Cellular-based Population to Enhance Genetic Algorithm for Assignment Problems

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1 Amerin Journl of Intelligent Systems. 0; (): -5 DOI: /j.jis Cellulr-se Popultion to Enhne Geneti Algorithm for Assignment Prolems Hossein Rjlipour Cheshmehgz *, Hiollh Hron, Mohmm Rez Myoi Fulty of Computer Siene n Informtion Systems, Universiti Teknologi Mlysis (UTM), 830, Skui, Johor, Mlysi Soft Computing Lortory, Computer Engineering n Informtion Tehnology Deprtment, Amirkir University of Tehnology (Tehrn Polytehni), Tehrn, Irn Astrt In this pper, we esrie new mehnism of ellulr seletion s n improve Geneti Algorithm for some optimiztion prolems like Cellulr Chnnel ssignment, whih hve multi fesile/optimum solution per one se. Consiering the prolems n the nture of reltionship mong iniviuls in popultion, we use -imension Cellulr Automt in orer to ple the iniviuls onto its ells to mke the lolity n neighorhoo on the Hmming istne sis. This ie s D Cellulr Automt Hmming GA hs introue lolity in Geneti Algorithms n glol knowlege for their seletion proess on Cells of D Cellulr Automt. The seletion se on D Cellulr Automt n ensure mintining popultion iversity n fst onvergene in the geneti serh. The ellulr seletion of iniviuls is ontrolle se on the struture of ellulr utomt, to prevent the fst popultion iversity loss n improve the onvergene performne uring the geneti serh. Keywors Geneti Algorithms, Cellulr Automt, Optimiztion, Assignment Prolems. Introution Geneti Algorithms (GAs) re use to solve iffiult optimiztion prolems like mny NP-Hr Prolems[] in sientifi n engineering res. Resoure istriution[], mngement[3] n Assignment: Chnnel Assignment Prolem[4], Grph Coloring Prolem[5]; Multi-ojetive Optimiztion of Systems: 0/ Multiple Knpsk Prolem[6], Assemly Line Blning Prolem[7], et. re onsiere to solve y GA. The min isvntges of geneti lgorithms re the isruption of goo su-solutions y rossover n muttion opertions n unesire popultion iversity loss y seletion opertions, whih onstntly ereses the vriety of its speimens. Popultion iversity shoul e preserve to prevent egenertion while mintining the generl tren of the evol uti on n som e sort of sel eti ve pressur e. Crossover opertions re pplie to proue new iniviuls from prents on the fitness whih hs een notie y seletion opertion n Rnom muttions re pplie to every new solution proposl in n ttempt to slow own egenertion n introue new hrteristis to the popultion. * Corresponing uthor: hrjlipour@uk..ir ( H. Rjlipour Cheshmehgz ) Pulishe online t Copyright 0 Sientifi & Aemi Pulishing. All Rights Reserve Severl mehnisms hve een integrte with geneti lgorithms in vrious wys to preserve the iversity of speies. Referene [8], Mtousek n Nolle hve presente inry GA with moifie muttion opertor, whih is se on the well-known Hill Climing Algorithm (HCA) n the seletion opertor preserves the est iniviul from the GA popultion uring the seletion proess while mintining the positive hrteristis of the stnr tournment seletion. Chen n Wng [9] hve presente seletion metho omining roulette seletion with tournment seletion is presente to reinfore the lol serh ility. In[0-], some moifie seletion methos re propose to inrese the gin of resoures, reliility n iversity; n erese the unertinty in the seletion proess. Lolity, in Selet opertion, hs een onsiere in some reserhes s one im to inrese the spee of fining solution (fesile or optimum)[]. In[4], the reserher hs introue the ellulr utomt to relize the lolity n neighorhoo in the popultion struture. Bse on the struture of ellulr utomt, the seletion of iniviuls is ontrolle to voi the fst popultion iversity loss uring the geneti serh. Mostly, the prolems whih re mentione in the eginning of this setion hve more thn one optimum/fesile solution. Thin seletion (uilt only on the fitness sis) woul isrupt the goo solutions (ifferent goo solutions) n proue not etter more offspring for new popultion. Therefore, onsiering the etil of the speifi prolem

2 H. Rjlipour Cheshmehgz et l. The Improve Geneti Algorithm for Assignment Prolems n then using thik seletion (uilt on more etil ses), on hmming istne etween prents n fitness ses, n improve the onvergene performne uring the geneti serh. In this pper, we introue the multi fesile/optimum fesile n ellulr utomt to evlute the lolity n neighorhoo in the popultion struture, whih is one y mpping se on the hmming istnes etween iniviuls. The seletion of iniviuls is ontrolle to voi the fst popultion iversity loss uring the geneti serh. We pply the new ies s improve GA, D Cellulr Automt Hmming GA (DCAHGA), on some well-known NP-hr Prolem like grph oloring, Chnnel Assignment Prolems n the outome shows s simultion results.. Geneti Algorithms n Multi solution Prolems.. Vriety of Solutions Geneti lgorithms re generl onept for solving omplex optimiztion prolems, whih re se on mnipulting popultion of solutions y geneti opertors like seletion, rossover (reomintion) n muttion [5]. For mnipulting solutions y geneti opertors, they hve to e enoe in form of so-lle iniviul (or hromosomes) eh of whih onsists of sequene of genes. For exmple, simple grph oloring (NP-hr) prolem is shown in Figure, with 3 olors, there is string of its s mtrix C tht illustrtes olors ssigne to noes of the grph (in eh row of mtrix C). In this solution (not extly orret), olor is ssigne to noes n," olor is ssigne to noe ; olor 3 is ssigne to noe." The generl ie of typil GA is est expline y the following sheme: Step : t= 0; (Strt with n initil time) Step : init popultion P (t); (Construt n initil popultion of iniviuls) Step 3: evlute P (t); (Evlute fitness of ll Iniviuls of initil popultion) Step 4: t: = t + ; (Inrese the time ounter) Step 5: P_:= selet prents P (t) (Selet su-popultion for offspring proution on their fitness sis) Step 6: rossover P_ (t); (Do rossover the genes of selete prents) Step 7: mutte P_ (t); (Pertur the mte popultion stohstilly) Step 8: evlute P_ (t); (Evlute its new fitness) Step 9: P: = survive P, P_ (t); (Selet the survivors from tul fitness) Step 0: while not o steps 4 through 9 (Test for termintion riterion (time, fitness, et.)) Step : en GA (Terminte the lgorithm) Figure. A Grph oloring Prolem with 3 olors. Consiering the grph oloring prolem, the most importnt thing in suh this se is hving more thn one fesile (or optimum) solution for the prolem. Figure. shows two ifferent fesile solutions for the previous se. However, there re some etter solutions with the minimum numer of olors s optimum solutions tht re shown in Figure.. Crossover point Crossover point. Color Assignment Mtrix C s two optimum solutions with olors Firuge. Two fesile/optimum solutions for prolem in figure Color Assignment Mtrix C s offspring (fter rossover) with 3 olors Color Assignment Mtrix C s two fesile solutions with 3 olors Grph with 4 noes. Color Assignment Mtrix C s offspring (fter rossover) with olors Firuge 3. Two fesile/optimum offspring fter o rossover on iniviuls in Figure. Unfortuntely, thin seletion in GAs without notie to similrity mong iniviuls my use popultion iversity n low onvergene in the geneti serh. Both pir solutions in Figure hve penlty with 0 n re est solutions for their own prolem n ojetives (with fesile solutions with 3-olors / optimum solutions with -olors), ut if GA tries to selet these n o rossover on them (step 5 mentione efore), the result woul not e Color Assignment Mtrix C By 3 olors (not extly orret solution)

3 Amerin Journl of Intelligent Systems. 0; (): -5 3 etter n the prents nnot rerete offspring etter thn themselves. Figure 3 shows offspring fter rossover opertion. As it is ler, the penlty for two new solutions is (Figure 3.) n for others (Figure 3.) is. This retrogression hs resulte from the seletion of the ouple to rete offspring... Hmming Distne etween Binry Solutions Some reserher trie to moify the mehnism of seletion in orer to improve the GA[-5]. In[4] Hmming istne is use for gring the ifferene etween inry strings tht hve n equl numer of its. They onsier popultion with m iniviuls, n eh iniviul inlues inry string of length l. For every pir of inry strings v i = ( i,..., i l ) n v j = ( j,..., j l ), where is equl to 0 or, the Hmming istne is efine y: l i j i j k k k H ( v, v ) () Where stisfies inry exusive-or opertion. To mke offspring, GA seletion opertion selets first oring to the fitness s est (y roulette wheel seletion n tournment seletion tehniques[5]), ut to selet seon prent, in ition to fitness, noties the mount of hmming istne etween new one n previous selete prent. Then, to mke eision to selet, Moifie new seletions hs two mehnisms: Seletion- opertion is onsiere for first prent n Seletion- for the seon prents. As it is following, eh seletion hs the own mehnism ut seon one epen to first prent. So the vlue of the seon prents to selet n moify s follows: Vlue for SeonPrent f (, ) () : Fitness of seon prent. : Amount of hmming istne etween the first prent (selete prent) n seon prents s the nite (seleting prent) f is n inrementl funtion epene on the prolem. For exmple, it n e exponent funtion. The new GA lle Hmming GA (HGA) n e hnging the GA s follows in steps 5: Step 5: // selet ouple of iniviuls for offspring proution on their fitness sis for first one n on the Vlue-for-Seon Prent vlue sis for the seon..3. Compring HGA n Clssi GA To ompring, HGA is pplie on the well-known NP-Prolem in ellulr moile networks, Chnnel Assignment Prolem (CAP)[4] tht hs multi fesile/optimum solution with three onstrints. All three onstrints re onsiere for the hnnel ssignments: the o-hnnel onstrint, the jent hnnel onstrint n the o-site hnnel onstrint. The gol of CAP is the ssignment of the hnnel frequenies whih stisfie these onstrints with the lower oun numer of hnnels (s the fesile/optimum solution). For instne, The CAP is onsierte with 6 sttions of BTS. The numer of hnnels is 48, n the popultion size of GA is 00. The results re gthere n hve een ompre s is shown in Figure 4. In Figure 4., there re 00 runs of eh lgorithm with the sme prmeters (rossover n muttion rte =90% n 5%, popultion size = 00, rossover type= vertil see Figure n 3). The perentge xis in the igrm shows the numer of runs tht the lgorithms hve rehe n optimum solution. Improvement of HGA in perentge of rehing n optimum solution in ifferent itertions is sensile s ompre to Clssi GA (GA with no notie out Hmming Distne mong iniviul). HGA oul e mostly suessful thn lssi GA to hve high hne to pproh n optimum solution, espeilly in low itertions. However, most importnt isvntge is rel time tken y the HGA. Both of lgorithms run on Pentium 4 CPU.6GHz omputer with GB RAM. As it is shown in Figure 4., unfortuntely, the rel time woul e high (for 50 itertions it tkes more thn 6 hours). In next setion, we pply -Dimensionl Cellulr Automt to reue the rel time of running HGA, n it keeps the improvement s well. Itertions %0 %50 %70 Itertions ) Tken rel time of running the HGA n GA HGA Clssi GA Firuge 4. Compring HGS with GA for CA Prolem (Perentge of suess to reh n optimum solution / Rel time). # # # # C # # # # Perentge ) Perentge of rehing n optimum solution in ifferent Itertions Figure 5. A prt of DCA C ell n its neighors with mrk #. 3. D CELLULAR AUTOMATA HAMMING GENETIC ALGORITHM (DCAHGA) 3.. Cellulr Automt Cellulr Automt (CA) is ynmil systems isrete in spe, time n stte vriles n hrterize y possession of exlusively lol mehnisms of intertion. They onstitute goo moels for the stuy of nonliner Se

4 4 H. Rjlipour Cheshmehgz et l. The Improve Geneti Algorithm for Assignment Prolems omplex systems. In spite of the simpliity in the efinition, the set of ellulr utomt ontins mny rules with very omplite ehvior. Formlly, CA is represente y the 4-tuple {Z, S, A, f} where: [] Z is the finite or infinite lttie (ll ells) [] S is finite set of ell sttes or vlues (for this rtile, they re iniviuls n their Fitness Vlue) [3] A is the finite neighorhoo [4] f is the lol trnsition funtion efine y the trnsition tle or the rule (for this rtile re seletion, rossover n muttion opertion) The lttie is finite or infinite isrete regulr gri of ells on finite numer of imensions. Eh ell is efine y its isrete position (n integer numer for eh imension) n y its isrete vlue. Time is lso isrete. The future stte of ell (time t + ) is funtion of the present stte (time t) of finite numer of ells surrouning the oserve ell lle the neighorhoo (see Figure 5, DCA). 3.. Mpping Popultion of GA onto CA Before eveloping DCAHGA, the iniviuls of eh popultion re mppe onto ll the ells of the ellulr utomton, se on the Hmming istnes. Consier popultion onsisting of m iniviuls, with inry string v. We rnomly hoose v i (i= size of Popultion), to e the element C (Figure 5), then the neighorhoo of C is onstrute s follows. The iniviuls in popultion re sorte in sening orer of their orresponing Hmming istnes to v i. Choose the first eight elements v k s the neighorhoo of C, with other elements sttere outsie its neighorhoo, n then the whole popultion hve een mppe onto ellulr utomton. The losest neighors of C element re mrke y # in Figure 5. They usully hve smller Hmming istne to C. This ie hs introue lolity in GAs n glol knowlege for their seletion proess. A ell looking for mte Numer of Cnites is 3 Firuge 6. One seletion phse for ells with their own nites Steps of DCAHGA For new GA, seletion opertion iniviully oes for iniviuls in eh ell n it woul hve limittion to selet mte from only ell s neighorhoos. Eh ell in CA, inepenently o seletion n selet one neighor on the seletion rules s own lol trnsition funtion. This opertion n e one in onurreny wy too s extr vntge (in some prllel omputers whih n o). Eh ell oring to own position on CA woul hve some nites to selet s mte. Figure 6 illustrtes one seletion phse for ll ells n own nites to selet (for 5 5 CA). The DCA se Hmming geneti lgorithm n, in essene, e given s follows: Phse (Mking initil popultion) [] Step : generte initil popultion. Phse (Mpping on CA) [] Step : inepenently selet n iniviul j from the urrent popultion. Mp the selete iniviul j onto ellulr utomton, s esrie in the ove susetion (previous su setion). [3] Step 3: rnomly hose one fille ell of CA suh tht hs t les n empty neighor, n then selet iniviuls n mp just onto empty its neighors s esrie efore (previous su setion). [4] Sept 4: if ll ells of CA re not full, go to step 3. Phse 3 (Cellulr geneti tions) [5] Step 5: inepenently, ut they n onurreny, eh ell selet one of its neighors s mte to prout new offspring on their fitness n seletion rules. [6] Step 6: in eh ell, o the rossover on iniviul in ell n its selete neighor, n then o muttion then, one of the results woul e kept inste of previous iniviul in the ell. Phse 4 (Cheking to finish) [7] Step 7: if the est solution is not rehe go to step. There re 3 most importnt phses to o for DCAHGA n t the en of ellulr geneti phse, there is new popultion on ells. To ontinue the lgorithm in orer to reh etter solution, the phses woul e ontinue (phse 4). Itertions %0 3 % % ) Perentge of rehing n optimum solution in ifferent Itertions Itertions Se ) Tken rel time of running DCAHGA HGA Clssi GA Figure 7. Property of DCAHGA, HGS n Clssi GA for CA Prolem (Perentge of suess to reh n optimum solution / Rel time). 4. Simultion Results Perentge The new lgorithm, DCAHGA is pplie for the sme CAP hs mentione in setion II. Whtever, DCAHGA is

5 Amerin Journl of Intelligent Systems. 0; (): -5 5 le to pply on mny multi solution prolems suh 0/ multi knpsk, grph oloring, ssemly line lning for mnufturing, plement, n, et. s well s CAP n the results re the sme. There is notiele improvement in the rel time tken y running DCAGHA s ompre to HGA n Clssi GA in Figure 7.. ut there is not extly speifi ehvior in low itertions (50 to ner 50 itertions) to emonstrte tht new mehnism of ellulr seletion woul e etter thn the seletion mehnism (in HGA) in perentge of rehing n optimum solution (see Figure 7.). Oviously, new mehnism works well (out 90%) in high itertions of running (50 itertions) n the rel time (out 000 se) too. Totlly, the results show tht the ellulr seletion is ple of serhing fesile/optimum solutions in resonle time n itertions. 5. Conlusions A two imensionl ellulr utomt se hmming geneti lgorithm (DCAHGA) hs een evelope to mintin popultion iversity uring the geneti serh, whih is essentil for geneti lgorithms to solve the iffiult multi fesile/optimum solution prolems suh s 0/ multiple knpsk, hnnel ssignment in ellulr moile network, grph oloring, ssemly line lning prolem n mny NP-hr prolems, whih hve more thn one fesile or optimum solution. The ellulr utomton is introue to relize the lolity n neighorhoo in the popultion struture. The seletion of iniviuls is ontrolle se on the struture of ellulr utomt. Applitions of CAHGA in optimiztion prolems hve shown tht CAHGA is ple of serhing for the glol optimum solution of iffiult nonliner optimiztion prolems. REFERENCES [] G. J. Woeginger, Ext Algorithms for NP-Hr Prolems: A Survey, Leture notes in omputer Siene, Springer-Verlg, Germny, vol. 570, 003, pp [] S. Wng, n C. Ni, Applition of Projetion Pursuit Dynmi Cluster Moel in Regionl Prtition of Wter Resoures in Chin, Wter Resoures Mngement, vol., 008, pp [3] L. C. Chng, Guiing rtionl reservoir floo opertion using penlty-type geneti lgorithm, Hyrology, vol. 354, 008, pp [4] L.M. Sn Jose-Revuelt, A new ptive geneti lgorithm for fixe hnnel ssignment, Informtion Sienes, vol. 77,007,pp [5] J. Q. Yu, n S. N. Yu, A novel prllel geneti lgorithm for the grph oloring prolem in VLSI hnnel routing, Pro. 3r Interntionl Conferene on Nturl Computtion, Chin, vol. 4, 007, pp [6] Y. Yoon, Y. Kim, n B. R. Moon, An evolutionry Lgrngin metho for the 0/ multiple knpsk prolems, Pro. Geneti n Evolutionry Computtion Conferene, South Kore, vol., 005, pp [7] S. O. Tsn n S. Tunli, A review of the urrent pplitions of geneti lgorithms in ssemly line lning, Intelligent Mnufturing, vol. 9, 008, pp [8] R. Mtousek n L. Nolle, GAHC: Improve Geneti Algorithm, Pro. Worl Congress on Engineering n Computer Siene, USA, 007, pp [9] Z. Q. Chen, R. L. Wng, n K. Okzki, An Effiient Geneti Algorithm Bse Approh for the Minimum Grph Bisetion Prolem, Computer Siene n Network Seurity, vol.8, 008. [0] O. A. Jn, D. Rjmni, n C. R. Ro, Improve Seletion Opertion for GA, Theoretil n Applie Informtion Tehnology, vol. 4, 008, pp [] R. Kumr, K. Izui, n Y. Mstk, Multilevel Reunny Allotion Optimiztion Using Hierrhil Geneti Algorithm, IEEE Trnstions on Reliility, vol. 57, 008, pp [] F. Jimes-Romero, n D. Munoz-Roriguez, Evolutionry serhing in ellulr rio systems plnning, Europen Trnstions on Teleommunitions, vol. 0, 999, pp [3] L. Di Gspero n A. Sherf, Multi-neighourhoo lol serh with pplition to ourse timetling, Pro. 4th Interntionl Conferene on Prtie n Theory of Automte Timetling, Belgium, vol. 740, 00, pp [4] Y. J. Co n H. Q. Wu, A Cellulr Automt Bse Geneti Algorithm n its Applition in Mehni Design Optimiztion, Interntionl Conferene on Control, 998. [5] Z. Ming n S. Shuong, Theory of Geneti Algorithm n Its Applition, Ntionl Defense Inustry pulishing Compny, Beijing; 999, pp. 5-7.