Implementation of digital pheromones for use in particle swarm optimization

Transcription

1 Mechancal Engneerng Conference Presentatons, Papers, and Proceedngs Mechancal Engneerng 006 Implementaton of dgtal pheromones for use n partcle swarm optmzaton Jung Leng Foo Iowa State Unversty Vjay Kalvarapu Iowa State Unversty, vkk@astate.edu Elot H. Wner Iowa State Unversty, ewner@astate.edu Follow ths and addtonal works at: Part of the Computer-Aded Engneerng and Desgn Commons Recommended Ctaton Foo, Jung Leng; Kalvarapu, Vjay; and Wner, Elot H., "Implementaton of dgtal pheromones for use n partcle swarm optmzaton" (006). Mechancal Engneerng Conference Presentatons, Papers, and Proceedngs Ths Conference Proceedng s brought to you for free and open access by the Mechancal Engneerng at Iowa State Unversty Dgtal Repostory. It has been accepted for ncluson n Mechancal Engneerng Conference Presentatons, Papers, and Proceedngs by an authorzed admnstrator of Iowa State Unversty Dgtal Repostory. For more nformaton, please contact dgrep@astate.edu.

2 Implementaton of dgtal pheromones for use n partcle swarm optmzaton Abstract Ths paper presents a new approach to partcle swarm optmzaton (PSO) usng dgtal pheremones to coordnate the movements of the swarm wthn an n-dmensonal desgn space. In tradtonal PSO, an ntal randomly generated populaton swarm propagates towards the global optmum over a seres of teratons. Each partcle n the swarm explores the desgn space based on the nformaton provded by prevous best partcles. Ths nformaton s used to generate a velocty vector ndcatng a search drecton towards a promsng desgn pont, and to update the partcle postons. Ths paper presents how dgtal pheromones can be ncorporated nto the velocty vector update equaton. Dgtal pheromones are models smulatng the real pheromones produced by nsects for communcaton to ndcate a source of food or a nestng locaton. Ths prncple of communcaton and organzaton between each nsect n a swarm offers substantal mprovement when ntegrated nto PSO. Partcle swarms search the desgn space wth dgtal pheromones adng communcaton wthn the swarm to mprove search effcency. Through addtonal nformaton from the pheromones, partcles wthn the swarm explorng the desgn space and locate the soluton more effcently and accurately than tradtonal PSO. In ths paper, the development of ths method s descrbed n detal along wth the results from several optmzaton test problems. Keywords Vrtual Realty Applcatons Center, nformaton analyss, nteratve methods, structural desgn, partcle postons, partcle swarm optmzaton Dscplnes Computer-Aded Engneerng and Desgn Mechancal Engneerng Comments Ths s a conference proceedng from Collecton of Techncal Papers - AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamcs and Materals Conference, (006): AIAA , do: 0.54/ Posted wth permsson. Ths conference proceedng s avalable at Iowa State Unversty Dgtal Repostory:

3 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamcs, and Materals Confere - 4 May 006, Newport, Rhode Island AIAA Implementaton of Dgtal Pheromones for Use n Partcle Swarm Optmzaton Jung Leng Foo *, Vjay K. Kalvarapu and Elot Wner Iowa State Unversty, Ames, IA, 500, USA Downloaded by IOWA STATE UNIVERSITY on Aprl, 05 DOI: 0.54/ C Ths paper presents a new approach to partcle swarm optmzaton (PSO) usng dgtal pheremones to coordnate the movements of the swarm wthn an n-dmensonal desgn space. In tradtonal PSO, an ntal randomly generated populaton swarm propagates towards the global optmum over a seres of teratons. Each partcle n the swarm explores the desgn space based on the nformaton provded by prevous best partcles. Ths nformaton s used to generate a velocty vector ndcatng a search drecton towards a promsng desgn pont, and to update the partcle postons. Ths paper presents how dgtal pheromones can be ncorporated nto the velocty vector update equaton. Dgtal pheromones are models smulatng the real pheromones produced by nsects for communcaton to ndcate a source of food or a nestng locaton. Ths prncple of communcaton and organzaton between each nsect n a swarm offers substantal mprovement when ntegrated nto PSO. Partcle swarms search the desgn space wth dgtal pheromones adng communcaton wthn the swarm to mprove search effcency. Through addtonal nformaton from the pheromones, partcles wthn the swarm explorng the desgn space and locate the soluton more effcently and accurately than tradtonal PSO. In ths paper, the development of ths method s descrbed n detal along wth the results from several optmzaton test problems. I. Introducton urrent heurstc optmzaton technques such as Genetc Algorthms (GA) and Smulated Annealng (SA) are capable of exhaustvely nvestgatng desgn spaces to locate optmal desgn ponts. The probablstc nature of heurstc methods gves dstnct advantages over determnstc methods n fndng a global optmum, partcularly n a mult-modal optmzaton problem. Thus, these types of methods have become qute popular when formal optmzaton s requred. However, these methods are hampered by ther computatonal expense. To obtan global optmal solutons, a large populaton of desgn ponts over much teraton must be evaluated. The ntroducton of Partcle Swarm Optmzaton (PSO) by Kennedy and Eberhart, offers capabltes to locate global solutons wth less computatonal resources and tme. Compared to GA and SA, PSO s smpler to mplement and has fewer parameters to adjust 3, 4. In a tradtonal PSO, an ntal randomly generated populaton swarm (a collecton of partcles) propagates towards an optmal pont n the desgn space, and reaches the global optmum over a seres of teratons. Each partcle n the swarm explores the desgn space based on the nformaton provded by prevous best partcles. A basc PSO algorthm uses ths nformaton to generate a velocty vector ndcatng a search drecton towards a promsng desgn pont, and updates the locatons of all partcles n the swarm. However, ths can be a drawback as all partcles are drected towards the current best pont as well as the overall best pont obtaned. Ths makes the method very ntal condton dependent for an effectve and effcent search of the desgn space. Ths paper focuses on mprovng the search and resultant soluton through the use of dgtal pheromones wthn the velocty update. Coupled wth statstcal analyss on the pheromones, an effcent move set s generated to update the search drecton of each partcle. Ths method s tested wth n-dmensonal problems and the results presented. * Research Assstant, Department of Mechancal Engneerng, Human Computer Interacton, Vrtual Realty Applcatons Center, 74 Howe Hall, Iowa State Unversty, Ames, IA, 500, USA, Student Member. Research Assstant, Department of Mechancal Engneerng, Human Computer Interacton, Vrtual Realty Applcatons Center, 74 Howe Hall, Iowa State Unversty, Ames, IA, 500, USA, Student Member. Assstant Professor, Department of Mechancal Engneerng, Human Computer Interacton, Vrtual Realty Applcatons Center, 74 Howe Hall, Iowa State Unversty, Ames, IA, 500, USA, Member. Amercan Insttute of Aeronautcs and Astronautcs Copyrght 006 by Elot H Wner. Publshed by the Amercan Insttute of Aeronautcs and Astronautcs, Inc., wth permsson.

4 II. Background A. Partcle Swarm Optmzaton The PSO algorthm s a recent addton to the lst of global search methods 5. It s a populaton based zero-order optmzaton method that portrays several evolutonary algorthm characterstcs smlar to Genetc Algorthms (GA) and Smulated Annealng (SA). These are: a) ntalzaton wth a populaton of random solutons, b) desgn space search for an optmum through updatng generatons of desgn ponts and c) update based on prevous generatons 6. Intal success of the algorthm has brought substantal attenton to further research 7, 8. The workng of the algorthm s based on a smplfed socal model smlar to the behavor exhbted by a swarm of bees or a flock of brds. In ths analogy, a bee (partcle) uses ts own memory and the behavor of the rest of the swarm to determne the sutable locaton of food (global optmum). The algorthm teratvely updates the drecton of the swarm movement toward the global optmum. Equatons () and () defne the mathematcal smulaton of ths behavor. Downloaded by IOWA STATE UNIVERSITY on Aprl, 05 DOI: 0.54/ V = V + c rand() * ( pbest []! X []) + c " rand() " ( gbest[]! X []) * + = X! V + () X + Equaton (), represents the ntally developed PSO method where rand() s a random number between zero and one, c and c are the confdence parameters. pbest represents the best poston attaned by the swarm n the current teraton and gbest represents the best poston attaned by the swarm n any prevous teraton. Equaton () denotes the updated swarm locaton n the desgn space. There were sgnfcant modfcatons and enhancements to the ntally developed PSO algorthm to cater for a multtude of problems, some of them beng: a) ntroducton of an nerta weght factor w multpled to V - n eq () 9, 0, b) mutaton factors for better desgn space exploraton,, c) methods for constrant handlng 6, 3, 4, d) parallel mplementaton 5, e) methods for solvng mult-objectve optmzaton problems 6, and f) methods for solvng mxed dscrete, nteger and contnuous varables 7. B. Dgtal Pheromones Pheromones are chemcal scents produced by nsects to communcate wth each other and serve as a stmulus to nvoke behavoral responses from creatures of ther own speces (e.g., food source, nestng locaton, etc). The stronger the pheromone, the more the nsects are attracted to the path. A dgtal pheromone s analogous to an nsect generated pheromone n that t can be used as a marker to determne whether or not an area of a desgn space s promsng for further nvestgaton. For example, dgtal pheromones have been used n the automatc adaptve swarm management of Unmanned Aeral Vehcles (UAVs) 8, 9, where the costs of human operators are greatly reduced. By releasng dgtal pheromones n a vrtual envronment, the UAVs can be ntellgently and automatcally guded towards a specfc zone or target. Other applcatons of dgtal pheromones nclude ant colony optmzaton for solvng mnmum cost paths n graphs 0,,, and solvng network communcaton problems 3. C. PSO and Dgtal Pheromones The benefts of dgtal pheromones from swarm ntellgence and adaptve applcatons can be merged nto the partcle swarm optmzaton method to better explore the desgn space and gude the partcles towards a desred optmal soluton. The concept of dgtal pheromones s consderably new 4 and has not yet been explored to ts full potental for nvestgatng n-dmensonal desgn spaces. The advantage s n the addtonal nformaton avalable to the swarm movng towards the optmum. In a basc PSO algorthm, the swarm movement s governed by the velocty vector computed n Eq (). The swarm s therefore, essentally presented wth nformaton obtaned from two specfc locatons from the desgn space at any teraton. However, multple pheromones released by the swarm members potentally provde the opportunty of explorng more promsng locatons wthn the desgn space when the nformaton obtaned from pbest and gbest are nsuffcent or neffcent. The research presented n ths paper explores the possblty of combnng PSO and dgtal pheromones. An addtonal pheromone component n the velocty vector update equaton s nvestgated and presented. The remanng sectons focus on the method development and evaluaton. () Amercan Insttute of Aeronautcs and Astronautcs

5 III. Methodology A. Method overvew The early stages of the method presented n ths paper are smlar to the basc PSO method. The addtonal steps usng dgtal pheromones are mplemented after the objectve functons for all partcles n the swarm are evaluated, to generate the thrd component of the velocty vector. Fgure summarzes the steps requred to mplement the method developed, wth steps usng dgtal pheromones hghlghted. Populate partcle swarm wth random ntal values Start Iteratons Downloaded by IOWA STATE UNIVERSITY on Aprl, 05 DOI: 0.54/ Evaluate ftness value of each swarm member Store best ftness value and desgn varables: - In the current teraton as pbest - All teratons untl the current as gbest Decay current dgtal pheromones n desgn space (f any) In the frst teraton, 50% of the partcles n the populaton are selected at random to release a pheromone each. For subsequent teratons, partcles mprovng the soluton wll release a pheromone Merge pheromones based on relatve dstance between each Fnd target pheromone toward whch the swarm moves Update velocty vector and poston of the swarm No Converged? Yes STOP Fgure. Flowchart of Partcle Swarm Optmzaton wth Dgtal Pheromones. 3 Amercan Insttute of Aeronautcs and Astronautcs

6 Intalzaton of the algorthm begns wth a user defned number of partcles beng placed wthn the bounds of the desgn space. A selected percentage of partcles from the swarm that fnd a better soluton release pheromones wthn the desgn space n the frst teraton. For subsequent teratons, each swarm member that fnds a better objectve functon releases a pheromone. Each pheromone locaton s then compared to the locatons of other exstng pheromones. From these comparsons, pheromones (from current as well as past teratons) that are close to each other n terms of desgn varable values are merged nto a sngle pheromone locaton. Ths effectvely creates a pheromone pattern across the desgn space whle stll keepng the number of pheromones manageable. Each dstnct pheromone s then gven a probablty based on ts pheromone level and ts poston relatve to a partcle. Ths probablty s then used n a rankng process to select a target pheromone for each partcle n the swarm. The target poston for each partcle wll be an addtonal component of the velocty vector update n addton to pbest and gbest. Followng ths, the objectve value for each partcle s recalculated and the entre process contnues untl the convergence crtera s satsfed. Downloaded by IOWA STATE UNIVERSITY on Aprl, 05 DOI: 0.54/ B. Dgtal Pheromones Placement and Decay In order to populate the desgn space wth an ntal set of dgtal pheromones, approxmately 50% of the partcles n the populaton are randomly selected to release pheromones. The method s ntated wth an exploraton characterstc n the early stage of the optmzaton process snce partcles wth poor objectve functon values are allowed to release pheromones. For subsequent teratons, the objectve functon value for each partcle n the populaton s evaluated and only partcles fndng an mprovement n ts current objectve functon (compared to the objectve functon value from the prevous teraton) wll release a dgtal pheromone. The partcle therefore marks ths locaton of the desgn space as promsng for mprovng the soluton and potentally contans an optmum. The level of the dgtal pheromone released, P, s allocated a value of.0. Just as natural pheromones produced by nsects decay n tme, a user defned decay rate, λ P, defaultng to 0.995, s assgned to the dgtal pheromones released by the partcle swarm. Dgtal pheromones are decayed as the teratons progress forward to allow the swarm to move toward a better desgn pont nstead of gettng attracted to an older pheromone wth a poorer objectve functon value. The decay process s shown n equatons (3) and (4). P =! P ) (3) new P ( current! = (4) P C. Mergng Dgtal Pheromones Snce a partcle wll release a dgtal pheromone whenever t fnds a soluton mprovement, a large number of pheromones are potentally generated durng the optmzaton process. Therefore, an addtonal step to reduce them to a manageable number, yet retanng the functonalty, was mplemented. Pheromones that are closely packed wthn a small regon of the desgn space are merged together. To check for mergng, each pheromone s assocated wth an addtonal property denoted ts Radus of Influence (ROI). For each dmenson of a pheromone, an ROI s computed and stored. The value of ths ROI s a functon of the pheromone level and the bounds of the desgn varables. If the dstance between two pheromones for a desgn varable s less than the sum of the ROIs, the pheromones are merged nto one. Ths s analogous to sayng that two spheres are merged nto one f the dstance between them s less than the sum of ther rad. A resultant pheromone level s then computed for the merged pheromones. Regons of the desgn space wth stronger resultant pheromone levels wll attract more partcles and deally, pheromones that are closely packed would ndcate a hgh chance of optmalty. Also smlar to the pheromone level decay, the ROI also has ts own decay factor, λ ROI, whose value s set equal to λ P as a default. Ths s to ensure that both the pheromone levels and the radus of nfluence decay at the same rate. Fgure llustrates ths process. 4 Amercan Insttute of Aeronautcs and Astronautcs

7 Check f ntersectng wth any other dgtal pheromones. Calculate new locaton of pheromone Create new merged pheromone Downloaded by IOWA STATE UNIVERSITY on Aprl, 05 DOI: 0.54/ Fgure. Flowchart of dgtal pheromones mergng process. D. Attracton to a Dgtal Pheromone Wth numerous dgtal pheromones placed wthn the desgn space, t s crucal to compute a sngle target dgtal pheromone for each partcle of the swarm. The crteron for ths computaton are a) a small magntude of dstance from the partcle and b) a hgh pheromone level. Therefore, n order to rank whch dgtal pheromone has the most nfluence and attracton, a target pheromone attracton factor P s computed. The value of P s a product of the normalzed dstance between that pheromone and the partcle, and the current pheromone. Also, the attracton factor must ncrease when a pheromone s closer to a partcle. Therefore, the normalzed dstance s subtracted from one as shown n equaton (5). Equaton (6) computes the dstance between the pheromone and each partcle n the swarm. Fgure 3 shows an example scenaro of a partcle beng attracted to a target pheromone. P' (! d)p = (5) k ' Xp $ k! X k d = ( % ", k = : n & rangek # Xp! Locaton of pheromone X! Locaton of partcle Repeat untl no pheromones can be merged # of desgn varables The partcle n Fgure 3 wll be more attracted to a pheromone wth a hgher P value, as opposed to pheromones that are closer but wth a lower P value. (6) 5 Amercan Insttute of Aeronautcs and Astronautcs

8 X Desgn Space P = 0.5 P =0.35 P = 0.4 P =0.45 d = 0.5 d = 0.35 d = 0.4 P = 0.87 P =0.5 TARGET Pheromones Partcle d = 0. Downloaded by IOWA STATE UNIVERSITY on Aprl, 05 DOI: 0.54/ Fgure 3. Illustraton of target pheromone selecton. E. Velocty Vector Update The dgtal pheromone computatons are added as a fourth component n addton to the prevous velocty, pbest and gbest components n the velocty vector update equaton. Ths s shown n equatons (7)-(9). V + = w * V X + V + = X! =! w w w * + c * rand() *( pbest []! X []) + c P = 0.65 P = " rand() " ( gbest[]! X []) + c * rand() *( Target[]! X []) + (9) where w s the nerta weght specfyng how much the current velocty vector wll affect the new velocty vector. The nerta weght s ntalzed at.0 and s gradually reduced wth a decay factor of λ w = as the number of teratons ncreases. c 3 s the confdence parameter for the pheromone component of the velocty vector, and s set to be larger than c and c, Ths s done n order to ncrease the nfluence of pheromones n the velocty vector. From expermentaton, t was found that a default value of 0.0 suffced for most problems. Table 8 n the results secton explans the effect of alterng values for c 3. F. Move Lmts Modfyng the velocty vector update equaton to nclude an addtonal component consderably ncreases the magntude of the velocty vector, especally f the weghtng constant c 3 s set to be a large value. In order to avod the velocty vector from becomng unmanageably large, a move lmt s mposed. The move lmt s set to an ntal value and reduced gradually as the teratons progress forward. Ths ensures a far amount of freedom n exploraton n the begnnng and as the method approaches a soluton, a smaller move lmt explots the current desgn pont of a partcle for a more constraned search towards an optmum. Although ths s a user defned parameter, an ntal set value of 0% of the desgn space for the move lmt showed good performance characterstcs. Equaton (0) shows ths mathematcal representaton. X (7) (8) ML * " ML =! ML ML (0) +! Move lmt decay factor, default Amercan Insttute of Aeronautcs and Astronautcs

9 IV. Results A. Test cases Four separate unconstraned problems of varyng dmensonalty were used as test cases to evaluate the performance of PSO wth dgtal pheromones. These problems were:. Sx-hump camelback problem Two desgn varables, mult-modal soluton, 0 ntal partcles.. Ackley s path Fve desgn varables, mult-modal soluton, 50 ntal partcles. 3. Ackley s path 0 desgn varables, mult-modal soluton, 00 ntal partcles. 4. Sprng-mass system 0 desgn varables, unmodal soluton, 00 ntal partcles. Downloaded by IOWA STATE UNIVERSITY on Aprl, 05 DOI: 0.54/ As a smple rule of thumb, the number of partcles allocated for a partcular test case problem s defned as 0 tmes the number of desgn varables. Other parameters for the a basc PSO method and PSO wth dgtal pheromones were set to the values dsplayed n Table for all test cases soluton runs. The results from the test cases were obtaned from 000 runs. test cases were evaluated on a PC wth an Intel Pentum 4,.66GHZ processor, the Wndows XP SP operatng system, and GB RAM. Table. Default values of parameters used for test cases Parameter Default value c.0 c.0 c Sze of ntal move lmt, ML 0.*range of desgn varables Move lmt decay factor, λ ML Inerta weght ntal value, w.0 Inerta weght decay factor, λ w Pheromone level decay factor, λ P Pheromone radus of nfluence decay factor, λ ROI Sx-hump Camel Back Functon Ths s a mult-modal problem of two desgn varable wth sx local mnma, two of whch are global mnma. The optmzaton problem statement s: Mnmze: F( x, x " 3! x Soluton: F mn ( x, x ( x, x 4 ( x % ) = & 4. # " x x x x ' $! 3 and "! x! ) =!.0368 ) = (0.0898,! 0.76), + (" 4 + 4x ) x (! ,0.76) From Table, the soluton accuracy and tmes from both methods are farly equal, but PSO wth dgtal pheromones generally solved the problem n fewer teratons. In terms of tme, both methods were equal. Thus, the computatonal overhead assocated wth dgtal pheromones does not apprecably detract from soluton effcency. Table. Summary of results for Sx-hump Camel Back Functon Soluton Iteraton Duraton (seconds) Accuracy Average Mn Max Std Dev Average Mn Max Std Dev Basc PSO 85.3% PSO wth Dgtal Pheromones 88.4% Amercan Insttute of Aeronautcs and Astronautcs

10 . Ackley s Path Fve Desgn Varable Ackley s Path problem s a scalable optmzaton problem. In two dmensons ts behavor shown n Fgure 4. Ths problem contans many local mnma and a sngle global mnma. For ths paper, two test cases have been based on Ackley s Path problem: ) a fve desgn varable problem and ) a 0 desgn varable problem. The optmzaton problem statement s formulated as: Downloaded by IOWA STATE UNIVERSITY on Aprl, 05 DOI: 0.54/ Mnmze: F( x) = " a # e a = 0; " 3.768! x Soluton: " b# b = 0.; 5 x 5 $ " e! F mn ( x) = 0.0, x = cos $ ( c# x ) 5 c = # PI; + a + e = : 5; Fgure 4. Illustraton of a two-dmensonal Ackley s Path functon. Bounds of desgn space: Left mage [-0, 0], Rght mage [-, ]. Table 3 shows the results from the 000 soluton runs performed. For ths problem, PSO wth dgtal pheromones performed better n terms of soluton accuracy when compared to basc PSO. However, for ths problem basc PSO performed better n terms of speed (.e. teratons and tme), whle not locatng the global optma n approxmately 5% of the soluton runs. Lookng at the complex nature of the desgn space from Fgure 4, t was concluded that the ncluson of pheromones spread the swarm around the desgn space and took longer to locate the optmal soluton. However, ths more thorough nvestgaton led to locatng the global optma nearly 00% of the tme. Table 3. Summary of results for Ackley s Path Fve Desgn Varable. Soluton Iteraton Duraton (seconds) Accuracy Average Mn Max Std Dev Average Mn Max Std Dev Basc PSO 76.0% PSO wth Dgtal Pheromones 99.9% Ackley s Path 0 Desgn Varable The optmzaton problem statement for the 0 desgn varable formulaton of Ackley s Path problem s: 8 Amercan Insttute of Aeronautcs and Astronautcs

11 Mnmze: F( x) = " a # e " 3.768! x 0 $ x " b# 0 " e! cos ( c# x ) a = 0; b = 0.; c = # PI; = : 0; Soluton: F mn ( x) = 0.0, x = 0.0 $ 0 + a + e Downloaded by IOWA STATE UNIVERSITY on Aprl, 05 DOI: 0.54/ Table 4. Summary of results for Ackley s Path 0 Desgn Varable. Soluton Iteraton Duraton (seconds) Accuracy Average Mn Max Std Dev Average Mn Max Std Dev Basc PSO 0.0% PSO wth Dgtal Pheromones 8.6% Table 4 shows the results from the soluton runs for ths test case. The advantage of usng dgtal pheromones wth PSO becomes extremely evdent n ths test case, where basc PSO faled to fnd the global optma on any of the 000 soluton runs. PSO wth dgtal pheromones located ths soluton approxmately 8% of the tme. In addton, t found the soluton much more effcently n terms of teratons. Even wth a lower number of teratons, the average soluton tme was stll hgher. Ths agan s due to havng an analytcal objectve functon on the same order of magntude as the comparsons and computatons nvolved wth the pheromone operatons. 4. Sprng-Mass System For ths test problem, a sprng-mass system as shown n Fgure 5 was consdered. The desgn varables X and Y represent the coordnates of the fve weghts, wth respect to the coordnate axes shown n Fgure 5. The goal s to mnmze the potental energy of the system, thus brngng the entre system to equlbrum. The standard optmzaton statement s: Mnmze: F = N + = where : ( L ) = K( L + = [( X ' X ) + ( Y ' Y )] + ) j= W Y j+ & N # K = $ '!, % 3 " W = 50 j, j = : 5 j N j j j N = 5, ' L o = : 6 9 Amercan Insttute of Aeronautcs and Astronautcs

12 Fgure 5. Sprng-mass system 5. Downloaded by IOWA STATE UNIVERSITY on Aprl, 05 DOI: 0.54/ Table 5. Summary of results for Sprng-Mass System. Soluton Iteraton Duraton (seconds) Accuracy Average Mn Max Std Dev Average Mn Max Std Dev Basc PSO 4.3% PSO wth Dgtal Pheromones 96.% The advantages of the proposed method are agan demonstrated n ths test case, as shown n Table 5. The soluton accuracy of basc PSO pales n comparson to that obtaned by PSO usng dgtal pheromones, when appled to a problem of hgh dmensonalty. 5. Compled results for all test cases Table 6. Compled summary of results from all test cases. (N: number of desgn varables). Soluton accuracy Mean number of teratons Mean duraton (seconds) N Basc Phrms Basc Phrms % mprovement Basc Phrms % mprovement 85.3% 88.4% % 99.9% % 96.% % 8.6% Table 6 shows a combned summary of all the test cases performed to evaluate the performance of PSO wth dgtal pheromones as compared to basc PSO. Despte havng a reducton n the number of teratons beng performed to reach a soluton, the tme taken to complete the optmzaton process s ncreased. Ths s because the addtonal computaton requred for pheromone calculatons s the same order of magntude of the tme t takes to evaluate the analytcal objectve functon n each test case. However, as the dmensonalty of the test case ncreased, basc PSO have great dffculty n locatng the global soluton. In the case of the 0 desgn varable Ackley Path problem, t was vrtually unable to ever reach the global soluton. B. Results for longer objectve functon evaluaton tme The test cases presented to ths pont were academc n nature wth easly computed analytcal objectve functons. Thus, they do not represent the types of problems solved n realstc engneerng scenaro where functon evaluatons can take consderably longer perods of tme. Wth the general decrease n teratons shown from the prevous test cases, sgnfcant savngs n tme was expected wth an objectve functon evaluaton of longer duraton. To test ths theory sleep tmes were added when evaluatng the objectve functon of the 0 desgn varable Ackley s Path functon. Ths provdes a smulated objectve functon wth a longer evaluaton tme. 0 Amercan Insttute of Aeronautcs and Astronautcs

13 Table 7. Summary of results for Ackley s Path functon (0 Desgn varables) wth varable sleep tmes. Sleep tme Duraton Basc Duraton Pheromones (mllseconds) (seconds) (seconds) % dfference Table 7 shows a sgnfcant mprovement n soluton tmes as the length of sleep tmes s ncreased. Fgure 6 compares the performance of basc PSO and PSO wth dgtal pheromones. The postve percentage dfference for sleep tmes of 5, 0 and 0 mllseconds proves that the break even pont for decreased soluton tmes wth ncreased objectve functon complexty les between 0 and 5 mllseconds. Thus, the benefts of usng PSO wth dgtal pheromones as opposed to basc PSO ncreases sgnfcantly as the objectve functon evaluaton tme ncreases. Downloaded by IOWA STATE UNIVERSITY on Aprl, 05 DOI: 0.54/ Soluton tme (seconds) Soluton tme VS Sleep tme Sleep tme (mllseconds) Basc Pheromones Fgure 6. Plot of resultng tme to solve for an optmum, wth varyng sleep tmes are ntroduced. C. Varyng parameter values The fnal test cases were run to examne some newly created parameters n the develop method. Specfcally, these were the confdence parameter c 3 and the sze of the move lmts. These tests were agan run on the 0 desgn varable Ackley s Path problem.. Confdence Parameter, c 3 The method was tested wth varous values for c 3 and checked for performance mprovement. Table 8 summarzes the results obtaned for c 3 =, 50, 00 and compared wth a default value of c 3 = 0 (hghlghted yellow). Varyng c 3 sgnfcantly affected the soluton accuracy, but only showed a slght effect on the number of teratons and the tme taken for a soluton. From Fgure 7, t s justfable to select c 3 = 0 as a suggested default value although t can be user defned dependng upon problem characterstcs. Table 8. Summary of results when usng varyng values of parameter c 3. c 3 Accuracy Iteratons Duraton (seconds) Avg Mn Max Std Dev Avg Mn Max Std Dev 64.5% % % % Amercan Insttute of Aeronautcs and Astronautcs

14 Soluton accuracy when usng varyng values of parameter C % 80.00% 70.00% 60.00% Accuracy (%) 50.00% 40.00% 30.00% Downloaded by IOWA STATE UNIVERSITY on Aprl, 05 DOI: 0.54/ % 0.00% 0.00% Fgure 9. Soluton accuracy when usng varyng values of parameter c 3.. Sze of Move Lmts Snce the move lmt controls the maxmum allowable dstance a partcle can travel wthn the desgn space, ths parameter s crtcal to the effcency and accuracy of the method. In Table 9, the summary of results obtaned wth varous ntal move lmt szes are tabulated. The accuracy of the solutons obtaned from the dfferent szes of move lmts dd not dffer sgnfcantly. However, there s a notceable lnear relatonshp between the ntal sze of the move lmt to the number of teratons and soluton tmes. Although the parameter s user defned, a 0% value was expermentally determned to be sutable for all the test cases performed. Table 9. Summary of results when usng varyng values of the ntal move lmt sze, ML. ML Accuracy Iteratons Duraton (seconds) Avg Mn Max Std Dev Avg Mn Max Std Dev 5% 8.8% % 8.6% % 8.9% % 80.3% V. Summary, Conclusons, and Future Work Ths paper proposes a new method of mplementng dgtal pheromones nto a PSO algorthm. The use of dgtal pheromones s drectly modeled from natural pheromones used n real nsect swarms to effcently and accurately search a gven doman. In ths method, dgtal pheromones were used to more effcently search an optmal desgn space and locate the global mnma. From the test cases presented, t s evdent that the method showed sgnfcant mprovement n terms of soluton effcency and accuracy on a wde range of optmzaton problems. Whle some mprovements exst for lower dmensonal problems, sgnfcant mprovements, partcularly n obtanng an accurate soluton, were observed for hgh dmensonal problems or those wth longer objectve functon evaluaton tmes. Future work wll nclude further method refnement. In addton, the method wll be tested on constraned problems and aganst other mult-modal soluton methods. C3 References Kennedy, J., and Eberhart, R. C., "Partcle Swarm Optmzaton", Proceedngs of the 995 IEEE Internatonal Conference on Neural Networks, Vol. 4, Inst. of Electrcal and Electroncs Engneers, Pscataway, NJ, 995, pp Amercan Insttute of Aeronautcs and Astronautcs

15 Downloaded by IOWA STATE UNIVERSITY on Aprl, 05 DOI: 0.54/ Eberhart, R. C., and Kennedy, J., "A New Optmzer Usng Partcle Swarm Theory", Proceedngs of the Sxth Internatonal Symposum on Mcro Machne and Human Scence, Inst. of Electrcal and Electroncs Engneers, Pscataway, NJ, 995, pp J.F. Schutte. Partcle swarms n szng and global optmzaton. Master s thess, Unversty of Pretora, Department of Mechancal Engneerng, A. Carlsle and G. Dozer. An off-the-shelf pso. In Proceedngs of the Workshop on Partcle Swarm Optmzaton, 00, Indanapols. 5 Russell C. Eberhart and Yuhu Sh, Partcle swarm optmzaton: Developments, applcatons, and resources, In Proceedngs of the 00 Congress on Evolutonary Computaton 00, Hu X H, Eberhart R C, Sh Y H., Engneerng Optmzaton wth Partcle Swarm, IEEE Swarm Intellgence Symposum, 003: G. Venter and J. Sobeszczansk-Sobesk, Multdscplnary optmzaton of a transport arcraft wng usng partcle swarm Optmzaton, In 9th AIAA/ISSMO Symposum on Multdscplnary Analyss and Optmzaton 00, Atlanta, GA. 8 P.C. Foure and A.A. Groenwold, The partcle swarm algorthm n topology optmzaton, In Proceedngs of the Fourth World Congress of Structural and Multdscplnary Optmzaton 00, Dalan, Chna. 9 Sh, Y., Eberhart, R., Parameter Selecton n Partcle Swarm Optmzaton, Proceedngs of the 998 Annual Conference on Evolutonary Computaton, March Sh, Y., Eberhart, R., A Modfed Partcle Swarm Optmzer, Proceedngs of the 998 IEEE Internatonal Conference on Evolutonary Computaton, pp 69-73, Pscataway, NJ, IEEE Press May 998 Natsuk H, Htosh I., Partcle Swarm Optmzaton wth Gaussan Mutaton, Proceedngs of IEEE Swarm Intellgence Symposum, Indanapols, 003:7-79. Hu, X., Eberhart, R., Sh, Y., Swarm Intellgence for Permutaton Optmzaton: A Case Study of n-queens Problem, IEEE Swarm Intellgence Symposum 003, Indanapols, IN, USA 3 Venter, G., Sobeszczansk-Sobesk, J., Partcle Swarm Optmzaton, AIAA Journal, Vol.4, No.8, 003, pp Hu, X., Eberhart, R., Solvng Constraned Nonlnear Optmzaton Problems wth Partcle Swarm Optmzaton, 6 th World Multconference on Systemcs, Cybernetcs and Informatcs (SCI 00), Orlando, USA 5 Schutte, J., Renbolt, J., Fregly, B., Haftka, R., George, A., Parallel Global Optmzaton wth the Partcle Swarm Algorthm, Int. J. Numer. Meth. Engng, Hu, X., Eberhart, R., Sh, Y., Partcle Swarm wth Extended Memory for Multobjectve Optmzaton, Proceedngs of 003 IEEE Swarm Intellgence Symposum, pp 93-97, Indanapols, IN, USA, Aprl 003, IEEE Servce Center 7 Tayal, M., Wang, B., Partcle Swarm Optmzaton for Mxed Dscrete, Integer and Contnuous Varables, 0 th AIAA/ISSMO Multdscplnary Analyss and Optmzaton Conference, Albany, New York, Aug 30-, Walter, B., Sanner, A., Reners, D., Olver, J., UAV Swarm Control: Calculatng Dgtal Pheromone Felds wth the GPU, The Interservce/Industry Tranng, Smulaton & Educaton Conference (I/ITSEC),Volume 005 (Conference Theme: One Team. One Fght. One Tranng Future). 9 Gaudano, P, Shargel, B., Bonabeau, E., Clough, B., Swarm Intellgence: a New C Paradgm wth an Applcaton to Control of Swarms of UAVs, In Proceedngs of the 8 th Internatonal Command and Control Research and Technology Symposum, Colorn, A., Dorgo, M., Manezzo, V., Dstrbuted Optmzaton by Ant Colones, In Proc. Europ. Conf. Artfcal Lfe, Edtors: F. Varela and P. Bourgne, Elsever, Amsterdam, 99. Dorgo, M., Manezzo, Colorn, A., Ant System: Optmzaton by a Colony of Cooperatng Agents, In IEEE Trans. Systems, Man and Cybernetcs, Part B, Vol. 6, Issue, pp 9-4, 996. Montgomery, J., Towards a Systematc Problem Classfcaton Scheme for Ant Colony Optmzaton, Techncal Report tr0-5, School of Informaton Technology, Bond Unversty, Australa, Whte, T., Pagurek, B., Towards Mult-Swarm Problem Solvng n Networks, cmas, p. 333, Thrd Internatonal Conference on Mult Agent Systems (ICMAS 98), Parunak, H., Purcell M., O Conell, R., Dgtal Pheromones for Autonomous Coordnaton of Swarmng UAV s. In Proceedngs of Frst AIAA Unmanned Aerospace Vehcles, Systems, Technologes, and Operatons Conference, Norfolk, VA, AIAA, Vanderplaats, G., Numercal Optmzaton Technques for Engneerng Desgn, 3 rd Edton, Vanderplaats Research and Development, Inc., Colorado Sprngs, CO, 999 pp Amercan Insttute of Aeronautcs and Astronautcs