On Generalized Fuzzy Goal Optimization for Solving Fuzzy Multi- Objective Linear Programming Problems. Abstract

Transcription

1 O Geerlzed Fuzzy Gol Optmzto or Solv Fuzzy Mult- Ojetve er Prormm Prolems Je u Feje Wu Guqu Zh Fulty o Iormto Teholoy Uversty o Teholoy Sydey PO Bo 3 Brodwy NSW 007 Austrl E-ml: jelu@t.uts.edu.u Astrt My orztol deso prolems e ormulted y mult-ojetve ler prormm (MOP) models. eerr to the mpreso heret hum judmets uertty my e orported the prmeters o MOP model whe t s estlshed whh s lled Fuzzy MOP (FMOP) prolem. Wht s optml soluto or FMOP prolem s the rst ssue o ths study. The seod ssue s how to eetvely derve optml soluto se uertty s lso releted soluto proess o FMOP deso prolem. By trodu three types o omprso o uzzy umers d djustle ststory deree ths study ew soluto oept o FMOP s ve. For hdl the seod ssue ths study develops tertve uzzy ol optmzto method whh mkes terto wth deso mkers dur soluto proess d llows deso mkers provd uzzy ols y orms o memershp utos. A llustrtve emple ves the detls o the soluto oept d the proposed method. Keywords: Optmzto; Fuzzy prormm; Mult-ojetve ler prormm; Itertve deso-mk method.

2 . Itroduto Mult-ojetve ler prormm (MOP) s eet tehque to model d solve deso prolems whh severl olt d ommesurle ojetves re to e optmzed smulteously sujet to speed ostrts [5]. However sett up sutle mthemtl model eet soluto proedure d pertet omput method or MOP prolem s qute dult euse there ests uertty d mpreso the judmets o oth model retors (eperts) d deso mkers d thereore the dulty s releted oth the proess o model mult-ojetve deso prolem d solv the prolem. I model relst MOP prolem the possle vlues o the prmeters the ojetve utos d ostrts my e ssed epermetl sttstl or sujetve mer throuh some eperts uderstd o the ture or the deso prolem. These methods re eet to model some prolems ut my use mportt ormto lost d sometmes l to the rellty o the models. The m reso s tht the vlues o the prmeters model re ote mpresely or muously uderstood to the eperts d dult to e determed us prese vlues. Wth ths oservto t my e more pproprte to terpret the eperts uderstd o these prmeters s uzzy umerl dt. A MOP model wth uzzy prmeters ts ojetve utos or ostrts or oth s lled uzzy MOP (FMOP) prolem. I order to derve ststo soluto o MOP prolem or deso mker sed o hs/her sujetve vlue judmet d preeree my methods hve ee proposed rom ltertures. Two m types o the methods re ol prormm d tertve prormm [5]. Cosder the mprese eture o deso mkers judmet d preeree t s ture to ssume tht the deso mkers my oly

3 hve mprese ols or the ojetve utos. Ths ssue s reereed s the uzzy ol prormm optmzto []. Fuzzy ol optmzto methods hve ee ppled solv lssl MOP prolem [8]. To solve FMOP prolem the uzzy ol prormm method eeds to e urther developed so tht t eetvely ompre the vlues etwee ojetve uto wth uzzy prmeters d ts uzzy ol. I eerl there does t est uque soluto or oth MOP d FMOP prolems. To ot ststory soluto o FMOP prolem or prtulr deso mker volves lot o terto to rry out the deso mker s preeree or soluto. Whe oth the prmeters the model d the ols ve y deso mker re wth uertty the tertve soluto proedure my eome very omple d thereore more eet FMOP methods re eeded. My optmzto methods d tehques or model d solv FMOP prolems hve ee proposed [] [6] [8] [8]. As uzzy umers well represet oth uzzy prmeters d uzzy ols us uzzy umers to model d solve FMOP prolem s prtulrly proms. My ppltos hve lso proved t s pplle or del wth hum deso-mk prolems most prtl stutos [] [9]. For emple Tk d As [3] ormulted FMOP prolems y us trulr uzzy umers to desre the uzzy prmeters oth ojetve utos d ostrts. d Hw [9] modeled FMOP prolems y us trulr uzzy umers s well d lso solved FMOP prolems y us the uzzy rk oept to hdle mprese ostrts d mmze the most possle vlues o ts ojetves. uhdjul [0] proposed the oepts o α-possle eslty d β-possle eey sed o the oept o uzzy umer d used the two oepts to solve FMOP prolems y trserr t to ulry rsp 3

4 MOP prolem. Furthermore Slowsk [] proposed tertve method (FIP) or solv MOP prolems wth uzzy prmeters the ojetve utos d o the oth sdes o the ostrts. ommeler [4] preseted method (FUPA) or solv (multrter) ler prorms where the rht-hd sdes s well s the prmeters the ostrts d/or the ojetve utos my e uzzy. Smlrly mk et l. [] proposed ued pproh sed o the uzzy equlty reltos or uzzy mthemtl prormm prolem whh uzzy prmeters my hve oler memershp utos. Espelly Iuuh et l. [6] revewed some uzzy ler prormm methods d tehques rom prtl pot o vew d trodued the eerl hstory d the pprohes o uzzy mthemtl prormm. I metme ol prormm [3] s eetve method hs ee suessully ppled solv FMOP prolems. Kuwo [7] ppled the oepts o the - optml soluto d the restrted -optml vlue t the -optml soluto to estlsh ol prormm pproh or solv FMOP prolems. Nshzk d Skw [] pushed the work orwrd sed o ther prevous results [8] [0] y de two soluto oepts or FMOP sed o uzzy ols. Oe s deed y mmz the mml uzzy ol d the other y mmz the sum o uzzy ols. They the developed two omputtol methods or ot the solutos or FMOP prolems. More mporttly mk [] eerted stdrd ol prormm prolem wth ltertves d ols e uzzy sets d the ststo o ol y uzzy ojetve uto s lso epressed y uzzy relto d proposed uy pproh over severl pprohes kow rom lterture. Althouh these methods re eet to solve FMOP prolems there est two 4

5 lmttos ther urret results. Oe s tht oly some spelzed orms o memershp utos suh s trulr orm were used to del wth the uzzy prmeters d the uzzy ols. Ths my restrt the use o other orms o memershp utos to desre the prmeters model FMOP prolem d to epress ther ols y deso mkers solv FMOP prolem. The seod lmtto s tht the vlues o ojetve utos orrespod to ststory soluto o FOMP prolem re oly desred y some rsp vlues whh s sometmes ot pproprte prte. Se deso prolem ws ormulted wth uertty d ts soluto s reeved wth uzzy vlues t s more resole to provde the vlues o the ojetve utos wth re vlues. Ths study thereore develops eerlzed uzzy ol optmzto method to ssst deso mkers to ot ststory solutos or FMOP prolem. The method solve FMOP prolem whtever the prmeters o oth ojetve d ostrtsre desred whh orm o memershp utos. The method lso llows the deso mkers provd ther uzzy ols or the ojetve utos o ther deso prolems y orms o memershp utos. By trodu three types o omprso o uzzy umers d djustle ststory deree ew oept o soluto o or FMOP prolem s deed. As result the oted vlues o ojetve utos orrespod to soluto e desred y uzzy vlues whh rel umer s s spel se. Moreover the eerlzed uzzy ol optmzto method hs the etures o terto wth deso mkers dur soluto proess. Ths pper s orzed s ollows. Some relted oepts d theorems out uzzy umer d MOP re revewed seto. Seto 3 ves the oept o djustle ststory deree α the deto o uzzy umer sed soluto 5

6 oept d relted theorems or solv FMOP prolem wth deret deree α. A eerlzed tertve uzzy ol optmzto method or solv FMOP prolems s desred Seto 4. A llustrtve emple s preseted Seto 5 to demostrte the method. Dsussos urther remrks d uture study pl re oluded Seto 6.. Prelmres The results show ths pper re s otu reserh o our prevous reports. Zh et l. [9] proposed methods to solve uzzy ler prormm prolem y trsorm t to orrespod our-ojetve ostred optmzto prolem. Follow tht Zh et l. [8] developed method to ormulte ler prormm prolems wth uzzy equlty d equlty ostrts. Bsed o these work Wu et l. [5-7] proposed relted tehques optml ppromte lorthm d uzzy ol sed optmzto method or solv FMOP prolems. Ths seto revews two roups o oepts uzzy umer d MOP d our prevous relted work o FMOP.. Fuzzy umer Deto. [6] et X deote uversl set. The uzzy suset A o X s deed y ts memershp uto : X [0] () A whh sss to eh elemet X rel umer the tervl [ 0] where the vlue o t represets the rde o memershp o A. Deto. [6] A uzzy set A s ove d oly ) m( ( ) ( )) () ( 6

7 or every X d [0]. Deto.3 [6] The -level set o uzzy set A s deed s ordry set A or whh the deree o ts memershp uto eeeds the level : A 0. (3) A By the oept o -level sets the reltoshps etwee ordry sets d uzzy sets e etured y the ollow deomposto theorem. Theorem. (Deomposto theorem) [6] A uzzy sets A e represeted y where A A (4) 0 A deotes the ler produt o slr wth the -level set A. I ths pper we del wth rel uzzy umers suh s out m or roud whh re ormlly deed s ollows. Deto.4 [4] et e the set o ll rel umers. The rel uzzy umer s deed y ts memershp uto whh stses: ) A otuous mpp rom to the losed terl [ 0] ) 0 or ll 3) Strtly res d otuous o 4) or ll 5) Strtly deres d otuous o d 6) 0 or ll d Aord to Deomposto theorem Theorem. uzzy umer e represeted s ollows: (5) 0 7

8 By the pplto o the eteso prple [6] o uzzy set we hve the ollow deto. Deto.5 [9] et e two uzzy umers wth the memershp utos d respetvely d 0 the ddto o two uzzy umers the sutrto o two uzzy umers d the slr produt o d re deed y the memershp utos z sup m (6) y z y z sup m (7) y z y z (8) sup z From Deomposto theorem d Deto.5 we hve the ollow theorem. Theorem. [9] et e two uzzy umers wth the memershp utos d respetvely d [0] [0] [0] 0 0 we hve (9) (0) () Deto.6 [9] et e uzzy umers wth the memershp utos we dee whose memershp uto s deed s () m where s -dmesol vetor. The s lled -dmesol uzzy umer o. 8

9 I del wth FMOP prolems the m questo to e swered ossts the omprsos o uzzy ojetve utos uzzy ols d let- d rht-hd-sde o uzzy ostrts. Suh omprsos re ll reled o the omprsos o - dmesol uzzy umers. There est my deret methods or ompr the uzzy umers [8] [-4] []. I ths pper we dopt deret rk pproh whh re more pproprte to the deret kds o the FMOP optml detos. Ths pproh s sutle or the -dmesol uzzy umers wth the oler memershp uto d t s pplle to the omprso o the o-uzzy vetors s ts spel se. et d y y y y e y two vetors where y. For y two vetors y we hve ) y y. ) y y d y 3) y y Here we dee the rk wy etwee two -dmesol uzzy umers s ollows. Deto.7 et e y two -dmesol uzzy umers we dee ) d 0 ) d 0 3) d 0 where or represets the let or rht etreme pot o the λ-level set o the uzzy umer the -dmesol uzzy umer respetvely; smlrly or represets the let or rht etreme pot o the λ-level set o the uzzy umer the -dmesol uzzy umer respetvely. 9

10 . Mult-ojetve ler prormm A lssl MOP prolem s speed y ler utos whh re to e mmzed sujet to set o ler ostrts. The stdrd orm o MOP e wrtte s ollows: (MOP) m C s.t. X A 0 (3) where C s k ojetve uto mtr A s m ostrt mtr s m-vetor o rht hd sde d s -vetor o deso vrles [5]. Whe the otos o sle ojetve ler prormm re dretly ppled to MOP the ollow otos o omplete optml soluto re oted. Deto.8 [7] s sd to e omplete optml soluto to the MOP prolem d oly there ests X suh tht k or ll X. However suh omplete optml soluto tht mmzes ll ojetve utos smulteously does ot lwys est. Thus the Preto-optml soluto s trodued to the MOP. Deto.9 [7] s sd to e Preto optml soluto to the MOP prolem d oly there does ot est other X suh tht or ll d j or t lest oe j. j I ddto to Preto optmlty the ollow wek Preto optml soluto s deed s slht wek soluto oept th Preto optmlty. Deto.0 [7] s sd to e wek Preto optml soluto to the MOP prolem d oly there does ot est other X suh tht k. 3. Fuzzy mult-ojetve ler prormm 0

11 I ths pper we osder the stuto whh ll prmeters o the ojetve utos d the ostrts re uzzy umers represeted y orms o memershp utos. Suh FMOP prolem e ormulted s ollows: (FMOP) m s.t. C X A 0 (4) where C s k mtr eh elemet o whh j s uzzy umer wth memershp uto A s m mtr eh elemet o whh j s uzzy j umer wth memershp uto j s uzzy umer wth memershp uto vrles. s m-vetor eh elemet o whh d s -vetor o deso Theorem. dees the ddto operto o two uzzy umers (9) d the slr produt operto o postve rel umer d uzzy umer (). eerr to t the proposed FMOP model (4) or eh rsp vlue o X eh ojetve uto d eh let-hd-sde o ostrts re uzzy umers d let-hd-sde d rht-hd-sde o ostrts re ompred. For deret rsp vlue o X deret s eed to e rked. Beuse there s o uversl oept o optml soluto to e epted wdely we propose the ollow detos out the optml soluto or FMOP prolem sed o Deto.7. Deto 3. s sd to e omplete optml soluto to the FMOP prolem d oly there ests X k or ll X. suh tht Deto 3. s sd to e Preto optml soluto to the FMOP prolem d oly there does ot est other X suh tht or ll.

12 Deto 3.3 s sd to e wek Preto optml soluto to the FMOP prolem d oly there does ot est other X suh tht or ll. Uder some rumstes deso mker my motor hs/her ststory deree or soluto o FMOP prolem. Tht mes whe ompr two uzzy umers ll vlues wth memershp rdes smller th re eleted. I some ppltos t e uderstood s rsk tke y deso mker who elets ll vlues wth the posslty o ourree smller th. et s osder typl uzzy prmeter j the FMOP model. As the posslty o suh prmeter j tk vlue the re [ j ] j s or ove whle the posslty o eyod [ j ] j j s less th the deso mker would eerlly e more terested solutos oted us prmeters j tk vlues [ j ] j wth 0. Furthermore or eh rsp vlue o X uzzy ojetve utos (4) re mult-dmesol uzzy umers d Deto.7 s employed or the omprsos o these uzzy ojetve utos uder deret solutos o FMOP prolem. However y us Deto.7 to rk uzzy ojetve utos d set up uzzy reltos ostrt some rel FMOP prolems ot e epressed y (4) euse t s so hrsh to ormulte the prolem. Thereore Deto.7 eeds to e eteded s ollows. Deto 3.4 [9] et e y two -dmesol uzzy umers we dee ) d ) d 3) d

13 Uder ert ststory deree the orl FMOP prolem (4) wll e redesred s ollow: (FMOP ) m s.t. X C A 0 (5) where d j wth lj k l j d l m. Bsed o Deto 3.4 whh supples rk wy or y two -dmesol uzzy umers uder the ststory deree we propose ollow detos out the optml solutos or the FMOP prolem (5). Deto 3.5 s sd to e omplete optml soluto to the FMOP prolem d oly there ests X suh tht k or ll X. Deto 3.6 s sd to e Preto optml soluto to the FMOP prolem d oly there does ot ests other X suh tht or ll. Deto 3.7 s sd to e wek Preto optml soluto to the FMOP prolem d oly there does ot ests other X suh tht or ll. A eet pproh or solv the FMOP prolem s to trsorm the orl prolem to ssotve rsp prormm oe. De ssotve ouzzy MOP d sett up the reltoshps etwee the soluto o FMOP d o ts ssotve MOP re rul to mplemet ths pproh. Thereore ssoted wth the FMOP prolem (5) let us osder the ollow multple ojetve ler prormm (MOP λ ) prolems: (MOP λ ) m s.t. C C X A A 0 (6) where 3

14 4 k k k C k k k C m m m A m m m A T m d T m. The ollow theorem shows the reltoshp etwee the soluto o the FMOP prolem d o the MOP λ prolem. Theorem 3. et X e esle soluto to the FMOP prolem. The. s omplete optml soluto to the FMOP prolem d oly s omplete optml soluto to the MOP λ prolem.. s Preto optml soluto to the FMOP prolem d oly s Preto optml soluto to the MOP λ prolem. 3. s wek Preto optml soluto to the FMOP prolem d oly s wek Preto optml soluto to the MOP λ prolem. Proo. The proo s ovous rom Deto 3.4. I del wth the FMOP prolem uder some rumstes deso mker my wt to spey some ols or the uzzy ojetve utos. The purpose o the ol prormm s to mmze the devtos rom ols set y the deso mker to the uzzy ojetve utos. Thereore most ses ol prormm wll yeld stsy soluto rther th optml oe [8]. Cosder the FMOP prolem or the uzzy multple ojetve utos T k the deso mker spey uzzy ols T k uder ststory deree whh relet the desred vlues o the

15 5 ojetve utos o the deso mker. These uzzy ols e represeted y eerl uzzy umers. By de uzzy dste uto D s uzzy deree etwee uzzy ojetve utos T k d uzzy ols T k the uzzy ol prormm (FGP α ) prolem my e deed s optmzto prolem: (FGP α ) 0 s.t. m A X D (7) Normlly the uzzy dste uto D s deed s mmum o devtos o dvdul ols k D D m. (8) By (8) the FGP α prolem (7) s overted s ollows: 0 s.t. m m A X D k (9) where D m m 0 0 k (0) From (0) the optml soluto o (9) e oted y solv the ollow ouzzy GP models: (GP λ - ) 0 s.t. m m k A A X () or (GP λ - ) 0 s.t. m m k A A X () where k k k k k k k k

16 A m m m A m m. T T m m The dopto o GP λ- () or GP λ- () or solv FGP α prolem depeds m o the reltoshp o d.e. the GP λ- () s used otherwse GP λ- () s dopted. 4. A tertve uzzy ol optmzto method or solv FMOP prolems It hs ee oud tht the most eetve wy or solv FMOP prolem s to use tertve pproh. A tertve pproh mkes deso mkers dretly ee the prolem solv proess. Desos tke y the deso mkers t eh tertve phse re otuously revsed d updted sed o ther perorme d ormto eom vlle dur the mplemetto [5] [4] [30]. The tertve eture s luded the proposed method. Eept the tl ste t eh tertve phse the deso mker spees uzzy ols (ould e lust term) the memershp uto spe. The FMOP prolem t the lol level s the solved sed o ths uzzy ols to provde the deso mker wth seres o solutos. Dur the tertve proess the deso mker selet the most sutle soluto o the ss o hs/her preeree. The m steps o the tertve uzzy ol optmzto method wth 5 steps uder stes re desred s ollows. et the tervl e deomposed to l me su-tervls wth (l+) odes 0 l tht re rred the order o. Wth the urret 0 l 6

17 deompos we dee the ostrt X l l X X A A 0. Ste : Itlzto Ths ste s to eerte tl optml soluto or the FMOP prolem. Bsed o the detos o the FMOP d the MOP λ prolems d Theorem 3. we mke oluso tht the soluto o the MOP λ prolem s equl to o the FMOP prolem. [Step ] Ask the deso mker to ve the memershp utos o uzzy prmeters to set wehts or eh d to selet tl vlue o ststory deree 0. [Step ] Assot the MOP λ model (6) we deote: (MOP λ ) l j m k; j 0 l j (3) j l s.t. X Set l the solve (MOP λ ) l (3) wth soluto l where l l d the soluto oted s sujet to ostrt l X. I ths step the tervl s ot splt tlly d oly 0 d re osdered. The eh o the uzzy ojetve utos s overted to our orrespod ouzzy ojetve utos: k. (4) Suh kd o overso s lso ppled o ostrts the sme wy d eh uzzy ostrt s overted to our orrespod ouzzy ostrts: s s s s s m. (5) s s s s 7

18 The soluto s sed o ew ouzzy ojetve utos (4) d ouzzy ostrts (5). [Step 3]: Solve (MOP λ ) l wth soluto l d the soluto oted s sujet to ostrt l X. I urret step the tervl s splt urther. Suppose there re l odes 0 4 l the tervl d l ew odes 3 l re serted. The reltoshp etwee the ew serted odes d the prevous odes s: 0 l. (6) l So eh o the uzzy ojetve utos s overted to orrespod ouzzy ojetve utos d the sme overso hppes o the ostrts. The soluto l s lso sed o ew ouzzy ojetve utos d ouzzy ostrts. [Step 4]: I the the l soluto o MOP λ prolem s l l l Otherwse updte l to l d o k to Step 3. [Step 5]: I the orrespod Preto optml soluto. ests o orwrd to the et step. Otherwse the deso mker must o k to Step to ress deree (res the deree ). [Step 6]: Ask the deso mker whether to e stsed wth the tl uzzy Preto optml soluto lulted Step 4. I stsed the whole tertve proess stops d the tl Preto optml soluto s the l stsy soluto to the deso mker. Otherwse o to Ste. Ste : Itertos I ths ste tertve proess wll proeed. At eh terto phse the deso mker s suppled wth the soluto oted Itlzto ste or the 8

19 prevous terto phse. I ot stsed wth the urret soluto the deso mker s sked to spey the uzzy ols or the uzzy ojetve utos d the ew ompromse soluto wll e eerted utl the deso mker stops the tertve proedure. [Step 7]: Spey ew uzzy ols T or the ojetve utos d/or k ew ststory deree sed o the urret soluto. The deso mker wll hve to mke the ompromse mo the uzzy ojetves. A mprovemet oe or more o the uzzy ojetves wll result the sres o the other uzzy ojetves. [Step 8]: Assot wth the GP λ- model () we deote: Set l ostrt m m j (GP λ- ) l k j j j j s.t. j 0 l (7) X l the solve (GP λ- ) l (7) wth soluto l whh s sujet to the l X. I ths step oly 0 d the tervl re osdered. The eh uzzy ojetve uto uder theuzzy ols T s orrespod to our ouzzy dste utos: k k. (8) Ths overso s lso ppled o ostrts d eh uzzy ostrt overted to our orrespod ouzzy ostrts s ollows: s s s s m. s s s s s The soluto o (7) s sed o (8) d (9). s (9) [Step 9]: Solve (GP λ- ) l wth soluto l d the soluto oted s sujet to the ostrt l X. 9

20 I ths step the tervl s splt urther. Suppose there re l odes 0 4 l the tervl t the urret step d l ew odes 3 l re serted. Thereore eh uzzy ojetve uto uder the uzzy ols T s overted to l orrespod k ouzzy dste utos d the sme overso hppes o the ostrts. The soluto l o the urret step s lso sed o ew ouzzy dste utos d ouzzy ostrts. [Step 0]: I the the soluto o GP λ- model () s l l Otherwse updte l to l d o k to Step 9. l [Step ]: I the soluto o GP λ- model oted Step 0 s resole the o to Step 5 otherwse move to Step. [Step ]: The proedure rom Step to 4 s smlr to the oe rom Step 8 to 0 ut s omed wth the deret GP λ model. Assot wth the GP λ- model () we deote: m m j (GP λ- ) l k j j j j s.t. j 0 l X l. (30) Set l the solve (GP λ- ) l (30) wth soluto l d the soluto l s sujet to the ostrt l X. I ths step oly 0 d the tervl re osdered. The eh uzzy ojetve uto uder uzzy ols T s orrespod to our ouzzy dste utos: k k. (3) Ad the smlr overso s ppled to eh uzzy ostrt. 0

21 s s s s s s s s. m s (3) [Step 3]: Solve (GP λ- ) l wth soluto l d the soluto oted s sujet to the ostrt l X. [Step 4]: I l l the the soluto o GP λ- prolem s l. Otherwse updte l to l d o k to Step. [Step 5]: I the deso mker s stsed wth the urret soluto the whole tertve proess stops d the urret soluto s the l stsy soluto o the FMOP prolem to the deso mker. Otherwse o k to Step 7. Drm shows the low hrt o the tertve uzzy ol optmzto method. 5. A llustrtve emple To llustrte the tertve uzzy ol optmzto method developed ths study let us osder the ollow FMOP prolem whh hs two uzzy ojetve utos d our uzzy ostrts: 4-4 m m m m (33) s.t. 0 0; The memershp utos o the prmeters o the ojetve utos d ostrts re set up s ollows: or or 4

22 / / / / or -.5 or or or or or or / / / / / or or or or or or or Ste : Itlzto Step : Iput the memershp utos o ll uzzy prmeters. For emple the memershp utos o s ve s show Fure. We suppose tl vlue o the ststory deree s set to 0. d the wehts o the two uzzy ojetve utos re equl. Step -4: Uder the urret ststory deree 0. lulte Preto optml soluto. Assoted wth the FMOP model o the llustrtve emple the orrespod MOP λ model the llustrtve emple s lsted:

23 st. M M (34) where. eer to the MOP λ prolem tlly the tervl s ot splt oly 0. d 0 re osdered the totlly eht ouzzy ojetve utos d 6 ouzzy ostrts re eerted. From (34) the result o overso s lsted s ollows: m s.t m (35) 3

24 The the tervl s splt urther. Flly the result o Ste s tht the deso vrles re: (36) d two uzzy ojetve utos re: whh re show s Fure Step 5-6: Suppose the deso mker s ot stsed wth the tl uzzy Preto optml soluto lulted Step 4 the tertve proess would proeed d o to Itertve ste. (37) Ste : Itertos Iterto No : Step 7: Bsed o the soluto Ste the deso mker spees the ew uzzy ols to e heved ths terto y res the rst uzzy y 30 ojetve uto peret d deres the seod uzzy ojetve uto 4

25 uzzy ols re: y 5 peret. The the ew.3. (38) 0.75 Step 8-0: Clulte the stsy soluto sed o the ew uzzy ols speed Step 7 d the ststory deree 0.. Uder the ew uzzy ols (38) the FMOP prolem wll e overted to orrespod ouzzy GP λ- whh s lsted s ollows: m m s.t where (39) The result o the urret terto s tht the deso vrles re: (40) d two uzzy ojetve utos re: whh re show s Fure Compr Fure wth Fure 3 sed o the ew uzzy ols (38) t e (4) oud tht the rst uzzy ojetve uto hs oted some remet d the seod uzzy ojetve uto ot some deremet. 5

26 Step : Se the solutos oted Step 0 s resole the proess wll o to Step 5. Step 5: Suppose the deso mker s lso ot stsed wth the soluto lulted the prevous step the tertve proess wll proeed d o k to Step 7. Iterto No : Step 7: At ths terto suppose the deso mker spees the ew uzzy ols y the orrespod memershp utos whh re set up s ollows: or or (4) (43) The memershp utos o ew uzzy ols re show Fure 4. Step 8-0: Clulte the stsy soluto sed o the ew uzzy ols speed (4) d (43) d deree 0. Uder the ew uzzy ols the FMOP prolem wll lso e overted to orrespod ouzzy GP λ- prolem whh s lsted s ollows: m m (44) s.t

27 where. The result o ths terto s tht the deso vrles re: (45) d two uzzy ojetve utos re: whh re show s Fure (46) Step : Se the solutos oted Step 0 s resole the proess wll o to Step 5. Step 5: Now the deso mker s stsed wth the soluto lulted Step 0 the whole tertve proess stops d the urret soluto s the l ststory soluto o the FMOP prolem to the DM whh s (48) By solv ths emple we see how ths proposed tertve uzzy ol optmzto method s ppled to solve FMOP prolem. (47) 6. Colusos Deso mk lud oth model deso prolem d solv the prolem omple d ll-strutured stuto s ote eted y uertty. Ths s essetlly due to the suet d mprese ture o dt evluted y oth model estlshers d deso mkers. For MOP prolem the uertty s eted oth the prmeters o the model d the ols o the deso mkers. Ths study eerlzes tertve uzzy ol optmzto method to solve FMOP prolem whh the uzzy prmeters oth the ojetve utos d the ostrts e represeted y y orms o memershp utos. Ths 7

28 method lso llows deso mkers to provde ther uzzy ols y y orms o memershp utos s well. I prtulr ths study hs deed ew soluto oept o FMOP d ppled the oept the developmet o our tertve uzzy ol optmzto method. Compr wth est methods the proposed method s more eerl wth the dvte del wth y type o uzzy umer d more pertet prte wth the dvte desrpto o soluto vlues o the ojetve utos us uzzy vlues whh rel umer s s spel se. Moreover the method hs the eture o terto wth deso mkers. A deso support system hs ee developed to pply the method to ssst deso mkers to solve relst FMOP prolems. Ths system hs ee tlly tested y umer o emples d results re very postve. Akowledemets Ths reserh s prtlly supported y Austrl eserh Coul (AC) uder dsovery rt DP070. eerees [] Bellm. E. d Zdeh. A. Deso-mk uzzy evromet Memet see 7 (970) [] Crlsso C. d Fuller. Fuzzy multple rter deso mk: eet developmets Fuzzy Sets d Systems 78 (996) [3] Chres A. d Cooper W. W. Gol prormm d multple ojetve optmztos Europe Jourl o Opertol eserh (977) [4] Duos D. d Prde H. Opertos o uzzy umer Itertol Jourl o Systems See (978) [5] Hw C.. d Msud A. S. Multple Ojetve Deso Mk: Methods d Appltos. Sprer-Verl Berl: 979. [6] Iuuh M. d mk J. Posslst ler prormm: re revew o uzzy mthemtl prormm d omprso wth stohst prormm portolo seleto prolem Fuzzy Sets d Systems (000) 3-8. [7] Kuwo H. O the uzzy mult-ojetve ler prormm prolem: Gol prormm pproh Fuzzy Sets d Systems 8 (996)

29 [8] Y. J. d Hw C.. Fuzzy Multple Ojetve Deso Mk: Methods d Appltos. Sprer-Verl Berl: 994. [9] Y. J. d Hw C.. A ew pproh to some posslst ler prormm prolems Fuzzy Sets d Systems 49 (99) -33. [0] uhdjul M. K. Multple ojetve prormm prolems wth posslst oeets Fuzzy Sets d Systems (987) [] mk J. Fuzzy ols d uzzy ltertves ol prormm prolems Fuzzy Sets d Systems (000) [] mk J. d ommeler H. Fuzzy mthemtl prormm sed o some ew equlty reltos Fuzzy Sets d Systems 8 (996) [3] mk J. d ommeler H. A sle- d mult-vlued order o uzzy umers d ts use ler prormm wth uzzy oeets Fuzzy Sets d Systems 57 (993) [4] ommeler H. FUPA - A tertve method or solv (Multojetve) uzzy ler prormm prolems : Stohst Versus Fuzzy Approhes to Multojetve Mthemtl Prormm uder Uertty. Slowsk d J. Tehem Eds. Kluwer Adem Pulshers Dordreht / Bosto / odo 990 pp [5] ommeler H. Itertve deso mk uzzy ler optmzto prolems Europe Jourl o Opertol eserh 4 (989) 0-7. [6] Skw M. Fudmetls o uzzy set theory : Fuzzy sets d tertve multojetve optmzto. Pleum Press New York 993. [7] Skw M. Fuzzy ler prormm : Fuzzy sets d tertve multojetve optmzto. Pleum Press New York 993. [8] Skw M. Fuzzy sets d tertve multojetve optmzto. Pleum Press New York: 993. [9] Skw M. Itertve multojetve ler prormm wth uzzy prmeters : Fuzzy sets d tertve multojetve optmzto. Pleum Press New York 993. [0] Skw M. d H. Y. Itertve deso mk or multojetve prormm prolems wth uzzy prmeters : Stohst Versus Fuzzy Approhes to Multojetve Mthemtl Prormm uder Uertty. Slowsk d J. Tehem Eds. Kluwer Adem Pulshers Dordreht / Bosto / odo 990 pp [] Skw M. d Nshzk I. Solutos sed o uzzy ols uzzy ler prormm mes Fuzzy Sets d Systems 5 (000) [] Slowsk. 'FIP': A tertve method or multojetve ler prormm wth uzzy oeets : Stohst Versus Fuzzy Approhes to Multojetve Mthemtl Prormm uder Uertty. Slowsk d J. Tehem Eds. Kluwer Adem Pulshers Dordreht / Bosto / odo 990 pp [3] Tk H. d As K. Fuzzy ler prormm prolems wth uzzy umers Fuzzy Sets d Systems 3 (984) -0. [4] Werers B. A tertve uzzy prormm system Fuzzy Sets d Systems 3 (987) [5] Wu F. u J. d Zh G. Q. "A -uzzy ol ppromte lorthm or uzzy multple ojetve ler prormm prolems" Proeeds o The Thrd Itertol Coeree o Iormto Tokyo Jp Nov 8 - De

30 [6] Wu F. u J. d Zh G. Q. "A uzzy ol ppromte lorthm or solv multple ojetve ler prormm prolems wth uzzy prmeters" Proeeds o FINS 004: 6th Itertol Coeree o Appled Computtol Itellee Blkeerhe Belum Sep [7] Wu F. u J. d Zh G. Q. A ew ppromte lorthm or solv multple ojetve ler prormm prolems wth uzzy prmeters Appled Mthemts d Computto (005) Artle I Press. [8] Zh G. Q. Wu Y. ems M. d u J. A -uzzy m order d soluto o ler ostred uzzy optmzto prolems Est-West Jourl o Mthemts Spel Volume (00) P84. [9] Zh G. Q. Wu Y. H. ems M. d u J. Formulto o uzzy ler prormm prolems s our-ojetve ostred optmzto prolems Appled Mthemts d Computto 39 (003) [30] Zot S. d Wlleus J. A tertve prormm method or solv the multple rter prolem Memet see (975)

31 Strt Set up FMOP model.e. put the memershp utos o or d or Set weht or eh Spey the tl vlue o deree 0 Clulte the uzzy Preto optml soluto or FMOP prolem wth (MOP λ ) model N Soluto ests? Y Stsy soluto? Y N Spey ew uzzy ols T k or ojetve utos sed o the urret uzzy Preto optml soluto Clulte the uzzy stsy soluto to FMOP prolem sed o the urret uzzy ols wth (MOP λm- ) model Y Soluto resole? N Clulte the uzzy stsy soluto to FMOP prolem sed o the urret uzzy ols wth (MOP λm- ) model N Stsy soluto? Y The tertve proess stops here d the l soluto s show Drm : Flow hrt or the tertve uzzy ol optmzto method Ed 3

32 3 Fure : Memershp uto o uzzy prmeter

33 Fure : Memershp utos or d Ste 33

34 Fure 3: Memershp utos or d. Iterto No 34

35 Fure 4: Memershp utos o the ew uzzy ols Iterto No 35

36 Fure 5: Memershp utos or d Iterto No 3 36