Shabnam Razavyan 1* ; Ghasem Tohidi 2

Transcription

1 J. Id. Eg. It., 7(5), 8-4, Fall 0 ISSN: IAU, Sth Teha Bach Shaba Razava ; Ghae Thd Atat Pfe, Det. f Matheatc, Ilac Azad Uvet, Sth Teha Bach, Teha-Ia Atat Pfe, Det. f Matheatc, Ilac Azad Uvet, Cetal Teha Bach, Teha-Ia Receved: 9 Al 009; Reved: 8 Jl 009; Acceted: 6 Agt 009 Abtact: Th ae e tegated Data Evelet Aal (DEA) del t ak all extee ad -extee effcet Dec Makg Ut (DMU) ad the ale tegated DEA akg ethd a a cte t df Geetc Algth (GA) f fdg Paet tal lt f a Mlt-Obectve Pgag (MOP) ble. The eeache have ed akg ethd a a htct wa t df GA t deceae the teat f GA. The dfed algth edce the ctatal efft t fd Paet tal lt f MOP ble ad ca be ed t fd Paet tal lt f MOP wth cvex ad -cvex effcet fte. A exale gve t lltate the dfed algth. Kewd: Data evelet aal (DEA); Rakg; Itegated DEA del; Mlt-bectve gag (MOP); Geetc algth (GA); Effcec. Itdct Data Evelet Aal (DEA) ad Mltle Obectve Pgag (MOP) ae tl that ca be ed aageet ctl ad lag. DEA tdced b Chae et al. 978, a atheatcal gag baed aach that evalate the effcec f a gazat, geeal, a Dec Makg Ut (DMU) elatve t a et f caable gazat. DEA cde ltle t ad tt ltael, eqg ethe a weght a fctal f f t/tt elath. DEA tlze atheatcal gag t ctct a ecal dct blt et, ad vde a gle effcec ce f that DMU b cag t a vtal dce the effcet fte. Afte tdcg the ft del DEA, the CCR del b Chae et al. (978), Bake et al. (984) develed the DEA techqe b vdg the BCC ad FDH del (Tlke, 993). A addtal chaactetc f DEA that ha challeged e t ve ght ad vale deved f the ethdlg vlve the effcet et f DMU,.e., the wth a ce f.0. Adee ad Petee (993) ed a e-effcec cede f akg effcet DMU. Th ethd ak l -extee effcet DMU. I e cae, the AP del feable. I addt t th dffclt, the AP del a be table becae f extee etvt t all vaat the data whe e DMU have elatvel all vale f e f the t. I a actcal ble ch a egeeg deg ble, ctea fct cat be gve exlctl te f deg vaable. Ude th cctace, vale f ctea fct f gve vale f deg vaable ae all btaed b e aale ch a tctal aal, thedacall aal fld echacal aal. Thee aale eqe cdeabl ch ctat te. Theefe, t ealtc t al extg teactve tzat ethd t the ble. MOP the ltae cdeat f tw e bectve fct that ae cletel atall cflct wth each the. The talt f ch tzat lagel defed thgh the Paet talt. Recetl, MOP ethd g geetc algth (GA) have bee tded actvel b a ath (Aakawa et al., 998), (Feca ad Fleg, 993), (Schafe, 985) ad (Taak et al., 996). GA ae efl f geeatg effcet fte wth tw thee bectve fct. Dec akg ca be eal efed the ba f valzed effcet fte. Hweve, thee ethd have e htcg; thee a tedec f Vect Evalated Geetc Algth (VEGA) (Schafe, 985) t geeate ch lt that e f the bectve fct exteel gd. Aakawa et al. (Aakawa et al., 998) ed the Cedg Ath Eal: h_azava@azad.ac. Tel.:

2 Sh. Razava ad Gh. Thd / Jal f Idtal Egeeg Iteatal 7(5) (0) thetc GA ad DEA t geeate Paet tal lt f MOP ble. Y et al. (00) ed a ethd b cbg geealzed data evelet aal (GDEA) ad GA t geeate effcet fte MOP ble. GDEA eve dated deg alteatve fate tha ethd baed l GA. The ed ethd ca eld deable effcet fte. Th ethd, hweve, ha t w defcece. The t tat defcec whe a MOP ble ha e tha tw thee bectve fct. I e tat a DMU ae effcet ad the dec ake wat t elect l e DMU ag the a the t effcet DMU. F tace, t detf the t effcet, Faclt Lat Deg (FLD) Eta et al. (006) ed a DEA/AHP ethdlg. A ad Tl (007) exteded the wk ad ed a tegated DEA del baed a c et f weght t elect the t DMU de- at f ctat et t cale (CRS). A (008) hwed e tat the t effcet DMU ag DMU wth CRS t gle ad ed a lea gag ble t bta a gle t CCR-effcet DMU. Th ae e A' del (008) a a tegated akg DEA del t ak all extee ad -extee effcet DMU. The, the eeache have ed tegated akg DEA del a a cte ad df GA t fd Paet tal lt f MOP ble. Th ethd ca be ed t fd Paet tal lt f a MOP ble wth e tha tee bectve fct ble wth -cvex effcet fte. It edce the ctatal efft t lve MOP ble. The et f th ae gazed a fllw: Sect vde a backgd f MOP ad DEA. I Sect 3 the eeache have ed tegated DEA akg del t ak all extee ad extee effcet DMU. I Sect 4, the eeache have ed tegated DEA akg ethd t df GA. Sect 5 dce a exale that ale the dfed GA. Fall, Sect 6 eet the ccldg eak.. Backgd.. The MOP ble A MOP ble defed a fllw: f ( x ) = ( f ( x ),..., f,( x )) { x R g ( x ) 0, =,...,l} x S = () whee, f (x),.,f l (x) ae bectve fct, x=(x,,x ) dec vaable vect ad S eeet the feable eg f Pble (). Deft. xˆ S a Paet tal lt f ble () f thee ext xˆ S ch that f(x) f( x ) ad f(x) f( x ). The Paet tal lt f Pble () ae fd b the aache ch a the aat level techqe, lexcgah (Sawaag et al., 985). Bt, thee aache eqe a lg te atclal whe the ble have a bectve fct... The DEA del Se we have a et f ee DMU, DMU ( =,, ), wth ltle t x =(,,), ad ltle tt (=,,). The DEA del f eag the elatve effcec f DMU de a at f ctat et t cale the CCR del, Chae et al. (978). Th del a factal lea ga whch ca be tafed t the fllwg lea ga: θ = ax = =, w x = = w ε, =,..., ε, =,...,.,..., () I e tat the effcet DMU t qe ad the dec ake wat t elect the t effcet DMU ag effcet DMU. A (008) ed the fllwg gag ble t bta a gle t CCR-effcet DMU: M M d 0, =,...,, =,..., w x + d = 0, =,..., =

3 0 Sh. Razava ad Gh. Thd / Jal f Idtal Egeeg Iteatal 7(5) (0) 8-4 = θ =, θ d β = 0, =,..., β,d θ w ε, =,..., ε, =,..., { 0, }, =,..., (3) whee, ε cted b the fllwg del (008): ε = axε =, =,..., w ε, =,..., ε, =,...,. =,..., (4) Aakawa et al. (998) ed the cbat f the DEA ad the GA ethd t fd effcet lt f a MOP ble. I th ethd dvdal vale ctat cdeed a the DEA del t ad cte fct vale f each dvdal cdeed a DEA del tt. See Fge, whee each DMU ha tw t ad e ae tt. DEA effcet fte ca be ed a a axat t the MOP effcet fte (ee Fge, (Aakawa et al., 998; Y et al., 00)). Y et al. (00) ed GDEA ethd whch, al, clde bac DEA del the effcec f DMU GDEA del btaed b the fllwg del. ax d ~ =,..., + v =,v α =, v ( F ( x ) + F ( x ε, =,...,, =,...,. k )), (5) whee α defed te f the ble vaable ad f =,,, we have; ~ d =,...,, Fge : GA wth DEA ethd ad DEA effcec. = ax { ( ), v ( x + x )} k =,...,. Effcet fte GDEA del baed dffeet α hwed Fge, (Y et al., 00). Y et al. (00) ed a ethd t elate the defect whch wee the Aakawa et al. (998) del. The defed t f GDEA del a fllw; F ( x ) = f ( x ) + l = k [{ g ( x )} ], =,..., + α (6) whee eeet a ealt ad α eeet ealt cet ad { g ( x )} = ax{,g ( x )}. 0 + Y et al. (00) ed (6) ad cveted del () t the fllwg del: F( x ) x E = ( F ( x ),...,F ( x )) T, (7) ad, theefe, the GDEA del cveted a fllw (Aakawa et al., 998): = ax =,...,, d ~ α v ( F ( x ) + F ( x )), (8) v =, v ε, =,...,.

4 Sh. Razava ad Gh. Thd / Jal f Idtal Egeeg Iteatal 7(5) (0) 8-4 Fge : Effcet fte wth dffeet α. We wll e the ete f GDEA del ad tegated DEA akg ethd (ee Sect 3) t geeate cvex ad -cvex effcet fte f the MOP ble. 3. Rakg all f effcet DMU g the tegated DEA del Th ect e the ed del b A (008), Mdel (3), t ak all extee ad extee effcet DMU. T th ed, let E = {,, }( {,, }) be the dce et q f CCR extee ad -extee effcet DMU. Becae, b CCR del at leat a DMU effcet, theefe E et. T detee t effcet DMU ag DMU,, DMU q, g c et f weght, we cde Mdel (3) wth CCR effcet DMU. Theefe, we have the fllwg del: M M d, E = θ = E, E β,d θ + d θ d β = 0, E w ε, =,..., ε, =,..., = 0, E { 0, }, E (9) whee E the cadalt f E ad ε cted b the fllwg del, whch a dfcat f Mdel (4): ε = axε =, E w ε, =,..., ε, =,...,. E (0) Let d =0 g Mdel (), whee E. A ted A (008), DMU the t effcet DMU ag DMU,,.DMU q. Theefe, t ha the hghet ak,.e. t ak. Nw, we eve DMU ag CCR effcet DMU ad defe E a: E = {,..., } { } = {,..., }. q q If we et E tead f E Mdel (9) ad (0), the the ecd t effcet DMU fded. Nw, e that: = {,..., } {,..., } E q = { k,...,k }. q + T detee th t effcet DMU ag DMU k,, DMU k q-+, we e Mdel (3) ad cde the fllwg del: M M d 0,, E E

5 Sh. Razava ad Gh. Thd / Jal f Idtal Egeeg Iteatal 7(5) (0) 8-4 = E θ = β,d E, + d θ d β = 0, E θ w ε, =,..., ε, =,..., = 0, E { 0, }, E () whee DMU,, DMU - ae t,,(-) th t effcet DMU g E, E,, E -, eectvel. Ad, ε cted g the fllwg del: ε = axε =, E w ε, =,..., ε, =,...,. E () If d =0 g Mdel (), whee E, the the ak f DMU. Theefe, t ak all f extee ad -extee effcet DMU, we have the fllwg algth. Stage : et = ad lve Mdel (9), ad let DMU be the t effcet DMU, Stage : et =+ ad E + =E -{ }, whee, Stage 3: f E =, the t, whee E the cadalt f E. Stage 4: lve Mdel () ad g t Stage. Thee: B Mdel (9) ad () ak ce f eve effcet DMU qe. Pf: Accdg t Thee A (008) each teat, a th teat, Mdel () fd a gle DMU, a DMU, a t effcet DMU ag DMU, DMUk,, DMUk q-+. S, the ak f DMU a a gle DMU. Theefe, g the abve algth we ca ak all extee ad -extee effcet DMU ad eve effcet DMU ha qe ak ce. 4. Alcat the tegated DEA akg del t df GA F each x S, g (6) we ca ctct a DMU wth t ad e tt, whee tt qatt. Theefe, whe ebe ae elected the a tage f the ed algth, we ca ctct DUM wth t ad e tt. The, we evalate thee DMU b Mdel () ( Mdel (8)) t bta θ ( ). If we ltle the t f DMU b θ, the t le the DEA effcet fte. I the wd we aach t the MOP effcet fte (ee Fge 4.a ad 4.b whch cedg t DEA effcet fte), whle wth Y et al. (00) Aakawa et al. (998) algth we hld eeat the abt 0 t 30 te, that the ebe the gated DMU ca be laced the fte. Theefe, we a e Mdel () a a cte t elect the ext lat teat. That b akg all DMU g Mdel (), extee ad -extee effcet DMU, whch le effcet fte DUM wth bette ak ae elected t geeate the ext lat. Th ethd a htct t deceae the teat f GA t fd the Paet effcet lt f MOP ble. Theefe, bef we have the fllwg algth whch a dfcat f ed algth b Y et al. (00) ad called the dfed geetc algth. Stage : (Italzat). Geeate -dvdal adl. Hee, the be f gve befe had. Stage : (Cve-Mtat). Make /- a adl ag the lat. Makg cve each a geeate a ew lat. Mtate the accdg t the gve bablt f tat. Stage 3: (Evalat f fte b GDEA). Evalate the GDEA-effcec b lvg the ble (8) ad ect all DMU DEA effcet fte. Stage 4: (Select). Select - dvdal f cet lat the thd tage, te f the btaed akg ce f Mdel (). The ce Ste -Ste 4 cted tl the be f geeat atta a gve be.

6 Sh. Razava ad Gh. Thd / Jal f Idtal Egeeg Iteatal 7(5) (0) Exale Cde the fllwg lea b-bectve ble. ( f ( x ), f ( x )) = ( x,x ) x,x ( x ) 0. + ( x ) 4 Nw we cae the btaed elt f the ed algth wth Aakawa et al. (998) ad Y et al. (00) algth. Fge 5.a ad 5.b hw the elt wth 30 eett g Aakawa et al. (998) ad Y et al. (00) algth, eectvel. Thee ext e defcet lt, whch the lead t a accate fte. Bt the dfed ethd (ee Fge 5.c) the effcet fte whch aate, wll ext afte a eveal eett, addg that 90% f the lt ae effcet. Meve, f we have e eett, t 30 te, the accac ehace t %99. It eakabl attactve t et that the ebe dated b the ebe the eett ae cled t the tzed Paet lt. f f f f Fge 3.a: Obectve fct vale ace. Fge 3.b: Cedg DEA effcet fte ad ect. f f f f Fge 4.a: Se Paet tal lt. Fge 4.b: All DMU the cedg DEA effcet fte. Fge 5.a: Aakawa et al.' (998) ethd. Fge 5.b: Y et al.' (00) ethd. Fge 5.c: Ug the ed ethd. 6. Ccl Th ae ed the tegated DEA del t ak all f effcet DMU. The, the eeache ed the ed tegated DEA akg ethd t df GA f fdg the Paet effcet lt f MOP ble. The dfed GA edce the ctatal efft ad ca be ed t geeate cvex ad -cvex Paet effcet fte f MOP ble, ad ca al be aled t lve the MOP ble wth tw eveal bectve fct. I the wd the

7 4 Sh. Razava ad Gh. Thd / Jal f Idtal Egeeg Iteatal 7(5) (0) 8-4 eeache have ed a htct de t deceae the teat f GA f fdg the Paet effcet lt f MOP ble. Refeece A, Gh. R., (008), Cet fdg the t effcet DMU DEA: A ved tegated del. Cte & Idtal Egeeg, 56(4), A, Gh. R.; Tl, M., (007), Fdg the t effcet DMU DEA: A ved tegated del. Cte & Idtal Egeeg, 5(), Adee, P.; Petee, N. C., (993), A cede f akg effcet t data evelet aal. Maageet Scece, 39(0), Aakawa, M.; Nakaaa, H.; Hagwaa, I.; Yaakawa, H., (998), Mlt-bectve tzat g adatve age geetc algth wth data evelet aal. I: A Cllect f Techcal Pae Seveth S Mltdcla Aal ad Otzat [TP ], AIAA, 3, Bake, R. D.; Chae, A.; Ce, W. W., (984), Se del f etatg techcal ad cale effcece data evelet aal. Maageet Scece, 30, Chae, A.; Ce, W. W.; Rhde, E., (978), Meag the effcec f dec akg t. Eea Jal f Oeatal Reeach,, Da, H. f. A.; Vaccel, J. A., (00), Mltbectve geetc algth aled t lve tzat ble. IEEE Taact Magetc, Eta, T.; Ra, D.; Tzkaa, U.R., (006), Itegatg data evelet aal ad aaltc heach f the faclt lat deg afactg te. Ifat Scece, 76, Feca, C. M.; Fleg, P. J., (993), Geetc algth f lt-bectve tzat: Flat, dc ad geealzat. Pceedg f the Ffth Iteatal Cfeece Geetc Algth, Sawaag, Y.; Nakaaa, H.; Ta, T., (985), The f lt-bectve tzat. Acadec Pe. Schafe, J. D., (985), Mltle bectve tzat wth vect evalated geetc algth. Pceedg f the Ft Iteatal Cfeece Geetc Algth, Taak, H.; Kta, H.; Kbaah, S., (996), Mlt-bectve tzat b geetc algth: A evew. Pceedg f the ICEC'96, 575. Tlke, H., (993), O FDH effcec: Se ethdlgcal e ad alcat t etal bakg, ct, ad ba tat. Jal f Pdctvt Aal, 4, Y, Y. B.; Nakaaa, H.; Ta, T.; Aakawa, M., (00), Geeat f effcet fte lt-bectve tzat ble b geealzed data evelet aal. Eea Jal f Oeatal Reeach, 9,