Experimental results are presented in two parts in this section. Firstly, different benchmark functions, as well as engineering problems, are investigated to assess the effects of improvement on GOA. Secondly, an evaluation is conducted of IGOA-based feature selection.

For the benchmark functions and engineering problem sections, we have compared Improved Gannet Optimization Algorithm (IGOA) with the Gannet Optimization Algorithm (GOA), Whale Optimization Algorithm (WOA) [41], Grey Wolf Optimizer (GWO) [42] Particle Swarm Optimization (PSO) [43], and Transient Search Optimization (TSO) [44]. These algorithms were selected to demonstrate the performance of IGOA in solving complex optimization tasks and to analyze the improvements made over the original GOA in various benchmark functions and engineering design problems.

In contrast, for the feature selection task, we utilized all the methods listed, namely GOA, Elephant Herding Optimization (EHO) [45], Snake Optimizer (SO) [46], Geometric Octal Zones Distance Estimation (GOZDE) [47], WOA, TSO, GWO, Dandelion Optimizer (DO) [48], Genetic Algorithm (GA) [49], Ant Colony Optimization (ACO) [50], Differential Evolution (DE) [51], and PSO. These algorithms were employed to assess IGOA’s effectiveness in selecting optimal feature subsets from various datasets, comparing the results on several classification metrics.

A brief introduction to these methods is provided below:

Gannet Optimization Algorithm (GOA): GOA simulates gannet foraging behavior, enabling effective exploration and exploitation. As a result of its sudden turns and random walks, the algorithm’s U-shaped and V-shaped diving patterns enable it to identify better solutions in the search space.

Elephant Herding Optimization (EHO): The EHO metaheuristic is based on the social behavior of elephant herds. When male elephants reach maturity, they leave their family to join another clan led by their matriarch. As a result, EHO emulates these behaviors by using a clan updating operator and a separating operator.

Snake Optimizer (SO): It was inspired by snake mating behavior. In a low-temperature environment, snakes compete for the best partner when food is abundant. To create an efficient optimization method, the SO algorithm mathematically mimics the foraging and reproductive behaviors of snakes.

Geometric Octal Zones Distance Estimation (GOZDE): GOZDE facilitates information sharing across the search space by using median values and the distance between geometric zones. Each zone represents a different search strategy that the algorithm divides the search space into.

Whale Optimization Algorithm (WOA): Humpback whales use bubble-nets to hunt. That’s the inspiration behind WOA’s hunting strategy. It simulates the search, encirclement, and bubble net foraging processes that occur during whale hunting.

Transient Search Optimization (TSO): Transient switching operating on electrical circuits with inductances and capacitances as storage elements is the basis of TSO.

Grey Wolf Optimizer (GWO): Grey wolves are characterized by hierarchical leadership and hunting strategies. A wolf pack hierarchy is simulated by the algorithm, with alpha, beta, delta, and omega wolves representing the different levels of wolves in the pack, and hunting is broken down into searching for prey, encircling it, and attacking it.

Dandelion Optimizer (DO): DO simulates dandelion seeds dispersed by wind over long distances. The optimization process is divided into three stages: the rising stage, when seeds are carried upward; the descending stage, when seeds adjust their direction while falling; and the landing stage, where seeds randomly settle and grow. The trajectory of seeds during descent and landing is described by Brownian motion and Levy random walks.

Genetic Algorithm (GA): GA repeatedly modifies an individual solution population. Next generation offspring are generated by selecting parents from the existing population.

Particle Swarm Optimization (PSO): It mimics the social behavior of birds flocking or fish schooling. In order to converge on the best solution, each particle adjusts its position in the search space according to its own experience and the experience of its neighbors.

Ant Colony Optimization (ACO): This simulation simulates the behavior of ants when they forage for food by depositing pheromones to guide others towards it. Positive feedback and path reinforcement lead to the discovery of optimal solutions through the accumulation of stronger pheromone trails over time.

Differential Evolution (DE): It is based on the concept of population evolution. New solutions are generated through mutation and crossover, and the fitness of each solution is evaluated to determine which solution should be evolved further, ensuring both exploration and exploitation in the search for optimal solutions.

It has been shown that classical algorithms are still useful in many contemporary applications in recent studies. For instance, Wan et al. [52] introduced a modified binary coded ACO for feature selection, and Ye et al. [53] demonstrated the advantages of integrating improved ACO variants with hybrid methodologies. Classical algorithms are just as significant when confronting complex challenges as modern approaches when addressing high-dimensional problems, as demonstrated in these examples. Therefore, we have compared both classical and modern algorithms.

Table 6 presents the common settings used for each of these algorithms.

Table 6 Parameter configurations for all methods usedComparisons of IGOA and GOA Based on Benchmark Functions

The presented algorithm is compared to its original version by examining a set of 20 benchmark test functions [54], including popular unimodal and multimodal functions. Sect. "Comparisons of IGOA and GOA Based on Benchmark Functions" describes the experimental settings and benchmark functions in detail.

Parameters Setting

During the comparative experiments, both algorithms were utilized under the same conditions to achieve fairness. To remove random effects from the results and to determine how stable IGOA is compared to GOA, each method was running 30 times for each benchmark function and averaged. There is a maximum number of function evaluations (MNFE) set to D × 1000 as the termination condition for all of them. Also, the population number is 30. MATLAB software (R2018a) and Windows 10 operating system are used for the implementation of all algorithms.

Table 7 shows the characteristics of each function used in this paper. M indicates a multimodal function, whereas U indicates a unimodal function. An abbreviation for separable functions is S. A non-separable function is denoted by N.

Table 7 Overview of testing function detailsPerformance Comparisons

After the maximum number of iterations are reached, both GOA and IGOA stop trying to find the optimal global solution. Therefore, the computational loads of these algorithms are not very different. In this case, a small number of iterations is enough to find the optimum solution. As shown in Fig. 14, GOA and IGOA have different abilities.

Fig. 14The alternative text for this image may have been generated using AI.The alternative text for this image may have been generated using AI.The alternative text for this image may have been generated using AI.

Convergence curve comparison

According to the red curves, the IGOA algorithm shows a significantly faster convergence curve than the original GOA algorithm. The performance of IGOA on Dixon-Price, Sphere, Step, and Trid10 was higher than that of GOA on these functions. As shown in Fig. 14, IGOA consistently outperforms GOA in most unimodal test functions. In its initial phase, IGOA exhibits strong exploratory capabilities with moderate exploitation, beginning with extensive exploration, and maintaining high levels of exploitation as iterations progress. According to Booth, IGOA’s curve initially shows rapid progress, followed by small adjustments around iterations 200 to 300, indicating that the approach is being fine-tuned near promising areas. In contrast, GOA is more erratic, with periods of stagnation. In addition, IGOA’s enhanced exploration mechanism, which scans the solution space thoroughly, allows it to achieve faster descent rates on multimodal functions such as Rosenbrock and Rastrigin. In simpler problems, PSO converges rapidly, while in high-dimensional ones, GA requires careful tuning of crossover and mutation rates to prevent premature convergence, WOA performs well on Rastrigin but sometimes spirals around suboptimal areas, and EHO’s clan-based updates can take longer to converge. As a result of IGOA’s dual memory matrix, the resulting convergence rates are superior or at least comparable to those of the other benchmarks.

For benchmark functions using fixed-dimensional functions, Fig. 15 illustrates IGOA exploration–exploitation behavior. An effective metaheuristic algorithm requires two factors: exploration and exploitation. Exploration is to search a broad area and exploitation is to focus the search on a smaller area. According to Hussain et al. [55], search mechanisms should be adjusted according to exploration and exploitation. The dimension-wise diversity measurement can be calculated using Eq. (35). A dimension becomes explored when its average distance increases. Meanwhile, a decreased distance indicates that the search space is being exploited.

$$Di_=\frac\sum_^median\left(^\right)-_^$$

(34)

$$Di^=\frac\sum_^Di_$$

(35)

where \(median\left(^\right)\) is the median of the j-th dimension for N individuals, and \(_^\) is the j-th dimension for the i-th individual in the population. \(Di^\) is the average diversity of the t-th iteration’s population, while \(Di_\) is the average diversity of the j-th dimension. In terms of exploration and exploitation, the percentage is calculated as follows:

Fig. 15The alternative text for this image may have been generated using AI.The alternative text for this image may have been generated using AI.The alternative text for this image may have been generated using AI.

Exploration and exploitation phases

$$Exploration\%=\frac^}_}}\times 100$$

(36)

$$Exploitation\%=\frac^-Di_}\right|}_}}\times 100,\hspacet=\text,...,_}$$

(37)

A maximum diversity is defined as \(Di_}\) when all iterations are considered.

Figure 15 illustrates IGOA exploration–exploitation behavior for benchmark functions using fixed-dimensional functions. Exploration (searching a large area) and exploitation (refining a smaller area) are both key components of an effective metaheuristic algorithm. For functions such as Rastrigin and Rosenbrock, IGOA maintains higher exploration early on, contrary to GOA. After 200 iterations, IGOA shows stronger exploration, reflected in larger fluctuations in diversity, which is crucial for escaping multiple local optima. The diversity of GOA sometimes drops abruptly, leading to premature convergence. According to Bohachevsky2 and Booth, IGOA converges around iteration 300 to 400, while GOA continues exploring without improving. Results from Easom and Griewank indicate that IGOA’s final broad search step occurs between iterations 400 and 500 before settling, which yields better results despite a slight increase in computational effort. Once the swarm converges on a best solution, PSO may lose diversity, GA relies heavily on mutation for variety conservation, WOA’s spiral updating may not always switch to exploitation smoothly, and EHO’s clans can merge prematurely. Achieving superior performance over GOA, IGOA maintains diversity and avoids stagnation through adaptively balancing these phases.

Figure 16 shows how the trajectory of the first particle changes in the first dimension, showing whether the initial iteration involves sudden movement and how it transitions thereafter. First gannets of IGOA typically make abrupt changes early on, covering more than half the search space for broad exploration, then gradually refine their steps, reflecting a shift from global exploration to local exploitation. The second memory matrix steers the algorithm to the global optimum while maintaining diversity of solutions. It is possible for GOA shifts to be less effective or more erratic without this second matrix. Generally, the first particle of a PSO converges rapidly to the global best, which can pose a risk when dealing with multimodal tasks. GA’s first chromosome may jump dramatically due to crossover, WOA’s spiral path can converge around suboptimal spots, and EHO’s matriarchal updates frequently result in stepwise changes. IGOA avoids premature convergence by preserving large movements initially and tapering them off later.

Fig. 16The alternative text for this image may have been generated using AI.The alternative text for this image may have been generated using AI.The alternative text for this image may have been generated using AI.The alternative text for this image may have been generated using AI.The alternative text for this image may have been generated using AI.The alternative text for this image may have been generated using AI.The alternative text for this image may have been generated using AI.The alternative text for this image may have been generated using AI.The alternative text for this image may have been generated using AI.The alternative text for this image may have been generated using AI.

First particle’s trajectory in the first dimension

Figure 17 presents a visualization that combines two key elements for understanding the methods’ search behavior: sampled points (shown in red) and a color gradient indicating the objective function values across the search space. During earlier iterations, the dark blue color bar represents higher function values, while in later iterations, the dark red color bar represents lower function values.

Fig. 17The alternative text for this image may have been generated using AI.The alternative text for this image may have been generated using AI.The alternative text for this image may have been generated using AI.The alternative text for this image may have been generated using AI.The alternative text for this image may have been generated using AI.The alternative text for this image may have been generated using AI.The alternative text for this image may have been generated using AI.The alternative text for this image may have been generated using AI.The alternative text for this image may have been generated using AI.The alternative text for this image may have been generated using AI.

Ackley function shows a funnel-like landscape where dark blues are concentrated at the global optimum. The sampling points of IGOA show strategic concentration in these darker blue regions, while exploring some lighter areas as well. It is evident that the algorithm is capable of identifying and focusing on promising low-objective-value regions. According to the IGOA algorithm, the "moths" explore promising regions of the search space and focus on exploiting near global optima across all test functions. IGOA’s ability to approximate the global optimization optimum is illustrated by these observations. When testing unimodal functions, the sample points are sparsely distributed in non-promising areas, while when testing multimodal and fixed-dimension multimodal functions, most sample points cluster around promising regions. IGOAs avoid being trapped in local optima by thoroughly exploring the search space. As a result of this distribution of sample points, IGOA demonstrates its ability to exploit the true optimal solution.

Figure 17 shows sampled points (in red) with a gradient of color indicating the objective function values, from dark blue (higher) to dark red (lower). According to Ackley, IGOA’s points cluster around the global optimum, but also appear in other regions, suggesting a balanced exploration–exploitation strategy. Initially, GOA’s points are more scattered, but in some runs they tend to converge prematurely. In multimodal problems such as Ackley and Schwefel, PSO often forms a single dense cluster when a best solution is identified. If mutation and crossover are used aggressively, GA can maintain scattered samples, but final convergence may be slowed. Spiral updates from WOA can produce small clusters that don’t always connect, and EHO shows clan-based clusters merging over time. In contrast, IGOA preserves multiple promising areas simultaneously, as seen on Griewank and Schwefel, offering broader coverage and a higher likelihood of finding the true global minima. Based on the trajectory analysis and diversity measurements, IGOA’s enhanced exploration avoids local traps and exploits the best regions more effectively. Additionally, the exploration–exploitation curves demonstrate that IGOA maintains a higher level of exploration during early iterations (as seen in functions like Rosenbrock and Schawefel) while transitioning smoothly to exploitation, avoiding stagnation. Its ability to consistently achieve better solutions across multiple benchmark functions, in addition to its superior convergence behavior, can be attributed to the synergy between its global exploration capabilities and local exploitation capabilities.

In different benchmark functions, the dual memory matrix mechanism of IGOA shows its effectiveness. The second memory matrix enables IGOA to maintain multiple search fronts simultaneously by utilizing functions like Ackley, and Griewank. This is visible as clusters of blue stars distributed across promising regions while maintaining coverage of the broader search space at the same time. The risk of getting trapped in local optima increases exponentially in high-dimensional spaces. By dynamically adjusting the balance between exploration and exploitation according to the current state of the search, fuzzy weight adaptation complements this search behavior. For instance, consider the Schawefel and Rosenbrock results, where the sample points form dense clusters in promising regions (indicating exploitation driven by the first memory matrix), while maintaining scattered points in unexplored areas (indicating exploration capability maintained by the second matrix).

Figure 14 shows a common qualitative outcome of objective optimization, which shows the most effective solutions over repeated iterations. Figure 18 shows the convergence curve related to average fitness that shows how all solutions approximate the global optimum over time and can be used to better understand their behavior.

Fig. 18The alternative text for this image may have been generated using AI.The alternative text for this image may have been generated using AI.The alternative text for this image may have been generated using AI.The alternative text for this image may have been generated using AI.

Convergence behavior illustrated by average fitness values

Figure 18 illustrates how IGOA balances exploration and exploitation over the course of optimization. In the first 100 iterations, IGOA achieves strong convergence on Sphere and Trid10. In the later stages, GOA lags behind IGOA, especially in higher-dimensional or more complex functions. PSO often shows a sharp drop in average fitness early on but may plateau if the swarm converges too early, GA convergence rates vary widely based on genetic parameters, WOA’s spiral updates can result in moderate improvement before leveling off, and EHO’s clan-based process may take longer to reach similar fitness levels. These results indicate that IGOA can outperform or match well-known metaheuristics on multimodal and unimodal benchmarks because of its second memory matrix. Next, we will review how the robust convergence behavior of IGOA applies to constrained engineering problems, illustrating how it has a number of practical advantages.

Real-World Engineering Problems

A comparison is made between IGOA and GOA results with two engineering problems [56].

Pressure Vessel Design

A pressure vessel design optimization problem is solved using the proposed IGOA and GOA methods. In this problem, the objective is to minimize the cost of a storage tank with \(3\times ^\) psi internal pressure and a minimum volume of 750 ft3. Two hemispheres are welded together and a longitudinal cylinder is formed with another weld. Design variables include X1 thickness for hemisphere, X2 thickness for shell, X3 radius of inner sphere, and X4 length for cylindrical shape (see Fig. 19).

Fig. 19The alternative text for this image may have been generated using AI.

According to Fig. 20 and Table 8, the IGOA performs better than the standard GOA for the pressure vessel design problem based on its convergence analysis and numerical results. IGOA achieved a significantly better optimal objective value of mean fitness equal to 5873.2938, while GOA achieved 5885.3328, representing a 0.21% improvement in solution quality. It can be attributed to IGOA’s modified exploration–exploitation mechanism, which uses two memory matrices (MX and MTX) to enable more effective search space navigation.

Fig. 20The alternative text for this image may have been generated using AI.

Convergence behavior on solving pressure vessel design problem

Table 8 Optimal value achieved for pressure vessel design parameters

As shown in Fig. 20, IGOA not only produces better final results but also exhibits faster convergence characteristics after the 100th iteration. According to this accelerated convergence, IGOA’s dual memory structure allows for more efficient knowledge transfer between search agents, resulting in better exploitation of promising regions. Based on the optimization variables obtained by IGOA (X1 = 0.7782, X2 = 0.4136, X3 = 41.0206, X4 = 196.31762) versus GOA (X1 = 0.7782, X2 = 0.3846, X3 = 40.3196, X4 = 196.23381), it can be concluded that while both algorithms identify the same regions of the search space, IGOA is more accurate in fine-tuning the design parameters. The performance of IGOA is significantly superior to that of GWO (f = 11988.2649), PSO (f = 6051.0194), TSO (f = 6696.7457), and WOA (f = 7243.5401). Using the results, it can be concluded that IGOA’s enhanced search mechanism provides a robust and efficient approach for solving complex engineering optimization problems.

$$f\left(x\right)=0.6224___+1.7781__^+3.1661_^_+19.84_^_$$

(38)

Subject to:

$$_\left(x\right)=0.0193_-_\le 0$$

(39)

$$_\left(x\right)=0.00954_-_\le 0$$

(40)

$$_\left(x\right)=1296000-\pi _^_^-\frac\pi _^\le 0$$

(41)

$$_\left(x\right)=_-240\le 0$$

(42)

Variable range: \(0.0625\le _\), \(_\le 0.61875\), \(10\le _\), \(_\le 200\)

Welded Beam Design

Designing this structure primarily aims to minimize fabrication costs and simultaneously reduce vertical deflection (see Fig. 21). In order to support the designated load F, the beam is welded to a rigid base made of low-carbon steel (C-1010). Design continuous variables are the weld thickness h, the length l of the welded joint, the beam width t, and the beam thickness b. There are different properties associated with steel, cast iron, aluminum, and brass, which make up the bulk of the beam. According to Ravindran et al. [57], the constant parameters are L = 14 in, δmax = 0.25 in, and F = 6,000 lbs. A continuous variable h, l, t, and b are included in the input x, while discrete variables w and m are present.

Fig. 21The alternative text for this image may have been generated using AI.

The IGOA displays better performance improvements than the original GOA and other metaheuristic algorithms based on the quantitative analysis of Table 9 and Fig. 22. As a result of the final objective values achieved, IGOA achieved a mean fitness of 5950000034.8570 vs. GOA’s mean fitness of 5950000034.8574. This represents a small but meaningful improvement in solution quality. There may be a small difference between the two, but it reflects IGOA’s superior ability to fine-tune solutions within highly constrained design spaces.

Table 9 Optimal value achieved for welded beam problemFig. 22The alternative text for this image may have been generated using AI.

Convergence behavior on solving welded beam design problem

The accelerated convergence of IGOA demonstrates its exploration capability. The optimal design variables obtained by IGOA (h = 0.2091, l = 0.2080, t = 8.8946, b = 5.0000) compared to GOA (h = 0.1468, l = 0.2099, t = 7.7148, b = 5.0000) show notable differences in geometric parameters, particularly the weld thickness (h) and beam width (t), indicating IGOA’s ability to better navigate the complex constraint of the problem.

The mean fitness displays a hierarchy of performance when compared to other compared algorithms:

IGOA: 5,950,000,034.8570 (best)

GOA: 5,950,000,034.8574

WOA: 5,950,000,034.8575

PSO: 5,950,000,034.8577

GWO: 5,950,000,034.8730