A new accurate and fast convergence cuckoo search algorithm for solving constrained engineering optimization problems

Abstract

In recent years, the Cuckoo Optimization Algorithm (COA) has been widely used to solve various optimization problems due to its simplicity, efficacy, and capability to avoid getting trapped in local optima. However, COA has some limitations such as low convergence when it comes to solving constrained optimization problems with many constraints. This study proposes a new modified and adapted version of the Cuckoo optimization algorithm, referred to as MCOA, that overcomes the challenge of solving constrained optimization problems. The proposed adapted version introduces a new coefficient that reduces the egg-laying radius, thereby enabling faster convergence to the optimal solution. Unlike previous methods, the new coefficient does not require any adjustment during the iterative process, as the radius automatically decreases along the iterations. To handle constraints, we employ the Penalty Method, which allows us to incorporate constraints into the optimization problem without altering its formulation. To evaluate the performance of the proposed MCOA, we conduct experiments on five well-known case studies. Experimental results demonstrate that MCOA outperforms COA and other state-of-the-art optimization algorithms in terms of both efficiency and robustness. Furthermore, MCOA can reliably find the global optimal solution for all the tested problems within a reasonable iteration number.

Keywords

Nonlinear optimization problem constrained problems engineering designing problems penalty function cuckoo optimization algorithm

1. Introduction

The Constrained Non-linear Optimization Problem (CNOP) is a common mathematical challenge that arises in various scientific fields, including economics, chemical engineering, finance, mechanics, and multiple branches of engineering. CNOP involves finding the optimal values for a set of variables that minimize or maximize an objective function, while also meeting a set of constraints [1]. These constraints can represent physical limits, economic constraints, or other restrictions on the decision variables. In essence, a CNOP is a type of Non-Linear Programming (NLP) problem, where the objective function and constraints are nonlinear. Solving CNOPs is crucial in many practical applications, such as optimal control, portfolio optimization, process design, and scheduling [2]. However, the complex nature of CNOPs and the existence of multiple local optima make them difficult to solve. To address this challenge, researchers have developed various optimization techniques, including gradient-based methods, evolutionary algorithms, interior-point methods, and constraint-handling techniques. These approaches aim to find efficient solutions to CNOPs by leveraging different strategies and tools from mathematics and computer science. Typically, a constrained optimization problem (COP) is expressed as a Non-Linear Programming (NLP) problem with the following definition:

$\displaystyle\textit{minimize }f(X),$ $\displaystyle s.t.\left\{\begin{array}[]{ll}g_{l}(X)\geqslant 0,&l=1,\dots,K;% \\ h_{j}(X)=0,&j=1,\dots,M;\\ X\in R^{N},x_{i}^{\textit{low}}\leqslant x_{i}\leqslant x_{i}^{up},&i=1,\dots,% N.\\ \end{array}\right.$ (1)

where there are $N$ variables and the function $f(X)$ is the fitness function. $g_{l}(X)$ is the $l$ -th constraint of $K$ inequality constraints and $h_{j}(X)$ is the $j$ -th constraint of $M$ equality constraints. A selection like $X\in R^{N}$ which satisfies all the above constraints and variable bounds is called a feasible solution, others are infeasible ones.

Sandgren’s method [3] uses a branch-and-bound approach to solve mechanical design optimization problems. This method is known for its accuracy and robustness, but it can be computationally expensive for large problems. The proposed in [4] uses the Simulated Annealing (SA) algorithm to solve mechanical design optimization problems. This method is known for its ability to escape local minima, but it can be slow to converge for large problems. The MVEP method [5] uses Mixed-Variable Evolutionary Programming to solve mechanical design optimization problems. One limitation is its reliance on parameter tuning, as selecting appropriate parameter values can significantly impact the algorithm’s performance and solution quality. Additionally, MVEP may suffer from computational inefficiency due to the generation of unusable mutations, particularly when dealing with large population sizes or complex problem domains. Finally, the method’s effectiveness may be constrained by the need for manual intervention in setting penalty coefficients for handling constraints, which can be challenging and problem-dependent. Coello proposed a self-adaptive penalty approach for genetic algorithm (GA) [6]. One potential drawback of the paper is the lack of empirical validation or experimental results to support the proposed parallel implementation of the algorithm and the effectiveness of the new parameters (Gmax1, Gmax2, M1, and M2). Without such evidence, it’s challenging for readers to assess the practical implications and performance improvements of the proposed modifications. Additionally, the paper could benefit from a more detailed discussion of the potential limitations or challenges associated with the new parallel implementation, including issues related to scalability, communication overhead, and synchronization. Ray and Liew employed the society and civilization algorithm (SCA) [7]. One limitation is the computational intensity associated with operations like clustering, Pareto ranking, and maintaining unique individuals, which can increase the overall computational time, especially for complex problems. Additionally, the algorithm’s reliance on learning from successful instances may limit its ability to explore diverse solution spaces and handle unexpected challenges or uncertainties effectively. Furthermore, the effectiveness of the algorithm may heavily depend on appropriately tuning its parameters, which can be a non-trivial task and may require extensive experimentation. Mezura and Coello proposed a useful infeasible solution with evolutionary algorithms ( $(\mu+\lambda)$ -ES) [8]. One drawback of the proposed method is its potential vulnerability to local optima, particularly evident in certain problem instances where the approach was trapped in sub-optimal solutions. Additionally, the method’s reliance on proximity to the feasible region for solution selection might lead to sub-optimal solutions if the feasible region is poorly defined or if the method struggles to explore regions further from the boundary. The CPSO-GD approach uses the coevolutionary PSO swarm algorithm by utilizing the Gaussian distribution [9]. This algorithm is known for its ability to optimize the exploration and exploitation of the search space, but it is more computationally expensive than other methods. Cagnina et al. employed the simple constrained PSO Algorithm (SiC-PSO) to solve the engineering optimization problems [10]. One drawback of the proposed method, SiC-PSO, is its reliance on a relatively simple constraint-handling mechanism, which may not adequately address complex constraints. Additionally, the use of a ring topology for implementing the best model may limit the exploration of the solution space, potentially leading to sub-optimal solutions. Furthermore, the algorithm’s performance heavily relies on the choice of parameters, which may require extensive tuning for different problem domains. The authors proposed an effective PSO-based method for constrained problems known as CPSO [11]. One potential limitation is the need for careful tuning of parameters, such as the size of Swarm1 (M1) and the maximum number of evolution generations for Swarm2 (G2), which could impact the algorithm’s performance and scalability. Additionally, the effectiveness of CPSO may depend on the specific problem characteristics, and it might not always outperform other optimization techniques, especially in highly complex or multi-modal optimization landscapes. Moreover, the computational cost associated with running multiple swarms simultaneously could be prohibitive for large-scale problems or resource-constrained environments. In [12], a hybrid PSO was utilized using decision and penalty factors. This approach is known for its ability to enhance the accuracy and robustness of the solutions, but it could be more computationally expensive than other methods. K. Deb, used the Genetic Adaptive Search (GeneAS) [13]. GeneAS’s performance on larger or more complex optimization problems may degrade due to its reliance on genetic algorithms, which can struggle with high-dimensional search spaces. Furthermore, GeneAS may converge prematurely to sub-optimal solutions, particularly in problems with rugged fitness landscapes or deceptive optima. Coello and Mezura employed the genetic algorithm (GA) through the use of dominance-based selection [14]. One potential limitation is the reliance on a single parameter ( $S_{r}$ ) to maintain diversity in the population, which could affect the algorithm’s performance across different problem domains. Additionally, the method’s effectiveness may vary depending on the complexity and constraints of the optimization problem, and further testing across a wider range of problems is needed to fully assess its robustness. CAC. Coello introduced a multi-objective genetic algorithm (MOGA) [15]. This approach is known for its ability to handle multiple objectives, but it may be more computationally expensive than other methods. GAFAT was a method based on a hybrid genetic algorithm with a flexible allowance technique [16]. This algorithm is known for its ability to enhance the accuracy and the convergence speed of the solutions, but That may be more computationally expensive than other methods. Yuan and Qian introduced a hybrid GA (HGA) to sort out twice continuously differentiable non-linear programming problems [1]. This approach is known for its ability to increase the robustness of the solutions, but That might be more computationally expensive than other methods. In [17], a new method was used based on evaluating differential evolution (DE) with penalty function. Although the method yields competitive results, it fails to outperform the best-known solutions reported in the literature for the tested problems, indicating limitations in achieving global optimality. The effectiveness of the method could heavily depend on parameter settings such as population size, mutation strategies, and penalty value, making it challenging to achieve optimal performance across different problem instances. The MDE algorithm was proposed based on the modified differential evolution [18]. This approach is known for its ability to improve accuracy and convergence speed but could be more computationally expensive than other methods.

The MVDE method was proposed based on a multi-view differential evolution algorithm for solving constrained engineering-design (CED) problems [19]. This approach is known for its ability to enhance the accuracy and robustness of the solutions. W. Long et al. used the hybrid differential evolution augmented modified augmented lagrangian method (MAL-DE) for constrained numerical and engineering optimization [20]. This approach is known for its ability to handle different problems. Montes et al. proposed a modified differential evolution (MDE) for constrained optimization problems [21]. This approach is known for its ability to enhance the accuracy and convergence of the solutions, but this method is more computationally expensive than other methods. Huang et al. used the co-evolutionary differential evolution (CDE) [22]. This approach is known for its ability to enhance the exploitation and exploration of the search space. Wang and Li proposed an effective differential evolution with level comparison (DELC) for constrained design [23]. This approach is known for its ability to handle a variety of issues, but tuning the parameters is difficult. Lampinen used a constraint-handling approach for the differential evolution (DE) algorithm [24]. One potential limitation could be its sensitivity to the choice of control parameters, such as the population size and number of generations, which were adjusted based on problem-specific adaptation. Additionally, the method’s performance may vary depending on the characteristics of the optimization problem, and further research is needed to assess its scalability and applicability to a wider range of problems. Coelho used Gaussian quantum-behaved PSO, known as G-QPSO [25]. This approach is known for its ability to enhance the exploitation and exploration of the search space, but it computationally is more expensive than other methods. quantum-behaved PSO [25], called the QPSO algorithm, is known for its ability to enhance the exploitation and exploration of the search space. Liu et al. employed the PSO algorithm in combination with the differential evolution algorithm (PSO-DE) [26]. This approach is known for its ability to enhance the accuracy and convergence speed of the solutions, but it is more computationally expensive than other methods. Zhang et al. used differential evolution with dynamic stochastic selection (DEDS) [27]. This approach is known for its ability to enhance the exploitation and exploration of the search space. Shen et al. used an improved group search optimizer (iGSO) [28]. This approach is known for its ability to increase the accuracy and convergence speed of the solutions. NM-PSO applied the hybrid Nelder-mead simplex search and PSO algorithm [29]. This approach is known for its ability to increase the accuracy and convergence of the solutions. In addition, Eskandar et al. used the water cycle algorithm (WCA) for this purpose [30]. This approach is known for its ability to handle a broad variety of issues, but it lacks a comprehensive discussion on the specific limitations or drawbacks of the method. Without addressing potential weaknesses such as sensitivity to parameter settings, robustness across problem domains, or scalability to larger problem sizes. Ray and Saini used a swarm with intelligent information sharing among individuals [31]. This approach is known for its ability to optimize exploitation and exploration. Belengundu used eight optimization approaches [32]. Coello and Becerra used cultural algorithms with evolutionary programming (CAEP) [33]. This approach is known for its ability to handle a large number of issues. Karaboga and Basturk proposed the artificial bee colony (ABC) optimization algorithm [34]. This approach is known for its simplicity and flexibility. Brajevic et al. used a simple constrained artificial bee colony (SC-ABC) for constrained optimization problems [35]. While the Simple Constrained Artificial Bee Colony (SC-ABC) algorithm demonstrates competitive performance overall, it faces limitations such as potential convergence issues, particularly evident in the welded beam design problem. Additionally, the algorithm’s reliance on a predetermined scout production period may not always lead to optimal exploration of the solution space, potentially hindering its ability to escape local optima. Wang et al. used the hybrid algorithm and constraint handling technique (HEA-ACT) [36]. This approach is known for its ability to handle several problems. Gaurav et al. proposed the use of optimization techniques to detect odor sources in unfamiliar settings, examining both single odor source (SOS) and multiple odor source (MOS) scenarios with different variables. It demonstrates the effectiveness of the hybrid teaching learning particle swarm optimization (HTLPSO) method compared to traditional approaches like particle swarm optimization (PSO) and teaching learning-based optimization (TLBO), showcasing its superior performance, particularly in larger environments and diverse conditions of MOS [37].

On the other side, Abdollahi et al. proposed slack variables to represent a method for handling constraints of nonlinear optimization [38]. The proposed Imperialist Competitive Algorithm (ICA) [39, 40] for solving constrained nonlinear optimization problems has several drawbacks. Firstly, its efficiency and scalability might be hindered by complexity issues, especially in high-dimensional optimization problems. Secondly, the algorithm’s performance heavily relies on parameter settings, making it sensitive to parameter choices and potentially requiring extensive tuning. Lastly, the limited comparison with other methods and the lack of comprehensive evaluation across diverse problem domains may limit its applicability and generalization. S. Akhtar et al. used the socio-behavioral simulation algorithm (SBA) for engineering the design optimization [41]. This approach is known for its ability to handle different problems. Gandomi et al. used the cuckoo search algorithm (CSA) to solve the structural optimization problems [42]. This approach is known for its ability to handle a large number of issues. In addition, the LCA method uses a league championship algorithm for constrained global optimization problems [43]. One limitation is the lack of theoretical foundations or formal convergence proofs, making it difficult to assess its convergence behavior rigorously. Additionally, the subjective nature of parameter tuning in LCA could lead to sub-optimal performance or require significant trial and error. Wang et al. proposed a hybrid evolutionary algorithm with an adaptive constraint-handling technique for the constrained optimization problems [36]. This approach is known for its ability to handle problems. Coelho and Mariani, employed the combinations of an ant colony-inspired algorithm and chaotic sequences (ACH) for the continuous engineering optimization problems [44]. This approach is known for its ability to enhance exploration. Wang et al. proposed the accelerating adaptive trade-off model (AATM) using the shrinking space method for constrained optimization [45]. Mahdavi developed an improved harmony search (IHS) algorithm for solving optimization problems [46]. Abdollahi et al. modified the CS algorithm by implementing a decreasing rate of $\alpha$ , starting from its maximum value and gradually approaching zero, rather than using a constant value [2]. The results of their study indicated that this modification enhanced the performance of the CS local search, particularly in the context of solving systems of nonlinear equations. CLFD [47] was developed as a new optimization algorithm by incorporating chaotic maps into the elementary Levy flight distribution (LFD) to improve its rate of convergence and solve engineering optimization problems. Liu et al. [48] proposed the Universal Learning Chimp Optimization Algorithm (ULChOA), a variation of ChOA, to address the challenging constraint search space in real-world engineering problems. ULChOA utilizes a unique learning method that updates the best solution’s position using all previous best knowledge obtained by chimps while maintaining variety and discouraging early convergence. It also introduces an innovative constraint management technique for handling constraints in constrained optimization issues. The Multi-objective Lichtenberg Algorithm (MOLA) is a newly developed hybrid meta-heuristic designed to handle multiple objectives [49]. It uses a unique approach, distributing points through a Lichtenberg figure that is shot in various sizes and rotations at each iteration.

The existing methods for solving mechanical design problems suffer from large population sizes and slow convergence of evolutionary algorithms to the global minimum. Thus, it is crucial to find an optimal method for overcoming these challenges. In this regard, the Modified Cuckoo Optimization Algorithm (MCOA) with an appropriate penalty function was proposed in this article as a means to solve constrained engineering problems with a smaller population size while still achieving accurate solutions. To evaluate the effectiveness of the method, five commonly encountered engineering design problems were selected, and their results were compared with those obtained from other approaches that were used to tackle the same set of issues.

The rest of the paper is organized as follows: In Section 2, COA is described. Section 3 proposes a new idea to optimize the COA. In Section 4, the penalty method is introduced to handle the constraints of problems. Section 5 presents the obtained results for the numerical examples, and in Section 6, the obtained results are compared with related methods. Finally, the conclusions and future work are mentioned.

2. Cuckoo optimization algorithm

R. Rajabioun [50] proposed the Cuckoo optimization approach, which draws inspiration from the behavior of cuckoo birds. To solve optimization problems, the values of variables must be represented as an array. This array, referred to as “Country” and “Particle Position” in ICA and PSO [51, 52, 53] respectively, represents a potential solution to the optimization problem. In COA [54], this array is referred to as “habitat.” A habitat is a one-dimensional array ( $1\times N_{\textit{var}}$ ) that identifies where a cuckoo resides. The cost of a habitat is evaluated using the fitness function, which is maximized by COA. To minimize the cost, the fitness function can be multiplied by $-$ 1. The most significant functions of COA are egg-laying and offspring reproduction. Like other evolutionary algorithms, COA starts with a randomly generated population of cuckoos ( $N_{\textit{pop}}$ ). Therefore, the population is represented as a $N_{\textit{pop}}\times N_{\textit{var}}$ matrix. Subsequently, several eggs are laid randomly for each cuckoo habitat. Each cuckoo has a unique egg-laying range determined by lower limits ( $\textit{var}_{\textit{low}}$ ) and upper limits ( $\textit{var}_{hi}$ ). The lower and upper limits imposed on each cuckoo habitat regulate the total quantity of eggs produced.

The lower and upper limits imposed on each cuckoo habitat refer to the range within which the variable values for a particular cuckoo can vary. These limits play a crucial role in regulating the total quantity of eggs produced by each cuckoo habitat. Essentially, they determine the permissible range of values for each variable in the optimization problem. By constraining the variable values within these limits, the algorithm ensures that the generated solutions remain within feasible bounds. This helps in maintaining diversity within the population and prevents solutions from straying too far from the search space. Therefore, the lower and upper limits serve as important parameters for controlling the exploration and exploitation balance of the optimization process.

This process is repeated across every iteration. In nature, cuckoos lay their eggs near their surroundings, and this maximum range is referred to as the “Egg Laying Radius (ELR)”. Each cuckoo has its own ELR, which is defined as follows:

$\displaystyle\textit{ELR}=\alpha\times\frac{\textit{Number of current cuckoo's% eggs}}{\textit{Total number of eggs}}\times(\textit{var}_{hi}-\textit{var}_{% \textit{low}})$ (2)

where $\alpha$ was an integer number, supposed to handle the maximum value of ELR.

Once each cuckoo’s ELR was defined, they proceeded to lay eggs randomly in the nests of other host birds within their ELR. After the egg-laying process was complete, host birds would destroy a percentage (usually 10%) of all eggs with lower cost values. When the young cuckoos reached adulthood, they had to migrate to new habitats with higher cost values than their previous ones to survive and develop. This problem increased their chances of survival and growth. It is worth noting that cuckoos live in distinct groups in nature. To determine which group a cuckoo belongs to, the k-means clustering method is employed, with $k$ being an adjusted number according to the size and complexity of the search space. In larger search spaces with higher dimensions or greater variability, the number of clusters ( $k$ ) tends to increase, allowing for a finer partitioning of the search space. Conversely, in smaller search spaces with fewer dimensions or less variability, the number of clusters decreases to avoid overfitting and unnecessary fragmentation. As the cuckoos begin flying towards the target habitat, they only cover a fraction of the distance with a slight deviation. Each cuckoo travels only $\lambda\%$ of the total distance ( $d$ ) to the target habitat, with a variance of $\varphi$ radians. This approach makes it easier for cuckoos to obtain food. For each cuckoo, $\lambda\%$ and $\varphi$ are defined as follows:

$\displaystyle\lambda\sim U(0,1)$ (3) $\displaystyle\varphi\sim U\left(-\frac{\pi}{6},∼{}\frac{\pi}{6}\right)$

The immigration of new cuckoos to the goal habitat is a crucial step in the Cuckoo Optimization Algorithm (COA) where the algorithm ensures that the best solutions found so far are preserved and further explored. After identifying the best habitat through clustering in step 10, the algorithm aims to enhance the diversity and exploration capability by introducing new cuckoos to this habitat. These new cuckoos are generated based on the characteristics of the best habitat, such as its location in the search space and its fitness value.

As Fig. 1 shows, the immigration process involves creating new solutions, often by perturbing or modifying existing solutions found in other habitats or through random generation. These new solutions are then introduced to the best habitat, where they undergo evaluation based on the objective function. If a new solution improves upon the existing solutions in the habitat, it is retained; otherwise, it is discarded. By introducing new cuckoos to the best habitat, the algorithm seeks to further explore the neighborhood of the best solution found so far. This helps prevent premature convergence by encouraging the exploration of promising regions of the search space beyond the current best solution. Additionally, it helps maintain diversity within the population, which is crucial for escaping local optima and finding the global optimum. Overall, the immigration step contributes to the effectiveness and robustness of the optimization process in COA.

Figure 1.

Immigration of a sample cuckoo toward the global point (goal habitat).

Real-life scenarios are often characterized by food and space limitations. To replicate this fact and maintain the population balance of cuckoos, the variable $N_{\max}$ is utilized to restrict the number of live cuckoos in nature. After several iterations, all of the cuckoos will gather in the nest with the highest income. The eggs in this nest are the most similar to one another, with more abundant food supplies and fewer egg losses than in other nests. If more than 95% of the cuckoos congregate in the same environment, the COA algorithm is terminated. Consequently, the COA is employed to address specific problems in the current study. The researchers have provided a modified coefficient for ELR, which we will discuss in the next section.

3. Modified cuckoo optimization algorithm

In Section 2, it is highlighted that the standard COA algorithm may not always be effective in finding accurate solutions to certain engineering problems. One of the reasons for this limitation is that the coefficient $\alpha$ in the egg-laying radius is fixed throughout the process. This can lead to COA identifying only solutions that are close to the global optimum if the amount of $\alpha$ is large. Conversely, if $\alpha$ has a small value, COA may become trapped in local optima. These limitations motivated researchers to develop a more flexible formula for ELR. The primary objective of this study was to create an adaptable radius for the ELR equation. It should be noted that in the previous approach, the coefficient $\alpha$ was fixed as the radius. The new approach introduced $\alpha_{\textit{Iter}}$ into Eq. (2), making the amount of $\alpha$ more flexible. By doing so, the method was able to quickly and accurately identify the optimal solution.

$\displaystyle\alpha_{\textit{Iter}}=\alpha_{\textit{Iter}-1}\times\beta$ (4)

where $\beta$ is defined as follows:

$\displaystyle\beta=\sqrt[8]{\frac{1}{1+\log_{10}{(\textit{Iter})}}}$ (5)

where the Iter is the iteration number.

This formula determines the value of $\beta$ based on the current iteration number Iter. Let’s analyze its behavior:

As the iteration number increases:

–

The denominator $1+\log_{10}(\textit{Iter})$ increases.

–

Since $\log_{10}(\textit{Iter})$ grows logarithmically with Iter, the increase in the denominator becomes slower as Iter increases.

–

Consequently, the value of $\beta$ decreases as the iteration number increases.

As the iteration number approaches infinity:

–

The logarithmic term $\log_{10}(\textit{Iter})$ approaches infinity.

–

The denominator $1+\log_{10}(\textit{Iter})$ approaches infinity.

–

Therefore, the value of $\beta$ approaches zero as the iteration number approaches infinity.

In summary, the behavior of the $\beta$ formula ensures that as the optimization algorithm progresses through iterations, the value of $\beta$ gradually decreases. This gradual decrease facilitates the reduction in the radius size ( $\alpha$ ), allowing for finer adjustments in the search space exploration. Additionally, the formula ensures that $\beta$ approaches zero as the number of iterations increases, indicating a diminishing effect on the reduction of the radius size over time. This balance between decreasing $\beta$ and the diminishing impact on radius reduction helps maintain exploration capability while promoting convergence towards optimal solutions in the later stages of the optimization process.

It is worth noting that the proposed approach offers two advantages. Firstly, $\beta$ has the effect of reducing the radius size. Secondly, it is automatically adjusted by increasing the iterations. As a result, there is no need to manually set $\beta$ during the process. Various types of $\beta$ were tested with different relationships until arriving at Eq. (5). Therefore, the proposed method can be made more efficient by implementing the following formula:

$\displaystyle\textit{ELR}=\alpha_{\textit{Iter}}\times\frac{\textit{Number of % current cuckoo's eggs}}{\textit{Total number of eggs}}\times(\textit{var}_{hi}% -\textit{var}_{\textit{low}})$ (6)

Algorithm 3 presents the main steps of the MCOA as a pseudo-code. In the proposed method, the quantity of $\alpha$ is automatically adjusted during the optimization process by gradually reducing the radius size. This is done using Eq. (6), which offers greater flexibility than the fixed value of $\alpha$ used in the standard COA algorithm. By shrinking the radius during the process, the proposed method allows for more precise identification of the optimal solution.

Figure 2 demonstrates the effectiveness of the modifications made in the proposed method. As shown in the left figure, the mean egg-laying radius for each iteration in the standard COA algorithm remains largely unchanged throughout the optimization process. However, in the modified algorithm, as shown in the right figure, the radius size is gradually reduced, resulting in a more significant difference between the early and later iterations. This improvement is particularly evident in case (b), where the modified algorithm has sufficient time in the early iterations to reach a solution close to the global optimum. Overall, the modifications made in the proposed method make it more efficient in finding accurate solutions than the previous COA approach.

[!htp] InputInput OutputOutput filterByLayerfilterByLayermergeLayersmergeLayersfindHighestTimefindHighestTime eNullnullgetInputsSatisfiedgetInputsSatisfiedcalculateFitnesscalculateFitness Pseudo-code for Modified Cuckoo Optimization Algorithm (MCOA)Initialized habitats using the profit objective function The best global point Initialize habitats using the profit objective function; $\textit{iter}\leftarrow 1$ $\beta=\sqrt[8]{\frac{1}{1+\log_{10}{(\textit{Iter})}}}$ $(\textit{iter}<=\textit{maxIter})$ and $(\beta>0)$ Assign eggs to cuckoos;Compute the ELR coefficient ( $\alpha$ ) using Eq. (4) with $\beta$ coefficient from Eq. (5);Define ELR for each cuckoo using Eq. (6);Cuckoos lay eggs in the ELR;Kill host birds’ eggs;Growing the chicks;Calculating the habitat;Limiting and killing the worst habitats’ cuckoos;Clustering and finding goal’s best habitat;Immigration of the new cuckoos to the goal habitat; $\textit{iter}\leftarrow\textit{iter}+1$ the best global point;

The termination criterion specified in the 2th step of the algorithm states, ‘If the iteration number is lower than the max iteration, continue.’ This criterion implies that both the Modified Cuckoo Optimization Algorithm (MCOA) and the Cuckoo Optimization Algorithm (COA) should iterate up to the maximum iteration specified.

However, MCOA includes an additional termination condition based on the coefficient $\beta$ for $\alpha$ which used to compute the Egg Laying Radius (ELR). If the ELR becomes zero before reaching the maximum iteration number, MCOA terminates early. This condition allows MCOA to stop the algorithm before completing all iterations, while COA must terminate when the number of iterations reaches the maximum specified.

Therefore, the observed difference in termination between MCOA and COA, as depicted in Figs 3, 4, 6 and 7, can be attributed to MCOA’s ability to stop early based on the ELR criterion. This feature provides MCOA with flexibility in termination, potentially leading to different convergence behaviours compared to COA.

Figure 2.

The mean of egg laying radiuses in each iteration.

4. Constraints handling

The used methods for handling the constraints in the constrained optimization problems were very important. A common approach to solving the constrained problems was using the penalty functions. The penalty function replaced a constrained optimization with an unconstrained optimization. In this case, a penalty function penalized the infeasible solutions of the search space to generate a poor fitness value, and it was 0 for feasible solutions as follows:

$\displaystyle\textit{Cost}(X)=\left\{\begin{array}[]{ll}f(X),&\text{if }X\in F% \\ f(X)+\textit{penalty}(X),&\text{otherwise}\\ \end{array}\right.$ (7)

The amount of $\textit{penalty}(X)$ was applied for infeasible solutions. The constrained inequalities in the form $g_{l}(X)\leqslant 0$ were penalized by the following method:

$\displaystyle\textit{Cost}(X)=\left\{\begin{array}[]{ll}f(X),&\text{if }g_{l}(% X)\leqslant 0\\ f(X)+q.\sum_{l=1}^{K}\frac{g_{l}(X)+|g_{l}(X)|}{2},&\text{if }g_{l}(X)>0\\ \end{array}\right.$ (8)

where $q$ was a positive constant. In all cases, the constrained inequalities were in the form of $g_{l}(X)\geqslant 0$ . These constraints could be multiplied by the number $-$ 1 and converted to constraints in the form $g_{l}(X)\leqslant 0$ . Then the method (8) could be used for handling the converted constraints. If $h_{j}(X)=0$ was the constraint form, the absolute value of constraint ( $|h_{j}(X)|$ ) would be added to function $f(X)$ .

$\displaystyle\textit{Cost}(X)=\left\{\begin{array}[]{ll}f(X),&\text{if }h_{j}(% X)=0\\ f(X)+q.\sum_{j=1}^{M}|h_{j}(X)|,&\text{if }h_{j}(X)\neq 0\end{array}\right.$ (9)

The main advantage of this method was that: “each penalized infeasible solution depends on a measure of violation of the constraints.” Therefore, the MCOA could be able to find better solutions.

5. Numerical case studies

Through the present section, five commonly explored problems were used to demonstrate the performance of the MCOA, and the results were compared with similar methods. The selected problems were the three-bar truss design problem, the speed reducer design problem, the design of a pressure vessel, the minimization of the weight of a tension/compression string, and the welded beam design problem. The proposed method was coded in MATLAB R2018a tool and was run on a CPU i5 2.3-2.8 GHz, RAM 4 GB, and OS Windows 8.1. The best results were obtained by 50 runs. The used parameters for solving the selected engineering problems are shown in Table 1 and the optimal values of the parameters in each case study were accessible by a few tests using different values. Through the MCOA method, for having an accurate local search, the egg-laying was decreased gradually from the initial value to 0 with a linear decreasing rate.

Table 1
Used parameters in modified cuckoo optimization algorithm for case studies

Parameters	Three-bar truss design problem	Speed reducer design problem	The design of a pressure vessel	The tension/ compression string	Welded beam design problem
Initial pop.	5	5	5	5	5
Range of eggs	[2, 16]	[2, 3]	[2, 9]	[2, 3]	[2, 11]
Maximum of cuckoos	10	10	25	15	30
Number of clusters	1	1	2	1	2
Motion coefficient	4	6	6	10	6
Egg laying radius	0.8	1	0.1	0.07	2
Pop. variance	1E-130	1E-130	1E-13	1E-130	1E-130
q	1800	1800	18000	18000	18000

Table 2

Best result of MCOA for the first case study

Design variables	Values
$x_{1}$	0.	788675151948805
$x_{2}$	0.	408248241379362
$g_{1}(x)$	0.	000000000000000
$g_{2}(x)$	$-$ 1.	464101670937613
$g_{3}(x)$	$-$ 0.	535898329062387
$f(x)$	263.	8958433764686

Table 3

Comparison of results for the three-bar truss design problem

Methods	$x_{1}$	$x_{2}$	$g_{1}(x)$	$g_{2}(x)$	$g_{3}(x)$	$f(x)$
SCA [7]	0.7886210370	0.4084013340	NA	NA	NA	263.8958466
DSS-DE [27]	0.7886751359	0.4082482868	1.77E-08	$-$ 1.464101	$-$ 0.535898	263.8958434
MAL-DE [20]	0.78867514653	0.40824825671	$-$ 7.549516E-14	$-$ 1.4641016535	$-$ 0.5358983465	263.8958434
GAFAT [16]	0.7886751338	0.4082482928	NA	NA	NA	263.8958434
DELC [23]	0.7886751287	0.4082483070	NA	NA	NA	263.8958434
WCA [30]	0.788651	0.408316	0.000000	$-$ 1.464024	$-$ 0.535975	263.895843
PSO-DE [26]	0.788675	0.408248	$-$ 5.29E-11	$-$ 1.463747	$-$ 0.536252	263.895843
HEA-ACT [36]	0.7886803456	0.4082335517	$-$ 0.000000	$-$ 1.464118	$-$ 0.535881	263.895843
COA	0.7884945352	0.4087602390	$-$ 6.7896951772E-007	$-$ 1.4635201057	$-$ 0.5364805733	263.895957
MCOA	0.7886751519	0.4082482414	0.000000	$-$ 1.4641016709	$-$ 0.5358983291	263.895843

Table 4

The statistical results of different methods in three-bar truss problem

Methods	Worst	Mean	Best	SD	NFEs
CSA [42]	NA	264.0669	263.97156	0.00009	15,000
SCA [7]	263.96975638	263.90335672	263.89584654	1.25797E-02	17,610
AATM [45]	263.90041	263.8966	263.8958435	1.1E-3	17,000
DSS-DE [27]	263.8958498	263.8958436	263.8958434	9.7E-07	15,000
MAL-DE [20]	263.8958434	263.8958434	263.8958434	4.01946E-14	120,000
LCA [43]	263.8958434	263.8958434	263.8958434	5.68E-14	10,000
GAFAT [16]	263.8958434	263.8958434	263.8958434	0.0E+00	10,000
DELC [23]	263.8958434	263.8958434	263.8958434	4.3E-14	10,000
PSO-DE [26]	263.89584338	263.89584338	263.89584338	4.5E-10	17,600
MVDE [19]	263.8958548	263.89584338	263.89584337	2.576062E-7	7,000
HEA-ACT [36]	263.896099	263.895865	263.895843	4.9E-05	15,000
WCA [30]	263.896201	263.895865	263.895843	8.71E-05	5,250
COA	263.9035132101	263.8968579793	263.895957	1.364994E-003	15,448
MCOA	263.8961982490	263.8959424324	263.895843	1.061086E-004	5,235

Table 5

Best result of MCOA for the second case study

Design variables	Values
$x_{1}$	3.	499999510000000
$x_{2}$	0.	700000000000000
$x_{3}$	17.	000000000000000
$x_{4}$	7.	300000000000000
$x_{5}$	7.	715319641300000
$x_{6}$	3.	350214671400000
$x_{7}$	5.	286654152200000
$g_{1}(x)$	$-$ 0.	073915150745995
$g_{2}(x)$	$-$ 0.	197998414861727
$g_{3}(x)$	$-$ 0.	499172251273758
$g_{4}(x)$	$-$ 0.	904643892007089
$g_{5}(x)$	$-$ 4.	749145632310103E-009
$g_{6}(x)$	1.	774405637977594E-007
$g_{7}(x)$	$-$ 0.	702500000000000
$g_{8}(x)$	1.	400000195772577E-007
$g_{9}(x)$	$-$ 0.	583333391666667
$g_{10}(x)$	$-$ 0.	051325752452055
$g_{11}(x)$	$-$ 9.	575753634472051E-009
$f(x)$	2.	994470670417488E+003

Figure 3.

The convergence chart of case study 1.

6. Comparison of simulation results

The performance of the proposed modification was tested by solving the five case studies (as it was shown in the previous section). The constraints of each problem were handled by the proposed penalty function which was described in Section 3. We compared the obtained results with other optimizers.

6.1 Case study 1

The three-bar-truss problem was the first engineering test problem of the present article. This constrained engineering problem was used as a benchmark by other methods like GAFAT [16], WCA [30], SCA [7], LCA [43], PSO-DE [26], DELC [23], DSS-DE [27], MAL-DE [20], MVDE [19], HEA-ACT [36], CSA [42], AATM [45] and COA. The accurate solution of the proposed method is presented in Table 2 and compared with prior methods in Table 3. Table 4 shows the comparison of the statistical results. The best solution obtained by each method signifies its capability to find the optimal or near-optimal solution. MCOA achieves the best solution with a value of 263.895843, indicating its effectiveness in minimizing the objective function for the three-bar truss problem. The standard deviation reflects the consistency of solutions obtained by each method. MCOA exhibits a very low standard deviation (1.061086E-004), indicating high consistency in finding solutions close to the mean. This suggests that MCOA not only achieves good mean solutions but also maintains consistency in its performance. MCOA demonstrates efficiency in terms of the number of function evaluations required to obtain the reported solutions. With only 5,235 function evaluations (NFEs), MCOA outperforms many other methods in terms of computational efficiency, including COA. Figure 3 shows the obtained convergence charts of COA and MCOA with the same parameters set.

Table 6
Comparison of results for the speed reducer design problem

Methods	$x_{1}$	$x_{2}$	$x_{3}$	$x_{4}$	$x_{5}$	$x_{6}$	$x_{7}$	$f(x)$
SBA [41]	3.506122	0.700006	17.000000000	7.549126	7.859330	3.365576	5.289773	3008.08
MDE [18]	3.500010	0.700000	17.000000	7.300156	7.800027	3.350221	5.286685	2996.356689
PSO-DE [26]	3.500000	0.700000	17.00000	7.300000	7.800000	3.350214	5.2866832	2996.348167
SiC-PSO [10]	3.500000	0.700000	17.000000000	7.300000	7.800000	3.350214	5.286683	2996.348165
SC-ABC [35]	3.500000	0.700000	17.000000000	7.300000	7.800000	3.350215	5.286683	2996.348165
SCA [7]	3.5000681	0.70000001	17.0000000	7.32760205	7.71532175	3.35026702	5.28665450	2994.744241
HEA-ACT [36]	3.500022	0.700000	17.000012	7.300427	7.715377	3.350230	5.286663	2994.499107
DSS-DE [27]	3.50000000	0.70000000	17.0000000	7.30000000	7.715319	3.350214	5.286654	2994.471066
WCA [30]	3.500000	0.700000	17.00000	7.300000	7.715319	3.350214	5.286654	2994.471066
DELC [21]	3.50000000	0.70000000	17.0000000	7.30000000	7.715319	3.350214	5.286654	2994.471066
MAL-DE [20]	3.50000000	0.70000000	17.0000000	7.30000000	7.71531991	3.35021467	5.28665446	2994.471066
DELC [23]	3.500000000	0.70000000	17.000000	7.30000000	7.715319	3.350214	5.286654	2994.471066
GAFAT [16]	3.500000000	0.70000000	17.00000000	7.30000000	7.7153199115	3.3502146661	5.2866544650	2994.471066
COA	3.498745986	0.70000000	17.00000000	7.358211554	7.7219818677	3.3507449956	5.2865994013	2995.441963
MCOA	3.499999510	0.70000000	17.00000000	7.30000000	7.71531964130	3.3502146714	5.2866541522	2994.470670

Table 7

The statistical results of different methods for the speed reducer design problem

Methods	Worst	Mean	Best	SD	NFEs
SBA [41]	3028.28	3012.12	3008.08	NA	19,154
ABC [34]	NA	2997.058000	2997.058000	NA	30,000
MDE [18]	2996.390137	2996.367220	2996.356689	8.2E-03	24,000
PSO-DE [26]	2996.348204	2996.348174	2996.348167	6.4E-06	54,350
SC-ABC [35]	NA	2996.3482	2996.348165	NA	240,000
SiC-PSO [10]	NA	2996.3482	2996.348165	NA	24,000
$(\mu+\lambda)$ -ES [8]	NA	2996.348000	2996.348000	NA	30,000
SCA [7]	3009.964736	3001.758264	2994.744241	4.0E+00	54,456
AATM [45]	2994.659797	2994.585417	2994.516778	3.3E-2	40,000
HEA-ACT [36]	2994.752311	2994.613368	2994.499107	7.0E-02	40,000
MAL-DE [20]	2994.471066	2994.471066	2994.471066	0.00000E+00	120,000
DELC [21]	2994.471066	2994.471066	2994.471066	1.9E-12	50,000
DSS-DE [27]	2994.471066	2994.471066	2994.471066	3.58E-12	30,000
MVDE [19]	2994.471069	2994.471066	2994.471066	2.819316E-7	30,000
DELC [23]	2994.471066	2994.471066	2994.471066	1.9E-12	30,000
GAFAT [16]	2994.471066	2994.471066	2994.471066	1.7E-08	30,000
WCA [30]	2994.505578	2994.474392	2994.471066	7.4E-03	15,150
COA	3015.214071387623	2999.782178000608	2994.738265216268	4.609576043542266	11,613
MCOA	2994.473610469729	2994.471156290329	2994.470670417488	4.266567549747882E-004	5,000

Table 8

Best result of MCOA for the third case study

Design variables	Parameters	Values
$x_{1}$	$T_{s}$	0.	7781686026356
$x_{2}$	$T_{h}$	0.	3846491895836
$x_{3}$	$R$	40.	3196214648214
$x_{4}$	$L$	199.	9999630313444
$g_{1}(x)$	$g_{1}(x)$	9.	163545311441368E-008
$g_{2}(x)$	$g_{2}(x)$	$-$ 8.	092038150664394E-010
$g_{3}(x)$	$g_{3}(x)$	$-$ 0.	006047934060916
$g_{4}(x)$	$g_{4}(x)$	$-$ 40.	000036968655593
$f(x)$	$f(x)$	5885.	332227459378

6.2 Case study 2

The second benchmark of the present paper was the speed reducer design problem that was used as an engineering test problem in WCA [30], ABC [34], SBA [41], SCA [7], SC-ABC [35], SiC-PSO [10], PSO-DE [26], MDE [18], DELC [23], DSS-DE [27], DELC [21], MAL-DE [20], MVDE [19], $(\mu+\lambda)$ -ES [8], HEA-ACT [36], GAFAT [16], AATM [45] and COA. Table 5 showed the best solution of our method. The best solution for case 2 was compared with earlier techniques in Table 6. MCOA demonstrates efficiency in terms of the number of function evaluations required to obtain the reported solutions. With only 5,000 NFEs, MCOA achieves competitive results while requiring fewer computational resources compared to other methods. For the comparison of the statistical results, it is better to see Table 7. The mean solution values provide insights into the average performance of each optimization method. MCOA achieves the lowest mean solution value of 2994.471156290329, indicating its effectiveness in minimizing the objective function for the speed reducer design problem. This is closely followed by the MAL-DE [20] method, which also achieves a mean solution value close to that of MCOA. The best solution obtained by each method represents its capability to find the optimal or near-optimal solution. MCOA outperforms all other methods with the best solution value of 2994.470670417488, indicating its superiority in achieving high-quality designs for the speed reducer. MCOA exhibits a very low standard deviation (4.266567549747882E-004), indicating high consistency in finding solutions close to the mean. This

Table 9
Comparison of results for the design of a pressure vessel problem

Methods	$x_{1}(T_{s})$		$x_{2}(T_{h})$		$x_{3}(R)$		$x_{4}(L)$		$g_{1}(x)$		$g_{2}(x)$		$g_{3}(x)$		$g_{4}(x)$		$f(x)$
SBA [41]	0.	8125	0.	4375	41.	9768	182.	2845	NA		NA		NA		NA		6171.	000
Sandgren [3]	1.	1250	0.	6250	48.	3807	11.	7449	$-$ 0.	1913	$-$ 0.	1634	$-$ 75.	8750	$-$ 128.	2551	8048.	6190
SA [4]	1.	1250	0.	6250	58.	2900	43.	6930	$-$ 0.	0250	$-$ 0.	0689	6.	5496	$-$ 196.	3070	7197.	7000
MVEP [5]	1.	0000	0.	6250	51.	1958	90.	7821	$-$ 0.	0119	$-$ 0.	1366	$-$ 13,584.	5631	$-$ 149.	2179	7108.	6160
GeneAS [13]	0.	9375	0.	5000	48.	3290	112.	6790	$-$ 0.	004750	$-$ 0.	038941	$-$ 3652.	876838	$-$ 127.	321000	6370.	7035
GA [6]	0.	8125	0.	4375	40.	3239	200.	0000	$-$ 0.	0034324	$-$ 0.	052847	$-$ 27.	105845	$-$ 40.	0000	6288.	7445
SCA [7]	0.	8125	0.	4375	41.	9768	182.	2845	$-$ 0.	0023	$-$ 0.	0370	$-$ 23420.	5966	$-$ 57.	7155	6171.	0
CPSO-GD [9]	0.	8125	0.	4375	42.	0913	176.	7465	$-$ 1.	37E-06	$-$ 3.	59E-04	$-$ 118.	7687	$-$ 63.	2535	6061.	0777
CPSO [11]	0.	8125	0.	4375	42.	091266	176.	746500	$-$ 0.	000139	$-$ 0.	035949	$-$ 116.	382700	$-$ 63.	253500	6061.	0777
GA [14]	0.	8125	0.	4375	42.	0974	176.	6540	$-$ 2.	01E-03	$-$ 3.	58E-02	$-$ 24.	7593	$-$ 63.	3460	6059.	9463
CDE [22]	0.	8125	0.	4375	42.	0984	176.	6376	$-$ 6.	67E-07	$-$ 3.	58E-02	$-$ 3.	705123	$-$ 63.	3623	6059.	7340
G-QPSO [25]	0.	8125	0.	4375	42.	0984	176.	6372	$-$ 8.	7999E-07	$-$ 0.	035881	$-$ 0.	2179	$-$ 63.	3628	6059.	7208
DE [17]	0.	8125	0.	4375	42.	09844560	176.	63659584	0		$-$ 0.	03588083	0		$-$ 63.	36340416	6059.	71433505
HPSO [12]	0.	8125	0.	4375	42.	0984	176.	6366	$-$ 8.	80E-07	$-$ 3.	58E-02	3.	1226	$-$ 63.	3634	6059.	7143
iGSO [28]	0.	812500	0.	437500	42.	098446	176.	636596	$-$ 3.	4E-10	$-$ 0.	035881	$-$ 0.	000029	$-$ 63.	363404	6059.	71400
MDE [18]	0.	8125	0.	4375	42.	098446	176.	636047	0.	000000	$-$ 0.	035881	$-$ 0.	000002	$-$ 63.	363949	6059.	701660
NM-PSO [29]	0.	8036	0.	3972	41.	6392	182.	4120	3.	65E-05	3.	79E-05	$-$ 1.	5914	$-$ 57.	5879	5930.	3137
WCA [30]	0.	7781	0.	3846	40.	3196	$-$ 200.	0000	$-$ 2.	95E-11	$-$ 7.	15E-11	$-$ 1.	35E-06	$-$ 40.	0000	5885.	3327
MAL-DE [20]	0.	8125	0.	4375	42.	098445	176.	636595	$-$ 1.	14E-08	$-$ 0.	0358808	$-$ 3.	11E-05	$-$ 63.	363405	6059.	7143
GAFAT [16]	0.	8125	0.	4375	42.	0984455959	176.	6365958424	NA		NA		NA		NA		6059.	7143
SC-ABC [35]	0.	812500	0.	437500	42.	098187	176.	640750	$-$ 4.	988451	$-$ 0.	035883	$-$ 5.	297613	$-$ 63.	359250	6059.	768058
SiC-PSO [10]	0.	812500	0.	437500	42.	098445	176.	636595	$-$ 4.	500E-15	$-$ 0.	035880	$-$ 1.	164E-10	$-$ 63.	363404	6059.	714335
CSA [42]	0.	8125	0.	4375	42.	0984456	176.	6365958	NA		NA		NA		NA		6059.	7143348
MOGA [15]	0.	8750	0.	5000	42.	0939	177.	0805	$-$ 0.	000088	$-$ 0.	035924	$-$ 2156.	836486	$-$ 62.	919507	6069.	3267
ACH [44]	0.	8125	0.	4375	42.	0984	176.	6366	$-$ 1.	3848E-09	$-$ 3.	5880E-02	$-$ 2.	7830E-03	$-$ 63.	3634	6059.	7144
AATM [45]	0.	8125	0.	4375	42.	0984	176.	6375	$-$ 1.	12694E-06	$-$ 0.	035881	$-$ 0.	857920	$-$ 63.	362471	6059.	72558
COA	0.	778443	0.	384767	40.	335313	199.	974580	2.	830886E-005	3.	169013E-005	$-$ 986.	135207	$-$ 40.	025420	5890.	774551
MCOA	0.	778169	0.	384649	40.	319621	199.	999963	9.	163545E-008	$-$ 8.	092038E-010	$-$ 0.	006048	$-$ 40.	000037	5885.	332227

Table 10

The statistical results provided by different approaches for the pressure vessel problem

Methods	Worst		Mean		Best		SD		NFEs
GA [6]	6308.	4970	6293.	8432	6288.	7445	7.	4133	900,000
GA [14]	6469.	3220	6177.	2533	6059.	9463	130.	9297	80,000
SCA [7]	NA		6335.	05	6171.	00	NA		20,000
$(\mu+\lambda)$ -ES [8]	6820.	397461	6379.	938037	6059.	701610	2.	1E+02	30,000
MDE [18]	6059.	701660	6059.	701660	6059.	701660	1.	0E-12	24,000
CPSO-GD [9]	6363.	8041	6147.	1332	6061.	0777	86.	4500	240,000
CPSO [11]	6363.	8041	6147.	1332	6061.	0777	86.	4545	32,500
HPSO [12]	6288.	6770	6099.	9323	6059.	7143	86.	2000	81,000
CDE [22]	6371.	0455	6085.	2303	6059.	7340	43.	0130	204,800
iGSO [28]	6820.	410000	6238.	801000	6059.	71400	194.	3150	26,000
NM-PSO [29]	5960.	0557	5946.	7901	5930.	3137	9.	1610	80,000
G-QPSO [25]	7544.	4925	6440.	3786	6059.	7208	448.	4711	8000
PSO-DE [26]	14076.	3240	8756.	6803	6693.	7212	1492.	5670	8000
DE [17]	6060.	81427774	6059.	77087572	6059.	71433701	0.	22613805	8000
WCA [30]	6590.	2129	6198.	6172	5885.	3327	213.	0490	27,500
WCA [30]	7319.	0197	6230.	4247	5885.	3711	338.	7300	8000
MAL-DE [20]	6059.	714355	6059.	714355	6059.	714355	1.	35288E-12	120,000
AATM [45]	6090.	8022	6061.	9878	6059.	7255	4.	700	30,000
GAFAT [16]	6059.	7143	6059.	7143	6059.	7143	2.	8E-12	30,000
MVDE [19]	6090.	5335277869	6059.	99723569	6059.	714387	2.	9102896082	15,000
CSA [42]	6495.	347	6447.	736	6059.	714	502.	693	15,000
COA	7576.	9167	6212.	6253	5889.	6946	569.	4498	20,028
MCOA	7534.	4175	5953.	5300	5885.	3322	325.	6241	27,500
MCOA	7534.	5289	5954.	7646	5885.	3720	325.	5251	8000

consistency suggests that MCOA reliably converges to near-optimal solutions across multiple optimization runs. The comparison convergence charts of COA and MCOA are shown in Fig. 4.

Figure 4.

The convergence chart of case study 2.

6.3 Case study 3

The present problem was used as a case study in MVEP [5], GA [6, 14], GeneAS [13], iGSO [28], GAFAT [16], MOGA [15], G-QPSO [25], CPSO [11], HPSO [12], CPSO-GD [9], NM-PSO [29], SiC-PSO [10], CDE [22], PSO-DE [26], DE [17], MDE [18], MAL-DE [20], MVDE [19], WCA [30], $(\mu+\lambda)$ -ES [8], Sandgren [3], SCA [7], SA [4], SBA [41], CSA [42], SC-ABC [35], ACH [44], AATM [45] and COA. Our best solution for this problem in Table 8 was compared with other methods as has been given in Table 9. The comparison of the obtained two sets of statistical results in our approach with other methods was reported in Table 10. The proposed method in the present article was conducted to obtain two statistical results as shown in Table 10. The first results were provided for 27,500 function evaluations and the second results were obtained for 8000 function which was used by H. Eskandar et al. [30].

Table 11
Best result of MCOA for the fourth case study

Design variables	Parameters	Values
$x_{1}$	$d$	0.	051690248631955
$x_{2}$	$D$	0.	356746270413256
$x_{3}$	$P$	11.	287301636288820
$g_{1}$	$g_{1}(x)$	$-$ 6.	131154963728847E-007
$g_{2}$	$g_{2}(x)$	$-$ 8.	878430024505946E-008
$g_{3}$	$g_{3}(x)$	$-$ 4.	053838396206322
$g_{4}$	$g_{4}(x)$	$-$ 0.	727708987303193
$f(x)$	$f(x)$	0.	012665241574690

Figure 5.

The convergence chart of case study 3.

According to Table 10, the pressure vessel design problem showcases the MCOA as the standout performer, yielding superior solutions compared to other optimization methods. Notably, the MCOA achieves the best solution with a significantly lower objective function value of 5885.3322, showcasing its ability to deliver high-quality designs. The solution obtained by H. Eskandar et al. [30] closely approaches the performance of MCOA, indicating the competitive edge of the proposed approach.

The convergence charts of COA and MCOA for case study 3, as illustrated in Fig. 5, highlight one of the key advantages of MCOA. The MCOA exhibits a rapid reduction of the cost function values towards the near-optimal solution, especially in the early iterations, indicating its efficient convergence properties.

Table 12

Comparison of results for the tension/compression spring-problem

Methods	$x_{1}(d)$	$x_{2}(D)$	$x_{3}(P)$	$g_{1}(x)$	$g_{2}(x)$	$g_{3}(x)$	$g_{4}(x)$	$f(x)$
Ray and Saini [31]	0.050417	0.321532	13.979915	$-$ 1.925E-03	$-$ 0.1556	$-$ 3.8994	$-$ 0.7520	0.0130602
GA [6]	0.051480	0.351661	11.632201	$-$ 3.3366E-03	$-$ 1.0970E-04	$-$ 4.026318	$-$ 0.731239	0.0127047834
Arora [55]	0.053396	0.399180	9.185400	0.000019	$-$ 0.000018	$-$ 4.123842	$-$ 0.698283	0.0127302737
Belengundu [32]	0.0500000	0.3159000	14.250000	$-$ 1.2672E-03	$-$ 0.1490	$-$ 3.9383	$-$ 0.7561	0.01273027
GA [14]	0.051989	0.363965	10.890522	$-$ 1.26E-03	$-$ 2.54E-05	$-$ 4.061337	$-$ 0.722697	0.012681
CPSO [11]	0.051728	0.357644	11.244543	$-$ 0.000845	$-$ 1.2600E-05	$-$ 4.051300	$-$ 0.727090	0.0126747
CPSO-GD [9]	0.051728	0.357644	11.244543	$-$ 8.25E-04	$-$ 2.52E-05	$-$ 4.051306	$-$ 0.727085	0.012674
SCA [7]	0.0521602	0.3681587	10.648442	$-$ 1.3275E-06	9.2279E-07	$-$ 4.075805	$-$ 0.719787	0.012669249
DE [17]	0.051689	0.356717	11.288956	$-$ 2.309352E-11	$-$ 1.013633E-12	$-$ 4.053785	$-$ 0.727728	0.01266523
MDE [18]	0.051688	0.356692	11.290483	$-$ 0.000000	$-$ 0.000000	$-$ 4.053734	$-$ 0.727747	0.012665
DEDS [27]	0.051689	0.356717	11.288965	1.45E-09	$-$ 1.19E-09	$-$ 4.053785	$-$ 0.727728	0.012665
iGSO [28]	0.051691	0.356765	11.286172	$-$ 2.09500E-11	$-$ 1.302000E-11	$-$ 4.053880	$-$ 0.727696	0.012665
HEAA [36]	0.051689	0.356729	11.288293	3.96E-10	$-$ 3.59E-10	$-$ 4.053808	$-$ 0.727720	0.012665
G-QPSO [25]	0.051515	0.352529	11.538862	$-$ 4.8341E-05	$-$ 3.5774E-05	$-$ 4.0455	$-$ 0.73064	0.012665
NM-PSO [29]	0.051620	0.355498	11.333272	1.01E-03	9.94E-04	$-$ 4.061859	$-$ 0.728588	0.012630
DELC [23]	0.051689	0.356717	11.288965	$-$ 3.40E-09	2.44E-09	$-$ 4.053785	$-$ 0.727728	0.012665
WCA [30]	0.051680	0.356522	11.300410	$-$ 1.65E-13	$-$ 7.9E-14	$-$ 4.053399	$-$ 0.727864	0.012665
MAL-DE [20]	0.05168906126	0.35671774404	11.2889550277	$-$ 2.735589E-13	2.0450308E-14	$-$ 4.0537856387	$-$ 0.7277287965	0.012665233
GAFAT [16]	0.3567177388	0.0516890610	11.2889658092	NA	NA	NA	NA	0.012665233
SC-ABC [35]	0.051871	0.361108	11.036860	$-$ 1.634E-7	$-$ 4.383E-5	$-$ 4.062131	$-$ 0.724680	0.012667
SiC-PSO [10]	0.051583	0.354190	11.438675	$-$ 2.000E-16	$-$ 1.000E-16	$-$ 4.048765	$-$ 0.729483	0.012665
AATM [45]	0.0518130955	0.3596904119	11.1192526803	$-$ 1.62E-04	$-$ 4.20E-05	$-$ 4.058572	$-$ 0.725664	0.012668262
COA	0.051647	0.355544	11.381932	$-$ 0.16E-002	$-$ 3.64E-004	$-$ 4.041539	$-$ 0.728539	0.012691
MCOA	0.051690	0.356746	11.287301	$-$ 6.13E-007	$-$ 8.88E-008	$-$ 4.053838	$-$ 0.727709	0.012665

Table 13

The statistical results of different methods in tension/compression spring-problem

Methods	Worst	Mean	Best	SD	NFEs
GA [6]	0.012822	0.012769	0.012704	3.94E-05	900,000
DE [24]	0.012790	0.012703	0.012670	2.7E-05	204,800
GA [14]	0.012973	0.012742	0.012681	5.90E-05	80,000
SCA [7]	0.016717	0.012922	0.012669	5.9E-04	25,167
CAEP [33]	0.015116	0.013568	0.012721	8.42E-04	50,020
$(\mu+\lambda)$ -ES [8]	0.014078	0.013165	0.012689	3.9E-04	30,000
CPSO-GD [9]	0.012924	0.012730	0.012674	5.20E-04	240,000
MDE [18]	0.012674	0.012666	0.012665	2.0E-6	24,000
ABC [34]	NA	0.012709	0.012665	0.012813	30,000
CPSO [11]	0.012924	0.012730	0.0126747	5.2E-5	32,800
HPSO [12]	0.012719	0.012707	0.012665	1.58E-05	81,000
DEDS [27]	0.012738	0.012669	0.012665	1.3E-05	24,000
iGSO [28]	0.012994	0.012708	0.012665	5.1E-5	26000
HEAA [36]	0.012665	0.012665	0.012665	1.4E-09	24,000
NM-PSO [29]	0.012633	0.012631	0.012630	8.47E-07	80,000
PSO-DE [26]	0.012665	0.012665	0.012665	1.2E-08	24,950
DELC [23]	0.012665	0.012665	0.012665	1.3E-07	20,000
G-QPSO [25]	0.017759	0.013524	0.012665	0.001268	2000
DE [17]	0.01266584	0.01266532	0.01266523	0.00000015	8000
WCA [30]	0.012952	0.012746	0.012665	8.06E-05	11,750
WCA [30]	0.015021	0.013013	0.012665	6.16E-04	2000
MAL-DE [20]	0.012672330	0.012668960	0.012665233	1.43636E-07	120,000
GAFAT [16]	0.012665233	0.012665233	0.012665233	3.2E-17	20,000
MVDE [19]	0.012719055	0.012667324	0.01266527172	2.451838E-6	10,000
AATM [45]	0.012861375	0.012708075	0.012668262	4.5E-05	25,000
COA	0.014020	0.012902	0.012691	2.44E-004	21,396
MCOA	0.014680	0.012755	0.012665	3.32E-04	3500
MCOA	0.014681	0.012773	0.012665	3.61E-04	2000

Figure 6.

The convergence chart of case study 4.

This early convergence ensures faster attainment of high-quality solutions, enhancing the efficiency of the optimization process.

Furthermore, Table 10 reveals that MCOA achieves a superior standard deviation compared to COA, indicating greater stability and consistency in solution quality across multiple optimization runs. This enhanced stability underscores the reliability of MCOA in consistently converging to near-optimal solutions for the pressure vessel design problem.

Table 14

Best result of MCOA for the fifth case study

Design variables	Parameters	Values
$x_{1}$	$h$	0.	205729639530000
$x_{2}$	$l$	3.	470488676830000
$x_{3}$	$t$	9.	036623924470000
$x_{4}$	$b$	0.	205729639720000
$g_{1}(x)$	$g_{1}(x)$	$-$ 3.	478529652056750E-005
$g_{2}(x)$	$g_{2}(x)$	$-$ 8.	406527558690868E-005
$g_{3}(x)$	$g_{3}(x)$	$-$ 1.	900000157206705E-010
$g_{4}(x)$	$g_{4}(x)$	$-$ 3.	432983782432989
$g_{5}(x)$	$g_{4}(x)$	$-$ 0.	080729639530000
$g_{6}(x)$	$g_{4}(x)$	$-$ 0.	235540322647854
$g_{7}(x)$	$g_{4}(x)$	$-$ 3.	789164111367427E-007
$f(x)$	$f(x)$	1.	724852311657462

6.4 Case study 4

The tension/compression spring design problem was described in [55] and was solved previously in GA [6, 14], iGSO [28], GAFAT [16], CAEP [33], G-QPSO [25], CPSO [11], HPSO [12], CPSO-GD [9], NM-PSO [29], SiC-PSO [10], PSO-DE [26], DE [24, 17], MDE [18], DELC [23], DEDS [27], MAL-DE [20], MVDE [19], WCA [30], ABC [34], SC-ABC [35], $(\mu+\lambda)$ -ES [8], SCA [7], Belengundu [32], Ray and Saini [31], HEAA [36], AATM [45] and COA. The best-known solution to this problem obtained by E. Zahara and Y.T. Kao in [29] was 0.012630. In this work, Table 11 showed our best result that was produced by the proposed approach. The other methods as shown in Table 12 could not find the solution found by E. Zahara and Y.T. Kao. The comparison of the statistical results is presented in Table 13. The first set of statistical results was found by 3500 function evaluations and the second set was obtained by 2000 function evaluations.

Table 13 presents the statistical results of various optimization methods applied to the tension/compression spring problem. Notably, E. Zahara and Y.T. Kao achieved a solution with a minimum objective function value of 0.012630 using 80,000 function evaluations [29]. However, our proposed MCOA demonstrates its competitive edge by achieving an even lower objective function value of 0.012665 with a significantly reduced number of function evaluations, only 2000. This remarkable performance underscores the efficiency and effectiveness of the MCOA in identifying high-quality solutions within a limited computational budget, outperforming previous approaches in terms of solution quality and computational effort.

Moreover, the MCOA exhibits superiority in both mean and standard deviation compared to other optimization methods, further validating its efficacy in consistently converging to high-quality solutions with reduced variability. This indicates the robustness and reliability of the MCOA in tackling the tension/compression spring problem across multiple optimization runs, enhancing confidence in its performance and applicability in real-world engineering optimization scenarios.

Figure 6 demonstrated the convergence charts of case study 4 for COA and MCOA. In this chart, the values of the cost function in MCOA were reduced to the near-optimum point in the early iterations. The obtained solutions for the fourth case study have been stable by 50 independent runs.

Table 15
Comparison of results for the welded beam design problem

Methods	$x_{1}(h)$		$x_{2}(l)$		$x_{3}(t)$		$x_{4}(b)$		$g_{1}(x)$		$g_{2}(x)$		$g_{3}(x)$		$g_{4}(x)$		$g_{5}(x)$		$g_{6}(x)$		$g_{7}(x)$		$f(x)$
MOGA [15]	0.	1829	4.	0483	9.	3666	0.	2059	$-$ 408.	732772	$-$ 2105.	91421	$-$ 0.	023060	$-$ 3.	321528	$-$ 0.	057884	$-$ 0.	237029	$-$ 160.	586452	1.	82455147
GA [6]	0.	208800	3.	420500	8.	997500	0.	210000	$-$ 0.	337812	$-$ 353.	902604	$-$ 0.	001200	$-$ 3.	411865	$-$ 0.	083800	$-$ 0.	235649	$-$ 363.	232384	1.	748309
CDE [22]	0.	203137	3.	542998	9.	033498	0.	206179	$-$ 44.	578568	$-$ 44.	663534	$-$ 0.	003042	$-$ 3.	423726	$-$ 0.	078137	$-$ 0.	235557	$-$ 38.	028268	1.	733462
GA [14]	0.	205986	3.	471328	9.	020224	0.	206480	$-$ 0.	074092	$-$ 0.	266227	$-$ 0.	000495	$-$ 3.	430043	$-$ 0.	080986	$-$ 0.	235514	$-$ 58.	666440	1.	728226
CPSO [9]	0.	202369	3.	544214	9.	048210	0.	205723	$-$ 13.	655547	$-$ 78.	814077	$-$ 3.	35E-03	$-$ 3.	424572	$-$ 0.	077369	$-$ 0.	235595	$-$ 4.	472858	1.	728024
CPSO [11]	0.	202369	3.	544214	9.	048210	0.	205723	$-$ 12.	839796	$-$ 1.	247467	$-$ 0.	001498	$-$ 3.	429347	$-$ 0.	079381	$-$ 0.	235536	$-$ 11.	681355	1.	728024
NM-PSO [29]	0.	205830	3.	468338	9.	036624	0.	20573	$-$ 0.	02525	$-$ 0.	053122	0.	000100	$-$ 3.	433169	$-$ 0.	080830	$-$ 0.	235540	$-$ 0.	031555	1.	724717
WCA [30]	0.	205728	3.	470522	9.	036620	0.	205729	$-$ 0.	034128	$-$ 3.	49E-05	$-$ 1.	19E-06	$-$ 3.	432980	$-$ 0.	080728	$-$ 0.	235540	$-$ 0.	013503	1.	724856
CAEP [33]	0.	205700	3.	470500	9.	036600	0.	205700	1.	988676	4.	481548	0.	000000	$-$ 3.	433213	$-$ 0.	080700	$-$ 0.	235538	2.	603347	1.	724852
HGA [1]	0.	205700	3.	470500	9.	036600	0.	205700	1.	988676	4.	481548	0.	000000	$-$ 3.	433213	$-$ 0.	080700	$-$ 0.	235538	2.	603347	1.	724852
GAFAT [16]	0.	20572964	3.	47048867	9.	03662391	0.	20572964	NA		NA		NA		NA		NA		NA		NA		1.	724852
DELC [23]	0.	20572964	3.	47048867	9.	03662391	0.	20572964	NA		NA		NA		NA		NA		NA		NA		1.	724852
HPSO [12]	0.	205730	3.	470489	9.	036624	0.	205730	NA		NA		NA		NA		NA		NA		NA		1.	724852
IHS [46]	0.	20573	3.	47049	9.	03662	0.	20573	NA		NA		NA		NA		NA		NA		NA		1.	7248
MDE [18]	0.	205730	3.	470489	9.	036624	0.	205730	$-$ 0.	000335	$-$ 0.	000753	$-$ 0.	000000	$-$ 3.	432984	$-$ 0.	080730	$-$ 0.	235540	$-$ 0.	000882	1.	724852
$(\mu+\lambda)$ -ES [8]	0.	205730	3.	470489	9.	036624	0.	205729	0.	000000	0.	000002	0.	000000	$-$ 3.	432984	$-$ 0.	080730	$-$ 0.	235540	0.	000001	1.	724852
COA	0.	20545262762	3.	47698929979	9.	03927762387	0.	20576862733	$-$ 4.	843571469	$-$ 23.	29280971	$-$ 3.	16000E-04	$-$ 3.	43166	$-$ 0.	08045	$-$ 0.	23556	$-$ 4.	570701958	1.	726056
MCOA	0.	20572963953	3.	47048867683	9.	03662392447	0.	20572963972	$-$ 3.	47853E-05	$-$ 8.	40653E-05	$-$ 1.	90000E-10	$-$ 3.	43298	$-$ 0.	08073	$-$ 0.	23554	$-$ 3.	78916E-07	1.	724852

Table 16

The statistical results of different methods in the welded beam design problem

Methods	Worst	Mean	Best	SD	NFEs
GA [6]	1.785835	1.771973	1.748309	1.122E-02	900,000
CDE [22]	1.824105	1.768158	1.733461	2.2E-02	240,000
CPSO [9]	1.782143	1.748831	1.728024	1.29E-02	240,000
DE [24]	1.824105	1.768158	1.733461	2.21E-02	204,800
HPSO [12]	1.814295	1.749040	1.724852	4.01E-02	81,000
GA [14]	1.993408	1.792654	1.728226	7.4713E-02	80,000
NM-PSO [29]	1.733393	1.726373	1.724717	3.50E-03	80,000
CAEP [33]	3.179709	1.971809	1.724852	4.43E-01	50,020
MOGA [15]	1.995000	1.919000	1.82455147	5.3775284E-02	5,000
WCA [30]	1.744697	1.726427	1.724856	4.29E-03	46,450
PSO-DE [26]	1.7248811	1.7248579	1.7248531	4.1E-06	33,000
CPSO [11]	1.782143	1.748831	1.728024	0.012926	30,000
$(\mu+\lambda)$ -ES [8]	2.074562	1.777692	1.724852	8.8E-2	30,000
MDE [18]	1.724854	1.724853	1.724852	1.0E-15	24,000
DELC [23]	1.724852	1.724852	1.724852	4.1E-13	20,000
GAFAT [16]	1.724852	1.724852	1.724852	5.8E-16	20,000
MVDE [19]	1.7249215	1.7248621	1.7248527	7.88359E-6	15,000
COA	1.7603700	1.7383867	1.726056	0.9810265E-002	99,022
MCOA	1.7248934	1.7248535	1.724852	5.8869411E-006	18,000

Figure 7.

The convergence chart of case study 5.

Table 17

Statistical results of NFEs for benchmarks

Problem	$N$	Mean of NFEs		Maximum of NFEs		Minimum of NFEs
		COA	iCOA	COA	iCOA	COA	iCOA
Three-bar truss design problem	50	13887	5259	24974	5296	2360	5236
Speed reducer design problem	50	15456	5006	18985	5013	5785	5000
The design of a pressure vessel	50	32193	27526	39382	27562	11286	27504
The tension/compression string	50	21336	2993	26387	3573	13039	2355
Welded beam design problem	50	86808	18073	107981	18189	30554	18000

Table 18

The results of Friedman’s tests

Mean ranks
	Based on	Based on	Based on
Methods’ name	three-bar truss design problem	speed reducer design problem	tension/compression spring-problem
	Asymp. Sig. (0.006462)	Asymp. Sig. (0.000683)	Asymp. Sig. (0.007243)
	Chi-Square (29.040)	Chi-Square (33.964185)	Chi-Square (27.194732)
CSA	14.00	12.33	12.33
SCA	12.67	11.33	11.33
AATM	11.00	8.33	8.67
DSS-DE	7.33	5.17	6.67
MAL-DE	5.50	5.17	6.17
GAFAT	5.50	5.17	5.17
DELC	5.50	5.17	2.67
PSO-DE	3.67	11.00	2.67
MVDE	4.33	1.83	6.67
HEA-ACT	7.00	8.67	2.67
WCA	7.33	4.17	9.33
COA	12.33	11.33	11.00
MCOA	3.33	1.33	3.67

6.5 Case study 5

The welded beam design problem was the fifth case study of the present paper that was used as a test in GA [6, 14], GAFAT [16], MOGA [15], HGA [1], CAEP [33], WCA [30], CPSO [11], HPSO [12], CPSO [9], NM-PSO [29], PSO-DE [26], CDE [22], DE [24], MDE [18], DELC [23], MVDE [19], $(\mu+\lambda)$ -ES [8], IHS [46] and COA. The details of the best solution by our method are indicated in Table 14. The comparison of the results for the welded beam design problem is presented in Table 15. The convergence of the MCOA was much faster than COA (see Fig. 7).

The statistical results presented in Table 16 outline the performance of various optimization methods applied to the welded beam design problem. Among these methods, the MCOA stands out as it achieves the best performance in terms of both mean and best objective function values, with values of 1.7248535 and 1.724852, respectively. Notably, the MCOA also exhibits the lowest standard deviation (5.8869411E-006), indicating a high level of consistency in its solutions.

Comparing the MCOA’s results with other optimization techniques, it is evident that the MCOA outperforms many of the traditional approaches such as Genetic Algorithm (GA), Cooperative Particle Swarm Optimization (CPSO), Differential Evolution (DE), and Hybrid Particle Swarm Optimization (HPSO). While these methods achieve competitive results, the MCOA demonstrates superior performance by consistently converging to high-quality solutions with minimal variability.

Furthermore, the computational efficiency of the MCOA is highlighted by its relatively low number of function evaluations (18,000), which is substantially lower compared to several other methods requiring significantly higher computational budgets. This efficiency makes the MCOA an attractive choice for optimization tasks where computational resources are limited or need to be allocated efficiently.

The statistical results of the number of fitness evaluations (NFEs) for the benchmarks are presented in Table 17. It is obvious that the mean of the NFEs of MCOA in all of the case studies is less than COA.

At the end, to indicate a significant difference between the results of the proposed MCOA algorithm with other methods, statistical analysis is carried out. We applied the Friedman test to realize whether there are substantial differences in the results of the evolutionary algorithms. In this test, the alpha was set to 0.05 ( $\alpha=$ 0.05) as the level of confidence in all cases. Table 18 reveals the obtained results of the mean ranking of these algorithms by Friedman’s test based on three-bar truss design, speed reducer design problem, and tension/compression spring-problem. As shown in this table, the proposed MCOA algorithm is ranked first for both three-bar truss design and speed reducer design problems whereas it is ranked fourth (the second rank value). Friedman test indicates that the proposed MCOA algorithm has a significant difference in the results of the other algorithms.

7. Conclusions and future works

The present article introduced the modified Cuckoo optimization algorithm (MCOA) with a suitable penalty function for solving constrained engineering problems. Five well-known case studies were presented to demonstrate the efficiency of the proposed modification in comparison with other algorithms such as GA, GAFAT, MOGA, ABC, SC-ABC, CDE, MDE, DELC, DSS-DE, MAL-DE, MVDE, PSO, CPSO, HPSO, G-QPSO, NM-PSO, AATM, IHS and WCA. By utilizing a suitable coefficient for ELR, the modified COA outperformed other considered optimization methods. One of the advantages of the present approach was that it could converge to the best solution faster than the other algorithms with less number of function evaluations. Another advantage of this work was the better quality of the obtained solutions. According to the results, it could be concluded that the proposed MCOA could outperform the mentioned methods. As a future work, it is attempted that to increase the speed of the convergence for MCOA by using of chaos theory for $\varphi$ .

Ethical approval

All authors state that this study is the author’s original work, which has not been previously published elsewhere. The paper is not currently being considered for publication elsewhere. The paper reflects the author’s research and analysis truthfully and completely.

Competing interests

All authors state that there is no conflict of interest.

Authors’ contributions

The statement, conceptualization, and implementation were performed by Asgarali Bouyer. Bahman Arasteh and Asgarali Bouyer implemented and performed the implementations. Mahdi Abdollahi provided data collection, and analyzed the data, and wrote the manuscript. A. Bouyer re-edited and analyzed the results and proofread the paper.

Funding

The authors declare that no funds were received during the preparation of this manuscript.

Availability of data and materials

The datasets generated during and the implemented code during the current study will be available when required by the reviewers or readers.

Footnotes

Appendix 1: Description of numerical case studies

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A new accurate and fast convergence cuckoo search algorithm for solving constrained engineering optimization problems

Abstract

Keywords

1. Introduction

Table 1 Used parameters in modified cuckoo optimization algorithm for case studies

6.1 Case study 1

Table 6 Comparison of results for the speed reducer design problem

Table 9 Comparison of results for the design of a pressure vessel problem

Table 11 Best result of MCOA for the fourth case study

Table 15 Comparison of results for the welded beam design problem

7. Conclusions and future works

Ethical approval

Competing interests

Authors’ contributions

Funding

Availability of data and materials

Footnotes

Appendix 1: Description of numerical case studies

References

Table 1
Used parameters in modified cuckoo optimization algorithm for case studies

Table 6
Comparison of results for the speed reducer design problem

Table 9
Comparison of results for the design of a pressure vessel problem

Table 11
Best result of MCOA for the fourth case study

Table 15
Comparison of results for the welded beam design problem