1 Introduction

Optimization challenges that involve two or more conflicting objectives are classified as multi-objective optimization problems (MOOPs). These stand distinct from single-objective optimization problems (SOOPs) due to their multifaceted nature and the imperative for decision-makers (DMs) to navigate and make judicious choices. Addressing MOOPs effectively necessitates the adoption of multi-objective optimization algorithms (MOOAs). The literature offers a rich array of such algorithms, warranting a structured overview. Broadly, MOOAs can be categorized into:

  1. 1.

    Pareto dominance oriented algorithms: These algorithms capitalize on Pareto dominance to select non-dominated solutions. Their ability to encompass the entire Pareto front in a singular iteration has made them a favorite among researchers. The early MOOAs, termed the first generation, were characterized by their simplicity. In contrast, the subsequent generation, the second, pivoted toward efficiency [1]. Together, these two generations carved out MOO as a distinct research niche. The third generation accentuated effectiveness, aiming to bolster convergence and diversity. It is anticipated that the forthcoming fourth generation will holistically address efficiency, effectiveness, productivity, and robustness. Beyond the conventional Pareto dominance, myriad dominance relations play a crucial role in MOOA design. Among the vast ensemble of Pareto dominance-based algorithms, a few stand out. For instance, the competitive mechanism-driven multi-objective particle swarm optimizer (CMOPSO) [2] is tailored for swift convergence to the true Pareto front. However, its reliance on random leader selection can compromise its convergence and diversity. Another noteworthy algorithm improves upon the original NSGA-II dominance relationship (NSGAIISDR) [3], juxtaposing various dominance relations against a robust one that melds angle-based dominance with niche creation. Yet, its efficacy can be constrained in limited iteration scenarios. The guided population archive whale optimization algorithm (GPAWOA) [4] underwent testing across multiple objectives and real-world scenarios. However, its reliance on archived entities did not consistently yield diverse solutions across the Pareto front, underscoring the need for a more versatile Pareto dominance-based algorithm.

  2. 2.

    Non-Pareto dominance algorithms: These can be further segmented into:

  3. 3.

    Trajectory or single point-based algorithms: These are tailored for exploration and are particularly adept for multi-objective integer optimization challenges. For instance, MOTS [5] addresses intricate scheduling and resource allocation, while MOSA/R-HH [6] augments NDS through a bifurcated heuristic approach.

  4. 4.

    Population-based algorithms: This category encompasses:

  5. 5.

    Scalarization algorithms: These metamorphose MOOPs into SOOPs. EGAM [7] and WGP [8] are illustrative, each grappling with its unique set of challenges.

  6. 6.

    Decomposition-based algorithms: Here, a MOOP is dissected into multiple SOOPs, each tethered to a distinct vector. MFEA/D-DRA [9] and MLIPMOEA/D [10] are representative, though they grapple with convergence and diversity.

  7. 7.

    Speciation-based algorithms: These focus on individual objectives, with VEGA being emblematic. MMACO_R [11] and SSMOPSO [12], while innovative, face challenges.

  8. 8.

    Algorithms supported by approximation models:

  9. 9.

    Such algorithms, like MOFEPSO [13], harness approximation models, notably Kriging, to pinpoint NDS. However, they often wrestle with convergence and diversity issues [14].

  10. 10.

    Algorithms relying on performance indicators: Relying on performance indicators:

  11. 11.

    These algorithms, such as MaOEAIGD [15], utilize performance metrics to guide the search. Yet, they often face challenges like extensive function evaluations [16].

  12. 12.

    Algorithms with reference point orientation: LMOCSO [17] and hpaEA [18], for instance, generate reference vectors to ensure convergence and diversity of NDS.

  13. 13.

    Combined MOOAs: These algorithms merge multiple complementary methods. Examples include WOFSMPSO [19] and MOEA/D-AMG [20]. The key challenge is the effective integration of diverse algorithmic techniques and the adept management of their complex configurations.

The realm of multi-objective (MO) optimization is further expanded by a variety of MO algorithms. This includes the MO ant lion optimizer (MOALO) [21], MO equilibrium optimizer (MOEO) [22], and MO slime mold algorithm (MOSMA) [23], MOAOA [24], NSIMO [25], MOMPA [26], MOEA/D [27], NSGA-II [28], MOMVO [29], NSGWO [30], MOGBO [31], MOPGO [32], NSHHO [33], MOTEO [34], MOHTS/D [35], MOSOS/D [36], MOGNDO [37], NSMFO [38], NSDA [39], and NSWOA [40] each offering unique contributions to the optimization field.

  1. 1.

    While each algorithm has distinct advantages, they also comes with inherent weaknesses. A common finding is that many algorithms lack extensive testing across varied benchmark sets or real-world applications. They often struggle with achieving fast convergence or in producing the desired number of non-dominated solutions within limited function evaluations. This highlights the critical need for an algorithm that embodies efficiency, effectiveness, and productivity and consistently excels across a wide range of challenges.

  2. 2.

    The foundational Runge–Kutta optimizer (RKO), along with its later iterations, has shown outstanding performance in a variety of benchmarks and practical applications, particularly in single-objective optimization across continuous, discrete, and combinatorial settings. This consistent success suggests that enhancing the RKO framework for multi-objective contexts could potentially establish new performance standards.

  3. 3.

    The no free lunch theorem (NFL) [41] postulates that no singular algorithm can claim universal supremacy. This theorem delineates three potential outcomes:

  • An algorithm excels in niche problems

  • It showcases commendable performance over a benchmark test suite

  • It consistently outperforms across diverse benchmark test suites.

A hyper-heuristic algorithm using proximal policy optimization for multi-objective truss problems using reinforcement learning to optimize decision making rules and improve search for Pareto-optimal solutions in structural engineering is one of the significant developments [42]. Adaptive differential evolution for collaborative path planning of multiple unmanned aerial vehicles (UAVs) introduces adaptive mechanisms to strike a balance between exploration and exploitation to generate optimized flight paths and enhanced mission efficiency [43]. In another approach, a hyper-heuristic framework is integrated with a deep Q network to solve the scheduling problem of multi-objective unmanned surface vehicles by dynamically selecting appropriate heuristics to adapt to changing conditions in complex scheduling environments [44]. Adaptive operator selection with a dueling deep Q network is also used in evolutionary multi-objective optimization to learn and adapt during optimization, improving convergence and producing more diversified Pareto front solutions [45]. The multi-objective equilibrium optimizer slime mold algorithm is a bio-inspired algorithm that uses the strengths of both equilibrium and slime mold mechanisms to solve engineering problems in the realm of bio-inspired algorithms [46]. An indicator-based multi-swarm slime mold algorithm, IBMSMA, is proposed for truss optimization using multi-swarm strategies and indicator-based selection to improve convergence and diversity of solutions [47]. Finally, an equilibrium optimizer slime mold algorithm is applied to the inverse kinematics of a seven degree of freedom robotic manipulator, to demonstrate its effectiveness in optimizing joint configurations to achieve precise end effector positioning [48].

Multi-objective Runge–Kutta optimizer (MORKO), is an algorithm that employs elitist non-dominated sorting and crowding distance techniques to overcome the main difficulties of efficiency, diversity, and robustness in multi-objective optimization. This approach is poised to fill the existing gaps in achieving well-balanced exploration and exploitation, while ensuring convergence to the Pareto-optimal front. Limitations in other state-of-the-art algorithms, such as multi-objective marine predicator algorithm (MOMPA), multi-objective gradient-based optimizer (MOGBO), non-dominated sorting genetic algorithm II (NSGA-II), and multi-objective evolutionary algorithm based on decomposition (MOEA/D), motivate the development of MORKO. While these are innovative algorithms, they tend to have specific weaknesses, such as slow convergence rates, insufficient diversity preservation, or being prone to getting stuck in local optima in the search of complex, high-dimensional tasks. The references cited within the literature review have been expanded upon to provide a comprehensive foundation for the contributions and innovations proposed by MORKO. For instance, the NSGA-II is well known for its non-dominated sorting, but it has problems in preserving diversity in highly constrained search spaces. On the other hand, the MOMPA and MOGBO algorithms, which introduce competitive mechanisms and gradient-based approaches, respectively, suffer from the difficulty of balancing leader selection with convergence efficiency in the presence of a complex Pareto front structure. In contrast to these approaches, the MORKO algorithm provides a robust framework which combines adaptive mechanisms based on the Runge–Kutta method, resulting in better convergence, spread, and solution distribution. An analysis of gaps in existing methodologies further strengthens the motivation for developing MORKO. More specifically, the Runge–Kutta optimizer (RKO) has been successfully used for single-objective optimization and is now extended to the multi-objective domain. MORKO attempts to overcome the long-standing problem of algorithmic stagnation by incorporating advanced sorting and diversity maintenance techniques to guide the optimization process toward a diverse, convergent set of solutions. The emphasis on optimizing complex, real-world engineering problems, like those that occur in constrained and multi-dimensional scenarios, shows that MORKO is adaptable and performs better in a wide range of optimization landscapes.

A comprehensive comparative overview of different multi-objective optimization algorithms along with their advantages and disadvantages is given in Table 1. The comparison highlights the unique strengths and contributions of the multi-objective Runge–Kutta optimizer (MORKO) among existing methods.

Table 1 Related work
Full size table

The proposed MORKO algorithm differentiates itself through the following unique contributions and advantages:

  1. 1.

    Enhanced diversity and convergence: A crowding distance mechanism in MORKO maintains the solution diversity and prevents premature convergence, which is a common problem in many existing algorithms.

  2. 2.

    Robustness across diverse problems: MORKO has been rigorously validated on benchmark test suites (ZDT, DTLZ, and constraint problems) and real-world engineering problems, and has consistently shown superior performance.

  3. 3.

    Dual strategy mechanism: Effective exploration and convergence is guaranteed even for complex and constrained problems by the novel integration of maturity levels (iterations) and non-dominated sorting solutions.

  4. 4.

    Comprehensive statistical validation: The non-parametric statistical evaluations demonstrate that MORKO is superior to existing algorithms in terms of generational distance (GD), inverted generational distance (IGD), hyper volume (HV), and spacing (SP).

These outcomes can be perceived as minor, moderate, and major triumphs, respectively. The NFL theorem underscores the significance of devising a versatile algorithm, proficient in addressing both specific and broad challenges. In this context, we introduce the multi-objective RKO (MORKO) algorithm, underpinned by elitist non-dominated sorting and crowding distance principles.

Building on these rationales, our research hypothesis postulates that the development of an elitist non-dominated sorting and crowding distance-based MORKO algorithm holds the potential to rival or even surpass existing algorithms in generating convergent and diverse Pareto-optimal solutions across an array of benchmark test suites and real-world challenges. Key contributions of this work are:

  • Augmented features in the MORKO algorithm to bolster efficiency, robustness, and productivity.

  • Integration of non-dominance sorting and crowding distance mechanisms to amplify selection pressure.

  • Comprehensive performance metrics complemented by a novel framework for correlation analysis.

  • Algorithm ranking through non-parametric tests to ensure validated results.

This paper is motivated by the increasing complexity and specialization of real-world optimization problems that demand innovative and very effective optimization methods. Many approaches for developing algorithms for specific multi-objective scenarios have been developed in the recent surge of algorithm development, but their applicability and robustness are rarely tested in broader, practical settings. In this regard, this work presents the multi-objective Runge–Kutta optimizer (MORKO) to fill the void in algorithms that perform well in a wide variety of problem domains. The existing literature shows that many multi-objective optimization algorithms (MOOAs) do not perform consistently across different benchmarks and real-world problems. Typically, these algorithms are not able to trade-off exploration of search spaces and their exploitation in optimal optima causing problems such as premature convergence, lack of solution diversity or getting stuck in local optima. In addition, no free lunch (NFL) theorem offers a well-established reinforcement that no single algorithm can dominate any other over all problem domains in the sense that there is no free lunch in the use of algorithms, indicating the requirement for versatile optimization solution which can solve a broad spectrum of problems in the robust and efficient way. To motivate these challenges, MORKO uses the Runge–Kutta optimizer (RKO) framework, which has been demonstrated to be effective in single-objective optimization, and extends it to the multi-objective case using elitist non-dominated sorting and crowding distance mechanisms. The main goal of this approach is to achieve a better balance between exploration and exploitation, and to converge to true Pareto-optimal fronts with diversity of the generated solutions. The MORKO algorithm was designed to overcome the shortcomings of existing algorithms, namely insufficient testing on benchmark sets and real-world applications, slow convergence, and limited ability to generate diverse non-dominated solutions within a constrained number of function evaluations.

Section 2 presents a brief of Runge–Kutta optimizer. Section 3 describes the proposed MORKO algorithm. Section 4 describes the performance of the MORKO algorithm across benchmarks and its application to real-world scenarios. Section 5 describes the research design attributes. Sections 6 and 7 describe the conclusion and future research directions.

2 Runge–Kutta Optimizer

The optimization algorithm inspired by the Runge–Kutta method [49] consists of two primary components: the Runge–Kutta search strategy (SM) and an advanced solution quality enhancer. The subsequent sections detail the core processes within this algorithm. To refine the current solution \(\left({x}_{m}\right)\) during each iteration, the algorithm employs the SM, drawing inspiration from the RK technique.

$${x}_{\text{new}1}=\left\{\begin{array}{ll}\left({x}_{b}+u\cdot AF\cdot e\cdot {x}_{\text{b}}\right)+AF\cdot SM+\omega .\text{ randn. }\left({x}_{\text{d}}-{x}_{\text{b}}\right)\text{ if rand }<0.5\\ \left({x}_{d}+u\cdot AF\cdot e\cdot {x}_{\text{d}}\right)+AF\cdot SM+\omega .\text{ randn. }\left({x}_{\text{a}1}-{x}_{\text{a}2}\right)\text{ otherwise}\end{array}\right.$$
(1)

Here, \(u\) is an integer (either 1 or −1); \(e\) is a random value between [0, 2]; \({x}_{\text{a}1}\) and \({x}_{\text{a}2}\) are two randomly selected solutions within \([1, N]\), where \(N\) represents the population count; \(\text{AF}\) is an adaptive coefficient, and \(\omega\) is a random value. The SM is defined as:

$$\begin{array}{ll}& \text{SM}=\frac{1}{6}\left({x}_{\text{RuK}}\right)\Delta x\\ & {x}_{\text{RuK}}={k}_{1}+2\times {k}_{2}+2\times {k}_{3}+{k}_{4}\\ & {k}_{1}=\frac{1}{2\Delta x}\left(\text{ rand. }{x}_{\text{ws}}-v\cdot {x}_{\text{bs}}\right)\\ & v=\text{round}(1+\text{rand})\cdot (1-\text{rand})\\ & {k}_{2}=\frac{1}{2\Delta x}\left(\text{ rand }\cdot \left({x}_{\text{ws}}+{r}_{1}\cdot {k}_{1}\cdot\Delta x\right)-\left(v\cdot {x}_{\text{bs}}+{r}_{2}\cdot {k}_{1}\cdot\Delta x\right)\right)\\ & {k}_{3}=\frac{1}{2\Delta x}\left(\text{ rand. }\left({x}_{\text{ws}}+{r}_{1}\cdot \left(\frac{1}{2}{k}_{2}\right)\cdot\Delta x\right)-\left(v\cdot {x}_{\text{bs}}+{r}_{2}\cdot \left(\frac{1}{2}{k}_{2}\right)\cdot\Delta x\right)\right)\\ & {k}_{4}=\frac{1}{2\Delta x}\left(r\cdot \left({x}_{\text{ws}}+{r}_{1}\cdot {k}_{3}\cdot\Delta x\right)-\left(v\cdot {x}_{\text{bs}}+{r}_{2}\cdot {k}_{3}\cdot\Delta x\right)\right)\end{array}$$
(2)

In this context, \({x}_{\text{ws}}\) and \({x}_{\text{bs}}\) represent the least and most optimal solutions, respectively.\({r}_{1}\) and \({r}_{2}\) denote two random numbers in the range of \([\text{0,1}].\) \(\Delta x\) is given by:

$$\Delta x=2.\text{rand}.|S|$$
$$S=\text{rand}.\left(\left({x}_{bs}-\text{ rand }.{x}_{\text{avg}}\right)+\delta \right)$$
(3)
$$\delta =\text{rand}.\left({x}_{\text{m}}-\text{rand}.(Ub-Lb)\right)\cdot \text{exp}\left(-4\cdot \frac{It}{\text{ MaxIt}}\right)$$

Here, \(S\) denotes step size; \(Ub\) and \(Lb\) are the problem upper and lower bounds, respectively; It represents the current iteration, and \(MaxIt\) is the maximum iteration count. \({x}_{avg}\) is the average solution at the current iteration. The best and worst solutions are determined as:

$$\begin{array}{cc}& \text{if }f\left({x}_{\text{m}}\right)<f\left({x}_{b\text{s},\text{m}}\right)\\ & {x}_{\text{bs}}={x}_{\text{m}}\\ & {x}_{\text{ws}}={x}_{\text{bs},\text{m}}\\ & \text{ else }\\ & {x}_{\text{bs}}={x}_{\text{bs},\text{m}}\\ & {x}_{\text{ws}}={x}_{\text{m}}\\ & \text{ end}\end{array}$$
(4)

where \({x}_{bs,m}\) denotes the best solution obtained from three random solutions ( \({x}_{a1},{x}_{a2}\), and \(\left.{x}_{a3}\right)\). \(AF\) is expressed as:

$$\begin{array}{cc}& AF=2.(0.5-\text{ rand })\times \mu \\ & \mu =10\times \text{exp}\left(-12.\text{ rand. }\left(\frac{\text{ It }}{\text{ MaxIt}}\right)\right)\end{array}$$
(5)

\({x}_{b}\) and \({x}_{d}\) are given by:

$$\begin{array}{cc}& {x}_{b}=\alpha \times {x}_{m}+(1-\alpha )\times {x}_{a1}\\ & {x}_{d}=\alpha \times {x}_{best}+(1-\alpha )\times {x}_{cbest}\end{array}$$
(6)

Here, \(\alpha\) is a random value between (0,1); \({x}_{\text{best}}\) is the optimal solution up to this point, and \({x}_{\text{cbest}}\) is the best solution in the current iteration. To refine solution quality and avoid local optima, the algorithm employs an efficient operator, termed ESQ. The algorithm applies the ESQ operator as follows:

$$\begin{array}{ll}\text{if rand }<0.5\\ \text{if }w<1\\ {x}_{ESQ}={x}_{\text{new }2}+\lambda \cdot \rho \cdot \mid \left({x}_{\text{new }2}-{x}_{\text{avg }}\right)+\text{ randn }\mid \\ {x}_{ESQ}=\left({x}_{\text{new }2}-{x}_{\text{avg }}\right)+\lambda \cdot \rho \cdot \mid \left(2\right..\text{ rand}. \left.{x}_{\text{new }2}-{x}_{\text{avg }}\right)+\text{ randn }\mid \\ \text{end}\\ \text{end}\end{array}$$
(7)
$$\rho =\text{rand}(\text{0,2})\cdot \text{exp}\left(-g\cdot \left(\frac{It}{\text{ MaxIt }}\right)\right)$$
$${x}_{\text{avg}}=\frac{{x}_{\text{a}1}+{x}_{\text{a}3}+{x}_{\text{a}3}}{3}$$
(8)
$${x}_{\text{new}2}=\phi \times {x}_{\text{avg }}+(1-\phi )\times {x}_{\text{best}}$$

where \(\phi\) is a random value in [0,1]; \(g\) is a random value, equivalent to \(5\times\) rand; and \(\lambda\) is an integer, taking values 1, 0, or −1. If the solution \({x}_{\text{ESQ}}\) does not improve the objective function compared to \({x}_{{\text{n}}_{3}}\) (i.e., \(f\left({x}_{\text{ESQ}}\right)>f\left({x}_{\text{m}}\right)\)), another solution, \({x}_{\text{new}3}\), is generated:

if rand < . \(\rho\)

$${x}_{\text{new}3}=\left({x}_{\text{ESQ}}-\text{rand}\cdot {x}_{\text{ESQ}}\right)+AF\cdot \left(\text{ rand. }{x}_{\text{RuK}}+\left(2\cdot \text{rand}.{x}_{\text{bs}}-{x}_{\text{ESQ}}\right)\right)$$
(9)

end.

3 Proposed Multi-objective Runge–Kutta Optimizer (MORKO)

3.1 Basic Definitions of Multi-objective Optimization

A multi-objective optimization is an approach that facilitates the simultaneous evaluation of multiple objectives within a given problem. Traditional relational operators prove inadequate for comparing solutions within MO problem spaces. Instead, Pareto optimality becomes essential for such evaluations. The basic definitions associated with MO problems emphasize the significance of the Pareto optimality operator in evaluating solutions.

Definition 1: Pareto optimality (PO)

The solution \(\overrightarrow{x}\in X\) is called PO if and only if:

$$\nexists \overrightarrow{y}\in X | F\left(\overrightarrow{y}\right)\prec F\left(\overrightarrow{x}\right)$$
(10)

Definition 2: Pareto dominance (PD)

Let two different vectors be represented as \(\overrightarrow{x}=({x}_{1},{x}_{2},\dots ,{x}_{j})\) and \(\overrightarrow{y}=({y}_{1},{y}_{2},\dots ,{y}_{j})\). The vector \(\overrightarrow{y}\) is said to dominate the vector \(\overrightarrow{x}\) (symbolized as \(\overrightarrow{y}\prec \overrightarrow{x}\)), if and only if:

$$\forall i\in \left\{\text{1,2},\dots ,j\right\}: {f}_{i}\left(\overrightarrow{y}\right)\le {f}_{i}\left(\overrightarrow{x}\right)\wedge \exists i\in \left\{\text{1,2},\dots ,j\right\}:{f}_{i}\left(\overrightarrow{y}\right)<{f}_{i}\left(\overrightarrow{x}\right)$$
(11)

Definition 3: Pareto-optimal set (POS)

The collection of all POS is termed the Pareto set, and it can be articulated in a subsequent manner.

$${P}_{s}=\{x,y\in X | \exists F\left(\overrightarrow{y}\right)\succ F(\overrightarrow{x})\}$$
(12)

Definition 4: Pareto-optimal front (POF)

The POF consists of a set of optimal solutions found within the POS, as indicated in Eq. (13).

$${P}_{f}=\left\{F\left(\overrightarrow{x}\right)|\overrightarrow{x}\in {P}_{s}\right\}$$
(13)

The relationship between the objective and search domains is depicted in Fig. 1.

Fig. 1
Fig. 1
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Objective space and search space in MO problem

3.2 Multi-objective Runge–Kutta Optimizer (MORKO)

The sophisticated MORKO optimization algorithm incorporates a blend of the crowding distance (CD) technique and the elitist non-dominated sorting (NDS) strategy. The steps of the NDS process are outlined as follows:

  1. i.

    Identification of non-dominated solutions.

  2. ii.

    Implementation of the NDS strategy.

  3. iii.

    Calculation of non-dominated ranking (NDR) for all non-dominated candidates.

The NDR process distinguishes between different layers or fronts. Solutions on the primary front receive a rank of “0,” indicating no dominance by other solutions, whereas solutions on subsequent fronts are dominated by at least one solution from the primary front. The rank of a solution reflects the number of solutions that dominate it. The CD mechanism, depicted in Fig. 2, ensures a diverse range of solutions.

Fig. 2
Fig. 2
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The procedure of the CD measurement

The CD process is elaborated upon in the following section.

$${\text{CD}}_{\text{j}}^{i} =\frac{{fobj}_{j}^{i+1}-{fobj}_{j}^{i-1}}{{fobj}_{j}^{\text{max}}-{fobj}_{j}^{\text{min}}}$$
(14)

where \({fobj}_{j}^{\text{max}}\) and \({fobj}_{j}^{\text{min}}\) are the max and min values of \({j}^{\text{th}}\) objective function. The diagrammatic illustration of an NDS-based approach is illustrated in Fig. 3.

Fig. 3
Fig. 3
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The procedure of the NDS approach

The pseudocode for the MORKO algorithm is outlined in Algorithm II. The process begins by setting essential parameters such as population size (Np), stopping conditions, the maximum number of generational cycles, and the ultimate iteration count (Maxit). Initially, the objective space vector F for Po is determined by evaluating each objective function with a spontaneously generated initial population Po within the viable search domain S. Po then undergoes refinement through the elitist-based CD and NDS. Following this, Po is merged with a newly formed population, Pj, to create a combined population Pi. This combined set Pi is organized based on insights from CD, NDR, and elitist non-dominance in Algorithm 1. From this organized set, the top Np solutions are selected to form the next generation of the parent population. This cycle repeats until the predefined stopping conditions are met. A visual representation of the MORKO process is depicted in Fig. 4.

Fig. 4
Fig. 4
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Flowchart of MORKO algorithm

In multi-objective optimization problems, the MORKO algorithm balances exploration and exploitation by means of a mechanism that adjusts search behavior according to the phase of the optimization. In particular, MORKO uses a dual-level approach based on adaptive Runge–Kutta-inspired updating scheme and elitist non-dominated sorting with a crowding distance mechanism. This structure allows for fine grained control of exploration and exploitation dynamics during the optimization process.

MORKO starts with exploration to spend time broadly in sampling the solution space. Global search and avoidance of premature convergence to local optima is done using large distribution of search points and solution updates with randomness. The Runge–Kutta-based search strategy introduces randomness and diverse step sizes, which leads to a broad search, while preserving feasible solution structures.

At intermediate stages of the optimization, MORKO shifts its focus to a balance between exploration and exploitation. The adaptive coefficient mechanism adjusts step size and direction according to the convergence state of the population, decreasing randomness and focusing more work on potentially high potential regions of the search space. The crowding distance mechanism ensures diversity by keeping the Pareto front well spread, whereas the elitist non-dominated sorting directs the search toward promising regions with a combination of diversified and focused search behavior.

Later on, MORKO shifts its priority toward exploitation in later stages. It reduces the search space of the algorithm, improving fine tuning and local convergence on the Pareto front. Adaptive coefficients and the elitist crowding distance are used as mechanisms to refine solutions with precision while avoiding the population from stagnating. MORKO achieves an adaptive balance between exploration and exploitation at all points of the optimization process by incrementally adjusting parameters including randomness and step size according to iteration counts and current solution distributions. The staged prioritization guarantees consistent convergence toward a diverse set of high-quality Pareto-optimal solutions.

Algorithm 1
Algorithm 1
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Pseudocode of multi-objective Runge Kutta optimizer (MORKO)

4 Performance Analysis of MORKO Algorithm

The research design is a deliberate process, demanding meticulous attention to various parameters. This section provides a breakdown and elaboration of the research structure, decision parameters, and results analysis.

4.1 Algorithmic Comparison and Settings

Our proposed MORKO algorithm efficacy is gauged against the latest algorithms that were selected for juxtaposition, such as MOMPA [26], MOGBO [31], NSGA-II [28], and MOEA/D [27]. Original paper-recommended parameter settings were retained for this study.

4.2 Test Suites

In line with Rationale 3, diverse benchmark test suites were employed to scrutinize the MORKO algorithm. The test suites, including ZDT [50], DTLZ [51] based on constraint [52, 53] (CONSTR, TNK, SRN, BNH, OSY, and KITA) and based on engineering design brushless DC wheel motor [54] (RWMOP1), safety isolating transformer [55] (RWMOP2), helical spring [53] (RWMOP3), two-bar truss [53] (RWMOP4), welded beam [56] (RWMOP5), disk brake [57] (RWMOP6), tool spindle [58] (RWMOP7) and cantilever beam [59] (RWMOP8) problems.

The benchmark tests chosen in this study were chosen because they can rigorously test the strengths and weaknesses of multi-objective optimization algorithms and offer a complete performance landscape. Historically significant and well-known benchmark test suites were selected for their difficulty levels, facilitating a robust analysis of the multi-objective Runge–Kutta optimizer (MORKO). The ZDT, DTLZ, constrained (CONSTR, TNK, SRN, BNH, OSY, and KITA) and real-world engineering design problems are all benchmarks that pose different challenges for an algorithm to converge, to be diverse, and to be computationally efficient. The ZDT and DTLZ test suites are well known for their varying Pareto front characteristics, dimensionalities and non-convex, discontinuous and multi-modal properties. In particular, these properties are important for testing the ability of an algorithm to capture diverse and convergent Pareto fronts in multi-objective optimization problems. For instance, the challenges presented by various Pareto front shapes with the ZDT problems (convex, concave, discontinuous distribution) can test the adaptability of the algorithm with regard to different solution landscapes. In a similar vein, the DTLZ problems increase the complexity of the problem by introducing higher dimensional objective spaces, non-linear constraints and multi-modal search spaces, which makes the problem more difficult to solve in terms of convergence and diversity metrics. To assess MORKO robustness in handling constraints and preserving diversity on the Pareto front subject to strict feasibility requirements, constrained benchmark problems, CONSTR, TNK, SRN, BNH, OSY, and KITA were included. The problem instances vary in complexity and constraint interactions, and the challenge is to find an acceptable trade-off between feasibility and optimality. To demonstrate the applicability of MORKO in real-world engineering design problems, the Brushless DC wheel motor, safety isolating transformer, helical spring, two-bar truss, welded beam, disk brake, tool spindle, and cantilever beam were selected. These problems are complex, introducing non-linear constraints, multiple objectives, and highly dynamic search spaces that reflect real-world engineering design optimization conditions. These real-world problems are important because they show that MORKO is useful in practice and can find feasible and diverse solutions in realistic conditions. By systematically selecting benchmark tests and real-world applications, MORKO can be thoroughly and fairly tested across a wide range of theoretical and practical challenges. The characteristics of each problem are then used in a holistic assessment of the algorithm convergence capabilities, diversity preservation, computational efficiency, and handling of constraints, which demonstrate the robustness and versatility of MORKO in solving complex optimization problems.

4.3 Evaluative Metrics

In multi-objective optimization (MOO), performance metrics can be segregated into 1) qualitative (visual) and 2) quantitative (numerical) measures. While qualitative metrics depict solution quality, they fall short in offering algorithmic comparative analysis. Hence, quantitative metrics supplement qualitative measures to holistically appraise algorithmic prowess. Utilizing a solitary metric might not provide comprehensive insights; thus, an assortment of performance metrics is harnessed to ensure a balanced evaluation. Numerous performance metrics exist [60], categorized into (1) accuracy (convergence), (2) spread (coverage), and (3) distribution. To check the performance of algorithms in terms of faster convergence generational distance (GD), and spacing (SP), combined uniformity-convergence-coverage hyper volume (HV) and inverted generational distance (IGD), computational complexity (RT), and coverage spread (SD) have been applied. The mathematical equations of these performance indicators are as follows:

$$\text{Generational distance }\left(\text{GD}\right)=\frac{\sqrt{\sum_{i=1}^{no}{d}_{i}^{2}}}{n}$$
(15)
$$\text{Inverted generational distance }\left(\text{IGD}\right)=\frac{\sqrt{\sum_{i=1}^{nt}{\left({d}_{i}^{\prime}\right)}^{2}}}{n}$$
(16)
$${\text{Spacing}}\,{ }\left( {SP} \right) \triangleq \sqrt {\frac{1}{n - 1}\mathop \sum \limits_{i = 1}^{n} \left( {\overline{d} - d_{i} } \right)^{2} }$$
(17)
$$\text{Spread }\left(\text{SD}\right)=\sqrt{\sum_{i=1}^{o}\text{max}\left(d\left({a}_{i},{b}_{i}\right)\right)}$$
(18)
$${\text{Hyper volume}}\left( {{\text{HV}}} \right) = {\Lambda }\left( {\mathop {\text{U}}\limits_{s \in PF} \left\{ {s^{\prime } s \prec s^{\prime } \prec s^{{{\text{nadir}}}} } \right\}} \right)$$
(19)
$$\text{RUNTIME metric }\left(\text{RT}\right)=\frac{{\sum }_{i=1}^{n} \text{T}}{n}$$
(20)

where no is the number of true Pareto solutions (PS), nt is the number of true Pareto-optimal solutions, o is the number of objectives, \(\overline{d}\) is the average of all \({d}_{i}\), \({d}_{i}\) and \({d}_{i}^{\prime}\) specifies the Euclidean distance, n is the number of obtained PS, \({d}_{i}=\underset{j}{\text{min}}\left({|f}_{1}^{i}\left(\overrightarrow{x}\right)-{f}_{1}^{j}(\overrightarrow{x})|+{|f}_{2}^{i}\left(\overrightarrow{x}\right)-{f}_{2}^{j}(\overrightarrow{x})\right)\) for all i,j = 1,2,…,n., \({a}_{i}\) and \({b}_{i}\) is the maximum and minimum value in the ith objective. “ + / − / ~ ” Wilcoxon signed-rank test (WSRT) was conducted at a significance level of 0.05 between the total amount of test problems on which the corresponding optimizers had a better performance, a worse performance, and an equal performance of MOMPA, MOGBO, MOEA/D and NSGA-II with respect to MORKO algorithm.

The parameter settings of the MORKO algorithm are critical for its performance optimization and for achieving adequate exploration and exploitation over the search space. The key parameters and their roles are detailed below:

  • Population size (Np): This means, in each iteration how many candidate solutions will be considered for further improvement. Diversity of solutions is not possible with smaller population size, thus the solutions do not display large coverage of the Pareto front; however, this leads to reduced computational overhead. On the contrary, a bigger population size can promote the convergence but may bring about the convergence in advance or low diversity.

  • Maximum iterations (MaxIt): It gives the maximum number of iterations or generational cycle this algorithm will run with before terminating. It is a stopping criterion, which means an algorithm terminates in a finite time. MaxIt is the value that is chosen usually to achieve the balance of computational cost and solution quality and based on the complexity of the problem.

  • Adaptive coefficients (AF and related terms): Adaptive coefficients are used by MORKO to control perturbations and to refine solutions during the search process. The goal of these coefficients is to dynamically change the magnitude of changes made to candidate solutions, in such a way that a balance between exploration (searching broadly in the solution space) and exploitation (refining known good solutions) is maintained. These coefficients are adaptive, so that as the optimization process proceeds, the algorithm changes its behavior as needed.

  • Crowding distance (CD) parameters: The crowding distance mechanism ensures that the solutions within the same Pareto front are diverse, by calculating the average distance between a solution and its neighboring solutions in objective space. This guarantees that the chosen solutions are well spread and diverse, and therefore less likely to cluster, and better represent the Pareto front.

  • Non-dominated sorting parameters: Solutions are ranked based on elitist non-dominated sorting mechanism, giving priority to lower rank non-dominated solutions. Selecting a better (Pareto) solution from a generation depends upon these parameters which directly affects the convergence and the diversity of the solutions.

These two parameters jointly govern behavior of the MORKO algorithm by balancing convergence toward the optimal solutions through difficulty of locating new solutions, and a need for diversity of solutions, i.e., to avoid convergence to the same region of the search space. These settings need to be correctly tuned and understood to achieve high performance across a wide range of multi-objective optimization problems.

4.4 Analysis and Observation

Simulations were conducted 30 times for each test issue on a system featuring: Windows 10 (64-bit), Intel i5 CPU, 8 GB RAM, and Matlab R2021a. This section delves into the results from distinct metrics and offers insights.

4.4.1 ZDT Benchmarks Analysis

Tables 2, 3, 4, 5, 6, 7, and Fig. 5 show the qualitative and quantitative outcomes of the MORKO optimizer on ZDT1-ZDT6 problems with different metrics results. In addition, these experimental results illustrate that MORKO has a better performance in solving ZDT problems, as it is best on 4/6 on GD, 4/6 on IGD, 4/6 on SP, 5/6 on SD, 4/6 on HV, and 5/6 on RT metrics. MOMPA, MOGBO, NSGA-II, and MOEA/D are, respectively, best on 0, 0, 2, and 0 cases for GD, best on 0, 0, 2, and 0 cases for IGD, best on 0, 2, 0, and 0 cases for SP, best on 1, 0, 0, and 0 cases for SD, best on 0, 1, 1, and 0 cases for HV and, respectively, best on 1, 0, 0, and 0 cases for RT metrics. For instance, ZDT4 has multiple optimal points, but MORKO never gets trapped in a local Pareto front and is better in terms of convergence (GD, SP), divergence (IGD, HV), computational burden (RT), and solution distribution (SD) compared to MOMPA, MOGBO, MOEA/D, and NSGA-II algorithms.

Table 2  Performance outcomes of GD metric across various multi-objective algorithms for ZDT problems
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Table 3  Performance outcomes of IGD metric across various multi-objective algorithms for ZDT problems
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Table 4  Performance outcomes of SP metric across various multi-objective algorithms for ZDT problems
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Table 5  Performance outcomes of SD metric across various multi-objective algorithms for ZDT problems
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Table 6  Performance outcomes of HV metric across various multi-objective algorithms for ZDT problems
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Table 7  Performance outcomes of RT metric across various multi-objective algorithms for ZDT problems
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Fig. 5
Fig. 5
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Optimal Pareto fronts achieved by the MORKO algorithm for ZDT1-ZDT6 with dual objectives

4.4.2 DTLZ 2-Objectives Benchmark Insights

Tables 8, 9, 10, 11, 12, 13, and Fig. 6 show the qualitative and quantitative outcomes of the MORKO optimizer on DTLZ1-DTLZ8 two objectives’ problems with different metrics results. In addition, these experimental results illustrate that MORKO has a better performance in solving DTLZ two objectives’ problems, as it is best on 6/8 on GD, 5/8 on IGD, 6/8 on SP, 6/8 on SD, 5/8 on HV, and 5/8 on RT metrics. MOMPA, MOGBO, NSGA-II, and MOEA/D are, respectively, best on 0, 0, 1, and 1 cases for GD, best on 0, 3, 0, and 0 cases for IGD, best on 1, 0, 0, and 1 case for SP, best on 1, 0, 0, and 1 cases for SD, best on 2, 1, 0, and 0 cases for HV and, respectively, best on 1, 1, 1, and 0 cases for RT metrics. DTLZ is a tier above in terms of difficulty, outpacing both classical and ZDT tests. Issues like DTLZ1 to DTLZ4 pose significant convergence challenges. Still, the MORKO algorithm showcased superior results in convergence and diversity compared to its peers. Remarkably, on problems like DTLZ7, where others stumbled, MORKO consistently outperformed and was better in terms of convergence (GD, SP), divergence (IGD, HV), computational burden (RT) and solution distribution (SD) compared to MOMPA, MOGBO, MOEA/D, and NSGA-II algorithms.

Table 8  Performance outcomes of GD metric across various multi-objective algorithms for DTLZ 2-objectives problems
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Table 9  Performance outcomes of IGD metric across various multi-objective algorithms for DTLZ 2-objectives problems
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Table 10  Performance outcomes of SP metric across various multi-objective algorithms for DTLZ 2-objectives problems
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Table 11  Performance outcomes of SD metric across various multi-objective algorithms for DTLZ 2-objectives problems
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Table 12  Performance outcomes of HV metric across various multi-objective algorithms for DTLZ 2-objectives problems
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Table 13  Performance outcomes of RT metric across various multi-objective algorithms for DTLZ 2-objectives problems
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Fig. 6
Fig. 6
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Optimal Pareto fronts achieved by the MORKO algorithm for DTLZ1-DTLZ8 with dual objectives

4.4.3 DTLZ 3-Objectives Benchmark Insights

Tables 14, 15, 16, 17, 18, 19 and Fig. 7 show the qualitative and quantitative outcomes of the MORKO optimizer on DTLZ1-DTLZ7 three objectives’ problems with different metrics results. In addition, these experimental results illustrate that MORKO has a better performance in solving DTLZ three objectives’ problems, as it is best on 4/7 on GD, 5/7 on IGD, 4/7 on SP, 4/7 on SD, 4/7 on HV, and 5/7 on RT metrics. MOMPA, MOGBO, NSGA-II, and MOEA/D are, respectively, best on 0, 1, 1, and 1 cases for GD, best on 1, 0, 1, and 0 cases for IGD, best on 0, 1, 1, and 1 case for SP, best on 1, 1, 1, and 0 cases for SD, best on 1, 0, 1, and 1 cases for HV and, respectively, best on 0, 0, 1, and 1 cases for RT metrics. DTLZ is a tier above in terms of difficulty, outpacing both classical and ZDT tests. Issues like DTLZ1 to DTLZ4 pose significant convergence challenges. Still, the MORKO algorithm showcased superior results in convergence and diversity compared to its peers. Remarkably, on problems like DTLZ7, where others stumbled, MORKO consistently outperformed. In conclusion, across a spectrum of benchmarks and challenges, the MORKO algorithm has demonstrated remarkable resilience and capability, often surpassing its competitors in critical areas.

Table 14  Performance outcomes of GD metric across various multi-objective algorithms for DTLZ 3-objectives problems
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Table 15  Performance outcomes of IGD metric across various multi-objective algorithms for DTLZ 3-objectives problems
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Table 16  Performance outcomes of SP metric across various multi-objective algorithms for DTLZ 3-objectives problems
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Table 17  Performance outcomes of SD metric across various multi-objective algorithms for DTLZ 3-objectives problems
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Table 18  Performance outcomes of HV metric across various multi-objective algorithms for DTLZ 3-objectives problems
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Table 19  Performance outcomes of RT metric across various multi-objective algorithms for DTLZ 3-objectives problems
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Fig. 7
Fig. 7
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Optimal Pareto fronts achieved by the MORKO algorithm for DTLZ1-DTLZ8 with 3-objectives

4.4.4 Evaluation of Constraint Benchmark

The constraint benchmarks represent a challenging set of large-scale tests. These tests are marked by a non-linear, multi-modal search space. A consistent challenge across these tests is achieving convergence and ensuring diversity in the non-dominated solution (NDS) across the Pareto front. The performance outcomes of algorithms for the constraint benchmark problems are detailed in Tables 20, 21, 22, 23, 24, 25, and Fig. 8 shows the qualitative and quantitative outcomes of MORKO optimizers on constraint problems with different metrics results. In addition, these experimental results illustrate that MORKO has a better performance in solving constraint (CONSTR, TNK, SRN, BNH, OSY, and KITA) problems, as it is best on 4/6 on GD, 4/6 on IGD, 4/6 on SP, 5/6 on SD, 4/6 on HV, and 4/6 on RT metrics. MOMPA, MOGBO, NSGA-II, and MOEA/D are, respectively, best on 0, 2, 0, and 0 cases for GD, best on 0, 2, 0, and 0 cases for IGD, best on 0, 1, 1, and 0 cases for SP, best on 0, 1, 0, and 0 cases for SD, best on 0, 1, 0, and 1 cases for HV and respectively best on 0, 0, 1, and 1 cases for RT metrics. Notably, several algorithms, such as MOMPA, MOGBO, NSGA-II, and MOEA/D, are tailored for handling such expansive problems. Yet, many struggled with finding an optimal solution within a restricted set of function evaluations. They often become ensnared in local optima, failing to approach the true Pareto front. In contrast, MORKO exhibited superior performance on all three performance measures: GD, IGD, SP, SD, HV, and RT. Figure 8 visually summarizes the average performance metrics for each algorithm. The GD metric illustrates the proximity of NDS to the true Pareto-optimal front. MORKO performance, as highlighted by the GD metric, shows its prowess in achieving NDS alignment with the Pareto-optimal front (see Fig. 8). The IGD and HV metrics further reinforce MORKO outstanding capability, especially in solution accuracy and SD.

Table 20  Performance outcomes of GD metric across various multi-objective algorithms for constrained problems
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Table 21  Performance outcomes of IGD metric across various multi-objective algorithms for constrained problems
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Table 22  Performance outcomes of SP metric across various multi-objective algorithms for constrained problems
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Table 23  Performance outcomes of SD metric across various multi-objective algorithms for constrained problems
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Table 24  Performance outcomes of HV metric across various multi-objective algorithms for constrained problems
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Table 25  Performance outcomes of RT metric across various multi-objective algorithms for constrained problems
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Fig. 8
Fig. 8
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Optimal Pareto fronts achieved by the MORKO algorithm for CONSTR, TANK, SRN, OSY, BIN, and KITA

4.4.5 Real-World Applications of MORKO

MORKO, having showcased impressive results across various benchmark test suites, is further tested on intricate real-world problems. All problems addressed incorporated constraints via the penalty function method, chosen for their varying complexity levels. The list includes: brushless DC wheel motor [54] (RWMOP1), safety isolating transformer [55] (RWMOP2), helical spring [53] (RWMOP3), two-bar truss [53] (RWMOP4), welded beam [56] (RWMOP5), disk brake [57] (RWMOP6), tool spindle [58] (RWMOP7) and cantilever beam [60] (RWMOP8) problems.

Tables 26, 27, 28, 29, 30, 31, and Fig. 9 show the qualitative and quantitative outcomes of the MORKO optimizer on RWMOP1-RWMOP8 problems with different metrics results. In addition, these experimental results illustrate that MORKO has a better performance in solving Brushless DC wheel motor (RWMOP1), safety isolating transformer (RWMOP2), helical spring (RWMOP3), two-bar truss (RWMOP4), welded beam (RWMOP5), disk brake (RWMOP6), tool spindle (RWMOP7), and cantilever beam (RWMOP8) problems, as it is best on 3/8 on GD, 5/8 on IGD, 5/8 on SP, 5/8 on SD, 6/8 on HV, and 5/8 on RT metrics. MOMPA, MOGBO, NSGA-II, and MOEA/D are, respectively, best on 1, 3, 0, and 1 cases for GD, best on 1, 1, 1, and 0 cases for IGD, best on 0, 1, 1, and 1 case for SP, best on 1, 1, 1, and 0 cases for SD, best on 2, 0, 0, and 0 cases for HV and, respectively, best on 3, 0, 0, and 0 cases for RT metrics. MORKO algorithm is better in terms of convergence (GD, SP), divergence (IGD, HV), computational burden (RT), and solution distribution (SD) compared to MOMPA, MOGBO, MOEA/D, and NSGA-II algorithms for solving real-world problems.

Table 26  Performance outcomes of GD metric across various multi-objective algorithms for RWMOP problems
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Table 27  Performance outcomes of IGD metric across various multi-objective algorithms for RWMOP problems
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Table 28  Performance outcomes of SP metric across various multi-objective algorithms for RWMOP problems
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Table 29  Performance outcomes of SD metric across various multi-objective algorithms for RWMOP problems
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Table 30  Performance outcomes of HV metric across various multi-objective algorithms for RWMOP problems
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Table 31  Performance outcomes of RT metric across various multi-objective algorithms for RWMOP problems
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Fig. 9
Fig. 9
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Optimal Pareto fronts achieved by the MORKO algorithm for RWMOP with dual objectives

Advanced mechanisms are combined to maintain computational efficiency and stability in the MORKO algorithm for solving complex problems. It first integrates elitist non-dominated sorting and crowding distance techniques to generate diverse and high-quality Pareto-optimal solutions without sacrificing computational speed. MORKO dynamically balances between exploration and exploitation during the optimization process, avoiding local optima and remains efficient at increasing problems complexity. The Runge–Kutta-based update mechanism also improves computational stability by allowing precise and adaptive updates to solution trajectories, guaranteeing convergence consistency across a wide variety of problem spaces.

The adaptive coefficient and efficient operator (ESQ) mechanism of MORKO makes it computationally efficient by effectively utilizing the computational resources. The dual-stage mechanism of combining crowding distance with non-dominated sorting optimizes the computational workload by filtering out solutions that are less promising early in the optimization cycle, and thus reducing the overall computational burden.

Indeed, adding complexity analyses and comparative experiments on computational efficiency would give further insight. We analyze the efficiency of the algorithm in terms of time complexity, space complexity and convergence rate on different test suites, e.g., ZDT and DTLZ benchmarks. MORKO robustness and computational stability would be highlighted by comparative studies with algorithms such as MOMPA, MOGBO, NSGAII, and MOEAD across a wide range of computational environments. The computational advantages of MORKO over other state-of-the-art algorithms would be quantified by running time performance, memory usage, and convergence rate metrics.

4.4.6 Significance Testing

For datasets that do not adhere to a normal distribution, it is vital to utilize non-parametric statistical evaluations. MORKO performance deviates from the comparator algorithms. Based on the WRST test was selected as the most suitable test for this purpose. The outcomes indicate MORKO premier ranking across all metrics. Given that all p values are below the 0.05 confidence threshold, there is a statistically significant difference in algorithm performance. MORKO consistently emerges as the top performer, both in individual metrics and in aggregated rankings.

5 Research Design Attributes

5.1 Comprehensive Test Suites Including Real-World Challenges

The MORKO algorithm demonstrated commendable efficacy across an array of benchmark test suites, ranging from classical ZDT, DTLZ, and constraint (CONSTR, TNK, SRN, BNH, OSY, and KITA) problems. The complexity of these problems varies, spanning different search spaces, objectives, and Pareto front shapes. In addition, MORKO adeptness is further confirmed by its successful application to real-world challenges (brushless DC wheel motor, safety isolating transformer, helical spring, two-bar truss, welded beam, disk brake, tool spindle, and cantilever beam) that introduce unique complexities such as multiple constraints and varied search domains.

5.2 Algorithm Benchmarking Set

Renowned algorithms, including MOMPA, MOGBO, NSGA-II, and WOEA/D, were chosen to evaluate MORKO performance. This comparison set is diverse, featuring NDS-based algorithms like NSGA-II, alongside decomposition-based ones like MOEA/D. Moreover, the latest MOMPA and MOGBO-rooted in a current optimization framework are also taken into account. MORKO superior outcomes, when pitted against these elite algorithms, establish it as a leading non-dominated sorting-based algorithm. It is worth noting that if too many algorithms were considered, their performances could blur into an indistinguishable average, a phenomenon akin to the no free lunch (NFL) theorem.

5.3 Dual Mechanism

One innovative aspect of our research and the MORKO algorithm is the introduction of a dual number of maturity levels (iterations) and the count of NDS solutions with CD mechanism, the algorithm efficiency is further tested. Although each algorithm has its strengths and limitations, with constrained resources and unique features, MORKO consistently identifies top-tier solutions in comparison to its competitors.

5.4 Evaluating Performance Metrics

Historical research often leans on a single metric for performance analysis, which may not capture an algorithm full capability. If an algorithm excels in one specific metric but struggles in others, it might provide a skewed understanding. For a more balanced view, multiple performance metrics have been employed. By conducting a correlation analysis across these metrics, we have determined that GD, IGD, SP, SD, HV, and RT are independent metrics that can truly reflect an algorithm performance. Recommendations include:

  1. 1.

    Utilizing independent performance metrics.

  2. 2.

    Integrating both quantitative metric results with qualitative Pareto front outcomes for performance evaluation.

5.5 Methodology for Ranking and Validation

Numerous quantitative metrics are added to the results section to validate the effectiveness of the proposed multi-objective Runge–Kutta optimizer (MORKO). Nevertheless, a thorough examination of these results, with a discussion of any disadvantages or anomalies that might have been observed, is required to achieve a balanced view. These points are addressed in more depth in this section.

While always superior across most benchmark sets, the behavior of MORKO exhibits some peculiarities and areas of improvement. For example, in the analysis of the ZDT benchmarks, MORKO showed strong convergence (better GD metrics) and good distribution characteristics (better SP and HV metrics). In a few cases, such as ZDT4, MORKO had difficulties maintaining diversity in densely populated areas of the Pareto front, and as a result, slightly lower coverage was achieved compared to other algorithms. The crowding distance mechanism could be limited to this due to its dependence on parameter tuning, which may be improved with further refinement.

MORKO outperformed the competitors in achieving a balance between convergence and diversity for DTLZ benchmarks. But the results for DTLZ6 and DTLZ7 showed a slightly less stable spread of solutions for some cases than NSGA-II and MOEA/D. This is due to the complex multi-modal landscapes and high-dimensional Pareto fronts of these benchmarks. Although robust, MORKO performance sometimes resulted in a marginal imbalance between exploration and exploitation. We suggest that this issue could be mitigated by adjusting MORKO parameter settings or using dynamic adaptation strategies.

MORKO was shown to be capable of handling diverse constraints on the constrained benchmarks (e.g., CONSTR, TNK, SRN, BNH, OSY, and KITA). MORKO delivered better results for the majority of constrained problems, but some cases showed convergence delays during the initial search phases. In particular, TNK and SRN converged slower at the beginning than MOMPA and MOEA/D because of the complexity of the constraint boundaries and the step size adjustment strategy of the algorithm. In this regard, a more flexible way for dealing with constraints within MORKO is emphasized as to accelerate convergence without sacrificing accuracy.

Overall, MORKO performed well in most scenarios with respect to real-world applications. However, RWMOP4 (two-bar truss design) was a challenge because of its complex trade-off structure and required slightly higher computational burden than MOMPA and MOGBO in the later iterations. These results indicate that additional improvements in computational efficiency, e.g., better operator design or fewer function evaluations, will enhance MORKO applicability to real-world problems with high computational cost.

The experimental results show that MORKO is robust in convergence, maintains diversity well, and is versatile enough to be applied to a variety of benchmarks. These performance nuances can be ameliorated in special instances of the problem, notably for constraint handling, and the high dimensionality of the Pareto front, as well as constraints in other scenarios. These improvements could make MORKO a leading optimization algorithm for a wide range of applications. This balanced analysis shows the impressive capabilities and areas for improvement, and is consistent with the metrics presented in this study.

MORKO robustness in evaluating the performance of optimization algorithms without assuming a specific data distribution, the non-parametric statistical methods were selected. Through this approach, comparisons across a different set of datasets and performance metrics remain valid and reliable. Optimization results, which may not conform to parametric assumptions of normality or homoscedasticity, are particularly well suited to non-parametric tests.

In this study, the Wilcoxon signed-rank test was applied to compare the performance of the MORKO algorithm to other algorithms on different benchmark test suites and real-world applications, taking into account paired observations. This test is very useful to check if the observed differences of performance metrics between MORKO and its competitors are statistically significant. A robust test statistic for this case is obtained by ranking the absolute differences between paired observations and adding the ranks of the positive and negative differences separately. Because the test does not assume normality of data, it is a more flexible and accurate performance evaluation method, particularly in the case of multi-objective optimization problems where non-linearities and complex interactions can exist.

The Wilcoxon signed-rank test is used to apply the test in an unbiased way, comparing MORKO performance to other algorithms in terms of solution quality, diversity, and computational efficiency over a variety of benchmarks. These tests results showed statistically significant differences (p values < 0.05 confidence level threshold) which validate that the MORKO algorithm is superior in generating consistent, high-quality Pareto fronts. Moreover, the non-parametric evaluations supported the outcomes of ranking on these measures, which have confirmed that MORKO was able to achieve better performance consistently across different metrics and test situations, validating the capability of the algorithm to produce reasonable and effective results in various situations of optimization.

5.6 Limitation and Potential Remedy

MORKO algorithm demonstrates robust performance on varied benchmark problems and viewed the real-world cases, there are several limitations in the algorithm that needs to be further explored and enhanced. The main limitations of these algorithms are due to the complex mechanisms required to converge and to achieve diversity, as well as their adaptability to different problem domains.

5.6.1 Computational Complexity

The elitist non-dominated sorting and crowding distance techniques on which MORKO relies inherently increase computational demands, especially for large-scale optimization problems with many objectives. The scalability and run time efficiency of the algorithm can be hindered when the population size or objective dimensions grow substantially, because of the computational burden. To alleviate this limitation, future improvements can entail developing efficient, though lighter weight, approximation schemes for sorting and diversity maintenance to decrease computational overhead.

5.6.2 Premature Convergence

Although MORKO includes a variety of exploration mechanisms, it may suffer from premature convergence in highly multi-modal optimization landscapes. In particular, this could make the algorithm stay confined in some local Pareto fronts, instead of uncovering a more extensive set of optimal solutions. Adaptive mechanisms can be introduced to dynamically balance exploration and exploitation or to hybridize with other metaheuristic approaches to improve capability in escaping local optima.

5.6.3 Parameter Sensitivity

MORKO is sensitive to a variety of algorithmic parameters including population size, adaptive coefficients, and step sizes used during iterations. However, tuning these parameters can be challenging, and context dependent, which may limit the robustness of the algorithm in different environments. This work focuses on the adaptation of the algorithm to changing, unanticipated problem instances by exploring automatic parameter tuning mechanisms, such as self-adaptative techniques or machine learning-based parameter control.

The performance of optimization algorithms, including MORKO, is highly sensitive to the choice of parameter settings. To be consistent and fair in benchmarking the performance of MORKO relative to its competitors, we carefully decided to use original parameter settings as suggested in previous literature for comparative algorithms (e.g., MOMPA, MOGBO, NSGA-II, and MOEA/D). These settings represent the best or near best configurations for these algorithms in various benchmark tests, such that comparisons accurately characterize algorithmic capabilities under realistic conditions.

Experimental evaluations of MORKO considered parameter sensitivity and its effect on performance. The values of the critical parameters like, population size, maximum number of generations, adaptive coefficients, randomness factors, have large impacts on the convergence rates, solution diversity, and the computational effort. MORKO is robust in different problem spaces because it can dynamically adjust these parameters using mechanisms such as adaptive coefficients and crowding distance control. MORKO adapts effectively to varying optimization landscapes due to the balance between exploration and exploitation phases which prevents MORKO from prematurely converging.

5.6.4 Dependency on Pareto Dominance

Although Pareto0based sorting techniques for non-dominated solutions are efficient for maintaining diverse fronts, they become less efficient for problems with many conflicting objectives (MaOPs) due to their heavy reliance on them. In these cases, the algorithm may not be able to maintain diverse Pareto fronts at low computational cost. Its capability in high-dimensional objective spaces could be enhanced by exploring alternative dominance relations, for example ε-dominance or reference point-based approaches.

5.6.5 Handling of Constraint-Based Problems

MORKO shows good performance on constraint-based problems via penalty functions, but its performance may be problem-dependent and constraint dependent. Tuning difficulties can arise when using penalty functions, sometimes because there rarely exists an optimal choice for a penalty function in complex, highly constrained scenarios. More consistent performance may be achieved by enhancing constraint handling techniques, for example hybrid penalty adaptive strategies, or by including constraint satisfaction mechanisms during search.

5.6.6 Scalability for Dynamic and Real-Time Applications

Further validation is needed to confirm that MORKO is suitable for dynamic and real-time optimization tasks where objectives and constraints can change over time. In case one needs to adapt the algorithm for such scenarios online learning mechanisms, rapid convergence criteria, or memory based techniques are likely to be required to monitor and respond to rapid variation in the search space.

Algorithmic refinements and parameter adjustments that implement judicious balancing mechanisms, and robust constraint handling ability provide the means to address these limitations. The strengths of MORKO applicability and performance using these enhancements should extend MORKO applicability and performance across an even broader spectrum of optimization challenges.

6 Conclusion and Future Scope

MORKO, leveraging advanced strategies such as elitist non-dominated sorting and crowding distance methodologies, has demonstrated its efficacy by outperforming several established algorithms such as MOMPA, MOGBO, MOEA/D, and NSGA-II. Its effectiveness is proven across a variety of benchmark challenges, ranging from the classical ZDT and DTLZ series to constraints (CONSTR, TNK, SRN, BNH, OSY, KITA) and specific engineering design problems (including brushless DC wheel motors, safety isolating transformers, Helical springs, two-bar trusses, welded beams, disk brakes, tool spindles, and cantilever beams). MORKO comprehensive performance analysis reveals its exceptional capability, shedding light on the limitations of current evaluation metrics and setting new standards for algorithmic efficiency. MORKO design integrates unique strategies such as an RKO update mechanism, which supports a balanced approach to exploration and exploitation during optimization tasks. This balance helps avoid common pitfalls like algorithmic stagnation, leading to its stellar performance in generating Pareto fronts that surpass those of its competitors. The algorithm robust framework has been consistently successful across diverse testing scenarios, proving its reliability and utility in optimizing complex systems.

Future research directions for the MORKO algorithm encompass several promising areas of development aimed at broadening its adaptability and efficiency in complex, real-world optimization scenarios:

  1. 1.

    Enhanced adaptability in multi-domain problems: A key target for continued expansion is MORKO ability to solve more difficult and diversified optimization problems in many other domains. Such refinements could include adaptive mechanisms for parameter adaptation with respect to different problem landscapes, making performance robust under possible disparities in constraints, objective contradictions, and domain specific challenges.

  2. 2.

    Hybridization with other metaheuristic strategies: Further improvements in convergence speed and solution diversity can be achieved by integrating complementary optimization strategies, e.g., by hybridizing MORKO with techniques from other algorithms, e.g., swarm intelligence or learning-based optimization. It could use the combination of strengths between different methods that could increase the quality of the generated Pareto fronts on complex search spaces.

  3. 3.

    Adaptive exploration and exploitation mechanisms: The risks of stagnation and optimization to globally optima are, however, mitigated by the introduction of advanced adaptive mechanisms that automatically balance the exploration and exploitation phases of the algorithm. This could be self-adaptive parameters or reinforcement learning inspired strategies that can lead the optimization process better.

  4. 4.

    Incorporation of real-time constraints and dynamic environments: Further research opportunity exists in extending MORKO capabilities to dynamic optimization problems in which constraints and objectives change over time. In such contexts, real-time adaptability is critical for applications in rapidly changing environments, e.g., industrial processes or evolving market conditions.

  5. 5.

    Scalability improvements for high-dimensional objectives: An important research goal is to improve the scalability of MORKO in terms of the number of objectives and size of the decision space. It covers the issues of computational efficiency and solution quality for solving high-dimensional multi-objective problems that are becoming common in modern engineering and management tasks.

  6. 6.

    Application-specific customizations and domain adaptations: The algorithm can be tailored for domain specific applications like large-scale engineering design, logistics and supply chain optimization, and may provide significant practical benefits. By incorporating problem specific heuristics and constraints, we can fine tune MORKO and improve performance in these areas.

  7. 7.

    Exploration of distributed and parallel implementations: The distributed and parallel implementations of MORKO could be further explored to take advantage of modern high-performance computing platforms. That would make processing faster, and allow one to handle computationally more intensive optimization problems. Through these focused directions, MORKO can be significantly adapted, scaled, and efficient on multi-domain optimization tasks, opening the door to more complex and practical applications.