An Improved Clustering Based Genetic Algorithm for Solving Complex NP Problems

Transcription

1 Journal of Computer Scence 7 (7): , 2011 ISSN Scence Publcatons An Improved Clusterng Based Genetc Algorthm for Solvng Complex NP Problems 1 R. Svaraj and 2 T. Ravchandran 1 Department of CSE, Velalar College of Engneerng and Technology, Erode, Inda 2 Hndusthan Insttute of Technology, Combatore, Inda Abstract: Problem statement: The selecton process s a major factor n genetc algorthm whch determnes the optmalty of soluton for a complex problem. The selecton pressure s the crtcal step whch fnds out the best ndvduals n the entre populaton for further genetc operators. The proposed algorthm tres to fnd out the best ndvduals wth reduced selecton pressure than standard genetc algorthm whch s commonly used. Approach: The selecton process s refned n the proposed algorthm by usng the concept of clusterng rather than tradtonal selecton mechansms lke Roulette wheel selecton, Rank selecton, Tournament selecton. Results: As the selecton process s mproved n our approach, the convergence velocty of the genetc algorthm s mproved by reachng the optmal soluton quckly and the optmalty of the soluton s also fne-tuned. Concluson: A new varant of the standard genetc algorthm s proposed whch reduces the executon tme of the algorthm by gearng up the selecton process to reach the most effcent soluton. The ft ndvduals are selected for crossover and mutaton n all generatons thereby reachng the soluton wthout much complex process. Keywords: Genetc algorthm, clusterng algorthm, selecton pressure, crossover and mutaton INTRODUCTION Genetc algorthms are adaptve heurstc search algorthms based on the evolutonary concept of natural selecton and genetcs. It follows the Darwn s prncples of Survval of the fttest where the ft best ndvduals retan ther postons overtakng the weaker ndvduals n a group whch they compete for the lmted resources. Genetc algorthm s robust when compared to other searchng mechansms. Even though t s generally sad to be random process, t s not actually random. Instead, t chooses the best ndvduals n each teraton thereby movng fast towards the stable optmal soluton from the ntal random populaton of ndvduals chosen. In genetc algorthm, the potental soluton to a problem can be represented by a set of parameters called genes and the genes are combned together to form a structure called chromosome. N chromosomes are collectvely called as populaton. In genetc terms, genes are called as genotype and chromosomes are called as phenotype. Intally, the chromosomes n the populaton are chosen at random. It then apples recombnaton genetc operators to these structures so as to proceed towards fnal soluton. By calculatng the ftness value for all chromosomes, t evaluates these structures and allocates reproductve opportuntes n the next generaton n such a way that those chromosomes whch provde a better soluton to the target problem are gven more chances to reproduce than those chromosomes whch represent poorer solutons. The goodness of a soluton s typcally determned wth respect to the current populaton. Genetc algorthms are usually seen as functon optmzers although the range of problems and areas to whch genetc algorthms have been appled s very broad. They are wdely used n solvng many complex Non-Determnstc Polynomal (NP) problems n dfferent domans whose tme effcency cannot be specfed n polynomal tme (Patvchachod, 2011; Kannaah et al., 2011). The general outlne of the standard genetc algorthm (Goldberg, 1989) s gven below: Choose the ntal populaton of ndvduals at random Evaluate the ftness of each ndvdual n that populaton usng objectve functon Repeat the followng steps n each generaton untl termnaton crteron (tme lmt, suffcent ftness acheved, fxed no of teratons) s acheved Select the best-ft ndvduals for crossover Breed new ndvduals through crossover and mutaton operatons to gve brth to offsprng Correspondng Author: R. Svaraj, Department of CSE, Velalar College of Engneerng and Technology, Erode, Taml Nadu, Inda 1033

2 Evaluate the ftness value of all new ndvduals Replace least-ft ndvduals wth new hgh ft ndvduals The process of representaton of chromosomes n terms of genes s called as encodng. There are many types of encodng technques lke bnary encodng, value encodng, permutaton encodng. They dffer n the way the genes are represented and used for processng. The basc steps nvolved n GA are Intalzaton, Selecton, Reproducton and Termnaton (Kalyanmoy, 2004). Ftness functon: A ftness functon must be formulated for each problem that s to be solved by genetc algorthm. For a partcular chromosome, a ftness functon or objectve functon returns a sngle numerc ftness value or fgure of mert whch s proportonal to the utlty or ablty of that chromosome n the entre populaton consstng of n chromosomes. For many problems lke functonal optmzaton, objectve functon value s enough to attan a soluton. But for complex combnatoral problems, a combnaton of performance measures relatng to that specfc problem wll only drve towards the optmal soluton. Selecton: Parents for ths reproducton (matng or crossover) are selected usng some mechansms (Goldberg, 1989) lke Roulette Wheel selecton, Rank selecton, Truncate selecton, Tournament selecton, Boltzmann selecton. Although all selecton mechansms (Svaraj and Ravchandran, 2011) have the fnal target of choosng the best chromosomes for the next crossover phase, they dffer n the way they use to evaluate them. They retan the best ndvduals over the teratons by replacng the worst ndvduals whch have the low probablty of beng carred over to the next generatons. Genetc recombnaton operators: Although many genetc recombnaton operators are avalable, the commonly used ones are crossover and mutaton Fg. 1 and 2. Reproducton (Crossover): Durng Reproductve phase, the chromosomes wth hgh ftness values are recombned to form new chromosomes whch consttute the ndvduals for the next generaton. Crossover takes two ndvduals, cuts them at random postons to gets head and tal segments. The tal segments are then swapped between two parents to form two new full length chromosomes or offsprngs. The offsprngs thus produced nhert the genes and ther characterstc from both parents whch may eventually turn out to be the optmal soluton when they are subjected to the same process n further generatons. J. Computer Sc., 7 (7): , 2011 Fg. 1: One pont crossover Mutaton s the process of randomly flppng a bt n the entre chromosome wth some fxed small probablty. It s done to ntroduce some form of dversty among the chromosomes wthout sacrfcng the characterstcs of the parents. In the process of genetc algorthm, sub-optmal solutons may be attrbuted to selecton pressure n tradtonal selecton mechansms whch gnore the weaker ndvduals to a larger extent. If they are gven some better chance of survval n the next generaton, there wll be mproved chance for them to converge to an optmal soluton. The proposed approach s a new varant of the standard genetc algorthm whch ncorporates the advantageous feature of clusterng n selecton process of genetc algorthm. MATERIALS AND METHODS In many studes, to cluster smlar objects usng k- means clusterng algorthm, genetc algorthm s used (Maulk and Bandyopadhyay, 2000; Twar et al., 2010). But our approach s proposed n a new drecton whch n order to mprove the effcency of the fnal global optmal soluton of genetc algorthm k-means clusterng s ncorporated wthn. K means clusterng: Clusterng s an unsupervsed learnng mechansm used to group smlar objects nto clusters. Although dfferent clusterng technques are avalable, there s no general strategy that works equally well n dfferent problem domans. However, t s better to use some smpler clusterng mechansm whch runs more number of tmes rather than complex mechansms whch needs to be run only once. K-means clusterng has been a very popular technque for parttonng large data sets wth numercal attrbutes. It s classfed as a parttonal or non-herarchcal clusterng method. It s defned as follows: Gven a set D = {X1,..., Xn} of n numercal data objects, a user defned natural number k n and a dstance measure d, the k-means algorthm s amed at fndng a partton C of D nto k non-empty dsjont clusters C 1,...,C k 1034

3 k where C C j = Φ and U= 1 C = D such that the overall sum of the squared dstances between data objects and ther cluster centers s mnmzed. The basc step of k-means clusterng s smple. In the begnnng, the number of clusters K s defned and the centrod or center of these clusters. Any random objects can be taken as the ntal centrods or the frst K objects n sequence can also serve as the ntal centrods. Then the K means algorthm wll do the three steps below untl convergence: Iterate the followng steps untl the clusters become stable (no objects move among groups): Determne the centrod coordnate Determne the dstance of each object to the centrods Group the object based on mnmum dstance The dstance between objects can be calculated by usng Hammng dstance, Manhattan dstance or Mnkowsk dstance. To show the performance of the proposed clusterng genetc algorthm, a complex NP hard 0/1 knapsack problem s chosen. The Knapsack problem s an example of a combnatoral optmzaton problem, whch seeks for a best soluton from among many solutons. It s concerned wth a knapsack that has postve nteger volume (or capacty) V. There are n dstnct tems that may potentally be placed n the knapsack. Item has a postve nteger volume (or capacty) V and postve nteger beneft (or proft) B. Let X denotes how many copes of tem are to be placed nto the knapsack. In 0/1 knapsack problem, only one copy of an tem can be placed n the knapsack. Hence X s always equal to 1 for all the tems selected. The prmary goal s to: Maxmze: N = 1 B X J. Computer Sc., 7 (7): , 2011 Table 1: Example of knapsack problem soluton Volume Beneft of Item A Item B Item C of the set the set or degrades) N = 1 Subject to the constrants: V X V Example of 0-1 knapsack problem: Suppose there s a knapsack that has a capacty of 14 cubc nches and several tems of dfferent volumes and dfferent benefts. The prmary objectve s to nclude n the knapsack only those tems that wll have the greatest total beneft that ft wthn the knapsack s capacty. There are three potental tems (labeled A, B, C ). Ther volumes and benefts are as follows: Item # A B C Beneft Volume For ths problem there are 2^3 =8 possble subsets of tems. In order to fnd the best soluton, a subset that meets the constrant and has the maxmum total beneft has to be dentfed. Table 1 clearly shows that n ths example, only 7 th row (110) satsfes the constrant. Hence, the optmal beneft for the gven constrant (V = 14) can only be obtaned wth one quantty of A, one quantty of B and zero quantty of C and t s 10.If number of tems s less, soluton can be easly found. If t s too large to apply smple algorthms to fnd the best soluton, some optmzaton technques lke Genetc algorthm has to be appled. Proposed methodology: The proposed Clusterng Genetc Algorthm (CGA) tres to reduce the selecton pressure of the genetc algorthm by combnng the features of k means clusterng. The ndvduals n the ntal populaton are taken at random. The ftness values of the ndvduals are calculated. Then k means clusterng s ncorporated. The ntal centrods are taken at random. The dstance between the ndvduals are calculated where the ftness value s taken to calculate the smlarty measure. The smlar ndvduals are grouped nto a cluster. Smlarly k clusters are formed wth each cluster havng ndvduals wth smlar ftness values. Tradtonal selecton mechansm s used wthn each cluster to select parents for crossover and mutaton. After the applcaton of these genetc recombnaton operators, the ndvduals for the next teraton are produced. In the next teraton, agan k means clusterng s used to change the cluster centers and the genetc algorthm steps are contnued untl termnaton crteron s satsfed. The ndvduals mgrate to other clusters whch has mnmum dstance to cluster center f ts ftness value changes (mproves

4 J. Computer Sc., 7 (7): , 2011 Fg. 3: Impact of varyng the number of clusters Fg. 2: Outlne of the proposed clusterng genetc algorthm RESULTS The CGA s run wth dfferent genetc parameters and the results are analyzed for 200 tems. The selecton mechansm chosen for mplementaton s Tournament selecton wth 10% eltsm and one pont crossover s chosen for matng. Tournament selecton s the process of conductng tournaments for n ndvduals and the wnner of each tournament s chosen as parent. Results n the lterature (Ztzler et al., 2000) show clearly that eltsm can speed up the performance of the GA sgnfcantly; also t helps to prevent the loss of good solutons once they have been found. Hence eltsm s combned wth selecton mechansm to prevent the loss of good solutons for the next generatons once they have been found. The Crossover probablty P c and Mutaton probablty P m are chosen as 80% and 1% respectvely. Fg. 4: Impact of varyng cluster sze Impact of varyng cluster sze: The number of chromosomes or ndvduals for each cluster s vared and the results are reported. The ndvduals wthn a cluster should have hgh coheson such that ther smlarty should be more. If number of ndvduals grouped n a cluster ncreases, ther smlarty decreases and ths wll eventually affect the performance of the fnal soluton. The performance mprovement of CGA comparng to SGA s shown n Fg. 4 whch shows that f cluster sze s low, CGA tends to perform better. Impact of varyng number of clusters: In k means Impact of varyng populaton sze: The populaton clusterng, the value of k s user defned and the value sze chosen wll also drectly mpact the optmalty of chosen for k wll have ts heavy mpact on the result of the soluton. If the populaton sze s set too low, t wll clusterng. If the number of clusters s very low, the soon converge to a local optmum. If t s set too hgh, t dversty of ndvduals cannot be acheved. If t s very wll unnecessarly process all the ndvduals whch large, t wll create an unnecessary overhead. So t take more tme to execute. So, the populaton sze should be chosen accordngly for the specfc problem. should be chosen carefully. As shown n Fg. 5, the In ths mplementaton, the number of clusters (k) s results clearly show the superor performance of CGA vared from 3 to 6 and the results are compared to show when the populaton sze s vared. In CGA, the the performance of CGA. As shown n Fg. 3, CGA populaton sze wll have lttle mpact on the effcency results n hgh proft when compared to mplementaton of the fnal soluton compared to SGA and t has overall of SGA wth dfferent number of clusters. better performance than SGA. 1036

5 J. Computer Sc., 7 (7): , 2011 REFERENCES Fg. 5: Impact of varyng populaton sze DISCUSSION The results clearly shows that the proposed Clusterng Genetc Algorthm has reached better proft by selectng optmal subset of tems satsfyng the constrants. The fnal soluton space of the Genetc Algorthm crtcally depends upon the genetc parameters chosen. In K-means clusterng also the value of k to be chosen s a crtcal factor to be determned. Hence CGA s run by changng these operators. Fgure 3-5 depcts that CGA shows better performance wth hgh proft than SGA wth varyng genetc parameters lke number of clusters, cluster sze populaton sze. CONCLUSION CGA shows much greater performance than standard genetc algorthm wth dfferent genetc parameters chosen. It ntroduces a checkpont pror to selecton mechansm n each generaton by the usage of k-means clusterng algorthm. It thus helps n selectng only the best chromosomes to be carred over to further generatons and also helps n reducng the pressure for selecton process. However, to gan better performance a compromse should be made for the tme t takes to complete a generaton because of the ncluson of clusterng. When the fnal optmalty of the soluton s consdered, ths s neglgble compared to the SGA performance whch operates on the entre populaton all the tmes. Hence wth mproved performance of CGA, many complex NP problems n dfferent domans and wth dfferent functonaltes can be easly solved. Goldberg, D.E., Genetc Algorthms n Search, Optmzaton and Machne Learnng. 1st Edn., Addson-Wesley, Readng, Massachusetts, ISBN- 10: , pp: 432. Kalyanmoy, D., Optmzaton for Engneerng Desgn: Algorthms and Examples. 1st Edn., Prentce-Hall of Inda, New Delh, ISBN- 10: X, pp: 396. Kannaah, S.K., J. Thangavel and D.P. Kothar, A genetc algorthm based mult objectve servce restoraton n dstrbuton systems. J. Comput. Sc., 7: DOI: /jcssp Maulk, U. and S. Bandyopadhyay, Genetc algorthm based clusterng technque. Patt. Recog., 33: &rep=rep1&type=pdf Patvchachod, S., An mproved genetc algorthm for the travelng salesman problem wth mult-relatons. J. Comput. Sc., 7: DOI: /jcssp Svaraj, R. and T. Ravchandran, A revew of selecton methods n genetc algorthms. Int. J. Eng. Sc. Technol., 3: Twar, A.K., L.K. Sharma and G.R. Krshna, Entropy weghtng genetc k-means algorthm for subspace clusterng. Int. J. Comput. Appl., 7: DOI: / Ztzler, E., K. Deb and L. Thele, Comparson of multobjectve evolutonary algorthms: Emprcal results. Evolut. Comput., 8: DOI: /