Int. J. Mach. Learn. & Cyber. DOI 10.1007/s13042-014-0277-6
ORIGINAL ARTICLE
A multiobjective cellular genetic algorithm based on 3D structure and cosine crowding measurement Hu Zhang • Shenmin Song • Aimin Zhou X. Z. Gao
•
Received: 8 March 2014 / Accepted: 11 June 2014 Springer-Verlag Berlin Heidelberg 2014
Abstract Multiobjective cellular genetic algorithms (MOcGAs) are variants of evolutionary computation algorithms by organizing the population into grid structures, which are usually 2D grids. This paper proposes a new MOcGA, namely cosine multiobjective cellular genetic algorithm (C-MCGA), for continuous multiobjective optimization. The CMCGA introduces two new components: a 3D grid structure and a cosine crowding measurement. The first component is used to organize the population. Compared with a 2D grid, the 3D grid offers a vertical expansion of cells. The second one simultaneously considers the crowding distances and location distributions for measuring the crowding degree values for the solutions. The simulation results show that C-MCGA outperforms two typical MOcGAs and two state-of-the-art algorithms, NSGA-II and SPEA2, on a given set of test instances. Furthermore, the proposed measurement metric is compared with that in NSGA-II, which is demonstrated to yield a more diverse population on most of the test instances.
H. Zhang S. Song (&) Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin 150001, China e-mail:
[email protected] H. Zhang e-mail:
[email protected] A. Zhou Department of Computer Science and Technology, East China Normal University, Shanghai 200235, China e-mail:
[email protected] X. Z. Gao Department of Electrical Engineering and Automation, Aalto University School of Electrical Engineering, Aalto, Finland e-mail:
[email protected]
Keywords Multiobjective optimization Cellular genetic algorithm 3D grid Crowding measurement
1 Introduction In practice, numerous optimization problems involve more than one objective. This kind of problems are named multiobjective optimization problems (MOPs). Different from the problems with the scalar-objective, there usually does not exist a single solution, which can optimize all the objectives at the same time for a MOP. Instead, a set of tradeoff solutions among different objectives, called Pareto set (PS) in the decision space and Pareto front (PF) in the objective space, are of great interests [15, 26]. As a matter of fact, we aim at finding an approximate set (nondominated front) as diverse as possible and as close as possible to the true PF of a MOP. The traditional deterministic optimization methods need to be executed for many times to find multiple solutions, which might be computationally expensive. The evolutionary algorithms usually maintain a population of candidate solutions in the optimization process and they have the potential to get multiple nondominated solutions to approximate the Pareto optimal solutions in a single run. For this reason, multiobjective evolutionary algorithms (MOEAs) have been attracting more and more attention in both scientific research and real-world applications [1, 12, 14, 20, 26, 32, 37, 41]. In the past decades, a variety of MOEAs have been proposed, such as NSGA-II [16], SPEA2 [27], PAES [22], cMOGA [8], MOCell [30], CellDE [18], aMOCell4 [29], DECell [40], MOEA/D [23, 39], IBEA [42], etc. However, most of these MOEAs are based on the framework of genetic algorithms (GAs). As we know that a GA has a fast convergence speed, but may lose the population diversity
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and get stuck in a local optimum. In order to overcome this drawback, decentralized GAs are studied to maintain the diversity of solutions [7, 10, 25]. In the decentralized GAs, distributed and cellular algorithms are two popular optimization schemes [6, 11]. In this paper, we focus on the second one: cellular genetic algorithms (cGAs). In a cGA [6, 36], the population is conceptually set in a topological structure (also called population grid). Based on this structure, for each solution (central individual), several individuals in its close cells of the grid (neighbor grid) are assigned as its neighbors (neighborhood). The central individual only interacts with its neighbors. Therefore, the overlapped small neighborhoods of cGAs help in exploring the search space, since the induced slow diffusion of solutions through the population provides a kind of exploration (diversification). The exploitation (intensification) occurs inside each neighborhood by genetic operations, which enables cGAs to maintain a proper balance between diversity and convergence. The scalar-objective cGAs (SOcGAs) are drawing a lot of research attention [19, 24, 33]. However, there have not been many reports on multiobjective cellular genetic algorithms (MOcGAs). Obviously, the grids are the highlights of the cGAs, which can be 1, 2, 3 dimensions, i.e., d = 1, 2, 3 (see Fig. 1) [6], and the SOcGAs with these three kinds of grid structures have been intensively investigated. However, in the present study on the MOcGAs, the individuals are always set in a two-dimensional (2D) grid. As discussed in the cases of scalar-objective optimization [2–5, 28], the short radius and dense neighborhood result from the vertical expansion of cells, and the 3D structure can facilitate faster spreading of solutions. Moreover, the high cellular dimensions can improve the algorithm performance. Thus, an enhanced optimization performance is expected, if the population grid adopts a 3D structure. In our previous work on 3D MOcGA, we proposed a simple MOcGA, named MCGA, by replacing 2D grid with 3D structure [38]. The aMOCell4, CellDE, and MCGA were used to test the DTLZ benchmark. The comparison results showed that MCGA has the best convergence, diversity and computational stability. However, the
proposed MCGA is only evaluated using DTLZ problems, and its diversity and convergence are not analyzed. In this paper, a new 3D MOcGA, cosine multiobjective cellular genetic algorithm (C-MCGA), is proposed by extending our preliminary work [38]. In C-MCGA, it employs a 3D grid to arrange population individuals. The most important difference between C-MCGA and MCGA is that C-MCGA introduces and utilizes a new cosine crowding measurement (cCM). The cCM takes the crowding distance and distribution information into consideration while measuring the crowding degree, which can deal with the drawback in the most widely used measurement approach proposed in [16]. To demonstrate the performance of our C-MCGA, it is compared with two advanced MOcGAs: aMOCell4 and CellDE as well as two popular MOEAs: NSGA-II and SPEA2. Furthermore, in order to evaluate the effect of cCM on diversity maintenance, the comparison between C-MCGA and its variant with the crowding measurement in NSGA-II is also performed. The rest of this paper is organized as follows: Sect. 2 summarizes a canonical 3D cellular genetic algorithm model for the scalar-objective optimization (3DSOcGA). Section 3 presents the cCM method in details. Based on the cCM, the concrete procedures of C-MCGA are depicted in Sect. 4. In Sect. 5, three groups of experiments and analysis are given so as to assess the performances of C-MCGA and cCM, respectively. Finally, the paper is concluded in Sect. 6.
2 3D Cellular Genetic Algorithm Since C-MCGA is based on the canonical 3DSOcGA, in this section, we first describe and explain the model of a 3DSOcGA. As aforementioned, the distinguishing feature of a cGA is that its individuals are arranged in a grid, and only neighboring individuals (i.e., the closest ones measured in Manhattan distance) are allowed to interact during the breeding loop. As a result, some kind of isolation in the population that depends on the distances among individuals
Fig. 1 An illustration of toroidal cellular grids
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is formed. Hence, the genetic information of a given individual can spread slowly through the grid (neighbors are overlapped), and it requires a large number of generations to reach the distant individuals, thus preventing the population from premature convergence. Structuring the population in this way can achieve a good exploration/ exploitation tradeoff in the search process for solving complex optimization problems [9].
Algorithm 1. Pseudo-code of a canonical 3D asynchronous ‘cGA’ 1. proc Steps Up(3DSOcGA) //Algorithm parameters in ‘3DSOcGA’ 2. while Terminal Condition is not met do 3. for i 1 to 3DSOcGA.pop.ROWS do 4. for j 1 to 3DSOcGA.pop.COLUMNS do 1 to 3DSOcGA.pop.LAYERS do 5. for k 6. neighbors GetNeighborhood (3DSOcGA.pop.position(i,j,k)); parent1 3DSOcGA.pop.position(i,j,k); 7. parent2 Selection (neighbors); 8. offspring Recombination (3DSOcGA.Pc, parent1, parent2); 9. offspring Mutation (3DSOcGA.Pm, offspring); 10. EvaluateFitness (offspring); 11. Replacement (pop.position(i,j,k), offspring, 3DSOcGA); 12. 13. end for 14. end for 15. end for 16. end while
Usually, the individuals are settled in a 2D structure. In our paper, we extend this structure to 3D space. The pseudo-code of a 3DSOcGA is given in Algorithm 1 [5], and its breeding loop is shown in Fig. 2. In this cGA, the population is structured in a regular 3D grid, and a neighborhood composed of the individuals in six directions (i.e., horizontal north and south, vertical north and south, east, and west) defined on it (line 6). The algorithm iteratively does the genetic operations on each individual in the grid (line 3–5). An individual may only interact with another individual chosen from its neighbors (line 8) based on a given criterion, and the two individuals build up the
Parents
1. Selection
Central individual
Neighbor individual
2. Crossover
4. Replacement
3. Mutation
Fig. 2 Breeding loop of a 3DSOcGA
parents. Crossover and mutation operators are applied to the parents in lines 9 and 10 with probabilities Pc and Pm, respectively. The algorithm calculates the fitness value of the new offspring individual (or individuals) (line 11), and inserts it (or one of them) into the equivalent place of the current individuals in the population (line 12) following a replacement policy. This iteration procedure is repeated until a termination condition is met (line 2). As discussed in [28], since the 3D grids give a vertical expansion of cells, the short radius and dense neighborhood resulting from this expansion can facilitate faster spreading of the solutions, and with the 3D grids, the size of neighborhood is larger than in the 2D case with regard to similar population sizes and distance steps. In other words, the larger the neighborhood size, the higher the selection intensity. Thus, the optimal solutions can be discovered in a faster way. In the above 3DSOcGA, the current individuals are immediately updated with the offspring individuals after generation, and the offspring individuals are likely to interact with the individuals belonging to their parent generation. This update strategy is an asynchronous model. Actually, cGA also has an alternative update policy, called synchronous model. Unlike the former one, it first creates all the offspring individuals of the entire population, and next updates all the parent individuals with the new individuals. Usually, the asynchronous cGAs may have a faster convergence speed than the synchronous ones.
3 Cosine Crowding Measurement A MOP solver aims at obtaining a set of nondominated solutions, whose objective values (nondominated front) have good diversity and convergence. The convergence means that the obtained nondominated solutions should approach to the PF as close as possible. The diversity (also called crowding degree) has two aspects: one is that the distances among the objective points should be as large as possible, i.e., a wide coverage area is always desirable. The other one indicates that these points should distribute in the objective space as uniformly as possible. In the MOEAs, crowding measurement is the key to select those solutions with good diversity. To acquire the ideal nondominated solutions, an appropriate measurement metric of crowding degree should be applied here. Several crowding measurement methods have been proposed and used in the regular MOEAs. For the PAES and MOPSO [13, 22], they use two kinds of adaptive hypercubes, where an appropriate depth parameter (PEAS) or the number of divisions (MOPSO) is needed to control the hypercube size. When a solution converges near the Pareto front, the hypercube is comparatively large. In the SPEA2
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[21], a density estimator is included in the fitness assignment, and it considers the inverse of the distance to the k-th nearest neighbor as the density estimation. The individuals with larger distances have better fitness to survive for the next generation. However, this technique needs a considerable computational effort. NSGA-II [16] also employs a density estimation strategy. To obtain the density estimation of a particular solution surrounded by several other ones, it calculates the average distance of two points on either side of this point along each of the objectives. The density estimation (crowding distance) of a solution is the sum of its average distance on each side. To our best knowledge, this crowding estimation is the most widely used method, since it is easy to implement and has a low computational complexity. However, this technique only considers the crowding distance instead of the distribution information, which cannot accurately reflect the crowding degree. In order to utilize the crowding distance and distribution, MOSADE [35] uses a crowding entropy-based diversity measurement to preserve the diversity of the Pareto optimality. Different from the strategy in NSGA-II, this new method uses entropy to replace distance to measure the crowding degree along each of the objectives, which, to some extent, may contain distribution information and crowding degree. Unfortunately, the entropybased method has its own drawbacks. If there are two or more the same points along any of the objectives, in the crowding entropy, there is 0 9 ? = NAN or the denominator equals to 0. Among the typical MOcGAs, cMOGA [8] uses a crowding procedure based on the adaptive grid mechanism in PAES, MOCell and aMOCell4 [29, 30] adopt the crowding measurement method in NSGA-II, and CellDE [18] deploys the same one in SPEA2. Inspired by the above analysis, a new cosine crowding measurement (cCM) method taking advantage of crowding distances and distribution of objective values is proposed in this section, which accurately depict the crowding degree accurately and therefore can be used to obtain the nondominated fronts with a satisfactory diversity. For a MOP with K objectives F = (f1,…fk,…,fK), suppose there are N solutions, and FV is the set of their objective values. The procedure of using a cCM to measure crowding degree for this problem is shown in Algorithm 2.
Algorithm 3. Pseudo-code of crowding assignment for the solutions along fk 1. proc CrowdingMeasurement (FVk) zeros(N,1); // Initialization 2. CDk [fv; seq] unique(FVk); // Unique and sorting 3. size(seq,1); n 4. cd = zeros(n, 1); 5. if (n==1) or (n==2) then 6. 0; cd(1:n) 7. 8. else for i = 1:n do 9. if (i==1) or (i==n) then 10. cd(1) K; 11. cd(n) K; 12. 13. else (fv(i+1)+fv(i-1))/2; // Calculate the midpoints m(i) 14. d(i) fv(i+1)-fv(i-1); // Calculate the distances between adjacent solutions 15. r(i) (fv(i)-m(i))/(d(i)/2); // Calculate the drift rates 16. r(i) DriftRateAdjustment(r(i)) 17. cd(i) cos(r(i)×pi/2)×d(i)/(fv(n)-fv(1)); 18. 19. end if 20. end for 21. end if cd; CDk(seq 22. end proc CrowdingMeasurement 23.
Similar to the popular NSGA-II method, cCM calculates the crowding degrees of a solution along each objective and add them together. Along the k-th (k = 1, 2, …,K) objective, the crowding measurement method is given in Algorithm 3. Zero is usually set to be the initial crowding degree value for each solution. cCM first removes the reduplicative individuals in FVk (the objective values along fk in FV). Next, the remaining values are sorted in the ascending order, and a new vector fv is obtained. In the fv, K is directly assigned as the crowding degree of the boundary solutions. For a non-boundary solution (the ith solution), cCM uses the following strategy. It calculates the distance, d(i), and the midpoint of the interval (see Fig. 3a) formed by two adjacent solutions, m(i), using Eqs. 1 and 2, respectively. r(i), the rate that the current solution drifts from m(i), is estimated by Eq. 3. For r(i), an adjustment mechanism is applied here. Based on r(i) and Eq. 4, the crowding degree value cd(i) is acquired. It is easy to discover that a cosine relationship exists between cd(i) and r(i) (see Fig. 3b). After the crowding degree measurement for every solution along each objective is available, a sum operation is performed, and the final crowding degree values CD(i) are obtained by Eq. 5. More details of cCM are given in Algorithms 2 and 3. fvði þ 1Þ þ fvði 1Þ 2
ð1Þ
dðiÞ ¼ fvði þ 1Þ fvði 1Þ
ð2Þ
mðiÞ ¼
rðiÞ ¼ Algorithm 2. Pseudo-code of a cosine crowding measurement (cCM) 1. proc cCM(FV) 2. CD = zeros(N,1); 3. if N == 1 or 2 then 4. CD(1:N) K; 5. else for k = 1:K do 6. CDk CrowdingMeasurement(FVk);//Calculate the crowding degree along fk 7. CD CD+CDk; 8. 9. end for 10. end if end proc cCM 11.
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2 ðfvðiÞ mðiÞÞ dðiÞ
cdðiÞ ¼
cosðrðiÞ pi2 Þ dðiÞ fvmax fvmin
CDðiÞ ¼
K X
CDk ðiÞ
ð3Þ ð4Þ ð5Þ
k¼1
A few key issues of the cCM are explained as follows:
Int. J. Mach. Learn. & Cyber. Fig. 3 A nondominated front (a) and the variation curve of crowding degree (b)
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f2
cd(i)
midpoint m(B)
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l C
r(D)>t r(E)= 2 and i t; 4. s2 r(i) < (-t); 5. if s1&&s2 then 6. r(i) 0; 7. 8. end if 9. end if 10. end proc DriftRateAdjustment
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-1
0
1
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Eq. 4 and Fig. 3b show the relationship between the drift rate r(i) and crowding degree value cd(i). We can discover that, when r(i) (for the ith solution) is near to 1 or -1, cd(i) approaches to 0. Obviously, if two adjacent solutions have the drift rates close to -1 and 1, respectively, they both have small crowding degree values near 0, and either of them is difficult to preserve for obtaining a uniformly distributed nondominated solutions. Therefore, we design an adjustment mechanism (see Algorithm 4) to tackle this issue. If two solutions are adjacent, i.e., r(i-1) [ t and r(i) \ (-t), the original value of r(i) is replaced with 0, where t is a predefined drift rate, and its value range is [0.8,1). 0.9 is usually chosen in most cases. According to Eq. 4, we observe that r(i-1) has a very small crowding distance value, which causes the (i-1)th solution to be easily removed in the next step. On the contrary, r(i) = 0 gives the ith solution a large probability to survive. Thus, we can retain an appropriate solution from these two adjacent ones, and this solution has good crowding distance in the next step so that a set of evenly distributed solutions are obtained. In NSGA-II, the complexity of the proposed crowding measurement method is O(KNlogN). With regard to cCM, it is a variant of NSGA-II, and its computational complexity is also O(KNlogN).
From the above descriptions, it can be observed that based on the cosine relationship, the cCM method simultaneously considers the crowding distance and distribution. This new approach is capable of accurately reflecting the crowding degree as well as obtaining the nondominated fronts with a good diversity.
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4 Algorithm Framework of C-MCGA In this section, we present the C-MCGA by embedding the cCM method and extending our previous work in [38]. Unlike other cellular algorithms for mutiobjective optimization, C-MCGA uses a 3D grid to organize the population. Algorithm 5 shows the pseudo-code of CMCGA. The Pareto front is an additional population (the external archive) composed of the nondominated solutions found. The archive has a size limitation, and, therefore, the insertion of solutions in the Pareto front has to be carefully managed so as to obtain a diverse set. A density estimator is needed to remove solutions from the archive, when it becomes full.
Algorithm 5. Pseudo-code of the C-MCGA 1. proc Steps Up(C-MCGA)//Algorithm parameters in ‘C-MCGA’ 2. ParetoFront = CreateFront() //Create an empty Pareto front 3. while not Termination Condition() do 4. for i 1 to C-MCGA.pop.ROWS do 5. for j 1 to C-MCGA.pop.COLUMNS do 6. for k 1 to C-MCGA.pop.LAYERS do 7. neighbors NeighborhoodAssignment(C-MCGA.pop.position(i,j,k)); [parent1,parent2] Selection(neighbors); 8. parent3 C-MCGA.pop.position(i,j,k); 9. offspring DifferentialEvolution(parent1,parent2,parent3,CR,F); 10. FitnessEvaluation (offspring); 11. Replacement(pop.position(i,j,k),offspring,C-MCGA); 12. Insertion (ParetoFront, offspring); 13. 14. end for 15. end for 16. end for C-MCGA.pop Feedback (C-MCGA.pop,ParetoFront); 17. 18. end while end proc Steps Up; 19.
Breeding Loop: C-MCGA starts by creating an empty Pareto front (line 2 in Algorithm 5). Individuals are arranged in a 3D toroidal grid, and the DE operations [18] are successively applied to them (lines 11) until the termination condition is met (line 3). The breeding loop of C-MCGA is illustrated in Fig. 4. For each individual, three parents are selected. Two of them are from the neighborhood by the binary tournament selection, and the third is the current individual. After selection, the algorithm applies a DE operation to generate a new offspring, and evaluates the resulting individual. Based on the results, it decides whether the new offspring replaces the current one (line 10–12). The next step (line 13) is to insert the appropriate offspring into the external archive. Finally, after each generation, a feedback procedure is invoked to replace certain randomly chosen individuals by the solutions selected from the archive (line 17). Replacement: In this algorithm, the offspring replaces the individual at the current position, if the former is better than the later. In other words, a better individual can replace the current individual, if it is dominated by the offspring or both the two are nondominated and the current individual has the worst crowding degree value assigned by cCM in a population consisting of the neighborhood and the offspring.
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Archive Maintenance: In order to insert offspring into the Pareto front (line 13 in Algorithm 5), the following mechanism is used: the dominance relationships between the offspring and the solutions in the archive are considered. All the archive solutions dominated by offspring are removed. If the offspring is dominated by any remaining archive solutions, it will not be inserted; otherwise, it will become one of the solutions in the archive. If the external archive is full, a fitness value will be assigned to each solution in the archive (including the offspring) based on the method in SPEA2 [21], and the solution with the maximal value is removed, because smaller fitness values are always desired in this fitness assignment strategy.
5 Simulations In this section, we evaluate the performance of C-MCGA using numerical simulations. The performance indicators used to assess the quality of the nondominated front of the test problems obtained by the MOEAs are first discussed. A comparison against two typical MOcGAs, aMOCell4 [29] and CellDE [18], is next presented. We further compare CMCGA with two state-of-the-art MOEAs, NSGA-II [16] and SPEA2 [21]. Finally, an evaluation of the influence on the diversity of the nondominated front caused by the new cCM is the performed. All the algorithms (C-MCGA, aMOCell4, CellDE, NSGA-II, and SPEA2) are programmed using the MATLAB software. A total of 30 trials have been made in our experiments. The median and interquartile range, IQR, of each performance indicator acquired in these runs are used as the measures of locations (or central tendency) and statistical dispersion, respectively. The one-way ANOVA statistical analysis with the significance level of 95 % is performed here. All the successful tests are marked with ‘‘(?)’’ in the last column of the tables, and ‘‘(-)’’ is used for the remaining ones. In the tables, the first and second best medians of the quality measures for each benchmark problem with statistical confidence are highlighted using gray and light grey backgrounds, respectively. The best IQR values are indicated by bold, underlines, and italics. Our numerical simulations are carried out on a PC with 8 GB RAM and a 3.2 GHz Intel quad core processor. 5.1 Test problems We select 12 well-known benchmark problems to evaluate C-MCGA, i.e., Schaffer, Fonseca, Kursawe [30], Viennet2, Viennet3 [31], and 7 DTLZ problems [17]. These testbeds have two (Schaffer, Fonseca, Kursawe) or three objectives (Viennet2, Viennet3, and DTLZ family), and they occupy different properties (separability, unimodality,
Int. J. Mach. Learn. & Cyber. Fig. 4 Breeding loop of the C-MCGA
Parents
Neighbor individual 1. Selection
Central individual
Neighbor individual 4. Replacement
2. Differential evolution 3. Add to archive if nondominated
Pareto front
offspring 5. Feedback
multimodality, convexity, linearity, nonconvexity, continuity, discontinuity, bias, Pareto many-to-one, etc.).
5.2 Quality metrics Convergence and diversity are usually two most important criteria for evaluation of MOEAs. The convergence refers to the distance from the nondominated front generated by the optimization algorithm to the true PF. The diversity involves coverage area and uniformity as explained in Sect. 3, and a front with wide coverage and good uniformity is always pursued. Thus, to demonstrate the performance of our C-MCGA, in this paper, the quality indicators for both convergence and diversity are employed. Note that a normalization of the objective values in the nondominated front obtained is performed beforehand. Inverted generational distance. (IGD) [43]. The IGD metric is used to assess both convergence and diversity. Let P* be a set of uniformly distributed Pareto optimal points in the true PF, andP be a nondominated front of the problems. The IGD metric is defined as: P dðv; PÞ IGDðP ; PÞ ¼ v2P ; ð6Þ jP j where d(v,P) is the minimum distance between v and any point in P, and |P*| is the cardinality of P*. Apparently, a lower IGD value is desirable. P must be close to the true PF, and cannot miss any part of the whole PF. Generalized Spread. (D) [16] The spread measurement is an indicator that measures the distribution and spread of the obtained nondominated front of the problems with two or more objectives:
Pm D¼
i¼1
P dðei ; SÞ þ X2S dðX; SÞ d ; Pm i¼1 dðei ; SÞ þ jS jd
ð7Þ
where S is a set of solutions, S* is the set of Pareto optimal solutions, (e1,…,em) are m extreme solutions in S*, m is the number of objectives, and dðX; SÞ ¼ min FðXÞ FðYÞ2 ; ð8Þ Y2S;Y6¼X
1 X d¼ dðX; SÞ: X2S jS j
ð9Þ
We emphasize that the fronts with lower values of D are desirable. Generational distance. (GD) [34]. This indicator can measure the distance between the generated nondominated front and the true PF: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 i¼1 di GD ¼ ; ð10Þ n where n is the number of the nondominated solutions, and di is the Euclidean distance between the ith solution and the nearest solution in the true PF in the objective space. It is obvious that all the obtained solutions are exactly the Pareto optimal solutions, if GD = 0. Therefore, the smaller GD is, the better it converges to the Pareto optimal solutions. The true PFs of these test problems are available and downloadable from the website of NEO at http://neo.lcc. uma.es/, and their sizes are 201 for schaffer, 434 for fonseca, 874 for kursawe, 8,122 for viennet2, 9,920 for viennet3, 1,000 for both DTLZ1 and DTLZ2, 4,000 for both DTLZ3 and DTLZ4, 166,500 for DTLZ5, 28,000 for DTLZ6, and 676 for DTLZ7.
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5.3 Comparison with aMOCell4 and CellDE C-MCGA is first compared with two typical MOcGAs: aMOCell4 and CellDE [18, 29]. As discussed in [29], aMOCell4 uses the same density estimator as NSGA-II, and it can deal with the problems with two and three objectives. On the other hand, CellDE utilizes a fitness assignment developed in SPEA2, and has superior performances in handling three objectives [18]. In this experiment, for the problems with two and three objectives, the above three MOcGAs are terminated after 25,000 and 50,000 evaluations are reached. The sizes of their external archives are 125 and 225. 25 and 45 individuals are randomly selected in the feedback populations, respectively. The other parameters are given as follows. In aMOCell4, for the problems with two and three objectives, the population sizes are popSize = 121 and popSize = 225, and the population individuals are arranged in 11 9 11 and 15 9 15 square grids, respectively. The sizes of archive are arcSize = 125 and arcSize = 225 for tackling MOPs with two and three objectives. A NEWS (i.e., north, south, west and east in Fig. 1b) neighbor structure is deployed here. The basic genetic operators are binary tournament selection, simulated binary crossover, and polynomial mutation. The crossover and mutation probabilities are Pc = 0.9 and Pm = 1/n, and the distribution index in crossover and mutation g = 20. In CellDE, the population size, archive size, grid shape, and neighbor structure are the same as in aMOCell4. The control parameters in DE are F = 0.5 and CR = 0.1. In C-MCGA, for the problems with two objectives, popsize = 125, and the population grid is a 5 9 5 9 5 cube; for the problems with three objectives, popSize = 216, and the shape of the grid is 6 9 6 9 6. The sizes of archive are the same as those in MOCell. A 3D NEWS (i.e., horizontal north and south, vertical north and south, east, and west in Fig. 1c) neighbor grid is used. The control parameters of DE are the same as those in CellDE. The test MOPs presented in Sect. 5.1 are attracted using by the three MOcGAs, and the indicators of IGD, D, and GD are shown in Tables 1, 2, 3, respectively. As can be observed from Table 1, C-MCGA successfully outperforms aMOCell4 and CellDE, because it can acquire six best IGD indicators while aMOCell4 and CellDE have only three and zero, respectively. Table 2 provides the indicator values of diversity. Among all the 12 problems, C-MCGA obtain five best and second best D values, which are significantly better than the other two algorithms. Moreover, it maintains the best solution diversity among the three methods. The GD values are presented in Table 3, which shows that C-MCGA has three best indicator values of convergence. To summarize, with better convergence and
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stability, C-MCGA gains six second best indicator values of convergence and six best IQR values. From these tables, it is obvious that compared with aMOCell4 and CellDE, C-MCGA yields an improved performance. To further illustrate the performance of C-MCGA, the approximated DTLZ3 fronts of each algorithm representing the best IGD values obtained in the 30 runs are demonstrated in Fig. 5. It is apparently visible that aMOCell4 only converges to three parts of the PF. Compared with aMOCell4, although CellDE has a better convergence area, it cannot provide the whole PF shape. There also exists a certain distance between the approximation and PF. On the other hand, the final solutions from the C-MCGA spread along the whole PF, and uniformly distribute on the front surface. 5.4 Comparison with NSGA-II and SPEA2 To illustrate the competitiveness of our algorithm against the existing popular MOEAs, we compare CMCGA with two state-of-the-art MOEAs: NSGA-II and SPEA2. Proposed by [16], NSGA-II is characterized by designing a density estimator for the individual consisting of a Pareto ranking and a crowding distance. SPEA2 was developed by [21], which assigns the fitness value to an individual by summing up the strength raw fitness and a density estimation based on the distance to the k-th nearest neighbor. In our simulations, C-MCGA has the same parameter settings with those in Sect. 5.3. Both NSGA-II and SPEA2 use the same parameters as in the original papers [16] and [21] except for the sizes of populations and external archives. Actually, the selection, crossover, mutation, and termination conditions with their parameters in NSGA-II and SPEA2 are also the same as those in aMOCell4. However, in SPEA2, to attack the problems with two and three objectives, the population sizes are set to be 125 and 225, respectively, and the sizes of the external populations are the same as the population sizes. There is no external population in NSGA-II, and it uses the same population sizes as those in SPEA2. The corresponding optimization results of the aforementioned test suite are presented in Tables 4, 5, 6. Let us first analyze the IGD indicator in Table 4. Obviously, for the test suite used, NSGA-II, SPEA2 and C-MCGA do not have clear differences on dealing with DTLZ1, DTLZ4 and DTLZ7; but for the remaining problems, C-MCGA obtains six best values, and SPEA2 obtains three ones, which indicates that C-MCGA is the winner algorithm and NSGA-II is the worst one. Next, the indicator D in Table 5 is examined, and the optimization results imply that C-MCGA clearly outperforms the other two algorithms on the diversity,
Int. J. Mach. Learn. & Cyber. Table 1 Median and interquartile range of inverted generational distance (IGD)
Table 2 Median and interquartile range of generalized spread (D)
Table 3 Median and interquartile range of generational distance (GD)
Algorithms Problems
aMOCell4 median IQR
CellDE median
IQR
C-MCGA median IQR
schaffer
1.0385E+00
8.5745E-01
1.4964E-02
4.0380E-04
1.4948E-02 4.2979E-03
3.4015E-04 (+) 2.3781E-04 (+)
fonseca
3.1276E-03
6.2760E-05
4.2331E-03
2.2107E-04
kursawe
2.6139E-02
4.0462E-04
3.0642E-02
1.6820E-03
3.0865E-02
1.6391E-03
(+)
viennet2
1.3220E-02
1.4241E-03
7.5362E-03
2.1351E-04
7.5095E-03
2.2863E-04
(+)
viennet3
2.3965E-02
1.0432E-03
1.8850E-02
9.3247E-04
1.8741E-02
DTLZ1
2.9962E-02
1.4557E-02
1.4014E-02
4.6297E-04
1.4141E-02
6.7851E-04 (+) 5.2018E-04 (-)
DTLZ2
4.7780E-02
2.3675E-03
3.7756E-02
8.9714E-04
3.7434E-02
DTLZ3
2.1871E+00 2.7932E+00
2.4406E+00
2.4003E+00
2.0257E+00
8.0268E-04 (+) 1.9863E+00 (-)
DTLZ4
4.4406E-02
5.8529E-03
3.2248E-02
3.3615E-03
3.1179E-02
4.4080E-03
(+)
DTLZ5
1.8099E-03
4.2475E-05
2.1351E-03
1.0214E-04
2.2122E-03
1.7008E-04
(+)
DTLZ6
2.3542E+00
2.2801E-01
2.0068E-03
1.8112E-04
1.9953E-03
DTLZ7
5.3473E-02
4.3763E-03
3.8064E-02
1.1754E-03
3.8672E-02
1.6571E-04 (+) 1.7523E-03 (-)
Algorithms Problems
aMOCell4 median IQR
CellDE median
IQR
C-MCGA median IQR
schaffer
6.2594E-01
6.7298E-01
2.5814E-01
1.9965E-02
2.4153E-01
2.4434E-02
(+)
fonseca
7.8614E-02
1.7724E-02
2.3623E-01
2.6170E-02
2.2683E-01
1.8468E-02
(+)
kursawe
2.4609E-01
1.0891E-02
4.0737E-01
2.9179E-02
4.0229E-01
2.0619E-02
(+)
viennet2
3.5542E-01
2.9956E-02
2.4949E-01
1.8033E-02
2.4433E-01
2.0910E-02
(+)
viennet3
2.3989E-01
2.7905E-02
5.5776E-01
4.0164E-02
5.5647E-01
3.6536E-02
(+)
DTLZ1
1.7073E+00
7.7991E-01
1.5259E-01
1.7915E-02
1.5302E-01
1.3578E-02
(+)
DTLZ2
4.7676E-01
3.8243E-02
1.5253E-01
1.1940E-02
1.5107E-01
8.9712E-03
(+)
DTLZ3
1.0095E+00
1.4438E-01
3.1318E-01
2.5204E-01
2.6865E-01
1.4896E-01
(+)
DTLZ4
4.3749E-01
3.9812E-02
1.5836E-01
1.7960E-02
1.5930E-01
1.6573E-02
(+)
DTLZ5
1.1566E-01
1.4430E-02
2.7217E-01
2.3326E-02
2.6817E-01
3.2041E-02
(+)
DTLZ6
6.2101E-01
5.5447E-02
2.5897E-01
2.8818E-02
2.5553E-01
1.7828E-02
(+)
DTLZ7
4.6078E-01
2.9453E-02
2.7619E-01
2.0303E-02
2.7868E-01
3.1816E-02
(+)
Algorithms Problems
aMOCell4 median IQR
CellDE median
IQR
C-MCGA median IQR
fonseca
1.2746E-04
3.9454E-06
2.6983E-04
2.2413E-05
2.6630E-04
8.7459E-06 (+) 3.5705E-05 (+)
kursawe
1.1713E-04
7.2599E-06
1.6647E-04
1.9223E-05
1.5877E-04
2.1448E-05
(+)
viennet2
3.5362E-04
2.2498E-04
4.1204E-04
7.4715E-05
3.8078E-04
5.4416E-05
(-)
viennet3
6.4359E-05
1.0858E-05
1.5383E-04
4.2394E-05
1.7364E-04
DTLZ1
2.1161E-01
7.7339E-01
5.3278E-04
1.3240E-04
5.8747E-04
DTLZ2
2.0561E-03
7.2433E-04
6.2678E-04
9.7941E-05
5.7573E-04
6.8470E-05 (+) 1.2550E-04 (+) 5.9266E-05 (+)
DTLZ3
4.2511E-01
3.6603E-01
2.0305E-01
2.5763E-01
1.4102E-01
DTLZ4
3.3334E-03
2.8015E-04
3.3028E-03
2.4015E-04
3.2253E-03
DTLZ5
1.7514E-04
1.1779E-05
1.8592E-04
1.3792E-05
1.9018E-04
1.2572E-05
(-)
DTLZ6
2.5449E-01
2.3719E-02
3.8710E-04
8.6494E-06
3.9052E-04
1.0915E-05
(+)
DTLZ7
2.4879E-03
7.6680E-04
1.3067E-03
1.2460E-04
1.3540E-03
1.1933E-04 (+)
schaffer
8.0939E-02
1.4545E-01
2.1310E-04
1.0316E-05
2.1312E-04
because it yields the best indicator values in seven of the 12 MOPs. For the DTLZ family, SPEA2 has the best performance of diversity, due to the fact that SPEA2 obtains five best D values in seven MOPs. NSGA-II is
1.4640E-01 (+) 1.4904E-04 (+)
again the worst one failing to yield any best indicator value. With regard to the GD indicator, C-MCGA achieves the best values in seven MOPs. However, different from the
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Int. J. Mach. Learn. & Cyber.
Fig. 5 The PF approximations with the lowest IGD values found by aMOCell4, CellDE and C-MCGA in 30 runs in the search space on DTLZ3
cases of indicator IGD and D, NSGA-II provides two best and four second best values, while SPEA2 obtains one best and 6 s best ones. Therefore, we can conclude that NSGAII has the similar convergence property with that of SPEA2. As in Sect. 5.3, the approximated DTLZ6 fronts of each algorithm with the lowest IGD values in 30 runs are shown in Fig. 6. It is clear that both NSGA-II and SPEA2 do not converge to the PF, and their final solutions have poor distributions. However, the solutions obtained by CMCGA widely and uniformly distribute along the PF, which illustrates that it has a better convergence and diversity compared with the other two algorithms. In summary, with IGD, D, and GD taken into consideration, among the three MOEAs, C-MCGA is the best algorithm to deal with the MOPs used in our simulations.
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5.5 Performance of cCM It is well known that the crowding measurement is a key issue for an MOEA to obtain nondominated fronts with good diversity. To assess the influences on the diversity of the nondominated fronts of our new crowding measurement method: cCM, we perform an analysis on the C-MCGA-tCM and C-MCGA-cCM in this section. As a matter of fact, C-MCGA-cCM is the C-MCGA, and CMCGA-tCM is variation of C-MCGA by replacing the cCM with the traditional crowding measurement (tCM) in NSGA-II. The reason of choosing the crowding measurement in NSGAII for comparison with is that it is the most widely employed method in the MOEAs. These two algorithms are examined using the above test suite, and the parameter settings used are the same as in the previous sections. The
Int. J. Mach. Learn. & Cyber. Table 4 Median and interquartile range of inverted generational distance (IGD)
Table 5 Median and interquartile range of generalized spread (D)
Table 6 Median and interquartile range of generational distance (GD)
Algorithms Problems
NSGA-II median IQR
SPEA2 median IQR
C-MCGA median IQR
schaffer
1.0734E+00
9.1645E-01
1.1841E+00 1.1653E+00
1.4948E-02
fonseca
5.7130E-03
9.1396E-04
7.2300E-03
2.3677E-03
4.2979E-03
kursawe
3.6524E-02
3.2077E-03
3.8978E-02
3.7162E-03
3.0865E-02
viennet2
1.2225E-02
9.8124E-04
7.2976E-03
3.7727E-04
7.5095E-03
viennet3
2.6618E-02
1.6114E-03
2.1502E-02
2.9538E-03
1.8741E-02
DTLZ1
1.8745E-02
3.1542E-03
1.4098E-02
8.6634E-04
1.4141E-02
DTLZ2
4.5455E-02
1.2620E-03
3.6789E-02
7.7171E-04 8.9447E-01
3.4015E-04 (+) 2.3781E-04 (+) 1.6391E-03 (+) 2.2863E-04 (+) 6.7851E-04 (+) 5.2018E-04 (-)
DTLZ3
5.4108E+00 3.9594E+00 1.2159E+00
DTLZ4
4.4048E-02
6.2743E-03
3.1687E-02
6.2928E-03
8.0268E-04 (+) 2.0257E+00 1.9863E+00 (+) 3.1179E-02 4.4080E-03 (-)
DTLZ5
2.3794E-03
1.3162E-04
2.2684E-03
1.0760E-04
2.2122E-03
DTLZ6
7.1735E-01
3.4262E-02
4.5653E-01
3.6226E-02
1.9953E-03
1.7008E-04 (+) 1.6571E-04 (+)
DTLZ7
4.8154E-02
2.2514E-03
3.9588E-02
1.6769E-03
3.8672E-02
1.7523E-03 (-)
Algorithms Problems
NSGA-II median IQR
SPEA2 median IQR
3.7434E-02
C-MCGA median IQR
schaffer
6.7689E-01
3.9813E-01
8.7772E-01
4.0798E-01
2.4153E-01
fonseca
4.1403E-01
7.2380E-02
3.7594E-01
7.8174E-02
2.2683E-01
kursawe
5.8322E-01
6.2981E-02
6.2020E-01
1.1609E-01
4.0229E-01
viennet2
5.4881E-01
4.1921E-02
3.5865E-01
4.9511E-02
2.4433E-01
viennet3
4.8493E-01
4.5062E-02
6.1019E-01
3.7702E-02
5.5647E-01
DTLZ1
5.9068E-01
8.1184E-01
1.4373E-01
1.0220E-01
1.5302E-01
DTLZ2
4.8020E-01
4.3409E-02
1.1132E-01
1.3400E-02
1.5107E-01
DTLZ3
1.3104E+00
4.7860E-01
1.0720E+00
3.2278E-01
2.6865E-01
DTLZ4
4.7984E-01
3.5388E-02
1.0904E-01
1.3448E-02
1.5930E-01
DTLZ5
4.0951E-01
2.9930E-02
2.2996E-01
2.6281E-02
2.6817E-01
DTLZ6
6.2488E-01
5.6486E-02
3.7857E-01
1.1040E-01
2.5553E-01
DTLZ7
4.7584E-01
2.9414E-02
2.6771E-01
2.7846E-02
2.7868E-01
Algorithms Problems
NSGA-II median IQR
SPEA2 median IQR
2.4434E-02 (+) 1.8468E-02 (+) 2.0619E-02 (+) 2.0910E-02 (+) 3.6536E-02 (+) 1.3578E-02 (+) 8.9712E-03 (+) 1.4896E-01 (+) 1.6573E-02 (+) 3.2041E-02 (+) 1.7828E-02 (+) 3.1816E-02 (+)
C-MCGA median IQR
schaffer
5.1265E-02
1.7315E-01
9.5515E-02
2.3388E-01
2.1312E-04
fonseca
2.8867E-04
4.9952E-05
3.1675E-04
4.4402E-05
2.6630E-04
kursawe
1.9013E-04
4.0561E-05
2.1249E-04
7.1588E-05
1.5877E-04
viennet2
3.6499E-04
1.1550E-04
3.2931E-04
7.2445E-05
3.8078E-04
viennet3 DTLZ1
8.5360E-05 1.1881E-02
1.4540E-05 6.2335E-02
1.4721E-04 1.0179E-03
5.5276E-05 4.9364E-03
1.7364E-04 5.8747E-04
DTLZ2
8.0030E-04
8.8968E-05
7.6448E-04
1.5450E-04
5.7573E-04
DTLZ3
3.5769E+00
3.1084E+00
3.0104E-01
5.0053E-01
1.4102E-01
DTLZ4
2.9773E-03
2.3893E-04
3.0994E-03
1.4259E-04
3.2253E-03
DTLZ5
1.9448E-04
1.8412E-05
2.0656E-04
1.1947E-05
1.9018E-04
8.7459E-06 (+) 3.5705E-05 (+) 2.1448E-05 (+) 5.4416E-05 (+) 6.8470E-05 (+) 1.2550E-04 (+) 5.9266E-05 (+) 1.4640E-01 (+) 1.4904E-04 (+)
DTLZ6
7.4379E-02
4.3037E-03
4.9219E-02
6.8454E-03
3.9052E-04
1.2572E-05 (-) 1.0915E-05 (+)
DTLZ7
1.4035E-03
1.2617E-04
1.3840E-03
1.6108E-04
1.3540E-03
1.1933E-04 (-)
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Int. J. Mach. Learn. & Cyber. 2
2.5 2
1.5 NSGA−II PF
1.5
SPEA2 PF
1 1 0.5
0.5
0 0
0 0
1
1
2
2 3
0
2
1.5
1
0.5
3
0.5
0
(a) NSGA-II
1
1.5
2
2.5
(b) SPEA2
1 0.8 0.6
C−MCGA PF
0.4 0.2 0 0 0.2 0.4 0.6 0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
(c) C-MCGA Fig. 6 The PF approximations with the lowest IGD values found by NSGA-II, SPEA2 and C-MCGA in 30 runs in the search space on DTLZ6
values of the diversity indicator D are given in Table 7. We can find out that the cCM is capable of considerably improving the diversity of the nondominated solutions, since the C-MCGA-tCM yields the best D indicator values in 11 of the 12 MOPs.
6 Conclusions and Future Work For the cellular genetic algorithms, it has been shown that 3D grid structures can accelerate the spreading of solutions in case of single-objective optimization. However, for the MOEAs, some popular crowding measurements cannot make full use of the population distribution information in the nondominated fronts. This paper introduces a 3D grid structure and a cosine crowding measurement into MOcGAs, and proposes the CMCGA. Unlike the popular 2D structures, the 3D grids give a
123
vertical expansion of cells, which leads to a shorter radius and denser neighborhood. When assigning the crowding degree for a solution, the new cosine crowding measurement links crowding distance with location distribution on the basis of a cosine function. Numerical simulation results show that, compared with two typical MOcGAs and two state-of-the-art MOEAs, NSGA-II and SPEA2, our C-MCGA can achieve significantly enhanced performances of convergence and diversity. Moreover, the new measurement method can lead to a better diversity of the obtained nondominated fronts than the crowding measurement in NSGA-II. It should be emphasized that in MOcGAs, the population grids are fixed. Thus, how to develop a self-adaption mechanism for these grids and neighbor shapes will be a promising topic in our future work. In addition, the MOcGAs do not preserve the nondominated solutions based on the grid structures. We are going to construct
Int. J. Mach. Learn. & Cyber. Table 7 Median and interquartile range of generational spread (D)
Algorithms Problems schaffer
C-MCGA-tCM median IQR 2.4553E-01 2.7289E-02
C-MCGA-cCM median IQR 2.4251E-01 3.0676E-02
fonseca
2.3086E-01
2.9574E-02
2.2644E-01
kursawe
4.0349E-01
2.4604E-02
viennet2
2.3938E-02 1.9838E-02
4.0206E-01
2.4756E-01
2.4508E-01
1.8229E-02
viennet3
5.6143E-01
4.2875E-02
5.5272E-01
3.2966E-02
DTLZ1
1.5478E-01
1.5450E-02
1.5307E-01
1.3219E-02
DTLZ2
1.5435E-01
1.8808E-02
1.5421E-01
1.2818E-02
DTLZ3
2.8462E-01
5.0404E-02
2.9436E-01
1.4028E-01
DTLZ4
1.5793E-01
8.6315E-03
1.5281E-01
1.4892E-02
DTLZ5
2.7050E-01
2.6375E-01
2.4640E-02
DTLZ6
2.5079E-01
1.5512E-02 2.4918E-02
2.4806E-01
3.3511E-02
DTLZ7
2.8270E-01
2.5764E-02
2.8111E-01
2.0727E-02
nondominated fronts by using the grid characteristic of cells. Acknowledgments This work was supported by the National Basic Research Program of China (Grant No. 2012CB821205), the Foundation for Creative Research Groups of the National Natural Science Foundation of China (Grant No. 61021002), the National Natural Science Foundation of China (Grant No. 51275274, 61174037, and 61273313), and the Innovation Funds of China Academy of Space Technology (Grant No. CAST20120602). The authors would like to thank the anonymous reviewers for their insightful comments and constructive suggestions that have improved the paper.
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