Transformation ofvariablesin Data EnvelopmentAnalysis

Transcription

1 DepartmentofInformation andserviceeconomy On theuseofanon-singularlinear Transformation ofvariablesin Data EnvelopmentAnalysis AbolfazlKeshvari Pekka Korhonen BUSINESS+ ECONOMY WORKINGPAPERS

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3 Aalto University publication series BUSINESS + ECONOMY 5/2013 On the Use of a Non-Singular Linear Transformation of Variables in Data Envelopment Analysis Abolfazl Keshvari Pekka Korhonen Aalto University School of Business Department of Information and Service Economy Management Science

4 Contact information: Aalto University publication series BUSINESS + ECONOMY 5/2013 Abolfazl Keshvari and Pekka Korhonen ISBN (pdf) ISSN-L ISSN (printed) ISSN (pdf) Unigrafia Oy Helsinki 2013 Finland

5 OntheUseofNon-SingularLinear TransformationofVariablesinData EnvelopmentAnalysis AbolfazlKeshvari,PekkaKorhonen AaltoUniversitySchoolofBusiness,Finland P.O.Box21220,00076Aalto,Helsinki,FINLAND,Tel Abstract Inthispaper,weconsidernon-singularlineartransformationoftheinput-andoutputvariablesintheDataEnvelopmentAnalysis(DEA).Thetransformationisusefulinselecting variables and dealing, for instance, with interval scale variables. We will developgeneral theory and show that the results are invariant due tonon-singular linear transformation providedtheconceptof dominance isdefinedaccordingly.theinvariancepropertyisvalid onlyfornon-singularlineartransformation.finally,webrieflydiscussinsingularlinear transformationandillustratesomepitfalls,whichmayleadtowrongresults. Keywords:DataEnvelopmentAnalysis,VariableReduction,LinearTransformation. 1 Introduction Performanceespeciallytoimproveperformanceisoneofthekeyissuesofmanagement inorganizations.the goodness ofoperations,orperformanceisclearlymultidimensionalof its nature. Several indicators (outputs) are required to capture all essential aspects of the performance.factors(inputs)neededtoproduceperformancearemultidimensionalaswell. In the sequel, we call them outputs/inputs or output-/input-variables. In practice, to find relevantvariablesisoneofthekeyproblems. Iftheessentialoutputsandinputscanbepresentedinquantitativeform(onratioscale) andifthereareavailablecomparativedata,thendataenvelopmentanalysis(dea)developed bycharnes etal.(1978,1979)providescommonlyusedwaytodoperformanceanalysis. PerformanceevaluationiscarriedoutrelativelybycomparingDecisionMakingUnits(DMUs) essentiallyperformingthesametask.indea,thereisnoneedtoexplicitlyknowrelationships betweeninputsandoutputs.thevaluesoftheinputsandoutputsoftheunitsinadditionto background assumptionsis the only requisite information needed for the analysis. That s whythechoiceofvariablesdeservesspecialattention.

6 Data Envelopment Analysis reveals the units which are supposed to be able to improve theirperformanceandtheunitswhichcannotberecognizedaspoor-performers.becausewe usemultidimensionalindicatorstomeasureperformance, goodness isnotfullydefined.dea identifiesso-calledtechnicallyefficientunits,butitisvalue-freeinthesensethatitdoesnot takeintoaccountimportanceofvariousaspects. IntheuseofDataEnvelopmentAnalysis,thereexiststhesameproblemasinperformance analysisgenerally:whicharerelevantoutputsandinputs,andhowtochoosethem.howthe outputsandinputsarechosenhassignificantimpactontheresultsoftheanalysis.inthis paper,wewillfirstconsiderthechoiceofoutputsandinputs.forinstance,ifwewouldliketo comparetheperformanceofstudentswithtwooutput-variables,itisimportanttorecognize how to use either the outputs the number of excellent grades and the number of good grades or thenumberofexcellentgrades and thenumberoftotalgrades insuchway that the results are the same. 1 That isnatural requirement, because any pair of those variablescarriesthesameinformation. Anotherexampleoftheneedoflineartransformationissimplifiedproblem,inwhich weassumethattheperformanceofunitsisevaluatedwithoneinput(cost)andoneoutput (Profit).However,Profitismeasuredontheintervalscale,andthereforeitcausesproblem indea.ifprofitsalescost,wemayusethevariablescostandsalesinsteadofcostand Profit.However,theproblemisnotthesameifwesimplyreplaceCostandProfitbyCostand Sales.Instead,wehavetore-definethewholeproblem,becausee.g.theefficientfrontierisnot thesameifweonlyreplacetheoldvariablesbythenewones. Furthermore, we establish the foundation of the linear transformations of input/output variables in DEA by introducing the relevant mathematical formulation. The proposed formulation is the natural extension of the DEA problem into the transformed spaces such thatthetransformedproblemisequivalenttotheoriginaldeaproblem.weshowthatnonsingulartransformedvariablesdonothaveaneffectontheoptimalsolutionoftheproblem. Thepaperispresentedinfoursections.Insection2,somebasicnotationanddefinitions aregiven,andinsection3,weconsidernon-singularlineartransformation,presentsome theory and motivate theoretical considerations with two examples. Singular linear transformationisdiscussedinsectionandconcludingremarksandgiveninsection5. 2 SomeTheory 2.1 Basics Inthissub-section,weintroducethebasicdefinitionsandconcepts.Denotetheindexsetof ndecision-makingunitsn{1,2,,n}.eachunitconsumesminputsandproducessoutputs. 1 Thenumberoftotalgrades refersheretothesumofexcellentandgoodgrades.

7 Letx andy standforthe(column)vectorofinputsandoutputs,respectively.we definetheproductionpossibilityset(pps)asfollows: = {(, ) canbeproducedfrom (2.1) where T consists of all feasible inputs and outputs. As usual, we assume more is better in outputs and less is better in inputs. We denote by = (,, )and = (,, )the matriceswiththeoutput-andinput-valuesoftheunitsoncolumns.furthermore,wedenote = 1,...,1. The traditional definitions for efficient and weaklyefficientpoints in set T are given as follows: Definition1.Point(, ) isefficient(non-dominated)iff(ifandonlyif)theredoesnot existanother(, ) suchthat, and(, ) (, ) Ifpoint(, ) isnotefficient,thenitissaidtobeinefficientordominatedhowever,if aninefficientpointisnotaninteriorpointintitmaystillbeweaklyefficient Definition2.Point(, ) isweaklyefficient(weaklynon-dominated)ifftheredoesnot existanother(, ) suchthat > and < To simplify notation, we occasionally refer to vector by and write = + Correspondingly,wedenote = Asthetransformationwillchangethenumericalvaluesoftheoriginalinput-andoutput- variables,definitionsandforefficiencyandweakefficiencyaretoorestrictive,becausethe new variables are not necessarily anymore maximized or minimized after linear transformation. In order to be able to define the dominance relationships in linearly transformedproblemweusethepointedpolyhedralconesindea. Definition3.Givensetofnon-zerovectorsc1c2,ck p k1,pointedpolyhedralcone Cisdefinedasconvexsetwhichconsistsofallnonnegativelinearcombinationsofvectorsc1 c2,ck = 0, =1,2,, (2.2) andforwhichc(-c{0}. 2 Directionsc1c2,ckarecalledthegeneratorsofconeCNotethatCcontainstheorigin andthedirectionscii1,2,,kemanatingfromtheorigin.whenitisnecessary,weuse notationc{0}toemphasizethattheoriginisthecone svertex.wemayalsoshifttheconecto 2Notation Creferstotheconewhichconsistsofallnonnegativelinearcombinationsofvectors-c 1-c 2,-c k

8 start from any point z p Then we write alternatively z C, z C{0} or C{z}. We occasionallyusethenotation-c{ztorefertheconezc Non-dominance(efficiency)andweaknon-dominance(weakefficiency)isnowdefinedas follows: Definition4.pointedpolyhedralcone generatedbysetofnon-zerovectorsd1d2,dk k1,iscalleddominatingconeifpoint issaidtobedominatedbyiff D{z0}andz0 Using the definition of pointed cones, the dominating conecan be written as = 0, =1,2,, and correspondingly = ) 0, = 1, 2,, Definition5vector isnon-dominatedinsetwithrespecttothedominating coneiffthesetd{z0}{z0}. Definition6vector isweaklynon-dominatedwithrespecttothedominating cone iff the set (z0int D){z0}, where int D refers to the interiorof cone D that is definedformally int = >0, = 1,2,,. (2.3) Ifpointzisnotweaklynon-dominated(weaklyefficient),thenitissaidtobestrongly dominated (strongly inefficient with respect to cone D. If point z is dominated (inefficient),butweaklynon-dominated,thenitissaidtobeweaklydominatedwithrespectto coned. Lemma1Assumez1z p, z1z2,are two points for which z1d{z2}.thenz2d{z1}. Proof.Becausez1D{z2i(atleastonei),i1,2,,ksuchthatz1z2 z2z1 )whichmeansthatz2-d{z1}.wedefinedthedominating conesuchthatd{z1(d{z1}){z1}.because z1z2hencez2d{z1}. Corollary1.Theassumptionthatconeispointedisnecessary.Otherwise,foreachpointz0 p, z1 p such that z0dominatespointz1andisdominatedbypointz1simultaneously. Proof. Assume that D is not pointed, i.e.{ } ({ }) { }.. Then z1z0such that z1 D{z0 and z1 (- D{z0}). Hence, z1 dominates z0 On the other hand, z1z0 )z0z1 z0dominatesz1 Remark.Theassumptionthatconeispointedmakesdominancewell-definedinthesense thatitisasymmetric.

9 2.2 LinearTransformation In this sub-section, we introduce some notation and present theoretical results, when non-singularlineartransformationisappliedtotheoriginaldataset.themainpointinthe considerations is that it is not enough to only transform the original variables (inputs and outputs),butitisalsonecessarytotransformthedominatingconeprovidedwewouldliketo preservetheoriginaldominanceinformation. Initially, we introduce some notation. The (1 p p2)linear transformation matrix is generally denoted by F and the production possibility set after transformation is T(F{z(Fz(F=FzzT Occasionally,wemaydenoteT(FFTwhereT(with boldletter)isdefinedas = where(, ) Weassumethatisoffullrowrank.Thus the non-singular F is and the determinant 0 The dominating cone after transformationisdenotedbyd(f 0, =1,2,,WeusenotationDto refertotheconeandthematrixwiththegeneratorsascolumns.thuswemaywrited(f FD Next, we will prove that applying non-singular transformation does not change the dominancerelationshipbetweenpoints. Lemma2.Ifpointz0Tisnon-dominated,weaklynon-dominated,orstronglydominated insettitsrolepreservesinnon-singularlineartransformation. Proof.LetFbenon-singularlineartransformation,andz0anarbitrarynon-dominated point,i.e.d{z0}{z0},whereisdominatingcone.assumethatfz0t(fisdominated insett(fafternon-singulartransformation.henceitfollowsthatzt(f),fz0such that (T(F) FD{Fz0}), where FD{Fz0 represents the transformed dominance cone D startingfromthepointfz0becausefisnon-singularlineartransformation, zt, z D{z0},and zz0thisisinconflictwiththeassumptionthatz0tisnon-dominated. Inthecorrespondingway,wemayprovetheresultsforweaklynon-dominatedandstrongly dominatedpoints. Lemmaproves that in applying non-singular transformations the status of DMU will be preserved.thisresultshowsthatefficientdmusshouldbeevaluatedasefficientaslongas thetransformationonthevariablesisnon-singular.weusethisresulttoshowthatvarious linearcombinationsofvariablescanbeconstructedfromoriginalvariables. 3 Non-SingularTransformationandSelectionofVariables Typically,theaimofDEAproblemistoestimatetheefficientfrontierofthegivendata, and also to compute the efficiency scores of DMUs relative to the frontier. Efficient DMUs build the frame of the efficient frontier and they have the property that there is no

10 combination 3 of DMUs that can dominate them. If the number of inputs and outputs are relativelylarge,thenmanyofdmusescapefrombeingdominatedbyotherdmusandwillbe recognizedefficient,andthusthediscriminationpoweroftheanalysisisweak.thiseffectis sometimes called the curse of dimensionality. The problem is the same as in regression analysis. The increasing of the number of independent variables never decreases the coefficientofmultipledetermination,buttheprediction(explanation)powerofthemodelis notnecessarilyimproves. InrealapplicationsofDEA,oneofthekeytasksofthedecisionmaker(DM)istochoose theminimalsetofinputsandoutputssuchthatallrelevantinformationistakenintoaccount, andnoessentialinformationislost.thedmmayfollowthebasicapproachesofaggregation andeliminationofinputsandoutputs,whicharecommonlyusedmethodsforimprovingthe discriminationpowerofdeaproblem(podinovskithanassoulis,2007). TheaggregationandeliminationofvariablesmakestheproblemdifferentinDEA.Clearly, thetransformationisverycriticalandchangesthefinalscoresofdmus.byeliminatingsome variables,welosetheirinformation,buttheaggregatedvariablesstillcarrytheinformationof originalvariables.ifthedmdoesnotremove anyofvariablesbutcarryoutnon-singular lineartransformationofthoseones,thenew variablescontainthesameinformationasthe originalvariablesandthusweexpecttogetthesameresultsfrombothproblems.however, usuallytheresultsdiffer,becausecommonpracticeisjusttoreplacetheoldvariablesbythe newonesandassumethatoutputsaremaximizedandinputsareminimizedsuchasinthe originalproblem. Dependingonthecontextoftheproblem,thedecisionmakeroftensubjectivelyselectsan acceptable set of variables, but if there are two different sets of variables with the same information but different representation, should the DM prefer one to another? In other words,iftwodatasetsare,basically,thesame,shouldwehavedifferentperformancescores forthedmus?thejustificationovervariablesshouldbedependentontheamountandtype ofinformationratherthanthewaytheydisplaythedata?wediscusstheissueinanexample below. Throughout this paper, we try to keep DEA considerations as simple as possible. That s whywedealwithanoutput-orientedvariablereturnstoscale(vrs)model(3.1)whichis definedin spaceandgiveninslightlymodifiedform(see,bankeretal.1984).even though in the following example we usevrs DEA model, since there issingle constant input,themodelisequivalenttothecorrespondingcrsdeamodel(knoxlovellpastor, 1999),thusbothmodelscanbeused,butinordertokeepthesamemodelinthediscussions ofthepaper,wepresentitasvrsdeaformulation. 3TheallowedcombinationofotherDMUsaredefinedbythereturnstoscaleassumptionofmodelandwhat isassumedabouttheproductionpossibilityset.

11 Inourformulation0,iftheunitisefficientorweaklyefficientand0,ifitisstrongly inefficient. max + ( + ).. =, =1,2,,, 4 + =, =1,2,,, (3.1a) =1,,, 0, where( Non-Archimedean ) 5 Notethatthedominatingconeof(original)model(3.1)isoftheform: = where I isunity matrix. The first I matrix is and stands for s outputs and the latter one is andtostandsforminputs. Themodel(3.1a)inthematrixformisasfollows: max +.. =, =1,,, where on-archimedean )and = and = (3.1b) Thusweseetheroleofthedominatingconeintheoriginal(nottransformed)DEAproblem. Inthenextsub-sectionwepresentan exampleexplaininghowselectinginput-andoutputvariablescanleadtoapplyingnon-singulartransformationonthevariables. 3.1 Howtoselectvariables? Let sfirstconsiderthesituationinwhichthevariables(inputsand/oroutputs)arelinearly dependentinsuchwaythateach variableoutofpvariables can bepresented aslinear combination of any other k variables. In this case, any set of variables carry the same information as all p variables. These types of variables do not cause any problem in some techniques like as in regression analysis. If those p variables are potential independent variables,wemayuseanykvariablesintheanalysisandthecoefficientofdetermination( isalwaysthesame.however,thesituationisnotthesameindea.evenifanysetofdifferent variables forms basis on k dimensional space and carry in the identical information, different variables produce different results (efficiency scores) provided that efficiency is Informula(3.1)subscript 0 referstotheunitunderconsideration. 5Formoredetailson Non-Archimedean,seeArnoldetal.(1998).

12 defined for each set intraditional way (see, Definition 1).Practical problems are not so simple, but considerations are applicable to the problems in which the assumption is approximatelytrue. Becauseeachsetofvariablescanbedefinedaslineartransformationfromanyotherset ofvariables,weshowthateachsetwilldefinethesameefficientfrontierprovidedthatwe apply the same transformation to the dominating cone (Definition 3) as well. We will first illustrate the problem and its solution by usingsimple example. We explain the problem withsomeexamplesandtheninsub-section3.3wepresenttherequisitetheory. Example1.AssumethatDMwouldliketoevaluatetheperformanceofstudentsbyusing theoutputs thenumberofexcellentgrades (EG), thenumberofgoodgrades (GG),and the number of total grades (TG). Those variables are clearly linearly dependent, because we assumetgeg+gg.thustwoofthemcarrynecessaryinformationweneed.considerthe sets{eg,tg}and{eg,gg}.thedataoftheexampleisshownintable1.weassumesingle constantinput,andtheoutputorientedvariablereturnstoscaledeamodel(3.1). Table1.Datasetwithoneinputandtwooutputs Variables DMUs Input EG GG TG (EG+GG) A B C D E Despite the fact that the two sets of outputs (EG and TG or EG and GG) have the same information about students, if the DM chooses = {, }as the output variables, the results differ from the case = {, }provided the traditional efficiency definition is used. We name the problems corresponding to the set of outputs and as and respectively. Figure shows the position of DMUs in and in panels (a) and (b), respectively. Since the input value of all DMUs is unity, we can illustrate DMUs and correspondingproductionpossibilitysetsintwodimensionalspaceusingonlytheoutput values(infigure1).theshadedareasaretheproductionpossibilitysetsandtheshowncones arethedominatingcones(definingtraditionaldominance).usingtheinterpretationofthe variables,weunderstandthatthetraditionalshapeoftheproductionpossibilitysetcontains an infeasible region (the region specified by gray dots). To understand this change in the productionpossibilityset,considerthattg=eg+gg(allarepositive)andthustgcannotbe lessthaneg.

13 (a) () Figure1.DMUsindifferentsettingsofoutputs.Inpanel(a)withEGandTGasoutputsandinpanel (b)withegandggasoutputs.thefrontiersaredeterminedbydeamodel.

14 ThesolidlinesinFigurearestandingfortheefficientfrontiersandthedashedlinesfor weakly efficient frontiers. As it can be seen from Figure 1, clearly by changing the set of outputsthestatusofdmuschange.forexample,dmuisweaklyefficientin whileitis efficientin anddmuisinefficientin butefficientin Consideringthefactthatboth problemshavethesameinformationaboutthestudents,weexpecttohavethesameresults for asfor Traditionally,theanalysisiscarriedoutwithonesetofvariablesanddifferent resultsareconsideredacceptable. ThereisreasontobelievethatDMprefersset = {, }fortheanalysis,because point(eg=8,gg=4,tg=12)isclearlybetterthanpoint(6,5,11).twoexcellentgrades moreitisbetterthanonegoodgrademore.moreover,point(9,3,12)isclearlybetterthan point(8,4,12).wewilldemonstratethatthechoiceof = {, }willleadtothesame results as using set = {, } provided that the relationship between variables (TG=EG+GG) is defined and applied on the mathematical programming of the problem, thereforetransformation{eg,tg{eg,ggshouldbetakenintoaccountinthedefinitionof thedominatingcone. Let sdenotethedatamatrixasmatrixconsistingofthevaluesofegandtgonthefirst tworows,andtheconstantinputonthethirdrowofmatrixz Z= Thedominatingconefortheproblemis = Let sconsidertheefficiencyofunite,i.e.thevaluesinthelastcolumn( inmatrixzwe presentthemodelforinthefollowingform: max = =1, (3.2) 0, 0, >0( Non-Archimedean ). The solution of problem 3.2 is = =1 = = = =0 =2.45 and = =0 It means that the unithas to improve proportionally its output values with 9.1% and in addition to improve the excellent grades with 2.45 to become efficient. The referencepointisunit(notd,whichisonlyweaklyefficient,butnotefficient)(figure1a). 10

15 Let snowanalyzethe effectofthelineartransformation{eg,tg{eg,gg}.ifweonly replacethevariablesegandtgbyegandgg,theproblemtobesolvedis max = =1, (3.3a) 0, 0, >0( Non-Archimedean ), where = = and = Thesolutionofthemodelis =0 =1 = = = =0and = = =0 Thusunitisdiagnosedefficient(see,Figure1b).Theresultisnotclearlyreasonableaswe explainedbefore.toseehowtheweightsareaffectedusingthedominatingconewewritethe multiplierformoftheproblem3.3abelow: min [6,5,1] , (3.3b) [6,5,0] =1, : where = and and refertoegandtg,respectively.thelastweight refersto theweightoftheinputvariable. Inproblem3.3bweseethattheweightsintheconstraint + 0areaffectedbythe transformation, but the transformation is not appeared in the objective function and the constraint[6,5,0] =1 Thus the result of this problem is not acceptable as the productionspossibilitysetistransformed,butthedmuunderassessmentisnot. Considernowthemodel,inwhichthedominatingconeisalsotransformedinadditionto thevariables.thetransformationmatrixfisasdefinedabove.thuswehavethetransformed data matrix =,the transformed dominating cone = = and the transformedproblemis max + s. t. 11

16 =1, (3.4a) 0, 0 >0( Non-Archimedean ). 6 6 andinthiscase 11 = Thesolutionofproblem(3.4)isexactlythesameasproblem(3.2).Figureillustratesthe situation. Figure2.DMUsandfrontierwithEGandGGasoutputs,consideringthetransformeddominating cone. Note we show the effect of applying the transformation by writing the dual model (multipliermodel)ofmodel(3.4a) min[6,11,1]. 0 (3.4b) [6,11,0] =1 12

17 , : The constraints for the multipliers are: and 6 Thus the multiplier of EG is required to be higher than GG in absolute values. It means that the variables{eg,gg}canbeusedinthedea-modelaswellprovidedthatthemultiplierofegis requiredtobehigherthanthatofgg,whichsoundsquitereasonable. We may use in the analysis either output variables EG and TG or EG and GG, but in the latter case, in the multiplier model the multiplier of EG is greater than that of GG. More generalconsiderationsaregiveninsub-section DealingwithIntervalScaleVariables Insomeproblems,non-singulartransformationispracticalwaytodealwithinterval scalevariables(halmeetal.,2002;dehnokhalajietal.,2010).transformationmaybeused toreplaceintervalvariablesbyratioscalevariables.aswehavedemonstratedintheprevious sub-section, the efficient frontier does not change provided the dominating cone is transformedaccordingly.weuseanexampletoillustratethetechnique. Example2.Let sconsidertheproblemconsistingofsixunitswhichareevaluatedwithone input(cost)andoneoutput(profit)(tableandfigure3a).salesisassumedtobecost Profit. Table2.Datasetforintervalscaleexample Variables DMUs Cost Profit Sales A B C D E F Profit is measured on an interval scale. It means that there is no theoretical basis to computeefficiencyscoresbasedonradialmeasurements(see,figure3a).pointsa,b,and are efficient andis only weakly efficient, and thus they cause no problem. The efficiency scoremaybedefinedtothemtoequalone.technically,wemaycomputetheefficiencyscore forpointaswell,butitsinterpretationisnotclear.instead,forpointwemaycomputethe 6It is important to note that since we need to incorporate the dominating cone in the formulation of problems, the weights of outputs are represented as negative values. This does not have any effect on the optimumresultsofthe problems.theabsolutevalues oftheweightsarecorrespondingtotheweightsinthe traditionalformulationofdeaproblems. 13

18 distancefromtheefficientfrontierandeventhereferencevalueontheefficiencyfrontier(not radial),butnotanefficiencyscoreasusually. (a) (b) Figure3.DMUs,dominatingconesandproductionsetsinpanel(a)Cost Profitspace,andinpanel (b)cost-salesspace. 14

19 However,wemaysimplycarryoutnon-singularlineartransformation.Profitisreplaced bysales(see,figure3b),butinadditiontothatwehavetotransformthedominatingcone.if wepresentourdatamatrixintheform,whichhastheinput(cost)inthefirstrow,andthe output(profit)inthesecondrow: = , then the transformation matrix is simply = 1 0 The dominating cone for the original 1 1 problem is = and hence = = and = = OurVRS-model 7 formeasuringtheefficiencyofunitisnowasfollows: max = 5 3 =1 (3.5), >0,( Non-Archimedean ). Thesolutionofthemodelis =2 =1 =0 = A,B,C,E,F =1and =0Note thatthereferenceunitisd,notvirtualunitonthelinesegmentstartingfromandpassing throughunitf,becausethelinesegmentisonlyweaklyefficient.becausesalesismeasured ontheratioscale,wemaycomputeitsefficiencyscore:1/(1+2)0.33.wemayalsocompute an inefficiency score for Profit by replacing vector 3 0 by vector3 The model is called 5 combinedmodel(see,joroetal.(1998))thesolutionofthismodelis = = = and =0 =,,, = =0 The solution means that unit has to increasesalesanddecreasecostby52.8%forbecomingefficient.thusfeasiblereference valueforprofitis andthecorrespondingvalueforcostis2.36. HerebywehaveobtainedmeasureforProfitandfoundreasonablereferencevalueand interpretationforit. There are also available other techniques to deal with interval variables. For instance Halmeetal.(2002)proposedmethodtoreplaceanoriginalintervalscalevariablebythe difference of two ratio scale variables.however, this approach may make an inefficient or weakly efficient (not efficient) unit efficient. We can demonstrate this feature with the followingexample(see,datasetintable3).assumethatwehaveinitiallyoneintervalscale outputandoneratioscaleinput,andouraimistoconsiderefficiencyoftheunitsbyusingan 7VariableReturnstoScale 15

20 output-orientedvrs-model.wesplittheoutputvariableintotwopartssuchthattheoriginal outputisthedifferenceoftworatioscalevariables.theoldvariableisreplacedbythefirst newvariableandthesecondoneisdefinedtobenewinput.thuse.g.theoldoutputofunit is received asdifference 4-3. By the original model, we obtain that unitsandare efficient, andweaklyefficient(notefficient) andisinefficient.withthenewmodel,we obtainthatallunitsareefficient.weassumethatweknowthatthenewouputandinputare measuredonratioscale. Table3.Splittinganintervalscaleoutputvariableintooneratioscaleoutputandoneratio scaleinputvariable OriginalVariables NewVariables DMUsOutput Input CurrentStatus OutputNew InputNew InputOrig. NewStatus A 1 1 Eff Eff. B 2 2 Eff Eff. C 2 3 WeakEff Eff. D 0 2 Ineff Eff. Dehnokhalaji et al proposed another way to measure efficiency, when variables are measuredoneitherintervalorordinalvariables.themethodisbasedontheideatolocate linearvaluefunctionpassingthroughtheunitunderconsiderationsuchthatthenumberof better units is minimal. The method recognize efficient, weakly efficient, and strongly inefficientunit.theonlyproblemisthattheefficiencymeasureisnotstandardone. 3.3 SomeTheory Inthissectionweprovethattheoriginalproblemandthetransformedproblemhavethe samesolution.let sconsiderclosertheformulation(3.1b): Theorem 1.The problem (3.6) hasfinite solution iff the problem (3.1b) hasfinite solution,wherematrixfisnon-singular -matrix.incasethesolutionisfinite,theyare identical. (3.6). By multiplying the constraints = by the inverse of non-singularmatrix,weobtainthecorrespondingconstraintsof(3.1).henceitfollowsthat 16 max +.. = (3.6) =1, >0,( Non-Archimedean ). Proof.If{,, }isthefiniteoptimalsolutionofproblem(3.1b),thenitisalsofeasible solutiontoproblem(3.6).thus where,, istheoptimalsolutionofproblem

21 andfurther = Moreover, = and = Thebothsolutionsarefiniteor thebothonesareunbounded. The result of Theoremshows that applyingnon-singular transformation does not changetheefficiencyscoresofdmus.thisisstrongerthanthelemmawhichconsidersonly thestatusofdmus. Asresultfromthistheoremweseethatiftwosetsofvariablescanbeobtainedfromeach otherbyapplyingnon-singulartransformation,thentheresultsoftheefficiencyevaluations arethesame.thisresultisinaccordancewiththecommonsensethatiftwosetsofvariables conveytheexactlythesameinformation,theresultoftheevaluationsshouldbethesame. 4 SingularLinearTransformationoftheVariables So far we discussed the case of applyingnon-singular transformation, which does not changethenumberofvariablesintheproblem.inotherwords,thedimensionoftheproblem doesnotchangeundernon-singulartransformation.butifthetransformationissingular,it meansthatsomeinformationwillbelostafterthetransformation,andthedimensionofthe problem will decrease. Despite the fact that the amount of information will be smaller, singular transformations builduseful technique for reducing the number of variables to overcomethecurseofdimensionality. Whentheproblemconsistsoftoomanyvariables,thenthenumberofvariableshastobe reduced.theyaretwotechniqueswhichareusuallyusedforthispurpose: 1. Selectingsubsetofvariableseitherobjectivelyorsubjectively 2. Constructingthelinearcombinationsofthevariableseithersubjectivelyorobjectively. Therearemanywaystocarryoutthoseselections.However,someofthemleadtowrong results.nextweconsidercloserthosetwodifferentmaintechniques.thebothcasescanbe consideredassingularlineartransformation. 4.1 Selectingsubsetofvariablesfromamongpotentialvariables Let sconsiderourstudents performanceexample.wenoticedthatthevariablesegandtg bestdescribetheperformanceofthestudents(figure1a).studentsandwerediagnosed efficient, studentsandweakly inefficient, and studentwas strongly inefficient. If we havedecidedtouseonlyoneobservedoutputvariable,wehavetochooseeitheregortg. ThedatacanbereadfromTablefromcolumnsEGandTG.Quitemanyofthechangeswill happen.forinstance,onvariableegweaklyinefficientstudentwillbecomeefficient(a), weaklyinefficientstudent(d)andanefficientunit(c)becomestronglyinefficient,anefficient student (B) remains efficient, and finally, strongly inefficient student (E) will stay strongly inefficient.thecorrespondingchangescanbeobservedonvariabletgaswell. 17

22 Todropvariablesistechnicallycorrectmethod,becausethedominatingconeremains pointedconeinsub-space,andthusitisfeasibledominatingcone.whichvariablesarethe bestonestocarryonthebestperformanceisanopenquestion.jenkinsanderson(2003) proposedmethodtoomitthevariablesthatarehighlycorrelated.themethodprovides systematicandobjectivemethodtochoosevariables.itbasedonthestatisticalpropertiesof thevariablesandthusitisnotperhapstobestwaytochoosevariablesfordataenvelopment Analysis.However,itistechnicallycorrectandthusdoesnotcausecompletelywrongresults such as thatstrongly inefficient unit becomes efficient. This may happen, when for the reductionofvariablesareusedprincipalcomponentanalysis,whichwewillconsiderinthe nextsub-section. 4.2 Reducingdimensionsbyusingprincipalcomponentanalysis AnothercommonlyusedtechniquetoreducethenumberofvariablesinDEAisPrincipal Component Analysis (PCA) (see, e.g. Adler Golany 2001, 2002). Principal Component Analysis is statistical multivariate method and it seeks the best standardized linear combinations of the original variables in the sense that best is defined by maximizing variance.largevariance separatesout theunitsindea,butnotnecessarilyonthebasisof efficiency. Actually, the purpose of PCA is suitable to DEA as well: PCA looksfew linear combinations which can be used to summarize the data, losing in the process as little information as possible. The attempt to reduce dimensionality can be described as parsinomous summarization of the data. (Mardia et al. 1988, p. 213). However, to lose information doesnotmeanindeathesameasinstatistics. TheproblemofusingPCAtoreducethedimensionisillustratedbyFigure4.Inpanels(a) and (b) of Figure 4, two different random DEA problems are illustrated and principle componentsareshownasarrows.thefirstandthesecondpcsareshowsaspc1andpc2.if weusethefirstprinciplecomponentintheanalysis,theresultsarequitesatisfactoryinpanel (b)andcompletelyuselessinpanel(a).inpanel(a),someefficientunitsarediagnosed very inefficient. Instead, in panel (b) efficient units are almost efficient and inefficient units inefficient. 18

23 a) b) Figure4.IllustrationofPCAoftwodifferentsettings. ThusweseethatsincethebasicfoundationsofapplyingsingulartransformationsonDEA problem is not laid mathematically, the available approaches may lead to unacceptable results. How to makesingular linear transformation in suchway that the results are reasonableisthetopicofourongoingresearchproject. 5 Conclusions In this paper, we have studied the use of the linear transformation of variables in DEA problems. We have introduceddominating cone concept,which plays an essential rolein transformingvariables.thedominatingconeisrequiredtobepointed.ifthispropertywill lose in transformation, the results may be completely misleading. non-singular transformationdoesnotchangethestatusofunit.hence,choosinganylinearcombination ofvariablesdoesnotchangetheresultofproblem,aslongasthedecisionmakerkeepsthe transformationnon-singular. An interesting topic for future research is study which kind of the projection of the dominating cone causes the loss of pointed property in singular transformation. Another interestingresearchquestionis:howtoreducethedimensionsoftheproblemwithlosingas littleinformationaspossible.whatisgoodmeasureforthisinformation: 19

24 Therankorderofefficiencyscores? Thenumberefficiencyunits? etc. EventhoughreducingthedimensionofDEAproblemisessentiallyinterestinganduseful, there isvery little mathematical foundationfor approaches. We presented this issue in simple example demonstrating risk to have useless results when used single linear transformation. As one interesting future research topic, the properties ofsingularlinear transformationanditseffectsonthedominatingconemustbestudiedandtheconditionsfor acceptablesingulartransformationsshouldbeestablished. References Adler, N., Golany, B Evaluation of deregulated airline networks using data envelopmentanalysiscombinedwithprincipalcomponentanalysiswithanapplication towesterneurope.europeanjournalofoperationalresearch Adler, N., Golany, B Including principal component weights to improve discriminationindataenvelopmentanalysis.journaloftheoperationalresearchsociety Arnold, V., Bardhan, I., Cooper, W.W.,Gallegos, A Primal and Dual Optimality in ComputerCodesUsingTwo-StageSolutionProceduresinDEA.p InAronson,J.E., Zionts,S.(eds.),OperationsResearch:Methods,Models,andApplications.QuorumBooks, Westport,CT. Banker, R.D., Charnes, A., & Cooper, W.W Some Models for Estimating Technical and Scale Inefficiencies in Data Envelopment Analysis. Management Science 30, Charnes, A., Cooper, W.W.,Rhodes, E Measuring the efficiency of decision making units.europeanjournalofoperationalresearch Charnes, A., Cooper, W.W.,Rhodes, E Measuring the efficiency of decision making units.econometricsjournal3339. Dehnokhalaji,A.,Korhonen,P.J.,Köksalan,M., Nasrabadi,N.,Wallenius,J.2010.Efficiency analysistoincorporateinterval-scaledata.europeanjournalofoperationalresearch Halme, M., Joro, T.,Koivu, M Dealingwith interval scale data in data envelopment analysis.europeanjournalofoperationalresearch Jenkins,L.,Anderson,M.2003.multivariatestatisticalapproachtoreducingthenumberof variables in data envelopment analysis. EuropeanJournalofOperationalResearch

25 Joro, T., Korhonen, P.J.,Wallenius, J Structural Comparison of Data Envelopment AnalysisandMultipleObjectiveLinearProgramming.ManagementScience KnoxLovell,C.A.,Pastor,J.T.1999.RadialDEAmodelswithoutinputsorwithoutoutputs. EuropeanJournalofOperationalResearch Mardia,K.V.,Kent,J.T.,Bibby,J.M.1988.MultivariateAnalysis.AcademicPress. Podinovski, V. V.,Thanassoulis, E Improving discrimination in data envelopment analysis:somepracticalsuggestions.journalofproductivityanalysis

26 Aalto-BE5/2013 ISBN (pdf) ISSN-L ISSN ISSN (pdf) AaltoUniversity SchoolofBusiness DepartmentofInformation andserviceeconomy BUSINESS+ ECONOMY ART+ DESIGN+ ARCHITECTURE SCIENCE+ TECHNOLOGY CROSSOVER DOCTORAL DISSERTATIONS