Enhanced Russell measure in fuzzy DEA

Transcription

1 140 It. J. Data Aaly Techque ad Statege, Vol. 2, No. 2, 2010 Ehaced Ruell eaue fuzzy DEA Meqag Wag* School of Maageet, Guzhou Uvety, Guyag , PR Cha Fax: E-al: *Coepodg autho Yogu L School of Maageet, Uvety of Scece ad Techology of Cha, He Fe, A Hu Povce , PR Cha E-al: lol@al.utc.edu.c Abtact: The adal eaue of clacal DEA odel (CCR, BCC) ae coplete, they ae oly epaate eaue of put ad output effcecy ad the effcecy dex ot the o-zeo put ad output lack. Ehaced Ruell gaph eaue (ERM) elate thee defcece. All of the extg fuzzy DEA odel ae exteo of CCR o BCC odel, effcece of DMU, ultately, ae oluto of CCR o BCC odel. Baed o ERM odel, a fuzzy DEA odel popoed to deal wth the effcecy evaluato poble wth the gve fuzzy put ad output data, by ug a akg ethod baed o the copao of -cut. The popoed faewok llutated though a applcato to pefoace aeet of flexble aufactug yte ad copaatve eult ae peeted. The effcecy eaue of the popoed appoach elatvely oe eaoable tha thoe of fuzzy DEA odel baed o CCR o BCC odel ad epeet oe eal-lfe pocee oe appopately. Keywod: o-adal; DEA; data evelopet aaly; fuzzy; FMS; flexble aufactug yte. Refeece to th pape hould be ade a follow: Wag, M. ad L, Y. (2010) Ehaced Ruell eaue fuzzy DEA, It. J. Data Aaly Techque ad Statege, Vol. 2, No. 2, pp Bogaphcal ote: Meqag Wag a Vce Pofeo of School of Maageet, Guzhou Uvety. H eeach aea of teet data evelopet aaly. Yogu L, cuetly, a Pot-Doctoal Fellow at the School of Maageet, Uvety of Scece ad Techology of Cha. H eeach aea of teet clude data evelopet aaly ad deco aaly. Copyght 2010 Idecece Etepe Ltd.

2 Ehaced Ruell eaue fuzzy DEA Itoducto Data evelopet aaly (DEA) a o-paaetc ethod fo evaluatg the elatve effcecy of deco-akg ut (DMU) o the ba of ultple put ad output. Tadtoal DEA odel uch a CCR odel (Chae et al., 1978), BCC odel (Bake et al., 1984) do ot deal wth pece data ad aue that all put ad output ae exactly kow; a key to the ucce of the DEA appoach the accuate eaue of all facto, cludg put ad output. I eal-wold tuato, howeve, put ad output data of DMU evaluated ofte ae ot exactly kow but ca be chaacteed by fuzzy ube whch wa toduced by Zadeh (1965) to deal wth poble whch a ouce of vaguee volved. Seveal appoache have bee popoed to deal wth fuzzy data the faewok of DEA (Wag ad Lag, 2009). Kao ad Lu (2000) uggeted tafog fuzzy data to teval data by applyg the -cut o that a faly of covetoal cp DEA odel could be utled, but th appoach ot effcet fo a coputatoal pot of vew. Letwoakul et al. (2003a,b) popoed a poblty appoach whch fuzzy cotat wee teated a fuzzy evet ad fuzzy DEA odel wa tafoed to poblty DEA odel by ug poblty eaue o fuzzy evet. Th appoach ot a geeal way to olve all fuzzy DEA odel ug poblty appoach fo olvg the fuzzy BCC odel ay eult a ubouded optal value fo DMU, ad caot be adapted fo othe odel. Guo ad Taaka (2001) popoed a fuzzy CCR odel whch fuzzy cotat cludg fuzzy equalte ad fuzzy equalte wee all coveted to cp cotat by pedefg a poblty level ad ug the copao ule fo fuzzy ube. The ethod excellet, but t ot a geeal appoach, becaue the odel ha a optal oluto ude a pecfc etctve codto. I a la ae, Leo et al. (2003) alo popoed oe fuzzy BCC odel, but explotg the ue of the evelopet foulato of the BCC odel tead of the dual ultple oe. The fuzzy DEA odel take the fo of fuzzy lea pogae whch eque akg of fuzzy et, but akg of fuzzy et ay lead to a poble of ug dffeet akg ethod ca eult dffeet eult. Both the appoach baed o the copao ule fo fuzzy ube ad the appoach tafog fuzzy data to teval data by applyg the -cut ae cooly ued appoache eal applcato. All of the above fuzzy DEA odel ae exteo of CCR o BCC odel, effcece of DMU, ultately, ae oluto of CCR o BCC odel. But the eaue of thee two odel ae coplete, they ae oly epaate eaue of put ad output effcece whch have eulted fo the adal eaue depedg o whethe a output-oeted o put-oeted appoach ued; ad that the effcecy dex ot the o-zeo put ad output lack ad, thu, fal to accout fo all effcece that the odel ca detfy. The Ruell gaph effcecy eaue (Fae et al., 1985) elate thee defcece but t pogae dffcult to olve. Ehaced Ruell gaph effcecy eaue (ERM) (Pato ad Ruz, 1999) utle a ato eaue place of the weghted aveage of athetc ad haoc ea of Ruell gaph effcecy eaue, ad t ca be tafoed to a oday lea pogae that geeate a optal oluto fo the coepodg ERM odel. The ERM odel otly detee put ad output effcece that epeet epaate etate of put ad output effcecy, ad ufy the ato effcecy ad lack to a cala eaue. Th pape develop a ew o-adal DEA odel wth fuzzy put output data by ug a ethod baed o the copao of

3 142 M. Wag ad Y. L -cut. Th ew odel ca be ee a a exteo of the ERM, that, oe poweful fo applcato. A flexble aufactug yte (FMS) deged to cobe the effcecy of a a-poducto le ad the flexblty of a ob hop to poduce wok pece o a goup of ache (Kaak, 2008; Kaak ad Kuzgukaya, 2002; Kaak ad Tolga, 2001). Recetly, the ue of DEA ha bee ecoeded a a dcete alteatve ultple ctea tool fo evaluato of aufactug techologe ad FMS (Kaak, 2008; Kaak ad Kuzgukaya, 2002; Kaak ad Tolga, 2001). But a obut DEA pocedue ued fo FMS electo hould be able to copoate quattatve a well a qualtatve data. A effcet way to expe facto uch a wok poce (wp) level, flexblty ad qualty of the poduct, whch ca ethe be aeed by cp value o ado pocee, ug lgutc vaable o fuzzy ube. The popoed fuzzy o-adal DEA odel appled to pefoace aeet of FMS, t eable the deco ake to deal wth peco heet the expeo of each ctea by talatg vague data to uecal oe. The et of th pape ogaed a follow: Secto 2 befly toduce the ogal ERM odel. Secto 3 cota oe eult of fuzzy aaly baed o the copao of -cut that wll be ued, the, develop a fuzzy o-adal DEA odel. Secto 4 llutate ou popoed fuzzy DEA ethod wth a applcato to FMS. Fally, Secto 5 coclude th pape. 2 Ehaced Ruell gaph effcecy eaue ERM (Pato ad Ruz, 1999) epeet the followg odel: (1 / ) M Re ( X0, Y0) (1 / ).t x x, 1,, 0 y y, 1,, 0 0, 1,, 1 1 (1) The obectve (1) otly e the put ad output effcece, the deoato axed to gve value fo ay choce of the ueato, ultaeouly, the ueato ed to gve value fo ay choce of the deoato, the ueato ad deoato ae otly opted to acheve a u value fo the ato, t ca be tepeted a the ato betwee the aveage effcecy of put ad the aveage effcecy of output (Coope ad Huag, 2007). O the othe had, by ea of

4 Ehaced Ruell eaue fuzzy DEA the followg chage of vaable: 1, 1,, ad 1, 1,,, x0 y0 t eay to tepet foulato of R e te of total lack, th povde a alteatve expeo of the ERM, the ultate odel lack-baed eaue (SBM) (fo futhe detal about SBM, pleae ee Toe, 2001). The followg tue fo R e (Pato ad Ruz, 1999): R e 2 e 1 DMU0 3 Re R beg evaluated Koopa-effcet, whee the coepodg adal effcecy eaue. Model (1) ca be tafoed to a lea pogae ug the Chae Coope tafoato (Chae ad Coope, 1962) the la way a the CCR odel. Let u ultply a cala vaable (>0) to both the deoato ad the ueato of (1). Th caue o chage R e. We adut o that the deoato becoe 1. The th te oved to cotat. The obectve to e the ueato. Let u, v, t the we have: M.t xt ux, 1,, y t v y, 1,, 0 u, 1,, v v, 1,, 0 t, 1,, 0 1 u (2) 3 No-adal fuzzy DEA odel ad t oluto 3.1 No-adal fuzzy DEA odel Suppoe that we ae teeted evaluatg the elatve effcecy of DMU whch ue put to poduce output, ad the data of put ad output caot be pecely eaued but ca be expeed a fuzzy ube. Let u aue that odel (2) ued to evaluate the elatve effcecy of th et of DMU. The, the exteded odel of odel (2) ca be expeed a the followg fuzzy LP poble.

5 144 M. Wag ad Y. L M.t xt ux, 1,, y t v y, 1,, 0 u, 1,, v v, 1,, 0 t, 1,, 0 1 u (3) I odel (3), defed a h, whee 0 h 1 a pe-defed poblty level by deco ake. The elatohp betwee effcecy coe of a DMU ad poblty level h odel (3) : Popoto 1: Effcecy coe of a DMU a deceag fucto of the poblty level h. Th cocluo obvou. If a feable oluto be a optal oluto of odel (3) at h h 0, t ut be a feable oluto of odel (3) fo h h0, o, h 0 h, whee hh, 0 [0,1]. 3.2 The oluto to the o-adal fuzzy DEA odel To get the oluto of odel (3), we ae ecallg how to pefo the bac opeato of athetc ad the copao of fuzzy teval fo akg pupoe Copao of fuzzy teval baed o -cut To be oe pece, we deal wth LR-fuzzy ube whoe defto a follow (Wag ad Lag, 2009): Defto 1: Suppoe that M a fuzzy ube, M () t ebehp fucto. Let f : 0, 0,1 be a appg ad atfy followg codto: 1 tctly deceag upp ( M ) { : ( ) 0} 2 uppe e-cotuou 3 f (0) 1. f ad to be a efeece fucto of fuzzy ube M. M

6 Ehaced Ruell eaue fuzzy DEA 145 Defto 2: A fuzzy ube M ad to be a LR-fuzzy ube, f t ebehp fucto ha the followg fo: L - L, L a L R () 1 M R R, R L R whee L ad R ae efeece fucto of fuzzy ube M. LR-fuzzy ube M L R L R deoted by M (,,, ) LR, Fo a gve et of LR-fuzzy ube L R L R, M,,,, x R, 1,, LR ad oe cala x 0, 1,,, we have that (4) L R L R M x x, x, x, x We have followg popoto wth epect to copao of fuzzy ube. Theoe 1: (Rak ad Raek, 1985) Let M, N be two fuzzy ube, ad h be a eal ube, h 0,1 h. The, M N f ad oly f k h,1 LR, the two tateet below hold: f { : () k } f { t : (t) k }, up{ : () k} up{ t : (t) k} M N Coollay 1: Let M, N L R L R be two LR-fuzzy ube, M (,,, ), L R L R N (,,, ), the L R L L L L h L ( k) L ( k) k h,1 M N R R R R R ( k) R ( k) k h,1 M N LR whee L ( k) up{ z: L( z) k} R ( k) up{ z: R( z) k} L * ( k) up{ z: L( z) k} R * ( k) up{ z: R( z) k} Coollay 2: Let M, N L R L R be two LR-fuzzy ube, M (,,, ), L R L R N (,,, ) L, R have bouded uppot, L L ad R R, the M N f, ad oly f h LR L L L L L L L ( h) L ( h) R R R R R R R ( h) R ( h) (5)

7 146 M. Wag ad Y. L Tafoato of odel (3) Suppoe that the data of put ad output ca be expeed a LR-fuzzy ube wth bouded uppot L R L R L, L R L R L, x x, x,,, 1,,, 1,, R y y, y,,, 1,,, 1,, R Satfyg L1 L L, 1,,, L1 L L, 1,, R1 R R, 1,, R1 R R, 1,, that, equg that fo all put ad output, the coepodg data ca be decbed by ea of LR-fuzzy ube of the ae type. By ug (4) ad (5), odel (3) ca be tafoed to be: M.t. v 1 u 1 L L tx ux 0, 1,, 1 R R tx ux 0, 1,, 1 L L L L R R R R L L ty vy 0, 1,, 1 R R ty vy0, 1,, 1 L L L L R R R R tx L( h) t ux L( hu ), 1,, tx R( h) t ux R( hu ), 1,, t y L ( h) t v y L ( h) v, 1,, t y R ( h) t v y R ( h) v, 1,,

8 Ehaced Ruell eaue fuzzy DEA 147 u, 1,, v, 1,, 0 t, 1,, 0 1 (6.1) I pactce, tagula fuzzy ube (a pecal cae of LR-fuzzy ube) ae vey ofte ued to odel a wde vaety of tuato, they appea a ueful ea of quatfyg the ucetaty deco akg due to the tutve appea ad coputatoal-effcet epeetato (Wag ad Lag, 2009). Fo tace, aug that a expet etate about a ceta vaable aoud 5, t ca be epeeted by a yetcal tagula fuzzy ube a (5, 2), whee 5 the cete ad 2 the pead (the cete of a yetcal tagula fuzzy ube epeet the ot geeal cae ad the pead eflect oe poblte). If put ad output ae ow aued to be yetcal tagula fuzzy ube (a pecal cae of LR-fuzzy ube) deoted by the pa cotg of the coepodg cete ad pead, x ( x, ), 1,,, 1,, ; y ( y, ), 1,,, 1,,. I th pecal tuato, ad L R L R L R L R x x x, y y y ;, L ( h) L ( h) R ( h) R ( h) 1 h,0 h1; 1,,, 1,,, 1,, Model (6.1) ca be plfed, t becoe (6.2) M.t. v 1 1 u tx ux, 1,, ty vy0, 1,, 1 tx (1 h) t ux (1 hu ), 1,, tx (1 h) t ux (1 hu ), 1,,

9 148 M. Wag ad Y. L t y (1 h) t v y (1 h) v, 1,, t y (1 h) t v y (1 h) v, 1,, u, 1,, v, 1,, 0 t, 1,, 0 1 (6.2) Note that odel (6.1), f we add cotat (4) by cotat (3), we ca get cotat (2); f we add cotat (7) by cotat (6), we ca get cotat (5). Cotat (2) ad cotat (5) ca be elated ce they ae edudat. The, odel (6.1) ca be futhe plfed to be M.t. 1 1 tx (1 h) t ux (1 hu ), 1,, tx (1 h) t ux (1 hu ), 1,, t y (1 h) t v y (1 h) v, 1,, v u t y (1 h) t 1 1 u, 1,, v, 1,, 0 t, 1,, 0 1 vy (1 hv ), 1,, 0 0 (7) By la tafoato, fuzzy DEA odel baed o put-oeted CCR ca alo be etablhed (put ad output ae alo aued to be yetcal tagula fuzzy ube).

10 Ehaced Ruell eaue fuzzy DEA 149 M 0 x (1 h) x (1 h), 1,, x (1 h) x (1 h), 1,, y (1 h) y (1 h), 1,, y (1 h) y (1 h), 1,, , 1,, (8) A copaatve eult wll be gve betwee odel (8) ad odel (7) Secto 4, to llutate the advatage of ou o-adal appoach ove pevou appoach baed o adal odel. Theoetcally, fuzzy o-adal DEA odel ca deal wth LR-fuzzy ube, the fuzzy o-adal DEA odel foulato whch ca deal wth LR-fuzzy ube ae ut lke odel (6). But pactce, t dffcult to obta the ebehp fucto of all LR-fuzzy put ad output, theefoe, odel (6) ad alke odel have oly theoetcal gfcace. Wth a vew to tuto ad coputatoal-effcecy, yetcal tagula fuzzy ube ad odel (7) ae hghlghted th pape. I fact, lteatue o fuzzy DEA uch a (Guo ad Taaka, 2001; Leo et al., 2003; Letwoakul, 2003a,b) do lke th way; epecally the ube exaple, alot all deged oly to deal wth yetcal tagula fuzzy ube (Wag ad Lag, 2009). 4 FMS pefoace evaluato I th ecto, we wll eploy the popoed appoach (odel (7)) fo electg a FMS. The data utled fo the llutatve aaly lghtly odfed fo the tudy of Kaak ad Kuzgukaya (2002). I the eeach, the put te clude captal ad ateace cot ad floo pace ued, the output te clude educto wok-poce, educto etup cot, poveet qualty, ceae aket epoe ad educto labou cot. The value fo each FMS DMU ae expeed ug tagula fuzzy ube. The FMS DMU ctea value fo captal ad ateace cot ad floo pace ued, educto wok--poce, educto etup cot, educto labou cot ae expeed by quattatve data. Howeve, FMS DMU value fo poveet qualty, ceae aket epoe ae qualtatve te that ae epeeted by lgutc expeo uch a weak, fa ad good (uecal etate eque oe etal effot tha lgutc decpto, people ae oe lkely to ba the evaluato f they ae foced to povde uecal etate of vague o pece te). The dataet ued the DEA gve Table 1. The ebehp fucto of the lgutc vaable ued to epeet poveet qualty ad ceae aket epoe ae defed the Fgue 1 (Kaak ad Kuzgukaya, 2002).

11 150 M. Wag ad Y. L Table 1 Data ued the aaly FMS Iput Output DMU Captal ad ateace Floo pace ued (q.ft) Reducto labou Reducto wp (%) cot ($) cot (%) A (14, 15, 18) (40, 50, 60) (25, 30, 35) (20, 23, 26) B (11, 13, 15) (55, 60, 65) (16, 18, 20) (7, 13, 16) C (7.5, 9.5, 11.5) (60, 70, 80) (10, 15, 20) (10, 12, 16) D (8, 12, 13) (35, 40, 45) (23, 25, 27) (12, 20, 22) E (8.5, 9.5, 10.5) (15, 35, 55) (12, 14, 16) (10, 18, 25) F (10, 12.5, 15) (35, 52.5, 70) (14, 17, 20) (13, 15, 20) G (9, 11, 13) (25, 30, 35) (17, 23, 27) (13, 18, 23) H (14, 15, 16) (20, 30, 40) (12, 16, 20) (15, 8, 12) Output Reducto Ipoveet DMU etup cot qualty Iceae aket epoe A (0, 5, 10) Good (0.6, 0.8, 1) Good (0.6, 0.8, 1) B (10, 15, 25) Good (0.6, 0.8, 1) Good (0.6, 0.8, 1) C (5, 10, 20) Fa (0.3, 0.5, 0.7) Fa (0.3, 0.5, 0.7) D (11, 13, 15) Good (0.6, 0.8, 1) Good (0.6, 0.8, 1) E (10, 14, 20) Good (0.6, 0.8, 1) Weak (0, 0.2, 0.4) F (5, 9, 15) Fa (0.3, 0.5, 0.7) Good (0.6, 0.8, 1) G (10, 20, 25) Good (0.6, 0.8, 1) Fa (0.3, 0.5, 0.7) H (10, 14, 20) Fa (0.3, 0.5, 0.7) Weak (0, 0.2, 0.4) Fgue 1 Mebehp fucto fo lgutc vaable Note: Weak: (0, 0.2, 0.4), fa: (0.3, 0.5, 0.7), good: (0.6, 0.8, 1).

12 Ehaced Ruell eaue fuzzy DEA 151 Table 2 The effcece of DMU obtaed by ug odel (7) fo the dffeet h value Poblty DMU level (h) A B C D E F G H h = h = h = h = h = h = h = h = h = h = h = The effcece of DMU obtaed by the odel (7) fo the dffeet h value ae llutated Table 2. Table 2 how that a the value of h ceae, the effcecy coe of a DMU deceae. Extee tuato h = 0 epeet all the poble poducto ceao ae codeed, h = 1 epeet cp cae. DMU D, E, G all ae effcet at ay h value. DMU H ae effcet at ay h value, o D, E, G ae bet choce of all eght FMS alteatve. DMU F, patculaly, etve to vaable eaueet, t ot effcet cp cae; but whe put ad output ae fuzzy ( 0 h 1), t becoe effcet. So, we ca coclude that t potat to ue a fuzzy appoach to evaluate the effcecy wth DEA odel whe put ad output ae fuzzy. Obvouly, the eao why the ube of effcet DMU at gve h value elatvely lage copag to the ube of DMU that the ube of DMU all copaed wth the ube of ctea eployed fo evaluato. The effcece of DMU obtaed by ug odel (8) fo the dffeet h value ae llutated Table 3. DMU D, E ad G all ae effcet Table 3 ut a Table 2 at ay poble h value. Whe DMU ae all effcet Table 2 ad 3, the effcece of thee DMU Table 2 ae alle tha the effcece of coepodg DMU Table 3 at gve h value, th becaue effcecy dex Table 3 oly put effcecy ad ot the o-zeo put lack ad, thu, fal to accout fo all effcece that the odel ca detfy. It woth otg that, whe h = 1 (cp cae), DMU F effcet Table 2 but effcet Table 3; t the elatohp Re betwee ERM odel ad CCR odel lead to th dffeece of effcecy (ee Secto 2), ad ottg the o-zeo put lack lead to DMU F whch hould be detfed to be effcet detfed to be effcet Table 3. We ca obeve that the ube of effcet DMU Table 2 elatvely lage tha the ube of effcet DMU Table 3 at gve h value, th becaue the ube of cotat odel (7) lage tha the ube of cotat odel (8).

13 152 M. Wag ad Y. L Table 3 The effcece of DMU obtaed by ug the odel (8) fo the dffeet h value Poblty DMU level (h) A B C D E F G H h = h = h = h = h = h = h = h = h = h = h = By ug copaatve eult whch ae peeted above, we caot coclude that ou appoach (odel (7)) oe effcet tha fuzzy DEA odel baed o CCR odel (odel (8)). But we ut pot out that the effcecy eaue of ou appoach (odel (7)) elatvely oe eaoable tha that of fuzzy DEA odel baed o CCR odel (odel (8)), that the foe otly detee put ad output effcece that epeet epaate etate of put ad output effcecy, ad ufy the ato effcecy ad lack to a cala eaue; whle the late oly put effcecy ad ot ozeo put lack ad, thu, fal to accout fo all effcece; theefoe, the popoed appoach epeet oe eal-lfe pocee oe appopately. Thee ae eveal effcet DMU ou exaple, t the ube of put ad output ctea, that, elatvely lage copag to the ube of DMU bg th tuato. Th ay bg dffculty to deco ake choce, but th a ubqutou poble heet DEA. Thee ae two way to deal wth th poble, oe to ake geat effot to educe the ube of put ctea ad ouput ctea, the othe to take accout of the akg of fuzzy DMU. 5 Cocluo I th pape, by ug a akg ethod baed o the copao of -cut, we tafo the fuzzy veo of ERM o-adal DEA odel to equvalet cp LP foulato. The popoed fuzzy DEA odel exted ERM odel to a oe geeal fo whch cp, fuzzy ad hybd data ca be hadled ealy. Ucetaty copoated the odel foulato. The effcecy eaue of ou appoach elatvely oe eaoable tha thoe of fuzzy DEA odel baed o CCR o BCC odel, that the foe otly detee put ad output effcece that epeet epaate etate of put ad output effcecy, ad ufy the ato effcecy ad lack to a cala eaue; whle the late oly put effcecy ad ot o-zeo put lack ad thu fal to accout fo all effcece. The popoed appoach epeet oe eal-lfe pocee oe appopately.

14 Ehaced Ruell eaue fuzzy DEA 153 The fuzzy DEA appoach that th pape peet ca facltate deco akg the electo of a FMS. I Secto 4, lgutc vaable ad tagula fuzzy ube ae ued to quatfy the vaguee heet deco paaete. The popoed decoakg faewok eable tagble a well a tagble apect to be take to accout the techology electo poce. It a vable deco-akg tool by ogaato codeg techology vetet ad ca detee the ot appopate FMS alteatve. It obvou that the deco faewok peeted th pape equally applcable to dvee deco-akg poble ecouteed aageet cece that copoate vaguee. Ackowledgeet The autho thak two aoyou efeee fo the cotuctve coet o eale veo of th pape. Meqag Wag thak Guzhou Uvety Scece Fud ued to attact talet (No ) fo the uppot. Yogu L thak the Natoal Natue Scece Fud of Cha (No ) ad Cha Pot-doctoal Scece Fud Specal Gat Poect (No ) fo the uppot. Refeece Bake, R.D., Chae, A. ad Coope, W.W. (1984) Soe odel fo etatg techcal ad cale effcece data evelopet aaly, Maageet Scece, Vol. 30, No. 9, pp Chae, A. ad Coope, W.W. (1962) Pogag wth lea factoal fuctoal, Naval Reeach Logtc Quately, Vol. 15, pp Chae, A., Coope, W.W. ad Rhoe, E. (1978) Meaug the effcecy of deco akg ut, Euopea Joual of Opeatoal Reeach, Vol. 2, pp Coope, W.W. ad Huag, Z. (2007) Effcecy aggegato wth Ruell eaue data evelopet aaly, Soco-Ecooc Plag Scece, Vol. 41, pp Fae, R., Gokoff, S. ad Lovell, C.A.K. (1985) The Meaueet of Effcecy of Poducto. Boto: Kluwe-Nhoff, p.162. Guo, P. ad Taaka, H. (2001) Fuzzy DEA: a peceptual evaluato ethod, Fuzzy Set ad Syte, Vol. 119, pp Kao, C. ad Lu, S.T. (2000) Fuzzy effcecy eaue data evelopet aaly, Fuzzy Set ad Syte, Vol. 119, pp Kaak, E.E. (2008) Ug data evelopet aaly fo flexble aufactug yte the peece of pece data, It. J. Advace Maufactug Techology, Vol. 35, pp Kaak, E.E. ad Kuzgukaya, O. (2002) A fuzzy ultple obectve pogag appoach fo the electo of a flexble aufactug yte, It. J. Poducto Ecooc, Vol. 79, pp Kaak, E.E. ad Tolga, E. (2001) Fuzzy ult-ctea deco-akg pocedue fo evaluatg advaced aufactug yte vetet, It. J. Poducto Ecooc, Vol. 69, pp Leo, T., Le, V., Ruz, J.L. ad Svet, I. (2003) A fuzzy atheatcal pogag appoach to the aeet of effcecy wth DEA odel, Fuzzy Set ad Syte, Vol. 139, pp

15 154 M. Wag ad Y. L Letwoakul, S. (2003a) Fuzzy data evelopet aaly (DEA): a poblty appoach, Fuzzy Set ad Syte, Vol. 139, pp Letwoakul, S. (2003b) Fuzzy BCC odel fo data evelopet aaly, Fuzzy Optzato ad Deco Makg, Vol. 2, pp Lu, S-T. (2008) A fuzzy DEA/AR appoach to the electo of flexble aufactug yte, Copute ad Idutal Egeeg, Vol. 54, pp Pato, J.T. ad Ruz, T.L. (1999) A ehaced DEA Ruell gaph effcecy eaue, Euopea Joual of Opeatoal Reeach, Vol. 115, pp Rak, J. ad Raek, J. (1985) Iequalty elato betwee fuzzy ube ad t ue fuzzy optzato, Fuzzy Set ad Syte, Vol. 16, pp Shag, J. ad Sueyoh, T. (1995) A ufed faewok fo the electo of a flexble aufactug yte, Euopea Joual of Opeatoal Reeach, Vol. 85, pp Toe, K. (2001) A lack-baed eaue of effcecy data evelopet aaly, Euopea Joual of Opeatoal Reeach, Vol. 130, pp Wag, M. ad Lag, L. (2009) Fuzzy cotext-depedet data evelopet aaly, It. J. Data Aaly Techque ad Statege, Vol. 1, No. 3, pp Zadeh, L.A. (1965) Fuzzy et, Ifoato ad Cotol, Vol. 8, pp