A statistical analysis of particle swarm optimization with and without digital pheromones

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1 Mechancal Engneerng Conference Presentatons, Papers, and Proceedngs Mechancal Engneerng 2007 A statstcal analyss of partcle swarm optmzaton wth and wthout dgtal pheromones Vjay Kalvarapu Iowa State Unversty, vkk2@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 Kalvarapu, Vjay and Wner, Elot H., "A statstcal analyss of partcle swarm optmzaton wth and wthout dgtal pheromones" (2007). 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 statstcal analyss of partcle swarm optmzaton wth and wthout dgtal pheromones Abstract Partcle Swarm Optmzaton (PSO) s a populaton based heurstc search method for fndng global optmal values n mult-dscplnary desgn optmzaton problems. PSO s based on smple socal behavor exhbted by brds and nsects. Due to ts smplcty n mplementaton, PSO has been ncreasngly ganng popularty n the optmzaton communty. Prevous work by the authors demonstrated superor desgn space search capabltes of partcle swarm through mplementng dgtal pheromones n a regular PSO. Although prelmnary results showed substantal performance gans, a quanttatve assessment has not yet been made to prove the clam. Through a formal statstcal hypothess testng, ths paper attempts to evaluate the performance characterstcs of PSO wth dgtal pheromones. Specfcally, the authors clam that the use of dgtal pheromones mproves the soluton qualty and soluton tmes are tested usng varous multdmensonal unconstraned optmzaton test problems. Conclusons are drawn based on the results from statstcal analyss of these test problems and presented n the paper. Keywords Vrtual Realty Applcatons Center, dgtal servces, problem solvng, qualty control, dgtal pheromones 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, (2007): AIAA , do: 0.254/ Posted wth permsson. Ths conference proceedng s avalable at Iowa State Unversty Dgtal Repostory:

3 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamcs, and Materals Conference<br>5th Aprl 2007, Honolulu, Hawa AIAA A Statstcal Analyss of Partcle Swarm Optmzaton Wth and Wthout Dgtal Pheromones Vjay Kalvarapu * and Elot Wner Iowa State Unversty, Ames, IA, 500, USA H Partcle Swarm Optmzaton (PSO) s a populaton based heurstc search method for fndng global optmal values n mult-dscplnary desgn optmzaton problems. PSO s based on smple socal behavor exhbted by brds and nsects. Due to ts smplcty n mplementaton, PSO has been ncreasngly ganng popularty n the optmzaton communty. Prevous work by the authors demonstrated superor desgn space search capabltes of partcle swarm through mplementng dgtal pheromones n a regular PSO. Although prelmnary results showed substantal performance gans, a quanttatve assessment has not yet been made to prove the clam. Through a formal statstcal hypothess testng, ths paper attempts to evaluate the performance characterstcs of PSO wth dgtal pheromones. Specfcally, the authors clam that the use of dgtal pheromones mproves the soluton qualty and soluton tmes are tested usng varous mult-dmensonal unconstraned optmzaton test problems. Conclusons are drawn based on the results from statstcal analyss of these test problems and presented n the paper. 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. Heurstc technques are prmarly zero order methods that nether requres desgn spaces to be contnuous nor need dervatve nformaton. As such, they are a very popular choce n place of tradtonal determnstc methods for solvng mult-modal optmzaton problems. A drawback to these methods however s ther computatonal expense and complexty. PSO,2 s a populaton based heurstc method retanng many characterstcs of evolutonary search algorthms such as GA and SA. It s a recent addton to global search methods 3 and one of ts key features s ts smplcty n mplementaton due to a small number of parameters to adjust 4, 5. In a regular 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 two members the best poston of a swarm member n ts hstory tral (pbest), and the best poston attaned by all partcles (gbest) untl that teraton. Ths nformaton s used to generate a velocty vector ndcatng a search drecton towards a promsng desgn pont, and the locaton of each swarm member s updated. However, the drawback of ths approach s that nformaton from these two members alone s not suffcent for the swarm to propagate toward the global optmum effcently. Ths ether could cause the swarm to lock nto a local mnmum or take very long tme to reach the global optmum. Prevous work by the authors demonstrated promsng performance mprovement of PSO n terms of ncreased soluton accuracy and decreased soluton tmes through mplementng dgtal pheromones n PSO 6, 7. A quanttatve assessment of the developed method has not yet been made to prove the clam. Usng standard mult-dmensonal and mult-modal benchmark problems, ths paper attempts to report the fndngs and conclusons obtaned through statstcal hypothess testng of PSO wth and wthout dgtal pheromones. A. Partcle Swarm Optmzaton II. Background * Research Assstant, Department of Mechancal Engneerng, Human Computer Interacton, Vrtual Realty Applcatons Center, 2274 Howe Hall, Iowa State Unversty, Ames, IA, 500, USA, Student Member. Assstant Professor, Department of Mechancal Engneerng, Human Computer Interacton, Vrtual Realty Applcatons Center, 2274 Howe Hall, Iowa State Unversty, Ames, IA, 500, USA, Member. Copyrght 2007 by Elot Wner and Vjay Kalvarapu. Publshed by the, Inc., wth permsson.

4 PSO shares many characterstcs of evolutonary search algorthms such as 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 8. The success of the algorthm has brought substantal attenton among the research communty n the recent past 9, 0. The workng of the algorthm s based on a smplfed socal model smlar to the swarmng behavor exhbted by nsects and brds. In ths analogy, a swarm member 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 (2). V+ = w * V + c * rand p () * ( pbest[]! X []) + c2 " rand g () " ( gbest[]! X []) () X X V (2) + = + + = w * w w! + (3) pbest represents the best poston attaned by a swarm member n ts hstory tral, and gbest represents the best poston attaned by the swarm n the entre teraton hstory. Equaton (), represents the velocty vector update of a tradtonal PSO method where rand p () and rand g () are random numbers generated between 0 and each for pbest and gbest. c and c 2 are confdence parameters. w s called as the nerta weght, 2 and decreases n every teraton by a factor of λ w, as represented n Equaton (3). Equaton (2) 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, 4, b) methods for constrant handlng 5, 6, c) parallel mplementaton 7, 8, d) methods for solvng mult-objectve optmzaton problems 9, e) methods for solvng mxed dscrete, nteger and contnuous varables 20. 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) 2, 22. 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 23, 24 applcatons of dgtal pheromones nclude ant colony optmzaton for solvng mnmum cost paths n graphs solvng network communcaton problems 25. The concept of dgtal pheromones s consderably new 26 and has not yet been explored to ts full potental for nvestgatng n-dmensonal desgn spaces for locatng an optmum. In a regular 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 and has a potental to reach the global optmum faster. C. Statstcal Hypothess Testng In statstcal terms, a populaton s a group or ndvdual that represents all members of a certan category of nterest. A sample s a subset drawn from the populaton. Descrptve statstcs apply only to the members of a sample of data collected from the populaton. Inferental statstcs, on the other hand refer to the use of sample data to reach conclusons about the characterstcs of the populaton that the sample represents. A hypothess s typcally a statement about the parameters n a populaton dstrbuton. It s called as hypothess because t s not known whether the statement s true or not. The prmary objectve of hypothess testng s to test whether or not the values of a random sample from the populaton s consstent wth the clamed hypothess or not. The hypothess s consdered as accepted f the random sample s consstent wth the hypothess under consderaton. Else, the hypothess s consdered as rejected 27, 28, 29, 30. Prevous research on dgtal pheromones for use n PSO has produced sgnfcant mprovement strdes n terms of soluton qualty and soluton tmes. Ths paper attempts to quanttatvely evaluate ths clam through formal hypothess testng. The rest of the paper s organzed as follows: An overvew of dgtal pheromones n PSO s provded followed by varous consderatons and steps observed for performng the hypothess testng. The fndngs 2

5 are reported and conclusons are then drawn based on testng the soluton qualty and tmngs of fve dfferent multmodal mult-dmensonal unconstraned optmzaton problems. III. Methodology A. Overvew 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 pbest and gbest Decay dgtal pheromones n the desgn space (f any) st teraton? No Only mproved partcles release pheromones Yes Merge pheromones based on relatve dstance between each Fnd target pheromone toward whch the swarm moves Update velocty vector and poston of the swarm 50% of swarm randomly chosen to release pheromones No Converged? Yes STOP! Fgure Overvew of PSO wth Dgtal Pheromones 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 3

6 (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 feld 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 a thrd 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 an ntal set of dgtal pheromones, 50% of the populaton s 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 0.95, 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 2 Illustraton of pheromone 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 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 2 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 4

7 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 2 # of desgn varables (6) 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. Fgure 3 Illustraton of target pheromone selecton 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). V + = w * V X 2 + c * rand P = 0.5 P =0.325 p P = 0.9 P =0.4 2 () *( pbest []! X d = 0.5 d = 0.35 d = P = P =0.50 []) + c + c 3 2 * rand d = 0.4 " rand T g Desgn Space P = 0.87 P =0.522 TARGET PHEROMONE () " ( gbest[]! X () *( Target[]! X c 3 s the confdence parameter for the pheromone component of the velocty vector, and s typcally set to be larger than or equal to c and c 2, 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 suffced for most problems. 4 X []) []) Pheromone s Partcle (7) Move Lmts, ML The addtonal pheromone term n the velocty vector update, 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 5

8 value of 0% of the desgn space for the move lmt showed good performance characterstcs. A default decay factor, λ ML of value 0.95 was used. B. Statstcal Hypothess Testng The prmary objectve of ths research s to test the clam that dgtal pheromones when mplemented n PSO perform better when compared to regular PSO n terms of soluton qualty (.e., better optmum values) and soluton tmes. The hypothess that specfes a partcular value for the parameter beng studed s called the null hypothess and s denoted by H 0. It represents the standard operatng procedure of a system or a known procedure. The hypothess that specfes those values of the parameter that represent an mportant change from standard operatng procedure or known procedure s called the alternatve hypothess or research hypothess, and s denoted by H a. Evdence from a sample of the results nconsstent wth the stated hypothess leads to rejecton of the hypothess, whereas evdence supportng the hypothess leads to ts acceptance. In statstcal hypothess testng, t s a norm that the acceptance of a proposed hypothess s the result of nsuffcent evdence to reject t. There are two ways that errors can be commtted n the decson process usng hypothess testng. A type I error s commtted f the null hypothess s rejected when t s actually true. A type II error s commtted f the null hypothess s not rejected when t s actually false. Table shows the truth table of decson makng whle performng hypothess testng. The probablty of commttng a type I error s called the level of sgnfcance of the test and s denoted by α, and the probablty of commttng a type II error s denoted by β. Table Decsons and Errors n Hypothess Testng Decson H 0 Accept H 0 H 0 True Type I Error Correct decson taken H 0 False Correct decson taken Type II error Hypothess testng can be one-taled or two-taled. For example, H 0 : µ = µ 0 and H a : µ µ 0 s called a two-taled hypothess where the equalty of µ and µ 0 are tested. On the other hand, H 0 : µ < µ 0 and H a : µ µ 0 (or) H 0 : µ > µ 0 and H a : µ µ 0 s called a one-taled test, where µ represents the populaton mean and µ 0 represents the sample mean. A t-test assesses whether the mean of two groups are statstcally dfferent from each other, and s an especally approprate tool when comparson of the means of two dfferent group of parameters s desred. The t-dstrbutons are affected by the sample sze, and they approach normal dstrbutons wth large sample szes. The followng s a fve-step procedure adopted for performng hypothess testng of regular and pheromone PSO:. The null and alternate hypotheses are to be defned. Hypothess testng can be sngle-sample based or mult-sample based. In a sngle sample, the null and alternate hypothess wll have parameters only from the problem under consderaton. A two-sample test on the other hand allows for comparson of means of two dfferent methods (e.g., regular PSO and PSO wth dgtal pheromones). Hypothess testng can be performed on more than two samples smultaneously usng Analyss of Varance (ANOVA) tests. Snce the objectve of the research presented n ths paper s to nvestgate the performance characterstcs of regular PSO and pheromone PSO, a two-sample hypothess testng s performed. Based on these consderatons, one-taled null hypothess and alternate hypotheses are defned as shown n eq. 8: H 0 : µ - µ 2 0 (null hypothess) H a : µ - µ 2 > 0 (research or alternate hypothess) Where, µ and µ 2 corresponds to the means obtaned from PSO wth and wthout pheromones respectvely (8) 2. A level of sgnfcance equal to α needs to be chosen. For ths research, hypothess test s performed at a popularly used confdence levels of 95%. Ths means the hypothess test s performed wth a 0.05 probablty for type I error. 3. An approprate test statstc (.e., t) s to be selected and ts correspondng crtcal value (t crtcal ) s to be obtaned from t dstrbuton tables. Dependng upon whether there s any dependency between the data samples obtaned for regular PSO and pheromone PSO, the test can be ether ndependent or pared. Snce the test runs for regular PSO and pheromone 6

9 PSO are performed ndependent of each other and have dfferent random seed values for each durng tral runs, ndependent two-sample hypothess testng s performed. Therefore, the test statstc (or the t-value) s calculated usng eq. 9 (a standard t-value estmator whose descrpton can be looked up n any standard statstcs textbook), where x and x 2 represents the means of regular and pheromone PSO respectvely. t = (x " x 2) S p n + n 2 (9) S 2 p = (n ")s (n 2 ")s 2 n + n 2 " 2 (0) Where, eq. (0) represents the square of the standard devaton or the varance of the sample data from regular and pheromone PSO, wth (n +n 2-2) degrees of freedom. 4. The value of the statstc (t) s to be computed from the random sample of sze, n. Most t-dstrbuton tables n standard statstcs textbooks consder degrees of freedom greater than 30 as a very near approxmaton to normal dstrbuton. For all test cases used n ths research, 35 tral runs are performed each for regular and pheromone PSO. The number of degrees of freedom for ths hypothess test s (n + n 2-2) = 68, where n represents the sample sze of results from regular PSO, and n 2 represents the sample sze of results obtaned from pheromone PSO. Ths means that the data can be consdered as normally dstrbuted for all statstcal testng purposes. 5. H 0 s to be rejected f the statstc has a value n the crtcal regon; otherwse H a s to be rejected. A hypothess s accepted not because there s evdence n support of t, but t means that there s no evdence to reject t. If the value of t calculated from eq. 9 s greater than t crtcal, H 0 needs to be rejected. If on the other hand, the value of t s less than t crtcal, H a needs to be rejected. The value of t crtcal s obtaned from t-dstrbuton tables correspondng to the probablty of error chosen n step 2. IV. Results A. Problem Settngs Fve unconstraned test problems of varyng dmensonalty are used for performng hypothess testng on ther soluton qualty and soluton tmes. These problems are solved usng PSO wth and wthout dgtal pheromones, 35 tmes each to ensure normal dstrbuton. Two hypotheses are tested for each test problem wth and wthout pheromones: a) Whether the soluton qualty of PSO wth dgtal pheromones compare better aganst regular PSO, and b) Whether the soluton tmes of PSO wth dgtal pheromones compare better aganst regular PSO. The hypothess tests are performed at a 95% confdence level. Ths means that the tests are performed wth a 0.05 probablty for error. As shown n eq. 8, the null hypothess (H 0 ) states that regular PSO fares better n comparson to PSO wth dgtal pheromones and the research hypothess (H a ) states that PSO wth dgtal pheromones has better performance characterstcs than regular PSO. The fve-step procedure for performng the hypothess test s outlned n the prevous secton, and s smlar for all the test cases. For smplcty n understandng, the hypothess test for one test case (camelback 2D for soluton qualty at a confdence level of 95% wth pheromone combnaton shown n eq. ) s explaned n detal. Results from all other test cases wll be summarzed n a table and dscussed. The pheromone parameter combnatons used n ths research are shown n eq. below. The t crtcal value for 0.05 probablty n error (95% confdence level) for 68 degrees of freedom obtaned from t-dstrbuton tables s Combnaton : c 3 =2.0, λ p = 0.85, λ ML = 0.85 Combnaton 2: c 3 =5.0, λ p = 0.95, λ ML = 0.95 Combnaton 3: c 3 =5.0, λ p = 0.85, λ ML = 0.95 Combnaton 4: c 3 =5.0, λ p = 0.85, λ ML = 0.85 () 7

10 The swarm sze used for each test problem s chosen as 0 tmes the number of desgn varables, although ths value s capped at 500. The computng platform for the tral runs s Red Hat Enterprse Lnux Operatng System, wth a processor speed of 3.2GHz and 2GB of system memory. B. Test Cases Results and Dscusson Test Case : 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 2 ) = 4 " 2.x + x 4 & % ( x x x 2 + ("4 + 4x 2 )x 2 $ 3 ' "3 ) x ) 3 and " 2 ) x 2 ) 2 Table 2 explans n detal the results obtaned from performng hypothess testng on a 2D camel back functon at a 95% confdence level. Pheromone parameter combnaton (from eq. ) s used for ths test. Table 2 Hypothess test results for Camelback functon x = (soluton mean of tral runs of regular PSO) x 2 = (soluton mean of tral runs of PSO wth dgtal pheromones) S p = (from eq. 0) df = 68 (n + n 2 2) t calculated = (usng eq. 9) t crtcal (α=0.05) = (from t-dstrbuton tables wth probablty of error = α) From the table, n and n 2 are the number of samples (tral runs) drawn from PSO wth and wthout dgtal pheromones respectvely. It can be seen from table that t calculated s greater than t crtcal, whch leads to the concluson that the null hypothess, H 0 can be rejected. Ths means that the soluton qualty of regular PSO s not better than PSO when mplemented wth dgtal pheromones. Snce there s no evdence to prove that regular PSO fares better than PSO wth dgtal pheromones, the research hypothess (H a ) that the soluton qualty of PSO wth dgtal pheromones s better than a regular PSO s consdered as accepted. The soluton qualty and soluton tmngs for all test problems are estmated usng the procedure lad out n table 2. Table 3 below summarzes the hypothess testng of camelback problem for soluton qualty and soluton tmngs. Table 3 Summary of hypothess testng for Camelback 2D problem Test Case Combnaton Combnaton 2 Combnaton 3 Combnaton 4 Camelback2D t calc H 0 t calc H 0 t calc H 0 t calc H 0 qualty Tmes (< t crtcal ) Accept (< t crtcal ) Accept Ths table shows that the null hypothess, H 0 s rejected n two dfferent nstances for soluton tmes when usng pheromone parameter combnaton and 2. Ths means that the soluton qualty of regular PSO s not better than PSO wth dgtal pheromones at a 95% confdence level. The results from hypothess testng of soluton tmes also concur wth the fact that dgtal pheromones has a postve nfluence n reducng the soluton tmes when compared to regular PSO. However, wth the pheromone parameter combnatons 3 and 4, the null hypothess s accepted at 95% confdence level. That means regular PSO performs better when compared to PSO wth dgtal 8

11 pheromones n terms of soluton qualty. However, the null hypothess s rejected for pheromone combnatons 3 and 4 for soluton tmes. Whle ths hypothess test demonstrates that not all suggested pheromone parameter combnatons can be benefcal, t ponts to the fact that slght changes n the values for pheromone parameters substantally affects the performance of PSO when mplemented wth dgtal pheromones, especally n twodmensonal optmzaton problems. Test Case 2: Hmmelblau s functon (2D) Ths s a mult-modal problem wth one global mnmum and three local mnmums. The objectve functon s: F(x,x 2 ) = (x 2 + x 2 ") 2 + (x + x 2 2 " 7) 2 "6 # x # 6 and " 6 # x 2 # 6 Table 4 summarzes the hypothess testng of hmmelblau problem for soluton qualty and soluton tmngs. Table 4 Summary of hypothess testng for Hmmelblau 2D problem Test Case Combnaton Combnaton 2 Combnaton 3 Combnaton 4 Hmmelblau 2D qualty Tmes t calc H 0 t calc H 0 t calc H 0 t calc H The results from table 4 shows that the calculated t value (t calculated ) s greater than t crtcal for all pheromone parameter combnatons. Ths means that the null hypothess statng that regular PSO s better than PSO wth dgtal pheromones can be rejected at an error probablty of Both the soluton qualty and soluton tmes suggest that H 0 can be rejected. Snce there s no other evdence showng that regular PSO performs better, the research hypothess that pheromone PSO performs better s accepted. Test Case 3: Rosenbrock s functon (5D) Rosenbrock s functon (also known as the banana functon) can be parameterzed to any number of desgn varables. For ths test case, a 5 desgn varable Rosenbrock functon s mnmzed, and s gven by the followng equaton: 4 # F(x ) = 00*(x + " x 2 ) 2 + (" x ) 2 = 0 "2.048 $ x $ Table 5 summarzes the hypothess testng of Rosenbrock 5D problem for soluton qualty and soluton tmngs. Table 5 Summary of hypothess testng for Rosenbrock 5D problem Test Case Combnaton Combnaton 2 Combnaton 3 Combnaton 4 Rosenbrock 5D qualty Tmes t calc H 0 t calc H 0 t calc H 0 t calc H Hypothess testng for soluton qualty and soluton tmes of Rosenbrock 5D problem shows that the null hypothess can be rejected. The table shows that the t calculated value s greater than t crtcal for all suggested pheromone parameter combnatons. Therefore, the null hypothess statng that regular PSO performs better when compared to PSO wth dgtal pheromones s rejected. Snce there s no other evdence to prove that regular PSO can perform 9

12 better, the research hypothess statng that PSO wth dgtal pheromones has better performance characterstcs n terms of soluton qualty and soluton tmngs. Test Case 4: Ackley s path functon (0D) Test Case 5: Ackley s path functon (00D) The problem s scalable to any number of dmensons. The problem statement for solvng the Ackley s path functon s as follows: Mnmze: F(x ) = "a # e "b# 5 2 x $ 5 " e 5 $ cos( c#x ) 5 + a + e a = 20; b = 0.2; c = 2 # PI; =: n; " % x % Table 6 summarzes the hypothess testng of Ackley 0D problem for soluton qualty and soluton tmngs. Table 6 Summary of hypothess testng for Ackley 0D problem Test Case Combnaton Combnaton 2 Combnaton 3 Combnaton 4 Ackley 0D t calc H 0 t calc H 0 t calc H 0 t calc H 0 qualty Tmes Ths table shows that the null hypothess, H 0 s rejected for all suggested combnatons of pheromone parameters. Ths means that the hypothess testng demonstrates that regular PSO s not better than PSO wth dgtal pheromones at a 0.05 probablty for error. The fact that t calculated value exceeds t crtcal value for both soluton qualty and soluton tmes suggest that the research hypothess H a can be accepted due to the lack of evdence to prove superor performance of regular PSO. That means that PSO wth dgtal pheromone PSO compares better aganst regular PSO n terms of both soluton qualty and soluton tmngs for ths test problem. Table 7 summarzes the hypothess testng of Ackley 00D problem for soluton qualty and soluton tmngs. Table 7 Summary of hypothess testng for Ackley 00D problem Test Case Combnaton Combnaton 2 Combnaton 3 Combnaton 4 Ackley 00D t calc H 0 t calc H 0 t calc H 0 t calc H 0 qualty Tmes (< t crtcal ) Accept (< t crtcal ) Accept Table 7 s the result of performng hypothess testng on a 00 desgn varable Ackley s path functon. The table shows that the t calculated values for soluton qualty s greater than t crtcal for all combnatons of suggested pheromone parameters. Ths means that the null hypothess statng that regular PSO fares better n comparson to pheromone PSO can be rejected. At the same tme, the hypothess testng of soluton tmes demonstrate that the null hypothess can be accepted for pheromone combnaton and 4. Ths means that the soluton tmes for regular PSO s faster when compared to PSO wth dgtal pheromones. However, t s to be noted that the null hypothess for combnatons and 4 come at a cost of compromse n soluton qualty. Ths means that although regular PSO compares better aganst PSO wth dgtal pheromones n terms of soluton tmes, the qualty of the soluton s better wth pheromone PSO rather than wth regular PSO. 0

13 C. General Trend Hypothess testng of fve benchmarkng test problems showed a general trend for mprovement n terms of soluton qualty and soluton tmngs n PSO when dgtal pheromones are mplemented. All test cases showed ths general trend wth the excepton of Camelback functon wth pheromone combnatons 3 and 4. Ths s possble because of a hgh value of c 3 for a two dmensonal problem. Another excepton to the general trend s Ackley s path functon wth 00 desgn varables, where pheromone parameter combnatons and 4 resulted n reduced soluton tmngs. However, ths came at the cost of soluton qualty wth regular PSO. Overall, t can be generalzed that dgtal pheromones n PSO tend to produce better qualty solutons at decreased soluton tmes. V. Concluson Ths paper presents a quanttatve assessment of the performance of PSO wth and wthout dgtal pheromones through statstcal hypothess testng. The methodology secton provded a detaled procedure of how ths s attaned followed by dscusson of results from testng fve benchmarkng problems. Hypothess testng results show a promsng tendency of dgtal pheromones to mprove the desgn space search especally when the dmensonalty of the problems ncreased. Comparson of performances s made between regular PSO and varous pheromone parameters ndvdually. Wth the excepton of camelback functon for combnaton 3 and 4 pheromone parameters, the remanng test problems showed that the soluton qualty has been consstently better when dgtal pheromones n PSO are used. Although the soluton tmes of regular PSO s better n Ackley s 00 dmensonal path problem, the solutons were not as effectve as demonstrated by dgtal pheromones n PSO. Snce mplementaton of dgtal pheromones s farly new, a substantal amount of future work s requred for further development and fne-tunng. Some of the future work ncludes performng hypothess testng of varous combnatons of pheromone parameters that could potentally throw lght on how senstve the method could become based on alterng them ether pror to begnnng actual optmzaton or durng run-tme. Mult-lnear regresson and ANOVA could show the extent of correlaton between the pheromone parameters. Also, a formal statstcal analyss of PSO wth and wthout pheromones can be performed for solvng constraned optmzaton problems, thereby provdng a more approprate opportunty to evaluate the performance of dgtal pheromones n PSO. 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 Russell C. Eberhart and Yuhu Sh, Partcle swarm optmzaton: Developments, applcatons, and resources, In Proceedngs of the 200 Congress on Evolutonary Computaton 200, 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, 200, Indanapols. 6 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, 2nd AIAA Multdscplnary Desgn Optmzaton Specalst Conference, Newport, RI, -4 May Kalvarapu, V., Foo, J., Wner, E., A Parallel Implementaton of Partcle Swarm Optmzaton Usng Dgtal Pheromones, th AIAA/ISSMO Multdscplnary Analyss and Optmzaton Conference, AIAA , Portsmouth, VA, September Hu X H, Eberhart R C, Sh Y H., Engneerng Optmzaton wth Partcle Swarm, IEEE Swarm Intellgence Symposum, 2003: 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 2002, Atlanta, GA. 0 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 200, Dalan, Chna. 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 Natsuk H, Htosh I., Partcle Swarm Optmzaton wth Gaussan Mutaton, Proceedngs of IEEE Swarm Intellgence Symposum, Indanapols, 2003:72-79.

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