Connected Directional Slack-Based Measure of. Efficiency in DEA

Transcription

1 Appled Mathematcal Scece, Vl. 6, 2012,. 5, Cected Dectal Slack-Baed Meaue f Effcecy DEA G. R. Jahahahl Faculty f Mathematcal Scece ad Cmpute Egeeg, Tabat Mallem Uvety, Teha, Ia F. Hezadeh Ltf Depatmet f Mathematc, Scece ad Reeach Bach, Ilamc Azad Uvety, Teha, Ia M. Mehdlzad Faculty f Mathematcal Scece ad Cmpute Egeeg, Tabat Mallem Uvety, Teha, Ia I. Rhd Depatmet f Mathematc, Scece ad Reeach Bach, Ilamc Azad Uvety, Teha, Ia Abtact I the hty f DEA, thee ae tw fudametal Radal ad N-adal appache f the effcecy meauemet. Each appach ha t w advatageu ad htcmg. Avka et al. (2008) pvded a ufed famewk, amed Cected-SBM, f lkg thee appache. I the peet pape, ft, baed the dectal dtace fuct we develp the Dectal Slack-Baed Meaue (DSBM) f effcecy ude Geealzed Retu t Scale Cepdg auth. E-mal addee: m.mehdlzad@gmal.cm (M. Mehdlzad),.hd@gmal.cm (I. Rhd)

2 238 G. R. Jahahahl et al (GRS) whch ha may attactve ppete. Secdly, emplyg the DSBM, we develp a geealzed fm f the Cected-SBM mdel, -called Cected-DSBM. Ou pped mdel me flexble tha the Cected-SBM mdel. I addt, the Cected-SBM ad may f the well-kw DEA mdel, e.g., the CCR, BCC, ERM ad SBM mdel, ae deved fm t. Keywd: DEA, Techcal Effcecy, Dectal Dtace Fuct, Dectal Slack-Baed Meaue f Effcecy, Bdgg Radal ad N-adal Meaue. 1 Itduct Data Evelpmet Aaly (DEA), gally develped by Chae et al. (1978) ad ubequetly exteded by Bake et al. (1984), a -paametc lea pgammg-baed methd t evaluate the elatve effcecy f a et f hmgeeu dec makg ut (DMU). Numeu applcat ecet yea have bee accmpaed by ew exte, mdfcat ad develpmet ccept ad methdlgy f DEA (See Sefd 1997 ad Emuzead 2008). Geeally peakg, the pevu eeach have bee pped baed the tw fudametal Radal ad N-adal appache f meaug the effcecy. The dea f adal pect date back t Debeu (1951) ad Faell (1957). Meawhle, the dea f -adal pect date back t Kpma (1951) ad Ruell (1985). The adal meaue (e.g., the CCR mdel: Chae et al. 1978) maly deal wth the pptal mpvemet put ad utput ad uffe fm the fllwg htcmg: Due t the pptal mpvg thee mdel, they ca t be emplyed f cae wth put uch a lab, mateal ad captal. They mt the -ze put ad utput lack ad theefe fal t accut f the -adal excee ad htfall. Thu, They d t ecealy lead t a effcet taget DEA. They d t allw ay flexblty f a DM t che a efeece ut f a effcet ut. They ae value fee,.e., effcecy evaluat baed the data avalable wthut takg t accut the dec-make (DM ) pefeece fmat. Depte the weakee, the adal meaue have a umbe f deable featue, e.g., the adal effcecy meaue ha a clea ecmc tepetat egadle f the pce. The -adal meaue cpate lack t cdeat. I the hty f DEA, thee ae eveal dffeet type f -adal mdel f techcal effcecy-baed pefmace evaluat. Chae et al. (1985) ft pped a addtve mdel a a -adal mdel. Cpe et al. (1999) al pped a -adal meaue, efeed t a Rage-Aduted Meaue (RAM), whch

3 Cected dectal lack-baed meaue 239 a exte f the addtve mdel (Cpe et al., 2000, 2001). Pat et al. (1999) deved a Ehaced Ruell Meaue (ERM) that cpated the aalytcal featue f Ruell meaue t the famewk f the SBM. Te (2001) pped the lack-baed meaue (SBM) f effcecy whch maxmze put ad utput lack. Althugh the -adal meaue the ptmal effcecy value accut f the -adal lack, the pected DMU may le the pptalty the gal. Theefe, they may uffe fm the fllwg htcmg: If the l f the gal pptalty apppate f the aaly, the th becme a htcmg f -adal. Whe we evaluate effcecy chage ve tme, the -ze patte f lack at tme ped t may gfcatly dffe fm that f tme ped t +1. Thu, we wll be uable t tell whch patte eaable. Athe htcmg f the -adal LP-baed meaue, e.g. the SBM mdel, that the ptmal lack ted t exhbt a hap ctat takg ptve ad ze value. (See Appedx A f Avka et al. 2008). I a efft t vecme the abve meted htcmg f the adal ad -adal mdel, Avka et al. (2008) tduced the Cected-SBM mdel whch clude tw cala paamete. They tated that a apppate chce f thee paamete, by elcatg the aaly aywhee betwee the adal ad -adal mdel, culd vecme the key htcmg the tw appache. I th pape, ft, baed the dectal dtace fuct we develp the Dectal Slack-Baed Meaue (DSBM) f effcecy ude Geealzed Retu t Scale (GRS) whch ha may attactve ppete. Secdly, emplyg DSBM, we develp a geealzed fm f the Cected-SBM mdel, -called Cected-DSBM. I huld be ted that u pped mdel me flexble tha the Cected-SBM mdel. I addt, th mdel ad may well-kw DEA mdel, e.g., the CCR, BCC, ERM ad SBM mdel, ae deved fm t. The emde f th pape gazed a fllw. I the ubequet ect, at ft, a bef evew f the dectal dtace fuct pvded. The, the Dectal Slack-Baed Meaue (DSBM) f effcecy tduced ad a detaled dcu abut the ppete ad featue f th meaue pvded. Sect 3 tduce the Cected Dectal Slack-Baed Meaue (Cected-DSBM) f effcecy. A llutatve example pvded Sect 4. Fally, the lat ect ummaze the eult ad cclude the pape ummay. 2 Radal ad N-Radal Dectal Dtace Fuct Thughut th pape, we deal wth DMU wth m put 1,..., m ad utput 1,...,. The put ad utput vect f DMU 1,...,, ae,..., T x x 1 x m ad y y y,..., T 1 whee x 0, x 0, y 0 ad y 0.The Pduct Pblty Set (PPS), T, the et f all feable put

4 240 G. R. Jahahahl et al ad utput vect ad t defed a fllw: T x, y : x ca pduce y. (1) The dectal dtace fuct, ecetly tduced by Chambe et al. (1996; 1998), a ve f Luebege htage fuct (Luebege 1992; 1995), whch geealze the tadtal Shephad dtace fuct (Shephad 1970) ad ad well-uted t the tak f pvdg a meaue f techcal effcecy the full put-utput pace. Th fuct pect a gve put-utput vect, x, y, adally fm telf t the fte f PPS, T, a pe-aged dect vect g g, g m T D x, y ; g, g Max x g, y g T., ad defed a: Th dtace fuct multaeuly eek t expad utput ad ctact put (See Fg.1). y (2) A g g, g A x Fg. 1 Dectal Dtace Fuct Ude the tadad aumpt f Iclu f bevat, cvexty, geeal etu t cale (GRS) 2 ad fee dpablty f put ad utput, the DMU = x, y, uque -empty PPS paed by beved DMU, 1, 2,...,, a fllw: G whee L 0 L 1 ad U 1 U. Ntce that L 0, U T x, y x X, y Y, L 1 U, 0. (3) ae uppe ad lwe bud f the um f cepd t the PPS wth Ctat Retu t Scale, T C, (Chae et al. 1978) ad L U 1 cepd t the PPS wth Vaable Retu t Scale, T V, (Bake et al. 1984). Nw, the DEA fmulat f the dectal dtace fuct elat t (3) becme Max. t. x x g, 1,2,..., m, 1 1 y y g, 1,2,...,, 2 F me detal abut GRS ee Cpe et al. (2007) pp

5 Cected dectal lack-baed meaue 241 L U, (4) 1 0, 1,2,...,. Althugh, pcple, uetcted g, t ptmal value wll eve be lve tha ze,.e. 0. Meve, f the dect vect g g, g ha bee elected uch that Max x g (5) 1, 1,...,,, e.g., e ca apply the fllwg dect vect: g x, g y,, f all,, (6) g x Max x, g y Max y, f all,, (7) the 1 ad 1 ca be tepeted a a effcecy meaue. A meted eale, the mdel (4) fal t take accut the extece f the -ze lack. T vecmg th pblem, we develp the dectal lack-baed meaue (DSBM) f effcecy, elat t T, a fllw: 1 1 m M m 1 1. t. x x g, 1,2,..., m, L y y g, 1,2,...,, U, 0, 1,2,...,, 0, 0, f all,. whee the dect vect g g, g atfe (5). Hee, ad epeet the ate f ctact ad expa the th put ad th utput f DMU whch ha bee pected t the effcet fte f T G the dect g. Futheme, the bectve fuct f (8) tly maxmze the value f 1,..., m ad 1,...,. Obvuly, e 1. I addt, ce x g 0, accdg t (5), we have 0e 1 ad, theefe, e ca be tepeted a a effcecy meaue. The ptmal value f (8), e, the effcecy ce f DMU ad baed t, we deteme a DMU a beg DSBM -effcet a fllw: Deft1. DMU ad t be DSBM-effcet f ad ly f 1. Th cdt equvalet t 0, f all, each ptmal lut f (8),.e., thee put effcecy (wate) ad utput effcecy (htfall) all put ad utput ay ptmal lut. G (8)

6 242 G. R. Jahahahl et al Remak1. The CCR mdel a pecal cae f the mdel (4) ad the ERM (Pat et al. 1999) ad the SBM (Te 2001) mdel ae pecal cae f the mdel (8) ad ca be ealy deved by agg the dect vect (6). Futhe, DMU DSBM-effcet f ad ly f t ERM-effcet. By electg a utable dect vect, the DSBM mdel wll have may attactve ppete that we utle them a fllw: (P1). Cmputatal apect Th mdel a factal pgammg pblem. Hweve, t ca be lved utlzg Chae Cpe tafmat (Chae ad Cpe 1962) the mla way a the SBM mdel. (P2). Cmpletee Th meaue cmplete, that t, ctat wth eted meaue, a -eted meaue ad cde all effcece acated wth the -ze lack that may be detfed by the mdel. (P3). Ut vaace By electg a dect vect uch that the th cmpet f g 1,..., m ad th cmpet f g 1,..., have the ame ut f meauemet a the th put ad th utput, epectvely, th mdel wll be ut vaat, e.g., the vect (6) ad (7) atfy th cdt. (P4). Icpatg Dec Make (DM ) pefeece fmat I me pactcal cae, f the DM de t equally pefe the effcet ut, the t eceay t cpate the DM udgmet a p kwledge t the cdeat. Accdg t the pefeece de f put/utput gve by DM, we ca flexbly mdfy vect g. Ideed, the value f the mdfed dect vect g cmpet decbe the elatve mptace f put/utput gve by DM. Let the -ze weght, w, 1,..., m ad v, 1,...,, ae acated wth the pte gve by DM t the put ad utput, epectvely uch that the lage the w ( v ), the me mptat the th put ( th utput). Afte cpatg thee weght (9), the ceffcet f vaable ad, the bectve fuct wll be w ad v, epectvely. Theefe, the cmpet f mdfed dect vect, g, huld be g g ad g g, whee 1/ w ad 1/ v. Th hw that f a put (utput) ha a lage mptace, t huld be attached a lage weght equvaletly mall dect cmpet. By cdeg (5), we mut have 1, 1,..., m, equvaletly w 1, 1,..., 3. (P5). Mtcty 3 If the gve weght d t atfy thee cdt, the malzed (dvdg by Max w : 1,..., m ) fm f them wll atfy thee cdt.

7 Cected dectal lack-baed meaue 243 The meaue tgly mte deceag each ad. 3 Cected Dectal Slack-Baed Meaue Smla the appach peeted Avka et al. (2008), we develp a ufcat f adal ad -adal dectal meaue a fllw: m 1 1 m 1 [Cected-DSBM] M t. x x g, 1,2,..., m, y y g, 1,2,...,, L U, 1,2,..., m, L U, 1,2,...,, L U, 1,2,...,, 1 I 0, 0, 1,2,...,, whee 0 L 1 U, 0 L 1 U, I 0,1 0, 0, 0, 0, f all,. ad g g, g (9) atfe (5). I th mdel, ug depedet -egatve vaable ad tgethe wth paamete L, L, U ad U, we ae able t ctl the pptalty f lack (effcece). I fact, cepd t the magtude whch L, L ae cle t U, U, the vaable, 1,..., m, ad, 1,...,, ted t be ufm. Specfcally, f L U 1 ad L U 1, the we have a full pptal mdel. Al, f 0 L U 0 L U, the a the paamete L ad U ( L ad U ) multaeuly deceae ad ceae, epectvely,.e. multaeuly L ad U ( L ad U ), the gal pptalty f put (utput) deceae accdgly ad the mdel ted t be -pptal. It huld be ted that the vaable ad play a develped le cmpa wth the le f vaable f the Cected-SBM. Meve, hee, the Paamete I emplyed a a ctlle f the pptalty ate betwee put ad utput. If I 1, the we wll have the ame ate f pptal educt ad cemet put ad utput, epectvely; thewe, f I 0, the the ate f pptal educt put may be dffe fm the ate f cemet utput. Cmpa wth the Cected-SBM: Ou appach a ueful geealzat f the Cected-SBM; ad t ca be deved eadly fm (9) by ettg, L 0,

8 244 G. R. Jahahahl et al U, L L I 0, 0 U, U. A ma dffeece betwee the tw appach that, ctat wth the Cected-SBM whch utlze a depedet vaable lack, ( f the aveage f malzed lack) a a etct fact f themelve, u appach emply tw depedet vaable, ad, a the etct fact put ad utput lack wheeby the flexblty f mdel ceae. Belw we ummaze me ppete ad bevat f the pped Cected-DSBM that ca be vefy wth eae: If L L 0, U U 0, I 1, the the Cected-DSBM mdel educe t a eved factal fm f the mdel (4) whee t bectve fuct 1 ad t ctat the mdel (4) ctat. 1 If L L I 0 ad U 0, U 0, the the Cected-DSBM mdel educe t the mdel (8). The adal put- utput-eted CCR ad BCC (Bake et al. 1984) mdel ad the -adal put-/utput-/-eted ERM ad SBM mdel ude GRS ae pecal cae f (9). Exte: Althugh we apply a cmm lwe bud acated wth all put/utput, e ca ue dvdual lwe bud L / U ad / L U acated wth put ad utput. If a put/utput allw ly mall vaat ppt, e ca et L ad U / L ad U cle t each the, ad vce vea. Thu, th exte allw me flexblty t the mdel. 4 Illutatve Example Th ect peet a umecal example de t pvde a llutat f the pped methd. By th example we hw that the Cected-SBM a pecal cae f the Cected-DSBM. Example1. Cde thee DMU A, B ad C wth tw put ad e utput a defed by Tab. 1. Thee data gally ha bee epted by Avka et al. (2008). DMU x 1 x 2 y A B C Tab. 1 DMU data (extacted fm Avka et al. 2008) Cde the mdel (9) the cae f L 0, U, I 0, L U 1, g x, y. The eult f evaluatg the ut C ug th mdel, wth dffeet value f L ad U, have bee epted Tab. 2.

9 Cected dectal lack-baed meaue 245 N-Radal Radal L U x x Tab. 2 The eult f example 1 A ca be ee Tab. 2, by allcatg dffeet value f the paamete L ad U, ad evaluatg ut C by Cected-DSBM, we ae able t pvde the ame epted eult the example f Avka et al. (2008), hweve, the cepdg value f the paamete the tw appach ae dffeet fm each the. Thu, the Cected-DSBM mdel a geealzed fm f the Cected-SBM mdel. 5 Cclu I th pape, we have pped a geealzed fm f the Cected-SBM mdel whch wa develped by Avka et al. (2008) t lk the tw fudametal adal ad -adal effcecy meauemet appache. I the pped methd, at ft, by emplyg the dectal dtace fuct, we develped a fmalzat f a cmplete effcecy dex, amed DSBM whch ha may attactve ppete. The, applyg the dea f bdgg adal ad -adal meaue, we exteded the DSBM mdel ad pped a geealzat f the Cected-SBM mdel, amed Cected-DSBM. A ted eale, u pped mdel me flexble tha the Cected-SBM mdel. I addt, Cected-SBM ad may well-kw DEA mdel, e.g, the CCR, BCC, ERM ad SBM mdel, ae deved fm t. Refeece [1] A. Chae, W. W. Cpe, B. Glay, L. Sefd, J. Stutz, Fudat f data evelpmet aaly f Paet-Kpma effcet empcal pduct fuct, Jual f Ecmetc, 30(1985), [2] A. Chae, W. W. Cpe, E. Rhde, Meaug the effcecy f dec makg ut, Eupea Jual f Opeatal Reeach, 2(1978), [3] A. Chae, W. W. Cpe, Pgammg wth lea factal fuctal, Naval Reeach Lgtc Quately, 15(1962), [4] A. Emuzead, B. R. Pake, G. Tavae, Evaluat f eeach effcecy ad pductvty: a uvey ad aaly f the ft 30 yea f chlaly lteatue DEA. Sc-Ecmc Plag Scece, 42(2008), [5] D. G. Luebege, Beeft fuct ad dualty, Jual f Mathematcal Ecmc, 21(1992), [6] D.G., Luebege, Mcecmc they, McGaw Hll, New Yk, 1995.

10 246 G. R. Jahahahl et al [7] G. Debeu, The ceffcet f euce utlzat, Ecmetca, 19(1951), [8] J. T. Pat, J. L. Ruz, I. Svet, A ehaced Ruell gaph effcecy meaue, Eupea Jual f Opeatal Reeach, 115(1999), [9] K. Te, A lack-baed meaue f effcecy data evelpmet aaly, Eupea Jual f Opeatal Reeach, 130(2001), [10] L. M. Sefd, A bblgaphy f data evelpmet aaly ( ), Aal f Opeat Reeach, 73(1997), [11] M.J. Faell, The meauemet f pductve effcecy, Jual f the Ryal Stattcal Scety, See A, 120 (Pat 3) (1957), [12] R. D. Bake, A. Chae, W. W. Cpe, Sme mdel f etmatg techcal ad cale effcece data evelpmet aaly, Maagemet Scece, 30(1984), [13] R. G. Chambe, Y. Chug, R. Fäe, Beeft ad dtace fuct, Jual f Ecmc They, 70(1996), [14] R. G. Chambe, Y. Chug, R. Fäe, Pft, Dectal dtace fuct, ad Nelva effcecy, Jual f Optmzat They ad Applcat, 98 (1998), [15] R. R. Ruell, Meaue f techcal effcecy, Jual f Ecmc They, 35(1985), [16] T. C. Kpma, A aaly f pduct a a effcet cmbat f actvte. I: Kpma TC, edt. Actvty aaly f pduct ad allcat. Wley, New Yk, [17] W. W. Cpe, K. S. Pak, J. T. Pat, Magal ate ad elatcte f ubttut wth addtve mdel DEA, Jual f Pductvty Aaly, 13(2000), [18] W. W. Cpe, K. S. Pak, J. T. Pat, RAM: a age aduted meaue f effcecy f ue wth addtve mdel ad elat t the mdel ad meaue DEA, Jual f Pductvty Aaly, 11(1999), [19] W. W. Cpe, K. S. Pak, J. T. Pat, The age aduted meaue (RAM) DEA: A epe t the cmmet by Stema ad Zwefel, Jual f Pductvty Aaly, 15(2001), [20] W. W. Cpe, L. M. Sefd, K. Te, Data Evelpmet Aaly: a cmpeheve text wth mdel, applcat, efeece ad DEA-lve ftwae (2d ed.), Spge, Bel, Receved: Augut, 2011