Generalized Super Efficiency Model for Ranking Efficient Decision Making Units in Data Envelopment Analysis

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1 Autala Joual of Bac ad Appled Scece, 5(12): , 2011 ISSN Geealzed Supe Effcecy Model fo Rakg Effcet Deco Makg Ut Data Evelopet Aaly 1 M. Fallah Jeloda, 2 G.R. Jahahahloo, 2 F. Hoezadeh Lotf, 1 A. Ghola Ab 1 Depatet of Matheatc, Ilac Azad Uvety, Foozkooh Bach, Foozkooh, Ia. 2 Depatet of Matheatc, Ilac Azad Uvety, Scece ad Reeach Bach Teha, Ia. Abtact: I evaluatg deco akg ut (DMU ) by ug Data Evelopet Aaly (DEA) techque, we ecoute the tuato whch oe tha oe ut take effcecy coe of oe. I uch a cae, oe ctea hould be codeed to ak the DMU. Soe effcet techque uch a AP, MAJ,etc ay be ued th way. Fo oe et of data, wth pecal tuctue odel that above etoed, ay be feable ad utable. I th pape, a ew odel developed that all the extg dawback of pevouly appled odel eove. Soe uecal exaple ae put fowad. Key wod: DEA, Effcecy, Rakg. INTRODUCTION Data Evelopet Aaly (DEA) a o-paaetc ethod fo evaluatg deco akg ut (DMU). It toduced by Chae et al., (1978), aeet of a educatoal cete the USA ad the exteded by Bake et al., (1984). Matheatcal pogag wa ued by the to eet the goal. I evaluatg the elatve effcecy of each deco akg ut by DEA, the coe betwee zeo ad oe ae obtaed. I th way, uually oe tha oe ut ay be effcet the DEA odel ad the coe wll be 1. It hould be codeed that a ube of effcet ut the Vaable Retu to Scale (VRS) odel, ae ot le tha the Cotat Retu to Scale (CRS) odel a well. So, the eeache have popoed oe ethod to dffeetate betwee the effcet ut. It called Rakg effcet ut DEA. Thee ae o ay akg ethod that each of the ha pecal faclte ad popete fo akg effcet ut. Chae et al., (1958a), have couted a ube of te whch, a effcet DMU play the ole of a bechakg ut fo othe, ad the o wll be ued to ak the. Becaue of dffculte fdg the efeece et of a DMU, the odel ot a appopate oe. Chae et al., (1985b), have popoed aothe ethod ode to fdout a bechak DMU. The ate of output ha bee chaged ad evaluated the vaato of effcecy coe by the. Howeve, they dd t ealze, how they fulfl t. Sexto et al., (1986), have uggeted the Co Effcecy ethod. I th ethod, the weght, whch ae obtaed by olvg each of -lea poble, ae ued. They have evaluated the effcecy of each DMU, fo eveal te ad the the data would be put a atx. Each ow of th atx cota the co effcecy coe of DMU. The aveage of each ow coputed ad the eult ae toed a a akg eaue. It ee that, t hould be a vald ethod, but oe dffculte would be faced. The ot potat poble coe to extece, whe the DEA odel have alteatve oluto. Fally, t hould be oted that, thee ae oe techque ad tatege DEA, whch ae poed o the akg. Fo exaple, Thopo et al., (1992), have ued the afe ego. I the techque, a ube of effcet DMU ay be educed. But t the ethod ot a utable oe becaue of dffculte fdg the appopate weght. Adle et al., (2005), have uggeted aothe ethod to ake a dffeece betwee DMU. I the odel they have deceaed a ube of put ad output a well, by ug copoet aaly. It caue, the ube of effcet DMU to be deceaed. But, geeal, the odel ay ot be ued fo a pefect akg. Adeo ad Petee (AP odel) (1993), have aked extee effcet ut by ottg the fo Poblty Poducto Set (PPS), ad the Mehaba et al., (MAG) (1999), have odfed the APE odel. I oe ccutace, the etoed odel ay be feable ad pecally the AP odel ay be utable becaue of extee etvty to all vaato data, whee oe DMU have elatvely all value fo Coepodg Autho: Mehd Fallah Jeloda, Depatet of Matheatc, Ilac Azad Uvety, Foozkooh Bach, Foozkooh, Ia. E-al: ehd.fallaheloda@yahoo.co, ehd.fallah@aufb.ac. 2952

2 Aut. J. Bac & Appl. Sc., 5(12): , 2011 oe of the put. Saat et al., (1999), have odfed MAJ odel ad olved t feablty ad Jahahahloo et al (2006), have chaged the type of data oalzato ode to eceve a uch bette eult. I ode to eove the dffculte fo AP ad MAJ odel, oe atheatca have ued pecfc o. Fo tace, Jahahahloo et al., (2004), have pactced L1 o fo akg effcet ut. Ateoo et al., (2001), have expeeced L2 o to fd the gap betwee evaluated effcet ut ad the ew PPS. Gadet le ad ellpod o have bee ued by Jahahahloo et al., (2004), ode to ak effcet ut. Toe, (2001) ad (2002), ha ued SBM odel th way. Shalg L et al., (2006), have developed the wok of Toe, (2001) ad (2002), fo t feablty by ug a kd of SBM odel. To evew akg ethod ee alo Adle et al., (2002). Rakg Model: I th pat, oe akg odel Data Evelopeet Aaly (DEA) ae uazed. Code, DMU wth put ad output. The put ad output vecto of DMU ( = 1,...,) ae X = (x 1,...,x ) t, Y = (y 1,...,y ) t whee X $ 0, X 0, Y $ 0,Y 0. By ug the cotat etu to cale, covexty ad poblty potulate, the o-epty poducto poblty et (PPS) defed a follow: Tc ( X, Y) : X X, Y X, 0, 1..., 1 1 I ode to ak the effcet ut, the evaluated DMUo fo Tc ae otted by Adeo ad Peteo (2005). The popoed odel CCR/ε odel a follow: ax 1 1 St. vx 1 uy vx 0, 1,..., o 1 1 u, 1,..., v, 1,..., uy o o (1) ad t dual : 1, o 1, o 1 1 [ ] St. x x, 1,..., o y y, 1,..., o 0, 1,...,, o 0, 1,..., 0, 1,..., (2) 2953

3 Aut. J. Bac & Appl. Sc., 5(12): , 2011 Fo effcet ut θ*$ 1 ad fo effcet ut 0 < θ* < 1.. Fo oe detal ee fgue 1: Fg. 1: AP odel Ut B a extee effcet oe. I ode to ak B by ug AP odel, at ft, t hould be otted * OB * fo T c. The ew fote AFCD ad t effcecy coe : B. It obvou that B 1. OB Ottg o extee effcet ut F would ot chage the fote, o t coe ea 1. That ea, AP odel ay ot be ued fo akg o extee effcet ut. It the poble of all DEA akg odel. AP odel ha two potat poble: (1) AP odel ay be feable fo pecal data put oeted cae. I fgue 2, Fo exaple, ut D ha the aout of zeo t ft put but oe of the othe ut ha zeo the ae tuato. Hece, AP odel feable th cae. It hould be codeed that the odel alway feable output oeted cae. Fg. 2: AP odel feable (2) AP odel utable fo oe DMU whch, oe of the data eleet cloed to zeo. To eove the poble that coe to accout AP odel, Mehaba et al., (2006) have offeed MAJ odel. Ft of all, they the evaluated ut fo poducto pobly et ae extcated, the all put by w ae ceaed. The popoed odel depcted fgue 3: 2954

4 Aut. J. Bac & Appl. Sc., 5(12): , 2011 Fg. 3: MAJ odel MAJ odel ay be wtte a follow: 1+w 1 w St. x x, w 1,..., 1, o 1, o o y y, 1,..., o 0, 1,...,, o (3) It ca be cocluded that the odel ot alway feable fo ay et of data. Theoe 1: The eceay ad uffcet codto fo feablty of MAJ odel : evaluatg of DMU o, o y o =0, = 1,..., o thee ext DMU, o uch that y 0. Poof. ee Mehaba et al., (1999). Popoed Model fo Rakg Effcet Ut: I ode to extcate the feablty of AP ad MAJ odel, t hould be codeed dffeet tep fo ceag put ad deceag output, tead of ug equal tep. The uggeted odel fo akg extee effcet ut a follow: 1 1 [ ] 1 1, o 1, o St. x x, w 1,..., y y z, 1,..., 0, 1,...,, o o o w 0, 1,..., z 0, 1,..., w z (4) 2955

5 Aut. J. Bac & Appl. Sc., 5(12): , 2011 whee α ad β ae the weght that aage tate, o they hould be foud fo the data et by the elatve potace. Theefoe, the odel 4 ay be opeated by aage hae, becaue of codeg aage opto a well. Suppog the etty of obectve fucto whch zato, the aout of α ad β ae potat. I th tatueque, the lttle α be ued, the oe ceag w be obtaed. Ad alo we ca llutate th fact fo β. So the aout of α ad β fluece upo the optal obectve value of odel 4 ad at lat, t wll poe o optal oluto. If the aage had o uggeto fo the weght, the followg ctea would be hadled: R f x o 0, xo 0 othewe. ad yo f yo 0, R 0 othewe. whee R ax { x } ad R { y } fo evey ad. By ug above foulato, the weght wll be y elatve potace of put ad output. If thee o elatve potace betwee put ad output, the all weght ca be codeed wth the aout of 1 the odel. Fally, t hould be pad atteto that all data odel 4 ae oalzed. Fo oalzg, the followg defto ae eeded whch would be ued fo akg: x y x ad y. It hould be udeled hee that odel 4 ak oly extee ax { x } ax { y } effcet ut. Obvouly, the optal obectve value of odel 4 geate tha 1 fo thee ut ad t wll be 1 fo the othe. Theoe 2: Model 4 alway feable ad t optal obectve value bouded. Poof: 1, o To poof feablty defe: y y o 0 1 1,...,,, o hece w x 1,..., 0 o the z =0 fo all. If y y 0 the z y y, o o 1, o 1, o 1, o y y 0 the z y y, o o 1, o 1, o =1,...,. If =1,..., If obvou that thee oluto ae feable fo odel 4 ad the poof of feablty co To poof the ecod pat, ug oalzed data, 0 # w # 1 ad becaue of o-egatve: 1 0 w. Clealy 0 the w Slaly 1 1 z. 2956

6 Aut. J. Bac & Appl. Sc., 5(12): , 2011 w 1 [ 1 1 ] z, ad the poof copleted. Theoe 3: If R* = (λ*, W*, Z*) be a optal oluto of the ecet odel, the all of the followg cotat ae actve o R* a): x x, w 1,..., 1, o b): y y z, 1,..., 1, o o o (5) e, thee o lack optalty. Poof. By cotadcto, uppoe that oe of the cotat of (a) (e.g. the ft cotat) ot actve o R*. * * * x xo w. 1 ( x1 o w1) x1 1, o 1, o * * * * * * * * * Alo: The defe Theefoe w w w theaw w z w z ad t volatg wth optalty of (λ*, W*, Z*). Theefoe all cotat (a) ad (b) ae bdg fo evey optal oluto. I odel4 f w = w, = 1,..., ad z = 0, = 1,...,, the the MAJ odel wllbe cocluded. Slaly, f w = w, = 1,..., ad z = w, = 1,...,, the the odfed MAJ odel [14] wll be the coequet. It a follow: M 1 w St. x x, w 1,..., 1, o 1, o o y y, w 1,..., o 0, 1,...,, o (6) 4 Exaple: The etoed appoach gog to be appled by two data et. I the ft exaple, a data et cotucted to how poble of tadtoal akg odel. AP ad MAJ odel (ee table 1). The ecod exaple cot of 15 Fotue top US cte 1996 (ee table 3), (2004). 4.1 Exaple1: Code 28 (DMU) wth 3 put ad 3 output. Thee data ae uazed table 1: The eult of akg ae how table 2. I th exaple, all weght ae codeed wth the aout of 1. I the followg table S.AP ea the coe of AP odel ad R.AP ea the ak of AP odel etc. Code the ft effcet ut. It ha oly a zeo put t data, theefoe, t would be feable AP odel. Ut 2 ha a all put ecod copoet, hece AP odel utable fo th DMU ad t effcecy coe ot a eal oe. Ut 1 o-zeo t ft output ad all the othe ut ae zeo the 2957

7 Aut. J. Bac & Appl. Sc., 5(12): , 2011 copoet, o ut 1 feable MAJ odel. Obvouly, thee o poble fo the ew odel ad t alway feable ad table fo pecal data uch a the exaple. 4.2 Exaple2: Rakg of 15 US cte: Hee, hgh-ed houg pce (1000 US dolla) lowe-ed houg othly etal pce (US dolla) ad ube of volet ce, a thee DEA put ad eda houehold coe (US dolla), ube of people wth bachelo degee (llo) ad ube of docto (thou-ad) a thee DEA output evaluatg 15 US cte. Thee data ae uazed table 3: Table 1: Data DMU Iput 1 Iput 2 Iput 3 Output 1 Output 2 Output Table 2: The eult of akg Eff Ut S.AP R.AP S.MAJ R.MAJ S.ew odel R.ew odel DMU 1 Ifeable. - Ifeable DMU E DMU DMU DMU Table 3: Data of 15 US cte. DMU Iput 1 Iput 2 Iput 3 Output 1 Output 2 Output

8 Aut. J. Bac & Appl. Sc., 5(12): , 2011 Table 4: Relatve potace of put ad output. EFF ut α 1 α 2 α 3 β 1 β 2 β 3 DMU DMU DMU DMU DMU DMU DMU Table 5: The eult of akg. Eff Ut S.AP R.AP S.MAJ R.MAJ S.ew odel R.ew odel DMU DMU DMU DMU DMU DMU DMU By ug the etoed ctea, the weght α ad β fo each effcet DMU ae fgued out the followg table: The eult of akg cocluded fo 15 US cte ae how table 4. I the followg table S.AP ea coe of AP odel ad R.AP ea the ak of AP odel etc. Cocluo: I th pape a ew odel popoed ode to ak the extee effcet ut whch ae alway etcted ad feable. Oe of the advatage of the offeed odel t coopeatve featue fo the aage a well a codeg the dea evaluatg the ut. Futhe-oe, the odel ha o lack optalty, ad ot pobably, t doe t take place AP ad MAJ odel. I the ft exaple, ut 1 feable AP ad MAJ odel but t eached the hghet coe the ew odel.the AP odel fo ut 2 ot utable. A a cocluo, the ew odel ha eoved the poble of AP ad MAJ odel. REFERENCES Adle, N., L. Feda, Z. Suay-Ste, Revew of akg ethod the data evelopet aaly cotext. Euopea ual of opeatoal eeach, 140: Adle, N., B. Golay, Evaluato of deegulated ale etwok ug data evelopet aaly cobed wth pcple copoet aaly wth a applcato Wete Euope. Euopea Joual of Opeato Reeach, 132(2): Ateoo, A., G.R. Jahahahloo, S. Kodota, Rakg of deco akg ut data evelopet aaly: A dtace-baed appoach. Appled atheatc ad coputato., 171: Adeo, P., N.C. Petee, A pdocedue fo akg effcet ut data evelopet aaly. Maageet cece., 39(10): Bake, R.D., A. Chae, W.W. Coope, Soe odel fo etatg techcal ad cale effcece data evelopet aaly. Maageet Scece, 30(9): Chae, A., C.T. Clak, W.W. Coope, B. Golay, 1985a. A developet tudy of data evelopet aaly eaug the effcecy of ateace ut US a foce. Aal of Opeato Reeach, 2: Chae, A., W.W. Coope, B. Golay, L. Sefod, J. Stutz, 1985b. Foudato of data evelopet aaly fo Paeto-Koopa effcet epcal poducto fucto. Joual of Ecooetc, 30: Chae, A., W.W. Coope, E. Rhode, Meaug the effcecy of deco akg ut. Euopea Joual of Opeato Reeach, 2: Coope, W., L. Sefod, K. Toe, Data evelopet aaly a copeheve text wth Model applcato efeece, DEA olved oftwae. Thd Ptg. By Kluwe acadec publhe. Jahahahloo, G.R., F. Hoezadeh Lotf, N. Shoa, G. Tohd, S. Razava, Rakg ug L1 o data evelopet aaly. Appled atheatc ad coputatoal., 153(1):

9 Aut. J. Bac & Appl. Sc., 5(12): , 2011 Jahahahloo, G.R., L. Pouka, M. Zaepheh, odfed MAJ odel fo akg deco akg ut data evelopet aaly. Appled atheatc ad coputato., 174: Jahahahloo, G.R., M. Sae, F. Hoezadeh Lotf, N. Shoa,, Ug the gadet le fo akg DMU DEA. Appled atheatc ad coputato., 151(1): Mehaba, S., M.R. Alezaee, G.R. Jahahahloo, A coplete effcecy akg of deco akg ut data evelopet aaly. Coputatoal optzato ad applcato., 4: Saat, M.S., M. Zaafat Agz, G.R. Jahahahloo, A odel fo akg deco akg ut data evelopet aaly. Recca opeatva., pp: 31. Sexto, T.R., R.H. Slka, A.J. Hoga, Data evelopet aaly: Ctque ad exteo : Slka, R.H., (Ed), Meaug effcecy: A Aeet of Data evelopet aaly. Joy-Bath, Sa Facco, CA, pp: Shalg, L.., G.R. Jahahahloo, M. Khodabakhh, A upe-effcecy odel fo akg effcet ut data evelopet aly. Appled atheatc ad coputato. Atcle pe. Thopo, R.G., E. Lee, R.M. Thall, DEA/AR effcecy of US depedet ol/ga poduce ove te. Copute ad Opeato Reeach, 19(5): Toe, K., A lack-baed eaue of effcecy data evelopet Aaly. Euopea oual of opeatoal eeach, 130: Toe, K., A lack-bed eaue of effcecy data evelopet Aaly. Euopea oual of opeatoal eeach, 143: Yao Che, Rakg effcet ut DEA. Oega., 32: