Inverse DEA Model with Fuzzy Data for Output Estimation
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1 Aaabe e at Iaa Ja Optat Iaa Ja Optat Iee DEA Mde wt F Data Otpt Etat A Mad Rad a Rea Dea a Faad Heade Lt b a Depatet Mateatc Iac Aad Uet Maedea Bac Ia b Depatet Mateatc Iac Aad Uet Scece & Reeac Bac Tea Ia Abtact I t pape we w tat ee Data Eepet Aa DEA de ca be ed t etate tpt wt data a Dec Mak Ut DMU we e a pt ae ceaed ad decec ee te t ea caed. Kewd: Data Eepet Aa Mt-becte Pa Iee DEA Mde F Nbe Cepd at Ea addee: aadad@a.c
2 389. Itdct Data eepet aa DEA a -paaetc tecqe ea ad eaat te eate ecece a et ette caed dec ak t DMU wt te c pt ad tpt. Sce te t tdct te tecqe b Cae et a.[] kw a te CCR de. Sce te a pbcat DEA a bece a ppa etd aa te ecec a aat t [34]. Iteet Cae ad cpe ae a ad a cat pact te deepet tpe becte ea pa MOLP t te deepet a pa. A t Cae ad Cpe ae paed a cat e te deepet DEA ad MOLP eeace tee tw cap ae eea t pad c attet t eeac peed te te cap e deta ee[560]. Recet We et a. [9] pped ee DEA t awe te qet a w: a a p DMU we ceae ceta pt t a patca t ad ae tat te DMU ata t cet ecec ee wt epect t te DMU w c d te tpt te DMU ceae te tpt eed t be ceaed t a ceta ee ad te ecec te t ea caed w c e pt d be pded t te t? I ecet ea et te a bee pped a a wa t qat pece ad ae data DEA de []. Te DEA de wt data F DEA Mde ca e eatca epeet ea-wd pbe ta te ceta DEA de. F et te a aw tc data t be ed dect wt te DEA de. F DEA de take te ea pa de. Te qet dced ee DEA ca be cdeed wt data tat ae tat e data ae be ad we ceae e a pt ee a e DMU ad ae tat te DMU ata t cet ecec ee w c d te tpt te DMU cae? I t pape we
3 390 cde abta ca pt ee wt taa be ad we pped a MOLP de tpt etat. Te ede te pape aed a w: I te w ect we eew et ad be ea pa pbe. We cde DEA pbe ad t ee DEA pbe ect 3. I ect 4 we pped a MOLP tpt etate. We cde weak ecec cae I ect 5. I ect 6 we e a exape t tate cptat etd. Cc ae e ect Peae Sce te ke et be et te w be ed eqe; we a cde a ew ecea det. Det 2. I X a cect bect deted eeca b x te a et X a et deed pa: A x A x x X Wee A x caed te ebep ct wc acate wt eac x X a be [0] dcat t wat deee x a be. Det 2.2 Let A be a be.e. A cex aed bet te ea e te ee tat: ax R Ax. b A 0 0 x0 I a pecewe ct ct. Te - ee et A te A x Ax wee [0 ].
4 39 Det 2.3 A taa be deted b A A A A Wee A A A ad A A ad A ae ea be. I t pape we dete te et a taa be b F R. Let A = be. Dee: A A A ad B = B B B x 0 x R : xa xa xa xa x 0 x R : xa xa xa xa A B A A B A B B A A B B A A bt be taa B B T tdce a ea de be we t exteded peat ad ax ea be t cepd peat be MIN ad MAX. F a tw be A ad B we dee MIN A B MAX A B p x p. ax x A x B A x B
5 F a 392 R ee7. Te F R MIN MAX ca be expeed a te pa F R wee a pata de deed a: A B A B MIN A B A MAX A B B a a a Det 2.4 A be a 0 0 cpet a a ad a 0 0. ea a t Tee 2.. F a tw taa be A = A A A ad B = B B B we ae A B A B A B A B. Tee 2.2. F a tw taa be te ext k > c tat k. ad P. I ad k k k 3 : 2 : : k k k 3 2
6 393 I k = {k k 2 k 3 } te ae k. A ea pa pbe LPP deed a: M = cx S.t Ax=b 2. x 0 Wee c = c c b = b... b A a. I pbe 2. a te paaete ae cp[2]. Nw te LPP e cecet te pbe te becte ct tecca cecet te t-ad de cecet be te be te we a pbe a be ea pa pbe. Hee we cde te LPP wt be tecca ad t-ad de cecet. A be ea pa pbe FNLPP deed a w: ax. t x A x c 0 x B Wee A a a a F R ad B b b b FR.
7 394 B tee 2. pbe 2.2 ca be ewtte a Max c x S.t. a x b =2 a x b =2 2.3 a x b =2 Hwee ce a be ed pbe 2.3 ae cp te pbe 2.3 a ea pa pbe. I t pape we a pbe 2.3 eqaet t pbe F DEA pbe ad t ee pbe Cde DMU: DMU DMU wt pt ad tpt. Ipt ad tpt DMU ae X = x x 2... x ad Y... 2 = 2... epecte wee x 2... ad 2... ae Nbe. A X R ady R ad a X 0 ad Y 0 = 2. Cde te w tpt eted CCR de wee pt ad tpt ae taa be: Max
8 395 S.t X X 3. Y Y B tee 2. pbe 3. edce t te w: Max S.t x x =2 x x =2 x x =2 3.2 =2 =2 =2 0 =2. Te pbe 3.2 a tpt eted CCR de wee pt ad tpt ae ea be.
9 396 Sppe tat DMU pt DMU te pta ae 3. X t X X ad ae ceaed wee te ect X 0 ad X 0.e. at eat e cpet ceae. We eed t etate te cepd tpt ee we te ecec dex DMU ea caed wee T... ad F R =2. F ceece ppe DMU epeet DMU ate ca te pt ad tpt. Hece t eae te ecec te DMU we e te w de: Max S.t. X 3.3 Y 0 =2 +. Wee X X X 0 X Te eated MOLP
10 397 T d tpt ax c tat ecec dex ea caed we cde te w MOLP: Max 2... S.t.. X Y 4. Y 0 =2. Wee 3.. deed a bee ad e a te pta ae pbe Det 4. Let a eabe t pbe 4.. I tee eabe t 3.3 c tat te we ca weak Paet t pbe 4.. Tee 4.. Let be te pta ae pbe 3.. Te w pbe: be a weak Paet t pbe 4. ad a te pta ae te Max S.t. X
11 398 Y =2. P. Becae pbe 4.2 a a pta ae ppe a pta t te pbe. A a weak Paet t 4. t ate te cdt:. X Y Y 0 =2. Becae a eabe t 4.2 we ae 0:. te I we wd ae Y Wee ad.
12 S 399 wd be a eabe t pbe 4. wc pbe becae a weak Paet t 4.. S we t ae.e. te pta ae 4.2. Tee 4.2. Let be eabe t pbe 4.. I te pta ae pbe te t be a weak Paet t P. I t tee wee t te te tee wd ext ate eabe t 4. c tat. A X Y F Eq. ad tee 2.2 k > c tat Y k k S te pta ae 4.2 at eat apt tat te pta ae 4.2. k wee k wc aat Cde te w pbe:
13 400 Max S.t X Y Tee 4.3. Y Ae. I te pta ae pbe 4.3 > te te pta ae 4.2 a ; cee te pta ae pbe 4.2 > te pta ae 4.3. P. Ft we ae tat te pta t 4.3 Cde te eqaet pbe 4.3: Max ad >. S.t x =2 x =2 x =2 4.4 =2
14 40 =2 =2 0 =2 We cde te da pbe 4.3: M S.t x x x Sppe Ad
15 402 Ae a pta t 4.4. S ad we ae = Accd t te cpeeta acke cdt LP pbe a pta t 4.3 te aabe λ wc cepd t te ctat te da pbe 4.5 t be λ = 0. Bt we λ = 0 pbe 4.3 t bece 4.2. S pbe 4.2 a ae pta ae. Cee te pta ae pbe 4.2 > cde te eqaet pbe 4.2: Max S.t x =2 x =2 x =2 4.6 =2 =2
16 403 =2 0 =2 Wte te da pbe 4.6 M S.t x x x A a a pta ae. Te deece betwee 4.5 ad 4.6 tat 4.5 cta e e ctat.e. ctat. Bt we ca w tat a pta t Ad pbe 4.5 t ctat t d a tct eqat. 0 I 0
17 404 Te te pta ae 4.5. Ad ece te pta ae 4.7 wd be wc ctadct te apt. S t addta ctat t be bd at a pta t ad ece pbe 4.5 eqaet t 4.7 wc pe tat a pta ae pbe 4.3 ad 4.5. Tee 4.4. Sppe tat te pta ae pbe 3. > ad te pt t DMU ae t ceae X t X X X 0 X 0. Let be a weak Paet t pbe 4. te te pta t 4.3 t. Cee Let be a eabe t pbe 4.. I te pta ae pbe 4.3 te t be a weak Paet t 4.. P. Ae a Paet t 4.. B tee 4. te pta ae 4.2. A > ad Y b tee 4.3 a te pta ae 4.3. Cee te pta ae 4.3 te b tee 4.3 a te pta ae 4.2. U tee 4.2 we kw tat a weak Paet t 4.. T det e Paet t 4. we cet t t a ebecte pa pbe b a p > 0 a te wet -t tpt =. Teee we w ae:
18 405 Max p p S.t. X Y 4.8 Y 0 =2. We kw a pta t 4.8 t be a weak Paet pbe 4.. Ca. Sppe te ecec dex DMU de de 3. > ad pt ae ceaed X t X X X 0 X 0. Let be a pta t pbe 4.8. Te we te tpt DMU ae ceaed t te ecec dex te DMU t. 5. Weak ecec cae We w t t te cae =. Cde te w LP pbe Max S.t. X 5.
19 406 Y Y 0 =2. Wee X X ad X 0 X 0. Let t pta ae be. I we dete te eabe e pbe 3. ad 5. b S 0 ad S epecte we ae: S 0 S. Teee =. Tee 5.. Sppe tat te pta ae pbe 3. DMU = ad te pt t DMU ae ceaed X t X X X 0 X 0. Te we te tpt DMU ae ceaed Y t Y wee te pta ae pbe 5. te pta ae pbe 5. wc cepd te ew DMU t. Y P. We te pt ad tpt DMU bece X X X 0 X 0 ad Y te ecec dex DMU eqa te pta ae te pbe bew: S.t. Max X 5.2 Y Y Y 0 =2.
20 407 Let te pta ae pbe 5.2 be ẑ. Wat we eed t pe tat ẑ =. I ẑ = te ẑ >. We wte te eqaet pbe 5.2: Max S.t x =2 x =2 x =2 5.3 =2 =2 =2 0 =2 We wte te da pbe 5.3 M S.t x x x
21 Sppe ad ae t pta t. B dat et we kw tat 3 S te ctat 2 ad 3 we ae: 0 B te cpeeta acke tee at eac pta t T ea ẑ a te pta ae te pbe Max S.t. X Y Y 0 =2. Ad teee te pta ae te pbe
22 409 Max S.t. X Y Y =2. I. Bt te abe pbe wt epac te b t pbe 5. te pta ae pbe 5. wd be ad we ae ẑ > te ẑ > wc ctadct tat ax ae Neca exape We cde tee DMU wt tee pt ad tw tpt. Te data pt ad tpt ae w te w tabe. DMU S I I 2 I 3 O O2 DMU DMU DMU B eaat DMU 3 tpt-eted de 3. we ae O =.88. We w ceae te pt DMU3
23 40 X 3 = t te b te de 4.8 wt 3 p = p 2 = we w ae I t cae te ecec ad X Y3 eqa. A b eaat DMU 2 de 3. we ae = we w ceae te pt DMU 2 X 2 = t = te b te de 5 2 we w ae =.4667 ad 2 2 Y I t cae te ecec eqa ad bt =. ad X Y Cc I t pape we dc tee pbe: I te peece data w d we ct te cae pt ee a e DMU c tat te ecec dex te DMU peeed. T e te pbe we pped a MOLP de. We pde te ecea ad cet cdt te pt cae de te ae ecec dex. Reeece [] Bake R. D. Cae A. Cpe W.W. Se de etat tecca ad cae ecece data eepet aa Maaeet cece
24 4 [2] Baaaa M. S. Ja J. J. ad Sea H. D. Lea pa ad etwk w J We New Yk ecd edt 990. [3] Cae A. Cpe W.W. ad Rde E. Mea te ecec dec ak t Epea Ja Opeata Reeac [4] Cae A. Cpe W.W. Lew A.Y. ad Sed L.M. Data eepet Aa- Te Metd ad Appcat Kwe Acadec Pbe Bt 994. [5] J R. Ke P. Wae J. Stcta cpa data eepet aa ad tpe becte ea pa Maaeet cece [6] J R. Ke P. Zt S. A teacte appac t pe etate ae ecec data eepet aa Epea Ja peata eeac [7] K G. J. Ya B. F et ad c: Te ad Appcat Petce- Ha Ida 200. [8] Maek H. R. Tata M. Mac M. Lea pa wt aabe F et ad te [9]We Q. Za L.J. ad Za X. A ee DEA de pt/tpt etate Epea Ja Opeata Reeac [0] W B. Y. H. Lqe M. Ya J. B. U teacte t becte etd t e DEA pbe wt ae det Cpte ad Opeat eeac [] Zea H. J. F et te ad t appcat Kwe Acadec pbe Ld 996.
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