Localization or positioning scheme in Wireless sensor networks (WSNs) is one of the most challenging and fundamental operations in various monitoring or tracking applications because the network deploys a large area and allocates the acquired location information to unknown devices. The metaheuristic approach is one of the most advantageous ways to deal with this challenging issue and overcome the disadvantages of the traditional methods that often suffer from computational time problems and small network deployment scale. This study proposes an enhanced whale optimization algorithm that is an advanced metaheuristic algorithm based on the siege mechanism (SWOA) for node localization in WSN. The objective function is modeled while communicating on localized nodes, considering variables like delay, path loss, energy, and received signal strength. The localization approach also assigns the discovered location data to unidentified devices with the modeled objective function by applying the SWOA algorithm. The experimental analysis is carried out to demonstrate the efficiency of the designed localization scheme in terms of various metrics, e.g., localization errors rate, converges rate, and executed time. Compared experimentalresult shows that the SWOA offers the applicability of the developed model for WSN to perform the localization scheme with excellent quality. Significantly, the error and convergence values achieved by the SWOA are less location error, faster in convergence and executed time than the others compared to at least a reduced 1.5% to 4.7% error rate, and quicker by at least 4% and 2% in convergence and executed time, respectively for the experimental scenarios.
A wireless sensor network (WSN) is constructed with various small devices called sensor nodes installed in an observing region for tracking some environmental or physical factors as a preeminent resource critical in multiple ramifications, including surveillance, military, healthcare, agriculture, and astronomy [
Moreover, it is challenging to manually offer every node its unique position information. Therefore, for performing this process, other practical approaches require GPS, the principal location method primarily utilized for designing WSNs. It is impractical to attach every single node with a GPS device owing to financial cost and other factors comprising their size and incapability of working in various applications [
Additionally, one of the excellent dealing ways with the localization challenges problem is the metaheuristic algorithms [
The WOA is a recent swarmbased intelligent optimization algorithm that vividly simulates whales’ behavior of finding and attacking prey to model its mathematical formulas [
This study also suggests an investigation strategy for enhancing WOA based on the siege mechanism, namely SWOA, for the complicated node localization problem in WSN. The method in the SWOA is figured out based on the siege mechanism, essential initial candidate locations, and inertia weighting parameters for avoiding the drawbacks of the local optimum and stagnation in the original one, which is referred to as escaping or allowing scoring goals in a difficult situation. Sieges enclose the target successfully applied in the Harris hawks optimization (HHO) [
Siege’s enhanced whale optimization algorithm (SWOA) is a suggested strategy aiming to improve the performance solution to the challenging multimodals with more complex, e.g., one of those issues like localization in WSN and prevent the drawback of the trapping local optima additional algorithm. This section presents the proposed SWOA.
Several highlights are considered investigation of this study.
An exploration capacity is increased by hybridizing it with HHO’s siege mechanism to make the shrinkage and encirclement whale mechanism robust and adaptable to the specific node localization problem.
A reinitialization of essential initial candidate locations employs inertia weighting parameters. It makes the closer expecting solution and avoids the local optimum’s drawbacks to increasing population variety.
The node localization model is figured out for the objective function with
The outcomes demonstrate that the suggested algorithms can significantly raise SWOA’s performance.
The remaining parts of the study include the following sections.
This section presents a statement of the node localization in the WSN problem and reviews the node localization situation developments and the original WOA algorithm.
The sensor nodes in a WSN gather information like humidity, temperature, and pressure dependent on the specific region [
Author [Citation]  Approach  Features  Challenges 

Ren et al. [ 
EGA  It was precision still only appropriate due to the fitness function independent units. The overlapping of rings was figured out by calculating the binary code sequence.  It quantized RSSI measurements from sensor nodes with irregular appearing areas that reduced the localization error. 
Gou et al. [ 
IWOA  It reduced the distance measurement error that was used in a network largeranging positioning accuracy to overcome the disadvantage of the traditional positioning algorithm.  It suffers from time consumption with a largeranging network. The modified RSSI by Gaussian in the fitness function. 
Phoemphon et al. [ 
IPSO  It does not undergo the local optima problem but ensures communication without obstructions between the anchor nodes and unknown nodes within the same group.  It provides less accuracy when observing the actual positions of the unknown nodes at the convex hull outside, making less precision for localizing the unknown nodes. 
Lakshmi et al. [ 
HPSO  It provides more accurate localization results and also decreases localization errors.  It suffers from handling certain challenging scenarios that require realworld node localization. 
Nithya et al. [ 
ABCBAT  It reduces the localization delay and localization error.  It does not consider the propagation error for further improvements in node localization. 
Shi et al. [ 
SGO  It achieves good convergence efficiency.  It does not perform efficiently when the anchors are randomly placed inside the networks. 
It is also appropriate for distributed optimization over the networks.  
Kulkarni et al. [ 
BIA  It performs faster and more accurate localization.  It is not applicable for centralized localization, which makes it particularly useful regarding energy awareness. 
It reduces the sensor nodes’ count in deploying terrains without interest.  
Al Shayokh et al. [ 
CSO  It is considered to be robust and efficient for determining the unknown nodes at a considerable rate of minimum error.  It secures lower precision on node localization since it does not improve their rooster behaviors for making the velocity update properly. 
Arora et al. [ 
BOA  It provides effective performance regarding the computation time, localizing the nodes, and localization error.  It does not consider the energy problems involved in the WSN and also needs to reduce the location estimation error. 
The elitist genetic algorithm (EGA) [
The improved particle swarm optimization (IPSO) [
Krill Herd Optimization Algorithm (KHA) [
Chicken Swarm Optimization (CSO) [
As is noted, a good monitoring and tracking application relies heavily on location accuracy. The existing works suffered from handling specific difficult conditions that need realworld node localization, tried to make it more accurate for localizing the unknown nodes, and did not consider propagation error for further advancements in node localization. The scheme positioning system’s executed time consumption should be considered when there is a largescale ranging network. It benefited energy awareness because it did not require centralized localization and functioned well when the anchors were dispersed throughout the networks. Additionally, they neglected to consider the WSN’s energyrelated issues and the need to lower location estimation errors. Creating a new metaheuristic technique or improving the existing algorithm for suitable localizing the unknown nodes in WSN is motivated in this study by these issues with the current localization scheme in WSN.
The problem definition of node localization in WSN possesses many sensor nodes, such as anchors or beacon nodes, unknown nodes or dumb nodes, and settled nodes, where every node has a communication range [
Let us assume the WSN is a symmetric type, illustrated as a Euclidean graph
The objective function is mainly designed with the fitness approaching value to validate the efficacy of the node localization approach in WSN. Once the optimal location is determined, it aids in reducing the error factor in locating the sensor nodes. Here, the localization error is mainly calculated by the distance estimation concerning anchor nodes and sensing ranges of the chosen dumb node and the beacon node. The mathematical expression of the objective function is given formula as follows:
where
Here,
The WOA simulates the whale’s predation action. It divides the whale’s predation process into three steps according to the whale’s predation characteristics, that is, three position update methods: shrinkage and encirclement, spiral position update, and random search [
Shrink and surround is a phase of the place whales can perceive the area where the prey cover it and is located for the position of the optimal design in the hunting or search space is inconsistent with the previous position. The WOA optimization algorithm assumes that solution
In the formula,
where
Updating whale location phase is figured out with two ways to update the whale’s position: a spiral position update and a random search. The whale’s position update method at a particular time ensures that the whale has an equal probability of choosing a spiral position update or random search at the same time. A random number
Spiral position update is carried out when
where
where,
Random search update is carried out when
A randomly selected whale position is formulated given the following equation:
where
Several variants of the WOA have been developed recently that, includes the AWOA [
Siege’s enhanced whale optimization algorithm (SWOA) is a suggested strategy aiming to improve the performance solution to the challenging multimodals with more complex, e.g., one of those issues like localization in WSN and prevent the drawback of the trapping local optima additional algorithm. This section presents the proposed SWOA based on the siege mechanism, essential initial candidate locations, and inertia weighting parameters for avoiding the drawbacks of the local optimum and stagnation in the WOA.
The whale position update process randomly selects the position update mechanisms, so the practical updating whale locations cannot be specified by changing the role of the leader
The harris eagle siege mechanism is applied to speed up the whale hunting; add an inertia weighting parameter at the end of each whale hunting iteration. The initial population location of the random generation algorithm is mapped by the chaotic Tent method so that the population is distributed more uniformly and the convergence speed of the algorithm would be accelerated; a new nonlinear parameter a is proposed so that the whale optimization algorithm can adapt to complex nonlinear problems; the fitness control is introduced mechanism, by controlling the update of the population position, to prevent the update from stagnation, and to improve the ability of the algorithm to jump out of the local optimum. The flowchart of the SWOA is shown in
On the other hand, for the intelligent optimization algorithm based on population iteration, the initial population’s quality affects the algorithm’s solution accuracy and convergence speed. It assumes that the current best candidate solution is the target prey or close to the optimal solution whale defines the best search agent; other search agents will then try to change positions and move closer to the best search agent.
The following formula describes the characteristics of the chaotic map that can make the algorithm effectively escape from the local optimum, thereby maintaining the diversity of the population and improving the global search ability. The Tent map is used to initialize the whale candidate location, and the reference [
Here
Here,
Let
Here,
Here
This siege mechanism has been successfully applied in the HHO to jump out of the trap of failing the local optimum in the complex optimization problem solution. The HHO simulates the predation action of the Harris eagle and simulates its action with mathematical formulas.
The algorithm vividly simulates the Harris eagle’s Siege and predation mechanism, which gives the algorithm a robust global search ability. In the WOA, finding the optimal position is often the random exploration of a single whale individual. The lack of communication between individuals and groups makes some individuals far from the prey to conduct many useless investigations.
The siege strategy in the HHO would be used to improve the position of the WOA to a certain extent. The formula of the HHO is applied to adapt for the WOA exploring phase updating equation as follows:
In the formula:
Here, Γ(x) is a Gamma function,
The exploring phase capability is the key for the group to use the position update method to explore a wide search area and avoid the algorithm falling into the local optimum. However, the exploiting phase capability is mainly to use the existing information of the group to analyze the solution space. The local search is carried out in some neighborhoods of the algorithm, which has a decisive influence on the convergence speed of the algorithm. The convergence factor with a significant change has a better global search ability and prevents the algorithm from falling into a local optimum; a smaller convergence factor with a more vital local search ability can speed up the convergence speed of the algorithm [
Here,
This subsection analyzes the qualified performance of the suggested SWOA algorithm by comparing it with the selected popular algorithms. A popular suit test with benchmark functions of the CEC2013 consists of 23 tasks (F1~F23) with variabledimensional parts used to test and evaluate the SWOA algorithm. Three sets of comparisonexperimental tests consist of with the original optimization algorithms, with different optimization algorithms, and with improved WOA algorithms for the selected benchmark functions of the CEC2013 test suit. The practical testing compared to the original optimization methods includes the WOA and HHO [
The experimental test with other optimization algorithms consists of the ALO [
Algorithm  Parameters settings 

SWOA  
WOA [ 

AWOA [ 

EWOA [ 

IWOA [ 

MWOA [ 

ALO [ 

PSO [ 

MFO [ 

GWO [ 

HHO [ 
The first test set of comparison with the original optimization algorithms is implemented by uniformly setting the same population size, number of iterations, and dimension, e.g., 30, 50, and 100D. The results of the SWOA algorithm are compared with WOA [
The second test set is implemented experiment and comparison with different optimization algorithms. Several popular developed optimization algorithms in recent years are selected, e.g., ALO, GWO, MFO, and PSO algorithms for the test benchmark functions.
The third test set is implemented experiment and comparison with variant improvement strategies for the WOA algorithm. Several developed improved WOAS algorithms are selected, e.g., AWOA, EWOA, IWOA, and MWOA algorithms, for the test functions with the same condition settings, population size, iteration, and the number of run times.
Stepwise presenting subsections are described as following stepwise execution in designing an optimal node localization with the SWOA strategy in WSN. The objective function frame as the efficient localization scheme in WSN would be stated via solving the objective functional derivation regarding factors like delay, path loss, energy, and RSS for optimizing the anchor nodes to reach the target nodes in sensor field simulation. A node localization schematic in optimizing node positions for anchor nodes towards the target nodes using the newly recommended SWOA algorithm. Update the objective function in optimizing using
The node localization strategy to reach the target nodes in the sensor field using a new SWOA algorithm for minimizing localization errors with the optimized anchor node provided a limited resource of the solution element. As a result of resolving the objective functional derivation concerning variables, including latency, path loss, energy, and RSS, are recognized as the effective localization strategy in WSN. The designed node localization model for WSN is derived in the following manner with the objective function
Here,
The energy function of anchor nodes is estimated as derived as follows:
Here,
The node localization strategy to reach the target nodes in the sensor field using a new SWOA algorithm for minimizing localization errors with the optimized anchor node where provided limited resources of the solution element. Once the optimal position is determined, it aids in reducing the error factors related to the distance estimation via energy
Here,
The derivation concerning the variable of the RSS is given as follows:
Here, the term
Here,
Here,
As mentioned, the sensor nodes in WSN are employed to gather the details, e.g., humidity, temperature, and pressure, which rely on the corresponding location to be collected with concern WSN for the node localization scheme owing to lesscost sensor nodes. The WSN deployment region must be divided into several virtual grid cells based on the node’s coordination and communication radius. The adjacent grid cells must guarantee direct communication between two nodes. In order to determine which cell the node would belong to, it is assumed that it knows the location coordination of its neighbor. For instance, a specific area has three subrings intersecting with r as a grid unit length, which means that the mesh is surrounded by three rings covered; multiple rings cover the actual location of the grid as equal to 3r; as a result, the more covered grids there are, the more likely it is that there will be unknown nodes in the area. The fundamental idea behind building an ideal model is establishing a boundary condition for the optimization algorithm constraint that must be specified to control the forward updated solution. It ensures that any two nodes in adjacent cells can interact with each other directly, removing the need for noisereducing device terminals and ensuring that cell radius requirements are met as
where
Considering these constraints, the metaheuristic optimization algorithms, e.g., bioinspired algorithms, swarm intelligence, and geneticbased heuristic approach, are applied for node localization and formulated the equations for reducing the localization error among the nodes in WSN. Over the iteration, the algorithm is deployed to find the position of unknown nodes that continues till the dumb nodes become settled nodes.
The prime intent of the suggested model is to resolve the node localization issue over the WSN sector and build the objective function of node localization based on the SWOA with distance computation and localization error. The distance measurement and range value among the nodes, the novel method can mitigate the localization error. Due to the attainment of fewer errors, the proposed procedure ensures effective localization performance. The simulation results are validated and compared against other existing heuristic optimizations, which is reviewed in the following subsection.
The obtained results of the node localization framework in WSN and simulation setup from the proposed SWOA are analyzed to evaluate performance. Based on convergence analysis and statistical analysis, the performance of the suggested model is compared with several previous schemes in the literature with model condiction. Here, the total iteration count is set at 1000, the number of populations is taken to be 40, and the number of dimensions is set to the number of anchors and unknown sensor nodes, along with the node’s
Moreover, the obtained results of the suggested SWOA method are compared with the previous scheme algorithms, including EGA [
Description  Parameter settings  Value settings 

Simulation area of the network of deployment  100 m × 100 m, 150 m × 150 m  
Initial node 

Transmission energy  0.00000000 001J  
Acquired energy  0.00000000 5J  
Coefficientsloss for free space  0.00000000 001J, and 4000  
Anchor noise and receiving power  −95 and −35 dBm  
Sensing radius ranges with directional antenna  20 m, 20 m, 30 m, 30 m  
Number of anchor nodes  15, 20, 30, 35  
Communication radius, with directional antenna  10 m, 10 m, 12 m, 20 m  
Number of unknown sensor nodes  25, 35, 45, 60, 80  
The number of iterationsNo. of rounds  500, 1000 
In most operations during the routing process, it becomes necessary to gather neighbor information to understand the nodes’ state, convey nodes’ parameters (energy, memory, and nodes’ id), and so forth, using probe messages. The protocol is used in networks to send beacon and probe messages by taking the control packet with piggybacking for updating the neighbor node about node status or querying some neighbor nodes. Additionally, communication networks commonly use broadcast, unicast, multicast, or any cast. Broadcasting or beacon messages are discouraged unless necessary because of the “always broadcast nature” of messaging in wireless communication.
Metrics  EGA [ 
IPSO [ 
KHA [ 
SCO [ 
IWOA [ 
SWOA 

Best score  4.48E+00  4.81E+00  4.76E+00  4.99E+00  4.90E+00  4.01E+00 
Worst score  2.70E+01  2.70E+01  2.91E+01  2.99E+01  2.31E+01  2.11E+01 
Mean  7.70E+00  7.62E+00  7.70E+00  7.62E+00  7.25E+00  7.88E+00 
Std. deviation  6.98E+00  6.87E+00  6.91E+00  7.38E+00  8.57E+00  6.54E+00 
Time (s)  5.84E+00  4.49E+00  6.37E+00  5.94E+00  6.98E+00  5.45E+00 
The performance of the routing protocol is also impacted by and dependent on the deployment of WSN applications. Because the sensor nodes are dispersed at random, an ad hoc infrastructure is produced. To enable connection and energyefficient network operation, optimum clustering is required if the resulting node distribution is not uniform. Intersensor communication typically takes place within small transmission ranges due to energy and bandwidth restrictions.
As a result, it is very possible that a route will have several wireless hops. In this work, Span [
Several metrics over iterations represent analysis for the node localization scheme based on the SWOA with the objective function, e.g., the best, worst, mean, standard deviation score values, and computation time of different optimization approaches. A statistical evaluation of the proposed SWOA for the node localization scheme in WSN over classical optimizations. The SWOA algorithm attains better quality performance in contrast with conventional algorithms such as EGA, IPSO, KHA, CSO, and IWOA approaches.
It means that it tends to attain a higher convergence rate. The enhanced model effectively determines the position of the unknown node in WSN. It depicts the convergence evaluation of the proposed node localization approach over specific optimizations. The most cases, the superior belongs to the SWOA scheme. Hence, the lower value convergence tends to significantly improve the convergence rate to locate the sensor nodes in WSN.
The localization error analysis of localization errors of the proposed method compared with traditional algorithms concerning the variation of anchor nodes and sensor ranges.
In most cases of setting net area deployments, the localization error analysis of the proposed SWOA scheme is smaller than the other schemes’ optimizations. In the error analysis with net deploying ranges of
Sensor nodes can exhaust their limited energy supply by performing computations and information transmission in a wireless environment without losing accuracy. Therefore, it is crucial to use energyefficient communication and analytic methods. The battery life determines the lifespan of each sensor node, which serves as both a router and a data emitter. Its power outages or runs out cause some sensor nodes to malfunction, might have a substantial topological impact, and need packet rerouting and network reorganization.
The energy consumption during the receiving, demodulating, decapsulating, processing, encapsulating, modulating, transmission, and routing processes negatively affects network efficiency, which causes congestion and increases delays. The energyaware routing involves the routing features, e.g., cluster formation, routing table, establish, and maintenance path. Due to its central significance in these functionalities, solutions are needed to reduce message broadcasting and beacon message exchange. The routing algorithm minimizes broadcast in environments with strict energy factor constraints. Packet sequencing is a popular method for resolving broadcast storm issues. The broadcast protocol should transport packets to all nodes in the network with the least amount of overhead, latency, and energy consumption possible.
In the Span method, the coordinator forms the messageforwarding backbone of the network. If two neighbors of a coordinator node cannot communicate directly or through one or more coordinators, the node should become a coordinator. Rotating coordinators show how localized node choices result in a connected, capacitypreserving global topology. As the ratio of idletosleep energy consumption rises and grows with network density, the improvement in system lifespan due to Span increases. For instance, the simulations demonstrate that the system lifetime of an 802.11 network in powersaving mode with Span is two times better with a realistic energy model than without one. When used with the 802.11 powersaving methods, Span seamlessly interacts with the latter and enhances system longevity, capacity, and communication latency.
Approach  Factor variables  60 m × 60 m  80 m × 80 m  100 m × 100 m  150 m × 150 m 

Localization errors  7.8%  9.0%  7.9%  7.1%  
Time execution (s)  2.26E+00  5.84E+00  8.01E+00  7.25E+00  
EGA [ 
Round iterations for convergence reaching  354  459  554  735 
Average optimal converges  3.42E+00  6.25E+00  8.57E+00  7.76E+00  
Localization errors  8.7%  7.9%  7.9%  9.2%  
Time execution (s)  2.56E+00  4.48E+00  7.36E+00  8.89E+00  
IPSO [ 
Round iterations for convergence reaching  145  458  336  781 
Average optimal converges  4.08E+00  7.48E+00  1.03E+01  1.18E+01  
Localization errors  7.6%  8.0%  1.9%  9.3%  
Time execution (s)  2.98E+00  6.32E+00  8.35E+00  8.45E+00  
KHA [ 
Round iterations for convergence reaching  379  485  468  719 
Average optimal converges  3.19E+00  6.76E+00  8.93E+00  9.04E+00  
Localization errors  7.7%  7.9%  2.0%  9.2%  
Time execution (s)  3.32E+00  5.94E+00  7.13E+00  8.19E+00  
CSO [ 
Round iterations for convergence reaching  445  555  665  776 
Average optimal converges  4.18E+00  1.06E+01  1.39E+01  1.41E+01  
Localization errors  7.8%  7.9%  8.9%  9.1%  
Time execution (s)  2.92E+00  6.98E+00  7.40E+00  8.24E+00  
IWOA [ 
Round iterations for convergence reaching  665  473  595  824 
Average optimal converges  4.23E+00  1.06E+01  2.39E+01  2.41E+01  
Localization errors  
Time execution (s)  
Round iterations for convergence reaching  
Average optimal converges 
From the results, the statistical estimation of the recommended node localization model of the SWOA offers better performance in the cases of setting deployments than the other schemes’ optimizations. Significantly, the error and convergence values achieved by the SWOA are less location error, faster in convergence and executed time than the others compared to at least a reduced 1.5% to 4.7% error rate, and quicker by at least 4% and 2.1% in convergence and executed time, respectively for the experimental scenarios.
Because of the unsatisfactory performance of the traditional whale optimization algorithm (WOA), this paper proposed a siege whale optimization algorithm (SWOA) for the node localization scheme in WSN. The siege mechanism learned from the Harris Eagle optimization (HHO) algorithm was utilized to speed up whale hunting. An inertia weighting parameter was added at the end of each whale hunting iteration to control the update of the population position to prevent the update from stagnation. The mapped chaotic method generated the random initial population location to improve the algorithm’s ability and jump out of the local optimum. The SWOA algorithm was analyzed through function tests and node localization, compared with the other algorithms in the literature, which proved that the SWOA made significantly differs from the original algorithm. The core objective of the localization model is to determine the location of the unknown node in the WSN, considering variables like delay, path loss, energy, and received signal strength. The obtained optimal unknown node localization is provided with the help of the optimal value of the optimal solution in terms of position by the SWOA. The graph estimates the localization error with the objective function mathematically derived based on optimization from the SWOA. The simulation and performance are measured as convergence, and statistical analysis in the mean value of the proposed SWOA has acquired a higher value than 1.5% to 4.6% for the EGA, 2.3% to 4.1% for the IPSO, 1.5% to 3.23% for the KHA, 1.7% to 4.1% for the SCO, and 2.3% to 4.1% for the IWOA algorithms, respectively for some deployed area networks in terms of the measured localization error. Thus, the novel method can appropriately estimate the location of unknown nodes. In future work, the proposed algorithm could be applied to the broader use of WSN localizations in cloud computing, autonomous driving, the Internet of Things (IoT), and vectorized mapping. The placement can be established using vectorized road network maps and sensor data. Alternatively, the positioning is used for cloudbased crowdsourcing today and offers the selfservices and other clients based on the information.
The authors would like to thank the editor and anonymous reviewers for their valuable suggestions and comments.
This study was partially supported by the VNUHCMUniversity of Information Technology’s Scientific Research Support Fund.
The authors confirm contribution to the paper as follows: conceptualization, T.T. Nguyen and T.K. Dao; methodology, T.T. Nguyen; software, T.K. Dao; validation, T.K. Dao, T.K. Dao and T.K. Dao; formal analysis, T.K. Dao; investigation, T.T. Nguyen; resources, T.K. Dao; writing—original draft preparation, T.K. Dao; writing—review and editing, T.T. Nguyen and T.K Dao; visualization, T.K. Dao; supervision, T.K. Dao; project administration, T.T. Nguyen; funding acquisition, T.K. Dao and T.T. Nguyen.
Not applicable.
The authors declare that they have no conflicts of interest to report regarding the present study.
Function number  Statistical results  WOA  HHO  SWOA  

30D  50D  100D  30D  50D  100D  30D  50D  100D  
F1  Mean  5.659E64  3.664E64  3.143E64  1.137E62  5.201E63  2.082E65  
Std  1.116E63  1.498E64  1.087E64  1.618E62  6.942E63  1.041E62  
F2  Mean  6.569E38  9.547E39  7.878E39  8.265E32  4.357E32  5.297E32  
Std  8.820E38  1.979E38  1.773E38  1.046E31  5.922E32  6.538E32  
F3  Mean  1.140E27  8.690E28  8.607E28  1.003E31  5.062E32  4.729E33  
Std  2.509E27  1.887E28  1.431E28  4.124E32  1.049E28  3.401E28  
F4  Mean  6.874E22  4.063E23  1.688E23  3.436E27  6.765E24  1.048E28  
Std  3.314E22  2.596E22  2.445E22  1.225E26  1.920E22  5.144E24  
F5  Mean  2.418E01  6.561E+00  4.029E01  5.021E+00  3.341E+00  1.290E+00  
Std  1.674E01  1.126E01  3.498E02  3.907E02  4.718E03  2.021E01  
F6  Mean  2.547E06  2.248E06  1.404E06  1.764E06  
Std  3.709E07  1.953E07  5.720E07  5.260E07  2.549E07  6.121E02  2.746E02  5.610E03  
F7  Mean  4.678E04  3.493E04  8.580E05  1.590E02  1.489E02  2.283E03  
Std  1.560E04  1.121E04  5.298E05  1.092E02  5.818E03  1.675E03  
F8  Mean  −2.689E+03  −9.446E+02  −7.513E+02  −4.798E+02  2.039E06  5.571E06  
Std  6.265E+01  2.352E+00  1.847E+00  1.890E+02  3.822E+01  1.876E+02  
F9  Mean  3.950E01  5.730E+01  5.820E+01  3.210E+01  1.240E+01  −8.664E01  
Std  7.970E+00  3.710E01  3.280E01  2.940E+00  3.120E01  2.845E02  
F10  Mean  5.862E15  2.046E15  1.359E15  7.826E16  4.043E16  1.178E15  
Std  1.946E15  1.572E15  7.726E16  6.941E16  8.715E17  0.000E+00  
F11  Mean  8.060E03  9.857E04  5.958E04  3.565E16  2.790E17  
Std  8.420E03  5.279E03  4.822E03  1.732E03  3.312E05  3.141E16  
F12  Mean  4.105E03  2.195E03  6.689E04  4.948E04  1.840E04  4.142E06  
Std  8.560E03  4.667E03  3.426E03  1.734E03  1.751E04  1.835E04  
F13  Mean  4.008E02  2.236E03  1.540E03  2.254E03  1.817E03  5.203E04  
Std  5.017E02  4.416E02  2.922E02  3.296E03  1.180E03  2.531E02  
F14  Mean  4.126E+00  1.332E+00  4.193E01  9.980E01  9.609E01  3.334E02  1.082E01  
Std  3.494E+00  2.680E+00  1.379E+00  6.695E02  3.676E02  3.696E04  
F15  Mean  8.370E03  2.989E04  6.554E04  2.390E04  5.362E02  1.001E03  
Std  7.096E03  5.947E04  4.538E04  1.569E04  1.144E04  4.445E03  
F16  Mean  −1.032E+00  −3.929E01  −2.666E02  −1.032E+00  −1.252E01  −1.017E01  
Std  1.724E08  5.614E09  1.518E10  5.594E13  −5.488E03  4.056E05  
F17  Mean  3.979E01  2.762E01  1.012E01  3.979E01  1.057E02  3.406E01  
Std  1.503E06  1.043E06  6.070E07  1.223E12  1.127E07  1.569E02  
F18  Mean  3.000E+00  1.398E+00  1.301E+00  3.000E+00  2.819E+00  2.977E+00  
Std  1.892E05  5.044E06  2.891E06  9.388E08  8.371E02  1.561E08  
F19  Mean  −7.191E01  −2.164E01  −3.863E+00  −1.635E01  8.190E08  2.750E02  
Std  7.492E05  5.293E05  1.024E05  9.410E07  1.998E06  −1.527E01  
F20  Mean  −3.322E+00  −6.330E01  −2.156E01  −1.232E+00  1.705E06  −5.943E02  
Std  3.576E06  1.377E06  6.602E02  1.599E02  7.949E03  −4.673E01  
F21  Mean  −6.084E+00  −3.155E+00  −1.088E+00  −1.808E+00  5.297E07  −3.908E01  
Std  5.852E01  1.421E01  6.737E01  5.286E01  6.008E01  −1.314E+00  
F22  Mean  −3.500E+00  −7.214E+00  −6.548E+00  −4.614E+00  −2.268E01  5.047E02  
Std  2.314E01  2.208E01  5.789E02  2.314E01  1.739E01  4.450E02  
F23  Mean  −7.800E01  −2.810E01  −9.455E+00  −4.840E02  1.318E02  −1.126E03  
Std  4.596E01  3.396E01  6.531E02  −3.011E02  1.685E03  
+/−/=  31/7/8  34/8/4  34/7/5  20/18/8  20/17/19  27/13/6  –  –  – 
Function number  Statistical results  ALO  GWO  MFO  PSO  

30D  100D  30D  100D  30D  100D  30D  100  30D  100D  
F1  Mean  4.033E09  9.142E11  6.420E64  3.105E65  2.246E13  4.155E15  2.327E+02  7.712E+00  1.234E97  4.057E100 
Std  8.600E20  2.315E21  1.055E09  9.642E12  3.804E03  3.731E04  9.721E+01  2.249E+00  5.690E66  2.129E99  
F2  Mean  9.132E02  1.777E03  2.935E37  2.277E38  3.344E09  3.140E10  8.585E+00  1.534E01  7.852E39  3.853E100 
Std  3.381E12  7.316E14  1.988E01  2.881E03  1.320E02  7.658E05  4.800E+00  3.815E01  8.934E39  3.727E67  
F3  Mean  2.218E04  3.102E06  7.260E29  4.054E30  1.887E02  1.967E04  9.961E+02  6.995E+01  3.894E28  5.375E40 
Std  1.584E+01  1.534E+00  2.364E04  1.496E06  7.640E02  2.090E03  1.285E+03  1.167E+02  1.049E28  1.626E39  
F4  Mean  4.494E04  4.168E05  1.432E20  2.005E22  1.945E01  1.180E02  1.278E+01  1.324E02  6.765E24  1.659E28 
Std  3.556E04  5.661E05  4.128E04  7.244E07  2.015E02  2.056E04  6.087E+00  4.100E01  1.920E22  3.138E29  
F5  Mean  4.768E+01  4.633E+00  6.643E+00  4.118E01  6.127E+02  3.998E+01  9.515E+03  3.125E+02  3.213E01  4.686E24 
Std  3.676E+00  3.528E02  8.943E+01  8.762E+00  5.609E+01  4.296E+00  7.125E+03  4.576E+02  2.137E02  3.222E23  
F6  Mean  5.178E09  4.687E10  3.567E06  1.471E07  1.164E13  1.151E14  4.617E+01  3.943E+00  3.398E08  5.202E02 
Std  6.699E20  6.270E22  2.362E09  1.475E10  6.031E03  1.550E05  6.005E+01  4.096E+00  6.121E09  5.610E03  
F7  Mean  2.177E02  7.798E04  8.192E04  5.031E05  4.562E03  2.380E05  6.240E01  1.596E02  4.665E05  2.175E08 
Std  2.203E03  1.007E04  8.534E03  5.003E04  1.074E03  2.619E06  9.874E01  1.721E02  6.906E06  7.912E11  
F8  Mean  −2.400E+03  −2.204E+02  −2.895E+03  −4.824E+01  −3.480E+03  −1.558E+02  −2.953E+03  −1.995E+02  −4.798E+02  5.571E06 
Std  1.381E+01  1.581E01  7.518E+01  1.969E+00  1.621E+02  8.374E+00  2.869E+02  9.149E+00  1.313E01  1.795E06  
F9  Mean  2.288E+01  2.570E01  4.780E+00  2.195E02  2.210E+01  1.998E+00  2.968E+01  7.819E01  1.240E+01  −8.664E01 
Std  2.939E10  4.791E12  4.223E+00  1.890E01  3.848E+00  2.195E01  3.371E+00  1.735E01  1.300E01  2.845E02  
F10  Mean  2.311E01  1.657E02  6.573E15  1.023E16  1.209E07  2.203E09  5.920E+00  1.262E01  1.178E15  0.000E+00 
Std  8.078E11  7.993E12  2.437E01  1.726E02  1.547E02  8.542E04  3.702E+00  2.170E01  6.941E16  0.000E+00  
F11  Mean  1.752E01  8.807E03  2.892E02  2.003E03  2.282E01  5.805E03  5.913E+00  8.178E02  2.446E04  2.790E17 
Std  3.570E03  2.420E04  2.133E02  1.993E04  1.630E02  9.992E04  4.726E+00  1.434E01  1.732E03  3.141E16  
F12  Mean  2.432E+00  1.015E01  9.561E07  7.404E08  6.220E02  5.072E03  1.259E+01  1.715E01  4.948E04  4.142E06 
Std  1.020E20  2.071E22  2.138E+00  3.983E03  6.859E05  3.685E06  4.522E+00  2.530E01  1.734E03  1.835E04  
F13  Mean  2.197E03  1.842E04  5.792E06  5.917E08  4.395E03  7.303E05  1.122E+04  9.946E+02  1.472E03  1.077E04 
Std  1.799E20  9.186E22  4.838E03  1.739E04  4.637E03  2.132E04  2.507E+04  1.998E+03  2.531E02  6.872E05  
F14  Mean  2.186E+00  5.466E02  4.126E+00  3.469E02  2.973E+00  4.658E03  2.778E+00  6.194E02  1.082E01  2.770E04 
Std  1.995E04  1.491E02  1.623E+00  1.034E01  9.592E12  4.849E13  1.684E+00  7.752E02  6.695E02  3.696E04  
F15  Mean  8.001E04  6.625E05  4.921E04  7.230E06  8.622E04  3.636E05  2.516E03  4.417E05  1.360E04  1.001E03 
Std  6.950E05  4.010E06  1.911E04  2.017E06  6.157E05  4.365E06  3.183E03  2.977E04  5.486E05  4.445E03  
F16  Mean  −1.032E+00  −6.431E02  −1.032E+00  −1.601E02  −1.032E+00  −2.690E02  −1.032E+00  −2.689E02  −1.013E02  5.509E05 
Std  3.524E11  1.190E12  4.059E14  1.387E15  2.425E07  1.850E08  1.850E08  1.850E08  5.594E13  4.056E05  
F17  Mean  3.979E01  3.979E02  3.979E01  3.544E02  3.979E01  2.308E02  3.979E01  1.708E02  3.972E02  −5.180E03 
Std  5.773E02  2.271E02  2.456E14  1.455E15  6.177E07  7.468E10  2.978E01  1.693E02  1.127E07  2.272E13  
F18  Mean  3.000E+00  2.695E01  3.000E+00  2.855E01  3.000E+00  4.780E02  3.000E+00  8.771E02  1.430E01  3.099E03 
Std  5.839E01  5.094E03  2.312E13  1.744E14  4.065E07  3.738E08  5.439E16  3.129E18  9.388E08  1.561E08  
F19  Mean  −3.863E+00  −1.644E01  −3.862E+00  −1.234E01  −3.863E+00  −3.696E02  −3.858E+00  −4.130E02  −1.635E01  2.750E02 
Std  2.567E+00  8.463E02  2.367E12  1.842E13  2.963E07  9.612E09  2.959E03  6.874E05  1.998E06  7.569E08  
F20  Mean  −3.251E+00  −1.412E01  −3.298E+00  −2.256E01  −3.214E+00  −4.205E02  −3.225E+00  −5.742E03  −1.232E+00  −5.943E02 
Std  1.239E02  5.441E04  2.373E02  1.119E03  3.482E02  1.045E03  8.004E02  1.366E03  7.994E07  5.716E07  
F21  Mean  −6.135E+00  −5.538E01  −9.142E+00  −1.989E02  −8.659E+00  −6.227E01  −5.671E+00  −2.434E01  −1.808E+00  −3.908E01 
Std  1.207E04  5.661E06  3.778E02  1.391E03  1.595E04  3.750E06  1.531E+00  1.238E01  7.497E02  4.285E07  
F22  Mean  −3.345E+00  −2.771E01  −1.040E+01  −2.652E02  −6.958E+00  −5.538E01  −4.586E+00  −3.400E01  −2.268E01  −6.184E01 
Std  1.147E03  6.107E05  2.777E01  2.136E02  1.319E01  9.475E03  6.466E01  3.143E02  3.066E02  7.656E03  
F23  Mean  −4.672E+00  −3.681E01  −1.054E+01  −4.921E01  −9.196E+00  −5.002E01  −8.913E+00  −4.321E01  −4.840E02  −1.126E03 
Std  8.581E03  3.468E03  1.747E01  6.538E03  2.385E01  6.875E03  3.614E01  3.375E02  6.594E06  1.685E03  
+/−/=  28/15/3  23/15/8  20/18/8  22/15/9  31/13/2  27/14/5  27/9/10  28/8/10  –  – 
Function number  Statistical metrics  AWOA  EWOA  IWOA  MWOA  

30D  50D  30D  50D  30D  50D  30D  50D  30D  50D  
F1  Mean  1.48E85  1.42E85  1.29E85  9.48E86  4.91E86  1.94E88  1.88E88  4.88E89  1.23E97  4.06E100 
Std  1.51E+03  1.38E+03  7.67E+02  5.73E+02  4.57E+02  2.36E+02  1.77E+02  1.05E+02  8.50E12  3.18E14  
F2  Mean  4.24E55  3.77E55  2.74E55  1.75E55  8.73E56  3.34E56  8.55E57  1.83E57  3.85E+00  2.92E02 
Std  1.42E+00  4.11E01  3.38E01  3.37E01  1.01E01  3.34E02  2.10E03  9.85E04  7.02E08  2.98E10  
F3  Mean  1.36E+02  5.82E+01  4.19E+00  1.95E+00  1.77E+00  1.59E+00  1.01E01  8.62E02  1.03E+02  9.03E01 
Std  1.25E+03  7.32E+02  6.77E+02  5.95E04  1.35E+02  3.31E+01  9.18E+00  8.09E+00  2.01E+02  1.02E05  
F4  Mean  6.47E+00  5.09E+00  3.12E+00  1.28E+00  1.13E+00  9.88E03  2.11E02  8.43E03  1.02E01  6.74E03 
Std  1.75E+00  1.38E01  1.12E01  1.05E01  6.33E02  6.02E02  1.48E02  1.78E03  1.06E03  8.31E06  
F5  Mean  6.46E+00  8.83E03  5.34E03  1.70E+00  7.54E01  8.95E03  2.52E+00  1.98E+00  1.36E+04  7.60E+01 
Std  1.24E+02  1.20E+02  1.14E+02  1.00E+02  9.51E+01  2.78E+01  2.58E+01  6.07E+00  4.46E02  1.43E03  
F6  Mean  6.61E04  5.15E04  2.31E04  1.92E04  7.64E05  3.53E05  7.68E07  4.58E07  4.65E+01  1.55E02 
Std  4.88E+00  5.20E01  7.87E02  5.26E02  9.94E07  9.17E08  4.71E03  3.11E03  3.71E02  8.90E03  
F7  Mean  6.26E03  6.59E04  5.56E05  1.96E05  1.39E06  5.11E07  4.58E07  2.91E09  4.88E02  6.39E04 
Std  1.41E02  1.02E02  1.92E03  9.84E04  4.53E04  1.73E05  3.32E06  1.71E06  8.11E04  3.90E07  
F8  Mean  −3.46E+03  −8.56E+02  −3.29E+02  −1.99E+02  −1.62E+02  −2.74E+01  −1.64E+01  −8.87E+00  −2.47E+01  −3.59E+01 
Std  8.20E+01  5.38E+01  5.16E+01  1.61E+01  1.43E+01  9.14E+00  8.12E+00  4.69E+00  5.62E+00  6.61E05  
F9  Mean  4.74E+00  1.38E+00  1.40E01  2.87E02  4.25E01  3.27E01  8.14E01  4.56E01  1.24E+01  9.14E01 
Std  5.79E+00  4.70E+00  4.47E+00  1.05E+00  1.01E+00  7.49E01  4.54E01  8.17E02  1.30E01  6.06E03  
F10  Mean  7.28E15  3.87E15  2.41E15  1.96E15  1.29E15  9.95E16  2.71E16  4.68E17  2.28E+00  9.15E02 
Std  1.52E01  6.82E03  1.53E03  6.90E04  6.46E04  2.15E04  1.07E04  2.44E05  7.56E07  2.41E08  
F11  Mean  2.58E02  1.85E02  1.85E02  3.34E03  3.18E03  1.93E04  1.68E04  1.47E04  2.86E+00  2.44E02 
Std  4.74E01  1.44E02  2.33E03  1.21E03  9.86E05  6.14E05  3.40E06  2.26E06  1.12E04  1.66E06  
F12  Mean  4.38E04  8.47E06  7.25E06  1.89E05  1.67E05  9.65E06  3.23E05  3.18E05  7.48E01  1.13E02 
Std  1.78E+00  1.02E+00  7.92E01  2.91E01  6.33E02  3.53E02  2.27E02  1.47E02  7.95E03  4.14E05  
F13  Mean  5.97E03  4.39E03  1.76E03  1.23E03  9.89E04  6.94E04  6.26E04  4.64E04  2.66E+02  2.24E+01 
Std  2.51E+00  5.25E01  9.90E02  3.79E02  3.30E02  2.09E02  1.68E02  1.35E03  2.22E02  9.35E04  
F14  Mean  1.20E+00  6.82E01  3.12E01  2.69E01  2.31E01  8.46E02  3.98E02  3.76E03  2.55E01  2.40E02 
Std  1.98E+00  1.19E+00  1.18E+00  4.34E04  3.99E04  1.33E02  4.48E02  1.49E02  7.84E02  1.14E03  
F15  Mean  8.64E04  2.91E05  2.62E05  2.37E05  5.44E06  5.41E06  4.46E06  1.12E06  2.43E06  1.01E06 
Std  1.29E04  4.59E05  2.38E06  3.80E07  1.13E05  1.11E05  1.76E05  1.20E05  1.36E03  8.00E05  
F16  Mean  −1.03E+00  −9.86E01  −1.82E02  −5.97E03  −3.77E03  −2.36E03  −1.46E03  −1.97E04  −1.81E01  −4.60E04 
Std  2.21E15  1.37E15  1.63E16  1.47E17  5.00E18  4.10E18  2.25E18  4.83E19  2.26E09  3.29E11  
F17  Mean  3.98E01  3.91E01  1.57E01  1.36E01  1.79E02  7.05E04  4.45E04  1.68E04  6.10E04  1.20E04 
Std  3.33E01  3.01E02  8.39E02  6.26E02  1.04E02  1.03E04  1.52E04  2.89E05  6.87E07  3.78E05  
F18  Mean  3.00E+00  4.60E01  2.22E01  2.00E02  2.71E05  7.01E06  9.71E03  1.00E03  1.94E+00  1.35E03 
Std  2.11E15  2.06E16  1.19E16  5.89E17  3.76E17  9.46E18  7.31E18  1.43E18  6.07E04  8.52E09  
F19  Mean  −3.86E+00  −2.91E+00  −1.19E+00  −2.81E01  −7.51E02  −1.69E02  −4.40E03  −2.03E03  −1.53E01  −1.58E02 
Std  3.76E03  5.48E05  2.61E05  1.54E06  1.26E06  9.94E07  5.96E07  4.80E07  3.68E04  1.89E08  
F20  Mean  −5.53E01  −2.82E01  −5.93E01  −3.21E+00  −7.75E01  −1.59E01  −1.02E01  −2.05E03  −5.17E01  −3.59E03 
Std  9.39E02  6.64E02  8.21E04  5.66E04  3.49E04  9.27E05  1.46E05  1.28E05  5.48E03  1.04E05  
F21  Mean  −9.13E+00  −4.50E+00  −1.32E+00  −7.78E01  −2.34E02  −8.21E03  −3.70E03  −1.59E03  −1.77E+00  −2.33E02 
Std  2.03E01  1.54E01  2.84E02  5.91E03  7.13E04  6.02E05  5.82E05  2.13E05  8.32E03  1.45E05  
F22  Mean  −2.03E01  −1.88E01  −7.68E01  −4.45E01  −8.27E+00  −3.85E+00  −2.07E02  −1.20E03  −4.44E+00  −2.34E01 
Std  5.59E01  1.12E01  1.11E01  2.46E02  1.14E02  5.57E03  3.49E03  1.21E03  4.85E03  1.38E03  
F23  Mean  −6.56E+00  −2.36E+00  −1.10E+00  −7.06E02  −1.28E02  −4.55E03  −3.96E04  −2.92E04  −1.51E+00  −4.53E+00 
Std  1.04E04  1.86E05  3.91E06  2.47E06  3.24E07  3.88E09  2.42E06  9.25E07  2.95E02  1.28E03  
+/−/=  34/6/6  21/19/6  27/16/3  21/20/5  23/20/3  18/27/2  22/19/5  16/28/2  –  – 