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Computers and Operations Research
journalhomepage:www.elsevier.com/locate/cor
On the time-consistent stochastic dominance risk averse measure for tactical supply chain planning under uncertainty R
Laureano F. Escudero
a,∗, Juan Francisco Monge
b, Dolores Romero Morales
ca Estadística e Investigación Operativa,Universidad Rey Juan Carlos, Móstoles (Madrid), Spain
b Centro de Investigación Operativa,Universidad Miguel Hernández, Elche (Alicante), Spain
c Copenhagen Business School,Frederiksberg, Denmark
a rt i c l e i n f o
Article history:
Received 21 February 2016 Revised 23 February 2017 Accepted 13 July 2017 Available online xxx Keywords:
Tactical supply chain planning Nonlinear separable objective function Multistage stochastic integer optimization Risk management
Time-consistency
Stochastic nested decomposition
a b s t r a c t
Inthiswork amodeling framework and asolution approachhave been presented fora multi-period stochasticmixed0–1problemarisingintacticalsupplychainplanning(TSCP).Amultistagescenariotree basedschemeis used torepresent theparameters’uncertainty and develop therelated Deterministic EquivalentModel.Acostriskreductionisperformedbyusinganewtime-consistentriskaversemeasure.
Giventhedimensionsofthisprobleminreal-lifeapplications,adecompositionapproachisproposed.It isbasedonstochasticdynamicprogramming(SDP).Thecomputationalexperienceistwofold,acompar- isonisperformedbetweentheplainuseofacurrentstate-of-the-artmixedintegeroptimizationsolver andtheproposedSDPdecompositionapproachconsideringtheriskneutralversionofthemodelasthe subjectforthebenchmarking.Theadd-valueofthenewriskaversestrategyisconfirmedbythecompu- tationalresultsthatareobtainedusingSDPforbothversionsoftheTSCPmodel,namely,riskneutraland riskaverse.
© 2017 Elsevier Ltd. All rights reserved.
1. Introduction
1.1. Theproblemtobeaddressed,itsimportanceanddifficultyto solve
Thedeterministic versionofthe tacticalsupplychainplanning problem(TSCP)thatisdiscussedinthisworkisbasedonthereal- lifecaseintheassemblysector.Ithasabroadapplicability,specif- ically, in sectors such as car, computer and domestic appliances manufacturing, among others.It is thecase inwhich a company withmultiplerawmaterialsuppliers,plants,products,tiersofpro- ductioninthebillofmaterial(BoM)andmarkets needstosatisfy a product demand vector over a given time horizon. The goalis todeterminearawmaterialsupplyingplanandamasterproduc- tion, inventory and distribution planning that best makes useof theavailableresourcesandtheircapacityextensionacquisitionsin the whole supply chainfor each period ofa given time horizon.
R This research has been partially supported by the projects MTM2015-63710-P (L.F. Escudero) and MTM2016-79765-P (Juan F. Monge) from the Spanish Ministry of Economy, Industry and Competitiveness and the European Regional Development Fund (AEI/FEDER, UE).
∗ Corresponding author.
E-mail addresses: [email protected] (L.F. Escudero), [email protected] (J.F.
Monge), [email protected] (D.R. Morales).
The resources’ best use consists of minimizing the raw material supplyingcommitmentcost,theproductionandinventorycostsin theplants,andtheproductbackloganddemandlostpenalization along thetime horizon. The rawmaterial supplying commitment costisfrequentlymodeledbyapiecewiselinear,concaveandnon- decreasingfunction ofthe totalvolume to commitforthe whole time horizon. Typical typesof constraints (someof them related to either-or decisions) are as follows: Balance equations of end- productsandcomponents,conditionallowerandupperboundsfor rawmaterialsupplyingandproductrelease,resourceconsumption boundsandcapacityextensionacquisitions,andbalanceequations oflostdemandandbacklogging,amongothers.Therearedifferent typesofresourcesatdifferentlevelsforgroupsofconsecutivepe- riods(so-calledstages)alongthetimehorizon.Thecostofthere- sources’capacityextensionacquisitionisexpressedasapiecewise discreteandnondecreasingfunction.Anotherimportantfeatureof theproblemisthatthat nowarehouses areencouragedforinter- mediatecomponentsandproducts stocking,althoughsome stock- ingisallowedintheplants.Eventheburdenofrawmaterialstock- ingisfrequentlytransferredtothesuppliers.
There are many contributions on TSCP for severalvariants to the casepresented above, seethe seminal worksCohen andLee (1989),Escudero (1994)andShapiro (1993),among others.How- ever,giventhevolatilityinthemarketsandthedynamicnatureof theplanningproblem,someparameters,besidesbeingnotknown
http://dx.doi.org/10.1016/j.cor.2017.07.011 0305-0548/© 2017 Elsevier Ltd. All rights reserved.
Pleasecitethisarticleas:L.F.Escuderoetal.,Onthetime-consistentstochasticdominanceriskaversemeasurefortacticalsupplychain
2 L.F. Escudero et al. / Computers and Operations Research 0 0 0 (2017) 1–17 withcertaintywhendecisionsaretobemade,havehighvariabil-
ityin their realizations. Those parameters are the coefficients in the function to minimize (e.g., production cost), the right-hand- sidevector(rhs)ofsome oftheconstraints(e.g.,productdemand andresourceavailability) andtheconstraint matrix(e.g.,product demandlostfraction).Usually,thoseuncertainparametersarerep- resentedby their expectedvalue (EV)and, thus, the parameters’
variabilityisnot considered inthemodel. Hence,the verypopu- larEVstrategy isfrequentlyinadequate forproblemsolving. This workdealswithstochastictacticalsupplychainplanning(STSCP), wheretheavailableinformationabouttheparameters’uncertainty isrepresentedintheformofafinitesetofscenarios.Theyarecon- sideredinthe constraintsof themodel andtheexpected costto beminimized.Theproblem’sformulationisso-calleddeterministic equivalentmodel(DEM),seeine.g.,BirgeandLouveaux(2011)the mainconceptsonstochasticoptimization.Animportantuncertain parameteristhelostdemandfractionfortheproductsineachpe- riodofthe time horizon.Giventhe highcompetitive character of themarkets,thedemandvolatilityandthepotentialunavailability ofrequired resources, by no means thenon-served demand ina periodcanbeconsideredwithcertaintyabacklogforthenextone.
So,thelost demandfractionisuncertain.Onthe other hand,the timelatencyintheavailabilityofrawmaterial andsubassemblies forproductionandend-productavailabilityinthemarkersshould betakenintoaccount,duetoitsstrongimplicationontherelated multistagescenariotree.
1.2.Currenttopicsinstochasticoptimization
In the presence of uncertain parameters, different approaches forsolvingnonlinearseparablemixed0–1problemscanbefound in the literature in the two-stage and multistage settings. A re- centreview ofdecompositionalgorithmsispresentedinAldasoro etal.(2017),mostofthealgorithmsareintendedforproblemsolv- ingwithmoderate modeldimensions. Forbiggerinstances, some typesof scenario cluster decomposition approachescan be used, suchasBranch-and-FixCoordination(Aldasoroetal.,2017;Alonso- Ayusoetal.,2003),two-stageLagrangeandecomposition(Carœand Schultz, 1999), Progressive Hedging algorithm (Gade etal., 2016) andmultistage clusterLagrangean decomposition(Escuderoetal., 2016b), among others. For instances with very large dimensions, such asreal-life STSCPinstances, matheuristic approachesshould beused,asthealgorithmsthatbelongtothestochasticnestedde- compositionmethodology,seeAldasoroetal.(2015),Cristobaletal.
(2009),Escuderoetal.(2015)andZouetal.(2016).Thosestochas- ticoptimizationapproachesdealwiththeminimizationoftheob- jective function expectedvalue alone, so-calledrisk neutral(RN) strategies.However,thevariabilityoftheSTSCPcostoverthesce- nariosisnot completelytakenintoaccount and,then, inducinga negativecostimpact ofthe RNsolutionsonlow-probabilityhigh- costscenarios.
Some approachesthat presentrisk reduction measures are as follows: scenario immunization,see Dembo (1991) and its treat- mentinEscudero(1995),semi-deviations(Ahmed,2006;Ogryczak andRuszczy´nski,1999),min-risk(i.e,excessprobabilities)(Ahmed, 2006; Schultz andTiedemann,2003), value-and-risk (Gaivoronski andPlug, 2005), conditional value-at-risk (CVaR) (Ahmed, 2006;
Pflug and Pichler, 2015b; Rockafellar and Uryasev, 2000; Schultz and Tiedemann, 2006) and stochastic dominance (SD) strategies (Escudero etal., 2016a; Gollmer etal., 2011; 2008). Forcomputa- tionalcomparisonofthetime inconsistentversionsofthosemea- sures,seeAlonso-Ayusoetal.(2014).
Recentstate-of-the-artsurveysonriskmanagement,specifically dealing with supply chains, can be found in Esmaeilikia (2013), Esmaeilikiaetal.(2016a);2016b),Fahimniaetal.(2015),Heckmann etal.(2015)andHoetal.(2015).However,thereareonlyafewre-
centSTSCPworksdealingwithrisk aversemeasures,mainlyCVaR inAlem andMorabito (2013) andNickel etal.(2012),andmean- risk and minmax for robust solutions in Govindan and Fattahi (2017) presenting real-life cases for validating the proposed ap- proaches,amongothers.
Inaddition,mostoftheapproachespresentedabovearetarget- ingstochastictwo-stageproblemsandspecificdecompositionalgo- rithmsaredeveloped.Inastochasticmulti-periodtwo-stageprob- lem,anodeinthescenariotreehasonlyoneimmediatesuccessor and,then, itsparameterscan onlyinfluenceontheparametersof thesamescenario.So,thenon-anticipativity (NA)principle isnot satisfiedforthevariablesinanynodebutthefirstone.
However, in a multistage setting the realizations in any node inthescenariotree(besides thefirstone)mayhavean influence ontheparametersofitssuccessornodesindifferentscenarios.So, thenodesthatbelongtoanystage(butthefirstone)haveacondi- tionalprobabilitybasedontheir ancestorcommonnode(i.e.,itis aMarkovianprocess).Hence,theNA principleshould besatisfied forthedecisionvariables ofthenodesinthetreewitha one-to- onecorrespondencewiththegroupofscenariosthathaveidentical realizationsuptothenode-relatedperiod.
Additionally,a SD risk averse measure is very appropriate for the STSCPcost risk reduction, since it controlstheexpected cost excessoverasetofmodeler-drivenincreasinglycostthresholdsin thescenarios.Thatcontrolismaterializedinapolicyforwhichthe higherthecostthresholdis,thesmallerthemodeler-drivenupper boundsshouldbefortheexpectedcostexcessoverthethreshold andthefailureprobabilitytosatisfyit.
1.3. MaincontributionsofthisworkinSTSCPproblemsolving Themaincontributionsofourworkareaimedtoreducingsome gaps in the literature on STSCP problem solving that have been identifiedabove.Theyareasfollows:
1. Presentingarealisticmulti-periodSTSCPproblemwithamulti- tierBoM fora multi-product,multi-supplierandmulti-market setting, a piecewise linear objective function and uncertainty indynamicproductdemand,demand lostfraction,production costandavailableresourcebasecapacity.
2. Modeling the STSCP via scenario analysis by using a multi- periodmixed0–1fullrecourse DeterministicEquivalentModel (DEM)withS2 setstorepresentthepiecewise lineartermsin theobjectivefunction.Becausethesupplyingorderingtimelag andtransportationtimebetweenvendorsandplants,originand destinationplants,andend-productplantsandmarkets,theor- deringanddeliveringcouldbelongtodifferenttimeperiods.As a consequence, different modeling objects are required, given thevarietyofscenariosthatmayoccurinthetimeintervalbe- tweentheorderingandthedelivering.
3. Modeling the risk reduction of the cost impact of the STSCP solutionsonlow-probabilityhigh-costscenarios(i.e.,theblack swans). For the reasons that will be fully formalized in Section4, the dynamic SD risk averse measure takes benefit from a new version so-called expected conditional stochastic dominance(ECSD),thatisintroducedinthiswork.Itisatime- consistentriskaversemeasure consistingofamixtureoffirst- andsecond-order SD functionals related to a set of modeler- drivenprofiles inthenodesofagivensubset ofperiods.Each profileisincludedbyacostthreshold,aboundontheexpected cost excess over the threshold in the scenario group with a one-to-onecorrespondencewiththenodesoftheselected pe- riods,andaboundontheprobabilityofanyofthosescenarios tofailonsatisfyingthe costthreshold.Therationalebehindit isthat thesolution ofanynode andits successornode setin thescenariotreeshouldnotconsiderthedata,constraintsand
Tier 0 Tier 1 Tier 2 Tier 3 Tier 4
1 4 7 10
13 14
2 5 8 11
15 16
3 6 9
12
17 18
Fig. 1. Bill of material in a supply chain.
variablesthat belong toscenarios that cannotoccur fromthat point.So,ECSDbelongstothetypeofriskaversemeasuresthat havethetimeconsistencypropertyaspresentedinHomem-de MelloandPagnoncelli(2016).
4. Introducingaspecializationofourstochasticnesteddecomposi- tion(SND)approachinCristobaletal.(2009)basedonstochas- ticdynamicprogramming(SDP)fordealingwiththeproposed time-consistentrisk aversemeasure,giventhehighnumberof cross-scenarioconstraintsand,asaconsequence,thenicesce- narionode-basedmodelingstructureislost.
5. Performinga computationalcomparisoninthe multistageset- ting between the time-consistent ECSD cost risk reduction strategyandtheRiskNeutral(RN)one,andbetweenthetime- consistent model and its time-inconsistent counterpart. Addi- tionally, the comparison is also performed betweenthe plain use of a state-of-the-art mixedinteger optimization solver of choiceandourmatheuristicSDPspecializationforSTSCPsolv- ing.
Theremainderofthepaperisorganizedasfollows.Section2in- troduces the TSCP problem and the deterministic data is pre- sented. Section3 presents the uncertain information as well as the RN based model for STSCP. Section4 deals with the time- inconsistentversion ofthestochasticdominance riskaverse mea- sure,so-calledTSD,anditintroducesthetime-consistentECSDver- sion.Section5introducestheSDPalgorithmtodealwiththeECSD measureforSTSCPproblemsolving.Section6reportsthemainre- sults of our computational experience. Finally, some conclusions aredrawn fromthisworkinSection7anditalsooutlinesthefu- tureresearch.
2. TheTSCPproblem
TheTSCPproblemofconcernispresentedindetail.Itisbased onareallifecasebut,infact,ithasabroadapplicability.Forthe sakeofpresentation,letusfixsomenotationtobeusedthrough- outthepaper.Theproblemisformalizednext.
A timehorizon is a set ofconsecutive time periods, whichdo not necessarilyhavethesamelength,wherethe planningis con- sidered.Theperiodsaregroupedintheso-calledstages.
A productis anyitem whose productionvolume, location and schedulingisdecidedbytheplanner.Anend-productisafinalout- put of the supply chain. A subassembly is a product that is as- sembled by the supply chain and is used to produce either an end-product oranother subassembly.Therefore,the termproduct referstobothend-productsandsubassemblies.Acomponentisany storableitemthatisrequiredfortheproduction.
A BoM of a product is the structuring of the set of compo- nentsthat are requiredforits manufacturing/assembly,see Fig.1. The BoM can be described asa set oftiers in the supply chain.
Aso-calledfirsttier componentofa productis acomponentthat is directlyrequired forthe manufacturing/assemblyofthat prod- uct. Note that a component can be used for the manufactur-
ing/assemblyofoneormoreproducts.Thecycletimeofaproduct isthesetofconsecutivetimeperiodsthatarerequiredforitscom- pletion fromits release in themanufacturing/assembly line until itsavailabilityforuse.
Arawmaterialisanystorableitemthatisrequiredtoproduce anyofthe products,butwhoseownBoMis notaconcern tothe planner. Therefore, subassemblies and raw materials are compo- nents. The stock of an item(either a product or a raw material) is its available volume at the end of a given time period. A re- sourceisany non-storableitem that isrequired forthemanufac- turing/assembly ofproducts andits availability is limitedforany periodalong the time horizon, although its basecapacity canbe expanded.
Avendorisanyexternalsourcesupplyingrawmaterials.Itisas- sumedthateachrawmaterialisonlysuppliedbyonevendor.We modelthesupplyingcost foragivenrawmaterialasapiecewise linear,concaveandnondecreasingfunction.Thesupplyingvolume commitment is performedat the beginning of the time horizon, andarewardisreceivedforeachunitinsurplusattheendofthe time horizon.We assume that thestockofraw materials ishan- dled by the vendor.There is usually a time interval between or- deringanddeliveringtherawmaterials,duetohandlingandtrans- portation,amongotherfactors.
A market is any external source having demand for end- products.Itisassumedthat,ifnecessary,end-productsarestocked attheshowcentersinthemarkets.
Thesupplychainconsistsofplants,whereproduction,assembly andstoragetakeplace,thatneedtosatisfythedemandofmarkets.
It is assumedthat each product is only manufactured/assembled inone plant,butplantscanhandlemorethanone product.Three typesof resources are considered forthe plants.In one type the extensionofthebasecapacitycanbeperformedatthefirstperiod andcanbeusedalongthewholetime horizon.Inanothertype it canbe extendedatthe firstperiodofanystage andcanbe used untilthe last period ofthe stage.And in thetheir type the base capacitycan beextendedindependentlyforeachperiod.Thecost ofcapacityexpansionforanyresourceismodeled asapiecewise discretenondecreasingfunction.
Themain TSCPdecisions are thoseinvolving theraw material volume whose supplying is to be committed from vendors, the product volume to be processed and the stock volume of prod- uctstobestoredintheplants,thecomponentvolumetobetrans- ported from the origin plants to the destination plants, the re- sources’ capacity expansion, if any, and the end-product volume tobe shipped fromtheplants to themarkets. While thesupply- ing commitmentofraw materialis done atthe beginningof the planninghorizon,theremainingdecisionshavetobetakenineach period.
ThenotationusedforthedeterministicTSCPisasfollows.
Sets
T, periodsalongthetimehorizon,lexicographi- callyordered,such thatt=1isthefirstone andT=
|
T|
isthelastperiod.E, stages inthe time horizon, lexicographically ordered,such that Te is theset ofconsecu- tiveperiodsinstagee,whereTe∩Te=∅for e,e∈E:e=e.Note:E=
|
E|
isthelaststage.JE, end-products.
JS, subassemblies.
J =JE
JS products.
DSj, demand markets for end-product j, for j∈ JE.Note:Withoutlossofgenerality,itisas- sumedthatnosubassemblyhasexternalde- mand.
4 L.F. Escudero et al. / Computers and Operations Research 0 0 0 (2017) 1–17
DS =
j∈JEDSj.
IR, rawmaterials.
I, components.Note:I=JS
IR.
Ij, firsttiercomponentsforproductj,for j∈J. R1, resources such that the capacity extension is performedat the first period of the time horizon, ifany,andit can be used untilthe endofthehorizon.
Re, resourcessuch thatthecapacityextensionis performedatthefirstperiodofstagee,ifany, anditcan onlybeused untilthelast period ofthestage,fore∈E.
R2, resourcessuch thatthecapacityextensionis performed at any period, if any, and it can onlybeusedatthatperiod.
R=R1∪e∈ERe∪R2. Note:Allthesetsaredisjoint.
Fi, segments in the cost function for supplying commitmentrawmateriali,fori∈IR. Hr, segments for the capacity expansion of re-
sourcer,forr∈R. BoMparameters
cj, cycletimeofproductj,forj∈J,andnumberofperiods inthetimeintervalfromorderingrawmaterialj untilit isreadyforstartingitsdelivering,for j∈IR.
Ni,j, volume offirst tiercomponenti that is requiredbythe BoMtoproduceoneunitofproductj,fori∈Ij,j∈J. Timelatencyforavailability
τ
i, number of time periods required to deliver component i from its origin plant to its destination plant, for i∈ JSIR.
τ
d, deliverytime,i.e.,numberoftimeperiodstodeliverend- productjtomarketdafteritscompletion,ford∈DSj,j∈ JE.Productionandproductstockbounds
Xi,t,Xi, minimum and maximum volume of raw material i that can be ordered at time period t, respectively, if any,fori∈IR,t∈T,andminimumandmaximumvol- ume whose manufacturing/assembly can be released forproduct i at time periodt,respectively, ifany, for i∈J,t∈T. Note: The upper bound Xi is stationary across the time horizon, but the assumption can be easilyremoved.
Sj,t,Sj, minimumandmaximumvolumeofitemj thatcanbe instockat(the endof)timeperiodt,respectively,for j∈J
IR,t∈T.
Sed,t,Sed, minimumandmaximumvolumeofend-productjthat can be in stock in (the show center of) market d at (theendof)timeperiodt,respectively,ford∈DSj, j∈ JE,t∈T.
Resourcecoefficients
or,j, amount of resource r required to produce one unit of productj,forr∈R, j∈J.
Or, basecapacityofresourcer,forr∈R1∪e∈ERe.
Ohr, base capacityexpansion of resource rrelated to its hth segment,forh∈Hr,r∈R1∪e∈ERe.
Ohr,t, base capacityexpansion of resource rrelated to its hth segmentattimeperiodt,forh∈Hr,r∈R2,t∈T. Costcoefficients
(SCif,Yif), datadefiningtheupperboundpairofthefthsegment ofthesupplyingcommitmentcostfunctionofrawma- teriali, for f∈Fi,i∈IR, whereSCif is the cost, and
Yif is the related volume. Note: Usually SC1i =0 and Y1i =0.
fcj, fixed cost incurred when product j is manufac- tured/assembled, for j∈J, and fixed cost incurred whenrawmaterialjisordered,for j∈IR.
f chr, fixed costassociatedwiththehthsegmentoftheca- pacity expansion cost function of resource r, for h∈ Hr,r∈R1.
f chr,t, fixed costassociatedwiththehthsegmentoftheca- pacity expansion cost function of resource r at time period t, forh∈Hr,r∈Re,e∈E, wheret isthe first periodinsetTe.
f chr,t, fixed costassociatedwiththehthsegmentoftheca- pacity expansion cost function of resource r at time periodt,forh∈Hr,r∈R2,t∈T.
hcj, unit holdingcostofitemj atanytimeperiod,for j∈ J
IR.
hced, unit holdingcostofend-productj inmarket datany timeperiod,ford∈DSj, j∈JE.
cd,t, (resp.bcd,t),unitlostdemand(resp.unit backlogging) costofend-productjformarketdattimeperiodt,for d∈DSj,j∈JE,t∈T.
ui, unitrewardofthesurplusvolumeofrawmaterialiat theendofthetimehorizon,fori∈IR.
Inthenextsection,theSTSCPmodelingtostudyispresentedas well asthechallengesposedby thetopology ofthesupplychain andthetimingoftheeventsinthere.
3. Uncertaintyintacticalsupplychainplanning 3.1. Introduction
The main uncertain parameters to consider in a realistic case alongagiventimehorizonareproductioncost,available resource capacity, product demand and demand lost fraction. As usual, it isassumedthat therealizationoftheuncertain parameterstakes place at periods. One of the important decisions to be made in TSCPconsistsofdeterminingtheorderingtimeperiodforrawmate- rial supplyingandsubassemblyreadinessalong thetime horizon.
There is usually a time interval betweenordering and delivering thecomponents(i.e.,making themavailable)inthesupplychain, duetohandlingoftheorderingandtransportation timesthat are required,amongotherfactors.Inthecasethatthetimeintervalis not subjectto specific constraintsand costs relatedto the items’
transportation andother factors, a deterministic model can con- siderthattheorderingtimeisthesameoneasthedeliveringtime.
However, inthestochasticsettingtherelatedtime intervalisim- portant, since the productionand market environments can vary alongthetimeintervalbetweenbothevents,seebelow.Thisisone ofthechallengestobefacedinthiswork.
Let scenario be one realization of the uncertain parameters along theperiods ofthetime horizon, andascenario groupfora given periodis the set ofscenarios withthe same realizationof theuncertainparametersuptotheperiod.Thisinformationstruc- ture can be visualized asthe tree depictedin Fig.2, where each root-to-leafpath representsone specificscenarioandcorresponds to one realizationof the whole set of the uncertain parameters.
Eachnodeinthetreecanbeassociatedwithascenariogroupina one-to-onecorrespondence,such thattwoscenariosbelongtothe same groupin a givenperiod,provided that they havethe same realization of the uncertain parameters up to the period. Letus point out that it is beyond the scope of this work to present a methodologyformulti-periodscenariotreegenerationandreduc- tion;seee.g.,Dupaˇcová etal.(2000),HeitschandRömisch(2009),
t = 1 t = 2 t = 3 t = 4
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17
Ω = Ω1= {10, 11, . . . , 17}; Ω2= {10, 11, 12}
G = {1, . . . , 17}; G2= {2, 3, 4}
A17= {1, 4, 9, 17}; σ(9) = 4
Fig. 2. Non-symmetric multi-period scenario tree.
Hoyland etal. (2003), Leövey and Römisch (2015) and Pflug and Pichler(2015a)forwaystoperformbothtasks.
Sets(lexicographicallyorderedinthescenariotree)
, scenarios.
G, nodesinscenariotree.
Ge⊆ G, nodesinstagee,fore∈E.
Gt⊆ G, nodes in period t, for t∈T. Note: By construction,
|
G1|
=1and1∈G1.g⊆, scenariosingroupg,forg∈G.
A˜g, node g and its ancestors, for g∈G. Notice that A˜1 is onlyincludedbynode1and,then,1∈G1.
Ag⊆ ˜Ag, nodesinA˜gwhosevariableshavenonzeroelementsin constraintsrelatedtonodeg(includingitself),forg∈G.
Sg, node gand itssuccessors,for g∈G.Note:Sg=
{
g}
forg∈GT.
Qe⊆ Ge, root nodes in the scenario subtrees starting in stage e, for e∈E. As an illustration, let in Fig.2 the set of stagesE=
{
1,2}
withT1={
1,2}
andT2={
3,4}
,such thatG1={
1,2,3,4}
andG2={
5,· · · ,17}
.So,e.g.,Q2={
5,6,7,8,9}
,wherenodeq=7istherootnodeofthe scenariosubtreewhosenodesareS2={
7,13,14}
.Otherelementsinthescenariotree
wg, weight factor representingthe likelihood that is associ- atedwithnodeg,forg∈G.Note:wg=
ω∈gwω,where wω gives the modeler-driven likelihood associated with scenario
ω
(wherew=g forg∈GT such thatitisasin- gletonset),forω
∈,andg∈Gtwg=1∀
t∈T.σ
(g), immediate ancestor node of node g, for g∈G. By con- struction,σ
(1)=0.t(g), periodtowhichnodegbelongsto,so,g∈Gt(g). Uncertainparametersinnodeg∈G
Dgd, demand ofend-productj frommarket dattime period t(g),ford∈DSj, j∈JE.
fdg, lostfractionofnon-servedaccumulateddemandofend- productjfrommarketd,ford∈DSj, j∈JE.
Ogr, basecapacityofresourcer,forr∈R2. pcgj, unitproductioncostforproductj,for j∈J.
Theso-calledriskneutralmeasureinSTSCPcanberepresented by the multi-period stochastic mixed 0–1 model (1)-(16) that is presentednext.
3.2.Decisionvariables
Continuous variables for raw material i∈IR. yi, volume of rawmaterialithat iscommittedatthebeginningofthe timehorizon.
sci, costofsupplyingcommittedvolumeyiofrawmateriali. xgj, volumeofrawmaterialjthatisorderedat(thebeginning
of)timeperiodt(g),for j∈IR.
zωi, surplusvolume ofrawmaterialiattheendofthetime horizon,for
ω
∈.β
if, [0,1]value definingthefractionusedofthefthsegment ofthepiecewiselinear,concave andnondecreasingsup- plyingcommitmentcostfunctionofrawmateriali.Itbe- longs tothe S2 setsfor modelingthat type of function.AnS2 set Bealeand Forrest (1976)is an ordered set of nonnegativevariables
{ β
if∀
f∈Fi}
,suchthatthesumof thevariablesinthesetmustbe1andnomorethantwo membersmaybenonzerowiththefurtherconditionthat ifthereareasmanyastwotheymustbeadjacent.Continuousvariablesinnodeg∈G
xgj, released volume of product j in the manufactur- ing/assemblylineat(thebeginning of) timeperiod t(g), forj∈J.
ygd, shippedvolumeofend-productjtomarketdat(theend of)timeperiodt(g),ford∈DSj, j∈JE.
sgj, stockvolumeof item j at(the endof) time period t(g), forj∈J
IR.
segd, stockvolumeofend-productjinmarketdat(theendof) timeperiodt(g),ford∈DSj, j∈JE.
gd, lostdemandofend-productjinmarketdattimeperiod t(g),ford∈DSj, j∈JE.
bgd, demandbacklogofend-productjinmarketdat(theend of)timeperiodt(g),ford∈DSj, j∈JE.
Note:Itisassumedthats0j,se0d,b0d areinputdataforanyj,d. 0–1variablesforresources’capacityexpansionandproductrelease
α
rh, itsvalue is 1ifthe hthsegment ofthecapacity expan- sion of resource r is performed at (the beginning of) thefirst periodofthetimehorizonandotherwise0,for h∈Hr,r∈R1.Note1:Itcanbeuseduntilthelastperiod ofthetimehorizon.Note2:Byconstruction,α
r0=1.α
rq,h, itsvalueis1ifthehthsegmentofthecapacityexpansion ofresource r is performedat (the beginning of) period t(q)ofnodeq instagee(q),forh∈Hr,r∈Re,q∈Qe,e∈6 L.F. Escudero et al. / Computers and Operations Research 0 0 0 (2017) 1–17 E.Note1:Itcanbeuseduntilthelastperiodofthestage.
Note2:Byconstruction,
α
rq,0=1.α
rg,h, itsvalueis1ifthehthsegmentofthecapacityexpansion ofresourceristoperformedat(thebeginningof)period t(g) of node g andotherwise 0, forh∈Hr,r∈R2,g∈G.Note1: Itcanonly beusedatthesame period.Note2:
Byconstruction,
α
rg,0=1.δ
gj, its value is1ifproduct jis releasedinthemanufactur- ing/assemblylinefor j∈J,andit is1ifrawmaterialj isorderedtovendorfor j∈IR,bothat(thebeginningof) timeperiodt(g)ofnodegandotherwise,0,forg∈G.3.3.Deterministicequivalentmodel,DEM
Themixed0–1DEMcanbeexpressedasfollows, min
i∈IR
sci+
r∈R1
h∈Hr
f chr
α
rh+
e∈E
q∈Qe
r∈Re
h∈Hr
wqfchr,t(q)
α
rq,h+g∈G
r∈R2
h∈Hr
wgfchr,t(g)
α
rg,h+
g∈G
wg
j∈J
pcgjxgj+
j∈J IR
hcjsgj+
d∈DS
(
hcedsegd+cd,t(g)gd +bcd,t(g)bgd)
+
g∈G
wg
j∈J IR
f cj
δ
gj−i∈IR
ω∈
wωuizωi (1)
subjectto sci≡
f∈Fi
SCif
β
if∀
i∈IRyi≡
f∈Fi
Yif
β
if∀
i∈IR( β
if∀
f∈Fi)
is anS2setforrawmateriali∀
i∈IRyi=
g∈Aω
xgi+zωi
∀ ω
∈,i∈IR.
(2) sσ (j g)+xqj=
d∈DSj
ygd+sgj
∀
j∈JE,g∈G,whereq∈Ag:t
(
q)
=t(
g)
− cj+1 (3)sσ (i g)+xqi =
j∈J:i∈Ij
Ni, jxhj+sgi
∀
h∈Sg:t(
h)
=t(
g)
+
τ
i,g∈G,i∈I,whereq∈Ag:t
(
q)
=t(
g)
− ci+1 (4) Sj,t(g)≤ sgj≤ Sj∀
j∈J∪IR,g∈G (5)Xj,t(g)
δ
gj≤ xgj≤ Xjδ
gj,δ
gj∈{
0,1} ∀
j∈J∪IR,g∈G (6)
j∈J
or, jxgj≤ Or+
h∈Hr
Ohr
α
rh∀
r∈R1,g∈G (7)α
hr ≤α
rh−1andα
hr ∈{
0,1} ∀
h∈Hr,r∈R1 (8)
j∈J
or, jxgj≤ Or+
h∈Hr
Ohr
α
qr,h∀
r∈Re,g∈Sq∩Ge,q∈Qe,e∈E(9)
α
qr,h≤α
qr,h−1 andα
qr,h∈{
0,1} ∀
h∈Hr,r∈Re,q∈Qe,e∈E(10)
j∈J
or, jxgj≤ Ogr+
h∈Hr
Ohr,t(g)
α
gr,h∀
r∈R2,g∈G (11)α
rg,h≤α
rg,h−1andα
rg,h∈{
0,1} ∀
h∈Hr,r∈R2,g∈G (12)seσ (d g)+yqd+gd+bgd=bσ (dg)+Dgd+segd
∀
d∈DS,g∈G.where q∈Ag:t
(
q)
=t(
g)
−τ
ed (13)gd≡ fdg
(
bσ (d g)+Dgd− seσ (d g)− yqd)
≥ 0∀
d∈DS,g∈G (14)bgd≡
(
1− fdg)(
bσ (d g)+Dgd− seσ (d g)− yqd)
≥ 0∀
d∈DS,g∈G (15)Sed,t(g)≤ segd≤ Sed
∀
d∈DS,g∈G, (16) whereitisassumedthatxqj=0andyqd=0fort(q)≤ 0,andxhj=0 fort(h)>T.Objectivefunction
Itconsistsofminimizingthesupplyingcommitmentrawmate- rial cost,the expectedcost ofproduction andstockofthe items, resourcecapacityexpansion,productstockatthemarketsandthe penaltyduetolostdemandandbacklogginginthescenariosalong the time horizon, and minus the rewarding of the raw material surplusvolumeattheendofthetimehorizon.
RawmaterialsupplyingcommitmentcostfunctionbasedonS2sets Theconstraintsystem(2)istheS2setsobjectmodelingforthe rawmaterial supplying commitmentcost function. Inaddition, it imposes that the total supplying volume ofa givenraw material alongthetimehorizon,
g∈Aωxgi,foranyscenario
ω
∈,plusthe surplus,zωi,isequaltothesupplyingvolume,yi,thatiscommitted atthebeginningofthetimehorizon.Balanceequationsofend-productsandcomponents
Constraint(3)isthebalanceequationforend-productjinnode g,whereitscycletimecjneedstobetakenintoaccount.Without lossofgeneralityitisassumedthatnodischargesaremadeduring theprocess.Constraint (4)isthebalanceequation forcomponent iinnodeg.Notethatthisinvolvesthesetofproducts(subassem- blies andend-products)
{
j∈J :i∈Ij}
that have componenti inthe firsttier of their BoM.Note: The cycletime andthe product releasetime are taken intoaccount. Constraint (5)impose upper andlowerboundsonstocklevels.
Upperandlowerboundsforrawmaterialsupplyingandproduct release
TheConstraint(6)forcetheconditionalbounds.
Resourceconsumption
Theconstraintsystem(7),(9),(11) boundstheresources’con- sumption,allowing theextensionofthecapacityoftheresources, whose costs are expressed by a piecewise discrete nondecreas- ing functions. Given thecharacter of thecapacity expansion cost functions, the variableupperbounding systemgivenby (8),(10), (12)forcesthelexicographicorderforthesegments’scommitment intheresources.
Balanceequationsofproductdemand,lostdemandandbacklogging Constraint (13)balance thetotal demandof marketdin node g,wheretheamountshippedtothemarketsistakenintoaccount as well asthe inventory, backlogged demand,and thepossibility oflostdemand.Constraints(14)and(15)ensurethatthelostand backlogged demand are well defined. Constraint (16) impose up- perandlowerboundsonstocklevelsin(the showcentersof)the markets.Notethatyqd isshippedattimeperiodt(q)=t(g)−
τ
dto satisfy the demandof end-productj frommarket din thenodes insetSq∩Gt(q)+τd.Observealsothatifthestockisnotenoughfor serving the backlog anddemand, then theshipment to the mar- ketwillbeused.Ifthisisnotstill enough,then itispartiallylost andpartiallybackloggedto beserved attheimmediatesuccessor scenarionodes.3.4. SynthesizedriskneutralSTSCPmodel
The model (1)-(16) can be synthesized in the following risk neutral (RN) model, whose aim consists of minimizing the ex- pectedtacticalsupplychainplanningcostalone,
zRN=min
g∈G
wg
(
agxg+bgyg)
[2mm]s.t.
q∈Ag
(
Agqxq+Bgqyq)
=hg∀
g∈Gxg∈
{
0,1}
nx(g),yg∈IRny(g)∀
g∈G,(17)
where,foreachnodegsuchthatg∈G,xgandygdenotethenx(g)- andny(g)-vectorsofthe0–1andcontinuousvariables,respectively, agandbgaretherelatedobjectivefunctionvectors,AgqandBgqare theconstraintsmatriceswhosenonzerocoefficientsgivetheinflu- enceoftheappropriate0–1andcontinuousvariablesfromances- tornodeqinnodeg,respectively,andhgistherhsvector.
4. Time-basedstochasticdominancemeasures
Given the highvariability on the uncertainparameters of the STSCP model (1)-(16), the time-inconsistent and time-consistent versions of risk averse measure are presented. It consists of a stochastic dominance functional as a multi-period oriented mix- tureofthefirst-andsecond-orderstochastic dominance(SD)risk aversemeasuresconsideredinSection1.
4.1. Time-inconsistentTSDmeasure
To perform the TSCP cost risk reduction, the multi-period stochastic dominance risk averse measure proposed in Escudero etal.(2016a)isconsidered.Inthefollowing,wewillrefer toitas TSD. Forthatpurpose,theoperational costfunction valueis con- trolledforasubsetofperiods,say,T˜⊆ T,suchthatasetofcost- relatedprofiles,sayPt,isconsidered,fort∈T˜.Aprofilep∈Pt is definedbya3-tuple(cp,ep,
θ
p),wherecpisthethresholdonthe TSCPcostup toanynodeinGt;ep istheupperboundoftheex- pectedTSCPcostexcessoverthresholdcp thatisallowed inthose nodes;andθ
p istheupperbound ontheprobability offailingto satisfythethreshold.Additionally,letctdenotethemaximumTSCP costthatisallowedforanyofthosenodes.Letegpdenotethenon-negativevariablethat definesthediffer- ence (if it is positive)between the TSCP cost up to node g and thresholdcp,and
θ
gpisa0–1variablesuchthatitsvalueis1ifthe costuptonodegisgreaterthanthethresholdandotherwise,0.TheTSDmeasurecanbeexpressed
zTSD=min
g∈G
wg(agxg+bgyg)
[2mm]s.t.
q∈Ag
(Agqxq+Bgqyq)=hg ∀g∈G
q∈A˜g
(aqxq+bqyq)− epg≤ cp ∀g∈Gt,p∈Pt,t∈T˜ 0≤ epg≤(ct− cp)θgp,θgp∈{0,1} ∀g∈Gt,p∈Pt,t∈T˜
g∈Gt
wgegp≤ ep ∀p∈Pt,t∈T˜
g∈Gt
wgθgp≤θp ∀p∈Pt,t∈T˜ xg∈{0,1}nx(g),yg∈IRny(g) ∀g∈G.
(18)
Theconceptoftheexpectedcostexcessoftheobjectivevalue on satisfyingagiventhresholdhaveitsrootsintheIntegratedChance ConstraintsconceptintroducedinKlein (1986),seealsoKleinand vanderVlerk(2006).
FollowingtherationaleinPflug(2000)fortheCVaRmeasure,it canbeshownthatTSDisacoherentriskmeasure,accordingtothe standardssetin Artzneretal.(1999); 2007). In other words,TSD satisfiestheproperties:translationinvariance,positivehomogene- ity,monotonicityandconvexity. Inaddition,itcan alsobeshown Homem-deMelloandPagnoncelli(2016)thatthetime-consistency ofTSD, asdefinedbelow, dependsontheboundsep and
θ
p. Thetightertheseboundsare,thelower theconsistencyprobability of TSD.Thus,ingeneral,itisatime-inconsistentmeasure.
4.2.Time-consistentECSDmeasure
The TSD risk reduction has an interesting add-value, since it controlstheperiod-basedpeakoftheTSCPcostingiveninterme- diate periods while minimizing the expected cost for the whole timehorizon,suchthatthelongerthetimehorizon,themoreuse- fulnessofthatstrategy. However,itstimeinconsistencyisadraw- back. Notice that the outlooks of the scenarios that do not be- longto the groupswitha one-to-onecorrespondencewithsome nodes in the scenario tree (and, then, they are not to occur in those situations)have influence on the solutionof those groups.
Toprovethestatementit canbeobserved inmodel(18)that the solution fornode g∈Gt,t∈T is affected by the satisfaction of therisk constraintsystemrelatedto periodtover allof itsnodes, fort∈T˜:t≥ t.
Inthis section,the expectedconditional stochastic dominance (ECSD)measure is introduced. Here again a subset of periods T˜ is given. However, the set of profiles, denoted by Pg, are asso- ciated with node g, where t(g)∈T˜, instead of been associated withperiodt(g).A profile p∈Pg,t(g)∈T˜ consistsofthe3-tuple (cp,ep,
θ
p),wherenowcp istheTSCPcostthresholdtobesatis- fiedbyanyscenarioingroupgfromperiod1uptothelastone, T;epistheupperboundoftheexpectedcostexcessover thresh- oldcpthatisallowedforthatgroupofscenarios;andθ
pistheup-perboundontheprobabilityoffailingtosatisfythethreshold.The profilesetPgcouldbedifferentforthenodesginsetGt,t∈T˜.(It issuggestedtousethisdeviceforproblemswheretherearehigh differencesinthe objectivefunction coefficientsbetweenthesce- nariosthatbelongtodifferentgroupsinthesameperiod,atleast).
Additionally,letcgdenotethemaximumTSCPcostthatisallowed foranyofscenario inset g.Note that periodT cannot be inT˜ and,forpracticalreasons,thecardinalityofT˜ shouldbesmall.
Letthecontinuousvariableeωp andthe0–1variable
