Measuring the efficiency of Portuguese hospitals with DEA: an approach using the General Algebraic Modeling System

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1 Measurng the effcency of Portuguese hosptals wth DEA: an approach usng the General Algebrac Modelng System ANTÓNIO XAVIER Faculdade de Cêncas e Tecnologas and CEFAGE-UE (Center For Advanced Studes n Management and Economcs) Unversdade do Algarve Gambelas Campus, Edf. 8, Faro, Portugal PORTUGAL amxav@sapo.pt Abstract: - The Data Envelopment Analyss (DEA) s a Lnear programmng based technque for evaluatng the relatve effcency of Decson Mang Unts (DMU's) provdng a new defnton of effcency for use n evaluatng the actvtes of not-for-proft enttes partcpatng n publc programs. Snce ts creaton the methodology has evolved to more complex framewors and, n a relatvely short perod of tme, t has grown nto a powerful quanttatve, analytcal tool for measurng and evaluatng performance. Smlarly, DEA software technology has emerged from ts academc roots nto producton usage. However, there are stll several optmzaton pacages that offer consderable possbltes. Therefore, n ths paper, we demonstrate and analyze some of the advantages of usng a partcular modelng system: GAMS (General Algebrac Modelng System). In order to do that, we apply the approach to a case study: the effcency analyss of several Portuguese Hosptals. The results seem promsng, snce the approach allowed us to obtan several results that the standard DEA pacages can t provde. Key-Words: - Data envelopment analyss, GAMS, effcency analyss, EMS, constant returns to scale model, varable returns to scale model. 1 Introducton Increasng health care costs are a maor concern of many governments n the world, ncludng the Portuguese government. Governments and the general publc are concerned that patents receve the approprate level of care and that the care s delvered as effcently as possble. Therefore, accordng to Parer and Newbrander cted by Al- Shammar [1], one ey component of health sector efforts to mprove operatng effcency has to do wth mang the best use of exstng resources. In ths context, Data Envelopment Analyss (DEA) s a Lnear programmng (LP) based technque for evaluatng the relatve effcency of Decson Mang Unts (DMU's). Ths methodology was created by Charnes et al. [8] who presented a programmng model provdng a new defnton of effcency for use n evaluatng actvtes of not-forproft enttes partcpatng n publc programs. Snce then, the methodology has evolved n more complex framewors and, n a relatvely short perod of tme, DEA has grown nto a powerful quanttatve, analytcal tool for measurng and evaluatng performance [9]. Smlarly, DEA software technology has emerged from ts academc roots nto producton usage, accompaned by expectatons of advanced modelng optons and professonal mplementatons, ncludng graphcal user nterfaces, nteroperablty wth other applcatons, and the ablty to qucly evaluate large populatons [6]. However, n spte of such developments there are several modelng systems sparsely used, but that present consderable possbltes. One of these systems s GAMS (General Algebrac Modelng System). Ths modelng system offers enormous possbltes for DEA problem solvng and has the man advantage of allowng the nserton of restrctons or combnng several analyses, and taclng some specfc ssues of ths methodology. Therefore, the obectve of ths paper s to exemplfy the potentaltes of ths modellng system n comparson to other systems, namely, Effcency Measurement System (EMS). To do so, a mathematcal programmng code wll be created, based on the prevous exstng nformaton and ts results wll be compared wth the ones presented by EMS n a set of hosptals n Portugal, n order to ISBN:

2 satsfy the current analyss needs for the decson maers of ths feld. The remander of ths paper s organzed as follows: n secton 2 the DEA models consdered are explaned; n secton 3 the prevous studes are descrbed; n chapter 4 the methodologcal approach s presented and explaned; n secton 5 the results are dscussed. Fnally, secton 6 presents the man conclusons of ths wor. 2 The DEA analyss The DEA analyss was developed by Charnes et al. (CCR) [8] who used t for the estmaton of multple nput, multple output producton correspondences and the evaluaton of the productve effcency of DMUs. The basc DEA model presented by Charnes et al. [8] (CCR) consders that we have a set of n DMUs each of whch utlzes nputs to produce outputs. The nputs and outputs for all of the DMUs are assumed to be strctly postve when calculatng the relatve effcency of each DMU. The multpler verson of the DEA model used to calculate the relatve effcency of DMU s as follows: u. y Max( eff ) = (1) v. x u. y v. x 1 (2) u, v (3) Where u s the weght for each output, v s the weght of each nput, y are the outputs produced by each unt ; x are the nputs used by each unt ; y are the outputs n unt ; x are the nputs n unt. Equaton 1 ntends to maxmze the effcency of DMU and t s a fractonal programmng model for computng techncal effcency whch can be solved wth nonlnear technques [14]; equaton 2 mples that the weghts calculated for each unt are lower or equal than one; and equaton 3 mples that the weghts calculated are strctly postve or. The above rato form has an nfnte number of solutons; f (u*, v*) s optmal, then (αu*, αv*) s also optmal for α > [1]. Ths equaton may be rewrtten nto an ordnary lnear programmng problem by proceedng as follows (assumng an nput orentaton): = Maxu v u y,. (4) v. x = 1 (5) u. y v. x (6) u, v (7) Where the obectve functon s equaton 4; equaton 5 determnes that the sum of the vrtual weghts of the nputs s one and together wth equaton 6 determne ts orentaton; equaton 7 guarantes that the weghts are postve or. Ths model s the Constant Returns to Scale model (CRS). We can also draw a dual of ths formulaton, whch may be preferable sometmes for modelng ssues: Mnλ = θ (8) zo y y λ, (9) θ x λ x (1).,, λ (11) Fnally, to obtan a Varable Returns to Scale (VRS) model one ust has to use the prevous model by addng the followng restrcton: λ =1 (12) The formulae of the VRS model may be also drawn as follows [5]: Maxz = u. y u (13) u. y v. x u (14) v. x = 1 (15) v ε (16) u ε (17) 3 Prevous studes The DEA lterature s qute extensve, and dfferent models have been proposed and several emprcal applcatons to dverse contexts have been documented. Wth regards to the programmng language used to calculate the effcency scores, one ISBN:

3 can conclude that very few studes have used GAMS. Below we dscuss the DEA studes that have used ths programmng language. Barr [6] presents the GAMS as a tool for DEA analyss. The GAMS/DEA module provdes language extensons specfcally for DEA. Walden and Krley [17] present several programs n a GAMS programmng language for modelng producton effcency and fshng capacty whch were prepared for the Natonal Marne Fsheres Servce Worshop on Capacty Estmaton n Marne Fsheres. Xue and Harer [19] present a GAMS language code whch allowed the use of bootstrappng to provde addtonal nformaton for statstcal nference. Fnally, Haas et al. [13] use the dual model for VRS and /CRS DEA model n order to analyze the man football teams n Germany. 4 The methodologcal approach For developng the methodologcal approach of mplementaton of the DEA models n GAMS, we consdered a code that could be oned n a sngle fle and that would dsplay a large amount of useful nformaton. The requstes were that the code to be developed should automatcally provde the followng nformaton: effcency scores for CCR and BCC models consderng nput and output orentaton; benchmarng values defnng the reference DMUs; the vrtual weghts; the scale effcency scores; and other statstcal ndcators (averages, medans, summares ). Therefore, the approach requres several formulatons n order to obtan the relevant nformaton. Frst, we conceve a model based on the Dual (envelopment form) for generatng automatcally effcency scores and ts benchmars values for the CCR and VRS models n nput and output, as presented below: Input orentaton Mnλ = θ (18) zo y y λ, (19) θ x λ x (2).,, λ =1 (21) λ (22) Output orentaton: Maxλ = θ (23) z. x,. x, λ (24) y, λ y, θ. (25) λ =1 (26) λ (27) The varaton for the BCC model, under VRS, s made smply by omttng or usng restrctons 21 and 26 respectvely. λ wll provde the ntensty value to be appled to each reference DMU. For defnng the vrtual weghts, the formulaton s presented as follows for an nput orented CCR model: = Maxu v EF, (28) EF = u. y (29) v. x = 1 (3) u. y v. x (31), u v (32) And for the VRS model, we may also rewrte t as follows: = Maxu v EF, (33) EF = u y + u (34). v. x = 1 (35) u. y + u v. x (36) ISBN:

4 , u v (37) Where u s a scale varable, whch resulted from ncludng the addtonal restrcton (equaton 26) n the envelopment model. We fnally add dsplayng routnes for the model to automatcally calculate the techncal and scale effcency, dsplayng the vrtual weghts and for calculatng some other statstcal parameters. may be better presented f ntended, for nstance). The followng fgure presents the results of the programmng codes created n GAMS envronment The emprcal and techncal mplementaton The adaptaton of the current methodology to the data used n the model and to the specfcs of such analyss was made, snce there was the necessty for carefully selectng the data of the healthcare sector used. A sample of 43 hosptals operatng n Portugal n 25 was selected. The nputs and outputs are partally substtutable between them followng the assumptons of ths methodology. In partcular, 3 nputs and 5 outputs were consdered for the DEA analyss. The nput varables consdered were: Doctors, Nurses and Other staff. The output varables consdered were: N.º of npatent epsodes; N.º of outpatent consultatons; N.º of emergency epsodes; N.º of npatent surgeres; N.º of baby delvery epsodes; N.º of day surgeres. In order to techncally mplement ths analyss, we codfed the data nto two fles whch were later unfed nto a sngle fle. We selected the best lcensed solvers consderng also the use of MINOS, OSL or CONOPT. Moreover, n ths emprcal applcaton several parallel models wth the same characterstcs were also developed n the Effcency Measurement System (EMS, verson 1.3) n order to obtan a term of comparson. Other mportant ssue s that these results are not subect to normalzaton and requre a calculaton of the normalzed vrtual weghts. In order to compare the results wth those provded by EMS, for the DMUs classfed as neffcent, we calculated the normalzed vrtual weghts by dvdng each vrtual weght by the total obtaned for each DMU. 5. Results The approach generated a complete set of effcency scores for all the approaches conceved, as well as other data regardng statstcal and orderng parameters that are not avalable n ordnary programs (questons regardng benchmars Fg. 1- The results from the GAMS program Results are also presented n table annex one for effcency scores, whch allowed also the elaboraton of table 1 and 2 presented bellow. Table 1-Synthess nformaton on the effcent unts n the nput orentated models tested nput Indc/models CRS/CCR VRS/BCC Scale effcency Average,839,91,913 St dev.,144,123,11 Mn,513,555,513 Max 1, 1, 1, Table 2- summary results of the model Models/orentaton 5.1. Analyss of the results All models Input All models 15 - CRS/CCR - 15 VRS/BCC - VRS/BCC - 25 Scale effcent (o) - Scale effcent () - 16 The results of the data provde valuable nformaton regardng the DMUs analyzed n ths paper. The frst mportant concluson s the fact that regardng the techncal effcency ndex there are 15 unts (34% of the DMUs) that are effcent both under CRS and VRS assumptons, suggestng that these hosptals show both pure techncal effcency and scale effcency. Moreover, we are also able to dentfy the 1 hosptals that present pure techncal effcency, whlst showng scale neffcency. The remanng 18 ISBN:

5 hosptals present both pure techncal neffcency and scale neffcency. The effcent hosptals that are references for learnng for each of the neffcent ones are also dentfed by the GAMS solver. The nformaton regardng the type of neffcency dentfed n each DMU and the references for learnng s very useful for polcy and practce. 5.2.Comparson wth EMS The GAMS program allows the establshment of the routnes and codes usng the theoretcal formulae, wth the use of powerful solvers that allow a good accuracy of the results. However, for users that are not famlar wth the mathematcal programmng, t may be dffcult to use. The man advantages and dsadvantages of usng the GAMS and the EMS solver are presented n table 3. Our comparson of the results provded by GAMS and EMS solver allowed us to confrm that t s possble to obtan dfferent sets of optmal weghts for the effcent DMUs, dependng on the solver used. Ths reveals the exstence of alternatve sets of optmal weghts for the DMUs classfed as effcent, as dscussed by other authors. Table 3- Comparson of GAMS wth EMS Dmenson of GAMS EMS analyss Solvers Statstcal functons Mathematcal Knowledge Several solvers for use n dfferent optmzaton problems Many extra-functons and statstcal measures may be added after runnng the models. Almost unlmted possbltes of modelng. Requres detaled nowledge of the Mathematcal formulaton Lmted use for the exploratory analyss of data. Several statstcal functons and t has a lmted varety of analyses No detaled nowledge of the Mathematcal formulaton s needed 6. Concluson Ths paper evaluated the use of GAMS programmng software for constructng several lnes of programmng codes for the DEA models. The GAMS allows the automatc calculaton of several DEA models followng a set of pre-defned lnes of programmng codes, beng able to do now more man routnes than the ones present n specfc DEA software programs. Moreover, the comparson wth an ordnary DEA EMS software also revealed mportant conclusons. In specfc, by usng GAMS one can wor wth nonstandard models and obtan nformaton that s not provded by DEA software pacages, such as EMS. Therefore, t s our convcton that GAMS has strong potental as a solver for DEA analyss References: [1] Al-Shammar, M., A mult-crtera data envelopment analyss model for measurng the productve effcency of hosptals, Internatonal Journal of Operatons & Producton Management, vol. 19, nº 9, 1999, pp [2] Appa, G., Wllams, H., A new framewor for the soluton of DEA models. European Journal of Operatonal Research, vol. 172, nº 2, 26, pp [3] Baner, R.D., Charnes, A., and Cooper, W., Models for the estmaton of techncal and scale neffcences n data envelopment analyss, Management Scence, nº 3, 1986, pp [4] Baner, R., Estmaton of returns to scale usng Data Envelopment Analyss. European Journal of Operatonal Research, nº 62, 1992, pp [5] Baner, R., Cooper, W. Seford, L., Thrall, R., Zhu, J., Returns to scale n dfferent DEA models. European Journal of Operatonal Research, nº 154, 24, pp [6] Barr, R., DEA software tools and technology: A State-of-the-Art Survey, 25. Accessed at [ do= ]. [7] Barros, C., Measurng effcency n the hotel sector, Annals of Toursm Research, vol. 32, nº 2, pp [8] Charnes, A., W. Cooper, & E., Rhodes, Measurng the effcency of decson-mang unts, European Journal of Operatonal Research nº 2, 1978, pp [9] Cooper, W., Seford, L., Zhu, J., Data envelopment analyss, W. D, Accessed at [ [1] Daz, V., Banng Effcency and Remttances: The Mexcan Case, 21 Conference Amercan Economc Assocaton, 21. Accessed at [ m/retreve.php?pdfd]. [11] Emrouznead A., Parer B., Tavares, A., Evaluaton of research n effcency and productvty: A survey and analyss of the frst 3 years of scholarly lterature n DEA. Soco- Economc Plannng Scences, vol. 42, nº 3, 28, pp ISBN:

6 [12] Glover F.; Sueyosh T., Contrbutons of Professor Wllam W. Cooper n Operatons Research and Management Scence. European Journal Of Operatonal Research, vol. 197, nº1, 29, pp [13] Haas, D. Kocher, M. and Sutter, M., Measurng Effcency of German Football Teams by Data Envelopment Analyss, Unversty of Innsbruc, 23. [14] Kalvelagen, E.,, Effcently solvng DEA models wth GAMS, W.D.. [15] Olesen, O.; Petersen, N. (1996). A presentaton of GAMS for DEA. Comput. Oper. Res. 23(4): [16] Scheel, H., EMS: Effcency Measurement System User s Manual, 2. [17] Seford, L., Thrall, R., Recent Developments n DEA: The Mathematcal Programmng Approach to Fronter Analyss, Journal of Econometrcs, nº 46, 199, pp [18] Walden, J. and Krley, J., Measurng Techncal Effcency and Capacty n Fsheres by Data Envelopment Analyss Usng the General Algebrac Modelng System (GAMS): A Worboo. U. S. Department of commerce, Washngton D.C, 2. [19] Xue, M., Harer, P., Overcomng the Inherent Dependency of DEA Effcency Scores: A Bootstrap Approach, Tech. Report, Department of Operatons and Informaton Management, The Wharton School, Unversty of Pennsylvana, 199. Table annex 1-The effcency ndex scores results for the studed DMUs Input orentaton Output orentaton CRS/CCR VRS/BCC Scale effcency CRS/CCR VRS/BCC Scale effcency H1 1, 1, 1, 1, 1, 1, H2 1, 1, 1, 1, 1, 1, H3 1, 1, 1, 1, 1, 1, H4,93,94,99 1,7 1,7 1, H5 1, 1, 1, 1, 1, 1, H6 1, 1, 1, 1, 1, 1, H7 1, 1, 1, 1, 1, 1, H8 1, 1, 1, 1, 1, 1, H9,86 1,,86 1,16 1, 1,16 H1,7 1,,7 1,43 1, 1,43 H11 1, 1, 1, 1, 1, 1, H12,73 1,,73 1,37 1, 1,37 H13,71,72,98 1,41 1,19 1,18 H14,82,82 1, 1,22 1,16 1,5 H15,81,84,96 1,24 1,14 1,8 H16,95,96,99 1,5 1,3 1,1 H17 1, 1, 1, 1, 1, 1, H18 1, 1, 1, 1, 1, 1, H19,78,85,92 1,28 1,14 1,12 H2,89,97,92 1,12 1,3 1,9 H21,99 1,,99 1,1 1, 1,1 H22,98 1,,98 1,2 1, 1,2 H23,91,99,92 1,11 1,1 1,9 H24,84,91,92 1,2 1,8 1,11 H25 1, 1, 1, 1, 1, 1, H26,98 1,,98 1,3 1, 1,3 H27 1, 1, 1, 1, 1, 1, H28,51 1,,51 1,95 1, 1,95 H29,6,64,94 1,66 1,48 1,12 H3 1, 1, 1, 1, 1, 1, H31,91 1,,91 1,1 1, 1,1 H32 1, 1, 1, 1, 1, 1, H33,75,76,99 1,33 1,21 1,1 H34 1, 1, 1, 1, 1, 1, H35,76 1,,76 1,32 1, 1,32 H36,83,95,87 1,2 1,4 1,15 H37,56,6,93 1,78 1,4 1,27 H38,85,85 1, 1,18 1,18 1, H39,54,55,97 1,85 1,67 1,11 H4,8 1,,8 1,25 1, 1,25 H41,72,75,96 1,39 1,31 1,6 H42,91 1,,91 1,1 1, 1,1 H43,72,73,98 1,39 1,26 1,1 (source: model results) ISBN:

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