A Modified Particle Swarm Optimization For Engineering Constrained Optimization Problems

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1 Inernaional Journal of Compuer Science and Elecronics Engineering (IJCSEE Volume 3, Issue 1 (015 ISSN (Online A Modified Paricle Swarm Opimizaion For Engineering Consrained Opimizaion Problems Choosak Pornsing, Kawinhorn Saichareon, and Thanahorn Karo Absrac This paper presens a modified paricle swarm opimizaion (PSO algorihm for solving engineering consrained opimizaion problem. The proposed PSO presens ime-varying ineria weigh swarm opology echnique. The novely of his proposed mehod is ha he whole swarm may divide ino many subswarms in order o find a good source of food or o flee from predaors. This behavior engages he paricles disperse hrough he search space and he sub-swarm wih he wors performance dies ou while ha wih he bes performance grows by producing offspring. Furhermore, he penaly funcion mehod was used o handle he consrains of he problem. The penaly facor is a self-adapive funcion in which based on he performance of he las ieraion. Numerical experimens based on well-known consrained engineering design problem show ha he modified mehod ouperforms oher compeiive algorihms from lieraure. Keywords consrained opimizaion problem, paricle swarm opimizaion, swarm inelligence. I. INTRODUCTION Generally, a consrained opimizaion problem can be saed as follows. x 1 x Find X = which minimize f(x (1 x n subjec o he consrains g j (X 0, j = 1,,, m ( l j (X = 0, j = 1,,, p (3 where X is an n-dimensional vecor called he design vecor, f(x is ermed he objecive funcion, and g j (Xand l j (X are known as inequaliy and equaliy consrains, respecively. In a common pracice an equaliy consrain l j (X can be replaced by a couple of inequaliy consrains l j (X δ and l j (X δ (δ is a small oleran amoun. Therefore all consrains can be ransformed o M = m + p inequaliy consrains. We Dr. ChoosakPornsing, Deparmen of Indusrial Engineering and Managemen, Faculy of Engineering and Indusrial Technology, Silpakorn Universiy, Thailand: choosak@su.ac.h Kawinhorn Saichareon, Deparmen of Indusrial Engineering and Managemen, Faculy of Engineering and Indusrial Technology, Silpakorn Universiy, Thailand: kawinho@gmail.com Thanahorn Karo, Deparmen of Indusrial Engineering and Managemen Faculy of Engineering and Indusrial Technology, Silpakorn Universiy Thailand: karo@gmail.com call X a feasible soluion if i saisfies all he consrains. Usually, a radiion mahemaical programming mehod such as he Lagrange muliplier mehods requires he gradien informaion of he objecive funcion and relevan. Furhermore, he obained soluion ofen ends o be a local opimum unless he search space is convex. Recenly, evoluionary compuaion (EC echniques have araced much aenion for a variey of complex opimizaion mehod especially nonlinear consrained opimizaion problem and combinaorial opimizaion problem due o ECs do no need he objecive funcion o be derivable or even coninuous. ECs perform as global opimizaion echniques in which he cooperaion of exploraion and exploiaion processes are well balanced. In his sudy, we made use of a novel EC echnique which founded by Eberhar and Kennedy [1] in 1995 o solve engineering consrained opimizaion problems, named Paricle Swarm Opimizaion (PSO. We proposed a modified PSO algorihm in order o miigae a premaure convergence problem in which yields a local opimum soluion. Beside, we proposed a self-adapive penaly funcion echnique which he penaly facors are adjusable based on he level of consrains violaion. Two engineering opimizaion problems were used o es he proposed mehod s performance compare o published resuls in lieraure. The res of his paper is organized as follows. Secion briefly describes convenional PSO. This secion expresses he concep of PSO and is mahemaical formulaion. Secion 3 describes he modified PSO in which aemping o alleviae he premaure convergence problem. Secion 4 repors he compuaional experimens wih wo well-known engineering opimizaion problems; minimizaion of he weigh of a ension/compression spring design problem and minimizaion of he oal cos of a pressure vessel design problem. A conclusion summarizing he conribuions of his paper is in Secion 5. II. PARTICLE SWARM OPTIMIZATION (PSO Paricle Swarm Opimizaion (PSO is an evoluionary compuing (EC echnique ha belongs o he field of Swarm Inelligence proposed by Eberhar and Kennedy [1] inspired by he social behavior of bird flocking or fish schooling. PSO is an ieraive algorihm ha engages a number of simple eniies he paricles ieraively over he search space of some funcions. The paricles evaluae heir finess values wih respec o he search funcion, a heir curren locaions. 1

2 Inernaional Journal of Compuer Science and Elecronics Engineering (IJCSEE Volume 3, Issue 1 (015 ISSN (Online Subsequenly, each paricle deermines is movemen hrough he search space by combining informaion abou is curren finess, is bes finess from previous locaions wih regards o one or more members of he swarm (social perspecive, wih some random perurbaions. The nex ieraion sars afer he posiions of all paricles have been updaed as in (4 and (5. V id = V 1 id + φ 1 β 1 (pbes id X id +φ β (gbes id X id (4 X id +1 = X id + V id (5 V id is he curren velociy of ih paricle of he d-h dimension, V id 1 is he previous velociy of ih paricle of he dh dimension, and X id is he curren posiion of ih paricle of he dh dimension. φ 1 and φ are weighed facors which are also called he cogniive and social parameer, respecively. β 1 and β are random variables uniformly disribued wihin [0,1]. X id +1 is he posiion of ih paricle of he d-h dimension in he nex ieraion. pbes id and gbes id are he bes personal posiion ever visied by he paricle and he bes global posiion ever visied by all paricles in he swarm. s pbes id = arg min s f i (6 gbes id = arg min i f(pbes i (7 Shi and Eberhar [] inroduced an ineria weigh facor, ω, o he original PSO. Then, Shi and Eberhar [3] suggesed ha in order o obain he bes performance he iniial seing ω is se a some high velue (e.g. 0.9, which corresponds o paricles moving in a low viscosiy medium and performing exensive exploraion and he value of ω is hen gradually reduced o some low value (e.g A his low value of ω he paricles move in a high viscosiy medium, perform exploiaion, and are beer a homing owheads local opima. Equaion (4 is modified as below. V id = ωv 1 id + φ 1 β 1 (pbes id X id +φ β (gbes id X id (8 Because of a number of advanages of PSO rapid convergence owards an opimum, is qualiy of being easy o encode or decode, and being fas and easy o compue i has been applied in many research areas such as global opimizaion, arificial neural nework raining, fuzzy sysem conrol, engineering design opimizaion, and logisics and supply chain managemen. Noneheless, a number of researchers have noed ha PSO ends o converge premaurely on local opima, especially in complex mulimodal funcions (Clerc and Kennedy [4], Parsopoulos and Vrahais [6]. Accordingly, many papers have been proposed o improve PSO in order o avoid he problem of premaure convergence. III. PROPOSED PSO A. The Modified PSO The idea of he modified PSO manipulae a swarm animal behavior such ha when he swarm is large, sub-swarms ofen form o find a good source of food or o flee from predaors. This mehod suppors a higher capabiliy o find food sources or o flee from predaion for he whole swarm. The swarm opology has been changed because he agens communicae wihin is sub-swarm insead of he whole swarm. As a resul, he sub-swarms enable he whole swarm o conduc an exensive exploraion of he search space. Insead of using he bes posiion ever visied by all paricles (gbes, he subswarms have heir own sbes he bes posiion ever visied by paricles in he sub-swarms. Equaion (5 and (8 are modified as: V sid = ωv 1 sid + φ 1 β 1 (pbes sid X sid +φ β (gbes sid X sid (9 X +1 sid = X sid + V sid (10 where he index s is he sub-swarm index. Subsequenly, he sub-swarm wih he capabiliy o find a sufficien food source has a higher probabiliy of survival. This example is more prominen in a school of fish in which he whole swarm fracions o sub-swarms in an effor o escape from a dangerous siuaion. The unlucky sub-swarm ha is sill suck among predaors will be exerminaed. Be exerminaed, wha would happen o he whole swarm if he wors sub-swarm disappeared from he flock? Reasonably, he animal swarm should sill be able o produce heir offspring and mainain is species in is environmen. The paricle swarm should also be able o mainain is swarm size in he sysem. Accordingly, we propose a process called an exincion and offspring reproducion process. In his process, he wors performance sub-swarm is periodically killed hroughou he evoluionary, named as exincion process. Then, a sub-swarm in he sysem should be seleced as an ancesor o produce offspring in wha is known as offspring reproducion process. We shall borrow a selecion mehod from he geneic algorihm. Alhough, here are many selecion mehods e.g. roulee wheel, sigma scaling, Bolzmann, ournamen selecion many researchers have found ha he eliism mehod significanly ouperforms he oher mehods (see Melanie [6] and is herefore wha has been used in his sudy. Furhermore, we applied a nonlinear decreasing ineria weigh approach (NLDIWA which inspired by he sudy of Xu [7] in which he ineria weigh facor was conrolled by he following nonlinear decreasing funcion. ω = ω ini 1+cos(π/T end (11 where ω is he ineria weigh of h ieraion, ω ini is he iniial ineria weigh, and T end is se as 0.95 T, T is he maximum number of ieraions. B. Consrains Handling Normally, in he evoluionary compuaion, he consrains are deal wih when evaluaing soluions. The violaed consrain of each soluion is considered separaely; he

3 Inernaional Journal of Compuer Science and Elecronics Engineering (IJCSEE Volume 3, Issue 1 (015 ISSN (Online relaionship beween infeasible soluions and feasible regions is exploied o guide search. However, he difficuly of his echnique is here is no cerain meric crierion o measure his relaionship [8]. In he sudy of Michalewicz [9], he poined ha here are a leas hree choices o cope wih his problems: coun he number of violaions for a given soluion; consider he amoun of infeasibiliy in erms of consrains violaion; and compue he effor of repairing he individual. In his sudy, we made use of he penaly funcion mehod in order o cope wih he consrains. Consrains are incorporaed ino he objecive funcion. Consequenly, i ransforms consrained problem ino unconsrained problem. Thus, he consrained opimizaion problem in (1, (, and (3 can be ransformed as. = (X, r = f(x + r m j=1 G j [g j (X] (1 where G j is some funcion of he consrain g j, and r is a posiive consan known as he penaly parameer. The soluion may be brough o converge o ha original problem saed in (1, (, and (3 if he unconsrained minimizaion of he funcion is repeaed for a sequence of he value of i k ( = 1,,, T. In his sudy, we made use of he inerior formulaion o form G j as (see [10]: G j = 1 g j (X (13 The penaly facors are he key performance of his echnique. We deployed an objecive funcion raio beween he feasible global bes and he infeasible paricle. The penaly facor can be calculaed as follows. r +1 = f(x f(x (14 where f, f(x is he objecive value of non-violaion paricle, (X is he objecive value of he violaion paricle. I is noiceable ha he paricle of which severely violae a consrain yields very low value of objecive funcion while he feasible global bes paricle yields higher value of objecive funcion; consequenly, a big number of r is received. Fig. 1 The raio beween an infeasible soluion and he feasible global bes soluion The idea of r can be shown in Fig. 1. The paricle B is he bes soluion of he h ieraion; however, i is infeasible because i violaes a consrain in consrained opimizaion problem. Thus, he global bes among all paricles in his ieraion is paricle A. Nex, r +1 is calculaed for paricle B in order o calculae (X, r +1 of he nex ieraion. In summary, a PSO algorihm is proposed based on he adapive parameers (nonlinear decreasing ineria weigh approach in which evaluae he performance of each subswarm is evaluaed, and he wors performing sub-swarm is periodically eliminaed. Offspring are produced form he bes performing sub-swarms. Addiionally, he consrains are handled by he penaly funcion mehod in which he penaly parameer is adjusable belong o he level of consrains violaion. The algorihm of he proposed PSO is described in Table 1. IV. COMPUTATIONAL EXPERIMENTS In his secion, wo consrained engineering opimizaion problems ha are commonly used in lieraure are performed. A. Minimizaion of he Weigh of a Tension/Compression Spring Design Problem This problem was inroduced by Belegundu [11] and Arora [1]. The objecive is o minimize he weigh of a ension/compression spring subjec o consrains on minimum deflecion, shear sress, surge frequency, limis on ouside and on design variables. The design variables are he mean coil diameer (D, he wire diameer (d, and he number of acive coils (N. The problem can be described as follows: Minimize f 1 (X = (x 3 + x x 1 (15 subjec o g 1 (X = 1 x 3 x x (16 g (X = 4x x 1 x 1566(x x 3 1 x x (17 g 3 (X = x 1 0 x x 3 (18 g 4 (X = x 1+x (19 where X = [x 1 x x 3 ] T and x 1 = D, x = d, x 3 = N, resepecively. The ranges of each design variables were se as: 0.05 x 1.0, 0.5 x 1.3,.0 x Fig. 3 shows he sysem of his problem. B. Minimizaion of he Toal Cos of a Pressure Vessel Design Problem The objecive of his problem is o minimize he oal cos f(x of he maerial, forming, and welding processes of a cylindrical vessel. The cylindrical vessel mus be capped by hemispherical heads a boh ends as shown in Fig. 3. The hickness of he shell (T s, he hickness of he head (T h, he inner radius (R, and he lengh of he cylindrical secion of he vessel (L are he design variables. The problem can be formulaed as follows: Minimize f (X = 0.64x 1 x 3 x x x 3 (0 subjec o g 1 (X = x x 3 0 (1 3

4 Inernaional Journal of Compuer Science and Elecronics Engineering (IJCSEE Volume 3, Issue 1 (015 ISSN (Online g (X = x x 3 0 ( g 3 (X = πx 3 x 4 4 πx (3 g 4 (X = x (4 where X = [x 1 x x 3 x 4 ] T and x 1 = T s, x = T h, x 3 = R, x 4 = L, resepecively. The ranges of each design variables were se as: 1 x 1 99, 1 x 99, 10.0 x , 10.0 x TABLE I PROPOSED PSO ALGORITHM 1. Iniializaion (a Se = 0 (ieraion couner (b Specify he parameer N (number of paricles in a sub-swarm and S (number of sub-swarm (c Randomly iniialize he posiion of he N paricles x 11, x 1,.., x SN wih x si S R N (d Randomly iniialize he velociies of he N paricles v 11, v 1,, v SN wih v si S R N (e For i = 1,,, N do pbes si = x si (f Se gbes = argmin s 1,,,S;i 1,,,N f(x si. Terminae Check. If he erminaion crieria hold sop. The final oucome of he algorihm will be gbes. 3. Calculae he ineria weigh ω using (11 4. For s = 1,,, S Do 4.1. For i = 1,,, N Do 4.. Updaing paricle swarm (a Updae he velociy V si using (9 (b Updae he posiion X +1 sid using (10 (c Evaluae he finess of he paricle i, f(x si (d If f(x si is feasible and f(pbes si hen pbes si = X si (e If f(pbes si f(sbes s he sbes = pbes si (f If f(x si +1 is infeasible, calculae r si using (14 End For End For 5. Se gbes = argmin s 1,,,N f(x si 6. If he ieraion number can be exacly divided by he predeermined inerval hen perform exincion and offspring reproducion process (a Delee paricles in he wors performance sub-swarm (b Copy vecor of he bes performance paricle Noe ha he number of he new paricles mus be equal o he number of he died-ou paricles Else go o sep Se = + 1, Go o sep Fig. 3 Pressure Vessel Design Problem C. Compeiive algorihms and Parameer seings Three oher opimizaion echniques; a classic-pso [3], an effecive co-evoluionary PSO [8], and GA wih consrainhandling mehod [13], have been seleced o be compared wih our proposed approach. The proposed PSO and a classic-pso were implemened in Pyhon language on Windows 7 Inel Xeon,.4 GHz processor wih 64 GB of RAM. The parameers of he proposed PSO were se as ω ini = 0.7, φ 1 = φ =.0, T = 500, s = 5, and he swarm size was 30 (ha means here were 6 sub-swarm in his sudy. The bes resul is aken from 40 independen runs. D. Resuls and Discussions The bes resul from 40 runs of our proposed PSO and a classic-pso are shown in Table and 3 for he spring design problem and he pressure vessel design problem, respecively. The resuls of an effecive co-evoluionary PSO and GA wih consrain-handling approach are excerp from [8] and [13], respecively. The bes resul of each design problem is highlighed in boldface. Design Variable TABLE II COMPARISON OF THE BEST RESULTS FOR THE TENSION/COMPRESSION SPRING DESIGN PROBLEM Proposed PSO Classic-PSO [3] He [8] Coello [13] x x x g g g g f 1 (X RPD % 7.38% 7.56% Fig. Tension/Compression Spring Design Problem 4

5 Inernaional Journal of Compuer Science and Elecronics Engineering (IJCSEE Volume 3, Issue 1 (015 ISSN (Online Design Variable TABLE III COMPARISON OF THE BEST RESULTS FOR THE PRESSURE VESSEL DESIGN PROBLEM Proposed PSO Classic-PSO [3] He [8] Coello [13] x x x x g g g g f (X RPD % 0.015% 0.019% From Table and Table 3, our proposed PSO ouperforms oher compeiive mehods. We made use of he relaive percenage deviaion (RPD o measure he deviaion from he bes resul of each design problem. The equaion of RPD is shown as. RPD = X O 100 (5 O where X is he resul which we need o compare and O is he bes resul. As shown in Table, oher algorihms yield he resuls higher han he proposed PSO wih RDP from 7.38% o7.56%. As shown in Table 3, oher algorihms yield he resuls higher ha he proposed PSO wih RDP from 0.015% o 0.05%. V. CONCLUSIONS This paper has inroduced a modified PSO wih selfadapive penaly facor in order o handle he consrains in consrained opimizaion problem. In our proposed PSO, he swarm behavior was uilized by ha he whole swarm may divide ino many sub-swarms in order o find a good source of food. The diversificaion of he paricles is our favorable. Furhermore, we modified he penaly funcion mehod as selfadapive penaly facor which dynamically changed belong o he performance of he las ieraion. The experimen resuls showed ha our proposed mehod ouperformed oher compeiive algorihms. Our fuure work is o invesigae he rajecory of paricles in he swarm in which affeced from he echnique in order o explain he algorihm s performance concisely. Furhermore, he parameer seings also need o be sudied. A saisical echnique, such as an analysis of variance, is needed in order o define he appropriae parameer seings. REFERENCES [1] RC. Eberhar and J. Kennedy, A new opimizer using paricle swarm heory, in Proc. 6 h Inernaional Symposium on Micro Machine and Human Science, 1995, pp [] Y. Shi and RC. Eberhar, A modified paricle swarm opimizaion, in Proc. The Evoluionary Compuaion, 1998, pp [3] Y. Shi and RC. Eberhar, Empirical sudy of paricle swarm opimizaion, in Proc. IEEE congress on Evoluionary Compuaion, 1999, pp [4] M. Clerc and J. Kennedy, The paricle swarm explosion, sabiliy, and convergence in a mulidimensional complex space, IEEE Transacions on Evoluionary Compuaion, vol. 6(1, pp. 58-7, 00. [5] KE. Parsopoulos and MN. Vrahais, Paricle Swarm Opimizaion and Inelligence: Advances and Applicaions. Hershey, PA: ISTE Ld., 010. [6] M. Melanie, An Inroducion o Geneic Algorihms. Cambridge, MA: MIT Press, [7] G. Xu, An adapive parameer uning of paricle swarm opimizaion algorihm, Applied Mahemaics and Compuaion, vol. 19, pp , 013 [8] Q. He and L. Wang, An effecive co-evoluionary paricle swarm opimizaion for consrained engineering design problems, Engineering Applicaion of Arificial Inelligence, vol. 0, pp , 007. [9] Z. Michalewicz, A survey of consrain handling echniques in evoluionary compuaion mehods, in Proc. 4 h Annual Conference on Evoluionary Programming, 1995, pp [10] S. S. Rao, Engineering Opimizaion: Theory and Pracice. Hoboken, NJ, 009, pp [11] A.D. Belegundu, A Sudy of Mahemaical Programming Mehods for Srucural Opimizaion. Iowa Ciy, IA: Deparmen of Civil and Environmenal Engineering, 198. [1] J.S. Arora, Inroducion o Opimum Design. New York, NY: McGraw- Hill, [13] C. A. Coello Coello and E. M. Mones, Consrain-handling in geneic algorihms hrough he use of dominance-based ournamen selecion, Advanced Engineering Informaics, vol. 16, pp , 00. Abou Auhor (s: Choosak Pornsing is currenly a lecurer of Deparmen of Indusrial Engineering and Managemen a Silpakorn Universiy, Thailand. He has a Ph.D. in indusrial and sysems engineering from he Universiy of Rhode Island, an M.S. in indusrial and sysems engineering from Lehigh Universiy, PA, USA. He also has M.Eng. in indusrial engineering from Thammasa Universiy, Thailand and B.Eng. in indusrial engineering from Naresuan Universiy Thailand. His research ineress are in he area of Compuaional Inelligence, Swarm Inelligence and is applicaions. He also conducs he research in area of Logisics and Supply Chain Managemen, Manufacuring Sysems Design, Applied Opimizaion, and Producion and Operaion Managemen. Kawinhorn Saicharoen is a lecurer of Deparmen of Indusrial Engineering and Managemen a Silapakorn Universiy, Thailand. He has a B.Eng. in Indusrial Engineering, M.Eng. in Indusrial Managemen Engineering from King Mongku s Insiue of Technology Norh Bangkok, Thailand. His research ineress are in he area of Logisics and Supply Chain Managemen. His curren researches focus on wo echelon invenory managemen wih fuzzy demand. Thanahorn Karo is a lecurer in logisics managemen and indusrial engineering a Silpakorn Universiy. Currenly, he is a eacher of informaion echnology for logisics and simulaion for Arena. He has a bachelor's degree in Civil Engineering a Chulalongkorn Universiy, Thailand, and a maser s degree in Logisics a Wollongong Universiy, Ausralia. He sared work a Perfec Companion Group, hen joined Silpakorn Universiy in 014. His research ineress are Indusrial Producion Managemen, Warehouse Managemen, Transporaion Managemen and Mainenance Managemen. ACKNOWLEDGMENT This research was suppored parly by Deparmen of Indusrial Engineering and Managemen, Faculy of Engineering and Indusrial Technology, Silpakorn Universiy, Thailand. 5