Efficient Fog Node Placement using Nature-Inspired Metaheuristic for IoT Applications

Managing the explosion of data from the edge to the cloud requires intelligent supervision such as fog node deployments, which is an essential task to assess network operability. To ensure network operability, the deployment process must be carried out effectively in terms of two main factors: connectivity and coverage. The network connectivity is based on fog node deployment which determines the physical topology of the network while the coverage determines the network accessibility. Both have a significant impact on network performance and guarantee the network QoS. Determining an optimum fog node deployment method that minimizes cost, reduces computation and communication overhead, and provides a high degree of network connection coverage is extremely hard. Therefore, maximizing coverage as well as preserving network connectivity is a non-trivial problem. In this paper, we proposed a fog deployment algorithm that can effectively connect the fog nodes and cover all edge devices. Firstly, we formulate fog deployment as an instance of multi-objective optimization problems with a large search space. Then, we leverage Marine Predator Algorithm (MPA) to tackle the deployment problem and prove that MPA is well-suited for fog node deployment due to its rapid convergence and low computational complexity compared to other population-based algorithms. Finally, we evaluate the proposed algorithm on a different benchmark of generated instances with various fog scenario configurations. The experimental results demonstrate that our proposed algorithm is capable of providing very promising results when compared to state-of-the-art methods for determining an optimal deployment of fog nodes.

computing and edge computing are regarded as an extension of cloud computing that can provide distributed computing, storage, and networking functions close to the IoT sensors at the edge network to serve the IoT applications needs which support low latency, real-time response, location awareness, devices mobility, scalability, heterogeneity, transient storage, high-speed data dissemination, decentralized computation, and security [3].The enormous resulting requests from IoT sensors and edge devices could lead to a data explosion through the network, thereby an efficient network deployment is required.In most Ad-hoc networks, the deployment of computing nodes is accomplished either in a pre-planned or in an ad-hoc method.The pre-planned deployment method is used when the deployment field access is limited and the deployment cost is not expensive.While the ad-hoc manner is used when the deployment field is large, costly, and consists of many computing nodes [4].
The deployment strategy is constructed on the basis of several considerations such as the IoT sensor and edge device location, system functionality, and the environments.Also, it may differ from one area to another according to its nature, whereas in an open area, the deployment tries to cover a wider area than the closed area such as inside a building.There are plenty of deployment possibilities for user coverage and connectivity needs, for example in critical situations such as natural disasters, oil rigs, mines, battlefield surveillance, high-speed mobile video gaming, or in public transport [5].Finding an optimal deployment for a wide area is a challenge that could suffer from user density and the unregulated deployment surface compared to small areas.Because the performance of the entire system may be impacted by the deployment of computing nodes which is considered a crucial concern [6].Therefore, in this work, we concentrate on providing an optimal fog node placement within the edge network to serve the boundary edge devices and IoT sensors in a specific area.
In order to deploy the fog computing nodes effectively, we need a full understanding of the relation between the network topology and node density, subjected to different factors such as location and density [7].Hence, some issues should be addressed to enhance the network performance including connectivity, coverage, reliability, accessibility, etc.If the fog nodes are installed without taking into account restricting factors of the real region of interest and the underlying topology, it can cause low network coverage and connectivity.
The node deployment problem has been investigated with different networks and in different environments.It remains a very challenging problem that has been proven to be NP-Hard [8].In order to address these issues, metaheuristic techniques have been developed, although they typically only provide optimal local solutions [9].
In comparison to other approaches, metaheuristic methods have experienced the greatest success and development in addressing a variety of real-world optimization issues.The computing scenario considered for this work consists of IoT sensors and edge devices located at the edge of the network and fog nodes located within the fog layer.As the problem showed its NP-Hard, we propose a MPO population-based metaheuristic algorithm to solve the fog node deployments within the edge network devices in an efficient way and evaluate the network coverage and connectivity effect on the performance of the network.Our contributions in this study are summarized as follows: • To effectively connect the fog nodes and cover all edge nodes, we propose a multi-objective aggregate objective function aimed at maximizing edge device coverage and fog node connectivity at the same time, ensuring that all devices and fog nodes are connected and covered.This will effectively enhance the network performance and guarantee the network QoS.• We provide an MPA metaheuristic nature-inspired optimization algorithm, where the objective function is set to achieve maximum coverage and connectivity for the network.We analyze how various parameters affect the network's performance and determine the most effective ways to optimize.• For seamless computation, the proposed algorithm is implemented and tested in various network settings.The results show a significant improvement in coverage and connectivity compared to other methods.The rest of the paper is organized as follows: Section II reviews existing research of fog node deployment.Section III describes the system modelling and problem formulation.Section IV details the proposed metaherictic algorithm for fog node deployment.Section V presents the evaluation details and experiment results.We conclude the paper and outline future research directions in Section VI.

II. R
Today networks require precise computing node placement within network architecture, for reliable measurement and efficient data transmission.For this purpose, node deployment procedures are frequently mentioned as one of the most crucial methods; Although several researchers have been conducted to address this issue, the existing solutions must be improved upon or replaced with new ones on a regular basis due to their limitations.To achieve this, it is possible to employ optimization algorithms to find the optimum node locations that meet the specified criteria.
Heuristics and metaheuristic methods have been used to solve the Fog Node deployment Problem (FND) with different objectives.Authors in [10] initially addressed the coverage problem in WSN and demonstrated that it is NPhard.They concentrated on the issue of improving the computing node's coverage for a given number of sensors.In [11] authors suggested an improved Artificial Bee Colony method to optimize the lifetime of a two-tiered wireless sensor network through optimum relay node deployment.The dimension of the problem is first integrated into the candidate discovery formula, and the local search is changed based on the fit of the issue and the number of iterations which assists in balancing the algorithm's exploration and exploitation capacity.In [12], the authors proposed a particle swarm optimization (PSO) algorithm to optimize the node network deployment and improve the adaptive ability of the network, however, the algorithm has the disadvantage of falling into the local optimum.In [13], the authors presumed that the network's topology is in hexagonal form and suggested an improved virtual spring force algorithm (VSFA) to deploy the network nodes within.This essentially decreases the area of vulnerability, but the proposed strategy has not been evaluated in a complex setting It has just been tested under perfect circumstances.As in [13], authors in [14] performed their evaluation with an ideal deployment environment presuming that nodes are homogeneous, where an enhanced fish swarm algorithm (AIFS) was adopted to optimize the node's deployment by targeting the coverage rate, which greatly increases the network coverage area and reduces the energy usage, Authors in [15] [16] concentrated on improving energy utilization, energy balance amount, and network coverage, whereas in [15] they combined three optimization objectives into a single objective in a linear weighted fashion, then used the whale group algorithm (WGA) to optimize the single objective function.Although the approach is simple, it consumes too long time for computation.In [16] authors only consider the deployment of homogeneous nodes and the presence of obstacles and they use two versions of a multi-objective evolutionary algorithm one based on decomposition and the other to jointly optimize the model.
Authors on [17] address fault tolerance and connectivity issues in WSNs.They concentrated their efforts on connection restoration and introduced a simple and fast algorithm for motion k-connectivity reconstruction, which separates nodes into critical and non-critical clusters based on their failure to lower k.When a crucial node fails, the algorithm grabs and transfers the non-critical nodes.The authors of [18] presented methods for estimating connectivity in IoT-enabled wireless sensor networks.They introduce a simple and efficient method (PINC) for movement-based k-connectivity restoration that splits nodes into critical and non-critical groups.When a crucial node fails, the PINC algorithm takes over and transfers the non-critical nodes.This approach transfers a non-critical node with the fewest of movement cost to the position of the failed mote.
Unlike previous works which focused on placing nodes in a discrete grid area, which limits the positions of nodes.Our approach gives us high flexibility to place fog nodes within a continuous region of interest.In addition, previous approaches optimized network connectivity and user coverage in two hierarchical ways, while in our work, we considered a multi-objective aggregate function to optimize both of them at the same time, because user coverage may also be crucial in some practical services.Some experimental results will be provided to show the advantage of simultaneous optimization.Our approach gives us high flexibility to place fog nodes within a continuous region of interest.In addition, previous approaches such as in [19,20] optimized network connectivity and user coverage in hierarchical ways, while in our work, we considered a multi-objective aggregate function to optimize both of them at the same time, because user coverage may also be crucial in some practical services.

III. S
In this section, we introduce a fog deployment model within the IoT-fog-cloud infrastructure.The studied system includes IoT sensors, devices, and fog nodes as shown in figure 1.In order to fulfill terminal IoT edge nodes requirements for high throughput and high quality of service while keeping costs low IoT-edge-based fog infrastructure has been presented in Figure 1.Which presents the core concepts of the model involving system architecture, coverage, and network connectivity, as well as multi-objective fog node deployment challenges.I.

B. System model assumptions
We investigate a deployment scenario where both fog nodes and IoT-edge nodes are considered static.To respond to a real network deployment scenario in practice, Each fog node has a different length of communications radius, and fog nodes can communicate with each other via their radio coverage [21].On the other hand, edge nodes only have the essential functions for network connectivity, but do not have the function of gateways or bridges [22].Hence, edge nodes must go through fog nodes to communicate with other nodes.Moreover, since the locations of fog nodes are determined on the basis of edge nodes' locations, it is necessary to know the locations of edge nodes in advance [23].Briefly, to ensure network connectivity and coverage, the following assumptions have been established: • The fog nodes and IoT-edge nodes are deployed uniformly and randomly over the entire network and are considered static inside the region of interest.
• Each fog node in the region of interest has access to the cloud center via cellular networks; the terminal nodes' placements are considered fixed.• To adapt to the heterogeneity of terminal nodes in practice, each fog node    is assumed to have a different transmission range     .• A fog node can only be attached to a certain number of edge nodes.Note that Euclidean distance is an important metric for determining network accessibility and connectivity.Therefore, two conditions must be satisfied: 1) An Iot-edge node    is considered connected, if and only if it is covered by at least a fog node    , as shown in eq(1: . 2) Two fog nodes,    , and    are regarded as connected.if and only if they are in the same transmission range, as shown in eq(2: Figure 2 demonstrates an instance of the introduced problem, in this example nodes consist of two types: fog and IoT-edge nodes, which consists of 10 fog nodes n = 10 and 80 IoT-edge devices m = 80, located within the deployment area, each fog node has a varied communication range.If two fog nodes are in the communication range of each other, they will be linked via a dark link (e.g., see the dark link between fog nodes 1 and 2).If an IoT-edge node is located within the communication range of a fog node, it will be connected by a thin black link to the nearest fog.Furthermore, the topology graph includes three sub-graph components, the largest of which is 5 (i.e.,  () = 5), and 68 IoT-edge devices are covered (i.e., Φ() == 68).If we shift certain fog nodes toward the most congested part of the IoT-edge device, as illustrated in Figure 2a, practically all of the sub-graphs will be combined into a single giant graph with size 10 (i.e., Φ() = 10), and 76 IoT-edge devices will be covered (i.e.,  ()= 76).Therefore, both metrics: edge coverage and network connectivity can be improved by changing the locations of some fog nodes.

C. Problem Formulation
We focus on determining near-optimal fog node placement by placing fog nodes in appropriate positions within the region of interest while ensuring connectivity and maximizing coverage.The network coverage refers to the number of covered edge nodes while the connectivity refers to connected fog nodes within the network.To address this issue, it is not practical to analyze the whole network, which is hard or even impossible to provide a connected network that covers all edge nodes due to the network dis-connectivity which can be treated separately instead.Therefore, in our study, we target the greatest sub-network i.e. largely connected subnetwork.By assuming L is the initially given location set of fog nodes  = { ( 1 ,  1 ) ,  ( 2 ,  2 ) , . . .,  (  ,   )}.
Our objective is to update the fog nodes' locations in such a way that the covered edge nodes and the size of the greatest sub-network of connectivity are maximized.We observe that these two targets are in dispute, which means that a large sub-component network does not necessarily indicate a wide coverage of edge nodes.We analyze our scenario problem with a network of n fog nodes and m edge devices deployed in a 2D area.We represent an undirected topology graph  = (, ), where  =  ∪  . refers to network nodes which include a set of fog nodes denoted by  and a set of edge nodes denoted by  , while  defines the edge node's connectivities.Two  As mentioned earlier, It is challenging to find a fully connected and covered graph based on analyzing the entire network due to network disconnectivity, Therefore, targeting a large sub-network could be a solution to improve network connectivity.We pointed out that the corresponding graph G of the target network may not be connected, i.e., G may consist of several sub-graph components.However, note that maximizing network connectivity of fog nodes may not be able to cover all edge nodes.In this situation, we aim to render the size of the largest sub-graph component as large as possible to maximize the connectivity of the network.
Network connectivity is defined as the size of the largest connected fog nodes subgraph.where G is a graph and  ℎ is a subset of G.The network connectivity is calculated as follows: The network coverage function   of an edge node i, relative to a fog node is defined by a binary value as follows:

D. Objective function
To assess the performance of the introduced topology graph, we consider two objectives to optimize: network where  is a weighting coefficient between zero and one that describes the ratio in which the objectives are prioritized.

IV. M P A
The MPA algorithm [24] was recently proposed to mimic the behavior of marine predators in pursuit of their food, where the predator trades off the Brown and Levy strategy by using the speed ratio between predator and prey, The major motivation for the MPA algorithm is the extensive foraging strategy of ocean predators, namely Lévy and Brownian motions [25], and encounter optimal rate policy in biological interactions between predator and prey.MPA adheres to the criteria that naturally regulate optimal foraging strategy and encounter rate policy in marine habitats [26].According to the MPA definition, the MPA-based optimization algorithm is presented as follows: At the initiation phase, a number of the prey initially dispersed uniformly around the search area of the optimization process.Equation 6 illustrates its mathematical model.
and   , respectively, define the search space's boundaries, where   is the upper bound and   is the lower bound, rand is a variable generated randomly between 0 and 1.
During the optimization phase, each distributed solution is evaluated using the objective function, and the solution with the best objective value is chosen as the top predator.This later is used to create a matrix called Elite, also referred to as E, which is based on the notion of the survival of the fittest.The elite matrix is demonstrated as follows : The variables X1, n, and d, respectively, represent the top predator vector which is repeated N times to construct Elite matrix, the number of search agents within the population, and the number of dimensions.The search agents predator and prey both their tasks are based on space exploration to locate food.The Elite matrix updates each iteration according to the better predator, in comparison to the top predator in the previous iteration.Typically, the construction of the prey occurs during the initiation phase, when the predators change their positions.Because of this, a matrix called prey has the same dimensions as the Elite matrix is defined.The fittest of the original prey is utilized to build the Elite matrix.This matrix is formulated as follows: . denotes the   ℎ dimension of the   ℎ prey.The two matrices described previously perform a significant role in the overall optimization problem.Based on the predator and prey movement patterns.The MPA optimization process is separated into the following stages: 1) HIGH-VELOCITY RATIO (Stage1): At this stage, the prey is traveling faster than the predator in relation to its speed when exploration is important.This stage usually occurs early in the initial stage of the iteration process, Therefore this stage is defined as an exploration level, in which the prey starts to explore the search space using the Brownian strategy to identify the prospective areas that could contain the ideal solution.Consequently, mathematical formalism can be defined as: stepsize  =   ⊗ Elite  −   ⊗ pre  prey  = prey  +  ×  ⊗ stepsize  (10) where  represents the current iteration count,   represents the algorithm's maximum cycle count, stepsize represents the motion scale factor,   represents a Brownian walk random vector that follows the normal distribution,   represents the elite matrix formed by the top predator,   represents the prey matrix, ⊗ stands for elementwise multiplication,  is a random variable with a range of [0, 1], whereas  is a fixed value set to 0.5.

2) RELATIVE UNIT VELOCITY(Stage2):
This stage is based on exploration and exploitation [27].After completing the first third of the exploration process, both predator and prey explore the search space at the same velocity in order to search for their own food.This is due to the proximity of the potential locations that may contain the ideal solution, therefore this stage is considered as an intermediate stage of optimization.In summary, in this stage, the exploration process will be progressively converted into the exploitation process.Specifically, the predators will be responsible for the exploration, while the prey for the exploitation.To model this stage mathematically, the population is divided into two parts: the first one is exploited by prey using a Lévy walk, while the second is explored by the predator using Brownian motion.
Where   is the random vector facilitating Lévy's mobility and  is an adaptive variable determined by equation 15; works on controlling the predator's path.
3) SLOW VELOCITY RATIO (Stage3): In the last third of the maximum iteration of the optimization process, the predator starts to approach the prey following Lévy's motion.This stage can be modeled as: stepsize The flexibility of MPA to mimic the habits of predators to maximize the chance of escape from local optima is another benefit [28].This effect is that predator movement would be impacted by environmental factors including eddy formation and fish aggregation devices (FADs).Its mathematical model is shown in equation 18: indicates the likelihood that FADs would affect the optimization process;  denotes a binary vector containing 0 and 1 value;  a distinct value falling between [0, 1]; and the values 1 and 2, produced at arbitrary from the overall population range which stand for the random indices of the prey matrix.
MPA may conserve data by preserving the prey's prior location.The fitness values of each current solution and each obsolete solution are compared once the existing solutions have been updated, and if the fitness of the previous solution is higher than that of the latter, the two are exchanged.The major steps of the applied marine predators optimization method (MFO) are outlined in algorithm 1.

V. EXPERIMENTAL AND RESULTS
In this section, we discuss the achieved results using the proposed architecture.A three-tier IoT-edge fog-based architecture with IoT-edge devices located at the first tier that is focused on producing data from various IoT-edge devices.The second comprises fog servers, and the last tier includes a resource-rich cloud.This architecture facilitates the deployment and the usage of resources while specifying the responsibilities of each tier.In order to assess the performance of the presented MPA algorithm.First, we describe the experience data and environment.Then, we conduct various scenarios under different settings and compare them to different algorithms.

A. Experimental setup
In the experimental setting, we implement a practical application using Matlab 2019Ra.This permits the calculations to be conducted as fast as possible.Based on the pseudo-code presented in algorithm 1, we conducted our experiment in a rectangular area of 1000m × 1000m with normal/uniform distribution to fog nodes using an AMD Ryzen 7 5700U with 8 cores with a 4.3 GHz clock speed, 8GB of memory, and a Windows 10 environment.The experimental setup enables us to repeat the experiment under various conditions and restrictions allowing us to conduct the experiment in the control environment with a desired set of settings.Table II

B. Experimental Results
We have conducted various to evaluate the connectivity and coverage of the proposed algorithm compared to the following baselines: harris hawks optimization (HHO) [29], particle swarm optimization (PSO) [30] and sine cosine algorithm (SCA) [31] algorithms.Due to the fact that our target function involves two competing priorities and that the network may experience inadequate connectivity and coverage as illustrated in Figures 2a, 2b.Therefore, the conducted tests have been carried out under different network parameters to establish the best weight  of the defined objective function for comparing the performance of the proposed algorithm to other methods.

C. Fitness Function Study Under Different 𝜔 Values
To determine an appropriate weight-coefficient , we analyze the user coverage and fog nodes connectivity simultaneously with different values of , and we employ the convergence curves which are considered the most critical analysis tool to understand the algorithm's behaviors while developing an optimal solution using metaheuristics algorithms.Therefore, to identify the optimum weight-coefficient , several iterations were conducted until the termination criteria in equation 5 are satisfied.We execute the experiments 10 times, each of which has 1000 iterations.And, each experiment was associated with different values of .In each experiment, 1000 instances are employed to determine the effect of changing the weight-coefficient  value.We observed in that when using a larger value of , the user coverage is low which means fog nodes have a tendency to crowd in a small area, leaving many IoTedge nodes uncovered, whereas when using a smaller value, the user coverage is high which mean the fog nodes are dispersed throughout the area and can cover the majority of IoT-edge nodes independently but cannot make up a larger component(i,e, High connectivity).Hence we conclude, to some measure, that the weight-coefficient  defines the tradeoff between our two concerned objectives ( and Φ).Therefore, we determine the weight coefficient , which is most likely to meet the two relevant criteria instantaneously.After numerous tests, figure 3 shows that the obtained results with the value of  = 0.5 give an excellent tradeoff.PSO, HHO, and SCA algorithms for the proposed objective function  as given in equation 5.Both the network and user coverage are examined.Figure 5 shows how the proposed algorithm can cover more IoT-edge nodes with fewer fog nodes, whereas in Figure 5a 40% percent of connected fog nodes can cover more than 90% of the target population as shown in Figure 5b, while with PSO, HHO, and SCA only connect (45,45,41)% of fog servers with covering (67,67,79)% of terminal edge devices, respectively.Hence, the results showed that our presented MPA algorithm outperforms better than other methods.

D. EVALUATION CRITERION
In order to have a better understanding of the adjustments that have been made to the optimization objectives, we have evaluated the performance of the proposed marine predator's optimization algorithm against the previous methods, in terms of the following evaluation metrics, namely; Network connectivity (Φ), user coverage (), and objective function value  , we assess our experiment by varying the number of edge nodes, the number of fog nodes, and the fog nodes' transmission range.
1) The impact of network density: To analyze the network coverage and connectivity, we establish a fog network by deploying a variety of network sizes (N), varying from 10 to 120 fog nodes as shown in Figures 6, 7 and 8.As can be seen in Figure 6a, the number of connected fog nodes increases as additional fog nodes are added to the network which reflects the network connectivity.As predicted, increasing the number of fog nodes increases network connectivity.This is due to the fact that by adding additional fog nodes to the network the isolated network connects to the rest of the network.As a direct consequence of this, a wider network will be formed.While, in terms of coverage, Figure 6b displays the relationship between the number of fog nodes as well as the total number of covered IoT-edge nodes.The results show that our proposed algorithm finds optimum coverage, with more IoT-edge nodes covered as predicted.
2) The impact of edge node density: Figure 7 shows the network coverage and connectivity results by varying the total number of IoT-edge nodes from 30 to 200.From figure 7a, it is clear that the number of connected fog nodes is always somewhere between 70% and 90%.This is due to the fact that our algorithm makes an effort to connect all of the accessible fog nodes in an attempt to encompass additional edge nodes.Furthermore, figure 7b shows the network's overall coverage which remains stable between 70% and 93%.We observed that when the number of IoT-edge nodes increased, the percentage of coverage and connectivity remained consistent.This is because the IoTedge node distribution is uniform, which implies that the likelihood of an IoT node falling into a covered or uncovered region is the same.
3) The impact of communication range: Figure 8, shows the communication range adjustments' impact on the network coverage and connectivity performance across multiple

4) Experimental objective function impact:
This part shows the impact of user coverage and network connectivity suggested in the objective function 5 in the performance of the proposed algorithm.Using the same experimental parameters as before, Table III  nodes and the value of the objective function increase in a proportional manner.Quite notably, our network structure is nearly always interconnected.However, the obtained results affirmed that the proposed approach provides better network coverage and better value to the fitness function whenever additional fog nodes are added to the network.By varying the number of terminal nodes from 30 to 195, results in Table IV show that the percentage of user coverage and network connectivity, as well as the objective function value f almost remained practically unchanged.This occurs because node placement distribution is totally randomly affected.Since the whole network is almost connected for all IoT-edge node values as shown, it is remarkable that even with the addition of terminal nodes, the coverage value will remain stable, which will result in the same proportion of all edge nodes.Similarly, statistics analysis demonstrates that the proposed algorithm performs better than other algorithms and achieves better coverage.The samples in Table V illustrate how the communication range of fog nodes affects both coverage and network connectivity.Simply explained, an increase in communication range will lead to an increase in coverage, which in turn will result in an improvement in the fitness value.Moreover, the fitness value that can be attained using the proposed algorithm is greater than that which can be obtained using the other methods.

VI. C
To improve network performance and provide a high user QoS at the edge of the network, efficient data and resource placement are required due to the IoT edge device resource limitation and user needs.Therefore, in this paper, we proposed a three-tier IoT edge-fog-cloud integration architecture and utilized Meta-heuristics algorithms for a seamless resource deployment process.Results show that with optimal resource deployment network performance can be enhanced.We propose an aggregated bi-objective function to maximize the network connectivity of the network measured by the number of connected fog nodes and to maximize the number of covered IoT-edge nodes simultaneously and we proposed

Figure 1 :
Figure 1: IoT-Edge based Fog infrastructure Maximum step movement which includes many IoT-edge terminal nodes.These nodes are linked to the second layer, which is referred to as the fog tiers and contains many fog server nodes.The third layer is the cloud layer.Computing nodes could conduct simple analytics on the data they acquire.Let the set of computing nodes within infrastructure be represented by  =  ∪  , where the fog layer consists of  fog nodes denoted by  = { 1 ,  2 ,  3 ...,   } with the corresponding set of transmission ranges denoted by (  ) = { 1 ,  2 ,  3 ...,   }, and the IoT-Edge layer has  edge nodes denoted by   = { 1 ,  2 ,  3 ...,   }.To facilitate the problem definition notations and symbols used in the system model are listed in table

Figure 2 :
Figure 2: Fog connectivity and Iot-Edge nodes coverage Figures 4  show obtained results of studied systems with respect to different network configuration values.As noticed at the beginning of an experiment, all algorithms undergo a significant change; nevertheless, this change gradually decreases as the iteration progresses.On the other hand, the HHO, PSO and SCA algorithm has trouble with premature convergence, which makes them more likely to get stuck in a local optimum than a global one.A crucial advantage of an algorithm's overall performance is its velocity of progression.As demonstrated in Figures 4, after a certain number of iterations, the proposed algorithm showed excellent potential and faster convergence than other algorithms while also achieving an optimal fitness function value.Hence, the results show that the proposed algorithm outperforms other algorithms in solving fog deployment problems for different aggregation values of .

Figure 4 :
Figure 4: Convergence curve of studied algorithms 1) Comparison of algorithms convergence: To assess how well the proposed algorithm performs in comparison to

Figure 3 :Figure 5 :
Figure 3: Fitness Function Study Under Different  Values

Figure 6 :
Figure 6: Effect of fog nodes density on coverage and connectivity

Figure 7 :Figure 8 :
Figure 7: Effect of IoT-edge nodes density on coverage and connectivity

Table I :
mathematical notation used in model development Symbol Meaning  Number of fog nodes  Number of IoT-edge clients  Location vector of fog nodes  (  ,   ) Node location  Region of interest width  Region of interest height  Fog node set   IoT-edge nodes set    The -th edge node client  Set of edge links between fog nodes   -th subgraph component |  | Size of the -th subgraph component ℎ Number of subgraph components  () Network connectivity Φ() Terminal IoT-edge node coverage   Boolean coverage function   The i-th particle  The Global best solution fog nodes are considered connected if an edge (    ,    ) ∈  exists and     ∩     ≠ ∅; an edge node    ∈  are considered covered if and only if fog node    ∈ ,     ∈   , and edge-fog (  ,    ) ∈  exists.
provides a list of parameters and their corresponding values, which were derived and established by various initial experiments.

Table III :
and Table IV outline accurate numerical results of the objective function obtained by the proposed algorithm.Results in Table III, show that when increasing the number of fog nodes from 30 to 230, the covered IoT-edge nodes percentage and value of objective function increase correspondingly.In addition, nearly Coverage, connectivity, and fitness value vs Fog node density all the time, we obtained a connected network topology.However, upon adding additional fog nodes to the network, our proposed algorithm gives greater IoT-edge node coverage and the best fitness value  .According to the statistics that are provided in Table V, when the number of fog nodes is changed from 30 to 230, both the percentage of covered edge

Table IV :
Coverage, connectivity, and fitness value vs IoT-

Table V :
Coverage, connectivity, and fitness value vs communication rang effective optimization algorithm.The results indicate that the proposed algorithm provided very promising results with fast convergence and low computational complexity compared to state-of-the-art baselines.Moreover, the experimental results demonstrated the efficiency of the proposed algorithm in finding network connections.A full network connection is always detected.However, network coverage is strongly influenced by the number and distribution of network IoT-edge nodes in the operational area.AThis work was supported by The National Natural Science Foundation of China under Grant 61872038. an