On the Consistency of Slacks-based Measure-Max Model and Super-Slacks-based Measure Model

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1 GRIPS Dicui Paper 6-24 O the Citecy Slack-baed Meaure-Max Mdel ad Super-Slack-baed Meaure Mdel Karu Te Nveber 206 Natial Graduate Ititute r Plicy Studie Rppgi, Miat-ku, Tky, Japa

2 O the Citecy Slack-baed Meaure-Max Mdel ad Super-Slack-baed Meaure Mdel Karu Te Natial Graduate Ititute r Plicy Studie Rppgi, Miat-ku, Tky , Japa te@grip.ac.jp Abtract. Slack-baed eaure (SBM) (Te (200), Patr et al. (999)) ha bee widely utilized a a repreetative -radial DEA del. Hwever, thi del, called SBM-Mi here, evaluate the eiciecy a ieiciet DMU reerrig t the urthet rtier pit withi a rage. I ctrat, the SBM-Max del lk r the earet rtier pit ad hece it cre i geerally greater tha the SBM-Mi cre. The Super-SBM del (Te (2002)) evaluate the eiciecy a eiciet DMU reerrig t the earet pit the rtier except itel. We ca reee a cle cecti betwee SBM-Max ad Super-SBM del, becaue the tivati behid the tw del are ae. I thi paper we detrate thi citecy uig a real dataet. Keywrd: Data Evelpet Aalyi, Slack-baed Meaure, SBM-Max, Super-SBM. Itrducti There are tw type del i DEA (Data Evelpet Aalyi); radial ad -radial. Radial del are repreeted by the CCR (Chare-Cper-Rhde) del (Chare et al. (978)). Baically they deal with prprtial chage iput r utput. A uch, the CCR cre relect the prprtial axiu iput (utput) reducti (expai) rate which i c t all iput (utput). Hwever, i real wrld buiee, t all iput (utput) behave i the prprtial way. Fr exaple, i we eply labr, aterial ad capital a iput, e the are ubtitutial ad d t chage prprtially. Ather hrtcig the radial del i the eglect lack i reprtig the eiciecy cre. I ay cae, we id a lt reaiig -radial lack. S, i thee lack have a iprtat rle i evaluatig aagerial eiciecy, the radial apprache ay ilead the decii whe we utilize the eiciecy cre a the ly idex r evaluatig perrace DMU.

3 I ctrat, the -radial SBM del put aide the aupti prprtiate chage i iput ad utput, ad deal with lack directly. Thi ay dicard varyig prprti rigial iput ad utput. The SBM del are deiged t eet the llwig tw cditi. () Uit ivariat: The eaure huld be ivariat with repect t the uit data (2) Mte: The eaure huld be te decreaig i each lack i iput ad utput. The rigial SBM (SBM-Mi) del evaluate eiciecy DMU reerrig t the urthet rtier pit withi a rage. Thi reult i the wrt cre r the bjective DMU ad the prjecti ay g t a rete pit the eiciet rtier which ay be iapprpriate a the reerece. Iput 2 Eiciet rtier A Q SBM-Mi P O B CCR R C S SBM-Max D Iput Figure : Cpari SBM-Mi, CCR ad SBM-Max del We depict the relatihip ag the rdiary SBM (SBM-Mi), CCR ad SBM-Max del by Fig.. Ieiciet DMU P prjecti are Q, R ad S repectively by SBM-Mi, CCR ad SBM-Max. Matheatically, idig S belg t a NP-hard prble, becaue it i a axiizati prble a cvex ucti ver a -cvex regi. Hwever, the prjected pit S idicate that we ca attai a eiciet tatu with le iput reducti ad le utput expai tha the rdiary SBM (Mi) del. We ca ay that the prjecti by the SBM-Max del repreet a practical Kaize (iprveet) by DEA. The ret thi paper i rgaized a llw. Secti 2 itrduce the rdiary SBM-Mi del briely. Secti 3 preet the SBM-Max del, while Secti 4 decribe the Super-SBM del. A uerical exaple i reprted i Secti 5. Secti 6 cclude thi paper. Althugh we preet the del i -rieted del, we ca treat iput- ad utput-rieted del a well. A t retur-t-cale characteritic, we 2

4 preet the ctat retur-t-cale (CRS) cae. Hwever we ca deal with the variable retur-t-cale (VRS) del a well. 2. The SBM-Mi Mdel The SBM del wa itrduced by Te (200) (ee al Patr et al. (999)). It ha three variati, i.e. iput-, utput- ad -rieted. The -rieted del idicate bth iput- ad utput-rieted. Let the et DMU be J,2,,, each DMU havig iput ad utput. We dete the vectr iput ad utput r DMUj by x 2 ( x, x,, x ) T j j j j ad y ( y, y,, y ) T j j 2j j, repectively. We deie iput ad utput atrice X ad Y by X ( x, x,, x ) ad Y ( y, y,, y ) R. () 2 R 2 We aue that all data are pitive i.e. X 0 ad Y Prducti Pibility Set The prducti pibility et i deied uig the -egative cbiati the DMU i the et J a: λ ( x, y ) x j x j, 0 y j y j, λ 0. (2) j j P T, 2,, i called the iteity vectr. The iequalitie i (2) ca be trared it equalitie by itrducig lack a llw: j j j x x j j y y 0, 0, j T T where (, 2,, ) R ad (, 2,, ) R are repectively called iput ad utput lack. (3) 2.2 N-rieted SBM N-rieted r bth-rieted SBM eiciecy i ( x, y ) i deied by 3

5 i [SBM-Mi] ( x, y) i λ,, ubject t i ij j j i r rj j j r i r x x ( i,, ) y y ( r,, ) x y i i r 0 ( j), 0( i), 0( r). j i r r (4) [Deiiti ] (SBM-eiciet) A DMU ( x, y) i called SBM-eiciet i i ( x, y ) hld. Thi ea 0ad * 0, i.e. all iput ad utput lack are zer. [SBM-Mi] ca be trared it a liear prgra uig the Chare-Cper trarati a llw: [SBM-Mi-LP] * i t, Λ, S, S ubject t t t r i i j ij j i S y r r tx x S ( i,, ) r j rj j r S x i ty y S ( r,, ) 0 ( j), S 0( i), S 0( r), t 0. j i r i (5) Let a ptial luti be * * * * * (, t,,, ) Λ S S. The, we have a ptial luti [SBM-Mi] a deied by ( x, y), λ Λ / t, S / t, S / t. (6) i * * * * * * * * * * 3. The Frtier Prducti Pibility Set ad the SBM-Max Mdel We deie the rtier F the prducti pibility et P a llw: 4

6 F i uch that ( x, y ) x Xλ, y Yλ, λ 0. x, y (7) I Figure, the et lie eget (AB, BC ad CD) i the rtier which i -cvex. Fr a ieiciet DMU ( x, y ), we deie the SBM-Max cre a llw: ax ( x, y) ax ubject t x, y F. Reerrig thee variati, everal authr publihed ew luti ethd. Ag the, we itrduce three paper. Fukuyaa et al. (204) develped a leat ditace eiciecy eaure with the trg/weak ticity the rati r eaure uder everal r icludig -r, 2-r ad -r. Thi del utilize ixed-iteger liear prgraig (MILP) t idetiy eiciecy rtier ad hece a cputatial diiculty arie r large-cale prble. Hadi-Vecheh et al. (205) develped a ew SBM del t id the earet pit the eiciet rtier. They utilize the ultiplier r del t id all upprtig hyperplae. It al utilize tware which ue ractial ceiciet (high precii arithetic) t avid l data. Hece, cputatial tie icreae r large-cale prble. Te (206) prped a chee r lvig the SBM-Max prble. Thi ethd require a liited uber additial liear prgra luti r each ieiciet DMU ad eed ixed-iteger liear prgra cde. Althugh the pit thu btaied i t alway the earet pit ad de t alway atiy Paret-Kpa eiciecy cditi, it i acceptable r practical purpe r lvig large cale prble ad r the pit cputatial lad. We utilized thi del r lvig the uerical data i Secti 5. i r x y i i r r (8) 5

7 4. The Super-SBM Mdel The SBM-Max del ai at gettig t the earet pit the eiciet rtier. Thi ccept i i lie with the uper-eiciecy SBM del (Te (2002)) which lve the llwig prgra r a eiciet DMU x, y t eure the iiu rati-cale ditace r the eiciet rtier excludig the DMU x, y. [Super-SBM] * i λ,, ubject t j, j j j j, j j j i r x x y y λ 0, 0, 0. x y i i r r (9) We ca lve the uper-eiciecy SBM del by applyig LP cde jut ce, becaue thi prble belg t a cvex prgraig, i.e., iiizati a cvex ucti ver a cvex regi. Hwever, SBM-Max prble cat be lved i thi aer, becaue it i a axiizati the bjective ucti ver a -cvex regi. See Fig. 2 where the eiciet B i prjected t E the rtier AC with the iiu ditace. Iput 2 A E B O C D Iput Figure 2: Super-SBM r B 6

8 5. A Nuerical Exaple We cpare the tw apprache, i.e. (SBM-Mi + Super-SBM) ad (SBM-Max + Super-SBM), uig the data Japaee uicipal hpital. 5. Data The data were cllected r the Aual Databk Lcal Public Eterprie publihed by the Miitry Iteral Aair ad Cuicati Japaee Gveret, (a) Nuber DMU: 700 hpital ( = 700). (b) Nuber iput: 5. () N. bed (Bed), (2) Expee r uturcig (Outurce), (3) N. dctr (Dctr), (4) N. ure (Nure) ad (5) Expee r ther edical aterial (Material). ( = 5) (c) Nuber utput: 4. () Reveue r perati per day (Operati), (2) Reveue r irt cultati per day (t tie), (3) Reveue r retur t cliic per day (Fllw-up) ad (4) Reveue r hpitalizati per day (Htel). ( = 4) Table exhibit tatitic the dataet. Table : Statitic dataet ( = 700) Bed Outurce Dctr Nure Material Operati t tie Fllw-up Htel Max Mi Average SD Mdel ad Methd We applied SBM-Max ad SBM-Mi del cupled with Super-SBM del, i -rieted cae uder the ctat-retur-t-cale aupti. Fr SBM-Max, we eplyed the ethd develped i Te (206). 5.3 SBM cre The SBM del ud that 66 hpital ag 700 are eiciet. Table 2 cpare tw cre, e SBM-Max cupled with Super-SBM ad the ther SBM-Mi cupled with Super-SBM.. Table 2: Cpari tw cre SBM-Max + Super-SBM SBM-Mi + Super-SBM Average Max Mi St Dev

9 DMU Fig. 3 (SBM-Max + Super-SBM) ad Fig. 4 (SBM-Mi + Super-SBM) exhibit repectively cre 700 hpital i acedig rder where we ca berve big dierece. The rer hw a th trait r ieiciet t eiciet, while the latter exhibit a -th trait. We ca reee a cle cecti betwee SBM-Max ad Super-SBM del, becaue the tivati behid the tw del are ae. H40 H245 H560 H426 H398 H552 H520 H397 H33 H623 H493 H377 H Eiciecy Figure 3: SBM-Max + Super-SBM 8

10 DMU H403 H242 H592 H62 H263 H643 H5 H673 H686 H485 H260 H377 H Eiciecy Figure 4: SBM-Mi + Super-SBM 6. Cclui I thi paper, we have cpared the SBM-Max ad SBM-Mi del cected with the Super-SBM del. The idig idicate that the SBM-Max del i thly cected with the Super-SBM del, althugh the rer eed 3 tie cputati tie tha the latter i ur cae (we utilized DEA-Slver Pr: Saitech-Ic). I e wihe the wrt cae aalyi, the SBM-Mi del i the chice. I ctrat, i iprveet t eiciet tatu i the ai ccer, the SBM-Max del i qualiied better. Ieiciet DMU ca be iprved t the eiciet tatu with le iputreducti ad le utput-elargeet. Thu, the SBM-Max del prpe a eiciet Kaize (iprveet) tl by DEA. Reerece Chare, A., Cper, W.W., Rhde, E., 978. Meaurig the eiciecy decii akig uit. Eurpea Jural Operatial Reearch 2, Fukuyaa, H., Maaki, H., Sekitai, K., Shi, J., 204. Ditace ptiizati apprach t rati-r eiciecy eaure i data evelpet aalyi. Jural Prductivity Aalyi 42, Hadi-Vecheh, A., Jablky, J., Eaeilzadeh, A., 205. The lack-baed eaure 9

11 del baed upprtig hyperplae prducti pibility et. Expert Syte with Applicati, di: Patr, J.T., Ruiz, J.L., Sirvet, I., 999. A ehaced DEA Ruell graph eiciecy eaure. Eurpea Jural Operatial Reearch 5, Te, K., 200. A lack-baed eaure eiciecy i data evelpet aalyi. Eurpea Jural Operatial Reearch 30, Te, K., A lack-baed eaure uper-eiciecy i data evelpet aalyi. Eurpea Jural Operatial Reearch 43, Te, K., 206. Data Evelpet Aalyi a a Kaize Tl: SBM Variati reviited. Bulleti Matheatical Sciece ad Applicati, 6, DOI: / 0