integrating production and distribution decisions in … · used and less often, train and tank...

September 24-28, 2012Rio de Janeiro, Brazil

INTEGRATINGPRODUCTIONANDDISTRIBUTIONDECISIONSINTHESPECIALITYOILSSUPPLYCHAIN

MarioGuajardo

DepartmentofFinanceandManagementScience,NHHNorwegianSchoolofEconomicsHelleveien30,N‐5045Bergen,Norway

[email protected]

MartinKylingerDepartmentofManagementandEngineering,LinköpingUniversity

SE‐58183Linköping,[email protected]

MikaelRönnqvist

DepartmentofFinanceandManagementScience,NHHNorwegianSchoolofEconomicsHelleveien30,N‐5045Bergen,Norway

[email protected]

ABSTRACT

Weproposelinearprogrammingmodelsfordecoupledandintegratedplanninginadivergentsupplychainforspecialtyoils.Theoptimizationprobleminvolvesdecisionsonproduction, inventory, internal transportation, sales and distribution to customers. Theintegrated objective is tomaximize contribution. In the decoupled approach, an internalprice systempursues toalign sellerswith thisobjective.Wesolvenumerical examplesofthemodels to illustrate potential effects of integration and coordination and discuss theadvantages and disadvantages of the integrated over the decoupled approach.While thetotal contribution is higher in the integrated approach, the sellers’ contribution may belower.Wesuggestcontributionsharingrulestomakeboththecompanyandsellersbetteroffundertheintegratedplanning.

KEYWORDS.Supplychainmanagement.Integratedplanning.Contributionsharing.

P&G‐POnaÁreadePetróleo&Gás

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1.IntroductionIntegratingdecisions aboutproductionwithother functions in the supply chain,

such as inventory and distribution, has proved to be of significant relevance inorganizations. An important body of Operations Research literature has been devoted tothisissue,asreviewedbyErengüçetal.(1999).Thebasicideaofanintegratedmodelistosimultaneously optimize decision variables of different functions that have traditionallybeenoptimizedinasequencewheretheoutputofonestagewasusedastheinputtootherstage (Sarmiento and Nagi, 1999). Aligning decisions under the same goal can bechallengingwhentheobjectivesofthedifferentfunctionsareinconflict.

In this paper, we address a problem of tactical planning in a divergent supplychain.Ourmotivationcomesfromaprojectinwhichweareworkingwithacompanyinthespecialityoils industry.The logisticsnetwork is composedof refineries,hubs,depotsandsalesoffices.Althoughownedbythecompany,thesalesofficesaremanagedindependentlyand thedecisiononhow to ship to customers is decentralized.According to thedemandtheyobserve,thesellersmakedecisionsontypeandamountofproductstoorder,andfromwhichstoragelocationtoorderfrom.Thisdecisionismainlydrivenbyaninternalpricesetbythecompanyandthedistributioncostcalculatedbytheseller.Thispriceissetforeachproductandeach locationwhere it is stored.Aftera sale is realized, the seller receivesapercentageofthecontributionmargin(revenueminustheinternalpriceandminusthecostofdistributiontocustomers),andtherestoftherevenueisreceivedbythecompanyitself.

We formulate linear programming models to represent this supply chain,considering decisions on production, inventory, internal transportation, sales anddistributiontocustomers. Ina firstapproach,weproposedecoupledmodels torepresentthesituationwheresalesanddistributiontocustomersaredecidedseparatefromtherestofthefunctionsinthesupplychain.Then,weintegrateallthedecisionsinthesamemodeland analyze its potential to improve the performance in comparison to the decoupledmodels.

Integratingplanninghasbeenoneofthemaintopicsstudiedbyrecentliteratureintheoilsupplychain.Pintoetal.(2000)workonplanningandschedulingapplicationsforrefineryoperations.NeiroandPinto(2004)proposeamodelforapetroleumsupplychaininthecontextoftheBraziliancompanyPetrobras,integratingsources,terminals,refineries,distribution centres and consumers. Bengtsson et al. (2011) integrate production andlogistics decisions under uncertainty in ship arrivals. Guyonnet et al. (2009) explore thebenefits of an integrated model involving unloading, oil processing, and distribution. Intheseworks, oneof themain challenges is givenby thenumerousnon‐linear constraintsappearing fromcomputing thepropertiesof theproductsafterbeingprocessed.A recentoverview of refinery planning and scheduling by Bengtsson and Nonås (2010) haveidentifiedthehandlingofnon‐linearitiesasoneofthemainissuesintheagendaforfuturework.Adistinctionoftheproblemwedealwithisthatfixedanduniquerecipesareusedtomix each final product from semi finished products. This characteristic allows us toapproach the problem by linear programming, in both the decoupled and the integratedapproach. A second distinction of our problem is the sales mechanism involved in thesupplychain.Normally,intheoilplanningliteratureithasbeenassumedthattheobjectivesof thesalesunitsarealignedwith theobjectivesof thewholecompany. In thedecoupledversionof theproblemapproached in thisarticle,wegive insights in thecasewhenbothparts are not aligned. This has been a research topic in other industrial contexts (e.g.Ouhimmou et al. 2008; Feng et al. 2008, 2010).As for the integratedplanning approach,while it results in higher total contribution, the sellers’ contribution may be lower. Theagreement among the actors in the supply chain has been identified by Erengüç et al.(1999)asaparticularlyimportantissueontheintegrationofproductionanddistribution,becausetheseagreementswilldeterminetoalargeextentwhethereachcomponentofthe

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chainwillbemotivated toachievethecostreductionsby integratingdecisionsacross thechain.Inthenumericalexamplesofourproblem,wediscusscontributionsharingrulesthatmakeboththesellersandthecompanybetteroffintheintegratedcase.

2.SpecialtyoilssupplychainTheoilindustryhasbeenidentifiedasatypicalexampleofdivergentsupplychain

(ViswanadhamandRaghavan,2000;LasschuitandThijssen,2004).This isthecaseofthesupply chain for speciality oils thatwe face in our problem,which is characterized by adivergentproductstructureaswellasadivergentphysicalstructure.ArepresentationofthesupplychainispresentedinFigure1.Next,wedescribeitsmainparts.

Figure1:Supplychainforspecialityoilproductsandplanninglevels.

Refineriesandproducts

The refineries are supplied with crude oil from external suppliers. There aredifferent types of crude oil, some of them containingmorepercentage of one or anothercomponent.Intherefineries,thecrudeoilsareexposedtoaseriesofprocesses,inordertogeneratesaleableproducts.Therearetwoproductsegments,thatwecallbasicoilproductsand speciality oil products (or simply basic oils and speciality oils). The processes in therefineries andhubsdiffer somewhat for different products, but they can be simplified tothreesteps:distillation,hydrotreatmentandblending.

Duringthedistillationprocess,thecrudeoil isdividedintoseveralfractions.Thecharacteristicsofthefractionsdependonwhichcrudeoilandrun‐modeareused.Therun‐modedefinesthedivisionbetweenthefractionsandthegenerationofdifferentdistillates.Thisdetermines thecharacteristicsof thedifferent fractions, for instance, in termsof thehydrocarbons thatwill be containedwithin them, viscosity and point of ignition. Given arun‐modeandatypeofcrudeoil,theproportionsbetweenthedistillatesobtainedfromtheprocessarefixed.Hence,ifitisdesiredtogeneratemoreofacertaindistillate,thenmoreoftheotherdistillatesobtainedinthisrun‐modewillalsobegenerated.

During the hydrotreatment process, the distillates obtained from the distillationreceivepropertiessuchasdensity,volatilityflashpoint,pourpointandcolour.Theproductsresulting from this stage correspond to the basic oil products. Some of them are alreadysaleable products, but they can also be used for blending, in order to generate themoresophisticatedspecialityoilproducts.

Theblendingprocessdoesnottakeplaceattherefineries,asthedistillationand

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hydrotreatmentprocessesdo,butinthehubslaterinthesupplychain.Here,thebasicoilsare mixed based on recipes in order to create desired properties for the speciality oilproducts.

StoragelocationsThe saleable products are transported to depots that serve as storage locations.

Thehubsalsoactasstoragelocationsofsaleableproducts.Therefineriesserveasstoragelocations, but only for crude oils. From the refineries, some few products will be sentdirectlytothedepots,whilemostwillgothroughoneofthehubs.

SalesSellersperformtheproducttransactionswiththecustomers,inanumberoflocal

markets. The sellers play an important role in the supply chain, since they decide fromwhichstoragelocationtoshipaproductinordertosatisfyacustomerrequirement.

CustomersanddemandThecustomers forbasicandspecialityoilproducts includeanumberof firms in

construction,roadbuilding,pipecoatingandautomotiveindustries.Dependingonthetypeofproduct,differentpatternsofsalesdemandareobserved;somepresenthighseasonalityin demand, with peaks during the northern hemisphere summer, while the demand forotherproductsismorestable.Inpractice,thereislittleflexibilitytocopewiththeseasonalvariations.Highlevelsofinventoryaretheresultoftryingtocounteracttheseasonality.

TransportationWedistinguishprimaryandsecondarytransportationordistribution.Theprimary

distributioncorrespondstothetransportationofoilswithinthefacilities(refineries,hubsand depots), while the secondary distribution corresponds to the transportation of thesaleableproductstothecustomers.Thetransportationofcrudeoilfromsupplysourcestorefineriesiscarriedoutbyships,thesameasfromrefineriestohubsanddepots.Fromhubstodepotsand fromthese locations to thecustomers, themeansof transportvariesmore,since the volumes are smaller and variable. For the transportation of products to thecustomers,tanktrucksareusedmoreoften.Onoccasion,acombinationofshipandtruckisusedandlessoften,trainandtanktrucksarealsoused.

SupplychainplanningThecurrentplanningandmanagementofthesupplychainisperformedinthree

mainlevels,asFigure1shows.Thestrategicplanningconsidersdecisionsonhowmuchofeachcrudeoilwillbeusedinayearandperformsaggregatedestimationsinordertocheckthat there will be a production balance between the different products. Our researchfocuses on the tactical level, which includes two stages. One stage is performed by theplannersattherefineriesandhubs.Theyperformaproductionplan,consideringahorizonofthreemonths.Decisionsinvolvedintheplanaretheamountofeachproducttoproduceineach locationandtheprimarydistribution.Asecondstage involvestheplanningof thesecondarydistribution,fromhubsanddepotstothecustomers.Thisplanningisbasedonamechanismwith internalpricing.Foreachdepot, eachproduct is givenan internalprice.The sumof this internal price plus the distribution cost from the storage location to thecustomerresultsinafigurethatwecallthevaluechaincost.

ValuechaincostdescriptionAn internal pricing mechanism considers the assignment of premiums to the

sellers, depending on their sales results. One significant part of the premiums is thedifferencebetweenthesalespriceandthevaluechaincost.Inconsequence,foreachsale,amain goal of the sellers is to maximize this difference so as to maximize their ownpremiums.Thevaluechaincostiscalculatedasfollows:

Valuechaincost=Costofgoodssold+Primarydistributioncost+Secondarydistributioncost.

The Cost of goods sold (COGS) includes raw material cost, cost for externally

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procured products, exchange rates and processing costs in refining and blending. ThePrimarydistributioncost isrelatedtothedistributiontostoragefacilities, includingdepotfreight and associated costs of running depots and hubs. The Secondary distribution costincludesthetransportcosttothecustomer,acostforfillingtheproductindrumsandothervariablecosts(suchasimporttaxes).Inpractice,thecompanycentralizesthecalculationofCOGSandthePrimarydistributioncost,resultinginwhatiscalledtheinternalprice.Hence,

Valuechaincost=Internalprice+Secondarydistributioncost.For each sale opportunity the secondary distribution cost is calculated by the

sellerandaddedtothe internalprice,thuscompletingthetotalvaluechaincost.Thesaleprice is based on a negotiation between the seller and the customer. When the sale isrealized,thesellerreceivesapremium,themainshareofwhichisproportionaltothegrossresultofthesale(revenueminustotalvaluechaincost).Thesellerhasachoicefromwhichdepottosupplythecustomerfrom(assumingavailability).Boththeinternalpriceandthesecondary distribution cost depend on which depot the product will be shipped from.Hence, it isnotnecessarilyconvenientforthesellertoordertheproductfromtheclosestdepot(or theonewithcheapest transportationcost),becausethesameproductcanhavedifferentinternalpricesindifferentdepots.Itisalsonotalwaysbestforthesellertoorderfrom the depot with the lowest internal price, because the transportation cost from thedepottothecustomermightbetoohigh.Theideafromthecompanyisthatthismechanismshouldbeselfregulatoryandmakethesellersactinsuchawaythat,whileactingintheirown interests, theyminimize the total long termcost of distribution for the company. Inpractice,however,thiscontrolmechanismisnotexemptfromimperfections.

3.PlanningmodelsInthissectionweformulatelinearprogrammingmodelstorepresentthetactical

planning including the refineriesandechelonsdownstream. InAppendixA,we introducethenotationofsetsandparametersthatareusedthroughtheremainderofthearticle.

FullydecoupledmodelWefirstconsiderafullydecoupledcase,wherethereisnocoordinationbetween

sales and operations units. While the sales units focus on their sales premiums, theoperationsunitsfocusonsupplyingatminimumcost.Forthiscase,wedevelopadecoupledmodel that is composedof two sub‐models: the sales sub‐model and theoperations sub‐model. In the sales sub‐model, the sales units do their planning separate from the otherechelonsofthesupplychain,byconsideringonlythesalespricesandthevaluechaincoststo maximize their premiums. In the operations sub‐model, production and primarydistribution are planned together and the decisions from the sales sub‐model areconsideredasinput.

Theformulationofthesalessub‐modelispresentedinAppendixB.Theobjectivefunction (1)maximizes the total premiumobtained by all the sellers, through thewholeplanninghorizon.Constraint(2)statesthateachsellerwillorderforeachcustomeratmosttheamountthatthiscustomerdemanded,consideringthatitispossibletoservethesamecustomer from different depots. Constraint (3) corresponds to the non‐negativity of thevariables.

The formulation of the operations sub‐model is presented In Appendix D.Objective function (6)minimizes the total cost through thewholeplanninghorizonup tothedepotlevel(i.e.,excludingdistributioncosttothecustomers).Thefirsttermisthecostofprocessingcrudeoilsat therefineries; thesecondtermis thecostofproductionat thehubs; the third term is theprimarydistribution transport costs; thenext three termsarethetotalcostsoftheaverageinventoryperperiod;thelasttermisthecostforunsatisfieddemand. Constraint (7) sets the initial level of inventories of crude oils, basic oils andspecialityoils.Constraint(8)setstheinitialvaluesofcrudeoilsrefinedineachmodeand

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refinery.Constraint(9)setstheinitialvaluesofbasicoilsutilizedineachhubforproducingeachtypeofspecialityoil.Constraint(10)representstheflowconservationofcrudeoilsattherefineries.Constraints(11),(12)and(13)statetheconservationofflowofbasicoilsatthe refineries, hubs and depots, respectively. Constraints (14) and (15) give theconservationofflowofspecialityoilsatthehubsanddepots,respectively.Constraint(16)statesthatthecompanysuppliesatmosttheamountrequestedbythesellers.Notethatthequantities wpagkt ordered by the sellers play the role of demand parameters in theoperations sub‐model (from the solution to the sales sub‐model, the sellershave alreadydecidedonthelocationfromwhichtoorder).Constraints(17)statenon‐negativityofthevariables.

DecoupledmodelwithcoordinationconstraintsIn practice, the company attempts to set conditions in order to achieve certain

balance between production and sales of different products from different depots. InAppendix C, we incorporate two coordination constraints into the sales sub‐model.Constraint (4) sets an upper boundα on the proportion between two different productsthat the same seller can order from the same depot. Constraint (5) imposes amaximumquantityλforeachproductthatcanbeorderedintotalfromsellersinthesameregion.

IntegratedplanningmodelInAppendixE,weproposeamodelthatintegratessalesandoperationsdecisions,

under a same objective function (18) of maximizing the resulting contribution of salesminus variable costs over the planning horizon. The decision on how to fulfil demand ismade centrally, as well as the decisions on production, inventory and primary andsecondarydistribution.

4.Numericalresults

WeprovidenumericalresultsfortheimplementationofthemodelsinaninstancewhosedimensionisgiveninTable1.Weconsideratimehorizonofthreemonthssplit in12‐weekperiodsandweeklydemandforecastsasgiven.Inordertokeepthearticlewithina reasonable extension, we do not provide details on the data. We remark that thedecoupledandintegratedversionsusethesameparametervaluesand,forfaircomparison,we have considered a penalization on unfulfilled demand ψ high enough to satisfy alldemand,thustherevenueresultremainsunaffectedinallcases.Also,thepricesconsideredaresuchthatitisconvenientforthesellerstoacceptalldemandfromtheircustomers.

Table1:Instancedescription.

Refineries:2 Hubs:2 Depots:3 Crudeoils:2 Basicoils:2Specialtyoils:4 Regions:3 Sellers:3 Customers:9 Periods:12

WeusedAMPLtocodethemodelsandCPLEX10.0tosolvethemonanIntelCore2

Duo 2.27GHz processorwith 2GB of RAM. It took less than a second to find the optimalsolution.TheresultsareshowninTable2.

Table2:Costs,revenueandcontributionoftheoptimalsolutiontoinstanceI1.

DM DCαΔ%DM DCλΔ%DM DCαλΔ%DM IntegratedΔ%DMDown‐to‐depotcosts 30,905 30,350 ‐1.80% 30,725 ‐0.58% 30,211 ‐2.25% 30,811 ‐0.31%2rydistributioncosts 4,079 3,930 ‐3.67% 3,997 ‐2.02% 3,856 ‐5.48% 908 ‐77.74%Totalcosts 34,985 34,280 ‐2.01% 34,722 ‐0.75 % 34,067 ‐2.62% 31,719 ‐9.33%Revenue 68,352 68,352 0.00% 68,352 0.00% 68,352 0.00% 68,352 0.00%Contribution 33,367 34,072 2.11% 33,630 0.79% 34,285 2.75% 36,633 9.79%

The second column of Table 2 shows the result obtained for the fully decoupledmodel(DM),expressedinmonetaryunits.Thenextcolumncorrespondstothesolutionofthe decoupled model with the coordination constraint (4). The percentage figurecorrespondstothedifferencebetweenthissolutionandtheDMsolution.Noteareduction

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of 2.01% in total costs is achieved when introducing this coordination constraint. ThecolumnDCλshowstheresultswhen thecoordinationconstraint(5) isconsidered. In thiscase,areductionof0.75%intotalcostsisobtainedcomparedtotheDMcase.ThecolumnDCαλ corresponds to the solution when both coordination constraints (4) and (5) areconsidered simultaneously, leading to a drop of 2.62% in total costs and an increase of2.75% in contribution.The last column shows the results of the integratedmodel,whichoutperformsallpreviousones.Areductionof9.33%intotalcostsandanincreaseof9.79%in contribution are achieved by the integrated compared to the DM case. Note thesecondarydistributioncost fromthe integratedsolution isdramatically lowerthan in theDMcase,witha77.74%reduction.WhencomparedtotheDCαλcase,theintegratedmodelleadstoareductionof6.89%intotalcostsandanincrementof6.85%incontribution. An observation concerns the premium amounts obtained by the sellers in eachmodel.Table3showsandcomparestheseamounts.

Table3:Premiumamountsobtainedbythesellersineachmodel. DM DCαΔ%DM DCλΔ%DM DCαλΔ%DM IntegratedΔ%DMPremiumseller1 607 607 0.00% 600 ‐1.09% 595 ‐1.96% 227 ‐62.62%Premiumseller2 647 626 ‐3.17% 641 ‐0.92% 626 ‐3.17% 453 ‐29.98%Premiumseller3 596 596 0.00% 596 0.00% 596 0.00% 105 ‐82.42%Totalpremium 1,850 1,829 ‐1.11% 1,837 ‐0.68% 1,817 ‐1.75% 784 ‐57.59%Thepremiumsobtainedby the sellers in the integratedmodelexhibithighdifferences incomparisontoall theothercases. Inparticular,whencomparingwiththefullydecoupledcase, in the integrated case the sellers receives between 29.98% and 82.42% lowerpremium.Itisthereforearguablewhether,underthepremiumsobtainedintheintegratedmodel, the sellerswould still be encouraged to sell high volumes or not.However, giventhattheintegratedmodelleadstohighertotalcontribution,findinganothermechanismtoshare the contribution among sellers and the company could keep the incentives for thesellerstoachievehighsalesvolumes.Then,thequestionarisesofhowtofindanallocationsuchthatallstakeholdersaremotivatedtousetheintegratedapproach. Table4showsthepercentageofthecontributionthatthepremiumsofthesellersrepresentineachproblem(i.e.,thepercentagethattheresultsofTable3representoverthecontributionresultspresentedinthelastrowofTable2).Thesharethatthesellersgetintheintegratedsolutionisconsiderablylowerthanintheothercases.

Table4:Premiumofthesellersaspercentageofthetotalcontribution. DM DCα DCλ DCαλ Integrated%P/CSeller1 1.82% 1.78% 1.78% 1.73% 0.62%%P/CSeller2 1.94% 1.84% 1.91% 1.83% 1.24%%P/CSeller3 1.79% 1.75 % 1.77% 1.74% 0.29%%P/CTotal 5.54% 5.37% 5.46% 5.30% 2.14%

Table 5 shows the equivalent premium, that we define as the premium amountobtainedbythesellersconsideringthesamepercentagetheyreceivedintheoriginalcaseof thecorrespondingproblem(DM,DCα,DCλ,DCαλ)butappliedtothecontribution fromtheintegratedsolution.NotethatalltheequivalentpremiumssoobtainedaregreaterthanthepremiumsreceivedbythesellersintheoriginalcasethatwerepresentedinTable3.

Table5:Equivalentpremiumamounts.DM DCα DCλ DCαλ

Equivalentpremiumseller1 666 652 654 635Equivalentpremiumseller2 710 673 698 669Equivalentpremiumseller3 655 641 650 637Totalequivalentpremium 2,031 1,967 2,001 1,942

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Table6showsthecontributionaftertotalpremiumforeachoftheotherproblemsaccordingtodifferentwaysofcalculatingthepremiums.Thecontributionafterpremiumsintheintegratedsolutioncorrespondsto36,633‐784=35,849.

Table6:Contributionafterpremiumallocations.DM DCα DCλ DCαλ

Originalcontributionafterpremium 31,518 32,243 31,793 32,468Integrated'scontrib.afterequivalentpremium 34,602 9.79% 34,666 7.52% 34,632 8.93% 34,691 6.85%Integrated'scontrib.afteridenticalpremium 34,783 10.36% 34,804 7.94% 34,796 9.44% 34,816 7.23%

ThefirstrowofTable6showsthecontributionafterpremiumintheoriginalcase,derivedfromsubtractingthetotalpremiumvaluesofTable3fromthecontributionvaluesofTable2.Thesecondrowshows theresultof thecontribution fromthe integratedcase(36,633) minus the total equivalent premiums of Table 5 and the percentage ofimprovementachievedbyusing theequivalentpremiumrulewith respect to theoriginalcase. The resulting contribution after premium computed by this rule outperforms theoriginalcaseswithanincreaserangingfrom6.85%to9.79%.ThelastrowofTable6showsthe result of the contribution from the integrated case (36,633) minus the identicalpremium, thatwedefineasthesameabsolutepremiumamountobtainedbythesellersintheoriginalcase(thetotalvaluesinTable3).Theresultingcontributionafterpremiuminthiscaseisbetween7.23%and10.36%higherthaninthecorrespondingoriginalcases. Reallocationrulesbasedonequivalentpremiumsand identicalpremiumsare twoexamplesofsimplewaysofreallocatingtheadditionalcontributionamongthesellersandthe company, such that all the sellers arebetteroff than in thedecoupled casewhile thecompanyalsoobtainsabetterresult. Moreanalysisandnumericalresultsreinforcingthecontentsof thisarticlecanbefoundinGuajardoetal.(2012).5.Concludingremarks By using linear programming, we have formulated decoupled and integratedplanningmodelsforadivergentsupplychainofspecialityoilproducts.Theadvantagesofthe integrated approach over the decoupled planning is that the decisions on secondarydistributionaremadetogetherwithpreviousechelonsinthesupplychain,thusprovidingabetter match with production and storage units. These advantages lead the integratedmodel to achieve important decreases in total costs and increases in contribution incomparisontothedecoupledmodels,asweillustratedinthenumericalresults. We also discussed the premiums obtained by the sellers. Theymay obtain lowerpremiums in the integrated solution, thus the practical situation may not allow anintegratedmodeltobeimplemented.Inordertomotivatethesellers,thedevelopmentofarevenue/contribution sharing principle might be required. This has successfully beendevelopedinotherindustries(e.g.Frisketal.,2010),but,toourknowledge,ithasnotbeenaddressedinoilrelatedliterature.Weexploredtworulesforreallocatingcontributionsuchthatallthesellersandthecompanywerebetteroff intheintegratedplanningthaninthedecoupledplanning.Developingpricingmechanismswithallocationofpremiumsispartofour future researchagenda. Ideally, such internalprices shouldgenerate solutionswherethedecoupledandintegratedmodelsprovidethesamesolutions.Intheliterature,thereareseveraldecompositionschemesthatcanbeuseful forthispurpose, forexample,basedonLagrangianrelaxation(LidestamandRönnqvist,2011;PirkulandJayaraman,1998). Wearecurrentlycollaboratingwithamaincompanyinthespecialityoilsindustry.Webelievetheproposedintegratedmodelhasthepotentialtoimprovethecurrentsupplychainplanning,forexample,inhowtoachieveabettermixofproducts,thetimingatwhichtheyareproducedandhowtodistributethem.Afurtherresearchissueisthepossibilityto

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delaymixingofsomeoilstothedepotsandincorporatingtheuncertaintyofdemand.Otherpossible extension is the integration of the procurement decision, which in our modelwouldbepossibletoachievebydefiningη(theamountofcrudeoilincomingtorefineries)as a decision variable instead of a parameter, and adding the corresponding cost in theobjectivefunction.Afurtherissueisthepossibilityofclosingdownoropeningdepots.Thisrequiresthemodeltoincludebinaryvariablesandtoassignfixedcostfortheuseofdepots.

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APPENDIX A: NOTATION

Indexes and setsa ∈ A: set of sellers.j ∈ J : set of geographic regions.a ∈ Lj : set of sellers that belong to region j.k ∈ K: set of customers.i ∈ I: set of crude oils.b ∈ B: set of basic oil products.s ∈ S: set of speciality oil products.p ∈ P : set of saleable products, the union of set B and set S (basic oils and speciality oils).r ∈ R: set of refineries.h ∈ H: set of hubs.d ∈ D: set of depots.f ∈ F : set of storage locations for basic oils, the union of sets R, H and D (refineries, hubs and depots).g ∈ G: set of storage locations for saleable products, the union of set H and set D (hubs and depots).m ∈M : set of run-modes in refining process.t ∈ T : set of periods.

Parametersαappg: maximum proportion between the amount of saleable products p and p possible to assign to sellera from location g.βa: fraction of the revenue that seller a receives as premium.δakpt: demand of customer k to seller a for product p in period t.ηirt: amount of crude oil i incoming to refinery r in period t.γbs: amount of basic oil b necessary for producing one unit of speciality oil s.λpgjt: maximum amount of product p that the company can sell from location g to region j in period t.ρbim: amount of basic oil b generated from one unit of crude oil i at run-mode m.θpkt: sale price of one unit of product p to customer k in period t.ζpgkt: value chain cost of one unit of product p if it is ordered from location g to be sold to customer kin period t.ψpkt: cost for unsatisfied demand of customer k for product p in period t.Ci

rmt: cost of refining one unit of crude oil i in mode m at refinery r in period t.Cs

ht: cost of producing one unit of speciality oil s at hub h in period t.Cfgt: unitary transport cost from location f to location g in period t.Cgkt: unitary transport cost from location g to customer k in period t.Ci

r: inventory cost from storing one unit of crude oil i in refinery r.Cb

f : inventory cost from storing one unit of basic oil b in location f .Cs

g : inventory cost from storing one unit of speciality oil s in location g.

Iir0: initial inventory of crude oil i at refinery r.Ibf0: initial inventory of basic oil b at location f .Isg0: initial inventory of speciality oil s at location g.

Y irm0: initial amount of crude oil i refined in mode m in refinery r.Y hs0: initial amount of speciality oil s produced at hub h.

APPENDIX B: SALES SUB-MODEL

Decision variableswp

agkt: amount of saleable product p ordered by seller a from location g to be shipped to customer k inperiod t.

Max-premium objective function

maxPremium =∑a∈A

∑p∈P

∑g∈G

∑k∈K

∑t∈T

βawpagkt(θpkt − ζpgkt) (1)

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Constraints

∑g∈G

wpagkt ≤ δakpt ∀a ∈ A, p ∈ P, k ∈ K, t ∈ T. (2)

wpagkt ≥ 0 ∀a ∈ A, p ∈ P, g ∈ G, k ∈ K, t ∈ T. (3)

APPENDIX C: COORDINATION CONSTRAINTS

∑k∈K

wpagkt ≤ αappg

∑k∈K

wpagkt ∀a ∈ A, p ∈ P, p ∈ P, g ∈ G, t ∈ T. (4)

∑k∈K

∑a∈Lj

wpagkt ≤ λpgjt ∀j ∈ J, p ∈ P, g ∈ G, t ∈ T. (5)

APPENDIX D: OPERATIONS SUB-MODEL

Decision variablesvpagkt: amount of saleable product p sold from location g to customer k through seller a in period t.

xpfgt: amount of saleable product p transported from location f to location g in period t.

yirmt: amount of crude oil i refined at refinery r in mode m in period t.ysht: amount of speciality oil s produced at hub h in period t.zirt: amount of crude oil i stored in refinery r at the end of period t.zbft: amount of basic oil b stored in location f at the end of period t.zsgt: amount of speciality oil s stored at location g at the end of period t.

Min-cost objective function

minCost =∑m∈M

∑r∈R

∑i∈I

∑t≥1

Cirmty

irmt +

∑h∈H

∑s∈S

∑t≥1

Cshty

sht

+∑p∈P

∑f∈F

∑g∈G

∑t≥1

Cfgtxpfgt +

∑r∈R

∑i∈I

∑t≥1

Cir(z

ir,t−1 + zirt)/2

+∑f∈F

∑b∈B

∑t≥1

Cbf (z

bf,t−1 + zbft)/2 +

∑g∈G

∑s∈S

∑t≥1

Csg(z

sg,t−1 + zsgt)/2

+∑a∈A

∑p∈P

∑k∈K

∑t≥1

ψpkt(δakpt −∑g∈G

vpagkt)

(6)

Constraints

zir0 = Iir0 ∀r ∈ R, i ∈ I; zbf0 = Ibf0 ∀f ∈ F, b ∈ B; zsg0 = Isg0 ∀g ∈ G, s ∈ S. (7)

yirm0 = Y irm0 ∀m ∈M, r ∈ R, i ∈ I. (8)

ysh0 = Y sh0 ∀b ∈ B, h ∈ H, s ∈ S. (9)

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zir,t−1 + ηirt = zirt +∑m∈M

yirmt ∀r ∈ R, i ∈ I, t ≥ 1. (10)

zbr,t−1 +∑i∈I

∑m∈M

ρbimyir,m,t−1 = zbrt +

∑h∈H

xbrht +∑d∈D

xbrdt ∀r ∈ R, b ∈ B, t ≥ 1. (11)

zbh,t−1 +∑r∈R

xbrht = zbht +∑d∈D

xbhdt +∑a∈A

∑k∈K

vbahkt +∑s∈S

γbsysht ∀h ∈ H, b ∈ B, t ≥ 1. (12)

zbd,t−1 +∑r∈R

xbrdt +∑h∈H

xbhdt = zbdt +∑a∈A

∑k∈K

vbadkt ∀d ∈ D, b ∈ B, t ≥ 1. (13)

zsh,t−1 + ysh,t−1 = zsht +∑d∈D

xshdt +∑a∈A

∑k∈K

vsahkt ∀h ∈ H, s ∈ S, t ≥ 1. (14)

zsd,t−1 +∑h∈H

xshdt = zsdt +∑a∈A

∑k∈K

vsadkt ∀d ∈ D, s ∈ S, t ≥ 1. (15)

vpagkt ≤ wpagkt ∀a ∈ A, p ∈ P, g ∈ G, k ∈ K, t ∈ T. (16)

vpagkt ≥ 0 ∀a ∈ A, g ∈ G, k ∈ K, t ∈ T ; xpfgt ≥ 0 ∀p ∈ P, f ∈ F, g ∈ G, t ∈ T ;

yirmt ≥ 0 ∀i ∈ I, r ∈ R,m ∈M, t ∈ T ; ysht ≥ 0 ∀s ∈ S, h ∈ H, t ∈ T ;

zirt ≥ 0 ∀i ∈ I, r ∈ R, t ∈ T ; zbft ≥ 0 ∀b ∈ B, f ∈ F, t ∈ T ; zsgt ≥ 0 ∀s ∈ S, g ∈ G, t ∈ T. (17)

APPENDIX E: INTEGRATED MODEL

We maintain the notation and definitions from the previous section for all parameters, sets and vari-ables, but for explicit differentiation in the decision variable on demand fulfilment between this integratedmodel and the previous cases, instead of using vpagkt we use the notation v

pagkt as decision variable for the

amount of saleable product p sold from location g to customer k through seller a in period t.

Objective Function

maxContribution =∑a∈A

∑p∈P

∑g∈G

∑k∈K

∑t≥1

vpagktθpkt −∑m∈M

∑r∈R

∑i∈I

∑t≥1

Cirmty

irmt

−∑h∈H

∑s∈S

∑t≥1

Cshty

sht −

∑p∈P

∑f∈F

∑g∈G

∑t≥1

Cfgtxpfgt

−∑a∈A

∑g∈G

∑k∈K

∑p∈P

∑t≥1

Cgktvpagkt −

∑r∈R

∑i∈I

∑t≥1

Cir(z

ir,t−1 + zirt)/2

−∑f∈F

∑b∈B

∑t≥1

Cbf (z

bf,t−1 + zbft)/2−

∑g∈G

∑s∈S

∑t≥1

Csg(z

sg,t−1 + zsgt)/2

(18)

Constraints

The constraints of the integrated model are the same as constraints (7) - (11); constraints (12) - (17)are also considered, but now the variables v in the formulations are changed to v and Constraint (16) isre-formulated as

∑g∈G v

pagkt ≤ δakpt ∀a ∈ A, p ∈ P, k ∈ K, t ∈ T .

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integrating production and distribution decisions in … · used and less often, train and tank...

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