Variations on the theme of slacks-based measure of efficiency in DEA

Transcription

1 GRIPS Plicy Ifrmati Ceter Dicui Paper : 8-4 Variati the theme f lac-baed meaure f efficiecy i DEA Karu Te Natial Graduate Ititute fr Plicy Studie Rppgi, Miat-u, Ty , Japa te@gripacp Abtract: I DEA, there are typically tw cheme fr meaurig efficiecy f DMU; radial ad -radial Radial mdel aume prprtial chage f iput/utput ad uually remaiig lac are t directly accuted fr iefficiecy O the ther had, -radial mdel deal with lac f each iput/utput idividually ad idepedetly, ad itegrate them it a efficiecy meaure, called lac-baed meaure (SBM) I thi paper, we pit ut hrtcmig f the SBM ad prpe 4 variat f the SBM mdel The rigial SBM mdel evaluate efficiecy f DMU referrig t the furthet frtier pit withi a rage Thi reult i the hardet cre fr the bective DMU ad the precti may g t a remte pit the efficiet frtier which may be iapprpriate a the referece I a effrt t vercme thi hrtcmig, we firt ivetigate frtier (facet) tructure f the prducti pibility et The we prpe Variati I that evaluate each DMU by the earet pit the ame frtier a the SBM fud Hwever, there exit ther ptetial facet fr evaluatig DMU Therefre we prpe Variati II that evaluate each DMU frm all facet We the emply cluterig methd t claify DMU it everal grup, ad apply Variati II withi each cluter Thi Variati III give mre reaable efficiecy cre with le effrt Latly we prpe a radm earch methd (Variati IV) fr reducig the burde f eumerati f facet The reult are apprximate but practical i uage Keywrd: DEA, SBM, facet, eumerati, cluterig, radm earch Reearch upprted by Grat-i-Aid fr Scietific Reearch, Japa Sciety fr the Prmti f Sciece

2 GRIPS Plicy Ifrmati Ceter Dicui Paper : 8-4 Itrducti I mt DEA mdel, the prducti pibility et i a plyhedral cvex et whe vertice crrepd t the efficiet DMU i the mdel A plyhedral cvex et ca be defied by it vertice r by it upprtig hyperplae (Simard [4]) I DEA literature, mai fcu i directed t vertice while cmparatively few reearche are ccered with the upprtig hyperplae Oe f the purpe f thi paper i t fill the gap betwee the tw apprache: vertex ad hyperplae We firtly dicu the characteritic f the upprtig hyperplae t the prducti pibility et i DEA The, baed thi hyperplae, we prpe everal variat f the lac-baed meaure f efficiecy Rughly peaig, we have tw type f meaure i DEA; radial ad -radial Radial meaure are repreeted by CCR [2] ad BCC [] mdel Their drawbac exit i that iput/utput are aumed t uderg prprtial chage ad remaiig lac are t accuted fr i the efficiecy cre N-radial mdel are repreeted by the lac-baed meaure (SBM) [5] The SBM evaluate efficiecy baed the lac-baed meaure t the efficiet frtier Hwever, ice it bective i t miimize thi meaure, the referet pit i apt t be far frm the bective DMU Hwever, there exit ther apprach; t fid the earet pit the frtier Fr thi purpe we firt mdify the SBM t catch the miimum lac-baed meaure pit the facet that the SBM fud fr the DMU We call thi Variati I The, after ivetigati f upprtig hyperplae (facet) t the prducti pibility et, we exted thi apprach t cider all facet, reultig i Variati II Sice the eumerati f facet eed maive cmputati, we prpe tw mre cveiet variati; e cluterig (Variati III) ad the ther radm earch (Variati IV) Thi paper ufld a fllw We itrduce the SBM ad everal prpertie f facet (hyperplae) i Secti 2 The we mdify the SBM i uch a way that itead f miimizati f the bective fucti we maximize it the facet explred by the SBM (Variati I) i Secti 3 We prpe a methd fr fidig all facet f the prducti pibility et i Secti 4 Uig thi reult we exted Variati I t emply all facet (Variati II) i Secti 5 The we implify Variati II t tw cheme; e cluterig (Variati III) ad the ther radm earch (Variati IV) i Secti 6 We mdify ur reult t cpe with the variable retur-t-cale (VRS) evirmet i Secti 7 We cmpare ur variati with the radial (CCR) mdel i Secti 8 Sme ccludig remar fllw i the lat ecti 2 Prelimiarie I thi ecti we itrduce the SBM ad dicu everal prpertie f the facet f prducti pibility et 2 Ntati ad Prducti Pibility Set We deal with DMU (=,,) each havig m iput { xi }( i =, K, m) ad utput { yi}( i =, K, ) We dete the DMU by (, y ) ( =, K, ) ad the iput/utput data matrice by m x X = ( x i ) R ad = ( y i ) R Y, repectively We aume X > ad Y > Uder the ctat retur-t-cale (CRS) aumpti the prducti pibility et i defied by P = {( x, y) x Xλ, y Yλ, λ } () where m λ R i the iteity vectr We itrduce -egative iput ad utput lac R ad R t expre x = Xλ ad y = Yλ (2) 22 Efficiecy ad SBM [Defiiti ] (Efficiet DMU) A DMU ( x, y ) P i called CRS-efficiet if ay luti f the ytem x = Xλ y = Yλ, λ,,, ( x, y ) ha = ad = Otherwie i called CRS-iefficiet, ie there exit -egative but -zer (emi-pitive) lac fr the abve ytem 2

3 GRIPS Plicy Ifrmati Ceter Dicui Paper : 8-4 Thi defiiti crrepd t the Paret-Kpma defiiti f efficiecy: A DMU i fully efficiet if ad ly if it i t pible t imprve ay iput r utput withut wreig me ther iput r utput (Cper et al [3], p 45) The SBM ([5]) lve the fllwig prgram fr DMU ( x, y)( =, K, ) [Theme -- Origial SBM] m i i= mi m xi ρ = mi (3) r r= yr ubect t = = λ ( ) x λ = x y λ = y (4) Thi fractial prgram ca be lved by trafrmig it a equivalet liear prgram (ee [5]) Let a ptimal luti f the SBM be ( λ,, ) [Defiiti 2](Referece et) The referece et fr DMU ( x, y ) i defied by R = { λ >, =, K, } (5) [Defiiti 3](Precti) The precti f DMU ( x, y ) i defied by x = x = x λ R y = y = y λ R (6) [Therem ] The prected DMU ( x, y ) i CRS-efficiet (See Appedix A fr a prf) A the bective fucti (3) ugget, the rigial SBM aim t fid the miimum (the wrt) cre aciated with the relatively maximum lac uder the ctrait (4) Thi might prect the DMU t a very remte pit the frtier (facet) ad metime it i hard t iterpret O the ther had, there i the ppite apprach, ie, t l fr the earet pit the facet, by miimizig the lac-baed meaure frm the frtier Fr thi purpe, we eed t ivetigate the facet f the prducti pibility et, a we hw i the fllwig ecti 23 Facet f Prducti Pibility Set Let ( ξ, η )( =, K, ) be DMU i P We mae a liear cmbiati f thee DMU with pitive 3

4 GRIPS Plicy Ifrmati Ceter Dicui Paper : 8-4 cefficiet a where > ( =, K, ) w ξ η = w ξ = w η L w ξ L w η, (7) [Lemma ] If ay member f ( ξ, η ) ( =, K, ) i CRS-iefficiet, the ξ, ) i CRS-iefficiet ( Prf: Withut lig geerality, we aume that ( ξ, η ) i CRS-iefficiet The, the ytem ξ = Xλ, η = Yλ, λ,, (8) ha a luti ( λ,, ) with (, ) (,) ad (, ) (,) We et By iertig (9) it (7), we have Let u defie Sice ξ = Xλ ad η = Yλ (9) = w w 2 w = ξ ξ ξ = w w 2 w = η η η ξ = wξ w ξ ξ, ) P, ad > ( =, K, ), we have ( η w Hece, we have = 2 η = w η w η = 2 () () ( ξ, η ) P (2) ξ = ξ w η = η w ( ξ, η ) Thu, ha -egative ad -zer lac (, ) agait ( ξ, η ) Hece it i CRS-iefficiet QED A a ctrapiti f Lemma, we have [Therem 2] If ξ, η ) defied by (7) i CRS-efficiet, the ( ξ, η ) ( =, K, ) mut be CRS-efficiet ( We tice that the revere f thi therem i t alway true Nw, we aume we demtrate the fllwig therem ( ξ, η ) i (7) i CRS-efficiet ad [Therem 3] If ( ξ, η ) defied by (7) i CRS-efficiet, the there exit a upprtig hyperplae t P at ( ξ, η ) which al upprt P at ( ξ, η ) ( =, K, ) Prf: By the trg therem f cmplemetarity, there exit dual variable uch that m v R,u R with v >, u > We ca btai uch a trg cmplemetary luti by uig the additive mdel r the -rieted lac-baed meaure (SBM) mdel [3, 5] 4

5 GRIPS Plicy Ifrmati Ceter Dicui Paper : 8-4 Iertig the defiiti f Taig te f w Hece, the hyperplae v ξ v ξ u u η η = v X u Y ( =, K, ) (3) ξ, η ) i (7) it the firt equality i (3), we have w ( v ξ u η ) L ( v ξ u η ) (4) ( w = > ( =, K, ) ad the ecd iequality i (3), the equality (4) hld if ad ly if v ξ u η = ( =, K, ) v x u y = pae thrugh ( ξ, η ) ( =, K, ) ad upprt P QED Thi therem i helpful i idetifyig the facet f P Sice the ytem f equati (5) i hmgeu, if (, u v ) i a luti t (5), the t( v, u ) ( t > ) i al a luti If the ra f the matrix ξ, K, ξ m ( ) R (6) η, K, η i t le tha m, the the cefficiet ( v, u ) i uiquely determied except fr the calar multiplier t, ice the hyperplae v x u y = pae thrugh the rigi ( x=, y = ) ad remaiig m liearly idepedet ( ξ, η ) determie the hyperplae Thi mea that the directi ( v, u ) i the uique rmal t the upprtig hyperplae If the ra f (6) i le tha m, the there exit multiple ( v, u ) fr the ytem (3) [Defiiti 3] (Facet) We call the upprtig hyperplae v x u y defied i Therem 3 a facet f P 3 Variati I Miimizig lac-baed meaure frm the facet The firt variati i a imple mdificati f the baic SBM i the precedig ecti We maximize the bective fucti rather tha miimizati Fr each DMU ( x, y )(, K, ), we lve the SBM mdel i (3-4) If it i iefficiet, we have it referece = et R defied by (5) The prected DMU i efficiet by Therem ad hece the DMU i the referece et are efficiet by Therem 2 Furthermre, by Therem 3, they frm a facet f P We evaluate the miimum lac-baed meaure ad hece the maximum cre the facet a fllw 2 (5) max m ρ = max ubect t m i= r= x y i i r r (7) 2 Simard (966) called uch hyperplae a extremal upprtig ray 5

6 GRIPS Plicy Ifrmati Ceter Dicui Paper : 8-4 R R λ ( ) x λ = x y λ = y (8) Sice we deal with the ame facet a the baic mdel, we have the relatihip: max mi (9) ρ ρ Thi variati demad e additial LP luti fr each iefficiet DMU ad i cmputatially rather eay Hwever, ice the facet defied by R i a itace f facet ad there may be ther facet f P t be cidered i evaluatig the maximum efficiecy f DMU ( x, y ), we eed t w all facet f P We dicu thi ubect i the ext ecti Nw we hw a example f the SBM ad Variati I [Example ] Table exhibit data fr 2 hpital havig tw iput ad tw utput Iput: Number f dctr ad ure Output: Number f utpatiet ad ipatiet << Iert Table here Table : Data f 2 hpital>> Firt, we lved thi cae by the SBM i (3-4) The, wig the referece et ad hece a facet f iefficiet DMU, we lved the Variati I i (7-8) The reult are diplayed i Table 2 Every iefficiet DMU imprved mi their efficiecy except H Fr example, Hpital C i iefficiet 8265 by the SBM ad it referece are B ρ = ad L We lved the maximum prblem (Variati I) the facet paed by B ad L, ad btaied with the referece B The differece i the gap betwee the max ad the mi bective value meaured by (7) << Iert Table 2 here Table 2: Reult f SBM ad Variati I>> 4 Eumerati f facet I thi ecti, we prpe a methd fr eumeratig all facet f P Let P = ( ξ, η )( =, K, K) be the CRS-efficiet DMU i P C ρ = 855 [Defiiti 3] (fried) A ubet P, K, } f { P } = {( ξ, η )} ( =, K, K) i called fried if a liear cmbiati with pitive cefficiet f { P { P, K, P } i CRS-efficiet max C [Defiiti 4] (maximal fried) A fried i called maximal if ay additi f P (t i the fried) t the fried i mre fried [Defiiti 5] (dmiated fried) A fried i dmiated by ther fried if the et f DMU i a ubet f ther 6

7 GRIPS Plicy Ifrmati Ceter Dicui Paper : 8-4 We prpe a algrithm fr fidig the maximal fried f P = ( ξ, η )( =, K, K) [Algrithm A] Begi Fr = t K Fid_Maximal_Fried f P Next Delete dmiated fried frm the et f fried Obtai the et f facet frm the fial et f fried Ed Subrutie Fid_Maximal_Fried f P Exclude P, K, P frm the cadidate f fried Eumerate all fried f P Remve dmiated fried frm the et f fried Exit ub Let the umber f facet thu geerated be H We have H facet t P: ( h) ( h) Facet( h) : v x u y ( h =, K, H) (2) Facet(h) pae thrugh it fried ad upprt P The abve facet cit f geuie efficiet frtier f the prducti pibility et P Hwever, P ha -efficiet budarie a we ee i Figure a example I the figure lie egmet AB ad BC are efficiet facet, while AD ad CE are -efficiet budarie f P WE tice that, i thi paper, we berve ad deal ly with the efficiet prti f the budary << Iert Figure Figure : Efficiet ad -efficiet frtier>> [Therem 4] Fr every efficiet frtier pit f P, there exit a Facet(h) that tuche the efficiet pit Prf: Every efficiet frtier pit ca be expreed by a pitive liear cmbiati f a et f efficiet vertice f P By ctructi f the maximal_fried i the Algrithm A, the et a well a the efficiet pit i me Facet(h) QED 5 Variati II Miimizig the SBM frm all facet We deal with a et f DMU defied i Secti 2 Step Fidig Efficiet DMU Slve the -rieted SBM mdel r the additive mdel ad fid the et f efficiet DMU Let the et be {( ξ, η ) =, K, K )} where K i the umber f efficiet DMU Step 2 Eumerati f Facet Eumerate all facet applyig Algrithm A i Secti 4 Let the umber f facet thu btaied be H We deal with ly facet i the maximal fried Step 3 Evaluati f Iefficiet DMU Fr a iefficiet DMU x, y ) we evaluate it efficiecy cre a fllw ( 7

8 GRIPS Plicy Ifrmati Ceter Dicui Paper : 8-4 Fr each Facet( h) ( h =, K, H), we lve the fllwig fractial prgram: ρ ( h) = ubect t m max R ( h) R ( h) λ ( ), ξ λ η λ where R(h) i the et f efficiet DMU that pa Facet(h) We btai the efficiecy cre f DMU x, y ) a ( m i= r= ( h) { } x y i i r r = x = y (2) (22) all ρ = max ρ (23) h We have the fllwig iequalitie amg the three cre: all max mi ρ ρ ρ (24) [Example 2] I the abve example, the et f fried cmped f tw DMU are fud t be AD, BD, AL, BL ad DL The et f fried cmped f three DMU are ADL ad BDL The et ABDL cat be a fried (facet) Hece the maximal fried are ADL ad BDL Uig ADL ad BDL a referece repectively, we lved the prgram (2-22), ad btaied the efficiecy cre fr iefficiet DMU a exhibited i Table 3 Fr example, fr DMU E, we have BDL ρ = 76823(with referece A) ad ρ = 75236(with referece D, L) Thu ρ all = ADL E E E <<Iert Table 3 here Table 3: Reult f SBM ad Variati II>> Cmpari with Table 2 reveal everal iteretig feature f Variati II A demtrated i (24), the efficiecy cre f Variati II i t le tha the f the SBM ad Variati I fr each DMU 6 Hw t reduce a maive eumerati I Variati II, the eumerati f facet eed a ermu cmputati time ad pace fr large cale prblem, eve thugh advace i recet IT techlgie are amazig i bth apect If we have m=6 (# f iput), =5 (# f utput) ad =2 (# f efficiet DMU), the i the wrt cae we might eumerate abut 2 C =84,756 cae Of cure, mt f them wuld be fud t be a iefficiet cmbiati I thi ecti we prpe tw mdified veri f Variati II which are le time ad pace cumig 6 Variati III Cluterig Step Cluterig DMU Uig me cluterig methd, we claify all DMU i cluter, ay, Cluter t Cluter L Step 2 Fidig efficiet DMU Thi tep i the ame a the Step f the Variati II Step 3 Evaluatig efficiecy cre fr a iefficiet DMU If the iefficiet DMU x, y ) belg t Cluter h, pic up the efficiet DMU i Cluter h If e f DMU i ( Cluter h i efficiet, we pic up the efficiet DMU i the adacet cluter 8

9 GRIPS Plicy Ifrmati Ceter Dicui Paper : 8-4 Let the ubet f efficiet DMU crrepdig t Cluter h be E( h) = {( ξ, η ), K, ( ξ J, η J )} We create the facet cmped f the efficiet DMU i E(h) uig the ame prcedure a decribed i Step 2 f the precedig ecti We evaluate the efficiecy f DMU x, y ) i referece t the facet thu btaied i the ( ame way a the Step 3 f the precedig ecti If the prgram (2-22) ha feaible luti, DMU ( x, y ) i udged t be efficiet i thi cluter, ie, it i glbally iefficiet but lcally efficiet The merit f thi mdificati are a fllw: () By itrducig a ciderable umber f cluter, we ca reduce the umber f the cadidate cmbiati (2) Fr iefficiet DMU, the efficiecy cre i btaied i referece t the efficiet DMU i the ame cluter Thu, the reult are mre acceptable ad udertadable [Example 3] We claified 2 hpital i Table it tw cluter depedig their ize (umber f dctr ad ipatiet) a decribed i the clum Cluter f Table 4 We lved -rieted SBM mdel ad fud 4 efficiet DMU (A,, B, D, L) ad 8 iefficiet DMU with their referece a exhibited i the left ide f Table 4 where we fud everal iapprpriate referece Fr example, C ha referece B ad L, wherea L i t i the ame cluter a C I the cluter, the maximal fried are AD ad BD, ad i the cluter 2 we have ly e facet L Fially, we lved the efficiecy f iefficiet DMU referrig t the facet i the ame cluter ad fud the reult recrded i the right half f Table 4 DMU C ha it referece D ad efficiecy cre which wa upgraded frm the SBM cre 826 DMU i the cluter 2 were all evaluated their efficiecy agait L We fud ifeaibility fr G ad J Hece, we udged them efficiet i thi cluter They are glbally iefficiet but lcally efficiet << Iert Table 4 here Table 4: SBM ad Cluterig reult (Variati III)>> 62 Variati IV Radm Search I thi ecti we prpe a apprximate methd fr fidig facet Step Fidig ceter f gravity f efficiet DMU Let the et f efficiet DMU be P = ( ξ, η )( =, K, K) We calculate their ceter f gravity G a x = ( ξ L ξ )/ K G y = ( η L η )/ K G Figure 2 illutrate a example We te that we ca utilize ay pitive liear cmbiati f efficiet DMU itead f the ceter f gravity fr ur purpe Step 2 Creatig radm directi arud efficiet DMU Fr each efficiet DMU P = ( ξ, η ) we cmpute the directi frm G t P = ( ξ, η ) a ( ξ x, η y ) ad the perturb the directi lightly uig radm umber Let the directi thu perturbed be G G ( d, d ) x y Step 3 Fidig a facet We lve the fllwig liear prgram i t Rad K λ R : K K (25) 9

10 GRIPS Plicy Ifrmati Ceter Dicui Paper : 8-4 max t ubec t x d t ξ λ L ξ λ G x K K y d t ηλ L η λ G y K t, λ K (26) Let a ptimal luti be ( t, λ ) If t =, the the ceter G i efficiet ad all P = ( ξ, η)( =, K, K) are fried Thi cae ha ly e efficiet facet by Therem 3 If t >, the the referece DMU crrepdig t pitive λ frm a facet f P, ice the ptimal luti i btaied a budary f P Step 4 Repeatig the radm earch We repeat the radm earch arud the K efficiet DMU util a ufficiet umber f facet i fud Step 5 Evaluatig iefficiet DMU We evaluate the efficiecy cre f iefficiet DMU uig the facet thu fud i the ame maer a the Variati II <<Iert Figure 2 here Figure 2: Radm earch arud efficiet DMU>> [Example 4] I the hpital example, DMU A, B, D ad L are efficiet Table 5 dete their ceter f gravity ad directi vectr frm the ceter t A, B, D ad L We diturb thee vectr radmly ad, fr example fr D, we have, dx=7, dx2=-3, dy=8, dy2=-3 Uig thi directi we lved the prgram (26) ad btaied λ = 3822, λ = 455, λ = Thu, ADL pa a facet f P I thi way we ca fid A D L facet f P apprximately Table 6 exhibit reult f radm earche We tried tw radm earche (perturbed directi) fr each efficiet DMU a diplayed i the table Evetually we fud the tw maximal fried (facet); ADL ad BDL The rea why we perturb the directi arud vertice i that everal facet are cected at a vertex ad we ca fid facet with high prbability <<Iert Table 5 here Table 5: Ceter ad directi>> Iert Table 6 here Table 6: Reult f radm earch>> 7 Variable retur-t-cale (VRS) cae S far we have dicued the ctat retur t cale cae We eed me alterati i the variable retur-t-cale (VRS) cae, which require the cvexity cditi the iteity vectr λ R : L = (27) λ λ I thi ecti, we preet ly imprtat addeda t the precedig ecti The prducti pibility et () ad the SBM mdel (4) have the additial ctrait (27) 2 Equati (7) i mdified t:

11 GRIPS Plicy Ifrmati Ceter Dicui Paper : Lemma tur ut t: ξ = wξ L w ξ η = wη L w η w L w =, w > ( ) (7A) [Lemma A] If ay member f ( ξ, η ) ( =, K, ) i VRS-iefficiet, the ξ, ) i VRS-iefficiet ( Prf: Withut lig geerality, we aume that ( ξ, η ) i VRS-iefficiet The, the ytem ξ = Xλ, η = Yλ eλ =, λ,, ha a luti ( λ,, ) with (, ) (,) ad (, ) (,), where e i the rw vectr with all elemet equal t We et ξ = Xλ ad η = Yλ (9A) By iertig (9) it (7), we have Let u defie Sice = w w 2 w = ξ ξ ξ = w w 2 w = η η η ξ = wξ w ξ = 2 η = w η w η = 2 ξ, ) P ad w =, w > ( =, K, ), we have ( η Hece, we have = (A) (A) ( ξ, η ) P (2A) ξ = ξ w η = η w ( ξ, η ) Thu, ha -egative ad -zer lac (, ) agait ( ξ, η ) Hece it i VRS-iefficiet QED 4 Therem 3 chage t: [Therem 3A] If ( ξ, η ) defied by (7A) i VRS-efficiet, the there exit a upprtig hyperplae t P at ( ξ, η ) which al upprt P at ( ξ, η ) ( =, K, ) Prf: By the trg therem f cmplemetarity, there exit dual variable v >, u > uch that v ξ v ξ u u η η u = v X u Y u e u m v R,u R, u R with ( =, K, ) (3A)

12 GRIPS Plicy Ifrmati Ceter Dicui Paper : 8-4 Iertig the defiiti f ξ, η ) i (7A) it the firt equality i (3A) ad tig =, we have Taig te f w ( w = w ( v ξ u η u ) L w ( v ξ u η u ) = (4A) > ( =, K, ) ad the ecd iequality i (3A), the equality (4A) hld if ad ly if (5A) v ξ u η = ( =, K, ) Hece the hyperplae v x u y u = pae thrugh ( ξ, η ) ( =, K, ) ad upprt P QED Sice the ytem f equati (5A) i hmgeu, if t( v, u, u ) ( t > ) i al a luti If the ra f the matrix ξ, K, ξ R η, K, η i t le tha m, the the cefficiet ( v, u, u ( m ) ( v, u, u ) i a luti t (5A), the (6A) ) i uiquely determied except fr the calar multiplier t Thi mea that the directi ( v, u ) i the uique rmal t the upprtig hyperplae If the ra f (6A) i le tha m, the there exit multiple ( v, u, u ) fr the ytem (3A) Defiiti (Facet) We call the upprtig hyperplae v x u y u defied i Therem A a facet f P I what fllw, we che the ceter f gravity f ( ξ, η )( =, K, ) a ( ξ, η ), ie w / ( ) = 5 We add the cvexity cditi eλ =t the liear prgram (26) 8 Cmpari with the radial mdel We cmpared the cre btaied by the SBM, Variati II ad the radial CCR mdel a diplayed i Table 7 The CCR cre i t le tha that f the SBM ([3, p ]) Hwever, Variati II ad the CCR are mixed We have theretical evidece betwee the tw The reult idicate vlatility f cre ad ra depedig the mdel, ad cte the imprtace f mdel electi a i alway the cae i DEA applicati 9 Ccludig remar <<Iert Table 7 here Table 7: Cmpari f SBM, Variati II ad CCR>> I thi paper, we have prped 4 variat f the SBM They have cmm characteritic a fllw: They are uit-ivariat, ie the cre are idepedet f the uit i which the iput ad utput are meaured prvided thee uit are the ame fr every DMU 2 We ca impe weight exgeuly t each iput/utput depedig their imprtace, eg ct hare Refer t Cper et al [4, p5] ad Tutui ad Gt [7] 3 Althugh we have develped ur mdel i the -called -rieted veri, ie bth iput ad utput iefficiecie are accuted i the efficiecy evaluati, we ca deal with the iput (utput) rieted mdel by taig the umeratr (demiatr) f the bective fucti (3, 7, 2) a the target 4 Future reearch ubect iclude (a) experimet real-wrld large cale prblem ad (b) extei t the uper-sbm mdel [6] Referece [] Baer RD, Chare A, Cper WW (984) Sme methd fr etimatig techical ad cale efficiecie i DEA, Maagemet Sciece 984; 3:

13 GRIPS Plicy Ifrmati Ceter Dicui Paper : 8-4 [2] Chare A, Cper WW, Rhde E (978) Meaurig the efficiecy f decii maig uit Eurpea Jural f Operatial Reearch, 2, [3] Cper WW, Seifrd LM, Te K (27) Data evelpmet aalyi: A cmpreheive text with mdel, applicati, referece ad DEA-Slver ftware, 2d Editi, Spriger [4] Simard M, (966) Liear prgrammig, tralated by Jewell WS, Pretice-Hall [5] Te K (2) A lac-baed meaure f efficiecy i data evelpmet aalyi, Eurpea Jural f Operatial Reearch, 3, [6] Te K (22) A lac-baed meaure f uper-efficiecy i data evelpmet aalyi, Eurpea Jural f Operatial Reearch, 43, 32-4 [7] Tutui M, Gt M (28) A multi-divii efficiecy evaluati f US electric pwer cmpaie uig a weighted lac-baed meaure, Sci Ecmic Plaig Sciece, i pre Appedix A Prf f Therem Suppe that ( x, y ) i CRS-iefficiet The there exit a ptimal luti ad -egative lac (, ) fr the prgram: (, λ,, ) with -zer ρ Iertig (6) t (4B) we have: ρ = mi m ubect t = = λ ( ) = = m i= r= x λ = x y λ = y x y i i r r = x λ x = y λ y Fr thi maipulati, we have the bective fucti value fr ( x, y ), Sice (, ) i emi-pitive, we have: m ρ = ρ < ρ mi m i i i= xi r r r= y r (3B) (4B) (5B) (6B) 3

14 GRIPS Plicy Ifrmati Ceter Dicui Paper : 8-4 Thi ctradict the ptimality f ρ QED mi 4

15 GRIPS Plicy Ifrmati Ceter Dicui Paper : 8-4 Table : Data f 2 hpital Iput Output DMU Dctr Nure Outpatiet Ipatiet A B C D E F G H I J K L Table 2: Reult f SBM ad Variati I DMU SBM Ref Variati I Ref A A A B B B C B,L B D D D E B,L L F A,L L G B,L B,L H L L I A,L L J B,L B K B,L L L L L 5

16 GRIPS Plicy Ifrmati Ceter Dicui Paper : 8-4 Table 3: Reult f SBM ad Variati II DMU SBM Ref Variati II Ref A A A B B B C B,L D D D D E B,L A F A,L D G B,L D H L 898 D I A,L A,D,L J B,L D K B,L A,D L L L Table 4: SBM ad Cluterig reult (Variati III) DMU SBM Ref Cluter Variati III Ref Remar A A A B B B C 826 B,L D D D D E 728 B,L A F 686 A,L L G 877 B,L 2 G lcally eff H 77 L 898 D I 92 A,L L J 765 B,L 2 J lcally eff K 862 B,L L L L 2 L 6

17 GRIPS Plicy Ifrmati Ceter Dicui Paper : 8-4 Table 5: Ceter ad directi DMU (I)Dctr (I)Nure (O)Outpatiet (O)Ipatiet A B D L Ceter directi dx dx2 dy dy2 A B D L Table 6: Reult f radm earch DMU dx dx2 dy dy2 Facet fud A A A AL B BL B BDL D ADL D BD L BL L ADL Table 7: Cmpari f SBM, Variati II ad CCR DMU SBM Ref Ra Variati II Ref Ra CCR Ref Ra A A A A B B B B C B,L D B,D 8 D D D D E B,L A A,D,L 2 F A,L D B,D G B,L D B,L 7 H L D A,D,L I A,L A,D,L B,L 5 J B,L D D 9 K B,L A,D A,D 6 L L L L 7

18 GRIPS Plicy Ifrmati Ceter Dicui Paper : 8-4 Iput 2 D A B C E Iput Figure : Efficiet ad -efficiet frtier Iput 2 A Ceter G B C Iput Figure 2: Radm earch arud efficiet DMU 8