Super Efficiency with 2- Stage DEA Model

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1 Sup Effccy wth 2- Stag DEA Md Sha Ea Put Dpatt f Mathatc, Uvty f Suata Utaa Mda, Ida Abtact DEA d tat a t f vauatd DMU ad u t tat th ffccy c by vauatg ach DMU a data t. Th ach dtd th w ch f 2-tag DEA d aay bta f w ffccy c, cad up ffccy DEA d. W th xtdd th DEA d that wa fuatd by cdg put-uput td f ach ud data. Th d fuatd by a a pga ad gv th a ut: () a atatv w ch f th 2-tag f DEA d, (2) up ffccy c f a gv data ad (3) DMU ag bad ach t up ffccy c. Kywd Data Evpt Aay (DEA), a pga, up ffccy, ag I. INTRODUCTION Data Evpt Aay (DEA) d a thd that ud t tat a ft t vauat th pfac th ffccy f a f th tt that a t b vauatd. I pvu tud 2-tag DEA d wa dvpd t f appcat. Ba ad Nataaa [] dvpd 2-tag DEA d by ug a g aay, Mt-Ca uat, that btad 2-tag DEA tat f a cta vaab ctxt wth dft ctat put vct d. Sa ad W [2] xtdd th axu hd thd d t dt 2-tag g t DEA d that pducd a DEA tat f 2-tag DEA d a a ut. Ad t a. [3] dvpd a ada up-ffccy au cad AP d. Th d wa cpag th DMU that vauatd wth a a cbat f th DMU, wh xcudg th bvat f th DMU bg vauatd. Th d th wa xtdd by T [4] by cdg th put-utput ac f a -ada up ffccy cad SBM d. Cat t a. [5] ppd a cphv catgzd vvw f thd ad d f dfft ut-tag pduct achtctu DEA d. Sfd ad Shu [6] wa tudd a pduct pc bag ct by tatg th tw tag, dpdty, wthut aug ay athp btw th tw tag. A v appach th dvpd by Ka ad Hwag [7] that cd a athp f th tw tag ad pvd a d that tat th va ffccy f th pduct pc. Th appach bad th aab aupt that th dat au f th vau a wthut cd whth a utput put th ft tag. Ch t a. [8] th tducd th addtv ffccy dcpt tw-tag pc ud th aupt f athp. Th dg appach factat th azat f a -a athatca pga ad bad th aupt that th wght f th tw tag xtdab t Vaab Rtu t Sca (VRS) aupt. I th pap w pt a atatv w ch f th 2-tag ud th aupt f th athp btw th tw tag. W th fuatd th d by cdg put-utput td f ach gv data. W ct put tat that pduc th ffccy c f th ft tag,. Th, utput tat f th cd tag that pduc 2, th ffccy c f th cd tag. Thu, th va ffccy c ca btad by a p dv u. Th pap ufd a fw. Sct II w vw bacgud fat th tudy aa ad ut that adptd th 2-tag aay. Sct III w pp a atatv w ch f 2-tag DEA d. I Sct IV, w pt th ut cpa f th w d wth a gv t data tudd by Wag t a. [9]. Fay, ccu ad futu ach a pvdd th at ct. Th ut f up ffccy c ad DMU ag a pvdd th Appdx ct. II. DATA ENVELOPMENT ANALYSIS (DEA) DEA d ca b hw tw ga f, a pga ad a g f. DEA d ug a pga thd whch th wghtd c a dc vaab ad pduc th ffccy c f ach Dc Mag Ut (DMU) a ut f DEA d ( Sfd ad Tha [0]; Lv []; Cp t a. [2] ad Thaau [3]). Cha t a. [4] hwd that DEA a ut-fact pductvty aay d that ud t tat th ffccy atv c (E) f hgy t f DMU that fuatd a u f utput wght E 00% (2.) u f put wght Au vauatd DMU, DMU ( =,, ) f ach ug put, x ( =,, ), that pduc utput, y ( =, ), pctvy. Th ffccy c f DMU ca b dfd a u f a utput wght dvdd by u 94

2 f a put wght wh g put, v ( =,, ) ad g utput, u ( =,, ). Mathatcay, t ca b fuatd a fw u y (2.2) vx DEA d tat a t f vauatd DMU ad u t tat th ffccy c by vauatg ach DMU a data t. It d by tatg a ft pt whch gv tva f ffccy c, 0 E t ach vauatd DMU. Th ffccy c wa btad by cpag DMU pfac t a f vauatd DMU pfac a cta data t. Th btad ffccy c by ug DEA d gv th hght ffccy atv t tva 0 E f ach DMU ( =,, ). f,, ;,, ;,, ad u, v 0. F (x,y) P, ay ptv actvty ( x, y) wth x x ad y y cudd P. Thu, ay actvty wth put f tha x ay cpt ad wth utput gat tha y ay cpt fab. B. Ba, Cha ad Cp (BCC) Md Th xtdd f CCR d th tudd by Ba t a. [5]. Th d ha t pduct ft pad by cvx hu f xtg DMU that ad t Vaab Rtu t Sca (VRS) aupt chaactzat. Ba t a. [5] pubhd th BCC d wh pduct pbty t whch dfd by P {( x, y) x X, y Y,, 0} A a g f, DEA d a paatc t aayzg ffccy c wth utp put ad utput that cd bth uatatv ad uattatv a data t. A, DEA a a pgag d that cacuat utp put ad utput ad vauat DMU bth uatatvy ad uattatvy by a a pga f. Gay, th a tw bac DEA d a fw. A. Cha, Cp ad Rhd (CCR) Md DEA d a gay ppd by Cha t a. [4] ay Cha, Cp ad Rhd (CCR) d t pduc th ffccy ft bad ccpt f Pat ptu. Th d wa but th aupt f Ctat Rtu t Sca (CRS) f th pduct ft th g put ad g uput ca. M gay, th d aud that th pduct pbty t wh X = (x ) R da Y = (y ) R a a gv data t, λ R ad a w vct wth a t ua t. Th BCC d dff f th CCR d that dvpd by addg cvxty ctat. Mathatcay, BCC d ca b fuatd a fw ax u y w (2.6). t vx (2.7) u y v x w 0 (2.8) P {( x, y) x X, y Y, 0} f,, ;,, ;,, ad u, v 0. wth th pa f ptv put ad utput vct, ( x, y )(,, ) bg t P f DMU. Thu, w ca au that uch a pa f ptv put-utput, x R ad y R. Cha t a. [4] dvpd CCR bac d put td DEA whch cta f bctv fuct, axzg DMU ffccy c wth ctat that ffccy c f a DMU tha ua t a fw ax uy (2.3) III. 2-STAGE DEA MODEL WITH INPUT-OUTPUT ORIENTED Th ach bad 2-tag DEA d dvpd by Dpt t a. [6]. W th xtdd th d that wa fuatd by cdg put-uput td f ach ud data. Th bctv f DEA d wth put td t z put whch pducd at at ua t tta utput f gv data. Wha th a f utput td d t axz th utput whch btad t gat tha tta put f a gv data. Dpt t a. [6] dvpd a addtv dcpt. t vx (2.4) d t 2-tag DEA d t tat th ffccy c. Th d ud th CRS aupt by cdg put-utput td f ach vauatd data. Fg. u y v x 0 (2.5) hw ch f 2-tag (2.6) DEA d by Dpt t a. [6] a fw 942

3 Fg. 2-tag DEA d by Dpt t a. [6] Au th xt DMU ( =,, ) wh ach u put, x ( =,, ), that pduc utput, y ( =,, ). F ach ffccy c f vauatd DMU ca b btad by 2-tag DEA d that pduc ad 2, pctvy Stag. Output td p vx (2.9). t wpzp (2.0) w z v x 0 (2.) p p p Stag 2. Iput td 2 p ax uy (2.2).t wp zp (2.3) u y w z 0 (2.4) p p p w, u 0 p Futh, 2-tag DEA d a btad f ach pututput td a fw: Stag. Output td vx (2.5) p.t wpzp (2.6) w z v x 0 (2.7) p p p u y w z 0 (2.8) p p p v, wp, u 0 p (2.2).t wpzp p Stag 2. Iput td ax uy (2.9).t wpzp (2.20) w z v x 0 (2.2) p p p v, wp, u 0 Md (2.9-2.) wa dvd th fwg cbd t b a b-bctv a pga wth th a t axz th va ffccy c f 2-tag DEA d that fuatd a fw ax F ax u y v x w z v x 0 (2.22) p p p u y w z p p p v, wp, u 0 0 Ad 2-tag DEA d f tat th va ffccy c a.t z z 0 p p zp y y (2.23), 0 f,, ; p,, ;,,. IV. 2-STAGE DEA MODEL WITH INPUT-OUTPUT ORIENTED Th ach hw th xtdd f ut-tag pc f DEA d, w btad a atatv 2-tag DEA d. Lt u cd that th a DMU ( =,, ) wh ach ug put, x ( =,, ). It pduc utput, z ( =,, ) f Stag. Th, cd that w hav h ( =,, ) a put f Stag 2 that pduc fa utput, y p (p =,, ). Fg. 2 hw th w ch f th xtdd f 2-tag DEA d that w ud. 943

4 Fg. 2 Extdd f 2-tag DEA d.t wpzp (2.32) p w z v x 0 (2.33) p p p t, u, v, wp 0 Ntc that Stag pduc utput, z, that b pcd by put, x. Sc th dft f ffccy c Euat (), w bta a th ffccy c f Stag. Aftwad f Stag 2, th ffccy c th tatd f at cpa btw th fa utput (y p ) ad put f Stag 2, (h ). W th btad 2 a th ffccy c f Stag 2. S that, h w btad DEA d Stag ad Stag 2 bad put-utput td a fw. Stag. Output td p p w z p p vx (2.24).t wpzp (2.25) w z v x 0 (2.26) p p p v, w 0 Stag 2. Iput td p 2 p uy th (2.27).t wpzp (2.28) u y t h 0 (2.29) p p p p t h w z 0 (2.30) t, u, v, wp 0 Th, ud th CRS aupt, w ca fuatd th 2- tag DEA d a ax u y v x (2.3) By a p dv u, w btad th va ffccy c by 2 (2.34) 2 that th fd a up ffccy c f 2-tag DEA d f ach vauatd DMU. V. COMPUTATIONAL RESULTS Th xtdd 2-tag DEA d ( ) btad th w appy t a data t that tudd by Wag t a. [8] that gv Tab ( Appdx). W ud th utput: Dpt (z ), Fxd at (z 2 ) ad IT data (z 3 ) that ha b pcd by a put: Nub f py (x ) whch pduc a th ffccy c f DEA d Stag. O Stag, w btad wghtd utput ad wghtd put pctvy that gv c. F DEA d Stag 2, w u a put: Pft (h ) that pduc a utput: La cvd (y ) ad gv 2 a th ffccy c f DEA d Stag 2. Thf, th va ffccy c btad by d (2.34) that fd a up ffccy c. Tab 2 ( Appdx) pt th ffccy c btad by appyg d ( ) th data f Tab. Tab 2 hw th cpa ut f th ffccy c btw d that dvpd by Dpt t a. [6] ad d ( ). F th ut, w btad put ad utput wght f DEA d Stag ad Stag 2, pctvy. F ach up ffccy c f ach vauatd DMU,, w dtd DMU ag that gv Tab 3 ( Appdx) that uaz th ut btad f d ( ). VI. CONCLUSIONS AND FUTURE RESEARCH Th ach tducd a w atatv ch f 2- tag DEA d t bta up ffccy c f vauatd DMU a data t. 2-tag DEA d th wa fuatd t a pga f a bad a w ch that hwd Fg. 2. Th d wa xtdd f CCR d by cdg put-utput td a data t. Th bac da f th d bad put-utput td Stag ad utput td Stag 2, that up ffccy c btad f d (2.34). Ttg u d wth data t ta f pvu tud [9], hw that ut btad a cpaab t th ptd tatu a gv Tab 2. I futu 944

5 ach, w w xtd a w atatv ch f 2- tag DEA d by cdg put ad utput tva a data t. REFERENCES [] Ba, R.D. ad Nataaa, R. Evauatg ctxtua vaab affctg pductvty ug Data Evpt Aay, Opat Rach, V. 56 (2008), pp [2] Sa, L. ad W, P.M. Tw-tag DEA: Cavat pt, Jua f Pductvty Aay, V. 35 (20), pp [3] Ad, P. ad Pt, N.C. A pcdu f ag ffct ut Data Evpt Aay, Maagt Scc, V. 39 (993), pp [4] T, K. A ac-bad au f up-ffccy Data Evpt Aay. Eupa Jua f Opata Rach, V. 43 (2002), pp [5] Cat, L., Pt, R., ad Uvch, W. A cafcat f DEA d wh th ta tuctu f th Dc Mag Ut cd, A. Opata Rach, V. 73 (200), pp [6] Sfd, L.M. ad Zhu, J. Pftabty ad atabty f th tp 55 US cca ba. Maag Scc, V. 45 (999), pp [7] Ka, C. ad Hwag, C.N. Dcpt f tchca ad ca ffcc tw-tag pduct yt. Eupa Jua f Opata Rach, V. 2 (200), pp [8] Ch, Y., C, W.D., ad Zhu, J. Dvg th DEA ft f tw-tag DEA pc. Eupa Jua f Opata Rach, V. 202 (200), pp [9] Wag, C.H., Gpa, R. ad Zt, S. U f Data Evpt Aay ag fat tchgy pact f pfac, A. Opata Rach, V. 73 (997), pp [0] Sfd, L.M.., ad Tha, R.M. Rct dvpt DEA, Jua f Ectc, V. 46 (990), pp [] Lv, C.A.K. La pgag appach t th aut ad aay f pductv ffccy, Jua f Spah Scty f Stattc ad Opat Rach, V. 2 (994), pp [2] Cp, W.W., Sfd, L.M. ad T, K. Data Evpt Aay, Kuw Acadc Pubh (2000), pp [3] Thaau, E. Th u f Data Evpt Aay th guat f UK wat utt, Eupa Jua f Opata Rach, V. 26 (200), pp [4] Cha, A., Cp, W.W., ad Rhd, E. Maug th ffccy f dc ag ut, Eupa Jua f Opata Rach, V. 2 (978), pp [5] Ba, R.D., Cha, A. ad Cp, W.W. S d f tatg, tchca ad ca ffcc Data Evpt Aay, Maagt Scc, V. 30 (984), pp [6] Dpt, D.K., Ka, G. ad St, D. Addtv dcpt tw-tag DEA: a atatv appach, ICOCBA (202). 945

6 APPENDIX Tab. IT data (Suc: Wag t a. [9]) DMU Fxd at IT budgt Nub f Dpt Pft La cvd ($ b) ($ b) Epy (thuad) ($ b) ($ b) ($ b) Tab 2. Rut f IT data Dpt t a. [6] Md ( ) DMU Θ Θ Θ utput put utput put 2 2 Wghtd Wghtd Wghtd Wghtd (Stag-) (Stag-) (Stag-2) (Stag-2)

7 Tab 3. DMU ag bad t up ffccy c DMU 2 0 Rag

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