A Genetic Algorithm for Fuzzy Shortest Path in a Network with mixed fuzzy arc lengths
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- Frederick Wheeler
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2 for fdg fuzz shortst pth d fuzz twork A pl s workd out to llustrt th ppllt of th proposd odl Sto 6 W olud Sto 7 Dftos Th α-ut d strog α-ut for fuzz ur r show d rsptvl d for [0 r dfd to : X A X A whr X s th uvrsl st Coputg α-uts for fuzz urs For th fuzz urs wth d vrtl futos th -uts r: [ [ For spf d futos th followg ss r dsussd α-uts for trpzodl fuzz urs t trpzodl fuzz ur A ut for s oputd s: 0 whr [ s th orrspodg α-ut α-uts for orl fuzz urs If s orl fuzz ur th s oputd s: 0 Fuzz pprot su oprtors Hr w us our rtl proposd pproh [ for sug vrous fuzz urs pprotl usg α- uts Th pproto s sd o fttg pproprt odl for th su usg α-uts of th ddto s th ftss dt t us dvd th α-trvl [0 to qul sutrvls lttg 0 0 = Ths w w hv st of + qudstt pots For th orl fuzz urs t s 809
3 propr to ssu α g qul to zro Thrfor ths s w osdr 0 d thus us th ozro Cut su t d th trpzodl d orl fuzz urs rsptvl Gv 0 th α-ut su of ths fuzz urs usg qutos d s otd s follows: whr [ [ Usg quto orrspodg to pots r otd for pots for th d pots for th Usg ths pots t s possl to pprot th su of th two fuzz urs A pprot rshp futo of th su s oputd fttg pproprt futo usg th α-ut pots t d d usg th pots osdr th fttg odl s Th ukow prtrs d ppr orl W rz th odl otg tht for s s th s hr for th rght hd odl d r pltl dtrd to : 5 Now slrl lt d d osdr th odl W th hv 6 7 Thus th rshp futo s dtrd to : 8 wth d s dfd 5 6 d 7 rsptvl 5 A lgorth for fuzz shortst pth twork 5 Dst tw fuzz urs 80
4 Assu tht d r two fuzz urs W ppl fuzz rkg thod for fuzz urs W hv usd ths rkg thod fftvl rt work [8t us osdr fuzz oprtos s follows: 9 It s vdt tht for o-oprl fuzz urs d th fuzz oprto rsults fuzz ur dffrt fro oth of th For pl for 509 w gt fro 9 fuzz d MV 599 whh s dffrt fro oth d To llvt ths drwk w us thod sd o th dst tw fuzz urs W us th dst futo trodud [ Th dvtgs of ths dst futo r th grlt of ts usg o vrous fuzz urs d ts rllt dstgushg uqul fuzz urs Idd th usg of th dst futo low workd out to qut pproprt for our pproh Th D p q -dst dd prtrs p d 0 q ogtv futo gv : p p p q d q d p D 0 0 p q q sup q f p 0 0 For two fuzz urs d wth orrspodg to : Dp q q If q p th th ov quto turs to: tw two fuzz urs d s 0 -uts th D p q dst s pprotl proportol p q p D To opr two fuzz r lgths d wth -uts s thr pprotos s th r supposd to rprst postv vlus w opr th wth MV 000 I ft w us forul to oput D MV d D MV d th us ths vlus for oprso of th two urs 5 Th Gt Algorth 5 Gt Algorth GA for solvg shortst pth prol wth fuzz r lgths Hr w dsr th odg sh usd th GAs Gt oprtors spf to ths odg sh r lso dfd Ths lud th tlzto rossovr d utto oprtors 5 GA Eodg Sh d Populto Itlzto How to od pth grph s rtl for dvlopg gt lgorth to ths prol Ths s ot s s s trvg sls prol to fd out turl rprstto Spl dffults rs fro pth ots vrl ur of ods d th l ur s for od grph d rdo squ of dgs usull dos ot orrspod to pth To ovro suh dffults w doptd drt pproh: od so gudg forto for ostrutg pth ut pth tslf hrooso Th pth s grtd th squtl od ppdg produr gg wth th spfd od d trtg t th spfd od A vtor p s usd to kp th tr of th ods th pth Algorth grtg tl populto Fd th vt tr Aof drtd twork G = N A dtr pop-sz d st k St l d pl= Slt r of j j A d ll t j t l l d pl=j whr j p 8
5 If j th lt =j d go to 5 Fd produd pth usg th lls th lg vtor p t k k 6 If k pop-sz th go to ls stop 5 Crossovr oprtor Crossovr os forto fro two prts suh tht th two hldr hv rs to h prt Stdrd rossovrs suh s o-pot two-pot d ufor r usd GA odls Two pths lld prts r rdol sltd fro th populto Th w slt o or two oo rs gs d rpl dffrt stos fro prts H two w hldr hroosos r grtd It s ovous tht th grtd pths r fsl 5 Mutto oprtor Th trdtol utto oprtor utts th gs vlu rdol ordg to sll prolt of utto; thus t s rl rdo wlk d dos ot gurt postv drto towrd th optl soluto Th proposd hurst utto rds ths df I ths sh th ur of pths or sltd hroosos q for th utto oprtor s dtrd usg utto oprtor s rt or proltp ; th ur of pths q s oputd to th produt of populto sz d p d q dffrt rdo urs r grtd tw d populto sz to usd s ur of sltd pths Th ur s r tw d pth lgth s sltd rdol Th opots to r- s kpt uhgd d opots r to pth lgth r rovd rplg th wth w ods otd usg Algorth 55 Evluto d slto strtg Cosdrg populto hrooso pth for h hrooso s dtrd d th pth lgth s osdrd s th hrooso vlu Th for oprso of hrooso vlus w us for fdg dsts Th w lgorth for fdg shortst pth follows hr Algorth fdg shortst pth Stp Fd th possl pths fro sour vrt s to dstto vrt d fro og produd populto h rptto d oput th orrspodg pth lgths = for th possl pths Stp Fd th fuzz shortst lgth th followg stps: Stp - St Stp - for do Stp - Clult MV M Eq 9 Stp - Fd th dst D p q of MV fro d usg Eq : D D p q MV D D p q MV rg D D 6 Nurl Illustrtos Epl :W osdr lrgr twork hvg d r lgths oto of orl d trpzodl fuzz urs d us our gt lgorth to fd th shortst pths Th r lgths r spfd Tl Tl Th r lgths r fuzz ur r fuzz ur r fuzz ur r fuzz ur r fuzz ur
6 Usg th dst futo D p q od s dtrd to : Fgur shows th ovrg urv for Epl for q=/ d p= th shortst pth fro th sour od to th dstto Shortst pth lgths Fgur Covrg urv for Epl Itrtos B ddto of vrous fuzz urs o th orrspodg pth th rshp futo s otd -7 s follows: μ = < < < 65 > 65 7 Colusos W osdrd th prol of fdg th ll-prs fuzz shortst pths wth th lgths of r gv fuzz urs Frst w proposd ordr rlto tw fuzz urs Th w dvlopd fuzz rkg thod to vod grtg th st of o-dotd pths or Prto optl pths us th ur of o-dotd pths drvd fro lrg twork too urous d t ould dffult for dso kr to hoos prfrl pth Th w vstgtd th posslt of usg gt lgorth to solv fuzz shortst pth prols Th proposd pproh ws llustrtd o rdol grtd prol hvg ods d 0 rs frs [ M Blu B Bush J Puktt Ufd pproh to fuzz grph prols Fuzz Sts d Ssts [ T-N Chug J-Y Kug Th fuzz shortst pth lgth d th orrspodg shortst pth twork Coput Opr s [ D Duos H Prd Fuzz Sts d Ssts: Thor d Appltos Ad Prss Nw York 980 [ D Eppst Fdg th k-shortst pths : Pro IEEE Sp o Foudtos of Coputr S [5 CM Kl Fuzz shortst pths Fuzz Sts d Ssts [6 C MS Chr Th fuzz shortst pth prol d ts ost vtl rs Fuzz Sts d Ssts [7 MT Tkhsh Cotruçõs o studo d grfos fuzz: Tor lgortos PhD Thss Fuldd d Eghr Elétr d Coputção UNICAMP 00 [8 I Mhdv Nourfr A Hdrzd N Mhdv Ar A d progrg pproh for fdg shortst hs fuzz twork Appld Soft Coputg [9 JA Moro JM Moro J Vrdg Fuzz loto prols o tworks Fuzz Sts d Ssts [0 SMA N M Pl Shortst pth prol o twork wth prs dg wght Fuzz Opt Ds Mkg [ S Okd Fuzz shortst pth prols orportg trtvt og pths Fuzz Sts d Ssts [ B Sdghpour Gldh D G Dst-Dpq t l Cofft d Corrélto tr du Vrls Alétors flous Ats d FA Mos-Blgu 00 [ A Sgupt TK Pl O oprg trvl urs Europ J Opr s [ A Tjd I Mhdv N Mhdv-Ar B Sdghpour-Gldh Coputg fuzz shortst pth twork wth d fuzz r lgths usg α-uts Coputrs d Mthts wth Appltos prss 00 8
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