Borgulya

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Borgulya

  1. 1. CEJOR manuscript No.(will be inserted by the editor)A Multi-objective Evolutionary Algorithmwith a Separate ArchiveIstv´n Borgulya aUniversity of P´cs, Faculty of Business and Economics, H 7621 P´cs R´k´czi ut e e a o ´80, Hungary, e-mail: borgulya@ktk.pte.huReceived: date 24.06.2004 / Revised version: 16.11.2004Abstract In this paper a new algorithm has been developed for solvingconstrained non-linear multi-objective optimization problems. This algo-rithm was acquired by the modification of an earlier algorithm, while weuse other selection and recombination operators from the earlier algorithm,we extended the algorithm with a separate archive (population). The archivecontinuously stores the new potential Pareto optimal solutions, when it be-comes full, the algorithm deletes a part of the archive. The new versiongives results which show a substantial improvement in quality over the ear-lier version and performs slightly better then NSGA II on the chosen testproblems.1 Multi-objective optimization problemWe can examine a constrained nonlinear multi-objective optimizationproblem (MOP) with the following form Minimize f (x) = (f 1 ( x),... ,f k ( x))where x=(x 1 ,...,x m )∈ F ⊆S , x is the decision vector, S the decision spaceand where we have k objective functions f i : m → . Let us suppose that the domain S of dimension m is m S = {x∈ | ai ≤ xi ≤ bi , ai , bi , xi ∈ ,i=1,2,...,m}whereas the feasible region F ⊆ S is defined by a set of r additional linear The Hungarian Research Program OTKA T 042448 supported the study.
  2. 2. 24 I. Borgulya and/or non-linear constraints (r ≥0): gj (x) ≤0, j = 0, 1, 2,..., r. Con-straints in terms of equations e.g. h( x)=0, can be replaced by a pair ofinequalities: h( x)- 0, - h( x)- 0 where an additional parameter δ is used todefine the tolerance of the algorithm. Let us review some corresponding concepts and definitions: Pareto dominance: A solution x =(x 1 , x 2 , ... ,x m )∈ S is said to dominatea solution z=(z 1 , z 2 , ... ,z m )∈ S (notation: x z ) if and only if ∀ i ∈{1,2,...,k} f i ( x ) ≤f i ( z )∧∃j ∈{1,2,...,k} f j ( x )< f j ( z). Pareto optimality: A solution x ∈S is Pareto optimal if and only if z∈ S: z x. Pareto optimal set: For a given multi-objective problem (MOP), thePareto optimal set (PS) is defined as: PS= {x ∈S | z ∈S: z x }. Pareto front: For a given MOP and Pareto optimal set PS, the Paretofront (PF) is defined as: PF= {f (x) | x ∈PS}. The solving of MOPs is a significantly more complex task than solvingscalar optimization problems, because their solution is usually not a singlevalue. The applicability of the best-known, classical methods for solvingMOPs is mostly restricted to linear objective functions [28]. In order to solvethe problem these methods reduce the objective functions to one function bymeans of weighting; for example, they use a lexicographic order or use linearprogramming procedures. In several cases the decision-maker’s possibilitiesare widened by interactive facilities to allow the results to be monitored, toset parameters and to influence the directions of the search [28], [20]. In the case of non-linear objective functions the solving of the MOPis generally more difficult. We may choose from families of methods basedon a fuzzy system (e.g. [25], [26]), based on methods applying parame-ter optimization (e.g. [17]) and we may choose heuristics methods. Variousheuristic methods are frequently used to solve MOP, such as simulated an-nealing, tabu search, evolutionary algorithm (EA) and versions (e.g. geneticalgorithm (GA), evolutionary strategy (ES)). EAs are well suited to MOP and a number of different algorithms hasbeen developed [30]. If we focus on the applied technique, we can dif-ferentiate aggregation-based methods, population-based methods, Pareto-dominance principle based techniques and niche-method- based techniques[30]. The majority of the methods use the Pareto-dominance principle. During the execution of an EA, a local set of Pareto optimal solutions isdetermined at each EA generation. Many algorithms also use a secondarypopulation storing all, or some Pareto optimal solutions found through thegenerations [30]. This is due to the EA method’s stochastic nature, whichdoes not guarantee that desirable solutions, once found, remain in the popu-lation until the algorithm terminates. This secondary population is regularlyupdated, and it can generally be used in two ways. The first of these waysis when the individuals of the secondary population take part in the evo-lutionary process (e.g. in the selection and recombination operations) [31],[22], whilst in the second way the secondary population is a separate archiveand stores only the potential Pareto optimal solutions found [5], [7].
  3. 3. A Multi-objective Evolutionary Algorithm 25 In this paper we show a new algorithm for MOP. In [3] we have demon-strated an EA to solve the MOP. This method, named MOSCA (M ulti-objective Optimum S earch with C luster-based Aalgorithm) is a cluster-based EA also using the Pareto-dominance principle and which producesthe approximation of PF with the decision maker. The MOSCA can han-dle nonlinear continuous objective functions and takes constraints into ac-count. The new method is an extended and improved version of MOSCA.The difference between the old and the new version (named MOSCA2) isas following:– MOSCA2 uses subpopulations instead of clusters,– MOSCA2 use truncation selection instead of random selection,– MOSCA2 is extended with recombination operator and– MOSCA2 is extended with a separate archive (population) also. The new algorithm is capable of solving unconstrained or constrainedcontinuous nonlinear MOPs, and by increasing the size of the archive itproduces more and more high quality solutions. It can apply to cases of1-100 numbers of dimensions. In the remainder of the paper we review the technique of the new algo-rithm, compare with other techniques, discuss the new algorithm in detail,and show the computation results with a comparison with other methods.2 About the used techniquesIn the opinion of researchers (e.g. [13], [4]) there are two important tasksfor solving the MOP: to reach a good convergence to the PF and to coverall points of PF with different solutions. Analyzing the methods we find dif-ferent techniques solve this task. Let us see the techniques of our algorithmwith a comparison of the techniques of three well known Pareto-based ap-proaches; with the PESA (the Pareto Envelope-based Selection Algorithm)[6], with the SPEA2 (Strength Pareto evolutionary algorithm 2) [33] andwith the NSGAII (Non-dominated sorting genetic algorithm II) [9]. By thiscomparison we analyze the convergence, the diversity preservation and thehandling of the archive. We can say in general that the methods reach a good convergence with aselection operator. They usually apply a fitness function based on the Paretodominance, and use a niching method (e.g. crowding) for the formation andmaintenance of a stable subpopulation within the population. We usuallycan’t separate the technique which has an influence on the convergence oran influence on the diversity preservation. Let us see the techniques of thechosen methods. The PESA is a conventional GA, in which selection and diversity main-tenance are controlled with a hyper-grid based scheme. PESA use a smallinternal population and a larger external population or archive for the ac-tually non-dominated solutions. The archive (and the objective space) is
  4. 4. 26 I. Borgulyadivided into hyper-box. Each individual in the archive is associated with ahyper-box, and has an attribute called a squeeze factor, which is the totalnumber of individuals in the same box (this is a crowding measure). Thesqueeze factor is used for selective fitness by tournament selection, wherethe individuals with the lowest squeeze factor are chosen. With this selec-tion the search is oriented to areas of the PF, where the population haslittle representation. This squeeze factor is also used for the archive update;if the archive fills, we can remove the individual with the highest squeezefactor from the archive. The SPEA2 is a conventional GA too, in which selection and diversitymaintenance is based on a fine grained fitness function. Like PESA, SPEA2uses a small internal population and a larger external population or archivefor the non-dominated solutions. The fitness function is based on two values:for each individual it takes into account how many individuals it dominatesand density information. The density estimation technique is an adaptationof the k-th nearest method, and the density at any individual is a function ofthe distance to the k-th nearest individuals. Based on this fitness functionthe SPEA2 uses a binary tournament selection with replacement on thearchive. For archive update the SPEA2 uses a truncation procedure. If thearchive fills, the truncation procedure removes iteratively individuals, whichhave the minimum distance to another individual. The NSGAII is a conventional GA with binary tournament selection,recombination and mutation operators. The idea behind NSGAII is thata ranking selection method is used to emphasize current non-dominatedpoints and a niching (crowding) method is used to maintain diversity in thepopulation. In this ranking method the NSGAII uses density information.To get an estimate of the density of a solution, NSGAII computes a crowdingdistance to every solution. The crowding distance serves as an estimate ofthe size of the largest cuboid enclosing the point i without including anyother point in the population. During the ranking process the algorithm usesthis crowding distance: between two solutions with the same non-dominatedranks it prefers the point where the size of the cuboid enclosing it is larger.To get good convergence the NSGAII also has an elitist strategy. After thechild population is formed from the parent population, both populationsare combined and the ranking method is performed. The new populationwill constructed from the best individuals based on the ranking. Finally the MOSCA2. Our algorithm is a steady-state ES. It uses asubpopulation structure in the population, a separate archive for the non-dominated solutions and a fitness function based on a Pareto-based rankingmethod. To obtain good convergence our algorithm uses a truncation selec-tion method, and to speed up the convergence it uses a deleting procedureThe deleting procedure deletes a given percentage of the most dominatedindividuals from the subpopulations periodically. The diversity maintenance is based on the subpopulation structure with-out density information or niching method. Our algorithm uses only a re-combination operator and the deleting procedure to reach uniformly dis-
  5. 5. A Multi-objective Evolutionary Algorithm 27tributed solutions along the PF. The selection method selects the parentsfrom two subpopulations, and the recombination operator chooses the twoparents from different subpopulations. The deleting procedure can deletecomplete subpopulation where elements are mostly dominated (The deletedsubpopulation will be replaced new ones). The subpopulation structure also helps to maintain the diversity. Themembers of the population are segregated into t subpopulations, each sub-population will approximate another part of the PF sought. Each subpop-ulation stores only non-dominated individuals of the possible members ofthe subpopulation (in a limited amount). The dominance of a new descen-dant which enters into the subpopulation is determined by comparing itto a designated non-dominated individual, the prototype. If it finds a non-dominated descendant superior to the previous prototype, it deletes theformer members of the subpopulation, and replaces the prototype by thenew descendant. In our algorithm we also use a separate archive. This separate archive(notation: SARC ) improves the quality of the solutions. The archive period-ically stores the new potential Pareto optimal solutions found. The archivehas a limited size and when it is full, the algorithm deletes a given percent-age of its elements. We select the most dominated individuals for deletion. Iffurther elements still need to be deleted from the archive, then a part of theremaining non-dominated solutions are deleted. This is a filtering process,which selects iteratively one of the solutions close to each other, the otherones are deleted. Our algorithm can solve constrained problems as well and only the NS-GAII has a similar technique. The principle of the constrained problem-solving is as follows: the constraints at a point of the search domain (S ) areregarded in the form of a measure that is a degree of violation of constraints.Each point of S is characterized with a D distance from the constraint space,on the grounds of this distance the points may be accepted or rejected andpoints outside the constraint space (infeasible points) are successfully uti-lized [2]. Let us see the structure of the algorithm. This is different from the struc-ture of the earlier methods. Regarding the function of the algorithm this canbe divided into two steady-state EAs. The first EA, or first stage, improvesthe quality of the initial population. In this stage the subpopulations haveonly one individual, the ”prototype”. The second EA, or second stage, isa steady-stage ES, and approximates the PF sought with more and morepoints. In each iteration it focuses on a subpopulation, looks for new, bettersolutions and inserts new non-dominated descendants into the subpopula-tion. It update the SARC and uses the deleting procedure periodically. In conclusion we can say, that the MOSCA2 represents a new technique.It has a different algorithm and population structure compared to the wellknown methods. With the help of the subpopulation, the deleting proce-dure and the recombination operator it can preserve the diversity of thepopulation and without density information or niching method we get a
  6. 6. 28 I. Borgulyauniformly distributed solution along the PF. The MOSCA2 therefore usesa new diversity preservation method.3 The new algorithm3.1 The main steps of MOSCA26 parameters affect the run of the algorithm: t, subt, arct, itend, , and .Their individual roles are: t - the number of subpopulations. subt- parameter of the subpopulations. The maximum size of each subpop- ulation may be subt. arcn - parameter of the SARC. arcn is the maximum size of the SARC. itend - parameter for the stopping condition. The procedure terminates if the value of it is more than itend. σ- parameter of the second stage. σ is the common standard deviation of N(0, σ) at the beginning. δ- is used to define the tolerance of the algorithm by equation constraints. Variables: it - the number of the iterations, itt - the number of iterations in thefirst stage, P - the population, SUBP - a subpopulation, kn - variable whichdetermines the timing of checks, p i - prototype. The main steps of MOSCA2 are as follows: Procedure MOSCA2 (t, subt, arcn, itend, σ, δ, SARC ). it=0, SUBP i = ∅ (i=1,2,...,t) /* The initial values. itt=1400, kn=100. Let p i ∈SUBP i (i=1,2, ..., t): SARC = ∅ /* Initial population *First stage* Ranking of P Do itt times it=it+1 Random selection. Reinsertion. od . * Second stage* Repeat Do kn times it=it+1. Truncation selection, recombination, mutation. Reinsertion. od Ranking of P, Update of SARC, Deleting Update of the parameters.
  7. 7. A Multi-objective Evolutionary Algorithm 29 until it > itend. end The results of the test examples show that the algorithm yields moreaccurate results if the elements are transformed to the [0, 1] interval (the[0, 1]m hypercube).3.2 The characteristics of the EAsThe main functions and characteristics of the EAs of the first and secondstage are as follows: Initial population. The p 1 , p 2 , ..., p t individuals of the P population arerandomly generated from S. These are the first prototypes. Population-subpopulations. Both EAs are using the same P populationand SUBP 1 , SUBP 2 ,..., SUBP t (P= ∪ SUBP i ) subpopulations. The p 1 ,p 2 ,..., p t individuals are the members of the SUBP 1 , SUBP 2 , ..., SUBP tsubpopulations, respectively. The sizes of the subpopulations vary, but theirmaximal size is given. Constraints. Let D(x) be the measure of violation of the constraints. (Ifx ∈F or no constraints are specified, THEN D(x)=0 ): 1/2 r D(x)= j=1 max{gj (x), 0}2 We utilize the value D(x) for the characterization of the decision vectorx in the following way: A vector x is better than a vector z if D(x)<D(z).In case D(x) = D( z) the characterizing depends on the Pareto dominance. Modified Pareto dominance. Based on the D measure, we use a modifiedPareto dominance in our algorithm. The definition of the modified Paretodominance is as follows: a decision vector x dominates the decision vector zbased on D (notation: x z) if D(x)<D(z), or in the D(x) = D(z) case ifx z. (This is a similar constraint handling technique as in NSGAII [11]). Fitness functions. The value of the fitness function is a rank number.The rank numbers are determined according to the Pareto ranking methodby Goldberg [16] using the modified Pareto dominance. A rank number of 1is assigned to the non-dominated individuals of the population, and a higherinteger value to the dominated ones. Selection operator. In the first stage descendants are randomly selectedfrom S, without the application of any further operators (recombination,mutation). In the second stage the algorithm selects parents from the ithand j th (i= j ) subpopulations. Truncation-selection is used for both sub-populations separately (according to the rank numbers), and the parentsoriginating from different subpopulations are stored separately (there aretwo selection pools). Recombination operator. In the second stage the algorithm selects on a p rprobability basis two parents from the selection pools, and the descendant is
  8. 8. 30 I. Borgulyaconstructed with single-point recombination. The two parents are randomlychosen from different selection pools. Otherwise it chooses a single parent,and the descendant is built by copy-making. Mutation operator. The values of continuous variables are modified bythe value of N(0, σ), where N(0, σ) is a normal distribution with mean0 and standard deviation. σ decreases parallel to the increasing number ofiterations. The values of discrete variables are modified by the step size withp md probability. There are some problems that we can solve more successfully, if we allowa few variables to take any possible ”noise” values. This ”noise” is a randomnumber from [a i ,b i ] and it is applied on the specified variables with a p mprobability basis. Reinsertion. On reinsertion in a subpopulation we do not use the ranknumbers: it is enough to examine the modified Pareto dominance betweena prototype and a descendant. In the first stage, the algorithm compares the descendant with the mostsimilar prototype. If the descendent dominates ( ) the former solution, itis replaced by the descendent. (The measure of similarity of two decisionvectors x and z is H(x, z)= 1/(1 + d(x, z)), where d(x, z) is the Euclideandistance of the vectors). In the second stage the algorithm compares the descendant to the pro-totype of the ith subpopulation. If the descendant dominates ( ) the pro-totype, the members of the subpopulation are deleted, and the prototypeis replaced by the new descendant (the rank number does not change). Ifboth the prototype and the descendant are non-dominated, the descendantis added to the ith subpopulation as a new individual (if the subpopulationsize can be still extended) Ranking. P, or SARC is ranked based on the modified Pareto dominanceperiodically. The procedure Ranking calculates the rank numbers of theindividuals. (This is a version of Goldberg’s Pareto ranking method [16]). Update of the SARC. A copy of the non-dominated individuals is storedin the SARC periodical. The use of the SARC is regulated in the followingway: – all the solutions in the SARC are different; – new, non-dominated solutions found are stored in SARC, if the maxi- mum SARC size has not been reached yet. – if a new non-dominated solution is found and the maximum SARC size has already been reached, a part of SARC is deleted first, and then the new non-dominated solution is stored. – When the SARC is full, sdp percentage of its elements are deleted as follows: 1. SARC is ranked with the Ranking procedure . 2. We select the most dominated individuals for deletion. 3. If further elements still need to be deleted then a part of the remaining non-dominated solutions are deleted. This is a filtering process, and we
  9. 9. A Multi-objective Evolutionary Algorithm 31 select only one of the solutions close to each other, the other ones are deleted (x and x’ are close to each other if the d(x,x’) is less than a predefined h value (initial value: h=min i |a i -b i |/1000)). If still further elements need to be deleted after the filtering procedure, we double the value of h and re-apply the filtering procedure. This is repeated until the necessary number of elements has been deleted. Deleting. In order to keep the diversity of the P, during the control agiven percentage of the most dominated individuals are deleted by the Delet-ing procedure: ddp percent of the most dominated individuals in P will bedeleted based on the rank number. The deleted subpopulations (prototypes)will be replaced with new ones. (We used ddp as an initial value. The valueof ddp decreases as the number of iterations increases). Stopping criteria. The algorithm is terminated if a pre-determined num-ber of iterations have been performed. The non-dominated individuals fromthe SARC will be the results, and the algorithm saves the results in a fileand prints a graph.4 Solutions of test problems4.1 Test problems.Deb [8] has identified several features that may cause difficulties for multi-objective EAs in converging to the PF and maintaining diversity within thepopulation. Based on these features, Ziztler et al. [32] constructed six testfunction (notation ZDT1,..., ZDT6). Each test function involves a particularfeature that is known to cause difficulty in the evolutionary optimizationprocess, mainly in converging to the PF (e.g. multimodality and deception).For some years the researchers use these test problems as benchmark prob-lems to demonstrate the possibilities of a new method. Other test problemsoften used are: e.g. the problem of Kursawe [19] (notation KUR), the prob-lem of Fonseca et al.[15] (notation FON) and the problem of Poloni [23](notation POL). For constrained problem we found less test problems. Deb et al. [12]proposed seven test problems to analyze the ability of the methods (notationCTP1,..., CTP7), and there are some other test problem in used in thepapers, e.g. the constrained test problem of Srinivas et al [27] (notationF3) and the welded beam design problem Deb [11] (notation WBD) frommechanical component design problems. To demonstrate the capabilities of MOSCA2, we chose the following12 benchmark problems: ZDT1, ZDT2, ZDT3, ZDT4, ZDT5, ZDT6, KUR,FON, POL, CTP7, F3 and WBD. All problems have two objective functionsand the CTP7, F3 and WBD have only constraints. The problems, showingthe number of variables, their bound, the Pareto optimal solution knownand the features are as follows: T1: n=30, variable bounds [0, 1]
  10. 10. 32 I. Borgulya f 1 (x)=x 1 ,f 2 (x)=g(x)(1- x1 /g(x)),g(x)=1+9( n xi )/(n − 1) i=2 Optimal solution: x 1 ∈ [0, 1] ,xi =0, i=2,...,n. Feature: PF is convex. T2: n=30, variable bounds [0, 1] n f 1 (x)=x 1 ,f 2 (x)=g(x)(1-( x1 /g(x))2 ),g(x)=1+9( i=2 xi )/(n − 1) Optimal solution: x 1 ∈ [0, 1] ,xi =0, i=2,...,n. Feature: PF is nonconvex. T3: n=30, variable bounds [0, 1] xi f 1 (x)=x 1 ,f 2 (x)=g(x)(1- x1 /g(x) − g(x) sin(10πx1 )), n g(x)=1+9( i=2 xi )/(n − 1) Optimal solution: x1 ∈ [0, 1] ,x i =0, i=2,...,n. Feature: PF consists of several noncontiguous convex parts. T4: n=10, variable bounds x 1 ∈ [0, 1], xi ∈ [−5, 5] i = 2, ..., n f1 (x)=x 1 ,f 2 (x)=g(x)(1- x1 /g(x)), n g(x)=1+10(n-1)+ i=2 (x2 − 10 cos(4πxi )) i Optimal solution: x 1 ∈ [0, 1] ,x i =0, i=2,...,n. Feature: There are 219 local Pareto fronts. T5: m=11, variable bounds x 1 ∈ {0, 1}30, x2 , ..., xm ∈ {0, 1}5 f 1 (x)=1+u(x 1),f 2 (x)=g(x)(1/f 1 (x)), m g(x)= i=2 v(u(xi )) 2 + u(xi ) if u(xi ) < 5 v(u(x i ))= 1 if u(xi ) = 5 Optimal solution: g(x)=10. Feature: There is a deceptive problem, the (discrete) PF is convex. T6: n=10, or 100, variable bounds [0, 1] f 1 (x)=1-exp(-4x 1 ) sin6 (4πx1 ),f 2 (x)=g(x)(1-( x1 /g(x))2 ), g(x)=1+9(( n xi )/(n − 1))0.25 i=2 Optimal solution: x 1 ∈ [0, 1] ,x i =0, i=2,...,n. Feature: The PF is nonconvex and non-uniformly distributed. KUR: n=3, variable bounds [−5, 5] n−1 2 f 1 (x)= i=1 (−10 exp(−0.2 xi + x2 )), i+1 n 0.8 f 2 (x)= i=1 (|xi | + 5 sin x3 ) i Feature: PF is nonconvex. FON: n=3, variable bounds [−4, 4] f 1 (x)=1-exp(- 3 (xi − √3 )2 ), i=1 1 3 1 f 2 (x)=1-exp(- √ )2 ) i=1 (xi + 3 1 1 Optimal solution: x 1 =x 2 = x3 ∈ − √3 , √3 Feature: PF is nonconvex. POL: n=2, variable bounds [−π, π] f 1 (x)= 1 + (A1 − B1 )2 + (A2 − B2 )2 f 2 (x)= (xi + 3)2 + (x2 + 1)2 A1 =0.5sin 1-2cos 1+sin 2-1.5cos 2 A2 =1.5sin 1-cos 1+2sin 2-0.5cos 2 B 1 =0.5sin x 1 -2cos x 1 +sin x 2 -1.5cos x 2
  11. 11. A Multi-objective Evolutionary Algorithm 33 B 2 = 1.5sin x 1 -cos x 1 +2sin x 2 -0.5cos x 2 Feature: PF is nonconvex and disconnected. CPT7: n=10, variable bounds [0, 1] n f 1 (x)=x 1 ,f 2 (x)=g(x)(1-( x1 /g(x))),g(x)=1+9( i=2 xi /(n − 1)0.5 subject: cos(θ)(f2 (x) − e) − sin(θ)f1 (x) ≥ d α [sin(bπ(sin(θ)(f2 (x) − e) + cos(θ)f1 (x))e ] where θ = −0.05π, α = 40, b = 5, c = 1, d = 6, e = 0 Optimal solution: g(x)=1. Feature: PF is disconnected. F3: n=2 variable bounds [−20, 20] f 1 (x)=(x 1 -2) 2 +(x 2 -1) 2 , f 2 (x)=9x 1 -(x 2 -1) 2 , 2 2 subj.: x 1 +x 2 -225 ≤ 0, x1 − 3x2 + 10 ≤ 0 Optimal solution: x 1 = 2.5,x 2 ∈ [2.5, 14.79] WBD: n=4, variable bounds x 1 , x 4 ∈ [0.125, 5.0] , x2 , x3 ∈ [0.1, 10.0] and x 1 , x 3 ,x 4 are discrete variables with a resolution of 0.0625. 2 f 1 (x)=1.10471 x 1 x 2 + 0.04811 x 3 x 4 (14 + x 2 ), f 2 (x)=2.1952 / (x 3 x 4 ) 3 subjects: g 1 (x)=13.600-τ (x) ≥ 0 g 2 (x)=30.000-σ(x) ≥ 0 g 3 (x)=x 4-x 1 ≥ 0 g 4 (x)=P c(x)-6.000 ≥ 0 τ (x) = d2 + e2 + (x2 de)/ 0.25x2 + (x1 + x3 )2 2 √ √ 6.000(14+0.5x2 ) 0.25(x2 +(x1 +x3 )2 ) d=6.000/ 2x1 x2 , e = 2 0.707x x (x2 /12+0.25(x +x )2 ) 2 [ 1 2 2 1 3 ] σ(x) = 504.000/(x2x4 ) 3 3 P c (x)=64.746022(1-0.0282346x 3 )x 3 x 4 . Feature: PF is convex.4.1.1 Parameter selection To achieve a quick and accurate solution weneed appropriate parameter values. Studying some of the test problems, weanalyzed how the parameter values were affecting the speed of the conver-gence and the finding of PF. Therefore we analyzed the number of subpopulations, the subpopulationsize, the value of the standard deviation σ, the p m , p md , p r probabilities,the value of the ddp, sdp percentage and the SARC size, which should beused in MOSCA2 to achieve a good trade-off between the probability offinding good solutions and the necessary running time to do so. In general,we have found: – Best behavior was obtained with a number of subpopulations between 10 and 60. With an inadequate number of subpopulations (t<10), only a few points of PF were found and those rather quickly. With a larger number of subpopulations MOSCA2 often achieves higher probabilities of finding more points of the PF, but with an increasing running time. Therefore, for the test problems (where the population size is generally
  12. 12. 34 I. Borgulya 100 in the different methods), the number of subpopulations of 10 was found appropriate (t=10).– The subpopulation size (subt) affects the approximation of the points of the PF similarly. Although good approximations can be achieved with small sizes (subt=2), by increasing the subpopulation size (subt= 10, 20 or 50) we can increase the number of the well approximating points, and so we chose a subpopulation size of 10 for the test problems (subt=10).– The value of σ is considerably task-dependent. Usually σ ∈ [0.001, 0.5] and we can obtain a good result with σ=0.1.– As a next step, we studied the p m , p md probabilities of the mutation, the p r probabilities of the recombination operator and the ddp percentage in the Deleting procedure. We found that we can use similar probabilities as by the GAs: p r =0.9 and p md =1/m for the discrete variable. We got good results with p m =0.1 in generally. There were many possible values found for the ddp percentage; the val- ues between 0.1 and 0.5 were usually satisfactory, but we had faster convergence to the PF with higher values. So we used the value of 0.5 for the test.– Finally, we studied the SARC sizes between 50 and 1000 and the value of the sdp percentage. We found that with the increasing size of the archive we obtained more and more solutions of better and better qual- ity without significantly lengthening the running time (because fewer removal steps are needed). There were many possible values found for the sdp percentage. When the sdp value is small many deletions are needed increasing computation time. High sdp values require less com- putation but can result in the loss of ”important” individual from the SARC. We searched a balance between the small and high sdp values and in the end we used the value 0.1 for the test.4.1.2 Performance metrics We use the following performance metrics in-troduced by Zitzler et al. [32]: 1 M ∗ (Y )= |Y | p ∈Y min{||p − p||; p ∈ P O} 1 M ∗ (Y ) = |Y 1 2 −1| ∗ p ∈Y |{ q’∈ Y ; |||p − q || > σ }| n M ∗ (Y ) = 3 i=1 max{||pi − qi ||; p , q ∈ Y } where Y’ is the set of objective vectors corresponding to the non-dominated solution found, and PO is a set of uniform Pareto-optimal ob-jective vectors. A neighborhood parameter σ*>0 is used in M ∗ to calculate 2the distribution of the non-dominated solutions. M ∗ (Y’) gives the average 1distance from Y’ to PO. M ∗ (Y’) describes how well the solution in Y’ are 2distributed. The value of M ∗ is in [0, |Y’|], and the higher value is better 2for the distribution. M ∗ (Y’) measures the spread of Y’. 3 (In our test we chose 500 points from a uniform distribution from thetrue Pareto front PO by all problems, and σ* was set to 0.01).4.1.3 Computation results When setting the values of the parameters, ouraim was to obtain results which were comparable to the results of other
  13. 13. A Multi-objective Evolutionary Algorithm 35methods. Therefore, we adopted the maximal number of function evalua-tions (itend ) from other publications. (itend =80000 at each problem, exceptF3 where itend =10000). The size of the population proved itself to be moredifficult to adapt, but we chose the maximal size of P (|P|=t*subt) in com-pliance with the usual sizes of populations in other publications. (The SARCis another population, different from P ). During the test we studied the SARC size. We tried different sizes (50,100, 250, 500 and 1000) and we compared the results. Besides the visualcomparisons, we used the M ∗ , M ∗ and M ∗ metric and the running times for 1 2 3a quantitative comparison of results (MOSCA2 was implemented in VisualBasic and ran on a Pentium 4, 1.8 GHz with 256 MB RAM). Table 1. Average values of the M ∗ metrics. 1 task/arcn 50 100 250 500 1000 ZDT1 7.0E-4 7.6E-4 7.4E-4 6.6E-4 6.2E-4 ZDT2 6.6E-3 6.8E-4 6.6E-4 6.3E-4 6.2E-4 ZDT3 3.8E-3 3.8E-3 3.1E-3 2.7E-3 2.7E-3 ZDT4 1.4E-2 1.2E-3 3.9E-4 9.6E-4 9.6E-4 ZDT5 0.81 0.78 0.77 0.72 0.74 ZDT610 0.25 0.21 0.17 0.14 0.08 ZDT6100 0.36 0.25 0.17 0.16 0.09 FON 1.1E-2 1.2E-2 5.8E-3 5.5E-3 9.2E-3 CPT7 2.7E-2 1.3E-3 8.1E-4 1.1E-3 7.5E-4 F3 5.4 5.9 5.4 6.1 4.9 ∗ Table 2. Average values of the M 2 metrics. task/arcn 50 100 250 500 1000 ZDT1 45.3 89.2 235.6 495.1 906.2 ZDT2 45.6 87.8 236.6 469.4 845.2 ZDT3 43.6 94.8 229.6 326.1 316.5 ZDT4 43.5 91.0 225.1 469.6 773.1 ZDT5 22.0 21.5 23.0 22.5 21.0 ZDT610 48.1 92.5 246.1 492.2 960.6 ZDT6100 46.1 95.1 237.0 482.6 931.0 FON 41.5 85.6 184.0 290.1 296.5 CPT7 23.2 55.3 148.5 239.2 477.3 F3 48.0 93.0 178.3 377.2 753.1 KUR 44.0 90.5 228.5 384.0 503.2 POL 38.3 77.0 218.5 435.2 758.3 WBD 46.0 85.0 222.5 435.5 879.0
  14. 14. 36 I. Borgulya Table 3. Average values of the M ∗ metrics. 3 task/arcn 50 100 250 500 1000 ZDT1 1.41 1.41 1.41 1.41 1.41 ZDT2 1.41 1.41 1.41 1.41 1.41 ZDT3 1.71 1.62 1.62 1.62 1.62 ZDT4 1.65 1.41 1.68 1.41 1.41 ZDT5 5.00 5.18 5.21 5.01 4.87 ZDT610 1.70 1.28 1.28 1.70 1.78 ZDT6100 102.10 113.10 106.10 123.20 166.20 FON 1.15 1.22 1.21 1.19 1.22 CPT7 1.46 1.41 1.41 1.33 1.41 F3 20.60 20.60 24.00 30.00 24.30 KUR 3.95 4.14 4.12 4.13 4.13 POL 6.60 6.37 6.45 6.59 6.55 WBD 5.80 5.70 11.90 21.50 12.10 Table 4. Average running times in CPU seconds. task/arcn 50 100 250 500 1000 ZDT1 39.6 41.5 32.5 34.2 34.4 ZDT2 36.8 40.8 30.8 31.0 32.2 ZDT3 31.0 26.8 26.1 28.9 28.0 ZDT4 34.2 41.2 28.2 30.0 41.0 ZDT5 52.3 51.8 52.4 51.0 52.0 ZDT610 67.5 99.5 65.8 73.1 111.8 ZDT6100 102.1 113.1 106.3 123.1 166.2 FON 22.0 23.5 20.2 21.7 26.8 CPT7 22.1 16.3 22.5 21.1 25.4 F3 96.4 99.3 37.5 23.9 11.9 KUR 32.2 45.1 28.1 25.9 33.2 POL 37.8 39.2 22.0 24.1 29.5 WBD 54.2 53.0 42.5 40.6 47.1 As MOSCA2 is a stochastic method, each problem was run ten timeswith the same parameter values. The results presented in the Table 1, 2,3 and 4 are mean values calculated from the 10 runs. In all tables we canshow the results by different SARC size (arcn). We present some typical non-dominated fronts visually also. Figure 1and 2 show problems with arcn=1000. On the graphs of the solutions (onthe pictures) we drew the PF with a thin line, except for the problems KUR,POL and WBD, where we do not know the PF; the bold lines or points onthe picture are the non-dominated solutions of the problems. E.g. the solu-tions of problems ZDT1, ZDT2, ZDT3, ZDT4, ZDT6 and F3 approximatethe PF extremely well, we do not see or we hardly see the thin lines of thePFs (by the problem ZDT3 we can see the boundary of the Pareto optimalregion too).The solutions of problem FON do not approximate all points ofPF well and the solutions of problem ZDT5 approximate only a local PF
  15. 15. A Multi-objective Evolutionary Algorithm 37of the problem. For the problem CPT7 the unconstrained Pareto optimalregion becomes infeasible in the presence of the constraint. The periodicnature of the constraint boundary makes the Pareto optimal region discon-tinuous, having a number of disconnected continuous regions. We can seethat our algorithm approximates the disconnected PF of the problem CPT7well (the sets of PF lie on the thin line). From the figures it can be clearly seen that the solutions of the MOSCA2fit very well and that they approximate the PF extremely well. On analyzingthe results we can state that, by increasing the size of SARC, we obtain more ∗and more good solutions: the number of point of PF increases (˜M 2 ) andthe distance from the true PF decreases (M ∗ ). Therefore the quality of the 1solutions continues to improve and we obtain the best solutions by highersize of SARC (arcn=500 or 1000). The running times, however, do not sharethis tendency, the running time itself is clearly problem dependent. For themajority of the problems studied (except problem F3 and T6) increasingthe SARC size the running times hardly change. However, it is a fact: therunning time has the higher value generally by arcn=100.4.2 Comparative resultsIn the section 2 we compared the MOSCA2 with other methods based onthe used techniques. Let us compare the computation results of differentmethods now. In our multiobjective topic the new method is compared with the state-of-art-methods; recently with the NSGAII and SPEA2. Generally the newmethod is compared with one method: with the NSGAII (e.g. [24], [1]) orwith the SPEA2 (e.g. [21]). But there are some papers, where both methods,the NSGAII and the SPEA2 are compared with other methods (e.g. [29],[33]). Based on [29] we can establish, that the results of NSGAII and SPEA2are similar, there is not significant difference between the results of the twomethods generally. Various performance measures have been proposed in the literature forevaluating a set of non-dominated solutions, so we find in the papers dif-ferent performance metrics generally. As a metric we find, e.g. the M ∗ , M ∗ 1 2and M ∗ metrics (e.g. in [1]), the hyperarea metric (e.g. in [21]), the aver- 3age quality metric (e.g. in [29]), the error ratio, generational distance andspacing metrics (e.g. in [24]). To compare the results we chose the M ∗ , M ∗ 1 2and M ∗ metrics from [1],where the NSGAII was compared with an other 3method based on the ZDT1, ZDT2, ZDT3, ZDT4 and ZDT6 problems. To compare the results of the MOSCA2, we chose first the MOSCA [3]and then the NSGAII [9] methods. With our comparison between MOSCAand MOSCA2 we can demonstrate the superior potential of the new ver-sion, and with our comparison of NSGAII and MOSCA2 we show the highquality of the results of MOSCA2. The results were also compared usingthe problems ZDT1, ZDT2, ZDT3, ZDT4 and ZDT610 .
  16. 16. 38 I. BorgulyaFig. 1 Non-dominated solutions on ZDT1, ZDT2, ZDT3, ZDT4, ZDT5, andZDT6100
  17. 17. A Multi-objective Evolutionary Algorithm 39Fig. 2 Non-dominated solutions on KUR, FON, POL, CPT7, WBD, and F3
  18. 18. 40 I. Borgulya Table 5 shows the result of the comparison between MOSCA andMOSCA2. Analyzing the results, we can state that– The values of the M ∗ metric by MOSCA2 are 3-4 times better than for 1 MOSCA,– The values of the M ∗ metric by MOSCA2 are better by 60-80 % than 2 for MOSCA,– The values of the M ∗ metric are similar in both of the methods. 3 We can, therefore, conclude that MOSCA2 produces substantially betterresults with SARC size 100 based on the performance metrics than MOSCA. Table 5. Results of the comparison between MOSCA (MO) and MOSCA2 (MO2) with arcn=100. M∗ 1 M∗2 M∗ 3 Problem MO MO2 MO MO2 MO MO2 ZDT1 4.7E-3 7.6E-4 20.1 89.2 1.41 1.41 ZDT2 4.1E-3 6.6E-4 22.4 87.8 1.41 1.41 ZDT3 6.2E-3 3.8E-3 13.4 94.8 1.60 1.62 ZDT4 9.4E-3 1.2E-3 19.9 91.0 1.41 1.41 ZDT610 0.69 0.21 20.1 92.5 1.28 1.28 Table 6 shows the result of the comparison between MOSCA2 and NS-GAII. A population size of 100 was used for the NSGAII, where 100 descen-dants were added to each generation. A crossover rate of 0.9 was used forNSGAII and it was real-coded [18]. In the Table 6 we can show MOSCA2with arcn=1000. Analyzing the results we can state that– The values of the M ∗ metric differ the most. MOSCA2 with arcn=100 1 has a better value in case of T1, the values are similar in case of T2 and T4, and NSGAII has better values in cases T3 and T610 (see Table 5). In case of MOSCA2 with arcn=1000, the results are better: MOSCA2 has better values in cases of T1, T2 and T4, the values are similar in case T610 , and NSGAII has better values in case T3.– The values of the M ∗ metric by NSGAII are 4-5% better than by 2 MOSCA2 with arcn=100. In the case of MOSCA2 with arcn=1000, the results are better, MOSCA2 has 4-8 times better values.– The values of the M ∗ metric are similar in both of the methods. 3 We can, therefore, conclude that MOSCA2 with arcn=1000 is similar,or little bit better as the NSGAII on a set of test problems.
  19. 19. A Multi-objective Evolutionary Algorithm 41 Table 6. Results of the comparison between NSGAII (NSG) and MOSCA2 (MO2) with arcn=1000. M∗1 M∗2 M∗ 3 Problem NSG MO2 NSG MO2 NSG MO2 ZDT1 1.0E-3 6.2E-4 99.4 906.1 1.41 1.41 ZDT2 6.7E-4 6.2E-4 86.1 845.2 1.23 1.41 ZDT3 2.8E-4 2.8E-3 98.1 316.7 1.60 1.62 ZDT4 1.2E-3 9.6E-4 96.2 773.1 1.37 1.41 ZDT610 7.0E-2 8.5E-2 99.1 960.6 1.25 1.785 ConclusionIn this paper, we have introduced a subpopulation based EA, namedMOSCA2, for non-linear constrained MOPs. This method is the modifi-cation of a previous algorithm, which enables the further resolution of con-strained continuous non-linear MOPs. Due to modifications in the searchoperations and due to introducing a separate archive the results of MOSCA2have substantially improved quality compared with the earlier version. Acomparison with a state-of-the-art method, the NSGAII, shows that it givessimilar, or slightly better results on a set of test problems. In general, by increasing the size of the archive we achieved more andmore results of higher and higher quality without significantly lengtheningthe running time. The size of the archive can be set in a flexible way and sothe decision-maker has the opportunity to decide if he wants more resultsof higher quality in spite of greater memory usage.References 1. Antony W.I., Xiaodong L.: A Cooperative Coevolutionary Multiobjective Al- gorithm Using Non-dominated Sorting. In: Deb K. et al. (Eds.): GECCO 2004, Springer-Verlag Berlin Heidelberg LNCS 3102, (2004) 537-548. 2. Borgulya, I.: Constrained optimization using a clustering algorithm Central European Journal of Operations Research. 8 (1) (2000) 13-34. 3. Borgulya, I.: A Cluster-based Evolutionary Algorithm for Multi-objective Op- timization. In.: Reusch (ed): Computation intelligence Theory and Applica- tions. LNCS 2206 Springer-Verlag Heidelberg, (2001) 357-368. 4. Coello Coello C.A., Van Veldhuizen D.A., Lamont G.B., Evolutionary Algo- rithms for Solving Multi-Objective Problems, Kluwer Academic Publishers, 2002. 5. Cort´s N.C., Coello Coello, C.A.: Multiobjective Optimization Using Ideas e from the Clonal Selection Principle. In: Cant´-Paz E. et al. (Eds): GECCO u 2003, LNCS 2723, Springer-Verlag Heidelberg, (2002) 158-170. 6. Corne D.W., Knowles J.D., Oates M.J.,: The Pareto envelope-based selection algorithm for multiobjective optimization. Proceedings of the Sixth Inter- national Conference on Parallel Problem Solving from Nature (PPSN VI). Springer, (2000) 839-848.
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