Improving Efficiency Scores of Inefficient Units. with Restricted Primary Resources

Transcription

1 Applied Matheatical Science, Vol. 3, 2009, no. 52, Iproving Efficienc Score of Inefficient Unit with Retricted Priar Reource Farhad Hoeinzadeh Lotfi * Departent of Matheatic, Science and Reearch Branch Ilaic Azad Univerit, Tehran, Iran Ghola Reza Ain Departent of Coputer Science, Potgraduate Engineering Center Ilaic Azad Univerit of South Tehran Branch, Iran Seed Ali Kharazi Departent of Matheatic, Science and Reearch Branch Ilaic Azad Univerit, Tehran, Iran Rain Enaati Departent of Matheatic, Science and Reearch Branch Ilaic Azad Univerit, Tehran, Iran Abtract An iportant iue in Data Envelopent Anali (DEA) i finding the efficienc core of Deciion Maing Unit (DMU). For iproving the efficienc core of inefficient unit, it i iportant to conider the priar reource fixed and retricted. In thi paper, we propoe a odel to iprove the efficienc core of inefficient unit with retricted priar reource. We alo ugget the perturbation on priar reource in uch a wa that the efficienc core of inefficient unit iproved and the efficient DMU reain unchanged. Keword: data envelopent anali, reallocation, priar reource, perturbation. *Correponding Author: Farhad Hoeinzadeh Lotfi, E-ail: Hoienzadeh_lotfi@ahoo.co

2 2596 F. Hoeinzadeh Lotfi et al. Introduction Data envelopent anali (DEA), originall introduced b Charne et al (978), nonparaetric ethod for evolution the efficienc core of deciion aing unit (DMU). B conidering different technolog auption, a erie of DEA odel, uch a CCR, BCC odel, have been propoed in the literature. Recentl Erouznead et al (2008) publiher a urve on the DEA odel and application. Reallocation i an iportant application of DEA, firt introduced b Golan and Tair (995) propoe an output oriented reource allocation odel in which there are contraint on the total input conuption. Since the affect on the aggregated efficiencie. In thi repect, an reearcher have developed new DEA odel for reallocation proble. For exaple, Coo and Kre (999) a DEA approach for cot allocation proble which are baed on two principle: invariance and pareto-inialit. Their approach cannot be ued directl to deterine a cot allocation aong the DMU. Coo and Zhu (2005) extend the Coo and Kre approach, and provide a practical approach to the cot allocation proble. Jahanhahloo et al (2004) propoe an approach in which without olving linear prograing proble onl uing iple forula, the euitable allocation i achieved. Their ethod i baed on invariance principle of Coo and Kre. Beale (2003) introduced a non-linear reource allocation odel baed on the ratio for that aied at ointl coputing input and output for each DMU for the next period with the obective of axiizing the average efficienc. Recentl Chen et al (2007) ue the relative efficienc of one aignent relative to other intead of eauring the cot or profit. Exiting retriction in reallocation DEA odel detect oe efficient unit a inefficient one. In thi paper we conider the proble of ipoing retriction on the reallocation DEA odel. We aue reallocation flow a the perturbed inforation on the priar reource aong the unit. Thi ean that where an input of an inefficient DMU ove on a the input of efficient unit. Our purpoe, it i to olve thi proble and iproving the efficienc core of inefficient DMU, with the condition in which the efficienc core of efficient DMU are reain unchanged, and alo the efficienc core of inefficient unit increaed a uch a poible. The ret of paper i organized a fallow: Section 2 review the necear preliinarie. Section 3 introduce our propoed odel. Then in ection 4 we illutrate the propoed odel with a nuerical exaple. The concluion rear and further reearch are given in Section Preliinarie Aue that we have n DMU, where DMU (,..., n) conue x (,..., ) t x x a input and produce (,..., ) t a output. For evaluation of DMU under contant return to cale technolog the following linear prograing odel, CCR odel, hould be olved.

3 Iproving efficienc core of inefficient unit 2597 θ ax u * t.. u vx 0, K, n vx u 0, v 0 () Where, u ( u, K, u ) and v ( v, K, v ) are defined a the vector of weight for output and input, repectivel. We partition all DMU into two group. We denote efficient unit a the firt and econd group contain inefficient unit, indicated b E and Ê repectivel. We define p a cardinal of E and a cardinal of Ê and we define aggregate * * * efficienc core a follow: θ θ + θ + + θ 2... n Y A x A C A B x C C x B B D xd D Figure : Before perturbation over the input X Y + x ( l ) A A A A x ( l ) C C C C + x ( l ) B B X Figure 2: After perturbation on input B D B x ( l ) D D D Figure indicate a two-dienional production poibilit et (PPS) coniting four DMU, A, B, C and D. A it i hown A and B are efficient unit, o E {A, B}, and C and D are inefficient unit, o Ê {C, D}. Figure 2 how the otivation of thi tud. A the figure indicate we are looing for a axiu increaing of the input of the unit A and B a follow: x( + l), A, B, and axiu decreaing of the input of unit C and D a x ( l ), C, D, without changing the output ide, in uch a wa that the efficient core of unit A and B reain unchanged and the efficienc core of the other unit are iproved, if it i poible. With thi perturbation, one can iprove the efficienc value of oe inefficient unit without worening the core of efficient unit.

4 2598 F. Hoeinzadeh Lotfi et al 3. Methodolog In thi ection, we deterine a new cale of change that increae a given input of all DMU in E and decreae the input of all DMU in Ê. Our purpoe i to find the axiu perturbation for which all of the efficient DMU reain unchanged. We propoe a odel that cobine p CCR odel b introducing the following contraint: r r r i, i o i i o o o r r r i, i o i i o o o u ( vx + v( x ( + l ))) 0 E u ( vx + v( x ( l ))) 0 Eˆ Where p E and o denote the index of under invetigation input. In addition, we inert the contraint preerve efficient DMU ( E) unchanged. We alo ue ipoe the contraint (2) u p E for all efficient unit, thi r r r p l l 0 for conidering the retricted priar reource. Furtherore aue the contraint xθ l α i introduced to that uppl reuiring input of DMU where α i deterine b deciion aer. Conidering all of the uggeted contraint we have: Max { l,..., l, l,..., l } p t. VX + vl t E t t to t t Er p UY ( v x + v ( x ( + l ))) 0,, t E t i, i o ti i to o o UY ( v x + v ( x ( l ))) 0, Eˆ & t E t i, i o ti i to o o pr pr x l α, Eˆ o u p l l 0 l, l 0, E& Eˆ U 0, V 0, t E. t t (3) Where U ( u,..., u ) t t t V ( v,..., v ) t t t Obvioul, odel (3) i a ulti obective prograing (MOP) and the odel i nonlinear. To linearize, we define,

5 Iproving efficienc core of inefficient unit 2599 v u ti tr Where v & u & v ( i,...,, i o & r,..., & t E ) ti tr to v v v to to to In addition, for tranfor odel (3) to an LP odel, we change obective function of odel (3) to axiization of Z odel: p Max Z l + l t. VX + l v t E t t t to t i, i o ti i o o t i, i o ti i o o u t E r tr tr v to p l 0 l p l + l. So, we have the following LP UY ( v x + ( x ( + l ))) 0,, t E UY ( v x + ( x ( l ))) 0, Eˆ & t E xo l α Eˆ l, l 0, E & Eˆ U 0, V 0 t E. t t * * * * * Where, L ( l,..., lp, l,..., l * ) i an optial olution odel (4), l i the value th reduced in the input o of inefficient DMU, and * th l i the added value to the o input of efficient DMU ( E ˆ, E ). Clearl p+ n. Note that if the priar reource of all DMU are variable, we conider the contraint p l l M. Now we how the following theore. Theore. Model (4) i feaible. Proof. The proof i clear, a that L (0,...,0) i a feaible olution, of the odel. In the following theore we aue the efficient unit reained unchanged. Theore 2. The axiu perturbation on a pecified input for iproving the * efficienc core of inefficient DMU (holding the efficient unit) i L. Proof. On the contrar, aue L * ( l * * * *,..., lp, l,..., l ) denote the axiu perturbation, for which * * * L L, L L (5) * Clearl L i a feaible olution of odel (4) and ineualitie given in (5) are a contradiction. (4)

6 2600 F. Hoeinzadeh Lotfi et al 4. Illutrative exaple In thi ection we deontrate the propoed odel (4) uing 49 chool, a real data et originall give in Charne et al. (98). The data conit five input and three output. The input eaure the education level of other, the highet occupation level of a fail eber, the freuenc of parental viit to the chool, parental involveent with their children, and the nuber of teacher at a given ite. The output are regarding to reading and ath abilit a well a elfetee. Now, we appl odel (4) and obtain following reult CCR odel (), we conclude that E 7& Eˆ 32, and the original aggregate core i θ The reult are uarized in Table. Table : Reult of new odel for chool data η η2 η3 η4 η To ee the reult of appling odel (4), the above table how that perturbation on the econd input for all DMU increae the aggregate efficienc core of all unit fro to Siilar dicuion i true for the reaining colun of Table. Alo we propoe the perturbation on the fourth input i better to be ipleented, a it i the axiu perturbation odel (4) aong the input uing. 5. Concluion In thi paper, we have invetigated the aggregate efficienc core of all DMU and propoed a new DEA odel for perturbation on a pecified input. Alo we have hown that thi perturbation i the axiu perturbation on a pecified input for iproving the efficienc core of inefficient DMU (holding the efficient unit). A a further reearch, we offer to extend our propoed odel that the odified odel could deal with all coponent of input vector. In real world ituation, oe reallocation of reource i poible but it uuall i retricted, epeciall in the hort run. Thu, it ight be ipoible to ove eploee acro the countr, and hiring new eploee (a well a firing old eploee) ight be ver cotl or even ipoible due to hortage of ualified labor (or exiting η Aggregate efficienc core after perturbation on the th input

7 Iproving efficienc core of inefficient unit 260 contract), therefore, we hould ention that our propoed odel In thi paper could be ue reallocation of reource cot for all DMU. Reference [] R. D. Baner, A. Charne, W. W. Cooper, Soe odel for etiating technical and cale inefficiencie in data envelopent anali. Manageent Science, 30(984), [2] A. Charne, W. W. Cooper, E. Rhode, Meauring the efficienc of deciion aing unit. European Journal Operation Reearch, 2(978), [3] A. Charne, W. W. Cooper, E. Rhode, Evaluating progra and anagerial efficienc: An application of data envelopent anali to progra follow through. Manageent Science, 27(98), 668±697. [4] A. Charne, W. W. Cooper, A. Y. Lewin, R. C. More, J. Roueau, Senitivit and tabilit anali in DEA. Annal of Operation Reearch, 2(985), [5] A. Charne, L. Neralic, Senitivit anali of the additive odel in data envelopent anali. European Journal Operation Reearch, 48(990), [6] A. Erouznead, B.R. Parer, and G. Tavare, Evaluation of reearch in efficienc and productivit: A urve and anali of the firt 30 ear of cholarl literature in DEA. Socio-Econoic Planning cience, 42(3): (2008), [7] B. Golan, E. Tair. Evaluating efficienc-effectivene-eualit trade-off: a data envelopent anali approach, Mngt Sci 4(995): [8] E. V. Pachova, Retricted reallocation of reource. European Journal of Operational Reearch, 96(2009), [9] F. Hoeinzadeh Lotfi, S. A. Kharazi, G. R. Ain, A generalized enitivit anali DEA odel. Applied Matheatical Science, [0] J. E. Beale, Allocating fixed cot and reource via data envelopent anali, European Journal of Operational Reearch 47 (2003) [] J. Zhu, DEA/AR anali of the perforance of Naning Textile Corporation. Annal of Operation Reearch, 66(996), [2] G. R. Jahanhahloo, F. Hoeinzadeh Lotfi, N. Shoa, M. Sanei, "An alternative approach for euitable allocation of hared cot b uing DEA, Applied Matheatic and Coputation 53 (2004) [3] L. H. Chen, H. W. Lu, "An extended aignent proble conidering ultiple input and output", Applied Matheatical Modeling, 3 (2007) [4] L. M. Seiford, J. Zhu, Senitivit anali of DEA odel for iultaneou change in all data. Journal of the Operational Reearch Societ, 49(998), 060±07. [5] R. Metter, V. Varga, C. Whbar, An invetigation of the enitivit of DEA to data error. Coputer and Indutrial Engineering, 4(200), 63±7. [6] T. Sowlati, J. C. Paradi, Etablihing the practical frontier in data envelopent anali. Oega, 32(2004), [7] W. D. Coo, M. Kre, " Characterizing an euitable allocation of hared cot: A DEA approach", European Journal of Operational Reearch 9 (999)

8 2602 F. Hoeinzadeh Lotfi et al [8] W. D. Coo, J. Zhu, "Allocation of hared cot aong deciion aing unit: a DEA approach", Coputer & Operation Reearch 32 (2005) Received: Ma, 2009