Evolutionary Many-Objective Optimization Using Ensemble Fitness ...

Evolutionary Many-Objective Optimization Using Ensemble Fitness Ranking Yuan Yuan, Hua Xu, Bo Wang State Key Laboratory of Intelligent Technology and Systems Tsinghua National Laboratory for Information Science and Technology Department of Computer Science and Technology, Tsinghua University, Beijing 100084, P.R.China

[email protected], [email protected], [email protected] ABSTRACT

1. INTRODUCTION

In this paper, a new framework, called ensemble fitness ranking (EFR), is proposed for evolutionary many-objective optimization that allows to work with different types of fitness functions and ensemble ranking schemes. The framework aims to rank the solutions in the population more appropriately by combing the ranking results from many simple individual rankers. As to the form of EFR, it can be regarded as an extension of average and maximum ranking methods which have been shown promising for many-objective problems. The significant change is that EFR adopts more general fitness functions instead of objective functions, which would make it easier for EFR to balance the convergence and diversity in many-objective optimization. In the experimental studies, the influence of several fitness functions and ensemble ranking schemes on the performance of EFR is fist investigated. Afterwards, EFR is compared with two state-of-the-art methods (MOEA/D and NSGA-III) on wellknown test problems. The computational results show that EFR significantly outperforms MOEA/D and NSGA-III on most instances, especially for those having a high number of objectives.

During the past several decades, multi-objective optimization problems (MOPs) have stirred much interest in the evolutionary computation community. In a MOP, there exist several objectives that should be optimized simultaneously. Generally, these objectives are in conflict to each other to some extent, and no single solution can be found to best satisfy all of them. When solving MOPs, we aim to seek for a set of solutions in the decision space achieving the best tradeoffs between each objective, known as Pareto Set (PS). The mapping of PS in the objective space is referred as Pareto Front (PF). Evolutionary algorithms (EAs) have been recognized as especially desirable for handing MOPs, mainly because they can obtain an approximation of PF in a single simulation run. Up to present, various of multi-objective evolutionary algorithms (MOEAs) have been proposed to effectively solve MOPs. However, most of these algorithms have been evaluated and applied to MOPs with only two or three objectives, despite the fact that optimization problems involving a large number of objectives indeed appear widely in real-world applications [13]. In the literature, such MOPs having more than three objectives are often termed as many-objective optimization problems. Unfortunately, experimental and analytical results [15, 24] have indicated that many-objective optimization problems would pose serious difficulties to the existing MOEAs, particulary for the popular Pareto-based MOEAs, such as NSGA-II [8] and SPEA2 [28]. The primary reason for this is that the proportion of non-dominated solutions in a population increases rapidly with the number of objectives, and it would lead to the severe loss of Pareto dominance-based selection pressure toward the PF. To overcome this shortage, it is key to devising alternative approaches to rank individuals in the population instead of using the original Pareto dominance. In the light of this direction, some efforts have been made and the proposed approaches can be roughly classified into three types. The first class of approaches focuses on adopting new preference relations that could induce finer grain order in the objective space than that induced by Pareto dominance relation, for example, weighted sum, average and maximum ranking [3], flavour relation [11] and -dominance [20]. The second class of methods decomposes a MOP into a set of single-objective subproblems through scalarizing functions, and then solves these subproblems simultaneously by evolving a population of solutions. MOEA/D [26, 21] is probably the most representative implementation of this class. The last class is based on the use of performance indicators. It is intended to perform

Categories and Subject Descriptors I.2.8 [Artificial Intelligence]: Problem Solving, Control Methods, and Search—Heuristic methods; G.1.6 [Numerical Analysis]: Optimization

General Terms Algorithms, Performance

Keywords Many-objective optimization, ensemble fitness ranking, average ranking, maximum ranking, fitness function, NSGAIII, MOEA/D

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3, the proposed EFR methods are described in detail. The experimental studies are conducted in Section 4. Finally, Section 5 concludes this study.

solution selection by directly optimizing a single indicator such as hypervolume [27, 4]. Besides the above mentioned three classes, it is necessary to note that there is another different type of research, aiming to promote the diversity in many-objective optimization, including the work in [1] and recently proposed NSGA-III [7], although there has not been much study in this respect yet. The average ranking (AR) and maximum ranking (MR) preference relations are both originally proposed by Bentley and Wakefield [3]. In spite of the simplicity, they have been found to perform well in converging to the PF in manyobjective optimization. Corne and Knowles [6] reported that AR is highly effective in comparison with other more complicated ranking schemes, covering problems with 5 to 20 objectives and differing amounts of inter-objective correlation. Kukkonen and Lampinen [19] specially examined AR and MR, and implemented them in an MOEA called GDE3 [18]. They concluded that AR and MR appear to be a promising alternative to Pareto dominance relation for many-objective problems. In several similar researches [23, 16, 14], the comparative studies of different preference relations or ranking methods,including AR and MR, are carried out to deal with problems with a high number of objectives. It is worth noting that although AR and MR have shown their potential, both of them often converge into a subset of PF due to the lack of diversity maintenance mechanism [19, 16]. As for MR, there also exists the particular downside that it tends to favor extreme solutions, i.e., it prefers solutions with high performance in some objectives, but without taking into account the especially poor performance in the rest of the objectives. To our knowledge, the efforts to improve AR or MR are rather limited. Recently, Li et al. [22] proposed a diversity enhancement strategy for AR to balance convergence and diversity during evolution process. But overall, the potential of AR and MR in many-objective optimization is still far from being fully explored. This paper proposes a new framework, called ensemble fitness ranking (EFR), for evolutionary many-objective optimization. EFR assigns the global rank to each solution by combining a set of ranking lists, each of which is collected according to a unique fitness function. Our motivation is to improve ranking reliability and rationality by integrating multiple ranking results produced by different simple individual rankers (fitness functions). With regard to the form, EFR can be viewed as the extension of AR and MR. The significant difference lies in that EFR employs more general fitness functions instead of objective functions. Unlike the single objective function, a suitable fitness function can indicate the “closeness” of the solution to the PF in a sense. A number of different fitness functions may have an active effect on the diversity maintenance. So, compared with AR and MR, it is easier for the proposed EFR to balance the convergence and diversity in many-objective optimization. To instantiate the proposed framework, we provide three kinds of alternative fitness functions and three kinds of ensemble ranking schemes, and several EFR variants are involved to investigate their influence to the convergence ability and overall performance. Further, we compare one EFR variant with MOEA/D and NSGA-III on many-objective problems, the computational results indicate that EFR performs significantly better than the other two in most cases. In the remainder of this paper, some background knowledge is first provided in Section 2. Thereafter, in Section

2. BACKGROUND In this section, some necessary definitions in multi-objective optimization are first given. Then, we will briefly introduce the average ranking (AR) and maximum ranking (MR) which are the important basis of our work.

2.1 Basic Definitions A MOP can be mathematically defined as follows: min f (x) = (f1 (x), f2 (x), . . . , fm (x))T subject to x ∈ Ω ⊆ Rn

(1)

where x = (x1 , x2 , ..., xn )T is a n-dimensional decision variable vector from the decision space Ω; f : Ω → Λ ⊆ Rm consists a set of m objective functions, and is a mapping from n-dimensional decision space Ω to m-dimensional objective space Λ. Definition 1. Given x, y ∈ Ω, x is said to Pareto dominate y, denoted by x ≺ y, if fi (x) ≤ fi (y), for every i ∈ {1, 2, . . . , m}, and fj (x) < fj (y), for at least one index j ∈ {1, 2, . . . , m}. Definition 2. A decision vector x∗ ∈ Ω is Pareto optimal if there is no x ∈ Ω such that x ≺ x∗ . Definition 3. The Pareto set, P S, is defined as: P S = {x ∈ Ω|x is Pareto optimal}

(2)

Definition 4. The Pareto front, P F , is defined as: P F = {f (x) ∈ Rm |x ∈ P S}

(3)

Definition 5. The ideal vector z∗ is a vector z∗ = (z1∗ , ∗ T z2∗ , . . . , zm ) , where zi∗ is the infimum of fi for every i ∈ {1, 2, . . . , m}. The goal of MOEAs is to move the non-dominated objective vectors towards PF (convergence), and also generate a good distribution of these vectors over the PF (diversity).

2.2 Average and Maximum Ranking The AR and MR methods consider each objective independently. For an objective fj , j ∈ {1, 2, . . . , m}, the solutions in the population are sorted in the non-decreasing order in accordance with values of such objective, and a ranking list is built where each solution has its own ranking position. When all the objectives are concerned, we totaly have m ranking lists corresponding to m objectives. For each solution x, it has m ranking positions, and can be represented by the vector R(x) = (r1 (x), r2 (x), . . . , rm (x))T , where rj (x) is the rank of x for the objective fj . Once R(x) is calculated, a single rank is obtained for each solution by combing the m ranking positions, which is used to reflect the overall quality of solutions. To be specific, AR gives the global rank to a solution x in this way: Ravg (x) =

m  j=1

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rj (x)

(4)

produced, where N is the population size. Second, K different fitness functions are generated. Finally, Steps 4-21 are iterated until the termination criterion is satisfied. In each generation t, the binary tournament selection based on Rg , the simulated binary crossover (SBX) operator and polynomial mutation [2] are first performed to produce the offspring population Qt . In Step 6, the populations Pt , Qt are merged as the population Rt . In Steps 7-10, R(x) is calculated for each solution x. In Step 11, the global ranks are assigned and Rt is further classified into some solution sets (analogous to “fronts” in NSGA-II), just as stated in the previous paragraph. In the remaining steps, exactly N solutions are selected from Fi in order. When reaching the last considered Fi , another selection criterion is used to sort it (see Step 18), and the best solutions needed are chosen to fill all population slots. In this paper, we mainly emphasize the ensemble ranking, so the random sort is indeed used in Step 18 for simplicity.

while MR computes the global rank as the following: m

Rmax (x) = min rj (x)

(5)

j=1

In Table 1, AR and MR are demonstrated with a small example of six 3-objective solutions. Table 1: An example of average ranking and maximum ranking.

Solution

(f1 , f2 , f3 )

r1

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MR

A B C D E F

(5, 2, 4) (2, 3, 6) (7, 1, 1) (8, 6, 5) (9, 0, 2) (4, 7, 3)

3 1 4 5 6 2

3 4 2 5 1 6

4 6 1 5 2 3

10 11 7 15 9 11

3 1 1 5 1 2

Algorithm 1 Framework of the proposed EFR methods

Based on Ravg or Rmax , we can further define preference relations. A solution x dominates solution y with respect to AR relation, denoted by x ≺avg y, if and only if Ravg (x) < Ravg (y). Similarly, a solution x dominates solution y with respect to MR relation, denoted by x ≺max y, if and only if Rmax (x) < Rmax (y). It is apparent that both ≺avg and ≺max are transitive relations, so both of them can be easily used to discriminate solutions within the same population unambiguously.

3.

1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21:

PROPOSED METHODS

3.1 Overview Instead of considering each objective fj , we introduce the fitness functions in our proposed ensemble fitness ranking (EFR) methods. For a fitness function Fj , it assigns the value Fj (x) to each solution x, which can reflect how well the solution is in a certain aspect. Without loss of generality, the smaller Fj (x) value means better quality of x according to Fj . Suppose we have K different fitness functions F1 , F2 , . . . , FK . Then similar to average and maximum ranking, for each solution x, it has K ranking positions, denoted by the vector R(x) = (r1 (x), r2 (x), . . . , rK (x))T , here rj (x) is the rank of x for the fitness function Fj . After obtaining R(x) for all solutions, the ensemble ranking is used to give the global rank Rg (x) to each solution x based on R(x). Obviously, we can compute Rg (x) in the same way with respect to AR or MR scheme as follows: Rg (x) =

K 

rj (x)

From Algorithm 1, it can be seen that there exist two critical issues to implement the framework. One is how to choose suitable fitness functions, and the other is how to perform the ensemble ranking. In Section 3.2, several possible fitness functions that can be used in our framework are introduced. In Section 3.3, we will also present another available ensemble ranking scheme in addition to AR and MR to assign global ranks, referred to as layering ranking (LR) in this paper.

(6)

3.2 Alternative Fitness Functions

j=1 K

Rg (x) = min rj (x) j=1

P0 ← InitializePopulation() {F1 , F2 , . . . , FK } ← GenerateFitnessFunctions() t←0 while the termination criterion is not met do Qt ← MakeOffspringPopulation(Pt ) Rt ← Pt ∪ Qt for i ← 1 to K do sort Rt according to Fi in non-decreasing order set ri for each individual in Rt end for {F1 , F2 , . . .} ← EnsembleRanking(Rt , R) Pt+1 ← ∅ i←1 while |Pt+1 | + |Fi | ≤ N do Pt+1 ← Pt+1 ∪ Fi i←i+1 end while Sort(Fi ) Pt+1 ← Pt+1 ∪ Fi [1 : (N − |Pt+1 |)] t←t+1 end while

The fitness function Fj used in our framework would better meet the condition that the calculation of Fj (x) for solution x should not depend on the other solutions in Rt . Otherwise, Fj values will make no sense in the selected next population Pt+1 from Rt . Hence, the assignment of Fj values would be aided by some external means. In this paper, three kinds of alternative fitness functions are to be investigated: Lp norm, Tchebycheff function, and penaltybased boundary intersection (PBI) function [26]. Let λj = (λ , λj,2 , . . . , λj,m )T , where λj,k ≥ 0, k = 1, 2, . . . , m and j,1 m k=1 λj,k = 1, then they can be respectively described as follows

(7)

It should be noted that some solutions may be assigned the same global rank. So, according to Rg , the considered population can be partitioned into a number of solution sets {F1 , F2 , . . .} , where the solutions in Fi have the i-th minimum Rg value. The basic framework of our proposed EFR methods is similar to NSGA-II [8], which is described in Algorithm 1. First, an initial population with N solutions is randomly

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f2

1) Lp Norm: m  1 Fj (x) = { (λj,k |fk (x) − zk∗ |p )} p

Ȝj

(8)

k=1

dj,2

where p ≥ 1. When p = 1, it degenerates to weighted sum, which has difficulty in capturing non-convex PFs. However, large p value would greatly increase the computation load, although it may work well on more complex PFs. 2) Tchebycheff Function: m

Fj (x) = max{ k=1

1 |fk (x) − zk∗ |} λj,k

(9)

PF z*

It is worth noting that the adopted form is a little different from that in [26, 21], where Fj is defined as Fj (x) = ∗ maxm k=1 {λj,k |fk (x) − zk |}. The modification is based on the following theorem:

Figure 1: Illustration of PBI function. Algorithm 2 Generate K uniformly distributed vectors 1: for j ← 1 to K do 2: s←0 3: for k ← 1 to m do 4: if k < m then 5: r ← rand(0, 1) 1 6: λj,k ← (1 − s)(1 − r m−k ) 7: s ← s + λj,k 8: else 9: λj,k ← 1 − s 10: end if 11: end for 12: end for

Proof. Let f (x) be the intersection point with the PF, then we have the following equality (10)

where C is a constant. Suppose f (x) is not the optimal solution to Fj , then ∃f (y) satisfies Fj (y) < Fj (x). According to Equation (10), Fj (x) = C. Then ∀k ∈ {1, 2, . . . , m}, we have fk (x) − zk∗ fk (y) − zk∗ ≤ Fj (y) < C = (11) λj,k λj,k

3.3 Ensemble Ranking

Hence, fk (y) < fk (x). This is is contradicted with the condition that f (x) is located on the PF and the supposition is invalid.

In this subsection, we will provide another ensemble ranking scheme, called layering ranking (LR), to assign global ranks. The procedure of LR can be depicted in Algorithm 3, where the population P is to be given global ranks. Take the solutions shown in Table 1 for instance, LR assigns the global ranks as follows: A(2), B(1), C(1), D(3), E(1), F (2). It can be seen that, different from AR and MR, LR gives the consecutive ranks to solutions.

According to Theorem 1, the form used in this paper can produce more uniformly distributed solutions in the objective space if uniformly distributed λj , j = 1, 2, . . . , are generated. 3) PBI Function: Fj (x) = dj,1 + θdj,2 ;

Attainable Objective Set

f1

Theorem 1. If the line going through z∗ with direction λj has an intersection with the PF, then the intersection point is the optimal solution to Fj defined in Equation (9).

∗ f1 (x) − z1∗ f2 (x) − z2∗ fm (x) − zm = = ... = =C λj,1 λj,2 λj,m

f(x)

dj,1

(12)

Algorithm 3 LayeringRanking(P , R)

where θ > 0 is a user-defined penalty parameter, dj,1 =

(f (x)−z∗ )T λj / λj and dj,2 = f (x)−z∗ −dj,1 (λj / λj ) . As shown in Figure 1, L is a line passing through z∗ with direction λj , and u is the projection of f (x) on L. dj,1 is the distance between z∗ and u, and dj,2 is the perpendicular distance between f (x) and L. The decreasing of Fj would push f (x) as low as possible so that it reaches the the boundary of attainable objective set. Certainly, K fitness functions can be comprised of all three kinds mentioned above. However, in this paper, we only investigate the situation that K fitness functions belong to the same type. To ensure the diversity, K uniformly distributed vectors λ1 , λ2 , . . . , λK are generated [17] so as to obtain K fitness functions F1 , F2 , . . . , FK , whose procedure is described Algorithm 2, where rand(0, 1) is a random function returning a real number between 0 and 1 with uniform distribution. Moreover, it should be mentioned that it is often very time-consuming to compute exact zk∗ , so it is indeed approximated by the minimum value found so far for objective fk in our methods.

1: rank ← 1 2: while P = ∅ do 3: Q←∅ 4: for j ← 1 to K do 5: choose solution x in P with minimum rj value 6: Rg (x) ← rank 7: Q ← Q ∪ {x} 8: end for 9: remove all solutions in Q from P 10: rank ← rank + 1 11: end while

4. EXPERIMENTAL STUDIES The proposed methods are all implemented in the jMetal framework [12]. The well-known DTLZ1, DTLZ2, DTLZ3, and DTLZ4 problems [9] are considered. We vary the number of objectives between 2 to 20. In order to assess the performance of the proposed algorithms, the generational distance (GD) [25] and inverted generational distance (IGD)

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From Figure 2, for DTLZ1 and DTLZ2, EFR-5 performs the best while EFR-2 seems to be the worst; EFR-3 is merely better than EFR-2 and they differs only a bit for DTLZ2. For DTLZ3, EFR-5 still exhibits the advantage for the instances that the number of objective is not very high; but for instances more than 8 objectives, EFR-1 and EFR-3 are the winners while EFR-5 becomes the worst except for the 14-objective instance, where EFR-5 outperforms EFR-1 and EFR-2; the results of EFR-1 and EFR-2 are distinguishable only for instances having more than 14 objectives, where EFR-2 is outperformed by EFR-1 very weakly. For DTLZ4, the situation is similar to that for DTLZ3; but EFR-5 performs worst for instances with more than 6 objectives all the while; for instances with more than 14 objectives, EFR2 appears sightly better than EFR-1 on the contrary to that in DTLZ3. From Figure 3, for DTLZ1, there is no obvious difference between EFR-1, EFR-3 and EFR-4 and they perform better overall than the other two; EFR-2 is better than EFR-5 for instances with less than 16 objectives, but is outperformed by EFR-5 when the number of objectives further increases. For DTLZ2, the performance of EFR-5 is almost the worst when the number of objectives is no more than 14, but when the objective gets larger, EFR-2 outperforms both EFR-1 and EFR-2; EFR-1 seems better than EFR-2 for instances with high number of objectives; EFR-3 and EFR-4 still show the superiority over the others. For both DTLZ3 and DTLZ4, the performance of EFR variants can be roughly ranked. Specially, for DTLZ3, EFR-3 and EFR-4 perform best, following by EFR-1 and EFR-2, while EFR5 is the worst. For DTLZ4, the variants can be separated into three groups, EFR-3 and EFR-4 form the first group which are ranked first; EFR-1 and EFR-2 have the similar performance and are ranked second; EFR-5 still ranks last. Because GD is a convergence metric and IGD is the metric considering both convergence and diversity, we can make the following summary based on the above results. For fitness functions, PBI seems to be the best choice, Lp norm and Tchebycheff function have no distinct difference on DTLZ 24, but for DTLZ1, Lp norm leads to better convergence. As for ensemble ranking schemes, MR and LR nearly show the equivalent overall performance and are better than AR, but AR achieves the best convergence in DTLZ1. It can be safely inferred that the diversity of AR in our framework still needs to be further enhanced.

[5] are involved as metrics in our experiments. They can be expressed as follows: Let P ∗ be a set of uniformly distributed points along the PF. Let A be an approximation to the PF. The metrics GD and IGD of the set A are respectively defined as:   |A|  1   GD(A, P ∗ ) = d2 (13) |A| i=1 i  |P ∗ |  1  ∗  IGD(A, P ) = d˜2 |P ∗ | i=1 i

(14)

where di (d˜i ) is the Euclidean distance between the i-th member in the set A (P ∗ ) and its nearest member in the set P ∗ (A). GD can only reflect the convergence of an algorithm, while IGD combines the information of convergence and diversity into a whole. For both of two metrics, smaller values means better quality. Because there are different fitness functions and ensemble ranking schemes that could be used, several variants of the implemented methods will be concerned in the experiments, which are listed in Table 2. All the algorithms are run 30 times independently for each problem, and average results are recoded. Table 2: Variants of the implemented methods. Variant

Fitness Function

EFR-1 EFR-2 EFR-3 EFR-4 EFR-5

Lp

Norm Tchebycheff PBI PBI PBI

Ensemble Ranking LR LR LR MR AR

4.1 Comparison between EFR methods In this subsection, we will make the comparison between EFR variants to examine the influence of alternative fitness functions and ensemble ranking schemes. The shared parameters of EFR methods are shown in Table 3. Specially, for Lp norm, p is set to be 3, and for PBI function, θ is set to be 5. To ensure a fair comparison, for each problem, we limit the number of objective function evaluations to a maximum of 100,000 for all algorithms.

4.2 Comparison with other methods

Table 3: Parameter settings of EFR variants Parameter Population size (N ) Number of fitness functions (K) Crossover probability (pc ) Mutation probability (pm ) Distribution index for crossover (ηc ) Distribution index for mutation (ηm )

In this subsection, we will compare the variant EFR-3 (in the following, call it EFR for short), with two established MOEAs, named as MOEA/D [21] and NSGA-III [7]. MOEA/D is the MOEA based on decomposing a MOP into a number of scalar optimization problems that are simultaneously optimized, which could be easily scaled up for many-objective problems. NSGA-III is a very recent MOEA particulary for many-objective optimization, whose framework is similar to NSGA-II with significant changes in its selection mechanism. The population size N of MOEA/D and NSGA-III are both controlled by an integer H, where m−1 N = CH+m−1 . The chosen value of H for problems with different number of objectives, the resulting population size, and the maximum objective function evaluations are given in Table 4. The settings of the other parameters of MOEA/D and NSGA-III are just according to the original papers. For

Value 200 200 0.9 1/n 20 20

Figures 2 and 3 present the average GD and IGD results for DTLZ 1-4 problems respectively. First, it is interesting to note that EFR-3 and EFR-4 nearly always have the same performance on all the considered cases. It may be because the preference of MR and LR is roughly consistent in the situation selecting half of the solutions from a population, although they lead to different subtle rankings.

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Figure 2: Comparison of the performance of EFR variants on GD metric for DTLZ 1-4 problems with varying number of objectives (dimension). The figures show the average of 30 independent runs each.

(with the significance level of 0.05) is marked in bold. From Table 5, it can be seen that the proposed EFR significantly outperforms MOEA/D and NSGA-III on all the considered problems with more than four objectives. NSGA-III shows superiority in problems with three and four objectives. To be specific, NSGA-III is significantly better than EFR and MOEA/D on all 3-objective problems, and it is still the winner on all 4-objective problems except for DTLZ3, where EFR is a little better than NSGA-III but there exists no significance. For 3-objective problems, although EFR is worse than NSGA-III, it achieves the competitive performance with respect to MOEA/D. EFR is significantly outperformed by NSGA-III for 4-objective DTLZ1, DTLZ2 and DTLZ4, but the gap between them is not so obvious. In summary, EFR is a very promising alternative for manyobjective problems compared with existing state-of-the-art methods.

the EFR, the population size and maximum objective function evaluations are reset to the same with those in the compared algorithms, the number of fitness functions is set equal to the population size, and the other parameters remain the same with those shown in Table 3. Since IGD could measure both the diversity and convergence in a sense, it is adopted as comparison metric here. Table 4: Parameters of MOEA/D and NSGA-III. m

H

N

Function Evaluations

3 4 5 6 8 10 12 14

19 9 6 5 4 3 3 3

210 220 210 252 330 220 364 560

99,960 100,100 99,960 100,044 99,990 100,100 100,100 100,240

5. CONCLUSION AND FUTURE WORK Table 5 provides the average IGD of each compared algorithm for DTLZ 1-4 having 3 to 14 objectives. In order to find significant differences, the Wilcoxon signed-rank test [10] is further carried out on the three average results respectively obtained by three algorithms for each instance, and the one that is significantly better than the other two

In this paper, we have presented an ensemble fitness ranking (EFR) framework for solving many-objective optimization problems. This framework allows to work with different kinds alternative fitness functions and ensemble ranking schemes. Through the experiments, PBI is found to be the most ideal fitness function, while MR and LR are ranking

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EFR-1 EFR-2 EFR-3 EFR-4 EFR-5

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(a) DTLZ1

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Figure 3: Comparison of the performance of EFR variants on IGD metric for DTLZ 1-4 problems with varying number of objectives (dimension). The figures show the average of 30 independent runs each.

6. REFERENCES

schemes resulting in similar performance. AR shows good convergence ability as a whole, but its maintenance of diversity needs to be further improved. To verify the effectiveness of our framework, the proposed EFR is compared with MOEA/D and NSGA-III on problems having varying number of objectives. The results show that EFR significantly outperforms MOEA/D and NSGA-III on most instances, especially for those with high number of objectives. In the future, we will continue this study in the following aspects. First, we would develop strategies to enhance the diversity of AR in our framework, where the work in [22] may be a useful reference. Second, we want to incorporate the preference into our framework by weighting raking results from different rankers. Lastly, EFR would be tested on broader range of problems to further demonstrate its effectiveness.

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Acknowledgment This work is supported by National Natural Science Foundation of China (Grant No. 61175110), National Basic Research Program of China (973 Program) (Grant No. 2012CB 316305), National S&T Major Projects of China (Grant No. 2011ZX02101-004) and National Banking Information Technology Risk Management Projects of China.

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Table 5: Comparison of EFR, MOEA/D and NSGA-III on IGD metric. Problem m

EFR

MOEA/D

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NSGA-III

DTLZ1

3 4.136483E-04 4.633166E-04 4 2.522413E-03 5.807972E-03 5 3.795216E-03 1.678922E-02 6 4.673733E-03 2.593178E-02 8 6.549155E-03 2.712590E-02 10 9.327998E-03 2.383760E-02 12 9.740158E-03 3.214111E-02 14 1.091462E-02 6.116184E-02

2.938980E-04 2.419834E-03 4.304645E-03 5.930539E-03 1.384594E-02 2.599091E-02 3.320180E-02 5.230736E-02

DTLZ2

3 5.036636E-04 5.009822E-04 4 1.152701E-03 1.739747E-03 5 1.797095E-03 3.423462E-03 6 2.256326E-03 3.751035E-03 8 3.061871E-03 4.696051E-03 10 4.065464E-03 5.611757E-03 12 4.552580E-03 6.117158E-03 14 4.908524E-03 6.358814E-03

3.775795E-04 1.100574E-03 2.013230E-03 2.806929E-03 4.321507E-03 6.315871E-03 7.338335E-03 8.367192E-03

DTLZ3

3 8.184213E-04 8.055299E-04 4 1.160800E-03 1.753353E-03 5 1.793648E-03 6.031817E-03 6 2.274413E-03 6.667546E-03 8 3.074188E-03 8.369069E-03 10 4.086756E-03 7.918472E-03 12 4.530450E-03 9.876872E-03 14 4.875834E-03 1.293029E-02

6.094970E-04 1.170746E-03 2.036958E-03 2.886499E-03 4.811574E-03 9.017786E-03 1.145193E-02 1.817348E-02

DTLZ4

3 8.228233E-04 1.005974E-03 4 1.147441E-03 2.509969E-03 5 1.780017E-03 4.127854E-03 6 2.266260E-03 4.587130E-03 8 3.091705E-03 5.988025E-03 10 4.128603E-03 7.270253E-03 12 4.734533E-03 8.459775E-03 14 5.267423E-03 9.364443E-03

6.085986E-04 1.100826E-03 2.014415E-03 2.808462E-03 4.317174E-03 6.161857E-03 7.074542E-03 7.829817E-03

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