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A Decomposition of Cost Efficiency under Weight Restrictions in DEA

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2000▲年度日本オペレーションズ・リサーチ学会 春季研究発表会

ADecompositionofCostE臨ciencyundeTⅥ毎ightR′eStrictionsin

DEA

O1302170N■ationalGradua七e‡ns七itu七eforPolicyStudies TONEKaoru

1.ht,rOduction

htheoptimalweightofDEAmodelsforinef一

色cientDMUs,WemaySeemanyZCrOS−Showing

thattheDMUhasaweakn飴Sinthecorresponding

iterhscomparedwithother(efRcient)DMUs・AIso,

1argedifrbrencesinweights鉦omitem七oitemmay

be aconcem.

Theimpositionofweightrestrictionsin】DEAhas

been recognized as one oftheimportant触tors

whenapplyingDEAtoac七ualsituationsandsev−

eralmodels are developedfor this purpose.See 【2,3,4,5,6ト

As instatistics or o七her empirical1y oriented

methodologies,thereisaprobleminvolvingdegre6

0ffreedom,WhichiscompoundedinDEÅbe ofitsorientationto柁ぬtiveeBiciency.Thenumber ofdegreesoffreedomw丑=血crea紀with七henumber

ofDMUsanddecreaふewiththenumberofinputs

andoutputs.Aroughruleofthumbwhichcanpro− videguidanCeisasfollows[1]: れ≧max(mxき,3(m+8)), wheren=numberofDMUs,m=numberofinputs ands=numberofoutpu七s. Theweightrestrictionapproachhelpstodiscrim−

inateamongDMUsineL五ciencyevenif七heaboⅧ

inequalityishardtohold・Thepurposeofthispa− peristodevelopade00mpOSitionofcoste伍ciency intoscale,pureteChnicalandallocativeoneswhen weightrest;ictionsareimposed・

2.WeightRestric七ions

Asimplerestric七ionsuchasvl≦2v2Canbeex▲

pressedusingPaswellasamorecomplicatedcon− Straintsuch8SVl+v2≦3v3CanbeincludedinP. Asfortheoutput,thematriⅩQhasasimilarrole forrestrictingtheregionofoutputweights.Here,

wedealwithhomogenousinequalityconstraints(4)

and(5),i・e・,COnStraintswiththeright−hand be− ing2;erO.nlrther,WeaSSumethattherestrictions (4)and(5)areconsistentand(DWR。)haLSafinite positiveoptimum.Thedualof(DWR。)is: (Ⅳ月。) minβ

St. ぬ。−∬Å+ダガーβ●=O

y入+QT−β+=y。 A≧0,8 ̄≧0,β+≧0, ︶ ︶ ︶ ︶ 7 00 9 0 ︵ ︵ ︵ l ′−■lヽ

Where the vectors訂and T are nOnnegative dual

Variablescorrespondingtotheconstraints(4)and

(5),reSpeCtively.

3.ProductionPossib止itySetofWRModel

Theproductionpossibilityset掬oftheWRmodel

isdefinedasasetofsemipositive(a;,y)asfo1lows: 昂〝=((雷,財)l‡≧ズ入一夕訂,封≦y入+QT),(11) WbereÅ≧0,河≧叫andT≧仇 4.SeveralE凪ciencyMeasl∬eS Wbde丘nethefechnicalqPiciencyofDMU($。,y。) underWR a$七heoptimalsolution O寧to(WR。)・ TheptL7Y teChnicalq9;ciencye妄Of(ェ。,y。)isde一

丘nedast.heoptimalsolutionoftheaboveLPunder

thev甜iablere七urnS−tO−SCalcassumption,i.e, eÅ=1.

of七he CCR Ⅵ屯consider thefo1lowlng eXtenSion

model: (1)Eviden七1y・WehaNe (2)恥叩①S紘五①皿且1≧埠≧β○・

(3)Thescale郷ciency。*unde,Weightres七ricti。nSis

(4)de丘nedas (5) α傘=β寧/埠・ (12) (6) Let七hecostvectorofinput3;。forDMU(3。,y。) (βⅣ月。) max z=叫‰ St. ℡∬。=1 一肌方+血y≦① 凱P≦① 髄Q≦0 り≧¢,%≧0, bec。.SoIvethefo1lowlngLP: mln Co芯

wherethe matrices P and Q are associated with

weightrestricもionsasdescribedbelow・ (13)

−112 −

(2)

︶ ︶ ︶ 4 5 6 1 1 1 ︵ ︵ ︵ st. 諾≧ズ入一夕汀

y。≦y入+QT

入≧0,汀≧0,丁≧0.

Tablel:ASampleDecomposition

DMU Cost−E仔 PTE Scale Allocative

Hl O.97 H2 1 H3 0.86 H4 0.50 H5 0.89 H6 0.81 H7 0.81 H8 1 H9 0.42 HlO O.98 Hll O.75 1 0.99 0.98 1 1 1 0.90 0.96 0.98 0,53 0.99 0.95 1 0.96 0.93 0.97 0.86 0.96 0.82 0.99 0.98 1 1 1 0.47 0.94 0.95 1 1 0.由

0・88 ・・

− Usingtheoptimalvaluec。X■fortheaboveLP,We definethecostq伊ciencyでOfDMU(3:。,y。)under

WRby

7●=C。エ∵c。諾。・ (17) Wthaveaproposition

Proposition2TYle00Sfq伊ciency7●under WR

れe〃ere〇Ceedβ仇efec九乃血Jq伊c由れぢyβ■. Theallocative(ガiciencyα■underWRisdefinedaB theratioofcostvs.tedlnicale瓜cienciesas H12 0.96 H13 0.89 1 0.97 0.98 H14 0.85 H15 0.74 H16 0.81 H17 0.97 1 0.91 0.97 1 0.86 0.98 0.87 0.89 0.95 1 0.82 0.99 1 1 0.97 α●=7●/β●. (18)

5.ADecomposition

UsingtheabovedefinitionswehaⅧadecomposi−

tionofthecoste瓜ciencyunderWRasfo1lows: 7書 = β●×α● (19) = 埠×J◆×α■・ (20) Ⅵ∋rballywehave

Cost Eff=PTEx ScaleEffxA1locativeEff.

6.Implication

C抑甲柁九e†ば血e 飛出 血統肋deヱβ,Appおcか

肋間月(水代nαβαmd上)且A一方Oれerβqβぴα陀−, KluwerAcademicPublishers.

【2】Pedrqja−Chaparro,F.,J.Salinu−Jiminezand P・Smith,1997,“On the Role of Weight RestrictionsinDataEnvelopment Analysis,竹 山umαJ毎夕和血c土油土yAn¢如由,8,215−230.

【3]Roll,Y.,W.D.Cook and B.Golany,1991,

“Contro11ingFhctorWbightsinDataEnvelop− mentAnalysis,”1mThnsactions,23(1),2−9. 【4】Roll,Y.,and B.Golany,1993,“Alternate MethodsofneatingFactorWもightsinDEA,” Om印αご九ねmα如乃αJ力むmαJ扉鵬乃叩eme乃f βcie乃α,21,99−109. 【5]Thompson,R.G.,L.N.Langemeier,C−T.Lee,

E.Lee and R.M.Thral1,1990,“The Role of MultiplierBoundsinEfRciencyAnalysiswith

Application toI(ansas Fhrming,”Joumalqf

助0乃8mefわcち46,93−108. 【6]Thompson, R.G., F.D. Single− ton,Jr・,R.M.Thral1,andB.A.Smith,1986, 以ComparativeSiteEvaluationsforLocatin写a High−EnergyPhysicsLabinTbx璽,”九terJhce, 16,35−49. 【7】Tbne,K.,1999,“OnReturnstoScaleunder

Wbight Restrictionsin DEA,”Research Re−

port,GRIPS. The above−mentioned decomposition,Whichis

unique,depicts the soufces ofine伍ciency,i.e.,

whetheritiscausedbyine瓜cientoperation(PTE) Or bydisadvantageous conditiondisplayd bythe

scalee用.ciencyorbyallocativeine氏ciency.

AIso,WeCaninterpretthevaluedefinedby(Cost Eff)/(ScaleEff)as“scale−adjusted”coste侃ciency which is the

product

Ofpuretechnicale氏ciencyandallocativee侃ciency.

7.AnExample

Tablelexhibits a sample of de00mpOSition.

We wulshow detai1ed data at the Conference.

J8.Cdncl血

Asane血ensionofresearchesinDEA−e伍ciencyun−

derweightrestrictionswehavedemonBtratedacost

e侃ciency modelunder WR circurhstanCeSalong

Withitsdecomposition.ThiswillenhanCepraCtical

usageOfDEAresults.Seealso【7]forcharacteriza−

tionsofreturnStOSCaleunder WR.

References

[1】Cooper W・W.,L・M・Seiford and K・Tbne (1999),加ゎ 助〃eJopme乃f A†相加由 一 A

−113 −

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