Data envelopment analysis (DEA) Thirty years on

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1 Avalable nlne at Eupean Junal f Opeatnal Reseach 19 (009) 1 17 Invted Revew Data envelpment analyss (DEA) Thty yeas n Wade D. Ck a, *, Lay M. Sefd b a Depatment f Opeatns Management and Infmatn Systems, Schulch Schl f Busness, Yk Unvesty, Tnt, Onta, Canada M3J 13 b Industal and Opeatns Engneeng, Cllege f Engneeng, Unvesty f Mchgan, Ann Ab, MI, Unted States Receved 19 Januay 008; accepted Januay 008 Avalable nlne 3 Febuay Abstact Ths pape pvdes a sketch f sme f the ma eseach thusts n data envelpment analyss (DEA) ve the thee decades snce the appeaance f the semnal wk f Chanes et al. (1978) [Chanes, A., Cpe, W.W., Rhdes, E.L., Measung the effcency f decsn makng unts. Eupean Junal f Opeatnal Reseach, 9 ]. The fcus heen s pmaly n methdlgcal develpments, and n n manne des the pape addess the many excellent applcatns that have appeaed dung that ped. Specfcally, attentn s pmaly pad t (1) the vaus mdels f measung effcency, () appaches t ncpatng estctns n multples, (3) cnsdeatns egadng the status f vaables, and () mdelng f data vaatn. Ó 008 Elseve B.V. All ghts eseved. Keywds: DEA; Mdels; Multple estctns; Data vaatn 1. Intductn Effcency measuement has been a subect f temendus nteest as ganzatns have stuggled t mpve pductvty. Reasns f ths fcus wee best stated ffty yeas ag by Faell (1957) n hs classc pape n the measuement f pductve effcency. The pblem f measung the pductve effcency f an ndusty s mptant t bth the ecnmc thest and the ecnmc plcy make. If the theetcal aguments as t the elatve effcency f dffeent ecnmc systems ae t be subected t empcal testng, t s essental t be able t make sme actual measuements f effcency. Equally, f ecnmc plannng s t cncen tself wth patcula ndustes, t s mptant t knw hw fa a gven ndusty can be expected t ncease ts utput by smply nceasng ts effcency, wthut absbng futhe esuces. Faell futhe stated that the pmay easn that all attempts t slve the pblem had faled, was due t a falue t cmbne the measuements f the multple nputs * Cespndng auth. Tel.: / E-mal addess: wck@schulch.yku.ca (W.D. Ck). nt any satsfacty measue f effcency. These nadequate appaches ncluded fmng an aveage pductvty f a sngle nput (gnng all the nputs), and cnstuctng an ndex f effcency n whch a weghted aveage f nputs s cmpaed wth utput. Respndng t these nadequaces f sepaate ndces f lab pductvty, captal pductvty, etc., Faell ppsed an actvty analyss appach that culd me adequately deal wth the pblem. Hs measues wee ntended t be applcable t any pductve ganzatn; n the wds, fm a wkshp t a whle ecnmy. Unftunately, he cnfned hs numecal examples and dscussn t sngle utput stuatns, althugh he was able t fmulate a multple utput case. Twenty yeas afte Faell s semnal wk, and buldng n thse deas, Chanes et al. (1978), espndng t the need f satsfacty pcedues t assess the elatve effcences f mult-nput mult-utput pductn unts, ntduced a pweful methdlgy whch has subsequently been ttled data envelpment analyss (DEA). The gnal dea behnd DEA was t pvde a methdlgy wheeby, wthn a set f cmpaable decsn makng unts (DMUs), thse exhbtng best pactce culd be dentfed, and wuld fm an effcent fnte /$ - see fnt matte Ó 008 Elseve B.V. All ghts eseved. d: /.e

2 W.D. Ck, L.M. Sefd / Eupean Junal f Opeatnal Reseach 19 (009) 1 17 Futheme, the methdlgy enables ne t measue the level f effcency f nn-fnte unts, and t dentfy benchmaks aganst whch such neffcent unts can be cmpaed. Snce the advent f DEA n 1978, thee has been an mpessve gwth bth n theetcal develpments and applcatns f the deas t pactcal stuatns. The pupse f the cuent pape s t pvde a sketch f the ma dectns n methdlgcal develpments (as ppsed t a dscussn f applcatns), n ths mptant feld dung the past thee decades. The cveage s by n means cmplete as the vlume f lteatue s enmus, and beynd the cuent scpe. Sectn evews the vaus DEA mdels, ncludng thse that g beynd the usual defntn f DEA, specfcally the fee dspsal hull (FDH) mdel, css evaluatn, and mnmum dstance mdels. Sectn 3 ges beynd the sngle level mdels f Sectn and examnes multlevel mdels. Sectn dscusses vaus fms f multple estctns used t cnstan the fnte. In Sectn 5 the status f dffeent types f vaables s evewed. These nclude nn-dscetnay, nn-cntllable, categcal, and dnal vaables. As well, we cnsde the ssue egadng uncetanty as t the nput vesus utput status f vaables. Data vaatn s expled n Sectn 6. Ths ncludes senstvty analyss, pbablty-based mdels, wndw analyss, Malmqust mdels f captung tmes sees mpacts n effcency, and statstcal nfeence ssues suundng the effcent fnte. Cncludng emaks fllw n Sectn 7.. The mdels.1. The cnstant etuns t scale (CRS) mdel Cnsde a set f n DMUs, wth each DMU, ( =1,...,n) usng m nputs x ( =1,...,m) and geneatng s utputs y ( =1,...,s). If the pces multples u ; v asscated wth utputs and nputs, espectvely, ae knwn, then bwng fm cnventnal beneft/cst they, ne culd expess the effcency ē f DMU as the at f weghted utputs t weghted nputs,.e. X u y = X v x : Ths beneft/cst at s, f cuse, the bass f the standad engneeng at f pductvty. In the absence f knwn multples, Chanes et al. (1978) ppsed devng apppate multples f a gven DMU by slvng a patcula nn-lnea pgammng pblem. Specfcally, f DMU s unde cnsdeatn, the Chanes et al mdel f measung the techncal effcency f that DMU s gven by the slutn t the factnal pgammng pblem: e ¼ max s:t: u y = v x u y v x 6 0; u ; v e; all ; : all ð:1þ whee e s a nn-achmedan value desgned t enfce stct pstvty n the vaables. We pnt ut that ths mdel nvlvng the at f utputs t nputs s efeed t as the nput-ented mdel. One culd, as well, nvet ths at and slve the cespndng utput-ented mnmzatn pblem. We wll geneally deal wth the nputented mdel heen. blem (.1) s efeed t as the CCR (Chanes, Cpe and Rhdes) mdel, and pvdes f cnstant etuns t scale (CRS). It s bseved that n the gnal 1978 pape, the auths smply estcted the vaables t be nn-negatve (e = 0); the mpstn f a stctly pstve lwe lmt (e > 0) was ntduced n a fllw-up pape, Chanes et al. (1981). F cnvenence we efe t (.1) as the gnal CCR mdel. Applyng the Chanes and Cpe (196) they f factnal pgammng, makng the change f vaables l = tu and t = tv, whee t ¼ð m x Þ 1, pblem (.1) can be cnveted t the lnea pgammng (L) mdel: e 0 ¼ max l y s:t: t x ¼ 1 l y ð:þ t x 6 0; 8 l ; t e; all ; : By dualty, ths pblem s equvalent t the lnea pgammng pblem: mn h e s þ þ s s:t: k x þ s ¼ h x ; ¼ 1;...; m k y s þ ð:3þ ¼ y ; ¼ 1;...; s k ; s ; s þ 0; 8; ; h uncnstaned: blem (.3) s efeed t as the envelpment pmal pblem, and (.) the multple dual pblem. The cnstant space f (.3) defnes the pductn pssblty set T. That s, ( T ¼ ðx ; Y ÞX X k X ; Y 6 X ) k Y ; k 0 : T get a gemetc appecatn f the CRS mdel, ne can epesent pblem (.3) n a fm such as pctued n Fg. 1. Ths fgue pvdes an llustatn f a sngle utput sngle nput case. If we slve (.3) f each f the DMUs, ths amunts t pectng that DMU t the left, t a pnt n the fnte. In the case f DMU #3, f example, ts

3 W.D. Ck, L.M. Sefd / Eupean Junal f Opeatnal Reseach 19 (009) DMU X Y Y A 3 * (9,8) 3 (6,7) 7 (10,7) B (3,5) 5 (5,3) 1 (,) 6 (,1) X Effcency Measue = A B Fg. 1. Cnstant etuns t scale pectn n the sngle nput sngle utput case. pectn t the fnte s epesented by the pnt 3 *. Intutvely, ne wuld easnably measue the effcency f DMU #3 as the at A/B =./6 =.70 70%. The slutn f (.3) f ths DMU esults n h 3 ¼ :70. It s useful t nte that had we defned pefmance by the ecpcal at f nputs t utputs, the esultng value f that at wuld be 1.3, and we wuld deem the effcency t be 1/1.3 =.70, whch s the same as aved at abve. ctally, slvng ths utput-ented mdel nvlves a vetcal pectn fm DMU 3 up t the fnte, athe than the hzntal pectn t the left as shwn n the fgue. An altenatve gemetc vew f mdel (.3) s pvded n Fg.. Hee, we have tw nputs (and we assume a sngle cmmn utput value f all DMUs). In slvng (.3) we fnd that DMUs A, B, C and D ae effcent,.e., h A = h B = h C = h D = 1. F DMU E h E ¼ð83:3%Þ, and INUT A B C D E F G BUDGET (X 1 ) COND (X ) the esultng pected value h E x E s smply the fnte DMU B. We may efe t B as a benchmak f DMU E. In the case f DMU G, ts pected (fnte) value s epesented by the pnt K, hence B and C ae apppate benchmaks f DMU G. As llustated by ths example, the CCR mdel s apppately efeed t as pvdng a adal pectn. Specfcally, each nput s educed by the same pptnalty fact h. In the example f Fg., n slvng (.3) f DMUs A, B, C, D, E, F, G, all slack vaables (n ths case s 1 ; s ) wll be ze. Fg. 3 s a edawn vesn f Fg. wth an addtnal DMU H added. F the DMU H, hweve, the pected pnt les n an extensn f the fnte at H *, and nt n the fnte ppe. In ths case, the slack cespndng t nput #1 ðs 1 Þ wll be pstve, and equal t the dstance epesented by the lne segment fm H * t D. We say that DMU H s mppely envelped. A DMU lcated at pnt H * wuld be deemed weakly effcent, as ppsed t pnts such as A, B, C, D whch ae stngly effcent smply effcent (DEA-effcent). F a cmplete dscussn f effcency classes, the eade s efeed t Chanes et al. (1986, 1991)... The vaable etuns t scale (VRS) mdel Banke et al. (198) (BCC), extended the eale wk f Chanes et al. (1978) by pvdng f vaable etuns t scale (VRS). Ths s pctued n the edawn vesn f Fg. 1 n the fm f Fg.. Shwn ae the gnal CRS fnte, and the VRS fnte, hee epesented by the lne segments 1, 3 and 3. The BCC at mdel dffes fm (.1), by way f an addtnal vaable,.e. e ¼ max u y u = v x s:t: u y u v x 6 0; ¼ 1;...; n u e; v e; 8; u unestcted n sgn: ð:þ X A B E K F C G Effcency f pnt E = OB OE Effcency f pnt F = OK OF D X A B E F C G D X X 1 Fg.. A tw nput llustatn f the DEA pectn. Fg. 0. Impact f assuance egn estctns.

4 W.D. Ck, L.M. Sefd / Eupean Junal f Opeatnal Reseach 19 (009) 1 17 X The lnea pgammng equvalent f (.) s e ¼ max s:t: A B E K F C X 1 Fg. 3. DEA pectn f an mppely envelped DMU. Y G l y l t x ¼ 1 l y l t x 6 0; ¼ 1;...; n l e; t e; 8; ; l unestcted: ð:5þ D 3 7 H* X Fg.. The vaable etuns t scale Fnte. H It s nted that (.6) dffes fm (.3) n that t has the addtnal cnvexty cnstant n the k, namely k ¼ 1. In efeence t Fg., that ptn f the fnte fm pnt 1 up t (but nt ncludng) pnt, cnsttutes the nceasng etuns t scale ptn f the fnte; pnt s expeencng cnstant etuns t scale; all pnts n the fnte t the ght f (.e., the segments fm t 3 and fm 3 t ) make up the deceasng etuns t scale ptn f the fnte. As wth the CRS mdel, a DMU s BCC-effcent n the VRS sense f thee exsts a slutn t (.6) such that h ¼ 1 and all slacks s ; s þ ae ze n value. Clealy, any CCR-effcent DMU s als BCCeffcent. The etuns t scale (RTS) classfcatn f DMUs has been the subect f study by numeus auths, ncludng Banke (198) (usng the mst pductve scale sze cncept and lettng the sum f lambda values dctate the RTS), Banke et al. (198) (usng the fee vaable n (.5)), and Fäe et al. (199), (applyng the scale effcency ndex methd). A pblem n classfyng RTS s the exstence f multple ptma, meanng that the classfcatn may be a functn f the patcula slutn selected by the ptmzatn sftwae. Vaus attempts have been made t pvde a me defntve RTS classfcatn assgnment f a gven DMU, ncludng develpng ntevals f the vaus fee vaables asng fm the multple ptma. Zhu and Shen (1995) suggest a emedy f the CCR RTS methd unde multple ptma. Sefd and Zhu (1997, 1999b) evew the vaus methds and suggest cmputatnally smple methds t chaacteze RTS, and t ccumvent the need f explng all altenate ptmal slutns..3. The addtve mdel The pevus tw effcency mdels ae adal pectn cnstucts. Specfcally, n the nput-ented case, nputs ae pptnally educed whle utputs eman fxed. (F the utput-ented case, utputs ae pptnally nceased whle nputs ae held cnstant.) Chanes et al. (1985b) ntduced the addtve aet Kpmans (K) mdel whch, t an extent, cmbnes bth entatns. Fg. 5 llustates ths dea wheen any dectn n the quadant fmed by B A C s pemtted. the dual f whch s gven by mn h e s þ s þ s:t: k x þ s ¼ h x ; ¼ 1;...; m k y s þ ¼ y ; ¼ 1;...; s: k ¼ 1 k ; s ; s þ 0 8; ; h unestcted: ð:6þ Y 10 C 8 (9,8) B 6 3 (6,7) A 7 (10,7) (3,5) 5 (5,3) 1 (,) 6 (,1) X Fg. 5. Addtve mdel pectn (cnstant etuns t scale).

5 W.D. Ck, L.M. Sefd / Eupean Junal f Opeatnal Reseach 19 (009) Thee ae seveal vesns f the addtve mdel, the mst basc beng gven by the lnea ptmzatn pblem shwn as (.7). The cnvexty cndtn n the k vaables mples that we ae usng the VRS technlgy. The fnte geneated by mdel (.7) s dentcal t that asng fm the cespndng VRS stuctue (.6), hence a DMU s addtve-effcent K effcent (all slacks equal t ze at the ptmum n (.7)) f and nly f t s VRS-effcent. Clealy, the CRS pductn pssblty set can be used as well (and s the ne llustated n Fg. 5). ¼ max s þ s þ s:t: k x þ s ¼ x ; ¼ 1;...; m k y s þ ¼ y ; ¼ 1;...; s k ¼ 1 k ; s ; s þ 0; 8; ; : ð:7þ Snce the vaus nputs and utputs may be measued n nn-cmmensuate unts, (Russell, 1988), t may nt be pactcal n cetan cntexts t use the smple sum f slacks as the bectve n (.7). Meve, mdel (.7) des nt pvde f an actual measue f neffcency as n the case f the BCC and CCR mdels. T vecme ths latte pblem, Chanes et al. (1985b) ppsed the use f Q, whee Q ¼ d X s =x þ X! s þ =y subect t the cnstants as n (.7). A suggested value f d was 1/(m + s). The dvsn f the s and s þ by x and y, espectvely, s ntended t ende these slacks unts nvaant (.e., cmmensuate), whle multplyng by d cntls the veall scale. In de t mantan cnsstency wth the sense f effcency n the CCR and BCC mdels, Sueysh (1990) ffes 1 Q as such a measue. The pblem, as acknwledged n a late pape by Chang and Sueysh (1991), s that Q 6 1 may nt necessaly hld, and may n fact be negatve... Slacks-based measues T addess the abve shtcmngs n the addtve mdel, Geen et al. (1997) ppse as a measue f effcency " R ¼ 1 X s =x þ X # s þ s þ =ðy þ s þ Þ ; and ecmmend slvng the pblem max R Subect t the cnstants f ð:7þ: ð:8þ Whle the nn-lneaty f R pses a cmputatnal ncnvenence, the esultng measue 1 R des pssess the ppety f beng n the unt scale [0,1], hence sevng as a legtmate effcency sce. Tne (001) ntduced the s-called slacks-based measue (SBM) whch s nvaant t the unts f measuement and s mntne nceasng n each nput and utput slack. The SBM s deved fm the slutn f the factnal pgammng pblem 1 m 1 s x mn p ¼ 1þ 1 s s þ =y ð:9þ Subect t the cnstants f ð:7þ: Clealy, 0 6 p 6 1, and s theefe a legtmate K effcency sce n the spt f the CCR and BCC mdels. It s shwn n Tne (001) that (.9) can be tansfmed nt a lnea pgammng pblem..5. The Russell measue The Russell measue mdel, named by Fäe and Lvell (1978), and late evsted by ast et al. (1999) (efeng t t as the enhanced Russell measue), s equvalent t Tne s SBM, as dscussed n Cpe et al. (006). Specfcally, the mdel s R ¼ mn ðh =mþ= ðu =sþ s:t: k x 6 h x ; k y u y ; k ¼ 1 ¼ 1;...; m ¼ 1;...; s k 0; 0 6 h 6 1; u 1; all ; ; : ð:10þ.6. Othe nn-adal mdels Thee ae seveal the nn-adal mdels. One f these s the RAM (ange adusted measue) mdel f Cpe et al. (1999a), and s smla t the addtve mdel wth the addtnal featue that the sce les n the [0,1] scale. Thee ae als nn-adal mdels emplyed as a secnd stage n a tw-stage effcency analyss, afte a pectn pnt has been dentfed f a gven DMU. See Tne (001), Cpe et al. (001), tela and Thanassuls (007), tela et al. (003), and thes..7. Altenatve vews.7.1. FDH The fee dspsal Hull mdel A pductn pssblty set (S) efeence technlgy can be thught f as a declaatn f the ttalty f pductn actvtes that mght plausbly have been bseved n the evdence f the actvtes actually bseved.

6 6 W.D. Ck, L.M. Sefd / Eupean Junal f Opeatnal Reseach 19 (009) 1 17 DEA uses the fnte f the S, defned n tems f bseved actvtes deemed effcent, and specfc lnea/cnvex cmbnatns theef, t evaluate the bseved actvtes. Depns et al. (198) and Tulkens (1993) take an altenate vew, wheeby the assumptn s that nly the bseved DMUs make up the fnte, nt lnea cnvex cmbnatns f thse bseved unts. The mdel that wll geneate ths fnte s smply the VRS mdel f Banke et al. (198), but wth the addtnal estctn that the k {0,1}. Whle Thall (1999) challenged ths cncept, fm an ecnmc they pespectve, ths was stngly ebutted by Chechye et al. (000), and FDH emans an attactve, but ptentally undeutlzed appach t effcency measuement. See als Geen and Ck (00)..7.. Css effcency The css effcency sce f a gven DMU s btaned by cmputng f that DMU the set f n techncal effcency sces (usng the n sets f ptmal weghts cespndng t the n DMUs), and then aveagng thse sces. Thus, css effcency ges beynd pue self-evaluatn nheent n cnventnal DEA analyss, and cmbnes ths wth the the (n 1) sces asng fm the ptmal pee multples. Ths appach was gnated by Sextn et al. (1986), and was futhe nvestgated by Dyle and Geen (199), and thes. Css effcency pvdes an effcency deng amng all the DMUs t dffeentate between gd and p pefmes. It elmnates the need f ncpatng addtnal weght estctns n multples, theeby avdng ptentally unealstc weghtng schemes (Andesn et al., 00). One can fnd many uses f css effcency f example, R&D pect selectn (Oal et al., 1991), pefeence vtng (Dyle et al., 1996) and thes. As pnted ut by Dyle and Geen (199), the nn-unqueness f the DEA ptmal weghts pssbly educes the usefulness f css effcency. T cmbat ths pblem, thse auths have ppsed vaus secnday gals such as gven by the aggessve and benevlent mdels. Lang et al. (008) futhe mpve n the dea f the css effcency sce, usng game theetc cnstucts. T mplement the dea, ne vews DMUs as playes n a game, and defnes the effcency sce as that asng fm (.1). In a game sense, suppse ne playe DMU d s gven an effcency sce a d, and that anthe playe DMU then tes t maxmze ts wn effcency, gven that a d cannt be deceased. The auths pesent an algthm that cntnually updates the a d, avng at a fnal set f sces that ae, n a cmpettve sense, best f the set f DMUs..8. Least dstance pectns A numbe f auths have examned the pblem f devng the least dstance pectn t the effcent fnte; nte that ths s the ppste cten t that f the addtve mdel that seaches f the geatest dstance. Fe and Hake (1999) ppsed usng the Eucldean nm t defne the clsest pnt. Chanes et al. (199, 1996) and Bec (1999) btan the mnmum cty blck dstance t the weak effcent fnte. Gnzalez and Alvaez (001) mnmze nput cntactns, whle tela et al. (003), Chechye and Van uyenbeck (001), appach ths by dentfyng all the effcent facets. In a ecent pape Apac et al. (007) pesent a set f mdels f btanng least dstance pectns..9. Invaance t data alteatns Ethe ut f necessty cnvenence, the mdele s smetmes called upn t alte tansfm the data t be used n a DEA analyss. F example, t may be me cnvenent f scale pupses t epesent a esuce n thusands f dllas athe than n dllas. If cetan pft fgues can take negatve values t may be desable t tanslate the data by addng a fxed numbe t the value f that vaable f each DMU (theeby endeng all values pstve). An mptant cnsdeatn s whethe nt such alteatns made t gnal data, nfluence the utcmes asng fm the applcatn f the vaus effcency measuement mdels dscussed abve. Ths questn f nvaance has been a subect f mptance n the DEA lteatue, and s dscussed, f example, n Al and Sefd (1990), Thall (1996), ast (1996) and Cpe et al. (006). In the case f a gven fact (e.g., x ), tw specfc fms f data alteatn ae f patcula sgnfcance: Table.1 DEA mdels and the espectve nvaance ppetes (adapted wth pemssn fm Cpe et al. (006), p. 105) Mdel CCR-I CCR-O BCC-I BCC-O ADD SBM Data X Sem-p Sem-p Sem-p Fee Fee Sem-p Y Fee Fee Fee Sem-p Fee Fee Tans. X N N N Yes Yes a N Invaance Y N N Yes N Yes a N Unts nvaance Yes Yes Yes Yes N Yes h * [0,1] [0,1] (0,1] (0,1] N [0,1] Tech. mx Tech. Tech. Tech. Tech. Mx Mx Retuns t scale CRS CRS VRS VRS C(V)RS b C(V)RS a The Addtve mdel s tanslatn nvaant nly when the cnvexty cnstant s added. b C(V)RS means Cnstant Vaable etuns t scale accdng t whethe nt the cnvexty cnstant s ncluded.

7 W.D. Ck, L.M. Sefd / Eupean Junal f Opeatnal Reseach 19 (009) (1) Scalng usng a cmmn multple a,.e., tansfmng x t ^x ¼ ax, () Tanslatn usng a cmmn ceffcent a, tansfmng x t x ¼ a þ x. If a data tansfmatn f the fm (1) s undetaken, and f the utcmes fm a mdel d nt change fm what they wuld have been unde the gnal data, the mdel n questn s sad t pssess the ppety f unts nvaance. If utcmes ae nt affected by tanslatn f the data (#), then the mdel exhbts tanslatn nvaance. Table.1 adapted fm Cpe et al. (006) pvdes a summay f the extent t whch these ppetes hld n the vaus mdels. 3. Multlevel mdels The mdels f the pevus sectn geneally petan t sngle level stuatns n whch we wsh t evaluate the effcency status f each membe f a gven set f decsn makng unts at a gven pnt n tme. A numbe f effcency measuement stuatns can nvlve havng t lk at what mght be egaded as multple levels. Sme f the mdel stuctues f such stuatns ae befly dscussed heen Multstage/seal mdels Netwk DEA Ths bdy f wk gnated by Fae and Gsskpf (1996) s bult aund the cncept f sub-technlges wthn the black bx f DEA. Ths appach allws ne t examne n me detal the nne wkngs f the pductn pcess, ptentally leadng t a geate undestandng f that pcess. Fäe et al pesent thee geneal netwk mdels: (1) A statc netwk mdel n whch a fnte set f subtechnlges actvtes ae cnnected t fm a netwk. Its man featue s that t allws ne t analyze the allcatn f ntemedate pducts. Fg. 6 llustates the cncept. () Dynamc netwk mdel: Ths stuctue pemts ne t examne a sequence f pductn technlges whee a decsn at ne stage (e.g. a tme ped) x 0 1 x 0 x x 0 3 y 1 3 y 3 y 1 y y 3 y mpacts late stages. Hee, ntemedate pducts ae accunted f, meanng that the utputs f ne stage becme nputs t a late stage. (3) Technlgy adptn: Ths mdel allws ne, f example, t examne pductn n dffeent pcesss (e.g. machnes). Inputs ae allcated amng the pcesss t allw ne t detemne whch technlgy t adpt Supply chans Seveal DEA-based appaches have been used t examne buye supple supply chan settngs, leadng t effcency evaluatn. The mptant ssue s that f devng a measue f veall effcency as ppsed t lkng nly at the effcences f ndvdual membes f the chan. Theen, f cuse, les the dffculty n that a ecmmended effcency mpvement f ne membe f the chan can lead t a decease n the effcency f the the membes. Sefd and Zhu (1999c) and Chen and Zhu (00) pvde tw appaches n mdelng tw-stage pcesses. Zhu (003b) pesents a DEA-based supply chan mdel t bth measue the veall effcency, and that f ts membes. Fg. 7 captues the type f stuctue examned by Zhu (003b). A numbe f supply chan appaches due t Lang et al. (006) ae bult n game theetc cnstucts. Lang s tw pncpal mdels ae (1) a nn-cpeatve mdel and () a cpeatve mdel. In the nn-cpeatve mdel, he vews the selle as the leade and buye as the fllwe. In the fst stage, ne ptmzes the leade s effcency sce and then maxmzes (n the secnd stage) that f the fllwe, wth the cnstant that the multples used must be such that the fst stage (leade) sce emans unchanged. The esultng mdel s a nn-lnea paametc pgammng pblem. In the cpeatve game mdel, n leade fllwe assumptn s made. 3.. Multcmpnent/paallel mdels The multlevel settngs f the pevus subsectn ae geneally dected twad what can be temed seal pcesses. Sme stuatns can nvlve multple cmpnents that can be egaded as peatng n paallel. In Ck et al. (000) a study f bank banch pefmance s dscussed but whee n each banch actvtes can be guped unde tw headngs sales actvtes and sevce actvtes. Whle ne culd cncevably cnsde lkng at the tw sets f actvtes as nvlvng sepaate analyses, the cmplcatn that ases s that f dvdng up the shaed nputs such as suppt staff. Ck et al. (000) nstead ppse an aggegate effcency X A Selle Y A X B Buye Y B Fg. 6. A netwk technlgy (adapted wth pemssn fm Fäe and Gsskpf (000)). Fg. 7. Selle buye supply chan (adapted wth pemssn fm Zhu (003b)).

8 8 W.D. Ck, L.M. Sefd / Eupean Junal f Opeatnal Reseach 19 (009) 1 17 mdel that allws ne t evaluate bth the cmpnent level effcences as well as the veall aggegate effcency. Anthe appach t multcmpnent stuatns n ganzatns s gven by tela et al. (007) Heachcal/nested mdels Sme multlevel effcency measuement stuatns can nvlve heachcal nested stuctues. Ck et al. (1998), Ck and Geen (005) examne a set f pwe plants, wheen each plant s made up f a set f ndvdually peatng pwe unts. At ne level, t s necessay t cnsde measung the effcency f each pwe unt elatve t the full set f pwe unts acss all plants; hee, the pwe unts ae the DMUs. At the same tme, at the next level up n the heachy, ne can evaluate the effcency f each plant gup f unts, aganst all the plants. Thee can be nputs and utputs at ne level that ae nt pat f the analyss at anthe level. Ck et al. (1990) gve anthe example f such a stuctue n cnsdeng the effcency f hghway mantenance cews patls. Hee, patls (level 1 DMUs) ae guped unde dstcts (level DMUs), whch ae futhe guped unde the dffeent egns (level 3 DMUs) wthn the pvnce state.. Multple estctns Wthn the DEA lteatue, thee s a bdy f eseach amed at addessng the pblem f unacceptable weghtng schemes. We efe t the cllectn f methdlges hee as nvlvng multple estctns, althugh as dscussed belw, sme f the tls d ths n an ndect athe than dect way (e.g., va cnstaned facet analyss)..1. Abslute multple estctns Sme f the ealest wk hee nvlved mpsng abslute lwe and uppe bunds n nput and utput multples, that s 1 6 l 6 ; Q 1 6 t 6 Q : Rll et al. (1991) examned the use f such abslute lmts n the cntext f evaluatng hghway mantenance unts (see als Ck et al. (1990)). In an eale pape by Dysn and Thanassuls (1988), a smla appach was ppsed. Implementng abslute bunds can pve dffcult, n that apppate levels f the k, Q k ae vey much a functn f the scales used f the vaables. Only afte unnng an unbund mdel wll the ange f pssble multple values be knwn n elatn t the scales adpted... Cne at estctns Chanes et al. (1990), n the study f lage ndustal banks, ecgnzed that undesable weghtng schemes ae a natual utcme n many DEA applcatns. T pvde f me ealstc multples, they ppsed mpsng a set f lnea estctns that defne a cnvex cne. Specfcally, the feasble egn f say the nput multple vect t =(t 1,...,t m ) s defned t be n the plyhedal cnvex cne spanned by a set f k admssble nn-negatve dectn vects (a ), =1,...,k. Thus, a feasble t can be expessed as t ¼ X a a ; wth a 0; 8 : ð3:1þ Let the esultng plyhedal cne be dented by V, the set f all t satsfyng (3.1). Lettng U be a smla cne defnng the set f feasble utput multple vects l =(l 1,...,l s ), the CCR cne at mdel s then gven by max l y s:t: t x ¼ 1 l y ð3:þ t x 6 0; ¼ 1;...; n tev ; leu: Thee s, f cuse, the cespndng dual f ths mdel whch can be fund n Chanes et al. (1990). An mptant genealzatn n (3.) s that gven by Thmpsn et al. (1995), wheen the spaces f l and t ae cnnected by way f lnked cnes. Thse auths apply the geneal cne at mdel t a pblem nvlvng Illns cal mnng..3. Assuance egns A specal case f the cne at dea s what Thmpsn et al. (1986, 1990) temed an assuance egn (AR). The AR cncept was develped t phbt lage dffeences n the values f multples, and mpses cnstants n the elatve magntudes f thse multples. F example, ne mght add a cnstant n the at f multples f a pa f nputs 1 and, n the fm: L 1 6 t =t 1 6 U 1 ; ð3:3þ whee L 1, U 1 ae lwe and uppe bunds, espectvely, n the at t /t 1. Geneally, the mpstn f multple estctns, whethe t s thugh abslute bunds, cne at cnstants, AR cnstants, leads t a wsenng f effcency sces. Refeng agan t Fg., a edawn vesn s shwn n Fg. 0. The slpes f the facets n ths fgue ae elated t the elatve values f t 1 and t. When estctns say f the fm (3.3) ae mpsed n the DEA mdel, cetan slpes may n lnge be admssble. Ths has the affect f bendng the fnte ut (gvng t less cuvatue), as llustated by the dashed lne n Fg. 0. Thus, DMUs that ae effcent n an unestcted settng, e.g. DMU D n Fg., may be endeed neffcent as n Fg. 0. In the ecent bk, Cpe et al. (006), p. 17 pvde a useful altenatve pctal vew f ths dea n the multple space. Many applcatns f the AR fm f the vaus DEA mdels can be fund n the lteatue. These help t

9 W.D. Ck, L.M. Sefd / Eupean Junal f Opeatnal Reseach 19 (009) enlghten the eade n the pactcaltes f devng apppate bunds. In sme cases, these bunds ae pesented by the auth as beng llustatve nly, and ne s left wth the questn as t hw the ght values culd be deved. In the ccumstances, bunds may fall natually ut f the avalable data. In Ck et al. (000), f example, n studyng bank banch effcency, utputs ae vaus classes f banch tansactns (depsts, wthdawals, etc.), and the multples ae tansactn pcessng tmes, n mnutes hus. Whle exact tmes ae nt gven, snce these can vay fm banch t banch, fm ne emplyee t anthe etc., anges wth establshed lwe and uppe lmts ae avalable. It s these anges that lead t cncete lmts f the fm llustated n (3.3). Vaus genealzatns n the AR cncept appea n the lteatue. Allen et al. (1997) and Thanassuls et al. (1998) pesent a glbal type f estctn n the weghted values f each DMU. F example, f nput #1 s t epesent at least 10%, and at mst 0% f the ttal weghted nput f any DMU, then cnstants such as :10 6 t 1 x 1 =ð t x Þ 6 :0 wuld be apppate. Nte that sepaate cnstants f each DMU wuld esult fm ths. Ck and Zhu (008) pesent a cntext-dependent assuance egn DEA (CAR-DEA) mdel whch pvdes f estctns f the fm (3.3) that may dffe fm ne subset f DMUs t anthe. Thus, f thee ae K gups f DMUs, and f we wsh t mpse estctns n say utputs f the fm c k L 6 l 1=l 6 c k U ; k ¼ 1;...; K; t s necessay t accunt f ptental ncnsstency (nfeasblty) f all K sets ae mpsed smultaneusly. If we assume that utput #1 s used as the numeae aganst whch the utputs ae cmpaed, the appach taken by Ck and Zhu (007) s t eplace the set f K gups f AR estctns by a sngle set f AR cnstants, applcable t all K classes f DMUs. As an example, the methd f Geen et al. (1996) s a thee-stage pcedue. In stage 1, the standad CCR mdel s slved t detemne the full set f effcent unts E [ E 0. Nte that E s cmpsed f exteme effcent DMUs (cnes f facets) whle E 0 ae the nn-exteme (nte t facets) effcent unts. Stage then fllws Chanes et al. (1986) t pattn ths full set f unts nt the tw subsets. Fnally, n the thd stage ne slves the mxed ntege pgammng pblem, max e ¼ l y s:t: l y e t x e 6 0; eee; l y e t x e þ Mz e 0; eee; t x ¼ 1; ð:1þ z e ¼ E ðmþs 1Þ; eee l 0 ¼ 1;...; s; t 0 ¼ 1;...; m; z e ef0; 1geeE; M 0: Ths mdel guaantees that exactly m + s 1 f the m + s cnstants wll be satsfed as equaltes. Ths means that at mst ne slack vaable wll be pstve, hence all l, t vaables wll be fced t be stctly pstve, meanng that the DMU n questn s pected aganst a full-dmensnal facet. The methd s llustated by Fg Geneatng unbseved DMUs It s nted that n extendng facets t elmnate weakly effcent pectns, new unbseved DMUs wll be geneated. In Fg. 8, pnts whee the ays fm the gn t mppely envelped DMUs ( and 7 ) ntesect the extended facet, defne such DMUs. Thanassuls and Allen (1998) pesent a fmal pcedue f pducng new unbseved DMUs, thus ceatng the means f extendng bseved facets. The appach amunts t btanng.. Facet mdels Sgnfcant wk has been dne elatng t facet extensn and facet dentfcatn, t addess the nheent pblem nvlvng the ccuence f ze weghts ( e-weghts) n the multple mdels, as ndcated abve. Ths s equvalent t pectn t weakly effcent facets nn-fulldmensnal facets. Bessent et al. (1988) wee the fst t ntduce the dea f cnstaned facet analyss (CFA). In the event that a gven unt s pected t a weakly effcent facet, CFA nvlves extendng a selected aet-effcent (full-dmensnal) facet, and then pectng the gven DMU n t that extended facet. Lang et al. (1995) mpved n ths dea by adptng a tw-stage appach whch ultmately amunts t fndng the clsest fulldmensnal facet t whch t pect the DMU n questn. Othe smla appaches have been suggested by Geen et al. (1996), and by Olesen and etesen (1996). Fg. 8. Facet extensn n DEA (adapted wth pemssn fm Geen et al. (1996)).

10 10 W.D. Ck, L.M. Sefd / Eupean Junal f Opeatnal Reseach 19 (009) 1 17 nfmatn fm the decsn make as t hs/he estmates f the tadeff between pas f facts. Anthe lne f eseach n a smla dectn s due t Glany and Rll (199) wh ntduced the dea f ncpatng standad DMUs nt the DEA stuctue. Ck and Zhu (005) extended ths wk by way f ncpatng pductn standads (as ppsed t standad DMUs). 5. Specal cnsdeatns egadng the status f vaables As gnally cnceved, the DEA mdel nvlves the geneatn f s utputs fy g s ¼1 usng m nputs fx g m ¼1.In a stuctue such as that n (.3) all nputs ae pected adally t the effcent fnte, all vaables ae assumed t be quanttatve, and the cllectn f DMUs unde evaluatn s assumed t fm a elatvely hmgeneus gup (all ae cmpaable t ne anthe). As new applcatns f DEA have asen, t has becme necessay t expand the gnal mdel stuctues t accmmdate new stuatns, hence elaxng a numbe f thse gnal assumptns. In ths sectn we dscuss sme f the me mptant develpments n ths egad Nn-dscetnay vaables In many applcatns f DEA, cetan f the nput vaables may nt be unde the dect cntl f management. In a DEA analyss f bank banch effcency, f example, an nput vaable such as fxed expendtues (ent, utltes, etc.) culd nt be pptnally educed as wuld be the case f vaable expendtues such as staff. Thus, t s mptant t dentfy thse vaables that ae dscetnay (staff) vesus nn-dscetnay (fxed csts). Banke and Mey (1986a) ntduced the fst DEA mdel that allwed f nn-dscetnay nputs by mdfyng the nput cnstants t dsallw nput eductn n the fxed fact. Lettng D dente the subset f nputs e{1,,..., m} that ae dscetnay, and ND, the nn-dscetnay nputs, the Banke and Mey mdel becmes mn h e s þ s þ D s:t: k x þ s ¼ h x ; D k x þ s ¼ x ; ND ð:1þ k y s þ ¼ y ; ¼ 1;...; s k ; s ; s þ 0; 8; ; ; h 0 unestcted: The cespndng dual pblem s max l y t x ND s:t: l y t x t x 6 0; ¼ 1;...n D ND l e 8; t e D; t 0; ND: ð:þ It s nted that f nn-dscetnay ND, t 0 athe than t e. Cespndngly, n (.1) t s nly thse nput slacks elated t dscetnay facts that appea n the bectve functn. See Cpe et al. (006), pp f a full dscussn f ths. Me ecently Rugge (1996) has pnted ut that n cetan cases, the Banke and Mey mdel can ve estmate techncal effcency by allwng pductn mpssbltes nt the efeent set. Rugge s appach estcts weghts t ze f pductn pssbltes wth hghe levels f the nn-dscetnay nputs, and as a esult, pductn mpssbltes ae apppately excluded fm the efeent set. Recent smulatn analyses n Syanen (00) and Muñz et al. (006) demnstate that Rugge s methd pefms elatvely well n evaluatng effcency. See als Rugge (1998, 007). 5.. Nn-cntllable vaables In the nn-dscetnay vaable mdel (.1) we nte that the slacks s ; ND ae pemtted t be stctly pstve. Ths means that at the ptmum, a DMU may end up beng cmpaed t a lnea cmbnatn f pees wheen the value f a nn-dscetnay vaable f that cmbnatn s less than ts value n DMU. In cetan stuatns thee can be nputs (and utputs) whse values must eman fxed, and can nly be cmpaed t DMUs whse lnea cmbnatns ae at the same levels as thse fxed facts. Such vaables have been labeled as nn-cntllable. Let N 1, N dente nn-cntllable nputs and utputs espectvely, and N 1 ; N dente egula (cntllable) nputs and utputs, espectvely. The nn-cntllable vaable mdel s then: mn h e þ! s:t: N 1 s N s þ k x þ s ¼ h x ; N 1 k x ¼ x ; N 1 k y s ¼ y ; N k y ¼ y ; N k 0; 8 ; s 0; N 1 ; s þ 0; N : 5.3. Categcal vaables (categcal DMUs) ð:3þ Thee ae stuatns n whch DMUs fall nt natual categes. An example wuld be when we ae evaluatng a set f etal establshments wheen dffeent levels f cmpettn exst fm ne establshment t anthe. T pvde a fa evaluatn f each DMU, t can be agued that a DMU n any gven categy shuld be cmpaed nly t thse the unts n the same less-advantaged

11 W.D. Ck, L.M. Sefd / Eupean Junal f Opeatnal Reseach 19 (009) categes. A DMU unde heavy cmpettn wuld be unfaly penalzed f cmpaed t unts n sgnfcantly me favable cmpettve envnments. Banke and Mey (1986b) pesented the fst mdel t deal wth such stuatns. The mdel saw the ntductn f categcal nputs x, = m 0 +1,...,m n addtn t egula cntllable nputs x, =1,...,m 0. In the case whee a categcal vaable x s nt cntllable by management, the teatment nvlves eplacng that vaable by a set f bnay vaables d ðkþ ; k ¼ 1;...; K, wth K beng the numbe f categes. Aangng the categes n deceasng de f favablty ne sets d ðkþ ¼ 1; k 6 k, and d ðkþ ¼ 0; k > k, f DMU s n categy k. The usual nn-dscetnay nput cnstant k x þ s ¼ x s then eplaced by a set f K cnstants k d ðkþ 6 d ðkþ. In ths way, a DMU s cmpaed nly t the DMUs n the same less favable categes. In the case f a cntllable categcal vaable, Banke and Mey (1986b) pesented a mxed ntege L fmulatn. Hweve, as pnted ut by Kamakua (1988), the Banke and Mey mdel was flawed due t a ms-specfed cnstant, and a evsed mdel was gven. The Kamakua mdel pesented ts wn dffcultes, hweve, and these wee addessed n a late pape by Russeau and Semple (1993). The latte auths dealt wth bth nput and utput categcal vaables, and wee able t educe the ntege pblem t a me cnventnal L appach. 5.. Odnal vaables/data DEA analyses ae geneally based n a set f quanttatve utput and nput facts. In cetan settngs, hweve, qualtatve vaables may be pesent. F a fact such as management cmpetence, f example, ne may be able t pvde nly a ankng f the DMUs fm best t wst. The capablty f pvdng a me pecse, quanttatve measue eflectng such a fact s ften nt feasble. In sme stuatns such facts can be quantfed, but ften such quantfcatn s supefcally fced as a mdelng cnvenence. The gnal DEA mdels ncpatng ank de vaables ae due t Ck et al. (1993, 1996). T captue such ank de vaables wthn the DEA stuctue, the auths pceed as fllws. F such an utput, f example, assume a DMU k can be assgned t ne f L ank pstns (L 6 n). One can vew the assgnment f DMU k t pstn d n dnal utput, as havng assgned that k an utput value wth y (d). Me ecently, Cpe et al. (1999b) examned the DEA stuctue n the pesence f what they temed mpecse data (IDEA). Zhu (003a) and thes have extended the Cpe et al. (1999a) mdel. Whle vaus fms f mpecsn ae lked at unde the umbella f IDEA, the pncpal fcus s n ank de data. In a ecent pape by Ck and Zhu (006), ank de vaables and IDEA ae evsted, and bth dscete and cntnuus pectn mdels ae dscussed. It s shwn that the IDEA appach f ank data s equvalent t the Ck et al. (1993, 1996) methdlgy Mdellng undesable facts The usual vaables n DEA ae such that me s bette f utputs, and less s bette f nputs. In sme stuatns, hweve, a fact can behave ppste t ths; cnsde, f example, a pllutn as ne f the utputs fm pwe plants. A numbe f auths have addessed ths ssue, n patcula, Scheel (001), Sefd and Zhu (00), Fae and Gsskpf (00) and Hua and Bn (007). Appaches ange fm lnea tansfmatns f the gnal data t the use f dectnal dstance functns Flexble measues Classfyng nputs and utputs In the standad applcatns f DEA t s assumed that the nput vesus utput status f each pefmance measue elated t the DMUs s knwn. In sme stuatns, hweve, the le f a vaable may be flexble. Cnsde the example f measung pwe plant effcency as dscussed n Ck et al. (1998) and Ck and Geen (005), whee ne f the utputs s a functn f what s temed utages. Ths measue s desgned t epesent the pecentage f tme that a plant s avalable t be n peatn, and can, theefe, be vewed as a type f accmplshment (utput) n the pat f management. At the same tme, t s easnable t vew ths vaable as an envnmental nput that has a dect nfluence n plant pefmance. The ncpatn f such flexble vaables nt the DEA stuctue pesent a pblem n that thee s a need t make allwance f them n bth the nput and utput sdes f the mdel. Beasley (1995) dealt wth a smla pblem, and pesented a fmulatn f a stuatn whee the vaable eseach fundng was cunted as bth an nput and an utput n evaluatng unvestes n the UK. Ck et al. (006) late shwed that Beasley s mdel was flawed and gave an altenatve, cected vesn. The flexble vaable pblem s nt ne f cuntng the nfluence n bth places, but athe cuntng t n the mst apppate place. Suppse thee exst L flexble measues, whse nput/ utput status we wsh t detemne. Dente the values assumed by these measues as w f DMU ( = 1,..,L). F each measue, ntduce the bnay vaables d {0,1}, whee d = 1 desgnates that fact s an utput, and d = 0 desgnates t as an nput. Let c be the weght f each measue. One appach, pesented by Ck and Zhu (007), t decdng n the apppate status f a flexble vaable, s t vew the pblem fm the pespectve f the ndvdual DMU. Specfcally, adpt the pstn that f any gven DMU the status shuld be that whch maxmzes that DMU s effcency sce. Ck and Zhu (007) establsh the fllwng mathematcal pgammng mdel (.). Hee each DMU s allwed t select the status f each vaable that wll cedt t wth the hghest pssble sce. The auths then suggest takng

12 1 W.D. Ck, L.M. Sefd / Eupean Junal f Opeatnal Reseach 19 (009) 1 17 a maty ule pstn, gvng each vaable the status (nput utput) pefeed by the maty f the DMUs. An altenate mdel s gven that vews the pblem fm the pespectve f the aggegate cmpste f all the DMUs. The status f the flexble vaable s that whch maxmzes the effcency sce f that cmpste DMU max l y þ d c w mxþ ð1 d Þc w s:t: l y þ d c w ð:þ 6 1 ¼ 1; ;...; n mxþ ð1 d Þc w d f0; 1g; 8 l ; m ; c 0; 8; ; : 6. Data vaatn In the methdlgy develpments dscussed abve, t s assumed that the data values ae fxed and knwn. Sgnfcant lteatue has, hweve, been dedcated t stuatns wheen the data may exhbt vaatn uncetanty ; we befly dscuss vaus lnes f eseach elatng t such stuatns Senstvty analyss Ths bdy f wk addesses the questn f cetan paametes/data wthn a mdel such as (.1) and (.), (.3) ae alteed, hw des ths nfluence the effcency status f DMUs? Seveal dectns have been taken hee blem sze ssues Vaus studes have examned the senstvty f DMU effcency t the addtn f DMUs t extactn f DMUs fm the analyss. See Wlsn (1995). A numbe f smulatn studes (e.g. Banke et al., 1996) have elated t the mpact n the effcency geneated, f vayng numbes f DMUs and f nputs and utputs Dect data petubatns Chanes and Nealc (199) and Nealc (1997, 00) addessed the subect f the mpact n effcency f petubatns t the values f nputs and utputs. We efe t ths as nvlvng dect data petubatn. That s, the eseach petans t the devatn f anges f vaatn n data ve whch matx nvesn n the smplex algthm s unaffected Indect data petubatn Radus f stablty An altenatve t the abve dect petubatn appaches, ae what we shuld call ndect appaches. A numbe f studes have fcused n the questn f a maxmum adus that wll mantan the effcency status. That s, f a gven DMU, what s the maxmum allwable ncease n utputs decease n nputs such that ts effcency status (effcent neffcent) s unalteed? The fst wk n ths dectn was ntated by Chanes et al. (199). It examned the fllwng pblem elatng t the addtve mdel, and nvlvng an neffcent DMU: max d s:t: : k x þ s ¼ x dd ; ¼ 1;...; m k y s þ ¼ y þ dd þ ; ¼ 1;...; s k ¼ 1; ð5:1þ d ; d þ ¼ 10; dependng n whethe a dmensn s t be ncluded excluded fm the petubatn d. Ths mdel lks at the maxmum mpvement n the status f an neffcent DMU (ncease n utputs/decease n nputs) befe t s endeed effcent. F the case f an effcent DMU, Chanes et al examne the mdel: mn d s:t: : k x þ s ¼ x dd ; ¼ 1;...; m 6¼ k y s þ ¼ y þ dd þ ; ¼ 1;...; s ð5:þ 6¼ k ¼ 1: 6¼ Hee, we emve that DMU unde evaluatn, fm the cnvexty cnsdeatns, much n the spt f supe-effcency (Andesn and etesen 1993). These pblems have been evsted by vaus auths ncludng Cpe et al. (001), Sefd and Zhu (1998a, 1999a), and thes Supe-effcency An mptant pblem n the DEA lteatue s that f ankng thse DMUs deemed effcent by the DEA mdel, all f whch have a sce f unty. One appach t the ankng pblem s that pvded by the supe effcency mdel f Andesen and etesen (1993), as mentned abve. See als Banke et al. (1989). The supe-effcency mdel nvlves executng the standad DEA mdels (CRS VRS), but unde the assumptn that the DMU beng evaluated s excluded fm the efeence set. In the nput-ented case, the mdel pvdes a measue f the pptnal ncease n the nputs f a DMU that culd take place wthut destyng the effcent status f that DMU elatve t the fnte ceated by the emanng DMUs. The supe-effcency sce can als be thught f as a measue f stablty. That s, f nput data f nstance, s subect t e change ve tme, the supe-effcency sce pvdes a means f evaluatng the extent t whch such changes culd ccu wthut vlatng that DMU s status as an effcent unt. Hence, the sce yelds a measue f stablty. In addtn t beng a tl f ankng, the supe-effcency cncept has been used n the stuatns; f example, tw-pesn at effcency games (Russeau and Semple, 1995), and acceptance decsn ules (Sefd and Zhu, 1998b), amng thes.

13 W.D. Ck, L.M. Sefd / Eupean Junal f Opeatnal Reseach 19 (009) It s well knwn that unde cetan cndtns, the supeeffcency DEA mdel may nt have feasble slutns f effcent DMUs (see, e.g., Zhu, 1996; Sefd and Zhu, 1998a,b; Dulá and Hckman, 1997; Sefd and Zhu, 1999a,b). As shwn n Sefd and Zhu (1999a,b), nfeasblty must ccu n the case f the vaable etuns t scale (VRS) supe-effcency mdel. Althugh nfeasblty mples a fm f stablty n DEA senstvty analyss (Sefd and Zhu, 1998b), lmted effts have been made t pvde numecal supe-effcency sces f thse effcent DMUs f whch feasble slutns ae unavalable n the VRS supe-effcency mdel. Lvell and Ruse (003) develped a standad DEA appach t the supe-effcency mdel by scalng up the nputs (scalng dwn the utputs) f a DMU unde evaluatn. As a esult, a feasble slutn can be fund f effcent DMUs that d nt have such (feasble) slutns n the standad VRS supe-effcency mdel. The supe-effcency sces f all effcent DMUs wthut feasble slutns ae then equal t the use-defned scalng fact. Chen (00, 005) suggests usng bth the nput- and utput-ented VRS supe-effcency mdels t quantfy the supe-effcency when nfeasblty ccus. Hweve, Chen s appach wll fal f bth the nput- and utput-ented VRS supe-effcency mdels ae nfeasble. Recently, Ck et al. (008) psed an altenatve appach t eslvng the ssue f nfeasblty. Unlke the standad nput-ented and utput-ented supe-effcency mdels, each f whch has a specfc entatn (nput utput), ths mdel pvdes f the mnmum mvement n bth dectns needed t each the fnte geneated by the emanng DMUs. Vewed anthe way, n the case f nfeasblty, the mdel deves the mnmum change needed t pect a data pnt, classfed as an extemty, t a nn-exteme pstn. 6.. Data uncetanty and pbablty-based mdels A numbe f eseaches have cncentated n the pblem f mdelng techncal effcency when the data f the nputs and utputs ae andm vaables. The (1987) and Land et al. (199, 199) lked at the applcatn f chance cnstaned pgammng (CC) t DEA. Cpe et al. (1996, 00) demnstated the use f CC n the fm f a satsfcng mdel. Specfcally, the CCR mdel (.1) s eplaced by the CC mdel! u ~y max b s:t: v ~x u ~y v ~x u ; v 0 8; :! 6 b 1 a ; ¼ 1;...; n ð5:3þ Hee the andm vaables ~y ; ~x ae assumed t have knwn pbablty dstbutns, and a s a scala n the unt ange [0, 1] that specfes the allwable lkelhd f falng t meet the cnstants. The value b s efeed t as an aspatn level, and specfes the desed effcency level f DMU Tme sees data Wndw analyss Tme sees data epesent an mptant fmat n whch data vaablty ccus. Specfcally, n many applcatns, data f a DMU ae avalable at dffeent pnts n tme, f example, n each f a set f quates ve seveal yeas. Whle ne can pefm statc DEA analyses n the data f each quate, and then apply standad egessn cncepts t study effcency changes, such an appach ften pves athe unsatsfacty, geneally falng t captue mptant nteactns fm ped t ped. Wndw analyss as ntduced by Chanes et al. (1985a) s a mdel stuctue that tes t bng a me bust teatment t effcency changes n a tme sees sense. The dea s t chse a wndw f k bsevatns f each DMU (say k = quates), and teat these as f they epesented k dffeent DMUs. Hence, n any analyss, a ttal f n k DMUs ae evaluated; kdffeent sces f each DMU ae then ceated. One then mves the wndw by ne ped (e.g. nstead f quates Q 1 t Q, ne uses Q t Q 5 ) and epeats the analyss. Wndw analyss allws the analyst t bseve bth the stablty f a DMU f any pnt n tme acss dffeent data sets, as well as tends acss the k bsevatns f a DMU, wthn the same data set. Cpe et al. (006) dscuss a numbe f weaknesses n cnventnal wndw analyss, ne f whch s the fact that begnnng and endng peds ae nt tested as fequently as s the case f the peds. Sueysh (199) has attempted t emedy ths stuatn by the use f a und bn appach. Ths pceeds by fst lkng at each DMU n ne ped, then tw,..., etc., up t k peds. Ths gves a me cmplete pctue f stablty and tends, but wth the dsadvantage f becmng cmputatnally budensme as the numbe f cmbnatns gws expnentally. 6.. Tme sees data The Malmqust ndex The Malmqust ndex was fst suggested by Malmqust (1953) as a quantty f use n the analyss f cnsumptn f nputs. Fäe et al. (199) develped a DEA-based Malmqust pductvty ndex whch measues pductvty change ve tme. The ndex can be decmpsed nt tw cmpnents, wth ne measung the change n the technlgy fnte and the the the change n techncal effcency. T descbe the methd, let x t ; yt dente the nput and utput levels f a DMU at any gven pnt n tme t. The Malmqust ndex calculatn eques tw sngle ped and tw mxed ped measues. The tw sngle ped measues can be btaned by usng the CRS DEA mdel. Thus, f ped t we slve fllwng CRS DEA mdel whch calculates the effcency n tme ped t, as dsplayed n (5.)

14 1 W.D. Ck, L.M. Sefd / Eupean Junal f Opeatnal Reseach 19 (009) 1 17 h t ðxt ; yt Þ¼mn h s:t: k x t 6 h x t ; k y t yt ; k 0; ¼ 1;...; n ð5:þ In a smla way, usng t + 1 nstead f t f the abve mdel, we get h tþ1 ðx tþ1 ; y tþ1 Þ the techncal effcency sce f DMU n tme ped t + 1. The fst f the mxed ped measues, whch s defned as h t ðxtþ1 ; y tþ1 Þ f each DMU, s cmputed as the ptmal value t the fllwng lnea pgammng pblem: h t ðxtþ1 ; y tþ1 Þ¼mn h ; s:t: k x t 6 h x tþ1 ; k y t ð5:5þ ytþ1 ; k 0; ¼ 1;...; n: Ths mdel cmpaes x tþ1 t the fnte at tme t. In a smla way, we can btan the the mxed ped measue, h tþ1 ðx t ; yt Þ whch cmpaes xt t the fnte at tme t +1. The (nput-ented) Malmqust pductvty ndex can be expessed as " h t M ¼ ðxt ; yt Þ h tþ1 #1 ðx t ; yt Þ : h t ðxtþ1 ; y tþ1 Þ h tþ1 ðx tþ1 ; y tþ1 Þ M measues the pductvty change between peds t and t + 1. ductvty declnes f M > 1, emans unchanged f M = 1 and mpves f M < 1. The fllwng mdfcatn f M makes t pssble t measue the change f techncal effcency, and the mvement f the fnte n tems f a specfc DMU. M ¼ ht ðxt ; yt Þ tþ1 h tþ1 ðx tþ1 ; y tþ1 Þ h ðx tþ1 ; y tþ1 Þ h tþ1 ðx t ; yt Þ 1 h t ðxtþ1 ; y tþ1 Þ h t ðxt ; yt Þ : The fst tem n the ght hand sde measues the magntude f techncal effcency change between peds t and t+1. TEC ¼ ht ðxt ;yt Þ h tþ1 ðx tþ1 ;y tþ1 Þ ¼< 1 ndcatng that techncal effcency mpves, emans the same, declnes. The secnd > h tem FS ¼ htþ1 ðx tþ1 ;y tþ1 Þ h tþ1 ðx t ;yt Þ 1 measues the shft n the h t ðxtþ1 ;y tþ1 Þ h t ðxt ;y t Þ fnte between peds t and t + 1. A value f FS geate than unty ndcates egess n the fnte technlgy, a value f FS less than unty ndcates pgess n the fnte technlgy, and a value f FS equal t unty ndcates n shft n the fnte technlgy Stchastc data-statstcal nfeence Anthe appach t teatng data vaatns nvlves the chaactezatn f the pductn functn by way f classcal statstcal nfeence methdlgy. Tw lnes f eseach have emeged aund ths ssue; stchastc fnte analyss, and a DEA appach. Stchastc fnte egessn dates back t Faell (1957); ths was subsequently extended by Agne and Chu (1968) wth the cected dnay least squaes mdel. Late ths appach was pesented n a me fmalzed statstcal fmat by Agne et al. (1977), and has been labeled the cmpsed e appach. Me ecently Banke and Mandatta (199), Banke (1993) and Banke and Nataasan (00) appached ths ssue fm a DEA pespectve. They shw that DEA pvdes a cnsstent estmat f abtay mntne and cncave pductn functns when the (ne-sded) devatns fm such a pductn functn ae egaded as stchastc vaatns n techncal neffcency. Cnvegence s slw, hweve, snce, as s shwn by Kstlev et al. (1995), the DEA lkelhd estmat n the sngle utput m nput case cnveges at the ate n /(1+m) and n the estmat can cnvege at a faste ate. The abve appaches teat nly the sngle utput multple nput case. Sma and Wlsn (1998) tun t btstap methds whch enable them t deal wth the case f multple utputs and nputs. In ths manne, the senstvty f h *, the effcency sce btaned fm the BCC mdel, can be tested by epeatedly samplng fm the gnal samples. A samplng dstbutn f h * values s then btaned fm whch cnfdence ntevals may be deved and statstcal tests f sgnfcance develped. In sme espects, the utput-ented VRS mdel s a nn-paametc vesn f the dnay least squaes mdel f Agne and Chu (1968). Ths was alluded t n Banke and Mandatta (199) and Kusmanen (006). F a thugh cveage f stchastc fnte analyss, and the appaches t effcency evaluatn, see Cell et al. (1998), Kumbhaka and Lvell (000). 7. Cnclusns Ths pape has attempted t pvde a bef sketch f sme f the mptant aeas f eseach n DEA that have emeged ve the past thee decades. The fcus hee s n thse tpcs that, n the auths estmatn, have attacted the mst attentn. At the same tme t s acknwledged that, due t lmted space, and pssbly t gnance n u pat, many mptant wks n DEA may nt have been hghlghted. Sme f these tpcs nclude the mdelng f ntege vaables, ssues f cngestn, handlng mssng data, allcatn f fxed nputs acss DMUs, esuce cnstaned DEA, analyss f cmpste DMUs, dectnal devatves, and the cnnectns elatng t DEA and geneal multple ctea decsn mdels. In a numbe f these stuatns the tpc has eceved sme, but nt sgnfcant attentn, and may be a dectn f futue wk. Acknwledgment The auths wsh t thank fess Rbet Dysn, Edt, Eupean Junal f Opeatnal Reseach, f