# Ls-II: An Improved Locust Search Algorithm for Solving Optimization Problems

Authors: Octavio Camarena; Erik Cuevas; Marco Pérez-Cisneros; Fernando Fausto; Adrián González; Arturo Valdivia
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

The Locust Search (LS) algorithm is a swarm-based optimization method inspired in the natural behavior of the desert locust. LS considers the inclusion of two distinctive nature-inspired search mechanism, namely, their solitary phase and social phase operators. These interesting search schemes allow LS to overcome some of the difficulties that commonly affect other similar methods, such as premature convergence and the lack of diversity on solutions. Recently, computer vision experiments in insect tracking methods have conducted to the development of more accurate locust motion models than those produced by simple behavior observations. The most distinctive characteristic of such new models is the use of probabilities to emulate the locust decision process. In this paper, a modification to the original LS algorithm, referred to as LS-II, is proposed to better handle global optimization problems. In LS-II, the locust motion model of the original algorithm is modified incorporating the main characteristics of the new biological formulations. As a result, LS-II improves its original capacities of exploration and exploitation of the search space. In order to test its performance, the proposed LS-II method is compared against several the state-of-the-art evolutionary methods considering a set of benchmark functions and engineering problems. Experimental results demonstrate the superior performance of the proposed approach in terms of solution quality and robustness.


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## Body

## 1. Introduction

For the last few decades, optimization approaches inspired by the natural collective behavior of insects and animals have captivated the attention of many researchers. These techniques, commonly referred to as swarm optimization methods, combine deterministic rules and randomness with the purpose of mimicking some kind of natural phenomena, typically manifested in the form of a swarm behavior. Search strategies based in swarm behaviors have demonstrated to be adequate to solve complex optimization problems, often delivering significantly better solutions than those produced by traditional methods. Currently, an extensive variety of swarm-based optimization techniques can be found on the literature. Some examples include Particle Swarm Optimization (PSO), which emulates the social behavior of flocking birds or fishes [1], the Artificial Bee Colony (ABC) approach, which considers the cooperative behavior manifested in bee colonies [2], the Cuckoo Search (CS) algorithm, which simulates the brood parasitism behavior manifested by cuckoo birds [3], the Firefly Algorithm (FA), which mimics the distinctive bioluminescence-based behavior observed in fireflies [4], among others.The Locust Search (LS) [5] algorithm is a swarm optimization approach inspired in the biological behavior of the desert locusts (Schistocerca gregaria). Biologically, locusts experiment two opposite phases: solitary and social. In the solitary phase, locusts avoid contact with others conspecifics in order to explore promising food sources. In opposition, in the social phase, locusts frantically aggregate around abundant foods sources (such as plantations) devastating them. This aggregation is carried on through the attraction to those elements that are found the best food sources. By integrating these two distinctive behaviors, LS maintains powerful global and local search capacities which enable it to solve effectively a wide range of complex optimization problems such as image processing [6], parameter estimation of chaotic systems [7], pattern recognition [8], among others.In spite of its interesting characteristics, LS has some shortcomings from the evolutionary computing point of view. In its social phase operator, LS performs a series of random walks around some of the best individuals with the objective of refining its original solution. This process seems to be appropriate for the purpose of local exploitation; however, the excessive production of candidate solutions adversely increases the algorithm computational overload. Other important flaws of LS are the switch between the solitary and social phase. In LS, both behaviors are performed in the same cycle; this means that each individual search agent in LS is subject to performing both global and local movements at each iteration of the algorithms search process. Although this mechanism could be useful in some contexts, it is known that a misbalance between exploration and exploitation of solutions is able to significantly degrade any algorithm’s performance, thus making this approach somewhat unreliable.Base on this premise, in this paper, a modification to the original LS is proposed in order to better and more efficiently handle global optimization problems. In our modified approach, coined Locust Search II (LS-II), a probabilistic criterion, is introduced in order to control how the solitary and social phase operators are addressed by the algorithm, essentially allowing the swarm of locusts to “decide” when to apply each behavioral phase. Also, individual decision making based on probabilities is introduced to the algorithm’s social phase operator as a mean to allow search agents to manifest an attraction toward prominent individuals within the population. Said modifications were devised with the purpose of providing a better balance between the exploration and exploitation of solutions, while also allowing an important reduction on the algorithms overall computational cost. To demonstrate the proficiency of our proposed approach, we performed a series of experiments over a set of benchmark test functions. The results of our approach are compared to those produced by the original LS, as well as some other state-of-the-art optimization techniques, including Particle Swarm Optimization (PSO) [1], Artificial Bee Colony (ABC) [2], Bat Algorithm (BA) [9], Differential Evolution (DE) [10], and Harmony Search (HS) [11]. Also, in order to enhance the analysis of our proposed method, we also performed comparative experiments over several popular engineering optimization problems, including the design of pressure vessels, gear trains [12], tension/compression springs [13], welded beams [14], three-bar truss [15], parameter estimation for FM synthesizers [16], and Optimal Capacitor Placement for Radial Distribution Networks [17, 18]. The experimental results for both sets shows that LS-II are superior not only over the original LS, but also over all other compared methods in terms of solution quality and robustness.The rest of this paper is organized as follows: in Section2, the original LS approach is described; in Section 3, the proposed LS algorithm modification (LS-II) is presented, and in Section 4, the experimental analysis is developed. Finally, in Section 5, our conclusions are drawn.


## 2. The Locust Search Algorithm

Locust Search (LS) [5] is a global optimization algorithm based on the gregarious behavior observed in swarms of desert locusts [19–24]. In LS, search agents are represented by a set of N individual locusts L={l1,l2,…,lN} (with N representing the total population size), interacting with each other while moving through a n-dimensional feasible solution space. Each individual position li=[li,1,li,2,…,li,n] is defined within a bounded space S={x∈Rn∣lbd≤xd≤ubd} (with x=x1,x2,…,xd and where lbd and ubd represent the lower and upper bounds at the d-th dimension, respectively) and represents a candidate solution for a specified optimization problem.Similar to other swarm-based optimization techniques, LS comprises an iterative scheme in which search agents change their positions at each generation of the algorithm during its evolution. The change of position applied to each individual is conducted by a set of operators inspired in the two behavioral phases observed in desert locusts: solitary phase and social phase.


### 2.1. LS Solitary Phase

In the solitary phase, individuals move in different locations searching for promising food sources (solutions) while they avoid to aggregate with other conspecifics. This scheme is modeled by considering attraction and repulsion forces manifested among individuals within the population. Therefore, for any iteration “k”, the total attraction and repulsions forces (collectively referred to as social force) experienced by a specific individual “i” are given by the following expression:(1)Sik=∑j=1j≠1Nsijkwhere sijk denotes the pairwise attraction (or repulsion) between individual “i” and some other individuals “j” and is given by(2)sijk=ρlik,ljksrijkdij+rand1,-1where the operator ρlik,ljk is known as dominance value between lik and ljk and the value srijk represents the so called social factor, where rijk=lik-ljk denotes the Euclidian distance between the elements “i” and “j”. Therefore, dij=ljk-lik/rijk stands for the unit vector pointing from lik to ljk, while rand(1,-1) is a random vector whose elements are drawn from the uniform distribution of -1,1. The value srijk is given by(3)srijk=Fe-rijk/L-e-rijkwhere the parameters F and L denote the attraction/repulsion magnitude and length scale, respectively. To apply the operator ρlik,ljk, it is first assumed that each individual lik∈Lkl1k,l2k,…,lNk is ranked with a number from 0 (best individual) to N-1 (worst individual) depending on their respective fitness value. Therefore, the dominance value may be given as follows:(4)ρlik,ljk=e-ranklik/Nifranklik≤rankljke-rankljk/Nifranklik>rankljkAs a result of the influence of total social force Sik, each individual “i” manifests a certain tendency to move toward (or away from) other members within the population. Under such conditions, the new position adopted by the individual “i” as a result of Sik may be expressed as(5)li∗=lik+SikThe result of applying the solitary movement operators to each individual lik∈Lk is a new set of candidate solutions L∗=l1∗,l2∗,…,lN∗, which represent the positions taken by each individual as a result of the influence exerted by all other members in the swarm.


### 2.2. LS Social Phase

The social phase operation is applied to refine some of the best candidate solutionsL∗=l1∗,l2∗,…,lN∗ generated by applying the solitary phase movement operator described in Section 2.1. For this purpose, a subset of solutions B=b1,…,bq, comprised by the q best solutions within the set L∗, is first defined. Then, for each solution li∗∈B, a set of h random solutions Mi=m1i,…,mhi is generated within a corresponding subspace Ci∈S, whose limits are given by(6)Ci,nlower=bi,n-rCi,nupper=bi,n+rwhere Ci,nlowerand Ci,nupper represent the upper and lower bounds of each subspace Ci at the n-th dimension, respectively, while bi,n stands for the n-th element (decision variable) from solution bi, and where(7)r=∑n=1dbnupper-bnlowerd∙βwith bdlowerand bdupper denoting the lower and upper bounds at the n-th dimension, respectively, whereas d stands for total number of decision variables. Furthermore, β∈[0,1] represents a scalar factor which modulates the size of Ci.Finally, the best solution fromli∗∈B and its respective h random solutions (m1i,m2i,…,mhi) is assigned as the position for individual “i” at the following iteration “k+1”. That is,(8)lik+1=bestli∗,m1i,m2i,…,mhiIt is worth nothing that any solution li∗ that is not grouped within the set of best solutions B is excluded by the social phase operator. As such, the final position update applied to each individual ‘i’ within the whole swarm may be summarized as follows:(9)lik+1=bestli∗,m1i,m2i,…,mhiifli∗∈Bli∗ifli∗∉B


## 2.1. LS Solitary Phase

In the solitary phase, individuals move in different locations searching for promising food sources (solutions) while they avoid to aggregate with other conspecifics. This scheme is modeled by considering attraction and repulsion forces manifested among individuals within the population. Therefore, for any iteration “k”, the total attraction and repulsions forces (collectively referred to as social force) experienced by a specific individual “i” are given by the following expression:(1)Sik=∑j=1j≠1Nsijkwhere sijk denotes the pairwise attraction (or repulsion) between individual “i” and some other individuals “j” and is given by(2)sijk=ρlik,ljksrijkdij+rand1,-1where the operator ρlik,ljk is known as dominance value between lik and ljk and the value srijk represents the so called social factor, where rijk=lik-ljk denotes the Euclidian distance between the elements “i” and “j”. Therefore, dij=ljk-lik/rijk stands for the unit vector pointing from lik to ljk, while rand(1,-1) is a random vector whose elements are drawn from the uniform distribution of -1,1. The value srijk is given by(3)srijk=Fe-rijk/L-e-rijkwhere the parameters F and L denote the attraction/repulsion magnitude and length scale, respectively. To apply the operator ρlik,ljk, it is first assumed that each individual lik∈Lkl1k,l2k,…,lNk is ranked with a number from 0 (best individual) to N-1 (worst individual) depending on their respective fitness value. Therefore, the dominance value may be given as follows:(4)ρlik,ljk=e-ranklik/Nifranklik≤rankljke-rankljk/Nifranklik>rankljkAs a result of the influence of total social force Sik, each individual “i” manifests a certain tendency to move toward (or away from) other members within the population. Under such conditions, the new position adopted by the individual “i” as a result of Sik may be expressed as(5)li∗=lik+SikThe result of applying the solitary movement operators to each individual lik∈Lk is a new set of candidate solutions L∗=l1∗,l2∗,…,lN∗, which represent the positions taken by each individual as a result of the influence exerted by all other members in the swarm.


## 2.2. LS Social Phase

The social phase operation is applied to refine some of the best candidate solutionsL∗=l1∗,l2∗,…,lN∗ generated by applying the solitary phase movement operator described in Section 2.1. For this purpose, a subset of solutions B=b1,…,bq, comprised by the q best solutions within the set L∗, is first defined. Then, for each solution li∗∈B, a set of h random solutions Mi=m1i,…,mhi is generated within a corresponding subspace Ci∈S, whose limits are given by(6)Ci,nlower=bi,n-rCi,nupper=bi,n+rwhere Ci,nlowerand Ci,nupper represent the upper and lower bounds of each subspace Ci at the n-th dimension, respectively, while bi,n stands for the n-th element (decision variable) from solution bi, and where(7)r=∑n=1dbnupper-bnlowerd∙βwith bdlowerand bdupper denoting the lower and upper bounds at the n-th dimension, respectively, whereas d stands for total number of decision variables. Furthermore, β∈[0,1] represents a scalar factor which modulates the size of Ci.Finally, the best solution fromli∗∈B and its respective h random solutions (m1i,m2i,…,mhi) is assigned as the position for individual “i” at the following iteration “k+1”. That is,(8)lik+1=bestli∗,m1i,m2i,…,mhiIt is worth nothing that any solution li∗ that is not grouped within the set of best solutions B is excluded by the social phase operator. As such, the final position update applied to each individual ‘i’ within the whole swarm may be summarized as follows:(9)lik+1=bestli∗,m1i,m2i,…,mhiifli∗∈Bli∗ifli∗∉B


## 3. The LS-II Algorithm

The LS algorithm proposes an interesting global and local search capacities in the form of the so called solitary and social phase operators. In spite of its interesting characteristics, LS has some shortcomings from an evolutionary computing point of view. Specifically, in its social phase operator, LS performs a series of random walks around some of the best individuals with the objective of refining their current solutions; while this process seems to be appropriate for the purpose of local exploitation, excessive evaluations of candidate solutions are known to adversely increase the algorithm computational load, a phenomenon that is undesirable to virtually any optimization approach. Another important flaw of the original LS approach is given by the way in which the solitary and social phase behavioral operators are addressed. In LS, both the solitary and social phases operators are coded to be performed in sequence at each iteration of the algorithms search process; this means that each individual is subject to undergone to both global and local operations without considering its individual decision. Although this mechanism could be useful in some contexts, it produces a misbalance between exploration and exploitation which could significantly degrade its performance.In this paper a modification to the original LS algorithm, referred to as LS-II, is proposed. In our approach, instead of applying both the solitary and social phase operators in the same iteration, we propose a simple probabilistic scheme to selectively apply either one of this two behaviors at each iteration of the algorithms search process. Furthermore, we have also proposed some changes the original LS’s social operators: essentially, instead of performing a series of local evaluations around promising solutions, LS-II’s social operator incorporates a probabilistic selection mechanism that allows locusts to be attracted toward some other prominent individuals within the population. This modification not only allows to reduce the algorithms overall computational cost, but also enhances LS-II’s convergence speed.


### 3.1. Selecting between Solitary and Social Phases

The first modification, considered in the original LS, involves the selection for any given iteration between the solitary phase and the social phase. In the proposed LS-II approach, we assume that, at the beginning of the search process, individuals have a tendency to perform intensive exploration over the feasible search space by means of their solitary phase behavior, whereas in later stages of the evolution process exploitation is favored by the social phase behavior. Also, it is considered that the individuals maintain a certain probability to behave as either solitary or social depending on the current stage (iteration) of the algorithm’s search process. Therefore, as the evolutionary process progresses, the probability that an element behaves in a social manner increases, while the probability that it acts in solitary behavior, is reduced. Under such circumstances, at each iteration “k”, the behavioral phase P(Lk) applied to the population Lk is chosen as follows:(10)PLk=Solitaryifrand≤pkSocialifrand>pkwhere rand denotes a random number sampled within the uniformly distributed interval 0,1, while value pk, called behavior probability, is given by(11)pk=1-kiternwhere itern stands for the maximum number of iterations considered in the search process. From the expression, it is clear that, as the iterative process increases, the value of the behavior probability pk is linearly decreased. Intuitively, other monotonically decreasing models (such as logarithmic or exponential functions) could also be considered to handle this value.It is important to remark that from both operators (solitary and social) only the social phase has been modified, and the solitary phase behavior proposed by the original LS remains unchanged. In Figure1, we show a comparison between the original LS algorithm and the proposed LS-II approach.Figure 1
Flowchart of (a) LS and (b) LS-II.
(a) (b)


### 3.2. Modified Social Phase Operator

The second modification considers several changes to the original LS algorithm’s social phase, each aimed at giving every individuallik the ability to move toward potentially good solution within the feasible solution space instead of performing a series of local function evaluations. This mechanism does not only allow taking advantage of the information provided by all individuals within the population, but also allows a significant reduction in computational load by taking away excessive and unnecessary function evaluations. Similarly to the original LS approach, the LS-II’s social phase starts by selecting a subset of solutions Bk=b1k,…,bqk that include the q best solutions from the total set Lk={l1k,l2k,…lNk} (therefore, Bk⊆Lk). However, instead of defining a set of random candidate solutions around each position bjk⊆Bk (as in the original LS), we propose a movement scheme that allows each individual lik to be attracted toward the direction of a randomly selected solution bjk. For this purpose, to each solution bjk is assigned to a probability of being selected by a given individual lik, which depends on both the quality of bjk and the distance separating both individuals (lik and bjk) as follows:(12)Plibjk=Abjke-lik-bjk∑n=1qAbnke-lik-bnkwhere lik-bjk denotes the Euclidian distance between the individual “i” lik and a member “j” from the set of best solutions Bkbjk, while Abjk stands for the attractiveness of solution bjk as given by(13)Abjk=fbjk-fworstBkfbestBk-fworstBk+εwhere fbjk denotes the fitness value (quality) related to bjk, while fbestBkfworstBk stand for the best and worst fitness value from among the members within the set of best solutions Bk. Finally, ε corresponds to a small value (typically between 1.0×10-4 and 1.0×10-5) used to prevent a division by cero.With the previous being said, under LS-II’s social phase behavior, each individuallik within the swarm population Lk updates their position as follows:(14)lik+1=lik+2brk-lik∙randwhere brk (with r∈[1,…,q]) is a randomly chosen solution bjk∈Bk, selected by applying the roulette selection method [25] with regard to their respective probabilities Pijk (relative to lik and bjk), while rand stands for a random number drawn from within the uniformly distributed interval 0,1.


## 3.1. Selecting between Solitary and Social Phases

The first modification, considered in the original LS, involves the selection for any given iteration between the solitary phase and the social phase. In the proposed LS-II approach, we assume that, at the beginning of the search process, individuals have a tendency to perform intensive exploration over the feasible search space by means of their solitary phase behavior, whereas in later stages of the evolution process exploitation is favored by the social phase behavior. Also, it is considered that the individuals maintain a certain probability to behave as either solitary or social depending on the current stage (iteration) of the algorithm’s search process. Therefore, as the evolutionary process progresses, the probability that an element behaves in a social manner increases, while the probability that it acts in solitary behavior, is reduced. Under such circumstances, at each iteration “k”, the behavioral phase P(Lk) applied to the population Lk is chosen as follows:(10)PLk=Solitaryifrand≤pkSocialifrand>pkwhere rand denotes a random number sampled within the uniformly distributed interval 0,1, while value pk, called behavior probability, is given by(11)pk=1-kiternwhere itern stands for the maximum number of iterations considered in the search process. From the expression, it is clear that, as the iterative process increases, the value of the behavior probability pk is linearly decreased. Intuitively, other monotonically decreasing models (such as logarithmic or exponential functions) could also be considered to handle this value.It is important to remark that from both operators (solitary and social) only the social phase has been modified, and the solitary phase behavior proposed by the original LS remains unchanged. In Figure1, we show a comparison between the original LS algorithm and the proposed LS-II approach.Figure 1
Flowchart of (a) LS and (b) LS-II.
(a) (b)


## 3.2. Modified Social Phase Operator

The second modification considers several changes to the original LS algorithm’s social phase, each aimed at giving every individuallik the ability to move toward potentially good solution within the feasible solution space instead of performing a series of local function evaluations. This mechanism does not only allow taking advantage of the information provided by all individuals within the population, but also allows a significant reduction in computational load by taking away excessive and unnecessary function evaluations. Similarly to the original LS approach, the LS-II’s social phase starts by selecting a subset of solutions Bk=b1k,…,bqk that include the q best solutions from the total set Lk={l1k,l2k,…lNk} (therefore, Bk⊆Lk). However, instead of defining a set of random candidate solutions around each position bjk⊆Bk (as in the original LS), we propose a movement scheme that allows each individual lik to be attracted toward the direction of a randomly selected solution bjk. For this purpose, to each solution bjk is assigned to a probability of being selected by a given individual lik, which depends on both the quality of bjk and the distance separating both individuals (lik and bjk) as follows:(12)Plibjk=Abjke-lik-bjk∑n=1qAbnke-lik-bnkwhere lik-bjk denotes the Euclidian distance between the individual “i” lik and a member “j” from the set of best solutions Bkbjk, while Abjk stands for the attractiveness of solution bjk as given by(13)Abjk=fbjk-fworstBkfbestBk-fworstBk+εwhere fbjk denotes the fitness value (quality) related to bjk, while fbestBkfworstBk stand for the best and worst fitness value from among the members within the set of best solutions Bk. Finally, ε corresponds to a small value (typically between 1.0×10-4 and 1.0×10-5) used to prevent a division by cero.With the previous being said, under LS-II’s social phase behavior, each individuallik within the swarm population Lk updates their position as follows:(14)lik+1=lik+2brk-lik∙randwhere brk (with r∈[1,…,q]) is a randomly chosen solution bjk∈Bk, selected by applying the roulette selection method [25] with regard to their respective probabilities Pijk (relative to lik and bjk), while rand stands for a random number drawn from within the uniformly distributed interval 0,1.


## 4. Experiments and Results

### 4.1. Benchmark Test Functions

In our experiments, we studied LS-II’s performance with regard to the optimization of 13 well-known benchmark tests functions (see AppendixA) operated in 30 dimensions. Furthermore, we have also compared out proposed method’s result against those produced by other popular optimization techniques, including Particle Swarm Optimization (PSO) [1], Artificial Bee Colony (ABC) [2], Bat Algorithm (BA) [9], Differential Evolution (DE) [10], Harmony Search (HS) [11], and the original Locust Search (LS) [5].The parameters setting for each of the methods has been performed by using the automatic algorithm configuration software known as iRace [26]. This method allows obtaining the best possible parameter configuration under a specific set of benchmark problems. After the iRace analysis, the following settings have been found:(1)
LS: the algorithm’s parameters were set toq=10, h=3 [5].(2)
PSO: the cognitive and social coefficients are set toc1=1.9460 and c2=1.8180, respectively. Also, the inertia weight factor ω is set to decrease linearly from 0.543254 to 0.33362 as the search process evolves [1].(3)
ABC: the algorithm was implemented by setting the parameterlimit=numOfFoodSources∗dims, where numOfFoodSources=N (population size) and dims=n (dimensionality of the solution space) [2]. The code has been obtained from [27] which corresponds to the optimized version (v2).(4)
BA: the parameters were set as follows: initial loudness rateA=0.311895, pulse emission rate r=0.70767, minimum frequency fmin=0.3272, and maximum frequency fmax=1.65815, respectively [9].(5)
DE: the crossover rate is set toCR=0.58143, while the differential weight is given as F=0.12470 [10].(6)
HS: the Harmony Memory Considering Rate is set toHCMR=0.98950 while the pitch adjustment rate is set to PA=0.812174 [11].(7)
LS-II: the number of best individuals considered for the social phase operator is set toq=10.The code for each algorithm has been obtained from their original sources. In the analysis, all of the compared methods were tested by considering a population size ofN=50 individuals and a maximum number of iterations itern=1000 as a stop criterion. Said tests setup was considered in order to keep consistency with other similar works currently found in the literature. All experiments were performed on MatLAB® R2016a, running on a computer with an Intel® Core™ i7-3.40 GHz processor, and Windows 8 (64-bit, 8GB of memory) as operating system.Our experimental setup aims to compare the performance of LS-II against those of PSO, ABC, BA, DE, HS, and LS. The averaged experimental results, corresponding to 30 individual runs, are reported in Table1 where the best outcomes for each test function are boldfaced. The results reported in this paper consider the following performance indexes: Average Best fitness (AB), Median Best fitness (MB), and the standard deviation of the best finesses (SD). As shown by our experiments, LS-II demonstrates to be superior not only to the original LS approach, but also to all other of the compared methods. Such a notorious performance is related to both the excellent global search capabilities provided LS-II’s solitary phase operators, the highly effective exploitation mechanism of our modified social phase operators, and naturally the better trade-off between exploration and exploitation that is achieved through the selective application of said behavioral phases. Furthermore, Wilcoxon’s rank sum test for independent samples [28] was conducted over the best fitness values found by each of the compared methods on 30 independent test runs (30 executions by each set). Table 2 reports the p-values produced by said test for the pairwise comparison over two independent sets of samples (LS-II versus LS, ABC, BA, DE, and HS), by considering a 5% significance level. For the Wilcoxon test, a null hypothesis assumes that there is a significant difference between the mean values of two algorithms. On the other hand, an alternative hypothesis (rejection of the null hypothesis) suggests that the difference between the mean values of both compared methods is insignificant. As shown by all of the p-values reported in Table 3, enough evidence is provided to reject the null hypothesis (all values are less than 0.05, and as such satisfy the 5% significance level criteria). This proves the statistical significant of our proposed method’s results and thus discard the possibility of them being a product of coincidence.Table 1
Minimization results for the benchmark test functions illustrated in AppendixA operated in 30 dimensions. Results were averaged from 30 individual runs, each by considering population size of N=50 and maximum number of iterations itern=1000.
PSO DE ABC HS BA LS LS-II f 1 f m e a n 4.05×10−13 1.30×10−02 2.44×10−11 1.50×10−04 4.48×1000 4.32×10−04 4.33×10 -32 f m e d i a n 2.99×10−13 1.25×10−02 7.07×10−12 1.53×10−04 4.43×1000 2.17×10−04 1.02×10 -33 f s t d 5.65×10−13 4.86×10−03 4.66×10−11 1.61×10−05 8.88×10−01 6.52×10−04 9.30×10 -32  f 2 f m e a n 8.54×1001 1.54×1000 7.80×10−07 7.71×10−01 3.23×1028 5.12×10−02 2.06×10 -16 f m e d i a n 8.94×1001 5.17×10−02 7.43×10−07 7.19×10−01 4.44×1025 5.50×10−02 1.69×10 -16 f s t d 4.95×1001 3.37×1000 3.11×10−07 1.66×10−01 7.16×1028 1.71×10−02 2.13×10 -16  f 3 f m e a n 3.31×10−17 2.67×1002 1.23×1004 9.86×1000 2.51×1006 1.82×10−03 3.86×10 -30 f m e d i a n 2.30×10−18 1.12×1002 1.25×1004 7.30×1000 2.71×1006 1.55×10−03 1.28×10 -31 f s t d 5.23×10−17 3.47×1002 2.57×1003 5.26×1000 1.12×1006 1.38×10−03 8.39×10 -30  f 4 f m e a n 2.02×1000 3.75×1001 2.15×1001 8.16×10−01 3.70×1001 1.05×10−02 7.43×10 -17 f m e d i a n 1.61×1000 3.86×1001 2.17×1001 6.57×10−01 3.80×1001 1.15×10−02 5.12×10 -17 f s t d 1.28×1000 6.43×1000 4.25×1000 2.42×10−01 2.57×1000 6.63×10−03 7.49×10 -17  f 5 f m e a n 7.90×1001 2.05×1001 2.48×10 01 9.76×1000 6.33×1001 4.27×1002 2.89×1001 f m e d i a n 5.87×1001 1.52×1001 2.49×10 01 1.02×1001 6.07×1001 3.74×1002 2.90×1001 f s t d 3.68×1001 1.22×1001 1.24×1000 1.69×1000 1.52×1001 2.81×1002 4.15×10 -02  f 6 f m e a n 2.40×1000 6.94×1000 2.17×10 -11 0.00×1000 9.23×1003 5.58×10−02 6.06×1000 f m e d i a n 1.00×1000 3.90×1000 1.58×10 -11 0.00×1000 9.53×1003 5.67×10−02 5.89×1000 f s t d 2.41×1000 7.74×1000 1.76×10 -11 0.00×1000 1.95×1003 1.07×10−02 5.16×10−01  f 7 f m e a n 9.31×10−03 2.47×1005 1.07×10−01 2.96×1001 5.54×1009 2.21×10−02 8.77×10 -03 f m e d i a n 8.91×10−03 1.32×1005 1.03×10−01 3.13×1001 6.25×1009 1.90×10−02 2.18×10 -03 f s t d 9.53×10−03 4.47×1005 2.58×10-02 3.56×1000 2.17×1009 1.12×10−02 9.65×10 -03  f 8 f m e a n -9.24×1003 -1.16×1004 -1.26×1004 -1.26×10 04 -1.89×10199 -3.80×1002 -3.45×1003 f m e d i a n -9.26×1003 -1.15×1004 -1.26×1004 -1.26×10 04 -3.19×10195 -3.69×1002 -3.32×1003 f s t d 4.45×1002 3.68×1002 1.35×1002 4.49×10 -02 6.55×1004 9.22×10−02 3.62×10−02  f 9 f m e a n 2.82×10−07 4.17×1001 2.68×10−01 1.25×1000 4.74×1002 2.69×10−02 0.00×10 00 f m e d i a n 1.82×10−08 2.32×1000 4.12×10−04 1.28×1000 2.31×1002 3.44×10−03 0.00×10 00 f s t d 5.26×10−07 8.73×1001 4.32×10−01 2.80×10−01 5.43×1002 4.00×10−02 0.00×10 00  f 10 f m e a n 1.43×1000 2.84×1000 1.49×10−05 5.62×10−02 1.28×1001 3.00×10−02 4.44×10 -15 f m e d i a n 1.50×1000 2.96×1000 1.28×10−05 5.80×10−02 1.22×1001 1.73×10−02 4.44×10 -15 f s t d 3.86×10−01 7.21×10−01 9.97×10−06 5.65×10−03 1.17×1000 3.56×10−02 0.00×10 00  f 11 f m e a n 1.52×10−02 1.71×1000 9.66×10−04 1.01×10−01 1.07×1002 1.42×10−04 0.00×10 00 f m e d i a n 8.88×10−16 1.08×1000 1.32×10−08 9.57×10−02 1.14×1002 1.49×10−05 0.00×10 00 f s t d 2.10×10−02 2.01×1000 3.92×10−03 2.74×10−02 2.42×1001 3.69×10−04 0.00×10 00  f 12 f m e a n 8.22×1001 6.98×1004 1.23×10 -12 8.20×1001 2.24×1007 5.08×10−03 1.34×1000 f m e d i a n 8.16×1001 3.53×1004 6.46×10 -13 8.22×1001 2.53×1007 5.12×10−03 1.32×1000 f s t d 4.06×1000 8.22×1004 3.09×10 -12 2.64×1000 7.32×1006 4.17×10−04 1.34×10−01  f 13 f m e a n 1.02×1002 5.08×1004 2.27×10 -11 8.73×1001 9.92×1006 1.08×10−01 3.15×1000 f m e d i a n 1.00×1002 9.69×1003 8.53×10 -12 8.91×1001 9.08×1006 1.05×10−01 3.00×1000 f s t d 3.77×1000 7.78×1004 3.13×10 -11 5.92×1000 2.53×1006 1.37×10−02 2.69×10−01Table 2
Wilcoxon’s test comparison for LS-II versus PSO, De, ABC, HS, BA, and LS. The table shows the resultingp-values for each pairwise comparison.
Function LS-II vs PSO LS-II vs DE LS-II vs ABC LS-II vs HS LS-II vs BA LS-II vs LS f 1 6.54×10−15 1.11×10−12 3.01×10−11 1.12×10−13 4.36×10−13 4.86×10−12 f 2 1.57×10−14 1.52×10−15 8.48×10−09 9.65×10−14 9.78×10−13 2.02×10−13 f 3 5.47×10−12 4.33×10−10 3.01×10−11 7.56×10−10 2.99×10−13 2.89×10−11 f 4 3.65×10−13 4.88×10−10 3.01×10−11 6.19×10−10 3.45×10−10 9.11×10−12 f 5 6.14×10−12 6.95×10−10 8.18×10−01 9.12×10−12 7.25×10−10 1.38×10−11 f 6 6.24×10−11 2.45×10−10 3.01×10−11 4.54×10−12 8.34×10−10 1.84×10−11 f 7 8.53×10−10 7.88×10−10 3.01×10−11 6.37×10−12 8.15×10−12 2.31×10−11 f 8 2.70×10−15 6.31×10−11 3.01×10−11 1.14×10−12 7.77×10−13 2.77×10−11 f 9 9.81×10−11 4.69×10−11 1.21×10−12 1.54×10−13 3.25×10−13 3.24×10−11 f 10 5.64×10−13 2.55×10−11 3.02×10−11 7.18×10−13 4.12×10−10 3.71×10−11 f 11 9.71×10−12 3.78×10−12 1.21×10−12 4.51×10−13 3.03×10−10 4.17×10−11 f 12 4.32×10−12 1.99×10−14 3.02×10−11 8.4710−13 7.21×10−11 4.64×10−11 f 13 2.21×10−12 1.19×10−12 3.02×10−11 9.10×10−12 3.32×10−11 5.10×10−11Table 3
Minimization results for the engineering optimization problems described in AppendixB. Results were obtained from 30 individual runs, each by considering population size of N=50 and maximum number of iterations itern=1000.
PSO DE ABC HS BA LS LS-II J 1 J w o r s t 7302.944 6590.18 6497.539 9702.135 400595 28910.08 6059.73 J b e s t 6876.679 6877.065 6138.001 6523.004 6683.271 6882.137 6059.72 J a v g 6876.679 6897.291 6311.334 7210.724 103291.8 7986.061 6059.72 J s t d 5.61×1002 1.33×1002 7.84×1001 6.82×1002 1.20×1005 5.87×1003 4.30×10-03  J 2 J w o r s t 1.17×10−01 1.18×10−02 3.85×10−02 3.82×10−02 5.14×10−02 6.47×10−02 9.61×10 -03 J b e s t 1.61×10−02 8.13×10−03 2.50×10−02 2.70×10−02 7.99×10−02 7.27×10−03 5.39×10 -03 J a v g 1.17×10−02 7.96×10−03 2.21×10−02 1.09×10−02 5.52×10−02 7.03×10−02 6.74×10 -03 J s t d 4.43×1005 7.94×10−04 1.31×10−18 4.79×1005 2.40×1005 1.16×10−03 1.03×10 -03  J 3 J w o r s t 1.9851 1.7511 1.7498 1.8012 1.7955 1.7449 1.72485 J b e s t 1.7421 1.725 1.7249 1.7248 1.7249 1.7277 1.72485 J a v g 1.7578 1.7298 1.7251 1.7562 1.7252 1.7311 1.72485 J s t d 7.41×10−03 9.87×10−02 3.50×10−04 4.40×10−04 2.50×10−03 8.50×10−01 7.16×10 -07  J 4 J w o r s t 347.763 347.2367 346.4102 347.5836 300.9182 320.2528 279.936 J b e s t 346.4523 346.4538 346.4102 345.3665 284.377 301.3875 279.725 J a v g 346.675 346.7444 346.4102 346.345 286.777 308.2089 279.743 J s t d 2.90×10−01 1.98×10−01 5.74×10 -14 1.98×10−01 3.44×1000 6.68×10−01 4.12×10−02  J 5 J w o r s t 28.51646 26.27407 27.60109 28.81973 30.08068 28.34164 25.7868 J b e s t 25.5127 24.56246 22.85514 23.59037 22.32354 21.05672 20.1103 J a v g 26.15824 25.82232 24.45296 25.59217 27.00966 25.42715 23.898 J s t d 2.19 6.46 3.50 1.72 2.14 2.55 1.56  J 6 J w o r s t 1.35×10−06 9.09×10−07 4.70×10−07 1.79×10−06 3.13×10−08 1.57×10−02 3.96×10 -08 J b e s t 2.22×10−06 1.78×10−09 1.33×10−10 2.67×10−09 8.90×10−10 4.45×10−07 5.10×10 -13 J a v g 7.17×10−03 5.73×10−03 4.30×10−03 8.60×10−03 2.87×10−03 1.43×10−03 1.27×10 -10 J s t d 2.06×10−03 1.65×10−03 1.23×10−03 2.47×10−03 8.23×10−05 4.12×10−03 7.77×10 -09  J 7 J w o r s t 1.58×1005 1.11×1005 8.66×1004 1.07×1005 2.60×1005 8.53×1004 8.44×10 04 J b e s t 8.49×1004 8.37×1004 8.37×1004 8.45×1004 1.12×1005 8.37×1004 8.27×10 04 J a v g 1.05×1005 8.95×1004 8.43×1004 9.35×1004 1.26×1005 8.42×1004 8.33×10 04 J s t d 1.58×1004 8.06×1003 8.30×1002 4.84×1003 2.64×1004 4.85×1002 4.80×10 02The analysis of the final fitness values cannot absolutely characterize the capacities of an optimization algorithm. Consequently, a convergence experiment on the seven compared algorithms has been conducted. The objective of this test is to evaluate the velocity with which an optimization scheme reaches the optimum. In the experiment, the performance of each algorithm is analyzed over a representative set of functions, includingf1, f3, f9, and f11. In the test, all functions are operated considering 30 dimensions. In order to build the convergence graphs, we employ the raw data generated in the simulations. As each function is executed 30 times for each algorithm, we select the convergence data of the run which represents the median final result. Figure 2 presents the fitness evolution of each compared method for functionsf1, f3, f9, and f11. According to Figure 2, the LS-II method presents the overall best convergence when compared to the other compared methods; once again, this performance is testimony of LS-II’s better trade-off between diversification and intensification, as well as the decision-making scheme incorporated to the algorithms social phase operators.Figure 2
Evolution curves of the run which represents the median final result. (a)f1, (b) f3, (c) f9, and (d) f11 represent the convergence graphs obtained by applying HS, DE, BA, ABC, PSO, LS, and LS-II. For each experiment, a population size of N=50 and maximum number of iterations itern=1000 have been considered.
(a) (b) (c) (d)


### 4.2. Engineering Optimization Problems

In order to further evaluate the performance of our proposed approach, we tested LS-II over several well-studied engineering design optimization problems, namely, the design of pressure vessels, gear trains [12], tension/compression springs [13] welded beams [14], three-bar trusses [15], parameter estimation for FM synthesizers [16], and Optimal Capacitor Placement for Radial Distribution Networks [17, 18]. Appendix B provides a detailed description of each engineering problem used in this study.Similarly to the experiments reported in Section4.1, we compared LS-II’s performance over the previously mentioned real problems with those of Particle Swarm Optimization (PSO) [1], Artificial Bee Colony (ABC) [2], Bat Algorithm (BA) [9], Differential evolution (DE) [10], Harmony Search (HS) [11], and the original Locust Search (LS) [5]. For each design problem a penalty function is implemented to penalize the fitness value of solutions that infringe any of the restrictions modeled on each optimization task. The penalty implemented of the fitness function may be illustrated by the following expression: (15)Ji∗x=Jix+Pwhere Ji(x) is one of the seven fitness functions (i∈1,…,7) defined by each engineering problem in terms of a given solution “x”, where P is a penalty value whose magnitude is set to 1.0×105 [29].All experiments described in this section have been performed by considering the same parameter setup and settings described in Section4.1. In Table 3, the experimental results corresponding to each of the compared methods are shown. All of the considered engineering optimization problems were tested a total of 30 times per method. The performance indexes shown in said table include the best and worst fitness value found over all 30 individual runs (Jbest and Jworst, respectively), as well as the average and standard deviation of all fitness values found over said each set of experiments (Javg and Jstd, respectively). For each individual problem, the best outcomes from among all of the compared methods are boldfaced.From the results of Table3, LS-II proves to be the superior choice when applied to handle each of the chosen engineering design problems. Naturally, this is easily attributed to LS-II’s to the effective exploration and exploitation mechanism provided by the selective use of both solitary and social phase operators. Finally, Tables 4 and 5 report the best set of design variables (solutions) found by LS-II for each of the considered engineering optimization problems.Table 4
Best parameter configurations obtained by LS-II for each of the engineering optimization problems illustrated in AppendicesB.1–B.6.
Problem x 1 x 2 x 3 x 4 x 5 x 6 J i ( x ) J 1 0.812500214 0.437500122 42.09844537 176.636599 - - 6059.71623 J 2 40 25 13 56 - - 5.10463×10−13 J 3 0.050664 0.329604 4.372104 - - - 0.0053911 J 4 0.205730336 3.470474331 9.0366238 0.205729651 - - 1.724851989 J 5 0.8696133 0.2169257 -7.2121×10−06 -1.70011996 -0.29988725 - 279.725029 J 6 0.419345 2.422595 4.930669 2.403819 4.247489 4.90245 20.11029Table 5
Best parameter configurations obtained by LS-II for the Optimal Capacitor Placement for the IEEE’s 69-Bus Radial Distribution Networks illustrated in AppendixB.7.
Problem x 1 - x 15 x 16 x 17 - x 59 x 60 x 61 - x 69 J i ( x ) J 7 0 350 0 1200 0 83708.21


## 4.1. Benchmark Test Functions

In our experiments, we studied LS-II’s performance with regard to the optimization of 13 well-known benchmark tests functions (see AppendixA) operated in 30 dimensions. Furthermore, we have also compared out proposed method’s result against those produced by other popular optimization techniques, including Particle Swarm Optimization (PSO) [1], Artificial Bee Colony (ABC) [2], Bat Algorithm (BA) [9], Differential Evolution (DE) [10], Harmony Search (HS) [11], and the original Locust Search (LS) [5].The parameters setting for each of the methods has been performed by using the automatic algorithm configuration software known as iRace [26]. This method allows obtaining the best possible parameter configuration under a specific set of benchmark problems. After the iRace analysis, the following settings have been found:(1)
LS: the algorithm’s parameters were set toq=10, h=3 [5].(2)
PSO: the cognitive and social coefficients are set toc1=1.9460 and c2=1.8180, respectively. Also, the inertia weight factor ω is set to decrease linearly from 0.543254 to 0.33362 as the search process evolves [1].(3)
ABC: the algorithm was implemented by setting the parameterlimit=numOfFoodSources∗dims, where numOfFoodSources=N (population size) and dims=n (dimensionality of the solution space) [2]. The code has been obtained from [27] which corresponds to the optimized version (v2).(4)
BA: the parameters were set as follows: initial loudness rateA=0.311895, pulse emission rate r=0.70767, minimum frequency fmin=0.3272, and maximum frequency fmax=1.65815, respectively [9].(5)
DE: the crossover rate is set toCR=0.58143, while the differential weight is given as F=0.12470 [10].(6)
HS: the Harmony Memory Considering Rate is set toHCMR=0.98950 while the pitch adjustment rate is set to PA=0.812174 [11].(7)
LS-II: the number of best individuals considered for the social phase operator is set toq=10.The code for each algorithm has been obtained from their original sources. In the analysis, all of the compared methods were tested by considering a population size ofN=50 individuals and a maximum number of iterations itern=1000 as a stop criterion. Said tests setup was considered in order to keep consistency with other similar works currently found in the literature. All experiments were performed on MatLAB® R2016a, running on a computer with an Intel® Core™ i7-3.40 GHz processor, and Windows 8 (64-bit, 8GB of memory) as operating system.Our experimental setup aims to compare the performance of LS-II against those of PSO, ABC, BA, DE, HS, and LS. The averaged experimental results, corresponding to 30 individual runs, are reported in Table1 where the best outcomes for each test function are boldfaced. The results reported in this paper consider the following performance indexes: Average Best fitness (AB), Median Best fitness (MB), and the standard deviation of the best finesses (SD). As shown by our experiments, LS-II demonstrates to be superior not only to the original LS approach, but also to all other of the compared methods. Such a notorious performance is related to both the excellent global search capabilities provided LS-II’s solitary phase operators, the highly effective exploitation mechanism of our modified social phase operators, and naturally the better trade-off between exploration and exploitation that is achieved through the selective application of said behavioral phases. Furthermore, Wilcoxon’s rank sum test for independent samples [28] was conducted over the best fitness values found by each of the compared methods on 30 independent test runs (30 executions by each set). Table 2 reports the p-values produced by said test for the pairwise comparison over two independent sets of samples (LS-II versus LS, ABC, BA, DE, and HS), by considering a 5% significance level. For the Wilcoxon test, a null hypothesis assumes that there is a significant difference between the mean values of two algorithms. On the other hand, an alternative hypothesis (rejection of the null hypothesis) suggests that the difference between the mean values of both compared methods is insignificant. As shown by all of the p-values reported in Table 3, enough evidence is provided to reject the null hypothesis (all values are less than 0.05, and as such satisfy the 5% significance level criteria). This proves the statistical significant of our proposed method’s results and thus discard the possibility of them being a product of coincidence.Table 1
Minimization results for the benchmark test functions illustrated in AppendixA operated in 30 dimensions. Results were averaged from 30 individual runs, each by considering population size of N=50 and maximum number of iterations itern=1000.
PSO DE ABC HS BA LS LS-II f 1 f m e a n 4.05×10−13 1.30×10−02 2.44×10−11 1.50×10−04 4.48×1000 4.32×10−04 4.33×10 -32 f m e d i a n 2.99×10−13 1.25×10−02 7.07×10−12 1.53×10−04 4.43×1000 2.17×10−04 1.02×10 -33 f s t d 5.65×10−13 4.86×10−03 4.66×10−11 1.61×10−05 8.88×10−01 6.52×10−04 9.30×10 -32  f 2 f m e a n 8.54×1001 1.54×1000 7.80×10−07 7.71×10−01 3.23×1028 5.12×10−02 2.06×10 -16 f m e d i a n 8.94×1001 5.17×10−02 7.43×10−07 7.19×10−01 4.44×1025 5.50×10−02 1.69×10 -16 f s t d 4.95×1001 3.37×1000 3.11×10−07 1.66×10−01 7.16×1028 1.71×10−02 2.13×10 -16  f 3 f m e a n 3.31×10−17 2.67×1002 1.23×1004 9.86×1000 2.51×1006 1.82×10−03 3.86×10 -30 f m e d i a n 2.30×10−18 1.12×1002 1.25×1004 7.30×1000 2.71×1006 1.55×10−03 1.28×10 -31 f s t d 5.23×10−17 3.47×1002 2.57×1003 5.26×1000 1.12×1006 1.38×10−03 8.39×10 -30  f 4 f m e a n 2.02×1000 3.75×1001 2.15×1001 8.16×10−01 3.70×1001 1.05×10−02 7.43×10 -17 f m e d i a n 1.61×1000 3.86×1001 2.17×1001 6.57×10−01 3.80×1001 1.15×10−02 5.12×10 -17 f s t d 1.28×1000 6.43×1000 4.25×1000 2.42×10−01 2.57×1000 6.63×10−03 7.49×10 -17  f 5 f m e a n 7.90×1001 2.05×1001 2.48×10 01 9.76×1000 6.33×1001 4.27×1002 2.89×1001 f m e d i a n 5.87×1001 1.52×1001 2.49×10 01 1.02×1001 6.07×1001 3.74×1002 2.90×1001 f s t d 3.68×1001 1.22×1001 1.24×1000 1.69×1000 1.52×1001 2.81×1002 4.15×10 -02  f 6 f m e a n 2.40×1000 6.94×1000 2.17×10 -11 0.00×1000 9.23×1003 5.58×10−02 6.06×1000 f m e d i a n 1.00×1000 3.90×1000 1.58×10 -11 0.00×1000 9.53×1003 5.67×10−02 5.89×1000 f s t d 2.41×1000 7.74×1000 1.76×10 -11 0.00×1000 1.95×1003 1.07×10−02 5.16×10−01  f 7 f m e a n 9.31×10−03 2.47×1005 1.07×10−01 2.96×1001 5.54×1009 2.21×10−02 8.77×10 -03 f m e d i a n 8.91×10−03 1.32×1005 1.03×10−01 3.13×1001 6.25×1009 1.90×10−02 2.18×10 -03 f s t d 9.53×10−03 4.47×1005 2.58×10-02 3.56×1000 2.17×1009 1.12×10−02 9.65×10 -03  f 8 f m e a n -9.24×1003 -1.16×1004 -1.26×1004 -1.26×10 04 -1.89×10199 -3.80×1002 -3.45×1003 f m e d i a n -9.26×1003 -1.15×1004 -1.26×1004 -1.26×10 04 -3.19×10195 -3.69×1002 -3.32×1003 f s t d 4.45×1002 3.68×1002 1.35×1002 4.49×10 -02 6.55×1004 9.22×10−02 3.62×10−02  f 9 f m e a n 2.82×10−07 4.17×1001 2.68×10−01 1.25×1000 4.74×1002 2.69×10−02 0.00×10 00 f m e d i a n 1.82×10−08 2.32×1000 4.12×10−04 1.28×1000 2.31×1002 3.44×10−03 0.00×10 00 f s t d 5.26×10−07 8.73×1001 4.32×10−01 2.80×10−01 5.43×1002 4.00×10−02 0.00×10 00  f 10 f m e a n 1.43×1000 2.84×1000 1.49×10−05 5.62×10−02 1.28×1001 3.00×10−02 4.44×10 -15 f m e d i a n 1.50×1000 2.96×1000 1.28×10−05 5.80×10−02 1.22×1001 1.73×10−02 4.44×10 -15 f s t d 3.86×10−01 7.21×10−01 9.97×10−06 5.65×10−03 1.17×1000 3.56×10−02 0.00×10 00  f 11 f m e a n 1.52×10−02 1.71×1000 9.66×10−04 1.01×10−01 1.07×1002 1.42×10−04 0.00×10 00 f m e d i a n 8.88×10−16 1.08×1000 1.32×10−08 9.57×10−02 1.14×1002 1.49×10−05 0.00×10 00 f s t d 2.10×10−02 2.01×1000 3.92×10−03 2.74×10−02 2.42×1001 3.69×10−04 0.00×10 00  f 12 f m e a n 8.22×1001 6.98×1004 1.23×10 -12 8.20×1001 2.24×1007 5.08×10−03 1.34×1000 f m e d i a n 8.16×1001 3.53×1004 6.46×10 -13 8.22×1001 2.53×1007 5.12×10−03 1.32×1000 f s t d 4.06×1000 8.22×1004 3.09×10 -12 2.64×1000 7.32×1006 4.17×10−04 1.34×10−01  f 13 f m e a n 1.02×1002 5.08×1004 2.27×10 -11 8.73×1001 9.92×1006 1.08×10−01 3.15×1000 f m e d i a n 1.00×1002 9.69×1003 8.53×10 -12 8.91×1001 9.08×1006 1.05×10−01 3.00×1000 f s t d 3.77×1000 7.78×1004 3.13×10 -11 5.92×1000 2.53×1006 1.37×10−02 2.69×10−01Table 2
Wilcoxon’s test comparison for LS-II versus PSO, De, ABC, HS, BA, and LS. The table shows the resultingp-values for each pairwise comparison.
Function LS-II vs PSO LS-II vs DE LS-II vs ABC LS-II vs HS LS-II vs BA LS-II vs LS f 1 6.54×10−15 1.11×10−12 3.01×10−11 1.12×10−13 4.36×10−13 4.86×10−12 f 2 1.57×10−14 1.52×10−15 8.48×10−09 9.65×10−14 9.78×10−13 2.02×10−13 f 3 5.47×10−12 4.33×10−10 3.01×10−11 7.56×10−10 2.99×10−13 2.89×10−11 f 4 3.65×10−13 4.88×10−10 3.01×10−11 6.19×10−10 3.45×10−10 9.11×10−12 f 5 6.14×10−12 6.95×10−10 8.18×10−01 9.12×10−12 7.25×10−10 1.38×10−11 f 6 6.24×10−11 2.45×10−10 3.01×10−11 4.54×10−12 8.34×10−10 1.84×10−11 f 7 8.53×10−10 7.88×10−10 3.01×10−11 6.37×10−12 8.15×10−12 2.31×10−11 f 8 2.70×10−15 6.31×10−11 3.01×10−11 1.14×10−12 7.77×10−13 2.77×10−11 f 9 9.81×10−11 4.69×10−11 1.21×10−12 1.54×10−13 3.25×10−13 3.24×10−11 f 10 5.64×10−13 2.55×10−11 3.02×10−11 7.18×10−13 4.12×10−10 3.71×10−11 f 11 9.71×10−12 3.78×10−12 1.21×10−12 4.51×10−13 3.03×10−10 4.17×10−11 f 12 4.32×10−12 1.99×10−14 3.02×10−11 8.4710−13 7.21×10−11 4.64×10−11 f 13 2.21×10−12 1.19×10−12 3.02×10−11 9.10×10−12 3.32×10−11 5.10×10−11Table 3
Minimization results for the engineering optimization problems described in AppendixB. Results were obtained from 30 individual runs, each by considering population size of N=50 and maximum number of iterations itern=1000.
PSO DE ABC HS BA LS LS-II J 1 J w o r s t 7302.944 6590.18 6497.539 9702.135 400595 28910.08 6059.73 J b e s t 6876.679 6877.065 6138.001 6523.004 6683.271 6882.137 6059.72 J a v g 6876.679 6897.291 6311.334 7210.724 103291.8 7986.061 6059.72 J s t d 5.61×1002 1.33×1002 7.84×1001 6.82×1002 1.20×1005 5.87×1003 4.30×10-03  J 2 J w o r s t 1.17×10−01 1.18×10−02 3.85×10−02 3.82×10−02 5.14×10−02 6.47×10−02 9.61×10 -03 J b e s t 1.61×10−02 8.13×10−03 2.50×10−02 2.70×10−02 7.99×10−02 7.27×10−03 5.39×10 -03 J a v g 1.17×10−02 7.96×10−03 2.21×10−02 1.09×10−02 5.52×10−02 7.03×10−02 6.74×10 -03 J s t d 4.43×1005 7.94×10−04 1.31×10−18 4.79×1005 2.40×1005 1.16×10−03 1.03×10 -03  J 3 J w o r s t 1.9851 1.7511 1.7498 1.8012 1.7955 1.7449 1.72485 J b e s t 1.7421 1.725 1.7249 1.7248 1.7249 1.7277 1.72485 J a v g 1.7578 1.7298 1.7251 1.7562 1.7252 1.7311 1.72485 J s t d 7.41×10−03 9.87×10−02 3.50×10−04 4.40×10−04 2.50×10−03 8.50×10−01 7.16×10 -07  J 4 J w o r s t 347.763 347.2367 346.4102 347.5836 300.9182 320.2528 279.936 J b e s t 346.4523 346.4538 346.4102 345.3665 284.377 301.3875 279.725 J a v g 346.675 346.7444 346.4102 346.345 286.777 308.2089 279.743 J s t d 2.90×10−01 1.98×10−01 5.74×10 -14 1.98×10−01 3.44×1000 6.68×10−01 4.12×10−02  J 5 J w o r s t 28.51646 26.27407 27.60109 28.81973 30.08068 28.34164 25.7868 J b e s t 25.5127 24.56246 22.85514 23.59037 22.32354 21.05672 20.1103 J a v g 26.15824 25.82232 24.45296 25.59217 27.00966 25.42715 23.898 J s t d 2.19 6.46 3.50 1.72 2.14 2.55 1.56  J 6 J w o r s t 1.35×10−06 9.09×10−07 4.70×10−07 1.79×10−06 3.13×10−08 1.57×10−02 3.96×10 -08 J b e s t 2.22×10−06 1.78×10−09 1.33×10−10 2.67×10−09 8.90×10−10 4.45×10−07 5.10×10 -13 J a v g 7.17×10−03 5.73×10−03 4.30×10−03 8.60×10−03 2.87×10−03 1.43×10−03 1.27×10 -10 J s t d 2.06×10−03 1.65×10−03 1.23×10−03 2.47×10−03 8.23×10−05 4.12×10−03 7.77×10 -09  J 7 J w o r s t 1.58×1005 1.11×1005 8.66×1004 1.07×1005 2.60×1005 8.53×1004 8.44×10 04 J b e s t 8.49×1004 8.37×1004 8.37×1004 8.45×1004 1.12×1005 8.37×1004 8.27×10 04 J a v g 1.05×1005 8.95×1004 8.43×1004 9.35×1004 1.26×1005 8.42×1004 8.33×10 04 J s t d 1.58×1004 8.06×1003 8.30×1002 4.84×1003 2.64×1004 4.85×1002 4.80×10 02The analysis of the final fitness values cannot absolutely characterize the capacities of an optimization algorithm. Consequently, a convergence experiment on the seven compared algorithms has been conducted. The objective of this test is to evaluate the velocity with which an optimization scheme reaches the optimum. In the experiment, the performance of each algorithm is analyzed over a representative set of functions, includingf1, f3, f9, and f11. In the test, all functions are operated considering 30 dimensions. In order to build the convergence graphs, we employ the raw data generated in the simulations. As each function is executed 30 times for each algorithm, we select the convergence data of the run which represents the median final result. Figure 2 presents the fitness evolution of each compared method for functionsf1, f3, f9, and f11. According to Figure 2, the LS-II method presents the overall best convergence when compared to the other compared methods; once again, this performance is testimony of LS-II’s better trade-off between diversification and intensification, as well as the decision-making scheme incorporated to the algorithms social phase operators.Figure 2
Evolution curves of the run which represents the median final result. (a)f1, (b) f3, (c) f9, and (d) f11 represent the convergence graphs obtained by applying HS, DE, BA, ABC, PSO, LS, and LS-II. For each experiment, a population size of N=50 and maximum number of iterations itern=1000 have been considered.
(a) (b) (c) (d)


## 4.2. Engineering Optimization Problems

In order to further evaluate the performance of our proposed approach, we tested LS-II over several well-studied engineering design optimization problems, namely, the design of pressure vessels, gear trains [12], tension/compression springs [13] welded beams [14], three-bar trusses [15], parameter estimation for FM synthesizers [16], and Optimal Capacitor Placement for Radial Distribution Networks [17, 18]. Appendix B provides a detailed description of each engineering problem used in this study.Similarly to the experiments reported in Section4.1, we compared LS-II’s performance over the previously mentioned real problems with those of Particle Swarm Optimization (PSO) [1], Artificial Bee Colony (ABC) [2], Bat Algorithm (BA) [9], Differential evolution (DE) [10], Harmony Search (HS) [11], and the original Locust Search (LS) [5]. For each design problem a penalty function is implemented to penalize the fitness value of solutions that infringe any of the restrictions modeled on each optimization task. The penalty implemented of the fitness function may be illustrated by the following expression: (15)Ji∗x=Jix+Pwhere Ji(x) is one of the seven fitness functions (i∈1,…,7) defined by each engineering problem in terms of a given solution “x”, where P is a penalty value whose magnitude is set to 1.0×105 [29].All experiments described in this section have been performed by considering the same parameter setup and settings described in Section4.1. In Table 3, the experimental results corresponding to each of the compared methods are shown. All of the considered engineering optimization problems were tested a total of 30 times per method. The performance indexes shown in said table include the best and worst fitness value found over all 30 individual runs (Jbest and Jworst, respectively), as well as the average and standard deviation of all fitness values found over said each set of experiments (Javg and Jstd, respectively). For each individual problem, the best outcomes from among all of the compared methods are boldfaced.From the results of Table3, LS-II proves to be the superior choice when applied to handle each of the chosen engineering design problems. Naturally, this is easily attributed to LS-II’s to the effective exploration and exploitation mechanism provided by the selective use of both solitary and social phase operators. Finally, Tables 4 and 5 report the best set of design variables (solutions) found by LS-II for each of the considered engineering optimization problems.Table 4
Best parameter configurations obtained by LS-II for each of the engineering optimization problems illustrated in AppendicesB.1–B.6.
Problem x 1 x 2 x 3 x 4 x 5 x 6 J i ( x ) J 1 0.812500214 0.437500122 42.09844537 176.636599 - - 6059.71623 J 2 40 25 13 56 - - 5.10463×10−13 J 3 0.050664 0.329604 4.372104 - - - 0.0053911 J 4 0.205730336 3.470474331 9.0366238 0.205729651 - - 1.724851989 J 5 0.8696133 0.2169257 -7.2121×10−06 -1.70011996 -0.29988725 - 279.725029 J 6 0.419345 2.422595 4.930669 2.403819 4.247489 4.90245 20.11029Table 5
Best parameter configurations obtained by LS-II for the Optimal Capacitor Placement for the IEEE’s 69-Bus Radial Distribution Networks illustrated in AppendixB.7.
Problem x 1 - x 15 x 16 x 17 - x 59 x 60 x 61 - x 69 J i ( x ) J 7 0 350 0 1200 0 83708.21


## 5. Conclusions

Locust Search (LS) is a swarm-based optimization method inspired in the natural behavior of the desert locust [5]. When it was first proposed, the LS method considered as basis the biological models extracted from observations of locust behaviors. In these models, the procedure to change from the solitary phase to the gregarious phase, together with the mechanism to select the element to which a locust will be attracted, is fixed and does not consider the decision of each locust in the behavioral process.Currently, new computer vision experiments in insect tracking methods have conducted to the development of more accurate locust motion models than those produced by simple behavior observations. The most distinctive characteristic of such new models is the use of probabilities to emulate the locust decision process.In this paper, a modification to the original LS algorithm, referred to as LS-II, has been proposed to better handle global optimization problems. In LS-II, the locust motion model of the original algorithm is modified incorporating the main characteristics of the new biological formulations. Under the new model, probability decisions are considered to emulate the behavior of the locust colony. Therefore, the decisions, to change of behavioral phase and to select the attraction elements, are probabilistically taken.The proposed LS-II method has been compared against several state-of-the-art evolutionary methods considering a set of benchmark functions and engineering problems. Experimental results demonstrate the superior performance of the proposed approach in terms of solution quality and robustness.



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