WORKING PAPER 2012/ Centralized resource reduction and target setting under DEA control

Transcription

1 WORKING PAPER 202/ Cetalzed eouce educto ad taget ettg ude DEA cotol Adel Hata-Mab, Pe J. Agell Louva School of Maageet Fahad Hoezadeh Lotf, Koba Ghola, Zaha Ghele Beg Ilac Azad Uvety, LOUVAIN SCHOOL OF MANAGEMENT RESEARCH INSTITUTE

2 Louva School of Maageet Wokg Pape See Edto : Pof. Jea Vadedockt Cetalzed eouce educto ad taget ettg ude DEA cotol Fahad Hoezadeh Lotf, Depatet of Matheatc, Ilac Azad Uvety Adel Hata-Mab, CORE, Louva School of Maageet, Pe J. Agell, CORE, Louva School of Maageet, Koba Ghola, Depatet of Matheatc, Ilac Azad Uvety, Zaha Ghele Beg, Depatet of Matheatc, Ilac Azad Uvety Suay Data evelopet aaly (DEA) a poweful tool fo eaug the elatve effcece of a et of deco akg ut (DMU) uch a chool ad bak bache that tafo ultple put to ultple output. I cetalzed deco-akg yte, aageet oally poe coo eouce cotat uch a fxed captal, budget fo opeatg captal ad taff cout. I coequece, the poft o et value added of the ut ubect to eouce educto wll deceae. I te of pefoace evaluato cobed wth eouce allocato, the teet of cetal aageet to etoe the geeal effcecy value of the DMU. The pape ake fou cotbuto to the lteatue: () we take to codeato the pefoace evaluato of the cetalzed budgetg of heachcal ogazato alog wth ale ad aket allocato wth aufactug ad dtbuto ogazato; (2) we adde the evaluato poble that the cetal deco ake doe ot dee to deteoate the effcecy coe of the DMU afte put ad/o output educto; (3) we develop a coo et of weght (CSW) ethod baed o the goal poga (GP) cocept to cotol the total weght flexblty the covetoal DEA odel; (4) we exted a ew appoach to optze the put ad/o output cotacto uch that the effcecy of all DMU wll get bgge tha o equal to the effcecy of pevou chage. The pape clude a uecal exaple volvg thee put ad two output to llutate the applcablty ad effcacy of the popoed appoach. Keywod : Data evelopet aaly; Iput ad output deteoato; Coo et of weght. JEL Clafcato: C4, M, C6. The autho ackowledge the Fech Couty of Belgu (ARC poect o aagg haed eouce upply cha) fo patal uppot of th eeach. Coepodg autho: Pof. Pe J. Agell Cete fo Supply Cha Maageet CESCM Louva School of Maageet L.03.0 Uveté catholque de Louva B-348 Louva-la-Neuve, BELGIUM Eal : pe.agell@uclouva.be The pape the WP ee have udegoe oly lted evew ad ay be updated, coected o wthdaw wthout chagg ubeg. Pleae cotact the coepodg autho dectly fo ay coet o queto egadg the pape. ILSM, UC Louva, place de doye, B-348 Louva-la-Neuve, BELGIUM ad

3 Louva School of Maageet Wokg Pape See CENTRALIZED RESOURCE REDUCTION AND TARGET SETTING UNDER DEA CONTROL Fahad HOSSEINZADEH LOTFI, Adel HATAMI-MARBINI Pe J. AGRELL Koba GHOLAMI Zaha GHELEJ BEIGI Accepted (Decebe 202) Data evelopet aaly (DEA) a poweful tool fo eaug the elatve effcece of a et of deco akg ut (DMU) uch a chool ad bak bache that tafo ultple put to ultple output. I cetalzed deco-akg yte, aageet oally poe coo eouce cotat uch a fxed captal, budget fo opeatg captal ad taff cout. I coequece, the poft o et value added of the ut ubect to eouce educto wll deceae. I te of pefoace evaluato cobed wth eouce allocato, the teet of cetal aageet to etoe the geeal effcecy value of the DMU. The pape ake fou cotbuto to the lteatue: () we take to codeato the pefoace evaluato of the cetalzed budgetg of heachcal ogazato alog wth ale ad aket allocato wth aufactug ad dtbuto ogazato; (2) we adde the evaluato poble that the cetal deco ake doe ot dee to deteoate the effcecy coe of the DMU afte put ad/o output educto; (3) we develop a coo et of weght (CSW) ethod baed o the goal poga (GP) cocept to cotol the total weght flexblty the covetoal DEA odel; (4) we exted a ew appoach to optze the put ad/o output cotacto uch that the effcecy of all DMU wll get bgge tha o equal to the effcecy of pevou chage. We ultately peet a uecal exaple volvg wth thee put ad two output to llutate the applcablty ad effcacy of the popoed appoach. Keywod: Data evelopet aaly; Iput ad output deteoato; Coo et of weght. JEL clafcato: C4, M, C6. Depatet of Matheatc, Scece ad Reeach Bach, Ilac Azad Uvety, Teha, Ia, E-al: fahad@hoezadeh.. Louva School of Maageet, Cete of Opeato Reeach ad Ecooetc (CORE), Uveté catholque de Louva, L.03.0, B-348 Louva-la-Neuve, Belgu, E-al: adel.hataab@uclouva.be. Louva School of Maageet, Cete of Opeato Reeach ad Ecooetc (CORE), Uveté catholque de Louva, L.03.0, B-348 Louva-la-Neuve, Belgu, E-al: pe.agell@uclouva.be. Depatet of Matheatc, Scece ad Reeach Bach, Ilac Azad Uvety, Teha, Ia, E-al: koba_ghola@yahoo.co. Depatet of Matheatc, Scece ad Reeach Bach, Ilac Azad Uvety, Teha, Ia, E-al: z.ghelebeg@au.ac..

4 2 Hoezadeh Lotf, Hata-Mab, Agell, Ghola, Ghele Beg. Itoducto No-paaetc fote aaly wa ft toduced by Faell (957) ad late developed a data evelopet aaly (DEA) by Chae et al. (978) to a lea pogag baed techque fo effcecy aeet ad akg of deco akg ut (DMU). DEA a apdly gowg aea of opeatoal eeach that deal wth the pefoace aeet of ogazato (cf. Eouzead et al. (2008)). Wheea the covetoal aaly plctly aue that all DMU eoy coplete autooy the acto ad acce to fee eouce ad poduct aket, pefoace aaly ceagly ued wth ogazato ude a coo aageet. A pcpal dffeece wth epect to the po aupto the that the DMU ae ubect to coo eouce ad aket cotat, poed by a cetal deco ake. Obvou exaple ae foud cetalzed budgetg of heachcal ogazato a well a ale ad aket allocato wth aufactug ad dtbuto ogazato. Hece, ay eal-wold poble we ut code gfcat chage put ad output eaue. Howeve, the cetal deco ake doe ot dee to deteoate the effcecy coe of the DMU afte put ad/o output educto, ule ethe the aageet acto cotay to the ogazatoal obectve, whch abud, o the evaluato uelated to the aageal obectve, whch would ede t eagle. Seveal eeache have appled the put ad/o output deteoato to DEA odel the DEA lteatue. Actvty plag DEA wa popoed by Bake et al. (989), Bogetoft (993, 994, 2000) ad Golay ad Ta (995). I the lteatue, a fa a we kow Cook ad Ke (999) wee the ft to toduce the dea of eouce allocato DEA by chaactezg a equtable way fo allocato of haed cot. Howeve, the appoach caot povde the cot allocato dectly fo DMU. Jahahahloo et al. (2004) ft dcated the hotcog of Cook ad Ke (999) appoach. The they appled a ple ethod to acheve a cot allocato wthout olvg ay lea poga. Cook ad Zhu (2005) alo exteded the ethod of Cook ad Ke (999) to dect cot allocato. L (20) exteded the ethod of Cook ad Zhu (2005) fo allocatg fxed eouce wth oe addtoal cotat. Athaaopoulo (995) popoed a ethod fo taget ettg ad eouce allocato ult-level plag poble ug goal pogag ad DEA. Sla to the faewok peeted by Athaaopoulo (995), Athaaopoulo (998) popoed a eouce allocato odel, called TARBA cotg of two tep: () deteg of the optal weght ug a ultple DEA odel (2) defg feable tade-off allocato. Athaaopoulo et al. (999) appled axu ad u boud o put fo each dvdual DMU that had to be atfed afte eallocato. Ito et al. (999) eallocated the aageet eouce to povde the axu output ug the cocept of poducto poblty et of DEA-BCC odel. Ya et al. (2002) developed a vee geealzed DEA odel ad they the dcued the applcato of the exteded odel to eouce eallocato poble. Cook ad Zhu (2003) developed a DEA odel fo effcecy eaueet of hghway ateace cew a axu achevable by educto put wthout pactg the output fo the poce. Bealey (2003) ued the cocept of DEA to axze the aveage of the

5 Cetalzed eouce educto ad taget ettg ude DEA cotol 3 effcecy coe of the DMU a well a allocatg fxed cot ad output taget by a o-lea poga poble. Ateoo ad Kodota (2005) odfed the cotat of Bealey (2003) odel to pevet feablty ay cae. Kohoe ad Syäe (2004) developed a eouce-allocato odel fo the cetalzed ogazato ug DEA ad ultple-obectve lea pogag to fd a equtable allocato pla. Jahahahloo et al. (2005) peeted a ethod fo allocatg a fxed output faly aog DMU wthout olvg ay lea poga. Ateoo ad Shafe (2006) popoed a DEA-baed ethod fo eovg a fx eouce fo all DMU a fa way uch that the effcecy of ut befoe ad afte educto ea uchaged. L ad Cu (2008) peeted a eouce allocato faewok cotg of vaou etu to cale odel, vee DEA odel, coo weght aaly odel, ad exta eouce allocato algoth. L et al. (2009) ft codeed the lkage betwee the effcecy coe ad the cot allocato ad they the developed a DEA appoach to allocatg the fxed cot betwee DMU. Pachkova (2009) popoed a odel baed o DEA to eallocate put, whee th odel wa tade-off betwee the axu allowed eallocato cot ad the hghet poble uato of effcecy of all DMU. Vaz et al. (200) ft aeed the effcecy of the etal toe wth eveal ellg ecto a etwok DEA odel ude VRS ad howed how eouce eallocato ad taget ettg ug the appoach popoed by Fäe et al. (997) pove the effcecy coe. Baed o the paallel DEA odel toduced by Kao (2009), B et al. (20) uggeted eouce allocato ad taget ettg fo paallel poducto yte. Ateoo ad Mohaghegh Taba (200) popoed a DEA appoach fo eouce allocato ad taget ettg poble. I the ettg, the deco ake dee to add a fxed addtoal eouce equtably to all DMU ad dead a fxed addtoal output to dtbute aog the DMU. Futhe, Ateoo ad Eouzead (200) peeted a DEAbaed appoach to detee the hghet poble put educto ad lowet poble output deteoato wthout educg the effcecy coe fo each DMU. Recetly, Lozao et al. (20) toduced a ube of o-adal, output-oeted ad cetalzed DEA odel fo eouce allocato ad taget ettg fo put wth tege cotat. I the ogal DEA odel, Chae et al. (978) popoed that the effcecy of a DMU ca be obtaed a the axu of a ato of weghted output to weghted put, ubect to the codto that the ae ato fo all DMU ut be le tha o equal to oe. I fact, thee ae o etcto o how uch weght (ultple) ca be placed o each put o output elatve to the othe. Thu, the edogeou weght fo each dvdual DMU ae choe uquely wth t ow effcecy d. Th chaactetc of DEA called the total weght flexblty. Obvouly, t poble that a patcula DMU oly take to accout weght o a few vaable. Moeove, the ettg wth a cetal deco ake evaluatg a et of tuctually copaable ut e.g. a ogazato o ecto, t hghly plauble ad ovely coevatve to aue that each DMU face uque agal cot ad beeft fo the coo techology. Coequetly, ay applcato volve deco ake povdg a po pefeece value fo put ad output weght to be take to accout the effcecy evaluato.

6 4 Hoezadeh Lotf, Hata-Mab, Agell, Ghola, Ghele Beg May eeache have focued o dealg wth the poble of uacceptable weghtg chee. Dyo ad Thaaoul (988) gve aothe appoach to abolute ultple etcto. Chae et al. (990) deotated that udeable weghtg pla ae uavodable eult ay DEA applcato. They thu popoed coe ato etcto odel to povde oe ealtc weght. Thopo et al. (995) odfed odel developed by Chae et al. (990).The auace ego toduced by Thopo et al. (988, 990) a pecal cae of the coe ato cocept. Thee ae oe exteo o the auace ego cocept the DEA lteatue uch a Alle et al. (997), Thaaoul et al. (998) ad Cook ad Zhu (2008). Beet et al. (988) peeted the cotaed facet aaly to deal wth the heet poble volvg the occuece of zeo weght. Lag et al. (995) poved th latte appoach by adoptg a two-tage appoach. Sla ethod have bee uggeted by Gee et al. (996) ad Olee ad Petee (996). The coo weght appoach DEA wa tally toduced by Cook et al. (990) ad developed by Roll et al. (99). Hoezadeh Lotf et al. (2000) ad Jahahahloo et al. (2005) ued a ult-obectve poble to pecfy a coo et of weght (CSW) fo all DMU ug a o-lea tafoato. A gae-theoetcal appoach to CSW a ettg whee the DMU ut agee upo a coo edogeou evaluato foud (Agell ad Bogetoft, 200). I the ecet tudy, Saat et al. (202) popoed a two-phae CSW appoach ug a deal vtual ut that coputatoally effcet. The ethod wa appled eegy egulato ug pael data fo 286 Dah dtct heatg plat. I th pape, we popoe a alteatve DEA-baed ethod fo a cetally poed eouce o output educto aco the efeece et. I othe wod, th tudy addee the followg queto: how uch hould the put ad output fo each DMU educed ubect to the codto that the effcecy coe of all DMU ceae? Cotet wth the ettg fo a cetal evaluato, we ue the DEA-baed ethod ode to get bette effcecy coe fo all DMU afte the educto aout of put ad output. The et of th pape ogazed to fve ecto. I Secto 2, we peet a bef evew of the covetoal DEA odel ad Secto 3 we popoe the coo-weght DEA odel. Secto 4 peet the detal of the odel popoed th tudy. I Secto 5 we how a uecal exaple to llutate the effcacy of the popoed ethod. I Secto 6, we cloe the pape wth cocluo ad futue eeach decto. 2. The Tadtoal DEA Model Data evelopet aaly (DEA) eetally etatg a covex hull coveg a et of deco akg ut (DMU) ad adally poectg the agat the hull a pecfed decto. Suppoe that thee ae DMU to be evaluated whee evey DMU, =,,, poduce output y R + +, =,...,., ug the put, x R, =,...,. The put-oeted odel (CCR o CRS fo cotat etu to cale) fo evaluatg the elatve effcecy of a gve DMU a follow (Chae et al. 978): o

7 Cetalzed eouce educto ad taget ettg ude DEA cotol 5 x a = = =.., =,...,, = wy µ x o o wy t µ x w, µ ε, =,.., ; =,...,. whee ε a potve o-achedea fteal ube. The odel () alo called the ultple odel. It clea that the odel () o-lea ad t ca be coveted to the lea poga poble va Chae ad Coope (962) ethod a how odel (2). θ o = ax uy o = t.. vx =, Defto : wth u ε, o = uy vx 0, =,...,, = = u, v ε, =,.., ; =,...,. DMU effcet f thee ext at leat oe optal o ε ad uy o =. Othewe, DMU effcet. o v = 3. The Coo-Weght DEA Model () (2) ( u, v ) of odel (2) The elatve effcecy ug the ultple DEA odel deteed by agg weght to the put ad output of DMU to axze the ato of the weghted u of output to the weghted u of put. The oly udelyg aupto fo the weght o put ad output o-egatvty (called total weght flexblty ). The calculato of DEA coe eque a lea poga (2) pe DMU ad obta a dvdual et of edogeou weght. We ecall that the dffeece aog the dvdual weght ay be uacceptable fo aageet eao, aket eao o by techcal o ecooc ecety. To cope wth thee dffculte, the coo et of weght (CSW) odel ca be ued to geeate a coo et of weght fo all DMU whch ae able to poduce the hghet effcecy coe at the ae te. I the eug ecto, we exae a CSW odel baed o the ult-obectve poga (MOP). May eeache have vetgated the elatohp betwee DEA ad MOP fo the dffeet apect (e.g., ee Hoezadeh Lotf et al. 200a, 200b; Yag et al. 2009). To puue ou a, we ca

8 6 Hoezadeh Lotf, Hata-Mab, Agell, Ghola, Ghele Beg equvaletly code the followg ult-obectve factoal poga (MOFP) fo eaug the effcecy of all DMU ultaeouly: uy uy 2 uy = = = θ = ax,,..., vx vx 2 vx = = = (3) uy = t.., =,...,, vx = u, v ε, =,.., ; =,...,. Sce ove two decade ago, ay ethod fo olvg ult-obectve poble have bee developed the optzato lteatue (ee e.g., Hwag ad Maud 979; Steue 986). I th pape, we adapt the MOFP odel (3) to a goal poga (GP) odel whch wa developed by Feed ad Glove (98) ad exteded (Feed ad Glove 986; Glove 990). I the axzato MOFP odel (3), uy vx the th = = obectve fucto whch, baed o the cotat, hould be a cloe a poble to effcecy coe uty. I othe wod, the goal o apato level fo each obectve fucto odel (3) to take the uty value o full techcal effcecy. The dffeece betwee th obectve fucto ad t goal defed a the egatve ad + potve devato, deoted by ( =,..., ) ad ( =,..., ) epectvely. The pupoe of the GP ethod to ze thee devato fo the peet goal fo each f, +, obectve fucto. Thee ext oe ethod to defe the obectve fucto ( ) whee each of the lead to dffeet GP ethod. We ue weghted GP, whch ze the weghted u of the devatoal vaable a f (, + ) = ( α ) + β + = whee α ad β peet the weght of the egatve ad potve devato o the th obectve fucto ad ae chaactezed by the deco ake. Wthout lo of geealty, we aue the detcal devato weght fo all DMU by codeg α = β =. To cloe the gap betwee the value of each obectve fucto ad the effcecy coe oe, we ca code thee cae: () uy vx, (2) = ug the egatve devatoal vaable = ; uy, vx = = ug the potve devatoal vaable + ; (3) uy vx, ad uy vx, at the ae te ug the = = = = egatve ad potve devatoal vaable. Th epeetato thu eult the followg odel wth a gle obectve fucto:

9 Cetalzed eouce educto ad taget ettg ude DEA cotol 7 t.. = ( = = = = ) + =, =,...,,, =,...,, 0, 0, =,...,, uy vx uy vx + u, v ε, =,.., ; =,...,. (4) + Notce that ( =,..., ) odel (4) ot allowed to take the potve value ce the + potve value of ( =,..., ) ad the zeo value of ( =,..., ) doe ot atfy the ecod et of cotat. We ca thu ot + odel (4) ad coequetly the ecod et of cotat edudat. Model (4) a o-lea poga ad t pupoe to ze the total gap to each goal. Baed o the GP cocept, we peet a ew lea pogag odel fo olvg the MOFP odel (3). I DEA, evey DMU ca ze the u of the total vtual gap to eceve the bechakg fote by addg + to uy ad takg = away fo vx. A a eult, the ult-obectve = factoal poga (3) ca be coveted to the followg lea odel: ( + ).. t = = = = = + =, =,...,,, =,...,, 0, 0, =,...,, u, v ε, =,.., ; uy vx uy vx + =,...,. (5) whee the o-lea poga (5) ca be ply chaged to the followg lea poga:

10 8 Hoezadeh Lotf, Hata-Mab, Agell, Ghola, Ghele Beg ( + ) = = = t.. uy vx + + =0, =,...,, uy vx 0, =,...,, = = 0, 0, =,...,, u, v ε, =,..,, =,...,. We futhe ply the poga by eplacg + to : =.. t uy vx + = 0, =,...,, (7a) = = uy vx 0, =,...,, (7b) = = u, v ε, =,.., ; 0, =,...,, =,...,. (6) (7) Fo cotat (7a), uy vx = =. If we ubttute the left had de of =, we obta 0 equvaletly 0. It how that the cotat (7b) wth cotat (7b) ae edudat ad coequetly the ecod cotat the odel (5) ad (6) ae edudat ad ca be otted fo odel. Ug the optal oluto ( u, v, ),, to (7), the effcecy coe fo DMU, =,...,, ae calculated a follow: uy = θ vx vx = = = =, =,...,. Defto 2: DMU, =,...,, o-doated f ad oly f the odel (7). I othe wod, (8). 4. A Popoed Method DMU o-doated f ad oly f (8) = 0, =,...,, θ =, equato I pefoace evaluato of cetalzed ogazato wth oe coo cotol, the teet to ata optal techcal effcecy goe had had wth the ecety to coply wth eouce ad aket cotat. Fo tace, a publc authoty taffg ad upplyg chool wth pecal eouce, a well a agg tudet to ego ad

11 Cetalzed eouce educto ad taget ettg ude DEA cotol 9 dtct. A budget educto to the ecto ut be pleeted aco the chool, a well a deogaphc chage ay lead to educto the ube of tudet both adtted ad gaduated. I both cae, t podal fo the cetal aage to ata o pove techcal effcecy o t ut afte the eouce eallocato. I th ecto, we popoe a alteatve data-baed appoach fo deteg the hghet poble put educto ad the lowet poble output deteoato wthout educg the effcecy coe fo each DMU deved fo CSW appoach. + Recall that we code a techology wth put, x R, =,...,, ad output, + y R, =,...,. Aue that I = {, 2,..., k }, I2 = {, 2,..., } I, O = {, 2,...,} t ad O2 = {, 2,..., } O, whee I ad O ae the ubet of put ad output, epectvely, whch the ogazato wllg to educe thee put ad output. The total educto of the th put ad the th output, deoted by C, I, ad P, O, ca be obtaed a: c = C, I, = p = P, O. = whee c ad p ae, epectvely, the th educed put ad the th educed output wth epect to th DMU. Let u θ be the effcecy coe of th DMU obtaed fo (8) wthout chagg the data. I ode to detee the adequate aged value to c ad p ad keep effcecy coe geate tha o equal to code the followg et of cotat: uy + u( y p ) O2 O (9 ) θ, =,...,, vx + v( x c) I2 I O2 O (9 ),,...,, I2 I (9 ) c = C, I, ( 9 v) uy + u( y p ) = vx + v( x c ) = p = P O =,, (9 v) c x, I, =,...,, (9 v) p y, O, =,...,, u, v ε, c, p 0, =,..,, =,...,, =,...,. θ fo DMU, we eque to (9)

12 0 Hoezadeh Lotf, Hata-Mab, Agell, Ghola, Ghele Beg whee c ad p ae deco vaable addto to u ad v, theeby, (9) a olea yte the peece of cotat (9) ad (9). The tepetato of each et of cotat (9) cocely expeed a Cotat (9) eue that the effcecy coe fo each DMU ae ot alle tha the pay effcecy θ calculated by (8). Cotat (9) ply that the effcecy coe fo each DMU ae le tha o equal to uty. Cotat (9) et the u of educed put equal to C. Cotat (9v) et the u of educed output equal to P. Cotat (9v) efoce the axu cotacto of th put x fo th DMU. Cotat (9v) efoce the axu cotacto of th output y fo th DMU. We ca ewte cotat (9) ad (9) a follow: uy up uy up = O = O θ, =,...,,, =,...,. vx vc vx vc = I = I Ug alteato vaable up = p ad vc = c, the o-lea yte (9) ca be tafoed to the followg lea yte: ( ) ( ) ( ) ( ) ( v) ( ) = O θ, =,...,, vx c = I = O, =,...,, = I c = v C, I, = uy uy vx v p = u P, O, = p p c c v x, I, =,...,, v p u y, O, =,...,, u, v ε, c, p 0, =,.., ; =,..., ; =,...,. At peet, the a to olve the above yte to detee the aout of educto the put ad output. Thee ext oe ethod to olve yte () uch a Gau- Joda ad Gaua elato ethod (See Datta 994 fo futhe detal). Iput ad output cotacto fo each DMU ut be popotoal to the peet put ad output to eue a equtable pact. To deal wth th poble, we ca take to (0) ()

13 Cetalzed eouce educto ad taget ettg ude DEA cotol accout the elatve potace o weght of the put ad output. Thee ae vaou ethod fo deteato of weght uch a egevecto ethod, weghted quae ethod ad etopy ethod. I th pape, we ue the followg ple calg foula ode to ake the odel oe eaoable: x y ρ =, =,...,, I, µ =, =,...,, O. x y t t= t= = = t whee ρ ad µ ae put ad output ultple fo the th I put ad th O output, epectvely, fo each DMU. Note that the above foulato ρ = µ =. I the peece of the ultple ρ ad µ yte () ght be feable, theefoe, we ae ot able to olve () ug coo appoache. I th ecto, we develop a ew ethod to olve () wth epect to the GP cocept. I cotat () t deable that the effcecy coe afte put-output educto wll be geate tha o equal to the effcecy coe befoe copoatg chage to data. Moeove, we ca pevet feablty fo cotat () ad (v) by defg the egatve ad potve devatoal vaable. We deote the egatve ad potve devato vaable byα ad α + fo c ( I), ad β ad β + fo p ( O ). Notce that ude the put-output effect, the goal of c () ad p (v) ae v( ρ C) ad u( µ P), epectvely. Accodg to GP cocept, we ze the u of the defed egatve ad potve devatoal vaable to acheve the goal. Theeupo, we ceate the followg odel: ( α + α ) + ( β + β ) = I = O uy p = O.. θ, =,...,, t vx c = I uy p = O vx c, = I α α ρ =,...,, c + = v ( C ), I, =,...,, p + β β = u ( µ P ), O, c v x, I, =,...,, =,...,, p uy, O, =,...,, c = vc, I ; p = u P, O, = = u, v ε, c, p, α, α, β, β 0, =,.., ; =,..., ; =,...,. (2)

14 2 Hoezadeh Lotf, Hata-Mab, Agell, Ghola, Ghele Beg I the above odel, ε a o-achedea cotat. Model (2) a factoal poga poble that caot be olved by tadad optzato ethod. Hece, we ply covet (2) to the followg lea poga poble: ( α + α ) + ( β + β ) = I = O t.. uy p θ vx c 0, =,...,, = O = I uy p vx c 0, =,...,, = O = I c + α α = v ( ρ C ), I, =,...,, p + β β + = u ( µ P), O, =,...,, c v x, I, =,...,, p u y, O, =,...,, = c = vc, I, = p = u P, O, u, v ε, c, p, α, α, β, β 0, =,.., ; =,..., ; =,...,. Theoe: Thee alway ext a feable oluto to odel (3). Poof. We have the followg feable oluto to (3): = = C P v = I ; u =, O, c γ v =, I, u = O, =,..., whee γ = θ x ad O2 I2 Theefoe, we have γ uy = y = γ ky O2 O2 2 2 ky, =,, I addto, we have = C = P p (3) = k. Note that f epeet a cadal ube of a et f. c = v x, I, =,...,, c = I, =,..., p = u y, O, =,...,, p = I, =,..., x y c p

15 Cetalzed eouce educto ad taget ettg ude DEA cotol 3 uy p O O vx = = c I I v( ρc) c v( ρc) c I,,...,, = α = 0 v( ρc) c I, =,...,, c v ( ρc ) v( ρc) c I,,...,, + = α = 0 v( ρc) c I, =,...,, u( µ P) p u( µ P) p O, =,...,, β = 0 u( µ P) p O, =,...,, p u ( µ P ) u( µ P) p O,,...,, + = β = 0 u( µ P) p O, =,...,, The above deteed feable oluto ca be thu atfed all the cotat of odel (3). The poof coplete. 5. A Nuecal Exaple I th ecto, we ue pael data fo a bakg applcato popoed by Kao ad Hwag (2009) ad alo ued by Ateoo ad Eouzead (200) to llutate the applcablty of the popoed odel. To ae the pact of foato techology (IT) o bak pefoace, we take to accout thee put ad two output decbed below: The put (I): IT budget (USD) The put 2 (I2): Fxed aet (USD) The put 3 (I3): Staff (headcout) The output (O): Depot (USD) The output 2 (O2): Poft (USD). The put ad output data fo 27 bak a peod of ae epoted Table. I the ft tep we apply the popoed odel (7) to obta the optal coo weght. The we eaue the effcecy coe ( θ ) of the bak ug the equato (8) epoted Table 2. I the ecod tep the bakg yte wa foced to educe the IT budget ad the poft value owg to oe exogeou facal cotat. Theefoe, the peet budget, bllo dolla, ut be educed by 3 bllo dolla (.e., C = 3 ). I uch cae, aageet expect that the bak poft wll hk fo.948 bllo dolla to bllo dolla (.e., P 2 = 5 ). To detee the adequate value of IT budgetpoft educto (deoted by c - p 2 ) fo each bak, we ft apply odel (3) to get the optal oluto of c ad p 2, =,,27. We the ue alteato vaable up = pad vc = c to obta the aout of IT budget ad poft educto deoted by c ad p 2, =,,27, epectvely. The optal value of c ad p 2 ae peeted Table 2. I the lat tep, to e-gauge the effcecy coe ( θ ) of the ew

16 4 Hoezadeh Lotf, Hata-Mab, Agell, Ghola, Ghele Beg bak we ue the popoed coo-weght DEA odel (7) ad equato (8) the peece of the ew value fo IT budget ad poft deteed fo the pecedg tep. The eult peeted Table 2. The a of th exaple to obta the pope educto the put (IT budget) ad output (poft) uch that the effcecy coe of each bak bach ataed geate tha o equal to the pevou value. A how Table 2, whe we apply the popoed ethod the ew effcecy coe of bak ae alway geate tha o equal to the effcecy coe befoe deceag the value of IT budget ad poft. It how that th bakg yte able to povg the opeatg effcecy of each bak bach. Ateoo ad Eouzead (200) toduced a DEA-baed ethod (heeafte aed a the AE ethod ) to ae the effcecy of a et of DMU afte educg the value of a gve put ad output. The pupoe of AE to peeve the effcecy coe of the DMU afte agg the apt educto to the put ad output of all DMU. I othe wod, the AE odel ata the effcecy coe of each DMU calculated fo the tadad DEA (CCR o CRS) odel befoe educg put ad output value. Hee we ake a copao betwee the eult of the popoed ethod ad the AE ethod. Table 2 dplay the effcecy coe ( θ odel (2) a well a the optal oluto of c ad the eewal effcecy coe ( θ Table. The put-output data fo 27 bak DMU I I 2 I 3 O O ewae AE p ) of the bak va the CCR DEA ug the AE odel. I addto, ) of the bak calculated ug odel (2) the peece of c ad p. A how Table 2, the educto value of ou odel alot la to the AE ethod but ou odel chaactezed by lowe coputatoal

17 Cetalzed eouce educto ad taget ettg ude DEA cotol 5 coplexty. The AE effcecy coe befoe educto ( θ ) ad afte educto ( ewae θ ) fo each DMU ae exactly detcal whle ou ethod the effcecy coe poved ay ut, excludg DMU 3, DMU 26 ad DMU 27. The AE ethod eque epetto of the ultple CCR odel (2) whee evey CCR odel volve cotat ad + vaable whle we eaue the effcecy coe of DMU by olvg gle coo-weght DEA odel (7) cotg of + + cotat ad + + vaable. I th exaple =27, =3, =2, k= ad h=, the AE ethod olve 27 odel whee evey odel ha 33 cotat ad 5 vaable whle we olve oly oe odel wth 32 cotat ad 32 vaable. The AE odel cota 2 ( k+ h+ ) + ( + )( k+ h) cotat whee k ad h ae the ube of educto dexe fo the put ad output, epectvely, wheea ou 2 k+ h+ + k+ h+ + cotat whee + popoed odel (3) clude ( ) AE coepodg wth u, v ε. Moeove, the AE ethod ha 2 ( k+ h) + ( k+ h+ ) + + vaable, epectvely, whle ou odel cota ( ) k+ h vaable. Table 2. The copao of the eult fo the popoed ethod ad the AE ethod AE ewae ew c p θ θ c p θ θ DMU AE ethod Popoed ethod Su

18 6 Hoezadeh Lotf, Hata-Mab, Agell, Ghola, Ghele Beg I th exaple, the AE ethod clude 28 cotat, 94 vaable whle the popoed odel cota 69 cotat, 72 vaable. Obvouly, the popoed odel coputatoally ecoocal becaue of 49 cotat fewe tha the AE ethod. Theefoe, the popoed ethod deceae the ube of cotat ad vaable, whch lead to a tog educto the coputatoal equeet. Fally, the AE ethod e-olve te the ultple CCR odel fo eaug the effcecy of DMU wth codeato of the educto, wheea ou ethod e-olve oly gle coo-weght DEA odel. Ideed, the popoed ethod olve ( ) lea poga fewe tha the AE ethod ad th cottute a coputatoal advatage. Natually, the educto the coputatoal bude paly lked to the adopto of the coo weght cocept all tep whle the AE ethod ue dffeet ethod. Befly, Fg. how the above-etoed copao betwee the AE ethod ad the popoed ethod. Step : Meaue the effcece of DMU The AE ethod olve odel Ou ethod olve oe odel Step 2: Calculate the value of put ad output educto AE odel ha 2 (k+h+) + (+) (k+h) cotat Ou odel odel ha 2 (k+h+) + k+h++ cotat Step 3: Meaue the effcece of DMU afte educto The AE ethod olve odel Ou ethod olve oe odel Fg.. Copao betwee the AE ad the popoed ethod 6. Cocluo ad Futue Reeach Decto The tegato of actvty plag, eouce allocato ad pefoace aageet ae cuet challege both the theoy ad the pactce of DEA. I th pape, we popoe a ew appoach to pove the effcecy of the ut whe oe gve put ad/o output ae educed the evaluato poce. Ou a to optze the eouce cotacto uch that the effcecy of all DMU wll get bgge tha o equal to the effcecy of pevou chage. I th pape, we ft toduce a coo weght ethod fo eaug effcecy of DMU befoe ad afte data chage. Thu, we acheve the effcece by olvg a lea poga whch coputatoally ecoocal. I addto, copao wth total weght flexblty the tadtoal DEA odel, the coo-weght DEA odel take to accout the coo weght. The, baed o the goal poga (GP) cocept we popoed a ew ethod to fd a adequate aget fo the educto aout of put ad output the peece of the cuet data effect the evaluato yte. The popoed odel ot oly cotet wth the

19 Cetalzed eouce educto ad taget ettg ude DEA cotol 7 outled aageal obectve; t alo gfcatly educe the coputatoal bude fo the aaly. The developed faewok th pape ca potetally led telf to ay pactcal applcato. Howeve, thee ae a ube of challege volved the popoed eeach that povde a geat deal of futful cope fo futue eeach. Fo exaple, thee o echa the popoed ethod to detee the adequate educto value fo tege put ad output. Aothe potetal fo futue eeach to detfy the uppe ad lowe boud fo C ad P, epectvely, uch that the aget yte fo educg put/output ea feable. Refeece Agell, P. J., Bogetoft, P., 200. Edogeou geealzed weght ude DEA cotol. Wokg Pape 200/02. Louva School of Maageet, Uvete catholque de Louva. Alle, R., Athaaopoulo, A., Dyo, R.G., Thaaoul, E., 997. Weght etcto ad value udget data evelopet aaly: Evoluto, developet ad futue decto, A. Ope. Re. 73, Ateoo, A., Eouzead, A., 200. Iput/output deteoato poducto pocee, Expet. Syt. Appl. 38 (5), Ateoo, A. Kodota, S., Allocatg fxed cot ad taget ettg: a DEA-baed appoach, Appl. Math. Coput. 7, Ateoo, A., Mohaghegh Taba, M., 200. Reouce allocato ad taget ettg data evelopet aaly, Expet. Syt. Appl. 37, Ateoo, A., Shafe, M., Chaactezg a equtable oo of haed eouce: a DEA-baed appoach, Appl. Math. Coput. 77, Athaaopoulo, A. D.,995. Goal pogag & data evelopet aaly (GoDEA) fo taget-baed ult-level plag: Allocatg cetal gat to the Geek local authote, Eu. J. Ope. Re. 87, Athaaopoulo, A. D., 998. Deco uppot fo taget-baed eouce allocato of publc evce ultut ad ult- level yte. Maage. Sc. 44(2), Athaaopoulo, A.D., Labouko, N., Sefod, L., 999. Data evelopet ceao aaly fo ettg taget to electcty geeatg plat. Eu. J. Ope. Re. 5, Bake, R. D., Chae, A., Coope, W. W., Clake, R., 989. Cotaed gae foulato ad tepetato fo data evelopet aaly, Eu. J. Ope. Re. 40(3), Bealey, J. E., Allocatg fxed cot ad eouce va DEA, Eu. J. Ope. Re. 47, Beet, A., Beet, W., Ela, J., Clak, T., 988. Effcecy fote deteato by cotaed facet aaly, J. Ope. Re. Soc. 36, B, G., Dg, J., Luo, Y., Lag, L., 20. Reouce allocato ad taget ettg fo paallel poducto yte baed o DEA, Appl. Math. Model.35 (9), Bogetoft, P., 993. Paallel foato poducto: Moal hazad ad elatve pefoace evaluato, Maage. Sc. 39(4), Bogetoft, P., 994. Icetve effcet poducto fote: A agecy pepectve o DEA, Maage. Sc. 40(8), Bogetoft, P., DEA ad actvty plag ude ayetc foato, J. Pod.

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21 Cetalzed eouce educto ad taget ettg ude DEA cotol 9 oetated CCR dual odel, Expet. Syt. Appl. 37 (200b) Hwag, C.L., Maud, A.S.M., 979. Multple obectve deco akg Method ad applcato: A State-of-the-at uvey. Spge, Bel. Ito, R., Naatae, T., Yaaguch, T., 999. Reouce allocato poble baed o the DEA odel, J. Ope. Re. Soc. Jap. 42(2), Jahahahloo, G.R., Hoezadeh Lotf, F., Moad, M., A DEA appoach fo fa allocato of coo eveue, Appl. Math. Coput Jahahahloo, G.R., Hoezadeh Lotf, F., Shoa, N., Sae, M., A alteatve appoach fo equtable allocato of haed cot by ug DEA, Appl. Math. Coput.53, Jahahahloo, G. R., Meaa, A., Hoezadeh Lotf, F., Rezae, H. Z., A ote o oe DEA odel ad fdg effcecy ad coplete akg ug coo et of weght, Appl. Math.Coput.66, Kao, C., Effcecy eaueet fo paallel poducto yte, Eu. J. Ope. Re. 96, Kao, C., Hwag, S. N., Effcecy eaueet fo etwok yte: IT pact o f pefoace, Expet. Syt. Appl. 47, Kohoe, P., Syäe, M., Reouce allocato baed o effcecy aaly, Maage. Sc. 50 (8), Lag, P., Yolala, O. R., Ketta, O., 995. Cotolled evelopet by face exteo DEA, J. Ope. Re. Soc. 46 (4), L, X., Cu, J., A copeheve DEA appoach fo the eouce allocato poble baed o cale ecooe clafcato, J. Syt. Sc. Coplex. 2, L,Y., Yag, F., Lag, L., Hua, Z., Allocatg the fxed cot a a copleet of othe cot put: A DEA appoach, Eu. J. Ope. Re.97, L, R., 20. Allocatg fxed cot o eouce ad ettg taget va data evelopet aaly, Appl. Math. Coput. 27, Lozao, S., Vlla, G., Caca, D., 20. Applcato of cetaled DEA appoach to captal budgetg Spah pot, Coput. Id. Eg. 60, Olee, O. B., Petee, N.C., 996. Idcato of ll-codtoed data et ad odel pecfcato data evelopet aaly: A exteded facet appoach, Maage. Sc. 42 (2), Pachkova, E. V., Retcted eallocato of eouce, Eu. J. Ope. Re. 96, Roll, Y., Cook, W.D., Golay, B., 99. Cotollg facto weght data evelopet aaly, IIE Taacto. 23, 2 9. Saat, S., Hata-Mab, A., Agell, P.J., Tavaa, M., 202. A coo et of weght appoach ug a deal deco akg ut data evelopet aaly, J. Id. Maag. Opt., 8 (3), Steue, R. E., 986. Multple ctea optzato: Theoy, coputato, ad applcato. Wley, New Yok. Thaaoul, E., Alle, R., 998. Sulatg weght etcto data evelopet aaly by ea of uobeved DMU, Maage. Sc. 44 (4), Thopo, R.G., Dhaapala, S., Thall, R.M., 995. Lked-coe DEA poft ato ad techcal effcece wth applcato to Illo coal e, It. J. Pod. Eco. 39, Thopo, R.G., Lageee, L.N., Lee, C., Lee, E., Thall, R.M., 990. The ole of ultple boud effcecy aaly wth applcato to Kaa fag, J.

22 20 Hoezadeh Lotf, Hata-Mab, Agell, Ghola, Ghele Beg Ecooetc. 46, Thopo, R.G., Sgleto J., F.D., Thall, R.M., Sth, B.A., 986. Copaatve te evaluato fo locatg a hgh-eegy phyc lab Texa, Iteface. 6, Vaz, C.B., Caaho, A.S., Guaãe, R.C., 200. The aeet of etalg effcecy ug etwok data evelopet aaly, A. Ope. Re. 73, Ya, H., We, Q. L., Hao, G., DEA odel fo eouce eallocato ad poducto put/output etato, Eu. J. Ope. Re. 36(), 9-3. Yag, J.B., Wog, B.Y.H., Xu, D.L., Stewat, T.J., Itegatg DEA-oeted pefoace aeet ad taget ettg ug teactve MOLP ethod. Eu. J. Ope. Re. 95,