A Tuning-Free Constrained Team-Oriented Swarm Optimizer (CTOSO) for Engineering Problems

Abstract

Constrained optimization problems (COPs) are frequent in engineering design yet remain challenging due to complex search spaces and strict feasibility requirements. Existing swarm-based optimizers often rely on penalty functions or algorithm-specific control parameters, whose performance is sensitive to problem-dependent tuning and may lead to premature convergence or infeasible solutions when feasible regions are narrow. This paper introduces the Constrained Team-Oriented Swarm Optimizer (CTOSO), a tuning-free metaheuristic that adapts the ETOSO framework by replacing linear exploiter movement with spiral search and integrating Deb’s feasibility rule. The population divides into Explorers, promoting diversity through neighbor-guided navigation, and Exploiters, performing intensified local search around the global best solution. Extensive evaluation on twelve constrained engineering benchmark problems shows that CTOSO achieves a 100% feasibility rate and attains the highest overall composite performance score among the compared algorithms under limited function-evaluation budgets. On the CEC 2017 constrained benchmark suite, CTOSO attains an average feasibility rate of 79.78%, generating feasible solutions on 14 out of 15 problems. Statistical analysis using Wilcoxon signed-rank tests and Friedman ranking with Nemenyi post hoc comparison indicates that CTOSO performs significantly better than several baseline optimizers, while exhibiting no statistically significant differences with leading evolutionary methods under the same experimental conditions. The algorithm’s design, requiring no tuning of algorithm-specific control parameters, makes it suitable for real-world engineering applications where tuning effort must be minimized.

1. Introduction

Constrained Optimization Problems (COPs) pose a fundamental challenge in engineering design, where optimal performance must be achieved without violating strict physical and functional limitations. These problems can be formally defined as the search for a vector of decision variables $\mathit{x}$ in a $D$-dimensional bounded decision space, which minimizes an objective function $f\left(\mathbf{x}\right)$ subject to a set of inequality and equality constraints.

In practice, the primary challenge of constrained optimization problems arises from the inherent tradeoff between optimality and feasibility. The feasible region is often disjoint, non-convex, and occupies only a small fraction of the decision space, making it difficult for traditional optimization methods to navigate effectively. Metaheuristic algorithms have been widely applied to this domain due to their flexibility and gradient-free nature. However, their effectiveness depends on two critical components: the constraint-handling technique (CHT) and the search algorithm’s ability to balance exploration and exploitation.

Among CHTs, penalty function methods are widely used but suffer from a critical flaw. Their performance is highly sensitive to the chosen penalty coefficients, which require problem-specific tuning. As a parameter-free alternative, Deb’s feasibility rule has emerged as a popular choice, prioritizing feasibility over objective value. While this rule is effective, it can lead to premature convergence if the search algorithm itself lacks a robust mechanism for maintaining population diversity and navigating complex landscapes.

A Team-Oriented Swarm Optimizer (TOSO) [1] proposed a novel population structure based on a fixed division of labor. This was followed by the Enhanced Team-Oriented Swarm Optimizer (ETOSO) [2], which refined the internal mechanics of each team by introducing a linear weight-based exploitation strategy for exploiters and a neighbor-guided strategy for explorers, significantly improving performance on unconstrained benchmark mathematical problems.

This paper presents the Constrained Team-Oriented Swarm Optimizer (CTOSO), a tuning-free constrained optimization algorithm obtained through a deliberate re-engineering of the Enhanced Team-Oriented Swarm Optimizer (ETOSO). Unlike ETOSO, which relies on linear exploiter movement and auxiliary recovery mechanisms, CTOSO introduces fundamental algorithmic modifications that improve robustness and usability in constrained search spaces. Within the existing literature, the team-oriented swarm paradigm has been represented by the original TOSO algorithm and its enhanced variant ETOSO. The present work introduces CTOSO as a further development within this framework.

The main contributions of this work are summarized as follows:

• Structurally simplified constrained optimization framework:
  CTOSO eliminates recovery-based operators and algorithm-specific control parameters, reducing algorithmic complexity and minimizing tuning effort in constrained optimization problems.
• Feasibility-driven spiral exploitation mechanism:
  The linear exploiter movement of ETOSO is replaced with a single-center, adaptive spiral contraction around the global best solution, enabling controlled intensification without introducing additional control parameters.
• Consistent feasibility integration via Deb’s rule:
  Deb’s feasibility rule is applied uniformly across all solution comparisons, selections, and updates, ensuring stable convergence and reliable feasibility preservation when feasible regions are narrow or highly constrained.
• Design tailored for low evaluation budgets:
  CTOSO is explicitly designed and evaluated under fixed and limited function-evaluation budgets, making it suitable for black-box engineering problems where evaluation cost is high and rapid convergence is required.

The remainder of this paper is organized as follows: Section 2 provides a review of related work in constrained optimization and metaheuristics. Section 3 details the formal mathematical model of the Constrained Team-Oriented Swarm Optimizer (CTOSO). Section 4 presents the initial validation of the proposed algorithm on the CEC 2017 constrained benchmark suite. Section 5 provides a focused comparative evaluation against the predecessor variants, TOSO and ETOSO, to isolate the impact of the structural modifications. Section 6 details the twelve constrained engineering benchmark problems, the comparative algorithms, and the experimental setup. Section 7 presents the comprehensive analysis of the empirical results. Section 8 discusses the findings in depth. Finally, Section 9 concludes the paper and suggests directions for future work.

2. Related Work

Constrained optimization problems (COPs) are fundamental across numerous engineering domains, where design variables must satisfy nonlinear inequalities, equalities, and bound constraints. Their feasible regions often form nonconvex, disjoint, or extremely narrow manifolds, making classical gradient-based methods inadequate for black-box or simulation-driven engineering models [3]. Recent reviews emphasize that evolutionary and swarm-based metaheuristics have become dominant for COPs due to their population diversity, robustness to noise, and ability to explore irregular landscapes [4].

Within this broad class of population-based optimizers, evolutionary and swarm intelligence algorithms have been widely adopted in structural, civil, mechanical, and energy engineering. Methods such as Genetic Algorithms (GA) [5], Differential Evolution (DE) [6], Particle Swarm Optimization (PSO) [7], and a wide range of nature-inspired approaches, including Grey Wolf Optimizer (GWO) [8], Moth Flame Optimizer (MFO) [9], Whale Optimization Algorithm (WOA) [10], and Teaching-Learning-Based Optimization (TLBO) [11], have been successfully applied to practical design problems involving discrete constraints, nonlinear material laws, and simulation-based objective functions [12,13,14]. It has been reported that these population-based optimizers have largely replaced classical mathematical programming techniques in real-world structural optimization, particularly in cases involving discontinuities or high-dimensional constraints. In parallel, significant progress has been documented in hybrid, machine-learning-assisted, and physics-informed metaheuristics for engineering optimization under constraints.

Despite these advances, constraint handling remains a central challenge across both evolutionary and swarm-based methods. Penalty functions continue to be widely used due to their conceptual simplicity; however, their performance is highly sensitive to penalty coefficients, which must be carefully tuned to balance feasibility and objective improvement [15]. To reduce this sensitivity, adaptive, dynamic, and co-evolutionary penalty mechanisms have been proposed, adjusting penalty parameters based on population statistics or online learning strategies [16,17]. Although such approaches alleviate manual tuning to some extent, several studies report that they remain problem-dependent and exhibit limited robustness on highly nonlinear COPs [18].

To overcome the limitations of penalty-based methods, feasibility-driven constraint-handling techniques have gained considerable attention. Among these, Deb’s feasibility rule [19] remains the most influential parameter-free approach and is extensively used in modern constrained evolutionary optimization. By deterministically prioritizing feasible solutions and ranking infeasible ones according to constraint violation magnitude, Deb’s rule eliminates the need for penalty coefficients entirely. Feasibility-based techniques have demonstrated strong robustness and ease of integration across a wide range of applications, including structural engineering, energy systems, biomedical optimization, and logistics design. Nevertheless, when feasible regions are extremely small or severely distorted, feasibility-driven selection may lead to premature convergence, motivating hybridizations with repair operators or multi-stage search schemes [20].

In parallel, relaxation-based constraint-handling strategies have evolved as an alternative means of balancing feasibility and exploration. ε-constrained methods, which progressively tighten feasibility tolerance during the search, have shown strong performance on COPs with complex feasible boundaries. An improved ε-constrained differential evolution method incorporating adaptive gradient-based repair has demonstrated superior performance on real-world constrained mechanical and industrial design problems [21]. Related approaches, including stochastic feasibility control and multi-objective reformulations of COPs, have also been explored to enhance exploration while maintaining constraint awareness [22].

Repair and decoder-based techniques remain particularly relevant in engineering contexts where feasibility is governed by domain-specific physical laws or geometric relationships. Recent studies have integrated local surrogate models, physics-based correctors, and boundary-informed mapping operators to enforce feasibility efficiently in simulation-driven design environments [23]. However, the reliance of these approaches on problem-specific knowledge and their tendency to distort search trajectories limit their general applicability.

More recently, adaptive and learning-driven constraint-handling frameworks have emerged as a major research trend. Reinforcement learning has been employed to guide penalty adjustment, operator selection, and strategy switching, leading to self-adaptive COP solvers that outperform fixed constraint-handling techniques on many benchmark problems [24,25]. Adaptive constraint-handling selection mechanisms have also been proposed for constrained multi-objective optimization, dynamically choosing the most effective strategy based on population-level metrics [26]. Ensemble-based approaches further extend this idea by maintaining and adaptively weighting multiple constraint-handling techniques throughout the optimization process to exploit complementary strengths [27,28].

Alongside developments in constraint-handling strategies, the swarm intelligence literature continues to expand with the introduction of new population interaction mechanisms and search dynamics. Recent swarm optimizers, including the Random Average Marine Predators Algorithm (RAMPA) [29], improved variants of the Tunicate Swarm Algorithm such as IMATSA [30], and competitive swarm-based frameworks, represent a growing trend toward adaptive and competition-driven search strategies. These approaches typically enhance exploration–exploitation balance through dynamic control of search operators, inter-agent competition, or adaptive parameter adjustment based on population feedback. While such mechanisms have shown promising performance on complex and high-dimensional problems, they introduce additional algorithmic layers and control logic that influence convergence behavior. In contrast, the present work deliberately avoids adaptive learning components and competitive selection schemes, focusing instead on a tuning-free, team-oriented swarm structure with feasibility-driven selection. This design choice prioritizes simplicity, robustness, and reproducibility under constrained optimization with limited evaluation budgets, which is the primary scope of this study.

In the same domain, modern swarm algorithms such as the Grasshopper Optimization Algorithm [31], Harris Hawks Optimization [32], Prairie Dog Optimization [33], Raven Roost Optimization [34], and the Slime Mould Algorithm [35] illustrate the breadth and continued evolution of swarm-based optimization research. Despite their diversity, many of these approaches still rely on algorithm-specific control parameters or require careful tuning, which limits their robustness in constrained engineering applications and motivates further investigation into tuning-free swarm optimizers.

The survey of the related work reveals three main gaps. First, many constrained optimization methods still rely on penalty functions or adaptive penalty rules, which require problem-dependent tuning and often behave inconsistently across different engineering problems. Second, several studies report that feasibility-rule approaches can suffer from premature convergence, because the population may be pushed too quickly toward the first feasible region and lose the diversity needed to reach better feasible areas. Third, most swarm and evolutionary algorithms were originally designed for unconstrained problems, so their update rules do not naturally fit constrained spaces where feasible regions are narrow or irregular. Together, these points highlight the need for an approach that avoids problem-dependent penalties, prevents premature convergence, and uses search operators that take feasibility into account during the search. These trends motivate the development of a tuning-free feasibility-driven framework designed specifically to address these observed limitations.

3. The CTOSO Algorithm

The Constrained Team-Oriented Swarm Optimizer (CTOSO) is a metaheuristic designed to efficiently solve complex COPs. CTOSO represents a substantial reworking of ETOSO’s core mechanics, replacing the original linear exploitation strategy with spiral-based local search and introducing constraint handling, while preserving the team-oriented population structure. It builds upon this foundation by integrating Deb’s feasibility rule for constraint handling throughout all selection processes. This structure balances the exploration and exploitation phases while requiring no tuning of algorithm-specific control parameters, making it particularly suitable for real-world engineering applications. In this section, x represents a D-dimensional vector, and f(x) and v(x) represent its objective and violation values, respectively.

3.1. Population Definition and Team Architecture

Let the swarm at iteration t be a population P(t) of ps candidate solutions (individuals):

P(t) = {x₁(t), x₂(t), …, x_ps(t)}. Each individual x_i(t) is a D-dimensional vector x_i(t) = (x_i,1(t), …, x_i,D(t)) within the bounds [l_b, u_b]. Associated with each individual is its objective value f_i(t) and its total constraint violation v_i(t). The population is deterministically partitioned into two equal-sized teams:

• Exploiters: X_E(t) = {x₁(t), …, x_ps/2(t)}
• Explorers: X_O(t) = {x_{ps/2 + 1}(t), …, x_ps(t)}

The update equation for an individual is determined strictly by its team membership.

3.2. Constraint Handling via Deb’s Rule

CTOSO employs Deb’s feasibility rule for all solution comparisons. The total constraint violation v(x) for a solution x is calculated as the L₂-norm (Euclidean distance) of constraint breaches:

$$v\left(\mathit{x}\right)=\sqrt{{\sum }_k{\left[max\left(0,g_k\left(\mathit{x}\right)\right)\right]}^2+{\sum }_l{\left[h_l\left(\mathit{x}\right)\right]}^2}$$

(1)

A solution x_i is considered superior to x_j if any of the following conditions holds:

• x_i is feasible and x_j is not.
• Both are feasible and f(x_i) < f(x_j).
• Both are infeasible and v(x_i) < v(x_j).

This dominance logic is applied consistently for updating the global best solution $x_{gbest}$ as well as for all internal selection decisions inside the explorer and exploiter teams, ensuring that feasible solutions are always prioritized during the entire search process.

3.3. Exploiter Update Model: Spiral-Based Search

Individuals in the exploiter team X_E(t) perform an intensified local search by executing a spiral movement around the current global best solution x_gbest(t). This spiral-based approach fundamentally differs from ETOSO’s linear movement strategy, providing more sophisticated local search capabilities around promising regions. This behavior is inspired by the Moth-Flame Optimizer (MFO). The position of each exploiter x_i in X_E(t) is updated in every dimension d in {1, …, D} as follows:

$$x_{i,d}^{\left(t+1\right)}=D_{i,d}{e}^{\left(b{r}_{e,d}\right)}\mathit{cos}\left(2\pi r_{e,d}\right)+x_{gbest,d}^{\left(t\right)}$$

(2)

where

• D_i,d = |x_gbest,d(t) − x_i,d(t)| is the distance from the individual to the global best in dimension d
• r_e,d is a random scalar uniformly distributed in [−1, 1]
• b = (t/FE_max) − 1 an internal adaptive state (not a user-tuned parameter) that deterministically transitions the search from global to local as the evaluation budget is consumed. At the beginning of a run, $b\approx -1$, so the exponent $b{r}_{e,d}$ lies in [−1, 1] and the radial factor $e^{b{r}_e,d}$ranges from approximately $0.37$ to $2.72$, allowing moderate contraction and expansion around $x_{gbest}$. As FE approaches its maximum, $b$ converges to 0 and thus $e^{b{r}_{e,d}}$ approaches 1, so the spiral reduces to small oscillations whose amplitude is governed mainly by the shrinking distance $D_{i,d}$. This schedule produces a smooth transition from broader spiral search in early iterations to fine-grained exploitation near convergence, without introducing additional control parameters. All candidate solutions are clamped to the bounds l_b and u_b after the position update to maintain variable feasibility.

CTOSO differs from spiral-based optimizers such as MFO and the Spiral Dynamic Algorithm (SDA). In CTOSO, the spiral operator is employed strictly as a local intensification mechanism restricted to the exploiter team and guided solely by the current global best solution, whereas MFO applies spiral motion uniformly across the population within a flame-sorting and flame-reduction hierarchy.

CTOSO also differs from the Spiral Dynamic Algorithm (SDA) in several structural aspects. While the exploiter update in (2) follows a spiral-shaped trajectory, its update mechanism is not the SDA. The two approaches differ in their core mathematical structure and update logic, in the following aspects:

• Independent dimension-wise stochastic update vs. matrix-based operator: In the canonical SDA, spiral motion is defined through a rotation–contraction operator, where a rotation matrix is applied to the solution vector. This matrix operation couples the decision variables through a global transformation of the state vector [36]. In contrast, CTOSO does not construct or apply any rotation matrix. The exploiter update is computed using a scalar expression inside a loop over dimensions, where each coordinate $x_{i,d}$ is updated directly based on its own value and the corresponding coordinate of the global best. This results in an independent dimension-wise update rather than a coupled matrix-based transformation.
• Distance-modulated spiral amplitude: In CTOSO, the spiral step size in each dimension is explicitly proportional to the distance $\mid x_{gbest,d}-x_{i,d}\mid .$ As a result, the spiral amplitude in each coordinate naturally decreases as that coordinate approaches the global best. In SDA, the spiral radius is controlled by predefined contraction parameters that are part of the rotation–contraction operator, rather than being derived from coordinate-wise distances to the current best solution [36,37].
• Decoupled stochastic spiral phase vs. deterministic spiral mapping: In CTOSO, stochasticity is introduced by independently sampling the spiral phase $r_{e,d}$ for each dimension during the exploiter update. Consequently, different dimensions of the position vector may follow distinct spiral trajectories around the global best solution. In canonical SDA formulations, once the rotation and contraction parameters of the spiral operator are fixed, the resulting spiral mapping is deterministic and applied as a single structured transformation of the position vector [36].
• Budget-driven spiral scheduling: CTOSO controls the spiral contraction strength through a budget-dependent schedule, where the spiral constant is updated based on the remaining number of function evaluations. This removes the need for user-defined spiral parameters. In SDA, spiral behavior is governed by explicitly defined rotation angles and contraction factors that must be selected in advance as part of the spiral operator [36,37].

For these reasons, CTOSO employs a distance-scaled, dimension-wise stochastic spiral exploitation operator, whereas SDA is defined by a matrix-based rotation–contraction spiral operator with explicitly parameterized dynamics.

3.4. Explorer Update Model: Adaptive Neighbor-Guided Movement

Individuals in the Explorer team X_O(t) are responsible for promoting population diversity and exploring new regions of the search space. Each explorer identifies its neighborhood best using a ring topology with wrap-around indexing. The neighborhood best selection uses a feasibility-based comparison for efficient rank-based comparison, consistent with standard metaheuristic implementations, while employing an advanced ring topology for coordinated movement. The explorer then moves toward this local leader using an adaptive scaling factor. This is a vector operation:

$${\mathrm{x}}_i^{\left(t+1\right)}={\mathrm{x}}_i^{\left(t\right)}+w{r}_o\odot \left({\mathrm{x}}_{nbest,i}^{\left(t\right)}-{\mathrm{x}}_i^{\left(t\right)}\right)$$

(3)

where

• $r_o$ is a D-dimensional vector of uniform random numbers in [0, 1].
• ⊙ denotes elementwise multiplication.
• w is an adaptive scaling factor that normalizes the movement based on the objective landscape of the explorer team: $w={\displaystyle \frac{\mid f\left({\mathit{x}}_{gbest}\right)\mid }{1+\underset{j\in X_O}{max\mid f\left({\mathit{x}}_j\right)\mid }}}$. This prevents excessively large moves that could destabilize the search, especially when objective values are large. Following the update, all new positions are clamped to the problem’s lower and upper bounds l_b and u_b.

3.5. Algorithmic Flow and Tuning-Free Design

The design of CTOSO is motivated by the points raised in the related work. The algorithm avoids the use of penalty functions and relies solely on Deb’s rule for selection, making it fully free of problem-dependent parameters. The movement rules of both the exploiter and explorer teams help reduce premature convergence by maintaining diversity and by using feasibility information without forcing the search to collapse toward the first feasible point. Feasibility is incorporated into the update logic, the spiral exploiter follows feasibility-aware guidance, and the explorer team selects neighborhood leaders using simple feasibility-based comparisons. Together, these elements shape CTOSO into a tuning-free method with feasibility naturally integrated into its search behavior.

In addition to these design choices, CTOSO maintains a simple and lightweight structure. It adapts its search behavior through the internal dynamics of the spiral constant $b$, the adaptive weight $w$, and the deterministic rules used for team selection and constraint handling. Because it does not rely on control coefficients such as crossover rates or mutation factors, the algorithm remains easy to configure and operate. This simplicity makes CTOSO well suited for black-box engineering scenarios where little prior information about the problem is available.

The core logic and flow of the CTOSO algorithm are formally detailed in the pseudocode in Algorithm 1. The main inputs are the problem definition (D, l_b, u_b), the computational budget (FE_max), and the population size (ps). The algorithm outputs are the final best feasible solution and its objective value. The global best solution (g_best) is updated immediately after each individual solution evaluation within both the exploiter and explorer loops. This real-time update mechanism allows newly discovered superior solutions to immediately influence the search direction of subsequent individuals within the same iteration, creating a dynamic and responsive optimization process.

We next highlight the key structural differences between CTOSO and ETOSO. While CTOSO retains ETOSO’s division into exploiter and explorer teams, it departs from ETOSO in two important ways. First, the linear, rank-weighted exploiter motion used in ETOSO is replaced with a full-dimensional spiral operator applied only to the exploiter team. This modification provides a smoother, evaluation-driven transition from broad to focused search without relying on weight schedules or repeated reinitialization at the global best. Second, the explorer team incorporates feasibility-based guidance during neighborhood selection, improving reliability on constrained landscapes without requiring personal-best archives or penalty-parameter tuning.

Algorithm 1. Pseudocode of the Constrained Team-Oriented Swarm Optimizer (CTOSO)

Input:$I,D,\mathbf{l}\mathbf{b},\mathbf{u}\mathbf{b},\max \text{\_}\mathrm{FE},\mathrm{ps}$ Output: $\mathrm{best}\text{\_}\mathrm{obj}\text{\_}\mathrm{found}, \mathrm{best}\text{\_}\mathrm{sol}\text{\_}\mathrm{found}, \mathrm{convergence}\text{\_}\mathrm{trace}, \mathrm{total}\text{\_}\mathrm{evals}\text{\_}\mathrm{used}, \mathrm{final}\text{\_}\mathrm{violation}$ Step 1: Initialization Initialize population positions:                               $\mathbf{p}\mathbf{o}\mathbf{s}={\mathit{l}}_{\mathit{b}}+\mathrm{rand}(\mathrm{ps},D)\odot ({\mathit{u}}_{\mathit{b}}-{\mathit{l}}_{\mathit{b}})$ Initialize objective values and constraint violations to $\mathrm{\infty }$ Initialize convergence trace to $\mathrm{\infty }$ Set total function evaluations $\mathrm{total}\text{\_}\mathrm{evals}=0$ Set best feasible objective so far to $\mathrm{\infty }$      Step 2: Initial Evaluation       For each individual $i=1$to $\mathbf{p}\mathbf{s}$ Evaluate objective and constraints:                               $\left(f_i,vi{o}_i)\leftarrow \mathrm{EngineeringBenchmarks}({\mathbf{p}\mathbf{o}\mathbf{s}}_i,I\right)$ Increment evaluation counter Store objective and violation values Update best feasible solution using Deb’s feasibility rule Record convergence information End For      Step 3: Global Best Selection Select the global best solution using Deb’s feasibility rule Initialize $\mathbf{g}\mathbf{b}\mathbf{e}\mathbf{s}\mathbf{t}\_\mathbf{p}\mathbf{o}\mathbf{s},\mathbf{g}\mathbf{b}\mathbf{e}\mathbf{s}\mathbf{t}\_\mathbf{o}\mathbf{b}\mathbf{j},\mathbf{g}\mathbf{b}\mathbf{e}\mathbf{s}\mathbf{t}\_\mathbf{v}\mathbf{i}\mathbf{o}$      Step 4: Population Division Assign first half of population as Exploiters Assign second half of population as Explorers      Step 5: Main Optimization Loop      While $\mathrm{total}\text{\_}\mathrm{evals}<\max \text{\_}\mathrm{FE}$do      Exploiter Update (Spiral Search)       For each exploiter do Update position using spiral movement toward $\mathbf{g}\mathbf{b}\mathbf{e}\mathbf{s}\mathbf{t}\_\mathbf{p}\mathbf{o}\mathbf{s}$ Evaluate objective and constraint violation Increment evaluation counter Update global best using Deb’s feasibility rule Record convergence information End For      Explorer Update (Neighborhood Search)       For each explorer do Select neighborhood leader in ring topology using Deb’s feasibility rule Update position using adaptive weighted movement toward neighborhood leader Evaluate objective and constraint violation Increment evaluation counter Update global best using Deb’s feasibility rule Record convergence information End For      End While      Step 6: Output Return best objective value and solution Return final constraint violation Return total evaluations used Return convergence trace

3.6. Rationale and Design Motivation of CTOSO

CTOSO is derived through a deliberate restructuring of the Team-Oriented Swarm Optimization framework to achieve stable constrained optimization under fixed and limited evaluation budgets, while eliminating the need for parameter tuning or recovery-based operators. In the original TOSO implementation, search behavior is governed by several auxiliary components, including personal-best memory, mutation of poorly performing explorers, random reinitialization, and rank-based weighting schemes. Although these mechanisms enhance exploration, they introduce additional control parameters and can repeatedly disrupt feasible solutions once discovered, particularly in constrained problems where feasibility regions are narrow. ETOSO reduces this complexity by removing personal-best memory and correcting global-best updates; however, it retains linear exploiter movements whose step magnitudes scale with the full decision-variable range and are independent of proximity to feasible regions.

CTOSO replaces this linear exploitation mechanism with a full-dimensional spiral contraction centered on the current global best. Unlike the spiral motion used in Moth–Flame Optimizer (MFO), which relies on multiple flames and spiral shape parameters to balance exploration and exploitation, the CTOSO spiral is single-center, feasibility-driven, and deterministically coupled to the remaining evaluation budget. The step size is proportional to the distance from the global best and decreases monotonically as the evaluation budget is consumed, without introducing additional spiral control parameters. This design enables controlled intensification near promising feasible regions while preserving the original team-oriented search structure. In parallel, all recovery-based operators present in TOSO and ETOSO (including mutation, random restarts, feasibility archives, and personal-best tracking) are removed. As a result, feasibility preservation is enforced exclusively through Deb’s feasibility rule during all solution comparisons and updates, ensuring that feasible solutions consistently dominate the search trajectory once identified. These changes yield a structurally simpler, tuning-free algorithm in which constrained search behavior is governed solely by population interactions, global-best geometry, and feasibility-based selection.

4. Evaluation on CEC 2017 Constrained Benchmark Problems

Before evaluating the proposed algorithm on real-world engineering design problems, an initial validation was conducted using the CEC 2017 constrained optimization benchmark suite (f01–f15) [38]. These benchmarks are widely regarded as a standardized reference for assessing constrained evolutionary and swarm-based optimizers, providing a controlled environment to evaluate feasibility handling, convergence behavior, and robustness under well-defined conditions. This evaluation establishes a baseline comparison against canonical reference optimizers prior to application-oriented engineering studies.

These problems involve nonlinear, multimodal, shifted, and rotated landscapes with both inequality and equality constraints, and are specifically designed to challenge constraint-handling mechanisms and search dynamics. All problems were treated as constrained minimization problems and were evaluated using the official CEC 2017 Pure MATLAB implementation, without re-deriving or modifying any benchmark equations. Internal CEC state handling, including rotation initialization, was correctly reset for each problem to ensure reproducibility. The dimensionality was fixed to $D=10$ for all benchmark functions, consistent with standard CEC experimental protocols.

All algorithms were evaluated under a fixed computational budget of 10,000 function evaluations (FE) per run. Each algorithm was executed for 30 independent runs per problem, and a population size of 100 was used for all population-based methods to ensure comparability. The same initialization bounds, stopping criteria, and random-seed control were applied uniformly across all algorithms. Constraint satisfaction was assessed using a feasibility tolerance of ${10}^{-6}$, and performance statistics were computed using feasible solutions only, in accordance with standard constrained optimization practice.

For the standard CEC benchmark evaluation, the proposed CTOSO was compared against a set of widely adopted reference optimizers. Classical baselines (GA and PSO) and evolutionary standards (DE and L-SHADE [39]) were selected due to their extensive use in CEC benchmarking and competition studies. In addition, representative population-based optimizers including GWO, TLBO, and MFO were included to reflect commonly used modern swarm, tuning-light, and spiral-based search paradigms. This selection ensures fair comparison against canonical baselines while avoiding novelty-driven algorithm proliferation. CTOSO is the only newly proposed algorithm evaluated in this section.

For each benchmark problem, the best feasible objective value, average feasible objective value, feasibility rate, and average computational time were recorded. Per-problem ranks were computed based on best feasible objective values, and average ranks across all benchmark problems were calculated. Non-parametric statistical tests were employed, including Friedman ranking with Nemenyi post hoc analysis and pairwise Wilcoxon signed-rank tests, using identical data sets to ensure consistency and fairness.

Table 1 reports the best feasible objective values obtained by all compared algorithms on the CEC2017 constrained benchmark suite, together with feasibility rates, number of feasible problems, average ranks, and win counts. As shown in the table, CTOSO achieves the lowest average rank (AvgRank = 2.4) among the compared methods and attains the highest feasibility rate (79.78%), producing feasible solutions on 14 out of 15 benchmark functions.

Table 1. Empirical performance on the CEC2017 constrained benchmark suite (f01–f15): empty entries indicate no feasible solution was found.

	CTOSO	GA	PSO	LSHADE	DE	MFO	TLBO	GWO
f01	2.07 × 10−5	21.15736	0.272625	1.33 × 10−9	1006.123	0.0003	3.445277	0.081434
f02	1.49 × 10−5	10.69442	0.325351	2.91 × 10−10	754.3456	0.001072	1.80341	0.063298
f03	3956.399	3178.555	—	—	—	29,388.24	—	—
f04	40.79311	34.70913	47.42172	16.28481	2014.517	20.23924	103.8898	19.35096
f05	0.039222	79.47138	2.090526	0.344004	—	0.000418	34.18892	1.944388
f06	903.1204	—	—	—	—	—	—	—
f07	−332.212	−157.042	—	—	—	−27.5699	—	—
f08	0.000111	—	—	−9.2 × 10−5	—	—	—	—
f09	0.156965	—	97.0689	−0.0005	—	0.320292	—	—
f10	−4.4 × 10−5	—	—	−4.8 × 10−5	—	−3.1 × 10−5	—	—
f11	—	—	—	−2.38898	—	—	—	—
f12	3.987955	20.23622	4.219791	4.039389	—	3.987948	13.60401	10.61497
f13	0.00437	142.7698	1.545034	0.000192	—	0.002884	96.98268	10.5451
f14	3.304776	—	—	3.39774	—	3.30349	—	—
f15	11.78097	8.639379	11.78097	11.78097	—	11.78097	18.06416	—
AvgFR (%)	79.77778	52.88889	41.55556	63.77778	20	66.22222	38.22222	38.66667
No. FeasibleProblems	14	9	8	12	3	12	7	6
AvgRank	2.4	5.133333	5.1	2.6	6.766667	2.9	5.933333	5.166667
Wins	2	2	0	8	0	3	0	0

To further assess statistical significance across all benchmark problems, rank-based analysis was conducted using the Friedman test followed by Nemenyi post hoc comparison. The resulting critical difference (CD) diagram is shown in Figure 1, where CTOSO and L-SHADE fall within the same statistical performance group, indicating no statistically significant difference between their average ranks at the selected significance level.

Figure 1. Critical difference (CD) plot for average ranks on the CEC2017 benchmark suite.

In addition, pairwise Wilcoxon signed-rank test results between CTOSO and each reference algorithm are summarized in Table 2. These results indicate that CTOSO performs significantly better than PSO and TLBO, while no statistically significant differences are observed in comparisons with GA, DE, MFO, GWO, or L-SHADE. Taken together, the empirical results and the rank-based visualization confirm that CTOSO exhibits competitive and reliable constrained optimization performance under standardized CEC benchmark conditions.

Table 2. Pairwise Wilcoxon signed-rank test results between CTOSO and reference algorithms.

	Versus	p	Verdict
CTOSO	GA	0.25	No significant difference
CTOSO	PSO	0.0078	CTOSO better
CTOSO	LSHADE	0.5771	No significant difference
CTOSO	DE	0.25	No significant difference
CTOSO	MFO	0.7002	No significant difference
CTOSO	TLBO	0.0157	CTOSO better
CTOSO	GWO	0.4375	No significant difference

5. Comparative Evaluation of TOSO, ETOSO, and CTOSO

To clarify the practical impact of the structural modifications introduced in CTOSO relative to its predecessor variants, a focused comparative evaluation was conducted between TOSO, ETOSO, and CTOSO using the same CEC 2017 constrained benchmark suite (f01–f15). This comparison isolates variant-level behavior within the TOSO family under a standardized constrained testbed, complementing the broader comparison against reference optimizers presented in Section 4.

All three algorithms were evaluated using an identical experimental protocol: dimensionality $D=10$, a budget of 10,000 function evaluations, and 30 independent runs per benchmark function. Constraint satisfaction was assessed using a feasibility tolerance of ${10}^{-6}$, and reported values correspond to best feasible objective values.

Table 3 reports the per-function best feasible objective values and aggregate feasibility indicators for the three variants. CTOSO produces feasible solutions on 14 out of 15 benchmark functions (AvgFR = 79.78%), whereas ETOSO and TOSO each produce feasible solutions on only 3 functions (AvgFR = 20%). The feasible outcomes of ETOSO and TOSO are limited to f01, f02, and f04, while CTOSO achieves feasible solutions across nearly the entire benchmark suite. For the subset of functions where all three algorithms report feasible solutions, CTOSO consistently attains smaller best feasible objective values than both predecessor variants.

Table 3. Best feasible objective values of CTOSO, ETOSO, and TOSO on the CEC2017: empty entries indicate no feasible solution was found.

Metric/Function	CTOSO	ETOSO	TOSO
f01	2.07106 × 10−5	5725.951149	888.2916486
f02	1.48756 × 10−5	6781.844391	598.8760425
f03	3956.399228	—	—
f04	40.79310902	608.4204647	772.7285459
f05	0.039222331	—	—
f06	903.1203672	—	—
f07	−332.2117729	—	—
f08	0.000110564	—	—
f09	0.156964729	—	—
f10	−4.35889 × 10−5	—	—
f11	—	—	—
f12	3.987954704	—	—
f13	0.004370377	—	—
f14	3.304775761	—	—
f15	11.78097174	—	—
AvgFR (%)	79.77777778	20	20
No. Feasible Problems	14	3	3
AvgRank	1.000000	2.535714286	2.464285714
Wins	14	0	0

To summarize performance across functions, per-problem rankings were computed based on best feasible objective values. CTOSO achieves the best overall average rank (AvgRank = 1.00), while ETOSO and TOSO obtain larger average ranks (2.54 and 2.46, respectively). The corresponding Friedman–Nemenyi critical difference diagram is illustrated in Figure 2, where CTOSO is separated from both ETOSO and TOSO by a margin exceeding the critical difference threshold, indicating statistically meaningful rank separation under this evaluation setup.

Figure 2. Critical difference (CD) diagram for average ranks of CTOSO, ETOSO, and TOSO across CEC2017 constrained benchmark functions (Friedman test with Nemenyi post hoc; CD = 0.856; lower rank is better).

Overall, this comparative evaluation demonstrates that CTOSO exhibits substantially stronger constrained optimization behavior than its predecessor variants, characterized by markedly broader feasibility coverage and improved objective quality on shared feasible functions. These results clarify the effect of the structural changes introduced in CTOSO under standardized benchmark conditions, prior to assessing performance on real-world engineering design problems.

6. Engineering Design Optimization Studies

The performance and robustness of CTOSO were rigorously evaluated against a comprehensive set of competitive metaheuristics. This section details the 12 constrained engineering design benchmarks used for testing, the performance metrics applied for quantitative assessment (including the composite score), and the statistical validation methods employed to confirm the significance of the results. The setup also identifies the 20 state-of-the-art algorithms utilized for comparative analysis. All the parameters for the comparative algorithms are obtained from [2]. It was ensured to apply the same constraint handling to the competitive algorithms for fair comparison.

6.1. Benchmark Problems

The performance of CTOSO was assessed across twelve widely used constrained engineering design problems derived from the literature. These benchmarks are selected for their complexity, diverse search spaces, and relevance to real-world applications, offering a versatile and robust platform for validating the effectiveness of constrained optimization algorithms. The benchmarks are well-established and frequently employed in studies on constrained engineering optimization. Using this benchmark set ensures that all algorithms are tested under standardized, well-defined formulations with known reference solutions, providing a fair and reproducible basis for comparative evaluation under identical computational budgets. Each benchmark is detailed below, including its objective function, constraints, and design variables. All engineering design optimization problems considered in this section are formulated and solved as constrained minimization problems in the form:

Minimize: f(x), Subject to:
g_k(x) ≤ 0, for k = 1, …, m
h_l(x) = 0, for l = 1, …, p
x ∈ [l_b, u_b]^D

(4)

The summary of the twelve constrained engineering benchmark problems is given in Table 4.

Table 4. Summary of Constrained Engineering Benchmark Problems (inequality constraint).

ID	Problem Name	Dimension	Complexity Type
1	Speed Reducer	7	Nonlinear, Mixed Terms
2	Tension/Compression Spring	3	Nonlinear
3	Pressure Vessel	4	Nonlinear
4	Welded Beam	4	Highly Coupled
5	Himmelblau’s Beam	5	Polynomial Mixed
6	Cantilever Beam	5	Rational Function
7	Tubular Column	2	Analytical, Simple
8	Piston Lever	4	Trigonometric, Nonlinear
9	Car Side Impact	11	Mixed Linear & Nonlinear
10	Corrugated Bulkhead	4	Nonconvex, Engineering
11	Three-Bar Truss	2	Rational Fractions
12	Reinforced Concrete Beam	3	Discrete, Quadratic

6.1.1. Speed Reducer Design [40]

Objective Function:

$$\begin{array}{c}f\left(\mathbf{x}\right)=0.7854x_1{x}_2^2\left(3.3333x_3^2+14.9334x_3-43.0934\right)+0.7854\left(x_4{x}_6^2+x_5{x}_7^2\right)\hfill \\ -1.508x_1\left(x_6^2+x_7^2\right)+7.477\left(x_6^3+x_7^3\right)\hfill \\ \mathrm{Constraints} (g_i\left(\mathbf{x}\right)\le 0):\hfill \\ g_1\left(\mathbf{x}\right)=27-x_1{x}_2^2{x}_3\\ g_2\left(\mathbf{x}\right)=397.5-x_1{x}_2^2{x}_3^2\\ g_3\left(\mathbf{x}\right)=1.93-\left(x_2{x}_3{x}_6^4\right)/\left(x_4^3\right)\\ g_4\left(\mathbf{x}\right)=1.93-\left(x_2{x}_3{x}_7^4\right)/\left(x_5^3\right)\\ g_5\left(\mathbf{x}\right)={\displaystyle \frac{10}{x_6^3}}\sqrt{{\left({\displaystyle \frac{745x_4}{x_2{x}_3}}\right)}^2+16.91\times {10}^6}-1100\\ g_6\left(\mathbf{x}\right)={\displaystyle \frac{10}{x_7^3}}\sqrt{{\left({\displaystyle \frac{745x_5}{x_2{x}_3}}\right)}^2+157.5\times {10}^6}-850\\ g_7\left(\mathbf{x}\right)=-40+x_2{x}_3\\ g_8\left(\mathbf{x}\right)=5-x_1/x_2\\ g_9\left(\mathbf{x}\right)=-12+x_1/x_2\\ g_{10}\left(\mathbf{x}\right)=1.5x_6-x_4+1.9\\ g_{11}\left(\mathbf{x}\right)=1.1x_7-x_5+1.9\end{array}$$

(5)

Variable Bounds:

$$x_1\in \left[2.6,3.6\right],x_2\in \left[0.7,0.8\right],x_3\in \left[17,28\right],x_4,x_5\in \left[7.3,8.3\right],x_6\in \left[2.9,3.9\right],x_7\in \left[5.0,5.5\right]$$

Optimal Value: $f^{\mathrm{*}}=2994.422564$

6.1.2. Tension/Compression Spring Design [41]

Objective Function:

$$f\left(\mathbf{x}\right)=\left(x_3+2\right)x_2{x}_1^2$$

(6)

Constraints ($g_i\left(\mathbf{x}\right)\le 0$):

$$\begin{array}{c}{g}_1\left(\mathrm{x}\right)=1-{\displaystyle \frac{x_2^3{x}_3}{71785x_1^4}}\\ g_2\left(\mathrm{x}\right)={\displaystyle \frac{4x_2^2-x_1{x}_2}{12566\left(x_2{x}_1^3-x_1^4\right)}}+{\displaystyle \frac{1}{5108x_1^2}}-1\\ g_3\left(\mathrm{x}\right)=1-{\displaystyle \frac{140.45x_1}{x_2^2{x}_3}}\\ g_4\left(\mathrm{x}\right)={\displaystyle \frac{x_1+x_2}{1.5}}-1\end{array}$$

Variable Bounds: $x_1\in \left[0.05,2.0\right],x_2\in \left[0.25,1.3\right],x_3\in \left[2,15\right]$

Optimal Value: $f^{\mathrm{*}}=0.012665$

6.1.3. Pressure Vessel Design [40]

Objective Function:

$$f\left(\mathbf{x}\right)=0.6224x_1{x}_3{x}_4+1.7781x_2{x}_3^2+3.1661x_1^2{x}_4+19.84x_1^2{x}_3$$

(7)

Constraints ($g_i\left(\mathbf{x}\right)\le 0$):

$$\begin{array}{c}{g}_1\left(\mathbf{x}\right)=-x_1+0.0193x_3\\ g_2\left(\mathbf{x}\right)=-x_2+0.00954x_3\\ g_3\left(\mathbf{x}\right)=-\left(\pi x_3^2{x}_4+{\displaystyle \frac{4\pi x_3^3}{3}}\right)+1296000\\ g_4\left(\mathbf{x}\right)=x_4-240\end{array}$$

Variable Bounds: $x_1,x_2\in \left[0.0625,99\right],x_3,x_4\in \left[10,200\right]$

Optimal Value: $f^{\mathrm{*}}=5885.3328$

6.1.4. Welded Beam Design [40]

Objective Function:

$$f\left(\mathbf{x}\right)=1.10471x_1^2{x}_2+0.04811x_3{x}_4\left(14+x_2\right)$$

(8)

Auxiliary Expressions:

$$\begin{array}{c}M=P\left(L+x_2/2\right)\\ R=\sqrt{x_2^2/4+\left(\right(x_1+x_3)/2{)}^2}\\ J=2\left[\sqrt{2}{x}_1{x}_2\left({\displaystyle \frac{x_2^2}{12}}+{\left({\displaystyle \frac{x_1+x_3}{2}}\right)}^2\right)\right]\\ \tau =\sqrt{{\tau }^{\prime 2}+2{\tau }^{\prime }{\tau }^{″}{\displaystyle \frac{x_2}{2R}}+{\tau }^{″2}}, \mathrm{where} {\tau }^{\prime }={\displaystyle \frac{P}{\sqrt{2}{x}_1{x}_2}}, {\tau }^{″}={\displaystyle \frac{MR}{J}}\\ \sigma ={\displaystyle \frac{6PL}{x_4{x}_3^2}}\\ \delta ={\displaystyle \frac{6P{L}^3}{E{x}_3^3{x}_4}}\\ P_c=\left[{\displaystyle \frac{4.013E\sqrt{x_3^2{x}_4^6/36}}{L^2}}\right]\left[1-{\displaystyle \frac{x_3}{2L}}\sqrt{{\displaystyle \frac{E}{4G}}}\right]\end{array}$$

Constraints ($g_i\left(\mathbf{x}\right)\le 0$):

$$\begin{array}{c}{g}_1\left(\mathbf{x}\right)=\tau -13600\\ g_2\left(\mathbf{x}\right)=\sigma -30000\\ g_3\left(\mathbf{x}\right)=x_1-x_4\\ g_4\left(\mathbf{x}\right)=0.10471x_1^2+0.04811x_3{x}_4\left(14+x_2\right)-5\\ g_5\left(\mathbf{x}\right)=0.125-x_1\\ g_6\left(\mathbf{x}\right)=\delta -0.25\\ g_7\left(\mathbf{x}\right)=P-P_c\end{array}$$

Optimal Value: $f^{\mathrm{*}}=1.724852$

6.1.5. Himmelblau’s Beam Design [40]

Objective Function:

$$f\left(\mathbf{x}\right)=5.3578547x_3^2+0.8356891x_1{x}_5+37.293239x_1-40792.141$$

(9)

Auxiliary Expressions:

$$\begin{array}{c}{y}_1=85.334407+0.0056858x_2{x}_5+0.0006262x_1{x}_4-0.0022053x_3{x}_5\\ y_2=80.51249+0.0071317x_2{x}_5+0.0029955x_1{x}_2+0.0021813x_3^2\\ y_3=9.300961+0.0047026x_3{x}_5+0.0012547x_1{x}_3+0.0019085x_3{x}_4\end{array}$$

Constraints ($g_i\left(\mathbf{x}\right)\le 0$):

$$\begin{array}{c}{g}_1\left(\mathbf{x}\right):-y_1\le 0\\ g_2\left(\mathbf{x}\right):y_1-92\le 0\\ g_3\left(\mathbf{x}\right):-y_2+90\le 0\\ g_4\left(\mathbf{x}\right):y_2-110\le 0\\ g_5\left(\mathbf{x}\right):-y_3+20\le 0\\ g_6\left(\mathbf{x}\right):y_3-25\le 0\end{array}$$

Optimal Value: $f^{\mathrm{*}}=-30665.539$

6.1.6. Cantilever Beam Design [41]

Objective Function:

$$f\left(\mathbf{x}\right)=0.0624\left(x_1+x_2+x_3+x_4+x_5\right)$$

(10)

Constraint: $g_1\left(\mathbf{x}\right)={\displaystyle \frac{61}{x_1^3}}+{\displaystyle \frac{37}{x_2^3}}+{\displaystyle \frac{19}{x_3^3}}+{\displaystyle \frac{7}{x_4^3}}+{\displaystyle \frac{1}{x_5^3}}-1\le 0$

Optimal Value: $f^{\mathrm{*}}=1.339956$

6.1.7. Tubular Column Design [40]

Objective Function:

$$f\left(d,t\right)=9.8dt+2d$$

(11)

Constraints: ($g_i\left(\mathbf{x}\right)\le 0$):

$$\begin{array}{c}{g}_1\left(d,t\right)={\displaystyle \frac{2500}{\pi dt\cdot 500}}-1\\ g_2\left(d,t\right)={\displaystyle \frac{8\cdot 2500\cdot {250}^2}{{\pi }^3\cdot 0.85\times {10}^6\cdot dt\left(d^2+t^2\right)}}-1\\ g_3\left(d\right)={\displaystyle \frac{2}{d}}-1\\ g_4\left(d\right)={\displaystyle \frac{d}{14}}-1\\ g_5\left(t\right)={\displaystyle \frac{0.2}{t}}-1\\ g_6\left(t\right)={\displaystyle \frac{t}{0.8}}-1\end{array}$$

Optimal Value: $f^{\mathrm{*}}=26.4863$

6.1.8. Piston Lever Design [42]

Objective Function:

$$f\left(\mathbf{x}\right)={\displaystyle \frac{\pi }{4}}{D}^2\left(l_2-l_1\right)$$

(12)

$$l_1=\sqrt{{\left(X-B\right)}^2+H^2},$$

$$l_2=\sqrt{{\left(X\sin \mathsf{\theta }+H\right)}^2+{\left(B-X\cos \mathsf{\theta }\right)}^2}$$

$$R={\displaystyle \frac{\left|-X\left(X\sin \mathsf{\theta }+H\right)+H\left(B-X\cos \mathsf{\theta }\right)\right|}{\sqrt{{\left(X-B\right)}^2+H^2}}}$$

$$F={\displaystyle \frac{\mathsf{\pi }}{4}}P{D}^2$$

Constraints ($g_i\left(\mathbf{x}\right)\le 0$):

$$\begin{gathered} \begin{array}{c}{g}_1\left(x\right)=\hspace{0.25em}QL\cos \mathsf{\theta }-RF\\ g_2\left(x\right)=\hspace{0.25em}Q\left(L-X\right)-M_{max}\\ g_3\left(x\right)=1.2\left(l_2-l_1\right)-l_1\\ g_4\left(x\right)=\hspace{0.25em}0.5D-B\end{array} \\ \theta ={\displaystyle \frac{\mathsf{\pi }}{4}}, \\ \mathit{P}=1500, \\ \mathit{Q}=10000, \\ \mathit{L}=240, \\ {M}_{max}=1.8\times {10}^6 \\ 0.05\le \mathrm{H}\le 500 \\ 0.05\le \mathrm{B}\le 500 \\ 0.05\le \mathrm{D}\le 500 \\ 0.05\le \mathrm{X}\le 120 \end{gathered}$$

Optimal Value: f* = 8.412698

6.1.9. Car Side Impact Design [43]

Objective Function:

$$f\left(\mathbf{x}\right)=1.98+4.9x_1+6.67x_2+6.98x_3+4.01x_4+1.78x_5+2.73x_7$$

(13)

Constraints ($g_i\left(\mathbf{x}\right)\le 0$):

$${\mathrm{g}}_1\left(\mathrm{x}\right)=1.16-0.3717{\mathrm{x}}_2{\mathrm{x}}_4-0.0092928{\mathrm{x}}_5{\mathrm{x}}_9-0.484{\mathrm{x}}_3{\mathrm{x}}_8+0.01343{\mathrm{x}}_6{\mathrm{x}}_{10}$$

$$\begin{gathered} {\mathrm{g}}_2\left(\mathrm{x}\right)=0.261-0.0159{\mathrm{x}}_1{\mathrm{x}}_2-0.188{\mathrm{x}}_1{\mathrm{x}}_8-0.019{\mathrm{x}}_2{\mathrm{x}}_7+0.0144{\mathrm{x}}_3{\mathrm{x}}_5+0.0008757{\mathrm{x}}_5{\mathrm{x}}_{10} \\ +0.08045{\mathrm{x}}_6{\mathrm{x}}_9+0.00139{\mathrm{x}}_8{\mathrm{x}}_{11}+0.00001575{\mathrm{x}}_{10}{\mathrm{x}}_{11} \end{gathered}$$

$$\begin{gathered} g_3\left(x\right)=0.214+0.00817x_5-0.131x_1{x}_8-0.0704x_1{x}_9+0.03099x_2{x}_6-0.018x_2{x}_7 \\ +0.0208x_3{x}_8+0.121x_3{x}_9-0.00364x_5{x}_6+0.0007715x_5{x}_{10} \\ -0.0005354x_6{x}_{10}+0.00121x_8{x}_{11}+0.00184x_9{x}_{10} \end{gathered}$$

$$g_4\left(x\right)=0.74-0.61x_2-0.163x_3{x}_8+0.001232x_3{x}_{10}-0.166x_7{x}_9+0.227x_2^2$$

Optimal Value: $f^{\mathrm{*}}=22.84296954$

6.1.10. Corrugated Bulkhead Design [41]

Objective Function:

$$f\left(\mathbf{x}\right)={\displaystyle \frac{5.885t\left(b+l\right)}{b+\sqrt{\mid l^2-h^2\mid }}}$$

(14)

Constraints ($g_i\left(\mathbf{x}\right)\le 0$):

$$\begin{array}{c}{g}_1\left(\mathbf{x}\right)=-th\left(0.4b+l/6\right)+8.94\left(b+\sqrt{\mid l^2-h^2\mid }\right)\\ g_2(\mathbf{x})=-t{h}^2(0.2b+l/12)+2.2(8.94(b+\sqrt{\mid l^2-h^2\mid }){)}^{4/3}\\ g_3\left(\mathbf{x}\right)=-t+0.0156b+0.15\\ g_4\left(\mathbf{x}\right)=-t+0.0156l+0.15\\ g_5\left(\mathbf{x}\right)=-t+1.05\\ g_6\left(\mathbf{x}\right)=-l+h\end{array}$$

Optimal Value: $f^{\mathrm{*}}=6.842958$

6.1.11. Three-Bar Truss Design [44]

Objective Function:

$$f\left(x\right)=\left(2\sqrt{2}\hspace{0.17em}{A}_1+A_2\right)\cdot 100$$

(15)

Constraints ($g_i\left(\mathbf{x}\right)\le 0$):

$$\begin{gathered} g_1\left(x\right)={\displaystyle \frac{2\left(\sqrt{2}{A}_1+A_2\right)}{\left(\sqrt{2}{A}_1^2+2A_1{A}_2\right)}}-2 \\ {g}_2\left(x\right)={\displaystyle \frac{2A_2}{\left(\sqrt{2}{A}_1^2+2A_1{A}_2\right)}}-2 \\ {\mathrm{g}}_3\left(\mathrm{x}\right)={\displaystyle \frac{2}{\sqrt{2}{\mathrm{A}}_2+{\mathrm{A}}_1}}-2 \end{gathered}$$

Optimal Value: $f^{\mathrm{*}}=263.8958$

6.1.12. Reinforced Concrete Beam Design [45]

Objective Function:

$$f\left(\mathbf{x}\right)=29.4A_s+0.6bh$$

(16)

Lookup Tables and Indexing Logic:

Steel area set $A_s$: [6, 6.16, 6.32, 6.6, 7, 7.11, 7.2, 7.8, 7.9, 8, 8.4]

Beam width set $b$: [28, 29, …, 40]

The normalized variables $x_1$ and $x_2$ are mapped to these discrete sets using:

$$\begin{gathered} \begin{array}{c}i{x}_1=\mathrm{m}\mathrm{i}\mathrm{n}\left(\mathrm{m}\mathrm{a}\mathrm{x}\left(1,\mathrm{round}\left(x_1\cdot \mathrm{length}\left(A_s\right)\right)\right),\mathrm{length}\left(A_s\right)\right)\\ i{x}_2=\mathrm{m}\mathrm{i}\mathrm{n}\left(\mathrm{m}\mathrm{a}\mathrm{x}\left(1,\mathrm{round}\left(x_2\cdot \mathrm{length}\left(b\right)\right)\right),\mathrm{length}\left(b\right)\right)\\ \mathrm{Then}\text{:} A_s=A_s\left[i{x}_1\right], b=b\left[i{x}_2\right]\end{array} \\ \mathrm{Constraints} (g_i\left(\mathbf{x}\right)\le 0) \\ \begin{array}{c}{g}_1\left(\mathbf{x}\right)=b/h-4\\ g_2\left(\mathbf{x}\right)=180+7.375{\displaystyle \frac{A_s^2}{h}}-A_{s}b\end{array} \\ \text{Optimal Value:} f^{\mathrm{*}}=359.20 \end{gathered}$$

6.2. Comparative Algorithms and Evaluation Metrics

CTOSO was compared against twenty state-of-the-art metaheuristic algorithms: the Bat Algorithm (BA) [46], Bees Algorithm (BEE) [47], Butterfly Optimization Algorithm (BOA) [48], Crow Search Algorithm (CSA) [49], Cuckoo Search (CS) [50], Differential Evolution (DE) [51], Elephant Herding Optimization (EHO) [52], Firefly Algorithm (FA) [53], Flower Pollination Algorithm (FPA) [54], Grasshopper Optimization Algorithm (GOA) [31], Gravitational Search Algorithm (GSA) [55], Grey Wolf Optimizer (GWO) [8], Harris Hawks Optimization (HHO) [32], Moth-Flame Optimization (MFO) [9], Monkey King Algorithm (MKA) [56], Prairie Dog Optimization (PDO) [33], Raven Roost Optimization (RRO) [34], Sine Cosine Algorithm (SCA) [57], Slime Mould Algorithm (SMA) [35], Teaching-Learning-Based Optimization (TLBO) [58], and Team-Oriented Swarm Optimizer (TOSO) [1]. The parameters for all the algorithms are used from [2]. The evaluation was conducted over 30 independent replications per problem (ensuring statistical reliability), with a population size of 100. A strict limit of 3000 function evaluations was imposed to emulate scenarios where objective or constraint evaluation is computationally expensive, as is typical in simulation-driven engineering design. Using a fixed and deliberately tight FE budget ensures that all algorithms face the same re-stricted-resource conditions and emphasizes their ability to produce feasible and competi-tive solutions when evaluation cost (not iteration count) is the dominant constraint in practice.

The 20 comparative metaheuristics were originally proposed for unconstrained optimization and do not define a unified constraint-handling method. To ensure fair and unbiased comparison, all algorithms were equipped with the same constraint-handling strategy, namely Deb’s feasibility rule, which is widely recognized as a standard, parameter-free CHT in constrained optimization research. Recent surveys on constraint handling for population-based and metaheuristic algorithms emphasize that such parameter-free feasibility rules are among the most widely used baselines, precisely because they avoid penalty tuning and provide a consistent feasibility criterion across different optimizers [4,59]. In addition, several recent algorithmic studies adopt Deb-type feasibility rules when extending unconstrained metaheuristics to constrained engineering design problems, treating them as a generic and reliable CHT [7]. Using a single, consistent CHT in this work therefore eliminates the confounding effect of penalty coefficients and ensures that performance differences arise primarily from the search dynamics of each algorithm rather than from differences in constraint treatment, in line with current recommendations for fair benchmarking in constrained engineering optimization.

7. Results and Comparative Analysis

This section presents a comprehensive evaluation of CTOSO against the twenty algorithms across the twelve constrained engineering design problems. The analysis is structured to provide a multi-faceted performance assessment. First, a composite ranking system is employed to objectively identify the top-performing algorithms for further detailed comparison. Subsequently, the performance of these top algorithms is examined on each individual benchmark problem. To statistically validate the observed performance differences, Wilcoxon signed-rank and Friedman–Nemenyi tests are conducted. Finally, the convergence behavior and computational complexity of the algorithms are analyzed to provide insights into their operational characteristics and efficiency.

7.1. Top Algorithm Selection via Composite Ranking

Table 5 presents the composite ranking results, showing for each algorithm its average rank (Avg. Rank), normalized Rank Score, feasibility rate (Feasibility %), normalized Feasibility Score, success rate (Success %), normalized Success Score, final Composite Score, and Top 10 selection status. In real-world constrained optimization, solution effectiveness must be judged not only by the quality of objective values but also by an algorithm’s capacity to consistently satisfy constraints and to do so robustly across multiple trials. As such, a multi-criteria selection framework is essential for fair and meaningful comparison. To accomplish this, we implemented a composite ranking mechanism synthesizing three essential facets of algorithm performance (across all problems):

Table 5. Algorithm Filtering (Top 10 Selection).

Algorithm	Avg. Rank	Rank Score	Feasibility %	Feasibility Score	Success%	Success Score	Composite Score	In Top 10
CTOSO	1.92	1.00	100.00	1.00	91.67	1.00	1.00	Yes
BEE	10.00	0.57	100.00	1.00	25.00	0.27	0.67	Yes
BOA	10.08	0.57	100.00	1.00	25.00	0.27	0.67	Yes
CSA	10.50	0.55	100.00	1.00	25.00	0.27	0.66	Yes
CS	15.00	0.31	100.00	1.00	8.33	0.09	0.49	No
DE	10.67	0.54	100.00	1.00	33.33	0.36	0.66	Yes
EHO	4.08	0.89	98.33	0.94	50.00	0.55	0.87	Yes
FA	18.83	0.11	91.11	0.67	0.00	0.00	0.26	No
FPA	7.25	0.72	100.00	1.00	33.33	0.36	0.77	Yes
GOA	10.58	0.54	97.78	0.92	41.67	0.45	0.65	No
GSA	13.83	0.37	91.11	0.67	16.67	0.18	0.44	No
GWO	5.67	0.80	100.00	1.00	41.67	0.45	0.83	Yes
HHO	10.00	0.57	93.33	0.75	33.33	0.36	0.61	No
MFO	2.58	0.96	100.00	1.00	58.33	0.64	0.94	Yes
MKA	16.33	0.24	100.00	1.00	0.00	0.00	0.44	No
PDO	20.83	0.00	73.06	0.00	0.00	0.00	0.00	No
RRO	14.25	0.35	100.00	1.00	8.33	0.09	0.52	No
SCA	11.83	0.48	91.67	0.69	25.00	0.27	0.52	No
SMA	13.83	0.37	100.00	1.00	25.00	0.27	0.55	No
TLBO	10.67	0.54	100.00	1.00	25.00	0.27	0.65	Yes
TOSO	12.42	0.44	99.72	0.99	25.00	0.27	0.59	No

• Solution Quality: measured Via average rank of best feasible solutions across problems (60% weight for normalized rank score)
• Feasibility Rate: quantifying reliability in constraint satisfaction (30% weight for feasibility score)
• Robustness: assessing how often an algorithm reaches near-optimal solutions within 0.1% of known optima (10% weight for Success Score).

The weighting scheme is aligned with the priority structure established in the constraint-handling studies [4,12,28,59], which consistently treat objective comparison among feasible solutions as the strongest basis for discrimination, constraint-satisfaction reliability as the next most informative factor and run-to-run stability as a useful but lower-impact indicator. For this reason, the largest weight is assigned to the metric that differentiates feasible solutions, a moderate weight is given to the metric that reflects constraint-handling reliability across runs, and a smaller complementary weight captures robustness effects. All metrics are normalized to the [0, 1] range, ensuring scale comparability, with higher values denoting superior performance. The weighting reflects the practical priorities of constrained optimization: achieving feasible, high-quality solutions consistently. Algorithms were then ranked based on their overall composite score, and the top 10 were retained for subsequent fine-grained analysis, including statistical testing and convergence behavior. This filtering step is not merely procedural; it is crucial for eliminating underperforming or unstable algorithms, which may excel in one metric but fail in others. CTOSO emerged as the standout performer, achieving a perfect composite score of 1.00, ranking first in all three normalized dimensions.

7.2. Detailed Performance on Benchmark Engineering Problems

The per-problem performance of the top 10 algorithms provides detailed insights into their strengths and weaknesses across different types of constrained engineering challenges. The following tables (Table 6, Table 7, Table 8, Table 9, Table 10, Table 11, Table 12, Table 13, Table 14, Table 15, Table 16 and Table 17) present the detailed results for each benchmark. The columns in these tables are defined as follows:

Table 6. Speed Reducer Design Results.

Algorithm	Known Best	Best Found	Average	Std Dev	Feasibility %	Time (s)
CTOSO	2994.42256	2994.43022	3009.98229	18.28088	100.00000	0.00799
BEE		3402.16526	4109.66421	828.96543	13.33333	0.00511
BOA		2998.60450	3013.54627	16.83084	100.00000	0.00642
CSA		3014.54906	3043.39636	12.24176	100.00000	0.00902
DE		3010.43154	3026.90970	9.62804	100.00000	0.00939
EHO		3009.45844	3084.07376	117.39808	100.00000	0.00606
FPA		3002.76888	3015.16978	9.85771	100.00000	0.00985
GWO		3025.16695	3044.75405	10.86458	100.00000	0.02451
MFO		2994.55111	2998.92267	9.89997	100.00000	0.00698
TLBO		3044.16490	3101.99308	45.33784	100.00000	0.00668

Table 7. Tension/Compression Spring Design Results.

Algorithm	Known Best	Best Found	Average	Std Dev	Feasibility %	Time (s)
CTOSO	0.01266	0.01268	0.01650	0.00508	100.00000	0.01179
BEE		0.03261	0.07289	0.03827	43.33333	0.00820
BOA		0.01272	0.04881	0.03045	100.00000	0.00971
CSA		0.01275	0.01471	0.00170	100.00000	0.01456
DE		0.01283	0.01377	0.00082	100.00000	0.01210
EHO		0.01267	0.01326	0.00140	100.00000	0.00899
FPA		0.01277	0.01326	0.00083	100.00000	0.01200
GWO		0.01274	0.01337	0.00100	100.00000	0.03478
MFO		0.01268	0.01378	0.00157	100.00000	0.03881
TLBO		0.01313	0.01636	0.00259	100.00000	0.03923

Table 8. Pressure Vessel Design Results.

Algorithm	Known Best	Best Found	Average	Std Dev	Feasibility %	Time (s)
CTOSO	5885.33280	5885.36543	6665.59614	561.98063	100.00000	0.04927
BEE		61,261.59527	286,910.42601	153,258.51664	96.66667	0.03783
BOA		6403.00266	27,150.86275	45,676.80253	100.00000	0.04274
CSA		10,438.84287	20,235.74230	6044.29053	100.00000	0.06959
DE		13,099.04062	22,977.19519	5439.74015	100.00000	0.05339
EHO		5922.95265	26,126.86180	51,467.62641	100.00000	0.04030
FPA		7779.07475	12,510.58836	2677.35331	100.00000	0.03897
GWO		5973.92289	6526.91214	411.40888	100.00000	0.07358
MFO		5924.55315	6693.85946	575.36999	100.00000	0.03811
TLBO		7187.54770	12,546.10337	4363.66476	100.00000	0.04009

Table 9. Welded Beam Design Results.

Algorithm	Known Best	Best Found	Average	Std Dev	Feasibility %	Time (s)
CTOSO	1.72485	1.72596	2.21985	0.52940	100.00000	0.05618
BEE		3.11375	5.37952	1.69375	40.00000	0.04848
BOA		1.84662	2.75130	0.50826	100.00000	0.05559
CSA		2.04664	2.73409	0.34587	100.00000	0.07470
DE		1.94475	2.18405	0.12798	100.00000	0.05872
EHO		1.75153	2.49167	0.62229	96.66667	0.04540
FPA		1.86269	2.16848	0.19531	100.00000	0.05769
GWO		1.73362	1.74921	0.00883	100.00000	0.11098
MFO		1.74370	2.06702	0.39453	100.00000	0.03879
TLBO		2.06438	2.18894	0.10541	100.00000	0.03249

Table 10. Himmelblau’s Beam Results.

Algorithm	Known Best	Best Found	Average	Std Dev	Feasibility %	Time (s)
CTOSO	−30665.53900	−30,665.53932	−30,635.43189	163.30138	100.00000	0.04606
BEE		−29,962.25656	−29,273.66802	303.73524	96.66667	0.03641
BOA		−30,662.58116	−30,493.40983	151.10401	100.00000	0.04828
CSA		−30,650.80310	−30,479.80552	92.05694	100.00000	0.07075
DE		−30,618.59428	−30,536.50208	56.47780	100.00000	0.05395
EHO		−30,665.53832	−30,439.96609	244.06583	100.00000	0.04613
FPA		−30,660.44229	−30,598.05113	30.68531	100.00000	0.04565
GWO		−30,659.87004	−30,625.75881	24.38995	100.00000	0.07824
MFO		−30,665.51035	−30,645.17326	87.53560	100.00000	0.04049
TLBO		−30,645.18615	−30,473.82089	121.25306	100.00000	0.03946

Table 11. Cantilever Beam Results.

Algorithm	Known Best	Best Found	Average	Std Dev	Feasibility %	Time (s)
CTOSO	1.33996	1.34016	1.34191	0.00175	100.00000	0.02628
BEE		3.41957	6.94688	1.47072	100.00000	0.01611
BOA		1.34067	1.36801	0.12802	100.00000	0.02046
CSA		1.83228	2.47104	0.29989	100.00000	0.02310
DE		2.03405	3.42562	0.61963	100.00000	0.03132
EHO		1.38933	3.19498	1.42424	100.00000	0.02205
FPA		1.58736	2.15174	0.31028	100.00000	0.02847
GWO		1.34004	1.34102	0.00062	100.00000	0.06784
MFO		1.34444	1.36886	0.01649	100.00000	0.01776
TLBO		1.62927	2.16094	0.32001	100.00000	0.01735

Table 12. Tubular Column Results.

Algorithm	Known Best	Best Found	Average	Std Dev	Feasibility %	Time (s)
CTOSO	26.48630	26.49949	26.50071	0.00480	100.00000	0.04424
BEE		26.85545	29.02514	1.56285	100.00000	0.03574
BOA		26.50111	26.50722	0.00375	100.00000	0.04733
CSA		26.49421	26.57613	0.06240	100.00000	0.05395
DE		26.50653	26.56346	0.03414	100.00000	0.04790
EHO		26.49950	26.49969	0.00047	100.00000	0.03854
FPA		26.50440	26.53293	0.03144	100.00000	0.04634
GWO		26.51061	26.53719	0.02009	100.00000	0.07977
MFO		26.49949	26.49977	0.00054	100.00000	0.03923
TLBO		26.51657	26.63795	0.08282	100.00000	0.04903

Table 13. Piston Lever Design Results.

Algorithm	Known Best	Best Found	Average	Std Dev	Feasibility %	Time (s)
CTOSO	8.41270	8.41270	86.62171	87.61831	100.00000	0.05120
BEE		555.62173	14,022.58125	19,172.81065	100.00000	0.03951
BOA		226.67109	552.76981	223.59413	100.00000	0.03993
CSA		74.78814	274.43840	71.45653	100.00000	0.04790
DE		24.79565	141.35404	67.24652	100.00000	0.05920
EHO		9.37145	330.96262	181.72972	100.00000	0.03993
FPA		14.77209	189.06532	83.15320	100.00000	0.05122
GWO		8.43224	110.63247	79.01639	100.00000	0.09884
MFO		8.41833	120.78746	74.73346	100.00000	0.02912
TLBO		14.73820	189.60583	81.56745	100.00000	0.03640

Table 14. Car Side Impact Results.

Algorithm	Known Best	Best Found	Average	Std Dev	Feasibility %	Time (s)
CTOSO	22.84297	27.92469	30.02881	1.92685	100.00000	0.05107
BEE		Inf	NaN	NaN	0.00000	0.03887
BOA		29.01693	31.22205	1.30467	100.00000	0.05152
CSA		29.15435	30.56327	0.68335	100.00000	0.07011
DE		28.62434	30.12540	0.57927	100.00000	0.06029
EHO		29.58585	32.42696	3.12805	66.66667	0.04442
FPA		28.69990	29.60509	0.62486	100.00000	0.05643
GWO		28.24233	29.10644	0.66774	100.00000	0.11014
MFO		28.11378	28.94354	0.74435	100.00000	0.04111
TLBO		28.99566	30.32304	0.78456	100.00000	0.03429

Table 15. Corrugated Bulkhead Results.

Algorithm	Known Best	Best Found	Average	Std Dev	Feasibility %	Time (s)
CTOSO	6.84296	6.84297	6.89546	0.27283	100.00000	0.04922
BEE		7.90937	11.12998	1.82991	100.00000	0.04833
BOA		6.85104	7.22045	0.39607	100.00000	0.04814
CSA		6.97449	7.38823	0.25567	100.00000	0.07832
DE		7.10367	7.60323	0.31234	100.00000	0.06253
EHO		6.86392	7.09830	0.38665	100.00000	0.05045
FPA		6.99358	7.25039	0.19522	100.00000	0.05481
GWO		6.86545	6.90424	0.04063	100.00000	0.09829
MFO		6.84477	6.85111	0.00631	100.00000	0.03887
TLBO		7.03625	7.41446	0.16978	100.00000	0.04257

Table 16. Three-Bar Truss Design Results.

Algorithm	Known Best	Best Found	Average	Std Dev	Feasibility %	Time (s)
CTOSO	263.89580	263.89874	271.60509	4.48839	100.00000	0.04946
BEE		329.23104	872.63628	334.28303	100.00000	0.03225
BOA		263.91885	264.26650	0.35916	100.00000	0.05173
CSA		264.06106	265.89242	1.88533	100.00000	0.07390
DE		263.89648	264.31322	0.29849	100.00000	0.04331
EHO		263.89589	264.24102	1.38562	100.00000	0.04396
FPA		263.91726	265.55814	2.21588	100.00000	0.04065
GWO		263.89730	263.94505	0.03789	100.00000	0.07332
MFO		263.89976	269.81182	5.16554	100.00000	0.03946
TLBO		263.98009	264.59121	0.53620	100.00000	0.04116

Table 17. Reinforced Concrete Beam Design Results.

Algorithm	Known Best	Best Found	Average	Std Dev	Feasibility %	Time (s)
CTOSO	359.20800	359.20796	360.05451	1.40914	100.00000	0.04526
BEE		363.11596	373.93852	6.58367	100.00000	0.02741
BOA		359.20834	364.49939	4.84366	100.00000	0.04557
CSA		359.16516	360.05412	0.96460	100.00000	0.06112
DE		359.21403	359.53435	0.35829	100.00000	0.05969
EHO		359.20800	360.39324	1.69635	100.00000	0.03257
FPA		359.21081	359.79091	1.07844	100.00000	0.04153
GWO		359.21043	360.04425	1.37822	100.00000	0.06801
MFO		359.20796	360.98368	1.58284	100.00000	0.02405
TLBO		359.22485	360.04466	0.91562	100.00000	0.03279

• Algorithm: The name of the metaheuristic algorithm.
• Known Best: The known global optimum value for the problem.
• Best Found: The best feasible objective value found by the algorithm across all runs.
• Average: The mean of the best feasible objective values across 30 independent runs.
• Std Dev: The standard deviation of the best feasible objective values.
• Feasibility %: The percentage of the 30 runs that produced a feasible solution.
• Time (s): The average computation time in seconds. The experiments were conducted using MATLAB 2024 on a system equipped with a relatively fast Intel(R) Core (TM) Ultra 9 185H processor (2.30 GHz), 32.0 GB RAM, and a 64-bit operating system.

7.3. Statistical Significance Analysis

To statistically validate CTOSO’s performance, Wilcoxon signed-rank tests and a Friedman test with Nemenyi post hoc analysis were conducted. The results provide robust evidence of its superiority. The Wilcoxon signed-rank tests (Table 18) confirm that CTOSO achieves a statistically significant performance advantage (p < 0.01) over every other leading algorithm in direct, head-to-head comparisons. This establishes that the observed improvements are consistent and reliable.

Table 18. Wilcoxon Signed-Rank Test: CTOSO vs. Top Algorithms.

	Versus	p-Value	Verdict
CTOSO	BEE	0.000488281	CTOSO better
	BOA	0.000488281	CTOSO better
	CSA	0.004882813	CTOSO better
	DE	0.001464844	CTOSO better
	EHO	0.009277344	CTOSO better
	FPA	0.000488281	CTOSO better
	GWO	0.004882813	CTOSO better
	MFO	0.000488281	CTOSO better
	TLBO	0.000488281	CTOSO better

Furthermore, the Friedman–Nemenyi post hoc test (Table 19) confirms CTOSO’s overall dominance by assigning it the best average rank of 1.58. While the Nemenyi test identifies a cluster of competitors (BOA, EHO, GWO, MFO) whose ranks are not statistically different from CTOSO’s based on the critical difference of 4.08, CTOSO still holds the absolute top rank among all algorithms. Interestingly, the test also confirms that five major algorithms (BEE, CSA, DE, FPA, and TLBO) are statistically placed in a significantly inferior performance cluster. This comprehensive analysis confirms CTOSO’s position as a top-performing algorithm in this comparative study.

Table 19. Friedman–Nemenyi Test Results.

	Versus	Avg Rank 1	Avg Rank 2	Abs Rank Diff	Critical Difference	Significant
CTOSO	BEE	1.58	10.00	8.42	4.08	TRUE
	BOA		5.00	3.42		FALSE
	CSA		6.42	4.83		TRUE
	DE		7.08	5.50		TRUE
	EHO		3.75	2.17		FALSE
	FPA		6.00	4.42		TRUE
	GWO		4.50	2.92		FALSE
	MFO		2.92	1.33		FALSE
	TLBO		7.75	6.17		TRUE

7.4. Convergence Behavior

Across the twelve engineering design benchmarks, CTOSO shows a consistent and favorable convergence pattern as shown in Figure 3 and Figure 4. In problems with relatively easier constraint structures, such as the Cantilever Beam, Reinforced Concrete Beam, and Corrugated Bulkhead, the CTOSO curve appears from the earliest evaluations, indicating immediate feasibility. In more restrictive problems, including the Speed Reducer, Tension/Compression Spring, Welded Beam, and Car Side Impact, the CTOSO curve appears shortly after the start, reflecting a brief initial infeasible phase; however, this delay is noticeably shorter than that of several competing algorithms, which often remain infeasible for much longer.

Figure 3. Convergence Plots for Top 10 Algorithms (Problems 1–6), 30 replications.

Figure 4. Convergence Plots for Top 10 Algorithms (Problems 7–12), 30 replications.

Once feasibility is reached, CTOSO often shows a clear initial decrease in the objective value, indicating effective early improvement on many of the benchmark problems, showing that it quickly moves toward high-quality regions of the search space. After this fast descent, the convergence profile becomes smooth and stable, with steady improvement and no oscillatory or divergent behavior. Across all twelve benchmarks, CTOSO reliably reaches a near-optimal region early and then refines the solution effectively throughout the remaining budget. Overall, CTOSO’s convergence behavior is characterized by early feasibility, quick improvement, and stable long-term refinement, making it a dependable and efficient optimizer across a variety of constrained engineering design problems.

7.5. Computational Complexity

As detailed in Table 20, The computational complexity of CTOSO is theoretically derived as O(FE (D + ps)), placing it in the moderate complexity tier alongside established algorithms like Differential Evolution. This efficient structure is driven by two key mechanisms: the spiral search strategy, which scales linearly with problem dimensionality (D), and the adaptive explorer weighting, which depends linearly on population size (ps). Unlike computationally expensive hierarchy-based algorithms such as GWO, which suffer from a high overhead of O(FE (D + ps log ps)) due to repetitive population sorting, CTOSO eliminates the need for sorting entirely. This strategic design choice allows the algorithm to allocate computational resources toward finding better solutions rather than maintaining internal hierarchy.

Table 20. Computational Complexity Analysis.

Algorithm	Big-O Complexity (ps = 100)	Key Overhead Drivers	Overall Complexity Tier
CTOSO	O(FE · (D + ps))	ps/2 spiral (exp, cos) per D; O(ps) max operation for weight w	Moderate
BEE	O(FE · D)	Sort ps bees each gen (ps log ps); Gaussian site search	Moderate
BOA	O(FE · D)	Lightweight fragrance calculation; no population sorting	Low
CSA	O(2FE · D)	Awareness move + memory copy per agent	High
DE	O(FE · (D + ps))	randperm on ps agents; mutation & crossover vectors	Moderate
EHO	O(FE · D)	Two rand calls + clan mean per update	Low
FPA	O(FE · D)	Lévy flight (three variables) per D	Low
GWO	O(FE · (D + ps log ps))	Sort ps wolves every eval; updates three variables	High
MFO	O(FE · D)	exp + cos per D; flame re-rank once/generation	Low
TLBO	O(FE · D)	Teacher/learner vector operations; one start-up sort	Low

Furthermore, CTOSO demonstrates robust scalability when applied to high-dimensional real-world problems. While simple algorithms like MFO or BEE operate with a baseline complexity of O(FE.D), the additional overhead introduced by CTOSO’s population-dependent term is mathematically negligible in complex engineering scenarios. As problem dimensionality grows significantly larger than the population size ($D\gg ps)$, the computational cost becomes dominated by the dimension term, effectively converging to the efficiency of the simplest competitors. Consequently, CTOSO offers a superior trade-off, providing the advanced search capabilities necessary to achieve 100% feasibility on constrained problems without incurring the prohibitive runtime penalties associated with high-complexity methods.

In addition to the population-based metaheuristics summarized in Table 20, CTOSO can be contrasted with single-agent based optimizers that operate on a single solution trajectory. In such approaches, the optimization process updates a single agent by manipulating either several elements or the full set of design parameters at each iteration, without maintaining a population of candidate solutions. Representative examples include improved smoothed functional algorithm approaches and related single-trajectory stochastic approximation methods, as well as sequential learning formulations often described in the context of game-theoretic approaches. Owing to the absence of population maintenance, sorting, or inter-agent coordination, these single-agent methods typically incur lower internal computational burden per iteration.

Recent studies in zeroth-order and smoothed functional optimization formalize this single-agent structure and emphasize its computational efficiency under fixed function-evaluation budgets through accelerated updates and estimator-efficient mechanisms [60,61,62]. When viewed against this class of single-agent methods, the additional population-dependent cost introduced by CTOSO remains within the moderate complexity tier identified in Table 20 and is dominated by the dimension-dependent term as the problem size increases. As demonstrated by the empirical results in Section 5, this moderate increase in computational effort enables cooperative exploration and systematic feasibility handling in constrained search spaces, resulting in consistently feasible solutions without approaching the higher overhead associated with hierarchy-based swarm algorithms.

7.6. Performance Visualization and Win Analysis

The performance trends observed in the numerical results are further confirmed through a series of visualizations. Figure 5a quantifies the number of benchmark problems where each algorithm achieved the best mean performance, with CTOSO securing the most wins. Figure 5b provides a rank-based heatmap of the best-found objectives, offering an immediate visual confirmation of CTOSO’s consistent top-tier performance across the problem suite, which aligns with the data in Table 3, Table 4, Table 5, Table 6, Table 7, Table 8, Table 9, Table 10, Table 11, Table 12, Table 13 and Table 14. The distribution of solution quality is detailed in Figure 6a, where the log-scaled boxplot of average objectives reveals that CTOSO not only achieves the lowest median but also the smallest interquartile range, highlighting its precision and reliability. Finally, Figure 6b presents the Critical Difference (CD) diagram from the Friedman–Nemenyi test conducted in Section 7.3, graphically validating that CTOSO’s average rank is statistically superior to a distinct group of its closest competitors.

Figure 5. Visualization of algorithms performance: (a) Algorithm Win Count Across the Benchmark Suite; (b) Performance Heatmap: Rankings of Best Feasible Objectives.

Figure 6. Statistical Algorithms Performance: (a) Log-Scaled Boxplots; (b) Critical Difference Diagram from the Friedman–Nemenyi Test.

8. Discussion

The standardized CEC2017 constrained benchmark evaluation established the feasibility-handling robustness and competitive ranking behavior of CTOSO under a fixed evaluation budget. This section concentrates on the comparative analysis conducted on the constrained engineering design problems. It provides a comprehensive, data-driven evaluation of the top 10 algorithms selected via composite ranking. The assessment draws on convergence behavior, numerical performance across 12 benchmark functions, feasibility robustness, statistical hypothesis testing, and computational complexity analysis. Across all benchmark functions, CTOSO exhibits the most favorable convergence dynamics, consistently achieving rapid objective reduction while maintaining feasibility throughout the search process. Its curves consistently demonstrate rapid descent in the initial search phase (first 500–1000 FEs), suggesting effective early exploitation. This is followed by smooth and consistent flattening, which indicates strong convergence stability and minimal noise or oscillation, reflecting stable constraint management. By contrast, MFO shows similarly steady convergence but at a slower rate. GWO demonstrates delayed takeoff but achieves competitive final values, while CSA and DE show late convergence with wider result spread. Algorithms such as FPA, BOA, and EHO display intermittent flat regions and jumps, and TLBO and BEE show erratic convergence with inconsistent improvement rates.

A critical observation regarding the low performing algorithms in highly constrained problems is not their constraint-handling method, as all use Deb’s Feasibility Rule, but their limited exploitation efficiency under this low FE budget. Competitors often rely on less directed, generalized search mechanisms for local refinement. In contrast, CTOSO’s dedicated exploiter team, guided by the spiral strategy, provides a highly intensified and focused search. This superior, structured exploitation plays a key role in rapidly converging to the precise feasible optimum when the search budget is strictly limited to 3000 FEs, providing a plausible explanation for CTOSO’s consistent statistical advantage.

Examining the average and standard deviation of feasible objectives provides crucial cross-problem consistency insights. CTOSO achieves near best or best average feasible objectives in the majority of the problems, often with low standard deviations. While MFO matches or approaches CTOSO in multiple functions, it suffers from higher variability. GWO stands out in precision and variance control.

The statistical validation confirms these performance findings. The Wilcoxon signed-rank tests between CTOSO and each of the nine other top-performing algorithms yielded statistically significant results (p < 0.01) across all comparisons, confirming that CTOSO’s superiority is not incidental. Furthermore, the Friedman–Nemenyi test stratified the algorithms, showing that CTOSO’s average rank of 1.58 is statistically better than algorithms like TLBO, DE, CSA, and FPA, although differences with GWO and MFO were not statistically significant per the critical difference (CD).

Although CTOSO incorporates a spiral exploitation mechanism inspired by the Moth-Flame Optimization (MFO) algorithm, a structural distinction is necessary. In canonical MFO, every agent moves towards a sorted elite solution (a flame). In contrast, CTOSO selectively applies this spiral function only to its exploiters (half the population), and it does not employ a global sorting mechanism to determine flame positions. This team-based role separation and selective application makes CTOSO’s exploitation more modular, less greedy, and benefits from cooperative balance with the explorer team.

CTOSO’s primary strengths include achieving the best overall composite score, maintaining 100% feasibility across all problems, and providing statistically significant superiority over competitors. Its simplicity, requiring no algorithm-specific hyperparameters, and its structurally clean architecture (based on just two roles and no memory archive) make it easy to implement and computationally efficient.

9. Conclusions and Future Work

This paper has introduced the Constrained Team-Oriented Swarm Optimizer (CTOSO), a metaheuristic designed specifically for constrained engineering problems, following an initial validation on the standardized CEC2017 constrained benchmark suite. CTOSO’s architecture is defined by three core components: a dual-role population that separates exploration and exploitation duties, a spiral-based search strategy for intensive local refinement, and the systematic application of Deb’s feasibility rule for constraint handling. A principal advantage of this design is that it operates without algorithm-specific control parameters, reducing the tuning effort required for practical application.

The algorithm’s performance was rigorously evaluated against twenty state-of-the-art metaheuristics across twelve established engineering benchmarks. The results demonstrate that CTOSO is highly effective and robust. It achieved the highest composite ranking, maintained a 100% feasibility rate across all problems and independent runs, and demonstrated competitive computational efficiency. This consistent feasibility is not accidental; once a feasible solution appears, Deb’s rule ensures that feasible candidates are always preferred over infeasible ones, which naturally biases the search toward maintaining feasibility across runs. Statistical analysis, including Wilcoxon signed-rank tests and the Friedman–Nemenyi post hoc procedure, confirmed that CTOSO’s performance is significantly superior to all other algorithms in the study. Beyond these aggregate metrics, CTOSO exhibited consistent behavioral strengths, including rapid initial convergence, stable performance with low variance across runs, and reliable constraint satisfaction throughout the search process.

In summary, CTOSO represents a reliable, parameter-free solution for constrained optimization. Its consistent top-tier performance across multiple dimensions—solution quality, feasibility, and robustness—makes it a compelling choice for engineering applications where tuning effort must be minimized and reliable results are critical. Future work will focus on extending CTOSO’s capabilities to more complex scenarios. Promising directions include rigorous evaluation on large-scale problems with hundreds of variables to test its scalability, and extension to handle mixed-integer design variables, which combine continuous and discrete parameters and remain difficult for most metaheuristics. In addition, future evaluations may consider broader constrained test environments such as recent benchmark suites that include equality-heavy, simulation-driven, or mixed-variable scenarios, which would further clarify CTOSO’s behavior across diverse constraint structures.