MANY real-world applications (e.g. production

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1 Barebones Parcle Swarm for Ineger Programmng Problems Mahamed G. H. Omran, Andres Engelbrech and Ayed Salman Absrac The performance of wo recen varans of Parcle Swarm Opmzaon (PSO) when appled o Ineger Programmng problems s nvesgaed. The wo PSO varans, namely, barebones Parcle Swarm (BB) and he eplong barebones Parcle Swarm (BBEp) are compared wh he sandard PSO and sandard Dfferenal Evoluon (DE) on several Ineger Programmng es problems. The resuls show ha he BBEp seems o be an effcen alernave for solvng Ineger Programmng problems. I. INTRODUCTION MANY real-world applcaons (e.g. producon schedulng, resource allocaon, VLSI crcu desgn, ec.) requre he varables o be opmzed o be negers. These problems are called Ineger Programmng problems. Opmzaon mehods developed for real search spaces can be used o solve Ineger Programmng problems by roundng off he real opmum values o he neares negers []. The unconsraned Ineger Programmng problem can be defned as N mn f ( ), S Z d () where Nd Z s an d N dmensonal dscree space of negers, and S represens a feasble regon ha s no necessarly a bounded se. Ineger Programmng problems encompass boh mamzaon and mnmzaon problems. Any mamzaon problem can be convered no a mnmzaon problem and vce versa. The problems ackled n hs paper are mnmzaon problems. Therefore, he remander of he dscusson focuses on mnmzaon problems. The Branch and Bound mehod [] s one common deermnsc approach o ackle he Ineger Programmng problems. Evoluonary algorhms (EAs) [3] are generalpurpose sochasc search mehods smulang naural Manuscrp receved November 3, 006. M. G. Omran s wh he Deparmen of Compuer Scence, Gulf Unversy for Scence and Technology, Kuwa (phone: 8866 (E. 55); e-mal: momran@gmal.com). A. Engelbrech s wh he Deparmen of Compuer Scence, Unversy of Preora, Preora, Souh Afrca. A. Salman wh he Deparmen of Compuer Engneerng, Kuwa Unversy, Kuwa. selecon and evoluon n he bologcal world. EAs have been used successfully o solve Ineger Programmng problems []. On he oher hand, Parcle Swarm Opmzaon (PSO) [5] s a populaon-based sochasc opmzaon algorhm modeled afer he smulaon of he socal behavor of brd flocks, where a swarm of ndvduals (called parcles) fly hrough he search space. PSO has also been used o solve Ineger Programmng problems wh promsng resuls [,6,7]. Kennedy [8] proposed a new PSO approach where he sandard PSO velocy equaon s removed and replaced wh samples from a normal dsrbuon. Ths approach, known as barebones Parcle Swarm (BB), requres no parameer unng. Kennedy [8] proposed a varaon of he BB PSO where appromaely half of he me velocy s based on samples from a normal dsrbuon, and for half of he me velocy s derved from he parcle's personal bes poson. Ths verson s called eplong barebones Parcle Swarm (BBEp). Ths paper nvesgaes he performance of BB and BBEp mehods on Ineger Programmng problems. The wo varans are compared o boh PSO (as proposed n []) and sandard Dfferenal Evoluon (DE) [9]. The remnder of he paper s organzed as follows: Secon II provdes an overvew of PSO, BB and BBEp. An overvew of DE s gven n Secon III. Benchmark funcons o measure he performance of he dfferen approaches are provded n Secon IV. Resuls of he epermens are presened n Secon V. Fnally, Secon VI concludes he paper. II. PARTICLE SWARM OPTIMIZATION AND ITS VARIANTS In PSO, each parcle n he swarm s represened by he followng characerscs: : The curren poson of he parcle; v : The curren velocy of he parcle; y : The personal bes poson of he parcle. The personal bes poson of parcle s he bes poson (.e. one resulng n he bes fness value) vsed by parcle so far. Le f denoe he obecve funcon. Then he personal bes of a parcle a me sep s updaed as y ) ) f f ( )) f ( y ) f f ( )) < f ( y ) y () /07/$ IEEE 70

2 If he poson of he global bes parcle s denoed by he vecor ŷ, hen { y, y, K, y } mn{ f ( y ), f( y ), K, f ( y ( ))} ˆy s 0 (3) 0 s where s denoes he sze of he swarm. For each eraon of a PSO algorhm, he velocy v updae sep s specfed for each dmenson,, N d, where N d s he dmenson of he problem. Hence, v, represens he h elemen of he velocy vecor of he h parcle. The velocy of parcle s updaed as v, c r ) wv, ( ŷ, + c r, ), ( y,, ) + where w s he nera wegh, c and c are he acceleraon consans and r,, ~ (0,). Equaon () consss of r, U hree componens, namely The nera wegh erm, w, whch serves as a memory of prevous veloces. The nera wegh conrols he mpac of he prevous velocy: a large nera wegh favors eploraon, whle a small nera wegh favors eploaon. The cognve componen, y, whch represens he parcle's own eperence as o where he bes soluon s. The socal componen, ˆy, whch represens he belef of he enre swarm as o where he bes soluon s. The poson of parcle,, s hen updaed usng ( + ) + v ( + ) (5) The reader s referred o [0,] for a sudy of he relaonshp beween he nera wegh and acceleraon consans, n order o selec values for hese conrol parameers whch wll ensure convergen behavor. Velocy updaes can also be clamped hrough a user defned mamum velocy, V ma, whch would preven hem from eplodng, hereby reducng he chances ha parcles wll leave he boundares of he search space. The PSO algorhm performs he updae equaons above, repeaedly, unl a specfed number of eraons have been eceeded, or velocy updaes are close o zero. The qualy of parcles s measured usng a fness funcon whch reflecs he opmaly of a parcular soluon. () A. Barebones Parcle Swarm approaches BB PSO replaces equaons () and (5) wh he followng equaon, y, + ŷ ) N, y - ŷ,, (6) Equaon (6) s based on heorecal sudes where formal proofs have shown ha each parcle converges o a weghed average of s personal bes and neghborhood bes posons [0,]. Therefore, he mean of y ŷ, + s used for he normal dsrbuon. The devaon of y, - ŷ allows parcles whose personal bes poson are far away from he global bes poson o make large sep szes owards he global bes poson. Ths may cause personal bes posons o move closer o he global bes poson. When a personal bes poson s close o he global bes poson, sep szes are small o lm eploraon n favor of eploaon. The BBEp, on he oher hand, replaces equaons () and (5) wh y, + ŷ N >, y ( ) - ( ), ŷ f U (0,) 0.5 ) (7), y, oherwse The fac ha poson updaes are se equal o he personal bes poson for 50% of he me causes he BBEp o eplo personal bes posons more han BB PSO, hereby lmng eploraon. Accordng o [8], BBEp generally ouperformed oher varans of PSO when appled o a se of benchmark funcons. III. DIFFERENTIAL EVOLUTION Dfferenal evoluon does no make use of a muaon operaor ha depends on some probably dsrbuon funcon, bu nroduces a new arhmec operaor whch depends on he dfferences beween randomly seleced pars of ndvduals. For each paren, (), of generaon, an offsprng, (), s creaed n he followng way: Randomly selec hree ndvduals from he curren populaon, namely ( ), 7

3 ( ) and ( ), wh 3 3 and,, 3 ~ U(,, s), where s s he populaon sze. Selec a random number r ~ U(,, N d ), where N d s he number of genes (parameers) of a sngle chromosome. Then, for all parameers,, N d, f U(0, ) < P r, or f r, le (8), F(, -, + 3 oherwse, le,, ( )), (9) In he above, P r s he probably of reproducon (wh P r [0, ]), F s a scalng facor wh F (0, 8), and () and,, () ndcae respecvely he -h parameer of he offsprng and he paren. Thus, each offsprng consss of a lnear combnaon of hree randomly chosen ndvduals when U(0,) < P r ; oherws e he offsprng nhers drecly from he paren. Even when P r 0, a leas one of he parameers of he offsprng wll dffer from he paren (forced by he condon r). The muaon process above requres ha he populaon consss of more han hree ndvduals. Afer compleon of he muaon process, he ne sep s o selec he new generaon. For each paren of he curren populaon, he paren s replaced wh s offsprng f he fness of he offsprng s beer, oherwse he paren s carred over o he ne generaon. Prce and Sorn [] proposed en dfferen sraeges for DE based on he ndvdual beng perurbed (.e. 3, () ), number of ndvduals used n he muaon process and he ype of crossover used. The sraegy shown n hs secon s known as DE/rand/. Ths sraegy s consdered o be he mos wdely used sraegy. IV. BENCHMARK FUNCTIONS S commonly used Ineger programmng benchmark problems [] were chosen o nvesgae he performance of he PSO varans. Elemens of he parcle poson vecors are rounded o he neares neger, afer he applcaon of he poson updae equaon. Tes Problem : F () N d where 0 and f ( ) 0. Ths problem was consdered for dmensons 5, 5 and 30. Tes Problem : F ( 9 + ) + ( 3 + ) ( ) 7, and ( ) 0 where ( ) T Tes Problem 3: F. ( + 0 ) + 5( ) + ( ) + ( ) F ( 3 ) where 0 and F ( ) 0. Tes Problem : 3 F ( ) where 0 and F ( ) 0. Tes Problem 5: F ( + 5 ) , and ( ) where ( ) T Tes Problem 6: F ( 6 ) T F. 5 where 0 and f ( ) 0. Ths problem was consdered for dmenson 5 as n []. For all he above problems, [ 00, 00] Z. V. EXPERIMENTAL RESULTS In hs secon, BB and BBEp are compared wh sandard gbes PSO and DE/rand/. For PSO, w 0.7, c c.9 (hese values were suggesed by [3]). For DE, F 0.5, P r 0.9 (hese values were suggesed by []). For BB and BBEp, no conrol parameer unng s needed (ecep for he swarm sze s). For all he algorhms used n hs secon, s 50. The resuls repored n hs secon are averages and sandard devaons over 30 smulaons. Each smulaon was allowed o run for evaluaons of he obecve funcon. Table I summarzes he resuls of he epermens. For all 7

4 he es problems, ecep for F wh N d 5 and F wh N d 30, all he algorhms found he global opmum soluons. For larger dmensonal problems (F wh N d 30), he BBEp performed bes, gvng sgnfcanly beer resuls han he oher algorhms. Wha s neresng o noe s ha he BB PSO performed worse han he oher algorhms for F wh N d 5 and N d 30. Ths may be due o larger sep szes caused by he devaon, y, - ŷ, of he normal dsrbuon. Eamnng he resuls, BBEp sgnfcanly ouperformed he oher approaches when appled o he 30-dmensonal F. BB performed worse han he oher approaches when appled o F. Eamnng Fgure can be shown ha BBEp converges faser han he oher approaches when appled o he 30-dmensonal F. Furhermore, Fgure shows ha BB suffers from premaure convergence. The epeced reason for he premaure convergence s ha BB los s dversy faser han he oher approaches as shown n Fg.. For all he oher funcons, all he approaches performed equally well and ehb smlar convergence characerscs. Alhough he algorhms acheved he same accuracy for he lower dmensonal problems, BBEp have shown he followng advanages: Beer accuracy for he hgher dmensonal problems. Faser convergence. Requres no parameer unng, whereas he performance of PSO and DE s dependen on values seleced for her conrol parameers. [6] T. Tsukada, T. Tamura, S. Kagawa and Y. Fukuyama. Opmal operaonal plannng for cogeneraon sysem usng parcle swarm opmzaon. In Proceedngs of he IEEE Swarm Inellgence Symposum 003 (SIS 003), Indanapols, Indana, USA. pp. 38-3, 003. [7] K. Parsopoulos and M. Vrahas. Recen approaches o global opmzaon problems hrough parcle swarm opmzaon. Naural Compung, vol., no. -3, pp , 00. [8] J. Kennedy. Bare bones parcle swarm, IEEE Swarm Inellgence Symposum, pp , 003. [9] R. Sorn and K. Prce. Dfferenal Evoluon A Smple and Effcen Adapve Scheme for Global Opmzaon over Connuous Spaces. Techncal Repor TR-95-0, Inernaonal Compuer Scence Insue, Berkeley, CA, 995. [0] F. Van den Bergh. An analyss of parcle swarm opmzers, Ph.D. dsseraon. Deparmen of Compuer Scence, Unversy of Preora, 00. [] F. Van den Bergh and A. P. Engelbrech. A Sudy of Parcle Traecores. Informaon Scences, vol. 76, no. 8, pp , 006. [] K. Prce and R. Sorn. DE Web se, hp:// (vsed 8 Aug 006), 006. [3] M. Clerc and J. Kennedy. The parcle swarm: Eploson, Sably and Convergence n a mul-dmenonal comple space. IEEE Transacons on Evoluonary Compuaon, vol. 6, pp , 00. VI. CONCLUSION Ths paper evaluaed he performance of wo verson of he barebones PSO n solvng Ineger Programmng problems. In comparson wh a gbes PSO and DE/rand/, he algorhms showed he same performance n erms of accuracy for he lower dmensonal problems. For hgher dmensonal problems, BBEp showed o be sgnfcanly beer han he oher algorhms. The resuls also showed ha BBEp converges faser o good soluons. REFERENCES [] E. Laskar, K. Parsopoulos and M. Vrahas. Parcle Swarm Opmzaon for Ineger Programmng. In: Proceedngs of he 00 Congress on Evoluonary Compuaon, vol., pp , 00. [] R. Hors and H. Tuy. Global Opmzaon, Deermnsc Approaches, Sprnger, 996. [3] A. Engelbrech. Compuaonal Inellgence: An Inroducon. John Wley and Sons, 00. [] G. Rüdolph. An Evoluonary Algorhm for Ineger Programmng. Y. Davdoe, H. Schwefel and R. Männer (eds.), Parallel Problem Solvng from Naure, vol. 3, pp. 39-8, Sprnger, 99. [5] J. Kennedy and R. Eberhar. Parcle Swarm Opmzaon. In Proceedngs of IEEE Inernaonal Conference on Neural Neworks, Perh, Ausrala, vol., pp. 9-98,

5 T ABLE I MEAN, STANDARD DEVIATION (SD) AND 95%CONFIDENCE INTERVAL OF THE TEST PROBLEMS Funcon Mehod Mean (SD) 95% Confdence Inerval (zdsrbuon) F ( N DE 0(0) [0,0] d 5) BBEp 0(0) [0,0] F ( N DE 0(0) [0,0] d 5) BB ( ) [0.0565,0.6650] BBEp 0(0) [0,0] F PSO ( ) [ ,7.500] ( N DE.5 (.97833) [.38808,.869] d 30) BB (6.398) [9.67,.73858] BBEp ( ) [0.656,0.6888] F DE 0(0) [0,0] BBEp 0(0) [0,0] F 3 DE 0(0) [0,0] BBEp 0(0) [0,0] F PSO -6(0) [-6,-6] DE -6(0) [-6,-6] BB -6(0) [-6,-6] BBEp -6(0) [-6,-6] F 5 PSO (0) [-3833.,-3833.] DE (0) [-3833.,-3833.] BB (0) [-3833.,-3833.] BBEp (0) [-3833.,-3833.] F 6 ( N DE 0(0) [0,0] d 5) BBEp 0(0) [0,0] F (30 dm) F gbes PSO DE/rand/ BB BBEp Fg. : Comparson beween PSO, DE, BB and BBEp for he 30-dmensonal F benchmark problem. The vercal as represens he funcon value and he horzonal as represens he number of generaons. 7

6 F (30 dm) 50 0 dversy 30 0 gbes PSO DE/rand/ BB BBEp Fg. : Comparson beween PSO, DE/rand/, BB and BBEp for he 30-dmensonal F benchmark problem. The vercal as represens he dversy and he horzonal as represens he number of generaons. 75