A novel metaheuristic for multi-objective optimization problems: The multi-objective vortex search algorithm

A novel metaheuristic for multi-objective optimization problems: The multi-objective vortex search algorithm

Accepted Manuscript A novel metaheuristic for multi-objective optimization problems: The Multi-Objective Vortex Search algorithm ¨ Ahmet OZKIS ¸ , Ah...

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Accepted Manuscript

A novel metaheuristic for multi-objective optimization problems: The Multi-Objective Vortex Search algorithm ¨ Ahmet OZKIS ¸ , Ahmet BABALIK PII: DOI: Reference:

S0020-0255(17)30616-3 10.1016/j.ins.2017.03.026 INS 12809

To appear in:

Information Sciences

Received date: Revised date: Accepted date:

21 July 2016 20 January 2017 21 March 2017

¨ Please cite this article as: Ahmet OZKIS ¸ , Ahmet BABALIK , A novel metaheuristic for multi-objective optimization problems: The Multi-Objective Vortex Search algorithm, Information Sciences (2017), doi: 10.1016/j.ins.2017.03.026

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ACCEPTED MANUSCRIPT Highlights VS algorithm is modified and multi-objective VS (MOVS) has been proposed MOVS algorithm has been tested on 36 different benchmark problems along with 4 other algorithms

Obtained results show that MOVS is a very successful and competitive algorithm

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ACCEPTED MANUSCRIPT A novel metaheuristic for multi-objective optimization problems: The Multi-Objective Vortex Search algorithm

First and corresponding author: Reserach Assistant Ahmet ÖZKIŞ Selcuk University, Faculty of Engineering, Department of Computer Engineering, 42075 Konya, Turkey Tel.: +90 332 223 37 39; fax: +90 332 241 0635. [email protected] Co-author: Assistant Professor Ahmet BABALIK

Tel.: +90 332 223 37 21; fax: +90 332 241 0635. [email protected]

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Selcuk University, Faculty of Engineering, Department of Computer Engineering, 42075 Konya, Turkey

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ABSTRACT This study investigates a multi-objective Vortex Search algorithm (MOVS) by modifying the single-objective Vortex Search algorithm or VS. The VS is a metaheuristic-based algorithm that uses a new adaptive step-size adjustment strategy to improve the performance of the search process. Search mechanism of the VS is inspired by the vortex pattern, so it is called a “Vortex Search” algorithm. The original VS is an improved way of solving single-objective continuous problems. To improve the MOVS algorithm, the VS algorithm is enhanced with added calculation approaches, such as fast-nondominated-sorting and crowding-distance, in order to identify the degree of non-dominance of the solutions and the densities of their occurrence. In addition, a crossover operation is added to the MOVS algorithm in order to enhance the Pareto front convergence capacity of the solutions. Finally, to spread the solutions more successfully over the Pareto front, it has been randomly produced using the inverse incomplete gamma function using a parameter between 0 and 1. The proposed MOVS algorithm is tested against 36 different benchmark problems together with NSGAII, MOCell, IBEA and MOEA/D algorithms. The test results indicate that the MOVS algorithm achieves a better performance on accuracy and convergence speed than any other algorithms when comparisons are made against several test problems, and they also show that it is a competitive algorithm.

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Keywords: Vortex Search algorithm, Multi-objective Optimization, Metaheuristics, Non-dominated Sorting Genetic Algorithm-II;

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Jel Code: C61 - • Optimization Techniques • Programming Models • Dynamic Analysis

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1. Introduction Many real-world problems necessitate optimization of multiple objectives, which generally conflict with each other, to be minimized or maximized synchronously. These kinds of problems are called Multi-Objective Optimization Problems or MOOPs [39]. A MOOP targeted minimization is mathematically symbolized as follows [33]:

(1)

Here f1, f2,…,fm state the objective functions to be minimized, m is the number of objective functions, X is the decision th variables vectors, p is the number of inequality constraints, s is the number of equality constraints, gi is the i inequality th constraint, hj is the j equality constraint , n is the number of decision variables, and Lk and Uk refer to the lower and upper th limit values of the k decision variable in the solution space. Multi-Objective Optimization (MOO) is different from Single-Objective Optimization (SOO). In SOO, only one global best solution is obtained from the population, whereas in MOO, a multiple solution set is obtained called Pareto-optimal solutions. In SOO, the objective function is scalar, and in order to compare two candidate solutions, it is enough to look at the objective function values. In a minimization problem, the solution whose objective function value is lowest, is the best. In MOO, the objective functions are a vector, and the comparison of two solutions is made with respect to the concept of Pareto dominance [44], which enables the comparison of multiple objective function values. Where X and Z are the

ACCEPTED MANUSCRIPT candidate solutions for a minimization problem, the set of all the solutions where the constraints are provided, is referred to as The four separate mathematical definitions expressing the concepts of Pareto are given as follows [9, 51]: Def.1 Pareto dominance:

{ {

If

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For the X solution to be better (dominance) than the Z solution, it is necessary not to be worse than Z in any objective function and at least to gain a better value than Z in an objective function. If this condition is met, the X solution dominates the Z solution (Pareto dominates) and is shown as . Def.2 Pareto optimal:

Def.3 Pareto optimal set (PS): The set of Pareto optimal solution vectors.

{

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Def.4 Pareto optimal Front (PF): The set of Pareto optimal objective vectors.

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If , is Pareto optimal. Each solution belonging to the set is compared with the rest of the set using Pareto dominance (given in Def.1). After this comparison, unless there is a Z solution that dominates the X solution, X is a Pareto optimal solution.

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The related work and main contribution of the study is explained in the introductory section. The other parts of the study are organized as follows: Section 2 shows the basic steps of a VS algorithm, and the detailed explanation of the proposed MOVS algorithm. In Section 3, the experimental results of the MOVS algorithm are presented and analyzed. Section 4 outlines the conclusions of the study and thoughts for further studies.

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1.1 Related Works Although the ideas related to the use of Evolutionary Algorithms (EAs) to solve the MOOPs were first raised in a doctoral thesis in 1967, a real Multi-Objective Evolutionary Algorithm (MOEA) could not be improved in this study (MOOPs were organized as SOOPs and solved with a genetic algorithm) [7]. Until the mid 1980s, the researchers working on computer science and applied mathematics used mathematical MOO methods (deterministic methods etc.) to solve MOOPs. While MOOPs are solved via mathematical methods, there was a problem of getting stuck in a local optimum solution. So, the researchers tended to study metaheuristic-based methods. In 1984, the Vector Evaluated Genetic Algorithm (VEGA) was improved and the first MOEA was proposed , which contained stochastic (evolutionary and intuitive) optimization techniques for MOOPs [40]. Goldberg (1989) criticized VEGA and suggested the use of nondominated sorting and selection methods for MOOPs [23]. This suggestion by Goldberg has contributed to the use of the Pareto optimal concept in EAs [7]. After these studies led to the field of MOEAs, many researchers adapted single-objective evolutionary and metaheuristicbased algorithms and also proposed new multi-objective algorithms. The most common of these are as follows: Nondominated Sorting Genetic Algorithm (NSGA) [42], Pareto Archived Evolution Strategy (PAES) [27], Strength–Pareto Evolutionary Algorithm (SPEA) [50, 54], Strength–Pareto Evolutionary Algorithm2 (SPEA2) [53], Pareto–frontier Differential Evolution (PDE) [2], Nondominated Sorting Genetic Algorithm II (NSGAII) [13], Multi-Objective Particle Swarm Optimization (MOPSO) [8], Indicator-based evolutionary algorithm (IBEA) [52], OMOPSO [41], Multi-Objective Evolutionary Algorithm based on Decomposition (MOEA/D) [49], Generalized Differential Evolution 3 (GDE3) [28], Multi-Objective Cellular genetic algorithm (MOCell) [36], Speed-constrained Multi-objective PSO (SMPSO) [35], Multi-Objective Artificial Bee Colony (MOABC) [24] and decomposition-based Multi-Objective Particle Swarm Optimizer (dMOPSO) [31]. In addition to these well-known algorithms, many studies have been done recently in the field of MOO. Some of the recent studies are as follows: [3] proposes three different multi-objective Artificial Bee Colony (ABC) algorithms called A-MOABC/PD, A-MOABC/NS and S-MOABC/NS. These algorithms are based on synchronous and asynchronous models using Pareto-dominance and nondominated sorting. [39] proposes an ABC-based hybrid algorithm called the Improved Bee Colony algorithm (IBMO). IBMO combines the main ideas of the ABC algorithm with nondominated-sorting strategies. [47] proposes a new elitism-based algorithm called eMOABC. In this algorithm, an elitism strategy is adopted to avoid premature convergence. Moreover, eMOABC uses a fixed-size archive to store nondominated solutions. In [9], a new decomposition based multi-objective PSO (MPSO/D) is proposed. MPSO/D decomposes the objective space of a problem into a set of sub regions, and each of them has a solution to maintain the diversity. Secondly, MPSO/D calculates the fitness value by using the crowding distance of the stored solutions for operator selection, and determines the global best (gbest) position of a particle – the algorithm uses neighboring particles to do this. Multi-objective algorithms are also used to solve multi-objective real-world problems. In [4], Multi-Objective Differential Evolution (MODE) is used to solve Multi-Objective Reactive Power Dispatch (MORPD) problems. MODE has been tested on

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IEEE’s 30-bus, 57-bus and 118-bus systems. In [32], the Multi-Objective Time-Dependent Orienteering Problem (MOTDOP) is solved successfully by using a Multi-Objective Memetic Algorithm (MOMA) and a Multi-objective Ant Colony System (MACS), which presents a multi-objective approach for selecting an optimal network of public transport (PT) priority lanes. In [19], a modified multi-objective artificial bee colony algorithm is used to design a low power, finite-impulse response (FIR) filter. The design intends to reduce the power consumption in addition to minimizing the pass band and stop band ripples. The approach has been validated experimentally by using field programmable gate array (FPGA). In [34], a differential evolution-based, multi-objective optimization algorithm is proposed to optimally size a photovoltaic water pumping system (PVPS). In the proposed algorithm, nondominated sorting and crowding distance strategies are used to enhance elitism and diversity. According to the results, loss of load probability of the system is around 0.5%. In [43], Genetic Algorithm [10] and Differential Evolution [11] are hybridized within multi-objective evolutionary algorithm based on decomposition (MOEA/D) framework and the effectiveness of hybridization is demonstrated on multi-objective unit commitment (UC) problem. In research literature, there have been many considerably successful algorithms for the solution of multi-objective optimization problems. However, it is not possible for any algorithms to be able to successfully solve all the optimization problems given their varying characters. The No Free Lunch Theorem [33, 46] is an algorithm that has gained extremely successful results in some kinds of problems, although it can fail in other kinds of problems. For this reason, a great number of scientists have conducted new studies in the multi-objective optimization field and have been proposing new algorithms in research literature. The proposed MOVS algorithm has been compared with 4 well known and different multi-objective algorithms in the field of multi-objective optimization. These are as follows:

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NSGAII [13]: This is one of the most well known multi-objective algorithm in literature. The members in the population (total number N) are separated into fronts with the nondominated-sorting strategy. The most qualified members are collected on the first front and named Pareto front (PF). Offspring solutions are created by using crossover and mutation th th operators. To create i offspring, SBX crossover operator [12] is applied between the selected parent solution and i member of the population. Parent solutions are selected from the population with the binary tournament selection. For each member of the main population is created a new offspring solution (totally N offsprings). After each iteration, main population and offsprings are combined (N+N = 2N). From this combined population (2N), N solutions are selected by using the fast-nondominated-sorting and crowding-distance calculation approaches, and selected ones form the next generation.

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MOCell [36] : This is a cellular model of Genetic Algorithms (cGAs). In cGAs, a concept called (small) neighborhood is used intensively. This concept means that an individual can only cooperate with its adjacent neighbors in a breeding loop. The overlapping (small) neighborhoods of cGAs ensure the search space is explored, while genetic operations provide exploitation inside each neighborhood.

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IBEA [52]: This is a performance, metric-based, evolutionary algorithm. It calculates the quality of the individuals in the population by using binary performance metrics and, by considering these calculations, carries out the selection process. It does not need any other mechanism to protect the diversity of the solutions on the Pareto front.

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MOEA/D [49]: This is a decomposition-based evolutionary algorithm. A MOOP is decomposed into N scalar suboptimization problems and MOEA/D minimizes all these objective functions simultaneously in a single run. The population consists of the best solution found so far for each sub problem. Only the current solutions to its neighboring sub problems are used for optimizing a sub problem in this algorithm.

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The performance of the algorithms is tested with a total of 36 different benchmark functions that consist of 4 different problem families and 8 different classical functions. These are: (1) ZDT problem family [51] (ZDT1 - ZDT4, ZDT6), (2) WFG problem family [25] (WFG1 - WFG7, WFG9), (3) DTLZ problem family [14] (DTLZ1, DTLZ2, DTLZ4 – DTLZ7), (4) LZ09 problem family [30] (LZ09_F1 – LZ09_F9) and OKA1 – OKA2 [37], Viennet1 – Viennet2 [45], Schaffer [40], Fonseca [20], Kursawe [29] and Poloni functions. 1.2. The Main Contribution of the Study In this study, the single-objective Vortex Search (VS) algorithm [16], which was proposed by Doğan and Ölmez – having gained inspiration from the vortex flow in liquids – was modified, and a Multi-Objective Vortex Search (MOVS) algorithm has been proposed. To the best of our knowledge, this is the first study that modifies VS algorithm to solve multi-objective problems. In order to adapt the VS algorithm to solve multi-objective optimization problems, the adaptations used with VS are listed below: 

The fast-nondominated-sorting and crowding-distance calculation approach proposed in NSGAII [13] has been added.

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The center solution that will be searched has been determined using binary tournament selection [13]. Detailed information about binary tournament selection is explained in section 2.2.2. The crossover operation used in [6] has been added to improve the convergence ability towards the Pareto front. In the inverse incomplete gamma function, the value of parameter “a” has been randomly produced between (0,1) in order to provide a better spread on the Pareto front.

2. The Main Study In this section, the single-objective Vortex Search algorithm and the basic steps of the proposed Multi-Objective Vortex Search algorithm are described.

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2.1 Vortex Search (VS) algorithm The Vortex Search algorithm, as proposed by Doğan and Ölmez [16], is used in the optimization of single-objective continuous numerical functions – having been inspired by the vortex shapes. The VS algorithm searches in the boundaries of a radius adaptively determined around the center of the vortex (gbest solution) to form a good balance between a global and local search. In the first iterations, a global search is used while the radius has an initially big value, whilst in the following iterations the radius is gradually restricted to increase the effect of the local search. Thus, the search around the global best solution is intensified. At first, the center of the search µ0, is determined as the middle of the lower and upper boundary values of decision variables. (4)

Here, the upperlimit and lowerlimit are d × 1 vectors which refer to the lower and upper limits of the decision variables in the d – dimensional search space. The candidate solutions C0(s) ={s1, s2, …, sk} are randomly formed around the initial center µ0 by using Gaussian distribution. Here k = 1,2,…,n states the solutions and n refers to the total number of solutions. The general formula for the multivariate Gaussian distribution is given in Eq. (5).

Here, d represents the dimension,

2

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is the d × 1 vector of a random variable, µ is the d × 1 vector of the sample mean

is the covariance matrix. The

value can be calculated by using equal variance and zero covariance as in Eq.

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(center), and (6).

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(6)

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In Eq. (6), σ refers to the variance of the distribution and represents the d × d unit matrix. The initial standard deviation (σ0) of the distribution is selected to be the initial radius (r0) (Eq. (7)). Thus, in the beginning, the whole search space is covered. (7)

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All the candidate solutions produced in the initial (t = 0) and later iterations must be within the boundaries of the search space. For this purpose, the solutions exceeding the boundaries are leveled down to the boundary values as in Eq. (8).

{

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, Here, k = 1,2,…,n and i = 1,2,…,d and rand are random numbers spread uniformly. In the first iteration, the best (s’ C0(s)) candidate solution is selected to specify the center point µ1 and it is moved from the circle center µ0 position to µ1. Around the newly selected center (µ1), the candidate solutions for the first iteration (C1(s)) are produced in the boundary of a circle whose starting radius (r1) is less than r0. Among the C1(s) solutions, the best one is selected and compared with the best one up to that point. The better solution is selected as the center point of the 2nd iteration (µ2). This process is repeated until the termination criterion is reached. In the VS algorithm, the effect of the

ACCEPTED MANUSCRIPT local search is increased by restricting the search radius in every successive iteration. Therefore, restricting the search radius forms a similar vortex shape. In the VS algorithm the restriction process of the radius is very important in terms of the success of the algorithm. This process is applied to conduct an initial global search in the first iterations and to conduct a local search in later ones. To enable this, the value of the radius is properly restricted as the search process proceeds. In the VS algorithm, in order to reduce the radius, the inverse of the incomplete gamma function is used in every successive iteration. The inverse incomplete gamma function computes this function with respect to the integration limit (Eq. (9)).

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Here, a > 0 is the shape parameter and x 0 is the random variable. So parameter “a” determines the resolution of the search of the inverse incomplete gamma function for the solution space and is calculated as in Eq. (10): (10)

Here, to be able to search the solution space completely in the first iteration, it is chosen to be a0 = 1 and t refers to the number of the iteration. MaxItr shows the maximum iteration number. As the iterations advance, the value of at approaches 0. At which radius width around the center point the search will be made is calculated by Eq. (11).

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( )

Here, t expresses the number of the iteration at that moment,

(11)

refers to the standard deviation of the distribution (Eq.

(7)), gammaincinv states the inverse incomplete gamma function. In the VS algorithm, the value is used as 0.1. The pseudo code for the VS algorithm is given in Fig. 1.

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[please insert Fig. 1. about here]

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2.2. Multi-Objective Vortex Search (MOVS) algorithm The MOVS algorithm is a novel meta-heuristic multi-objective algorithm using the basic strategies of NSGAII [13]. In the MOVS algorithm, there is a main population consisting of N solutions. First of all, the proposed fast-nondominated-sorting approach in NSGAII is applied to the solutions in the main population, and the solutions are separated into fronts according to their qualities. Secondly, the crowding-distance is applied, and the distance from each other of the solutions that are on the same front is calculated. A parent solution is selected from the main population by the binary tournament to form each offspring solution. The selected parent solution refers to the center point in the Vortex Search algorithm. Around this point, for each decision variable, a new solution is formed within the limits of radii determined by using the Eq. (11). The solution created is crossed with the i. solution cyclically selected from the main population, and so the offspring solution (Ctsi) is formed (i=1,…,N; t=0,…,MaxItr). If there are decision variables of the offspring solution that exceed the boundary values, they are restricted to the boundary values and the th objective functions are calculated. With the help of this method, after forming the N offspring solution, the solutions of the main population and the offspring population are combined (N+N=2N). In the combined population, the solutions of number N are chosen from the 2N individuals, using the fast-nondominatedsorting and crowding-distance calculation method. The selected N solutions constitute the main population for the next iteration. This process is continued until the termination criterion is met. When the criterion is met, the first front that comprises the nondominated solutions gives the Pareto front (PF). In order to improve the MOVS algorithm the adaptations made to the VS algorithm are explained below. 2.2.1. The NSGAII-based Strategies NSGAII is one of the most well-known multi-objective algorithms in literature. NSGAII’s concepts, called as fast-nondominated-sorting and crowding-distance calculation, are still very popular and in many recent works such as [3, 26, 33, 38, 47], at least one of the concepts are still used. So, non-dominated-sorting and crowding distance concepts were used in the MOVS algorithm and successful results were obtained. The strategies of the fast-nondominated-sorting approach and the crowding-distance calculation are first added to the MOVS algorithm. For detailed information about these strategies, see [13]. (i) Fast-nondominated-sorting approach

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This is used to determine the non-domination degrees of the solutions in a population. Each of the N solutions in the population is compared with the remaining (N−1) solutions, in terms of Pareto domination. The nondominated solutions that cannot be dominated by any solutions within the population, create the first nondominated front. To find the second front, the nondominated solutions are found among the remaining ones by overlooking the members of the first front. In the same way, all the N solutions are placed in a front according to their non-domination degree. (ii) Crowding-distance Calculation This is used to calculate the density of the location of the solutions in the population. For each of the solutions, the average distance value is calculated for all the objective functions in accordance with the two solutions located on both the nearest neighborhoods. To calculate the crowding-distance, it is necessary for the solutions to be ranked according to their objective function values. The end solutions (having the smallest and the biggest value) of each objective function are assigned an infinite distance value – since we always need to protect the end solutions. All the central ones are assigned the average distance value, as explained above. The fact that the crowding-distance calculation is small means that this region has been sufficiently searched, and if it is big, it means that this region must undergo further searching. 2.2.2 Tournament Selection Two solutions are randomly taken from the main population. The one on the better front (1.front is the best) is selected as the center point. If both of them are at the same front, the one whose crowding-distance is bigger is selected. If their crowding-distances are the same, one of them is randomly taken and used to determine the center point (Eq. (12)).

In Eq. (12), Fr() refers to the front numbers of the solutions, Cd refers to the crowding-distance of the solutions.

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2.2.3 The arrangement of the parameter “a” As the iterations progress, single-objective meta-heuristic algorithms aim at gathering all the agents around the best solution (center). For this purpose, the VS algorithm iteratively minimizes the parameter “a” that is used in the calculation of the inverse gamma function and so restricts the search radius (Eq. (10)). Multi-objective heuristic algorithms aim at reaching all the agents on the Pareto optimal front and spreading them out on that front. To spread the algorithm on the Pareto front more successfully, the parameter “a” used in the calculation of the inverse gamma function is randomly created between the range (0,1) for each candidate solution. Thus, a strategy that provides much bigger radii and searches the whole search space successfully up to the last iteration is also able to spread the selected agents on the Pareto front. 2.2.4 Crossover Operation Another adaptation, to allow better convergence on the Pareto front for the proposed algorithm, is the addition of the crossover operation used in the MODE-RMO [6] algorithm. Crossover operation is shown in Eq. (13).

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Here, the offSpringCt(Si,j), which is in the input position, refers to the new solution, which is formed within the radii boundaries calculated according to the Eq. (11) around the center solution determined by the binary tournament selection in the Vortex Search algorithm. Ct(Si,j) shows a solution cyclically selected from the main population. The offspring in the output position refers to the offspring solution formed as a result of the crossover of offSpringCt(Si,j) as the input and Ct(Si,j). Furthermore, rand indicates a randomly produced value between (0,1); CR shows the crossover constant and sn refers to a whole number randomly selected from the range ,1, 2,…,D- and provides at least one parameter coming from the offSpringCt(si) solution. 2.2.5 Explanation of the MOVS Algorithm In this section, pseudo code of the MOVS algorithm given in the Fig.2 is explained. Each line of the pseudo code was explained separately and line number was given parenthetically. The initial center μ0 of the search is calculated by using the Eq. (4) (Line 1). The initial radius r0 (or the standard deviation, σ0) is calculated by using the Eq. (7) (Line 2). The first candidate solutions C0(s) are randomly formed around the initial center µ0 by using Gaussian distribution in the range of the boundary values (Line 3). The fast-nondominated-sorting and crowding-distance strategies are applied to C0(s) solutions to evaluate their quality (Line 4). Iteration counter “t” is set at 1 (Line 5) and iteration loop (between Line 6 and Line 18) is set. Another loop (between Line 8 and Line 14) is set to select parent solutions of each offspring separately. Offspring counter “i” is set at 1 (Line 7). Binary tournament selection is made between any two solutions of the population by using Eq. (12) (Line 9). Winner solution becomes the parent of i. offspring solution in t. iteration and generates offSpringCt(si) (Line 10).

ACCEPTED MANUSCRIPT Generated offSpringCt(si) and i.member of the main population are crossovered by using Eq. (13) (Line 11). Boundary control is made to newly generated offSpringCt(si) by using Eq. (8) (Line 12). Counter “i” is increased by one (Line 13). Main population Ct(s) and offspring solutions offSpringCt(s) are combined (N+N = 2N) (Line15). The fast-nondominated-sorting and crowding-distance strategies are applied to combined population (2N) and best N solutions are transferred to the next iteration (Line 16). Counter “t” is increased by one (Line 17). This process repeats until the maximum iteration number (Line 18). When reached the maximum iteration, nondominated solutions of the population give estimated Pareto Front of the MOVS algorithm (Line 19).

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1: Inputs: Calculate the initial center (μ0) 2: Calculate the initial radius (r0) 3: Create first candidate solutions (C0(s)) 4: Apply fast-nondominated-sorting & crowding-distance (N) 5. t = 1; 6: Repeat 7: i = 1; 8: Repeat 9: Binary tournament selection 10: Generate(offSpringCt(si)) 11: offSpringCt(si) = Crossover(offSpringCt(si), Ct(si)); 12: If exceeded, then shift the offSpringCt(si) values into the boundaries (Eq. (8)). 13: i = i + 1; 14: Until i > N (number of Cs solutions) 15: Combine(Ct(s), offSpringCt(s)) (N+N = 2N) 16: Apply fast-nondominated-sorting & crowding-distance (2N) 17: t = t + 1; 18: Until Maximum iteration 19: Output Pareto front solutions Fig. 2. The pseudo code of MOVS algorithm [please insert Fig. 3. about here]

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In order to determine the ideal value of the CR crossover constant, all the CR values starting from 0 to 1 in the range of 0.1 have been run 5 times on the problem set used in the study. The results obtained are analyzed according to a quality metric called hypervolume (HV). Detailed information about HV is given in section 3.2.1. The average HV metric values obtained for all the problems after 5 runs are given in Table 1. In Table 1 the best average values are shown highlighted. When highlighted results of each CR value is numbered, it is shown that the maximum highlighted results belong to a CR value of 0.5 used on 12 different problems. So, it was decided to fix the value of the CR constant at 0.5.

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[please insert Table 1 about here]

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Four different MOVS versions have been run on problems in order to test how the crossover operation and the random production method of parameter “a” between (0, 1) – which are added to the MOVS algorithm – affect the convergence and diversity. In Fig. 4, we give the Pareto fronts of the DTLZ1 problem for 4 different MOVS versions as an illustration. In Fig. 4 (a), the crossover operation has not been used, and the parameter “a” has been originally calculated by Eq. (10). In Fig. 4 (b), the crossover operation has not been used and the parameter “a” has been randomly produced between (0, 1). In Fig. 4 (c), the crossover operation has been used and the parameter “a” has been originally calculated by Eq. (10). In Fig. 4 (d), the crossover operation has been used and the parameter “a” has been randomly produced between (0, 1). Because of the reason given in section 3.3, agent numbers are set at 100, and the maximum function evaluation number (maxFES) is set at 25,000. As mentioned in section 2.2.4, using of crossover operation increases the convergence performance of the MOVS algorithm. Otherwise, convergence performance of the algorithm gets worse. Because first and second versions of the MOVS algorithm have not used the crossover operation, estimated Pareto fronts in Fig. 4(a) and (b), are quite far than true Pareto front (PFt, blue points). In third version of the MOVS algorithm, estimated Pareto front in Fig. 4 (c) has converged to the PFt very well thanks to crossover operation. However, it has a poor diversity performance on PFt because of parameter “a”. As mentioned in 2.2.3, VS algorithm decreases the value of parameter “a” every single iteration, therefore the search radius restricts step by step. As a result of this restriction, new agents are created very close to parent solution. To avoid this density, parameter “a” has been randomly produced between (0, 1) in fourth version of the MOVS algorithm. In Fig. 4 (d), It is clearly shown that the crossover operation and the random production method of the parameter “a” between (0, 1) have made a significant contribution to convergence and diversity performance of the proposed algorithm.

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Crossover no, a original

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Crossover yes, a original

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According to experimental results, similar to DTLZ1, other problems also have generally better Pareto fronts with this version of the algorithm. For this reason, fourth version of the MOVS algorithm has been used in experimental studies.

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Corssover yes, a rand (0,1)

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Fig. 4. The Pareto fronts that MOVS versions have obtained for the DTLZ1 problem

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3. Experimental Results 3.1 Benchmarks In this study, the performance of the algorithms has been tested against a total of 36 different benchmark functions consisting of 4 different problem families and 8 different classical functions. These are (1) ZDT problem family [51] (ZDT1 ZDT4,ZDT6), (2) WFG problem family [25] (WFG1 - WFG7, WFG9), (3) DTLZ problem family [14] (DTLZ1, DTLZ2, DTLZ4 – DTLZ7), (4) LZ09 problem family [30] (LZ09_F1 – LZ09_F9) and OKA1 – OKA2 [37], Viennet1 – Viennet2 [45], Schaffer [40], Fonseca [20] and Kursawe [29] and Poloni functions. All the DTLZs, LZ09_F6, Viennet1 and Viennet2 functions have three objectives, the others have two objectives. This problem set is selected because it is quite well known in research literature. So, many researchers have used the same problems to test their own algorithms [9, 13, 35, 47]. 3.2 The Performance Metrics P* is a set of points scattered uniformly along the true Pareto front (PFt) in the objective space; A is a set of points that estimates the PFt. How successful set A is, is calculated with the performance metrics (indicators) defined below. 3.2.1 Hypervolume (HV) HV [17, 47] calculates the volume, in the objective space, covered by members belonging to set A. Mathematically, for each i A solution, a vi hypercube is formed with W the reference point and where the i solution is the diagonal of the hypercube. The reference point can be found by forming a vector from the worst values of the objective functions. The combination of all the hypercubes found and their HV values are calculated as in Eq. (14).

HV

volume(⋃

)

(14)

ACCEPTED MANUSCRIPT HV provides us with information about both the convergence and the diversity performance of the set A and larger HV values indicate a better algorithm. 3.2.2 SPREAD SPREAD [17] presents the extent of spread of the computed set A and is symbolized as . In [17], two versions of spread indicator are described. The first version of the SPREAD works only for two-objective problems. The second one (Generalized SPREAD) can be used for all multi-objective problems. Here, the second version of the SPREAD indicator is given as follows: ∑



(15)

∑ *

min ∑

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Here, e1,… em refers to the extreme points in set P ; d(ei , A) refers to the minimum Euclid distance between ei and the set A; refers the number of the objective function and (16)

3.2.3 EPSILON When a computed front for a problem is A, epsilon is the indicator that measures the smallest distance that is required to

inf Here, ⃗ number. ⃗ ⃗ only and only

{ ⃗



, ⃗



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convert every solution in A so that it is able to dominate the optimal Pareto front of the problem.

, m refers the number of the objective function and .

(17) expresses a small positive



(18)

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3.2.4 Inverted Generational Distance (IGD) * IGD [41, 48] is used to measure the average from P to A.

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Here, d(v,A) is the minimum Euclid distance between v and the set A. If has a magnitude that represents the Pareto front well, both the diversity and the convergence of set A can be measured by using . . As the metrics become small, this states that the set A has reached the PFt.

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3.3 Parameter Settings In existing literature, the MOVS algorithm has been compared with the well-known NSGAII, MOCell, IBEA and MOEA/D algorithms by using HV, SPREAD, EPSILON and IGD metrics. In this study, the other algorithms – except the MOVS algorithm – have been obtained from the jMetal 4.5 software package [17, 18], which has been improved by Durillo and Nebro. The MOVS algorithm has been coded in Java by the developers to be compatible with the jMetal 4.5 software package. In jMetal4.5, agent numbers of most of the algorithms are set to 100, and the maximum function evaluation number (maxFES) is set to 25,000. Similarly to jMetal4.5, we set the agent numbers of all the algorithms as 100, and the maxFES as 25,000. Thus the maximum cycling number of all the algorithms is 25,000/100 = 250. The other parameters belonging to the algorithms have been used with the default values used in the jMetal 4.5 software package [1]. 3.4 Experimental Results for Benchmark Functions For the each of the benchmark problems, all the algorithms have been run 30 times, and afterwards the HV, SPREAD, EPSILON and IGD metric values obtained are given in Tables 2 to 5. In order to be able to increase the readability in the tables, the best and the second-best metric values have been respectively shown as dark grey and light grey colors. For a more successful solution the HV metric is a bigger value, while the other metric values are smaller. [please insert Table 2 about here] In Table 2, when we look at the HV metric values, it is clear that the MOVS algorithm has become the most competitive algorithm, as it has the best or the second-best value in 20 (11+9) problems out of 36. MOCell in 15 (8+7) problems, NSGAII in 15 (7+8), MOEA/D in 14 (6+8) and IBEA in 8 (4+4) obtained the best or the second-best values.

ACCEPTED MANUSCRIPT [please insert Table 3 about here] When examining the SPREAD metric values in Table 3, it is seen that MOCell is clearly the most competitive algorithm by obtaining 30 (22+8) of the best or the second-best values. While the proposed MOVS algorithm has the best or the secondbest values in 18 (7+11) problems, NSGAII in 11 (2+9), MOEA/D in 7 (3+4) and IBEA in 6 (2+4) have the best or the secondbest values. Thus MOEA/D and IBEA have displayed a much worse performance than MOCell and MOVS using the SPREAD metric. [please insert Table 4 about here]

[please insert Table 5 about here]

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When examining the Epsilon metric values in Table 4, the MOVS algorithm is seen to be the most competitive algorithm with the best or the second-best value in 22 (9+13) problems. NSGAII is the second best algorithm with the best or the second-best values in 17 (9+8) problems. While MOCell has the best or the second-best values in 15 (11+4) problems, MOEA/D in 12 (5+7) problems has the best or the second-best values. Finally, IBEA has the best or the second-best values in 6 (2+4) problems, so its performance has become weaker than the other algorithms.

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When examining the IGD metric values in Table 5, it shows that the MOVS algorithm has become the most competitive algorithm with the best or the second-best values in 21 (9+12) problems. NSGA2 was the closest to the MOVS algorithm with the best and second-best values in 19 (5+14) problems. While MOCell has the best or the second-best values in 16 (13+3) problems, MOEA/D gained the best or the second-best values in 13 (9+4) problems. Finally, IBEA with the best or the second-best values in 3 (0+3) problems displayed a weak performance using the IGD metric. [please insert Table 6 about here]

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Table 6 lists the average rankings calculated by applying the Friedman test [15] [21] to all the algorithms. Results of the Friedman test show that MOVS is the most successful algorithm. The proposed MOVS algorithm obtained the best results with the HV, EPSILON and IGD metrics, and MOCell was the most successful algorithm with the SPREAD metric. NSGAII obtained the second best results with the HV, EPSILON and IGD metrics, and it is the second most successful algorithm.

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In recent years, experimental analysis has had growing interest in the field of EAs [22]. By using experimental analysis, researchers try to prove that their proposal is different from the ones they compared their proposal to. Similar to [5], a non-parametric statistical test, called Wilcoxon’s rank sum test for independent samples, was conducted at the 5% significance level in order to decide whether the results obtained with the MOVS algorithm differed from the final results of the rest of the competitors in a statistically significant way. P-values obtained through the rank sum test over indicators of all the problems are presented in Tables 7 to 8. If the p-values are less than 0.05 (5% significance level), signed with “+”, there is strong evidence that the results of the MOVS algorithm is statistically significant and has not occurred by chance.

[please insert Table 7 about here]

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When examining the p-value obtained for HV in Table 7, MOVS is statistically different from NSGAII, MOCell, IBEA and MOEA/D in 26, 29, 32 and 29 problems, respectively, out of 36. When examining the p-value of SPREAD, MOVS is different from NSGAII, MOCell, IBEA and MOEA/D in 22, 29, 29 and 34 problems, respectively, out of 36.

[please insert Table 8 about here]

When examining the p-values obtained for EPSILON in Table 8, MOVS is statistically different from NSGAII, MOCell, IBEA and MOEA/D in 17, 29, 28 and 26 problems, respectively, out of 36. When examining the p-value for IGD, MOVS is different from NSGAII, MOCell, IBEA and MOEA/D in 23, 34, 34 and 28 problems, respectively, out of 36. The results in Tables 7 to 8 show that the MOVS algorithm is significantly different from the algorithms it has been compared to. In Figs. 5 to 7, in an attempt to observe how much the algorithms converge on the Pareto fronts in order to reach PFt, it has been drawn in a light blue color. Some Pareto fronts have been selected among the problems in which the proposed MOVS algorithm is the best, the second best and unsuccessful, in terms of HV metric values.

ACCEPTED MANUSCRIPT In Figs. 8 to 9, box plot drawings that were formed after 30 runs for the HV, SPREAD EPSILON and IGD metrics of the problems have also been shown. When both the Pareto fronts and the box plot drawings are examined, it is clearly seen that the MOVS algorithm has achieved better results in many situations than the other algorithms.

[please insert Fig. 5. about here] [please insert Fig. 6. about here] [please insert Fig. 7. about here] [please insert Fig. 8. about here] [please insert Fig. 9. about here]

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Table 9 lists, for each multi-objective problem, the number of evaluations required by all the algorithms to reach 98% the HV value of the PFt. The “-” sign in some cells indicate that the given algorithm has been not able to reach such HV value during 25,000 evaluations. For each problem, minimum and second minimum evaluation numbers have been respectively shown as dark grey and light grey colors. [please insert Table 9 about here]

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When examining the evaluation values, the proposed MOVS algorithm has been able to reach 98% the HV value of the PF t in 14 problems as best or second best. MOCell, IBEA, MOEA/D and NSGAII have been able to reach 98% the HV value of the PFt respectively in 11, 5, 5 and 4 problems. This results shows that the MOVS algorithm has remarkable convergence performance on the test problems. Additionally, in Figs. 10 for stated problems, evaluation of the HV values were given iteratively. This figure shows that the MOVS algorithm has successful convergence ability on stated problems.

[please insert Fig. 10. about here]

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4. Conclusion This study proposed a novel multi-objective heuristic algorithm called MOVS, which is based on a single-objective Vortex Search (VS) algorithm. In order to create an improved MOVS algorithm, the VS algorithm has been enhanced with fastnondominated-sorting and crowding-distance calculation approaches – in order to determine the nondominance degrees of the solutions and the density of their locations. Moreover, to enhance the convergence ability of the solutions on the Pareto front, the crossover operation has also been added to the MOVS algorithm. Finally, the parameter “a” in the inverse incomplete gamma function has been randomly produced between 0 and 1 for the solutions to be able to be successfully spread on the Pareto front. The proposed algorithm was tested on 36 different benchmark problems, along with NSGAII, MOCell, IBEA, MOEA/D algorithms, by calculating HV, SPREAD, EPSİLON and IGD performance metrics. The test results and statistical analyses show that the proposed MOVS algorithm showed a more outstanding performance than any other comparable algorithms in numerous test problems, and thus, it is a competitive algorithm. Additionally, convergence speeds of the algorithms were analyzed and MOVS algorithm had outstanding convergence ability on stated problems. Although the MOVS algorithm has been remarkably successful, the algorithm could be improved for the diversity (SPREAD) indicator. In future work, the proposed MOVS algorithm can be tested on new benchmark sets, including constrained, dynamic, multimodal and discrete multi-objective problems. Additionally, to improve performance, the MOVS algorithm can hybridize with other successful algorithms or be modified with other accomplished techniques. Finally, the MOVS algorithm can be applied to real-world problems such as circular antenna design, optimal power flow, aerodynamic design, industrial neural-network design and others.

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Inputs: The initial center μ0 is calculated by using the Eq. (4). The initial radius r0 (or the standart deviation, σ0) is calculated by using the Eq. (7). The fitness of the best solution ever f(sbest) = infinite t=0; Repeat /* The candidate solutions are produced with the standart deviation of the radius rt by using Gaussian distribution around the center μt */ Generate(Ct(s)); If the Ct(s) boundary values exceed, there is a restriction to the boundary values as in the Eq.(8). /*The best solution in Ct(s) is selected to update the current center μt */ s’ = Select(Ct(s)); if f(s’) < f(sbest) sbest = s’ f(sbest) = f(s’) else Keep the best solution ever in sbest end /* The best solution ever is always assigned to the center */ μt+1 = sbest /*Reduce the standart deviation(radius) for the next iteration*/ rt+1 = Reduce(rt) t=t+1 Until the maximum iteration number is reached Output: The best solution ever sbest Fig. 1. The pseudo code of VS algorithm [13]

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Fig. 2. The pseudo code of MOVS algorithm

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1: Inputs: Calculate the initial center (μ0) 2: Calculate the initial radius (r0) 3: Create first candidate solutions (C0(s)) 4: Apply fast-nondominated-sorting & crowding-distance (N) 5. t = 1; 6: Repeat 7: i = 1; 8: Repeat 9: Binary tournament selection 10: Generate(offSpringCt(si)) 11: offSpringCt(si) = Crossover(offSpringCt(si), Ct(si)); 12: If exceeded, then shift the offSpringCt(si) values into the boundaries (Eq. (8)). 13: i = i + 1; 14: Until i > N (number of Cs solutions) 15: Combine(Ct(s), offSpringCt(s)) (N+N = 2N) 16: Apply fast-nondominated-sorting & crowding-distance (2N) 17: t = t + 1; 18: Until Maximum iteration 19: Output Pareto front solutions

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Fig. 3. The flow diagram of MOVS algorithm

Crossover no, a original

g)

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h)

Corssover yes, a rand (0,1)

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Fig. 4. The Pareto fronts that MOVS versions have obtained for the DTLZ1 problem

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Fig. 5. The Pareto fronts and the box plots produced by algorihtms for the DTLZ1, DTLZ2, DTLZ4, DTLZ6, DTLZ7 problems.

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Fig. 6. The Pareto fronts produced by algorihtms for the LZ09F2, LZ09F7, LZ09F9, OKA1 and Viennet3 problems.

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Fig. 7. The Pareto fronts produced by algorihtms for the Schaffer, WFG1, WFG2, ZDT3 and ZDT4 problems.

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Fig. 8. The box plots produced by algorihtms for stated problems(I).

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Fig. 9. The box plots produced by algorihtms for stated problems(II).

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Fig. 10. Convergence speeds of the algorihtms for stated problems.

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Table 1. The average HV metric values for different CR values from 0 to 1. 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

6.60E-01

6.60E-01

6.60E-01

6.60E-01

6.59E-01

6.59E-01

6.59E-01

6.57E-01

6.54E-01

4.70E-01

3.24E-01

3.26E-01

3.27E-01

3.27E-01

3.26E-01

3.26E-01

3.26E-01

3.25E-01

3.24E-01

3.19E-01

1.08E-01

5.14E-01

5.15E-01

5.15E-01

5.15E-01

5.15E-01

5.15E-01

5.14E-01

5.14E-01

5.13E-01

5.11E-01

3.42E-01

6.15E-01

5.66E-01

5.18E-01

3.59E-01

2.62E-01

6.28E-02

2.30E-02

7.74E-03

0.00E+00

0.00E+00

0.00E+00

3.96E-01

3.97E-01

3.97E-01

3.98E-01

3.98E-01

3.99E-01

3.97E-01

3.94E-01

3.84E-01

3.52E-01

2.78E-01

4.51E-01

5.07E-01

5.09E-01

4.74E-01

5.11E-01

4.91E-01

4.73E-01

4.49E-01

4.10E-01

3.39E-01

2.24E-01

5.56E-01

5.57E-01

5.57E-01

5.57E-01

5.57E-01

5.57E-01

5.57E-01

5.57E-01

5.57E-01

5.55E-01

5.52E-01

4.89E-01

4.91E-01

4.92E-01

4.92E-01

4.93E-01

4.93E-01

4.92E-01

4.92E-01

4.91E-01

4.89E-01

4.79E-01

2.09E-01

2.09E-01

2.10E-01

2.09E-01

2.10E-01

2.10E-01

2.09E-01

2.09E-01

2.08E-01

2.08E-01

1.98E-01

2.20E-01

2.21E-01

2.19E-01

2.21E-01

2.18E-01

2.17E-01

2.18E-01

2.17E-01

2.17E-01

2.17E-01

2.16E-01

2.03E-01

2.00E-01

2.03E-01

2.07E-01

2.04E-01

2.05E-01

2.07E-01

2.08E-01

2.07E-01

2.05E-01

2.00E-01

2.08E-01

2.08E-01

2.09E-01

2.09E-01

2.09E-01

2.09E-01

2.09E-01

2.08E-01

2.08E-01

2.06E-01

1.99E-01

2.22E-01

2.22E-01

2.24E-01

2.22E-01

2.24E-01

2.23E-01

2.23E-01

2.22E-01

2.23E-01

2.21E-01

2.19E-01

7.61E-01

7.64E-01

7.63E-01

7.39E-01

4.86E-01

6.10E-01

5.98E-02

1.07E-02

0.00E+00

1.85E-03

0.00E+00

3.85E-01

3.84E-01

3.83E-01

3.82E-01

3.75E-01

3.75E-01

3.68E-01

3.61E-01

3.57E-01

3.47E-01

2.52E-01

3.88E-01

3.87E-01

3.88E-01

3.85E-01

3.85E-01

3.85E-01

3.80E-01

3.82E-01

3.82E-01

3.79E-01

3.40E-01

9.24E-02

9.27E-02

9.29E-02

9.29E-02

9.28E-02

9.30E-02

9.26E-02

9.23E-02

9.22E-02

9.10E-02

8.52E-02

9.31E-02

9.33E-02

9.35E-02

9.37E-02

9.39E-02

9.40E-02

9.30E-02

9.12E-02

7.83E-02

5.21E-02

0.00E+00

2.86E-01

2.90E-01

2.87E-01

2.84E-01

2.78E-01

2.71E-01

2.66E-01

2.56E-01

2.43E-01

2.28E-01

6.91E-02

6.48E-01

6.51E-01

6.52E-01

6.51E-01

6.51E-01

6.50E-01

6.49E-01

6.49E-01

6.49E-01

6.49E-01

6.48E-01

5.25E-01

5.23E-01

5.32E-01

5.14E-01

5.26E-01

5.19E-01

5.13E-01

5.25E-01

5.25E-01

5.19E-01

3.75E-01

5.91E-01

5.91E-01

5.90E-01

5.86E-01

5.79E-01

5.73E-01

5.79E-01

5.84E-01

5.81E-01

5.78E-01

5.25E-01

5.94E-01

6.05E-01

5.97E-01

5.95E-01

5.95E-01

5.92E-01

5.94E-01

5.96E-01

5.98E-01

6.02E-01

5.31E-01

6.02E-01

6.04E-01

6.02E-01

6.01E-01

5.98E-01

5.96E-01

5.98E-01

5.96E-01

6.01E-01

6.01E-01

5.34E-01

2.85E-01

2.64E-01

2.70E-01

2.55E-01

2.34E-01

2.54E-01

2.03E-01

1.96E-01

1.90E-01

9.93E-02

1.63E-02

4.58E-01

4.42E-01

4.31E-01

4.71E-01

3.81E-01

3.93E-01

3.54E-01

2.56E-01

2.43E-01

2.47E-01

2.23E-02

3.90E-01

4.29E-01

4.22E-01

4.03E-01

3.14E-01

2.96E-01

3.07E-01

3.06E-01

2.94E-01

3.00E-01

1.48E-01

2.17E-01

2.07E-01

2.08E-01

2.03E-01

2.03E-01

1.96E-01

1.94E-01

1.95E-01

1.94E-01

1.96E-01

9.11E-02

5.67E-01

5.78E-01

5.80E-01

5.96E-01

6.05E-01

6.09E-01

6.08E-01

6.11E-01

6.11E-01

6.14E-01

6.19E-01

1.31E-01

9.88E-02

9.95e-02

1.02E-01

1.16E-01

1.42E-01

1.12E-01

1.33E-01

1.20E-01

1.49E-01

1.42E-01

9.18E-01

9.20E-01

9.17E-01

9.19E-01

9.20E-01

9.20E-01

9.20E-01

9.19E-01

9.20E-01

9.19E-01

9.20E-01

8.31E-01

8.31E-01

8.31E-01

8.31E-01

8.32E-01

8.32E-01

8.31E-01

8.33E-01

8.32E-01

8.32E-01

8.32E-01

9.12E-01

9.12E-01

9.12E-01

9.12E-01

9.12E-01

9.13E-01

9.12E-01

9.13E-01

9.13E-01

9.13E-01

9.13E-01

8.26E-01

8.27E-01

8.27E-01

8.27E-01

8.27E-01

8.27E-01

8.28E-01

8.28E-01

8.28E-01

8.28E-01

8.29E-01

3.06E-01

3.07E-01

3.07E-01

3.08E-01

3.08E-01

3.08E-01

3.08E-01

3.08E-01

3.08E-01

3.08E-01

3.08E-01

3.99E-01

3.99E-01

3.99E-01

3.99E-01

3.99E-01

4.00E-01

3.99E-01

3.99E-01

3.99E-01

3.99E-01

3.97E-01

AC

AN US

M

ED

PT

CR IP T

6.57E-01

CE

CR ZDT1 ZDT2 ZDT3 ZDT4 ZDT6 WFG1 WFG2 WFG3 WFG4 WFG5 WFG6 WFG7 WFG9 DTLZ1 DTLZ2 DTLZ4 DTLZ5 DTLZ6 DTLZ7 LZ09F1 LZ09F2 LZ09F3 LZ09F4 LZ09F5 LZ09F6 LZ09F7 LZ09F8 LZ09F9 OKA1 OKA2 Viennet2 Viennet3 Poloni Schaffer Fonseca Kursawe

ACCEPTED MANUSCRIPT Table 2. The mean and the standard deviation values of the HV metric for all the algorithms.

8.4e-04 3.4e-03 1.5e-03 9.6e-02 2.4e-03 6.8e-04 4.1e-04 1.7e-03 1.1e-02 2.7e-04 1.5e-03 1.8e-01 5.9e-03 4.6e-03 2.3e-04 0.0e+00 4.4e-03 8.7e-04 4.4e-02 6.2e-03 4.7e-03 8.6e-03 4.1e-02 5.0e-02 4.2e-02 4.3e-02 5.3e-03 2.5e-02 1.2e-03 7.3e-04 6.9e-05

CE AC

Std 2.9e-04 2.9e-04 1.4e-03 3.0e-03 7.8e-04 9.4e-02 1.2e-02 5.0e-04 2.6e-04 3.5e-05 2.6e-02 9.7e-05 3.3e-03 3.0e-01 4.4e-03 1.8e-01 3.2e-05 0.0e+00 2.2e-02 5.0e-03

2.3e-01 3.9e-04 2.1e-04

Mean 6.62e-01 3.27e-01 5.10e-01 1.90e-01 3.96e-01 4.88e-01 5.50e-01 4.94e-01 2.09e-01 2.17e-01 1.95e-01 2.08e-01 2.24e-01 1.61e-01 4.12e-01 2.85e-01 9.17e-02 6.97e-02 2.33e-01 6.54e-01 4.91e-01 5.90e-01 6.01e-01 6.07e-01 5.81e-02 3.68e-01 3.96e-01 1.52e-01 5.65e-01 6.44e-02 9.08e-01 8.27e-01 9.11e-01 5.06e-01 3.11e-01 3.95e-01

MOEA/D Std 6.4e-05 1.5e-04 2.7e-04 6.7e-02 6.3e-04 1.1e-01 7.3e-04 4.1e-04 1.8e-04 1.4e-04 7.1e-03 1.1e-04 1.1e-03 6.8e-02 8.4e-04 1.3e-01 1.5e-04 1.4e-02 4.8e-02 1.5e-03

Mean 6.41e-01 3.10e-01 4.49e-01 3.60e-01 4.01e-01 3.62e-01 5.55e-01 4.93e-01 2.04e-01 2.18e-01 2.09e-01 2.09e-01 2.22e-01 2.36e-01 7.55e-02 8.88e-02 1.43e-02 1.45e-02 7.19e-02 6.61e-01 5.22e-01 5.96e-01 6.19e-01 6.13e-01 7.38e-02 4.38e-01 3.38e-01 1.85e-01 6.01e-01 6.26e-02 7.26e-01 5.66e-01 9.11e-01 4.20e-01 3.12e-01 4.00e-01

Std 8.1e-03 6.6e-03 2.3e-02 1.7e-01 2.9e-03 6.8e-02 3.0e-04 1.7e-04 1.3e-03 6.6e-05 9.0e-05

9.7e-02 2.2e-02 3.5e-03 9.7e-03 4.1e-02 6.2e-02 9.0e-02 4.7e-02 1.2e-02 6.0e-02 7.6e-04 3.7e-04 1.7e-05 2.5e-01 9.6e-05 9.9e-05

4.7e-02 1.7e-02 5.6e-03 1.6e-02 7.5e-02 5.8e-02 4.4e-02 6.7e-02 1.6e-02 5.3e-02 7.2e-03 3.7e-03 7.7e-04 2.4e-01 1.2e-04 7.3e-04

MOVS Mean 6.60e-01 3.26e-01 5.15e-01 1.10e-01 3.99e-01 4.90e-01 5.57e-01 4.93e-01 2.10e-01 2.20e-01 2.07e-01 2.09e-01 2.23e-01 6.37e-01 3.75e-01 3.82e-01 9.28e-02 9.40e-02 2.70e-01 6.50e-01 5.27e-01 5.78e-01 5.93e-01 5.97e-01 2.27e-01 3.78e-01 3.32e-01 2.05e-01 6.04e-01 1.23e-01 9.20e-01 8.32e-01 9.13e-01 8.27e-01 3.08e-01 4.00e-01

Std 2.2e-04 3.3e-04 2.6e-04 1.5e-01 4.2e-04 5.7e-02 1.3e-04 3.5e-04 2.1e-04 4.0e-03 2.1e-03

CR IP T

4.0e-04

Mean 6.61e-01 3.28e-01 5.14e-01 6.57e-01 3.95e-01 3.66e-01 5.51e-01 4.94e-01 2.10e-01 2.19e-01 1.76e-01 2.10e-01 2.24e-01 3.13e-01 3.76e-01 2.90e-01 9.40e-02 0.00e+00 2.57e-01 6.50e-01 4.38e-01 5.69e-01 5.93e-01 5.94e-01 1.47e-01 3.32e-01 3.16e-01 1.67e-01 5.78e-01 5.62e-02 9.22e-01 8.35e-01 9.13e-01 5.13e-01 3.12e-01 4.01e-01

IBEA

M

Std 3.7e-04

ED

Mean 6.59e-01 3.26e-01 5.14e-01 6.49e-01 3.88e-01 5.28e-01 5.54e-01 4.92e-01 2.09e-01 2.18e-01 1.96e-01 2.09e-01 2.22e-01 6.53e-01 3.74e-01 3.75e-01 9.27e-02 0.00e+0 2.79e-01 6.52e-01 4.98e-01 5.96e-01 6.10e-01 6.09e-01 1.66e-01 4.66e-01 4.14e-01 1.64e-01 5.92e-01 1.42e-01 9.20e-01 8.32e-01 9.13e-01 5.16e-01 3.08e-01 4.00e-01

PT

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6 WFG1 WFG2 WFG3 WFG4 WFG5 WFG6 WFG7 WFG9 DTLZ1 DTLZ2 DTLZ4 DTLZ5 DTLZ6 DTLZ7 LZ09F1 LZ09F2 LZ09F3 LZ09F4 LZ09F5 LZ09F6 LZ09F7 LZ09F8 LZ09F9 OKA1 OKA2 Viennet2 Viennet3 Poloni Schaffer Fonseca Kursawe

MOCell

1.1e-04 6.0e-04 1.2e-01 2.6e-06 2.4e-02 1.7e-07

0.0e+00 5.0e-02 1.4e-04

AN US

NSGAII

4.4e-02 2.9e-02 5.9e-03 1.5e-02 5.5e-03 1.6e-01 6.7e-02 5.6e-02 7.8e-03 2.6e-02 4.0e-04 3.0e-05 1.2e-03 2.2e-01 9.8e-06 1.1e-04

2.1e-04 1.0e-03 1.6e-01 3.5e-03 4.3e-03 2.1e-04 1.6e-04 5.1e-03 1.1e-03 1.5e-02 6.4e-03 4.5e-03 3.8e-03 3.0e-02 6.0e-02 4.5e-02 1.4e-02 6.5e-03 2.9e-02 1.1e-03 6.2e-04 1.1e-04 3.4e-04 4.1e-04 2.4e-04

ACCEPTED MANUSCRIPT Table 3. The mean and the standard deviation values of the SPREAD metric for all the algorithms.

CE AC

Std 1.9e-02 1.2e-02 4.7e-03 3.3e-02 4.6e-02 9.9e-02 3.1e-03 8.1e-03 1.5e-02 1.5e-02 3.0e-02 1.6e-02 1.4e-02 2.3e-01 4.4e-02 1.5e-01 4.2e-02 3.5e-02 6.5e-02 1.9e-01 2.1e-01 1.1e-01 1.2e-01 8.9e-02 7.6e-02 1.7e-01 1.8e-01 1.7e-01 8.1e-02 1.8e-01 7.0e-02 7.1e-02 1.8e-02 2.4e-01 1.0e-02 4.2e-03

Mean 2.91e-01 3.36e-01 1.19e+00 1.10e+00 4.02e-01 8.94e-01 1.26e+00 2.55e-01 5.05e-01 5.78e-01 5.21e-01 5.18e-01 5.18e-01 1.63e+00 5.55e-01 6.61e-01 6.73e-01 9.78e-01 8.16e-01 7.76e-01 1.54e+00 1.15e+00 1.03e+00 1.09e+00 1.67e+00 1.06e+00 1.20e+00 1.70e+00 1.61e+00 1.47e+00 9.61e-01 7.83e-01 1.17e+00 6.50e-01 4.08e-01 8.66e-01

Std 1.9e-02 2.0e-02 7.2e-02 5.4e-02 4.0e-02 7.4e-02 8.7e-02 2.3e-02 2.9e-02 7.2e-02 2.6e-02 2.6e-02 4.3e-02 1.3e-01 4.0e-02 1.1e-01 4.4e-02 1.7e-01 1.2e-01 4.1e-02 1.3e-01 9.0e-02 3.8e-02 7.6e-02 4.2e-01 1.0e-01 2.2e-01 1.2e-01 9.2e-02 3.3e-01 5.1e-02 7.4e-02 6.0e-02 2.4e-01 2.2e-02 2.9e-02

MOEA/D Mean 3.74e-01 3.31e-01 9.93e-01 9.67e-01 1.53e-01 1.09e+00 1.11e+00 3.45e-01 5.18e-01 4.53e-01 4.13e-01 4.14e-01 4.49e-01 1.07e+00 1.00e+00 1.07e+00 1.00e+00 1.00e+00 9.90e-01 3.13e-01 9.34e-01 7.15e-01 9.76e-01 6.85e-01 9.89e-01 1.20e+00 1.25e+00 9.62e-01 1.16e+00 1.60e+00 1.00e+00 1.00e+00 1.27e+00 1.15e+00 1.46e-01 7.32e-01

Std 3.9e-02 1.1e-01 2.1e-02 1.2e-01 2.8e-03 1.6e-01 8.5e-03 7.9e-04 4.9e-02 2.9e-03 4.3e-03 7.6e-03 1.3e-02 6.8e-02 1.5e-05 1.6e-01 2.4e-06 0.0e+00 2.6e-02 6.5e-02 1.3e-01 9.6e-02 1.6e-01 9.9e-02 2.7e-02 1.3e-01 7.3e-02 1.5e-01 7.2e-02 2.4e-01 5.1e-04 2.1e-05 4.5e-02 4.3e-02 4.0e-04 2.7e-03

MOVS Mean 4.64e-01 5.19e-01 7.55e-01 1.23e+00 5.70e-01 6.67e-01 7.88e-01 3.24e-01 3.55e-01 3.98e-01 3.49e-01 3.75e-01 3.31e-01 7.58e-01 6.58e-01 6.54e-01 4.98e-01 5.35e-01 7.19e-01 4.13e-01 1.51e+00 8.11e-01 6.18e-01 6.89e-01 8.70e-01 1.29e+00 1.19e+00 1.66e+00 1.03e+00 1.44e+00 8.55e-01 7.33e-01 8.12e-01 7.39e-01 4.10e-01 5.65e-01

Std 3.1e-02 3.9e-02 9.0e-03 1.1e-01 4.5e-02 1.9e-01 8.7e-03 3.1e-02 2.3e-02 2.8e-02 2.7e-02 2.7e-02 2.9e-02 5.9e-02 4.1e-02 4.0e-02 5.8e-02 3.9e-02 3.8e-02 7.5e-02 1.2e-01 1.0e-01 7.9e-02 7.4e-02 7.8e-02 1.7e-01 1.6e-01 1.9e-01 5.7e-02 8.7e-02 9.8e-02 4.5e-02 2.1e-02 3.5e-02 3.6e-02 2.5e-02

CR IP T

Mean 8.45e-02 8.43e-02 7.07e-01 1.29e-01 1.11e-01 6.74e-01 7.58e-01 5.58e-02 1.23e-01 1.26e-01 1.49e-01 1.18e-01 1.14e-01 1.15e+00 6.85e-01 7.38e-01 1.50e-01 7.45e-01 7.21e-01 5.37e-01 1.34e+00 6.66e-01 5.53e-01 5.73e-01 8.64e-01 1.22e+00 1.29e+00 1.54e+00 1.07e+00 1.15e+00 8.51e-01 6.73e-01 7.52e-01 6.25e-01 7.92e-02 4.16e-01

IBEA

AN US

Std 3.2e-02 2.3e-02 1.5e-02 3.9e-02 3.2e-02 5.0e-02 1.1e-02 2.2e-02 3.0e-02 2.8e-02 3.5e-02 3.5e-02 3.2e-02 2.0e-01 4.8e-02 4.5e-02 5.0e-02 4.9e-02 6.4e-02 1.1e-01 9.9e-02 9.2e-02 4.4e-02 5.8e-02 9.1e-02 9.9e-02 1.0e-01 1.5e-01 7.0e-02 7.1e-02 9.5e-02 5.1e-02 1.9e-02 2.3e-01 3.2e-02 2.9e-02

ED

Mean 3.71e-01 3.78e-01 7.43e-01 4.03e-01 3.54e-01 7.26e-01 7.89e-01 3.69e-01 3.62e-01 4.00e-01 3.88e-01 3.84e-01 3.65e-01 8.95e-01 7.06e-01 6.68e-01 4.65e-01 8.13e-01 7.29e-01 5.20e-01 1.50e+00 7.12e-01 5.69e-01 6.42e-01 9.37e-01 1.39e+00 1.26e+00 1.64e+00 1.13e+00 1.52e+00 8.25e-01 7.34e-01 8.00e-01 6.05e-01 3.94e-01 5.59e-01

PT

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6 WFG1 WFG2 WFG3 WFG4 WFG5 WFG6 WFG7 WFG9 DTLZ1 DTLZ2 DTLZ4 DTLZ5 DTLZ6 DTLZ7 LZ09F1 LZ09F2 LZ09F3 LZ09F4 LZ09F5 LZ09F6 LZ09F7 LZ09F8 LZ09F9 OKA1 OKA2 Viennet2 Viennet3 Poloni Schaffer Fonseca Kursawe

MOCell

M

NSGAII

ACCEPTED MANUSCRIPT Table 4. The mean and the standard deviation values of the EPSILON metric for all the algorithms.

CE AC

Std 4.5e-04 4.0e-04 1.2e-01 1.3e-02 5.2e-04 2.2e-01 3.4e-01 1.2e-03 6.8e-04 2.9e-02 4.7e-02 6.0e-04 3.0e-03 2.3e-01 1.4e-02 3.7e-01 4.7e-04 1.1e-01 8.4e-01 2.6e-02 1.1e-01 8.2e-02 1.7e-02 5.3e-02 1.9e-01 1.5e-01 1.6e-01 1.6e-01 1.5e-01 2.3e-01 4.8e-03 1.1e-02 7.4e-03 1.4e+00 5.9e-04 4.5e-03

Mean 8.82e-03 1.64e-02 7.34e-02 7.59e-01 1.22e-02 4.02e-01 6.20e-01 2.35e-02 4.38e-02 4.91e-02 6.35e-02 4.54e-02 3.33e-02 3.00e-01 8.92e-02 4.03e-01 3.24e-02 6.33e-02 1.24e+00 4.01e-02 2.42e-01 1.77e-01 2.08e-01 1.46e-01 4.74e-01 5.27e-01 4.20e-01 2.82e-01 5.29e-01 7.13e-01 2.62e-02 7.35e-02 4.36e-01 1.83e+00 8.29e-03 2.67e-01

Std 8.6e-04 1.4e-03 1.4e-01 1.2e-01 1.2e-03 2.1e-01 3.2e-01 1.8e-03 4.8e-03 1.9e-02 2.3e-02 4.8e-03 5.4e-03 2.5e-02 5.1e-03 2.9e-01 1.8e-03 2.2e-02 9.6e-01 3.0e-02 6.2e-02 6.5e-02 2.6e-02 3.9e-02 1.4e-01 1.1e-01 1.1e-01 7.6e-02 1.2e-01 2.0e-01 9.2e-03 2.0e-02 5.7e-02 1.6e+00 6.3e-04 2.5e-02

MOEA/D Mean 2.44e-02 4.92e-02 1.28e-01 4.60e-01 5.09e-03 6.19e-01 5.64e-02 2.82e-02 5.95e-02 7.74e-02 2.41e-02 2.51e-02 3.11e-02 2.63e-01 5.77e-01 6.46e-01 5.77e-01 5.77e-01 2.14e+00 8.54e-03 2.03e-01 2.07e-01 1.68e-01 1.57e-01 7.04e-01 3.80e-01 3.79e-01 2.22e-01 3.07e-01 6.58e-01 1.26e-01 4.78e-01 2.50e-01 2.04e+00 6.33e-03 7.87e-02

Std 5.6e-03 2.3e-02 3.1e-02 2.5e-01 3.3e-04 1.4e-01 1.4e-01 1.8e-03 1.2e-02 1.3e-04 6.5e-04 6.1e-04 1.1e-03 1.8e-01 2.4e-05 6.5e-02 2.1e-06 0.0e+00 6.9e-01 7.0e-04 8.1e-02 9.8e-02 2.7e-02 6.1e-02 9.2e-02 2.4e-01 9.2e-02 8.5e-02 4.2e-02 1.4e-01 5.2e-05 1.9e-04 1.5e-01 1.2e+00 2.8e-05 5.0e-03

MOVS Mean 1.25e-02 1.51e-02 7.83e-03 1.11e+00 1.19e-02 2.55e-01 1.45e-02 3.28e-02 3.09e-02 4.55e-02 3.16e-02 3.31e-02 3.61e-02 1.83e-01 1.25e-01 1.11e-01 1.03e-02 8.24e-03 1.56e-01 2.39e-02 1.97e-01 1.63e-01 1.81e-01 1.37e-01 2.97e-01 3.76e-01 3.80e-01 2.02e-01 3.24e-01 5.16e-01 3.47e-02 4.89e-02 1.54e-01 5.51e-02 1.32e-02 8.01e-02

Std 2.9e-03 3.5e-03 1.3e-03 4.5e-01 2.1e-03 9.6e-02 2.8e-03 3.9e-03 4.4e-03 2.3e-02 4.3e-03 4.3e-03 5.0e-03 1.6e-01 1.5e-02 1.6e-02 1.9e-03 1.1e-03 3.5e-02 4.6e-03 2.9e-02 2.7e-02 1.9e-02 1.5e-02 7.5e-02 7.8e-02 5.6e-02 2.9e-02 7.3e-02 7.6e-02 7.8e-03 1.2e-02 4.3e-02 1.0e-02 2.2e-03 1.5e-02

CR IP T

Mean 6.61e-03 5.86e-03 6.84e-02 1.40e-02 8.20e-03 1.02e+00 6.92e-01 1.66e-02 1.49e-02 4.33e-02 7.25e-02 1.47e-02 1.79e-02 3.23e-01 1.30e-01 3.11e-01 4.71e-03 1.71e+00 8.83e-01 4.73e-02 3.44e-01 2.29e-01 1.91e-01 1.82e-01 4.80e-01 5.67e-01 5.51e-01 3.75e-01 4.66e-01 7.89e-01 3.25e-02 5.07e-02 7.25e-02 1.66e+00 6.62e-03 4.42e-02

IBEA

AN US

Std 2.1e-03 1.9e-03 1.6e-03 2.1e-02 2.0e-03 2.5e-01 3.9e-01 5.7e-03 6.5e-03 2.0e-02 1.9e-02 6.5e-03 8.5e-03 1.0e-01 1.8e-02 1.4e-02 2.6e-03 8.7e-02 3.7e-02 2.1e-03 5.8e-02 4.1e-02 2.4e-02 2.2e-02 3.5e-02 1.2e-01 8.1e-02 6.0e-02 8.3e-02 9.9e-02 9.7e-03 1.1e-02 2.8e-02 7.7e-01 1.6e-03 1.0e-02

ED

Mean 1.36e-02 1.31e-02 8.42e-03 2.05e-02 1.52e-02 4.12e-01 3.82e-01 3.81e-02 3.32e-02 6.13e-02 5.17e-02 3.56e-02 3.60e-02 1.28e-01 1.29e-01 1.11e-01 1.13e-02 8.64e-01 1.40e-01 1.81e-02 2.24e-01 1.50e-01 1.64e-01 1.24e-01 3.14e-01 3.20e-01 3.31e-01 2.67e-01 3.52e-01 4.79e-01 3.37e-02 5.20e-02 1.42e-01 1.64e+00 1.29e-02 7.47e-02

PT

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6 WFG1 WFG2 WFG3 WFG4 WFG5 WFG6 WFG7 WFG9 DTLZ1 DTLZ2 DTLZ4 DTLZ5 DTLZ6 DTLZ7 LZ09F1 LZ09F2 LZ09F3 LZ09F4 LZ09F5 LZ09F6 LZ09F7 LZ09F8 LZ09F9 OKA1 OKA2 Viennet2 Viennet3 Poloni Schaffer Fonseca Kursawe

MOCell

M

NSGAII

ACCEPTED MANUSCRIPT Table 5. The mean and the standard deviation values of the IGD metric for all the algorithms.

CE AC

Std 2.0e-06 3.0e-06 9.7e-04 1.6e-04 1.2e-05 1.8e-03 1.8e-03 2.0e-06 1.5e-06 1.1e-05 8.4e-04 1.4e-06 1.4e-05 5.5e-03 3.6e-05 3.5e-03 3.1e-07 9.7e-04 8.2e-03 3.4e-04 4.8e-03 3.1e-03 5.1e-04 1.9e-03 2.5e-03 6.2e-03 6.2e-03 3.4e-03 2.7e-03 1.1e-02 2.8e-05 1.3e-05 1.0e-06 1.9e-02 3.1e-06 2.1e-06

Mean 1.64e-04 5.42e-04 1.53e-03 2.25e-02 2.52e-04 3.40e-03 3.63e-03 1.31e-04 5.20e-04 3.58e-04 8.86e-04 5.21e-04 4.74e-04 4.24e-03 1.39e-03 4.28e-03 1.03e-04 4.72e-04 1.64e-02 8.00e-04 7.74e-03 5.59e-03 6.09e-03 4.23e-03 1.66e-02 2.17e-02 1.59e-02 9.63e-03 5.48e-03 1.91e-02 1.41e-03 4.49e-03 1.11e-03 2.75e-02 2.58e-04 1.25e-03

Std 5.5e-06 3.4e-05 9.6e-04 3.3e-03 9.6e-06 2.3e-03 1.1e-03 2.6e-06 3.5e-05 2.4e-05 2.8e-04 3.2e-05 5.8e-05 3.2e-04 2.4e-05 2.9e-03 5.1e-06 1.3e-04 8.6e-03 6.5e-04 2.1e-03 2.6e-03 1.1e-03 1.5e-03 2.3e-03 5.1e-03 5.1e-03 2.3e-03 2.4e-03 9.1e-03 3.0e-04 2.5e-04 8.2e-04 2.3e-02 5.4e-06 2.1e-04

MOEA/D Mean 5.21e-04 4.86e-04 1.54e-03 1.11e-02 1.40e-04 6.01e-03 5.59e-04 1.36e-04 2.29e-04 1.24e-04 2.28e-04 1.48e-04 2.41e-04 6.58e-03 5.72e-03 1.13e-02 1.48e-03 3.82e-03 2.88e-02 2.33e-04 4.81e-03 5.04e-03 2.98e-03 3.66e-03 1.93e-02 9.51e-03 1.06e-02 4.56e-03 2.47e-03 1.45e-02 2.91e-03 7.50e-03 7.47e-04 2.86e-02 2.04e-04 1.77e-04

Std 1.1e-04 1.8e-04 7.3e-04 6.6e-03 4.4e-07 1.7e-03 6.6e-04 8.2e-07 3.0e-05 3.3e-07 8.4e-07 7.6e-07 5.4e-06 5.0e-03 1.4e-07 1.7e-03 3.7e-09 0.0e+00 1.1e-03 1.5e-05 2.1e-03 3.2e-03 5.5e-04 2.1e-03 1.3e-03 6.9e-03 3.4e-03 1.9e-03 3.1e-04 7.2e-03 2.2e-06 1.4e-06 1.0e-03 1.7e-02 3.9e-07 1.3e-06

MOVS Mean 1.99e-04 4.74e-04 1.30e-04 2.92e-02 2.19e-04 2.81e-03 1.31e-04 1.44e-04 1.56e-04 1.32e-04 2.38e-04 1.53e-04 2.38e-04 4.01e-03 7.61e-04 1.19e-03 2.05e-05 5.10e-05 2.22e-03 5.09e-04 5.67e-03 3.98e-03 3.84e-03 3.05e-03 5.78e-03 1.23e-02 1.20e-02 5.75e-03 2.65e-03 9.57e-03 3.52e-04 1.83e-04 1.05e-04 5.84e-04 3.19e-04 1.76e-04

Std 2.1e-05 1.0e-04 4.4e-06 1.3e-02 1.5e-05 1.2e-03 4.5e-06 6.2e-06 4.3e-06 1.9e-05 2.7e-05 6.0e-06 1.2e-05 4.2e-03 3.8e-05 1.7e-04 1.2e-06 3.4e-06 1.0e-04 6.0e-05 6.8e-04 9.1e-04 6.6e-04 3.4e-04 1.3e-03 4.0e-03 2.3e-03 5.7e-04 9.6e-04 1.2e-03 4.8e-05 2.0e-05 6.7e-06 2.2e-05 1.4e-05 9.0e-06

CR IP T

Mean 1.40e-04 1.42e-04 5.91e-04 2.33e-04 2.08e-04 7.31e-03 3.59e-03 1.11e-04 1.21e-04 9.56e-05 1.13e-03 1.13e-04 1.77e-04 6.08e-03 7.63e-04 3.12e-03 1.42e-05 1.50e-02 1.02e-02 7.25e-04 1.20e-02 6.41e-03 3.89e-03 4.56e-03 8.99e-03 2.04e-02 2.00e-02 9.98e-03 4.83e-03 2.19e-02 3.01e-04 1.62e-04 7.04e-05 2.63e-02 2.17e-04 1.25e-04

IBEA

AN US

Std 1.1e-05 8.4e-06 6.7e-06 6.8e-04 3.8e-05 2.0e-03 2.0e-03 6.7e-06 8.9e-06 1.1e-05 3.0e-04 9.3e-06 1.9e-05 2.4e-03 4.5e-05 1.1e-04 1.2e-06 6.7e-04 1.7e-04 2.2e-05 1.9e-03 1.4e-03 6.0e-04 6.0e-04 9.4e-04 4.2e-03 3.6e-03 2.6e-03 1.3e-03 1.7e-03 3.3e-05 1.9e-05 4.6e-06 1.2e-02 1.1e-05 8.9e-06

ED

Mean 1.88e-04 1.91e-04 1.33e-04 3.83e-04 3.27e-04 2.85e-03 1.97e-03 1.54e-04 1.61e-04 1.33e-04 4.82e-04 1.57e-04 2.39e-04 2.11e-03 7.79e-04 1.20e-03 2.06e-05 7.17e-03 2.25e-03 4.24e-04 6.62e-03 3.66e-03 3.19e-03 2.60e-03 6.84e-03 1.01e-02 9.63e-03 8.22e-03 3.03e-03 9.28e-03 3.27e-04 1.76e-04 9.80e-05 2.56e-02 3.15e-04 1.77e-04

PT

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6 WFG1 WFG2 WFG3 WFG4 WFG5 WFG6 WFG7 WFG9 DTLZ1 DTLZ2 DTLZ4 DTLZ5 DTLZ6 DTLZ7 LZ09F1 LZ09F2 LZ09F3 LZ09F4 LZ09F5 LZ09F6 LZ09F7 LZ09F8 LZ09F9 OKA1 OKA2 Viennet2 Viennet3 Poloni Schaffer Fonseca Kursawe

MOCell

M

NSGAII

ACCEPTED MANUSCRIPT Algorithm NSGAII MOCell IBEA MOEA/D MOVS

Table 6. The average rankings of the algorithms for all the metrics Ranking (HV) Ranking (SPREAD) Ranking (EPSILON) Ranking (IGD) 3.250 2.861 2.5690 2.569 3.055 1.694 3.111 2.861 2.500 3.972 3.583 4.083 2.638 3.722 3.361 3.153 3.555 2.750 2.375 2.333

Table 7. Wilcoxon’s Rank Sum Test Results of MOVS vs other algorithms for HV and SPREAD indicators Indicator HV

MOCell

AC

MOEAD

NSGAII

MOCell

IBEA

MOEAD

Sig.

p-value

Sig.

p-value

Sig.

p-value

Sig.

p-value

Sig.

p-value

Sig.

p-value

Sig.

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + +

3,02E-11 3,02E-11 3,02E-11 1,38E-08 3,02E-11 0,970516 3,02E-11 2,37E-10 1,33E-10 3,02E-11 1,11E-06 1,33E-10 3,32E-06 1,76E-10 3,02E-11 0,379036 3,02E-11 3,02E-11 0,077272 1,60E-07 0,000178 0,000158 1,29E-06 2,39E-08 9,24E-09 0,122352 1,17E-05 2,00E-06 3,34E-11 6,50E-06 9,76E-10 6,01E-08 4,08E-11 3,01E-11 3,02E-11 3,02E-11

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

3,02E-11 3,02E-11 3,02E-11 5,31E-08 3,02E-11 5,19E-07 3,02E-11 3,37E-05 3,02E-11 0,149448 4,69E-08 0,446419 0,599689 1,68E-09 3,02E-11 3,02E-11 3,02E-11 2,10E-11 3,02E-11 3,02E-11 0,108689 0,005569 3,02E-11 5,09E-08 3,02E-11 0,000398 0,539510 0,899995 0,059427 2,18E-08 3,02E-11 3,02E-11 3,02E-11 3,02E-11 3,02E-11 3,26E-07

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + +

8,99E-11 3,02E-11 3,83E-05 3,02E-11 3,02E-11 0,006668 6,63E-01 5,60E-07 1,96E-01 0,761828 2,28E-05 0,222572 0,000117 6,74E-06 0,000253 2,28E-01 0,035136 3,02E-11 3,87E-01 2,13E-04 0,923442 2,84E-04 6,67E-03 1,22E-02 6,38E-03 3,64E-02 2,77E-01 0,569220 8,20E-07 0,000526 0,340288 0,739398 5,01E-02 5,57E-03 0,115362 0,283778

+ + + + + + + + + + + + + + + + + + + + + + -

3,02E-11 3,02E-11 3,02E-11 3,02E-11 3,02E-11 5,55E-02 3,02E-11 3,02E-11 3,02E-11 3,02E-11 3,02E-11 3,02E-11 3,02E-11 1,07E-09 0,040595 2,60E-02 3,02E-11 3,02E-11 0,935192 0,045146 1,27E-02 4,12E-06 0,004033 1,02E-05 6,52E-01 0,190730 0,051877 0,096262 4,68E-02 7,62E-08 7,73E-01 1,68E-03 9,76E-10 1,08E-02 3,02E-11 3,02E-11

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + +

3,02E-11 3,02E-11 3,02E-11 3,32E-06 3,02E-11 3,37E-05 3,02E-11 1,41E-09 3,02E-11 3,02E-11 3,02E-11 3,02E-11 3,02E-11 3,02E-11 4,20E-10 0,911709 4,50E-11 3,02E-11 9,21E-05 3,02E-11 0,176127 8,15E-11 3,02E-11 3,02E-11 3,83E-06 1,87E-05 9,94E-01 5,01E-01 3,02E-11 7,45E-02 2,43E-05 1,44E-02 3,02E-11 9,33E-02 8,42E-01 3,02E-11

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + +

4,18E-09 1,43E-08 3,02E-11 3,50E-09 3,02E-11 1,56E-08 3,02E-11 2,71E-02 3,02E-11 4,62E-10 3,02E-11 5,00E-09 3,02E-11 3,02E-11 3,02E-11 3,02E-11 3,02E-11 9,34E-12 3,02E-11 1,25E-07 3,02E-11 0,000300 1,86E-09 7,17E-01 4,31E-08 0,016954 0,347827 3,02E-11 4,31E-08 5,86E-06 8,48E-09 3,02E-11 3,02E-11 3,02E-11 3,02E-11 3,02E-11

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

AN US

p-value 3,02E-11 3,02E-11 0,0076171 1,79E-11 3,02E-11 4,12E-06 7,69E-08 2,20E-07 4,98E-11 0,0013017 0,0002126 3,02E-11 2,28E-05 5,67E-07 0,2707053 4,66E-06 3,02E-11 1,21E-12 0,2009489 0,3632223 2,83E-08 0,1494487 0,1259702 0,8418015 1,41E-09 0,0020523 0,5105979 0,0003006 1,55E-09 2,92E-05 1,21E-10 3,02E-11 3,02E-11 3,02E-11 3,02E-11 3,02E-11

PT

+ + + + + + + + + + + + + + + + + + + + + + + + + + -

CE

0,206205 0,026077 0,115362 1,79E-11 3,02E-11 0,105469 7,09E-08 2,60E-08 2,20E-07 0,087710 1,36E-07 0,520144 0,028128 0,000470 0,684322 1,87E-07 0,387099 1,21E-12 8,10E-10 2,67E-09 0,222572 1,86E-09 3,02E-11 9,76E-10 6,53E-07 1,60E-07 3,96E-08 0,006097 3,32E-06 0,015014 0,000526 0,001679 5,09E-06 3,02E-11 0,379036 0,841801

ED

p-value Sig. ZDT1 ZDT2 ZDT3 ZDT4 ZDT6 WFG1 WFG2 WFG3 WFG4 WFG5 WFG6 WFG7 WFG9 DTLZ1 DTLZ2 DTLZ4 DTLZ5 DTLZ6 DTLZ7 LZ09F1 LZ09F2 LZ09F3 LZ09F4 LZ09F5 LZ09F6 LZ09F7 LZ09F8 LZ09F9 OKA1 OKA2 Viennet2 Viennet3 Poloni Schaffer Fonseca Kursawe

IBEA

M

NSGAII

Indicator SPREAD

CR IP T

MOVS vs

ACCEPTED MANUSCRIPT Table 8. Wilcoxon’s Rank Sum Test Results of MOVS vs other algorithms for EPSILON and IGD indicators MOVS vs

Indicator EPSILON MOCell

AC

MOEAD

NSGAII

MOCell

IBEA

MOEAD

Sig.

p-value

Sig.

p-value

Sig.

p-value

Sig.

p-value

Sig.

p-value

Sig.

p-value

Sig.

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

5,49E-11 0,036438 3,02E-11 0,000471 0,176127 0,015014 3,02E-11 4,98E-11 6,72E-10 0,022252 6,07E-11 9,76E-10 0,050120 1,61E-06 3,34E-11 0,318304 3,02E-11 3,02E-11 0,003183 0,026077 0,001442 0,706171 9,21E-05 0,387099 2,87E-10 8,84E-07 0,190730 5,86E-06 2,67E-09 6,36E-05 0,000471 1,86E-06 3,02E-11 3,02E-11 3,02E-11 3,02E-11

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

2,15E-10 1,78E-10 3,02E-11 6,53E-08 3,02E-11 2,37E-10 3,02E-11 1,49E-06 4,08E-11 4,41E-11 3,02E-11 3,69E-11 2,60E-05 0,053685 3,02E-11 3,02E-11 3,02E-11 1,92E-11 3,02E-11 3,02E-11 0,482516 0,589451 0,055545 0,739398 1,33E-10 0,099257 0,830255 0,482516 0,473346 6,71E-06 3,02E-11 3,02E-11 9,26E-09 3,02E-11 3,02E-11 0,559230

+ + + + + + + + + + + + + + + + + + + + + + + + + + -

0,002379 3,02E-11 0,021506 3,02E-11 3,02E-11 0,355472 5,26E-04 3,83E-06 0,006668 0,599689 5,53E-08 0,115362 0,830255 0,876634 0,206205 0,332854 0,473346 3,02E-11 3,02E-11 1,07E-09 2,34E-01 0,011227 0,000812 2,28E-05 4,74E-06 1,70E-02 1,87E-05 5,27E-05 4,06E-02 0,455296 2,71E-02 0,251880 7,66E-05 3,02E-11 0,171450 0,195790

+ + + + + + + + + + + + + + + + + + + + + + + -

3,02E-11 3,02E-11 6,77E-05 3,02E-11 4,03E-03 2,15E-10 9,51E-06 3,02E-11 3,02E-11 2,87E-10 0,00137 3,02E-11 3,02E-11 0,010763 0,684323 0,026077 3,02E-11 3,02E-11 3,02E-11 2,32E-02 2,67E-09 1,17E-02 0,599689 0,000446 2,02E-08 8,84E-07 1,25E-07 9,06E-08 3,83E-06 2,38E-07 3,16E-05 4,35E-05 3,02E-11 3,02E-11 3,02E-11 3,02E-11

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

3,02E-11 0,002157 3,02E-11 0,019883 1,17E-09 0,464273 3,02E-11 1,33E-10 3,02E-11 3,02E-11 3,02E-11 3,02E-11 3,02E-11 6,63E-01 3,02E-11 1,34E-05 3,02E-11 3,02E-11 3,02E-11 3,15E-02 7,22E-06 5,83E-03 1,07E-09 9,21E-05 3,02E-11 2,02E-08 5,56E-04 8,89E-10 4,57E-09 5,57E-10 3,02E-11 3,02E-11 3,02E-11 3,02E-11 3,02E-11 3,02E-11

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

3,02E-11 8,30E-01 3,02E-11 9,83E-08 3,02E-11 1,01E-08 3,02E-11 2,39E-08 3,02E-11 6,41E-01 1,62E-01 4,46E-04 4,55E-01 0,070127 3,02E-11 3,02E-11 3,02E-11 4,11E-12 4,11E-12 3,02E-11 7,96E-03 0,911709 7,22E-06 2,90E-01 3,02E-11 4,23E-03 0,012732 0,003183 0,818746 2,02E-08 3,02E-11 3,02E-11 3,02E-11 3,02E-11 3,02E-11 9,88E-03

+ + + + + + + + + + + + + + + + + + + + + + + + + + + +

PT

AN US

CR IP T

p-value 3,02E-11 3,02E-11 0,0169549 3,02E-11 3,02E-11 3,69E-11 3,09E-06 3,02E-11 3,02E-11 0,728062 0,0046371 3,02E-11 3,02E-11 0,0022658 0,17145 0,3255266 3,02E-11 3,02E-11 0,0198831 1,64E-05 9,83E-08 0,0083146 0,0270863 0,0027548 1,25E-07 1,19E-06 1,39E-06 2,60E-08 5,61E-05 6,28E-06 0,3632223 0,5996895 3,02E-11 3,02E-11 3,02E-11 3,02E-11

CE

+ + + + + + + + + + + + + + + + + + + -

ED

p-value Sig. 0,006668 ZDT1 0,028128 ZDT2 0,096262 ZDT3 3,02E-11 ZDT4 7,04E-07 ZDT6 0,042066 WFG1 4,64E-05 WFG2 0,000445 WFG3 0,166866 WFG4 0,000130 WFG5 3,81E-07 WFG6 0,185766 WFG7 0,411910 WFG9 0,773119 DTLZ1 0,379036 DTLZ2 0,830255 DTLZ4 0,111986 DTLZ5 3,02E-11 DTLZ6 0,070126 DTLZ7 1,25E-07 LZ09F1 0,195790 LZ09F2 0,007288 LZ09F3 0,007288 LZ09F4 0,000654 LZ09F5 0,000318 LZ09F6 0,006668 LZ09F7 0,002052 LZ09F8 4,64E-05 LZ09F9 0,118817 OKA1 0,180899 OKA2 Viennet2 0,395267 Viennet3 0,355472 0,355472 Poloni Schaffer 3,02E-11 Fonseca 0,662734 Kursawe 0,099257

IBEA

M

NSGAII

Indicator IGD

ACCEPTED MANUSCRIPT Table 9. Number of Evaluations Required by the Algorithms to Reach 98% the HV of the True Pareto Front (PFt) MOEA/D

MOVS

14300 24400 4500 21100 5500 6300 15500 -

14000 11300 12500 18000 3200 7000 14000 5200 12000 19000 15200 -

10300 19600 5000 5200 4300 -

15400 17100 23500 11300 5600 5200 17700 -

9500 19200 3500 14600 4100 5000 11100 1800 18000 16200 19300 -

ED PT CE AC

LZ09F1 LZ09F2 LZ09F3 LZ09F4 LZ09F5 LZ09F6 LZ09F7 LZ09F8 LZ09F9 OKA1 OKA2 Viennet2 Viennet3 Poloni Schaffer Fonseca Kursawe

NSGAII

MOCell

IBEA

MOEA/D

MOVS

24000 600 800 500 3000

400 500 300 5200 2500

16400 700 700 600 4900 13500

4300 400 2400 4600

400 1000 400 3600 3600

CR IP T

IBEA

AN US

MOCell

M

ZDT1 ZDT2 ZDT3 ZDT4 ZDT6 WFG1 WFG2 WFG3 WFG4 WFG5 WFG6 WFG7 WFG9 DTLZ1 DTLZ2 DTLZ4 DTLZ5 DTLZ6 DTLZ7

NSGAII