Finding Optimal Wiring for Hardware Components, using Fuzzy PSO and Traveling Salesperson Problem

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1 Ida Joural of Scece ad Techology Vol 8(), DOI:.7485/js/5/v8/7789, Jue 5 ISSN (Pr) : ISSN (Ole) : Fdg Opmal Wrg for Hardware Compoes, usg Fuzzy PSO ad Travelg Salesperso Problem Alreza Rezaee * ad Farba Jahaddeh Shekealgourab Deparme of Sysem ad Mecharocs Egeerg, Faculy of New Sceces ad Techologes, Uversy of Tehra, Tehra , Ira; arrezaee@u.ac.r Compuer of Scece, Uversy of Tehra, Tehra , Ira; farba.jahaddeh4@gmal.com Absrac Ths sudy ams o fd a way for accessg he opmal coeco ework, usg he Parcle Swarm for problems opmzao as well as fuzzy heory for expadg he problem space o a fuzzy space as a more real space he decsos. Havg bee esed o he radom daa, hs mehod has proved o ejoy a relave algorhm solvg he problems. Keywords: Fuzzy, Geec Algorms, Marce, Parcle Swarm Opmzao, Travelg Salesperso, Wrlg. Iroduco Desgg he opmal wrg pror o he acual wrg ad lks creao has caugh he aeo of expers may areas. Due o problem rcacy, such a desg s of umos mporace. The volume of he cable used by he ework, parcular he bg oes, s oe of he compoes eeded o be deermed he archecure ad durg he creao of compuer wrg. Provdg a opmal map dcave of he requred wrg ca resul he cosderable savgs he orgazaoal expedures. However, as for compuer wrg, hs requres ha he varous compoes of he wrg be cosa ad clear 3. The opmal wrg desg plays a mpora role he desg of dgal crcus ad VLSI crcus, wh s accuracy beg measured o he aomeer scale. Ths shows he exe o whch he accuracy ad he sesvy are esseal for developg he opmal wrg map ad he esmao requred for he wrg volume 4 7. The developme of wrg map for a buldg whch s uder cosruco s aoher area whch he opmal wrg s very esseal 8. The applcaos meoed above as well as he oher possble applcaos wll eal dffere requremes, depedg o he problem space. Cosequely, he process of fdg opmal wrg wll be subjec o varous compoes ad lmaos 9,. As a maer of fac, fdg he opmal wrg as he case of Travelg Salesperso Problem akes o dffere forms as he problem space vares. Ths sudy seeks o model opmal wrg fdg usg Travelg Salesperso Problem sadard sae. Ths s followed by seekg a opmal respose for he problem, usg fuzzy mehod ad bo audg. I s assumed ha he dsaces bewee coeco ps are deermed ad he cos of creag a lk s raslaed o he legh of he wre.. Travelg Salesperso Problem: a Opmal Wrg Model Alhough o he face of Travelg Salesperso Problem (TSO) s smple; due o s dffculy ad complexy has caugh he aeo of mahemacas ad compuer researchers. Ths problem s saed as follow: havg ake accou of he cos C requred for ravelg from a cy I o he cy j, Travelg Salesperso Problem eds o pass m gve ces oly oce ad back o s ow cy. The desrable roue s oe mposg he leas cos ad hs s wha TSP ams o do. TSO s mpora as s a sace of sgfca swarms of problems called Combaoal Opmzao Problems. *Auhor for correspodece

2 Fdg Opmal Wrg for Hardware Compoes, Usg Fuzzy PSO ad Travelg Salesperso Problem Drawg o Graph heory, aoher expresso of TSP s as follows: I graph G = (V, E) V equals he combao of odes ad E equals he combao of cres ( ) as E = {(a, b):a, b v} Dab dcaes he Euacldus dsace bewee a ad b as D ab + D ba TSP ams o fd a closed roue wh mmal legh whch passes each ode oly oce. Ths closed roue s he Hamloa cycle Graph. As oe ou of may problems, opmal wrg ca smply be modeled by TSP. The problem s saed as below: A gve se of ps s assumed. The problem ams o fd a wrg for all ps so ha he mmum volume of wres s used ad each p s aached o oly wo wres. The cos hs case equals he legh of wre used for wrg whose mmum volume s obaed, usg opmzao algorhm. The wrg o be foud ou he above-meoed problem s he Hamla cycle TSP. TSPs a suable crero agas whch may avalable opmzao mehods ca be compared cludg geec algorhm, smulaed aealg, eural ad coloal eworks. Cosderg as a model, hs sudy uses PSO fuzzy o solve TSP.. Parcle Swarm Opmzao Iroduced by Keedy ad Eberhar Parcle Swarm Opmzao (PSO) are used almos every applcao ad every dscple of egeerg rece years. PSO s a evoluoary compuaoal algorhm spred by he aure based o he erao. I descrpo of PSO algorhm, a swarm of gve parcles s formed, recevg he prmary values as hey are subjec o a se of radom soluos. Each parcle wll be defed erms of poso ad velocy whch are modeled by poso verex ad velocy verex respecvely. These parcles move recurrely -dmeso space, searchg for ew soluos by calculag he level of fess as a assessme crero. The umber of compoes avalable he fuco used opmzao equals he problem space dmeso 4. PSO algorhm for solvg combaoral opmzao problems s o have good performace. A memory s allocaed for savg he bes poso of each parcle he pas ad aoher memory s allocaed o savg he bes poso occurrg amog all he parcles. Drawg o he expereces bul up by hese memores, he parcles make a decso as how o make he ex moveme. I each erao, all parcles move -dmeso space of he problem ad fally fd he geeral opmal po. The followg equaos yeld he poso ad velocy of each parcle: + v = wv + C Rad() + C Rad() ( p + = + T g Idex ad dex yeld he parcle h ad he umber of eraos of algorhm up o ow respecvely. V ad show he velocy verex ad h poso verex of parcle h respecvely. v = ( v, v,..., V ) = (,,..., ) I Equao (), w deoes era/wegh ad R ad R are he radom dgs corbug o he dversy of parcle ad has a eve dsrbuo rage (,) for he dmeso of h parcle. C ad C are he posve cosas called self recogo compoe ad socal compoe coeffces respecvely. Combed, hese cosas are called cogve cofdece coeffces. P s he bes local poso obaed by parcle h up o ow ad R g s he bes poso a parcle amog all parcles have acheved. Fuco rad ca geerae a radom dg bewee ad.. Fuzzy PSO I fuzzy PSO model, he poso ad velocy of parcles show by real verexes wll be expaded ad refleced by fuzzy marces. Ths wll be poed ou he defo of problem. As a resul of hs chage, ew sgs ad Operaors wll be roduced so ha he equaos cocerg PSO algorhm are mplemeed fuzzy mode as well. A fuzzy equao o he fe ses ad Y as equao domas ca be expressed by a fuzzy merc. Assume ses ad Y are Y = {Y, Y Y m }, = {, m } respecvely. The equao from o Y s saed as r r R= ( rv m = ) r r m m (5) Vol 8 () Jue 5 Ida Joural of Scece ad Techology

3 Alreza Rezaee ad Farba Jahaddeh Shekealgourab Here r j [,] dcaes he membershp degree of h compoe ad jh compoe Y equao R. To model he problem, some defos are ecessary 3. Descrpo of he Problem 3. Defo of Problem Sae Space Defo : Se S s a cluser of ces or pu oher way a cluser of ps as a soluo for TSP. If S = {S, S S }, s he umber of ps ad f S ( {, }) s he h ode of obaed p wrg. Defeo : se N s acually he seral umber of a soluo for TSP, formed by covergece of he seral umber of each se. If, N = {N, N N } s he umber of ces or ps ad N I (I {,, }) dcaes a cy or a p he problem space. Through he covergece of ps umbers, he umber of wrg ework s obaed. Each S ε S dcaes ha he curre pah has udergoe - p ad h ps wll be me he ex sep. The space of sae (SS) s as follows: SS = S N = { (S, N I )}: S S, N N} The fuzzy equao R whch s defed from S o N dcaes ha for each eleme marce R r = m ( S, N ),( r ) (6) v R J v m r s he fuco of equao membershp. The value of r j s he degree of membershp of hs eve whch h p (S ) he cluser of possble respose S s he p h seral umber N j. 3. Defos of Symbols The bes poso of each parcle sce he begg of algorhm mplemeao up o ow s defed as follows: p p p p = p p Ad he curre poso of he parcle s as follows (7) The elemes of he above-meoed marces are defed ad erpreed smlarly o ha of he elemes of marc (6). The velocy of he parcles s defed as v v v V = v v 3.3 Defo of Operaor (9) As he poso ad velocy of parcles have bee covered o he marces, he ages equaos () ad () eed o be redefed. Symbol f s he modfed operaor of mulpler. Gve ha α s a real dg, he expresso α f V or α f equals he mulplcao of all elemes of marces V or by α. wo symbols f ad Θ are he modfed operaors of addo ad subraco respecvely. Gve ha A ad B are wo marces of poso ad velocy, AØB ad AΘB are he ormal addo ad subracos operaos bewee wo marces respecvely. Beg so, he modfed equaos PSO of he problem s obaed as + v = w v ( c Rad()) ( p Θ ( C Rad()) ( p Θ ) g g + = V + I 4. Ivesgao of Algorhm 4. The Prmary Value Allocao () The prmary value allocaed for poso as well as he bes poso of each parcle s p p = P = p p () The elemes of above marc are radomly geeraed ad should mee he followg codos x x x = x x (8),, v { } j= ) p,,,..., ( ) ) pv, (3) (4) Vol 8 () Jue 5 Ida Joural of Scece ad Techology 3

4 Fdg Opmal Wrg for Hardware Compoes, Usg Fuzzy PSO ad Travelg Salesperso Problem The same s rue for he prmary values allocaed for velocy as follows v v v = v v (5) The elemes of hs marce are also radomly geeraed ad should mee he followg codos vv =, {,,..., } (6) j= The followg heory shows he ecessy of he above equao. If codos (3) ad (6) are me equaos () ad (), follows ha afer several eraos, he velocy wll coue o mee he codo (6) ad poso wll coue o mee codo (3). The above heory s proved as follows: Ially 4 Lemma are saed: Lemma : Gve ha µ s a real dg, f Velocy V mees codo (6), he he same codo would be me by µ f V as well. Lemma : If P ad P mee he equao (3) he P f P s me by hs equao as well. Lemma 3: If P mees equao (3) ad V mees equao (6) he equao (3) wll mee PØV as well. Lemma 4: If V ad V mee equao (6), he V f V s me by he same equao as well. The above heory ca be duced as follows: I equaos () ad () ad sae = v w v c Rad p Θ ( c Rad pg o ( )) ( Θ ) = ( ) ( ) = ( ) V (7),(8) Here V ad are he prmary values of velocy ad poso. Three values A, B ad C are defed as follows: ( ) ( ) = ( ) ( g ) B C Rad p Θ A w V = ( ). C C Rad P Θ = ( ) V ad mee equao (6) ad (3) respecvely. Gve Lemma, A mees equao (6) so ( P Θ ( )) mees he same equao, usg Lemma. So B mees equao (6) accordg o Lemma. Fally V = AfBC f mees (6), usg Lemma 4. Moreover, gve (8) follows ha mees Equao (3), usg Lemma 3. So f = he velocy ad poso mee equaos (6) ad (3). Gve ha +g, (g>) he velocy V g g ad poso mee equaos (6) ad (3). Smlarely gve =, g + V ad g + mee he same equao whe = g +. Gve he above heory follows ha f equaos (3) ad (6) are me value allocao phase, excep for ormalzao; here wll be o eed for reseg he values of velocy ad poso whle he algorhm s beg mplemeed. Ths reders he compuaos smple. 4. The Normalzao of Poso Marc Poso marc (7) may volae codo (4) afer several eraos. Ths requres he marc o be ormalzed. To do so, ally all egave values are covered o. The he marc s rasformed provded equao (3) s o volaed. P / p p / p P / P P / p P / p (9) There s o maxmal value of velocy velocy marc. The prevous experece shows ha hs algorhm requres o use of maxmal value for velocy (V_Max). 4.3 Defuzzcao Sce marc of poso (7) shows he poeal TSP respose, hs marc eeds o be decoded so ha possble respose ca be obaed. Ths s called defuzzcao whch s made possble by usg Max Number mehod: I hs mehod a preseao of flags s used for savg he seleco or o-seleco of marc colums ad aoher row whch deoes he pah s used for savg he respose pah. Ially, o colum s seleced. The a eleme havg he hghes value ad o seleced by prevous rows s seleced for each row of marc. Aferwards, he flag of colum relag o ha eleme s marked as seleced. I addo, he umber of ha colum 4 Vol 8 () Jue 5 Ida Joural of Scece ad Techology

5 Alreza Rezaee ad Farba Jahaddeh Shekealgourab s saved pah preseao. The respose wrg s obaed from he pah preseao afer all rows havg bee processed. The legh of wre requred s also calculaed. The cos mposed by he poso marc s he cos of TSP pah, obaed by usg maxmal umber as well as defuzzcao of he marc. 4.4 Descrpo of Algorhm Sep : Prmary value allocao Sep. he umber of parcles Max_Num ad he hghes umber of eraos Max_Ierao are deermed. Sep. for each parcle, a radom poso marc ad a radom velocy marc V are allocaed, usg a prmary allocao mehod. The bes local poso P = ad he bes geeral poso make up he bes P. Sep : If he umber of prese eraos () equals Max-erao, sep 5 eeds o be mplemeed. Sep 3: For all (s) ragg from o Max_Num-, he poso ad velocy of h eeds o be compued. Sep 3.. The ew velocy should be compued, usg equao (7). Sep 3.. The ew poso eeds o be compued, usg equao (8) ad poso marc should be ormalzed, usg equao (9). Sep 3.3 he ew poso should be defuzzcaed ad he poso coss should be calculaed. If he cos mposed by ew poso s less ha he cos mposed by he bes local poso of he parcle, he he bes local poso should be replaced by he ew poso. Sep 4: If he legh of he wre he bes local poso of some parcles s less ha he wre legh he bes geeral poso, he he bes geeral poso should be replaced by he bes local poso. Sep should be mplemeed. Sep 5: The bes respose wrg ad s legh should appear he oupu. 5. Solvg a Sample Problem The daa of sample problem wh 4 ps ad radomly dsrbued dsaces are gve fle Fuzzy_PSO.m. The proposed ral was operaed o a compuer wh hese specfcaos: Premum IV, CPU GHz, M RAM 5, Operag Sysem WINDOWS P, verso 7 MATLAB. The ral shows ha Gve parcles problem space ad whe he compoes of equaos () ad () are as follows: c = c = ad w =.5 The, eraos of he algorhm wll yeld a raher opmal respose. 6. Dscusso Ths sudy combed PSO mehod ad fuzzy heory as a rapd mehod for solvg opmzao problems for purpose of fdg he lkg loop of shores legh. TSP was used o model he problem of fdg a loop wh he leas cosumpo of wre. The evaluao of hs mehod ad rals showed ha alhough hs combed mehod s o much beer ha he algorhm offered by L- Kergha, s a sarg po for solvg he combed opmzao problems. The algorhm used hs sudy ca also be appled varous problems of roug ad oher combaoal opmzao problems. 7. Refereces. Pag W, Wag K, Zhou C-G, Dog L-J. Fuzz dscree parcle swarm opmzao for solvg ravelg salesma problem. The 4 h IEEE Ieraoal Coferece o Compuer ad Iformao Techology; 4.. Keedy J, Eberhar R. Parcle swarm opmzao. IEEE Ieraoal Coferece o Neural Neworks (Perh, Ausrala); Pscaaway, JN: IEEE Servce Ceer; 995. p Abraham A, Guo H, Lu H. Swarm ellgece: foudaos, perspecves ad applcaos. A Swarm chaper o he Iere Bejam KBA. Geerc alogorhms ad he ravelg salesma problem [Thess]. Havey Mudd Uversy, Deparme of Mahemacs;. 5. Eberhar RC, Sh Y. Comparg era weghs ad cosrco facors parcle swarm opmzao. Cogress o Evoluoary Compug. ; : Moha CK, Al-kazem B. Dscree parcle swarm opmzao. Proceedgs Workshop o Parcle Swarm Opmzao; Ida pols, IN: Purdue School of Egeerg ad Techology, IUPUI;. 7. L S, Kergha BW. A effecve heursc algorhm for he ravelg salesma problem. MD, USA; Operaos Res, Is Operaos Research Maageme Sceces; 973. p Vol 8 () Jue 5 Ida Joural of Scece ad Techology 5

6 Fdg Opmal Wrg for Hardware Compoes, Usg Fuzzy PSO ad Travelg Salesperso Problem 8. Hua Z, Huag F. A varable-groupg based geec algorhm for large-scale eger programmg. If Sc. 6; 76(9): Acampora G, Gaea M, Loa V. Combg mul age paradgm ad memec compug for persoalzed ad adapve learg expereces. Compu Iell J. ; 7(): Guvec U, Duma S, Saracoglu B, Ozurk A. A hybrd GA PSO approach based o smlary for varous ypes of ecoomc dspach problems. Elecro Elecr Eg Kauas: Techologja. ; (8):9 4.. Ya S, Wu QH, Hu CY, Lag QZ. Crcu desg based o parcle swarm opmzao algorhms. Key Egeerg Maerals. ; : Maraks Y, Marak M. A hybrd mul-swarm parcle swarm opmzao algorhm for he probablsc ravelg salesma problem. Compu Oper Res, ; 37(3):43 4. Do:.6/j.cor Nkou HR, Semsar M. Dgal crcu desg usg chaoc parcle swarm opmzao asssed by geec algorhm. Ida Joural of Scece ad Techology. 3 Sep; 6(9): Solemaa GF, Ebrahm L, Malek I, Gourab SJ. A ovel PSO based approach wh hybrd of fuzzy C-meas ad learg auomaa sofware cos esmao. Ida Joural of Scece ad Techology. 4 Ju; 7(6): Vol 8 () Jue 5 Ida Joural of Scece ad Techology

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