Cross Efficiency of Decision Making Units with the Negative Data in Data Envelopment Analysis

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1 Pceedg f the 202 Iteatal Cfeece Idutal Egeeg ad Opeat Maageet Itabul, Tuey, July 3 6, 202 C Effcecy f Dec Mag Ut wth the Negatve Data Data Evelpet Aaly Ghae Thd Depatet f Matheatc Ilac Azad Uvety - Cetal Teha Bach Teha, Ia Shaba Razavya Depatet f Matheatc Ilac Azad Uvety - Suth Teha Bach Teha, Ia Abtact The c-effcecy evaluat a effectve ethd t evaluate the dec ag ut (DMU) wth the egatve put ad utput data evelpet aaly (DEA) ad ca be pefed wth the dffeet fulat. But, e tuat the DMU ecute wth the egatve put ad utput. Th pape geealze the c effcecy evaluat f the DMU wth the egatve put ad utput DEA evet. T th ed, ft t defe the c effcecy f the DMU wth the egatve data ad the ppe a del ug deal pt ad age dectal del (RDM). Fally, the pped del dfed t educe the ze weght. T llutate the pped del a uecal exaple gve. Keywd Data evelpet aaly (DEA); Rage dectal del (RDM); C effcecy; Negatve data.. Itduct Data evelpet aaly (DEA) a paaetc ethdlgy f eaug the bet elatve effcece f a gup f the pee dec ag ut (DMU) that cue ultple put t pduce ultple utput. DEA del, ay CCR (Chae et al. (978)), ae ued t detee a et f weght that ptal the ee that t eult the bet effcecy ce f a patcula DMU ude evaluat, whle the c effcecy ce f a DMU btaed by cputg that DMU et f ce, ad the aveagg the ce. The a dea f the c effcecy t ue the DEA a pee evaluat (ee f exaple the Sext et al. (986) ad Dyle ad Gee (994)). The c effcecy ued t dffeetate betwee gd ad p pefe ad a pce wth the ccept f pee appaal, a pped t elf appaal pled by ple effcecy. A detated Dyle ad Gee (984), becaue f the -uquee f the DEA ptal weght, the ethd f aggeve ad beevlet wee pped. Depedg whch f the alteatve ptal lut t the DEA lea pga ued, t ay be pble pve a DMU' peface atg, but geeally ly by weg the atg f the. The tadtal DEA del that appled f cputg c effcecy cat deal wth the egatve data. The appach aed age dectal del (RDM) wa develped by Ptela et al. (2004) t eaue the effcecy ude the egatve data. They tduced a del baed a deal pt ad a dectal vect. The the appache ae al tduced t eaue the effcecy f a DMU Shap et al. (2007) ad Euzead et al. (200) f the cae whee the data ae egatve. F exaple Shap et al. (2007) peeted a dfed lacbaed eaue (MSBM) del that ca deal wth bth egatve put ad egatve utput. The dectal vect aed the SP age the MSBM del ad the effcecy ce f a DMU that calculated by th appach cat be geate tha the RDM effcecy f Ptela et al. (2004). Th pape defe the c effcecy f DMU wth the egatve put ad utput the DEA evet ad ppe a del by ug the deal pt ad the RDM t detee the c effcecy f the DMU wth the egatve data. The, the pped del dfed t educe the ze weght. 42

2 Th pape gazed a fllw. Sect 2 peet a bacgud f the c effcecy ad the RDM. Sect 3 defe the c effcecy the peece f the egatve data ad ppe a dfed del t bta the c effcecy f the DMU wth the egatve data. Sect 4 pvde a uecal exaple. Sect 5 cclude. 2. DEA C Effcecy Aue thee ae DMU wth put ad utput. We dete the th put ad th utput f DMU ( =,..., ) a x ( =,..., ) ad y ( =,..., ), epectvely. The effcecy atg f ay gve DMU d cputed ug the CCR del that wa edeed by Chae et al. (978): ax θ = µ y d d =.t. ω x µ y 0, =,...,, = = ωx d = =, () ω 0, =,...,, µ 0, =,...,. A et f the ptal weght btaed f the DMU ( d d =,..., ). The c effcecy f DMU ( =,..., ) calculated by ug the ptal lut f del () (Dyle ad Gee (994)) a fllw: µ d y = Ed =, d, =,...,. ω x = d The aveage f all E d ( d =,..., ) a ew effcecy eaue f DMU ( =,..., ). Th effcecy called the c effcecy f DMU ad a fllw: E = E d d =. (3) C effcecy pvde a eaue f the effcecy that t ly the bet ultple budle f DMU d ude evaluat, but al the bet budle f all the DMU. The RDM tduced by Ptela et al. (2004) ca be ued f cpag DMU whe e put ad/ utput ae egatve. Aue that th ut, =,,, ue a put vect x = ( x, x2,, x ) t pduce a utput vect y = ( y, y2,, y ). Cde a pt wth axu utput ad u put a a deal pt (f each put, =,,, x ad f each utput, =,,, ax y ) ad the IP a { } IP a { } dectal vect a ( g, g ) = ( g,, g, g,, g ). I the RDM del th dectal vect f DMU a, y x y x x y y = = ax { } ; =,, ad g R x { x } g R y y y x = = ; =,,, that eflect age f pble pveet f th DMU. The RDM del f DMU a fllw (Ptela et al. (2004)): x (2) { β β β } DR ( x, y, R, R ) = up ( x R, y + R ) T, x y x y (4) 422

3 whee ( ) {, ca pduce } T= xy x y the pduct techlgy ad exhbt the vaable etu t cale (VRS). Th pape apple the put-eted f the RDM that a patcula type f del (4). I th cae ze vect. F DMU the put eted f the RDM a fllw (Ptela et al. (2004)): R et t the y β = ax β.t. λ x x β R, =,,, = = = λ y y, =,,, λ =, λ 0, t x =,,. (5) The value f β del (5) a effcecy eaue ad equal t β = DR ( x, y, R,0). Theefe, the x eaue f the put effcecy f ut RDM ( x, y, R,0) = β. Al, f the th put we have x β = x { x } { x } x ad th effcecy eaue eflect the dtace betwee the beved ad the taget put level ( x ). I fact the beved ad the taget level ae cpaed by ug the ubeved deal DMU. 3. The C Effcecy f the Dec Mag Ut wth the Negatve Data T etate the c effcecy f the DMU wth the egatve data we eed the ptal weght value. Theefe, we cde the dual f the RDM a fllw: θ = vx uy + u = =.t. vx u y + u 0, =,, + ur = = v 0, u 0, u fee, =,,, =,, vr (6) whee R = x { x } ad R = ax{ y } y. The effcecy f DMU btaed a ( vx uy + u ) ug the abve del, whee ( v, u, u ) the ptal lut f del (6) whe the DMU evaluated. Theefe, we defe the c effcecy f the DMU ( =,..., ) ug the ptal lut f del (6) a fllw: E = v x u y + u = (7) ( ),,,...,. The aveage f all E ( =,..., ) a ew effcecy eaue,.e. the c effcecy ce, f DMU ( =,..., ) wth the egatve data ad a fllw: 423

4 E = = E. (8) The ze weght ae a pble the c effcecy evaluat. T avd ze weght f the put ad utput f DMU wth the egatve data we ca pve del (6) by ug the ecday gal. T th ed, we cde the fllwg pveet f del (6): ax δ = =.t. vx u y + u 0, =,, vx uy + u = θ + ur = = vx v 0, u 0, u fee, =,,, =,, vr δ (9) whee the ctat vx uy + u = θ ad vx δ have bee added t pve del (6). Mdel (9) che a ptal lut ag the alteatve ptal lut f del (6) wth the hghet value f the put f the ude evaluat DMU,. e. DMU. It ca be ee, del (9) che a ptal lut f del (6) wth the ptve put weght. Theefe, t educe the ze weght. 4. Nuecal Exaple T llutate the c effcecy f DMU wth the egatve data ad the dfed del, we cde 0 DMU wth e put ad tw utput (Euzead et al. (200)). Table hw the data. It ca be ee, the ft utput f e DMU egatve. Hece, the del f the egatve data cat be ued t detee the c effcecy. Table. The put utput data f 0 DMU DMU Iput Output Output2 DMU 2 5 DMU DMU DMU DMU DMU DMU DMU DMU DMU Table 2. The c effcecy f the DMU wth the egatve data d, C eff

5 The lat w f Table 2 hw the eult f the c effcecy f DMU ug del (9). It ca be ee, the DMU 4 effcet by ug the weght f all DMU, whle DMU 7 effcet ug the weght f the the DMU. By ug the c effcecy we ca bta e fat abut the peface f the DMU wth the egatve data. F tace the DMU ca be aed by ug the c effcecy ce. Accdg the c effcece, the DMU 4 ha the hghet a ad the DMU 7 ha the lwet a. 5. Cclu I e tuat thee ae e DMU wth the egatve put ad utput. Hece, the DEA del f egatve data DEA lteatue cat be ued t detee the c effcecy f thee DMU. T th ed, th pape cdeed the dual f the RDM t etate the weght f deteg the c effcecy f DMU ad pped a dfed del. The, dfed the pped del t educe the ze weght ad che a ptal lut ag the alteatve ptal lut f del (6) wth the hghet value f the put f the ude evaluat DMU. Refeece Chae, A., Cpe, W. W., ad Rhde, E., Meaug the effcecy f dec ag ut, Eupea Jual f Opeatal Reeach, vl. 2, pp , 978. Dyle, J., ad Gee R., Effcecy ad c effcecy DEA: devat, eag ad the ue, Jual f the peatal Reeach Scety, vl. 45, pp , 994. Euzead, A., Auze, A.L., ad Thaaul, E., A e-eted adal eaue f eaug the effcecy f dec ag ut wth egatve data, ug DEA, Eupea Jual f Opeatal Reeach, vl. 200, , 200. Ptela, M.C.A.S., Thaaul, E., ad Sp, G.P.M., Negatve data DEA: A dectal dtace appach appled t ba bache, Jual f the Opeatal Reeach Scety, vl. 55, pp. -2, Sext, T.R., R.H. Sla, ad Hga, A.J., Data evelpet aaly: ctque ad exte. I: Sla RH, edt. Meaug effcecy: a aeet f data evelpet aaly, vl. 32, Sa Facc: Jey-Ba, pp , 986. Shap, J.A., Meg, ad W., Lu, W., A dfed lac-baed eaue del f data evelpet aaly wth 'atual' egatve utput ad put, Jual f the Opeatal Reeach Scety, vl. 58, pp ,