A parallel implementation of particle swarm optimization using digital pheromones

Transcription

1 Mechancal Engneerng Conference Presentatons, Papers, and Proceedngs Mechancal Engneerng 006 A parallel mplementaton of partcle swarm optmzaton usng dgtal pheromones Vjay Kalvarapu Iowa State Unversty, vkk@astate.edu Jung Leng Foo Iowa State Unversty 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 Kalvarapu, Vjay; Foo, Jung Leng; and Wner, Elot H., "A parallel mplementaton of partcle swarm optmzaton usng dgtal pheromones" (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 A parallel mplementaton of partcle swarm optmzaton usng dgtal pheromones Abstract A parallel mplementaton of Partcle Swarm Optmzaton (PSO) usng dgtal pheromones to coordnate the movements of the swarm wthn an n-dmensonal desgn space s presented n ths paper. Dgtal pheromones are models smulatng real pheromones emtted 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 a Partcle Swarm Optmzaton algorthm. Dgtal swarms are used to search a desgn space wth dgtal pheromones adng communcaton wthn the swarm to mprove search effcency. Wth statstcal analyss, the pheromone strength n a regon of the desgn space s determned. The swarm then reacts accordngly based on the probablty that ths regon may contan an optmum. When mplemented n a parallel computng archtecture, sgnfcant performance ncreases were observed. Ths paper presents the method development and results from several test cases. Keywords Vrtual Realty Applcatons Center, algorthms, communcaton systems, computer smulaton, parallel processng systems, probablty, search engnes, statstcal methods, dgtal pheromones, Partcle Swarm Optmzaton (PSO) Dscplnes Computer-Aded Engneerng and Desgn Mechancal Engneerng Comments Ths s a conference proceedng from Collecton of Techncal Papers - th AIAA/ISSMO Multdscplnary Analyss and Optmzaton Conference (006): AIAA , do: 0.54/ Posted wth permsson. Ths conference proceedng s avalable at Iowa State Unversty Dgtal Repostory:

3 th AIAA/ISSMO Multdscplnary Analyss and Optmzaton Conference 6-8 September 006, Portsmouth, Vrgna AIAA A Parallel Implementaton of Partcle Swarm Optmzaton Usng Dgtal Pheromones Vjay Kalvarapu *, Jung Leng Foo and Elot Wner Iowa State Unversty, Ames, IA, 500, USA H A parallel mplementaton of Partcle Swarm Optmzaton (PSO) usng dgtal pheromones to coordnate the movements of the swarm wthn an n-dmensonal desgn space s presented n ths paper. Dgtal pheromones are models smulatng real pheromones emtted 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 a Partcle Swarm Optmzaton algorthm. Dgtal swarms are used to search a desgn space wth dgtal pheromones adng communcaton wthn the swarm to mprove search effcency. Wth statstcal analyss, the pheromone strength n a regon of the desgn space s determned. The swarm then reacts accordngly based on the probablty that ths regon may contan an optmum. When mplemented n a parallel computng archtecture, sgnfcant performance ncreases were observed. Ths paper presents the method development and results from several test cases. I. Introducton eurstc optmzaton technques such as Genetc Algorthms (GA) and Smulated Annealng (SA) are capable of exhaustvely nvestgatng desgn spaces to locate global optmal desgn ponts. The probablstc nature of these heurstc methods gves dstnct advantages over determnstc methods n fndng a global optmum and hence are popular choces n solvng mult-dscplnary optmzaton problems. A drawback to these methods s ther computatonal expense and complexty. PSO, s a dervatve free, populaton based heurstc method smlar to GA and SA. Its nherent advantage s ts smplcty n mplementaton due to a small number of parameters to adjust 3, 4. In a tradtonal PSO, an ntal randomly generated populaton swarm (a collecton of partcles) 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. A basc PSO uses ths nformaton to generate a velocty vector ndcatng a search drecton towards a promsng desgn pont, and updates the locatons of the partcles. PSO s one of the recent addtons to the global search methods 5, and the fact that t s evolutonary n nature makes t partcularly sutable for a parallel mplementaton. Ths paper focuses on the mplementaton of dgtal pheromones wthn a parallel PSO. Coupled wth a statstcal analyss on the pheromones, an effcent moveset s generated to update the search drecton of each partcle. Prevous work by the authors 6 on mplementng dgtal pheromones wthn PSO proved successful n decreasng the: a) number of functon evaluatons, b) number of teratons, c) soluton tme, and d) soluton consstency. When mplemented n parallel, the method produced further performance mprovement when compared to both seral mplementaton as well as tradtonal PSO. Ths method s tested wth a seres of n-dmensonal problems and the results are 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. Copyrght 006 by Elot H. Wner. Publshed by the, Inc., wth permsson.

4 II. Background A. Partcle Swarm Optmzaton PSO s a populaton based zero-order optmzaton method that portrays several evolutonary algorthm characterstcs smlar to Genetc Algorthms (GA) and Smulated Annealng (SA) a) Intalzaton wth a populaton of random solutons, b) Desgn space search for optmum through updatng generatons and c) Update based on prevous generatons 7. The success of the algorthm has brought substantal attenton among the research communty n the recent past 8, 9. The workng of the algorthm s based on a smplfed socal model smlar to the swarmng 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. The mathematcal formulaton of the method s gven n Equatons () and (). V+ = w * V + c * rand() * ( pbest[]! X []) + c " rand() " ( gbest[]! X []) () X X V () + = + + = w * w w! + (3) Equaton (), represents the velocty vector update of a tradtonal PSO method where rand() s a random number between 0 and. c and c are 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 the entre teraton hstory. w s called as the nerta weght 0, and decreases n every subsequent teraton by a factor of λ w, as represented n Equaton (3). Equaton () denotes the updated swarm locaton n the desgn space. In addton to the orgnally developed PSO algorthm, sgnfcant enhancements have been proposed such as: a) mutaton factors for better desgn space exploraton, 3, b) methods for constrant handlng 4, 5, c) parallel mplementaton 6, 7, d) methods for solvng mult-objectve optmzaton problems 8, e) methods for solvng mxed dscrete, nteger and contnuous varables 9. B. PSO and Dgtal Pheromones Pheromones are chemcal scents produced by nsects to communcate wth each other to fnd a sutable 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 they are the markers to determne whether or not an area s promsng for further nvestgaton. One of the well-known applcatons of dgtal pheromones s ts use n the automatc adaptve swarm management of Unmanned Aeral Vehcles (UAVs) 0,. In ths research, the UAVs are automatcally guded towards a specfc zone or target through releasng dgtal pheromones n a vrtual envronment, thereby reducng the requrement of humans physcally controllng from ground statons. Other, 3, applcatons of dgtal pheromones nclude ant colony optmzaton for solvng mnmum cost paths n graphs 4, solvng network communcaton problems 5. The concept of dgtal pheromones s consderably new 6 and has not yet been explored to ts full potental for nvestgatng n-dmensonal desgn spaces for locatng an optmum. In a basc PSO algorthm, the swarm movement obtans desgn space nformaton from only two components pbest and gbest. When coupled wth an addtonal pheromone component, the swarm s essentally presented wth more nformaton for desgn space exploraton that has a potental to reach the global optmum faster. Ths dea was prevously tested and mplemented by the authors wth a substantal amount of success. C. Parallelzaton The prmary requrement for parallelzaton s the ablty of the method to decompose nto segments for multprocessor operaton. In addton, the two hghly desrable characterstcs for parallelzaton are: a) scalablty the ablty to adapt to any number of processors wth no/mnmal changes and b) processor load balancng use of the avalable number of processors to the full extent wthout any processor substantally runnng dle. Populaton based optmzaton methods such as GA and PSO are a natural ft for parallelzaton because the method parameters do not lmt the number of processors that can be used for solvng the problem. Parallelzaton can be synchronous or asynchronous. Synchronous parallelzaton facltates a step wse parallel executon of tasks. Coarse decomposton schemes are examples of synchronous parallelzaton where each processor has ts own swarm explorng the desgn space. Solutons obtaned from dfferent processors are synchronzed and gathered on a common processor (usually, the root processor) to evaluate the fnal global optmum. Ths s acheved through the use of a barrer functon n the Message Passng Interface (MPI), the most

5 commonly used nterface for parallel programmng. Asynchronous parallelzaton s the dvdng of a sequental algorthm nto autonomous tasks each of whch can be carred out on dfferent processors. Dependences among the tasks are modeled by message passng or through shared memory 7, dependng upon the hardware confguraton. The research presented n ths paper explores the use of dgtal pheromones n PSO through mplementng two parallelzaton schemes: a) synchronous coarse gran parallel mplementaton, and b) synchronous shared pheromone parallelzaton method usng MPI (MPICH mplementaton) on a dstrbuted memory system over a Myrnet connecton. III. Methodology A. Overvew of seral mplementaton of dgtal pheromones n PSO Fgure summarzes the procedure for PSO, wth steps nvolvng dgtal pheromones hghlghted n blue. Populate partcle swarm wth random ntal values Start Iteratons 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 Algorthm. 3

6 The ntalzaton of pheromone-based PSO s smlar to a basc PSO except that 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. Pheromones (from current as well as past teratons) that are close to each other n terms of desgn varable values are merged nto a new pheromone locaton. Ths effectvely creates a pheromone pattern across the desgn space whle stll keepng the number of pheromones manageable. Based on the pheromone level and ts poston relatve to a partcle, a 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. Dgtal Pheromones and Mergng In order to populate the desgn space wth a user-defned ntal set of dgtal pheromones (default s 50% of the partcles n the populaton) are randomly selected to release pheromones, regardless of the objectve functon value. Ths s done so as to ensure a good desgn space exploraton by the partcle swarm n the ntal stages of the optmzaton process. For subsequent teratons, the objectve functon value for each partcle n the populaton s evaluated and only partcles fndng an mprovement n the objectve functon value wll release a pheromone. Any newly released pheromone s assgned a level P, wth 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 pheromones released by the partcle swarm. Dgtal pheromones are decayed as the teratons progress forward to allow the swarm to propagate toward a better desgn pont nstead of gettng attracted to an older pheromone whch may not be a good desgn pont. Check f ntersectng wth any other dgtal pheromones. Calculate new locaton of pheromone Create new merged pheromone Repeat untl no pheromones can be merged Fgure. Flowchart of dgtal pheromones mergng process. Every partcle that fnds a soluton mprovement releases a pheromone potentally makng the pheromone pool unmanageably large. Therefore, an addtonal step to reduce them to a manageable number, yet retanng the functonalty, s 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 called Radus of Influence (ROI). For each desgn varable 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. Any two pheromones for a desgn varable less than the sum of the ROIs are merged nto one. Ths s analogous to two spheres mergng 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. Through ths approach, regons of the desgn space wth stronger resultant pheromone levels wll attract more partcles and therefore, 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 4

7 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 the pheromone mergng process. Attracton to a Target Dgtal Pheromone Wth numerous dgtal pheromones generated wthn the desgn space, a swarm member needs to dentfy whch pheromone t wll be attracted too most. The crtera for generatng ths target pheromone are: a) small magntude of dstance from the partcle and b) hgh pheromone level. To rank whch dgtal pheromone from the pheromone pool fts ths crtera, 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 ts pheromone level. Also, the attracton factor must ncrease when the pheromones are closer to the partcles. Therefore, the attracton factor s computed 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 X P = 0.5 P =0.35 # of desgn varables P = 0.9 P =0.45 d = 0.5 d = 0.35 P = 0.65 P =0.50 d = 0. d = 0.4 P = 0.87 P =0.5 TARGE Desgn Space X Pheromones Partcle (6) Fgure 3. Illustraton of target pheromone selecton. In the fgure, the partcle 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. Velocty Vector Update The velocty vector update mplements the pheromone component as a thrd term n addton to the pbest and gbest components n a tradtonal PSO. Ths s shown n Equaton (7). 5

8 V + = w * V + c * rand() *( pbest []! X []) + c 3 " rand() " ( gbest[]! X []) + c * rand() *( Target[]! X []) 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. (7) Move Lmts, ML The addtonal pheromone term n the velocty vector update, especally wth a large c 3 value, can consderably ncrease the computed velocty. To avod ths value 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. A default decay factor, λ ML of value was used. B. Synchronous Coarse Gran Parallel Implementaton Seral mplementaton on processor 0 Parallel ntalzaton Seral mplementaton on processor Barrer Synchronzaton Gather Results Sort from results to select best objectve functon Report Results Seral mplementaton on processor p- STOP! Fgure 4. Schematc of synchronous coarse gran parallel mplementaton of dgtal pheromones n PSO The schematc shown n Fgure 4, detals the varous steps nvolved n the coarse gran synchronous parallelzaton scheme employed. In ths decomposton approach, each processor proceeds wth ts own copy of the seral PSO code wth ts own randomzed populaton swarm. Upon calculatng the velocty vector and partcle poston update, each processor checks for ts own convergence crtera and then arrves at the optmal pont. Usng barrer synchronzaton, optmal ponts from all the processors are gathered on the root processor and the overall best objectve functon value and ts correspondng desgn varable values are sorted and selected. 6

9 The larger the number of processors used the greater the chances of fndng the global optmum. In addton, data communcaton between the processors takes place only toward the end when gatherng results from each processor, avodng network latences the prmary bottleneck n parallelzaton. Whle ths s a desrable feature n the performed coarse gran parallelzaton, t s also true that each processor s unaware of the progress of each other processor. A potental good pheromone locaton ponted out by processor A s completely obscure to the partcle swarm n processor B. A communcaton of some sort between the processors durng an teraton could mprove the qualty of the search drecton, and hence the chances of fndng the global optmum. Ths dea s explored through the mplementaton of shared pheromones across processors. C. Shared Pheromone Parallel Implementaton Fgure 5 shows the schematc of varous steps nvolved n the method. Optmzaton processors Intalze and Start teratons - Calculate ftness value - Store best as processor-pbest - ~ 50% of populaton release pheromones n st teraton - Improved partcles release pheromones from nd teraton - Send pheromone and processor-pbest nfo to phrm processor Fnd target pheromone toward whch the swarm moves - Compute velocty vector - Update partcle postons - Create Pheromone - Check for mergng - Merge pheromones based on relatve dstance between each - Decay pheromones f teraton number > 0 Broadcast pheromone lst Store best ftness value and DVs: Current teraton: sort from processor-pbest to fnd pbest All teratons: store gbest No Pheromone processor (root) Intalze and Start teratons Broadcast gbest and check for convergence Converged? Barrer Synchronzaton Fgure 5. Schematc of shared pheromone parallel mplementaton of dgtal pheromones n PSO In ths method, the avalable processors are dvded nto two categores based on ther assgned functons, and are desgnated as the optmzaton processor(s) and a pheromone processor. Each of the optmzaton processors perform: ) random populaton swarm generaton, ) ftness value evaluaton, 3) pheromone release, 4) calculaton of target pheromones, 5) calculaton of velocty vector, and 6) partcle poston update. However, access to the common pheromone lst, pbest and gbest s obtaned only through communcaton wth the pheromone processor. The optmzaton processors are scalable to any number whle the pheromone computatons are currently performed on a sngle processor. Ths pheromone computatons are: 7 Yes STOP!

10 a. Create, merge and decay pheromones as and when released by the optmzaton processors. b. Mantan a global pheromone lst made avalable to the optmzaton processors through a broadcast. c. Gather processor-pbest (pbest on each processor) values from the optmzaton processors and determne the overall pbest and gbest. d. Perform convergence check and broadcast pbest, gbest to the optmzaton processors. Ths scheme ensures that a promsng desgn pont located by an optmzaton processor s transparent to the rest of the processors through communcaton wth the pheromone processor. Snce the pheromone and the optmzaton processors perform two dstnctly dfferent tasks, they need to be synchronzed when accessng and sendng nformaton such as the global pheromone lst for target pheromone calculaton, pbest and gbest values for velocty vector calculaton and proceedng wth subsequent teratons. Ths s acheved through the use of a barrer synchronzaton, depcted as blue dotted lnes n Fgure 5. Snce pbest and gbest are avalable on the pheromone processor, the convergence check s performed on the pheromone processor and the outcome of whether or not subsequent teratons are necessary s broadcast to the optmzaton processors. Pheromones are created and the global lst s updated every teraton on the pheromone processor. As the teratons progress toward convergence, the pheromone actvty ncreases substantally. Gven that the thrd component of the velocty vector s stll actve and wth a default c 3 value of 0.0, the magntude of the velocty vector becomes substantally large thereby causng the partcle poston update locaton to move away from the global optmum nstead of convergng towards t. Therefore, the value of c 3 was decayed as the number of teratons ncreased. Ths serves two purposes ) a hgh value of c 3 provdes a greater spread over the desgn space n the ntal stages of optmzaton, causng a better desgn space exploraton ) a low value of c 3, and hence a lower velocty vector magntude, towards the close of convergence reduces the spread of the partcle swarm thereby propagatng toward the global optmum nstead of movng away from t. Although a default decay value of 0.95 provded best results when solvng the test cases, the factor can be user defned dependng upon the problem parameters. IV. Results A. Test Cases Three unconstraned problems of varyng dmensonalty are used as test cases to evaluate the performance of PSO wth dgtal pheromones n a parallel computng envronment. The swarm sze for each test case was expermentally determned to be 0 tmes the number of desgn varables. Ths parameter s user-defned and can be adjusted to any value. The test cases used are descrbed below.. Sx-hump camelback functon Ths s a mult-modal optmzaton problem wth two desgn varables wth sx local mnma, two of whch are global mnma. The optmzaton problem statement s: Mnmze: F( x, x " 3! x 4 ( x % ) = & 4. # " x x x x ' $! 3 and "! x! + (" 4 + 4x ) x Publshed soluton: F mn ( x, x ( x, x ) =!.0368 ) = (0.0898,! 0.76), (! ,0.76) Ths problem was solved usng a swarm sze of 0 partcles per processor on both coarse gran mplementaton ands shared pheromone mplementaton. Although ths s just a two desgn varable problem, t was chosen to evaluate the scalablty characterstcs of the developed method. The solutons obtaned from the parallelzaton were tested for % accuracy wth the publshed soluton values. 8

11 . Ackley s path The problem statement for solvng the Ackley s path functon s as follows: Mnmze: F( x) = " a # e a = 0; " 3.768! x Publshed soluton: " b# b = 0.; 5 x 5 $ " e! F mn ( x) = 0.0, x = cos $ ( c# x ) 5 c = # PI; + a + e = : n; The problem s scalable to any number of dmensons. Fgure 6 llustrates a D Ackley s path functon. The fgure on the left mplcates that the functon s un-modal but a closer look at the desgn space shows that t s a mult-modal problem (fgure on the rght). Two cases were chosen to test the developed method a 5 desgn varable problem wth a swarm sze of 50 per processor and a 0 desgn varable problem wth a swarm sze of 00 per processor. The publshed mnmum value for Ackley s path functon s 0.0. Therefore, a percentage accuracy cannot be used for evaluatng the solutons obtaned from parallelzaton. Hence, the solutons obtaned were regarded as accurate f the mnmum objectve functon values were less than 0.. Fgure 6. Illustraton of a two-dmensonal Ackley s Path functon. Bounds of desgn space: Left mage [-0, 0], Rght mage [-, ]. The followng parameters were used for solvng the optmzaton problem for all test cases: 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 c 3 decay (shared pheromone mplementaton only) 0.5% decrease per teraton 9

12 Both the parallel mplementatons were coded n C++ and the results were noted from performng 00 runs for each test case. Due to the unavalablty of dentcal computng platforms, the evaluaton of coarse gran and shared pheromone parallelzaton was made on two dfferent systems and the computng envronment specfcatons are descrbed below n table. Table. Computng platforms for Coarse Gran and Shared Pheromone parallelzaton Coarse Gran Parallelzaton: Shared Pheromone Parallelzaton: - Intel Pentum 4, Xeon-MP processor,.8 GHz - Intel Pentum 4, Xeon, 3.0 GHz (hyper-threaded) - Red Hat Enterprse Lnux 4 - Red Hat Enterprse Lnux 4 - GB Memory/node wth CPUs per node - GB system Memory B. Results from the two schemes of parallelzaton. Sx-hump Camel Back Functon Table 3 summarzes the results from solvng the sx-hump camelback functon usng coarse gran approach. Table 3. Summary of results for Sx-hump Camel Back Functon usng Coarse Gran approach Number Soluton Iteraton Duraton (seconds) Seconds/ of proc. Accuracy Average Mn Max Average Mn Max Iteraton Basc PSO 00% % % PSO wth 00% Dgtal 4 00% Pheromones 8 00% The soluton accuracy for both basc and pheromone mplementaton were farly equal regardless of the number of processors used. However, the average number of teratons for the pheromone mplementaton was consderably lower compared to a basc PSO method. The overhead due to pheromones was not substantal as evdent from the tme taken per teraton n both the basc and pheromone mplementaton. There s no sgnfcant dfference n the average performance data from usng varous numbers of processors, snce each processor s performng the same functons n the same code. Gven that the problem s two-dmensonal, savngs due to lesser number of teratons through usng dgtal pheromones was not readly evdent. Table 4. Summary of results for Sx-hump Camel Back Functon usng Shared Pheromones Number Soluton Iteraton Duraton (seconds) Seconds/ of proc. Accuracy Average Mn Max Average Mn Max Iteraton PSO wth 00% Dgtal 4 00% Pheromones 8 00% Table 4 shows the results obtaned from the shared pheromone mplementaton. The soluton accuracy s the same as that used for the coarse gran mplementaton. At least one optmzaton processor and one pheromone processor are requred to solve the problem thereby makng the mnmum number of processors two. Although the dfference n the number of seconds per teraton can be partly attrbuted to dfferent computng systems used, the addtonal processng nvolved n the nformaton flow between the pheromone and the optmzaton processors s an addtonal cause for the ncreased tme per teraton.. Ackley s Path Fve Desgn Varable Table 5 summarzes the results of solvng the Ackley s 5 desgn varable problem usng the coarse gran approach. 0

13 Table 5. Summary of results for Ackley s Path Fve Desgn Varable usng Coarse Gran approach Number Soluton Iteraton Duraton (seconds) Seconds/ of proc. Accuracy Average Mn Max Average Mn Max Iteraton Basc PSO 8.00% % % PSO wth 00.00% Dgtal % Pheromones % The results ndcate that the soluton accuracy s superor to the solutons obtaned from basc PSO although there s an ncrease n the number of seconds per teraton n the thrd decmal place. Ths ncrease can be attrbuted to the overhead due to the pheromone actvty before reachng the soluton. The soluton accuracy from usng the shared pheromone approach, as shown n table 6, s almost 00% n all the cases of usng, 4 and 8 processors although the number of seconds per teraton ncreased. Ths ncrease s partly attrbuted to the ncreased nformaton flow between the optmzaton and pheromone processors. An addtonal reason s due to the barrer synchronzaton that causes all processors to wat before proceedng to the subsequent teraton. Table 6. Summary of results for Ackley s Path Fve Desgn Varable usng Shared Pheromones Number Soluton Iteraton Duraton (seconds) Seconds/ of proc. Accuracy Average Mn Max Average Mn Max Iteraton PSO wth 00% Dgtal 4 98% Pheromones 8 00% Ackley s Path 0 Desgn Varable Ths problem s the same as test case #, but wth 0 desgn varables. The results for the coarse gran approach are summarzed n table 7. Table 7. Summary of results for Ackley s Path 0 Desgn Varable Number Soluton Iteraton Duraton (seconds) Seconds/ of proc. Accuracy Average Mn Max Average Mn Max Iteraton Basc PSO 0.00% % % PSO wth 85.00% Dgtal % Pheromones % The advantage of usng dgtal pheromones wth PSO s also evdent n ths test case. Whle the basc PSO faled to fnd a soluton wthn the pre-set accuracy tolerance, the PSO wth dgtal pheromones solved the problem about 83% of the tme. The number of teratons decreased to about 90 whle the hgh average for basc PSO was about 33 (wthout locatng the soluton). The overhead to attan ths accuracy level s approxmately an addtonal 0.0 seconds per teraton. Table 8 represents the results from solvng the 0 desgn varable problem usng the shared pheromone approach. Wthn the pre-set tolerance lmts, the accuracy attaned was 00% wth an overhead of 0. seconds more per teraton when compared to PSO wth and wthout pheromones. Table 8. Summary of results for Ackley s Path 0 Desgn Varable usng Shared Pheromones Number Soluton Iteraton Duraton (seconds) Seconds/ of proc. Accuracy Average Mn Max Average Mn Max Iteraton PSO wth 00% Dgtal 4 00% Pheromones 8 00%

14 C. Varyng Objectve Functons Computaton Tme The test cases thus far are academc n nature and they do not properly scale to the type of problems solved n ndustral settngs. Evaluatng objectve functons wth longer computatonal tmes was performed n an attempt to model these types of problems. Ths stuaton was smulated by addng sleep tmes when evaluatng objectve functons. For coarse gran parallelzaton, three dfferent sleep tmes 5, 0 and 0 mllseconds were ntroduced whle calculatng the objectve functon. The results are summarzed n table 9. The results show that there s a sgnfcant mprovement n soluton tmes as the lengths of a sngle objectve functon evaluaton were ncreased. Ths means that coarse gran parallel PSO wth dgtal pheromones provdes sgnfcant tme savngs when solvng problems wth complex objectve functons that take consderable amount of computng tme. Table 9. Summary of results for Ackley s Path functon (0 Desgn varables) wth varable sleep tmes. Sleep tme Duraton Basc Duraton Pheromones (mllseconds) (seconds) (seconds) % mprove Fgure 7 compares the performance of basc PSO and PSO wth dgtal pheromones. When appled to realstc objectve functons, the benefts of usng PSO wth dgtal pheromones wll be more notceable. Snce coarse gran parallelzaton s smlar on multple processors, results from usng only processor are presented n Table 9. Soluton tmes (sec) Soluton Tmes VS Sleep Tmes Sleep Tmes (msec) Basc Pheromones Fgure 7. Plot of resultng tme to solve for an optmum, wth varyng sleep tmes are ntroduced. Solved usng one processor. In addton to the above, the Ackley s 0 desgn varable problem was solved usng the shared pheromone parallelzaton scheme wth a 0 mllsecond sleep tme, and s portrayed n fgure 8. The fgure shows that, when sleep tme s ntroduced, the soluton tme decreased wth ncreased number of processors. Ths demonstrates that the shared pheromone approach wll mprove the soluton tmes for a problem wth longer objectve functon evaluaton tmes.

15 Soluton tme VS # of processors Soluton tme (sec) # processors Fgure 8. Plot of resultng tme to solve for an optmum usng Parallel PSO wth shared pheromones, varyng the number of processors. V. Concluson Ths paper presents two dfferent parallelzaton schemes for mplementng dgtal pheromones n PSO. From the test results presented, t s evdent that the developed methods consstently found the global optmum soluton. The solutons showed mproved accuracy, especally when the complexty of the problem ncreases. The substantal ncrease n performance can be attrbuted to the ncreased transparency of the soluton progress n each processor. That s, through accessng the global pheromone lst n each teraton, the partcle swarm s drected to a better locaton n the desgn space. Havng common overall pbest and gbest values also mproved the qualty of the soluton as opposed to each processor solvng the PSO wthout any sort of communcaton as n the case of coarse gran parallelzaton. The scalablty ssue was addressed by not lmtng the number of processors that can be used for solvng the optmzaton problem, wth the excepton that at least two processors are requred for the shared pheromone parallelzaton scheme. The advantages of the proposed methods became qute sgnfcant when the complexty of the objectve functon ncreased. Ths was demonstrated when artfcal complextes were smulated, by addng sleep tmes durng objectve functon evaluaton. PSO wth dgtal pheromones performed sgnfcantly faster, whle also provdng more accurate solutons. 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 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, Kalvarapu, V., Foo, J. L., Wner, E. H., Implementaton of Dgtal Pheromones for Use n Partcle Swarm Optmzaton, 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamcs, and Materals Conference, nd AIAA Multdscplnary Desgn Optmzaton Specalst Conference, Newport, RI, -4 May

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