A Novel Deluge Swarm Algorithm for Optimization Problems

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1 A Novel eluge Swarm Algorthm for Optmzaton Problems Anahta Samad,* - Mohammad Reza Meybod Scence and Research Branch, Islamc Azad Unversty, Qazvn, Iran Soft Computng Laboratory, Computer Engneerng and Informaton Technology epartment, Amrkabr Unversty of Technology, Tehran, Iran * Correspondng author. E-mal addresses:anahta.samad@gmal.com. ABSTRACT In ths study, a novel populaton based algorthm whch s based on great deluge, s presented for solvng optmzaton problems. Optmzaton problems due to the vast applcaton n real lfe and scence are so mportant. By applyng more lmtaton they are classfed as NP-hard problems. NP-hard problems, because of hgh complexty, can be solved by Meta-heurstc methods. In ths regards populaton-based approaches are consdered as a good opton. Accordng to the proposed algorthm mechansm, local and global search can be done. Some of the advantages of ths algorthm nclude smplcty, avod trappng n local optma, approprate balance between local and global search. The approach s examned on standard functons. Ackley and sphere, for nstance are takng nto account. The results are compared wth G-PSO, G-HS, GSA, CM-AFSA and M-ABC whch n all cases proposed algorthm shows better results. Keywords: eluge Swarm algorthm; Optmzaton; NP-hard problems. INTROUCTION In today's world, optmzaton plays crucal role for solvng varous problems especally scentfc ssues []. Accordng to problem complexty, dfferent solutons and algorthms have been suggested. The goal of all optmzaton algorthms s maxmze or mnmze the cost functon []. Real world problems are most classfed as NP-hard problems whch need complcated computaton [3]. Meta-heurstc technques have shown great ablty n solvng ths knd of problems. In ths approach, optmal soluton can be ganed by teratng process whch tres to enhance the result n each evaluaton by dfferent mechansms [4].One of the most popular approaches n Meta-heurstc algorthms s populaton based whch works accordng to swarm ntellgent methods. Swarm ntellgent s nspred by assocatve behavor of organsms such as fshes, nsects and brds. The rules between members of the swarm lead to effectve use of resources therefore optmal soluton for specal problem can be acheved [5]. Ths paper presents an effcent algorthm nspred by the great deluge that was ntroduced n 993 by ueck [6]. Great deluge algorthm nspred by people clmbng up to avod snkng. Great deluge algorthm ntates ts parameters based on estmaton of envronment and consder them as a prmary answer, n the next teratons t produces more answers based on prmary answer, afterwards new parameter called water level engaged as a boundary for acceptng result n whch one wth greater A Novel eluge Swarm Algorthm for Optmzaton Problems

2 water level s accepted. Water level rses gradually wth new ran speed parameter [6]. Great deluge algorthm s a robust, easy and flexble algorthm. It has been used n solvng dfferent problems ncludng, record-to-record travel [7], examnaton tme tablng Problem [8], buffer allocaton problem n unrelable producton lnes [9] and effcently dynamc layout [0]. In the proposed algorthm to explot swarm ntellgence, agents just as members of swarm are engaged and follow the rules of great deluge algorthm. Moreover new mechansms are used for agent s nteracton to set them n search space. In ths way hgh dversty of results are provded, thus proposed algorthm can fnd better peaks and avods beng trapped n local optma. Proposed algorthm whch s called eluge swarm optmzaton has smple structure and uses low numbers of parameters. In ths study, eluge Swarm Optmzaton have been tested on eght standard benchmark functons, such as Zakharov, Schwefel, Sphere, Ackley then It s results compared to the well-known algorthms ncludng global-best partcle swarm optmzaton(gpso) [], modfed artfcal bee colony algorthm (M-ABC) [], gravtatonal search algorthm(gsa) [3], modfed artfcal fsh swarm algorthm wth communcaton behavor(cm-afsa) [4], and global-best harmony search [5]. The rest of the paper s organzed as follows. etal of the proposed algorthm and flow chart are presented n secton. The results and comparson wth other algorthms are presented n secton 3. The concluson s presented n sectons 4.. ELUGE SWARM OPTIMIZATION Proposed algorthm s a populaton- based namely eluge Swarm Optmzaton. Smlar to standard Great eluge, n ths novel algorthm the water level s supposed to ncrease due to probable floods and a group of people try to save ther lves by clmbng up to hgh alttudes. Generally, n suggested algorthm, the best poston of ndvduals s the hgher one, and all agents try to poston themselves n sutable places near agents n hgher alttudes. Smultaneously the water level s ncreasng accordng to ranng rate; populaton at the same tme should clmb up hgher than water levels. In a standard Great eluge, the ran s velocty parameter s a small constant value. In each teraton, ranng wth constant rate leads to an ncrease n the water level constantly. etermnng the approprate ranng velocty amount s a challengng problem n the algorthm. In our suggested algorthm, the velocty value s consdered equal to zero n ntaton, from the second teraton the parameter s value s set to the dfference of average ftness value of the populaton category n the current and prevous teraton. Thus, the value of the parameter s determned based on the range of ftness. The determned agents values n early teratons where they move wth longer steps for a global search are consderable and gradually that wth ncreasng the ablty of local search after then agents steps shortened so the ftness values decreases. Ths decrease n the values contnues untl the water level rses be n proporton to the populaton movng step n transformng a global search capablty to local search durng the process of optmzaton. In real world, people gettng stuck n a stuaton smlar to eluge algorthm, usually try to reach themselves to safer postons by evaluatng the surroundngs. Ths reacton s consdered as a local search and smulated n proposed algorthm. For ths am, people beng postoned n reasonable places (.e. hgher alttudes) search the surroundngs for better postons. A search-space wth the length Vsonradus s consdered for agents and each agent can search ths space for better postons. Vson-radus amount wll be decreased gradually so that the capablty of local search s ncreased and the steps are n proporton wth the space, the algorthm s gong to search n, for better postons. To mplement agents search process, n proposed algorthm, each agent postonng above water level s set to assess neghborng dfferent postons n Vson-radus doman equal to try number tmes. If the assessed poston s more sutable than the current poston, the agent moves to that poston. Implementng the mentoned process s as follows: A Novel eluge Swarm Algorthm for Optmzaton Problems

3 . Frst, the th agent placed n X poston consders a new poston Xtarget by the followng equaton: XTarget, d X, d rand, Vson Radus () In the above equaton rand (-,) creates a random unformly dstrbuted number n the nterval (-,). The ftness of calculated Xtarget n all dmensons s then assessed, f the ftness s proven to be better than the X, then the agent moves forward to t. The above mentoned process s repeated try-number tmes n each teraton.. In addton to ndvdual search, group search s performed by the populaton except the agent wth the best poston. Indvdual agents try to reach themselves to a better locaton followng the others wth better locatons. For ths purpose, each agent randomly chooses the agent wth better poston. Accordng to equaton the new locaton s: XTemp, d X, d X j, d X, d rand 0, () Xtemp n all dmensons s calculated based on the poston of the chosen random agent (.e. jth agent). The new locaton s assessed for ftness. Whle ts locaton s better than the current poston of th agent and placed above water level, the th agent moves toward t. Pseudo code of proposed algorthm s presented n Table. Table. Pseudo code of proposed algorthm eluge Swarm Optmzaton: : ntalze N(populaton sze),vson Radus : for each Person do 3: ntalze X randomly n -dmensonal search space 4: Endfor 5: evaluate all People 6: Repeat 7: Calculate Ran Speed based on dfference of ftness average n tr T and tr T- 8: Water Level = Water Level + Ran Speed 9: Update Vson Radus 0: for each Person do : f ftness of Person s better than Water Level then : for = to Try Number do 3: obtan X Target by Eq. () 4: check search space boundary for X Target 5: If f(x Target ) s better than f(x ) then 6: X = X Target 7: endf 8: endfor 9: Endf 0: Endfor : foreach Person do A Novel eluge Swarm Algorthm for Optmzaton Problems 3

4 : determne Person j randomly whch f(x j )<f(x ) 3: Obtan X Temp by Eq. () 4: check search space boundary for X Temp 5: f f(x Temp )<f(x ) & f(x Temp )<Water Level then 6: X = X Temp 7: endf 8: Endfor 9: Untl stoppng crteron s met 3. EXPERIMENTAL RESULTS In ths secton, the proposed algorthm has been tested over 8 standard functon benchmark whch generally appled as acceptable test tools for optmzaton algorthm n contnuous statc envronment. Functon propertes and dmenson of the problem are defnes n table [6]. Optmal value for tested functons s equal to zero. Table. Test functons that used n ths study Name Test Functon Zakharov 4 4 x x x x Search range f 0 0 [-5,0] 0 Schwefel. f x x Hyperellpsod Sphere x x 6 30 [-0,0] 0 f8 x 30 [-5.,5.] 0 sn x x Schaffer f 6 f x x 30 [-00,00] x 0.5 [-00,00] x x 9 Ackley f x 0 exp 0. x exp cos x 0 e Penalzed f x 0 sn y y 0 sn y x ux k,0,00,4 y 4 m x a, ux, a, k, m 0, m k x a, a x y x a a x a f Mn 30 [-3,3] 0 30 [-50,50] 0 4 A Novel eluge Swarm Algorthm for Optmzaton Problems

5 Rotated Penalzed f y x 0sn y w 0sn y 7 y 4 w x M w ux k,0,00,4 w a, uw, a, k, m 0, m k w a, m w a a w a w a 30 [-50,50] 0 * M s an orthogonal Matrx eluge Swarm Optmzaton and fve well-known algorthms ncludng G-PSO [], G-HS [5], GSA [3], CM-AFSA [4] and M-ABC [] are examned over functons that are mentoned n Table. Proposed algorthm confguraton and other algorthms are shown n Table 3. Settngs of other algorthms are taken from related references. Confguraton of proposed algorthm s the result of dfferent experments whch lead to best performance of ths algorthm. Mentoned algorthms Average and standard devaton from 30 replcaton are demonstrated n Table 4. Table 3. Parameter settng of algorthms Algorthm SO G-PSO G-HS CM-AFSA GSA M-ABC Parameter Settng Populaton sze = 0 try-number=5 W=0.98 Populaton sze =0 c =c = W mn =0.4 W max =0.9 neghbor topology=global star Inerta weght = lnear decrement Populaton sze(hms) =5 HMCR=0.9 PAR mn =0.0 PAR max =0.99 Populaton sze =50 try-number=5 crowd factor=0.75 Vsual=Par[] Step = Vsual/5 c 3 =c 6 = Populaton sze =50 α=0 G0=00 k0=50 Populaton sze =40 Modfcaton rate=0.8 Lmt = 0.5 menson Populaton sze Scalng factor = A Novel eluge Swarm Algorthm for Optmzaton Problems 5

6 Table 4. Average and standard devaton results of 6 algorthms of functons showed n table FUNCTION Zakharov Schwefel. Hyper-ellpsod Sphere Schaffer Ackley Penalzed Rotated Penalzed G-PSO 9.07 (37.7) (444.4) 0.76 (8.9).48e-50 (3.8e-5) 0 (0) 0.98 (.79) 0.09 (0.) (83.94) G-HS (0.0003) 0.0 (0.0) 3.43e-05 (6.7e-05) (0.00) 0.0 (0.0) (0.004) 5.36e-06 (6.44e-06) 0.73 (.9) GSA 3.4e-8 (.9e-8).8e-08 (.09e-09).e-6 (.99e-7).9e-7 (.6e-8) 0.0 (0.0036).77e-09 (.90e-0) 0.00 (0.0).0e-9 (3.e-0) M-ABC.4e-5 (.59e-5).43e-08 (5.07e-09) 3.68e- (.e-).35e-09 (6.55e-0) (0.000).63e-05 (5.30e-06) 0.00 (0.008) 0.09 (0.08) CM-AFSA 3.87e-4 (.8e-3) (0.0008) 5.08e-08 (8.5e-08) 7.9e-08 (4.57e-08) 9.79e-5 (0.0004).49 (0.9).07 (.83) 4.6 (.83) SO.60e-37 (.06e-37).33e- (.58e-) 8.00e-7 (4.9e-7) 6.60e-75 (3.84e-75) 0 (0) 5.56e-0 (3.05e-0).39e-7 (3.04e-8) 7.0e-4 (.33e-4) As t can be seen n Tables4, eluge Swarm Optmzaton acheves best results n all functons except Schaffer one,n that case proposed algorthm and G-PSO gan the best results among the tested algorthm. a. Sphere b. Rotated Penalzed c. Schwefel. d. Ackley Fg.. Convergence behavor of Sphere, Rotated Penalzed, Schwefel. and Ackley functons 6 A Novel eluge Swarm Algorthm for Optmzaton Problems

7 4. CONCLUSION In ths study, novel algorthm namely eluge Swarm Optmzaton based on great eluge algorthm has been proposed. Ths new algorthm takng advantage of the mult agents, enables proposed algorthm to make dversty n search space. Agents n proposed algorthm mprove ther result by comparson to water level parameter ndvdually. On the other hand wth the ablty to comparson between agents, eluge Swarm Optmzaton s capable to local and global search leadng to reasonable results, and acceptable convergence rate. Some of the prvlege of ths algorthm s ncludng smplcty, avodng local optma, approprate balance between local and global search. To prove the effcency of proposed algorthm the results were compared wth G-PSO, G-HS, GSA, CM-AFSA, M-ABC usng some standard functons ncludng Ackley and sphere. Comparng results showed that the proposed algorthm performs better than all other algorthms wth mentoned functons. REFERENCES [] Cheng, H., & Yang, S. (00, Aprl). Mult-populaton genetc algorthms wth mmgrants scheme for dynamc shortest path routng problems n moble ad hoc networks. In European Conference on the Applcatons of Evolutonary Computaton (pp ). Sprnger Berln Hedelberg. [] Lung, R. I., & umtrescu,. (007, July). A new collaboratve evolutonary-swarm optmzaton technque. In Proceedngs of the 9th annual conference companon on Genetc and evolutonary computaton (pp ). ACM. [3] L, X., Branke, J., & Blackwell, T. (006, July). Partcle swarm wth specaton and adaptaton n a dynamc envronment. In Proceedngs of the 8th annual conference on Genetc and evolutonary computaton (pp. 5-58). ACM. [4] Yang, S., & L, C. (00). A clusterng partcle swarm optmzer for locatng and trackng multple optma n dynamc envronments. IEEE Transactons on Evolutonary Computaton, 4(6), [5] Yazdan,., Sadegh-Ivrgh, S., Yazdan,., Sepas-Moghaddam, A., & Meybod, M. R. (05). Fsh Swarm Search Algorthm: A New Algorthm for Global Optmzaton. Internatonal Journal of Artfcal Intellgence, 3(), [6] ueck, G. (993). New optmzaton heurstcs: The great deluge algorthm and the record-torecord travel. Journal of Computatonal physcs, 04(), [7] ueck, G. (993). New optmzaton heurstcs: The great deluge algorthm and the record-torecord travel. Journal of Computatonal physcs, 04(), [8] Burke, E., Bykov, Y., Newall, J., & Petrovc, S. (004). A tme-predefned local search approach to exam tmetablng problems. Ie Transactons, 36(6), [9] Nahas, N., At Kad,., & Nourelfath, M. (006). A new approach for buffer allocaton n unrelable producton lnes. Internatonal journal of producton economcs, 03(), [0] Nahas, N., Kad,. A., & El Fath, M. N. (00). Iterated great deluge for the dynamc faclty layout problem. CIRRELT. [] Sh, Y., & Eberhart, R. (998, May). A modfed partcle swarm optmzer. In Evolutonary Computaton Proceedngs, 998. IEEE World Congress on Computatonal Intellgence., The 998 IEEE Internatonal Conference on (pp ). IEEE. [] Akay, B., & Karaboga,. (0). A modfed artfcal bee colony algorthm for real-parameter optmzaton. Informaton Scences, 9, 0-4. [3] Rashed, E., Nezamabad-Pour, H., & Saryazd, S. (009). GSA: a gravtatonal search algorthm. Informaton scences, 79(3), A Novel eluge Swarm Algorthm for Optmzaton Problems 7

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