A sinusoidal differential evolution algorithm for numerical optimisation

A sinusoidal differential evolution algorithm for numerical optimisation

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ARTICLE IN PRESS

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Applied Soft Computing xxx (2014) xxx–xxx

Contents lists available at ScienceDirect

Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc

A sinusoidal differential evolution algorithm for numerical optimisation

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Amer Draa ∗ , Samira Bouzoubia, Imene Boukhalfa MISC Laboratory, Constantine 2 University, Algeria

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a r t i c l e

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i n f o

a b s t r a c t

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Q3 Article history: Received 18 January 2013 Received in revised form 4 November 2014 Accepted 4 November 2014 Available online xxx

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Keywords: Differential evolution Sinusoidal parameter adjustment Exploration Exploitation Optimisation

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1. Introduction

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This paper presents a new variant of the Differential Evolution (DE) algorithm called Sinusoidal Differential Evolution (SinDE). The key idea of the proposed SinDE is the use of new sinusoidal formulas to automatically adjust the values of the DE main parameters: the scaling factor and the crossover rate. The objective of using the proposed sinusoidal formulas is the search for a good balance between the exploration of non visited regions of the search space and the exploitation of the already found good solutions. By applying it on the recently proposed CEC-2013 set of benchmark functions, the proposed approach is statistically compared with the classical DE, the linearly parameter adjusting DE and 10 other state-of-the-art metaheuristics. The obtained results have proven the superiority of the proposed SinDE, it outperformed other approaches especially for multimodal and composition functions. © 2014 Published by Elsevier B.V.

Differential Evolution (DE) is a powerful optimisation metaheuristic that was first proposed by Price and Storn in 1995 [1,2]. Since then, this population-based metaheuristic has attracted the attention of researchers in many fields. It was intensively used to solve academic and real life problems, especially in the fields of engineering and sciences. This evolutionary algorithm is considered as one of the most reliable and versatile optimisation techniques available today [3]. Differential evolution uses rather a greedy selection and less stochastic approach to solve optimisation problems than other classical evolutionary algorithms. It differs from other evolutionary algorithms in the mutation and recombination phases. Unlike some metaheuristic techniques such as genetic algorithms and evolution strategies, where perturbation occurs in accordance with random quantities, DE uses weighted differences between solution vectors to perturb a given population [1,4]. According to Price [5], differential evolution has the ability to find the true global optimum regardless of the initial parameter values, is fast and simple with regard to application and modification, requires few control parameters, has a parallel processing nature and offers fast convergence, capable of providing multiple

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∗ Corresponding author. Tel.: +213 791844353 E-mail addresses: draa [email protected] (A. Draa), [email protected] (S. Bouzoubia), [email protected] (I. Boukhalfa).

solutions at a single run, effective on integer, discrete and mixed parameter optimisation and able to find the optimal solution for a non-linear constrained optimisation problem with penalty functions. It is clear that conventional DE is a bit overestimated by its inventors; new optimisation problems proved that DE sometimes fails at finding the global optimum, when failing at achieving a good balance between the exploitation of the already-found good solutions and the exploration of the non-visited regions of the search space. This weakness has led to the emergence of many variants of the basic algorithm. Basic variants of DE were first defined and used by its inventors, Price and Storn, to solve optimisation problems [2,5–7]. Later on, other variants have been proposed and thoroughly studied, by other researchers, to analyse their explorative and exploitive capabilities. For example, in [8], Lampinen proposed a DE algorithm for handling non-linear constraint functions. In [9], Gamperle et al. initiated the series of works studying the influence of parameters’ values on the performance of the DE algorithm; the authors gave some guidance about parameter values setting. Through comparing differential evolution with other evolutionary techniques, such as Particle Swarm Optimisation (PSO) [10], and with the emergence of many new variants of the DE algorithm, the problem of parameters setting and its impact on the overall performance of the algorithm appeared as a serious challenge for researchers interested in exploiting this metaheuristic (DE) for solving daily life problems. Hence, new variants working on the adjustment and adaptation of DE parameters, especially those based on hybridising DE with other evolutionary techniques, have

http://dx.doi.org/10.1016/j.asoc.2014.11.003 1568-4946/© 2014 Published by Elsevier B.V.

Please cite this article in press as: A. Draa, et al., A sinusoidal differential evolution algorithm for numerical optimisation, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.11.003

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emerged in recent years [11–16]. A rich review of some of these automatically parameter adjusting and adaptive variants of the DE algorithm can be found in the recent survey of Das and Suganthan about differential evolution [17]. In its original version [1], the DE algorithm uses a special type of mutation in which a given individual is perturbed using the weighted difference between two other individuals chosen randomly from the population. The weight of this difference is known as mutation scaling factor or scaling parameter. In early generations of the algorithm, the individuals are relatively different from each others. So, small values of the scaling factor will be able to offer good diversity to the population. However, in later generations, the individuals would become very similar to each others, thanks to the crossover operator. So, small values of the scaling factor will be virtually incapable of offering diversity to the populations. Thus, the algorithm is likely to fall in local optima, especially in the case of high-dimension problems, where a big number of generations is generally needed to get good solutions. As a result, more freedom should be allowed to the algorithm to switch from exploration into exploitation, and vice versa, through an automatic adjustment of the values of parameters and so the change of search direction. In the same way, the crossover rate can dramatically influence the quality of solutions; a small value would lead to not getting profit from the mutant, while a big value will make big perturbations. In this perspective, we propose here a novel automatically parameter adjusting version of differential evolution: the Sinusoidal Differential Evolution (SinDE). This variant uses a sinusoidal framework to define the value of the mutation scaling factor and/or the crossover rate at each generation. Different patterns are proposed; some of them use the sinusoidal adjustment of only one parameter, the other parameter is fixed to the recommended value; other patterns adjust both parameters. This new scheme of calculating DE parameter values, as will be shown in the numerical results section, allows a good balance between exploration and exploitation, since the parameter value increases and decreases periodically, thanks to the periodicity of the Sine function. Moreover, the gradually-increasing (or decreasing) interval between maximum and minimum allowed values of the parameter being adjusted offers a good possibility to escape local optima. The proposed SinDE algorithm has been tested on the recently defined set of reference benchmark functions: the CEC-2013 testbed [18]. First, It has been compared using the Wilcoxon rank-sum test against the classical DE, the linearly parameteradjusting DE (LADE), other four recent powerful variants of the DE algorithm (the SMADE, the DEcfbLS, the b6e6rl and the TLBSaDE), and the recently proposed CMA-ES-RIS algorithm. Next, the Holm–Bonferroni [19] statistical procedure has been used to further compare SinDE to these algorithms and five other state-of-the art algorithms. The obtained results show the superiority of the proposed SinDE in both statistics; it has largely outperformed the majority of these algorithms and has been slightly better than the others. The remainder of this paper is organised as follows. In Section 2, basic concepts of differential evolution and its standard variants are presented. The proposed sinusoidal DE, SinDE, and its different patterns are described in Section 3. Section 4 validates the performances of the proposed approach through comparing it against classical DE and other state-of-the-art metaheuristics. The obtained experimental results are discussed in the same section. Finally, conclusions and future directions of research are drawn up in Section 5.

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2. Differential evolution

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The Differential Evolution (DE) algorithm [1], follows the general procedure of an evolutionary algorithm: an initial population of individuals is created by random selection and evaluated; then

the algorithm enters a loop of generating offspring, evaluating offspring, and selection to create the next generation [20], till a stop condition is met. Generating offspring consists of two operations: differential mutation and differential crossover. In basic DE, the mutation operation creates a new individual vi called the mutant by adding the weighted difference between two individuals (vectors) chosen randomly from the population to a third one, also chosen randomly, as shown in Eq. (1) below [1]. In the equation, i, i1 , i2 , i3 ∈ {1, . . . , NP} are mutually different indices, NP is the size of population, and F is a real positive scaling factor of the difference di = xi2 − xi3 .

vi = xi1 + F.(xi2 − xi3 )

(1)

The crossover operator implements a discrete recombination of the mutant, vi , and the parent vector, xi , to produce the offspring ui . This operator is formulated as shown in Eq. (2) below [21,22]; where: Uj (0, 1) is a random number in the interval [0,1], CR ∈ [0, 1] is the crossover rate, and k ∈ {1, 2, . . . , d} is a random parameter index, chosen once for each individual i to be sure that at least one parameter is always selected from the mutant. Popular values for CR are in the range [0.4, 1].



ui (j) =

vi (j), if Uj (0, 1) ≤ CR or j = k, xi (j),

(2)

xi,G+1 =

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otherwise

This type of crossover is called binomial crossover. Another type of crossover used in DE is the exponential crossover. Its principle is as follows. The starting position of crossover is chosen randomly  from 1, ..., d , d is the problem dimension; then, L consecutive elements (counted in circular manner) are taken from the mutant vector v [16]. In the present work, we are interested in binomial crossover. After applying differential crossover, the obtained individual, called trial vector, is compared to the parent individual (also called target vector). A greedy selection takes place at this point, i.e. the fittest of them will become the target vector. This selection operation, in the case of a minimisation problem, can be expressed as shown in Eq. (3); in the formula, xi,G+1 is the new target vector (at generation G+1), ui,G+1 is the trial vector obtained from the crossover operation, xi,G is the target vector at generation G, and fit(*) is the fitness function. Of course, in the case of a maximisation problem, the opposite comparison is used, i.e. a ‘≥’ replaces the ‘≤’ in the equation.



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ui,G+1 ,

if eval(ui,G+1 ) ≤ eval(xi,G ),

xi,G ,

otherwise,

(3)

As already stated, the differential evolution algorithm repeats these three operations of mutation, crossover and selection till a stop condition is met. The stop condition is generally chosen to be a predetermined number of generations. Another stop condition that can be used is achieving the global optimum, when its fitness is already known. The non-evolution of the best fitness among population individuals is an other widely-used stop condition. Generally, it is recommended to use a combination of two or three of these stop conditions. 2.1. The ‘DE/x/y/z’ notation A number of variations to the basic DE algorithm have been developed in literature, mainly by its inventors [2,5–7]. DE strategies differ in the way the target vector is selected, the number of difference vectors used to perturb it, and haw to determine crossover points. In order to characterise these variations, a notation was adopted in differential evolution literature, namely DE/x/y/z notation [2,7,21]. In this notation, x refers to the method

Please cite this article in press as: A. Draa, et al., A sinusoidal differential evolution algorithm for numerical optimisation, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.11.003

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of selecting the target vector, y indicates the number of difference vectors used, and z indicates the type of crossover. The DE strategy already presented above is referred to as DE/rand/1/bin for binomial crossover. In addition to the DE/rand/1/bin variant, Storn and Price [2] propose four other DE variants which are expressed in Eqs. (4)–(7) below. DE/best/1 : Vi,G = Xbest,G + F.(Xr i ,G − Xr i ,G ) 1

(4)

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DE/target − to − best/1 : Vi,G = Xi,G + F.(Xbest,G − Xi,G ) + F.(Xr i ,G − Xr i ,G ) 1

(5)

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DE/best/2 : Vi,G = Xbest,G + F.(Xr i ,G − Xr i ,G ) + F.(Xr i ,G − Xr i ,G )

(6)

DE/rand/2 : Vi,G = Xr i ,G + F.(Xr i ,G − Xr i ,G ) + F.(Xr i ,G − Xr i ,G )

(7)

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2.2. DE parameter setting One of the advantages of differential evolution is the limited number of parameters to be adjusted: the scaling parameter F, the crossover rate CR, and the population size NP. However, these control parameters and the learning strategies presented above (Eqs. (4)–(7)) are highly dependent on the optimisation problem to be solved [23]. Consequently, some new variants of DE have been proposed for the sake of finding a relationship between the problem dimension and the values of parameters [13]. Zaharie in [16] gives an interesting study of this relationship. Though, such a relationship cannot cover all types of problem requirements. For instance, one does not expect that the DE algorithm would perform similarly using the same parameter values for uni-modal and multi-modal problems for the same dimensions; the nature of the search space makes the difference. Hence, differential evolution, F and CR, need to be dynamic to cope with both the requirements of the problem at hand and the current state of population. With regard to the adopted parameter setting philosophy, DE variants can be categorised into three classes. In the first class, parameter values are set by the designer, the same values of parameters are used throughout the search process. In the second kind, the parameter values are continuously adapted or self-adapted according to feedback from the environment, which can be a fitness quality or other environmental constraints. Finally, in the automatically parameter-adjusting variants, some predefined formulas are used to automatically change the parameter value. Our proposed SinDE belongs to this third category. In the following paragraphs, we briefly present basic works that dealt with DE dynamic parameter setting. Chang and Xu [11] proposed one of the first, and most used, dynamic-setting variants of the DE algorithms. In their proposal, the authors use a linear adjustment of the parameters to make the crossover rate decrease from 1 to 0.7 and inversely the scaling factor increase from 0.3 to 0.5. This linear variant of the DE, called LADE in the following, is detailed in the next section. Another initiating work for dynamic setting of DE parameters is that of Abbass et al. [12]. The idea of this setting is that in each generation a new value of the scaling factor F is sampled from a Gaussian distribution, F ∼ N(0, 1). The same idea has been used in [24] with shifting the mean of the distribution to 0.5 and its standard deviation to 0.3 (i.e. F ∼ N(0.5, 0.3)). Later on, Abbass [25] applied the same formulation on the crossover parameter, i.e. CR ∼ N(0, 1). It is worth to mention that even if Abbass refers to these DE variants as being adaptive [12,25], we clearly see that they belong to the third category of parameter-adapting DE approaches: the automatically parameter-adjusting variants [21], since no information about the

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the already found solutions neither about current population individuals is exploited. As far as self-adaptation of DE parameters is concerned, Ali and Torn [26] proposed the use of information about the fitness of individuals to determine the value of the scaling parameter F at each generation. They have been motivated by the fact that the amount of difference between the best and worst individuals’ fitness is a key to know about population diversity, and so to compute the value of the F parameter, in order to favour either exploitation or exploration. In [27], Liu and Lampinen exploited the power of Fuzzy logics, on the basis of information about the population, to adapt the values of DE parameters. They proved that their variant of DE, called FADE, converges much faster than the classical DE. Brest et al. [28] went farther and devised new DE variants where the parameters are integrated to the individual structure. The basic idea was that good individuals are those found using good parameter values through generations, so it would be a good idea to define for each individual i at each generation t its specific parameters Fti and CRti , so that the parameters also evolve with the other variables of the individual. The authors proved that their variant, called jDE, is better than both classical DE and the FADE algorithm. In [29], Price et al. draw the values of DE parameters from a Gaussian distribution. However, unlike the works in [12,24,25], the mean and standard deviation of the used distribution are not constant; they are themselves selfadapted through generations. The just-cited approaches are fundamental works on which other parameter-adjusting or adaptive DE variants have been based. A detailed review of other works related to dynamic parameter setting of the DE algorithm is given by Das and Suganthan in [17]. 3. Sinusoidal adjustment of DE parameters

Fmax − Fmin ∗ it it max

CRit = CRmax −

CRmax − CRmin ∗ it it max

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As mentioned above, one basic scheme of dynamic setting of parameters in the DE algorithm is the linear adjustment. Eqs. (8) and (9) present the way these parameters are adjusted. In the equations, Fmin , Fmax , CRmin and CRmax are the lower and upper bounds of the scaling factor F and crossover rate CR, respectively. The inverse scheme, in which the scaling parameter F decreases and the crossover rate CR increases is also common [11]. Fit = Fmin +

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(8)

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(9)

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In this linear adjustment of parameter values, only one direction of value change is permitted, either increasing or decreasing. This scheme also suffers from the relatively big number of parameters: Fmin , Fmax , CRmin and CRmax ; which implies additional effort from the user for their setting. In order to overcome these limits, the present work proposes the use of a sinusoidal-based pattern to calculate the parameter values. This allows changing the direction of adjustment: from increasing to decreasing and vice versa. In the following, we detail this new sinusoidal adjustment. 3.1. The proposed sinusoidal differential evolution: SinDE The key idea of the proposed sinusoidal differential evolution is to use a new formula which does not only permit the adjustment of parameter values, but also permits to this adjustment to change its direction; from increasing to decreasing and vice versa. Such a possibility is well offered by the Sine, or the Cosine, function, see Fig. 1.

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As seen from these equations, the parameters F and CR are automatically adjusted in a way which allows them to change the direction of change with limited flexibility at first generations of the algorithm and more flexibility at final generations; and vice versa. The different configurations of parameter adjustment and setting are summarised in Fig. 2 below. In order to find the best combination of formulas, all these combinations are compared according to their performances on the chosen set of benchmark functions, as will be seen in experimental design section. For the case of the sub-variants adjusting only one parameter, the value of the other parameter is set to the value recommended in literature [22].

Fig. 1. The sine and cosine functions.

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What is remarkable in the case of sinusoidal functions is that the value of a given parameter increases and decreases periodically. This is exactly what is needed for permitting a certain flexibility in changing the direction of change for a given parameter. In the context of the present work, we use a sine-based formulation of parameters with different configurations. These configurations are presented in the sets of Eqs. (10)–(15). In the formulas, freq represents the frequency of the sinusoidal function, itmax is the maximum number of generations and it is the current generation. According to the notation system used to identify DE variants, seen in Section 2.1, our new SinDE approach can be called SinDE/rand/1/bin.

Configuration 1:

⎧  ⎨ Fit = 1 ∗ sin (2 ∗ freq ∗ it) ∗ ⎩

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it +1 itMax

(10)

CRit = 0.9

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Configuration 2:

 ⎧ 1 it ⎪ ⎨ Fit = 2 ∗ sin (2 ∗ freq ∗ it) ∗ itMax + 1  ⎪ 1 it ⎩ CRit =

2

∗ sin (2 ∗ freq ∗ it + ) ∗

itMax

(11)



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Configuration 3:

 ⎧ 1 itMax − it ⎪ ⎨ Fit = 2 ∗ sin (2 ∗ freq ∗ it) ∗ itMax + 1  ⎪ 1 itMax − it ⎩ CRit =

2

∗ sin (2 ∗ freq ∗ it + ) ∗

itMax



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Configuration 4:



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itMax

(13)

CRit = 0.9

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Configuration 5:

⎧ Fit = 0.5 ⎪ ⎪ ⎪  ⎨ 1



it CRit = ∗ sin (2 ∗ freq. ∗ it+) ∗ +1 2 itMax ⎪

(14)

⎪ ⎪ ⎩

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Configuration 6:

⎧ ⎨ Fit = 0.5





⎩ CRit = 1 ∗ sin (2 ∗ freq. ∗ it+) ∗ itMax − it + 1 2 itMax

4.1. Benchmark functions In order to test the performances of the proposed sinusoidal differential evolution, we opted for using a recent and complete set of benchmark functions, known in the field of real-parameter single objective optimisation; the CEC-2013 benchmarks [18]. The benchmark set consists of 28 functions: 5 unimodal functions, 15 basic multimodal functions and 8 composition functions. The CEC-2013 testbed expands its CEC-2005 counterpart by adding new test functions, introducing oscillation and symmetric breaking transformations and modifying the formula of composition functions. In fact, these functions span a diverse set of problem features, including multimodality, ruggedness, noise in fitness, illconditioning, non-separability and interdependence (rotation). The whole set of 28 functions are tested with three dimensions D = 10, D = 30 and D = 50; as recommended by the designers of the test suite. These functions exhibit several characteristics of problems with different levels of difficulty. It is this multitude of characteristics which makes this set of functions a reference test suite. Further details about these functions can be found in [18]. 4.2. Best SinDE variant

+1

⎧  ⎨ Fit = 1 ∗ sin (2 ∗ freq. ∗ it) ∗ itMax − it + 1

In this section, the numerical results obtained from testing the proposed SinDE on the chosen set of benchmark functions, introduced in the following, are presented. Detailed statistical comparisons against classical DE and its variants, the CMA-ES and two of its variants, and two variants of the PSO are also discussed.

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(12)

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4. Experimental results

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(15)

Before advanced setting of the SinDE parameters (population size and frequency value), the six configurations of our algorithm formulated above have been compared through applying them on the CEC-2013 benchmark functions, as starting point we set the population size to 50 and the frequency value to 0.05. These parameters are just used to compare the six variants of the proposed SinDE. As recommended in [18], the mean function error from the optimal solution (f(x0 ) − f(x* )) and error standard deviation are reported in all comparison tables. Since all 28 cases are minimisation problems, the aim is to minimize the error; the best results are set in bold in the coming tables. The mutual comparison between the variants of the proposed SinDE and between the SinDE and other approaches is hereafter achieved through the Wilcoxon rank-sum statistics; where the ‘+’ in the tables means the reference algorithm performed better than the other approach, the ‘–’ means it performed worse and the ‘=’ means that both approaches performed equally. In all Wilcoxon rank-sum tests done in this paper, we consider a confidence level of 0.95 (˛ = 0.05). Tables 1–3 show, for the three dimensions respectively, the average error from the optimum, the error standard deviation and the statistical tests (of the 51 runs) of the six variants of the

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Fig. 2. Classical DE, LADE and the proposed configurations of Sinusoidal DE parameter values.

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Table 1 Q4 error, standard deviation of error and Wilcoxon rank-sum statistical comparison, Comparison between different configurations of the proposed SinDE for D = 10; average reference: Configuration 2. Func

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Config 2

Config 1

Config 3

Mean

Std

Mean

Std

0.00E+00 3.94E+04 9.10E−01 6.73E+02 0.00E+00 7.25E+00 1.78E−04 2.04E+01 7.59E−01 5.40E−02 0.00E+00 4.39E+00 6.62E+00 1.79E+00 6.48E+02 1.05E+00 1.08E+01 2.78E+01 6.15E−01 1.94E+00 4.00E+02 3.31E+01 3.95E+02 1.99E+02 2.00E+02 1.10E+02 3.00E+02 3.00E+02

0.00E+00 4.13E+04 1.40E+00 5.73E+02 0.00E+00 4.00E+00 2.98E−04 6.18E−02 8.14E−01 2.79E−02 0.00E+00 1.94E+00 3.13E+00 1.63E+00 2.41E+02 1.87E−01 2.04E−01 3.70E+00 1.31E−01 3.42E−01 1.72E−13 3.65E+01 2.33E+02 1.12E+01 4.88E−04 1.94E+01 0.00E+00 0.00E+00

0.00E+00 7.85E+02 1.10E+00 1.29E+01 0.00E+00 1.67E+00 2.76E−04 2.04E+01 5.94E−01 4.21E−02 7.41E−01 6.56E+00 1.01E+01 1.78E+01 5.41E+02 1.10E+00 1.64E+01 2.91E+01 5.44E−01 1.37E+00 3.92E+02 3.91E+01 2.75E+02 1.97E+02 2.00E+02 1.30E+02 3.04E+02 2.96E+02

0.00E+00 1.60E+03 2.25E+00 2.85E+01 0.00E+00 3.55E+00 4.60E−04 5.94E−02 6.84E−01 2.34E−02 7.13E−01 3.01E+00 5.39E+00 2.30E+01 2.71E+02 1.90E−01 3.65E+00 5.28E+00 1.84E−01 4.03E−01 3.92E+01 3.31E+01 2.16E+02 1.89E+01 1.32E+00 4.03E+01 1.96E+01 2.80E+01

= = = = = = + = + + = + = + + + + + + + = + + = − + + =

Config 4

Mean

Std

5.15E−02 1.52E+05 9.70E+06 9.14E+03 4.43E−01 1.10E+01 1.74E+01 2.04E+01 3.47E+00 1.02E+00 4.88E−01 1.56E+01 2.22E+01 2.28E−01 7.25E+02 1.12E+00 1.09E+01 2.97E+01 5.11E−01 2.74E+00 3.96E+02 5.38E+01 8.99E+02 1.95E+02 2.01E+02 1.35E+02 3.39E+02 2.87E+02

3.68E−01 1.28E+05 1.93E+07 4.86E+03 2.15E+00 1.74E+01 1.44E+01 7.31E−02 1.39E+00 1.88E+00 7.28E−01 7.50E+00 1.04E+01 6.55E−01 2.19E+02 1.96E−01 5.09E−01 6.57E+00 1.98E−01 4.99E−01 2.80E+01 5.21E+01 2.60E+02 3.14E+01 2.04E+01 3.39E+01 5.74E+01 1.37E+02

= = = = = = = = = + = = = + + + + + + + = + = = + = = =

Config 5

Mean

Std

0.00E+00 1.88E+04 4.88E+05 2.10E+02 2.65E−01 5.25E+00 4.49E+00 2.04E+01 2.53E+00 1.26E−01 2.73E+00 1.17E+01 2.07E+01 3.89E+01 6.25E+02 8.29E−01 1.18E+01 1.73E+01 6.13E−01 2.67E+00 3.90E+02 1.30E+02 8.10E+02 2.07E+02 2.03E+02 1.49E+02 3.19E+02 2.74E+02

0.00E+00 2.42E+04 2.66E+06 2.17E+02 1.84E+00 4.71E+00 8.32E+00 6.89E−02 1.28E+00 7.76E−02 2.79E+00 5.06E+00 1.15E+01 4.84E+01 2.56E+02 4.78E−01 1.91E+00 4.85E+00 1.97E−01 6.81E−01 5.01E+01 1.00E+02 2.82E+02 5.61E+00 7.22E+00 4.91E+01 4.29E+01 8.68E+01

= + = = = = − = = = = = = + + + + + + = = + = = = = = =

Config 6

Mean

Std

0.00E+00 4.88E+02 4.26E−01 9.90E+00 0.00E+00 7.28E+00 6.47E−04 2.04E+01 3.49E+00 2.83E−01 0.00E+00 1.58E+01 1.50E+01 2.79E−01 1.17E+03 1.04E+00 1.01E+01 3.27E+01 7.43E−01 2.55E+00 4.00E+02 8.67E+01 1.19E+03 2.00E+02 2.00E+02 1.26E+02 3.00E+02 3.00E+02

0.00E+00 9.85E+02 9.13E−01 2.06E+01 0.00E+00 4.16E+00 7.54E−04 5.93E−02 2.49E+00 6.10E−02 0.00E+00 3.42E+00 3.51E+00 5.13E−01 1.61E+02 2.05E−01 8.97E−15 3.35E+00 1.09E−01 2.84E−01 1.72E−13 6.17E+01 1.83E+02 1.79E−04 2.72E−05 2.74E+01 0.00E+00 0.00E+00

= = = = = = = − = = = = = + + + + + + = = + = = = = = =

Mean

Std

0.00E+00 2.59E+03 4.04E+00 1.15E+01 0.00E+00 2.79E+00 1.25E−03 2.04E+01 5.30E+00 3.63E−01 0.00E+00 2.04E+01 2.01E+01 2.55E−01 1.31E+03 1.15E+00 1.01E+01 3.50E+01 9.38E−01 2.74E+00 4.00E+02 2.18E+02 1.39E+03 1.94E+02 1.99E+02 1.29E+02 3.12E+02 2.73E+02

0.00E+00 4.00E+03 5.85E+00 1.67E+01 0.00E+00 4.15E+00 1.60E−03 5.81E−02 2.16E+00 7.35E−02 0.00E+00 3.82E+00 3.98E+00 5.13E−01 1.42E+02 2.06E−01 8.97E−15 3.73E+00 1.47E−01 3.27E−01 1.72E−13 8.44E+01 1.50E+02 2.06E+01 1.04E+01 2.28E+01 3.25E+01 6.95E+01

= = = = = = = = = = = = = + + + + + + + = + = = = = = =

sinusoidal increasing adjustment, largely outperforms all other configurations. Thus, it will be compared against other state of the art algorithms and considered to be representative of the proposed approach and so it will be simply referred to as SinDE.

proposed SinDE. Table 4 summarises the number of cases where Configuration 2 was better, worse than or performed equally to other configurations in terms of Wilcoxon rank-sum statistics. As seen from the tables, especially from Table 4, Configuration 2 in which both F and CR parameters are subject to

Table 2 Comparison between different configurations of the proposed SinDE for D = 30; average error, standard deviation of error and Wilcoxon rank-sum statistical comparison, reference: Configuration 2. Func

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

Config 2

Config 1

Config 3

Mean

Std

Mean

Std

4.46E−15 3.14E+06 2.24E+04 7.32E+03 1.09E−13 1.48E+01 9.70E−02 2.09E+01 1.49E+01 1.74E−02 1.78E−01 3.11E+01 7.18E+01 6.80E+02 3.75E+03 2.27E+00 5.72E+01 1.43E+02 4.20E+00 1.06E+01 2.69E+02 7.81E+02 3.54E+03 2.00E+02 2.50E+02 2.00E+02 3.01E+02 3.00E+02

3.18E−14 1.30E+06 6.70E+04 2.22E+03 2.23E−14 2.37E+00 1.27E−01 5.78E−02 3.28E+00 1.18E−02 3.84E−01 8.26E+00 1.68E+01 1.62E+02 8.24E+02 2.93E−01 3.40E+00 2.81E+01 9.70E−01 6.69E−01 5.34E+01 2.82E+02 6.96E+02 5.03E−03 5.90E+00 3.18E−02 1.35E+00 0.00E+00

1.20E−13 1.54E+05 9.44E+05 3.87E+02 1.68E−08 2.77E+01 1.85E+00 2.09E+01 1.11E+01 6.13E−02 1.50E+01 2.53E+01 5.79E+01 4.37E+02 6.69E+03 2.39E+00 4.56E+01 1.90E+02 2.52E+00 1.10E+01 3.05E+02 4.16E+02 5.83E+03 2.05E+02 2.49E+02 2.13E+02 3.63E+02 3.00E+02

1.15E−13 8.78E+04 2.30E+06 2.73E+02 1.19E−07 2.45E+01 2.20E+00 4.63E−02 3.08E+00 3.35E−02 4.01E+00 6.81E+00 1.90E+01 2.19E+02 7.35E+02 3.13E−01 3.67E+00 1.82E+01 6.72E−01 6.40E−01 7.12E+01 1.65E+02 1.32E+03 4.11E+00 8.74E+00 3.47E+01 5.54E+01 0.00E+00

= = = = = = + = + + = + = + + + + + + + = + + = − + + =

Config 4

Mean

Std

2.64E+01 1.84E+06 3.57E+07 2.75E+04 6.08E+01 2.93E+01 7.42E+01 2.09E+01 2.53E+01 2.50E−01 1.91E+01 8.51E+01 1.61E+02 7.71E+02 3.95E+03 2.43E+00 7.66E+01 1.05E+02 5.78E+00 1.20E+01 3.48E+02 4.71E+02 4.53E+03 2.56E+02 2.89E+02 2.12E+02 9.25E+02 3.24E+02

8.44E+01 8.70E+05 5.00E+07 9.15E+03 1.50E+02 2.74E+01 2.24E+01 4.15E−02 4.24E+00 1.64E−01 1.03E+01 2.67E+01 3.67E+01 3.34E+02 6.47E+02 2.56E−01 7.92E+00 4.04E+01 3.17E+00 5.98E−01 1.09E+02 3.01E+02 6.15E+02 1.39E+01 7.73E+00 4.21E+01 1.09E+02 2.02E+02

= = = = = = = = = + = = = + + + + + + + = + = = + = = =

Config 5

Mean

Std

1.37E+01 1.33E+05 8.99E+06 5.11E+02 6.79E+01 3.30E+01 4.40E+01 2.10E+01 1.97E+01 2.10E−01 4.02E+01 6.64E+01 1.38E+02 7.01E+02 3.50E+03 2.42E+00 7.39E+01 1.15E+02 4.99E+01 1.09E+01 3.15E+02 9.72E+02 3.55E+03 2.42E+02 2.78E+02 2.25E+02 7.60E+02 2.96E+02

5.87E+01 6.23E+04 1.14E+07 3.70E+02 1.06E+02 2.53E+01 1.72E+01 5.23E−02 3.05E+00 1.23E−01 1.50E+01 2.06E+01 3.05E+01 3.04E+02 7.64E+02 3.96E−01 1.88E+01 6.54E+01 1.44E+02 1.01E+00 7.68E+01 3.91E+02 9.05E+02 1.12E+01 7.95E+00 5.36E+01 1.07E+02 2.80E+01

= + = = = = − = = = = = = + + + + + + = = + = = = = = =

Config 6

Mean

Std

3.12E−14 3.63E+06 2.84E+02 5.99E+03 1.09E−13 1.39E+01 4.11E−01 2.09E+01 3.36E+01 6.91E−03 1.45E+01 1.45E+02 1.60E+02 1.05E+03 6.78E+03 2.40E+00 4.95E+01 2.03E+02 5.65E+00 1.21E+01 2.81E+02 1.50E+03 7.08E+03 2.00E+02 2.86E+02 2.00E+02 3.04E+02 3.00E+02

7.90E−14 1.50E+06 1.63E+03 1.66E+03 2.23E−14 1.09E−01 4.43E−01 5.80E−02 1.15E+00 6.60E−03 2.90E+00 1.24E+01 1.30E+01 1.50E+02 3.21E+02 2.85E−01 2.10E+00 1.09E+01 3.89E−01 2.27E−01 6.06E+01 3.57E+02 2.71E+02 1.21E−02 1.52E+01 3.44E−02 7.47E+00 0.00E+00

= = = = = = = − = = = = = + + + + + + = = + = = = = = =

Mean

Std

4.46E−15 6.65E+06 2.91E+04 1.38E+04 1.07E−13 1.48E+01 1.23E+00 2.09E+01 3.59E+01 1.50E−02 3.76E+01 1.80E+02 1.82E+02 2.11E+03 7.13E+03 2.52E+00 7.15E+01 2.12E+02 8.29E+00 1.24E+01 3.08E+02 2.85E+03 7.35E+03 2.00E+02 3.07E+02 2.00E+02 7.58E+02 3.00E+02

3.18E−14 2.26E+06 1.42E+05 4.90E+03 2.70E−14 7.67E+00 1.85E+00 4.77E−02 1.07E+00 1.19E−02 3.28E+00 9.64E+00 1.19E+01 1.84E+02 2.75E+02 2.80E−01 3.71E+00 1.12E+01 8.90E−01 2.31E−01 7.38E+01 2.68E+02 4.04E+02 1.79E−02 4.47E+00 1.03E−01 3.96E+02 0.00E+00

= = = = = = = = = = = = = + + + + + + + = + = = = = = =

Please cite this article in press as: A. Draa, et al., A sinusoidal differential evolution algorithm for numerical optimisation, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.11.003

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Table 3 Comparison between different configurations of the proposed SinDE for D = 50; average error, standard deviation of error and Wilcoxon rank-sum statistical comparison, reference: Configuration 2. Func

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

409

410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433

Config 2

Config 1

Config 3

Mean

Std

Mean

Std

2.23E−13 4.56E+06 9.99E+03 1.03E+04 1.14E−13 4.34E+01 5.81E−01 2.11E+01 3.36E+01 8.37E−02 8.53E+00 5.67E+01 1.46E+02 2.00E+03 7.59E+03 3.07E+00 1.25E+02 2.24E+02 7.87E+00 2.03E+01 5.85E+02 1.94E+03 7.15E+03 2.00E+02 2.95E+02 2.46E+02 4.11E+02 4.00E+02

3.18E−14 1.30E+06 2.59E+04 2.64E+03 7.65E−29 1.44E−14 8.96E−01 3.76E−02 5.69E+00 4.71E−02 2.53E+00 1.01E+01 3.50E+01 3.45E+02 1.11E+03 2.81E−01 8.59E+00 4.56E+01 2.33E+00 9.62E−01 4.35E+02 4.64E+02 9.93E+02 8.06E−02 9.69E+00 5.81E+01 1.28E+02 0.00E+00

2.85E−13 5.76E+05 1.83E+07 1.82E+03 5.32E−05 4.42E+01 9.85E+00 2.11E+01 2.98E+01 1.32E−01 3.68E+01 5.51E+01 1.36E+02 1.17E+03 1.39E+04 3.36E+00 8.81E+01 3.79E+02 5.47E+00 2.09E+01 7.77E+02 1.15E+03 1.34E+04 2.32E+02 2.99E+02 2.85E+02 7.40E+02 4.60E+02

1.35E−13 2.24E+05 2.43E+07 9.32E+02 2.63E−04 1.49E+00 4.78E+00 3.61E−02 5.60E+00 6.21E−02 9.26E+00 1.10E+01 3.28E+01 3.77E+02 3.48E+02 3.06E−01 8.62E+00 1.64E+01 1.46E+00 4.72E−01 4.08E+02 3.93E+02 1.17E+03 9.62E+00 9.06E+00 6.66E+01 1.20E+02 4.30E+02

= = = = = = + = + + = + = + + + + + + + = + + = − + + =

Config 4

Mean

Std

3.07E+01 2.59E+06 1.31E+08 3.69E+04 4.39E+01 4.84E+01 9.09E+01 2.11E+01 5.29E+01 2.11E−01 6.35E+01 1.88E+02 3.51E+02 1.95E+03 7.55E+03 3.23E+00 1.66E+02 2.08E+02 2.70E+01 2.11E+01 8.14E+02 1.35E+03 9.02E+03 3.20E+02 3.77E+02 3.17E+02 1.61E+03 8.88E+02

1.09E+02 1.01E+06 2.07E+08 1.01E+04 2.96E+02 1.32E+01 1.65E+01 4.56E−02 6.23E+00 1.38E−01 1.99E+01 4.67E+01 6.20E+01 7.52E+02 8.71E+02 4.95E−01 2.04E+01 4.35E+01 1.68E+01 9.58E−01 3.82E+02 7.92E+02 1.05E+03 1.70E+01 1.48E+01 1.16E+02 1.21E+02 1.24E+03

= = = = = = = = = + = = = + + + + + + + = + = = + = = =

4.3. Parameter setting As far as the benchmarks are concerned, the parameters used are those dictated by the designers of the bench suite [18]; three dimensions are considered: D = 10, D = 30 and D = 50. For each test function 51 independent runs are done, each run is continued for T = 10000 × D fitness evaluations. As said above, there are two objectives of the proposed SinDE; improving the DE performances and reducing the number of parameters at the same time. The latter has been achieved through adopting the self-adjustment approach. Consequently, for the different configurations of the SinDE presented above, there are at most two parameters to be set: the frequency of the sinusoidal equations freq and the other parameter to be set in the configurations adjusting only one parameter, Configurations 1, 4, 5 and 6. For all dimensions, functions and all configurations, different population sizes and frequency values are tested, each time we fix one parameter and test different values for the other. So, we started by fixing the frequency value to 0.05 and the population size to 50, and each time, different values for the other parameter are tested. Tables A.1–A.3 and A.4–A.6 of the appendix summarise the results obtained for these settings. The Wilcoxon rank-sum test, is achieved for all results with taking as reference the best parameter configuration. It is clear from the tables that the best settings are popSize = 40 and freq = 0.25. It is worth to mention that 30 individuals offered almost the same values of 40 individuals in the

Config 5

Mean

Std

7.20E+01 5.12E+05 5.75E+07 2.15E+03 3.23E+02 5.22E+01 6.61E+01 2.11E+01 4.22E+01 1.82E−01 1.24E+02 1.59E+02 3.08E+02 2.01E+03 1.29E+04 3.33E+00 2.00E+02 4.21E+02 1.91E+02 2.12E+01 8.40E+02 2.38E+03 9.73E+03 2.98E+02 3.55E+02 3.54E+02 1.32E+03 6.84E+02

3.01E+02 1.92E+05 5.59E+07 8.83E+02 5.51E+02 1.75E+01 1.31E+01 3.27E−02 4.29E+00 1.02E−01 3.04E+01 3.14E+01 4.86E+01 5.86E+02 2.70E+03 3.07E−01 5.58E+01 8.26E+01 3.30E+02 4.80E−01 3.59E+02 7.19E+02 3.67E+03 1.40E+01 1.20E+01 8.21E+01 1.34E+02 9.84E+02

= + = = = = − = = = = = = + + + + + + = = + = = = = = =

Config 6

Mean

Std

2.23E−13 1.35E+07 4.79E+02 1.51E+04 1.16E−13 4.34E+01 9.36E+00 2.11E+01 6.44E+01 2.81E−02 7.30E+01 3.31E+02 3.58E+02 3.74E+03 1.34E+04 3.30E+00 1.28E+02 4.17E+02 1.36E+01 2.20E+01 5.35E+02 4.16E+03 1.38E+04 2.01E+02 3.82E+02 2.53E+02 1.14E+03 4.00E+02

3.18E−14 4.74E+06 1.13E+03 3.53E+03 1.59E−14 1.44E−14 4.93E+00 4.00E−02 2.02E+00 1.44E−02 6.29E+00 1.69E+01 1.56E+01 2.91E+02 4.30E+02 2.40E−01 6.59E+00 1.29E+01 7.45E−01 2.35E−01 4.14E+02 7.13E+02 4.12E+02 2.22E−01 8.82E+00 9.68E+01 5.66E+02 0.00E+00

= = = = = = = − = = = = = + + + + + + = = + = = = = = =

Mean

Std

2.05E−13 1.96E+07 5.27E+05 4.00E+04 1.14E−13 4.34E+01 2.03E+01 2.11E+01 6.80E+01 4.27E−02 1.28E+02 3.81E+02 3.71E+02 5.64E+03 1.39E+04 3.27E+00 1.86E+02 4.21E+02 2.04E+01 2.23E+01 4.18E+02 6.78E+03 1.44E+04 2.14E+02 4.08E+02 2.53E+02 1.95E+03 4.00E+02

6.83E−14 8.21E+06 2.50E+06 4.95E+03 7.65E−29 1.44E−14 1.05E+01 3.02E−02 1.95E+00 2.09E−02 7.12E+00 1.44E+01 1.86E+01 3.84E+02 3.86E+02 3.40E−01 1.10E+01 1.49E+01 1.34E+00 2.36E−01 3.82E+02 5.98E+02 3.70E+02 3.48E+01 4.74E+00 1.00E+02 1.06E+02 0.00E+00

= = = = = = = = = = = = = + + + + + + + = + = = = = = =

population, but we opted for choosing popSize = 40 since when combined with a frequency value equal to 0.25, it gave better results. The parameters used for the basic DE algorithm and the LADE algorithms, with which we have compared our algorithm, are set as follows. In the case of the basic DE, the scaling parameter F is set to 0.5 and the crossover rate CR is chosen to be 0.9. These values are in fact those recommended in literature [22]. For the LADE variant, the parameters Fmin , Fmax , CRmin and CRmax are set to 0.3, 0.5, 0.7 and 0.9 respectively. These values have been extracted from literature analyses of the LADE variants [11]. 4.4. Numerical results

D = 10 D = 30 D = 50 Total

Config 3

This section presents the numerical results obtained from applying the proposed SinDE on the set of CEC-2013 benchmarks. The parameters are set to popSize = 40 and freq = 0.25, proven to be the best as seen above. Tables 5–7 report the best, worst, median, mean and standard deviation of error from optimum solution of the proposed SinDE over 51 runs for all 28 benchmark functions. Tables 8–10 compare, for the three dimensions, our SinDE algorithm against conventional DE with recommended values of parameters, F = 0.5 and CR = 0.9, and against the LADE algorithm. It is clear from the three tables that the proposed SinDE is much more powerful than both other approaches. In fact, it significantly outperforms the calssical DE in 75 cases out of 84 and the LADE in 55

Config 4

435 436 437 438 439 440 441 442 443

444

Table 4 Summarised Wilcoxon rank-sum comparisons between different configurations of SinDE, reference: Configuration 2. Config 1

434

Config 5

Config 6

Worse

Equal

Better

Worse

Equal

Better

Worse

Equal

Better

Worse

Equal

Better

Worse

Equal

Better

6 9 7 22

15 4 6 25

7 15 15 37

4 3 3 10

6 5 4 15

18 20 21 59

5 4 2 11

6 4 1 11

17 20 25 62

5 5 2 12

11 5 5 21

12 18 21 51

7 1 2 10

7 5 5 17

14 22 21 57

Please cite this article in press as: A. Draa, et al., A sinusoidal differential evolution algorithm for numerical optimisation, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.11.003

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Table 5 Best, worst, median, mean et standard deviation of error over 51 runs for D = 10.

458 459 460 461 462 463 464 465 466 467

Function

Best

Worst

Median

Mean

Std

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

0.00E+00 1.98E+03 4.63E−04 9.14E+01 0.00E+00 4.68E−01 7.40E−08 2.02E+01 1.61E−07 0.00E+00 0.00E+00 1.99E+00 1.99E+00 0.00E+00 1.55E+01 4.79E−01 1.01E+01 1.31E+01 1.62E−01 8.07E−01 4.00E+02 1.14E−13 2.50E+01 1.11E+02 2.00E+02 1.01E+02 3.00E+02 3.00E+02

0.00E+00 2.55E+05 9.03E+01 3.97E+03 0.00E+00 9.82E+00 1.23E−03 2.05E+01 2.79E+00 1.55E−01 0.00E+00 1.09E+01 1.93E+01 4.37E−01 9.21E+02 1.22E+00 1.03E+01 2.45E+01 5.62E−01 3.25E+00 4.00E+02 1.03E+02 8.36E+02 2.07E+02 2.00E+02 2.00E+02 3.00E+02 3.00E+02

0.00E+00 3.24E+04 1.44E−01 4.70E+02 0.00E+00 9.81E+00 5.59E−05 2.04E+01 3.94E−01 5.66E−02 0.00E+00 4.97E+00 7.46E+00 1.25E−01 4.52E+02 9.93E−01 1.01E+01 1.78E+01 3.43E−01 1.76E+00 4.00E+02 1.10E+01 3.43E+02 2.00E+02 2.00E+02 1.05E+02 3.00E+02 3.00E+02

0.00E+00 4.41E+04 2.23E+00 7.17E+02 0.00E+00 7.74E+00 1.12E−04 2.04E+01 7.45E−01 5.94E−02 0.00E+00 5.36E+00 7.62E+00 1.27E−01 4.39E+02 9.46E−01 1.01E+01 1.75E+01 3.50E−01 1.76E+00 4.00E+02 1.67E+01 3.45E+02 1.96E+02 2.00E+02 1.09E+02 3.00E+02 3.00E+02

0.00E+00 4.43E+04 1.26E+01 7.43E+02 0.00E+00 3.78E+00 1.92E−04 7.60E−02 8.57E−01 3.12E−02 0.00E+00 1.95E+00 3.25E+00 8.77E−02 1.76E+02 1.76E−01 2.29E−02 2.48E+00 8.87E−02 4.65E−01 1.72E−13 2.29E+01 1.96E+02 1.68E+01 2.29E−04 1.87E+01 0.00E+00 0.00E+00

cases out of 84. On the other hand, the DE outperformed the SinDE algorithm in only 3 cases out of 84, while the LADE was better in only 13 cases out of 84. In addition to its comparison to the classical variants, the proposed SinDE is compared against stronger recent variants of the DE algorithm, namely: The Super-fit Multicriteria Adaptive Differential Evolution (SMADE) [30], the Differential Evolution with Concurrent Fitness Based Local Search (DEcfbLS) [31], a powerful variant of the competitive differential Evolution [32] called b6e6rl [33] and the Teaching and Learning Based Self-adaptive DE (TLBSade) [34].

Such comparisons are basically motivated by the fact that these approaches gave very interesting results and proved to be superior to other metaheuristics in recent publications; they have also been tested for CEC-2013 testbed and gave very good results. Tables 11–13 present the results of comparing our SinDE algorithm against these state-of-the-art DE algorithms. The tables give the average error from the optimal solution (f(x0 ) − f(x* )), the standard deviation of error and the statistical Wilcoxon rank-sum test, taking as reference the SinDE algorithm. From the tables, it is clear that the SinDE algorithm outperforms the four DE algorithms

Table 6 Best, worst, median, mean et standard deviation of error over 51 runs for D = 30. Function

Best

Worst

Median

Mean

Std

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

0.00E+00 9.43E+05 4.51E−05 2.71E+03 1.14E−13 1.39E+01 4.87E−04 2.08E+01 8.94E+00 0.00E+00 0.00E+00 1.59E+01 1.70E+01 1.31E+01 1.89E+03 1.14E+00 3.21E+01 5.35E+01 1.39E+00 8.53E+00 2.00E+02 7.95E+01 1.81E+03 2.00E+02 2.34E+02 2.00E+02 3.00E+02 3.00E+02

2.27E−13 3.61E+06 1.17E+06 1.27E+04 1.14E−13 2.64E+01 7.20E−01 2.10E+01 2.11E+01 6.16E−02 9.95E−01 6.28E+01 1.17E+02 1.08E+02 3.84E+03 2.24E+00 3.61E+01 1.18E+02 3.09E+00 1.11E+01 4.44E+02 1.81E+02 4.15E+03 2.00E+02 2.62E+02 3.00E+02 3.17E+02 3.00E+02

0.00E+00 2.11E+06 1.40E+03 6.14E+03 1.14E−13 1.41E+01 6.24E−02 2.09E+01 1.53E+01 2.22E−02 5.68E−14 2.90E+01 7.43E+01 4.42E+01 2.96E+03 1.79E+00 3.36E+01 7.79E+01 2.21E+00 9.99E+00 3.00E+02 1.49E+02 3.11E+03 2.00E+02 2.48E+02 2.00E+02 3.00E+02 3.00E+02

2.23E−14 2.16E+06 8.49E+04 6.38E+03 1.14E−13 1.46E+01 1.21E−01 2.09E+01 1.52E+01 2.04E−02 1.95E−02 3.02E+01 7.33E+01 5.04E+01 2.95E+03 1.74E+00 3.37E+01 7.86E+01 2.24E+00 9.99E+00 2.87E+02 1.49E+02 3.14E+03 2.00E+02 2.49E+02 2.02E+02 3.01E+02 3.00E+02

6.83E−14 6.15E+05 2.11E+05 2.00E+03 7.65E−29 2.42E+00 1.57E−01 4.96E−02 3.05E+00 1.30E−02 1.39E−01 8.65E+00 2.07E+01 1.92E+01 4.86E+02 2.52E−01 7.97E−01 1.42E+01 3.79E−01 5.50E−01 6.40E+01 1.76E+01 5.31E+02 7.16E−03 6.85E+00 1.40E+01 2.36E+00 0.00E+00

Please cite this article in press as: A. Draa, et al., A sinusoidal differential evolution algorithm for numerical optimisation, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.11.003

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Table 7 Best, worst, median, mean et standard deviation of error over 51 runs for D = 50.

478 479 480 481 482 483 484 485

Function

Best

Worst

Median

Mean

Std

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

2.27E−13 1.04E+06 1.01E−02 5.36E+03 1.14E−13 4.34E+01 1.23E−02 2.10E+01 2.41E+01 2.46E−02 9.95E−01 1.90E+01 7.52E+01 9.86E+01 4.46E+03 1.15E+00 5.87E+01 7.39E+01 3.37E+00 1.76E+01 2.00E+02 1.20E+02 4.56E+03 2.00E+02 2.68E+02 2.00E+02 3.03E+02 4.00E+02

2.27E−13 5.13E+06 2.58E+06 1.13E+04 1.14E−13 4.34E+01 3.25E+00 2.12E+01 4.36E+01 1.85E−01 1.19E+01 8.86E+01 2.05E+02 5.53E+02 8.74E+03 2.79E+00 7.31E+01 1.87E+02 7.15E+00 2.06E+01 1.12E+03 1.50E+03 8.28E+03 2.01E+02 3.28E+02 3.63E+02 8.94E+02 4.00E+02

2.27E−13 2.58E+06 1.44E+03 8.09E+03 1.14E−13 4.34E+01 4.15E−01 2.11E+01 3.54E+01 7.39E−02 4.99E+00 5.48E+01 1.47E+02 2.26E+02 6.69E+03 2.09E+00 6.48E+01 1.41E+02 4.72E+00 1.93E+01 2.00E+02 2.59E+02 6.59E+03 2.00E+02 2.96E+02 3.11E+02 4.34E+02 4.00E+02

2.27E−13 2.66E+06 1.01E+05 8.28E+03 1.14E−13 4.34E+01 6.10E−01 2.11E+01 3.48E+01 7.93E−02 5.92E+00 5.61E+01 1.39E+02 2.34E+02 6.80E+03 2.08E+00 6.52E+01 1.41E+02 4.85E+00 1.92E+01 5.84E+02 3.51E+02 6.59E+03 2.00E+02 2.97E+02 2.76E+02 4.75E+02 4.00E+02

1.53E−28 8.33E+05 3.77E+05 1.54E+03 7.65E−29 1.44E−14 5.97E−01 3.59E−02 4.34E+00 3.57E−02 2.86E+00 1.41E+01 3.41E+01 9.23E+01 1.00E+03 3.66E−01 3.47E+00 2.27E+01 8.82E−01 7.52E−01 4.22E+02 2.72E+02 8.47E+02 1.34E−01 1.33E+01 5.96E+01 1.55E+02 0.00E+00

we compared it to, especially for high dimensions. The statistical tests are summarised in Table 14. As seen from the tables, the power of the SinDE algorithm is specially observed for composition functions, known to be harder to optimise. The Covariance Matrix Adaptation Evolution Strategy (CMAES) is one of the most powerful metaheuristics that emerged the last decade. This approach is known for its unique ability of being invariant to landscape transformations, scaling modulation

and linear transformation of the search space [35–37]. Many efficient variants of the CMA-ES have recently appeared and proven their efficiency in solving continuous optimisation problems; the Restart CMA Evolution Strategy (CMA-ES-RIS) [38] is one of these variants. Tables 15–17 compare the proposed SinDE to the CMA-ES-RIS. As seen from the tables, the SinDE algorithm largely outperformed the CMA-ES-RIS algorithm. It was better in 46 cases out of 84 and

Table 8 Comparison between the proposed SinDE, conventional DE and the LADE algorithm for D = 10, reference: SinDE. Func

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

SinDE

Basic

LADE

Mean

Std

Mean

Std

0.00E+00 4.41E+04 2.23E+00 7.17E+02 0.00E+00 7.74E+00 1.12E−04 2.04E+01 7.45E−01 5.94E−02 0.00E+00 5.36E+00 7.62E+00 1.27E−01 4.39E+02 9.46E−01 1.01E+01 1.75E+01 3.50E−01 1.76E+00 4.00E+02 1.67E+01 3.45E+02 1.96E+02 2.00E+02 1.09E+02 3.00E+02 3.00E+02

0.00E+00 4.43E+04 1.26E+01 7.43E+02 0.00E+00 3.78E+00 1.92E−04 7.60E−02 8.57E−01 3.12E−02 0.00E+00 1.95E+00 3.25E+00 8.77E−02 1.76E+02 1.76E−01 2.29E−02 2.48E+00 8.87E−02 4.65E−01 1.72E−13 2.29E+01 1.96E+02 1.68E+01 2.29E−04 1.87E+01 0.00E+00 0.00E+00

7.96E+01 1.20E+05 5.95E+07 3.60E+03 7.44E+01 2.10E+01 2.34E+01 2.03E+01 3.51E+00 2.49E+01 1.35E+01 1.64E+01 2.73E+01 6.49E+01 5.98E+02 1.02E+00 1.46E+01 2.52E+01 8.56E+00 2.50E+00 3.99E+02 1.23E+02 4.93E+02 2.12E+02 2.11E+02 1.37E+02 4.35E+02 5.73E+02

1.08E+02 1.27E+05 2.91E+08 2.65E+03 8.20E+01 2.85E+01 1.60E+01 7.97E−02 1.02E+00 3.06E+01 6.85E+00 7.05E+00 1.31E+01 8.33E+01 2.79E+02 3.08E−01 4.06E+00 8.62E+00 2.30E+01 6.46E−01 2.14E+01 1.09E+02 2.42E+02 1.39E+01 3.92E+00 4.16E+01 5.49E+01 1.97E+02

+ − + + + + + = + = + + + + + + + + + + + + + + + + + +

Mean

Std

0.00E+00 1.82E+03 5.59E−01 9.22E+00 0.00E+00 2.27E+00 8.80E−05 2.04E+01 4.05E−01 4.59E−02 1.03E+00 8.90E+00 1.13E+01 2.67E+02 1.22E+03 1.02E+00 1.72E+01 3.07E+01 1.09E+00 2.11E+00 3.88E+02 1.95E+02 9.84E+02 1.97E+02 2.00E+02 1.36E+02 3.08E+02 2.76E+02

0.00E+00 5.43E+03 1.17E+00 2.60E+01 0.00E+00 4.00E+00 4.09E−04 6.87E−02 7.17E−01 3.23E−02 1.52E+00 5.56E+00 6.66E+00 3.50E+02 2.44E+02 2.65E−01 5.42E+00 4.20E+00 5.11E−01 3.47E−01 4.76E+01 2.84E+02 3.67E+02 1.78E+01 6.36E−01 4.22E+01 2.72E+01 6.51E+01

+ − + − + + + = = + + + + + + + + + + + = + + + = + + =

Please cite this article in press as: A. Draa, et al., A sinusoidal differential evolution algorithm for numerical optimisation, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.11.003

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Table 9 Comparison between the proposed SinDE, conventional DE and the LADE algorithm for D = 30, reference: SinDE. Func

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

SinDE

Basic Std

Mean

Std

2.23E−14 2.16E+06 8.49E+04 6.38E+03 1.14E−13 1.46E+01 1.21E−01 2.09E+01 1.52E+01 2.04E−02 1.95E−02 3.02E+01 7.33E+01 5.04E+01 2.95E+03 1.74E+00 3.37E+01 7.86E+01 2.24E+00 9.99E+00 2.87E+02 1.49E+02 3.14E+03 2.00E+02 2.49E+02 2.02E+02 3.01E+02 3.00E+02

6.83E−14 6.15E+05 2.11E+05 2.00E+03 7.65E−29 2.42E+00 1.57E−01 4.96E−02 3.05E+00 1.30E−02 1.39E−01 8.65E+00 2.07E+01 1.92E+01 4.86E+02 2.52E−01 7.97E−01 1.42E+01 3.79E−01 5.50E−01 6.40E+01 1.76E+01 5.31E+02 7.16E−03 6.85E+00 1.40E+01 2.36E+00 0.00E+00

9.96E+01 9.78E+05 4.04E+07 1.85E+04 4.36E+02 3.93E+01 4.76E+02 2.09E+01 2.95E+01 1.05E−01 2.58E+02 2.82E+02 3.82E+02 1.80E+03 3.51E+03 2.39E+00 3.20E+02 3.57E+02 6.34E+01 1.20E+01 3.57E+02 2.18E+03 3.85E+03 3.16E+02 3.31E+02 2.48E+02 1.13E+03 3.33E+03

3.34E+02 5.38E+05 5.89E+07 6.00E+03 9.34E+02 2.89E+01 5.72E+02 4.87E−02 2.07E+00 1.82E−01 7.19E+01 6.53E+01 5.99E+01 5.48E+02 9.28E+02 3.75E−01 6.60E+01 7.00E+01 2.65E+02 1.06E+00 1.47E+02 5.79E+02 7.90E+02 1.90E+01 1.24E+01 7.83E+01 6.46E+01 5.99E+02

495

the two algorithms performed equally in 6 cases out of 84. The CMA-ES-RIS was better, on the other hand, in 31 cases out of 84.

496

4.5. Holm–Bonferroni based comparison

494

497 498

LADE

Mean

As a further step of statistical validation of the proposed SinDE, the Holm–Bonferroni statistical ranking procedure [19,39] is used

+ − + + + + + = + = + + + + + + + + + + + + + + + + + +

Mean

Std

6.24E−14 1.16E+05 7.07E+05 3.66E+02 2.50E−13 2.13E+01 8.78E−01 2.10E+01 1.36E+01 6.46E−02 1.41E+01 1.17E+02 1.49E+02 6.55E+02 7.12E+03 2.46E+00 5.45E+01 2.00E+02 4.22E+00 1.17E+01 3.23E+02 6.19E+02 7.29E+03 2.05E+02 2.43E+02 2.00E+02 3.50E+02 3.00E+02

1.02E−13 5.61E+04 1.19E+06 2.31E+02 6.65E−13 1.94E+01 1.07E+00 4.64E−02 1.11E+01 3.63E−02 3.89E+00 6.34E+01 4.18E+01 4.18E+02 3.19E+02 3.19E−01 1.01E+01 9.80E+00 3.55E+00 3.15E−01 9.07E+01 6.15E+02 3.22E+02 3.69E+00 1.48E+01 3.81E−03 4.59E+01 0.00E+00

+ − + − + + + = = + + + + + + + + + + + = + + + = + + =

here to compare the proposed SinDE to the above mentioned state-of-the-art approaches and other recently proposed ones. This procedure is used in the same way as described in [40]. This statistics is based on the notion of average ranking; so only mean values of best errors to the optimum over 51 runs are needed. We compare then our SinDE algorithm, in addition to all the approaches seen previously, against the Self-Adaptive Differential

Table 10 Comparison between the proposed SinDE, conventional DE and the LADE algorithm for D = 50, reference: SinDE. Func

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

SinDE

Basic

LADE

Mean

Std

Mean

Std

2.27E−13 2.66E+06 1.01E+05 8.28E+03 1.14E−13 4.34E+01 6.10E−01 2.11E+01 3.48E+01 7.93E−02 5.92E+00 5.61E+01 1.39E+02 2.34E+02 6.80E+03 2.08E+00 6.52E+01 1.41E+02 4.85E+00 1.92E+01 5.84E+02 3.51E+02 6.59E+03 2.00E+02 2.97E+02 2.76E+02 4.75E+02 4.00E+02

1.53E−28 8.33E+05 3.77E+05 1.54E+03 7.65E−29 1.44E−14 5.97E−01 3.59E−02 4.34E+00 3.57E−02 2.86E+00 1.41E+01 3.41E+01 9.23E+01 1.00E+03 3.66E−01 3.47E+00 2.27E+01 8.82E−01 7.52E−01 4.22E+02 2.72E+02 8.47E+02 1.34E−01 1.33E+01 5.96E+01 1.55E+02 0.00E+00

1.75E−03 1.55E+06 1.02E+08 2.46E+04 5.92E+01 6.52E+01 1.75E+02 2.11E+01 5.86E+01 1.04E−01 5.05E+02 5.22E+02 7.35E+02 4.73E+03 8.00E+03 3.01E+00 7.90E+02 7.94E+02 2.46E+02 2.17E+01 7.94E+02 5.75E+03 7.72E+03 4.63E+02 4.72E+02 3.87E+02 2.09E+03 6.11E+03

1.20E−02 5.33E+05 1.24E+08 6.74E+03 2.17E+02 3.26E+01 7.91E+01 3.97E−02 3.11E+00 6.53E−02 8.25E+01 8.90E+01 8.09E+01 9.75E+02 2.41E+03 8.31E−01 1.28E+02 1.39E+02 1.11E+03 1.18E+00 3.39E+02 9.16E+02 1.79E+03 4.37E+01 1.92E+01 1.16E+02 1.10E+02 4.84E+02

+ − + + + + + = + = + + + + + + + + + + + + + + + + + +

Mean

Std

2.72E−13 4.47E+05 1.36E+07 2.12E+03 3.70E−13 4.37E+01 7.86E+00 2.11E+01 5.03E+01 1.13E−01 3.68E+01 3.16E+02 3.52E+02 1.28E+03 1.40E+04 3.30E+00 9.63E+01 3.93E+02 5.78E+00 2.16E+01 6.56E+02 1.15E+03 1.42E+04 2.31E+02 2.98E+02 2.96E+02 7.26E+02 4.00E+02

9.12E−14 1.44E+05 2.14E+07 8.49E+02 1.45E−13 1.11E+00 3.81E+00 4.17E−02 2.54E+01 6.39E−02 8.55E+00 9.04E+01 1.39E+01 4.53E+02 3.10E+02 3.04E−01 9.39E+00 1.39E+01 2.24E+00 3.13E−01 4.29E+02 4.20E+02 3.54E+02 9.25E+00 8.30E+00 6.00E+01 1.27E+02 0.00E+00

+ − + − + + + = = + + + + + + + + + + + = + + + = + + =

Please cite this article in press as: A. Draa, et al., A sinusoidal differential evolution algorithm for numerical optimisation, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.11.003

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Table 11 Comparing the proposed SinDE against the SMADE [30], the DEcfbLS [31], the b6e6rl [33] and the TLBSade [34] algorithms for D = 10, reference: SinDE. Func

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

506 507 508 509 510 511

SinDE

SMADE

DEcfbLS

Mean

Std

Mean

Std

0.00E+00 4.41E+04 2.23E+00 7.17E+02 0.00E+00 7.74E+00 1.12E−04 2.04E+01 7.45E−01 5.94E−02 0.00E+00 5.36E+00 7.62E+00 1.27E−01 4.39E+02 9.46E−01 1.01E+01 1.75E+01 3.50E−01 1.76E+00 4.00E+02 1.67E+01 3.45E+02 1.96E+02 2.00E+02 1.09E+02 3.00E+02 3.00E+02

0.00E+00 4.43E+04 1.26E+01 7.43E+02 0.00E+00 3.78E+00 1.92E−04 7.60E−02 8.57E−01 3.12E−02 0.00E+00 1.95E+00 3.25E+00 8.77E−02 1.76E+02 1.76E−01 2.29E−02 2.48E+00 8.87E−02 4.65E−01 1.72E−13 2.29E+01 1.96E+02 1.68E+01 2.29E−04 1.87E+01 0.00E+00 0.00E+00

0.00E+00 0.00E+00 2.48E−01 0.00E+00 0.00E+00 5.41E+00 2.27E+00 2.03E+01 2.29E+00 1.42E−02 9.75E−02 7.80E+00 1.21E+01 3.64E+00 7.36E+02 4.04E−01 1.03E+01 2.46E+01 3.95E−01 2.65E+00 3.83E+02 4.93E+01 5.78E+02 2.02E+02 2.02E+02 1.26E+02 3.37E+02 3.17E+02

0.00E+00 0.00E+00 1.24E+00 0.00E+00 0.00E+00 4.81E+00 4.50E+00 1.04E−01 7.26E−01 9.67E−03 2.99E−01 4.14E+00 6.47E+00 4.44E+00 2.63E+02 3.17E−01 1.56E−01 4.73E+00 1.26E−01 4.52E−01 5.56E+01 5.38E+01 3.20E+02 1.78E+01 1.93E+00 3.73E+01 5.29E+01 6.94E+01

− − + − − = + = + − + + + + + − = + + = + = + + + = + +

rf33

Mean

Std

0.00E+00 1.01E+02 1.14E+00 8.19E−01 0.00E+00 1.35E+00 8.71E−01 2.03E+01 3.54E+00 3.29E−02 1.95E−02 6.15E+00 1.19E+01 1.84E−02 5.27E+02 2.79E−01 9.80E+00 1.65E+01 2.90E−01 2.56E+00 4.00E+02 3.08E+01 6.56E+02 1.13E+02 1.82E+02 1.10E+02 3.90E+02 2.41E+02

0.00E+00 6.15E+02 2.02E+00 4.61E+00 0.00E+00 3.41E+00 8.29E−01 1.13E−01 1.10E+00 1.72E−02 1.39E−01 1.90E+00 4.44E+00 3.13E−02 1.40E+02 2.01E−01 1.53E+00 2.69E+00 6.27E−02 4.05E−01 4.59E−13 1.90E+01 1.61E+02 2.21E+01 3.74E+01 1.32E+01 3.00E+01 9.20E+01

Evolution with pBX crossover (MDE−pBX) [41], the CMA-ES algorithm [42,43], the Restart CMA Evolution Strategy With Increasing Population Size (G-CMA-ES) [44] and against two recent variants of the PSO: the Comprehensive Learning Particle Swarm Optimizer (CLPSO) [45] and the Cooperatively Coevolving Particle Swarms Optimizer (CCPSO2) [46].

− − + − − + + = + − − + + = − − = + − + = + + + + − + =

TLBSade

Mean

Std

0.00E+00 1.82E−04 3.94E−01 2.12E−09 0.00E+00 4.23E+00 2.96E−03 2.04E+01 1.78E+00 3.51E−02 0.00E+00 8.07E+00 9.57E+00 7.10E−02 6.99E+02 9.48E−01 1.01E+01 2.49E+01 3.38E−01 2.31E+00 3.83E+02 1.98E+01 7.05E+02 2.04E+02 2.00E+02 1.55E+02 3.25E+02 2.92E+02

0.00E+00 1.28E−03 1.50E+00 1.07E−08 0.00E+00 4.91E+00 6.41E−03 7.90E−02 1.87E+00 2.54E−02 0.00E+00 2.25E+00 3.41E+00 6.37E−02 1.72E+02 2.49E−01 1.26E−14 4.75E+00 4.13E−02 2.98E−01 6.24E+01 2.53E+01 2.02E+02 3.52E+00 1.31E+01 4.64E+01 6.25E+01 3.92E+01

− − + − − − + = + − − + + − + + − + − + = − + + + = + =

Mean

Std

0.00E+00 0.00E+00 9.58E−01 0.00E+00 0.00E+00 0.00E+00 4.54E−01 2.02E+01 4.28E+00 4.88E−03 0.00E+00 5.44E+00 5.67E+00 1.61E+00 5.61E+02 7.75E−01 1.05E+01 1.90E+01 4.82E−01 2.59E+00 1.65E+02 1.01E+02 7.95E+02 1.75E+02 2.09E+02 1.09E+02 4.06E+02 9.61E+01

0.00E+00 0.00E+00 5.79E−01 0.00E+00 0.00E+00 0.00E+00 1.56E−01 4.07E−02 3.34E−01 3.74E−03 0.00E+00 6.53E−01 1.79E+00 6.30E−01 6.16E+01 8.93E−02 9.59E−02 1.09E+00 4.77E−02 2.07E−01 5.53E+01 1.67E+01 1.21E+02 2.03E+01 1.48E+01 2.32E+00 5.63E+00 1.96E+01

− − + − − − + − + − − + + + + − + + + = − + + + + − + =

The Holm–Bonferroni procedure is used with the same parameters used in [40]. The null hypothesis is set to: ‘the proposed SinDE has the same performance as the j-th algorithm’, where j is the algorithm in question (SMADE, DEcfbLS, CMAES-RIS, etc.). Table 18 displays the ranks, zj values, pj values, and corresponding /j values for all 12 algorithms with which our SinDE is compared. The rank

Table 12 Comparing the proposed SinDE against the SMADE [30], the DEcfbLS [31], the b6e6rl [33] and the TLBSade [34] algorithms for D = 30, reference: SinDE. Func

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

SinDE

SMADE

DEcfbLS

Mean

Std

Mean

Std

2.23E−14 2.16E+06 8.49E+04 6.38E+03 1.14E−13 1.46E+01 1.21E−01 2.09E+01 1.52E+01 2.04E−02 1.95E−02 3.02E+01 7.33E+01 5.04E+01 2.95E+03 1.74E+00 3.37E+01 7.86E+01 2.24E+00 9.99E+00 2.87E+02 1.49E+02 3.14E+03 2.00E+02 2.49E+02 2.02E+02 3.01E+02 3.00E+02

6.83E−14 6.15E+05 2.11E+05 2.00E+03 7.65E−29 2.42E+00 1.57E−01 4.96E−02 3.05E+00 1.30E−02 1.39E−01 8.65E+00 2.07E+01 1.92E+01 4.86E+02 2.52E−01 7.97E−01 1.42E+01 3.79E−01 5.50E−01 6.40E+01 1.76E+01 5.31E+02 7.16E−03 6.85E+00 1.40E+01 2.36E+00 0.00E+00

0.00E+00 0.00E+00 9.82E+03 0.00E+00 0.00E+00 2.67E+00 3.25E+01 2.10E+01 2.23E+01 1.84E−02 1.09E+01 5.72E+01 1.28E+02 1.33E+02 4.10E+03 1.31E−01 3.48E+01 8.33E+01 2.55E+00 1.05E+01 3.27E+02 1.79E+02 4.22E+03 2.32E+02 2.78E+02 2.15E+02 6.47E+02 3.88E+02

0.00E+00 0.00E+00 4.99E+04 0.00E+00 0.00E+00 7.92E+00 1.63E+01 4.85E−02 3.61E+00 1.35E−02 4.23E+00 1.72E+01 3.53E+01 1.28E+02 8.55E+02 7.65E−02 1.54E+00 2.08E+01 5.23E−01 8.15E−01 8.73E+01 4.54E+01 8.83E+02 2.60E+01 1.00E+01 5.30E+01 1.39E+02 3.27E+02

− − + − − = + = + − + + + + + − = + + = + = + + + = + +

rf33

Mean

Std

0.00E+00 1.99E+05 2.11E+06 3.82E+02 0.00E+00 7.08E+00 5.68E+01 2.09E+01 2.40E+01 2.01E−02 5.85E−02 5.42E+01 1.01E+02 3.29E+01 3.43E+03 7.27E−01 3.53E+01 7.94E+01 1.50E+00 1.17E+01 3.36E+02 2.56E+02 3.59E+03 2.64E+02 2.83E+02 2.00E+02 9.38E+02 3.00E+02

0.00E+00 1.08E+05 4.69E+06 5.17E+02 0.00E+00 4.21E+00 1.68E+01 9.54E−02 2.89E+00 1.74E−02 2.36E−01 8.89E+00 1.88E+01 1.36E+01 1.09E+03 9.02E−01 1.15E+00 1.11E+01 1.76E−01 6.59E−01 9.88E+01 9.22E+01 5.04E+02 9.24E+00 5.85E+00 6.68E−03 5.81E+01 0.00E+00

− − + − − + + = + − − + + = − − = + − + = + + + + − + =

TLBSade

Mean

Std

0.00E+00 7.00E+04 4.36E+03 1.81E−02 0.00E+00 5.25E+00 2.44E+01 2.09E+01 2.86E+01 1.91E−02 0.00E+00 8.36E+01 1.14E+02 2.37E−02 4.66E+03 1.98E+00 3.04E+01 1.72E+02 1.84E+00 1.18E+01 2.97E+02 1.23E+02 5.01E+03 2.52E+02 2.75E+02 2.10E+02 1.01E+03 3.00E+02

0.00E+00 4.41E+04 1.36E+04 2.88E−02 0.00E+00 9.90E+00 8.96E+00 4.73E−02 1.15E+00 1.34E−02 0.00E+00 1.30E+01 1.40E+01 2.50E−02 3.39E+02 3.85E−01 4.15E−05 1.35E+01 1.46E−01 3.24E−01 8.55E+01 1.63E+01 4.06E+02 1.38E+01 1.76E+01 3.94E+01 7.44E+01 0.00E+00

− − + − − − + = + − − + + − + + − + − + = − + + + = + =

Mean

Std

st

0.00E+00 6.73E+03 3.91E+01 2.34E+00 0.00E+00 1.04E−02 1.54E+01 2.08E+01 2.69E+01 1.62E−02 0.00E+00 4.99E+01 8.58E+01 3.07E+01 3.61E+03 1.48E+00 3.25E+01 7.68E+01 2.67E+00 1.06E+01 2.67E+02 2.90E+02 4.34E+03 3.03E+02 2.96E+02 2.00E+02 1.19E+03 2.96E+02

0.00E+00 2.55E+03 6.89E+01 1.99E+00 0.00E+00 2.14E−02 3.35E+00 5.14E−02 3.92E+00 6.93E−03 0.00E+00 3.49E+00 8.65E+00 1.14E+01 2.07E+02 2.08E−01 7.03E−01 7.19E+00 2.31E−01 3.66E−01 4.76E+01 1.22E+02 3.57E+02 2.66E+00 2.01E+00 0.00E+00 1.20E+02 2.80E+01

− − + − − − + − + − − + + + + − + + + = − + + + + − + =

Please cite this article in press as: A. Draa, et al., A sinusoidal differential evolution algorithm for numerical optimisation, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.11.003

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Table 13 Comparing the proposed SinDE against the SMADE [30], the DEcfbLS [31], the b6e6rl [33] and the TLBSade [34] algorithms for D = 50, reference: SinDE. Func

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

SinDE

SMADE

DEcfbLS

Mean

Std

Mean

Std

2.27E−13 2.66E+06 1.01E+05 8.28E+03 1.14E−13 4.34E+01 6.10E−01 2.11E+01 3.48E+01 7.93E−02 5.92E+00 5.61E+01 1.39E+02 2.34E+02 6.80E+03 2.08E+00 6.52E+01 1.41E+02 4.85E+00 1.92E+01 5.84E+02 3.51E+02 6.59E+03 2.00E+02 2.97E+02 2.76E+02 4.75E+02 4.00E+02

1.53E−28 8.33E+05 3.77E+05 1.54E+03 7.65E−29 1.44E−14 5.97E−01 3.59E−02 4.34E+00 3.57E−02 2.86E+00 1.41E+01 3.41E+01 9.23E+01 1.00E+03 3.66E−01 3.47E+00 2.27E+01 8.82E−01 7.52E−01 4.22E+02 2.72E+02 8.47E+02 1.34E−01 1.33E+01 5.96E+01 1.55E+02 0.00E+00

0.00E+00 0.00E+00 3.81E+05 0.00E+00 0.00E+00 4.30E+01 4.32E+01 2.11E+01 4.36E+01 2.47E−02 4.81E+01 1.57E+02 3.35E+02 3.41E+02 8.54E+03 8.96E−02 6.57E+01 1.93E+02 5.43E+00 1.92E+01 8.46E+02 3.39E+02 9.89E+03 3.00E+02 3.68E+02 2.91E+02 1.18E+03 1.07E+03

0.00E+00 0.00E+00 1.36E+06 0.00E+00 0.00E+00 6.34E+00 1.68E+01 3.89E−02 4.10E+00 1.50E−02 1.50E+01 4.56E+01 5.69E+01 2.07E+02 9.87E+02 4.28E−02 5.33E+00 3.49E+01 1.08E+00 8.94E−01 3.47E+02 2.27E+02 1.91E+03 1.21E+01 1.38E+01 9.80E+01 1.68E+02 1.29E+03

− − + − − = + = + − + + + + + − = + + = + = + + + = + +

rf33

Mean

Std

0.00E+00 6.55E+05 2.20E+08 1.21E+03 0.00E+00 4.34E+01 1.05E+02 2.11E+01 4.71E+01 3.21E−02 4.64E+00 1.34E+02 2.47E+02 2.31E+02 6.25E+03 1.63E+00 6.58E+01 1.57E+02 2.95E+00 2.17E+01 5.24E+02 6.89E+02 7.77E+03 3.31E+02 3.60E+02 2.00E+02 1.55E+03 4.00E+02

0.00E+00 3.77E+05 2.16E+08 1.96E+03 0.00E+00 5.02E−14 9.62E+00 9.61E−02 3.20E+00 1.68E−02 4.42E+00 3.58E+01 4.92E+01 8.73E+01 1.41E+03 1.39E+00 2.33E+00 2.11E+01 2.92E−01 8.59E−01 4.02E+02 1.51E+02 2.20E+03 7.47E+00 1.00E+01 3.40E−02 9.59E+01 1.96E−10

− − + − − + + = + − − + + = − − = + − + = + + + + − + =

TLBSade

Mean

Std

0.00E+00 3.23E+05 8.61E+06 2.32E−01 0.00E+00 4.34E+01 8.26E+01 2.11E+01 5.67E+01 3.54E−02 0.00E+00 1.89E+02 2.53E+02 3.65E−02 9.27E+03 2.30E+00 5.08E+01 3.28E+02 3.43E+00 2.15E+01 4.60E+02 3.60E+01 9.78E+03 3.33E+02 3.64E+02 3.28E+02 1.73E+03 4.00E+02

0.00E+00 1.56E+05 2.33E+07 3.12E−01 0.00E+00 1.44E−14 1.56E+01 4.65E−02 2.57E+00 1.85E−02 0.00E+00 2.40E+01 3.08E+01 2.06E−02 4.57E+02 6.41E−01 6.46E−14 3.84E+01 1.88E−01 3.86E−01 4.10E+02 2.44E+01 5.33E+02 1.54E+01 2.09E+01 1.21E+02 1.07E+02 0.00E+00

− − + − − − + = + − − + + − + + − + − + = − + + + = + =

Mean

Std

0.00E+00 1.68E+05 7.08E+05 6.58E+02 0.00E+00 4.07E+01 4.96E+01 2.11E+01 6.09E+01 1.76E−02 0.00E+00 1.20E+02 2.19E+02 8.25E+02 7.69E+03 1.80E+00 7.95E+01 1.81E+02 7.57E+00 1.93E+01 3.12E+02 2.59E+03 9.68E+03 3.98E+02 3.79E+02 2.01E+02 2.17E+03 4.00E+02

0.00E+00 2.85E+04 3.59E+05 1.76E+02 0.00E+00 6.96E+00 5.31E+00 2.71E−02 1.66E+00 6.85E−03 0.00E+00 8.18E+00 1.74E+01 1.07E+02 2.86E+02 2.00E−01 1.81E+00 7.64E+00 4.78E−01 3.55E−01 2.45E+02 3.82E+02 3.99E+02 2.27E+00 2.88E+00 1.72E+00 3.22E+01 0.00E+00

− − + − − − + − + − − + + + + − + + + = − + + + + − + =

Table 14 Summarised Wilcoxon rank-sum comparisons between SinDE and other state of the art DE algorithms: SMADE, DEcfbLS, b6e6rl and TLBSade; reference: SinDE. SMADE

D = 10 D = 30 D = 50 Total

DEcfbLS

b6e6rl

TLBSade

Worse

Equal

Better

Worse

Equal

Better

Worse

Equal

Better

Worse

Equal

Better

7 7 6 20

5 5 6 16

16 16 16 48

12 10 10 32

6 3 5 14

10 15 13 38

8 11 11 30

8 4 4 16

12 13 13 38

11 12 11 34

4 4 2 10

13 12 15 40

Table 15 Comparing the proposed SinDE against the CMA-ES-RIS algorithm [38] for D = 10, reference: SinDE. Func

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

SinDE

CMA-ES-RIS

Mean

Std

Mean

Std

0.00E+00 4.41E+04 2.23E+00 7.17E+02 0.00E+00 7.74E+00 1.12E−04 2.04E+01 7.45E−01 5.94E−02 0.00E+00 5.36E+00 7.62E+00 1.27E−01 4.39E+02 9.46E−01 1.01E+01 1.75E+01 3.50E−01 1.76E+00 4.00E+02 1.67E+01 3.45E+02 1.96E+02 2.00E+02 1.09E+02 3.00E+02 3.00E+02

0.00E+00 4.43E+04 1.26E+01 7.43E+02 0.00E+00 3.78E+00 1.92E−04 7.60E−02 8.57E−01 3.12E−02 0.00E+00 1.95E+00 3.25E+00 8.77E−02 1.76E+02 1.76E−01 2.29E−02 2.48E+00 8.87E−02 4.65E−01 1.72E−13 2.29E+01 1.96E+02 1.68E+01 2.29E−04 1.87E+01 0.00E+00 0.00E+00

0.00E+00 0.00E+00 7.04E−01 0.00E+00 0.00E+00 1.10E+00 5.33E+01 2.03E+01 3.59E+00 1.24E−02 3.57E+00 1.29E+01 2.56E+01 1.02E+02 6.17E+02 1.64E−01 1.04E+01 2.98E+01 8.14E−01 4.16E+00 1.61E+02 2.44E+02 8.35E+02 1.19E+02 1.93E+02 1.61E+02 3.13E+02 2.06E+02

0.00E+00 0.00E+00 4.61E+00 0.00E+00 0.00E+00 2.88E+00 4.68E+01 1.37E−01 1.04E+00 1.35E−02 1.48E+00 5.42E+00 1.08E+01 7.39E+01 1.74E+02 7.56E−02 3.73E+00 6.16E+00 2.74E−01 3.99E−01 6.03E+01 1.09E+02 1.90E+02 5.69E+00 3.42E+01 4.06E+01 2.30E+01 1.07E+02

+ − + + + + + = + = + + + + + + + + + + + + + + + + + +

Please cite this article in press as: A. Draa, et al., A sinusoidal differential evolution algorithm for numerical optimisation, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.11.003

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Table 16 Comparing the proposed SinDE against the CMA-ES-RIS algorithm [38] for D = 30, reference: SinDE. Func

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

518 519 520 521 522 523 524

SinDE

CMA-ES-RIS

Mean

Std

Mean

Std

2.23E−14 2.16E+06 8.49E+04 6.38E+03 1.14E−13 1.46E+01 1.21E−01 2.09E+01 1.52E+01 2.04E−02 1.95E−02 3.02E+01 7.33E+01 5.04E+01 2.95E+03 1.74E+00 3.37E+01 7.86E+01 2.24E+00 9.99E+00 2.87E+02 1.49E+02 3.14E+03 2.00E+02 2.49E+02 2.02E+02 3.01E+02 3.00E+02

6.83E−14 6.15E+05 2.11E+05 2.00E+03 7.65E−29 2.42E+00 1.57E−01 4.96E−02 3.05E+00 1.30E−02 1.39E−01 8.65E+00 2.07E+01 1.92E+01 4.86E+02 2.52E−01 7.97E−01 1.42E+01 3.79E−01 5.50E−01 6.40E+01 1.76E+01 5.31E+02 7.16E−03 6.85E+00 1.40E+01 2.36E+00 0.00E+00

0.00E+00 0.00E+00 2.24E+03 0.00E+00 0.00E+00 6.94E−04 4.48E+01 2.09E+01 2.37E+01 8.31E−03 2.54E+01 7.94E+01 1.56E+02 7.92E+02 3.13E+03 1.07E−01 5.50E+01 1.89E+02 2.80E+00 1.43E+01 1.86E+02 1.17E+03 4.03E+03 2.59E+02 2.82E+02 1.97E+02 7.49E+02 5.39E+02

0.00E+00 0.00E+00 1.10E+04 0.00E+00 0.00E+00 2.01E−03 2.96E+01 8.19E−02 1.95E+00 5.46E−03 6.36E+00 4.39E+01 5.42E+01 2.21E+02 4.57E+02 6.78E−02 5.24E+00 2.73E+01 6.42E−01 5.75E−01 4.01E+01 2.93E+02 5.43E+02 1.76E+01 8.50E+00 1.21E+01 1.87E+02 1.33E+03

of our SinDE is shown in parentheses in the table caption. Also, it is indicated whether the null hypothesis is rejected or not. As seen from the table the SinDE algorithm came in first position among the whole set composed of 13 algorithms. The null hypothesis is rejected in 7 cases out of 12. So, the SinDE is significantly better than 7 of them, but its superiority to the five other approaches is not really significant.

+ − + + + + + = + = + + + + + + + + + + + + + + + + + +

For further analysis of the behaviour of the proposed approach, we illustrate the evolution of the fitness error for an independent run for some functions. As examples, we will take the case of f4 for unimodal functions, f9 for multimodal functions and f25 for composition functions. Fig. 3 illustrates the evolution of the best, worst, median and mean error through generations for these functions in the case of the three tested dimensions.

Table 17 Comparing the proposed SinDE against the CMA-ES-RIS algorithm [38] for D = 50, reference: SinDE. Func

SinDE

CMA-ES-RIS

Mean

Std

Mean

Std

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

2.27E−13 2.66E+06 1.01E+05 8.28E+03 1.14E−13 4.34E+01 6.10E−01 2.11E+01 3.48E+01 7.93E−02 5.92E+00 5.61E+01 1.39E+02 2.34E+02 6.80E+03 2.08E+00 6.52E+01 1.41E+02 4.85E+00 1.92E+01 5.84E+02 3.51E+02 6.59E+03 2.00E+02 2.97E+02 2.76E+02 4.75E+02 4.00E+02

1.53E−28 8.33E+05 3.77E+05 1.54E+03 7.65E−29 1.44E−14 5.97E−01 3.59E−02 4.34E+00 3.57E−02 2.86E+00 1.41E+01 3.41E+01 9.23E+01 1.00E+03 3.66E−01 3.47E+00 2.27E+01 8.82E−01 7.52E−01 4.22E+02 2.72E+02 8.47E+02 1.34E−01 1.33E+01 5.96E+01 1.55E+02 0.00E+00

0.00E+00 0.00E+00 2.83E+05 0.00E+00 3.54E−08 9.51E+00 4.81E+01 2.10E+01 4.72E+01 8.60E−03 5.36E+01 2.66E+02 4.56E+02 1.49E+03 6.30E+03 8.66E−02 1.02E+02 4.18E+02 5.04E+00 2.43E+01 2.85E+02 2.39E+03 8.37E+03 3.22E+02 3.66E+02 2.00E+02 1.25E+03 1.24E+03

0.00E+00 0.00E+00 7.87E+05 0.00E+00 1.46E−08 1.43E+01 2.18E+01 6.08E−02 2.79E+00 5.92E−03 1.32E+01 1.03E+02 8.62E+01 3.10E+02 6.81E+02 4.25E−02 1.05E+01 5.29E+01 9.43E−01 5.55E−01 2.22E+02 3.71E+02 1.02E+03 1.97E+01 1.00E+01 3.28E−02 2.10E+02 1.54E+03

+ − + + + + + = + = + + + + + + + + + + + + + + + + + +

Please cite this article in press as: A. Draa, et al., A sinusoidal differential evolution algorithm for numerical optimisation, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.11.003

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Table 18 Holm–Bonferroni procedure for: SinDE, DECFBLS [31], SMADE [30], CMAES-RIS [38], CLPSO [45], classical DE, MDe-PBX [41], CCPSO2 [46], CMAES [42], GCMAES [44] and LADE algorithms (reference: SinDE, Rank = 9.61E+00). j

Optimiser

Rank

zj

pj

ı/j

Hypothesis

1 2 3 4 5 6 7 8 9 10 11 12

TLBSade DECFBLS SMADE b6e6rl CMAES-RIS CLPSO LADE MDe-PBX CCPSO2 CMAES GCMAES DE

9.58E+00 9.42E+00 9.05E+00 8.85E+00 8.51E+00 7.18E+00 6.90E+00 6.57E+00 5.31E+00 5.02E+00 4.12E+00 3.74E+00

−4.65E−02 −3.72E−01 −1.09E+00 −1.49E+00 −2.14E+00 −4.75E+00 −5.28E+00 −5.93E+00 −8.40E+00 −8.96E+00 −1.07E+01 −1.15E+01

4.81E−01 3.55E−01 1.37E−01 6.83E−02 1.62E−02 1.04E−06 6.44E−08 1.50E−09 2.28E−17 1.68E−19 3.93E−27 9.53E−31

5.00E−02 2.50E−02 1.67E−02 1.25E−02 1.00E−02 8.33E−03 7.14E−03 6.25E−03 5.56E−03 5.00E−03 4.55E−03 4.17E−03

Not rejected Not rejected Not rejected Not rejected Not rejected Rejected Rejected Rejected Rejected Rejected Rejected Rejected

Fig. 3. Best (*), median (), worst (.-.) and mean (..) error evolution through generations for functions f4 , f9 and f25 for the three dimensions.

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Table 19 Complexity of the proposed SinDE for the three dimensions. Dimension

T0 (s)

T1 (s)

T 2 (s)

(T 2-T1)/T0

D = 10 D = 30 D = 50

0.5049

1.5017 1.9898 2.5064

3.1534 3.8457 4.5625

3.2714 3.6756 4.0722

Table A.1 The impact of different population sizes on the SinDE performances; average error, standard deviation of error and statistical comparison (reference: popSize = 40) for CEC 2013 benchmarks in 10 dimensions. Func

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

popSize = 40

popSize = 50

popSize = 80

Mean

Std

Mean

Std

0.00E+00 3.64E+04 3.43E+00 7.86E+02 0.00E+00 8.77E+00 1.69E−04 2.04E+01 8.15E−01 6.05E−02 0.00E+00 5.28E+00 8.01E+00 2.49E−01 4.85E+02 1.04E+00 1.03E+01 2.47E+01 5.39E−01 1.79E+00 4.00E+02 2.13E+01 3.22E+02 1.99E+02 2.00E+02 1.16E+02 3.00E+02 3.00E+02

0.00E+00 3.48E+04 1.08E+01 7.54E+02 0.00E+00 2.87E+00 4.10E−04 6.31E−02 8.11E−01 3.09E−02 0.00E+00 2.24E+00 3.25E+00 6.70E−01 2.54E+02 1.92E−01 1.45E−01 3.86E+00 1.24E−01 3.76E−01 1.72E−13 2.90E+01 2.10E+02 1.14E+01 7.29E−05 2.85E+01 7.84E−05 0.00E+00

0.00E+00 3.16E+04 1.06E+00 6.82E+02 0.00E+00 8.63E+00 1.83E−04 2.03E+01 5.59E−01 4.85E−02 0.00E+00 5.41E+00 7.62E+00 1.81E+00 5.38E+02 1.07E+00 1.08E+01 2.79E+01 6.11E−01 1.87E+00 3.92E+02 4.42E+01 4.25E+02 1.99E+02 2.00E+02 1.07E+02 3.00E+02 3.00E+02

0.00E+00 3.35E+04 2.00E+00 8.31E+02 0.00E+00 2.99E+00 3.71E−04 7.29E−02 7.01E−01 2.57E−02 0.00E+00 2.10E+00 3.67E+00 2.13E+00 2.56E+02 2.00E−01 2.52E−01 3.67E+00 1.22E−01 3.43E−01 3.92E+01 4.61E+01 2.66E+02 9.25E+00 2.28E−04 1.35E+01 0.00E+00 0.00E+00

= + = + = = = = = = = = = + + = + + + + = + = = = = = =

popSize = 100

Mean

Std

0.00E+00 4.25E+04 4.20E−01 6.57E+02 0.00E+00 7.26E+00 3.75E−04 2.04E+01 7.87E−01 4.30E−02 0.00E+00 4.93E+00 6.43E+00 2.45E+01 9.11E+02 1.15E+00 1.22E+01 3.07E+01 7.81E−01 2.32E+00 4.00E+02 1.01E+02 7.05E+02 1.99E+02 1.99E+02 1.09E+02 3.00E+02 3.00E+02

0.00E+00 5.03E+04 6.07E−01 5.87E+02 0.00E+00 4.00E+00 5.02E−04 7.27E−02 7.41E−01 2.26E−02 0.00E+00 2.06E+00 3.17E+00 9.51E+00 1.96E+02 2.04E−01 4.10E−01 3.61E+00 1.08E−01 3.05E−01 1.72E−13 6.82E+01 3.01E+02 8.67E+00 7.52E+00 1.88E+01 2.80E−05 0.00E+00

= + = + = − = = = = + = = + + = + + + + = + + + = − + =

Mean

Std

0.00E+00 3.81E+04 5.48E−01 6.16E+02 0.00E+00 7.79E+00 9.14E−04 2.04E+01 9.65E−01 4.01E−02 3.92E−10 6.15E+00 7.67E+00 5.11E+01 9.87E+02 1.17E+00 1.31E+01 3.26E+01 8.44E−01 2.49E+00 4.00E+02 1.65E+02 8.53E+02 1.96E+02 1.99E+02 1.06E+02 3.00E+02 3.00E+02

0.00E+00 3.82E+04 7.31E−01 3.91E+02 0.00E+00 3.69E+00 7.52E−04 7.70E−02 8.63E−01 1.88E−02 2.08E−09 2.07E+00 3.68E+00 1.68E+01 1.70E+02 1.57E−01 6.40E−01 3.35E+00 1.21E−01 2.45E−01 1.72E−13 7.27E+01 2.26E+02 1.46E+01 7.98E+00 2.06E+00 1.46E−04 0.00E+00

= + = + = − + = = = + + = + + = + + + + = + + + + − + =

Table A.2 The impact of different population sizes on the SinDE performances; average error, standard deviation of error and statistical comparison (reference: popSize = 40) for CEC 2013 benchmarks in 30 dimensions. Func

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

popSize = 40

popSize = 50

popSize = 80

Mean

Std

Mean

Std

1.34E−14 2.60E+06 1.07E+05 5.71E+03 1.03E−13 1.65E+01 7.84E−02 2.10E+01 1.53E+01 2.05E−02 6.40E−02 3.05E+01 6.58E+01 4.31E+02 3.47E+03 2.32E+00 4.97E+01 1.12E+02 3.22E+00 1.01E+01 2.69E+02 4.25E+02 3.45E+03 2.00E+02 2.48E+02 2.04E+02 3.01E+02 3.00E+02

5.40E−14 9.95E+05 2.81E+05 1.63E+03 3.41E−14 9.17E+00 1.17E−01 5.39E−02 3.81E+00 1.42E−02 2.36E−01 9.77E+00 2.02E+01 1.11E+02 7.56E+02 2.59E−01 2.74E+00 2.09E+01 7.85E−01 6.62E−01 5.85E+01 1.55E+02 5.94E+02 9.88E−03 7.60E+00 1.98E+01 1.15E+00 0.00E+00

4.46E−15 3.14E+06 2.24E+04 7.32E+03 1.09E−13 1.48E+01 9.70E−02 2.09E+01 1.49E+01 1.74E−02 1.78E−01 3.11E+01 7.18E+01 6.80E+02 3.75E+03 2.27E+00 5.72E+01 1.43E+02 4.20E+00 1.06E+01 2.69E+02 7.81E+02 3.54E+03 2.00E+02 2.50E+02 2.00E+02 3.01E+02 3.00E+02

3.18E−14 1.30E+06 6.70E+04 2.22E+03 2.23E−14 2.37E+00 1.27E−01 5.78E−02 3.28E+00 1.18E−02 3.84E−01 8.26E+00 1.68E+01 1.62E+02 8.24E+02 2.93E−01 3.40E+00 2.81E+01 9.70E−01 6.69E−01 5.34E+01 2.82E+02 6.96E+02 5.03E−03 5.90E+00 3.18E−02 1.35E+00 0.00E+00

= + = + = = = = = = = = = + + = + + + + = + = = = = = =

popSize = 100

Mean

Std

1.78E−14 3.90E+06 6.86E+03 1.10E+04 1.11E−13 1.51E+01 7.88E−02 2.10E+01 1.46E+01 1.73E−02 1.45E+00 2.97E+01 7.47E+01 1.50E+03 5.59E+03 2.36E+00 7.32E+01 1.86E+02 6.30E+00 1.16E+01 2.71E+02 1.67E+03 5.23E+03 2.00E+02 2.49E+02 2.00E+02 3.01E+02 3.00E+02

6.17E−14 1.43E+06 1.46E+04 3.09E+03 1.59E−14 2.16E−01 1.12E−01 5.03E−02 3.15E+00 1.34E−02 1.43E+00 8.58E+00 2.56E+01 1.56E+02 7.40E+02 3.08E−01 4.08E+00 1.40E+01 8.65E−01 3.47E−01 4.55E+01 3.79E+02 9.53E+02 8.37E−03 9.06E+00 5.70E−02 1.13E+00 0.00E+00

= + = + = − = = = = + = = + + = + + + + = + + + = − + =

Mean

Std

4.46E−15 4.98E+06 4.28E+03 1.34E+04 1.07E−13 1.56E+01 1.51E−01 2.09E+01 1.69E+01 1.57E−02 8.29E+00 3.38E+01 7.51E+01 1.94E+03 6.26E+03 2.32E+00 8.19E+01 1.92E+02 7.24E+00 1.17E+01 2.59E+02 2.14E+03 6.15E+03 2.00E+02 2.51E+02 2.00E+02 3.01E+02 3.00E+02

3.18E−14 1.66E+06 1.26E+04 3.37E+03 2.70E−14 4.26E−01 1.57E−01 5.98E−02 3.88E+00 9.95E−03 2.91E+00 8.31E+00 2.09E+01 2.40E+02 4.29E+02 2.30E−01 5.03E+00 1.25E+01 7.15E−01 3.24E−01 4.96E+01 3.28E+02 4.76E+02 2.51E−02 6.33E+00 5.83E−02 9.72E−01 0.00E+00

= + = + = − + = = = + + = + + = + + + + = + + + + − + =

Please cite this article in press as: A. Draa, et al., A sinusoidal differential evolution algorithm for numerical optimisation, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.11.003

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Table A.3 The impact of different population sizes on the SinDE performances; average error, standard deviation of error and statistical comparison (reference: popSize = 40) for CEC 2013 benchmarks in 50 dimensions. Func

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

532 533 534 535 536

537

538 539 540 541 542 543 544 545 546 547 548 549 550 551 552

553

554 555 556 557 558 559 560 561 562

popSize = 40

popSize = 50

popSize = 80

Mean

Std

Mean

Std

2.27E−13 3.78E+06 4.47E+05 7.93E+03 1.14E−13 4.34E+01 5.47E−01 2.11E+01 3.55E+01 7.87E−02 7.69E+00 5.86E+01 1.46E+02 1.25E+03 7.47E+03 3.01E+00 1.06E+02 1.93E+02 5.53E+00 1.99E+01 6.53E+02 1.15E+03 7.79E+03 2.00E+02 2.95E+02 2.41E+02 3.01E+00 4.00E+02

1.53E−28 1.32E+06 2.87E+06 1.80E+03 7.65E−29 1.44E−14 5.63E−01 4.10E−02 5.25E+00 3.66E−02 3.05E+00 1.35E+01 3.37E+01 2.79E+02 8.94E+02 3.43E−01 6.90E+00 3.31E+01 9.65E−01 7.59E−01 4.22E+02 4.25E+02 1.02E+03 8.44E−02 1.40E+01 5.55E+01 3.43E−01 0.00E+00

2.23E−13 4.56E+06 9.99E+03 1.03E+04 1.14E−13 4.34E+01 5.81E−01 2.11E+01 3.36E+01 8.37E−02 8.53E+00 5.67E+01 1.46E+02 2.00E+03 7.59E+03 3.07E+00 1.25E+02 2.24E+02 7.87E+00 2.03E+01 5.85E+02 1.94E+03 7.15E+03 2.00E+02 2.95E+02 2.46E+02 3.07E+00 4.00E+02

3.18E−14 1.30E+06 2.59E+04 2.64E+03 7.65E−29 1.44E−14 8.96E−01 3.76E−02 5.69E+00 4.71E−02 2.53E+00 1.01E+01 3.50E+01 3.45E+02 1.11E+03 2.81E−01 8.59E+00 4.56E+01 2.33E+00 9.62E−01 4.35E+02 4.64E+02 9.93E+02 8.06E−02 9.69E+00 5.81E+01 2.81E−01 0.00E+00

= + = + = = = = = = = = = + + = + + + + = + = = = = = =

From the different plots on the figure, we can see certain homogeneity of evolution of fitness error, the mean, median and even worst error in the population keep getting better (decreasing) with preserving at the same time population diversity, especially for multimodal and composition functions. 4.6. Algorithm complexity As dictated in [18], algorithmic complexity is to be calculated on the basis of some computation on algorithmic performance for the function f14 . Using the same process, described in [18], Table 19 below summarises the algorithmic complexity of our approach. As seen from the table, the proposed SinDE inherits the linear complexity of classical DE; this is a real plus when comparing it to powerful recent metaheuristics such as: the DEcfbLS [31] and the CMAES-RIS [38] algorithms. This is mainly justified by the fact that the SinDE algorithm does not use any local search procedures, known to be greedy in terms of the needed number of objective function evaluations to get local optima. All experiments, complexity computation included, have been conducted with a machine having as configuration an Intel Core i73770K 3.5GHz CPU and 4 Gb of Memory. As software configuration, we used Matlab 2013 on a Windows 7 system.

popSize = 100

Mean

Std

2.01E−13 6.90E+06 8.20E+03 1.82E+04 1.14E−13 4.34E+01 6.61E−01 2.11E+01 3.29E+01 8.97E−02 1.54E+01 5.78E+01 1.46E+02 3.86E+03 1.07E+04 3.21E+00 1.68E+02 3.47E+02 1.38E+01 2.14E+01 5.63E+02 3.95E+03 9.74E+03 2.00E+02 2.96E+02 2.27E+02 3.21E+00 4.00E+02

7.40E−14 2.29E+06 2.73E+04 2.98E+03 7.65E−29 5.60E−04 5.06E−01 3.38E−02 6.00E+00 4.79E−02 5.55E+00 1.13E+01 3.72E+01 4.32E+02 1.50E+03 2.97E−01 9.35E+00 3.25E+01 2.46E+00 3.60E−01 4.54E+02 4.25E+02 1.99E+03 1.34E−01 8.59E+00 4.85E+01 2.97E−01 0.00E+00

= + = + = − = = = = + = = + + = + + + + = + + + = − + =

Mean

Std

2.03E−13 9.24E+06 7.65E+03 2.32E+04 1.14E−13 4.35E+01 8.16E−01 2.11E+01 3.50E+01 1.02E−01 3.09E+01 6.28E+01 1.53E+02 4.78E+03 1.19E+04 3.09E+00 1.90E+02 3.71E+02 1.68E+01 2.16E+01 4.97E+02 4.87E+03 1.15E+04 2.00E+02 2.98E+02 2.16E+02 3.09E+00 4.00E+02

7.09E−14 3.42E+06 2.11E+04 3.62E+03 7.65E−29 6.19E−02 5.74E−01 3.33E−02 5.64E+00 4.03E−02 7.23E+00 1.61E+01 4.13E+01 3.56E+02 9.74E+02 3.20E−01 1.13E+01 2.50E+01 1.57E+00 3.84E−01 4.16E+02 5.94E+02 1.52E+03 1.69E−01 9.94E+00 4.01E+01 3.20E−01 0.00E+00

= + = + = − + = = = + + = + + = + + + + = + + + + − + =

Six different configurations of this sinusoidal adjustment have been analysed revealing that the best configuration is one in which both F and CR parameters are sinusoidally adjusted with an increasing interval between the upper and lower bounds of the parameter. The SinDE algorithm has been compared against 12 other stateof-the-art algorithms including the classical variant of the DE algorithm, the linearly parameter-adjusting DE, four adaptive DE variants, the CMA-ES algorithm and two of its variants, and finally two variants of the PSO. Two types of statistics have been used: the Wilcoxon rank-sum test and the Holm–Bonferroni procedure. As a summary of results, we have found that the proposed SinDE has outperformed all other approaches in the majority of functions and for different dimensions; especially for composition and multimodal functions. It has been significantly better than 7 of these approaches and slightly better than the others. In addition to its very promising performances, the SinDE algorithm has the advantage of inheriting the linear algorithmic complexity of the basic DE. In fact, many of the state-of-the-art algorithms are quite complex thanks to their use of advanced local search procedures. A future direction of research in the context of this work would be including some local search procedures in the SinDE process or hybridising the SinDE algorithm with other powerful metaheuristics, such as the CMA-ES. Besides, exploring other mutation strategies and exponential crossover are thought to be promising perspectives.

563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587

5. Conclusion This paper has presented a new variant of the differential evolution algorithm. This new variant, called SinDE (for Sinusoidal Differential Evolution), is based on the use of a new sinusoidal framework for automatically adjusting the parameters of the DE algorithm: the scaling factor F and the crossover rate CR. In addition to permitting a periodic change of the adjusted parameter, some flexibility of the range of permitted parameter values is offered by giving the chance to the parameter to increase or decrease the interval between higher and lower values throughout generations.

Appendix A. Detailed numerical tests for parameter setting

588

A.1. Population size

589

Different population size values are tested on the same benchmarks, CEC-2013. Tables A.1–A.3 show the average error and standard deviation to the optimum, in addition to the statistical comparison between different configurations using different population sizes equal to 40, 50, 80 and 100; taking as reference

Please cite this article in press as: A. Draa, et al., A sinusoidal differential evolution algorithm for numerical optimisation, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.11.003

590 591 592 593 594

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Func

freq = 0.01

freq = 0.02

Mean

Std

Mean

Std

0.00E+00 3.41E+04 4.30E+00 7.43E+02 0.00E+00 7.68E+00 2.94E−04 2.04E+01 6.86E−01 4.15E−02 0.00E+00 4.84E+00 7.56E+00 2.20E−01 3.68E+02 9.31E−01 1.01E+01 1.82E+01 3.96E−01 1.78E+00 3.97E+02 1.64E+01 3.84E+02 2.00E+02 2.00E+02 1.06E+02 3.00E+02 3.00E+02

0.00E+00 4.21E+04 2.63E+01 5.99E+02 0.00E+00 3.89E+00 6.08E−04 6.23E−02 6.58E−01 2.21E−02 0.00E+00 2.01E+00 3.40E+00 1.37E−01 1.59E+02 1.56E−01 1.25E−02 2.62E+00 9.16E−02 3.52E−01 2.23E+01 1.87E+01 1.88E+02 1.49E−04 1.49E−04 6.58E+00 0.00E+00 0.00E+00

0.00E+00 3.92E+04 8.84E−01 7.15E+02 0.00E+00 7.90E+00 4.38E−04 2.04E+01 1.64E+00 6.86E−02 9.75E−03 6.81E+00 8.34E+00 7.13E+00 6.53E+02 1.13E+00 1.30E+01 3.17E+01 7.81E−01 2.03E+00 3.98E+02 4.13E+01 5.57E+02 1.98E+02 1.99E+02 1.11E+02 3.00E+02 3.00E+02

0.00E+00 3.94E+04 1.52E+00 6.25E+02 0.00E+00 3.66E+00 8.32E−04 7.20E−02 1.29E+00 3.40E−02 9.85E−02 2.59E+00 3.63E+00 8.62E+00 2.19E+02 2.04E−01 7.02E−01 4.43E+00 1.86E−01 3.50E−01 1.98E+01 3.31E+01 2.08E+02 1.30E+01 8.08E+00 1.86E+01 6.72E−04 0.00E+00

= = = = = = + = + + = + = + + + + + + + = + + = − + + =

freq = 0.05

Mean

Std

0.00E+00 4.08E+04 8.69E−01 7.83E+02 0.00E+00 7.52E+00 2.09E−04 2.03E+01 8.88E−01 5.27E−02 0.00E+00 4.94E+00 6.67E+00 9.34E−01 5.81E+02 1.09E+00 1.08E+01 2.34E+01 5.25E−01 1.93E+00 4.00E+02 2.99E+01 3.96E+02 1.99E+02 2.00E+02 1.13E+02 3.00E+02 3.00E+02

0.00E+00 4.43E+04 1.93E+00 5.56E+02 0.00E+00 3.96E+00 3.08E−04 7.31E−02 8.04E−01 2.27E−02 0.00E+00 1.81E+00 3.34E+00 9.84E−01 1.60E+02 2.21E−01 2.51E−01 4.52E+00 1.07E−01 3.62E−01 1.72E−13 3.15E+01 2.19E+02 7.13E+00 3.01E−03 2.61E+01 1.96E−05 0.00E+00

= = = = = = = = = + = = = + + + + + + + = + = = + = = =

freq = 0.1

Mean

Std

0.00E+00 5.18E+04 6.72E−01 6.08E+02 0.00E+00 7.73E+00 1.78E−04 2.04E+01 7.25E−01 4.68E−02 0.00E+00 4.63E+00 7.46E+00 1.31E+00 5.53E+02 1.12E+00 1.09E+01 2.65E+01 5.83E−01 1.86E+00 4.00E+02 3.80E+01 3.76E+02 2.00E+02 2.00E+02 1.09E+02 3.00E+02 3.00E+02

0.00E+00 5.03E+04 9.87E−01 4.55E+02 0.00E+00 3.80E+00 4.78E−04 6.89E−02 6.47E−01 2.65E−02 0.00E+00 1.87E+00 3.92E+00 1.01E+00 2.24E+02 2.15E−01 2.60E−01 3.67E+00 1.35E−01 4.52E−01 1.72E−13 4.13E+01 1.89E+02 8.83E−01 1.96E−04 1.86E+01 0.00E+00 0.00E+00

= + = = = = − = = = = = = + + + + + + = = + = = = = = =

freq = 0.15

Mean

Std

0.00E+00 3.16E+04 1.06E+00 6.82E+02 0.00E+00 8.63E+00 1.83E−04 2.03E+01 5.59E−01 4.85E−02 0.00E+00 5.41E+00 7.62E+00 1.81E+00 5.38E+02 1.07E+00 1.08E+01 2.79E+01 6.11E−01 1.87E+00 3.92E+02 4.42E+01 4.25E+02 1.99E+02 2.00E+02 1.07E+02 3.00E+02 3.00E+02

0.00E+00 3.35E+04 2.00E+00 8.31E+02 0.00E+00 2.99E+00 3.71E−04 7.29E−02 7.01E−01 2.57E−02 0.00E+00 2.10E+00 3.67E+00 2.13E+00 2.56E+02 2.00E−01 2.52E−01 3.67E+00 1.22E−01 3.43E−01 3.92E+01 4.61E+01 2.66E+02 9.25E+00 2.28E−04 1.35E+01 0.00E+00 0.00E+00

= = = = = = = − = = = = = + + + + + + = = + = = = = = =

freq = 0.2

Mean

Std

0.00E+00 2.88E+04 7.21E−01 6.40E+02 0.00E+00 7.19E+00 1.27E−04 2.04E+01 7.10E−01 5.08E−02 0.00E+00 4.74E+00 6.76E+00 1.84E+00 8.56E+02 1.14E+00 1.07E+01 3.01E+01 7.29E−01 2.00E+00 4.00E+02 5.62E+01 3.76E+02 2.00E+02 2.00E+02 1.10E+02 3.00E+02 3.00E+02

0.00E+00 2.96E+04 1.73E+00 5.19E+02 0.00E+00 4.12E+00 1.81E−04 7.77E−02 6.32E−01 2.59E−02 0.00E+00 1.85E+00 3.58E+00 1.70E+00 2.55E+02 2.05E−01 2.35E−01 3.26E+00 9.22E−02 4.12E−01 1.72E−13 5.32E+01 2.87E+02 5.79E−05 1.49E−04 2.27E+01 0.00E+00 0.00E+00

= = = = = = = = = = = = = + + + + + + + = + = = = = = =

freq = 0.25

Mean

Std

0.00E+00 4.19E+04 9.33E−01 5.10E+02 0.00E+00 5.76E+00 1.98E−04 2.04E+01 6.30E−01 4.88E−02 0.00E+00 4.95E+00 6.57E+00 1.63E+00 5.99E+02 1.11E+00 1.08E+01 2.81E+01 6.32E−01 1.98E+00 4.00E+02 3.51E+01 3.02E+02 1.99E+02 2.00E+02 1.07E+02 3.00E+02 3.00E+02

0.00E+00 5.56E+04 1.71E+00 3.51E+02 0.00E+00 4.52E+00 2.83E−04 7.09E−02 7.20E−01 2.44E−02 0.00E+00 1.80E+00 3.28E+00 2.06E+00 2.50E+02 2.04E−01 2.31E−01 3.72E+00 1.20E−01 4.00E−01 1.72E−13 3.23E+01 1.59E+02 1.04E+01 3.59E−04 1.38E+01 0.00E+00 0.00E+00

= = = = = = = = = = = = = + + + + + + + = + − = = = = =

Mean

Std

0.00E+00 3.48E+04 8.21E−01 6.12E+02 0.00E+00 8.23E+00 1.57E−04 2.04E+01 6.38E−01 4.19E−02 0.00E+00 4.86E+00 7.55E+00 2.27E+00 7.15E+02 1.13E+00 1.07E+01 3.10E+01 7.38E−01 2.04E+00 3.96E+02 5.49E+01 3.25E+02 1.99E+02 2.00E+02 1.07E+02 3.00E+02 3.00E+02

0.00E+00 4.05E+04 1.53E+00 4.45E+02 0.00E+00 3.46E+00 2.96E−04 6.06E−02 6.43E−01 2.24E−02 0.00E+00 1.77E+00 3.93E+00 1.93E+00 2.74E+02 2.21E−01 2.47E−01 3.85E+00 1.09E−01 4.44E−01 2.80E+01 4.66E+01 2.07E+02 9.22E+00 2.38E−05 1.35E+01 0.00E+00 0.00E+00

= = = = = = − = = = = = = + + + + + + + = + = = = = = =

ARTICLE IN PRESS

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

freq = 0.001

A. Draa et al. / Applied Soft Computing xxx (2014) xxx–xxx 17

Please cite this article in press as: A. Draa, et al., A sinusoidal differential evolution algorithm for numerical optimisation, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.11.003

Table A.4 The impact of different frequency values on the SinDE performances; average error, standard deviation of error and statistical comparison (reference: freq = 0.25) for CEC 2013 benchmarks in 10 dimensions.

G Model

ASOC 2602 1–27

18

Func

freq = 0.01

freq = 0.02

Mean

Std

Mean

Std

1.78E−14 2.42E+06 8.21E+04 7.80E+03 1.14E−13 1.54E+01 1.17E−01 2.09E+01 1.48E+01 1.64E−02 1.56E−01 3.06E+01 6.64E+01 8.33E+01 2.96E+03 1.86E+00 3.63E+01 7.92E+01 2.30E+00 9.95E+00 2.84E+02 1.88E+02 3.12E+03 2.00E+02 2.48E+02 2.02E+02 3.01E+02 3.00E+02

6.17E−14 9.93E+05 2.68E+05 2.31E+03 7.65E−29 8.20E+00 1.55E−01 5.33E−02 2.92E+00 1.24E−02 4.16E−01 7.15E+00 1.80E+01 2.31E+01 5.71E+02 3.25E−01 1.28E+00 1.42E+01 4.58E−01 6.54E−01 4.55E+01 4.63E+01 5.91E+02 7.28E−03 5.15E+00 1.40E+01 1.44E+00 0.00E+00

4.90E−14 4.15E+06 6.23E+04 8.05E+03 1.11E−13 1.47E+01 2.20E−01 2.09E+01 2.13E+01 2.49E−02 7.94E+00 7.89E+01 1.05E+02 9.35E+02 5.37E+03 2.48E+00 7.40E+01 1.57E+02 5.08E+00 1.10E+01 2.75E+02 1.03E+03 5.48E+03 2.00E+02 2.55E+02 2.00E+02 3.12E+02 3.00E+02

9.44E−14 1.32E+06 1.31E+05 2.00E+03 1.59E−14 3.26E−01 1.97E−01 5.00E−02 3.57E+00 1.80E−02 3.27E+00 2.30E+01 2.10E+01 3.31E+02 5.72E+02 2.85E−01 6.21E+00 1.68E+01 8.22E−01 3.65E−01 5.09E+01 3.98E+02 4.73E+02 3.35E−02 9.41E+00 4.70E−02 3.75E+01 0.00E+00

= = = = = = + = + + = + = + + + + + + + = + + = − + + =

freq = 0.05

Mean

Std

3.57E−14 4.01E+06 4.77E+04 7.05E+03 1.11E−13 1.46E+01 7.08E−02 2.09E+01 1.65E+01 1.93E−02 8.38E−01 3.21E+01 7.04E+01 6.20E+02 4.07E+03 2.37E+00 5.55E+01 1.50E+02 4.31E+00 1.07E+01 2.67E+02 5.83E+02 3.99E+03 2.00E+02 2.49E+02 2.02E+02 3.01E+02 3.00E+02

8.35E−14 1.44E+06 1.48E+05 1.86E+03 1.59E−14 3.46E−01 8.86E−02 5.24E−02 2.42E+00 1.42E−02 9.11E−01 8.64E+00 2.15E+01 1.66E+02 8.44E+02 2.91E−01 3.90E+00 1.97E+01 8.61E−01 5.99E−01 5.68E+01 2.15E+02 8.88E+02 1.07E−02 5.72E+00 1.41E+01 1.86E+00 0.00E+00

= = = = = = = = = + = = = + + + + + + + = + = = + = = =

freq = 0.1

Mean

Std

1.34E−14 3.63E+06 2.58E+04 6.81E+03 1.09E−13 1.47E+01 1.02E−01 2.09E+01 1.58E+01 1.69E−02 3.15E−01 2.98E+01 7.38E+01 6.86E+02 4.01E+03 2.30E+00 5.80E+01 1.48E+02 4.27E+00 1.06E+01 2.75E+02 7.09E+02 3.81E+03 2.00E+02 2.49E+02 2.02E+02 3.01E+02 3.00E+02

5.40E−14 1.58E+06 6.77E+04 1.97E+03 2.23E−14 3.61E−01 1.38E−01 5.36E−02 2.76E+00 1.12E−02 5.80E−01 8.75E+00 2.21E+01 1.71E+02 8.05E+02 2.93E−01 3.59E+00 2.34E+01 1.14E+00 6.10E−01 5.08E+01 2.57E+02 6.26E+02 4.29E−03 5.60E+00 1.40E+01 1.77E+00 0.00E+00

= + = = = = − = = = = = = + + + + + + = = + = = = = = =

freq = 0.15

Mean

Std

4.46E−15 3.14E+06 2.24E+04 7.32E+03 1.09E−13 1.48E+01 9.70E−02 2.09E+01 1.49E+01 1.74E−02 1.78E−01 3.11E+01 7.18E+01 6.80E+02 3.75E+03 2.27E+00 5.72E+01 1.43E+02 4.20E+00 1.06E+01 2.69E+02 7.81E+02 3.54E+03 2.00E+02 2.50E+02 2.00E+02 3.01E+02 3.00E+02

3.18E−14 1.30E+06 6.70E+04 2.22E+03 2.23E−14 2.37E+00 1.27E−01 5.78E−02 3.28E+00 1.18E−02 3.84E−01 8.26E+00 1.68E+01 1.62E+02 8.24E+02 2.93E−01 3.40E+00 2.81E+01 9.70E−01 6.69E−01 5.34E+01 2.82E+02 6.96E+02 5.03E−03 5.90E+00 3.18E−02 1.35E+00 0.00E+00

= = = = = = = − = = = = = + + + + + + = = + = = = = = =

freq = 0.2

Mean

Std

1.78E−14 2.74E+06 1.27E+04 7.05E+03 1.11E−13 1.43E+01 1.04E−01 2.09E+01 1.38E+01 1.40E−02 1.76E−01 2.81E+01 6.50E+01 1.37E+03 5.95E+03 2.40E+00 6.63E+01 1.95E+02 6.44E+00 1.15E+01 2.79E+02 1.76E+03 4.90E+03 2.00E+02 2.48E+02 2.00E+02 3.01E+02 3.00E+02

6.17E−14 1.11E+06 4.14E+04 2.05E+03 1.59E−14 2.40E−01 1.38E−01 4.22E−02 3.10E+00 1.06E−02 3.83E−01 8.01E+00 1.72E+01 1.75E+02 8.94E+02 2.59E−01 2.95E+00 1.01E+01 5.76E−01 5.26E−01 4.87E+01 4.10E+02 1.30E+03 5.92E−03 5.56E+00 3.08E−02 8.56E−01 0.00E+00

= = = = = = = = = = = = = + + + + + + + = + = = = = = =

freq = 0.25

Mean

Std

3.12E−14 2.93E+06 6.26E+04 7.18E+03 1.11E−13 1.43E+01 1.20E−01 2.10E+01 1.46E+01 1.72E−02 2.35E−01 3.05E+01 7.35E+01 7.09E+02 3.56E+03 2.33E+00 5.67E+01 1.39E+02 4.23E+00 1.05E+01 2.73E+02 7.84E+02 3.48E+03 2.00E+02 2.49E+02 2.00E+02 3.01E+02 3.00E+02

7.90E−14 1.14E+06 2.02E+05 2.11E+03 1.59E−14 3.23E−01 1.47E−01 4.62E−02 3.68E+00 1.32E−02 5.11E−01 6.64E+00 2.21E+01 1.26E+02 8.33E+02 3.41E−01 3.26E+00 2.56E+01 1.07E+00 7.41E−01 4.46E+01 2.65E+02 7.29E+02 1.18E−02 5.26E+00 2.97E−02 1.78E+00 0.00E+00

= = = = = = = = = = = = = + + + + + + + = + − = = = = =

Mean

Std

3.12E−14 2.31E+06 2.44E+04 6.39E+03 1.09E−13 1.54E+01 1.20E−01 2.09E+01 1.43E+01 1.65E−02 2.54E−01 2.92E+01 6.85E+01 1.35E+03 5.79E+03 2.36E+00 6.59E+01 1.96E+02 6.45E+00 1.15E+01 2.82E+02 1.63E+03 4.94E+03 2.00E+02 2.48E+02 2.00E+02 3.01E+02 3.00E+02

7.90E−14 1.05E+06 6.13E+04 1.88E+03 2.23E−14 8.50E+00 2.25E−01 5.34E−02 2.86E+00 1.32E−02 5.92E−01 9.42E+00 1.95E+01 1.82E+02 1.05E+03 2.63E−01 3.32E+00 1.00E+01 6.30E−01 4.80E−01 6.55E+01 3.93E+02 1.28E+03 8.63E−03 6.31E+00 2.56E−02 2.18E+00 0.00E+00

= = = = = = − = = = = = = + + + + + + + = + = = = = = =

ARTICLE IN PRESS

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

freq = 0.001

A. Draa et al. / Applied Soft Computing xxx (2014) xxx–xxx

Please cite this article in press as: A. Draa, et al., A sinusoidal differential evolution algorithm for numerical optimisation, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.11.003

Table A.5 The impact of different frequency values on the SinDE performances; average error, standard deviation of error and statistical comparison (reference: freq = 0.25) for CEC 2013 benchmarks in 30 dimensions.

G Model

ASOC 2602 1–27

Func

freq = 0.01

freq = 0.02

Mean

Std

Mean

Std

2.18E−13 3.26E+06 3.56E+04 1.14E+04 1.14E−13 4.34E+01 5.38E−01 2.11E+01 3.40E+01 7.69E−02 7.53E+00 5.65E+01 1.29E+02 3.33E+02 6.49E+03 2.32E+00 7.07E+01 1.50E+02 4.76E+00 1.92E+01 6.05E+02 3.81E+02 6.50E+03 2.00E+02 2.97E+02 2.44E+02 4.63E+02 4.00E+02

4.46E−14 1.15E+06 2.01E+05 1.87E+03 7.65E−29 1.44E−14 5.31E−01 3.82E−02 5.20E+00 4.28E−02 2.97E+00 1.23E+01 3.17E+01 1.17E+02 8.13E+02 3.80E−01 3.83E+00 2.27E+01 8.12E−01 5.91E−01 4.54E+02 1.45E+02 9.55E+02 1.60E−01 1.18E+01 5.82E+01 1.31E+02 0.00E+00

2.14E−13 7.20E+06 5.88E+04 1.31E+04 1.14E−13 4.34E+01 1.27E+00 2.11E+01 4.83E+01 1.03E−01 2.60E+01 7.55E+01 1.90E+02 2.79E+03 1.16E+04 3.36E+00 1.42E+02 3.23E+02 1.35E+01 2.08E+01 6.08E+02 3.36E+03 1.17E+04 2.01E+02 3.13E+02 2.55E+02 5.50E+02 4.00E+02

5.40E−14 2.29E+06 2.28E+05 2.26E+03 7.65E−29 6.34E−05 8.32E−01 5.58E−02 5.63E+00 8.97E−02 8.28E+00 4.19E+01 6.81E+01 6.64E+02 6.11E+02 2.45E−01 1.44E+01 2.70E+01 1.27E+00 4.92E−01 4.41E+02 9.33E+02 7.00E+02 4.09E−01 1.52E+01 6.63E+01 1.61E+02 0.00E+00

= = = = = = + = + + = + = + + + + + + + = + + = − + + =

freq = 0.05

Mean

Std

2.27E−13 5.46E+06 2.02E+04 1.10E+04 1.14E−13 4.34E+01 5.48E−01 2.11E+01 3.72E+01 7.96E−02 1.22E+01 6.26E+01 1.46E+02 2.06E+03 8.20E+03 3.13E+00 1.27E+02 2.68E+02 9.32E+00 2.04E+01 5.85E+02 1.75E+03 8.33E+03 2.00E+02 3.00E+02 2.51E+02 4.63E+02 4.00E+02

1.53E−28 1.55E+06 6.64E+04 2.22E+03 7.65E−29 5.88E−06 5.44E−01 3.49E−02 5.68E+00 5.15E−02 4.06E+00 1.45E+01 3.03E+01 3.97E+02 1.30E+03 2.78E−01 9.07E+00 4.24E+01 2.33E+00 8.21E−01 4.32E+02 4.82E+02 1.19E+03 1.16E−01 1.39E+01 6.08E+01 1.74E+02 0.00E+00

= = = = = = = = = + = = = + + + + + + + = + = = + = = =

freq = 0.1

Mean

Std

2.23E−13 6.01E+06 4.06E+03 1.10E+04 1.14E−13 4.34E+01 4.64E−01 2.11E+01 3.63E+01 6.97E−02 1.03E+01 6.31E+01 1.54E+02 2.21E+03 8.20E+03 3.24E+00 1.31E+02 2.40E+02 8.44E+00 2.05E+01 6.08E+02 1.87E+03 7.99E+03 2.00E+02 2.95E+02 2.31E+02 4.61E+02 4.00E+02

3.18E−14 2.31E+06 8.42E+03 2.40E+03 7.65E−29 2.72E−06 4.21E−01 3.12E−02 5.61E+00 4.70E−02 3.58E+00 1.15E+01 3.60E+01 3.96E+02 1.22E+03 2.58E−01 9.90E+00 5.06E+01 2.18E+00 8.03E−01 4.30E+02 6.35E+02 1.06E+03 8.98E−02 1.30E+01 5.23E+01 1.29E+02 0.00E+00

= + = = = = − = = = = = = + + + + + + = = + = = = = = =

freq = 0.15

Mean

Std

2.23E−13 4.56E+06 9.99E+03 1.03E+04 1.14E−13 4.34E+01 5.81E−01 2.11E+01 3.36E+01 8.37E−02 8.53E+00 5.67E+01 1.46E+02 2.00E+03 7.59E+03 3.07E+00 1.25E+02 2.24E+02 7.87E+00 2.03E+01 5.85E+02 1.94E+03 7.15E+03 2.00E+02 2.95E+02 2.46E+02 4.11E+02 4.00E+02

3.18E−14 1.30E+06 2.59E+04 2.64E+03 7.65E−29 1.44E−14 8.96E−01 3.76E−02 5.69E+00 4.71E−02 2.53E+00 1.01E+01 3.50E+01 3.45E+02 1.11E+03 2.81E−01 8.59E+00 4.56E+01 2.33E+00 9.62E−01 4.35E+02 4.64E+02 9.93E+02 8.06E−02 9.69E+00 5.81E+01 1.28E+02 0.00E+00

= = = = = = = − = = = = = + + + + + + = = + = = = = = =

freq = 0.2

Mean

Std

2.14E−13 3.53E+06 1.05E+04 1.01E+04 1.14E−13 4.34E+01 5.57E−01 2.11E+01 3.25E+01 7.12E−02 8.75E+00 5.46E+01 1.37E+02 4.35E+03 1.25E+04 3.33E+00 1.62E+02 3.91E+02 1.61E+01 2.17E+01 5.38E+02 4.65E+03 1.12E+04 2.00E+02 2.97E+02 2.34E+02 4.99E+02 4.00E+02

5.40E−14 1.33E+06 3.23E+04 2.21E+03 7.65E−29 1.44E−14 5.30E−01 5.11E−02 5.23E+00 3.68E−02 3.53E+00 1.21E+01 3.00E+01 3.37E+02 1.23E+03 2.64E−01 7.50E+00 1.84E+01 9.08E−01 2.99E−01 4.24E+02 6.68E+02 2.38E+03 1.65E−01 1.19E+01 5.25E+01 1.78E+02 0.00E+00

= = = = = = = = = = = = = + + + + + + + = + = = = = = =

freq = 0.25

Mean

Std

2.18E−13 3.85E+06 1.17E+04 1.06E+04 1.14E−13 4.34E+01 4.75E−01 2.11E+01 3.28E+01 7.25E−02 8.26E+00 5.43E+01 1.43E+02 1.95E+03 7.69E+03 3.11E+00 1.26E+02 2.09E+02 7.78E+00 2.01E+01 5.27E+02 1.89E+03 7.56E+03 2.00E+02 2.95E+02 2.54E+02 4.13E+02 4.00E+02

4.46E−14 1.23E+06 2.91E+04 2.57E+03 7.65E−29 1.44E−14 3.87E−01 3.80E−02 5.80E+00 3.37E−02 3.27E+00 1.31E+01 3.07E+01 3.04E+02 1.27E+03 3.14E−01 7.74E+00 4.71E+01 2.08E+00 7.87E−01 4.33E+02 4.54E+02 9.93E+02 8.10E−02 1.09E+01 5.98E+01 1.31E+02 0.00E+00

= = = = = = = = = = = = = + + + + + + + = + − = = = = =

Mean

Std

2.27E−13 3.24E+06 8.11E+03 1.03E+04 1.14E−13 4.34E+01 4.43E−01 2.11E+01 3.30E+01 8.89E−02 8.27E+00 5.69E+01 1.40E+02 4.27E+03 1.25E+04 3.35E+00 1.61E+02 3.88E+02 1.58E+01 2.16E+01 5.71E+02 4.69E+03 1.10E+04 2.00E+02 2.94E+02 2.44E+02 4.63E+02 4.00E+02

1.53E−28 1.42E+06 1.93E+04 2.24E+03 7.65E−29 1.44E−14 4.07E−01 3.60E−02 4.89E+00 3.75E−02 2.82E+00 1.38E+01 3.34E+01 2.91E+02 1.28E+03 2.31E−01 8.53E+00 1.88E+01 1.48E+00 5.48E−01 4.24E+02 5.67E+02 2.33E+03 1.03E−01 1.03E+01 5.57E+01 1.65E+02 0.00E+00

= = = = = = − = = = = = = + + + + + + + = + = = = = = =

ARTICLE IN PRESS

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

freq = 0.001

A. Draa et al. / Applied Soft Computing xxx (2014) xxx–xxx 19

Please cite this article in press as: A. Draa, et al., A sinusoidal differential evolution algorithm for numerical optimisation, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.11.003

Table A.6 The impact of different frequency values on the SinDE performances; average error, standard deviation of error and statistical comparison (reference: freq = 0.25) for CEC 2013 benchmarks in 50 dimensions.

G Model

ASOC 2602 1–27

20

Func

 = /2

 = /3

Mean

Std

Mean

Std

0.00E+00 3.94E+04 9.10E−01 6.73E+02 0.00E+00 7.25E+00 1.78E−04 2.04E+01 7.59E−01 5.40E−02 0.00E+00 4.39E+00 6.62E+00 1.79E+00 6.48E+02 1.05E+00 1.08E+01 2.78E+01 6.15E−01 1.94E+00 4.00E+02 3.31E+01 3.95E+02 1.99E+02 2.00E+02 1.10E+02 3.00E+02 3.00E+02

0.00E+00 4.13E+04 1.40E+00 5.73E+02 0.00E+00 4.00E+00 2.98E−04 6.18E−02 8.14E−01 2.79E−02 0.00E+00 1.94E+00 3.13E+00 1.63E+00 2.41E+02 1.87E−01 2.04E−01 3.70E+00 1.31E−01 3.42E−01 1.72E−13 3.65E+01 2.33E+02 1.12E+01 4.88E−04 1.94E+01 0.00E+00 0.00E+00

2.53E+02 7.18E+06 2.00E+09 1.09E+04 7.87E+01 4.94E+01 4.00E+01 2.05E+01 4.81E+00 8.43E+01 2.39E+01 3.79E+01 3.79E+01 5.91E+02 1.48E+03 1.59E+00 4.24E+01 5.97E+01 1.85E+01 3.18E+00 4.34E+02 7.34E+02 1.52E+03 2.10E+02 2.10E+02 1.48E+02 4.14E+02 4.97E+02

4.66E+02 1.18E+07 3.12E+09 1.66E+04 1.33E+02 6.37E+01 5.85E+01 1.92E−01 4.66E+00 1.29E+02 3.46E+01 3.57E+01 3.71E+01 8.45E+02 4.86E+02 7.28E−01 4.64E+01 3.99E+01 3.59E+01 1.05E+00 5.18E+01 9.82E+02 6.60E+02 1.46E+01 1.47E+01 4.74E+01 1.64E+02 2.83E+02

= + − = = = = + + = − + + − + + − + + + = − + − + = = =

 = /4

Mean

Std

0.00E+00 2.58E+03 5.71E−01 4.35E+01 0.00E+00 8.14E+00 4.20E−04 2.04E+01 3.54E+00 2.43E−01 0.00E+00 1.57E+01 1.46E+01 7.79E−02 1.11E+03 1.12E+00 1.03E+01 3.14E+01 7.02E−01 2.59E+00 3.92E+02 4.87E+01 1.09E+03 1.97E+02 2.00E+02 1.21E+02 3.00E+02 3.00E+02

0.00E+00 1.84E+03 7.62E−01 2.22E+01 0.00E+00 3.64E+00 6.04E−04 6.96E−02 2.17E+00 7.33E−02 0.00E+00 2.89E+00 3.34E+00 6.48E−02 1.83E+02 1.75E−01 1.13E−01 3.66E+00 9.89E−02 2.60E−01 3.92E+01 3.94E+01 1.88E+02 1.50E+01 6.23E−05 2.03E+01 0.00E+00 0.00E+00

= + − + = = + = + − − + + − + + − + + + = − + = + + + =

 = /5

Mean

Std

0.00E+00 1.37E+04 1.18E+00 2.26E+02 0.00E+00 7.52E+00 1.17E−03 2.03E+01 3.92E+00 2.51E−01 0.00E+00 1.52E+01 1.56E+01 1.38E−01 1.14E+03 1.09E+00 1.03E+01 3.25E+01 6.68E−01 2.69E+00 3.96E+02 3.86E+01 1.15E+03 2.00E+02 2.00E+02 1.24E+02 3.00E+02 3.00E+02

0.00E+00 7.09E+03 1.49E+00 1.03E+02 0.00E+00 4.17E+00 1.50E−03 7.33E−02 1.90E+00 5.72E−02 0.00E+00 3.35E+00 3.48E+00 4.73E−01 1.41E+02 2.05E−01 8.94E−02 3.61E+00 1.33E−01 3.26E−01 2.80E+01 3.36E+01 1.92E+02 2.29E−03 2.14E−04 2.31E+01 0.00E+00 0.00E+00

= + = + = = + + + = − + + − + + − + + + = − + + + + + =

 = /6

Mean

Std

0.00E+00 2.58E+04 1.48E+00 4.68E+02 0.00E+00 6.17E+00 1.83E−03 2.04E+01 3.78E+00 2.44E−01 0.00E+00 1.49E+01 1.50E+01 6.97E−02 1.16E+03 1.18E+00 1.03E+01 3.18E+01 6.92E−01 2.65E+00 4.00E+02 4.39E+01 1.12E+03 1.99E+02 2.00E+02 1.18E+02 3.00E+02 3.00E+02

0.00E+00 1.43E+04 1.78E+00 2.26E+02 0.00E+00 4.77E+00 2.12E−03 6.31E−02 2.07E+00 5.29E−02 0.00E+00 3.16E+00 3.09E+00 6.80E−02 1.38E+02 1.99E−01 9.55E−02 3.47E+00 9.46E−02 2.77E−01 1.72E−13 3.79E+01 2.12E+02 1.26E+01 7.20E−04 4.26E+00 0.00E+00 0.00E+00

= + − + = = + = + = − + + − + + − + + + = − + + + + + =

=0

Mean

Std

0.00E+00 4.18E+04 2.96E+00 6.06E+02 0.00E+00 7.31E+00 1.50E−03 2.04E+01 4.14E+00 2.52E−01 0.00E+00 1.47E+01 1.52E+01 4.93E−02 1.15E+03 1.08E+00 1.02E+01 3.22E+01 6.87E−01 2.66E+00 4.00E+02 3.48E+01 1.16E+03 1.98E+02 2.00E+02 1.22E+02 3.00E+02 3.00E+02

0.00E+00 2.25E+04 3.61E+00 2.61E+02 0.00E+00 4.32E+00 1.72E−03 6.81E−02 1.65E+00 5.78E−02 0.00E+00 3.01E+00 3.31E+00 5.88E−02 1.66E+02 1.79E−01 7.70E−02 3.11E+00 9.52E−02 2.50E−01 1.72E−13 3.06E+01 2.39E+02 1.27E+01 1.78E−04 2.04E+01 0.00E+00 0.00E+00

= + = + = = + + + = − + + − + + − + + + = − + + + + + =

Mean

Std

0.00E+00 9.43E+04 4.71E+00 1.29E+03 0.00E+00 7.89E+00 3.34E−03 2.03E+01 4.75E+00 2.54E−01 0.00E+00 1.48E+01 1.45E+01 8.03E−02 1.18E+03 1.12E+00 1.03E+01 3.24E+01 6.74E−01 2.67E+00 3.96E+02 3.83E+01 1.16E+03 1.97E+02 2.00E+02 1.23E+02 3.00E+02 3.00E+02

0.00E+00 4.15E+04 8.18E+00 4.66E+02 0.00E+00 3.91E+00 3.73E−03 7.78E−02 1.44E+00 6.77E−02 0.00E+00 3.13E+00 3.12E+00 6.68E−02 1.44E+02 2.31E−01 9.08E−02 3.88E+00 9.28E−02 2.67E−01 2.79E+01 3.44E+01 2.05E+02 1.52E+01 2.61E−03 2.20E+01 1.39E−05 0.00E+00

= + = + = = + = + − − + + − + + − + + + = − + + + + + =

ARTICLE IN PRESS

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A. Draa et al. / Applied Soft Computing xxx (2014) xxx–xxx

Please cite this article in press as: A. Draa, et al., A sinusoidal differential evolution algorithm for numerical optimisation, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.11.003

Table A.7 The impact of different phase shift values on the SinDE performances; average error, standard deviation of error and statistical comparison (reference:  = ) for CEC 2013 benchmarks in 10 dimensions.

G Model

ASOC 2602 1–27

Func

 = /2

 = /3

Mean

Std

Mean

Std

4.46E−15 3.14E+06 2.24E+04 7.32E+03 1.09E−13 1.48E+01 9.70E−02 2.09E+01 1.49E+01 1.74E−02 1.78E−01 3.11E+01 7.18E+01 6.80E+02 3.75E+03 2.27E+00 5.72E+01 1.43E+02 4.20E+00 1.06E+01 2.69E+02 7.81E+02 3.54E+03 2.00E+02 2.50E+02 2.00E+02 3.01E+02 3.00E+02

3.18E−14 1.30E+06 6.70E+04 2.22E+03 2.23E−14 2.37E+00 1.27E−01 5.78E−02 3.28E+00 1.18E−02 3.84E−01 8.26E+00 1.68E+01 1.62E+02 8.24E+02 2.93E−01 3.40E+00 2.81E+01 9.70E−01 6.69E−01 5.34E+01 2.82E+02 6.96E+02 5.03E−03 5.90E+00 3.18E−02 1.35E+00 0.00E+00

2.23E−14 3.13E+06 1.36E+04 4.82E+03 1.09E−13 1.43E+01 9.50E−02 2.09E+01 3.32E+01 1.48E−02 9.75E−03 1.44E+02 1.54E+02 3.67E+02 6.78E+03 2.43E+00 5.02E+01 2.00E+02 5.50E+00 1.20E+01 2.77E+02 6.43E+02 6.98E+03 2.00E+02 2.61E+02 2.00E+02 3.01E+02 3.00E+02

6.79E−14 1.45E+06 6.02E+04 1.61E+03 2.22E−14 1.23E+00 1.51E−01 4.92E−02 1.60E+00 1.14E−02 9.85E−02 1.14E+01 1.18E+01 7.13E+01 3.29E+02 2.66E−01 2.43E+00 1.01E+01 4.85E−01 2.91E−01 4.96E+01 3.06E+02 3.42E+02 2.57E−03 1.78E+01 3.04E−02 4.83E+00 0.00E+00

= + − = = = = + + = − + + − + + − + + + = − + − + = = =

 = /4

Mean

Std

2.23E−14 1.20E+07 4.03E+02 1.72E+04 1.14E−13 1.43E+01 2.84E−01 2.10E+01 3.38E+01 1.38E−02 1.95E−02 1.42E+02 1.51E+02 1.69E+02 6.73E+03 2.48E+00 4.46E+01 1.98E+02 5.14E+00 1.20E+01 2.79E+02 3.91E+02 6.86E+03 2.00E+02 2.81E+02 2.01E+02 3.02E+02 3.00E+02

6.83E−14 3.97E+06 1.39E+03 4.34E+03 7.65E−29 1.73E+00 3.86E−01 3.87E−02 1.69E+00 1.20E−02 1.39E−01 1.16E+01 1.14E+01 3.24E+01 3.22E+02 2.93E−01 1.59E+00 1.26E+01 4.90E−01 2.44E−01 6.15E+01 1.99E+02 4.78E+02 3.81E−03 1.30E+01 1.98E−01 6.03E+00 0.00E+00

= + − + = = + = + − − + + − + + − + + + = − + = + + + =

 = /5

Mean

Std

3.12E−14 2.07E+07 2.81E+02 2.37E+04 1.14E−13 1.49E+01 5.98E−01 2.09E+01 3.36E+01 1.15E−02 1.95E−02 1.42E+02 1.52E+02 1.26E+02 6.81E+03 2.42E+00 4.25E+01 2.01E+02 4.97E+00 1.21E+01 2.73E+02 4.13E+02 6.92E+03 2.00E+02 2.88E+02 2.01E+02 3.03E+02 3.00E+02

7.90E−14 4.72E+06 1.25E+03 4.04E+03 7.65E−29 7.92E+00 6.69E−01 4.18E−02 1.46E+00 8.97E−03 1.39E−01 1.07E+01 1.08E+01 3.11E+01 2.79E+02 3.43E−01 1.94E+00 1.10E+01 4.63E−01 2.52E−01 5.19E+01 2.35E+02 3.40E+02 9.28E−03 1.10E+01 3.06E−01 8.36E+00 0.00E+00

= + = + = = + + + = − + + − + + − + + + = − + + + + + =

 = /6

Mean

Std

4.46E−14 2.33E+07 7.39E+03 2.37E+04 1.14E−13 1.38E+01 8.05E−01 2.09E+01 3.36E+01 1.15E−02 1.95E−02 1.40E+02 1.53E+02 1.22E+02 6.77E+03 2.50E+00 4.23E+01 1.99E+02 5.06E+00 1.21E+01 2.58E+02 3.72E+02 6.99E+03 2.00E+02 2.88E+02 2.01E+02 3.03E+02 3.00E+02

9.12E−14 5.09E+06 4.96E+04 4.19E+03 7.65E−29 2.65E+00 8.59E−01 3.88E−02 1.94E+00 6.50E−03 1.39E−01 1.23E+01 1.16E+01 2.83E+01 3.30E+02 2.79E−01 1.78E+00 1.01E+01 3.80E−01 2.74E−01 5.65E+01 2.25E+02 3.69E+02 9.24E−03 1.07E+01 4.05E−01 5.25E+00 0.00E+00

= + − + = = + = + = − + + − + + − + + + = − + + + + + =

=0

Mean

Std

3.12E−14 2.34E+07 1.67E+03 2.38E+04 1.07E−13 1.28E+01 7.27E−01 2.09E+01 3.37E+01 9.62E−03 5.85E−02 1.43E+02 1.53E+02 1.14E+02 6.73E+03 2.44E+00 4.12E+01 1.99E+02 5.01E+00 1.21E+01 2.79E+02 3.01E+02 7.07E+03 2.00E+02 2.89E+02 2.03E+02 3.15E+02 3.00E+02

7.90E−14 5.83E+06 7.70E+03 3.46E+03 2.70E−14 5.75E−01 7.82E−01 6.00E−02 1.41E+00 7.67E−03 2.36E−01 1.05E+01 1.31E+01 2.84E+01 3.11E+02 3.01E−01 1.56E+00 9.92E+00 4.88E−01 2.82E−01 6.15E+01 1.58E+02 3.28E+02 7.15E−03 7.43E+00 1.39E+01 7.24E+01 0.00E+00

= + = + = = + + + = − + + − + + − + + + = − + + + + + =

Mean

Std

3.12E−14 2.67E+07 4.58E+03 2.46E+04 1.09E−13 1.25E+01 1.32E+00 2.09E+01 3.35E+01 1.04E−02 2.34E−13 1.42E+02 1.51E+02 9.25E+01 6.84E+03 2.48E+00 4.12E+01 2.02E+02 4.86E+00 1.20E+01 2.82E+02 3.56E+02 7.03E+03 2.00E+02 2.89E+02 2.04E+02 3.28E+02 3.00E+02

7.90E−14 6.21E+06 2.16E+04 4.55E+03 2.23E−14 5.37E−01 1.00E+00 6.56E−02 1.33E+00 7.25E−03 5.36E−13 1.14E+01 1.17E+01 2.36E+01 2.54E+02 3.09E−01 1.54E+00 1.13E+01 4.26E−01 2.47E−01 3.85E+01 2.23E+02 3.58E+02 2.34E−02 8.06E+00 1.39E+01 1.03E+02 0.00E+00

= + = + = = + = + − − + + − + + − + + + = − + + + + + =

ARTICLE IN PRESS

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

=

A. Draa et al. / Applied Soft Computing xxx (2014) xxx–xxx 21

Please cite this article in press as: A. Draa, et al., A sinusoidal differential evolution algorithm for numerical optimisation, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.11.003

Table A.8 The impact of different phase shift values on the SinDE performances; average error, standard deviation of error and statistical comparison (reference:  = ) for CEC 2013 benchmarks in 30 dimensions.

G Model

ASOC 2602 1–27

22

Func

 = /2

 = /3

Mean

Std

Mean

Std

2.23E−13 4.56E+06 9.99E+03 1.03E+04 1.14E−13 4.34E+01 5.81E−01 2.11E+01 3.36E+01 8.37E−02 8.53E+00 5.67E+01 1.46E+02 2.00E+03 7.59E+03 3.07E+00 1.25E+02 2.24E+02 7.87E+00 2.03E+01 5.85E+02 1.94E+03 7.15E+03 2.00E+02 2.95E+02 2.46E+02 4.11E+02 4.00E+02

3.18E−14 1.30E+06 2.59E+04 2.64E+03 7.65E−29 1.44E−14 8.96E−01 3.76E−02 5.69E+00 4.71E−02 2.53E+00 1.01E+01 3.50E+01 3.45E+02 1.11E+03 2.81E−01 8.59E+00 4.56E+01 2.33E+00 9.62E−01 4.35E+02 4.64E+02 9.93E+02 8.06E−02 9.69E+00 5.81E+01 1.28E+02 0.00E+00

2.21E−13 8.23E+06 7.88E+03 9.85E+03 1.14E−13 4.34E+01 7.29E−01 2.11E+01 6.46E+01 7.31E−02 3.71E−01 3.10E+02 3.34E+02 1.19E+03 1.35E+04 3.27E+00 1.05E+02 3.99E+02 1.32E+01 2.20E+01 6.27E+02 1.33E+03 1.37E+04 2.00E+02 3.26E+02 2.43E+02 4.32E+02 4.00E+02

3.86E−14 3.38E+06 2.84E+04 2.03E+03 1.90E−28 2.86E−14 1.01E+00 3.45E−02 2.08E+00 4.05E−02 6.25E−01 1.64E+01 1.81E+01 1.82E+02 4.23E+02 2.84E−01 4.54E+00 1.57E+01 8.93E−01 2.68E−01 4.34E+02 8.12E+02 4.51E+02 7.39E−02 3.86E+01 5.47E+01 1.56E+02 0.00E+00

= + − = = = = + + = − + + − + + − + + + = − + − + = = =

 = /4

Mean

Std

2.23E−13 5.73E+07 4.00E+03 3.07E+04 1.14E−13 4.34E+01 1.75E+00 2.11E+01 6.46E+01 5.92E−02 2.93E−01 3.14E+02 3.41E+02 5.68E+02 1.35E+04 3.26E+00 8.77E+01 4.06E+02 1.23E+01 2.19E+01 5.74E+02 8.74E+02 1.37E+04 2.00E+02 3.70E+02 2.59E+02 6.06E+02 4.00E+02

3.18E−14 1.85E+07 1.12E+04 5.43E+03 7.65E−29 1.44E−14 1.47E+00 3.53E−02 1.71E+00 2.99E−02 5.37E−01 1.65E+01 1.44E+01 9.17E+01 3.32E+02 2.51E−01 4.27E+00 1.32E+01 8.24E−01 2.68E−01 4.53E+02 8.67E+02 4.75E+02 7.01E−02 1.85E+01 7.66E+01 4.32E+02 0.00E+00

= + − + = = + = + − − + + − + + − + + + = − + = + + + =

 = /5

Mean

Std

2.27E−13 8.91E+07 1.13E+04 3.63E+04 1.14E−13 4.34E+01 4.25E+00 2.11E+01 6.48E+01 7.27E−02 1.17E−01 3.08E+02 3.41E+02 4.38E+02 1.35E+04 3.20E+00 8.29E+01 4.02E+02 1.17E+01 2.20E+01 6.02E+02 6.68E+02 1.36E+04 2.00E+02 3.75E+02 2.58E+02 8.03E+02 4.00E+02

1.53E−28 1.74E+07 2.47E+04 5.12E+03 7.65E−29 1.44E−14 3.57E+00 4.63E−02 1.65E+00 2.83E−02 3.24E−01 2.05E+01 1.77E+01 9.42E+01 4.24E+02 2.65E−01 3.16E+00 1.37E+01 7.95E−01 2.70E−01 4.25E+02 6.54E+02 4.61E+02 1.12E−01 1.09E+01 8.53E+01 4.98E+02 0.00E+00

= + = + = = + + + = − + + − + + − + + + = − + + + + + =

 = /6

Mean

Std

2.27E−13 8.78E+07 3.15E+03 3.65E+04 1.16E−13 4.34E+01 6.27E+00 2.11E+01 6.46E+01 6.68E−02 1.17E−01 3.10E+02 3.41E+02 4.12E+02 1.35E+04 3.30E+00 8.01E+01 3.97E+02 1.15E+01 2.19E+01 6.80E+02 5.03E+02 1.37E+04 2.00E+02 3.73E+02 2.64E+02 9.87E+02 4.00E+02

1.53E−28 1.55E+07 8.03E+03 4.27E+03 1.59E−14 1.44E−14 4.21E+00 4.11E−02 1.71E+00 3.79E−02 3.80E−01 1.78E+01 1.69E+01 6.69E+01 3.59E+02 2.98E−01 3.16E+00 1.77E+01 8.15E−01 3.20E−01 4.33E+02 4.56E+02 5.68E+02 2.26E−01 9.30E+00 9.73E+01 5.24E+02 0.00E+00

= + − + = = + = + = − + + − + + − + + + = − + + + + + =

=0

Mean

Std

2.23E−13 9.77E+07 7.99E+03 3.55E+04 1.14E−13 4.34E+01 8.07E+00 2.11E+01 6.46E+01 6.55E−02 1.95E−01 3.14E+02 3.41E+02 3.76E+02 1.34E+04 3.26E+00 7.97E+01 4.07E+02 1.15E+01 2.20E+01 6.14E+02 6.32E+02 1.37E+04 2.00E+02 3.75E+02 2.70E+02 8.85E+02 4.00E+02

3.18E−14 1.67E+07 2.10E+04 5.04E+03 7.65E−29 1.44E−14 4.32E+00 3.47E−02 2.03E+00 2.89E−02 4.46E−01 1.72E+01 1.93E+01 6.25E+01 4.13E+02 2.48E−01 3.31E+00 1.48E+01 8.51E−01 3.18E−01 4.46E+02 7.54E+02 5.28E+02 1.41E−01 1.27E+01 1.02E+02 5.40E+02 0.00E+00

= + = + = = + + + = − + + − + + − + + + = − + + + + + =

Mean

Std

2.23E−13 9.17E+07 1.50E+04 3.81E+04 1.20E−13 4.34E+01 9.48E+00 2.11E+01 6.48E+01 5.83E−02 2.54E−01 3.17E+02 3.43E+02 3.69E+02 1.33E+04 3.25E+00 7.89E+01 4.04E+02 1.12E+01 2.19E+01 6.25E+02 5.81E+02 1.37E+04 2.00E+02 3.77E+02 2.61E+02 1.04E+03 4.00E+02

3.18E−14 1.96E+07 2.94E+04 4.08E+03 2.70E−14 1.44E−14 3.76E+00 3.07E−02 2.07E+00 2.67E−02 4.81E−01 1.71E+01 1.38E+01 6.16E+01 4.30E+02 2.99E−01 3.27E+00 1.43E+01 7.23E−01 3.29E−01 4.32E+02 5.17E+02 4.72E+02 5.01E−01 7.75E+00 9.19E+01 5.16E+02 0.00E+00

= + = + = = + = + − − + + − + + − + + + = − + + + + + =

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A. Draa et al. / Applied Soft Computing xxx (2014) xxx–xxx

Please cite this article in press as: A. Draa, et al., A sinusoidal differential evolution algorithm for numerical optimisation, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.11.003

Table A.9 The impact of different phase shift values on the SinDE performances; average error, standard deviation of error and statistical comparison (reference:  = ) for CEC 2013 benchmarks in 50 dimensions.

G Model

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A. Draa et al. / Applied Soft Computing xxx (2014) xxx–xxx

23

Table A.10 Summarised Wilcoxon rank-sum comparisons for setting the population size, reference pop = 40. PopSize = 50

D = 10 D = 30 D = 50 Total

PopSize = 80

PopSize = 100

Worse

Equal

Better

Worse

Equal

Better

Worse

Equal

Better

2 0 2 4

21 19 18 58

5 9 8 22

4 2 4 10

13 13 10 36

11 13 14 38

4 2 2 8

11 10 11 32

13 16 15 44

Table A.11 Summarised Wilcoxon rank-sum comparisons for setting the frequency value, reference freq = 0.25. freq = 0.001

freq = 0.01

freq = 0.02

freq = 0.05

freq = 0.1

freq = 0.15

freq = 0.2

Worse Equal Better Worse Equal Better Worse Equal Better Worse Equal Better Worse Equal Better Worse Equal Better Worse Equal Better D = 10 D = 30 D = 50 Total

1 3 0 4

12 6 7 25

15 19 21 55

0 1 0 1

18 14 14 46

10 13 14 37

1 2 0 3

19 13 13 45

8 13 15 36

1 1 3 5

20 16 13 49

7 11 12 30

0 1 1 2

20 17 18 55

8 10 9 27

1 1 3 5

19 16 14 49

8 11 11 30

1 2 1 4

19 16 16 51

8 10 11 29

Table A.12 Summarised Wilcoxon rank-sum comparisons for setting the phase, reference  = .  = /2

D=10 D=30 D=50 Total

 = /3

 = /4

 = /5

Equal

Better

Worse

Equal

Better

Worse

Equal

Better

Worse

Equal

Better

Worse

Equal

Better

Worse

Equal

Better

0 8 6 14

1 9 10 20

27 11 12 50

6 6 6 18

12 8 7 27

11 14 15 40

4 6 4 14

10 4 7 21

13 18 17 48

11 7 5 23

14 4 7 25

4 17 16 37

9 7 4 20

15 4 7 26

3 17 17 37

3 7 5 15

10 4 7 21

15 17 16 48

601

602

A.2. Frequency value setting

596 597 598 599 600

=0

Worse

the configuration with 40 individuals as population size. From the tables, we see that the configuration with a population value equal to 40 outperformed all other configurations. Table A.10 summarises these comparisons in terms of numbers of cases where the best configuration, having population size equal to 40, was worse, equal and better, in terms of optimisation results, than the other configurations.

595

 = /6

following three tables show the impact of using different phase values on the performance of the SinDE algorithm, we have set the phase parameter  to the values , (/2), (/3), (/4), (/5), (/6) and 0 respectively and compared the obtained results using the Wilcoxon rank-sum statistics (Tables A.7–A.11). It is well seen from the tables that the best phase shift is  = ; this is also summarised in Table A.12.

610 611 612 613 614 615 616

606

Tables A.4–A.6 give detailed comparisons of different frequency configurations and the corresponding Wilcoxon rank-sum test, taking as reference the configuration with frequency value 0.25. The statistical comparisons are summarised in Table A.11.

607

A.3. Phase value setting

603 604 605

608 609

As well seen from the formulas of the configurations of the SinDE algorithm proposed in this paper, the phase difference between the sine formulas of the two parameters (F and CR) is fixed to  = . The

A.4. Configuration 3 and phase setting Configuration 3 has been also tested using different values of phase. The same values as with Configuration 2 have been used: , (/2), (/3), (/4), (/5), (/6) and 0, then the average error, the standard deviation of error and the Wilcoxon rank-sum test are calculated to compare these configurations it with Configuration 2 using a phase equal to . Tables A.13−A.15 summarise these comparisons. It is clear from the tables that Configuration 2 is considerably better than Configuration 3 (Table A.16).

Please cite this article in press as: A. Draa, et al., A sinusoidal differential evolution algorithm for numerical optimisation, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.11.003

617

618 619 620 621 622 623 624 625

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ASOC 2602 1–27

24

Config 2.  = 

Config 3.  = 

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

0.00E+00 3.94E+04 9.10E−01 6.73E+02 0.00E+00 7.25E+00 1.78E−04 2.04E+01 7.59E−01 5.40E−02 0.00E+00 4.39E+00 6.62E+00 1.79E+00 6.48E+02 1.05E+00 1.08E+01 2.78E+01 6.15E−01 1.94E+00 4.00E+02 3.31E+01 3.95E+02 1.99E+02 2.00E+02 1.10E+02 3.00E+02 3.00E+02

5.15E−02 1.52E+05 9.70E+06 9.14E+03 4.43E−01 1.10E+01 1.74E+01 2.04E+01 3.47E+00 1.02E+00 4.88E−01 1.56E+01 2.22E+01 2.28E−01 7.25E+02 1.12E+00 1.09E+01 2.97E+01 5.11E−01 2.74E+00 3.96E+02 5.38E+01 8.99E+02 1.95E+02 2.01E+02 1.35E+02 3.39E+02 2.87E+02

0.00E+00 4.13E+04 1.40E+00 5.73E+02 0.00E+00 4.00E+00 2.98E−04 6.18E−02 8.14E−01 2.79E−02 0.00E+00 1.94E+00 3.13E+00 1.63E+00 2.41E+02 1.87E−01 2.04E−01 3.70E+00 1.31E−01 3.42E−01 1.72E−13 3.65E+01 2.33E+02 1.12E+01 4.88E−04 1.94E+01 0.00E+00 0.00E+00

3.68E−01 1.28E+05 1.93E+07 4.86E+03 2.15E+00 1.74E+01 1.44E+01 7.31E−02 1.39E+00 1.88E+00 7.28E−01 7.50E+00 1.04E+01 6.55E−01 2.19E+02 1.96E−01 5.09E−01 6.57E+00 1.98E−01 4.99E−01 2.80E+01 5.21E+01 2.60E+02 3.14E+01 2.04E+01 3.39E+01 5.74E+01 1.37E+02

Config 3.  = /2 = = = = = = + = + + = + = + + + + + + + = + + = − + + =

0.00E+00 1.29E+04 1.38E+03 3.58E+01 0.00E+00 6.77E+00 5.93E−02 2.04E+01 1.12E+00 1.31E−01 4.88E−02 1.49E+01 1.47E+01 3.47E−01 1.28E+03 1.17E+00 1.02E+01 3.32E+01 8.30E−01 2.69E+00 3.98E+02 5.97E+01 1.35E+03 1.98E+02 1.97E+02 1.28E+02 3.12E+02 2.87E+02

0.00E+00 2.68E+04 1.32E+04 4.86E+01 0.00E+00 4.42E+00 2.65E−01 7.04E−02 1.01E+00 1.12E−01 2.57E−01 5.51E+00 5.18E+00 9.05E−01 1.72E+02 1.93E−01 1.04E+00 3.65E+00 1.39E−01 2.94E−01 1.98E+01 6.57E+01 1.52E+02 1.67E+01 1.77E+01 2.70E+01 3.24E+01 6.02E+01

Config 3.  = /3 = = = = = = = = = + = = = + + + + + + + = + = = + = = =

0.00E+00 1.55E+04 4.37E+00 1.18E+02 0.00E+00 5.27E+00 1.97E−03 2.03E+01 3.99E+00 3.58E−01 1.95E−02 2.06E+01 2.13E+01 2.27E−01 1.29E+03 1.16E+00 1.02E+01 3.43E+01 8.35E−01 2.82E+00 3.96E+02 6.05E+01 1.36E+03 1.99E+02 1.95E+02 1.30E+02 3.10E+02 2.84E+02

0.00E+00 8.09E+03 6.15E+00 7.59E+01 0.00E+00 4.87E+00 4.05E−03 7.39E−02 2.66E+00 8.57E−02 1.39E−01 4.31E+00 3.52E+00 6.54E−01 1.36E+02 1.78E−01 1.02E−01 3.32E+00 1.46E−01 3.17E−01 2.80E+01 5.04E+01 1.79E+02 1.19E+01 1.96E+01 2.17E+01 3.00E+01 5.43E+01

Config 3.  = /4 = + = = = = − = = = = = = + + + + + + = = + = = = = = =

0.00E+00 7.59E+04 1.12E+01 5.38E+02 0.00E+00 5.20E+00 2.01E−03 2.04E+01 6.02E+00 3.72E−01 1.95E−02 2.18E+01 2.02E+01 3.90E−01 1.30E+03 1.16E+00 1.02E+01 3.37E+01 8.13E−01 2.81E+00 3.96E+02 7.24E+01 1.34E+03 2.00E+02 1.97E+02 1.30E+02 3.04E+02 2.73E+02

0.00E+00 4.68E+04 2.60E+01 3.33E+02 0.00E+00 4.94E+00 2.57E−03 8.17E−02 1.22E+00 8.90E−02 1.39E−01 3.39E+00 4.14E+00 1.69E+00 1.42E+02 1.91E−01 1.25E−01 3.62E+00 1.42E−01 2.78E−01 2.80E+01 6.56E+01 1.97E+02 9.37E+00 1.38E+01 1.77E+01 1.96E+01 6.95E+01

Config 3.  = /5 = = = = = = = − = = = = = + + + + + + = = + = = = = = =

0.00E+00 1.06E+05 2.34E+01 8.71E+02 0.00E+00 5.33E+00 3.15E−03 2.04E+01 5.94E+00 3.97E−01 0.00E+00 2.16E+01 2.13E+01 2.16E−01 1.29E+03 1.17E+00 1.02E+01 3.45E+01 8.21E−01 2.85E+00 3.92E+02 8.07E+01 1.39E+03 1.99E+02 1.98E+02 1.35E+02 3.02E+02 2.76E+02

0.00E+00 5.16E+04 3.65E+01 3.54E+02 0.00E+00 4.88E+00 6.03E−03 8.18E−02 1.60E+00 8.55E−02 0.00E+00 3.79E+00 3.93E+00 6.63E−01 1.55E+02 1.96E−01 1.06E−01 4.35E+00 1.30E−01 2.36E−01 3.91E+01 6.26E+01 1.84E+02 1.23E+01 1.18E+01 2.41E+01 1.39E+01 6.48E+01

Config 3.  = /6 = = = = = = = = = = = = = + + + + + + + = + = = = = = =

0.00E+00 1.49E+05 4.29E+01 1.15E+03 0.00E+00 5.04E+00 2.34E−03 2.04E+01 5.96E+00 4.12E−01 0.00E+00 2.33E+01 2.09E+01 1.42E−01 1.29E+03 1.17E+00 1.01E+01 3.49E+01 8.24E−01 2.88E+00 3.92E+02 6.79E+01 1.38E+03 1.96E+02 2.00E+02 1.32E+02 3.04E+02 2.76E+02

0.00E+00 8.58E+04 7.83E+01 3.92E+02 0.00E+00 4.92E+00 2.18E−03 6.72E−02 1.63E+00 7.79E−02 0.00E+00 4.84E+00 3.84E+00 4.72E−01 1.50E+02 2.04E−01 3.92E−02 4.22E+00 1.46E−01 2.30E−01 3.92E+01 5.00E+01 1.87E+02 1.85E+01 5.45E−04 1.86E+01 1.96E+01 6.51E+01

Config 3.  = 0 = = = = = = = = = = = = = + + + + + + + = + − = = = = =

0.00E+00 3.01E+05 3.52E+01 1.96E+03 0.00E+00 5.02E+00 5.01E−03 2.04E+01 6.26E+00 4.32E−01 0.00E+00 2.31E+01 2.32E+01 7.84E−02 1.26E+03 1.13E+00 1.01E+01 3.54E+01 8.12E−01 2.96E+00 3.96E+02 7.36E+01 1.41E+03 1.98E+02 1.98E+02 1.35E+02 3.00E+02 2.84E+02

0.00E+00 1.52E+05 7.05E+01 8.68E+02 0.00E+00 4.93E+00 4.78E−03 6.45E−02 7.07E−01 7.47E−02 0.00E+00 4.11E+00 4.01E+00 5.56E−02 1.63E+02 1.88E−01 6.12E−02 3.81E+00 1.20E−01 2.07E−01 2.80E+01 5.74E+01 1.70E+02 1.47E+01 1.53E+01 2.05E+01 1.40E−05 5.43E+01

= = = = = = − = = = = = = + + + + + + + = + = = = = = =

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Please cite this article in press as: A. Draa, et al., A sinusoidal differential evolution algorithm for numerical optimisation, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.11.003

Table A.13 Comparing Configuration 3 with different phase values against Configuration 2; average error, standard deviation of error and statistical comparison (reference:  = ) for CEC 2013 benchmarks in 10 dimensions.

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ASOC 2602 1–27

Config 2.  = 

Config 3.  = 

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

4.46E−15 3.14E+06 2.24E+04 7.32E+03 1.09E−13 1.48E+01 9.70E−02 2.09E+01 1.49E+01 1.74E−02 1.78E−01 3.11E+01 7.18E+01 6.80E+02 3.75E+03 2.27E+00 5.72E+01 1.43E+02 4.20E+00 1.06E+01 2.69E+02 7.81E+02 3.54E+03 2.00E+02 2.50E+02 2.00E+02 3.01E+02 3.00E+02

2.64E+01 1.84E+06 3.57E+07 2.75E+04 6.08E+01 2.93E+01 7.42E+01 2.09E+01 2.53E+01 2.50E−01 1.91E+01 8.51E+01 1.61E+02 7.71E+02 3.95E+03 2.43E+00 7.66E+01 1.05E+02 5.78E+00 1.20E+01 3.48E+02 4.71E+02 4.53E+03 2.56E+02 2.89E+02 2.12E+02 9.25E+02 3.24E+02

3.18E−14 1.30E+06 6.70E+04 2.22E+03 2.23E−14 2.37E+00 1.27E−01 5.78E−02 3.28E+00 1.18E−02 3.84E−01 8.26E+00 1.68E+01 1.62E+02 8.24E+02 2.93E−01 3.40E+00 2.81E+01 9.70E−01 6.69E−01 5.34E+01 2.82E+02 6.96E+02 5.03E−03 5.90E+00 3.18E−02 1.35E+00 0.00E+00

8.44E+01 8.70E+05 5.00E+07 9.15E+03 1.50E+02 2.74E+01 2.24E+01 4.15E−02 4.24E+00 1.64E−01 1.03E+01 2.67E+01 3.67E+01 3.34E+02 6.47E+02 2.56E−01 7.92E+00 4.04E+01 3.17E+00 5.98E−01 1.09E+02 3.01E+02 6.15E+02 1.39E+01 7.73E+00 4.21E+01 1.09E+02 2.02E+02

Config 3.  = /2 = = = = = = + = + + = + = + + + + + + + = + + = − + + =

4.46E−15 3.17E+06 3.42E+06 8.53E+03 4.03E−09 1.93E+01 7.50E+00 2.10E+01 3.39E+01 1.40E−01 2.59E+00 1.65E+02 1.68E+02 7.55E+02 7.06E+03 2.44E+00 6.79E+01 1.99E+02 6.92E+00 1.20E+01 3.01E+02 9.05E+02 7.32E+03 2.11E+02 2.57E+02 2.00E+02 5.11E+02 3.00E+02

3.17E−14 1.26E+06 6.68E+06 3.10E+03 4.07E−08 1.65E+01 7.25E+00 4.66E−02 4.30E+00 7.13E−02 2.13E+00 1.43E+01 1.19E+01 2.47E+02 3.19E+02 2.89E−01 4.75E+00 9.70E+00 7.02E−01 3.06E−01 8.28E+01 5.84E+02 3.44E+02 8.57E+00 1.62E+01 5.10E−02 2.21E+02 0.00E+00

Config 3.  = /3 = = = = = = = = = + = = = + + + + + + + = + = = + = = =

1.34E−14 2.30E+07 6.53E+05 2.40E+04 1.11E−13 1.59E+01 2.56E+00 2.09E+01 3.57E+01 4.11E−02 9.75E−01 1.72E+02 1.74E+02 5.21E+02 7.19E+03 2.45E+00 6.48E+01 2.07E+02 6.81E+00 1.23E+01 3.04E+02 1.19E+03 7.45E+03 2.01E+02 3.02E+02 2.01E+02 8.21E+02 3.00E+02

5.40E−14 6.48E+06 1.10E+06 4.07E+03 1.59E−14 8.67E+00 2.55E+00 5.79E−02 1.26E+00 2.88E−02 9.64E−01 1.27E+01 1.05E+01 1.60E+02 2.93E+02 2.63E−01 4.20E+00 7.11E+00 7.28E−01 2.44E−01 7.68E+01 6.23E+02 2.75E+02 2.67E+00 5.71E+00 4.46E−01 3.90E+02 0.00E+00

Config 3.  = /4 = + = = = = − = = = = = = + + + + + + = = + = = = = = =

0.00E+00 4.67E+07 8.60E+05 2.60E+04 1.09E−13 1.65E+01 2.61E+00 2.09E+01 3.59E+01 2.72E−02 6.05E−01 1.81E+02 1.79E+02 4.30E+02 7.17E+03 2.48E+00 6.30E+01 2.04E+02 6.90E+00 1.24E+01 3.12E+02 1.21E+03 7.48E+03 2.01E+02 3.03E+02 2.03E+02 1.10E+03 3.00E+02

0.00E+00 1.28E+07 1.06E+06 3.42E+03 2.23E−14 1.12E+01 3.14E+00 4.70E−02 1.21E+00 1.82E−02 7.47E−01 1.07E+01 8.15E+00 1.47E+02 2.93E+02 2.38E−01 3.96E+00 1.41E+01 8.09E−01 2.45E−01 7.46E+01 5.80E+02 2.59E+02 4.76E+00 5.04E+00 7.44E−01 2.02E+02 0.00E+00

Config 3.  = /5 = = = = = = = − = = = = = + + + + + + = = + = = = = = =

3.12E−14 5.51E+07 1.10E+06 2.58E+04 1.11E−13 1.54E+01 2.93E+00 2.10E+01 3.60E+01 2.37E−02 4.68E−01 1.84E+02 1.82E+02 4.27E+02 7.19E+03 2.45E+00 6.29E+01 2.11E+02 6.85E+00 1.23E+01 3.01E+02 1.27E+03 7.44E+03 2.01E+02 3.04E+02 2.03E+02 1.13E+03 3.00E+02

7.90E−14 1.23E+07 1.45E+06 4.36E+03 1.59E−14 8.16E+00 3.74E+00 4.46E−02 1.34E+00 1.32E−02 7.54E−01 9.36E+00 9.16E+00 1.30E+02 3.14E+02 2.86E−01 3.93E+00 7.97E+00 6.40E−01 2.43E−01 7.41E+01 5.44E+02 3.15E+02 6.80E+00 4.42E+00 6.27E−01 1.40E+02 0.00E+00

Config 3.  = /6 = = = = = = = = = = = = = + + + + + + + = + = = = = = =

1.34E−14 5.15E+07 8.21E+05 2.71E+04 1.11E−13 1.67E+01 2.82E+00 2.09E+01 3.63E+01 2.37E−02 3.51E−01 1.81E+02 1.83E+02 3.85E+02 7.16E+03 2.47E+00 6.22E+01 2.13E+02 6.97E+00 1.23E+01 3.27E+02 1.25E+03 7.52E+03 2.02E+02 3.05E+02 2.04E+02 1.12E+03 3.00E+02

5.40E−14 1.32E+07 9.52E+05 4.49E+03 1.59E−14 1.13E+01 2.79E+00 5.96E−02 1.44E+00 1.21E−02 5.91E−01 1.12E+01 1.09E+01 1.23E+02 2.60E+02 2.88E−01 3.40E+00 7.80E+00 6.47E−01 2.91E−01 8.40E+01 4.68E+02 2.70E+02 1.20E+01 3.88E+00 9.19E−01 1.67E+02 0.00E+00

Config 3.  = 0 = = = = = = = = = = = = = + + + + + + + = + − = = = = =

4.90E−14 6.16E+07 5.71E+05 2.63E+04 1.11E−13 1.38E+01 4.26E+00 2.09E+01 3.59E+01 2.02E−02 1.95E−01 1.87E+02 1.85E+02 3.75E+02 7.14E+03 2.47E+00 6.18E+01 2.09E+02 6.91E+00 1.24E+01 3.14E+02 1.21E+03 7.44E+03 2.07E+02 3.04E+02 2.04E+02 1.16E+03 3.00E+02

9.44E−14 1.28E+07 7.69E+05 4.22E+03 1.59E−14 4.05E−01 3.88E+00 6.55E−02 1.52E+00 1.27E−02 4.46E−01 1.08E+01 1.11E+01 9.06E+01 2.72E+02 3.20E−01 3.28E+00 1.16E+01 6.10E−01 2.70E−01 6.87E+01 4.96E+02 2.68E+02 1.94E+01 3.96E+00 1.07E+00 5.15E+01 0.00E+00

= = = = = = − = = = = = = + + + + + + + = + = = = = = =

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Please cite this article in press as: A. Draa, et al., A sinusoidal differential evolution algorithm for numerical optimisation, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.11.003

Table A.14 Comparing Configuration 3 with different phase values against Configuration 2; average error, standard deviation of error and statistical comparison (reference:  = ) for CEC 2013 benchmarks in 30 dimensions.

G Model

ASOC 2602 1–27

26

Config 2.  = 

Config 3.  = 

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

2.23E−13 4.56E+06 9.99E+03 1.03E+04 1.14E−13 4.34E+01 5.81E−01 2.11E+01 3.36E+01 8.37E−02 8.53E+00 5.67E+01 1.46E+02 2.00E+03 7.59E+03 3.07E+00 1.25E+02 2.24E+02 7.87E+00 2.03E+01 5.85E+02 1.94E+03 7.15E+03 2.00E+02 2.95E+02 2.46E+02 4.11E+02 4.00E+02

3.07E+01 2.59E+06 1.31E+08 3.69E+04 4.39E+01 4.84E+01 9.09E+01 2.11E+01 5.29E+01 2.11E−01 6.35E+01 1.88E+02 3.51E+02 1.95E+03 7.55E+03 3.23E+00 1.66E+02 2.08E+02 2.70E+01 2.11E+01 8.14E+02 1.35E+03 9.02E+03 3.20E+02 3.77E+02 3.17E+02 1.61E+03 8.88E+02

3.18E−14 1.30E+06 2.59E+04 2.64E+03 7.65E−29 1.44E−14 8.96E−01 3.76E−02 5.69E+00 4.71E−02 2.53E+00 1.01E+01 3.50E+01 3.45E+02 1.11E+03 2.81E−01 8.59E+00 4.56E+01 2.33E+00 9.62E−01 4.35E+02 4.64E+02 9.93E+02 8.06E−02 9.69E+00 5.81E+01 1.28E+02 0.00E+00

1.09E+02 1.01E+06 2.07E+08 1.01E+04 2.96E+02 1.32E+01 1.65E+01 4.56E−02 6.23E+00 1.38E−01 1.99E+01 4.67E+01 6.20E+01 7.52E+02 8.71E+02 4.95E−01 2.04E+01 4.35E+01 1.68E+01 9.58E−01 3.82E+02 7.92E+02 1.05E+03 1.70E+01 1.48E+01 1.16E+02 1.21E+02 1.24E+03

Config 3.  = /2 = = = = = = + = + + = + = + + + + + + + = + + = − + + =

2.25E−13 4.39E+06 1.43E+07 2.64E+04 1.66E−08 4.46E+01 3.25E+01 2.11E+01 6.72E+01 2.23E−01 1.13E+01 3.48E+02 3.56E+02 2.37E+03 1.39E+04 3.35E+00 1.52E+02 3.96E+02 1.78E+01 2.20E+01 6.74E+02 2.61E+03 1.42E+04 2.49E+02 3.13E+02 2.41E+02 1.04E+03 4.00E+02

3.91E−14 1.93E+06 1.21E+07 6.68E+03 8.56E−08 6.63E+00 1.42E+01 4.01E−02 2.23E+00 8.07E−02 4.53E+00 4.16E+01 1.74E+01 4.17E+02 3.74E+02 2.69E−01 1.09E+01 1.24E+01 2.39E+00 2.37E−01 4.29E+02 1.59E+03 4.47E+02 1.29E+01 3.25E+01 8.31E+01 4.33E+02 0.00E+00

Config 3.  = /3 = = = = = = = = = + = = = + + + + + + + = + = = + = = =

2.14E−13 1.12E+08 4.12E+06 3.96E+04 1.27E−13 4.36E+01 1.64E+01 2.11E+01 6.79E+01 1.32E−01 5.19E+00 3.65E+02 3.62E+02 1.90E+03 1.40E+04 3.30E+00 1.44E+02 4.03E+02 1.61E+01 2.21E+01 6.61E+02 2.81E+03 1.44E+04 2.27E+02 3.99E+02 2.43E+02 1.86E+03 4.00E+02

5.40E−14 3.22E+07 5.86E+06 5.83E+03 3.70E−14 8.01E−01 7.40E+00 4.50E−02 1.95E+00 5.79E−02 2.26E+00 1.90E+01 1.56E+01 3.37E+02 3.27E+02 3.09E−01 8.21E+00 1.44E+01 1.22E+00 2.69E−01 4.33E+02 1.40E+03 3.71E+02 3.50E+01 8.40E+00 8.20E+01 3.38E+02 0.00E+00

Config 3.  = /4 = + = = = = − = = = = = = + + + + + + = = + = = = = = =

2.14E−13 1.83E+08 5.15E+06 4.18E+04 1.20E−13 4.34E+01 1.62E+01 2.11E+01 6.84E+01 1.16E−01 3.43E+00 3.79E+02 3.71E+02 1.79E+03 1.40E+04 3.32E+00 1.41E+02 4.08E+02 1.62E+01 2.22E+01 6.85E+02 3.43E+03 1.43E+04 2.34E+02 3.99E+02 2.29E+02 1.98E+03 4.00E+02

5.40E−14 3.26E+07 1.29E+07 4.18E+03 2.70E−14 1.44E−14 9.55E+00 3.75E−02 1.88E+00 5.14E−02 2.20E+00 1.34E+01 1.41E+01 3.47E+02 3.89E+02 3.09E−01 6.84E+00 1.28E+01 1.27E+00 3.46E−01 4.37E+02 1.20E+03 3.98E+02 5.14E+01 5.16E+00 5.18E+01 6.43E+01 0.00E+00

Config 3.  = /5 = = = = = = = − = = = = = + + + + + + = = + = = = = = =

2.18E−13 2.05E+08 5.54E+06 4.31E+04 1.16E−13 4.34E+01 1.72E+01 2.11E+01 6.78E+01 1.07E−01 2.15E+00 3.77E+02 3.74E+02 1.71E+03 1.39E+04 3.37E+00 1.40E+02 4.11E+02 1.58E+01 2.23E+01 6.93E+02 2.92E+03 1.44E+04 2.54E+02 4.00E+02 2.46E+02 1.99E+03 4.00E+02

4.46E−14 3.00E+07 1.99E+07 4.89E+03 1.59E−14 1.44E−14 9.46E+00 3.47E−02 1.98E+00 4.45E−02 1.38E+00 1.42E+01 1.52E+01 3.42E+02 3.71E+02 2.92E−01 6.85E+00 1.49E+01 1.04E+00 2.32E−01 4.15E+02 1.11E+03 3.27E+02 6.13E+01 5.91E+00 7.74E+01 6.42E+01 0.00E+00

Config 3.  = /6 = = = = = = = = = = = = = + + + + + + + = + = = = = = =

2.23E−13 2.05E+08 5.09E+06 4.39E+04 1.14E−13 4.34E+01 1.92E+01 2.11E+01 6.84E+01 9.73E−02 1.70E+00 3.83E+02 3.75E+02 1.71E+03 1.40E+04 3.31E+00 1.35E+02 4.15E+02 1.62E+01 2.22E+01 5.41E+02 2.96E+03 1.44E+04 2.88E+02 4.01E+02 2.59E+02 1.99E+03 4.00E+02

3.18E−14 2.77E+07 1.46E+07 5.17E+03 7.65E−29 6.04E−04 8.86E+00 3.50E−02 1.99E+00 4.40E−02 1.23E+00 1.49E+01 1.31E+01 3.00E+02 3.42E+02 2.84E−01 7.59E+00 1.44E+01 1.20E+00 3.18E−01 4.20E+02 9.24E+02 4.36E+02 5.91E+01 6.26E+00 8.76E+01 5.76E+01 0.00E+00

Config 3.  = 0 = = = = = = = = = = = = = + + + + + + + = + − = = = = =

2.23E−13 2.18E+08 1.57E+07 4.37E+04 1.14E−13 4.34E+01 2.49E+01 2.11E+01 6.84E+01 7.18E−02 9.36E−01 3.89E+02 3.78E+02 1.70E+03 1.40E+04 3.35E+00 1.33E+02 4.24E+02 1.58E+01 2.24E+01 5.82E+02 3.17E+03 1.44E+04 3.07E+02 4.02E+02 2.51E+02 2.00E+03 4.00E+02

3.18E−14 3.22E+07 7.31E+07 4.24E+03 7.65E−29 5.60E−06 9.11E+00 3.39E−02 1.80E+00 3.24E−02 1.25E+00 1.61E+01 1.60E+01 2.96E+02 3.98E+02 2.80E−01 5.97E+00 1.09E+01 1.10E+00 2.27E−01 4.34E+02 1.01E+03 4.12E+02 5.54E+01 6.07E+00 7.64E+01 5.55E+01 0.00E+00

= = = = = = − = = = = = = + + + + + + + = + = = = = = =

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Table A.15 Comparing Configuration 3 with different phase values against Configuration 2; average error, standard deviation of error and statistical comparison (reference:  = ) for CEC 2013 benchmarks in 50 dimensions.

G Model

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27

Table A.16 Summarised Wilcoxon rank-sum comparisons between Configuration 2 and Configuration 3 with different phases. Config 3.  = 

Config 3.  = /2

Config 3.  = /3

Config 3.  = /4

Config 3.  = /5

Config 3.  = /6

Config 3.  = 0

Worse Equal Better Worse Equal Better Worse Equal Better Worse Equal Better Worse Equal Better Worse Equal Better Worse Equal Better D = 10 D = 30 D = 50 Total

626

627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690

3 0 0 3

3 3 3 9

22 25 25 72

5 2 0 7

8 5 5 18

15 21 23 59

7 2 0 9

8 8 6 22

13 18 22 53

8 2 1 11

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16 19 19 54

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691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757