SDN-IoT: SDN-based efficient clustering scheme for IoT using improved Sailfish optimization algorithm

Sailfish algorithm

The SFO algorithm has advantages such as a balance between exploitation and exploration and the avoidance of local optimization. In the SFO, the sailfish represent the candidate solutions. This algorithm has two kinds of search agents called sailfish and sardine. The initial population in the solution space is generated randomly. In a next d search space, the ith member in the kth round contains the current position SF_i,k ∈ R(i = 1 , 2, …, m). The SF matrix is designed to maintain the position of all the sailfish. (1)${\displaystyle S{F}_{position}=\left[\begin{array}{cccc}\hfill S{F}_{1,1}\hfill & \hfill S{F}_{1,2}\hfill & \hfill \cdots \hfill & \hfill S{F}_{1,d}\hfill \\ \hfill S{F}_{2,1}\hfill & \hfill S{F}_{2,2}\hfill & \hfill \cdots \hfill & \hfill S{F}_{2,d}\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill \\ \hfill S{F}_{m,1}\hfill & \hfill S{F}_{m,2}\hfill & \hfill \cdots \hfill & \hfill S{F}_{m,d}\hfill \end{array}\right]}$

In Eq. (1), m, d and SF_i,j represent the number of sailfish, variables, and the next j value of the i sail. Equation (2) represents the fitness function of each sail as follows: (2)${\displaystyle \text{Fitness value of sailfish}=f\left(\text{sailfish}\right)=f\left(S{F}_1,S{F}_2,\dots ,S{F}_m\right)}$

To evaluate each sailfish, a matrix is defined for the suitability of all solutions as follows: (3)${\displaystyle S{F}_{\mathrm{fitness}}=\left[\begin{array}{c}\hfill f\left(S{F}_{1,1},S{F}_{1,2},\dots ,S{F}_{1,d}\right)\hfill \\ \hfill f\left(S{F}_{2,1},S{F}_{2,2},\dots ,S{F}_{2,d}\right)\hfill \\ \hfill ⋮⋮⋮⋮\hfill \\ \hfill f\left(S{F}_{m,1},S{F}_{m,2},\dots ,S{F}_{m,d}\right)\hfill \end{array}\right]=\left[\begin{array}{c}\hfill F_{S{F}_1}\hfill \\ \hfill F_{S{F}_2}\hfill \\ \hfill ⋮\hfill \\ \hfill F_{S{F}_m}\hfill \end{array}\right]}$

In Eq. (3), m saves the number of sailfish, SF_i,j after jth of the sail ith, f stores the fitness function and SF_fitness stores the fitness value of each factor based on the target function. The first row of the SF_position matrix is passed to the fitness function, and the output shows the fitness value of each factor in the SF_fitness.

The sardine consensus is another important component of the SFO algorithm and sardines also swim in the search space. Therefore, the location and suitability value of each sardine are defined according to Eq. (4). (4)${\displaystyle S_{\mathrm{position}}=\left[\begin{array}{cccc}\hfill S_{1,1}\hfill & \hfill S_{1,2}\hfill & \hfill \cdots \hfill & \hfill S_{1,d}\hfill \\ \hfill S_{2,1}\hfill & \hfill S_{2,2}\hfill & \hfill \cdots \hfill & \hfill S_{2,d}\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill \\ \hfill S_{n,1}\hfill & \hfill S_{n,2}\hfill & \hfill \cdots \hfill & \hfill S_{n,d}\hfill \end{array}\right]}$

where n, and S_i,j and S_position are the number of sardines, the jth dimension of sardine i, the position of all sardines in the matrix, respectively. The fitness value of each sardine is defined according to Eq. (5). (5)${\displaystyle S_{\mathrm{fitness}}=\left[\begin{array}{c}\hfill f\left(S_{1,1},S_{1,2},\dots ,S_{1,d}\right)\hfill \\ \hfill f\left(S_{2,1},S_{2,2},\dots ,S_{2,d}\right)\hfill \\ \hfill ⋮⋮⋮⋮\hfill \\ \hfill f\left(S_{n,1},S_{n,2},\dots ,S_{n,d}\right)\hfill \end{array}\right]=\left[\begin{array}{c}\hfill F_{S_1}\hfill \\ \hfill F_{S_2}\hfill \\ \hfill ⋮\hfill \\ \hfill F_{S_m}\hfill \end{array}\right]}$ where S_i,j represents the jth dimension, f is the target function, and S_fitness stores the fitness value of each sardine. In the SFO, the new position of the sailfish ( $X_{ne{w}_{SF}}^i$) is updated according to Eq. (6). In this phase, search agents provide the exploration phase, which involves searching a large part of the search space for promising solutions that have not yet been updated. Sailfish does not attack only from top to bottom or from right to left and vice versa. They can attack in all directions and in a shrinking circle. As a result, the sailfish updates its position in a circle around the best solution. (6)${\displaystyle X_{ne{w}_{SF}}^i=X_{elit{e}_{SF}}^i-{\lambda }_i\times \left(rand\left(1,0\right)\times \left(\frac{X_{elit{e}_{SF}}^i-X_{injure{d}_S}^i}{2}\right)-X_{ol{d}_{SF}}^i\right)}$

In Eq. (6), $X_{elit{e}_{SF}}^i$ elite sailfish position, $X_{injure{d}_S}^i$ best sardine position, $X_{ol{d}_{SF}}^i$ sail current position, rand(1, 0) is a random number, and λ_i is a factor in the ith iteration that is produced by Eq. (7). (7)${\displaystyle {\lambda }_i=2\times rand\left(0,1\right)\times PD-PD}$

Fluctuation of λ and the update position of the sailfish can lead to the divergence of the sailfish and their convergence around the prey. This method leads to discovery and search for solutions at the global level. PD indicates the number of baits per repetition. The adaptive formula for the PD parameter is defined according to Eq. (8). Since the number of prey is reduced during group hunting by sailfish, the PD parameter is an important parameter to update the position of the sailfish around the prey. (8)${\displaystyle PD=1-\left(\frac{N_{SF}}{N_{SF}+N_S}\right)}$ where N_SF and N_S are the number of Sailfish and sardines per phase of the algorithm, respectively. In the SFO algorithm, the new position of sardine $X_{newS}^i$ is defined according to Eq. (9). (9)${\displaystyle X_{ne{w}_S}^i=r\times \left(X_{elit{e}_{SF}}^i-X_{ol{d}_S}^i+AP\right)}$ where $X_{elit{e}_{SF}}^i$ is the best position for an elite sail, $X_{ol{d}_S}^i$ is the current position for sardines, r is a random number, and Attack Power (AP) is the rate of attack of a sail’s fish per Represents the iteration defined by Eq. (10). The sardines update rule depends on the AP of the sail. (10)${\displaystyle AP=A\times \left(1-\left(2\times Itr\times \varepsilon \right)\right)}$ where A and ɛ are coefficients to reduce the AP value linearly from A to 0. Using the AP, the number of sardines updating α and β are defined as follows:

(11)${\displaystyle \alpha =N_s\times AP}$ (12)${\displaystyle \beta =d_i\times AP}$ where d_i and NS are the number of variables and the number of sardines, respectively. In the SFO, bait hunting is assumed to occur when sardines become more suitable than sailfish. In this case, the position of the sailfish replaces the last location of the sardine being hunted to increase the chances of catching new prey (Eq. 13): (13)${\displaystyle X_{SF}^i=X_s^{i}iff\left(S_i\right)<f\left(S{F}_i\right)}$ where $X_s^i$indicates the current location of the sardine in the i iteration and $X_{SF}^i$ indicates the current location of the sailfish in the i iteration.

Proposed model

This article proposes an SDN-IoT-based framework for ISFO clustering of IoT sensor nodes. In the ISFO model, a set of sensor nodes SN = s₁, s₂, …, s_n are located in an area of size L × L, where L is the size of the area. The purpose of sensor nodes is to collect IoT devices data. In the proposed model, the role of the SDN network is as a designer. SDN collects various network information and automatically sets up the network. The proposed model is installed inside the SDN controller and the network clustering operation is performed by SDN. The SDN controller is responsible for programming the sink, and the sink sends the appropriate message to the IoT nodes. The SDN controller creates the network topology. The clustering operations on an IoT sensor node set are done by the proposed model, which is based on the ISFO model. In this article, the residual energy of the node, the center of the neighbor and the distance between the node and the sink are considered in which the set of CHs selected as CH = CH₁, CH₂, …, CH_s are defined. By selecting CH, the process of sending packets is possible with confidence. This is because the sensors communicate directly with CH instead of directly with base station, resulting in a reduction in energy when the packets are transmitted by CH to BS. Figure 2 indicates the flowchart of the proposed scheme. Algorithm 1 indicates the pseudocode of the Improved Sailfish Optimization (ISFO). In the pseudocode, the steps of the ISFO model are shown.

Improved SFO

The SFO algorithm uses two factors to search, so it can effectively improve population diversity and prevent population diversity shortcomings. Sailing populations of fish and sardines are randomly initialized. s^min and d^min represent the minimum number of fish and sardine sails, respectively. The value of both is equal to the minimum number of sensor nodes. Similarly, s^max and d^max represent the maximum values of sail and sardines, both of which are equal to the maximum sensor nodes.

${\displaystyle s_{i,j}=s^{\text{min}}+rand\times \left(s^{\text{max}}-s^{\text{min}}\right)}$ (14)${\displaystyle d_{i,j}=d^{\text{min}}+rand\times \left(d^{\text{max}}-d^{\text{min}}\right)}$

The best solutions are regarded as vectors that CHs have a high residual energy. The dynamic weight coefficient (w) is continuously entered in the formula for updating the position of the sailfish, which in the early stages is greater than the value of w and leads to global exploration. Finally, the value of w decreases comparatively, which is more useful for local search. Therefore, by Eq. (15) in iteration i, the new position of the sailfish ( $X_{ne{w}_{SF}}^i$) is updated. Weighting strategies are common in most swarm intelligence algorithms (Ouyang, Qiu & Zhu, 2021; Wu et al., 2022). In general, meta-heuristic algorithms partially reduce getting stuck in local optima by adaptively shifting between maximum and minimum values. Weighting in the initial stage weakens the effect of randomness and balances the search mechanism in the problem space. Adaptive weighting improves the quality of agents’ positions. It enables agents to converge to optimal positions faster and overall accelerates the rate of convergence. (15)${\displaystyle X_{ne{w}_{SF}}^i=\omega \times X_{elit{e}_{SF}}^i-{\lambda }_i\times \left(rand\left(1,0\right)\times \left(\frac{X_{elit{e}_{SF}}^i-X_{injure{d}_S}^i}{2}\right)-X_{ol{d}_{SF}}^i\right)}$ where $X_{elit{e}_{SF}}^i$ position of elite sail, $X_{injure{d}_S}^i$ best position of damaged sardine, $X_{ol{d}_{SF}}^i$ current position of sail, and λ_i is a coefficient in the ith iteration that is generated according to Eq. (16). (16)${\displaystyle {\lambda }_i=2\times rand\left(0,1\right)\times PD-PD}$

PD indicates the number of baits per repetition. Since the number of baits decreases during group hunting by the sailfish, the PD parameter is an important parameter to improve the location of the sailfish around the bait. The formula of PD metric is defined according to Eq. (17). (17)${\displaystyle PD=1-\left(\frac{N_{SF}}{N_{SF}+N_S}\right)}$

In the SFO algorithm, the new location of sardine $X_{newS}^i$ is defined according to Eq. (18). By the parameter w, the search balance is created in the environment and the optimal solution is reached in the shortest possible time. (18)${\displaystyle X_{ne{w}_S}^i=r\times \left(\omega \times X_{elit{e}_{SF}}^i-X_{ol{d}_S}^i+AP\right)}$

The value of ω is defined by Eq. (19). In the Eq. (19) i_max represents the maximum iteration. (19)${\displaystyle \omega =\frac{e^{2\times \left(1-i/i_{\text{max}}\right)}-e^{-2\left(1-i/i_{\text{max}}\right)}}{e^{2\times \left(1-i/i_{\text{max}}\right)}+e^{-2\left(1-i/i_{\text{max}}\right)}}}$

Figure 3 shows the impact of w on the other factors. The weight factor (w) changed the position of X_injured and X_elite. That is, the new position of the agents in the problem space is closer to the optimal solution. Like other population-based optimization algorithms, the SFO must strike a balance between exploration and exploitation. Fundamental SFO tends toward exploration, since the position-updated equation ignores the target point’s location information and only utilizes it to determine the distance to the next searching zone at random. Additionally, the SFO’s results for multi-modal issues show that its exploitation capability is lacking (Zhang & Mo, 2022).

The ISFO model uses elite sailfish in the current group to complete learning. The ISFO model has a different approach to updating sail and sardine that directly affects agent learning.

Problem coding

In the ISFO, agents are considered as sensor nodes while their position indicates their performance. Agents can change the information about the multidimensional search space by their position. In ISFO with factor N, the position of the nth sensor in the iteration t for i = 1, 2, …, N is defined according to Eq. (20), where $X_i^d\left(t\right)$ indicates the position of the i factor in the repetition of t in the dimension d and m refers to the dimension of the search space. (20)${\displaystyle X_i\left(t\right)=\left(X_i^1\left(t\right),\dots ,X_i^d\left(t\right),\dots ,X_i^m\left(t\right)\right);i=1,2,\dots ,N}$

The sail of the fish moves in the search space to find its best local solution, and the quality of the solution produced by the sail is evaluated using the fit function. A change in the search behavior of sailfish leads to a change in the location of other sardines in the search space. To achieve the optimal answer, the sails of the fish in the search space change their position. The movement of each sailfish is influenced by the best sardines. As a result, the sailfish are guided to the best possible position (global solution). In the proposed model, each node has a chance to participate in the CH selection process, and the role of CH among all network nodes is to balance the consumption of energy. Selecting the thread in rotation mode among all nodes leads to uniform mode.

For example, if 100 sensor nodes are located in the IoT network, they are marked with node IDs from 1 to 100. Therefore, the value of each sailfish X_i,d is determined as 1 ≤ X_i,d ≤ 100. The proposed model starts with the process of validating the energy of the nodes, in which the nodes are considered as a set of factors. Figure 4 shows that in a 10-member vector, nodes 15, 23, 5, 58, 92, 71, 35, 11, 87, and 46 were selected as CH nodes. The selected CHs are the position of the sailfish. The position of the sailfish changes when it reaches the sardine. Every change is creating an optimal solution.

Fitness function

The fitness function is considered as the main factor for selecting the CH nodes in the proposed model. Since the solutions of the agents depend on the fit function, the fitness function of the proposed model is defined based on the residual energy parameters, the distance between the clusters and the distance between the sensors and the sink. The rate of weights (ω₁, ω₂, ω₃) is determined based on different tests. If the weights are not adjusted, the value of the fitness function moves towards the maximum.

${\displaystyle \text{minimize}={\omega }_1\times f_1+{\omega }_2\times f_2+{\omega }_3\times f_3}$ (21)${\displaystyle \text{where}{\omega }_1+{\omega }_2+{\omega }_3=1and0<{\omega }_1,{\omega }_2,{\omega }_3<1.}$

Residual energy (f1): The remaining energy prevents dead nodes from being selected as CH. CH involves collecting data from non-CH nodes and transferring the data to the sink after aggregation. The best headwaters are selected based on the residual energy and the distance to the neighboring nodes to determine the optimal paths to the sink node. Hence, the sensor node with maximum remaining energy (f1) is considered as CH. In Eq. (22), E_CHi represents the residual energy of CH_i, and m determines the number of threads. (22)${\displaystyle f_1=\sum _{i=1}^m\frac{1}{E_{C{H}_i}}}$

Intra-cluster distance (f2): Indicates the Euclidean distance between the members of the cluster and the CH nodes. If this distance is minimal, then the energy required to process the data is also reduced. Therefore, the goal of f2 is to achieve the minimum distance based on the distance within the cluster. I_mj Specifies the nodes within a cluster of j cluster. (23)${\displaystyle f_2=\sum _{j=1}^m\left(\sum _{i=1}^{I_{m_j}}d\left(s_i,C{H}_j\right)/I_{m_j}\right)}$

The distance between the sensor nodes and CH is determined using the distance matrix D (L × M) according to Eq. (24). The Euclidean distance between CH and a normal node is described as CH_c, and the sensor nodes are defined as s₁, s₂, …, s_n. Two nodes, like a normal node, are denoted by i and a thread by j, and their positions are denoted by u and v. Eq. (25) is used to calculate the Euclidean distance between the sensor node and the CH node.

(24)${\displaystyle D\left(L\mathrm{\ast }M\right)=\left[\begin{array}{cccc}\hfill C{H}_{c_1,s_1}\hfill & \hfill C{H}_{c_1,s_2}\hfill & \hfill \cdots \hfill & \hfill C{H}_{c_1,s_n}\hfill \\ \hfill C{H}_{c_2,s_1}\hfill & \hfill C{H}_{c_2,s_2}\hfill & \hfill \cdots \hfill & \hfill C{H}_{c_2,s_n}\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill \\ \hfill C{H}_{c_k,s_1}\hfill & \hfill C{H}_{c_k,s_2}\hfill & \hfill \cdots \hfill & \hfill C{H}_{c_k,s_k}\hfill \end{array}\right]}$ (25)${\displaystyle d_{i,j}=\sqrt{{\left(j_u-i_u\right)}^2+{\left(j_v-i_v\right)}^2}}$

In data transferring phase, a time slot is provided for each node. The main purpose of each node is to collect data and send it to the header. Once data has been collected from all nodes within the cluster, the header node transmits the associated data to the BS.

Distance between CH and sink (f3): Indicates Euclidean distance between CH and sink. The sensor node with more residual energy, less distance with neighboring nodes and the sink node is selected as the CH node using ISFO. Consequently, the strategy of the proposed model for selecting CH is the average distance between nodes to send packets to BS. As a result, the shortest distance to the sink is considered. SN indicates the sink node. (26)${\displaystyle f_3=\sum _{i=1}^{m}d\left(C{H}_j,SN\right)}$

Terms of completion

The iterative phase is continued until the halting requirements are met or the algorithm has completed the maximum number of cycles. The population of sailfish is updated during this operation. The location of the entire sardine must be updated, or the attack power determines the position of the selected sardines. When there are no other assailants present, the sailfish alter their position against the victim. Other parameter values have been analyzed, and the ISFO algorithm determines the attack alteration method throughout hunting. The fitness value of all sardines is evaluated and sardine in solution is replaced. Finally, the best sailfish and sardine have been upgraded. The entire process is repeated till the maximum number of iterations is reached.

Computational complexity

The increase in computational complexity in meta-heuristic algorithms is dependent on the number of iterations, changes in the initial population, and changes in updating the agents’ positions. The increase in the number of repetitions and population diversity in the proposed model is insignificant. In the SFO algorithm, due to the proper balance between exploration and exploitation, the increase in computational complexity is low. Compared with the standard SFO, the proposed model increases the complexity of executing the new position of the sailfish and computing individual position. Other operators do not add complexity.

Model of consuming energy

The energy consumption model is evaluated by the radio energy dissipation model according to the distance between the receiver and the transmitter. The effectiveness of clustering models is frequently assessed using the metric of energy usage. The entire of energy required by sensor nodes during network operation is known as energy consumption. The distance between the source and the destination, the retransmission rate, and the control messages all have an impact on this measure. This ratio should be kept to a minimum to reflect efficient energy use. The energy allocated to send an L-bit packet at distance d is defined according to Eq. (27). (27)${\displaystyle {\mathit{E}}_{\mathit{TX}}\left(\mathit{L},\mathit{D}\right)=\left\{\begin{array}{c}\begin{array}{cc}\hfill \mathit{L}.{\mathit{E}}_{\mathit{elec}}+\mathit{L}.{\mathit{E}}_{\mathit{fs}}\times {\mathit{d}}^2\hfill & \hfill \left(\mathit{d}<{\mathit{d}}_0\right)\hfill \end{array}\hfill \\ \begin{array}{cc}\hfill \mathit{L}.{\mathit{E}}_{\mathit{elec}}+\mathit{L}.{\mathit{E}}_{\mathit{mp}}\times {\mathit{d}}^2\hfill & \hfill \left(\mathit{d}\ge {\mathit{d}}_0\right)\hfill \end{array}\hfill \end{array}\right.}$ (28)${\displaystyle E_{RX}\left(l\right)=l\times E_{elec}}$ (29)${\displaystyle d_0=\sqrt{E_{fs}/E_{mp}}}$ When a sensor node receives l-bit packet, its energy consumption is calculated according to Eq. (28). where E_elec,  E_mp and E_fs are constant parameters. E_TX(L, d), E_RX(l) and d representing the energy consumed to send and receive a 1-bit data packet and, the distance between the transmitter and receiver respectively. d₀ is the threshold value. The E_fs parameter expresses the energy consumption of the transmission amplifier for free routing and cooperation between nodes. The E_mp parameter expresses the power consumption of the transmission amplifier for multi-route routing.

Number of alive nodes

Figure 5 shows the number of alive nodes for the 150 and 300 nodes. The number of alive nodes for the ISFO model in 500 and 1,000 rounds with 150 nodes included 138 and 94 nodes. For 300 sensor nodes, the number of alive nodes for ISFO and SFO per 1,000 rounds is 179 and 167, respectively. For 300 sensor nodes, the number of alive nodes for ISFO and SFO per 1,000 rounds is 179 and 167, respectively. In the ISFO model, the alive nodes are still stable after the simulation round is completed because the ISFO model has achieved the best degree of convergence by improving the solutions, and the agents in the environment have chosen the best cluster. From the results of the diagrams, it is noteworthy that the ISFO model has a good performance compared to the LEACH and LEACH-E in term of network lifetime.

In Fig. 6 the results are shown for 2,500 rounds. The chart results show that the ISFO model has more live nodes compared to other models. The purpose of 2,500 rounds are to show the convergence chart of the models. The convergence graph of the number of live nodes with 1,700 rounds is not clear. But the comparison between 1,700 rounds and 2,500 rounds shows that the convergence of dead nodes in the ISFO model is done in a smoother form. The number of live nodes after 2,500 rounds for 150 and 300 nodes mode by ISFO model is two nodes and five nodes, respectively. The number of live nodes after 2,500 rounds for 150 and 300 nodes mode by SFO model is equal to 0.

Table 2 shows the number of clusters for each model based on different runs. The proposed model decreases the number of cluster when compared to LEACH, LEACH-E, SFO, and ISFO. According to the Table 2, it is clear that the number of clusters is different in each run. Therefore, the results will be different based on the number of clusters. If the number of clusters is less, then the energy consumption between the cluster heads will be less. Therefore, the lifetime of the network increases. A large number of clusters leads to network interference and lost packets. In the ISFO model, the number of clusters is optimal.

Number of delivered packets

Figure 7 shows the number of packets delivered to the sink for the ISFO model and the LEACH, LEACH-E and SFO models. The number of received packets in 1,000 rounds with 150 nodes by ISFO and SFO models is equal to 2,856 and 3,523, respectively. The number of received packets in 1,500 rounds with 150 nodes by ISFO and SFO models is 5,326 and 4,156, respectively. The number of received packets in 500 rounds with 300 nodes by ISFO and SFO models is equal to 2,059 and 1,847, respectively. The number of received packets in 1,200 rounds with 300 nodes in ISFO and LEACH-E models is equal to 5,287 and 3,055, respectively. The number of received packets in 1,200 rounds with 150 nodes in LEACH and LEACH-E models is equal to 1,502 and 1,594, respectively. Comparisons show that the ISFO model has been able to increase the number of packages due to the selection of optimal clusters.

Packet delivery rate

Figure 8 shows a comparison of package delivery rates for the ISFO model and the LEACH, LEACH-E and SFO models. Package delivery rates are defined according to Eq. (30) (Guleria et al., 2021). (30)${\displaystyle PDR=\frac{\text{Number of Packets Received}}{\text{Number of Packets Transmitted}}\times 100}$

If PDR is too low, these models will not be able to send the packets to the sink completely. In Fig. 8, the X-Axis indicates the simulation rounds and the Y axis indicates the packet delivery rate (%). Figure 8 shows that the PDR of the proposed model is higher than other models.

Table 3 shows the number of nodes with respect to PDR. The ISFO increases the PDR when compared to LEACH, LEACH-E, and SFO. It is observed that the PDR increases as the number of nodes increases. It is mainly due to consideration of ISFO. For 150 nodes, the PDR of the ISFO model is 89.44%. PDR of the ISFO model is 94.47% on 300 nodes. PDR of the SFO is 87.40% on 300 nodes.

Throughput

Transmitting the volume of data packets throughout the simulation period is considered as throughput. Operating power is defined according to Eq. (31). (31)${\displaystyle Throughput=\frac{\text{number of data packets sent (bits)}}{\text{Time period (seconds)}}.}$

In Fig. 9, the X and Y axes represent the number of rounds and throughput in bits per second, respectively. The ISFO model provides 3.5 × 10 4 bit/s throughput compared to other models up to the end of 1700 rounds with 150 nodes. LEACH, LEACH-E models achieved 2.2 × 104, 2.4 × 104 bps in the last simulation round with 150 nodes. LEACH, LEACH-E models achieved 2.1 × 104, 2.3 × 104 bps in the last simulation round with 300 nodes, respectively. According to the results, the ISFO model has more throughput than the LEACH, LEACH-E models.

Remaining energy

Figure 10 shows the average residual energy for the 150 and 300 sensor nodes. It can be seen that at 1,700 rounds with 150 and 300 nodes, the residual energy of the ISFO scheme is higher than other schemes. LEACH, LEACH-E models are weaker than ISFO and SFO. Because traditional models do not benefit from the function of fit and selection of optimal solutions. The average residual energy in 900 rounds with 150 nodes by ISFO and SFO models is 51 and 47 joules, respectively. The average residual energy in 900 rounds with 150 nodes by ISFO and LEACH models is 51 and 23 joules, respectively. The average residual energy at 1,300 rounds with 150 nodes by LEACH-E and SFO is 19 and 31 joules, respectively. The average residual energy at 1,700 rounds with 300 nodes by the ISFO and SFO models is 42 and 37 joules, respectively. The average residual energy in 500 rounds with 300 nodes by ISFO and LEACH-E is 147 and 119 joules, respectively.

The statistical analysis of the proposed model and other models based on residual energy is shown in Table 4. This may prove the efficiency of the ISFO model over the other models in terms of remaining energy. The metrics of minimum value, mean, maximum value and standard deviation (SD) are specified and computed for the models based on the results of the remaining energy of number of nodes. It was found that the LEACH, LEACH-E, and SFO models all performed significantly worse than the ISFO.

In Table 5, the statistical significance of the residual energy for one round is shown using the paired t test. In all cases, P < 0.05, so the null hypothesis is rejected at the 5% significance level and the alternative hypothesis is accepted at the 95% confidence level. CH selection in the proposed model reduces energy consumption based on the distance between clusters.

In addition, the significance of the proposed model’s performance was evaluated using statistical tests, and it was found to be significant with a confidence level of 95%. During this procedure of testing, the samples connected to the amount of leftover energy were chosen for each algorithm that was being investigated. This was done in order to ensure accurate results. Figures 11 and 12 highlights the descriptive investigation of the proposed model and benchmarked algorithms with respect to residual energy. It can be seen that the ISFO showed the best performance for 150 sensors, with median value of 50. Also, it can be seen that the ISFO showed the best performance for 300 sensors, with median value of 124.