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Optimization Models for Sourcing and Distribution Systems

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Title:
Optimization Models for Sourcing and Distribution Systems
Creator:
Pérez Sigüenza, Cinthia C.
Place of Publication:
[Gainesville, Fla.]
Publisher:
University of Florida
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english
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1 online resource (173 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Industrial and Systems Engineering
Committee Chair:
GEUNES,JOSEPH PATRICK
Committee Co-Chair:
RICHARD,JEAN-PHILIPPE P
Committee Members:
GUAN,YONGPEI
PAUL,ANAND ABRAHAM
Graduation Date:
5/3/2014

Subjects

Subjects / Keywords:
Average cost ( jstor )
Business orders ( jstor )
Cost allocation ( jstor )
Customers ( jstor )
Delivery costs ( jstor )
Delivery services ( jstor )
Fees ( jstor )
Geographical distribution ( jstor )
Transportation costs ( jstor )
Warehouses ( jstor )
Industrial and Systems Engineering -- Dissertations, Academic -- UF
competition -- delivery-fee -- distribution-system -- dual-sourcing -- fee-schedule -- inventory
Genre:
Electronic Thesis or Dissertation
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
Industrial and Systems Engineering thesis, Ph.D.

Notes

Abstract:
The first part of this dissertation examines the trade-offs involved in dual sourcing in a supply chain. Although splitting procurement orders may increase shipping costs as a result of diseconomies of scale, dual sourcing may offer certain benefits that can more than offset these costs. We consider a single-stage inventory replenishment model that includes two delivery modes: a cheaper, less reliable mode, and another, more expensive but perfectly reliable mode. The high-reliability mode is only utilized in replenishment intervals in which the less-reliable mode's lead time exceeds a certain value. This permits substituting the high-reliability mode for safety stock, to some degree. We characterize optimal replenishment decisions, and potential benefits of simultaneously using two delivery modes. In the second part of this dissertation we study the USDA food distribution programs administered by Florida Department of Agricultural and Consumer Services, where distributors are contracted to provide warehousing and delivery services for recipient agencies. To reduce the imbalance in the average expense per case for the agencies, and to ensure the attractiveness of the invitation to bid to potential contractors, we propose and evaluate different fee scenarios that result from the combination of different distribution system and fee schedule design parameters. As a result, we identify fee scenarios that reduce the imbalance in the agencies' expenses and guarantee a minimal increase for those agencies with higher expenses under the new fee structures. Finally, we consider a problem in which two carriers compete for the distribution services for a shipper who wishes to supply a set of retail points. The shipper uses a single-item auction for contracting the delivery services for each retail point. Each carrier places a bid for serving each of the delivery points, with the goal of maximizing the number of retail points assigned. We consider the carrier's problem of determining the bid prices it should submit to achieve this goal. Using three cost allocation mechanisms specified by the shipper, we evaluate the conditions under which the associated game has an equilibrium solution, and how the different cost parameters affect the model's results and the retail point allocation to carriers. ( en )
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.
Thesis:
Thesis (Ph.D.)--University of Florida, 2014.
Local:
Adviser: GEUNES,JOSEPH PATRICK.
Local:
Co-adviser: RICHARD,JEAN-PHILIPPE P.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-05-31
Statement of Responsibility:
by Cinthia C. Pérez Sigüenza.

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UFRGP
Rights Management:
Applicable rights reserved.
Embargo Date:
5/31/2015
Resource Identifier:
907295064 ( OCLC )
Classification:
LD1780 2014 ( lcc )

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OPTIMIZATIONMODELSFORSOURCINGANDDISTRIBUTIONSYSTEMSByCINTHIAC.PEREZSIGUENZAADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2014

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2014CinthiaC.PerezSiguenza 2

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Idedicatethistomyfamily.Youaremyinspirationandmotivation. 3

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ACKNOWLEDGMENTS IwouldliketorstthankGodforthemanyblessingsIhavereceivedduringmylifeandforbeingmystrengthandguidethroughoutthedurationofthiswork.Iwouldalsoliketoexpressmydeepestgratitudetomyadvisor,Dr.JosephGeunes.AllIhaveaccomplishedduringtheprogramwouldhavenotbeenpossiblewithouthisguidanceandtremendousamountofpatience.Icannotthinkofabetteradvisorforme,andIwillalwaysbegratefultohimforacceptingmeashisstudent.IwouldalsoliketothankmycommitteemembersDr.JeanPhilippeRichard,Dr.YongpeiGuanandDr.AnandPaul.Igreatlyappreciatetheirgenerosity,notonlywhensharingtheirexperienceandknowledge,butalsotheirtime.Iwouldalsoliketothankmyparents,mybrotherandmysisterfortheinniteloveandunconditionalsupporttheyalwaysgiveme.AllIhaveandallIwanttobeisbecauseofthem.IalsowanttoexpressmygratitudetomydearCarlos,whohasproventobeoneofmybiggestsupporters,evenwhenmydreamsmadehislifeharder.Finally,Iwouldalsoliketothankmyfriendsbackhome,whodespitethedistancealwaysfeltclose,andmynewfriendsinGainesvillewhosharedthisjourneywithmeandmadeitmorepleasant. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 9 LISTOFFIGURES ..................................... 11 ABSTRACT ......................................... 14 CHAPTER 1INTRODUCTION ................................... 16 2A(Q,R)INVENTORYREPLENISHMENTMODELWITHMULTIPLEDELIVERYMODES ........................................ 20 2.1MotivationandLiteratureReview ....................... 20 2.2ProblemDescription .............................. 27 2.2.1Single-ModeModel ........................... 27 2.2.1.1Shippingmodewithuniformlydistributedleadtime ... 27 2.2.1.2Deterministicleadtimemode ................ 33 2.2.2Dual-ModeModel ............................ 34 2.2.2.1Case1:2[L2,l+L2) ................... 35 2.2.2.2Case2:2[l+L2,u) ................... 41 2.3NumericalAnalysis ............................... 48 2.3.1ImpactofIncreasingl ......................... 50 2.3.2ImpactofIncreasingu ........................ 51 2.3.3ImpactofIncreasingMode2LeadTime,L2 ............. 53 2.3.4ChangeinDemandRate: ...................... 53 2.3.5ChangeinHoldingCostperUnitperTime:h ............ 55 2.3.6ChangeinIncrementalFixedOrderCostforMode2:A ....... 56 2.3.7ChangeinPurchasingCostWhenUsingShippingMode2:c2 ... 57 2.3.8ChangeintheLeadTimeDistributionforSupplyMode1 ...... 58 2.3.9ComparingSupplyModes ....................... 60 3ANALYSISANDIMPROVEMENTOFTHEUSDASCHOOLLUNCHANDFOODBANKDISTRIBUTIONPROGRAMSINTHESTATEOFFLORIDA ....... 64 3.1IntroductionandCurrentStatus ........................ 64 3.2RegionalCongurations ............................ 68 3.3ScopeofParametersStudied ......................... 70 3.3.1RegionalConguration ......................... 71 3.3.2DeliveryFeePolicies .......................... 71 3.3.3StorageFeePolicies .......................... 73 3.3.4DistributionNetworkStructure ..................... 74 3.3.5LimitsontheQuantityofProductSenttoFoodProcessors ..... 74 5

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3.4AnalysisofResults ............................... 75 3.4.1SeparateNSLPandTEFAPNetworks ................ 77 3.4.2CombinedDistributionNetworks ................... 78 3.4.3LimitsonVolumeSenttoFoodProcessors .............. 78 4DESIGNINGDISTRIBUTIONFEESUNDERCOMPETITION .......... 81 4.1Introduction ................................... 81 4.2ProblemDescription .............................. 87 4.2.1CostAllocationMechanisms ...................... 88 4.2.1.1Distance-basedcostallocation ............... 89 4.2.1.2Uniformcostallocation ................... 90 4.2.1.3Branch-basedcostallocation ................ 90 4.2.2GameDenition ............................. 92 4.2.2.1Equilibriumsolutions ..................... 94 4.2.3SolutionProcedure ........................... 95 4.3NumericalAnalysis ............................... 96 4.3.1Distance-BasedCostAllocation .................... 98 4.3.1.1Impactofxedtransportationcost ............. 99 4.3.1.2Impactofvariabletransportationcost ........... 102 4.3.1.3Impactofthegeographicaldistributionofcarriersanddeliverypoints ........................ 103 4.3.2UniformCostAllocation ........................ 104 4.3.2.1Impactofxedtransportationcost ............. 106 4.3.2.2Impactofvariabletransportationcost ........... 107 4.3.2.3Impactofthegeographicaldistributionofcarriersanddeliverypoints ........................ 110 4.3.3Branch-BasedCostAllocation ..................... 112 4.3.3.1Impactofxedtransportationcost ............. 113 4.3.3.2Impactofvariabletransportationcost ........... 115 4.3.3.3Impactofthegeographicaldistributionofcarriersanddeliverypoints ........................ 117 5CONCLUSIONSANDFUTURERESEARCH ................... 119 APPENDIX AORDERCROSSINGCONDITION ......................... 126 BSINGLE-MODESOLUTIONWITHQUANTITYBASEDUPPERBOUND .... 127 CCONVEXITYCONDITIONSFORGI(Q,) .................... 128 DALGORITHMFORDUAL-MODEMODELWHEN2[L2,l+L2) ........ 131 EALGORITHMFORDUAL-MODEMODELWHEN2[l+L2,u) ........ 132 6

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FVALIDITYOFAVERAGEINVENTORYEXPRESSIONFORDUAL-MODEMODELWHEN2[l+L2,u] ........................... 133 GNSLP:ANNUALDEMANDANDLOCATIONOFRECIPIENTAGENCIES ... 134 HTEFAP:ANNUALDEMANDANDLOCATIONOFRECIPIENTAGENCIES ... 135 ICOSTESTIMATIONMETHODOLOGY ...................... 136 JPROPOSEDREGIONALCONFIGURATIONS .................. 140 KNSLP:CURRENTREGIONALCONFIGURATION ................ 142 LPROPOSEDREGIONALCONFIGURATION1 .................. 143 MPROPOSEDREGIONALCONFIGURATION2 .................. 144 NPROPOSEDREGIONALCONFIGURATION3 .................. 145 OPROPOSEDREGIONALCONFIGURATION4 .................. 146 PPROPOSEDREGIONALCONFIGURATIONS.COSTANDDEMAND ..... 147 QFEESCENARIOSWITHTHESMALLESTAVERAGEOVERPAYMENTWHENNSLPANDTEFAPUSETWODISTRIBUTIONNETWORKS .......... 149 RFEESCENARIOSWITHTHESMALLESTAVERAGEOVERPAYMENTWHENNSLPANDTEFAPUSEONEDISTRIBUTIONNETWORK ........... 152 SMAXIMUM30%OFENTITLEMENTSENTTOFOODPROCESSORS:EFFECTONPROPOSEDFEESWHENNSLPANDTEFAPUSETWODISTRIBUTIONNETWORKS ..................................... 155 TMAXIMUM30%OFENTITLEMENTSENTTOFOODPROCESSORS:EFFECTONPROPOSEDFEESWHENNSLPANDTEFAPUSEONEDISTRIBUTIONNETWORK ...................................... 157 UMAXIMUM50%OFENTITLEMENTSENTTOFOODPROCESSORS:EFFECTONPROPOSEDFEESWHENNSLPANDTEFAPUSETWODISTRIBUTIONNETWORKS ..................................... 159 VMAXIMUM50%OFENTITLEMENTSENTTOFOODPROCESSORS:EFFECTONPROPOSEDFEESWHENNSLPANDTEFAPUSEONEDISTRIBUTIONNETWORK ...................................... 161 WDISTRIBUTIONFEEDESIGNINGMODELSOLUTIONPROCEDURE ..... 163 XDISTANCE-BASEDCOSTALLOCATION:TYPEOFNON-TRIVIALEQUILIBRIUMSOLUTIONS ..................................... 164 YUNIFORMCOSTALLOCATION:NON-TRIVIALEQUILIBRIUMSOLUTIONS 166 7

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ZBRANCH-BASEDCOSTALLOCATION:TYPEOFNON-TRIVIALEQUILIBRIUMSOLUTIONS ..................................... 168 REFERENCES ....................................... 170 BIOGRAPHICALSKETCH ................................ 173 8

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LISTOFTABLES Table page 2-1Baseparametervaluesfornumericalanalysis. .................. 49 2-2Effectoflonthedual-modemodel. ........................ 51 2-3Effectofuonthedual-modemodel. ........................ 52 2-4EffectofL2onthedual-modemodel. ........................ 53 2-5Effectofdemandrateonthedual-modemodel. .................. 54 2-6Effectofholdingcostonthedual-modemodel. .................. 55 2-7Effectofholdingcostaspercentageofpurchasingcost. ............. 56 2-8EffectofAonthedual-modemodel. ........................ 57 2-9Dual-modemodelforL1fl+Beta(a,b)g. .................. 59 3-1Summaryofestimatedstorageanddeliveryexpenses,Schoolyear2011-2012. 66 3-2Increasedinannualdemandwhenusingapolicythatlimitstheamountofproductsenttofoodprocessors ........................... 79 4-1Carriersandcustomerareasforgeographicaldistributiontype1. ........ 97 I-1NSLP:Summaryofestimateddeliveryandstoragecostsforthestate-contractedwarehouses,Schoolyear2011-2012 ........................ 139 I-2TEFAP:Summaryofestimateddeliveryandstoragecostsforthestate-contractedwarehouses,Schoolyear2011-2012 ........................ 139 J-1Proposedregionalcongurations .......................... 140 P-1Currentregionalconguration.Costsanddemand ................ 147 P-2Proposedregionalconguration1.Costsanddemand .............. 147 P-3Proposedregionalconguration2.Costsanddemand .............. 148 P-4Proposedregionalconguration3.Costsanddemand .............. 148 P-5Proposedregionalconguration4.Costsanddemand .............. 148 Q-1Feescenarioswithsmallestaverageoverpayment.TwodistributionnetworksforNSLPandTEFAP ................................ 149 Q-2Uniformdistributionandstoragefeesforthe25feescenarioswiththesmallestaverageoverpayment.TwodistributionnetworksforNSLPandTEFAP ..... 150 9

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Q-3Order-quantity-anddelivery-frequency-basedfeescheduleforthe25feescenarioswiththesmallestaverageoverpayment.TwodistributionnetworksforNSLPandTEFAP ...................................... 151 R-1Feescenarioswithsmallestaverageoverpayment.OnedistributionnetworksforNSLPandTEFAP ................................ 152 R-2Uniformdistributionandstoragefeesforthe25feescenarioswiththesmallestaverageoverpayment.OnedistributionnetworkforNSLPandTEFAP ..... 153 R-3Order-quantity-anddelivery-frequency-basedfeescheduleforthe25feescenarioswiththesmallestaverageoverpayment.OnedistributionnetworkforNSLPandTEFAP ...................................... 154 S-1Uniformdistributionandstoragefeesforthe25feescenarioswiththesmallestaverageoverpayment.TwodistributionnetworksforNSLPandTEFAPandmaximum30%ofentitlementsenttofoodprocessors .............. 155 S-2Order-quantity-anddelivery-frequency-basedfeescheduleforthe25feescenarioswiththesmallestaverageoverpayment.TwodistributionnetworksforNSLPandTEFAPandmaximum30%ofentitlementsenttofoodprocessors ..... 156 T-1Uniformdistributionandstoragefeesforthe25feescenarioswiththesmallestaverageoverpayment.OnedistributionnetworkforNSLPandTEFAPandmaximum30%ofentitlementsenttofoodprocessors .............. 157 T-2Order-quantity-anddelivery-frequency-basedfeescheduleforthe25feescenarioswiththesmallestaverageoverpayment.OnedistributionnetworkforNSLPandTEFAPandmaximum30%ofentitlementsenttofoodprocessors ..... 158 U-1Uniformdistributionandstoragefeesforthe25feescenarioswiththesmallestaverageoverpayment.TwodistributionnetworksforNSLPandTEFAPandmaximum50%ofentitlementsenttofoodprocessors .............. 159 U-2Order-quantity-anddelivery-frequency-basedfeescheduleforthe25feescenarioswiththesmallestaverageoverpayment.TwodistributionnetworksforNSLPandTEFAPandmaximum50%ofentitlementsenttofoodprocessors ..... 160 V-1Uniformdistributionandstoragefeesforthe25feescenarioswiththesmallestaverageoverpayment.OnedistributionnetworkforNSLPandTEFAPandmaximum50%ofentitlementsenttofoodprocessors .............. 161 V-2Order-quantity-anddelivery-frequency-basedfeescheduleforthe25feescenarioswiththesmallestaverageoverpayment.OnedistributionnetworkforNSLPandTEFAPandmaximum50%ofentitlementsenttofoodprocessors ..... 162 10

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LISTOFFIGURES Figure page 2-1Single-modemodel,whereL1U(l,u). .................... 28 2-2Expectedinventoryasafunctionoftimeforthesingle-modemodel,whereL1U(l,u). .................................... 29 2-3Leadtimeupperboundasastepwisefunctionoftheorderquantity. ...... 31 2-4Averageinventorycostwhenleadtimeupperboundisafunctionofquantityordered. ........................................ 32 2-5Dual-modemodelrealization.Case1:2[L2,l+L2). ............. 36 2-6ExpectedInventoryasfunctionoftime.Case1:2[L2,l+L2). ........ 37 2-7Dual-modemodelrealization(Case2.1:2[l+L2,u)andL1)]TJ /F3 11.955 Tf 11.95 0 Td[(L2). ... 42 2-8Dual-modemodelrealization(Case2.2:2[l+L2,u)andL1>)]TJ /F3 11.955 Tf 11.96 0 Td[(L2). ... 42 2-9Expectedinventoryasfunctionoftime(Case2:2[l+L2,u)). ........ 44 2-10%CRforsupplymodeswithdifferentL2andc. ................. 60 2-11%CRforsupplymodeswithdifferent. ...................... 61 2-12%CRforsupplymodeswithdifferentE[L1]. .................... 62 3-1FloridaFoodDistributionProgramNetwork .................... 65 3-2Summaryofestimatedstorageanddeliveryexpenses,2011-2012 ....... 67 3-3SummaryofproductnetvaluesentbyUSDAperdestination,2011-2012 ... 68 3-4Parameterstobestudied .............................. 70 4-1Instancewithuniformlydistributedcustomers ................... 100 4-2Distance-basedcostallocation:numberofnon-trivialequilibriumsolutions ... 101 4-3Instancewithfavorablelocationforcarrier1. ................... 101 4-4Instancewithtwoclustersofcustomers. ...................... 102 4-5Instancewheredeliverypointsarenotsubstantiallyclosertoanycarrier. .... 103 4-6Instancewhendeliverypointsandcarriersarelocatedwithinthesameregion. 105 4-7Uniformcostallocation:numberofnon-trivialequilibriumsolutions. ....... 108 4-8Instancewithnoclusterofcustomers. ....................... 109 11

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4-9Branch-basecostallocation:numberofnon-trivialequilibriumsolutions. .... 115 4-10Instancewithisolateddeliverypoint. ........................ 117 G-1AnnualdemandsandlocationsofNSLPrecipientagencies ........... 134 H-1AnnualdemandsandlocationsofTEFAPrecipientagencies .......... 135 K-1NSLP:Currentregionalconguration ........................ 142 L-1Proposedregionalconguration1 ......................... 143 M-1Proposedregionalconguration2 ......................... 144 N-1Proposedregionalconguration3 ......................... 145 O-1Proposedregionalconguration4 ......................... 146 X-1Distance-basedcostallocationandgeographicaldistributiontype1:typeofnon-trivialequilibriumsolutions ........................... 164 X-2Distance-basedcostallocationandgeographicaldistributiontype2:typeofequilibriumsolutions. ................................. 164 X-3Distance-basedcostallocationandgeographicaldistributiontype3:typeofnon-trivialequilibriumsolutions. ........................... 165 X-4Distance-basedcostallocationandgeographicaldistributiontype4:typeofnon-trivialequilibriumsolutions. ........................... 165 Y-1Uniformcostallocationandgeographicaldistributiontype1:typeofnon-trivialequilibriumsolutions. ................................. 166 Y-2Uniformcostallocationandgeographicaldistributiontype2:typeofnon-trivialequilibriumsolutions. ................................. 166 Y-3Uniformcostallocationandgeographicaldistributiontype3:typeofnon-trivialequilibriumsolutions. ................................. 167 Y-4Uniformcostallocationandgeographicaldistributiontype4:typeofnon-trivialequilibriumsolutions. ................................. 167 Z-1Branch-basedcostallocationandinstancegeographicaldistribution1:typeofnon-trivialequilibriumsolutions. ......................... 168 Z-2Branch-basedcostallocationandinstancegeographicaldistribution2:typeofnon-trivialequilibriumsolutions. ......................... 168 Z-3Branch-basedcostallocationandgeographicaldistributiontype3:typeofnon-trivialequilibriumsolutions. ........................... 169 12

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Z-4Branch-basedcostallocationandgeographicaldistributiontype4:typeofnon-trivialequilibriumsolutions. ........................... 169 13

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyOPTIMIZATIONMODELSFORSOURCINGANDDISTRIBUTIONSYSTEMSByCinthiaC.PerezSiguenzaMay2014Chair:JosephGeunesMajor:IndustrialandSystemsEngineeringTherstpartofthisdissertationexaminesthetrade-offsinvolvedindualsourcinginasupplychain.Althoughsplittingprocurementordersmayincreaseshippingcostsasaresultofdiseconomiesofscale,dualsourcingmayoffercertainbenetsthatcanmorethanoffsetthesecosts.Weconsiderasingle-stageinventoryreplenishmentmodelthatincludestwodeliverymodes:acheaper,lessreliablemode,andanother,moreexpensivebutperfectlyreliablemode.Thehigh-reliabilitymodeisonlyutilizedinreplenishmentintervalsinwhichtheless-reliablemode'sleadtimeexceedsacertainvalue.Thispermitssubstitutingthehigh-reliabilitymodeforsafetystock,tosomedegree.Wecharacterizeoptimalreplenishmentdecisions,andpotentialbenetsofsimultaneouslyusingtwodeliverymodes.InthesecondpartofthisdissertationwestudytheUSDAfooddistributionprogramsadministeredbyFloridaDepartmentofAgriculturalandConsumerServices,wheredistributorsarecontractedtoprovidewarehousinganddeliveryservicesforrecipientagencies.Toreducetheimbalanceintheaverageexpensepercasefortheagencies,andtoensuretheattractivenessoftheinvitationtobidtopotentialcontractors,weproposeandevaluatedifferentfeescenariosthatresultfromthecombinationofdifferentdistributionsystemandfeescheduledesignparameters.Asaresult,weidentifyfeescenariosthatreducetheimbalanceintheagencies'expensesandguaranteeaminimalincreaseforthoseagencieswithhigherexpensesunderthenewfeestructures. 14

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Finally,weconsideraprobleminwhichtwocarrierscompeteforthedistributionservicesforashipperwhowishestosupplyasetofretailpoints.Theshipperusesasingle-itemauctionforcontractingthedeliveryservicesforeachretailpoint.Eachcarrierplacesabidforservingeachofthedeliverypoints,withthegoalofmaximizingthenumberofretailpointsassigned.Weconsiderthecarrier'sproblemofdeterminingthebidpricesitshouldsubmittoachievethisgoal.Usingthreecostallocationmechanismsspeciedbytheshipper,weevaluatetheconditionsunderwhichtheassociatedgamehasanequilibriumsolution,andhowthedifferentcostparametersaffectthemodel'sresultsandtheretailpointallocationtocarriers. 15

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CHAPTER1INTRODUCTIONThisdissertationconsistsoftwomainresearchtopics:therstisrelatedtotheuseofdualsourcingtoreduceinventoryrelatedcostsandthesecondisconcernedwiththeanalysisanddesignofdistributionsystems.Whenselectingsuppliers,ingeneral,thereisaconictbetweenservicecostandreliability,wherethelatterismeasuredintermsofdeliveryperformance.Whenalow-costshippingmodeispreferred,abuyermayneedtodealwithhighuncertaintylevelsinthedeliverytime.Assumingthatinventorystockoutsareundesirable,suchabuyerwillrequireincreasedsafetyinventorytoavoidshortagesforthecaseswhentheleadtimeisunusuallylong.Insteadofusingsafetyinventoryasabufferforlongerdeliverytimes,thebuyercanuseafaster,morereliable(andconsequently,moreexpensive)shippingmodeasanemergencysupplier,onlywhenstockoutsarelikelytooccur.However,whenusingadualsourcingstrategytoreduceinventoryrelatedcost,itisnotevidentwhatcombinationofsupplymodesandreplenishmentparametersoneshouldusethatresultinlowtotalinventorycostandreducedprobabilityofstockouts,comparedwithasinglesourceoption.Inrecentyearstheuseofmultiplesourcinghasbeenfacilitatedbythedevelopmentofinformationtechnologythatallowustohavenearreal-timeinformationaboutthedeliverystatusoftheordersplacedwithdifferentsuppliers.Additionally,multiplesourcinghasbeenstudiedinthecontextofmanagingsupplychaindisruptions,asintheworkpresentedbyYuetal.[ 40 ]andBabichetal.[ 2 ].Althoughusingmorethanonesupplymodeincreasesoperationalandshippingcostsduetodiseconomiesofscale,itmayoffercertainbenetsthatcanmorethanoffsetthesecosts.Intherstpartofthisdissertation,wewillevaluatethebenetsofdeliverymodediversicationasastrategytoreduceinventory-relatedcostsandidentifycircumstancesunderwhichhavingdualsourcingisrecommendedoverthesinglesourceoption. 16

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ThesecondpartofthisdissertationfocusesontheanalysisanddesignofthedistributionsystemoftwooftheUnitedStateDepartmentofAgriculture(USDA)fooddistributionprogramsinthestateofFlorida:theNationalSchoolLunchProgram(NSLP)andtheEmergencyFoodAssistantProgram(TEFAP).TheFloridaDepartmentofAgricultureandConsumerServices(FDACS)istheresponsiblefortheadministrationoftheprogramsinthestateofFlorida.Usingabiddingprocess,FDACSinvitescontractorstosubmitproposalsforreceipt,storageanddeliveryoftheproductssentbytheUSDAandfoodsupplierstotherecipientagencieslocatedinthestate.Therecipientagencieshavemorethan5,000orderchoicesthatcanbeplacedtoUSDAfoodsuppliersthroughtheFDACS.However,theordersshouldfollowthedeliverypoliciesdenedbytheUSDAandfoodsuppliersregardingminimumorderquantity,numberofstopsperdeliverytruckandinsomecasestypeofproduct.ThecomplexityofthedistributionsystemisincreasedbytheservicerequirementsdenedintheInvitationtoBid(ITB)designedbytheFDACS,whichdecreasesthenumberofpotentialcontractorsinterestedinthebid.Additionally,asaresultofthehighservicefeesofferedbycurrentcontractors,severalrecipientagenciesprefertoleavethestatedistributionnetworkandcontractcommercialfooddistributorsornegotiatedirectlywithfoodprocessors,whichdecreasesthevolumeofproductsentthroughthestatedistributionnetwork,andincreasestheaveragecostpercaseoftheservice.Thesefactors,inadditiontothehighlycompetitiveenvironmentofthefooddistributionbusiness,whichuseseconomiesofscaleasleveragetoincreaseprotability,presentachallengefortheFDACStoofferanattractiveITBthatresultsincompetitivedeliveryfeeschedulesfortheagencies.Weproposenewdistributionsystemandfeescheduledesignparametersthatnotonlyresultinareductionofstorageanddeliveryexpensesfortherecipientagencies,butalsoincreasetheattractivenessoftheITBtopotentialbiddersforallregionsinthestate. 17

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Continuingwiththestudyofdistributionsystems,forourlastproblemwedesignadistributionfeesettingstrategyunderaprice-basedcompetitionmodel,wheretwocompetitorcarrierswanttomaximizethenumberofdeliverypointsservedbytheminagivenregion.DuringthedevelopmentofourprojectwiththeFDACS,oneofthereasonsthatpotentialcontractorshadtojustifytheincreaseindeliveryfeesofferedtotherecipientagencieswasthehighgeographicaldispersionoftheagencies,notonlyamongtheagenciesthatbelongtoeachregion,butalsoinrelationtotheircurrentsetofcustomers.Thisisbecausethedeliveryfeesthatacarriermayoffertoanewpotentialcustomerdonotdependonlyontheparticularcharacteristicsofthatcustomer(e.g.,location,demand,ordesireddeliveryfrequency);theyalsodependonthecharacteristicsofthesetofcustomersthecarrieriscurrentlyserving.Forexample,assumingthereexistssufcienttruckcapacity,ifthenewpotentialcustomerislocatedalongtheexistingdeliveryrouteofthecurrentsetofcustomers,thispotentialcustomer'smarginaltransportationcostislowercomparedwiththecostofaddinganewcustomerwhoislocatedfartherawayfromthecurrentsetofcustomers,forwhichalargerincreaseintheroutewouldbeneeded.Basedontheaboveobservations,itseemsthatagoodstrategyforacarrierthatwantstoreduceitsaveragedeliverycost,andtherefore,beabletooffercompetitivedeliveryfeestopotentialcustomers,wouldbetoservealargesetofcustomersconcentratedinarelativelysmallgeographicarea,suchthatthedeliveryroutescanbesharedwhileincreasingtheutilizationofitsresources(e.g.,truckcapacityanddrivers).However,thisstrategyleadstohighcompetitionbetweencarriersthatwanttomaximizemarketshare,sincecustomerswillchoosethecarrierthatoffersthelowestdeliveryfee,allelsebeingequal.Weexaminethecaseofashipperusingasingle-itemauctionschemetocontractdeliveryservicesforasetofretailpoints.Weassumethatthereisanincumbentcarrierthatiscurrentlyservingthesetofretailpoints,anewcarrierinterestedinparticipatingin 18

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thebiddingprocess,andthatbothcarrierswanttomaximizethenumberofretailpointsassignedtothem.Giventhatthedistributioncostsdependonthesetofcustomersserved,andthatthesetofcustomersthatchoosesacarrierdependsonthatcarrier'sfeesandthecompetitor'sfees,weareinterestedincharacterizingthepricingandcustomerselectiondecisionsthatguaranteeawinningbidforagivencarrier.Thisdissertationproceedsasfollows.InChapter 2 wedevelopa(Q,R)inventoryreplenishmentmodelthatusesdualsourcingasstrategytoreduceinventoryrelatedcosts.InChapter 3 wepresentanalysisandimprovementrecommendationsfortheUSDASchoolLunchandFoodBankdistributionprogramsinthestateofFlorida.InChapter 4 westudyadeliveryfeesettingproblemusingaprice-basedcompetitionmodelwhereeachplayerwantstomaximizethenumberofcustomersserved.WesummarizeourworkandpresentfutureresearchdirectionsinChapter 5 19

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CHAPTER2A(Q,R)INVENTORYREPLENISHMENTMODELWITHMULTIPLEDELIVERYMODES 2.1MotivationandLiteratureReviewIninventoryreplenishment,thethreemaincriteriaofcost,quality,anddeliveryreliabilityareofparamountimportanceinevaluatingsupplierperformance,wheredeliveryreliabilityistypicallyafunctionofthesupplier'slead-timeperformance,asnotedbyMinner[ 25 ].Allelsebeingequal,lowercostshippingmodesimplybothlongerandlessreliabledeliverylead-timeperformance.Usingsuchsuppliersthereforetendstonecessitateatleastanoccasionaluseofquick-responsesupplierswhocanllthegapswhenthelow-costsupplier'sleadtimeistoolong.Evenifasinglesupplierispreferredforsourcing,avarietyofdifferentdeliverymodesaretypicallyavailabletoshippers,dependingontheproducttype.Thatis,awiderangeofpackagecarriersandthird-partylogisticsprovidersmaybeusedtofulllorders.Becauseoftheuncertaintyindeliveryperformanceofdifferentmodes,itisnotimmediatelyclearwhichmode,orwhatmixofdifferentmodes,shouldbeusedwhenorderingfromasupplierorfrommultiplesuppliers.Thisresearchfocusesonthebenetsofdeliverymodediversicationasastrategytoreduceinventory-relatedcosts.Weconsiderasysteminwhichabuyerusestwodifferentdeliverymodesforinventoryreplenishment.Thesemodesmaycorrespondtoentirelydifferentsuppliers,ortheymayimplydifferentdeliverymodesemployedbyasinglesupplier.Inparticular,weconsideraninventoryreplenishmentmodelwherethebuyerusesacontinuousreview(Q,r)replenishmentpolicyandhasaconstant,deterministicdemandrate,.Ourmodelassumesthatthebuyerdoesnotpermitshortages,andmayusetwodifferentshippingmodeswithineachinventoryreplenishmentcycle.Oneofthesemodescomesatalowcost,butislessreliable,reectedbyastochasticdeliveryleadtime(withaniteupperbound).Inaddition,ahighercost,perfectlyreliableshippingmodeisavailableifneeded. 20

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Weassumethattheuncertaintyinthedeliverytimeofthelessreliablesupplymoderesultstoalargeextentfromahighutilizationofthismode.Ourmodelconsiderstheeffectthatanincreaseintheutilizationofastressedshippingmodehasondeliveryperformance.Thisisdonebyexpressingthemaximumvalueofthelessreliablemode'sleadtimeasastepwise,nondecreasingfunctionoftheordersize.Wecanthinkofthelessreliablesupplymodeastraversinganalreadycongestedroute,wheresendingalargerquantityviathismodemightrequireincreasingthenumberoftrucksusingtheroute(orthefrequencyatwhichtrucksutilizetheroute).Asaresult,morecapacityisconsumedandtheexpectedvalueandvariabilityofthetraveltimefortheentireorderquantityalongtherouteincrease.Alternatively,anincreaseinorderquantitywouldalsolikelyincreasetherequiredproductiontime,whichleadstoanincreasethedeliveryleadtimefortheorder.Fortheperfectlyreliablemode,weassumethatithasasufcientdegreeofexcesscapacity,andtherefore,itwillnotbeaffectedbythequantityordered.Usingmultipletransportationmodesandmakingdynamicdecisionsfordeliverieswithinasupplychainhasbecomemoreprevalentinrecentyearswiththeabilitytouseinformationtechnologytoobtainnearreal-timeinformationonthestatusofordersplacedwithsuppliers.Forexample,LizClaiborneusesaproductfromTradeBeamthatenablesmonitoringthestatusofordersplacedwithsuppliersandexpeditingtheseorderswhenneededusinganonlinetool(Jainetal.[ 17 ]).Theuseofmultiplesourcingofsupplyhasalsobeenstudiedinthecontextofmanagingsupplychaindisruptions.Yuetal.[ 40 ]discussseveralpracticalsituationsinwhichrmshavebeenabletomitigatetheimpactsofsupplydisruptionsthroughtheuseofmultiplesourcingstrategies.Theyevaluatetheimpactofusingsingleordualsourcingstrategiestomanagetheimpactsofsupplychaindisruptions.Theirmodelalsoassumesthepresenceofalessreliableandcheapersupplier(identiedasaforeignsupplier)andamorereliablebutmoreexpensivesupplier(characterizedasalocalsupplier).Babichetal.[ 2 ]statethatoneofthemainreasonsforrmstodiversifytheirsupplybaseistomanagesupplyrisk.Their 21

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workreviewstherelationshipamongnancingconstraints,trade-creditloansandthenumberofsuppliers.DongandTomlin[ 13 ]mentiontheuseofinventory,emergencysourcing,dualsourcing,demandmanagementandprocessimprovementasoperationalmeasurestomanagedisruptionrisk,andtheystudytheinteractionbetweenbusinessinterruptioninsuranceandoperationalmeasurestomanagerisk(specicallytheuseofinventoryandemergencysourcing).Ourworkfallswithinastreamofliteratureonmultiplesourcinginventorymodels.FollowingthecharacterizationthatMinner[ 25 ]presentedinhisreviewofmultiple-supplierinventorymodels,weidentifytwomainlinesofresearchbasedontheassumptionsregardingsupplierleadtimes.First,wewillreviewinventorymodelswithnsupplymodeswithdeterministicleadtimes,wherethegeneralassumptionisthatL1c2...>cn,whereLiandciaretheleadtimeandunitpurchasingcostforsupplymodei,respectively.Themainfocusofmostoftheworksthatemploythisassumptionistodeneoptimalreplenishmentpoliciesortodenepolicyparametersforaspecicreplenishmentpolicy.Afterdiscussingtheseworks,wewillreviewmodelsthatassumemultipledeliveryoptionswithstochasticleadtimes.Thosemodelsareprimarilyconcernedwithordersplitting,whereanorderisplacedatthebeginningofareplenishmentcycleandissplitamongdifferentsuppliers.Amongthestudiesthatassumetheuseofsupplymodeswithdeterministicleadtimes,MoinzadehandNahmias[ 26 ]consideramodelwithacontinuousreviewpolicy,twosupplymodesandrandomdemand;theirgoalistondtheorderquantitiesandreorderpointsforeachsupplymodeinordertominimizetotalinventorycosts.Later,MoinzadehandSchmidt[ 27 ]presentedamodelbasedonaone-for-one(S)]TJ /F2 11.955 Tf 12.8 0 Td[(1,S)inventorypolicywithtwosupplyoptions,wherethedecisiononwhethertoplaceanorderwiththeregularoremergencysupplierdependsontheageoftheoutstandingorder.Jainetal.[ 17 ]considera(Q,r)policyinwhichtwotransportationmodesexist,eachwithadeterministicleadtimeandxedplusvariableshippingcoststructure.After 22

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themanufacturingleadtime,thebuyermustdecideonhowmuchoftheordertoshipviaeachtransportationmodeunderstochasticdemand.Incontrastwithourmodel,theseworksassumedeterministicleadtimesforbothsupplymodesandallowinventoryshortages.Severalpastworksonmultiplesourcingconsiderafastemergencysupplier,whichcanbeusedtoavoidshortagesbyplacinganemergencyorder(basedonsometriggeringcondition).Forexample,TagarasandVlachos[ 37 ]assumeanemergencyorderisplacedwhentheprobabilityofstockoutishigh;ChiangandGutierrez[ 8 ]proposean`indifferencelevel'forinventorythattriggersasecondorder;inJohansenandThorstenson[ 18 ],thereorderpointfortheemergencysupplierisafunctionofthetimeuntiltheregularorderarrives.Ourmodelissimilarinspirittotheseworks,exceptthatanorderisplacedwiththe`emergencysupplier'onlywhenreachingagiventimepointinthereplenishmentcycle,andonlyiftheorderfromtheprimary(cheaper)supplymodehasnotarrivedbythistimepoint.Insteadofusingmultiplesupplymodes,ChiangandChiang[ 7 ]proposeaninventorymodelwithonesupplierandmultipledeliveries.TheyassumetheuseofacontinuousreviewpolicywithNormallydistributeddemand,constantleadtimeandapredeterminedservicelevel.IntheirmodelanorderofsizeQisplacedatthebeginningofthereplenishmentcycle,whichissplitbetweenndeliverieswithinterarrivaltimesLifori=1,2,..n.WendasimilarapproachinChiang[ 5 ],whoproposestheuseofonesupplymodeandmultipledeliveriesforperiodicreview(R,S)inventorysystems.Inbothofthesepapers,thecostreduction,whencomparedwithasingledeliverymodel,resultsfromareductionincyclestock.Unlikethepreviousworksintheliterature,weproposetheuseofmultiplesupplymodeswithdifferentorderingandpurchasingcostsanddifferentlevelsofdeliveryreliability.Ourmodeldoesnotplaceorderswiththetwosupplymodesatthebeginningofthereplenishmentcycle;onthecontrary,wedecidewhethertoplaceasecondorderwiththemorereliableandexpensivesupplymodeata 23

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specictimeduringthereplenishmentcycle,onlyiftherstorderhasnotarrivedbythattime.Pastworksonmodelsthatassumestochasticleadtimesandmultipledeliveryoptionsareprimarilyconcernedwithordersplitting.Inordersplitting,anorderisplacedatthebeginningofthereplenishmentcycle,andissplitamongdifferentsuppliers.Themainbenetofusingordersplittingisthereductionintheeffectiveleadtime(thetimeuntiltherstorderarrives),whichleadstoareductioninthesafetystocklevelrequiredtomeetagivenservicelevel.ThiseffectwasdemonstratedbySculliandWu[ 31 ]andKelleandSilver[ 19 ],wheneachorderissplitbetweentwosuppliers.Later,Panetal.[ 29 ],basedonorderstatistics,presentedexpressionstoestimatetheparametersforthedistributionsoftheeffectiveleadtimeandtimebetweenarrivalswhentwosupplymodeswithidenticalleadtimesdistributionsareused.Asanexample,theyconsideredthreeleadtimedistributions:Normal,Uniform,andExponential,andobservedareductioninthemeanandvarianceofthersteffectiveleadtime,comparedwiththeuseofasinglesupplymode.Mostoftherelevantmodelsthatassumestochasticleadtimesalsoassume(Q,r)inventorypolicies,andseektheoptimalorderquantity,reorderpointandtheproportionofordersplitting,asinLauandLau[ 21 ]andLauandZhao[ 22 ],thelatteraccountingforstochasticdemandsintheirmodel.Theseworksshowedthatmostofthesavingsfromordersplittingresultfromareductionincyclestock,whichoftenexceedsthecostsavingsduetoassociatedsafetystockreduction.Ramaseshetal.[ 30 ]showedthat,undertheassumptionofidenticalleadtimedistributionsforthedifferentsupplymodes,anoptimalsolutionisfoundwhentheorderissplitequally,andthatthesavingscomefromreducedholdingandbackorderingcosts.AdifferentapproachwaspresentedbyGaneshan,TyworthandGuo[ 15 ],whoproposedadiscountingoptionforthelessreliablesupplier,andpresentedexchangecurvestodenewhenitisbenecialtosplittheorder,aswellasthepercentagediscountnecessarytomakethemodelattractive. 24

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Ourmodeldiffersfrompreviousliteratureasitconsidersone(cheaper,lessreliable)supplymodewithastochasticleadtimeandanotherwithaperfectlyreliable(deterministic)leadtime.Thismorecloselyreectstypicalcasesinwhichabuyerusesaprimarydeliverymodethatisnotperfectlyreliable,butmayplaceexpeditedorders(via,e.g.,overnightshipping)whenstockoutsbecomeimminent.Incontrastwithordersplittingmodels,ourworkdoesnotplaceanorderwithtwoshippingmodesatthebeginningofeveryreplenishmentcycle.Instead,weseektheoptimalreorderpointandorderquantitysuchthatthelong-runexpectedinventoryordering,holdingandpurchasingcostsareminimizedwithoutanyshortages(i.e.,witha100%servicelevel).Ourassumptionofa100%servicelevelisintendedtoapproximatesystemswithextremelyhighstockoutcosts,andwerecognizethatmanypracticalsystemsoperateatnear100%,althoughtheyarenotperfectlyreliable(GravesandWillems[ 16 ]forasimilarapplicationofa100%servicelevelasanapproximationtoensuremodeltractability).Notethatadistinguishingfeatureofourmodelisreectedinthefactthatweneednotalwaysplaceanorderwiththereliabledeliverymode.Thatis,iftheorderfromtheunreliablesupplierarrivessufcientlyearlyinthecycle,weneednotplaceanorderwiththereliablesupplymode.Inotherwords,wecan`wait-and-see'whetherasecondorderisneededineachcycle,allowingustoincorporatespecicinformationabouttheunreliablesupplier'sperformanceineachcyclebeforedeterminingwhetherasecondorderisneeded.Thisfeatureresemblestheuseofexpeditingpolicies,asintheworkpresentedbyChiang[ 6 ],whoconsidersasingle-itemcontinuousreviewexpeditedorderingpolicywithstochasticdemand,orderquantityQandreorderpoints.Inhismodel,ifattheendofthesupplier'smanufacturingtime,theinventorylevelfallsbelowtheexpedite-up-to-levelR(R
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tousethesecondsupplymodemayoccuratanytimeduringthereplenishmentcycle,Chiang'smodelassumesthatthisexpeditingdecisionmustbemadeaftertheorderhasbeenmanufacturedbutbeforeithasbeenshipped.Inaddition,theexpeditedorderservesasafractionoftheoriginalorder,wherewepermitanadditionalorderquantityabovetheamountthatwasorderedviatheregularsupplymode.Inparticular,ourmodelsetsareorderpointr=,wherecorrespondstoanamountoftimethatmaybelessthananupperlimitontheleadtimeofthelowercostshippingmode(assumingthisleadtimedistributionisboundedfromabove).Ifanorderisplacedattimezero,thenisthetimeatwhichthecurrenton-handinventorywillreachzero,i.e.,theinventory`run-out'time.Supposethatatthebeginningofeachreplenishmentcyclethebuyerplacesanorderusingthelessreliableshippingmode.Iftheorderhasnotbeenreceivedbytime)]TJ /F3 11.955 Tf 12.72 0 Td[(L2,whereL2istheleadtimeoftheperfectlyreliableshippingmode,thebuyerwillneedtoplaceasecondorderwiththefaster,reliableshippingmode,whichwillbereceivedattime,therebyavoidinganyshortages.Anotherimportantattributethatdifferentiatesourmodelfrompreviousworksisthatweaccountfortheimpactthatavariationintheutilizationlevelofacongestedsupplymodehasondeliveryperformance.Weassumethattheleadtimeofthelessreliablesupplymodefollowsacontinuousdistributionthatissupportedonaboundedinterval,andthatasweincreasethesizeoftheorderplacedwiththissupplier,theupperboundonitsdeliverytimemayalsoincrease;therefore,thereisacorrespondingincreaseinthevarianceofthedeliverytime.Weareinterestedinthedegreetowhichusingthereliable,quick-responsesupplymodecanreduceinventory-relatedcostswhencomparedwiththeuseofasingleshippingmode.Wearealsointerestedincharacterizingthecircumstancesandsystemparametervaluesunderwhichtheuseofbothsupplymodesisbenecial.Thereminderofthischapterisorganizedasfollows.Section 4.2 presentsourmodeldescription,whereexpressionsforthecostfunctionsarederived,andSection 4.3 presentsa 26

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numericalanalysisandshowsthecircumstancesunderwhichourproposedmodelispreferabletousingonlyasingleshippingmode. 2.2ProblemDescriptionThissectionrstdescribesthecostsassociatedwiththeuseofasingleshippingmodeandthenwiththeuseoftwoshippingmodesforinventoryreplenishment. 2.2.1Single-ModeModelWerstconsiderthecostofusingeachshippingmodeindependently,whichwillpermitbenchmarkingtheperformanceofthemodelwithtwodeliverymodes. 2.2.1.1ShippingmodewithuniformlydistributedleadtimeWeconsideraninventoryreplenishmentmodelwhereabuyerusesacontinuousreview(Q,r)policy,withconstantdemandrate,wherenoordercrossingispermitted.Thebuyerordersfromasupplierwithunlimitedcapacitywhoshipsviaamodewithastochasticleadtime,calledshippingmode1.Giventhatwetreattimeascontinuous,weinitiallyassumethatthisdeliveryleadtimefollowsaUniformdistribution,whereL1istherandomvariableformode1leadtime,andlandudenotethelowerandupperlimitsforL1(notethatourlaternumericaltestswillconsideramoregeneralclassoflead-timedistribution,namelyaBetadistribution).Weassumethatthebuyerdesiresazeroprobabilityofstockout;therefore,Qrandthereorderpoint,r,mustequalu,whenthesupplierusesmode1exclusively.Figure 2-1 showsarealizationofthemodel,assumingwithoutlossofgeneralitythatanorderisplacedattimezero.SincetheleadtimeisUniformlydistributed,theprobabilityofanorderarrivalbeforeorattimetisPfL1tg=t)]TJ /F10 7.97 Tf 6.59 0 Td[(l u)]TJ /F10 7.97 Tf 6.59 0 Td[(lwheret2[l,u].Basedonthisprobabilitywecancomputetheexpectedinventorylevelatanytimet2[0,T],wheret=0isthetimewhenanorderisplaced,i.e.,whentheinventorylevelisequaltor,andT=Q isthelengthofthereplenishmentcycle. 27

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Figure2-1. Single-modemodel,whereL1U(l,u). Tondtheexpectedinventorylevelatanytimet2[0,T],wedividethecycleintothreedifferentintervals:[0,l),[l,u),[u,T].Therstintervalstartsattimezero,whentheinventorylevelisequaltorandanorderofsizeQisplaced.Fromthistimeuntiltheendofthecycletheinventorylevelwillbedepletedataconstantrateperunitoftimeduetodemand.Sincetheorderwillarriveatoraftertimel,wehavethatPfL1>>>>><>>>>>>:u)]TJ /F4 11.955 Tf 11.96 0 Td[(t0t
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AsshowninAppendixA,weneedQ(u)]TJ /F4 11.955 Tf 11.96 0 Td[(l)topreventordercrossing.Also,inordertopreventshortages,thesizeoftheorderhastobeatleastthesizeofthereorderpoint,i.e.,Qu.BasedonthepreviousconditionsforthesizeofQ,wecanseethattheexpectedinventoryduringt2[l,u]willhaveapositiveslope.Figure 2-2 showstheexpectedinventorylevelasafunctionoftime. Figure2-2. Expectedinventoryasafunctionoftimeforthesingle-modemodel,whereL1U(l,u). Wenextdeterminetheaverageinventorylevelinareplenishmentcycle.Inordertondtheaveragetotalinventorypercycle,denotedbyIT,weintegratetheexpectedinventoryleveloverthecycle,whichisequalto:IT=Q2 2+Q(u)]TJ /F4 11.955 Tf 11.96 0 Td[(l) 2.Toobtaintheaverageinventorylevelperunittime,denotedby^I,wedividetheaveragetotalinventorybyT,whichgives^I=Q 2+1 2(u)]TJ /F4 11.955 Tf 11.96 0 Td[(l). 29

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Aswecansee,thismodelhasasafetystockof1 2(u)]TJ /F4 11.955 Tf 11.96 0 Td[(l)since,onaverage,orderswillbereceivedattime1 2(u+l)andtheinventorypositionatthattimeisI)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(t=u)]TJ /F10 7.97 Tf 6.59 0 Td[(l 2=r)]TJ /F6 7.97 Tf 13.15 4.71 Td[(1 2(u+l),whichisequalto1 2(u)]TJ /F4 11.955 Tf 11.96 0 Td[(l).Tocomputethetotalrelevantcostpercycleweconsideraxedordercost,inventoryholdingcost,andvariablepurchasingcost.WedeneA1asthecostofplacinganorderusingshippingmode1,hastheinventoryholdingcostperunitperunittime,andc1astheunitpurchasingcostwhenusingmode1.Forthismodel,theaveragetotalcostperreplenishmentcycleisTC1(Q)=A1+hQ2 2+Q(u)]TJ /F4 11.955 Tf 11.96 0 Td[(l) 2+c1Q.Theaveragecostpercycleisequaltothetotalcostdividedbythelengthofthecycle,whichequals G1(Q)=A1 Q+hQ 2+1 2(u)]TJ /F4 11.955 Tf 11.96 0 Td[(l)+c1.(2)Sincethesecondderivativeoftheaveragecostfunction,@2G1 @Q2=2A1 Q3,isnon-negative8Q>0,weconcludethatthefunctionisconvexforQ>0andwillreachitsminimumatQsuchthat@f @Q=0,whichimplies Q1=r 2A1 h.(2)Theaboveequationimpliesthattheorderquantitycorrespondstotheeconomicorderquantity(EOQ).UsingQ1inG1(Q)wehaveaminimumaveragecostpercycleof G1=p 2A1h+h 2+c1,(2)where=u)]TJ /F4 11.955 Tf 12.13 0 Td[(l.Wemayviewthesecondtermabove,h 2,astheincrementalcostoverthatofthestandardEOQModel,duetotheuncertaintyinthedeliverymode1leadtime. 30

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Wenextconsiderthepossibilitythatthedegreeofuncertaintyinthedeliverytimeofshippingmode1maybeaffectedbythelevelofutilizationofthismode.Therefore,wewouldliketoaccountforthepotentialeffectthatachangeintheutilizationofmode1hasondeliveryperformance.Toapproximatethiseffect,wedenetheupperlimitonmode1leadtime,u,asastepfunctionoftheordersizeQ,suchthat,u=iuifqiQ
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Figure2-4. Averageinventorycostwhenleadtimeupperboundisafunctionofquantityordered. ( 2 ),isfeasible,oratoneofthebreakpointstotheleftofQ1.AppendixBpresentsadetaileddescriptionofthesolutionproceduretondQ1whenshippingmode1isusedexclusively.Analternativeapproachtoaccountfortheimpactthatutilizationhasonthedeliveryperformanceofmode1wouldbetodenetheleadtimeupperboundofmode1asalinearfunctionofQ,suchthatu=^u+Q,where^uisthebaselevelfortheleadtimeupperboundofmode1,andisthefactorbywhichtheleadtimeincreasesperunitordered.TheproceduretondtheaverageinventorycostperunittimewhenuisalinearfunctionofQ,representedbyG1,issimilartotheoneusedwhenuisxed.Asresultweobtain:G1(Q)=A1 Q+hQ 2+1 2(^u+Q)]TJ /F4 11.955 Tf 11.96 0 Td[(l)+c1.ObservethatwhentheleadtimeupperboundisalinearfunctionoftheordersizeQ,thenasweincreasethequantityorderedwewillhaveanincreaseintheaverageinventoryholdingcostequalto1 2hQ,duetotheincreaseinsafetystock. 32

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TheaverageinventorycostperunittimeisaconvexfunctioninQ,since@2G1 @Q2=2A1 Q3>08Q>0,theordersizethatminimizestheaverageinventorycostisequaltoQ=s 2A1 h(1+),andtheminimumaverageinventorycostisG1=p 2A1h(1+)+h 2+c1.Fortheremainder,wewillassumethattheleadtimeupperboundfordeliverymode1,u,isastepfunctionofQ,sincewebelievethisprovidesabetterapproximationoftheeffectthattheorderquantityhasonthedeliverytimeofastressedsupplymode.Thatis,astepfunctionallowsforchangesatdiscretequantities,asmaybethecasewhenanadditionaltruckisrequiredforahighlycongestedrouteoranadditionalproductionbatchisrequiredonacapacitatedproductionline. 2.2.1.2DeterministicleadtimemodeWenextconsiderthecostofusingashippingmode,calledmode2,thathasadeterministicleadtimeL2.InthiscasewecanusetheEOQmodelwhereQ2isequaltoQ2=r 2A2 h,A2isthexedcostofplacinganorderwhenusingmode2,histheinventoryholdingcostperunitperunittime,andistheconstantdemandrate.Werecognizethatamoregeneralmodelwouldpermitholdingcoststhatdependonthevariablepurchasecost.However,thisdifferenceinholdingcostistypicallyverysmallinpractice,andtheresultingmodelinwhichholdingcostsdonotdependonpurchasecostpermitsobtainingmuchmoreinthewayofanalyticalresults(ourcomputationaltestspresentedlaterconsidertheimpactofholdingcoststhatdependonvariablecosts,andshowthatthisimpactisquitesmall). 33

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Lettingc2denotetheunitpurchasingcostformode2andc=c2)]TJ /F3 11.955 Tf 12.02 0 Td[(c1,theaveragecostperreplenishmentcycleisequalto: G2=p 2A2h+(c1+c).(2)Inordertocomparetheaveragecostofexclusivelyusingshippingmode1versusmode2,wesubtract( 2 )from( 2 ).AssumingthatA1A2andc0weobtain: G1)]TJ /F3 11.955 Tf 11.96 0 Td[(G2h 2)]TJ /F4 11.955 Tf 11.95 0 Td[(c=!,(2)where!=h 2)]TJ /F4 11.955 Tf 11.99 0 Td[(c.Notethatwhen!0mode1shouldbechosenasthepreferredsinglemode.Laterwewillusethisexpressionduringthecomputationoftheaverageinventorycostperreplenishmentcycleforthedual-modemodel. 2.2.2Dual-ModeModelWenextpresentamodelwherethebuyermayusetwodifferentshippingmodes,implyingapositive(expected)orderquantityforbothmodes.Wewillthencompareourproposeddual-modemodelwiththesingle-modemodelandchoosetheonewiththeminimumexpectedcostperunittime.Ourproposedmodelassumesthatinadditiontomode1,thebuyermayplaceasecondorderthatwilluseanexpeditedshippingmode,calledmode2,withdeterministicleadtime,L2.Inthiscase,inordertoreducetherequiredsafetystock,thebuyermaydeneadifferentreorderpointr=,where2[L2,u),andwhereQrinordertoavoidshortages.Notethatforthedual-modemodel,isstrictlylessthanu(inotherwords,theexpectedorderquantityviamode2isstrictlypositive;thecaseinwhichtheexpectedorderquantityiszerocorrespondstothesingle-modecasewithonlymode1).UsingtodenethereorderpointincreasestheprobabilityofstockouttoPfL1g=u)]TJ /F6 7.97 Tf 6.59 0 Td[(maxfl,g u)]TJ /F10 7.97 Tf 6.59 0 Td[(l,whichisstrictlygreaterthanzero,since)]TJ /F3 11.955 Tf 12.12 0 Td[(L2.Notethat)]TJ /F3 11.955 Tf 12.13 0 Td[(L2is 34

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thetimeatwhichthebuyermustplaceasecondorderusingmode2ifthemode1orderhasnotarrived.Ifthisisthecase,thenthebuyermustplaceanorderfor(u)]TJ /F2 11.955 Tf 12.35 0 Td[()unitsthatwillbesentviamode2.Thissecondorderwillarriveattime,whichistheinventoryrun-outtime,andmustcontributeenoughinventorytoavoidstockingoutinthecasethatL1=u,whichistheworst-casedeliverytimeformode1.NotethatweassumethatuisastepfunctionofQ,asshowninFigure 2-3 .Thismeansthattheleadtimeupperboundformode1willincreaseforincreasingvaluesofthequantityordered.Inthefollowingsectionwerstdeveloptheexpressionsfortheaverageinventorycostforthedual-modemodel,andafterthatwewillpresentaproceduretondanoptimalsolutionforourproposedmodelwhenuisastepfunctionofQ.Observethatif)]TJ /F3 11.955 Tf 12.71 0 Td[(L2
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Figure2-5. Dual-modemodelrealization.Case1:2[L2,l+L2). Toanalyzetheexpectedinventorylevelatanytimet,wedividethereplenishmentcycleintofourintervals:[0,l),[l,),[,u),[u,T].Assumewithoutlossofgeneralitythatthecyclestartsattimet=0,whentheinventorylevelisandanorderofsizeQisplacedwithmode1.Thisorderwillbereceivedatsometimet2[l,u];therefore,theprobabilityofreceivingtheorderduringtherstintervalisPfL1
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Asummaryoftheexpectedinventorylevelasafunctionoftimeispresentedbelow.E[I(t)]=8>>>>>>>>>><>>>>>>>>>>:)]TJ /F4 11.955 Tf 11.95 0 Td[(t0t
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Forthedual-modemodel,thexedordercosthastwocomponents:A1,whichisthexedordercostincurredbythebuyerwhenplacingtherstorderviamode1;andA,whichistheadditionalordercostincurredwhenplacinganexpeditedorderviamode2.Notethattheexpeditedordermaybeplacedwiththesamesupplierormaybeplannedwhenplacingtherstorder;wethereforeassumethatAA2,sincepartoftheassociatedxedordercostisincurredwhenplacingtherstorderinthecycle.Thetotalcostpercycleincludesthexedordercost,A1+A,theinventoryholdingcostperunitperunittime,h,andthepurchasingcostperunitshippedbymodes1and2,denotedbyc1andc2,respectively.Thisimpliesanaveragetotalcostpercycleequalto:TCI=)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(A1+A+hQ2 2+Q(u)]TJ /F4 11.955 Tf 11.95 0 Td[(l) 2+ 2(u)]TJ /F2 11.955 Tf 12.35 0 Td[()2+(Qc1+(u)]TJ /F2 11.955 Tf 12.35 0 Td[()c2).TheaveragecostpercycleisequaltothetotalcostpercycledividedbythecyclelengthT,whichequalsGI(Q,)=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(A1+A+h 2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Q2+Q(u)]TJ /F4 11.955 Tf 11.95 0 Td[(l)+2(u)]TJ /F2 11.955 Tf 12.36 0 Td[()2+c1Q+c22(u)]TJ /F2 11.955 Tf 12.35 0 Td[() Q+(u)]TJ /F2 11.955 Tf 12.35 0 Td[().WithoutlossofgeneralityweletA=A1+Aand=(u)]TJ /F2 11.955 Tf 12.35 0 Td[().Observethat=Q2,i.e.,thequantitydeliveredviamode2.Thus,thetotalorderquantityinacycleequalsQ+.NotingthatQandaredecisionvariables,theaveragecostperunittimeisgivenby GI(Q,)=A (Q+)+h(Q+) 2+ (Q+)(c1Q1+c2)+hQ (Q+))]TJ /F17 10.909 Tf 12.11 7.38 Td[(u+l 2.(2)Therstthreetermsof( 2 ),aretheaverageannualxedordercost,holdingcostandpurchasingcostoftheEOQmodelifweweretoorderQ+unitsandwithadeterministicleadtimeequalto.ThelasttermrepresentsacorrectionfactorfortheaverageannualholdingcosttermtoaccountforthefactthattheorderofsizeQfacesastochasticleadtime.Toillustratethis,notethatinourmodel,theorderplacedwithsupplymode2willalwaysarriveattime;however,theorderplacedwithsupply 38

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mode1mayarriveatanytimebetweenlandu.Inthecasethattheexpectedleadtimefortheorderplacedwithmode1,u+l 2,islessthan,thesecondtermof( 2 )underestimatestheaverageannualholdingcost,andtherefore,thecorrectionfactorwillbepositive;ontheotherhand,iftheexpectedleadtimeofsupplymode1isgreaterthan,thecorrectionfactorwillbenegative,sinceweareoverestimatingtheaverageannualholdingcost.FromAppendixC,wehavethatGI(Q,)isconvex8Q>0and>0if4Ah>!2,where!=h 2)]TJ /F4 11.955 Tf 12.19 0 Td[(c(AppendixCalsoprovidesanargumentastowhythisconditionislikelytobemildandnon-restrictiveinpractice).ThereforewendastationarypointforGI(Q,)usingtheconditions@GI @=0and@GI @Q=0.Asresult,wehavethatforagiven,assumingtheconvexityconditionismet,theoptimalQandminimumaverageinventorycostpercycleare:Q()=)]TJ /F4 11.955 Tf 9.3 0 Td[(+r 222)]TJ /F4 11.955 Tf 11.96 0 Td[(2+22c h+2A h (2)GI()=hr 222)]TJ /F4 11.955 Tf 11.96 0 Td[(2+22c h+2A h+h 2)]TJ /F4 11.955 Tf 11.96 0 Td[(+c1 (2)Using@GI @=0,wearriveatthefollowingstationarypointsolutionfor:=1 2h24s 4Ah)]TJ /F12 11.955 Tf 11.96 16.86 Td[(h 2)]TJ /F4 11.955 Tf 11.96 0 Td[(c2+h 2)]TJ /F4 11.955 Tf 11.96 0 Td[(c35.Substituting!=h 2)]TJ /F4 11.955 Tf 11.96 0 Td[(cand=4Ah,wehave =1 2hhp )]TJ /F4 11.955 Tf 11.95 0 Td[(!2+!i,(2)andtheoptimalvalueforQcanthenbeexpressedas Q=1 2hhp )]TJ /F4 11.955 Tf 11.95 0 Td[(!2)]TJ /F4 11.955 Tf 11.95 0 Td[(!i.(2)Assumingthestationarypointsolutionisfeasible,becausethetotalorderquantitypercycleequalsQ+,Equations( 2 )and( 2 )indicatethatthetotalorder 39

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quantitypercycleatoptimalityequals Q+=p )]TJ /F4 11.955 Tf 11.95 0 Td[(!2 h.(2)Basedon( 2 )through( 2 ),weobservethatthequantitiesshippedviamodes1and2(andthetotalorderquantitypercycle)areconcavefunctionsof!.Thetotalorderquantitypercyclewillattainitsmaximumwhen!=0,whichresultsinQ=.Thismeansthatwhentheincrementalholdingcostduetotheuncertaintyintheleadtimeofmode1isequaltotheadditionalpurchasingcostperunitofmode2,thedual-modemodelwillorderthesamequantityfrombothsupplymodes,assumingthestationarypointsfor( 2 )arefeasible,i.e.,Q>0and2[u)]TJ /F4 11.955 Tf 11.96 0 Td[(l)]TJ /F3 11.955 Tf 11.96 0 Td[(L2,u)]TJ /F3 11.955 Tf 11.95 0 Td[(L2).Wealsoobservethatwhen!<0wewillhaveQ>.Thereasonforthisisthatanegativevalueof!indicatesthattheaverageinventorycostwhenusingshippingmode1exclusivelyislessthantheaverageinventorycostofusingshippingmode2,andtherefore,thedualmodelmodelwillincreasetheuseofshippingmode1.Theoppositehappenswhen!>0,whentheincrementalcostofusingshippingmode2overthestandardEOQmodelissmallerthantheincrementalcostofusingshippingmode1,andasaresult>Q.Using( 2 )and( 2 ),theminimumaverageinventorycostis GI=1 2hp )]TJ /F4 11.955 Tf 11.96 0 Td[(!2+!i+c2.(2)Notethatwhenthestationarypointsolutionisfeasible,i.e.,2(u)]TJ /F4 11.955 Tf 12.42 0 Td[(l)]TJ /F3 11.955 Tf 12.41 0 Td[(L2,u)]TJ /F3 11.955 Tf 12.02 0 Td[(L2]wewilluse( 2 )and( 2 )forQandGI,respectively;otherwisetheoptimalvalueofwillbeatanendpointoftheintervalandwewilluse( 2 )and( 2 )attheappropriatevalueoftodetermineQandGI,respectively.Nowthatwehaveanexpressionfortheaverageinventorycostforthedual-modemodelwhen2[L2,L2+l),wecanderiveamethodfordeterminingoptimalorderquantitiesformodes1and2whenuisastepfunctionofQ,suchthatu=iuif 40

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qiQqi+1orQ
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Figure2-7. Dual-modemodelrealization(Case2.1:2[l+L2,u)andL1)]TJ /F3 11.955 Tf 11.96 0 Td[(L2). WhenL1>)]TJ /F3 11.955 Tf 12.4 0 Td[(L2,asecondorderofsize(u)]TJ /F2 11.955 Tf 12.35 0 Td[()isplacedattimet=)]TJ /F3 11.955 Tf 12.4 0 Td[(L2,whichwillbeshippedusingmode2andwillbereceivedattime.ThisoccurswithprobabilityPfL1>()]TJ /F3 11.955 Tf 11.95 0 Td[(L2)g=u)]TJ /F6 7.97 Tf 6.58 -.11 Td[(()]TJ /F7 7.97 Tf 6.59 0 Td[(L2) u)]TJ /F10 7.97 Tf 6.59 0 Td[(l.Figure 2-8 showsanexampleofthiscase. Figure2-8. Dual-modemodelrealization(Case2.2:2[l+L2,u)andL1>)]TJ /F3 11.955 Tf 11.96 0 Td[(L2). Basedontheprobabilityofeachofthesesub-cases,wecancomputetheexpectedinventoryasafunctionoftime.Asintheprevioussectionwewilldividethereplenishmentcycleintofourintervals:[0,l),[l,),[,u),[u,T],whereT,thecyclelength,isnowarandomvariable,andE[T]istheexpectedcyclelength,i.e.,E[T]=Q PfL1)]TJ /F3 11.955 Tf 11.96 0 Td[(L2g+Q +(u)]TJ /F2 11.955 Tf 12.36 0 Td[()PfL1>)]TJ /F3 11.955 Tf 11.95 0 Td[(L2g, 42

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where,aftersubstitutingPfL1)]TJ /F3 11.955 Tf 11.95 0 Td[(L2g=)]TJ /F7 7.97 Tf 6.58 0 Td[(L2)]TJ /F10 7.97 Tf 6.59 0 Td[(l u)]TJ /F10 7.97 Tf 6.58 0 Td[(l,wehave E[T]=Q +(u)]TJ /F2 11.955 Tf 12.35 0 Td[()u)]TJ /F2 11.955 Tf 12.35 0 Td[(+L2 u)]TJ /F4 11.955 Tf 11.96 0 Td[(l.(2)Thereplenishmentcyclestartsattimet=0,whentheinventorylevelisandanorderofsizeQisplaced,whichwillbesentviamode1andreceivedatsometimet2[l,u];theprobabilityofreceivingtheorderduringtherstintervalisPfL1)]TJ /F3 11.955 Tf 11.96 0 Td[(L2g,however,thebuyerplacesasecondorderviamode2attimet=)]TJ /F3 11.955 Tf 12.18 0 Td[(L2for(u)]TJ /F2 11.955 Tf 12.36 0 Td[()units,whichwillbereceivedattime.Forthelattercase,whenthesecondorderwithmode2wasplacedattimet=)]TJ /F3 11.955 Tf 11.95 0 Td[(L2,wecanhavetwosituationsfortheintervalt2[,u): 1. L1>t,wheretheinventorylevelis+(u)]TJ /F2 11.955 Tf 12.35 0 Td[())]TJ /F4 11.955 Tf 11.95 0 Td[(t,fort2[,u); 2. L1t,wheretheinventorylevelis+(u)]TJ /F2 11.955 Tf 12.35 0 Td[()+Q)]TJ /F4 11.955 Tf 11.95 0 Td[(t,fort2[,u).Therefore,fort2[,u),theexpectedinventorylevelis:E[I(tjttg(+(u)]TJ /F2 11.955 Tf 12.35 0 Td[())]TJ /F4 11.955 Tf 11.95 0 Td[(t)+Pf)]TJ /F3 11.955 Tf 11.96 0 Td[(L2
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Thelastintervalofthecycleist2[u,T].Bytimet=utheorderfromsupplier1hasarrivedwithprobabilityone,andwehavetwocaseswithrespecttothesecondorder: 1. IfL1)]TJ /F3 11.955 Tf 10.84 0 Td[(L2,theinventorylevelisQ+)]TJ /F4 11.955 Tf 10.83 0 Td[(t,aftertimet=u,andT=Q=+; 2. IfL1>)]TJ /F3 11.955 Tf 12.02 0 Td[(L2,theinventorylevelisQ++(u)]TJ /F2 11.955 Tf 12.35 0 Td[())]TJ /F4 11.955 Tf 12.02 0 Td[(t,aftertimet=u,andT=Q=+u.Thereforetheexpectedinventoryfort2[u,T]is:E[I(t)]=Q++(u)]TJ /F2 11.955 Tf 12.35 0 Td[()u)]TJ /F2 11.955 Tf 12.35 0 Td[(+L2 u)]TJ /F4 11.955 Tf 11.96 0 Td[(l)]TJ /F4 11.955 Tf 11.96 0 Td[(t.Asummaryoftheexpectedinventorylevelasafunctionoftimeispresentedbelow.Figure 2-9 showstheexpectedinventorylevelasafunctionoftime.E[I(t)]=8>>>>>>>>>><>>>>>>>>>>:)]TJ /F4 11.955 Tf 11.95 0 Td[(t0t
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Inordertocharacterizetheaveragetotalinventoryperreplenishmentcycle,weintegratetheexpectedinventoryfunctionoverthefourintervals,andweobtain:E[I]=Q2 2+Q 2(u)]TJ /F17 10.909 Tf 10.91 0 Td[(l)h(u)]TJ /F16 10.909 Tf 11.35 0 Td[()2+2(u)]TJ /F16 10.909 Tf 11.35 0 Td[()L2+()]TJ /F17 10.909 Tf 10.91 0 Td[(l)2i+h(u)]TJ /F16 10.909 Tf 11.35 0 Td[()2+(u)]TJ /F16 10.909 Tf 11.35 0 Td[()L2i2 2(u)]TJ /F17 10.909 Tf 10.91 0 Td[(l)2 (2)Ourmodelcanbeanalyzedasarenewalrewardprocess(AppendixF),andtherefore,theaverageinventoryperunittime,E[I(t)] tast!1,isequaltoE[I] E[T].Using( 2 )and( 2 )wehavethattheaverageinventoryperunittimeis:^I=Q2(u)]TJ /F24 9.963 Tf 9.96 0 Td[(l)2+(u)]TJ /F24 9.963 Tf 9.96 0 Td[(l)Qh(u)]TJ /F21 9.963 Tf 10.37 0 Td[()2+2(u)]TJ /F21 9.963 Tf 10.36 0 Td[()L2+()]TJ /F24 9.963 Tf 9.97 0 Td[(l)2i+2h(u)]TJ /F21 9.963 Tf 10.37 0 Td[()2+(u)]TJ /F21 9.963 Tf 10.37 0 Td[()L2i2 2hQ(u)]TJ /F24 9.963 Tf 9.96 0 Td[(l)2+(u)]TJ /F24 9.963 Tf 9.96 0 Td[(l)h(u)]TJ /F21 9.963 Tf 10.37 0 Td[()2+(u)]TJ /F21 9.963 Tf 10.36 0 Td[()L2iiTheexpectedtotalcostpercycleiscomposedoftheexpectedxedordercost,expectedholdingcostandexpectedpurchasingcost.UsingA1asthecostofplacinganorderviamode1andAastheincrementalcostofplacingasecondorderviamode2,theexpectedxedordercostis:A1PfL1)]TJ /F3 11.955 Tf 11.95 0 Td[(L2g+)]TJ /F2 11.955 Tf 6.37 -7.03 Td[(A+A1PfL1>)]TJ /F3 11.955 Tf 11.96 0 Td[(L2g.Substitutingtheprobabilityvalues,weobtainanexpectedxedordercostofA1+Au)]TJ /F2 11.955 Tf 12.35 0 Td[(+L2 u)]TJ /F4 11.955 Tf 11.96 0 Td[(l.Theexpectedtotalinventoryholdingcostpercycleish8><>:Q2 2+Q 2(u)]TJ /F17 10.909 Tf 10.9 0 Td[(l)h(u)]TJ /F16 10.909 Tf 11.35 0 Td[()2+2(u)]TJ /F16 10.909 Tf 11.35 0 Td[()L2+()]TJ /F17 10.909 Tf 10.91 0 Td[(l)2i+h(u)]TJ /F16 10.909 Tf 11.35 0 Td[()2+(u)]TJ /F16 10.909 Tf 11.35 0 Td[()L2i2 2(u)]TJ /F17 10.909 Tf 10.9 0 Td[(l)29>=>;.Theexpectedpurchasingcostinacycleisgivenbyc1QPfL1)]TJ /F3 11.955 Tf 11.96 0 Td[(L2g+(c1Q+c2(u)]TJ /F2 11.955 Tf 12.35 0 Td[())PfL1>)]TJ /F3 11.955 Tf 11.96 0 Td[(L2g. 45

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Substitutingtheprobabilityvalues,weobtainanexpectedvariablepurchasecostpercycleofQc1+(u)]TJ /F2 11.955 Tf 12.35 0 Td[()c2u)]TJ /F2 11.955 Tf 12.35 0 Td[(+L2 u)]TJ /F4 11.955 Tf 11.96 0 Td[(l.Addingtheexpectedxedordercost,expectedholdingcostandexpectedpurchasecost,weobtainthefollowingexpressionforthetotalcostpercycle:TCII=A1+Au)]TJ /F16 10.909 Tf 11.36 0 Td[(+L2 u)]TJ /F17 10.909 Tf 10.91 0 Td[(l+Qc1+(u)]TJ /F16 10.909 Tf 11.35 0 Td[()c2u)]TJ /F16 10.909 Tf 11.35 0 Td[(+L2 u)]TJ /F17 10.909 Tf 10.91 0 Td[(l+h264Q2 2+Q 2(u)]TJ /F17 10.909 Tf 10.91 0 Td[(l)h(u)]TJ /F16 10.909 Tf 11.35 0 Td[()2+2(u)]TJ /F16 10.909 Tf 11.35 0 Td[()L2+()]TJ /F17 10.909 Tf 10.91 0 Td[(l)2i+h(u)]TJ /F16 10.909 Tf 11.35 0 Td[()2+(u)]TJ /F16 10.909 Tf 11.35 0 Td[()L2i2 2(u)]TJ /F17 10.909 Tf 10.9 0 Td[(l)2375.Theexpectedcostperunittime,whichwedenotebyGII,isequaltotheexpectedtotalcostpercycledividedbytheexpectedcyclelength,andwethereforehave: GII(Q,)=2A12+2A(+L2)+22Qc1 2[Q2+(2+L2)]+22(+L2)c2+hQ22+2h)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(2+L22 2[Q2+(2+L2)]+hQ2+2L2+()]TJ /F4 11.955 Tf 11.95 0 Td[()2 2[Q2+(2+L2)],(2)where=u)]TJ /F2 11.955 Tf 12.62 0 Td[(,=u)]TJ /F4 11.955 Tf 12.23 0 Td[(l,andc=c2)]TJ /F3 11.955 Tf 12.23 0 Td[(c1.Notethatisthequantityorderedviamode2whenthisorderisplaced.ThedecisionvariablesintheaboveexpectedcostequationarethereforeQand.InordertoshowtheconvexityofGIIasfunctionofQ,weconsideritssecondderivative,whichisgivenby@2GII @Q2=2h)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(2+L2)]TJ /F2 11.955 Tf 12.95 -9.68 Td[(2)]TJ /F2 11.955 Tf 11.95 0 Td[(2+222)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(2+L2c [Q+(2+L2)]3+22)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(A1+A(+L2) [Q+(2+L2)]3. 46

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Itisstraightforwardtoshowthatthedenominatorof@2GII @Q2ispositive8Q>0and>0,andthenumeratorisnon-negativeif: A1+Ap p!)]TJ /F4 11.955 Tf 11.95 0 Td[(h,(2)wherep=u)]TJ /F6 7.97 Tf 6.95 0 Td[(+L2 istheprobabilitythatanorderisplacedviamode2.Notethat( 2 )presentsaconvexityconditionforGIIasafunctionofQforany,andthereforeweneedtondthevalueQforminimizingGII(Q)foranygiven,where2(0,)]TJ /F3 11.955 Tf 11.95 0 Td[(L2](using,e.g.,gradientsearchtechniques).Then,inordertondtheoptimalvaluesofQandwedeneGII=minQ>0GII)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(Q,82(0,)]TJ /F3 11.955 Tf 11.95 0 Td[(L2].Althoughisacontinuousvariablethatcorrespondstothelengthoftimethattheordershippedthroughmode2shouldcover,wewillassumethatitissufcienttoconsideradiscretesetofcandidatevaluesfor(typicallyinpracticethevalueofwillbemeasuredinsomediscretemeasureoftime,suchasdays).WecanseethattheconvexityconditionforGII(Q)isnotrestrictivebyobservingthatwhenever!0(fromequation( 2 )),i.e.,whenmode1isthepreferredsinglemode,then( 2 )willalwayshold.Wealsonotethattheleft-handsideof( 2 )istheexpectedordercostinacycledividedbytheexpectedextratimeinthecycleifanorderisplacedviamode2;theleft-handsideisthereforeanupperboundonthexedordercostperunittime.Theright-handsideoftheinequalityislessthan!,theamountbywhichthecostofbufferingtheuncertaintyusingmode1exceedsthecostofbufferingtheuncertaintyusingmode2.Therefore,evenifmode1isnotthepreferredsinglemodeofoperation,itisnotunlikelyforcondition( 2 )toholdforabroadrangeofpracticalcases.SinceweassumethatuisastepfunctionofQsuchthat,u=iuifqiQ
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section.InAlgorithm2(AppendixE)wepresenttheproceduretondtheminimumaverageinventorycostforthedual-modemodelwhen2[l+L2,u]. 2.3NumericalAnalysisThissectionreportsresultsofasetofnumericalteststhatcharacterizethebenetsofthedual-modemodel.Whileweshownumerouscasesinwhichthedual-modemodeloutperformsthebetterofthetwosinglemodesolutions,wealsoanalyzehowchangesinmodelparametersaffecttheaverageinventorycostofthedual-modemodel(recallthattheminimumdual-modeaverageinventorycostequalsG=minfGI,GIIg).Sincetheaveragepurchasingcostperyearisatleastc1(whetherweusethesingleordual-modemodel),tocomparetheresultsofthedual-modemodelagainstthesingle-modemodel,wesubtractc1fromtheaverageinventorycostforbothmodels.Thereforewewillcomparethepercentagecostreductionovertheinventorycoststhatcanbemodiedwhenusinganalternativeorderingpolicy.Hence,thepercentagecostreductionisgivenby:%CR=minfG1,G2g)]TJ /F3 11.955 Tf 20.59 0 Td[(G minfG1,G2g)]TJ /F4 11.955 Tf 20.59 0 Td[(c1100%,whereGiistheminimumaverageinventorycostperunittimewhentheorderisshippedbymodeiexclusively,i=f1,2gandGistheminimumaverageinventorycostforthedual-modemodel.Weareofcourseparticularlyinterestedincharacterizingconditionsunderwhich%CR>0.Forournumericalanalysis,wecreatedprobleminstanceswithparametervaluessimilartothecoststructuresthatcanbeobservedinpractice,andwherewecandemonstratethedifferenteffectsthatchangesinaparticularparametermayhaveforthedual-modemodel.Table 2-1 showsthebaseparametervaluesusedforthenumericalanalysisofourproposedmodel.Forthexedordercost,weassumethatthexedcostformode1,A1,islessthanorequaltothexedordercostofmode2,A2.Also,assumingthatorderscanbeplacedwiththesamesupplier,butusingdifferentshippingmodes,wehavethat 48

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Table2-1. Baseparametervaluesfornumericalanalysis. ParameterUnitsValueParameterUnitsValue units=yr10,0001udays50ldays142udays55L2days53udays60h$yr1.54udays65c1$=unit10q1units0A1$=order100q2units1,000A$=order70q3units2,000A2$=order170q4units3,000 A1+AA1+A2,whereAistheadditionalordercostofplacingasecondexpeditedorder,sinceaportionofthexedordercostwasalreadyincurredwhentherstorderwasplaced.Beforeanalyzingtheeffectsofkeyparametersonthedual-modemodel,wediscusspropertiesofoptimalsolutionsforourproposedmodel.Aswecansee,basedontheresultsinTables 2-2 through 2-8 ,weobservethatforincreasingvaluesofc,thevalueofisnon-decreasing(andtheordersizewithsupplymode2isnon-increasing),untilitreachesitsmaximumvalue,=u,atwhichpointtheordersizewithsupplymode2iszero.Thismeansthatincaseswheretheadditionalunitpurchasingcostforusingthemorereliablemodeissufcientlysmall,itisoptimaltoalwaysutilizebothsuppliersduringeachreplenishmentcycle,andconsequently2[L2,l+L2).Ascincreases,theuseofshippingmode2becomesmoreexpensive,andtherefore,themodelwillincreasethevalueof(decreasingthemode2ordersize),suchthattheprobabilityofplacinganorderwithdeliverymode2isstrictlylessthanone,andtherefore,2(l+L2,u].However,evenwhensupplymode2isthepreferredsinglesuppliermode,i.e.,G2
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Thisoccursbecausewhenissufcientlylarge,placingalargeorderwithsupplier2impliesasmallervalueof(sincetheorderwithsupplymode2isequalto(u)]TJ /F2 11.955 Tf 12.52 0 Td[()),andthereforeanearlierdecisiontime,)]TJ /F3 11.955 Tf 12.02 0 Td[(L2,onwhethertoplacethesecondorder.Asaresult,theprobabilityofplacinganorderwithsupplymode2increases,aswellastheexpectedcostofholdingthisorderininventory.Sincewedonothaveclosed-formexpressionsfortheoptimalvaluesofQandforthedual-modemodel,weperformnumericalteststoevaluatehowchangesinkeymodelparametersformodes1and2mayaffecttheoptimalsolutionofthedual-modemodelanditsperformancerelativethesingle-modemodel.Sections 2.3.1 through 2.3.8 showhowchangesinasingleparameterofinterestinuencetherelativeperformanceofthedual-modemodel,allelsebeingequal.Followingthis,inSection 2.3.9 wecomparetheresultsofthedual-modemodelundervariouscombinationsofmodedeliveryleadtimeandvariablecost. 2.3.1ImpactofIncreasinglInordertoanalyzetheeffectsofchangesinthevalueofthelowerlimitontheleadtimefordeliverymode1,l,weperformednumericaltestswhereweincreasedthevalueoflwhileholdingallotherparametersequal.Whenlincreases(foraxedu),themode1leadtimeuncertaintydecreases,andtherefore,theneedforanemergencysupplieralsodecreases.Asaresult,forincreasinglweexpecttoobserveadecreaseintheordersizewithdeliverymode2,,andtherefore,anincreasein.Sinceanincreaseinlrepresentsadecreasein,wealsoexpecttoseeadecreaseinaverageinventorycost.Table 2-2 showstheoptimalsolutionsforthedual-modemodelandsingle-modemodelsand%CR,forincreasingvaluesofl.Asexpected,weseeadecreaseintheorderplacedwithdeliverymode2,exceptincaseswhenc=0.Inthisexample,whenc=0itisbenecialtoincreasetheuseofmode2,sincewecanbenetfromusingamorereliablemodewithoutpayingapremiuminpurchasingcost,andwethusobserveincreasingvaluesforthesizeofthe 50

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Table2-2. Effectoflonthedual-modemodel. Dual-ModeModelSingle-ModeModelCRlQGQ1G1Q2G2%daysdaysunitsunits$=yrunits$=yrunits$=yr c=07229041,315102,0501,507102,7801,506102,2589.221451,233986101,9371,507102,6361,506102,25814.222051,233839101,8751,507102,5131,506102,25816.97c=0.57491641,343102,7301,507102,7801,506107,2581.8014501371,370102,6301,507102,6361,506107,2580.242052821,425102,5401,507102,5131,506107,258-1.08c=1753551,452102,8001,507102,7801,506112,258-0.721453551,452102,6801,507102,6361,506112,258-1.662054271,480102,5801,507102,5131,506112,258-2.67 secondorder,.Wealsoseeadecreaseintheaverageinventorycostperunittime,althoughtheeffecton%CRwilldependonwhichsupplymodehastheloweraverageinventorycost.Theaveragecostofmode1decreasesasweincreasel,sincethereisareductionintheuncertaintyintheorderarrivaltime,andtherefore,inthesafetystockrequired.ThereductioninG1isgreaterthanthereductioninaverageinventorycostofthedual-modemodel,G,andtherefore,whenG1
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model,althoughitwillnotaffecttheinventorycostassociatedwithusingmode2exclusively.Table 2-3 showstheeffectthatincreasinguhasonthedual-modemodelandthesingle-modemodel,foraspecicinstance.Asexpected,forincreasingvaluesofuweobserveanincreaseintheorderplacedwithmode2andanincreaseintheaverageinventorycostofthedual-modemodel,G.Notethatwhenc=0,theincreaseinu(andtherefore,in)hasadifferenteffectonthequantityorderedbymodes1and2.Asmentionedpreviously,whenc=0andforlargevaluesofwhenG2
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2.3.3ImpactofIncreasingMode2LeadTime,L2WenextdiscusstheeffectofincreasingL2(whilekeepingtheotherparametersequal)onthedual-modemodel.AsL2increases,thetimeatwhichwemustdecideonwhethertoplacethesecondorder,)]TJ /F3 11.955 Tf 12.71 0 Td[(L2,willbeearlierinthecycle,andtherefore,theprobabilityofplacingthesecondorderwillincrease.WecanviewtheincreaseinL2asadecreaseintheresponsivenessondeliverymode2,andasaresult,theaverageinventorycostofthedual-modemodelwillincrease,andthemodelwillbelessbenecial.Table 2-4 showstheeffectoftheincreaseinL2ontheperformanceofthedual-modemodel.Asexpected,theaverageinventorycostofourproposedmodelincreasesforincreasingvaluesofL2,andsincetheaverageinventorycostofthesingle-modemodeisindependentofL2,weobserveadecreasein%CR. Table2-4. EffectofL2onthedual-modemodel. Dual-ModeModelSingle-ModeModelCRL2QGQ1G1Q2G2%daysdaysunitsunits$=yrunits$=yrunits$=yr c=0551,233986101,9371,507102,6361,506102,25814.221414986986101,9711,507102,6361,506102,25812.7130306851,123102,1811,507102,6361,506102,2583.40c=0.55501371,370102,6301,507102,6361,506107,2580.241454271,480102,7901,507102,6361,506107,258-5.83305501,507102,9801,507102,6361,506107,258-13.04c=15501371,452102,6801,507102,6361,506112,258-1.661454271,507102,7901,507102,6361,506112,258-5.83305501,507102,9801,507102,6361,506112,258-13.04 2.3.4ChangeinDemandRate:Inordertoevaluatehowthedemandrateaffectstheperformanceofthedual-modemodelweperformednumericaltestswhereweincreasedthevalueof,whileholdingotherparametersequal. 53

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Asthedemandrateincreasesweexpecttoobserveanincreaseinthetotalquantityorderedthroughmodes1and2,i.e.,Q+,andanincreaseintheaverageinventorycostforthedual-modemodelaswellasforthesingle-modemodel.Table 2-5 showstheoptimalsolutionandaverageinventorycostforthedual-modemodelandsingle-modemodelforincreasingvaluesofdemandrate.Notethatfortheseinstances,becausethedemandratesusedwerelessthanthebasevalueof10,000,wemodiedthebreakpointsatwhichuincreasesforincreasingvaluesofQ(refertothecaptionofTable 2-5 ).Thispermittedanalyzingsolutionswithdifferentvaluesofu,andshowinghowthevalueofaffectsthevalueofuintheoptimalsolution. Table2-5. Effectofdemandrateonthedual-modemodel.q1=0unitsq2=500unitsq3=1,500unitsq4=3,000units Dual-ModeModelSingle-ModeModelCRQGQ1G1Q2G2%units/yrdaysunitsunits$=yrunits$=yrunits$=yr c=05005002705,4432585,4243375,505-4.343,000541152731,04350031,19782531,23712.877,000595978671,6081,05572,0451,26071,88914.859,00059591,01191,8731,35692,4391,42892,14212.57c=0.55005002705,4432585,4243375,755-4.343,000513368131,23650031,19782532,737-3.277,00051771,05372,0791,05572,0451,26075,389-1.699,00051771,25892,4471,35692,4391,42896,642-0.33c=15005002705,4432585,4243376,005-4.343,00055065931,24150031,19782534,237-3.687,00054191,03672,1001,05572,0461,26078,889-2.729,00054191,33292,4881,35692,4391,428101,140-2.01 Weobservedthatforincreasingvaluesof,whenG1
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2.3.5ChangeinHoldingCostperUnitperTime:hToevaluatetheeffectoftheholdingcostperunitperunittimeinthedual-modemodel,weperformednumericaltestswhereweincreasedthevalueofhandheldtherestoftheparametersxed.Table 2-6 showstheorderquantitiesandaverageinventorycostforthesingle-modemodelandthedual-modemodel.Weobservethatwhenhincreases,thetotalquantityorderedinthedual-modemodel,Q+,decreasesandtheaverageinventorycostincreases.Throughournumericaltests,wecannotpredicttheeffectsoftheincreaseofhon%CR,sinceitwilldependontherelativevalueofhwithrespecttotheotherparameters. Table2-6. Effectofholdingcostonthedual-modemodel. Dual-ModeModelSingle-ModeModelCRhQGQ1G1Q2G2%$/unityrdaysunitsunits$=yrunits$=yrunits$=yr c=00.851,3701,173101,3871,507101,7162,062101,64915.881.251,233986101,7031,507102,2421,683102,02015.692.526658986102,6601,507103,9511,166102,9168.76c=0.50.854271,670101,7701,507101,7162,062106,649-3.171.252821,425102,2801,507102,2421,683107,020-1.712.5452741,233103,6501,507103,9511,166107,9167.63c=10.85501,647101,7701,507101,7162,062111,649-3.171.25501,507102,3001,507102,2421,683112,020-2.602.5501371,370103,8601,507103,9511,166112,9162.31 Next,wewishtounderstandhowcomputingtheholdingcostasapercentageoftheitem'svariablepurchasecostmightaffectourresults.Todothis,werstcalculatedtheholdingcostperunittimeasapercentageofthepurchasecostofsupplymode1exclusively.Wethencomparedthistothecaseinwhichtheholdingcostiscomputedasapercentageofthevalueofeachitem,whichdependsonwhethertheitemwaspurchasedusingsupplymode1orsupplymode2,assumingarst-in,rst-out 55

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(FIFO)inventorypolicy.Inordertoestimatethesecosts,weusedasimulation-basedoptimizationmodeltoestimatetheaverageinventorycostsperunittimeforourmodel.Table 2-7 showstheresultsforthesetwocasesandthedifferenceintheiraverageinventorycostsaspercentageoftheaverageinventorycostforourdual-modemodel,wheretheholdingcostiscalculatedaspercentageofthepurchasingcostofmode1.Asexpected,thedifferenceincreasesforincreasingvaluesofc2.Foreveryinstance,however,thisdifferenceislessthan0.05%,andtherefore,weconcludethatourassumptionofaholdingcostthatisindependentofthesupplymodehaslittleimpactontheresultsofthemodel. Table2-7. Effectofholdingcostaspercentageofpurchasingcostonthedual-modemodel. Holdingcostaspercentageofc1Holdingcostaspercentageofc1andc2 hQGQG%%daysunitsunits$=yrdaysunitsunits$=yr c=0851,3701,000100,92651,3701,000100,9230.001261,205894101,23961,205894101,2360.002551,233867102,190121,041641102,1620.03c=0.5854271,606101,5765501,692101,581-0.0112501371,382102,128501371,382102,1240.0025462471,263103,500443011,332103,517-0.02c=1854271,627101,5885501,641101,5640.021254271,546102,11554271,591102,1200.002552821,457103,717491641,378103,6780.04 2.3.6ChangeinIncrementalFixedOrderCostforMode2:AWhentheincrementalxedordercostforusingshippingmode2inthedual-modemodelincreases,whileholdingallotherparametersxed,weexpecttoobserveanincreaseintheaverageinventorycostofthedual-modemodel,sinceitismoreexpensivetoplaceasecondorder. 56

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Sincetheaverageinventorycostforthesingle-modemodelisindependentoftheextraxedordercostofplacingasecondorder,asweincreaseAthevalueof%CRwilldecrease,makingthedual-modemodellessbenecial.Table 2-8 showstheresultsofournumericaltestswhenweincreasethevalueofA. Table2-8. EffectofAonthedual-modemodel. DualModeModelSingleModeModelCRAQGQ1G1Q2G2%$/orderdaysunitsunits$=yrunits$=yrunits$=yr c=07051,233986101,9371,507102,6361,506102,25814.2210051,233986102,0731,507102,6361,506102,2588.2315051,2331,000102,2971,507102,6361,506102,258-1.73c=0.570501371,370102,6301,507102,6361,506107,2580.24100511101,397102,6801,507102,6361,506107,258-1.6615053551,452102,7401,507102,6361,506107,258-3.94c=17053551,452102,6801,507102,6361,506112,258-1.6610054271,480102,7101,507102,6361,506112,258-2.801505501,507102,7601,507102,6361,506112,258-4.69 2.3.7ChangeinPurchasingCostWhenUsingShippingMode2:c2Whenweincreasethepurchasingcostperunitforsupplymode2,andkeepallotherparametersxed,weexpecttoobserveanincreaseintheaverageinventorycostofthedual-modemodel.Tables 2-2 to 2-6 andTable 2-8 showtheoptimalsolutionforthedual-modemodelandsingle-modemodelfordifferentvaluesofc2.Weobservethatalthoughitmaybeintuitivetoconcludethatasweincreasec2theuseofthedual-modemodelwillbecomelessattractive,thisisnotalwaysthecase.Theimpactofthechangeinc2dependsonwhichshippingmodehastheminimumcostwhentheyareusedindependently.Whenshippingmode2isthepreferredsupplymode,asweincreasec2,thecostofusingitexclusivelyalsoincreases,butatafasterratethantheincreaseintheaverageinventorycostofthedual-modemodel,G,andthereforethepercentagecostreductionobtainedbyusingthedual-modemodelincreaseswithincreasingvaluesofc2. 57

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Thisincreasingtrendin%CRstopswhenc2islargeenoughtomakemode1thenewpreferredshippingmode.Whenmode1isthepreferredsinglesupplymode,asweincreasec2,thevalueofG1doesnotchange,and%CRdecreases.Thereforethemaximum%CRforincreasingvaluesofc2isreachedatthemaximumvalueofc2suchthatG2G1. 2.3.8ChangeintheLeadTimeDistributionforSupplyMode1Forthedevelopmentofourmodelweassumedthattheleadtimeforsupplymode1isUniformlydistributed.Whenthemode1leadtimefollowsaUniformDistribution,L1U(l,u),theprobabilitydensityofanorderarrivalattimetisequalforanytimet2[l,u],andtheexpectedarrivaltimeisE[L1]=l+u 2,whichisequaltothemidpointoftheinterval.Weareinterestedintheeffectsthatusinganon-Uniformdistributionforthemode1leadtimehasonthedual-modemodel.Inparticular,weassumethatL1fl+Beta(a,b)g1wherea=2andb>2,andthereforethedistributionisunimodalandpositivelyskewed.Underthisassumptiontheprobabilityofhavinganearlyarrival(beforethemiddlepointoftheinterval)ishigherthanwhenaUniformlydistributedleadtimeisassumed.SincewecannolongerusetheexpressionsfoundinSection 4.2 ,inordertondtheoptimalsolutionforthedual-modemodelwhenweuseaBetadistributionforL1,weneedtouseasimulation-basedoptimizationapproach;inparticular,weusedthecommercialpackageOptQuestfromArena.Table 2-9 showstheoptimalvaluesandaverageinventorycostsforthesingle-modemodelanddual-modemodelwhenweassumethatL1fl+Beta(a,b)g.Thelast 1WedeneL1asascaledBetavariable,sincetheBetadistributionisdenedintheinterval[0,1]andweareinterestedingeneralcaseswhereL12[l,u]. 58

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Table2-9. Dual-modemodelforL1fl+Beta(a,b)g.q1=0unitsq2=500unitsq3=1,500unitsq4=3,000units Dual-ModeModelSingle-ModeModelCRQGQ1G1Q2G2%%units/yrdaysunitsunits$=yrunits$=yrunits$=yrL1fl+Beta(a,b)L1U(l,u) c=05004572415,4342585,4403375,5051.51-4.343,000541158530,98850031,29282531,23720.1412.877,000792160471,4511,05572,2971,26071,88923.2314.85c=0.55,0043102385,4282585,4403375,7552.69-4.343,000439966931,19650031,29282532,7377.42-3.277,0004224919771,9801,05572,2971,26075,38913.80-1.69c=150043102415,4312585,4403376,0052.10-4.343,000417450031,18250031,29282534,2378.54-3.687,00042741,01372,0011,05572,2971,26078,88912.90-2.72 twocolumnsofTable 2-9 showthe%CRvaluesforthedifferentassumptionsforthedistributionofL1.Weobservedthatweobtainedhigher%CRvalueswhenweassumethatL1followsaBetadistribution,andthereasonsforthisaretwofold:rst,theaverageinventorycostofusingsupplymode1exclusivelyishigherwhenweuseapositiveskewedBetadistributionforL1,comparedwiththeaverageinventorycostofassumingaUniformlydistributedleadtime.ThisisbecausetheexpectedarrivaltimewhenL1fl+Beta(a,b)gislessthanl+u 2,andtherefore,theholdingcostduetosafetystock(u)]TJ /F4 11.955 Tf 12.63 0 Td[(E[L1])isincreasedundertheBetadistribution(whencomparedtotheUniform);second,sincetheprobabilityofanearlyarrivalishigherwhenweassumethatL1followsaBetadistribution,comparedwiththeuseoftheUniformdistribution,theprobabilityofplacingthesecondorderwithsupplymode2decreases,andtherefore,theaveragetotalcostofusingthedual-modemodelwilldecrease.InpracticeweexpectthatL1followsaprobabilitydistributionwithsomepositiveskew,sincetheunusuallylongcasestypicallyleadtotheskewinthedistribution.Hence,basedonthepreviousanalysis,wecanexpecttohavehigher%CRvalues 59

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thantheonesobservedundertheassumptionofaUniformlydistributedleadtime,andtherefore,thedual-modemodelmaybemorebenecialinpractice. 2.3.9ComparingSupplyModeswithDifferentCostandLeadTimeParametersTheanalysisdiscussedintheprevioussectionsshowshowcertainparameterchangesaffectthebenetsofthedual-modemodel.Thissectionconsidershowabuyermightevaluatemultiplesupplymodeoptions,whereagivenmodeimpliescertaincostsandleadtimedistributioncharacteristics.Thatis,allelsebeingequal,wewanttoknowhowtochoosefromdifferentsupplyalternatives.Forexample,usingtheinstancesinFigure 2-10 ,wecanseehowthebuyermightcomparedifferentdeterministicsupplymodesthatcanbeusedassecondarysourcing,withdifferentcombinationsofpurchasingcostsandleadtimes. Figure2-10. PercentagecostreductionfordifferentvaluesofL2andc.A2=$150=orderA=$50=order InFigure 2-10 weobservethatchoosingthesupplymodewiththeshortestleadtime,whilekeepingtheotherparametersequal,willprovidethebiggest%CRvalue.However,inpractice,areductionintheleadtimecorrespondstoanincreaseintheunitpurchasingcost.Asaresultitmaybemorebenecialtochooseashippingmodewithalongerleadtimebutwithareducedunitpurchasingcost.Usingourmodelandwiththistypeofanalysis,abuyercannegotiatepurchasingcostsbasedondeliverytimes. 60

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Nextwewouldliketocomparedifferentsupplymodeswithdifferentuncertaintylevels.ThecurvesinFigure 2-11 representdifferentshippingmodeswithUniformlydistributedleadtimes,eachwiththesamemean,butwithdifferentvaluesofvariance(where=41daysforthebasecase).Thismeansthatwewillcompareshippingmodeswithequalvaluesfortheexpectedarrivaltime,butwithdifferentvaluesofuncertainty,. Figure2-11. PercentagecostreductionfordifferentvaluesofL2,cand.A2=$150=orderA=$50=order Althoughintuitionsuggeststhatthedual-modemodelwillbemorebenecialastheuncertaintyintheleadtimeofshippingmode1increases,thisisnotalwaysthecase.AswecanobserveinFigure 2-11 ,whenc=0weobtainthehighest%CRvaluefortheinstanceswiththesmallestvaluesofuncertainty(=29days).Thereasonforthisisthat,fortheseinstances,whenc=0,mode2isthepreferredsupplier(G2
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deliverytime.Thismeansthatwearecomparingshippingmodeswithequalvaluesof,butincreasingvaluesofE[L1],whileholdingallotherparametersequal.Figure 2-12 showstheresults,whereE[L1]=34.5daysforthebasecase. Figure2-12. PercentagecostreductionfordifferentvaluesofL2,candE[L1].A2=$150=orderA=$50=order Notethattheaverageinventorycostofusingmode2exclusivelyisindependentofE[L1];andthereforeforinstanceswithdifferentexpectedleadtimesforsupplymode1,thevalueofG2willnotchange.Forthesingle-modemodelwhenweusemode1exclusively,inordertoavoidstockoutsweneedanordersizeatleastasgreatasthereorderpoint,i.e.,Qu;thereforeforincreasingvaluesofE[L1],therighthandsideoftheinequalityincreasesandwewillhaveincreasingvaluesofG1.Asexpected,wehaveincreasingvaluesof%CRforincreasingE[L1].Notethatwhenc=0wehavesomecaseswhenthe%CRvaluesareequalfordifferentE[L1].Therearetworeasonsforthisresults:rst,whenc=0supplymode2ischosenasregularsupplier(G2
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valuesofE[L1]andequalvaluesof,thevaluesofQandareequalfordifferentinstances,whichresultsinthesameaverageinventorycostforthedual-modemodel.Aswecansee,theproposedmodelpresentssavingopportunitiesforthebuyer,anditcanbeusedinajointmannerwiththesuppliertoachieveinventorycostreductionsforthebuyerandbetterresourceallocationforthesupplier. 63

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CHAPTER3ANALYSISANDIMPROVEMENTOFTHEUSDASCHOOLLUNCHANDFOODBANKDISTRIBUTIONPROGRAMSINTHESTATEOFFLORIDA 3.1IntroductionandCurrentStatusThroughFoodDistributionProgramsliketheNationalSchoolLunchProgram(NSLP)andtheEmergencyFoodAssistantProgram(TEFAP),theUSDAseekstostrengthentheU.S.foodsupplynetworkbysendingfoodandnutritionassistancedestinedtorecipientagencies(RAs),andbyensuringagriculturalmarketstabilizationthroughthedistributionofhighquality,100%American-grownUSDAFoods( http://www.fns.usda.gov/food-distribution/food-distribution-programs/ ).IntheStateofFlorida,thereceiptoftheUSDAfoodandstorageanddeliverytotheagenciesisaccomplishedviaacommercialdeliveryprogramadministeredbytheFloridaDepartmentofAgricultureandConsumerServices(FDACS).Usingabiddingprocess,FDACSinvitescontractorstosubmitproposalsforreceipt,storageanddeliveryoftheproductssentbytheUSDAandfoodsupplierstotheRAslocatedinthestate.FDACSissuesanInvitationtoBid(ITB)thatincorporatesrequirementssetbytheUSDAwithrespecttotechnicalspecicationsforservicestobeprovided,specialrequirements,biddinginstructionsandacommitmenttomeetthedeliveryschedulesdeterminedbytheRAs.AmongtherequirementssetbyFDACSisaminimumordersizeof20casesperdelivery,a60-dayfreestorageperiod,andamaximummonthlystoragefeeof25%ofthedeliveryfeepercase.Henceforthwewillrefertothesuppliersthatareawardedthecontractasstate-contractedwarehouses.WhilenumerousRAsarepartoftheNSLP,anumberofhigh-demandRAsprefernottousethestate-contractedwarehouses.Instead,theseRAsusecommercialfooddistributorsthatofferthemmoreadvantageousstorageanddeliveryfees.Additionally,individualagenciescanrequestdirectshippingofUSDAfoodtofoodprocessors,insteadofusingstate-contractedwarehouses.Insuchcasesfoodisshippeddirectlytothefoodprocessorandthesubsequentdeliveryoftheproducttotheagencyisnotpart 64

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oftheFoodDistributionProgramadministeredbyFDACS.Althoughitisnotcommon,asmallnumberofagenciesprefertoreceivetheirproductdirectlytotheirowninternalwarehouse,inthisway,reducingthecostassociatedwiththeuseofthestate-contractedwarehouse.Figure 3-1 showsadiagramoftheFloridaFoodDistributionNetwork. Figure3-1. FloridaFoodDistributionProgramNetwork FortheNSLP,theStateofFloridaisdividedintoveregions,andeachregionisservedbyastate-contractedwarehousewithaparticulardeliveryfeeschedule.TheagenciesthatbelongtoTEFAPuseacommonstate-contractedwarehouse,withitsowndeliveryfeeschedule.BasedontheinformationprovidedbyFDACSandfourstated-contractedwarehousesfromtheschoolyear2011-2012,weestimatedtheannualquantityofUSDAfooddeliveredtotheagenciesusingthecommercialdistributionprogramandthestorage 65

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anddeliveryexpensesincurredbytheRAsasresultofusingthestate-contractedwarehouses.Table 3-1 showsasummaryoftheestimatedstorageanddeliveryexpensesforeveryregionfortheNSLPandforTEFAP.Aswecanseethereislittlehomogeneityinthecharacteristicsoftheregionsintermsofthenumberofagencies,totaldemandandtotalexpenses. Table3-1. Summaryofestimatedstorageanddeliveryexpenses,Schoolyear2011-2012. RegionRAEstimatedcasesdeliveredEstimatedstorageex-pensesEstimateddeliveryex-pensesEstimatedtotalex-pensesAverageexpensepercaseEstimatedproductnetvalueCases$$$$/case$ 13276,71634,385200,225234,6103.062,576,18524281,32961,131276,647337,7774.152,680,77336589,710141,430325,363466,7935.202,983,75541525,68965,039116,431181,4707.06854,41652713,74516,56445,27261,8364.50457,159TEFAP17370,67753,490576,440629,9291.704,340,155 WecanobserveanimportantdifferencebetweenthecoststructuresfortheagenciesthatbelongtoTEFAPandNSLP,andthereareseveralreasonsforthis.First,fortheNSLP,astate-contractedwarehousemayservebetween15and65agenciesperregion,andtheagenciesmayhaveseveraldeliverypoints,whichincreasesthedeliverycostsforthestate-contractedwarehouse.Incontrast,astate-contractedwarehousethatservesTEFAPagenciesprovidesstorageanddeliveryservicesfor17agencieswithonlyonedeliverystopeach.Additionally,thelargevarietyofproductreceivedbytheNSLPagenciesdecreasestheaverageordersizeperproductperdeliverypoint,andthereforeincreasestheoperationalcostsforpickinganddeliveryforthestate-contractedwarehouse.TEFAPagencies,ontheotherhand,havelessvarietyofproductsandplacemostlyfull-palletorders.Finally,sincetheagenciesbelongingtotheTEFAPprogrampossesswarehousesandtrucks,theycanpickupproductsfromthestate-contractedwarehouseandincuralowerfeewhencomparedtoadeliveryservice.Thisalso 66

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reducestheaveragestoragetimeinthestate-contractedwarehouseandthereforethetotalexpensesfortheagencies.Anotherimportantdifferenceisintheannualdemandfortheagenciesandtheirgeographicaldispersion.AswecanseeinAppendix G ,76.61%oftheagenciesthatbelongtotheNSLPhaveanannualdemandlessthanorequalto1,800casesandaredispersedthroughthestate;howeveronlytwoofthe17agenciesthatbelongtoTEFAPhaveannualdemandlessthan2,500cases,asshowninAppendix H .Duetothelackofhomogeneitybetweentheregionsandamongtheagenciesthatbelongtothesameregion,intermsofdemandrate,storagecapacityandgeographicaldemanddistribution,thedeliveryfeespercaseofferedbythestate-contractedwarehousesdifferbetweenregions,asdothestorageanddeliveryexpensesoftheagencies.Figure 3-2 illustratestheinconsistencyinthecoststructuresanddemandsamongthedifferentregionsandFoodPrograms. Figure3-2. Summaryofestimatedstorageanddeliveryexpenses,schoolyear2011-2012 AnotherfactorthatincreasesthetotalexpensesforstorageanddeliveryofUSDAfoodfortheagenciesisthereductioninthevolumeofproductsenttothe 67

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state-contractedwarehousesasaresultofincreasinguseofcommercialfooddistributorsandfoodprocessors.Figure 3-3 presentsasummaryofthetotalnetvalueofproductsentbytheUSDAperregionandthepercentageofproductthatwassenttothestate-contractedwarehousesduringtheschoolyear2011-2012. Figure3-3. SummaryofproductnetvaluesentbyUSDAperdestination,schoolyear2011-2012 ThepurposeofouranalysisistopresentdifferentregionalcongurationsandstorageanddeliverypoliciesforthedesignoftheITB,whichresultinamorehomogeneouscoststructureamongtheregionsandattractpotentialbiddersforalloftheregionsinthestate.Additionally,sinceanincreasedvolumeofproductsenttofoodprocessorsisalsoanimportantfactorintermsoftheagencies'expenses,andbecausetheuseoffoodprocessorsreducestheattractivenessofthecontracttopotentialbidders,wewillconsidertheimpactofpoliciesthatlimitthequantityofproductsenttofoodprocessors. 3.2RegionalCongurationsInthissectionwepresentasetofcandidateregionalcongurationsforwhichwewillevaluatenewstorageanddistributionpoliciesdescribedinthenextsection.Themotivationbehindthedesignofthecandidateregionalcongurationswastocreategreaterhomogeneityamongtheregions,intermsoftotalannualdemandandestimatedstorageanddeliverycostpercaseperregion.Asaresult,weexpecttoreducetheinconsistencyinthedeliveryandstoragefeesofferedbythestate-contracted 68

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warehousesbetweenthedifferentregionsandensuretheinterestofpotentialbiddersforalloftheregionsinthestate.Fortheestimationofthecostsincurredbythestate-contractedwarehousesthatservetheagenciesbelongingtotheNSLPandTEFAPprograms,wefollowedthemethodologydescribedinAppendix I .Werecognizethattheactualcostoftheserviceprovidedtotheagencieswillvaryfordifferentwarehouses,astheirwarehouselocation,coststructure,andgeographicaldistributionofcurrentcustomerswillimpacttheactualcostofservingcustomers.Forthisreason,wedonotattempttodesignamodelthatpreciselyestimatesthecostofthereceipt,storageanddeliveryofproductstotheagenciesforaspecicwarehouseataspeciclocation;rather,weareinterestedindesigningamodelthatcanpermitustoestimatetheeffects,andtherefore,approximateservicesfees,associatedwithdifferentstorageanddeliverypolicyparameters.InAppendix J wehavedetailedthenewassignmentsofcountiestoregionsforeachofthecandidateregionalcongurationsweproposed,andAppendices K through O showthesameinformationusingtheFloridastatemap.Forcandidateregionalcongurations1,2and3,wewillanalyzethesituationinwhichtheagenciesthatbelongtotheNSLPandTEFAPusetwodifferentdistributionnetworks,aswellasthesituationinwhichtheyuseacommondistributionnetwork.Forregionalconguration4,weassumethattheagenciesthatarelocatedinHillsborough,Orange,Osceola,Pasco,Pinellas,Polk,Broward,DadeandPalmBeachcountiesshareadistributionnetworkwiththeTEFAPagencies,andtherefore,areservedbythesamesate-contractedwarehouse.UsingthecostmethodologydetailedinAppendix I weestimatedthecostofservingtheagenciesunderthenewlyproposedregionalcongurations,assumingthattheagenciescurrentlyusingtheprogramwillcontinuetobeservedbystate-contractedwarehouses.Appendix P showstheestimatedcostsandannualdemandsforthecandidateregionalcongurations. 69

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3.3ScopeofParametersStudiedInthissectionwepresentadescriptionoftheparametersandpoliciesofthedistributionnetworkthatwewillstudy,inordertoevaluatehowtheseparametersaffectthedeliveryandstoragefeesofferedbythestate-contractedwarehousesaswellasthetotalestimatedexpensesfortheagencies.Figure 3-4 showsasummaryoftheparametersstudiedandthefollowingsectionsdescribetheseparametersingreaterdetail. Figure3-4. Parameterstobestudied 70

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3.3.1RegionalCongurationTheregionalcongurationconsiderstheassignmentofcountiestodifferentregions,suchthatthecountiesassignedtoaregionareservedbythesamestate-contractedwarehouse.Asnotedpreviously,thegoalduringthedesignofthecandidateregionalcongurationswastobalancethetotaldemandsamongtheregions,suchthattherearenotregionsthatarelessattractivetopotentialbiddersorforwhichitishardertoconsolidateafulltruckloadorderfromtheUSDA.Additionally,weattemptedtobalancetheestimatedaveragecostpercasefortheproposedregionsand,asaresult,weexpecttohavemoreuniformityoffeesamongthedifferentregions.WeconsideredfourdifferentcandidateregionalcongurationsasnotedintheprevioussectionandillustratedinAppendices L to O 3.3.2DeliveryFeePoliciesForthedesignoftheITB,weneedtodenedeliveryfeepoliciesfortheserviceprovidedbythestate-contractedwarehouses,whichwillaffectthefeestheyareabletooffer.Inparticular,weareinterestedinanalyzinghowachangeintheminimumordersizeandinthedesignofthefeeschedulewillaffecttheestimatedcoststotheagencies.Currentlythereisaminimumordersizeof20cases,whichmeansthatwhenanagencyorderslessthantheminimumordersize,itwillbechargedasifitordered20cases.Althoughdecreasingtheminimumordersizemayreducethedeliveryexpensesforagencieswithlowaverageordersizes,thiswilllikelyleadtoanincreaseinthefeerequiredbythestate-contractedwarehouses,astheywillneedtoguaranteeaminimumprotmarginforapotentiallysmallordersize.Therefore,adecreaseintheminimumordersizeislikelytoultimatelynegativelyaffecttheagenciesthatalreadymeettheminimumordersize,astheywilllikelyhavetopayahigherdeliveryfee.Wewillevaluatetheeffectsofvaryingtheminimumordersizebetween10,20,30and40cases,byestimatingtheassociatedfeesthatastate-contractedwarehousemayproposeandtheexpensestotheagencies. 71

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AnotherimportantfactorinthedeliveryfeepolicyisthetypeoffeerequestedintheITB.First,wewillevaluatetheresultsobtainedassumingthatthestate-contractedwarehouserequiresauniformdeliveryfeefortheagencieslocatedinthesameregion.Sinceusingauniformfeeforallagencieswithinaregionmayseemunfairtoagencieswithhighdemand,wealsowanttoanalyzethesituationinwhichbiddersareaskedtoprovideanorder-quantity-anddelivery-frequency-basedfeeschedule.Inthislattercase,theorderquantitiesanddeliveryfrequenciesaregroupedbyranges,andthefeetablecontainsthedeliveryfeeforeachpairoforder-quantity-anddelivery-frequency-range.Forourproposedfeetable,weassumeaminimumordersizeof20casesandtherangesfortheordersizesare:20-60cases,61-150cases151-300casesandmorethan300cases.Therangesforthedeliveryfrequencyare:weekly,bi-weekly(everytwoweeks),monthlyandlessthanmonthly.Usingasimilarmodeltotheonepresentedin[ 3 ],wewanttoensurethatthefeepercasedoesnotincreaseasdeliveryfrequencydecreasesorasorderquantityincreases.Thismeansthatthedeliveryfeepercasewillbelowerforlargerandlessfrequentordersthanforsmallerandmorefrequentorders,sincethosetypesofordersarelessexpensiveforthestate-contractedwarehouse.Wealsodenedupperandlowerlimitsonthepercentagefeedecreasebetweentwoconsecutiverangesinthetable.Forthedeliveryfrequencies,wedenedaminimumandmaximumdecreaseof(0%,10%),(1%,20%),and(2%,20%)betweenconsecutiveranges,i.e.,whengoingfromweeklytobi-weekly(everytwoweeks),bi-weeklytomonthly,andmonthlytolessthanmonthlydeliveryfrequencies,respectively.Fortheordersize,wedenedaminimumandmaximumdecreaseof(5%,25%),(5%,20%),and(1%,10%)whengoingfrom20-60casesto61-150cases,61-150casesto151-300cases,andfrom151-300casestomorethan300cases,respectively.Theboundswere 72

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denedbasedonthefeeschedulesofferedbythestate-contractedwarehousesduringtheschoolyear2011-2012,fordifferentdeliveryfrequencies. 3.3.3StorageFeePoliciesWearealsointerestedinhowchangingthestoragefeepoliciesmayaffectagencies'expenses.Inparticular,weevaluatetheimpactofchangingthelengthoffreestoragetimeatthestate-contractedwarehouseintheITB,andtherelationshipbetweenthestoragefeeanddeliveryfee.CurrentlyintheITB,FDACSrequeststhatthersttwomonthsofstoragearefreeofchargeforalloftheagencies.Althoughthestate-contractedwarehousesdonotchargedirectlyforthersttwomonthsofstorage,theyarealmostcertaintoincludeintheirproposedfeesthecosttocoveratleasttwomonthsofinventoryfortheproducthandledfortheagencies.Therefore,theagencieswithlowstoragetimesmightbechargedahigherthannecessarydeliveryfee,sincetheactualfeewaslikelycalculatedassumingaminimumstoragetimeoftwomonths.Anotherimportanteffectofthispolicyisonthebehavioroftheagenciesthatstoretheirproductsformorethantwomonths,astheyassumethattheyarenotbeingchargedforthisstorage(althoughitislikelytobeembeddedintheirdeliveryfees).Sinceweareinterestedintheeffectthatchangingthelengthofstoragetimefreeofchargewillhaveonagencyexpenses,weevaluatethesituationsinwhichweassume0,1and2monthsofstoragefreeofcharge.Additionally,thecurrentITBrequiresthatthestoragefeecannotbegreaterthan25%ofthedeliveryfee.Asaresult,thestate-contractedwarehousesdeneafeeequalto25%ofthedeliveryfee,andtherefore,thisincreasesthestorageexpensesfortheagenciesthatbelongtoregionswithhighdeliveryfees.Sincewewanttoanalyzetheeffectonthedeliveryfeeandtotalexpensesfortheagencieswhenthestoragefeeisindependentofthevalueofthedeliveryfee,wewillevaluatethescenariosinwhichthestoragefeeisequalto$0.13/casemonth, 73

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$0.17/casemonth,and$0.20/casemonth.Whenweassumethatthestate-contractedwarehouseusesanorder-quantity-anddelivery-frequency-basedfeeschedule,weevaluatethescenarioswhenthestoragefeeisequalto$0.13/casemonthand$0.17/casemonth. 3.3.4DistributionNetworkStructureCurrentlytheagenciesthatbelongtotheNSLPandTEFAPformtwodifferentdistributionnetworks,andbecauseofthecharacteristicspreviouslydescribedSection 4.1 ,thecoststructureforTEFAPallowsthemtoreceivelowerfeescomparedwiththeagenciesintheNSLP.IfweaggregatetheagenciesthatbelongtotheNSLPandTEFAPinasingledistributionnetwork,wewillreducethecostpercasefortheagenciesinNSLPandwillmaketheITBmoreattractivetobidders.Ontheotherhand,thiswillincreasethecorrespondingfeefortheTEFAPagencies.Sinceweareinterestedinndingasolutionthatcanreducetheoveralldeliveryandstorageexpensesforbothprograms,whilereducingpotentialnegativeimpactstosomeoftheagencies,wewanttoevaluatetheproposedregionalcongurationswhenweuseoneortwoseparatedistributionnetworksfortheagenciesintheNSLPandTEFAP. 3.3.5LimitsontheQuantityofProductSenttoFoodProcessorsAsmentionedinSection 4.1 ,theagenciesintheNSLPmaysendthefoodprovidedbytheUSDAdirectlytofoodprocessors,whichreducesthevolumeofproductshippedtothestate-contractedwarehouses.Thecontinuousdecreaseinthevolumeofproductsenttothestate-contractedwarehouseshasanegativeimpactontheper-casedeliveryandstorageexpensesfortheagencies.Thisleadstoanincreaseinthecostpercaseforthedeliveryandstorageservicesofferedbythestate-contractedwarehouses(duetodiseconomiesofscale).ItalsoincreasesthestorageexpensesforagenciesthatrequireplacinglargerorderstotheUSDAinordertoachievetheminimumordersizefordeliveryfromtheUSDAfood 74

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suppliers.ThecombinationoftheseeffectsleadstolowinterestintheITBandrequiresbidderstoofferhighfees,particularlyincaseswherenocompetitorsexistforthesamebid.Weareinterestedintheeffectthatapolicythatlimitstheamountofproductsenttothefoodprocessorsmayhaveontheproposeddeliveryandstoragefeesthatpotentialbiddersmayoffer.Afterselectingthefeescenarioswiththeleastnegativeimpactontheagencies,wewillevaluatetheirimpactsassumingalimitof30%and50%onthetotalamountofproductsenttothefoodprocessors.ThedenitionoftheselimitsisbasedonpreviousconversationswithFDACSpersonnel.Forouranalysisweareinterestedinestimatingthecorrespondingdeliveryandstoragefeesthatabiddermayproposeforeachcombinationoftheparameterswehavediscussedthroughoutthissection,assumingthatthebidderaimsforaprotmarginof25%overitscost.Intotalwehave399feescenarios:171whentheagenciesintheNSLPandTEFAPusetwodifferentdistributionnetworks,and228whenthetwoprogramsuseacommondistributionnetwork. 3.4AnalysisofResultsInanalyzingthe399scenariosthatresultfromthecombinationsoftheparametersdescribedintheprevioussection,wedenetheoverpaymentforeachrecipientagencyasthedifferencebetweenthecurrenttotalexpenseandtheestimatedexpenseunderaspecicfeescenario.Inthisway,anegativeoverpaymentmeansareductioninthedeliveryandstorageexpensefortheagencyandapositiveoverpaymentistheresultofanincreaseinitsexpenses.Therefore,whencomparingtwodifferentfeescenarios,wewillchoosetheonewiththesmallestaverageoverpayment.Thereareafewagenciesthatseeanoverpaymentvaluegreaterthan$5,000/yrundermostofthe399feescenarios,anditisimportantthatwespecicallydiscussthesecases.Foragencies01003,01017,01020,01037,01046and01057fromthe 75

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currentregion1,and01035and01059fromthecurrentregion3,theseagenciesseehighoverpaymentvaluesinpartbecausetheybelongtoregionsfromwhich,basedonourcostestimationmodel,thecurrentprotmarginislessthan25%,andtherefore,underournewmodel,whereweaimfora25%protmargin,theirexpenseswillincrease.Additionally,sincetheseagenciesproducethehighesttotalexpensevaluesfromtheircurrentregions,theimpactoftheincreasedprotmarginishighwhencomparedwithotheragenciesfromthesameregion.Wealsonotethatthelimiteddatatowhichwehadaccessforregion1ledtoanunderestimateofthepreviousfeespaidfortheagenciesinthatregion.Thus,thecombinationoftheassumptionofa25%marginandtheunderestimationofpreviousfeespaidbytheagenciesimpliesthattheoverpaymentvaluesforregion1areoverestimates.Thehighoverpaymentvaluesforagencies01043,01053and04001isduetothefactthattheycurrentlypickuptheirproductfromthestate-contractedwarehouses,andtherefore,theirdeliveryexpensesarelowcomparedwiththeestimateddeliveryexpensefromourproposedfeescenarios,whereweassumethatalltheagenciesreceivedeliveriesfromthestate-contractedwarehouse.Inourmodelwedidnotincludethecasewhentheagenciespickuptheirproductsfromthestate-contractedwarehouses,sincethenumberofagenciesintheNSLPthatrequestthisoptionislow.Thegoalofouranalysisistoidentifythefeescenarioswiththeleastnegativeimpactontheagencies.Inordertoevaluateeachfeescenario,wecomputetheaverageoverpaymentfortheagencieswithanoverpaymentgreaterthan$5,000/yr.Thelistofthe25feescenarioswiththeleastaverageoverpayment(inexcessof$5,000/yr)amongthe171scenarioswhenweusetwodifferentdistributionnetworksforNSLPandTEFAPisdetailedinAppendix Q ,alongwithdetailedestimatedofthefeesforeachregion.Thesameinformationforthe25bestfeescenariosamongthe228scenarioswhenweuseonedistributionnetworkforNSLPandTEFAPispresentedinAppendix R 76

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3.4.1SeparateNSLPandTEFAPNetworksWhenweusetwodifferentdistributionprogramsfortheNSLPandTEFAPagencies,allofthetop25resultsarefromcandidateregionalconguration3.Thisdemonstratesthatthisregionalcongurationissuperiortotheothertwocandidates,notonlybecauseitresultsinthesmallestaverageoverpayment,butalsobecauseitismorerobustinthesensethatwithdifferentdistributionpolicyparameters,itstillhastheleastnegativeimpactontheagencies.AnotherimportantobservationisthatwhenweusetwodifferentdistributionnetworksforNSLPandTEFAP,theuseofauniformdeliveryfeeforagenciesinthesameregiongenerallydominatesorder-size-anddelivery-frequencybasedfeeschedules.However,usinganorder-size-anddelivery-frequency-basedfeeschedulewith0or1monthsoffreestoragealsoproducesgoodresults.Wenotethatincreasingtheminimumordersizeto40casesperdeliverywillencouragebidderstoofferlowerfees,andbasedontheresultsobtained,thisalsoreducesthepotentialaverageincreaseinexpensesfortheagencies(note,however,thatsomeagenciesmayrequireinvestinginon-sitestoragespaceunderthisrequirement).Wealsoobservethatanotherfactorthatreducestheaverageoverpaymentfortheagenciesisreducingthetimeoffreestoragetozero.Thisnewpolicynotonlywillresultinlowerfeesfortheagencies,butalsomaychangethebehavioroftheagencies,astheywillseedirecteconomicbenetsfromreducingtheirstoragetime.Asaresult,thebidwilllikelybemoreattractiveforpotentialbidders,andanincreaseinthenumberofcompetitorsforthebidwillresultinfavorablefeesfortheagencies.Finally,weobservethatoneofthemostfrequentlyattractiveoptionsrelatedtostoragepoliciesliesindeningthestoragefeetoequal25%ofthedeliveryfee.Usingotherpoliciesresultsinlowervaluesforstoragefees,andsincethestate-contractedwarehousesneedtoachieveaminimumprotmargin,thecorrespondingdeliveryfeesincrease.Anincreaseindeliveryfeeshasamorenegativeimpactontheaverage 77

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overpaymentoftheagencies,comparedwithanincreaseinthestoragefees,sincethiswillaffecteveryagency(notonlytotheoneswithhighstoragetimes). 3.4.2CombinedDistributionNetworksWithrespecttothecaseinwhichweuseonedistributionprogramfortheNSLPandTEFAPagencies,candidateregionalconguration4isalwaysthebestoption.Thereasonforthisisthatforthiscandidateconguration,oneregioniscreatedthatcontainsalloftheagenciesthatbelongtoTEFAPaswellastheNSLPagenciesfrom9counties.Asaresult,theimpactontheagenciesthatbelongtoTEFAPisreducedwhencomparedwiththeotherthreecandidateregionalcongurations(wherewesplitupTEFAPagenciesandassignthemtotheregioninwhichtheyfall).UndertheassumptionthatweuseonedistributionprogramforNSLPandTEFAP,theagencieswiththehigheroverpaymentvaluesaretheonesthatbelongtoTEFAP(ineffect,theTEFAPsubsidizestheNSLP).Ifweuseanorder-quantity-anddelivery-frequency-basedfeeschedule,wereducethenegativeimpactontheagenciesinTEFAP,sincethedifferenceintheaverageordersizeforagenciesintheNSLPisconsiderablylarge.Forthisreasonthetop6resultsinTable R-1 ,whichshowsthe25feescenarioswiththelowestaverageoverpayment,applyavariablefeeschedule.SincetheagenciesthatbelongtoTEFAPhavealoweraveragestoragetimepercaseandlargerordersizewhencomparedwiththeagenciesfromNSLP,thepoliciesrelatedtotheminimumordersizeandfreestoragetimedonothaveaconsiderableimpactontheoverpaymentoftheTEFAPagencies.ForthisreasonwedonotobserveaspecictrendforthesefactorswhenweuseonedistributionnetworkforNSLPandTEFAP. 3.4.3LimitsonVolumeSenttoFoodProcessorsInordertoevaluatetheimpactofapolicythatlimitstheamountofproductthattheagenciescansendtothefoodprocessors,weestimatedthedemandfortheagenciesundertheassumptionthattheycansendatmost30%and50%oftheirentitlementto 78

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foodprocessors.Table 3-2 showsthedetailedinformationontheincreaseintheannualdemandwhenweapplysuchpolicy(onlytheaffectedagenciesareshowninthetable). Table3-2. Increasedinannualdemandwhenusingapolicythatlimitstheamountofproductsenttofoodprocessors RACurrentannualdemandAdditionaldemand30%ofentitlementAdditionaldemand50%ofentitlementRACurrentannualdemandAdditionaldemand30%ofentitlementAdditionaldemand50%ofentitlementCasesCasesCasesCasesCasesCases 010013,9688,5074,943010433,0514,2792,185010041,332137010453,3722,9341,133010071,205119010468,4442,044010086,286612010471454,5743,226010094,6593,9491,490010496,70717,59510,652010105,02513,5058,211010511,95035,05324,481010121,5994,8543,010010531,83951,51136,268010141,792505010543,1964,4892,2930101646156,34540,115010554,6433,9231,476010178,83712,7996,617010578,4434,091510010185,1428260105810,33011,9275,568010199401110105916,6958,3901,2230102342718010601,1682,0721,14701024740180106413,29019,0529,812010265442,8761,89902021337242010287,941520204924920010312,0267,7114,92902052683105010322,8769500206080150103374870206511670103425434117102067539640103521,3402,12902070824903270103620033,79824,085020712661,9481,3160103712,3377050250313112351010382,26417324083172614010397552152002494109010426,60017,67310,738 InAppendices S through V weshowtheestimatedfeesthatabiddermayofferwhenwelimittheamountofproductsenttothefoodprocessorto30%and50%oftheentitlementforthe25bestfeescenariosassumingtheuseofbothseparateandcommondistributionnetworksforNSLPandTEFAP.Asexpected,weobservethatforallfourscenarios,thestorageanddeliveryfeescandramaticallydecreasewhenweincreasetheannualdemandoftheagenciesaccordingtoTable 3-2 .Thereasonforthisisthatthecostpercaseforthestate-contractedwarehouseswilldecreasedue 79

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toeconomiesofscale,andtherefore,theywillrequirelowerper-casefeesfortheirservices.Asaresultofouranalysiswehaveidentiedfeescenariosthatnotonlyreducetheimbalanceintheaverageexpensepercasefortheagenciesindifferentregions,butalsothatleadtoalownegativeimpactfortheagencieswhoseexpensesmayincreaseundertheproposedfeescenarios.Ourrecommendationswithrespecttothedistributionsystemandfeescheduleparametersarelimitedbyexternalitiesthataffectwhethercertainstructuralchangesmaybeimplemented.However,ifwewanttodesignanITBthatisattractivetopotentialcontractbidders,werecommendcombiningthetwofooddistributionprogramsintoonedistributionnetwork,increasingtheminimumordersizeto40cases,andeliminatingthefreestoragetime.Anotherrecommendationthatcandramaticallydecreasestorageanddeliveryfeesfortheagenciesisenforcingalimitonthepercentageofeachagency'sentitlementthatcanbesenttofoodprocessors.Althoughthisrestrictionreducestheexibilityfortherecipientagenciesintermsofhowtheyallocatetheirfoodentitlement,itwillresultinlowerservicefeesfortheNSLPagenciesandinamoreattractivecontractforwhichpotentialcontractorsarelikelytocompete. 80

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CHAPTER4DESIGNINGDISTRIBUTIONFEESUNDERCOMPETITION 4.1IntroductionDeningfeesfordeliveryservicesinwhichcustomerssharetruckcapacitytypicallyrequiressolvingsomevariationofthevehicleroutingproblem(VRP)inordertodeterminetheminimumassociatedroute-baseddeliverycostsacarrierwillincur(whereweusethetermcarriertorefertoanyrmthatprovidesdeliveryservices).Inpractice,deningfeesfortransportationservicesinvolvescomplexitiesbeyondthatofsolvingaVRP,whichisanextremelycomplexproblembyitself(see[ 38 ]foranoverviewoftechniquesforsolvingtheVRPanditsvariations).Thecomplexityassociatedwithdesigningsuchfeestructuresliespartlyinthefactthatthecostsacarrierincursinservingapotentialcustomerdonotdependonlyontheparticularcharacteristicsofthatcustomer(e.g.,location,demand,ordesireddeliveryfrequency).Theyalsodependtosomedegreeonthecharacteristicsoftheentiresetofcustomersthecarrierserves.Forexample,assumingacarrierhasavailabletruckcapacity,ifapotentialnewcustomerislocatedalongtheexistingdeliveryrouteforacurrentsetofcustomers,thecorrespondingmarginaltransportationcostassociatedwiththenewcustomerissmallcomparedwiththecostofaddinganewcustomerlocatedfarfromtheexistingsetofcustomers.Basedontheeconomiesofscaleassociatedwithdeliveryoperations,carriersoftenwishtoservealargesetofcustomersconcentratedinarelativelysmallgeographicarea,sothatdeliveryroutescanbesharedwhileincreasingtheutilizationofresources(e.g.,trucksanddrivers).Thisstrategyoftenleadstointensecompetitionbetweenrmsseekingtomaximizemarketshare,ascustomerswillchoosethecarrierthatoffersthelowestdeliveryfee,allelsebeingequal.Weconsiderasettinginwhichashipperusesabiddingprocesstocontractdeliveryservicesforshipmentstoasetofretailpoints.Acarrierwhoisinterestedintheshipper'sbusinessmustthereforepreparea 81

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bidforprovidingdeliveryservicetooneormoreoftheretailpoints.Suchacarrierisnotunlikelytosubmititsbid(s)basedonthecostassociatedwithservingananticipatedsetofcustomers.Ifthecontractforeachdeliverypointisawardedtothebidderwiththelowestfee,thenthecarrierfacesthepossibilityofendingupwithasetofdeliverypointsthatisdifferentfromtheoriginallyanticipatedset.Thisdisconnectbetweentheanticipatedsetofdeliverypointsandtheactualrealizationwillaffecttheaveragecostofthecarrier'sservice,andtherefore,thecarrier'sprotability.Giventhefactthatacarrier'scostsdependonthesetofcustomersserved,andthatthesetofcustomerswhoselectacarrierdependsonthecarrier'sfeesandthatofitscompetitors,weareinterestedindetermininghowtodesignafeeschedulethatmaximizesthenumberofcustomersacarrierserves,whileensuringthatthecarrierrealizesadesiredprotmargin.Theobjectiveofmaximizingthesetofcustomersacarrierservesisjustiedbytheeconomicsoftransportationservices.CarriersinthetransportationservicesindustryfaceacompetitivemarketwithlowprotmarginsaccordingtoSongandReagan[ 35 ],andthereforefocusonstrategiesthatincreasemarketsharebyexploitingeconomiesofscale.Balakrishnanetal.[ 3 ]consideradeliveryfeesettingproblemfordistributorsofconsumerproductsinalarge-scaledistributionnetwork.Usingalinearoptimizationmodelthatminimizesthesupplier'sdeliverycostsandguaranteesequitablecompensationtoserviceproviders,theydesignafeeschedulethatdependsondeliveryweightsanddistances.Thismodelassumesthatthesuppliersetsthecompensationlevelsfordistributionservicesforanexistingnetworkofdistributors.Weareinterestedinadifferentcontextinwhichashipperwishestocontractdeliveryservicesforasetofretailpoints,andcarriersproposeservicefeesforaselectedsetoftheseretailpoints.Inpractice,whenshipperswishtocontracttransportationservices,theyoftenusearequestforproposal(RFP)process,which,aspointedoutbyCapliceandShef[ 4 ],correspondstoanauction.Insuchcases,theshipperwhoneedstomoveproductfrom 82

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anorigintoadestinationistheauctioneer,andthecarriersinterestedinprovidingtheservicesactasthebidders.Inregularauctions,bidsareplacedforindividualitemsforsale.Intransportation,eachitemmaycorrespondtoaspeciclane(denedbyanoriginanddestinationpair,withthepossibilityofstopsinbetween)oradeliverypoint(denedasaretailpointinourmodel).Thedisadvantageofusingsingleunitauctionsforcontractingdeliveryservicesisthattheydonotexplicitlyaccountfortheinterdependenciesofallofthefactorsthataffectthevaluetothebidder,inourcaselanesorretailpoints.Acarrier,forexample,maybeinterestedinservingaspecicsetofretailpoints,butnotinservingastrictsubsetofthissetofretailpoints.Insuchacase,becauselosingthebidforatleastoneoftheretailpointsinthesetmaymakethedeliveryservicefortheotherretailpointsunattractive,thecarriermaydecidenottoplaceabidforanyoftheretailpoints,ormayneedtoincreasetherequiredprotmargin(and,therefore,bidprice)inordertoreducetheassociatedriskofwinningonlyasubsetofthebids.Inordertoallowbidderstoexpressinterestinagroupofitems,ratherthaneachitemindividually,theauctioneermayuseacombinatorialauction(CA),wherebidderscansubmitbidsforsetsofitemscalledbundles.DeVriesandVohra[ 12 ]discusshowCAscanbeusedwhencomplementaryorsubstitutioneffectsexistbetweenitemsinabid,citingexamplesoftheuseofCAsintheauctionofradiospectrumrights,airporttime-slotallocation,andthepurchaseoftransportationservices.TheypresentanoverviewofCAsfromtheauctioneer'sperspective,focusingonbiddesign,thewinnerdeterminationproblem,iterativeauctions,andincentives.Theauthorsstudyaformulationforthewinnerdeterminationproblemasaninstanceoftheset-packingproblem(inthecaseofpurchasingauction,aset-partitioningorset-coveringproblemwouldbeusedinstead).Fortransportationservicesinparticular,theinterdependenciesintheoperationalcostsofdifferentservicerequestsmaketheproblemofefcientlypurchasingand 83

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pricingtransportationserviceschallenging.CapliceandShef[ 4 ]proposetheuseofoptimization-basedtechniquesforprocuringtransportationservices,exploitingtheeconomiesofscopefordistributorsintheassignmentoflanes.Thisapproachisjustiedbasedonthefactthatefcientlaneassignmentsreducecustomertransportationcostsandresultinlessexpensivecarrierservicecosts.CasesofsuccessfulimplementationofCAsforcontractingtransportationservicesarediscussedbyLedyardetal.[ 23 ],whodiscussadetailedimplementationofacombinatorialauctionbySearsLogisticsServices,wherethecompanysaved13%inservicecosts,andElmaghrabyandKeskinocak[ 14 ],whoprovideanoverviewoftheimplementationofCAsbyTheHomeDepot.Mostoftheliteratureoncombinatorialauctionsfocusesonthebenetstotheauctioneerandontheauctiondesignandwinnerdeterminationproblems.However,weareinterestedinthebidconstructionproblem,i.e.,fromthebidder'sperspective,howtoselectthebundleofitemsonwhichtoplaceabid.Inparticular,weareinterestedinidentifyingconditionsunderwhichitispossibletoidentifyabiddingstrategyforacarrierthatmaximizesthenumberofretailpointswhoselectthecarrier.Thecomplexityofthebidconstructionproblemliesintheexponentialnumberofpossiblebundlesthatabiddermustevaluate.Kwonetal.[ 20 ]presentedapricingandbidconstructionmechanismforaCAthatoperatessimilarlytotheDantzing-Wolfedecompositiontechnique.Theyproposetheuseofaniterativeascendingauction,wheretheauctioneersolvesarevenue-maximizingallocationproblem(whichisthemasterproblem).Theythendenesingle-itempricesthatareusedbybidderstosolvesubproblemsinwhichtheymaximizetheutilityassociatedwithbiddingonabundle.Theproposedmechanismwasnotdesignedfortransportationservicesprocurement,andtherefore,doesnotaccountforthecostevaluationcomplexityoftransportationservices.Additionally,thebestresponseobtainedfromeachbidderdoesnotaccountfortheeffectsofcompetitionandisbasedonautilitymaximizationobjective. 84

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ThebiddingconstructionproblemforCAs,specicallyforpurchasingtransportationservices,hasbeenanalyzedinseveralpapersbySongandReagan.Initiallyin[ 33 ],theyevaluatedthebenetsofCAsforacarrierandproposedabidconstructionmethodthatminimizesthetotalcostofoperatingemptytrucks(whattheyrefertoasemptycost).Theyuseasearchmethodtondallpossibleatomicbundles,i.e.,bundlesofitemswithanall-or-nothingrelationship,startingwithbundlesthatinvolvefourloadedlanesandnoemptylanes,andnishingwithbundlesthatinvolveoneloadedlaneandoneemptylane.Inordertodenethepriceofeachbundle,theyaddedthedistance-basedtransportationcostmultipliedbyaprotmargin,plustheemptycostofthebundle.Usingsimulation,theydemonstratedareductioninthecarrier'stotalemptycostwhenusingCAsascomparedtosingleitemauctions.Later,SongandReagan[ 34 ]presentedanapproximationmethodforthebidconstructionproblem.Usingasearchmethod,theyndthepossiblebundlesonwhichthecarriercanbid,andchoosetheoptimalsetofbundlesusingasetpartitioningformulationthatminimizesthecarrier'semptycost.In[ 36 ],theyusesimulationtoshowtheeffectivenessofthisnewmethodoverthemoresimplesearchmethodoriginallyproposed.Insteadofminimizingthecostofemptytrucks,Leeetal.[ 24 ]formulatedaquadraticintegeroptimizationproblemfortheselectionofbidsinaCAthatmaximizesacarrier'sprot.Theyproposeadecompositionapproachtosolvetheproblembasedoncolumngeneration.Thenoveltyoftheirapproachliesinthefactthattheyintegratecustomerselectionandroutegenerationdecisions,whilebalancingoperationalconstraints.Althoughtheirapproachaddressesthecomplexityoftheroutingproblem,itdoesnotincludecompetitiveeffectsintheselectionofbundles,anddoesnotguaranteethebundleassignmentforacarrier.Basedonthecomplementaryeffectsofdifferentlanes,Anetal.[ 1 ]proposeabiddingstrategythatndsthevalueofanybundlebasedonpairwisesynergies,whereapairoflanesexhibitsynergieswhentheirtotalvalueislargerthanthesumoftheir 85

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independentvalues.Theyevaluatetheeffectoftheirstrategyunderdifferentmarketconditions,whichmayincludethevaluationoftheircompetitor.Usingsimulation,theydemonstratedthebenetstothecarrierandshipperwhenusingCAsinsteadofsinglebidauctions.However,intheirmodel,whencomparingthevalueofacompetitor,theydidnotaccountforhowthesynergisticeffectofthebundlesaffectsthecompetitor'sbehavior.WangandXia[ 39 ]alsostudiedthebidgenerationproblemfortransportationserviceprocurement.Theyrstexaminethesynergiesbetweennewlanesandexistinglanes,anddenetheoptimalsetofbundlesasthesetoflanesthatminimizetheexpectedemptydistancetraveled.Theyassumethatalllaneshavethesameprobabilityofbeingwonintheauction,incontrastwithourmodel,whereweacknowledgethatthedifferentcoststructuresbetweencarriersaffecttheabilitytowinaretailpoint'sbusiness.Inthelatterpartofthepaper,WangandXia[ 39 ]proposedtwoheuristicmethodsforsolvingthebidgenerationproblem,onebasedonaroutingandschedulingoptimizationproblemformulation,andthesecondbasedonanearestinsertionmethod.Theirnumericalanalysisdoesnotshowthateithermethodstrictlyoutperformstheother,andtheyadvisebiddersthatsolvinganoptimaltruckassignmentproblemdoesnotalwaysresultinasolutionwiththeminimumcostofoperatingemptytrucks.Asintheworksmentionedearlier,theroleofacompetitorwasnotaccountedforintheirmodel,andtherefore,thereisnoguaranteethatacarrierwillultimatelyserveadesiredsetofretailpoints.Incontrastwiththepreviouslydiscussedworks,westudyhowtheinteractionbetweendifferentagentsaffectsthedecisionstheymake.Specically,weconsideracarrier'spricingandretailpointselectiondecisionsinthepresenceofacompetitorthatiscurrentlyservingthesetentireofcustomers.Wepresentaprice-basedcompetitionmodelwheretheincumbentcarrieristheleaderandthenewcarrieristhefollower;wemodelthenewcarrier'sentrydecisionasasequentialgameinwhicheachplayeroffers 86

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deliveryfeestothesetofretailpointsservedbytheincumbent,withanobjectiveofmaximizingthenumberofdeliverypointsserved.Ourgoalistocharacterizethepricingandcustomerselectiondecisionsthatguaranteeawinningbidforthenewcarrier.Inparticular,wewishtodevelopastrategythatenablesacarriertoplanentryintoanewmarket,withanobjectiveofmaximizingmarketshare,measuredbythenumberofretailpointswhochoosethecarrierfordeliveryservice.Weassumethatasingleshipperusesasingleitemauctionschemeforeachretailpoint,whichcanbeviewedasaworst-casescenarioforthebidder,sincelosingatleastonebidwillaffectitsexpectedprot.Weanalyzethreedifferentcostallocationmechanismsforsettingprices,andattempttocharacterizeacarrier'sbeststrategy,basedontheircoststructureandthatofacompetitor.Wesaythatanequilibriumsolutionexistsforthegameifwecanndasolutionforacarrieranditscompetitorfromwhichneitherhasanincentivetodeviate.Weinvestigateconditionsunderwhichanequilibriumsolutionmayexist,aswellaswhetherornotsuchasolutioncanbefound.Forcasesinwhichanequilibriumsolutiondoesnotexist,anewcarrierwillnothaveanincentivetocompete,andtherefore,theexistingcarrierwillcontinueservingallofthecustomers.Theremainderofthispaperisorganizedasfollows.Section 4.2 introducesourmodel,wherewepresentdifferentcostallocationschemes,deneourgame,andprovideasolutionprocedureforourmodel.Section 4.3 presentsanumericalanalysisofourmodelandtheeffectsthatthedifferentcostparametershaveonthecarrierschosenbyeachdeliverypoint,whileSection??summarizesourworkanddiscussesfutureresearchdirections. 4.2ProblemDescriptionConsiderashipperwhoisinterestedincontractingthedeliveryservicesofasetofproductstoasetofcustomersC.Theshipperusesasingleitembiddingprocessfordeliveryservice,whereinterestedbidders(e.g.,carriers)proposedeliveryfeesforeachcustomer,andwhereeachcustomerisassignedtothebidderwhoproposesthe 87

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lowestfee.Foreachcarrier,theproposedfeeswilldependonthecostofservingeachcustomer.However,acarrier'smarginalcostforservingacustomerdependsnotonlyonthecustomer'scharacteristics(locationanddemand),butalsoonthecharacteristicsofthecollectivesetofcustomersthecarrierserves.Giventhattheservicefeesthatthecarriercanproposetoeachcustomerdependontotalcostsinthepresenceofeconomiesofscale,areasonablecarrierstrategyisoftenoneofattemptingtoserveasmanycustomersaspossiblewithinaspecicregion.Thechallengeassociatedwiththisstrategyisthatacompetingcarrierisnotunlikelytoalsousesuchastrategy,andcustomerswillultimatelychoosethecarrierwiththelowestproposedfee.Sincecarriersinterestedinofferingdeliveryservicestypicallyestimatecostsandfeesbasedonanexpectedsetofcustomers,iftheyendupwithasetofcustomersthatdiffersfromtheirexpectations,theircostsandprotsmaydiffersubstantiallyfromwhattheyanticipated.Wemodelthecarriers'decisionsusingaStackelberggameinwhichtheincumbentcarrieristheleader.Theleaderbeginsbyproposingasetoffees,andthefollowercounterswithitsproposedsetoffees.Thissequentialgamecontinuesuntilanequilibriumsolutionisachieved,oruntilitisconcludedthatnoequilibriumsolutionexists.Sincedifferentcostallocationschemeswillresultindifferentproposedfeesfordeliverypoints,andwillalsoproducedifferentresultsinourmodel,inthenextsectionwewillcharacterizethreedifferentcostallocationschemesthattheshippermayrequirecarrierstofollowinsettingdeliveryfees.Wewillthendeneourgame,thenecessaryconditionsfortheexistenceofanequilibriumsolutionandthesolutionprocedureforsolvingourmodel. 4.2.1CostAllocationMechanismsConsiderasetJ=f1,2goftwocarriers,eachwithadepot,suchthateachcarrierwishestodenefeesforservinganysubsetofashipper'sdeliverypointsinthesetC=f1,2,...,ng.Letdijdenotethedistancefromdeliverypointitodepotjforeach 88

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i2Candj2J.Weassumethateachcarrierhassufcienttruckcapacitytoservealldeliverypointsandthatalldeliverypointshavethesamequantityrequirementsfordelivery.Considerasubset^CC,andletRj(^C)denotetheminimumcostincurredbycarrierj2Jwhenservingonlythecustomersintheset^C.WedeneDj(^C)astheminimumroutedistanceforcarrierjwhenservingcustomersin^C.LetjdenotethevariabletransportationcostpermileandletFjdenotethexedtransportationcostperrouteforcarrierj2J.Thenthetotalcostofservingtheset^CforcarrierjisgivenbyRj(^C)=Fj+jDj(^C).Whencomputingdeliveryservicefeesforacarrier,weassumethateachdeliverypointwillbechargedsomefractionofthetotalcostofdeliveryservices,ensuringthatthesumofallfeesequalsthetotaldeliverycostforacarrier.Acommonpracticeforthird-partylogisticscompaniesinNorthAmericaistouseacost-pluspricingstrategytocomputedeliveryservicefees,asdiscussedbySmyrlis[ 32 ].Weassumethatbothcarriersusethesamedesiredprotmargin;therefore,usingonlythetotalservicecostinourmodelwillgiveequivalentresultstoaddingaxedprotmarginforeachcarrier.Nextweprovideadetailedexplanationofthethreedifferentcostallocationschemesthattheshipperrequirescarrierstouseinsettingdeliverypointservicefees. 4.2.1.1Distance-basedcostallocationAssumingadistance-basedcostallocation,letpdij(^C)bethefractionofthetotaltransportationcostallocatedtodeliverypointi2^Cwhencarrierjservesonlydeliverypointsin^C,wherepdij(^C)=dij Pc2^Cdcj.Thenthedeliveryfeeofferedbycarrierjtodeliverypointi2^C,isequaltofdij(^C)=pdij(^C)Rj(^C), 89

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whichimplies fdij(^C)=Fj+jDj(^C) Pc2^Cdcjdij,(4)andXi2^Cfdij(^C)=Xi2^Cpdij(^C)Rj(^C)=Rj(^C).Therefore,thefeeschargedtothedeliverypointsin^Ccoverthetotaltransportationcostoftheservice.Thistypeofcostallocationmechanismcanbeusedwhentheshipperrequiresadistance-basedfeeschedule,whichmaybeperceivedasafaircostallocationfordeliverypoints,sincetheclosertheyaretothedepot(relativetotheotherdeliverypoints),thelowertheassociatedfee. 4.2.1.2UniformcostallocationAuniformcostallocationschemewillassignanequalproportionofthetotalcostforservingdeliverypointsinCtoeachdeliverypoint,independentoflocation.Therefore,thefeeforanydeliverypointi2^Cisequalto fuij=Fj+jDj(^C) j^Cj.(4)Thiscostallocationcanbeusedwhenashipperrequiresauniformfeescheduleforallcustomers.Thismaybethecasewhenashipperrequiresnopricediscriminationamongitsdeliverypoints. 4.2.1.3Branch-basedcostallocationAbranch-basedcostallocationusestheresultsofatravelingsalesmangametoallocatecostsforagivenminimumcostrouteand,thereforedeterminedeliveryfees.Inusingsuchacostallocationscheme,notethatwearenotnecessarilytryingtoestablishafaircostallocationforthedeliverypoints,astheyallbelongtothesameshipper,andthushavenointerestinformingcoalitions.Forthisreason,wearenotrequiredtosatisfy 90

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anyconditionthatguaranteesanon-emptycoreinthegame,asintheonepresentedbyCurielin[ 10 ].Assumethatcarrierj2Joffersdeliveryservicestoasetofretailpoints^CCusingaminimumlengthroutethatfollowsthesequence=f(1),(2),...,(j^Cj)g.Thatis,thetourstartsatthedepotofdistributorj,visitsdeliverypoint(1)rst,thendeliverypoint(2)andsoon,untilthelastdeliverypoint,(j^Cj),isvisited,beforereturningtodepotj.Wedene!j(i),(i+1)asthedistancetraveledfromdeliverypoint(i)todeliverypoint(i+1)inthetheminimumlengthrouteon^C[fjg.Then,forthebranch-basedcostallocationschemeandforagivenminimumlengthrouteon^C[fjg,weuniformlydistributethexedtransportationcostandtheroute-distancecostoftherstandlastbranchesoftherouteamongalldeliverypointsontheroute.Wethenassigntodeliverypointi2^Cthetransportationcostassociatedwiththebranchoftheroutethatstartsatlocationi.Asaresult,thefeechargedtodeliverypointi2^Cwhencarrierjservestheset^Cusingtheminimumlengthrouteisgivenbyfbij=Fj j^Cj+j(!jj,(1)+!j(j^Cj),j) j^Cj+j!j(i),(i+1),where!jj,(1)and!j(j^Cj),jarethedistancesoftherstandlastbranchesoftheminimumlengthtouron^C[fjg,respectively.Sincetheminimumlengthroutewhenservingasetofretailpointsmaynotbeuniqueandeachalternativeoptimalsolutionwillresultinadifferentdeliveryfeescheduleforthesetofretailpointsserved,wedenethefeeofferedbycarrierjtoretailpointi2^Castheaverageofthedifferentservicefeesthatresultfromeachminimumlengthroute2j(^C),wherej(^C)isthesetofalternativeoptimalsolutionsfortheminimumlengthrouteproblemon^C[fjg.Therefore,wehavethatthefeeofferedbycarrierjtodeliverypointi2^C,whenabranch-basedcostallocationisused,isequal 91

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to fbij=Pjj(^C)jk=1fbkij jj(^C)j,(4)wherethesumofallthefeeschargedtothesetofretailpointsin^Ccoversthetotalcostoftheservice,sincefb1j+fb2j+...+fbj^Cjj=Pjj(^C)jk=1fbk1j jj(^C)j+Pjj(^C)jk=1fbk2j jj(^C)j+...+Pjj(^C)jk=1fbkj^Cjj jj(^C)j,whichimpliesfb1j+fb2j+...+fbj^Cjj=Fj+jDj(^C).Wecaninterpretthiscostallocationapproachaschargingeachdeliverypointanamountproportionaltoitsmarginalcost,sincethedeliverypointsthatarelocatedwithinageographicalclusterhavealowermarginalcostcomparedwithanisolateddeliverypoint(assumingthattheyarelocatedwithinacomparableradiusfromthedepot).Thiscostallocationapproachcanbeusedwhentheshipperrequiresafeeschedulethatrewardsdeliverypointsthatarelocatedwithinacluster(e.g.,adenselypopulatedcity)andpenalizesgeographicallyisolateddeliverypoints. 4.2.2GameDenitionSupposeacarrierwishestomaximizethenumberofdeliverypointsitserves,andthatacompetitorexistswhowouldalsoliketoserveamaximalsetwithinthesamesetofdeliverypoints.Sincetheshipperwillassigneachdeliverypointtothecarrierthatoffersthelowestfee,itiscriticalforthecarrierstoidentifythesetofdeliverypointstowhichtheycanofferfeesthatcannotbeunderpricedbytheircompetitor.Otherwise,acarriermaybeassignedadifferentsetofdeliverypointsthananticipated,resultinginunanticipatedcostandprotlevels.Consideragivencarrierandassumethatthiscarrieranditscompetitorusethesamecostallocationscheme.Supposefurtherthatthiscarrierknowsitscompetitor'sxedandvariabletransportationcostvalues,and,therefore,isabletocomputethefees 92

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thecompetitorwilloffertoanysubsetofdeliverypoints.Denexi=1ifcustomerichoosescarrier1,andxi=0ifcustomerichoosesthecompetitor,carrier2,8i2C.Assumethatfj(^C)forj=f1,2gisthefeevectorofferedbycarrierjwhenservingthesetofdeliverypointsin^CC,wherefij(^C)isthefeeofferedbycarrierjtodeliverypointi2^C.Sincedeliverypointi2Cwillchoosethecarrierthatoffersthelowestfee,foragivenfeevectorfj(C)proposedbycarrier2forservingdeliverypointsinthesetCC,theoptimalresponseforcarrier1isdeterminedbysolvingthefollowingproblem. maxi2CXi2Cxi (4a)s.t.fi1(X)
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neithercarriercanimproveitsresponsetotheother'sproposedfees.Byassumption,ifanequilibriumsolutionresults,carrier1proposesdeliveryfeestothosedeliverypointssuchthatxi=1intheequilibriumsolution,andthesecustomerswillacceptthefeeofferedbecausecarrier2cannotofferabetterrate.Recallthatthedifcultyassociatedwiththisoptimizationproblemliesinthefactthatitrequiressolvingatravelingsalesmanproblem,whichisNP-hard. 4.2.2.1EquilibriumsolutionsWehaveanequilibriumsolutionifcarrier1servesdeliverypointsinasetC1,carrier2servesdeliverypointsinthesetC2=CnC1,andeachofthesesetsisthebestresponseforeachcarrier.Thismeansthatifcarrier1servesthesetofdeliverypointsC1,thenthemaximumnumberofdeliverypointsthatcarrier2canserveequalsjC2j.Also,whencarrier2servesthesetofdeliverypointsC2,themaximumnumberofdeliverypointsthatcarrier1canserveequalsjC1j.Inotherwords,thesolutioninwhichcarrier1servesdeliverypointsinthesetC1andcarrier2servesdeliverypointsinC2=CnC1,isanequilibriumsolutionifforeachsetofdeliverypointsSCsuchthatS\C16=;,andforatleastonek2S\C1, fk2(S)fk1(C1);(4)inaddition,foranysetTsuchthatT\C26=;,andforatleastonel2T\C2,wemusthave fl1(T)fl2(C2).(4)Forinstance,ifcarrier2canofferalowerfeetodeliverypointk2C1,thenthisenablesservingthesetC2[fkg.However,thiswouldcontradictcondition( 4 ),sinceC2[fkgCand(C2[fkg)\C16=;,andasaresult,carrier2cannotofferalowerfeetodeliverypointk2C1.Thatis,neithercarriercanndasubsetSCcontainingatleastonedeliverypointservedbyitscompetitor,suchthattheirpositionisimprovedbyservingalldeliverypointsinS.NotethatsinceweareinterestedinaStackelberggame 94

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wheretheincumbentcarrieristheleader,thesolutionthatsatisesconditions( 4 )and( 4 )isasubgameperfectequilibriumforoursequentialgameasdenedin[ 28 ]. 4.2.3SolutionProcedureInthissectionwewilldenethesolutionprocedureforourmodelusingtheresultspresentedinSection 4.2.2 .Weconsidertheexistenceofanincumbentcarrierthatiscurrentlyservingallthedeliverypoints,andtherefore,istheleaderinoursequentialgame.Then,acompetitorndsitsbestresponsebysolving( 4 )withthefeesofferedbytheincumbentcarrierasaninput.Wecontinuewithasequentialgame,whereeachcarrierndsitsbestresponsetothefeesofferedbyitscompetitor,untilwendasolutionthatsatisesconditions( 4 )and( 4 )orwedetectthenonexistenceofanequilibriumsolution.Sincethesolutionforourmodelwilldependoftheresultsofthegame,wenextdiscussthepossibletypesofoutcomesthatmayresult. 1. Thereisatleastonepairofsets(C1,C2)thatsatisesconditions( 4 )and( 4 ):Thesolutionforourmodelisthepairofsets(C1,C2),thatsatisesconditions( 4 )and( 4 ),identiedbythefollower.Notethatsincethefolloweridentiestheleader'sbestresponseinadvance,ifmultiplesolutionsexistthesolutionforourmodelisgivenbythefeesproposedbythefollower. 2. Therstbestresponsefromthenewcarrieristheemptyset:Inthiscase,thereisnochangeintheinitialassignmentofthedeliverypoints,andtheincumbentcarriercontinuesservingtheentiresetofcustomers.Notethatitispossiblethatthebestresponseofeachcarriertoitscompetitor'sinitialfeeproposalistheemptyset;inthiscase,theleaderhastherstmoveradvantageoverthenewentrant,andwillcontinueservingtheentiresetofretailpoints. 3. Nopairsofsets(C1,C2)thatsatisfyconditions( 4 )and( 4 )canbefound:Sinceanequilibriumsolutioncannotbereached,thecurrentcarriercontinuestoserveallofthecustomers.Algorithm 4 inAppendix W showsthesolutionprocedureforsolvingourmodelassuming,withoutlossofgenerality,thatcarrier1istheincumbentcarrier.Werstcomputethedeliveryfeesforcarrier1whenservingtheentiresetofcustomersC;thenwesequentiallyndthebestresponseforeachcarrieruntilweeitherndanequilibrium 95

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solutionthatsatisestheconditionsdenedinSection 4.2.2.1 orwedetectthatsuchasolutioncannotbefound. 4.3NumericalAnalysisInthissectionwewouldliketocharacterizeconditionsunderwhichanequilibriumsolutionmayexist,andprovideintuitiononthebeststrategythatacarriershouldfollowwhenplacingbidsfordeliveryservices(assumingthegoalofbothcarriersistomaximizemarketshare).Werecognizethatourmodelisnottractableforlargeprobleminstances,notonlyduetotheembeddedTSP,butalsobecauseoftheexponentialnumberofinstancesoftheTSPthatwemayneedtoevaluate.Ourgoalisnot,therefore,toprovideadecisionsupporttool;instead,wewishtoutilizethemodelinordertounderstandthestructuralpropertiesofoptimalfeesettingsolutionsunderdifferentcostallocationmechanisms,aswellasconditionsunderwhichequilibriumsolutionsarelikelytoexist.Inordertogainsomeinsightintothesefactors,weconsideredabroadsetofrandomlygeneratedprobleminstances.Theprobleminstanceswegeneratedcontaintencustomersandtwocarriers.Althoughthenumberofcustomersisnotlarge,itpermitsatractablenumberofpossibleresponsesforeachcarrier(210)]TJ /F2 11.955 Tf 12.59 0 Td[(1possibleresponsesintheworstcase)whileallowingustogainsomeintuitionaboutthestructuralresultsofthegame.Forthecarrieranddeliverypointlocationswegeneratedfourgeographicaldistributiontypeswithveinstanceseach.Thefollowingdescribeseachgeographicaldistributiontype: Geographicaldistributiontype1:deliverypointsandcarriersarerandomlylocatedintwoindependentadjacentregions.ThesizeofeachoftheseregionsforeachoftheveassociatedprobleminstancesisshowninTable 4-1 Geographicaldistributiontype2:deliverypointsandcarriersarerandomlylocatedwithinthesamesquareregionwith40,000sqmiofarea. Geographicaldistributiontype3:deliverypointsandcarriersarerandomlylocatedwithinthesamesquareregionwith90,000sqmiofarea. Geographicaldistributiontype4:deliverypointsandcarriersarerandomlylocatedwithinthesamesquareregionwith250,000sqmiofarea. 96

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Table4-1. Carriersandcustomerareasforgeographicaldistributiontype1. CarriersareaDeliverypointsareaWidthHeightWidthHeight(miles)(miles)(miles)(miles) 2001,2006001,2001,0001,0004,0003,0005005002,0003,0006002,1002,0003,0006003,0002,0003,000 Forthexedandvariabletransportationcosts,sincewedonotassumethateithercarrierislessexpensivethantheother,weconsidervariouscostvaluesinordertoevaluateadiversityofpossiblerelativecostvalues.Forthexedtransportationcostswedenesevenpairsofvaluesfor(F1;F2),in$=route,equalto(1,000;1,500),(3,000;3,500),(5,000;5,500),(1,500;1,000),(3,500;3,000),(5,500;5,000)and(0;0).Forthevariabletransportationcost,j,forj2f1,2g,wedenethreebaseparametervaluesof$1.5/mile,$2/mileand$5/mile.Forournumericaltests,whenweassignedthebaseparametervalueforthevariabletransportationcosttooneofthecarriers,weallocatedacostthatis20%highertoitscompetitor.Intotalwehave42totalcostcombinationsforeachlocationinstancetype.Wewanttoanalyzehowchangesineachcostparameterandgeographicaldistributionaffecttheresultsofourmodelunderthethreedifferentcostallocationmechanisms.Weareparticularlyinterestedinthecaseswheretheequilibriumsolutionresultsinapartitionofthesetofcustomersintotwodisjointsets(C1,C2),whereneitherofthesetsisempty,orwhentheequilibriumsolutionresultsintheassignmentofallcustomerstoonecarrier,independentoftheidentityoftheleader,i.e.,whenthereisadominantplayer.Ingeneralwerefertothesetypesofsolutionsasnon-trivialequilibriumsolutions,sinceallotherresultsdonotmodifythecurrentassignmentofdeliverypoints,i.e.,theleadercannotbeunderpricedandretainsallcustomers,asexplainedinSection 4.2.3 .Specically,wedeneanon-trivial-partitionequilibriumsolutionaresultwhereeachcarrierisassignedadisjointstrictsubsetofdeliverypoints,anda 97

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non-trivial-dominantequilibriumsolutionaresultwheretheentiresetofdeliverypointsisassignedtothedominantplayer,regardlessoftheleader'sidentity.ForeachcostallocationschemewepresentasummaryoftheresultsofournumericaltestsinFigures 4-2 4-7 and 4-9 ,whereweshowthepercentageofinstanceswithanon-trivial(eithernon-trivial-partitionornon-trivial-dominant)equilibriumsolutionforeachcostparameterandgeographicaldistributiontypecombination.Sincewegenerateveinstancesforeachcostparameterandgeographicaldistributiontypecombination,aresultof100%correspondstohavingveoutofveinstanceswithnon-trivialequilibriumsolutions.InFigures 4-2 4-7 and 4-9 ,foreachgeographicaldistributiontypewepresenttwocharts,onefortheinstanceswhenF1F2andanotherfortheinstanceswhenF1F2(notethatbothchartscontaintheinstanceswhereF1=F2=0).Eachchartshowsthepercentageofinstanceswithnon-trivialequilibriumsolutionsfordifferentxedcostvalues,andforincreasingvaluesofthedifferenceinvariabletransportationcostbetweenthecarriers(=1)]TJ /F4 11.955 Tf 12.51 0 Td[(2),representedonthehorizontalaxisofeachchart. 4.3.1Distance-BasedCostAllocationInthissectionweanalyzetheeffectsofthecostandgeographicaldistributionparametersontheresultsofourmodelwhencarriersuseadistance-basedcostallocationschemetosetfeesfordeliveryservices.AssumethatC1andC2aretheequilibriumresponsesforcarriers1and2,respectively.Then,asaresultof( 4 ),foreachsetSCsuchthatS\C16=;,thefollowingconditionmustholdforatleastonek2S\C1: F2+2D2(S) Pi2Sdi2dk2F1+1D1(C1) Pi2C1di1dk1.(4)Inaddition,forthefeesofferedbydepot2,foreachsetTCsuchthatT\C26=;andforatleastonel2T\C2,wemusthave F1+1D1(T) Pi2Tdi1dl1F2+2D2(C2) Pi2C2di2dl2.(4) 98

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Sincewewanttocharacterizetheoptimalfeesettinganddeliverypointselectionforthecarriersthatresultfromourmodel,inSections 4.3.1.1 and 4.3.1.2 weevaluatehowchangesinasinglecostparameterinuencetheresultsofourmodel,allelsebeingequal.Followingthis,inSection 4.3.1.3 weanalyzehowthegeographicaldistributionofthedeliverypointsandcarriersaffecttheretailpointassignmentwhenadistance-basedcostallocationisused. 4.3.1.1ImpactofxedtransportationcostInthissectionwewanttoanalyzetheeffectofchangesinthexedtransportationcostontheresultsofourmodel.Weareparticularlyinterestedincharacterizingconditionsunderwhichanon-trivialequilibriumsolutioncanbefound,i.e.,instancesthatresultinanon-trivial-partitionsolutionoranon-trivial-dominantsolution.Toevaluatetheimpactofthexedtransportationcostonourmodelwerstconsiderthecasewhenthexedtransportationcostforeachcarrierisrelativelylarge.Underthiscondition,acarrierinterestedinofferingcompetitivefeeswillwanttoincreasethenumberofcustomersserved,sincethiswillleadtolowerfees,regardlessofthecustomerlocations.However,whentheretailpointsarenotparticularlyclosetoonecarrierandfarfromtheother,theremaynotbesufcientdifferentiationbetweenthesetoffeesofferedbytheplayerstoensureanon-trivialequilibriumsolution.InthecaseinwhichbothcarrierssettheirfeesassumingthattheywillservetheentiresetofretailpointsC,thenneithercarrierisabletounderpricethecompetitor'sbidforallthedeliverypointsintheset,andtherefore,theleadercontinuesservingallretailpoints.Toillustratethissituation,considertheinstanceinFigure 4-1 ,andassumethatforbothcarriersthexedtransportationcostis$100/routeandthatthevariabletransportationcostiszero.Assumefurtherthatcarrier1istheincumbentcarrier,andtherefore,thecurrentfeesfordeliverypoints1,2and3are$16.67,$33.33and$50,respectively.Sincethebeststrategyforcarrier2istoserveallthedeliverypointsintheset,thelowestfeesthatcarrier2canofferare$50,$33.33and$16.67fordelivery 99

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points1,2and3respectively.Giventhatcarrier2cannotunderpricethefeesfordeliverypoints1and2,andthattheservicefeeswillincreaseifasmallersubsetofretailpointsisselected,thebestresponsefromcarrier2isnottoparticipateinthebid,andtherefore,carrier1continuesservingtheentireset.Underthesecircumstances,therstmoveradvantageissuchthatwecanincreasethexedtransportationcostforcarrier1upto$300/route,andcarrier2isstillnotabletoparticipateinthebid(notethatifthexedtransportationcostforcarrier1islargerthan$300/route,carrier2willdominatethegameandservealldeliverypoints). Figure4-1. Instancewithuniformlydistributedcustomers Figure 4-2 summarizestheresultsofournumericaltestswhenadistance-basedcostallocationisusedtocalculateservicefees,whereeachchartshowsthepercentageofnon-trivialequilibriumsolutionsfordifferenttransportationcostsandgeographicaldistributiontypecombinations.Asthechartsillustrate,forsufcientlylargexedtransportationcostvalues,thepercentageofinstanceswithnon-trivialequilibriumsolutionsdecreasesduetothereasonsexplainedabove.However,evenforsufcientlylargexedtransportationcostvaluesthatarenotsufcientlydifferentiatedbetweenthetwocarriers,ourmodelmayresultinanon-trivial-dominantequilibriumsolutionifoneofthecarriershasamorefavorablelocationcomparedwiththecompetitor.Insuchacase,thecarrierlocatedclosertothesetofcustomerswilldominatethegame,regardlessoftheleader'sidentity.Consider,forexample,theinstanceinFigure 4-3 andassumethatthetransportationcostsvaluesarethesameforbothcarriers.Asaresultofourgame,carrier1willunderpricethefeesofferedbythecompetitor,andwillservetheentiresetofdeliverypoints,evenifcarrier2istheleader.Thesesituationsaremorelikelytooccurwhendeliverypointsand 100

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Figure4-2. Distance-basedcostallocation:numberofnon-trivialequilibriumsolutions carriersarerandomlylocatedintwoindependentadjacentregions,whichisthecaseforinstanceswithgeographicaldistributiontype1. Figure4-3. Instancewithfavorablelocationforcarrier1. 101

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Whencarriershavesufcientlysmallxedtransportationcostvalues,changesinthevariabletransportationcostwillhaveagreaterimpactontheresultsofourmodel.Wewillstudytheeffectofthevariabletransportationcostinthenextsection. 4.3.1.2ImpactofvariabletransportationcostInthissectionweevaluatetheeffectofchangesinthedifferenceinthevariabletransportationcostbetweenthecompetitorsontheresultsofourmodel,whileholdingallotherparametersequal.Inordertocharacterizetheeffectsofthedifferenceinthevariabletransportationcost,weassumethatthexedtransportationcostvaluesaresufcientlysmall;otherwise,theeffectsofthexedtransportationcostmayoutweightheimpactofthevariabletransportationcostontheoptimalfeesettingsolutions.Whenthedifferenceinvariablecostbetweenthecarriers(jj=j1)]TJ /F4 11.955 Tf 12.8 0 Td[(2j)issmallenough,thebeststrategyforeachcarrieristoserveonlythecustomersthataresufcientlyclosetotheirdepot.Inthiscase,anon-trivial-partitionequilibriumsolutioncanbeobtainedifthesetofcustomerscanbedividedintwo,whereeachsubsetisclosetoonecarrierandfarfromtheother.Thistypeofnon-trivial-partitionequilibriumsolutionmayresultinsuchcasesbecause,foradeliverypointthatisclosetoacarrierandfarfromitscompetitor,thefractionofthetotaltransportationcostthatwillbeallocatedtosuchadeliverypointwillbesmallerthanthefractionthecompetitorallocates,andtherefore,thenearbycarrier'sfeecannotbeunderpriced.Forexample,supposethat,fortheinstanceshowninFigure 4-4 ,thexedtransportationcostiszeroandthevariabletransportationcostis$1/mileforbothcarriers.Asaresult,independentlyofwhichcarrieristheleader,theequilibriumsolutionwillresultinC1=f1gandC2=f2g. Figure4-4. Instancewithtwoclustersofcustomers. 102

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Wheneachcustomerisnotsubstantiallyclosertooneofthecarriers,alargerdifferenceinthevariabletransportationcost(jj)isoftenneededinorderforanon-trivialequilibriumsolutiontobefound.Insuchcases,ourgamewillresultinanon-trivial-dominantequilibriumsolution.Toobservetheeffectofjjontheresultsofourmodel,considertheinstanceinFigure 4-5 .Assumethat,forbothcarriers,thexedtransportationcostiszeroandthevariabletransportationcostis$1/mile,andthatcarrier1iscurrentlyservingthesetwithdeliveryfees$1.77,$2.66and$3.56todeliverypoints1,2and3,respectively.Inthiscasecarrier2willnotplaceabidandtheincumbentcarrierwillretainallofthecustomers.However,ifthevariabletransportationcostforcarrier1increasesto$2/mile(and,therefore,jjincreases),weobtainanon-trivial-dominantequilibriumsolution,wherecarrier2isthedominantplayer.Asaresult,whenthedifferenceinthevariabletransportationcostsissufcientlylarge,theequilibriumsolutionclearlyfavorsthecarrierwiththelowertotaltransportationcostforservingallcustomers,asthiscarriercanunderpriceitscompetitorformanyorallofthecustomers,independentoftheidentityoftheleader. Figure4-5. Instancewheredeliverypointsarenotsubstantiallyclosertoanycarrier. 4.3.1.3ImpactofthegeographicaldistributionofcarriersanddeliverypointsInthissectionwewanttocharacterizehowdifferentgeographicaldistributionsforcarriersanddeliverypointsaffecttheassignmentofdeliverypointstothecarriers,whenadistance-basedcostallocationisrequired.Appendix X providesadetailedillustrationofthedeliverypointassignmentsobtainedfortheinstanceswithnon-trivialequilibriumsolutionsforeachgeographicaldistributiontypeandcostparametercombination.EachchartinAppendix X showsthepercentageofinstanceswithnon-trivial-dominantequilibriumsolutions,wheretheentiresetofcustomersisassignedtothedominantplayer(carrier1orcarrier2),andwithnon-trivial-partitionequilibriumsolutions,where 103

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theassignmentresultsinapartitionofthesetofdeliverypointsintotwodisjointsets(C1,C2)andneitherofthesesubsetsisempty.Whenthedeliverypointsarelocatedinanindependentregionfromthecarriers,asinthegeographicaldistributiontype1,weobservethat41%oftheinstancesresultinanon-trivialequilibriumsolution,where96%ofthemresultinanon-trivial-dominantequilibriumsolution.Forthistypeofgeographicaldistribution,itislikelythatoneofthecarriershasamoreadvantageouslocationcomparedwithitscompetitor,andtherefore,suchacarriercanofferlowerfees.Asanexample,considertheinstancefromFigure 4-3 where,assumingequalcostparametersforbothcarriers,carrier1hasanadvantageovercarrier2.Inthecasethatthedeliverypointsandcarriersarelocatedwithinthesameregion,asingeographicaldistributiontypes2to4,weobservethat20%ofthoseinstancesresultinnon-trivialequilibriumsolutions.Thereasonforthereductioninnon-trivialequilibriumsolutions,comparedwithgeographicaldistributiontype1,isthatapproximately85%oftheinstanceshavesufcientlylargexedtransportationcosts,andasexplainedinSection 4.3.1.1 ,whenthereisnoclusterofcustomersclosertooneofthecarriers,asinFigure 4-6 ,itmaynotbepossibletondanon-trivialequilibriumsolution.Howeverfortheinstanceswithsufcientlysmallxedtransportationcost,itispossibletondanon-trivial-partitionequilibriumsolution,wherebothcustomersserveastrictsubsetofdeliverypoints.Forthisreason,forgeographicaldistributiontypes2to4,fromthetotalnumberofinstancewithnon-trivialequilibriumsolution,49%ofthemresultinnon-trivial-partitionequilibriumsolutions. 4.3.2UniformCostAllocationInthissectionwewanttocharacterizetheoptimalfeesettingdecisionsforeachcarrier,underdifferenttransportationcostparametervaluesandgeographicaldistributionforthecarriersanddeliverypoints,whencarriersuseauniformcostallocationschemetosetdeliveryfees.AssumethatC1andC2aretheequilibrium 104

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Figure4-6. Instancewhendeliverypointsandcarriersarelocatedwithinthesameregion. responsesforcarrier1and2,respectively.Then,weneedtosatisfythefollowingconditionforeachsetSCforthefeesofferedbycarrier1: F2+2D2(S) jSjF1+1D1(C1) jC1j;(4)and,foreachsetTCforthefeesofferedbycarrier2: F1+1D1(T) jTjF2+2D2(C2) jC2j.(4)Beforecontinuingwithouranalysis,wewouldliketoexplainhowusingasinglefeeforalldeliverypoints(independentofthedeliverypointlocation)contributestondingnon-trivialequilibriumsolutionsforourmodel.Fortheuniformcostallocation,sincethecarriersofferthesamefeetoallcustomersinaset,anequilibriumsolutionisreachedwhenacarrieroffersa(single)feetoasetofcustomersthatcannotbeunderpricedbythecompetitor.Asaresult,whenusingauniformcostallocationmechanismweobservethat89%oftheinstancesfromournumericaltestsresultinanon-trivialequilibriumsolution.Thereasonforhavingalargerpercentageofinstanceswithnon-trivialequilibriumsolutions,comparedwiththeothercostallocationmechanisms,isbecausethecarrierwiththelowertotaltransportationcostforservingtheentiresetCofretailpointswillhaveanadvantageoveritscompetitor,evenfordeliverypointslocatedclosertothecompetitor'sdepot.Forthisreason,inournumericaltests99% 105

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oftheinstanceswithnon-trivialequilibriumsolutionsresultinanon-trivial-dominantequilibrium,i.e.,withoutregardoftheidentityoftheleader,theentiresetofdeliverypointsisassignedtothecarrierwiththelowertotaltransportationcostforservingallofthecustomers.Inordertocharacterizethecarriers'optimalfeesettingdecisionsthatresultfromourmodel,inSections 4.3.2.1 and 4.3.2.2 weevaluatehowchangesinasinglecostparameterinuencetheresults,allelsebeingequal.Followingthis,inSection 4.3.1.3 weanalyzehowthegeographicaldistributionofthedeliverypointsandcarriersaffectstheassignmentofretailpoints,whenauniformfeescheduleisused. 4.3.2.1ImpactofxedtransportationcostForsufcientlylargexedtransportationcostvalues,increasingthenumberofretailpointsservedwilldecreasedeliveryservicefees,regardlessoftheretailpoint'slocation.GiventhatbothcarrierswillattempttoservejCjdeliverypoints,thecarrierwiththelowertotalcostforservingtheentiresetofcustomerswillbeabletounderpricethecompetitor,andasaresult,theentiresetofcustomerswillbeassignedtosuchcarrier.Notehowever,thatinordertohaveanon-trivial-dominantequilibriumsolutionforsufcientlylargexedtransportationcostvalues,thecostservingallofthecustomersforbothcarrierscannotbeequal.Otherwise,bothcarrierswillofferthesamefeesforthesetCofdeliverypoints,andthebestresponsefromthefollowerwillbenottoplaceabid.Toillustratetheeffectofthexedtransportationcostonourmodel,considertheinstanceinFigure 4-1 ,whenthexedtransportationforbothcarriersis$120/routeandthevariabletransportationcostiszero.Inthiscase,sincethesinglefeeofferedbyeachcarrierwhenservingtheentiresetofcustomersis$40,thenewcarrierwillnotplaceabidandtheincumbentcarrierwillretainallofthecustomers.Suppose,however,thatthexedtransportationcostforcarrier1is$150/route.Asaresult,thedifferenceintotaltransportationcostforservingalldeliverypointsallowscarrier2tounderpriceitscompetitor,andcarrier2willdominatethegame. 106

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Figure 4-7 reportsthepercentageofinstanceswithnon-trivialequilibriumsolutionsfordifferentcostparametersandgeographicaldistributioncombinations,whenauniformdeliveryfeeisrequiredbytheshipper.Fromournumericaltests,weobservethat89%oftheinstanceswithxedtransportationcostsstrictlygreaterthanzeroresultinnon-trivialequilibriumsolutions.Thereasonisthatthecarrierwiththelowertotaltransportationcostforservingallofthecustomerswilldominatethegameandwillwinthebidfortheentiresetofdeliverypoints.Asaresult,fromthetotalnumberofinstanceswithnon-trivialequilibriumsolution,99%ofthemareclassiedasnon-trivial-dominantsolutions.Theremaining1%resultsinanon-trivial-partitionequilibriumsolution,andtheseuncommoncaseswillbeanalyzedinSection 4.3.2.3 .Whenthexedtransportationcostvaluesaresufcientlysmall,thevariabletransportationcosthasamorerelevanteffectonourmodelresults.Inthenextsectionwewillcharacterizehowchangesinthedifferenceinthevariabletransportationcostimpactthedeliveryfeesettingdecisionforthecarriers. 4.3.2.2ImpactofvariabletransportationcostInordertoevaluatetheimpactofvariabletransportationcostonourmodel,wewillassumeinthissectionthatthevaluesforxedtransportationcostsaresufcientlysmallforbothcarriers.Inthiscase,thetotalroute-distancewhenservingasetofdeliverypointshasamoresignicanteffectontheoptimaldeliverypointselectionforthecarriers.And,therefore,thecarriers'locationsrelativetothedeliverypointswillinuencetheequilibriumsolutionforourgame.Considerinitiallythecasewhenthesetofcustomerscannotbesplitintwodisjointstrictsubsets,suchthateachsubsetisclosertoaparticularcarrier.Insuchcases,whenthedifferenceinvariabletransportationcostbetweenthecarriersisnotsufcientlylarge,gettinganon-trivialequilibriumsolutionmaynotbepossible.Undertheseconditionsthedifferentiationbetweenthesetoffeesofferedbyeachcarriermaynotbelargeenough,andtherefore,eachplayerwillbeabletorespondtothecompetitor's 107

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Figure4-7. Uniformcostallocation:numberofnon-trivialequilibriumsolutions. beststrategy,resultinginaninniteloopofresponses.ConsidertheexamplefromFigure 4-8 ,whenthexedtransportationcostiszeroforbothcarriers,andthevariabletransportationcostis$1/mileand$1.2/mileforcarrier1and2,respectively.Inthiscase,whencarrier2isthefollower,itsresponseswillloopbetweenC2=f1,2gandC2=C.TheinitialbestresponsefromthefolloweristoservethesetC2=f1,2g,andbasedonthestrategyfromcarrier2theleader'sbestresponseistoserveC1=f3,4g.Afterthat,carrier2canunderpriceitscompetitorbyservingallthedeliverypoints,butthe 108

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responsefromcarrier1willbetoservetheentireset,whichistheinitialstateofthegame. Figure4-8. Instancewithnoclusterofcustomers. Inthecasethattherearetwostrictdisjointsubsetsofdeliverypoints,whereeachsubsetislocatedclosertoaparticularcarrierandfarfromtheother,wemayobtainnon-trivialequilibriumsolutionsforourgame.Inthiscase,thedifferenceinthevariablecostneededtohaveanon-trivialequilibriumsolutiondependsontheroutedistancesofeachcarrier,andtherefore,onthepositionsofthecustomersrelativetoeachcarrier.Ifthedistancebetweenthecarriersislargeenoughandtherearetwodistinctsetsofdeliverypointsclosetoeachcarrier,thenthedifferenceinthevariabletransportationcostcanbehigh,becausetheaddeddistanceofservingadistantcustomerwillincreasetheproposedfeeinawaythatmaynotbeattractiveenoughtotakecustomersfromthecompetitor.Incaseswherethecarriersortheclusterofcustomersarelocatedclosetoeachother,thedifferenceinvariabletransportationcostbetweenthecarriersshouldbesufcientlysmall(dependingontherelativelocationbetweenthecarriersandretailpoints);otherwisethecarrierwiththelowertotaltransportationcostforservingallcustomerscanoutperformthefeesofferedbythecompetitor,evenwhenitsroute-distanceislarger.Take,forexample,thecaseinFigure 4-4 ,andassumethat 109

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thexedtransportationcostforbothcarriersiszero.Whenthevariabletransportationcostforcarrier1and2is$1/mileand$2/mile,respectively,theequilibriumsolutionresultsinC1=f1gandC2=f2g.Ifweincreasethevariabletransportationcostforcarrier2to$3/mile,carrier1willdominatethegameandwillservetheentiresetofcustomers.Forthereasonsexplainedaboveandforsignicantlysmallvaluesforthexedtransportationcost,thepercentageofinstanceswithnon-trivialequilibriumsolutiondecreasescomparedwiththeresultsfrominstanceswithlargerxedtransportationcostvalues,asshowninFigure 4-2 .Notealsothatthedifferenceinvariabletransportationcostneededforanon-trivialequilibriumresultdependsonthegeographicaldistributionofthedeliverypointsandcarriers,andtherefore,theeffectofaspecicvaluecannotbegeneralized.However,largerdifferencesinthevariabletransportationcostsmaybenetthecarrierwiththelowertotaltransportationcostforservingallcustomers,andasaresult,allthedeliverypointswillbeassignedtosuchacarrier. 4.3.2.3ImpactofthegeographicaldistributionofcarriersanddeliverypointsInthissectionwewanttoanalyzetheeffectofthegeographicaldistributionofcarriersanddeliverypointsontheequilibriumsolutionforourgame,whenauniformcostallocationschemeisusedbythecarriers.InAppendix Y wepresentasummaryofthepercentageofnon-trivial-partitionandnon-trivial-dominant(identiedbyeithercarrier1orcarrier2)equilibriumsolutionsfordifferentcostparametersandgeographicaldistributionsoftheretailpoints.Whenthexedtransportationcostforthecarriersissufcientlylargeorwhentherearenoclustersofcustomersclosertoonecarrierandfarfromtheother,bothcarrierswillwanttoincreasethenumberofdeliverypointsserved,sincethiswillresultinlowerservicefees,independentoftheroutedistance.Since,underauniformfeecostallocationscheme,asinglefeewillbeofferedbyeachcarriertoalldeliverypoints(independentofthedeliverypoint'slocation),thecarrierwiththelowertotal 110

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transportationcostforservingallcustomerscannotbeunderpricedbythecompetitorandwillwinthebid,regardlessofwhichcarrierwasidentiedastheleader.Inournumericaltests,whenthexedcostisstrictlygreaterthanzerowehavethat96%oftheinstancesresultinanon-trivial-dominantequilibriumsolution,wherethesetofdeliverypointsisassignedtothecarrierwiththelowertotaltransportationcostforservingallcustomers.Notethattheseresultsareconsistentunderdifferentgeographicaldistributions,asshowninAppendix Y .Forsufcientlysmallvaluesofthexedtransportationcost,theeffectofthegeographicaldistributiononourmodelbecomesmorerelevant.Forexample,considerthecasewhencarriersanddeliverypointsarelocatedrandomlyinindependentadjacentregions(similartogeographicaldistributiontype1)andthecarriershavecomparabletransportationcostparameters.Sinceoneofthecarriersmaybelocatedclosertothesetofcustomers,asinFigure 4-3 ,thatcarrierwillhavealowertotaltransportationcostforservingalldeliverypointsinthesetC,andtherefore,willdominatethegame.Forthisreason,whenthexedtransportationcostiszero,weobserveinFigure 4-7 that80%oftheinstancesthathaveageographicaldistributiontype1resultinnon-trivialequilibriumsolutions,comparedwithanaverageof53%ofinstanceswithnon-trivialequilibriumsolutionsforgeographicaldistributiontypes2to4.Notealsothat,whenthecarriersanddeliverypointsarerandomlylocatedinthesameregion,asweincreasethesizeoftheregion,thepercentageofinstanceswithnon-trivialequilibriumsolutiondecreases,duetothehighdispersionofthedeliverypoints.Forexample,forgeographicaldistributiontype2weobservethat60%oftheinstanceswithzeroxedtransportationcosthavenon-trivialequilibriumsolutionscomparedwith30%theinstanceswithgeographicaldistributiontype4,wheretheareaofthelatterismorethansixtimeslarger.Althoughanon-trivial-partitionequilibriumsolutionisnotlikelytooccurundertheuseofauniformfeeschedule,havingsufcientlysmallxedcostvaluesandhaving 111

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carriersanddeliverypointslocatedrandomlyinthesameregionmayallowthistypeofresult.Forthisreason,only10instances(representing1.3%oftheinstanceswithnon-trivialequilibriumsolutions)thatresultinthistypeofassignmentareobservedforgeographicaldistributiontypes2to4,asshowninAppendix Y 4.3.3Branch-BasedCostAllocationInthissectionwewanttocharacterizetheeffectsthatchangesinspecictransportationcostandgeographicaldistributionparametershaveonthefeesettingdecisionsforcarriersthatuseabranch-basedcostallocationfordeningdeliveryfees,whileholdingallotherparametersequal.AssumethatC1andC2,whereC1=CnC2,arethesetsofcustomersservedbycarrier1and2,respectively.Then,wesaythatC1andC2areequilibriumsolutionsifforeachsetSCandforatleastonecustomerk2(S\C1)wesatisfythefollowingconditionforthefeesofferedbycarrier1: F2 jSj+Pj2(S)jm=12(!22,m(1)+!2m(jSj),2) jSj j2(S)j+Pj2(S)jm=12!2m(i),m(i+1) j2(S)jF1 jC1j+Pj1(C1)jm=11(!11,m(1)+!1m(jC1j),1) jC1j j1(C1)j+Pj1(C1)jm=11!1m(i),m(i+1) j1(C1)j(4)and,foreachsetTCandforatleastonecustomerl2(T\C2)wesatisfythefollowingconditionforthefeesofferedbycarrier2: F1 jTj+Pj1(T)jm=11(!11,m(1)+!1m(jTj),1) jTj j1(T)j+Pj1(T)jm=11!1m(i),m(i+1) j1(T)jF2 jC2j+Pj2(C2)jm=12(!22,m(1)+!2m(jC2j),2) jC2j j2(C2)j+Pj2(C2)jm=12!2m(i),m(i+1) j2(C2)j,(4)wherej(^C)isthesetofalternativeoptimalsolutionsfortheminimumlengthrouteproblemon^C[fjg.Beforeanalyzingtheeffectsofkeycostparametersontheresultsofourmodel,wewilldiscussthestrategythatacarrierwillfollowtobeabletooffercompetitivefees, 112

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whenabranch-basedcostallocationisusedinadenselypopulatedarea.Whenabranch-basedcostallocationschemeisusedtosetdeliveryfees,thefractionofthetotaldistance-basedroutecostforservingasetofcustomersthatisallocatedtoaparticulardeliverypointdependsonitspositioninrelationtotheotherdeliverypointsintheserviceroute.Consequently,ifcustomersarelocatedinaclusteri.e.,deliverypointsarenotgeographicallydispersedwithintheserviceregion,thebeststrategyforeachcarrieristoserveallthecustomersinthatcluster,sinceaddingstopsinadenselypopulatedserviceregionreducesthefractionoftheroute-distancecostassignedtoeachdeliverypoint(denedasj!j(i),(i+1)8j=1,2inSection 4.2.1.3 ).Inordertocharacterizetheoptimalfeesettinganddeliverypointselectionstrategiesforthecarriers,inSections 4.3.3.1 and 4.3.3.2 wewillanalyzetheeffectthatchangesincostparametersvalueshaveonourmodelresults,whileholdingeverythingelseequal.Followingthis,inSection 4.3.3.3 weanalyzehowthegeographicaldistributionofthedeliverypointsandcarriersaffectsthedeliverypointselectionforthecarrierswhenabranch-basedcostallocationisused. 4.3.3.1ImpactofxedtransportationcostWhencarriersinterestedinofferingdeliveryfeesforasetofretailpointshavesufcientlylargexedcostvalues,increasingthenumberofretailpointsservedwillresultinlowerservicefees,independentofthecustomerlocations.Sincethesequenceforstopsatthedeliverypointswillbesimilarforbothcarrierswhentheyservethesamesetofcustomersi.e.,customeriwillbelocatedbeforecustomerkintherouteforbothcarriers,thefractionofthecostassociatedwiththelengthofthetourbranchallocatedtoeachcustomerwillbesimilarforbothcarriers.Basedonthepreviousobservation,andsincethecarrierswanttoservethesamenumberofcustomersjCj,wehavethatthedifferenceinthetotaltransportationcostforservingallofcustomersintheset,willsignicantlyinuencetheresultofthegame. 113

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Whenthedifferenceintotaltransportationcostforservingtheentiresetofdeliverypointsbetweenthetwocarriersisnotsufcientlyhigh,thegamemayresultinatrivialequilibriumsolution(wheretheincumbentcarriercontinuesservingtheentiresetofdeliverypoints),sincesmalldifferencesinthevariableandxedtransportationcostsmayallowthenewcarriertounderpricethefeesforsomeofthedeliverypoints,butnotfortheentireset,andtherefore,thenewcarrier'sbestresponsewillbenottoplaceabid.Forthisreason,asignicantlylargedifferenceinthetotaltransportationcostforservingallthecustomersinthesetisneeded,sothatthecarrierwiththelowertotaltransportationcostforservingalldeliverypointscandominatethegame,withoutregardtowhichcarrieristheincumbent.Figure 4-9 showsthepercentageofinstanceswithnon-trivialequilibriumsolutionsfordifferentcostparametersandgeographicaldistributioncombinationswhenabranch-basedcostallocationisused.NotethatonthechartslocatedontheleftofFigure 4-9 ,carrier1hasthelowestxedtransportationcost,andweobservethataswemovetotherightonthehorizontalaxisthenumberofinstanceswithnon-trivialequilibriumsolutionsdecreases.Thereasonisthatthevalueof=1)]TJ /F4 11.955 Tf 13.35 0 Td[(2decreasesaswemovetotheright,andtherefore,thelevelofdominanceofcarrier1decreasesaswell.Ontheotherhand,ontheplotsontherightinFigure 4-9 ,carrier2hasthelowerxedtransportationcost,andaswemovetotherightonthehorizontalaxisitsdominancelevelincreases,andtherefore,thenumberofinstanceswithnon-trivialequilibriumsolutionsalsoincreases.NotealsothatinFigure 4-9 ,asthexedtransportationcostincreases,thenumberofnon-trivialequilibriumsolutionsdecreases.ThereasonforthisisthatinournumericaltestsweassumedthatjF1)]TJ /F3 11.955 Tf 12.26 0 Td[(F2j=500,andtherefore,asweincreasethexedtransportationcostforthecarriers,whilekeepingeverythingelseequal,thepercentagedifferenceintotaltransportationcostforservingallcustomersbetweencarriersdecreases,andtherefore,thereisnodominantcarrier. 114

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Figure4-9. Branch-basecostallocation:numberofnon-trivialequilibriumsolutions. Whenthexedtransportationcostvaluesforthecarriersaresufcientlysmall,theeffectofthevariabletransportationcostinourmodelbecomesmorerelevant.Inthenextsectionwewillevaluatehowchangesinthedifferenceofthevariabletransportationcostsbetweencarriersaffectsourmodelresults. 4.3.3.2ImpactofvariabletransportationcostInthissectionwewanttoanalyzetheeffectthatchangesinthedifferenceinthevariabletransportationcostbetweencarriershasontheoptimalfeesettinganddeliverypointselectiondecisionsforthecarriers,whileholdingallelseequal.Notethatinthis 115

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sectionweassumethatthexedtransportationcostforbothcarriersissufcientlysmall,sowecanevaluatetheimpactofthevariabletransportationcostonourresults.AsexplainedinSection 4.3.3 ,whencustomersarelocatedwithinaclusterthebeststrategyforthecarriersistoservethemaximumnumberofdeliverypointsintheserviceregion,sincethiswillleadtolowerdeliveryfeesforthecustomers.Underthosecircumstances,sincebothcarrierswanttoservetheentireregion,inordertoobtainanon-trivial-dominantequilibriumsolutionthatresultsinalldeliverypointsassignedtoone(dominant)carrier,weneedasufcientlylargedifferenceinthetotaltransportationcostforservingalldeliverypointsbetweencarriers.Therefore,thedominantcarriershouldhaveashorterroute-distancecomparedwithitscompetitor,orasufcientlysmallervariabletransportationcost(notethattherequireddifferenceinthevariabletransportationcostwilldependonthedifferenceinthetourlengthofthecarriers).AsobservedinFigure 4-9 ,whenthexedtransportationcostforthecarriersiszero,wehavealargerpercentageofinstancesthatresultinanon-trivialequilibriumsolutioncomparedwiththeinstanceswithaxedtransportationcoststrictlygreaterthanzero.Whenthexedtransportationcostisnottakenintoconsideration,thepercentagedifferenceintotaltransportationcostforservingallcustomersislargerthanwhenthexedtransportationcostisadded(assumingthecarriershaveequalxedcosts).Therefore,thedominantcarriercannotbeunderpriced,andthiswillresultinmoreinstanceswithanon-trivial-dominantequilibriumsolution.Considernowthecasewhenthedeliverypointsarehighlydispersedaroundtheserviceregion.Ifwecandividethesetofdeliverypointsintwodisjointsubsets,suchthateachsetisclosertoonecarrieranddistantfromtheother,eachcarrierwillonlybeinterestedinservingthesetofcustomersclosertoitsdepot(assumingalowdifferenceinvariabletransportationcosts).Toillustratethiscase,considertheinstanceinFigure 4-10 ,andassumethatthexedtransportationcostiszeroandvariabletransportationcostis$1/mileforbothcarriers.Thenon-trivial-partitionequilibriumsolutionisC1=f1g 116

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andC2=f2,3g,sinceaddingcustomer1toC2resultsinalargeincreaseinthefeesfortheotherdeliverypoints,makingthatbidnolongercompetitive,andthesameoccurswhenwewanttoincreasethesizeofC1.Therefore,incaseswherethedifferenceinthevariabletransportationcostislow,inordertohaveanon-trivial-partitionequilibriumsolution,thecustomersassignedtoaspeciccarriershouldbefarawayfromthecompetitor'sdepotandtheothercustomersservedbythecompetitor.Thismeansthat,fromacarrier'sperspective,forlowvaluesofthexedtransportationcostandjj,acarrierwillprefertoserveasetofcustomersthatarelocatedwithinaclusterclosetoitsdepot,andleavetheisolatedcustomerstothecompetitor. Figure4-10. Instancewithisolateddeliverypoint. 4.3.3.3ImpactofthegeographicaldistributionofcarriersanddeliverypointsInthissectionwewanttostudyhowdifferentgeographicaldistributionsforcarriersanddeliverypointsimpacttheequilibriumsolutionthatresultsfromourmodel,whenabranch-basedcostallocationisused.Appendix Z providesdetailedinformationonthepercentageofnon-trivial-partitionandnon-trivial-dominant(identiedbyeithercarrier1orcarrier2)equilibriumsolutionsfordifferentcostparametersandgeographicaldistributioncombinations.AsmentionedinSection 4.3.3.1 ,forsufcientlylargexedcostvalues,increasingthenumberofdeliverypointsservedbyacarrierleadstolowerservicefees,regardlessthedeliverypointlocation.However,weneedasufcientlylargedifferenceinthetotaltransportationcostforservingalldeliverypointsbetweenthecarriers,inordertondanon-trivial-dominantequilibriumsolution.Asaresult,forournumericaltestswhenthexedtransportationcostisstrictlygreaterthanzero,21%oftheinstancesresultinanon-trivial-dominantequilibriumsolution.Notethatthisresultisconsistentamongthedifferentgeographicaldistributionsstudied,asdescribedinAppendix Z 117

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Whenthexedtransportationcostislow,thepercentagedifferenceinthetotaltransportationcostforservingallcustomersislarger,comparedwithinstanceswithlargervaluesofxedtransportationcost.Therefore,insuchcaseswendmoreinstanceswithadominantcarrierthatcannotbeunderpriced.Forthisreason,inournumericaltests,48%oftheinstanceswithxedtransportationcostequaltozeroresultinanon-trivialequilibriumsolution.AlthoughinFigure 4-9 weobservethatthepercentageofinstanceswithnon-trivialequilibriumsolutionsisconsistentforthedifferentgeographicaldistributiontypes,Appendix Z showsadifferenceintheretailpointassignmentsfordifferentgeographicaldistributiontypes.Whenthecarriersanddeliverypointsarerandomlylocatedintwoindependentregions(asingeographicaldistributiontype1),bothcarrierswilltrytoservetheentiresetofcustomers,andtherefore,wewillobtainanon-trivial-dominantequilibriumsolution,asshowninAppendix Z .Considernowthecasewhencarriersanddeliverypointsarerandomlylocatedwithinthesameregion,asingeographicaldistributiontypes2to4.Undersuchgeographicaldistributionswemayndtwodisjointsubsetsofdeliverypointssuchthatasubsetthatisclosetoonedepotisfarawayfromthecompetitor'sdepotandtheothersubsetofdeliverypoints..Insuchcases,wecanndanon-trivial-partitionequilibriumsolution.InAppendix Z weobservethat,inournumericalanalysis,9instanceswithanon-trivial-partitionequilibriumsolutionwerefound,whichrepresent4.3%oftheinstanceswithnon-trivialequilibriumsolutions,andtheyallbelongtogeographicaldistributiontypes2to4whencarriershavezeroxedcost. 118

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CHAPTER5CONCLUSIONSANDFUTURERESEARCHWhenallocatingorderstosuppliers,thereisoftenatradeoffbetweencostanddeliveryperformance.Theuseofacheapersupplymodeoftenimpliesdealingwithhigheruncertaintylevelswhencomparedwithamoreexpensivemode.Assumingthatshortagesarehighlyundesirable,ifacheapandlessreliabledeliverymodeisused,itmaybebenecialtoconsidersupplementingthiswithasecondarysupplymodethatismorereliable(andthereforemoreexpensive).Chapter 2 focusedonthebenetsofthisdual-modeorderingstrategybymodelingabuyer'sorderingdecisionswithtwodeliverymodeoptions.Ourmodelallowedustocharacterizethebenetsofthedual-modesupplystrategyandcomparetheoptimaldual-modepolicywiththebetterofthetwosingle-suppliersolutions.Usingnumericalanalysis,wecharacterizedsituationsunderwhichtheuseofthedual-modemodelispreferredoverthesingle-modemodel.Weobservedincreasingbenetsinourmodelforincreasingvaluesinthemeanandvarianceintheleadtimeofthelessreliablesupplymode(assumingthatsupplymode1isusedasthepreferredsingle-modesupplier).Wealsoobservedthatwhenthesupplymode1leadtimefollowsapositivelyskewedprobabilitydistribution,whichistypicallyamorerealisticassumptionthanaUniformleadtime,thepercentagecostreductionduetothedual-modeoperationisgreatercomparedwiththeresultswhenweassumethatsupplymode1leadtimeisUniformlydistributed.Ourmodelcanbeusedbyamanagerwhowantstoanalyzedifferentcombinationsofshippingmodesinordertominimizeinventorycostswhileguaranteeingzerostockouts.Ourworkshowsthatasupplierthatoffersdifferentshippingmodeoptionswithdifferentdeliverycostsandreliabilitylevelsmayprovideadditionalvaluetopotentialbuyers.Furtherresearchmayconsiderdifferentsupplierpricingandincentives.Forexample,inordertoincreasetheutilizationofshippingmodeswithavailablecapacity, 119

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asuppliermightofferquantitydiscountsfortheuseofthosemodes,makingshippingdiversicationmoreattractiveforthebuyer.Finally,wenotethatonedimensionourmodeldoesnotaccountforistheriskpreferenceofthedecisionmaker,effectivelyassumingthatthebuyerisriskneutral.Aninterestingextensionofourworkmayincludeaccountingforthewayinwhichdifferentbuyerriskprolesinuencetheallocationoforderquantitiestodifferentsupplymodes.Inchapter 3 weanalyzedandpresentedrecommendationsforthecurrentfooddistributionnetworkoftheUSDAfooddistributionprogramsadministeredbyFDACS.Inparticular,wefocusedourstudyintheNationalSchoolLunchProgram(NSLP)andtheEmergencyFoodAssistantProgram(TEFAP).BasedontheinformationprovidedbyFDACSandfourstate-contractedwarehousesfortheschoolyear2011-2012,weobservedthatduetothelackofhomogeneitybetweentheregionsandamongtheagenciesthatbelongtothesameregion,intermsofdemandrates,storagecapacity,andgeographicaldemanddistribution,thedeliveryfeespercaseofferedbythestate-contractedwarehousesdiffersignicantlybetweenregions,asdothestorageanddeliveryexpensespercaseoftheagencies.Inthisstudy,wepresentedananalysisoftheeffectofmodifyingparametersrelatedtothedesignofthedistributionnetworkandtheInvitationtoBidonthedeliveryandstoragefeesrequiredbythestate-contractedwarehousesandontheestimatedannualexpensefortheagencies.Inparticular,weevaluatedtheeffectsofmodifyingregionalcongurations,storageanddeliveryfeepolicies,foodprogramrelationshipsandregulationinthequantityofproductsenttofoodprocessors.Toevaluatethe399feescenariosthatresultedfromthedifferentcombinationsoftheparametersstudied,wecalculatedtheaverageagencyoverpaymentforeachscenario,wheretheoverpaymentforeachagencyisequaltothedifferencebetweenthecurrentestimatedannualexpensesandtheestimatedannualexpensesunderaparticularfeeandpolicyscenario. 120

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Sincedifferentfeescenariosincreasetheestimatedtotalexpensesforsomeagenciesandreduceitforothers,wewouldliketoselectthescenariosthatresultintheleastnegativeimpactsontheagencies.Bycomputingtheaverageoverpaymentfortheagencieswithanoverpaymentgreaterthan$5,000/yr,weselectedthe25feescenarioswiththeleastaverageoverpaymentfromthe171scenariosthatassumetheuseofseparatedistributionnetworksforNSLPandTEFAP,andthe25feescenarioswiththeleastaverageoverpaymentfromthe228scenariosthatassumetheuseofacommondistributionnetworkforNSLPandTEFAP.UndertheassumptionthatweusetwodistributionnetworksfortheagenciesinNSLPandTEFAP,weobservethatselectingtheproposedregionalconguration3resultsinthelowestaverageoverpaymentvalues,comparedwiththeothercongurations.Wealsoobservethattheuseofauniformfeefortheagenciesinthesameregionmaybeadvantageousinreducingthenegativeimpactsofthechangesofthedistributionnetworkparameters.Finally,itisalsorecommendedtoincreasetheminimumordersizeanddecreasethelengthoftimeoffreestorage,whichnotonlyreducesthenegativeimpactofchangingthesystemparameters,butisalsolikelytoincreasetheinterestofpotentialbidders.WhenweuseonedistributionnetworkfortheagenciesinNSLPandTEFAP,theagencieswiththelargestoverpaymentvaluesaretheonesthatbelongtoTEFAP,sinceservingtheagenciesintheNSLPismoreexpensiveforthestate-contractedwarehouses,andtherefore,theproposedfeeswillbehigherthanwhentheagenciesfromTEFAPuseanindependentdistributionnetwork.SincewewanttoreducethenegativeimpactontheTEFAPagencieswhentheNSLPandTEFAPusethesamedistributionnetwork,candidateregionalconguration4isthebestoptionamongthecandidates.ThereasonforthisisthatwearesharingtheTEFAPdistributionnetworkwithonly9counties,andtherefore,theimpactoftheadditionalexpensesontheTEFAPagenciesisreduced.AnotherparameterthatreducesthenegativeimpactontheTEFAP 121

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agenciesistheuseofanorder-quantity-anddelivery-frequency-basedfeeschedule,sincethedifferencebetweentheaveragestoragetimepercaseandordersizeissignicantbetweentheagenciesfromTEFAPandtheNSLP.Finally,sincetheincreaseinthevolumeofproductsenttofoodprocessorsincreasesthecostofthestorageanddeliveryservicesofferedbythestate-contractedwarehouses,anddecreasestheinterestofpotentialbidders,weevaluatedtheeffectsthatapolicylimitingtheamountoffooddivertedfromthedistributionnetworkhasonthefeesproposedbythestate-contractedwarehouses.Specically,weanalyzedcasesinwhichwelimittheamountofproductsenttothefoodprocessorsto30%and50%oftheagencyentitlement.Asexpected,anincreaseintheannualdemandoftheagenciesreducestheestimatedper-casecostoftheserviceprovidedbythestate-contractedwarehouses(duetoeconomiesofscale)and,therefore,resultsinlowerfeesfortherecipientagencies.Sincetheprocessofchangingthedifferentregionalcongurationsandstorageanddeliveryparametersmayneedtobedonegradually,duetoadministrativeandresourcelimitations,wedonotaimtorecommendasinglespeciccombinationofparameterlevelsthatwillimprovetheefciencyofthesystem;rather,wehaveattemptedtodemonstratewhichdistributionandstoragepolicyparametersmayhaveagreaterimpactonthedistributionnetwork,inordertosupporttheFDACSplanningprocessinthedesignofamoreefcientdistributionnetworkandInvitationtoBid.InChapter 4 weconsideredthecasewhenashipperisinterestedinpurchasingdeliveryservicesforasetofretailpointsusingasingleitemauction,i.e.,eachcustomerwillbeassignedtothecarrierthatoffersthelowestservicefee.Sincethemarginalcostofservingacustomerdecreaseswhenthatcustomerisclosetoacarrier'scurrentserviceregion(assumingequalcustomerdemandquantities),ifacarrierisinterestedinofferingcompetitivedeliveryfees,suchacarrierwillwanttomaximizethenumberofdeliverypointsservedinaserviceregion. 122

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Giventhattheservicecostsforeachcarrierdependonthesetofcustomersserved,andthateachcustomerwillbeassignedtothecarrierthatoffersthelowestfee,thecarriersthatintendtoparticipateinthebiddingprocessneedtondacustomerselectionanddeliveryfeedesignstrategythatguaranteestheassignmentofasetofdeliverypointsofmaximumsize.Weproposeadeliveryfeedesignmodelbasedonapricecompetitiongamewithtwocompetitorsthatwanttomaximizethenumberofcustomerstheyserve.Weassumethatthereisanincumbentcarrierthatiscurrentlyservingthesetofcustomers,andtherefore,istheleaderinoursequentialgame.Wesaythatanequilibriumsolutionhasbeenreachedwhenwendasubsetofcustomersthatcanbeassignedtoeachcarrier,suchthatneitherofthecompetitorscanndadifferentsetofcustomersoflargersizewhichtheycanserve.Incaseswhenanequilibriumsolutiondoesnotexistorwhenthebestresponseofeachcarriertoitscompetitor'sinitialfeeproposalistheemptyset,weassumethatthefollowerwillnotbeinterestedinplacingabid,andtherefore,theincumbentcarrierwillcontinueservingtheentiresetofdeliverypoints.Usingthreedifferentcostallocationmechanismsweidentiedtheconditionsunderwhichwecanexpecttondequilibriumsolutionsandcharacterizedtheretailpointassignmentsfordifferentcostparametersandgeographicaldistributioncombinations.Whencarriersuseadistance-basedcostallocation,anequilibriumsolutioncanbefoundinthecaseswhenthecarriers'xedtransportationcostissufcientlysmall.Inthatcase,asaresultofourmodel,thesetofcustomerswillbesplitbetweenthetwocarriersbasedonthevariabletransportationcostandlocationofeachcompetitor.Inthecasethatthexedtransportationcostforthecarriersissufcientlyhigh,havingalargedifferenceinthevariabletransportationcostwillresultinasolutionwherealldeliverypointsareassignedtoonecarrier.However,asthexedcostincreases,thepercentageofinstanceswithequilibriumsolutionsdecreases,andtherefore,newcarrierswillnotbeinterestedinthebiddingprocess. 123

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Iftheshipperrequiresauniformfeeschedule,weobservedthat,in89%oftheinstancesfromournumericaltests,theequilibriumsolutionresultsintheassignmentoftheentiresetofdeliverypointstothecarrierwiththelowertotaltransportationcostforservingallcustomers,andthepossibilityofhavinganequilibriumsolutiondecreasesasthexedtransportationcostdecreases.Althoughusingauniformfeeschedulemayresultinalowertotalcostfordeliveryservicesfortheentiresetofdeliverypointsfortheshipper,itdecreasestheattractivenessofthebidtopotentialcontractors,sincethelowcostcarrierwillhaveanadvantageoverothercompetitors,evenifitsroute-distanceislarger.Underabranch-basedcostallocationscheme,whenthecustomersarechargedthecostofthebranchoftheroutethatstartsattheirlocation,theresultswilldependonthedifferenceinthetotaltransportationcostforservingallcustomersbetweencarriers.Ifthereisadominantcarrierwithalowertotaltransportationcostforservingallcustomers,comparedwithitscompetitor,theequilibriumsolutionwillresultintheassignmentofalldeliverypointstothatcarrier.However,ifthedifferenceinthetotaltransportationcostforservingalldeliverypointsbetweencarriersislowandthereisnocleardominantcarrier,thefollowerwillnotplaceabid.Forabranch-basedcostallocation,theequilibriumsolutionwillresultinapartitionofthesetiftherearetwodisjointsubsetsofdeliverypointssuchthateachsubsetisclosetooneofthecarriers,andfarfromthecompetingcarrierandtheothersubsetofdeliverypoints.Asaresult,neitherofthecarrierswilltrytotakecustomersfromthecompetitor,sinceservingthosecustomerswillreducetheircompetitivenessintheauction.Usingthiscostallocationschemecanbeadvantageouswhenthereexistsomeisolatedcustomersclosertooneoftheinterestedcontractors;otherwise,theexistenceofacarrierwithalowertotalcostforservingalldeliverypointswilldecreasetheattractivenessoftheauctiontopotentialcontractors. 124

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Althoughourmodelisnottractableforlargeinstances,theresultscanbeusedforcarriersasaframeworktoplanbiddingstrategieswhentheyareinterestedinincreasingtheirmarketshare.Additionallyourmodelcanbeusedbyshipperstoidentifythetypeoffeeschedulethatcanincreasecompetitionordecreaseservicecostunderdifferentmarketscenarios.Oneaspectthatwewouldliketoanalyzeinthefutureistheuseofcapacitatedtrucksandtheassumptionofdifferentcustomerdemandquantities,thatwillallowustoanalyzetheresultsforourgameundertheuseofdistanceandquantity-basedfees.Additionallywewouldliketocomparetheresultsofourgamewhenusingapproximatevaluesforthecomputationoftheminimumcostroute,asanapproachtoincreasethetractabilityofourmodel. 125

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APPENDIXAORDERCROSSINGCONDITIONAssumingthatweuseasupplymodewithastochasticleadtime,L1U(l,u)andaconstantdemandrate,ifweuseacontinuousreview(Q,r)policy,wherer=u,weneedQ(u)]TJ /F4 11.955 Tf 11.96 0 Td[(l)topreventordercrossing.Inordertoprovethisassertion,weassumeforacontradictionthatQ<(u)]TJ /F4 11.955 Tf 11.95 0 Td[(l).Supposeweplaceanorderattimet=0.Notethattheinventorypositionatanytimet0isequaltoQ+u)]TJ /F4 11.955 Tf 12.45 0 Td[(t.Whentheinventorypositionreachesthereorderpointr=u,anotherorderisplaced.Thatis,whenQ+u)]TJ /F4 11.955 Tf 12.45 0 Td[(t=u,anorderisplaced.Thisimpliesthatanorderisplacedattimet=Q andbytheassumptionthatQ<(u)]TJ /F4 11.955 Tf 11.96 0 Td[(l),wehavethat:t=Q
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APPENDIXBSINGLE-MODESOLUTIONWITHQUANTITYBASEDUPPERBOUNDThissectionpresentsthesolutionproceduretondanoptimalorderquantityfrommode1,Q1,andminimumaverageinventorycost,G1forthesingle-modemodel,whentheleadtimeupperboundforsupplymode1,u,isastepfunctionoftheordersizeQsuchthat,u=iuifqiQ1do 4. FindGk)]TJ /F6 7.97 Tf 6.59 0 Td[(11)]TJ /F3 11.955 Tf 5.47 -9.68 Td[(qk,k)]TJ /F6 7.97 Tf 6.59 0 Td[(1uusing(2.1),whichistheaverageinventorycostperunittimeusingtherightbreakpointforQinthe(k)]TJ /F2 11.955 Tf 11.96 0 Td[(1)thinterval; 5. ifG1Gk)]TJ /F6 7.97 Tf 6.59 0 Td[(11then 6. Q1istheoptimalorderquantityandtheminimumaverageinventorycostisG1.Setk=1; 7. else 8. SetQ1=qk,G1=Gk)]TJ /F6 7.97 Tf 6.59 0 Td[(11,k=k)]TJ /F2 11.955 Tf 11.95 0 Td[(1; 9. endif 10. endwhile 127

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APPENDIXCCONVEXITYCONDITIONSFORGI(Q,)Thissectionpresentsconvexityconditionsfortheaverageinventorycostfunctionforthedual-modemodelwhen2[L2,L2+l),representedby:GI(Q,)=22h+2c22+hQ2+2c1Q+hQ(u)]TJ /F4 11.955 Tf 11.95 0 Td[(l)+2A 2(Q+)First,weanalyzetheaverageinventorycostasafunctionofQandindependently,andthenwewillconsidertheconvexityconditionforGIasajointfunctionofQand.ToanalyzetheaverageinventorycostasafunctionofQweneedtondtherstandsecondderivativesofGI:@GI @Q=hQ2+2hQ+h2)]TJ /F2 11.955 Tf 11.95 0 Td[(22c)]TJ /F4 11.955 Tf 11.95 0 Td[(2h2)]TJ /F2 11.955 Tf 11.96 0 Td[(2A 2(Q+)2@2GI @Q2=22h2)]TJ /F4 11.955 Tf 11.96 0 Td[(2h+22c+2A (Q+)3SinceQandarestrictlypositive,thedenominatorof@2GI @Q2isstrictlypositiveaswell.Thenumeratorcanbeanalyzedasaquadraticfunctionofoftheformh()=a2+b+c,wherea=2h2,b=22c)]TJ /F3 11.955 Tf 11.96 0 Td[(h2andc=2A.Sincea>0,h()isconvex,andthevaluesofh()willdependonitsdiscriminant=b2)]TJ /F2 11.955 Tf 12.42 0 Td[(4ac.Inparticular,weareinterestedintheconditionsunderwhichh()0,sinceweneed@2GI @Q20forGI(Q)tobeconvex.Note,thatwhen0,h()hasatmostonerealroot,andtherefore,h()08,andwhen>0,h()willhavetworealroots,1and2,andh()willbepositivefor2(,1]S[2,1),andnegativeotherwise.Inordertohave0weneed:!24hA, (C)where!=h 2)]TJ /F4 11.955 Tf 11.95 0 Td[(c.Thisconditionrequiresthatthesquareoftheupperboundonthedifferencebetweentheaverageinventorycostofusingshippingmode1andtheaverageinventorycostofusingshippingmode2,mustbelessthanorequaltotwicethe 128

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squareoftheorderingandholdingcostoftheEOQmodelwithholdingcostperunitperunittimehandorderingcostA=A1+A.Therefore,whencondition( C )holds,wehavethat@2f=@Q2isnon-negativeforQ0and0,andhence,GIisconvexforQ0and0.Ifcondition( C )doesnothold,wehavethat>0;thereforeh()willhavetworealroots:1=!)]TJ /F5 11.955 Tf 6.59 8.69 Td[(p ()]TJ /F10 7.97 Tf 6.59 0 Td[(!)2)]TJ /F6 7.97 Tf 6.58 0 Td[(4hA 2hand2=!+p ()]TJ /F10 7.97 Tf 6.58 0 Td[(!)2)]TJ /F6 7.97 Tf 6.58 0 Td[(4hA 2h,andh()willbepositivefor2(,1]S[2,1).Wecanseethatthesignof1and2dependsonthevalueof!.When!<0,wehave1<2<0,whichmeansthatthetworealrootsofh()arenegative,andthereforeh()>080,andconsequently@2f @2>0.If!>0,wehavethat2>1>0,andtherefore,@2f @20when2[0,1]S[2,1).FinallywecanconcludethatGI(Q)isconvex8Q>0and>0when( C )holdsorwhen!<0.If( C )doesnotholdand!>0,thenGI(Q)isconvex8Q>0and2[0,1]S[2,1).Wenextanalyzetheaverageinventorycostasfunctionofusingitsrstandsecondderivatives:@GI @=22hQ+22Qc+32h)]TJ /F4 11.955 Tf 11.95 0 Td[(hQ2)]TJ /F4 11.955 Tf 11.95 0 Td[(2hQ)]TJ /F2 11.955 Tf 11.96 0 Td[(22A 2(Q+)2@2GI @2=22hQ2)]TJ /F2 11.955 Tf 11.96 0 Td[(23Qc+3hQ+23A (Q+)3Weobservethatthedenominatorof@2GI @2ispositive8Q>0and>0andthatthenumeratorisaquadraticfunctionofQoftheformk(Q)=aQ2+bQ+c,wherea=22h,b=h3)]TJ /F2 11.955 Tf 11.95 0 Td[(23candc=23Awithroots:Q1=)]TJ /F10 7.97 Tf 6.58 0 Td[(!)]TJ 6.59 6.71 Td[(p !2)]TJ /F6 7.97 Tf 6.59 0 Td[(4Ah 2handQ2=)]TJ /F10 7.97 Tf 6.59 0 Td[(!+p !2)]TJ /F6 7.97 Tf 6.59 0 Td[(4Ah 2hFollowingthesameanalysisusedforGI(Q),wehavethatGI()isconvex8Q>0and>0ifcondition( C )holdsorif!>0.Ifcondition( C )doesnotholdand!<0,GI()isconvexforQ2[0,Q1]S[Q2,1)and>0. 129

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Notethat( C )itisnotrestrictive,sincetheordercostistypicallybiggerthancandthantheholdingcostperunitduring.Fortherestofthissectionwewillassumethat( C )holds,andthereforeGI()andGI(Q)areconvex8>0andQ>0.ToestablishconvexityconditionsforGIasajointfunctionofQandweneedtoanalyzeitsHessian:H=264@2GI @Q2@2GI @Q@@2GI @@Q@2GI @2375,where:@2GI @@Q=!(Q)]TJ /F4 11.955 Tf 11.96 0 Td[())]TJ /F2 11.955 Tf 11.96 0 Td[(22hQ+22A (Q+)3Since@2GI @2and@2GI @Q2arenon-negativewhencondition( C )holds,weneedtotransformHtotheform:Hnew=264h11h120hnew22375InordertodothetransformationweuseF=)]TJ /F10 7.97 Tf 10.5 5.59 Td[(!(Q)]TJ /F10 7.97 Tf 6.59 0 Td[())]TJ /F6 7.97 Tf 6.59 0 Td[(22hQ+22A 22h2)]TJ /F6 7.97 Tf 6.59 0 Td[(2!+2A,andwehavethat,hnew22=Fh12+h22=2)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(4Ah)]TJ /F4 11.955 Tf 11.96 0 Td[(!2 22h2)]TJ /F2 11.955 Tf 11.96 0 Td[(2!+2ASubstitutingh()=2h2)]TJ /F2 11.955 Tf 11.96 0 Td[(2!+2A,wehave:hnew22=2)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(4Ah)]TJ /F4 11.955 Tf 11.95 0 Td[(!2 h().Becauseweassumethat( C )holds,hnew220,andthisimpliesthatHispositivesemidenite,andthereforeGI(Q,)isconvex8Q>0and>0. 130

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APPENDIXDALGORITHMFORDUAL-MODEMODELWHEN2[L2,L+L2)Algorithm 2 showsadetaileddescriptionofthesolutionproceduretondtheoptimalorderquantitiesfrommodes1and2,andminimumaverageinventorycostforthedual-modemodelwhentheleadtimeupperboundofmode1,u,isastepfunctionofQand2[L2,l+L2).NotethatAlgorithm 2 usestheexpressionsdescribedinSection2.2.1. Algorithm2Minimumaverageinventorycostforthedual-modemodelwhen2[L2,l+L2). 1. i 1 2. whileindo 3. FindiandQiforiu 4. ifQi2(qi,qi+1]then 5. FindtheminimumaverageinventorycostfortheithintervalGiI 6. elseifQiqi+1 13. ifGiI(qi+1,(qi+1))Gi+1I(qi+1,(qi+1))then 14. TheminimumaverageinventorycostfortheithintervalisGiI(qi+1,(qi+1)) 15. else 16. Thesolutionisnotintheithinterval 17. endif 18. endif 19. endwhile 20. GI=minGiI8i2[1,n] 131

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APPENDIXEALGORITHMFORDUAL-MODEMODELWHEN2[L+L2,U)Algorithm 3 showsadetaileddescriptionofthesolutionproceduretondtheoptimalorderquantitiesfrommodes1and2,andminimumaverageinventorycostforthedual-modemodelwhentheleadtimeupperboundofmode1,u,isastepfunctionofQand2[l+L2,u].NotethatAlgorithm 3 usestheexpressionsdescribedinSection2.2.2. Algorithm3Minimumaverageinventorycostforthedual-modemodelwhen2[l+L2,u]. 1. i 1 2. whileindo 3. FindiandQiforiu 4. ifQi2(qi,qi+1]then 5. FindtheminimumaverageinventorycostfortheithintervalGiII 6. elseifQiqi+1 13. ifGiII(qi+1,(qi+1))Gi+1II(qi+1,(qi+1))then 14. TheminimumaverageinventorycostfortheithintervalisGiII(qi+1,(qi+1)) 15. else 16. Thesolutionisnotintheithinterval 17. endif 18. endif 19. endwhile 20. GII=minGiII8i2[1,n] 132

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APPENDIXFVALIDITYOFAVERAGEINVENTORYEXPRESSIONFORDUAL-MODEMODELWHEN2[L+L2,U]WeconsiderourinventorymodelasaRenewalRewardProcess,wheretheinterarrivaltimesTn,n1,areequaltothelengthofthereplenishmentcycle,andeachtimearenewaloccurswereceivearewardIn,n1,equaltotheaverageinventoryinthecycle.Weobservethatthepair(Tn,In),n1areindependentandidenticallydistributedandthatE[Tn]=E[T]from(2.13),andE[In]=E[I]from(2.14).UsingtheresultshowedbyRoss[16],wehaveaRenewalRewardProcesswhereE[T]<1andE[I]<1,andtherefore,withprobability1,E[I(t)] t!E[I] E[T]ast!1. 133

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APPENDIXGNSLP:ANNUALDEMANDANDLOCATIONOFRECIPIENTAGENCIES FigureG-1. AnnualdemandsandlocationsofNSLPrecipientagencies 134

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APPENDIXHTEFAP:ANNUALDEMANDANDLOCATIONOFRECIPIENTAGENCIES FigureH-1. AnnualdemandsandlocationsofTEFAPrecipientagencies 135

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APPENDIXICOSTESTIMATIONMETHODOLOGYToestimatethecostincurredbyastate-contractedwarehouseforthereceipt,storageanddistributionofUSDAfoodtotherecipientagenciesthatbelongtoNSLPandTEFAP,weidentiedfourmaincostfactors:transportationcosts,costperstop,pickingcostandstoragecost.Toestimatethetransportationcosts,weaggregatedtheagenciesbycountyandassumedthattheagenciesinthesamecountywouldbeservedusingacommondistributionroute.Fortheestimationofthedistancetraveledforthetrucksservingtheagenciesinonecountyweusetheroute-distanceapproximationmethodpresentedbyDaganzoin[ 11 ],whereheprovidesanexpressiontoestimatethedistancetraveledtovisitNcustomers,whenamaximumofCstopspertruckareallowed.Forouranalysis,Nisthenumberofdeliverystopsoftheagenciesthatarelocatedinonecountyand,basedontheinformationgivenbythestate-contractedwarehouses,weassumethatatruckmakesatmost15stopsinaroute,andtherefore,Cisatmost15.Theexpressionusedassumesthatthecustomersareuniformlydistributedthroughoutthecountyandthattheroutestartsinthecenteroftheservicearea.However,thoseassumptionsdonotapplyfortheagenciesthatarepartoftheNSLP,sincetheyareusuallyconcentratedneartheareaswithhighpopulationsintheircounties;additionally,thebeginningofthedistributionroutewilldependonthelocationofthewarehouseandtheothercustomersitserves.Althoughtheseassumptionsmayoverestimatetheactualdistancestraveled,weexpectthatthiswillcountertheeffectofnotincludingthedistancetraveledbythetrucktogettothecounties(orserviceregions)fromthestate-contractedwarehouse,sincewecannotmakeanyassumptionaboutthelocationofthewarehouses.Frominformationobtainedbythestate-contractedwarehouses,weknowthatthemarginalcostofservingtheagenciesinthecurrentregions3and4ishigherthanfor 136

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theotherregionsinthestate.Sinceoneofthereasonsisthesparsityoftheagenciesandcountiesinthoseregions,weincludedadditionaltraveledmilesintheroutesforthoseregions.Onceweestimatedthetotaldistancetraveltoservetheagenciesinaregion,wecomputedthetransportationcostbymultiplyingthetotaldistancetraveledbytheaveragecostpermileintheUnitedStates.Forourmodelweuse$2.53012/mile,whichisthevalueestimatedbywww.freightrateindex.cominMarch2013,andincludesthecostsfor:fuel,wages,equipment,depreciation,nancing,administration,complianceandinsurance.BasedontheinformationpublishedbyCNNmoney,weincreasedby10%thecostpermileforthecountieslocatedinthecurrentregion5,sincethetransportationcostsarehigherinthatregion.Thecostperstopisthecostincurredeverytimeatruckhastostopinadeliverydestinationandunloadproduct,andthepickingcostisthecostrelatedtothelaborandequipmentcostofreceivingtheproductinthewarehouse,locatingitintheracks,andthenloadingtheproductontotrucks,whenthecustomerrequestsadelivery.Toestimatethesecosts,webenchmarkedthemwithdifferentwarehousesindifferentstatesintheUnitedStates,andthenusedinformationfromtheDepartmentofLabortoadjustthesevaluestothestateofFlorida.Asaresult,weestimateacostof$60/stopand$0.0778/case,fordeliverystopsandpickingrespectively.Toestimatedthestoragecosts,webenchmarkedthestoragecostsofdifferentwarehouseslocatedintheUnitedStates.Thestoragecostperunittimeisnormallyafunctionoftheweightoftheproduct,andsincetheweightoftheproductsdeliveredtoagenciesvaries,weuseanapproximatevalueof50pounds/case.UsingtheinformationfromtheDepartmentofLaborweadjustedthestoragecostfortheStateofFlorida,andobtainedacostof$0.126/casemonth.FortheagenciesinthecurrentRegion5,wheretherealestateandlaborcostsarehigher,weincrementedthestoragecostby20%. 137

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Sincetherearefourtypesofproductswithdifferentstoragerequirements,i.e.,dry,specialdry,coolerandfreezer,weusedacorrectionfactorthattakesintoaccounttheadditionalcostofspecialstoragerequirementsandthepercentageofeachtypeofproductforthecurrentregions.Inordertoevaluatetheaccuracyofourmodel,wecomparedourresultswiththeinformationfromtheagenciesinregion2,sincethatwastheregionfromwhichweobtainedthemostcompleteanddetailedinformation.AswecanseeinTable I-1 ,usingourcostestimationmodel,thestate-contractedwarehousefromregion,hasaprotof24.49%overitscost,whichwethinkisareasonableresult.Forthecaseofregion1,thewarehousehasalossof10%,butthisisbecauseweareunderestimatingitsdeliverycost.Sincewedonothavedetailedinformationontheordersizeforeverydeliverypoint,butrather,wehaveinformationonlyonthetotalordersizeofeachagency,wehaveunderestimatedthepenaltyfeesforordersoflessthan20cases.Also,basedontheinformationprovidedbyFDACS,thefeeschargedbythewarehousearelowerthanthewarehousewouldprefertocharge,butitcannotchangethesefeesduetocontractlimitations;thereforeweknowthattheprotmarginforthisregionislowerthanthestate-contractedwarehousewouldlike.Regions3,4and5,shouldbeanalyzedinanaggregatedmanner,sincetheyareservedbythesamestate-contractedwarehouse.Also,animportantnotefortheseregionsisthatthestate-contractedwarehousehadanadditionalrevenueof$105,662duringtheschoolyear2011-2012payedbytheFDACS,duetothestorageofunassignedproductanddeliverytoagenciesoutsidethedistributionnetwork,increasingtheprotmarginforthatstate-contractedwarehouse.TEFAPismoreprotable,forthereasonsexplainedinSection 4.1 .SincewedonothavedetailedinformationaboutthetypeofservicerequestedbytheTEFAPagencies,wedonotknowtheproportionofordersthatweredeliveredbythestate-contractedwarehouseorpickedupbytheagencies.Sincethefeesforthepickupservicesare 138

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approximate20%lowerthanthedeliveryservicefees,inTable I-2 wepresenttheprotmarginsforthetwotypesofdeliveries. TableI-1. NSLP:Summaryofestimateddeliveryandstoragecostsforthestate-contractedwarehouses,Schoolyear2011-2012 Region EstimatedAnnualDemand Estimatedannualrevenue Estimatedannualcost Protmarginperregion Protmarginperstate-contractedwarehouse Cost/caseperregion Cost/caseperstate-contractedwarehouse Cases $ $ % % $/case $/case 1 76,716 234,610 261,875 (10.41) (10.41) 3.41 3.41 2 81,329 337,777 271,328 24.49 24.49 3.34 3.34 3 123,998 466,793 563,323 (17.14) 8.68 4.54 4.64 4 25,689 181,470 106,733 70.02 4.15 5 13,745 68,260 62,125 9.88 4.52 FDACS 105,662 26,476 TableI-2. TEFAP:Summaryofestimateddeliveryandstoragecostsforthestate-contractedwarehouses,Schoolyear2011-2012 Region EstimatedAnnualDemand Estimatedannualdeliveryassumingdeliveryservice Estimatedannualdeliveryassumingpickupservice Estimatedannualcost Protmarginassumingdeliveryservice Protmarginassumingpickupservice Cost/case Cases $ $ $ % % $/case TEFAP 370,677 629,929 503,944 389,954 61.54 29.23 1.05 139

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APPENDIXJPROPOSEDREGIONALCONFIGURATIONS TableJ-1. Proposedregionalcongurations Proposedregionalcongurations Counties Currentregions Option1 Option2 Option3 Option4 Bay 1 1 1 1 1 Calhoun 1 1 1 1 1 Escambia 1 1 1 1 1 Franklin 1 1 1 1 1 Gadsden 1 1 1 1 1 Gulf 1 1 1 1 1 Holmes 1 1 1 1 1 Jackson 1 1 1 1 1 Jefferson 1 1 1 1 1 Leon 1 1 1 1 1 Liberty 1 1 1 1 1 Okaloosa 1 1 1 1 1 SantaRosa 1 1 1 1 1 Wakulla 1 1 1 1 1 Walton 1 1 1 1 1 Washington 1 1 1 1 1 Alachua 2 2 2 2 2 Baker 2 2 2 2 2 Bradford 2 2 2 2 2 Clay 2 2 2 2 2 Columbia 2 2 2 2 2 Dixie 2 2 2 2 2 Duval 2 2 2 2 2 Flagler 2 2 2 2 2 Gilchrist 2 2 2 2 2 Hamilton 2 2 2 2 2 Lafayette 2 2 2 2 2 Levy 2 2 2 2 2 Madison 2 2 2 1 2 Marion 2 2 2 2 2 Nassau 2 2 2 2 2 Putnam 2 2 2 2 2 St.Johns 2 2 2 2 2 Suwannee 2 2 2 2 2 140

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Table J-1 .Continued Proposedregionalcongurations Counties Currentregions Option1 Option2 Option3 Option4 Taylor 2 2 2 1 2 Union 2 2 2 2 2 Volusia 2 2 2 2 2 Brevard 3 3 5 3 4 Citrus 3 3 2 3 4 Hardee 3 3 4 4 4 Hernando 3 3 4 3 4 Highlands 3 3 5 3 4 Hillsborough 3 4 4 4 3withTEFAP IndianRiver 3 3 5 4 4 Lake 3 3 2 3 4 Manatee 3 4 4 4 4 Okeechobee 3 3 5 4 4 Orange 3 3 5 3 3withTEFAP Osceola 3 3 5 3 3withTEFAP Pasco 3 3 4 3 3withTEFAP Pinellas 3 3 4 4 3withTEFAP Polk 3 4 5 4 3withTEFAP Seminole 3 3 2 3 2 Sumter 3 3 2 3 4 Charlotte 4 4 4 4 4 Collier 4 4 4 4 4 DeSoto 4 4 4 4 4 Glades 4 4 5 4 4 Hendry 4 4 4 4 4 Lee 4 4 4 4 4 Sarasota 4 4 4 4 4 Broward 5 4 5 4 3withTEFAP Dade 5 4 5 4 3withTEFAP Leon-DOC 5 4 5 4 4 Martin 5 4 5 4 4 Monroe 5 4 5 4 4 PalmBeach 5 4 5 4 3withTEFAP St.Lucie 5 4 5 4 4 141

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APPENDIXKNSLP:CURRENTREGIONALCONFIGURATION FigureK-1. NSLP:Currentregionalconguration 142

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APPENDIXLPROPOSEDREGIONALCONFIGURATION1 FigureL-1. Proposedregionalconguration1 143

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APPENDIXMPROPOSEDREGIONALCONFIGURATION2 FigureM-1. Proposedregionalconguration2 144

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APPENDIXNPROPOSEDREGIONALCONFIGURATION3 FigureN-1. Proposedregionalconguration3 145

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APPENDIXOPROPOSEDREGIONALCONFIGURATION4 FigureO-1. Proposedregionalconguration4 146

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APPENDIXPPROPOSEDREGIONALCONFIGURATIONS.COSTANDDEMAND TableP-1. Currentregionalconguration.Costsanddemand Region Cost Percentage Annualdemand Percentage Cost/case $ % Cases % $/case Twodistributionnetworks:NSLPandTEFAP 1 261,875 27 76,716 27 3.41 2 271,328 28 81,329 28 3.34 3 284,168 29 89,710 31 3.17 4 106,733 11 25,689 9 4.15 5 56,722 6 13,745 5 4.13 TEFAP 389,954 100 370,677 100 1.05 Onedistributionnetwork 1 274,944 24 99,209 15 2.77 2 289,027 25 114,452 17 2.53 3 372,045 32 277,945 42 1.34 4 130,830 11 77,222 12 1.69 5 88,865 8 89,038 14 1.00 TableP-2. Proposedregionalconguration1.Costsanddemand Region Cost Percentage Annualdemand Percentage Cost/case $ % Cases % $/case Twodistributionnetworks:NSLPandTEFAP 1 261,875 27 76,716 27 3.41 2 271,328 28 81,329 28 3.34 3 262,402 27 81,051 28 3.24 4 185,222 19 48,093 17 3.85 5 TEFAP 389,954 100 370,677 100 1.05 Onedistributionnetwork 1 274,944 24 99,209 15 2.77 2 289,027 25 114,452 17 2.53 3 312,868 27 190,279 29 1.64 4 278,872 24 253,926 39 1.10 5 147

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TableP-3. Proposedregionalconguration2.Costsanddemand Region Cost Percentage Annualdemand Percentage Cost/case $ % Cases % $/case Twodistributionnetworks:NSLPandTEFAP 1 261,875 27 76,716 27 3.41 2 408,650 42 129,237 45 3.16 3 4 137,134 14 36,473 13 3.76 5 173,168 18 44,763 16 3.87 TEFAP 389,954 100 370,677 100 1.05 Onedistributionnetwork 1 274,944 24 99,209 15 2.77 2 426,349 37 162,360 25 2.63 3 4 210,612 18 189,616 29 1.11 5 243,806 21 206,681 31 1.18 TableP-4. Proposedregionalconguration3.Costsanddemand Region Cost Percentage Annualdemand Percentage Cost/case $ % Cases % $/case Twodistributionnetworks:NSLPandTEFAP 1 272,148 28 84,138 29 3.23 2 261,055 27 73,907 26 3.53 3 226,662 23 73,438 26 3.09 4 220,962 23 55,706 19 3.97 5 TEFAP 389,954 100 370,677 100 1.05 Onedistributionnetwork 1 285,217 25 106,631 16 2.67 2 298,338 26 112,018 17 2.66 3 294,533 25 178,518 27 1.65 4 277,624 24 260,699 40 1.06 5 TableP-5. Proposedregionalconguration4.Costsanddemand Region Cost Percentage Annualdemand Percentage Cost/case $ % Cases % $/case Onedistributionnetwork 1 261,875 19 76,716 12 3.41 2 324,612 24 98,024 15 3.31 3 522,270 38 401,410 61 1.30 4 262,024 19 81,716 12 3.21 5 148

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APPENDIXQFEESCENARIOSWITHTHESMALLESTAVERAGEOVERPAYMENTWHENNSLPANDTEFAPUSETWODISTRIBUTIONNETWORKS TableQ-1. Feescenarioswithsmallestaverageoverpayment.TwodistributionnetworksforNSLPandTEFAP Minimumordersize Feetype Freestoragetime Storagefee Ranking Averageoverpay-ment Proposedregionalconguration 10cases 20cases 30cases 40cases Uniformfee Feeschedule 0months 1month 2months 25%ofdeliveryfee $0.13/casemonth $0.17/casemonth $0.20/casemonth 1 1,378 3 X X X X 2 1,587 3 X X X X 3 1,461 3 X X X X 4 1,487 3 X X X X 5 1,599 3 X X X X 6 1,824 3 X X X X 7 1,577 3 X X X X 8 1,332 3 X X X X 9 1,406 3 X X X X 10 1,649 3 X X X X 11 1,644 3 X X X X 12 1,326 3 X X X X 13 1,470 3 X X X X 14 1,874 3 X X X X 15 1,648 3 X X X X 16 1,448 3 X X X X 17 1,609 3 X X X X 18 1,923 3 X X X X 19 1,370 3 X X X X 20 1,670 3 X X X X 21 1,519 3 X X X X 22 1,289 3 X X X X 23 1,678 3 X X X X 24 1,426 3 X X X X 25 1,464 3 X X X X 149

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TableQ-2. Uniformdistributionandstoragefeesforthe25feescenarioswiththesmallestaverageoverpayment.TwodistributionnetworksforNSLPandTEFAP Rank Deliveryfeepercase($/case) Storagefeepercase($/casemonth) Region1 Region2 Region3 Region4 TEFAP Region1 Region2 Region3 Region4 TEFAP 1 2.42 2.23 1.70 2.45 0.88 0.60 0.56 0.42 0.61 0.22 2 3.49 3.09 2.41 4.11 0.92 0.20 0.20 0.20 0.20 0.20 3 2.86 2.85 2.14 3.08 1.05 0.71 0.71 0.54 0.77 0.26 4 3.25 3.42 2.51 3.54 1.20 0.81 0.86 0.63 0.88 0.30 5 3.69 3.27 2.58 4.29 1.12 0.20 0.20 0.20 0.20 0.20 7 3.25 3.44 2.53 3.58 1.20 0.81 0.86 0.63 0.89 0.30 8 2.42 2.35 1.79 2.58 0.88 0.61 0.59 0.45 0.64 0.22 9 2.84 2.57 1.93 2.75 1.05 0.71 0.64 0.48 0.69 0.26 10 3.74 3.31 2.65 4.38 1.15 0.17 0.17 0.17 0.17 0.17 11 3.82 3.44 2.76 4.47 1.24 0.20 0.20 0.20 0.20 0.20 12 2.85 2.74 2.05 2.92 1.05 0.71 0.68 0.51 0.73 0.26 13 3.24 3.05 2.24 3.15 1.20 0.81 0.76 0.56 0.79 0.30 15 3.85 3.46 2.80 4.54 1.25 0.17 0.17 0.17 0.17 0.17 16 2.85 2.83 2.13 3.05 1.05 0.71 0.71 0.53 0.76 0.26 17 3.57 3.17 2.51 4.23 0.98 0.17 0.17 0.17 0.17 0.17 20 3.67 3.26 2.63 4.39 1.06 0.13 0.13 0.13 0.13 0.13 21 3.50 3.37 2.64 4.57 0.92 0.20 0.20 0.20 0.20 0.20 22 3.24 3.28 2.40 3.36 1.20 0.81 0.82 0.60 0.84 0.30 23 3.80 3.38 2.74 4.51 1.19 0.13 0.13 0.13 0.13 0.13 24 2.43 2.42 1.85 2.67 0.88 0.61 0.60 0.46 0.67 0.22 25 2.43 2.43 1.86 2.70 0.88 0.61 0.61 0.46 0.67 0.22 150

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TableQ-3. Order-quantity-anddelivery-frequency-basedfeescheduleforthe25feescenarioswiththesmallestaverageoverpayment.TwodistributionnetworksforNSLPandTEFAP 151

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APPENDIXRFEESCENARIOSWITHTHESMALLESTAVERAGEOVERPAYMENTWHENNSLPANDTEFAPUSEONEDISTRIBUTIONNETWORK TableR-1. Feescenarioswithsmallestaverageoverpayment.OnedistributionnetworksforNSLPandTEFAP Minimumordersize Feetype Freestoragetime Storagefee Ranking Averageoverpay-ment Proposedregionalconguration 10cases 20cases 30cases 40cases Uniformfee Feeschedule 0months 1month 2months 25%ofdeliveryfee $0.13/casemonth $0.17/casemonth $0.20/casemonth 1 2,922 4 X X X X 2 2,839 4 X X X X 3 2,185 4 X X X X 4 2,164 4 X X X X 5 2,951 4 X X X X 6 2,800 4 X X X X 7 2,147 4 X X X X 8 1,696 4 X X X X 9 2,178 4 X X X X 10 2,168 4 X X X X 11 2,167 4 X X X X 12 1,806 4 X X X X 13 2,109 4 X X X X 14 2,022 4 X X X X 15 2,086 4 X X X X 16 1,901 4 X X X X 17 2,062 4 X X X X 18 2,086 4 X X X X 19 2,082 4 X X X X 20 2,058 4 X X X X 21 2,064 4 X X X X 22 2,069 4 X X X X 23 2,069 4 X X X X 24 2,093 4 X X X X 25 2,075 4 X X X X 152

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TableR-2. Uniformdistributionandstoragefeesforthe25feescenarioswiththesmallestaverageoverpayment.OnedistributionnetworkforNSLPandTEFAP Rank Deliveryfeepercase($/case) Storagefeepercase($/casemonth) Region1 Region2 Region3 Region4 Region1 Region2 Region3 Region4 7 3.81 3.44 1.25 3.03 0.17 0.17 0.17 0.17 8 2.59 2.45 1.06 1.86 0.65 0.61 0.26 0.47 9 4.13 3.79 1.55 3.45 0.13 0.13 0.13 0.13 10 4.05 3.67 1.46 3.33 0.13 0.13 0.13 0.13 11 3.92 3.55 1.33 3.20 0.13 0.13 0.13 0.13 12 3.46 3.46 1.45 2.45 0.87 0.87 0.36 0.61 13 3.74 3.57 1.19 3.06 0.20 0.20 0.20 0.20 14 4.08 3.97 1.53 3.47 0.20 0.20 0.20 0.20 15 3.94 3.77 1.39 3.26 0.20 0.20 0.20 0.20 16 3.46 3.48 1.45 2.47 0.87 0.87 0.36 0.62 17 3.75 3.59 1.20 3.09 0.20 0.20 0.20 0.20 18 3.82 3.65 1.26 3.19 0.17 0.17 0.17 0.17 19 4.10 3.99 1.54 3.55 0.17 0.17 0.17 0.17 20 4.08 3.99 1.53 3.51 0.20 0.20 0.20 0.20 21 3.99 3.82 1.43 3.37 0.17 0.17 0.17 0.17 22 3.95 3.79 1.40 3.30 0.20 0.20 0.20 0.20 23 3.83 3.67 1.26 3.22 0.17 0.17 0.17 0.17 24 4.11 4.01 1.54 3.58 0.17 0.17 0.17 0.17 25 4.00 3.84 1.43 3.40 0.17 0.17 0.17 0.17 153

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TableR-3. Order-quantity-anddelivery-frequency-basedfeescheduleforthe25feescenarioswiththesmallestaverageoverpayment.OnedistributionnetworkforNSLPandTEFAP 154

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APPENDIXSMAXIMUM30%OFENTITLEMENTSENTTOFOODPROCESSORS:EFFECTONPROPOSEDFEESWHENNSLPANDTEFAPUSETWODISTRIBUTIONNETWORKS TableS-1. Uniformdistributionandstoragefeesforthe25feescenarioswiththesmallestaverageoverpayment.TwodistributionnetworksforNSLPandTEFAPandmaximum30%ofentitlementsenttofoodprocessors Rank Deliveryfeepercase($/case) Storagefeepercase($/casemonth) Region1 Region2 Region3 Region4 TEFAP Region1 Region2 Region3 Region4 TEFAP 1 1.98 1.06 1.15 0.93 0.88 0.50 0.27 0.29 0.23 0.22 2 2.81 1.27 1.40 1.06 0.92 0.20 0.20 0.20 0.20 0.20 3 2.33 1.32 1.48 1.14 1.05 0.58 0.33 0.37 0.29 0.26 4 2.66 1.56 1.74 1.33 1.20 0.67 0.39 0.44 0.33 0.30 5 3.01 1.44 1.57 1.24 1.12 0.20 0.20 0.20 0.20 0.20 7 2.66 1.56 1.77 1.34 1.20 0.67 0.39 0.44 0.34 0.30 8 1.99 1.11 1.21 0.96 0.88 0.50 0.28 0.30 0.24 0.22 9 2.32 1.21 1.32 1.06 1.05 0.58 0.30 0.33 0.27 0.26 10 3.06 1.51 1.63 1.32 1.15 0.17 0.17 0.17 0.17 0.17 11 3.16 1.61 1.74 1.43 1.24 0.20 0.20 0.20 0.20 0.20 12 2.33 1.28 1.40 1.10 1.05 0.58 0.32 0.35 0.28 0.26 13 2.65 1.41 1.54 1.23 1.20 0.66 0.35 0.38 0.31 0.30 15 3.19 1.65 1.77 1.48 1.25 0.17 0.17 0.17 0.17 0.17 16 2.33 1.32 1.46 1.13 1.05 0.58 0.33 0.37 0.28 0.26 17 2.90 1.36 1.48 1.17 0.98 0.17 0.17 0.17 0.17 0.17 20 3.01 1.48 1.59 1.31 1.06 0.13 0.13 0.13 0.13 0.13 21 2.82 1.38 1.54 1.12 0.92 0.20 0.20 0.20 0.20 0.20 22 2.66 1.50 1.66 1.29 1.20 0.66 0.38 0.41 0.32 0.30 23 3.14 1.59 1.71 1.43 1.19 0.13 0.13 0.13 0.13 0.13 24 1.99 1.15 1.26 0.99 0.88 0.50 0.29 0.32 0.25 0.22 25 1.99 1.15 1.27 1.00 0.88 0.50 0.29 0.32 0.25 0.22 155

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TableS-2. Order-quantity-anddelivery-frequency-basedfeescheduleforthe25feescenarioswiththesmallestaverageoverpayment.TwodistributionnetworksforNSLPandTEFAPandmaximum30%ofentitlementsenttofoodprocessors 156

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APPENDIXTMAXIMUM30%OFENTITLEMENTSENTTOFOODPROCESSORS:EFFECTONPROPOSEDFEESWHENNSLPANDTEFAPUSEONEDISTRIBUTIONNETWORK TableT-1. Uniformdistributionandstoragefeesforthe25feescenarioswiththesmallestaverageoverpayment.OnedistributionnetworkforNSLPandTEFAPandmaximum30%ofentitlementsenttofoodprocessors Rank Deliveryfeepercase($/case) Storagefeepercase($/casemonth) Region1 Region2 Region3 Region4 Region1 Region2 Region3 Region4 7 3.04 1.51 1.02 1.65 0.17 0.17 0.17 0.17 8 2.08 1.18 0.91 1.19 0.52 0.29 0.23 0.30 9 3.37 1.89 1.33 2.08 0.13 0.13 0.13 0.13 10 3.28 1.76 1.23 1.95 0.13 0.13 0.13 0.13 11 3.15 1.64 1.10 1.82 0.13 0.13 0.13 0.13 12 2.78 1.61 1.25 1.55 0.69 0.40 0.31 0.39 13 2.96 1.50 0.96 1.61 0.20 0.20 0.20 0.20 14 3.30 1.89 1.31 2.03 0.20 0.20 0.20 0.20 15 3.16 1.70 1.16 1.82 0.20 0.20 0.20 0.20 16 2.78 1.61 1.26 1.57 0.69 0.40 0.31 0.39 17 2.96 1.50 0.97 1.63 0.20 0.20 0.20 0.20 18 3.04 1.60 1.03 1.75 0.17 0.17 0.17 0.17 19 3.33 1.94 1.32 2.10 0.17 0.17 0.17 0.17 20 3.30 1.90 1.32 2.05 0.20 0.20 0.20 0.20 21 3.21 1.77 1.20 1.92 0.17 0.17 0.17 0.17 22 3.16 1.70 1.17 1.84 0.20 0.20 0.20 0.20 23 3.05 1.60 1.04 1.77 0.17 0.17 0.17 0.17 24 3.34 1.94 1.33 2.13 0.17 0.17 0.17 0.17 25 3.22 1.77 1.21 1.95 0.17 0.17 0.17 0.17 157

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TableT-2. Order-quantity-anddelivery-frequency-basedfeescheduleforthe25feescenarioswiththesmallestaverageoverpayment.OnedistributionnetworkforNSLPandTEFAPandmaximum30%ofentitlementsenttofoodprocessors 158

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APPENDIXUMAXIMUM50%OFENTITLEMENTSENTTOFOODPROCESSORS:EFFECTONPROPOSEDFEESWHENNSLPANDTEFAPUSETWODISTRIBUTIONNETWORKS TableU-1. Uniformdistributionandstoragefeesforthe25feescenarioswiththesmallestaverageoverpayment.TwodistributionnetworksforNSLPandTEFAPandmaximum50%ofentitlementsenttofoodprocessors Rank Deliveryfeepercase($/case) Storagefeepercase($/casemonth) Region1 Region2 Region3 Region4 TEFAP Region1 Region2 Region3 Region4 TEFAP 1 2.24 1.29 1.37 1.08 0.88 0.56 0.32 0.34 0.27 0.22 2 3.22 1.67 1.78 1.33 0.92 0.20 0.20 0.20 0.20 0.20 3 2.64 1.61 1.75 1.32 1.05 0.66 0.40 0.44 0.33 0.26 4 3.00 1.89 2.06 1.53 1.20 0.75 0.47 0.51 0.38 0.30 5 3.42 1.84 1.96 1.52 1.12 0.20 0.20 0.20 0.20 0.20 7 3.01 1.90 2.08 1.55 1.20 0.75 0.47 0.52 0.39 0.30 8 2.24 1.35 1.44 1.11 0.88 0.56 0.34 0.36 0.28 0.22 9 2.62 1.48 1.57 1.22 1.05 0.66 0.37 0.39 0.31 0.26 10 3.47 1.91 2.02 1.60 1.15 0.17 0.17 0.17 0.17 0.17 11 3.56 2.02 2.13 1.71 1.24 0.20 0.20 0.20 0.20 0.20 12 2.63 1.56 1.66 1.27 1.05 0.66 0.39 0.42 0.32 0.26 13 2.99 1.72 1.83 1.42 1.20 0.75 0.43 0.46 0.35 0.30 15 3.59 2.06 2.17 1.76 1.25 0.17 0.17 0.17 0.17 0.17 16 2.63 1.60 1.73 1.31 1.05 0.66 0.40 0.43 0.33 0.26 17 3.30 1.76 1.87 1.45 0.98 0.17 0.17 0.17 0.17 0.17 20 3.41 1.88 1.99 1.59 1.06 0.13 0.13 0.13 0.13 0.13 21 3.23 1.81 1.95 1.42 0.92 0.20 0.20 0.20 0.20 0.20 22 3.00 1.83 1.96 1.48 1.20 0.75 0.46 0.49 0.37 0.30 23 3.54 2.00 2.10 1.72 1.19 0.13 0.13 0.13 0.13 0.13 24 2.24 1.39 1.49 1.14 0.88 0.56 0.35 0.37 0.29 0.22 25 2.25 1.39 1.51 1.15 0.88 0.56 0.35 0.38 0.29 0.22 159

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TableU-2. Order-quantity-anddelivery-frequency-basedfeescheduleforthe25feescenarioswiththesmallestaverageoverpayment.TwodistributionnetworksforNSLPandTEFAPandmaximum50%ofentitlementsenttofoodprocessors 160

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APPENDIXVMAXIMUM50%OFENTITLEMENTSENTTOFOODPROCESSORS:EFFECTONPROPOSEDFEESWHENNSLPANDTEFAPUSEONEDISTRIBUTIONNETWORK TableV-1. Uniformdistributionandstoragefeesforthe25feescenarioswiththesmallestaverageoverpayment.OnedistributionnetworkforNSLPandTEFAPandmaximum50%ofentitlementsenttofoodprocessors Rank Deliveryfeepercase($/case) Storagefeepercase($/casemonth) Region1 Region2 Region3 Region4 Region1 Region2 Region3 Region4 7 3.51 1.95 1.08 2.01 0.17 0.17 0.17 0.17 8 2.38 1.43 0.95 1.36 0.59 0.36 0.24 0.34 9 3.83 2.32 1.39 2.43 0.13 0.13 0.13 0.13 10 3.74 2.20 1.30 2.31 0.13 0.13 0.13 0.13 11 3.62 2.08 1.17 2.18 0.13 0.13 0.13 0.13 12 3.17 1.95 1.30 1.78 0.79 0.49 0.33 0.45 13 3.43 1.95 1.02 1.98 0.20 0.20 0.20 0.20 14 3.77 2.35 1.37 2.40 0.20 0.20 0.20 0.20 15 3.63 2.15 1.23 2.19 0.20 0.20 0.20 0.20 16 3.17 1.96 1.31 1.79 0.79 0.49 0.33 0.45 17 3.44 1.96 1.03 2.01 0.20 0.20 0.20 0.20 18 3.51 2.05 1.09 2.12 0.17 0.17 0.17 0.17 19 3.80 2.39 1.38 2.47 0.17 0.17 0.17 0.17 20 3.77 2.36 1.37 2.43 0.20 0.20 0.20 0.20 21 3.68 2.22 1.26 2.30 0.17 0.17 0.17 0.17 22 3.64 2.16 1.23 2.22 0.20 0.20 0.20 0.20 23 3.52 2.06 1.10 2.15 0.17 0.17 0.17 0.17 24 3.80 2.40 1.39 2.51 0.17 0.17 0.17 0.17 25 3.69 2.23 1.27 2.33 0.17 0.17 0.17 0.17 161

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TableV-2. Order-quantity-anddelivery-frequency-basedfeescheduleforthe25feescenarioswiththesmallestaverageoverpayment.OnedistributionnetworkforNSLPandTEFAPandmaximum50%ofentitlementsenttofoodprocessors 162

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APPENDIXWDISTRIBUTIONFEEDESIGNINGMODELSOLUTIONPROCEDURE Algorithm4Distributionfeedesigningmodel 1. i 1 2. C1 C 3. C2 CnC1 4. L1 ; 5. L2 ; 6. L1 L1[fC1g 7. L2 L2[fC2g 8. sol 0 9. loop 0 10. Findf1(C1) 11. whilei2nORsol=0ORloop=0do 12. FindC2bysolving( 4 )andusingf1(C1)astheRHSfor( 4b ) 13. ifC2=C2ANDC2=CnC1then 14. sol 1 15. else 16. ifC22L2then 17. loop 1 18. else 19. C2 C2 20. L2 L2[fC2g 21. endif 22. endif 23. FindC1bysolving( 4 )andusingf2(C2)astheRHSfor( 4b ) 24. ifC1=C1ANDC1=CnC2then 25. sol 1 26. else 27. ifC12L1then 28. loop 1 29. else 30. C1 C1 31. L1 L1[fC1g 32. endif 33. endif 34. endwhile 35. ifsol=0then 36. C1 C 37. C2 CnC1 38. else 39. C1 C1 40. C2 C2 41. endif 163

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APPENDIXXDISTANCE-BASEDCOSTALLOCATION:TYPEOFNON-TRIVIALEQUILIBRIUMSOLUTIONS FigureX-1. Distance-basedcostallocationandgeographicaldistributiontype1:typeofnon-trivialequilibriumsolutions FigureX-2. Distance-basedcostallocationandgeographicaldistributiontype2:typeofequilibriumsolutions. 164

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FigureX-3. Distance-basedcostallocationandgeographicaldistributiontype3:typeofnon-trivialequilibriumsolutions. FigureX-4. Distance-basedcostallocationandgeographicaldistributiontype4:typeofnon-trivialequilibriumsolutions. 165

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APPENDIXYUNIFORMCOSTALLOCATION:TYPEOFNON-TRIVIALEQUILIBRIUMSOLUTIONS FigureY-1. Uniformcostallocationandgeographicaldistributiontype1:typeofnon-trivialequilibriumsolutions. FigureY-2. Uniformcostallocationandgeographicaldistributiontype2:typeofnon-trivialequilibriumsolutions. 166

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FigureY-3. Uniformcostallocationandgeographicaldistributiontype3:typeofnon-trivialequilibriumsolutions. FigureY-4. Uniformcostallocationandgeographicaldistributiontype4:typeofnon-trivialequilibriumsolutions. 167

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APPENDIXZBRANCH-BASEDCOSTALLOCATION:TYPEOFNON-TRIVIALEQUILIBRIUMSOLUTIONS FigureZ-1. Branch-basedcostallocationandinstancegeographicaldistribution1:typeofnon-trivialequilibriumsolutions. FigureZ-2. Branch-basedcostallocationandinstancegeographicaldistribution2:typeofnon-trivialequilibriumsolutions. 168

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FigureZ-3. Branch-basedcostallocationandgeographicaldistributiontype3:typeofnon-trivialequilibriumsolutions. FigureZ-4. Branch-basedcostallocationandgeographicaldistributiontype4:typeofnon-trivialequilibriumsolutions. 169

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BIOGRAPHICALSKETCH CinthiaPerezholdsaB.S.degreeinIndustrialEngineeringfromtheEscuelaSuperiorPolitcnicadelLitoral(ESPOL)inEcuador,wheresheobtainedtheGoldenDiplomaforhavingthebestscoreamongtheEngineeringSchools.Afterworkingforseveralyearsinthemanufacturingindustryindifferentareas,suchas:qualityassurance,productionmanagementandoperationalexcellence,sheearnedaFulbrightscholarshipandcompletedamaster'sdegreeinindustrialandsystemsengineeringfromUniversityofFlorida.Duetoherinterestinsupplychain,inventorymanagement,logisticsanddistributionsystemsshecontinuedwiththePh.D.programinIndustrialandSystemsEngineeringatUniversityofFloridaandearnedherdegreeonMay2014. 173