On the Estimation of Directional Returns to Scale via DEA models

Transcription

1 On the Etaton of Dectonal Retun to Scale va DEA odel Guolang ang 1, Wenbn Lu 2, Wanfang Shen 3 and Xaoxuan L 1 1 Inttute of Polcy and Manageent, Chnee Acadey of Scence, Beng 119, Chna (Eal: glyang@cap.ac.cn) 2 Kent Bune School, Cantebuy C2 7PE, UK 3 School of Matheatc and Quanttatve Econoc, Shandong Unvety of Fnance and Econoc, Jnan 2414, Chna Abtact. Data envelopent analy (DEA) one of the ot coonly ued ethod to etate the etun to cale (RS) of the publc ecto (e.g., eeach nttuton). Extng tude ae all baed on the tadtonal defnton of RS n econoc and aue that ultple nput and output change n the ae popoton, whch the tatng pont to detene the qualtatve and quanttatve featue of RS of decon akng unt (DMU). Howeve, fo oe coplex poduct, uch a the centfc eeach n nttute, change of vaou type of nput o output ae often not n popoton. heefoe, the extng defnton of RS n the faewok of DEA ethod ay not eet the need to etate the RS of eeach nttuton wth ultple nput and output. h pape popoe a defnton of dectonal RS n the DEA faewok and etate the dectonal RS of eeach nttuton ung DEA odel. Futhe n-depth analy conducted fo an llutatve exaple of 16 bac eeach nttute n Chnee Acadey of Scence (CAS) n 21. Keywod: Data envelopent analy; Retun to cale; Dectonal etun to cale 1 Intoducton In he Quately Jounal of Econoc, Panza and Wllg (1977) popoed a ethod to detene the etun to cale (RS) of decon-akng unt (DMU) baed on the poducton functon. he etaton of RS of DMU ung the data envelopent analy (DEA) ethod wa nvetgated ft by Banke (1984) and Banke et al. (1984). Banke (1984) ntoduced the defnton of the RS fo clacal econoc nto the faewok of the DEA ethod, and he ued the CCR-DEA odel wth adal eaue to etate the RS of evaluated DMU. Soon afte that, Banke et al. (1984) popoed the BCC-DEA odel unde the aupton of vaable RS and nvetgated how to apply the BCC-DEA odel to etate the 1

2 RS of DMU. hu fa, n addton to the cot-baed eaueent of RS of DMU (e.g., Fäe and Gokopf, 1985; Fäe et al., 1994; Setz, 197; Sueyoh, 1999), DEA-baed tude of DMU RS can be oughly dvded nto fou categoe: (1) RS eaueent ung CCR-DEA odel, (2) RS eaueent ung BCC-DEA odel, (3) RS eaueent ung FGL-DEA odel and quanttatve eaueent of cale elatcty (SE) and (4) RS eaueent ung non-adal DEA odel. Pleae ee the lteatue evew n Secton 2 fo detal. he extng tude on the RS eaueent n DEA odel ae all baed on the defnton of RS n the DEA faewok ade by Banke (1984), whch extended the applcaton aea of DEA fo elatve effcency evaluaton to RS eaueent. he RS a clac econoc concept decbng the elatonhp between change n the cale of poducton and output. he tadtonal defnton of RS n econoc baed on the dea of eaung adal change n output caued by all nput. Fo exaple, the SE (ght-hand) of 1.5 tell u that the nceae of all nput by, ay, 1% coepond to the nceae of the output by 1.5%. Followng th concept, Banke (1984) defned RS n the DEA faewok ung the adal change n output caued by all nput. In tadtonal ndutal poducton, the popoton of labou and captal nput ae often fxed, o ung a adal dea to defne RS n econoc pactcal. Howeve, n eeach oganaton, t often can be obeved that poducton facto ae not necealy ted togethe popotonally becaue of the coplexty of eeach actvte and nput change non-popotonally, a llutated by the followng Exaple 1. Exaple 1: In the peod of , both the S& nput and output nceaed gnfcantly n the Chnee Acadey of Scence (CAS). We elected taff and fundng a nput ndcato and ntenatonal pape a one of the output. he full-te equvalent (FE) of R&D peonnel at the CAS gew fo 3,611 n 1998 to 44,37 n 27. he total fundng fo the CAS gew fo 4, llon RMB n 1998 to 17,39.71 llon RMB n 27. he nube of ntenatonal pape gew fo 5,478 n 1998 to 24,45 n 27. he popoton of annual change of the two nput ndcato ae vey dffeent a hown n able 1. able 1: he change of oe ndcato fo the CAS fo 1998 to 27 ea Input ndcato Output ndcato 2

3 otal fundng R&D peonnel Intenatonal pape otal aount (llon RMB) Gowth popoton FE Gowth popoton Nube Gowth popoton , % 3, % 5, % , % 28, % 8, % 2 7, % 28, % 9, % 21 8, % 25, % 1, % 22 1, % 27, % 11, % 23 9, % 3, % 14, % 24 12, % 34, % 15, % 25 12, % 37, % 22, % 26 14, % 38, % 23, % 27 17,39.71 N/A 44,37 N/A 24,45 N/A Data ouce: Stattcal eabook of Chnee Acadey of Scence, Fo able 1, we can ee that the nput do not change popotonally. In fact, n adal eaueent, the nput dd not even nceae fo oe yea. Howeve, t alo clea that the output have geatly nceaed dung the peod. he tadtonal defnton of RS baed on adal eaue tay condeably fo the ealty of the nput change. heefoe, we need to conde dectonal etun to cale (dectonal RS) wth non-popotonal change n nput (o output), a een n Fg 1-2 below. In uch a cae of non-popotonal change, fo exaple, the dectonal ght-hand SE (See defnton on dectonal SE and dectonal RS n Secton 3) of 1.5 tell u that the nceae of all nput by 1% n cetan nput decton coepond to the nceae of the output by 1.5% n cetan output decton. It hould be noted that the dectonal RS tll baed on the Paeto pefeence. Radal decton Radal decton Non-adal decton Fgue 1: adtonal RS Fgue 2: Dectonal RS h pape a to nvetgate the RS eaueent of DMU on the effcent fonte of poducton poblty et (PPS) n the cae of nput and output changng n unequal popoton, whch eentally dffeent wth the RS eaueent ung 3

4 non-adal odel. he et of th pape oganed a follow. Secton 2 povde the clacal RS n the DEA faewok. Secton 3 popoe the defnton of dectonal RS and dectonal SE. wo appoache, the fnte dffeence ethod (FDM) and uppe and lowe bound ethod (ULBM), ae popoed n Secton 4 to detene the dectonal RS of effcent DMU on the effcent fonte. Secton 5 povde an llutatve exaple of the n-depth analy of 16 bac CAS eeach nttute n 21. he lat ecton offe the concluon. 2 Clacal RS n DEA faewok We conde a et of n obevaton of actual poducton poblte X,, 1,..., n. he output vecto can be poduced fo the nput vecto Ft, we povde the followng defnton: follow. Defnton 1: he PPS unde the aupton of vaable RS defned a n n n PPS X, X X,, 1,, 1,..., n (1) Defnton 2: he weakly and tongly effcent fonte of PPS can be defned a follow. (1) Weakly effcent fonte: weak, thee no, uch that,, EF X PPS X PPS X X (2) (2) Stongly effcent fonte:, thee no, uch that,, and,, EF X PPS X PPS X X X X (3) tong Defnton 3: A uppotng hypeplane of PPS can be defned a follow. If a hypeplane atfe (1) H V, U, u X, U V X (4) X, X, U V X PPS ; fo all X, (2) U V X (3) VU,, whee,,..., and,,..., U u1 u2 u paaete fee of gn. V v1 v2 v PPS ; X. ae vecto of ultple, a 4

5 pont hen, we ay that,, X,, whch efeed a,, X, H V U a uppotng hypeplane of PPS on the H V U. Defnton 4: A ubet of PPS efeed to a a Face on the pont X f thee ext a uppotng hypeplane,, X, to an nteecton between PPS and,, X,, H V U uch that the ubet dentcal H V U. Banke (1984) defned the clacal RS baed on PPS n DEA faewok. A entoned n the ft ecton, n addton to the cot-baed eaueent of RS of DMU, DEA-baed tude of DMU RS can be oughly dvded nto fou categoe: (1) RS eaueent ung CCR-DEA odel. he ft and well-known n appoach to detene the RS of DMU to calculate the value of 1 n CCR-DEA odel, whee denote the weght of DMU (Note: In oe cae ay be not unque). Reeach effot elated to th appoach can be found n Banke (1984), Chang and Guh (1991), Banke and hall (1992), Zhu and Shen (1995), Banke et al. (1996a, 1996b), Sefod and Zhu (1998, 1999), aong othe. In th appoach, the type of RS can be detened a nceang RS, contant RS o deceang RS, but the agntude of RS cannot be detened. (2) RS eaueent ung BCC-DEA odel. Banke et al. (1984) popoed the ethod to exane the ntecept of the uppotng hypeplane on the poducton poblty et (PPS) unde a vaable RS aupton. h ntecept coepond to a dual vaable egadng the convex contant n BCC-DEA. he type of RS can be detened by the gn of th ntecept (potve, negatve o zeo). Note that the ntecept nteval need to be condeed when the ntecept not unque. In addton, we can analye the popete of DMU wthn a all neghbouhood to detene the RS on th pont. h type of eeach effot nclude Banke and hall (1992), one (1996), Golany and u (1997), Sueyoh (1999), Coope et al. (2), one and Sahoo (23). he an contbuton of th type of eeach le n povdng a theoetcal ba fo not only the type of RS but alo the qualtatve eaueent of RS. (3) RS eaueent ung FGL-DEA odel and quanttatve eaueent of cale elatcty (SE). h type of eeach can be taced to the effot of Fäe and Gokopf (1985) and Fäe et al. (1983, 1985, 1994). hey exaned the cale effcency to detene whethe the DMU beng evaluated acheve contant RS. 5

6 he appoach dentfe RS though the ato of a ee of elatve effcence obtaned by dffeent DEA odel wth adal eaue, whch ha dffeent contant. hee no poble of ultple oluton fo RS, a occu n the ft two type of eeach. Howeve, the appoach lted n detenng the type of RS. In th context, Føund (1996) dcued the quanttatve eaueent of SE and RS, whch extended futhe to atheatcal chaacteaton of SE fo both fonte and non-fonte unt by Fukuyaa (2). Huang et al. (1997), Keten and Vanden Eeckaut (1998) and Read and hanaoul (2) alo nvetgated the quanttatve eaueent of SE n DEA odel. (4) RS eaueent ung non-adal DEA odel. hee ae a vaety of DEA odel. he ot well-known DEA odel ae often efeed to a adal odel, ncludng the CCR-DEA odel and BCC-DEA odel wth adal eaue. he CCR-DEA odel and BCC-DEA odel ake aupton of contant RS and vaable RS, epectvely. he ft thee type of appoache ae all baed on adal odel, whch wll lack when evaluatng DMU. heefoe, chola have popoed dozen of non-adal odel (e.g., Zhu, 1996, 21; one, 21, 22; Chen, 23) to elnate th poble, uch a DEA odel wth Ruell eaue and addtve odel, aong othe. It natual to exploe the RS eaueent ung non-adal odel. Fo exaple, Banke et al. (24) dcued the RS eaueent ung an addtve odel and ultplcatve odel. Sueyoh and Sektan (25) exploed the RS eaueent ung dynac DEA whoe poducton chee nclude a feedback poce. Zaepheh et al. (21) dcued the RS ue n ultplcatve odel, whch a ngle odel n one tage and dffeent wth the two-tage ethod popoed by Banke et al. (24). Lozano and Guteez (211) analyed the RS of 41 Spanh apot ung DEA odel wth Ruell eaue. Khodabakhh et al. (21) dcued the RS ue n vague DEA odel. Sueyoh and Sektan (27) theoetcally exploed the eaueent of RS ung a non-adal odel wth a ange-aduted eaue. A new lnea pogang RAM/RS appoach wa popoed to adde a ultaneou occuence of ultple efeence et, ultple uppotng hypeplane and ultple poecton. Solean-daaneh et al. (26) exploed the RS eaueent n FDH odel. In fact, n the RS eaueent, the poecton on the effcent fonte of DMU wthn the poducton poblty et (PPS) ae dffeent between adal DEA odel and non-adal DEA odel, but th dffeence doe not affect the RS of DMU on the effcent fonte. 6

7 Now we ecall oe bac fact on the clac RS n DEA faewok. Aue DMU X whee, PPS XR, R, let ax 1, 1 t, ae nput and output calng facto, epectvely. In the cae that t e X, at any X, t 1 X, 1 PPS dffeentable, the clacal SE t t X PPS, defned a the ato of t agnal poductvty (whee t ext) to t aveage poductvty, whee agnal and aveage poductvte ae defned a and t 1 t 1 epectvely (ee, e.g., Podnovk et al., 29). hat e X, d t t 1 dt 1 t When t t, we have the clacal SE at DMU, e X d t, t dt X a follow. d t dt Banke (1984) ftly defned the clacal RS baed on PPS n DEA faewok. Podnovk et al. (29), Podnovk and Føund (21) and Atc and Podnovk (212) ponted out that the devatve n the above clacal defnton of SE (RS) ay not alway ext, and thu they eplaced the clacal devatve by the dectonal devatve, and defned left-hand and ght-hand SE a follow. Defnton 5 and 6 (Left-hand and ght-hand SE): he left and ght hand cale elatcty of DMU ae defned, epectvely, a follow. e e X d t, t X d dt t, t dt (5) (6) hen we can defne: (a) f e X, 1 (o e X, 1) hold, then nceang RS peval on the left-hand (o ght-hand) de of th pont; (b) f e X, 1 (o e X, 1) hold, then contant RS peval on the left-hand (o ght-hand) de of th pont; (c) f e X, 1 (o e X, 1) hold, then deceang RS peval on the left-hand (o ght-hand) de of th pont. 7

8 Reak 1: It poble to cobne left and ght de cale elatcty to defne an oveall RS a n Banke and hall (1992) and Podnovk and Føund (21). hey defned that: () nceang RS peval at DMU X f and only f e X, e X, 1, () contant RS peval at DMU X, e X, 1 e X,, and () deceang RS peval at DMU X, only f 1,,, f and only f f and e X e X. In th pape we wll ue epaately left and ght dectonal RS (ee Secton 3) to keep oe nfoaton. If neceay, eade can cobne the laly a above. 3 Dectonal SE and dectonal RS 3.1 Defnton of dectonal SE and dectonal RS fo explct poducton functon ang (212) popoed the defnton of dectonal RS and dectonal SE fo explct poducton functon. In th pape, we etate befly and extend thee defnton. Let the nput and output vecto be,,..., and,,..., X x1 x2 x y1 y2 y epectvely. Aue we have a contnuouly dffeentable appng F : R gven a follow: whee, F X,, F X F X,, X R, R,,, 1,...,, 1,..., x y F X, efe to the vecto,,,,...,, contnuouly dffeentable. Reak 2: he above equaton F X, f1 X f2 X f X, wth R f beng wll lead to an plct appng =g(x), whch the undelyng poducton functon. Howeve the above condton ae geneally not enough to enue the oothne of the poducton functon. o th end, one need to apply the plct functon theoe. Fo exaple, Kantz and Pak (22) how that f F(X,) atfe the followng condton:.e., F(X,) a contnuouly dffeentable functon, and t Jacoban atx nvetble, whee the Jacoban atx defned a follow: 8

9 f1 f1 f1 f1 X, X, X, X, x 1 x y1 y X f f f f X, X, X, X, x1 x y1 y whee X the atx of patal devatve n the vaable x 1,..., the atx of patal devatve n the vaable y 1,..., contuct a appng g: R R whoe gaph, X, uch that FX, g( X ), and that 9, and, and then we can X g X pecely the et of all g X contnuouly dffeentable. In condeng the dea of nput and output change non-popotonally, we can expe the nput-output change a the followng equaton: whee,..., 1 t t and F y,..., y, t x,..., t x (7) ,..., epeent the vecto of change n nput coponent of X, and the coepondng output coponent of, epectvely. Suppoe we have 1 t, 1,..., t 1 t, 1,..., whee and t epeent the aount of dectonal change of output and nput, epectvely. Paaete, 1,...,, 1 and, 1,...,, 1 ae fxed nube epeentng the decton of nput and output (See Reak 3), epectvely, and t, 1,..., ae the hghe ode nfnteal and atfy t ' t and t l l, epectvely. t We defne t ax : F 1y1,..., y, t1x1,..., tx and aue that the t defned n a vey all neghbouhood of t. Ftly, let u t ooth. Sla to clacal defnton of SE (ee, functon exane the cae whee e.g., Podnovk et al., 29), at any pont let whee dag, 1,...,1,,..., X dag t t X dag denote the dagonal atx. 1 1 We defne t agnal poductvty and aveage poductvty a the output ganed n decton,,..., by addng one unt of nput n decton,,..., (denoted a d t dt ) and the ato of change of output n

10 decton 1, 2,..., and nput n decton 1, 2,..., 1 t 1 t (denoted a ), epectvely. hu we can defne t dectonal SE a the ato of t agnal poductvty to t aveage poductvty. hu we have In patcula, at dectonal SE: e X, d t 1 t dt 1 X, whee t t, e X t, we have the followng foula fo t d (8) dt, t he atonale behnd foula (8) a follow: If the quantty of the nput agnally nceaed by a facto t n decton,,...,, then the axu quantty of the output poble n the technology wll nceae by a facto t e X, n decton,,...,. We can alo dffeentate (7) w..t. the nput calng facto t and obtan: t F d d F dt y x y d t dt x dt 1 1 (9) Fo Equaton (9), we obtan: d t F F ex, x y 1 1 dt x y t X, (1) hen, Equaton (1) the foula of dectonal SE at pont X fo the cae whee the poducton functon contnuouly dffeentable n the decton of,,..., and, 1,...,, and,,...,, whee we aue, 1,...,, 1., 1 Moeove, t clea that n the ooth cae fo the dagonal decton (.e., 1, 1,..., ; 1, 1,..., ), Equaton (1) a follow: d t F F ex, x y 1 1 dt x y t X, (11) whch the ae a the foula of tadtonal SE n econoc (ee, e.g., Føund 1996). 1

11 Howeve, nce t a axu functon, n geneal t ay not be alway dffeentable even f F(X,) ooth. In th cae, applyng the eult n Bowen and Lew (26), thee alway ext the dectonal devatve at t=. Motvated by Equaton (8) (a n Podnovk and Føund (21)), we can ply defne the left and ght-hand dectonal SE, epectvely, a follow. Left: e X Rght: e X d t, t dt d t, t dt (12) (13) t he tadtonal defnton of cale elatcty aue that the epone functon dffeentable at t. Howeve, Podnovk and Føund (21) and Atc and Podnovk (212) ponted out that t often not dffeentable at t. Futheoe, fo the cae whee the poducton gven by DEA, they deontated that t ght-hand and left-hand devatve alway ext wthn the doan of Reade ae efeed to thee two pape fo the detal. heefoe, we defne: (a) f e X, 1 (o e X, 1 peval left-hand (o ght-hand) of pont, 1, 2,..., and,,..., ; (b) f e X, 1 (o e X, 1 peval left-hand (o ght-hand) of pont, 1, 2,..., and,,..., ; (c) f e X, 1 (o e X, 1 peval left-hand (o ght-hand) of pont, 1, 2,..., and,,...,. Reak 3: Hee we aue paaete we ay aue 1 A and 1 11 t. ) hold, then nceang dectonal RS X n the decton of ) hold, then contant dectonal RS X n the decton of ) hold, then deceang dectonal RS X n the decton of 1 and 1. In fact B, whee A and B ae abtay potve nube epectvely. In the gven nput and output decton, we can alway have the dectonal SE, whch depend on the vecto of nput and output decton. heefoe, wthout lo of genealty and fo plcty, we aue 1 and. In th cae, fo the dagonal decton, the dectonal SE exactly the 1 ae a the foula of tadtonal SE n econoc.

12 3.2 Dectonal SE and RS n DEA faewok In th ubecton, we wll ntoduce the defnton of dectonal RS nto DEA faewok, gven the dectonal SE defned above fo an explct poducton functon. Baed on the defnton n Secton 2 and Secton 3.1, we popoe the defnton of left-hand and ght-hand dectonal SE on DMU X baed on PPS a follow. Defnton 7 and 8 (Left-hand and ght-hand dectonal SE n DEA): AungDMU X,, PPS and X R, R, we let ax t, t X PPS whee t dag1 1t,...,1 t and dag1 1,...,1, dag the dagonal atx. Vecto,..., 1 (, 1,...,,..., 1 (, 1,..., atfy ; 1 1 whee, epectvely. he left-hand and ght-hand dectonal SE on DMU, follow: denote ) and ) epeent nput and output decton, epectvely, and t ae nput and output calng facto, X ae a e e X d t, t X d dt t, t dt (14) (15) hen we have (a) f e X, 1 (o e X, 1) hold, then nceang dectonal RS peval left-hand (o ght-hand) of pont X, n the decton of 1, 2,..., and 1, 2,..., ; (b) f e X, 1 (o e X, 1) hold, then contant dectonal RS peval left-hand (o ght-hand) of pont X, n the decton of 1, 2,..., and 1, 2,..., ; (c) f e X, 1 (o e X, 1) hold, then deceang dectonal RS peval left-hand (o ght-hand) of pont X, n the decton of 1, 2,..., and 1, 2,...,. It hould be noted that thee ay ext oe tongly effcent X, PPS whoe nput cannot be futhe educed n a decton of,,..., egadle of 12

13 output decton,,...,. hee tongly effcent DMU ae defned a the dectonal allet cale ze (DSSS, See Defnton 9). On the contay, becaue of the aupton of fee dpoal, the nput of any tongly effcent X, PPS could alway be futhe expanded n a decton of,,...,. heefoe, we hould adde thee two cae dffeently. hu, we obtan the followng Defnton 9 on the dectonal allet cale ze n the decton of 1, 2,..., and,,...,. Defnton 9: he tongly effcent X, of the dectonal allet cale ze f and only f t X, PPS fo any and t. Banke and hall (1992) defned the extee cale ze (ethe the allet cale ze o the laget cale ze) fo weakly effcent unt but wthout futhe elaboaton on the laget cale ze unt. In ou Defnton 9, the dectonal allet cale ze defned on the tongly effcent DMU. Pleae note that the defnton n ou pape copatble wth the defnton of extee cale ze n Banke and hall (1992) n the ene that the defnton efe to the adal decton, the dectonal allet cale ze unde Defnton 9 alo the allet cale ze defned n Banke and hall (1992). 4 Meaueent of dectonal RS he DMU ae often dvded nto two categoe n the eaueent of RS ung DEA odel. he two categoe ae efeed to a the tongly effcent DMU 1 on the effcent fonte and weakly effcent o neffcent DMU. he RS of weakly effcent o neffcent DMU can be eaued though the poecton onto the tongly effcent fonte. h pape follow the above two categoe and conduct RS eaueent baed on the PPS poduced by an nput-baed BCC-DEA odel unde the aupton of vaable RS and focue on the RS eaueent of tongly effcent DMU. he followng Model (16) nput-baed BCC-DEA wth adal eaue. 1 Unle t expely tated, the effcent DMU efe to the tongly effcent DMU n th pape. 13

14 ,,, n 1 1 x x, 1,...,. t. y y, 1,..., 1,,,, 1,..., ; 1,..., ; 1,..., n (16) he dual fo of Model (16) ead ax u, v, 1 uy u 1 y v 1 x, 1,..., n. t. v 1 x 1 u, v, 1,...,, 1,...,, fee (17) 4.1 Fnte dffeence ethod (FDM) Golany and u (1997) ued FDM to etate RS fo each DMU by tetng the extence of oluton n fou egon defned n the neghbouhood of the analyed unt. hey povded a pocedue to detene the RS to the ght and left of the DMU beng evaluated. Roen et al. (1998) etated the dectonal devatve of DMU on tongly effcent fonte ung FDM. he bac dea of FDM to exane the ato of the aount of change of output on the effcent fonte n the pecfed decton caued by the nceae (o deceae) n a all enough aount of nput t n the pecfed decton becaue RS a local popety of DMU Dectonal RS eaueent of tongly effcent DMU It well known that the weakly o tongly effcent fonte of BCC-DEA pecewe lnea. hu, we detene the dectonal RS to the ght and left of DMU beng evaluated. Fgue 3 how the dectonal RS to the ght and left of the pont E, whch on the tongly effcent fonte. 14

15 Fgue 3: Dectonal RS to the ght and left of the pont E Dectonal RS eaueent to the ght of tongly effcent DMU Baed on the Defnton 8 and the FDM popoed by (Roen et al., 1998; Golany and u, 1997), let t ght. We have the followng Model (18) to detene the ght-hand dectonal RS: ax t, ght n 1 x 1 tght x, 1,..., n. t. 1 1 y y, 1,..., n 1,, 1,..., 1 n (18) whee, 1,..., and, 1,..., epeent the decton facto of nput and output, epectvely, and atfy ; 1 1. Model (18) appea to be a nonlnea pogang. Howeve, a to be een below, t obectve ndependent of t ght afte tght all enough. hu, we wll undetand that t ght a all potve quantty, whch epeent the aount of dectonal change of nput. Vaable t actually becoe a lnea pogang. whee epeent the aount of dectonal change of output. hen, Let t dag1 1t ght,...,1 tght, and dag 1,...,1 1, the optal oluton of Model (18). We have the followng heoe 1: heoe 1. hee ext t atfyng (1) when tght, t and X t, PPS X t, located on the weakly effcent fonte EF weak, and (2) when tght, t, X, and X t, hypeplane, whch ay be dffeent fo dffeent t ght. have the ae uppotng 15

16 Poof. Pleae ee Appendx A. be the optal obectve of Model (18). hu, we can detene the dectonal RS to the ght of DMU X a follow. Defnton 1: Let X,, 1 hold, then nceang dectonal RS peval n the decton of,,..., and,,..., ; (a) f X (b) f X, 1 hold, then contant dectonal RS peval n the decton of,,..., and,,..., ; (c) f X, 1 hold, then deceang dectonal RS peval n the decton of,,..., and,,...,. Hee, we how that fo vey all potve t ght, the obectve value n Model (18) a contant fo any gven nput and output decton., heoe 2. hee ext a all enough quantty value of Model (18) contant fo all tght, t. Poof. Pleae ee Appendx A. t uch that the optal (19): Now, we dcu how to elect tght ax, n pactce. Conde the followng Model n 1 x 1 tght x, 1,..., n. t. 1 1 y y, 1,..., n 1,, 1,..., 1 n Let, be optal oluton of Model (19). Conde the followng Model (2): (19) ax = U UV,, U V X, 1,..., n V X 1 t.. U V t X U, V, fee (2) atfe Fo the poof of heoe 2, f the optal obectve value of Model (2) 1, potve contant t ght all enough to guaantee that both 16

17 X, and X t, ae located on the weakly effcent fonte EF weak, and they have the ae uppotng hypeplane. hu, n pactce, we ft elect a all nube t ght n (19) and olve (2) to ee f the optal the unt. If not, we wll attept alle nube. Fo the contnuty, t wll be the unt when enough. tght Dectonal RS eaueent to the left of tongly effcent DMU Ft, we need to detene whethe the tongly effcent X, all of the dectonal allet cale ze. Accodng to Defnton 9, we conde the followng Model (21): ax,, n 1 x 1 x, 1,..., n 1 y 1 y, 1,..., t.. n 1,, 1,..., 1 n ; fee (21) heoe 3. he optal obectve value of Model (21) zeo f and only f tongly effcent X of the dectonal allet cale ze., Poof. Pleae ee Appendx A. We ft dcu the cae n whch tongly effcent X dectonal allet cale ze n the decton of,,...,, and 1, 2,..., not of the Baed on the Defnton 7 and the FDM popoed by (Roen et al., 1998; Golany and. u, 1997), we let t left be a all potve contant and have the followng Model (22) to detene the left-hand dectonal RS: n t, left n 1 x 1 tleft x, 1,..., n. t. 1 1 y y, 1,..., n 1,, 1,..., 1 n (22) whee, 1,..., and, 1,..., epeent the decton facto of nput and output, epectvely, and atfy ; 1 1. Contant left t a all potve quantty, whch epeent the aount of dectonal change of 17

18 nput. Vaable epeent the aount of dectonal change of output. We let t dag1 1t left,...,1 tleft, dag 1,...,1 1, and the optal oluton of Model (22). hu, we have the followng heoe 4: heoe 4. hee ext t atfyng (1) when tleft, t and X t, PPS X t, located on the weakly effcent fonte EF weak, and (2) when tleft, t, X, and X t, hypeplane, whch ay be dffeent fo dffeent Poof. Pleae ee Appendx A. Defnton 11: We let X, have the ae uppotng be the optal obectve of Model (22). Accodngly, we can detene the dectonal RS to the left of DMU X a follow:, 1 hold, then nceang dectonal RS peval n the decton of,,..., and,,..., ; (a) f X (b) f X, 1 hold, then contant dectonal RS peval n the decton of,,..., and,,..., ; (c) f X, 1 hold, then deceang dectonal RS peval n the decton of,,..., and,,...,. Next, we dcu how to chooe t left t. left. Agan, we ft have, heoe 5. hee ext a all enough quantty value of Model (22) contant fo all tleft, t. Poof. Pleae ee Appendx A. Now, agan, conde the followng Model (23): t uch that the optal ax, n 1 x 1 tleft x, 1,..., n. t. 1 1 y y, 1,..., n 1,, 1,..., 1 n Let, t dag1 1t left,...,1 tleft and dag 1,...,1 1 (23) be the optal oluton of Model (23) and. Conde the 18

19 followng Model (24): ax = U UV,, U V X, 1,..., n V X 1 t.. U t V X U, V, fee (24) Agan, f the optal obectve value of Model (24) atfe all enough contant to guaantee that both located on the weakly effcent fonte EF weak hypeplane. hu, we wll elect tleft laly. Now, we tun to the cae that the tongly effcent X allet cale ze n the decton of,,..., 1, left t a X, and X t, ae, and they have the ae uppotng, and 1, 2,..., cae, we cannot fnd a feable oluton n Model (22) when tleft of the dectonal. In th a all potve contant. hu, we povde the followng Defnton 12 to adde the left-hand dectonal RS of X :, Defnton 12: If tongly effcent X, of the dectonal allet cale ze n the decton of 1, 2,..., and 1, 2,...,, then nceang dectonal RS peval at the left-hand of X. Sla to Equaton (A-5), we have, v 1 x t left u 1 y whee U u1, u2,..., u and V v 1, v2,..., v (24), and U, V DMU (25) ae the optal oluton of Model the noal vecto of the uppotng hypeplane on the X, and DMU X t, A pocedue fo etatng dectonal RS of tongly effcent DMU Baed on the above analy, we now popoe a pocedue fo etatng dectonal RS to the ght and left of tongly effcent DMU X on the tongly effcent fonte Pocedue 1. EFtong a follow., 19

20 Step 1: Detene the dectonal RS to the ght of DMU X Step 1-1: Chooe a all enough quantty tght, baed on Model (19) and Model (2), to guaantee that both, ae located on the weakly effcent fonte EF weak X, and X t,, and they have the ae uppotng hypeplane. Step 1-2: Solve Model (18) to detene the dectonal RS to the ght of DMUX :,, 1 hold, then nceang dectonal RS peval n the decton of,,..., and,,..., ; (a) f X (b) f X, 1 hold, then contant dectonal RS peval n the decton of,,..., and,,..., ; (c) f X, 1 hold, then deceang dectonal RS peval n the decton of,,..., and,,...,. Step 2: Detene the dectonal RS to the left of DMU X Step 2-: Solve Model (21) to ee whethe t optal obectve value zeo. If o, X of the dectonal allet cale ze. Othewe, we have the, followng two tep to detene the left-hand dectonal RS. Step 2-1: Chooe a all enough quantty (24), to guaantee that both, effcent fonte EF weak t left X and X t,,, baed on Model (23) and Model ae located on the weakly, and they have the ae uppotng hypeplane. Step 2-2: Solve Model (22) to detene the dectonal RS to the left of DMUX :,, 1 hold, then nceang dectonal RS peval n the decton of,,..., and,,..., ; (a) f X (b) f X, 1 hold, then contant dectonal RS peval n the decton of,,..., and,,..., ; (c) f X, 1 hold, then deceang dectonal RS peval n the decton of,,..., and,,..., Dectonal RS eaueent of neffcent o weakly effcent DMU Fo etatng dectonal RS to the ght and left fo neffcent o weakly 2

21 effcent DMU, we can pefo the followng two tep: Step 1: Ft, we poect the neffcent o weakly effcent DMU onto the tongly effcent fonte EFtong ung BCC-DEA wth adal eaue (See Model (16) fo detal), and the foula fo poecton the followng Equaton (26): x x, 1,..., y y, 1,..., (26) Step 2: When we detene the poected pont on the tongly effcent fonte EF tong, we can etate the dectonal RS to the ght and left fo the neffcent o weakly effcent DMU ung Pocedue 1 n Secton Reak 4: Hee we only ue an nput-baed adal poecton. Howeve, dffeent poecton ay geneate dffeent RS (See, e.g., Sueyoh and Sektan, 27). In th pape we concentate on dcuon of the dectonal RS of tongly effcent DMU ntead of neffcent o weakly effcent one o we only how one poble way to etate the dectonal RS by ung the nput-baed adal poecton. 4.2 Uppe and lowe bound ethod (ULBM) In th ubecton, we dcu how to etate the dectonal RS of the tongly effcent DMU unde the aupton of vaable RS fo anothe vewpont. Accodng to Equaton (A-5) and (25), we know that the followng foula hold when tght and tleft ae all enough potve contant: v 1 x ght 1 t u y and v 1 x left 1 t u y (27) hu, we can ue the followng Model (28) to calculate the uppe and lowe bound of the dectonal SE and then detene the type of dectonal RS of DMU X. ax n u, v, u, v, 1 1 vx u y u 1 y v 1 x, 1,..., n u 1 y v 1 x t.. vx 1 1 u, v, 1,...,, 1,...,, fee (28), 21

22 heoe 6. Suppoe that X not of the dectonal allet cale ze., he uppe and lowe bound ( X, and X, the optal obectve value X, value X, n Model (18). Poof. Pleae ee Appendx A. ) n Model (28) ae equal to n Model (22) and the optal obectve heoe 7. If the axal optal obectve functon X, of Model (28) unbounded (+ ), the tongly effcent X of the dectonal allet cale ze. Poof. Pleae ee Appendx A. heefoe, we have the pocedue la to Pocedue 1 fo etatng dectonal RS. Model (28) a factonal pogang that dffcult to olve, o we tanfo Model (28) nto an equvalent lnea pogang though Chane-Coope tanfoaton (Chane et al., 1962). Ft, we ewte Model (28) ung the followng Model (29):, ax n UV,, UV,, V WX U U V X, 1,..., n U V X t.. V X 1 U, V, fee whee U u1, u2,..., u and V v1, v2,..., v dag and W dag,,..., nput and output. We let (29) ae vecto of ultple, and 1, 2,..., ae atxe of decton of 1 U VW U (3) We let, whee a non-achedean contuct to aue the 22

23 nvee atx of W ext. In th cae, we have the followng Equaton (31) fo Equaton (3): 1 U 1 V W 1 U hu, Model (29) can be tanlated nto the followng Model (32): ax n,,,,,, W X, 1,..., n 1 1 W X 1 t.. W X 1,,, fee X (31) (32) We let ', and Model (32) can be conveted nto the followng lnea pogang: ax n ' ',,,,,, X 1 1 ' W X, 1,..., n 1 1 ' W X 1 t.. W X 1 ',,, fee (33) Solvng Model (33), we can obtan the optal obectve value of Model (28) o Model (29). 5 A Cae Study In th ecton, we conduct a cae tudy to analye the dectonal RS of 16 bac eeach nttute n the Chnee Acadey of Scence (CAS) n 21. Snce the Plot Poect of Knowledge Innovaton (KIPP) n 1998 at the CAS, nttute evaluaton ha becoe nceangly potant, and the equeent fo the evaluaton poce have dvefed. Snce 25, CAS headquate ha bult up the Copehenve Qualty 23

24 Evaluaton (CQE) yte fo nttute evaluaton n CAS. he eult of evaluaton ae expeed a ult-denonal feedback data and ued a the tool to povde ba of copehenve analy and decon-akng and to povde nttute wth tageted evaluaton nfoaton and dagnotc coent. In the faewok of CQE, ultple nput and output of the bac eeach nttute of the CAS ae ontoed ung eveal quanttatve ndcato. In th pape, we ue the ae ndex ndcato a popoed n Lu et al. (211) fo 16 bac eeach nttute n CAS: Input: (1) Staff denote the nube of full-te eeach taff, and (2) Re. Expen. denote the total eeach expendtue; Output: (1) SCI Pub. denote the publcaton, ncludng the ntenatonal pape ndexed by Scence Ctaton Index; (2) Hgh Pub. denote hgh-qualty pape publhed n top eeach ounal; (3) Exte. Fund denote the extenal eeach fundng; (4) Gad. Enoll. denote gaduate tudent enolent. Colun 2-7 of able 2 how the detaled data of thee ndcato of 16 bac CAS eeach nttute n 21. Inttute able 2: Detaled data of 16 bac CAS eeach nttute n 21 SCI Pub. (Nube) Hgh Pub. (Nube) Output Input Effcency Gad. Enoll. (Nube) Exte. Fund. (RMB llon) Staff (FE) Re. Expen. (RMB llon) DMU DMU coe (Model DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU DMU ) 24

25 DMU Data ouce: (1) Quanttatve ontong epot of eeach nttute n CAS, 211; (2) Stattcal eabook of CAS, 211. fonte Next, we analye the dectonal RS of 16 bac CAS eeach nttute n 21. Step 1: We detene the tongly effcent fonte EF weak ung the nput-baed BCC-DEA odel (16). EFtong and weakly effcent he lat colun of able 2 how the elatve effcence of 16 bac CAS eeach nttute n 21. Fo able 2, we can ee that DMU 1, DMU 2, DMU 4, DMU 13 and DMU 14 ae effcent DMU. Colun 2-6 of able 3 how the poecton of thee DMU on the tongly effcent fonte EFtong ung the followng foula: x x, 1,..., y y, 1,..., (34) Step 2: Fo Model (18), (21), (22) and (33), we can obtan the dectonal RS to the ght and left of each DMU. Fo copaon pupoe, we et the decton of output a and the decton of nput a 1.5, 2 1.5, 1 1.5, 2.5 and 1 1, 2 1 (tadtonal RS), epectvely. We ue Model (21) to detene whethe DMU ae of the dectonal allet cale ze. We fnd that DMU 2 of the dectonal allet cale ze n the above thee nput decton. heefoe, we let t t E 6 ght left 1, whch can pa the tet of Equaton (19) - (2) and Equaton (23) - (24) unde the above thee decton. Colun 8-1 of able 3 how the dectonal RS of each DMU n the cae of 1.5, (Cae 1), 1 1.5, 2.5 (Cae 2) and the tadtonal cae 1, 1(Cae 3), epectvely. In able 3, I, C and D denote nceang, contant and deceang dectonal RS, epectvely. Baed on the eult of analy, we fnd that the dectonal RS of DMU ay change due to dffeent decton of nput, and dectonal RS dffeent than tadtonal RS. akng DMU 16 a an exaple, we can ee that (1) deceang dectonal RS peval to both the left and the ght of th DMU n the decton of 1 1, 2 1 and ; (2) nceang dectonal RS peval to the left and deceang dectonal RS peval to the ght of th DMU n the decton of 1.5, and ; and (3) contant dectonal RS peval to the left and deceang dectonal RS peval to the ght of th DMU n the decton of 1 1.5,

26 and See able 3 fo detal. able 3: he poecton of thee DMU on the tongly effcent fonte and dectonal RS n thee cae Inttute Poecton (Output) Poecton SCI Pub. (Nube) Hgh Pub. (Nube) Gad. Enoll. (Nube) Exte. Fund. (RMB llon) Staff (FE) (Input) Re. Expen. (RMB llon) Cae 1 (Left /Rght) Cae 2 (Left /Rght) DMU I/D I/D I/D Cae 3 (Left /Rght) DMU I(DSSS)/D I(DSSS)/D I(DSSS)/D DMU I/D D/D D/D DMU I/D I/D I/D DMU I/D I/D I/D DMU D/D I/D I/D DMU D/D I/D I/D DMU I/D I/D I/D DMU I/D I/D C/D DMU I/D I/D D/D DMU D/D I/I D/D DMU I/D I/D D/D DMU I/D I/D I/D DMU I/D I/D C/D DMU I/D I/D D/D DMU I/D C/D D/D Note: DSSS denote that DMU 2 of the dectonal allet cale ze, and nceang dectonal RS peval to the left of DMU 2 accodng to Defnton 12. Step 3: We take tongly effcent DMU 1 and DMU 5 a exaple to analye the dectonal RS n ultple nput decton. A befoe, we et the decton of output a We apply Model (21) to thee two DMU and fnd that the optal obectve value ae all lage than zeo n dffeent decton of nput. We let t t E 6 ght left 1, whch can pa the tet of Equaton (19) - (2) and Equaton (23) - (24) n dffeent decton of nput. hu, we can obtan the dectonal SE and RS of DMU 1 and DMU 5 n dffeent decton of nput ung FDM ethod. Addtonally, we ue Model (33) to obtan the uppe and lowe bound of obectve functon value. See able 4 fo detal. able 4: he dectonal RS of DMU 1 and DMU 5 n dffeent nput decton DMU 1 2 (Rght) (Left) (Lowe (Uppe Dectonal RS(Rght) Dectonal RS(Left) 26

27 DMU 1 DMU 5 bound) bound) Deceang Inceang Deceang Inceang Deceang Inceang Deceang Inceang Deceang Inceang Deceang Inceang Deceang Inceang Deceang Inceang Deceang Inceang Deceang Inceang Deceang Contant Deceang Inceang Deceang Inceang Deceang Inceang Deceang Inceang Deceang Inceang Deceang Inceang Deceang Inceang Deceang Inceang Deceang Inceang Deceang Inceang Deceang Inceang Deceang Inceang Deceang Inceang Deceang Inceang Deceang Inceang Deceang Inceang Deceang Inceang Deceang Inceang Deceang Deceang Deceang Contant Deceang Inceang Deceang Inceang Deceang Inceang Deceang Inceang Deceang Inceang Deceang Inceang Deceang Inceang Fo able 4, we can ee and. Baed on the above analy, we have the followng uae, ee Fgue 4 to 7 fo detal. 27

28 Regon 1: Inceang ω 2 ω 1 =ω 2 ω 2 ω 1 =ω 2 Regon 1: Deceang Regon 2: Contant Regon 3: Inceang ω 1 ω 1 Fgue 4: Dectonal RS to the ght of DMU 1 Fgue 5: Dectonal RS to the ght of DMU 5 ω 2 ω 1 =ω 2 Regon 1: Inceang ω 2 Regon 1: Inceang ω 1 =ω 2 Regon 2: Deceang Regon 3: Contant Regon 4: Inceang ω 1 ω 1 Fgue 6: Dectonal RS to the left of DMU 1 Fgue 7: Dectonal RS to the left of DMU 5 (a) he dectonal RS to the ght of DMU 1 and DMU 5 (a-1) Fo DMU 1, on the ba of extng nput, f Staff and Re. Expen. nceae n any popoton (unde Paeto pefeence), deceang dectonal RS peval on DMU 1,.e., DMU 1 locate to the egon wth deceang dectonal RS n any decton of nput nceae. See Fgue 4. (a-2) Fo DMU 5, on the ba of extng nput, f Staff and Re. Expen. nceae n adal popoton, nceang dectonal RS peval on DMU 5. If the popoton of Staff and Re. Expen. nceae locate n Regon 1 and Regon 3, nceang dectonal RS peval. Othewe, f the popoton of nput nceae locate n Regon 2 n Fgue 5, contant dectonal RS peval. See Fgue 5. (b) he dectonal RS to the left of DMU 1 and DMU 5 (b-1) Fo DMU 1, on the ba of extng nput, f Staff and Re. Expen. deceae n any popoton (unde Paeto pefeence), nceang dectonal RS peval on DMU 1. See Fgue 6. (b-2) Fo DMU 5, on the ba of extng nput, f Staff and Re. Expen. deceae n adal popoton, nceang dectonal RS peval. If the popoton of Staff and Re. Expen. locate n Regon 1 and Regon 4 n Fgue 7, nceang dectonal RS peval. If the popoton of nput deceae locate n Regon 2, deceang 28

29 dectonal RS peval. If the popoton of nput deceae locate n Regon 3, contant dectonal RS peval. See Fgue 7. 6 Concluon In eeach nttuton, gven the coplexty of eeach actvte, change of vaou type of nput o output ae often not popotonal. heefoe, the extng defnton of RS n the faewok of the DEA ethod ay not eet the need fo etaton of the RS of eeach nttuton wth ultple nput and output ung the DEA ethod. h wok extend the defnton of RS n the DEA faewok, popoe the defnton of dectonal RS n DEA and etate the dectonal RS of eeach nttuton ung DEA odel. he tadtonal RS a pecal cae of dectonal RS when change of nput-output ae n the dagonal decton. he dectonal RS can be ued fo analyng thoe nput change decton that ae utable fo a patcula DMU and thu ueful fo decon-ake (DM) to decde atonal cobnaton of eouce. Acknowledgeent. We would lke to acknowledge the uppot of the Natonal Natual Scence Foundaton of Chna (No ) and Gean Acadec Exchange Sevce (DAAD, No. A139433). We wh to expe ou ncee thank to the efeee and edto, whoe uggeton geatly poved ou pape qualty. 29

30 Appendx A. Poof of heoe 1. (1) When t t ght and, X PPS t, X t, located on the weakly effcent fonte weak EF. Aung X t, not located on the weakly effcent fonte EF weak, we exploe the followng Model (A-1): Let ax n 1 x 1 tght x, 1,..., n. t. 1 1 y y, 1,..., n 1,, 1,..., 1 n be the optal obectve of the above odel. Becaue not located on the weakly effcent fonte have X t, PPS, whee In addton, becaue Hence, contadct the fact that 1, we have EF weak, we have dag,...,. 1 y 1 y, 1,..., a feable oluton fo Model (18) and (2) When ght all quantty t, X t, t ght the optal oluton of Model (18). convege to X, atfyng that both, X and X, t (A-1) X t, 1. heefoe, we, whch. hu, thee ext a obvouly havng the ae uppotng hypeplane. h uppotng hypeplane ay be dffeent fo dffeent t ght. Q.E.D. Poof of heoe 2. We let, be the optal oluton of Model (18). Becaue tght a all enough quantty, we know that both X, and X t, ae located on the weakly effcent fonte EF weak, and they have the ae uppotng hypeplane, whee t dag1 1t ght,...,1 tght dag 1,...,1 1 and. In th cae, we know that the optal obectve value of Model (2) atfe 1. hu, we have U V X U V t X (A-2) 3

31 whee U u1, u2,..., u and V v 1, v2,..., v (2). Accodng to Equaton (A-2), we know Fo Equaton (A-3), we have u 1 y v 1 tght x ae the optal oluton of Model (A-3) v 1 x ght 1 t u y (A-4) heefoe, when tght a all enough quantty, we can obtan the optal obectve value of Model (18) a follow: v 1 x t ght u 1 y whee U u1, u2,..., u and V v 1, v2,..., v (2), and U, V DMU t ght (A-5) ae the optal oluton of Model the noal vecto of the uppotng hypeplane on the X, and DMU X t,. A the lt of Equaton (15) alway ext, we know when tght t and, both,, ae located on the ae Face of the X and X t weakly effcent fonte, o the value of the Equaton (A-5) ean unchanged. Othewe, the lt of Equaton (15) doe not ext. hu, fo Equaton (A-5), we know that the optal obectve value of Model (18) contant wth epect to t ght when tght a all enough quantty. Q.E.D. Poof of heoe 3. Accodng to Defnton 9, we can ealy ee that heoe 3 hold. Q.E.D. Poof of heoe 4. he poof la to that of heoe 1 and otted hee. Poof of heoe 5. he poof la to that of heoe 2 and otted hee. Poof of heoe 6. We ft dcu the equalty between the optal obectve value X, and the lowe bound, X. Fo Equaton (A-5), we know that the dectonal RS to the ght of DMUX ead, 31

32 1 v x X, t ght u 1 y (A-6) whee tght 1, 2,...,, 1, 2,...,, a all enough quantty and U u u u V v v v the optal oluton of Model (2). (1) If X, X,, the optal oluton of Model (2) 1, 2,...,, 1, 2,...,, U u u u V v v v atfe whch contadct the fact that X, (2) If X, X, u 1 y v 1 x, 1,..., n u 1 y v 1 x vx 1 1 Equaton (A-5) and Equaton (27): A tght, we have. the lowe bound n Model (28). (A-7), we can deduce the followng foula (A-8) fo X, X, tght (A-8) t ght Fo Model (28), we know that DMU X, atfe U V X (A-9) U V X (A-1) whee U, V the noal vecto of a cetan Face of weakly effcent fonte on the DMU U, V the noal vecto of a uppotng hypeplane on DMUX., X,, and Fo (A-6) and (A-9), we have U V X (A-11) t whee t dag1 1t ght,...,1 tght and dag 1,...,

33 We pck up a pont X t, on the uppotng hypeplane wth the noal vecto U, V on the DMUX,. hu, we obtan whee dag1 1,...,1 ext at leat one et U, V U V X (A-12) t. A DMU, that atfe context, we can obtan the pont X t,. X tongly effcent, thee U V X. In th A, we can obtan the followng foula fo Equaton (A-12): U V X (A-13) t We know that X t, on the weakly effcent fonte EF weak, and th fact contadct the uppotng hypeplane U V X n Model (28). Slaly, we can pove that the optal obectve value of Model (22) equal to the uppe bound X, n Model (28). Q.E.D. Poof of heoe 7. We uppoe that the axal optal obectve functon X of Model (28) unbounded, but tongly effcent X, not of the, dectonal allet cale ze. Accodng to Model (21), we know that we can fnd the optal oluton of Model (21), denoted by,,, n whch effcent, we know that. Becaue X, tongly. Accodng to heoe 4 and 5, we know that we can fnd a all enough potve contant t left that can enue that the optal value of Model (22) contant. Accodng to heoe 6, we know that the uppe bound X of Model (28) equal to the optal obectve value X,, Model (22). h fact contadct the uppoton that the axal optal obectve functon X of Model (28) unbounded. Q.E.D., of 33

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