Research Article Heuristic-Based Firefly Algorithm for Bound Constrained Nonlinear Binary Optimization

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1 Advaces i Operatios Research, Article ID , 12 pages Research Article Heuristic-Based Firefly Algorithm for Boud Costraied Noliear Biary Optimizatio M. Ferada P. Costa, 1 AaMariaA.C.Rocha, 2 Rogério B. Fracisco, 1 ad Edite M. G. P. Ferades 2 1 Departmet of Mathematics ad Applicatios, Cetre of Mathematics, Uiversity of Miho, Braga, Portugal 2 Algoritmi Research Cetre, Uiversity of Miho, Braga, Portugal Correspodece should be addressed to M. Ferada P. Costa; mfc@math.umiho.pt Received 30 May 2014; Accepted 20 September 2014; Published 8 October 2014 Academic Editor: Imed Kacem Copyright 2014 M. Ferada P. Costa et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Firefly algorithm (FA) is a metaheuristic for global optimizatio. I this paper, we address the practical testig of a heuristicbased FA (HBFA) for computig optima of discrete oliear optimizatio problems, where the discrete variables are of biary type. A importat issue i FA is the formulatio of attractiveess of each firefly which i tur affects its movemet i the search space. Dyamic updatig schemes are proposed for two parameters, oe from the attractiveess term ad the other from the radomizatio term. Three simple heuristics capable of trasformig real cotiuous variables ito biary oes are aalyzed. A ew sigmoid erf fuctio is proposed. I the cotext of FA, three differet implemetatios to icorporate the heuristics for biary variables ito the algorithm are proposed. Based o a set of bechmark problems, a compariso is carried out with other biary dealig metaheuristics. The results demostrate that the proposed HBFA is efficiet ad outperforms biary versios of differetial evolutio (DE) ad particle swarm optimizatio (PSO). The HBFA also compares very favorably with agle modulated versio of DE ad PSO. It is show that the variat of HBFA based o the sigmoid erf fuctio with movemets i cotiuous space is the best, i terms of both computatioal requiremets ad accuracy. 1. Itroductio This paper aims to aalyze the merit, i terms of performace, of a heuristic-based firefly algorithm (HBFA) for computig the optimal ad biary solutio of boud costraied oliear optimizatio problems. The problem to be addressed has the form mi f (x) subject to x Ω R (acompactcovexset) x l {0, 1} for l=1,...,, where f is a cotiuous fuctio. Due to the compactess of Ω,wealsohaveLb l x l Ub l, l=1,...,,wherelb ad Ub are the vectors of the lower ad upper bouds, respectively. We do ot assume that f is differetiable ad covex. Istead of searchig for ay local (oglobal) solutio we wat the (1) globally best biary poit. Direct search methods might be suitable sice we do ot assume differetiability. However, they are oly local optimizatio procedures ad therefore thereisoguarateethataglobalsolutioisreached.for global optimizatio, stochastic methods are geerally used ad aim to explore the search space ad coverge to a global solutio. Metaheuristics are higher-level procedures or heuristics that are desiged to search for good solutios, kow as ear-optimal solutios, with less computatioal effort ad time tha more classical algorithms. They are usually odetermiistic ad their behaviors do ot deped o problem s properties. Populatio-based metaheuristics have bee used to solve a variety of optimizatio problems, from cotiuous to the combiatorial oes. Metaheuristics are commo for solvig discrete biary optimizatio problems [1 10]. May approaches have bee developed aimig to solve oliear programmig problems

2 2 Advaces i Operatios Research with mixed-discrete variables by trasformig the discrete problem ito a cotiuous oe [11]. The most used ad simple approach solves the cotiuous relaxed problem ad the discretizes the obtaied solutio by usig a roudig scheme. This type of approach works well o simple ad small dimesio academic ad bechmark problems but may be somehow limited o some real-world applicatios. Recetly, a metaheuristic optimizatio algorithm, termed firefly algorithm (FA), that mimics the social behavior of fireflies based o the flashig ad attractio characteristics of fireflies, has bee developed [12, 13]. This is a swarm itelligece optimizatio algorithm that is capable of competig with the most well-kow algorithms, like at coloy optimizatio, particle swarm optimizatio, artificial bee coloy, artificial fish swarm, ad cuckoo-search. FA performace is cotrolled by three parameters: the radomizatio parameter α, the attractiveess β, ad the absorptio coefficiet γ. Authors have argued that its efficiecy is due to its capability of subdividig the populatio ito subgroups (sice local attractio is stroger tha logdistace attractio) ad its ability to adapt the search to problem ladscape by cotrollig the parameter γ [14, 15]. Several variats of the firefly algorithm do already exist i the literature. Based o the settigs of their parameters, a classificatio scheme has appeared. Gaussia FA [16], hybrid FA with harmoy search [17], hybrid geetic algorithm with FA [18], self-adaptive step FA [15], ad modified FA i [19] are just a few examples. Further improvemets have bee made aimig to accelerate covergece (see, e.g., [20 22]).A practical covergece aalysis of FA with differet parameter sets is preseted i [23]. FA has become popular ad widely used i recet years i may applicatios, like ecoomic dispatch problems [24] ad mixed variable optimizatio problems [25]. The extesio of FA to multiobjective cotiuous optimizatio has already bee ivestigated [26]. A recet review of firefly algorithms is available i [14]. Based o the effectiveess of FA i cotiuous optimizatio, it is predicted that it will perform well whe solvig discrete optimizatio problems. Discrete versios of the FA are available for solvig discrete NP hard optimizatio problems [27, 28]. The mai purpose of this study is to icorporate some heuristics aimig to deal with biary variables i the firefly algorithm for solvig oliear optimizatio problems with biary variables. The biary dealig methods that were implemeted are adaptatios of well-kow heuristics for defiig 0 ad 1 bit strigs from real variables. Furthermore, a ew sigmoid fuctio aimig to costrai a real valued variable to the rage [0, 1] is also proposed. Three differet implemetatios to icorporate the heuristics for biary variables ad adapt FA to biary optimizatio are proposed. We apply the proposed heuristic strategies to solve a set of bechmark oliear problems ad show that the ewly developedhbfaiseffectiveibiaryoliearprogrammig. The remaiig part of the paper is orgaized as follows. Sectio 2 reviews the stadard FA ad presets ew dyamic updates for some FA parameters, ad Sectio 3 describes differet heuristic strategies ad reports o their implemetatios to adapt FA to biary optimizatio. All the heuristic approaches are validated with tests o a set of well-kow boud costraied problems. These results ad a compariso with other methods i the literature are show i Sectio 4. Fially,thecoclusiosadideasforfutureworkarelistedi Sectio Firefly Algorithm Firefly algorithm is a bioispired metaheuristic algorithm that is able to compute a solutio to a optimizatio problem. It is ispired by the flashig behavior of fireflies at ight. Accordig to [12, 13, 19], the three mai rules used to costruct the stadard algorithm are the followig: (i) all fireflies are uisex, meaig that ay firefly ca be attracted to ay other brighter oe; (ii) the attractiveess of a firefly is determied by its brightess which is associated with the ecoded objective fuctio; (iii) attractiveess is directly proportioal to brightess but decreases with distace. Throughout this paper, we let represet the Euclidea orm of a vector. We use the vector x = (x 1,x 2,...,x ) T to represet the positio of a firefly i the search space. The positio of the firefly j will be represeted by x j R.We assume that the size of the populatio of fireflies is m. I the cotext of problem (1), firefly j is brighter tha firefly i if f(x j )<f(x i ) Stadard FA. First, i the stadard FA, the positios of the fireflies are radomly geerated i the search space Ω as follows: x i l =Lb l + rad (Ub l Lb l ), for l=1,...,, (2) where rad is a uiformly distributed radom umber i [0, 1], hereafter represeted by rad U[0,1]. The movemet of a firefly i is attracted to aother brighter firefly j ad is give by x i =x i +β(x j x i )+α(rad 0.5) S, (3) where α [0,1] is the radomizatio parameter, rad U[0, 1], S R is a problem depedet vector of scalig parameters, ad β=β 0 exp ( γ xi x j p ) for p 1 (4) gives the attractiveess of a firefly which varies with the light itesity/brightess see by adjacet fireflies ad the distace betwee themselves ad β 0 is the attractio parameter whe the distace is zero [12, 13, 22, 29]. Besides the preseted exp fuctio, ay mootoically decreasig fuctio could be used. The parameter γ which characterizes the variatio of the attractiveess is the light absorptio coefficiet ad is crucial

3 Advaces i Operatios Research 3 Data: k max, f, η Set k=0; Radomly geerate x i Ω,,...,m; Evaluate f(x i ),,...,m,rakfireflies(fromlowesttolargestf); while k k max ad f(x 1 ) f >ηdo forall x i such that i=2,...,m do forall x j such that j=1,...,i 1 do Compute radomizatio term; Compute attractiveess β; Move firefly i towards j usig (3); Evaluate f(x i ),,...,m,rakfireflies(fromlowesttolargestf); Set k=k+1; Algorithm1:StadardFA. to determie the speed of covergece of the algorithm. I theory, γ couldtakeayvalueitheset[0, ).Wheγ 0, the attractiveess is costat β=β 0, meaig that a flashig firefly ca be see aywhere i the search space. This is aidealcaseforaproblemwithasigle(usuallyglobal) optimum sice it ca easily be reached. O the other had, whe γ, the attractiveess is almost zero i the sight of other fireflies ad each firefly moves i a radom way. I particular, whe β 0 = 0,thealgorithmbehaveslikea radom search method [13, 22]. The radomizatio term ca be exteded to the ormal distributio N(0, 1) or to ay other distributio [15]. Algorithm 1 presets the mai steps of the stadard FA (o cotiuous space) Dyamic Updates of α ad γ. The relative value of the parameters α ad γ affects the performace of FA. The parameter α cotrols the radomess or, to some extet, the diversity of solutios. Parameter γ aims to scale the attractio power of the algorithm. Small values of γ with large values of α ca icrease the umber of iteratios required to coverge to a optimal solutio. Experiece has show that α must take large values at the begiig of the iterative process to eforce the algorithm to icrease the diversity of solutios. However, small α values combied with small values of γ i the fial iteratios icrease the fie-tuig of solutios sice the effort is focused o exploitatio. Thus, it is possible to improve the quality of the solutio by reducig the radomess. Covergece ca be improved by varyig the radomizatio parameter α so that it decreases gradually as the optimum solutio is approachig [22, 24, 26, 29]. I order to improve covergece speed ad solutio accuracy, dyamic updates of the parameters α ad γ of FA, which deped o the iteratio couter k of the algorithm, are implemeted as follows. Similarly to the factor which cotrols the amplificatio of differetial variatios, i differetial evolutio (DE) metaheuristic [5], the iertial weight, i particle swarm optimizatio (PSO) [29, 30], ad the pitch adjustig rate, i the harmoy search (HS) algorithm [31], we allow the value of α to decrease liearly with k,fromaupperlevelα max to a lower level α mi : α (k) =α max k α max α mi k max, (5) where k max is the maximum umber of allowed iteratios. To icrease the attractiveess with k, the parameter γ is dyamically updated by γ (k) =γ max exp ( k l ( γ mi )), (6) k max γ max where γ mi ad γ max are the miimum variatio ad maximum variatio of attractiveess, respectively Lévy Depedet Radomizatio Term. We remark that our implemetatio of the radomizatio term i the proposed dyamic FA cosiders the Lévy distributio. Based o the attractiveess β, i(4), the equatio for the movemet of firefly i towards a brighter firefly j cabewritteas follows: x i =x i +y i with y i =β(x j x i )+αl(x 1 )σ i x, (7) where L(x 1 ) is a radom umber from the Lévy distributio cetered at x 1, the positio of the brightest firefly, with a uitary stadard deviatio. The vector σ i x represets the variatio aroud x 1 (adbasedorealpositiox) σ i x =( xi 1 x1 1,..., xi x1 )T. (8) 3. Dealig with Biary Variables The stadard FA is a real-coded algorithm ad some modificatios are eeded to eable it to deal with discrete optimizatio problems. This sectio describes the implemetatio of some heuristics with FA for biary oliear optimizatio problems. I the cotext of the proposed HBFA, three differet heuristics to trasform a cotiuous real

4 4 Advaces i Operatios Research variable ito a biary oe are preseted. Furthermore, to exted FA to biary optimizatio, differet implemetatios to icorporate the heuristic strategies ito FA are described. We will use the term discretizatio to defie the process that trasforms a cotiuous real variable, represeted, for example, by x, ito a biary oe, represeted by b Sigmoid Logistic Fuctio. This discretizatio methodology is the most commo i the literature whe populatiobased stochastic algorithms are cosidered i biary optimizatio, amely, PSO [6, 8, 9], DE [3], HS [1, 32], artificial fish swarm [33], ad artificial bee coloy [4, 7, 10]. Whe x i moves towards x j, the likelihood is that the discrete compoets of x i chage from biary umbers to realoes.totrasformarealumberitoabiaryoe,the followig sigmoid logistic fuctio costrais the real value to the iterval [0, 1]: Sigmoid fuctio /(1 + exp( x)) 0.5(1 + (x)) erf Figure 1: Sigmoid fuctios. x sig (x i l )= 1 1+exp ( x i l ), (9) where x i l, i the cotext of FA, is the compoet l of the positio vector x i (of firefly i) after movemet recall (7)ad (4). Equatio (9) iterprets the floatig-poit compoets of a solutio as a set of probabilities. These are the used to assig appropriate biary values by usig b i l ={ 1, if rad sig (xi l ) 0, otherwise, (10) where sig(x i l ) gives the probability that the compoet itself is 0 or 1 [28] ad rad U[0,1].Weotethatdurigthe iterative process the firefly positios, x, wereotallowedto move outside the search space Ω Proposed Sigmoid erf Fuctio. The error fuctio is a special fuctio with a shape that appears maily i probability ad statistics cotexts. Deoted by erf, the mathematical fuctio defied by the itegral, erf (x) = 2 x π exp ( t 2 )dt, (11) satisfies the followig properties erf (0) =0, erf ( ) = 1, erf (+ ) =1, erf ( x) = erf (x) 0 (12) ad it has a close relatio with the ormal distributio probabilities. Whe a series of measuremets are described by a ormal distributio with mea 0 ad stadard deviatio σ, the erf fuctio evaluated at (x/σ 2), for a positivex, gives the probability that the error of a sigle measuremet lies i the iterval [ x, x]. The derivative of the erf fuctio follows immediately from its defiitio: d 2 erf (t) = dt π exp ( t2 ), for t R. (13) The good properties of the erf fuctio are thus used to defie a ew sigmoid fuctio, the sigmoid erf fuctio: sig (x i l ) = 0.5 (1 + erf (xi l )), (14) which is a bouded differetiable real fuctio defied for all x R ad has a positive derivative at each poit. A compariso of both fuctios (9) ad(14) is depicted i Figure 1. Note that the slope at the origi of the sigmoid fuctio i (14) is aroud , while that of fuctio (9) is 0.25, thus yieldig a faster growig from 0 to Roudig to Iteger Part. The simplest discretizatio procedure of a cotiuous compoet of a poit ito 0/1 bit uses the roudig to the iteger part fuctio, kow as floor fuctio, ad is described i [34]. Each cotiuous value x i l R is trasformed ito a biary oe, 0 bit or 1 bit, b i l, for l=1,...,i the followig way: b i l = xi l mod 2, (15) where z represets the floor fuctio of z ad gives the largest iteger ot greater tha z. The floatig-poit value x i l is first divided by 2 ad the the absolute value of the remaider is floored. The obtaied iteger umber is the bit value of the compoet Heuristics Implemetatio. I this study, three methods capable of computig global solutios to biary optimizatio problems usig FA are proposed Movemet o Cotiuous Space. I this implemetatio of the previously described heuristics, deoted by movemet o cotiuous space (mcs), the movemet of each firefly is made o the cotiuous space ad its attractiveess term is updated cosiderig the real positio vector.therealpositiooffireflyi is discretized oly after all movemets towards brighter fireflies have bee carried out. We ote that the fitess evaluatio of each firefly, for firefly

5 Advaces i Operatios Research 5 Data: k max, f, η Set k=0; Radomly geerate x i Ω,,...,m; Discretize positio of firefly i: x i b i,,...,m; Compute f(b i ),,...,m, rak fireflies (from lowest to largest f); while k k max ad f(b1 ) f >ηdo forall x i such that i=2,...,m do forall x j such that j=1,...,i 1 do Compute radomizatio term; Compute attractiveess β; Move positio x i of firefly i towards x j usig (7); Discretize positios: x i b i,,...,m; Compute f(b i ),,...,m, rak fireflies (from lowest to largest f); Set k=k+1; Algorithm 2: HBFA with mcs. Data: k max, f, η Set k=0; Radomly geerate x i Ω,,...,m; Discretize positio of firefly i: x i b i,,...,m; Compute f(b i ),,...,m, rak fireflies (from lowest to largest f); while k k max ad f(b1 ) f >ηdo forall b i such that i=2,...,m do forall b j such that j=1,...,i 1 do Compute radomizatio term; Compute attractiveess β based o distace bi b j p ; Move biary positio b i of firefly i towards b j usig x i =b i +β(b j b i )+αl(b 1 )σ i b ; Discretize positio of firefly i: x i b i ; Compute f(b i ),,...,m,rakfireflies(fromlowesttolargestf); Set k=k+1; Algorithm 3: HBFA with mbs. rakig, is always based o the biary positio. Algorithm2 presets the mai steps of HBFA with mcs Movemet o Biary Space. This implemetatio, deoted by movemet o biary space (mbs), moves the biary positio of each firefly towards the biary positios of brighter fireflies; that is, each movemet is made o the biary space although the correspodig positio may fail to be0or1bitstrigadmustbediscretizedbeforetheupdatig of attractiveess. Here, fitess is also based o the biary positios. Algorithm 3 presets the mai steps of HBFA with mbs Probability for Biary Compoet. For this implemetatio, amed probability for biary compoet (pbc), we borrow the cocept from the biary PSO [6, 9, 35] where each compoet of the velocity vector is directly used to compute the probability that the correspodig compoet oftheparticlepositio,x i l,is0or1.similarly,ithefa algorithm, we do ot iterpret the vector y i i (7) asastep size, but rather as a mea to compute the probability that each compoet of the positio vector of firefly i is0or1.thus,we defie b i l ={ 1, if rad sig (yi l ) (16) 0, otherwise, where sig() represets a sigmoid fuctio. Algorithm 4is the pseudocode of HBFA with pbc. 4. Numerical Experimets I this sectio, we preset the computatioal results that were obtaied with HBFA Algorithms 2, 3,ad4,usig(9), (14), or (15) aimig to ivestigate its performace whe solvig a set of biary oliear optimizatio problems. Two small 0-1 kapsack problems are also used to test the algorithms behavior o liear problems with 0/1 variables. The umerical experimets were carried out o a PC Itel Core 2 Duo Processor E7500 with 2.9 GHz ad 4 Gb of memory. The algorithms were coded i Matlab Versio (R2012b).

6 6 Advaces i Operatios Research Data: k max, f, η Set k=0; Radomly geerate x i Ω,,...,m; Discretize positio of firefly i: x i b i,,...,m; Compute f(b i ),,...,m, rak fireflies (from lowest to largest f); while k k max ad f(b1 ) f >ηdo forall b i such that i=2,...,m do forall b j such that j=1,...,i 1 do Compute radomizatio term; Compute attractiveess β based o distace bi b j p ; Compute y i usig biary positios (see (7)); Discretize y i ad defie b i usig (16); Compute f(b i ),,...,m,rakfireflies(fromlowesttolargestf); Set k=k+1; Algorithm 4: HBFA with pbc Experimetal Settig. Each experimet was coducted 30 times. The size of the populatio is made to deped o the problem sdimesioadissettom = mi{40, 2 }.Some experimets have bee carried out to tue certai parameters ofthealgorithms.itheproposedfawithdyamicα ad γ, they are set as follows: β 0 =1, p=1, α max = 0.5, α mi = 0.01, γ max =10,adγ mi = 0.1. I Algorithms 2 (mcs), 3 (mbs), ad 4 (pbc), iteratios were limited to k max = 500 ad the tolerace for fidig a good quality solutio is η= Results reported are averaged (over the 30 rus) of best fuctiovalues,umberoffuctioevaluatios,adumber of iteratios Experimetal Results. First,weuseasetoftebechmark oliear fuctios with differet dimesios ad characteristics. For example, five fuctios are uimodal ad the remaiig multimodal [3, 9, 10, 36]. They are displayed i Table 1. Although they are widely used i cotiuous optimizatio, we ow aim to coverge to a 0/1 bit strig solutio. First, we aim to compare with the results reported i [3, 9, 10]. Due to poor results, the authors i [10] do ot recommed the use of ABC to solve biary-valued problems. The other metaheuristics therei implemeted are the followig: (i) agle modulated PSO (AMPSO) ad agle modulated DE (AMDE) that icorporate a trigoometric fuctio as a bit strig geerator ito the classic PSO ad DE algorithms, respectively; (ii) biary DE ad PSO based o the sigmoid logistic fuctio ad (10), deoted by bide ad bipso, respectively. We oticed that the problems Foxholes, Griewak, Rosebrock, Schaffer, ad Step are ot correctly described i [3, 9, 10]. Table 2 shows both the averaged best fuctio values (obtaied durig the 30 rus), f avg,withthest.d.i paretheses, ad the averaged umber of fuctio evaluatios, fe, obtaied with the sigmoid logistic fuctio (see i (9)) ad (10),whileusigthethreeimplemetatios:mCS, mbs, ad pbc. Results obtaied for these te fuctios idicate that our proposal HBFA produces high quality solutios ad outperforms the biary versios bipso ad bide, as well as AMPSO ad AMDE. We also ote that mcs has the best fe values o 6 problems, mbs is better o 3 problems (oe is a tie with mcs), ad pbc o 2 problems. Thus, the performace of mcs is the best whe comparedwiththoseofmbsadpbc.thelatteristhe least efficiet of all, i particular for the large dimesioal problems. To aalyze the statistical sigificace of the results we perform a Friedma test. This is a oparametric statistical test to determie sigificat differeces i mea for oe idepedet variable with two or more levels also deoted as treatmets ad a depedet variable (or matched groups take as the problems). The ull hypothesis i this test is that the mea raks assiged to the treatmets uder testig are thesame.siceallthreeimplemetatiosareabletoreachthe solutios withi the η error tolerace o 9 out of 10 problems, the statistical aalysis is based o the performace criterio fe. I this hypothesis testig, we have three treatmets adtegroups.friedma schi-squarehasavalueof2.737 (with a P value of 0.255). For 2 degrees of freedom referece χ 2 distributio, the critical value for a sigificace level of 5% is Hece, sice , the ull hypothesis is ot rejected ad we coclude that there is o evidece that the three mea raks values have statistically sigificat differeces. To further compare the sigmoid fuctios with the roudig to iteger strategy, we iclude i Table 3 the results obtaied by the erf fuctio i (14), together with (10), ad the floor fuctio i(15). Oly the implemetatios mcs ad mbs are tested. The table also shows the averaged umber of iteratios, it. The results illustrate that implemetatio mcs (Algorithm 2) works very well with strategies based o (14), together with (10), ad (15). The success rate for all the problems is 100%, meaig that the algorithms stop because thef value at the positio of the best/brightest firefly is withi

7 Advaces i Operatios Research 7 Ackley Foxholes Griewak Quartic Rastrigi Rosebrock2 Rosebrock Table 1: Problems set. f (x) = 20exp ( x 2 i ) exp ( 1 cos (2πx i )) +20+e =30,Ω = [ 30, 30] 30, f =0at x = (0,...,0) 1 f(x) = j=1 (1/ (j + (x 1 a 1j ) 6 +(x 2 a 2j ) 6 )) [a ij ]=[ ] =2,Ω = [ , ] 2, f 13at x = (0, 0) f (x) = x 2 i cos ( x i i ) =30,Ω = [ 300, 300] 30, f =0at x = (0,...,0) f(x) = ix 4 i +U[0,1] =30,Ω = [ 1.28, 1.28] 30, f =0+ oise at x = (0,...,0) f(x) = 10 + (x 2 i 10cos (2πx i )) =30,Ω = [ 5.12, 5.12] 30, f =0at x = (0,...,0) f(x) = 100 (x 2 1 x 2) 2 +(1 x 1 ) 2 =2,Ω = [ 2.048, 2.048] 2, f =0at x = (1, 1) f(x) = 1 100(x 2 i x i+1 ) 2 +(1 x i ) 2 =30,Ω = [ 2.048, 2.048] 30, f =0at x =(1,...,1) Schaffer f(x) = (si( x2 1 +x2 2 )) ( (x1 2 +x2 2 ))2 =2,Ω = [ 100, 100] 2, f =0at x = (0, 0) Spherical Step f (x) = x 2 i =3,Ω = [ 5.12, 5.12] 3, f =0at x = (0, 0, 0) f(x) = 6 + x i =5,Ω = [ 5.12, 5.12] 5, f =30at x = (0, 0, 0, 0, 0) atoleraceη of the optimal solutio f,iallrus.further, mbs (Algorithm 3) works better whe the discretizatio of the variables is carried out by (15). Overall, mcs based o (14) produces the best results o 6 problems, mcs based o (15) gives the best results o 7 problems (icludig 4 ties with the former case), mbs based o (14) wisolyooeproblem, admbsbasedo(15) wis o 3 problems (all are ties with mcs based o (15)). Further, whe performig the Friedma test o the four distributios of fe values, the chi-square statistical value is (ad the P value is ). From the χ 2 distributio table, the critical value for a sigificace level of 5% ad 3 degrees of freedom is Sice > 7.81, the ull hypothesis is rejected ad we coclude that the observed differeces of the four distributios are statistically sigificat. We ow itroduce i the statistical aalysis the results reported i Tables 2 ad 3 cocered with both implemetatios mcs ad mbs. Six distributios of fe values are ow i compariso. Friedma s chi-square value is (P value =0.0027).Thecriticalvalueofthechi-squaredistributio for a sigificace level of 5% ad 5 degrees of freedom is Thus, the ull hypothesis of o sigificat differeces o mea raks is rejected ad there is evidece that the six distributios of fe values have statistically sigificat differeces. Multiple comparisos (two at a time) may be carried out to determie which mea raks are sigificatly differet. The estimates of the 95% cofidece itervals are show i the graph of Figure 2 for each case uder testig. Two compared distributios of fe are sigificatly differet if their itervals are disjoit ad are ot sigificatly differet if their itervals overlap. Hece, from the six cases, we coclude that the mea raks produced by mcs based o (15) are sigificatly differet from those of mbs based o (9) admbsbasedo(14). For the remaiig pairs of compariso there are o sigificat differeces o the mea raks.

8 8 Advaces i Operatios Research Table 2: Compariso with AMPSO, bipso, bide, ad AMDE based o f avg ad St.D. (show i paretheses). Prob. Ackley Foxholes Griewak Quartic Rastrigi Rosebrock2 Rosebrock Schaffer Spherical Step mcs based o (9) mbsbasedo(9) pbcbasedo(9) AMPSO bipso bide AMDE f avg fe f avg fe f avg fe f avg f avg f avg f avg 8.88e e e e e e e () () () (0.57e 01) (0.49e 01) (0.31e01) (0.76e00) 0.53e e e01 5.0e (9.03e 15) (9.03e 15) (9.03e 15) () (0.97e 14) (0.86e00) (0.0e00) 1.06e e e e () () () (0.44e01) (0.98e01) (0.10e02) (0.39e01) 4.55e e e e e e e (3.01e 01) (3.03e 01) (2.71e 01) (0.19e01) (0.19e01) (0.66e00) (0.81e00) 2.25e e e e () () () (0.35e02) (0.28e01) (0.45e02) (0.31e02) 0.49e e e e () () () (0.11e 03) (0.88e 04) (0.15e 04) (0.17e 04) 2.20e e e e () () () (0.86e02) (0.77e02) (0.16e02) (0.42e02) 0.24e e e00 1.0e () () () (0.42e 02) (0.11e 01) (0.27e 06) (0.0e00) 0.30e e e e () () () () (0.0e00) (0.41e 04) (0.15e 04) 0.17e e e () () () () (0.15e 04) (0.0e00) (0.60e 01) mcs based o (9) mbs based o (9) mcs based o (14) mbs based o (14) mcs based o (15) mbs based o (15) mbs based o (9) ad (14) have mea raks sigificatly differet from mcs based o (15) Figure 2: Cofidece itervals for mea raks of fe. For comparative purposes we iclude i Table 4 the resultsobtaiedbyusigtheproposedlévy (L)distributio i the radomizatio term, as show i (7), ad those produced by the Uiform (U) distributio, usig rad U[0,1] as show i (3). The reported tests use implemetatio mcs (described i Algorithm 2)withthetwoheuristicsforbiary variables: (i) the erf fuctio i (14), together with (10), ad (ii) the floor fuctio i (15). It is show that the performace of HBFA with Uiform distributio is very sesitive to the dimesio of the problem, sice the efficiecy is good whe is small but gets worse whe is large. Thus, we have show that the Lévy distributio is a very good bid. We add to some problems with = 30 from Table 1 Ackley, Griewak, Rastrigi, Rosebrock, ad Spherical three other fuctios Schwefel 2.22, Schwefel 2.26, ad Sum of Differet Power to compare our results with those reported i [1]. Schwefel 2.22 is uimodal ad for Ω = [ 10, 10] 30, the biary solutio is (0,...,0) with f =0;Schwefel2.26 is multimodal ad i Ω = [ 500, 500] 30,thebiarysolutio is (1,1,...,1) with f = ; SumofDifferet Power is uimodal ad i Ω = [ 1, 1] 30,themiimumis0 at (0,...,0). For the results of Table 5, we use HBFA based o mcs, with both erf fuctio i (14), together with (10), ad the floor fuctio (15). The table reports o the average

9 Advaces i Operatios Research 9 Prob. Table3:ComparisoofmCSversusmBSad(14)versus(15), based o f avg,fe,adit. mcs based o (14) mbsbasedo(14) mcsbasedo(15) mbsbasedo(15) f avg fe it f avg fe it f avg fe it f avg fe it Ackley 8.88e e e e Foxholes e Griewak Quartic 2.38e e e e 01 Rastrigi Rosebrock e Rosebrock e Schaffer e Spherical Step Table 4: Compariso betwee Lévy ad Uiform distributios i the radomizatio term, based o f avg,fe,adit(withst.d.i paretheses). Prob. Ackley Foxholes Griewak Quartic Rastrigi Rosebrock2 Rosebrock Schaffer Spherical Step mcs based o (14)+L mcs based o (14)+U mcs based o (15)+L mcs based o (15)+U f avg fe it f avg fe it f avg fe it f avg fe it 8.88e e e e () () () (1.26e 02) (9.03e 15) (9.03e 15) (9.03e 15) (9.03e 15) 1.27e () () () (2.76e 02) 2.38e e e e (1.70e 01) (1.16e01) (1.03e 01) (2.90e 01) 1.00e () () () (4.03e 01) () () () () 8.22e e () (1.02e02) () (8.43e01) () () () () () () () () () () () () Table 5: Compariso of HBFA (with mcs) with ABHS i [1]basedof avg,fe,adsr(%). Prob. mcs based o (14) mcsbasedo(15) ABHSi[1] f avg fe SR (%) f avg fe SR (%) f avg fe SR (%) Ackley 8.88e e e Griewak e Rastrigi e Rosebrock e Schwefel Schwefel e e e Spherical Sum of Differet Power

10 10 Advaces i Operatios Research Table 6: Results for varied dimesios ( = 50, 100, 200), cosiderig m=40. Ackley Griewak Quartic Rosebrock Spherical Step f mcs based o (14) mcsbasedo(15) f avg St.D. fe it f avg St.D. fe it e e e e e e oise e e e e 01 + oise e e e e 01 + oise e e e e e e e e e e e e e03 fuctio values, average umber of fuctio evaluatios, ad success rate (SR). Here, 50 idepedet rus were carried outtocomparewiththeresultsshowi[1]. The maximum umber of fuctio evaluatios therei used was It is show that our HBFA outperforms the proposed adaptive biary harmoy search (ABHS) Effect of Problem s Dimesio o HBFA Performace. We ow cosider six problems with varied dimesios from the previous set to aalyze the effect of problem s dimesio o the HBFA performace. We test three dimesios: =50, = 100, ad = 200. The algorithm s parameters are set as previously defied. We remark that the size of the populatio for all the tested problems ad dimesios is 40 poits. Table 6 cotais the results for compariso based o averaged values of f, umber of fuctio evaluatios, ad umber of iteratios. The St.D. of the f values are also displayed. Sice the implemetatio mcs, show i Algorithm2, performs better ad shows more cosistet results tha the other two, we tested oly mcs based o (14) admcsbasedo(15). Besides testig sigificat differeces o the mea raks produced by the two treatmets, mcs based o (14)admCS based o (15), we also wat to determie if the differeces o mea raks produced by problem s dimesio 50, 100, ad 200 are statistically sigificat at a sigificace level of 5%. Hece, we aim to aalyze the effects of two factors A ad B. A is the HBFA implemetatio (with two levels) ad B is the problem s dimesio (with three levels). For this purpose, the results obtaied for the six problems for each combiatio of the levels of A ad B are cosidered as replicatios. Whe performig the Friedma test for factor A, the chi-square statistical value is (P value = ) with 1 degree of freedom. The critical value for a sigificace levelof5%ad1degreeoffreedomitheχ 2 distributio table is 3.84, ad there is o evidece of statistically sigificat differeces. From the Friedma test for factor B, we also coclude that there is o evidece of statistically sigificat differeces, sice the chi-square statistical value is (P value = ) with 2 degrees of freedom. (The critical value of the χ 2 distributio table for a sigificace level of 5% ad 2 degrees of freedom is 5.99.) Hece, we coclude that the dimesiooftheproblemdoesotaffectthealgorithm sperformace. Oly with problem Quartic, the efficiecy of mcs based o (14) gets worse as dimesio icreases. Overall, both tested strategies are rather effective whe biary solutios are required o small as well as o large oliear optimizatio problems Solvig 0-1 Kapsack Problems. Fially,weaimtoaalyze the behavior of our best tested strategies whe solvig well-kow biary ad liear optimizatio problems. For this prelimiary experimet, we selected two small kapsack problems. The 0-1 kapsack problem (KP) ca be described as follows. Let be the umber of items, from whichwehavetoselectsomeofthemtobecarriedia kapsack. Let w l ad V l be the weight ad the value of item l, respectively, ad let W be the kapsack s capacity. It is usually assumed that all weights ad values are oegative. The objective is to maximize the total value

11 Advaces i Operatios Research 11 of the kapsack uder the costrait of the kapsack s capacity: max x s.t. V (x) l=1 l=1 w l x l W, V l x l x l {0, 1}, l=1,...,. (17) If item l is selected, x l =1;otherwise,x l =0.Usigapealty fuctio,thisproblemcabetrasformedito mi x l=1 V l x l +μmax {0, l=1 w l x l W}, (18) where μ is the pealty parameter which was set to be 100 i this experimet. Case 1 (a istace of a 0-1 KP with 4 items). Kapsack s capacity is W=6ad the vectors of values ad weights are V = (40, 15, 20, 10) ad w = (4,2,3,1).Basedotheabovemetioed parameters, the HBFA with mcs based o (14) was ru 30 times ad the averaged results were the followig. With a success rate of 100%, items 1 ad 2 are icluded i the kapsack ad items 3 ad 4 are excluded, with a maximum value of 55 (St.D. = 0.0e00). O average, the rus required 0.8 iteratios ad 29.3 fuctio evaluatios. With a success rate of 23%, the heuristic based o the floor fuctio, thus mcs based o (15), reached f avg =49(St.D. = 4.0e00) after a average of fuctio evaluatios ad a average of iteratios. Case 2 (aistaceofa0-1kpwith8items). Themaximum capacity of the kapsack is set to 8 ad the vectors of values ad weights are V = (83, 14, 54, 79, 72, 52, 48, 62) ad w = (3,2,3,2,1,2,2,3). The results are averaged over the 30 rus. After 8.7 iteratios ad fuctio evaluatios, the maximum value produced by the strategy mcs based o (14) is 286 (St.D. = 0.0e00), with a success rate of 100%. Items 1, 4, 5, ad 6 are icluded i the kapsack ad the others are excluded. The heuristic mcs based o (15)didotreachthe optimal solutio. All rus required 500 iteratios ad fuctio evaluatios ad the average fuctio values were f avg = 227 with St.D. = 3.14e Coclusios ad Future Work I this work we have implemeted several heuristics to compute a global optimal biary solutio of boud costraied oliear optimizatio problems, which have bee icorporated ito FA, yieldig the herei called HBFA. The problems addressed i this study have bouded cotiuous search space. Our FA proposal uses dyamic updatig schemes for two parameters, γ from the attractiveess term ad α from the radomizatio term, ad cosiders the Lévy distributio to create radomess i firefly movemet. The performace behavior of the proposed heuristics has bee ivestigated. Three simple heuristics capable of trasformig real cotiuous variables ito biary oes are implemeted. A ew sigmoid erf fuctio is proposed. I the cotext of the firefly algorithm, three differet implemetatios aimig to icorporate the heuristics for biary variables ito FA areproposed(mcs,mbs,adpbc).basedoasetof bechmark problems, a compariso is carried out with other biary dealig metaheuristics, amely, AMPSO, bipso, bide, ad AMDE. The experimetal results show that the implemetatio deoted by mcs whe combied with either the ew sigmoid erf fuctio or the roudig scheme basedothefloorfuctioisquiteefficietadsuperior to the other methods i compariso. The statistical aalysis carried out o the results shows evidece of statistically sigificat differeces o efficiecy, measured by the umber of fuctio evaluatios, betwee the implemetatio mcs basedothefloorfuctioapproachadthembsbased o both tested sigmoid fuctios schemes. We have also ivestigated the effect of problem s dimesio o the performace of our algorithm. Usig the Friedma statistical test we coclude that the differeces o efficiecy are ot statistically sigificat. Aother simple experimet has show that the implemetatio mcs with the sigmoid erf fuctio is effective i solvig two small 0-1 KP. The performace of this simple heuristic strategy will be further aalyzed to solve large ad multidimesioal 0-1 KP. Future developmets cocerig the HBFA will cosider its extesio to deal with iteger variables i oliear optimizatio problems. Differet heuristics to trasform cotiuous real variables ito iteger variables will be ivestigated. Challegig mixediteger ocovex oliear problems will be solved. Coflict of Iterests The authors declare that there is o coflict of iterests regardig the publicatio of this paper. Ackowledgmets The authors wish to thak two aoymous referees for their valuable suggestios to improve the paper. This work has bee supported by FCT (Fudação para a Ciêcia e Tecologia, Portugal) i the scope of the Projects PEst- OE/MAT/UI0013/2014 ad PEst-OE/EEI/UI0319/2014. Refereces [1]L.Wag,R.Yag,Y.Xu,Q.Niu,P.M.Pardalos,adM. Fei, A improved adaptive biary harmoy search algorithm, Iformatio Scieces,vol.232,pp.58 87,2013. [2] M. A. K. Azad, A. M. A. C. Rocha, ad E. M. G. P. Ferades, Improved biary artificial fish swarm algorithm for the 0-1 multidimesioal kapsack problems, Swarm ad Evolutioary Computatio, vol. 14, pp , [3] A.P.EgelbrechtadG.Pampará, Biary differetial evolutio strategies, i Proceedigs of the IEEE Cogress o Evolutioary Computatio (CEC 07), pp , September [4] M. H. Kasha, N. Nahavadi, ad A. H. Kasha, DisABC: a ew artificial bee coloy algorithm for biary optimizatio, Applied Soft Computig Joural,vol.12,o.1,pp ,2012.

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