An Indian Journal FULL PAPER ABSTRACT KEYWORDS. Trade Science Inc. Super-efficiency infeasibility and zero data in DEA: An alternative approach

Transcription

1 [Type text] [Type text] [Type text] ISSN : Volue 0 Iue 7 BoTechology 204 A Ida Joual FULL PAPER BTAIJ, 0(7), 204 [ ] Supe-effcecy feablty ad zeo data DEA: A alteatve appoach Wag Q, Guo Dog,2 School of Maageet, Uvety of Scece ad Techology of Cha, Hefe , (CHINA) 2 Depatet of Matheatc, Hefe Electoc Egeeg Ittute, Hefe, , (CHINA) E-al : gdd777@al.utc.edu.c ABSTRACT The ptfall of feablty poble ot VRS adal upe-effcecy odel a hot ue data evelopet aaly (DEA) tude. Lee et al. (202) [Euopea Joual of Opeatoal Reeach 26 (202) ] popoed a ethod to adde the poble of feablty ae fo zeo put data VRS upe-effcecy DEA odel. I th pape, we pot out that the ethod ca be eplaced wth a alteatve appoach ad the a eult ae obtaed detcally fo two ethod. The popoed alteatve ethod ca alo ovecoe the feablty poble caued by zeo data upe-effcecy DEA odel ad have eveal advatage copaed to Lee odel. Meawhle, two uecal exaple ae utlzed to llutate valdty ad applcablty of ou alteatve odel. KEYWORDS Data evelopet aaly (DEA); Ifeablty; Supe-effcecy; Evaluato. Tade Scece Ic.

2 774 Supe-effcecy feablty ad zeo data DEA: A alteatve appoach BTAIJ, 0(7) 204 INTRODUCTION The clac adal upe-effcecy odel ft developed by Adee ad Petee [] povde a effectve ea fo ehacg the dcato powe of futhe ecogzg effcet DMU. It opeate wth the DMU ude evaluato excluded fo the efeece et whch paed by the eag DMU. Howeve, the adal upe-effcecy odel ay uffe fo the ptfall of feablty poble ude the codto of vaable etu to cale (VRS). I lght of the poble, a ube of atcle have bee ted to ettle th poble. Lovell ad Roue [2] odfy the covetoal odel by agg a ue-defed calg facto to fd a feable oluto fo thoe effcet DMU fo whch feable oluto ae uavalable the VRS upeeffcecy odel. Che [3] obta the upe-effcecy value wth the coepodg vtual effcet poecto fo the DMU whoe obevato data ae feable. Cook et al [4]. develop a odfed VRS upe-effcecy odel that yeld optal oluto ad upe-effcecy coe ca chaacteze the extet of upe-effcecy both put ad output level. Lee et al [5]. develop a two-tage poce to adde the VRS feablty ue. I the ft tage, they tet whethe a VRS upe-effcecy odel feable by vetgatg the extece of output uplu (put avg) whe feablty occu the put-oeted (output-oeted) VRS upe-effcecy odel. I the ecod tage, they popoed a odfed VRS upe-effcecy odel to yeld a upe-effcecy coe that chaacteze both the adal effcecy ad put avg/output uplu. Che ad Lag [6] futhe pove that the two-tage poce ca be olved a gle lea poga. Howeve, whe a DMU ha zeo data, thee odel ay tll be feable. Lee et al. [7] pot out that zeo output data wll ot lead to feablty of the outputoeted upe-effcecy odel etoed above becaue the output de of the cotat ca alway be atfed. Theefoe, they oly aue that oe put ae zeo. The, they popoed a eved odel wll be feable whe zeo data ext put. The cuet pape pot out that the ethod Lee et al. [7] ca be eplaced wth a alteatve appoach ad the a eult ae obtaed detcally fo two ethod. The ut-vaat popety ad o-zeo popety ca alo be utable to ou ethod ad eveal late ad dffeece betwee two ethod ae copaed. The eade of th pape ogazed a follow. Secto 2 look back upo eveal adal upe-effcecy odel befoe ad the poble of upe-effcecy feablty. I Secto 3 we how a alteatve appoach to ovecoe the feablty poble eultg fo zeo data. Meawhle, two exaple ae ued to deotate the uefule of ou appoach. Secto 4 copae ou alteatve appoach wth the ethod Lee et al. [7] ad how eveal late ad dffeece betwee two ethod. I the ed, oe cocluo wll follow Secto 5. SUPER-EFFICIENCY MODELS Suppoe thee ae DMU, { DMU ( =,2, L, ) }. Let { x, y } deote the put ad output k k vecto of the kth DMU. The th put of the kth DMU deoted a x k ad the th output of the kth DMU deoted a y k. The ogal put-oeted VRS upe-effcecy odel fo effcet DMU k ca be expeed a: t.. λ x θx, =, L, θ k

3 BTAIJ, 0(7) 204 Wag Q ad Guo Dog 775 λ y y, =, L, λ = λ 0, k k () Obvouly, Model () feable at leat the followg two tuato: oe that the DMU ude codeato ha the laget output. A poted out Lee et al. [5], whe the output of the evaluated DMU outde the poducto poblty et paed by the output of the eag DMU, the feablty of put-oeted upe-effcecy wll occu,.e., the ecod cota codto caot be et. The othe cae attbute to the fact that zeo data ext put of the evaluated DMU,.e., the ft cota codto caot be et. Baed o the wok developed Cook et al. [4] ad Lee et al. [5], Che et al. [6] popoed oe odel appoach, whee oly oe odel (2) eed to be olved to obta the upe-effcecy coe f odel () feable. =.. t λ x ( + τ) x, =, L, k λ y ( β ) y, =, L, k λ = λ 0, k, β 0. τ + M β (2) whee a ue-defed lage potve ube. (I Cook et al. [4] ad Lee et al. [5] 5 M, M et equal to.) Next, Lee et al. [7] 0 popoed the followg odel (3) to adde the feablty eultg fo that whe oe put ae zeo fo oe effcet DMU. Hee, x = { x }. k k = =.. λ ( + τ), =, L, k τ + M β + t x t x x λ y ( β ) y, =, L, ; k λ = λ 0, k, t 0, β 0 t (3)

4 776 Supe-effcecy feablty ad zeo data DEA: A alteatve appoach BTAIJ, 0(7) 204 They deducted the te tx fo the left de of the put cota o that thee cota wll ot be volated whe zeo put occu. Meawhle, they explaed two eao fo uch coduct: the ft ut-vaat (ut-vaat popety); the ecod that tx wll ot be zeo whe x k zeo (o-zeo popety). Ad they poved that odel (2) ad odel (3) yeld the ae eult whe data ae potve. THE ALTERNATIVE APPROACH I fact, to eet the put cota whe zeo put data ext, we ca alo ubttute { k} x { k } x, whee. The, baed o odel (2), a alteatve odel ca be laly x = { x }, cotucted a follow: fo { k}.. λ ( + τ), =, L, k λ y ( β ) y, =, L, ; k λ = τ+ M β + λ 0, t 0, 0, β 0, = = t t x t x x (4) { k} { k} Note that a a eult of λ =, we have. λ x t x λ x t x { k } { } ( t ) x k Whe t lage eough, eve f x = 0, odel (4) alway feable. k Model () feable f ad oly f oe β > 0 odel (4) (It ca be eadly deduced fo Cook et al. [4], Lee et al. [5] ad Che et al. [6] ). Futhe, the ae cocluo ca be ealy pove that odel (2) ad odel (4) yeld the ae eult whe data ae potve. Copehevely, odel (4) ca alo hadle the feablty occug fo thoe two tuato etoed above. Model (3) ad odel (4) have a cloe elato that evealed the ubequet theoe. Theoe Model (3) ad odel (4) yeld the ae a eult optalty. Poof { k} Ft ote that x equal to ethe x o x fo DMU. k k ( k =, L, ) { } ) whe x = x k, odel (3) detcal wth odel (4). { 2) whe k } x x,.e. x = x 0, Fo a potve x 0, whe uffcetly lage k k ( + τ ) k eough, wll be zeo. The ae thg happe whe ext. So odel (3) ad odel (4) yeld the ae a eult upe-effcecy. { } t x

5 BTAIJ, 0(7) 204 Wag Q ad Guo Dog 777 { k} Th theoe dcate that whe x = x fo the evaluated DMU,.e., whe the cuet DMU do ot hold the laget th put, odel (3) ad odel (4) yeld the ae eult optalty. { k} Howeve, whe x x fo the evaluated DMU,.e., whe the cuet DMU hold the laget th put, oe dffcult optal value have bee obtaed by two odel. Nevethele, t woth otg that oe τ, t ad acqued dffeetly fo two odel ae too ty to ( =, L, ) β ( =, L, ) eglect, oly leavg the ae value of t ad β that lage eough to code. Th why the expeo that the ae a eult called. Th eult elated to the wokg blackbox lea optalty, but t ca be uecally tetfed (Two exaple ae lted TABLE -6). We alo check the alteatve appoach to the epcal cae of llo tp e fo Lee et al. [7], ad the ae a eult ae alo acheved. Futhe, the upe-effcecy coe ca be defed a ( θ = + τ + ˆ + oˆ Slaly, the put avg dex î ad output avg dex ô ca be defed the followg ae. 0, f I = φ 0, f R = φ { k} { k} ˆ x + t x + t = { k} I x oˆ = I = R β,f I φ,fr φ I I R = { > } { } whee R β 0 ad I = t > 0. { k} Becaue that ad do ot fluece the put avg dex, the detcal upeeffcecy coe ( x x î θ gaed fo each DMU baed o odel (3) ad odel (4) epectvely. COMPARING WITH LEE ET AL. Ft, ut-vaat popety ad o-zeo popety ca alo be utable fo the ue of deductg tx { k} ou odel (4) obvouly. The what we wat to ephaze ae thee: ) Fo thoe DMU ude aeet who have the laget data oe put, they get a qute { } dffeet x k copaed to the othe DMU ad the ethod of Lee et al. [7]. Fo tace, TABLE, accodg to the ethod of Lee et al. [7], all DMU have the ae x = 3, x = 4. Whle followg ou 2 { C} appoach, DMU C wth the laget X = 4 ha x = 3, whch ot equal to x = 4, ad DMU E wth { E} the laget X ha x =, whch ot equal to x =. 2 3 TABLE : Nuecal exaple fo Lee et al. [7] DMU X X 2 Y odel () odel (2) A 2 B C 4 2 Ifeable 3 D E 3 0 Ifeable Ifeable

6 778 Supe-effcecy feablty ad zeo data DEA: A alteatve appoach BTAIJ, 0(7) 204 t TABLE 2 : Reult of uecal exaple baed o odel (3) β Iput avg dex Output uplu dex Supe-effcecy coe A.3E- 2.45E E E B E E-2.87E C 7.3E- 2.68E E D E E E E E E TABLE 3 : Reult of uecal exaple baed o odel (4) t β Iput avg dex Output uplu dex Supe-effcecy coe A.3E- 2.45E E E B E E-2.87E C.22E E E D E E E E E E TABLE 4 : Nuecal exaple 2 t DMU X X 2 Y A 2 B 2 C 4 2 D 0 0 E 0 8 F 0 0 TABLE 5 : Reult of uecal exaple 2 baed o odel (3) β Iput avg dex Output uplu dex Supe-effcecy coe A E E-20.89E B E E-8 7.0E C E-4.0E D E E E E E F E E E TABLE 6 : Reult of uecal exaple 2 baed o odel (4) t β Iput avg dex Output uplu dex Supe-effcecy coe A E E-20.89E B E E-8 7.0E C E-4.0E D E E E E E F E E-7 4.3E

7 BTAIJ, 0(7) 204 Wag Q ad Guo Dog 779 { 2) The put cota codto ou alteatve appoach toge va ubttutg k } fo k. Due to x x, thee ae λ x tx { k}. λ x t x ( + τ ) x k { } x 3) Whe the DMU k get zeo data th put, the coepodg t wll be gally lage (fo 0 { k} exaple, t > 0 ). At th oet, x = x 0, o tx { k } = tx, The te t x { k} aloeflect how fa the DMU k below the hozotal effcet bouday a tx ea Lee et al. [7]. 4) A llutated above, thee two ethod yeld the ae a eult except that oe tety coeffcet ae dffeet fo DMU wth the laget put data ceta put. Ad thee dffeet tety coeffcet ae too ty to eglect. 5) By the way, the left de of both put ad output cotat elate the cluo of the { put x k ( x ay be equal to x ) fo the DMU k ude evaluato by ubttutg k } x fo, whch k x foally accod wth the ogal dea that the DMU ude evaluato hould be excluded fo the efeece et. CONCLUSIONS The covetoal VRS adal upe-effcecy odel ut uffe fo the poble of feablty. A lot of wok have bee doe a effot to ovecoe th poble. To adde th ue, the cuet pape povde a alteatve appoach fo the ethod developed Lee et al. [7]. We have how eveal late ad dffeece by copag two ethod ad two exaple ae ued to deotate that the ae a eult ae obtaed by thee two ethod. REFERENCES [] P.Adee, N.C.Petee; A pocedue fo akg effcet ut data evelopet aaly. Maage.Sc., 39, (993). [2] C.A.K.Lovell, A.P.B.Roue; Equvalet tadad DEA odel to povde upe-effcecy coe. J.Ope.Re.Soc., 54, 0 08 (2003). [3] Y.Che; Meaug upe-effcecy DEA the peece of feablty. Eu.J.Ope.Re., 6, (2005). [4] W.D.Cook, L.Lag, Y.Zha et al.; A Modfed Supe-effcecy DEA Model fo Ifeablty. J.Ope.Re.Soc., 60, (2009). [5] H.S.Lee, C.W.Chu, J.Zhu; Supe-effcecy DEA the peece of feablty. Eu.J.Ope.Re., 22, 4 47 (20). [6] Y.Che, L.Lag; Supe-effcecy DEA the peece of feablty: Oe odel appoach. Eu.J.Ope.Re., 23, (20). [7] H.S.Lee, J.Zhu; Supe-effcecy feablty ad zeo data DEA. Eu.J.Ope.Re., 26, (202). x