DATA ENVELOPMENT ANALYSIS WITH FUZZY RANDOM INPUTS AND OUTPUTS: A CHANCE-CONSTRAINED PROGRAMMING APPROACH. 1. Introduction
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1 Iaa Joa of Fzz Sstes Vo. No DAA ENVEOPMEN ANAYSIS WIH FUZZY ANDOM INPUS AND OUPUS: A CHANCE-CONSAINED POGAMMING APPOACH S. AMEZANZADEH M. MEMAIANI AND S. SAAI ABSAC. I ths ae we dea wth fzz ado vaabes fo ts ad otts Data Eveoet Aass DEA. hese vaabes ae cosdeed as fzz ado fat bes wth kow dstbto. he obe s to fd a ethod fo covetg the ecse chace-costaed DEA ode to a cs oe. hs ca be doe b fst defzzfcato of ecse obabt b costctg a stabe ebesh fcto secod defzzfcato of the aaetes sg a -ct ad fa covetg the chace-costaed DEA to a cs ode sg the ethod of Cooe [4].. Itodcto Most eseach wok DEA deas wth ecse ad detestc foato. he aes eated to cetat ae ethe dea wth ado foato see [ &8] o fzz foato fo ts ad otts see [9 & ]. Howeve we have ot otced a cotbto that a cooate the hbd cetat.e. fzzess ad adoess DEA. Whe the aaetes ts ad otts of a DEA ode ae fzz ado vaabes we have a fzz chace-costaed DEA obe. heefoe we cosde a DEA ode to evaate the effcec of DMUs whe data ae fzz ado vaabes. I addto the obabt of the costats a aso be descbed as fzz eato. Kawakeaak [3] ad P & aesc [9] todced fzz ado vaabes ad Gaga & Ye [7&8] [4&5] Chakabot [&3] ad hada [6&7] eseted soe otat ethods fo sovg atheatca ogag wth fzz ado vaabe coeffcets. I ths ae we cosde the CC ode of DEA wth chace-costaed ogag aoach whch ts ad otts ae fzz ado vaabes; we asse that these fzz ado vaabes ae fat fzz bes. O obectve s to covet the fzz chace-costat DEA to cs DEA. Fo ths ose fst of a the ode s defzzfed b sg a stabe ebesh fcto fo the fzz eato of the obabt. I the secod stage the fzzess of the aaetes s eoved b a -ct aoach ad fa the adoess s ectfed b cassca ea-vaace ethod of Cooe [4]. he stcte of ths ae s as foows: Secto esets the fzz eceved: Novebe 4; Acceted: Je 5 Ke wods ad hases: Data eveoet aass Chace-costaed DEA Fzz ado vaabe aga fzz be.
2 S. aezazadeh M. Meaa ad S. Saat chace-costaed DEA ad the ocede of ts coveso to a cs DEA ode. o deostate the ocess a eca eae s gve secto 3. Secto 4 cossts of a cocso.. Fzz Chace Costaed DEA We foate the fzz chace-costaed DEA as foows: et... ad... eeset s a s fzz ado t ad ott vectos fo each DMU whee ; & ; s ae fat fzz ado vaabes eated to a ado vaabe ad ae the eft ad ght seads. We eset the fzz chace-costaed CC ode as foows: M s.t : Pob Pob > > s... "> " sgfes that the costats ae fzz satsfed wth obabt. Net we deostate a vew of the effcet fote of sch a ode fo sge t ad sge ott. Fo sct of esetato we cosde to be a ado vaabe ad to be a taga fzz ado vaabe. I Fge the es ad 3 eeset the e at ea ad the ote at of the effcec fote. Each -ct tesects ths fote ado vaabes. Fo stace the abscssa of the ot fo s a ado vaabe wth dstbto N σ ad ts odate s a ado vaabe wth dstbto N σ. I what foows the ocess of coveso has bee deveoed thee hases. I hase I the ecse obabt s defzzfed. I hase II defzzfcato of the aaetes s caed ot theeb covetg the obe to a chace-costaed
3 Data Eveoet Aass wth Fzz ado Its ad Otts: A Chace-costaed Pogag Aoach 3 DEA. Fa the coveso of ths chace-costaed DEA to a cs ode s efoed hase III. DMU N µ σ N σ 3 FIGUE. Effcec fote of a fzz chace-costaed CC ode. Phase I: It s cea that evets. Fo sct et ad C deote the evet ae fzz. he ebesh fcto µ eas the gade to whch... ob fts the costats. he obabt of ths fzz evet s the ease of the degee to whch s eabe. Accodg to [] we defe the ebesh fcto fo the Pob C > as foows: whee µ Pob f Pob C Pob C f Pob C othewse s the toeace of the obabt. It s evdet that: Pob C
4 S. aezazadeh M. Meaa ad S. Saat 4 heefoe... Pob ad sa...s Pob So ca be coveted to:... s... Pob... Pob : s.t M Phase II: I ths hase the fzzess of the coeffcets ae deat wth. Fo ths we a the cocet of -ct as []. B todcg the -ct of costats ad sato of fat fzz bes we w have the foowg obe: : s.t M [ ]... Pob [ ] s... Pob... 3
5 Data Eveoet Aass wth Fzz ado Its ad Otts: A Chace-costaed Pogag Aoach 5 Accodg to Saat et a. [] to evaate the effcec of DMUs the owe eve of ts ad e eve of otts that s the best at of DMUs fo each DMU ae coaed wth the e at of effcec fote. he best at of a DMU s ad the e at of the fote s. heefoe ode 3 ca be wtte as foows: 4 hs obe s a aaetc chace-costaed DEA whe ] s a aaete. Phase III: Now we ca covet the chace-costaed DEA 4 to the foowg cs oea ogag b Cooe [4]: s s.t : M σ ϕ σ ϕ Pob... Pob : s.t M s
6 6 S. aezazadeh M. Meaa ad S. Saat whee : s the ea of the ado vaabe ϕ s the stadad oa dstbto fcto ϕ s vese of ϕ σ σ k Va Va Cov Cov k Cov Cov. k k k We have a ota soto fo each. hs fo dffeet ] the ota sotos ca be obtaed based o the choce of decso ake wth egad to ad the aoate soto a be seected. he o-eat 5 s de to σ adσ. he σ ad [... ] V [... ] t σ [... ] V [... ] t σ σ ae as foows: whee V ad V ae vaace-covaace atces fo each set of the costats esectve. Sce these atces ae ostve defte so σ ad σ ae cove []. heefoe we ca ca that the obe 5 s cove ogag obe. As a seca case whe the data ae taga fzz ado bes.e. ad ; s the ode 5 s as foows:
7 Data Eveoet Aass wth Fzz ado Its ad Otts: A Chace-costaed Pogag Aoach 7 M s.t : ϕ σ ϕ σ s whee σ σ k Cov Va Va Cov k Cov Cov. k k 6 3. Istato Eae he effcec of 4 fas D D D 3 ad D 4 wth aeas of ad 7 aces esectve ae to be evaated. I a the fas the co ctvated s wheat. he aot of the ed s a ado vaabe oa dstbted wth ea ad 3.5. he vaace s fo a. he aot of afa s estated as a fzz ado vaabe wth aaetes ρ ρ ρ ρ γ ad η η η η whee N6 ρ N γ N4 ad η N.5. he ed s the ott of the ode ad the aea ad afa ae ts. he data ae sted tabe ad the effceces of DMUs wth the oosed ethod fo dffeet vaes ae sted abe.
8 8 S. aezazadeh M. Meaa ad S. Saat I I O D D D 3 D 4 5 N.5 ρ 5 N 3 ABE. Data fo Neca Eae γ 4 N 5 η 7 N 3.5 D D D 3 D ABE. he Effceces b Poosed Method As see the effceces ae deceased b ceasg bt D 3 s effcet fo a. I case of 6 s evaet to the chace-costaed CC ode. Ftheoe DMUs ae aked as D 3 D D ad D Cocso I ths ae a CC ode s sggested fo chace-costaed DEA wth fzz ado data. We asse that the fzz ado vaabes ae fat fzz bes. Fo o-ea cases afte -ct the eato 3 ad 4 st be odfed accodg to the dstbto of the fzz be. We oose a ethod fo covetg ths obe to a cs chace-costaed DEA ode based o -ct ad fzz obabt ease. I ths ethod the owe eve of ts ad e eve of otts ae coaed wth the e at of effcec fote. he statve eae shows the acabt of the ode. It s sggested that the effcec of ths agoth be stded fo age obes. Ackowedget. he athos wod ke to thak the aoos efeee whose coets oved the at of the ae. EFEENCES [] A. Chaes W. W. Cooe ad G. Y Modes fo deag wth ecse data DEA Maaget Scece [] D. Chakabot J.. ao ad. N. wa Mtobectve ecse chace-costaed ogag obe J. Fzz Math Coged to: Mtobectve ecse-chace costaed ogag obe J. Fzz Math [3] D. Chakabot edefg chace-costaed ogag fzz evoet Fzz Sets ad Sstes [4] W. W. Cooe H. Deg Z. M. Hag ad S. X. Satsfg DEA odes de Chace costats he Aas of Oeatos eseach a
9 Data Eveoet Aass wth Fzz ado Its ad Otts: A Chace-costaed Pogag Aoach 9 [5] W. W. Cooe H. Deg Z. M. Hag ad S. X. Chace costaed ogag aoaches to techca effceces ad effceces stochastc data eveoaet aass Joa of the Oeatoa eseach Socet 53 a [6] W. W. Cooe H. Deg Z. M. Hag ad S. X. Chace costaed ogag aoaches to cogesto stochastc data eveoaet aass Eoea Joa of Oeatoa eseach [7] W. Gagva ad Z. Ye he theo of fzz stochastc ocesses Fzz Sets ad Sstes [8] W. Gaga ad Q. Zhog ea ogag wth fzz ado vaabe coeffcets FSS [9] P. Gao ad H. aaka Fzz DEA : A eeceta evaato ethod Fzz Sets ad Sstes [] J.. Hogaad Fzz scoes of techca effcec Eoea Joa of Oeato eseach [] P. Ka ad S. W. Waace Stochastc Pogag Joh We &Sos New Yok 994. [] C. Kao ad S.. Fzz Effcec Meases Data Eveoet Aass Fzz Sets ad Sstes [3] H. Kwakeaak Fzz ado vaabes deftos ad theoes If. Sc [4] B. Fzz ado chace-costaed ogag IEEE asactos o Fzz Sstes [5] B. Fzz ado deedet-chace ogag IEEE asactos o Fzz Sstes [6] M.K. hada Fzzess ad adoess a otzato faewok Fzz Sets ad Sstes [7] M. K. hada ad M. M.Gta O fzz stochastc otzato Fzz Sets ad Sstes [8] O.B. Oese ad N. C. Petese Chace costaed effcec evaato Maageet Scece [9] M.. P ad D.A. aesc Fzz ado vaabes J.Math. Aa. A [] S. Saat A. Meaa ad G.. Jahashahoo Effcec aass ad akg of DMUs wth fzz data Fzz Otzato ad Decso Makg [] B. Seave ad K. ats A fzz csteg aoach sed evaatg techca effcec eases afactg Joa of odctvt Aass [] J. K. Segta A Fzz Sste Aoach Data Eveoet Aass Cotes Math. Ac SAEED AMEZANZADEH* DEPAMEN OF MAHEMAICS POICE UNIVESIY EHAN IAN E-a addess: aezazadeh_s@ahoo.co AZIZOAH MEMAIANI DEPAMEN OF INDUSIA ENGINEEING BU-AI SINA UNIVESIY HAMEDAN IAN E-a addess: a_eaa@ahoo.co SABE SAAI DEPAMEN OF MAHEMAICS EHAN NOH BANCH ISAMIC AZAD UNIVESIY EHAN IAN E-a addess: ssaat@ahoo.co * COESPONDING AUHO O.. GOUP INSIUE FO FUNDAMENA ESEACH IMAM-HOSSEIN UNIVESIY EHAN IAN
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