Enhanced Puzzle Optimization Algorithmfor Complex Engineering Design Problems

Abstract

This paper introduced the Enhanced Puzzle Optimization Algorithm (EPOA), a hybrid metaheuristic that augmented the original Puzzle Optimization Algorithm (POA) with uniform crossover, random-resetting mutation, and explicit elitism. The contribution does not lie in inventing these operators individually, since they are classical search components, but in integrating them into POA’s two-phase search dynamics to address premature convergence, diversity loss, and best-solution preservation in a targeted manner. This paper formalized EPOA’s update rules, provided pseudocode and flow diagrams, and enforced bound handling for box-constrained problems. Comprehensive tests on the CEC2022 single-objective benchmark suite (F₁–F₁₂) showed that EPOA attained rank 1 on 11 of 12 functions and rank 3 on the remaining case, with large error reductions relative to baseline POA (e.g., on F₁, the mean error dropped from 62.836 to 0.004; on F₆, the mean error dropped from 2370.962 to 7.239). The method was further evaluated on six classical constrained engineering design problems (welded beam, tension/compression spring, speed reducer, pressure vessel, three-bar truss, and cantilever beam). Statistical indicators such as the mean and standard deviation were used to assess robustness. The results showed that EPOA delivered a strong exploration–exploitation balance and robust solution quality across rugged landscapes and real-world constraints.

1. Introduction

The use of metaheuristic algorithms in the computational sciences has received significant attention following their capability to address complex optimization problems that the traditional gradient-based or exhaustive search techniques cannot provide viable solutions to them [1,2]. These are population-based or single-agent methods that simulate natural or artificial processes to search and exploit high-dimensional landscapes, using randomness, selection mechanisms, and adaptive operators to escape local optima. Examples of such representative families of metaheuristics are evolutionary algorithms (based on biological evolution) [3], swarm intelligence algorithms (based on animal collective behavior) [4], and physics-based optimizers (based on natural laws) [5]. Their major strength is that they strike a balance between exploration of a large search space and exploitation of the most promising solutions, which makes them powerful and adaptable in solving diverse types of problems.

The design tasks that involve the use of engineering design are usually multi-variable, non-linear goals, with complex constraints, like safety margin, cost limits, or performance requirements. Conventional optimization techniques, such as linear or non-linear programming techniques, may not fare well in situations where there are discontinuities or where the black-box functionality of interest is costly to obtain analytical gradients or approximate such gradients [6]. Metaheuristics provide a very strong alternative: they do not make use of derivative information but instead are based on the assessment of the objective function of candidate solutions, and hence are general-purpose and relatively easy to implement. Adaptive metaheuristics can also explore rough, multi-modal surfaces in the parameter space by stochastically sampling the parameter space, finding better design configurations. In the last ten years, engineers have been able to embrace the techniques in structural optimization, aerodynamic design, energy distribution, among other real-life problems of the real world [7].

The Puzzle Optimization Algorithm (POA) is one of the numerous metaheuristic frameworks that have been developed, which is characterized by a cooperative and adaptive model of search agents [8]. Based on the concept of natural puzzle-like interactions, or more metaphorically, the behavior of specific foraging animals, POA places multiple agents in a solution space and coordinates their movements based on two high-level phases: a global exploration phase (moving towards prey) and a local exploitation phase (winging on the water surface) [9]. In the exploration, the agents approach and detach themselves from a specified agent of food according to their fitness levels. They do small local adjustments to optimize promising solutions in the exploitation phase, which involves less information than in the exploration phase [10]. This two-stage strategy enables POA to explore new areas, as well as increase its search in those areas that have high probabilities of having optima. Its ability to combine global and local search heuristics has been successful in many benchmark tasks and real-life optimization problems [11].

Although effective, the baseline POA can be ineffective in a situation where the optimization problem is very nonlinear, multi-modal, or has strong constraints. Such environments regularly occur in engineering design problems, where an orthodox methodology may be susceptible to local optima or not adequately diversified in searching the larger search space. In order to overcome these weaknesses, the Enhanced Puzzle Optimization Algorithm (EPOA) supplements POA with new operators based on evolutionary computation, i.e., crossover, mutation, and an elitism mechanism. These genetic-inspired improvements make EPOA create a more diverse population and ensure the best answer at any point, therefore, reducing premature convergence and enhancing general robustness. It is not the invention of crossover, mutation, and elitism as such that contributes to scientific contributions, but rather their deployment towards POA in such a way that each operator corrects a particular vulnerability of the original update process.

The driving problems of EPOA are the engineering design issues of a wide range. Engineers are often required to trade off several parameters and meet non-linear constraints that indicate safety, cost, or performance. Techniques based strongly on gradient information or local searching may not be adequate in such situations, particularly in cases where precise analytical gradients are not known or discontinuities occur. EPOA, in its turn, takes advantage of the strength of stochastic sampling, movement in the direction of the promising solutions, and repeated refinements (which are based on mutation) to exhaust the design space. This functionality enables it to move over complicated objective surfaces and determine the best design configurations.

The key contributions are as follows:

• In this paper, EPOA was suggested that incorporated the ideas of uniform crossover, random-resetting mutation, and explicit elitism into POA in order to mitigate the drawbacks of premature convergence and loss of diversity while maintaining the two-phase search logic.
• The proposed method was given a full mathematical formulation, a bound-handling policy, readable pseudocode, and more detailed workflow diagrams in this paper.
• The proposed EPOA ranked 1st on 11 CEC2022 functions and ranked 3rd on the rest of the functions, with a significant decrease in mean error compared with baseline POA and most current optimizers.
• In this paper, the applicability of EPOA to six canonical constrained engineering design problems was discussed using standard performance statistics along with convergence and boxplot analyses.

The remainder of this paper is organized as follows. Section 2 reviews representative metaheuristic families and clarifies the research gap addressed in this study. Section 3 presents the baseline POA, the motivation for enhancement, and the full formulation of EPOA. Section 4 describes the experimental design and comparison protocol. Section 5 reports the benchmark and engineering results, Section 6 interprets the main findings and limitations, and Section 7 concludes the paper.

2. Related Work and Research Gap

Metaheuristic optimization can be summarized into four broad families: (i) evolutionary and population-based methods that rely on selection, variation, and explicit or implicit population models; (ii) swarm and bio-inspired methods that model decentralized interactions among agents; (iii) physics-based methods that abstract natural laws and dynamical processes; and (iv) socio-cognitive or human-inspired methods that emulate learning, competition, or collective decision-making. Figure 1 summarizes this grouping for readability and provides representative examples from the literature reviewed in this section.

Population-based evolutionary computation begins with Evolutionary Programming [12], Evolution Strategies [13], and Genetic Algorithms [14], and later extends to coevolution [15], Cultural Algorithms [16], Genetic Programming [17], Estimation of Distribution [18], and Differential Evolution [19]. Grammar- and expression-driven variants include Grammatical Evolution [20] and Gene Expression Programming [21]; hybrid and quantum-inspired refinements broaden the design space [22,23], as do competitive and imperialist models [24]. More recent directions refine sampling and variation via Differential and Backtracking searches and fractal mechanisms [25,26,27], alongside task-specific operators such as Synergistic Fibroblast Optimization [28].

Swarm intelligence abstracts collective behavior into decentralized search. Foundational exemplars include Ant Colony Optimization [29] and Particle Swarm Optimization (continuous and binary forms) [30,31,32], with immune-system and clonal models adding selection and memory effects [33,34]. A wide ecosystem of animal and microbial swarms followed–fish, bacteria, birds, bees, wolves, whales, butterflies, grasshoppers, salps, and more–yielding algorithms such as Artificial Fish Swarm and Bacterial Foraging [35,36]; bee-inspired heuristics and Shuffled Frog Leaping [37,38,39,40,41,42,43,44]; and later packs and herds including Krill Herd, Wolf/Grey Wolf, Ant Lion, Dragonfly, Crow, Whale, Butterfly, Grasshopper, and Salp swarms [45,46,47,48,49,50,51,52,53,54,55]. Additional species metaphors continue to appear (e.g., Chicken/Bird swarms, Virus Colony, Shark Smell, Squirrel, Mouth-Brooding Fish, Selfish Herd) [56,57,58,59,60,61,62].

Physics-inspired heuristics map search to thermodynamic, dynamical, or cosmological processes: microcanonical and simulated annealing [63,64]; diffusion/particle interactions and variable neighborhoods [65,66,67]; Harmony Search [68]; gravitational, central-force, and black-hole families [69,70,71,72]; water and river dynamics and intelligent water drops [73,74,75]; charged, electromagnetic, spiral, and ray trajectories [76,77,78,79]; and models of curved or galactic spaces and atmospheric or mine-blast phenomena [80,81,82,83].

Socio-cognitive and human-inspired methods emulate learning, competition, and organization in human systems. Examples include society and civilization models, human-inspired optimization, league championships, social-emotional interaction, brainstorming, teaching-learning, anarchic societies, and sports leagues [84,85,86,87,88,89,90,91]. Big Bang–Big Crunch [92] and photosynthesis-inspired search [93] further illustrate the breadth of analogies that have shaped the development of metaheuristics.

Overall, the reviewed literature shows that improvements in metaheuristics often come not from inventing entirely new low-level operators, but from integrating well-understood mechanisms into an existing search backbone in a way that resolves identifiable failure modes. In the case of POA, the most relevant open question is therefore whether classical operators such as crossover, mutation, and elitism can be coupled with POA’s search dynamics in a way that measurably improves diversity retention, convergence stability, and performance on constrained engineering problems. This gap motivated the EPOA formulation developed in the next section.

3. Proposed Method

This section first summarizes the baseline Puzzle Optimization Algorithm (POA), then explains the design rationale behind the enhancement, and finally presents the mathematical formulation, pseudocode, and workflow of the proposed Enhanced Puzzle Optimization Algorithm (EPOA).

3.1. Baseline Puzzle Optimization Algorithm

POA is a population-based, game-inspired metaheuristic that models each candidate solution as a “puzzle”, where the decision variables are the puzzle pieces and the objective function evaluates how well the puzzle is assembled. A notable feature of POA is that it is parameter-free in the sense that it does not rely on a large set of algorithm-specific control parameters. The search proceeds in two stages per iteration: (i) guided imitation, where a solution adjusts its pieces by following a randomly chosen guiding member with acceptance conditioned on fitness improvement, and (ii) piece suggestion, where a solution replaces a subset of its pieces with pieces suggested by other population members.

Although POA has shown competitive performance on several benchmark suites and practical optimization problems, it may still suffer from premature convergence and limited population diversity in highly multimodal or high-dimensional landscapes [94,95]. Moreover, because the basic formulation does not include an explicit elitism mechanism, a high-quality solution may be lost after a stochastic update [96]. These limitations motivated the development of EPOA.

3.2. Design Rationale for EPOA

The proposed change was motivated by two feasible deficiencies. To begin with, there are a few agents in hard multimodal landscapes that could be drawn to common areas of the search space, and then diversity in the population is reduced, and the search can get stuck in a good but globally optimum basin. This tendency is evident in the later reported results: the means of the errors generated in baseline POA are 62.836 on F₁ and 2370.962 on F₆, and the EPOA decreases the numbers to 0.004 and 7.239, respectively. The scale of these variations is in line with premature convergence and lack of diversity in the baseline approach.

Second, limited engineering problems typically have small regions of feasibility. In the design of welded beams or pressure vessels, such as one a candidate may have an attractive objective value and then become infeasible following a stochastic perturbation that breaks stress, deflection, or thickness. And even in cases where there is no explicit elitism mechanism, the best feasible solution identified to date may not make it through such updates. It was due to this reason that the suggested EPOA assumed crossover, mutation, and elitism into POA such that new candidate solutions would be produced while the best-so-far solution would be maintained.

3.3. Mathematical Formulation of EPOA

EPOA was developed as an enhancement of POA. The original POA already balances exploration and exploitation through two complementary phases, but it may still become trapped in local optima when it is applied to highly nonlinear or constrained problems. EPOA therefore preserved the basic POA structure while adding mechanisms that increase diversity and strengthen global search.

Specifically, EPOA retained the two POA phases: a global exploration step, in which each agent updates its position relative to a designated “food” agent, and a local exploitation step, in which agents refine their positions through smaller perturbations. To reduce the risk of premature convergence, EPOA then incorporated genetic operators that generate new candidate solutions and broaden the sampling of the search space over time. In addition, an explicit elitism step was added so that the best solution discovered so far could not be overwritten by weaker offspring.

The reason for these classical operators is unique to POA. A large population of agents can be directionally concentrated around the identical food-directed movement pattern in a baseline POA, reducing the effective search directions of the agents in the population. Uniform crossover combats this effect by a process of recombining the coordinates of different promising agents and hence generating candidate points not restricted to an individual food-centered path. Random-resetting mutation plays a complementary role: when the shrinking exploitation step becomes too local, a reset of one coordinate can reopen a blocked search direction and help the algorithm jump from a poor basin into a more promising one. Elitism makes these more aggressive variations practical, because the best-so-far feasible solution is copied forward even if several offspring deteriorate. In that sense, the improvement in EPOA should not be interpreted as a “secret” in the operators themselves, but as a consequence of coupling those common operators with POA’s exploration/exploitation backbone in a way that directly addresses POA’s observed failure modes.

Exploration phase. In each iteration t, a random agent, referred to as the “food”, is chosen from the population. All other agents update their positions in relation to this food according to

$${\mathbf{X}}_i^{(t+1)}=\left\{\begin{array}{cc}{\mathbf{X}}_i^{\left(t\right)}+\alpha \left({\mathbf{X}}_{\mathrm{food}}^{\left(t\right)}-\beta {\mathbf{X}}_i^{\left(t\right)}\right),\hfill & \mathrm{if}f\left({\mathbf{X}}_i^{\left(t\right)}\right)>f\left({\mathbf{X}}_{\mathrm{food}}^{\left(t\right)}\right),\hfill \\ {\mathbf{X}}_i^{\left(t\right)}+\alpha \left({\mathbf{X}}_i^{\left(t\right)}-{\mathbf{X}}_{\mathrm{food}}^{\left(t\right)}\right),\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(1)

In Equation (1), ${\mathbf{X}}_i^{\left(t\right)}$ denotes the position of the i-th agent at iteration t, ${\mathbf{X}}_{\mathrm{food}}^{\left(t\right)}$ is the randomly selected food agent, $\alpha$ scales the global movement, and $\beta$ is a small factor (often 1 or 2). Agents with worse fitness move toward the food agent, whereas better agents move away to probe new regions.

Exploitation phase. After exploration, each agent refines its position locally by applying a smaller perturbation:

$${\mathbf{X}}_i^{(t+1)}={\mathbf{X}}_i^{\left(t\right)}+\gamma \left(1-{\displaystyle \frac{t}{\mathrm{MaxIt}}}\right)\left(2\mathbf{r}-1\right)\odot {\mathbf{X}}_i^{\left(t\right)},$$

(2)

where $\gamma$ controls the perturbation intensity, $\mathbf{r}$ is a random vector in ${[0,1]}^D$, ⊙ denotes elementwise multiplication, and MaxIt is the maximum number of iterations. The factor $\left(1-t/\mathrm{MaxIt}\right)$ decreases over time, enabling progressively finer local refinement.

Genetic operators. To enrich the population and avoid local stagnation, uniform crossover and random mutation were incorporated after the POA-specific phases. If ${\mathbf{P}}_1$ and ${\mathbf{P}}_2$ denote two parent solutions, the offspring ${\mathbf{C}}_1$ and ${\mathbf{C}}_2$ are generated as

$${\mathbf{C}}_1\left[j\right]=\left\{\begin{array}{cc}{\mathbf{P}}_2\left[j\right],\hfill & \mathrm{if}{\rho }_j<0.5,\hfill \\ {\mathbf{P}}_1\left[j\right],\hfill & \mathrm{otherwise},\hfill \end{array}\right.{\mathbf{C}}_2\left[j\right]=\left\{\begin{array}{cc}{\mathbf{P}}_1\left[j\right],\hfill & \mathrm{if}{\rho }_j<0.5,\hfill \\ {\mathbf{P}}_2\left[j\right],\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

(3)

where ${\rho }_j\sim \mathrm{Uniform}(0,1)$ for each index j. Offspring then undergo mutation. For example, a randomly selected coordinate k in ${\mathbf{C}}_1$ is updated as

$${\mathbf{C}}_1\left[k\right]=L_k+\eta (U_k-L_k),$$

(4)

where $\eta \sim \mathrm{Uniform}(0,1)$ and $[L_k,U_k]$ is the feasible range of the k-th variable. Every newly generated or updated position is clipped to remain in $[\mathbf{L},\mathbf{U}]$.

The combined use of exploration (1), exploitation (2), crossover (3), mutation (4), and elitism provided EPOA with a stronger ability to maintain diversity, intensify local refinement, and preserve the best solution found so far, the pseudocode is shown in Algorithm 1.

Algorithm 1 Pseudocode of the Enhanced Puzzle Optimization Algorithm (EPOA)

1: Input: Population size N, maximum iterations MaxIt, bounds $\mathbf{L}$ and $\mathbf{U}$, dimension D, objective function $f\left(\mathbf{x}\right)$, crossover probability $P_c$, mutation probability $P_m$.  2: Randomly initialize the population $\{{\mathbf{X}}_1,\dots ,{\mathbf{X}}_N\}$ in $[\mathbf{L},\mathbf{U}]$.  3: for $i=1$ to N do  4:      Evaluate $f\left({\mathbf{X}}_i\right)$.  5: end for  6: Determine the global best solution ${\mathbf{X}}_{\mathrm{best}}$ and its fitness $f_{\mathrm{best}}={min}_{i}f\left({\mathbf{X}}_i\right)$.  7: for $t=1$ to MaxIt do  8:      Randomly select an agent ${\mathbf{X}}_{\mathrm{food}}$ as the food position.  9:      for $i=1$ to N do 10:           Update ${\mathbf{X}}_i$ using the exploration rule in Equation (1). 11:           Apply bound handling and keep the new position only if it improves fitness. 12:      end for 13:      for $i=1$ to N do 14:           Update ${\mathbf{X}}_i$ using the exploitation rule in Equation (2). 15:           Apply bound handling and keep the new position only if it improves fitness. 16:      end for 17:      Shuffle the population and form parent pairs. 18:      for each pair do 19:           Apply crossover with probability $P_c$. 20:           Apply mutation with probability $P_m$. 21:           Evaluate offspring and replace parents if the offspring are better. 22:      end for 23:      Reinsert the best-so-far solution (elitism). 24:      Update ${\mathbf{X}}_{\mathrm{best}}$ and $f_{\mathrm{best}}$ if a better solution is found. 25: end for 26: Output: $\text{Best\_score}\leftarrow f_{\mathrm{best}}$ and $\text{Best\_pos}\leftarrow {\mathbf{X}}_{\mathrm{best}}$.

3.4. Algorithmic Workflow and Flowchart

Figure 2 shows the EPOA flowchart in a more detailed manner compared with the earlier manuscript version. The flowchart clearly indicates that there is a flow of the initializations, evaluation, two POA phases, bound handling, greedy replacement, genetic operators, elitism, and the iteration/termination logic.

3.5. Search Dynamics in EPOA

The two POA phases still provide the main search dynamics of EPOA. The exploration operator in Equation (1) promotes longer moves relative to the selected food agent, which helps the population cover distant regions of the search space. The exploitation operator in Equation (2) scales the perturbation amplitude by $\left(1-t/\mathrm{MaxIt}\right)$, so local refinements naturally become more conservative near the end of the run.

What changed in EPOA is the feedback around these two phases. After the POA-based movement, crossover and mutation reintroduce structural diversity, while elitism prevents regression by retaining the best-so-far solution. This interaction is significant: the POA steps lead to search direction, the genetic operators to the diversification of the search and the stabilization of the search is achieved by elitism. All these elements justify why EPOA is able to spend time in the initial iterations exploring widely and still converging reliably in the late iterations without having to repeat the exploration equation in the second form.

4. Experimental Design

4.1. CEC2022 Benchmark Suite

The single-objective benchmark suite of the CEC2022 utilized in this paper includes 12 functions, which represent basic, hybrid, and composition landscapes [97]. Functions F₁–F₅ are shifted and rotated basic functions, F₆–F₈ represents hybrid functions, and F₉–F₁₂ represents composition functions. This combination present optimizers with unimodal, multimodal, hybrid, and composition search spaces and hence gives a general test of convergence behavior, search robustness, and adaptability.

4.2. Comparison Protocol and Fairness Considerations

The benchmark study must be understood as a common-protocol comparison, and not as an assertion that each competing algorithm has been fine-tuned to its absolute optimum configuration on each of the individual functions [98]. All the methods were tested on the same definition of CEC2022 problems, and the same summary statistics is reported in the manuscript. In the case of algorithms with control parameters, the comparison was done using the parameterizations used in the original literature studies or standard settings, instead of an independent per-function hyperparameter search per baseline. This explanation forms the scope of the comparison and does not exaggerate the degree of parameter tuning.

4.3. Compared Algorithms and Evaluation Metrics

The study compared EPOA against a broad set of recent optimization algorithms.

Table 1. Compared optimization algorithms.

Algorithm	Abbr.	Year	Inspiration/Class	Brief Idea	Ref.
Genetic Algorithm	GA	1975	Evolutionary	Selection, crossover, mutation, and fitness-based survival drive search	[14]
Particle Swarm Optimization	PSO	1995	Swarm intelligence	Particles update velocities using personal and global best information	[31]
Henry Gas Solubility Optimization	HGSO	2019	Physics-based (Henry’s law)	Gas solubility equilibrium guides exploration–exploitation	[99]
Sea-Horse Optimizer	SHO	2023	Bio-inspired (marine behavior)	Seahorse social/foraging behavior drives search	[100]
Sine Cosine Algorithm	SCA	2016	Math-based	Sine/cosine position updates balance exploration and exploitation	[101]
Horned Lizard Optimization Algorithm	HLOA	2024	Bio-inspired (anti-predator tactics)	Horned lizard behaviors model avoidance and target pursuit	[102]
Butterfly Optimization Algorithm	BOA	2019	Bio-inspired (foraging)	Fragrance-based attraction/dispersion controls movement	[103]
Multi-Trial Vector-Based Differential Evolution	MTDE	2020	Evolutionary (DE variant)	Multiple trial vectors per target enhance DE’s search	[104]
Four Vector Intelligent Optimizer	FVIM	2024	Math-based (vector-guided)	Four directional vectors coordinate global–local moves	[2]
Moth-Flame Optimization	MFO	2015	Bio-inspired (navigation)	Moth spiral motion around flames steers exploitation	[48]
Whale Optimization Algorithm	WOA	2016	Bio-inspired (hunting)	Bubble-net encircling and spiral attack mechanisms	[51]
Multi-Verse Optimizer	MVO	2016	Physics/cosmology-inspired	White/black holes and wormhole tunneling for selection	[105]
Equilibrium Optimizer	EO	2020	Physics-based (mass balance)	Equilibrium pool and concentration updates guide search	[106]
Artificial Hummingbird Algorithm	AHA	2022	Bio-inspired (foraging)	Flight/fueling strategies emulate hummingbird nectar search	[107]
Reptile Search Algorithm	RSA	2022	Bio-inspired (predation)	Encircling/ambush behaviors emulate reptile hunting	[108]
Dragonfly Algorithm	DA	2016	Bio-inspired (swarm)	Alignment–cohesion–separation rules with prey attraction	[49]
Dandelion Optimizer	DO	2022	Bio-inspired (dispersal)	Seed dispersal and wind drift promote exploration	[109]
Geometric Mean Optimizer	GMO	2023	Math-based	Operators built around the geometric mean for updating	[110]
Grey Wolf Optimizer	GWO	2014	Bio-inspired (pack hunting)	Leadership hierarchy and encircling prey	[47]
Ant Lion Optimizer	ALO	2015	Bio-inspired (predation)	Random walks and trap building mimic antlion hunting	[55]
Arithmetic Optimization Algorithm	AOA	2021	Math-based	Arithmetic operators (add/sub/mul/div) govern moves	[111]
White Shark Optimizer	WSO	2022	Bio-inspired (predation)	Apex predator pursuit and exploitation dynamics	[112]
Golden Jackal Optimization	GJO	2022	Bio-inspired (cooperative hunting)	Pair and pack hunting strategies coordinate updates	[113]
Red-Tailed Hawk Algorithm	RTH	2023	Bio-inspired (raptor attack)	Soaring, scouting, and dive attack patterns	[114]
Mountain Gazelle Optimizer	MGO	2022	Bio-inspired (herd/escape)	Herding and escape maneuvers maintain diversity	[115]
Flow Direction Algorithm	FDA	2021	Physics/Hydrology-inspired	Flow-direction fields steer solutions downhill/uphill	[116]
Giant Trevally Optimizer	GTO	2022	Bio-inspired (ambush)	Schooling/ambush tactics of giant trevally fish	[117]
Grasshopper Optimization Algorithm	GOA	2017	Bio-inspired (swarm)	Attraction–repulsion with social interaction terms	[53]

lists the compared methods together with their abbreviations, inspiration classes, and references. To position EPOA relative to widely used classical optimizers, Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) were also included through a separate contextual CEC2022 comparison in Section 5.2. For the benchmark tables reported in this study, the summary statistics include the mean, error, standard deviation, and rank.

4.4. Engineering Design Evaluation

Beyond the CEC2022 benchmark suite, EPOA was also evaluated on six classical constrained engineering design problems in order to examine practical performance under nonlinear feasibility restrictions. These cases were the welded beam, tension/compression spring, speed reducer, pressure vessel, three-bar truss, and cantilever beam problems. Their problem-specific descriptions and quantitative results are reported in Section 5.5.

5. Results

5.1. Main Results on CEC2022

Table 2. Comparison results over CEC2022 part 1.

Function	Statistics	EPOA	POA	GWO	ROA	SMA	GBO	FLA	RSA	SHO
F1	Mean	300.002	456.882	694.710	7072.734	36,773.533	12,169.269	1047.869	7474.758	3966.171
	Error	0.004	62.836	602.955	2097.590	11,644.288	4730.090	936.248	1125.154	2773.308
	Std	0.002	156.882	394.710	6772.734	36,473.533	11,869.269	747.869	7174.758	3666.171
	Rank	1	5	6	18	27	23	7	19	15
F2	Mean	406.809	422.344	424.552	661.665	506.962	458.456	420.220	1318.078	442.806
	Error	3.821	29.406	19.202	160.371	52.097	42.003	31.934	958.160	33.704
	Std	6.809	22.344	24.552	261.665	106.962	58.456	20.220	918.078	42.806
	Rank	1	6	8	21	19	16	5	24	12
F3	Mean	600.003	619.456	601.397	640.247	636.883	630.799	600.230	644.868	612.178
	Error	0.004	7.503	1.456	13.634	21.036	11.118	0.044	5.499	2.879
	Std	0.003	19.456	1.397	40.247	36.883	30.799	0.230	44.868	12.178
	Rank	1	12	4	18	17	15	3	22	6
F4	Mean	812.138	821.300	817.012	846.635	850.243	853.630	828.532	848.350	825.986
	Error	6.068	3.588	8.274	14.693	5.006	12.064	13.179	5.342	5.354
	Std	12.138	21.300	17.012	46.635	50.243	53.630	28.532	48.350	25.986
	Rank	1	4	2	20	24	25	8	21	7
F5	Mean	900.454	1074.874	913.666	1338.405	2219.851	1366.908	1006.594	1553.335	1077.129
	Error	0.450	116.742	18.349	166.234	379.333	115.894	80.826	195.202	116.478
	Std	0.454	174.874	13.666	438.405	1319.851	466.908	106.594	653.335	177.129
	Rank	1	11	4	17	27	18	9	24	12
F6	Mean	1807.481	3459.706	6974.726	439,707.967	46,208.047	1,433,932.346	4121.856	80,543,277.999	3884.579
	Error	7.239	2370.962	2690.943	622,856.593	30,375.356	1,036,737.289	1301.366	25,298,593.595	2140.241
	Std	7.481	1659.706	5174.726	437,907.967	44,408.047	1,432,132.346	2321.856	80,541,477.999	2084.579
	Rank	1	6	15	19	18	20	10	25	8
F7	Mean	2004.416	2027.786	2027.714	2112.478	2047.912	2112.391	2017.182	2155.713	2038.572
	Error	8.762	3.628	5.774	37.395	12.913	45.070	8.519	48.257	11.961
	Std	4.416	27.786	27.714	112.478	47.912	112.391	17.182	155.713	38.572
	Rank	1	5	4	22	11	21	3	27	7
F8	Mean	2215.626	2223.174	2222.881	2239.683	2256.307	2241.467	2221.109	2256.762	2224.521
	Error	8.479	1.107	10.639	19.104	55.281	3.322	0.386	19.611	2.537
	Std	15.626	23.174	22.881	39.683	56.307	41.467	21.109	56.762	24.521
	Rank	1	5	4	17	20	18	3	21	7
F9	Mean	2529.284	2529.523	2562.003	2706.310	2623.554	2608.399	2533.724	2736.844	2588.019
	Error	0.000	0.397	38.354	24.019	45.077	36.420	5.348	45.103	26.028
	Std	229.284	229.523	262.003	406.310	323.554	308.399	233.724	436.844	288.019
	Rank	1	4	9	22	18	16	5	24	14
F10	Mean	2500.376	2545.652	2544.571	2551.087	2596.819	2626.910	2549.176	2630.949	2549.812
	Error	0.041	61.354	60.550	75.905	84.244	69.366	66.516	160.654	65.723
	Std	100.376	145.652	144.571	151.087	196.819	226.910	149.176	230.949	149.812
	Rank	1	10	9	16	20	22	13	23	15
F11	Mean	2630.093	2714.601	2804.409	3215.240	3002.326	2905.480	2694.500	3255.158	2903.304
	Error	67.291	63.103	164.181	505.580	209.447	272.056	138.576	256.376	234.282
	Std	30.093	114.601	204.409	615.240	402.326	305.480	94.500	655.158	303.304
	Rank	1	7	13	22	19	17	5	24	16
F12	Mean	2863.468	2862.915	2866.247	2926.937	2902.116	2875.886	2864.790	2999.695	2893.475
	Error	1.520	2.933	4.600	42.776	40.022	8.437	1.462	90.910	23.123
	Std	163.468	162.915	166.247	226.937	202.116	175.886	164.790	299.695	193.475
	Rank	3	1	6	20	18	12	4	24	16

,

Table 3. Comparison results over CEC2022 part 2.

Function	Statistics	FLO	FOX	SCA	RIME	SCSO	DOA	TSO	AO	SSOA
F1	Mean	8170.391	2956.075	1614.750	300.546	2472.746	2822.042	5709.491	1850.017	11,060.046
	Error	1670.501	3405.452	685.390	0.396	1458.216	3144.511	5376.758	877.387	1913.289
	Std	7870.391	2656.075	1314.750	0.546	2172.746	2522.042	5409.491	1550.017	10,760.046
	Rank	20	13	8	2	11	12	17	10	22
F2	Mean	1707.979	428.080	481.222	408.567	450.121	482.071	423.327	419.824	1289.714
	Error	873.297	37.210	6.884	0.783	27.066	92.089	32.114	19.243	464.505
	Std	1307.979	28.080	81.222	8.567	50.121	82.071	23.327	19.824	889.714
	Rank	25	10	17	2	14	18	7	4	23
F3	Mean	652.959	655.950	618.410	600.127	612.408	626.841	644.161	616.362	657.760
	Error	9.650	6.621	2.541	0.046	14.780	7.299	14.102	2.426	6.039
	Std	52.959	55.950	18.410	0.127	12.408	26.841	44.161	16.362	57.760
	Rank	24	25	10	2	7	13	21	9	26
F4	Mean	845.471	828.854	845.584	829.461	828.977	824.990	848.492	824.816	863.310
	Error	7.373	9.698	4.224	8.746	7.668	17.808	12.796	9.021	6.996
	Std	45.471	28.854	45.584	29.461	28.977	24.990	48.492	24.816	63.310
	Rank	18	9	19	11	10	6	22	5	27
F5	Mean	1464.708	1507.550	996.803	900.770	993.050	1122.962	1216.991	985.952	1693.860
	Error	274.828	109.835	36.143	1.283	36.694	109.250	139.998	66.625	264.650
	Std	564.708	607.550	96.803	0.770	93.050	222.962	316.991	85.952	793.860
	Rank	22	23	8	2	7	13	14	6	25
F6	Mean	19,706,744.924	2292.826	3,378,343.339	2717.828	5416.774	2211.786	4632.724	9642.914	154,133,536.142
	Error	21,224,374.955	648.820	3,907,529.105	860.726	3040.543	761.909	1998.965	4770.662	143,152,473.171
	Std	19,704,944.924	492.826	3,376,543.339	917.828	3616.774	411.786	2832.724	7842.914	154,131,736.142
	Rank	23	3	22	4	14	2	13	16	26
F7	Mean	2097.324	2136.301	2058.482	2017.061	2051.517	2062.340	2089.368	2039.575	2132.961
	Error	32.884	43.113	4.196	8.288	32.654	51.247	60.505	10.882	30.031
	Std	97.324	136.301	58.482	17.061	51.517	62.340	89.368	39.575	132.961
	Rank	20	25	14	2	12	15	17	9	23
F8	Mean	2237.363	2428.873	2233.108	2220.959	2225.940	2225.963	2232.611	2231.843	2332.172
	Error	8.288	193.900	3.655	0.443	3.112	11.886	7.421	6.477	82.496
	Std	37.363	228.873	33.108	20.959	25.940	25.963	32.611	31.843	132.172
	Rank	16	27	14	2	8	9	13	11	25
F9	Mean	2735.540	2580.773	2583.431	2529.285	2586.200	2557.890	2561.015	2566.081	2792.114
	Error	34.225	20.919	22.424	0.000	41.619	60.136	70.165	24.305	54.281
	Std	435.540	280.773	283.431	229.285	286.200	257.890	261.015	266.081	492.114
	Rank	23	11	12	2	13	6	8	10	26
F10	Mean	2615.252	2539.164	2531.330	2548.651	2500.919	2559.714	2580.650	2549.535	2636.320
	Error	107.757	84.883	64.879	65.930	0.790	63.130	73.814	66.596	94.731
	Std	215.252	139.164	131.330	148.651	100.919	159.714	180.650	149.535	236.320
	Rank	21	8	7	11	2	17	19	14	24
F11	Mean	3292.855	2750.684	2784.678	2631.058	2725.890	2819.974	2873.354	2681.061	3223.648
	Error	318.721	106.081	11.489	66.759	134.303	184.961	197.199	64.722	210.883
	Std	692.855	150.684	184.678	31.058	125.890	219.974	273.354	81.061	623.648
	Rank	25	10	11	2	8	14	15	3	23
F12	Mean	2995.161	2948.844	2871.121	2863.357	2867.364	2875.152	2883.171	2867.238	3119.348
	Error	38.716	50.548	1.413	1.607	7.554	5.971	23.966	1.398	68.032
	Std	295.161	248.844	171.121	163.357	167.364	175.152	183.171	167.238	419.348
	Rank	23	21	10	2	8	11	13	7	26

and

Table 4. Comparison results over CEC2022 part 3.

Function	Statistics	GJO	HLOA	WOA	BOA	SHIO	OHO	AOA	HGSO	AVOA
F1	Mean	1707.302	301.832	23,526.990	8778.329	3262.291	15,254.459	12,320.878	5032.466	315.907
	Error	1267.871	2.182	12,891.493	1283.198	2608.867	6100.263	4843.207	1981.574	13.738
	Std	1407.302	1.832	23,226.990	8478.329	2962.291	14,954.459	12,020.878	4732.466	15.907
	Rank	9	3	26	21	14	25	24	16	4
F2	Mean	453.693	408.920	448.791	1803.492	425.309	2451.766	1246.062	508.673	430.848
	Error	16.007	0.006	31.816	852.260	29.060	799.223	533.760	19.527	36.180
	Std	53.693	8.920	48.791	1403.492	25.309	2051.766	846.062	108.673	30.848
	Rank	15	3	13	26	9	27	22	20	11
F3	Mean	613.404	643.002	630.247	640.879	603.533	663.482	646.685	631.299	618.576
	Error	9.341	17.424	8.360	7.969	2.753	4.090	10.502	5.650	10.717
	Std	13.404	43.002	30.247	40.879	3.533	63.482	46.685	31.299	18.576
	Rank	8	20	14	19	5	27	23	16	11
F4	Mean	835.479	831.242	840.942	849.473	818.913	853.751	830.899	838.496	833.509
	Error	8.095	8.634	14.980	8.973	7.785	5.433	10.300	3.526	11.324
	Std	35.479	31.242	40.942	49.473	18.913	53.751	30.899	38.496	33.509
	Rank	15	13	17	23	3	26	12	16	14
F5	Mean	927.241	1396.390	1444.078	1251.170	913.211	1733.258	1313.899	1047.064	1380.416
	Error	31.814	181.371	217.076	85.654	26.733	91.857	122.468	22.608	120.116
	Std	27.241	496.390	544.078	351.170	13.211	833.258	413.899	147.064	480.416
	Rank	5	20	21	15	3	26	16	10	19
F6	Mean	12,794.797	3365.492	4340.455	39,613,711.328	3945.650	1,122,784,054.173	4510.580	2,418,857.903	3617.885
	Error	5578.562	2664.991	2139.378	47,147,891.821	2637.977	433,350,248.065	1510.089	1,467,188.924	2481.498
	Std	10,994.797	1565.492	2540.455	39,611,911.328	2145.650	1,122,782,254.173	2710.580	2,417,057.903	1817.885
	Rank	17	5	11	24	9	27	12	21	7
F7	Mean	2038.989	2133.855	2057.704	2093.078	2047.303	2144.577	2096.789	2075.145	2036.157
	Error	16.559	38.936	26.301	12.391	27.423	27.599	28.764	11.212	12.572
	Std	38.989	133.855	57.704	93.078	47.303	144.577	96.789	75.145	36.157
	Rank	8	24	13	18	10	26	19	16	6
F8	Mean	2227.302	2280.754	2231.933	2303.480	2249.565	2356.776	2314.441	2233.823	2223.984
	Error	2.375	62.670	7.642	101.647	54.278	94.140	97.642	4.326	7.048
	Std	27.302	80.754	31.933	103.480	49.565	156.776	114.441	33.823	23.984
	Rank	10	22	12	23	19	26	24	15	6
F9	Mean	2589.784	2558.671	2650.555	2751.208	2610.548	2845.611	2704.691	2649.866	2529.286
	Error	36.010	65.710	57.284	23.700	37.480	81.133	45.807	28.261	0.003
	Std	289.784	258.671	350.555	451.208	310.548	545.611	404.691	349.866	229.286
	Rank	15	7	20	25	17	27	21	19	3
F10	Mean	2549.021	2858.596	2530.633	2502.549	2521.696	2917.466	2665.152	2561.688	2528.059
	Error	66.367	560.635	66.477	0.813	47.639	199.619	119.455	75.974	61.147
	Std	149.021	458.596	130.633	102.549	121.696	517.466	265.152	161.688	128.059
	Rank	12	26	6	3	4	27	25	18	5
F11	Mean	2749.515	2690.467	2972.053	3210.399	3056.688	4349.899	3611.770	2794.816	2710.087
	Error	13.777	134.137	222.447	295.765	310.480	337.397	299.148	16.523	174.666
	Std	149.515	90.467	372.053	610.399	456.688	1749.899	1011.770	194.816	110.087
	Rank	9	4	18	21	20	27	26	12	6
F12	Mean	2870.260	2892.288	2888.284	2957.390	2905.389	3200.141	3005.036	2895.506	2865.521
	Error	4.424	28.954	26.022	64.724	29.548	94.622	59.349	6.192	2.275
	Std	170.260	192.288	188.284	257.390	205.389	500.141	305.036	195.506	165.521
	Rank	9	15	14	22	19	27	25	17	5

summarize the quantitative results obtained on the CEC2022 functions. EPOA achieved rank 1 on F₁–F₁₁ and rank 3 on F₁₂. Relative to baseline POA, the largest improvements were observed on functions such as F₁ and F₆, where the mean error was reduced substantially.

5.2. Contextual Comparison with Classical GA and PSO Baselines

To position EPOA relative to widely used classical optimizers,

Table 5. Contextual comparison with classical GA and PSO baselines on the 10-dimensional CEC2022 functions. Lower mean values are better.

Function	EPOA	POA	PSO	GA
F1	300.002	456.882	$3.004\times {10}^2$	$3.29\times {10}^4$
F2	406.809	422.344	$4.025\times {10}^2$	$5.07\times {10}^2$
F3	600.003	619.456	$6.034\times {10}^2$	$6.51\times {10}^2$
F4	812.138	821.300	$8.194\times {10}^2$	$8.55\times {10}^2$
F5	900.454	1074.874	$9.00\times {10}^2$	$1.03\times {10}^3$
F6	1807.481	3459.706	$3.069\times {10}^3$	$8.70\times {10}^3$
F7	2004.416	2027.786	$2.028\times {10}^3$	$2.09\times {10}^3$
F8	2215.626	2223.174	$2.223\times {10}^3$	$2.26\times {10}^3$
F9	2529.284	2529.523	$2.486\times {10}^3$	$2.70\times {10}^3$
F10	2500.376	2545.652	$2.555\times {10}^3$	$2.64\times {10}^3$
F11	2630.093	2714.601	$2.672\times {10}^3$	$3.14\times {10}^3$
F12	2863.468	2862.915	$2.857\times {10}^3$	$3.01\times {10}^3$
Average rank	1.42	2.83	1.83	3.92
Wins (rank $=1$)	8	0	4	0
Top-2 count	11	3	10	0

Note: The GA baseline was taken from [118] and the PSO baseline from [119]. Because those values come from independently published implementations on the 10-dimensional CEC2022 suite, they are reported as a contextual comparison rather than merged into the main same-study benchmark tables.

reports a contextual comparison of EPOA and baseline POA against representative GA and PSO results from published 10-dimensional CEC2022 studies. These values were not generated inside the present code base and are therefore shown separately from Table 2, Table 3 and Table 4; nevertheless, they provide a useful classical reference point for interpreting the proposed enhancement.

Table 5 shows that EPOA retained the best overall average rank and the highest number of function-wise wins among the four methods. PSO was competitive particularly inF₂, F₅, F₉, and F₁₂ and GA was continuously weaker on this benchmark. This is significant to compare them: since GA already utilizes crossover and mutation as fundamental mechanisms, the acquisition of EPOA cannot be ascribed to these operators alone. Instead, this improvement in performance seems to be due to the incorporation of these classical variation operators into the search dynamics of POA using food as guidance and stabilizing the resultant search using elitism.

5.3. Convergence Analysis

The average convergence of EPOA over F₁–F₁₂ is reported in Figure 3, FirstOptimizer AvgConvergence. Comparative convergence curves of the optimizers being tested on the same functions are plotted in Figure 4. Overall, there is quick initial improvement and a level effect in late stages in most functions in EPOA.

5.4. Robustness Analysis via Boxplots

The boxplots of the compared optimizers over F₁–F₁₂ are presented in Figure 5. The number recaps the median, interquartile range, and outliers of each optimizer, and thus gives an active perspective of the strength and dispersion of recurring executions.

5.5. Results on Engineering Design Problems

In this subsection, EPOA was evaluated on six classical constrained engineering design problems. The presentation below reports the problem setting, the design variables, and the statistics listed in the corresponding tables. Runtime is discussed only for the tension/compression spring case, because computation time is explicitly reported only in

Table 6. Optimization results for the Welded beam design problem.

Optimizer	Min	Mean	Max	Std	Best_Score	X1	X2	X3	X4
EPOA	1.729963	1.790268	1.887251	0.045333	1.729963	0.204801	3.497639	9.057125	0.205642
ROA	1.794014	3.842021	7.242874	1.735509	1.794014	0.214953	3.29096	9.08834	0.215075
SMA	1.755108	2.288533	3.677828	0.484321	1.755108	0.196857	3.74533	9.027943	0.206914
GBO	1.835159	2.684118	4.844368	0.71897	1.835159	0.17506	4.30554	9.365214	0.204831
FLA	2.183717	3.101547	4.949606	0.785252	2.183717	0.320175	2.553082	7.103688	0.334902
SCA	1.816348	1.933675	2.076164	0.060474	1.816348	0.205103	3.704041	8.938069	0.215977
FOX	1.865269	2.232396	3.152754	0.331473	1.865269	0.189925	4.067233	8.573929	0.228538
FLO	2.413134	3.883307	7.0137	1.077117	2.413134	0.199937	7.680737	8.449632	0.235316

.

5.5.1. Welded Beam Design Problem

The welded beam design problem, widely used in constrained engineering optimization and reported by Coello [120], seeks the least expensive welded connection between a vertical support and a horizontal beam, as shown in Figure 6. The design variables are the weld throat size, the weld length, the beam-web thickness, and the beam height. The objective is to select these dimensions so that the material and manufacturing cost of the joint is minimized while satisfying the problem constraints. In addition to geometric bounds on each variable, the design must satisfy stress-related constraints: the shear stress in the weld and the bending stress in the beam must not exceed allowable values, the vertical deflection of the beam must remain within an acceptable limit, and the design must avoid buckling of the welded bar. These constraints ensure that the connection is both safe and stiff while remaining inexpensive.

The results of the performance of various optimization algorithms in solving the welded beam design problem are summarized in Table 6. EPOA has the highest results and the smallest minimum fitness (1.729963), mean fitness value (1.790268) of the compared methods. SMA is also competitive and has decent performance. In comparison, FLO is the most objective giving, which means that the quality of solutions to this problem is worse.

5.5.2. Tension/Compression Spring Design Problem

This problem aims to minimize the weight of a tension/compression spring (TCS), shown in Figure 7, subject to standard geometric and stress constraints. The design variables are the wire diameter $d\left(x_1\right)$, mean coil diameter $D\left(x_2\right)$, and number of active coils $N\left(x_3\right)$.

Table 7. Optimization results for the tension/compression spring design problem.

Optimizer	Min	Mean	Max	Std	Time	Rank	Best_Score	Dim	X1	X2	X3
EPOA	4,000,000	4,000,000	4,000,000	0.000705	9.329241	4	4,000,000	3	0.051469	0.350941	11.73782
ROA	4,000,000	4,000,000	4,000,000	0.00136	34.24322	7	4,000,000	3	0.054606	0.430902	8.039721
SMA	4,000,000	4,000,000	4,000,000	0.002038	0.702198	2	4,000,000	3	0.051	0.33973	12.40638
GBO	4,000,000	4,000,000	4,000,000	0.003399	0.615262	6	4,000,000	3	0.051	0.338868	12.5387
FLA	4,000,000	4,000,000	4,000,000	0.001461	8.606973	8	4,000,000	3	0.051	0.337728	12.6979
SCA	4,000,000	4,000,000	4,000,000	0.000202	8.235249	3	4,000,000	3	0.051	0.339495	12.44138
FOX	4,000,000	4,000,000	4,000,000	0.001284	8.893275	1	4,000,000	3	0.051	0.340215	12.33267
FLO	4,000,000	4,659,271	5,907,972	753,275	17.07674	5	4,000,000	3	0.054394	0.425268	8.170715

presents the results of various optimization algorithms for the tension/compression spring design problem. All algorithms except FLO achieve the same minimum objective value, indicating that the optimum is reached consistently across most methods. In addition, the table reports the computation time for this case. EPOA attains the target solution with a runtime of 9.329241 s, while SMA and GBO are faster. FLO again exhibits the weakest robustness, with a substantially larger mean and maximum fitness value.

5.5.3. Speed Reducer Design Problem

This problem seeks to minimize the weight of a speed reducer subject to numerous constraints [121]. A schematic of the speed reducer is shown in Figure 8. The variables $x_1$ to $x_7$ represent the principal geometric dimensions of the reducer. Because the problem includes several coupled design variables and nonlinear constraints, it is considered challenging for many optimization algorithms.

$$0.7\le x_2\le 0.8$$

(5)

$$17\le x_3\le 28$$

(6)

$$7.3\le x_4\le 8.3$$

(7)

$$7.3\le x_5\le 8.3$$

(8)

$$2.9\le x_6\le 3.9$$

(9)

$$5.0\le x_7\le 5.6$$

(10)

Table 8. Optimization results for the speed reducer design problem.

Optimizer	Min	Mean	Max	Std	Best_Score	X1	X2	X3	X4	X5	X6	X7
EPOA	3001.377	3009.469	3020.542	6.021796	3001.377	3.503791	0.700206	17.00064	7.441767	7.730021	3.361127	5.28666
ROA	3059.622	564,875	1,112,806	534,435.8	3059.622	3.546564	0.7	17	7.3	7.936809	3.350275	5.351832
SMA	3004.506	3026.801	3102.77	21.44836	3004.506	3.500337	0.700021	17	7.556039	7.904454	3.353513	5.290662
GBO	3000.954	3062.249	3154.282	38.90149	3000.954	3.500001	0.7	17	7.428327	7.736012	3.363289	5.289083
FLA	3021.132	3498.344	4880.238	468.7076	3021.132	3.529494	0.7	17	7.3	7.775103	3.376688	5.297608
SCA	3059.538	3141.727	3225.092	45.64241	3059.538	3.582584	0.7	17	7.3	8.3	3.400927	5.297153
FOX	2998.601	3044.992	3479.469	106.1178	2998.601	3.500281	0.7	17.00056	7.50607	7.722803	3.354966	5.287796
FLO	4712.278	1,018,921	1,099,671	193,440.6	4712.278	3.569144	0.713717	24.39535	7.586502	7.984192	3.365742	5.286026

presents the results of the compared optimizers for the speed reducer design problem. FOX performs best, with the lowest minimum fitness value (2998.601) and the lowest mean fitness value (3044.992). EPOA is highly competitive, achieving a minimum value of 3001.377 and a mean value of 3009.469 with a small standard deviation. By contrast, FLO performs poorly, with substantially larger mean and maximum fitness values.

5.5.4. Pressure Vessel Design Problem

The pressure vessel design problem shown in Figure 9 involves minimizing the total fabrication cost of a cylindrical pressure vessel. The decision variables are the shell thickness, head thickness, inner radius, and length of the cylindrical section, represented by $x_1$, $x_2$, $x_3$, and $x_4$, respectively. The shell and head thicknesses are fixed to multiple sizes of available plates, but the inner radius and the length are continuous. The goal is a combination of the material and welding cost. Use of constraints guarantees that the thicknesses are adequate to withstand the internal pressure, that the vessel volume is adequate to accommodate the storage need, and that the total length is not more than a given maximum.

The findings of the pressure vessel design problem using various optimization algorithms are given in

Table 9. Optimization results for the pressure vessel design problem.

Optimizer	Min	Mean	Max	Std	Best_Score	X1	X2	X3	X4
EPOA	5894.705	6268.949	6863.476	283.9167	5894.705	12.52892	6.194416	40.57199	196.5434
ROA	6793.112	9796.239	14181.82	2145.013	6793.112	16.93783	9.043746	54.75769	64.65558
SMA	5976.057	6978.33	11439.9	1256.706	5976.057	12.93051	6.453479	41.74479	181.4563
GBO	6373.913	7992.479	11639.25	1263.803	6373.913	14.02068	6.76105	44.21846	159.3755
FLA	6236.567	9170.934	16733.65	3001.2	6236.567	14.77887	7.406824	47.74829	117.7936
SCA	6348.207	7444.238	8636.877	633.1883	6348.207	12.93176	6.908451	40.92991	200
FOX	5905.472	10247.2	28985.57	6708.422	5905.472	12.6351	6.24535	40.91543	191.8689
FLO	7887.666	13522.91	18895.83	2926.167	7887.666	20.75721	10.51484	62.6062	21.77952

. The performance of EPOA is the highest, and the minimum fitness (5894.705) and the mean fitness (6268.949) are the lowest values. FOX comes right behind when it comes to the best value, yet its standard deviation is significantly higher, which means that its behavior is not as consistent. ROA, SMA, GBO, and SCA are competitive, but do not correspond to the general consistency of EPOA. The solution quality is the lowest in FLO compared with other methods.

5.5.5. Three-Bar Truss Design Problem

In this structural optimization problem, the goal is to minimize the weight (or equivalently the volume) of a three-member truss while meeting stress requirements. The truss has three bars arranged in a triangular configuration, as shown in Figure 10, with the cross-sectional areas of two unique members as decision variables: $A_1=A_3=x_1$ and $A_2=x_2$. The objective is proportional to the total member length, so choosing smaller areas reduces weight. Constraints require that axial stresses in each bar do not exceed an allowable level, and bounds on $x_1$ and $x_2$ keep the areas within feasible limits.

Table 10. Optimization results for the three-bar truss design problem.

Optimizer	Min	Mean	Max	Std	Best_Score	X1	X2
EPOA	6,000,264	6,000,264	6,000,264	0.00827	6,000,264	0.7881	0.409878
ROA	6,000,264	6,000,266	6,000,271	2.795681	6,000,264	0.791167	0.401246
SMA	6,000,264	6,000,271	6,000,291	6.185368	6,000,264	0.784041	0.421826
GBO	6,000,264	6,000,264	6,000,265	0.287699	6,000,264	0.789199	0.406823
FLA	6,000,264	6,000,266	6,000,278	3.139559	6,000,264	0.794192	0.393109
SCA	6,000,264	6,000,269	6,000,283	8.4262	6,000,264	0.788955	0.407649
FOX	6,000,264	6,000,264	6,000,264	0.002923	6,000,264	0.788859	0.407729
FLO	6,000,264	6,000,265	6,000,271	1.745493	6,000,264	0.787707	0.411307

presents the results of the compared optimization algorithms for the three-bar truss design problem. FOX achieves the best performance, with the lowest standard deviation (0.002923) while attaining the same minimum fitness value as EPOA and several other methods. EPOA also performs strongly, reaching the same minimum value with a very small standard deviation (0.00827). By contrast, SMA and FLA show larger variability, indicating less stable performance across runs.

5.5.6. Cantilever Beam Design Problem

The problem in the cantilever beam, as in Figure 11, reduces the weight of a beam with constant wall thickness and different heights and is made of five hollow sections nested together. The number of heights of the segment is a decision variable, and the goal is to minimize the material use and maintain the structural integrity. The beam should be able to carry a tip load, but the allowable bending stress and deflection levels must not be surpassed. Extra side constraints impose geometric ratios between successive segments to make it stable. It is a problem to determine the combination of the five heights that gives the lightest beam, which still meets all mechanical and geometrical criteria.

Table 11. Optimization results for the cantilever beam design problem.

Optimizer	Min	Mean	Max	Std	Best_Score	X1	X2	X3	X4	X5
EPOA	8,000,001	8,000,001	8,000,001	0.000236	8,000,001	5.996026	5.290975	4.515814	3.508951	2.162565
ROA	8,000,001	8,000,001	8,000,002	0.062824	8,000,001	6.111266	5.490347	4.27695	3.406179	2.249534
SMA	8,000,001	8,000,001	8,000,001	0.012975	8,000,001	6.023155	5.444688	4.429739	3.499314	2.124597
GBO	8,000,001	8,000,001	8,000,001	0.020085	8,000,001	6.024015	5.367952	4.638712	3.337791	2.153299
FLA	8,000,001	8,000,001	8,000,002	0.076507	8,000,001	5.649543	5.154543	4.771313	3.611915	2.467864
SCA	8,000,001	8,000,001	8,000,001	0.019143	8,000,001	5.838884	5.740438	4.688059	3.225127	2.138083
FOX	8,000,001	8,000,001	8,000,001	3.33E-05	8,000,001	6.022235	5.309476	4.491685	3.504452	2.145924
FLO	8,000,001	8,000,001	8,000,002	0.064764	8,000,001	5.120259	5.750428	4.751561	3.963126	2.535949

presents the results of various optimization algorithms for the cantilever beam design problem. FOX performs best, achieving the lowest minimum fitness value (8000001) with the smallest standard deviation (3.33E-05). EPOA follows closely, reaching the same minimum value with a similarly competitive mean fitness value. SMA and GBO also produce competitive results, whereas ROA, FLA, and FLO exhibit larger variability.

6. Discussion

6.1. Interpreting the CEC2022 Results

The benchmark outcomes reflect the fact that the suggested changes were not cosmetic. Compared with baseline POA, EPOA yielded significantly smaller errors on a few challenging functions, with F₁ and F₆ being among them. This trend confirms the design logic presented in Section 3: the additional crossover and mutation processes assisted the search to avoid population collapse, whereas elitism ensured the maintenance of good candidate solutions following stochastic updates.

The findings also indicate that EPOA was especially effective in functions where search-space coverage as well as late-stage refinement were significant. It was not necessarily the first method, but F12 was not a dominant one. This finding is practical as it demonstrates that the suggested improvement increased performance significantly without suggesting an overall excellence across all landscapes.

6.2. Interpreting the Convergence and Robustness Analyses

The convergence curves are additional evidence for the last summary statistics. EPOA fell very fast in the early stages of numerous functions, then leveled off, which is expected to be the case in the presence of an efficient balance between exploration and exploitation. The comparison curves also demonstrate that some of the competing methods either progressed more slowly or stagnated at a lower fitness.

This picture is supplemented by the boxplots that demonstrate the distribution of results at repeated runs. EPOA has a lower median and a relatively small interquartile range on a number of functions, which is evidence of not only excellent average performance but also excellent repeatability. One can see wider spreads of certain competing methods show sensitivity to both initialization and search dynamics. In the more difficult cases, it is also evident from the boxplots that there is variability; hence, the discussion cannot be based on best-case results alone, but on the central tendency as well as the dispersion.

6.3. Interpreting the Engineering Design Results

The more detailed pattern is revealed in the engineering-design experiments, compared with the benchmark suite. On the welded-beam and pressure-vessel problems, EPOA was most competitive, and on the speed-reducer, three-bar truss, and cantilever-beam ones, it was very competitive. This is a positive finding since these issues can be characterized in terms of nonlinear goals and constraints on feasibility, and this is exactly where diversity maintenance and elitism come into play.

Simultaneously, the engineering outcomes reveal that EPOA failed to be the most competent technique in all design issues. On the speed reducer, FOX scored the highest minimum value, and in the cases of the three-bar truss and cantilever beam, FOX scored the same or slightly higher than EPOA. This result must be viewed as a positive one, as opposed to being a secret: it indicates that EPOA is strong when faced with extremely dissimilar constrained problems, yet that performance remains conditional on the geometry and the structure of the task constraints. Lastly, the only tension/compression spring table reports runtime, meaning the discussion of computational time is restricted to that case and does not speculate on how things would work otherwise.

6.4. Limitations and Future Directions

The current research is also limited. Their crossover and mutation rates and the chosen constraint-handling strategy can still make EPOA’s performance dependent. Future research needs to explore adaptive operator control, specific rules for feasibility or repair, multi-objective extensions, and ablation studies that measure the contribution of each improvement separately.

7. Conclusions

EPOA, as proposed in this paper, is a specific improvement of POA by use of uniform crossover, random-resetting mutation, and explicit elitism. The approach enhanced diversity conservation and convergence security without losing the original search logic by integrating these classical operators within POA’s exploration–exploitation backbone.

The experimental findings indicated that EPOA provided the dominant performance in the CEC2022 suite, was competitive with classic GA and PSO references, and generated robust feasible solutions of six constrained engineering design problems. These results, combined with the previous ones, are an indication that EPOA is a valid and strong extension of POA to complex continuous optimization and engineering design.