Evaluating decision making units under uncertainty using fuzzy multi-objective nonlinear programming

Transcription

1 Evaatng decson akng nts nde ncetanty sng fzzy t-objectve nonnea pogang M. Zeafat Angz L 1 *, M. K. M. Nawaw 1, R. Khad 1, A. Mstafa 2, A. Eoznejad 3, R. John 5, G. Kenda 4, 5 1 Schoo of Qanttatve Scences, Unvest Utaa Maaysa, Sntok, Kedah, Maaysa. 2 Schoo of Matheatca Scences, Unvest Sans Maaysa, Penang Maaysa. 3 Aston Bsness Schoo, Aston Unvesty, Bngha, UK. 4 Schoo of Copte Scence, Unvesty of Nottngha Maaysa Caps, Maaysa 5 ASAP Reseach Gop, Schoo of Copte Scence, Unvesty of Nottngha, UK Abstact Ths pape poposes a new ethod to evaate Decson Makng Unts (DMUs) nde ncetanty sng fzzy Data Enveopent Anayss (DEA). In the poposed t-objectve nonnea pogang ethodoogy both the objectve fnctons and the constants ae consdeed fzzy. The coeffcents of the decson vaabes n the objectve fnctons and n the constants, as we as the DMUs nde assessent ae assed to be fzzy nbes wth tanga ebeshp fnctons. A copason between the cent fzzy DEA odes and the poposed ethod s stated by a neca eape. Keywods: Fzzy DEA; ebeshp fncton; fzzy t-objectve nea pogang; possbty pogang. * Coespondng atho. Te: , Ea addess: adjd@.ed.y 1

2 1. Intodcton Data Enveopent Anayss (DEA) s a eatvey ecent appoach to the assessent of pefoance of oganzatons and the fnctona nts. DEA s abe to evaate the Decson Makng Unts (DMUs) based on tpe npts and otpts. Snce the fst deveopent of DEA [1, 2], thee have been any appcatons of DEA n a vaety of dffeent contets [3, 4]. Howeve n any ea appcatons, npt o otpt vaabes ae not aways epesented by csp vaes. Hence, the tadtona DEA odes cannot be sed fo evaatng sch DMUs. Sevea attepts have been ade to deveop fzzy DEA odes that ae powef toos fo copang the pefoance of a set of actvtes o oganzatons nde ncetanty. Fo nstance, Sengpta [5] consdeed the objectve fncton to be fzzy when tzng a standad DEA. He then obtaned ests sng Zeann s ethod [6, 7]. Leon et a. [8] tansfoed the fzzy DEA nto csp DEA [9]. Takeda and Satoh [10] sed both tctea decson anayss and DEA wth ncopete data. Letwoask et a. [11, 12] apped the possbstc appoach [13] to teat the constants of the DEA as fzzy events. Sevea othe fzzy odes [14] have been poposed to evaate DMUs wth fzzy data, sng the concept of copason of fzzy nbes. The α-ct appoach [15, 16] fo fzzy DEA, whch s based on a coon set of weghts n fzzy DEA, s one of the ost feqenty sed ethods. Ths ethod fst soves a nea poga to detene the ppe bond of the weghts, then a coon set of weghts ae obtaned by sovng anothe nea pogang pobe. We popose a tobjectve pogang ode [17] that can etan the ncetanty n any aspects ncdng objectve fnctons, coeffcents of the decson at and the DMUs nde assessent. The pape s oganzed as foows. A bef descpton of standad DEA and fzzy DEA s gven n Secton 2. A specfc tobjectve ode w be dscssed n secton 3. Sbseqenty, n Secton 4 we popose an atenatve fzzy DEA ode nde ncetanty. Ths s foowed by a neca staton n Secton 5. In 2

3 secton 6 the ethodoogy s dscssed, and echans of the poposed appoach s pesented n secton 7. Concsons ae dawn n Secton DEA and Fzzy DEA DEA s a eatvey new appoach to the assessent of pefoance of oganzatons and the fnctona nts. It s a nonpaaetc technqe fo easng the eatve effcency of a set of DMUs wth tpe npts and tpe otpts. Today, DEA s adopted n any dscpnes as a powef too fo assessng effcency and podctvty. Hence any appcatons of DEA ae epoted, fo eape hospta effcency [18], bankng [19, 20], easeent effcency of heath centes [21], anfactng effcency [20, 21], podctvty of Ogansaton fo Econoc Coopeaton and Deveopent (OECD) contes [22 24]. Many oe appcatons can be fond n the teate [3] whch ndcates that ost of these stdes gnoed the ncetanty n npt and otpt vaes. Ths ncetanty cod have an affect on the bode defned by the standad DEA; hence the CCR-DEA ode ay not obtan the te effcency of DMUs. Theoetcay, the standad CCR-DEA ode has ts podcton fonte spanned by the nea cobnaton of the obseved DMUs. The podcton fonte nde ncetanty s dffeent. The dea poposed n ths eseach s to aow soe febty n defnng the fontes wth ncetan DMUs, sng a fzzy concept. 2.1 Fzzy nbe Defnton 1. A tanga fzzy nbe s defned as foows - fo - μ( ) = - fo - (1) 3

4 , and ae the ean vae, the owe bond and the ppe bond of the nteva of fzzy nbe. The nteva of fzzy nbe [, ] s the egon whee the vae of fctates. Sybocay, s denoted by (,, ). 2.2 Fzzy DEA The technqe poposed evaates the eatve effcency of a set of hoogenos DMUs by sng a ato of the weghted s of otpts to the weghted s of npts. It geneazes the sa effcency easeent fo a snge-npt, snge-otpt ato to a tpe-npt, tpe-otpt ato. Let npts j ( = 1,2,..., ) and otpts y ( =1,2,..., s ) be gven fo DMU j ( j=1,2,..., n ). The factona pogang stateent fo the CCR ode s foated as foows: a s.t. s = 1 = 1 s y 1 v j = 1,v 0 = 1 y v p p j, whee v and espectvey. ae the weght vaabes fo th and th npt and otpt, 4

5 The above ode s tansfoed to the foowng nea pogang pobe by soe sbstttons: Mode 1: CCR-DEA ode a s.t. s = 1 = 1 s p v = 1 p y - v 0 j j = 1 =1,v y 0, At the tn of the pesent centy, edcng cope ea-wod systes nto pecse atheatca odes was the an tend n scence and engneeng. Unfotnatey, ea-wod statons ae feqenty not deang wth eact data. Ths pecse atheatca odes ae not enogh to tacke a pactca pobes. In pactce thee ae any pobes n whch, a (o soe) npt otpt eves ae fzzy nbes. It s dffct to evaate DMUs n an accate anne to ease the effcency. Fzzy DEA s a powef too fo evaatng the pefoance of a set of oganzatons o actvtes nde an ncetan envonent. Sppose that thee ae n DMUs denoted by j=1,,n, each of whch podces a fzzy ~ nonzeo otpt vecto ( ~, ~,... ~ t Y y y y ) 0 sng a fzzy nonzeo vecto j 1 j 2 j sj ~ ( ~, ~,... ~ t X ) 0 whee the spescpt t ndcates the tanspose of a j 1 j 2 j vecto. Consde ~ X, Y ~ ae atces of fzzy npt and otpt vaabes of a DMUs. Then, the CCR ode wth fzzy coeffcents fo assessng foows. DMU p s foated as 5

6 Mode 2: Fzzy CCR-DEA, tpe ode a, v t ~ y p t v ~ p 1 t ~ t ~ v X Y 0 t t, v 0 v t R 1, t R s1 con ae vectos of npts and otpts weghts, espectvey and R n1 s con vecto of a nea cobnaton of n DMUs. Saat et a. [15] poposed a fzzy DEA by consdeng the α-ct of objectve fncton and the α-ct of constants; hence the foowng ode s obtaned. Mode 3: Fzzy CCR-DEA, sng α-ct appoach 5 a ( y (1 ) y, y (1 ) y ) 1 s. t. v ( (1 ), (1 ) ) ( (1 ), (1 ) ) ( y (1 ) y, y (1 ) y ) v ( (1 ), (1 ) ) 0 j, v p p p p p p p p j j j j 0,. If we sbsttte (,, ), (,, y y y y ) and 1 (1,1,1 ), Mode (3) s wtten as foows. j j j j j j j j 6

7 Mode 4: Fzzy CCR-DEA, sng α-ct appoach, nteva pogang a yˆ s.. t v ˆ L p p yˆ v ˆ 0 j 1 1 y (1 ) y yˆ y (1 ) y (1 ) ˆ (1 ) j j j j j (1 ) L (1 ), v 0,. As t s shown n Saat et a. [15] we have (1 ) L 1. One an dawback n Mode 4 s that the opt effcency eve occs when the otpts of the evaated DMU and the npts of othe DMUs ae set to the ppe bonds, whe the npts of the evaated DMU and the otpts of othe DMUs ae set to the owe bonds. As a est the evaated DMU w have the agest possbe effcency vae; hence Mode 4 does not obtan the te effcency scoe. In the net secton we popose an atenatve fzzy DEA to tacke ths pobe. In the sggested ethod the evaated DMU w have the effcency vae between the saest and the agest possbe vaes. 3. Mt--objectve pogang Snce we st sove a patca t-objectve ode, a shot dscsson eated to ths knd of pobe s pesented. Consde the foowng t-objectve pobe a f ( ), f ( ),..., f ( ) 1 2 s.t. X n 7

8 In the above ode, fnctons f1( ), f2( ),..., fn( ) ae objectve fnctons and X s consdeed as a feasbe egon. To sove the above atheatca pobe, a two stage pocede s poposed. 1. Goa of fncton f( ) 1,2,...,n s obtaned by the foowng atheatca pogang: f * a f ( ) s.t. X f ( ) 2. In ths stage scae s ntodced to ove fnctons 1towads the * f optaty. Fo ths ppose the foowng atheatca pogang pobe shod be soved: a s.t. X f ( ) * f 3.1. A t-objectve fzzy DEA ode nde ncetanty Ths secton poposes an atenatve fzzy DEA ode. The an dea of the sggested ethod s based on the ebeshp fnctons of the coeffcents. We consde the coeffcents as tanga fzzy nbes (,, ). Hence, the ebeshp fnctons of the coeffcents can be defned as foows. j j j j j j j ( ), j j j (1) j j j j j j j 8

9 y j y j y j y j y j y y y ( y ), j j (2) y j y j y y j y j y j y j Vaabes jand y, n foas (1) and (2), ae epesentatve of vaes n the coespondng ntevas of fzzy nbes. We sggest the foowng t-objectve nonnea poga that azes both the objectve fncton and the ebeshp fnctons of technca coeffcent staneosy. Mode 5: A t-objectve nonnea pogang Fzzy CCR-DEA a ( ), ( y ) j a 1 1 s. t. v 1 s s j j y p p y v 0 j( j p) j 1 1 y y y, j p p p p p p j j j y y y, j, v 0, y Vaabes, v ndcate the coeffcents of fzzy otpts and npts. Ftheoe, vaabes j and y epesent the ntevas of fzzy nbes j and y, espectvey. Ths s a t-objectve nonnea fzzy ode that we sggest to sove n two stages as epaned n the est of ths pape. 9

10 Let s gnoe the objectve fnctons coespondng to ebeshp fnctons n Mode 5, that s, a ( ), ( ) j j y y ode w be as foows:. Then, the opta soton of the odfed j p * * j j p p y y j p y y * * p p Ths s becase each DMU wth npts geate than and otpts ess than npts and otpts DMU espectvey, w not be bette than DMU. So the opta vae of p p Mode (5) s eqas to effcency of DMU. p Ignong the ast objectve fncton n Mode (5), the opta soton w be as foows: j p * * j j p p y y j p y y * * p p Inteacton between two opposed objectve fnctons specfy the opta soton. Lea1: Let s consde the optstc pont of vew that s the best condton fo DMU nde evaaton and the wost condton fo othe DMUs. a. The opta soton fo ( ), ( y ) ae obtaned n the second j j yp p condton of the ebeshp fnctons (1) and (2), espectvey. b. The opta soton fo ( ), ( y )( j p ) ae obtaned n the fst p p y condton of the ebeshp fnctons (1) and (2), espectvey. Poof: Sppose that objectve fncton n Mode (5) be ony ( a entoned above, de the nate of the ode the opta soton w be: s y p 1 ), as 10

11 n a, j( j p) p a y n y, j( j p) p j When consdeng the effect of the ebeshp fncton, the vaes of, j( j p) and y w be deceased and the vaes of and j p y, j( j p ) w be nceased (ebeshp nbes w be zeo fo the above entoned vaes). So, to obtan the opta soton of ( ), ( y ) the p j j yp p second condton of the ebeshp fnctons (1) and (2) ae sffcent, espectvey. Say to obtan the opta vae fo ( ), ( y )( j p ) the p p y fst condton of the ebeshp fnctons (1) and (2) ae sffcent, espectvey,.e. p p ( ) [, ] p p p p p p p )3( y p yp ( y ) y [ y, y ] y y yp p p p p p p )4( j j ( ) [, ], ( ) j j j j j j j p j j )5( y y y ( y ) [, ], ( ) y y y j j p y y )6( * * Let, y ( j p) and *, y * be the opta soton fo, y ( j p) and, y. It s j p p cea that thee est two vaes n the ntevas [, ],[ y, y ] ( j p) and [, ],[ y, y ] wth the sae ebeshp fncton, say, p p p p j j j p p [, ], y [ y, y ] * * j1 j j 1 [, ], y [ y, y ] * * j2 j j 2 (7) 11

12 [, ], y [ y, y ] * * p1 p p p1 p p [, ], y [ y, y ]. * * p2 p p p2 p p In ths vew, the sae j sa to the npt vaes and the ys ae sa to the otpt vaes n the DEA odes, so by consdeng constant vaes fo sand j ys, Mode (5) w be conveted to Mode (4). Asse that npts and otpts ofdmu1and DMU 2 ae (,, y, y )( j p) * * * * p1 j1 p2 2 * * * * and ( p2 j2, y p1, y 1)( j p), espectvey. ObvosyDMU1 s oe effcent than DMU 2. Ths eans ony the second condton of the ebeshp fnctons (1) and (2) ae sffcent to obtan the opta soton fo ( ), ( y ). Say the j j yp p fst condton of ebeshp fncton (1) and (2) ae sffcent to obtan the opt vae fo ( ), ( y)( j p). p p yj j Hence, to sove Mode (5), the ethodoogy pesented n secton 3 s apped, and t-objectve pogang pobe (5) s conveted to the foowng nonnea pogang pobe: 12

13 Mode 6: A new Fzzy CCR-DEA, non-nea pogang a st.. Z h 1 p s 1 1 h ( y ) / z s * p p y v 0 j( j p) j 1 1 j j h, j( j p) j j y j y j h, j( j p) y y j p p p p p p p p, j( j p) 6.1 y j j j j v h h y y y j y y y, j( j p) 6.2 p p p y y y p p p, v 0, In Mode (6), * z p s obtaned wth the best staton of the DMUs as foows: Mode 7: A new Fzzy CCR-DEA, estaton of Z * p z 5 p a yp 1 s. t v 1 1 p s p 1 1 y v 0 j( j p), v 0, 13

14 Obvosy, fctatng between 0 and 1, the objectve fnctons coespondng to ebeshp fnctons do not need to foow the fst stage of secton 3. The vaabe h n Mode (6) s sed to convet the t-objectve pobe Mode (5) to a nonnea pogang pobe. Ths vaabe s wthn the nteva [0,1]. Addng the concept of α-ct to Mode (6), t s sffcent to epace the foowng constants nstead of 6-1, 6-2, 6-3 and 6-4. (1 ), j( j p) j j j j y (1 ) y y y, j( j p) j j j (1 ) p p p p y y y (1 ) y p p p p Ths s dffeent fo the standad α-ct sed n the fzzy DEA Mode (4), becase n each α-eve the ode st etans ncetanty nfoaton nteo of the nteva that was geneated by α. Net secton copaes o ests wth the cent fzzy DEA ode. 4. An staton wth a neca eape In ths secton, a neca eape s pesented to state the dffeence between the ests obtaned sng the poposed appoach and the cent fzzy DEA odes. Consde the data n Tabe 1 that s etacted fo Go and Tanaka [14] and sed by Letwoask et a. [11] and Saat et a. [15]. Thee ae 5 DMUs wth two syetca tanga fzzy npts and 2 syetca tanga fzzy otpts. 14

15 Tabe 1: Data fo neca eape DMU Vaabe D1 D2 D3 D4 D5 I1 (4.0, 3.5, 4.5) (2.9, 2.9, 2.9) (4.9, 4.4, 5.4) (4.1, 3.4, 4.8) (6.5, 5.9, 7.1) I2 (2.1, 1.9, 2.3) (1.5, 1.4, 1.6) (2.6, 2.2, 3.0) (2.3, 2.2, 2.4) (4.1, 3.6, 4.6) O1 (2.6, 2.4, 2.8 (2.2, 2.2, 2.2) (3.2, 2.7, 3.7) (2.9, 2.5, 3..3) (5.1, 4.4, 5.8) O2 (4.1, 3.8, 4.4) (3.5, 3.3, 3.7) (5.1, 4.3, 5.9) (5.7, 5.5, 5.9) (7.4, 6.5, 8.3) Usng fzzy CCR Mode (4), the effcency scoes ae sazed n the Tabe 2. Tabe 2: The effcences sng Mode (4) DMU Α D1 D2 D3 D4 D Consdeng the above Lea1, obvosy, the opta soton gven n Tabe 2 s eqvaent to the opta soton eated to the optstc pat of Kao and L [27] appoach n ts sppe effcency fo. As t s known, the ethods based on the α- ct appoach jst etent nbe of ebeshp vaes consdeed n the evaaton; theefoe the ajo pat of the fzzy concept s gnoed. Dffeences between the poposed ethod and the α-ct based appoach can be copaed wth dffeences between ntegaton and neca ethods fo ntegas. The neca ethods don t cove the whoe aea nde cve n ntegaton. 15

16 Rests fo the possbty appoach of Letwoask [11] ae shown n Tabe 3. As can be seen, the effcency vaes n the above two odes ae vey sa. Tabe 3: The effcences sng Letwoask [11] ode DMU α D1 D2 D3 D4 D Usng the poposed Mode (6), the ests ae shown n Tabe 4. Tabe 4: The effcences sng the poposed ode n ths pape DMU α D1 D2 D3 D4 D De to the nate of the fzzy CCR Mode (4) the a effcency occs when the otpts of the evaated DMU and the npts of othe DMUs ae set to the ppe bonds. It s obvos that the ests n Tabe 2 ae aways geate than the ests that we obtaned n Tabe 4 snce Mode 4 aways captes the effcency nde pessstc ccstances. The ests obtaned sng the poposed ode n ths pape have the effcency vaes between the saest and the agest possbe vaes, hence they ae oe cose to the te effcency. 16

17 6. Epca stdy To state the fzzy DEA appoach, we consde data gven n [28] whch has pesented fo an acaft seecton. Fve types of acaft (B , A-321, B , MD-82, and A ) ae to be evaated. Fo npts and two otpts ae ntodced n Tabe 5 as foows: Tabe 5: Inpts and otpts fo acafts evaaton Data Inpt1(I1) Inpt2(I2) Inpt3(I3) Inpt4(I4) Otpt1(O1) Otpt2(O2) Descpton Mantenance eqeents (Sbjectve assessent) Pot adaptabty (Sbjectve assessent) Ma ange (Koete) Pchasng pce (US ons) Passenge pefeence(sbjectve assessent) Opeatona podctvty (Seat-koete pe ho) The fst npt s the acaft antenance capabty (I1) whch s concened wth the avaabty and the eve of standadzaton of spae pats and post-sae sevces. The second npt, pot adaptabty (I2) s eated to the sks of avaabe pots and the specfc feates of the acaft. The thd npt a ange (I3) of an acaft s detened by the a koetes that the acaft can tave at the a payoad and the foth npt, pchasng pce (I4) s the pce to be pad fo a new acaft whch coeates wth eabty of the acaft. On the othe hand fo the otpts, passenges pefeence (O1) efects the soca esponsbty of the ane n ode to estabsh a postve age n pbc and of the eqeents posed by vaos envonent potecton aws and egatons whst opeatona podctvty (O2) s detened by the nbe of seats avaabe, the oad ate, the tave feqency, and the acaft tave speed. 17

18 In ths eseach, the eght decson akes stated the opnon abot 3 sbjectve npts and otpts. They sed a set of fve ngstc tes {vey ow, ow, ed, hgh, vey hgh} whch ae assocated wth the coespondng nbes 1, 2, 3, 4 and 5, espectvey, as n a 5-pont Lket scae. Tabe 6 shows the npts and otpts of the fve acafts. Fo eape, B type of acaft has two sbjectve npts (I1 and I2) and one sbjectve otpt (O1), wth tanga fzzy nbes. Fo othe two npts and one otpt, the vaes ae csps. Tabe 6: Data fo neca eape DMU Vaabe B A-321 B MD-82 A I1 (2.0, 3.064, 4) (4, 4.229,5) (3, 3.224, 4) (1, 1.929, 3) (3,3.464, 4) I2 (2, 2.852, 3) (2,2.000,2) (2, 2.852, 3) (4, 4.113, 5) (2,2.000,2) I I O1 (4, 4.000, 4) (2, 2.852, 3) (4, 4.000, 4) (3, 3.591, 4) (3, 3.342, 4) O Usng Mode (6), the vaes of h*, the effcency scoes and ank of each acaft ae gven n Tabe 7. The MD-82 acaft type gves the hghest effcency scoe of and s anked fst, whst B gves owest scoe of and s anked ast. 18

19 Tabe 7: The ank of fve types of acafts DMU * h Eff. scoes Rank B A B MD A Dscsson Accodng to theoe 2, f the objectve fnctons coespondng to ebeshp fnctons n Mode (5) ae gnoed, the opta soton fo npts and otpts w be asen n endponts of nteva of fzzy nbes. Ftheoe, f the ast objectve fncton ( a s p 1 y ) n Mode (5) s enated, Lea1 adopted the opta soton w be n the an vae of fzzy nbe. Fge 1 states the above entoned concept fo evaatng DMU. The nteo aows epesent the opta soton when the ast objectve fncton ( a P s p 1 y ) s absent n Mode (5) and the aows ocated nde fzzy nbes constct the opta soton Mode (5) when ony the objectve fncton ( a s p 1 y ) s pesent. Fge 1: Concepts of evaatng DMUs 19

20 Inteacton between the objectve fnctons coespondng to objectve fnctons and the ast objectve fncton ( a soton. 6. Concson s p 1 y ) n Mode (5), case the fzzy opta In evaatng DMUs nde ncetanty sevea fzzy DEA odes have been poposed n the teate. The α-ct appoach s one of the ost feqenty sed odes. Howeve, de to the nate of the α-ct appoach the ncetanty n npts and otpts s effectvey gnoed. Ths pape poposed a t-objectve fzzy DEA ode to etan fzzness of the ode by azng the ebeshp fncton of npts and otpts. In the poposed ethod, both the objectve fnctons and the constants ae consdeed fzzy. A neca eape s sed to show the dffeence between the poposed and the cent fzzy DEA odes. Fo fthe stdes, t s sggested that an epoaton be done on: a) edcng the sze of the conveted (csp eqvaent) pobe, b) possbe neazaton of the nonnea ode. Refeences [1] A. Chanes, W.W. Coope, E. Rhodes, Measng the effcency of decson akng nts, E. J. Ope. Res. 2 (1978) [2] R.D. Banke, A. Chanes, W.W. Coope, Soe odes fo estatng technca and scae neffcences n data enveopent anayss, Manage. Sc. 30 (1984) [3] A. Eoznejad, B.R. Pake, G. Tavaes, Evaaton of eseach n effcency and podctvty: A svey and anayss of the fst 30 yeas of schoay teate n DEA, Socoecon. Pann. Sc. 42 (2008) [4] A. Eoznejad, K. De Wtte, COOPER-faewok: A nfed pocess fo non-paaetc pojects, E. J. Ope. Res. 207 (2010)

21 [5] J.K. Sengpta, A fzzy systes appoach n data enveopent anayss, Copt. Math. wth App. 24 (1992) [6] H.-J. Zeann, DESCRIPTION AND OPTIMIZATION OF FUZZY SYSTEMS, Int. J. Gen. Syst. 2 (1975) [7] H.-J. Zeann, Fzzy pogang and nea pogang wth sevea objectve fnctons, Fzzy Sets Syst. 1 (1978) [8] T. León, V. Len, J.L. Rz, I. Svent, A fzzy atheatca pogang appoach to the assessent of effcency wth DEA odes, Fzzy Sets Syst. 139 (2003) [9] J.L. Hogaad, A spe appoaton of podctvty scoes of fzzy podcton pans, Fzzy Sets Syst. 152 (2005) [10] E. Takeda, J. Satoh, A data enveopent anayss appoach to tctea decson pobes wth ncopete nfoaton, Copt. Math. wth App. 39 (2000) [11] S. Letwoask, S.-C. Fang, J.A. Jones, H.L.W. Ntte, Fzzy data enveopent anayss (DEA): a possbty appoach, Fzzy Sets Syst. 139 (2003) [12] S. Letwoask, S.-C. Fang, H.L.W. Ntte, J.A. Jones, Fzzy BCC ode fo data enveopent anayss, Fzzy Opt. Decs. Mak. 2 (2003) [13] M. Zaafat Angz, S. Saat, a Meaan, M.M. Movahed, Sovng possbstc nea pogang pobe consdeng ebeshp fncton of the coeffcents, Adv. Fzzy Sets Syst. 1 (2006) [14] P. Go, H. Tanaka, Fzzy DEA: a pecepta evaaton ethod, Fzzy Sets Syst. 119 (2001)

22 [15] S.M. Saat, A. Meaan, G.R. Jahanshahoo, Effcency anayss and ankng of DMUs wth fzzy data, Fzzy Opt. Decs. Mak. 1 (2002) [16] S. Saat, A. Meaan, Redcng weght febty n fzzy DEA, App. Math. Copt. 161 (2005) [17] M.G. Iskande, A coptatona copason between two evaaton ctea n fzzy tobjectve nea pogas sng possbty pogang, Copt. Math. wth App. 55 (2008) [18] J.M. Kga, A. Eoznejad, R. Gaa Vaz, H. Bastene, J. Padayachy, A copaatve assessent of pefoance and podctvty of heath centes n Seychees, Int. J. Podct. Pefo. Manag. 57 (2007) [19] H.D. Shean, F. God, Bank banch opeatng effcency: Evaaton wth data enveopent anayss, J. Bank. Fnanc. 9 (1985) [20] M. Vassogo, D. Gokas, A stdy of the eatve effcency of bank banches: an appcaton of data enveopent anayss, J. Ope. Res. Soc. (1990) [21] K. Fed, A. Eoznejad, Measng the pefoance of neonata cae nts n Scotand, J. Med. Syst. 27 (2003) [22] R. Mwa, A. Eoznejad, L. Mhaad, Econoc effcency of sahode aze podces n Westen Kenya: a DEA eta-fonte anayss, Int. J. Ope. Res. 4 (2009) [23] M.R. Mwa, A. Eoznejad, F.M. Mth, Ipact of beazaton on effcency and podctvty of sga ndsty n Kenya, J. Econ. Std. 36 (2009)

23 [24] A. Eoznejad, An atenatve DEA ease: a case of OECD contes, App. Econ. Lett. 10 (2003) [25] A. Eoznejad, E. Thanassos, A atheatca ode fo dynac effcency sng data enveopent anayss, App. Math. Copt. 160 (2005) [26] A. Eoznejad, E. Thanassos, Measeent of podctvty nde wth dynac DEA, Int. J. Ope. Res. 8 (2010) [27] C. Kao, S.-T. L, Fzzy effcency eases n data enveopent anayss, Fzzy Sets Syst. 113 (2000) [28] C.-H. Yeh, Y.-H. Chang, Modeng sbjectve evaaton fo fzzy gop tctea decson akng, E. J. Ope. Res. 194 (2009)