Hiroyuki Sato. Minami Miyakawa. Keiki Takadama ABSTRACT. Categories and Subject Descriptors. General Terms

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1 Controlling election Area of Useful Infeasible olutions and Their Archive for Directed Mating in Evolutionary Constrained Multiobjective Optimization Minami Miyakawa The University of Electro-Communications -5- Chofugaoka, Chofu,Tokyo, Japan Keiki Takadama The University of Electro-Communications -5- Chofugaoka, Chofu,Tokyo, Japan Hiroyuki ato The University of Electro-Communications -5- Chofugaoka, Chofu,Tokyo, Japan ABTRACT As an evolutionary approach to solve constrained multi-objective optimization problems (CMOPs), recently a MOEA using the twostage non-dominated sorting and the directed mating (TNDM) has been proposed. In TNDM, the directed mating utilizes infeasible solutions dominating feasible solutions to generate offspring. Although the directed mating contributes to improve the search performance of TNDM in CMOPs, there are two problems. First, since the number of infeasible solutions dominating feasible solutions in the population depends on each CMOP, the effectiveness of the directed mating also depends on each CMOP. econd, infeasible solutions utilized in the directed mating are discarded in the selection process of parents (elites) population and cannot be utilized in the next generation. To overcome these problems and further improve the effectiveness of the directed mating in TNDM, in this work we propose an improved TNDM introducing a method to control selection area of infeasible solutions and an archiving strategy of useful infeasible solutions for the directed mating. The experimental results on m objectives k knapsacks problems shows that the improved TNDM improves the search performance by controlling the directionality of the directed mating and increasing the number of directed mating executions in the solution search. Categories and ubject Descriptors I.2.8 [Artificial Intelligence]: Problem olving, Control Methods, and earch Heuristic methods; G..6 [Numerical Analysis]: Optimization General Terms Algorithms, Design, Performance Keywords multi-objective optimization, constraint-handling, directed mating. INTRODUCTION Multi-objective evolutionary algorithms (MOEAs) try to find Pareto optimal solutions (PO) showing the trade-off among objective func- 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. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. GECCO 4, July 2 6, 24, Vancouver, BC, Canada. Copyright is held by the owner/author(s). Publication rights licensed to ACM. ACM /4/7...$5.. tions in multi-objective optimization problems (MOPs) []. MOEAs are particularly suited to solve MOPs since they can obtain a set of Pareto optimal solutions (PO) from the population in a single run of the algorithm. When we address constrained MOPs (CMOPs) involving several constraints, we need to consider how to handle infeasible solutions in MOEAs. o far, several constraint-handling methods studied for singleobjective optimization have been extended for solving CMOPs [2]. As an approach to avoid special handling of infeasible solutions in the process of evolution, death penalty methods [3, 4] eliminating infeasible solutions from the population have been introduced in MOEAs [5]. Repairing methods modifying infeasible solutions to satisfy all constraints by using problem specific procedures have also been investigated for multi-objective problems [6, 7]. On the other hand, there is another approach to evolve infeasible solutions into feasible ones. The representative penalty methods [8, 9, ] have been extended for MOEAs [, 2, 3]. In these methods, a penalty value (e.g., constraint violation values multiplied by a penalty parameter) is added to each objective function value, and the combined values are used for a parent selection. However, generally, an appropriate penalty parameter depends on each CMOP [4]. Other methods evolving infeasible solutions into feasible ones by independently treating objective and constraint violation values have been studied, and the constrained NGA-II (CNGA- II) [5] has been known as a representative MOEA employing this approach. Recently, as the same approach, a MOEA using the twostage non-dominated sorting and the directed mating (TNDM) [6] has been proposed. TNDM introduces a parents selection based on the two-stage non-dominated sorting of solutions and the directed mating to improve the convergence of solutions toward the Pareto front. In the parents selection, first, we classify the entire population into several fronts by the non-dominated sorting based on constraint violation values. Then, we re-classify each obtained front by the nondominated sorting based on objective function values, and select the parents population from upper fronts. In this way, the superiority of solutions on the same non-dominance level of constraint violation values is determined by the non-dominance level of objective function values. It leads to find feasible solutions having better objective function values in the evolutionary process of infeasible solutions. Also, to generate one offspring, after we select a primary parent, we pick solutions M dominating the primary parent from the entire population including infeasible solutions. Then we select a secondary parent from the picked solutions M and apply genetic operators. In this way, the directed mating utilizes valuable genetic information of infeasible solutions to enhance the con- 629

2 Two-tage Non-Dominated orting Non-dominated sorting based on. Violation values v 2. Objective values a Parents election Directed Mating Crossover & Mutation olutions dominating by the conventional dominance area in the objective space Figure : The block diagram of the conventional TNDM [6] vergence of each primary parent toward its search direction in the objective space. The search performance of TNDM has been verified on several benchmark CMOPs [6]. The results showed that the directed mating significantly contributed to improve the search performance of TNDM in CMOPs. However, the conventional directed mating has two problems. First, since the number of infeasible solutions dominating feasible solutions in the population depends on the feasibility of solutions in each CMOP, the effectiveness of the directed mating also depends on each CMOP. econd, infeasible solutions utilized in the directed mating are discarded in the selection process of parents (elites) population and cannot be utilized in the next generation. To overcome the two problems in the conventional directed mating and further improve the effectiveness of the directed mating in TNDM, in this work we propose an improved TNDM introducing a method to control selection area of solutions M and an archiving strategy of useful infeasible solutions for the directed mating. These methods introduced in the improved TNDM encourage further utilization of infeasible solutions in evolutionary constrained multi-objective optimization. In this work, we focus on combinatorial constrained multi-objective optimization problems, and we verify the search performance of the improved TNDM and compare its search performance with the conventional constrained CNGA-II [5] on m objectives k knapsack problems [7] with m = {2, 4, 6} objectives and feasibility ratio of solutions ϕ = {.,.3,.5} in the solution space. 2. EVOLUTIONARY CONTRAINED MULTI- OBJECTIVE OPTIMIZATION 2. Constrained MOPs Constrained MOPs (CMOPs) are concerned with finding solution(s) x maximizing (or minimizing) m kinds of objective functions f i (i =, 2,..., m) subject to satisfy k kinds of constraints g j (j =, 2,..., k). CMOP is defined as { Maximize/Minimize fi (x) (i =, 2,..., m) () subject to g j(x) (j =, 2,..., k). olutions satisfying all k constraints are said to be feasible, and solutions satisfying not all k constraints are said to be infeasible. The constraint violation vector v(x) is defined as { gj (x), if g v j (x) = j (x) < (j =, 2,..., k). (2), otherwise Also, the sum of constraint violation values is Ω(x) = k j= v j(x). Next, Pareto dominance between x and y in maximization problems is defined as follows: If i : f i (x) f i (y) i : f i (x) > f i (y) (i =, 2,..., m) (3) is satisfied, x dominates y on objective function values, which is denoted by x f y in the following. In the case of minimization problems, the inequalities of Eq. (3) are reversed. Also, a feasible solution x not dominated by any other feasible solutions is said to be a non-dominated solution. A set of non-dominated solutions is called Pareto optimal solutions (PO), and the trade-off among objective functions represented by PO in the objective space is called the Pareto front. 2.2 MOEAs for olving CMOPs To solve CMOPs by using MOEAs, we need to introduce a mechanism to obtain feasible solutions from infeasible ones in the evolutionary process. In this work, we focus on an approach to evolve infeasible solutions into feasible ones, and we pick the constrained NGA-II (CNGA-II) [5] as a representative constrained MOEA employing this approach. CNGA-II is an extended NGA-II for solving CMOPs. CNGA-II uses constraint-dominance [5] instead of Pareto dominance using only objective function values defined in Eq. (3). ince CNGA-II uses constraint-dominance, only the sum of constraint violation values is considered in the evolutionary process of infeasible solutions. Consequently, objective function values of obtained feasible solutions would be worse. Also, since infeasible solutions have less chance to generate offspring than feasible ones, valuable genetic information of infeasible solutions would not be utilized in the solution search. To overcome the problems in the conventional CNGA-II, a MOEA using the two-stage non-dominated sorting and the directed mating (TNDM) has been proposed [6]. 3. TNDM The block diagram of the conventional TNDM [6] is shown in Fig.. TNDM is designed based on the framework of NGA- II [5] where the parents (elites) population P and the offspring population Q construct the entire population R (= P Q). 3. Two-tage Non-Dominated orting To select the parents population P from the entire population R, TNDM classifies R into several fronts by using the two-stage non-dominated sorting based on constraint violation values and objective function values. TNDM employs dominance based on 63

3 f 2 (Maximize) olutions dominating aa in the objective space f 2 (Maximize) olutions dominating aa in the objective space f 2 (Maximize) olutions dominating aa in the objective space f 2( ) φ 2 f 2 ( ) φ 2 f 2 ( ) f 2 ( ) r r f 2( ) φ 2 r ω 2 ω φ Feasible ω 2 Infeasible ω φ Feasible ω 2 Infeasible ω φ Feasible Infeasible f ( ) f (Maximize) f ( ) f ( ) f (Maximize) f ( ) f ( ) f (Maximize) Figure 2: The directed mating (the conventional selection area with =.5) [6] constraint violation values [8, 9]. If the following equation is satisfied, x dominates y on constraint violation values (x v y). j : v j(x) v j(y) j : v j(x) < v j(y) (j =, 2,..., k) (4) Figure 3: Expanded selection area with <.5 First the entire population R is classified into several fronts (F v, F v 2,... ) based on the non-dominance level of constraint violation values by using Eq. (4). ince all constraint violation values of feasible solutions are zero (v(x) = {,,..., }), feasible solutions are always classified into the uppermost front F v. Then, each front F v i (i =, 2,... ) is re-classified into sub-fronts (F f, F f 2,... ) based on the non-dominance level of objective function values by using Eq. (3). Fig. shows an example of that F v is re-classified into F f, F f 2 and F f 3, and F v 2 is re-classified into F f 4, F f 5 and F f 6. Thus, the superiority of solutions decided by the non-dominance level of constraint violation values is maintained even after the reclassification of solutions by the non-dominance level of objective function values. Next, similar to the conventional NGA-II, TNDM selects the half of solutions in the entire population R as the parents population P from upper fronts whilst considering the crowding distance (CD) [5]. In this way, the superiority of solutions on the same non-dominance level of constraint violation values is determined by the non-dominance level of objective function values. It leads to find feasible solutions having better objective function values from infeasible ones. 3.2 Directed Mating TNDM introduces the directed mating to improve the convergence of each solution toward its search direction in the objective space. Fig. 2 shows a conceptual figure of the directed mating. In this figure, all solutions in the entire population R are distributed in the objective space, and feasible solutions belonging to F f are the parents population P. First, we select a primary parent p a from the parents population P by using the crowded tournament selection used in [5]. In the tournament, two solutions are randomly chosen from P, and the solution belonging to an upper front becomes parent p a. If both of them belong to the same front, the solution having a larger CD becomes parent p a. Next, we pick a set of solutions M (= {x R x f p a}) dominating p a in the objective space from the entire population R including infeasible solutions. If p a is infeasible or the size of M is less than two ( M < 2), we cannot perform the directed mating, and a secondary parent p b is selected from P by using the crowded tournament in the same way of NGA-II. Otherwise, we perform the directed mating. In Figure 4: Contracted selection area with >.5 this case, a secondary parent p b is selected from M dominating the primary parent p a. To select p b from M, the crowded tournament selection [5] is also applied in this work. In the example of Fig. 2, two solutions belonging to F f 4 and F f 5 are randomly chosen from M, and the solution belonging to F f 4 becomes p b to mate with p a. In the case of the conventional CNGA-II, all mating are performed in the parents population P, and all parents are feasible after the total number of feasible solutions exceeds the half size of the entire population R. On the other hand, in the directed mating, all primary parents are selected from P but secondary parents are selected even from infeasible solutions discarded in the selection of P if they dominate their primary parents in the objective space. As example in Fig. 2, although secondary p b is infeasible, there is a possibility that p b has valuable genetic information to enhance the convergence of primary p a toward the true Pareto front since p b dominates p a in the objective space. 3.3 Two Problems in Directed Mating The search performance of the conventional TNDM has been verified on several benchmark CMOPs [6]. The results showed that the directed mating significantly contributed to improve the search performance of TNDM in CMOPs. However, the conventional directed mating has two problems. First, since the number of solutions M dominating each primary parent depends on the feasibility of solutions in each CMOP, the effectiveness of the directed mating also depends on each CMOP. In CMOPs with high feasibility, the number of infeasible solutions in the population is low. In this case, the most of primary parents cannot perform the directed mating. Consequently, the effectiveness of the directed mating cannot be obtained. Therefore, for CMOPs with high feasibility, it is desirable to expand the selection area of M in order to perform the directed mating. Contrary, in CMOPs with low feasibility, the number of infeasible solutions in the population is high. In this case, since the number of solutions M dominating each primary parent become large, the selection pressure to select its secondly parent by the crowded tournament selection in the directed mating will become low. Therefore, for CMOPs with low feasibility, it is desirable to contract the selection area of M in order to restrict the size of M by selecting only solutions having similar search direction. econd, when the number of feasible solutions exceeds the half size of the entire population, since most of solutions in M are infeasible solutions, they are discarded in the selection process of parents (elites) population P and cannot be utilized in the next gen- 63

4 Two-tage Non-Dominated orting Non-dominated sorting based on. Violation values v 2. Objective values a Parents election Directed Mating Crossover & Mutation. elect solutions dominating by controlled dominance area using in the objective space 2. Mark the best α solutions in to archive useful solutions for directed mating Archive marked solutions not members of parents population Figure 5: The block diagram of the improved TNDM using the archive population A eration. Therefore, the framework of TNDM has a potential to further improve the search performance by archiving useful solutions selected in M to the next generation and repeatedly utilizing them in the directed mating. 4. PROPOAL: IMPROVED TNDM To overcome the two problems described in ection 3.3 and further improve the effectiveness of the directed mating in TNDM, in this work we propose an improved TNDM introducing a method to control selection area of solutions M and an archiving strategy of useful solutions for the directed mating. These methods introduced in the improved TNDM encourage further utilization of infeasible solutions in evolutionary constrained multi-objective optimization. 4. Controlling election Area of olutions M To overcome the first problem described in ection 3.3, the improved TNDM introduces a method to control selection area of M by controlling dominance area of solutions (CDA) [2]. In the improved TNDM, we pick solutions M based on the dominance area controlled by CDA. In CDA, we modify each objective function value by using the user defined parameter in the following equation. f i (x) = r sin(ω i + π) sin( π) (i =, 2,, m), (5) where, = φ i /π. Figs. 2-4 show examples of the modification of the objective function values of x (= p a ) with different, where r is the norm of f(x), f i (x) is the i-th objective function value, ω i is the declination angle between f(x) and f i (x), and φ i is the controlled angle by. In the following, x dominates y based on the controlled dominance area is denoted by x f y. Fig. 2 shows a case of =.5. In this case, the dominance area is equivalent to the conventional dominance area, and the selection area of solutions M is equivalent to the conventional directed mating [6]. In the example of Fig. 2, four solutions are picked as M. Fig. 3 shows a case of <.5. In this case, the selection area of solutions M is expanded. Although the directionality of the directed mating is deteriorated, more solutions can be picked as M. In the example of Fig. 3, six solutions are picked as M. In this case, we can expect to increase the number of the directed mating exe- Algorithm The improved TNDM : Randomly generate initial solutions (R) 2: A = 3: for t = to T do 4: F v = F f = 5: F v (= {F v, F2 v,...}) =Non-dominated sort (R A, v) 6: for i = to F v do 7: j = + F f 8: F f = F f {F f j, Fj+, f... } =Non-dominated sort (Fi v, f ) 9: end for : Crowding distance (F f ) : P = Truncation (R A, R /2) 2: for j = to Q do 3: p a = Tournament selection (P) 4: M = {x R A x f p a } 5: Mark best α solutions selected in M ({x R A x f p a }) 6: if p a is infeasible or M < 2 then 7: p b = Tournament selection (P) 8: else 9: p b = Tournament selection (M) 2: end if 2: Q j = Crossover and Mutation (p a, p b) 22: end for 23: A = {x R A x is a marked solution x P} 24: R = P Q 25: end for 26: PO = {x R feasible y R feasible : y f x} cutions especially in problems with high feasibility. Fig. 4 shows a case of >.5. in this case, the selection area of solutions M is contracted. ince only solutions having similar search direction to the primary parent p a are selected as M, we can expect to emphasize the directionality of the solution search in the directed mating. However, in this case, since the number of solutions in M is decreased, some primary parents cannot perform the directed mating, then the effectiveness of the directed mating cannot be obtained. 4.2 Archive of Useful Infeasible olutions To overcome the second problem described in ection 3.3, the improved TNDM introduces an archiving strategy of useful solutions for the directed mating. Algorithm and Fig. 5 show the 632

5 Table : ymbols used in the pseudo code P The parents population. The size is P (= R /2). Q The offspring population. The size is Q (= R /2). Offspring are {Q, Q 2,..., Q Q } (= Q). R The entire population, P Q. A The proposed archive population for directed mating. F v A set of fronts classified by violation values v. F v = {F v, F2 v,..., F F v v }. F f A set of fronts classified by objective values f. F f = {F f, F f 2,..., F f F f }. p a, p b A pair of parents to generate one offspring. M olutions dominating p a in the objective space. T The total number of generation. R feasible Feasible solutions in R. R feasible R. PO Pareto optimal solutions (PO), the output of MOEA. pseudo-code and the block diagram of the improved TNDM including the archiving strategy, respectively. Also, Table shows symbols used in Algorithm and Fig. 5. The improved TNDM introduces the archive population A to maintain useful solutions for the directed mating. In Algorithm, codes of the conventional TNDM are written in black, and additional codes for the improved TNDM are written in red. First, the improved TNDM classifies the combined solutions R A into several fronts by the two-stage non-dominated sorting, and selects the parents population P. In the directed mating, a primary parent p a is selected by the crowded tournament selection in the same manner of the conventional TNDM. Then, in the improved TNDM, solutions M dominating p a based on the controlled dominance area are selected from the combined solutions R A (M = {x R A x f p a}). Also, in the improved TNDM, to archive useful solutions for the directed mating, we mark the best α solutions in M in order of solution belonging to upper front. The superiority of solutions belonging to the same front are decided by CD. This mark on solutions in R A is used to select the archive population A. Next, we select a secondly parents p b from M, and generate an offspring by applying crossover and mutation. This process is repeated until filling up the offspring population Q. Next, we select the archive population A. In the combined solutions R A, we select marked solutions not members of the parents population P as the archive population A (the 23th line of Algorithm ). In other words, A is all marked solutions in the discarded solutions by the selection process of the parents population P. In the improved TNDM, we can control the size of archive by varying the parameter α. The size of archive is increased by increasing α. 5. EXPERIMENTAL ETUP 5. Benchmark Problem In this work, we use m objectives k knapsacks problems (mk- KP) [7]. mk-kp is different from multi-objective / knapsack problems [6] often used as a benchmark problem of MOEAs in that mk-kp can independently vary the number of objectives m and knapsacks (constraints) k. mk-kp is defined as { Maximize fi(x) = n l= p li x l (i =, 2,..., m) subject to n l= w lj x l c j (j =, 2,..., k). (6) In this problem, there are n items and k knapsacks (constraints). Each item l has m kinds of profits p li (i =, 2,..., m) and k kinds of weights w lj (j =, 2,..., k). The task is to find combinations of items x = {x, x 2,..., x n } {, } n which maximizes the total of profits on m kinds of objectives subject to the total of weights does not exceed k kinds of knapsack capacities c j. The capacities of knapsacks c j are defined as c j = ϕ j n l= w lj (j =, 2,..., k), (7) where, ϕ j is the feasibility ratio for each knapsack (constraint), we can control the difficulty of each constraint by varying ϕ j. In this work, we use a constant ϕ for all knapsacks (i.e., ϕ = ϕ = ϕ 2 =... = ϕ k ). 5.2 Parameters and Metrics We use mk-kp with n = 5 items (bits), m = {2, 4, 6} objectives, k = 6 knapsacks (constraints) and feasibility ϕ = {.,.3,.5}. We set profits and weights of each item to random integers in the interval [,]. As genetic parameters, we use uniform crossover with crossover ratio P c =., bit-flip mutation with mutation ratio P m = /n, and the population size is set to R = 2 ( P = Q = ). As the termination criterion of optimization, the total number of generations is set to T = 4 for each run. Although the computational cost of the improved TNDM using the archive is increased by increasing its parameter α, to clarify the effects of the proposed archive, we compare the search performance of each algorithm under the condition of the same total number of generations T (the same total number of evaluations of solutions) in this work. In the following experiments, we show average (mean) results of 5 runs. To evaluate the obtained PO, we use Hypervolume ( ) [2] as a comprehensive metric evaluating both the convergence and the diversity of the obtained PO. measures m-dimensional volume covered by obtained PO and a reference point r in the objective space. Obtained PO showing a higher value of can be considered as a better set of solutions in term of both the convergence and the diversity toward the true Pareto front. In this work, r is set to the origin point in the objective space (r = {,,..., }). 6. REULT AND DICUION 6. The Increase of Directed Mating Executions Before we verify the search performance of the improved TNDM, here we observe the number of directed mating executions in the improved TNDM. Figs. 6-8 show the percentage of directed mating executions in all matings during the solution search in mk-kps with m = {2, 4, 6} objectives and feasibilities ϕ = {.,.3,.5}. In each figure, we plot results of the improved TNDMs when the parameters and α are varied. The selection area of M is expanded by decreasing from.5. Contrary, the selection area of M is contracted by increasing from.5. In addition, the archive size of the improved TNDM is increased by increasing α. In each figure, the case using =.5 and is equivalent to the conventional TNDM [6] using the conventional selection area of M and no archive. 6.. Effects of Controlling election Area of M First, we focus on the effects of the controlling selection area of M in the improved TNDM. As general tendency in Figs. 6-8, we can see that the number of directed mating executions is decreased by increasing. This is because the number of primary parents being able to perform the directed mating is decreased by contracting the selection area of M. Contrary, the number of primary parents being able to perform the directed mating is increased by increasing and expanding the selection area of M. Also, we can see the tendency that the number of directed mating executions is decreased by increasing feasibility ϕ. This is because the number of infeasible 633

6 Conventional TNDM (a) Feasibility ϕ =. Conventional TNDM 6 2 Conventional TNDM (b) Feasibility ϕ = Conventional TNDM (c) Feasibility ϕ =.5 Figure 6: Percentage of directed mating executions in all matings in m = 2 objective problems (a) Feasibility ϕ =. Conventional TNDM 6 2 Conventional TNDM (b) Feasibility ϕ = Conventional TNDM (c) Feasibility ϕ =.5 Figure 7: Percentage of directed mating executions in all matings in m = 4 objective problems (a) Feasibility ϕ =. 6 2 Conventional TNDM (b) Feasibility ϕ = Conventional TNDM (c) Feasibility ϕ =.5 Figure 8: Percentage of directed mating executions in all matings in m = 6 objective problems solutions in the population is decreased by increasing ϕ. However, even in problems with high feasibility, we can see that the number of directed mating executions can be improved by expanding the selection area of M with < Effects of Archiving Useful Infeasible olutions Next, we focus on the effects of the archive in the improved TNDM. As we can see in Figs. 6-8, the number of directed mating executions is increased by increasing the size of archive α. These results reveal that the archive strategy in the improved TNDM contributes to increase the number of directed matings by maintaining more candidates of secondary parents in the archive A. 6.2 The earch Performance Verification To verify the search performance of the improved TNDM, Figs. 9- show the results of achieved by the improved TNDM at the final generation in mk-kps with m = {2, 4, 6} objectives and feasibilities ϕ = {.,.3,.5}. In each figure, we plot the results of the improved TNDMs when the parameters and α are varied. imilar to Figs. 6-8, the improved TNDM using =.5 and is equivalent to the conventional TNDM [6], and its is shown as horizontal red line in each figure. All the results are normalized by the results obtained by the conventional CNGA-II [5] Effects of Controlling election Area of M First, we focus on the effects of the controlling selection area of M in the improved TNDM. To discuss the effects of only the controlling selection area of M, here we will focus on the results achieved by the improved TNDMs without the archive (the solid line and the rectangle marker) in Figs

7 Conventional TNDM.4.2 Conventional TNDM.4.2 Conventional TNDM CNGA-II (a) Feasibility ϕ =. CNGA-II (b) Feasibility ϕ =.3 Figure 9: Results of in m = 2 objective problems CNGA-II (c) Feasibility ϕ = Conventional TNDM CNGA-II (a) Feasibility ϕ = Conventional TNDM CNGA-II (b) Feasibility ϕ =.3 Figure : Results of in m = 4 objective problems.5..5 Conventional TNDM CNGA-II (c) Feasibility ϕ = Conventional TNDM. Conventional TNDM. Conventional TNDM CNGA-II CNGA-II CNGA-II (a) Feasibility ϕ =. (b) Feasibility ϕ =.3 Figure : Results of in m = 6 objective problems (c) Feasibility ϕ =.5 First, we discuss the results of Fig. 9 in m = 2 objective problems. From the results of Fig. 9 (a) in the problem with feasibility ratio ϕ =., we can see that values of are monotonically increased by increasing. In this case, as shown in Fig. 6 (a), almost % of offspring are generated by the directed mating. Thus, in case of the number of infeasible solutions in the population is large in problems with low feasibility ϕ, we can see that the directed mating emphasizing the directionality of the solution search by contracting the selection area of solutions M achieves high. Next, from the results of Fig. 9 (c) in the problem with feasibility ϕ =.5, we can see that there is the optimal parameter =.55 to maximize. ince is decreased by setting >.55, we can see that too large deteriorates the search performance. In this case, as shown in Fig. 6 (c), the number of directed mating executions is decreased by increasing since the selection area of solutions M is contracted. These results suggest that the deterioration of in >.55 is caused by the decrease of the number of directed mating executions. These results reveal that a large has a positive effect to emphasize the directionality of the solution search in the directed mating and a negative effect to reduce the number of directed mating executions in the solution search. Also, although methods with <.5 maintain almost % of directed mating executions, values of are deteriorated. This is caused by deterioration of the directionality of the directed mating using too large selection area of M with too small. Therefore, there is an appropriate maximizing depending on the feasibility of each problems. Next, we discuss the results of Figs. and in m = {4, 6} objectives problems. From the results in the problems with high feasibility ratio ϕ = {.3,.5}, we can see that the optimal pa- 635

8 rameters to maximize become <.5. That is, in these problems, values of are improved when the selection area of solutions M is expanded. As shown in Figs. 7 and 8, the number of directed mating executions is decreased by increasing the number of objectives m and feasibility ratio ϕ. Therefore, the expanded selection area of solutions M contributes to increase the number of directed mating executions, and it leads to improve the search perfromance Effects of Archiving Useful Infeasible olutions To verify the effects of the archive of useful solutions in the improved TNDM, we focus on the parameter of archive size α. From all results in Figs. 9-, the improved TNDMs with α > achieves higher than the one without the archive (), and values of are improved by increasing the archive size α. Also, in some problems, when the archive size α is increased, the optimal parameter of the controlling selection area is moved to a larger value, and it allows to further emphasize the directionality of the solution search in the directed mating. These results reveal that the archive of useful solutions for the directed mating contributes to improve the search performance of the improved TNDM. Thus, we can see that both the controlling selection area of M and the archive improve the search performance of TNDM, and the highest search performance is achieved when we combined them in the improved TNDM. Also, the search performance improvement from CNGA-II becomes more significant by increasing the number of objectives. This may be caused by the performance deterioration of the conventional dominance-based MOEAs such as CNGA-II in many-objective optimization. 7. CONCLUION To further improve the effectiveness of the directed mating in TNDM for solving CMOPs, in this work we proposed the improved TNDM introducing the controlling selection area of solutions M and the archive of useful solutions for the directed mating. The experimental results showed that is improved by controlling selection area of solutions M and archiving useful solutions for the directed mating. Although each of them contribute to improve the search performance of TNDM, the highest search performance is achieved when we combine them in the improved TNDM. As future works, we will verify the search performance of the improved TNDM on continuous CMOPs. Also, we will study an adaptive control of in the controlling selection area of M by detecting a characteristic of CMOP during the solutions search. 8. REFERENCE [] K. Deb, Multi-Objective Optimization using Evolutionary Algorithms, John Wiley & ons, 2. [2] E. 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