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1 Proceedngs of the 007 IEEE Swarm Intellgence Symposum (SIS 007) Probablstcally rven Partcle Swarms for Optmzaton of Mult Valued screte Problems : esgn and Analyss Kalyan Veeramachanen, Lsa Osadcw, Ganapath Kamath epartment of Electrcal Engneerng and Computer Scence 77, Lnk Hall Syracuse Unversty, Syracuse NY 344 kveerama, laosadc,gkamathh@syr.edu Abstract A new partcle swarm optmzaton (PSO) algorthm that s more effectve for dscrete, mult-valued optmzaton problems s presented. The new algorthm s probablstcally drven snce t uses probablstc transton rules to move from one dscrete value to another n the search for an optmum soluton. Propertes of the bnary dscrete partcle swarms are dscussed. The new algorthm for dscrete mult-values s desgned wth the smlar propertes. The algorthm s tested on a sute of benchmarks and comparsons are made between the bnary PSO and the new dscrete PSO mplemented for ternary, quaternary systems. The results show that the new algorthm s performance s close and even slghtly better than the orgnal dscrete, bnary PSO desgned by Kennedy and Eberhart. The algorthm can be used n any real world optmzaton problems, whch have a dscrete, bounded feld. I. INTROUCTION Many real world optmzaton problems, lke desgn optmzaton n transstor szng problems, are dscrete and mult-valued. Any feld, whose values are dscrete, need a dscrete, mult-valued optmzaton algorthm. There s an ncreasng need to develop these types of algorthms as the uses for optmzaton algorthms grow. In ths paper, a new algorthm based on partcle swarm optmzaton s presented. We analyze the performance of the dscrete, bnary PSO developed by Kennedy, et al.,[8] to entfy the crtcal propertes needed by dscrete PSOs. We desgn a partcle swarm for the dscrete, mult-valued optmzaton problems that exhbts the same crtcal propertes as the orgnal dscrete, bnary PSO. The new algorthm s performance s tested usng benchmarks adapted from Lang et al. []. The partcle swarm optmzaton algorthm, orgnally ntroduced n terms of socal and cogntve behavor by Kennedy and Eberhart n 995 [], can effectvely solve optmzaton problems n many felds, especally engneerng and computer scence. The power n the technque s ts farly smple computatons and sharng of nformaton wthn the algorthm as t derves ts nternal communcatons from the socal behavor of ndvuals. The ndvuals, called partcles henceforth, are flown through the multmensonal search space wth each partcle representng a possble soluton to the multmensonal problem. Each soluton s ftness s based on a performance functon related to the optmzaton problem beng solved. The movement of the partcles s nfluenced by two factors usng nformaton from teraton-to-teraton as well as partcle-to-partcle. As a result of teraton-toteraton nformaton, the partcle stores n ts memory the best soluton vsted so far, called pbest, and experences an attracton towards ths soluton as t traverses through the soluton search space. As a result of the partcle-to-partcle nformaton, the partcle stores n ts memory the best soluton vsted by any partcle, and experences an attracton towards ths soluton, called gbest, as well. The frst and second factors are called cogntve and socal components, respectvely. After each teraton the pbest and gbest are updated for each partcle f a better or more domnatng soluton (n terms of ftness) s found. Ths process contnues, teratvely, untl ether the desred result s converged upon, or t s determned that an acceptable soluton cannot be found wthn computatonal lmts. The PSO formulae defne each partcle n the - dmensonal space as X x, x,..., x ), where ( the subscrpt represents the partcle number, and the second subscrpt s the dmenson. The memory of the prevous best poston, pbest, s represented as P p, p,..., p ) and velocty as ( ( v, v,..., v ) V. After each teraton, the velocty term s updated nfluenced by both ts own best poston, P, and the global best poston, P g. The velocty update equaton s ( t ) V + ( t ) ( t) ( t) ω V [0,] + U ( p x ) + () ( t ) ( t ) U[0,] ( p x ) gd /07/$ IEEE 4

2 Proceedngs of the 007 IEEE Swarm Intellgence Symposum (SIS 007) where U[0,] s a sample from a unform random number generator, t represents a relatve tme ndex, s a weght determnng the mpact of the prevous best soluton, and s the weght on the global best soluton s mpact on partcle velocty. The next soluton to test s ( t+ ) ( t ) ( t+ ) X X + V. () The next soluton n () s defned n a contnuous valued soluton space. Kennedy and Eberhart [8] desgned a dscrete verson of the algorthm, whch s sgnfcantly dfferent from the contnuous verson. The algorthm uses the same velocty update equaton but the values for the soluton or X are now dscrete and bnary. Kennedy, et al., preserved the socal and cogntve learnng components n the algorthm but changed the partcle soluton updatng. In the next secton, the motvaton for desgnng a partcle swarm algorthm for dscrete mult-valued optmzaton problems s presented. Analyss of probablstc transtons made n the bnary PSO s presented n Secton 3. The new PSO algorthm for dscrete mult-valued optmzaton problem s presented n Secton 4. In Secton 5, the benchmarks and the expermental settngs are descrbed. Results are presented n Secton 6 followed by conclusons and future work n Secton 7. II. MOTIVATION : ISCRETE MULTI VALUE PARTICLE SWARM OPTIMIZATION Recently, there has been an ncreasng nterest n developng partcle swarm optmzaton based algorthm for dscrete mult-valued optmzaton problems [, 3]. Many real world optmzaton problems have dscrete varable values. It can be argued that dscrete varables can be transformed nto an equvalent bnary representaton, and the bnary PSO can be used. However, the range of the dscrete varable often does not match the upper lmt of the equvalent bnary representaton. For example, a dscrete varable of range [0,,,3,4,5] requres a three bt bnary representaton, whch ranges between [0-7]. Thus, specal condtons are requred to manage the values past the orgnal range of the dscrete varable. Secondly, the Hammng dstance between two dscrete values undergoes a nonlnear transformaton when an equvalent bnary representaton s used nstead. Ths often adds complexty to the search process. The thrd reason s that the bnary representaton ncreases the dmensons of the partcle. For these reasons, an extenson to the orgnal dscrete, bnary PSO convertng t to a dscrete multvalued PSO s necessary. Prevously researchers have attempted to enhance the performance of the bnary PSO. Al Kazem, et al. [0], mproves the orgnal bnary PSO algorthm by modfyng the way partcles nteract. The research on algorthms that optmze dscrete, mult-valued problems, however, s sparse. Wth the new algorthm, any dscrete mult-valued problem can be optmzed usng partcle swarms wthout convertng the problem nto equvalent bnary representatons. The algorthm s propertes and probablstc rules are ntroduced and dscussed n Secton IV. The movement of the partcles through soluton value updates remans probablstc for performance reasons. Results are presented for a sute of 5 benchmark problems adapted from []. III. PROPERTIES AN ANALYSIS OF ISCRETE BINARY PSO Kennedy and Eberhart desgned a dscrete verson of the PSO algorthm. The algorthm uses the same velocty update equaton as n () but the values of X are now dscrete and bnary. For poston update, frst the velocty s transformed nto a [0, ] nterval usng the sgmo functon gven by S sg( V ) (3) + where V e V s the velocty of the th partcle s d th dmenson. A random number s generated usng a unform dstrbuton whch s compared to the value generated from the sgmo functon and a decson s made about the X n the followng manner. X u( S U[0,]) (4) u s a unt step functon. The decson regardng now probablstc, mplyng that hgher the value of the V, hgher the value of the S, makng probablty of decng for X hgher. As V S makng t unlkely that X s, then X wll become zero agan. Fgure shows ths property of the probablty, X ncreases as V ncreases, n the bnary PSO. However, P( X ) s almost equal to for V >0 but not equal to. Ths s the key to the desgn of the dscrete bnary PSO, snce ths prevents partcles from gettng stuck once. However, t rases some mportant questons about the velocty. Wll the velocty of a partcle tend toward nfnty f the local optmum s at (,) for (P, P gd )? What parameter values result for hgher veloctes? Can we analyze swarm s behavor for such condtons? Fnally, can we desgn a smlar model for a dscrete mult-valued optmzaton problem? 4

3 Proceedngs of the 007 IEEE Swarm Intellgence Symposum (SIS 007) Fgure. Probablty of X and X 0 gven the V In the velocty update functon () n the Bnary PSO, one or both terms of nfluence become equal to zero dependng on the dfference between the current soluton, local best soluton, P, and the global best soluton, P gd. When the nfluences are not to zero, the number added to the prevous velocty ( ω V ) comes from standard dstrbutons. Unlke the contnuous PSO, these dstrbutons do not get scaled, nor are ther propertes changed over teratons. Ths makes analyss of bnary PSO easer than the contnuous PSO. In the next secton, four dfferent cases and dstrbutons used n the velocty equaton () are descrbed. Ozcan, et al., [4], presented the theoretcal framework for the analyss of a contnuous PSO. A smple partcle s behavor n one-dmensonal space s analyzed leadng to a concluson that the partcle follows a snusoal path, when gbest and pbest are fxed. In ths paper, the dscrete bnary partcle swarm optmzaton algorthm s analyzed for fxed gbest and pbest. Snce the values of the pbest and gbest are bnary, there are four cases. We generate the probablty densty of velocty for these dfferent cases analyzng the behavor of a sngle partcle n a one-dmensonal space. The probablty densty of velocty s formed through Monte Carlo runs. Frst the algorthm s run by startng the algorthm at dfferent veloctes [-0,0] and s run for 000 teratons each tme for a fxed case.e., fxed gbest and pbest. Ths captures the steady state behavor of the partcle. Ths set of runs can be repeated formng an average probablty densty of the velocty for each specfc case. The probablty densty of velocty s analyzed for dfferent values of, and ω n the four cases. A. 4 Case Analyss Conserng a sngle dmenson and a sngle partcle. Let us drop the notaton n () as mples partcle number and d dmenson number. Four cases occur n the dscrete bnary PSO wth each causng certan behavors n the velocty. Pbest s now ( t) P; gbest s now G for smplcty. X s the current poston of the partcle. The analyss assumes parameter values of,, 0 ω. The generalzed form of velocty update equaton for dscrete bnary PSO s gven by ( t+ ) ( t ) V ω V + (5) s gven by U[0,] ( P X ) + U[0,] ( G X ) (6) and s a random number that comes from dfferent dstrbutons for dfferent cases. ) Case : P0; G; ependng upon the current poston of the partcle one of the two nfluences become zero. Thus, only one nfluence remans n the velocty update equaton. When ( t) X, s a random number generated from the ( t ) unform dstrbuton U[,0]. When X 0, t s a random sample generated from unform dstrbuton U[0, ]. In ths case, the partcle wll swng between values 0 and untl G0. The partcle then falls nto case 4. Fgure shows the probablty densty of velocty when gbest s G and pbest s P0. The two varables, [, ], are vared to analyze the affect. The probablty densty of the velocty s centered around zero, and an ncrease n [, ] ncreases the standard devaton of the probablty densty of the velocty but stll centers around zero. The statstcs collected for the poston of the partcle revealed a 50-50% dstrbuton for a value of X and X0. Change n, d not affect ths dstrbuton. ) Case : P; G0; Ths case s smlar to Case wth the partcle wll swngng between values of 0 and untl G. The partcle then behaves as n 3 rd case. In case P0 the partcle behaves as n 4 th case. 3) Case 3 : P; G The velocty changes when the partcle value s 0, t X 0 and U[0,] + U[0,] (7) Hence, the convoluton of two unform dstrbutons yelds the trangular dstrbuton (when ) that s 0 < f ( ) (8) + < + ( t), when X, the random term s smply, 0 and velocty gets scaled by ω. 43

4 Proceedngs of the 007 IEEE Swarm Intellgence Symposum (SIS 007) Fgure. Probablty densty of the velocty for dfferent, for Case. Ths case s mportant to analyze snce due to the sgmo functon there s always a probablty that the partcle s current poston wll become 0 ncreasng the velocty of the partcle whose P stays at and G at. The probablty densty for the velocty s plotted for dfferent, n Fgure 3. It can be seen as, ncreases the probablty densty of the velocty spreads over a wer range of veloctes. Hence by controllng the values of, one can control the values of the velocty. Three dfferent cases of, are shown n the fgure. The postons of the partcle had a 75% and 5 % dstrbuton for X and X0 respectvely, for, as shown n Fgure 4. When, are ncreased to 4 the postons of the partcles had a dstrbuton of 86.8% and 3.% for X and X0 respectvely. Increasng the, stablzes the postons of the partcles snce there are hgher veloctes that are possble. Smlar analyss can be done to demonstrate the affect of ω on the probablty densty of velocty. 4) Case 4: P0; G0 The velocty s only changed n ths case when the t current value of the partcle poston,.e., X and s a random number generated from the trangular dstrbuton n (8). When 0. ( t ) X 0, the random value s Ths analyss shows that the probablty of hgher veloctes s very low even for hgh values of. 4 Fgure 3. Probablty densty of veloctes for the Case 4 for dfferent,. IV. ISCRETE MULTI VALUE PARTICLE SWARM OPTIMIZATION For dscrete mult valued optmzaton problems the range of the dscrete varable values between [0 M-], where M mples the M-ary number system. The same velocty update and partcle representaton are used n the algorthm as for the bnary valued PSO. The poston update equaton s however changed n the followng manner. The velocty s transformed nto a number between [0, M] usng the sgmo transformaton, M S (9) + V e Fgure 4. Hstogram of the current poston X for Case 3 for ; ω 0.8 A number s generated usng the normal dstrbuton wth parameters N ( S, σ ( M )). The result s rounded to X round ( S + ( M ) σ randn()) (0) If X M, X > M 0, X < 0 44

5 Proceedngs of the 007 IEEE Swarm Intellgence Symposum (SIS 007) The velocty update equaton remans the same as (). The postons of the partcles are dscrete values between [0, M-]. Note that for any gven S there s a probablty for choosng a number between [0, M-]. However, n ths paper, the probablty of selectng a number decreases based on ts dstance from S. In the followng subsecton the relatonshp between S and the probablty of pckng dscrete values s gven. A. Probablty of a dscrete value m For a partcular S, the probablty of a dscrete varable becomng assgned to a value of m s dscussed n ths secton. For m 0, the probablty s 0.5 P( X 0 S ) g( x) dx () 0.5 S Q σ ( M ),where, Q s the error functon and the functon, g, s g ( x ) exp ( ) πσ ( M ) σ ( M ) x S () For m n the range to M, the probablty s m+ 0.5 P( X m S ) g( x) dx (3) or m 0.5 m 0.5 S m S Q Q σ ( M ) σ ( M ) For m M-, the probablty s P( X ( M ) S ) g( x) dx (4) or ( M ) 0.5 ( M ) 0.5 S Q σ ( M ) Of course, the sum of the probablty s always M P( X m / S ) (5) m 0 Fgure 5 shows the probablty of varous dscrete varables for dfferent S values. The fgure s for a ternary system usng a σ of 0.5 or standard devaton for the normal dstrbuton of. In Fgure 6, the plot s shown for σ 0.or equvalently standard devaton of 0.. As σ value decreases, one gets curves wth sharper peaks for the probablty of dscrete value gven S. As σ 0 the algorthm wll smply round the S value to determne the dscrete value. If a hgher σ s used, the algorthm approaches a unform dstrbuton. In the new algorthm, an addtonal parameter σ s ntroduced. The settng of ths σ s crtcal to the algorthms performance. Emprcal results show that a sgma of 0. s a good choce for ternary system. B. Case Analyss for a Ternary System For a ternary system, 9 cases exst for dfferent sets of pbest and gbest. ue to lmtatons of space we show the prelmnary analyss for one case, (P, G). Snce P, G, the velocty update equaton becomes ( t+ ) ( t ) V ω V + (6) The dstrbuton s trangular for both X values, X and X0. However, the trangular dstrbuton for X0 has twce the support range when compared to the trangular dstrbuton for X. The probablty densty of the velocty s plotted for dfferent values of (, ) n the Fgure 7. The probablty densty of the velocty spkes near and, eventually, decreases as the velocty ncreases. Wth an ncrease n (, ), a longer tal appears on the dstrbuton ndcatng hgher veloctes beng selected. Smlar to the bnary PSO, one controls the veloctes usng the parameters, (, ) n the new algorthm. Fgure 6. Affect of Selecton of Omega on the Probablty of dfferent values for a dscrete varable gven S. Ths example s shown for a trenary system, M3, 0. σ Fgure 5. Probablty of dfferent dscrete varables as S vares between the lmts of [0 M]. Ths example s shown for a ternary system, M3, and σ 0.5 The new algorthm desgned for dscrete multvalued optmzaton problems s an extenson of the bnary PSO so has smlar propertes and reacts to 45

6 Proceedngs of the 007 IEEE Swarm Intellgence Symposum (SIS 007),. In the next secton, the new algorthm s tested usng dfferent benchmark problems. Fgure 7. Probablty densty of veloctes for the ternary system for dfferent,. V. EXPERIMENTAL SETUP AN BENCHMARKS Fve benchmark problems are used from Lang, et al., [] for comparng the bnary dscrete PSO algorthm wth the mult-valued dscrete PSO. For each benchmark, the algorthm executes for 5 trals wth 0 dmensons of each problem. For more detals about the benchmarks, the reader s referred to []. A bref descrpton of the benchmark problems s gven n the followng subsecton. A. Benchmark Problems The fve-benchmark functons are defned n ths secton wth nformaton concernng ther optma as n []. ) Shfted Rotated Ackley s Functon wth Global Optma on Bounds f ( x ) 0 exp( 0. x ) exp( cos( π x )) e + f _ bas ) Shfted Rastrgn s Functon ( ) ( 0cos( π ) + 0) + _ f x x x f bas 3) Shfted Rotated Rastgrn s Functon 3( ) ( 0cos( π ) + 0) + _ f x x x f bas 4) Shfted Rotated Weerstrass Functon f ( x ) 4 k max k k ( a cos( π b ( x + 0.5)) ) k 0 k max k k a cos(π b 0.5) + f _ bas k 0 5) Schwefel s Problem.3 5( ) ( ( )) + _ f x A B x f bas Where, and A ( a snα + b cos α ) j j j j j B ( a sn x + b cos x ) j j j j j For,, A and B are two x matrces, a, b are nteger random numbers generated n the j j range [-00, 00], α [ α, α... α ], α j are random numbers generated from [ π, π ]. B. Samplng the Search Space for fferent screte omans The dscrete mult-valued PSO s appled to dfferent bases, bnary, ternary and quaternary, and tested usng standard ftness functons orgnally desgned for contnuous functons. Each dmenson n the orgnal functon has a recommended search space range. A procedure s used to transform these functons nto dscrete doman. In the followng table, we defne terms used n ths and followng sectons. Table : Term efntons for Benchmarks Term efnton menson menson of the orgnal contnuous benchmark problem Base The base of the number system (e.g. bnary base ) gts The length of the numercal character strng used n the number system (e.g. bnary dgts,.e., [0,]) Each orgnal dmenson s represented usng 6 bts for a bnary based system or 8 quaternary dgts whch results n dscrete values each. In a ternary based system, 0 dgts would only result n steps. Thus, these representatons have resulted n samples of search space for bnary and quaternary bases, and samples for ternary. Hgher samplng of ftness landscape proves more nformaton about the landscape leadng to better performance of the algorthm. As the samples tend toward nfnty, we approach the contnuous doman agan. Hence, t s pertnent for comparsons that the same samples of the ftness landscape are proved to all the algorthms. Farness s acheved by samplng the exact same ponts for all bases. The base that yelds the smallest number of steps, whch s 3 fxes the sample step sze. The addtonal samples for the other bases are adjusted to have the value at the upper or lower range of the problem. It s assumed that the partcles won t dwell sgnfcantly n these regons of the landscape. C. Partcle Swarm Settngs The benchmarks are compared usng the same parameter settngs. An equal weght of and 46

7 Proceedngs of the 007 IEEE Swarm Intellgence Symposum (SIS 007) s used. A tme varyng nerta, ω, s used as descrbed n [4, 5]. For an teraton, the value s gven by ( ω 0.4) ( no. of Iteratons ) ω( ) (7) no. of Iteratons 0.4, where, ω 0.9. The number of partcles used n the smulatons s 0 and the number of teratons s equal to VI. RESULTS A. Affect of σ on the screte PSO algorthm One parameter that controls the new algorthm s σ. It s shown n fgures, 5 and 6, that hgher σ flattens out the probabltes of the three dscrete values. The resultng probabltes for σ 0.5 are gven n fgure 5. If the σ s further ncreased, the probablty of each dscrete value for a gven S wll eventually become 0.33 for a ternary system. Ths makes the curves appear nearly flat and algorthm completely random and defeats the purpose of usng a swarm-based algorthm. A σ 0.4 or less s a better choce for the new algorthm. In ths secton, results acheved for ternary system usng σ 0.4 and σ 0. are shown. The results are shown for Functon descrbed n prevous secton. Smlar performance results occur wth the other functons descrbed n secton V.A.. Fgure 8 shows that a σ 0. produced better results for functon. Hence n ths paper, a σ 0. s used for a ternary system. B. Results wth σ 0. In ths secton, the results for the fve-benchmark problems are presented. The new algorthm, desgned for hgher number systems, performed better than the bnary PSO for all the benchmark problems as can be seen n fgures 9, 0,,, 3. Varyng sgma sgnfcantly affects the performance of the algorthm. 0. seems to be reasonable choce for sgma, for a ternary system. The sgma has to be vared for dfferent number systems,.e., ternary, quaternary and so on. A sgma of 0. has been used for quaternary system n ths paper. Table presents the statstcs for dfferent algorthms for the 5 trals that were performed. Ternary PSO performed equvalent to the Bnary PSO or even slghtly better. Quaternary PSO performed better than the bnary and ternary PSO. These results are sgnfcant and allow the new algorthm to be used for number systems other than bnary. Fgure 8. Comparson of mnma acheved for dfferent σ used for an algorthm desgned for a ternary system for Functon. Fgure 9. Mnma acheved (averaged over 5 trals) for a 0 dmensonal Shfted Rotated Ackley s Functon Fgure 0. Mnma acheved (averaged over 5 trals) for a 0 dmensonal Shfted Rastrgn s Functon 47

8 Proceedngs of the 007 IEEE Swarm Intellgence Symposum (SIS 007) The new algorthm wll be tested on deceptve functons; trap functons desgned for hgher ordered number systems. Table : Averaged Results for fferent Functons Bnary PSO Ternary PSO Quaternary PSO f Mean Std Mean Std Mean Std f f Fgure. Mnma acheved (averaged over 5 trals) for a 0 dmensonal Shfted Rotated Rastrgn s Functon f f f Fgure. Mnma acheved (averaged over 5 trals) for a 0 dmensonal Shfted Rotated Weerstrass Functon VII. CONCLUSIONS AN FUTURE WORK In ths paper we proposed a new algorthm for dscrete mult valued optmzaton problems. A theoretcal framework employng probablstc analyss for bnary PSO s presented. Specfcally, probablty densty functon for the velocty s modeled to nvestgate the affects of dfferent parameters of the PSO algorthm on the velocty. The new algorthm s analyzed under ths framework to examne ts stablty and affects of the parameters of the algorthm. The algorthm s appled to fve benchmark problems and the results presented show the performance benefts of the new algorthm. The algorthm can be successfully used for any number system. In future work, we want to formalze the analytc framework for the dscrete PSO and analyze the convergence and search behavor. Specfcally, the closed form expressons for the probablty densty functons of the velocty wll be derved. Ths s frst attempt to analyze a bnary PSO and ts behavor. Further analyss for the bnary PSO wll also be done. Fgure 3. Mnma acheved (averaged over 5 trals) for a 0 dmensonal Schwefel s Problem.3 References [] Eberhart, R. and Kennedy, J., A New Optmzer Usng Partcles Swarm Theory, Sxth Internatonal Symposum on Mcro Machne and Human Scence, 995, Nayoga, Japan. [] James Kennedy, Russell Eberhart and Sh, Y.H., Swarm Intellgence, Morgan Kaufman Publshers, 00. [3] Evolutonary Computaton : Bascs, Algorthms and Operators, Insttute of Physcs Publshng, 000. [4] E. Ozcan and C. K. Mohan, Partcle Swarm Optmzaton: Surfng the Waves, Proceedngs of Congress on Evolutonary Computaton (CEC 99), Washngton. C., July 999, pp [5] Sh. Y, R. C. Eberhart, Emprcal Study of Partcle Swarm Optmzaton, 999 Congress on Evolutonary Computng, Vol III, pp [6] Sh Y. H., Eberhart R.C., A Modfed Partcle Swarm Optmzaton Algorthm, IEEE Internatonal Conference on Evolutonary Computaton, 998, Anchorage, Alaska. [7] C. K. Mohan, B. Al-Kazem, screte Partcle Swarm Optmzaton, Proc. Workshop on Partcle Swarm Optmzaton, Indanapols, IN,

9 Proceedngs of the 007 IEEE Swarm Intellgence Symposum (SIS 007) [8] J. Kennedy and R. C. Eberhart, A screte Bnary Verson of Partcle Swarm Optmzaton, Proceedngs of the 997 Conf. on Systems, Man, and Cybernetcs, pp IEEE servce center, Pscataway, NJ. [9] Maurce Clerc, James Kennedy, The Partcle Swarm Exploson, Stablty, and Convergence n a Multmensonal Complex Space, IEEE Transactons on Evolutonary Computaton, Vol. 6, No., February, 00. [0] B. Al-Kazem and C. K. Mohan, Mult-Phase screte Partcle Swarm Optmzaton, Proc. The Fourth Internatonal Workshop on Fronters n Evolutonary Algorthms, 00. [] J. J. Lang, P. N. Suganthan and K. eb, Novel Comparson Test Functons for Numercal Global Optmzaton, IEEE Swarm Intellgence Symposum, pp , June 005. [] Elon S. Correa, Alex A. Fretas, Coln G. Johnson, A New screte Partcle Swarm Algorthm Appled to Attrbute Selecton n a Bonformtcs ata Set, GECCO 06, Seattle, Washngton, USA, July 8-, 006. [3] Jm Pugh, Alchero Martnol, screte Mult-Valued Partcle Swarm Optmzaton, IEEE Swarm Intellgence Symposum 06, Indanapols, Indana, USA, May -4, 006. [4] Kalyan Veeramachanen, Thanmaya Peram, Chlukur Mohan, lsa Osadcw, Optmzaton Usng Partcle Swarm Usng Near Neghbor Interactons, GECCO 03, Chcago, Illnos, USA, July, 003. [5] Partcle Swarm Optmzaton Code, Yuhu Sh, 49