A Method for Designing Optimal Systems for the Centralized Structures in DEA

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Available olie at http://.rbiau.ac.ir Vol.1, No.

Razava Departet of Matheatic, South Tehra Brach, Ilaic Azad Uiverit, Tehra, Ira Received Sprig 015, Accepted Suer 015 Abtract Thi tud deig optial te for the cetralized tructure i

1 Available olie at Vol.1, No.1, Sprig 015 Joural of New Reearche i Matheatic Sciece ad Reearch Brach (IAU) A Method for Deigig Optial Ste for the Cetralized Structure i DEA Sh.Razava Departet of Matheatic, South Tehra Brach, Ilaic Azad Uiverit, Tehra, Ira Received Sprig 015, Accepted Suer 015 Abtract Thi tud deig optial te for the cetralized tructure i data evelopet aali (DEA). It aue that a collectio of deciio akig uit (DMU) with a ater deciio aker ad a certai budget for the i available ad itroduce a optial te for each DMU b axiizig their total reveue. A erical exaple i ued to illutrate the propoed odel. Keword: DEA, reveue, cetralized tructure, budget, optial te. Eail: h_razava@azad.ac.ir

2 Sh Razava /JNRM Vol.1, No.1, Sprig Itroductio Data evelopet aali (DEA), developed b Chare et al. [1], provide a oparaetric ethodolog for evaluatig the efficiec of each of a et of coparable deciio akig uit (DMU), relative to oe aother [4]. I the origial odel of Chare et. al. [1], efficiec i repreeted b the ratio of weighted output to weighted iput. I a real world applicatio, there are ituatio (uch a bak brache, hopital, uiverit departet) i which all the DMU are uder a cetralized Deciio aker (DM) [5]. The cetralized deciio-aker wat to iiize the overall iput couptio (or to axiize the overall output productio) b all DMUS ([,3]). I recet ear, Wei ad Chag [7] developed optial te deig (OSD) etwork DEA odel to optiall deig a DMU optial etwork. Thee odel ca help DMU deig their optial iput ad output portfolio i ter of profit axiizatio, give their total available budget [6]. Thi paper develop optial te deig DEA odel to cetralized te to deig their optial te. It propoe odel to optiize reource allocatio i ter of reveue axiizatio give the DMU ' total available budget B. The paper ufold a follow. Sectio provide a brief of preliiarie. Sectio 3 propoe a odel for deigig of the optial te for the cetralized tructure. Exaple ad it reult are how i ectio 4 while Sectio 5 cotai cocluio.. Preliiarie Suppoe there are DMU ad each DMU j, j 1,, produce differet output fro differet iput. Let the oberved iput ad output vector of DMU j j 1j j be X ( x,, x ) ad Yj ( 1 j,, j ), repectivel, that all copoet of vector X j ad Y j for all DMU are poitive. The, a geeral productio poibilit et (T ) ca be repreeted a follow: T {( x, ) ca produce b x }. I additio, it i aued that the cot ad price for data are available, ad let C c1 c (,, ) 0 be the iput price, ad P p1 p (,, ) 0 be the output price. We coider the followig potulate to

3 A Method for Deigig Optial Ste for the Cetralized Structure i DEA 17 cotruct a productio poibilit et to defie OSD [7]: 1.Evelopet potulate: The oberved ( x, ) T, j 1,,. j j.covexit potulate: If ( x, ) T, ad ( x, ) T, ( x(1 ) x, (1 ) ) T, [0,1]. the 3.Output iefficiec potulate: If ( x, ) T, ad, the ( x, ) T. 4.Miiu extrapolatio potulate:t i the iterectio et of all T atifig Potulate 1-3. The productio poibilit et that atifie the above defied potulate ca be repreeted a follow: The ext ectio ue T to propoe a ethod to deterie optial te. 3. Deigig of the optial te for the cetralized tructure I cetralized ceario the DM wat to axiize the efficiec of idividual uit at the ae tie that total iput couptio i iiized or total output productio i axiized [7]. Accordig to the defiitio of T, the radial cetralized reource allocatio i iput orieted i a follow: i.t. x x, i 1,, ij ij kj kj, k 1,, 1, r 1,, 0, r, j 1,,. (1) Uig the defied T i ectio, the followig liear prograig i propoed to deterie the correpodig optial reveue-axiizatio te for the cetralized tructure: i k k1 j1.t. x xˆ, i 1,, B ij ij kj kj ˆ, k 1,, i ij i1 j1 1, r 1,, 0, r, j 1,,. p ˆ kj c x B j1 () ˆ k kj,( k 1,, ) ad x ˆ i xij,( i 1,, ) j1 odel (3) i coverted the followig odel:

4 Sh Razava /JNRM Vol.1, No.1, Sprig i p k k k1.t. x x, i 1,, ij i, k 1,, kj k kj i i i1 1, r 1,, 0, r, j 1,,. d c x B (3) Model (3) idetifie the optial reource allocatio with reveue axiizatio whe the total available budget for all DMU i kj j1 B We add the cotrait kj, k 1,, to odel (3) to preerve the total value of output of all DMU. th k k {1,, } Defiitio 1: Uig the optial olutio of odel (3) the optial te for DMU r,( r 1,, ) i defied a follow: xir rj xij, i 1,, j1 kr rj kj, k 1,, p. j1 (4) Theore 1: The cotrait kj k, k 1,, p are bidig r 1 j 1 at the optial olutio of odel (3). Proof: B cotradictio, let rj tj t ad et t {1,, }, t rj tj t 0. The, we have ad it icreae the t t t t optial value, which i a cotradictio. Uig Theore 1 ad the cotrait xij xi, i 1,, odel (3) ca be odified a follow: i.t. p k kj k 1 i ij i1 r 1 j1 kj j1 j1 d c x B 1, r 1,,, k 1,, kj 0, r, j 1,,. (5) Sice odel (3) ad (5) are equivalet, therefore odel (5) ca be ued to deterie the optial te, too, accordig to (4). 4. Nuerical Exaple To illutrate the capabilit of the propoed odel, coider 1 hopital brache (DMU) with two iput ( x 1, x ) ad two output ( 1, ). The data et ha bee how i Table 1. Suppoe that B=4000 available a a total budget for 1 DMU. Accordig to the above dicuio, b odel (5) the optial te for thee DMU are obtaied. The

5 A Method for Deigig Optial Ste for the Cetralized Structure i DEA 19 reult are how i Table. The d ad 3 th colu of Table how the optial value of iput for the optial te ad the colu 4 th ad 5 th how the optial value of output. Table1. The iput ad output of DMU DMU x 1 x Table. The target of DMU accordig to available budget DMU x 1 x

6 Sh Razava /JNRM Vol.1, No.1, Sprig Cocluio Thi paper propoed a odel to deterie optial te for the cetralized tructure i DEA b auig that a collectio of DMU with a ater deciio aker ad a certai budget for the i available. Nuerical exaple ued to illutrate the capabilit of the propoed odel. Ackowledgeet: The fiacial upport of thi paper i ackowledged b Ilaic Azad Uiverit South Tehra Brach i a project etitled: cetralized reource allocatio with iiu cot ad deteriig their optial budget i data evelopet aali.

7 A Method for Deigig Optial Ste for the Cetralized Structure i DEA 1 Referece [1] Chare, A., W.W. Cooper, E. Rhode, Meaurig the efficiec of deciio akig uit. Europea Joural of Operatioal Reearch, (1978) [] Fag, Lei., Zhag, C.-Q., Reource allocatio baed o the DEA odel. Joural of the Operatioal Reearch Societ 59 (008), [3] Korhoe, P., Srjäe, M., Reource allocatio baed o efficiec aali. Maageet Sciece, 50 (004), [4] J., Liag, L., Che, Y., Bi, G., DEAbaed productio plaig. Oega, 38, (010) [5] Lozao, S., Villa, G., Cetralized reource allocatio uig data evelopet aali. Joural of Productivit Aali,, (004) [6] Wei QL ad Chag TS, Optial te deig erie-etwork DEA odel. Joural of the Operatioal Reearch Societ, 6((011)), [7] Weia Q., Chag, DEA-tpe odel for deigig optial te ad deteriig optial budget. Matheatical ad Coputer Modellig, 54 (011)

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Selecting the best of Portfolio in cross efficiency evaluation with negative data

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