Improved S-CDAS using Crossover Controlling the Number of Crossed Genes for Many-objective Optimization

Improved S-CDAS using Crossover Controlling the Number of Crossed Genes for Many-objective Optimization Hiroyuki Sato Faculty of Informatics and Engineering, The University of Electro-Communications -5- Chofugaoka, Chofu, Tokyo, 82-8585 JAPAN sato@hc.uec.ac.jp Hernán E. Aguirre Faculty of Engineering, Shinshu University 4-7- Wakasato, Nagano, 380-8553 JAPAN ahernan@shinshuu.ac.jp Kiyoshi Tanaka Faculty of Engineering, Shinshu University 4-7- Wakasato, Nagano, 380-8553 JAPAN ktanaka@shinshu-u.ac.jp ABSTRACT Self-controlling dominance area of solutions (S-CDAS) reclassifies solutions in each front obtained by non-domination sorting to realize fine-grained ranking of solutions and improve the search performance of multi-objective evolutionary algorithms (MOEAs) in many-objective optimization problems (MaOPs). In this work, we further improve search performance of S-CDAS in MaOPs by analyzing genetic diversity in many-objective problems and enhancing crossover operators. First, we analyze genetic diversity in the population and the contribution of the conventional genetic operators when we increase the number of objectives, showing that the genetic diversity in the population significantly increases and offspring created by conventional crossover come to be not selected as parents because the operator becomes too disruptive and its effectiveness decrease. To overcome this problem, we implement crossover controlling the number of crossed genes (CCG) in S- CDAS and verify its effectiveness. Through performance verification using many-objective knapsack problems with 4 0 objectives, we show that the search performance of S-CDAS noticeably improves when we restrict the number of crossed genes. Also, we show that the effectiveness of CCG operator becomes significant as we increase the number of objectives. Furthermore, we show that offspring created by CCG are selected as parents more often than conventional crossover. Categories and Subject Descriptors I.2.8 [Artificial Intelligence]: Problem Solving, Control Methods, and Search Heuristic methods; G..6 [Numerical Analysis]: Optimization General Terms Algorithms, Design, Performance Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. GECCO, July 2 6, 20, Dublin, Ireland. Copyright 20 ACM 978--4503-0557-0//07...$0.00. Keywords Evolutionary many-objective optimization, self-control of dominance area of solutions, control of the number of crossed genes. INTRODUCTION The research interest of the multi-objective evolutionary algorithm (MOEA) [] community has rapidly shifted to develop effective algorithms for many-objective optimization problems (MaOPs) because more objective functions should be considered and optimized in recent complex applications. However, in general, MOEAs noticeably deteriorate their search performance as we increase the number of objectives to more than 4 [2, 3], especially Pareto dominancebased MOEAs such as NSGA-II [4] and SPEA2 [5]. This is because these MOEAs meet difficulty to rank solutions in the population, i.e., most of the solutions become non-dominated and the same rank is assigned to them, which seriously spoils proper selection pressure required in the evolution process[6, 7]. To overcome this problem, CDAS [8] relaxes the concepts of Pareto dominance by controlling dominance area of solutions using an user-defined parameter S to induce appropriate selection pressure in MOEA. CDAS achieves higher search performance in MaOP than NSGA- II [4] due to convergence by the effects of fine grained ranking of solutions using S < 0.5 [9]. Also, as a modification of CDAS, self-controlling dominance area of solutions (S-CDAS) has been proposed [0]. S-CDAS self-controls dominance areas without the need of an external parameter. Through performance verification using many-objective 0/ knapsack problems with 0 objectives, S-CDAS has shown that well-balanced search performance on both convergence and diversity compared to NSGA-II [4], IBEA ϵ+ [], MSOPS [2] and conventional CDAS with optimal parameter S [0]. In this work, we further improve search performance of S-CDAS in MaOPs by analyzing genetic diversity in many-objective problems and enhancing crossover operators. First, we analyze genetic diversity in the population and the contribution of the conventional genetic operators when we increase the number of objectives, showing that the genetic diversity in the population significantly increases and offspring created by conventional crossover come to be not selected as parents because the operator becomes too disruptive and its effectiveness decrease. To overcome this problem and enhance the search performance of S-CDAS in MaOPs, we apply controlling the number of crossed genes (CCG) [2] as crossover operator in S-CDAS. CCG TX, extension of two-point crossover, controls the maximum length of crossed genes by using the user-defined parameter α t. Also, CCG UX, extension of uniform crossover, controls the number of crossed genes by parameter α u. We verify the effec- 753

p 2 E 2 φ 2 (x) l 2 (x) rank += x p 2 F F F 2 F f 2 Solutions dominated by x rank += l (x) f 2 F 2 ω 2 (x) r(x) rank += F 3 F ω (x) φ (x) O=(O, O 2 ) f p O=(O, O 2 ) f E F p Figure : Reclassification of solutions in F by S-CDAS [0] Figure 2: Front classification by S-CDAS [0] tiveness of CCG TX and CCG UX in S-CDAS on many-objective 0/ knapsack problems with m = {4, 6, 8, 0} objectives. Also, we analyze the contribution of CCG for solutions search by observing the number of offsprings selected in parent population. 2. SELF-CONTROLLING DOMINANCE AREA OF SOLUTIONS (S-CDAS) S-CDAS [0] reclassifies solutions in each front F j (j =, 2, ) obtained by non-domination sorting [4] to realize fine-grained ranking, achieving well-converged solutions while keeping well-distributed solutions in the population. S-CDAS self-controls dominance area for each solution without using an external parameter using the procedure [0] described below. Fig. shows the illustration of the reclassification by S-CDAS. Step : Move the origin to O = (O, O 2,, O m ) in objective space. In this work, we set O i = fi min δ (i =, 2,, m), where fi min is the minimum value of the i-th objective function in F j and δ is a tiny constant value. Step 2: Create a set of landmark vectors L = {p, p 2,, p m }, where p i = (O, O 2,, fi max δ,, O m ), and fi max denotes the maximum value of the i-th objective function, which is derived from the extreme solution E i (i =, 2,, m) having the maximum fitness value for each objective function in F j. Step 3: Repeat the following calculation for all solutions in F j. Step 3-: For a single solution x, calculate φ(x) = (φ (x), φ 2(x),, φ m(x)) by Eq.() derived from sine theorem. Here, φ i(x) is the angle determined by the solution x and the landmark vector p i in the i-th objective function. φ(x) determines the individual dominance area of x, which does never dominate extreme solutions E i. φ i (x) = sin { r(x) sin(ωi(x)) l i(x) } (i =, 2,, m), where l i(x) is Euclidean distance between the solution x and the landmark vector p i. Step 3-2: Modify fitness values of all other solutions y F j by the following equation f i (y) = r(y) sin(ω i(y) + φ i (x)) sin(φ i(x)) (i =, 2,, m). Step 3-3: Check dominance relations between the solution x and all other solutions y F j. If a solution y F j is dominated by x, the counter (rank) of y is incremented. Step 4: Finally, reclassify all the solutions in F j based on the accumulated rank values, i.e., smaller rank corresponds to higher front, and larger rank corresponds to lower front. When multiple solutions have a same rank value, they are included in the same front. In S-CDAS, reclassification procedure is performed for each nondominated front F j (j =, 2, ) obtained by NSGA-II. That is, S-CDAS makes front distribution fine-grained, but superiority of each solution is never overturned by S-CDAS. In other words, superiority of solutions in fronts obtained by NSGA-II is maintained even after the reclassification by S-CDAS. Fig.2 shows that an example of reclassified fronts by S-CDAS, where all solutions area included in F by the classification of NSGA-II. We can see that different three fronts F, F 2 and F 3 are obtained by reclassification of S-CDAS. Also, S-CDAS always guarantees the inclusion of extreme solutions in F. In other words, in S-CDAS, not only highly converged solutions but also widely distributed ones are classified into higher front. () (2) 754

Average Hamming Distance 50 00 50 n= 00 n= 250 n= 500 n= 750 n=000 Number of offspring selected in P t 60 40 20 m= 2 m= 4 m= 6 m= 8 m=0 0 2 4 6 8 0 m (Number of objectives) 0 00 000 Figure 3: Average hamming distance of solutions in the population obtained by S-CDAS using the conventional two-point crossover Figure 4: Number of offsprings selected in P t by S-CDAS using the conventional two-point crossover (n = 500) After the reclassification of all fronts, S-CDAS selects parent solutions P t from higher fronts until filling up the half size of entire population similar to NSGA-II [4]. Also, to create offspring solutions, the crowded tournament selection is applied [4]. 3. ANALYSIS ON CONTRIBUTIONS OF GE- NETIC OPERATORS IN S-CDAS To further enhance the search performance of S-CDAS in MaOPs, here we analyze the genetic diversity in the population and contributions of genetic operator for solutions search when we perform S-CDAS with conventional two-point crossover. Here, we use the population size N = 200 (parent population P t = 00, offspring population Q t = 00), crossover ratio P c =.0, and apply bit-flipping mutation with probability P m = /n. We compare the average in 30 runs and each run spend 2,000 generations. Also, we use many-objective knapsack problem [5] with m = {2, 4, 6, 8, 0} objectives and n = {00, 250, 500, 750, 000} items (bits). First, Fig.3 shows average hamming distance of solutions in the population evolved by S-CDAS with two-point crossover at the final generation. From this result, we can see that the average hamming distance of solutions in the population significantly increases by increasing the number of objectives m. That is, the genetic diversity in the population becomes noticeably diverse, and the population come to be widely distributed in the solution space by increasing the number of objectives m. In the case of m = 0 objectives and n =, 000 bits, note that the average hamming distance becomes around 75 bits at the final generation. In this case, if we randomly select two solutions from the population as parents, they will be different in 75 bits out of,000 bits. Thus, since diversity of genes in the population obtained by S-CDAS is significantly high in MaOPs, the likelihood that the conventional recombination becomes too disruptive is also high, making it an inefficient genetic operator for solutions search. Next, Fig.4 shows that transition of the number of offsprings selected as parent population P t when we perform S-CDAS with conventional two-point crossover. From the result, we can see that the number of offsprings survived through extinctive selection substantially decreases by increasing the number of objectives m. This result suggests that the contribution of conventional genetic operation for solutions search substantially deteriorates in MaOPs due to increasing the genetic diversity in the population as shown in Fig.3. 4. IMPROVED S-CDAS USING CONTROL- LING CROSSED GENES FOR CROSSOVER 4. Motivation In S-CDAS, if genes of solutions in the population become noticeably diverse in MaOPs, conventional crossover become too disruptive and inefficient genetic operation which create inferior offsprings. To solve this problem and realize efficient genetic operation, in this work we propose an improved S-CDAS which apply controlling the number of crossed genes (CCG) [2] as crossover operator. CCG crossover can control the number of crossed genes by using the user-defined parameter α, which maintain most of parent s gene structure in offspring and avoid too disruptive genetic operation. For the population having well-diverse gene structure, improved S-CDAS introduces CCG crossover to create offsprings can be survived through parent selection. In this section we introduce CCG for two-point crossover (CCG TX ) and uniform crossover (CCG UX ). 4.2 CCG for Two-point Crossover (CCG TX) When we apply the conventional one- or two-point crossover for individuals with n genes, the length of crossed genes vary in the range [0, n] by randomly chosen the crossover point(s). To restrict the variation of genes in crossover for parents having large difference in gene structure, in this work we propose controlling crossed genes (CCG) for crossover. In this section we explain CCG for twopoint crossover (CCG TX ). CCG TX controls the length of crossed genes by using a user-defined parameter α t. Fig.5 shows the conceptual diagram of CCG TX. First we randomly select parents A and B from the parent population P t, and randomly choose the st crossover point p. Then, we randomly determine the length of the crossed genes l in the range [0, α t n]. In the case of p + l n, the second crossover point is set to p 2 = p + l. In the case of p + l > n, the second crossover point is set to p 2 = p + l n. Here, the possible range of the parameter α t is [0.0,.0]. In this method, when we utilize a small α t, the length of crossed segment becomes short. On the other hand, a large α t indicates a 755

A B Parents Offsprings p p 2 p p 2 l α n t bits n bits A B Figure 5: CCG for two-point crossover (CCG TX ) A Mask B Parents set with probability α t 0 0 0 0 0 0 n bits A B Offsprings Figure 6: CCG for uniform crossover (CCG UX ) long crossed segment. In the case of α t = 0.0, since the length of the crossed segment becomes α t n = 0, the solutions search is equivalent to only mutation without crossover. Also, in the case of α t =.0, the maximum length of the crossed segment become α t n = n. This case is equivalent to the conventional two-point crossover. In this work we verify the effects of CCG TX in MOEA varying α t in the range α t [0.0,.0]. 4.3 CCG for Uniform Crossover (CCG UX ) Next, we explain a method for CCG in uniform crossover (CCG UX ). As shown in Fig.6, for uniform crossover we randomly select two parents from the parent population and generate a n bit mask [3, 4]. For offspring A, if mask bit is 0, the gene is copied from parent A. If mask bit is, the gene is copied from parent B. Similarly, for offspring B, if mask bit is 0, the gene is copied from parent B. If mask bit is, the gene is copied from parent A. To control the number of crossed genes, in this work we control the probability of in the mask by using the parameter α u. The possible range of α u is [0, ], and α u = 0.5 indicates typical uniform crossover [3]. In this method, when we utilize a small α u, the number of crossed genes becomes small. On the other hand, when we utilize a large α u, the number of crossed genes becomes large. α u = 0.0 is equivalent to only mutation without crossover. Also, α u =.0 is equivalent to α u = 0.0 because all gene are exchanged in this crossover. In this work we verify the effects of CCG UX in MOEA varying α u in the range α u [0.0, 0.5] 5. PREPARATION 5. Algorithms and Selection Methods In this work, we implement CCG in S-CDAS [2] and compare the search performance with NSGA-II [4] and IBEA ϵ+ []. NSGA-II is dominance based MOEAs that use Pareto dominance to determine the superiority of solutions in parent selection. IBEA ϵ+ (Indicator-based Evolutionary Algorithm) introduces fine grained ranking of solutions by calculating fitness value based on the indicator I ϵ+ which measure the degree of superiority for each solution in the population []. 5.2 Problems, Parameters and Metrics In this paper we use many-objective 0/ knapsack problems [5] as benchmark problem. We generate problems with m = {2, 4, 6, 8, 0} objectives, n = 500 items (bits), and feasibility ratio ϕ = 0.5. For all algorithms to be compared, we adopt crossover with a crossover rate P c =.0, and apply bit-flipping mutation with a mutation rate P m = /n. In the following experiments, we show the average performance with 30 runs, each of which spent T = 2, 000 generations. Population size is set to N = 200 ( P t = Q t = 00). In IBEA ϵ+, scaling parameter κ is set to 0.05 similar to []. In this work, to evaluate the search performance of MOEAs we use HV, which measures the m-dimensional volume of the region enclosed by the obtained non-dominated solutions and a dominated reference point r in objective space. In this work we use r = (0, 0,, 0) as the reference point, while varying the reference point could lead to different absolute values of hypervolume. Obtained Pareto optimal solutions (POS) showing a higher value of hypervolume can be considered as a better set of solutions from both convergence and diversity viewpoints. To calculate the hypervolume, we use the improved dimension-sweep algorithm proposed by Fonseca et al. [6], which significantly reduces computational time especially for large m. To provide additional information separately on convergence and diversity of the obtained POS, in this work we also use Norm [7] and Maximum Spread (MS) [8], respectively. Higher value of Norm generally means higher convergence to true POS. Although Norm cannot precisely reflect (catch up) local features of the distribution of the obtained POS, we can observe the general tendency of POS from their values. On the other hand, higher MS indicates better diversity in POS, i.e. a widely spread Pareto front. 6. EXPERIMENTAL RESULTS AND DISCUS- SION 6. Effects of CCG TX in S-CDAS First, we verify the search performance of S-CDAS using CCG TX as we vary the parameter α t and the number of objectives m. Fig.7 shows results of HV as comprehensive metric, Norm as measure of convergence, and MS as measure of diversity of obtained POS. CCG TX with α t =.0 is equivalent to the conventional two-point crossover. The maximum length of crossed genes become short by decreasing α t. In these figures, all the plots are normalized by the results of NSGA-II using the conventional two-point crossover. First, from the result of Fig.7 (a), we can see that values of HV noticeably improve by decreasing α t to restrict the maximum number of crossed genes short. Also, the effectiveness becomes more significant by increasing the number of objectives m. Note that the optimal parameter to maximize the value of HV is α t = 0.03 for all objectives problem. In this case, the length of crossed genes is randomly determined in the range [0, α t n = 5]. Since the maximum length of crossed genes in conventional two-point crossover is n = 500 bits, this result reveals that the search performance of S- CDAS noticeably improves when we restrict the maximum length of crossed genes extremely short in MaOPs. The value of HV depends on the location of r. When we set r near from the obtained solutions, convergence is emphasized in HV calculation. On the other hand, when we set r far from the obtained solutions, diversity is emphasized. 756

.5.4 m = 0.04. m = 0 Hypervolume.3.2 Norm.02 MS 0.9. m = 0 0.8 0 0.5 0 0.5 0 0.5 α t α t α t (a) HV (b) Norm (c) MS Figure 7: Performance obtained by improved S-CDAS using CCG TX (n = 500) Hypervolume.5.4.3.2 m = 0 Norm.04.03.02 MS. 0.9 m = 0..0 m = 0 0.8 0.7 0 0. 0.2 0.3 0.4 0.5 0 0. 0.2 0.3 0.4 0.5 0 0. 0.2 0.3 0.4 0.5 α u α u α u (a) HV (b) Norm (c) MS Figure 8: Performance obtained by improved S-CDAS using CCG UX (n = 500) Next, from result of Fig.7 (b), values of Norm improve and the convergence of obtained POS is enhanced by decreasing α t in m 6 objectives problem. Also, similar to the result of HV, the improvement of convergence by CCG TX becomes significant by increasing m. Furthermore, from the result of Fig.7 (c), values of MS improve and the diversity of obtained POS enhances by decreasing α t. Summarizing, S-CDAS using CCG TX operator with a small α t improves convergence and diversity of obtained POS in MaOPs. Consequently, appreciably high HV is obtained and their effectiveness becomes larger as m increases. 6.2 Effects of CCG UX in S-CDAS Next, we verify the search performance of S-CDAS using CCG UX as we vary the parameter α u and the number of objectives m. Fig.8 show results of HV, Norm and MS. CCG UX with α u = 0.5 is equivalent to typical uniform crossover. The number of crossed genes becomes small by decreasing α u. Similar to the previous section, all the plots are normalized by the results of NSGA-II using the conventional two-point crossover. From Fig.8, results obtained by CCG UX have similar tendency to results obtained by CCG TX discussed in previous section. From results of Fig.8 (a), we can see that values of HV noticeably improve by decreasing α u. The optimal parameter to maximize the value of HV is α u = 0.0 for all objectives problem. In this case, the number of crossed genes becomes α u n = 5 bits for each crossover operation. Similar to CCG TX, values of HV significantly improve by restricting the number of crossed genes and their effectiveness becomes large by increasing m. Entirely, CCG UX achieves higher HV with higher convergence than CCG TX while diversity of POS obtained by CCG TX and CCG UX is nearly equivalent. 6.3 Comparison with Conventional MOEAs Next, we compare the search performance of S-CDAS using CCG operators with S-CDAS, NSGA-II and IBEA ϵ+ using conventional two-point and uniform crossover. Fig.9 shows results of HV, Norm and MS when we apply conventional two-point crossover and CCG TX. Also, Fig.0 shows results when we apply typical uniform crossover and CCG UX. For these figures, we plot results of S-CDAS using CCG with the optimal αt and αu maximizing the value of HV. Also, similar to previous sections, all the results are normalized by the results obtained by NSGA-II using conventional two-point crossover. First, from the result of Fig.9 (a), when we use the conventional 757

.7.6 Conv. NSGA-II (Two-point crossover) Conv. IBEA ε+ (Two-point crossover). Conv. NSGA-II (Two-point crossover) Conv. IBEA ε+ (Two-point crossover) Hypervolume.5.4.3.2 Norm.05 Maximum Spread 0.8 0.6 Conv. NSGA-II (Two-point crossover) Conv. IBEA ε+ (Two-point crossover). 0.4 4 6 8 0 4 6 8 0 4 6 8 0 (a) HV (b) Norm (c) MS Figure 9: Performance comparison of MOEAs using the conventional two-point crossover and improved S-CDAS using CCG TX.7.6 Conv. NSGA-II (Uniform crossover) Conv. IBEA ε+ (Uniform crossover). Conv. NSGA-II (Uniform crossover) Conv. IBEA ε+ (Uniform crossover) Hypervolume.5.4.3.2. Norm.05 Maximum Spread 0.8 0.6 0.4 Conv. NSGA-II (Uniform crossover) Conv. IBEA ε+ (Uniform crossover) 4 6 8 0 4 6 8 0 4 6 8 0 (a) HV (b) Norm (c) MS Figure 0: Performance comparison of MOEAs using the conventional uniform crossover and improved S-CDAS using CCG UX two-point crossover, we can see that S-CDAS and IBEA ϵ+ realizing fine-grained ranking achieve higher HV than NSGA-II by increasing the number of objectives m. IBEA ϵ+ achieves extremely high convergence of obtained POS but least diversity instead. Although NSGA-II shows highest diversity, convergence becomes the lowest. Consequently, NSGA-II shows the lowest HV in MOEAs compared in Fig.9. On the other hand, since S-CDAS using conventional two-point crossover achieves a well-balanced search between convergence and diversity, S-CDAS shows better HV than NSGA-II and IBEA ϵ+. Moreover, we can see that S-CDAS using CCG TX significantly improves HV. Note that the improvement by introducing CCG TX is higher than the one achieved by enhancing selection alone either through S-CDAS using conventional twopoint crossover or through IBEA ϵ+. Next, from Fig.9 (b), values of Norm improve by applying CCG TX in m 8 objectives problems. Furthermore, from Fig.9 (c), we can see that CCG TX work to improve MS and diversity of POS in the objective space. Next, Fig.0 obtained by CCG UX shows similar tendency to Fig.9 obtained by CCG TX. Although values of Norm obtained by S- CDAS using CCG UX are lower than S-CDAS using typical uniform crossover, S-CDAS using CCG UX achieves higher MS and obtain well-distributed solutions than S-CDAS using CCG TX. Consequently, we can see that S-CDAS using CCG UX achieves higher HV than S-CDAS using CCG TX. 7. ANALYSIS OF SOLUTIONS SEARCH BY IMPROVED S-CDAS USING CCG TX AND CCG UX Here, we analyze solutions search using CCG TX and CCG UX operators. To observe the variation of genes by conventional twopoint, uniform crossover and CCG, Fig. shows transition of average hamming distance between parent and offspring on the problem with objectives and n = 500 items (bits). For CCG TX and CCG UX, we use the optimal parameter α t and α u. When we compare conventional crossovers, we can see that average hamming distance between parent and offspring by two-point crossover is longer than uniform crossover. This is because, although typical uniform crossover exchanges around n/2 bits for each crossover, the length of crossed genes vary in the range [0, n] in the conventional two-point crossover. On the other hand, by applying CCG operators controlling the number of crossed genes, we can see that average hamming distance between parent and offspring 758

Average hamming distance 50 00 50 Average hamming distance 300 250 200 50 00 0 0 00 000 50 0 00 000 Figure : Transition of average hamming distance between parent and offspring (n = 500 items, objectives) Figure 2: Transition of average hamming distance of solutions in the population (n = 500 items, objectives) Number of offspring selected in P t 60 40 20 Number of offspring selected in P t 60 40 20 0 00 000 0 00 000 Figure 3: Number of offspring selected in P t (n = 500 items, objectives) Figure 4: Number of offspring selected in P t (n = 500 items, objectives) become significantly short and variation of genes in crossover become small. Next, Fig.2 shows transition of average hamming distance of solutions in the population when we apply conventional two-point, uniform crossover and CCG operators in S-CDAS. From this result, we can see that the average hamming distance of the population obtained by conventional two-point and uniform crossover decreases and deteriorate the genetic diversity in early stage of evolution. On the other hand, the average hamming distance of solutions in the population by CCG operators increases. From this result, we can see that CCG operators enhance the search performance of S-CDAS widely search in the solutions space by decreasing the number of crossed genes. Finally, to analyze the contribution of CCG TX and CCG UX for solutions search, we observe transition of the number of offsprings selected in parent population P t. Fig.3 shows the result obtained by S-CDAS using the conventional two-point crossover and CCG TX with optimal α t. Also, Fig.4 shows the result when we perform typical uniform crossover and CCG UX with optimal αu. From these results, we can see that the number of offsprings created by CCG operators is larger than conventional two-point and uniform crossover. These results reveal that CCG TX and CCG UX can create well-selected offsprings as parent population compare to conventional two-point and uniform crossovers. 8. CONCLUSIONS In this work, to further improve the search performance of MOEA using self-controlling dominance area of solutions (S-CDAS) in many-objective optimization problems, first we analyzed the genetic diversity in the population evolved by S-CDAS and the contribution of the conventional genetic operators when we increase the number of objectives m. As result, we showed that genetic diversity in the population significantly increases and offspring created by conventional crossover come to be not selected as parent by increasing the number of objectives m. In MOEAs, if genes of solutions in the population become noticeably diverse, conventional recombination might become too disruptive and decrease its effectiveness. To overcome this problem and enhance the evolutionary many-objective optimization by S-CDAS, we implemented CCG TX and CCG UX which control the number of crossed genes (CCG) in S-CDAS and verified its effectiveness. Through performance verification using many-objective knapsack problems with 759

4 0 objectives, we showed that the search performance of S- CDAS noticeably improves when we restrict the number of crossed genes short. Also, we showed that the effectiveness of CCG operator becomes significant as we increase the number of objectives m. Furthermore, we showed that offsprings created by CCG are selected as parents more times than conventional crossover. As future work, we want to design an algorithm to adaptively control parameters αt and α u during solutions search. Also, we are planning to study the effective crossover operators on manyobjective continuous optimization problems. Furthermore, we will enhance the algorithm of S-CDAS through the performance verification on problems having continuous variables or non-convex Pareto front. 9. REFERENCES [] K. Deb, Multi-Objective Optimization using Evolutionary Algorithms, John Wiley & Sons, 200. [2] E. J. Hughes, Evolutionary Many-Objective Optimisation: Many Once or One Many?, Proc. 2005 IEEE Congress on Evolutionary Computation (CEC2005), pp.222-227, 2005. [3] H. Aguirre and K. Tanaka, Working Principles, Behavior, and Performance of MOEAs on MNK-Landscapes, European Journal of Operational Research, Vol. 8, Issue 3, pp.670-690, 2007. [4] K. Deb, S. Agrawal, A. Pratap and T. Meyarivan, A Fast Elitist Non-Dominated Sorting Genetic Algorithm for Multi-Objective Optimization: NSGA-II, KanGAL report 20000, 2000. [5] E. Zitzler, M. Laumanns and L. Thiele: SPEA2: Improving the Strength Pareto Evolutionary Algorithm, TIK-Report, No.03, 200. [6] H. Ishibuchi, N. Tsukamoto, and Y. Nojima, Evolutionary many-objective optimization: A short review, Proc. 2008 IEEE Congress on Evolutionary Computation (CEC2008), pp. 2424-243, 2008. [7] T. Wagner, N. Beume and B. Naujoks, Pareto-, Aggregation-, and Indicator-Based Methods in Many-Objective Optimization, Proc. 4th Intl. Conf. on Evolutionary Multi-Criterion Optimization (EMO2007), pp.742 756, 2007. [8] H. Sato, H. Aguirre and K. Tanaka, Controlling Dominance Area of Solutions and Its Impact on the Performance of MOEAs, Proc. 4th Intl. Conf. Evolutionary Multi-Criterion Optimization (EMO 2006), LNCS, Vol.4403, pp.5 20, 2007. [9] H. Sato, H. Aguirre and K. Tanaka, Effect of Controlling Dominance Area of Solutions in MOEAs on Convex Problems with Many Objectives, Proc. 7th Intl. Conf. on Optimization: Techniques and Applications (ICOTA7), in CD-ROM, 2007. [0] H. Sato, H. Aguirre and K. Tanaka, Self-Controlling Dominance Area of Solutions in Evolutionary Many-objective Optimization, Proc. 8th Intl. Conf. on Simulated Evolution and Learning (SEAL200), LNCS, Vol.6457, pp.455 465, 200. [] E. Zitzler, S. Kunzili, Indicator-Based Selection in Multiobjective Search, Proc. 8th Intl. Conf. on Parallel Problem Solving from Nature (PPSN-VIII), LNCS Vol.3242, pp.832 842, 2004. [2] H. Sato, H. Aguirre and K. Tanaka, Genetic Diversity and Effective Crossover in Evolutionary Many-objective Optimization, Proc. 5th Intl. Conf. on Learning and Intelligent Optimization (LION 5), LNCS, 20, (to appear). [3] Syswerda, G., Uniform Crossover in Genetic Algorithms, Proc. 3rd Intl. Conf. on Genetic Algorithms (ICGA89), pp. 2 9, 989. [4] W. Spears and K.A. De Jong, An analysis of multi-point crossover, Proc. Foundations of Genetic Algorithms, 990. [5] E. Zitzler and L. Thiele, Multiobjective optimization using evolutionary algorithms a comparative case study, Proc. 5th Intl. Conf. on Parallel Problem Solving from Nature (PPSN-V), LNCS Vol.498, pp.292-304, 998. [6] C. Fonseca, L. Paquete, and M. López-Ibáñez, An Improved Dimension-sweep Algorithm for the Hypervolume Indicator, Proc. 2006 IEEE Congress on Evolutionary Computation (CEC2006), pp.57-63, 2006. [7] M. Sato, H. Aguirre, K. Tanaka, Effects of δ-similar Elimination and Controlled Elitism in the NSGA-II Multiobjective Evolutionary Algorithm, Proc. 2006 IEEE Congress on Evolutionary Computation (CEC2006), pp. 3980 398, 2006. [8] E. Zitzler, Evolutionary Algorithms for Multiobjective Optimization: Methods and Applications, PhD thesis, Swiss Federal Institute of Technology, Zurich, 999. 760