pathproblem'sconstraints,itsdualconstraints,andaconstraintequatingtheprimalanddualobjectivefunctions.Weconstructasimilarnetworkforthefollowerasdiscussedbefore,thistimewithsourcenodev00,nodesvijforj=0,...,QFandi2SnfNg,andsinknodevNQF.WedenesetsFi,Iik,andOikthesamewayasFi,Iik,andOikweredened,exceptthatQFreplacesQLinthedenitionofFi.(NotethatFi)]TJ /F7 7.97 Tf 6.58 0 Td[(1Fi,8i2S.)Thecostforarc(vi)]TJ /F7 7.97 Tf 6.59 0 Td[(1j,vik)ispi(k)]TJ /F3 11.955 Tf 12.24 0 Td[(j)fori2S,k2Fi,andj2Iik.Forj6=k,thecapacityonthisarcis(1)]TJ /F3 11.955 Tf 12.04 0 Td[(zi).Further,weletfijkrepresenttheowonarc(vi)]TJ /F7 7.97 Tf 6.59 0 Td[(1j,vik).Givenavectorz,theshortest-pathformulationofthefollower'sproblemisasfollows: PF:minXi2SXk2FiXj2Iikpi(k)]TJ /F3 11.955 Tf 11.96 0 Td[(j)fijk (3a) s.t.Xk2F1f10k=1, (3b) 52

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Xj2Iikfijk=Xj2Oikfi+1kj,8i2SnfNg,k2Fi, (3c) )]TJ /F14 11.955 Tf 17.25 11.36 Td[(Xj2FN)]TJ /F16 5.978 Tf 5.76 0 Td[(1fNjQF=)]TJ /F4 11.955 Tf 9.29 0 Td[(1, (3d) Xk2Fi)]TJ /F16 5.978 Tf 5.75 0 Td[(1fikkzi8i2S, (3e) fijk0,8i2S,k2Fi,j2Iik. (3f) Allconstraintsareanalogoustothosein( 3 ),exceptforConstraints( 3e ),whichensurethatthefollowermustpurchase0unitsfromsupplieriifsupplieriisprotected.WerefertothefeasibleregionofPFasF(PF)anditsobjectivefunctionasgPF(f).Let00,ik,NQL,andidenotedualvariablesassociatedwithConstraints( 3b ),( 3c ),( 3d ),and( 3e ),respectively.ThedualofPFisgivenas: DF:max00)]TJ /F4 11.955 Tf 12.9 0 Td[(NQF+Xi2Szii (3a) i)]TJ /F7 7.97 Tf 6.59 0 Td[(1k)]TJ /F4 11.955 Tf 12.9 0 Td[(ik+i0,8i2S,k2Fi)]TJ /F7 7.97 Tf 6.59 0 Td[(1, (3b) i)]TJ /F7 7.97 Tf 6.59 0 Td[(1j)]TJ /F4 11.955 Tf 12.9 0 Td[(ikpi(k)]TJ /F3 11.955 Tf 11.96 0 Td[(j),8i2S,k2Fi,j2Iiknfkg, (3c) 0. (3d) WeagainrestrictNQF=0,andrefertothefeasibleregionofDFasF(DF)anditsobjectivefunctionasgDF(,)(suppressingthedependenceonzaswedidthedependenceofgDLony).BecausePFandDFareLPs,bystrongduality,wehavegPF(f)=gDF(,)foranypairofoptimalsolutionsfand(,).Thisallowsustoreformulate( 3f )asfollows: f2F(PF), (3a) (,)2F(DF), (3b) gPF(f)=gDF(,), (3c) yi=Xk2FiXj2Iik(k)]TJ /F3 11.955 Tf 11.96 0 Td[(j)fijk,8i2S, (3d) y2ZN+, (3e) 53

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where( 3d )isusedtomapftoy.Formulation( 3 ),with( 3f )replacedby( 3a )( 3e ),isamixed-integernonlinearprogram(MINLP)becauseoftheconcaveobjectivefunctionandbilinearterms,zii,8i2S,ingDF(,).UsingtheresultsfromSection 3.2 ,weemploytheshortest-pathrepresentationoftheleader'sprocurementproblemtolinearizetheobjective.Accordingly,weintroducethefollowingset,whichmapsthefollower'sowvector,f,tothey-vectorusedintheleader'sshortest-pathproblem: =((f,y)jminfui,QFgXj=1yi,j=Xk2FiXj2Iik(k)]TJ /F3 11.955 Tf 11.96 0 Td[(j)fijk,8i2S,yi,j+1yi,j8i2S,j=1,...,minfui,QFg)]TJ /F4 11.955 Tf 20.59 0 Td[(1,yi,j2f0,1g8i2S,j=1,...,minfui,QFg).Giventhisset,weobtainthefollowingformulationfortheoptimisticcase: minrTz+gPL(f) (3a) s.t.f2F(PL), (3b) z2f0,1gN, (3c) Constraints( 3a ){( 3c ) (3d) (f,y)2 (3e) yi,1(1)]TJ /F3 11.955 Tf 11.95 0 Td[(zi)8i2S. (3f) NotethatConstraints( 3f )areunnecessarybuttightenthecontinuousrelaxationof( 3 ).Forexample,consideranLPrelaxationsolutioninwhich,forsomei2S,wehavezi=0.5,fikk=0.5forsomek2Fi,andfijk0=0.5forsomek02Fi,andwherej=k0)]TJ /F4 11.955 Tf 12.44 0 Td[(2(assumingthatui2).WithoutConstraints( 3f ),itisfeasibleforyi,1=1,whereas,whenConstraints( 3f )areincluded,itwouldonlybefeasibletohaveyi,1=yi,2=0.5. 54

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Wenoweliminatethenonlinearitiespresentin( 3 )duetothebilinearzii-termsingDF(,).LetEirepresentthecostofthemostexpensivepathonthefollower'sshortest-pathnetworkinwhichnounitsarepurchasedfromsupplieri(i.e.,themostexpensivepathinwhicharc(vi)]TJ /F7 7.97 Tf 6.59 0 Td[(1k,vik)isused,forsomek2Fi)]TJ /F7 7.97 Tf 6.58 0 Td[(1),andletCdenotethecostoftheleastexpensivepathinthefollower'snetwork.Proposition A.1 ofAppendixAshowsthatanoptimalsolutionexiststoDFinwhichi=(Ei)]TJ /F3 11.955 Tf 13.1 0 Td[(C)zi,8i2S.Thissubstitutionisthereforealsovalidfortheoptimisticmodelbecausef,f,andyareonlyrelatedtothefollower'sdualvariablesbythevalueofgDF(,),whichisthesameforallalternativeoptimalsolutionstoDF.Hence,wemakethesubstitutioni=(Ei)]TJ /F3 11.955 Tf 12.08 0 Td[(C)zi,8i2S,ingDF(,)andF(DF).Becausezisbinary,thebilineartermsbecomezii=(Ei)]TJ /F3 11.955 Tf 11.96 0 Td[(C)z2i=(Ei)]TJ /F3 11.955 Tf 11.95 0 Td[(C)zi,8i2S,whicharenowlinear.ObservethatweareabletoformulatetheoptimisticversionoftheproblemasasingleMIPbecausethefollowerisconcernedsolelywithminimizingitsownobjectiveasopposedtohurtingtheleader'sobjective.Ouranalysisinthenextsectionrevealsthatformulatingandsolvinggeneralversionoftheproblemisamorecomplextask. 3.4GeneralizedBilevelProblemWenowdiscussthegeneralizedbilevelproblem,whichisdescribedasfollows.Givensomenonnegativeparameter,thefollowerseekstomakeaprocurementdecisionthatmaximizestheleader'sprocurementcosts,subjecttotherestrictionthatthefollower'scostiswithinofitsminimumpossiblecostgiventheleader'sdecision,z.Onewayofinterpretingthisproblemisthatthefollowerhasabudget,,thatitiswillingtopayinadditiontoitsminimumpossibleprocurementcosts,inordertomaximizetheleader'sminimum-procurementcosts.Forthepessimisticmodel(i.e.,=0),thefollowerisrequiredtochooseanoptimalsolution,butchoosesonethatisworstfortheleader(asopposedtochoosingonethatisbestfortheleaderintheoptimisticcase).When=1,thenthefollower'sobjectiveissimplytomaximizethefollower'sobjective,whichispreciselythecasestudiedinChapter 2 .(Itisalsopossibletoformulatethe 55

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problemsothefollower'ssolutioncanbesuboptimalbyapercentageoftheiroptimalsolution'sobjectivevalue.Theanalysisinthefollowingsectioncanbereadilyadaptedtoaccommodatethiscase.)InSection 3.4.1 wepresentthegeneralizedbilevelformulation,andthenreformulatethismodelinSection 3.4.2 toremovethedependencyofitsfeasibleregionontheleader'srst-stagez-vector.Thisreformulationallowsustoprescribecutting-planealgorithmsfortwoalternativeformulations,whichwedescribeinSections 3.4.3 and 3.4.4 .WeclosethissectionbyprovidingtwoalgorithmicenhancementsinSection 3.4.5 3.4.1GeneralizedBilevelModelWebeginthissectionbynotingthatinthismodel,thefollower(ratherthantheleader)choosesaprimalfeasiblesolutionf2F(PF)anddualfeasiblesolution(,)2F(DF).Also,inthismodel,werequirethatgPF(f)gDF(,)+.Thisconstraintensuresthatthefollower'ssolutionissuboptimalbynomorethan,becausethevalueofgDF(,)cannotexceedtheoptimalobjectivefunctionvalue.Hence,wedenethefollower'sfeasibleregionbythesetW=nf2F(PF),(,)2F(DF),gPF(f)gDF(,)+o,andformulatethegeneralizedbilevelmodelasfollows: minz2f0,1gNrTz+max(,,f)2WmingPL(f) (3a) s.t.f2F(PL), (3b) (f,y)2. (3c) Constraint( 3b )ensuresthattheleaderchoosesafeasibleprocurementplan,giveny,andconstraint( 3c )performsthenecessarymappingfromftothey-vectorusedinF(PL).Bydualizingthethird-stageproblemandcombiningitwiththesecond,weobtainthefollowingformulation,wherewithaslightabuseofnotationweuseF(DL)todenotethefeasibleregionofDLinwhichyisavariable. 56

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Bydualizingthethird-stageproblemandcombiningitwiththesecond-stageproblemweobtainthefollowingformulation: minz2f0,1gNrTz+maxgDL() (3a) s.t.(,y)2F(DL), (3b) f2F(PF), (3c) (,)2F(DF), (3d) gPF(f)gDF(,)+, (3e) (f,y)2. (3f) Constraints( 3b )ensurethattheleader'ssolutionisdualfeasible,Constraints( 3c )( 3e )statethefollower'srestrictionsgiveninW,and( 3f )mapsthefollower'sprimalsolutionftoy. 3.4.2InvariantFeasibleRegionModelNext,weseekaformulationinwhichthesecond-stagefeasibleregionisindependentofz,whichassistsusinderivingacutting-planeapproachtosolvethegeneralizedbilevelproblem.NotethatConstraints( 3c )and( 3e )aretheonlyconstraintsin( 3 )thatdependonz.WedenoteproblemPFasthesameproblemasPFexceptwithConstraints( 3e )relaxed,i.e.,solvingPFyieldsthe(follower's)minimumprocurementcostforthefollowerwhentheleaderprotectsnosuppliers.WerefertothefeasibleregionofPFasF(PF)anditsobjectivefunctionasgPF(f)(whichisidenticaltogPF(f)).Wepresentthefollowingreformulationof( 3 ),whichremovesthez-variablesfromthefollower's(primal)feasibleregion,andobtainthefollowingformulation: minz2f0,1gNrTz+maxgDL() (3a) s.t.(,y)2F(DL), (3b) f2F(PF), (3c) 57

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(,)2F(DF), (3d) gPF(f)+M1 Xi2Sziyi,1!gDF(,)+, (3e) (f,y)2. (3f) Notethatbecause( 3c )doesnotrequireftoobey( 3e ),weseekavalueofM1thatislargeenoughtoensurethatPi2Sziyi,1=0.LettingEdenotethecostofthemostexpensivepathinthefollower'sshortest-pathnetwork,andconsiderthefollowingresult. Proposition3.1. Givenarst-stagesolution,^z,lettingM1=E)]TJ /F3 11.955 Tf 12.05 0 Td[(C+1+ensuresthat^ziyi,1=0,8i2S,inanyfeasiblesolutiontothesecondstageof( 3 ). Proof. Givenavector^z,let(,,,f,y)befeasibletothesecondstagein( 3 ).Assumebycontradictionthat9i2Ssuchthat^ziyi,1=1.Therefore,wemusthavegDF(,))]TJ /F3 11.955 Tf 10.89 0 Td[(gPF(f)(E)]TJ /F3 11.955 Tf 10.9 0 Td[(C+1+))]TJ /F5 11.955 Tf 10.89 0 Td[(.However,thisinequalitycannothold,becausegDF(,)E,8(,)2F(DF),andgPF(f)C,8f2F(PF).Therefore,wehavereachedacontradictionandtheproofiscomplete. Therefore,Pi2Sziyi,1=0foranyoptimalsolutionto( 3 ),andmodels( 3 )and( 3 )areequivalent.Next,notethatinModel( 3 ),z-variablesexistonlyinConstraints( 3e ).Toremovezfromthefeasibleregion,werelax( 3e )andpenalizeitsinfeasibilityintheobjectivefunctionasfollows: minz2f0,1gNrTz+maxgDL())]TJ /F3 11.955 Tf 9.3 0 Td[(M2max(0,gPF(f)+M1 Xi2Sziyi,1!)]TJ /F3 11.955 Tf 11.95 0 Td[(gDF(,))]TJ /F5 11.955 Tf 11.95 0 Td[()s.t.(,y)2F(DL), (3a) f2F(PF), (3b) (,)2F(DF), (3c) (f,y)2, (3d) 58

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whereM2isascalarwhosevalueislargeenoughtoguaranteethatgPF(f)+M1 Xi2Sziyi,1!)]TJ /F3 11.955 Tf 11.95 0 Td[(gDF(,))]TJ /F5 11.955 Tf 11.96 0 Td[(0atoptimality.WediscussamechanismforcomputingM2inSection 3.4.3 .Hence,( 3 )isalsoavalidformulationforthegeneralizedbilevelmodel,andhasasecond-stagefeasibleregionthatisindependentofz,whichallowsustodecomposetheproblemandsolveitusingacutting-planealgorithm,whichweexploreinthenextsubsection. 3.4.3Cutting-PlaneAlgorithm:Big-MModelTowardthedevelopmentofacutting-planealgorithmforsolvingthegeneralizedbilevelproblem,supposethatweenumerateallsolutions(0,0,0,f0,y0)thatsatisfytheconstraintsof( 3 ),andletdenotethecollectionofthesesolutions.Westatethefollowingformulation,whichisequivalentto( 3 ),as: minrTz+w (3a) s.t.wgDL(0))]TJ /F3 11.955 Tf 11.95 0 Td[(M2max(0,gPF(f0)+M1 Xi2Sziy0i,1!)]TJ /F3 11.955 Tf 11.96 0 Td[(gDF(0,0))]TJ /F5 11.955 Tf 11.96 0 Td[(),8(0,0,0,f0,y0)2, (3b) w0, (3c) z2f0,1gN. (3d) Observethat( 3b )neednotbewrittenforeverysolutionin;instead,weneedonlytoincludethosevectorsinthatoptimizethesecondstageof( 3 )forsomebinaryvectorz.Callthecollectionofsuchsolutions .Foreachbinaryvectorz0,let(0,0,0,f0,y0)beacorrespondingsolutionin thatoptimizes( 3 )withz=z0.Inoursubsequentconceptualanalysis,itisusefultoobtainaone-to-onecorrespondencebetweenz-vectorsandsolutionsin ,eventhoughsomesolutionsin mayoptimizethesecond-stageproblemof( 3 )forseveraldifferentz-vectors.Inthisevent,we 59

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simplyaddcopiesof(0,0,0,f0,y0)to ,onecorrespondingtoeachz-vectorforwhich(0,0,0,f0,y0)optimizesthesecond-stageproblemof( 3 ).Foreaseofnotation,wedenote(0,0,0,f0,y0)2 by 02 whereconvenient.Tolinearize( 3 ),weintroduceadummycontinuousvariablep 0andadummybinaryvariablev 0forevery 02 .ThisgivesusthefollowinglinearMIPformulation: BLG-MP:minrTz+w (3a) s.t.wgDL(0))]TJ /F3 11.955 Tf 11.95 0 Td[(M2p 08 02 (3b) p 0M3v 08 02 (3c) p 0M3(1)]TJ /F3 11.955 Tf 11.96 0 Td[(v 0)+gPF(f0)+M1 Xi2Sziy0i,1!)]TJ /F3 11.955 Tf 11.96 0 Td[(gDF(0,0))]TJ /F5 11.955 Tf 11.96 0 Td[(,8 02 (3d) v 02f0,1g8 02 (3e) w0, (3f) z2f0,1gN, (3g) whereM3isalargeconstantforwhichweprovideasufcientlylargevalueinProposition 3.2 Proposition3.2. WhenM3=E)]TJ /F3 11.955 Tf 12.84 0 Td[(C+1+,formulations( 3 )and( 3 )areequivalent. Proof. Asjustiedearlier,( 3b )needonlybeincluded8 02 ;hence,weshowthatforeach 02 ,(z,w)satises( 3b )ifandonlyif(z,w)satises( 3b )( 3e )forsomechoiceofp 0andv 0.First,supposethat(z,w)satises( 3b ).IfgPF(f0)+M1)]TJ 5.48 -.72 Td[(Pi2Sziy0i,1)]TJ /F3 11.955 Tf 13.06 0 Td[(gDF(0,0))]TJ /F5 11.955 Tf 13.06 0 Td[(0,thensupposethatwesetv 0=p 0=0.Constraints( 3b ),( 3c ),and( 3e )aresatised,and( 3d )issatisedsolongas(E)]TJ /F3 11.955 Tf 11.69 0 Td[(C+1+)+gPF(f0)+M1)]TJ 5.48 -.71 Td[(Pi2Sziy0i,1)]TJ /F3 11.955 Tf 11.69 0 Td[(gDF(0,0))]TJ /F5 11.955 Tf 11.69 0 Td[(isnonnegative.RewritingthisexpressionasE)]TJ /F3 11.955 Tf 11.95 0 Td[(gDF(0,0)+gPF(f0))]TJ /F3 11.955 Tf 11.96 0 Td[(C+1+M1)]TJ 5.48 -.72 Td[(Pi2Sziy0i,1,thisvalueisclearlypositivebecauseeachparentheticaltermisnonnegative.Ontheotherhand,if 60

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gPF(f0)+M1)]TJ 5.48 -.72 Td[(Pi2Sziy0i,1)]TJ /F3 11.955 Tf 11.95 0 Td[(gDF(0,0))]TJ /F5 11.955 Tf 11.95 0 Td[(>0,thensupposewesetv 0=1andp 0=min(E)]TJ /F3 11.955 Tf 11.96 0 Td[(C+1+,gPF(f0)+M1 Xi2Sziy0i,1!)]TJ /F3 11.955 Tf 11.95 0 Td[(gDF(0,0))]TJ /F5 11.955 Tf 11.96 0 Td[(),whichsatisesConstraints( 3c ),( 3d ),and( 3e ).Notethat( 3b )issatisedifitsright-hand-sidevalueisnogreaterthanCL.First,ifp 0=E)]TJ /F3 11.955 Tf 12.74 0 Td[(C+1+,thengDL(0))]TJ /F3 11.955 Tf 11.5 0 Td[(M 02p 0CLsinceM 02gDL(0))]TJ /F3 11.955 Tf 11.5 0 Td[(CLandp 0=E)]TJ /F3 11.955 Tf 11.5 0 Td[(C+1+1.Second,ifp 0=gPF(f0)+M1)]TJ 5.48 -.71 Td[(Pi2Sziy0i,1)]TJ /F3 11.955 Tf 12.09 0 Td[(gDF(0,0))]TJ /F5 11.955 Tf 12.09 0 Td[(,thengDL(0))]TJ /F3 11.955 Tf 12.09 0 Td[(M 02p 0CLsinceM 02wascomputedspecicallytoaccountforthisscenario.Now,considersomesolution(z,w),alongwithv 0andp 0,whichsatises( 3b )( 3e )forsome 02 .Ifv 0=0,thenp 00,andso( 3b )implies( 3b ).Else,v 0=1andthefollowinginequalitieshold: wgDL(0))]TJ /F3 11.955 Tf 11.95 0 Td[(M2p 0gDL(0))]TJ /F3 11.955 Tf 11.95 0 Td[(M2min(M3,gPF(f0)+M1 Xi2Sziy0i,1!)]TJ /F3 11.955 Tf 11.95 0 Td[(gDF(0,0))]TJ /F5 11.955 Tf 11.96 0 Td[()gDL(0))]TJ /F3 11.955 Tf 11.95 0 Td[(M2max(0,gPF(f0)+M1 Xi2Sziy0i,1!)]TJ /F3 11.955 Tf 11.96 0 Td[(gDF(0,0))]TJ /F5 11.955 Tf 11.95 0 Td[().Therstinequalityholdsdueto( 3b ),andthesecondholdsdueto( 3c )and( 3d )whenv 0=1.Thelastinequalityistruebecausethelattertermiscommontotheminimandinthesecondexpressionandtothemaximandinthethirdexpression.Thisinequalitychainveriesthat( 3b )isagainsatised,andthiscompletestheproof. Ourcutting-planemethoditerativelysolvesarestrictedversionof( 3 ),whichwecallBLG-RMP,inwhichConstraints( 3b )( 3e )areonlypresentforasubset^ofthesecond-stagesolutionsin .GivenaBLG-RMPoptimalsolution(z0,w0),wemustthensolveasubproblemthatidentiesanoptimalvector 02^correspondingtoz0, 61

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whichweachievebysolvingthefollower'sproblemgivenz0: BLG-SP:maxgDL() (3a) s.t.(,y)2F(DL), (3b) f2F(DF), (3c) gPF(f)gDF(,)+, (3d) (f,y)2 (3e) yi,1(1)]TJ /F3 11.955 Tf 11.96 0 Td[(z0i)8i2S. (3f) NotethatConstraints( 3f )areunnecessaryintheintegersense,buttightentheLPrelaxationof( 3 ).Afterwecomputeanoptimalsolutionto( 3 ),wecutoff(z0,w0)ifnecessary(i.e.,ifw0islessthantheoptimalobjectivefunctionvalueofBLG-SP)byaddinginequalitiesoftheform( 3b )( 3e ),alongwithnewvariablesp 0andv 0,toBLG-RMP.Giventhissetup,Algorithm 3 providesaformaldescriptionofthecutting-planealgorithmforsolvingBLG-MP,inwhich( 3b )( 3e )(andassociatedp-andv-variables)areonlypresentforsomesubset^ .InSteps 1 and 2 westartwith^=;,andinitializethelowerboundto0andtheupperboundtoinnity.InStep 4 ,wesolveBLG-RMP,yieldingalowerboundofw0+rTz0ontheoptimalobjectivevalue(sinceonlyasubsetofthecutshavebeengenerated).WethensolveBLG-SPinStep 5 toobtainv(z0)+rTz0.InStep 8 ,ifthisvalueislessthanthecurrentupperbound,weupdatetheincumbentsolution,z,andupperbound,UB.InStep 10 weadd(0,0,0,f0,y0)to^,requiringtheadditionofthreenewstructuralinequalitiesandtwonewvariablestoBLG-RMP.IfLB=UB,thenthealgorithmterminateswithanoptimalsolutioninStep 12 ;otherwise,itreturnstoStep 4 Remark7.NotethatthereexistsanoptimalsolutiontoBLG-SPinwhichgDF(,)assumesitslargestpossiblevalue.Further,otherthaninConstraint( 3d ),thereis 62

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Algorithm3Cutting-PlaneAlgorithmforSolving( 3 ) 1: CreateBLG-RMPwithrespecttotheemptysetofsolutions^=;. 2: SetLB=0andUB=1 3: whileLB0.(3)Intuitively,inordertodetermineavalueforM2,wewishtoknowthesmallestpossiblepositivevaluethattheleft-hand-sideof( 3 )cantake.However,insteadofusingthesamevalueofM2foreachInequality( 3b ),wedetermineaspecicvalue,M 02,associatedwitheach 02 .Lettingg( 0)denotethesecondargumentin( 3 ),wedeterminethesmallestpossiblepositivevalueforg( 0).WethensetM 02sothatgDL(0))]TJ /F3 11.955 Tf 12.88 0 Td[(M 02g( 0)CLforanypositiveg( 0)-value.Letz0denotetherst-stagesolutioncorrespondingtotheinstanceofBLG-SPforwhich 0wasoptimalandletzbe 63

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apotentialrst-stagesolutionwhichweusetoanalyzethevalueforg( 0).Notethatifzi=1forsomei2Ssuchthaty0i,1=1,theng( 0)1(sinceM1=E)]TJ /F3 11.955 Tf 12.49 0 Td[(C+1+,000+Pi2Szii0E,andgPF(f0)C).Next,considerthecaseinwhichPi2Sziy0i,1=0.Wehavethatg( 0)>0ifandonlyifthefollowingexpressionisstrictlypositive: gPF(f0))]TJ /F14 11.955 Tf 11.96 20.44 Td[( 000+Xi2Szii0+!.(3)Therefore,g( 0)isminimizedinthiscasewhenPi2Szii0takesonitslargestvaluesuchthat( 3 )ispositive.Todeterminethislargestvalue,wesolvethefollowingknapsackproblem: maxXi2Si0zi (3a) s.t.Xi2Si0zigPF(f0))]TJ /F14 11.955 Tf 11.96 9.69 Td[()]TJ /F4 11.955 Tf 6.43 -9.69 Td[(000++ (3b) z2f0,1gN, (3c) whereissomearbitrarilysmallvaluewhosepurposeistoensurethatthez-solutionto( 3 )ensuresthat( 3 )issatisedasastrictinequality.Lettingdenotetheoptimalobjectivefunctionvalueof( 3 ),where=if( 3 )isinfeasible,wesetM 02=gDL(0))]TJ /F3 11.955 Tf 11.96 0 Td[(CL=minn1,gPF(f0))]TJ /F14 11.955 Tf 11.96 9.68 Td[()]TJ /F4 11.955 Tf 6.42 -9.68 Td[(000++o.2 3.4.4Cutting-PlaneAlgorithm:M-FreeModelTheforegoingcutting-planealgorithmaddedthreestructuralconstraints(( 3b )( 3d )),plusonecontinuousandonebinaryvariable,totherelaxedmasterproblemaftereachsubproblemsolution.WealternativelyobtainaformulationofBLG-RMPthatobviatestheneedfortheseextravariables.WithM2takentobelarge,theright-hand-sideof( 3b )isredundantwhen gPF(f0)+M1 Xi2Sziy0i,1!>gDF(0,0)+=000+Xi2Szii0+.(3) 64

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First,notethattheleft-hand-sideof( 3 )isnotmorethantheright-hand-sideof( 3 )whenz=z0,i.e.,atsomez-vectortowhichthesolution(0,0,0,f0,y0)correspondsin .Hence,anecessaryconditiontosatisfytheinequalityin( 3 )isthatzsatisesatleastoneoftwoproperties:(a)zi=1foranyi2Ssuchthaty0i,1=1,or(b)zi=0foranyi2Ssuchthatz0ii0>0.Intuitively,theleadercanpotentiallyalterthefollower'ssolutionbyprotectingasupplierfromwhichthefollowerpurchasesgoodsinf,orunprotectingsomesupplierfromwhomthefollowermaywishtopurchasegoods.Basedoncase(b),deneS0=fi2Sjz0ii0>0g.Wealternativelyformulate( 3 )asfollows: BLG-MP2:minrTz+w (3a) s.t.wgDL(0))]TJ /F14 11.955 Tf 11.96 13.27 Td[(gDL(0))]TJ /F3 11.955 Tf 11.95 0 Td[(CL Xi2Sziy0i,1+Xi2S0(1)]TJ /F3 11.955 Tf 11.95 0 Td[(zi)!,8(0,0,0,f0,y0)2 (3b) w0, (3c) z2f0,1gN. (3d) Therestrictedversionofthisproblem,BLG-RMP2,isdenedonlywithrespecttovectorsin^ ,asdoneinSection 3.4.3 .Thecutting-planeapproachdescribedinAlgorithm 3 canbeemployeddirectlyforsolving( 3 ),exceptthat(a)therestrictedmasterproblemformulationusedinthealgorithmbecomesBLG-RMP2(insteadofBLG-RMP),and(b)theadditionofavector 0to^entailstheadditionofonlyasingleconstraintoftheform( 3b )andnoadditionalvariables,asopposedtotheadditionofthreeconstraintsandtwonewvariablesrequiredwhen 0isaddedto^inSection 3.4.3 .Asanalnotetothissection,examinethevalidinequalitiesoftheform( 3b )(whichareequivalentlycapturedinBLG-MP)and( 3b ).Notethatforanyintegervector^z,theright-hand-sideof( 3b )isnotmorethanCLifany^zi=1foranyi2Ssuchthaty0i,1=1,orifany^zi=0suchthati2S0.Similarly,theright-hand-sideof( 3b )isalsonotmorethanCLwhen^zi=1foranyi2Ssuchthaty0i,1=1.However, 65

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when^zi=0forsomei2S0,theright-hand-sideof( 3b )mayexceedCL.Considerforinstanceasolution^zinwhich^zi=0foreveryi2Ssuchthaty0i,1=1.Notethatforthesolutionz0thatinducedthisinequality,theright-hand-sideof( 3b )isgDL(0).IfgPF(f0))]TJ /F14 11.955 Tf 12.36 9.69 Td[()]TJ /F4 11.955 Tf 6.43 -9.69 Td[(000+Pi2S^zii0)]TJ /F5 11.955 Tf 12.37 0 Td[(0(i.e.,theM2coefcientofthisinequalityremains0inthesolution^z),thentheright-hand-sideof( 3b )stillevaluatestogDL(0).Hence,intheintegersense(i.e.,consideringonlythosesolutionsinwhichztakesonintegervalues),inequality( 3b )isatleastasstrongasthatgivenby( 3b ).However,inthecontinuoussense(whichisofimportanceinefcientlysolvingrestrictedmasterprobleminstanceswithinthecutting-planealgorithm),thetwoinequalitiesdonotdominateoneanother.Wehenceexaminetheefcacyofusingbothcutting-planealgorithmsinourcomputationalexperiments. 3.4.5AlgorithmEnhancementsInthissectionwediscussthreeenhancementstoourcutting-planemethods.TherstandthirdenhancementsapplytobothBLG-MPandBLG-MP2,whilethesecondappliesonlytoBLG-MP.Fortherstenhancement,aftersolvingBLG-SPtoobtain 0,wechecktoseeif000+gPF(f0).Ifso,then( 3 )isinfeasibleandthemaxfunctionin( 3b )ispositiveifandonlyifzi=1forsomei2Ssuchthaty0i,1=1.Therefore,insteadofaddingInequalities( 3b )( 3e )toBLG-RMPor( 3b )toBLG-RMP2,weadd wgDL(0))]TJ /F14 11.955 Tf 11.95 13.27 Td[(gDL(0))]TJ /F3 11.955 Tf 11.95 0 Td[(CLXi2Sziy0i,1.(3)ForBLG-MP,thiseliminatestwovariablesandtheneedtosolve( 3 ).Also,forbothBLG-MPandBLG-MP2,( 3 )providesastrongervalidinequalitythantheinequalitiesproposedforBLG-MPorBLG-MP2becausetheright-hand-sideof( 3 )isunaffectedwhen^zi=1forsomei2^S.Forthesecondenhancement,aftersolvingBLG-SPtoobtain 0,wechecktoseeifmini2S0i0>000+Pi2Sz0ii0+)]TJ /F3 11.955 Tf 12.73 0 Td[(gPF(f0).Ifso,unprotectinganysupplieri2S0 66

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automaticallymakes( 3b )redundant(i.e.,theright-hand-sideof( 3b )becomesnomorethanCL).Since( 3b )isalsoredundantwhenprotectingsomesupplieri2Ssuchthaty0i,1=1,insteadofsolving( 3 )andaddingtwoadditionalvariables,wereplaceInequalities( 3b )( 3e )withInequality( 3b ). Remark9.Whenapplyingtheseenhancementstothepessimisticmodel(=0),thereisnodifferencebetweenusingBLG-MPorBLG-MP2asthemasterproblem.GivenasolutiontoBLG-SP,wehave000+Pi2Sz0ii0=gPF(f0).IfPi2Sz0ii0=0,thentherstenhancementcanbeapplied.Otherwise,S06=;whichimpliesmini2S0i0>000+Pi2Sz0ii0)]TJ /F3 11.955 Tf 11.96 0 Td[(gPF(f0)=0.2ThethirdenhancementdealswiththeimplementationofbothBLG-MPandBLG-MP2.NotethatinBLG-SP(( 3 )),decreasingthevalueofrestrictstheproblem.Therefore,whensolvingthegeneralizedbilevelmodelforagiven,wecanobtainvalidinequalitiestothemasterproblembysolvingBLG-SPwithany^.Therefore,inourimplementationofthegeneralizedbilevelmodelforsome>0,werstsolvethepessimisticmodeltooptimality(i.e.,with=0),retainingallvalidinequalitiesbeforecontinuingtosolvethemodelwith>0.Theadvantageofthisstrategyisthatmodelshaving=0tendtobeeasiertosolvethanthosehaving>0;hence,thisstrategyallowsustocomputecuttingplanesforBLG-MP(orBLG-MP2)usinglesscomputationaleffort. 3.5ComputationalResultsToexaminetheefciencyofouralgorithms,werancomputationaltestsonthesetofinstancesfromChapter 2 .ForN=5,6,7,vesetsofsuppliercapacitiesandfoursetsofleaderandfollowerdemandswereusedtocreateatotalof354=60problemproles.Table 3-1 presentsthesetsofsuppliercapacitiesandTable 3-2 providestheleaderandfollowerdemandsasapercentageoftotalsuppliercapacity(withfractionalcapacitiesroundeddown).WerefertoarowfromTable 3-2 asademandlevel. 67

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Table3-1. Suppliercapacitysets CapacitySet# u1 u2 u3 u4 u5 u6 u7 u8 u9 u10 1 25 25 25 25 25 25 25 25 25 25 2 10 25 15 20 40 45 35 20 50 20 3 25 24 22 23 26 26 27 21 49 25 4 12 17 20 21 25 23 25 24 16 19 5 30 47 50 45 36 46 27 32 43 36 Table3-2. Buyerdemandasapercentageoftotalsuppliercapacity Leader Follower 20% 20% 40% 40% 20% 70% 70% 20% Foreachofthe60proles,teninstancesofsuppliercostfunctionswererandomlygenerated.Eachfunctionispiecewiselinearandconcavewithvesegments.Theunitcostoneachsegmentbelongstotheinterval[1,10],andforeachsupplieri2S,theprotectioncostwassetequaltothreetimesthesupplier'scheapestunit(i.e,ri=3(pi(ui))]TJ /F3 11.955 Tf 12.46 0 Td[(pi(ui)]TJ /F4 11.955 Tf 12.46 0 Td[(1))).Weusedprotectioncostsofthismagnitudebecausetheyresultedinsolutionsinwhichitisoptimaltoprotectsome,butnotall,ofthesuppliers.Wesolvedtheoptimisticandpessimisticmodelsforeachoftheproblemproles,andreporttheaveragecomputationtimeinseconds(CPU),andnumberofiterations(forthepessimisticmodel)inTable B-3 ofAppendixB.ThistablealsoprovidestheaverageCPUtimeandnumberofiterationsfortheworst-casemodelfromChapter 2 (usingStrategy2),inwhichthefolloweressentiallysets=1tosimplymaximizetheleader'sminimumprocurementcosts.ForN=8,9,and10,thesamevecapacitysetswereused,butinsteadofvaryingthedemandlevels,theprotectioncostswerevariedtocreate60moreproles.Boththeleaderandfollowerdemandswerexedto20%ofthetotalcapacity,andtheprotectioncostsweresettori=xi(pi(ui))]TJ /F3 11.955 Tf 12.39 0 Td[(pi(ui)]TJ /F4 11.955 Tf 12.39 0 Td[(1)),8i2S,wherexiisanintegerparameterusedtocalculatetheprotectioncostforsupplieri.Givenacapacityset,fourproleswerecreatedbyvaryingxi.Therstthreeprolessetallxito1,3,or5,andthelast 68

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prolerandomlyselectedeachxi-valuefromthesetofintegersf1,...,5g.Forthese60proles,resultsfortheaverage(overteninstances)CPUtimeandnumberofiterationsareshowninTable B-4 ofAppendixB.Ofthe1200instances,456(or38%)haddifferent(optimal)objectivefunctionvaluesfortheoptimisticandworst-casemodels.Ofthese456instances,onlysixhaddifferentobjectivefunctionvaluesfortheoptimisticandpessimisticmodels,and454haddifferentobjectivefunctionvaluesforthepessimisticandworst-casemodels.Thisimpliesthatthefollower'scostminimizationstrategyisalmostalwaysunique.Forthe456instanceswithdifferentobjectivefunctionvaluesfortheoptimisticandworst-casemodels,thelargestpercentagedifferenceintheobjectivefunctionvaluewas7.34%andtheaveragepercentagedifferenceintheobjectivefunctionvaluewas1.43%.Fromthisresultweconcludethatthefollower'scostminimizationprocurementstrategyisoftenverysimilar(andover60%ofthetimeactuallyequivalent)toastrategythatmaximizestheleader'sobjectivefunction.TheoptimisticmodelhadthefastestaverageCPUtimeon73ofthe120proles,thepessimisticmodelhadthefastestaverageCPUtimeonjusttwoofthe120prolesandtheworst-casemodelhadthefastestaverageCPUtimeon42ofthe120proles(theaverageCPUtimesfortheremainingthreeinstancesresultedinatiebetweentwoofthethreemodels).Thepessimisticmodelrequiredfewercutsonaveragethantheworst-casemodelon102ofthe120proles.Thisisduetothespecializedalgorithmdevelopedfortheworst-casemodelinwhichalargenumberofcutsareobtainedbysolvingsmallersubproblemsthatareeasiertosolve.WenowanalyzethecomputationaleffortrequiredbytheBLG-MPandBLG-MP2implementationstosolvethegeneralizedbilevelproblem.Table B-5 showstheaverage(overteninstances)CPUtimesandnumberofiterationsrequiredforeachoftherst60proles(withN=5,6,7)usingeachoftheseimplementations,giventwodifferentvaluesof.Thevaluesofwerecalculatedbymultiplying(E)]TJ /F3 11.955 Tf 12.73 0 Td[(C)by0.25and0.5. 69

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Ofthese600instancesusedinthiscomparison,234hadadifferentobjectivefunctionvaluefortheoptimisticandworst-casemodels,andonlyoneinstancehadadifferentobjectivefunctionvaluefortheoptimisticandpessimisticmodels.Also,232instanceshadadifferentvalueforthepessimisticmodelandgeneralizedbilevelmodelwith=0.25(E)]TJ /F3 11.955 Tf 13.09 0 Td[(C),vehadadifferentvalueforthegeneralizedbilevelmodelwith=0.25(E)]TJ /F3 11.955 Tf 11.97 0 Td[(C)and=0.50(E)]TJ /F3 11.955 Tf 11.97 0 Td[(C),andnonehadadifferentvalueforthegeneralizedbilevelmodelwith=0.5(E)]TJ /F3 11.955 Tf 12.9 0 Td[(C)andtheworst-casemodel.Theseresultsagainimplythatwhentryingtomaximizetheleader'sprocurementcosts,theleaderdoesnotgenerallyneedtosignicantlyincreaseitsownprocurementcosts.Asexpected,solvingthegeneralizedbilevelmodels(witheitherpositive-value)requiresmoreCPUtimeanditerationsthansolvingthepessimisticmodel.InTable 3-3 weprovide,forboththeBLG-MPandBLG-MP2implementationsandbothvaluesof,theaverageabsoluteandpercentageincreaseinCPUtimeandthenumberofiterationsrequiredtosolvethegeneralizedbilevelmodeloverthepessimisticmodel.WhencomparingtheBLG-MPandBLG-MP2implementations,weseethattheBLG-MPimplementationperformsslightlybetter.Thisisduetothefactthatinbothimplementations,themajorityofvalidinequalitiesareaddedbysolvingthesubproblemofthepessimisticmodel(seethethirdenhancementofSection 3.4.5 ),afterwhichonlyroughly50%morecutsareneededfromthesubproblemwith>0.Becauseofthisbehavior,themasterproblemforBLG-MP,inwhichextravariablesmustbeadded,doesnotbecomesignicantlymoredifculttosolvethanthemasterproblemforBLG-MP2. Table3-3. AverageincreaseinCPUtimeandnumberofiterationswhen>0 BLG-MP BLG-MP2 =0.25(E)]TJ /F6 7.97 Tf 8.47 0 Td[(C) =0.5(E)]TJ /F6 7.97 Tf 8.47 0 Td[(C) =0.25(E)]TJ /F6 7.97 Tf 8.46 0 Td[(C) =0.5(E)]TJ /F6 7.97 Tf 8.47 0 Td[(C) Absolute Percentage Absolute Percentage Absolute Percentage Absolute Percentage CPUTime(seconds) 4.8 80.53% 4.7 84.25% 5.2 84.98% 5.1 86.57% #ofIterations 3.2 48.17% 3.2 48.01% 3.3 48.44% 3.3 48.37% 70

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CHAPTER4CAPACITATEDFACILITYINTERDICTIONPROBLEMSWITHFORTIFICATIONUNDERMEDIANANDCENTEROBJECTIVEFUNCTIONS 4.1MotivationandLiteratureSurveyArecentslateofpapershasmadesignicantprogressinthestudyoffacilityinterdictionoptimization.ThesepapersexamineasetFoffacilitiesandasetNofcustomersordemandpoints,whichwesimplycallnodes.Betweeneachpairofnodesi2Nandfacilitiesj2Fisapositivedistancedij.Withrespecttothemedianobjective,adefenderminimizesthesumofnode-to-facilityassignmentdistances.Theinterdictionmedian(IM)problemseekstoremoveasetoffacilitiestomaximizetheminimumobjectivethatcanbeattainedbythedefender.Expandingthissituationfurther,theinterdictionmedianproblemwithfortication(IMF)allowsthedefendertoactrstbyfortifyingasubsetoffacilities;theattacker(alsoreferredtoasaninterdictor,oradversary)thensolvestheIMproblem,butisrestrictedtoremoveonlythosefacilitiesthatarenotfortiedbythedefender.Thischapterexpandsthescopeoffacilityinterdictionandforticationproblemsthatcanbesolvedusingexactmathematicalprogrammingalgorithms.Incontrasttopreviousstudies,wealsoconsidertheforegoingproblemsunderthecenterobjective,whichseekstominimizethelargestdistanceassignment.WerefertotheinterdictionproblemwiththecenterobjectiveasIC,andtotheICproblemwithforticationasICF.Furthermore,weproposemethodsinthischaptertoaccommodatethesituationinwhichfacilitiesareconstrainedonthenumberofdemandnodesthatcanbeassignedtothem(andinwhichnodescannotbepartiallyassignedtothefacilities).Asaresultofthecapacitylimitsplacedonfacilities,thedefendermayhavenofeasiblewaytoassignallnodestofacilitiesifthenumberofnodesexceedsthetotalcapacityofthefacilitiesremainingaftertheattacker'saction.Hence,anadditionalchallengefacedbythedefenderistoprotectasetoffacilitiessuchthatthereexistsafeasiblenode-to-facilityassignmentsolutionregardlessoftheattacker'sactions. 71

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Thep-medianandk-centerproblemsaretraditionalcombinatorialoptimizationproblems,whichseeklocationsforasetoffacilitiesrelativetoasetofnodes.Inthep-median(k-center)problem,thegoalistolocatep(k)facilitiestominimizethemedian(center)objective.Inbothcases,thesetofpotentialfacilitylocationsisnite,whichinducesthecombinatorialnatureoftheproblems.Agreatamountofliteratureexistsregardingthesetwoclassicalproblems,datingbacktotheclassicalworkofHakimi[ 32 ].Forareviewofliteratureregardingtheseandotherrelatedfacilitylocationproblemssee[ 24 28 38 40 42 46 ].Theproblemsanalyzedinthischapteraremodeledasthree-stagegames.Oursolutionapproachtransformsthesegamesintobilevelprogrammingformulations,whichwesolvethroughacutting-planealgorithm(see[ 50 51 53 58 ]forrelatedalgorithms).Alargeamountofnetworkinterdictionandinterdictionwithforticationresearchhasbeencontributedwithapplicationstofacilitylocationandsupplychains,specicallyonthep-medianproblem.Churchetal.[ 22 ]introduceIMandformulateitasasingle-stageintegerprogram.ChurchandScaparra[ 23 ]builduponthisresearchbyconsideringaforticationstageinwhichthedefendercanfortifyuptoqfacilities.Undertheassumptionthatthesetofpossibleinterdictionscenariosisrelativelysmall,theauthorsformulateIMFasasingle-stagelinearmixed-integerprogram.Next,in[ 49 ],theauthorsremovetherestrictiononthenumberofinterdictionscenariosandformulateIMFasabilevelmixed-integerprogram,forwhichtheydevelopanimplicitenumeration(IE)solutionmethod.Thevariableconsolidationtechniquesdevelopedin[ 21 ]helptosubstantiallyreducethesizeofthemodelsin[ 23 49 ].In[ 47 ],theauthorsimproveupontheiroriginalsingle-stageformulationforIMFbyreformulatingitasamaximalcoveringproblem.Eachofthesemodelsassumesthatfacilitieshaveinnitecapacity,whichmeansthatinthethirdstage,eachnodeisassignedtoitsclosestuninterdictedfacility.TowardthestudyofthecapacitatedIMF,ScaparraandChurch[ 48 ]allowdemandfromnodestobefractionallyassignedtofacilities,andtheunmetdemandispenalized 72

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inthethird-stageobjective.Aksenetal.[ 4 ]introducetwofeaturestoIMFproblems.First,theauthorsgeneralizetheconstraintintheforticationstageasageneralknapsackconstraint.Second,theyplacethenode-to-facilityassignmentdecisionsintherststage,andthenintroduceapenaltycostifthenodesmustbereassignedafterinterdictionhasoccurred.Theyusetheimplicitenumerationalgorithmdevelopedin[ 23 ]foritssolution.Aksenetal.[ 3 ]introducerst-stagedecisionsonwherethefacilitiesshouldbelocatedanddevelopanexhaustivesearchmethodandtwoheuristicsforsolvingthemodel.StochasticityisintroducedtoIMproblemsin[ 37 ]andtoIMFproblemsin[ 36 ].Inthelatter,theauthorsassumethatthenumberofpossibleinterdictionscenariosisrelativelysmall,enablingtheuseofthemaximalcoveringproblemformulationmentionedearlier.Thecontributionswemakeinthischapterareasfollows.One,weshowthataninner-dualizationprocedurethatiscommontonetworkinterdictionstudiesiscapableofproducingthedirectmodelsusedforsolvingIM,andcanbeextendedtomodelallfourproblemsthatweexamineinthischapter.Assuch,weconsiderbroaderclassesoffacilityforticationandinterdictionthanwhathasbeenexploredintheliteraturetodate.Inparticular,tothebestofourknowledge,ICFhasnotbeenexaminedthusfar,norhascaseinwhichthethird-stageproblemrequiresbinarynode-to-capacitated-facilityassignmentsundereitherobjective.Two,thischapterprovidesarobust-optimizationapproachforensuringthatthedefenderfortiesasetoffacilitiessuchthatafeasiblethird-stagesolutionmustexistregardlessoftheattacker'sactions.Three,asopposedtotheimplicitenumerationstrategymentionedpreviously,thealgorithmusedinourforticationproblemisbasedoncuttingplanesthatprovidelowerboundsonthedefender'sobjectivefunction.Theremainderofthischapterisorganizedasfollows.InSection 4.2 weprovideaformaldescriptionofthedifferentmodels.InSection 4.3 wegivebilevelreformulationsofthemodels,andinSection 4.4 weprescribeacutting-planemethodologyusedto 73

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solvethebilevelprograms.Section 4.5 providescomputationalresultsfromapplyingouralgorithmsoninstancesfromtwodifferentdatasets. 4.2ProblemDescriptions,Notation,andModelsWebeginbypresentingageneralthree-stagemodelthatencompassestheproblemsdiscussedinthischapter.Wedenethreesetsofbinarydecisionvariables,z,s,andx,wherezj=1ifandonlyif(iff)facilityj2Fisfortied,sj=1ifffacilityj2Fisinterdicted,andxij=1iffnodei2Nisassignedtofacilityj2F.Thesevariablesareconstrainedtobelongtofeasibilitysetsdescribedasfollows. Intherststage,thedefendermakesitsforticationdecision,z,fromaset,Z,whichwouldtypicallybeabudget-constrainedsetalongwithbinaryrestrictionsonthez-variables.However,ouranalysisreadilyextendstothecaseinwhichZisageneral0/1-linear-programmingfeasibleregion. Basedonthedefender'srst-stagedecision,theattackerchoosesanactions2S(z),whichisgivenbytheintersectionofasetSalongwiththerestrictionthatj2Fcannotbeattackedifjwasfortiedbythedefender(i.e.,sj(1)]TJ /F3 11.955 Tf 12.14 0 Td[(zj),forallj2F).ThesetSenforcesbinarinessofthes-variablesalongwithageneralsetoflinearconstraints,whichforinstancemaylimitthenumberoffacilitiesthatcanberemoved.Oneassumptionthatwemakeisthatifs02S,thens002Saswellifs00s0.Thatis,giventhats0isafeasibleinterdictionsolution(withoutconsideringz),thentheattackercanremoveattacksandretainfeasibility.Also,02S(z)and1=2S,i.e.,theattackerisalwaysallowedtomakenoattacks,andisneverallowedtoattackeveryfacility. Thedefenderthenminimizesitsobjective,(x),bychoosingsomex2X(s),whereX(s)ensuresthateachnodeisassignedtoanuninterdictedfacility,andthatthex-variablesarebinary-valued.Alternatively,wecanaddtoX(s)therestrictionthateachfacilityislimitedbya(facility-specic)numberofnodesthatcanbeassignedtothefacility;thesecardinality-constrainedversionsofproblemsIMFandICFarecalledIMF-CandICF-C,respectively.ForproblemsIMF-CandICF-C,letmjdenotethemaximumnumberofnodesthatcanbeassignedtofacilityj2F.Giventhisnotation,thegeneralmodelforourforticationproblemsisgivenas: minz2Zmaxs2S(z)minx2X(s)(x).(4) 74

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Remark10.PreviouspapersdealingwithIMandIMFincludenodeweights,ai,8i2N.Ouranalysisomitstheseweightswithoutlossofgenerality:lettingdij=ai~dij(where~dijaretheoriginaldistances)resultsinanequivalentformulation.2TheformulationforIMF-Cisasfollows: minz2Zmaxs2S(z)minXi2NXj2Fdijxij (4a) s.t.Xj2Fxij=18i2N, (4b) xij1)]TJ /F3 11.955 Tf 11.96 0 Td[(sj8i2N,j2F, (4c) Xi2Nxijmj8j2F, (4d) x0, (4e) whereObjective( 4a )minimizesthetotalcostofnodeassignments,Constraints( 4b )ensurethateachnodeisassignedtoafacility,Constraints( 4c )prohibittheassignmentofnodestointerdictedfacilities,and( 4d )restrictthenumberofnodesthatcanbeassignedtoeachfacility.(WhenConstraints( 4d )arenotpresent,modelIMFresults.)Notethatitisunnecessarytorestrictthex-variablestobebinary-valued,becausetheconstraintcoefcientmatrixresultingfromConstraints( 4b )( 4e )istotallyunimodular,withallright-handsidevaluesbeingintegers(becausesisabinaryvector).Thus,thereexistsanoptimalintegersolutiontothelinearrelaxationofthethird-stageproblem.Similarly,wemodelICF-Casfollows: minz2Zmaxs2S(z)minW (4a) s.t.Xj2Fxij=18i2N, (4b) WXj2Fdijxij8i2N, (4c) xij1)]TJ /F3 11.955 Tf 11.96 0 Td[(sj8i2N,j2F, (4d) 75

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Xi2Nxijmj8j2F, (4e) xij2f0,1g8i2N,j2F, (4f) whereWisanauxiliaryvariablethat,atoptimality,takesthevalueofthemaximumdistanceassignment.Constraints( 4c )enforcetheconditionthatWisatleastaslargeasalldistancescorrespondingtothenode-to-facilityassignments,whilethedefender'sobjectiveofminimizingWensuresthatWequalsthelargestsuchdistance.Theremainingconstraintscorrespondtothosein( 4 ).(WhenConstraints( 4e )arenotpresent,modelICFresults.)Notethatbinaryrestrictionsonxarenecessaryinthiscase,aswediscussinSection 4.3 Remark11.ForthecaseinwhichSconsistsonlyofacardinalityconstraintinadditiontobinaryrestrictionsonthes-variables(i.e.,S=fs2f0,1gjFjjPj2Fsjrg),ICFcanbeformulatedasasingle-stagelinearmixed-integerprogram(MIP).Letlidenotethedistancefromnodei2Ntoitslthclosestfacilityandleti(z)denotethedistancefromnodeitoitsclosestfortiedfacility,givenz.Inanyoptimalsolution,nodeiwillbeassignedtoafacilitynofartherthanminfr+1i,i(z)gunits.Therefore,givenavectorz,theattacker'soptimizationproblemissimple:selectanodeihavingthelargestvalueofminfr+1i,i(z)gandinterdicttheclosestunfortiedfacilitiestoi.Sincethisdecisionisimpliedoncezisselected,thedefenderdeterminestheattacker'sdecisionbyitschoiceofz.LetFli=fj2Fjdij
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facilityj2Fliisfortied,thentheright-handsideof( 4b )isnomorethandijwherej2argminj2Flifdijjzj=1g,andhence( 4b )remainsvalid.2 4.3BilevelModelsInthissectionwereformulateeachoftheforegoingproblemsasbilevelprograms.InSection 4.3.1 ,wepresentabilevelprogrammingreformulationforIMF-C,andconsiderthespecialuncapacitatedcase(IMF)inSection 4.3.2 .WethenpresentabilevelprogrammingreformulationofICF-CinSection 4.3.3 ,andabilevelreformulationforthespecialcaseICFinSection 4.3.4 4.3.1IMF-CBilevelModelLeti,)]TJ /F5 11.955 Tf 9.3 0 Td[(ij,and)]TJ /F5 11.955 Tf 9.3 0 Td[(jdenotethedualvariablescorrespondingtoConstraints( 4b ),( 4c ),and( 4d ),respectively.BydualizingthethirdstageoftheIMF-CFormulation( 4 )andcombiningitwiththesecondstage,weobtainthefollowingbilevelformulation: minz2ZmaxXi2Ni)]TJ /F14 11.955 Tf 11.95 11.35 Td[(Xi2NXj2F(1)]TJ /F3 11.955 Tf 11.95 0 Td[(sj)ij)]TJ /F14 11.955 Tf 11.95 11.35 Td[(Xj2Fmjj (4a) s.t.i)]TJ /F5 11.955 Tf 11.95 0 Td[(ij)]TJ /F5 11.955 Tf 11.95 0 Td[(jdij8i2N,j2F, (4b) ,0, (4c) s2S(z). (4d) Observethatifzischosensuchthattheattackercanmakethethird-stageprobleminfeasible,thenthethird-stagedualwillbeunbounded(notingthatanoptimalsolutionalwaysexiststothisdualbysetting===0).Therefore,theattacker'ssecond-stageproblemmustbeunboundedwhensischosensothatthethird-stageproblemisinfeasible.WenowsimplifyModel( 4 )byconsideringthefollowingproposition: Proposition4.1. Ifthereexistsanoptimalsolutiontothesecond-stage(maximization)problemof( 4 ),thenthereexistsanoptimalsolutionsuchthatij=(jFji)]TJ /F3 11.955 Tf 12.11 0 Td[(dij)sj,8i2N,j2F. 77

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Proof. Givenabinaryvectorz,let(^s,^,^,^)beanoptimalsolutionto( 4 ).Werstestablishthreeclaims.One,fori2Nandj2F,anoptimalsolutionexistssuchthatifsj=0thenij=0.Supposethatforsomei2Nandj2F,wehavethat^sj=0and^ij>0.Consideranalternativesolutioninwhichthevaluefori=^i)]TJ /F4 11.955 Tf 13.62 2.66 Td[(^ij,ij=0,andallothervariablevaluesremainthesame.Theobjectivefunctionvaluedoesnotchange,theleft-hand-sideofeachconstraintin( 4b )isnonincreasing,andall-and-valuesremainnonnegative.Hence,analternativeoptimalsolutionexistswith^sj=^ij=0.Two,foranyj2Fsuchthatsj=1,wehavethatj=0inanyoptimalsolution.Bycontradiction,if^sj=1and^j>0atoptimality,thenanimprovedsolutionwouldexistinwhichij-valuesareincreasedby^jforalli2N,andjissetto0:Allconstraintsremainsatisedandtheobjectiveincreasesbymj^j.Three,thereexistsanoptimalsolutioninwhichj=0forsomej2F:sj=0.Toseethis,dene=minj2F:sj=0^jandsupposethat>0.Analternativesolutionwouldexistinwhichj=^j)]TJ /F4 11.955 Tf 12.53 0 Td[(forallj2F:sj=0,andi=^i)]TJ /F4 11.955 Tf 12.53 0 Td[(foralli2N,withallothervaluesremainingthesame.Theleft-hand-sideofallConstraints( 4b )isnonincreasing,and( 4c )issatisedbecauseall-variablesremainnonnegative.Theobjectivechangeis(Pj2F:sj=0(mj))-313(jNj).However,byassumptionthatthesecond-stageproblemhasanoptimalsolution(andsothethird-stageproblemisfeasible),wemusthavethatPj2F:sj=0mjjNj,andsotheobjectivedoesnotdecrease.Hence,analternativeoptimalsolutionmustexistinwhichj=0forsomej2F:sj=0.Giventhesethreeclaims,letkbeanyfacilitysuchthatsk=k=0.Constraint( 4b )fori2Nandkimpliesthatidik(bytherstclaim).Nowconsideranyj2F:^sj=1(andthus^j=0bythesecondclaim).Constraint( 4b )impliesthatiji)]TJ /F3 11.955 Tf 12.16 0 Td[(dij.BecauseidikjFji,settingij=jFji)]TJ /F3 11.955 Tf 12.16 0 Td[(dijsatises( 4b )and( 4c )withoutaffectingtheobjectivefunctionvalue.Thus,thepropositionholdswhensj=1.Also,therstclaimestablishesthatthepropositionholdswhensj=0,whichcompletestheproof. 78

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Proposition 4.1 permitsustosimplify( 4 )asfollows,notingthattheobjectivetermPi2NPj2F(1)]TJ /F3 11.955 Tf 11.95 0 Td[(sj)ijbecomeszeroafterreplacingijasprescribedbytheproposition: minz2ZmaxXi2Ni)]TJ /F14 11.955 Tf 11.95 11.36 Td[(Xj2Fmjj (4a) s.t.i)]TJ /F5 11.955 Tf 11.95 0 Td[(jdij+(jFji)]TJ /F3 11.955 Tf 11.96 0 Td[(dij)sj8i2N,j2F, (4b) 0, (4c) s2S(z). (4d) 4.3.2IMFBilevelModelsInthespecialcaseofanuncapacitatedmodel,denerasthemaximumnumberoffacilitiesthatcouldbeinterdictedbytheadversary.(ThisvaluecanbedeterminedbysimplymaximizingPj2Fsjovers2Sinapreliminarystep.)Notethatanyvariablexijsuchthatdij>r+1icanberemovedfrommodelIMFbecause,atworst,nodeiwillbeassignedtoits(r+1)stfarthestfacility.Foreaseofnotation,letFi=fj2Fjdijr+1igdenotethesetoffacilitiestowhichnodeicouldpossiblybeassignedinanoptimalsolution.Theanalysisofthismodelfollows( 4 )forIMF-CwiththemodicationthatFisreplacedwithFiinthemodel,andtheabsenceof( 4d )impliesthatall-variablesequalzero.ThemodelcanbefurthersimpliedasprescribedinProposition 4.2 Proposition4.2. ConsiderthebilevelformulationofIMF,givenby( 4 )with=0andFreplacedbyFiin( 4a )and( 4b ).Thereexistsanoptimalsolutiontothesecond-stageprobleminwhichij=(r+1i)]TJ /F3 11.955 Tf 11.95 0 Td[(dij)sj,8i2N,j2Fi. Proof. TheproofisanalogoustotheproofofProposition 4.1 ,withtheexceptionofthefactthatall-valuesequalzerointhismodel,andtheassumptionthatanoptimalsolutiontothesecond-stageproblemexists(becausethethird-stageproblemin( 4 )musthaveafeasiblesolutionregardlessofsintheunconstrained-facilitycase).For 79

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eachi2N,thereexistsak2Fisuchthatsk=0,andsoidik.Notingthatdikr+1i,theremainingdetailsoftheprooffollowidenticallyfromthatofProposition 4.1 UsingtheresultsofProposition 4.2 ,thebilevelformulationofIMFbecomes: minz2ZmaxXi2Ni (4a) s.t.idij+(r+1i)]TJ /F3 11.955 Tf 11.95 0 Td[(dij)sj8i2N,j2Fi, (4b) s2S(z). (4c) Remark12.Observethat( 4a )and( 4b )canbeequivalentlyreplacedwiththefollowingobjective:minz2Zmaxs2S(z)Xi2Nminj2Fidij+(r+1i)]TJ /F3 11.955 Tf 11.96 0 Td[(dij)sj.Onewaytoremovetheinnerminfunctionistoutilizeasetofvariables~xij,whichtakeonavalueof1ifj2Fiistheclosestuninterdictedfacilitytonodei2N.Notethat~xandxhavethesameinterpretation,withthesubtledifferencethat~xischosenbytheattackerratherthanthedefender.Theattackermustthenbeconstrainedtoforce~xtocorrespondtothedefender'sactions.Constraintsthataccomplishthisstatethateachnodei2Nmustbeassignedtoexactlyonefacility,andforeachi2Nandj2Fi,iffacilityjisnotinterdicted,then~xik=0forallfacilitieskthatarefartherfromithanjis.DeningTij=fk2Fijdik>dijgasthesetofallfacilitiesthatarefartherthanjisfromnodei,( 4 )canberestatedas: minz2ZmaxXi2NXj2Fidij~xij (4a) s.t.Xj2Fi~xij=18i2N, (4b) Xk2Tij~xiksj8i2N,j2Fi, (4c) ~x0, (4d) s2S(z). (4e) 80

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Model( 4 )isidenticaltothemodelofChurchetal.[ 23 ]forIMF(thatis,whenS=fPj2Fsjrgconsistsonlyofacardinalityconstraint).2 4.3.3ICF-CBilevelModelOurprioranalysisregardsthird-stageproblemsthatareguaranteedtoyieldintegeroptimalsolutions.However,withoutthebinaryrestrictiononx,thethird-stageproblemformulatedin( 4 )doesnotnecessarilyyieldanintegersolution.Asasimpleexample,consideraproblemwithtwonodesandtwo(uninterdicted)facilities,wherem1=m2=1,d11=d21=10,andd12=d22=20.Anoptimalintegersolutionwouldassignnode1tofacility1andnode2tofacility2,withanobjectiveof20.However,the(unique)optimallinearprogrammingsolutionwouldsetx11=x12=x21=x22=0.5,withanobjectiveof15.Therefore,inordertoemploytheinnerdualizationstrategyutilizedforIMF-C,itisnecessarytoreformulatethethird-stageproblem.Accordingly,letDbethesetofalldistinctdistancesinfdijg(i,j)2NF,letK=f1,...,jDjg,andletDk2DbethekthlargestdistanceinD.Further,letE(k)=f(i,j)2NFjdijDkgbethesetofnode/facilitypairsthathaveadistancenogreaterthanDk.Denethebinaryvariableyk=1iffDkisthemaximumdistancelabelassigned,8k2K,anddenethebinaryvariablewijk=1iffnodei2Nisassignedtofacilityj2FwhenDk,k2K,isthemaximumdistancelabelassigned.AnalternativeformulationforICF-Cisgivenasfollows: minz2Zmaxs2S(z)minXk2KDkyk (4a) s.t.Xj2Fwijk=yk8i2N,k2K, (4b) Xi2Nwijkmjyk8j2F,k2K, (4c) Xk2Kyk=1, (4d) wijk(1)]TJ /F3 11.955 Tf 11.95 0 Td[(sj)8k2K,(i,j)2E(k), (4e) w,y0. (4f) 81

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TheObjective( 4a )minimizesthevalueofthelargestdistanceassignment.Constraints( 4b )statethatallwijk-variablesequalzerowhenyk=0,andwhenyk=1,enforcestheconditionthateachnodei2Nmustbeassignedtosomefacility.Constraints( 4c )statethatthenumberofnodesthatcanbeassignedtofacilityj2Fdoesnotexceedmj.Constraint( 4d )forcesonemaximumdistancevaluetobeselected.Finally,Constraints( 4e )prohibitanodefrombeingassignedtofacilityj2Fwhentheattackerhasremovedfacilityj2F(bysettingsj=1),and( 4f )requiresthatallvariablesarenonnegative.Wexvariableswijk=0,8k2K,(i,j)=2E(k),sincetheycannotbepositivebydenitionofE(k).WenowshowthatunlikeFormulation( 4 ),ifthereexistsafeasiblesolutiontothethirdstageof( 4 ),thenthereexistsanintegeroptimalsolutiontothisproblem. Proposition4.3. Ifthereexistsafeasiblesolutiontothethirdstageof( 4 ),anoptimalsolutionexistsinwhichallvariablesarebinary-valued. Proof. Webeginbyobservingthatifthethirdstageof( 4 )isfeasible,thenanoptimalsolutiontothisproblemexistsbecausetheobjectivefunctioncannotbenegative.Givenafractionalsolution,(w,y),tothethirdstageof( 4 ),weconstructafeasibleintegralsolutionwithatleastassmallanobjectivefunctionvalue.First,bycontradiction,supposethatyisnotanintegervector.Denel=argmink2KfDkjyk>0g,let=1=yl,andconsiderthefollowingalternativesolution:^wijk=8>><>>:wijkifi2N,j2F,k=l0ifi2N,j2F,k2Knflg^yk=8>><>>:ykifk=l0ifk2Knflg. 82

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Theobjectivefunctionvalueforthissolution,Dl,issmallerthantheobjectivefunctionvaluefor(w,y).Constraints( 4b )and( 4c )aresatisedbecausefork=l,bothsidesoftheconstraintsaremultipliedby(>0)andfork6=l,bothsidesequal0.Constraint( 4d )issatisedbecause^yl=1and^yk=0forallk6=l.Constraints( 4e )areclearlysatisedforallk6=l;fork=l,wehavethateach(i,j)2E(l)satises^wijl^yl=1,and^wijl>0iffwijl>0,whichcanonlybetrueifsj=0.Therefore,notingthat^w0and^y0,satisfying( 4f ),wehaveconstructedasolutioninwhich^yisintegralandtheobjectivefunctionvalueissmallerthantheobjectivefunctionvaluefor(w,y).Thiscontradictstheassumptionthat(w,y)wasoptimal,andhencealloptimalsolutionsmusthaveintegery.Now,deneLP(^y)asthethird-stageproblemof( 4 )withyxedto^y.SinceconstraintmatrixofLP(^y)istotallyunimodularanditsright-handsideisintegral,thereexistsanoptimalsolution,w,toLP(^y)inwhichwijk2f0,1g,8k2K,(i,j)2E(k).Thus,wehaveconstructedanintegraloptimalsolution,(w,^y)tothethird-stageproblemof( 4 ),whichcompletestheproof. Letik,)]TJ /F5 11.955 Tf 9.3 0 Td[(jk, ,and)]TJ /F5 11.955 Tf 9.3 0 Td[(ijkdenotethedualvariablescorrespondingtoConstraints( 4b ),( 4c ),( 4d ),and( 4e ),respectively.Bydualizingthethirdstageof( 4 )andcombiningitwiththesecondstage,weobtainthefollowingbilevelproblem: minz2Zmax )]TJ /F14 11.955 Tf 12.01 11.36 Td[(Xk2KX(i,j)2E(k)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(sj)ijk (4a) s.t.)]TJ /F14 11.955 Tf 11.29 11.36 Td[(Xi2Nik+Xj2Fmjjk+ Dk8k2K, (4b) ik)]TJ /F5 11.955 Tf 11.95 0 Td[(jk)]TJ /F5 11.955 Tf 11.96 0 Td[(ijk08k2K,(i,j)2E(k), (4c) ,0, (4d) s2S(z). (4e) 83

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Figure4-1. IllustrationofaugmentationnetworkGkintheproofofProposition 4.4 ,wheredij-valuesarelistedalongsidetheedgesandm1=m2=m3=2. Again,ifsischosensothatthethird-stageproblemof( 4 )isinfeasible,thenthesecond-stageproblemof( 4 )willbeunbounded.WeutilizethefollowingpropositiontosimplifyModel( 4 ). Proposition4.4. Ifthereexistsanoptimalsolutiontothesecond-stageproblemof( 4 ),thenthereexistsanoptimalsolutionsuchthatijk=(DjKj)]TJ /F3 11.955 Tf 13.05 0 Td[(Dk)sj,8k2K,(i,j)2E(k). Proof. Givens,let(w,y)beanoptimalinteger-feasibleprimalsolutiontothethirdstageof( 4 ).Weprovethepropositionbyestablishinganoptimalsolution(,, ,)tothesecond-stageproblemof( 4 ),suchthatthe-variablessatisfytheconditionsoftheproposition.First,fori2N,letj(i)bethefacilitytowhichiisassignedintheobtainedoptimalsolution.Giventhissolution,foreveryk2K,deneadirectedaugmentationnetworkGk,whichhasjNj+jFjvertices,onecorrespondingtoeverynodeinNandeveryfacilityinF.AnarcexistsinGkfromavertexcorrespondingtoi2Ntoavertexcorrespondingtofacilityj2FifdijDk,andforeveryi2N,anarcexistsinGkfromthevertexcorrespondingtofacilityj(i)tothevertexcorrespondingtoi.SeeFigure 4-1 foranillustrationofGk. 84

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Next,weprovidesomenotationanddenitionswithrespecttotheoptimalthird-stagesolutionto( 4 ). Facilityjissaidtobesaturatedifithasmjnodesassignedtoit. ^F=fj2FjPk2KPi2Nwijk=mjgdenotesthesetofsaturatedfacilities. Fork2K,letN(k)=fi2Njdij(i)>DkgdenotethesetofnodesthatareassignedtoafacilityfartherthanDkunits. Fori2N,j2F,andk2K,denebinaryparametersijk,whichequal1iffi2N(k),sj=0,andthereexistsadirectedpathfromitojinGk. Fork2K,let^F(k)=fj2^FjPi2N(k)ijk1g.Theimplicationofthelasttwodenitionsisasfollows.Ifijk=1,thendij(i)=1inthecurrentsolution.Expandingthecapacityoffacilityjbyonewouldallowasolutiontoexistinwhichiisassignedtoafacilityj0suchthatdij0Dk,andforeverynode`2NnN(k),`isstillassignedtosomefacilitythatisnomorethanDkunitsawayfrom`.Toseethis,examineapathfromitojinGk.Foreveryarcinthispathfromavertexcorrespondingtosomeh2Ntoavertexcorrespondingtoafacility`2F,reassignhtofacility`.ByfollowingthepathfromitojinGkandmakingthenode-to-facilityassignmentrevisionsdescribedabove,thenumberofnodesassignedtoj(i)decreasesbyone,thenumberofnodesassignedtojincreasesbyone,andthenumberofnodesassignedtoallotherfacilitiesremainsthesame.Moreover,allverticesinGkcorrespondingtoh2Nonthispath(includingtheonecorrespondingtonodei)arenowassignedtofacilitieswithanassignmentcostofnomorethanDk.Becausedij(i)>Dkoriginally,thismodicationdecreasesthenumberofnode-to-facilityassignmentshavingacostexceedingDk.Thus,inthelastdenition,^F(k)isthesetofsaturatedfacilitiesj,suchthatincreasingmjbyonepermitsareductioninnode-to-facilityassignmentcostsexceedingDk.Weconsideraprimaloptimalsolutionto( 4 )thatmeetstwocriteria. 85

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Criterion1.Foralli2Nandj2Fsuchthatdij><>>:Dk)]TJ /F3 11.955 Tf 11.96 0 Td[(Dkifi2N(k)orj(i)2^F(k)0otherwise8i2N,jk=8>><>>:Dk)]TJ /F3 11.955 Tf 11.95 0 Td[(Dkifj2^F(k)0otherwise8j2F,ijk=8>><>>:DjKj)]TJ /F3 11.955 Tf 11.95 0 Td[(Dkifsj=10otherwise8(i,j)2E(k).Clearly,theobjectivefunctionvalueofthissolutionisequaltotheprimalobjectivefunctionvalueandConstraints( 4d )aresatised.Also,Constraints( 4b )and( 4c )aresatisedforallksuchthatDkDk.Foreachk2KsuchthatDk
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j(i)2^F(k)).Hence,Xi2Nik=Xi2N:j(i)2^F(k)ik+Xi2N:j(i)=2^F(k)ik=0@Xj2^F(k)mj1A(Dk)]TJ /F3 11.955 Tf 11.95 0 Td[(Dk)+Xi2N:j(i)=2^F(k)ik.Wethereforemustestablishthatthereexistsan^2Nsuchthatj(^)=2^F(k)andd^j(^)>Dk(i.e.,^2N(k)),sothat^k=Dk)]TJ /F3 11.955 Tf 11.95 0 Td[(Dk,establishingthat( 4b )issatised.If^F(k)=;,thenanyi2Nsuchthatdij(i)>Dk(knowntoexistbecausetheprimalobjectivefunctionexceedsDk)canserveas^.Otherwise,supposebycontradictionthat^F(k)6=;,butthattheredoesnotexistsuchan^.Considerafacilityj12^F(k),andnotethatthereexistsanodei2N(k)currentlyassignedtoj22Fnfj1gsuchthatij1k=1.Byassumptionthatno^exists,j2mustalsobelongto^F(k).Createanauxiliarynetworkwithnodescorrespondingtofacilitiesin^F(k)andincludeadirectedarcfromj1toj2.Repeatingthisprocess,wecreatearcs(j2,j3),...,(j0,j00)intheauxiliarynetwork,wherefacilitiesj1,...,j0aredistinctandj002fj1,...,j0)]TJ /F4 11.955 Tf 12.28 0 Td[(1g.Hence,thereexistssomedirectedcycle,~j1,...,~jc,where~ji2fj1,...,j0g,8i=1,...,c,amongfacilitiesinthedirectedcycle.SeeFigure 4-2 foranillustration.Now,supposethatforeveryarc(u,v)inthedirectedcycle(correspondingtotheexistenceofsomei2N(k)assignedtovdespitethefactthatiuk=1),werevisethenode-to-facilityassignmentsalongthepathfromnodeitofacilityuinGkasdescribedabove.Thetotalnumberofnodesassignedtoeachfacilityremainsthesame,andhencethisrevisedsolutionremainsfeasible.However,all-valuesarenonincreasinginthisrevisedsolution,andinparticular,ijkdecreases,forsomei2N,from1to0foreveryj2f~j1,...,~jcg.Theresultingreductionincontradictscriterion2above.Hence,an^mustexist,and( 4b )issatisedbysomedualoptimalsolutionasdesired.Wenowshowthat( 4c )issatisedfork2K:Dk
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Figure4-2. IllustrationofauxiliarynetworkintheproofofProposition 4.4 andsoj2^F(k).Thisimpliesthatjk=Dk)]TJ /F3 11.955 Tf 12.97 0 Td[(Dk,whichsatisestheconstraint.Otherwise,jisinterdicted,andsoijk=DjKj)]TJ /F3 11.955 Tf 12.34 0 Td[(Dk,whichsatisestheconstraint.Nowsupposethati=2N(k)butj(i)2^F(k).Thenthereexistsanh2Nwithh,j(i),k=1.Becauseiiscurrentlyassignedtoj(i),anddijDk(since(i,j)2E(k)),apathexistsfromhtojinGk.Thisinturnimpliesthathjk=1,andbycriterion2,jmustbelongto^F(k).Thereforejk=(Dk)]TJ /F3 11.955 Tf 9.89 0 Td[(Dk),whichensuresthat( 4c )issatised.Thiscompletestheproof. Wenowsimplify( 4 )usingtheresultsofProposition 4.4 minz2Zmax (4a) s.t.Xi2N)]TJ /F5 11.955 Tf 9.3 0 Td[(ik+Xj2Fmjjk+ Dk8k2K, (4b) ik)]TJ /F5 11.955 Tf 11.95 0 Td[(jk(DjKj)]TJ /F3 11.955 Tf 11.95 0 Td[(Dk)sj8k2K,(i,j)2E(k), (4c) 0, (4d) s2S(z). (4e) 4.3.4ICFBilevelModelsInthespecialcaseforwhich( 4e )isnotpresent,thethirdstageof( 4 )willindeedhaveanoptimalintegersolutioninwhicheachnodeissimplyassignedtoa 88

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closestuninterdictedfacility.Hence,oneapproachdualizesthethird-stageproblemof( 4 )andcombinesitwithitssecond-stageproblem.Leti,i,)]TJ /F5 11.955 Tf 9.3 0 Td[(ijdenotethedualvariablescorrespondingtoConstraints( 4b ),( 4c ),and( 4d ),respectively.Weobtainthefollowingbilevelprogram: minz2ZmaxXi2Ni)]TJ /F14 11.955 Tf 11.95 11.36 Td[(Xi2NXj2Fi(1)]TJ /F3 11.955 Tf 11.95 0 Td[(sj)ij (4a) s.t.i)]TJ /F3 11.955 Tf 11.95 0 Td[(diji)]TJ /F5 11.955 Tf 11.95 0 Td[(ij08i2N,j2Fi, (4b) Xi2Ni=1, (4c) ,0, (4d) s2S(z). (4e) NotethatFireplacesFinthisformulationbecauseasinIMF,eachnodei2Nwillbeassignedtonoworsethanits(r+1)thfarthestfacility. Proposition4.5. Thereexistsanoptimalsolutiontothesecond-stageproblemof( 4 )inwhichij=(r+1i)]TJ /F3 11.955 Tf 11.95 0 Td[(dij)sj,8i2N,j2Fi. Proof. First,notethatanoptimalsolutionexiststothesecond-stageproblemof( 4 )becausethethird-stageproblemof( 4 )alwayshasanoptimalsolutionintheunconstrained-facilitycase.Consideranoptimalsecond-stagedecision,s,alongwithanoptimalthird-stagesolutionforthedefender.Foreveryi2N,letj(i)bethefacilitytowhichiisassignedinthethird-stageoptimalsolution.Leti2argmaxi2Nfdij(i)g,andseti=dij(i),i=1,andij=(r+1i)]TJ /F3 11.955 Tf 12.85 0 Td[(dij)sj,8j2Fi.Also,fori6=i,seti=0,i=0,andij=(r+1i)]TJ /F3 11.955 Tf 12.58 0 Td[(dij)sj,8j2Fi.ClearlyConstraints( 4c )and( 4d )aresatised,aswellas( 4b )foralli6=i.Therefore,wemustshowthatdij(i))]TJ /F3 11.955 Tf 12.25 0 Td[(dijij,8j2Fi.Ifjisinterdicted,thenij=r+1i)]TJ /F3 11.955 Tf 12.25 0 Td[(dijandtheconstraintissatisedbecausedij(i)r+1i.Ifjisnotinterdicted,thenbecauseiisassignedtoitsclosestuninterdictedfacility,weknowthatdij(i))]TJ /F3 11.955 Tf 12.68 0 Td[(dij0,andtheconstraintisagainsatised.Thereforewehaveconstructedafeasiblesolutionwithanobjective 89

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functionvalueofPi2Ni=i=dij(i)=maxi2Nfdij(i)g,matchingtheprimaloptimalobjectivefunctionvalue.Theconstructeddualsolution,whichsatisestheconditionsoftheproposition,mustbeoptimal.Thiscompletestheproof. BasedonProposition 4.5 ,oneformulationforICFisgivenby: minz2ZmaxXi2Ni (4a) s.t.i)]TJ /F3 11.955 Tf 11.95 0 Td[(diji(r+1i)]TJ /F3 11.955 Tf 11.95 0 Td[(dij)sj8i2N,j2Fi, (4b) Xi2Ni=1, (4c) 0, (4d) s2S(z). (4e) Tointerpretthisformulation,notethatiequalszerounlessi2Nisthenodethatisassignedfarthestfromitsassignedfacility(withtiesbrokenarbitrarily),inwhichcaseiequalstoitsassignmentdistancedij(i).Variablesiserveasindicatorvariablesthatarebinaryinsomeoptimalsolution,wherei=1ifiispositiveandotherwiseequal0.Asanalternative,considerFormulation( 4 )butwithConstraints( 4c )removed.ExaminethesimplicationstoProposition 4.4 thatwouldresultfromthisomission.First,notethatall-valueswouldequalzero,because^F(k)mustalwaysbeemptyforeveryk2K.Letting=maxi2Nfr+1ig,notethatProposition 4.4 canberevisedbysettingijk=()]TJ /F3 11.955 Tf 11.99 0 Td[(Dk)sjforeachk2K,(i,j)2E(k),usingthesameproofasbefore(exceptbysettingijk=()]TJ /F3 11.955 Tf 11.95 0 Td[(Dk)whensj=1,for(i,j)2E(k)).Formulation( 4 )becomes minz2Zmax (4a) s.t.Xi2N)]TJ /F5 11.955 Tf 9.3 0 Td[(ik+ Dk8k2K:Dk, (4b) ik()]TJ /F3 11.955 Tf 11.96 0 Td[(Dk)sj8k2K:Dk,(i,j)2E(k), (4c) (4d) s2S(z). (4e) 90

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Constraints( 4b )and( 4c )canbeinterpretedasfollows.First,notethat stillrepresentstheoptimalobjectivefunctionvalueasbefore.Theadversarythenexamineseachk2K,andnotesfrom( 4b )thattheobjectivewillnotexceedDkunlesssomeik-valueispositive.Constraints( 4c )thenstatethatikcannotbepositiveunlessallfacilitiesthatareatleastascloseasDkunitstonodeihavebeeninterdicted.Ifso,thenikisallowedto(andwillinsomeoptimalsolution)takeonitslargestpossiblevalueof)]TJ /F3 11.955 Tf 12.34 0 Td[(Dk.Asaresult,forthatk,theupperboundon from( 4b )becomesatleast(sinceikmaybepositiveforotheri2Naswell). Remark13.Becausethethird-stageoptimalsolutionforICFistrivialoncezandsarechosen,itcanalternativelybeformulatedinawaysimilartothesecondFormulation( 4 )forIMF.Letbibeabinaryvariablethattakesavalueof1ifnodeiisthefarthestassignednodeand0otherwise.WealternativelyformulateICFasfollows: minz2ZmaxW (4a) s.t.Xj2Fi~xij=18i2N, (4b) Xk2Tij~xiksj8i2N,j2Fi, (4c) Xi2Nbi=1, (4d) WXj2Fidij~xij+()]TJ /F5 11.955 Tf 11.95 0 Td[(1i)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(bi)8i2N, (4e) ~x0, (4f) bi2f0,1g8i2N, (4g) s2S(z). (4h) Asbefore,Constraints( 4b )ensurethateachnodeisassignedtosomefacilityandConstraints( 4c )forceeachnodetobeassignedtoitsclosestuninterdictedfacility.Constraint( 4d )allowstheadversarytoselectanodewhosedistanceassignmentwilldetermineW,andConstraints( 4e )statethatWisnomorethanthenodei 91

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assignmentcostwhenbi=1,andotherwiseprovidesaredundantupperboundofonWwhenbi=0.BecauseWwilltakeonitslargestpossiblevalueallowedby( 4e ),theformulationcorrectlymodelsICF. 4.4SolutionMethodInthissection,wepresentacutting-planealgorithmthatcanbeusedtosolveallofthebilevelprogramsfromtheprevioussection.ThegenericalgorithmpresentedinSection 4.4.1 isbasedoniterativelysolvingamasterproblemandsubproblemuntilthemasterproblemyieldsaprovablyoptimalsolution.InSection 4.4.2 ,wederiveamodelingconstructtoguaranteethatthedefender'sforticationstrategywillinduceafeasiblethird-stageproblemforcapacitatedproblems,basedontheformulationofarobustequivalentlinearprogrammingproblem. 4.4.1Cutting-PlaneApproachWedeviseacutting-planealgorithmsimilartoBenders'decompositionthatappliestoanyoftheforticationmodelsdiscussedinSection 4.3 .First,deneaninterdictionpatternasasetoffacilitiesthatcanbesimultaneouslyinterdictedbytheattacker.WedeneHtobethesetofallinterdictionpatterns,indexedbyh,whereshrepresentstheattacker'sdecisionvectorforpatternh.RecallthatforIMF-CandICF-C,itispossiblethatsomeinterdictionpatternsshcausethedefender'sthird-stageproblemtobeinfeasible.LetHIbethesetofpatternsforwhichthedefender'sthird-stageproblemisinfeasible,andletHFbethesetofpatternsforwhichthereexistsafeasiblesolutiontothedefender'sthird-stageproblem.Foreveryh2HF,letVhbethedefender'soptimalobjectivefunctionvaluewhentheattackerchoosesinterdictionpatternh.Now,letMhjdenoteanarbitrarilylargeconstant,whosevaluewewilllaterexamineinmoredetail.Thegenericcutting-planereformulation,whichappliestoanyofthebilevelmodels,isgivenasfollows: minv (4a) 92

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s.t.vVh)]TJ /F14 11.955 Tf 11.96 11.35 Td[(Xj2FMhjshjzj8h2HF, (4b) Xj2Fshjzj18h2HI, (4c) z2Z. (4d) NotethatFormulation( 4 )isa(linear)MIPwhenZisrepresentedbylinearconstraintsandbinaryrestrictions,becauseshj,forh2Handj2F,areknownparameters.Theobjectiveistominimizev,whichrepresentstheworst-caseobjectiveresultingfromadversarialattack.Constraints( 4b )statethattheadversarywillmaketheobjectivefunctionvalueatleastVh,unlessattackhispreventedbyinterdictingsomefacilityj2F:shj=1.Constraints( 4c )statethatthedefendermustpreventtheadversaryfrommakinganyattackthatwouldcausethethird-stageproblemtobeinfeasible.Unfortunately,solvingModel( 4 )directlyisnotpracticalbecauseoftheexponentialnumberofConstraints( 4b )and( 4c ).Thereforewegeneratetheinterdictionpatternsasneededinacutting-planefashion.Wedosobyiterativelysolvingarelaxationof( 4 ),calledtherelaxedmasterproblem(RMP),inwhichonlyasubsetofConstraints( 4b )and( 4c )arepresent.Hence,RMPisformulatedexactlyas( 4 ),exceptthatConstraints( 4b )arewrittenonlyforsomeHFHFand( 4c )arewrittenonlyforsomeHIHI.Asaresult,theoptimalobjectivefunctionvalueofRMPisalowerboundontheoptimalobjectivefunctionvalueoftheforticationproblembeingsolved.AfterhavingsolvedRMP,wethensolveaseparationproblem(referredtoasthesubproblem(SP))toidentifysomeConstraint( 4b )or( 4c )thatisviolatedbytheRMPsolution.OncetheSPindicatesthatthecurrentRMPsolutionsatisesallConstraints( 4b )and( 4c ),thecurrentRMPsolutionalsoprovidesanupperboundontheoptimalobjectivefunctionvalueandmustthereforebeoptimal.Algorithm 4 providestheformalstatementofthecutting-planemethod. 93

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Algorithm4Cutting-planealgorithm 1: SetHF=HI=;,LB=0,andUB=1 2: whileLB
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updated.Ifv
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( 4b ),andsomefacilityjsuchthatshj=1.Byfortifyingfacilityj,thedefendercanpotentiallyimproveitsobjectivefunctionvaluebyatmostMhj(fromthepreviousvalueofVh).InProposition 4.6 andCorollary 1 ,thiscoefcientrepresentsanupperboundonthesumofassignmentdistancedecreasesthatarepossibleoncefacilityjbecomesavailableinthethirdstage.Whenfortifyingmultiplefacilitiesthatwerepreviouslyinterdicted,thepotentialimprovementiscalculatedinanadditive,andconservative,manner(e.g.,ifshj0=shj00=1,thenfortifyingbothj0andj00couldpotentiallyimprovethetotaldistanceassignmentcostbyatmostMhj0+Mhj00). Proposition4.7. Considerthereformulationsof( 4 ),( 4 ),and( 4 )as( 4 ).Constraint( 4b )correspondingtoh2HFisvalidforMhj=(maxi2Nfdij(i))]TJ /F3 11.955 Tf 11.76 0 Td[(dijg)+,8j2F. Corollary2. Considerthereformulationof( 4 )as( 4 ).Constraint( 4b )correspondingtoh2HFisvalidforMhj=(maxi2Nfdij(i))]TJ /F3 11.955 Tf 11.96 0 Td[(dijg)+,8j2F.InProposition 4.7 andCorollary 2 ,Mhjrepresentsthepotentialdecreaseinthefarthestdistanceassignmentthatthedefendercanobtainbyfortifyingfacilityj. Remark15.Whensolvingthemodelsviathecutting-planealgorithm,theMhj-valuesusedinStep 17 ofAlgorithm 4 requirethenode-to-facilityassignmentcostscomputedinthedefender'sthird-stageproblem.ForIMFandICFmodels,theseassignmentsaretrivialtocomputebecauseeachnodeisassignedtoitsclosestuninterdictedfacility.Also,forIMF-Cweknowdij(i)=hi,8i2N.ForICF-C,wesolvealinearprogram(LP)todeterminetheassignmentcosts.GiventheoptimalobjectivefunctionvalueofSP,V(z),andoptimalinterdictionvector,sh,let)-312(=f(i,j)2NFjdij>V(z)_sj=1g.WesolvethefollowingLPtoobtainthenode-to-facilityassignments: min0 (4a) s.t.Xj2Fxij=18i2N, (4b) Xi2Nxijmj8j2F, (4c) 96

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xij=08(i,j)2, (4d) x0. (4e) ThesolutiontothisLPyieldstheindicesj(i)(andcostsdij(i)),8i2N,requiredfortheMhj-values.2 4.4.2RobustModelforThird-StageInfeasibilityRecallthat( 4 )requiresanexponentialsetofinequalitiesoftheform( 4c )toforcethedefendertochooseanactionz,suchthattherewillexistafeasiblesolutiontothethird-stageproblemregardlessoftheattacker'saction.WeexamineanalternativeapproachthatguaranteestheexistenceofsuchasolutionwithoutresortingtoFeasibilityCuts( 4c ),forthespecialcaseinwhichtheattackercandestroyanyruninterdictedfacilities,i.e.,S(z)isdenedasfollows: Xj2Fsjr (4a) sj1)]TJ /F3 11.955 Tf 11.96 0 Td[(zj8j2F, (4b) sj2f0,1g8j2F. (4c) Inthiscase,notethattheattackercanmakethethirdstageinfeasibleifandonlyifthereexistsans2S(z)suchthatPj2Fmjsj>Pj2Fmj)-235(jNj;hence,thedefender'schoiceofzmustsatisfythefollowing(exponentially-sized)setofconstraints: Xj2FmjsjXj2Fmj)-221(jNj8s2S(z).(4)Insteadofgeneratingthislargesetofconstraints,wedeviseacompactrobustformulationfortherststagethatincorporatestheserestrictions(see,e.g.,[ 6 ]forrobustoptimizationfoundations).Givenavectorz,theoptimalobjectivefunctionvalueofthefollowingLPindicatesiftheattackercanmakethethird-stageprobleminfeasible: maxXj2Fmjsj (4a) 97

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s.t.Xj2Fsjr, (4b) sj1)]TJ /F3 11.955 Tf 11.96 0 Td[(zj8j2F, (4c) s0. (4d) Iftheoptimalobjectivefunctionvalueto( 4 )isgreaterthanPj2Fmj)]TJ /F3 11.955 Tf 13.07 0 Td[(N,thentheattackercanmakethethirdstageinfeasible.Since( 4 )isanLP,itsdualcanequivalentlybesolvedtoobtaintheoptimalobjectivefunctionvalue.Associating!with( 4b )andjwiththejthconstraintof( 4c ),thisdualisgivenas: minr!+Xj2F(1)]TJ /F3 11.955 Tf 11.95 0 Td[(zj)j (4a) s.t.!+jmj8j2F, (4b) !,0. (4c) Therefore,forIMF-CandICF-Cformulations,weaddconstraints( 4b )and( 4c )totheRMPinlieuof( 4c ),andrequirethattheObjective( 4a )notexceedPj2Fmj)]TJ /F3 11.955 Tf -456.26 -23.9 Td[(N.However,theformulationisnonlinearwhenzjandjarebothtreatedasvariables,sowelinearizebyreplacingeachzjjtermwithj.Notingthatjneverneedstoexceedmjinanyoptimalsolutionto( 4 ),itissufcienttoenforcejmjzjandjj,foreachj2F.Toseethis,notethattheminimizationof( 4a )pushesj=zjjtoitslargestpossiblevalue,andsoifzj=1thenj=j(notingthatjmj),andifzj=0thenj=0.Insummary,therobustversionofRMPreplacesConstraints( 4c )with: r!+Xj2F(j)]TJ /F5 11.955 Tf 11.95 0 Td[(j)Xj2Fmj)-222(jNj, (4a) jmjzj8j2F, (4b) jj8j2F, (4c) Constraints( 4b )and( 4c ). (4d) 98

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Notethatthespecialstructuregivenby( 4 )isimportanthere,becauseitallows( 4 )tobesolvedasacontinuousproblemratherthananintegerprogrammingproblem.Ifanintegerprogramneededtobesolvedinstead,thenstrongdualitydoesnotholdandtheprecedingrobustequivalentformulationwouldnotbeobtainablebythemethodweprescribed. 4.5ComputationalExperimentsInthissection,weperformseveralcomputationalexperimentstotesttheefciencyofouralgorithm.First,forthesolutionofIMF,weimplementedourcutting-planealgorithmaswellastheimplicitenumeration(IE)methodpresentedin[ 49 ].Next,weimplementedthecutting-planealgorithmforthesolutionofICF,IMF-C,andICF-Cinstances.Allexperimentswererunonthe150-nodeLondon,Ontariodatasetfrom[ 30 ]usingCPLEX12.3withConcert2.9Technology,onanIBMsystemx3650withtwoIntelE5640Xeonprocessorsand24GBmemory.Tocompareourresultstothoseobtainedin[ 49 ],weimplementedtheirIEalgorithm,includingvariableconsolidation(COBRA)andwarmstartswhensolvingeachsubproblem.TheCOBRAformulation,developedin[ 21 ],usesthefollowingprinciple:Iffacilityjisthekclosestsiteforbothdemandsanddemandt,andifthesetofk)]TJ /F4 11.955 Tf 12.58 0 Td[(1closestsitesforsandfortisthesame,thenatoptimalityxsj=xtj.ThisconstructgreatlyreducesthemodelsizeusedfortheIEimplementation,aswellasthethird-stagemodelsforbothIMFandICF.Therefore,theadvantagesgainedfromCOBRAwerepropagatedintoourmodelsaswell.PerhapstheonlymajordiscrepancybetweenourIEimplementationandthatfrom[ 49 ]arethespecicdirectivestoimproveperformancementionedintheirpaper(thesedirectiveswerenotprovidedinthepaperandthuscouldnotbeincludedinourimplementation).Also,wenotethatinmostcases,over99%ofthecomputationtimewasspentsolvingthesubproblemimplyingthatthespecicimplementationofthetreestructureandbranchingstrategyarerelativelyinsignicant.Overall,ourimplementationoftheIEalgorithmproducedruntimesthat 99

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wereroughly2.5timesaslargeasthosereportedin[ 49 ].Asaresult,wereportinthischapter(faster)runtimesgivenin[ 49 ]ratherthantheresultsofourimplementation.Table B-6 ofAppendixBprovidestheCPUtimesandnumberofsubproblemssolvedfortheIMFmodel.Therstthreecolumnsprovidethevaluesofp,q,andr,respectively.ThefourthcolumnprovidestheCPUtimesgivenin[ 49 ],thefthcolumnprovidestheCPUtimesforthecutting-planealgorithmusingFormulation( 4 ),andthesixthcolumnprovidestheCPUtimesforthecutting-planealgorithmusingFormulation( 4 ).ThelastthreecolumnsprovidethetotalnumberofiterationsforourimplementationofIE(whichwouldmatchthenumbersolvedbytheimplementationin[ 49 ])andourtwoimplementationsofthecutting-planealgorithm.(NotethatFrmstandsforFormulationineachtablepresented.)ThistabledemonstratesthatthecomputationtimerequiredtosolveIMFinstancesbythecutting-planealgorithmscalesfarmoreefcientlythanthatrequiredbytheIEalgorithm.Asp,q,andrgrow,thepossiblenumberofforticationandinterdictionpatternsgrowsquickly,whichcausesthenumberofsubproblemsthatmustbesolvedbytheIEschemetogrowquicklyaswell.Fewersubproblemsmustbesolvedbythecutting-planealgorithm,becausethevalidinequalitiesaddedtothemasterproblemeliminatemanyofthesubproblemsthatmustbesolvedintheIEmethod.Table 4-1 providesasummaryofthedatainTable B-6 .(Asasidenote,thereasonthatcomputationtimedoesnotincreasemonotonicallyinpisbecausethistableaveragesdatafromsolvinginstanceshavingdifferentvaluesofqandr.)Figure 4-3 providesavisualillustrationofthealgorithms'performanceontheseinstances.Thehorizontalaxisspansthecomputationtimesfrom0to108seconds,whileaseparatecurveforeachapproachdepictsthenumberofinstancessolvedwithinthecorrespondingtimeonthehorizontalaxis.Allinstancesaresolvedusingthecutting-planemodelswithin108seconds,whereasonly39ofthe66instancesaresolvedwithin108secondsbytheIEmethod. 100

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Betweenthetwocutting-planemodels,Formulation( 4 )requireslesscomputationtimethan( 4 )inallbut6instances.Moreover,notingthatthecutting-planealgorithmsolvesthesamenumberofsubproblemsforeachofthetwoformulationsinallbuttwoinstances,thedifferenceinsolutiontimeisduetothetimerequiredtosolvethesubproblems.(Thesourceforanydiscrepancyinthenumberofsubproblemssolvedistheexistenceofalternativeoptimalsolutionstothesubproblem.)Basedontheseresults,weconcludethatthesubproblemformulationoriginallyprovidedbyScaparraandChurchisslightlyeasiertosolvethanthesubproblemformulationobtainedbydualizingthethirdstage. Table4-1. AveragecomputationtimesandnumberofiterationsforIMF AverageCPUTimeAverage#ofIterations IECutting-PlaneAlgorithmsIECutting-PlaneAlgorithms pFrm( 4 )Frm( 4 )Frm( 4 )Frm( 4 ) 25166.3219.4013.935787.4054.4054.4030641.0612.337.2012228.2029.2729.2740416.181.921.425591.8311.6711.6750851.777.256.2516369.0833.5033.50601485.0518.7517.6736037.3367.0868.50 Next,wesolvedICFusingFormulations( 4 ),( 4 ),and( 4 )(preliminarycomputationalresultsshowedthatsolving( 4 )wasmuchmorecomputationallyexpensivethantheotherthreeformulations).Table B-7 ofAppendixBprovidestheCPUtimesaswellasthenumberofsubproblemssolvedforeachofthersttwomodels.(Recallthat( 4 )canbesolvedasasingle-stageproblem,andhencerequiresthesolutionofnosubproblems.)Inalmosteverycase,thesingle-stageFormulation( 4 )outperformsthetwocutting-planemodels(substantiallyinmanycases).However,recallthatthismodelcanonlybeusedwhentheinterdictor'sonlyconstraintisacardinalityconstraint.Also,inmostcases,Formulation( 4 )outperformsFormulation( 4 ).ThisresultiscontrasttoIMF,wheretheformulationusingthird-stagedualityperformsworsethanthedirecttwo-stageformulation.Therearefouranomaliestothistrend,however.Interestingly,allfouroccurwhenthevaluesofqandrareroughlythesame 101

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Figure4-3. Numberofsubproblemssolved andq+rp 2.Asignicantlygreaternumberofcutsarerequiredtosolve( 4 )intheseinstances.Table 4-2 providesasummaryofthedatainTable B-7 ,andTable 4-3 providesthelargestcomputationtimesaswellasthepercentagesofinstancesthatsolveinunder1,5,and10seconds,foreachapproach. Table4-2. AveragecomputationtimesandnumberofiterationsforICF AverageCPUTimeAverage#ofIterations pFrm( 4 )Frm( 4 )Frm( 4 )Frm( 4 )Frm( 4 ) 2530.7322.470.4063.6032.803013.8723.470.6039.8728.47400.839.580.0815.5015.25501.2512.170.2517.8317.00601.8312.920.2521.3319.75 WesolvedIMF-CandICF-Cusingboththefeasibilitycutmethodandtherobustmodelforthird-stageinfeasibility.Becauseoftheadditionalcomplexityofthethirdstage,IMF-CandICF-CrequiremuchmoretimetosolvethanIMFandICF.(ThisisespeciallytrueforICF-Cbecauseofthesizeofthesetf(k,i,j)jk2K,(i,j)2E(k)g.)Therefore, 102

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Table4-3. SummaryofCPUtimesforICF Frm( 4 )Frm( 4 )Frm( 4 ) MaximumCPUTime266612%SolvedinUnder1second36.360.0071.21%SolvedinUnder5seconds78.7916.67100.00%SolvedinUnder10seconds84.8543.94100.00 inordertotestourcutting-planealgorithm,weconstructedinstancesusingsmaller(p,q,r)values.ForIMF-Cwestillusedall150nodesfromtheLondon,Ontariodataset,butforICF-Cweselectedtherst50ofthesenodes.Inbothcases,werandomlygeneratedthevaluesofmjfromauniformdistributionsothattheexpectedvalueofmjisroughlyequalto1+r+1 pN p.Inordertoensurethatthevaluesofmjareinteresting(i.e.,theproblemisfeasibleforsome,butnotallforticationpatterns),wesortthem-valuesinnonincreasingorderandcheckthatthetwofollowingconditionshold: (a) Thesumoftheqlargestm-valuesandthep)]TJ /F4 11.955 Tf 11.99 0 Td[((q+r)smallestm-valuesisatleastjNj; (b) Thesumofthep)]TJ /F3 11.955 Tf 11.96 0 Td[(rsmallestm-valuesislessthanjNj.Condition(a)ensuresthattheproblemisfeasibleandcondition(b)ensuresthatthereexistsaninterdictionpatternforwhichtheproblemisinfeasible.Onceasetofvaluesformj,8j2F,isgeneratedthatsatisesconditions(a)and(b),theinstanceissolved.Thisprocessisrepeatedtwoadditionaltimesforeachinstance(choiceof(p,q,r)).Tables B-8 and B-9 ofAppendixBprovidetheaverageCPUtimes,averagenumberofiterations,andaveragenumberoffeasibilitycuts(forthefeasibilitycutimplementation,whichweabbreviateFC)requiredtosolveIMF-CandICF-C,respectively.ForIMC-C,about73%oftheinstancessolvefasterwhenusingtherobustformulation.Whilethisstatisticmaynotnecessarilyindicatedominanceoftherobustformulationoverthefeasibilitycutimplementation,itisimportanttonotethatfortheinstancesinwhichtherobustformulationisfaster,thereisoftenalargedifferenceincomputationtime(withthelargestbeing6995.67seconds),whilefortheinstancesinwhichthefeasibilitycutimplementationisquicker,thedifferencesincomputationtime 103

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aremuchsmaller(thelargestis18.67seconds).Foreachvalueofp,Table 4-4 showstheaverageCPUtime,numberofiterations,andnumberoffeasibilitycuts,labeledNumCuts(fortheFCimplementation)requiredtosolveIMF-C.(Forp=25,wesplittheinstancesintotwogroups:therowlabeled25includestherst9instances,whichareconsistentwiththeinstancesforothervaluesofp;andtherowlabeled25includesthe15instanceshavingp=25from[ 49 ].Notethatinstance(25,3,4)fallsintobothofthesegroups.)Asseeninthistable,forallvaluesofp,themajorityofcutsrequiredtosolvethefeasibilitycutimplementationareinfactfeasibilitycuts.ThissuggeststhatwhenSconsistsofmorethanjustarestrictiononthenumberofinterdictions,theproblemmaybemuchmoredifculttosolve.Thereasonforthisistwofold:First,therobustformulationwillnolongerbevalid,andsecond,obtainingthefeasbilitycutswillmostlikelyrequiresolvingSP.Aspincreases,theaverageCPUtimetendstoincreaseaswell,butthenumberofiterationsremainssomewhatconsistent(exceptforinstancesinset25).Thisindicatesthatthenumberofiterationsrequiredforsolutionismoreafunctionofqandrthanofp,andthatthetimerequiredtosolveSPdependsheavilyonthevalueofp.Theincreaseddifcultyinsolvinginstancesinthegroup25,asopposedto25,isduetothelargervaluesofqandrin25.Fortherobustformulation,thealmosteleven-foldincreaseinaverageCPUtimeisaresultofboththeseven-foldincreaseinaveragenumberofiterationsandtheincreaseindifcultyofsolvingSP.Forthefeasibilitycutimplementation,thenumberoffeasibilitycutsincreasesbyafactorof83.Hence,thealmostthirty-foldincreaseinaverageCPUtimeforthefeasibilitycutimplementationisonlypartiallyduetotheincreasednumberofcuts,andismostlyrelatedtotheincreasedeffortrequiredtosolveSP.Finally,werepeatedthisexperimentofsolvingICF-C,usingtherobustformulationandfeasibilitycutbasedapproach.DuetothelargeCPUtimesrequiredtosolveICF-C,wesolvedonlythe18smallestinstancesforthisexperiment.Inallcases,the 104

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Table4-4. AveragecomputationtimesandnumberofiterationsforIMF-C AverageCPUTimeAverage#ofIterationsNumCuts pFCRobustFCRobust 102.152.1519.526.9312.63154.223.8917.934.2613.372011.8911.3716.564.8511.702545.4447.7013.746.227.70251263.89523.62676.7141.67635.56 robustformulationoutperformedthefeasibilitycutimplementation.Table 4-5 showstheaverageCPUtime,numberofiterations,andnumberoffeasibilitycutsrequiredtosolveICF-C.Asseeninthetable,whenincreasingpfrom10to15,theaverageCPUtimeincreasessubstantially,whiletheaveragenumberofiterationsandfeasibilitycutsremainsroughlyconstant.ThisbehaviorisaresultoftheincreasedsizeofSPandtheresultingincreaseinCPUtimerequiredforitssolution. Table4-5. AveragecomputationtimesandnumberofiterationsforICF-C AverageCPUTimeAverage#ofIterationsNumCuts pFCRobustFCRobust 10214.37167.4821.265.3316.37156546.334445.6321.154.7016.37 105

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CHAPTER5CONCLUSIONSANDFUTURERESEARCHInChapter 2 weconsideredathree-stageprocurementgamebetweentwocompetingbuyers(wherethesecondbuyermaysimplyrepresentuncertaintyinsuppliercapacity).Wemotivatedtheneedfortheleadertoconsiderthesituationinwhichthefollower'sactions(ortherealizationoftheuncertainsuppliercapacities)resultintheworst-casescenario.Afterreformulatingtheproblemasatwo-stageproblem,wedevelopedacutting-planemethodology.Improvementsweremadebytighteningtheformulationforthesubproblemandbydevelopinganalgorithmtoobtainvalidinequalitiesfromsmallersubproblems.InChapter 3 westudiedathree-stageprocurementgamebetweentwocompetingbuyers,whoactinturnsinaStackelberggame.WeextendedtheworkfromChapter 2 toincludescenariosinwhichthefollowereitherwishestominimizeitsownprocurementcosts,ortomaximizetheleader'sprocurementcostswhileensuringthattheirownprocurementcostiswithinsomespeciedtoleranceoftheirownminimumpossibleprocurementcost.Weformulatedoneofthesemodelsasasingle-levellinearMIPandtheotherasabilevellinearMIP.Wedevelopedtwocutting-planemethodsforthebilevelprogramandimprovedourmethodsthroughafewenhancements.Thereexistseveralpossibleavenuesforfutureresearchinthiseld.Onequestionregardsthenatureofthisgamewhenbothplayersmustmakeprocurementdecisionssimultaneously.Dependingonhowconictswouldbemanagedinsuchasituation(e.g.,whatprotocolsarefollowedwhenasuppliercannotlltheordersofbothplayerssimultaneously),itmaybepossibletoestablishanequilibriumthatwouldguidethedecisionsofthetwoplayers.Anotherinvestigationmightexaminethesituationinwhichdataparticularlythesuppliercapacitiesisstochastic.Suchaproblemmayseektominimizeexpectedcosts,givenadistributionofuncertainsupplierinventorylevels,wherethefollowerobservestheinventorylevelsbeforeprocuringitsgoods.Hence,the 106

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protectionschemewouldguaranteeavailableproductionlimitsinsuchamannerthatminimizestheoverallexpectedcosts,giventhatthefollowerminimizesprocurementcostsbasedonactualavailableinventoryvalues.Chapter 4 presentsacutting-planemethodforsolvingabroadclassoffacilityinterdictionandforticationproblems.OuralgorithmsyieldanimprovedmethodforsolvingproblemIMF,whichhasbeenthesubjectofseveralpriorstudies.Perhapsmoreimportantly,ourdualizationandcutting-planeapproachespermitthesolutionofamuchbroaderclassofinterdictionandforticationproblems.Inparticular,forthemedianproblem,weexpandthescopeofproblemsthatcanbesolvedbygeneralizingtheleader'sandinderdictor'sfeasibleregions,andformulateatwo-stagemodeltoaddressthesituationinwhichtherearerestrictionsonthenumberofnodesassignedtoeachfacility,andinwhichnode-to-facilityassignmentsmustremainintegral(IMF-C).Further,weformulateanalogousproblemswiththecenterobjective(i.e.,minimizationofthemaximumassigneddistance),underassumptionsofunlimitedandlimitednumbersofnodeassignmentsforeachfacility(ICFandICF-C,respectively).Theadditionofsuchrestrictionsintroducedanaddedcomplicationthepossibilitythatthethirdstageisinfeasibleforcertaininterdictionpatterns.Wepresenttwomethodsfordealingwiththissituation:Oneusingfeasbilitycuts,andanotherthatemploysarst-stagerobustformulation.Therobustformulationisanovelwayofobviatingtheneedforfeasibilitycuts,andourcomputationalresultsshowthatitismoreeffectivethanrelyingontheiterativegenerationoffeasibilitycuts.Forfutureresearch,wenotethatifthethirdstageofIMF-CorICF-Ccontainsgeneralknapsackconstraints(oftheformPi2Naijxijbj,forj2F),thenthethird-stageproblembecomesNP-hard.Inthiscase,thethird-stagedualizationprocedurecannotbedirectlyappliedasdoneinourresearch,becauseastrongdualformulationwould(presumably)notbeavailableforthethird-stageproblem.Asaresult,afutureresearchdirectionmightfocusondevelopingalternativesolutionmethodsforIMF-CandICF-C 107

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modelshavingmoregeneralfacilitycapacityconstraints.Wealsorecommendthestudyoftheseproblemsunderasymmetricinformationscenarios(inwhichtheattackermayperceivetheproblemdatadifferentlythanthedefender),andwhensuccessfuldefenseandattackactionsarestochastic. 108

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APPENDIXAPROOFSProofofProposition 2.1 .First,observethatBijk0,8i2S,k2Fi,andj2Iik.Ourproofconstructsanoptimalsolutionto( 2 )withnonnegativevaluesofikandijkthatsatisfy: (i) ijk=Bijkyi,ui)]TJ /F6 7.97 Tf 6.59 0 Td[(k+j+1,8i2S,k2Fi,j2Iik (ii) i)]TJ /F7 7.97 Tf 6.59 0 Td[(1j)]TJ /F5 11.955 Tf 11.95 0 Td[(ik)]TJ /F5 11.955 Tf 11.96 0 Td[(ijkpi(k)]TJ /F3 11.955 Tf 11.95 0 Td[(j),8i2S,k2Fi,j2Iik (iii) 00=SPI(v00,vNQL)andNQL=0.Notethat(ii),alongwithnonnegativityofthe-variables(followingfromthenonnegativityofallB-valuesandy-variables)veriesfeasibilityofthesolution,while(iii)provestheoptimalityofthesolution(since(i)impliesijk=0whenthearcisavailable).LetC1bethesetofnodesthatbelongtoCase1andC2bethesetofnodesthatbelongtoCase2.Givenavectory,letRbethesetofallnodesfromwhichvNQLisreachable,andRbethesetofallothernodes.(NotethatC1RandRC2.)ForeverynodevikinR,letik=SPI(vik,vNQL)(so(iii)issatised).ForeverynodevikinR,letik=LP(v00,vNQL))]TJ /F3 11.955 Tf 12.39 0 Td[(SP(v00,vik).Givenanarc(vi)]TJ /F7 7.97 Tf 6.58 0 Td[(1j,vik),wexthevaluesofijkasin(i)andconsiderthefollowingfourcasestoshowthat(ii)issatised:CaseA:Bothvi)]TJ /F7 7.97 Tf 6.59 0 Td[(1jandvikareinRWeshowinthiscasethati)]TJ /F7 7.97 Tf 6.59 0 Td[(1j)]TJ /F5 11.955 Tf 11.97 0 Td[(ikpi(k)]TJ /F3 11.955 Tf 11.97 0 Td[(j)whichthenallowsijk=Bijkyi,ui)]TJ /F6 7.97 Tf 6.58 0 Td[(k+j+1totakeanynonnegativevalue.Wehavei)]TJ /F7 7.97 Tf 6.58 0 Td[(1j)]TJ /F5 11.955 Tf 12.85 0 Td[(ik=LP(v00,vNQL))]TJ /F3 11.955 Tf 12.86 0 Td[(SP(v00,vi)]TJ /F7 7.97 Tf 6.59 0 Td[(1j))]TJ /F3 11.955 Tf -427 -23.91 Td[(LP(v00,vNQL)+SP(v00,vik)=SP(v00,vik))]TJ /F3 11.955 Tf 11.06 0 Td[(SP(v00,vi)]TJ /F7 7.97 Tf 6.58 0 Td[(1j)pi(k)]TJ /F3 11.955 Tf 11.07 0 Td[(j).(Thelastinequalitymusthold,orelseSP(v00,vik)couldnotbetheshortest-pathdistancefromv00tovik.)CaseB:Bothvi)]TJ /F7 7.97 Tf 6.59 0 Td[(1jandvikareinRRecallinthiscasethati)]TJ /F7 7.97 Tf 6.59 0 Td[(1j=SPI(vi)]TJ /F7 7.97 Tf 6.59 0 Td[(1j,vNQL)andik=SPI(vik,vNQL).Ifarc(vi)]TJ /F7 7.97 Tf 6.59 0 Td[(1j,vik)isavailable,thenbytheshortest-pathdenitionsofi)]TJ /F7 7.97 Tf 6.58 0 Td[(1jandik,wehavethati)]TJ /F7 7.97 Tf 6.59 0 Td[(1j)]TJ /F5 11.955 Tf 10.58 0 Td[(ikpi(k)]TJ /F3 11.955 Tf 10.59 0 Td[(j),whichthusallowsijk=0asdesired.Ifarc(vi)]TJ /F7 7.97 Tf 6.59 0 Td[(1j,vik)isnotavailable 109

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(i.e.,yi,ui)]TJ /F6 7.97 Tf 6.58 0 Td[(k+j+1=1),thenwemustshowthati)]TJ /F7 7.97 Tf 6.59 0 Td[(1j)]TJ /F5 11.955 Tf 11.95 0 Td[(ik)]TJ /F3 11.955 Tf 11.96 0 Td[(pi(k)]TJ /F3 11.955 Tf 11.95 0 Td[(j)ijk=Bijk.Wehave:i)]TJ /F7 7.97 Tf 6.58 0 Td[(1j)]TJ /F5 11.955 Tf 11.96 0 Td[(ik)]TJ /F3 11.955 Tf 11.95 0 Td[(pi(k)]TJ /F3 11.955 Tf 11.96 0 Td[(j)=SPI(vi)]TJ /F7 7.97 Tf 6.58 0 Td[(1j,vNQL))]TJ /F3 11.955 Tf 11.96 0 Td[(SPI(vik,vNQL))]TJ /F3 11.955 Tf 11.95 0 Td[(pi(k)]TJ /F3 11.955 Tf 11.96 0 Td[(j)LP(vi)]TJ /F7 7.97 Tf 6.59 0 Td[(1j,vNQL))]TJ /F4 11.955 Tf 11.96 0 Td[([SPI(vik,vNQL)+pi(k)]TJ /F3 11.955 Tf 11.95 0 Td[(j)].ThelatterexpressionisnotmorethantheCase1valueofBijkbecauseSPI(vik,vNQL)SP(vik,vNQL).Also,theCase1valueofBijkisnotmorethantheCase2valueofBijkbecauseLP(v00,vNQL))]TJ /F3 11.955 Tf 11.96 0 Td[(SP(v00,vi)]TJ /F7 7.97 Tf 6.59 0 Td[(1j)LP(vi)]TJ /F7 7.97 Tf 6.59 0 Td[(1j,vNQL).Hencei)]TJ /F7 7.97 Tf 6.59 0 Td[(1j)]TJ /F5 11.955 Tf 11.95 0 Td[(ik)]TJ /F3 11.955 Tf 11.96 0 Td[(pi(k)]TJ /F3 11.955 Tf 11.95 0 Td[(j)Bijk.CaseC:vi)]TJ /F7 7.97 Tf 6.58 0 Td[(1j2Randvik2RFirst,notethatarc(vi)]TJ /F7 7.97 Tf 6.59 0 Td[(1j,vik)cannotbeavailable.Toshowthati)]TJ /F7 7.97 Tf 6.58 0 Td[(1j)]TJ /F5 11.955 Tf 10.91 0 Td[(ik)]TJ /F3 11.955 Tf 10.91 0 Td[(pi(k)]TJ /F3 11.955 Tf 10.91 0 Td[(j)ijk=Bijk,wehavei)]TJ /F7 7.97 Tf 6.59 0 Td[(1j)]TJ /F5 11.955 Tf 11.96 0 Td[(ik)]TJ /F3 11.955 Tf 11.95 0 Td[(pi(k)]TJ /F3 11.955 Tf 11.96 0 Td[(j)=LP(v00,vNQL))]TJ /F4 11.955 Tf 11.95 0 Td[([SP(v00,vi)]TJ /F7 7.97 Tf 6.59 0 Td[(1j)+pi(k)]TJ /F3 11.955 Tf 11.96 0 Td[(j)+SPI(vik,vNQL)]LP(v00,vNQL))]TJ /F4 11.955 Tf 11.95 0 Td[([SP(v00,vi)]TJ /F7 7.97 Tf 6.59 0 Td[(1j)+pi(k)]TJ /F3 11.955 Tf 11.96 0 Td[(j)+SP(vik,vNQL)]=Bijk,wherethelatterequalityistruebecauseBijkisdeterminedbyCase2.CaseD:vi)]TJ /F7 7.97 Tf 6.58 0 Td[(1j2Randvik2RWeshowthati)]TJ /F7 7.97 Tf 6.59 0 Td[(1j)]TJ /F5 11.955 Tf 11.95 0 Td[(ik)]TJ /F3 11.955 Tf 11.95 0 Td[(pi(k)]TJ /F3 11.955 Tf 11.96 0 Td[(j)0.Wehavei)]TJ /F7 7.97 Tf 6.59 0 Td[(1j)]TJ /F5 11.955 Tf 11.96 0 Td[(ik)]TJ /F3 11.955 Tf 11.95 0 Td[(pi(k)]TJ /F3 11.955 Tf 11.96 0 Td[(j)=SPI(vi)]TJ /F7 7.97 Tf 6.58 0 Td[(1j,vNQL))]TJ /F3 11.955 Tf 11.96 0 Td[(LP(v00,vNQL)+SP(v00,vik))]TJ /F3 11.955 Tf 11.96 0 Td[(pi(k)]TJ /F3 11.955 Tf 11.96 0 Td[(j)SPI(vi)]TJ /F7 7.97 Tf 6.58 0 Td[(1j,vNQL))]TJ /F3 11.955 Tf 11.96 0 Td[(LP(v00,vNQL)+SP(v00,vi)]TJ /F7 7.97 Tf 6.58 0 Td[(1j)0.Therefore,i)]TJ /F7 7.97 Tf 6.59 0 Td[(1j)]TJ /F5 11.955 Tf 12.13 0 Td[(ik)]TJ /F3 11.955 Tf 12.12 0 Td[(pi(k)]TJ /F3 11.955 Tf 12.13 0 Td[(j)Bijkyi,ui)]TJ /F6 7.97 Tf 6.59 0 Td[(k+j+1,satisfyingcondition(ii).Thiscompletestheproof.2 PropositionA.1. Givenabinaryvector,^z,thereexistsanoptimalsolutiontoDFinwhichi=(Ei)]TJ /F3 11.955 Tf 11.95 0 Td[(C)^zi,8i2S. Proof. Leti=(Ei)]TJ /F3 11.955 Tf 12.36 0 Td[(C)^zi,8i2S,andlet SP(vik,v^^k,^z)denotetheshortest-pathcostfromnodeviktonodev^^kinthefollower'snetworkinwhichPm2Fl)]TJ /F16 5.978 Tf 5.75 0 Td[(1flmm^zl,8i
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(i.e.,thefollowerpurchasesonlyfromunprotectedsuppliers).Thatis, SPmeasurestheshortestpathdistancesinthefollower'snetwork,giventheprotectiondecisionsthathavebeenmadebytheleader.Webeginbyshowingbyinductiononithatafeasiblesolutionexistsinwhich ij= SP(vij,vNQF,^z))]TJ /F14 11.955 Tf 15.88 11.36 Td[(Xi
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If^zi=0,theni=0and( A )becomes i)]TJ /F7 7.97 Tf 6.59 0 Td[(1j=mink2Oi)]TJ /F16 5.978 Tf 5.76 0 Td[(1jpi(k)]TJ /F3 11.955 Tf 11.95 0 Td[(j)+ik=mink2Oi)]TJ /F16 5.978 Tf 5.76 0 Td[(1jnpi(k)]TJ /F3 11.955 Tf 11.96 0 Td[(j)+ SP(vik,vNQF,^z)o)]TJ /F14 11.955 Tf 15.87 11.35 Td[(Xi
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xij=0,8j2Fnf(i)gandforanyi2NnN,letxij=xhij,8j2F.Solution(s,)isfeasibleto( 4 )and(s,x)isfeasibleto( 4 ),becauseeachnodeisassignedtoitsclosestuninterdictedfacilityinbothsolutions.Theobjectivefunctionvaluesforthesetwosolutionsareequivalent)]TJ 5.48 -.72 Td[(Pi2Ni=Pi2NPj2Fdijxijandequaltothefollowing: =Vh)]TJ /F14 11.955 Tf 11.95 11.36 Td[(Xi2N)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(dij(i))]TJ /F3 11.955 Tf 11.96 0 Td[(di(i) (Aa) Vh)]TJ /F14 11.955 Tf 11.95 11.36 Td[(Xi2NXj2F)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(dij(i))]TJ /F3 11.955 Tf 11.96 0 Td[(dij+ (Ab) =Vh)]TJ /F14 11.955 Tf 11.95 11.35 Td[(Xj2FXi2N)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(dij(i))]TJ /F3 11.955 Tf 11.96 0 Td[(dij+ (Ac) =Vh)]TJ /F14 11.955 Tf 11.95 11.36 Td[(Xj2FMhj (Ad) =Vh)]TJ /F14 11.955 Tf 11.95 11.36 Td[(Xj2FMhjshjzj. (Ae) Equality( Aa )holdsbecauseforeachi2N,thetotalassigneddistanceisreducedbyexactlydij(i))]TJ /F3 11.955 Tf 12.29 0 Td[(di(i).Inequality( Ab )holdsbecause(i)2F,8i2N;( Ac )holdsbecause(dij(i))]TJ /F3 11.955 Tf 12.42 0 Td[(dij)+=0,8i2Nn^Nandj2F;and( Ad )and( Ae )holdbythedenitionsofMhjandF,respectively.Becauseoursolutions,(s,)and(s,x),haveobjectivefunctionvaluesgreaterthanorequaltotheright-hand-sidevalueofthecut( 4b ),theoptimalobjectivefunctionvaluetotheSPmustaswell,andtheinequalityisthereforevalid.2ProofofCorollary 1 .Notethatthesecondstageof( 4 )isarelaxationofthesecondstageof( 4 )(Constraints( 4b )forj2FnFiareredundantandwerethereforeremoved).Thecuts( 4b )arecalculatedinanidenticalfashionwhensolving( 4 )astheyarewhensolving( 4 ).Theseinequalitiesprovidealowerboundontheoptimalsolution'sobjectivevaluefor( 4 ),andthereforealsoprovidealowerboundontheoptimalsolution'sobjectivevaluefor( 4 ).Thiscompletestheproof.2 113

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ProofofProposition 4.7 .Denez,s,N,and(i)asintheproofofProposition 4.6 .Forthisproof,wealsoletdi=minfdij(i),di(i)gg,anddenei2argmaxi2Nfdig.Weconstructthefollowingfeasiblesolutionstothesecondstagesof( 4 ),( 4 ),and( 4 ): Leti=di,i=1,andi=i=0,8i6=i.Constraints( 4c ),( 4d ),and( 4e )areallclearlysatised.Constraints( 4b )foreachi6=iaresatisedbecausei)]TJ /F3 11.955 Tf 12.44 0 Td[(diji=0forallj2Fi,andbecausetheright-handsideof( 4b )isnonnegative.Fori=i,Constraint( 4b )reducestoir+1iifsj=1,andidijifsj=0.Recallthatiequalsthedistancebetweennodeianditsclosestuninterdictedfacility.Whensj=1,theseconstraintsaresatisedbecauseiisneverassignedtoafacilitythatismorethanr+1iunitsaway.Whensj=0,facilityjisuninterdicted,andsoidijbyconstructionofi. LetkbetheelementsuchthatDk=diand =Dk.Further,forallk2KsuchthatDk
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=Vh)]TJ /F4 11.955 Tf 11.95 0 Td[(maxj2Fn)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(dihj(ih))]TJ /F3 11.955 Tf 11.95 0 Td[(dihj+o (Ae) Vh)]TJ /F4 11.955 Tf 11.95 0 Td[(maxj2F(maxi2Nfdij(i))]TJ /F3 11.955 Tf 11.95 0 Td[(dijg+) (Af) Vh)]TJ /F14 11.955 Tf 11.95 11.36 Td[(Xj2Fmaxi2Nfdij(i))]TJ /F3 11.955 Tf 11.96 0 Td[(dijg+ (Ag) =Vh)]TJ /F14 11.955 Tf 11.95 11.35 Td[(Xj2FMhj (Ah) =Vh)]TJ /F14 11.955 Tf 11.95 11.36 Td[(Xj2FMhjshjzj, (Ai) Equality( Aa )andInequality( Ab )holdbydenitionofi.Equalities( Ac )and( Ad )holdbysubstitution.Equality( Ae )holdsbecausenodeihwilleitherbereassignedtoitsclosestfacilityinForstayassignedtofacilityj(ih).Inequality( Af )holdsbecauseih2N,andishenceacandidatetomaximizetheinnermaxterminthatexpression.Inequality( Ag )holdsbecausethejthatmaximizesthetermin( Af )isincludedinthesummation,andEqualities( Ah )and( Ai )holdbythedenitionsofMhjandF,respectively.Oursolutions,(s,,),(s, ,),and(s,W,x,b),haveobjectivefunctionvaluesgreaterthanorequaltotheright-hand-sidevalueofthecut( 4b ),andsotheoptimalobjectivefunctionvaluetotheSPmustaswell.TheInequality( 4b )isthereforevalid.2ProofofCorollary 2 .Notethatthesecondstageof( 4 )isarelaxationofthesecondstageof( 4 ).Thecuts( 4b )arecalculatedinanidenticalfashionwhensolving( 4 )astheyarewhensolving( 4 ).Theseinequalitiesprovidealowerboundontheoptimalsolution'sobjectivevaluefor( 4 ),whichprovidealowerboundontheoptimalsolution'sobjectivevaluefor( 4 ).Thiscompletestheproof.2 115

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APPENDIXBDETAILEDCOMPUTATIONALRESULTS TableB-1. ComputationaltimeforN=5,6,7withvaryingbuyerdemands Strategy0 Strategy1 Strategy2 Strategy3 NQLQFCapacitySet# Time(s)#Cuts Time(s)#Cuts Time(s)#Cuts Time(s)#Cuts 550501 335 0.26.8 0.36.8 0.46.8 525251 02 0.14.2 0.14.2 0.14.2 525871 77 0.518.1 0.510.7 0.510.3 587251 177 3.534.7 4.123.9 2.224 544442 137 1.429.8 0.922.3 1.320.5 522222 14 0.419.9 0.311 0.312.9 522772 14 0.723.4 0.515.8 118.2 577222 1310 1.835.7 1.623.5 1.827 548483 235 228.3 3.222.6 2.510.6 524243 02 0.517.2 0.310.9 0.35.5 524843 44 0.519.5 0.812.6 0.87.2 584243 107 3.834.6 429.5 2.416.9 538384 96 1.126.5 1.220.7 1.310.7 519194 04 0.23.6 0.23.6 0.33.6 519664 13 0.417.4 0.515.4 0.59.9 566194 66 1.525.6 2.224.6 1.417.1 583835 1117 14.435.6 20.127.2 8.118.9 541415 23 1.717.7 111.9 1.69.6 5411455 184 1.722.9 1.518.1 2.413.2 5145415 708 9.934.9 10.426.9 7.622.1 660601 12511 7.544.3 11.433.8 4.623.3 630301 14 1.623.7 1.413.8 1.711 6301051 2026 1.332.7 1.124.3 1.221.3 6105301 11820 8.242.7 7.430.2 429.1 662622 617 10.840.6 734.5 8.126.4 631312 15 121.7 0.815.5 1.914 6311082 107 1.227.3 1.221.9 215.6 6108312 18521 7.846.3 8.736.8 4.426 658583 11310 7.444.9 8.131.8 6.822.7 629293 24 1.620.6 1.314.2 212.3 6291023 264 2.134.9 227.6 4.918.7 6102293 14017 1444.4 9.237.8 15.423.6 647474 478 5.637.7 6.331.2 42.111.9 623234 14 0.920.2 0.814.2 0.94.3 623824 104 1.126.2 0.819.4 1.512.4 682234 5915 11.244.3 10.939.3 5.823.1 61011015 10629 49.135.9 31.932.4 24.820.7 650505 94 6.125.7 5.916.6 713.3 6501775 784 6.229.7 8.121 717.6 6177505 145719 39.857.5 84.644.5 2430.5 770701 81810 11.245.6 1537.4 6.627.3 735351 96 2.224 2.116 214.8 7351221 1106 1.936.8 1.928.3 1.422.6 7122351 116913 16.762.8 27.651.2 6.643.4 776762 91221 12.748.9 12.937.3 12.826.1 738382 206 225.9 2.118.6 2.218.7 7381332 10913 2.942 2.736.7 3.236.1 7133382 105323 18.154 23.652.9 14.244.8 769693 84010 24.751 32.335.6 10.126.4 734343 53 2.726.1 2.716.2 2.57.6 7341213 976 2.736.2 2.926.6 8.412.3 7121343 109714 24.663.3 35.953.7 53.732.4 757574 33610 1555.1 13.344.8 1530.2 728284 38 1.622 1.716 1.512.2 7281004 316 235.9 1.929.3 2.827.2 7100284 44313 13.856.7 22.152.8 26.639.8 71121125 -46.949.1 93.239.2 73.922.9 756565 706 10.727.9 10.517.6 6.710.2 7561965 6536 10.535 14.125.1 11.916.3 7196565 -92.272.2 81.165.7 7943.9 116

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TableB-2. ComputationaltimeforN=8,9,10withvaryingprotectioncosts Strategy0 Strategy1 Strategy2 Strategy3 NxiCapacitySet# Time(s)#Cuts Time(s)#Cuts Time(s)#Cuts Time(s)#Cuts 811 -20.269.3 19.955.8 10.437.5 831 644 5.524.5 3.517.2 3.112.4 851 5572 5.029.0 3.423.6 2.518.1 8[1,5]1 -29.581.1 36.780.6 16.776.9 812 -24.964.7 38.659.5 22.940.8 832 529 4.823.9 3.119.5 2.912.1 852 10394 8.434.6 6.933.8 6.118.8 8[1,5]2 -86.878.5 131.271.8 30.065.1 813 -54.866.2 44.448.4 219.936.6 833 528 3.926.9 3.917.1 4.412.1 853 5703 5.733.2 5.227.2 7.317.3 8[1,5]3 -68.297.8 66.085.1 121.459.0 814 172225 29.973.0 21.753.7 22.743.0 834 616 4.025.2 3.718.3 3.08.5 854 5243 7.537.2 8.726.8 6.226.8 8[1,5]4 -40.483.6 42.382.0 34.964.3 815 -256.368.6 128.056.1 321.637.2 835 6157 36.830.3 20.721.7 18.112.1 855 53389 28.538.4 20.731.1 36.016.1 8[1,5]5 -288.599.9 528.791.8 500.767.6 911 -43.782.8 44.171.3 37.958.9 931 777 7.527.0 5.216.3 4.013.2 951 -8.932.5 8.823.8 5.018.3 9[1,5]1 -101.6110.4 128.098.6 248.7114.3 912 -105.578.7 108.177.1 88.252.5 932 8216 19.133.8 15.324.5 11.614.9 952 -43.146.6 29.439.0 76.234.9 9[1,5]2 -320.1116.4 236.3117.2 215.182.5 913 -112.883.8 72.368.5 352.350.1 933 790 12.024.4 14.119.8 5.611.0 953 -44.135.4 24.626.3 36.614.4 9[1,5]3 -295.1110.7 365.1109.1 296.665.0 914 -88.580.0 82.872.7 124.654.1 934 933 5.526.0 4.619.6 4.115.2 954 24992 8.037.6 12.728.6 5.823.0 9[1,5]4 -97.9124.4 209.4122.8 78.8112.9 915 -322.7986.8 302.176.5 1054.4959.7 935 8472 64.131.3 74.922.5 33.912.6 955 -264.650.9 36.635.1 123.124.8 9[1,5]5 -647.3140.7 504.9129.5 1031.483.0 1011 -5.926.6 5.126.6 5.726.6 1031 8159 1.76.4 1.66.4 1.76.4 1051 -2.110.7 1.910.7 2.110.7 10[1,5]1 -5.332.9 4.532.9 5.232.9 1012 -500.1108.4 263.49110.2 617.7961.4 1032 11312 23.436.5 24.725.8 22.713.9 1052 -148.257.1 87.744.0 258.626.5 10[1,5]2 -309.99155.1 4819143.6 232.0981.3 1013 -594.2150.0 494.3136.8 1230.5102.4 1033 9304 41.933.0 46.225.6 59.217.2 1053 -248.051.9 360.244.8 398.536.7 10[1,5]3 -786.3222.0 650.0191.9 1126.4143.3 1014 -136.8102.2 200.5105.3 71.075.5 1034 994 19.825.7 6.621.1 6.418.2 1054 -14.940.9 29.243.5 16.730.4 10[1,5]4 -530.0196.1 882.1173.9 245.88149.6 1015 -802.0110.3 813.4100.8 1197.2959.1 1035 132121 76.036.6 133.626.4 47.018.3 1055 -98.065.1 101.1957.7 352.3827.3 10[1,5]5 -1129.9178.2 927.18185.0 1068.9994.0 117

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TableB-3. AverageCPUtimesandnumberofiterationsforN=5,6,7withvaryingbuyerdemands Optimistic Pessimistic Worst-case NQLQFCapacitySet# Time(s) Time(s)#Cuts Time(s)#Cuts 550501 1 2.16.1 0.36.8 525251 0.2 0.22.8 0.14.2 525871 2.2 1.94 0.510.7 587251 0.8 1.35.1 523.9 544442 1.7 1.97.3 1.622.3 522222 0.2 0.22.9 0.411 522772 1.6 1.45.5 1.115.8 577222 0.6 0.96.5 2.623.5 548483 1.4 2.97.2 3.322.6 524243 0.2 0.32.8 0.410.9 524843 2.1 1.64.1 0.912.6 584243 0.9 1.56.6 3.929.5 538384 1.1 1.46.7 120.7 519194 0.2 0.22.7 0.33.6 519664 1 0.84.2 0.615.4 566194 0.6 16.9 2.324.6 583835 12.3 19.29.9 19.727.2 541415 0.9 12.9 0.811.9 5411455 12.4 8.84.7 1.418.1 5145415 4.3 7.97.5 10.426.9 660601 5.4 13.611.2 11.533.8 630301 0.8 13.1 1.513.8 6301051 11.1 7.310.6 2.424.3 6105301 3.5 8.713.2 7.230.2 662622 5.6 8.39.5 6.834.5 631312 0.7 0.74.2 115.5 6311082 5.8 5.36.7 1.421.9 6108312 2.7 5.810.3 8.436.8 658583 6.4 11.611.3 8.331.8 629293 0.7 0.83.3 1.814.2 6291023 8.1 6.19.6 2.227.6 6102293 3.1 7.611.6 937.8 647474 3.2 5.411.9 6.531.2 623234 0.4 0.33.2 1.114.2 623824 3.2 2.46.5 2.119.4 682234 1.4 2.99.8 10.939.3 61011015 35.8 71.611.6 31.432.4 650505 3.2 4.23.6 5.916.6 6501775 43.5 34.17.8 8.321 6177505 13.1 4213.6 83.744.5 770701 13 29.115.5 15.237.4 735351 1.5 2.53.9 1.916 7351221 19.7 17.711.4 1.728.3 7122351 7 23.516.9 27.451.2 776762 13.4 3316.2 12.737.3 738382 1.2 23.5 218.6 7381332 15.7 17.311.4 2.936.7 7133382 6.7 21.415.3 23.352.9 769693 14.4 27.815.7 32.335.6 734343 1.6 2.13.6 4.116.2 7341213 17.1 15.911.9 2.726.6 7121343 6.6 2115.8 35.653.7 757574 8.3 1615.3 13.344.8 728284 0.7 1.13.6 2.416 7281004 8.4 7.210.1 229.3 7100284 4 10.814.9 21.952.8 71121125 70 185.318.8 92.839.2 756565 5.6 10.14.4 10.217.6 7561965 91.3 88.910.1 14.225.1 7196565 26 96.816.3 8065.7 118

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TableB-4. AverageCPUtimesandnumberofiterationsforN=8,9,10withvaryingprotectioncosts Optimistic Pessimistic Worst-case NxiCapacitySet# Time(s) Time(s)#Cuts Time(s)#Cuts 811 40.9 125.340.8 55.819.7 831 3.7 6.15.2 17.23.5 851 40.2 48.913.4 23.63.3 8[1,5]1 15.9 51.124.3 80.636.9 812 47.1 77.421.6 59.538.6 832 2.7 4.24 19.53.1 852 38.9 36.89.6 33.86.6 8[1,5]2 17.8 51.322.6 71.8131.7 813 44.2 88.828.7 48.444.1 833 3 4.74.6 17.13.8 853 41.1 44.512 27.25.2 8[1,5]3 10.4 42.419.8 85.166.3 814 22.1 50.931.4 53.721.6 834 1.8 2.64.7 18.33.6 854 28.7 19.712.7 26.88.8 8[1,5]4 9.7 28.922.4 8242.1 815 153.6 561.825.4 56.1128.1 835 11.1 24.25.4 21.720.9 855 170.6 203.213.1 31.121.3 8[1,5]5 41.2 242.322 90.4312.3 911 122.7 283.552.6 71.344.3 931 5 9.24.9 16.35.1 951 47.1 100.914.9 23.89 9[1,5]1 21.2 93.126.3 98.6128.2 912 134.3 291.433.4 77.1108.4 932 6.8 12.24.4 24.515.3 952 177.8 122.616.1 3929.6 9[1,5]2 64.1 185.438.6 116.6197.1 913 56.5 245.237.1 68.572.3 933 3.6 8.84.2 19.814.1 953 41.8 57.511 26.324.6 9[1,5]3 23 127.630.7 110.1275.2 914 51.2 122.441.4 72.782.7 934 2.5 4.85 19.64.6 954 46.4 47.217.6 28.612.7 9[1,5]4 14.2 6134 122.6175.3 915 591.2 163845.1 76.5301.9 935 15.4 49.65.5 22.575 955 307.7 505.417.8 35.136.7 9[1,5]5 138.6 629.133.8 129.5511.2 1011 65.6 391.441.4 26.65.3 1031 3.6 14.85.3 6.41.5 1051 49.9 65.911.7 10.71.8 10[1,5]1 24.4 165.832.9 32.94.5 1012 367.8 856.362.5 850.2105.2 1032 15.3 22.96.5 25.824.5 1052 301 228.220.8 4487.7 10[1,5]2 130.5 418.462.9 474.3139.1 1013 290.4 907.279.5 490.8136.8 1033 9.1 24.77 44.725.6 1053 226.7 283.723.9 324.644.8 10[1,5]3 90.5 466.263.6 550.8191.9 1014 113.1 288.466.4 169.7105.3 1034 4.9 9.77.2 5.821.1 1054 137.9 120.326.3 2443.5 10[1,5]4 25.8 130.450.1 927.6173.9 1015 1258.3 456068.9 658.1100.8 1035 24.9 87.75.4 96.126.4 1055 912.3 100923.8 526.655.2 10[1,5]5 363.7 1581.259.9 2332.1203.4 119

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TableB-5. AverageCPUtimesandnumberofiterationsforN=5,6,7withvaryingbuyerdemands BLG-MP BLG-MP2 0.25(E)]TJ /F6 7.97 Tf 8.47 0 Td[(C) 0.5(E)]TJ /F6 7.97 Tf 8.46 0 Td[(C) 0.25(E)]TJ /F6 7.97 Tf 8.47 0 Td[(C) 0.5(E)]TJ /F6 7.97 Tf 8.47 0 Td[(C) NQLQFCapacitySet Time(s)#Cuts Time(s)#Cuts Time(s)#Cuts Time(s)#Cuts 550501 2.48.1 2.28.1 2.18.1 2.38.1 525251 0.55.0 0.65.0 0.55.0 0.55.0 525871 2.36.0 2.26.0 2.36.0 2.36.0 587251 4.88.1 4.18.0 4.48.1 4.58.1 544442 5.211.0 5.510.9 5.211.1 5.410.9 522222 0.35.3 0.45.3 0.35.3 0.35.3 522772 1.88.0 1.88.0 1.98.0 1.78.0 577222 1.910.6 1.910.6 1.910.6 1.710.6 548483 2.58.8 2.78.8 2.78.8 2.88.8 524243 0.45.1 0.65.1 0.55.1 0.55.1 524843 2.46.3 2.36.3 2.36.3 2.46.3 584243 4.29.5 3.99.4 4.49.6 4.49.6 538384 2.29.4 2.19.4 2.19.4 2.09.4 519194 0.35.2 0.35.2 0.35.2 0.45.2 519664 1.16.5 1.16.5 1.16.5 1.16.5 566194 3.410.5 3.810.5 3.610.5 3.610.6 583835 19.012.1 19.012.1 18.912.1 19.112.1 541415 1.75.2 1.75.3 1.75.2 2.05.3 5411455 9.76.8 10.16.8 10.06.8 10.06.8 5145415 8.710.1 8.910.1 9.010.1 8.910.1 660601 23.717.6 24.017.4 24.817.7 25.317.5 630301 2.95.7 2.95.7 3.05.7 3.05.7 6301051 9.413.4 9.013.4 9.613.4 9.313.4 6105301 12.616.1 13.116.1 13.416.3 14.116.3 662622 26.814.1 28.614.1 27.414.2 29.214.2 631312 1.16.9 1.06.8 1.16.9 0.96.9 6311082 9.610.8 9.510.8 10.010.8 9.710.8 6108312 9.316.2 9.616.2 9.316.2 9.416.2 658583 20.017.9 19.917.9 20.117.9 20.017.9 629293 2.35.9 2.65.8 2.55.9 2.65.8 6291023 9.212.4 9.212.4 9.512.4 9.512.4 6102293 12.515.9 13.216.2 13.916.4 14.516.5 647474 6.114.3 6.014.3 5.914.3 6.014.3 623234 0.95.7 1.05.7 1.05.7 1.05.7 623824 3.19.3 3.29.3 3.29.3 3.29.3 682234 4.112.9 3.512.8 3.913.0 4.213.0 61011015 89.715.5 90.115.6 90.815.6 90.915.6 650505 9.76.2 10.26.3 9.76.2 9.96.3 6501775 43.210.7 44.310.7 43.210.6 43.610.6 6177505 52.017.8 51.617.8 52.217.8 52.117.8 770701 29.517.5 29.217.5 29.317.5 29.117.5 735351 8.57.1 8.67.1 8.57.1 8.57.1 7351221 23.413.8 22.413.8 23.613.8 22.913.8 7122351 24.319.4 24.519.4 24.519.4 24.319.4 776762 42.521.3 42.421.2 42.821.4 42.621.4 738382 9.07.7 9.77.7 9.17.7 9.77.7 7381332 19.614.1 19.314.1 19.414.1 19.414.1 7133382 30.021.5 28.321.5 29.621.7 28.321.7 769693 27.817.7 28.017.7 27.917.7 28.017.7 734343 7.37.0 7.26.8 7.47.0 7.16.8 7341213 21.814.3 22.414.3 21.914.3 22.214.3 7121343 26.020.0 25.520.0 25.920.0 25.620.0 757574 35.322.7 33.922.7 50.222.9 46.822.9 728284 3.87.0 3.87.0 3.87.0 4.07.0 7281004 11.913.5 12.113.5 12.113.5 12.113.5 7100284 18.819.0 18.819.0 19.919.3 19.919.3 71121125 202.221.3 203.821.3 202.321.3 203.521.3 756565 18.67.3 17.87.3 18.57.3 17.97.3 7561965 118.412.3 111.412.3 118.512.3 111.412.3 7196565 112.421.0 112.521.0 112.521.0 112.321.0 120

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TableB-6. ComputationtimesandnumberofiterationsforIMF Parameters CPUTime #ofIterations IE Cutting-PlaneAlgorithms IE Cutting-PlaneAlgorithms pqr Frm( 4 )Frm( 4 ) Frm( 4 )Frm( 4 ) 2534 2 11 51 1111 2535 5 21 91 1414 2536 10 22 122 1313 2537 27 54 291 1717 2538 42 157 371 2929 2554 7 32 324 2323 2555 23 54 646 3232 2556 56 86 1298 3737 2557 171 149 3830 4949 2558 246 2312 5204 6262 2574 19 44 1111 3434 2575 81 1510 3596 6262 2576 210 3526 8199 107107 2577 617 5137 23715 129129 2578 981 10884 37962 197197 3034 3 10 54 66 3035 9 11 106 88 3036 24 12 189 55 3037 44 73 291 99 3038 72 92 366 1010 3054 16 22 379 1515 3055 71 44 1200 2020 3056 196 117 2384 2929 3057 437 1912 4611 3939 3058 693 3215 6887 5050 3074 61 22 1930 1717 3075 357 65 9598 2929 3076 994 1511 21135 5050 3077 2382 2817 47737 6666 3078 4256 4725 86556 8686 4042 0 00 22 66 4043 1 10 65 77 4044 1 11 166 66 4045 3 23 339 1010 4062 11 00 64 99 4063 52 10 335 99 4064 15 11 1530 99 4065 111 43 3810 1414 4082 798 00 201 1111 4083 59 21 1967 1616 4084 476 32 15011 1717 4085 3467 86 43592 2626 5052 0 00 36 88 5053 1 11 191 99 5054 2 21 481 1212 5055 4 32 1142 1313 5082 27 01 142 1212 5083 77 22 1629 2222 5084 23 76 5362 3535 5085 198 109 23178 3939 50102 665 11 294 2222 50103 117 76 6125 5858 50104 1666 2117 21745 7777 50105 7441 3329 136104 9595 6062 1 10 94 1111 6063 1 12 437 1414 6064 3 22 1050 1515 6065 10 88 2432 3232 6092 66 12 196 3939 6093 254 33 3696 2929 6094 45 1512 14013 6969 6095 351 1614 38177 5757 60122 1568 32 300 4444 60123 204 1618 21259 93103 60124 2229 6665 78356 195202 60125 13088 9384 272438 207207 121

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TableB-7. ComputationtimesandnumberofiterationsforICF Parameters CPUTime #ofIterations pqr Frm( 4 )Frm( 4 )Frm( 4 ) Frm( 4 )Frm( 4 ) 2534 050 98 2535 060 77 2536 050 55 2537 1131 1414 2538 1210 1716 2554 190 1815 2555 2170 2423 2556 4250 3731 2557 13340 5844 2558 22500 7657 2574 3100 3125 2575 15191 6540 2576 41301 11954 2577 92421 18072 2578 266512 29481 3034 030 55 3035 050 77 3036 091 77 3037 1111 1012 3038 1130 1412 3054 050 1010 3055 1130 1418 3056 2210 2625 3057 4301 2931 3058 8470 3840 3074 1140 2526 3075 7201 4231 3076 19431 7254 3077 50612 11975 3078 114572 18074 4042 020 88 4043 030 77 4044 041 77 4045 0110 1010 4062 030 1111 4063 150 1212 4064 170 1212 4065 1250 2223 4082 040 1313 4083 180 1920 4084 1160 2727 4085 5270 3833 5052 020 77 5053 150 1111 5054 060 77 5055 0141 99 5082 030 1010 5083 160 1515 5084 1130 1414 5085 2310 2726 50102 030 1212 50103 1100 2423 50104 1151 2526 50105 8381 5344 6062 020 99 6063 060 1011 6064 081 1111 6065 180 910 6092 030 1313 6093 071 1616 6094 1140 1718 6095 3240 3125 60122 050 1717 60123 1110 2727 60124 4240 4033 60125 12431 5647 122

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TableB-8. ComputationtimesandnumberofiterationsforIMF-C Parameters Avg.CPUTime Avg.#ofIterations NumCuts pqr FCRobust FCRobust FC 1022 0.330.67 5.672.33 3.67 1023 1.331.67 8.333.67 4.67 1024 0.670.67 14.672.33 12.33 1032 0.330.33 8.333.00 5.67 1033 2.331.67 14.676.33 8.33 1034 3.003.00 30.338.00 22.33 1042 0.670.67 15.002.67 12.33 1043 4.674.67 24.0014.67 10.00 1044 6.006.00 54.6719.33 34.33 1522 1.671.67 5.332.67 2.67 1523 4.335.00 8.005.00 3.33 1524 3.002.00 11.002.67 8.00 1532 2.672.33 6.673.67 3.00 1533 9.007.67 15.337.00 6.67 1534 3.673.67 26.003.67 22.67 1542 3.332.67 13.674.00 8.67 1543 3.673.67 30.004.00 25.67 1544 6.676.33 45.335.67 39.67 2022 2.672.33 5.002.67 2.33 2023 10.339.33 10.003.33 6.33 2024 5.675.67 11.332.00 9.33 2032 5.336.00 7.335.00 3.00 2033 10.338.00 15.334.00 11.00 2034 25.3327.33 20.335.33 14.67 2042 3.674.33 12.004.00 8.33 2043 16.0013.00 21.007.67 12.33 2044 27.6726.33 46.679.67 38.00 2522 8.338.33 5.674.67 1.00 2523 25.6728.67 6.333.67 3.00 2524 39.6738.00 9.333.33 6.00 2532 5.674.67 8.003.33 4.33 2533 24.0027.67 13.674.67 9.00 2534 67.6773.33 18.676.33 13.00 2542 10.6710.67 14.675.67 9.00 2543 28.0028.00 16.337.67 8.67 2544 199.33210.00 31.0016.67 15.33 2534 65.3364.67 17.335.00 12.33 2535 183.67186.33 26.0011.00 15.33 2536 128.67133.00 71.007.00 64.00 2537 70.0053.67 160.003.00 156.33 2538 211.33215.33 116.336.67 109.67 2554 127.67126.67 82.6716.00 67.00 2555 249.33268.00 106.6722.67 85.33 2556 132.33132.00 326.0010.33 316.67 2557 564.67122.67 909.007.67 900.67 2558 1576.001463.00 486.0075.67 407.67 2574 288.00274.33 94.0050.33 43.33 2575 1371.0060.00 1786.3312.33 1776.00 2576 1006.67203.67 1477.0024.33 1454.33 2577 2623.671180.33 1685.33102.67 1579.67 2578 10357.673362.00 2805.67269.00 2544.33 123

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TableB-9. ComputationtimesandnumberofiterationsforICF-C Parameters Avg.CPUTime Avg.#ofIterations NumCuts pqr FCRobust FCRobust FC 1022 171.00144.67 6.333.00 3.67 1023 227.33136.00 10.673.33 7.33 1024 245.33173.00 14.334.33 10.67 1032 198.00127.67 7.673.67 4.00 1033 158.67120.00 18.333.67 14.67 1034 206.67197.00 25.676.67 20.67 1042 73.3367.33 12.673.00 10.00 1043 399.00342.00 33.3310.33 23.67 1044 250.00199.67 62.3310.00 52.67 1522 5243.003762.67 6.004.00 2.33 1523 5852.674293.33 8.673.33 5.33 1524 14621.009361.33 6.334.00 2.33 1532 3928.002576.67 10.334.33 5.67 1533 3079.002461.00 19.332.67 16.33 1534 13849.678979.00 26.007.00 19.00 1542 2727.671603.00 17.004.67 12.00 1543 4876.334142.67 28.337.00 22.00 1544 4739.672831.00 68.335.33 62.33 124

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BIOGRAPHICALSKETCH MikePrincewasborninColumbus,Ohioandspentallofhischildhoodindifferentareasofthestate.HegraduatedfromWestervilleNorthHighSchoolin2004.HethenattendedTheOhioStateUniversityandcompleteda5-yearcombinedbachelor'sandmaster'sprogramintheDepartmentofIndustrialandSystemsEngineering.InAugust2009,hebeganhisPh.D.inindustrialandsystemsengineeringattheUniversityofFlorida.Hisresearchfocusedonnetworkoptimization,decompositionmethods,andthree-stageinterdictionandforticationmodels.HewillreceivehisdegreeinApril2013andmoveonworkatBNSFRailwayasaSeniorOperationsResearchSpecialist. 130