A decision support system prototype for fuzzy multiple objective optimization

Transcription

1 EUSFLAT - LFA A eision support system prototype for fuzzy multiple ojetive optimiztion Fengjie Wu Jie Lu n Gungqun Zhng Fulty of Informtion Tehnology University of Tehnology Syney Austrli E-mil: {fengjiewjieluzhngg}@it.uts.eu.u Astrt An intertive fuzzy multiple ojetive optimiztion metho ws propose for solving fuzzy multiple ojetive liner progrmming prolems where fuzzy prmeters in oth ojetive funtions n onstrints n fuzzy gols of ojetives n e in ny form of memership funtion. Bse on the metho in this pper fuzzy multiple ojetive eision support system prototype is evelope. A etile esription of the metho n system re then supplie. Keywors: Fuzzy numer Optimiztion Fuzzy multiple ojetive liner progrmming Deision support systems. Fuzzy Multiple Ojetive Liner Progrmming Multiple ojetive liner progrmming (MOLP) is one of the populr pprohes use to el with some omple n ill-struture eision-mking situtions. Due to the unvoile influene of unertinty MOLP moels must tke into ount vgue informtion impreise requirements n moifitions of the originl input t from the moeling phse. During the MOLP prolem formultion proess the possile vlues of the prmeters in the ojetive funtions n onstrints my e ssigne in n eperimentl sttistil or sujetive mnner through some eperts unerstning to the eision prolem. With this oservtion the vlues of these prmeters re often impreisely or miguously unerstoo to the eperts n re iffiult to e etermine y preise vlues rther thn fuzzy numers [] []. In this pper the Fuzzy MOLP (FMOLP) prolem in whih ll oeffiients of the ojetive funtions n onstrints re rel fuzzy numers is onsiere. Then the FMOLP prolems n e formulte s follows: m f C s.t. () n X R A t ^ ` ¾ ½ where C is n k u n mtri eh element of whih is fuzzy numer with memership funtion ij P A is n m u n mtri eh element of whih ij is fuzzy numer with memership funtion ij is n m-vetor eh element of whih i is P ij fuzzy numer with memership funtion P i n is n n-vetor of eision vriles R. n An Intertive Fuzzy Multiple Ojetive Optimiztion Metho Mny optimiztion methos n tehniques for moeling n solving FMOLP prolems hve een propose [-6]. Wu et l. [7] [8] lso evelope some lgorithms n methos for solving FMOLP prolems. Bse on these lgorithms n methos n intertive fuzzy multiple ojetive optimiztion (IFMOO) metho for solving FMOLP prolems with fuzzy prmeters n fuzzy gols uner the ifferent stisidfwru\ GHJUHH ws evelope. The min struture of the IFMOO metho whih inlues two stges with si steps is outlines s following: Stge : Initiliztion Step : Set up FMOLP moel inluing inputting the memership funtion of eh fuzzy prmeter in the moel. Step : Ask the eision mker to selet stisftory egree D D n the iniviul weight for 8

2 EUSFLAT - LFA fuzzy ojetive funtions. Uner the urrent egree D n weights solve the FMOLP prolem.. Step : Ask the eision mker whether to e stisfie with the initil fuzzy Preto optiml solution lulte in Step. If stisfie the whole intertive proess stops n the initil solution is to e the finl stisfying solution. Otherwise go to Stge. Stge : Itertions Step : Speify new fuzzy gols g g g g T k for the fuzzy ojetive funtions se on the urrent fuzzy Preto optiml solution n new egree D if neee. Step : Clulte the fuzzy Preto optiml solution se on the urrent fuzzy gols of ojetive funtions n egree D speifie in Step. Step 6: If the eision mker is stisfie with the solution lulte in Step the whole intertive proess stops n the urrent fuzzy Preto optiml solution is the finl stisfying solution of FMOLP prolem. Otherwise go k to Step for nother itertion. Fuzzy Multiple Ojetive Deision Support System From the system esigner s point of view the fuzzy multiple ojetive eision support system (FMODSS) evelope onsists of four mjor softwre omponents whih re input-n-isply omponent moel mngement omponent optimiztion omponent n t mngement omponent. All the flows of t within the system suh s moels riteri n lterntive efinition n vlues intermeite n/or finl results even the t from the eternl soures will e store in the tse y the t mngement omponent. Aoring to the FMOLP moel () in Setion n the metho in Setion through the input-nisply omponent the following ommon t nee to e input from the user for setting up moel n other initil t for system. The numer of eision vriles the numer of fuzzy ojetive funtions n the numer of fuzzy onstrints; The oeffiients of fuzzy ojetive funtions the m/min for iniviul fuzzy ojetive funtion s showe in Figure ; The oeffiients of fuzzy onstrints the oeffiients of right-hn-sies the reltion sign of iniviul fuzzy onstrint s showe in Figure ; A Dilog Bo s showe in Figure is esigne n generte speilly for entering fuzzy numers. Referring to the fuzzy numer to e entere the forms of left ontinuous inresing funtion n right ontinuous eresing funtion of fuzzy numers n e selete s liner qurti ui eponentil logrithmi n other form from the ropown lists n four en-points of left n right funtion of fuzzy numers re entere in the tetoes simultneously. Figure : Input winow for fuzzy ojetive funtions n fuzzy onstrints Figure : Input winow for the memership funtion of fuzzy numer An importnt omponent of the system is moel mngement. Generlly the moel mngement omponent is omine with the tse mngement omponent n provies filities for the efinition storge retrievl n eeute of wie rnge of moels. It lso gives the opportunity for the user to speify or uil entirely new moel 86

3 y using moel-uiling fility ssoite with input-n-isply omponent. The optimiztion omponent of FMODSS presently inlues IFMOO metho to stisfy the fleiility require of the system. In winow s shown in Figure the ifferent weights for fuzzy ojetive funtions n e entere in FleGri n the egree of ll of the memership funtions of the fuzzy numers n lso e set y the slier s well. Following the IFMOO metho esrie in Setion lik Button Initite the initil solution of the prolem will e shown s eision vriles n fuzzy ojetive funtions in FleGri n respetively. To isply the memership funtions of the fuzzy ojetive funtion output lik the orresponing gri in the first row Ojetives of FleGri then nother winow will e pop up similrly s shown in Figure. When not stisfie with the solution lulte in the previous tril the eision mker n speify the fuzzy gols to e hieve in the net tril y two wys. One is y inresing or eresing the previous iniviul fuzzy ojetive funtion solution y perentge through liking the orresponing gri in the seon row By of FleGri in Figure. The other is y inputting the new fuzzy gols. In orer to input the fuzzy gols whih re represente y fuzzy numers lik the orresponing gri in the thir row By vlue of FleGri in Figure then nother Winow Fuzzy Gol Input similr s shown in Figure will pper. After the new fuzzy gols eing rete lik on Button Continue for getting new solution t the urrent tril. The optiml solutions for eh tril uring the intertive eision-mking proess re reore n liste in FleGri in Figure. At the time pressing Button Stisfy the finl solution of the prolem will isply n e store in the tse through the t mngement omponent. The intertive eision-mking proess ens t this itertion. An illustrtive emple To illustrte the system evelope let us onsier the following FMOLP prolem with three fuzzy ojetive funtions n five fuzzy onstrints: m m m f f f () s.t. t t ; 6 ( ( In this moel the unifie form of ll memership funtions of the prmeters of re ssume to e s following: P or () where left n right memership funtion re ll qurti. For simpliity we will represent the ove form of memership funtion s quruple pir ( ) in ltter on. For the FMOLP moel ll memership funtions of the fuzzy prmeters re to e represente in quruple pir form n liste in Tle n. Aoring to the esription of IFMOO metho liste in Setion the IFMOO metho involves two stges with si steps. The intertive proess for solving the prolem y FMODSS is suppose to e s following: Stge : Initiliztion Step : Initilly the FMOLP moel () of the prolem will e input into the system. The winow for entering the moel is showe in Figure. The informtion inlues the m/min of iniviul ojetive funtions tht re ll set to m n the reltionship involve in onstrints tht re ll set s EUSFLAT - LFA 87

4 EUSFLAT - LFA equl or less thn. In orer to enter fuzzy oeffiients lik on the orresponing gri in the tle n then nother Dilog Bo will pop up s showe in Figure. All fuzzy oeffiients s liste in Tle n re to e entere through Winow s in Figure n Dilog Bo s in Figure sequentilly. Tle : The memership funtions of fuzzy ojetive funtions prmeters the memership funtions of the fuzzy ojetive funtions solution f f n f () re shown s in Figure n the initil solution is logge n liste in the first row Tril of FleGri. Step : Suppose the eision mker is not stisfie with the initil solution lulte in Step the intertive proess will ontinue to Stge. ij (.) (8) () () () (8) () () (677) (6) (6) (668) (677) (6) (788) Tle : The memership funtions of fuzzy onstrints prmeters ij () (8) (8) (7) () (-6-- -) () (----8) (7) (8) (----.) () (----) (--- -) (-6---) () (.) (--- -8) ( ) (--- -) Tle : The memership funtions of righthn-sie s prmeters i (8 ) ( 6) ( ) ( 6) ( ) Step : After finishe estlishing FMOLP moel the eision mker will swith to the winow s shown in Figure to solve the prolem. Suppose the sdwlvidfwru\ghjuhh LVVHWWR(DFKZHLJKW for three fuzzy ojetive funtions is ll eqully set to.. Clik the Button Initite the initil ompromise solution to the FMOLP moel is generte. The eision vriles re: n the fuzzy ojetive funtions re f () f f By liking the orresponing gris in the first row Ojetives of FleGri in Figure one y one Figure : The memership funtions of f f n f t Tril Stge : Itertions Itertion No : Step : In this step the eision mker speifies new fuzzy gols for the fuzzy ojetive funtions to e hieve. Suppose these new fuzzy gols re ssigne y eresing oth of the first n thir fuzzy ojetive funtions y s the first n thir fuzzy gols respetively n inresing the seon fuzzy ojetive funtion y s the seon fuzzy gols se on the initil solution t Stge. Tht is: 88

5 EUSFLAT - LFA g. f () g. f g. f By liking the orresponing gri in the seon row By of FleGri in Figure one teto will pper n the inresing n eresing numers re e fille in these tetoes one y one. Step : Pressing Button Continue the new solution to the FMOLP prolem se on hieving the fuzzy gols () is generte. Consequently the eision vriles re n the fuzzy ojetive funtions re f (6) f f The memership funtions of f f n f (6) re isplye s in Figure. The new solution is lso logge n liste in the seon row Tril of FleGri s in Figure. Step 6: Suppose the eision mker is lso not stisfie with the solution lulte in Step the intertive proess will rry on to net itertion n go k to Step. the memership funtions of whih in quruple pir formt re liste s: u g (8) u g 8 u g By liking on the orresponing gris in the thir row By vlue of FleGri in Figure one Button Memership will pper then lik on the Button Memership nother Dilog Bo Fuzzy Gol Input similr s shown in Figure will pop up. An the memership funtions u g u g n u g (8) n e input y this Dilog Bo sequentilly. Figure : Min winow with IFMOLP metho for solving FMOLP prolem intertively Itertion No : Step : Now suppose the eision mker retes some new fuzzy gols s followings: g g g 7 8 (7) Figure : The memership funtions of f f n f t Tril Step : As the result of liking Button Continue the solution to the FMOLP prolem se on hieving the fuzzy gols (7) (8) is generte. The eision vriles re

6 EUSFLAT - LFA n the fuzzy ojetive funtions re f f f The memership funtions of f () f n f () re shown s in Figure 6. The solution is lso logge n liste in the seon row Tril of FleGri s in Figure. Step 6: Now the eision mker is stisfie with the solution lulte in Step the whole intertive proess stops n the urrent solution is the finl stisftory solution of the FMOLP prolem to the eision mker. Figure 6: The memership funtions of f f n f in Tril Aknowlegments This reserh is prtilly supporte y Austrlin Reserh Counil (ARC) uner isovery grnt DP7. Referenes [] Li Y. J. n Hwng C. L. "Fuzzy Multiple Ojetive Deision Mking: Methos n Applitions". Springer- Verlg Berlin:. [] Luhnjul M. K. "Multiple ojetive progrmming prolems with possiilisti oeffiients" Fuzzy Sets n Systems vol [] Rmik J. n Rommelfnger H. "Fuzzy mthemtil progrmming se on some new inequlity reltions" Fuzzy Sets n Systems vol [] Rommelfnger H. n Slowinski R. "Fuzzy liner progrmming with single n multiple ojetive funtions" in Fuzzy Sets in Deision Anlysis Opertions Reserh n Sttistis -The Hnook of Fuzzy Sets Series R. Slowinski E. Dorreht: Kluwer Aemi Pulishers 8 pp [] Skw M. "Fuzzy sets n intertive multiojetive optimiztion". Plenum Press New York:. [6] Slowinski R. "'FLIP': An intertive metho for multiojetive liner progrmming with fuzzy oeffiients" in Stohsti Versus Fuzzy Approhes to Multiojetive Mthemtil Progrmming uner Unertinty R. Slowinski n J. Teghem Es. Dorreht / Boston / Lonon: Kluwer Aemi Pulishers pp. -6. [7] Wu F. Lu J. n Zhng G. Q. "A fuzzy gol pproimte lgorithm for solving multiple ojetive liner progrmming prolems with fuzzy prmeters" Proeeings of FLINS : 6th Interntionl Conferene on Applie Computtionl Intelligene Blnkenerghe Belgium Sep [8] Wu F. Lu J. n Zhng G. Q. "A new pproimte lgorithm for solving multiple ojetive liner progrmming prolems with fuzzy prmeters" Applie Mthemtis n Computtion Aepte.