CTL.SC0x - Supply Chain Analytics - Amazon Web … Key Concept... · ... Supply Chain Analytics Key Concepts・MITx ... and the SCOR model (Supply Chain ... The Supply Chain Operations

V1 Spring 2017・CTL.SC0x – Supply Chain Analytics Key Concepts・MITx MicroMasters in Supply Chain Management MIT Center for Transportation & Logistics・Cambridge, MA 02142 USA ・[email protected] This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. 1

CTL.SC0x-SupplyChainAnalytics

KeyConceptsDocumentThisdocumentcontainstheKeyConceptsfortheSC0xcourse.Thedocumentisbasedonapastrunofthecourseanddoesnotincludeallthematerialthatwillbeincludedinthiscourse;pleasecheckbackfornewerversionsofthedocumentthroughoutthecourse.Thesearemeanttocomplement,notreplace,thelessonvideosandslides.Theyareintendedtobereferencesforyoutousegoingforwardandarebasedontheassumptionthatyouhavelearnedtheconceptsandcompletedthepracticeproblems.ThedraftwascreatedbyDr.AlexisBatemanintheSpringof2017.Thisisadraftofthematerial,sopleasepostanysuggestions,corrections,orrecommendationstotheDiscussionForumunderthetopicthread“KeyConceptDocumentsImprovements.”Thanks,ChrisCaplice,EvaPonceandtheSC0xTeachingCommunitySpring2017v1


TableofContentsSupplyChainIntro................................................................................................................................3

Models,Algebra,&Functions...............................................................................................................6Models.....................................................................................................................................................6Functions..................................................................................................................................................6QuadraticFunctions.................................................................................................................................7ConvexityandContinuity.........................................................................................................................8

Optimization........................................................................................................................................9UnconstrainedOptimization....................................................................................................................9ConstrainedOptimization......................................................................................................................11LinearPrograms.....................................................................................................................................11IntegerandMixedIntegerPrograms.....................................................................................................14

AdvancedOptimization......................................................................................................................18NetworkModels.....................................................................................................................................18Non-LinearOptimization........................................................................................................................20

AlgorithmsandApproximations.........................................................................................................23Algorithms..............................................................................................................................................23ShortestPathProblem...........................................................................................................................24VehicleRoutingProblem........................................................................................................................25ApproximationMethods........................................................................................................................29

DistributionsandProbability..............................................................................................................36Probability..............................................................................................................................................36Summarystatistics.................................................................................................................................37ProbabilityDistributions.........................................................................................................................40

Regression..........................................................................................................................................47MultipleRandomVariables....................................................................................................................47InferenceTesting....................................................................................................................................49OrdinaryLeastSquaresLinearRegression.............................................................................................54

Simulation..........................................................................................................................................58Simulation..............................................................................................................................................58StepsinaSimulationStudy....................................................................................................................58

References.........................................................................................................................................61


SupplyChainIntro

SummarySupplyChainBasicsisanoverviewoftheconceptsofSupplyChainManagementandlogistics.Itdemonstratesthatproductsupplychainsasvariedasbananastowomen’sshoestocementhavecommonsupplychainelements.Therearemanydefinitionsofsupplychainmanagement.Butultimatelysupplychainsarethephysical,financial,andinformationflowbetweentradingpartnersthatultimatelyfulfillacustomerrequest.Theprimarypurposeofanysupplychainistosatisfyacustomer’sneedattheendofthesupplychain.Essentiallysupplychainsseektomaximizethetotalvaluegeneratedasdefinedas:theamountthecustomerpaysminusthecostoffulfillingtheneedalongtheentiresupplychain.Allsupplychainsincludemultiplefirms.

KeyConceptsWhileSupplyChainManagementisanewterm(firstcoinedin1982byKeithOliverfromBoozAllenHamiltoninaninterviewwiththeFinancialTimes),theconceptsareancientanddatebacktoancientRome.Theterm“logistics”hasitsrootsintheRomanmilitary.Additionaldefinitions:

• Logisticsinvolves…“managingtheflowofinformation,cashandideasthroughthecoordinationofsupplychainprocessesandthroughthestrategicadditionofplace,periodandpatternvalues”–MITCenterforTransportationandLogistics

• “SupplyChainManagementdealswiththemanagementofmaterials,informationandfinancialflowsinanetworkconsistingofsuppliers,manufacturers,distributors,andcustomers”-StanfordSupplyChainForum

• “Callitdistributionorlogisticsorsupplychainmanagement.Bywhatevernameitisthesinuous,gritty,andcumbersomeprocessbywhichcompaniesmovematerials,partsandproductstocustomers”–Fortune1994

Logisticsvs.SupplyChainManagementAccordingtotheCouncilofSupplyChainManagementProfessionals…

• Logisticsmanagementisthatpartofsupplychainmanagementthatplans,implements,andcontrolstheefficient,effectiveforwardandreverseflowandstorageofgoods,servicesandrelatedinformationbetweenthepointoforiginandthepointofconsumptioninordertomeetcustomers'requirements.

• Supplychainmanagementencompassestheplanningandmanagementofallactivitiesinvolvedinsourcingandprocurement,conversion,andalllogisticsmanagementactivities.Importantly,italsoincludescoordinationandcollaborationwithchannelpartners,whichcanbesuppliers,intermediaries,thirdpartyserviceproviders,andcustomers.Inessence,supplychainmanagementintegratessupplyanddemandmanagementwithinandacrosscompanies.


SupplyChainPerspectivesSupplychainscanbeviewedinmanydifferentperspectivesincludingprocesscycles(Chopra&Meindl2013)andtheSCORmodel(SupplyChainCouncil).TheSupplyChainProcesshasfourPrimaryCycles:CustomerOrderCycle,ReplenishmentCycle,ManufacturingCycle,andProcurementCycle,Noteverysupplychaincontainsallfourcycles.TheSupplyChainOperationsReference(SCOR)Modelisanotherusefulperspective.Itshowsthefourmajoroperationsinasupplychain:source,make,deliver,plan,andreturn.(SeeFigurebelow)

Additionalperspectivesinclude:

• GeographicMaps-showingorigins,destinations,andthephysicalroutes.• FlowDiagrams–showingtheflowofmaterials,information,andfinancebetween

echelons.• Macro-ProcessorSoftware–dividingthesupplychainsintothreekeyareasof

management:SupplierRelationship,Internal,andCustomerRelationship.• TraditionalFunctionalRoles–wheresupplychainsaredividedintoseparatefunctional

roles(Procurement,InventoryControl,Warehousing,MaterialsHandling,OrderProcessing,Transportation,CustomerService,Planning,etc.).Thisishowmostcompaniesareorganized.

• SystemsPerspective–wheretheactionsfromonefunctionareshowntoimpact(andbeimpactedby)otherfunctions.Theideaisthatyouneedtomanagetheentiresystemratherthantheindividualsiloedfunctions.Asoneexpandsthescopeofmanagement,therearemoreopportunitiesforimprovement,butthecomplexityincreasesdramatically.

SupplyChainasaSystemItisusefultothinkofthesupplychainasacompletesystem.Thismeansoneshould:

Figure:SCORModel.Source:SupplyChainCouncil


• Looktomaximizevalueacrossthesupplychainratherthanaspecificfunctionsuchastransportation.

• Notethatwhilethisincreasesthepotentialforimprovement,complexityandcoordinationrequirementsincreaseaswell.

• Recognizenewchallengessuchas:o Metrics—howwillthisnewsystembemeasured?o Politicsandpower—whogainsandlosesinfluence,andwhataretheeffectso Visibility—wheredataisstoredandwhohasaccesso Uncertainty—compoundsunknownssuchasleadtimes,customerdemand,

andmanufacturingyieldo GlobalOperations—mostfirmssourceandsellacrosstheglobe

Supplychainsmustadaptbyactingasbothabridgeandashockabsorbertoconnectfunctionsaswellasneutralizedisruptions.

LearningObjectives• Gainmultipleperspectivesofsupplychainstoincludeprocessandsystemviews.• Identifyphysical,financial,andinformationflowsinherenttosupplychains.• Recognizethatallsupplychainsaredifferent,buthavecommonfeatures.• Understandimportanceofanalyticalmodelstosupportsupplychaindecision-making.


Models,Algebra,&Functions

SummaryThisreviewprovidesanoverviewofthebuildingblockstotheanalyticalmodelsusedfrequentlyinsupplychainmanagementfordecision-making.Eachmodelservesarole;italldependsonhowthetechniquesmatchwithneed.First,aclassificationofthetypesofmodelsoffersperspectivesonwhentheuseamodelandwhattypeofoutputtheygenerate.Second,areviewofthemaincomponentsofmodels,beginningwithanoverviewoftypesoffunctions,thequadraticandhowtofinditsroot(s),logarithms,multivariatefunctions,andthepropertiesoffunctions.These“basics”willbeusedcontinuouslythroughouttheremainderofthecourses.

KeyConcepts

ModelsDecision-makingisatthecoreofsupplychainmanagement.Analyticalmodelscanaidindecision-makingtoquestionssuchas“whattransportationoptionshouldIuse?”or“HowmuchinventoryshouldIhave?”Theycanbeclassifiedintoseveralcategoriesbasedondegreeofabstraction,speed,andcost.Modelscanbefurthercategorizedintothreecategoriesontheirapproach:

• Descriptive–whathashappened?• Predictive–whatcouldhappen?• Prescriptive–Whatshouldwedo?

FunctionsFunctionsareonethemainpartsofamodel.Theyare“arelationbetweenasetofinputsandasetofpermissibleoutputswiththepropertythateachinputisrelatedtoexactlyoneoutput.”(Wikipedia)

y=f(x)

LinearFunctionsWithLinearfunctions,“ychangeslinearlywithx.”Agraphofalinearfunctionisastraightlineandhasoneindependentvariableandonedependentvariable.(Seefigurebelow)


Figure:ComponentsofaLinearFunction**Typicallyconstantsaredenotesbylettersfromthestartofthealphabet,variablesarelettersfromtheendofthealphabet.

QuadraticFunctions

Figure:ComponentsofaQuadraticFunction Figure:GraphofQuadraticFunctions

RootsofQuadraticAroot,orsolution,satisfiesthequadratic.Theequationcanhave2,1,or0roots.Therootsmustbearealnumber.Therearetwomethodsforfindingroots:

Quadratic Functions

• When a>0, the function is convex (or concave up)• When a<0, the function is concave down

13

y =ax2 +bx+c

dependent variable

This is the function, f(x).

constantsindependent variable

Parabola - Polynomial function of degree 2 where a, b, and c are numbers and a≠0

a>0

x

y

x

y a<0

Aquadraticfunctiontakestheformofy=ax2+bx+c,wherea,b,andcarenumbersandarenotequaltozero.Inagraphofaquadraticfunction,thegraphisacurvethatiscalledaparabola,formingtheshapeofa“U”.(SeeFigures)

Quadraticequation

Factoring:Findr1andr2suchthatax2+bx+c=a(x-r1)(x-r2)


OtherCommonFunctionalFormsPowerFunctionApowerfunctionisafunctionwherea≠ zero,isaconstant,andbisarealnumber.Theshapeofthecurveisdictatedbythevalueofb.

y=f(x)=axbExponentialFunctionsExponentialfunctionshaveveryfastgrowth.Inexponentialfunctions,thevariableisthepower.

y=abx

MultivariateFunctionsFunctionwithmorethanoneindependentxvariable(x1,x2,x3).

ConvexityandContinuityPropertiesoffunctions:

• Convexity:Doesthefunction“holdwater”?• Continuity:Functioniscontinuousifyoucandrawitwithoutliftingpenfrompaper!

LearningObjectives• Recognizedecision-makingiscoretosupplychainmanagement.• Gainperspectivesonwhentouseanalyticalmodels.• Understandbuildingblocksthatserveasthefoundationtoanalyticalmodels.

Euler’snumber,ore,isaconstantnumberthatisthebaseofthenaturallogarithm.e=2.7182818…

Y=ex

Logarithms:Alogarithmisaquantityrepresentingthepowertowhichafixednumbermustberaisedtoproduceagivennumber.Itistheinversefunctionofanexponential.


Optimization

SummaryThisisanintroductionandoverviewofoptimization.Itstartswithanoverviewofunconstrainedoptimizationandhowtofindextremepointsolutions,keepinginmindfirstorderandsecondorderconditions.Italsoreviewsrulesinfunctionssuchasthepowerrule.Nextthelessonreviewconstrainedoptimizationthatsharessimilarobjectivesofunconstrainedoptimizationbutaddsadditionaldecisionvariablesandconstraintsonresources.Tosolveconstrainedoptimizationproblems,thelessonintroducesmathematicalprogramsthatarewidelyusedinsupplychainformanypracticessuchasdesigningnetworks,planningproduction,selectingtransportationproviders,allocatinginventory,schedulingportandterminaloperations,fulfillingorders,etc.Theoverviewoflinearprogrammingincludeshowtoformulatetheproblem,howtographicallyrepresentthem,andhowtoanalyzethesolutionandconductasensitivityanalysis.Inrealsupplychains,youcannot.5bananasinanorderorshipment.Thismeansthatwemustaddadditionalconstraintsforintegerprogrammingwhereeitherallofthedecisionvariablesmustbeintegers,orinamixedintegerprogrammingwheresome,butnotall,variablesarerestrictedtobeaninteger.Wereviewthetypesofnumbersyouwillencounter.Thenweintroduceintegerprogramsandhowtheyaredifferent.Wethenreviewthestepstoformulatinganintegerprogramandconcludewithconditionsforworkingwithbinaryvariables.

KeyConcepts

UnconstrainedOptimizationUnconstrainedoptimizationconsiderstheproblemofminimizingormaximizinganobjectivefunctionthatdependsonrealvariableswithnorestrictionsontheirvalues.Extremepoints

• Extremepointsarewhenafunctiontakesonanextremevalue-avaluethatismuchsmallerorlargerincomparisontonearbyvaluesofthefunction.

• Theyaretypicallyaminoramax(eitherglobalorlocal),orinflectionpoints.• Extremepointoccurwhereslope(orrateofchange)offunction=0.• TestforGlobalvs.Local

o Globalmin/max–forwholerangeo Localmin/max–onlyincertainarea

FindingExtremePointSolutionsUsedifferentialcalculustofindextremepointsolutions,lookforwhereslopeisequaltozero


Tofindtheextremepoint,thereisathree-stepprocess:

1. Takethefirstderivativeofyourfunction2. Setitequaltozero,and3. SolveforX*,thevalueofxatextremepoint.

ThisiscalledtheFirstOrderCondition.Instantaneousslope(orfirstderivative)occurswhen:

• dy/dxisthecommonform,wheredmeanstherateofchange.

TheProductRule:Iffunctionisconstant,itdoesn’thaveanyeffect.

y = f (x) = a à y ' = f '(x) = 0 PowerRuleiscommonlyusedforfindingderivativesofcomplexfunctions.

y = f (x) = axn à y ' = f '(x) = anxn−1

FirstandSecondOrderConditionsInordertodeterminex*atthemax/minofanunconstrainedfunction

• FirstOrder(necessary)condition–theslopemustbe0f’(x*)=0

• Secondorder(sufficiency)condition-determineswhereextremepointisminormaxbytakingthesecondderivative,f”(x).

o Iff”(x)>0extremepointisalocalmino Iff”(x)<0extremepointisalocalmaxo Iff”(x)=0itisinconclusive

• Specialcaseso Iff(x)isconvex–>globalmino Iff(x)isconcave–>globalmax

δ (delta)=rateofchange.


ConstrainedOptimizationSimilaritieswithunconstrainedoptimization

• Requiresaprescriptivemodel• Usesanobjectivefunction• Solutionisanextremevalue

Differences• Multipledecisionvariables• Constraintsonresources

MathProgramming:Mathprogrammingisapowerfulfamilyofoptimizationmethodsthatiswidelyusedinsupplychainanalytics.Itisreadilyavailableinsoftwaretools,butisonlyasgoodasthedatainput.Itisthebestwaytoidentifythe“best”solutionunderlimitedresources.SometypesofmathprogramminginSCM:

• LinearProgramming(LPs)• IntegerProgramming• MixedIntegerandLinearProgramming(MILPs)• Non-linearProgramming(NLPs)

LinearPrograms1.DecisionVariables

• Whatareyoutryingtodecide?• Whataretheirupperorlowerbounds?

2.Formulateobjectivefunction• Whatarewetryingtominimizeormaximize?• Mustincludethedecisionvariablesandtheformofthefunctiondeterminesthe

approach(linearforLP)3.Formulateeachconstraint

• Whatismyfeasibleregion?Whataremylimits?• Mustincludethedecisionvariableandwillalmostalwaysbelinearfunctions


SolutionThesolutionofalinearprogramwillalwaysbeina“corner”oftheFeasibleRegion:

• Linearconstraintsformaconvexfeasibleregion.• Theobjectivefunctiondeterminesinwhichcorneristhesolution.

TheFeasibleRegionisdefinedbytheconstraintsandtheboundsonthedecisionvariables(SeeFigure).

AnalysisoftheResultsSometimestheoriginalquestionistheleastinterestingone,itisoftenmoreinterestingtodivealittledeeperintothestructureoftheproblem.AdditionalQuestions:

• AmIusingallofmyresources?• WheredoIhaveslack?

Figure:LinearProgramExample

Figure:GraphicalRepresentationoftheFeasibleRegion


• WhereIamconstrained?• Howrobustismysolution?

SensitivityAnalysis:whathappenswhendatavaluesarechanged.

• ShadowPriceorDualValueofConstraint:Whatisthemarginalgainintheprofitabilityforanincreaseofoneontherighthandsideoftheconstraint?

• SlackConstraint–Foragivensolution,aconstraintisnotbindingiftotalvalueofthelefthandsideisnotequaltotherighthandsidevalue.Otherwiseitisabindingconstraint

• BindingConstraint–Aconstraintisbindingifchangingitalsochangestheoptimalsolution

AnomaliesinLinearProgramming

• AlternativeorMultipleOptimalSolutions(seeFigure)

• RedundantConstraints-DoesnoteffecttheFeasibleRegion;itisredundant.• Infeasibility-TherearenopointsintheFeasibleRegion;constraintsmaketheproblem

infeasible.

Figure:AlternativeofMultipleOptimalSolutions


IntegerandMixedIntegerProgramsAlthoughinsomecasesalinearprogramcanprovideanoptimalsolution,inmanyitcannot.Forexampleinwarehouselocationselection,batchorders,orscheduling,fractionalanswersarenotacceptable.Inaddition,theoptimalsolutioncannotalwaysbefoundbyroundingthelinearprogramsolution.Thisiswhereintegerprogramsareimportant.However,integerprogramsolutionsareneverbetterthanalinearprogramsolution,theylowertheobjectivefunction.Ingeneral,formulatingintegerprogramsismuchharderthanformulatinglinearprogram.

• Toidentifythesolutioninintegerprograms–theFeasibleRegionbecomesacollectionofpoints,itisnolongeraconvexhull(seeFigure)

• Inaddition,cannotrelyon“corner”solutionsanymore–thesolutionspaceismuchbigger

FormulatingIntegerProgramsToformulateanintegerprogram,wefollowthesameapproachforformulatinglinearprograms–variables,constraintsandobjective.Theonlysignificantchangeistoformulatingintegerprogramsisinthedefinitionofthevariables.SeeexampleformulationinFigurebelowwithintegerspecification.

Numbers• N=Natural,WholeorCounting

numbers1,2,3,4• Z-Integers=-3,-2,-1,0,1,2,3• Q=RationalNumber,continuous

numbers=Anyfactionofintegers½,-5/9

• R=RealNumbers=allRationalandIrrationalNumbers,ex:e,pie,e

• BinaryIntegers=0,1

Massenumeration-Unlikelinearprograms,integerprogramscanonlytakeafinitenumberofintegervalues.Thismeansthatoneapproachistoenumerateallofthesepossibilities–calculatingtheobjectivefunctionateachoneandchoosingtheonewiththeoptimalvalue.Astheproblemsizeincreases,thisapproachisnotfeasible.


BinaryVariablesSupposeyouhadthefollowingformulationofaminimizationproblemsubjecttocapacityatplantsandmeetingdemandforindividualproducts:

Wecouldaddbinaryvariablestothisformulationtobeabletomodelseveraldifferentlogicalconditions.Binaryvariablesareintegervariablesthatcanonlytakethevaluesof0or1.Generally,apositivedecision(dosomething)isrepresentedby1andthenegativedecision(donothing)isrepresentedbythevalueof0.Introducingabinaryvariabletothisformulation,wewouldhave:

Min z = cij xijj∑i∑s.t.

xiji∑ ≤C j ∀j

xijj∑ ≥ Di ∀i

xij ≥ 0 ∀ij

Maxz(XHL,XSL)=8XHL+20XSL

XHL+XSL≤11

3XHL+2XSL≤30

XHL+3XSL≤28

XHL,XSL≥0Integers

s.t.

Plant

Add.A

Add.B

Figure:Formulatinganintegerprogram

where:xij=Numberofunitsofproductimadeinplantjcij=CostperunitofproductimadeatplantjCj=CapacityinunitsatplantjDi=Demandforproductiinunits


Notethatthenotonlywehaveaddedthebinaryvariableintheobjectivefunction,wehavealsoaddedanewconstraint(thethirdone).Thisisknownasalinkingconstraintoralogicalconstraint.Itisrequiredtoenforceanif-thenconditioninthemodel.Anypositivevalueofxijwillforcetheyjvariabletobeequaltoone.The“M”valueisabignumber–itshouldbeassmallaspossible,butatleastasbigasthevaluesofwhatsumofthexij’scanbe.Therearealsomoretechnicaltricksthatcanbeusedtotightenthisformulation.WecanalsointroduceEither/OrConditions-wherethereisachoicebetweentwoconstraints,onlyoneofwhichhastohold;itensuresaminimumlevel,Lj,ifyj=1.

xiji∑ −My j ≤ 0 ∀j xiji∑ − Lj y j ≥ 0 ∀j

Forexample:xiji∑ ≤C j ∀j

xijj∑ ≥ Di ∀i

xiji∑ −My j ≤ 0 ∀j

WeneedtoaddaconstraintthatensuresthatifweDOuseplantj,thatthevolumeisbetweentheminimumallowablelevel,Lj,andthemaximumcapacity,Cj.ThisissometimescalledanEither-Orcondition.

Min z = cij xijj∑i∑ + f j y jj∑s.t.

xiji∑ ≤C j ∀j

xijj∑ ≥ Di ∀i

xiji∑ −My j ≤ 0 ∀j

xij ≥ 0 ∀ij

y j ={0,1}

where:xij=Numberofunitsofproductimadeinplantjyj=1ifplantjisopened;=0otherwisecij=Costperunitofproductimadeatplantjfj=FixedcostforproducingatplantjCj=CapacityinunitsatplantjDi=Demandforproductiinunits M=abignumber(suchasCjinthiscase)

where: xij=Numberofunitsofproductimadeinplantj yj=1ifplantjisopened;=0o.w. M=abignumber(suchasCjinthiscase) Cj=Maximumcapacityinunitsatplantj Lj=Minimumlevelofproductionatplantj Di=Demandforproductiinunits


xiji∑ ≤My j ∀j

xiji∑ ≥ Lj y j ∀j

where: xij=Numberofunitsofproductimadeinplantj yj=1ifplantjisopened;=0o.w. M=abignumber(suchasCjinthiscase) Lj=Minimumlevelofproductionatplantj

Finally,wecancreateaSelectFromCondition–thatallowsustoselectthebestNchoices.Notethatthiscanbeformulatedas“chooseatleastN”or“choosenomorethanN”bychangingtheinequalitysignonthesecondconstraint.

xiji∑ −My j ≤ 0 ∀j y jj∑ ≤ N

DifferencebetweenLinearProgramsandIntegerPrograms/MixedIntegerPrograms• Integerprogramsaremuchhardertosolvesincethesolutionspaceexpands.

o Forlinearprograms,acorrectformulationisgenerallyagoodformulation.o Forintegerprogramsacorrectformulationisnecessarybutnotsufficientto

guaranteesolvability.• Integerprogramsrequiresolvingmultiplelinearprogramstoestablishbounds–relaxing

theIntegerconstraints.• Whileitseemsthemoststraightforwardapproach,youoftencan’tjust“round”the

linearprogramssolution–itmightnotbefeasible.• Whenusinginteger(notbinary)variables,solvethelinearprogramfirsttoseeifitis

sufficient.

LearningObjectives• Learntheroleofoptimization.• Understandhowtooptimizeinunconstrainedconditions.• IdentifyhowtofindExtremePointSolutions.• Understandhowtoformulateproblemswithdecisionvariablesandresourceconstraints

andgraphicallypresentthem.• Reviewhowtointerpretresultsandconductsensitivityanalysis.• Understandthedifferent“types”ofnumbersandhowtheychangetheapproachto

problems.• Reviewtheapproachofformulatingintegerandmixedintegerprogramproblemsand

solvingthem.


AdvancedOptimization

SummaryThisreviewconcludesthelearningportiononoptimizationwithanoverviewofsomefrequentlyusedadvancedoptimizationmodels.Networkmodelsarekeyforsupplychainprofessionals.Thereviewbeginsbyfirstdefiningtheterminologyusedfrequentlyinthesenetworks.ItthenintroducescommonnetworkproblemsincludingtheShortestPath,TravelingSalesmanProblem(TSP),andFlowproblems.Theseareusedfrequentinsupplychainmanagementandunderstandingwhentheyariseandhowtosolvethemisessential.Wethenintroducenon-linearoptimization,highlightingitsdifferenceswithlinearprogramming,andanoverviewofhowtosolvenon-linearproblems.Thereviewconcludeswithpracticalrecommendationsofforconductingoptimization,emphasizingthatsupplychainprofessionalsshould:knowtheirproblem,theirteamandtheirtool.

KeyConcepts

NetworkModelsNetworkTerminology

• Nodeorvertices–apoint(facility,DC,plant,region)• Arcoredge–linkbetweentwonodes(roads,flows,etc.)maybedirectional• Networkorgraph–acollectionofnodesandarcs

CommonNetworkProblemsShortestPath–Easy&fasttosolve(LPorspecialalgorithms)Resultofshortestpartproblemisusedasthebaseofalotofotheranalysis.Itconnectsphysicaltooperationalnetwork.

• Given:One,origin,onedestination.• Find:Shortestpathfromsingleorigintosingledestination,• Challenges:Timeordistance?Impactofcongestionorweather?Howfrequentlyshould

weupdatethenetwork?• Integralityisguaranteed.• Caveat:Otherspecializedalgorithmsleveragethenetworkstructuretosolvemuch

faster.TravelingSalesmanProblem(TSP)–Hardtosolve(heuristics)

• Given:Oneorigin,manydestinations,sequentialstops,onevehicle.• Objective:Startingfromanoriginnode,findtheminimumdistancerequiredtovisit

eachnodeonceandonlyoneandreturntotheorigin.


• Importance:TSPisatthecoreofallvehicleroutingproblems;localroutingandlastmiledeliveriesarebothcommonandimportant.

• Challenges:Itisexceptionallyhardtosolveexactly,duetoitssize;possiblesolutionsincreaseexponentiallywithnumberofnodes.

• Primaryapproach:specialalgorithmsforexactsolutions(smallerproblems)–Heuristics(manyavailable).

o Twoexamples:NearestNeighbor,CheapestInsertion

FlowProblems(Transportation&Transshipment)–Widelyused(MILPs)• Given:Multiplesupplyanddemandnodeswithfixedcostsandcapacitiesonnodes

and/orarcs.• Objective:Findtheminimumcostflowofproductfromsupplynodesthatsatisfy

demandatdestinationnodes.• Importance:Transportationproblemsareeverywhere;transshipmentproblemsareat

theheartoflargersupplychainnetworkdesignmodels.Intransportationproblems,shipmentsarebetweentwonodes.Fortransshipmentproblems,shipmentsmaygothroughintermediarynodes,possiblychangingmodeoftransport.Transshipmentproblemscanbeconvertedintotransportationproblems.

• Challenges:datarequirementscanbeextensive;difficulttodrawthelineon“realism”vs.“practicality”.

• Primaryapproaches:mixedintegerlinearprograms;somesimulation–usuallyafteroptimization.

NearestNeighborHeuristicThisalgorithmstartswiththesalesmanatarandomcityandvisitsthenearestcityuntilallhavebeenvisited.Ityieldsashorttour,buttypicallynottheoptimalone.

• Selectanynodetobetheactivenode.

• Connecttheactivenodetotheclosestunconnectednode;makethatthenewactivenode.

• Iftherearemoreunconnectednodesgotostep2,otherwiseconnecttotheoriginandend.

CheapestInsertionHeuristicOneapproachtotheTSPistostartwithasubtour–tourofsmallsubsetsofnodes,andextendthistourbyinsertingtheremainingnodesoneaftertheotheruntilallnodeshavebeeninserted.Thereareseveraldecisionstobemadeinhowtoconstructtheinitialtour,howtochoosenextnodetobeinserted,wheretoinsertchosennode.

• Formasubtourfromtheconvexhull.

• Addtothetourtheunconnectednodethatincreasesthecosttheleast;continueuntilallnodesareconnected.


Non-LinearOptimizationAnonlinearprogramissimilartoalinearprograminthatitiscomposedofanobjectivefunction,generalconstraints,andvariablebounds.Thedifferenceisthatanonlinearprogramincludesatleastonenonlinearfunction,whichcouldbetheobjectivefunction,orsomeoralloftheconstraints.

• Manysystemsarenonlinear–importanttoknowhowtohandlethem.• Hardertosolvethanlinearprograms–lose‘corner’solutions(SeeFigure).• Shapeofobjectivefunctionandconstraintsdictateapproachanddifficulty.

Figure:ExampleofNLPwithlinearconstraintandnon-linearobjectivefunction(z=xy).

PracticalTipsforOptimizationinPractice• Knowyourproblem:

o Determiningwhattosolveisrarelyreadilyapparentoragreeduponbyallstakeholders.

o Establishanddocumenttheover-ridingobjectiveofaprojectearlyon.• Levelofdetail&scopeofmodel:

o Modelscannotfullyrepresentreality,modelswillneverrepresentallfactors,determineproblemboundariesanddataaggregationlevels.

• Inputdata:o Collectingdataishardest,leastappreciated,andmosttimeconsumingtaskinan

optimizationproject.o Datanevercompleteclean,ortotallycorrect.o Everhourspentondatacollection,cleaningandverificationsavesdayslateron

intheproject.• SensitivityandRobustnessAnalysis

o Thesearealldeterministicmodels–dataassumedperfect&unchanging.o Optimizationmodelswilldoanythingforadollar,yuan,peso,euro,etc.


o Runmultiple“what-if”scenarioschanginguncertaininputvaluesandtestingdifferentconditions.

• Modelsvs.People(modelsdon’tmakedecisions,peopledo!)o Optimizationmodelsaregoodatmakingtrade-offsbetweencomplicated

optionsanduncoveringunexpectedinsightsandsolutions.o Peoplearegoodat:

§ Consideringintangibleandnon-quantifiablefactors,§ Identifyingunderlyingpatterns,and§ Miningpreviousexperienceandinsights.§ ModelsshouldbeusedforDecisionSUPPORTnotforthedecision.

LearningObjectives• Introductiontoadvancedoptimizationmethods.• Understandtheconditionsandwhentoapplynetworkmodels.• Differentiatenonlinearoptimizationandwhenitshouldbeused.• Reviewrecommendationsforrunningoptimizationinpractice–emphasizing

importanceofknowingtheproblem,teamandtool.



AlgorithmsandApproximations

SummaryInthislessonwewillbereviewingAlgorithmsandapproximations.Thefirsthalfofthelessonwillbeareviewofalgorithms–whichyoutechnicallyhavealreadybeintroducedto,butperhapsnotintheseterms.Wewillbereviewingthebasicsofalgorithms,theircomponents,andhowtheyareusedinoureverydayproblemsolving!TodemonstratethesewewillbelookingatafewcommonsupplychainproblemssuchastheShortestPathproblem,TravelingSalesmanProblem,andVehicleRoutingProblemwhileapplyingtheappropriatealgorithmtosolvethem.Inthisnextpartofthelesswewillbereviewingapproximations.Approximationsaregoodfirststepsinsolvingaproblembecausetheyrequireminimaldata,allowforfastsensitivityanalysis,andenablequickscopingofthesolutionspace.Recognizinghowtouseapproximationmethodsareimportantinsupplychainmanagementbecausecommonlyoptimalsolutionsrequirelargeamountsofdataandaretimeconsumingtosolve.Soifthatlevelofgranularityisnotneeded,approximationmethodscanprovideabasistoworkfromandtoseewhetherfurtheranalysisisneeded.

AlgorithmsAlgorithm-aprocessorsetofrulestobefollowedincalculationsorotherproblem-solvingoperations,especiallybyacomputer.DesiredPropertiesofanAlgorithm

• shouldbeunambiguous• requireadefinedsetofinputs• produceadefinedsetofoutputs• shouldterminateandproducearesult,alwaysstoppingafterafinitetime.

AlgorithmExample:find_maxInputs:

• L=arrayofNintegervariables• v(i)=valueoftheithvariableinthelist

Algorithm:1. setmax=0andi=12. selectitemiinthelist3. ifv(i)>max,thensetmax=v(i)4. ifi<N,thenseti=i+1andgotostep2,otherwisegotostep55. end

Output:


maximumvalueinarrayL(max)

ShortestPathProblemObjective:FindtheshortestpathinanetworkbetweentwonodesImportance:Itsresultisusedasbaseforotheranalysis,andconnectsphysicaltooperationalnetworkPrimaryapproaches:

• StandardLinearProgramming(LP)• SpecializedAlgorithms(Dijkstra’sAlgorithm)

Minimize: cijj∑i∑ xij

Subject to:x jii∑ =1 ∀ j = s

x jii∑ − xiji∑ = 0 ∀ j ≠ s, j ≠ t

xiji∑ =1 ∀ j = t

xij ≥ 0

where: xij=Numberofunitsflowingfromnodeitonodej cij=Costperunitforflowfromnodeitonodej s=Sourcenode–whereflowstarts t=Terminalnode–whereflowendsDijkstra’sAlgorithmDjikstra'salgorithm(namedafteritsdiscover,E.W.Dijkstra)solvestheproblemoffindingtheshortestpathfromapointinagraph(thesource)toadestination.L(j)=lengthofpathfromsourcenodestonodejP(j)=precedingnodeforjintheshortestpathS(j)=1ifnodejhasbeenvisited,=0otherwised(ij)=distanceorcostfromnodeitonodejInputs:

• Connectedgraphwithnodesandarcswithpositivecosts,d(ij)• Source(s)andTerminal(t)nodes


Algorithm:1. forallnodesingraph,setL()=∞,P()=Null,S()=02. setstoi,S(i)=1,andL(i)=03. Forallnodes,j,directlyconnected(adjacent)tonodei;ifL(j)>L(i)+d(ij),thensetL(j)=

L(i)+d(ij)andP(j)=i4. ForallnodeswhereS()=0,selectthenodewithlowestL()andsetittoi,setS(i)=15. Isthisnodet,theterminalnode?Ifso,gotoend.Ifnot,gotostep36. end–returnL(t)

Output:L(t)andParrayTofindpathfromstot,startattheend.

• FindP(t)–sayitisj• Ifj=sourcenode,stop,otherwise,findP(j)• keeptracingprecedingnodesuntilyoureachsourcenode

TravelingSalesmanProblem(TSP)Startingfromanoriginnode,findtheminimumdistancerequiredtovisiteachnodeonceandonlyonceandreturntotheorigin.NearestNeighborHeuristic

1. Selectanynodetobetheactivenode2. Connecttheactivenodetotheclosestunconnectednode,makethatthenewactive

node.3. Iftherearemoreunconnectednodesgotostep2,otherwiseconnecttothestarting

nodeandend.

2-OptHeuristic1. Identifypairsofarcs(i-jandk-l),whered(ij)+d(kl)>d(ik)+d(jl)–usuallywherethey

cross2. Selectthepairwiththelargestdifference,andre-connectthearcs(i-kandj-l)3. Continueuntiltherearenomorecrossedarcs.

VehicleRoutingProblemFindminimumcosttoursfromsingleorigintomultipledestinationswithvaryingdemandusingmultiplecapacitatedvehicles.


Heuristics• RoutefirstClustersecond

o AnyearlierTSPheuristiccanbeused

Optimal

• MixedIntegerLinearProgram(MILP)• Selectoptimalroutesfrompotentialset

ClusterfirstRoutesecond

• SweepAlgorithm• Savings(Clarke-Wright)

VRPSweepHeuristicFindminimumcosttoursfromDCto10destinationswithdemandasshownusingupto4vehiclesofcapacityof200units.SweepHeuristic

1. FormarayfromtheDCandselectanangleanddirection(CWvsCCW)tostart2. Selectanewvehicle,j,thatisempty,wj=0,andhascapacity,cj.3. Rotatetherayinselecteddirectionuntilithitsacustomernode,i,orreachesthe

startingpoint(gotostep5).4. Ifthedemandati(Di)pluscurrentloadalreadyinthevehicle(wj)islessthanthevehicle

capacity,addittothevehicle,wj=Di+wjandgotostep3.Otherwise,closethisvehicle,andgotostep2tostartanewtour.

5. SolvetheTSPforeachindependentvehicletour.


Differentstartingpointsanddirectionscanyielddifferentsolutions!Besttouseavarietyorastackofheuristics.

Clark-WrightSavingsAlgorithmTheClarkeandWrightsavingsalgorithmisoneofthemostknownheuristicforVRP.Itappliestoproblemsforwhichthenumberofvehiclesisnotfixed(itisadecisionvariable)

• Startwithacompletesolution(outandback)• Identifynodestolinktoformacommontourbycalculatingthesavings:

Example:joiningnode1&2intoasingletour

Currenttourscost=2cO1+2cO2Joinedtourcosts=cO1+c12+c2O

So,if2cO1+2cO2>cO1+c12+c2OthenjointhemThatis:cO1+c2O–c12>0

• Thissavingsvaluecanbecalculatedforeverypairofnodes• Runthroughthenodespairingtheoneswiththehighestsavingsfirst• Needtomakesurevehiclecapacityisnotviolated• Also,“interiortour”nodescannotbeadded–mustbeonend


SavingsHeuristic1. Calculatesavingssi,j=cO,i+cO,j-ci,jforeverypair(i,j)ofdemandnodes.2. Rankandprocessthesavingssi,jindescendingorderofmagnitude.3. Forthesavingssi,junderconsideration,includearc(i,j)inarouteonlyif:

• Norouteorvehicleconstraintswillbeviolatedbyaddingitinarouteand• Nodesiandjarefirstorlastnodesto/fromtheoriginintheircurrentroute.

4. Ifthesavingslisthasnotbeenexhausted,returntoStep3,processingthenextentryinthelist;otherwise,stop.

SolvingVRPwithMILPPotentialroutesareaninputandcanconsiderdifferentcosts,notjustdistance.Mixedintegerlinearprogramusedtoselectroutes:

• Eachcolumnisaroute• Eachrowisanode/stop• Totalcostofeachrouteisincluded

Min C jYjj∑

s.t.

aijY jj=1

J

∑ ≥ Di ∀i

Yjj=1

J

∑ ≤ V

Yj = {0,1} ∀j

Indices Demandnodesi VehicleroutesjInputData Cj=Totalcostofroutej($) Di=Demandatnodei(units) V=Maximumvehicles aij=1ifnodeiisinroutej; =0otherwiseDecisionVariables Yj=1ifroutejisused, =0otherwise


ApproximationMethodsInthissecondhalfwewilldiscussexamplesofapproximationandestimation.InparticularwewillreviewestimationofOne-to-ManyDistributionthroughlinehauldistance,travelingsalesmanandvehicleroutingproblems.Approximation:avalueorquantitythatisnearlybutnotexactlycorrect.

Estimation:aroughcalculationofthevalue,number,quantity,orextentofsomething.synonyms:estimate,approximation,roughcalculation,roughguess,evaluation,back-of-theenvelope

Whyuseapproximationmethods?• Fasterthanmoreexactorprecisemethods,• Usesminimalamountsofdata,and• Candetermineifmoreanalysisisneeded:GoldilocksPrinciple:Toobig,Toolittle,Just

right.

Alwaystrytoestimateasolutionpriortoanalysis!

QuickEstimationSimpleEstimationRules:1.Breaktheproblemintopiecesthatyoucanestimateordeterminedirectly2.Estimateorcalculateeachpieceindependentlytowithinanorderofmagnitude3.CombinethepiecesbacktogetherpayingattentiontounitsExample:HowmanypianotunersarethereinChicago?"

• Thereareapproximately9,000,000peoplelivinginChicago.• Onaverage,therearetwopersonsineachhouseholdinChicago.• Roughlyonehouseholdintwentyhasapianothatistunedregularly.• Pianosthataretunedregularlyaretunedonaverageaboutonceperyear.• Ittakesapianotunerabouttwohourstotuneapiano,includingtraveltime.• Eachpianotunerworkseighthoursinaday,fivedaysinaweek,and50weeksinayear.

TuningsperYear=(9,000,000ppl)÷(2ppl/hh)×(1piano/20hh)×(1tuning/piano/year)=225,000TuningsperTunerperYear=(50wks/yr)×(5day/wk)×(8hrs/day)÷(2hrstotune)=1000NumberofPianoTuners=(225,000tuningsperyear)÷(1000tuningsperyearpertuner)=225ActualNumber=290


EstimationofOnetoManyDistributionSingleDistributionCenter:

• Productsoriginatefromoneorigin• Productsaredemandedatmanydestinations• AlldestinationsarewithinaspecifiedServiceRegion• Ignoreinventory(samedaydelivery)

Assumptions:

• Vehiclesarehomogenous• Samecapacity,QMAX• Fleetsizeisconstant


Findingtheestimatedtotaldistance:• DividetheServiceRegionintoDeliveryDistricts• Estimatethedistancerequiredtoserviceeachdistrict

Routetoserveaspecificdistrict:

• Linehaulfromorigintothe1stcustomerinthedistrict• Localdeliveryfrom1sttolastcustomerinthedistrict• Backhaul(empty)fromthelastcustomertotheorigin

dTOUR ≈ 2dLineHaul + dLocal

dLineHaul=Distancefromorigintocenterofgravity(centroid)ofdeliverydistrictdLocal=Localdeliverybetweencustomersinonedistrict


Howdoweestimatedistances?• PointtoPoint• RoutingorwithinaTour

EstimatingPointtoPointDistancesDependsonthetopographyoftheunderlyingregionEuclideanSpace: dA-B=√[(xA-xB)2+(yA-yB)2]Grid: dA-B=|xA-xB|+|yA-yB|RandomNetwork:differentapproach

ForRandom(real)Networksuse:DA-B=kCFdA-BFinddA-B-the“ascrowflies”distance.

• Euclidean:forreallyshortdistanceso dA-B=SQRT((xA-xB)2+(yA-yB)2)

• GreatCircle:forlocationswithinthesamehemisphereo dA-B=3959(arccos[sin[LATA]sin[LATB]+cos[LATA]cos[LATB]cos[LONGA-LONGB]])

• Where:o LATi=Latitudeofpointiinradianso LONGi=Longitudeofpointiinradianso Radians=(AngleinDegrees)(π/180o)

Applyanappropriatecircuityfactor(kCF)

• Howdoyougetthisvalue?• Whatdoyouthinktherangesare?• Whataresomecautionsforthisapproach?


EstimatingLocalRouteDistancesTravelingSalesmanProblem

• Startingfromanorigin,findtheminimumdistancerequiredtovisiteachdestinationonceandonlyonceandreturntoorigin.

• TheexpectedTSPdistance,dTSP,isproportionalto√(nA)wheren=numberofstopsandA=areaofdistrict

• Theestimationfactor(kTSP)isafunctionofthetopology

OnetoManySystemWhatcanwesayabouttheexpectedTSPdistancetocovernstopsindistrictwithanareaofA?Agoodapproximation,assuminga"fairlycompactandfairlyconvex"area,is:A=Areaofdistrictn=Numberofstopsindistrictδ=Density(#stops/Area)kTSP=TSPnetworkfactor(unitless)dTSP=TravelingSalesmanDistancedstop=Averagedistanceperstop

dTSP ≈ kTSP nA = kTSP n nδ

⎛

⎝⎜⎞

⎠⎟ = kTSP

nδ

⎛

⎝⎜

⎞

⎠⎟

dstop ≈kTSP nAn

= kTSPAn


WhatvaluesofkTSPshouldweuse?• LotsofresearchonthisforL1andL2networks-dependsondistrictshape,approachto

routing,etc.• Euclidean(L2)Networks

o kTSP=0.57to0.99dependingonclustering&sizeofN(MAPE~4%,MPE~-1%)o kTSP=0.765commonlyusedandisagoodapproximation!

• Grid(L1)Networkso kTSP=0.97to1.15dependingonclusteringandpartitioningofdistrict

EstimatingVehicleTourDistancesFindingthetotaldistancetraveledonalltours,where:

• l=numberoftours• c=numberofcustomerstopspertourand• n=totalnumberofstops=c*l

dTOUR = 2dLineHaul +ckTSPδ

dAllTours = ldTOUR = 2ldLineHaul +nkTSPδ

Minimizenumberoftoursbymaximizingvehiclecapacity

l = DQMAX

⎡

⎣⎢

⎤

⎦⎥

+

dAllTours = 2DQMAX

⎡

⎣⎢

⎤

⎦⎥

+

dLineHaul +nkTSPδ

[x]+=lowestintegervalue>x.ThisisastepfunctionEstimatethiswithcontinuousfunction: [x]+~x+½


KeyPoints

• Reviewthebasisandcomponentsofalgorithms• Recognizedesiredpropertiesofanalgorithm• Reviewdifferentnetworkalgorithms• RecognizehowtosolvetheShortestPathProblem• RecognizewhichalgorithmstousefortheTravelingSalesmanProblem• RecognizehowtosolveaVehicleRoutingProblem(ClusterFirst–RouteSecond)• Reviewhowtouseapproximations• Recognizestepstoquickestimation


DistributionsandProbability

SummaryWereviewtwoveryimportanttopicsinsupplychainmanagement:probabilityanddistributions.Probabilityisanoften-reoccurringthemeinsupplychainmanagementduetothecommonconditionsofuncertainty.Onagivenday,astoremightsell2unitsofaproduct,onanother,50.Toexplorethis,theprobabilityreviewincludesanintroductionofprobabilitytheory,probabilitylaws,andpropernotation.Summaryordescriptivestatisticsareshownforcapturingcentraltendencyandthedispersionofadistribution.Wealsointroducetwotheoreticaldiscretedistributions:UniformandPoisson.Wethenintroducethreecommoncontinuousdistributions:Uniform,Normal,andTriangle.Thereviewthengoesthroughthedifferencebetweendiscretevs.continuousdistributionsandhowtorecognizethesedifferences.Theremainderofthereviewisaexplorationintoeachtypeofdistribution,whattheylooklikegraphicallyandwhataretheprobabilitydensityfunctionandcumulativedensityfunctionofeach.

KeyConceptsProbabilityProbabilitydefinestheextenttowhichsomethingisprobable,orthelikelihoodofaneventhappening.Itismeasuredbytheratioofthecasetothetotalnumberofcasespossible.

ProbabilityTheory• Mathematicalframeworkforanalyzingrandomeventsorexperiments.• Experimentsareeventswecannotpredictwithcertainty(e.g.,weeklysalesatastore,

flippingacoin,drawingacardfromadeck,etc.).• Eventsareaspecificoutcomefromanexperiment(e.g.,sellinglessthan10itemsina

week,getting3headsinarow,drawingaredcard,etc.)


ProbabilityLaws1.Theprobabilityofanyeventisbetween0and1,thatis0≤P(A)≤12.IfAandBaremutuallyexclusiveevents,thenP(AorB)=P(AUB)=P(A)+P(B)3.IfAandBareanytwoevents,then

P(A | B) = P(A and B)P(B)

=P A∩B( )P(B)

WhereP(AIB)istheconditionalprobabilityofAoccurringgivenBhasalreadyoccurred.4.IfAandBareindependentevents,then

P(A | B) = P(A)

P(A and B) = P(A∩B) = P A | B( )P(B) = P A( )×P B( )

WhereeventsAandBareindependentifknowingthatBoccurreddoesnotinfluencetheprobabilityofAoccurring.

SummarystatisticsDescriptiveorsummarystatisticsplayasignificantroleintheinterpretation,presentation,andorganizationofdata.Itcharacterizesasetofdata.Therearemanywaysthatwecancharacterizeadataset,wefocusedontwo:CentralTendencyandDispersionorSpread.

CentralTendencyThisis,inroughterms,the“mostlikely”valueofthedistribution.Itcanbeformallymeasuredinanumberofdifferentwaystoinclude:

• Mode–thespecificvaluethatappearsmostfrequently• Median–thevalueinthe“middle”ofadistributionthatseparatesthelowerfromthe

higherhalf.Thisisalsocalledthe50thpercentilevalue.• Mean(μ)–thesumofvaluesmultipliedbytheirprobability(calledtheexpectedvalue).

Thisisalsothesumofvaluesdividedbythetotalnumberofobservations(calledtheaverage).

Notation• P(A)–theprobabilitythateventAoccurs• P(A’)=complementofP(A)–probabilitysomeothereventthatisnotAoccurs.This

isalsotheprobabilitythatsomethingotherthanAhappens.


E[X]= x = µ = pixii=1

n∑


DispersionorSpreadThiscapturesthedegreetowhichtheobservations“differ”fromeachother.Themorecommondispersionmetricsare:

• Range–themaximumvalueminustheminimumvalue.• InnerQuartiles–75thpercentilevalueminusthe25thpercentilevalue-capturesthe

“centralhalf”oftheentiredistribution.• Variance(σ2)–theexpectedvalueofthesquareddeviationaroundthemean;also

calledtheSecondMomentaroundthemean

Var[X]=σ 2 = pi xi − x( )i=1

n∑

2= pi xi −µ( )

i=1

n∑

2

• StandardDeviation(σ)–thesquarerootofthevariance.Thisputsitinthesameunitsastheexpectedvalueormean.

• CoefficientofVariation(CV)–theratioofthestandarddeviationoverthemean=σ/μ.Thisisacommoncomparablemetricofdispersionacrossdifferentdistributions.Asageneralrule:

o 0≤CV≤0.75,lowvariabilityo 0.75≤CV≤1.33,moderatevariabilityo CV>1.33,highvariability

PopulationversusSampleVarianceInpractice,weusuallydonotknowthetruemeanofapopulation.Instead,weneedtoestimatethemeanfromasampleofdatapulledfromthepopulation.Whencalculatingthevariance,itisimportanttoknowwhetherweareusingallofthedatafromtheentirepopulationorjustusingasampleofthepopulation’sdata.Inthefirstcasewewanttofindthepopulationvariancewhileinthesecondcasewewanttofindthesamplevariance.Theonlydifferencesbetweencalculatingthepopulationversusthesamplevariances(andthustheircorrespondingstandarddeviations)isthatforthepopulationvariance,σ2,wedividethesumoftheobservationsbyn(thenumberofobservations)whileforthesamplevariance,s2,wedividebyn-1.

σ 2 =xi −µ( )

i=1

n∑

2

ns2 =

xi − x( )i=1

n∑

2

n−1

Notethatthesamplevariancewillbeslightlylargerthanthepopulationvarianceforsmallvaluesofn.Asngetslarger,thisdifferenceessentiallydisappears.Thereasonfortheusen-1isduetohavingtouseadegreeoffreedomincalculatingtheaverage(xbar)fromthesamesamplethatweareestimatingthevariance.Itleadstoanunbiasedestimateofthepopulationvariance.Inpractice,youshouldjustusethesamplevarianceandstandarddeviationunlessyouaredealingwithspecificprobabilities,likeflippingacoin.


SpreadsheetFunctionsforSummaryStatisticsAllofthesesummarystatisticscanbecalculatedquiteeasilyinanyspreadsheettool.Thetablebelowsummarizesthefunctionsforthreewidelyusedpackages.

Table:SpreadsheetFunctionsforDescriptiveStatistics

ProbabilityDistributionsProbabilitydistributionscaneitherbeempirical(basedonactualdata)ortheoretical(basedonamathematicalform).Determiningwhichisbestdependsontheobjectiveoftheanalysis.Empiricaldistributionsfollowpasthistorywhiletheoreticaldistributionsfollowanunderlyingmathematicalfunction.Theoreticaldistributionsdotendtoallowformorerobustmodelingsincetheempiricaldistributionscanbethoughtofasasamplingofthepopulationdata.Thetheoreticaldistributioncanbeseenasbetterdescribingtheassumedpopulationdistribution.Typically,welookforthetheoreticaldistributionthatbestfitsthedataWepresentedfivedistributions.Twoarediscrete(UniformandPoisson)andthreearecontinuous(Uniform,Normal,andTriangle).Eachissummarizedinturn.

DiscreteUniformDistribution~U(a,b)Finitenumber(N)ofvaluesobservedwithaminimumvalueofaandamaximumvalueofb.Theprobabilityofeachpossiblevalueis1/NwhereN=b–a+1


PoissonDistribution~P(λ)Discretefrequencydistributionthatgivestheprobabilityofanumberofindependenteventsoccurringinafixedtimewheretheparameterλ =mean=variance.Widelyusedtomodelarrivals,slowmovinginventory,etc.Notethatthedistributiononlycontainsnon-negativeintegersandcancapturenon-symmetricdistributions.Asthenumberofobservationsincrease,thedistributionbecomes“belllike”andapproximatestheNormalDistribution.

Table:SpreadsheetFunctionsforPoissondistribution

SummaryMetrics• Mean=λ• Median≈ ⎣(λ + 1/3 – 0.02/λ)⎦• Mode=⎣λ⎦• Variance=λ

ProbabilityMassFunction(pmf):

where• e=Euler’snumber~2.71828...• λ=meanvalue(parameter)• x!=factorialofx,e.g.,3!=3×2×1=6and0!=1

ProbabilityMassFunction(pmf):

SummaryMetrics• Mean=(a+b)/2• Median=(a+b)/2• ModeN/A(allvaluesareequallylikely)• Variance=((b-a+1)2-1)/12


ProbabilityDensityFunction(pdf)Thepdfisfunctionofacontinuousvariable.TheprobabilitythatXliesbetweenvaluesaandbisequaltoareaunderthecurvebetweenaandb.Totalareaunderthecurveequals1,buttheP(X=t)=0foranyspecificvalueoft.

CumulativeDensityFunction(cdf)

• F(t)=P(X≤t)ortheprobabilitythatXdoesnotexceedt• ≤F(t)≤1.0• F(b)≥F(1)ifb>a–itisincreasing

Simplerules• P(X≤t)=F(t)• P(X>t)=1–F(t)• P(c≤X≤d)=f(d)–F(c)• P(X=t)=0

ContinuousUniformDistribution~U(a,b)Sometimesalsocalledarectangulardistribution

• “Xifuniformlydistributedovertherangeatob,orX~U(a,b)”.

pdf:

SummaryMetrics• Mean=(a+b)/2• Median= (a+b)/2• ModeN/Aallvaluesequallylikely• Variance=(b-a)2/12

cdf:


NormalDistribution~N(μ,σ)Widelyusedbell-shaped,symmetriccontinuousdistributionwithmeanμandstandarddeviationσ.Mostcommonlyuseddistributioninpractice.

Commondispersionvalues~N(μ,σ)

• P(Xw/in1σaroundμ)=0.6826• P(Xw/in2σaroundμ)=0.9544• P(Xw/in3σaroundμ)=0.9974• +/-1.65σaroundμ=0.900• +/-1.96σaroundμ=0.950• +/-2.81σaroundμ=0.995

UnitorStandardNormalDistributionZ~N(0,1)• Thetransformationfromany~N(μ,σ)totheunitnormaldistribution=Z=(x-μ)/σ• Zscore(standardscore)givesthenumberofstandarddeviationsawayfromthemean• Allowsforuseofstandardtablesandisusedextensivelyininventorytheoryforsetting

safetystock

Table:SpreadsheetFunctionsforNormalDistribution

SummaryMetrics• Mean=μ• Median= μ• Mode=μ• Variance=σ2

pdf:


TriangleDistribution~T(a,b,c)Thisisacontinuousdistributionwithaminimumvalueofa,maximumvalueofb,andamodeofc.Itisagooddistributiontousewhendealingwithananecdotalorunknowndistribution.Itcanalsohandlenon-symmetricdistributionswithlongtails.

Figure1TriangleDistribution

Triangle DistributionWe would say,

“X follows a triangle distribution with a minimum of a, maximum b, and a mode of c, ~T(a, b, c)”

22

f (x) =

0 x < a

2 x - a( )b- a( ) c- a( )

a £ x £ c

2 b- x( )b- a( ) b- c( )

c £ x £ b

0 x > b

ì

í

ïïïï

î

ïïïï

a bc

2(b- a)

x

E xéë ùû=a+b+ c

3

Var xéë ùû=118

æ

èç

ö

ø÷ a2 +b2 + c2 - ab-ac-bc( )

P x > déë ùû=b- d( )2

b- a( ) b- c( )

æ

è

ççç

ö

ø

÷÷÷

for c £ d £ b

d = b- P x > déë ùû b- a( ) b- c( ) for c £ d £ b

Characteristics• Good way to get a sense of an unknown distribution• People tend to recall extreme and common values• Handles asymmetric distributions

pdf:

cdf:


DifferencesbetweenContinuousandDiscreteDistributionsJustlikevariables,distributionscanbeclassifiedintocontinuous(pdf)anddiscrete(pmf)probabilitydistributions.Whilediscretedistributionshaveaprobabilityforeachoutcome,theprobabilityforaspecificpointinacontinuousdistributionmakesnosenseandiszero.Insteadforcontinuousdistributionswelookfortheprobabilityofarandomvariablefallingwithinaspecificinterval.Continuousdistributionsuseafunctionorformulatodescribethedataandthusinsteadofsumming(aswedidfordiscretedistributions)tofindtheprobability,weintegrateovertheregion.

DiscreteDistributions ContinuousDistributions

LearningObjectives• Understandprobabilities,importanceandapplicationindailyoperationsandextreme

circumstances.• Understandandapplydescriptivestatistics.• Understanddifferencebetweencontinuousvs.discreterandomvariabledistributions.• Reviewmajordistributions:Uniform(discreteandcontinuous),Poisson,Normaland

Triangle.• Understandthedifferencebetweendiscretevs.continuousdistributions.• Recognizeandapplyprobabilitymassfunctions(pmf),probabilitydensityfunctions

(pdf),andcumulativedensityfunctions(cdf).

SummaryMetrics



Regression

SummaryInthisreviewweexpandourtoolsetofpredictivemodelstoincludeordinaryleastsquaresregression.Thisequipsuswiththetoolstobuild,runandinterpretaregressionmodel.Wearefirstintroducedwithhowtoworkwithmultiplevariablesandtheirinteraction.Thisincludescorrelationandcovariance,whichmeasureshowtwovariableschangetogether.Aswereviewhowtoworkwithmultiplevariables,itisimportanttokeepinmindthatthedatasetssupplychainmanagerswilldealwitharelargelysamples,notapopulation.Thismeansthatthesubsetofdatamustberepresentativeofthepopulation.Thelaterpartofthelessonintroduceshypothesistesting,whichallowsustoanswerinferencesaboutthedata.Wethentacklelinearregression.Regressionisaveryimportantpracticeforsupplychainprofessionalsbecauseitallowsustotakemultiplerandomvariablesandfindrelationshipsbetweenthem.Insomeways,regressionbecomesmoreofanartthanascience.Therearefourmainstepstoregression:choosingwithindependentvariablestoinclude,collectingdata,runningtheregression,andanalyzingtheoutput(themostimportantstep).

KeyConcepts

MultipleRandomVariablesMostsituationsinpracticeinvolvetheuseandinteractionofmultiplerandomvariablesorsomecombinationofrandomvariables.WeneedtobeabletomeasuretherelationshipbetweentheseRVsaswellasunderstandhowtheyinteract.

CovarianceandCorrelationCovarianceandcorrelationmeasureacertainkindofdependencebetweenvariables.Ifrandomvariablesarepositivelycorrelated,higherthanaveragevaluesofXarelikelytooccurwithhigherthanaveragevaluesofY.Fornegativelycorrelatedrandomvariables,higherthanaveragevaluesarelikelytooccurwithlowerthanaveragevaluesofY.Itisimportanttorememberastheold,butnecessarysayinggoes:correlationdoesnotequalcausality.Thismeansthatyouarefindingamathematicalrelationship–notacausalone.

CorrelationCoefficient:isusedtostandardizethecovarianceinordertobetterinterpret.Itisameasurebetween-1and+1thatindicatesthedegreeanddirectionoftherelationshipbetweentworandomvariablesorsetsofdata.


SpreadsheetFunctionsFunction MicrosoftExcel GoogleSheets LibreOffice->Calc

Covariance =COVAR(array,array) =COVAR(array,array) =COVAR(array;array)

Correlation =CORREL(array,array) =CORREL(array,array) =CORREL(array;array)

LinearFunctionofRandomVariablesAlinearrelationshipexistsbetweenXandYwhenaone-unitchangeinXcausesYtochangebyafixedamount,regardlessofhowlargeorsmallXis.Formally,thisis:Y=aX+b.ThesummarystatisticsofalinearfunctionofaRandomVariableare:

SumsofRandomVariablesIFXandYareindependentrandomvariableswhereW=aX+bY,thenthesummarystatisticsare:

TheserelationsholdforanydistributionofXandY.However,ifXandYare~N,thenWis~Naswell!

CentralLimitTheoremCentrallimittheoremstatesthatthesampledistributionofthemeanofanyindependentrandomvariablewillbenormalornearlynormal,ifthesamplesizeislargeenough.Largeenoughisbasedonafewfactors–oneisaccuracy(moresamplepointswillberequired)aswellastheshapeoftheunderlyingpopulation.Manystatisticianssuggestthat30,sometimes40,isconsideredlargeenough.Thisisimportantbecauseisdoesn’tmatterwhatdistributionstherandomvariablefollows.

Covariance

Expectedvalue:E[W]=aμX+bμYVariance:VAR[W] =a2σ2X+b2σ2Y+2abCOV(X,Y)

=a2σ2X+b2σ2Y+2abσXσYCORR(X,Y)StandardDeviation:σW=√VAR[W]

Expectedvalue:E[Y]=μY=aμX+bVariance:VAR[Y]=σ2Y=a2σ2XStandardDeviation:σY=|a|σX


Canbeinterpretedasfollows:• Xi,..Xnareiidwithmean=µandstandarddeviation=σ

o ThesumofthenrandomvariablesisSn=ΣXio ThemeanofthenrandomvariablesisX�=Sn/n

• Then,ifnis“large”(say>30)o SnisNormallydistributedwithmean=nµandstandarddeviationσ√no X�isNormallydistributedwithmean=µandstandarddeviationσ/√n

InferenceTesting

SamplingWeneedtoknowsomethingaboutthesampletomakeinferencesaboutthepopulation.Theinferenceisaconclusionreachedonthebasisofevidenceandreasoning.Tomakeinferencesweneedtoasktestablequestionssuchasifthedatafitsaspecificdistributionoraretwovariablescorrelated?Tounderstandthesequestionsandmore–weneedtounderstandsamplingofapopulation.Ifsamplingisdonecorrectly,thesamplemeanshouldbeanestimatorofthepopulationmeanaswellascorrespondingparameters.

• Population:istheentiresetofunitsofobservation• Sample:subsetofthepopulation.• Parameter:describesthedistributionofrandomvariable.• RandomSample:isasampleselectedfromthepopulationsothateachitemisequally

likely.

Thingstokeepinmind• Xisarv~?(µ,σ,...)fortheentirepopulation• X1,X2,...Xnareiid• X�isanestimateofthepopulationparameter,themeanorµ• RememberthatX�isalsoarvbyitself!• x1,x2,...xn,aretherealizationsorobservationsofrvX• xisthesamplestatistic–themean• WewanttofindhowxrelatestoX�relatestoµ

Whydowecare?• WecanshowthatE[X�]=μ andthatS=σ/√n• Note:standarddeviationdecreasesassamplesizegetsbigger!• Also,theCentralLimitTheoremsaysthatsamplemeanX�is~N(μ,σ/√n)

ConfidenceIntervalsConfidenceintervalsareusedtodescribetheuncertaintyassociatedwithasampleestimateofapopulationparameter.


CalculatingConfidenceIntervalsWhenthen>30Wecanassume:X�~N(μ,σ/√n)Thelevelcofaconfidenceintervalgivestheprobabilitythattheintervalproducedincludesthetruevalueoftheparameter.


Wherezisthecorrespondingz-scorecorrespondingtotheareaaroundthemean:

z=1.65forβ=.90, z=1.96forβ=.95, z=2.81forβ=.995Forspreadsheetsuse: z=NORM.S.INV((1+β)/2)Whenn≤30Thenweneedtousethet-distribution,whichisbell-shapedandsymmetricaround0.

• Mean=0,butStdDev=√(k/k-2)• Wherekisthedegreesoffreedomand,generally,k=n-1• Thevalueofcisafunctionofβandk

Wherecisthecorrespondingt-statisticcorrespondingtotheareaaroundthemean.Forspreadsheets,use: c=T.INV.2T(1-β,k)Therearesomeimportantinsightsforconfidenceintervalsaroundthemean.Therearetradeoffsbetweeninterval(l),samplesize(n)andconfidence(b):

• Whennisfixed,usingahigherconfidencelevelbleadstoawiderinterval,L.• Whenconfidencelevelisfixed(b),increasingsamplesizen,leadstosmallerinterval,L.• Whenbothnandconfidencelevelarefixed,wecanobtainatighterinterval,L,by

reducingthevariability(i.e.smallsands).

Wheninterpretingconfidenceintervals,afewthingstokeepinmind:• Repeatedlytakingsamplesandfindingconfidenceintervalsleadstodifferentintervals

eachtime,• Butb%oftheresultingintervalswouldcontainthetruemean.• Toconstructab%confidenceintervalthatiswithin(+/-)Lofμ,therequiredsamplesize

is:n=z2*s2/L2

x − zs

n, x + zs

n

⎡

⎣⎢

⎤

⎦⎥

x − cs

n,x + cs

n

⎡

⎣⎢

⎤

⎦⎥


HypothesisTestingHypothesistestingisamethodformakingachoicebetweentwomutuallyexclusiveandcollectivelyexhaustivealternatives.Inthispractice,wemaketwohypothesesandonlyonecanbetrue.NullHypothesis(H0)andtheAlternativeHypothesis(H1).Wetest,ataspecifiedsignificancelevel,toseeifwecanRejecttheNullhypothesis,orAccepttheNullHypothesis(ormorecorrectly,“donotreject”).TwotypesofMistakesinhypothesistesting:

• TypeI:RejecttheNullhypothesiswheninfactitisTrue(Alpha)• TypeII:AccepttheNullhypothesiswheninfactitisFalse(Beta)• WefocusonTypeIerrorswhensettingsignificancelevel(.05,.01)

Threepossiblehypothesesoroutcomestoatest

• Unknowndistributionisthesameastheknowndistribution(AlwaysH0)• Unknowndistributionis‘higher’thantheknowndistribution• Unknowndistributionis‘lower’thantheknowndistribution


ExampleofHypothesisTestingIamtestingwhetheranewinformationsystemhasdecreasedmyordercycletime.Weknowthathistorically,theaveragecycletimeis72.5hours+/-4.2hours.Wesampled60ordersaftertheimplementationandfoundtheaveragetobe71.4hours.Weselectalevelofsignificancetobe5%.

1. Selecttheteststatisticofinterest meancycletimeinhours–useNormaldistribution(z-statistic)2.Determinewhetherthisisaoneortwotailedtest Onetailedtest3.Pickyoursignificancelevelandcriticalvalue alpha=5percent,thereforez=NORM.S.INV(.05)=-1.6448

4. FormulateyourNull&Alternativehypotheses H0:Newcycletimeisnotshorterthantheoldcycletime H1:Newcycletimeisshorterthantheoldcycletime

5. Calculatetheteststatistic z=(Xb-μXb)/σXb=(Xb-μ)/(σ/√n)=(71.4–72.5)/(4.2/√60)=-2.0287

6. Comparetheteststatistictothecriticalvaluez=-2.0287<-1.6448theteststatistic<criticalvalue,therefore,werejectthenull

hypothesisRatherthanjustreportingthatH0wasrejectedata5%significancelevel,wemightwanttoletpeopleknowhowstronglywerejectedit.Thep-valueisthesmallestlevelofalpha(levelofsignificance)suchthatwewouldrejecttheNullhypothesiswithourcurrentsetofdata.Alwaysreportthep-valuep-value=NORM.S.DIST(-2.0287)=.0212ChisquaretestChiSquaretestcanbeusedtomeasurethegoodnessoffitanddeterminewhetherthedataisdistributednormally.Touseachisquaretest,youtypicallywillcreateabucketofcategories,c,counttheexpectedandobserved(actual)valuesineachcategory,andcalculatethechi-squarestatisticsandfindthep-value.Ifthep=valueislessthanthelevelofsignificant,youwillthenrejectthenullhypothesis.


SpreadsheetFunctions:Function Returnsp-valueforChi-SquareTestMicrosoftExcel =CHISQ.TEST(observed_values,expected_values)GoogleSheets =CHITEST(observed_values,expected_values)LibreOffice->Calc =CHISQ.TEST(observed_values;expected_values)

OrdinaryLeastSquaresLinearRegressionRegressionisastatisticalmethodthatallowsuserstosummarizeandstudyrelationshipsbetweenadependent(Y)variableandoneormoreindependent(X)variables.ThedependentvariableYisafunctionoftheindependentvariablesX.Itisimportanttokeepinmindthatvariableshavedifferentscales(nominal/ordinal/ratio).Forlinearregression,thedependentvariableisalwaysaratio.Theindependentvariablescanbecombinationsofthedifferentnumbertypes.

LinearRegressionModelThedata(xi,yi)aretheobservedpairsfromwhichwetrytoestimatetheΒcoefficientstofindthe‘bestfit’.Theerrorterm,ε,isthe‘unaccounted’or‘unexplained’portion.LinearModel:

ResidualsBecausealinearregressionmodelisnotalwaysappropriateforthedata,youshouldassesstheappropriatenessofthemodelbydefiningresiduals.Thedifferencebetweentheobservedvalueofthedependentvariableandpredictedvalueiscalledtheresidual.

OrdinaryLeastSquares(OLS)RegressionOrdinaryleastsquaresisamethodforestimatingtheunknownparametersinalinearregressionmodel.Itfindstheoptimalvalueofthecoefficients(b0andb1)thatminimizethesumofthesquaresoftheerrors:

MultipleVariables

yi = β0 +β1xiYi = β0 +β1xi +εi for i =1,2,...n

yi = b0 +b1xi for i =1,2,...nei = yi − yi = yi −b0 +b1xi for i =1,2,...n

ei2( )i=1

n∑ = yi − yi( )

2

i=1

n∑ = yi −b0 −b1xi( )

2

i=1

n∑

= y −b1x b1 =(xi − x )(yi − y)i=1

n∑

(xi − x )2

i=1

n∑


Theserelationshipstranslatealsotomultiplevariables.

ValidatingaModelAllstatisticalsoftwarepackageswillprovidestatisticsforevaluation(namesandformatwillvarybypackage).Butthemodeloutputtypicallyincludes:modelstatistics(regressionstatisticsorsummaryoffit),analysisofvariance(ANOVA),andparameterstatistics(coefficientstatistics).OverallFitOverallfit=howmuchvariationinthedependentvariable(y),canweexplain?TotalvariationofCPL–findthedispersionaroundthemean.

Yi = β0 +β1x1i + ...+βk xki +εi for i =1,2,...nE(Y | x1,x2 ,...,xk ) = β0 +β1x1 +β2x2 + ...+βk xkStdDev(Y | x1,x2 ,...,xk ) =σ

ei2( )i=1

n∑ = yi − yi( )

2

i=1

n∑ = yi −b0 −b1x1i − ...−bk xki( )

2

i=1

n∑

TotalSumofSquares


MakeestimateforofYforeachx.

Modelexplains%oftotalvariationofthedependentvariables.

AdjustedR2correctsforadditionalvariables

IndividualCoefficientsEachIndependentvariable(andb0)willhave:

• Anestimateofcoefficient(b1),• Astandarderror(sbi)

o se=Standarderrorofthemodel

o sx=Standarddeviationoftheindependentvariables=numberofobservations

• Thet-statistic

o k=numberofindependentvariableso bi=estimateorcoefficientofindependent

variable

ErrororResidualSumofSquares

CoefficientofDeterminationorGoodnessofFit(R2)


Correspondingp-value–TestingtheSlopeo Wewanttoseeifthereisalinearrelationship,i.e.wewanttoseeiftheslope

(b1)issomethingotherthanzero.So:H0:b1=0andH1b1≠0

o Confidenceintervals–estimateanintervalfortheslopeparameter.

Multi-Collinearity,AutocorrelationandHeterscedasticityMulti-Collinearityiswhentwoormorevariablesinamultipleregressionmodelarehighlycorrelated.ThemodelmighthaveahighR2buttheexplanatoryvariablesmightfailthet-test.Itcanalsoresultinstrangeresultsforcorrelatedvariables.Autocorrelationisacharacteristicsofdatainwhichthecorrelationbetweenthevaluesofthesamevariablesisbasedonrelatedobjects.Itistypicallyatimeseriesissue.Heterscedasticityiswhenthevariabilityofavariableisunequalacrosstherangeofvaluesofasecondvariablethatpredictsit.Sometelltalesignsinclude:observationsaresupposedtohavethesamevariance.Examinescatterplotsandlookfor“fan-shaped”distributions.

LearningObjectives

• Understandhowtoworkwithmultiplevariables.• Beawareofdatalimitationswithsizeandrepresentationofpopulation.• Identifyhowtotestahypothesis.• Reviewandapplythestepsinthepracticeofregression.• Beabletoanalyzeregressionandrecognizeissues.


Simulation

SummaryThisreviewprovidesanoverviewofsimulation.Afterareviewofdeterministic,prescriptivemodeling(optimization,mathprogramming)andpredictivemodels(regression),simulationoffersanapproachfordescriptivemodeling.Simulationsletyouexperimentwithdifferentdecisionsandseetheiroutcomes.Supplychainmanagerstypicallyusesimulationstoassessthelikelihoodofoutcomesthatmayfollowfromdifferentactions.Thisreviewoutlinesthestepsinsimulationfromdefiningthevariables,creatingthemodel,runningthemodel,andrefiningthemodel.Itoffersinsightintothebenefitofusingsimulationforopenendedquestionsbutwarnsofitsexpensiveandtimeconsumingnature. Overthedurationofthecoursewehavereviewedseveraltypesofmodelsincludingoptimization,regression,andsimulation.Optimization(LP,IP,MILP,NLP)isaprescriptiveformofmodelthatfindsthe“bestsolution”and/orprovidesarecommendation.Regressionisapredictiveformofmodelthatmeasurestheimpactofindependentvariablesondependentvariables.Wenowcoversimulation,whichcapturestheoutcomesofdifferentpolicieswithanuncertainorstochasticenvironment.

SimulationSimulationcanbeusedinavarietyofcontexts;itismostusefulincapturingcomplexsysteminteractions,modelingsystemuncertainty,orgeneratingdatatoanalyze,describeandvisualizeinteractions,outcomes,andsensitivities.Thereareseveralclassesofsimulationmodelsincluding:systemdynamics;MonteCarloSimulation;discretetimesimulation;andagentbasedsimulations.Therearefivemainstepsindevelopingasimulationstudy.Formulateandplanthestudy;collectdataanddefineamodel;constructmodelandvalidate;makeexperimentalruns;andanalyzeoutput.Thefollowingwillreviewhoweachofthosestepscanbeconducted.

StepsinaSimulationStudyFormulate&planthestudyOnceithasbeendeterminedthatsimulationistheappropriatetoolfortheproblemunderinvestigation,thenextstepistoformulatetheplanandstudy.Thisinvolvesafewmainsteps:

• Definethegoalsofthestudyanddeterminewhatneedstobesolved• Developamodelwheredailydemandvaries,a“productionpolicy”willbeapplied• Basedondemandandpolicy–calculateprofitability• Assessprofitabilityandperformancemetricsofdifferentpolicies

Collectdata&defineamodel


Oncetheplanhasbeenformulated,thedataneedstobecollectedandamodeldefined.Ifyouarefacedwithalackofsampledata–youwillneedtodeterminethe“range”ofthevariable(s)bytalkingtostakeholdersorexpertstoidentifypossiblevalues.SomedatacanbederivedfromknowndistributionssuchasPoissonorUniform/TriangularDistributionswhenlittletonoinformationisavailable.Ifsampledataisavailable,conductstepsaswehavepreviouslyreviewedsuchashistograms,calculatingsummarystatistics.ThenconductaChi-Squaretesttofitthesampleto“traditional”distributions.Orusea“custom”empiricaldistributionsuchasdiscreteempirical(use%ofobservationasprobabilities),orcontinuous–usehistogramtocomputeprobabilitiesofeachrangeandthen“uniform”withintherange.InExcel:

=CHITEST(Observed_Range,Actual_Range)–returnsp-value=1-CHIDIST(Chi-square,Degreesoffreedom)–returnsthep-value

Nextstepsareto:

• Determinerelationshipbetweenvariousvariables• Determineperformancemetrics• Collectdata&estimateprobability

Constructmodel&validateMakenecessaryinputsrandom,addadatatabletoautomaterunsofmodel,addsummarystatisticsbasedonresultsfromdatatable.Generatingrandomvariableswiththeunderlyingprincipleofgeneratingarandom(orpseudo-random)numberandtransformittofitthedesireddistribution:

• ManualTechniques:rollingdie,turningaroulettewheel,randomnumbertables• Excel

o RAND()=continuousvariablebetween0and1§ Generaterandomnumberu§ Foreachrandomu,calculateavalueywhosecumulativedistribution

functionisequaltou;assignvalueyasthegeneratednumber:F(y)=P(y)=u

• UniformDistribution~U(a,b)o ~U(a,b)=a+(b–a)*RAND()

• NormalDistribution~N(μ,σ)o ~N(μ,σ)=NORMINV(RAND(),μ,σ)

Chi-SquareTest


ValidationofModel–istheprocessofdeterminingthedegreetowhichamodelanditsassociateddataareanaccuraterepresentationoftherealworldfortheintendeduseofthemodel.Differentwaysofvalidatingmodelincludingcomparingtohistoricaldataorgettingexpertinput.Oneprimarymethodisparametervariabilityandsensitivityanalysis:

• Generatestatisticalparameterswithconfidenceintervals• HypothesisTesting(seeWeek6)

MakeexperimentalrunsYouwillneedtomakemultiplerunsforeachpolicy;usehypothesistestingtoevaluatetheresults.Ifspreadsheetscontainarandominput,wecanuseourdatatabletorepeatedlyanalyzethemodel.Anadditionalcolumnforrunscanbemade.

AnalyzeoutputAnalyzingoutputdealswithdrawinginferencesaboutthesystemmodelbasedonthesimulationoutput.Needtoensurethatthemodeloutputhasmaintainedaproperbalancebetweenestablishingmodelvalidationandstatisticalsignificance.Dependingonthesystemexamined,simulationwillpotentiallybeabletoprovidevaluableinsightwiththeoutput.Abilitytodrawinferencesfromresultstomakesystemimprovementswillbediscussedfurtherinfuturecourses.

LearningObjectives• Reviewthestepstodevelopingasimulationmodel• Understandwhentouseasimulation,andwhentonot• Recognizedifferentkindsofsimulationsandwhentoapplythem


ReferencesAPICSSupplyChainCouncil-http://www.apics.org/sites/apics-supply-chain-council

Chopra,Sunil,andPeterMeindl."Chapter1."SupplyChainManagement:Strategy,Planning,andOperation.5thedition,PearsonPrenticeHall,2013.

CouncilofSupplyChainManagementProfessionals(CSCMP)https://cscmp.org/

HillierandLieberman(2012)IntroductiontoOperationsResearch,McGrawHill.Law&Kelton(2000).SimulationModeling&Analysis,McGrawHill.Taha,H.A.(2010).OperationsResearch.Anintroduction.9thedition.PearsonPrenticeHall.Winston(2003)OperationsResearch:ApplicationsandAlgorithms,CengageLearning.TherearemanydifferentbooksbyWayneWinston-theyareallprettygood.