The Modification of BCC Model Using Facet Analysis

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1 RECENT ADVANCES i APPLIED MATHEMATICS The Modificatio of BCC Model Usig Facet Aalysis SAHAND DANESHVAR Applied Mathematics Departmet Islamic Azad Uiversity Tabriz Brach, Tabriz, IRAN sahaddaeshvar@yahoo.com Abstract: I the most of Data Evelopmet Aalysis (DEA) models for avoidig the compariso of Decisio Maig Uits (DMUs) with wea part of efficiet frotier, sally a oarachimedea ifiitesimal is sed as a lower bod for the mltiplier. I BCC model these bods are defied for V i ad U r correspodig to the ipts ad otpts weights. A modificatio of BCC model i ipt orieted is preseted i this paper, where the bod is reqired oly for. It is illstrated throgh a example. Key-Words: Data Evelopmet Aalysis, Facet Aalysis, No Archimedea Nmber Itrodctio The techiqe of DEA itrodced by Chares, Cooper ad Rhodes (CCR) [8] is ow widely employed for the estimatio of mltiple ipt, mltiple otpt prodctive correspodeces ad evalatio of the prodctio efficiecy of DMUs. They provided a liear programmig formlatio to measre the prodctive efficiecy of a DMU relative to a set of referet DMUs. I 979 Chares, Cooper, ad Rhodes [9] gave a short commicatio abot CCR model ad itrodced a o archimedea ifiitesimal ε as a lower bod for ipts ad otpts weights i CCR model. Baer, Chares ad Cooper (BCC) [6] showed that the CCR measre ca be regarded as the prodct of a techical efficiecy (BCC measre) ad a scale efficiecy measre. BCC also provides a modificatio of the CCR liear programmig formlatio (via additioal covexity costrait) to estimate techical efficiecy ad the Retrs to Scale (RTS). As a theoretical costrct, i two latter models ε performs a lower bod for weights to move them from zero ad i fact modified wea parts of frotier ad efficiecy of DMUs which belog to or compared with these parts of frotier. Ali [,] ad Ali ad Seiford [3] preseted a pper bod of ε i mltiplier side sch that bodedess of the evelopmet side for the CCR ad BCC models is maitaied. Related to the BCC measre, the ecoomic otio of RTS is defied. Estimatio of RTS was cotied by Baer ad Thrall [7] based o Baer [4,5] ad Thrall []. They developed rigoros framewor to allow the possibility of mltiple optimal soltios, ad cosider the coseqet problems i estimatig RTS. They preseted some methods to empirically estimatio of bods for variable ( ) of spportig hyperplaes, which pass throght the efficiet poits. As oted i all of the papers after [9], the No- Archimadea Nmber is sed as the lower bod of factor weights i DEA models specially BCC model. These bods pertrbed the wea parts of frotier ad i this maer wea efficiet DMUs were appeared ad wea efficiet DMUs tae a efficiecy vale less the bt these vales are ot real efficiecy vales. Therefore sig ε as the lower bod of factor weights is ot sitable. Here the variatios of tae the feasibility of problem ad the above variatio is the reasoe of iefficiecy of DMU der cosideratio. O the other had to evalate the exact efficiecy of DMUs whicih belog to wea parets of frotier or ISSN: ISBN:

2 RECENT ADVANCES i APPLIED MATHEMATICS DMUs which compare with these parts of frotier it is sfficiet to pertrbe them. To do this first a prosecher is sggested to compte the vale of ε ad the it is sed as the pper bod of. I first sectio, we illstrate the BCC model ad basic otios of facet aalysis, ext pay attetio to how pper bod o effect o wea parts of frotier. Admissible spportig hyperplaes will defied i ext sectio the effect of ε o BCC model efficiecy scores of observed DMUs. Fially we provide a way for sig ε ad sggest modified model for BCC formlatio. Prelimiary Cosider the observed otpt Y = ( y,..., y ), Y, ad the observed ipt s = ( x,..., xm ), for DMU, =,...,. Geerally i DEA Prodctio Possibility Set (PPS), is defied as: T = {(, Y ) otpt vectory ca be prodced from ipt vector } As oted i [6], the PPS correspodig to variable retrs to scale is as follows: T = {(, Y ) λ Y λ Y, λ =, λ, } () = = = For evalatig efficiecy of DMU related to prodctio possibility set T,i ipt orietaio we have the followig liear program. b = mi b sbect to λ b = = = λ = λ, =,...,. b λ Y Y () The optimal vale of () is called radial techical efficiecy of a observatio (, Y ) T. The dal of () is give by: z = max UY sbect to UY =,..., (3) = U, V Or focs here is o ipt orietatio, bt similer reslts ca be developed for the case of otpt orietatio. The set of poits i the frotier for BCC model (ipt or otpt orietatio) ca be partitioed ito three classes, the strogly efficiet poits (SEP), the efficiet poits (EP), ad the wea efficiet poits (WEP). The SEP cosists of the poits which are located at the vertices of the frotier, the EP cosists of the set of efficiet poits which are efficiet at both ipt ad otpt orietatios ad are ot vertices, ad the WEP cosists of the set of poits which are efficiet i the ipt orietatio ad iefficiet i the otpt orietatio or vice versa []. Ecoomic textboos have typically defies RTS oly for sigle otpt sitatio. The extesio of the otio of RTS to the mltiple otpts sitatio is cosidered by [7]. Baer ad Thrall [7] explicitly cosider RTS oly at the poit that is radial techical efficiet. They cosidered proportioate icreases i ipt ad otpt while eepig the ipt ad otpt mix the same as for (, Y ). Assme that (, Y ) is a DMU o the frotier, which is beig cosidered for evalatio. We ted or attetio o the itersectio of the prodctio possibility set T ad the plae P= P(, Y ) = {(, Y ) = α, Y = β Y, α, β } as follows: (see Figre ) P T = {(, Y ) = α λ Y = βy λ Y, = λ =, λ,, α, β } = = If this plae is cosidered i the ew axes, α ad β, the eqivalet set will be defied as: T (, Y ) = {( α, βy ) α λ, Y λ Y, = = = λ =, λ,, α, β } ISSN: ISBN:

3 RECENT ADVANCES i APPLIED MATHEMATICS Let U,V ad represet a optimal soltio for the dal of BCC formlatio for, ). This ( Y poit is radial techical efficiet that is b, so = U Y = = V. I ipt ad otpt space the spportig hyperplae U Y = V passes throgh, ). The itersectio of this ( Y spportig hyperplae ad T is the lieβ ( U Y ) = α ( V ). If the it of measremet ad Y are spposed to α ad β axes, respectively, the this lie will pass throgh ( α, β ) = (, ) (for DMU der evalatio). As oted i [7] there may be prodctio possibilities where there are more tha oe (i fact, ifiite) spportig (tagetial) hyperplaes. For example, as see i Figre, there are ifiite bidig hyperplaes i, ). Hece, the vale of at sch poits. We eed to compte ( Y is ot iqely determied ad as the pper ad lower bods o variables of all spportig hyperplaes which passes throgh sch poits. Baer ad Thrall also described how the liear programmig formlatio i (3) ca be modified to determie these bods. This modifcatio is as follows: max UY for =,..., (4) sbect to UY = = U, V mi UY for =,..., (5) sbect to UY = = U, V The optimal vale for (4) is ad similarly, the optimal vale for (5) is. It is, of corse, possible that = bt the liear programmig algorithm will detect this. The, for ay optimal soltio U,V ad of system (3) we will have: (6) Y P T A (,Y ) β α Fig. P ad T for two ipts oe otpt space ISSN: ISBN:

4 RECENT ADVANCES i APPLIED MATHEMATICS The geerated spportig hyperplaes by, which pass throgh, ) (i. e. ( Y U Y V = ) are admissible spportig hyperplaes for T. As see i Figre, for example i poit (, Y ), = ad <. I the ext sectio we will cosider this id of DMUs. I other words, or modificatio is based o efficiet DMUs with = ad <. These DMUs belog to itersectio of efficiet parts ad wea parts of frotier. 3 No - Archimedea Nmber ε i BCC Model The o-archimeda mber ε was itrodced first i CCR model i [8] by Chares, Cooper ad Rohodes. I 984 the priciples of basic DEA models were sggested. Explorig ε i BCC model gives the followig model: z = max UY sbect to UY =,..., (7) = U ε V ε As oted i litertre, ε chages the wea parts of frotier []. I previse sectio we see that these parts of frotier i (3) are correspodig to the hyperplaes with = ad <, the let s A cotais all of the efficiet DMUs with = ad <. The meas of pertrbatio i these parts of frotier is doot exists ay hyperplae with = i frotier which correspod to (7). For prove the above metio, cosider followig liear programmig: = max ε sbect to UY = UY =,..., (8) = U ε V ε Lemma. The optimal vale of (8) is less tha oe. ε ε ε Proof. Sppose that U, V, ) is the ( optimal soltio of (8) for (, Y ) A. If =, the respect to (8b) we have Uε Y =. Bt with cosiderig the defiatio of PPS ad (8e) we mst have Uε Y > ad this is a cotradictio. The above lemma shows the relatio betwee the chage of wea parts of frotier by sig ε as the lower bods of weights, ad the variable. 4 Modified BCC Model I this sectio we try to modify PPS for BCC model by restrictig variable oly. As metioed before, for efficiet DMUs, correspods to spportig hyperplaes that passes thogh these DMUs which has the miimm slope. Now if for wea parts of frotier hyperplae caot be eqal to oe, the PPS will be modified. I order to do this, first cosider (6) for all efficiet DMUs the ε defies as follow: ε = Max{, for efficiet DMUs} By placig ε as pper bod for variable of reglar BCC model (4), this model is modified as follow: ISSN: ISBN:

5 RECENT ADVANCES i APPLIED MATHEMATICS z = max UY sbect to UY =,..., (9) U V = ε To complete or discssio we prove the followig theorem. Theorem. Model () dose ot chage i efficiet ad strog efficiet DMUs bt oly chages efficiecy of wea efficiet DMUs ad DMUs which are compared with wea parts of frotier. Proof. By cosiderig the defiitio of ε, after sig modified BCC model () for evalatio of observed DMUs for efficiet ad strog efficiet DMUs, i reglar BCC model. As a example, for DMU we have: ε The correspodig to which is satisfied i the above ieqalities there exist U adv where U Y V for =,,,. Also we have V = ad U Y =, so, U Y V = is a admissible spportig hyperplae which passes throgh (, Y ). The this DMU remais efficiet. Now sppose that DMU is wea efficiet with respect of (3). This DMU will be located o the hyperplae with i PPS. I modified model = with respect the costrait ε ( ε < ), we have ot ay hyperplae with = i ew PPS, so DMU caot be efficiet i modified BCC model. Clearly, sice wea parts of frotier are modified the the efficiecy of DMUs, which are compared with these parts of frotier, will be modified too. () is sed. I Table efficiecy vales ad optimal vector ( v,,, ) are show. Also Figre 3 presets the PPS correspodig to DMUs i Table. Table 3 shows the optimal soltio v,,, ) of (4) ad, Table 4 shows the ( optimal soltio ( v,,, ) of (5). Table. Ipt ad otpts of DMUs DMU Ipt Otpt Otpt DMU DMU DMU DMU DMU DMU DMU DMU8... DMU DMU Table. Optimal soltio of BCC model for DMUs DMU Eff v. DMU.... DMU DMU DMU4.... DMU DMU DMU DMU DMU9.... DMU Y Fig. PPS correspod to data i Table The followig example, idicates o advatage of modified BCC model. Example. Cosider Table with data for observed DMUs which have oe ipt ad two otpts. To evalate these DMUs, the BCC model Y ISSN: ISBN:

6 RECENT ADVANCES i APPLIED MATHEMATICS Table 3. The optimal soltio of (4) for DMUs DMU v DMU.... DMU.... DMU3.... DMU4.... DMU DMU DMU DMU8.... DMU9.... DMU Table 4. The optimal soltio of (5) for DMUs DMU v DMU DMU DMU DMU DMU DMU DMU DMU8.... DMU9.... DMU Cosider DMU4. Table 3 ad Table 4 show that this DMU belogs to A. Table 6 shows the reslts of sig () o data i Table. With refrece to Table 5 ad defiitio of ε, ε =.5. As see i Table, DMU4 remaied efficiet i modified model. Also the efficiecy of DMU9 which was a wea efficiet DMU i Table, is iefficiet with efficiecy score eqal to.83 i Table 5, ad the efficiecy of DMU which was compared with wea parts of frotier i previos PPS decreased from.8 to.76. Figre 5 shows ew PPS for modified BCC model. 5 Coclsio This paper first discsses abot BCC model ad the costrcts the bases of facet aalysis o this model. Sectio 3 explais the existece philosophy of ε ad cosiders the sage of ε i BCC model. Sectio 4 illstrates the ew sggestio for sig ε i BCC model ad applies the modified model o data i Example, ad fially we ca see that i modified model the effect of wea parts of frotier was removed withot ay lateral iflece. Table 5.Optimal soltio for modified BCC model DMU Eff. V U U DMU DMU DMU DMU DMU DMU DMU DMU8.... DMU DMU Y Referece [] Ali Agha Iqbal, Data Evelopmet Aalysis: Comptatioal Isses, Compters, Eviromet ad Urba System 4 (994) [] Ali Agha Iqbal, Streamlied Comptatio for Data Evelopmet Aalysis, Eropea Joral of Operatioal Research 64 () (993) [3] Ali Agha Iqbal., Lawrece M. Seiford, Comptatioal Accracy ad Ifiitesimal i Data Evelopmet Aalysis, INFOR 3 (4) (993) [4] R. D. Baer, Estimatig Most Prodctive Scale Size Usig Data Evelopmet Aalysis, Eropea Joral of Operatioal Research 7 (984) [5] R. D. Baer, Retrs to Scale, Scale Efficiecy ad Data Average Cost Miimizatio Y Fig 3. The ew PPS for modified BCC model ISSN: ISBN:

7 RECENT ADVANCES i APPLIED MATHEMATICS i Mlti-otpt Prodctio, Worig Paper, (986) Caregie Mello Uiversity. [6] R. D. Baer, A. Chares, ad W. W. Cooper, Some Methods for Estimatig Techical ad Scale Iefficiecies i Data Evelopmet Aalysis, Maagemet Sciece 3 (9) (984) [7] R. D. Baer, R. M. Thrall, Estimatio of Retrs to Scale Usig Data Evelopmet Aalysis, Eropea Joral of Operatioal Research 6 (988) [8] A. Chares, W. W. Cooper, ad E. Rhodes, Short Commicatio Measrig the Efficiecy of Decisio Maig Uits Eropea Joral of Operatioal Research 3, (979) 339. [9] A. Chares, W. W. Cooper, ad E. Rhodes, Measrig the Efficiecy of Decisio Maig Uits, Eropea Joral of Operatioal Research, (6) (978) [] Chares A., W. W. Cooper,ad Thrall R. M., A Strctre for Classifiyig ad Characterizig Efficiet ad Iefficiet i DEA The Joral of Prodctivity Aalysis, (99), Vol, pp [] W. W. Cooper, L. M. Seiford, ad K. Toe., Data Evelopmet Aalysis Klwer Academic Pblishers, (). [] R. M. Thrall, Overview ad Recet Developmet i DEA: The Mathematical Programmig Approach, Worig Paper 66 (988) Rice Uiversity. ISSN: ISBN: