Makespan Minimization of Machines and Automated Guided Vehicles Schedule Using Binary Particle Swarm Optimization

Transcription

1 Makespan Minimizaion of Machines and Auomaed Guided Vehicles Schedule Using Binary Paricle Swarm Opimizaion Muhammad Hafidz Fazli bin Md Fauadi and Tomohiro Muraa Absrac An efficien and opimized Auomaed Guided Vehicles (AGVs) operaion plays a criical role in improving he performance of a Flexible Manufacuring Sysem (FMS). Among he main elemens in he implemenaion of AGV is ask scheduling. This is due o he fac ha efficien scheduling would enable he incremen of produciviy and reducing delivery cos whils opimally uilizes he enire flee. In his research, Binary Paricle Swarm Opimizaion (BPSO) is used o opimize simulaneous machines and AGVs scheduling process wih makespan minimizaion funcion. I is proven ha he mehod is capable o provide beer soluion compared o ohers. Index Terms Flexible Manufacuring Sysem, Auomaed Guided Vehicle, Paricle Swarm Opimizaion I. INTRODUCTION Rapid developmen of informaion echnology has made he compeiion in manufacuring indusry becoming more complex and siff. Manufacurers are able o deliver produc in relaively shorer ime han ever. Thus, o win marke share, managing informaion of a manufacuring company is very crucial in order o ensure ha he informaion could be use when he company needs hem. Over he las few years, researchers had been inensely discussing abou he implemenaion of Flexible Manufacuring Sysem (FMS). While here are cerain scienific advancemens made, i is obvious ha for he implemenaion o be a success, here are many problems need o be resolved. One of hem is regarding simulaneous scheduling of machines and AGV operaion. There are many elemens of FMS scheduling. However, he more imporan facor ha should be considered is scheduling of muliple AGV. This is due o he fac ha in ypical shop floor environmen, AGV is shared by more han one machine. Assigning a non-opimal delivery would pu oher machines in longer idle ime han i should be. On he oher hand, delaying a delivery means delaying he processing chain of he maerial. Furhermore, efficien AGV Manuscrip submied on December 16, This work was parially suppored by he Public Services Deparmen of Malaysia (JPA) and Universii Teknikal Malaysia Melaka (UTeM). Muhammad Hafidz Fazli bin Md Fauadi is a PhD suden in Graduae School of Informaion, Producion and Sysems Engineering, Waseda Universiy. (Corresponding auhor s address: 2-7 Hibikino, Wakamasu-ku, Kiakyushu-shi, Fukuoka, Japan. hafidz.waseda@yahoo.com). ask allocaion mehod would be able o increase produciviy and reduce delivery cos whils opimally uilizes he enire flee. I is especially imporan when dealing wih large flee of AGVs. Among he researches conduced on AGV scheduling discipline are on hybrid approach o address scheduling and rouing of AGV [1-3], muli-aribue dispaching rules [4-6] and deadlock-resoluion [7, 8]. II. MACHINES AND AGV SCHEDULING FOR RESOURCE-CONSTRAINED FMS A. Inroducion o FMS FMS is a highly auomaed machine cell, consising of a group of processing worksaions (usually CNC machine ools), inerconneced by an auomaed maerial handling, auomaed sorage sysem and conrolled by a disribued compuer sysem (Groover, 2004). FMS is he key o an auomaed facory. Since he erm FMS was coined, various numbers of researches had been done in order o increase he capabiliy and o explore he poenial i could bring. Alhough here are significan advancemen had been achieved, here are sill pleny rooms for improvemen o ensure ha he benefi could be fully gained. B. Resource-Consrain FMS Resource-consrained FMS scheduling problem inheris he characerisics of combinaorial problem. Uilizing limied number of machines and auomaed vehicles, he main goal is o search for he bes soluion o solve a given se of problems. Over he years, i has araced aenions from worldwide researchers. Typically mahemaical opimizaion or heurisic mehods had been applied o solve he problem raher han heoreical mehod. This is due o he reason ha hey are more applicable o be applied in acual environmen. One of he approaches normally used o solve he problem is consrained opimizaion echnique. This sudy is based on single objecive funcion where oal operaion compleion ime is he parameers ha need o be minimized. Toal operaion compleion ime, O ij = T ij + P ij, (1) where i= job, j= operaion, T ij = raveling ime, P ij = operaion processing ime. Tomohiro Muraa is serving as a Professor in Graduae School of Informaion, Producion and Sysems Engineering, Waseda Universiy, Japan. ( -muraa@waseda.jp)

2 Job compleion ime, C i = O ij n i= 1 Makespan = Max (C1, C2, C3, Cn). As he scheduling involves combinaorial problem, i is imporan o ensure ha a suiable mehodology is seleced o opimize he problem. In addiion o he abiliy of finding opimal soluion, he mehod also has o be capable o find he soluion as quick as possible. Paricle Swarm Opimizaion (PSO) possesses boh crieria menioned. III. PARTICLE SWARM OPTIMIZATION A. Sandard PSO PSO is caegorized as swarm inelligence algorihm. I is a populaion based algorihm ha is inspired by he social dynamics and emergen behavior ha arises in socially organized colonies [12-14]. I explois a populaion of paricles o search for promising regions of he search space (swarm). While each paricle randomly moves wihin he search space wih a specified velociy, i sores daa of he bes posiion i ever encounered. This is known as personal bes (Pbes) posiion. Upon finishing each ieraion, he Pbes posiion obained by all individuals of he swarm is communicaed o all of he paricles in he populaion. The bes value of Pbes will be seleced as he global bes posiion (Gbes) o represen he bes posiion wihin he populaion. Each paricle will search for bes soluion unil i find sopping crieria. The movemen of he paricles owards he opimum is governed by equaions similar o he following: V = W V + C Rand ( P X + (3) i( d + 1) id 1 bes id ) C2 Rand ( G bes X id ) X i( d + 1) = X id + Vid (4) Where W is inerial weigh, c 1 and c 2 are consans (usually c 1 = c 2 = 2), r 1 and r 2 are uniform random numbers in [0,1], P i is he bes posiion vecor of paricle i h unil ieraion, P bes is he bes posiion vecor of all paricles so far, x id is he curren posiion vecor of paricle i h, and v id is he curren velociy parameer assigned for paricle i h. For Eq. (2), he firs par represens he inerial weigh of he previous velociy. The second par corresponds o he cogniion par, which represens he personal achievemen of he paricle. The hird par is for he social par, which represens he cooperaion among paricles. B. Binary PSO (BPSO) In solving binary/ discree problems, Kennedy and Eberhar [12] have deployed he PSO o search in binary spaces by applying a sigmoid ransformaion o he velociy componen Eq. (5). I employs he concep of velociy as a probabiliy ha a bi (posiion) akes on one or zero. In he BPSO, Eq. (3) for updaing he velociy remains unchanged, bu Eq. (4) for updaing he posiion is replaced by Eq. (6). 1 sigmoid ( vid ) = (5) k 1 v + e id (2) x id 1, rand < sigmoid( v = 0, oherwise C. Uilizing BPSO o Solve Scheduling Problem This secion describes how BPSO is implemened o solve he simulaneous machines and AGVs scheduling problem. Among he mos imporan maers of conenion when designing any PSO algorihm lies on how o represen he soluions, of which paricles bear he necessary informaion relaed o he problem-o-be solved. In order o map he relaionship beween he PSO paricles and he problem domain, each paricle will corresponds o a candidae soluion of he scheduling problem. In he proposed mehod each paricle represens a feasible soluion for ask assignmen using a vecor of r elemens, and each elemen is an ineger value beween 1 o n. Fig. 1 shows an illusraive example where each row represens he paricles which correspond o a ask assignmen ha assigns five asks o hree processors, and Paricle paricle3,t4 =P2 means ha in paricle 3 he Task 4 is assigned o Processor 2. Differences beween he curren posiion of he k h paricle, X k; and he posiion wih global bes value P k (or P g) can be presened by an array of n elemen. Each elemen shows ha wheher he conen of he resuling elemen in X k is differen from he desired one (bes global value) or no. If yes, he elemen ges is value from P k (or P g). For hose elemens ha have he same conen in X k and P k (or P g), heir corresponding jobs are lised based on specified rules and are assigned o machines successively, whenever a machine becomes free. For his research, he well-known longes processing ime (LPT) was uilized as he main rule. I is due o he reason ha in erm of minimizing makespan, LPT proved o perform beer han oher convenional mehod [3, 4 and 13]. Apar from LPT, precedence and machine assignmen consrains had also been considered during he scheduling process. Fig. 1 illusraes he operaion s working principle. Paricle Number T1 T2 T3 T4 T n Paricle 1 P1 P2 P3 P4 P5 Paricle 2 P3 P2 P2 P3 P5 Paricle 3 P1 P1 P1 P2 P5 Paricle 4 P2 P2 P3 P3 P1 Paricle n P n1 P n2 P n3 P n4 P n5 Fig. 1 Mapping represenaion of BPSO FMS scheduling k id Job1 Job2 Job3 Job4 A (P g) Job1 Job2 Job3 Job4 B (X k) Subrac Job1 Job2 Job3 Job4 A - B ) (6)

3 Le he number of asks be T and number of machines and AGVs available be M. The proposed BPSO algorihm for he ask allocaion process is summarized as he following: Le M be he number of machines and AGVs. Le T be he number of asks. Le P be he size of BPSO populaion. Le PSO[i] be he posiion of he i h paricle in he enire populaion wih T-dimensional vecor, whose enries values belong o he se {1,, M} Then PSO[i][j] be he processor number o which he j h ask in he i h paricle is assigned. Le finess[i] be he objecive funcion of he i h paricle according o (1) Le V[i] be he raveled disance (or velociy) of a i h paricle represened as an M-dimensional real-coded vecor. Le G bes be an index o global-bes posiion. Le P bes [i] be he posiion of he local-bes posiion. Le P bes _finess[i] be he local-bes finess for he bes posiion visied by he i h paricle. Iniializaion: For each paricle i in he populaion: i) For each ask j, iniialize PSO[i][j] randomly from he se {1,,N} ii) Iniialize V[i] randomly iii) Evaluae finess[i] iv) Iniialize G bes wih he index of he paricle wih he bes finess (lowes cos) among he populaion. v) Iniialize P bes [i] wih a copy of PSO[i] P Opimizaion Process: Repea unil a number of generaions, equal o wice he oal number of asks, are passed: i) Find Gbes such ha finess[g bes ] finess[i] P ii) For each paricle i: P bes [i] = PSO[i] if finess[i] P bes _finess [i] P Updae V[i] according o (3) Updae PSO[i] according o (5) and (6) iii) Evaluae finess[i] P IV. EXPERIMENTAL SETUP The FMS seleced as he case in his work has he configuraion as shown in Fig. 2. The case and daa se is adoped from [11] was originaed by [9]. In he case sudy, here are 10 job ses wih each possessing four o eigh differen job sequences, dedicaed machines and numbers were specified wihin he parenhesis is he processing ime of a paricular job (refer Table V). Based on he job ses and four differen layous, 82 problems are generaed. The problems are grouped ino wo caegories. The firs caegory conain problem ses which i /p i raios are greaer han 0.25 while second caegory consiss problems whose i /p i raios are lesser han A code is used o represen he example problems. The digis succeeding EX indicae he job se and he layou respecively. Meanwhile, for second caegory, anoher digi is appended o he code. In his case, having a 0 or 1 as he las digi implies ha he process imes had been doubled or ripled, respecively. Furhermore, ravel imes are halved. There are four machines consis of compuer numerical machines (CNCs) and wo AGVs for maerial delivery purpose. While he ypes and number of machines is fixed, he speed of he vehicles is consan a 40 m/min. Furhermore, loading and unloading imes are consan a 0.5 min each. Layou 1 Layou 2 Layou 3 Layou 4 Fig. 2 Layous for he case sudy I is assumed ha here is sufficien buffer space for inpu/oupu operaions a each machine. Loading/ unloading equipmens such as palles are sufficienly allocaed. Furhermore, he machine-o-machine disance and he disance beween loading/ unloading machines are known. The disance marix of load/unload saions o machines and machine-o-machine disances for all layous are shown in Appendix (Table IV). The load/unload (L/U) saion acs as he disribuion cener for incoming raw maerials and as he collecion cener for ougoing finished pars. All vehicles sar from he L/U saion iniially hough i does no need o reurn o L/U saion in beween delivery job. V. SIMULATION RESULTS AND DISCUSSION Analysis had been conduced using MATLAB sofware. For he BPSO algorihm, V max = 4, V min = -4, c 1 = 2, c 2 = 2, swarm size is se o be 70 and he maximum of ieraive generaions I max is se o be 400. For he reason ha PSO and i varians inheri heurisic aribues, 10 runs had been conduced for every se of problem in he sudy. Average compleion ime of all he run had been aken as he compleion ime for he se. Opimized ask assignmen of machine and AGV has been conduced. The research possesses offline scheduling behavior where complee se of ask, number of machines and number of vehicles are esablished prior o he ask assignmen process. This is differen o he online scheduling ha is based on real-ime scheduling where ask assignmen is mainly based on he delivery aribues. The oucomes discussed in his paper are compared o STW [9], UGA [10] and AGA [11]. While deailed resul obained based on he proposed mehodology for he described FMS environmen is given in Appendix (Table VI and Table VII), he conribuion of BPSO in minimizing average makespan is depiced in Table I and Table II. For i /p i raio >0.25 caegory, BPSO managed o improve he makespan for Layou 2 and Layou 3 as depiced in Table

4 I. Meanwhile, for i /p i raio <0.25, alhough BPSO couldn improved average makespan for any layou, resuls from oher cases as shown in Table I and Table II proved ha BPSO is able o provide opimal soluion in minimizing scheduling makespan paricularly for FMS. Furhermore, comparison of acual makespan among all of he mehodologies had also been analysed. Based on Table VI, ou of 40 ses of problem, BPSO proved o be beer in 15 cases when compared o he oher mehods while he resuls were on par for he oher 12 problems. On he oher hand, referring o he problem caegory wih i /p i raio <0.25 as shown in Table VII, BPSO improved hree resuls and equals he soluion of 31 problems. The comparison of improvemens made is lised shown in Table III. The numbers represen oal number of problem ses eiher caegorized as I Improved makespan, E Equal o exising bes makespan or Y ye o be improved. From he resuls obained, i is clear ha BPSO successfully conribued o he minimizaion of makespan ime. I is found ha BPSO is able o ouperform oher opimizaion mehods for i /p i raio > However, for i /p i raio < 0.25, BPSO only managed o improve soluion of hree cases. There is possibiliy ha he algorihm migh be rapped in local minima. This is corresponding o he searching mechanism of BPSO where upon having a P g value; paricles end o move surrounding he posiion due o he social elemen characerisics. This will be one of he aspecs for fuure improvemen. In general, BPSO sill beered oher mehodologies noiceably. In order o ensure he resuls obained are saisically accepable, analysis on makespan minimizaion characerisic over ieraion had been conduced. To furher explain abou he minimizaion characerisic, wo graphs are included as in Fig. 3 and Fig. 4. As i will be edious o represen makespan minimizaion behavior for all of he 82 problem ses, we had normalized makespan minimizaion daa ino percenage value. TABLE I COMPARISON OF AVERAGE MAKESPAN FOR T I /P I RATIO >0.25 STW UGA AGA BPSO Layou Layou Layou Layou TABLE II COMPARISON FOR AVERAGE MAKESPAN FOR T I /P I RATIO <0.25 STW UGA AGA BPSO Layou Layou Layou Layou TABLE III COMPARISON OF IMPROVEMENTS MADE BY BPSO Mehod i /p i raio > 0.25 i /p i raio < 0.25 I E Y I E Y BPSO AGA UGA STW This is o enable he calculaion of mean average for all of he cases. The makespan value afer firs generaion is used as he maximum value while he final acceped makespan value is used o represen 100% convergence. Firs quarile and hird quarile values are used o represen he convergence variaion beween problem cases. Referring o Fig. 3 and Fig. 4, boh graphs illusrae BPSO convergence rae ( i /p i > 0.25) and ( i /p i < 0.25) respecively. I is shown ha convergence variaion for i /p i raio < 0.25 is smaller han i /p i raio > According o [15], he variaion is a normal oucome for any BPSO uilizing saic opology. Since BPSO is more suiable for large search space, i becomes more sable for i /p i < 0.25 caegory. However, boh graphs also indicae ha on average, 100% convergence could be achieved afer 200 ieraions for mos of he cases. VI. CONCLUSION Based on he analysis conduced, i is found ha BPSO managed o provide a beer opimizaion soluion paricularly for simulaneous scheduling of machines and auomaed vehicles in producion environmen. For fuure sudy, more consideraion would be given on esablishing unique BPSO opimizaion mehod. Oher BPSO variaions would be considered no only o shoren he asks compleion ime bu also o shoren calculaion ime. Anoher limiaion of he work is ha i deals wih single objecive problem. Fuure work would consider muliple objecives so as o reflec acual indusrial applicaions. Convergence Percenage Convergence Percenage BPSO Convergence Rae (/p >0.25) Mean Average Mean 1s Quarile Mean 3rd Quarile Ieraion Fig. 3 BPSO Convergence Rae (/p > 0.25) BPSO Convergence Rae (/p <0.25) Mean Average Mean 1s Quarile Mean 3rd Quarile Ieraion Fig. 4 BPSO Convergence Rae (/p < 0.25)

5 APPENDIX TABLE IV MACHINE-TO-MACHINE DISTANCE CHART Layou 1 Layou 2 Layou 3 Layou 4 From - To LU M1 M2 M3 M4 LU M1 M2 M3 M4 LU M1 M2 M3 M4 LU M1 M2 M3 M4 L/U M M M M TABLE V PROBLEM SETS (JOB SEQUENCE WITH MACHINE PROCESSING TIME DETAIL) Job Se 1 Job Se 4 Job Se 7 Job Se 9 Job 1: M1(8); M2(16); M4(12) Job 1: M4(11); M1(10); M2(7) Job 1: M1(6); M4(6) Job 1: M3(9); M1(12); M2(9); M4(6) Job 2: M1(20); M3(10); M2(18) Job 2: M3(12); M2(10); M4(8) Job 2: M2(11); M4(9) Job 2: M3(16); M2(11); M4(9) Job 3: M3(12); M4(8); M1(15) Job3:M2(7);M3(10); M1(9); M3(8) Job 3: M2(9); M4(7) Job 3: M1(21); M2(18); M4(7) Job 4: M4(14); M2(18) Job4:M2(7);M4(8); M1(12); M2(6) Job 4: M3(16); M4(7) Job 4: M2(20); M3(22); M4(11) Job 5: M3(10); M1(15) Job 5: M1(9); M2(7); M4(8); M2(10); M3(8) Job 5: M1(9); M3(18) Job 6: M2(13); M3(19); M4(6) Job 5: M3(14); M1(16); M2(13); M4(9) Job Se 2 Job 7: M1(10); M2(9); M3(13) Job 1: M1(10); M4(18) Job Se 5 Job 8: M1(11); M2(9); M4(8) Job Se 10 Job 2: M2(10); M4(18) Job 3: M1(10); M3(20); Job 1: M1(6); M2(12); M4(9) Job 2: M1(18); M3(6); M2(15) Job 1: M1(11); M3(19); M2(16); M4(13) Job 4: M2(10); M3(15); M4(12) Job 3: M3(9); M4(3); M1(12) Job Se 8 Job 2: M2(21); M3(16); M4(14) Job 5: M1(10); M2(15); M4(12) Job 4: M4(6); M2(15) Job 1: M2(12); M3(21); M4(11) Job 3: M3(8); M2(10); M1(14); Job 6: M1(10); M2(15); M3(12) Job 5: M3(3); M1(9) Job 2: M2(12); M3(21); M4(11) M4(9) Job 3: M2(12); M3(21); M4(11) Job 4: M2(13); M3(20); M4(10) Job Se 3 Job Se 6 Job 4: M2(12); M3(21); M4(11) Job 5: M1(9); M3(16); M4(18) Job 1: M1(16); M3(15) Job 1: M1(9); M2(11); M4(7) Job 5: M1(10); M2(14); M3(18); Job 6: M2(19); M1(21); M3(11); Job 2: M2(18); M4(15) Job 2: M1(19); M2(20); M4(13) M4(9) M4(15) Job 3: M1(20); M2(10) Job 3: M2(14); M3(20); M4(9) Job 6: M1(10); M2(14); M3(18); Job 4: M3(15); M4(10) Job 4: M2(14); M3(20); M4(9) M4(9) Job 5: M1(8); M2(10); M3(15); 4(17) Job 6: M2(10); M3(15); M4(8); M1(15) Job 5: M1(11); M3(16); M4(8) Job 6: M1(1O); M3(12); M4(10) TABLE VI RESULT COMPARISON OF JOB MAKESPAN FOR T I /P I RATIO >0.25 Problem i /p i raio STW UGA AGA BPSO EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX

6 TABLE VII RESULT COMPARISON OF JOB MAKESPAN FOR T I /P I RATIO <0.25 Problem i /p i raio STW UGA AGA BPSO EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX EX REFERENCES [1] E. K. Bish, F. Y. Chen, Y. T. Leong, B. L. Nelson, J. W. C. Ng, and D. Simchi-Levi, "Dispaching vehicles in a mega conainer erminal OR Specrum, vol. 27, Number 4 pp [2] M. Grunow, H. Günher, and M. Lehmann, "Dispaching muli-load AGVs in highly auomaed seapor conainer erminals, OR Specrum, vol. 26, Number 2, [3] A. I. Corréa, A. Langevin, and L.-M. Rousseau, "Scheduling and rouing of auomaed guided vehicles: A hybrid approach, Compuers & Operaions Research, [4] T. Le-Anh and M. B. M. De Koser, "On-line dispaching rules for vehicle-based inernal ranspor sysems, Inernaional Journal of Producion Research, vol. 43, Number 8 / April 15, 2005 pp [5] G. Desaulniers, A. Langevin, D. Riopel, and B. Villeneuve, "Dispaching and Conflic-Free Rouing of Auomaed Guided Vehicles: An Exac Approach, Inernaional Journal of Flexible Manufacuring Sysems vol. 15, Number 4, pp , [6] Iris F.A. Vis, René de Koser, Kees Jan Roodbergen, Leon W.P. Peeers (2001). Deerminaion of he number of AGVs required a a semi-auomaed conainer erminal. Journal of he Operaional Research Sociey. Vol 52 pp [7] NaiQi Wu and MengChu Zhou (2007).Shores Rouing of Bidirecional Auomaed Guided Vehicles Avoiding Deadlock and Blocking. IEEE/ASME Transacions on Mecharonics. Vol. 12 Issue 1 pp [8] Mariagrazia Dooli and Maria Pia Fani (2007). Deadlock Deecion and Avoidance Sraegies for Auomaed Sorage and Rerieval Sysems. IEEE Transacions on Sysems, Man and Cyberneics Par C: Applicaions and Reviews, Vol 37, No 4, July [9] Ümi Bilge and Gündüz Ulusoy (1995). A Time Window Approach o Simulaneous Scheduling of Machines and Maerial Handling Sysem in an FMS. Journal of Operaions Research. Vol. 43, No 6, pp [10] Ulusoy, Gündüz, Sivrikaya-Serifoglu, Funda and Bilge, Ümi (1997). A geneic algorihm approach o he simulaneous scheduling of machines and auomaed guided vehicles. Journal of Compuers Operaional Research. Vol 24, No 4, pp Elsevier Ld [11] Tamer F. Abdelmaguid, Ashraf O Nassef, Badawia A. Kamal and Mohamed F. Hassan (2004). A Hybrid GA/ Heurisic approach o he Simulaneous Scheduling of Machines and Auomaed Guided Vehicle. Inernaional Research of Producion Research. Vol 42, No 2, Taylor and Francis Group. [12] Kennedy J, Eberhar RC. A discree binary version of he paricle swarm algorihm. In: Proceedings of he World Muli-Conference on Sysemic, Cyberneics and Informaics. NJ: Piscaawary; [13] J.Jerald, P.Asokan, G.Prabaharan and R.Saravanan (2005). Scheduling opimisaion of flexible manufacuring sysems using paricle swarm opimisaion algorihm. The Inernaional Journal of Advanced Manufacuring Technology, Volume 25, Numbers pp Springer London. [14] Ayed Salman, Imiaz Ahmad and Sabah Al-Madani (2002). Paricle swarm opimizaion for ask assignmen problem. Microprocessors and Microsysems. Vol 26, Issue 8. P.p Elsevier Science B.V. [15] Riccardo Poli, James Kennedy and Tim Blackwell (2007). Paricle Swarm opimizaion - An overview. Swarm Inelligence Journal. Volume 1, Number 1. Pages Springer New York