A Credibility Approach for Fuzzy Stochastic Data Envelopment Analysis (FSDEA)

Transcription

1 Poceedg of the 7 th A Pcfc Idutl Egeeg d Mgeet Ste Cofeece 6 7- Decee 6 Bgkok hld A Cedlt Appoch fo Fuzz Stochtc Dt Evelopet Al FSDEA Vtho Pug Deptet of Idutl Egeeg Skhwot vet Ogkk Nkhook 6 HAIAND El: v_pou@hoo.co Ptchpo Ypt d Peeuth Chethkul Deptet of Idutl Egeeg Kett vet M Cpu Bgkok 9 HAIAND El: {fegpp fegpc}@ku.c.th Sowee etwokul Deptet of Poduct Developet Fcult of Ago-Idut Kett vet M Cpu Bgkok 9 HAIAND El: pl@ku.c.th Atct. It well kow tht Dt Evelopet Al DEA eltve effcec eueet tool whch ue optzto techque to utotcll clculte the weght ged to the cp detetc ultple put d output of et of the Deco Mkg t DM eg eed. Howeve cp detetc dt equeet delt pplcto to the el wold pole whee oe put o output eue lkel e ed o the vlue udget of the deco ke. I th ppe the Fuzz Stochtc Dt Evelopet Al FSDEA odel the ce of tpezodl fuzz ue dtuted wth ol dtuto dul te popoed whch olved two tep of tfoto. Ft the fuzz detetc DEA FDDEA odel coveted the cocept of chce-coted pogg. Next the cedlt ppoch ued to covet FDDEA odel to well-defed cedlt pogg odel whch fuzz vle e eplced expected cedt. Kewod: Chce coted pogg Cedlt eue Dt evelopet l Fuzz thetcl pogg Pefoce evluto Polt heo. INRODCION he tdtol DEA wdel ppled opetc thetcl pogg techque the ee tht o choce of petc fuctol fo eeded the etto of the fote fo eug d echkg the eltve effcec of hoogeou deco kg ut DM wth ultple put d ultple output. Che et l. 978 Coope et l. Zhu 3 h techque cope evluted tget DM o DM o wth othe DM tht utlze the e ultple put to poduce the e ultple output ed o opeto eech techque to utotcll clculte the weght ged to put d output of ech DM. Nueou eech ppe o effcec eueet of el lfe pole ug DEA hve ee coducted. Fo exple Chlge 995 ued DEA d ult-fcto ot l to tud of the clcl effcec of 36 phc gle hoptl. Sueoh 995 popoed ew pplcto of DEA fo copg pefoce dffeet te peod. Nh d Ste- Kw 996 ppled DEA to eue fcl poduct co ellg effectvee t the k ch level. ofllt 997 ued DEA to evluted o tetol pege ce fo the e 99. Hog et l. 999 ued DEA to evlute the effcec of the te tegto SI poect d uggeted the ethodolog whch ovecoe the ltto of DEA though hd l utlzg DEA log wth che leg. Al-Sh 999 evluted the opetol effcec of 55 ufctug ogzto Jod odfed DEA odel. Zhu ued DEA to detf the et pctce of 5 cope whch w lzed Fotue gze 995. Mt d Roå ppled DEA odel to lze the techcl effcec d pefoce of ech dvdul Sph pot. : Coepodg Autho 633

2 Fuku d Wee ued lloctve DEA odel to evluted Jpee k opetg dug Fçh d Reede ppled the DEA cocept ed o flexle ppoch to evluted Bzl locl telepho ove the peod uo d Dothu 5 ued petc ppoch clled tochtc fote l SF coped wth DEA odel to e the dvetg effcec fo ple of kete elected fo top dvete. he foulto ued th ppe ed o the ot cll odel of DEA o clled the CCR odel tll toduced Che et l. 978 ude the upto of cott etu to cle CRS of the effcet poducto fote. hee e two odel vlle to ette the effcet fote. Oe delg wth zg put whle poducg t let the gve output level whch clled the put oeted CCR odel CCR-I. he othe ttept to xze output whle ug o oe th the oeved out of put whch clled the output oeted CCR odel CCR-O. I ddto to the CCR odel othe well-kow DEA odel ech of whch e exteo of CCR odel clude the BCC odel whch ult o the upto of vle etu to cle VRS of the effcet poducto fote Bke et l. 98. hee e two BCC odel oe delg wth put BCC-I d the othe the output oeted BCC odel BCC-O. Che et l. 985 developed the ddtve DEA odel ADD whch codeed pole put decee well output cee ulteoul the e poducto polt et the BCC d CCR odel.. DAA ENVEOPMEN ANAYSIS Fo the CCR odel uppoe tht thee e DM e evluted ech of whch coue the e tpe of put d poduce the e tpe of output fo DM. All put d output e ued to e oegtve ut t let oe put d oe output e potve. If the put d output dt fo DM e x x d epectvel the the effcec of tget DM o DM o whee o ge ove eued olvg the fctol pogg pole FP to ot vlue of put weght v d output weght u fo d deco vle. u FP-I x θ v x u v x o o < fo u v > 3 I the ce of put oeted the FP odel -3 eplced wth le pogg pole P ltg deoto of the oectve fucto to d ovg t to cott. Afte ultplg oth de of deoto the the FP odel equvlet to the followg P odel. CCR-I x θ u o v 5 x o u v x < fo 6 u v > 7 Fo pl pole of CCR-I -7 whch clled ultple odel the DM o evluted to e CCR-effcet f θ d thee ext t let oe optl v u wth v > d u >. Othewe the DM o CCR-effcet. Fo eve P thee the dul pole whch the ole of vle d cott e eveed. Suppoe tht the dul pole of the CCR odel expeed wth el vle θ d dul vle λ fo the the dul pole of CCR-I DCCR-I the followg P odel. DCCR-I θ 8 θx λ x > fo 9 λ o o > fo θ etcted λ > Fo DCCR-I 8- whch clled 63

3 evelopet odel the DM o deteed to e dl o techcl effcec o CCR-effcec f d ol f optl oluto tfe θ d the DM o deteed to e CCR-effcec f d ol f optl oluto tfe θ d ll lck e zeo. heefoe DCCR-I c e exteded follow. DCCR-I θ ε defed E o { / λ > } fo { }. heefoe the CCR-I poecto of the evelopet odel e the followg foul: x opoved θ x o θx λ x 3 o opoved o. 3 λ o θ etcted ε > λ > > > 5 Whee e lck of the put oeted evelopet odel d ε foulted o- Achede ftel whch dded the oectve fucto effectvel llow the zto ove θ to peept the optzto volvg ll lck. Howeve ug pecfc vlue fo ε dffcult pctce Coope et l. Zhu 3 theefoe the two-tge o twophe pocedue ued. I the ft tge P pole 8- olved to fd optl θ. he the ecod tge fdg oluto tht xze the u of ll lck wth uce tht θ wll ot e pole to pove put o output wth woeg oe othe put o output. he two-phe pocedue fo DCCR the followg fo: Phe I-DCCR-I θ 6 Phe II-DCCR-I x 7 I the ce of output oeted let put weght p d output weght q fo d e deco vle. he followg FP odel wll e olved fo evluted DM o fo o ge ove. px FP-O x η q px q o o > 5 p q > 6 Sl to the CCR-I the FP-O equvlet to the followg P odel. x o CCR-O x η p 7 o q 8 θx λ x 8 o q p x < 9 λ 9 o θ etcted λ > > >. A optl oluto λ fo 6- clled the x-lck oluto d f the x-lck oluto tfe oth of d the t clled zeo lck. Fo effcet DM o the effcec of x fo DM o c e poved poectg DM o to t efeece et E o ed o the x lck oluto whch o o p q > 3 he DM o evluted to e CCR-effcet f η d thee ext t let oe optl p q wth p > d q >. Othewe DM o CCR-effcet. o uld the evelopet odel fo the output oeted odel DCCR-O let η d μ fo e dul vle d t t e lck of the output oeted evelopet odel. he the two-phe pocedue of the dul pole of DCCR-O the followg P odel. Phe I-DCCR-O η 3 635

4 Phe II-DCCR-O x t t 3 x μ x t 33 o μ η t 3 o η etcted μ > t > t > 35 Sll the DM o deteed to e CCReffcet f d ol f optl oluto ut oth tfed θ d zeo lck t d effcet DM o the effcec of x o t o. Fo fo DM o c e poved poectg DM o to t efeece et E o ed o the x lck oluto whch defed E o { / μ > } fo { } 36 heefoe the CCR-O poecto of the evelopet odel e the followg foul. x opoved opoved x o t 37 η o t 38 hee e thee eo fo olvg the DCCR. Ft the ue of DM lge th the ue of put d output d hece t tke oe te d lge eo to olve CCR wth cott th to olve DCCR wth cott. Secod ug CCR the effcet DM cot e poved to pove ctvt ecue efeece et d x lck oluto cot e foud. Fll the tepetto of DCCR e oe tghtfowd th thoe of CCR. Coope et l. he evelopet odel focued th ppe d thee o dffeet etwee put d output oeted evelopet odel P tuctue thu DCCR-O wll e ot how th ppe. 3. FZZY SOCHASIC DAA ENVEOPMEN ANAYSIS I the pecedg ecto the ogl DCCR-I odel e dcued ed upo cp detetc dt equeet ut DEA odelg the el wold ed deco o foto whch oth of fuzzl pece d poltcll ucet. Soe of the thetcl odelg techque tht odel the heet oth of doe d vguee of the te clude fuzz d tochtc pogg. I th ecto the fuzz d tochtc put oeted CCR odel FSDCCR-I popoed follow. Phe I- FSDCCR-I θ 39 Phe II- FSDCCR-I x θ λ o λ ˆ ˆ o θ etcted λ > > > 3 Whee ˆ e fuzz do vle. I th ppe the cocept of chce-coted pogg CC whch w toduced Che d Coope 959 ued to odel FSDCCR-I. CC kd of tochtc optzto ppoche. It utle fo olvg optzto pole wth do vle cluded cott d oete the oectve fucto well. he cott e guteed to e tfed wth pecfed polt o cofdece level t the optl oluto foud. Suequetl oe eeche lke Che d Coope 96 Segupt d et l. 99 Olee d Petee 995 Coope et l. 996 Coope et l Sueoh Coope et l. d othe etlhed oe theoetcl eult feld of tochtc pogg. Stcu-M d Wet 976 peeted evew ppe o tochtc pogg wth gle oectve fucto. he FSDCCR-I odel fo 8- CC pogg CC-FSDCCR-I foed CC-FSDCCR-I θ P λ θ o > 5 P ˆ o λ ˆ > β 6 θ etcted λ > 7 whee P e polt d β e pecfed polte. DM o fo -7 deteed to e tochtc effcet f d ol f optl oluto tfe θ d ll lck e zeo fo ll optl oluto. et ρ d τ e lck whch c e eted to the eqult outde ce to cheve eqult fo 5-6 CC-FSDCCR-I. 636

5 P λ θ o ρ 8 P ˆ o λ ˆ β τ 9 et d e potve ue uch tht Phe I- CC-FSDCCR-I θ 5 Phe II- CC-FSDCCR-I x 5 P λ θ o 5 P ˆ o λ ˆ β 53 θ etcted λ > > >. 5 Fo 5 d 53 the level of polt e dcted the vlue of d epectvel pet futhe decee o d cee ˆ o fo ple of oevto wthout woeg othe put o output.. HE EQIVAEN FZZY DEERMINISIC DCCR-I MODE I th ecto the ethod to covet FSDCCR-I to fuzz detetc equvlet of the fuzz tochtc odel how. Aue tht the fuzz put d output e dtuted wth ol dtuto. Suppoe tht z d z epectvel e fuzz put d output dtuted wth tdd ol dtuto. z z λ ˆ o θx ˆ o λ Ex ˆ λ ˆ [θλ] Cov [θλ] E ˆ o [λ] Cov [λ] θe λ E ˆ o whee [θλ] θ λ λ λ [λ] λ λ λ Cov d Cov e x d x tce epectvel whch dcte vce d covce of fuzz output d put do vle fo the th DM. heefoe 5 d 53 e olzed followg foul. P P z z λ Ex ˆ θe [θλ] Cov [θλ] E ˆ [λ] o Cov λ E ˆ [λ] Refoulte 57 d 58 to e o β λ Ex ˆ θe Φ θλ] Cov [θλ] o [ E ˆ λ E ˆ Φ β λ] Cov [λ] o [ whee Φ epeet the ol cuultve dtuto fucto d Φ t vee. Sce d e expeed the expected vlue d the vcecovce tce Cov d Cov whch e foulted qudtc te. hu olvg two-phe pole of FSCCR o-tvl tk. I th ppe the lezto ppoch to ot le detetc equvlet odel ued Coope et l. 996 Coope et l Code the putoutput dt tuctue wth do dtuce fo ; d. follow: x ζ d ˆ ξ 6 whee x o x o d epectvel epeet fuzz e of fuzz put d output vle. o o d epectvel epeet fuzz tdd devto of fuzz put d output vle. o ζo ζ o ξo d ξ epeet etc dtuce o eo te of fuzz put d output. Geell eo tuctue ζ o ζ ξ o d ξ e ued to e tdd ol dtuto N the fuzz expected vlue d fuzz vce of put d output vle e epectvel defed 637

6 E o x o d E x E ˆ o o d E ˆ V o d V o V ˆ o o d V ˆ Sce vce d covce tce e o Cov o M o o Cov o M o heefoe o M o M O O o M o M θλ] Cov [θλ] θ o λ 68 [ λ] Cov [λ] o λ [ 69 he equvlet fuzz detetc odel of the twophe pole fo dul CCR odel FDDCCR the te of fuzz P odel Phe I-FDDCCR-I θ 7 Phe II-FDDCCR-I x 7 λ x θ x o Φ θ o λ 7 o λ θ etcted λ > Φ β o λ > 73 > HE EQIVAEN CRISP DEERMINISIC DCCR-I MODE 5. Polt d Necet Meue Polt theo the cotext of the fuzz et theo w toduced Zdeh 978 whch delg wth otochtc peco d vguee. A good efeece o polt theo Duo d Pde 98 d Ze 996. Polt d ecet eue e uzed d dopted to olve fuzz DEA etwokul et l Suppoe tht Θ PΘ π fo e polt pce wth Θ eg the oept et of teet PΘ the collecto of ll uet of Θ d π the polt eue fo PΘ to [ ] the πφ d πθ d 75 π A up {π A } wth ech A PΘ. 76 et Ψ e fuzz vle el-vlued fucto defed ove Θ theefoe the eehp fucto of gve μ Ψ up{ π{θ }/Ψ θ } R. 77 θ Θ et Θ PΘ π e poduct polt pce uch tht Θ Θ Θ the πa {π A /A A... A A PΘ. 78 o cope fuzz vle Duo d Pde 98 let... e fuzz vle d ƒ : R R e elvlued fucto fo. he polt eue of fuzz evet gve π ƒ... fo up { {μ }/ƒ... ;... } R he ecet eue N of fuzz evet defed the polt of oppote evet. If evet A d A c e oppote evet Θ the NA πa c. heefoe the ecet eue of fuzz evet gve N ƒ... fo {... up R μ }/ {... }ƒ... > }. 8 { 5. Expected Cedt of Nol Covex Fuzz Vle he cedlt eue C of fuzz evet defed u whch the vege of t polt d ecet eue.e. 638

7 π N C. 8 A expected cedt opeto of fuzz vle Ψ o polt pce defed u E Ψ CΨ tdt CΨ tdt. 8 Code fuzz vle ϕ o polt pce Θ PΘ π whch clled ol covex fuzz vle ee Fg. f ϕ tfe the followg popete. he fuzz vle ϕ ol f up {μ }. R he -level et of the fuzz vle ϕ defed the et of eleet tht elog to ϕ wth eehp of t let.e. ϕ { R/μ ϕ }. he fuzz vle ϕ covex f μ ϕ λ λ > { μ ϕ μ ϕ } fo ll R d λ []. μ ϕ ϕ ϕ ϕ ϕ ϕ ϕ Fg.: Nol covex fuzz vle ϕ ϕ I th ppe the fuzz vle Ψ ued to e the ol covex fuzz vle thu the expected cedt of Ψ c e deved follow. M E Ψ l CΨ tdt CΨ tdt 83 M M Bed o the ecet d the cedlt defto the expected cedt of Ψ defed M M E Ψ l dt dt πψ tdt M M M π Ψ > tdt πψ < tdt πψ tdt. 8 M M Sce Ψ the fuzz vle wth ol covex fucto the vlue of π Ψ < t d π Ψ > t e ve cloe to π Ψ < t d π Ψ > t. heefoe π Ψ < t d π Ψ > t e ppoxted π Ψ < t d π Ψ > t epectvel. et d deote lowe d uppe oud of - level et of μ Ψ. he the expected cedt of the ol covex fuzz vle E Ψ Ψ Ψ Ψ πψ tdt πψ tdt Ψ Ψ Ψ 85 o ot explct fo of the expected cedt te π Ψ t d π Ψ t 85 e evluted ppe oud; Mx t. Ψ > t 86 owe oud; Mx t. Ψ < t. 87 Note tht wee popoed d poved etwokul et l. 3. he eult lo ppled fo fuzz vle of the R tpe Duo d Pde 98. he expected cedt of ol covex fuzz vle e epeetg fuzz pete FSDEA odel ecue e d tdd devto of fuzz put-output vle uull hve th tpe of eehp fucto. 5.3 Mthetcl Modelg Fuzz Evoet Deco kg fuzz evoet w ft developed Bell d Zdeh 97. A pplcto of fuzz thetcl pogg pole w popoed Ze d howed tht the oluto oted fuzz le pogg w lw effcet. Suequetl oe eeche lke Segupt 99 uhdul u d Iwu 998 u 998 etwokul et l d othe hve etlhed oe theoetcl eult the feld of fuzz pogg. I th ppe the cocept of cedlt ppoch dopted ltetve w fo olvg FDDCCR odel. 5.. CDDCCR-I Model I th uecto the cedlt ppoch ued to covet the FDDCCR-I odel to e CDDCCR-I. h ppoch tet fuzz ucet the oectve fucto d cott tkg expected cedt opeto. et E[ ] e expected cedt opeto of fuzz vle the 7 d 73 ecoe E λ x Φ θ x o Φ o 88 E o Φ β o λ Φ β

8 Sce E[ X Y ] E[ X ] E[Y ] fo el ue d u d u 3 etwokul et l. 3 the the FDDCCR-I odel equvlet to the followg cp detetc P odel. Phe I-CDDCCR-I θ 9 Phe II-CDDCCR-I x 9 λ E[ x ] Φ E[ ] θe[ x ] Φ E[ ] 9 o E[ o ] Φ β E[ o ] o λ E[ ] Φ β E[ ] 93 θ etcted λ > > > 9 Sl to tdtol DCCR-I the equvlet cp detetc efeece et E o ed o optl oluto of 9-9 defed E o { / λ > } fo { }. 95 he effcet DM o c e poved poectg DM o to t efeece et E o theefoe CDDCCR-I poecto e gve followg foul. ˆ θ E[ x ] Φ E[ ] 96 x opoved opoved o ˆ E[ ] Φ β E[ ] 97 o 5.. CDDCCR-I Model wth pezodl Fuzz Iput d Output I th uecto ll of put d output vle e ued to e tpezodl fuzz ue. lke the ce of cp put d output fuzz e d tdd devto e coputed the fuzz thetc Ze 996 Kl et l. 997 d expected cedt of fuzz e d tdd devto e coputed olvg et fuzz ue c o o c c c c fo e tpezodl fuzz ue oeved ple whch c c c c e cp dt of lowe d uppe oud of -level et t d. Ad let μ c e eehp fucto of c ee Fg. whch e defed fo < c o > c c fo c < c c c μ c 98 fo c c c fo c < c c c μ c c c c c c c c Fg.: pezodl fuzz ue d t eehp fucto et c d c epectvel e the lowe d uppe oud of the -level et of fuzz ue c the cloed cp tevl of c e defed c c < < c c. 99 Bed o cocept of the fuzz thetc the fuzz ple e of c c e clculted c c c c c c c c c c whee c c c c c the fuzz ple e of c c d c e the lowe d uppe oud of the -level et of fuzz ple e 6

9 epectvel. 86 d 87 e ued to fd the xu c c c > t c c c < t 3 Sce c < c d c > c the c t c t < < c c c c heefoe π c t c > t 5 c c π c t c < t 6 c c 5 d 6 e tegted to e c c c c tdt 7 c c c c c tdt 8 c Suttute 7 d 8 85 thu the expected cedt of c c e clculted follow. c c c c E c 9 I l e the fuzz vce of c deote tht whch clculted c c c c c c c c c c c c c c c c c c et c c c d c e cp ple vce t d lowe d uppe oud c c d c c e cp ple covce the uppe d lowe oud of the -level et of fuzz tdd devto e epectvel gve c c c c c c. 3 c c c c Now we code coelto of two do vle whch e defed ρ XY σ σ X XY σ Y fo < ρ XY < whee X d Y e do vle ρ XY the coelto of X d Y σ XY the covce of X d Y d σ X σ Y e tdd devto of X d Y epectvel. he ge ove [ ] e tht two do vle hghet coelted whe coelto ρ XY equl to o. Suequetl f the eltohp of X d Y le equto the σ XY σ X σ Y. Sce c c ξ d c c ζ whee ξ d ζ e cott ped of fuzz ue e le equto the ple covce d 3 e gve c c c 5 c. 6 c c c c o fd the xu of d 3 e efoulted d uttuted 86 d 87 epectvel. c c > t 7 c c c < t 8 c B tegtg 7 d 8 d uttutg thee eult 85 the expected cedt of gve E c c c c. 9 Sce the expected cedt of c d 9 d 9 e volved cp uto te of coe pot of ech tpezodl fuzz vle theefoe cp clculto ple e d tdd devto t coe pot e ued the CDDCCR-I odel. 6

10 Suppoe tht thee e t d w oevto fo put d output fo DM fo epectvel.e. ˆ ˆ x k... x t fo ; k t d ˆ ˆ l ˆ... w fo ; l w. he expected cedt of x d e epectvel defed x E x E E E x x x. 3 Fo the CDDCCR-I odel wth tpezodl fuzz put d output the expected cedt of fuzz vle the CDDCCR-I odel e eplced -3 theefoe Phe I-CDDCCR-I θ Phe II-CDDCCR-I x 5 λ x x x x Φ θ x o x o x o x o Φ o o o o 6 o o o o Φ β o o o o λ Φ β 7 θ etcted λ > > > 8 Fo CDDCCR-I poecto e gve followg foul. θ opoved x o x o x o x o Φ o o o o 9 ˆ opoved o o o o 3 Φ β o o o o Sce -3 e cp vlue the the CDDCCR- I odel -8 d CDDCCR-I poecto 9 d 3 c e olved tdd P. 6. CONCSION lke the tdtol DEA FSDEA odel ed deco o foto whch oth doe d vguee of the te. h ppe h peeted two tep of tfoto. Ft the FDDEA odel coveted the cocept of chce-coted pogg ed o put d output vle whch e dtuted ol dtuto. Next the cedlt ppoch ued to covet FDDEA odel to welldefed cedlt pogg odel whch fuzz vle e eplced expected cedt. he eult of th pocedue how fo pecl ce whch fuzz vle e tpezodl fuzz ue d we could uze tht the fuzz ple e d tdd devto of put d output e tpezodl fuzz ue. B the expected cedt ppoch expected cedt of fuzz ple e d tdd devto e el clculted ug the cp coe pot of fuzz ue. Afte eplcg the the tdd P c e ued to olve the CDDCCR-I odel d CDDCCR-I poecto. 6

11 REFERENCES Al-Sh M. 999 Optzto odelg fo ettg d ehcg eltve effcec wth pplcto to dutl cope Euope Joul of Opetol Reech Bke R.D. A. Che W.W. Coope 98 Soe odel fo ettg techcl d cle effcec dt evelopet l Mgeet Scece Bell R.E. d.a. Zdeh 97 Deco kg fuzz evoet Mgeet Scece 7-6. Che A. d W.W. Coope 959 Chce coted pogg Mgeet Scece Che A. d W.W. Coope 96 Chcecott d ol devte Joul of the Aec Stttcl Aocto Che A. W.W. Coope B. Gol.M. Sefod d J. Stutz 985 Foudto of dt evelopet l fo Peto-Koop effcet epcl poducto fucto J. of Ecooetc 3-7. Che A. W.W. Coope d E. Rhode 978 Meug the effcec of deco kg ut Euope Joul of Opetol Reech 9-. Chlge J.A. 995 Evlutg phc effcec hoptl: A ultvte l of et pctce Euope Joul of Opetol Reech Coope W.W. H. Deg Z.M. Hug d S.X Chce coted pogg ppoche to techcl effcece d effcece tochtc dt evelopet l Joul of Opetol Reech Socet Coope W.W. Z.M. Hug V. el S.X d O.B. Olee 998 Chce coted pogg foulto fo tochtc chctezto of effcec d doce DEA Joul of Poductvt Al Coope W.W. Z.M. Hug d S.X. 996 Stfcg DEA odel ude chce cott Al of Opeto Reech Coope W.W..M. Sefod d K. oe Dt evelopet l: A copeheve text wth odel pplcto efeece d DEA-olve oftwe Kluwe Acdec Pulhe odo. Duo D. H. Pde 98 Fuzz et d te: heo d pplcto Acdec Pe New Yok. Fçh.O. d M. Reede Pce cp egulto cetve d qult: he ce of Bzl telecoucto It. J. Poducto Ecooc Fuku H. d W.. Wee Ettg output lloctve effcec d poductvt chge: Applcto to Jpee k Euope Joul of Opetol Reech Hog H.K. S.H. H C.K. Sh S.C. Pk d S.H. K 999 Evlutg the effcec of te tegto poect ug dt evelopet l DEA d che leg Expet Ste wth Applcto Kl G.J..S. Cl d B. Yu 997 Fuzz Set heo: Foudto d Applcto Petce-Hll PR SA. d K.C. C.A.K. ovell d S. hoe 993 Chce-coted dt evelopet l Mgel d Deco Ecooc etwokul S. S.C. Fg J. A. Joe d H..W. Nuttle 3 Fuzz dt evelopet l DEA: A polt ppoch Fuzz Set d Ste etwokul S. S.C. Fg J. A. Joe d H..W. Nuttle 3 Fuzz dt evelopet l DEA: A cedlt ppoch I J.. Vedeg ed. Fuzz Set ed Heutc fo Optzto Spge- Velg Bel Hedeleg -58. S.X. 998 Stochtc odel d vle etu to cle dt evelopet l Euope Joul of Opetol Reech u B. 998 Mx chce coted pogg odel fo fuzz deco te Ifoto Scece u B. cet pogg: A ufg optzto theo vou ucet evoet Appled Mthetc d Coputto 7-3. u B. d K. Iwu 998 Chce coted pogg wth fuzz pete Fuzz Set d Ste u Y.K. d B. u 3 A Cl of fuzz do optzto: expected vlue odel Ifoto Scece uhdul M.K. 996 Fuzze d doe optzto fewok Fuzz Set d Ste uhdul M.K. 3 Mthetcl pogg the peece of fuzz qutte d do vle Joul of Fuzz Mthetc uhdul M.K. Optzto ude hd ucett Fuzz Set d Ste uo X. d N. Dothu 5 Aeg dvetg ed pedg effcece geetg le Joul of Bue Reech Mt J.C. d C. Roå A pplcto of DEA to eue the effcec of Sph pot po to pvtzto Joul of A pot Mgeet 7 63

12 9-57 Olee O.B. d N.C. Petee 995 Chce coted effcec evluto Mgeet Scece -57. Pug V. P. Ypt P. Chethkul d S. etwokul 6 Fuzz Stochtc Dt Evelopet Al FSDEA Model: A Polt Appoch. Poceedg of the Stttc d Appled Stttc Cofeece Chou hld C88-C93. Sueoh. 995 Poducto l dffeet te peod: A pplcto of dt evelopet l Euope Joul of Opetol Reech Sueoh. Stochtc DEA fo Retuctue Stteg: A Applcto to Jpee Petoleu Cop Oeg Nh D. d A. Ste-Kw 996 A pplcto of DEA to eue ch co ellg effcec Copute Op Re Segupt J.K. 97 Stochtc Pogg: Method d Applcto Noth-Holld Ated. Segupt J.K. 98 Effcec eueet tochtc put-output te Itetol Joul of Ste Scece Segupt J.K. 987 Dt evelopet l fo effcec eueet the tochtc ce Copute d Opeto Reech 7-9. Segupt J.K. 99 foto tochtc DEA odel Joul of Ecooetc J.K. Segupt 99 A fuzz te ppoch dt evelopet l Coput. Mth. Appl Stcu-M I.M. d M.J. Wet 976 A eech logph tochtc pogg Opeto Reech ofll C. 997 Iput effcec poflg: A pplcto to le Copute Op Re Zdeh.A. 978 Fuzz et fo theo of polt Fuzz Set d Ste 3-8. Zhu J. Mult-fcto pefoce eue odel wth pplcto to Fotue 5 cope Euope Joul of Opetol Reech 3 5- Zhu J. 3 Qutttve odel fo pefoce evluto d echkg: dt evelopet l wth pedheet d DEA excel olve Kluwe Acdec Pulhe Boto. Ze H.J. 978 Fuzz pogg d le pogg wth evel oectve fucto Fuzz Set d Ste Ze H.J. 985 Applcto of fuzz et theo to thetcl pogg Ifo. Sc Ze H.J. 996 Fuzz Set heo d It Applcto Kluwe Acdec Pulhe odo. AHOR BIOGRAPHIES Vtho Pug ectue the Deptet of Idutl Egeeg t Skhwot vet. he eceved M. Eg Mufctug Ste Egeeg fo Kg Mogkut vet of echolog hou KM hld. Sce he gduted tudet the Deptet of Idutl Egeeg Kett vet. M topc of h eech clude ucett optzto pefoce eueet d evluto d DEA odel ue evoet. H el dde v_pou@hoo.co Ptchpo Ypt Aocte Pofeo d cuetl Decto of Egeeg Mgeet pecl Pog the Deptet of Idutl Egeeg t Kett vet. She eceved M. Sc. Idutl Egeeg d geet d D. ech. Sc. Idutl Egeeg fo A Ittute of echolog AI hld. He cuet eech teet clude tochtc veto odel effcec eueet d deco l egeeg poect. He el dde fegpp@ku.c.th Peeuth Chethkul cuetl Aocte Pofeo the Deptet of Idutl Egeeg t Kett vet. He eceved M.S. d Ph.D. Idutl Egeeg fo ex echologcl vet. H eech teet the feld of opeto eech d coputtol ethod. H el dde fegpc@ku.c.th Sowee etwokul Att Pofeo the Deptet of Poduct Developet Fcult of Agodut Aduct pofeo Deptet of Idutl Egeeg Fcult of Egeeg d Aduct Pofeo Deptet of Copute Egeeg Fcult of Egeeg Kett vet. She eceved M.S. Idutl Egeeg d Opeto Reech fo vet of Clfo t Bekele 998 d eceved Ph.D. Idutl Egeeg fo Noth Col Stte vet. He techg d eech teet clude DEA odel poduct d poce odelg d pedcto wth oft coputg d uppl ch geet. He el dde pl@ku.c.th 6