Revenue Efficiency Measurement With Undesirable Data in Fuzzy DEA

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1 06 7th teatioal Cofeece o telliet Stem, Modelli ad Simulatio Reveue Efficiec Meauemet With deiale Data i Fuzz DEA Nazila Ahai Depatmet of Mathematic Adail Bach, lamic Azad iveit Adail, a. azila.ahai@mail.com Atact - Mot DEA model wee iitiall coideed ol fo cip deiale iput ad output; ut i the eal wold data have impecie value; fuzz cocept i too impotat a impecie ad udeiale data. Moeove, the deciio mai uit ma have ee ome iput ad output idex uch udeiale data. heefoe, i thi tud, we itoduce a method to evaluate eveue efficiec of DM whe the data ae deiale ad udeiale i fuzz DEA. Fo thi pupoe, we appl cut method ad exteio piciple ad otai lowe ad uppe oud pe each [ 0,] povide fo aemet of eveue efficiec; hu, the amout of fuzz eveue efficiec i eate tha oe. Fiall, a umeical example i peeted fo applicatio of popoed method. Kewod - Data Evelopmet Aali, Memehip fuctio, Paametic poammi, Reveue efficiec, cut. NRODCON Data Evelopmet Aali (DEA i a opaametic method fo meaui the elative efficiec of a et of deciio mai uit (DM which ue eveal iput fo eeati eveal output. Fo the fit time Faell [] mea of oevatio ad piciple that ove DEA, ceated a Poductio Poiilit Set (PPS ad amed it fotie poductio fuctio. Each deciio mai uit i iefficiet, ule it tad o thi fotie. While DEA i a elative aemet techiue of DM, at leat oe of the uit i located o the fotie ad the othe ae located i PPS. Chae et. al., [] expaded Faell' method. he peeted a aloithm which had the ailit of meaui efficiec with eveal iput ad output. hi aloithm wa omiated CCR ude the title of DEA. Accodil, efficiec wa otaied fom dividi um of weihted output um of weihted iput. Afte Chae et. al., [], Bae et. al., [3] peeted a ew model chai ule, ad amed it BCC. Bellma ad Zadeh [4] ad Zade [5] itoduced fuzz cocept to wo with ucetai data. he coveted DEA model, which i a liea poammi model, to a fuzz liea poammi model alo with fuzz limitatio fo ome taet fuctio. Seupta [6] did ome eeach aout the ue of fuzz et theo. He coveted DEA model to fuzz liea poammi mea of memehip fuctio. Kao ad i [7] utilized - cut method to meaue efficiec of BCC model. he tafomed DEA fuzz model to a famil of cetai DEA model with a et of Alpha level. Wa et. al., [8] ivetiated DEA fuzz model accodi to fuzz loic. he peeted lowe, middle, ad uppe oud of efficiec ume fo DM. We ad i [9] aeed DEA fuzz model mea of cocave memehip fuctio. he tadad DEA model ad ome of thei exteio dicued Coope et al., [0]. hee DEA model wee iitiall fomulated ol fo deiale iput ad output, ad ae uuall aed o the aumptio that iput have to e miimized ad output have to e maximized. Howeve, i eal life polem, udeiale iput ad/o output ma e peet i the poductio poce which alo eed to e miimized. hi ca e udetood tai a example i the ai idut. a, the i of loa to ecome o-pefomi loa/aet (NPA alwa exit. NPA diectl affect the tailit, aet ualit, cedit ceatio ad pofitailit of a a. he aumetatio i NPA lead to itailit, poo aet ualit, le cedit ceatio ad mot impotat eductio i pofitailit of a a. hu, NPA ca e teated a udeiale output i the ai idut of a cout ad thee eed to e miimized i ode to iceae the aet ualit ad pofitailit of a a. thi tud, a ew model i popoed to meaue the eveue efficiec coe of DM whe the iput ad output data ae fuzz, deiale ad udeiale. Moeove, the pice of output i coideed fixed ad cip. he followi ectio, Sectio, povide eveue efficiec cocept with cetai data. ectio, the pima cocept of fuzz ae how. ectio V, eveue efficiec model i dicued ui fuzz, deiale ad udeiale data, that the aumptio i cotat etu to cale. ectio V, a umeical example fo calculati eveue efficiec i popoed. Ad the the cocluio i ive.. REVENE EFFCENCY MODE Suppoe thee ae DM that m iput i coide to poduce output a x,..., x m ad,..., uch that x 0,x ad DM i ude 0, 0, epectivel. Aume p (p,..., p evaluatio DM ad i the output pice vecto that i coideed ame fo all DM. he PPS i a follow: /6 $ EEE DO 0.09/SMS

2 x x,, 0,,..., heefoe, the eveue efficiec model i popoed a the followi poammi polem: max i,,...,, ( p.t. x x, i,...,m, 0,,...,. Whee ad ae vaiale. Suppoe, i the optimal olutio of model (, o the eveue efficiec coe of DM i a follow: whee OE.OE. OE p p * ad DM i eveue efficiet if. PRMARY CONCEPS OF FZZY Claic et i defied a adiet of iediet which i x X. Each meme could elo to et A X. Howeve, fo a fuzz et, each meme of X elo to A et with a ade of memehip. Defiitio. f X i a adiet of meme, which i deoted x, the fuzz et of A i X i ive : A x, μ x x X {( A ( } ( μ A i called memehip fuctio. Memehip fuctio domai i [0,]. Defiitio. f up μ ( x x, A A fuzz et i called omal. Defiitio 3. A fuzz et i covex, if: { } μ ( x + ( x mi μ ( x, μ ( x, A A A x, x X, [0,] Defiitio 4. Suppot of A i a et of data which ae 0 < μ x A o: ( ( { μ ( > 0 A } S A x X x Defiitio 5. cut of A fuzz et i defied : A { x X μa ( x } Whee i a cala i [0,]. V. HE REVENE EFFCENCY MEASREMEN WH DESRABE, NDESRABE AND FZZY DAA Suppoe thee ae DM that m deiale fuzz iput a x,..., x m, x 0,x ad m udeiale fuzz iput uch a x,...,x m,x 0,x ae coideed to poduce deiale fuzz output a,...,, 0, ad udeiale,...,, 0,. fuzz output a Moeove, aume DM i ude evaluatio DM ad p (p,...,p ad p (p,...,p ae the deiale ad udeiale output pice vecto, epectivel, that ae coideed cip ad ame fo all DM. heefoe, the followi model i popoed to meaue the eveue efficiec of DM: (OE p + p p + p Whee,.t. x x,, i,...,m, (a i x x,, p,...,m, ( p p,,...,, (c,,...,, (d 0,,...,. ad ae vaiale. Cotait (a ad (c ae coepodi to deiale iput ad output ad alo cotait ( ad (e ae elated to udeiale iput ad output. Model ( i a fuzz poammi polem. So the eveue efficiec coe hould e fuzz that hould e foud the memehip fuctio. Suppoe S, S, S, ad S e uppot of x,x ad,, epectivel. cut of them a the cetai et ae defied a follow: 0

3 {x S μ }, i,, X {x S μ }, i,, X { S μ },,, { S μ },,. Coeuetl, the fuzz DEA model i coveted to the cip DEA model ui diffeet cut a { 0 < },{ 0 < } ad { 0 < },{ 0 < }. the othe wod, we ca how the followi et: x x x x x x x x [mi{x S μ },max{x S μ }], [mi{x S μ },max{x S μ }], [mi{ S μ },max{ S μ }], [mi{ S μ },max{ S μ }], So the memehip fuctio of eveue efficiec coepodi to DM i otaied a the followi elatio ecaue of the exteio piciple. (3 μ OE (z up mi μ, μ, μ, μ i,, z OE,. x x x, ode to deive the memehip fuctio of μ OE, the μ OE i lowe ad uppe oud of -cut coepodi to calculated a follow: (OE mi x x i,,,p, OE p + p p + p.t. x x i, i,..., m, x x, p,...,m, (4 p p,,...,,,,...,,,,...,. (OE max x x i,,,p, OE p + p p + p.t. x x i, i,...,m, x x, p,...,m, p p,,...,,,,...,,,,...,. Model (4 ad (5 ae how the lowe ad uppe oud of eveue efficiec, epectivel. model (4, the deiale iput ad udeiale iput of ude evaluatio DM ae lie i the uppe ad lowe oud, epectivel, wheea the deiale iput ad udeiale iput of othe DM ae i the lowe ad uppe oud, epectivel. Alo the deiale output ad udeiale output of ude evaluatio DM ae lie i the lowe ad uppe oud, epectivel, wheea the deiale output ad udeiale output of othe DM ae i the uppe ad lowe oud, epectivel. A well, we have aout model (5. heefoe, model (4 ad (5 ae ewitte a follow: (OE p + p p + p.t. i + i, i,...,m, p + p, p,...,m p, +,,...,, +,,...,, 0,...,. (5 (6

4 (OE p + p p + p i i p p p.t. +, i,...,m, +, p,...,m, (7 +,,...,, +,,...,,,,...,. Whee, ad ae vaiale i oth model (6 ad (7. Sice the deiale ad udeiale iput ad output vale ae chaed fo [0,], the the eveue efficiec coe will e diffeece. f (OE,(OE ae eveile eadi, eveue efficiec memehip fuctio could e otaied (Z [(OE ] Z Z Z, μ (Z Z OE Z Z 3, R(Z [(OE ] Z3 Z Z 4. Othewie, eveue efficiec of DM will e otaied i a iteval lie (OE,(OE [0,]. { } heefoe, DM could e divided, accodi to level, ito the followi level: Cla. DM that ae eveue efficiet i lowe oud o level. Sice ( OE ( OE, the ae eveue efficiet uppe oud o level. Hece, { DM } ++ ( E ( OE. Cla. DM that ae eveue iefficiet i uppe oud o level, cetail ae eveue iefficiet i lowe { DM } oud. Hece ( E ( OE <. Cla3. hee i o cocluio aout uppe oud of DM, which ae eveue iefficiet i lowe + E DM OE <. { } oud,( ( V. NMERCA EXAMPE Aume thee ae fou DM with two deiale ad udeiale fuzz iput ad alo two deiale ad udeiale fuzz output. Futhemoe, two cip output pice vecto. he data ae how i ale. DM ABE. HE DAA OF FOR DMS. O O P O O P A (,,4 (,3,4 40 (0,,4 0 (,3,6 (0,, B 30 (0,,3 5 0 (,3,4,6 5 (7,9, C (,,4 0 0 (5,7, D (0,,5,7 (45,47,5,55 40 (,3,4,5 0 (5,6,8 (4,6,8, B appli model (6 ad (7, the eveue efficiec coe of DM ae otaied a a iteval fo each DM that ae ive i ale. ABE. HE REVENE EFFCENCY SCORES ( OE A,( OE A ( OE B,( OE B ( OEC,( OE C ( OE D,( OE D 0.0 [.0,.0] [.0,.0] [.0,.0] [.00,.477] 0. [.0,.0] [.0,.0] [.0,.0] [.07,.469] 0. [.0,.0] [.0,.0] [.0,.0] [.5,.46] 0.3 [.0,.0] [.0,.0] [.0,.0] [.3,.453] 0.4 [.0,.0] [.0,.0] [.0,.0] [.30,.445] 0.5 [.0,.0] [.0,.0] [.0,.0] [.38,.437] 0.6 [.0,.0] [.0,.0] [.0,.0] [ ] 0.7 [.0,.0] [.0,.0] [.0,.0] [.54,.4] 0.8 [.0,.0] [.0,.0] [.0,.0] [.6,.43] 0.9 [.0,.0] [.0,.0] [.0,.0] [.70,.405].0 [.0,.0] [.0,.0] [.0,.0] [.79,.397] ale illutate eveue efficiec of DM A, DM B ad DM C i eual to oe i eve level of fo oth the lowe ad uppe oud. So the ae amed eveue efficiet. Alo DM D i eveue iefficiet. Fi. Show that the iteval eflect the memehip fuctio if DM. fiue. memehip fuctio V. CONCSON hi pape i the ucetait. fact, coide the fuzz DEA ad the compute eveue efficiec coe of DM i cae the iput ad output data ae divided two

5 clae a deiale ad udeiale. he popoed model wa aed o -cut ad exteio piciple of coveti fuzz DEA method ito cetai DEA model o that, ue of -cut, it i poile to et uppe oud ad lowe oud of eveue efficiec coe. hi eeach ucoveed that thee i ot a oe-to-oe coepodece etwee memehip fuctio of eveue efficiec coe ad fuzz data. REFERENCES [] Faell, M., (957. he meauemet of poductive efficiec. Joual of the Roal Statitical Societ, 0 (3, [] Chae, A., Coope, W. W., & Rhode, E., (978. Meaui the efficiec of deciio mai uit. Euopea Joual of Opeatioal Reeach,, [3] Bae, R.D., Chame, A., & Coope, W.W., (984. Some model fo etimati techical ad cale efficiecie i data evelopmet aali. Maaemet Sciece, 30, [4] Bellma, R.E., Zadeh,.A., (970. Deciio-mai i a fuzz eviomet, Maaemet Sciece, 7B, [5] Zadeh,.A., (978. Fuzz et a a ai fo a theo of poiilit, Fuzz Set ad Stem, 3-8. [6] Seupta, J. K., (99. A fuzz tem appoach i data evelopmet aali, Compute & Mathematic with Applicatio, 4, [7] Kao, C. i, C.C. & Che, S.P.,(999. Paametic poammi to the aali of fuzz ueue, Fuzz Set ad Stem [8] Wa, Y.M., Geata, R., & Ya, J.B., (005.teval efficiec aemet ui data evelopmet aali, Fuzz Set ad Stem, 53, [9] We, M., i, H.,(009. Fuzz data evelopmet aali (DEA: Model ad ai method, Joual of Computatioal ad Applied Mathematic, 3, [0] Coope, W. W., Seifod,. M., & oe, K. (007. Data evelopmet aali: A compeheive text with model, applicatio, efeece ad DEA-olve oftwae (d ed.. New Yo: Spie. 3