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Grey wolf optimizer with cellular topological structure
Accepted Manuscript
Grey Wolf Optimizer with Cellular Topological Structure Chao Lu , Liang Gao , Jin Yi PII: DOI: Reference:
S0957-4174(18)30243-4 10.1016/j.eswa.2018.04.012 ESWA 11926
To appear in:
Expert Systems With Applications
Received date: Revised date: Accepted date:
15 December 2017 21 March 2018 9 April 2018
Please cite this article as: Chao Lu , Liang Gao , Jin Yi , Grey Wolf Optimizer with Cellular Topological Structure, Expert Systems With Applications (2018), doi: 10.1016/j.eswa.2018.04.012
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Highlights The cellular automata concept is embedded into the GWO CGWO with a topological structure can help to improve diversity of population The proposed CGWO can solve multimodal problems well The CGWO outperforms the other state-of-the-art algorithms on function and engineering problems
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Grey Wolf Optimizer with Cellular Topological Structure Chao Lua, Liang Gaob, Jin Yic Hubei Key Laboratory of Intelligent Geo-Information Processing (China University of Geosciences (Wuhan)), Wuhan 430074, China b. State Key Lab of Digital Manufacturing Equipment & Technology, Huazhong University of Science and Technology, Wuhan, China c. Department of Industrial Systems Engineering and Management, National University of Singapore, Singapore
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a.
Abstract: Grey wolf optimizer (GWO) is a newly developed metaheuristic inspired by hunting mechanism of grey wolves. The paramount challenge in GWO is that it is prone to stagnation in local optima. This paper proposes a cellular grey wolf optimizer with a topological structure (CGWO). The proposed CGWO has two characteristics. Firstly, each wolf has its own topological neighbors, and interactions among wolves are restricted to their neighbors, which favors exploitation of CGWO. Secondly, information diffusion mechanism by overlap among neighbors can allow to maintain the population diversity for
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longer, usually contributing to exploration. Empirical studies are conducted to compare the proposed algorithm with different metaheuristics such as success-history based adaptive differential evolution with linear population size reduction (LSHADE), teaching-learning based optimization algorithm (TLBO), effective butterfly optimizer with covariance matrix adapted retreat phase (EBOwithCMAR), novel dynamic harmony search (NDHS), bat-inspired algorithm (BA), comprehensive learning particle swarm optimizer (CLPSO), evolutionary algorithm based on decomposition (EAD), ring topology PSO (RPSO), crowding-based differential evolution (CDE), neighborhood based crowding differential evolution (NCDE), locally informed
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particle swarm (LIPS), some improved variants of GWO and GWO. Experimental results show that the proposed method performs better than the other algorithms on most benchmarks and engineering problems.
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Keywords: Grey wolf optimizer; Cellular automata; Metaheuristics; Engineering optimization; Global optimization Introduction
Metaheuristics have received widespread attention over the last two decades due to their simplicity, flexibility and
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derivation-free mechanism. A variety of metaheuristics, such as Genetic Algorithm (GA) (Goldberg & Holland, 1988), Particle Swarm Optimization (PSO) (J & R, 1995), Differential Evolution (DE) (Storn & Price, 1997), Cognitive Behavior Optimization Algorithm (CBO) (M. Li, Zhao, Weng, & Han, 2016) and Moth-flame Optimization Algorithm (MFO)
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(Seyedali Mirjalili, 2015), have been proposed and successfully applied in many engineering fields. Metaheuristics can usually be classified into two categories: (1) single solution-based algorithms like Simulated Annealing (SA) (Kirkpatrick, Gelatt, & Vecchi, 1983). It begins with a candidate solution, and then the quality of this candidate solution is improved during
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the search progress. (2) population-based metaheuristics such as Biogeography-Based Optimizer (BBO) (Simon, 2008), Teaching-Learning Based Optimization algorithm (TLBO) (R. V. Rao, Savsani, & Balic, 2012; R. V. Rao, Savsani, & Vakharia, 2012), Grey Wolf Optimizer (GWO) (S. Mirjalili, Mirjalili, & Lewis, 2014), Water Evaporation Optimization (WEO) (Kaveh & Bakhshpoori, 2016), Multi-Verse Optimizer (MVO) (S. Mirjalili, Mirjalili, & Hatamlou, 2016), and Yin-Yang-Pair Optimization (YYPO) (Punnathanam & Kotecha, 2016). The characteristic of population-based metaheuristics is that the optimization search is performed on a set of solutions. Compared with single solution-based metaheuristics, population-based metaheuristics have some advantages as follows:
Corresponding author.
E-mail address: [email protected] (Chao Lu)
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A set of trial solutions can share information about the search space which guides the trial solutions toward the promising areas of the search space.
A set of trial solutions can help each other to avoid local optimum (S. Mirjalili, et al., 2014).
Population-based metaheuristics usually have a greater exploration ability than single solution-based metaheuristics. Although different metaheuristics have various search manners, most of them are based on the common conceptualization
which balances diversification (exploration of the search space) and intensification (exploitation of already found approximate solutions). Thus, exploration and exploitation are two cornerstones of metaheuristics. Exploration is the process of visiting new regions of the search space, whilst exploitation is the process of searching those areas of the search space
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within the neighborhood of previously visited points (C. Lu, Li, Gao, Liao, & Yi, 2017; Matej, Črepinšek, Liu, & Mernik, 2013).
GWO (S. Mirjalili, et al., 2014) is one recently developed population-based algorithm inspired by hunting mechanism of grey wolves. Compared with other population-based algorithms such as PSO and GA, GWO presents a powerful search ability (Chao Lu, Gao, Li, & Xiao, 2017). Some efforts on the GWO have been made in terms of application and theory. From the applicable perspective, GWO has been utilized to address human recognition (Sanchez, Melin, & Castillo, 2017),
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and unmanned combat aerial vehicle path planning (Zhang, Zhou, Li, & Pan, 2016). From a theoretical perspective, Rodriguez et al. (2016; 2017; 2017) proposed a new GWO with a hierarchical operator and an improved GWO with fuzzy logic, respectively. Joshi and Arora (2017) proposed an enhanced grey wolf optimizer with a better hunting mechanism to balance between exploration and exploitation. Heidari and Pahlavani (2017) proposed an efficient GWO where Lévy flight (LF) and greedy selection strategies are integrated with the modified hunting phases of GWO. In GWO, the search process is guided by the three best wolves at each iteration. This search scheme promotes exploitation since all candidate wolves (solutions) are attracted toward the three best wolves, thereby converging faster toward these wolves. However, as a result of
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such a strong exploitation effect, the search diversity would be hampered in a sense. Finally, the GWO is prone to stagnation in local optimum (Chao Lu, Xiao, Li, & Gao, 2016). Meanwhile, according to the previous experiments (Qu, Liang, Wang, Chen, & Suganthan, 2016), many metaheuristics like GWO is not suitable for solving multimodal optimization problems
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where there exist multiple global optimum solutions because it is originally designed to address single global optimization problems. To address the above issues, cellular automata (CA) is embedded into GWO to maintain population diversity and locate multiple optimal peaks. The main motivations for the GWO combined with CA are as follows: (1) CA provides a
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neighborhood structure for GWO. In CA, all the individuals (solutions) are arranged in a toroidal mesh and each individual has its own neighborhood. Due to the isolation by using neighborhoods, the population diversity can be preserved well. (2)
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The shared information from overlap areas between neighborhoods can contribute to exploration. The information in different neighborhoods can transit from one area to another by diffusing good solutions in overlap areas between neighborhoods. Moreover, since individual’s interaction is restricted to its neighborhood, each neighbor has a good local search ability.
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Whereas all the individuals in different neighborhoods can find promising solutions in a parallel search way, which improves global search ability and helps to locate multiple optimal peaks. Therefore, GWO with CA can effectively balance exploitation and exploration. Some efforts on metaheuristics with a topological structure have been made in recent years. For example, Shi et al. (2011) proposed a cellular PSO by incorporating the cellular topological structure into the PSO to improve the performance of PSO. Gao et al. (2012) successfully applied the cellular PSO to parameters optimization of a multi-pass milling process. Yi et al. (2016) proposed a modified harmony search algorithm with a cellular local search to enhance the exploitation capability. Alba and Dorronsoro et al. (2005) developed a cellular GA by combining GA with cellular automata in order to enhance the performance of the basic GA. Li (2010) proposed a simple yet effective niching PSO with a ring neighborhood topology, which showed a more effective performance than some existing PSO niching algorithms. Das et al. (2009) developed an effective DE using a neighborhood-based mutation operator, which is competitive or superior to several existing DEs.
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Piotrowski (2013) extended the work of Das et al. (2009) by considering the additional strategies. Obviously, metaheuristics with a topological structure have a significant improvement in the search ability. CA can provide such a topological structure. Therefore, we propose a hybrid GWO with CA called CGWO for continuous optimization problems. This paper uses three groups of benchmarks and engineering application problems to evaluate the behavior of the proposed CGWO by comparing with other algorithms including success-history based adaptive differential evolution with linear population size reduction (LSHADE) (Tanabe & Fukunaga, 2014), effective butterfly optimizer with covariance matrix adapted retreat phase (EBOwithCMAR) (Kumar, Misra, & Singh, 2017), teaching-learning based optimization algorithm (TLBO) (Črepinšek, Liu, & Mernik, 2012), novel dynamic harmony search (NDHS) (J. Chen, Pan, & Li, 2012), bat-inspired algorithm (BA) (Yang & Gandomi, 2012), comprehensive learning particle swarm optimizer (CLPSO) (Liang, Qin,
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Suganthan, & Baskar, 2006), evolutionary algorithm based on decomposition (EAD) (Gu, Cheung, & Luo, 2015), ring topology PSO (RPSO) (X. D. Li, 2010), crowding-based differential evolution (CDE) (Thomsen, 2004), neighborhood based crowding differential evolution (NCDE) (Qu, Suganthan, & Liang, 2012), locally informed particle swarm (LIPS) (Qu, et al., 2016), island-based harmony search (iHS) (Al-Betar, Awadallah, Khader, & Abdalkareem, 2015), some improved versions of GWO (Joshi & Arora, 2017; Malik, Mohideen, & Ali, 2016; Yu, Liu, Wang, & Gao, 2017), and grey wolf optimizer (GWO) (S. Mirjalili, et al., 2014). Experimental results show the CGWO can improve the performance of the basic
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GWO.
Additionally, there are three crucial differences between our proposal and the previous researches on optimization approaches with a special topological structure.
In general, various cellular structures have different impacts on the behavior of the algorithms. Thus, this paper incorporates six different cellular structures into the GWO to test the performance of algorithms with different structures. The most appropriate one is chosen as our proposed cellular GWO as in Section 4.3. Actually, metaheuristics with the CA are a kind of fine-grained structure algorithms. In this case of the algorithm, the
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population is divided into many subpopulations that arranged in a given structure. Then, the search process is performed
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within the neighborhood of the current individual. Metaheuristics with CA are similar to niching metaheuristics, but traditional niching methods usually require extra parameters. One important advantage of the proposed CGWO algorithm is that there is no need to specify any parameters.
The previous cellular algorithms were primarily adopted to locate a single global optimum, rather than multiple global
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optima. However, the proposed CGWO can locate all global optima in Section 4.4.2.
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The rest of this paper is organized as follows. Firstly, a brief overview of the basic GWO algorithm and cellular automata (CA) is given in Section 2. The detailed presentation of the CGWO approach is provided in Section 3. Section 4 gives the experiment and results. Section 5 presents the conclusions and future work. Overview of GWO and CA
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Before we introduce our proposed CGWO algorithm, we give an overview of the basic GWO and CA, respectively. 2.1 GWO
In this subsection, we describe GWO. GWO is a new metaheuristic inspired by the grey wolf hunting for the prey. The main steps of GWO are provided in the following subsections (S. Mirjalili, et al., 2014).
2.1.1 Social hierarchy To establish a social hierarchy of wolves, all the grey wolves are classified into four kinds of wolf according to the fitness value. The best wolf (solution) in GWO is denoted as the alpha (α). Similarly, the second and third best wolves are called beta
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(β) and delta (δ) respectively. The rest of wolves are considered to be omega (ω). In GWO the search process is mainly guided by α, β and δ. The ω wolves obey these three wolves (S. Mirjalili, et al., 2014). 2.1.2 Encircling prey Grey wolves encircle the prey when the hunt. To mathematically model encircling behavior, the equation is defined as follows:
𝑫 = 𝑪 ∘ 𝑿𝑝 (𝑡) − 𝑿(𝑡) 𝑿(𝑡 + 1) = 𝑿𝑝 (𝑡) − 𝑨 ∘ 𝑫
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(1) (2)
where 𝑡 is the current generation number, ∘ is the hadamard product operation, 𝑿𝑝 and 𝑿 represents the position vector of the prey and a grey wolf. The vectors 𝑨 and 𝑪 are formulated as follows (S. Mirjalili, et al., 2014): 𝑨 = 2𝒂 ∘ 𝒓1 − 𝒂 𝑪 = 2𝒓2
(3) (4)
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Where the value of 𝒂 is linearly decreased from 2 to 0 during the optimization process, and it is used to emphasize exploration and exploitation respectively. 𝒓1 and 𝒓2 are random vectors in range [0,1]. 2.1.3 Hunting
The hunt is often guided by the alpha. The beta and delta might also join in the hunting. However, the position of prey (optimum) is unknown in an abstract search space. To simulate the hunting behavior of grey wolves, assume that the alpha, beta, and delta have better knowledge about the potential location of the prey. Therefore, the three best solutions found so far
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are maintained and guide the other wolves towards the potential location of the prey. The equation of hunting is calculated as follows (S. Mirjalili, et al., 2014):
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𝑫𝛼 = 𝑪1 ∘ 𝑿𝛼 − 𝑿, 𝑫𝛽 = 𝑪2 ∘ 𝑿𝛽 − 𝑿, 𝑫𝛿 = 𝑪3 ∘ 𝑿𝛿 − 𝑿 𝑿1 = 𝑿𝛼 − 𝑨1 ∘ 𝑫𝛼 , 𝑿2 = 𝑿𝛽 − 𝑨2 ∘ 𝑫𝛽 , 𝑿3 = 𝑿𝛿 − 𝑨3 ∘ 𝑫𝛿
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𝑿(𝑡 + 1) =
𝑿1 +𝑿2 +𝑿3 3
(5) (6) (7)
Fig. 1 presents a search process of updating a candidate’s position based on alpha, beta, and delta in 2-D search space. It
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can be observed that the three best wolves (alpha, beta, and delta) can estimate the position of the prey, and other wolves
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update the positions randomly around the prey.
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α
α wolf δ
β wolf
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the prey
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δ wolf
move β
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Position of candidate wolf
Fig. 1. Position updating in GWO
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2.1.4 Attacking prey
To mathematically model attacking the prey, the value of a vector 𝒂 is decreased. 𝑨 is also decreased by 𝒂. When random values of 𝑨 are in [-1, 1], the next position of a search agent can be in any position between its current position and
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the position of the prey. With the operators proposed so far, the GWO algorithm allows its search agents to update their positions based on the α, β, δ, and ω; and attack toward the prey. However, the GWO algorithm tends to premature with these
2.1.5 Search for prey
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operators (S. Mirjalili, et al., 2014).
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As stated previously, the search direction of grey wolves mainly depends on the position of the alpha, beta, and delta. They diverge from each other to search for prey and converge to attack prey. To simulate divergence, 𝑨 is used to oblige the search agent to diverge from the prey. This emphasizes exploration and allows the GWO algorithm to search globally.
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Another element of GWO that favors exploration is 𝑪 in Eq.(4). The random value of 𝑪 is in [0, 2]. It provides random weights for prey to stochastically emphasize or deemphasize the effect of prey in defining the distance in Eq.(1), which can help to improve exploration and avoid local optimum. It is noted that 𝑪 is not linearly decreased in relation to 𝑨 but random values over the course of iteration which is helpful in case of local optima stagnation in the latter phase of search.
2.2 CA In this subsection, we describe CA. CA has become a popular tool in the scientific research since the concept of CA was first proposed by Von Neumann and Ulam (Neumann, 1966). One of the most well-known CA rules, the “game of life” was conceived by Conway in the late 1960s (E. R. Berlekamp, J. H. Conway, & Guy, 1982). Furthermore, CA has also been widely applied in a variety of fields such as physics, biology, computer science and traffic. CA is a set of cells distributed in a
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special topological structure. Each cell in the topological structure has its own state, which is associated with its surrounding cells in discrete time steps. The state of each cell at the next time step is determined by the current states of a surrounding neighborhood of cells. In general, CA consists of five key elements as follows: cell, cell state, cell space, neighborhood, and transition rule. As previously stated, cell state is information about the current cell which can determine the next state of the cell. Cell space represents a set of cells. It often has one-dimensional, two-dimensional, and three-dimensional space structure in real applications. It is noted that the boundary of cell space needs to be defined since we usually simulate real-world systems by finite grids. The boundary is usually a ring grid. More precisely, the left boundary connects to the right boundary and the top boundary connects to the bottom boundary. Neighborhood is a set of cells surrounding a center cell. It plays a states of its neighborhood. Some definitions of CA can be described as follows.
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critical role in the next state of a center cell. Transition rule is used to determine the next state of a given cell based on the A d-dimensional CA contains a d-dimensional grid of cells, each of which can take on a value. The cells update their states automatically according to a given rule. Formally, a cellular automaton 𝑸 is a quadruple as follows: 𝑸 = (𝑆, 𝐺, 𝑑, 𝑓) where 𝑆 is a finite set of states, 𝐺 is the neighborhood, 𝑑 ∈ 𝑍
+
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is the dimension of 𝑸, and 𝑓 is the interaction rule, also
referred to as the transition rule.
Given the position of a cell 𝑖 ∈ 𝑍 𝑑 in a d-dimensional lattice or grid, its neighborhood 𝐺 is defined by:
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𝐺𝑖 = (𝑖, 𝑖 + 𝒓1 , ⋯ , 𝑖 + 𝒓𝑛 )
(9)
where 𝑛 represents the neighborhood size, and 𝒓𝑗 is a fixed vector in the d-dimensional space. Six kinds of neighborhood shapes in CA are presented in Fig. 2. Please note the names of these neighborhoods: the label Ln represents neighborhoods composed of the n nearest neighbors in a given axial direction (north, south, west and east), while the label Cn (compact) denotes the neighborhoods consisting of the n-1 nearest cell to the center one (in horizontal, vertical, and diagonal directions). Two most commonly used neighborhoods are called Von Neumann neighborhood (C5) and Moore neighborhood (C9),
Fig. 2. Structure of neighborhood
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respectively.
Transition rule can determine the next state of a given cell according to its current state and states of its neighbors. The transition rule 𝑓 is written as follow: 𝑓: 𝑆𝑛 → 𝑆
(10)
This transition rule maps the state 𝑠𝑖 ∈ 𝑆 of a given cell 𝑖 into another state from the set 𝑆, as a function of the states of the cells in the neighborhood 𝐺𝑖 . In addition, some features of CA are stated as follows: (1) Homogeneity and regularity: homogeneity means the change of each cell in cell space obeys the same rule (i.e., CA rule or called transformation function). However, regularity indicates the identical distribution pattern, size, shape as
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well as the orderly distribution regulation of all the cells. (2) Space discretization: space discretization denotes that cells are arranged in a discrete cell grids defined by specific rules. (3) Synchronous calculation (parallelism): the state change of each cell at time step t+1 is independent behavior. If the configuration change of CA is considered as computation or processing of data or information, the disposing process of CA will be synchronous, which is suitable for parallel computation.
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The proposed CGWO
3.1
The framework of the CGWO algorithm
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In this section, we first give an overview of the proposed algorithm, and then explain the improvement strategy in detail.
The goal of this work is to develop a cellular GWO called CGWO for continuous optimization problems. A flow chart of the proposed CGWO is shown in Fig. 3. One crucial strategy is that CA is utilized to improve the performance of the GWO. The improvement strategy is motivated by the following three advantages.
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(1) CA can help to improve the local search ability since cell in CA only interacts with its neighbors for exploitation. Meanwhile, information diffusion mechanism contributes to exploration. Finally, a collection of such cells congregates to solve the problem.
(2) We can take the advantages of the two approaches. The candidate solutions are attracted toward the good solutions in GWO, therefore, the convergence speed of the GWO is very fast but it is also easy to trap into local optima. While CA provides slow diffusion through the population regarding the good solutions of each neighborhood. Subsequently, the attraction to the good points is weaker, which avoids local optima. Thus, we can integrate these
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different techniques to strengthen their advantages and make up the respective shortcomings.
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(3) In addition, CA and GWO are easy to be implemented due to their simple mechanism. The proposed CGWO algorithm incorporates the concept of CA. It consists of five steps summarized as follows.
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Input. SOP (1);
A stopping condition;
N: size of wolves;
NS: neighborhood size;
Input other parameters;
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Output. The optimal (or near-optimal) results found so far. Step 1) Initialization. Generate an initial population (wolves) of N solutions X1,…,XN ∈ Ω. Step 2) Evaluation. Once the wolf population is initialized, compute the fitness value of the solutions (wolves). Step 3) Create the neighborhood. For i=1,…,N, define a set of indexes Bi={i(1),…,i(NS)}, where {Xi(1),…, Xi(NS)} are the wolves with size NS closest to xi in the topological structure. Step 4) Stopping condition. If the stopping criterion is satisfied, then output the alpha wolf (the optimal solution found so far). Otherwise, go to step 5. Step 5) Update. Step 5.1) Selection. First, compute the fitness of solutions, and then select the three best fitness solutions from the current neighbors to guide other solutions within neighborhood to perform the search optimization.
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Step 5.2) Hunter operation. A new solution is generated from three rules. (a) encircling prey, (b) hunting and (c) attacking, as described in Section 2.1. Note that the search optimization is only implemented in its neighborhood. Step 5.3) Replacement strategy. The replacement method is based on fitness values of solutions. In detail, replace the current solution if the current solution is worse than the new solution after updating, and vice versa. Note that we have incorporated a constraint handling mechanism in CGWO when dealing with constrained problems. This mechanism is the same as in (Deb, 2000). Overlap area
yes Stop condition is met?
no
Evaluate function
α β δ
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Select the first best three wolves from its current neighbors
end
Search the prey by the above three wolves
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Update position of wolves
the current wolf is replaced if the position of the wolf is worse than that of the current one after update
Wolves and the positions
Fig. 3. Flow chart of the proposed CGWO algorithm
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The proposed CGWO is discussed in the following parts. In CGWO, all information inherits inside the cell. Six typical cell structures are employed to construct cell space in Fig. 2. One wolf (solution) is defined as a cell, and all cells represent
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wolves in CGWO. Consequently, the size of cell structure is the same with that of the wolves. In the initialization phase, grey wolf population is randomly created; subsequently, each wolf is randomly assigned to one grid of the lattice structure one by one. Note that the index of each wolf in the lattice structure remains unchanged during the search process. Neighborhood
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could be defined based on the lattice structures. Taking C9 structure in Fig. 3 as an example, suppose that wolf size is 49, and they are randomly generated in the initialization, then 49 grids are created to construct a lattice structure. Each wolf is assigned to each cell. The gray cells surrounding around the current cell are its neighbors, and deep gray grids represent the
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overlapping neighbors belonging to two consecutive cell structures. The interaction among cells is restricted to the neighborhood. This could help each cell to perform exploitation inside their neighbors. Meanwhile, the overlapping neighbors provide a migration mechanism from one neighborhood to another one. It could enable each cell information to diffuse to the
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whole wolves, which is favorable for exploring the search space (Shi, et al., 2011). In CGWO, the center cell state is determined by its surrounding neighbors. The state of neighbors’ best position 𝑋𝑖 in the current wolf is denoted by 𝑆𝑖𝑡 (𝑋𝑛 ). For simplification, the transition rule is defined as: 𝑆𝑖𝑡+1 (𝑋𝑛 ) = 𝑓 (𝑆𝑖𝑡 (𝑋𝑖 ), 𝑆𝑖𝑡 (𝑋𝑖+𝒓1 ), ⋯ , 𝑆𝑖𝑡 (𝑋𝑖+𝒓𝑛 )) = 𝑚𝑖𝑛 (𝑓𝑖𝑡𝑛𝑒𝑠𝑠(𝑆𝑖𝑡 (𝑋𝑖 )), 𝑓𝑖𝑡𝑛𝑒𝑠𝑠 (𝑆𝑖𝑡 (𝑋𝑖+𝒓1 )) ⋯ , 𝑓𝑖𝑡𝑛𝑒𝑠𝑠 (𝑆𝑖𝑡 (𝑋𝑖+𝒓𝑛 ))) (11) where 𝑓𝑖𝑡𝑛𝑒𝑠𝑠(𝑆𝑖𝑡 (𝑋𝑖 )) is the fitness of the current solution 𝑋𝑖 . Eq. (11) defines that the neighbor with the best fitness value is chosen for updating cell state.
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CA of the CGWO algorithm
Two important factors affect the performance of metaheuristics when solving complex problems: (1) communication structures. (2) information inheriting and diffusing scheme. As stated above, CA has also been successfully applied in many fields due to its unique topology structures and communication interaction. Although CA and GWO stem from different domains, some similar characteristics of the two techniques can be observed. First, both of them contain a set of individuals, which are called cells in CA and wolves in GWO. Second, every individual interacts with each other to transmit certain intrinsic information. In CA, each cell communicates with its neighbors and updates its cell state. Similarly, each wolf communicates with other wolves to update such intrinsic information as position,
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fitness, α, β and δ wolf in GWO. Third, a transition rule is used to manage the evolution of cells in CA, while some search operators are used to update information of wolf in GWO. Finally, CA and GWO both run in a discrete time step. To enhance the behavior of the interaction, we utilize the idea of CA with the communication structure to investigate information diffusing mechanism of GWO. Wolves can only interact with their neighbors in the CA. It could help to exploit every cell’s local information inside the neighborhood and explore the search space due to the slow information diffusion
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through the entire wolf population. Based on the view above, we introduce the concept of CA into the GWO to enhance the performance of GWO. The CA model for GWO is defined as follows: (a) cell: A candidate wolf or solution. (b) cell space: A set of all solutions.
(c) cell state: The wolf’s information such as the neighbor’s best position Xn at time t. It can be denoted by 𝑆𝑖𝑡 . (d) neighborhood: A set of solutions surrounding the current solution.
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(e) transition rule: According to fitness value, if the new position of the current wolf is better than the old one of the current wolf before update, the current wolf moves to the new position and the information of wolf is also updated.
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Then, the local best wolf is chosen from the neighborhood, and guides its neighbors converge to its local optimal areas.
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(f) discrete time step: Iteration in GWO.
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Based on the above definitions, the pseudo code of CGWO is given in Fig. 4.
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1. Input Parameters(a); // Parameters of the CGWO 2. wolf population Xi(i=1,...N) initialization(wolves_Size); 3. evaluate (wolf population) ; 4. while (evaluation_number < max_evaluation_number) 5. for i 1 to wolves_Size 6. neighbors calculateNeighborhood(wolves(i)); 7. Xα, Xβ, Xδ select the first best three wolves(neighbors); 8. new position of the wolf update(the current wolf);// update the position based on Eq.(7) 9. update(a, A and C); 10. evaluate (new position of the wolf) ; 11. if new position of the wolf is better than that of the current wolf 12. replacement(new position, position of the current wolf); 13. end if 14. evaluation_number++; 15. end for 16. end while 17. Xα select the best wolf(the entire wolves); 18. return Xα Fig. 4. Pseudo code of the proposed CGWO algorithm
Fig. 4. shows the pseudo code of the proposed CGWO algorithm. The detailed steps of the CGWO are as follows. Firstly, input the parameters like wolf population size and neighborhood structure. The initial population is usually composed of randomly generated wolves in the line 2 of Fig.4. The population is distributed in n (n=2 in this paper) dimensional lattice grid according to the indices of the population members (as obtained during initialization). After initialization, the fitness
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values of wolf population are calculated as in the line 3, and then the CGWO algorithm starts an update loop. This loop step is to generate new and promising wolf population through update operators as mentioned in Section 2.1. The first best three wolves (alpha (α), beta (β) and delta (δ)) from the current wolf’s neighbors are selected out in the line 7 of Fig.4 due to
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adoption of CA. Fig. 3 also presents the process of the update loop of an individual in CGWO (the considered neighborhood is Moore neighborhood in Fig. 3). Note that the overlap among the neighborhoods provides an implicit migration mechanism to the CGWO. Since the best solutions move smoothly through the whole population, diversity in the population is kept
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longer than that in non-structured GWO. This soft diffusion of the best solution through the population is one of the main reasons of the good balance between exploration and exploitation. Another important character of the CGWO is that these
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update operators are limited within the neighborhood of the current individual, therefore individuals belonging to other neighborhoods are not allowed to interact. Fig. 5 further illustrates the search process of a candidate wolf in wolf population (population size = 57). The solution
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indices are sorted only randomly in order to maintain the diversity of population. The solutions are located in the lattice gird with respect to their indices and each solution has its own neighborhood. For a candidate solution 𝑋⃗3,4 with index (3, 4), its Moore structure neighbors contain solutions {𝑋⃗2,3 , 𝑋⃗2,4 , 𝑋⃗2,5 , 𝑋⃗3,3 , 𝑋⃗3,5, 𝑋⃗4,3, 𝑋⃗4,4 , 𝑋⃗4,5 } besides itself. The candidate solution or wolf is guided by the three best wolves found so far in its neighbors but not in the entire population. The circle represents the range of the current wolf’s neighbors and the wolves only communicate with each other in its neighborhood. In other words, the whole population is divided into many subpopulations and the update operation is performed independently on a set of subpopulations. But these independent subpopulations can occasionally transmit information by overlap areas among neighborhoods. The overlap area, containing the solutions 𝑋⃗3,3 and 𝑋⃗4,3 , is constructed by the neighborhoods of 𝑋⃗3,4 and 𝑋⃗4,2 . Since the search process among neighborhoods is based on sequence, there is a delay in the information spread through the population regarding the best position of each neighborhood. Therefore, the attraction towards good solutions is weaker, which prevents the population from getting trapped in local minima. If stop condition is satisfied, the algorithm returns the
ACCEPTED MANUSCRIPT best wolf (α) obtained over the course of search.
1
2
3
4
5
6
7
α 1
δ 2
α wolf
C3 β wolf
R
3
move
δ wolf
β C2
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4
5
Overlap area
the prey
Circle represents the range of the neighborhood of the current wolf
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Neighborhood of
Fig. 5. Position updating in CGWO
4.
Experiments
This section is devoted to measuring the performance of the proposed CGWO. In this section the experimental studies
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contain the following five aspects:
The best choice of neighborhoods (six kinds of neighborhoods are shown in Fig. 2) in Section 4.3.
2.
Performance comparisons with other metaheuristics on benchmarks in Section 4.4.
3.
Statistical test on results obtained by different algorithms in Section 4.5.
4.
CPU-time cost study on CGWO and GWO in Section 4.6.
5.
Real-world applications of the CGWO in engineering optimization in Section 4.7.
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1.
In the following subsections, the benchmarks and parameter settings are described at first, and then the experimental
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studies are further investigated step by step.
4.1 Benchmark functions
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Three test suites are used to evaluate the success of CGWO and the comparison algorithms. Table 1 records the common unimodal functions (Yao, Liu, & Lin, 1999). Table 2 lists the common multimodal functions (Yao, et al., 1999). Table 3 involves 15 benchmark instances used in CEC2015. Details about these problems are provided in technical report (Qu, Liang, Suganthan, & Chen, 2015). Table 1 Unimodal benchmark functions Name Sphere Schwefel’s problem 2.22
Test Functions F1 x
n
xi
[-100,100]
i 1
F2 x
n
i 1
n
xi
f min
S 2
xi i 1
30
[-10,10]30
0 0
ACCEPTED MANUSCRIPT i xj i 1 j 1 max xi , 1
F3 x
Schwefel’s problem 1.2 F4 x
Schwefel’s problem 2.21
F5 x
Rosenbrock
n
i
100x n 1
F6 x
Step
xi2
i 1
i 1
x i 1
f 7 x
Noise
ix i 1
x
1 2
i
0.5
n
n
2
2
i n
2
i
0
[-100,100]30
0
[-30,30]30
0
[-100,100]
random 0, 1
4 i
[-100,100]30
30
[-1.28,1.28]30
0 0
Table 2 Multimodal benchmark functions Test Functions i 1
x
F9 x
Rastrigin
n
F12 x
i 1 n
xi2
i 1
xi 1 i
n
i
u( x ,10,100,4) i
F13 x 0.1sin 2 3x1
n
u( x ,5,100,4) i
i 1
i
25
ED F15 x
11
j 1
ai i
1
Shekel 2
0
[-50,50]30
0
[-50,50]30
0
2 j i 1 ( xi a ij )
[-65,65]2
1
[-5,5]4
0.0003
-1
1
x 1 bi2 bi x 2 bi2 bi x 3 x 4
2
1 6 x1 x1 x2 4 x22 4 x24 [-5,5]2 3 2 5.1 2 5 1 cos x1 10 F17 x x 2 x1 x1 6 101 2 [-5,10], [0,15] 8 4
27 x )
F18 x 1 ( x1 x2 1)2(19 14 x1 3 x12 14 x2 6 x1 x2 3 x22 )
30 (2 x1 3 x2 ) (18 32 x1 12 x 48 x2 36 x1 x2 2
2 1
4 F19 x - ci exp i 1 4 F20 x - ci exp i 1
3
j 1
ij
j
x - X - a X - a x - X - a X - a
c c
F21 x - X - ai X - ai ci T
i 1
7
F22
T
i
i
7
F23
i 1
2 2
2 pij 6 2 a x p ij j ij j 1
a x
5
i 1
Shekel 3
[-600,600]30
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Shekel 1
0
F16 x 4 x12 2.1x14
CE
AC
Hartman 2
[-32,32]30
1)2 1 sin 2(3xi 1) ( xi 1)2 1 sin 2(2xn )
n
(x
1 F14 x 500
Kowalik’s Function
Hartman 1
0
k( xi a) xi a u xi , a, k , m 0 a xi a m xi a k( xi a)
Shekel’s Foxholes Function
Goldstein-Price Function
[-5.12,5.12]30
m
i 1
Six-hump camel back
-12569.5
1)2 1 10 sin 2(yi 1 ) ( yi 1)2
( y
n
i 1
cos i 1
n 1 i 1
i 1
2 i
20 e cos2xi
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Generalized Penalized Function 2
1 4000
10 sin y1 n
Generalized Penalized Function 1
n
n
x
[-500,500]30
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F11 x
Griewank
xi
1 F10 x -20 exp 0.2 n 1 exp n
f min
S
10 cos2xi 10
2 i
i 1
Ackley
Branin
F8 x - xi sin n
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Name Generalized Schwefel’s problem
T
i
i
-1.0316 0.39788
[-2,2]2
3
[0,1]3
-3.86
[0,1]3
-3.32
[0,10]4
-10.1532
[0,10]4
-10.4028
[0,10]4
-10.5363
1
1
i
1
i
Table 3 CEC2015 benchmark functions Name Shifted and Rotated Expanded two-Peak Trap
Test Functions
F24( x) f1( x) f M 1( x o1 ) f f = Expanded Two-Peak Trap function
* 1
S
f min
[-100,100]10
100
ACCEPTED MANUSCRIPT F25( x) f 2( x) f M 2( x o2 ) f 2*
Shifted and Rotated Expanded Five-Uneven-Peak Trap
[-100,100]8
200
[-100,100]4
300
[-100,100]10
400
[-100,100]4
500
f = Expanded Five-Uneven-Peak Trap
x o3 f 3* F26( x) f 3( x) f M 3 20
Shifted and Rotated Expanded Equal Minima
f = Expanded Equal Minima
x o4 F27( x) f 4( x) f M 4 20
Shifted and Rotated Expanded Decreasing Minima
f 4*
f = Expanded Decreasing Minima
f = Expanded Uneven Minima
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x o5 f 5* F28( x) f 5( x) f M 5 20
Shifted and Rotated Expanded Uneven Minima
x o6 f 6* F29( x) f 6( x) f M 6 5
Shifted and Rotated Expanded Himmelblau’s Function
f = Expanded Himmelblau’s Function
F30( x) f 7( x) f M 7 x o7 f7*
Shifted and Rotated Expanded Six-Hump Camel Back
f = Expanded Six-Hump Camel Back
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x o8 f 8* F31( x) f 8( x) f M 8 5
Shifted and Rotated Modified Vincent function
[-100,100]8
600
[-100,100]10
700
[-100,100]4
800
[-100,100]10
900
[-100,100]10
1000
[-100,100]10
1100
f = Modified Vincent function
N
F32( x) f 9( x)
i 1
i
N 10
* i g i( x) bias i f 9*
[10,20,10,20,10,20,10,20,10,20] [1,1,1e 6, ,1e 4, ,1e 5,1e 5] bias [0, ,0] f 9*
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Composition Function 1
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g1 2( x): Rotated Sphere Function g3 4( x): Rotated High Conditioned Elliptic Function g5 6( x): Rotated Bent Cigar Function g7 8( x): Rotated Discus Function g910(x): Rotated Different Powers Function
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F33( x) f10( x)
N
i 1
i
* i g i( x) bias i f10*
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N 10 [10,20,30,40,50,60,70,80,90,100]
[1e 5,1e - 5,1e 6, ,1e 4,1,1]
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Composition Function 2
bias [0,10,20,30,40,50,60,70,80,90] f10*
g1 2( x): Rotated High Conditioned Elliptic Function g3 4( x): Rotated Different Powers Function g5 6( x): Rotated Bent Cigar Function g7 8( x): Rotated Discus Function g910(x): Rotated Sphere Function N
F34( x) f11( x)
* g (x) bias f i
i i
i
* 11
i 1
Composition Function 3
N 10 [10, ,10] [0.1,0.1,10,10,10,10,1e 3,1e - 3,1,1] bias [0,0,0,0,0,0,0,0,0,0] f11*
g12( x): Rotated Rosenbrock’s Function
ACCEPTED MANUSCRIPT g3 4( x): Rotated Rastrigin’s Function g5 6( x): Rotated HappyCat Function g 7 8(x): Rotated Scaffer’s Function g9 10( x): Rotated Expanded Modified Schwefel’s Function
F35( x) f12( x)
N
i 1
i
* i g i( x) bias i f12*
N 10 [10,10,20,20,30,30,40,40,50,50]
[0.1,0.1,10,10,10,10,1e 3,1e - 3,1,1] bias [0,0,0,0,0,0,0,0,0,0] f12* g1 2( x): Rotated Rosenbrock’s Function g3 4( x): Rotated Rastrigin’s Function g5 6( x): Rotated HappyCat Function g7 8( x): Rotated Scaffer’s F6 Function g910(x): Rotated Expanded Modified Schwefel’s Function
i 1
1200
* i g i( x) bias i f13*
N
i
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F36( x) f13( x)
[-100,100]10
CR IP T
Composition Function 4
N 10 [10,20,30,40,50,60,70,80,90,100]
[0.1,10,10,0.1,2.5,1e 3,1e - 3,1e - 3,2.5,10] bias [0,0,0,0,0,0,0,0,0,0] f13*
M
[-100,100]10
1300
[-100,100]10
1400
[-100,100]10
1500
PT
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Composition Function 5
g 1(x ): Rotated Rosenbrock’s Function g2( x): Rotated HGBat Function g3( x): Rotated Rastrigin’s Function g4(x): Rotated Ackley’s Function g5( x): Rotated Weierstrass Function g6( x): Rotated Katsuura Function g7( x): Rotated Scaffer’s F6 Function g8( x): Rotated Expanded Griewank’s plus Rosenbrock’s Function g9( x): Rotated HappyCat Function g10( x): Rotated Expanded Modified Schwefel’s Function
CE
F37( x) f14( x)
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Composition Function 6
N
i 1
* i g i( x) bias i f14*
N 10 [10,10,20,20,30,30,40,40,50,50]
[10,1,10,1,10,1,10,1,10,1]
bias [0,20,40,60,80,100,120,140,160,180] f14* g1,3,5,7,9( x): Rotated Rastrigin’s Function
g2,4,6,8,10( x): Rotated Expanded Modified Schwefel’s Function
F38( x) f15( x)
N
i 1
Composition Function 7
i
i
* i g i( x) bias i f15*
N 10 [10,20,30,40,50,60,70,80,90,100]
[0.1,10,10,0.1,2.5,1e - 3,1e 3,1e 3,2.5,10] bias [0,0,0,0,0,0,0,0,0,0] f15*
ACCEPTED MANUSCRIPT
CR IP T
g1( x): Rotated Rosenbrock’s Function g2( x): Rotated HGBat Function g3( x): Rotated Rastrigin’s Function g4(x): Rotated Ackley’s Function g5( x): Rotated Weierstrass Function g6( x): Rotated Katsuura Function g7( x): Rotated Scaffer’s F6 Function g8( x): Rotated Expanded Griewank’s plus Rosenbrock’s Function g 9(x): Rotated HappyCat Function g10( x): Rotated Expanded Modified Schwefel’s Function
4.2 Parameter settings
To make a fair comparison, all algorithms are implemented in java on jMetal software (Durillo & Nebro, 2011). Experimental studies are conducted on Intel Core i5-4210U CPU @1.70GHz, 4GB RAM, with Microsoft Windows 8 operating system. Table 4 presents the initial values of the relevant control parameters for the different metaheuristics
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including LSHADE, TLBO, EBOwithCMAR, NDHS, BA, CLPSO, GWO, and CGWO. The parameter settings of these metaheuristics are obtained from the original literature. The relevant common parameter settings for the algorithms are as follows:
The population size is 30 (except for LSHADE and EBOwithCMAR).
The maximal number of function evaluation (NFEmax) is 100,000.
30 independent runs are conducted for each algorithm on each test problem.
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Table 4 Parameter setting of algorithms Control parameters =1.8; 𝑟 𝑎𝑟𝑐 = 2.6; 𝑝=0.11; 𝐻=6 No control parameters 𝑃𝑆1,𝑚𝑎𝑥 =18D, 𝑃𝑆1,𝑚𝑖𝑛 =4, 𝑃𝑆2,𝑚𝑎𝑥 =46.8D, 𝑃𝑆2,𝑚𝑖𝑛 =10, 𝐻=6, 𝑃𝑆3 =4+(3log(D)), 𝑝𝑟𝑜𝑏𝑙𝑠 =0.1, 𝑐𝑓𝑒𝑙𝑠 =0.25* NFEmax HMCR=0.99, PAR_max=0.99, PAR_min = 0.01, tournament size =2 Loudness: A=0.25, plus rate: r=0.5, minimum frequency: Qmin=0, maximum frequency: Qmax=2
CLPSO
ED
𝑖𝑛𝑖𝑡
c= 1.49445; m=0; 𝑝𝑐 = 0.5
𝑒 𝑡 −𝑒 𝑡(1) 𝑒 𝑡(𝑝𝑠 )−𝑒 𝑡(1)
where, 𝑡 = 0.5 × (0:
1 𝑝𝑠 −1
: 1)
a decreases linearly from 2 to 0 Neighborhood style:C25, a decreases linearly from 2 to 0
CE
GWO CGWO
𝑟𝑁
PT
Algorithms LSHADE TLBO EBOwithCMAR NDHS BA
4.3 The performance of CGWO with different neighborhood structures
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In this subsection we generate six variations of the CGWO by adopting six different neighborhood structures in Fig. 2 and compare the performance of each variation on the benchmarks from Tables 1-2. Tables 5-6 show the statistical results (best, worst, mean, and standard deviation values) on different variations over all runs. We must point out that the marks L9, C9, C13, C21, and C24 in the Tables 5-6 are short for the six corresponding neighborhoods in the Section 2.2. More precisely, CGWO (L9) denotes CGWO with L9 neighborhood. Similarly, CGWO (C9) represents CGWO using C9 neighborhood. From Table 5, it can be observed that CGWO (C25) has a better performance for the unimodal benchmarks except F7 with comparison to the other versions of CGWO since CGWO (C25) obtains the best mean values on 6 out of 7 unimodal benchmarks. For multimodal benchmarks in Table 6, it is also found that none of the neighborhood structures has a dominant performance for all test problems, which implies the neighborhood size is sensitive to the behavior of the CGWO and the setting of neighborhood size depends on specific problems. However, on the whole, CGWO with C25 are slightly better than its other versions on these multimodal benchmarks. Consequently, C25 is regarded as an appropriate neighborhood structure
ACCEPTED MANUSCRIPT
to form the best CGWO that will be studied in more detail in the following subsections. Table 5 Results by CGWO with different neighborhoods on unimodal benchmark functions Problem
Statistics
CGWO(L5)
CGWO(L9)
CGWO(C9)
CGWO(C13)
CGWO(C21)
CGWO(C25)
Best
1.2E-171
6.4E-210
0
0
0
0
Worse
3.2E-165
3.8E-204
0
0
0
0
Mean
1.9E-166
2.8E-205
0
0
0
0
0
0
F1 0
0
0
0
1.02E-96
3.0E-120
1.5E-277
3.5E-239
2.4E-306
0
Worse
4.57E-95
1E-116
9.4E-271
4.1E-234
6.6E-299
0
Mean
1.76E-95
1.7E-117
5.7E-272
4.6E-235
3.5E-300
0
std
1.50E-95
2.5E-117
0
0
Best
1.83E-14
7.59E-23
4.37E-61
2.68E-47
8.42E-65
3.56E-74
Worse
7.66E-09
4.34E-14
9.75E-43
9.9E-32
2.25E-45
6.21E-55
Mean
5.69E-10
1.82E-15
4.41E-44
4.16E-33
7.79E-47
4.18E-56
std
1.59E-09
7.91E-15
1.82E-43
1.81E-32
4.11E-46
1.31E-55
Best
7.26E-25
1.02E-33
2.72E-70
3.62E-73
Worse
3.95E-22
5.95E-31
Mean
3.23E-23
1.3E-31
std
7.22E-23
1.7E-31
Best
25.05527
24.81441
Worse
27.08967
27.00348
Mean
25.89631
25.43051
std
4.91E-01
Best
0
F2
1.08E-61
1.85E-51
4.6E-64
5.17E-65
7.75E-63
1.46E-52
2.63E-65
4.38E-66
2.32E-62
3.79E-52
8.58E-65
1.32E-65
24.24317
24.86651
24.66267
24.26925
27.06747
27.044
28.71735
26.2198
25.64893
25.46054
25.55428
25.42201
5.30E-01
7.68E-01
5.63E-01
8.03E-01
5.26E-01
0
0
0
0
0
ED
Mean
Best Worse F7
CE
Mean std
M
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
PT
std
0
2.53E-58
F5
Worse
0
3.46E-67
F4
F6
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F3
CR IP T
std Best
1.34E-04
7.73E-05
1.56E-04
9.66E-05
1.21E-04
1.21E-04
7.01E-04
7.52E-04
9.21E-04
6.93E-04
9.25E-04
8.07E-04
3.33E-04
3.84E-04
4.48E-04
3.69E-04
4.40E-04
4.30E-04
1.32E-04
1.82E-04
1.86E-04
1.41E-04
1.92E-04
1.83E-04
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Table 6 Results by CGWO with different neighborhood on multimodal benchmark functions
Problem
Statistics
CGWO(L5)
CGWO(L9)
CGWO(C9)
CGWO(C13)
CGWO(C21)
CGWO(C25)
Best
-5704.235
-7359.57
-7447.62
-7066.52
-6794.18
-6702.907
Worse
-3131.247
-3349.67
-3386.02
-3255.19
-3330.91
-3256.599
Mean
-3591.132
-4319.03
-4772.63
-4956.15
-4685.34
-5074.777 1226.6235
F8
std
442.26618
1104.816
1288.144
1261.855
1274.428
Best
0
0
0
0
0
Worse
15.40262
4.64472
11.30515
7.471893
5.27715
16.699364
Mean
1.3744344
0.292568
1.968907
1.159881
2.873349
4.2002956
std
3.5298336
0.974567
3.190097
2.315817
4.413376
5.4555579
Best
3.997E-15
4E-15
4E-15
4E-15
4E-15
4E-15
0
F9
F10
ACCEPTED MANUSCRIPT
Worse
3.997E-15
4E-15
7.55E-15
4E-15
4E-15
4E-15
Mean
3.997E-15
4E-15
6.96E-15
4E-15
4E-15
4E-15
std
0
0
1.35E-15
0
0
0
Best
0
0
0
0
0
0
Worse
0
0
1.81E-02
2.28E-02
4.01E-02
5.44E-02
Mean
0
0
3.13E-03
1.92E-03
3.15E-03
5.43E-03
std
0
0
5.89E-03
5.50E-03
8.47E-03
1.20E-02
Best
7.899E-08
5.57E-08
6.01E-08
6.13E-08
3.46E-08
6.01E-08
Worse
6.71E-03
6.47E-03
1.98E-02
1.33E-02
1.34E-02
1.98E-02
Mean
4.39E-04
2.16E-04
5.55E-03
1.73E-03
3.21E-03
5.55E-03
std
1.66E-03
1.18E-03
6.23E-03
3.40E-03
4.39E-03
6.23E-03
Best
7.045E-07
1.08E-06
1.49E-06
8.13E-07
6.01E-07
9.26E-07
Worse
3.64E-01
1.21E-01
3.64E-01
2.59E-01
4.85E-01
6.06E-01
Mean
2.56E-02
1.62E-02
1.33E-01
6.24E-02
1.29E-01
1.88E-01
std
7.61E-02
4.19E-02
1.07E-01
8.52E-02
1.46E-01
1.69E-01
9.98E-01
9.98E-01
F11
F13
Best
9.98E-01
9.98E-01
Worse
9.98E-01
9.98E-01
Mean
9.98E-01
9.98E-01
std
3.08E-14
2.26E-14
Best
3.07E-04
3.07E-04
Worse
4.12E-04
1.22E-03
Mean
3.11E-04
3.38E-04
std
1.90E-05
1.67E-04
Best
-1.031628
Worse
-1.031628
9.98E-01
9.98E-01
9.98E-01
9.98E-01
9.98E-01
9.98E-01
9.98E-01
9.98E-01
1.55E-14
4.00E-14
1.35E-14
0
3.07E-04
3.07E-04
3.07E-04
3.07E-04
2.04E-02
1.59E-03
2.04E-02
2.04E-02
2.34E-03
3.50E-04
3.05E-03
1.71E-03
6.11E-03
2.35E-04
6.91E-03
5.08E-03
-1.03163
3.07E-04
-1.03163
-1.03163
-1.03163
-1.03163
2.04E-02
-1.03163
-1.03163
-1.03163
ED
-1.031628
-1.03163
2.34E-03
-1.03163
-1.03163
-1.031623
std
1.312E-10
1.27E-10
6.11E-03
2.43E-10
9.75E-11
8.47E-11
Best
3.98E-01
3.98E-01
3.98E-01
3.98E-01
3.98E-01
3.98E-01
Worse
3.98E-01
3.98E-01
3.98E-01
3.98E-01
3.98E-01
3.98E-01
3.98E-01
3.98E-01
3.98E-01
3.98E-01
3.98E-01
3.98E-01
2.37E-09
1.96E-09
2.22E-09
1.71E-09
1.59E-09
0
Best
3
3
3
3
3
3
Worse
3.0000008
3.000001
3.000001
3.000001
3.000001
3.0000006
Mean
3.0000001
3
3
3
3
3
std
1.661E-07
7.16E-08
1.3E-07
3.19E-07
2.79E-07
2.26E-15
Mean
AC
CE
std
PT
Mean
F17
F18
M
F15
AN US
9.98E-01
9.98E-01
F14
F16
CR IP T
F12
Best
-3.862782
-3.86278
-3.86278
-3.86278
-3.86278
-3.862782
Worse
-3.86278
-3.86278
-3.86278
-3.86277
-3.86278
-3.86278
Mean
-3.862782
-3.86278
-3.86278
-3.86278
-3.86278
-3.862781
std
5.41E-07
6.63E-07
1.09E-06
2.41E-06
1.05E-06
1.09E-16
Best
-3.301995
-3.322
-3.322
-3.322
-3.322
-3.322
Worse
-3.301994
-3.32199
-3.20275
-3.20274
-3.20273
-3.201948
Mean
-3.301995
-3.322
-3.29026
-3.2982
-3.29026
-3.302106
std
2.856E-07
4.21E-08
5.35E-02
4.84E-02
5.35E-02
4.52E-02
Best
-10.1532
-10.1532
-10.1532
-10.1532
-10.1532
-10.1532
F19
F20
F21
ACCEPTED MANUSCRIPT
Worse
-10.15319
-10.1532
-5.0552
-10.1532
-10.1532
-5.10077
Mean
-10.1532
-10.1532
-9.98326
-10.1532
-10.1532
-9.984783
std
2.485E-06
3.52E-06
9.31E-01
1.11E-06
2.38E-06
9.22E-01
Best
-10.4029
-10.40294
-10.4029
-10.4029
-10.4029
-10.40294
Worse
-10.40292
-10.4029
-5.08767
-5.08767
-5.08767
-5.087672
Mean
-10.4029
-10.40294
-10.2258
-10.2258
-10.2258
-10.22576
std
3.92E-06
3.72E-06
9.70E-01
9.70E-01
9.70E-01
9.70E-01
Best
-10.53641
-10.5364
-10.5364
-10.5364
-10.5364
-10.53641
Worse
-10.5364
-10.5364
-5.12848
-10.5364
-10.5364
-10.5364
Mean
-10.5364
-10.5364
-10.3561
-10.5364
-10.5364
-10.53641
std
3.41E-06
3.42E-06
9.87E-01
1.85E-06
2.05E-06
1.39E-06
F22
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F23
Table 7 Wilcoxon sign rank test on the solution by CGWO with different structures for benchmarks in Tables 1-2 (a level of significance α=0.05). CGWO(C25) vs.
CGWO(C25) vs.
R
+
R
-
CGWO(L9)
p-value
R
+
R
-
CGWO(C9)
p-value
R
CGWO(C25) vs.
+
R
-
CGWO(C25) vs.
CGWO(C13)
AN US
CGWO(L5)
case
CGWO(C25) vs.
p-value
R
+
R
-
CGWO(C21)
p-value
R
+
R-
p-value
465
0
1.7E-06 +
465
0
1.7E-06 +
232.5
232.5
1=
232.5
232.5
1=
232.5
232.5
1=
F2
465
0
1.7E-06 +
465
0
1.7E-06 +
465
0
1.7E-06 +
465
0
1.7E-06 +
465
0
1.7E-06 +
F3
465
0
1.7E-06 +
465
0
1.7E-06 +
457
8
3.9E-06+
465
0
1.7E-06 +
432
33
4.1E-05 +
F4
465
0
1.7E-06 +
465
0
1.7E-06 +
465
0
1.7E-06 +
465
0
1.7E-06 +
347
118
1.9E-02 +
F5
411
54
2.4E-04 +
340
125
3.4E-02 +
397
68
7.2E-04 +
345
120
2.1E-02 +
360
105
8.7E-03 +
F6
232.5
232.5
1=
232.5
232.5
1=
232.5
232.5
1=
232.5
232.5
1=
232.5
232.5
1=
F7
129
336
3.3E-02 -
178
287
2.6E-01 =
345
120
3.3E-02 +
177
288
2.5E-01 =
350
115
1.6E-02 +
F8
447
18
1.0E-05 +
341
124
2.6E-02 +
351
114
1.5E-02 +
271
194
4.3E-02 +
359
106
9.3E-03 +
125
340
3.9E-02 -
77
232.5
232.5
1=
232.5
F11
126.5
338.5
7.8E-03 -
126.5
F12
50
415
1.7E-04 -
41
F13
42
423
8.9E-05 -
F14
270
195
4.2E-01 =
F15
313
152
9.8E-02 =
F16
362
103
F17
268
197
1.8E-04 -
157.5
307.5
9.3E-02 =
108
357
5.0E-02 =
181
284
2.3E-01 =
1=
457.5
7.5
5.7E-07 +
232.5
232.5
1=
232.5
232.5
1=
338.5
7.8E-03 -
226
239
7.2E-01 =
171
294
2.4E-01 =
196.5
268.5
4.9E-01 =
424
8.2E-05 -
233
232
9.9E-01 =
42
423
8.9E-05 -
115
350
1.6E-02 -
35
430
4.9E-05 -
155
310
1.1E-01 =
76
389
1.3E-03 -
161
304
1.4E-01=
258
207
5.6E-01 =
237
228
9.1E-01 =
244.5
220.5
8.6E-01 =
247
218
7.1E-01 =
150
315
8.9E-02 =
352
113
2.9E-02 +
99
366
6.0E-03 -
325
140
2.1E-02 +
7.7E-03 +
322
143
6.6E-02 =
269
196
4.5E-01 =
285
180
2.8E-01 =
257
208
6.1E-01 =
4.7E-01 =
259
206
5.9E-01 =
281
184
3.2E-01 =
278
187
3.5E-01 =
290
175
2.7E-01 =
CE
PT
388
232.5
AC
F18
ED
F9 F10
M
F1
299
166
1.7E-01 =
334
131
3.7E-02 +
237
228
9.3E-01 =
232
233
9.9E-01 =
265
200
5.0E-01 =
190
275
3.8E-01 =
325
140
5.7E-02 +
234
231
9.8E-01 =
255
210
6.4E-01 =
284
181
2.9E-01 =
254
211
6.6E-01 =
176
289
2.4E-01 =
345
120
2.1E-02 +
360
105
8.9E-03 +
345
120
2.1E-02 +
F21
366
99
6.0E-03 +
393
72
9.6E-04 +
244
221
8.1E-01 =
282
183
3.1E-01 =
255
210
6.4E-01 =
F22
365
100
6.4E-03 +
321
144
6.9E-02 =
199
266
4.9E-01 =
226
239
8.9E-01 =
179
286
2.7E-01 =
F23
442
23
1.6E-05 +
408
57
3.1E-04 +
325
138
1.6E-02 +
238
227
9.1E-01 =
365
100
6.4E-03 +
F19 F20
+/=/-
10/8/5
10/9/4
10/13/0
6/14/3
9/13/1
ACCEPTED MANUSCRIPT
4.4 Comparison with other algorithms This subsection is divided into three parts. The first part presents the comparison experiment between CGWO and other metaheuristic algorithms such as LSHADE, TLBO, EBOwithCMAR, NDHS, BA, CLPSO on unimodal functions and multimodal functions. The second part shows the comparison results between CGWO and other metaheuristics on CEC2015 test suite with multiple global peaks. The third part gives a comparison experiment using some improved versions of the GWO and multi-niche metaheuristics on multi-niche problems from CEC2015 test suite. 4.4.1 Experiment one To assess the performance of CGWO, it is compared with seven other metaheuristics on benchmark problems. These
CR IP T
algorithms in the study are LSHADE (winner in CEC2014 competition on Real-Parameter Single Objective Optimization Benchmarks), TLBO (it is a very efficient optimization algorithm), EBOwithCMAR (winner in CEC 2017 competition on bound constrained benchmark set), NDHS (it is a novel HS approach, and performs better than HS and its other variants), BA (BA is recent metaheuristic and shows better performance than basic PSO, GA, etc.), CLPSO (it is popular PSO variant and is usually used to compare performance of one algorithm), and GWO. Tables 8-9 report the statistical results for each algorithm on benchmarks.
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According to the results in Table 8, CGWO can provide very competitive results compared to the other algorithms. In detail, both of TLBO and CGWO can achieve the best average results on 4 out of 7 problems, whereas LSHADE, EBOwithCMAR, NDHS, BA, CLPSO, and GWO can provide the best average ones on 2, 1, 2, 0, 2, and 1 test functions, respectively. Note that the unimodal benchmarks are suitable for testing exploitation ability of the metaheuristics (S. Mirjalili, et al., 2014). Therefore, regarding exploiting the optimum, CGWO and TLBO are in the first rank; LSHADE, NDHS and CLPSO come in the second place; EBOwithCMAR and GWO are in the third place; the BA ranks at the rear. In addition, CGWO outperforms the basic GWO on 6 of 7 unimodal benchmarks in terms of the average value. The reason seems
M
straightforward: each wolf in CGWO has its own neighbors and only interacts with its neighbors. The three best wolves in its neighbors can be used to guide the movement of other candidate wolves within neighborhood toward their good areas.
ED
Obviously, CA can contribute to exploitation or local search of CGWO. Table 8 Results by different algorithms on unimodal benchmark functions EBOwithC
TLBO
NDHS
BA
CLPSO
GWO
CGWO
MAR
Best
8.56E-30
0
1.42E-24
0
1.22E-04
5.76E-12
1.5E-206
0
Worst
5.28E-26
0
2.87E-22
0
4.91E-01
6.05E-11
7.3E-201
0
2.94E-27
0
6.67E-23
0
8.07E-02
3.37E-11
3.7E-202
0
9.50E-27
0
6.79E-23
0
1.03E-01
1.40E-11
1.40E-89
0
3.11E-14
0
2.67E-12
0
1.61E-02
6.41E-08
1.68E-45
0
F1 Mean std
AC
Best F2
LSHADE
PT
Statistics
CE
Problem
Worst
1.26E-12
0
4.97E-11
0
10.12344
3.14E-07
1.41E-34
0
Mean
3.26E-13
0
1.63E-11
0
5.1E-01
1.95E-07
4.96E-36
0
std
3.41E-13
0
1.45E-11
0
1.822092
5.70E-08
2.58E-35
0
Best
1.00002
0
4.65748
842.1160
6.3E-02
57.4581
1.68E-45
3.57E-74
Worst
1.00993
0
33.85209
4805.8723
10196.4934
319.3870
1.41E-34
6.24E-55
Mean
1.00145
0
16.91642
2036.922
1090.884
165.6838
4.96E-36
4.18E-56
std
2.25E-03
0
8.814452
893.3847
2552.026
65.5480
2.58E-35
1.82E-43
Best
1.24E-07
9.34E-47
6.89E-06
2.01E-57
1.20E-02
8.1276
3.36E-41
3.62E-73
Worst
1.73E-06
4.72E-46
2.18E-04
2.11E-46
5.89E-01
14.0159
2.43E-36
5.17E-65
Mean
6.61E-07
3.12E-46
4.06E-05
7.02E-48
1.43E-01
10.3689
2.39E-37
4.38E-66
F3
F4
ACCEPTED MANUSCRIPT
std
4.08E-07
4.27E-45
3.90E-05
3.85E-47
1.18E-01
1.3238
5.66E-37
1.32E-65
Best
8.46335
28.72192
12.63520
23.57316
28.72252
2.2130
24.38665
24.24317
Worst
11.8763
28.94249
15.43227
123.4207
90029.3932
72.4817
27.0883
27.06747
Mean
10.2642
28.87576
14.21070
27.52616
3039.519
23.2816
25.75554
25.64893
std
9.87E-01
6.04E-02
6.49E-01
18.11298
16429.8
16.8490
7.32E-01
7.68E-01
F5
Best
0
0
0
0
0
0
0
0
Worst
0
0
0
1
2
0
0
0
Mean
0
0
0
1.67E-01
1.00E-01
0
0
0
std
0
0
0
3.79E-01
4.03E-01
0
0
0
Best
1.00E-03
2.20E-04
5.28E-04
4.76E-04
1.32E-03
1.25E-24
1.90E-04
1.21E-04
Worst
2.41E-03
1.48E-01
3.18E-03
8.59E-04
1.95E-01
4.69E-23
1.20E-03
8.07E-04
Mean
1.61E-03
3.39E-02
1.91E-03
4.68E-04
7.48E-02
1.03E-23
5.49E-04
4.30E-04
std
3.46E-04
3.69E-02
6.17E-04
3.57E-04
5.06E-02
9.78E-24
2.31E-04
1.83E-04
F7
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F6
Table 9 presents the statistical values obtained by different algorithms on multimodal benchmarks from Table 2. In
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contrast to unimodal benchmarks, multimodal benchmarks are suitable for measuring exploration ability of the algorithm (S. Mirjalili, et al., 2014). It can be obviously seen from this Table 9 that the LSHADE performs better than other seven methods on 9 out of 16 test functions (i.e., F11-13, F16-18, F20, and F22-23). Thus, the LSHADE is the most effective one among these compared algorithms. The CGWO is the second most effective approach, as it can obtain the best average result on 5 benchmarks (i.e., F10-11, F14, F17, and F19). EBOwithCMAR is the third most effective, since it generates the optimal average result on 4 problems (i.e., F11, F15, F17, and F21). TLBO can obtain the best average one on F9 and F11, consequently, TLBO is the fourth most effective algorithm. CLPSO obtains the best result on only one problem (i.e., F8). The
M
rest metaheuristics including NDHS, BA and GWO cannot get any one the best result on these benchmarks. Therefore, concerning exploration, CGWO is competitive with LSHADE and superior to the compared algorithms for most multimodal
ED
benchmarks. It means that CGWO with CA model has a better global search ability than its counterparts except for LSHADE on most multimodal functions. The major reason for the good performance of CGWO is that the overlapping neighbors provide an information diffusion scheme, and they enable each cells information to diffuse toward the whole wolf population,
PT
which is favorable for exploring the search space. Meanwhile, CA provides a niching mechanism for GWO, where wolf population is divided into many subpopulations and update operation is executed independently on these subpopulations. That is, different subpopulation may have different search directions in solution space and help to explore new promising
Statistics
LSHADE
AC
Problem
CE
areas of the search space.
Table 9 Results by different algorithms on multimodal benchmark functions EBOwithC TLBO
NDHS
BA
CLPSO
GWO
CGWO
MAR
Best
-1.26E+04
-5.29E+04
-9.93E+03
-1.26E+04
-1.07E+04
-1.26E+04
-8.35E+03
-6.70E+03
Worst
-1.26E+04
-2.63E+03
-7.77E+03
-1.26E+04
-4.20E+03
-1.26E+04
-3.86E+03
-3.26E+03
Mean
-1.26E+04
-3.88E+03
-9.18E+03
-1.26E+04
-8.28+03
-1.26E+04
-6.86E+03
-5.07E+03
std
9.63E-02
5.71E+02
7.98E-01
5.06E-06
1.58E+03
6.84E-08
1.02E+03
1.23E+03
Best
6.19E-03
0
3.87E-02
0
2.23E+01
4.44E-05
0
Worst
1.2211
0
3.04331
1.10E-01
1.41E+02
1.32E-03
2.50E+01
1.66E+01
Mean
9.78E-02
0
7.98E-01
1.80E-02
5.65E+01
3.65E-04
5.68E+00
4.20E+00
std
2.26E-01
0
8.08E-01
2.15E-02
2.25E+01
2.81E-04
6.42E+00
5.46E+00
Best
2.84E-14
4.44E-15
4.22E-13
4.44E-15
8.75E-03
1.56E-06
7.55E-15
4.00E-15
F8
0
F9
F10
ACCEPTED MANUSCRIPT
Worst
1.45E-13
4.46E-15
1.08E-11
4.44E-15
4.66E-01
5.53E-06
1.47E-14
4.00E-15
Mean
5.35E-14
4.44E-15
2.14E-12
4.44E-15
9.22E-02
3.15E-06
8.62E-15
4.00E-15
std
2.54E-14
0
1.91E-12
0
9.36E-02
1.07E-06
1.90E-15
0
Best
0
0
0
0
1.52E-03
5.88E-09
0
0
Worst
0
0
0
1.91E-01
7.04E-01
1.09E-06
1.79E-02
0
Mean
0
0
0
7.45E-02
2.78E-01
1.17E-07
1.96E-03
0
std
0
0
0
5.07E-02
2.18E-01
2.18E-07
4.69E-03
0
Best
0
3.45E-01
9.96E-01
5.26E-07
5.99E-03
3.28E-13
1.39E-07
6.01E-08
Worst
0
1.11E+00
9.96E-01
1.18E-06
9.49E-01
9.57E-12
2.01E-02
1.98E-02
Mean
0
5.99E-01
9.96E-01
8.73E-07
1.15E-01
1.90E-12
6.86E-03
5.55E-03
std
0
2.06E-01
5.82E-16
1.60E-07
2.08E-01
1.93E-12
6.38E-03
6.23E-03
Best
7.18E-30
2.21E+00
1.15E-24
2.29E-05
4.49E-01
8.59E-12
2.15E-06
9.26E-07
Worst
9.36E-27
3.68E+00
1.22E-21
1.07E-01
2.55E+00
6.76E-11
4.85E-01
6.06E-01
Mean
1.40E-27
2.92E+00
9.80E-23
2.97E-02
1.48E+00
2.63E-11
2.07E-01
1.88E-01
std
2.21E-27
3.67E-01
2.39E-22
2.82E-02
4.64E-01
1.24E-11
1.41E-01
1.69E-01
9.98E-01
9.98E-01
F11
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F12
F13
1
9.98E-01
N.A
1
4.98E+02
N.A
9.98E-01
9.98E-01
9.98E-01
9.98E-01
9.98E-01
9.98E-01
9.98E-01
9.98E-01
Mean
1
4.51E+01
N.A
9.98E-01
9.98E-01
9.98E-01
9.98E-01
9.98E-01
std
0
1.23E+02
N.A
2.06E-16
2.89E-16
2.15E-22
1.28E-13
0
Best
9.04E-04
3.55E-04
3.07E-04
3.07E-04
3.07E-04
3.85E-04
3.07E-04
3.07E-04
Worst
9.04E-04
5.26E-03
3.21E-04
2.04E-02
2.97E-02
2.04E-02
2.04E-02
2.04E-02
Mean
9.04E-04
1.63E-03
3.17E-04
5.76E-03
1.03E-02
1.71E-03
3.04E-03
5.71E-04
std
2.81E-19
1.50E-03
3.57E-06
8.96E-03
1.02E-02
7.08E-03
6.91E-03
7.75E-05
Best
-1.0316
-1.0316
1.63E+08
-1.0316
-1.0316
-1.0316
-1.0316
-1.03163
Worst
-1.0316
-1.0219
2.09E+08
-1.0316
-1.0316
-1.0316
-1.0316
-1.03163
F14
-1.0316
-1.0319
1.99E+08
-1.0316
-1.0316
-1.0316
-1.0316
-1.03163
std
6.66E-16
1.99E-03
1.12E+07
2.50E-10
2.81E-10
6.72E-10
7.38E-10
8.46E-11
Best
0.397887
0.397887
0.397887
3.98E-01
3.98E-01
3.98E-01
0.397887
0.397887
Worst
0.397887
0.729213
0.397887
3.98E-01
3.98E-01
3.98E-01
0.397887
0.397887
Mean
0.397887
0.423606
0.397887
3.98E-01
3.98E-01
3.98E-01
0.397887
0.397887
std
0
7.80E-02
0
1.60E-11
1.01E-14
0
1.59E-08
0
CE
PT
Mean
F17
Best
3
3
3
3
3
3
3
3
Worst
3
33.28821
3.000
30
84
3.0000
3.0000
3
Mean
3
7.5885
3.000
5.7
5.7
3.0000
3.0000
3
std
0
9.1882
1.44E-14
8.2385
1.48E+01
3.24E-15
3.77E-07
2.26E-15
AC
F18
ED
F16
M
F15
AN US
Best Worst
Best
-3.86
-3.8624
N.A
-3.8627
-3.8627
-3.8627
-3.8627
-3.8628
Worst
-3.86
-3.0819
N.A
-3.8627
-3.0898
-3.8627
-3.8551
-3.8628
Mean
-3.86
-3.7604
N.A
-3.8627
-3.8113
-3.8627
-3.8624
-3.8628
std
0
1.99E-02
N.A
1.63E-09
1.96E-01
2.71E-15
1.64E-03
1.09E-16
Best
-3.3220
-3.3016
N.A
-3.3220
-3.3220
-3.3220
-3.3220
-3.3220
Worst
-3.3220
-2.4316
N.A
-3.2031
-1.7061
-3.3220
-3.0867
-3.2020
Mean
-3.3220
-3.0465
N.A
-3.2943
-3.1976
-3.3220
-3.2759
-3.3021
std
2.22E-15
2.03E-01
N.A
5.11E-02
3.06E-01
1.77E-12
6.99E-02
4.52E-02
Best
-10.1532
-5.0535
-10.1532
-10.1532
-10.1532
-10.1532
-10.1532
-10.1532
F19
F20
F21
ACCEPTED MANUSCRIPT
Worst
-10.1532
-3.8736
-10.1532
-2.6305
-2.6305
-10.1419
-5.0552
-5.1007
Mean
-10.1532
-4.6398
-10.1532
-4.4102
-5.6498
-10.1526
-9.3096
-9.9848
std
3.55E-15
3.58E-01
0
3.2225
3.5591
2.10E-03
1.9185
9.22E-01
Best
-10.4029
-9.6104
-10.0634
-10.4029
-10.4029
-10.4029
-10.4029
-10.4029
Worst
-10.4029
-3.7112
-10.0634
-2.7519
-2.7519
-10.4029
-10.4029
-5.0877
F22 Mean
-10.4029
-4.9199
-10.0634
-8.3653
-5.0491
-10.4029
-10.4029
-10.2258
std
8.88E-15
9.67E-01
0
3.4373
3.1047
4.41E-06
2.29E-05
9.70E-01
Best
-10.5364
-10.5345
-10.0747
-10.5364
-10.5364
-10.5364
-10.5364
-10.5364
Worst
-10.5364
-2.3923
-10.0747
-2.4217
-1.8595
-10.5361
-10.5363
-10.5364
Mean
-10.5364
-4.7705
-10.0747
-8.6883
-5.0996
-10.5362
-10.5363
-10.5364
std
0
1.5789
8.70E-16
3.4090
3.2158
1.23E-05
2.16E-05
1.39E-06
CR IP T
F23
Table 10 Wilcoxon sign rank test on the solutions by different algorithms for benchmarks in Tables 1-2 (a level of significance α=0.05). CGWO vs. CGWO vs. LSHADE
CGWO vs. TLBO
0
R+
R-
232.5
232.5
0
232.5
F5
465
0
232
+ 1.7E-06
0
0
1.7E-6 465
1
1 232.5
=
=
0
232
232
465
360
.5
0
465
F14
0
465
465
465
0
=
1.7E-06
465
1.7E-06
+
1.7E-06
+
1.7E-06 465
0
+
1.7E-06 465
0
+
1.7E-06
0
0
+
1.7E-06 465
0
p-value
1.7E-06 465
0
+
+
+
+
2.2E-04
1.7E-06
1.7E-06
1.7E-06
412
465
0
465
0
465
0
+
+
+
1.7E-06
3.4E-05
1.7E-06
1.2E-02
5.8E-01
434
31
465
0
111
354
-
+
+
1
5.7E-04
1.7E-06
232
232
+
.5
.5
0
465
400
65
=
1.7E-06 465
0
+ 465
192
1
232
232
1
=
.5
.5
=
324
141
=
397
5.9E-01
-
7.2E-04 68
206
1.7E-06
+
1.7E-06 0
259 -
0
3.7E-05 32
1.7E-6
465
465
+ 433
465
=
1.7E-06 0
465
2.8E-05 29
436
-
-
-
1.1E-01
1.7E-06
1.7E-06
364
100
273
465
0
0
2.8E-05
465 .5
+
1.7E-6
1.7E-06
1.7E-06
1.7E-06
1.7E-06
379
85,
4.9E-04
+
.5
5
+
433
32
465
0
465
232
232
.5
.5
465
0
1
+ 1.7E-6 0
360
465
0
2.1E-02 120
345
+ 0
465
111
+
1.7E-06
418
-
+
-
+
232
232
1
1.7E-06
1.6E-02
2.0E-05
.5
.5
=
465
0
0
327
+
465
1.2E-02
47
NA
0
1.2E-02 354
-
1.7E-06 465
+
1.7E-06 0
+
1.4E-04
3.7E-05
-
1.7E-06 465
-
NA
0
1.7E-06 465
8.7E-03 105
465
+
+
1.7E-06
NA
+
= +
0
0
1.9E-06 1
1.7E-06
1.7E-6
465
+ 464
0
0
+
+
465
.5
465
1.7E-06
+
0
R-
.5
+
.5
0
R+
-
1.7E-06 0
1
+
1.7E-6
465
232
465
1.7E-06
0
p-value
+
0
1.7E-06 F13
232
R-
=
1.6E-2
-
=
R+
-
-
465
.5
p-value
-
327
1.7E-06 F12
R-
1
.5
1.7E-06
+
=
232
1.7E-6 0
R+
CGWO vs. GWO
-
0
1
138
.5
0
232
53
0
+
+
F11
.5
1.2E-4
AC
465
465
107
1.7E-06
F10
.5
232
9.8E-3
105
=
232
p-value
CGWO vs. CLPSO
+
465
+
358
225
0
CE
465
0
1.7E-6
465
210
0
+
6.4E-01 F9
0
-
1.7E-6 0
465
1.7E-06
+ F8
465
-
232.5
465
1.7E-06 0
-
1.7E-06 F7
+
1.7E-6
+
R-
465
+
F6 .5
0
1.7E-06
465
.5
1.7E-06
465
R+
465
+ 465
465
p-value 1.7E-06
1.7E-6 0
0
232
0
PT
F4
0
465
=
1.7E-06 465
R-
1 232.5
+ F3
R+
=
1.7E-06 465
p-value
1
+ F2
CGWO vs. BA
AN US
465
p-value
M
R-
ED
R+
1.7E-06 F1
CGWO vs. NDHS
EBOwithCMAR
case
355
138
440
+
110
25
+
ACCEPTED MANUSCRIPT
2.8E-3 378
88
+
+
-
232
232
1
5.3E-5
.5
.5
=
+
232
232
1
1.8E-5
232
232
1
.5
.5
=
+
.5
.5
=
0
465
420
45
F16
429
F17
441
87
0
36
465
465
0
435
461
4
1
1.2E-05
.5
=
+
.5
.5
=
+
1.9E-06
1.7E-06
1.4E-04
4.3E-06
1
465
101
465
465
0
146
465
1.7E-06 NA
NA
465
0
+ 0
2.6E-05
465
437
28
465
+
-
+
3.1E-05
1.7E-6
1.0E-03
1.7E-06
465
0
392
73
465
+ 1.7E-6
435
465
0
465
+
1.7E-06 465
0
+
16/3/4
276
232
11/7/5
9
1
+
.5
.5
=
1.7E-06
232
232
1
1.5E-02
.5
.5
351
114
114
+ 2.8E-03 378
87
=
+
1.5E-02
1.7E-06 465
0
+
+
+
1.7E-06
8.7E-03
2.8E-05
0
105
360
436
29
+
-
+
1.7E-06
5.3E-05
3.1E-05
0
36
429
+
456
20
+ 351
16
435
30
-
4.3E-06
9
=
11/3/6
232
0
3.7E-01
189
+
10/7/6
0
+
456
2
+
8.5E-06
449
1.7E-6
47
0
+
=
30
1.7E-06 465
7.7E-02 319
418
+
+
NA
445
0
6.8E-03 364
463
0
+
+
-
+
3.7E-05 433
32
3.1E-05 435
+
30
+
22/0/1
10/4/9
+ 19/3/1
To investigate the convergence rate of the CGWO algorithm, some convergence curves of benchmark functions selected at random are plotted in Fig. 6. To evaluate the quality of the solution obtained by each algorithm in our experiment, we employ fitness value to assess the performance of each algorithm and use the number of function evaluation (NFE) to track the trend
ED
of fitness value in a random run.
F3
F1
10
10
LSHADE TLBO EBOwithCMAR NDHS BA CLPSO GWO CGWO
fitnes value
-100
PT
10
0
CE
fitnes value
10
-200
AC
+/=/-
232
465
NA
1.7E-6
3.1E-05 30
+
232
+
NA
+ F23
+
1.7E-06
75
M
F22
.5
2.6E-6
212
= 0
+
2.1E-06
0
+
1.7E-6 F21
1
1.2E-04
NA
6.7E-01 253
232
1.7E-6
0
+ F20
+
232
1.7E-06 465
390
AN US
465
-
1.7E-06
+
1.7E-06 F19
58
464
1
1.2E-03
407
0
1.9E-6 464
3.3E-04
465
24
1.7E-6 F18
1.7E-06
377
CR IP T
3.0E-3 F15
0
2
4 6 8 number of function evaluation
10
10
10 x 10
4
0
LSHADE TLBO EBOwithCMAR NDHS BA CLPSO GWO CGWO
-100
-200
0
2
4 6 8 number of function evaluation
10 x 10
4
ACCEPTED MANUSCRIPT
10
10
10
fitnes value
10
10
10
10
6
4
10
10
10
2
0
0
2
4 6 8 number of function evaluation
10
10 x 10
0
-5
-10
-15
10 LSHADE TLBO EBOwithCMAR NDHS BA CLPSO GWO CGWO
0
-5
-10
2
4 6 8 number of function evaluation
2
4 6 8 number of function evaluation
10
x 10
10
10
x 10
4
F15
2
0
LSHADE TLBO EBOwithCMAR NDHS BA CLPSO GWO CGWO
-2
-4
4
0
2
4 6 8 number of function evaluation
10 x 10
4
M
0
10
10
-15
0
4
F11
5
LSHADE TLBO EBOwithCMAR NDHS BA CLPSO GWO CGWO
CR IP T
10
8
fitnes value
10
F9
LSHADE TLBO EBOwithCMAR NDHS BA CLPSO GWO CGWO
fitnes value
fitnes value
10
F5
10
AN US
10
Fig. 6. Convergence curve of these algorithms when solving benchmarks
ED
Fig. 6 presents the evolution of fitness value with the increase of NFE on these problems for each algorithm. X axis represents NFE and Y axis denotes the fitness value. These curves indicate the convergence speed of each algorithm. In terms of the NFE, the convergence speed of CGWO’s curve is much faster than those of its counterparts in minimizing the fitness
PT
value for F1 and F5. CGWO is slightly faster than its compared algorithms for F3, F9, F11 and F15. CGWO is a bit slower than the other algorithms for the rest of benchmarks but still shows strong competitiveness. The reasons for the good
CE
performance of the CGWO have two main aspects as follows. Firstly, each solution in CGWO has its own topological neighbors. Thus, the interaction among solutions is restricted to their neighborhood, which favors exploitation performance of CGWO. Secondly, information diffusion mechanism by overlap among neighborhoods can maintain the population diversity
AC
for longer, usually contributing to exploration performance. We can conclude from this observation that the result of plots is consistent with our view that CGWO with CA effectively improves the convergence of the algorithm. The following subsection reveals that CGWO can locate the whole global optimums.
4.4.2 Experiment two Empirical study on benchmarks with multiple global peaks is conducted in this experiment. Success rate is used to test performance of algorithm for CEC2015 benchmarks with multiple global optimums. Note that the success rate is the percentage of runs in which all the desired peaks are successfully located in this experiment, whereas in the experiment one the convergence only denotes that the algorithm may find a single global solution rather than all desired peaks. The level of accuracy is set to 0.1 in this competition. This parameter is used to measure how close the obtained solutions are to the known global/local peaks. If the distance between an obtained solution and the true optimal solution is smaller than a
ACCEPTED MANUSCRIPT
tolerance value (level of accuracy), it will be accepted that the optimal solution has been found. When all the desired peaks are found in one single run, this run is deemed to be successful. In addition, when all the algorithms satisfy stop condition in this experiment, they return the final population in the last iteration rather than the only one best solution. Table 11 gives the corresponding parameters of problems and summarizes the success rate on CEC2015 with multiple global peaks (i.e., F25, F26, F28, F29, F30, F31, F32, F34, F35 and F36). 30 independent runs are conducted in this experiment. Table 12 presents that CGWO algorithm gives a significantly better performance than the other algorithms in terms of success rate. Fig. 7 shows that CGWO can locate all 4 global peaks on F25 (decision dimension=2) at number of function evaluation (NFE) 40,000 in a single run. Multiple emerged niches are clearly visible at NFE equal to 80,000. Figs.
CR IP T
8-10 show global peaks on F26, F28 and F30 (decision dimension=2) in a single run. We can observe that CGWO is able to find all 25 global optimums on the F26 and F28, and 2 global peaks on the F30, respectively. It means that CGWO can successfully develop stable niches on majority of the global peaks without niching radius parameters. It also implies that CGWO has good diversity of population.
Table 11 Success rate on CEC2015 with multiple global optimum maximum Population
EBOwithC
evaluation dimension
CLPS
AN US
Decision Problem
LSHADE
size
TLBO
NDHS
BA
MAR
function number
GWO
CGWO
O
2
400
100,000
0%
0%
0%
0%
0%
0%
53.3%
100%
F26
2
400
100,000
0%
0%
0%
0%
0%
33.3%
16.7%
96.7%
F28
2
400
100,000
0%
0%
0%
0%
0%
0%
26%
80%
F29
2
526
100,000
0%
0%
0%
3.3%
3.3%
0%
6.7%
10%
F30
2
400
100,000
0%
0%
0%
0%
0%
3%
16%
100%
F31
2
400
500,000
0%
0%
0%
0%
0%
0%
3.3%
6.7%
F32
10
100
300,000
0%
0%
0%
0%
0%
0%
0%
0%
F34
10
100
300,000
0%
0%
0%
0%
0%
0%
0%
10%
F35
10
400
500,000
0%
0%
0%
0%
0%
0%
0%
23.3%
F36
10
400
300,000
0%
0%
0%
0%
0%
0%
0%
0%
PT
ED
M
F25
Table 12 Wilcoxon sign rank test on the solution by different algorithms for CEC2015 with multiple global optimum (a level of significance α=0.05). CGWO vs.
case R+
R-
p-value
465
R+
0
R-
465
0
+
F28
F29
464
455
276
1
464
262
CGWO vs. GWO
465
0
p-value
R+
R-
465
0
1.7E-06
R-
465
0
2
R+
R-
465
0
1
R+
R-
397
68
+
4.1E-05 432
33
p-value 1.2E-04
+
1.9E-06 464
p-value 1.7E-06
+
2.1E-06 462
p-value 1.7E-06
+
2.1E-06 2
R+
1.7E-06
+ 463
p-value
4.7E-06 455
10
+
+
+
+
+
+
4.7E-06
4.7E-06
4.7E-06
4.7E-06
4.7E-06
1.6E-02
10
455
10
455
10
455
10
455
10
455
10
348
117
2.5E-01
247
217
1
=
.5
.5
=
385
80
+
+
+
+
+
+
2.5E-01
2.5E-01
2.5E-01
5.0E-01
5.0E-01
189
276
0
203
CGWO vs. CLPSO
+
189
189
0
203
5.0E-01
203
0
203
5.0E-01
203
0
0
203
5.0E-01
1.7E-06 465
0
+ 262
203
5.0E-01
+
189
1.7E-06 465
+ 262
276 =
1.7E-06 465
+ 262
262
=
1.7E-06 465
+ 262
262
=
1.7E-06 465
5.0E-01
276
=
+ F31
CGWO vs. BA
4.7E-06
1.7E-06 465
R-
1.9E-06 1
= F30
R+
+
1.9E-06
F26
p-value
CGWO vs. NDHS
EBOwithCMAR
1.7E-06
AC
1.7E-06 F25
CGWO vs. TLBO
CE
CGWO vs. LSHADE
5.7E-06
+ 262
203
5.0E-01
+ 247
217
1
ACCEPTED MANUSCRIPT
=
=
2.5E-01
2.5E-01
189
276
2.5E-01
189
276
276
189
2.5E-01 276
2.5E-01
189
=
=
=
=
=
=
1.6E-02
1.6E-02
1.6E-02
1.6E-02
1.6E-02
325
138
325
138
+
325
138
325
+
5/3/0
5/3/0
80
70 60
-50
30 -100
-40
(a) number of function evaluation =4,000
40
-90
-80
-70
-60
-50
-40
(b) number of function evaluation =40,000
30 -100
-90
-80
-70
-60
-50
-40
(c) number of function evaluation =80,000
AN US
-60
global optimum CGWO
50
40
-70
5/3/0
60
50
Fig. 7 Niching behavior of CGWO (with a population of 400) on F25 over a run
5
10 global optimum CGWO
5
global optimum CGWO
0
0
-5 -10
-10
-20
-15 -20
-40 0
10
20
30
40
50
-25 20
(a) number of function evaluation =4,000
global optimum CGWO
-5
-10
-30
0
25
-15 -20
30
35
40
-25 20
(b) number of function evaluation =20,000
25
30
35
40
(c) number of function evaluation =40,000
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Fig. 8 Niching behavior of CGWO (with a population of 400) on F26 in a single run
-40
-40
global optimum CGWO
-60
AC
-70
-30
-20
-10
global optimum CGWO -45
-50 -50 -55 -55 -60
-80
-90 -40
-40 gobal optimum CGWO
-45
CE
-50
-60
-65
0
10
20
(a) number of function evaluation =4,000
-70 -30
-25
-20
-15
-10
-5
(b) number of function evaluation =20,000
-65 -30
138
+
5/3/0
70
50
-80
325
+
90 global optimum CGWO
60
30
138
5/3/0
70
40
325
+
90 80
-90
138
+
5/3/0
global optimum CGWO
20 -100
189
=
1.6E-02
5/3/0
80
276
.5
=
+
90
2.5E-01
189
.5
1.6E-02 138
+/=/-
2.5E-01
189
=
CR IP T
325
276
=
M
F35
276
=
ED
F34
=
-25
-20
-15
-10
-5
(c) number of function evaluation =80,000
Fig. 9 Niching behavior of CGWO (with a population of 400) on F28 in a single run
ACCEPTED MANUSCRIPT
40
35 global optimum CGWO
30
30
12 global optimum CGWO
global optimum CGWO 10
25
20
20
8
10 15 0
6
10 -10
5
4 -20
0
-30 40
50
60
70
80
90
-5 50
100
(a) number of function evaluation =4,000
60
70
80
90
2 50
100
(b) number of function evaluation =8,000
60
70
80
90
(c) number of function evaluation =80,000
CR IP T
Fig. 10 Niching behavior of CGWO (with a population of 400) on F30 in a single run
Therefore, CGWO shows a good performance over other metaheuristics. It is also obvious that CA has a positive effect on the performance improvement of the proposed CGWO. The reason behind is that the CGWO makes fully use of the cellular structure in CA to update solutions only within its neighborhood for local search and adopts an implicit migration mechanism implemented by overlap of neighborhoods to enhance information diffusion to improve the search diversity. More precisely,
AN US
CA provides different neighborhoods for the whole population by using a cellular topological structure. The population is divided into many subpopulations. Each solution is arranged in a given cellular grid, and each subpopulation is form of an independent neighborhood. Consequently, each subpopulation is updated independently in its own neighborhood. That is, different subpopulation may search for promising solutions in different search directions. Obviously, it has a stronger search ability to locate all the peaks compared to the other algorithms. Meanwhile, the migration mechanism implemented by overlap of neighborhoods can help to share valuable information among different independent subpopulations. This can improve the search efficiency. Thus, we can draw a conclusion that CA has a good effect on the behavior of the GWO
M
algorithm.
ED
4.4.3 Experiment three
To further investigate the performance of the proposed CGWO, it is compared to other improved variants of GWO and multiswarm metaheuristic, which include GWOCLS, EGWO, WdGWO, and iHS. This comparison experiment is conducted
PT
on benchmarks in Tables 1 and 2. These parameter settings of the compared algorithms are used as in their original literature (Al-Betar, et al., 2015; Joshi & Arora, 2017; Malik, et al., 2016; Yu, et al., 2017). In this subsection, we also compare the proposed CGWO to the well-known multi-niche metaheuristics including EAD,
CE
RPSO, CDE, NCDE, and LIPS for multi-niche optimization problems from CEC2015. These parameter settings of the compared algorithms are also used as in their original literature (Gu, et al., 2015; X. D. Li, 2010; Qu, et al., 2016; Qu, et al.,
AC
2012; Thomsen, 2004).
Problem
F1
F2
Table 13 Results by different improved versions of GWO and iHS on benchmarks in Tables 1 and 2
Statistics
GWOCLS
EGWO
WdGWO
iHS
CGWO
Best Worst Mean std Best Worst Mean
7.96E-94 2.35E-90 2.85E-91 5.97E-91 2.69E-54 1.13E-51 1.14Ee-52
3.01E-71 1.39E-64 1.21E-65 3.29E-65 9.94E-47 4.36E-43 3.66E-44
5.32E-70 3.91E-66 3.10E-67 7.21E-67 6.36E-40 1.93E-38 3.99E-39
2.93E-03 9.77E-03 5.27E-03 1.87E-03 1.41E-02 2.86E-02 2.18E-02
0 0 0 0 0 0 0
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F7
F8
F9
AC
CE
F10
F11
F12
F13
4.09E-03 1.07E-12 1.14E-07 4.74E-09 2.07E-08 3.58E-01 6.44E-01 5.15E-01 6.84E-02 10.1053 161.7552 78.7063 33.2843 0 0 0 0 1.97E-04 1.85E-03 8.82E-04 4.75E-04 -8.55E+03 -4.13E+03 -4.85E+03 1.03E+03 1.74E-03 8.36E-03 3.27E-03 1.48E-03 1.41E-02 2.91E-02 1.98E-02 4.44E-03 0 1.76E-02 2.45E-03 4.89E-03 4.46E-06 1.31E-04 2.75E-05 2.82E-05 2.05E-04 2.40E-03 4.89E-04
0 3.57E-74 6.24E-55 4.18E-56 1.82E-43 3.62E-73 5.17E-65 4.38E-66 1.32E-65 24.2431 27.0674 25.6489 7.68E-01 0 0 0 0 1.21E-04 8.07E-04 4.30E-04 1.83E-04 -6.70E+03 -3.26E+03 -5.07E+03 1.23E+03 0 1.66E+01 4.20E+00 5.46E+00 4.00E-15 4.00E-15 4.00E-15 0 0 5.44E-02 5.43E-03 1.20E-02 6.01E-08 1.98E-02 5.55E-03 6.23E-03 9.26E-07 6.06E-01 1.88E-01
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4.51E-39 7.16E-13 1.59E-08 1.01E-09 3.21E-09 1.53E-14 2.21E-12 4.40E-13 6.14E-13 24.1463 26.0961 25.0443 3.54E-01 0 0 0 0 2.59E-04 1.81E-03 9.57E-04 4.16E-04 -8.22E+03 -4.12E+03 -4.85E+03 9.35E+02 2.00E+00 1.63E+02 2.08E+01 2.81E+01 7.54E-15 2.03E+01 1.35E+01 9.71E+00 0 1.75E-02 1.43E-03 4.00E-03 4.88E-08 2.86E-06 1.16E-06 4.75E-07 1.38E-05 2.42E-01 1.58E-02
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F6
9.05E-44 3.08E-20 1.06E-13 7.70E-15 1.95E-14 1.84E-13 4.42E-09 4.65E-10 1.05E-09 24.13 26.18 25.29 5.82E-01 0 0 0 0 5.78E-04 2.80E-03 1.52E-03 6.23E-04 -9.94E+03 -6.77E+03 -7.64E+03 6.86E+02 0 4.20E+00 1.40E-01 7.68E-01 7.54E-15 2.01E+01 2.67E+01 6.93E+00 0 2.98E-02 1.90E-03 6.38E-03 4.68E-06 2.62E-02 9.06E-03 8.36E-03 1.22E-04 4.80E-01 1.47E-01
M
F5
ED
F4
2.28E-52 1.66E-25 2.32E-19 1.65E-20 5.41E-20 1.12E-20 1.34E-17 6.91E-19 2.44E-18 24.24 26.18 25.26 5.53E-01 0 0 0 0 1.61E-04 1.27E-03 5.95E-04 2.88E-04 -9.01E+03 -7.26E+03 -8.95E+03 3.19E+02 0 1.98E+01 7.70E+00 5.87E+00 7.54E-15 1.46E-14 1.22E-14 3.14E-15 0 2.29E-02 4.63E-03 7.67E-03 1.70E-07 1.68E-02 2.96E-03 5.10E-03 3.31E-06 3.63E-01 6.18E-02
PT
F3
std Best Worst Mean std Best Worst Mean std Best Worst Mean std Best Worst Mean std Best Worst Mean std Best Worst Mean std Best Worst Mean std Best Worst Mean std Best Worst Mean std Best Worst Mean std Best Worst Mean
ACCEPTED MANUSCRIPT
F18
F19
F20
AC
CE
F21
F22
F23
4.01E-04 9.98E-01 9.98E-01 9.98E-01 1.55E-13 4.21E-04 1.31E-03 6.93E-04 1.75E-04 -1.03163 -1.03163 -1.03163 5.24E-09 3.97E-01 3.97E-01 3.97E-01 3.91E-09 3.000 3.000 3.000 5.95E-08 -3.8627 -3.8627 -3.8627 4.06E-09 -3.3220 -3.2301 -3.3022 4.51E-02 -10.1532 -5.0552 -9.1396 2.06 -10.4029 -3.7243 -9.0304 2.55 -10.5364 -2.8711 -9.7432 2.0894
1.69E-01 9.98E-01 9.98E-01 9.98E-01 0 3.07E-04 2.04E-02 5.71E-04 7.75E-05 -1.03163 -1.03163 -1.03163 8.46E-11 3.97E-01 3.97E-01 3.97E-01 0 3.000 3.000 3.000 2.26E-15 -3.8627 -3.8627 -3.8627 1.09E-16 -3.3220 -3.2020 -3.3021 4.52E-02 -10.1532 -5.1007 -9.9848 9.22E-01 -10.4029 -5.0877 -10.2258 9.70E-01 -10.5364 -10.5364 -10.5364 1.39E-06
CR IP T
5.40E-02 9.98E-01 9.98E-01 9.98E-01 1.53E-13 3.07E-04 1.22E-03 4.29E-04 3.16E-04 -1.03163 -1.03163 -1.03163 1.92E-06 3.97E-01 3.97E-01 3.97E-01 1.42E-08 3.000 3.000 3.000 1.02E-08 -3.8627 -3.8627 -3.8627 2.95E-07 -3.3220 -3.1361 -3.2017 3.96E-02 -10.1532 -2.6828 -9.0605 2.26 -10.4029 -5.0876 -10.2256 9.70E-01 -10.5364 -10.5364 -10.5364 7.12E-05
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F17
1.25E-01 9.98E-01 9.98E-01 9.98E-01 2.75E-13 3.07E-04 1.22E-03 4.29E-04 3.16E-04 -1.03163 -1.03163 -1.03163 4.99E-10 3.97E-01 3.97E-01 3.97E-01 2.29E-08 3.000 3.000 3.000 1.77E-08 -3.8627 -3.8627 -3.8627 4.07E-07 -3.3220 -3.0866 -3.2296 8.19E-02 -10.1532 -5.0551 -7.9575 2.55 -10.4029 -5.0876 -9.6968 1.83 -10.5364 -5.1283 -9.2753 2.32
M
F16
ED
F15
9.52E-02 9.98E-01 9.98E-01 9.98E-01 2.32E-13 3.07E-04 2.03E-02 2.46E-03 6.07E-03 -1.03163 -1.03163 -1.03163 9.05E-10 3.97E-01 3.97E-01 3.97E-01 2.24E-08 3.000 3.000 3.000 1.18E-07 -3.8627 -3.8627 -3.8627 1.86E-06 -3.3220 -3.1974 -3.2699 6.06E-02 -10.1532 -10.1530 -10.1531 4.27E-05 -10.4029 -10.4027 -10.4029 4.68E-05 -10.5364 -10.5364 -10.5364 4.00E-05
PT
F14
std Best Worst Mean std Best Worst Mean std Best Worst Mean std Best Worst Mean std Best Worst Mean std Best Worst Mean std Best Worst Mean std Best Worst Mean std Best Worst Mean std Best Worst Mean std
Table 13 shows the statistic results (i.e., best, worst, mean, and std) obtained by different improved versions of GWO and iHS on benchmarks in Tables 1 and 2. 30 independent experiments of each metaheuristic are conducted on each problem.
ACCEPTED MANUSCRIPT
The optimal values in these tables are highlighted in boldface. It can be clearly observed that the proposed CGWO can find promising results on most problems compared to other improved versions of GWO and iHS. More precisely, the CGWO performs the best among these compared 5 algorithms. The good performance of CGWO is due to the CA technique. The CA technique can provide many different neighborhoods during the optimization process. As the CGWO can update its own subpopulation in each neighborhood, the search process of the entire population can be updated in different search directions. This will effectively maintain the search diversity. Table 14 Wilcoxon sign rank test on the solution by different improved versions of GWO for benchmarks in Tables 1-2 (a level of significance α=0.05). CGWO vs. EGWO
CGWO vs. WdGWO
CGWO vs. iHS
R+
R-
p-value
R+
R-
p-value
R+
R-
F1
465
0
1.7E-06 (+)
465
0
1.7E-06 (+)
465
0
F2
465
0
1.7E-06 (+)
465
0
1.7E-06 (+)
465
0
F3
465
0
1.7E-06 (+)
465
0
1.7E-06 (+)
465
0
F4
465
0
1.7E-06 (+)
465
0
1.7E-06 (+)
465
0
F5
126
339
2.8E-02 (-)
156
309
1.1E-01 (=)
86
379
CR IP T
CGWO vs. GWOCLS
F6
316
149
8.6E-02 (=)
233
232
1.0E+0 (=)
233
F7
316
149
8.6E-02 (=)
463
2
2.1E-06 (+)
F8
0
465
1.7E-06 (-)
1
464
F9
327
138
1.2E-04 (+)
3
F10
253
212
3.9E-01 (=)
F11
67
398
F12
184
F13
case
R+
R-
p-value
1.7E-06 (+)
465
0
1.7E-06 (+)
1.7E-06 (+)
465
0
1.7E-06 (+)
1.7E-06 (+)
465
0
1.7E-06 (+)
1.7E-06 (+)
465
0
1.7E-06 (+)
2.5E-03 (-)
452
13
6.3E-06 (+)
232
1.0E+0 (=)
233
232
1.0E+0 (=)
444
21
1.3E-05 (+)
465
0
1.7E-06 (+)
1.9E-06 (-)
235
230
9.5E-01 (=)
0
465
1.7E-06 (-)
462
2.3E-06 (-)
461
4
2.6E-06 (+)
0
465
1.7E-06 (-)
420
45
3.9E-05 (+)
406
59
3.7E-06 (+)
465
0
1.7E-06 (+)
2.4E-05 (-)
10
455
4.7E-06 (-)
9
456
4.2E-06 (-)
25
440
1.9E-05 (-)
281
3.1E-01 (=)
312
153
1.0E-01 (=)
105
360
8.7E-02 (=)
105
360
8.7E-03 (=)
162
303
1.4E-01 (=)
266
199
4.9E-01 (=)
65
400
5.7E-04 (-)
36
429
5.30E-05 (-)
F14
460
5
2.8E-06 (+)
447
18
1.0E-05 (+)
415
50
1.9E-05 (+)
284
181
6.4E-02 (=)
F15
292
173
2.2E-01 (=)
275
190
3.8E-01 (=)
319
146
7.5E-02 (=)
353
112
1.3E-02 (+)
F16
456
9
4.2E-06 (+)
446
19
1.1E-05 (+)
461
4
2.6E-06 (+)
456
9
4.2E-06 (+)
F17
456
9
4.2E-06 (+)
462
3
2.3E-06 (+)
461
4
2.6E-06 (+)
221
244
8.1E-01 (=)
F18
246
219
7.8E-01 (=)
284
181
6.4E-02 (=)
235
230
9.5E-01 (=)
278
187
3.5E-01 (=)
F19
232
233
9.9E-01 (=)
168
297
1.8E-01 (=)
103
362
7.7E-03 (-)
0
465
1.7E-06 (-)
F20
346
119
1.9E-02 (+)
433
32
3.7E-05 (+)
444
21
1.3E-05 (+)
69
396
7.7E-06 (-)
F21
30
435
3.1E-05 (-)
439
26
2.1E-05 (+)
440
25
1.9E-05 (+)
226
239
8.9E-01 (=)
F22
30
435
3.1E-05 (-)
437
28
2.6E-05 (+)
436
29
2.8E-05 (+)
263
202
5.3E-01 (=)
3.1E-05 (+)
441
24
1.8E-05 (+)
435
30
3.1E-05 (+)
465
0
1.7E-06 (+)
435
+/=/-
M
ED
PT
CE
AC F23
AN US
p-value
30
10/8/5
13/7/3
14/5/4
10/7/6
Table 15 Results by different algorithms on average number of optima found (F24-F31)
Problem
F24
Statistics
EAD
RPSO
CDE
NCDE
LIPS
CGWO
Best
8
1
0
0
11
0
Worst
1
0
0
0
5
0
Mean
4.17
1.17E-01
0
0
8.24
0
std
1.99
3.22E-01
0
0
1.26
0
ACCEPTED MANUSCRIPT
F29
F30
F31
26
60
Worst
33
16
0
0
14
35
Mean
50.21
15.8
0
0
23
53.2
std
4.25
2.91
0
0
3.04
3.51
Best
77
89
4
13
45
90
Worst
59
72
0
9
31
68
Mean
68.17
74.05
2
10.4
37.92
75.4
std
3.86
4.82
1.25
1.67
3.58
5.05
Best
1
0
0
0
0
0
Worst
0
0
0
0
0
0
Mean
2.15E-01
0
0
0
0
0
std
4.15E-01
0
0
0
0
0
Best
59
130
0
0
0
73
Worst
42
117
0
0
0
57
Mean
48.07
118.35
0
0
0
64.21
std
3.86
4.96
0
0
0
5.36
Best
85
0
63
90
Worst
83
0
Mean
84.66
0
std
5.88E-01
0
Best
8
0
Worst
1
0
Mean
3.66
0
std
1.49
Best
60
Worst
45
Mean std
CR IP T
0
AN US
F28
0
0
2
0
0
50
85
0
0.8
55.96
87.6
0
0.83
3.18
3.32
0
1
15
18
0
0
9
10
0
0.04
12.04
14.31
0
0
0.2
1.45
4.03
9
12
40
96
105
1
4
30
78
80
50.72
4.90
6.9
36.4
88.52
91.56
4.00
2.88
2.30
3.78
4.56
5.86
M
F27
22
ED
F26
57
PT
F25
Best
Table 16 Results by different algorithms (statistic values of 5 best solutions separated at least by 10 on F32-F38)
EAD
PSO (ring)
CDE
NCDE
LIPS
CGWO
Best
900.48
909.97
952.14
900.03
900
901.75
Worst
938.54
931.11
990.52
974.83
920.67
901.75
Mean
911.58
919.62
970.566
935.86
911.95
901.75
std
9.15
7.41
14.25
26.54
7.60
4.58E-05
Best
5081.11
1070.00
2701
1139.20
6287.1
1070
Worst
7607.26
1070.00
12551
12267
1070.4
1070
Mean
6812.02
1070.00
9622.02
7524.30
4079.02
1070
std
757.76
5.30E-04
4062.19
5618.14
2749.6
0
Best
1100.533
1100.81
1104.6
1100.30
1102
1100
Worst
1104.028
1103.32
1107.3
1111.80
1103.3
1100.00
Mean
111.993
1102.12
1105.7
1104.26
1102.74
1100.00
std
7.56E-01
5.09E-01
1.23
4.38
4.72E-01
4.17E-04
Best
1537.06
1511.40
1313.90
1221.80
1201.2
1200
AC
F32
Statistics
CE
Problem
F33
F34
F35
ACCEPTED MANUSCRIPT
F37
2808.51
1992.80
2129.70
1476.6
1201.09
Mean
1660.58
2281.25
1623.24
1652.64
1296.68
1200.07
std
101.96
298.65
286.42
392.61
132.95
2.50E-01
Best
1519.41
5261.5
1488.1
1503
1437.6
1300
Worst
1711.13
225790
1658
1656.60
1539.9
1300
Mean
1595.73
150498.9
1594.62
1597.10
1490.24
1300
std
39.75
1.09+E05
67.83
57.28
45.67
0
Best
2050.04
1440.00
2469
1950.30
1480
1520
Worst
2607.84
2523.49
3041
2959.60
2203.9
1726.77
Mean
2401.33
2044.92
2681.24
2382.08
1820.34
1569.78
std
135.55
286.21
253.17
375.00
236.13
43.29
Best
1640.96
1640.00
1686.40
1640.50
1640
1500
Worst
1758.05
1640.00
2069.70
2073.70
1824.1
2116.28
Mean
1665.74
1640.00
1918.70
1908.20
1742.1
1680.40
std
33.164
1.2E-09
142.28
162.26
68.67
113.29
AN US
F38
1831.72
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F36
Worst
The proposed CGWO is also compared to the well-known multi-niche metaheuristics such as EAD, RPSO, CDE, NCDE, and LIPS on multi-niche optimization problems in CEC2015. These multi-niche optimization problems contain 8 extended simple functions (i.e., F24-F31) and 7 composition functions (i.e., F32-F38). For 8 extended simple problems, the performance metric like the average number of optima found (ANOF) is used to test the behavior of these metaheuristics. If the gap between one obtained solution and one known global optimum is below 𝜀 (𝜀 = 0.1), then the peak is deemed to have been found. For the 7 composition problems, a different performance metric is adopted due to its difficulty in finding
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any good solutions. The statistic results of 5 best solutions of each algorithm are found for the 7 composition problems. Table 15 reports the best, worst, mean and standard deviation (i.e., std) of the number of optima found by different
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multi-niche metaheuristics for problems F24-F31. The optimal values in these tables are highlighted in boldface. We can find from this table that the proposed CGWO can provide promising results on most problems. Meanwhile, Table 16 presents statistic results of 5 best solutions found by each algorithm. CGWO is the best one of these compared 6 algorithms.
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Compared to other well-known multi-niche metaheuristics, it is obvious to see that CGWO shows the best performance. These results confirm our previous conclusion that the CA can contribute to balance between exploration and exploitation. The superior performance of CGWO is due to its CA mechanism. CA provides an effective topology structure for the whole
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population. As solution in CA only interacts with its neighbors for exploitation, the search can be enhanced in the niche. Meanwhile, information diffusion mechanism contributes to exploration. Consequently, a collection of such solutions
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congregates to solve the problem.
4.5 Wilcoxon sign-sum test analysis Due to the stochastic nature of these algorithms, the statistical test is necessary for providing confidential comparisons. A nonparametric statistical test called Wilcoxon sign-rank test should be carried out in this paper. This statistical test is used to detect the significant difference between the results obtained by different algorithms. The confidence level for all tests is set to 95% (corresponding to a level of significance α= 0.05). The symbol +(-) indicates that our proposed CGWO algorithm is significantly better (worse) than its counterpart. The sign = denotes that there is no significant difference between the proposed algorithm and its compared algorithm. R+ is the sum of ranks for the problems in which the first algorithm outperformed the second, and R− the sum of ranks for the opposite (Derrac, García, Molina, & Herrera, 2011). Table 7 records the CGWO with C25 is the best choice among six variants using Wilcoxon sign rank test on these results
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obtained by CGWO with different cell topology structures. Table 10 and Table 14 show that the proposed algorithm provides statistically better results compared with the other metaheuristics for benchmarks in Tables 1-2, based on the Wilcoxon sign rank test. Table 12 presents the multi-problem-based pairwise statistical comparison results using the averages of the global minimum values obtained through 30 runs of CGWO and the comparison algorithms to solve the CEC2015 with multiple global optimum. Wilcoxon sign rank test is not conducted on F32 and F36 as that all algorithms are unable to find all peaks on these problems according to Table 12. These results show that CGWO was statistically more successful than compared algorithms, with a statistical significance value α= 0.05. 4.6 CUP-time cost
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This subsection studies CPU-time cost on CGWO and GWO. According to the above experiments, concerning average values CGWO outperforms the basic GWO on 31 of 38 benchmark problems with 29 significant results. The mean CPU-time consumed values by CGWO and GWO algorithms for all test problems are plotted in Fig. 11, and they are divided into three categories in Table 17. The last column of Table 17 provides the increase rates of CPU-time brought by the incorporation of CA model. It is observed from Fig. 11 and Table 17 that good performance achieved by the combination of CA does not come for free. The CPU-time consumption of CGWO is larger than that of the basic GWO for most instances. The main reason for the large time cost of the CGWO can be analyzed as follows. CGWO searches for the promising solution in the
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current solution’s neighbors, whereas this neighborhood structure provides an information diffusion mechanism to CGWO in which population diversity is maintained longer due to the slow information diffusion through the wolf population regarding the best position of each neighborhood. Thus, there exists a delay in the information spread through the population, which escapes from a local optimum. Although CPU-time consumption of CGWO is a little larger than that of the original GWO, CGWO is better than GWO for most instances and it is well worth introducing a CA model to improve the behavior of
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CGWO.
Fig 11. CPU-Time consumption on all test functions
Table 17 CPU-time costs on three different test problems Problems
GWO
CGWO
Increase rate (%)
Unimodal problems(F1-F7)
7.026
9.96
41.8
Multimodal problems(F8-F23)
10.525
16.574
57.5
Composition problems(F24-F38)
64.484
71.892
11.5
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4.7 CGWO for engineering problems This subsection is devoted to evaluating the performance of the proposed CGWO on the real-world engineering problems, these constrained problems including tension/compression spring design, overspeed protection of a gas turbine and rolling element bearing are studied in this subsection. Since these problems contain constraints, the constraint handling mechanism as in Section 3.1 is also adopted. Compared with gradient-based optimization approaches, most metaheuristics have a derivation-free mechanism. For metaheuristics, there is no need to calculate the derivative of search spaces to find the optimal solutions. This makes metaheuristics suitable for real applications with expensive or unknown derivative information. Additionally, metaheuristics
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have superior abilities to avoid local optima over conventional optimization approaches. This is due to the stochastic nature of metaheuristics which allows them to avoid stagnation in local optima. The search space of these problems is usually unknown and very complex with a number of local optima. Therefore, metaheuristics are good options for solving these challenging real-world problems. Note that the significance test technique is not used in this subsection as that the compared results of these applications are from the existing literatures which just offer simple statistical results. 4.7.1 Tension/compression spring design
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The aim of this problem is to minimize the weight of a tension/compression spring subject to constraints on minimum deflection, shear stress, surge frequency and limits on outside diameter. This problem has three design variables defined as: the wire diameter (0.05≤x1≤2.0), the mean coil diameter (0.25≤x2≤1.3) and the number of active coils (2.00≤x3≤15). The mathematical formulation of the problem can be defined as follows:
min f x x3 2x2 x12 1
x 23 x3
71785 x14
12566 x 2 x13 x14
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1
0
4 x 22 x1 x 2
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g 1 x g 2 x s.t . g 3 x g 4 x
140.45 x1 x 22 x3
(12)
1
5108 x12
-1 0
0
x1 x 2 1 0 1.5
This problem has been regarded as a classical benchmark for evaluating different optimization algorithms. The best
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solutions among several algorithms such as GWO (S. Mirjalili, et al., 2014), HPSO (Q. He & L. Wang, 2007), CDE (Huang, Wang, & He, 2007), CPSO (Qie He & Ling Wang, 2007) and SSO-C (Cuevas & Cienfuegos, 2014) are recorded in Table 18.
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We can observe from Table 18 that CGWO, HPSO, and SSO-C can find the best result. However, among them, only CGWO satisfies all the constraints while others violate the constraints to some extent. Table 19 presents statistical results obtained by different algorithms such as CPSO, CDE, SSO-C, G-QPSO (Coelho, 2010), UPSO (Parsopoulos & Vrahatis, 2005), PSO-DE
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(Liu, Cai, & Wang, 2010), MBA (Sadollah, Bahreininejad, Eskandar, & Hamdi, 2013), HEAA (Wang, Cai, Zhou, & Fan, 2009), and TLBO. It can be seen from Table 19 that the CGWO shows a good performance and maintains an acceptable stability since it obtains the best result with comparison to most compared algorithms in terms of NFE. The proposed CGWO achieves the best average result except HEAA, PSO-DE, and TLBO metaheuristics. However, CGWO provides competitive mean results in much less NFEs than offered by the HEAA and PSO-DE method. CGWO is also competitive to TLBO for solving this problem. Table 18 Comparison of best solution obtained from various previous studies for tension/compression spring design. variable
CPSO
GWO
HPSO
SSO-C
CDE
CGWO
x1
0.051728
0.05169
0.051706
0.051689
0.051609
0.051689066186
x2
0.357644
0.356737
0.357126
0.35671775
0.354714
0.35671786256
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x3
11.244543
11.28885
11.265083
11.288965
11.410831
11.28895856
g1
−8.25E−04
-7.9065e-05
−3.06E−06
-4.746E-06
-3.864E-05
-4.51E−10
g2
−2.52E−05
-7.5056e-06
1.39E−06
3.337E-06
-1.8289E-04
−2.234E−11
g3
−4.051306
−4.053383
−4.054583
-4.0537858
-4.048627
−4.05378
g4
−0.727085
−0.7277153
−0.727445
-0.7277287
-0.7291179
−0.727728
F
0.0126747
0.012666
0.0126652
0.0126652
0.0126702
0.0126652
Table 19 Comparison of statistical results given by different optimizers for tension/compression spring design. worst
Mean
Best
std
NFEs
CPSO
0.0129240
0.0127300
0.0126747
5.20E−04
240,000
HPSO
0.0127190
0.0127072
0.0126652
1.58E−05
81,000
G-QPSO
0.017759
0.013524
0.012665
1.26E-03
2,000
DE
0.012790
0.012703
0.0126702
2.7E−05
204,800
HEAA
0.012665240
0.012665234
0.012665233
1.4E−09
24,000
PSO-DE
0.012665304
0.012665244
0.012665233
1.2E−08
24,950
0.012669249
0.012922669
N.A
0.02294
CDE
N.A
0.012703
TLBO
N.A
0.01266576
MBA
0.012900
0.012713
CGWO
0.0127173
0.01267444
5.9E−04
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0.016717272
25,167
0.01312
7.20E−03
100,000
0.01267
N.A
240,000
0.012665
N.A
10,000
0.012665
6.30E−05
7,650
0.0126652
1.26E-05
2,000
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SC UPSO
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Algorithms
4.7.2 Overspeed protection system for a gas turbine
This problem is to maximize reliability of this system and is a mixed-integer nonlinear reliability design optimization issue
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(T. C. Chen, 2006). Overspeed detection is continuously provided by the electrical and mechanical systems. When an overspeed happens, it is necessary to cut off the fuel supply using control values (Coelho, 2009). The problem has two types of design variables (i.e. positive integers ni and real number value ri). The mathematical formulation of the problem can be
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written below:
max f r,n
g 1 s .t .g 2 g 3
1 1 r m
ni
i
i 1
m
v i ni i
2
(13)
-V 0
1
C ri ni i m
e
1
m
w i nie i
0.25 ni
0.25 ni
- C
0
-W 0
1
C ri i T lnri i
1 n i 10, n i Z ,0.5 ri 1 10 6 , ri
where V is the upper bound on the sum of the subsystems’ products of volume and weight, C is the upper bound on the cost of the system, C(ri) is the cost of each component with reliability ri at subsystem i, T is the operating time during which the component must not fail, W is the upper limit on the weight of the system. The input parameters of this system are listed in Table 20. Table 20 Data used in overspeed protection system of a gas turbine.
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stage
𝛼𝑖
𝛽𝑖
𝑣𝑖
𝜔𝑖
𝑉
𝐶
𝑤
𝑇
1
1.0E-05
1.5
1
6
250
400
500
1000
2
2.3E-05
1.5
2
6
3
0.3E-05
1.5
3
8
4
2.3E-05
1.5
2
7
It is noted that any improvement in the objective of this problem is very important for reliability engineering and system safety. Table 21 compared the results found in this work for the complex system with those of other research reported. The
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best result obtained by CGWO and CS (Valian & Valian, 2013) can reach at 0.99995468, which has a slight advantage over the other solvers reported. However, CS violates a constraint whereas CGWO satisfies all constraints. Therefore, CGWO outperforms other compared algorithms in terms of the best result. Statistical results for this overspeed protection system are summarized in Table 22. According to the results in Table 22, with regard to the best results, the solutions of CGWO are just slightly better than the solution found by PSO-GC (Coelho, 2009) and GA-PSO (Sheikhalishahi, Ebrahimipour, Shiri, Zaman, & Jeihoonian, 2013) on this problem. Moreover, the standard deviation of the results by CGWO is much smaller than that by
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CS, which denotes CGWO is more stable when solving this problem.
Table 21 Comparison of best solution obtained from various previous studies for overspeed protection system of a gas turbine GA-PSO
CS
CGWO
n1
5
5
5
5
n2
6
5
5
6
n3
4
4
4
4
n4
5
6
6
5
r1
0.902231
0.901628
0.90161460
0.9016347
r2
0.856325
0.888230
0.88822337
0.8499661
r3
0.948145
0.948121
0.94814103
0.94812205
r4
0.883156
0.849921
0.84992090
0.8881983
g1
-55
-55
-55
-55
-0.978964
-0.000006
1.04E-05
-1.8383E-04
g3
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g2
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PSO-GC
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Variable
-24.80188
-15.363463
-15.363
-24.8018827
0.999953
0.99995467
0.99995468
0.99995468
Table 22 Comparison of statistical results given by different optimizers for the overspeed protection system of a gas turbine Worst
mean
Best
std
NFEs
PSO-GC
0.99993800
0.99990700
0.99995300
1.1E-05
NA
GA-PSO
0.99995467
0.99995467
0.99995467
1.00E−16
NA
CS
0.99991922
0.99995336
0.99995468
4.5576E-06
10,000
CGWO
0.99994674
0.99995419
0.99995468
2.16E−09
5,000
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algorithms
4.7.3 Rolling element bearing This problem is proposed in (Gupta, Tiwari, & Nair, 2007; B. R. Rao & Tiwari, 2007) and its objective is to maximize the dynamic load carrying capacity of a rolling element bearing. The design variables are as follows: the ball diameter Dm, pitch diameter Db, the number of balls Z, the inner and outer raceway curvature coefficients fi and f0, KDmin, KDmax, ε, e, and ζ. The latter five parameters appear in constraints and affect the internal geometry. Z is the discrete design variable and the
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remainder are continuous design variables. According to the literatures (Eskandar, Sadollah, Bahreininejad, & Hamdi, 2012; Sadollah, et al., 2013) , decision variables fi and Db are assumed to be independent variables. Actually there exists a relationship between fi and Db by equations fi = ri/Db and f0=r0/Db where r0 and ri are constant values. Therefore, the decision variables fi and f0 in the optimization procedure can be eliminated. Meanwhile, the upper bound of Db is determined by 0.45(D-d) and minimum value between ri/min(fi) and r0/min(f0), which can result in range of Db from [0.15(D-d), min(0.45(D-d), min(ri/min(fi), r0/min(f0))]. Consequently, the mathematical formulation of the problem can be reconstructed as follows: Db fc Z max f Dm , Db , Z , K D min , K D max , , e, 33 . 647 f c Z 2 / 3 Db1.4
g1 g 2 g 3 g 4 s.t . g 5 g 6 g 7 g 8 g 9 g 10 g 11
1.72
f i 2 f 0 1 f 0 2 f i 1
K D min D d 2Db 0
2Db K D max D d 0 B w Db 0
Db 25.4 Db 25.4
(14)
0
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0.5D d Dm 0
Dm 0.5 e D d 0
Db 0.5D Dm Db 0 0.515 f i 0 f i 0.6 0
0.515 f 0 0 f 0 0.6 0
0.41
10 3
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1 f c 37.911 1.04 1
0
2 sin 1 Db Dm
if if
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where
Z 1
1.8
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2/3
-0.3
0.3 1 - 1.39 2 f 0.41 i 13 1 2 f i 1
D d 2 3T 4 2 D 2 T 4 D 2 d 2 T 4 2 b 2D d 2 3T 4D 2 - T 4 - Db D r r b , f i i , f 0 0 , T D d 2Db , D 160, d 90, Bw 30, ri r0 11.033 Dm Db Db
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0 2 - 2 cos 1
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0.5D d Dm 0.6D d , 0.15D d Db min0.45D d ,min[ r0 min( f 0 ),ri min( f i )], 4 Z 50, 0.4 K D min 0.5, 0.6 K D max 0.7, 0.3 0.4, 0.02 e 0.1, 0.6 0.85.
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The CGWO and its competitive algorithms such DE and GWO are run 30 times on this problem and the best values found are recorded in Table 23. Obviously, the results obtained by CGWO are better than that found by the DE and GWO. Furthermore, Table 24 gives statistical results obtained through these algorithms and shows that CGWO outperforms DE and GWO in terms of the mean, best, and standard deviation values. It means CGWO is a very effective method for parameters optimization of a rolling element bearing. Table 23 Comparison of best solution obtained by DE, GWO, and CGWO for rolling element bearing Variable
DE
GWO
CGWO
Dm
125.69547638
125.71678875
125.69088021
Db
21.4174747008
21.4174513046
21.417475728
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11
11
11
KDmin
0.42433415670
0.48394163540
0.4
KDmax
0.64383088983
0.67494041047
0.7
ε
0.30028179945
0.30028688211
0.30051
e
0.05635192733
0.05297465987
0.02
ζ
0.63988721158
0.62514555525
0.606519856
g1
-1.1732E-03
-2.8946E-03
-4.1907E-10
g2
-13.13155
-8.95898
-14.83495
g3
-2.23321
-4.41093
-6.165049
g4
-2.22086
-2.66308
-3.22188
g5
-0.69547
-0.71678
-13.3925
-12.52688
g7
-0.012246
-1.5003E-03
g8
-2.4704E-08
-5.8728E-07
g9
-0.0850
-0.0850
g10
-2.4704E-08
-5.8728E-07
g11
-0.0850
F
81803.125
-0.68088 -4.31912
-0.014656
-3.7352E-12 -0.0850
-3.7352E-12
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g6
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Z
-0.0850
-0.0850
81801.065
81803.647
Table 24 Comparison of statistical results given by DE, GWO, and CGWO for rolling element bearing Worst
Mean
DE
76787.252
78403.648
GWO
76771
79525.009
CGWO
76789.100
81336.829
Best
std
NFEs
81803.125
2343.593
10,000
81801.065
2473.211
10,000
81803. 647
1417.442
10,000
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algorithms
To sum up, the above empirical studies indicate that the proposed CGWO has superiority over the current meta-heuristics considered. First, the statistical results for the unconstrained benchmarks show the good performance of CGWO in terms of exploitation and exploration, and then regarding convergence CGWO is also superior or competitive to its rivals. Finally, the
Conclusions and future work
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5.
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results of engineering problems demonstrate that CGWO is a promising approach.
In this paper we have proposed a CGWO by integrating CA model and GWO. The search process in CGWO is guided by
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three good wolves (solutions) but this interaction is restricted within its neighborhood, which can keep a good balance between diversity and convergence. CGWO has been compared to the other seven current metaheuristics such as LSHADE, TLBO, EBOwithCMAR, BA, NDHS, CLPSO and GWO on unimodal, multimodal and CEC2015 benchmarks. Furthermore, CGWO has been compared to some recent improved versions of GWO on unimodal and multimodal benchmarks. Then, CGWO is compared to state-of-art multi-niche metaheuristics such as EAD, CDE, NCDE, RPSO, and LIPS on multi-niche benchmarks in CEC2015. A nonparametric statistical test called Wilcoxon sign-rank test is used to detect the significant difference between the results obtained by different algorithms. CGWO is also applied successfully into engineering problems. Empirical results reveal that CGWO is an effective approach for optimization problems. Additionally, the main contributions of this work are as follows. (1) The cellular automata (CA) concept is embedded into the GWO. CA provides a given topological neighborhood
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structure. The population can be divided into many subpopulations by this given neighborhood structure. Each subpopulation forms an independent niche, which can improve its local search. (2) CGWO with a topological structure can also help to improve diversity of population. Each subpopulation can update itself in its own neighborhood. Therefore, different subpopulations have different search directions, which helps to the search diversity. The overlap between consecutive neighborhoods can provide a recessive migration mechanism without any complex mathematical equation. The information of good solutions can be shared to some extent. It can enhance its convergence of the proposal. (3) The performance of the CGWO is sensitive to neighborhood size for most instances. An appropriate neighborhood size depends on specific problems. It can be found by numerical experiments that the neighborhood size named C25 is the best
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choice for CGWO.
(4) Compared with other metaheuristics, the proposed CGWO is able to find all the optimal solutions rather than a single optimal solution for multiple peaks benchmarks, especially for CEC2015 test suit. It thus indicates that CGWO can maintain a good diversity of population.
In this study, the CGWO is proposed to solve single-objective and continuous optimization problems well. However, many
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optimization problems involve multiple objectives and discrete decision variables in real-world application. Additionally, only GWO is incorporated into the CA, while other metaheuristics are not hybridized with CA. With respect to future work, firstly, research can be extended to a multi-objective grey wolf optimization algorithm. Secondly, more metaheuristics and problems are considered to test the performance of the proposed CGWO in the comparison experiment. Thirdly, it is important to understand how to set an adaptive neighborhood size for CGWO, although a recommend neighborhood size is given in our work. Fourthly, it would be valuable to apply the CGWO into many applications such as production scheduling
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in industry.
Acknowledges
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The project was supported by the fundamental research funds for the central universities, China University of Geosciences (Wuhan) (No. CUG170688), National Natural Science Foundation of China (NSFC) under Grant nos. 51775216, 51375004
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and 51505439 References
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Caption lists Fig. 1. Position updating in GWO Fig. 2. Structure of neighborhood Fig. 3. Flow chart of the proposed CGWO algorithm Fig. 4. Pseudo code of the proposed CGWO algorithm
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Fig. 5. Position updating in CGWO Fig. 6. Convergence curve of these algorithms when solving benchmarks Fig. 7. Niching behavior of CGWO (with a population of 400) on F25 in a single run Fig. 8. Niching behavior of CGWO (with a population of 400) on F26 in a single run Fig. 9. Niching behavior of CGWO (with a population of 400) on F28 in a single run Fig. 10. Niching behavior of CGWO (with a population of 400) on F30 in a single run Fig. 11. CPU-Time consumption on all test functions
Table 2. Multimodal benchmark functions Table 3. CEC2015 benchmark functions Table 4. Parameter setting of algorithms Table 5. Results by CGWO with different neighborhood on unimodal benchmark functions Table 6. Results by CGWO with different neighborhood on multimodal benchmark functions
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Table 1. Unimodal benchmark functions
Table 7. Wilcoxon sign rank test on the solution by CGWO with different structures for benchmarks in Tables 1-2 (a level of significance α=0.05)
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Table 8. Results by different algorithms on unimodal benchmark functions Table 9. Results by different algorithms on multimodal benchmark functions
Table 10. Wilcoxon sign rank test on the solution by different algorithms for benchmarks in Tables 1-2 (a level of significance α=0.05) Table 11. Success rate on CEC2015 with multiple global optimum
Table 12. Wilcoxon sign rank test on the solution by different algorithms for CEC2015 with multiple global optimum (a level of significance α=0.05) Table 13. Results by different improved versions of GWO and iHS on benchmarks in Tables 1 and 2
Table 14. Wilcoxon sign rank test on the solution by different improved versions of GWO for benchmarks in Tables 1-2 (a level of significance α=0.05)
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Table 15. Results by different algorithms on average number of optima found (F24-F31)
Table 16. Results by different algorithms (statistic values of 5 best solutions separated at least by 10 on F32-F38)
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Table 17. CPU-time costs on three different test problems
Table 18. Comparison of best solution obtained from various previous studies for tension/compression spring design. Table 19. Comparison of statistical results given by different optimizers for tension/compression spring design Table 20. Data used in overspeed protection system of a gas turbine.
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Table 21. Comparison of best solution obtained from various previous studies for overspeed protection system of a gas turbine Table 22. Comparison of statistical results given by different optimizers for the overspeed protection system of a gas turbine
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Table 23. Comparison of best solution obtained by DE, GWO, and CGWO for rolling element bearing
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Table 24. Comparison of statistical results given by DE, GWO, and CGWO for rolling element bearing
Grey Wolf Optimizer with Cellular Topological Structure Chao Lu , Liang Gao , Jin Yi PII: DOI: Reference:
S0957-4174(18)30243-4 10.1016/j.eswa.2018.04.012 ESWA 11926
To appear in:
Expert Systems With Applications
Received date: Revised date: Accepted date:
15 December 2017 21 March 2018 9 April 2018
Please cite this article as: Chao Lu , Liang Gao , Jin Yi , Grey Wolf Optimizer with Cellular Topological Structure, Expert Systems With Applications (2018), doi: 10.1016/j.eswa.2018.04.012
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Highlights The cellular automata concept is embedded into the GWO CGWO with a topological structure can help to improve diversity of population The proposed CGWO can solve multimodal problems well The CGWO outperforms the other state-of-the-art algorithms on function and engineering problems
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Grey Wolf Optimizer with Cellular Topological Structure Chao Lua, Liang Gaob, Jin Yic Hubei Key Laboratory of Intelligent Geo-Information Processing (China University of Geosciences (Wuhan)), Wuhan 430074, China b. State Key Lab of Digital Manufacturing Equipment & Technology, Huazhong University of Science and Technology, Wuhan, China c. Department of Industrial Systems Engineering and Management, National University of Singapore, Singapore
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a.
Abstract: Grey wolf optimizer (GWO) is a newly developed metaheuristic inspired by hunting mechanism of grey wolves. The paramount challenge in GWO is that it is prone to stagnation in local optima. This paper proposes a cellular grey wolf optimizer with a topological structure (CGWO). The proposed CGWO has two characteristics. Firstly, each wolf has its own topological neighbors, and interactions among wolves are restricted to their neighbors, which favors exploitation of CGWO. Secondly, information diffusion mechanism by overlap among neighbors can allow to maintain the population diversity for
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longer, usually contributing to exploration. Empirical studies are conducted to compare the proposed algorithm with different metaheuristics such as success-history based adaptive differential evolution with linear population size reduction (LSHADE), teaching-learning based optimization algorithm (TLBO), effective butterfly optimizer with covariance matrix adapted retreat phase (EBOwithCMAR), novel dynamic harmony search (NDHS), bat-inspired algorithm (BA), comprehensive learning particle swarm optimizer (CLPSO), evolutionary algorithm based on decomposition (EAD), ring topology PSO (RPSO), crowding-based differential evolution (CDE), neighborhood based crowding differential evolution (NCDE), locally informed
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particle swarm (LIPS), some improved variants of GWO and GWO. Experimental results show that the proposed method performs better than the other algorithms on most benchmarks and engineering problems.
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Keywords: Grey wolf optimizer; Cellular automata; Metaheuristics; Engineering optimization; Global optimization Introduction
Metaheuristics have received widespread attention over the last two decades due to their simplicity, flexibility and
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derivation-free mechanism. A variety of metaheuristics, such as Genetic Algorithm (GA) (Goldberg & Holland, 1988), Particle Swarm Optimization (PSO) (J & R, 1995), Differential Evolution (DE) (Storn & Price, 1997), Cognitive Behavior Optimization Algorithm (CBO) (M. Li, Zhao, Weng, & Han, 2016) and Moth-flame Optimization Algorithm (MFO)
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(Seyedali Mirjalili, 2015), have been proposed and successfully applied in many engineering fields. Metaheuristics can usually be classified into two categories: (1) single solution-based algorithms like Simulated Annealing (SA) (Kirkpatrick, Gelatt, & Vecchi, 1983). It begins with a candidate solution, and then the quality of this candidate solution is improved during
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the search progress. (2) population-based metaheuristics such as Biogeography-Based Optimizer (BBO) (Simon, 2008), Teaching-Learning Based Optimization algorithm (TLBO) (R. V. Rao, Savsani, & Balic, 2012; R. V. Rao, Savsani, & Vakharia, 2012), Grey Wolf Optimizer (GWO) (S. Mirjalili, Mirjalili, & Lewis, 2014), Water Evaporation Optimization (WEO) (Kaveh & Bakhshpoori, 2016), Multi-Verse Optimizer (MVO) (S. Mirjalili, Mirjalili, & Hatamlou, 2016), and Yin-Yang-Pair Optimization (YYPO) (Punnathanam & Kotecha, 2016). The characteristic of population-based metaheuristics is that the optimization search is performed on a set of solutions. Compared with single solution-based metaheuristics, population-based metaheuristics have some advantages as follows:
Corresponding author.
E-mail address: [email protected] (Chao Lu)
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A set of trial solutions can share information about the search space which guides the trial solutions toward the promising areas of the search space.
A set of trial solutions can help each other to avoid local optimum (S. Mirjalili, et al., 2014).
Population-based metaheuristics usually have a greater exploration ability than single solution-based metaheuristics. Although different metaheuristics have various search manners, most of them are based on the common conceptualization
which balances diversification (exploration of the search space) and intensification (exploitation of already found approximate solutions). Thus, exploration and exploitation are two cornerstones of metaheuristics. Exploration is the process of visiting new regions of the search space, whilst exploitation is the process of searching those areas of the search space
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within the neighborhood of previously visited points (C. Lu, Li, Gao, Liao, & Yi, 2017; Matej, Črepinšek, Liu, & Mernik, 2013).
GWO (S. Mirjalili, et al., 2014) is one recently developed population-based algorithm inspired by hunting mechanism of grey wolves. Compared with other population-based algorithms such as PSO and GA, GWO presents a powerful search ability (Chao Lu, Gao, Li, & Xiao, 2017). Some efforts on the GWO have been made in terms of application and theory. From the applicable perspective, GWO has been utilized to address human recognition (Sanchez, Melin, & Castillo, 2017),
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and unmanned combat aerial vehicle path planning (Zhang, Zhou, Li, & Pan, 2016). From a theoretical perspective, Rodriguez et al. (2016; 2017; 2017) proposed a new GWO with a hierarchical operator and an improved GWO with fuzzy logic, respectively. Joshi and Arora (2017) proposed an enhanced grey wolf optimizer with a better hunting mechanism to balance between exploration and exploitation. Heidari and Pahlavani (2017) proposed an efficient GWO where Lévy flight (LF) and greedy selection strategies are integrated with the modified hunting phases of GWO. In GWO, the search process is guided by the three best wolves at each iteration. This search scheme promotes exploitation since all candidate wolves (solutions) are attracted toward the three best wolves, thereby converging faster toward these wolves. However, as a result of
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such a strong exploitation effect, the search diversity would be hampered in a sense. Finally, the GWO is prone to stagnation in local optimum (Chao Lu, Xiao, Li, & Gao, 2016). Meanwhile, according to the previous experiments (Qu, Liang, Wang, Chen, & Suganthan, 2016), many metaheuristics like GWO is not suitable for solving multimodal optimization problems
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where there exist multiple global optimum solutions because it is originally designed to address single global optimization problems. To address the above issues, cellular automata (CA) is embedded into GWO to maintain population diversity and locate multiple optimal peaks. The main motivations for the GWO combined with CA are as follows: (1) CA provides a
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neighborhood structure for GWO. In CA, all the individuals (solutions) are arranged in a toroidal mesh and each individual has its own neighborhood. Due to the isolation by using neighborhoods, the population diversity can be preserved well. (2)
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The shared information from overlap areas between neighborhoods can contribute to exploration. The information in different neighborhoods can transit from one area to another by diffusing good solutions in overlap areas between neighborhoods. Moreover, since individual’s interaction is restricted to its neighborhood, each neighbor has a good local search ability.
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Whereas all the individuals in different neighborhoods can find promising solutions in a parallel search way, which improves global search ability and helps to locate multiple optimal peaks. Therefore, GWO with CA can effectively balance exploitation and exploration. Some efforts on metaheuristics with a topological structure have been made in recent years. For example, Shi et al. (2011) proposed a cellular PSO by incorporating the cellular topological structure into the PSO to improve the performance of PSO. Gao et al. (2012) successfully applied the cellular PSO to parameters optimization of a multi-pass milling process. Yi et al. (2016) proposed a modified harmony search algorithm with a cellular local search to enhance the exploitation capability. Alba and Dorronsoro et al. (2005) developed a cellular GA by combining GA with cellular automata in order to enhance the performance of the basic GA. Li (2010) proposed a simple yet effective niching PSO with a ring neighborhood topology, which showed a more effective performance than some existing PSO niching algorithms. Das et al. (2009) developed an effective DE using a neighborhood-based mutation operator, which is competitive or superior to several existing DEs.
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Piotrowski (2013) extended the work of Das et al. (2009) by considering the additional strategies. Obviously, metaheuristics with a topological structure have a significant improvement in the search ability. CA can provide such a topological structure. Therefore, we propose a hybrid GWO with CA called CGWO for continuous optimization problems. This paper uses three groups of benchmarks and engineering application problems to evaluate the behavior of the proposed CGWO by comparing with other algorithms including success-history based adaptive differential evolution with linear population size reduction (LSHADE) (Tanabe & Fukunaga, 2014), effective butterfly optimizer with covariance matrix adapted retreat phase (EBOwithCMAR) (Kumar, Misra, & Singh, 2017), teaching-learning based optimization algorithm (TLBO) (Črepinšek, Liu, & Mernik, 2012), novel dynamic harmony search (NDHS) (J. Chen, Pan, & Li, 2012), bat-inspired algorithm (BA) (Yang & Gandomi, 2012), comprehensive learning particle swarm optimizer (CLPSO) (Liang, Qin,
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Suganthan, & Baskar, 2006), evolutionary algorithm based on decomposition (EAD) (Gu, Cheung, & Luo, 2015), ring topology PSO (RPSO) (X. D. Li, 2010), crowding-based differential evolution (CDE) (Thomsen, 2004), neighborhood based crowding differential evolution (NCDE) (Qu, Suganthan, & Liang, 2012), locally informed particle swarm (LIPS) (Qu, et al., 2016), island-based harmony search (iHS) (Al-Betar, Awadallah, Khader, & Abdalkareem, 2015), some improved versions of GWO (Joshi & Arora, 2017; Malik, Mohideen, & Ali, 2016; Yu, Liu, Wang, & Gao, 2017), and grey wolf optimizer (GWO) (S. Mirjalili, et al., 2014). Experimental results show the CGWO can improve the performance of the basic
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GWO.
Additionally, there are three crucial differences between our proposal and the previous researches on optimization approaches with a special topological structure.
In general, various cellular structures have different impacts on the behavior of the algorithms. Thus, this paper incorporates six different cellular structures into the GWO to test the performance of algorithms with different structures. The most appropriate one is chosen as our proposed cellular GWO as in Section 4.3. Actually, metaheuristics with the CA are a kind of fine-grained structure algorithms. In this case of the algorithm, the
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population is divided into many subpopulations that arranged in a given structure. Then, the search process is performed
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within the neighborhood of the current individual. Metaheuristics with CA are similar to niching metaheuristics, but traditional niching methods usually require extra parameters. One important advantage of the proposed CGWO algorithm is that there is no need to specify any parameters.
The previous cellular algorithms were primarily adopted to locate a single global optimum, rather than multiple global
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optima. However, the proposed CGWO can locate all global optima in Section 4.4.2.
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The rest of this paper is organized as follows. Firstly, a brief overview of the basic GWO algorithm and cellular automata (CA) is given in Section 2. The detailed presentation of the CGWO approach is provided in Section 3. Section 4 gives the experiment and results. Section 5 presents the conclusions and future work. Overview of GWO and CA
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2.
Before we introduce our proposed CGWO algorithm, we give an overview of the basic GWO and CA, respectively. 2.1 GWO
In this subsection, we describe GWO. GWO is a new metaheuristic inspired by the grey wolf hunting for the prey. The main steps of GWO are provided in the following subsections (S. Mirjalili, et al., 2014).
2.1.1 Social hierarchy To establish a social hierarchy of wolves, all the grey wolves are classified into four kinds of wolf according to the fitness value. The best wolf (solution) in GWO is denoted as the alpha (α). Similarly, the second and third best wolves are called beta
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(β) and delta (δ) respectively. The rest of wolves are considered to be omega (ω). In GWO the search process is mainly guided by α, β and δ. The ω wolves obey these three wolves (S. Mirjalili, et al., 2014). 2.1.2 Encircling prey Grey wolves encircle the prey when the hunt. To mathematically model encircling behavior, the equation is defined as follows:
𝑫 = 𝑪 ∘ 𝑿𝑝 (𝑡) − 𝑿(𝑡) 𝑿(𝑡 + 1) = 𝑿𝑝 (𝑡) − 𝑨 ∘ 𝑫
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(1) (2)
where 𝑡 is the current generation number, ∘ is the hadamard product operation, 𝑿𝑝 and 𝑿 represents the position vector of the prey and a grey wolf. The vectors 𝑨 and 𝑪 are formulated as follows (S. Mirjalili, et al., 2014): 𝑨 = 2𝒂 ∘ 𝒓1 − 𝒂 𝑪 = 2𝒓2
(3) (4)
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Where the value of 𝒂 is linearly decreased from 2 to 0 during the optimization process, and it is used to emphasize exploration and exploitation respectively. 𝒓1 and 𝒓2 are random vectors in range [0,1]. 2.1.3 Hunting
The hunt is often guided by the alpha. The beta and delta might also join in the hunting. However, the position of prey (optimum) is unknown in an abstract search space. To simulate the hunting behavior of grey wolves, assume that the alpha, beta, and delta have better knowledge about the potential location of the prey. Therefore, the three best solutions found so far
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are maintained and guide the other wolves towards the potential location of the prey. The equation of hunting is calculated as follows (S. Mirjalili, et al., 2014):
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𝑫𝛼 = 𝑪1 ∘ 𝑿𝛼 − 𝑿, 𝑫𝛽 = 𝑪2 ∘ 𝑿𝛽 − 𝑿, 𝑫𝛿 = 𝑪3 ∘ 𝑿𝛿 − 𝑿 𝑿1 = 𝑿𝛼 − 𝑨1 ∘ 𝑫𝛼 , 𝑿2 = 𝑿𝛽 − 𝑨2 ∘ 𝑫𝛽 , 𝑿3 = 𝑿𝛿 − 𝑨3 ∘ 𝑫𝛿
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𝑿(𝑡 + 1) =
𝑿1 +𝑿2 +𝑿3 3
(5) (6) (7)
Fig. 1 presents a search process of updating a candidate’s position based on alpha, beta, and delta in 2-D search space. It
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can be observed that the three best wolves (alpha, beta, and delta) can estimate the position of the prey, and other wolves
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update the positions randomly around the prey.
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α
α wolf δ
β wolf
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the prey
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δ wolf
move β
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Position of candidate wolf
Fig. 1. Position updating in GWO
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2.1.4 Attacking prey
To mathematically model attacking the prey, the value of a vector 𝒂 is decreased. 𝑨 is also decreased by 𝒂. When random values of 𝑨 are in [-1, 1], the next position of a search agent can be in any position between its current position and
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the position of the prey. With the operators proposed so far, the GWO algorithm allows its search agents to update their positions based on the α, β, δ, and ω; and attack toward the prey. However, the GWO algorithm tends to premature with these
2.1.5 Search for prey
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operators (S. Mirjalili, et al., 2014).
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As stated previously, the search direction of grey wolves mainly depends on the position of the alpha, beta, and delta. They diverge from each other to search for prey and converge to attack prey. To simulate divergence, 𝑨 is used to oblige the search agent to diverge from the prey. This emphasizes exploration and allows the GWO algorithm to search globally.
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Another element of GWO that favors exploration is 𝑪 in Eq.(4). The random value of 𝑪 is in [0, 2]. It provides random weights for prey to stochastically emphasize or deemphasize the effect of prey in defining the distance in Eq.(1), which can help to improve exploration and avoid local optimum. It is noted that 𝑪 is not linearly decreased in relation to 𝑨 but random values over the course of iteration which is helpful in case of local optima stagnation in the latter phase of search.
2.2 CA In this subsection, we describe CA. CA has become a popular tool in the scientific research since the concept of CA was first proposed by Von Neumann and Ulam (Neumann, 1966). One of the most well-known CA rules, the “game of life” was conceived by Conway in the late 1960s (E. R. Berlekamp, J. H. Conway, & Guy, 1982). Furthermore, CA has also been widely applied in a variety of fields such as physics, biology, computer science and traffic. CA is a set of cells distributed in a
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special topological structure. Each cell in the topological structure has its own state, which is associated with its surrounding cells in discrete time steps. The state of each cell at the next time step is determined by the current states of a surrounding neighborhood of cells. In general, CA consists of five key elements as follows: cell, cell state, cell space, neighborhood, and transition rule. As previously stated, cell state is information about the current cell which can determine the next state of the cell. Cell space represents a set of cells. It often has one-dimensional, two-dimensional, and three-dimensional space structure in real applications. It is noted that the boundary of cell space needs to be defined since we usually simulate real-world systems by finite grids. The boundary is usually a ring grid. More precisely, the left boundary connects to the right boundary and the top boundary connects to the bottom boundary. Neighborhood is a set of cells surrounding a center cell. It plays a states of its neighborhood. Some definitions of CA can be described as follows.
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critical role in the next state of a center cell. Transition rule is used to determine the next state of a given cell based on the A d-dimensional CA contains a d-dimensional grid of cells, each of which can take on a value. The cells update their states automatically according to a given rule. Formally, a cellular automaton 𝑸 is a quadruple as follows: 𝑸 = (𝑆, 𝐺, 𝑑, 𝑓) where 𝑆 is a finite set of states, 𝐺 is the neighborhood, 𝑑 ∈ 𝑍
+
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is the dimension of 𝑸, and 𝑓 is the interaction rule, also
referred to as the transition rule.
Given the position of a cell 𝑖 ∈ 𝑍 𝑑 in a d-dimensional lattice or grid, its neighborhood 𝐺 is defined by:
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𝐺𝑖 = (𝑖, 𝑖 + 𝒓1 , ⋯ , 𝑖 + 𝒓𝑛 )
(9)
where 𝑛 represents the neighborhood size, and 𝒓𝑗 is a fixed vector in the d-dimensional space. Six kinds of neighborhood shapes in CA are presented in Fig. 2. Please note the names of these neighborhoods: the label Ln represents neighborhoods composed of the n nearest neighbors in a given axial direction (north, south, west and east), while the label Cn (compact) denotes the neighborhoods consisting of the n-1 nearest cell to the center one (in horizontal, vertical, and diagonal directions). Two most commonly used neighborhoods are called Von Neumann neighborhood (C5) and Moore neighborhood (C9),
Fig. 2. Structure of neighborhood
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respectively.
Transition rule can determine the next state of a given cell according to its current state and states of its neighbors. The transition rule 𝑓 is written as follow: 𝑓: 𝑆𝑛 → 𝑆
(10)
This transition rule maps the state 𝑠𝑖 ∈ 𝑆 of a given cell 𝑖 into another state from the set 𝑆, as a function of the states of the cells in the neighborhood 𝐺𝑖 . In addition, some features of CA are stated as follows: (1) Homogeneity and regularity: homogeneity means the change of each cell in cell space obeys the same rule (i.e., CA rule or called transformation function). However, regularity indicates the identical distribution pattern, size, shape as
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well as the orderly distribution regulation of all the cells. (2) Space discretization: space discretization denotes that cells are arranged in a discrete cell grids defined by specific rules. (3) Synchronous calculation (parallelism): the state change of each cell at time step t+1 is independent behavior. If the configuration change of CA is considered as computation or processing of data or information, the disposing process of CA will be synchronous, which is suitable for parallel computation.
3.
The proposed CGWO
3.1
The framework of the CGWO algorithm
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In this section, we first give an overview of the proposed algorithm, and then explain the improvement strategy in detail.
The goal of this work is to develop a cellular GWO called CGWO for continuous optimization problems. A flow chart of the proposed CGWO is shown in Fig. 3. One crucial strategy is that CA is utilized to improve the performance of the GWO. The improvement strategy is motivated by the following three advantages.
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(1) CA can help to improve the local search ability since cell in CA only interacts with its neighbors for exploitation. Meanwhile, information diffusion mechanism contributes to exploration. Finally, a collection of such cells congregates to solve the problem.
(2) We can take the advantages of the two approaches. The candidate solutions are attracted toward the good solutions in GWO, therefore, the convergence speed of the GWO is very fast but it is also easy to trap into local optima. While CA provides slow diffusion through the population regarding the good solutions of each neighborhood. Subsequently, the attraction to the good points is weaker, which avoids local optima. Thus, we can integrate these
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different techniques to strengthen their advantages and make up the respective shortcomings.
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(3) In addition, CA and GWO are easy to be implemented due to their simple mechanism. The proposed CGWO algorithm incorporates the concept of CA. It consists of five steps summarized as follows.
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Input. SOP (1);
A stopping condition;
N: size of wolves;
NS: neighborhood size;
Input other parameters;
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Output. The optimal (or near-optimal) results found so far. Step 1) Initialization. Generate an initial population (wolves) of N solutions X1,…,XN ∈ Ω. Step 2) Evaluation. Once the wolf population is initialized, compute the fitness value of the solutions (wolves). Step 3) Create the neighborhood. For i=1,…,N, define a set of indexes Bi={i(1),…,i(NS)}, where {Xi(1),…, Xi(NS)} are the wolves with size NS closest to xi in the topological structure. Step 4) Stopping condition. If the stopping criterion is satisfied, then output the alpha wolf (the optimal solution found so far). Otherwise, go to step 5. Step 5) Update. Step 5.1) Selection. First, compute the fitness of solutions, and then select the three best fitness solutions from the current neighbors to guide other solutions within neighborhood to perform the search optimization.
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Step 5.2) Hunter operation. A new solution is generated from three rules. (a) encircling prey, (b) hunting and (c) attacking, as described in Section 2.1. Note that the search optimization is only implemented in its neighborhood. Step 5.3) Replacement strategy. The replacement method is based on fitness values of solutions. In detail, replace the current solution if the current solution is worse than the new solution after updating, and vice versa. Note that we have incorporated a constraint handling mechanism in CGWO when dealing with constrained problems. This mechanism is the same as in (Deb, 2000). Overlap area
yes Stop condition is met?
no
Evaluate function
α β δ
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Select the first best three wolves from its current neighbors
end
Search the prey by the above three wolves
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Update position of wolves
the current wolf is replaced if the position of the wolf is worse than that of the current one after update
Wolves and the positions
Fig. 3. Flow chart of the proposed CGWO algorithm
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The proposed CGWO is discussed in the following parts. In CGWO, all information inherits inside the cell. Six typical cell structures are employed to construct cell space in Fig. 2. One wolf (solution) is defined as a cell, and all cells represent
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wolves in CGWO. Consequently, the size of cell structure is the same with that of the wolves. In the initialization phase, grey wolf population is randomly created; subsequently, each wolf is randomly assigned to one grid of the lattice structure one by one. Note that the index of each wolf in the lattice structure remains unchanged during the search process. Neighborhood
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could be defined based on the lattice structures. Taking C9 structure in Fig. 3 as an example, suppose that wolf size is 49, and they are randomly generated in the initialization, then 49 grids are created to construct a lattice structure. Each wolf is assigned to each cell. The gray cells surrounding around the current cell are its neighbors, and deep gray grids represent the
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overlapping neighbors belonging to two consecutive cell structures. The interaction among cells is restricted to the neighborhood. This could help each cell to perform exploitation inside their neighbors. Meanwhile, the overlapping neighbors provide a migration mechanism from one neighborhood to another one. It could enable each cell information to diffuse to the
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whole wolves, which is favorable for exploring the search space (Shi, et al., 2011). In CGWO, the center cell state is determined by its surrounding neighbors. The state of neighbors’ best position 𝑋𝑖 in the current wolf is denoted by 𝑆𝑖𝑡 (𝑋𝑛 ). For simplification, the transition rule is defined as: 𝑆𝑖𝑡+1 (𝑋𝑛 ) = 𝑓 (𝑆𝑖𝑡 (𝑋𝑖 ), 𝑆𝑖𝑡 (𝑋𝑖+𝒓1 ), ⋯ , 𝑆𝑖𝑡 (𝑋𝑖+𝒓𝑛 )) = 𝑚𝑖𝑛 (𝑓𝑖𝑡𝑛𝑒𝑠𝑠(𝑆𝑖𝑡 (𝑋𝑖 )), 𝑓𝑖𝑡𝑛𝑒𝑠𝑠 (𝑆𝑖𝑡 (𝑋𝑖+𝒓1 )) ⋯ , 𝑓𝑖𝑡𝑛𝑒𝑠𝑠 (𝑆𝑖𝑡 (𝑋𝑖+𝒓𝑛 ))) (11) where 𝑓𝑖𝑡𝑛𝑒𝑠𝑠(𝑆𝑖𝑡 (𝑋𝑖 )) is the fitness of the current solution 𝑋𝑖 . Eq. (11) defines that the neighbor with the best fitness value is chosen for updating cell state.
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CA of the CGWO algorithm
Two important factors affect the performance of metaheuristics when solving complex problems: (1) communication structures. (2) information inheriting and diffusing scheme. As stated above, CA has also been successfully applied in many fields due to its unique topology structures and communication interaction. Although CA and GWO stem from different domains, some similar characteristics of the two techniques can be observed. First, both of them contain a set of individuals, which are called cells in CA and wolves in GWO. Second, every individual interacts with each other to transmit certain intrinsic information. In CA, each cell communicates with its neighbors and updates its cell state. Similarly, each wolf communicates with other wolves to update such intrinsic information as position,
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fitness, α, β and δ wolf in GWO. Third, a transition rule is used to manage the evolution of cells in CA, while some search operators are used to update information of wolf in GWO. Finally, CA and GWO both run in a discrete time step. To enhance the behavior of the interaction, we utilize the idea of CA with the communication structure to investigate information diffusing mechanism of GWO. Wolves can only interact with their neighbors in the CA. It could help to exploit every cell’s local information inside the neighborhood and explore the search space due to the slow information diffusion
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through the entire wolf population. Based on the view above, we introduce the concept of CA into the GWO to enhance the performance of GWO. The CA model for GWO is defined as follows: (a) cell: A candidate wolf or solution. (b) cell space: A set of all solutions.
(c) cell state: The wolf’s information such as the neighbor’s best position Xn at time t. It can be denoted by 𝑆𝑖𝑡 . (d) neighborhood: A set of solutions surrounding the current solution.
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(e) transition rule: According to fitness value, if the new position of the current wolf is better than the old one of the current wolf before update, the current wolf moves to the new position and the information of wolf is also updated.
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Then, the local best wolf is chosen from the neighborhood, and guides its neighbors converge to its local optimal areas.
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(f) discrete time step: Iteration in GWO.
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Based on the above definitions, the pseudo code of CGWO is given in Fig. 4.
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1. Input Parameters(a); // Parameters of the CGWO 2. wolf population Xi(i=1,...N) initialization(wolves_Size); 3. evaluate (wolf population) ; 4. while (evaluation_number < max_evaluation_number) 5. for i 1 to wolves_Size 6. neighbors calculateNeighborhood(wolves(i)); 7. Xα, Xβ, Xδ select the first best three wolves(neighbors); 8. new position of the wolf update(the current wolf);// update the position based on Eq.(7) 9. update(a, A and C); 10. evaluate (new position of the wolf) ; 11. if new position of the wolf is better than that of the current wolf 12. replacement(new position, position of the current wolf); 13. end if 14. evaluation_number++; 15. end for 16. end while 17. Xα select the best wolf(the entire wolves); 18. return Xα Fig. 4. Pseudo code of the proposed CGWO algorithm
Fig. 4. shows the pseudo code of the proposed CGWO algorithm. The detailed steps of the CGWO are as follows. Firstly, input the parameters like wolf population size and neighborhood structure. The initial population is usually composed of randomly generated wolves in the line 2 of Fig.4. The population is distributed in n (n=2 in this paper) dimensional lattice grid according to the indices of the population members (as obtained during initialization). After initialization, the fitness
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values of wolf population are calculated as in the line 3, and then the CGWO algorithm starts an update loop. This loop step is to generate new and promising wolf population through update operators as mentioned in Section 2.1. The first best three wolves (alpha (α), beta (β) and delta (δ)) from the current wolf’s neighbors are selected out in the line 7 of Fig.4 due to
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adoption of CA. Fig. 3 also presents the process of the update loop of an individual in CGWO (the considered neighborhood is Moore neighborhood in Fig. 3). Note that the overlap among the neighborhoods provides an implicit migration mechanism to the CGWO. Since the best solutions move smoothly through the whole population, diversity in the population is kept
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longer than that in non-structured GWO. This soft diffusion of the best solution through the population is one of the main reasons of the good balance between exploration and exploitation. Another important character of the CGWO is that these
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update operators are limited within the neighborhood of the current individual, therefore individuals belonging to other neighborhoods are not allowed to interact. Fig. 5 further illustrates the search process of a candidate wolf in wolf population (population size = 57). The solution
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indices are sorted only randomly in order to maintain the diversity of population. The solutions are located in the lattice gird with respect to their indices and each solution has its own neighborhood. For a candidate solution 𝑋⃗3,4 with index (3, 4), its Moore structure neighbors contain solutions {𝑋⃗2,3 , 𝑋⃗2,4 , 𝑋⃗2,5 , 𝑋⃗3,3 , 𝑋⃗3,5, 𝑋⃗4,3, 𝑋⃗4,4 , 𝑋⃗4,5 } besides itself. The candidate solution or wolf is guided by the three best wolves found so far in its neighbors but not in the entire population. The circle represents the range of the current wolf’s neighbors and the wolves only communicate with each other in its neighborhood. In other words, the whole population is divided into many subpopulations and the update operation is performed independently on a set of subpopulations. But these independent subpopulations can occasionally transmit information by overlap areas among neighborhoods. The overlap area, containing the solutions 𝑋⃗3,3 and 𝑋⃗4,3 , is constructed by the neighborhoods of 𝑋⃗3,4 and 𝑋⃗4,2 . Since the search process among neighborhoods is based on sequence, there is a delay in the information spread through the population regarding the best position of each neighborhood. Therefore, the attraction towards good solutions is weaker, which prevents the population from getting trapped in local minima. If stop condition is satisfied, the algorithm returns the
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1
2
3
4
5
6
7
α 1
δ 2
α wolf
C3 β wolf
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3
move
δ wolf
β C2
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4
5
Overlap area
the prey
Circle represents the range of the neighborhood of the current wolf
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Neighborhood of
Fig. 5. Position updating in CGWO
4.
Experiments
This section is devoted to measuring the performance of the proposed CGWO. In this section the experimental studies
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contain the following five aspects:
The best choice of neighborhoods (six kinds of neighborhoods are shown in Fig. 2) in Section 4.3.
2.
Performance comparisons with other metaheuristics on benchmarks in Section 4.4.
3.
Statistical test on results obtained by different algorithms in Section 4.5.
4.
CPU-time cost study on CGWO and GWO in Section 4.6.
5.
Real-world applications of the CGWO in engineering optimization in Section 4.7.
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1.
In the following subsections, the benchmarks and parameter settings are described at first, and then the experimental
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studies are further investigated step by step.
4.1 Benchmark functions
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Three test suites are used to evaluate the success of CGWO and the comparison algorithms. Table 1 records the common unimodal functions (Yao, Liu, & Lin, 1999). Table 2 lists the common multimodal functions (Yao, et al., 1999). Table 3 involves 15 benchmark instances used in CEC2015. Details about these problems are provided in technical report (Qu, Liang, Suganthan, & Chen, 2015). Table 1 Unimodal benchmark functions Name Sphere Schwefel’s problem 2.22
Test Functions F1 x
n
xi
[-100,100]
i 1
F2 x
n
i 1
n
xi
f min
S 2
xi i 1
30
[-10,10]30
0 0
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F3 x
Schwefel’s problem 1.2 F4 x
Schwefel’s problem 2.21
F5 x
Rosenbrock
n
i
100x n 1
F6 x
Step
xi2
i 1
i 1
x i 1
f 7 x
Noise
ix i 1
x
1 2
i
0.5
n
n
2
2
i n
2
i
0
[-100,100]30
0
[-30,30]30
0
[-100,100]
random 0, 1
4 i
[-100,100]30
30
[-1.28,1.28]30
0 0
Table 2 Multimodal benchmark functions Test Functions i 1
x
F9 x
Rastrigin
n
F12 x
i 1 n
xi2
i 1
xi 1 i
n
i
u( x ,10,100,4) i
F13 x 0.1sin 2 3x1
n
u( x ,5,100,4) i
i 1
i
25
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11
j 1
ai i
1
Shekel 2
0
[-50,50]30
0
[-50,50]30
0
2 j i 1 ( xi a ij )
[-65,65]2
1
[-5,5]4
0.0003
-1
1
x 1 bi2 bi x 2 bi2 bi x 3 x 4
2
1 6 x1 x1 x2 4 x22 4 x24 [-5,5]2 3 2 5.1 2 5 1 cos x1 10 F17 x x 2 x1 x1 6 101 2 [-5,10], [0,15] 8 4
27 x )
F18 x 1 ( x1 x2 1)2(19 14 x1 3 x12 14 x2 6 x1 x2 3 x22 )
30 (2 x1 3 x2 ) (18 32 x1 12 x 48 x2 36 x1 x2 2
2 1
4 F19 x - ci exp i 1 4 F20 x - ci exp i 1
3
j 1
ij
j
x - X - a X - a x - X - a X - a
c c
F21 x - X - ai X - ai ci T
i 1
7
F22
T
i
i
7
F23
i 1
2 2
2 pij 6 2 a x p ij j ij j 1
a x
5
i 1
Shekel 3
[-600,600]30
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Shekel 1
0
F16 x 4 x12 2.1x14
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Hartman 2
[-32,32]30
1)2 1 sin 2(3xi 1) ( xi 1)2 1 sin 2(2xn )
n
(x
1 F14 x 500
Kowalik’s Function
Hartman 1
0
k( xi a) xi a u xi , a, k , m 0 a xi a m xi a k( xi a)
Shekel’s Foxholes Function
Goldstein-Price Function
[-5.12,5.12]30
m
i 1
Six-hump camel back
-12569.5
1)2 1 10 sin 2(yi 1 ) ( yi 1)2
( y
n
i 1
cos i 1
n 1 i 1
i 1
2 i
20 e cos2xi
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Generalized Penalized Function 2
1 4000
10 sin y1 n
Generalized Penalized Function 1
n
n
x
[-500,500]30
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F11 x
Griewank
xi
1 F10 x -20 exp 0.2 n 1 exp n
f min
S
10 cos2xi 10
2 i
i 1
Ackley
Branin
F8 x - xi sin n
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Name Generalized Schwefel’s problem
T
i
i
-1.0316 0.39788
[-2,2]2
3
[0,1]3
-3.86
[0,1]3
-3.32
[0,10]4
-10.1532
[0,10]4
-10.4028
[0,10]4
-10.5363
1
1
i
1
i
Table 3 CEC2015 benchmark functions Name Shifted and Rotated Expanded two-Peak Trap
Test Functions
F24( x) f1( x) f M 1( x o1 ) f f = Expanded Two-Peak Trap function
* 1
S
f min
[-100,100]10
100
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Shifted and Rotated Expanded Five-Uneven-Peak Trap
[-100,100]8
200
[-100,100]4
300
[-100,100]10
400
[-100,100]4
500
f = Expanded Five-Uneven-Peak Trap
x o3 f 3* F26( x) f 3( x) f M 3 20
Shifted and Rotated Expanded Equal Minima
f = Expanded Equal Minima
x o4 F27( x) f 4( x) f M 4 20
Shifted and Rotated Expanded Decreasing Minima
f 4*
f = Expanded Decreasing Minima
f = Expanded Uneven Minima
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x o5 f 5* F28( x) f 5( x) f M 5 20
Shifted and Rotated Expanded Uneven Minima
x o6 f 6* F29( x) f 6( x) f M 6 5
Shifted and Rotated Expanded Himmelblau’s Function
f = Expanded Himmelblau’s Function
F30( x) f 7( x) f M 7 x o7 f7*
Shifted and Rotated Expanded Six-Hump Camel Back
f = Expanded Six-Hump Camel Back
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x o8 f 8* F31( x) f 8( x) f M 8 5
Shifted and Rotated Modified Vincent function
[-100,100]8
600
[-100,100]10
700
[-100,100]4
800
[-100,100]10
900
[-100,100]10
1000
[-100,100]10
1100
f = Modified Vincent function
N
F32( x) f 9( x)
i 1
i
N 10
* i g i( x) bias i f 9*
[10,20,10,20,10,20,10,20,10,20] [1,1,1e 6, ,1e 4, ,1e 5,1e 5] bias [0, ,0] f 9*
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Composition Function 1
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g1 2( x): Rotated Sphere Function g3 4( x): Rotated High Conditioned Elliptic Function g5 6( x): Rotated Bent Cigar Function g7 8( x): Rotated Discus Function g910(x): Rotated Different Powers Function
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F33( x) f10( x)
N
i 1
i
* i g i( x) bias i f10*
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N 10 [10,20,30,40,50,60,70,80,90,100]
[1e 5,1e - 5,1e 6, ,1e 4,1,1]
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Composition Function 2
bias [0,10,20,30,40,50,60,70,80,90] f10*
g1 2( x): Rotated High Conditioned Elliptic Function g3 4( x): Rotated Different Powers Function g5 6( x): Rotated Bent Cigar Function g7 8( x): Rotated Discus Function g910(x): Rotated Sphere Function N
F34( x) f11( x)
* g (x) bias f i
i i
i
* 11
i 1
Composition Function 3
N 10 [10, ,10] [0.1,0.1,10,10,10,10,1e 3,1e - 3,1,1] bias [0,0,0,0,0,0,0,0,0,0] f11*
g12( x): Rotated Rosenbrock’s Function
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F35( x) f12( x)
N
i 1
i
* i g i( x) bias i f12*
N 10 [10,10,20,20,30,30,40,40,50,50]
[0.1,0.1,10,10,10,10,1e 3,1e - 3,1,1] bias [0,0,0,0,0,0,0,0,0,0] f12* g1 2( x): Rotated Rosenbrock’s Function g3 4( x): Rotated Rastrigin’s Function g5 6( x): Rotated HappyCat Function g7 8( x): Rotated Scaffer’s F6 Function g910(x): Rotated Expanded Modified Schwefel’s Function
i 1
1200
* i g i( x) bias i f13*
N
i
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F36( x) f13( x)
[-100,100]10
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Composition Function 4
N 10 [10,20,30,40,50,60,70,80,90,100]
[0.1,10,10,0.1,2.5,1e 3,1e - 3,1e - 3,2.5,10] bias [0,0,0,0,0,0,0,0,0,0] f13*
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[-100,100]10
1300
[-100,100]10
1400
[-100,100]10
1500
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Composition Function 5
g 1(x ): Rotated Rosenbrock’s Function g2( x): Rotated HGBat Function g3( x): Rotated Rastrigin’s Function g4(x): Rotated Ackley’s Function g5( x): Rotated Weierstrass Function g6( x): Rotated Katsuura Function g7( x): Rotated Scaffer’s F6 Function g8( x): Rotated Expanded Griewank’s plus Rosenbrock’s Function g9( x): Rotated HappyCat Function g10( x): Rotated Expanded Modified Schwefel’s Function
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F37( x) f14( x)
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Composition Function 6
N
i 1
* i g i( x) bias i f14*
N 10 [10,10,20,20,30,30,40,40,50,50]
[10,1,10,1,10,1,10,1,10,1]
bias [0,20,40,60,80,100,120,140,160,180] f14* g1,3,5,7,9( x): Rotated Rastrigin’s Function
g2,4,6,8,10( x): Rotated Expanded Modified Schwefel’s Function
F38( x) f15( x)
N
i 1
Composition Function 7
i
i
* i g i( x) bias i f15*
N 10 [10,20,30,40,50,60,70,80,90,100]
[0.1,10,10,0.1,2.5,1e - 3,1e 3,1e 3,2.5,10] bias [0,0,0,0,0,0,0,0,0,0] f15*
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g1( x): Rotated Rosenbrock’s Function g2( x): Rotated HGBat Function g3( x): Rotated Rastrigin’s Function g4(x): Rotated Ackley’s Function g5( x): Rotated Weierstrass Function g6( x): Rotated Katsuura Function g7( x): Rotated Scaffer’s F6 Function g8( x): Rotated Expanded Griewank’s plus Rosenbrock’s Function g 9(x): Rotated HappyCat Function g10( x): Rotated Expanded Modified Schwefel’s Function
4.2 Parameter settings
To make a fair comparison, all algorithms are implemented in java on jMetal software (Durillo & Nebro, 2011). Experimental studies are conducted on Intel Core i5-4210U CPU @1.70GHz, 4GB RAM, with Microsoft Windows 8 operating system. Table 4 presents the initial values of the relevant control parameters for the different metaheuristics
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including LSHADE, TLBO, EBOwithCMAR, NDHS, BA, CLPSO, GWO, and CGWO. The parameter settings of these metaheuristics are obtained from the original literature. The relevant common parameter settings for the algorithms are as follows:
The population size is 30 (except for LSHADE and EBOwithCMAR).
The maximal number of function evaluation (NFEmax) is 100,000.
30 independent runs are conducted for each algorithm on each test problem.
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Table 4 Parameter setting of algorithms Control parameters =1.8; 𝑟 𝑎𝑟𝑐 = 2.6; 𝑝=0.11; 𝐻=6 No control parameters 𝑃𝑆1,𝑚𝑎𝑥 =18D, 𝑃𝑆1,𝑚𝑖𝑛 =4, 𝑃𝑆2,𝑚𝑎𝑥 =46.8D, 𝑃𝑆2,𝑚𝑖𝑛 =10, 𝐻=6, 𝑃𝑆3 =4+(3log(D)), 𝑝𝑟𝑜𝑏𝑙𝑠 =0.1, 𝑐𝑓𝑒𝑙𝑠 =0.25* NFEmax HMCR=0.99, PAR_max=0.99, PAR_min = 0.01, tournament size =2 Loudness: A=0.25, plus rate: r=0.5, minimum frequency: Qmin=0, maximum frequency: Qmax=2
CLPSO
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𝑖𝑛𝑖𝑡
c= 1.49445; m=0; 𝑝𝑐 = 0.5
𝑒 𝑡 −𝑒 𝑡(1) 𝑒 𝑡(𝑝𝑠 )−𝑒 𝑡(1)
where, 𝑡 = 0.5 × (0:
1 𝑝𝑠 −1
: 1)
a decreases linearly from 2 to 0 Neighborhood style:C25, a decreases linearly from 2 to 0
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GWO CGWO
𝑟𝑁
PT
Algorithms LSHADE TLBO EBOwithCMAR NDHS BA
4.3 The performance of CGWO with different neighborhood structures
AC
In this subsection we generate six variations of the CGWO by adopting six different neighborhood structures in Fig. 2 and compare the performance of each variation on the benchmarks from Tables 1-2. Tables 5-6 show the statistical results (best, worst, mean, and standard deviation values) on different variations over all runs. We must point out that the marks L9, C9, C13, C21, and C24 in the Tables 5-6 are short for the six corresponding neighborhoods in the Section 2.2. More precisely, CGWO (L9) denotes CGWO with L9 neighborhood. Similarly, CGWO (C9) represents CGWO using C9 neighborhood. From Table 5, it can be observed that CGWO (C25) has a better performance for the unimodal benchmarks except F7 with comparison to the other versions of CGWO since CGWO (C25) obtains the best mean values on 6 out of 7 unimodal benchmarks. For multimodal benchmarks in Table 6, it is also found that none of the neighborhood structures has a dominant performance for all test problems, which implies the neighborhood size is sensitive to the behavior of the CGWO and the setting of neighborhood size depends on specific problems. However, on the whole, CGWO with C25 are slightly better than its other versions on these multimodal benchmarks. Consequently, C25 is regarded as an appropriate neighborhood structure
ACCEPTED MANUSCRIPT
to form the best CGWO that will be studied in more detail in the following subsections. Table 5 Results by CGWO with different neighborhoods on unimodal benchmark functions Problem
Statistics
CGWO(L5)
CGWO(L9)
CGWO(C9)
CGWO(C13)
CGWO(C21)
CGWO(C25)
Best
1.2E-171
6.4E-210
0
0
0
0
Worse
3.2E-165
3.8E-204
0
0
0
0
Mean
1.9E-166
2.8E-205
0
0
0
0
0
0
F1 0
0
0
0
1.02E-96
3.0E-120
1.5E-277
3.5E-239
2.4E-306
0
Worse
4.57E-95
1E-116
9.4E-271
4.1E-234
6.6E-299
0
Mean
1.76E-95
1.7E-117
5.7E-272
4.6E-235
3.5E-300
0
std
1.50E-95
2.5E-117
0
0
Best
1.83E-14
7.59E-23
4.37E-61
2.68E-47
8.42E-65
3.56E-74
Worse
7.66E-09
4.34E-14
9.75E-43
9.9E-32
2.25E-45
6.21E-55
Mean
5.69E-10
1.82E-15
4.41E-44
4.16E-33
7.79E-47
4.18E-56
std
1.59E-09
7.91E-15
1.82E-43
1.81E-32
4.11E-46
1.31E-55
Best
7.26E-25
1.02E-33
2.72E-70
3.62E-73
Worse
3.95E-22
5.95E-31
Mean
3.23E-23
1.3E-31
std
7.22E-23
1.7E-31
Best
25.05527
24.81441
Worse
27.08967
27.00348
Mean
25.89631
25.43051
std
4.91E-01
Best
0
F2
1.08E-61
1.85E-51
4.6E-64
5.17E-65
7.75E-63
1.46E-52
2.63E-65
4.38E-66
2.32E-62
3.79E-52
8.58E-65
1.32E-65
24.24317
24.86651
24.66267
24.26925
27.06747
27.044
28.71735
26.2198
25.64893
25.46054
25.55428
25.42201
5.30E-01
7.68E-01
5.63E-01
8.03E-01
5.26E-01
0
0
0
0
0
ED
Mean
Best Worse F7
CE
Mean std
M
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
PT
std
0
2.53E-58
F5
Worse
0
3.46E-67
F4
F6
AN US
F3
CR IP T
std Best
1.34E-04
7.73E-05
1.56E-04
9.66E-05
1.21E-04
1.21E-04
7.01E-04
7.52E-04
9.21E-04
6.93E-04
9.25E-04
8.07E-04
3.33E-04
3.84E-04
4.48E-04
3.69E-04
4.40E-04
4.30E-04
1.32E-04
1.82E-04
1.86E-04
1.41E-04
1.92E-04
1.83E-04
AC
Table 6 Results by CGWO with different neighborhood on multimodal benchmark functions
Problem
Statistics
CGWO(L5)
CGWO(L9)
CGWO(C9)
CGWO(C13)
CGWO(C21)
CGWO(C25)
Best
-5704.235
-7359.57
-7447.62
-7066.52
-6794.18
-6702.907
Worse
-3131.247
-3349.67
-3386.02
-3255.19
-3330.91
-3256.599
Mean
-3591.132
-4319.03
-4772.63
-4956.15
-4685.34
-5074.777 1226.6235
F8
std
442.26618
1104.816
1288.144
1261.855
1274.428
Best
0
0
0
0
0
Worse
15.40262
4.64472
11.30515
7.471893
5.27715
16.699364
Mean
1.3744344
0.292568
1.968907
1.159881
2.873349
4.2002956
std
3.5298336
0.974567
3.190097
2.315817
4.413376
5.4555579
Best
3.997E-15
4E-15
4E-15
4E-15
4E-15
4E-15
0
F9
F10
ACCEPTED MANUSCRIPT
Worse
3.997E-15
4E-15
7.55E-15
4E-15
4E-15
4E-15
Mean
3.997E-15
4E-15
6.96E-15
4E-15
4E-15
4E-15
std
0
0
1.35E-15
0
0
0
Best
0
0
0
0
0
0
Worse
0
0
1.81E-02
2.28E-02
4.01E-02
5.44E-02
Mean
0
0
3.13E-03
1.92E-03
3.15E-03
5.43E-03
std
0
0
5.89E-03
5.50E-03
8.47E-03
1.20E-02
Best
7.899E-08
5.57E-08
6.01E-08
6.13E-08
3.46E-08
6.01E-08
Worse
6.71E-03
6.47E-03
1.98E-02
1.33E-02
1.34E-02
1.98E-02
Mean
4.39E-04
2.16E-04
5.55E-03
1.73E-03
3.21E-03
5.55E-03
std
1.66E-03
1.18E-03
6.23E-03
3.40E-03
4.39E-03
6.23E-03
Best
7.045E-07
1.08E-06
1.49E-06
8.13E-07
6.01E-07
9.26E-07
Worse
3.64E-01
1.21E-01
3.64E-01
2.59E-01
4.85E-01
6.06E-01
Mean
2.56E-02
1.62E-02
1.33E-01
6.24E-02
1.29E-01
1.88E-01
std
7.61E-02
4.19E-02
1.07E-01
8.52E-02
1.46E-01
1.69E-01
9.98E-01
9.98E-01
F11
F13
Best
9.98E-01
9.98E-01
Worse
9.98E-01
9.98E-01
Mean
9.98E-01
9.98E-01
std
3.08E-14
2.26E-14
Best
3.07E-04
3.07E-04
Worse
4.12E-04
1.22E-03
Mean
3.11E-04
3.38E-04
std
1.90E-05
1.67E-04
Best
-1.031628
Worse
-1.031628
9.98E-01
9.98E-01
9.98E-01
9.98E-01
9.98E-01
9.98E-01
9.98E-01
9.98E-01
1.55E-14
4.00E-14
1.35E-14
0
3.07E-04
3.07E-04
3.07E-04
3.07E-04
2.04E-02
1.59E-03
2.04E-02
2.04E-02
2.34E-03
3.50E-04
3.05E-03
1.71E-03
6.11E-03
2.35E-04
6.91E-03
5.08E-03
-1.03163
3.07E-04
-1.03163
-1.03163
-1.03163
-1.03163
2.04E-02
-1.03163
-1.03163
-1.03163
ED
-1.031628
-1.03163
2.34E-03
-1.03163
-1.03163
-1.031623
std
1.312E-10
1.27E-10
6.11E-03
2.43E-10
9.75E-11
8.47E-11
Best
3.98E-01
3.98E-01
3.98E-01
3.98E-01
3.98E-01
3.98E-01
Worse
3.98E-01
3.98E-01
3.98E-01
3.98E-01
3.98E-01
3.98E-01
3.98E-01
3.98E-01
3.98E-01
3.98E-01
3.98E-01
3.98E-01
2.37E-09
1.96E-09
2.22E-09
1.71E-09
1.59E-09
0
Best
3
3
3
3
3
3
Worse
3.0000008
3.000001
3.000001
3.000001
3.000001
3.0000006
Mean
3.0000001
3
3
3
3
3
std
1.661E-07
7.16E-08
1.3E-07
3.19E-07
2.79E-07
2.26E-15
Mean
AC
CE
std
PT
Mean
F17
F18
M
F15
AN US
9.98E-01
9.98E-01
F14
F16
CR IP T
F12
Best
-3.862782
-3.86278
-3.86278
-3.86278
-3.86278
-3.862782
Worse
-3.86278
-3.86278
-3.86278
-3.86277
-3.86278
-3.86278
Mean
-3.862782
-3.86278
-3.86278
-3.86278
-3.86278
-3.862781
std
5.41E-07
6.63E-07
1.09E-06
2.41E-06
1.05E-06
1.09E-16
Best
-3.301995
-3.322
-3.322
-3.322
-3.322
-3.322
Worse
-3.301994
-3.32199
-3.20275
-3.20274
-3.20273
-3.201948
Mean
-3.301995
-3.322
-3.29026
-3.2982
-3.29026
-3.302106
std
2.856E-07
4.21E-08
5.35E-02
4.84E-02
5.35E-02
4.52E-02
Best
-10.1532
-10.1532
-10.1532
-10.1532
-10.1532
-10.1532
F19
F20
F21
ACCEPTED MANUSCRIPT
Worse
-10.15319
-10.1532
-5.0552
-10.1532
-10.1532
-5.10077
Mean
-10.1532
-10.1532
-9.98326
-10.1532
-10.1532
-9.984783
std
2.485E-06
3.52E-06
9.31E-01
1.11E-06
2.38E-06
9.22E-01
Best
-10.4029
-10.40294
-10.4029
-10.4029
-10.4029
-10.40294
Worse
-10.40292
-10.4029
-5.08767
-5.08767
-5.08767
-5.087672
Mean
-10.4029
-10.40294
-10.2258
-10.2258
-10.2258
-10.22576
std
3.92E-06
3.72E-06
9.70E-01
9.70E-01
9.70E-01
9.70E-01
Best
-10.53641
-10.5364
-10.5364
-10.5364
-10.5364
-10.53641
Worse
-10.5364
-10.5364
-5.12848
-10.5364
-10.5364
-10.5364
Mean
-10.5364
-10.5364
-10.3561
-10.5364
-10.5364
-10.53641
std
3.41E-06
3.42E-06
9.87E-01
1.85E-06
2.05E-06
1.39E-06
F22
CR IP T
F23
Table 7 Wilcoxon sign rank test on the solution by CGWO with different structures for benchmarks in Tables 1-2 (a level of significance α=0.05). CGWO(C25) vs.
CGWO(C25) vs.
R
+
R
-
CGWO(L9)
p-value
R
+
R
-
CGWO(C9)
p-value
R
CGWO(C25) vs.
+
R
-
CGWO(C25) vs.
CGWO(C13)
AN US
CGWO(L5)
case
CGWO(C25) vs.
p-value
R
+
R
-
CGWO(C21)
p-value
R
+
R-
p-value
465
0
1.7E-06 +
465
0
1.7E-06 +
232.5
232.5
1=
232.5
232.5
1=
232.5
232.5
1=
F2
465
0
1.7E-06 +
465
0
1.7E-06 +
465
0
1.7E-06 +
465
0
1.7E-06 +
465
0
1.7E-06 +
F3
465
0
1.7E-06 +
465
0
1.7E-06 +
457
8
3.9E-06+
465
0
1.7E-06 +
432
33
4.1E-05 +
F4
465
0
1.7E-06 +
465
0
1.7E-06 +
465
0
1.7E-06 +
465
0
1.7E-06 +
347
118
1.9E-02 +
F5
411
54
2.4E-04 +
340
125
3.4E-02 +
397
68
7.2E-04 +
345
120
2.1E-02 +
360
105
8.7E-03 +
F6
232.5
232.5
1=
232.5
232.5
1=
232.5
232.5
1=
232.5
232.5
1=
232.5
232.5
1=
F7
129
336
3.3E-02 -
178
287
2.6E-01 =
345
120
3.3E-02 +
177
288
2.5E-01 =
350
115
1.6E-02 +
F8
447
18
1.0E-05 +
341
124
2.6E-02 +
351
114
1.5E-02 +
271
194
4.3E-02 +
359
106
9.3E-03 +
125
340
3.9E-02 -
77
232.5
232.5
1=
232.5
F11
126.5
338.5
7.8E-03 -
126.5
F12
50
415
1.7E-04 -
41
F13
42
423
8.9E-05 -
F14
270
195
4.2E-01 =
F15
313
152
9.8E-02 =
F16
362
103
F17
268
197
1.8E-04 -
157.5
307.5
9.3E-02 =
108
357
5.0E-02 =
181
284
2.3E-01 =
1=
457.5
7.5
5.7E-07 +
232.5
232.5
1=
232.5
232.5
1=
338.5
7.8E-03 -
226
239
7.2E-01 =
171
294
2.4E-01 =
196.5
268.5
4.9E-01 =
424
8.2E-05 -
233
232
9.9E-01 =
42
423
8.9E-05 -
115
350
1.6E-02 -
35
430
4.9E-05 -
155
310
1.1E-01 =
76
389
1.3E-03 -
161
304
1.4E-01=
258
207
5.6E-01 =
237
228
9.1E-01 =
244.5
220.5
8.6E-01 =
247
218
7.1E-01 =
150
315
8.9E-02 =
352
113
2.9E-02 +
99
366
6.0E-03 -
325
140
2.1E-02 +
7.7E-03 +
322
143
6.6E-02 =
269
196
4.5E-01 =
285
180
2.8E-01 =
257
208
6.1E-01 =
4.7E-01 =
259
206
5.9E-01 =
281
184
3.2E-01 =
278
187
3.5E-01 =
290
175
2.7E-01 =
CE
PT
388
232.5
AC
F18
ED
F9 F10
M
F1
299
166
1.7E-01 =
334
131
3.7E-02 +
237
228
9.3E-01 =
232
233
9.9E-01 =
265
200
5.0E-01 =
190
275
3.8E-01 =
325
140
5.7E-02 +
234
231
9.8E-01 =
255
210
6.4E-01 =
284
181
2.9E-01 =
254
211
6.6E-01 =
176
289
2.4E-01 =
345
120
2.1E-02 +
360
105
8.9E-03 +
345
120
2.1E-02 +
F21
366
99
6.0E-03 +
393
72
9.6E-04 +
244
221
8.1E-01 =
282
183
3.1E-01 =
255
210
6.4E-01 =
F22
365
100
6.4E-03 +
321
144
6.9E-02 =
199
266
4.9E-01 =
226
239
8.9E-01 =
179
286
2.7E-01 =
F23
442
23
1.6E-05 +
408
57
3.1E-04 +
325
138
1.6E-02 +
238
227
9.1E-01 =
365
100
6.4E-03 +
F19 F20
+/=/-
10/8/5
10/9/4
10/13/0
6/14/3
9/13/1
ACCEPTED MANUSCRIPT
4.4 Comparison with other algorithms This subsection is divided into three parts. The first part presents the comparison experiment between CGWO and other metaheuristic algorithms such as LSHADE, TLBO, EBOwithCMAR, NDHS, BA, CLPSO on unimodal functions and multimodal functions. The second part shows the comparison results between CGWO and other metaheuristics on CEC2015 test suite with multiple global peaks. The third part gives a comparison experiment using some improved versions of the GWO and multi-niche metaheuristics on multi-niche problems from CEC2015 test suite. 4.4.1 Experiment one To assess the performance of CGWO, it is compared with seven other metaheuristics on benchmark problems. These
CR IP T
algorithms in the study are LSHADE (winner in CEC2014 competition on Real-Parameter Single Objective Optimization Benchmarks), TLBO (it is a very efficient optimization algorithm), EBOwithCMAR (winner in CEC 2017 competition on bound constrained benchmark set), NDHS (it is a novel HS approach, and performs better than HS and its other variants), BA (BA is recent metaheuristic and shows better performance than basic PSO, GA, etc.), CLPSO (it is popular PSO variant and is usually used to compare performance of one algorithm), and GWO. Tables 8-9 report the statistical results for each algorithm on benchmarks.
AN US
According to the results in Table 8, CGWO can provide very competitive results compared to the other algorithms. In detail, both of TLBO and CGWO can achieve the best average results on 4 out of 7 problems, whereas LSHADE, EBOwithCMAR, NDHS, BA, CLPSO, and GWO can provide the best average ones on 2, 1, 2, 0, 2, and 1 test functions, respectively. Note that the unimodal benchmarks are suitable for testing exploitation ability of the metaheuristics (S. Mirjalili, et al., 2014). Therefore, regarding exploiting the optimum, CGWO and TLBO are in the first rank; LSHADE, NDHS and CLPSO come in the second place; EBOwithCMAR and GWO are in the third place; the BA ranks at the rear. In addition, CGWO outperforms the basic GWO on 6 of 7 unimodal benchmarks in terms of the average value. The reason seems
M
straightforward: each wolf in CGWO has its own neighbors and only interacts with its neighbors. The three best wolves in its neighbors can be used to guide the movement of other candidate wolves within neighborhood toward their good areas.
ED
Obviously, CA can contribute to exploitation or local search of CGWO. Table 8 Results by different algorithms on unimodal benchmark functions EBOwithC
TLBO
NDHS
BA
CLPSO
GWO
CGWO
MAR
Best
8.56E-30
0
1.42E-24
0
1.22E-04
5.76E-12
1.5E-206
0
Worst
5.28E-26
0
2.87E-22
0
4.91E-01
6.05E-11
7.3E-201
0
2.94E-27
0
6.67E-23
0
8.07E-02
3.37E-11
3.7E-202
0
9.50E-27
0
6.79E-23
0
1.03E-01
1.40E-11
1.40E-89
0
3.11E-14
0
2.67E-12
0
1.61E-02
6.41E-08
1.68E-45
0
F1 Mean std
AC
Best F2
LSHADE
PT
Statistics
CE
Problem
Worst
1.26E-12
0
4.97E-11
0
10.12344
3.14E-07
1.41E-34
0
Mean
3.26E-13
0
1.63E-11
0
5.1E-01
1.95E-07
4.96E-36
0
std
3.41E-13
0
1.45E-11
0
1.822092
5.70E-08
2.58E-35
0
Best
1.00002
0
4.65748
842.1160
6.3E-02
57.4581
1.68E-45
3.57E-74
Worst
1.00993
0
33.85209
4805.8723
10196.4934
319.3870
1.41E-34
6.24E-55
Mean
1.00145
0
16.91642
2036.922
1090.884
165.6838
4.96E-36
4.18E-56
std
2.25E-03
0
8.814452
893.3847
2552.026
65.5480
2.58E-35
1.82E-43
Best
1.24E-07
9.34E-47
6.89E-06
2.01E-57
1.20E-02
8.1276
3.36E-41
3.62E-73
Worst
1.73E-06
4.72E-46
2.18E-04
2.11E-46
5.89E-01
14.0159
2.43E-36
5.17E-65
Mean
6.61E-07
3.12E-46
4.06E-05
7.02E-48
1.43E-01
10.3689
2.39E-37
4.38E-66
F3
F4
ACCEPTED MANUSCRIPT
std
4.08E-07
4.27E-45
3.90E-05
3.85E-47
1.18E-01
1.3238
5.66E-37
1.32E-65
Best
8.46335
28.72192
12.63520
23.57316
28.72252
2.2130
24.38665
24.24317
Worst
11.8763
28.94249
15.43227
123.4207
90029.3932
72.4817
27.0883
27.06747
Mean
10.2642
28.87576
14.21070
27.52616
3039.519
23.2816
25.75554
25.64893
std
9.87E-01
6.04E-02
6.49E-01
18.11298
16429.8
16.8490
7.32E-01
7.68E-01
F5
Best
0
0
0
0
0
0
0
0
Worst
0
0
0
1
2
0
0
0
Mean
0
0
0
1.67E-01
1.00E-01
0
0
0
std
0
0
0
3.79E-01
4.03E-01
0
0
0
Best
1.00E-03
2.20E-04
5.28E-04
4.76E-04
1.32E-03
1.25E-24
1.90E-04
1.21E-04
Worst
2.41E-03
1.48E-01
3.18E-03
8.59E-04
1.95E-01
4.69E-23
1.20E-03
8.07E-04
Mean
1.61E-03
3.39E-02
1.91E-03
4.68E-04
7.48E-02
1.03E-23
5.49E-04
4.30E-04
std
3.46E-04
3.69E-02
6.17E-04
3.57E-04
5.06E-02
9.78E-24
2.31E-04
1.83E-04
F7
CR IP T
F6
Table 9 presents the statistical values obtained by different algorithms on multimodal benchmarks from Table 2. In
AN US
contrast to unimodal benchmarks, multimodal benchmarks are suitable for measuring exploration ability of the algorithm (S. Mirjalili, et al., 2014). It can be obviously seen from this Table 9 that the LSHADE performs better than other seven methods on 9 out of 16 test functions (i.e., F11-13, F16-18, F20, and F22-23). Thus, the LSHADE is the most effective one among these compared algorithms. The CGWO is the second most effective approach, as it can obtain the best average result on 5 benchmarks (i.e., F10-11, F14, F17, and F19). EBOwithCMAR is the third most effective, since it generates the optimal average result on 4 problems (i.e., F11, F15, F17, and F21). TLBO can obtain the best average one on F9 and F11, consequently, TLBO is the fourth most effective algorithm. CLPSO obtains the best result on only one problem (i.e., F8). The
M
rest metaheuristics including NDHS, BA and GWO cannot get any one the best result on these benchmarks. Therefore, concerning exploration, CGWO is competitive with LSHADE and superior to the compared algorithms for most multimodal
ED
benchmarks. It means that CGWO with CA model has a better global search ability than its counterparts except for LSHADE on most multimodal functions. The major reason for the good performance of CGWO is that the overlapping neighbors provide an information diffusion scheme, and they enable each cells information to diffuse toward the whole wolf population,
PT
which is favorable for exploring the search space. Meanwhile, CA provides a niching mechanism for GWO, where wolf population is divided into many subpopulations and update operation is executed independently on these subpopulations. That is, different subpopulation may have different search directions in solution space and help to explore new promising
Statistics
LSHADE
AC
Problem
CE
areas of the search space.
Table 9 Results by different algorithms on multimodal benchmark functions EBOwithC TLBO
NDHS
BA
CLPSO
GWO
CGWO
MAR
Best
-1.26E+04
-5.29E+04
-9.93E+03
-1.26E+04
-1.07E+04
-1.26E+04
-8.35E+03
-6.70E+03
Worst
-1.26E+04
-2.63E+03
-7.77E+03
-1.26E+04
-4.20E+03
-1.26E+04
-3.86E+03
-3.26E+03
Mean
-1.26E+04
-3.88E+03
-9.18E+03
-1.26E+04
-8.28+03
-1.26E+04
-6.86E+03
-5.07E+03
std
9.63E-02
5.71E+02
7.98E-01
5.06E-06
1.58E+03
6.84E-08
1.02E+03
1.23E+03
Best
6.19E-03
0
3.87E-02
0
2.23E+01
4.44E-05
0
Worst
1.2211
0
3.04331
1.10E-01
1.41E+02
1.32E-03
2.50E+01
1.66E+01
Mean
9.78E-02
0
7.98E-01
1.80E-02
5.65E+01
3.65E-04
5.68E+00
4.20E+00
std
2.26E-01
0
8.08E-01
2.15E-02
2.25E+01
2.81E-04
6.42E+00
5.46E+00
Best
2.84E-14
4.44E-15
4.22E-13
4.44E-15
8.75E-03
1.56E-06
7.55E-15
4.00E-15
F8
0
F9
F10
ACCEPTED MANUSCRIPT
Worst
1.45E-13
4.46E-15
1.08E-11
4.44E-15
4.66E-01
5.53E-06
1.47E-14
4.00E-15
Mean
5.35E-14
4.44E-15
2.14E-12
4.44E-15
9.22E-02
3.15E-06
8.62E-15
4.00E-15
std
2.54E-14
0
1.91E-12
0
9.36E-02
1.07E-06
1.90E-15
0
Best
0
0
0
0
1.52E-03
5.88E-09
0
0
Worst
0
0
0
1.91E-01
7.04E-01
1.09E-06
1.79E-02
0
Mean
0
0
0
7.45E-02
2.78E-01
1.17E-07
1.96E-03
0
std
0
0
0
5.07E-02
2.18E-01
2.18E-07
4.69E-03
0
Best
0
3.45E-01
9.96E-01
5.26E-07
5.99E-03
3.28E-13
1.39E-07
6.01E-08
Worst
0
1.11E+00
9.96E-01
1.18E-06
9.49E-01
9.57E-12
2.01E-02
1.98E-02
Mean
0
5.99E-01
9.96E-01
8.73E-07
1.15E-01
1.90E-12
6.86E-03
5.55E-03
std
0
2.06E-01
5.82E-16
1.60E-07
2.08E-01
1.93E-12
6.38E-03
6.23E-03
Best
7.18E-30
2.21E+00
1.15E-24
2.29E-05
4.49E-01
8.59E-12
2.15E-06
9.26E-07
Worst
9.36E-27
3.68E+00
1.22E-21
1.07E-01
2.55E+00
6.76E-11
4.85E-01
6.06E-01
Mean
1.40E-27
2.92E+00
9.80E-23
2.97E-02
1.48E+00
2.63E-11
2.07E-01
1.88E-01
std
2.21E-27
3.67E-01
2.39E-22
2.82E-02
4.64E-01
1.24E-11
1.41E-01
1.69E-01
9.98E-01
9.98E-01
F11
CR IP T
F12
F13
1
9.98E-01
N.A
1
4.98E+02
N.A
9.98E-01
9.98E-01
9.98E-01
9.98E-01
9.98E-01
9.98E-01
9.98E-01
9.98E-01
Mean
1
4.51E+01
N.A
9.98E-01
9.98E-01
9.98E-01
9.98E-01
9.98E-01
std
0
1.23E+02
N.A
2.06E-16
2.89E-16
2.15E-22
1.28E-13
0
Best
9.04E-04
3.55E-04
3.07E-04
3.07E-04
3.07E-04
3.85E-04
3.07E-04
3.07E-04
Worst
9.04E-04
5.26E-03
3.21E-04
2.04E-02
2.97E-02
2.04E-02
2.04E-02
2.04E-02
Mean
9.04E-04
1.63E-03
3.17E-04
5.76E-03
1.03E-02
1.71E-03
3.04E-03
5.71E-04
std
2.81E-19
1.50E-03
3.57E-06
8.96E-03
1.02E-02
7.08E-03
6.91E-03
7.75E-05
Best
-1.0316
-1.0316
1.63E+08
-1.0316
-1.0316
-1.0316
-1.0316
-1.03163
Worst
-1.0316
-1.0219
2.09E+08
-1.0316
-1.0316
-1.0316
-1.0316
-1.03163
F14
-1.0316
-1.0319
1.99E+08
-1.0316
-1.0316
-1.0316
-1.0316
-1.03163
std
6.66E-16
1.99E-03
1.12E+07
2.50E-10
2.81E-10
6.72E-10
7.38E-10
8.46E-11
Best
0.397887
0.397887
0.397887
3.98E-01
3.98E-01
3.98E-01
0.397887
0.397887
Worst
0.397887
0.729213
0.397887
3.98E-01
3.98E-01
3.98E-01
0.397887
0.397887
Mean
0.397887
0.423606
0.397887
3.98E-01
3.98E-01
3.98E-01
0.397887
0.397887
std
0
7.80E-02
0
1.60E-11
1.01E-14
0
1.59E-08
0
CE
PT
Mean
F17
Best
3
3
3
3
3
3
3
3
Worst
3
33.28821
3.000
30
84
3.0000
3.0000
3
Mean
3
7.5885
3.000
5.7
5.7
3.0000
3.0000
3
std
0
9.1882
1.44E-14
8.2385
1.48E+01
3.24E-15
3.77E-07
2.26E-15
AC
F18
ED
F16
M
F15
AN US
Best Worst
Best
-3.86
-3.8624
N.A
-3.8627
-3.8627
-3.8627
-3.8627
-3.8628
Worst
-3.86
-3.0819
N.A
-3.8627
-3.0898
-3.8627
-3.8551
-3.8628
Mean
-3.86
-3.7604
N.A
-3.8627
-3.8113
-3.8627
-3.8624
-3.8628
std
0
1.99E-02
N.A
1.63E-09
1.96E-01
2.71E-15
1.64E-03
1.09E-16
Best
-3.3220
-3.3016
N.A
-3.3220
-3.3220
-3.3220
-3.3220
-3.3220
Worst
-3.3220
-2.4316
N.A
-3.2031
-1.7061
-3.3220
-3.0867
-3.2020
Mean
-3.3220
-3.0465
N.A
-3.2943
-3.1976
-3.3220
-3.2759
-3.3021
std
2.22E-15
2.03E-01
N.A
5.11E-02
3.06E-01
1.77E-12
6.99E-02
4.52E-02
Best
-10.1532
-5.0535
-10.1532
-10.1532
-10.1532
-10.1532
-10.1532
-10.1532
F19
F20
F21
ACCEPTED MANUSCRIPT
Worst
-10.1532
-3.8736
-10.1532
-2.6305
-2.6305
-10.1419
-5.0552
-5.1007
Mean
-10.1532
-4.6398
-10.1532
-4.4102
-5.6498
-10.1526
-9.3096
-9.9848
std
3.55E-15
3.58E-01
0
3.2225
3.5591
2.10E-03
1.9185
9.22E-01
Best
-10.4029
-9.6104
-10.0634
-10.4029
-10.4029
-10.4029
-10.4029
-10.4029
Worst
-10.4029
-3.7112
-10.0634
-2.7519
-2.7519
-10.4029
-10.4029
-5.0877
F22 Mean
-10.4029
-4.9199
-10.0634
-8.3653
-5.0491
-10.4029
-10.4029
-10.2258
std
8.88E-15
9.67E-01
0
3.4373
3.1047
4.41E-06
2.29E-05
9.70E-01
Best
-10.5364
-10.5345
-10.0747
-10.5364
-10.5364
-10.5364
-10.5364
-10.5364
Worst
-10.5364
-2.3923
-10.0747
-2.4217
-1.8595
-10.5361
-10.5363
-10.5364
Mean
-10.5364
-4.7705
-10.0747
-8.6883
-5.0996
-10.5362
-10.5363
-10.5364
std
0
1.5789
8.70E-16
3.4090
3.2158
1.23E-05
2.16E-05
1.39E-06
CR IP T
F23
Table 10 Wilcoxon sign rank test on the solutions by different algorithms for benchmarks in Tables 1-2 (a level of significance α=0.05). CGWO vs. CGWO vs. LSHADE
CGWO vs. TLBO
0
R+
R-
232.5
232.5
0
232.5
F5
465
0
232
+ 1.7E-06
0
0
1.7E-6 465
1
1 232.5
=
=
0
232
232
465
360
.5
0
465
F14
0
465
465
465
0
=
1.7E-06
465
1.7E-06
+
1.7E-06
+
1.7E-06 465
0
+
1.7E-06 465
0
+
1.7E-06
0
0
+
1.7E-06 465
0
p-value
1.7E-06 465
0
+
+
+
+
2.2E-04
1.7E-06
1.7E-06
1.7E-06
412
465
0
465
0
465
0
+
+
+
1.7E-06
3.4E-05
1.7E-06
1.2E-02
5.8E-01
434
31
465
0
111
354
-
+
+
1
5.7E-04
1.7E-06
232
232
+
.5
.5
0
465
400
65
=
1.7E-06 465
0
+ 465
192
1
232
232
1
=
.5
.5
=
324
141
=
397
5.9E-01
-
7.2E-04 68
206
1.7E-06
+
1.7E-06 0
259 -
0
3.7E-05 32
1.7E-6
465
465
+ 433
465
=
1.7E-06 0
465
2.8E-05 29
436
-
-
-
1.1E-01
1.7E-06
1.7E-06
364
100
273
465
0
0
2.8E-05
465 .5
+
1.7E-6
1.7E-06
1.7E-06
1.7E-06
1.7E-06
379
85,
4.9E-04
+
.5
5
+
433
32
465
0
465
232
232
.5
.5
465
0
1
+ 1.7E-6 0
360
465
0
2.1E-02 120
345
+ 0
465
111
+
1.7E-06
418
-
+
-
+
232
232
1
1.7E-06
1.6E-02
2.0E-05
.5
.5
=
465
0
0
327
+
465
1.2E-02
47
NA
0
1.2E-02 354
-
1.7E-06 465
+
1.7E-06 0
+
1.4E-04
3.7E-05
-
1.7E-06 465
-
NA
0
1.7E-06 465
8.7E-03 105
465
+
+
1.7E-06
NA
+
= +
0
0
1.9E-06 1
1.7E-06
1.7E-6
465
+ 464
0
0
+
+
465
.5
465
1.7E-06
+
0
R-
.5
+
.5
0
R+
-
1.7E-06 0
1
+
1.7E-6
465
232
465
1.7E-06
0
p-value
+
0
1.7E-06 F13
232
R-
=
1.6E-2
-
=
R+
-
-
465
.5
p-value
-
327
1.7E-06 F12
R-
1
.5
1.7E-06
+
=
232
1.7E-6 0
R+
CGWO vs. GWO
-
0
1
138
.5
0
232
53
0
+
+
F11
.5
1.2E-4
AC
465
465
107
1.7E-06
F10
.5
232
9.8E-3
105
=
232
p-value
CGWO vs. CLPSO
+
465
+
358
225
0
CE
465
0
1.7E-6
465
210
0
+
6.4E-01 F9
0
-
1.7E-6 0
465
1.7E-06
+ F8
465
-
232.5
465
1.7E-06 0
-
1.7E-06 F7
+
1.7E-6
+
R-
465
+
F6 .5
0
1.7E-06
465
.5
1.7E-06
465
R+
465
+ 465
465
p-value 1.7E-06
1.7E-6 0
0
232
0
PT
F4
0
465
=
1.7E-06 465
R-
1 232.5
+ F3
R+
=
1.7E-06 465
p-value
1
+ F2
CGWO vs. BA
AN US
465
p-value
M
R-
ED
R+
1.7E-06 F1
CGWO vs. NDHS
EBOwithCMAR
case
355
138
440
+
110
25
+
ACCEPTED MANUSCRIPT
2.8E-3 378
88
+
+
-
232
232
1
5.3E-5
.5
.5
=
+
232
232
1
1.8E-5
232
232
1
.5
.5
=
+
.5
.5
=
0
465
420
45
F16
429
F17
441
87
0
36
465
465
0
435
461
4
1
1.2E-05
.5
=
+
.5
.5
=
+
1.9E-06
1.7E-06
1.4E-04
4.3E-06
1
465
101
465
465
0
146
465
1.7E-06 NA
NA
465
0
+ 0
2.6E-05
465
437
28
465
+
-
+
3.1E-05
1.7E-6
1.0E-03
1.7E-06
465
0
392
73
465
+ 1.7E-6
435
465
0
465
+
1.7E-06 465
0
+
16/3/4
276
232
11/7/5
9
1
+
.5
.5
=
1.7E-06
232
232
1
1.5E-02
.5
.5
351
114
114
+ 2.8E-03 378
87
=
+
1.5E-02
1.7E-06 465
0
+
+
+
1.7E-06
8.7E-03
2.8E-05
0
105
360
436
29
+
-
+
1.7E-06
5.3E-05
3.1E-05
0
36
429
+
456
20
+ 351
16
435
30
-
4.3E-06
9
=
11/3/6
232
0
3.7E-01
189
+
10/7/6
0
+
456
2
+
8.5E-06
449
1.7E-6
47
0
+
=
30
1.7E-06 465
7.7E-02 319
418
+
+
NA
445
0
6.8E-03 364
463
0
+
+
-
+
3.7E-05 433
32
3.1E-05 435
+
30
+
22/0/1
10/4/9
+ 19/3/1
To investigate the convergence rate of the CGWO algorithm, some convergence curves of benchmark functions selected at random are plotted in Fig. 6. To evaluate the quality of the solution obtained by each algorithm in our experiment, we employ fitness value to assess the performance of each algorithm and use the number of function evaluation (NFE) to track the trend
ED
of fitness value in a random run.
F3
F1
10
10
LSHADE TLBO EBOwithCMAR NDHS BA CLPSO GWO CGWO
fitnes value
-100
PT
10
0
CE
fitnes value
10
-200
AC
+/=/-
232
465
NA
1.7E-6
3.1E-05 30
+
232
+
NA
+ F23
+
1.7E-06
75
M
F22
.5
2.6E-6
212
= 0
+
2.1E-06
0
+
1.7E-6 F21
1
1.2E-04
NA
6.7E-01 253
232
1.7E-6
0
+ F20
+
232
1.7E-06 465
390
AN US
465
-
1.7E-06
+
1.7E-06 F19
58
464
1
1.2E-03
407
0
1.9E-6 464
3.3E-04
465
24
1.7E-6 F18
1.7E-06
377
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3.0E-3 F15
0
2
4 6 8 number of function evaluation
10
10
10 x 10
4
0
LSHADE TLBO EBOwithCMAR NDHS BA CLPSO GWO CGWO
-100
-200
0
2
4 6 8 number of function evaluation
10 x 10
4
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10
10
10
fitnes value
10
10
10
10
6
4
10
10
10
2
0
0
2
4 6 8 number of function evaluation
10
10 x 10
0
-5
-10
-15
10 LSHADE TLBO EBOwithCMAR NDHS BA CLPSO GWO CGWO
0
-5
-10
2
4 6 8 number of function evaluation
2
4 6 8 number of function evaluation
10
x 10
10
10
x 10
4
F15
2
0
LSHADE TLBO EBOwithCMAR NDHS BA CLPSO GWO CGWO
-2
-4
4
0
2
4 6 8 number of function evaluation
10 x 10
4
M
0
10
10
-15
0
4
F11
5
LSHADE TLBO EBOwithCMAR NDHS BA CLPSO GWO CGWO
CR IP T
10
8
fitnes value
10
F9
LSHADE TLBO EBOwithCMAR NDHS BA CLPSO GWO CGWO
fitnes value
fitnes value
10
F5
10
AN US
10
Fig. 6. Convergence curve of these algorithms when solving benchmarks
ED
Fig. 6 presents the evolution of fitness value with the increase of NFE on these problems for each algorithm. X axis represents NFE and Y axis denotes the fitness value. These curves indicate the convergence speed of each algorithm. In terms of the NFE, the convergence speed of CGWO’s curve is much faster than those of its counterparts in minimizing the fitness
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value for F1 and F5. CGWO is slightly faster than its compared algorithms for F3, F9, F11 and F15. CGWO is a bit slower than the other algorithms for the rest of benchmarks but still shows strong competitiveness. The reasons for the good
CE
performance of the CGWO have two main aspects as follows. Firstly, each solution in CGWO has its own topological neighbors. Thus, the interaction among solutions is restricted to their neighborhood, which favors exploitation performance of CGWO. Secondly, information diffusion mechanism by overlap among neighborhoods can maintain the population diversity
AC
for longer, usually contributing to exploration performance. We can conclude from this observation that the result of plots is consistent with our view that CGWO with CA effectively improves the convergence of the algorithm. The following subsection reveals that CGWO can locate the whole global optimums.
4.4.2 Experiment two Empirical study on benchmarks with multiple global peaks is conducted in this experiment. Success rate is used to test performance of algorithm for CEC2015 benchmarks with multiple global optimums. Note that the success rate is the percentage of runs in which all the desired peaks are successfully located in this experiment, whereas in the experiment one the convergence only denotes that the algorithm may find a single global solution rather than all desired peaks. The level of accuracy is set to 0.1 in this competition. This parameter is used to measure how close the obtained solutions are to the known global/local peaks. If the distance between an obtained solution and the true optimal solution is smaller than a
ACCEPTED MANUSCRIPT
tolerance value (level of accuracy), it will be accepted that the optimal solution has been found. When all the desired peaks are found in one single run, this run is deemed to be successful. In addition, when all the algorithms satisfy stop condition in this experiment, they return the final population in the last iteration rather than the only one best solution. Table 11 gives the corresponding parameters of problems and summarizes the success rate on CEC2015 with multiple global peaks (i.e., F25, F26, F28, F29, F30, F31, F32, F34, F35 and F36). 30 independent runs are conducted in this experiment. Table 12 presents that CGWO algorithm gives a significantly better performance than the other algorithms in terms of success rate. Fig. 7 shows that CGWO can locate all 4 global peaks on F25 (decision dimension=2) at number of function evaluation (NFE) 40,000 in a single run. Multiple emerged niches are clearly visible at NFE equal to 80,000. Figs.
CR IP T
8-10 show global peaks on F26, F28 and F30 (decision dimension=2) in a single run. We can observe that CGWO is able to find all 25 global optimums on the F26 and F28, and 2 global peaks on the F30, respectively. It means that CGWO can successfully develop stable niches on majority of the global peaks without niching radius parameters. It also implies that CGWO has good diversity of population.
Table 11 Success rate on CEC2015 with multiple global optimum maximum Population
EBOwithC
evaluation dimension
CLPS
AN US
Decision Problem
LSHADE
size
TLBO
NDHS
BA
MAR
function number
GWO
CGWO
O
2
400
100,000
0%
0%
0%
0%
0%
0%
53.3%
100%
F26
2
400
100,000
0%
0%
0%
0%
0%
33.3%
16.7%
96.7%
F28
2
400
100,000
0%
0%
0%
0%
0%
0%
26%
80%
F29
2
526
100,000
0%
0%
0%
3.3%
3.3%
0%
6.7%
10%
F30
2
400
100,000
0%
0%
0%
0%
0%
3%
16%
100%
F31
2
400
500,000
0%
0%
0%
0%
0%
0%
3.3%
6.7%
F32
10
100
300,000
0%
0%
0%
0%
0%
0%
0%
0%
F34
10
100
300,000
0%
0%
0%
0%
0%
0%
0%
10%
F35
10
400
500,000
0%
0%
0%
0%
0%
0%
0%
23.3%
F36
10
400
300,000
0%
0%
0%
0%
0%
0%
0%
0%
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ED
M
F25
Table 12 Wilcoxon sign rank test on the solution by different algorithms for CEC2015 with multiple global optimum (a level of significance α=0.05). CGWO vs.
case R+
R-
p-value
465
R+
0
R-
465
0
+
F28
F29
464
455
276
1
464
262
CGWO vs. GWO
465
0
p-value
R+
R-
465
0
1.7E-06
R-
465
0
2
R+
R-
465
0
1
R+
R-
397
68
+
4.1E-05 432
33
p-value 1.2E-04
+
1.9E-06 464
p-value 1.7E-06
+
2.1E-06 462
p-value 1.7E-06
+
2.1E-06 2
R+
1.7E-06
+ 463
p-value
4.7E-06 455
10
+
+
+
+
+
+
4.7E-06
4.7E-06
4.7E-06
4.7E-06
4.7E-06
1.6E-02
10
455
10
455
10
455
10
455
10
455
10
348
117
2.5E-01
247
217
1
=
.5
.5
=
385
80
+
+
+
+
+
+
2.5E-01
2.5E-01
2.5E-01
5.0E-01
5.0E-01
189
276
0
203
CGWO vs. CLPSO
+
189
189
0
203
5.0E-01
203
0
203
5.0E-01
203
0
0
203
5.0E-01
1.7E-06 465
0
+ 262
203
5.0E-01
+
189
1.7E-06 465
+ 262
276 =
1.7E-06 465
+ 262
262
=
1.7E-06 465
+ 262
262
=
1.7E-06 465
5.0E-01
276
=
+ F31
CGWO vs. BA
4.7E-06
1.7E-06 465
R-
1.9E-06 1
= F30
R+
+
1.9E-06
F26
p-value
CGWO vs. NDHS
EBOwithCMAR
1.7E-06
AC
1.7E-06 F25
CGWO vs. TLBO
CE
CGWO vs. LSHADE
5.7E-06
+ 262
203
5.0E-01
+ 247
217
1
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=
=
2.5E-01
2.5E-01
189
276
2.5E-01
189
276
276
189
2.5E-01 276
2.5E-01
189
=
=
=
=
=
=
1.6E-02
1.6E-02
1.6E-02
1.6E-02
1.6E-02
325
138
325
138
+
325
138
325
+
5/3/0
5/3/0
80
70 60
-50
30 -100
-40
(a) number of function evaluation =4,000
40
-90
-80
-70
-60
-50
-40
(b) number of function evaluation =40,000
30 -100
-90
-80
-70
-60
-50
-40
(c) number of function evaluation =80,000
AN US
-60
global optimum CGWO
50
40
-70
5/3/0
60
50
Fig. 7 Niching behavior of CGWO (with a population of 400) on F25 over a run
5
10 global optimum CGWO
5
global optimum CGWO
0
0
-5 -10
-10
-20
-15 -20
-40 0
10
20
30
40
50
-25 20
(a) number of function evaluation =4,000
global optimum CGWO
-5
-10
-30
0
25
-15 -20
30
35
40
-25 20
(b) number of function evaluation =20,000
25
30
35
40
(c) number of function evaluation =40,000
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Fig. 8 Niching behavior of CGWO (with a population of 400) on F26 in a single run
-40
-40
global optimum CGWO
-60
AC
-70
-30
-20
-10
global optimum CGWO -45
-50 -50 -55 -55 -60
-80
-90 -40
-40 gobal optimum CGWO
-45
CE
-50
-60
-65
0
10
20
(a) number of function evaluation =4,000
-70 -30
-25
-20
-15
-10
-5
(b) number of function evaluation =20,000
-65 -30
138
+
5/3/0
70
50
-80
325
+
90 global optimum CGWO
60
30
138
5/3/0
70
40
325
+
90 80
-90
138
+
5/3/0
global optimum CGWO
20 -100
189
=
1.6E-02
5/3/0
80
276
.5
=
+
90
2.5E-01
189
.5
1.6E-02 138
+/=/-
2.5E-01
189
=
CR IP T
325
276
=
M
F35
276
=
ED
F34
=
-25
-20
-15
-10
-5
(c) number of function evaluation =80,000
Fig. 9 Niching behavior of CGWO (with a population of 400) on F28 in a single run
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40
35 global optimum CGWO
30
30
12 global optimum CGWO
global optimum CGWO 10
25
20
20
8
10 15 0
6
10 -10
5
4 -20
0
-30 40
50
60
70
80
90
-5 50
100
(a) number of function evaluation =4,000
60
70
80
90
2 50
100
(b) number of function evaluation =8,000
60
70
80
90
(c) number of function evaluation =80,000
CR IP T
Fig. 10 Niching behavior of CGWO (with a population of 400) on F30 in a single run
Therefore, CGWO shows a good performance over other metaheuristics. It is also obvious that CA has a positive effect on the performance improvement of the proposed CGWO. The reason behind is that the CGWO makes fully use of the cellular structure in CA to update solutions only within its neighborhood for local search and adopts an implicit migration mechanism implemented by overlap of neighborhoods to enhance information diffusion to improve the search diversity. More precisely,
AN US
CA provides different neighborhoods for the whole population by using a cellular topological structure. The population is divided into many subpopulations. Each solution is arranged in a given cellular grid, and each subpopulation is form of an independent neighborhood. Consequently, each subpopulation is updated independently in its own neighborhood. That is, different subpopulation may search for promising solutions in different search directions. Obviously, it has a stronger search ability to locate all the peaks compared to the other algorithms. Meanwhile, the migration mechanism implemented by overlap of neighborhoods can help to share valuable information among different independent subpopulations. This can improve the search efficiency. Thus, we can draw a conclusion that CA has a good effect on the behavior of the GWO
M
algorithm.
ED
4.4.3 Experiment three
To further investigate the performance of the proposed CGWO, it is compared to other improved variants of GWO and multiswarm metaheuristic, which include GWOCLS, EGWO, WdGWO, and iHS. This comparison experiment is conducted
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on benchmarks in Tables 1 and 2. These parameter settings of the compared algorithms are used as in their original literature (Al-Betar, et al., 2015; Joshi & Arora, 2017; Malik, et al., 2016; Yu, et al., 2017). In this subsection, we also compare the proposed CGWO to the well-known multi-niche metaheuristics including EAD,
CE
RPSO, CDE, NCDE, and LIPS for multi-niche optimization problems from CEC2015. These parameter settings of the compared algorithms are also used as in their original literature (Gu, et al., 2015; X. D. Li, 2010; Qu, et al., 2016; Qu, et al.,
AC
2012; Thomsen, 2004).
Problem
F1
F2
Table 13 Results by different improved versions of GWO and iHS on benchmarks in Tables 1 and 2
Statistics
GWOCLS
EGWO
WdGWO
iHS
CGWO
Best Worst Mean std Best Worst Mean
7.96E-94 2.35E-90 2.85E-91 5.97E-91 2.69E-54 1.13E-51 1.14Ee-52
3.01E-71 1.39E-64 1.21E-65 3.29E-65 9.94E-47 4.36E-43 3.66E-44
5.32E-70 3.91E-66 3.10E-67 7.21E-67 6.36E-40 1.93E-38 3.99E-39
2.93E-03 9.77E-03 5.27E-03 1.87E-03 1.41E-02 2.86E-02 2.18E-02
0 0 0 0 0 0 0
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F7
F8
F9
AC
CE
F10
F11
F12
F13
4.09E-03 1.07E-12 1.14E-07 4.74E-09 2.07E-08 3.58E-01 6.44E-01 5.15E-01 6.84E-02 10.1053 161.7552 78.7063 33.2843 0 0 0 0 1.97E-04 1.85E-03 8.82E-04 4.75E-04 -8.55E+03 -4.13E+03 -4.85E+03 1.03E+03 1.74E-03 8.36E-03 3.27E-03 1.48E-03 1.41E-02 2.91E-02 1.98E-02 4.44E-03 0 1.76E-02 2.45E-03 4.89E-03 4.46E-06 1.31E-04 2.75E-05 2.82E-05 2.05E-04 2.40E-03 4.89E-04
0 3.57E-74 6.24E-55 4.18E-56 1.82E-43 3.62E-73 5.17E-65 4.38E-66 1.32E-65 24.2431 27.0674 25.6489 7.68E-01 0 0 0 0 1.21E-04 8.07E-04 4.30E-04 1.83E-04 -6.70E+03 -3.26E+03 -5.07E+03 1.23E+03 0 1.66E+01 4.20E+00 5.46E+00 4.00E-15 4.00E-15 4.00E-15 0 0 5.44E-02 5.43E-03 1.20E-02 6.01E-08 1.98E-02 5.55E-03 6.23E-03 9.26E-07 6.06E-01 1.88E-01
CR IP T
4.51E-39 7.16E-13 1.59E-08 1.01E-09 3.21E-09 1.53E-14 2.21E-12 4.40E-13 6.14E-13 24.1463 26.0961 25.0443 3.54E-01 0 0 0 0 2.59E-04 1.81E-03 9.57E-04 4.16E-04 -8.22E+03 -4.12E+03 -4.85E+03 9.35E+02 2.00E+00 1.63E+02 2.08E+01 2.81E+01 7.54E-15 2.03E+01 1.35E+01 9.71E+00 0 1.75E-02 1.43E-03 4.00E-03 4.88E-08 2.86E-06 1.16E-06 4.75E-07 1.38E-05 2.42E-01 1.58E-02
AN US
F6
9.05E-44 3.08E-20 1.06E-13 7.70E-15 1.95E-14 1.84E-13 4.42E-09 4.65E-10 1.05E-09 24.13 26.18 25.29 5.82E-01 0 0 0 0 5.78E-04 2.80E-03 1.52E-03 6.23E-04 -9.94E+03 -6.77E+03 -7.64E+03 6.86E+02 0 4.20E+00 1.40E-01 7.68E-01 7.54E-15 2.01E+01 2.67E+01 6.93E+00 0 2.98E-02 1.90E-03 6.38E-03 4.68E-06 2.62E-02 9.06E-03 8.36E-03 1.22E-04 4.80E-01 1.47E-01
M
F5
ED
F4
2.28E-52 1.66E-25 2.32E-19 1.65E-20 5.41E-20 1.12E-20 1.34E-17 6.91E-19 2.44E-18 24.24 26.18 25.26 5.53E-01 0 0 0 0 1.61E-04 1.27E-03 5.95E-04 2.88E-04 -9.01E+03 -7.26E+03 -8.95E+03 3.19E+02 0 1.98E+01 7.70E+00 5.87E+00 7.54E-15 1.46E-14 1.22E-14 3.14E-15 0 2.29E-02 4.63E-03 7.67E-03 1.70E-07 1.68E-02 2.96E-03 5.10E-03 3.31E-06 3.63E-01 6.18E-02
PT
F3
std Best Worst Mean std Best Worst Mean std Best Worst Mean std Best Worst Mean std Best Worst Mean std Best Worst Mean std Best Worst Mean std Best Worst Mean std Best Worst Mean std Best Worst Mean std Best Worst Mean
ACCEPTED MANUSCRIPT
F18
F19
F20
AC
CE
F21
F22
F23
4.01E-04 9.98E-01 9.98E-01 9.98E-01 1.55E-13 4.21E-04 1.31E-03 6.93E-04 1.75E-04 -1.03163 -1.03163 -1.03163 5.24E-09 3.97E-01 3.97E-01 3.97E-01 3.91E-09 3.000 3.000 3.000 5.95E-08 -3.8627 -3.8627 -3.8627 4.06E-09 -3.3220 -3.2301 -3.3022 4.51E-02 -10.1532 -5.0552 -9.1396 2.06 -10.4029 -3.7243 -9.0304 2.55 -10.5364 -2.8711 -9.7432 2.0894
1.69E-01 9.98E-01 9.98E-01 9.98E-01 0 3.07E-04 2.04E-02 5.71E-04 7.75E-05 -1.03163 -1.03163 -1.03163 8.46E-11 3.97E-01 3.97E-01 3.97E-01 0 3.000 3.000 3.000 2.26E-15 -3.8627 -3.8627 -3.8627 1.09E-16 -3.3220 -3.2020 -3.3021 4.52E-02 -10.1532 -5.1007 -9.9848 9.22E-01 -10.4029 -5.0877 -10.2258 9.70E-01 -10.5364 -10.5364 -10.5364 1.39E-06
CR IP T
5.40E-02 9.98E-01 9.98E-01 9.98E-01 1.53E-13 3.07E-04 1.22E-03 4.29E-04 3.16E-04 -1.03163 -1.03163 -1.03163 1.92E-06 3.97E-01 3.97E-01 3.97E-01 1.42E-08 3.000 3.000 3.000 1.02E-08 -3.8627 -3.8627 -3.8627 2.95E-07 -3.3220 -3.1361 -3.2017 3.96E-02 -10.1532 -2.6828 -9.0605 2.26 -10.4029 -5.0876 -10.2256 9.70E-01 -10.5364 -10.5364 -10.5364 7.12E-05
AN US
F17
1.25E-01 9.98E-01 9.98E-01 9.98E-01 2.75E-13 3.07E-04 1.22E-03 4.29E-04 3.16E-04 -1.03163 -1.03163 -1.03163 4.99E-10 3.97E-01 3.97E-01 3.97E-01 2.29E-08 3.000 3.000 3.000 1.77E-08 -3.8627 -3.8627 -3.8627 4.07E-07 -3.3220 -3.0866 -3.2296 8.19E-02 -10.1532 -5.0551 -7.9575 2.55 -10.4029 -5.0876 -9.6968 1.83 -10.5364 -5.1283 -9.2753 2.32
M
F16
ED
F15
9.52E-02 9.98E-01 9.98E-01 9.98E-01 2.32E-13 3.07E-04 2.03E-02 2.46E-03 6.07E-03 -1.03163 -1.03163 -1.03163 9.05E-10 3.97E-01 3.97E-01 3.97E-01 2.24E-08 3.000 3.000 3.000 1.18E-07 -3.8627 -3.8627 -3.8627 1.86E-06 -3.3220 -3.1974 -3.2699 6.06E-02 -10.1532 -10.1530 -10.1531 4.27E-05 -10.4029 -10.4027 -10.4029 4.68E-05 -10.5364 -10.5364 -10.5364 4.00E-05
PT
F14
std Best Worst Mean std Best Worst Mean std Best Worst Mean std Best Worst Mean std Best Worst Mean std Best Worst Mean std Best Worst Mean std Best Worst Mean std Best Worst Mean std Best Worst Mean std
Table 13 shows the statistic results (i.e., best, worst, mean, and std) obtained by different improved versions of GWO and iHS on benchmarks in Tables 1 and 2. 30 independent experiments of each metaheuristic are conducted on each problem.
ACCEPTED MANUSCRIPT
The optimal values in these tables are highlighted in boldface. It can be clearly observed that the proposed CGWO can find promising results on most problems compared to other improved versions of GWO and iHS. More precisely, the CGWO performs the best among these compared 5 algorithms. The good performance of CGWO is due to the CA technique. The CA technique can provide many different neighborhoods during the optimization process. As the CGWO can update its own subpopulation in each neighborhood, the search process of the entire population can be updated in different search directions. This will effectively maintain the search diversity. Table 14 Wilcoxon sign rank test on the solution by different improved versions of GWO for benchmarks in Tables 1-2 (a level of significance α=0.05). CGWO vs. EGWO
CGWO vs. WdGWO
CGWO vs. iHS
R+
R-
p-value
R+
R-
p-value
R+
R-
F1
465
0
1.7E-06 (+)
465
0
1.7E-06 (+)
465
0
F2
465
0
1.7E-06 (+)
465
0
1.7E-06 (+)
465
0
F3
465
0
1.7E-06 (+)
465
0
1.7E-06 (+)
465
0
F4
465
0
1.7E-06 (+)
465
0
1.7E-06 (+)
465
0
F5
126
339
2.8E-02 (-)
156
309
1.1E-01 (=)
86
379
CR IP T
CGWO vs. GWOCLS
F6
316
149
8.6E-02 (=)
233
232
1.0E+0 (=)
233
F7
316
149
8.6E-02 (=)
463
2
2.1E-06 (+)
F8
0
465
1.7E-06 (-)
1
464
F9
327
138
1.2E-04 (+)
3
F10
253
212
3.9E-01 (=)
F11
67
398
F12
184
F13
case
R+
R-
p-value
1.7E-06 (+)
465
0
1.7E-06 (+)
1.7E-06 (+)
465
0
1.7E-06 (+)
1.7E-06 (+)
465
0
1.7E-06 (+)
1.7E-06 (+)
465
0
1.7E-06 (+)
2.5E-03 (-)
452
13
6.3E-06 (+)
232
1.0E+0 (=)
233
232
1.0E+0 (=)
444
21
1.3E-05 (+)
465
0
1.7E-06 (+)
1.9E-06 (-)
235
230
9.5E-01 (=)
0
465
1.7E-06 (-)
462
2.3E-06 (-)
461
4
2.6E-06 (+)
0
465
1.7E-06 (-)
420
45
3.9E-05 (+)
406
59
3.7E-06 (+)
465
0
1.7E-06 (+)
2.4E-05 (-)
10
455
4.7E-06 (-)
9
456
4.2E-06 (-)
25
440
1.9E-05 (-)
281
3.1E-01 (=)
312
153
1.0E-01 (=)
105
360
8.7E-02 (=)
105
360
8.7E-03 (=)
162
303
1.4E-01 (=)
266
199
4.9E-01 (=)
65
400
5.7E-04 (-)
36
429
5.30E-05 (-)
F14
460
5
2.8E-06 (+)
447
18
1.0E-05 (+)
415
50
1.9E-05 (+)
284
181
6.4E-02 (=)
F15
292
173
2.2E-01 (=)
275
190
3.8E-01 (=)
319
146
7.5E-02 (=)
353
112
1.3E-02 (+)
F16
456
9
4.2E-06 (+)
446
19
1.1E-05 (+)
461
4
2.6E-06 (+)
456
9
4.2E-06 (+)
F17
456
9
4.2E-06 (+)
462
3
2.3E-06 (+)
461
4
2.6E-06 (+)
221
244
8.1E-01 (=)
F18
246
219
7.8E-01 (=)
284
181
6.4E-02 (=)
235
230
9.5E-01 (=)
278
187
3.5E-01 (=)
F19
232
233
9.9E-01 (=)
168
297
1.8E-01 (=)
103
362
7.7E-03 (-)
0
465
1.7E-06 (-)
F20
346
119
1.9E-02 (+)
433
32
3.7E-05 (+)
444
21
1.3E-05 (+)
69
396
7.7E-06 (-)
F21
30
435
3.1E-05 (-)
439
26
2.1E-05 (+)
440
25
1.9E-05 (+)
226
239
8.9E-01 (=)
F22
30
435
3.1E-05 (-)
437
28
2.6E-05 (+)
436
29
2.8E-05 (+)
263
202
5.3E-01 (=)
3.1E-05 (+)
441
24
1.8E-05 (+)
435
30
3.1E-05 (+)
465
0
1.7E-06 (+)
435
+/=/-
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AC F23
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p-value
30
10/8/5
13/7/3
14/5/4
10/7/6
Table 15 Results by different algorithms on average number of optima found (F24-F31)
Problem
F24
Statistics
EAD
RPSO
CDE
NCDE
LIPS
CGWO
Best
8
1
0
0
11
0
Worst
1
0
0
0
5
0
Mean
4.17
1.17E-01
0
0
8.24
0
std
1.99
3.22E-01
0
0
1.26
0
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F29
F30
F31
26
60
Worst
33
16
0
0
14
35
Mean
50.21
15.8
0
0
23
53.2
std
4.25
2.91
0
0
3.04
3.51
Best
77
89
4
13
45
90
Worst
59
72
0
9
31
68
Mean
68.17
74.05
2
10.4
37.92
75.4
std
3.86
4.82
1.25
1.67
3.58
5.05
Best
1
0
0
0
0
0
Worst
0
0
0
0
0
0
Mean
2.15E-01
0
0
0
0
0
std
4.15E-01
0
0
0
0
0
Best
59
130
0
0
0
73
Worst
42
117
0
0
0
57
Mean
48.07
118.35
0
0
0
64.21
std
3.86
4.96
0
0
0
5.36
Best
85
0
63
90
Worst
83
0
Mean
84.66
0
std
5.88E-01
0
Best
8
0
Worst
1
0
Mean
3.66
0
std
1.49
Best
60
Worst
45
Mean std
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0
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F28
0
0
2
0
0
50
85
0
0.8
55.96
87.6
0
0.83
3.18
3.32
0
1
15
18
0
0
9
10
0
0.04
12.04
14.31
0
0
0.2
1.45
4.03
9
12
40
96
105
1
4
30
78
80
50.72
4.90
6.9
36.4
88.52
91.56
4.00
2.88
2.30
3.78
4.56
5.86
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22
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F26
57
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F25
Best
Table 16 Results by different algorithms (statistic values of 5 best solutions separated at least by 10 on F32-F38)
EAD
PSO (ring)
CDE
NCDE
LIPS
CGWO
Best
900.48
909.97
952.14
900.03
900
901.75
Worst
938.54
931.11
990.52
974.83
920.67
901.75
Mean
911.58
919.62
970.566
935.86
911.95
901.75
std
9.15
7.41
14.25
26.54
7.60
4.58E-05
Best
5081.11
1070.00
2701
1139.20
6287.1
1070
Worst
7607.26
1070.00
12551
12267
1070.4
1070
Mean
6812.02
1070.00
9622.02
7524.30
4079.02
1070
std
757.76
5.30E-04
4062.19
5618.14
2749.6
0
Best
1100.533
1100.81
1104.6
1100.30
1102
1100
Worst
1104.028
1103.32
1107.3
1111.80
1103.3
1100.00
Mean
111.993
1102.12
1105.7
1104.26
1102.74
1100.00
std
7.56E-01
5.09E-01
1.23
4.38
4.72E-01
4.17E-04
Best
1537.06
1511.40
1313.90
1221.80
1201.2
1200
AC
F32
Statistics
CE
Problem
F33
F34
F35
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F37
2808.51
1992.80
2129.70
1476.6
1201.09
Mean
1660.58
2281.25
1623.24
1652.64
1296.68
1200.07
std
101.96
298.65
286.42
392.61
132.95
2.50E-01
Best
1519.41
5261.5
1488.1
1503
1437.6
1300
Worst
1711.13
225790
1658
1656.60
1539.9
1300
Mean
1595.73
150498.9
1594.62
1597.10
1490.24
1300
std
39.75
1.09+E05
67.83
57.28
45.67
0
Best
2050.04
1440.00
2469
1950.30
1480
1520
Worst
2607.84
2523.49
3041
2959.60
2203.9
1726.77
Mean
2401.33
2044.92
2681.24
2382.08
1820.34
1569.78
std
135.55
286.21
253.17
375.00
236.13
43.29
Best
1640.96
1640.00
1686.40
1640.50
1640
1500
Worst
1758.05
1640.00
2069.70
2073.70
1824.1
2116.28
Mean
1665.74
1640.00
1918.70
1908.20
1742.1
1680.40
std
33.164
1.2E-09
142.28
162.26
68.67
113.29
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F38
1831.72
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F36
Worst
The proposed CGWO is also compared to the well-known multi-niche metaheuristics such as EAD, RPSO, CDE, NCDE, and LIPS on multi-niche optimization problems in CEC2015. These multi-niche optimization problems contain 8 extended simple functions (i.e., F24-F31) and 7 composition functions (i.e., F32-F38). For 8 extended simple problems, the performance metric like the average number of optima found (ANOF) is used to test the behavior of these metaheuristics. If the gap between one obtained solution and one known global optimum is below 𝜀 (𝜀 = 0.1), then the peak is deemed to have been found. For the 7 composition problems, a different performance metric is adopted due to its difficulty in finding
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any good solutions. The statistic results of 5 best solutions of each algorithm are found for the 7 composition problems. Table 15 reports the best, worst, mean and standard deviation (i.e., std) of the number of optima found by different
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multi-niche metaheuristics for problems F24-F31. The optimal values in these tables are highlighted in boldface. We can find from this table that the proposed CGWO can provide promising results on most problems. Meanwhile, Table 16 presents statistic results of 5 best solutions found by each algorithm. CGWO is the best one of these compared 6 algorithms.
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Compared to other well-known multi-niche metaheuristics, it is obvious to see that CGWO shows the best performance. These results confirm our previous conclusion that the CA can contribute to balance between exploration and exploitation. The superior performance of CGWO is due to its CA mechanism. CA provides an effective topology structure for the whole
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population. As solution in CA only interacts with its neighbors for exploitation, the search can be enhanced in the niche. Meanwhile, information diffusion mechanism contributes to exploration. Consequently, a collection of such solutions
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congregates to solve the problem.
4.5 Wilcoxon sign-sum test analysis Due to the stochastic nature of these algorithms, the statistical test is necessary for providing confidential comparisons. A nonparametric statistical test called Wilcoxon sign-rank test should be carried out in this paper. This statistical test is used to detect the significant difference between the results obtained by different algorithms. The confidence level for all tests is set to 95% (corresponding to a level of significance α= 0.05). The symbol +(-) indicates that our proposed CGWO algorithm is significantly better (worse) than its counterpart. The sign = denotes that there is no significant difference between the proposed algorithm and its compared algorithm. R+ is the sum of ranks for the problems in which the first algorithm outperformed the second, and R− the sum of ranks for the opposite (Derrac, García, Molina, & Herrera, 2011). Table 7 records the CGWO with C25 is the best choice among six variants using Wilcoxon sign rank test on these results
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obtained by CGWO with different cell topology structures. Table 10 and Table 14 show that the proposed algorithm provides statistically better results compared with the other metaheuristics for benchmarks in Tables 1-2, based on the Wilcoxon sign rank test. Table 12 presents the multi-problem-based pairwise statistical comparison results using the averages of the global minimum values obtained through 30 runs of CGWO and the comparison algorithms to solve the CEC2015 with multiple global optimum. Wilcoxon sign rank test is not conducted on F32 and F36 as that all algorithms are unable to find all peaks on these problems according to Table 12. These results show that CGWO was statistically more successful than compared algorithms, with a statistical significance value α= 0.05. 4.6 CUP-time cost
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This subsection studies CPU-time cost on CGWO and GWO. According to the above experiments, concerning average values CGWO outperforms the basic GWO on 31 of 38 benchmark problems with 29 significant results. The mean CPU-time consumed values by CGWO and GWO algorithms for all test problems are plotted in Fig. 11, and they are divided into three categories in Table 17. The last column of Table 17 provides the increase rates of CPU-time brought by the incorporation of CA model. It is observed from Fig. 11 and Table 17 that good performance achieved by the combination of CA does not come for free. The CPU-time consumption of CGWO is larger than that of the basic GWO for most instances. The main reason for the large time cost of the CGWO can be analyzed as follows. CGWO searches for the promising solution in the
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current solution’s neighbors, whereas this neighborhood structure provides an information diffusion mechanism to CGWO in which population diversity is maintained longer due to the slow information diffusion through the wolf population regarding the best position of each neighborhood. Thus, there exists a delay in the information spread through the population, which escapes from a local optimum. Although CPU-time consumption of CGWO is a little larger than that of the original GWO, CGWO is better than GWO for most instances and it is well worth introducing a CA model to improve the behavior of
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CGWO.
Fig 11. CPU-Time consumption on all test functions
Table 17 CPU-time costs on three different test problems Problems
GWO
CGWO
Increase rate (%)
Unimodal problems(F1-F7)
7.026
9.96
41.8
Multimodal problems(F8-F23)
10.525
16.574
57.5
Composition problems(F24-F38)
64.484
71.892
11.5
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4.7 CGWO for engineering problems This subsection is devoted to evaluating the performance of the proposed CGWO on the real-world engineering problems, these constrained problems including tension/compression spring design, overspeed protection of a gas turbine and rolling element bearing are studied in this subsection. Since these problems contain constraints, the constraint handling mechanism as in Section 3.1 is also adopted. Compared with gradient-based optimization approaches, most metaheuristics have a derivation-free mechanism. For metaheuristics, there is no need to calculate the derivative of search spaces to find the optimal solutions. This makes metaheuristics suitable for real applications with expensive or unknown derivative information. Additionally, metaheuristics
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have superior abilities to avoid local optima over conventional optimization approaches. This is due to the stochastic nature of metaheuristics which allows them to avoid stagnation in local optima. The search space of these problems is usually unknown and very complex with a number of local optima. Therefore, metaheuristics are good options for solving these challenging real-world problems. Note that the significance test technique is not used in this subsection as that the compared results of these applications are from the existing literatures which just offer simple statistical results. 4.7.1 Tension/compression spring design
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The aim of this problem is to minimize the weight of a tension/compression spring subject to constraints on minimum deflection, shear stress, surge frequency and limits on outside diameter. This problem has three design variables defined as: the wire diameter (0.05≤x1≤2.0), the mean coil diameter (0.25≤x2≤1.3) and the number of active coils (2.00≤x3≤15). The mathematical formulation of the problem can be defined as follows:
min f x x3 2x2 x12 1
x 23 x3
71785 x14
12566 x 2 x13 x14
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1
0
4 x 22 x1 x 2
M
g 1 x g 2 x s.t . g 3 x g 4 x
140.45 x1 x 22 x3
(12)
1
5108 x12
-1 0
0
x1 x 2 1 0 1.5
This problem has been regarded as a classical benchmark for evaluating different optimization algorithms. The best
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solutions among several algorithms such as GWO (S. Mirjalili, et al., 2014), HPSO (Q. He & L. Wang, 2007), CDE (Huang, Wang, & He, 2007), CPSO (Qie He & Ling Wang, 2007) and SSO-C (Cuevas & Cienfuegos, 2014) are recorded in Table 18.
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We can observe from Table 18 that CGWO, HPSO, and SSO-C can find the best result. However, among them, only CGWO satisfies all the constraints while others violate the constraints to some extent. Table 19 presents statistical results obtained by different algorithms such as CPSO, CDE, SSO-C, G-QPSO (Coelho, 2010), UPSO (Parsopoulos & Vrahatis, 2005), PSO-DE
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(Liu, Cai, & Wang, 2010), MBA (Sadollah, Bahreininejad, Eskandar, & Hamdi, 2013), HEAA (Wang, Cai, Zhou, & Fan, 2009), and TLBO. It can be seen from Table 19 that the CGWO shows a good performance and maintains an acceptable stability since it obtains the best result with comparison to most compared algorithms in terms of NFE. The proposed CGWO achieves the best average result except HEAA, PSO-DE, and TLBO metaheuristics. However, CGWO provides competitive mean results in much less NFEs than offered by the HEAA and PSO-DE method. CGWO is also competitive to TLBO for solving this problem. Table 18 Comparison of best solution obtained from various previous studies for tension/compression spring design. variable
CPSO
GWO
HPSO
SSO-C
CDE
CGWO
x1
0.051728
0.05169
0.051706
0.051689
0.051609
0.051689066186
x2
0.357644
0.356737
0.357126
0.35671775
0.354714
0.35671786256
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x3
11.244543
11.28885
11.265083
11.288965
11.410831
11.28895856
g1
−8.25E−04
-7.9065e-05
−3.06E−06
-4.746E-06
-3.864E-05
-4.51E−10
g2
−2.52E−05
-7.5056e-06
1.39E−06
3.337E-06
-1.8289E-04
−2.234E−11
g3
−4.051306
−4.053383
−4.054583
-4.0537858
-4.048627
−4.05378
g4
−0.727085
−0.7277153
−0.727445
-0.7277287
-0.7291179
−0.727728
F
0.0126747
0.012666
0.0126652
0.0126652
0.0126702
0.0126652
Table 19 Comparison of statistical results given by different optimizers for tension/compression spring design. worst
Mean
Best
std
NFEs
CPSO
0.0129240
0.0127300
0.0126747
5.20E−04
240,000
HPSO
0.0127190
0.0127072
0.0126652
1.58E−05
81,000
G-QPSO
0.017759
0.013524
0.012665
1.26E-03
2,000
DE
0.012790
0.012703
0.0126702
2.7E−05
204,800
HEAA
0.012665240
0.012665234
0.012665233
1.4E−09
24,000
PSO-DE
0.012665304
0.012665244
0.012665233
1.2E−08
24,950
0.012669249
0.012922669
N.A
0.02294
CDE
N.A
0.012703
TLBO
N.A
0.01266576
MBA
0.012900
0.012713
CGWO
0.0127173
0.01267444
5.9E−04
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0.016717272
25,167
0.01312
7.20E−03
100,000
0.01267
N.A
240,000
0.012665
N.A
10,000
0.012665
6.30E−05
7,650
0.0126652
1.26E-05
2,000
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SC UPSO
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Algorithms
4.7.2 Overspeed protection system for a gas turbine
This problem is to maximize reliability of this system and is a mixed-integer nonlinear reliability design optimization issue
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(T. C. Chen, 2006). Overspeed detection is continuously provided by the electrical and mechanical systems. When an overspeed happens, it is necessary to cut off the fuel supply using control values (Coelho, 2009). The problem has two types of design variables (i.e. positive integers ni and real number value ri). The mathematical formulation of the problem can be
AC
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PT
written below:
max f r,n
g 1 s .t .g 2 g 3
1 1 r m
ni
i
i 1
m
v i ni i
2
(13)
-V 0
1
C ri ni i m
e
1
m
w i nie i
0.25 ni
0.25 ni
- C
0
-W 0
1
C ri i T lnri i
1 n i 10, n i Z ,0.5 ri 1 10 6 , ri
where V is the upper bound on the sum of the subsystems’ products of volume and weight, C is the upper bound on the cost of the system, C(ri) is the cost of each component with reliability ri at subsystem i, T is the operating time during which the component must not fail, W is the upper limit on the weight of the system. The input parameters of this system are listed in Table 20. Table 20 Data used in overspeed protection system of a gas turbine.
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stage
𝛼𝑖
𝛽𝑖
𝑣𝑖
𝜔𝑖
𝑉
𝐶
𝑤
𝑇
1
1.0E-05
1.5
1
6
250
400
500
1000
2
2.3E-05
1.5
2
6
3
0.3E-05
1.5
3
8
4
2.3E-05
1.5
2
7
It is noted that any improvement in the objective of this problem is very important for reliability engineering and system safety. Table 21 compared the results found in this work for the complex system with those of other research reported. The
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best result obtained by CGWO and CS (Valian & Valian, 2013) can reach at 0.99995468, which has a slight advantage over the other solvers reported. However, CS violates a constraint whereas CGWO satisfies all constraints. Therefore, CGWO outperforms other compared algorithms in terms of the best result. Statistical results for this overspeed protection system are summarized in Table 22. According to the results in Table 22, with regard to the best results, the solutions of CGWO are just slightly better than the solution found by PSO-GC (Coelho, 2009) and GA-PSO (Sheikhalishahi, Ebrahimipour, Shiri, Zaman, & Jeihoonian, 2013) on this problem. Moreover, the standard deviation of the results by CGWO is much smaller than that by
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CS, which denotes CGWO is more stable when solving this problem.
Table 21 Comparison of best solution obtained from various previous studies for overspeed protection system of a gas turbine GA-PSO
CS
CGWO
n1
5
5
5
5
n2
6
5
5
6
n3
4
4
4
4
n4
5
6
6
5
r1
0.902231
0.901628
0.90161460
0.9016347
r2
0.856325
0.888230
0.88822337
0.8499661
r3
0.948145
0.948121
0.94814103
0.94812205
r4
0.883156
0.849921
0.84992090
0.8881983
g1
-55
-55
-55
-55
-0.978964
-0.000006
1.04E-05
-1.8383E-04
g3
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F
PT
g2
M
PSO-GC
ED
Variable
-24.80188
-15.363463
-15.363
-24.8018827
0.999953
0.99995467
0.99995468
0.99995468
Table 22 Comparison of statistical results given by different optimizers for the overspeed protection system of a gas turbine Worst
mean
Best
std
NFEs
PSO-GC
0.99993800
0.99990700
0.99995300
1.1E-05
NA
GA-PSO
0.99995467
0.99995467
0.99995467
1.00E−16
NA
CS
0.99991922
0.99995336
0.99995468
4.5576E-06
10,000
CGWO
0.99994674
0.99995419
0.99995468
2.16E−09
5,000
AC
algorithms
4.7.3 Rolling element bearing This problem is proposed in (Gupta, Tiwari, & Nair, 2007; B. R. Rao & Tiwari, 2007) and its objective is to maximize the dynamic load carrying capacity of a rolling element bearing. The design variables are as follows: the ball diameter Dm, pitch diameter Db, the number of balls Z, the inner and outer raceway curvature coefficients fi and f0, KDmin, KDmax, ε, e, and ζ. The latter five parameters appear in constraints and affect the internal geometry. Z is the discrete design variable and the
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remainder are continuous design variables. According to the literatures (Eskandar, Sadollah, Bahreininejad, & Hamdi, 2012; Sadollah, et al., 2013) , decision variables fi and Db are assumed to be independent variables. Actually there exists a relationship between fi and Db by equations fi = ri/Db and f0=r0/Db where r0 and ri are constant values. Therefore, the decision variables fi and f0 in the optimization procedure can be eliminated. Meanwhile, the upper bound of Db is determined by 0.45(D-d) and minimum value between ri/min(fi) and r0/min(f0), which can result in range of Db from [0.15(D-d), min(0.45(D-d), min(ri/min(fi), r0/min(f0))]. Consequently, the mathematical formulation of the problem can be reconstructed as follows: Db fc Z max f Dm , Db , Z , K D min , K D max , , e, 33 . 647 f c Z 2 / 3 Db1.4
g1 g 2 g 3 g 4 s.t . g 5 g 6 g 7 g 8 g 9 g 10 g 11
1.72
f i 2 f 0 1 f 0 2 f i 1
K D min D d 2Db 0
2Db K D max D d 0 B w Db 0
Db 25.4 Db 25.4
(14)
0
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0.5D d Dm 0
Dm 0.5 e D d 0
Db 0.5D Dm Db 0 0.515 f i 0 f i 0.6 0
0.515 f 0 0 f 0 0.6 0
0.41
10 3
ED
1 f c 37.911 1.04 1
0
2 sin 1 Db Dm
if if
M
where
Z 1
1.8
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2/3
-0.3
0.3 1 - 1.39 2 f 0.41 i 13 1 2 f i 1
D d 2 3T 4 2 D 2 T 4 D 2 d 2 T 4 2 b 2D d 2 3T 4D 2 - T 4 - Db D r r b , f i i , f 0 0 , T D d 2Db , D 160, d 90, Bw 30, ri r0 11.033 Dm Db Db
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0 2 - 2 cos 1
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0.5D d Dm 0.6D d , 0.15D d Db min0.45D d ,min[ r0 min( f 0 ),ri min( f i )], 4 Z 50, 0.4 K D min 0.5, 0.6 K D max 0.7, 0.3 0.4, 0.02 e 0.1, 0.6 0.85.
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The CGWO and its competitive algorithms such DE and GWO are run 30 times on this problem and the best values found are recorded in Table 23. Obviously, the results obtained by CGWO are better than that found by the DE and GWO. Furthermore, Table 24 gives statistical results obtained through these algorithms and shows that CGWO outperforms DE and GWO in terms of the mean, best, and standard deviation values. It means CGWO is a very effective method for parameters optimization of a rolling element bearing. Table 23 Comparison of best solution obtained by DE, GWO, and CGWO for rolling element bearing Variable
DE
GWO
CGWO
Dm
125.69547638
125.71678875
125.69088021
Db
21.4174747008
21.4174513046
21.417475728
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11
11
11
KDmin
0.42433415670
0.48394163540
0.4
KDmax
0.64383088983
0.67494041047
0.7
ε
0.30028179945
0.30028688211
0.30051
e
0.05635192733
0.05297465987
0.02
ζ
0.63988721158
0.62514555525
0.606519856
g1
-1.1732E-03
-2.8946E-03
-4.1907E-10
g2
-13.13155
-8.95898
-14.83495
g3
-2.23321
-4.41093
-6.165049
g4
-2.22086
-2.66308
-3.22188
g5
-0.69547
-0.71678
-13.3925
-12.52688
g7
-0.012246
-1.5003E-03
g8
-2.4704E-08
-5.8728E-07
g9
-0.0850
-0.0850
g10
-2.4704E-08
-5.8728E-07
g11
-0.0850
F
81803.125
-0.68088 -4.31912
-0.014656
-3.7352E-12 -0.0850
-3.7352E-12
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g6
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Z
-0.0850
-0.0850
81801.065
81803.647
Table 24 Comparison of statistical results given by DE, GWO, and CGWO for rolling element bearing Worst
Mean
DE
76787.252
78403.648
GWO
76771
79525.009
CGWO
76789.100
81336.829
Best
std
NFEs
81803.125
2343.593
10,000
81801.065
2473.211
10,000
81803. 647
1417.442
10,000
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algorithms
To sum up, the above empirical studies indicate that the proposed CGWO has superiority over the current meta-heuristics considered. First, the statistical results for the unconstrained benchmarks show the good performance of CGWO in terms of exploitation and exploration, and then regarding convergence CGWO is also superior or competitive to its rivals. Finally, the
Conclusions and future work
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5.
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results of engineering problems demonstrate that CGWO is a promising approach.
In this paper we have proposed a CGWO by integrating CA model and GWO. The search process in CGWO is guided by
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three good wolves (solutions) but this interaction is restricted within its neighborhood, which can keep a good balance between diversity and convergence. CGWO has been compared to the other seven current metaheuristics such as LSHADE, TLBO, EBOwithCMAR, BA, NDHS, CLPSO and GWO on unimodal, multimodal and CEC2015 benchmarks. Furthermore, CGWO has been compared to some recent improved versions of GWO on unimodal and multimodal benchmarks. Then, CGWO is compared to state-of-art multi-niche metaheuristics such as EAD, CDE, NCDE, RPSO, and LIPS on multi-niche benchmarks in CEC2015. A nonparametric statistical test called Wilcoxon sign-rank test is used to detect the significant difference between the results obtained by different algorithms. CGWO is also applied successfully into engineering problems. Empirical results reveal that CGWO is an effective approach for optimization problems. Additionally, the main contributions of this work are as follows. (1) The cellular automata (CA) concept is embedded into the GWO. CA provides a given topological neighborhood
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structure. The population can be divided into many subpopulations by this given neighborhood structure. Each subpopulation forms an independent niche, which can improve its local search. (2) CGWO with a topological structure can also help to improve diversity of population. Each subpopulation can update itself in its own neighborhood. Therefore, different subpopulations have different search directions, which helps to the search diversity. The overlap between consecutive neighborhoods can provide a recessive migration mechanism without any complex mathematical equation. The information of good solutions can be shared to some extent. It can enhance its convergence of the proposal. (3) The performance of the CGWO is sensitive to neighborhood size for most instances. An appropriate neighborhood size depends on specific problems. It can be found by numerical experiments that the neighborhood size named C25 is the best
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choice for CGWO.
(4) Compared with other metaheuristics, the proposed CGWO is able to find all the optimal solutions rather than a single optimal solution for multiple peaks benchmarks, especially for CEC2015 test suit. It thus indicates that CGWO can maintain a good diversity of population.
In this study, the CGWO is proposed to solve single-objective and continuous optimization problems well. However, many
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optimization problems involve multiple objectives and discrete decision variables in real-world application. Additionally, only GWO is incorporated into the CA, while other metaheuristics are not hybridized with CA. With respect to future work, firstly, research can be extended to a multi-objective grey wolf optimization algorithm. Secondly, more metaheuristics and problems are considered to test the performance of the proposed CGWO in the comparison experiment. Thirdly, it is important to understand how to set an adaptive neighborhood size for CGWO, although a recommend neighborhood size is given in our work. Fourthly, it would be valuable to apply the CGWO into many applications such as production scheduling
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in industry.
Acknowledges
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The project was supported by the fundamental research funds for the central universities, China University of Geosciences (Wuhan) (No. CUG170688), National Natural Science Foundation of China (NSFC) under Grant nos. 51775216, 51375004
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and 51505439 References
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Caption lists Fig. 1. Position updating in GWO Fig. 2. Structure of neighborhood Fig. 3. Flow chart of the proposed CGWO algorithm Fig. 4. Pseudo code of the proposed CGWO algorithm
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Fig. 5. Position updating in CGWO Fig. 6. Convergence curve of these algorithms when solving benchmarks Fig. 7. Niching behavior of CGWO (with a population of 400) on F25 in a single run Fig. 8. Niching behavior of CGWO (with a population of 400) on F26 in a single run Fig. 9. Niching behavior of CGWO (with a population of 400) on F28 in a single run Fig. 10. Niching behavior of CGWO (with a population of 400) on F30 in a single run Fig. 11. CPU-Time consumption on all test functions
Table 2. Multimodal benchmark functions Table 3. CEC2015 benchmark functions Table 4. Parameter setting of algorithms Table 5. Results by CGWO with different neighborhood on unimodal benchmark functions Table 6. Results by CGWO with different neighborhood on multimodal benchmark functions
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Table 1. Unimodal benchmark functions
Table 7. Wilcoxon sign rank test on the solution by CGWO with different structures for benchmarks in Tables 1-2 (a level of significance α=0.05)
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Table 8. Results by different algorithms on unimodal benchmark functions Table 9. Results by different algorithms on multimodal benchmark functions
Table 10. Wilcoxon sign rank test on the solution by different algorithms for benchmarks in Tables 1-2 (a level of significance α=0.05) Table 11. Success rate on CEC2015 with multiple global optimum
Table 12. Wilcoxon sign rank test on the solution by different algorithms for CEC2015 with multiple global optimum (a level of significance α=0.05) Table 13. Results by different improved versions of GWO and iHS on benchmarks in Tables 1 and 2
Table 14. Wilcoxon sign rank test on the solution by different improved versions of GWO for benchmarks in Tables 1-2 (a level of significance α=0.05)
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Table 15. Results by different algorithms on average number of optima found (F24-F31)
Table 16. Results by different algorithms (statistic values of 5 best solutions separated at least by 10 on F32-F38)
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Table 17. CPU-time costs on three different test problems
Table 18. Comparison of best solution obtained from various previous studies for tension/compression spring design. Table 19. Comparison of statistical results given by different optimizers for tension/compression spring design Table 20. Data used in overspeed protection system of a gas turbine.
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Table 21. Comparison of best solution obtained from various previous studies for overspeed protection system of a gas turbine Table 22. Comparison of statistical results given by different optimizers for the overspeed protection system of a gas turbine
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Table 23. Comparison of best solution obtained by DE, GWO, and CGWO for rolling element bearing
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Table 24. Comparison of statistical results given by DE, GWO, and CGWO for rolling element bearing