Multi-objective optimization for 3D heterogeneous WSN deployment using an enhanced Genghis Khan shark algorithm.

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Learn more: PMC Disclaimer | PMC Copyright Notice Sci Rep . 2026 Apr 12;16:12075. doi: 10.1038/s41598-026-45399-z Search in PMC Search in PubMed View in NLM Catalog Add to search Multi-objective optimization for 3D heterogeneous WSN deployment using an enhanced Genghis Khan shark algorithm Essam H Houssein Essam H Houssein 1 Faculty of Computers and Information, Minia University, Minia, Egypt 2 Minia National University, Minia, Egypt Find articles by Essam H Houssein 1, 2, ✉ , Ibrahim E Ibrahim Ibrahim E Ibrahim 3 Faculty of Computers and Information, Luxor University, Luxor, Egypt Find articles by Ibrahim E Ibrahim 3 , Yaser M Wazery Yaser M Wazery 1 Faculty of Computers and Information, Minia University, Minia, Egypt Find articles by Yaser M Wazery 1 , Marwa M Emam Marwa M Emam 1 Faculty of Computers and Information, Minia University, Minia, Egypt Find articles by Marwa M Emam 1 Author information Article notes Copyright and License information 1 Faculty of Computers and Information, Minia University, Minia, Egypt 2 Minia National University, Minia, Egypt 3 Faculty of Computers and Information, Luxor University, Luxor, Egypt ✉ Corresponding author. Received 2026 Feb 12; Accepted 2026 Mar 18; Collection date 2026. © The Author(s) 2026 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ . PMC Copyright notice PMCID: PMC13077042  PMID: 41968147 Abstract Heterogeneous three-dimensional (3D) wireless sensor network (WSN) deployment requires balancing sensing coverage, communication connectivity, and deployment cost under coupled K -coverage and C -connectivity constraints. This setting yields a constrained mixed discrete optimization landscape where many conventional multi-objective methods lose diversity or handle feasibility inconsistently. We formulate heterogeneous 3D WSN deployment as a constrained multi-objective problem and propose the Enhanced Multi-Objective Genghis Khan Shark Optimizer (EnMOGKSO). The core novelty is the integration of leader-pursuit dynamics with (i) dual archive-guided selection (elite and neighborhood memories), (ii) bounded external archive diversity control, and (iii) feasibility-first environmental selection for fragmented feasible regions. On the Congress on Evolutionary Computation (CEC) 2020 suite, EnMOGKSO obtains the best Friedman mean ranks in hypervolume (HV) (2.04) and inverted generational distance (IGD) (2.38), with statistically significant differences against most competitors ( , Wilcoxon/Friedman). In heterogeneous 3D WSN deployment, EnMOGKSO yields higher coverage/connectivity values (typically coverage means around 11–12 and connectivity around 7) than weaker baselines (often coverage 5–7 and connectivity 4–5), with higher but stable deployment cost. Overall, the results indicate a stronger convergence-diversity balance and more reliable feasibility-aware search under tight constraints, with practical applicability to 3D monitoring tasks such as industrial facilities, smart buildings, and environmental sensing. Keywords: Wireless sensor network, Wireless sensor network deployment, Connectivity, Multi-objective, Optimization Subject terms: Energy science and technology, Engineering, Mathematics and computing Introduction Wireless sensor networks (WSNs) are increasingly deployed in volumetric and heterogeneous environments, including urban infrastructure, multi-story buildings, forest canopies, and industrial facilities. In such settings, planar (2D) assumptions are often insufficient for modeling sensing and communication. Deployment must therefore be optimized in three-dimensional (3D) space while accounting for heterogeneity in sensing range, communication radius, and deployment cost. We consider the heterogeneous 3D WSN deployment problem , where sensor locations and sensor types are jointly selected to (i) maximize sensing coverage, (ii) maintain reliable multi-hop connectivity, and (iii) satisfy cost constraints. These objectives are inherently conflicting: improving coverage often requires additional nodes or larger sensing radii, both of which increase cost. Enforcing stronger connectivity may also induce spatial clustering, which can reduce coverage efficiency. In 3D settings, connectivity depends on volumetric inter-node distances, while heterogeneity introduces discrete type-selection decisions in addition to spatial placement. The resulting optimization problem is constrained, nonlinear, combinatorial, and NP-hard in practical formulations, making exact methods impractical at realistic scales. Differences in sensing radius, communication behavior, and installation cost change both objective values and feasibility boundaries. Consequently, homogeneous approximations can yield overly optimistic deployment plans that violate practical connectivity or budget constraints once real sensor classes are used. This motivates explicit heterogeneous modeling in both the decision space and constraint set. The deployment formulation combines mixed decision variables, non-convex objective interactions, and fragmented feasible regions under coupled K -coverage and C -connectivity constraints. Under these conditions, deterministic methods are difficult to scale, while Pareto metaheuristics provide a practical way to recover multiple high-quality trade-off deployments in a single optimization run. Multi-objective evolutionary and swarm-based algorithms are natural candidates for this task because they approximate trade-off sets within a single run. However, under heterogeneous 3D deployment with mixed discrete–continuous variables and strict feasibility constraints, conventional Pareto-based methods (e.g., the Nondominated Sorting Genetic Algorithm II (NSGA-II), Multi-Objective Particle Swarm Optimization (MOPSO), and decomposition-based approaches) often exhibit premature convergence, archive crowding, and unstable feasibility behavior in sparse feasible regions. In addition, generic operators do not explicitly exploit deployment structure, where small spatial perturbations can trigger discontinuous changes in coverage or connectivity. To address these limitations, we introduce the Enhanced Multi-Objective Genghis Khan Shark Optimizer (EnMOGKSO) for heterogeneous 3D WSN deployment. EnMOGKSO integrates leader-pursuit swarm dynamics with Pareto-archive guidance and constraint-domination selection to preserve diversity and improve feasibility-aware search in discontinuous deployment landscapes. Unlike generic multi-objective evolutionary algorithms (MOEAs), the framework explicitly couples mixed encoding (3D positions and sensor types) with archive-driven diversity control, enabling exploration of constrained trade-off regions while preserving structural feasibility. The main contributions are as follows: We formulate heterogeneous 3D WSN deployment as a constrained multi-objective optimization problem that jointly models sensing coverage, communication connectivity, and deployment cost using unified notation. We develop EnMOGKSO as a multi-objective extension of GKSO, combining leader-pursuit dynamics, dual-archive management, and feasibility-first constraint domination to improve convergence stability and diversity preservation. We design a mixed encoding-decoding strategy for joint optimization of 3D placement and sensor-type selection, with explicit mapping between search operators and WSN objectives/constraints. We validate the method on CEC 2020 and heterogeneous 3D WSN scenarios using HV, IGD, and PSP with Wilcoxon/Friedman tests, with the base station fixed outside the deployment area. The remainder of the paper is organized as follows. “ Literature review ” reviews related work and positions the proposed method. “ Proposed EnMOGKSO and mathematical foundations ” details EnMOGKSO and its mathematical foundations. “ WSN system model and problem formulation ” presents the WSN system model and formal problem formulation. “ Experimental evaluation of EnMOGKSO ” reports benchmark validation, and “ Optimal WSN deployment results ” reports heterogeneous WSN deployment results. “ Conclusion and future work ” concludes the paper. Literature review This section reviews prior work on heterogeneous 3D WSN deployment in four categories: deployment geometry (2D/3D), objective modeling (energy/coverage/connectivity/cost), optimization strategies, and constraint handling. Across these categories, a recurring limitation is weak feasibility-aware diversity control when K -coverage and C -connectivity are enforced jointly. Review of WSN deployment strategies in 2D vs. 3D environments WSN deployment research has progressed from simplified 2D assumptions to heterogeneous 3D settings that account for terrain and obstacle effects 1 – 3 . Recent studies increasingly combine domain-aware modeling with metaheuristic search to optimize coverage, connectivity, and cost simultaneously 4 – 6 . Comparable trends are observed in underwater and irregular environments, where hybrid strategies can improve practical performance 7 – 14 . However, many methods still degrade when feasible regions are fragmented or objective conflicts are severe, especially in heterogeneous 3D deployment. Energy constraints Energy optimization in WSNs has been studied through protocol design, routing, clustering, energy harvesting, and AI-assisted control 15 – 38 . Despite this progress, energy behavior remains strongly coupled with deployment feasibility and multi-objective trade-offs, motivating methods that optimize coverage, connectivity, and cost jointly under constraints. Coverage models: k-coverage and its importance Coverage models typically target robust k -coverage under energy and redundancy limits, often with hybrid metaheuristics 30 , 39 – 41 . Remaining challenges include scalability and stable solution quality under tight feasibility constraints. Connectivity models: C-connectivity and fault tolerance Connectivity research spans clustering, relay placement, fault recovery, and learning-based adaptation 42 – 50 . These approaches improve reliability, but strict C -connectivity combined with simultaneous coverage/cost optimization remains challenging in heterogeneous 3D deployment. Cost model Cost-aware deployment studies emphasize low-power hardware and platform efficiency to reduce installation and maintenance burden 51 , 52 . In practice, cost must be optimized jointly with coverage and connectivity constraints to avoid low-quality network designs. Existing optimization techniques Deterministic optimization can provide exact solutions under restrictive assumptions, but it is generally impractical for large, nonlinear, and dynamic WSN deployment problems 53 . As a result, most current methods rely on metaheuristic and swarm-based approaches (e.g., GA, PSO, ACO, DE, WOA, GWO) for scalable search in constrained spaces 54 , 55 . Multi-objective frameworks (e.g., NSGA-II, the Multi-Objective Evolutionary Algorithm based on Decomposition (MOEA/D), SPEA2, and hybrids) are now standard for balancing coverage, connectivity, energy, and cost 50 , 56 . However, diversity preservation and stable feasibility handling remain difficult in heterogeneous 3D settings. Table 1 summarizes representative methods and their limitations. Relative to this literature, EnMOGKSO is positioned as a feasibility-aware, dual-archive swarm framework for constrained heterogeneous 3D deployment. Table 1. Review of literature: coverage, connectivity, and cost optimizations. Ref Objectives 2-D/3-D Methodology/limitations Conn. Cov. Cost Other 35 Scalability 2-D Heuristic Algorithm: Limited scalability in heterogeneous environments. 16 Sustainability 2-D Heuristic Algorithm: Increased computational costs for large-scale deployments. 17 Energy Efficiency 2-D Machine Learning: Relies on high-quality data for accurate predictions. 18 X Energy Savings 3-D Nano-Technology: High implementation costs due to advanced hardware. 19 X Reliability 2-D Heuristic Algorithm: Limited energy optimization for large-scale logistics networks. 20 X Decentralized Energy 2-D Blockchain: High overhead due to blockchain transaction costs. 21 X Sustainability 3-D RF Energy Harvesting: Requires reliable energy sources for consistent performance. 22 X Scalability 3-D Bayesian Optimization: Computational complexity in dynamic aerial environments. 23 Cost Efficiency 2-D Heuristic Algorithm: Limited range and data rate for large-scale monitoring. 24 X Adaptability 3-D Evolutionary Game Theory: High energy consumption for UAVs in long-duration tasks. 25 X X Latency Reduction 2-D Adaptive MAC Protocol: Inefficient for sparse deployments with low traffic. 26 X X Reliability 2-D Adaptive Backoff Algorithm: Performance decreases with increasing network density. 27 X X Collision Resolution 3-D Game Theory-Based Protocol: Not suitable for highly dynamic underwater environments. 28 X X Load Balancing 2-D Swarm Intelligence: High computational cost for large-scale networks. 29 X X Scalability 2-D Hybrid Swarm Optimization: Limited adaptability in real-time topology changes. 30 X Network Awareness 2-D Deep Reinforcement Learning: Training requires significant computational resources. 31 X X Reliability 2-D Joint Routing and Storage: Limited performance in highly dynamic environments. 32 Energy Efficiency 2-D Stochastic PSO: Convergence speed is slow for complex networks. 39 X Adaptability 2-D Ant Colony Optimization: High convergence time in large-scale scenarios. 41 X Efficiency 2-D Simulated Annealing: Computationally expensive for high-density deployments. 42 X X Robustness 2-D Clustering Algorithm (LEACH): Frequent cluster reformation increases energy costs. 57 X Scalability 2-D Pareto-Based Optimization: High complexity in balancing multiple objectives. 46 X Scalability 3-D Heterogeneous Resource Allocation: Increased computational overhead in heterogeneous deployments. 49 X X Connectivity Restoration 3-D AI Restoration Techniques: Requires pre-trained models for effective restoration. Open in a new tab Proposed EnMOGKSO and mathematical foundations Genghis Khan shark optimizer (GKSO) GKSO is a swarm metaheuristic inspired by hunting and survival behaviors. Its search process contains four components: exploration, exploitation, foraging transition, and defense 58 . Wandering hunting stage (exploration) This stage samples broad candidate positions within variable bounds to emphasize global exploration. where , N is population size, D is dimensionality, and t is the current iteration. Moving towards the best hunting position (exploitation) This stage moves each agent toward the best-known position, with attraction strength proportional to detected signal intensity: 1 with 2 where is random, I is fitness-related intensity, and m is a control constant. The final update averages attraction and historical motion: 3 Parabolic foraging This phase models cooperative exploitation around the current best position: 4 where p controls exploration-exploitation step size over iterations: 5 where denotes the weight coefficient at time and is computed as follows: 6 Self protection mechanism This stage introduces defensive perturbations around elite solutions to diversify local search and avoid stagnation. 7 where , , and are random coefficients. 8 Here, is a random solution, is a sampled population member, and is an adaptive coefficient: 9 10 11 This mechanism perturbs candidates around the current best and keeps the strongest trial, improving local escape and refinement. Basic concepts of multi-objective optimization problems (MOPs) Multi-objective optimization addresses problems with conflicting objectives and seeks a set of non-dominated trade-off solutions rather than a single optimum. Mathematical formulation A general MOP is defined by an objective vector and a feasible set. For consistency, we use the standard minimization form. 12 Pareto optimality Pareto dominance is used to compare conflicting objective vectors and define non-dominated trade-off solutions. A solution is said to dominate another solution , denoted , if and only if the following two conditions are satisfied: 13 A solution is Pareto optimal if no other feasible solution dominates it. The corresponding non-dominated set in decision space is the Pareto set (PS), and its image in objective space is the Pareto front (PF). WSN system model and problem formulation The deployment problem is defined over a finite set of 3D candidate locations and heterogeneous sensor types, with three coupled criteria: sensing coverage, network connectivity, and deployment cost. We first define decision variables, then present compact sensing and communication models, and finally state the constrained multi-objective formulation. Deployment domain, targets, and sensor types Let denote the set of candidate deployment locations, where . Let denote the set of target points, . Sensors belong to classes indexed by ; class has sensing radius and unit cost . Each candidate site i has a location-dependent deployment factor . Decision variables. Let indicate whether a sensor of class is deployed at site . At most one sensor can be deployed at each site: 14 Notation summary Table 2 lists the main symbols used in the WSN model and deployment formulation. Table 2. Notation used in the WSN system model and deployment formulation. Symbol Description , Candidate deployment sites and their count , Target points and their count Number of sensor classes Binary decision variable (place class at site i ) , Sensing radius and unit cost of class Location-dependent deployment cost factor at site i K , C Minimum coverage multiplicity and nodal connectivity thresholds Cost, coverage, and connectivity objectives Total constraint violation , Decoder threshold and site activation indicator , Parameters of the probabilistic link model Open in a new tab Coverage model and K-coverage constraint For a target and site , define the sensing indicator 15 The coverage multiplicity of target is 16 We enforce K -coverage for every target: 17 As a scalar coverage objective, we use the average coverage multiplicity: 18 Communication/connectivity model and C-connectivity constraint We adopt a probabilistic communication model between two (candidate) locations and : 19 where and are communication-model parameters. To obtain a deterministic adjacency relation, we define the binary link indicator 20 where is a connectivity threshold. The nodal connection degree (number of neighbors) of site i induced by a deployment is defined as 21 We impose a minimum C -connectivity requirement for each deployed node using a standard activation trick: 22 As a scalar connectivity objective, we report the average nodal degree over deployed sensors: 23 Cost model The total deployment cost combines the sensor hardware cost and the location-dependent deployment factor: 24 Unlike uniform-cost assumptions, captures spatial differences in accessibility, installation effort, and site preparation burden. This improves deployment realism in irregular terrains and built environments where physically difficult locations induce higher practical cost. Multi-objective deployment optimization problem The deployment of heterogeneous sensors is formulated as a multi-objective optimization problem that jointly considers several conflicting design objectives. In practical wireless sensor network (WSN) deployments, improving sensing performance or network connectivity typically requires additional sensors or higher-capability devices, which increases deployment cost. Therefore, an appropriate deployment strategy must carefully balance these competing requirements. We simultaneously (i) minimize deployment cost, (ii) maximize coverage multiplicity, and (iii) maximize connectivity degree, which together represent the key performance indicators of a reliable WSN deployment, subject to explicit K -coverage and C -connectivity constraints. To maintain consistency with Pareto-based multi-objective optimization methods, the problem is expressed in a minimization form. Consequently, the maximization objectives related to coverage and connectivity are transformed into minimization terms by taking their negative values. Using a minimization form consistent with Pareto dominance, the deployment problem is: 25 In this formulation, the binary decision variable indicates whether a sensor of class is deployed at candidate location i . The proposed formulation explicitly models sensor heterogeneity through the class-dependent deployment variable . This allows different sensor types to contribute differently to coverage performance, communication connectivity, and deployment cost. Such modeling reflects realistic deployment conditions where sensors may differ in sensing range, communication capability, and hardware cost. Feasibility of candidate deployment solutions is guaranteed by enforcing the K -coverage and C -connectivity constraints, ensuring that every monitored location is covered by at least K sensors while maintaining sufficient network connectivity for reliable data transmission. Accurate modeling of sensor heterogeneity is particularly important because homogeneous assumptions may significantly overestimate network feasibility. In practice, ignoring hardware diversity can result in deployment strategies that appear optimal in simulation but become difficult to implement due to physical device limitations or budget constraints. Experimental framework and statistical interpretation This section presents the evaluation protocol for two tracks: CEC 2020 benchmarks and constrained heterogeneous 3D WSN deployment (“ WSN system model and problem formulation ”), with emphasis on reproducibility and statistical validity. Algorithms compared and common experimental settings EnMOGKSO is compared with representative Pareto-based baselines (e.g., NSGA-II, MOPSO, MOGWO, MOEA/D) under a common budget (identical maximum function evaluations, run count, and stopping rule). Randomization and reproducibility Each algorithm is executed for R independent runs with different random seeds. Metrics are computed from the final non-dominated set of each run and summarized by mean/standard deviation (SD), best, and worst values. Parameter configuration Table 3 reports the parameter settings for EnMOGKSO and the compared algorithms. To ensure fairness, values were adopted from source papers when available; otherwise, they were tuned on a small validation subset and then fixed for all tests. Table 3. Parameter settings used in the experiments (all algorithms share the same population size and evaluation budget unless stated otherwise). Component Symbol / name Setting / description Common Population size Common Iteration budget Common Independent runs EnMOGKSO External archive size (maximum number of non-dominated solutions retained) EnMOGKSO Dual-archive design PBA (elite) + NBA (diversity/novelty) with pruning by crowding/density EnMOGKSO Leader selection Diversity-aware leader sampling from PBA/NBA (probability biased to sparse regions) EnMOGKSO Constraint handling Feasibility-first dominance + penalty/repair for constraint violations WSN model Coverage requirement K -coverage constraint in ( 17 ) WSN model Connectivity requirement C -connectivity constraint in ( 22 ) WSN model Link threshold in ( 20 ) WSN model Comm. parameters in ( 19 ) Open in a new tab All algorithms are run under the same common budget for fair comparison. Scenario descriptions CEC 2020 multi-objective benchmark scenarios The CEC 2020 suite contains 24 multi-objective test functions with varying Pareto-front shapes, separability, modality, and variable interactions. These properties diagnose optimizer behavior: multimodality stresses global exploration, nonseparability stresses coordinated high-dimensional search, and irregular fronts test archive-level diversity maintenance. For each function, we follow the standard suite settings (dimension and objective count) and evaluate all algorithms under the same evaluation budget. 3D heterogeneous WSN deployment scenarios We consider a 3D monitoring volume with candidate deployment sites and target points . Each site can host at most one sensor type (constraint ( 14 )). The objectives are cost minimization ( 24 ) and maximization of average coverage multiplicity ( 18 ) and average nodal degree ( 23 ), subject to K -coverage ( 17 ) and C -connectivity ( 22 ). Scenario factors To analyze scalability and constraint difficulty, scenarios systematically vary: (i) the number of targets (coverage density), (ii) the number of candidate sites (search space size), (iii) heterogeneity level (type-choice combinatorics), and (iv) constraints ( K , C ) (feasibility tightness). In all WSN scenarios, the base station (BS) is placed outside the deployment area. Under tight constraints (large K and/or C ), feasible regions become fragmented; effective optimizers must preserve feasible candidates while exploring trade-offs. Performance indicators and metric definitions We use three standard multi-objective quality indicators to quantify convergence and diversity of the approximation set P relative to a reference Pareto set . Hypervolume (HV) HV measures the dominated volume between the approximation set P and a user-defined reference point in objective space. For minimization objectives, a larger HV indicates a better combination of proximity to the Pareto front and coverage/spread along the front: 26 where is the Lebesgue measure and M is the number of objectives. Inverted generational distance (IGD) IGD measures the average distance from each point on the reference front to the approximation set: 27 Smaller IGD indicates that the approximation set is both close to and well distributed along the reference front. Pareto set proximity (PSP) PSP measures how close the obtained solutions are to the reference set. In this study, PSP is computed as the average distance from solutions in P to their nearest counterparts in : 28 Smaller PSP suggests stronger convergence toward the true Pareto-optimal set. The HV indicator quantifies both the dominance quality and the spread of the obtained Pareto solutions. In contrast, The IGD measures the average distance from the reference Pareto front to the set of solutions produced by the algorithm, thereby reflecting convergence and coverage of the true front. Meanwhile, the PSP metric evaluates the distance from the obtained solution set to the reference front, providing complementary information regarding the proximity of the generated solutions to the optimal Pareto boundary. Reporting these three indicators jointly mitigates the bias associated with relying on a single performance metric and enables a more balanced evaluation of optimization performance in terms of convergence, diversity, and approximation accuracy of the Pareto front. Statistical validation Because indicator distributions are often non-normal and heteroscedastic, we use nonparametric tests. Pairwise Wilcoxon signed-rank test We compare EnMOGKSO with each competitor using the Wilcoxon signed-rank test ( ) on paired runs. One-sided alternatives are used: HV (higher is better) and IGD/PSP (lower is better). Friedman test and ranking Across all problems, the Friedman test ranks algorithms (rank 1 is best; HV descending, IGD/PSP ascending) and tests overall differences. When significant, mean ranks summarize global performance. Quantitative comparison protocol For each CEC function and WSN scenario, we report mean, standard deviation, best, and worst values of HV/IGD/PSP over R runs. Best row values are highlighted, and statistical significance is indicated by Wilcoxon results. Scientific interpretation: why EnMOGKSO performs better Relative to classical Pareto swarms, the observed gains are explained by four design choices that jointly improve diversity and feasibility in constrained mixed-variable search. Mechanism 1: dual-archive selection reduces dominance pressure and prevents diversity collapse EnMOGKSO uses an elite archive (PBA) and a novelty archive (NBA). PBA preserves high-quality solutions, while NBA preserves sparse or bridging solutions, improving front coverage (HV) and reducing missed regions (IGD). Mechanism 2: feasibility-aware selection stabilizes search under tight K/C constraints Feasibility-first dominance prioritizes feasible solutions and then lower violation among infeasible ones. This guides search toward feasible regions without collapsing diversity and improves robustness. Mechanism 3: mixed representation benefits from continuous search plus deterministic decoding A continuous score matrix is decoded into a discrete placement with “at most one sensor per site.” This keeps search smooth while enforcing combinatorial validity, reducing wasted evaluations and accelerating convergence. Mechanism 4: leader sampling biased to sparse regions improves spread without sacrificing convergence Density-aware leader sampling emphasizes under-represented regions, improving spread and hypervolume while keeping convergence competitive. Experimental summary Results across CEC 2020 and heterogeneous 3D WSN scenarios indicate a better convergence-diversity balance and stronger feasibility handling, with nonparametric support from Wilcoxon and Friedman tests. Problem-specific encoding, decoding, and constraint handling (A) Solution encoding To enable the application of the continuous EnMOGKSO search operators to the discrete WSN deployment problem, each candidate solution (agent) is represented as a continuous vector that is later decoded into a feasible deployment configuration. Each EnMOGKSO agent is a continuous vector with 29 where indexes heterogeneous sensor classes. This representation allows the optimization algorithm to explore a continuous search space while still producing discrete deployment decisions after decoding. We reshape y into a score matrix , where denotes the selection score of placing class at site i . Higher scores indicate stronger preference for selecting a particular sensor class at the corresponding deployment location. The corresponding discrete deployment is represented by the binary placement variables defined in “ WSN system model and problem formulation ”. Continuous-to-binary transformation and “at most one sensor per site” To transform the continuous score representation into a valid deployment configuration, a site-wise decoding rule is applied. This rule ensures that each candidate location hosts at most one sensor, while still allowing heterogeneous sensor types to be selected. For each site i , we select a single sensor class (or none) by 30 and then set 31 where is an activation threshold and is the indicator function. The activation threshold controls whether a sensor is deployed at site i , thereby allowing the algorithm to deactivate locations when doing so improves the objective values or reduces constraint violations. Equations ( 30 )–( 31 ) enforce Eq. ( 14 ) by construction and encode heterogeneity through the selected class index . (B) Decoding and objective/constraint evaluation Once a continuous solution y is decoded into the binary deployment matrix , the resulting configuration is evaluated using the problem objectives and constraints defined in “ WSN system model and problem formulation ”. Given y , decoding yields via Eqs. ( 30 )–( 31 ). We then evaluate: Coverage: compute using Eq. ( 16 ) and using Eq. ( 18 ). Communication graph: build using Eqs. ( 19 )–( 20 ) and compute degrees using Eq. ( 21 ). Connectivity: compute using Eq. ( 23 ). Cost: compute using Eq. ( 24 ). Constraint violation: detect K -coverage and C -connectivity violations using 32 where . The violation function quantifies the degree to which a candidate deployment fails to satisfy the required coverage and connectivity constraints. A value of indicates a feasible deployment configuration. (C) Constraint handling mechanism (constraint-domination) To effectively guide the search toward feasible high-quality solutions, EnMOGKSO adopts a feasibility-first constraint-domination strategy, which is widely used in constrained multi-objective optimization. EnMOGKSO uses a feasibility-first constraint-domination relation. For two solutions and , with violations and , the relation is: 33 This rule prioritizes feasible solutions over infeasible ones and promotes solutions with smaller constraint violations when feasibility has not yet been achieved. This relation is used in sorting, selection, and archive updates. Consequently, the external archive maintains only feasible non-dominated solutions, ensuring that the final Pareto front represents practically deployable WSN configurations. (D) Interaction between EnMOGKSO operators and WSN structure The interaction between the EnMOGKSO search operators and the WSN deployment structure occurs through the encoding–decoding mechanism. PBA/NBA store continuous vectors y but are evaluated after decoding to . This design allows the optimization process to operate in a smooth continuous space while the evaluation phase respects the discrete deployment constraints of the WSN problem. Levy perturbations diversify Y by activating/deactivating sites and switching sensor classes, while NBA-guided refinement biases updates toward feasible deployments with lower cost and stronger coverage/connectivity. As a result, the algorithm maintains a balance between structural diversity in the deployment configurations and progressive improvement of the multi-objective performance metrics. Computational complexity This subsection analyzes the computational cost of the proposed EnMOGKSO framework with respect to the main problem parameters. Let denote the population size, the number of candidate deployment sites, the number of heterogeneous sensor classes, and the number of monitoring targets. During each iteration, every candidate solution must be decoded and evaluated using the coverage, connectivity, and cost objectives. The computational effort is therefore dominated by the objective and constraint evaluation procedures. Coverage evaluation costs because the sensing contribution of each sensor class at every deployment location must be assessed with respect to all targets. Connectivity evaluation costs since the communication graph requires pairwise distance checks among active deployment sites. Considering that these evaluations are performed for each individual in the population, the dominant per-iteration complexity becomes 34 which corresponds to the total cost of evaluating all candidate deployments in one iteration of the algorithm. In addition to objective evaluation, EnMOGKSO performs archive maintenance operations for storing elite non-dominated solutions. These operations include dominance comparisons and archive updates, which typically incur a computational cost of plus archive maintenance, typically . Since the archive size is bounded in practice, this overhead remains moderate compared with the evaluation cost of the coverage and connectivity objectives. Consequently, the overall computational complexity of EnMOGKSO is primarily governed by the population size and the structural parameters of the WSN deployment problem. Architecture of the enhanced GKSO algorithm (EnMOGKSO) EnMOGKSO extends GKSO with archive-based elitism and hybrid exploration–exploitation: In the proposed EnMOGKSO framework, the search process is organized through a coordinated balance between exploration and exploitation mechanisms. Global exploration is primarily promoted through Lévy-flight perturbations and stochastic global sampling, which allow agents to escape local optima and explore diverse regions of the decision space. Conversely, exploitation is guided by elite solutions stored in external archives, namely the Personal Best Archive (PBA) and the Neighborhood Best Archive (NBA). These archives act as leader repositories that provide high-quality search directions for the population. Local refinement operators further intensify the search around promising regions of the Pareto front, thereby improving convergence accuracy. This explicit separation between exploration-oriented and exploitation-oriented operators follows established principles in swarm intelligence and multi-objective optimization. By maintaining strong exploration capability while preserving elitist guidance through the archives, the algorithm mitigates premature convergence and population stagnation. At the same time, the archive-based leadership mechanism preserves high-quality non-dominated solutions, which helps maintain diversity and ensures steady convergence toward a well-distributed approximation of the Pareto-optimal front. 1. Personal best archive (PBA) and neighborhood best archive (NBA) To preserve high-quality solutions during the search process, each agent maintains a Personal Best Archive (PBA) , which stores the best solution encountered by that agent according to Pareto dominance. Each agent keeps a Personal Best Archive (PBA) : 35 In addition to individual memory, a global Neighborhood Best Archive (NBA) is maintained to store elite non-dominated solutions discovered by the population. This archive provides guidance for the collective search and promotes information sharing among agents. A global Neighborhood Best Archive (NBA) stores elite solutions: 36 To prevent uncontrolled archive growth and to maintain computational efficiency, the NBA size is restricted by a predefined capacity constraint: 37 The combination of PBA and NBA memories provides stronger search guidance compared with traditional single global-best strategies, as it simultaneously preserves individual experience and population-level elite solutions. 2. Advanced search mechanisms During the hunting stage of the algorithm, Lévy-flight perturbations are introduced to enhance global exploration capability and to reduce the risk of premature convergence. Lévy flights generate occasional long jumps in the search space, enabling agents to escape local optima and investigate unexplored regions. 38 3. Elite-guided local search strategy To balance intensification and diversification, EnMOGKSO applies a probabilistic search strategy that alternates between local exploitation and global exploration around the current leader (Silverback). With probability 60%, local search: 39 With probability 40%, global sampling: 40 The search scale is controlled by a time-varying scale factor SF , which gradually decreases during the optimization process to shift the search behavior from exploration toward exploitation. 41 After generating candidate solutions, EnMOGKSO performs an elite-guided refinement step using either NBA or PBA information to improve solution quality. Candidate positions are then refined using NBA or PBA guidance: NBA-guided refinement: 42 PBA-guided refinement: 43 The refined candidate solutions are evaluated using the multi-objective fitness function, and the better solution is retained for the next iteration. This refinement mechanism enhances exploitation while preserving diversity through multiple elite guidance sources. EnMOGKSO workflow for WSN deployment (archive and selection) EnMOGKSO updates offspring in continuous space y , decodes them to , and applies selection via (Eq. ( 33 )). The external archive stores feasible non-dominated deployments and is pruned by crowding when full. Maintaining archive diversity is essential for deployment decision-making because practitioners often require multiple feasible cost-coverage-connectivity trade-offs rather than a single operating point. Experimental evaluation of EnMOGKSO We evaluate EnMOGKSO against MOPSO 59 , NSGA-II 60 , MOEA/D 61 , MOGWO 62 , MOWOA 63 , MOSCA 64 , MOSMA 65 , and IMOMRFO 66 . All comparative methods are executed under equivalent population and iteration budgets with multiple random seeds, and results are reported with dispersion statistics to support reliable comparisons. Benchmarking is performed on the CEC 2020 multi-objective suite and on the defined 3D heterogeneous WSN deployment scenarios in “ WSN system model and problem formulation ”; no additional external dataset is introduced in this revision. In this revision, we provide mechanism-level interpretation and broad algorithmic comparisons; a full component-wise ablation of all EnMOGKSO operators is explicitly treated as future work (“ Conclusion and future work ”). Evaluation metrics We use HV, IGD, and PSP as defined in “ Performance indicators and metric definitions ”. For HV, reference points are fixed to (two-objective cases) and (three-objective cases) across all runs. Experimental configuration All runs used the same hardware/software environment (Table 4 ). Table 4. Hardware and software configuration used in the experiments. Configurations Various components Characterization Hardware CPU Core (TM) i7 Frequency 1.70 GHz RAM 32 GB Hard Drive 500 GB Software Operating system Windows 8 Language MATLAB R2013a (8.1.0.604) Open in a new tab Table 5 lists algorithm settings; shared values are , , and 20 runs. Table 5. Parameter settings for EnMOGKSO and competing algorithms. Approaches Settings configuration shared configurations Agents size: Number of iterations: Independent runs: MOGKSO & EnMOGKSO , MOEA/D , , , , MOSMA , NSGA-II , IMOMRFO , , MOSCA , , , , , MOWOA , , , , , , MOGWO , , , MOGTO , , , , MOPSO , , , , Open in a new tab Hyperparameters follow source papers with limited preliminary tuning, then remain fixed for all MMF tests. CEC 2020 benchmark results The CEC 2020 multi-objective suite is used to test robustness under diverse front geometries, multimodality, and variable interactions. This helps verify whether algorithmic improvements generalize beyond a single deployment instance and remain stable in irregular, non-convex landscapes. Tables 6 , 7 , 8 show that EnMOGKSO is the most consistent high-performing method under the same budget, with top/near-top ranks and generally lower dispersion . Table 6. Comparative HV results for EnMOGKSO and competing multi-objective algorithms. Algorithm MMF1 MMF2 MMF4 MMF5 Mean SD Rank Mean SD Rank Mean SD Rank Mean SD Rank EnMOGKSO 0.873 0.000297 1 0.874 0.0000659 1 0.54 0.000356 1 0.873 0.000171 1 MOGKSO 0.872 0.000198 2 0.872 0.000549 2 0.54 0.0000821 2 0.872 0.000153 2 IMOMRFO 0.865 0.00077 7 0.865 0.00108 5 0.53 0.0013 7 0.865 0.00136 7 MOEA/D 0.843 0.00963 10 0.709 0.0793 11 0.513 0.00598 11 0.844 0.00479 9 MOGTO 0.87 0.000292 3 0.843 0.0108 8 0.537 0.000541 3 0.87 0.000381 3 MOGWO 0.842 0.00781 11 0.861 0.11 7 0.522 0.0545 10 0.847 0.096 11 MOPSO 0.869 0.000312 4 0.868 0.000802 3 0.535 0.000367 4 0.869 0.000329 4 MOSCA 0.864 0.00112 8 0.776 0.265 9 0.525 0.00918 8 0.865 0.00127 6 MOSMA 0.868 0.00195 5 0.866 0.000631 4 0.532 0.00145 6 0.864 0.00692 8 MOWOA 0.737 0.0775 12 0.653 0.121 12 0.413 0.0946 12 0.781 0.0214 12 MO_Ring_PSO_SCD 0.867 0.000823 6 0.862 0.00178 6 0.533 0.000893 5 0.868 0.00066 5 NSGA-II 0.844 0.00619 9 0.726 0.0486 10 0.523 0.00466 9 0.84 0.0103 10 Algorithm MMF7 MMF8 MMF10 MMF11 Mean SD Rank Mean SD Rank Mean SD Rank Mean SD Rank EnMOGKSO 0.873 0.000541 1 0.422 0.000263 1 12.9 0.00103 1 14.5 0.00352 1 MOGKSO 0.873 0.000094 2 0.422 0.0000791 2 12.9 0.000355 2 14.5 0.000518 2 IMOMRFO 0.865 0.0014 7 0.411 0.00516 8 12.8 0.012 6 14.5 0.00777 6 MOEA/D 0.854 0.00358 9 0.372 0.0138 10 12.4 0.536 9 14.4 0.0291 8 MOGTO 0.87 0.000383 3 0.419 0.000343 3 12.8 0.0908 8 14.5 0.00468 4 MOGWO 0.833 0.0125 11 0.408 0.575 11 12.8 0.00332 3 14.3 0.021 11 MOPSO 0.868 0.000415 5 0.418 0.000458 4 12.8 0.00396 4 14.5 0.00263 3 MOSCA 0.862 0.00138 8 0.414 0.00149 7 12.8 0.00509 5 14.4 0.0135 7 MOSMA 0.869 0.00147 4 0.415 0.00132 5 12.3 0.204 10 14.4 0.0505 9 MOWOA 0.788 0.0355 12 0.357 0.0986 12 12 0.424 11 13.3 0.63 12 MO_Ring_PSO_SCD 0.865 0.00135 6 0.415 0.00145 6 12.8 0.0242 7 14.5 0.00528 5 NSGA-II 0.849 0.00683 10 0.408 0.00212 9 11.8 0.345 12 14.4 0.0962 10 Algorithm MMF12 MMF13 MMF14 MMF15 Mean SD Rank Mean SD Rank Mean SD Rank Mean SD Rank EnMOGKSO 1.57 0.000153 4 18.4 0.0021 1 2.96 0.246 2 4.24 0.181 2 MOGKSO 1.57 0.000118 1 18.4 0.00225 2 2.86 0.15 4 4.13 0.165 4 IMOMRFO 1.57 0.0034 7 18.4 0.0118 5 2.67 0.282 9 3.9 0.276 8 MOEA/D 1.47 0.125 9 18.2 0.0552 10 2.78 0.168 7 4.12 0.251 5 MOGTO 1.57 0.00013 2 18.4 0.0129 6 2.82 0.102 5 4.19 0.114 3 MOGWO 1.55 0.00329 8 18.2 0.0406 9 1.29 0.0223 12 2.08 0.0237 12 MOPSO 1.57 0.000139 3 18.4 0.00335 3 2.83 0.255 6 4.09 0.159 6 MOSCA 1.57 0.00278 6 18.3 0.0117 8 2.64 0.32 11 3.9 0.301 9 MOSMA 1.41 0.164 11 18.4 0.045 7 3.1 0.351 1 4.4 0.551 1 MOWOA 1.02 0.31 12 17 0.579 12 3 1.31 3 3.69 1.36 11 MO_Ring_PSO_SCD 1.57 0.00122 5 18.4 0.00551 4 2.65 0.237 10 3.95 0.26 7 NSGA-II 1.47 0.154 10 17.9 0.262 11 2.73 0.142 8 3.86 0.405 10 Algorithm MMF1_e MMF14_a MMF15_a MMF10_I Mean SD Rank Mean SD Rank Mean SD Rank Mean SD Rank EnMOGKSO 0.874 0.000296 1 2.84 0.0854 8 4.24 0.0865 4 12.9 0.00116 1 MOGKSO 0.872 0.000576 2 2.9 0.135 7 4.35 0.252 2 12.9 0.000374 2 IMOMRFO 0.865 0.00253 4 2.81 0.266 9 3.99 0.394 7 12.8 0.0117 7 MOEA/D 0.803 0.1 8 2.93 0.237 6 4.18 0.193 5 12.3 0.564 10 MOGTO 0.87 0.000542 3 3.1 0.204 1 4.37 0.46 3 12.9 0.00388 3 MOGWO 0.844 0.7 9 1.44 0.0432 12 2.43 0.0752 12 12.8 0.00311 4 MOPSO 0.865 0.00648 5 2.95 0.167 4 4.09 0.133 6 12.8 0.00435 5 MOSCA 0.862 0.00676 6 2.98 0.406 3 3.94 0.401 9 12.8 0.00443 6 MOSMA 0.11 0.72 11 2.97 0.254 2 4.37 0.22 1 12.4 0.147 9 MOWOA 0.611 0.27 10 2.83 1.21 10 3.36 1.92 11 12.1 0.388 11 MO_Ring_PSO_SCD 0.858 0.0102 7 2.95 0.304 5 3.95 0.258 8 12.8 0.0171 8 NSGA-II 0.0899 1.51 12 2.48 0.567 11 3.69 0.474 10 11.4 0.117 12 Algorithm MMF11_I MMF12_I MMF13_I MMF15_I Mean SD Rank Mean SD Rank Mean SD Rank Mean SD Rank EnMOGKSO 14.5 0.00116 1 1.57 0.0000935 1 18.4 0.00374 1 4.22 0.0715 3 MOGKSO 14.5 0.00047 2 1.57 0.0000681 2 18.4 0.00085 2 4.27 0.154 2 IMOMRFO 14.5 0.00768 6 1.57 0.00255 6 18.4 0.00935 6 3.89 0.193 9 MOEA/D 14.4 0.0665 10 1.45 0.143 9 18.2 0.0768 11 4.16 0.138 5 MOGTO 14.5 0.00453 4 1.57 0.00016 4 18.4 0.00717 5 4.19 0.135 4 MOGWO 14.4 0.00764 7 1.55 0.00216 8 18.2 0.0221 10 2.12 0.0261 12 MOPSO 14.5 0.00331 3 1.57 0.000139 3 18.4 0.00352 3 4.08 0.199 6 MOSCA 14.4 0.0153 8 1.57 0.00394 7 18.3 0.0147 8 3.94 0.247 8 MOSMA 14.4 0.0349 9 1.44 0.145 11 18.4 0.0249 7 4.32 0.331 1 MOWOA 13.2 0.636 12 1.1 0.325 12 16.7 1.49 12 3.51 1.65 11 MO_Ring_PSO_SCD 14.5 0.00625 5 1.57 0.00155 5 18.4 0.00472 4 3.84 0.313 10 NSGA-II 14.1 0.105 11 1.45 0.156 10 18.3 0.0937 9 3.99 0.152 7 Algorithm MMF15_a_I MMF16_I1 MMF16_I2 MMF16_I3 Mean SD Rank Mean SD Rank Mean SD Rank Mean SD Rank EnMOGKSO 4.25 0.06 4 4.24 0.0528 2 4.23 0.063 2 4.27 0.0532 3 MOGKSO 4.34 0.229 2 4.22 0.147 4 4.24 0.167 3 4.22 0.156 4 IMOMRFO 3.96 0.225 8 3.94 0.364 8 4.08 0.27 7 4.03 0.206 7 MOEA/D 4.18 0.227 5 4.22 0.2 5 4.21 0.119 4 4.17 0.234 5 MOGTO 4.36 0.493 3 4.23 0.149 3 4.21 0.126 5 4.28 0.119 2 MOGWO 2.44 0.129 12 2.12 0.0192 12 2.12 0.0152 12 2.18 0.0173 12 MOPSO 4.1 0.194 6 4.07 0.231 6 4.11 0.209 6 4.1 0.258 6 MOSCA 3.98 0.471 9 3.91 0.395 10 3.86 0.421 10 3.9 0.365 9 MOSMA 4.45 0.293 1 4.53 0.532 1 4.39 0.312 1 4.59 0.298 1 MOWOA 3.62 2.17 11 4.04 1.52 9 3.86 1.22 11 3.81 1.78 11 MO_Ring_PSO_SCD 4.01 0.288 7 3.88 0.253 11 3.93 0.151 9 3.93 0.349 8 NSGA-II 3.89 0.349 10 3.97 0.176 7 4.01 0.183 8 3.89 0.288 10 Open in a new tab Table 7. Comparative IGD results for EnMOGKSO and competing multi-objective algorithms. Algorithm MMF1 MMF2 MMF4 MMF5 Mean SD Rank Mean SD Rank Mean SD Rank Mean SD Rank EnMOGKSO 0.0444 0.00198 1 0.00941 0.00242 1 0.0285 0.0028 1 0.0779 0.00422 1 MOGKSO 0.08 0.00672 4 0.0395 0.0141 6 0.154 0.0482 8 0.251 0.0493 8 IMOMRFO 0.0834 0.00716 6 0.0165 0.00262 4 0.0611 0.0111 4 0.153 0.0217 4 MOEA/D 0.213 0.03 9 0.271 0.112 9 0.124 0.017 6 0.313 0.0404 9 MOGTO 0.0847 0.0222 5 0.124 0.0354 8 0.125 0.0484 7 0.161 0.0144 6 MOGWO 0.594 0.185 10 0.343 0.218 10 0.509 0.247 10 1.06 0.286 11 MOPSO 0.0636 0.00218 2 0.0113 0.000892 2 0.0423 0.00276 2 0.113 0.00625 2 MOSCA 0.0952 0.00843 7 0.0181 0.00942 5 0.0978 0.0214 5 0.158 0.0173 5 MOSMA 0.136 0.017 8 0.0699 0.0272 7 0.264 0.0608 9 0.213 0.0254 7 MOWOA 0.594 0.185 10 0.343 0.218 10 0.509 0.247 10 1.06 0.286 11 MO_Ring_PSO_SCD 0.0734 0.00716 3 0.0158 0.00239 3 0.0454 0.00313 3 0.122 0.00621 3 NSGA-II 0.852 0.00378 12 0.768 0.0604 12 0.523 0.00228 12 0.853 0.0063 10 Algorithm MMF7 MMF8 MMF10 MMF11 Mean SD Rank Mean SD Rank Mean SD Rank Mean SD Rank EnMOGKSO 0.0245 0.002 1 0.0727 0.0224 1 0.00171 0.000131 1 0.00368 0.000203 2 MOGKSO 0.0799 0.0179 6 0.665 0.166 7 0.00188 0.0000539 2 0.00325 0.000137 1 IMOMRFO 0.069 0.00855 5 0.254 0.123 5 0.0047 0.000252 4 0.00808 0.000437 6 MOEA/D 0.104 0.0223 8 1.13 0.514 8 0.149 0.146 11 0.0202 0.00732 9 MOGTO 0.0546 0.0142 4 1.87 1.05 12 0.00787 0.0145 7 0.0054 0.000763 3 MOGWO 0.364 0.111 10 1.44 0.686 10 0.0588 0.0518 9 0.0957 0.0666 10 MOPSO 0.0357 0.00266 2 0.0819 0.00609 2 0.00295 0.0000947 3 0.00646 0.000293 4 MOSCA 0.0801 0.00959 7 0.211 0.0825 4 0.00536 0.000645 5 0.0106 0.000938 7 MOSMA 0.142 0.0362 9 1.34 0.588 9 0.0345 0.0397 8 0.0146 0.00525 8 MOWOA 0.364 0.111 10 1.44 0.686 10 0.0588 0.0518 9 0.0957 0.0666 10 MO_Ring_PSO_SCD 0.0423 0.00308 3 0.0993 0.0121 3 0.00628 0.000804 6 0.0077 0.000785 5 NSGA-II 0.854 0.00285 12 0.415 0.000981 6 12.8 0.0151 12 14.4 0.0291 12 Algorithm MMF12 MMF13 MMF14 MMF15 Mean. SD . Rank Mean. SD . Rank Mean. SD . Rank Mean. SD . Rank EnMOGKSO 0.00156 0.000128 2 0.0436 0.00482 3 0.0701 0.00682 1 0.05 0.00397 1 MOGKSO 0.0015 0.00011 1 0.104 0.0332 7 0.108 0.0178 5 0.0576 0.00609 2 IMOMRFO 0.00333 0.00031 5 0.0447 0.00404 3 0.104 0.0092 4 0.0735 0.0049 5 MOEA/D 0.0166 0.00885 9 0.0843 0.0202 5 0.255 0.0534 8 0.0851 0.00634 8 MOGTO 0.00218 0.000428 3 0.155 0.0441 8 0.281 0.0116 9 0.0663 0.0078 3 MOGWO 0.0895 0.0712 10 0.212 0.0497 9 0.371 0.141 10 0.293 0.114 10 MOPSO 0.00282 0.000146 4 0.0385 0.00115 1 0.0949 0.00565 2 0.0703 0.00307 4 MOSCA 0.0035 0.000332 6 0.059 0.00966 4 0.127 0.00851 7 0.0983 0.0081 9 MOSMA 0.00878 0.00253 8 0.087 0.0153 6 0.116 0.0144 6 0.0825 0.0116 7 MOWOA 0.0895 0.0712 10 0.212 0.0497 9 0.371 0.141 10 0.293 0.114 10 MO_Ring_PSO_SCD 0.00352 0.00026 7 0.0413 0.00204 2 0.0994 0.0052 3 0.0744 0.00405 6 NSGA-II 1.52 0.0448 12 18.2 0.0435 11 2.68 0.142 12 3.77 0.294 12 Algorithm MMF1_e MMF14_a MMF15_a MMF10_I Mean SD Rank Mean SD Rank Mean SD Rank Mean SD Rank EnMOGKSO 0.195 0.0444 1 0.0809 0.00318 1 0.0556 0.00238 1 0.0201 0.0000643 1 MOGKSO 1.17 0.358 6 0.122 0.0226 5 0.0691 0.00836 2 0.0201 0.000026 2 IMOMRFO 3.19 0.465 9 0.115 0.00588 4 0.0817 0.00503 5 0.163 0.00206 4 MOEA/D 3.3 0.844 10 0.233 0.0559 7 0.107 0.00933 6 0.228 0.0225 11 MOGTO 3.05 0.682 8 0.289 0.0981 8 0.159 0.034 8 0.202 0.000401 9 MOGWO 3.57 1.27 11 0.391 0.157 9 0.308 0.17 9 0.198 0.0481 7 MOPSO 0.2 0.0424 2 0.102 0.0043 2 0.0792 0.005 3 0.186 0.0193 6 MOSCA 1.5 0.911 7 0.573 0.00555 11 0.334 0.00612 11 0.203 0.000272 10 MOSMA 1.09 0.447 5 0.231 0.0544 6 0.136 0.0125 7 0.0615 0.0281 3 MOWOA 3.57 1.27 11 0.391 0.157 9 0.308 0.17 9 0.198 0.0481 7 MO_Ring_PSO_SCD 0.299 0.0667 3 0.109 0.00665 3 0.0801 0.00526 4 0.164 0.00622 5 NSGA-II 0.884 0.027 4 2.98 0.268 12 4.13 0.401 12 12.8 0.0443 12 Algorithm MMF11_I MMF12_I MMF13_I MMF15_I Mean SD Rank Mean SD Rank Mean SD Rank Mean SD Rank EnMOGKSO 0.249 0.000484 6 0.245 0.000347 6 0.269 0.003 4 0.243 0.0239 4 MOGKSO 0.249 0.000312 6 0.245 0.000372 6 0.334 0.0212 8 0.243 0.0233 4 IMOMRFO 0.233 0.0223 2 0.198 0.0438 1 0.265 0.0119 3 0.205 0.0196 3 MOEA/D 0.259 0.00354 9 0.253 0.00765 9 0.31 0.0319 7 0.301 0.00424 9 MOGTO 0.253 0.000956 8 0.247 0.000445 8 0.376 0.032 9 0.286 0.00559 8 MOGWO 0.292 0.0552 10 0.299 0.113 10 0.394 0.0539 10 0.406 0.166 10 MOPSO 0.246 0.0134 4 0.244 0.000432 5 0.264 0.00746 2 0.174 0.0148 1 MOSCA 0.246 0.0135 5 0.243 0.00307 4 0.286 0.0122 5 0.243 0.024 4 MOSMA 0.244 0.0226 3 0.221 0.0342 3 0.291 0.0199 6 0.273 0.0189 7 MOWOA 0.292 0.0552 10 0.299 0.113 10 0.394 0.0539 10 0.406 0.166 10 MO_Ring_PSO_SCD 0.216 0.0272 1 0.207 0.0441 2 0.246 0.0135 1 0.178 0.0182 2 NSGA-II 14.5 0.00489 12 1.57 0.000742 12 18.4 0.00667 12 4.27 0.372 12 Algorithm MMF15_a_I MMF16_l1 MMF16_l2 MMF16_l3 Mean SD Rank Mean SD Rank Mean SD Rank Mean SD Rank EnMOGKSO 0.215 0.0082 4 0.164 0.00772 3 0.295 0.0246 6 0.212 0.00992 4 MOGKSO 0.225 0.00835 5 0.184 0.00727 5 0.294 0.0254 5 0.229 0.015 5 IMOMRFO 0.204 0.0133 3 0.164 0.018 3 0.245 0.0335 3 0.2 0.0182 3 MOEA/D 0.261 0.00501 6 0.256 0.0269 8 0.369 0.00506 9 0.309 0.0381 8 MOGTO 0.289 0.052 8 0.269 0.0429 9 0.361 0.00541 8 0.315 0.0586 9 MOGWO 0.365 0.215 9 0.356 0.139 10 0.43 0.0805 10 0.343 0.183 10 MOPSO 0.174 0.0105 1 0.147 0.00774 1 0.204 0.0164 1 0.175 0.00991 1 MOSCA 0.574 0.0033 11 0.188 0.0127 6 0.277 0.0273 4 0.232 0.0155 6 MOSMA 0.273 0.0166 7 0.203 0.0138 7 0.325 0.0325 7 0.252 0.0189 7 MOWOA 0.365 0.215 9 0.356 0.139 10 0.43 0.0805 10 0.343 0.183 10 MO_Ring_PSO_SCD 0.19 0.0116 2 0.156 0.00762 2 0.217 0.0173 2 0.183 0.00907 2 NSGA-II 4.2 0.41 12 3.99 0.228 12 4.08 0.206 12 4.15 0.147 12 Open in a new tab Table 8. Comparative PSP results for EnMOGKSO and competing multi-objective algorithms. Algorithm MMF1 MMF2 MMF4 MMF5 Mean SD Rank Mean SD Rank Mean SD Rank Mean SD Rank EnMOGKSO 22.5 0.978 1 113 27.5 1 35.3 3.3 1 12.8 0.675 1 MOGKSO 12.3 1.1 4 28 8.81 6 6.78 2.55 8 3.76 0.904 8 IMOMRFO 11.9 0.979 6 61.2 9.47 4 16.5 2.84 4 6.52 0.845 4 MOEA/D 4.39 0.666 9 3.26 2.35 11 7.94 1.18 7 2.89 0.482 9 MOGTO 12.2 2.39 5 8.67 2.35 9 9.17 3.67 6 6.19 0.579 6 MOGWO 2.81 0.868 11 8.86 9 8 1.03 0.125 12 0.78 0.167 11 MOPSO 15.6 0.55 2 87.8 7.17 2 23.5 1.44 2 8.81 0.479 2 MOSCA 10.5 0.915 7 48 20.1 5 10.4 1.99 5 6.33 0.701 5 MOSMA 7.32 1.01 8 16.8 7.21 7 4.13 2.29 9 4.63 0.567 7 MOWOA 1.25 0.672 12 3 1.42 12 2.31 2.63 11 0.477 0.295 12 MO_Ring_PSO_SCD 13.5 0.62 3 63 9.09 3 21.7 1.45 3 8.13 0.414 3 NSGA-II 3.78 0.598 10 3.59 1.59 10 3.81 0.964 10 2.46 0.367 10 Algorithm MMF7 MMF8 MMF10 MMF11 Mean SD Rank Mean SD Rank Mean SD Rank Mean SD Rank EnMOGKSO 40.6 3.13 1 14.4 3.34 1 589 46.8 1 308 12.9 1 MOGKSO 12.2 2.74 5 1.16 0.309 6 533 15.4 2 273 14.9 2 IMOMRFO 12 2.51 6 4.23 1.56 5 212 11.4 4 123 6.59 6 MOEA/D 7.73 2.16 8 0.877 0.492 7 11.3 8.31 11 52.7 20.4 10 MOGTO 19 4.86 4 0.579 0.572 11 212 57.5 4 189 28.1 3 MOGWO 0.21 0.12 12 0.106 0.0853 12 85.3 21 8 48.7 25.5 11 MOPSO 27.9 2.11 2 12.1 0.885 2 339 10.6 3 155 6.95 4 MOSCA 11.3 2.24 7 5.18 1.71 4 189 20.5 6 95.1 8.01 7 MOSMA 6.65 2.5 9 0.856 0.606 8 65.8 48.8 9 76.1 23.2 9 MOWOA 0.633 0.43 11 0.606 0.353 10 23.1 9.29 10 13.4 8.13 12 MO_Ring_PSO_SCD 23.3 1.66 3 10.1 1.17 3 160 19 7 131 12.9 5 NSGA-II 5.19 0.795 10 0.659 0.169 9 4.7 2.5 12 85.1 15.9 8 Algorithm MMF12 MMF13 MMF14 MMF15 Mean SD Rank Mean SD Rank Mean SD Rank Mean SD Rank EnMOGKSO 644 49 2 23 2.43 3 14.4 1.2 1 20.1 1.43 1 MOGKSO 669 46.4 1 9.23 4.2 8 9.52 1.46 5 17.5 1.65 2 IMOMRFO 303 26.7 5 22 1.94 4 9.63 0.829 4 13.6 0.917 5 MOEA/D 73 32 10 11.4 2.65 7 2.8 2.14 9 11.6 1 8 MOGTO 475 84.1 3 1.99 1.65 11 1.59 1.02 11 15.3 1.73 3 MOGWO 197 82.1 8 1.01 0.512 12 0.137 0.0873 12 0.733 0.182 12 MOPSO 355 17.9 4 25.8 0.738 1 10.6 0.603 2 14.2 0.634 4 MOSCA 288 28.5 6 16.8 2.82 5 7.92 0.519 7 10.1 0.856 9 MOSMA 121 26.3 9 11.8 2 6 8.71 1.04 6 12.3 1.68 7 MOWOA 17.3 12 12 3.01 1.93 10 2.67 1.8 10 3.5 2.62 11 MO_Ring_PSO_SCD 286 20.9 7 24.1 1.17 2 10.1 0.513 3 13.5 0.71 6 NSGA-II 63 39 11 6.09 1.57 9 5.46 0.541 8 6.67 1.19 10 Algorithm MMF1_e MMF14_a MMF15_a MMF10_I Mean SD Rank Mean SD Rank Mean SD Rank Mean SD Rank EnMOGKSO 5.3 1.13 1 12.4 0.48 1 18 0.76 1 0.133 0.02 11 MOGKSO 0.599 0.214 6 8.44 1.57 5 14.6 1.51 2 0.156 0.00698 10 IMOMRFO 0.107 0.0738 10 8.6 0.44 4 12.2 0.731 5 6.06 0.262 2 MOEA/D 0.0874 0.129 11 4.09 1.06 8 9.32 0.841 6 0.359 0.496 7 MOGTO 0.147 0.182 9 3.86 1.37 9 6.55 1.35 8 0.0925 0.0404 12 MOGWO 0.0858 0.106 12 0.691 0.509 11 1.32 0.806 11 0.192 0.0295 9 MOPSO 5.04 0.698 2 9.73 0.396 2 12.6 0.76 3 2.6 2.88 6 MOSCA 0.752 0.654 5 0.0328 0.147 12 0 0 12 0.23 0.0174 8 MOSMA 0.958 0.394 4 4.56 1.17 6 7.44 0.731 7 18.7 6.7 1 MOWOA 0.173 0.332 8 2.19 1.61 10 1.86 1.95 10 3.34 3.23 5 MO_Ring_PSO_SCD MO_Ring_PSO_SCD 3.31 0.761 3 9.12 0.52 3 12.5 0.8 4 6.04 0.195 3 NSGA-II 0.47 0.128 7 4.16 0.219 7 5.1 0.881 9 5.27 2.25 4 Algorithm MMF11_I MMF12_I MMF13_I MMF15_I Mean SD Rank Mean SD Rank Mean SD Rank Mean SD Rank EnMOGKSO 0.552 0.0483 8 0.465 0.0358 9 1.74 0.0319 5 3.35 1.32 7 MOGKSO 0.526 0.0385 9 0.438 0.0525 10 1.13 0.294 9 3.66 1.06 6 IMOMRFO 2.09 1.65 3 3.61 2.67 1 2.12 0.543 2 4.88 0.476 3 MOEA/D 0.213 0.0693 12 0.717 0.289 7 1.33 0.315 8 0.61 0.0968 11 MOGTO 0.259 0.197 11 0.283 0.0803 12 0.426 0.409 11 1.17 0.362 10 MOGWO 0.327 0.154 10 0.413 0.142 11 0.206 0.0787 12 0.128 0.0705 12 MOPSO 1.01 1.01 7 0.614 0.047 8 1.86 0.278 3 5.77 0.518 1 MOSCA 1.11 0.977 5 0.748 0.273 6 1.63 0.216 7 3.88 0.749 5 MOSMA 1.19 1.15 4 2.32 2.28 3 1.69 0.12 6 2.4 0.976 8 MOWOA 1.1 1.17 6 1.83 2.88 4 0.79 0.363 10 1.55 1.53 9 MO_Ring_PSO_SCD 3.32 1.99 1 2.96 2.67 2 2.68 0.659 1 5.67 0.554 2 NSGA-II 2.15 1.51 2 0.847 0.275 5 1.76 0.242 4 3.94 0.458 4 Algorithm MMF15_a_I MMF16_l1 MMF16_l2 MMF16_l3 Mean SD Rank Mean SD Rank Mean SD Rank Mean SD Rank EnMOGKSO 4 0.512 4 5.41 0.949 4 2.88 0.63 7 3.9 0.54 6 MOGKSO 3.71 0.359 6 5.15 0.64 6 3.09 0.61 6 3.83 0.727 7 IMOMRFO 4.69 0.467 3 6.13 0.717 3 4.01 0.713 3 4.86 0.613 3 MOEA/D 3.01 0.0614 8 1.48 1.01 10 0.611 0.0582 11 1.52 0.835 10 MOGTO 3.11 0.741 7 1.41 0.89 11 0.867 0.23 10 1.03 0.34 11 MOGWO 0.85 0.602 11 0.103 0.0343 12 0.115 0.0536 12 0.0842 0.0562 12 MOPSO 5.72 0.352 1 6.8 0.355 1 4.93 0.417 1 5.73 0.313 1 MOSCA 0 0 12 5.28 0.382 5 3.33 0.593 5 4 0.471 4 MOSMA 3 0.255 9 4 0.738 8 1.94 0.804 8 3.09 0.635 8 MOWOA 1.34 1.12 10 2.12 1.58 9 1.62 0.828 9 2.31 1.72 9 MO_Ring_PSO_SCD 5.14 0.415 2 6.42 0.306 2 4.63 0.351 2 5.44 0.312 2 NSGA-II 3.79 0.594 5 4.6 0.293 7 3.77 0.499 4 3.97 0.278 5 Open in a new tab The boxplots in Figs. 1 , 2 , 3 , 4 , 5 , 6 are consistent with the tabulated results: EnMOGKSO maintains high HV, low IGD, and competitive PSP, with tighter spread on most functions, indicating stronger run-to-run stability. Fig. 1. Open in a new tab HV boxplots for MMF1 to MMF15 functions. Fig. 2. Open in a new tab HV boxplots for MMF1_e to MMF16_l3 functions. Fig. 3. Open in a new tab PSP boxplots for MMF1 to MMF15 functions across algorithms. Fig. 4. Open in a new tab PSP boxplots for MMF1_e to MMF16_l3 functions across algorithms. Fig. 5. Open in a new tab IGD boxplots for MMF1 to MMF15 functions across algorithms. Fig. 6. Open in a new tab IGD boxplots for MMF1_e to MMF16_l3 functions across algorithms. Representative PF/PS visualizations (Figs. 7 , 8 , 9 , 10 , 11 and 12 ) show more uniform front tracking for EnMOGKSO, whereas competitors are less consistent on disconnected or multimodal cases. Fig. 7. Open in a new tab PF comparison against reference points on across algorithms. Fig. 8. Open in a new tab PF comparison against reference points on MMF 12 across algorithms. Fig. 9. Open in a new tab PF comparison against reference points on across algorithms. Fig. 10. Open in a new tab PS comparison against reference points on MMF1 across algorithms. Fig. 11. Open in a new tab PF comparison against reference points on MMF14 across algorithms. Fig. 12. Open in a new tab PS comparison against reference points on across algorithms. Pairwise tests align with the metric tables: EnMOGKSO is significant against most competitors, with only a few non-significant comparisons. Statistical Significance Analysis Table 9 ranks EnMOGKSO first in HV (2.04) and IGD (2.38), and third in PSP (3.29). Table 9. Scores and Friedman mean ranks for compared algorithms based on HV, IGD, and PSP. Algorithms EnMOGKSO MOGKSO IMOMRFO MOEA/D MOGTO MOGWO MOPSO MOSCA MOSMA MOWOA MO_Ring_PSO_SCD NSGA-II Scores of HV 2.04 2.46 6.92 7.96 3.83 9.92 4.54 7.71 5.25 10.96 6.63 9.79 Ranking 1 2 7 9 3 11 4 8 5 12 6 10 Scores of IGD 2.38 4.83 4.08 8.25 7.29 9.75 2.46 6.46 6.67 9.75 3.17 11.29 Ranking 1 5 4 9 8 10 2 6 7 11 3 12 Scores of PSP 3.29 5.79 4.13 8.96 8.21 11 2.75 6.63 6.92 9.67 3.13 7.5 Ranking 3 5 4 10 9 12 1 6 7 11 2 8 Open in a new tab Table 10 shows that 27 of 33 pairwise comparisons are significant across HV, IGD, and PSP. The reported p -values and Friedman ranks jointly indicate both pairwise and overall differences across algorithms. Table 10. Wilcoxon signed-rank test results for HV, IGD, and PSP when comparing EnMOGKSO with competing algorithms ( indicates significance). Algorithm HV IGD PSP EnMOGKSO vs. MOPSO EnMOGKSO vs. MOGWO EnMOGKSO vs. MOWOA EnMOGKSO vs. MOSCA EnMOGKSO vs. MOSMA EnMOGKSO vs. MO_Ring_PSO_SCD EnMOGKSO vs. MOEA/D EnMOGKSO vs. IMOMRFO EnMOGKSO vs. MOGTO EnMOGKSO vs. NSGA-II EnMOGKSO vs. MOGKSO Open in a new tab Optimal WSN deployment results Evaluation criteria Spacing metric (SP) SP measures how uniformly solutions are distributed on the Pareto front 67 ; lower values indicate better diversity. Mathematical definition 44 where is the distance between consecutive solutions, is the mean distance, and N is the number of solutions on the front. Table 11 shows the best SP behavior for EnMOGKSO across most ( c , k ) settings, with MOPSO usually second. Table 11. SP evaluation (mean, best, worst, and SD) for different ( C , K ) settings. C,k Metric EnMOGKSO MOGKSO MOGTO MOPSO MOSMA C=1, k=1 Mean 3.89E+01 1.20E+02 6.27E+01 5.69E+01 7.68E+01 Best 2.01E+01 1.02E+02 4.34E+01 3.99E+01 6.18E+01 Worst 7.41E+01 1.56E+02 9.08E+01 7.04E+01 9.48E+01 SD 1.84E+01 2.05E+01 1.75E+01 1.23E+01 1.42E+01 C=1, k=2 Mean 4.17E+01 1.22E+02 5.62E+01 4.29E+01 8.55E+01 Best 1.70E+01 9.52E+01 2.72E+01 2.98E+01 7.17E+01 Worst 9.89E+01 1.63E+02 9.40E+01 5.97E+01 1.00E+02 SD 2.93E+01 2.32E+01 2.37E+01 1.32E+01 1.10E+01 C=1, k=3 Mean 4.44E+01 1.14E+02 5.94E+01 5.78E+01 9.13E+01 Best 1.34E+01 8.67E+01 4.81E+01 4.01E+01 7.55E+01 Worst 1.27E+02 1.45E+02 7.13E+01 7.97E+01 1.40E+02 SD 4.45E+01 2.60E+01 9.75E+00 1.67E+01 2.47E+01 C=2, k=1 Mean 4.84E+01 1.06E+02 5.21E+01 4.70E+01 9.62E+01 Best 2.84E+01 8.00E+01 4.23E+01 2.26E+01 6.25E+01 Worst 9.75E+01 1.37E+02 6.13E+01 6.13E+01 1.42E+02 SD 2.61E+01 2.06E+01 6.84E+00 1.36E+01 2.63E+01 C=2, k=2 Mean 2.83E+01 1.31E+02 6.29E+01 5.09E+01 8.24E+01 Best 1.57E+01 1.01E+02 4.94E+01 3.41E+01 6.49E+01 Worst 4.91E+01 1.76E+02 9.64E+01 9.01E+01 1.11E+02 SD 1.30E+01 2.60E+01 1.88E+01 2.03E+01 1.79E+01 C=2, k=3 Mean 3.34E+01 1.11E+02 5.96E+01 5.00E+01 7.45E+01 Best 1.31E+01 7.72E+01 3.40E+01 3.17E+01 6.35E+01 Worst 4.44E+01 1.38E+02 9.87E+01 7.25E+01 9.21E+01 SD 1.11E+01 2.39E+01 2.15E+01 1.91E+01 9.92E+00 C=3, k=1 Mean 4.08E+01 1.04E+02 5.07E+01 4.71E+01 8.19E+01 Best 9.61E+00 8.21E+01 3.90E+01 2.85E+01 6.38E+01 Worst 8.33E+01 1.31E+02 6.68E+01 6.42E+01 1.16E+02 SD 2.71E+01 1.90E+01 1.05E+01 1.51E+01 1.99E+01 C=3, k=2 Mean 3.61E+01 1.15E+02 1.22E+02 4.60E+01 6.78E+01 Best 1.50E+01 9.63E+01 6.13E+01 3.18E+01 5.23E+01 Worst 6.56E+01 1.41E+02 2.12E+02 6.44E+01 8.70E+01 SD 2.01E+01 1.74E+01 7.11E+01 1.36E+01 1.13E+01 C=3, k=3 Mean 4.86E+01 1.12E+02 5.55E+01 4.38E+01 8.48E+01 Best 2.74E+01 7.62E+01 4.04E+01 2.90E+01 7.17E+01 Worst 7.20E+01 1.53E+02 6.91E+01 5.48E+01 1.07E+02 SD 1.64E+01 3.00E+01 1.11E+01 9.75E+00 1.32E+01 Open in a new tab Performance evaluation of the EnMOGKSO-based deployment method Figure 13 shows the initial 3D sensor-target layout. Fig. 13. Open in a new tab Initial 3D spatial distribution of sensors and targets in the WSN deployment space. The deployment study jointly optimizes k -coverage, C -connectivity, and cost in the same 3D environment, comparing MOGKSO, EnMOGKSO, MOPSO, MOGTO, and MOSMA. Table 12 reports higher and more stable coverage and connectivity for EnMOGKSO (often with MOPSO), at higher cost (typically around 19k). Table 12. Coverage, connectivity, and cost metrics for all algorithms across all ( c , k ) settings. C, K Objectives Metric MOGKSO EnMOGKSO MOPSO MOGTO MOSMA C=1, K=1 Coverage degree Mean 6 11 11 7 5 SD 4.8 1.56 1.63 4.21 4.24 Connectivity degree Mean 4 7 7 6 4 SD 2.38 0.07 0.08 0.69 2.21 Cost Mean 10546 19396 19126 13422 10320 SD 7898.21 1808.46 1721.22 5216.76 7474.44 C=1, K=2 Coverage degree Mean 6 12 11 8 6 SD 4.65 1.01 1.54 3.47 4.12 Connectivity degree Mean 5 7 7 7 4 SD 2.31 0.02 0.1 0.15 1.96 Cost Mean 12111 19670 19501 15194 10522 SD 7035.49 1305.72 1538.22 3864.03 7078.38 C=1, K=3 Coverage degree Mean 6 11 11 8 6 SD 4.7 1.46 1.87 3.46 3.83 Connectivity degree Mean 5 7 7 7 4 SD 2.07 0.05 0.1 0.14 2.06 Cost Mean 10911 19391 19098 15148 11298 SD 7220.41 1411.65 2018.79 4129.97 6656.06 C=2, K=1 Coverage degree Mean 6 12 11 7 6 SD 4.5 0.91 1.84 4.02 3.91 Connectivity degree Mean 5 7 7 7 4 SD 2.16 0.13 0.1 0.16 1.95 Cost Mean 10956 19884 19452 14281 11298 SD 7542.51 1146.12 1841.41 4863.33 6630.05 C=2, K=2 Coverage degree Mean 6 11 12 8 6 SD 4.78 0.87 1.6 4.04 4.21 Connectivity degree Mean 5 7 7 7 4 SD 2.31 0.13 0.1 0.13 2.01 Cost Mean 11884 19395 19475 15919 11047 SD 8113.22 1159.47 1974.11 4855.43 7183.61 C=2, K=3 Coverage degree Mean 6 12 11 7 6 SD 4.38 0.82 1.67 4 4.2 Connectivity degree Mean 5 7 7 6 4 SD 2.02 0.02 0.07 0.26 2.06 Cost Mean 11925 19702 19744 14346 10511 SD 6565.55 981.11 1790.66 4569.14 7222.49 C=3, K=1 Coverage degree Mean 6 11 11 8 5 SD 4.45 1.17 1.9 3.37 3.83 Connectivity degree Mean 5 7 7 7 4 SD 2.2 0.05 0.11 0.15 1.93 Cost Mean 12047 19221 19459 15138 9506 SD 7210.54 1380.37 1922.07 4086.29 7123.75 C=3, K=2 Coverage degree Mean 7 12 11 7 6 SD 4.34 1.01 1.54 3.85 3.812 Connectivity degree Mean 5 7 7 7 4 SD 2.31 0.13 0.1 0.13 2.01 Cost Mean 12899 19304 18978 14251 10608 SD 7234.85 832.81 1664.73 4436.48 6597.84 C=3, K=3 Coverage degree Mean 6 12 11 7 5 SD 4.74 1.22 1.6 3.46 4.08 Connectivity degree Mean 5 7 7 7 4 SD 2.19 0.09 0.08 0.1 2.18 Cost Mean 11077 19319 19037 14879 10438 SD 7401.22 1043.11 1822.56 4275.66 7076.94 Open in a new tab Figures 14 and 15 show that increasing c / k tightens feasibility and shifts trade-offs toward higher cost. Fig. 14. Open in a new tab EnMOGKSO objectives for different values of c and k . Fig. 15. Open in a new tab Deployment outcomes for different combinations of c and k . Figure 16 shows the same trade-off: EnMOGKSO and MOPSO provide higher quality at higher cost, while MOSMA has the lowest cost and the weakest quality. Fig. 16. Open in a new tab Bar-plot view of objective values for different combinations of c and k . Connectivity plots (Figs. 17 , 18 , 19 , 20 , 21 , 22 ) place EnMOGKSO and MOPSO at the top, with a tighter spread for EnMOGKSO. Fig. 17. Open in a new tab k=1. Fig. 18. Open in a new tab k=2. Fig. 19. Open in a new tab k=3. Fig. 20. Open in a new tab k=1. Fig. 21. Open in a new tab k=2. Fig. 22. Open in a new tab k=3. Figures 23 , 24 , 25 , 26 , 27 , 28 show dominant coverage for EnMOGKSO and MOPSO, again with tighter dispersion for EnMOGKSO. Fig. 23. Open in a new tab c=1. Fig. 24. Open in a new tab c=2. Fig. 25. Open in a new tab c=3. Fig. 26. Open in a new tab c=1. Fig. 27. Open in a new tab c=2. Fig. 28. Open in a new tab C=2. Figures 29 , 30 , 31 , 32 , 33 , 34 confirm the expected quality-cost trade-off. Fig. 29. Open in a new tab k=1. Fig. 30. Open in a new tab k=2. Fig. 31. Open in a new tab k=3. Fig. 32. Open in a new tab k=1. Fig. 33. Open in a new tab k=2. Fig. 34. Open in a new tab k=3. Table 13 corroborates Table 11 : EnMOGKSO yields the smallest SP values in most ( c , k ) settings. Table 13. SP metrics for all algorithms across all ( c , k ) settings. c, k Metric MOGKSO EnMOGKSO MOPSO MOGTO MOSMA C=1, K=1 Mean 4.59E+02 0.00E+00 2.22E+02 4.01E+02 4.57E+02 SD 1.08E+02 0.00E+00 1.17E+02 7.32E+01 1.34E+02 Best 2.97E+02 0.00E+00 1.14E+02 2.93E+02 3.21E+02 Worst 5.80E+02 0.00E+00 3.90E+02 5.15E+02 6.63E+02 C=2, K=1 Mean 5.97E+02 1.18E+01 1.57E+02 2.95E+02 4.94E+02 SD 1.22E+02 2.88E+01 5.72E+01 5.62E+01 1.06E+02 Best 4.24E+02 0.00E+00 7.46E+01 2.25E+02 3.33E+02 Worst 7.30E+02 7.05E+01 2.41E+02 3.84E+02 6.59E+02 C=3, K=1 Mean 5.45E+02 1.93E+01 2.47E+02 3.56E+02 4.29E+02 SD 1.37E+02 2.97E+01 1.37E+02 5.35E+01 1.46E+02 Best 3.08E+02 0.00E+00 1.41E+02 2.95E+02 2.08E+02 Worst 6.68E+02 7.57E+01 4.33E+02 4.46E+02 6.47E+02 C=1, K=2 Mean 6.38E+02 0.00E+00 1.75E+02 3.16E+02 5.59E+02 SD 1.21E+02 0.00E+00 6.07E+01 6.24E+01 1.01E+02 Best 5.00E+02 0.00E+00 1.01E+02 2.59E+02 4.92E+02 Worst 8.54E+02 0.00E+00 2.74E+02 4.32E+02 7.53E+02 C=2, K=2 Mean 6.40E+02 4.02E+00 1.93E+02 3.57E+02 4.68E+02 SD 1.66E+02 9.84E+00 5.51E+01 8.38E+01 8.79E+01 Best 3.69E+02 0.00E+00 1.21E+02 2.50E+02 3.48E+02 Worst 8.13E+02 2.41E+01 2.82E+02 4.71E+02 5.94E+02 C=3, K=2 Mean 5.53E+02 3.94E+00 1.97E+02 3.40E+02 5.36E+02 SD 8.12E+01 9.65E+00 4.27E+01 9.72E+01 3.58E+02 Best 4.43E+02 0.00E+00 1.58E+02 2.02E+02 6.70E+01 Worst 6.71E+02 2.36E+01 2.74E+02 4.94E+02 4.78E+02 C=1, K=3 Mean 5.01E+02 5.82E+00 2.35E+02 3.69E+02 4.98E+02 SD 7.50E+01 1.43E+01 5.60E+01 1.12E+02 6.12E+01 Best 4.17E+02 0.00E+00 1.59E+02 2.29E+02 4.11E+02 Worst 6.14E+02 3.49E+01 3.04E+02 4.89E+02 5.94E+02 C=2, K=3 Mean 6.54E+02 1.74E+01 1.96E+02 3.39E+02 3.87E+02 SD 1.04E+02 2.98E+01 4.90E+01 7.29E+01 9.29E+01 Best 5.02E+02 0.00E+00 1.04E+02 2.19E+02 2.20E+02 Worst 7.47E+02 7.25E+01 2.35E+02 4.01E+02 4.80E+02 C=3, K=3 Mean 6.48E+02 1.37E+01 1.81E+02 3.67E+02 4.82E+02 SD 1.83E+02 3.36E+01 4.47E+01 1.25E+02 8.72E+01 Best 3.70E+02 0.00E+00 1.40E+02 1.92E+02 4.16E+02 Worst 9.02E+02 8.22E+01 2.66E+02 5.76E+02 6.42E+02 Open in a new tab Table 14 shows moderate runtime overhead for EnMOGKSO relative to lower-cost baselines. Table 14. Algorithm runtimes for different ( k , c ) settings (in seconds). K C MOGKSO EnMOGKSO MOPSO MOGTO MOSMA 1 1 10.56 11.89 109.8 9.43 9.44 1 2 10.30 11.04 116.1 9.31 9.37 1 3 10.47 11.44 113.9 9.54 9.36 2 1 10.15 11.48 125.6 9.73 9.52 2 2 10.99 11.12 113.0 9.75 9.42 2 3 10.72 11.04 115.0 9.53 9.26 3 1 10.42 11.08 116.5 9.65 9.47 3 2 10.76 11.08 114.8 9.39 9.28 3 3 10.60 11.26 108.8 9.28 9.26 Open in a new tab Conclusion and future work EnMOGKSO shows competitive and stable performance on CEC 2020 and the heterogeneous 3D WSN deployment task, with improved convergence-diversity balance under constrained settings. Despite the improvements introduced in EnMOGKSO, several limitations remain that provide directions for future research. First, although mechanism-level analyses and partial operator studies are presented, a comprehensive component-wise ablation of all enhancement mechanisms under strictly controlled computational budgets remains limited and should be further investigated. Second, the current study focuses on a three-objective deployment formulation; extending the framework to many-objective optimization settings may require additional diversity preservation strategies and archive management mechanisms. Third, the experiments are conducted under static deployment assumptions. In practical Internet-of-Things (IoT) and WSN environments, dynamic conditions such as node failures, environmental changes, or time-varying coverage demands may require adaptive or online optimization strategies. Future work will therefore focus on developing stronger adaptive parameter control mechanisms, extending the algorithm to many-objective and dynamic deployment scenarios, and conducting more extensive ablation analyses under matched computational budgets. Additionally, large-scale real-world deployment studies will be explored to evaluate the robustness, feasibility, and scalability of the proposed framework when applied to practical sensing infrastructures where coverage reliability, communication stability, and cost efficiency must be maintained simultaneously. Author contributions Essam H. Houssein: Supervision, Methodology, Software, Formal analysis, Writing - review & editing. Ibrahim E. Ibrahim: Software, Investigation, Conceptualization, Resources, Data curation, Writing - review & editing. Yaser M. Wazery: Supervising. Marwa M. Emam: Methodology, Software, Visualization, Writing - review & editing.All authors read and approved the final article. Funding Open access funding provided by The Science, Technology & Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB). Data availability The datasets used and/or analysed during the current study available from the corresponding author on reasonable request. Declarations Competing interests The authors declare no competing interests. Footnotes Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. References 1. Boufares, N., Ben Saied, Y. & Azouz Saidane, L. A lightweight three dimensional redeployment algorithm for distributed mobile wireless sensor networks. Wirel. Pers. Commun. 135 (2), 835–873 (2024). [ Google Scholar ] 2. Kaur, G., Jyoti, K., Shorman, S., Alsoud, A. R. & Salgotra, R. An efficient approach for localizing sensor nodes in 2d wireless sensor networks using whale optimization-based naked mole rat algorithm. Mathematics (2227-7390) 12 (15). 3. De, S. 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