GRIPS Plicy Ifrmati Ceter Discussi Paper : 09-2 A epsil-based measure f efficiecy i DEA revisited -A third ple f techical efficiecy- Karu Te Natial Graduate Istitute fr Plicy Studies 7-22- Rppgi, Miat-ku, Tky 06-8677, Japa Miki Tsutsui Cetral Research Istitute f Electric Pwer Idustry 2-- Iwad Kita, Kmae-shi, Tky 20-85, Japa Abstract I DEA, we have tw measures f techical efficiecy with differet characteristics: radial ad -radial. I this paper we cmpile them it a cmpsite mdel called epsil-based measure (EBM). Fr this purpse we itrduce tw parameters which cect radial ad -radial mdels. These tw parameters are btaied frm the ewly defied affiity ide betwee iputs r utputs alg with pricipal cmpet aalysis the affiity matri. Thus, EBM takes it accut diversity f iput/utput data ad their relative imprtace fr measurig techical efficiecy. Keywrds: Data evelpmet aalysis, Radial, N-radial, CCR, SBM, EBM, Pricipal cmpet aalysis. Itrducti DEA (data evelpmet aalysis) is a data drive tl fr measurig efficiecy f decisi makig uits (DMU) ad shws a sharp ctrast t s-called parametric methds such as SFA. The latter methds assume specific prducti fucti frms t be idetified. This assumpti is t s reasable i several istaces ad aspects. Sice DEA ca deal with multiple iput vs. multiple utput relatis i a sigle framewrk, it has bee becmig a methd f chice fr efficiecy evaluati i recet days. Hwever, DEA has several shrtcmigs t be eplred further. I DEA, we have tw measures f techical efficiecy with differet characteristics: radial ad -radial. Histrically, the radial measure, represeted by the CCR mdel (Chares, Cper ad Rhdes [5]), was the first DEA mdel, whereas the -radial mdel, represeted by the SBM mdel (slacks-based measure by Te [8], see als Cper et al. [6]) was a latecmer. Fr istace, i the iput-rieted case, the CCR deals maily with prprtiate reducti f iput resurces. I ther wrds, if the rgaisatial uit uder study, als kw as a DMU, has tw iputs, this mdel aims at btaiig the maimum rate f reducti with the same prprti, i.e. a radial ctracti i the tw iputs that ca prduce the curret utputs. I ctrast, the -radial mdels put aside
GRIPS Plicy Ifrmati Ceter Discussi Paper : 09-2 the assumpti f prprtiate ctracti i iputs ad aim at btaiig maimum rates f reducti i iputs that may discard varyig prprtis f rigial iput resurces. I this paper, after itrducig radial ad -radial mdels briefly, we prpse a cmpsite mdel which cmbies bth mdels i a uified framewrk. This mdel has tw parameters: e scalar ad e vectr. I rder t determie these tw parameters, we itrduce a ew affiity ide assciated with iputs r utputs. We apply pricipal cmpet aalysis t thus defied affiity matri. This paper uflds as fllws. I Secti 2, we briefly survey radial ad -radial mdels i DEA. I Secti 3, we prpse the epsil-based measure f efficiecy (EBM). EBM eeds tw parameters. After bservig tw etreme diversities f dataset, we itrduce a ew crrelati cefficiet called affiity ide i Secti 4. We utilize this ide fr defiig affiity matri amg iput/utput data. Frm this matri we derive tw parameters fr EBM i Secti 5. We discuss ratiality f the scheme i Secti 6. We demstrate illustrative eamples i Secti 7. I Secti 8 we eted the mdel t ther rietatis ad variable returs-t-scale evirmet. We cclude this paper i Secti 9. 2. Radial ad -radial measures f efficiecy I this secti we itrduce the CCR ad SBM mdels as represetative radial ad -radial measures f efficiecy respectively, ad pit ut their shrtcmigs. Thrughut this paper, we deal with DMUs ( =, K, ) havig m iputs ( i =, K, m) ad s utputs ( r =, K, s). The iput m ad utput matrices are deted by X = { } R s ad Y = { y } R X >0 ad Y>0. 2. The CCR ad SBM Mdels We briefly eplai the CCR ad SBM mdels, ad cmpare their iefficiecy status. (a) The CCR Mdel i r, respectively. We assume The iput-rieted CCR mdel evaluates the techical efficiecy the fllwig liear prgram: θ f DMU (, y ) by slvig [CCR-I] θ = mi θ θ, λ, s () subect t 2
GRIPS Plicy Ifrmati Ceter Discussi Paper : 09-2 θ = Xλ + s y Yλ λ 0s, 0, where λ represets the itesity vectr ad s detes the -radial slacks. Usually, we slve [CCR-I] i a tw phase prcess. I the first phase, we slve [CCR-I] ad btai θ (weak efficiecy). The, i the secd phase, we maimize m si i (2) i terms f λ ad s, subect t (2) ad θ = θ. (b) The SBM Mdel Here, we ctiue with iput rietati csistet with ur epsiti f the CCR mdel i the precedig paragraph. The iput-rieted SBM mdel uder the cstat returs-t-scale assumpti evaluates the efficiecy τ f DMU (, y) by slvig the fllwig liear prgram where the abbreviatis I ad C idicate Iput-rieted ad Cstat-returs-t-scale, respectively. [SBM-I-C] τ = mi m subect t m s i i i iλ i = = + s ( i =, K, m) i iλ = y y ( i =, K, s) λ 0( ), s 0( i), i (3) whereis the itesity vectr, ad s represets -radial iput slacks vectr. Let a ptimal sluti f [SBM-I-C] be ( λ, s ). The, the bective fucti ca be rewritte as m i s i i τ =. m (4) Hece the SBM scre τ is the average f the cmpet-wise reducti rates which may vary frm e iput t ather. The SBM mdel is -radial. O the ther had, as ted i (2), the CCR scre θ satisfies the relatiship θ = Xλ + s = + s. Hece, we have i + si i θ = ( i). (5) The cmpet-wise reducti rates are the same fr all iputs. This same prprtial reducti rate, i.e. radial reducti rate, is the CCR scre. Betwee the SBM τ ad the CCR θ we have the iequality τ θ. See Te [8] fr mre details f their cmpariss. 3
GRIPS Plicy Ifrmati Ceter Discussi Paper : 09-2 2.2 Shrtcmigs f the radial ad -radial mdels I this secti we pit ut shrtcmigs f the radial ad -radial DEA mdels. (a) Shrtcmigs f the CCR Mdel The mai shrtcmig f the CCR mdel is the eglect f -radial slacks s i reprtig f the efficiecy scre θ. I may cases, we fid a lt f remaiig -radial slacks. S, if these slacks have a imprtat rle i evaluatig maagerial efficiecy, the radial appraches may mislead the decisi whe we utilize the efficiecy scre θ as the ly ide fr evaluatig perfrmace f DMUs. Furthermre, as t the prprtial chage θ, if we emply labr, materials ad capital as iputs, sme f them are substitutial ad d t chage prprtially. The radial (CCR) mdel cat cpe with such cases prperly. (b) Shrtcmigs f the SBM Mdel Sice mdels such as SBM capture the -radial slacks directly, the ptimal efficiecy value τ accuts fr the -radial slacks which are t csidered i the radial mdels. The SBM-precti t the efficiet frtier is defied by = s. Thus, the prected DMU may lse the prprtiality i the rigial because s is t ecessarily prprtial t. This is characteristic f the -radial mdels, ad if the lss f the rigial prprtiality is iapprpriate fr the aalysis, the this becmes a shrtcmig fr -radial mdels. Yet ather equally sigificat shrtcmig f SBM arises frm the ature f the liear prgrammig sluti, where the ptimal slacks ted t ehibit a sharp ctrast i takig psitive ad zer values. See Avkira et al. [4] fr mre detailed cmpariss f the zer ad -zer patters i the ptimal slacks i the SBM mdel. 3. A epsil-based measure f efficiecy (EBM) As pited ut i the precedig secti, bth radial ad -radial mdels have merits ad demerits regardig the prprtiality f the iputs/utputs chage. I this secti, we prpse a cmprmised mdel called epsil-based measure (EBM) which has bth radial ad -radial features i a uified framewrk. We defie the primal ad dual pair [EBM-I-C] ad [Dual] as fllws: [EBM-I-C] γ = mi θ ε θ, λ, s subect t θ m i i ws i (6) Xλ s = 0 (7) Yλ y λ 0 s 0. (8) [Dual] 4
GRIPS Plicy Ifrmati Ceter Discussi Paper : 09-2 γ = ma uy v,u (9) where subect t v (0) = vx + uy 0 () v i ε w i ( i =, K, m) (2) i u 0, wi is the weight (relative imprtace) f iput i ad satisfies m i w = ( w 0 i), ad i ε is a key parameter which cmbies the radial θ ad the -radial slacks terms. Parameters ε ad ws the term i i i w must be supplied prir t the efficiecy measuremets. As ca be see frm i the bective fucti f [EBM-I-C], i i s is uits-ivariat ad s wi shuld be a uits-ivariat value reflectig the relative imprtace f resurce i. We will discuss this subect i the succeedig sectis. [Prpsiti ] γ i [EBM-I-C] satisfies γ 0ad is uits-ivariat, i.e. γ is idepedet f the uits i which the iputs ad utputs are measured. [Prpsiti 2] If we set ε = 0 i [EBM-I-C], the it reduces t the iput-rieted CCR mdel. [Prpsiti 3] If we set θ = ad ε = i [EBM-I-C], the it reduces t the iput-rieted SBM mdel. Thus, [EBM-I-C] icludes the radial CCR ad the -radial SBM mdels as special cases, but it is basically -radial. The cstraits (0) ad (2) lead t m v i i ε = v =. Thus, ε must be t greater tha uity. [Prpsiti 4] [EBM-I-C] ad [Dual] have a fiite ptima fr ε [0,]. [Prpsiti 5] Fr ε >, [Dual] has feasible sluti ad [EBM-I-C] has ubuded sluti. [Prpsiti 6] γ is -icreasig i ε. [Defiiti ] (EBM iput-efficiecy) DMU is called EBM iput-efficiet if γ =. 5
GRIPS Plicy Ifrmati Ceter Discussi Paper : 09-2 [Defiiti 2] (EBM precti) Let a ptimal sluti t (6)-(8) be ( θ, λ, s ). We defie the precti f DMU (, y ) as fllws. [Prpsiti 7] = = θ = Yλ. Xλ s y The prected DMU (, y ) is EBM iput-efficiet. (See Appedi A fr a prf.) (3) [EBM-I-C] ca be maipulated i ather frm by itrducig a variable γ = mi ( ε ) θ + ε θ,, λ, s subect t Xλ = 0 = θ s Yλ y λ 0s, 0. m wi i i = θ s as fllws. This frmulati idicates that γ is btaied as the ptimal iterally dividig value betwee the radial θ ad the -radial term m wi i / i. Sice θ is t restricted, its ptimal value ca be greater tha, ad hece the ptimal is t ecessarily less tha r equal t. We tice that the cmpsite sigle stage apprach like the EBM was cmmeted i Ali ad Seifrd [3] ad further develped by Jhs ad Ruggier [7]. Hwever, ur bective is quite differet frm the precedig es as ca be see i the fllwig sectis. 4. Hw t determie epsil ad weights I EBM, the values f ε ad (4) w play the cetral rle fr evaluatig efficiecy f DMUs. We wuld like t determie them frm the data set (X, Y), sice DEA is a data drive methd. I this secti firstly we bserve tw etreme cases. The we itrduce a affiity ide betwee tw vectrs which replaces the Pears s crrelati cefficiet. 4. Tw etreme cases () Narrw rage case Figure plts a eample f data ccerig iputs ad 2 ccetratig i a arrw rage. If all iputs ad utputs g alg with the similar behavir, the assumpti f prprtial (radial) mdel ca be effected. Thus, i such case, we have ε 0ad the CCR mdel is a valid chice. θ 6
GRIPS Plicy Ifrmati Ceter Discussi Paper : 09-2 2 0 Figure : Narrw rage case (2) Widely scattered case I the ther etreme case, if the bserved data scatters widely as eemplified i Figure 2, the -radial mdel ca be applied. Thus, we have ε ad the SBM mdel with θ = is a chice althugh we d t shut ff the assumpti f radial mdels depedig the characteristics f prblems. 2 0 Figure 2: Widely scattered case These etreme cases suggest that ε ca be determied i the ctet f the degree f crrelatis amg iputs (utputs). Several authrs, e.g. Ueda ad Hshiai [9] ad Adler ad Glay [, 2] amg thers, utilized crrelati matri f iputs (utputs) ad applied pricipal cmpet aalysis (PCA) t DEA. Their mai bectives were itegrati f iputs (utputs) t ther represetative idicatrs. 7
GRIPS Plicy Ifrmati Ceter Discussi Paper : 09-2 I this paper, we emply similar but differet crrelati matri as described i the et secti i rder t gauge affiity amg iputs which will be utilized t estimate parameters ε ad i the EBM. 4.2 Diversity ide ad affiity ide Let R ad R a b be tw psitive vectrs with dimesi. They represet bserved values fr + + certai iput items ver DMUs. We defie a affiity ide S(, ab) betwee a ad b with the fllwig prperties. (P) S ( aa, ) = ( a ) Idetical (P2) S( ab, ) = S( ba, ) Symmetric (P3) St ( ab, ) = S( ab, )( t> 0) Uits-ivariat ad (P4) S( ab, ) 0 ( ab, ). The usual Pears s crrelati cefficiet itrduces the traslati f rigi i calculatig crrelatis. I ur mdel, we wish t evaluate affiity f tw vectrs withut traslati f rigi. Therefre, we itrduce ather crrelati cefficiet called affiity ide. Let us defie w b c = l ( =, K, ) a c= ma c = { } mi { } c = ma c ad c = mi c. (5) [Defiiti 3] (diversity ide) We defie the diversity ide f vectrs a ad b as the deviati f { c } frm the average c i the fllwig way. c c = D( a,b ) = (6) c ( c ) ma ma mi = 0 if c = c. mi [Prpsiti 8] 2 0 D( a,b) = D( b,a ). (7) See Appedi B fr a prf. D ( a,b) = 0ccurs if ad ly if tw vectrs a ad b are prprtial. [Defiiti 4] (affiity ide) We defie the affiity ide S( a,b) betwee tw vectrs a ad b by S( a,b) = 2 D( a,b ). (8) [Prpsiti 9] It hlds S( a,b ) 0. S( a,b ) satisfies prperties (P), (P2), (P3) ad (P4). 8
GRIPS Plicy Ifrmati Ceter Discussi Paper : 09-2 The reas why we emply the affiity ide (5) istead f the Pears s crrelati cefficiet is the fllwig:. Pears s crrelati cefficiet is defied by ( a a)( b b) = r( a,b ) =, 2 2 ( a a) ( b b) = = where aad bare respectively averages f { a} ad { b }. I this frmula, the abslute magitude f a ad b b effects r( a,b) strgly. I ctrast, i DEA, the relative measure, e.g. a, is a mai ccer. 2. Pears s crrelati cefficiet results i the rage r( a,b ). Hece, i the pricipal cmpet aalysis we will utilize i the et secti, it is t guarateed that the pricipal vectr csists f -egative cmpets. Althugh it is pssible t adust r( a,b) it [0, ], this might brig a skew distributi, sice mst f r( a,b ) are -egative i DEA applicatis. 3. We emply the lgarithmic fucti l b / a istead f b / a, because the latter vilates the prperty (P2). 5. Use f affiity matri i EBM I this secti, we measure the diversity f prducti pssibility set by meas f the affiity matri derived frm the bserved iputs ad utputs. Althugh we describe the methd i the iput-rieted mdel uder the cstat returs-t-scale (CRS) assumpti, we ca mdify it t the utput-rieted ad -rieted mdels uder cstat r variable returs-t-scale (VRS) assumptis. We discuss this subect i Secti 8. Step. Creati f prected VRS-efficiet DMUs I mst DEA mdels, the prducti pssibility set is spaed by the efficiet DMUs which usually csist f a small prti f the etire DMUs. I rder t icrease the accuracy f ur estimati, we first prect all DMUs t the VRS (variable returs-t-scale)-efficiet frtiers usig the Additive mdel r -rieted SBM mdel belw. We ca emply the bserved data (X, Y) istead f the prected DMUs i this step. Hwever, we utilized the prected DMUs, because ur mai ccers are the shape f frtiers. 9
GRIPS Plicy Ifrmati Ceter Discussi Paper : 09-2 [ADD] ma subect t m s + si s + i i yi i iλ i = = + s ( i =, K, m) + i iλ i = y = y s ( i =, K, s) = λ = + i i λ 0 ( ), s 0 ( i), s 0( i). (9) [SBM] m mi + s subect t m s i i s + si yi i iλ i = = + s ( i =, K, m) + i iλ i = y = y s ( i =, K, s) = λ = + i i λ 0( ), s 0( i), s 0( i). (20) Usig the ptimal slacks s ad s + we defie the prected iput ad utput fr DMU by i i i + i i i = s ( i =, K, m) y = y + s ( i =, K, s). We tice that [ADD] ad [SBM] may prduce differet prectis but they are the efficiet frtiers f the prducti pssibility set. Thus, we have VRS-efficiet DMUs deted by (2) L LLLL L X ml m m = = y y Y L y LLLL L ysl y s ys (22) All CRS (cstat-returs-t-scale) efficiet DMUs are icluded i this set alg with VRS-efficiet DMUs. Step 2. Frmati f affiity matri I the iput-rieted case, we calculate the affiity matri m m S = si R with the elemets 0
GRIPS Plicy Ifrmati Ceter Discussi Paper : 09-2 All elemets f the matri S satisfy the buds: s = S(, ) ( i, =, K, m) (23) i i s 0( ( i)). (24) i Step 3. Slvig the largest eigevalue ad eigevectr f the affiity matri By the frmati rule, S is symmetric ad -egative with the diagal elemets equal t uity. It has m pairs f eigevalue ad eigevectr. By the Perr-Frbeius therem fr -egative matrices, S has the largest eigevalue ρ with its assciated -egative eigevectr w ( 0 ). The -egative w crrespds t the weight f iput factrs. Sice S is -egative defiite, we have m ρ. Step 4. Calculati f ε ad w fr the EBM We defie ε ad w i the EBM as fllws. m ρ ε = (if m > ) m = 0 (if m = ). (25) w =. m wi w (26) The thus defied ε ad w satisfy the relatiship 0 ε ad ew =. Step 5. Use f ε ad w i the EBM These parameters are utilized i the EBM mdel [EBM-I-C]. 6. Ratiale f the prpsed EBM I this secti we demstrate the ratiale f the scheme prpsed i the precedig secti. Befre gig it theretical discussis, we shw sme real wrld data ccerig iput/utput items. Figure 4 depicts 84 samples f. f dctrs (as iput) vs.. f beds (as iput) f Japaese muicipally-wed hspitals. Figure 5 shws. f dctrs (as iput) vs. reveue/day (as utput) i the same 84 sample hspitals. Sice the muicipal hspitals are, t sme etet, stadardized uder the ctrl f respective admiistrative ffices, may iputs ad utputs have psitive relatiship ad hece the affiity matri is epected t have high affiity values. Csequetly, its pricipal eigevalue will be large ad hece ε will be small.
GRIPS Plicy Ifrmati Ceter Discussi Paper : 09-2 Figure 4: Empirical data Figure 5: Empirical data 2 Figures 6 ad 7 ehibit tw pltted data ccerig. f emplyees vs.. f visitrs ad area vs.. f visitrs fr Japaese regial museums. Sice museum busiess is t stadardized cmpared with regial hspitals, they are distributed widely. I this case, the affiity matri is epected t csist f lw values with relatively small pricipal eigevalue ad hece large ε. 2
GRIPS Plicy Ifrmati Ceter Discussi Paper : 09-2 Figure 6: Empirical data 3 Figure 7: Empirical data 4 Figures 8 ad 9 plt data f 273 electric pwer plats i the U.S. ccerig the geeratig pwer capacity (GW) (iput) vs.. f emplyees (iput) ad the csumed fuel (milli BTU) (iput) vs.. f emplyees (iput). They are psitively crrelated but csiderably diversified. 3
GRIPS Plicy Ifrmati Ceter Discussi Paper : 09-2 N. f emplyees GW Figure 8: Empirical data 5 N. f emplyees milli BTU Figure 9: Empirical data 6 T The ellipsid w Sw = has the pricipal ais i the first (psitive) quadrat as eemplified i Figure 0. As the degree f affiity becmes higher ad higher, the shape f the ellipsid cmes t be flat ad the largest eigevalue ρ teds t m. Thus, ε i (25) teds t 0. This cmes clse t the CCR mdel, i.e. all iputs ad utputs fllw prprtial chages. 4
GRIPS Plicy Ifrmati Ceter Discussi Paper : 09-2 2 0 Figure 0: Ellipsid f affiity matri Depedig the degree f affiity amg iputs, the pricipal eigevalue Cversely, the mre the data scatters widely, the mre ρ teds t ad the mre ρ icreases up t m. ε grws up. Hece, the mdel behaves SBM-like. Therefre, it ca safely be said that ε cdeses the affiity matri i a sigle value reflectig the scatterig f the data set. We w tur t the psitive eigevectr w crrespdig t the eigevalue ρ. First f all, we tice that w is uits-ivariat, sice the affiity matri is uits-ivariat. Suppse that, i the affiity matri = ( s i ) S, s > s2 0( = 3, K, m), the it hlds that w > w2. This idicates that the item which has higher affiity with thers has a large prti i the eigevectr, whereas item i with urelated t thers, i.e., s i 0( i) has wi 0. Thus, the magitude f elemets f w idicates imprtace f the item amg the whle items. We ca stregthe the discrimiati pwer efficiecy by impsig weight t slacks i prprti t applicati f the pricipal cmpet aalysis (PCA) t DEA. We te here that, i the iput-rieted mdel, we estimate w. Thus, this scheme is a ε depedig ly the iput data X, but t the utput data Y. This meas that the bective fucti i [EBM-I-C] relates t the radial factrθ ad the diversity idicatr ε. The frmer represets the radial feature f iputs ad the later implies the -radial characteristics f iputs. The iteractis betwee iput X ad utput Y are described i the cstraits f [EBM-I-C] thrugh the itermediary f the itesity vectr 7. Illustrative eamples I this secti, we eplai the EBM usig three eamples ad cmpare the results with the radial 5
GRIPS Plicy Ifrmati Ceter Discussi Paper : 09-2 (CCR) ad -radial (SBM) scres. 7. Eample Table reprts cmpariss f CCR-I, SBM-I-C ad EBM-I-C scres fr si DMUs A, B, C, D, E ad F with tw iputs (, 2 ) ad a sigle utput (y = ). Figure plts them graphically. This figure idicates that the data are ccetrated i a arrw gauge. See als Figure 2. Table : 2 y CCR-I SBM-I-C EBM-I-C A B 2 3 0.500 0.47 0.500 C 3 2 0.500 0.47 0.500 D 4 3 0.333 0.292 0.333 E 5 6 0.200 0.83 0.200 F 7 6 0.67 0.55 0.67 Figure : Eample Figure 2: Cmparis f scres As ca be see, EBM scres are the same with the CCR scres. We illustrate the EBM scheme i rder. Step : We used [ADD] fr fidig slacks ad prected DMUs t efficiet frtiers, as shw i 6
GRIPS Plicy Ifrmati Ceter Discussi Paper : 09-2 Table 2. They are all prected t the ly e efficiet DMU A. Table 2: Prected DMUs 2 y A B C D E F Step 2: We calculated the diversity matri by the frmula (6). See Table 3. Sice the set f efficiet DMUs csists f ly e DMU A, diversity eists. Table 3: Diversity matri fr Eample 2 0 0 2 0 0 Step 3: The affiity matri is calculated by the frmula (23) ad displayed i Table 4. Table 4: Affiity matri fr Eample 2 2 Step 4: The largest eigevalue ad eigevectr f the affiity matri are ρ = 2 ad w = (0.5, 0.5). Hece we have ε I this case, the ellipsid is perfectly flat. = ( mρ) / ( m ) = 0, w = 0.5, w2 = 0.5. Step 5: Usig these parameter values we applied EBM-I-C t the si DMUs. Sice we have ε = 0, the scres are idetical with the CCR scres. 7.2 Eample 2 This eample has diversified DMUs as ehibited i Table 5 ad Figure 3. Table 5: Eample 2 2 y CCR-I SBM-I-C EBM-I-C 7
GRIPS Plicy Ifrmati Ceter Discussi Paper : 09-2 A 2 6 B 6 3 C 0 3 0.8 0.8 D 2 0 0.8 0.8 Figure 3: Eample 2 The prected data are ehibited i Table 6. Table 6: Prected data fr Eample 2 2 y A 2 6 B 6 3 C 6 3 D 2 6 The diversity matri fr the EBM-I-C mdel is displayed i Table 7 alg with the affiity matri i Table 8. Table 7: Diversity matri fr Eample 2 2 0 0.5 2 0.5 0 8
GRIPS Plicy Ifrmati Ceter Discussi Paper : 09-2 Table 8: Affiity matri fr Eample 2 2 0 2 0 The largest eigevalue ad eigevectr f the affiity matri are ρ = ad w = (0.5,0.5). Hece we have ε = ( mρ )/( m ) = w = 0.5, = 0.5. w2 The value ε = is the largest e shwig the diversity f the data set ad EBM-I-C results are idetical with the SBM results. 7.3 Eample 3 Table 9 reprts efficiecy scres f 2 hspitals. We utilized umbers f dctrs ad urses as iputs, ad umbers f utpatiets ad ipatiets per mth as utputs. Figure 4 displays cmpariss f three scres: CCR-I, SBM-I-C ad EBM-I-C. Table 9: Hspital data ad efficiecy scres (I)Dctr (I)Nurse (O)Outpatiet (O)Ipatiet CCR-I SBM-I-C EBM-I-C A 20 5 00 90 B 9 3 50 50 C 25 60 60 55 0.883 0.852 0.868 D 27 68 80 72 0.986 E 22 58 94 66 0.763 0.756 0.76 F 55 255 230 90 0.835 0.704 0.77 G 33 235 220 88 0.902 0.895 0.898 H 3 206 52 80 0.796 0.774 0.788 I 30 244 90 00 0.960 0.905 0.93 J 50 268 250 00 0.87 0.78 0.829 K 53 306 260 47 0.955 0.866 0.92 L 38 284 250 20 0.958 0.936 0.946 9
GRIPS Plicy Ifrmati Ceter Discussi Paper : 09-2 Figure 4: Cmpariss f scres (hspital) We utilized [ADD] fr prectig the dataset t the VRS-efficiet frtiers ad btaied the ew dataset ehibited i Table 0. Table 0: Prected DMUs (hspital) Dctr Nurse Outpatiet Ipatiet A 20.00 5.00 00.00 90.00 B 9.00 3.00 50.00 50.00 C 24.59 60.00 60.00 72.98 D 27.00 68.00 80.00 72.00 E 22.00 56.79 58.26 66.00 F 35.06 255.00 230.00 08.90 G 33.00 235.00 220.00 88.04 H 27.44 206.00 62.03 02.4 I 30.00 223.45 90.00 02.29 J 50.00 268.00 250.00 00.00 K 53.00 306.00 260.00 47.00 L 38.00 284.00 250.00 20.00 The diversity matri is displayed i Table alg with the affiity matri i Table 2. Table : Diversity matri fr Eample 3 Dctr Nurse 20
GRIPS Plicy Ifrmati Ceter Discussi Paper : 09-2 Dctr 0 0.265 Nurse 0.265 0 Table 2: Affiity matri fr Eample 3 Dctr Nurse Dctr 0.47 Nurse 0.47 This affiity matri has the largest eigevalue ad eigevectr: ρ =.47, w = (0.5,0.5) Hece we have: ε = ( mρ) / ( m ) = 0. 529 w = 0.5, w2 = 0.5. The EBM scres were btaied usig these ε ad w values. Table 3 ehibits θ ad slacks s, s 2, s + ad s + i the sluti f the EBM-I-C mdel. It is 2 iterestig t tice that hspital D is iefficiet with the scre 0.986, i ctrast t the CCR ad SBM scre (efficiet). The EBM mdel impses restricti θ, ad D has a ptimalθ =.06( > ). Thus the ptimal sluti isists that all iputs are multiplied by.06 ad further. f dctr is decreased by the slacks s = 3.078. The prected iputs fr D are 27.06 3.078 = 24.35 fr Dctr ad 68.06 = 70.68 fr Nurse. D s refereces are A ( λ A = 0.28 ) ad B ( λ B =.0588 ). D is recmmeded t reduce dctrs frm 27 t 24 ad icrease urses frm 68 t 7 i rder t imprve efficiecy. This is e f characteristics f the cmpsite mdel EBM, whereas such substituti f iputs cat ccur i the CCR r the SBM mdels. Table 3: θ ad slacks DMU Scre Rak θ s s2 s+ s 2 + A 0 0 0 0 B 0 0 0 0 C 0.868 8 0.885.644 0 0 0 D 0.986 3.06 3.078 0 0 0 E 0.76 2 0.766 0.46 0 0 0 F 0.77 0.846 5.696 0 0 0 G 0.898 7 0.902 0 3.349 0 0 2
GRIPS Plicy Ifrmati Ceter Discussi Paper : 09-2 H 0.788 0 0.804.887 0 0 0 I 0.93 5 0.960 0 27.206 0 0 J 0.829 9 0.885 0.404 0 0 0 K 0.92 6 0.964 0.328 0 0 0 L 0.946 4 0.958 0 2.6 0 0 8. Etesis t ther rietatis ad variable returs-t-scale mdels S far, we have develped the EBM i the iput-rietati uder the cstat returs-t-scale evirmet. Hwever, we ca eted it t ther rietatis ad returs-t-scale evirmet as fllws. I every variati we fllw Step i Secti 5 ad btai the set f prected VRS-efficiet DMUs (22). 8. Output-rieted EBM Step 2. Frmati f affiity matri I the utput-rieted case, we calculate the affiity matri s s S = si R with the elemets si = S( yi, y ) ( i, =, K, s) (27) Step 3. Slvig the largest eigevalue ad eigevectr f the affiity matri We slve the largest eigevalue ρ y ad eigevectr w y f the affiity matri S i (27). + Step 4. Calculati f ε y ad w. We defie s ρ y ε y = (if s > ), = 0 (if s = ) s + w y w =. s wyi (28) Usig ε y [EBM-O-C] + ad w we slve the fllwig liear prgram: subect t = ma / τ η εy η, λ, s+ + s + + i i w s y i (29) Xλ (30) y Yλ + s λ 0 + s 0. η + = 0 (3) 8.2 N-rieted (bth-rieted) EBM We apply Steps 2 ad 3 fr the iput-rieted ad the utput-rieted affiity matri separately, ad 22
GRIPS Plicy Ifrmati Ceter Discussi Paper : 09-2 + btai ε, ε, w, ad w. The -rieted EBM ca be frmulated i the fllwig fractial y prgram which ca be slved as a liear prgram usig the Chares-Cper trasfrmati. (See Cper et al. [6].) κ = mi + θη,, λ,s, s subect t θ Xλ s = 0 θ ε η + ε m ηy Yλ + s = 0 λ 0s, 0s, 0. y + + 8.3 Variable returs-t-scale EBM All mdels ca be mdified t variable returs-t-scale (VRS) es by addig the cditi: λ+ λ2 + L + λ =. (33) 9. Ccludig remarks I this paper, we have prpsed EBM as a third ple f techical efficiecy i DEA by cmbiig radial ad -radial mdels i a uified framewrk. Sice DEA is a data drive methd, we eed t measure techical efficiecy frm the bserved data uder less assumptis its distributi. Fr this purpse we itrduced a ew ide called affiity ide fr measurig similarity betwee tw vectrs fr use i DEA. Usig this ide, we defied a scalar measure epsil ( ε ) that represets the diversity r the scatterig f the bserved dataset. We prpsed a scheme fr settig weights t slacks based the pricipal cmpet aalysis. We als eteded it t ther rietatis ad returs-t-scale assumptis. Future research subects iclude search fr ther measure f affiity ide that satisfies the prperties (P) t (P4), etesis t Super-EBM ad idetificatis f returs-t-scale ad scale efficiecy uder this mdel. Refereces [] Adler N, Glay B. (200) Evaluati f deregulated airlie etwrks usig data evelpmet aalysis cmbied with pricipal cmpet aalysis with a applicati t Wester Eurpe. Eurpea Jural f Operatial Research, 32, 260-273. [2] Adler N, Glay B. (2002) Icludig pricipal cmpet weights t imprve discrimiati i data evelpmet aalysis. Jural f the Operatial Research Sciety, 53, 985-99. [3] Ali AL, Seifrd LM. (993) Cmputatial accuracy ad ifiitesimals i data evelpmet aalysis. INFOR, 3, 290-297. [4] Avkira N, Te K, Tsutsui M. (2008) Bridgig radial ad -radial measures f efficiecy i DEA. Aals f Operatis Research, 64, 27-38. [5] Chares A, Cper WW, Rhdes E. (978) Measurig the efficiecy f decisi makig uits. Eurpea Jural f Operatial Research, 2, 429-444. s i i ws i + + i i w s y i (32) 23
GRIPS Plicy Ifrmati Ceter Discussi Paper : 09-2 [6] Cper WW, Seifrd LM, Te K. (2007) Data Evelpmet Aalysis: A Cmprehesive Tet with Mdels, Applicatis, Refereces ad DEA-Slver Sftware, Secd Editi, Spriger. [7] Jhs AL ad Ruggier J. (2009) - Substitutability, slacks ad data evelpmet aalysis. Wrkig paper. [8] Te K. (200) A slacks-based measure f efficiecy i data evelpmet aalysis. Eurpea Jural f Operatial Research, 30, 498-509. [9] Ueda T, Hshiai Y. (997) Applicati f pricipal cmpet aalysis fr parsimius summarizati f DEA iputs ad/r utputs. Jural f Operatis Research Sciety f Japa, 40, 466-478. Appedi A. Prf f Prpsiti 7 Sice (, y ) is EBM iput-iefficiet, it hlds that Let a ptimal sluti fr (, y ) be The crrespdig cstraits fr (, y ) are: This reduces t: θ i i m ws γ = θ ε <. (A) i ( γ, θ, λ, s ). The EBM bective fucti value is: i i i m ws γ = θ ε. (A2) = Xλ + s, y Yλ. (A3) = Xλ + s + θ s, y y Yλ. (A4) θ θ This is ather epressi fr (, y ) ad its bective fucti value is: m w ( ) m i si + θ si wi si i i f = θ θ ε = θ γ ε. (A5) We have three pssibilities as fllws: i) The case θ <. I this case, it hlds that f < γ. This ctradicts the ptimality f γ fr (, y ). Thus, this case ever ccurs. ii) The case θ =. I this case, by the ptimality f γ fr (, y ), we have s = 0( i). Thus, γ = ad (, y ) is EBM iput-efficiet. iii) The case θ >. Frm the ptimality f γ fr (, y ), it hlds that Hece we have m ws i i i θ γ ε γ m ws i i i ε θ +. (A6) γ i 24
GRIPS Plicy Ifrmati Ceter Discussi Paper : 09-2 Suppse that (, y ) is EBM-iefficiet, i.e. have: i i i m ws γ = θ ε <. The we We cmpare the terms i i i m ws θ < + ε. (A7) γ i i (A6) ad i i (A7). Sice i = θ i si, we have m ws ( ) k k γ i i = γ θ i si =εi si 0. k = k Thus, it hlds that γ i i. (A8) Cmparig (A6) ad (A7), we have m m ws i i ws i i + + > i γ = i i (, ) (A9) θ ε ε θ. This cat ccur. Thus, i this case, y is EBM iput-efficiet. Q.E.D. Appedi B. Prf f Prpsiti 8 (a) Prf f D(a,b)=D(b,a) Let d l( a / b ) (,, )) = = K, dma ma { d}, dmi mi{ d} ma mi mi ma = =, ad d = d. The it hlds that d = c ( =, K, ), d = c, d = c,ad d =c. Hece, we have d d c + c D( b,a) = = = D( a,b ). (B) d ( ma dmi ) ( cmi + cma ) (b) Prf f D(a,b) /2 If a ad b are prprtial, the it hlds that c ma = c mi ad D( a,b ) = 0. Otherwise if a ad b are t prprtial, the c ma > c mi ad D( a,b ) > 0. Let N ad N 2 be respectively the set f such that c cad c > c, ad = N ad 2 = N 2. We have = + 2. The umeratr f D(a,b) ca be trasfrmed it the fllwig: c c = ( c + c) + ( c c) = N N2 ( cmi + c) + 2( cma c) 2 2 = ( cma cmi )( ). 2 The last term i the last epressi attais the maimum /4at = /2 Hece, we have (B2) c c = Dab (, ) =. c ( c ) 2 ma mi (B3) D( a,b) /2 hlds whe { c } distributes as eemplified i Figure B. 25
GRIPS Plicy Ifrmati Ceter Discussi Paper : 09-2 c ma c c mi Figure B: The case D( a,b) /2 26