Super-efficiency and stability intervals in additive DEA

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1 This is PDF file of n unedited mnusript tht hs been epted for publition in Journl of the Opertionl Reserh Soiety. The mnusript will undergo opyediting, typesetting, nd review of the resulting proof before it is published in its finl form. Plese note tht during the prodution proess errors my be disovered whih ould ffet the ontent, nd ll legl dislimers tht pply to the journl pertin. The finl version will be vilble t: Super-effiieny nd stbility intervls in dditive DEA M C Gouvei ISCAC, Quint Agríol, Bennt, Coimbr, Portugl, nd INESC Coimbr, Ru Antero de Quentl, 199; Coimbr, Portugl mgouvei@is.pt C Dis Fuldde de Eonomi, Univ. Coimbr, Av. Dis d Silv 165, Coimbr, Portugl, nd INESC Coimbr, Ru Antero de Quentl, 199; Coimbr, Portugl ldis@ines.pt C H Antunes DEEC-FCT Universidde de Coimbr-Pólo II, 3030 Coimbr, Portugl, nd INESC Coimbr, Ru Antero de Quentl, 199; Coimbr, Portugl ntunes@ines.pt ABSTRACT. This study ddresses the problem of finding the rnge of effiieny for eh Deision Mking Unit (DMU) onsidering unertin dt. Unertinty in the DMU oeffiients in eh ftor (input or output) is ptured through intervl oeffiients (i.e., these re unertin but bounded). A twophse dditive Dt Envelopment Anlysis (DEA) model for performne evlution is used, whih is dpted to inlude the onept of super-effiieny to provide robustness nlysis of the DMUs in fe of unertin informtion, ssessing whether eh DMU is surely effiient, potentilly effiient, or surely ineffiient for the unertinty intervls speified. Another ontribution is to present how mximl stbility hyper-retngle n be omputed for eh DMU suh tht its effiieny sttus does not hnge when the oeffiients vry within tht intervl. KEYWORDS. Dt Envelopment Anlysis; Multi-Criteri Anlysis; Super-effiieny; Robustness Anlysis; Unertinty; Stbility intervls. 1

2 Introdution Dt Envelopment Anlysis (DEA), originlly developed by Chrnes et l. (1978), is nonprmetri pproh bsed on liner progrmming to evlute observtions representing the performnes of ll units (Deision Mking Units - DMUs) under evlution. Eh DMU is hrterized by the "onsumption" of multiple inputs for the "prodution" of multiple outputs. The different DEA models seek to determine whih of n DMUs form the effiient frontier (or envelopment surfe) in Preto-Koopmns sense. These evlutions result in performne sore tht rnges between zero nd unity tht represents the degree of effiieny obtined by DMUs. In rel-world evlution problems, informtion is generlly subjet to severl soures of unertinty, resulting from dt srity, diffiulties of dt estimtion or dt olletion, or even to ontrditory informtion from distint soures. Therefore, deision id models must ope with this dt unertinty by using models pble of providing robust onlusions, i.e., reommendtions tht re somehow immune to plusible dt instntitions. The use of intervl oeffiients is very flexible modelling tool for pturing this type of dt unertinty (i.e., the preise performnes of the DMUs re unknown but bounded within n intervl) sine it does not impose stringent reuirements bout probbility or possibility distributions. There re two min perspetives to del with unertinty in the ontext of our work. One, whih is generlly enompssed under the designtion impreise DEA (IDEA) (Cooper et l., 1999, 2001, 2001b), studies how to del with impreise dt suh s bounded dt, ordinl dt nd rtio bounded dt in DEA nd results in non-liner nd non-onvex DEA model. By using sle trnsformtions nd vrible hnges (Zhu, 2003), or only vrible trnsformtions (Despotis nd Smirlis, 2002), this non-liner model n be trnsformed into n euivlent liner progrmming problem. The other perspetive, whih is more relted to the pproh proposed in this pper, dels with omputing stbility intervls for unertin oeffiients so tht results do not hnge (Zhu, 1996, 2001; Seiford nd Zhu, 1998, 1998b). This pper ddresses this type of unertinty in the ontext of two-phse method developed by Gouvei et l. (2008), whih ws inspired on the dditive DEA model proposed by Chrnes et l. (1985) s well s the dditive model with oriented projetions presented by Ali et l. (1995). In Gouvei et l. s model, the DMUs re treted s lterntives of multiple riteri deision model (n dditive multi-ttribute utility model), eh lterntive being evluted in number of distint riteri. The two-phse method provides n effiieny mesure of eh DMU by lulting riterion weighting vetor nd, if neessry, obtining the projeted point. There re two min ontributions in this work. One ontribution is to present how the rnge of effiieny for eh DMU n be omputed in presene of intervl vlues for the DMU oeffiients in eh input/output. This llows performing robustness nlysis of eh DMU under evlution, ssessing whether eh DMU is surely effiient, potentilly effiient, or surely ineffiient for the unertinty intervls speified. Another ontribution is to present how mximl stbility hyper-retngle n be omputed for eh DMU suh tht its effiieny sttus does not hnge when the oeffiients vry within 2

3 tht intervl. This is onfronted with the ide of tking super-effiieny sore s proxy of robustness, fter dpting the originl model of Gouvei et l. (2008) to onsider the onept of super-effiieny. Following this introdution, setion 2 briefly desribes DEA models, with emphsis on the dditive DEA model nd the weighted dditive model. In setion 3, the onept of super-effiieny is reviewed. Setion 4 introdues the two-phse method of Gouvei et l. (2008) with the modifitions to inlude the supereffiieny onept. In setion 5, the rnges of effiieny re omputed nd the robustness of eh DMU is nlyzed by onsidering two-dimensionl pedgogil exmple. In setion 6, n exmple with rel world dt is provided for exploiting the insights from this robustness nlysis. Conluding remrks re presented in setion 7. Dt Envelopment Anlysis The set of n DMUs to be evluted is { DMU j : ( j = 1,..., n)}. Eh DMUj onsumes m different inputs x ij ( i = 1,..., m) to produe p different outputs y rj ( r = 1,..., p). X j (the jth olumn of X m n) denotes the vetor of inputs onsumed by DMUj. A similr nottion is used for outputs, Y j. The input dt is represented by mtrix X nd the output dt by mtrix Y p n. 1 denotes the summtion vetor( 1,...,1) T. m n There re two min types of DEA models tht provide mesure of reltive effiieny for eh DMU ording to the returns to sle onsidered (Seiford nd Zhu, 1999b): Constnt Returns-to-Sle (CRS) models, suh s the CCR model (see Chrnes et l., 1978), nd Vrible Returns to Sle (VRS) models, suh s the BCC model (see Bnker et l., 1984) nd the dditive model (ADD) of Chrnes et l. (1985). The dditive model In CCR nd BCC models we need to distinguish between input-oriented nd output-oriented models. The dditive model ombines both orienttions in single model, whih n be formulted s follows: ( ADD ( X k,y k )) min z k = - 1s+1e s.t. Yl - s= Y k, - Xl - e= -X k, 1l =1, l ³ 0, e ³ 0, s³ 0. The ADD model returns non-positive vlue k z, whih llows heking the reltive effiieny of the DMU k under nlysis. If the vlue obtined is negtive, then the DMU under nlysis is operting ineffiiently in some ftors. This vlue is the symmetri of the sum of the distnes in eh dimension to the envelopment surfe ( 1 distne). If DMU k is ineffiient (i.e., it does not lie on the effiient frontier defined by the set of DMUs) the model 3

4 identifies projeted point ( X, ˆ ) ˆ on the effiient frontier. If the optiml vlue of the priml ADD k Y k model is zero the point ( X, ) belongs to the effiient frontier, tht is ( Xˆ, ˆ ) = ( X, ) k Y k neessry nd suffiient ondition for effiieny, Chrnes et l. (1985). The projeted point n be hrterized, in n lterntive wy, s Xˆ, Yˆ X e, Y s k k k Y k k k Y k k. This is, from the priml onstrints. This ADD model mesures the exess of inputs, e, nd the defiit of outputs, s, in whih the DMU k opertes when onfronted with the DMUs tht operte on the effiient frontier. Ali et l. (1995) presented vrint of dditive model with oriented projetions, whih is heneforth lled weighted dditive model. The envelopment formultion for this model is: ( ADD W ( X k,y k, u k, v k )) min z k = - u k s+ v k e s.t. Yl - s= Y k, - Xl - e= -X k, 1l =1, l ³ 0, e ³ 0, s³ 0. The prmeters u k, v k (whih re fixed before the model is solved) hve ply n importnt role tht is lerly seen in the priml formultion. The vetors ( u k, v k ) re the oeffiients of the objetive funtion nd thus define the reltive weight ttributed to one unit of eh slk (for disussion on the role of weights nd vlue judgments in DEA (see Thnssoulis et l., 2004)). The weight vetors provide nd determine the diretions of the projetion. Setting u k = 1nd v k = 1 in the priml weighted dditive problem leds to the originl dditive model. Super-effiieny nd sensitivity nlysis in DEA Andersen nd Petersen (1993) developed n extended DEA mesure in whih the bsi ide is to ompre the DMU under evlution with liner ombintion of ll other DMUs in the referene set. This mens tht the prodution possibility set is redued by not onsidering the DMU being evluted, whih llows effiient DMUs to beome super-effiient nd hve different super-effiieny sores. In other words, the DMUs n inrese the input vetor (or derese the output) to some extent while preserving effiieny. We n lso pereive DEA models s projetion mehnisms nd the projetions of the ineffiient DMUs on the effiient frontier depend on the sles used to mesure eh input or output. Super-effiieny is very sensitive to the projetion mehnisms nd tends to fvour extreme solutions (Bouyssou, 1999). The set of DMUs n be prtitioned into two groups: frontier (effiient) DMUs nd non-frontier (ineffiient) DMUs. The frontier DMUs onsist of DMUs in the set E (extreme effiient), set E (effiient but not n extreme point) nd the set F (wekly effiient or frontier point but with non-zero slks). The 4

5 super-effiieny model identifies the lssifition of given DMU nd, using some extensions of the super-effiieny model, sensitivity nlysis of the onlusions bout effiieny n lso be performed. However, under ertin onditions the proess of determining the super-effiieny sore n led to n infesible liner progrm. Bsed on Thrll (1996), neessry, but not suffiient, ondition for infesibility is tht n exluded DMU is extreme effiient. Dulá nd Hikmn (1997) nd Seiford nd Zhu (1999) reported neessry nd suffiient ondition for infesibility in n input-oriented CCR supereffiieny model: the exluded DMU hs the only zero vlue for ny input, or the only positive vlue for ny output, mong ll DMUs in the referene set. Infesibility nnot rise in n output-oriented CCR super-effiieny model. Infesibility lso ours in the BCC super-effiieny model, when n effiient DMU under evlution nnot reh the frontier formed by the remining DMUs vi inresing the inputs (or deresing the outputs). Infesibility rises in either orienttion whenever there is no referene DMU for the exluded one. Mny DEA reserhers hve ddressed the sensitivity of the results to dt perturbtions nd the robustness of the effiieny sores resulting from these perturbtions, bsed on super-effiieny DEA pprohes. Continuing the work of Zhu (1996), Seiford nd Zhu (1998) developed sensitivity nlysis proedure to determine stbility regions for possible inreses in ll inputs nd for possible dereses in ll outputs within whih the effiieny of speifi effiient DMU remins unhnged. Seiford nd Zhu (1998b) extended the method by Zhu (1996) nd Seiford nd Zhu (1998) to the worst-se senrio, where the sme mximum perentge dt hnges for deteriorting the effiieny of DMU under nlysis nd the dt hnges for improving the effiienies of the other DMUs simultneously re lulted. In this work, the uthors lso onluded tht the reltionship between the infesibility nd stbility of effiieny lssifition, disovered in Seiford nd Zhu (1998), remins for the simultneous dt vritions se nd for ll bsi DEA models. Generlizing these results, Zhu (2001) onsidered tht the dt perturbtion in the DMU under nlysis nd the dt perturbtions in the remining DMUs n be different when ll the remining DMUs improve their effiienies t the expense of deteriorting the effiieny of the effiient DMU under nlysis. Neessry nd suffiient onditions for preserving effiieny were provided. Our pproh differs from the sensitivity nd robustness nlysis presented by previous uthors in severl spets. It hs been developed for the dditive two-phse model of Gouvei et l (2008). The unertinty in the oeffiients in eh ftor (input or output) is ptured through intervl oeffiients nd onverted into utility sles (whih re lwys to be mximized). An optimisti effiieny mesure nd pessimisti effiieny mesure re omputed. Additionlly, tolerne threshold for eh effiient DMU is determined, tht is mximum tolerne in the ftor sores for whih the DMU s effiieny sttus hnges. Using these two types of effiieny mesures, we n lssify the DMUs s surely effiient, potentilly effiient, or surely ineffiient. Unlike the stndrd super-effiieny models, with speifi orienttions, the model proposed in this pper projets the DMUs in ny diretion in wy tht minimizes the distne of the unit under evlution to the best of ll units (exluding the one in evlution), nd 5

6 therefore no infesibility onerns rise. Adpttion of the two-phse method to ompute super-effiieny sores The two-phse method developed by Gouvei et l. (2008) is vrint of the dditive DEA model with oriented projetions (Ali et l., 1995), whih uses onepts developed in the field of multiple riteri deision nlysis (MCDA) under impreise informtion (Athnssopoulos nd Podinovski, 1997; Dis nd Clímo, 2000). We onsider the DMUs s lterntives of multiple riteri evlution model, eh one being evluted in number of distint riteri. Eh riterion orresponds to n input or n output ftor in DEA models. A diretion of preferene is ssoited with eh riterion: inresing for outputs nd deresing for inputs. The method uses n dditive utility funtion to ggregte the utilities ssoited with eh lterntive, bsed on the Multi-Attribute Utility Theory (MAUT) (see Keeney nd Riff, 1976). Adpting to this ontext, the purpose of MAUT will be to ssess the utility of eh lterntive, onsidering tht the lrger the utility the better. MAUT is lso imed t simplifying the tsk of building the utility funtions when evluting the lterntives tht re desribed by multiple ttributes. In ddition the deision mker's tsk is filitted beuse he n fous the ttention on one ttribute t time nd then mke the ggregtion of ttributes, rther thn mke judgments diretly on the globl utility (the onept of ttribute in this theory is euivlent to our onept of riterion). This overomes the problem of the sles ssoited with the ADD model, sine ll the input nd output mesures re trnslted into utility units. Moreover, the weights used in the ggregtion gin speifi mening: they re the sle oeffiients of the utility funtions. Weights re hosen to benefit eh DMU s muh s possible, rther thn being fixed beforehnd s in the model by Ali et l. (1995). Finlly, the effiieny mesure ssigned to eh DMU gins n intuitive mening: it orresponds to min-mx regret (utility loss) mesure. Considering tht the lterntives re the DMUs to be evluted ording to riteri, we ssume tht the utility of eh lterntive is given by n dditive MAUT model udmu w u DMU j 1 j, where w ³ 0," =1,..., nd w =1(by onvention). The sle oeffiients w 1,...,w re the weights of the =1 utility funtions. The use of this model reuires tht the originl input nd output sles hve to be onverted into utility sles nd there re severl tehniues for uestioning the deision mker, in order to onstrut the utility funtions omptible with their nswers (see von Winterfeldt nd Edwrds, 1986). Hene, fter being onverted into utilities ll riteri re treted s outputs. Gouvei et l. (2008) proposed two-phse method to inorporte preferenes in the ADD model. In this study the two-phse method is dpted to onsider the super-effiieny onept. For tht purpose the following problem is solved: 6

7 min d,w s.t. w u DMU j - w u DMU k =1 w =1, =1 w ³ 0," =1,..., =1, j =1,..., n, j ¹ k (1) The sore denotes the distne defined by the utility differene to the best of ll lterntives (exluding the one under evlution). The im of our pproh is, for DMU k, to lulte the vetor w of utility funtion weights tht minimizes the distne (the utility differene) of this unit to the best one (note tht the best lterntive will lso depend on w), exluding itself from the referene set. Regrding the model of Gouvei et l. (2008), the only hnge is to exlude one onstrint in whih the DMU k under evlution ws ompred to itself (whih is hieved by introduing j k in problem (1)). This hnge implies tht is now llowed to beome negtive. This pproh is identil to tht desribed in Gouvei et l (2008), nd strts by finding the weights (the vribles of problem (1)) tht most benefit the DMU under onsidertion to hve the worst utility loss (lso vrible of problem (1)). Then, the weighted dditive problem n be solved using the optiml weighting vetor w, resulting from (1), to ompute the projeted point in se of the DMU is ineffiient. Phse 1: Convert inputs nd outputs into utility sles. Compute the effiieny mesure,, of eh DMU, k = 1,,n, nd the orresponding weighting vetor. Phse 2: If d k ³ 0 then solve the weighted dditive problem (2), using the optiml weighting vetor resulting from phse 1, w, nd determine the orresponding projeted point of the DMU under evlution. min l,s s.t. z k = - n j=1, j¹k 1l = 1, w s =1 l j u DMU j l ³ 0, s³ 0. - s = u ( DMU k ), = 1,..., (2) If the optiml vlue of the objetive funtion in (1) is not positive, then the DMU k under evlution is effiient. Otherwise it is ineffiient nd is the minimum differene of utility to the best DMU (i.e., the DMU with higher globl utility). However, with the dpttion mde to the originl method we n lso disriminte the effiient units. If < 0, then the DMU is in the set E (extreme effiient); if = 0 nd ll the slks re null in phse 2, then the DMU belongs to E (effiient but not n extreme point); if = 7

8 0 nd not ll the slks re null in phse 2, then the DMU belongs to set F (wekly effiient or frontier point but with non-zero slks). Using this mesure, we n ssess the extent to whih n effiient DMU my worsen its utility while remining effiient. This llows nlyzing the robustness of the lssifition of DMU s n effiient unit in fe of unertin informtion regrding ftor oeffiients. As expet the effiient DMU to be more robust to hnges in input nd output levels. beomes more negtive, we Robustness nlysis nd stbility intervls Consider tht the vlue p j (performne of DMU j in ftor ) is unertin but bounded within the rnge j j U j U ( j j j p p p. This implies u DMU ) u ( DMU ) u ( DMU ), if the ftor is n output, or u U ( DMU j j j ) u ( DMU ) u ( DMU ), if the ftor is n input. Given intervl performnes on eh ftor (input or output), it is possible to ompute n optimisti nd pessimisti effiieny mesure for eh DMU, k =1,,n, using the first phse of the two-phse method. As n exmple, let us onsider tht ll performnes re bound to intervls p U j = p j ( 1-d) p j p j ( 1+d) = p j, with d for instne eul to 5%, 10%, or 20%. For this work, we onsider tht ll performnes re pplied the sme tolerned, but we my onsider tht the tolerne is pplied only to subset of these (inputs or outputs). To present the methodology proposed in this pper, let us onsider s n illustrtion the dt in Tble 1 (displyed in Fig. 1). Tken from Gouvei et l. (2008), these dt hve been modified by dding DMU9, whih is wekly effiient, to portry more possibilities. For this illustrtion let us ssume tht inputs nd outputs re onverted into utilities in liner wy (in prtie these funtions my be onstruted with the lients of the study, refleting their vlue system, see Almeid nd Dis (2012)). So, for the plusible higher tolerne vlue onsidered (in this se = 20%), nd for eh =1,,, we hoose vlue M < min{ p j, j =1,..., n} nd M U > mx{ p U j, j =1,...,n }, nd then we ompute the utilities for eh unit using: u ì ( DMU j ) = í î p j - M M U - M, M U - p j M U - M, if ftor is n output if ftor is n input, j =1,..., n, =1,...,. 8

9 Tble 1: Test dt DMU Y 1 Y M 1 0 M U Considering the nominl vlues for eh DMU nd pplying the two-phse method (1) nd (2), we n build Tble 2 nd rnk the DMUs in terms of optiml utility loss : DMU1 DMU2 DMU3 DMU4 DMU9 DMU8 DMU7 DMU6 DMU5. The lower the vlue of the better, nd if is negtive then the DMU is effiient (DMUs 1 to 4). DMU9 hs = 0 but it is not stritly effiient, sine one of the slks is not null. The remining DMUs hve > 0 nd hene re not effiient. The projetions of ineffiient DMUs were obtined onsidering the weighting vetor tht resulted from phse 1, in whih the liner progrm (1) is solved for eh DMU. Among the effiient DMUs (DMUs 1-4) the first two hve lrger mrgin to derese their performne thn the remining ones. Tble 2. Effiieny mesure nd other results for eh DMU DMU w 1 w 2 s 1 s = = =0.5; 3= = =1 To ompute the optimisti effiieny mesure we onsider the best vlue of the intervls for the DMU 9

10 being evluted nd the worst vlue of the intervls for ll other DMUs. The reverse is onsidered to ompute the pessimisti effiieny mesure. et DMU ttins in ftor given its unertin performne, note tht { } M U > mx p j U, j =1,...,n in the next expressions: u denote the minimum utility tht DMU j j M < min{ p j, j =1,..., n} nd u ì ( DMU j ) = í î p j - M M U - M, M U U - p j M U - M, if ftor is n output if ftor is n input, j =1,..., n, =1,..., U et DMU u denote the mximum utility tht DMU j ttins in ftor given its unertin performne: j u U ì ( DMU j ) = í î p U j - M M U - M, M U - p j M U - M, if ftor is n output if ftor is n input, j =1,..., n, =1,..., To ompute the optimisti effiieny mesure opt for DMU k the following P similr to (1) is solved: min opt s.t. w u DMU U j - w u DMUk =1 w = 1, =1 =1 opt, j = 1,...,n, j ¹ k w ³ 0, " = 1,..., To ompute the pessimisti effiieny mesure d pes for DMU k the following P similr to (1) is k solved: pes min U s.t. w u DMU j - w u DMUk =1 w = 1, =1 w ³ 0, " = 1,..., =1 pes, j = 1,...,n, j ¹ k With this nlysis we n ssess the robustness of the DMU under onsidertion beuse despite being ssessed in pessimisti wy it my keep its effiieny sttus. A DMU is sid to be robust to hnges in its ftors if the DMU remins in the sme sttus fter the hnge. Therefore, the DMU is robustly 10

11 effiient in the rnge of unertinty onsidered. These results re displyed in Tble 3. é Tble 3. ower nd upper limits for the utility loss d opt k, dk pes ù, for eh DMU ëê ûú DMU 5% 10% 20% 1 [-0.177;-0.005] [-0.264;0.082] [-0.436;0.255] 2 [-0.138;-0.009] [-0.203;0.056] [-0.333;0.185] 3 [-0.095;0.031] [-0.158;0.094] [-0.284;0.219] 4 [-0.087;0.048] [-0.160;0.116] [-0.320;0.251] 5 [0.199;0.301] [0.148;0.352] [0.046;0.454] 6 [0.097;0.229] [0.018;0.295] [-0.145;0.428] 7 [0.037;0.155] [-0.024;0.213] [-0.146;0.331] 8 [0.024;0.147] [-0.038;0.209] [-0.161;0.332] 9 [-0.080;0.080] [-0.160;0.154] [-0.320;0.283] DMUs 1 nd 2 re effiient nd remin in this stte when tolerne of 5% is onsidered. DMUs 3 nd 4 do not remin effiient for this tolerne vlue. For this tolerne, DMUs 1 nd 2 re surely effiient, DMUs 3, 4, nd 9 re potentilly effiient, nd DMUs 5-8 re surely ineffiient. If the intervls of unertinty re defined by tolerne of 10% or 20%, then there re no surely effiient DMUs. Figure 1 shows the effiient frontier, given the nominl vlues. The pessimisti evlution of effiieny for DMU1 onsidering the 5% tolerne is portryed by the dshed line. The retngles define the region of tolerne. In this se, DMU1 is in its worst performne level while the remining units re t their best performne levels, nd it remins effiient. u u 1 Figure 1. The unit isount spnned by the pessimisti evlution of DMU1 Figure 2 shows DMU7's optimisti ssessment onsidering tolerne of 10%, whih is portryed by the dshed line. Hene, DMU7 is in its best performne while the remining units re t their worst for this tolerne vlues nd, with this type of nlysis, thedmu7 tht ws in the set of ineffiient units (onsidering the nominl vlues of DMUs) is now in the set of effiient ones. 11

12 u u 1 Figure 2. The unit isount spnned by the optimisti evlution of DMU7 A different type of nlysis tht n be rried out is to ompute the mximum tolerne suh tht the effiient DMUs mintin effiieny, i.e., to ompute stbility intervl in terms of prmeter ffeting ll the ftors: d mx = { d : d pes k (d) = 0} This n be esily omplished by bisetion tehniue (see Tble 4) if the utility funtions re monotonous. Tble 5 shows tht for tolerne greter thn 5.2% the DMU1 is no longer gurnteed to be effiient. For DMU2 the tolerne threshold for being robustly effiient is 5.6%. Tble 4: Computing δ mx using bisetion tehniue et d denote the mximum tolerne vlue (in this pper d = 0.2) et e denote the preision used (in this pper e = 0.001). d := 0; while (d - d ) <e do d ' := d +d 2 if pes ³ 0 then d := d ' end if end while δ mx :=d else d :=d ' 12

13 Tble 5. Effiieny threshold for the DMUs 1, 2, 3 nd 4 DMU δ mx 5.2% 5.6% 2.5% 1.4% It is noteworthy tht DMU1 hd mesure of effiieny highest thn DMU2, but in this nlysis this DMU emerges s being the most robust s it mintins the rnge of effiieny for higher level of unertinty on performnes. Conerning DMUs 3 nd 4, the tolerne threshold for whih they re surely effiient is lower. An exmple with rel world dt This setion revisits n empiril exmple of Cooper et l. (2006) displyed in Tble 6, iming to illustrte the insights tht n be obtined from the pproh proposed in this pper. There re 12 hospitls nd 6 ftors: number of dotors nd nurses nd the reltive unit osts of dotors nd nurses in terms of inputs, nd two outputs identified s number of outptients nd inptients (eh in units of 100 persons/month). In this se we lso onsider tht the higher tolerne vlue is = 20%, nd for eh =1,,, we hoose vlue M < min{ p j, j =1,..., n} nd M U > mx{ p U j, j =1,...,n }. Tble 6: Test dt Inputs Outputs Dotor Nurse Outptients Inptients DMU Number Cost Number Cost Number Number M M U For the purpose of this illustrtion, the performnes of the 12 hospitls re onverted into utilities, using 13

14 liner trnsformtion, s desribed in the previous setion, nd onsidering the tolerne d = 5%, 10%, or 20%. Tble 7 shows the effiieny mesure, weights, slk nd λ vlues, for eh DMU onsidering the nominl vlues, resulting from the pplition of the two-phse method (setion 4). From these results we n onlude tht only three hospitls re working ineffiiently. This effiieny mesure llows to disriminte the effiient units using the super-effiieny onept nd rnk ll DMUs: DMU11 DMU5 DMU1 DMU2 DMU12 DMU10 DMU4 DMU9 DMU7 DMU6 DMU3 DMU8. The DMUs 11, 5, 1, 2, 12, 10, 4, 9 nd 7 re effiient. With the slk vlues obtined by solving problem (2) nd introduing the weight vetor resulting from stge (1), we get the projetions of ineffiient DMUs. Tble 7. Effiieny mesure nd other results, for eh DMU DMU w 1 w 2 w 3 w 4 w 5 w 6 s 1 s 2 s 3 s 4 s 5 s =0.76; 10=0.17; 12= =0.10; 10= =0.63; 11=0.07; ;λ 12= Tble 8 displys the results for eh DMU onsidering the optimisti (for lower limits of the rnges) nd pessimisti (for upper limits of the rnges) perspetives, using the first phse of the two-phse method with = 5%, 10%, or 20%. 14

15 é Tble 8. ower nd upper limits for the utility loss d opt k, dk pes ù, for eh DMU ëê ûú DMU 5% 10% 20% 1 [-0.149;-0.043] [-0.206;0.007] [-0.321;0.092] 2 [-0.140;-0.040] [-0.191;0.007] [-0.306;0.089] 3 [-0.035;0.082] [-0.095;0.131] [-0.220;0.220] 4 [-0.075;0.049] [-0.138;0.106] [-0.270;0.218] 5 [-0.162;-0.037] [-0.229;0.021] [-0.362;0.137] 6 [-0.066;0.108] [-0.156;0.192] [-0.336;0.360] 7 [-0.081;0.075] [-0.164;0.142] [-0.331;0.268] 8 [-0.018;0.115] [-0.084;0.181] [-0.217;0.285] 9 [-0.072;0.051] [-0.135;0.111] [-0.280;0.230] 10 [-0.118;0.072] [-0.275;0.217] [-0.407;0.327] 11 [-0.288;-0.097] [-0.384;-0.002] [-0.574;0.189] 12 [-0.163;0.002] [-0.247;0.082] [-0.420;0.231] Aording to the rnk previously mde nd for tolerne of 5%, DMUs 11, 5, 1 nd 2 re surely (robustly) effiient. The DMUs 12, 10, 4, 9 nd 7 do not remin lwys effiient for the sme tolerne vlue. For tolerne of 10% only DMU11 mintins the effiieny sttus. Using bisetion tehniue we ompute the mximum tolerne vlue for whih the effiient DMUs mintin the effiieny sttus. Observing the stbility intervl limits shown in Tble 9,nd ompring with the rnk estblished bsed on the nominl vlues of DMUs, we n onlude tht the most robust DMU is DMU11, whih oinides with the nlysis bsed on super-effiieny. But the seond most robust one is DMU1, whih ws rnked fter DMU5 in the super-effiieny rnking. Tble 9. Effiieny threshold for the effiient DMUs DMU δ mx 9.2% 9.0% 1.0% 8.2% 0.0% 0.8% 1.1% 10.1% 4.9% This type of nlysis n be better understood when ompnied by the grph in Figure 3 where the behviour of d pes (pessimisti ssessment) is illustrted. In ft DMU11hs better effiieny mesure k nd mintins the effiieny for greter level of unertinty in performnes (δ mx = 10.1%). It is lso possible to see tht despite some DMUs hd better level of effiieny, with respet to the nominl sitution δ = 0%, they re overtken by others with lower effiieny mesure when more unertinty in the performne of these units is ssumed. This is the se of DMU10, whih hs lower d pes for k tolerne of less thn 5.4% nd it is surpssed by DMU7 for higher tolerne vlues. 15

16 Figure 3. Vlues pproximted by liner interpoltion of pes for DMUs 1, 2, 4, 5,7,9,10,11 nd 12 We n lso observe tht for liner utility funtions the urve tht represents the optiml vlue pes is onve. In Appendix Awe prove tht if ll utility funtions re liner, then the funtion tht desribes how pes hnges with is onve. Conluding remrks nd future work This work provides robustness nlysis of eh DMU in presene of intervl dt. A preliminry ssessment of the robustness of eh DMU is obtined using the first phse of two-phse method with modifition to inlude the super-effiieny of effiient units. This method projets the DMUs in ny diretion in wy tht minimizes the distne of this unit to the best of ll (exluding the one under evlution), nd therefore no infesibility ours. Assuming tht the vlues of the DMU performnes in eh ftor (inputs nd outputs) re not known extly, but n intervl of vlues for these performnes n be estblished, it is possible to lulte n effiieny rnge for eh DMU. The effiieny sores for the DMU under nlysis re omputed onsidering its oeffiients in the most unfvourble/fvourble bounds nd ll the other DMU s oeffiients in their most fvourble/unfvourble bounds, in order to ssess the DMU s robustness. This proess enbles to lssify the DMUs s surely effiient, potentilly effiient, or surely ineffiient. The mximum tolerne d suh tht the effiient DMUs mintin effiieny is lso omputed, by using bisetion tehniue. An illustrtive exmple shows tht, ording to this robustness mesure, the DMU with the highest super-effiieny sore is not neessrily the most robust one, i.e., the one with widest stbility intervls. 16

17 In future work we intend to inlude in the model wy to lulte this tolerne given different types of utility funtions. Appendix A et us suppose tht the under nlysis, DMU k, is improved by d z % while ll other DMUs re hnged by d z % in the opposite diretion. The problem in phse 1 is, for this onstnt d z : Considering tht =1,..., re output ftors nd = +1,...,re input ftors, we hve: min d dz k p s.t. w j (1-d z )- M p M U - w kj (1+d z )- M + - M M U - M =1 =1 M U w - p j (1-d z ) M U M U - w - p kj (1+d z ) - M M U - M =+1 w.1=1 w ³ 0, d dz k free =+1 d dz k,jk (1 ) EMMA 1. et d dz k, w be n optiml solution to (1 ). If the performne of DMU k in ftor, p kj, is ltered in ( d z +)%, with in the set of rel, then the optiml vlue to (1 ) is t most - w d z PROOF M U - M p j + p kj + =1 w M U ( - M p j + p kj ). =+1 Problem with DMU k ltered in ( d z +)%: min d dz k s.t. w p j (1-d z -)- M M U - w p kj (1+d z +)- M - M M U + - M =1 =1 w M U - p j (1-d z -) M U - w M U - p kj (1+d z +) - M M U d - M k ( dz ), jk (2 ) =+1 w.1=1, w ³ 0, d dz k free =+1 et dz, w be n optiml solution to (1 ). et B 1,..., k 1, k 1,.., n be the set of indies of 17

18 DMUs for whih there is no slk in (1 ). Note tht B, otherwise d z, w would not be the optiml solution (it would be possible to hve smller d dz k. So, w p j w p j d =1 M U - z w p kj w p kj d - M =1 M U - - M =1 M U - z - M =1 M U - - M w p j w p j d =+1 M U + z w - M =+1 M U + p kj w p kj d - M =+1 M U + z - M =+1 M U - M = d k ( d z ), "j Î B w p j w p j d =1 M U - z w p kj w p kj d - M =1 M U - - M =1 M U - z - M =1 M U - - M w p j w p j d =+1 M U + z w - M =+1 M U + p kj w p kj d - M =+1 M U + z - M =+1 M U - M < d k ( d z ), "j Ï B et D = j w p j (1-d z -)- M M U - w p kj (1+d z +)- M - M M U + - M =1 w M U - p j (1-d z -) M U - w M U - p kj (1+d z +) - M M U - M =+1 = d z - w =1 =+1 M U - M p j + p kj + =1 w M U ( - M p j + p kj ). =+1 Therefore, "j Î B, D j = d z "j Ï B, D j < d z - w M U - M p j + p kj + =1 - w M U - M p j + p kj + =1 w M U ( - M p j + p kj ). =+1 w M U ( - M p j + p kj ). =+1 Thus, d dz k = d k d z - w M U - M p j + p kj + =1 =+1 w M U - M p ( j + p kj ) nd w = w is n dmissible solution to (2 ), nd it is found n upper bound for the optiml vlue of (2 ). Wheres the optiml vlue to (2 ) is denoted by (dz +), then, (dz +) d z - w M U - M p j + p kj + =1 w M U ( - M p j + p kj ). =+1 18

19 Using EMMA 1. we re now ble to prove tht the d dz k is onve funtion of d z, s it ws our purpose. PROPOSITION 1. d dz k, the solution to P (1 ), is onve funtion of d z PROOF et,, be possible vlues for the tolerne pplied to the performnes of eh ftor, x y z onsidering ny DMU k, suh tht A = d y -d x nd d z = td x +t( 1- t)d y, with t Î ] 0,1[. Given proposition 1, if we onsider =-(1-t)A, we hve: + (dz - (1- t)a) d z A w 1- t M U ( - M p j + p kj ) - =1 if we do=t.a, we hve: (dz + t.a) d z =+1 - w t.a with A = d y -d x nd d z = td x +t( 1- t)d y, t Î ] 0,1[. A ( p j + p kj ) w 1- t M U - M M U - M p j + p kj + =1 nd, on the other hnd, w t.a M U ( - M p j + p kj ), So, onsidering d x = d z -( 1- t) A nd d y = d z + t.a nd given wht ws stted erlier, we get + (dx ) d z (dy ) d z A w 1- t M U ( - M p j + p kj ) - =1 - w t.a M U - M p j + p kj + =1 =+1 A ( p j + p kj ) w 1- t M U - M w t.a M U ( - M p j + p kj ). =+1 nd =+1 Considering now the definition of onve funtion nd the upper limits obtined, we hve: t.d k ( d x ) + ( 1- t).d k d y é (1- t) êd k d z ë é t ê d z - w t.a ëê + A w 1- t M U ( - M p j + p kj ) - =1 M U - M p j + p kj + =1 =+1 w t.a M U - M =+1 A ( p j + p kj ) w 1- t M U - M ù ( p j + p kj ) ú = d k d z û ù ú ûú + It follows tht the funtion d dz k is onve. 19

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