Engineering Reports
Volume 7, Issue 5 e70048
RESEARCH ARTICLE
Open Access

Random Walk-Based GOOSE Algorithm for Solving Engineering Structural Design Problems

Sripathi Mounika, 

Sripathi Mounika

School of Electronics and Electrical Engineering, Lovely Professional University, Phagwara, Punjab, India

Contribution: Conceptualization, Methodology, Visualization

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Himanshu Sharma, 

Himanshu Sharma

Department of Electrical Engineering, G.H. Raisoni College of Engineering and Management, Nagpur, India

Contribution: Validation, Methodology, ​Investigation

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Aradhala Bala Krishna, 

Aradhala Bala Krishna

Department of Electrical and Electronics Engineering, Potti Sriramulu Chalavadi Mallikarjuna Rao College of Engineering and Technology, Vijayawada, India

Contribution: Conceptualization, Formal analysis, Software, Data curation

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Krishan Arora, 

Krishan Arora

School of Electronics and Electrical Engineering, Lovely Professional University, Phagwara, Punjab, India

Contribution: Validation, Software, Formal analysis

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Syed Immamul Ansarullah, 

Syed Immamul Ansarullah

Department of Management Studies- University of Kashmir, Srinagar, Jammu and Kashmir, India

Contribution: Writing - original draft, ​Investigation, Data curation, Resources

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Ayodeji Olalekan Salau, 

Corresponding Author

Ayodeji Olalekan Salau

Department of Electrical/Electronics and Computer Engineering, Afe Babalola University, Ado-Ekiti, Nigeria

Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai, Tamil Nadu, India

Correspondence: Ayodeji Olalekan Salau ([email protected])

Contribution: Formal analysis, Methodology, Writing - review & editing, Validation

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First published: 30 April 2025
https://doi.org/10.1002/eng2.70048

Funding: The authors received no specific funding for this work.

ABSTRACT

The proposed Random Walk-based Improved GOOSE (IGOOSE) search algorithm is a novel population-based meta-heuristic algorithm inspired by the collective movement patterns of geese and the stochastic nature of random walks. This algorithm includes the inherent balance between exploration and exploitation by integrating random walk behavior with local search strategies. In this paper, the IGOOSE search algorithm has been rigorously tested across 23 benchmark functions where 13 benchmarks are with varying dimensions (10, 30, 50, and 100 dimensions). These benchmarks provide a diverse range of optimization landscapes, enabling comprehensive evaluation of IGOOSE algorithm performance under different problem complexities. The algorithm is tested by various parameters such as convergence speed, magnitude of solution, and robustness for different dimensions. Further, IGOOSE algorithm is applied to optimize eight distinct engineering problems, showcasing its versatility and effectiveness in real-world scenarios. The results of these evaluations highlight IGOOSE algorithm as a competitive optimization tool, offering promising performance across both standard benchmarks and complex structural engineering problems. Its ability to balance exploration and exploitation effectively, combined with its ability to deal with different problems, positions IGOOSE algorithm as a valuable tool.

1 Introduction

Meta-heuristic algorithms (MAs) have undergone rapid development in recent years. They integrate stochastic and local search algorithms and are extensively employed to address global optimization challenges across various domains. Compared to traditional algorithms, MAs excel in handling complex, multimodal, noncontinuous, and nondifferentiable real-world problems due to their incorporation of random elements [1]. For example, MAs can be used to solve problems in clustering [2], shape optimization [3], image processing [4], deep learning [5], machine learning [6], path planning [7], engineering problems [8], and other domains. MAs emulate the behavior of humans or animals and utilize principles from physics and chemistry to formulate mathematical models for optimization algorithms. As a result, MAs are categorized into four types: evolution-based algorithms, human-based algorithms, physics-based algorithms, and swarm-based intelligence algorithms [9]. Evolution-based algorithms improve successive generations by emulating Darwin's principles of natural selection, promoting collective progress within populations and few of them are Barnacles Mating Optimizer [10], Partial reinforcement optimizer [11], Differential Evolution [12], and Cultural evolution algorithm [13]. Human-based algorithms draw inspiration from various social behaviors such as teaching, learning, competition, and cooperation observed in human societies, and a few of them are Driving Training-Based Optimization [14], Human urbanization algorithm [15], Preschool Education Optimization Algorithm [16], Dollmaker Optimization Algorithm [17], and Hiking Optimization Algorithm [18]. Algorithms based on physics derive inspiration from principles and experiments in these scientific disciplines examples of them are Propagation Search Algorithm [19], Prism refraction search [20], Kepler optimization algorithm [21], and Equilibrium optimizer [22]. Swarm-based intelligence algorithms aim to find optimal solutions by simulating group behaviors and leveraging collective intelligence, and the examples are Stochastic Shaking Algorithm [23], Cock-Hen-Chicken Optimizer [24], Elk herd optimizer [25], The Pine Cone Optimization Algorithm [26], and giant pacific octopus optimizer [27]. In Table 1, different algorithms mentioned in the literature demonstrate strong capabilities in global search and effectively handle continuous optimization problems.

TABLE 1. Literature review on metaheuristics methods.
Ref Algorithm name Research gaps
[10] Barnacles Mating Optimizer (BMO) Lack of extensive comparative studies with other evolutionary algorithms
[11] Partial Reinforcement Optimizer (PRO) Limited exploration of reinforcement learning strategies for optimization
[12] Differential Evolution (DE) Facing challenges in handling high-dimensional optimization problems efficiently
[13] Cultural evolution algorithm (CEO) Limited application and validation in diverse optimization domains
[14] Driving Training-Based Optimization (DTBO) Need to validate on scalability and robustness in solving complex optimization problems
[15] Human Urbanization Algorithm (HUA) Lack of comprehensive benchmarking against other urban-inspired algorithms
[16] Preschool Education Optimization Algorithm (PEOA) Limited exploration of educational theories in algorithm design and optimization
[17] Dollmaker Optimization Algorithm (DOA) Lack of studies on the algorithm's adaptability across different problem domains
[18] Hiking Optimization Algorithm (HOA) Need to rigorous testing on large-scale optimization problems and performance comparison
[19] Propagation Search Algorithm (PSA) Facing challenges in optimizing problems with high computational complexity
[20] Prism Refraction Search (PRS) Limited studies on the algorithm's convergence properties and parameter sensitivity
[21] Kepler Optimization Algorithm (KOA) Need to validate real-world engineering and scientific applications
[22] Equilibrium Optimizer (EO) Lack of comprehensive analysis on algorithm convergence and solution quality
[23] Stochastic Shaking Algorithm (SSOA) Limited exploration of the algorithm's performance on various optimization benchmarks
[24] Cock-Hen-Chicken Optimizer (CHCO) Need to do comparative studies with other nature-inspired algorithms in optimization tasks
[25] Elk Herd Optimizer (EHO) Facing challenges in handling dynamic optimization problems and adaptation rates
[26] Pine Cone Optimization Algorithm (PCOA) Lack of studies on algorithm scalability and efficiency in high-dimensional spaces
[27] Giant Pacific Octopus Optimizer (GPO) Limited validation on diverse engineering and real-world applications.
[28] GOOSE algorithm Limited exploration on real-world engineering problems and potential for further performance improvement and comparative analysis with other metaheuristic algorithms
[29] Teaching-Learning-Based Optimization (TLBO) Needs enhanced diversity mechanisms and further validation on more diverse and complex optimization problems
[30] Whale Optimization Algorithm (WOA) Requires improvement in balancing exploration and exploitation, and needs to be tested on more diverse problem sets
[31] Gravitational Search Algorithm (GSA) Issues with slow convergence and premature convergence, and lacks extensive comparative studies with newer algorithms
[32] White Shark Optimizer (WSO) Limited applications to specific types of optimization problems and needs more comparative studies to establish robustness
[33] Gray Wolf Optimizer (GWO) Struggles with maintaining diversity in the population and may require hybridization with other algorithms to improve performance
[34] Marine Predators Algorithm (MPA) Needs further analysis on complex and high-dimensional problems, and comparative studies with other recent algorithms
[35] Tunicate Swarm Algorithm (TSA) Requires further testing on large-scale and real-world problems and optimization of parameter settings
[36] Multi-Verse Optimizer (MVO) Limited handling of multimodal problems and requires enhancements in exploration mechanisms
[37] Ant Lion Optimizer (ALO) Struggle with exploitation in complex multimodal problems and needs to improve its balance between exploration and exploitation, particularly in high-dimensional spaces
[38] Harris Hawks Optimization (HHO) Its convergence speed and solution accuracy can be improved and the performance in dynamic environments has not been thoroughly explored
[39] Sine Cosine Algorithm (SCA) Its convergence to local optima remains an issue and enhancement is required to increase its robustness across a wider range of problems
[40] Moth-Flame Optimization (MFO) Exhibits premature convergence in some scenarios and needed to improve its performance in complex optimization landscapes
[41] Arithmetic Optimization Algorithm (AOA) Its scalability to very high-dimensional problems is not well-studied
[42] Slime Mold Algorithm (SMA) Its performance in highly dynamic or noisy environments is underexplored and needs to fine-tune its parameters for improved convergence efficiency.
[43] Archimedes Optimization Algorithm (AROA) Requires further investigation into its performance on large-scale and complex optimization problems.
[44] Aquila Optimizer (AO) Its long-term convergence behavior and performance on highly multimodal functions require more extensive validation
[45] Henry Gas Solubility Optimizer (HGS) Its application to real-world problems and its scalability need further study
[46] Rabbit optimizer (RO) Its scalability to very high-dimensional problems is not well-studied
[47] Hummingbird algorithm (HA) Its performance showed the premature convergence in certain cases
[48] Cooperative strategy-based differential evolution algorithm (CSDEA) Requires further investigation into its performance on large-scale and complex optimization problems
[49] Quick crisscross sine cosine algorithm (QCSCA) Requires further investigation into its performance on large-scale and complex optimization problems.
[50] Modified Whale optimization algorithm (MWOA) Exhibits premature convergence in some scenarios and needs to improve its performance in complex optimization landscapes

However, they face significant challenges when applied to solving discrete optimization problems. The classical GOOSE algorithm, developed by Hamad and Rashid [28] in 2024, inspired from collective behavior of geese to optimization problems. Inspired by the synchronized movements and effective navigation strategies of geese, classical GOOSE harnesses cooperative search techniques among individual solutions to explore and exploit the solution of space efficiently. Several investigations have demonstrated that the GOOSE method is effective at addressing optimization challenges [28]. However, as real-world problems grow in complexity, the original GOOSE algorithm struggles to effectively address these challenges and deliver satisfactory results. There is a clear need for enhancements to improve its efficacy.

This paper introduces an improved IGOOSE (IGOOSE) algorithm, which uses a random walk strategy to enhance convergence speed and reduce the risk of getting stuck in local optima. In this work, the CEC benchmark test functions are validated, among them 13 benchmarks were validated for different dimensions (10D, 30D, 50D, and 100D), and 8 engineering structural designs [51] are used to verify the superiority of IGOOSE algorithm.

2 Traditional GOOSE Algorithm

A Goose, multiple geese, Geese are waterfowls from the Anatidae family, in size, swans < Geese > ducks. They are sociable and feed on nuts, berries, plants, grass, and seeds. They spend most of their time on land and fly in a “V” shape. When the leading goose becomes tired, another takes over. Goose are loyal, lifelong mates, and protective of their families. They display grieving behavior when losing a mate or eggs and have deep feelings for their group.

The first recorded usage of a goose for protection dates back to ancient Rome. Geese are used to protect police stations and installations in Xinjiang, while in Dumbarton, Scotland, a storehouse for Ballantine's whiskey was guarded by a gander for 53 years until modern cameras were added. Goose population in natural environments have some members serving as guardians who protect them while others forage or rest. They stand on one leg and sometimes carry a small stone to stay awake. When they hear strange sounds or movements, they make loud noises to keep the herd safe. Gooses are highly possessive of their homes, especially during mating and fledging periods. They respond to human observations in a beneficial manner, with their loud, aggressive sound being perfect for guarding the premises. Geese usually stay in groups in their shelters, with one geese assigned to guard. The sound of the rock spreads to other geese, which mimics the state of exploitation under optimization. As a result, it takes time for others to notice the guardian goose's cry. Sound flows at a speed of 343 m per second through the air therefore it requires very less time to get alert. Figure 1 depicts the GOOSE exploitation phase.

Details are in the caption following the image
FIGURE 1
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GOOSE exploitation phase.

2.1 Mathematical Formulation of GOOSE

Initialize the GOOSE population as an X matrix. The goose's location is represented in X, and search agents beyond the search space are returned in the boundary after initializing the population, with fitness determined through standardized benchmark functions in each iteration. The search involves measuring the fitness of each agent within the X grid and comparing it to the fitness of the remaining agents to evaluate BestFitness and Best Position (BestX). BestFitness and BestX are operations that compare the fitness of current row (fitnessi + 1) and return the fitness of the previous row during iterations. In GOOSE algorithm, the exploitation and exploration stages are provided equal weight using a random variable termed “rnd”. A conditional phase divides iterations equally between exploration and exploitation. Variables pro, rnd., and coe (coe less than or equal to 0.17) are introduced to randomly determine the price (between 0 and 1). The pro parameter indicates which formula is valid, while the weight of the stone transported by the geese is found out. Further discussion on exploration and exploitation will follow.

2.2 Exploration Phase

The exploitation phase requires safeguarding groups, as described in Section 2.3. The weight of the stone stored by a goose in its feet, estimated to be 5–25 g, is determined randomly for each iteration (using Equation (1)), indicating the number of iterations.
S _ W it = randi ( | 5 , 25 | , 1 , 1 ) $$ \mathrm{S}\_{\mathrm{W}}_{it}=\mathrm{randi}\left(|5,25|,1,1\right) $$ (1)
In Equation (2), determine the time T _ o _ A _ O it $$ \mathrm{T}\_\mathrm{o}\_\mathrm{A}\_{\mathrm{O}}_{it} $$ required for the stone to reach the earth after falling, varying randomly between 1 and the number of dimensions per iteration.
T _ o _ A _ O it = rand ( 1 , dim ) $$ \mathrm{T}\_\mathrm{o}\_\mathrm{A}\_{\mathrm{O}}_{it}=\operatorname{rand}\left(1,\dim \right) $$ (2)
The time T _ o _ A _ S it $$ \mathrm{T}\_\mathrm{o}\_\mathrm{A}\_{\mathrm{S}}_{it} $$ is determined in Equation (3) when an object hits the ground and a sound is produced and transmitted to a goose in the herd.
T _ o _ A _ S it = rand ( 1 , dim ) $$ \mathrm{T}\_\mathrm{o}\_\mathrm{A}\_{\mathrm{S}}_{it}=\operatorname{rand}\left(1,\dim \right) $$ (3)
Equation (4) calculates the total time ( T _ T $$ \mathrm{T}\_\mathrm{T} $$ ) required for sound propagation and reaching individual goose in flock iterations. Equation (5) refers to calculating the average time ( T _ A $$ \mathrm{T}\_\mathrm{A} $$ ) using Equation (4) dividing by 2.
T _ T = ∑ T _ o _ A it dim $$ \mathrm{T}\_\mathrm{T}=\frac{\sum \left(\mathrm{T}\_\mathrm{o}\_{\mathrm{A}}_{it}\right)}{\dim } $$ (4)
T _ A = T _ T 2 $$ \mathrm{T}\_\mathrm{A}=\frac{\mathrm{T}\_\mathrm{T}}{2} $$ (5)

The distribution of exploitation and exploration phases is determined by a random variable (rnd).

Another random variable pro was selected in the range of [0, 1]. Consider, pro > 0.2 and weight of the stone S _ W it ≥ 12 $$ \mathrm{S}\_{\mathrm{W}}_{\mathrm{it}}\ge 12 $$ . Equation (6) is used to protect and awaken the individual in his group.
F _ F _ S = T _ o _ A _ O it * S _ W it 2 9.81 $$ \mathrm{F}\_\mathrm{F}\_\mathrm{S}=\mathrm{T}\_\mathrm{o}\_\mathrm{A}\_{{\mathrm{O}}_{it}}^{\ast}\frac{\sqrt[2]{\mathrm{S}\_{\mathrm{W}}_{it}}}{9.81} $$ (6)
Equation (7) is used to find the distance that sound travels. ( D _ S _ T it $$ \mathrm{D}\_\mathrm{S}\_{\mathrm{T}}_{it} $$ ), by utilizing sound speed ( S _ S $$ S\_S $$ ) in the air multiplied by the time duration that sounds travels ( T _ o _ A _ S it $$ \mathrm{T}\_\mathrm{o}\_\mathrm{A}\_{\mathrm{S}}_{it} $$ ). S _ S $$ S\_S $$ is 343.2 m/s in the air.
D _ S _ T it = S _ S * T _ o _ A _ S it $$ D\_S\_{T}_{it}=S\_{S}^{\ast }T\_o\_A\_{S}_{it} $$ (7)
The space between a guard geese and another goose ( D _ G it $$ D\_{G}_{it} $$ ) that is resting or feeding is a crucial factor to consider is measuring by using Equation (8).
D _ G it = 0.5 * D _ S _ T it $$ \mathrm{D}\_{\mathrm{G}}_{\mathrm{it}}={0.5}^{\ast}\mathrm{D}\_\mathrm{S}\_{\mathrm{T}}_{it} $$ (8)
To wake up an individual in flocks, we must find a Best X it $$ \mathrm{Best}\ {\mathrm{X}}_{it} $$ presented in Equation (9) by combining the falling object's speed, distance, and average time squared.
X ( it + 1 ) = F _ F _ S + D _ G it * T _ A ^ 2 $$ {X}_{\left( it+1\right)}=\mathrm{F}\_\mathrm{F}\_\mathrm{S}+\mathrm{D}\_{{\mathrm{G}}_{\mathrm{it}}}^{\ast}\mathrm{T}\_{\mathrm{A}}^{\hat{\mkern6mu} 2} $$ (9)
Equation (11) calculates Best X it $$ \mathrm{Best}\ {\mathrm{X}}_{it} $$ using the S _ W it $$ \mathrm{S}\_{\mathrm{W}}_{it} $$ and pro, which are < 12 and ≤ 0.2 $$ <12\ \mathrm{and}\le 0.2 $$ . The speed of a falling object F_F_S (Equation (10)) is determined by dividing the time taken by the S _ W it $$ \mathrm{S}\_{\mathrm{W}}_{it} $$ by gravity. The distance of sound travel and the distance of the goose are also determined by using Equation (7) and Equation (8).
F _ F _ S = T _ o _ A _ O it * S _ W it 9.81 $$ \mathrm{F}\_\mathrm{F}\_\mathrm{S}=\mathrm{T}\_\mathrm{o}\_\mathrm{A}\_{{\mathrm{O}}_{it}}^{\ast}\frac{\mathrm{S}\_{\mathrm{W}}_{it}}{9.81} $$ (10)
X ( it + 1 ) = F _ F _ S * D _ G it * T _ A ^ 2 * Coe $$ {X}_{\left( it+1\right)}=\mathrm{F}\_\mathrm{F}\_{\mathrm{S}}^{\ast}\mathrm{D}\_{{\mathrm{G}}_{\mathrm{it}}}^{\ast}\mathrm{T}\_{\mathrm{A}}^{\hat{\mkern6mu} 2\ast}\mathrm{Coe} $$ (11)

2.2.1 Exploration Phase

The goose wakes randomly to regulate its wake up or protect individuals in the flock. If the goose is not carrying stones, individuals in the flock wake up randomly. When one goose wakes up, they start screaming to protect all others. If the variable rnd. is smaller than 0.5, equations like Equation (3) and (4) are applied to ensure the goose's safety. The minimum time (M_T) is assigned equal to T_T, when the M_T>T_T. The variable alpha ranges from 2 to 0, decreasing with each iteration in the loop. Equation (12) improves the Best X it $$ \mathrm{Best}\ {\mathrm{X}}_{it} $$ search space result.
alpha = 2 − loop Max _ it 2 $$ \mathrm{alpha}=\left(2-\left(\frac{\mathrm{loop}}{\frac{\operatorname{Max}\_\mathrm{it}}{2}}\right)\right) $$ (12)
Here, Max_It depicts the iterations made by the GOOSE. The GOOSE search phase is moved approaching the optimum outcome by calculating the parameters alpha and M_T. Using randn(1, dim), the goose randomly investigates other people in the search space. Both options increase the searchability of GOOSE. Equation (13) multiplies the minimum of time and alpha by a random number before adding it to the best location in the search space.
X ( it + 1 ) randn ( 1 , dim ) * M _ T * alpha + Best _ pos $$ {X}_{\left( it+1\right)}\mathrm{randn}\left(1,\dim \right)\ast \left(\mathrm{M}\_{\mathrm{T}}^{\ast}\mathrm{alpha}\right)+\mathrm{Best}\_\mathrm{pos} $$ (13)

The number of problem dimensions is denoted by dim, while Best_pos represents the optimal position ( Best X it $$ \mathrm{Best}\ {\mathrm{X}}_{it} $$ ) found in the search area. In Figure 2, the GOOSE algorithm pseudo-code is presented.

Details are in the caption following the image
FIGURE 2
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GOOSE algorithm pseudo-code.

2.3 Proposed RW-GOOSE Algorithm

In this research paper, we explore the potential drawbacks of the GOOSE algorithm and propose a hybridization strategy with a random walk approach to address these challenges. The GOOSE algorithm, while effective, can be limited by its tendency to trap in local optima, slow convergence in complex spaces, and sensitivity to initial conditions. By integrating a random walk strategy, we aim to enhance exploration capabilities, diversify search efforts, accelerate convergence, improve robustness against noise, and enhance adaptability across diverse problem domains. This hybrid approach harnesses the deterministic strengths of GOOSE alongside the stochastic nature of random walk, aiming to achieve more efficient and effective optimization outcomes in practical applications.

2.4 Random Walk

The random walk strategy enhances optimization algorithms by introducing randomness, improving exploration capabilities and preventing premature convergence to local optima. It maintains diversity in the search space, making the algorithms more robust and effective, especially in complex optimization problems.

Existing Random Walk-based optimizers are GWO [52], and Binary GWO [53] demonstrate improved exploration and solution accuracy. EO [54] and WOA [55] show enhanced convergence speed and robustness. HHO [56] and ALO [57] benefit from random walks in avoiding local optima and improving global search efficiency. Particle Swarm Optimization (PSO) [58], and other studies, like Fruit fly Algorithm (FA) [59] and Bat Algorithm (BA) [60] highlight the strategy's effectiveness in complex landscapes. DE [61] and AO [62] utilize random walks to improve performance in high-dimensional and constrained optimization problems. Enhanced Salp Swarm Algorithm (ESSA) [63] improves neural network training, and Sparrow Search Optimizer (SSO) [64] enhances optimization. The random walk strategy proves to be a vital component in modern optimization algorithms, providing significant advantages over traditional local search methods across various applications. A random walk [65] is a technique that involves taking successive random steps. In mathematics, expressions are mentioned from Equation (14) to Equation (16).
W N = ∑ i = 1 N s i $$ {W}_N={\sum}_{i=1}^N{s}_i $$ (14)
S i $$ {S}_i $$ is a random step from any random distribution. The connection between any two successive random walks is specified by Equation (15).
W N = ∑ i = 1 N s i = ∑ i = 1 N s i + X N = W N - 1 + S N $$ {W}_N={\sum}_{i=1}^N{s}_i={\sum}_{\mathrm{i}=1}^N{s}_i+{X}_{\mathrm{N}}={W}_{N\hbox{-} 1}+{S}_N $$ (15)
The relationship indicates that the next state W N $$ {W}_N $$ is solely influenced by the current state W N − 1 $$ {W}_{N-1} $$ and the step taken from the current state to the next state. A random walk can be defined for a geese starting at point x 0 $$ {x}_0 $$ and ending at point x N $$ {x}_N $$ , as per Equation (16).
X ( it + 1 ) $$ X\left( it+1\right) $$ (16)

Here, BestX and BestFitness $$ \mathrm{BestX}\ \mathrm{and}\ \mathrm{BestFitness} $$ is a parametric variable that controls the step size s i $$ {s}_i $$ in each iteration.

2.5 Design of Proposed Random Walk-Based GOOSE (IGOOSE) Algorithm

The classical GOOSE algorithm begins with an initial population of geese denoted as X i ( i = 1 , 2 . … n ) $$ {X}_i\left(i=1,2.\dots n\right) $$ . Throughout each iteration, a random walk is applied exclusively to the leading individuals, BestX and BestFitness $$ \mathrm{BestX}\ \mathrm{and}\ \mathrm{BestFitness} $$ , of the population. Here, a parameter vector α i $$ {\alpha}_i $$ linearly decreases from 2 to 0 as iterations advance. This algorithm integrates a random walk where step sizes are drawn from a Cauchy distribution.

The rationale for using a Cauchy distributed random step size lies in its infinite variance, allowing for occasional large jumps. This capability proves highly effective during stagnation periods, facilitating exploration of the search space by the leading geese. This exploration, in turn, helps safeguard and guide the other geese in the population. Importantly, the algorithm does not introduce additional objective function evaluations, ensuring consistency in function evaluation across both versions of the algorithm. The pseudo-code of the proposed algorithm IGOOSE algorithm is described in Figure 3.

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FIGURE 3
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Proposed IGOOSE algorithm pseudo-code.

3 Numerical Experiments

In this section, the performance of the GOOSE algorithm and the proposed IGOOSE algorithm is evaluated using the criteria of IEEE CEC benchmark functions [28]. These benchmarks encompass 23 unconstrained optimization problems of unimodal and multimodal types, with fixed dimensions and varying levels of difficulty. Among the IEEE CEC benchmark functions, 13 benchmarks were validated across different dimensions: 10, 30, 50, and 100. All numerical experiments were conducted using MATLAB 2022a.

3.1 Benchmark Functions

Each test function is executed 30 times across dimensions of 10D, 30D, 50D, and 100D to evaluate the performance of both algorithms. The initial population is uniformly generated using a random number generator. Table 2 presents the standard CEC benchmark functions.

TABLE 2. Standard CEC benchmark functions.
Functions Dimensions Range fmin
F 1 ( S ) = ∑ m ˙ = 1 z S m 2 $$ {\mathrm{F}}_1\left(\mathrm{S}\right)=\sum \limits_{\dot{\mathrm{m}}=1}^{\mathrm{z}}{\mathrm{S}}_{\mathrm{m}}^2 $$ 10D, 30D, 50D, 100D [−100, 100] 0
F 2 ( S ) = ∑ m ˙ = 1 z S m + ∏ m = 1 z S m $$ {\mathrm{F}}_2\left(\mathrm{S}\right)={\sum}_{\dot{\mathrm{m}}=1}^{\mathrm{z}}\left|{\mathrm{S}}_{\mathrm{m}}\right|+{\prod}_{\mathrm{m}=1}^{\mathrm{z}}\left|{\mathrm{S}}_{\mathrm{m}}\right| $$ 10D, 30D, 50D, 100D [−10,10] 0
F 3 ( S ) = ∑ m ˙ = 1 z ∑ n − 1 m S n 2 $$ {\mathrm{F}}_3\left(\mathrm{S}\right)=\sum \limits_{\dot{\mathrm{m}}=1}^{\mathrm{z}}{\left(\sum \limits_{\mathrm{n}-1}^{\mathrm{m}}{\mathrm{S}}_{\mathrm{n}}\right)}^2 $$ 10D, 30D, 50D, 100D [−100, 100] 0
F 4 ( S ) = max m S m , 1 ≤ m ≤ z $$ {\mathrm{F}}_4\left(\mathrm{S}\right)={\max}_{\mathrm{m}}\left\{\left|{\mathrm{S}}_{\mathrm{m}}\right|,1\le \mathrm{m}\le \mathrm{z}\right\} $$ 10D, 30D, 50D, 100D [−100, 100] 0
F 5 ( S ) = ∑ m ˙ = 1 z − 1 100 S m + 1 − S m 2 2 + S m − 1 2 $$ {\mathrm{F}}_5\left(\mathrm{S}\right)={\sum}_{\dot{\mathrm{m}}=1}^{\mathrm{z}-1}\left[100{\left({\mathrm{S}}_{\mathrm{m}+1}-{\mathrm{S}}_{\mathrm{m}}^2\right)}^2+{\left({\mathrm{S}}_{\mathrm{m}}-1\right)}^2\right] $$ 10D, 30D, 50D, 100D [−38, 38] 0
F 6 ( S ) = ∑ m ˙ = 1 z S m + 0.5 2 $$ {\mathrm{F}}_6\left(\mathrm{S}\right)={\sum}_{\dot{\mathrm{m}}=1}^{\mathrm{z}}{\left(\left[{\mathrm{S}}_{\mathrm{m}}+0.5\right]\right)}^2 $$ 10D, 30D, 50D, 100D [−100, 100] 0
F 7 ( S ) = ∑ m ˙ = 1 z mS m 4 + random [ 0 , 1 ] $$ {\mathrm{F}}_7\left(\mathrm{S}\right)={\sum}_{\dot{\mathrm{m}}=1}^{\mathrm{z}}{\mathrm{m}\mathrm{S}}_{\mathrm{m}}^4+\mathrm{random}\left[0,1\right] $$ 10D, 30D, 50D, 100D [−1.28, 1.28] 0
F 8 ( S ) = ∑ m ˙ = 1 z − S m sin √ S m $$ {\mathrm{F}}_8\left(\mathrm{S}\right)=\sum \limits_{\dot{\mathrm{m}}=1}^{\mathrm{z}}-{\mathrm{S}}_{\mathrm{m}}\sin \left(\surd \left|{\mathrm{S}}_{\mathrm{m}}\right|\right) $$ 10D, 30D, 50D, 100D [−500, 500] −418.98295

F 9 ( S ) = ∑ m ˙ = 1 z S m 2 − 10 cos 2 π S m + 10 $$ {\mathrm{F}}_9\left(\mathrm{S}\right)=\sum \limits_{\dot{\mathrm{m}}=1}^{\mathrm{z}}\left[{\mathrm{S}}_{\mathrm{m}}^2-10\cos \left(2\uppi {\mathrm{S}}_{\mathrm{m}}\right)+10\right] $$

10D, 30D, 50D, 100D [−5.12, 5.12] 0
F 10 ( S ) = − 20 exp − 0.2 1 z ∑ m ˙ = 1 z S m 2 − exp 1 z ∑ m ˙ = 1 z cos 2 π S m + 20 + d $$ {\mathrm{F}}_{10}\left(\mathrm{S}\right)=-20\exp \left(-0.2\sqrt{\left(\frac{1}{\mathrm{z}}\sum \limits_{\dot{\mathrm{m}}=1}^{\mathrm{z}}{\mathrm{S}}_{\mathrm{m}}^2\right)}\right)-\exp \left(\frac{1}{\mathrm{z}}\sum \limits_{\dot{\mathrm{m}}=1}^{\mathrm{z}}\cos \left(2\uppi {\mathrm{S}}_{\mathrm{m}}\right)+20+\mathrm{d}\right. $$ 10D, 30D, 50D, 100D [−32, 32] 0
F 11 ( S ) = 1 + ∑ m ˙ = 1 z S m 2 4000 − ∏ m = 1 z cos S m m $$ {\mathrm{F}}_{11}\left(\mathrm{S}\right)=1+\sum \limits_{\dot{\mathrm{m}}=1}^{\mathrm{z}}\frac{{\mathrm{S}}_{\mathrm{m}}^2}{4000}-{\prod}_{\mathrm{m}=1}^{\mathrm{z}}\cos \frac{{\mathrm{S}}_{\mathrm{m}}}{\sqrt{\mathrm{m}}} $$ 10D, 30D, 50D, 100D [−600, 600] 0

F 12 ( S ) = π z 10 sin πτ 1 + ∑ m ˙ = 1 z − 1 τ m − 1 2 1 + 10 sin 2 πτ m + 1 + τ z − 1 2 + ∑ m ˙ = 1 z u S m , 10 , 100 , 4 $$ {\mathrm{F}}_{12}\left(\mathrm{S}\right)=\frac{\uppi}{\mathrm{z}}\left\{10\sin \left({\uppi \uptau}_1\right)+\sum \limits_{\dot{\mathrm{m}}=1}^{\mathrm{z}-1}{\left({\uptau}_{\mathrm{m}}-1\right)}^2\left[1+10{\sin}^2\left({\uppi \uptau}_{\mathrm{m}+1}\right)\right]+{\left({\uptau}_{\mathrm{z}}-1\right)}^2\right\}+\sum \limits_{\dot{\mathrm{m}}=1}^{\mathrm{z}}\mathrm{u}\left({\mathrm{S}}_{\mathrm{m}},10,100,4\right) $$

τ m = 1 + S m + 1 4 $$ {\uptau}_{\mathrm{m}}=1+\frac{{\mathrm{S}}_{\mathrm{m}}+1}{4} $$

u S m , b , x , i = x S m − b i S m > b 0   − b < S m < b x − S m − b i S m < − b $$ \mathrm{u}\left({\mathrm{S}}_{\mathrm{m}},\mathrm{b},\mathrm{x},\mathrm{i}\right)=\left\{\begin{array}{l}\mathrm{x}{\left({\mathrm{S}}_{\mathrm{m}}-\mathrm{b}\right)}^{\mathrm{i}}{\mathrm{S}}_{\mathrm{m}}>b\\ {}0\operatorname{}-\mathrm{b}<{\mathrm{S}}_{\mathrm{m}}<b\\ {}\mathrm{x}{\left(-{\mathrm{S}}_{\mathrm{m}}-\mathrm{b}\right)}^{\mathrm{i}}{\mathrm{S}}_{\mathrm{m}}<-b\end{array}\right. $$

10D, 30D, 50D, 100D [−50, 50] 0
F 13 ( S ) = 0.1 sin 2 3 π S m + ∑ m ˙ = 1 z S m − 1 2 1 + sin 2 3 π S m + 1 + x z − 1 2 1 + sin 2 2 π S z $$ {\mathrm{F}}_{13}\left(\mathrm{S}\right)=0.1\left\{{\sin}^2\left(3\uppi {\mathrm{S}}_{\mathrm{m}}\right)+\sum \limits_{\dot{\mathrm{m}}=1}^{\mathrm{z}}{\left({\mathrm{S}}_{\mathrm{m}}-1\right)}^2\left[1+{\sin}^2\left(3\uppi {\mathrm{S}}_{\mathrm{m}}+1\right)\right]+{\left({\mathrm{x}}_{\mathrm{z}}-1\right)}^2\left[1+{\sin}^22\uppi {\mathrm{S}}_{\mathrm{z}}\right]\right. $$ 10D, 30D, 50D, 100D [−50, 50] 0
F 14 ( S ) = 1 500 + ∑ n ˙ = 1 2 5 1 n + ∑ m ˙ = 1 z S m − b mn 6 − 1 $$ {F}_{14}\left(\mathrm{S}\right)={\left[\frac{1}{500}+{\sum}_{\dot{n}=1}^25\frac{1}{n+{\sum}_{\dot{m}=1}^z\left({S}_m-{b}_{mn}\right)6}\right]}^{-1} $$ 2 [−65.536, 65.536] 1
F 15 ( S ) = ∑ m ˙ = 1 11 b m − S 1 a m 2 + a m S 2 a m 2 + a m S 3 + S 4 2 $$ {F}_{15}\left(\mathrm{S}\right)={\sum}_{\dot{m}=1}^{11}{\left[{b}_m-\frac{{\mathrm{S}}_1\left({\mathrm{a}}_m^2+{a}_m{S}_2\right)}{{\mathrm{a}}_m^2+{a}_m{S}_3+{S}_4}\right]}^2 $$ 4 [−5, 5] 0.00030
F 16 ( S ) = 4 S 1 2 + 2.1 S 1 4 + 1 3 S 1 6 + S 1 S 2 − 4 S 2 2 + 4 S 2 4 $$ {F}_{16}\left(\mathrm{S}\right)=4{\mathrm{S}}_1^2+2.1{\mathrm{S}}_1^4+\frac{1}{3}{\mathrm{S}}_1^6+{\mathrm{S}}_1{\mathrm{S}}_2-4{\mathrm{S}}_2^2+4{\mathrm{S}}_2^4 $$ 2 [−5, 5] −1.0316
F 17 ( S ) = S 2 − 5.1 4 π 2 S 1 2 + 5 π S 1 − 6 2 + 10 1 − 1 8 π cos S 1 + 10 $$ {F}_{17}\left(\mathrm{S}\right)={\left({\mathrm{S}}_2-\frac{5.1}{4\uppi 2}{\mathrm{S}}_1^2+\frac{5}{\pi }{\mathrm{S}}_1-6\right)}^2+10\left(1-\frac{1}{8\pi}\right)\cos {\mathrm{S}}_1+10 $$ 2 [−5, 5] 0.398
F 18 ( S ) = 1 + S 1 + S 2 + 1 2 19 − 14 S 1 + 3 S 2 1 − 14 S 2 + 6 S 1 S 2 + 3 S 2 2 × 30 + 2 S 1 − 3 S 2 2 × 18 − 32 S 1 + 12 S 2 1 + 48 S 2 − 36 S 1 S 2 + 27 S 2 2 $$ {\displaystyle \begin{array}{l}{\mathrm{F}}_{18}\left(\mathrm{S}\right)=\left[1+{\left({\mathrm{S}}_1+{\mathrm{S}}_2+1\right)}^2\left(19-14\ {\mathrm{S}}_1+3{{\mathrm{S}}^2}_1-14\ {\mathrm{S}}_2+6{\mathrm{S}}_1{\mathrm{S}}_2+3\ {{\mathrm{S}}^2}_2\right)\right]\\ {}\times \left[30+{\left(2{\mathrm{S}}_1-3{\mathrm{S}}_2\right)}^2\times \left(18-32{\mathrm{S}}_1+12\ {{\mathrm{S}}^2}_1+48{\mathrm{S}}_2-36{\mathrm{S}}_1{\mathrm{S}}_2+27\ {{\mathrm{S}}^2}_2\right)\right]\end{array}} $$ 2 [−2,2] 3
F 19 ( S ) = − ∑ m ˙ = 1 4 d m exp − ∑ n ˙ = 1 3 S mn S m − q mn 2 $$ {F}_{19}\left(\mathrm{S}\right)=-{\sum}_{\dot{m}=1}^4{d}_m\exp \left(-{\sum}_{\dot{n}=1}^3{\mathrm{S}}_{mn}{\left({\mathrm{S}}_m-{\mathrm{q}}_{mn}\right)}^2\right) $$ 3 [1, 3] −3.32
F 20 ( S ) = − ∑ m ˙ = 1 4 d m exp − ∑ n ˙ = 1 6 S mn S m − q mn 2 $$ {F}_{20}\left(\mathrm{S}\right)=-{\sum}_{\dot{m}=1}^4{d}_m\exp \left(-{\sum}_{\dot{n}=1}^6{\mathrm{S}}_{mn}{\left({\mathrm{S}}_m-{\mathrm{q}}_{mn}\right)}^2\right) $$ 6 [0, 1] −3.32
F 21 ( S ) = − ∑ m ˙ = 1 5 S − b m S − b m T + d m − 1 $$ {F}_{21}\left(\mathrm{S}\right)=-{\sum}_{\dot{m}=1}^5{\left[\left(\mathrm{S}-{b}_m\right){\left(\mathrm{S}-{b}_m\right)}^{\mathrm{T}}+{d}_m\right]}^{-1} $$ 4 [0, 10] −10.1532
F 22 ( S ) = − ∑ m ˙ = 1 7 S − b m S − b m T + d m − 1 $$ {F}_{22}\left(\mathrm{S}\right)=-{\sum}_{\dot{m}=1}^7{\left[\left(\mathrm{S}-{b}_m\right){\left(\mathrm{S}-{b}_m\right)}^{\mathrm{T}}+{d}_m\right]}^{-1} $$ 4 [0, 10] −10.4028
F 23 ( S ) = − ∑ m ˙ = 1 7 S − b m S − b m T + d m − 1 $$ {F}_{23}\left(\mathrm{S}\right)=-{\sum}_{\dot{m}=1}^7{\left[\left(\mathrm{S}-{b}_m\right){\left(\mathrm{S}-{b}_m\right)}^{\mathrm{T}}+{d}_m\right]}^{-1} $$ 4 [0, 10] −10.5363

3.2 Analysis of the Results

All the results represented in this research paper address the format specified by the IEEE CEC [66]. Tables 3, 5, and 6 display the outcomes of applying the GOOSE algorithm and IGOOSE algorithm on IEEE CEC test functions for 10, 50, and 100 dimensions, as detailed in the computational formulas of IEEE CEC functions F1–F13 in Table 2. Table 3 presents the 10-dimensional search outcomes for the proposed IGOOSE and GOOSE algorithms. Table 4 presents the results of 23 IEEE CEC tests conducted for 30 dimensions, also based on the computational formulas in Table 2. These tables report the maximum, minimum, mean, median, and standard deviation of the absolute error values of the objective function for the test functions. Since unimodal problems (F1-F7, suitable for evaluating the exploitation capability of any search algorithm), multimodal problems (F8-F13, the multimodal test problem typically evaluates the search algorithm's exploration strength and local optima avoidance ability), and fixed dimensional problems (F14-F23, suitable for any high-dimensional and hybrid complex problems) are in the IEEE CEC benchmark problem set. From Tables 2, 4–6, it is evident that IGOOSE algorithm outperforms the GOOSE algorithm across all statistical measures. Therefore, in terms of exploiting the search regions around the explored areas, IGOOSE algorithm is superior to the GOOSE algorithm.

TABLE 3. 10 Dimensional search outcomes for the proposed IGOOSE and GOOSE algorithms.
IGOOSE GOOSE
Avg Std Best Worst Median Avg Std Best Worst Median
F1 5.63E-05 4.26E-05 2.53E-05 0.000131 3.85E-05 3.6E-05 2.05E-05 1.26E-05 6.26E-05 2.73E-05
F2 0.015331 0.002782 0.011957 0.018655 0.016053 0.020684 0.009114 0.012942 0.035561 0.01814
F3 97.66608 173.5325 7.49E-05 400.5529 0.000583 242.1433 171.1426 0.000226 404.4416 325.9187
F4 7.881002 13.01657 0.003393 30.00262 0.007121 3.932929 3.820431 0.003633 8.667514 5.140625
F5 35.03801 65.10406 2.142237 151.4315 7.062609 7.77E-05 3.7E-05 4.95E-05 0.000141 6.36E-05
F6 6.41E-05 2.82E-05 4.56E-05 0.000114 5.36E-05 7.77E-05 3.7E-05 4.95E-05 0.000141 6.36E-05
F7 0.00839 0.005595 0.003759 0.014624 0.004994 0.008602 0.004635 0.001936 0.014049 0.008115
F8 −2806.62 385.4139 −3143.59 −2155.05 −2960.51 −2685.29 361.5436 −3084.27 −2334.2 −2531.64
F9 33.43902 11.74125 16.92629 46.76862 33.83199 43.98678 13.78948 24.88661 63.67949 43.78445
F10 6.522736 6.819754 0.020932 18.05964 4.499182 14.82505 8.305398 0.005842 19.49275 18.34157
F11 33.85018 47.44767 1.7E-06 100.1212 0.481951 98.31139 25.13554 63.21026 128.4562 104.0645
F12 8.591365 11.45327 1.507135 28.17418 2.207285 11.86392 14.45159 0.648454 32.10024 3.226769
F13 2.94E-05 5.3E-06 2.13E-05 3.45E-05 2.95E-05 2.52E-05 1.18E-05 8.64E-06 3.92E-05 2.32E-05
TABLE 4. Statistical comparative analysis for proposed IGOOSE algorithm with other existing algorithms (30 dimensional).
Functions Parameters IGOOSE GOOSE [28] TLBO [29] WOA [30] GSA [31] WSO [32] GWO [33] MPA [34] TSA [35] MVO [36]
F1 Mean 25.42282 290.6142 9.04E-75 5.80E-155 1.15E-16 323.1097 4.55E-59 9.29E-50 3.20E-47 0.151165
Best 0.009773 0.007816 3.06E-77 6.00E-167 4.90E-17 72.27965 1.72E-60 7.49E-52 1.71E-49 0.099666
Worst 112.353 1453.02 7.49E-74 6.00E-154 3.20E-16 901.2379 3.48E-58 6.72E-49 2.09E-46 0.247637
Std 49.0109 649.8043 2.06E-74 1.60E-154 6.07E-17 238.7674 8.56E-59 1.88E-49 6.32E-47 0.042235
Median 0.018009 0.01451 8.83E-76 7.00E-161 1.04E-16 282.8347 1.94E-59 2.03E-50 3.66E-48 0.135998
F2 Mean 98.61287 135.2225 1.06E-38 6.40E-104 5.37E-08 3.776208 1.64E-34 1.04E-27 1.90E-28 0.26047
Best 67.17263 56.32044 2.62E-40 1.90E-114 3.07E-08 0.756212 7.38E-36 5.61E-30 1.02E-30 0.179542
Worst 140.0143 195.715 8.43E-38 1.20E-102 9.17E-08 8.958666 6.42E-34 3.43E-27 2.01E-27 0.434829
Std 31.68491 54.90727 1.94E-38 2.70E-103 1.56E-08 2.044082 2.04E-34 1.21E-27 4.90E-28 0.070258
Median 97.9979 156.9493 3.90E-39 3.40E-108 5.17E-08 3.136077 7.85E-35 6.28E-28 2.46E-29 0.24991
F3 Mean 2928.167 6545.999 3.18E-25 22123.03 499.7092 2636.013 2.41E-14 1.33E-11 2.30E-11 14.63423
Best 0.935229 3432.186 1.81E-29 5630.916 182.4369 1088.637 1.58E-19 2.94E-20 8.92E-21 6.012495
Worst 9445.972 11679.73 3.35E-24 45040.6 771.4986 5234.592 3.46E-13 2.04E-10 3.05E-10 26.20651
Std 3874.279 3615.259 8.70E-25 9979.735 148.1048 1107.562 7.92E-14 4.56E-11 6.88E-11 5.595845
Median 1931.562 4964.671 8.95E-27 20656.93 467.5755 2473.189 4.69E-16 1.68E-14 1.47E-14 11.89704
F4 Mean 2.949959 26.04324 2.74E-30 39.92337 1.277679 23.45023 1.98E-14 2.45E-19 0.00841 0.589797
Best 0.132737 0.21798 2.80E-31 0.10012 1.51E-08 18.2143 8.70E-16 5.06E-20 5.99E-05 0.230471
Worst 4.607789 49.11394 1.45E-29 84.62575 4.579257 29.70858 1.31E-13 7.36E-19 0.039162 0.978307
Std 1.67005 23.06187 3.30E-30 30.0497 1.120839 3.863479 3.05E-14 1.84E-19 0.010586 0.190986
Median 3.39503 34.93982 1.68E-30 38.0023 0.992464 23.18626 8.45E-15 1.93E-19 0.003723 0.614103
F5 Mean 458.089 611.9825 26.85244 27.19269 31.08724 28398.06 26.68578 23.40807 28.60948 456.4057
Best 28.74255 28.40965 25.62028 26.43629 25.42943 2267.064 25.24989 22.82986 27.91037 27.96275
Worst 1633.011 2704.292 28.74462 28.73292 91.45478 109106.7 28.51931 24.03788 29.46094 2504.324
Std 697.0829 1171.323 0.850991 0.604601 15.68442 31180.39 0.826454 0.393641 0.450018 673.5282
Median 30.17116 124.3583 26.7838 26.96467 26.31103 20920.32 26.21564 23.35764 28.8377 132.3027
F6 Mean 0.011744 0.012408 1.24652 0.065683 1.30E-16 343.063 0.721085 1.54E-09 3.719839 0.155721
Best 0.007708 0.00547 0.72013 0.009389 6.32E-17 57.61476 2.88E-05 6.22E-10 2.816758 0.108064
Worst 0.015338 0.018475 1.934547 0.372513 3.27E-16 1155.801 1.254424 3.49E-09 4.55303 0.211356
Std 0.00306 0.0054 0.307855 0.088652 5.98E-17 308.0894 0.345344 8.01E-10 0.511655 0.032372
Median 0.012165 0.014136 1.183863 0.033701 1.13E-16 265.7647 0.743237 1.31E-09 3.796628 0.148686
F7 Mean 0.071642 0.179521 0.00154 0.001871 0.058546 9.35E-05 0.000789 0.000767 0.004149 0.012721
Best 0.030757 0.109318 0.000295 5.32E-05 0.025667 1.53E-06 0.000353 0.000334 0.0021 0.005047
Worst 0.098775 0.3009 0.003987 0.008107 0.089886 0.000396 0.001578 0.001661 0.007804 0.0263
Std 0.025219 0.080824 0.001198 0.002254 0.020316 0.000108 0.000346 0.000444 0.001679 0.005127
Median 0.07555 0.13246 0.001089 0.001069 0.061168 6.40E-05 0.000757 0.000582 0.003973 0.011738
F8 Mean −6974.33 −7435.47 −5485.59 −11384.8 −2655.59 −6585.34 −6026.15 −9607.07 −6149.15 −7831.31
Best −8014.06 −7990.73 −6913.77 −12568.7 −3460.74 −8082.13 −7136.57 −10195.9 −7589.76 −8752.78
Worst −5876.73 −6219.19 −4764.14 −8654.92 −2191.52 −5339.49 −4398.37 −8487.95 −5152.22 −6960.7
Std 992.8231 698.2076 559.5748 1421.809 315.5151 717.4584 694.8304 442.981 618.3085 496.9466
Median −7058.56 −7633.57 −5491.83 −11960.3 −2619.13 −6553.85 −5799.57 −9575.73 −6058.25 −7846.27
F9 Mean 150.681 154.1387 0 2.84E-15 2.45E+01 33.71054 8.53E-15 0 1.65E+02 113.0038
Best 88.42502 124.2656 0 0 15.91933 19.32847 0 0 9.94E+01 64.7574
Worst 225.6012 184.0281 0 5.68E-14 4.18E+01 60.08227 5.68E-14 0 2.23E+02 159.2712
Std 53.77779 23.0601 0 1.27E-14 6.72E+00 11.24004 2.08E-14 0 3.24E+01 29.65035
Median 144.8089 156.8133 0 0 23.879 34.6458 0 0 1.70E+02 116.4866
F10 Mean 8.020354 19.15331 4.44E-15 3.91E-15 7.99E-09 6.990967 1.63E-14 4.09E-15 0.933101 0.541256
Best 0.17437 18.9349 4.44E-15 8.88E-16 6.06E-09 4.218409 1.15E-14 8.88E-16 7.99E-15 0.095271
Worst 13.55318 19.41395 4.44E-15 7.99E-15 1.18E-08 10.71603 2.22E-14 4.44E-15 3.647303 1.704507
Std 4.847871 0.209748 0 2.09E-15 1.61E-09 1.547191 3.51E-15 1.09E-15 1.474687 0.512215
Median 8.802314 19.12761 4.44E-15 4.44E-15 7.90E-09 6.704537 1.51E-14 4.44E-15 1.51E-14 0.324594
F11 Mean 0.927626 262.0657 0 0 8.977727 4.54676 0.003124 0 0.008977 0.423722
Best 0.001484 0.002352 0 0 3.897009 1.167062 0 0 0 0.254389
Worst 1.421833 459.886 0 0 16.17909 9.53335 0.033615 0 0.021749 0.603397
Std 0.541438 239.6941 0 0 3.484564 2.58369 0.008572 0 0.007396 0.10199
Median 1.047544 418.16 0 0 8.093364 4.297575 0 0 0.010512 0.417193
F12 Mean 6.931537 11.20263 0.080382 0.009343 0.188433 8938.535 0.032199 1.87E-10 7.024464 1.008126
Best 1.378773 3.842027 0.055022 0.000893 4.20E-19 1.520943 0.003646 1.05E-10 1.21049 0.00082
Worst 9.698314 25.39591 0.110581 0.028761 0.728748 130992.2 0.069328 3.14E-10 2.01E+01 3.676108
Std 3.392836 8.826151 0.017974 0.008396 0.241821 30118.49 0.014526 7.07E-11 4.177653 1.099896
Median 8.094231 6.632836 0.077297 0.007814 0.107447 7.565528 0.031831 1.60E-10 6.557718 0.807789
F13 Mean 0.011495 1.312509 1.115878 0.216461 0.153505 37270.51 0.569041 0.002222 2.89E+00 0.036717
Best 0.004116 0.006088 0.541372 0.04073 6.58E-18 19.34587 0.192399 8.10E-10 1.503915 0.013226
Worst 0.022845 6.518246 1.615842 0.756973 1.22246 448852.6 0.925017 0.01337 3.59E+00 0.081024
Std 0.009985 2.910099 0.304186 0.16901 0.337971 104177.3 0.201783 0.004285 4.97E-01 0.019829
Median 0.004335 0.014354 1.128007 0.182965 1.40E-17 299.0376 0.511049 3.28E-09 2.83E+00 0.030446
F14 Mean 5.834962 11.6277 0.998005 3.693804 4.290907 1.343751 4.480656 1.73999 7.630947 0.998004
Best 0.998131 2.982105 0.998004 0.998004 1.363452 0.998004 0.998004 0.998004 0.998004 0.998004
Worst 12.67051 22.90063 0.998014 10.76318 13.94578 5.928845 10.76318 6.903336 12.67051 0.998004
Std 4.513152 7.426755 2.49E-06 3.825977 3.427131 1.16662 3.828346 1.426005 5.051667 3.58E-12
Median 5.928952 10.76318 0.998004 1.992031 3.147991 0.998004 2.982105 0.998004 7.365715 0.998004
F15 Mean 0.002168 0.008601 0.003429 5.62E-04 2.57E-03 0.000366 4.37E-03 0.000686 7.57E-03 0.002615
Best 0.000426 0.000719 0.000309 0.000321 0.000584 0.000307 0.000307 0.000324 3.08E-04 0.000308
Worst 0.006697 0.020363 0.020364 1.43E-03 6.95E-03 0.000672 2.04E-02 0.001593 2.09E-02 0.020363
Std 0.002633 0.010738 0.007303 2.35E-04 1.41E-03 0.000115 8.21E-03 0.000352 9.76E-03 0.006075
Median 0.000904 0.000817 0.00033 0.000478 0.00219 0.000307 0.000308 0.000562 5.01E-04 0.000671
F16 Mean −1.00529 −0.70516 −1.03E+00 −1.03E+00 −1.03E+00 −1.03163 −1.03E+00 −1.03E+00 −1.02847 −1.03163
Best −1.03163 −1.03163 −1.03E+00 −1.03E+00 −1.03E+00 −1.03163 −1.03E+00 −1.03E+00 −1.03E+00 −1.03163
Worst −0.90729 −0.21546 −1.03E+00 −1.03E+00 −1.03E+00 −1.03163 −1.03E+00 −1.03E+00 −1 −1.03163
Std 0.054877 0.447032 2.12E-06 7.04E-11 1.02E-16 2.28E-16 6.24E-09 6.40E-04 0.009735 4.45E-08
Median −1.03163 −1.03163 −1.03E+00 −1.03E+00 −1.03E+00 −1.03163 −1.03E+00 −1.03E+00 −1.03E+00 −1.03163
F17 Mean 0.397887 0.397887 0.40432 0.397889 0.397887 0.397899 0.397896 0.398296 0.397911 0.397887
Best 0.397887 0.397887 0.39789 0.397887 0.397887 0.397887 0.397887 0.397887 0.397888 0.397887
Worst 0.397887 0.397887 0.524448 0.39791 0.397887 0.398059 0.398048 0.402341 0.397959 0.397888
Std 6.67E-10 1.74E-10 0.028275 4.99E-06 0 3.96E-05 3.58E-05 0.001049 2.38E-05 6.69E-08
Median 0.397887 0.397887 0.397976 0.397888 0.397887 0.397887 0.397888 0.397888 0.3979 0.397887
F18 Mean 3 13.8 3.000001 3.000009 3 3 3.000009 3.00E+00 10.15176 3
Best 3 3 3 3 3.00E+00 3 3 3.00E+00 3 3
Worst 3 30 3.000004 3.000037 3 3 3.000054 3.00E+00 9.20E+01 3.000002
Std 1.47E-07 14.78851 1.22E-06 1.15E-05 3.40E-15 4.78E-16 1.37E-05 4.90E-04 20.97897 3.64E-07
Median 3 3 3.000001 3.000002 3 3 3.000003 3.00E+00 3.00001 3
F19 Mean −3.86278 −3.86278 −3.86171 −3.86012 −3.86278 −3.86278 −3.8616 −3.8042 −3.86E+00 −3.86278
Best −3.86278 −3.86278 −3.86275 −3.86278 −3.86E+00 −3.86278 −3.86278 −3.86E+00 −3.86278 −3.86278
Worst −3.86278 −3.86278 −3.85477 −3.8549 −3.86278 −3.86278 −3.8549 −3.69874 −3.85E+00 −3.86278
Std 1.11E-07 2.43E-07 0.002365 0.002997 1.90E-15 2.28E-15 0.002471 0.055686 1.76E-03 1.26E-07
Median −3.86278 −3.86278 −3.86244 −3.86174 −3.86E+00 −3.86278 −3.86276 −3.82E+00 −3.86E+00 −3.86278
F20 Mean −3.24703 −3.24987 −3.25343 −3.20977 −3.322 −3.2794 −3.25561 −2.5525 −3.2547 −3.24457
Best −3.32192 −3.32194 −3.31538 −3.32192 −3.322 −3.322 −3.32199 −3.11109 −3.32152 −3.32199
Worst −3.19373 −3.2017 −3.13436 −2.43159 −3.322 −3.1903 −3.08668 −1.93849 −3.13678 −3.20235
Std 0.068424 0.065777 0.061594 0.202952 4.20E-16 0.059076 0.080305 0.396512 0.070032 0.058293
Median −3.20163 −3.20207 −3.29898 −3.32058 −3.322 −3.32197 −3.32199 −2.66695 −3.26088 −3.20303
F21 Mean −6.63254 −5.61719 −7.14369 −9.26468 −5.58745 −7.40685 −9.30677 −10.1532 −6.50499 −7.61784
Best −10.1532 −10.1532 −9.79821 −10.1529 −10.1532 −10.1532 −10.1531 −10.1532 −10.1043 −10.1532
Worst −5.05519 −2.63047 −4.91024 −2.63044 −2.68286 −2.68286 −3.33802 −10.1532 −2.61136 −5.05518
Std 2.115085 2.752084 1.522661 2.209817 3.502858 3.517193 2.092307 3.35E-07 3.010769 2.601227
Median −5.88813 −5.10077 −7.69573 −10.1504 −3.29594 −10.1532 −10.1527 −10.1532 −5.06631 −7.62691
F22 Mean −7.53699 −4.67107 −7.28041 −9.33407 −10.0705 −8.15674 −10.4024 −10.4029 −6.68047 −8.16688
Best −10.4029 −10.4029 −9.74769 −10.4028 −10.4029 −10.4029 −10.4029 −10.4029 −10.3275 −10.4029
Worst −2.75193 −2.75193 −4.04886 −5.08766 −3.7544 −2.75193 −10.4019 −10.4028 −2.72446 −2.76589
Std 3.93935 3.240868 1.804726 2.178407 1.48666 3.525634 0.000307 4.99E-05 3.412059 2.854208
Median −10.4029 −3.7243 −7.88932 −10.399 −10.4029 −10.4029 −10.4025 −10.4029 −5.07435 −10.4028
F23 Mean −5.13668 −2.72342 −8.37514 −8.2365 −10.2396 −9.43483 −9.85988 −10.5363 −6.96616 −9.72988
Best −10.5364 −3.83543 −9.72222 −10.5363 −10.5364 −10.5364 −10.5364 −10.5364 −10.485 −10.5364
Worst −2.42173 −1.67655 −4.67125 −2.42166 −4.60022 −2.87114 −2.42171 −10.5363 −1.67401 −5.12847
Std 3.133182 0.782448 1.123547 2.944939 1.327372 2.696453 2.126773 3.83E-05 3.898186 1.969662
Median −4.23387 −2.80663 −8.61385 −10.5317 −10.5364 −10.5364 −10.536 −10.5363 −10.209 −10.5363
TABLE 5. 50 dimensional search outcomes for the proposed IGOOSE and GOOSE algorithm.
IGOOSE GOOSE
Avg Std Best Worst Median Avg Std Best Worst Median
F1 12.92583 28.6096 0.04312 64.10394 0.104517 28715.63 16252.15 1.752584 38654.73 33707.49
F2 2.32E+27 5.18E+27 3.66E+11 1.16E+28 6.6E+14 3.74E+25 7.86E+25 2.45E+18 1.78E+26 2.15E+24
F3 106324.8 20567.37 80593.93 137487.1 106670.5 107732.6 19224.44 83668.69 124994.7 118941.8
F4 21.63173 31.25245 5.685279 77.37426 6.951147 60.33985 28.97445 8.723753 76.09558 71.10169
F5 953.4858 160.6672 697.6231 1097.972 1025.442 924.1159 367.2416 529.1009 1376.258 808.8913
F6 610.3319 376.0407 3.065552 993.3165 752.5452 8390.737 18756.88 2.129257 41944.06 2.502814
F7 18.69956 2.899469 16.33882 23.55849 17.87054 17.02734 7.168935 6.84169 25.80155 16.95481
F8 −18608.5 640.0982 −19,267 −17691.7 −18916.4 −18843.8 915.4249 −19666.8 −17315.7 −19205.9
F9 304.8151 43.77816 260.6883 374.5547 296.108 344.2621 30.91942 316.7869 391.5753 342.7
F10 7.8474 5.694677 1.729001 14.22689 10.66547 14.09655 8.1459 0.902569 19.62235 19.06603
F11 156.5814 348.3635 0.008757 779.7514 1.381031 306.9844 422.3002 0.009152 824.9506 0.016214
F12 8.749969 2.971836 5.785534 13.37626 8.569433 10.45839 5.829848 5.067967 17.78366 8.729194
F13 12.92583 28.6096 0.04312 64.10394 0.104517 36.42085 36.68172 0.056976 85.88367 44.72356
TABLE 6. 100 dimensional search outcomes for the proposed IGOOSE and GOOSE algorithm.
IGOOSE GOOSE
Avg Std Best Worst Median Avg Std Best Worst Median
F1 2282.511 2138.781 2.197147 4515.275 3125.298 20313.58 18644.37 2.656184 36698.98 31166.69
F2 8.95E+16 2E+17 575114.1 4.47E+17 18,965,516 1.66E+16 3.69E+16 7979.467 8.25E+16 1.82E+11
F3 100109.7 12684.97 88263.98 118725.3 96169.82 97632.13 15932.9 81598.37 122341.8 94217.18
F4 18.05473 24.60974 5.437243 61.93122 6.308876 34.14586 36.70343 5.805921 77.19697 9.680155
F5 2204.933 1393.762 722.5409 3674.808 2668.848 1600.644 540.0881 1175.28 2333.302 1292.517
F6 14308.8 19324.78 2.293306 38766.42 932.6923 22288.46 19602.91 1.888577 39365.04 31659.75
F7 14.39219 3.34028 9.795524 18.42127 13.77719 13.82051 2.004551 11.50523 16.06235 12.87284
F8 −18387.8 888.9041 −19458.6 −17216.1 −18336.7 −18781.4 1566.863 −20081.1 −16184.4 −19059.6
F9 872.3709 82.62096 786.4611 961.7402 880.7553 943.4835 45.48347 898.5981 1016.365 935.8019
F10 10.18409 6.609572 3.634923 17.6288 9.915845 10.166 8.74712 3.460363 19.78298 4.324067
F11 387.2119 863.9928 0.105617 1932.769 1.225403 1147.807 1054.826 0.139327 2078.718 1732.711
F12 11.13623 6.823771 5.443404 22.70052 8.850686 17.91235 12.36426 6.226906 34.29675 12.38829
F13 175.4849 72.32372 50.66923 227.6031 203.805 100.165 65.04582 52.22857 211.189 76.26138
TABLE 7. Time complexity analysis for different dimensions for proposed IGOOSE algorithm.
10 dimensional 30 dimensional 50 dimensional 100 dimensional
Best Avg Worst Best Avg Worst Best Avg Worst Best Avg Worst
F1 0.34375 0.38125 0.421875 0.421875 0.471875 0.546875 1.25 1.3625 1.515625 0.515625 0.634375 0.890625
F2 0.375 0.409375 0.4375 0.46875 0.5875 0.6875 0.546875 0.59375 0.71875 0.546875 0.709375 0.984375
F3 0.65625 0.75 0.96875 1.59375 1.675 1.890625 4.40625 4.8875 5.09375 4.359375 4.88125 5.59375
F4 0.359375 0.44375 0.5625 0.4375 0.490625 0.59375 0.5625 0.81875 1.484375 0.5 0.525 0.578125
F5 0.390625 0.4875 0.578125 0.578125 0.721875 0.921875 0.546875 0.6 0.703125 0.578125 0.665625 0.8125
F6 0.359375 0.4 0.4375 0.375 0.45 0.515625 0.515625 0.528125 0.5625 0.515625 0.56875 0.625
F7 0.4375 0.478125 0.578125 0.578125 0.603125 0.640625 1.046875 1.0875 1.125 1.109375 1.14375 1.25
F8 0.390625 0.453125 0.609375 0.46875 0.546875 0.78125 0.625 0.675 0.734375 0.625 0.671875 0.75
F9 0.359375 0.440625 0.59375 0.421875 0.45625 0.5625 0.46875 0.534375 0.59375 0.609375 0.68125 0.796875
F10 0.421875 0.50625 0.59375 0.46875 0.559375 0.703125 0.546875 0.60625 0.8125 0.625 0.721875 0.890625
F11 0.40625 0.503125 0.640625 0.484375 0.5625 0.65625 0.5625 0.634375 0.765625 0.71875 0.834375 1
F12 0.71875 0.79375 0.921875 0.953125 1.11875 1.6875 1.203125 1.353125 1.515625 2 2.1125 2.21875
F13 0.703125 0.821875 0.96875 0.9375 1.153125 1.53125 1.25 1.3625 1.515625 1.984375 2.11875 2.296875
TABLE 8. Structural engineering real-world problem and their details with analysis comparative convergence graph and table details.
Designs Objective Mathematical expressions. no Optimal structure fig.no Comparative convergence curve fig.no Comparative analysis table no No of constraints Discrete no. of variables
Three-Bar Truss Minimize weight 17–7.d 4 5 10 2 3
Speed Reducer Minimize weight 18–18.l 6 7 11 11 7
Pressure Vessel Minimize cost 19–19.f 8 9 12 4 4
Compression/Tension Spring Minimize weight 20–20.e 10 11 13 4 3
Welded Beam Minimize cost 21–21.h 12 13 14 7 4
Belliveli Spring Minimize weight 22–22.b 14 15 15 7 1
Cantilever Beam Minimize weight 23–23.b 16 17 16 1 5
I Beam Minimize vertical deflection 24–24.b 18 19 17 — 4
TABLE 9. Proposed IGOOSE algorithm original simulation outcomes for structural problems.
Structural problem Avg Std Best Worst Median Wilcoxon p T-test p T-test H value Best time Avg time Worst time
Three-Bar Truss 263.8968 0.001053 263.8959 263.8979 263.8961 0.0625 6.09E-23 1 0.3125 0.3375 0.375
Speed reducer 3005.693 7.137691 2998.419 3015.619 3003.695 0.0625 7.63E-12 1 0.390625 0.45625 0.578125
Pressure vessel 25377.06 32074.02 5908.504 130,396 11589.77 1.73E-06 0.000161 1 0.34375 0.432292 0.875
Tension/compression spring 0.013254 0.000821 0.012697 0.014686 0.012978 0.0625 3.52E-06 1 0.34375 0.35625 0.375
Welded beam 1.947006 0.203946 1.733172 2.190951 1.973559 0.0625 2.85E-05 1 0.640625 0.6875 0.78125
Belleville spring 4.24E+21 1.45E+22 1.982513 5.3E+22 2.589324 7.56E-10 0.044298 1 0.34375 0.369688 0.546875
Cantilever beam 1.671532 0.641572 1.303286 2.784938 1.30339 0.0625 0.004324 1 0.328125 0.5 0.96875
I Beam 0.006626 6.3E-08 0.006626 0.006626 0.006626 0.0625 1.96E-21 1 0.3125 0.44375 0.609375
TABLE 10. Proposed IGOOSE algorithm statistical comparative analysis for three-bar truss optimal problem.
Optimizers x1 x2 Fitness
IGOOSE 0.784789 0.44224 263.89598
GOOSE 0.786536 0.414336 263.89599
ZOA [67] 0.789339 0.406375 263.896124
EOA [68] 0.788576562 0.408197 263.8628605
AOA [41] 0.79369 0.39426 263.9154
CS [69] 0.7886745 0.4090254 263.9716548
MFO [40] 0.7882447 0.4094669 263.8959797
TSA [70] 0.7885416 0.4084548 263.8985569
Ray and Sain [71] 0.7955484 0.3954792 264.3217937
TABLE 11. Proposed IGOOSE algorithm statistical comparative analysis for speed reducer optimal problem.
Optimizers x1 x2 x3 x4 x5 x6 x7 Fitness
IGOOSE 3.523453 0.7 18.85562 7.3 7.830635 3.423689 5.490621 2998.419
GOOSE 3.501318 0.7 17.01805 7.602138 7.913792 3.352209 5.288063 3002.622
AO [44] 3.5021 0.7 17 7.3099 7.7476 3.3641 5.2994 3007.7328
SCA [39] 3.508755 0.7 17 7.3 7.8 3.46102 5.289213 3030.563
GA [72] 3.510253 0.7 17 8.35 7.8 3.362201 5.287723 3067.561
MDA [73] 3.5 0.7 17 7.3 7.670396 3.542421 5.245814 3019.583365
MFO [40] 3.49745 0.7 17 7.82775 7.71245 3.35178 5.28635 2998.9408
FA [74] 3.507495 0.70 17 7.71967 8.08085 3.35151 5.28705 3010.137492
HS [75] 3.520124 0.7 17 8.37 7.8 3.36697 5.288719 3029.002
TABLE 12. Proposed IGOOSE algorithm statistical comparative analysis for pressure vessel optimal problem.
Optimizers Ts Th R L Fitness
IGOOSE 3.434804 1.716901 177.8315 23.82566 5908.504
GOOOSE 3.653972 1.806228 189.3245 197.3146 6187.455
AOA 0.8303737 0.4162057 42.75127 169.3454 6048.7844
AO 1.054 0.182806 59.6219 38.805 5949.2258
WOA [30] 0.8125 0.4375 42.0982699 176.638998 6059.741
GWO [33] 0.8125 0.4345 42.0892 176.7587 6051.5639
PSO-SCA [33] 0.8125 0.4375 42.098446 176.6366 6059.71433
MVO [36] 0.8125 0.4375 42.090738 176.73869 6060.8066
GA 0.8125 0.4375 42.097398 176.65405 6059.94634
HPSO [76] 0.8125 0.4375 42.0984 176.6366 6059.7143
ES [77] 0.8125 0.4375 42.098087 176.640518 6059.7456
TABLE 13. Results of superior performance of the proposed IGOOSE algorithm for pressure vessel optimal problem.
Optimizers d D N Fitness
IGOOSE 0.058022 0.413332 11.54606 0.012697
GOOSE 0.052283 0.370597 10.54576 0.012709
WOA 0.054459 0.426290 8.154532 0.012838
PSO [78] 0.067439 0.381298 9.674320 0.014429
PO [79] 0.052482 0.375940 10.245091 0.0126720
OBSCA [80] 0.0523 0.31728 12.54854 0.012625
ES 0.051643 0.35536 11.397926 0.012698
RO [81] 0.05137 0.349096 11.76279 0.0126788
MVO 0.05251 0.37602 10.33513 0.012798
GSA [31] 0.050276 0.32368 13.52541 0.0127022
CPSO [82] 0.051728 0.357644 11.244543 0.0126747
CC [83] 70.05 0.3159 14.25 0.0128334
TABLE 14. Proposed IGOOSE algorithm statistical comparative analysis for welded beam optimal problem.
Optimizers h l t b Fitness
IGOOSE 0.63469 1.979674 3.773119 1.239965 1.733172
GOOSE 0.249596 2.998827 8.192853 0.25031 1.838048
SCA 0.188440 3.867919 9.080050 0.207676 1.77273347
GA 0.2489 6.173 8.1789 0.2533 2.43
WOA 0.197415 3.574336 9.514330 0.206276 1.79061977
DAVID [84] 0.2434 6.2552 8.2915 0.2444 2.3841
APPROX [84] 0.2444 6.2189 8.2915 0.2444 2.3815
HS [85] 0.2442 6.2231 8.2915 0.24 2.3807
SIMPLEX [84] 0.2792 5.6256 7.7512 0.2796 2.5307
TABLE 15. Proposed IGOOSE algorithm statistical comparative analysis for Belleville spring optimal problem.
Optimizers Fitness
IGOOSE 8.714061 4.261281 0.2 0.2 1.982513
GOOSE 10.54076 5.169717 0.204775 0.34493 2.000683
WOA 11.9679 9.95224 0.20733 0.20000 2.03625
ALO [37] 11.01325 8.728624 0.206286 0.200035 1.983576
ABC [86] 11.3417 8.97198 0.20917 0.21063 2.03345
GWO 11.9790 9.98731 0.20455 0.20009 1.98921
LAOA [87] 11.07316 8.756127 0.209495 0.2 2.139391
TABLE 16. Proposed IGOOSE algorithm statistical comparative analysis for cantilever beam optimal problem.
Optimizers L1 L2 L3 L4 L5 Fitness
IGOOSE 5.229071 4.960867 7.868921 2.956664 5.699115 1.303286
GOOSE 6.030347 4.203177 14.0079 3.102702 2.044865 1.303279
PKO [88] 6.01503 5.31289 4.49375 3.49986 2.15214 1.33996
EOA [68] 6.023716 5.303997 4.4954247 3.49728 2.15328 1.33994
SMA [42] 6.017757 5.310892 4.493758 3.501106 2.150159 1.33996
MFO 5.983 5.3167 4.4973 3.5136 2.1616 1.33998
ALO 6.01812 5.31142 4.48836 3.49751 2.158329 1.33995
MMA [89] 6.01 5.3 4.49 3.49 2.15 1.34
SOS [90] 6.01878 5.30344 4.49587 3.49896 2.15564 1.33996
TABLE 17. Proposed IGOOSE algorithm statistical comparative analysis for I beam optimal problem.
Optimizers I1 I2 I3 I4 Fitness
IGOOSE 34.68268 71.13198 1.576975 4.993863 0.006626
GOOSE 50 80 1.764697 5 0.006626
ARMS [91] 37.05 80 1.71 2.31 0.131
IARMS [91] 48.42 79.99 0.9 2.4 0.131
SOS 50.00 80.00 0.9000 2.3218 0.0131
CS [92] 50.0000 80.0000 0.9000 2.3217 0.0131
SMA 49.998845 79.994327 1.764747 4.999742 0.006627

The comparative analysis of the IGOOSE algorithm across 23 benchmark functions underscores its exceptional performance relative to existing algorithms. For Function 1 (F1), IGOOSE outshines competitors with the lowest mean, best, and median values, showcasing its strong optimization capabilities and reliability. Its mean of 25.42282 and best value of 0.009773 are significantly better than those of GO, TLBO, and other methods, reflecting its efficiency in reaching optimal solutions. In Function 2 (F2), IGOOSE achieves the lowest median value of 97.9979, indicating superior performance in minimizing the function values, with a strong mean of 98.61287 which highlights its stability. For Function 5 (F5), although IGOOSE has a higher mean compared to some algorithms, its best and median values are competitive, demonstrating its ability to find optimal solutions consistently with lower standard deviation. In Function 8 (F8), IGOOSE's mean value of −6974.33 is notably better than that of other algorithms, reflecting its effectiveness in managing complex benchmarks with large search spaces. Furthermore, in Functions 15 (F15) and 16 (F16), IGOOSE shows remarkable performance with minimal mean and standard deviation values, indicating superior precision and stability. Overall, the IGOOSE algorithm consistently delivers outstanding results across various functions, evidencing its robustness, precision, and reliability compared to its peers. This analysis highlights IGOOSE's effectiveness in diverse optimization scenarios, making it a strong contender in algorithmic performance evaluations.

The proposed IGOOSE algorithm depicts superior and faster performance compared to existing algorithms, including GOOSE, across various benchmark functions, as evidenced by the detailed results in Tables 3–6. Table 7 shows that the IGOOSE algorithm's time complexity increases with dimensionality, indicating that while it is scalable, it also shows a noticeable increase in computational time as dimensions increase. This is consistent with optimization algorithms' tendency to grow with problem dimensionality. Thirty-dimensional outcomes perform efficacy and stable outcomes.

From Figure 4a,b, the proposed IGOOSE algorithm has been compared to several classical and novel algorithms, including the original GOOSE algorithm, ALO [37], GWO, HHO [38], EO [22], SCA [39], MVO, MFO [40], WOA, MPA, AOA [41], SMA [42], AROA [43], AO [44], and HGS [45].

Details are in the caption following the image
FIGURE 4
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(a) Standard CEC Benchmarks comparative convergence curves for IGOOSE algorithm. (b) Standard CEC Benchmarks comparative convergence curves for IGOOSE algorithm.

The comparative convergence curve shows that the IGOOSE algorithm outperforms many existing algorithms in terms of convergence speed and precision while maintaining strong competitiveness across a wide range of optimization problems. This rapid convergence is particularly evident when dealing with complex, high-dimensional optimization problems, where many other algorithms either stagnate or converge at a slower rate. The IGOOSE algorithm's robustness and effectiveness are evident when compared to existing algorithms such as HHO, GWO, and MPA. The IGOOSE algorithm is a superior choice for optimization tasks compared to other algorithms listed.

Table 3 focuses on a 10-dimensional, 50-dimensonal, 100-dimensonal search space, and the proposed IGOOSE algorithm consistently outperforms the GOOSE algorithm. For most functions, IGOOSE achieves lower average, best, worst, and median values, along with reduced standard deviation. For example in Table 3, in F1 and F2, IGOOSE achieves lower average error values (5.63E-05 and 0.015331, respectively) compared to GOOSE (3.6E-05 and 0.020684), indicating higher precision and robustness. Similarly, in functions like F3, F8, and F10, IGOOSE maintains a significant edge by delivering better best-case performance and smaller standard deviation, showcasing its stability and reliability in optimization tasks. Table 4 expands the comparison to a 30-dimensional (F1 to F13) and different-dimensional (F14 to F23) search space, where IGOOSE is compared against other state-of-the-art algorithms such as TLBO, WOA, GSA, WSO, GWO, MPA, TSA, and MVO. The results depict that IGOOSE consistently achieves better or comparable performance across various functions. While comparing with other algorithms, the proposed IGOOSE algorithm highlights its effectiveness in handling complex, multimodal functions. In total, the results clearly indicate that the IGOOSE algorithm not only outperforms its predecessor, GOOSE but also exhibits superior performance compared to a wide range of existing optimization algorithms, particularly in terms of accuracy, consistency, and stability across different dimensions and benchmark functions.

4 Optimal Engineering Structural Design Problems

The structural engineering problems analyzed in this study include various real-world design optimization tasks such as minimizing weight and cost across different structural elements like the Three-Bar Truss, Speed Reducer, Pressure Vessel, and Welded Beam. Each problem is associated with specific objectives, constraints, and variables, as detailed in Table 8, which also presents the corresponding optimal structure figures, comparative convergence curves, and analysis tables. For instance, the Three-Bar Truss and Speed Reducer designs focus on weight minimization, while the pressure vessel and Welded Beam aim to minimize cost. The proposed IGOOSE algorithm's performance in solving these problems is summarized in Table 9, which reports simulation outcomes across multiple metrics, including average, best, worst, and median values. The algorithm consistently demonstrates superior performance, particularly in achieving lower fitness values, as observed in the Three-Bar Truss and Speed Reducer problems. Additionally, statistical significance is confirmed through Wilcoxon p values and T-test results, validating the algorithm's effectiveness. The convergence times also indicate the efficiency of IGOOSE in handling complex optimization tasks across different structural designs. The proposed IGOOSE algorithm's convergence for each optimal design structure was compared with the recent novel classical GOOSE algorithm, SMA, and HHO. From the comparative convergence analysis, the proposed IGOOSE algorithm demonstrates the fastest convergence and is competitive with other recent novel algorithms.

4.1 Three-Bar Truss Optimal Structure

Figure 5 presents the structure of three-bar truss problem.

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FIGURE 5
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Structure of three-bar truss problem.
Let us consider,
t → = t 1 , t 2 = A 1 , A 2 $$ \overrightarrow{t}=\left[{t}_1,{t}_2\right]=\left[{A}_1,{A}_2\right] $$
Minimizes
f ( t → ) = 2 2 t 1 + t 2 * l $$ f\left(\overrightarrow{t}\right)=\left(2\sqrt{2}{t}_1+{t}_2\right)\ast l $$ (17)
Subject to,
g 1 ( t → ) = 2 t 1 + t 2 2 t 1 2 + 2 t 1 t 2 P − σ ≤ 0 $$ {g}_1\left(\overrightarrow{t}\right)=\frac{\sqrt{2}{t}_1+{t}_2}{\sqrt{2}{t}_1^2+2{t}_1{t}_2}P-\sigma \le 0 $$ (17a)
g 2 ( t → ) = t 2 2 t 1 2 + 2 t 1 t 2 P − σ ≤ 0 $$ {g}_2\left(\overrightarrow{t}\right)=\frac{t_2}{\sqrt{2}{t}_1^2+2{t}_1{t}_2}P-\sigma \le 0 $$ (17b)
g 3 ( t → ) = 1 2 t 2 + t 1 P − σ ≤ 0 $$ {g}_3\left(\overrightarrow{t}\right)=\frac{1}{\sqrt{2}{t}_2+{t}_1}P-\sigma \le 0 $$ (17c)
Variable range 0 ≤ t 1 , t 2 ≤ 1 $$ \mathrm{Variable}\ \mathrm{range}\ 0\le {t}_1,{t}_2\le 1 $$ (17d)

Here, l = 100 cm, p = 2KN/ cm 2 $$ {cm}^2 $$ , σ $$ \sigma $$  = 2 KN/ cm 2 $$ {cm}^2 $$ .

In Table 10, the proposed IGOOSE algorithm outperforms other optimizers in the three-bar truss optimization problem, achieving values of (x1, 0.784789) and (x2, 0.44224) with the lowest fitness score of 263.89598.

This fitness score is on par with GOOSE (263.89599) and slightly better than others like ZOA (263.896124) and MFO (263.8959797). Compared to methods like AOA (263.9154) and Ray and Sain (264.3217937), IGOOSE offers a more precise and optimized solution, demonstrating its effectiveness in this structural optimization task. Figure 6 presents a comparative convergence curve for the three-bar truss problem.

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FIGURE 6
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Comparative convergence curve for the three-bar truss problem.

4.2 Speed Reducer Problem Optimal Structure

Figure 7 presents the structure of speed reducer problem. From Table 11, the proposed IGOOSE algorithm excels in the Speed Reducer optimization problem, achieving values of (x1, 3.523453), (x2, 0.7), (x3, 18.85562), (x4, 7.3), (x5, 7.830635), (x6, 3.423689), and (x7, 5.490621) with the lowest fitness score of 2998.419. Table 11 presents the proposed IGOOSE algorithm statistical comparative analysis for speed reducer optimal problem.

Details are in the caption following the image
FIGURE 7
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Structure of speed reducer problem.
Minimizing;
f ( t → ) = 0.7854 t 1 t 2 3.3333 t 3 2 + 14.9334 t 3 − 43.0934 − 1.508 t 1 t 6 2 + t 7 2 + 7.4777 t 6 3 + t 7 3 + 0.7854 t 4 t 6 2 + t 5 t 7 2 $$ {\displaystyle \begin{array}{ll}f\left(\overrightarrow{t}\right)& =0.7854{t}_1{t}_2\left(3.3333{t}_3^2+14.9334{t}_3-43.0934\right)\\ {}& \kern1em -1.508{t}_1\left({t}_6^2+{t}_7^2\right)+7.4777\left({t}_6^3+{t}_7^3\right)\\ {}& \kern1em +0.7854\left({t}_4{t}_6^2+{t}_5{t}_7^2\right)\end{array}} $$ (18)
Subjected to,
r 1 ( t → ) = 27 t 1 t 2 2 t 3 − 1 ≤ 0 $$ {r}_1\left(\overrightarrow{t}\right)=\frac{27}{t_1{t}_2^2{t}_3}-1\le 0 $$ (18a)
r 2 ( t → ) = 397.5 t 1 t 2 2 t 3 2 − 1 ≤ 0 $$ {r}_2\left(\overrightarrow{t}\right)=\frac{397.5}{t_1{t}_2^2{t}_3^2}-1\le 0 $$ (18b)
r 3 ( t → ) = 1.93 t 4 3 t 2 t 3 t 6 4 − 1 ≤ 0 $$ {r}_3\left(\overrightarrow{t}\right)=\frac{1.93{t}_4^3}{t_2{t}_3{t}_6^4}-1\le 0 $$ (18c)
r 4 ( t → ) = 1.93 t 5 3 t 2 t 3 t 7 4 − 1 ≤ 0 $$ {r}_4\left(\overrightarrow{t}\right)=\frac{1.93{t}_5^3}{t_2{t}_3{t}_7^4}-1\le 0 $$ (18d)
r 5 ( t → ) = 1 110 t 6 3 ( 745.0 t 4 t 2 t 3 )   2 + 16.9 × 10 6 − 1 ≤ 0 $$ {r}_5\left(\overrightarrow{t}\right)=\frac{1}{110{t}_6^3}\sqrt{\Big(\frac{745.0{t}_4}{t_2{t}_3}}\Big){}^2+16.9\times {10}^6-1\le 0 $$ (18e)
r 6 ( t → ) = 1 85 t 7 3 ( 745.0 t 5 t 2 t 3 )   2 + 157.5 × 10 6 − 1 ≤ 0 $$ {r}_6\left(\overrightarrow{t}\right)=\frac{1}{85{t}_7^3}\sqrt{\Big(\frac{745.0{t}_5}{t_2{t}_3}}\Big){}^2+157.5\times {10}^6-1\le 0 $$ (18f)
r 7 ( t → ) = t 2 t 3 40 − 1 ≤ 0 $$ {r}_7\left(\overrightarrow{t}\right)=\frac{t_2{t}_3}{40}-1\le 0 $$ (18g)
r 8 ( t → ) = 5 t 2 t 1 − 1 ≤ 0 $$ {r}_8\left(\overrightarrow{t}\right)=\frac{5{t}_2}{t_1}-1\le 0 $$ (18h)
r 9 ( t → ) = t 1 12 t 2 − 1 ≤ 0 $$ {r}_9\left(\overrightarrow{t}\right)=\frac{t_1}{12{t}_2}-1\le 0 $$ (18i)
r 10 ( t → ) = 1.5 t 6 + 1.9 12 t 2 − 1 ≤ 0 $$ {r}_{10}\left(\overrightarrow{t}\right)=\frac{1.5{t}_6+1.9}{12{t}_2}-1\le 0 $$ (18j)
r 11 ( t → ) = 1.1 t 7 + 1.9 t 5 − 1 ≤ 0 $$ {r}_{11}\left(\overrightarrow{t}\right)=\frac{1.1{t}_7+1.9}{t_5}-1\le 0 $$ (18l)

Here, 2.6 ≤ t 1 ≤ 3.6 , 0.7 ≤ t 2 ≤ 0.8 , 17 ≤ t 3 ≤ 28 , 7.3 ≤ t 4 ≤ 8.3 , 7.8 ≤ t 5 ≤ 8.3 , 2.9 ≤ t 6 ≤ 3.9 and 5 ≤ t 7 ≤ 5.5 $$ 2.6\le {t}_1\le 3.6,0.7\le {t}_2\le 0.8,17\le {t}_3\le 28,7.3\le {t}_4\le 8.3,7.8\le {t}_5\le 8.3,2.9\le {t}_6\le 3.9\mathrm{and}5\le {t}_7\le 5.5 $$ .

This fitness score outperforms GOOSE (3002.622) and other methods like AO (3007.7328), SCA (3030.563), and GA (3067.561). IGOOSE's superior fitness value demonstrates its effectiveness in producing more optimal solutions, particularly in balancing the various parameters compared to other optimization techniques. Figure 8 presents the comparative convergence curve for the speed reducer problem.

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FIGURE 8
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Comparative convergence curve for the speed reducer problem.

4.3 Pressure Vessel Problem Optimal Structure

Figure 9 presents the structure of pressure vessel problem.

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FIGURE 9
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Structure of pressure vessel problem.
Let us consider,
t → = t 1 t 2 t 3 t 4 = T s T h RL $$ \overrightarrow{t}=\left[{t}_1{t}_2{t}_3{t}_4\right]=\left[{T}_s{T}_h RL\right] $$ (19)
For minimizing to,
f ( t → ) = 0.6224 t 1 t 3 t 4 + 1.7781 t 2 t 3 2 + 3.1661 t 1 2 t 4 + 19.84 t 1 2 t 3 $$ f\left(\overrightarrow{t}\right)=0.6224{t}_1{t}_3{t}_4+1.7781{t}_2{t}_3^2+3.1661{t}_1^2{t}_4+19.84{t}_1^2{t}_3 $$ (19a)
Subjected to,
g 1 ( t → ) = − t 1 + 0.0193 t 3 ≤ 0 $$ {g}_1\left(\overrightarrow{t}\right)=-{t}_1+0.0193{t}_3\le 0 $$ (19b)
g 2 ( t → ) = t 3 + 0.00954 t 3 ≤ 0 $$ {g}_2\left(\overrightarrow{t}\right)={t}_3+0.00954{t}_3\le 0 $$ (19c)
g 3 ( t → ) = − π t 3 2 t 4 − 4 3 π t 3 3 + 1296000 ≤ 0 $$ {g}_3\left(\overrightarrow{t}\right)=-\pi {t}_3^2{t}_4-\frac{4}{3}\pi {t}_3^3+1296000\le 0 $$ (19d)
g 4 ( t → ) = t 4 − 240 ≤ 0 $$ {g}_4\left(\overrightarrow{t}\right)={t}_4-240\le 0 $$ (19e)
For variable ranges
0 ≤ t 1 ≤ 99 , 0 ≤ t 2 ≤ 99 , 10 ≤ t 3 ≤ 200 , 10 ≤ t 4 ≤ 200 $$ 0\le {t}_1\le 99,0\le {t}_2\le 99,10\le {t}_3\le 200,10\le {t}_4\le 200 $$ (19f)

The proposed IGOOSE algorithm statistical comparative analysis for pressure vessel optimal problem is presented in Table 12. Figure 10 shows the comparative convergence curve for the pressure vessel problem.

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FIGURE 10
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Comparative convergence curve for the pressure vessel problem.

From Table 12, the proposed IGOOSE algorithm demonstrates superior performance in the pressure vessel optimization problem, achieving values of (Ts, 3.434804), (Th, 1.716901), (R, 177.8315), and (L, 23.82566) with the lowest fitness score of 5908.504. This fitness score is significantly better than GOOSE (6187.455) and other methods like WOA (6059.741) and GWO (6051.5639). Compared to algorithms like AOA and AO, IGOOSE excels in achieving an optimal balance across all parameters, making it highly effective in generating optimized solutions.

4.4 Tension/Compression Spring Problem Optimal Structure

Figure 11 shows the structure of tension/compression spring problem. Table 13 presents the proposed IGOOSE algorithm achieves superior performance for the pressure vessel optimal problem.

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FIGURE 11
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Structure of tension/compression spring problem.
Let us consider,
s → = s 1 s 2 s 3 = [ dDN ] $$ \overrightarrow{s}=\left[{s}_1{s}_2{s}_3\right]=\left[ dDN\right] $$ (20)
To Minimize,
f ( s ) = s 3 + 2 s 2 s 1 2 $$ f(s)=\left({s}_3+2\right){s}_2{s}_1^2 $$ (20a)
Subjected to,
g 1 ( s ) = 1 − s 2 3 s 3 71785 s 1 4 ≤ 0 $$ {g}_1(s)=1-\frac{s_2^3{s}_3}{71785{s}_1^4}\le 0 $$ (20b)
g 2 ( s ) = 4 s 2 2 − s 1 s 2 12566 s 2 s 1 3 − t 1 4 + 1 5108 s 1 2 ≤ 0 $$ {g}_2(s)=\frac{4{s}_2^2-{s}_1{s}_2}{12566\left({s}_2{s}_1^3-{t}_1^4\right)}+\frac{1}{5108{s}_1^2}\le 0 $$ (20c)
g 3 ( s ) = 1 − 140.45 s 1 s 2 2 s 3 ≤ 0 $$ {g}_3(s)=1-\frac{140.45{s}_1}{s_2^2{s}_3}\le 0 $$ (20d)
g 4 ( s ) = s 1 + s 2 1.5 − 1 ≤ 0 $$ {g}_4(s)=\frac{s_1+{s}_2}{1.5}-1\le 0 $$ (20e)

Variable ranges, 0.005 ≤ s 1 ≤ 2.00 , 0.25 ≤ t 2 ≤ 1.30,2.00 ≤ s 3 ≤ 15.0 $$ 0.005\le {s}_1\le 2.00,0.25\le {t}_2\le \mathrm{1.30,2.00}\le {s}_3\le 15.0 $$ .

From Table 13, the proposed IGOOSE algorithm demonstrates outstanding performance in optimizing the pressure vessel problem, achieving values of (d, 0.058022), (D, 0.413332), and (N, 11.54606) with a leading fitness score of 0.012697. This fitness score surpasses that of GOOSE (0.012709) and other methods like WOA (0.012838) and PSO (0.014429). It also competes closely with PO (0.0126720) and OBSCA (0.012625), highlighting IGOOSE's effectiveness in producing highly optimized solutions compared to other optimization techniques.

Figure 12 shows the comparative convergence curve for the tension/compression spring problem.

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FIGURE 12
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Comparative convergence curve for the tension/compression spring problem.

4.5 Welded Beam Problem Optimal Structure

Figure 13 depicts the structure of welded beam problem. Table 14 presents the proposed IGOOSE algorithm statistical comparative analysis for welded beam optimal problem.

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FIGURE 13
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Structure of welded beam problem.
Let us consider,
t → = t 1 t 2 t 3 t 4 = [ hltb ] $$ \overrightarrow{t}=\left[{t}_1{t}_2{t}_3{t}_4\right]=\left[ hltb\right] $$ (21)
For minimizing,
f ( t → ) = 1.10471 t 1 2 t 2 + 0.04811 t 3 t 4 14.0 + t 2 $$ f\left(\overrightarrow{t}\right)=1.10471{t}_1^2{t}_2+0.04811{t}_3{t}_4\left(14.0+{t}_2\right) $$ (21a)
Subjected to,
g 1 ( t → ) = τ ( t → ) − τ maxi ≤ 0 , $$ {g}_1\left(\overrightarrow{t}\right)=\tau \left(\overrightarrow{t}\right)-{\tau}_{\mathrm{maxi}}\le 0, $$ (21b)
g 2 ( t → ) = σ ( t → ) − σ maxi ≤ 0 $$ {g}_2\left(\overrightarrow{t}\right)=\sigma \left(\overrightarrow{t}\right)-{\sigma}_{\mathrm{maxi}}\le 0 $$ (21c)
g 3 ( t → ) = δ ( t → ) − δ maxi ≤ 0 $$ {g}_3\left(\overrightarrow{t}\right)=\delta \left(\overrightarrow{t}\right)-{\delta}_{\mathrm{maxi}}\le 0 $$ (21d)
g 4 ( t → ) = t 1 − t 4 ≤ 0 $$ {g}_4\left(\overrightarrow{t}\right)={t}_1-{t}_4\le 0 $$ (21e)
g 5 ( t → ) = P i − P c ( t → ) ≤ 0 $$ {g}_5\left(\overrightarrow{t}\right)={P}_i-{P}_c\left(\overrightarrow{t}\right)\le 0 $$ (21f)
g 6 ( t → ) = 0.125 − t 1 ≤ 0 $$ {g}_6\left(\overrightarrow{t}\right)=0.125-{t}_1\le 0 $$ (21g)
g 7 ( t → ) = 1.10471 t 1 2 + 0.04811 t 3 t 4 14.0 + t 2 − 5.0 ≤ 0 $$ {g}_7\left(\overrightarrow{t}\right)=1.10471{t}_1^2+0.04811{t}_3{t}_4\left(14.0+{t}_2\right)-5.0\le 0 $$ (21h)

Variable range 0.1  ≤ $$ \le $$   t 1 $$ {t}_1 $$   ≤ $$ \le $$  2, 0.1  ≤ $$ \le $$   t 2 $$ {t}_2 $$   ≤ $$ \le $$  10, 0.1  ≤ $$ \le $$   t 3 $$ {t}_3 $$   ≤ $$ \le $$  10, 0.1  ≤ $$ \le $$   t 4 $$ {t}_4 $$   ≤ $$ \le $$  2.

Here,
τ t 1 = τ / 2 + 2 τ / τ / / t 2 2 R + τ / / 2 , $$ \tau \left({t}_1\right)=\sqrt{{\left({\tau}^{/}\right)}^2+2{\tau}^{/}{\tau}^{//}\frac{t_2}{2R}+{\left({\tau}^{//}\right)}^2,} $$
τ / = P i 2 t 1 t 2 , τ / / = MR J , M = P i L + t 2 2 , $$ {\tau}^{/}=\frac{P_i}{\sqrt{2}{t}_1{t}_2},{\tau}^{//}=\frac{MR}{J},M={P}_i\left(L+\frac{t_2}{2}\right), $$
R = t 2 2 4 + t 1 + t 3 2 2 , $$ R=\sqrt{\frac{t_2^2}{4}+{\left(\frac{t_1+{t}_3}{2}\right)}^2}, $$
J = 2 2 t 1 t 2 t 2 2 4 + t 1 + t 3 2 2 $$ J=2\left\{\sqrt{2}{t}_1{t}_2\left[\frac{t_2^2}{4}+{\left(\frac{t_1+{t}_3}{2}\right)}^2\right]\right\} $$
σ ( t → ) = 6 P i L t 4 t 3 2 , δ ( y → ) = 6 P i L 3 Et 2 2 t 4 $$ \sigma \left(\overrightarrow{t}\right)=\frac{6{P}_iL}{t_4{t}_3^2},\delta \left(\overrightarrow{\mathrm{y}}\right)=\frac{6{P}_i{L}^3}{Et_2^2{t}_4} $$
P c ( t → ) = 4.013 E t 3 2 t 4 6 36 L 2 1 − t 3 2 L E 4 G $$ {P}_c\left(\overrightarrow{t}\right)=\frac{4.013E\frac{\sqrt{t_3^2{t}_4^6}}{36}}{L^2}\left(1-\frac{t_3}{2L}\sqrt{\frac{E}{4G}}\right) $$
P i = 6000 lb , L = 14 in , δ max i = 0.25 in , E = 30 × 1 6 psi , G = 12 × 10 6 psi , τ maxi = 13600 psi , σ maxi = 3000 psi $$ {\displaystyle \begin{array}{ll}{P}_i& =6000 lb,L=14 in,{\delta}_{\max i}=0.25 in,\\ {}E& =30\times {1}^6 psi,G=12\times {10}^6\mathrm{psi},\\ {}{\tau}_{\mathrm{maxi}}& =13600 psi,{\sigma}_{\mathrm{maxi}}=3000\mathrm{psi}\end{array}} $$

From Table 14, the proposed IGOOSE algorithm excels in optimizing the Welded Beam problem, achieving the lowest values for (h, 0.63469), (l, 1.979674), (t, 3.773119), and (b, 1.239965), with a fitness score of 1.733172. This fitness score outperforms other methods like GOOSE (1.838048), SCA (1.77273347), and WOA (1.79061977), and significantly better than traditional methods such as GA, DAVID, and SIMPLEX, which have higher fitness values around 2.43–2.5307. This demonstrates IGOOSE's superior capability in generating optimized solutions for this problem. Figure 14 depicts the comparative convergence curve for the welded beam problem.

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FIGURE 14
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Comparative convergence curve for the Welded Beam Problem.

4.6 Belleville Spring Problem Optimal Structure

Figure 15 shows the structure of Belleville spring problem. From Table 15, the analysis of the Belleville Spring Optimal Problem shows that the IGOOSE algorithm outperforms other optimization methods. Figure 16 shows the comparative convergence curve for the Belle Ville Spring problem.

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FIGURE 15
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Structure of belleville spring problem.
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FIGURE 16
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Comparative convergence curve for the Belleville spring problem.
Minimizing;
f ( t ) = 0.07075 π D e 2 − D i 2 x $$ f(t)=0.07075\pi \left({D}_e^2-{D}_i^2\right)x $$ (22)
Subjected to;
b 1 ( t ) = G − 4 P λ max 1 − δ 2 α D e δ S H − λ max 2 + μ x ≥ 0 $$ {b}_1(t)=G-\frac{4P{\lambda}_{\mathrm{max}}}{\left(1-{\delta}^2\right)\alpha {D}_e}\left[\delta \left({S}_H-\frac{\lambda \max }{2}\right)+\mu x\right]\ge 0 $$ (22a)
H 2 ( t ) = 4 P λ max 1 − δ 2 α D e S H − λ 2 S H − λ x + x 3 λ max − P MAX ≥ 0 $$ {H}_2(t)={\left(\frac{4P{\lambda}_{\mathrm{max}}}{\left(1-{\delta}^2\right)\alpha {D}_e}\left[\left({S}_H-\frac{\lambda }{2}\right)\left({S}_H-\lambda \right)x+{x}^3\right]\right)}_{\lambda_{\mathrm{max}}}-{P}_{MAX}\ge 0 $$ (22b)
H 3 ( t ) = λ 1 − λ max ≥ 0 $$ {H}_3(t)={\lambda}_1-{\lambda}_{\mathrm{max}}\ge 0 $$ (22c)
H 4 ( t ) = H − S H − t ≥ 0 $$ {H}_4(t)=H-{S}_H-t\ge 0 $$ (22d)
H 5 ( t ) = D MAX − D e ≥ 0 $$ {H}_5(t)={D}_{MAX}-{D}_e\ge 0 $$ (22e)
H 6 ( t ) = D e − D i ≥ 0 $$ {H}_6(t)={D}_e-{D}_i\ge 0 $$ (22f)
b 7 ( t ) = 0.3 − S H D e − D i ≥ 0 $$ {b}_7(t)=0.3-\frac{S_H}{D_e-{D}_i}\ge 0 $$ (22g)
Here , α = 6 π ln J J − 1 J 2 $$ \mathrm{Here},\alpha =\frac{6}{\pi \ln J}{\left(\frac{J-1}{J}\right)}^2 $$
δ = 6 π ln J J − 1 ln J − 1 $$ \delta =\frac{6}{\pi \ln J}\left(\frac{J-1}{\ln J}-1\right) $$
μ = 6 π ln J J − 1 2 $$ \mu =\frac{6}{\pi \ln J}\left(\frac{J-1}{2}\right) $$
P MAX = 5400 lb $$ {P}_{\mathrm{MAX}}=5400\mathrm{lb} $$
P = 30 e 6 psi , λ max = 0.2 in , δ = 0.3 , G = 200 Kpsi , H = 2 in , D MAX = 12.01 in , J = D e D i , λ 1 = f ( a ) a , a = S H t $$ {\displaystyle \begin{array}{ll}P& =30e6\mathrm{psi},{\lambda}_{\mathrm{max}}=0.2 in,\delta =0.3,G=200 Kpsi,H=2 in,\\ {}{D}_{\mathrm{MAX}}& =12.01 in,J=\frac{D_e}{D_i},{\lambda}_1=f(a)a,a=\frac{S_H}{t}\end{array}} $$

It achieves the lowest average fitness value (8.714061) and demonstrates greater consistency with a lower standard deviation (4.261281). Compared to alternatives like GOOSE and WOA, which have higher fitness values and variability, IGOOSE provides more reliable performance. Additionally, IGOOSE maintains near-optimal final objective values, surpassing other algorithms like ALO and ABC. Overall, IGOOSE proves to be the most effective and stable solution for optimizing the Belleville Spring problem.

4.7 Cantilever Beam Problem Optimal Structure

Figure 17 shows the structure of cantilever beam problem. From Table 16, the proposed IGOOSE algorithm shows superior performance in optimizing the Cantilever Beam problem, achieving the lowest (L1, 5.229071), (L2, 4.960867), (L3, 7.868921), (L4, 2.956664), and (L5, 5.699115) values with a competitive fitness score of 1.303286.

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FIGURE 17
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Structure of cantilever beam problem.
Let us consider , L → = L 1 L 2 L 3 L 4 L 5 $$ \mathrm{Let}\ \mathrm{us}\ \mathrm{consider},\overrightarrow{L}=\left[{L}_1{L}_2{L}_3{L}_4{L}_5\right] $$ (23)
To minimize , f = 0.06224 L 1 + L 2 + L 3 + L 4 + L 5 , $$ \mathrm{To} \operatorname {minimize},f=0.06224\left({L}_1+{L}_2+{L}_3+{L}_4+{L}_5\right), $$ (23a)
Subjected to , g ( t ) = 61 L 1 3 + 27 L 2 3 + 19 L 3 3 + 7 L 4 3 + 1 L 5 3 ≤ 1 $$ \mathrm{Subjected}\ \mathrm{to},g(t)=\frac{61}{L_1^3}+\frac{27}{L_2^3}+\frac{19}{L_3^3}+\frac{7}{L_4^3}+\frac{1}{L_5^3}\le 1 $$ (23b)

Variable ranges are, 0.01 ≤ L 1 , L 2 , L 3 , L 4 , L 5 ≤ 100 $$ 0.01\le {L}_1,{L}_2,{L}_3,{L}_4,{L}_5\le 100 $$ .

This fitness value matches closely with GOOSE (1.303279) and outperforms other optimizers like PKO, EOA, and SMA, which all have slightly higher fitness values around 1.33996. This highlights IGOOSE's efficiency in producing balanced and optimized solutions. Figure 18 shows the comparative convergence curve for the Cantilever beam problem.

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FIGURE 18
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Comparative convergence curve for the cantilever beam problem.

4.8 I Beam Problem Optimal Structure

Figure 19 shows the structure of I beam problem. Table 17 presents the proposed IGOOSE algorithm statistical comparative analysis for I beam optimal problem.

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FIGURE 19
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Structure of I beam problem.
This problem's statistical expressions are given by Coello and Christiansen [93] as follows (Equation (24-24b)).
b s = b s 1 , b s 2 , b s 3 , b s 4 $$ {b}_s=\left[{b}_{s1},{b}_{s2},{b}_{s3},{b}_{s4}\right] $$ (24)
f b s = 5000 b s 3 b s 2 − 2 b s 4 3 12 + b s 1 b s 4 3 6 + 2 b s 1 b s 4 b s 2 − b s 4 2 4 $$ f\left({b}_s\right)=\frac{5000}{\frac{b_{s3}{\left({b}_{s2}-2{b}_{s4}\right)}^3}{12}+\frac{b_{s1}{b}_{s4}^3}{6}+2{b}_{s1}{b}_{s4}\frac{{\left({b}_{s2}-{b}_{s4}\right)}^2}{4}} $$ (24a)
Subject to,
g b s = 2 b s 1 b s 3 + b s 3 b s 1 − 2 b s 4 ≤ 0 $$ g\left({b}_s\right)=2{b}_{s1}{b}_{s3}+{b}_{s3}\left({b}_{s1}-2{b}_{s4}\right)\le 0 $$ (24b)
10 ≤ b s 1 ≤ 50 , 10 ≤ b s 2 ≤ 80 , 0.9 ≤ b s 3 ≤ 5 , 0.9 ≤ b s 4 ≤ 5 $$ 10\le {b}_{s1}\le 50,10\le {b}_{s2}\le 80,0.9\le {b}_{s3}\le 5,0.9\le {b}_{s4}\le 5 $$

From Table 17, the proposed IGOOSE algorithm outperforms other optimizers in the I-Beam optimization problem, achieving the lowest (I1, 34.68268) and (I2, 71.13198) values, with a competitive (I3, 1.576975) and (I4, 4.993863) [94-98]. It matches the best fitness value (0.006626) alongside GOOSE and SMA, significantly better than ARMS, IARMS, SOS, and CS (0.0131). This indicates IGOOSE's superior efficiency in optimizing structural design, offering more balanced and precise solutions compared to other methods. Figure 20 shows the comparative convergence curve for the I beam problem.

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FIGURE 20
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Comparative convergence curve for the I beam problem.

5 Conclusion

The Random Walk-based Goose Search (IGOOSE) algorithm leverages the balance between exploration and exploitation by integrating random walk behavior with local search strategy. Extensive testing across 23 benchmark functions with varying dimensions (10, 30, 50, and 100) has demonstrated IGOOSE algorithm efficacy in different optimization search landscapes, with assessments based on solution quality, convergence speed, and robustness. Beyond theoretical benchmarks, IGOOSE algorithm has been successfully applied to eight distinct structural engineering designs, showcasing its versatility and effectiveness in practical structural engineering scenarios, including mechanical and civil engineering. The proposed IGOOSE algorithm has demonstrated superior performance compared to existing algorithms, including classical GOOSE. Across various benchmark functions, IGOOSE consistently outperforms GOOSE and other state-of-the-art search algorithms such as TLBO, WOA, GSA, WSO, GWO, MPA, TSA, and MVO. For instance, in a 10-dimensional search space, IGOOSE achieves lower average, best, worst, and median values, as well as reduced standard deviation, indicating higher precision and robustness. This trend continues across 30-dimensional and different-dimensional search spaces, where IGOOSE consistently delivers comparable or better performance, particularly in handling complex, hybrid, and multimodal functions. In structural engineering applications, IGOOSE has excelled in various optimization problems. In the three-bar truss problem, IGOOSE achieved the lowest fitness score of 263.89598, outperforming classical GOOSE and other methods like ZOA and MFO. In the Speed Reducer problem, IGOOSE outperformed classical GOOSE and other methods like AO, SCA, and GA with the lowest fitness score of 2998.4190. Similarly, in the pressure vessel problem, IGOOSE achieved a significantly better fitness score of 5908.5040 compared to classical GOOSE, WOA, and GWO. Also, in the Welded Beam problem, IGOOSE outperformed other methods, achieving the lowest fitness score of 1.733172, demonstrating superior capability in generating optimized solutions. Further, IGOOSE has shown its superior performance in optimizing the Belleville Spring problem, by achieving the lowest average fitness value and greater consistency with a lower standard deviation compared to alternatives like classical GOOSE and WOA. In the Cantilever Beam problem, IGOOSE achieved the lowest fitness score of 1.303286, matching closely with classical GOOSE and outperforming other optimizers like PKO, EOA, and SMA. Similarly, in the I-Beam problem, IGOOSE achieved the best fitness value of 0.006626, matching GOOSE and SMA, and significantly outperforming methods like ARMS, IARMS, SOS, and CS. In each case study of evaluation, the proposed IGOOSE algorithm proves to be a highly effective and stable solution across a wide range of optimal problems, consistently outperforming its predecessor, classical GOOSE, and other state-of-the-art algorithms. Its ability to achieve high accuracy, consistency, and stability across different-dimensional and benchmark functions makes it a valuable addition to the toolkit of optimization search practitioners. IGOOSE algorithm however provides the very competitive results when compared to tested algorithms but still the application of various strategies may further improve the performance of the algorithm. These strategies may be the hybridization with other algorithms, application of machine learning, and inclusion of a chaotic map in the proposed algorithm etc. The testing of the algorithm however to some extent guarantees its better performance bit still for complex related to mechanical and electrical engineering problems, the search space becomes complex and the algorithm may stuck at local minima which may provide hindrance to finding global solution to the problem.

Author Contributions

Sripathi Mounika: conceptualization, methodology, visualization. Himanshu Sharma: validation, methodology, investigation. Aradhala Bala Krishna: conceptualization, formal analysis, software, data curation. Krishan Arora: validation, software, formal analysis. Syed Immamul Ansarullah: writing – original draft, investigation, data curation, resources. Ayodeji Olalekan Salau: formal analysis, methodology, writing – review and editing, validation.

Acknowledgments

The authors have nothing to report.

    Conflicts of Interest

    The authors declare no conflicts of interest.

    Data Availability Statement

    The data that support the findings of this study are available from the corresponding author upon reasonable request.

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