DEA Models with Interval Scale Inputs and Outputs

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1 Available olie at It. J. Data Evelopet Aalyi (ISSN X) Vol.2, No.4, Year 2014 Article ID IJDEA-00245,7 page Reearch Article Iteratioal Joural of Data Evelopet Aalyi Sciece ad Reearch Brach (IAU) DEA Model with Iterval Scale Iput ad Output M.Mohaadpour * Departet of Matheatic, Bouka Brach, Ilaic Azad Uiverity, Bouka, Ira. Received 18 Augut 2014, Revied 25 Noveber 2014, Accepted 21 Deceber 2014 Abtract Thi paper propoe a alterative approach for efficiecy aalyi whe a et of DMU ue iterval cale variable i the productive proce. To tet the ifluece of thee variable, we preet a geeral approach of derivig DEA odel to deal with the variable. We ivetigate a uber of perforace eaure with uretricted-i-ig iterval ad/or iterval cale variable. Keyword: data evelopet aalyi; iterval cale variable; ratio cale variable. 1. Itroductio Data evelopet aalyi (DEA) i a o-paraetric ethod for evaluatig the relative efficiecy of deciio-akig uit (DMU) o the bai of ultiple iput ad output. The origial DEA odel ue (poitive) ratio cale iput ad output variable but thee odel do ot apply to variable i which iterval cale data ca appear. With the widepread ue of iterval cale variable, uch a profit or the icreae/decreae i bak accout ephai ha led to the developet of alterative odel aiig at aeig efficiecy i preece of ratio ad iterval cale data i DEA odel.however, iterval cale iput/output caot be ued widely i DEA odel. The proble with iterval cale variable arie fro the fact that ratio of eaureet o uch a cale are eaigle. Coequetly, a DEA odel ca be ued to hadle iterval cale variable i which the ratio of virtual effect iput ad/or output ad oberved iput ad/or output doe ot have ay role i the calculatio. Thi paper deal with the proble of iterval cale data i DEA, the ivetigate a uber of differet perforace eaure. The eaure are odified i uch a way that they alo are able to hadle variable coitig * Correpodig author: : aour_ohaadpour@yahoo.co, Tel.: ; fax:

2 552 M.Mohaadpour /IJDEA Vol.2, No.4, (2014) of ratio cale for oe ad iterval cale for other aple DMU. Thi paper ufold a follow. I ectio 2, we uarize three of the iitial DEA odel: the CCR odel (Chare et al., 1978), the BCC odel (Baker et al., 1984), the additive odel (Chare et al., 1978). I ectio 3 ad 4, we exaie a odified lack-baed eaure odel (Toe, 2001) ad weighted-additive odel of DEA propoed by Pator ad Ruiz (2005), repectively. Cocluio are uarized i ectio The iitial DEA odel Suppoe we have peer oberved deciio-akig uit (DMU) where every DMU j (j = 1,2,, ) produce ultiple output y rj, (r = 1,, ) by utilizig ultiple iput x ij, (i = 1,, ). The iput ad output vector of DMU j are deoted by x j ad y j, repectively, ad we aue x j ad y j are ei poitive, i.e., (x j, y j ) (0, 0)εR, x j 0, y j 0, j = 1,,. We ue by (x j, y j ) to decript DMU j, ad pecially ue (x o, y o ) (oε{1,2,, }) a the DMU uder evaluatio. Throughout thi paper, vector will be deoted by bold letter. Let i (iε{ 1,, } j(jε{1,, } & x ij > 0)), r (rε{ 1,, } j(jε{1,, } & y rj > 0)) The efficiecy of each DMU o ca be evaluated by the evelopet for of the iput-orieted CCR odel by olvig the followig liear progra, i=1 ) i θ ε( i r=1 r λ j x ij i = θx io ; i = 1,,, λ j y ij r = y ro ; r = 1,,, λ j 0, i 0, r 0; j = 1,,, i = 1,,, r = 1,,, where θ i a calar, ad ε > 0 i the o-archiedea eleet. I odel (1) the ratio of virtual effect iput ad oberved iput (poitive data) play a cetral role i the calculatio, thu iput data ut be eaured o ratio cale (Mohaadpour et al., 2015). However, i odel (1) the ratio of virtual effect output ad oberved output ha't ay role i the calculatio; therefore, the odel ca be applied with iterval cale output data. But, ice a auptio of CRS i ot poible i techologie where egative data ca exit (ee Portela et al., 2004), thu the odel (1) caot be ued with the egative iterval cale output data. If oe iput are iterval cale data, a odified ratio cale iput-orieted CCR odel i propoed a follow: (1)

3 M.Mohaadpour /IJDEA Vol.2, No.4, (2014) i θ ε(i=1 i r=1 r ) λ j x ij i = θx io ; iεr ip, λ j x ij i = x io ; iεi ip, λ j y rj r = y ro ; rεr out I out, θ 0, i = 1,, R ip, λ j 0, j; i 0, i; r 0, r. (2) where R ip = {i: iε{1,, } & j(jε{1,, } x ij i a ratio cale variable)} I ip = {i: iε{1,, } & j(jε{1,, } x ij i a iterval cale variable)} R out = {r: rε{1,, } & j(jε{1,, } y rj i a ratio cale variable)} I out = {r: rε{1,, } & j(jε{1,, } y rj i a iterval cale variable)}, (3) ad for iplificatio i uig the defied et let R ip = {1,, R ip }, R out = {1,, R out }, where ybol of. i the cardiality of et. Sice DMU j ( j = 1,, ) are peer, the R ip I ip = {1,, }, R out I out = {1,, }. Model (2), at firt evaluate the radial efficiecie of ratio cale iput data θ i (iεr ip ), the it take accout of the iterval cale iput excee ad output hortfall that are repreet by o-zero lack. Thu odel (2) detect all iefficiecie of iput ad output, if ay. Therefore, odel (2) idicate DMU o with the iterval cale iput ad/or output data i CCR-efficiet. Alo, auig that ((λ 1,, λ ), θ, ( 1,, ), ( 1,., )) be a optial olutio of odel (2), the productio poibility ( λ j x j, λ j y j ) i CCR-efficiet. Model (1) ad (2) are ot tralatio ivariat with repect to either output or iput; however thee odel with covexity cotrait λ j = 1 are tralatio ivariat with repect to iterval cale output. Therefore, odel (2) with covexity cotrait λ j = 1 ca be ued with o-egative iterval cale iput ad egative iterval cale output. Alo, odel (2) with cotrait λ j = 1 both idicate DMU o with iterval iput / output data i BCC-efficiet ad provide a perforace eaure. I odel (2) we aue that R ip. Now, if R ip =, the the followig odel i obtaied: ax i=1 i r=1 r λ j x ij i = x io ; i = 1,,, λ j y rj r = y ro ; r = 1,,, λ j 0, i 0, r 0; j, i, r. (4)

4 554 M.Mohaadpour /IJDEA Vol.2, No.4, (2014) Which i the odel additive odel propoed by Chare et al. (1985) ad it ca be ued with oegative cale iterval iput ad/or output. Model (4) with covexity cotrait λ j = 1 i the additive odel with the variable retur to cale auptio a follow: ax i i=1 r=1 r λ j x ij i = x io ; i = 1,,, λ j y ij r = y ro ; r = 1,,, λ j = 1, λ j 0, i 0, r 0; i, j, k. Model (5) which i tralatio ivariat a deotrated by Ali ad Seiford ca be ued with the iterval cale ad/or egative iterval cale iput ad output data. Alo odel (5) idicate DMU o i BCC- efficiet. I other word, the optial olutio odel of (5) i zero if ad oly if DMU o be BCC-efficiet [2]. The odel (5), however, doe ot provide a efficiecy eaureet for DMU which are ot BCC-efficiet. (5) 3. A odified lack-baed eaure odel to deal egative iterval output ad iput The followig odified lack-baed eaure (MSBM), propoed by Sharp et al (2007), alo provide efficiecy eaure ad obviate the proble aociated with zero weight beig aiged to iput or output. i 1 1 w i i i=1 R i0 1 1 vr r r=1 R r0 λ j x ij i = x io i = 1,, λ j y ij r = y ro r = 1,, λ j = 1 i=1 w i = 1, r=1 v r = 1 λ j 0, i 0, r 0 j, i, r. Where R io = x io i{x ij }, i = 1,, ; R ro = ax{y rj } y ro, r = 1,,. j j To avoid proble with zero, the lack correpodig to variable with R io = 0, for oe i = 1,,, ad/or R ro = 0, for oe r = 1,,, are igored. The ratio of virtual effect iterval cale data ad oberved iterval cale data play a cetral role i the calculatio whoe provide efficiecy eaure i the objective fuctio of odel (5), thu we caot ue the odel with iterval cale data. (5)

5 M.Mohaadpour /IJDEA Vol.2, No.4, (2014) Followig Toe (2001), the fractioal prograig proble repreeted by odel (5) ca be trafored to the followig proble ax λ w iα i i=1 R io v r β r R out r=1 y io r=1 λ j x ij α i = x io i = 1,, λ j y rj β r = y ro r = 1,, λ j = 1 λ 1 i=1 w i = 1, v r = 1 λ > 0; μ j 0, α i 0, β r 0, j, i, r. (6) Alo, a odified verio of odel (6) i propoed a follow: i λ 1 R ip w i α i iεr ip ε (iεi ip α i β r R io λ j x ij α i = x io i = 1,, λ j y rj β r = y ro r = 1,, λ j = 1 λ 1 R out v r β r rεr out = 1 y io i=1 w i = 1, r=1 v r = 1 λ > 0; μ j 0, α i 0, β r 0, j, i, r, rεi out ) (7) to deal with uretricted-i-ig iterval cale iput ad/or output data a well a to take accout of all iefficiecie of iput ad output data, if ay. 4. DEA weighed-additive odel The followig weighted-additive odel, propoed by Pator ad Ruiz (2005), allow u alo idicate DMU o i CCR-efficiet ad provide efficiecy eaure ax where 1 ( i i=1 R i0 r r=1 R ) r0 λ j x ij i = x io ; i = 1,,, λ j y ij r = y ro ; r = 1,,, λ j 0, i 0, r 0; j, i, r, (12)

6 556 M.Mohaadpour /IJDEA Vol.2, No.4, (2014) R io = x io i{x ij } i = 1,,, j = ax{y rj } y ro r = 1,,. R ro j To avoid proble with zero, the lack correpodig to variable with R io = 0, for oe i = 1,,, ad/or R ro = 0, for oer = 1,,, are igored. The ratio of virtual effect iterval cale data ad oberved iterval cale data play a cetral role i the calculatio whoe provide efficiecy eaure i the objective fuctio of odel (12), thu we caot ue the odel with iterval cale data. A odified verio of weighted-additive odel (12) propoed a follow, allow u alo deal with oegative iterval cale data ad provide efficiecy eaure. ax 1 i R i0 ( R ip R out iεr ip rεr out R ) ε (iεi ip i r0 r λ j x ij i = x io ; i = 1,,, λ j y ij r = y ro ; r = 1,,, λ j 0, i 0, r 0; j, i, r, r rεi out ) Model (13) with covexity cotrait λ j = 1 ca be ued with uretricted-i-ig iterval cale iput ad/or output data. Alo, odel (13) take accout of all iefficiecie of iput ad output data (if ay) both to idicate the efficiecy eaure ad to provide the efficiet projectio of DMU o. I additio, auig that (λ 1,, λ ), ( 1,, ), ( 1,, ) be a optial olutio of odel (13) productio poible ( λ j x j, λ j y j ) i BCC-efficiet. (13) 5. Cocluio I thi paper it i preeted a yteatic ivetigatio of the proble of iterval cale data i DEA. The proble with iterval cale variable arie fro the fact that ratio of eaureet o uch a cale are eaigle. Coequetly, a odel of DEA ca be ued to deal with iterval cale output (iput) i which the ratio of virtual effect output (iput) ad oberved output (iput) doe ot have ay role i the calculatio. Alo, it i poited out that the iterval cale variable are ivariat uder ay tralatio traforatio; therefore, a tralatio ivariat DEA odel ca be ued to hadle egative iterval cale variable. It i alo dicued a geeral approach of derivig DEA odel to deal with uretricted-i-ig iterval cale iput ad/or output data.

7 M.Mohaadpour /IJDEA Vol.2, No.4, (2014) Referece [1] A. I. Ali., L. M. Seiford,Tralatio Ivariace i Data Evelopet Aalyi, Operatio Reearch Letter 9 (1990), [2] A. Chare, W.W. Cooper, E. Rhode, Meaurig the efficiecy of deciio akig uit, Europea Joural of Operatioal Reearch 2 (1978) [3] J. Pator, J. Ruiz, I. Sirvet, A Ehaced DEA Ruell Graph EfficiecMeaure, Europea Joural of Operatioal Reearch, 115 (1999) [4] J. Sharp, W Meg, W. Liu, A odified lack-baed eaure odel for data evelopet aalyi with atural egative output ad iput. J Opl Re Soc 58 (2007) [5] M. C. A. S. Portela, E. Thaaouli, G. Sipo, Negative data i DEA: A directioal ditace approach applied to bak brache, Joural of the Operatioal Reearch Society 55 (2004) [6] M Hale, T Joro, M Koivu, Dealig with iterval cale data i data evelopet aalyi. Europea Joural of Operatioal Reearch 137(1) (2002), [7] M. Mohaadpour, F. Hoeizadeh-Lotfi, G.R. Jahahahloo, A exteded lack-baed eaure odel for data evelopet aalyi with egative data, Joural of the Operatioal Reearch Society 66(7) (2015), [8] P.F. Vellae, L. Wilkio, Noial, ordial, iterval, ad ratio typologie are ileadig, The Aerica Statiticia 47 (1993), [9] R.D. Baker, A. Chare, W.W. Cooper, Soe odel for etiatig techical ad cale iefficiecie i data evelopet aalyi, Maageet Sciece 30 (1984) [10] S.S. Steve, O the theory of cale of eaureet. Sciece 103 (1946) [11] W.W. Cooper, L.M. Seiford, K. Toe, Data Evelopet Aalyi: A Copreheive Text with Model, Applicatio, Referece ad DEA-Solver Software, ecod ed., Spriger, 2007.