Combinatorial Optimization of Multi-agent Differential Evolution Algorithm

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1 Send Orders or Reprnts to 0 The Open Cybernetcs & Systemcs Journal, 04, 8, 0-06 Open Access Combnatoral Optmzaton o Mult-agent Derental Evoluton Algorthm Fahu Gu,,*, Kangshun L, Le Yang and Yan Chen School o Inormaton, South Chna Agrcultural Unversty, Guangzhou, Guangdong 50006, Chna Department o Electronc Inormaton Engneerng, Jangx Appled Technology Vocatonal College, Ganzhou, Jangx 34000, Chna Abstract: Combnatoral optmzaton s oten wth the local extreme pont n large numbers. It s usually dscontnuous, multdmensonal, non-derentable, constrant condtons, hghly nonlnear NP problem. In ths paper, accordng to the characterstcs o combnatoral optmzaton problem, we put orward the combnaton optmzaton o mult-agent derental evoluton algorthm (COMADE through combnng the mult-agent and derental evoluton algorthm, n whch we desgned the competton behavor and sel-learnng behavor o agent. Through perormance testng o strong connected, weak connected and overlap connected deceptve uncton on the COMADE algorthm, the results show that the COMADE algorthm s eectve and practcal value. Keywords: Combnatoral optmzaton, competton behavor, derental evoluton algortthm, mult-agent, sel- learnng behavor.. INTRODUCTION Combnatoral optmzaton s oten wth the local extreme pont n large numbers. It s usually dscontnuous, multdmensonal, non-derentable, constrant condtons, hghly nonlnear NP problems. The combnatoral optmzaton s always hot subject n the elds o scence and engneerng. The travelng salesman problem (TSP s a amous problem n the combnatoral optmzaton. The combnatoral optmzaton s soluton has not only great academc value but mportant practcal value. It has many solutons [- 3], such as Ant colony Algorthm (ACA, Partcle Swarm Optmzaton (PSO and so on, and the method o ntellgent optmzaton s qute ecent. In ths paper, a knd o combnaton optmzaton o mult-agent derental evoluton algorthm (COMADE s proposed. Through perormance testng o strong connected, weak connected and overlap connected decepton uncton on the COMADE algorthm, the results show that the COMADE algorthm s eectve and practcal value.. DESIGN OF AGENT FOR COMBINATORIAL OP- TIMIZATION Combnatoral optmzaton can be descrbed as [4]: ( S,, where S s the search space, and s the objectve uncton: : S R. The purpose o solvng s to nd x * S or (x * (x and x S. Thereore, we can use an agent to represent a state o search space. Now we dene that an agent s a canddate soluton o problems to be optmzed, whch s expressed as a model one: a = (a, a,, a n S, a = 0or( n ( where n s the scale o the problem, the energy o agent s equal to the value o the objectve uncton, that s Energy ( a = ( a. In order to calculate the energy o each agent, we put the agents nto a xed grd L expressed Fg. (. Fg. (. Agent grd model. The agents whch can nteract wth L are determned by the parameter o percepton range, whch can be denoted as R s. Thus, the agents whch can nteract wth L are expressed as the ollow model tow. L k,l,( R s k ( + R s,( j R s ( j + R s ( 874-0X/4 04 Bentham Open

2 Combnatoral Optmzaton o Mult-agent Derental Evoluton Algorthm The Open Cybernetcs & Systemcs Journal, 04, Volume 8 where k and l can be denoted as ollows: j,(ll, LL j, or(ll, LL j, k+l,k< l+l,l< k = {k Lsze,k>L, l = {l Lsze,l>L sze sze sze The neghborhood o L, whch s the range o nteracton wth L, s denote as N. lnes o the table (LL n( n+ In competton behavor, the percepton scope o each agent s set, thus, there are 8 agents n the neghborhood, whch can be denoted as N c. When the energy o L s not less than the others o neghborhood, L wll contnue to survve, otherwse t wll de. The procedure o competton behavor can be descrbed as ollowng [4, 5]: Where L = (l, l,, ln, amax = (a, a,, an N,c j and a N, Thereore, the learnng table s (LL n( n+, the several.. Competton Behavor c n(n + p sze the sub meter. The learnng procedure o the agent (L = (l, l,..., ln s descrbed as the ollowng: Frst learnng way: : q osprng generate a new agent (a = (a, a,..., an rom the ollowng the model ve: l,(<llq or (>LLqj, j, a = { l,otherwse Frst way: when the set D represents the derental bt number o L and amax, we establsh the model three: a,( D or (( D and ( Random(=0 where Random(,( n (3 s used to randomly generate 0 or. Second way: we buld the model our: n a,otherwse a,random> c = { can be composed o a learnng 3: choose row rom (LLq, suppose no. j row, c = (c, c,, cn can be created by amax n the ways. c = { a,otherwse : generate (LLq Energy(a Energy(amax, Energy(L Energy(amax, 03, ( n 4: (4 The number o set D s equal to Hammng Dstance between L and amax. When the number s small, t shows that L and amax are smlar, t s hard to generate the osprng by the rst way. Thereore, we choose the way to generate the osprng c by the parameter o D h (0,, D > D h, the rst way s used, otherwse the second way. n.. Sel-learnng Behavor Each agent can ncrease energy by sel-learnng behavor. But only when energy o an agent s not less than the any one o the learnng scope, can the agent can get a chance to learn. We dene the learnng table as matrx (LL p, whch has p rows and columns, thereore the learnng table meets the ollowng condtons [4-6]: (5 Energy(a > Energy(L, then Learnng(a False, L a, stop 5: delete the j row rom (LLq, (LLq s null, then q q + where Random s used to randomly generate the real number between 0 and. LL, j n ( and LL, LL,, p, j = or, ( n 6: q LLw, turn to step, otherwse Learnng(L True, stop, where LLw s expressed the learnng sub meter number o the total learnng table, every n(n + sub meter table has / LLw, denote the table as LL LL, LL, LL w Second learnng way: : randomly generate a sequence o ntegers rom to n, denote t as ( p, p,, pn, set q : generate (LLq 3: choose row rom (LLq, suppose no. j row, generate a new agent (a = (a, a,..., an rom the ollowng the model ve: l a p = { lp,(<llqj, or (>LLqj, p,otherwse 4:, ( n Energy(a > Energy(L, (6 then Learnng(a False, L a, stop 5: delete the j row rom (LLq, (LLq s null, then q q +

3 04 The Open Cybernetcs & Systemcs Journal, 04, Volume 8 Gu et al. Table. Comparson o the average uncton evaluaton tmes o the COMADE test on strong connected uncton or 50 tmes ndependent and other algorthm. Functon Dmenson COMADE Paper [4] Paper [5] n= n= n= n= n= n= : q LL w, turn to step, otherwse Learnng(L True, stop. Generally speakng, t s good to take the rst learnng way. When t s alure to learn n the rst way and Learnng(L True, we wll take the second way to learn. 3. DESIGN OF COMADE The algorthm o COMADE s descrbed as the ollowng [4-7]: : ntalze populaton L 0, randomly generate L sze L sze agents, set Learnng(L False, where =,,, L sze t : evaluate Energy o every agent n L by the competton behavor method, a N c ', Energy(a Energy(L, L t+/ ' L, otherwse, generate a new agent c by D h, Learnng(c False, L t+/ c 3: evaluate Energy o every agent n L t+/ by the sel-learnng behavor method, a N, Energy(a Energy(L t+/ and Learnng(L t+/ = False, take the rst sel-learnng behavor method or L t+/ to learn, a N, Energy(a Energy(L t+/ and Learnng(L t+/ = True, take the second sel-learnng behavor method or L t+/ to learn, then L t+ t+/ L 4: derental and crossover operaton or each agent o L t+, generate L t+ 5: evaluate Energy o every agent n L t+, Energy(L t+ t > Energy(a max, then a t+ max = L t+. j, otherwse, a t+ t max = a max 6: meet the termnaton condton, ext, otherwse, t=t+, turn to step. 4. SIMULATION EXPERIMENT There are many practcal problems o combnatoral optmzaton. In order to test the algorthm perormance o COMADE, we choose the decepton uncton to test [4, 7- ]. 4.. Experment o Strong Connected Functon We use the ollowng tow strong connected unctons to n/3 (a = deceptve3 (a 3, a 3, a 3 n/5 (a = trap5 (a 5 4, a 5 3, a 5, a 5, a 5 From the above Table, we can know that the calculaton amount o COMADE s about 0% o the others, the perormance o COMADE s very good. 4.. Experment o Weak Connected Functon We use the ollowng tow weak connected unctons to n/3 3 (a = deceptve3 (a, a +n/, a +n/3 4 (a = n/6 bpolar6 (a, a +n/6, a, a +3n/6, a +4n/6, a +5n/6 + n/6 From the above Table, we can know that t s harder to solve the weak connected deceptve uncton than to solve the strong connected deceptve uncton, but we can see that t s only mllons o evaluaton to solve the 90 dmensonal weak connected cheat uncton through the COMADE algorthm 4.3. Experment o Overlap Connected Functon We use the ollowng tow overlap connected unctons to

4 Combnatoral Optmzaton o Mult-agent Derental Evoluton Algorthm The Open Cybernetcs & Systemcs Journal, 04, Volume 8 05 Table. Comparson o perormance based on COMADE through strong connected and weak connected uncton testng. Functon Average Tmes o Functon Evaluaton Rato Index Number n= n= n= n= n= n= n= n=0 764 n= n= n= n= n= n= n= n= O(n O(n O(n O(n 4.06 Table 3. Comparson o perormance based on COMADE through overlap connected and weak connected uncton testng. Functon Average Tmes o Functon Evaluaton Rato Index Number n= n= n= n= n= n= n= n= n= n= n= n= n= n= n=90 73 n= n= n= O(n O(n O(n.6

5 06 The Open Cybernetcs & Systemcs Journal, 04, Volume 8 Gu et al. Table 3. Contd. Functon Average Tmes o Functon Evaluaton Rato Index Number n= n= n= n= n= n= O(n.9 n 5 (a = deceptve3 (a, a +, a + n 4 6 (a = trap5 (a 4 3, a 4, a4, a, a 4 4+ From the above Table 3, we can know that t s only mllons o evaluaton to solve the 900 overlap connected cheat uncton through the COMADE algorthm. CONCLUSION From the Tables -3, we can see that the COMADE algorthm has good perormance, especally or solvng largescale complex combnatoral optmzaton problem. CONFLICT OF INTEREST The authors conrm that ths artcle content has no conlct o nterest. ACKNOWLEDGEMENTS Ths work s supported by the Educaton the Educaton Scence and technology und o the Educaton Department o Jangx, Chna (No. GJJ4807, and the Fund o Natural Scence Foundaton o Guangdong Provnce, Chna (No. 04A REFERENCES [] C. Ln, A. Qng, and Q. Feng, Synthess o unequally spaced antenna arrays by usng derental evoluton, IEEE Transactons on Antennas and Propagaton, vol. 58, no. 8, pp , 00. [] F. Peng, K. Tang, G. Chen, and X. Yao, Populaton-based algorthm portolos or numercal optmzaton, IEEE Transactons on Evolutonary Computaton, vol. 4, no. 5, pp , 00. [3] K. Tang, Y. Me, and X. Yao, Memetc algorthm wth extended neghborhood search or capactated arc routng problems, IEEE Transactons on Evolutonary Computaton, vol. 3, no. 5, pp. 5-66, 009. [4] Z. We-Ca, J. Lu, F. Lu, and L. Jao, Combnatoral optmzaton usng mult-agent evolutonary algorthm, Chnese Journal o Computers vol. 6, pp , 004. [5] M. Pelkan, D. E. Goldberg, BOA: the Bayesan optmzaton algorthm, IIIGAL Report No. 9803, Urbana, IL:Unversty o IIInos at Urbana-Champagn, IIInos Genetc Algorthms Laboratory, 998. [6] K. Tang, Y. Me, and X. Yao, Memetc algorthm wth extended neghborhood search or capactated arc routng problems, IEEE Transactons on Evolutonary Computaton, vol. 3, no. 5, pp. 5-66, 009. [7] K. L. Brown, E. Nudelman, and Y. Shoham, Emprcal hardness models: Methodology and a case study on combnatoral auctons, Journal o ACM, vol. 56, no. 4, pp. -5, 009. [8] K. D. Jong. Evolutonary Computaton: a uned Approach, MIT Press, Cambrdge, MA 006, pp [9] M. Gaglolo and J. Schmdhuber, Learnng dynamc algorthm portolos, Annals o Mathematcs and Artcal Intellgence, vol. 47, no. 3-4, pp , 007. [0] F. Peng, K. Tang, G. Chen, and X. Yao, Populaton-based algorthm portolos o rnumercal optmzaton, IEEE Transactons on Evolutonary Computaton, vol. 4, no. 5, pp , 00. [] K. H. Han, K. H. Park, C. H. Lee, and J. H. Km, Parallel quantum nspred genetc algorthm or combnatoral optmzaton problem, In: Proceedngs 00 Congress on Evolutonary Computaton, Pscataway, NJ:IEEE Press, vol., 00, pp [] S. Delucca, and D. H. Werner, Nature-based desgn o aperodc lnear arrays wth broadband elements usng a combnaton o rapd neural-network estmaton technques and genetc algorthms, IEEE Antennas and Propagaton Magazne, vol. 49, no. 5, pp. 3-3, 007. Receved: September 6, 04 Revsed: December 3, 04 Accepted: December 3, 04 Gu et al. Lcensee Bentham Open. Ths s an open access artcle lcensed under the terms o the Creatve Commons Attrbuton Non-Commercal Lcense ( lcenses/by-nc/3.0/ whch permts unrestrcted, non-commercal use, dstrbuton and reproducton n any medum, provded the work s properly cted.