A Fuzzy Mathematical Programming Approach to DEA Models

Transcription

1 Aeca Ja f Apped Scece 5 (0): , 2008 ISSN Scece Pbcat A Fzzy Matheatca Pgag Appach t DEA Mde A. Azadeh, S.F.Ghade, Z. Javahe ad 2 M. Sabe Depatet f Idta Egeeg ad Cete f Eceece f Iteget Baed Epeeta Mechac ad Reeach Ittte f Eegy Maageet ad Pag, Cege f Egeeg, Uvety f Teha, P.O. B , Ia 2 Depatet f Idta Egeeg, Uvety f Tafeh, Ia Abtact: Evaatg the peface f actvte gazat by tadta Data Evepet Aay (DEA) a effcecy fte aay de eqe cp pt/tpt data. Hweve, ea-wd pbe pt ad tpt ae fte pece. Th tdy devep DEA de g pece data epeeted by fzzy et. A ptat tce f aeg eatve effcece wth a gp f Dec Makg Ut (DMU) fzzy data evepet aay t detee effcet DMU. We fd effcecy eae wth fzzy pt ad tpt va pped de. A eape g fzzy data peeted f tatve ppe. We appy th ethd the appcat t the pwe geeat ect f Ia. Key wd: Fzzy gc, ptzat, data evepet aay, pwe pat INTRODUCTION Effcecy fte aay ha bee a ptat appach f evaatg f peface pvate ad pbc ect. Thee have bee ay effcecy fte aay ethd epted the teate. Hweve, the apt ade f each f thee ethd ae etctve. Each f thee ethdge ha t tegth a we a aj tat epecay etvty f fte de t data cae t e a fzzy atheatca pgag appach t the aeet f effcecy wth DEA de deveped. The ft tdy fzzy DEA wa wtte 992 [24]. The ath f tdy eped the e f fzzy et they dec akg [24]. I th tdy, thee type f fzzy de (fzzy atheatca pgag, fzzy ege ad fzzy etpy) wee peeted t tate the type f dec ad t that wee achevabe. Meve, the tdy akg ethd ed f deteg the effcecy f DMU pteted CCR de wth fzzy pt ad fzzy tpt [3]. A the eathp betwee DEA ad Rege Aay (RA) tded. The CCR de ad RA wee cdeed a tw peca cae f the fwg ga pgag pbe: DEARA(cbat f DEA ad RA) : G = (aρ + bη ) T T v y = ρ η T 0 =,v 0 ρ, η 0 =,..., () The the devep e fzzy ve f the caca DEA de by g e akg ethd baed the cpa f -ct [26]. Th ake a appach whch abe t dea wth eact be be age, deabe. T dea qattatvey wth pec dec pce, the t f fzze tdced [24]. I the cveta DEA appach, a et f weght whch atfe a et f ctat eected t gve the hghet pbe effcecy eae f each DMU. Whe e bevat ae fzzy, the ga ad ctat the dec pce bece fzzy a we. Sce the DEA de eetay a ea pga, e taghtfwad dea t appy the etg fzzy Lea Pgag (LP) techqe t the fzzy DEA pbe [3,4,3,6,8,9,26]. Uftatey, t f the etg techqe y pvde cp t ad the ae y tabe f pecfc pbe, athgh they ae abe t pdce pbty dtbt f the pta bjectve vae [3,4,3,68,9,26]. Thee ae atce Cepdg Ath: A. Azadeh, Depatet f Idta Egeeg, Cege f Egeeg, Uvety f Teha, P.O. B , Ia 352

2 A. J. Apped Sc., 5 (0): , 2008 dcg effcecy eae whe the bevat ae ad, yet t fzzy, ate. A tchatc DEA de va pecfc ebehp fct t gve a fzzy pgag tepetat tafed [2,23,25]. Tw DEA de ae fated: e de that gve ppe t (bet cae) effcecy ad e de that gve we t (we cae) effcecy [8]. The a teva-vaed effcecy ca be ctcted f thee tw etee effcece. F the ppe t cae, the de the ae a the CCR de. Hweve, y cp effcecy eae ae pvded. O the whe, thee ae the thee pcede that have bee dced vg fzzy DEA pbe. The ft the pcede that ve the fzzy DEA by the teace appach. The et, vg the fzzy DEA by the akg appach ad the at t ve the fzzy DEA by the paaetc pgag. I th tdy we devep a ethd whch abe t pvde fzzy effcecy eae f DMU wth fzzy bevat. Retated, the ebehp fct, athe tha cp eae, f effcece w be deved. The bac dea t appy the -ct t taf the fzzy DEA de t a ee f cveta cp DEA de. The cveta DEA de ae the ved by the LP ethd. DEA AND FUZZY DEA Data Evepet Aay (DEA) a ethdgy baed a Lea Pgag (LP) de f evaatg eatve effcece f Dec Makg Ut (DMU) wth c pt ad tpt. It ed t akg ad aay f Dec- Makg Ut (DMU), ch a dteete, hpta, cte, facte ayt, etc. [3]. The tw bac DEA de ae CCR ad BCC wth ctat et t cae ad vaabe et t cae, epectvey [3,6]. Each DMU k aged the hghet pbe effcecy ce ( h k ) that the ctat aw f the avaabe data, by chg the pta weght f the tpt ad pt. If DMU k eceve the aa vae h k =, the t effcet, bt f h k, t effcet, ce wth t pta weght, athe DMU eceve the aa effcecy Eq. (). Bacay, the de dvde the DMU t tw gp, effcet ( h k = ) ad effcet ( h k ), by detfyg the effcet f the data. The ga DEA de t capabe f akg effcet t Theefe; the de dfed awg f a akg f the effcet t theeve. The ga facta CCR de () evaate the eatve effcece f DMU (j =,, ), each wth pt ad tpt deted by j, 2j,, j ad y j, y 2j,, y j, epectvey, by azg the at f weghted f tpt t the weghted f pt. (CCR at de) Ma y e j = = v j = y = v j =, j =,..., 0, =,...,, =,...,. (2) I de t cptata cveece the facta pgag de (2) e-epeed LP f a fw: (CCR-LP de) Ma e j = y = y v j 0, j =,..., = = v j = = 0, =,...,, =,...,. (4) Sppe that thee ae DMU, each f whch ce the ae type f pt ad pdce the ae type f tpt. Let be the be f pt ad et be the be f tpt. A pt ad tpt ae aed t be egatve, bt at eat e pt ad e tpt ae ptve. The fwg tat w be ed thght th tdy. DMU the th DMU, DMU taget DMU, NOTATIONS 353

3 A. J. Apped Sc., 5 (0): , 2008 R the c vecte f pt cedby DMU R the c vecte f pt f ced by taget DMU R the at f pt f a DMU y R the c vecte f pt f ced by DMU y R the c vecte f pt f ced by taget DMU y R the at f tpt f a tpt λ = ( λ ), λ R the c vecte f a ea cbat f DMU θ the bjectve vae (effcecy) f the CCR de R the c vect f pt eweght v R the c vect f tpt weght Mde (5) a ea pgag. Thee ae va ethd t ve t.i t f thee ethd f vg t cvet the pbtc pgag pbe g -ct, the teva bth de f the ctat ae cpaed wth each the. Thee ae ay ethd f cpag the teva; hece ay ethd ay be ggeted f vg tevapgag pbe. THE PROPOSED MODEL The bac dea t taf the fzzy CCR de t a cp ea pgag pbe by appyg a ateatve -ct appach. Theeby, the pbe cveted t a teva pgag. Dffeet ethdge have bee ggeted f the cpa f the teva th tdy baed Tag Cheg Methd wked. At ft, we e -ct t cvet fzzy DEA t teva pgag a fw: The CCR de wth fzzy data ca be wtte a: Ma e j = y = y v j 0, j =,..., (5) = = v j = = 0, =,...,, =,...,. whee, dcate the fzze. Thee ae dffeet type f fzzy be, bt taga fzzy be ae e ef that we cde the pt ad tpt f DMU a taga fzzy be. Theefe, (3) ca be wtte a fw: Let j = (j, j, j )ad y = (y, y, y ) Ma e j = (y, y, y ) = ( y,y, y ) v ( j, j, j ) 0, = = j =,..., v (,, ) = = 0, =,...,, =,...,. (6) 354 Ma e j = ( y + ( - )y, y + ( - )y ) = ( y + ( )y, y + ( )y ) = v ( ( ), ( ) j + j j + j ) 0, = j =,..., v ( ( ), ( ) + + ) = = 0, =,...,, =,..., (7) Wth cdeg ethd de chage t a fw th de gve ppe bd f effcecy ad et de gve we bd f effcecy. a e j = ( y + ( )y ) = v ( j + ( ) j ) ( y + ( )y ) = v ( + ( ) ) = a e j = ( y + ( )y ) = v ( j + ( )j ) ( y + ( )y ) = v ( + ( ) ) = (8) (9)

4 A. J. Apped Sc., 5 (0): , 2008 The abve de eqvaet t a the fzzy ea pgag pbe wth (0, ]. It ted that f each, we have a pta t. Tabe : Lwe bd et Pwepat pdct = 0 = 0.25 = 0.5 = 0.75 = Mtaze Beat Fz Sa 0 Shazad 0 Rajae 0 Beheht Tabz Mfateh Bt Ra 0 Medhaj Bada Zaad Efeha 0 Mtaze T Mahhad Iahah 0 Th, we ca pvde the dec ake a t tabe wth dffeet [0, ). THE CASE STUDY eae fe cpt te f Tea Je (TJ). I the wd, fge have aeady adjted f the qaty f fe ed dffeet pat. Itea pwe the at f eegy ced ( egawatt h) wth the te (f eectcay pweed eqpet etc.). We ppe a ethd t evaate the peface f pwe pat ad fd the effcece. Lwe Tabe 2: Uppe bd et Pwe pat pdct = 0 = 0.25 = 0.5 = 0.75 = Mtaze Beat Fz Sa Shazad Rajae Beheht Tabz Mfateh Bt Ra Medhaj Bada Zaad Efeha Mtaze T Mahhad Iahah Evaat f cveta thea tea-eectc peface ay be decbed cveety wth a egeeg faewk. I th faewk, petet pt ae the fe qatty ced ad taed pwe, whch the a a pwe the pat ae tay deged. O the the had ab pt ctbte t pdct thgh ct ad ateace evce, whch a eqe e capta. The tpt, f ce, eectca eegy pdct. Bt by tce f tde abt effcecy eaeet f thea pwe geeat Ia whch dcate that ab 't a effectve fact [30]. tdy, eectc pwe ( egawatt h) geeated f thea pwe pat each DMU (P) ed a the tpt vaabe, whe capta (C), fe (F) ad tea pwe (Ic) ae thee pt ed f pwe geeat. Capta eaed te f taed thea geeatg capacty egawatt (MW) Va ata eeet have bee ed a fe the pdct f eectc pwe va tea pat Ia (ata ga, ga ad azte). The chce f fe deped ay fact ch a avaabty, ct ad eveta cce ad each fe ha t tat. O fge 355 Tabe 3: Ret f tw ethd Pwe pat Oday ethd Pped ethd Mtaze Beat 0.88 Fz 0.66 Sa Shazad Rajae Beheht Tabz Mfateh Bt 0.99 Ra Medhaj Bada Zaad 0.83 Efeha 0.88 Mtaze T Mahhad Iahah 0.87 bd epeed Tabe. A, ppe bd epeed Tabe 2. The de t vadate th appach we cpae thee et = wth et day DEA de. Wth egad t bjectve fct f pped de azg that we cde ppe bd f vae. We ca ee Tabe 3 f cpag tw ethd.

5 A. J. Apped Sc., 5 (0): , 2008 CONCLUSION We tafed the fzzy CCR de t a cp ea pgag pbe by appyg a ateatve -ct appach. Theeby, the pbe wa cveted t a teva pgag. Dffeet ethdge have bee ggeted f the cpa f the teva th tdy baed Tag Cheg Methd wked. We ed -ct t cvet fzzy DEA t teva pgag. I pped de, tw ea pgag pbe wee ved t bta the effcecy f a gve DMU wth yetca taga fzzy be. Th de a appcat f fzzy et they DEA. REFERENCE. Aade, C.A., 994. Ug data evepet aay t eae teata agcta effcecy ad pdctvty, Uted State Depatet f Agcte. Ecc Reeach Sevce, Tech. B., 83: Bake, R.D., A. Chae ad W.W. Cpe, 984. Se de f etatg techca ad cae effcecy data evepet aay. Maage. Sc., 30: Bake, R.D., H. Chag ad W.W. Cpe, 996. Sat tde f effcecy, et t cae ad pecfcat wth ea fct DEA. A. Ope. Re., 66: Chae, A. ad W.W. Cpe, 959. Chacectaed pgag. Maage. Sc., 6: Chae, A., W.W. Cpe ad E. Rhde, 978. Meag the effcecy f dec-akg t. E. J. Ope. Re., 2: Chae, A., W.W. Cpe, B. Gay ad L. Sefd, 985. Fdat data evepet aay f Paet-Kpa effcet epca pdct fct. J. Ec., 30: Chae, A., W.W. Cpe, A.Y. Lew ad L.M. Sefd, 994. Data Evepet Aay: They, Methdgy ad Appcat. Kwe Acadec Pbhe, Ld. 8. Cpe, W.W., K.S. Pak ad J.T. Pate, 999. RAM: A age adjted eae f effcecy f e wth addtve de ad eat t the de ad eae DEA. J. Pd. Aa., : Cpe, W.W., L.M. Sefd ad K. Te, Data Evepet Aay: A Cpeheve Tet wth Mde, Appcat. Refeece ad DEA-Sve Sftwae, Kwe Acadec Pbhe, Ld. 0. Cpe, W.W. ad K. Te, 997. Meae f effcecy data evepet aay ad tchatc fte etat. E. J. Ope. Re., 99: Db, D. ad H. Pade, 988. Pbty They: A Appach t Cptezed Pceg f Ucetaty. Pe Pe, New Yk. 2. Fag, S.C. ad S. Pthepa, 993. Lea Optzat ad Ete: They ad Agth. Petce-Ha, Egewd C/, NJ. 3. G, P. ad H. Taaka, 200. Fzzy DEA: Apecepta evaat ethd. Fzzy Set Syt., 9: Kahaa, C. ad E. Tga, 998. Data evepet aay g fzzy ccept. 28th Iteata Syp Mtpe-Vaed Lgc., Ka, C. ad S.T. L, Fzzy effcecy eae data evepet aay. Fzzy Set Syt., 3: Letwak, S., 200. Fzzy data evepet aay f ppy cha deg ad aay. Detat Ppa Idta Egeeg, Nth Caa State Uvety. 7. L, B., 999. Uceta Pgag. A Wey- Itecece Pbcat, New Yk. 8. Meada, Y., Eta, T., Taaka, H., 998. Fzzy DEA wth teva effcecy. 6th Epea Cge Iteget Techqe ad Sft Cptg, 2: Sefd, L.M. ad R.M. Tha, 990. Recet devepet DEA: The atheatca pgag appach t fte aay. J. Ec., 46 : Segpta, J.K., 992. A fzzy yte appach data evepet aay. Cpt. Math. App., 24: Segpta, J.K., 995. Dyac f Data Evepet Aay: They f Syte Eacecy. Kwe Acadec Pbhe, Ld. 22. Wag, X., Kee, E.E., 200. Reaabe ppete f the deg f fzzy qatte (I). Fzzy Set Syt., 8: Wag, X. ad E.E. Kee, 200. Reaabe ppete f the deg f fzzy qatte (II). Fzzy Set Syt., 8: Zadeh, L.A., 978. Fzzy et a a ba f a they f pbty, Fzzy Set Syt., : Zea, H.J., 996. Fzzy Set They ad It Appcat. Kwe Acadec Pbhe, Ld. 356

6 A. J. Apped Sc., 5 (0): , Le, T., V. Le ad J.L. Rz, A fzzy atheatca pgag appach t the aeet f effcecy wth DEA de. Fzzy Set Syt., 39 (2): Letwak, S. ad S.C. Fag, Fzzy data evepet aay (DEA): A pbty appach. Fzzy Set Syt., Bckey, J.J., 989. Svg pbtc ea pgag pbe. Fzzy Set Syt., 3: Degad, M., J.L. Vedegay ad M.A. Va, 990. Reatg dffeet appache t ve ea pgag pbe wth pece ct. Fzzy Set Syt., 37: EaMebd, A., 998. Effcecy cdeat the eectcty ppy dty: The Cae f Ia. Ph.D. The, Uvety f Sey. 3. Azadeh, A., S.F. Ghade, M. Ava ad M. Sabe, Meag peface eectc pwe geeat g atfca ea etwk ad fzzy cteg. I Pceedg f the 2006 IEEE Iteata Cfeece Idta Eectc - IECON 06 (Pa). 357