Multi-objective optimization has gained the attention of researchers in the last decade with the development of improved nature-inspired metaheuristic algorithms. Generally, there are two methodologies for addressing multi-objective optimization problems. The first ones convert the optimization problem with several objectives into a single objective optimization problem by summing all normalized objectives which are then weighted according to a set of predefined values. Despite the extensive application of these classical approaches in addressing multi-objective optimization problems, they exhibit limits in real-world scenarios. These strategies need convex Pareto fronts to obtain a solution, and show significant drawbacks on complex optimization problems [1,2]. Furthermore, the pre-algorithm expert-provided weights have a significant impact on the final solution and biased in their favor [3]. In order to circumvent these limits, researchers developed methods that solve all objectives directly without combining them. Determining a set of solutions that specify the optimal compromise between opposing objectives is one way to tackle such situations. Regarding this, multi-objective Evolutionary Algorithms (MOEAs) can provide solutions that are adequate for practical purposes within an acceptable runtime.

Evolutionary algorithms, or bio-inspired algorithms, are prominent and extensively utilized methods under the category of population-based metaheuristics. The Genetic Algorithm (GA) and its variations are a category of evolutionary algorithms extensively utilized on the practical optimization problems, due to its wide availability in different software packages, and well understood mechanism. The Non-dominated Sorting Genetic Algorithm II (NSGA-II) [4] is a popular multi-objective implementation of GA widely applied in various fields, including engineering design, which is designed to solve optimization problems with multiple conflicting objectives. NSGA-II employs a fast non-dominated sorting approach to classify solutions into different Pareto fronts, and it uses a crowding distance mechanism to ensure diversity among solutions by preserving a spread across the Pareto front. However, NSGA-II has some disadvantages, such as its computational complexity, which can become significant for large populations or problems with many objectives, and its sensitivity to parameter settings, which may require careful tuning for optimal performance.

The majority of practical engineering optimization problems necessitate the presence of multiple constraints of equality and inequality types. Conventional meta-heuristic optimization algorithms are presented for solving unconstrained optimization problems. In order to deal with constraints, various methods have been suggested in the literature. These methods can be grouped into three main categories: penalty functions, addressing constraints using the Multi-Objective Optimization (MOO) problem, and addressing constraints with separate considerations of feasible and infeasible solutions [5]. By adding a penalty term to the objective function associated with the constraints, the most typical method for dealing with constraints is to convert a constrained optimization problem into an unconstrained one. One alternative strategy takes a two-step approach to dealing with constraints: first, the algorithm optimizes the constraints independently of the objective functions; subsequently, once a large number of viable solutions have been found, the optimization of the objective functions is the algorithm’s primary concern. The most up-to-date method involves turning the constrined optimization issue into an unconstrained MOO problem by considering the constraints as additional objectives.

Another widely applied evolutionary optimization algorithm is the Differential Evolution (DE), which, unlike GAs, has not taken inspiration from nature but is designed purely based on mathematical concepts [6]. DE is highly regarded for its robustness and simplicity in solving optimization problems. A key benefit of DE is its effectiveness in tackling intricate optimization problems that involve non-linear relationships and multiple local optima. DE is particularly significant for its strong exploration capabilities, which enable it to effectively search global optimization landscapes and avoid premature convergence to local optima [7]. However, DE has some limitations. The performance of DE is also sensitive to the choice of its control parameters, such as the mutation factor, crossover rate, and population size. Fine-tuning these parameters is often necessary to achieve optimal performance, which can be time-consuming.

The multi-objective version of the DE algorithm extends its optimization capabilities to problems with multiple conflicting objectives, directly incorporating constraints and multi-objectiveness into the evolutionary process [8]. Notable variants include the Multi-Objective Differential Evolution Algorithm (MODEA), which utilizes Opposition-Based Learning for initial population generation and introduces a new selection mechanism for better Pareto front distribution [9], and the Adaptive Chaotic DE, which combines adaptive and chaotic principles for enhanced efficiency [10]. A significant strength of multi-objective DE is its robust exploration ability, efficiently navigating the search space to identify a diverse set of Pareto-optimal solutions. Applications of this algorithm span various fields, including engineering design, power plant control, and chemometrics, where it optimizes trade-offs among multiple objectives effectively [11].

Another category of metaheuristics are swarm optimization algorithms (SO), among which Ant Colony Optimization (ACO), Firefly Algorithm (FA) and Particle Swarm Optimization (PSO) are two approaches commonly used to solve real-world problems. These algorithms are inspired by the collective behavior of decentralized, self-organized systems, typically found in nature, such as bird flocking, fish schooling, and ant colonies. More recently, authors proposed a novel nature-inspired optimization algorithm that belongs to the group of SO algorithms, the Sailfish Optimization (SFO) algorithm, that draws inspiration from the collective hunting behavior of sailfish and sardines. The algorithm maintains two populations simultaneously, e.g., sailfish (predators) and sardines (prey). During the optimization process, SFO alternates between exploration and exploitation phases by having sailfish update their positions based on the elite sailfish and the worst-performing sardines. The movement of sailfish is directed towards the best-known positions, thereby exploiting the search space, while sardines move randomly to explore new areas. This approach ensures a balance between exploration and exploitation, with the aim to efficiently find optimal solutions for various optimization problems.

The SFO algorithm has been widely applied to various optimization problems. The authors in [12] introduced SFO as a novel metaheuristic algorithm for solving constrained engineering optimization problems, demonstrating superior performance in terms of exploration, exploitation, and convergence speed compared to existing algorithms. In the realm of power systems [13] applied SFO for optimal placement and sizing of static synchronous compensator, resulting in improved voltage profiles and reduced power losses. In [14], authors optimized the coverage and connectivity of sensor nodes in a 3D wireless sensor network using SFO, ensuring robust network performance.

These studies collectively highlight the versatility and effectiveness of the SFO algorithm in solving diverse optimization problems. However, the majority of these studies focused on single objective optimization. In [15], authors introduced a multi-objective Sailfish Optimizer combined with a genetic algorithm for expert recommendation in community-based question-answering, demonstrating superior performance in matching questions to experts. In order to design a frequency selective surface for 5G applications, the authors in [16] introduced a hybrid multi-objective optimization technique utilizing the SFO combined with a General Regression Neural Network. However, unlike the multi-objective implementation of SFO algorithm presented in this paper, the proposed algorithm in [16] employed the weighted sum method to balance multiple objectives, which cannot provide satisfactory result on non-convex Pareto fronts.

Despite much investigation using various algorithms, the inquiry into identifying the most appropriate one for a certain challenge remains inadequately addressed. Moreover, preserving the diversity of optimal solutions and preventing premature convergence to local optima remain essential for population-based algorithms. Moreover, the No Free Lunch theorem [17], which established that enhancements in performance for one category of optimization problems result in diminished performance for another, has spurred the advancement and comparative analysis of metaheuristic algorithms for addressing diverse optimization challenges. In general, it is not possible to provide a single algorithm that will successfully solve all problems that can be encountered in practice.

In this regard, researchers proposed a number of hybrid optimization algorithms, with the aim of strategic combination of different optimization techniques to leverage their respective strengths and mitigate their weaknesses, thereby enhancing overall problem-solving efficacy. This approach is performed to improve convergence speed, maintain solution diversity, and adapt to a wide range of problem characteristics, making the optimization process more robust and efficient. In [18], the authors developed a hybrid of the salp swarm algorithm and DE, which improves feature exploitation and demonstrated superior results in big data optimization compared to traditional methods. In [19], the authors introduced an adaptive DE algorithm incorporating a memory mechanism from PSO to adaptively select mutation operators, resulting in competitive performance on benchmark problems. In [20], the authors proposed a hybrid approach combining DE, PSO, and nondominated sorting, improving distributed optimization quality in complex multi-objective problems. Regarding the SFO algorithm, literature reveals numerous studies addressing its enhanced performance, although improvement through various hybridization techniques is seldom. Regarding hybridization, in [21], the authors developed a hybrid model combining SFO with the Whale Optimization Algorithm (WOA) termed SWO-DLSTM (A Hybrid Sailfish Whale Optimization and Deep Long Short-Term Memory). This hybrid model leverages the optimization capabilities of SFO and WOA for feature selection and parameter tuning, which are then utilized by DLSTM (Deep Long Short-Term Memory) for accurate time series predictions in energy demand forecasting, showcasing significant improvements in considered performance metrics. These hybrid methods effectively balance exploration and exploitation, leading to improved solution quality and convergence rates in multi-objective optimization tasks.

The mechanical engineering design problems often involves complex objective functions with numerous variables, a number of complex non-linear constraints. This results into the complex optimization problems, which often require multiple conflicting objectives. The modern requirements, posed on the designer, which require long service life, lightweight but strong construction as well as increased energy efficiency make these design problems even complex. In the area of gearbox design, the objectives are inherently non-linear and non-convex thus making the problem complex to solve. The background part presents a literature review on the development and implementation of several optimization techniques to address difficult gearbox design problems.

The aim of this paper is to explore the suitable choosing of Evolutionary Algorithms (EAs) for hybridization, with the aim to provide an improved optimization algorithm for the established multi-objective planetary gearbox optimization problem. Therefore, it is on researchers to explore the possibilities of application of different algorithms, to establish each of their weaknesses and strengths, and provide the possibilities to hybridize each of the algorithms, in order to obtain the solutions of high accuracy for the considered optimization problem. Based on the literature review and extensive research, we can see that the DE algorithm shows strong exploration capability, if applied with appropriate mutation operator, while the SFO algorithm shows fast convergence towards local optimum. This work presents the HMODESFO algorithm, a hybridization of the DE and SFO algorithms, to enhance the exploration capabilities of the SFO method. The hybridization was executed by integrating the mutation operators DE/rand/1 and DE/rand/2 from the DE algorithm into the equations for updating the locations of the sailfish (predator) and sardine (prey), respectively. The performance of the proposed algorithm is validated against well-known industry benchmarks, e.g., a set of ZDT (Zitzler-Deb-Thiele) and DTLZ (Deb-Thiele-Laumanns-Zitzler) benchmark problems, where the proposed algorithm showed improved performance. This work employs a series of well-recognized multi-objective optimization problems in literature that deal with mechanical engineering to further validate the created hybrid HMODESFO method. Finally, the performance of this algorithm has been validated on the developed multi-objective planetary gear optimization problem, where compared to industry standard it showed improved results.

The primary contributions of the paper can be summarized as follows:

The previously developed single stage planetary gearbox optimization problem has been extended to include additional constraints, and increase physical realization of obtained results.

The paper is structured as indicated: Section 1 presents the introduction and a survey of the existing literature. Section 2 gives the outline of the background and related work of applying EAs in the area of mechanical design, especially optimal gearbox design. In Section 3, the mathematical formulations of the The Multi-Objective Sailfish Optimization algorithm, Differential evolution algorithm as well as the proposed hybrid algorithm are outlined, with the appropriate pseudo-code that accompanies each algorithm. In Section 4, the formulations of used benchmark test functions as well as the formulated planetary gearbox optimization model are provided. Furthermore, Section 4 also gives the results of the numerical experimental simulations. In Section 5, the conclusions are drawn.

In mechanical engineering, optimizing gearbox design is a complex and essential task. It requires addressing multiple objectives, such as enhancing performance, improving efficiency, reducing weight, and ensuring long service life. These factors are critical to meeting the stringent requirements of modern gearbox design, which must balance these often-conflicting goals to achieve optimal functionality. The complexity of this task underscores the need for advanced optimization techniques that can handle the intricate trade-offs involved in creating high-performing, efficient, and durable gearboxes. Single-objective optimization has been a fundamental approach in the design and enhancement of gearboxes. Reference [22] proposed a powerful optimization method for the minimal dimensional design of gearboxes. By implementing a PSO algorithm, they achieved significant reductions in gearbox volume, which directly translated to cost savings and material efficiency. In another study [23], authors utilized single-objective optimization to minimize the volume and weight of gearboxes. Their research provided practical graphs and guidelines for deriving optimal gearbox parameters based on different input power, gear ratio, and material hardness. This approach facilitated the design of lightweight and compact gearboxes, which are essential for various industrial applications. In [24], the authors examined an innovative approach to optimizing the transmission efficiency of spiral bevel gears. By integrating PSO with the Gravitational Search Algorithm, they demonstrated significant improvements in gear design performance and efficiency. In [25], the authors explored the use of seven different nature-inspired optimization algorithms to minimize the volume of a two-stage planetary gearbox. This study applied algorithms such as Grey Wolf Optimizer, Artificial Bee Colony, and Multi-verse Optimizer, achieving optimal volume values significantly better than those previously recorded in the literature. The findings highlight the effectiveness of these meta-heuristic approaches in addressing complex engineering optimization challenges.

Given the inherent complexity of gearbox optimization, considering only one objective is often unsatisfactory, especially when aiming to optimize multiple conflicting objectives simultaneously. Multi-objective optimization has emerged as a robust approach to address these conflicting goals, such as minimizing weight while maximizing efficiency and durability. This method effectively balances the trade-offs necessary for optimal gearbox design. Multi-objective optimization techniques have been extensively applied to various types of gearboxes, including spur, helical, and planetary gearboxes. For instance, reference [26] explored the multi-objective optimization of a two-stage spur gearbox by incorporating tribological aspects to minimize weight and power losses. Their study utilized the NSGA-II to achieve significant improvements over traditional single-objective optimization approaches. In another study, the same authors extended their research to a two-stage helical gearbox, again focusing on minimizing volume and power losses while considering tribological constraints [27]. Their findings demonstrated a considerable reduction in power loss and a higher probability of avoiding wear failures when using a multi-objective optimization framework compared to single-objective methods. Additionally, in [28], authors applied the Taguchi method and Grey Relational Analysis to optimize a two-stage helical gearbox, focusing on maximizing efficiency and minimizing mass. Their study identified optimal design parameters, showcasing the effectiveness of combining single-objective and multi-objective optimization phases to achieve superior performance. In [29], the authors developed a multi-objective uncertainty optimization design framework to enhance the performance and reliability of planetary gear trains in electric vehicles. This study incorporates uncertainties in manufacturing size, material properties, and load inputs, optimizing both volume and transmission efficiency using an improved NSGA-II algorithm. The findings demonstrate that the proposed framework offers superior reliability and performance compared to traditional deterministic optimization methods under varying uncertainty. The paper [30] presents a comprehensive framework for enhancing the design of planetary gear sets. By employing advanced optimization algorithms, the authors aim to simultaneously minimize mass and power loss while reducing transmission error. This study achieves significant improvements in efficiency and reliability, offering a balanced trade-off among competing design objectives.

Furthermore, the use of hybrid optimization algorithms has shown promise in optimizing planetary gearboxes. In [31], authors proposed a hybrid algorithm combining particle swarm optimization and differential evolution to tackle the complex multi-objective non-linear optimization problems in planetary gearboxes. Their approach yielded significant improvements in efficiency, weight reduction, and prevention of premature gear failure. A recent research [31] suggested a modified hybrid approach called the Multi-objective Hybrid Butterfly Optimization and Particle Swarm Optimization approach (HMOBPSO), which combines PSO with the Butterfly Optimization Algorithm (BOA). This strategy sought to improve performance by tackling several competing objectives in the optimization of planetary gearboxes. The research has shown notable advancements in gearbox dimensions, efficiency, and spacing relative to traditional techniques.

These studies underscore the advancements and effectiveness of multi-objective optimization in the design and enhancement of gearboxes. By integrating various optimization algorithms and considering multiple performance metrics, researchers continue to push the boundaries of gearbox technology, leading to more efficient, durable, and lightweight designs.

In this section we will first introduce the Multi-objective SFO and DE algorithms, as the constituents of the proposed hyrbid HMODESFO optimization algorithm.

To assess the effectiveness of the proposed algorithm and contrast it with established algorithms in the literature, several benchmark tests have been conducted. Firstly, we compare the performance of the proposed method on seven DTLZ and five ZDT benchmark functions for multi-objective optimization [32,33]. Furthermore, two established multi-objective mechanical design optimization problems are used to validate the performance of the proposed modifications. Finally, we employed a multi-objective planetary gearbox optimization model [31], to validate performance. Three well-established performance metrics have been used in this paper including Hypervolume (HV), Spread (SP) and Inverted Generational Distance (IGD) [34] in order to measure the suggested algorithm’s convergence and coverage (diversity).

The HV measures the volume of the objective space dominated by a set of non-dominated solutions, bounded by a reference point r. Formally, the hypervolume indicator for a set of solutions S={s1,s2,…,sn} in the objective space is defined as: (26)HV(S)=vol(⋃s∈S{z∈Rm∣s⪯z⪯r}),where s⪯z indicates that solution s dominates or is equal to z, and vol represents the Lebesgue measure of the union of the hyperrectangles formed by each solution s and the reference point r. In this context, z is a point in the objective space, m is the number of objectives, and r is a reference point that is dominated by all solutions in the set S.

The hypervolume metric provides a scalar value that reflects both the convergence and diversity of the solution set by capturing the extent of the dominated objective space.

The SP metric is a crucial measure used to evaluate the distribution and spread of solutions along the Pareto front. It quantifies how well the solutions cover the extent of the Pareto front by considering both the distance between consecutive solutions and the overall extent of the front. The spread metric is formally defined as: (27)Δ=df+dl+∑i=1n−1|di−d¯|df+dl+(n−1)d¯,where df and dl are the Euclidean distances from the extreme solutions to the boundary solutions of the Pareto front, di is the distance between consecutive solutions, d¯ is the average of these distances, and n is the number of solutions. This metric ensures that solutions are evenly distributed and cover the entire range of the Pareto front, reflecting both convergence and diversity in the solution set.

The Inverted Generational Distance is a widely used metric in multi-objective optimization for assessing the convergence and diversity of a set of solutions generated by an algorithm with respect to a reference Pareto front. The IGD provides an average distance from the points on the reference front to their closest points in the obtained solution set, thus reflecting how well the solution set covers the true Pareto front. The IGD metric is calculated as: (28)IGD(S,Pref)=1|Pref|∑y∈Prefminx∈Sd(y,x),where S is the set of solutions obtained by the algorithm, Pref is the reference Pareto front, |⋅| denotes the cardinality of a set, y is a solution vector in the reference Pareto front Pref, x is a solution vector in the obtained solution set S, d(y,x) is the Euclidean distance between vectors y and x, calculated as: (29)d(y,x)=∑i=1m(yi−xi)2,where m is the number of objectives, and yi and xi are the i-th objective values of y and x, respectively. A lower IGD value indicates that the obtained solution set S is closer to the reference Pareto front and provides better coverage of the true Pareto-optimal solutions.

In contrast to the conventional IGD metric, the Inverted Generational Distance Plus (IGD+) [35] metric addresses the shortcomings of the IGD by incorporating a modified distance function that exclusively considers non-dominated vectors. This modification enhances the metric’s sensitivity to both convergence towards the genuine Pareto front and the dispersion of solutions along it, rendering IGD+ a more reliable tool for assessing the efficacy of diverse multi-objective optimization algorithms. The IGD+ metric is defined as: (30)IGD+(P∗,P)=1|P∗|∑v∈P∗minu∈Pd+(v,u),where P∗ represents the reference Pareto front (typically the true Pareto-optimal front), P is the set of solutions obtained by the algorithm under evaluation, and |P∗| denotes the number of solutions in P∗. The distance function d+(v,u) calculates the directional distance between vectors v and u, considering only the positive differences in each objective dimension: (31)d+(v,u)=∑i=1m(max{0,vi−ui})2,where m is the number of objectives, and vi and ui are the values of the i-th objective for solutions v and u, respectively. The use of the max{0,vi−ui} term ensures that only the distances where vi exceeds ui contribute to the metric, effectively penalizing solutions that are dominated or not well-converged. By averaging the minimal positive distances from each point in the reference set to the obtained solution set, IGD+ provides a comprehensive assessment of how closely and uniformly the algorithm’s solutions approximate the true Pareto front.

In order to perform the statistical comparison between obtained results the Wilcoxon signed rank test and Friedman rank test have been employed.

The Wilcoxon signed-rank test [36] is a non-parametric statistical technique utilized to compare two correlated samples or repeated measurements on a single sample in order to evaluate if their population mean ranks exhibit any differences. The test process entails computing the disparities between paired observations, ordering these disparities ac-cording to their absolute values, and subsequently aggregating the rankings for the positive and negative disparities individually. The lesser of these sums is utilized as the test statistic. The null and alternative hypotheses for the Wilcoxon signed-rank test are stated as follows: H0 states that the median difference between the paired observations is equal to zero, while H1 states that the median difference between the paired observations is not equal to zero. The determination to reject the null hypothesis relies on comparing the test statistic with crucial values obtained from the Wilcoxon signed-rank table or by generating a p-value. If the p-value is less than or equal to the selected significance level (α=0.05), the null hypothesis is rejected, suggesting a statistically significant distinction between the paired observations.

The Friedman test [36], a non-parametric statistical approach, is commonly used in evolutionary optimization to assess the efficiency of various algorithms across several problem occurrences. It is essential in benchmark studies to ascertain whether there are statistically significant disparities in performance indicators among different algorithms. The procedure involves ranking the performance of each algorithm on each problem instance, summing these ranks, and using these sums to compute the test statistic. This statistic follows a chi-squared distribution with k−1 degrees of freedom, where k is the number of algorithms. The null and alternative hypotheses evaluated by the test are: H0: “There are no disparities in the distributions of the performance measures among the algorithms.” H1: “At least one algorithm exhibits significant dissimilarity.” The choice to reject the null hypothesis is determined by comparing the test statistic to the crucial value obtained from the chi-squared distribution table at the selected significance level (α=0.05). If the value of the test statistic exceeds the critical value, it leads to the rejection of the null hypothesis, indicating a significant performance difference among the algorithms.

In the next subsections we present the benchmark problems employed to perform numerical analysis.