MODIFIED PREDATOR-PREY (MPP) ALGORITHM FOR CONSTRAINED MULTI-OBJECTIVE OPTIMIZATION

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1 EVOLUTIONARY METHODS FOR DESIGN, OPTIMIZATION AND CONTROL T. Burczynsk and J. Péraux (Eds.) CIMNE, Barcelona, Span 29 MODIFIED PREDATOR-PREY () ALGORITHM FOR CONSTRAINED MULTI-OBJECTIVE OPTIMIZATION Souma Chowdhury, George S. Dulkravch and Ramon J. Moral Department of Mechancal and Materals Engneerng, MAIDROC Lab., Florda Internatonal Unversty, 1555 West Flagler St., EC 3474, Mam, Florda 33174, U.S.A. Emal: web page: Abstract. In ths work, an evolutonary mult-objectve optmzaton algorthm based on the dynamcs of predator-prey nteractons exstng n nature s presented. Ths algorthm s comprsed of a relatvely small number of predators and a much larger number of prey, randomly placed on a two dmensonal lattce wth connected ends. The predators are partally or completely based towards one or more objectves, based on whch each predator klls the weakest prey n ts neghborhood. A stronger prey created through evoluton replaces ths prey. The prey reman statonary, whle the predators move around n the lattce durng the course of generatons. Modfcatons of the basc predator-prey algorthm have been mplemented n ths study regardng the selecton procedure, apparent movement of the predators, mutaton strategy, dynamcs of the Pareto convergence and a capablty of handlng equalty and nequalty constrants. The fnal modfed algorthm was tested on standard constraned and unconstraned multobjectve optmzaton problems. Key words: constraned optmzaton, evolutonary algorthms, mult-objectve optmzaton, predator-prey algorthm. 1 INTRODUCTION In 1998, Hans Paul Schwefel and co-workers 1 proposed a new optmzaton algorthm to search for Pareto-optmal solutons from a randomly generated ntal populaton of canddate solutons. Ths algorthm mtates the natural phenomena that a predator klls the weakest prey n ts neghborhood, and the next generatons of prey that evolve are relatvely stronger and more mmune to such predator attacks. However, ths ntal Predator-Prey (PP) optmzaton algorthm seemed to have dffculty n producng well dstrbuted non-domnated solutons along the Pareto front. Deb 2 suggested certan new features, namely, the elte preservaton operator, the recombnaton operator and the dversty preservaton operator. Further, L 3 ntroduced the movement of both predators and prey and changng populaton szes of prey. Some other versons of the algorthm have been presented by Grmme et al. 4 and Slva et al. 5. The former used a modfed recombnaton and mutaton model. The latter, predomnantly a partcle swarm optmzaton algorthm, ntroduces the concept of predator-prey nteractons n

2 S. CHOWDHURY AND G.S. DULIKRAVICH/ Modfed Predator-Prey Algorthm the swarm to control the balance between exploraton and explotaton, hence mprovng both dversty and rate of convergence. However, most of the avalable versons of PP fnd t dffcult to produce a well dstrbuted set of Pareto optmal solutons n a lmted number of functon evaluatons. Ths necesstates optmzaton algorthms capable of producng solutons that are both optmum as well as feasble wth respect to the system constrants. These system constrants can be modeled as mathematcal constrant functons. Dfferent constrant handlng technques have been developed, some of the most popular ones beng penalty functon methods 6,7, flterng methods and constrant domnaton method ntroduced by Deb et al. 8. The Modfed Predator-Prey () algorthm presented here s an evolutonary mult-objectve optmzaton algorthm, capable of handlng complex desgn optmzaton problems. It has been developed through the assmlaton of specal features of exstng PP models, modfcatons of the same and addton of certan new features; the most sgnfcant one beng the ablty to handle both lnear/non-lnear equalty and nequalty constrants. The constrant handlng method used n s a combnaton of flterng technque and constrant domnance technque. 2 MODIFIED PREDATOR-PREY () ALGORITHM The fundamental steps followed by are outlned n the ntal verson developed by Chowdhury et al. 9. The sgnfcant features of that dstngushes t from prevous versons of predator-prey algorthms are as follows: 2.1 Evoluton The generaton of new solutons n each actve localty (localty contanng a predator) s ntated by the crossover of the strongest two local prey. The blend crossover (BLX-α ), ntally proposed by Eshelman and Schaffer 6 for real-coded genetc algorthms (later mproved by Deb 2 ), s used n ths algorthm. BLX-α s defned by the followng equatons. Here, (1, t ) ( 1 γ ) ( 1 2 ) u x = x + γ x (1, t+ 1) (1, t ) (2, t) γ = + α α (2, t) x and x are the parent solutons, (1, t+ 1) x (1) s the chld soluton and u s the random number between. and 1.. A value of.5 s used for α as suggested by Deb 2. Ths crossover chld prey s then subjected to non-unform mutaton orgnally ntroduced by Mchalewcz 1 and modfed by Chowdhury et al. 9 to allow for dynamc adjustment of the extent of mutaton as shown below. (1, t 1) where y β = 1 t t max b 1 t (1, t 1) (1, t 1) ( ) ( ) max ( U L ) t + + y = x + τβ x x 1 r + s the chld soluton produced from the parent soluton (1, t + 1), by mutaton of the th ( U ) varable, x and x are upper and lower lmts of the th varable, τ takes a ( L ) Boolean value -1 or 1, each wth a probablty of.5, r s a random number between and 1, t and t max are the number of functon evaluatons performed untl then and maxmum allowed number of functon evaluatons, respectvely, (β = 1.5 determned x (2)

3 EVOLUTIONARY METHODS FOR DESIGN, OPTIMIZATION AND CONTROL (EUROGEN 29) emprcally) and β s the scalng parameter. The latter two factors montor the order of magntude, or n other words, the extent of mutaton. Both the crossover and mutaton technques employed here establsh an adaptve search, whch makes the algorthm more economcal wth respect to functon evaluatons. 2.2 Domnance and Constrant Handlng The constrant domnance s used as one of the qualfyng crteron for new prey when compared wth the survvng local prey. Ths method has been derved from the constrant domnance used n NSGA-II 8, whch s as follows, Soluton s sad to constrant-domnate soluton j f Soluton s feasble and soluton j s not or Solutons and j are both nfeasble, whle soluton has a smaller net constrant j volaton than soluton j,.e. f Nf + 1 < f (consderng functon mnmzaton) or Nf + 1 Solutons and j are both feasble, whle soluton weakly domnates soluton j. It s evdent that ths crteron gves feasblty preference over optmalty. 2.3 Dversty Preservaton The concept of objectve space hypercube s used as a qualfyng crteron for new prey to assure dversty preservaton. Each old local prey s consdered to be at the centre of ts hypercube, the sze of whch s dynamcally updated wth generatons. Ths proves to a very promsng approach to mantan dversty wthout posng any nhbton to Pareto convergence. In addton to ths, an nnovatve concept of sectonal convergence has been ntroduced to deal wth ths possble lack of effectve varaton n the prey populaton. Instead of the runnng the algorthm throughout for the same ntal specfed dstrbuton of weghts, there s redstrbuton of weghts wthn a small based range (<1.) after a certan number of functon evaluatons. Ths drects the ntermedate solutons to dfferent regons of the Pareto front dependng on the dstrbuton of weghts. Ths s essentally dfferent from the technque used by gradent based algorthms when solvng mult-objectve problems snce the weghted Pareto approach s ntated after partal convergence. However, the concept of sectonal convergence needs to be used only n case of problems wth dscontnuous domans or hgh dmensonalty. 2.4 Eltsm In a secondary set s mantaned, whch contans the non domnated solutons from each generaton. Ths set s known as the elte set, the sze of whch s preserved usng the clusterng technque desgned by Deb 2. After each generaton, a certan number of randomly selected solutons/preys (from the man populaton), f found to be domnated, are replaced from the 2D lattce by randomly selected elte solutons. Ths new attrbute boosts the speed of convergence of ths algorthm. The number of such replacements should be carefully allocated. The results shown n ths paper have been accomplshed by allowng replacement of 2% of the man populaton after each generaton. 2.5 Constrant handlng technque Optmzaton algorthms when dealng wth constraned mult-objectve problems need to ensure that the searchng mechansm employed fnd solutons that are both optmal and feasble. The feasble doman represents the regon n the search space where every soluton satsfes the problem constrants. In, feasblty s

4 S. CHOWDHURY AND G.S. DULIKRAVICH/ Modfed Predator-Prey Algorthm accomplshed and sustaned by a combnaton of the flterng technque and the constrant domnance crteron 7. For a general mult-objectve optmzaton problem Mnmze f = f ( X ), = 1,2,..., Nf subject to g, = 1,2,3,..., p h =, = p + 1, p + 2,..., p + q p, q N where X = ( x1, x2, x3,..., xm), x R. The constrants are added up to form the ( Nf + 1) th objectve p p+ q ( ) (( ) ) Mnmze f = max g, + max h ε, Nf + 1 = 1 = p+ 1 where ε s the tolerance for equalty objectves. Ths addtonal constrant objectve s treated lke any other objectve by the usual predator prey approach. Hence the selecton pressure n the actve localtes s nether based towards feasblty nor optmalty. On the other hand the chld prey produced has to be at least non-domnated, wth respect to the other local prey, to be accepted. Ths brngs n the constrant domnance mechansm whch s based towards achevng feasblty. Both these factors workng together provde a balance between selectons of prey (solutons) based on the actual objectve values as well as ther dstances from the feasble doman. 3 RESULTS AND DISCUSSION was mplemented usng C++ programmng language. To examne the constrant handlng capablty of, t was tested wth a well known constraned 2-objectve test case TNK studed by Deb et al. 8. Two standard test cases wth known analytcal solutons, Bnh 11 mult-objectve optmzaton problem no. 2 and the Osyczka 12 multobjectve optmzaton problem no. 2, have also been used. The fnal Pareto fronts n case of these constraned test cases were constructed of only those elte set solutons that do not volate any of the problem constrants. The user defned parameters n the algorthm pertnent to the constraned test cases are presented n Table 1. Parameter TNK values Bnh and Osyczka values Populaton sze (number of prey) 1 1 Number of predators 1 1 Elte strength 4 1 Crossover probablty Mutaton probablty.5.5 Number of prmary teratons (sectons), 3, 6 Table 1: General parameters for TNK, Bnh and Ozyczka two-objectves test cases A hgher number of prmary teratons and greater elte set strength were used n case of the Bnh and the Osyczka problems as seen from Table 1. Ths s to counteract the relatvely greater dffculty n coverng the whole Pareto front n these two test problems. The converged Pareto fronts computed by n each of these test cases are shown n Fgures 1 to 6. The fnal Pareto fronts computed for TNK and Bnh constraned mult-objectve problems (Fgures 1-4) are farly accurate and well dstrbuted. In the Bnh problem, there s a sgnfcant mprovement n performance when usng the sectonal convergence scheme (Fgures 3 and 4). In case of Osyczka (3) (4)

5 EVOLUTIONARY METHODS FOR DESIGN, OPTIMIZATION AND CONTROL (EUROGEN 29) constraned mult-objectve problems (Fgures 5 and 6), though solutons converge to the global Pareto front, ther dstrbuton on the fnal Pareto front s not unform, even wth the sectonal convergence scheme Fgure 1: Constraned 2-objectve test case TNK wthout sectonal convergence Fgure 2: Constraned 2-objectve test case TNK wth sectonal convergence 5 4 Analytcal 5 4 Analytcal Fgure 3: Constraned 2-objectve test case Bnh wthout sectonal convergence Fgure 4: Constraned 2-objectve test case Bnh wth sectonal convergence Fgure 5: Constraned 2-objectve test case Osyczka wthout sectonal convergence Fgure 6: Constraned 2-objectve test case Osyczka wth sectonal convergence

6 S. CHOWDHURY AND G.S. DULIKRAVICH/ Modfed Predator-Prey Algorthm Overall, compares well n performance, wth other popular algorthms such as NSGA II 8 n solvng smlar constraned mult-objectve problems. Nevertheless, approprate mplementaton of the sectonal convergence scheme s necessary for certan problems, n order to attan a reasonable spread of solutons along the fnal Pareto front. 4 SUMMARY The modfed predator-prey algorthm follows the basc concept of the predator-prey evolutonary strategy and combnes new features, some orgnally developed and others derved from NSGA II algorthm. Nevertheless, s qute dfferent from the latter due to the selecton mechansm, localzed mprovement of solutons, adaptve hypercube szng and adaptve extent of mutaton. The most promsng attrbute of s ts ablty to consstently produce feasble Pareto solutons, rrespectve of the number or nature (lnear or non-lnear) of problem constrants nvolved. Ths s accomplshed wthout normalzaton of objectve functons or constrant functons, or applcaton of computatonally costly penalty functon methods. 5 REFERENCES [1] M. Laumanns, G. Rudolph and H.P. Schwefel, A spatal predator-prey approach to mult-objectve optmzaton: A prelmnary study, n Proceedngs of the Parallel Problem Solvng from Nature, V, (1998). [2] K. Deb, Mult-objectve Optmzaton Usng Evolutonary Algorthms, Chchester, UK, Wley (22). [3] X. L, A real-coded predator-prey genetc algorthm for mult-objectve optmzaton, Lecture Notes n Computer Scence, Vol. 2632, pp. 69 (23). [4] C. Grmme and K. Schmtt, Insde a predator-prey model for multobjectve otpmzaton A second study, In Proc. Genetc and Evolutonary Computaton Conf.(GECCO 26), Seattle WA, ACM Press, New York, (26). [5] A. Slva, E. Neves and E. Costa, An emprcal comparson of partcle swarm and predator-prey optmzaton, Lecture Notes n Computer Scence; 2464: [6] L.J. Eshelman and J.D. Schaffer, Real coded genetc algorthms and nterval schemata, n Foundatons of Genetc Algorthms 2 (FOGA 2), (1993). [7] C. Coello Coello, Theoretcal and numercal constrant-handlng technques used wth evolutonary algorthms: A survey of the state of the art, Computer Methods n Appled Mechancs and Engneerng 191(11 12), (22). [8] K. Deb, A. Pratap, S. Agarwal and T. Meyarvan, A fast and eltst multobjectve genetc algorthm: NSGA-II, IEEE Transactons on Evolutonary Computaton, Vol. 6, No. 2, (22). [9] S. Chowdhury, R.J. Moral and G.S. Dulkravch, Predator-prey evolutonary mult-objectve optmzaton algorthm: performance and mprovements, 7 th ASMO-UK/ISSMO Internatonal Conference on Eng. Desgn Optmzaton, Bath, UK, July 7-8 (28). [1] J. Mchalewcz, Genetc Algorthms + Data Structures = Evolutonary Programs, Berln, Sprnger-Verlag (1992). [11] T.T. Bnh and U. Korn, MOBES: A multobjectve evoluton strategy for constraned optmzaton problems, In The Thrd Internatonal Conference on Genetc Algorthms (Mendel 97), Brno, Czech Republc, (1997). [12] A. Osyczka and S. Kundu A new method to solve generalzed multcrtera optmzaton problems usng the smple genetc algorthm, Structural Optmzaton, 1:94-99 (1995).