A Goal Programming Method for Finding Common Weights in DEA with an Improved Discriminating Power for Efficiency.

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1 Joral of Idtral ad Ste Egeerg Vol., No. 4, pp Wter 008 A Goal Prograg Method for Fdg Coo Weght DEA wth a Iproved Dcratg Power for Effcec A. Mak, A. Alezhad, R. Ka Mav 3,M. Zohrehbada 4 Departet of Idtral Egeerg, Ira Uvert of Scece & Techolog, Tehra, Ira. (E-al: aak@t.ac.r,3 Departet of Idtral Egeerg, Ilac Azad Uvert-Scece & Reearch Brach, Tehra, Ira. alezhad_r@ahoo.co rezakaav@ahoo.co 4 Departet of Matheatc, Ilac Azad Uvert-Kara P.O.Box , Kara, Ira. zohrebada@ahoo.co ABSTRACT A charactertc of data evelopet aal (DEA) to allow dvdal deco akg t (DMU) to elect the ot advatageo weght calclatg ther effcec core. Th flexblt, o the other had, deter the coparo aog DMU o a coo bae. For dealg wth th dffclt ad aeg all the DMU o the ae cale, th paper propoe g a ltple obectve lear prograg (MOLP) approach for geeratg a coo et of weght the DEA fraework. Keword: MOLP, Goal prograg, DEA, Effcec, Rakg, Weght retrcto.. INTRODUCTION Data evelopet aal (DEA) ha bee wdel appled to eare the relatve effcec of a grop of hoogeeo deco akg t (DMU) wth ltple pt ad ltple otpt. It charactertc to foc o each dvdal DMU to elect the weght attached to the pt ad otpt, ad to calclate ther effcec core. A the atheatcal odel DEA are r eparatel for each DMU, the et of weght wll be dfferet for the varo DMU, ad oe cae, t acceptable that the ae factor accorded wdel dfferg weght. Th flexblt electg the weght, deter the coparo aog DMU o a coo bae. A poble awer to th dffclt le the pecfcato of a coo et of weght, whch wa frt trodced b Roll et al.(99). I other word, the aor prpoe for geeratg a coo et of weght to provde a coo bae for rakg the DMU. Reearch o the dea of a coo et of weght ad ther rakg ha grow gradall recet ear. Kao ad Hg (005), baed o ltple obectve olear prograg ad b g Correpodg Athor

2 94 Mak, Alezhad, Ka Mav ad. Zohrehbada coproe olto approach, propoed a ethod to geerate a coo et of weght for all DMU whch are able to prodce a vector of effcec core cloet to the effcec core calclated fro the tadard DEA odel (deal olto). Lkewe, Jahahahloo et al.(005) baed o ltple obectve olear prograg ad Maxzato of the vale of the effcec core, propoed a ethod to geerate a coo et of weght for all DMU. Soe of the other tde th feld are attrbted to Dole ad Gree (994), Karak ad Ahka (005), Roll ad Gola (993). The pla for the ret of th paper a follow. I ecto we preet a bref dco abot DEA odel ad the ltple obectve lear prograg (MOLP). The atheatcal fodato of or ethod for fdg a coo et of weght ad the ethod telf are dced Secto 3. A Nercal exaple preeted ecto 4 ad fall, ecto 5 draw the coclve reark.. DEA AND MOLP PRELIMINARIES Thrt ear after the pblcato of the poeerg paper b Chare et al.(978), DEA ca afel be codered a oe of the recet cce tore Operato Reearch. Iteretgl, Chare ad Cooper have alo developed Goal Prograg (GP) that a ltple obectve lear prograg techqe (Chare ad Cooper, 96). Sce the 970, MOLP ha becoe a poplar approach for odelg ad aalzg certa tpe of ltple crtera deco akg (MCDM) proble. Soe work o the teracto betwee MCDM ad DEA, are a follow: Boo (999), Etellta et al.(004), Goka (997), Gola (988), Joro et al.(998), Stewart (996), ad Xao ad Reeve (999). Data Evelopet Aal Coder prodcto t, or DMU, each of whch coe a varg aot of pt to prodce otpt. Sppoe 0 deote the aot coed b the th pt ad 0 x deote the aot prodced b the rth otpt for the th deco akg t. The, the followg et the prodcto poblt et (PPS) of obvol the ot wdel ed DEA odel, CCR, wth cotat retr to cale charactertc: T c = λ x = = ( x, ) x, λ λ 0,, =,,..., Defto : DMU, =,,..., called effcet ff there doe ot ext aother ( x, ) T c ch that x < x ad >, ad called Pareto effcet ff there doe ot ext aother ( x, ) T c ch that x x ad ad ( x, ) ( x, ). r I DEA, the eare of effcec of a DMU defed a a rato of a weghted of otpt to a weghted of pt bect to the codto that correpodg rato for each DMU are le tha or eqal to oe. The odel chooe oegatve weght for a DMU a ot favorable wa. The orgal odel propoed b Chare et al.(978), for earg the effcec of t p, a fractoal lear progra a follow:

3 A Goal Prograg Method for Fdg Coo Weght 95 () bect to 0,,,,,, 0,,, where r ad v are the weght to be appled to the otpt ad pt, repectvel. The above odel ca be trafored to a lear progra b ettg the deoator the obectve fcto eqal to a arbtrar cotat (e.g., t) ad axzg the erator. The reltg odel, called a pt oreted CCR ltpler odel (CCR ), a follow: CCR ) bect to Max v x r v r = r= r r = v x = = p r rp 0, =,,..., 0, r =,,..., 0, =,,..., The opt olto of the proble aocated to a oralzed coeffcet (, ) () v of a pportg hperplae (a hperplae that cota the PPS ol oe of the halfpace ad pae throgh at leat oe of t pot). The dal proble of CCR odel called pt oreted CCR evelopet odel (CCR e ), wll alo be ed. Th odel ha a trog ttve appeal ad tpcall the oe ed to expla ad valze DEA. If θ p repreet the CCR effcec of DMU p the the CCR e odel CCR e ) bect to M = λ θ p x λ = r rp λ θ px 0, =,,..., p, r =,,..., 0, =,,..., (3) A DMU effcet f ad ol f the obectve fcto vale aocated wth the optal olto of the proble () above eqal to t; otherwe t effcet. Moreover, f the forer odel all varable take a trctl potve vale or a t coterpart the latter odel all lack varable are eqal to zero, the DMU Pareto effcet. Accordg to Kao ad Hg (005) ad baed o the olto of odel (3), we preet the followg lea.

4 96 Mak, Alezhad, Ka Mav ad. Zohrehbada Lea : If θ p the opt olto of odel (3), the ( ) DMU p o the effcet froter, a effcet vrtal DMU. θ x p p, Lea : DMU p effcet ff there ext a oegatve coeffcet ( ) R R to the gradet vector of a pportg hperplae where we have: r = r = p = rp v x 0 p, called proecto of, aocated v We ow preet a bref trodcto of MOLP Mltple Obectve Lear Prograg The MOLP proble ca be wrtte the geeral for a follow: Max f ( x ) = Cx bect to : { g ( ) 0,,,..., } x X = x x = g,..., Where x R, the obectve fcto atrx R C ad ( x) 0, =,, are lear fcto. I MOLP, a effcet olto trodced a follow: Defto : x X called a effcet olto (or o-doated olto) ff there doe ot ext aother x X ch that Cx Cx ad Cx Cx. I order to olve odel (4) ad detfg the effcet olto, there are a dfferet ethod the lteratre. Oe of thee ethod Goal Prograg whch developed b Chare ad Cooper (96). Th ethod reqre the deco aker (DM) to et goal for each obectve that he/he whe to atta. A preferred olto the defed a the oe whch ze the devato fro the et of goal. Th a ple GP forlato gve b: M ST. : g f k = ( ) p p + d, p ( x ) 0 =,,..., ( x ) + = =,,..., k d d d d, 0 =,,..., k d = 0 =,,..., k d d b (4) (5)

5 A Goal Prograg Method for Fdg Coo Weght 97 Where f, =,, k are obectve; b, =,...,k are the goal et b the DM for the obectve, d ad d + are the der-acheveet ad over-acheveet of the th goal repectvel. The vale of p baed po the tlt fcto of the DM. Now, b cobg DEA ad MOLP we preet a ew ethod for fdg a coo et of weght. 3. A METHOD FOR FINDING A COMMON SET OF WEIGHTS Korblth (99) otced that the DEA odel cold be expreed a a lt-obectve lear fractoal prograg proble. The obectve fcto of the odel the ae a the CCR odel () whch attept to axze the effcec of all DMU collectvel, tead of oe at a te b the ae cotrat. However, the propoed odel olear. Baed o Korblth approach oe other ethod alo have bee propoed the lteratre, all of whch are olear. I th ecto, we preet a proveet to Korblth approach b trodcg a MOLP for fdg coo weght DEA. The followg odel whch provde the ae relt a the CCR ltpler odel trodced to fd the effcec vale of DMU p. Th odel ha oe advatage copared to foregog odel that wll be dced later. bect Max to ( v x p ) r= r rp p = r v θ v θ ( v x ) r= r r = + r= r = = 0, =,,..., 0, r =,,..., 0, =,,..., Where θ, =,,..., the opt vale obtaed fro the CCR e odel, whe DMU der coderato. We preet the followg theore to addre the optal olto of the odel (6). Theore : The opt vale of the odel (6) zero ad for t optal olto, a,, we have ( ) v (6) Proof : Sce ( ) effcet. r = r rp θ = p. = v xp θ p x p, pt oreted proecto of DMU p p o the effcet froter, hece t

6 98 Mak, Alezhad, Ka Mav ad. Zohrehbada Accordg to the above odel ad the propoed approach b Korblth (99), The dea behd the detfcato of the coo weght forlated a the axzg the rato of otpt to pt for all proected DMU ltaeol. So we preet the followg MOLP proble. Max bect to v θ x v θ x,..., r= r r = r= r r = r v v θ r= r r = v x + r= r = = 0, =,,..., 0, r =,,..., 0, =,,..., Frtherore, order to olve the above MOLP odel, we et p= ad e a goal prograg wth all goal eqal to zero. bect to M = ( d + d ) r= r r = v x r= r r = v θ x d d r + = r v = = d d 0, =,,..., = 0, =,,..., v r θ, 0, =,,..., 0, r =,,..., 0, =,,..., Here, de to the fact that p= ad odel lear, the lat et of cotrat odel (5) doe ot appear the above odel. Moreover, the frt ad the ecod et of cotrat odel (8) force + d to take vale zero. However, olvg the above GP odel gve a coo et of weght ad the the effcec core of DMU, =,...,, ca be obtaed b g thee coo weght r = a r r ( ) = v x. If for r = r r, v we have =, the DMU p called effcet. = v x Baed o the work of Kao ad Hg (005) we ca propoe the followg lea. (7) (8) Lea 3: A DMU p whch how to be effcet b odel (8), alo effcet the pt oreted CCR odel. O the ba of the olto of odel (8) we preet the followg theore.

7 A Goal Prograg Method for Fdg Coo Weght 99 Theore : There ext a DMU, =,..., whch characterzed a the effcet DMU b odel (8). Proof:. There a DMU p, p {,,...,} for whch the frt eqalt (8) bdg. Becae, f that ot the cae, there ext a ffcetl all vale ε > 0 for whch T T (, v ) = ( + [ ε, 0,...,0], [,0,...,0] ) v ε atfe the et of retrcto (8). O the other had, the vale of d whch aocated wth (, v ) ad the ecod retrcto wll ted to decreae whch r cotrar to the optalt of d. Therefore, there a DMUp, P {,,...,} for whch we have: r= r = p p = 0 rp v θ x We kow that, ( ) θ p x p, effcet. Therefore, (,v) aocated wth the gradet vector of a p pportg hperplae. Frtherore, th pportg hperplae t pport the PPS at oe extree effcet DMU. Therefore, ch a DMU dcated to be effcet b the odel (8). Roll et al.(99) ad Gola ad Y (995) how that a geeral reqreet for the coo et of weght that t expla a hgh porto of a DMU perforace. Th reqreet ple that at leat oe DMU t atta effcec wth the coo weght. If there o DMU wth effcec core, the t obvo that the effcec core are der-etated baed o relatve coparo wth the hghet effcec actall oberved. More portatl, there o wa to kow whether the prodcto froter appropratel repreet the apled DMU. I th ee, the effcec core obtaed fro the propoed ethod are ot der-etated ad wll atf the + geeral reqreet. If the frt et of cotrat odel (8) are elated, ad we let d take a vale greater tha zero, a coplete rakg of DMU wll be obtaed. I other word, b g the r = r r effcec core = v for each DMU p, p=,...,, the DMU ca be characterzed three x grop: Sper effcet, effcet ad effcet. 4. NUMERICAL EXAMPLE To lltrate the ert of the propoed approach, we chooe a exaple fro Kao ad Hg (005). I that exaple, 7 foret dtrct (DMU); for pt (I-I4): bdget ( US dollar), tal tockg ( cbc eter), labor ( ber of eploee), ad lad ( hectare); ad three otpt (O-O3): a prodct ( cbc eter), ol coervato ( cbc eter), ad recreato ( ber of vt) are codered for earg the effcec. Table cota the orgal data, whle Table how the coo et of weght geerated b the propoed ethod (GP), wth repect to pt ad otpt. Frtherore, Table 3 how the effcec core of the 7 foret dtrct calclated fro the CCR Model, effcec core of the coproe olto approach b Kao ad Hg (005), ad the effcec core of the GP approach th paper, repectvel.

8 300 Mak, Alezhad, Ka Mav ad. Zohrehbada Table. Ipt ad otpt data for the 7 foret dtrct Tawa. DMU I I I3 I4 O O O Table. A coo et of weght geerated fro GP ethod. v v v 3 v The CCR effcec core are the hghet vale that the dtrct ca atta, ad there are e effcet t whch caot be dfferetated. Regardg the coproe olto approach (Kao ad Hg, 005) three vale of p, vz.,,, ad, have bee codered ad the relt are referred to a MAD, MSE ad MAX. The coo et of weght geerated fro thee for odel, o whoe ba the effcec core of ever dtrct are calclated, are dfferet et of weght de to the fact that the are obtaed fro dfferet vewpot. Therefore, t approprate to a whch weght are correct ad whch are ot. Bt, a Kao ad Hg (005) eto, the propert that the dtace betwee the vector of effcec core calclated fro the coproe olto approach to the effcec core calclated fro the tadard DEA odel, the hortet the Ecldea pace, gget that p= the ot table choce coproe olto approach. Therefore, to be coervatve, p= a better choce tha p= ad. We ca alo e correlato to obta Speara ρ (rak

9 A Goal Prograg Method for Fdg Coo Weght 30 correlato coeffcet). Lke the Pearo prodct oet correlato coeffcet, Speara ρ a eare of the relatohp betwee two varable. However, Speara ρ calclated o raked data. Table 3. Effcec core ad the aocated rakg ( parethee) calclated fro the CCR rato odel for dfferet ethod of coo weght. Relt obtaed fro Kao ad Hg (005). DMU CCR MAD MSE MAX GP.0000().0000().0000().0000().0000().0000().0000().0000().0000().0000() ().0000() (3) 0.73().0000() ().0000() 0.997(4) (4).0000() () (5) (5).0000().0000() () 0.854(9) 0.93(6) 0.869(7) (6) () 0.944(6) (7) 0.743(9) (8) () (7) (9) (5) (9) () 0.669(4) (4) 0.730() (3) (0) 0.87(8) (8) 0.876(6) (7) () (5) 0.658(5) (3) 0.656(5) 0.890() (0) 0.78(0) (8) 0.775() (3) 0.69(7) 0.660(6) (4) 0.67(6) (4) 0.740() 0.74() (0) 0.76() (5) 0.745() 0.70() 0.640(5) 0.75(0) (6) (3) 0.68(3) (7) (4) (7) 0.630(6) (7) (6) 0.595(7) Average For calclatg peara ρ we ca e the followg forlato whch d the dfferece betwee rak for the ae obervato (DMU) ad the ber of DMU. r = = ( ) A a alteratve, we ca copte the Pearo correlato o the col of raked data. The relt of th forlato too cloe to the exact Speara ρ. I th forlato x, are the rak for the ae DMU. Ad =,,3,,. d r = = x = x x x =. Eprcall, th exaple the peara correlato betwee the et of effcec core of the GP ethod ad MSE (where p=), greater tha 95%. However, GP approach eed to olve a lear proble ad th t advatage of t over the Kao ad Hg approach, whch ha to olve a olear proble. I geeral, the rakg of thee for ethod, a how parethee

10 30 Mak, Alezhad, Ka Mav ad. Zohrehbada Table 3, are cotet wth thoe of the CCR odel, dcatg that the relt are reaoable. I addto, the are ore foratve. Not ol do the dfferetate the effcet t, bt alo detect oe aboral effcec core calclated fro the CCR odel. The effcec core obtaed for dtrct 9 ad are two of ch exaple. 5. CONCLUSIONS The flexblt the choce of weght both a weake ad a atter of tregth for DEA approach. It a weake becae t ted to deter the coparo aog DMU o a coo ba. Th flexblt alo a g of tregth, however, for f a t tr ot to be effcet eve whe the ot favorable weght have bee corporated t effcec eare, the th a trog tateet ad partclar the arget that, the weght are correct ot table. For dealg wth th dffclt ad aeg all the DMU o the ae cale, th paper propoe the applcato of goal prograg approach for geeratg coo et of weght. There are other ethod the lteratre whch are alo able to geerate coo weght. A cae take fro Kao ad Hg (005) olved to vetgate the dfferece aog thee ethod ad oe coclo are derved. Solvg lear proble a advatage of the propoed approach agat geeral approache the lteratre whch are baed o olvg olear proble. Whe weght of the pt/otpt factor are avalable, effcec core ca be eared. Moreover, all DMU ca be raked ter of a coo ba. Copared to the orgal DEA odel, th approach dcrate a better wa aog DMU order to eld the le effcet oe. A the covetoal DEA odel, t doe ot reqre the forlato of odel. I fact, the effcece of all DMU ca be calclated b olvg a gle odel, eablg oe to evalate the relatve effcec of ever DMU o a coo weght ba. Fall, wth approprate odfcato, the propoed ethod, ca pl be geeralzed to other DEA odel. REFERENCES [] Boo D. (999), Ug DEA a a tool for MCDM: oe reark; Joral of the Operatoal Reearch Socet 50(9); [] Chare A., Cooper W.W. (96), Maageet Model ad Idtral Applcato of Lear Prograg; Joh Wle, New York. [3] Chare A., Cooper W.W., Rhode E. (978), Mearg the effcec of deco akg t; Eropea Joral of Operatoal Reearch ; [4] Dole J.R., Gree R.H. (994), Effcec ad cro-effcec DEA: dervatve, eag ad e; Joral of the Operatoal Reearch Socet 45; [5] Etellta L M.P., Aglo Meza L., Morera da Slva A.C. (004), A lt-obectve approach to detere alteratve target data evelopet aal; Joral of the Operatoal Reearch Socet 55; [6] Goka D. (997), The e of goal prograg ad data evelopet aal for etatg effcet argal cot of otpt; Joral of the Operatoal Reearch Socet 48(3);

11 A Goal Prograg Method for Fdg Coo Weght 303 [7] Gola B. (988), A teractve MOLP procedre for the exteo of DEA to effectvee aal; Joral of the Operatoal Reearch Socet 39(8); [8] Gola B., Y G. (995), A goal prograg-dcrat fcto approach to the etato of a eprcal prodcto fcto baed o DEA relt; Joral of Prodctvt Aal 6; [9] Jahahahloo G.R., Meara A., Lotf F.H., Reza H.Z. (005), A ote o oe of DEA odel ad fdg effcec ad coplete rakg g coo et of weght; Appled Matheatc ad Coptato 66; [0] Joro T., Korhoe P., Walle J. (998), Strctral coparo of data evelopet aal ad ltple obectve lear prograg; Maageet Scece 44; [] Kao C., Hg H.T. (005), Data evelopet aal wth coo weght: the coproe olto approach; Joral of the Operatoal Reearch Socet 56; [] Karak E.E., Ahka S.S. (005), Practcal coo weght lt-crtera deco-akg approach wth a proved dcratg power for techolog electo; Iteratoal Joral of Prodcto Reearch 43(8); [3] Korblth J. (99), Aalg polc effectvee g coe retrcted data evelopet aal; Joral of the Operatoal Reearch Socet 4; [4] Roll Y., Cook W.D., Gola B. (99), Cotrollg factor weght data evelopet aal; IIE Traacto 3(); -9. [5] Roll Y., Gola B. (993), Alterate ethod of treatg factor weght DEA; Oega (); [6] Stewart T.J. (996), Relatohp betwee data evelopet aal ad ltcrtera decoaal; Joral of the Operatoal Reearch Socet 47(5); [7] Xao Ba L., Reeve G.R. (999), A ltple crtera approach to data evelopet aal; Eropea Joral of Operatoal Reearch 5;