Solving the mesh router nodes placement in wireless mesh networks using coyote optimization algorithm

Wireless Mesh Networks (WMNs) have rapid real developments during the last decade due to their simple implementation at low cost, easy network maintenance, and reliable service coverage. Despite these properties, the nodes placement of such networks imposes an important research issue for network operators and influences strongly the WMNs performance. This challenging issue is known to be an NP-hard problem, and solving it using approximate optimization algorithms (i.e. heuristic and meta-heuristic) is essential. This motivates our attempts to present an application of the Coyote Optimization Algorithm (COA) to solve the mesh routers placement problem in WMNs in this work. Experiments are conducted on several scenarios under different settings, taking into account two important metrics such as network connectivity and user coverage. Simulation results demonstrate the effectiveness and merits of COA in finding optimal mesh routers locations when compared to other optimization algorithms such as Firefly Algorithm (FA), Particle Swarm Optimization (PSO), Whale Optimization Algorithm (WOA), Genetic Algorithm (GA), Bat Algorithm (BA), African Vulture Optimization Algorithm (AVOA), Aquila Optimizer (AO), Bald Eagle Search optimization (BES), Coronavirus herd immunity optimizer (CHIO), and Salp Swarm Algorithm (SSA).


I. INTRODUCTION
W IRELESS Mesh Network (WMN) is considered as an emerging communication technology that has seen an accelerated deployment since 2005 all over the world [1]- [3]. The main reasons behind this accelerated deployment are due to its easy implementation at a low cost, dynamic selforganization, self-configuration, and self-adaptive nature. In addition, WMN becomes a very attractive technology for several applications such as broadband home networking, education, healthcare, corporate networks, industrial automa-tion, disaster management, military, and rescue operations [4], [5]. WMN is composed of three types of nodes (i.e. Mesh Routers (MRs), Mesh Gateways (MGs), and Mesh Clients (MCs)) as illustrated in Figure 1. MCs such as desktops, mobiles, laptops, PDAs, and Pocket-PCs connect to the internet through MRs, MRs relay traffic to and from MGs which are connected to internet infrastructure.
Despite the desirable features of WMNs, there are still a number of problems and issues that prohibit them from being broadly deployed on a big scale. The most important issues are namely nodes placement problem [7]- [9] and security [10]- [14]. These issues have gotten a lot of attention in the literature.
The performance of WMN is mostly influenced by the bad positioning of mesh nodes (MR and/or MG) [15]. Consequently, many interferences and congestion are resulted causing considerable packet loss, high delays, and low throughput. To overcome these drawbacks, network operators must adopt efficient optimization techniques for WMN nodes placement.
WMN nodes placement is known to be an NP-hard problem [9]. So meta-heuristics have been presented as successful optimization algorithms to solve it providing acceptable solutions in a reasonable execution time.
Several works based on meta-heuristics have been proposed in the literature to solve the nodes placement problem in WMNs. Most of the proposed works considered stationary topology [3], [16]- [25] while others investigated the dynamic placement of mesh nodes subject to client mobility [26]- [29].
To deal with the static variant of the WMNs nodes placement problem, three algorithms have been proposed by Xhafa et al., including Simulated annealing (SA) [16], Hill Climbing (HC) [17], and Tabu Search (TS) [18]. The three algorithms were evaluated in terms of user coverage and network connectivity. The evaluation is done based on 48 benchmark instances using different mesh clients distributions with different grid sizes.
In the work of Nouri et al. [3], an accelerated PSO algorithm (APSO) was also proposed to tackle the mesh routers placement problem in a static environment. APSO was validated in terms of coverage and connectivity and results confirmed its effectiveness when compared with the linearly decreasing weight PSO algorithm.
In the same context, Sayad et al. proposed three new algorithms based on Chemical Reaction Optimization (CRO) algorithm [19], Firefly optimization (FA) algorithm [20], and Electromagnetism-like Mechanism (EM) meta-heuristic [21]. These algorithms were validated using many generated instances with various number of mesh clients and mesh routers, taking into account the coverage and connectivity metrics. Obtained results confirmed the superiority and effectiveness of these algorithms in terms of user coverage and connectivity.
Evolutionary algorithms (e.g. Genetic Algorithm (GA)) have been popular optimization algorithms in this area too [22]- [25]. For instance, the mesh router nodes placement problem was solved by Xhafa et al. [22] as a facility location problem using GA. It took into account user coverage and network connectivity metrics. In [23], an improved GA based on coupling GA with Minimum Spanning Tree (MST) was proposed to optimize cost and coverage metrics. Authors in [24] have proposed an enhanced GA, called MOGAMESH, for optimizing WMN topology by maximizing the user coverage percentage and minimizing the nodes degree. Similarly, two other variants of GA were applied in [25] namely the Non-dominated Sorting Genetic Algorithm-II (NSGA-II) and Multi-Objective Genetic Algorithm (MOGA), where cost, coverage, and reliability are taken into account as performance metrics. These have been some of the best attempts to employ the multi-objective algorithm to simultaneously optimize multiple objectives in this area [30]- [32].
In [26]- [29], several methods have been proposed to tackle the dynamic variant of the mesh nodes placement issue. An improved PSO algorithm based on the integration of restriction coefficient into PSO was proposed in [26] to deal with this problem. In another similar work, Lin et al. [27] proposed an improved BA based on the incorporation of the dynamic search scheme into the original BA. The improved BA was validated based on 10 instances, taking into account the coverage and connectivity parameters. In [28], authors focused on the so-called social-aware dynamic router nodes placement in WMNs. They solved this problem using an enhanced PSO including a social-supporting vector, called a social-based-PSO. SA approach was applied in [29] to find the dynamic placement of mesh routers. In addition to user coverage and network connectivity, this approach reduces the average distance traveled by routers. Table 1 summarizes some representative works using meta-heuristics for solving the WMNs nodes placement problem. According to the No Free Lunch (NFL) theorem [33], [34], there is no optimization algorithm that can solve all optimization problems effectively. In fact, most optimization algorithms are able to solve some optimization problems while they fail to solve other problems.
COA is precisely, one of the recent meta-heuristics introduced by Pierezan and Coelho in 2018 [35]. It is characterized by the use of few tuning parameters and the strong capability of exploration and exploitation phases. It was applied to solve a large variety of difficult optimization problems such as image segmentation [36], feature selection [37], economic load dispatch [38], wireless sensor networks [39], and heavy-duty gas turbine operation [40]. For each one of these applications, COA proved its successfulness and merits. All these advantages motivate us to apply COA for solving the mesh routers placement problem in WMNs, which is, to the best of our knowledge, never solved with this This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.  meta-heuristic.
In this paper, we present the implementation of COA for solving the mesh router nodes placement issue. We evaluate the performance of COA using many generated instances under various settings, taking into account the coverage and the connectivity metrics. The major contribution of this paper is given below: • Implementation of COA, FA, PSO, WOA, GA, BA, AVOA, AO, BES, CHIO, and SSA to tackle the mesh router nodes placement issue; • Assessment and comparison of the performance of the proposed COA with FA, PSO, WOA, GA, BA, AVOA, AO, BES, CHIO, and SSA using various scenarios under different settings. The remainder of this paper is organized as follows. We describe the formulation of the mesh router nodes placement problem in section 2. In section 3, We propose the new bio-inspired meta-heuristic COA to solve the formulated problem. Furthermore, we evaluate the performance of our approach and we provide results in section 4. Finally, we give the main findings and the potential perspectives in section 5.

II. MESH ROUTER NODES PLACEMENT PROBLEM FORMULATION
In this section, we describe the system model and the formulation of the mesh router nodes placement problem. For better readability, the main notations used in this paper are summarized in Table 2.

A. SYSTEM MODEL
WMN can be described mathematically as an undirected graph G = (V, E) where V is the set of network vertices (nodes) and E is the set of edges (links) between these vertices. The network G is formed by a set of disjoint subnetworks. In this paper, we consider the WMN with two types of nodes such as mesh clients and mesh routers. Thus  The i-th mesh router mc i The i-th mesh client CR i Coverage radius of the i-th mesh router Network connectivity ψ(G) Alpha coyote in the p-th group at iteration t cultr p,t j The cultural tendency of the p-th group at iteration t R p,t The ranked social condition of the coyote in the p-th group at iteration t age p,t c Age of the c-th coyote in p-th group at iteration t Pα Probability of leaving a group Ps Scatter probability Pz Association probability r1,r2,r j ,K j Random numbers in the range [0 1] with radio interface with the same coverage radius CR 1 = CR 2 = ... = CR m . Two mesh routers mr i and mr j can be connected if and only if the distance VOLUME 4, 2016 between them does not exceed two time the coverage radius CR i.e. d(mr i , mr j ) ≤ 2CR. • M C is the set of n mesh clients M C = {mc 1 , mc 2 , ...., mc n }, we assume that mesh clients are randomly distributed in 2D rectangle area of dimension W xH. A mesh client mc i is said covered by a mesh router mr j if it is within the coverage radius of this router: d(mc i , mr j ) ≤ CR. It can be associated at most to one router. It can be within coverage radius of various routers but it is associated with the closest router.

B. PROBLEM FORMULATION
As per the nature of studied environments (static or dynamic) and the nature of deployment spaces (discrete or continuous), several variants of the WMN router nodes placement problem can be found. In this paper, the static continuous mesh routers nodes placement problem was considered. Therefore, the main goal is to find the optimal placement of m mesh routers in a 2D area of dimensions W xH, depending on the location of n mesh clients.
The problem studied in this work considers two main objectives that need to be optimized: • User coverage: It represents the number of covered users by at least one mesh router according to the following equation: where σ ij defines the coverage variable represented as follows: (2) • Network connectivity: It is defined as the geant subnetwork among k formed sub-networks with regard to the number of mesh nodes (mesh routers and mesh clients). It is calculated as follows:

III. COYOTE OPTIMIZATION ALGORITHM
The Coyote Optimization Algorithm (COA) was introduced by Pierezan and Coelho [35] as a population-based metaheuristic for solving global optimization problems. This optimization algorithm is inspired by the social behaviors of the Canis Latrans species that reside mainly in North America. The population of coyotes is partitioned into G p groups with C c coyotes in each group. Therefore, the population size can be defined by the multiplication of G p *C c . Every coyote is considered as a possible (candidate) solution and its social condition refers to the objective function or decision variable. In this regard, the social condition of the c-th coyote in the pth group at iteration t can be represented mathematically as follows: where D is the search space dimension. This social behavior involves adapting coyotes to the environment named f t p,t c ∈ R. The adaptation of the coyote to its respective current social condition is validated using the following equation: At the beginning of the process, the coyotes are randomly assigned to the groups. However, coyotes have sometimes a tendency to abandon their groups in order to join other groups or to become solitary. The probability P a that a coyote leaves its group is given as follows: As the value of P a cannot exceed unity, the number of coyotes per group is limited to 14. This mechanism increases the cultural exchange between all the coyotes in the global population. In each group, the leader of the coyotes, also called the alpha coyote, is the coyote that achieves the best objective function. Considering a minimization problem, the alpha coyote can be determined mathematically by the following equation: The cultural tendency of each group is expressed as follows: where R p,t is the ranked social condition of the coyote inside the p-th group at iteration t for every j in the search space D.
After determining the global best alpha p,t and the cultural tendency cultr p,t of each group, new social conditions of coyotes are updated using the following equation: where SC p,t cr1 and SC p,t cr2 are social conditions of random coyotes cr 1 and cr 2 , respectively. r 1 and r 2 are random numbers in the range [0, 1].
The fitness value of the new social condition is calculated as follows: The selection of final social conditions is based on the computation of new solutions. Each one is compared with the worst solution in each group and the best one is then kept as it is shown in the following equation: In COA, the two main biological events of life (i.e. the birth and the death) are modeled considering the age of each coyote age p,t c ∈ N in the group p. The new coyote is born by combining two parents SC p,t r1,j and SC p,t r2,j chosen randomly. The process of birth can be mathematically given as follows: where r 1 and r 2 are the parents coyotes selected randomly from the group p. j 1 and j 2 are two random dimensions of the problem. r j and K j represent random numbers in the range [0, 1]. P s and P z denote the scatter and the association probabilities, respectively. They are calculated as follows: The main steps of COA are given below: Step 1: Generate randomly G p *C c solutions and evaluate the adaptive function.
Step 2: Execute the following operations inside each group • Detect the alphas of the group.
• Compute the cultural tendency of the group.
• Update the social conditions of the coyotes in the group.
• Evaluate the new social conditions. • Do the adaptation.
• Develop birth and death life cycle.
Step 3: Transition of coyotes among random groups.
Step 4: Repeat step 2 until the stop criterion is reached.
Step 5: Output the best social condition of the adapted coyote.
The corresponding flowchart of COA is given in Figure 2.

IV. COA ALGORITHM APPLIED FOR SOLVING THE MESH ROUTER NODES PLACEMENT PROBLEM
The mesh router nodes placement problem is known to be an NP-hard problem, and several meta-heuristics have proven their success in this field. As part of the permanent search for the optimal solution to cope with this issue, we choose the application of COA on WMN for the following reasons:  [36], feature selection [37], economic load dispatch [38], and wireless sensor networks [39]. • COA is a population-based meta-heuristic that is easy to implement with only two control parameters. • COA has shown good results when solving several placement problems such as sizing and location of renewable distributed generations [41]- [43], optimal placement of photovoltaic distributed generations [44]- [47], and perfect position of static compensators [48]. • Presence of some similarities between the mesh routers placement problem and the COA theory as illustrated in Table 3.

A. OBJECTIVE FUNCTION
Our approach aims to find the optimal placement of a given number of mesh routers that maximizes simultaneously the user coverage and network connectivity metrics. As defined earlier, user coverage refers to the number of mesh clients covered by at least one router, whereas, network connectivity is defined by the size of the biggest sub-network among k formed sub-networks in terms of the mesh nodes number. These metrics are involved to define the fitness function f used to assess the quality of solutions. The objective function is given as follows: Where G is the graph corresponding to the solution SC p c and λ is a floating parameter in the range [0, 1] which is used to control the importance of metrics. So, according to this objective function formulation, the tackled problem is then considered as a problem of maximization of f (SC p c ). However, the original version of COA given by Pierezan and Coelho [35] was defined for problem of minimization. Thus, it is essential to turn our objective function into a minimization function. Consequently, we have defined another function g as follows:

B. SOLUTION REPRESENTATION
The resolution of a mesh routers placement problem consists of determining the placement of all mesh routers. In this paper, the solution of each coyote, which is a mesh routers placement solution, is represented by an array SC p c = {x p c1 , y p c1 , x p c2 , y p c2 , .....x p cm , y p cm }, where (x p ck , y p ck ) are the  Table 4 represents the solution of deploying four mesh routers in an area of 2000 × 2000.

C. INSTANTIATING OF COA TO THE MESH ROUTERS PLACEMENT PROBLEM
As described earlier, COA is a population-based metaheuristic for solving global optimization problems. It has proven to be an effective method for solving several placement problems. So in this section, we will describe how COA is adapted to solve the mesh routers placement problem as illustrated in Algorithm 1:

V. SIMULATION RESULTS AND DISCUSSION
In this section, we study the performance of the proposed COA in solving the mesh router nodes placement problem in WMNs. Thus, COA is compared with ten algorithms namely FA [20], GA [49], PSO [28], WOA [50], BA [27], AVOA [51], AO [52], BES [53], CHIO [54], and SSA [55]. Its performance is assessed considering three metrics: i) user coverage; ii) network connectivity; iii) objective function value. The proposed COA and the ten compared algorithms are implemented in MATLAB and all experiments are carried out on a Core i7 2.5 GHz-CPU machine. A rectangle deployment area of 2000m x 2000m is considered in all simulations. The number of mesh routers is varied from 5 to 40 for covering 50 to 300 mesh clients. Mesh clients are randomly distributed in the deployment area. The total number of iterations is 1000. The results presented in this section are obtained after an average of 50 runs. The parameters used during the simulation are summarized in Table 6, and algorithms' parameters are described in Table 5. We evaluate and analyze the performance results of the proposed COA by investigating the impact of varying the number of mesh   of 4km 2 ). Yellow circles represent installed mesh routers whereas purple circles represent mesh clients. A solid line between two mesh routers indicates that these routers are  within the transmission of each other.

A. IMPACT OF VARYING THE NUMBER OF MESH CLIENTS
In this scenario, we varied the number of mesh clients from 50 to 300 with a fixed number of mesh routers. Table 7 illustrates the impact of increasing the number of mesh clients on user coverage, network connectivity, and fitness function. Its graphical representation is shown in Figure 14. Figure 14(a) shows the users' coverage while varying the number of mesh clients. It is observed that the users' coverage increases when increasing the number of mesh clients. It is also shown that our approach covers up to 10.6%, 9%, 14.22%, 32.42%, 41.48%, 13.62%, 37.59%, 20.42%, 7.69%, and 20.56% more clients than FA, GA, PSO, WOA, BA,  AVOA, AO, BES, CHIO, and SSA, respectively.
In terms of connectivity, Figure 14(b) shows that the connectivity increases when increasing the number of mesh clients. It is demonstrated that the network connectivity is increased considerably by our approach. More precisely, connectivity is increased by up to 8.74%, 6.4%, 10%, 35%, 37.40%, 13.35%, 33.65%, 15%, 6.85%, and 16.81% when compared to FA, GA, PSO, WOA, BA, AVOA, AO, BES, CHIO, and SSA, respectively. Results illustrated in Figure  14(c) demonstrate that the fitness value decreases with the increase of the number of mesh clients. In fact, more routers are needed to cover the added mesh clients. However, the number of mesh routers is fixed. Consequently, the new added mesh clients may be not covered by the deployed   routers leading to a decrease in coverage and connectivity that constitute the fitness value. The obtained results revealed that COA gives better results than FA, GA, PSO, WOA, BA, AVOA, AO, BES, CHIO, and SSA.

B. IMPACT OF VARYING THE NUMBER OF MESH ROUTERS
The impact of varying the number of mesh routers (from 5 to 40 mesh routers) on coverage, connectivity, and fitness value are given in Table 8 and Figure 15. Figure 15(a) shows the evolution of users' coverage when varying the number of mesh routers. It is observed that  The effect of varying the number of mesh routers on network connectivity is shown in Figure 15(b). It is clearly seen that the network connectivity increases when increasing the number of mesh routers. In fact, when adding more routers, the number of formed sub-networks will be decreased by connecting some sub-networks to form bigger sub-networks. Consequently, the size of the biggest sub-network will be increased until including all mesh nodes. It is also demonstrated that COA forms the biggest sub-network in all cases. More precisely, the network connectivity using COA is increased up to 10.39%, 10.43%, 8%, 25.71%, 39.2%, 17.47%, 35%, 15.41%, 15.41%, 7.77%, and 17.37% more than FA, GA, PSO, WOA, BA, AVOA, AO, BES, CHIO, and SSA, respectively.
Results illustrated in Figure 15(c) show that the fitness value is proportional to the number of mesh routers. Thus, as the number of mesh routers increases, the fitness value will increase for all algorithms. Again, the proposed COA outperforms FA, GA, PSO, WOA, BA, AVOA, AO, BES, CHIO, and SSA algorithms when the number of mesh routers exceeds 5.

C. IMPACT OF VARYING THE ROUTER COVERAGE RADIUS
The impact of varying the router coverage radius (from 50 m to 400 m) on coverage, connectivity, and fitness value is FIGURE 11. The optimal placement of mesh routers obtained using AVOA.

FIGURE 12.
The optimal placement of mesh routers obtained using SSA. illustrated in Table 9. Its graphical representation is shown in Figure 16. Figure 16(a) illustrates the influence of varying the mesh router coverage radius on coverage. Results presented in this figure show that when increasing the radius coverage of every mesh router, the coverage metric will increase automatically. In fact, when increasing the radius coverage, mesh routers cover a large area until covering approximately all mesh clients (when coverage radius exceeds 300 m for most of the algorithms). Moreover, COA outperforms other algorithms for all cases. More precisely, it covers up to 9%, 17.15%, 13.25%, 33%, 41.12%, 27.62%, 33.75%, 30.45%, 9%, and 27.25% more clients than FA, GA, PSO, WOA, BA,AVOA, AO, BES, CHIO, and SSA, respectively.
Results presented in Figure16(b) demonstrated that the network connectivity is proportional to the mesh router coverage radius. When increasing the radius coverage of every mesh router, the network connectivity will increase too. In fact, when increasing the radius coverage, each mesh router has the capability to cover more clients and to connect to other mesh routers. Consequently, the size of the biggest sub-network will increase until connecting approximately all mesh nodes. COA performs better than other algorithms when the coverage radius exceeds 50 m. More precisely, it increases the network connectivity up to 7.89% (8.52%, 8.46%, 29.26%, 53.12%, 17%, 40.86%, 19%, 11% and 26.12%) than FA (GA, PSO, WOA, BA, AVOA, AO, BES, CHIO, and SSA, respectively).
Results described in Figure16(c) showed that the fitness value is increased when increasing the mesh router coverage radius. It is revealed that our approach COA outperforms FA, GA, PSO, WOA, BA, AVOA, AO, BES, CHIO, SSA algorithms.  The convergence analysis of COA, FA, GA, BA, WOA, PSO, AVOA, AO, BES, CHIO, and SSA algorithms is shown in Figure [ [17][18][19][20] and Table 11. We considered four network instances of different sizes (i.e. (a) Instance 1 , (b) Instance 2 , (c) Instance 3 , (d) Instance 4 ) as described in Table 10. The convergence process is done according to two parame-ters: convergence speed and convergence efficiency (fitness value). Each obtained result is an average of 50 experiments. Results reported in Table 11 show that WOA converges more quickly than other algorithms. However, it provides less interesting solutions when compared to COA, GA, FA, PSO, AVOA, AO, BES, CHIO, and SSA algorithms. The fitness value obtained by COA is the best. This is due to its unique structure and search mechanisms. For instance, the random transition of coyotes among groups promotes the population diversity. In addition, using alphas dictates the movement of other coyotes during the cultural interaction phase providing a good balance between exploration and exploitation phases. Furthermore, the biological events (birth and death) serve as a mechanism to retain good solutions and eliminate weaker ones as the algorithm progresses.  For one year, he held a postdoctoral position at E.T.S., University of Quebec. He was working on developing an application for a newborn cry-based diagnosis system with the integration of interaction context, supported by the Bill and Melinda Gates Foundation. He is currently an Associate Professor with the College of Computers and Information Technology, Taif University, Saudi Arabia. He has published papers in national and international conferences and journals. His research interests include multimodal systems, pervasive and ubiquitous computing, IoT, AI and context-aware systems.
SEYEDALI MIRJALILI (Senior Member, IEEE) is the director of the Centre for Artificial Intelligence Research and Optimization at Torrens University Australia. He is internationally recognized for his advances in Swarm Intelligence and Optimization, including the first set of algorithms from a synthetic intelligence standpoint -a radical departure from how natural systems are typically understood -and a systematic design framework to reliably benchmark, evaluate, and propose computationally cheap robust optimization algorithms. He has published over 200 publications with over 25,000 citations and an H-index of 55. As the most cited researcher in Robust Optimization, he is in the list of 1% highlycited researchers and named as one of the most influential researchers in the world by Web of Science since 2019. He is a senior member of IEEE and an associate editor of several journals including Neurocomputing, Applied Soft Computing, Advances in Engineering Software, Applied Intelligence, and IEEE Access. His research interests include Robust Optimization, Engineering Optimization, Multi-objective Optimization, Swarm Intelligence, Evolutionary Algorithms, and Artificial Neural Networks. He is working on the application of multi-objective and robust meta-heuristic optimization techniques as well.
AMAR RAMDANE-CHERIF received his Ph.D. degree from Pierre and Marie university of Paris in 1998. He is currently Since 2000, he is currently a Professor at the University of Versailles, Saint Quentin en Yvelines, France. His research interests include Software architecture, dynamic architecture, architectural quality attributes, architectural styles and design patterns. VOLUME 4, 2016