Research Article Efficiency in the Worst Production Situation Using Data Envelopment Analysis

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1 Advaces i Decisio Scieces Volume 2013, Article ID , 9 pages Research Article Efficiecy i the Worst Productio Situatio Usig Data Evelopmet Aalysis Md. Kamrul Hossai, 1 Ato Abdulbasah Kamil, 1 Adli Mustafa, 2 ad Md. Azizul Bate 3 1 School of Distace Educatio, Uiversiti Sais Malaysia, Pulau Piag, Malaysia 2 School of Mathematical Scieces, Uiversiti Sais Malaysia, Pulau Piag, Malaysia 3 Departmet of Decisio Sciece, School of Quatitative Scieces, Uiversiti Utara Malaysia, Sitok, Kedah, Malaysia Correspodece should be addressed to Md. Kamrul Hossai; kamrulstat@gmail.com Received 28 Jue 2012; Accepted 10 Jauary 2013 Academic Editor: Graham Wood Copyright 2013 Md. Kamrul Hossai et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Data evelopmet aalysis (DEA) measures relative efficiecy amog the decisio makig uits (DMU) without cosiderig oise i data. The least efficiet DMU idicates that it is i the worst situatio. I this paper, we measure efficiecy of idividual DMU wheever it losses the maximum output, ad the efficiecy of other DMUs is measured i the observed situatio. This efficiecy is the miimum efficiecy of a DMU. The cocept of stochastic data evelopmet aalysis (SDEA) is a DEA method which cosiders the oise i data which is proposed i this study. Usig bouded Pareto distributio, we estimate the DEA efficiecy from efficiecy iterval. Small value of shape parameter ca estimate the efficiecy more accurately usig the Pareto distributio. Rak correlatios were estimated betwee observed efficiecies ad miimum efficiecy as well as betwee observed ad estimated efficiecy. The correlatios are idicatig the effectiveess of this SDEA model. 1. Itroductio Producer performace 1 is iflueced by three pheomea: efficiecy with which maagemet orgaizes productio, the effect of evirometal factors, ad radom error [1]. If all of these pheomea are iflueced positively to productio, the it is the best situatio for productio of the DMU, ad if all of these pheomea are iflueced the productio egatively, the it is the worst situatio. Several models of sigle stage, or multistage have bee proposed to icorporate evirometal factors i DEA. Baker ad Morey [2] proposed a sigle-stage DEA method for evirometal variables. A obstacle of this method is that the directio of impact o productio of the evirometal factors must be kow i advace. I some research, a two-stage approach is used to describe the effect of evirometal factors. First stage calculates efficiecy from DEA based o iputs ad outputs ad a secod-stage regressio aalysis tries to explai variatio i first-stage efficiecy score. A step further, developmet of the two-stage method was doe by McCarty ad Yaisawarg [3] adbhattacharyyaetal.[4] byusigthesecod-stage regressio residuals to adjust the first-stage efficiecy scores. Iput orieted DEA to iputs ad evirometal factors or output orieted DEA to outputs ad the evirometal factors applied i the first stage. The, the iputs or outputs were replaced by their residual projectios. I the secod stage of this method agai applied DEA to expaded data set cosistig of the origially efficiet observatios, the origially iefficiet observatios, ad the radial projectios of the origially iefficiet observatios. Camaho et al. [5] suggested a method of DEA cosiderig iteral ad exteral odiscretioary factors. I this method, they geeralized models of Baker ad Morey [2] ad their extesio to both odiscretioary iputs ad outputs by Golay ad Roll [6], the model of Ruggiero [7] ad its extesio described by Ruggiero [8]. Lotfi et al. [9] measured the relative efficiecy of decisio makig uits with odiscretioary iputs ad iterval discretioary data. They showed upper boud by assumig the best performace of a DMU agaist the worst performace of the rest of other DMU ad similarly lower

2 2 Advaces i Decisio Scieces boud by assumig the worst performace of a DMU agaist the best performace of the rest of other DMU. Sadjadi ad Omrai [10]presetedDEAmodelwithucertaidata. Simar ad Wilso [11 13] proposed bootstrap algorithm to study statistical properties of DEA models. I bootstrap method, it is difficult to fid a appropriate value of a smoothig parameter. Aother problem of this method is the large umber of iteratios. Etai et al. [14] itroduce a method to calculate iterval efficiecy. Kao ad Liu [15] used the iterval data to measure the iterval efficiecy. Our method will provide a iterval efficiecy, the ovelty we will be takig ito cosideratio is that the study approach lies with a assumptio of oise data i the existig DEA method ad bouded Pareto distributio is used to estimate efficiecy scores at a time. Themaifocusofthispaperistodevelopamethodfor DEA icludig oise i data. Toward the ed, we formulate amethodofsdeausigbccmodel.theobjectiveofthe paperistomeasuremiimumefficiecyofdmus.therest of the study is orgaized as follows. Sectio 2 deals with the backgroud of this study. Sectio 3 develops a methodology for SDEA ad characteristics of bouded Pareto distributio. Sectio 4 presets ad discusses a empirical example. Ad coclusios are icluded i the fial sectio. 2. Backgroud Data evelopmet aalysis is a oparametric method i operatios research for measure productio efficiecy of DMUs. It is a popular method to measure performace of DMUs. Farrell [16] was motivated to develop better methods ad models for evaluatig productivity performace: Not icorporatig oise DEA (1) CCR DEA model (2) BCC DEA model Data evelopmet aalysis Icorporatig oise DEA (1) Chace costraied DEA (2) Fuzzy DEA (3) Bootstrap DEA (4) Imprecise DEA (5) Robust DEA The iitial DEA model, as preseted by Chares et al. [17] ad built o the earlier work of Farrell [16], is kow as CCR model. It is a costat returs to scale model. CCR model is as follow: Max θ,λ θ i x i θy i λ i x i, λ i y i, (1) where x i 0is the iput ad y i 0is the output for the ith firm. λ i 0represets itesity of variable. θ i is the efficiecy of the ith firm. I the CCR model, if a costrait λ i =1is adjoied, the it is kow as BCC models [18]. Chace Costrait DEA (CCDEA) [19] cosiders two sets of costraits. There is a set of chace costraits that probability of efficiet of all DMUs is oly 0.05 or less. The CCDEA model is Mi θ Prob ( λ i y i y i ) 0.05, θx i λ i x i, θ urestricted i sig. Fuzzy DEA is a extesio of the CCR DEA model icorporatig fuzzy umbers [20]. The Fuzzy DEA model is Max μ t Y 0 v t X 0 I, μ t Y j v t X j (2) (j=1,2,...,), μ 0, v 0, (3) where X j ad Y j are s-dimesioal L-L fuzzy iput vector ad m dimesioal fuzzy output vector of jth DMU. v ad μ are coefficiet vectors of X j ad Y j, respectively. The idex o idicates the evaluated DMU. The bootstrap method itroduced by Efro [21] is used i Bootstrap DEA [11 13]. Algorithm of Bootstrap DEA is as follows [11]. Step 1. Trasform the iput-output vectors usig the origial efficiecy estimates { θ i,,2,...,}as ( x i,y i )=(x i, θ i,y i ). Step 2. Give a set of estimated efficiecies { θ i },useh = 0 1 mi{ σ, R/1.34} to obtai badwidth parameter h. σ deotes the stadard deviatio estimate of efficiecy estimates ad R deotes the iterquartile rage of empirical distributio, respectively. Step 3. Geerate {δ } by resamplig, with replacemet, from the empirical distributio of estimated efficiecies. δ is the osmoothed resample of origial efficiecies. Step 4. Geerate sequece { δ } usig δ ={ δ +hε, if δ +hε 1, 2 (δ +hε ), otherwise. Step 5. Geerate the smoothed pseudo efficiecies {γ } usig γ = δ+( δ δ)/ 1+h 2 / σ 2. γ is the smoothed resample efficiecies, δ=( δ )/. Step 6. Let the bootstrap pseudo data be give by ( x i, y i )= ( x i /γ,y i ). (4)

3 Advaces i Decisio Scieces 3 Step 7. Estimate the bootstrap efficiecies usig the pseudo data ad the liear program θ =mi{θ : y Yz, θx Xz, z=1,z R + }. Step 8. Repeat Steps 2 to 7 B times to create a set of B firm specific bootstrapped efficiecy estimates. Imprecise DEA (IDEA) model is a extesio of the CCR model icorporatig imprecise data iformatio ad this DEA is oliear ad ocovex program [22]. Sometimes, outputs ad iputs are imprecise data i the forms of bouded data, ratio bouded data, weak ordial data, ad strog ordial data [23, 24]. If data follow ay of the above forms, the DEA is IDEA as follows: Max π o = μ r y r0 r r μ r y rj i w i x ij =1, i w i x ij 0, (x ij ) Θ i, (y rj) Θ + i, μ r,w i 0, where (x ij ) Θ i ad (y rj ) Θ + i represet ay form of imprecise data. μ r ad w i are the weights of output ad iput, respectively. Robust structure proposed by Be-Tal ad Nemirovski [25] ad Bertsimas ad Sim [26] is used i DEA by Sadjadi ad Omrai [10]. They metioed it as robust DEA. The robust DEA model is expressed as follows [10]: (5) 3. Method We propose a method of DEA to kow the performace of a DMU i the worst situatio. I the first stage, we apply DEAwithiputadoutputdatatokowtheefficiecylevel of DMU. Thereafter, i the secod stage, we maximize the gap betwee frotier output ad observed output to assess what will be the maximum loss if the DMU is i the worst situatio. I the fial stage, we will agai use DEA for the worst coditio of idividual DMU output with observed output of other DMUs ad will determie the efficiecy level DEA Icorporatig Error Stage 1. DEA is familiar with iput-orieted method ad output-orieted method. We choose the output orieted DEA method of Baker et al. [18] which ca be expressed as the liear programmig problem as follows: (a) Max θ,λ θ a i x i θ a i y i λ i x i, λ i =1, λ i y i, (7) Max z 0 v i x i0 =1, i u r y r0 z 0 λ 0 p 0 q r0 0, r j J i i v i x ij u r y rj λ j p j q rj 0, r j J i p j +q rj ey rj t r, t r u r t r, r r,j p j,q rj 0, u r,v i 0, j = 1, 2,..., (6) where θ a i is a fuctio of output y. This LP problem is solved times ad is the umber of firms. Stage 2. Model ca be classified ito two groups as eoclassical model ad frotier model [27]. This classificatio is depedig o the iterpretatio of the deviatio terms ε i R. Cosiderig the assumptio of eoclassical model, all firms are efficiet, ad deviatios ε i R areseeasradom, ucorrelated oise terms that satisfy the Gauss-Markov assumptios. But i the frotier models, all deviatios from the frotier are attributed to iefficiecy which implies that, ε i 0,forall,2,...,[28]. A fuctio y i =g( ) +ε i (8) where z is the efficiecy, y rj ad x ij are the rth output ad ith iput of jth DMU. λ j is the budget of ucertaity for costrait j. u r ad v i are the weights of output ad iput, respectively. e is the utmost probability to violate costrait. p j ad q rj are the dual variables. I this paper we describe a methodology based o BCC model icorporatig oise i data. Is a frotier model if ε i R are iterpreted as composite error terms that iclude both iefficiecy ad oise compoets where g( ) is the productio frotier Aiger et al. [29]. The, output due to evirometal factor ad iefficiecy ε i is ε i = g( ) y i 0 sice frotier output is always greater tha or equal to the observed output. Now, to maximize the output ε i,weuseoutput-orieteddeamethod.themethod

4 4 Advaces i Decisio Scieces ca be expressed as the liear programmig problem as follows: (b) Max λ where θ b i is a fuctio of ε. θ b i x i ε i θ b i λ i x i, λ i =1, λ i ε i, Stage 3. Let us suppose that the maximized quatity of ε i is ε imax. To calculate efficiecy of ith firm i worst productio situatio take ito accout the output of the ith firm is g( ) ε imax whereasiputsofallformsadoutputofotherfirmsare observed data. Thiscabemathematicallyshowasthefollowig: (c) Max θ,λ θ c i x i λ i x i, (g ( ) ε i )θ c i λ i (g ( ) ε i ) max max λ i =1, + j=1,j = i λ j y j, (9) (10) where θ c i is a fuctio of frotier output ad output due to evirometal factor ad shows the miimum level of efficiecy of the ith firm Estimatio of DEA Efficiecy. Usig the above methodology, we the get miimum efficiecy of DMU ad the highest kow efficiecy is 1. Pareto distributio is usually used to describe the allocatio of wealth amog idividuals, but i this paper we use bouded pareto distributio because of its properties. Pareto distributio is applicable if the rage of a variable is a certai value to ifiity. The rage of efficiecy of BCC model to maximize the output is a certai value to 1. This situatio justifies the use of bouded pareto distributio. The probability desity fuctio of bouded Pareto distributio is f (x) =( αlα x α 1 1 (L/H) α ), (11) where L x H,α>0with. Mea 1 (L/H) α ( α α 1 ) ( 1 L α 1 1 ), Hα 1 α=1. (12) Variace L α L α 1 (L/H) α ( α α 2 ) ( 1 L α 2 1 ), α=2. (13) Hα 2 The, the probability desity fuctio of bouded Pareto distributio for efficiecy is f(θ) = (α(θ c i )α θ α 1 /(1 (θ c i )α )) as upper boud for efficiecy is 1, where θ c i is the miimum efficiecy. Usig pdf f(θ), we geerate radom umber of θ. 4. Empirical Example Table 1 shows the efficiecy from observed outputs ad iputs metioed as observed efficiecy. Firm 1, Firm 2, Firm 4, Firm 12, Firm 13, ad Firm 20 are showig to be perfectly efficiet amog the 20 firms. Firm 16 has the less efficiecy amog the firms. Frotier output is the projected output obtaied from the software DEAP versio 2.1. Gap betwee observed ad frotier outputs is the amout of output lost at DMU because of iefficiecy. Iefficiecy is iflueced by some evirometal factors. If a DMU ca cotrol these factors, the it will be efficiet DMU. Maximum loss of output is calculated from Stage 2 of methodology. We calculated miimumoutputas miimum output =(frotier output maximum loss of output). (14) Miimum output ad observed output are foud the same for Firm 7 ad Firm 16 sice from the begiig these firms are i the worst positio amog firms. Efficiecy i the worst situatio is the efficiecy of idividual firm whe it will follow the situatio of Firm 7 ad Firm 16. Maximum loss is the loss of output due to completely egative effect of uexplaied factors o productio. Maximum loss of firm 1will be 273 uits if all uexplaied factors effect as they effect i firm 7 or firm 16 because firm 7 ad firm 16 are i worst productio situatio. To calculate the efficiecy i the worst situatio, we cosider the miimum output of the firm with observed output of other firms. For efficiecy i the worst situatio of Firm 1, use the miimum output 2202 uit with observed output of Firm 2 to Firm 20 as output ad observed iputs of all firms. Observed efficiecy of Firm 16 (0.823) is the lowest. Ad i the worst situatio, Firm 18 ad Firm 19 are also showig the lowest efficiecy level (0.823) with Firm 16. Table 2 shows results of estimated efficiecy from bouded Pareto distributio. We arbitrarily choose 0.5, 1.5, 2.5, 3.5, ad 4.5 as shape value, efficiecy i the worst situatio as lower boud, ad 1 as highest value to draw 10 radom umbers. Mea of radom umbers is the estimated efficiecy. Bias is the differece betwee observed efficiecy ad estimated efficiecy. I Robust Bootstrap DEA, origial

5 Advaces i Decisio Scieces 5 Firm umber Observed output Table 1: Efficiecy level ad projected output usig observed iputs ad outputs. Frotier output Gap betwee observed ad frotier outputs Observed efficiecy Maximum loss of output Miimum output Efficiecy i worst situatio We use oe istead of zero i the gap betwee observed ad frotier outputs which will ot ifluece the result. Efficiecy Observed efficiecy Miimum efficiecy Firm umber Figure 1: Efficiecy iterval of firms. efficiecy is greater tha the estimated efficiecy [30], but i our case the bias is both positive ad egative. Mea of bias is the lowest for shape 0.5. Mea of bias is icreasig with icreasedthevalueofshape. Efficiecy Observed efficiecy Miimum efficiecy Shape 0.5 Shape 1.5 Firm umber Shape 2.5 Shape 3.5 Shape 4.5 Figure 2: Observed, miimum ad estimated efficiecy compariso. Figure 1 presets iterval efficiecy of firms. Amog the perfectly efficiet firm, firm 12 is highly cosistet ad firm 4ismosticosistet.ThatisFirm12willbelessaffected i worst situatio. Rage of efficiecy is almost the same for Firm 7, Firm 10, Firm 16, Firm 17, Firm 18, ad Firm 19. Figure 2 idicates the compariso amog the observed, miimum, ad estimated efficiecies. Estimated efficiecy is followig almost the same patter for differet shape parametric values. Estimated efficiecy curve is foud smooth ad it might be due to a parametric estimatio. Observed ad miimum efficiecy lies are showig the same poit for

6 6 Advaces i Decisio Scieces Firm Lower boud Observe efficiecy Table 2: Bias i estimated efficiecy usig bouded Pareto distributio. Estimate efficiecy Shape 0.5 Shape 1.5 Shape 2.5 Shape 3.5 Shape 4.5 Bias Estimate efficiecy Bias Estimate efficiecy Bias Estimate efficiecy Bias Estimate efficiecy Mea bias Bias Table 3: Rak correlatio matrix of miimum efficiecy, observed efficiecy, ad estimated efficiecy for differet values of shape parameter. Miimum efficiecy Observed efficiecy Estimated efficiecy Shape 0.5 Shape 1.5 Shape 2.5 Shape 3.5 Shape 4.5 Miimum efficiecy 1 Observed efficiecy Shape Shape Estimated efficiecy Shape Shape Shape Sigificat at 1 percet level of sigificace. Firm 7 ad Firm 16. Agai miimum efficiecy i the worst situatio occurred i cases of Firm 16, Firm 18 ad Firm 19. Correlatio (0.842) is high for shape parameter 4.5 betwee observed ad estimated efficiecies. Table 3 presets rak correlatios matrix amog observed efficiecy, miimum efficiecy, ad estimated efficiecy with differet values of shape parameter. All the correlatios are recorded sigificatly high at 1 percet level of sigificace, which idicates the effectiveess of the developed SDEA method. ThisresultsupportstheresultofYuichiro[31]. Correlatio betwee miimum efficiecy ad observed efficiecy is Miimum efficiecy is foud relatively more correlated with estimated efficiecy (03) whe shape parameter is 2.5. Though estimated efficiecy with shape parameter 0.5 has miimum bias, correlatio with miimum efficiecy as well as observed efficiecy is relatively low. Figure 3(a) shows the scatter diagram betwee miimum efficiecy ad observed efficiecy. The efficiecy correlatio coefficiet is calculated as Scatter diagram of observed efficiecy ad estimated efficiecy with differet values of shape parameter is show i Figures 3(b) 3(f),ad correlatio coefficiets of efficiecy are 0.521, 0.667, 0.592, 0.759, ad for shape parameters 0.5, 1.5, 2.5, 3.5, ad 4.5,respectively.Thecorrelatiobetweeobservedefficiecy ad estimated efficiecy is icreased as the value of shape parameter is icreased. All the correlatios are sigificat at 5

7 Advaces i Decisio Scieces 7 Miimum efficiecy Estimated efficiecy (shape 0.5) Observed efficiecy (a) Observed efficiecy (b) 1 Estimated efficiecy (shape 1.5) Estimated efficiecy (shape 2.5) Observed efficiecy (c) Observed efficiecy (d) Estimated efficiecy (shape 3.5) 8 Estimated efficiecy (shape 4.5) Observed efficiecy Observed efficiecy (e) (f) Figure 3: (a) Observed ad miimum efficiecies. (b), (c), (d), (e), ad (f) are observed ad estimated efficiecies for differet values of shape parameter of farms. percet level of sigificace. Estimated efficiecy with shape parameter 2.5 has the highest correlatio with miimum efficiecy. Ad the lowest correlatio betwee miimum efficiecy ad estimated efficiecy is foud whe shape parameter is 0.5. To select the appropriate value of shape parameter, we ca setupthefollowighypotheses. (Ha) There is sigificat differece betwee observed efficiecy ad estimated efficiecy.

8 8 Advaces i Decisio Scieces Table 4: Appropriate shape parameter for estimate efficiecy. Compariso of efficiecy score Paired Differeces Mea Stadard deviatio P value Observe versus shape Observe versus shape Observe versus shape Observe versus shape Observe versus shape Table 4 shows that there is o sigificat differece betwee observed efficiecy ad estimated efficiecies for shape parameters (0.5, 1.5, 2.5, ad 3.5) but there is differece betwee observed ad estimated efficiecies for shape parameter 4.5 at 10% level of sigificace. I Table 4,the P value is decreasig for higher value of shape parameter. To estimate efficiecy, the shape parameters 0.5, 1.5, 2.5, ad 3.5 cabeusedbutthesmallvalueofshapeparameterismore appropriate. 5. Coclusio This paper presets a method of the stochastic data evelopmet aalysis which hadles data with oise i DEA. The method cosists of three stages. I the first stage, we use BCC DEA model to see the gap betwee observed ad frotier productio. From the secod stage, we ca assess miimum productio level. I policy implicatio, it will help to determie the size of ivetory. The third stage measures lower boud of efficiecy for each DMU. It provides a iterval efficiecy to use statistical properties. For the small value of shape parameter, mea of bias is also small. Small value of shape parameter is more appropriate to estimate efficiecy usig the Pareto distributio. The result shows a almost similar patter of observed efficiecy, miimum efficiecies ad estimated efficiecies. Sigificatly high correlatios i betwee observed ad miimum efficiecy as well as betwee observed ad estimated efficiecy are recorded. Rak correlatios are showig the effectiveess of the developed SDEA method. Refereces [1] H. O. Fried, C. A. K. Lovell, S. S. Schmidt, ad S. Yaisawarg, Accoutig for evirometal effects ad statistical oise i data evelopmet aalysis, JouralofProductivityAalysis, vol. 17, o. 1-2, pp , [2] R. D. Baker ad R. C. Morey, Efficiecy aalysis for exogeously fixed iputs ad outputs, Operatios Research, vol. 34, o. 4, pp , [3] T. McCarty ad S. Yaisawarg, Techical efficiecy i New Jersey school districts, i The Measuremet of Productive Efficiecy, H. Fried, C. Lovell, ad S. Schmidt, Eds., pp , Oxford Uiversity Press, [4] A. Bhattacharyya, A. K. Lovell, ad P. Sahay, The impact of liberalizatio o the productive efficiecy of Idia commercial baks, Europea Joural of Operatioal Research, vol. 98, o. 2, pp , [5] A.S.Camaho,M.C.Portela,adC.B.Vaz, Efficiecyaalysis accoutig for iteral ad exteral o-discretioary factors, Computers ad Operatios Research,vol.36,o.5,pp , [6] B. Golay ad Y. Roll, Some extesios of techiques to hadle o-discretioary factors i data evelopmet aalysis, Joural of Productivity Aalysis,vol.4,o.4,pp ,1993. [7] J. Ruggiero, O the measuremet of techical efficiecy i the public sector, Europea Joural of Operatioal Research,vol.90, o.3,pp ,1996. [8] J. Ruggiero, Performace evaluatio whe o-discretioary factors correlate with techical efficiecy, Europea Joural of Operatioal Research,vol.159,o.1,pp ,2004. [9] F. H. Lotfi, G. R. Jahashahloo, ad M. Esmaeili, Nodiscretioary factors ad imprecise data i DEA, Iteratioal Joural of Math Aalysis,vol.1,o.5,pp ,2007. [10] S. J. Sadjadi ad H. Omrai, Data evelopmet aalysis with ucertai data: a applicatio for Iraia electricity distributio compaies, Eergy Policy, vol. 36, o. 11, pp , [11] L. Simar ad P. W. Wilso, Sesitivity aalysis of efficiecy scores: how to bootstrap i oparametric frotier models, Maagemet Sciece, vol. 44, o. 1, pp , [12] L. Simar ad P. W. Wilso, Statistical iferece i oparametric frotier models: the state of the art, Joural of Productivity Aalysis,vol.13,o.1,pp.49 78,2000. [13] L. Simar ad P. Wilso, A geeral methodology for bootstrappig o parametric frotier models, Joural of Applied Statistics,vol.27,o.6,pp ,2000. [14] T. Etai, Y. Maeda, ad H. Taaka, Dual models of Iterval DEA ad its extesio to iterval data, Europea Joural of Operatioal Research,vol.136,o.1,pp.32 45,2002. [15] C. Kao ad S. Liu, Stochastic data evelopmet aalysis i measurig the efficiecy of Taiwa commercial baks, Europea Joural of Operatioal Research, vol.196,o.1,pp , [16] M. J. Farrell, The measuremet of productive efficiecy, JouraloftheRoyalStatisticalSocietyA,vol.120,pp , [17] A. Chares, W. W. Cooper, ad E. Rhodes, Measurig the efficiecy of decisio makig uits, Europea Joural of Operatioal Research,vol.2,o.6,pp ,1978. [18] R. D. Baker, A. Chares, ad W. W. Cooper, Some models for estimatig techical ad scale iefficiecies i data evelopmet aalysis, Maagemet Sciece, vol. 30, o. 9, pp , [19] C.K.Lad,C.A.K.Lovell,adS.Thore, Chace-costraied data evelopmet aalysis, Maagerial ad Decisio Ecoomics,vol.14,pp ,1993.

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