Solving the Economic Load Dispatch Problem by Attaining and Refining Knowledge-Based Optimization

Abstract

The Static Economic Load Dispatch (SELD) problem is a paramount optimization challenge in power engineering that seeks to optimize the allocation of power between generating units to meet imposed constraints while minimizing energy requirements. Recently, researchers have employed numerous meta-heuristic approaches to tackle this challenging, non-convex problem. This work introduces an innovative meta-heuristic algorithm, named “Attaining and Refining Knowledge-based Optimization (ARKO)”, which uses the ability of humans to learn from their surroundings by leveraging the collective knowledge of a population. The ARKO algorithm consists of two distinct phases: attaining and refining. In the attaining phase, the algorithm gathers knowledge from the population’s top candidates, while the refining phase enhances performance by leveraging the knowledge of other selected candidates. This innovative way of learning and improving with the help of top candidates provides a robust exploration and exploitation capability for this algorithm. To validate the efficacy of ARKO, we conduct a comprehensive evaluation against eleven other established meta-heuristic algorithms using a diverse set of 41 test functions of the CEC-2017 and CEC-2022 test suites, and then, three real-life applications also verify its practical ability. Subsequently, we implement ARKO to optimize the SELD problem considering several instances. The examination of the numerical and statistical results confirms the remarkable efficiency and potential practical ability of ARKO in complex optimization tasks.

1. Introduction

Optimization involves identifying the most effective combination of decision variables to address a specific problem. It is a concept used across various fields, disciplines, and numerous applications. A canonical form of a constrained optimization problem can be defined as follows:

$$\begin{array}{l}\mathrm{M}inf(Y):Y=[y_1,y_2\cdots y_n]\\ s.t.:g_j(Y)=0;h_k(Y)\le 0;j=1,2,\cdots s;k=1,2,\cdots r;\\ \&y_{lower}\le y_i\le y_{upper}\forall i=1,2,\cdots n\end{array}$$

Here, s and r are the number of equality and inequality constrained, respectively.

The Static Economic Load Dispatch (SELD) optimization problem has an essential role in power system engineering. The objective is to determine the optimal power distribution among generating units to reduce fuel generation costs while adhering to all system inequality and equality restrictions. Enhancements in scheduling unit output could yield substantial annual savings in production expenses.

This issue has previously been thoroughly examined using many traditional optimization techniques, including linear programming [1], quadratic programming [2], the Lambda iteration method [3], the Newton–Raphson method [3], dynamic programming [4], and so on.

The Lambda iteration is the most familiar method to solve SELD problems due to its simple functioning; however, it requires a continuous objective formulation for effective execution. The other well-known method, dynamic programming, works effectively on a small power system only, but increases the computational complexity significantly while working on a large power system.

In general, the conventional approaches typically assume that the objective function of the generator’s fuel cost is convex and smooth; however, due to several limitations, including transmission line capacity, ramp rate restrictions, outlawed operating regions, and valve points, the problem is typically non-convex [5]. Therefore, it is difficult for these methods to address non-convex optimization issues and, consequently, researchers rely on meta-heuristic algorithms (MAs) to solve non-convex and complicated optimization problems [6].

The term “meta-heuristics” refers to a broad class of heuristic methods that have become increasingly valued due to their ease, portability, capability of avoiding local optima, and efficiency when applied to non-derivative situations. A common feature of MAs is that they merge rules and randomness to imitate some natural phenomena. The classifications of a few reputable and well-known meta-heuristic techniques are provided below.

Evolutionary-based MAs: These algorithms are inspired by several natural biological evolution processes. The effective exploring and exploiting capacity of complex solution spaces makes them useful tools for optimization in a variety of contexts. The evolution strategy (ES) [7], the genetic algorithm (GA) [8], and differential evolution (DE) [9] are the most common methods in the evolutionary-based MA category. The ES relies on some natural phenomena and self-adaptive mutation while the GA is inspired by natural and genetic selection. The GA relies on its chromosome crossover operation. Similarly, the DE algorithm is inspired by natural selection and Darwin’s survival of the fittest theory. Unlike the GA, DE relies on its mutation and crossover operation. Some other algorithms in the evolutionary-based MA category are the quantum evolutionary (QE) algorithm [10], the differential search (DS) algorithm [11], the backtracking optimization (BO) algorithm [12], etc.

Swarm-based MAs: These algorithms are modeled after the cooperative behavior of decentralized, self-organizing structures, such as swarms of insects, schools of fish, and flocks of birds. The fundamental idea behind these algorithms is that a collection of agents, or particles, can work together to efficiently search for a solution space and identify an ideal or almost ideal solution. The ant colony optimization (ACO) algorithm [13] is one of the popular algorithms under this category. It simulates the social behavior of ants in finding food by using the optimal path. The particle swarm optimization (PSO) algorithm [14] is another very popular algorithm that is inspired by the social behavior of fish and birds. It creates a swarm of particles that move in the search domain to discover the best solution. The particles update their position based on their own and nearby candidates’ experiences. The other well-known algorithms that fall into the swarm-based MA category are the bacterial foraging optimization (BFO) algorithm [15], the artificial bee colony (ABC) algorithm [16], the cuckoo search (CS) algorithm [17], the bat algorithm [18], the firefly algorithm (FA) [19], the gray wolf optimization (GWO) algorithm [20], the whale optimization algorithm (WOA) [21], manta ray foraging optimization (MRFO) [22], the reptile search algorithm (RSA) [23], etc.

Physics-based MAs: These methods draw inspiration from physics concepts and phenomena and can efficiently search solution spaces by utilizing ideas from thermodynamics, dynamic systems, and energy minimization. By imitating physical processes, they seek to attain the finest solutions. Some popular methods that fall within these categories include: simulated annealing (SA) [24], the variable neighborhood (VN) algorithm [25], the harmony search (HS) algorithm [26], central force optimization (CFO) [27], the gravitational search algorithm (GSA) [28], charged system search (CSS) [29], the black hole (BH) algorithm [30], the water cycle (WC) algorithm [31], the Sine–Cosine Algorithm (CSA) [32], thermal exchange optimization (TEO) [33], atom search optimization (ASO) [34], etc.

Human-based MAs: These methods draw inspiration from social and cognitive aspects of human behavior. These algorithms investigate and optimize solutions to complicated issues by utilizing human problem-solving techniques, teamwork, and decision-making. Some popular algorithms that fall within these categories include the following: society and civilization [35], the league championship algorithm [36], the brain storm optimization (BSO) algorithm [37], the teaching–learning-based optimization (TLBO) algorithm [38], the gaining sharing knowledge (GSK) algorithm [39], the driving training optimization (DTBO) algorithm [40], etc.

In the past, several studies have utilized different MAs to address SELD problems. Walters and Sheble [41] implemented GA to solve the SELD problem by utilizing the payoff information of candidates’ solutions and comparing the result with dynamic programming methods. Sinha et al. [42] also showed that the EP can also be an alternative method for solving ELD problems. Later, Gaing [43] applied PSO to the SELD problem with a non-smooth objective function with constraints such as prohibited operating zone and ramp rate limits and found better solutions than the GA. Noman and Iba applied DE [44] to solve this problem and found better results than other algorithms such as PSO and the GA. Later, Kumar et al. [45] implemented a modified DE variant and found impressive results compared with the GA, PSO, AIS, etc.

Some recent progress in the application of advanced MAs for solving SELD problems can be found in [46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63].

Although SELD problems have been successfully handled by these algorithms, our intensive study on this subject concludes that this type of problem can be solved more precisely by the establishment of new methods. In addition, many researchers are still motivated to build new algorithms for specific optimization problems, influenced by the No Free Lunch (NFL) theorem [64], which states that a method cannot be expected to perform optimally across all optimization problems.

The above research motivated us to develop a novel optimization technique for resolving optimization challenges in this study.

This study presents a fresh contribution through the construction of a new MA termed “Attaining and Refining Knowledge-based Optimization (ARKO)”, which is based on the ability of humans to learn from their surroundings. The proposed method aims to search the solution space as efficiently as possible by attaining knowledge from individuals of the elite group and then making improvements by exchanging the gained knowledge with other individuals in the subject population.

The working structure of ARKO distinguishes it from other well-known human-based MAs, such as TLBO or GSK. In TLBO, first, all candidates are trained by the best candidate, called the “teacher”, and subsequently improve themselves by learning from another selected candidate in the second phase. Similarly, GSK works in two phases called the junior and senior phases. During the junior phase, an individual gains knowledge from neighboring and trustworthy individuals and then updates this knowledge in the senior phase by utilizing the available information of the individuals selected from the best, medium, and worst categories, whereas in ARKO, a candidate first attains initial knowledge from two selected people from the elite group from the best and worst categories only, and then revises the old knowledge accordingly. After that, the candidates refine themselves by sharing their previously gained knowledge with two randomly selected candidates.

The following are the contributions of this study:

• The ARKO algorithm is presented in this study to attain and refine people’s knowledge using information from more knowledgeable people in a society.
• A set of 41 highly complicated test functions of the CEC-2017 and CEC-2022 test suites, three real-life applications, and 11 well-known MAs are employed to examine and compare the performance of the proposed ARKO algorithm.
• The efficiency of ARKO is assessed in terms of its ability to address a highly non-linear and non-convex SELD problem in power engineering.

The remainder of this article is structured as follows: Section 2 provides the mathematical description of the SELD problem. Section 3 explains the mathematical formulation and operations of the proposed ARKO algorithm. Section 4 analyzes the study results and comparisons of ARKO with other MAs based on the IEEE CEC test suites, real-life applications, and the SELD problem. Finally, this study’s conclusions and future directions are discussed in Section 5.

2. Static Economic Load Dispatch Problem [65]

The fundamental target of the SELD problem is to determine the best generator combination to minimize the total fuel cost of a power structure even while satisfying numerous related constraints. Symbolically, the objective function is represented by Equation (1):

$$\min C_{total}={\displaystyle \sum _{i=1}^{n_g}{f}_i(p_i)}$$

(1)

where C_total represents the total fuel cost, f_i(p_i) is the cost function for the power output p_i of the ith generator, and n_g is the generator quantity to be deployed in the power structure. The generator’s fuel cost is typically stated as a polynomial of second order, as provided by Equation (2):

$$f_i(p_i)={\lambda }_i{p}_i^2+{\eta }_i{p}_i+{\gamma }_i$$

(2)

where λ_i, η_i, and γ_i are the ith generation unit’s cost coefficients.

Additionally, the SELD problem is also subjected to several constraints, which are discussed below.

Generator Constraints: Every generating unit’s output power always falls between the bounds, i.e.,

$$p_i^L\le p_i\le p_i^U$$

(3)

Power Balance Constraints: This constraint is founded on the idea that the system’s losses and total power and load should be balanced. If p_Load and p_Loss are the system’s power load and transmission loss, respectively, then this constraint is defined by Equation (4).

$${\displaystyle \sum _{i=1}^{n_g}{p}_i}=p_{Load}-p_{Loss}$$

(4)

Next, the loss p_Load is attained using the B-coefficient method as specified in Equation (5):

$$p_{Load}={\displaystyle \sum _{i=1}^{n_g}{\displaystyle \sum _{j=1}^{n_g}{p}_i{B}_{ij}{p}_j+}}{\displaystyle \sum _{j=1}^{n_g}{B}_{0j}{p}_j+}{B}_{00}$$

(5)

where B_i,j is the ith-jth generator’s element of the B-matrix.

Ramp Rate Limit: One unrealistic presumption that simplified the issue is that power output adjustments are unbounded. However, in reality, all online unit operating ranges are limited by the ramp rate limit for altering the operation between two specific operating stages. The ramp rate constraints are defined in Equation (6):

$$\begin{array}{l}\mathrm{When}\mathrm{generation}\mathrm{increases}:p_i-p_i(G-1)\le R{R}_i^U\\ \mathrm{When}\mathrm{generation}\mathrm{decreases}:p_i(G-1)-p_i\le R{R}_i^L\end{array}$$

(6)

where p_i (G − 1) is the ith generator’s power generation at the preceding hour; $R{R}_i^L$ and $R{R}_i^U$ are the lower and upper ramp limits.

Hence, the ramp rate constraints change the generator restrictions using Equation (7):

$$\max (p_i^L,R{R}_i^U-p_i)\le p_i\le \min (p_i^U,p_i(G-1)-R{R}_i^L)$$

(7)

Prohibited Operating Zone:

Problems such as vibrations in the shaft bearings or constraints on the steam valves may cause the generating units to operate in restricted zones. The cost curves have discontinuities because of these zones. Thus, avoiding operations in these prohibited regions can result in optimal cost reduction.

The reasonable operation zones for the ith generating unit are specified in Equation (8):

$$p_i\le p_Z^L\&p_i\ge p_Z^U$$

(8)

where $[p_Z^L,p_Z^U]$ are the bounds of the ith generators, respectively, in the prohibited operating regions.

To ensure readability, the SELD problem can be written in a standard optimization form by including all the above constraints in Equation (9):

$$\begin{array}{l}\min C_{total}={\displaystyle \sum _{i=1}^{n_g}({\lambda }_i{p}_i^2+{\eta }_i{p}_i+{\gamma }_i}):p_i^L\le p_i\le p_i^U;\\ s.t.(a){\displaystyle \sum _{i=1}^{n_g}{p}_i}=p_{Load}-p_{Loss};\\ (b)p_{Load}={\displaystyle \sum _{i=1}^{n_g}{\displaystyle \sum _{j=1}^{n_g}{p}_i{B}_{ij}{p}_j+}}{\displaystyle \sum _{j=1}^{n_g}{B}_{0j}{p}_j+}{B}_{00}\\ (c)\left\{\begin{array}{l}{p}_i-p_i(G-1)\le R{R}_i^U:\mathrm{When}\mathrm{generation}\mathrm{increases}\\ p_i(G-1)-p_i\le R{R}_i^L:\mathrm{When}\mathrm{generation}\mathrm{decreases}:\end{array}\right.\\ (d)\max (p_i^L,R{R}_i^U-p_i)\le p_i\le \min (p_i^U,p_i(G-1)-R{R}_i^L)\\ (e)p_Z^U\le p_i\le p_Z^L\end{array}$$

(9)

3. Proposed Attaining and Refining Knowledge-Based Optimization (ARKO)

3.1. Inspiration

It is commonly believed that humans are influenced by the environment around them, and they learn a lot from individuals with greater knowledge in their society. In addition, they also continuously make efforts to improve their knowledge to make themselves more influential in society. Inspired by this human learning nature, we propose our Attaining and Refining Knowledge-based Optimization (ARKO) algorithm, where an individual attains knowledge from individuals with higher knowledge, and then refines it by adopting the knowledge from other selected persons of the updated society.

3.2. Mathematical Foundation of ARKO

The mathematical design and operations of the proposed ARKO algorithm are explained below:

Initialization: Similarly to other population-based meta-heuristics, we generate a uniformly distributed random population of n-dimensional M individuals, for example, $Y_i=\{(y_{i,1},y_{i,2},\cdots y_{i,n}):i=1,2,\cdots M\}$, by using Equation (10):

$$y_{i,j}=rand\times (y_{upper,j}-y_{lower,j})+y_{lower,j}$$

(10)

where rand is a uniform random number in [0, 1] and [y_lower,j, y_upper,j] are the jth component’s bounds, respectively.

The key difference between the MAs is the approach used for updating candidate positions. In ARKO, the positions of the individuals are updated in two different phases: (i) attaining knowledge and (ii) refining knowledge.

• Phase-1: Attaining Knowledge

Attaining knowledge is the primary phase of ARKO, where the candidates learn the behavior of other candidates and gain their initial knowledge. It contains three different stages: attaining, refining, and updating knowledge, which are defined below:

Attaining: At this stage, an individual gains knowledge with the help of the finest selected candidates. For this purpose, first, we divide the whole region into two groups based on their performances: group-I and group-II. Group-I, known as the elite group, contains the M_α best candidates of the population, while group-II contains the remaining (M − M_α) candidates. Now, a candidate $Y_i$ can attain and exchange their knowledge with randomly selected good candidates from each group. For this purpose, we select one candidate, for example, $Y_{\alpha }$, from group-I and another candidate, for example, $Y_{\beta }$, from group-II, and perform Equation (11):

$$Y_i^{new}=Y_i^{old}+ran{d}_1\times S_1\times (Y_{\alpha }-Y_i^{old})+ran{d}_2\times S_2\times (Y_{\beta }-Y_i^{old})$$

(11)

where rand₁ and rand₂ are uniform random numbers and S₁ and S₂ are sign functions defined in Equation (12).

$$S_1=\left\{\begin{array}{l}+1,iff(Y_{\alpha })\le f(Y_i^{old})\\ -1,otherwise\end{array}\right.;S_2=\left\{\begin{array}{l}+1,iff(Y_{\beta })\le f(Y_i^{old})\\ -1,otherwise\end{array}\right.$$

(12)

These sign functions play an essential role here. They help to improve the knowledge of a weak candidate toward the best candidate during their discussion. For example, there may be a possibility that $Y_i$ (especially for group-I candidates) will be better than the selected candidates $Y_{\alpha }$ or $Y_{\beta }$; in this case, the sign function will change the moving direction towards the $Y_i$ by giving a minus value.

Revising: In order to maintain diversity, the candidate revises its gained knowledge by restoring some information from old candidates. First, we fix a transfer ratio $T_R\in (0,1)$, which decides the number of components, for example, n₁, to be replaced where $n_1=round(n\times T_R)$. Next, we select n₁ indexes, for example, j₁, j₂, …j_n1, randomly from j =1, 2,…, n and replace these components with the old candidate’s information as defined in Equation (13):

$$y_{i,j}^{(new)}=\left\{\begin{array}{ccc}{y}_{i,j}^{(old)}& if& j\in (j_1,j_2,\cdots j_{n_1})\\ y_{i,j}^{(new)}& else\end{array}\right.$$

(13)

where $y_{i,j}^{(new)}$ and $y_{i,j}^{(old)}$ are the jth components of the corresponding ith candidates, respectively.

Updating: A tournament selection operation is performed to generate an upgraded candidate’s population for the next phase using Equation (14):

$$X_i=\left\{\begin{array}{ccc}{Y}_i^{(new)}& if& f(Y_i^{(new)})\le f(Y_i^{(old)})\\ Y_i^{old)}& else\end{array}\right.$$

(14)

• Phase-2: Refining Knowledge

This phase plays an essential role in improving the performance of ARKO. There are two steps to refining the knowledge of the candidates. The first step is refining, where two selected candidates discuss among themselves and move towards better knowledge, and then share this collective knowledge with the target person to improve their previously acquired knowledge. The second step is updating, in which better candidates are selected for the next generation based on their newly gained knowledge.

Refining: At this phase, an individual $X_i$ improves by gaining knowledge from other randomly selected candidates, for example, $X_a^{(G)}$ and $X_b^{(G)}$, from the updated population. First, the selected candidates, $X_a^{(G)}$ and $X_b^{(G)}$, exchange their knowledge and then transfer to $X_i$ as defined by Equation (15):

$$X_i^{(new)}=X_i^{(old)}+ran{d}_4\times S_3\times (X_a^{(old)}-X_b^{(old)})$$

(15)

where rand₄ is a uniform random number within [0, 1]; S₃ is a sign function and selects the direction to move between $X_a^{(old)}$ and $X_b^{(old)}$ as defined in Equation (16).

$$S_3=\left\{\begin{array}{l}+1,iff(X_a^{(old)})\le f(X_b^{(old)})\\ -1,otherwise\end{array}\right.$$

(16)

Updating: Finally, we select the best candidates between $X_i^{(new)}$ and $X_i^{(old)}$ to upgrade the next generation population using Equation (17):

$$X_i^{(new)}=\left\{\begin{array}{ccc}{X}_i^{(new)}& if& f(X_i^{(new)})\le f(X_i^{(old)})\\ X_i^{(old)}& else\end{array}\right.$$

(17)

A graphical presentation of working steps for the proposed ARKO algorithm and position updating of the individuals is provided in Figure 1. Figure 1a represents the working flow of both phases of the proposed ARKO, while Figure 1b signifies the movement of any individual Y_i to a new location Y_new during the attaining phase. However, transferring the information in this way may push the algorithm into local minima, which can be considered a disadvantage of the proposed approach. The issue of becoming stuck in local optima has been a common deficiency of almost all Mas, and various efforts have been made to overcome this limitation. Some of the prevalent approaches are hybridization with other algorithms, adjusting the algorithm’s control parameters, designing new escaping methods, etc. In ARKO, the refining phase was designed to maintain population diversity, and hence, it also helps the algorithm prevent the problem of local minima. The yellow dots represent the possible revising positions after transferring the knowledge gained to the old one. Hence, we can observe that the refining operation provides additional positions and recovers the exploration ability of the attaining phase. In addition, an appropriate setting of transfer rate (T_R) can improve the work efficiency of the algorithm (Algorithm 1).

Apart from this, the implementation of the above mentioned methods with ARKO may be helpful to effectively overcome the deficiency of local optima.

Algorithm 1: Attaining and Refining Knowledge-Based Optimization (ARKO)
1	Input: M, n, Max-iteration, Mα, TR, $n_1=round(n\times T_R)$;
2	Generate initial population $Y_i=\{(y_{i,1},y_{i,2},\cdots y_{i,n}):i=1,2,\cdots M\}$ using Equation (10);
3	Calculate function value f(Yi) for each I;
4	While iteration ≤ Max_Iteration;
5	Sort population by their fitness;
6	For i = 1: M //start phase-1;
7	Select $Y_{\alpha }$ randomly from best Mα candidates (group-I) and $Y_{\beta }$ randomly from remaining M − Mα candidates (group-II);
8	Perform the attaining phase and find $Y_i^{new}$ using Equations (11) and (12);
9	Find n1 indexes randomly from j = 1, 2, 3,…n and revise $Y_i^{new}$ using Equation (13):
10	if $f(Y_i^{new})\le f(Y_i^{old})$: $X_i=Y_i^{new}$; else: $X_i=Y_i^{old}$; Update population for the second phase using Equation (14);
11	End For //end phase-1;
12	For i = 1: M //start phase-2;
13	Select two candidates, for example, $X_a^{(old)}$ and $X_b^{(old)}$, randomly from the updated population and refine knowledge of $X_i^{(old)}$ using Equations (15) and (16)
	$X_i^{(new)}=X_i^{(old)}+rand\times S_3\times (X_a^{(old)}-X_b^{(old)})$ //Refine knowledge of $X_i$
14	if $f(X_i^{new})\le f(X_i^{old})$: $X_i^{new}=X_i^{new}$; else: $X_i^{new}=X_i^{old}$; Update population for next generation using Equation (17);
15	End For  //end phase-2;
16	iteration = iteration +1;
17	End While.

3.3. Computational Complexity

The computational complexity of ARKO will be equal to the sum of the complexity of the initialization and position updating phases. The computational complexity for initialization of the population of M candidates with n-decision variables will be equal to $O(M\times n)$. Now, the proposed ARKO algorithm requires two phases to update the individual’s position at each iteration. Hence, the computational complexity of the proposed algorithm at this stage is $O(G\times (M\times n+M\times n))$, i.e., $O(2G\times (M\times n))$, where G denotes the maximum generations. Therefore, finally, $O(M\times n\times (2G+1))$ is the total complexity of ARKO.

4. Results Examination

4.1. Performance Evaluation of IEEE CEC Functions

This section tests the algorithm’s efficiency using the IEEE CEC-2017 and CEC-2022 test suite benchmark functions.

4.1.1. IEEE CEC-2017 Test Functions

The IEEE CEC-2017 test suite is a popular benchmark suite made up of 29 (tf₀₁, tf₀₃–tf₃₀) rotated and shifted functions. These functions fall into four groups: unimodal (tf₀₁–tf₀₃), multimodal (tf₀₄–tf₁₀), hybrid (tf₁₁–tf₂₀), and composite (tf₂₁–tf₃₀) functions. For each function, the preliminary bounds are set to −100 and 100, while the optimum value is predetermined using $100\times t{f}_{no}$ [66]. More information about these functions can be found in [67].

4.1.2. IEEE CEC-2022 Test Functions

The IEEE CEC-2022 test suite has 12 (tc₀₁–tc₁₂) rotated and shifted benchmark functions. These functions fall into four groups: one basic unimodal (tc₀₁), four basic multimodal (tc₀₂–tc₀₅), three hybrid (tc₀₆–tc₀₈), and four composite (tc₀₉–tc₁₂) functions. For each function, the preliminary bounds are set to −100 and 100, while the optimum values are 300, 400, 600, 800, 900, 1800, 2000, 2200, 2300, 2600, and 2700, respectively. More information codes and input can be found at https://github.com/P-N-Suganthan?tab=repositories [68] (accessed on 12 January 2024).

4.1.3. Experimental Settings

The system configuration and parameter settings are as follows:

• System Configuration: OS—Windows 11 (64 bit), processor—i5 (Intel Core 2.6 GHz), and RAM—8 GB.
• M = 100, n = 30, M_α = 20, T_R = 0.5, Run = 50, Max-iteration = 100 × n.

4.1.4. Sensitivity Analysis of ARKO on Transfer Ratio

First, a sensitivity analysis was carried out by setting T_R = 0.3, 0.5, 0.7, and 0.9 on the CEC-2017 functions for dimensions n = 10 and 30. The best, mean, and standard deviation (SD) values were compiled in

Table 1. Sensitivity analysis of ARKO on the CEC-2017 functions (n = 10) at T_R = 0.3, 0.5, 0.7, and 0.9.

Fun	TR = 0.3			TR = 0.5			TR = 0.7			TR = 0.9
	Best	Mean	SD	Best	Mean	SD	Best	Mean	SD	Best	Mean	SD
tf01	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100
tf03	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100
tf04	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100
tf05	9.950 × 10−1	2.776 × 100	1.389 × 100	5.240 × 10−8	1.064 × 100	1.254 × 100	3.256 × 100	4.512 × 100	1.328 × 100	3.625 × 100	4.467 × 100	1.010 × 100
tf06	0.000 × 100	6.821 × 10−14	6.227 × 10−14	0.000 × 100	4.547 × 10−14	6.227 × 10−14	0.000 × 100	9.095 × 10−14	5.084 × 10−14	1.091 × 10−11	1.476 × 10−10	2.474 × 10−10
tf07	1.364 × 101	1.485 × 101	1.158 × 100	1.205 × 101	1.378 × 101	1.150 × 100	1.188 × 101	1.451 × 101	2.013 × 100	1.344 × 101	1.651 × 101	1.943 × 100
tf08	1.536 × 100	3.733 × 100	1.379 × 100	1.009 × 100	2.110 × 100	1.283 × 100	2.700 × 100	4.128 × 100	1.025 × 100	4.134 × 100	6.110 × 100	2.461 × 100
tf09	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100
tf10	1.904 × 102	3.641 × 102	1.105 × 102	1.078 × 102	2.938 × 102	1.101 × 102	1.321 × 102	2.526 × 102	8.402 × 101	1.608 × 102	2.373 × 102	6.343 × 101
tf11	4.027 × 10−9	1.473 × 10−4	2.990 × 10−4	0.000 × 100	0.000 × 100	0.000 × 100	3.829 × 10−3	3.077 × 10−1	2.307 × 10−1	1.400 × 10−1	7.073 × 10−1	5.013 × 10−1
tf12	1.140 × 101	2.442 × 101	1.173 × 101	2.550 × 100	6.082 × 100	4.183 × 100	3.978 × 100	1.629 × 101	7.505 × 100	1.105 × 101	1.798 × 101	7.661 × 100
tf13	3.587 × 100	5.087 × 100	1.101 × 100	3.801 × 100	4.940 × 100	9.225 × 10−1	3.443 × 100	4.507 × 100	7.370 × 10−1	2.564 × 100	5.270 × 100	1.791 × 100
tf14	1.293 × 10−8	5.229 × 10−2	5.332 × 10−2	0.000 × 100	3.980 × 10−1	5.450 × 10−1	8.738 × 10−4	2.710 × 10−2	2.149 × 10−2	8.962 × 10−2	5.753 × 10−1	4.120 × 10−1
tf15	3.188 × 10−2	5.028 × 10−2	1.332 × 10−2	7.559 × 10−3	1.198 × 10−2	4.309 × 10−3	5.542 × 10−2	9.599 × 10−2	3.672 × 10−2	9.162 × 10−2	2.883 × 10−1	1.668 × 10−1
tf16	7.712 × 10−1	1.089 × 100	3.019 × 10−1	2.340 × 10−1	7.041 × 10−1	4.234 × 10−1	1.251 × 100	1.554 × 100	2.706 × 10−1	1.448 × 100	1.918 × 100	3.553 × 10−1
tf17	3.568 × 100	1.067 × 101	5.720 × 100	1.970 × 10−1	7.573 × 100	9.386 × 100	4.093 × 100	5.925 × 100	1.473 × 100	1.695 × 100	3.562 × 100	1.699 × 100
tf18	7.787 × 10−3	1.527 × 10−2	7.470 × 10−3	7.312 × 10−3	3.673 × 10−2	6.393 × 10−2	1.845 × 10−2	3.918 × 10−2	2.627 × 10−2	2.256 × 10−1	3.568 × 10−1	1.301 × 10−1
tf19	5.690 × 10−4	1.945 × 10−2	2.381 × 10−2	8.322 × 10−10	3.182 × 10−2	3.295 × 10−2	3.137 × 10−2	8.594 × 10−2	3.317 × 10−2	2.586 × 10−1	3.000 × 10−1	3.643 × 10−2
tf20	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	6.243 × 10−2	1.396 × 10−1	0.000 × 100	4.320 × 10−12	9.533 × 10−12	4.279 × 10−3	4.459 × 10−2	2.446 × 10−2
tf21	1.000 × 102	1.000 × 102	2.491 × 10−13	1.000 × 102	1.416 × 102	5.693 × 101	1.000 × 102	1.000 × 102	0.000 × 100	1.000 × 102	1.000 × 102	1.415 × 10−3
tf22	0.000 × 100	7.892 × 101	4.418 × 101	1.000 × 102	1.000 × 102	7.007 × 10−2	4.547 × 10−13	6.524 × 101	4.849 × 101	4.729 × 10−11	2.004 × 101	4.471 × 101
tf23	3.046 × 102	3.058 × 102	1.544 × 100	3.000 × 102	3.025 × 102	1.871 × 100	3.048 × 102	3.064 × 102	9.573 × 10−1	3.054 × 102	3.075 × 102	1.470 × 100
tf24	1.000 × 102	2.398 × 102	1.277 × 102	3.252 × 102	3.289 × 102	3.175 × 100	1.000 × 102	1.000 × 102	0.000 × 100	1.000 × 102	1.214 × 102	4.777 × 101
tf25	3.977 × 102	3.979 × 102	1.541 × 10−1	3.980 × 102	4.167 × 102	2.552 × 101	3.977 × 102	3.977 × 102	0.000 × 100	3.977 × 102	3.977 × 102	5.617 × 10−5
tf26	3.000 × 102	3.000 × 102	0.000 × 100	3.000 × 102	3.000 × 102	0.000 × 100	3.000 × 102	3.000 × 102	0.000 × 100	5.139 × 10−11	2.400 × 102	1.342 × 102
tf27	3.890 × 102	3.892 × 102	2.807 × 10−1	3.890 × 102	3.894 × 102	2.599 × 10−1	3.873 × 102	3.886 × 102	7.283 × 10−1	3.869 × 102	3.873 × 102	9.341 × 10−1
tf28	3.000 × 102	3.000 × 102	0.000 × 100	3.000 × 102	3.817 × 102	1.353 × 102	3.000 × 102	3.000 × 102	0.000 × 100	3.000 × 102	3.000 × 102	0.000 × 100
tf29	2.428 × 102	2.496 × 102	7.520 × 100	2.377 × 102	2.419 × 102	3.838 × 100	2.362 × 102	2.411 × 102	4.346 × 100	2.400 × 102	2.434 × 102	2.385 × 100
tf30	3.949 × 102	3.950 × 102	8.423 × 10−2	3.948 × 102	3.950 × 102	9.876 × 10−2	3.950 × 102	3.952 × 102	1.491 × 10−1	3.946 × 102	3.949 × 102	1.750 × 10−1
1/2/3/4	9/9/7/4			12/7/3/7			13/7/7/2			11/2/5/11
Rank	2.21			2.17			1.97			2.55

and

Table 2. Sensitivity analysis of ARKO on the CEC-2017 functions (n = 30) at T_R = 0.3, 0.5, 0.7, and 0.9.

Fun	TR = 0.3			TR = 0.5			TR = 0.7			TR = 0.9
	Best	Mean	SD	Best	Mean	SD	Best	Mean	SD	Best	Mean	SD
tf01	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100
tf03	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100
tf04	0.000 × 100	4.796 × 101	2.692 × 101	0.000 × 100	1.410 × 101	2.491 × 101	0.000 × 100	1.595 × 100	2.184 × 100	1.137 × 10−13	2.509 × 101	3.060 × 101
tf05	2.093 × 101	2.628 × 101	6.393 × 100	2.288 × 101	2.905 × 101	4.186 × 100	1.990 × 101	2.925 × 101	7.264 × 100	4.378 × 101	5.731 × 101	1.295 × 101
tf06	1.127 × 10−13	1.127 × 10−13	0.000 × 100	3.080 × 10−7	1.120 × 10−6	8.746 × 10−7	3.182 × 10−5	1.176 × 10−2	2.116 × 10−2	1.245 × 10−1	4.036 × 10−1	2.674 × 10−1
tf07	5.459 × 101	5.945 × 101	4.850 × 100	5.175 × 101	5.597 × 101	4.062 × 100	5.373 × 101	6.650 × 101	1.006 × 101	6.482 × 101	8.671 × 101	2.011 × 101
tf08	2.885 × 101	3.506 × 101	5.413 × 100	2.089 × 101	2.607 × 101	4.186 × 100	2.686 × 101	2.965 × 101	3.255 × 100	3.184 × 101	4.597 × 101	9.528 × 100
tf09	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	2.175 × 10−1	3.889 × 10−1	8.953 × 10−2	9.610 × 10−1	9.497 × 10−1	7.540 × 100	2.623 × 101	1.587 × 101
tf10	3.150 × 103	3.679 × 103	4.276 × 102	3.106 × 103	3.201 × 103	4.833 × 102	3.594 × 103	3.933 × 103	2.839 × 102	2.171 × 103	2.805 × 103	4.472 × 102
tf11	4.527 × 100	2.079 × 101	3.005 × 101	3.956 × 100	2.055 × 101	2.826 × 101	1.592 × 101	2.169 × 101	3.749 × 100	7.263 × 101	9.981 × 101	2.289 × 101
tf12	3.185 × 101	2.930 × 102	1.642 × 102	1.347 × 102	2.278 × 102	1.713 × 102	3.576 × 101	3.000 × 102	1.479 × 102	4.168 × 102	8.865 × 102	2.809 × 102
tf13	1.901 × 101	2.982 × 101	6.952 × 100	1.779 × 101	3.736 × 101	1.743 × 101	1.463 × 101	2.687 × 101	7.916 × 100	2.630 × 101	3.542 × 101	1.549 × 101
tf14	1.934 × 101	2.735 × 101	5.249 × 100	2.590 × 101	2.876 × 101	2.442 × 100	2.263 × 101	2.598 × 101	2.178 × 100	5.031 × 101	5.947 × 101	9.268 × 100
tf15	7.515 × 100	1.039 × 101	2.195 × 100	8.294 × 100	9.940 × 100	2.200 × 100	3.508 × 100	5.041 × 100	1.931 × 100	1.437 × 101	3.840 × 101	2.917 × 101
tf16	7.203 × 101	2.965 × 102	1.807 × 102	6.827 × 101	2.863 × 102	1.947 × 102	1.580 × 102	1.801 × 102	3.411 × 101	1.254 × 102	3.918 × 102	1.761 × 102
tf17	8.731 × 101	1.028 × 102	1.600 × 101	4.600 × 101	6.695 × 101	1.333 × 101	4.681 × 101	5.338 × 101	6.408 × 100	8.382 × 101	1.303 × 102	4.616 × 101
tf18	2.274 × 101	2.381 × 101	8.486 × 10−1	2.456 × 101	2.527 × 101	1.053 × 100	2.165 × 101	2.394 × 101	1.537 × 100	2.420 × 101	3.269 × 101	7.469 × 100
tf19	9.089 × 100	1.196 × 101	1.728 × 100	1.207 × 101	1.391 × 101	1.684 × 100	6.099 × 100	8.151 × 100	2.021 × 100	1.052 × 101	3.331 × 101	1.916 × 101
tf20	6.569 × 101	8.154 × 101	2.298 × 101	6.063 × 101	8.325 × 101	2.665 × 101	3.294 × 101	7.294 × 101	5.321 × 101	3.386 × 101	1.039 × 102	7.405 × 101
tf21	2.232 × 102	2.253 × 102	1.765 × 100	2.130 × 102	2.194 × 102	4.162 × 100	2.143 × 102	2.227 × 102	7.641 × 100	2.276 × 102	2.395 × 102	1.132 × 101
tf22	1.000 × 102	1.000 × 102	0.000 × 100	1.000 × 102	1.000 × 102	0.000 × 100	1.000 × 102	1.000 × 102	0.000 × 100	1.000 × 102	1.005 × 102	1.097 × 100
tf23	3.621 × 102	3.825 × 102	1.162 × 101	3.688 × 102	3.755 × 102	5.948 × 100	3.697 × 102	3.858 × 102	1.139 × 101	3.770 × 102	3.977 × 102	1.903 × 101
tf24	4.349 × 102	4.548 × 102	1.244 × 101	4.404 × 102	4.458 × 102	4.264 × 100	4.406 × 102	4.568 × 102	1.100 × 101	4.425 × 102	4.697 × 102	2.152 × 101
tf25	3.867 × 102	3.868 × 102	1.390 × 10−1	3.834 × 102	3.863 × 102	1.635 × 100	3.837 × 102	3.882 × 102	4.982 × 100	3.837 × 102	3.866 × 102	1.660 × 100
tf26	1.162 × 103	1.251 × 103	6.587 × 101	1.246 × 103	1.335 × 103	1.130 × 102	1.168 × 103	1.305 × 103	9.760 × 101	1.497 × 103	1.744 × 103	2.083 × 102
tf27	4.646 × 102	4.890 × 102	1.399 × 101	4.807 × 102	4.954 × 102	1.204 × 101	4.931 × 102	5.040 × 102	6.368 × 100	5.038 × 102	5.181 × 102	1.697 × 101
tf28	3.000 × 102	3.000 × 102	0.000 × 100	3.000 × 102	3.000 × 102	0.000 × 100	3.000 × 102	3.000 × 102	0.000 × 100	3.000 × 102	3.000 × 102	0.000 × 100
tf29	4.842 × 102	5.397 × 102	3.869 × 101	4.680 × 102	5.372 × 102	5.420 × 101	4.310 × 102	4.967 × 102	8.661 × 101	4.953 × 102	5.713 × 102	1.067 × 102
tf30	1.944 × 103	1.950 × 103	6.601 × 100	1.946 × 103	1.951 × 103	8.838 × 100	1.949 × 103	1.980 × 103	4.560 × 101	1.974 × 103	2.039 × 103	7.049 × 101
1/2/3/4	11/10/7/1			12/10/5/2			13/4/11/1			4/1/2/22
Rank	1.93			1.89			2			3.44

based on 50 independent runs. A position-wise comparison is also counted in the tables.

For n = 10, the algorithm takes the first, second, third, and fourth positions on 9/9/7/4 functions at T_R = 0.3. Similarly, for T_R = 0.5, 0.7, and 0.9, the position-wise performances are 12/7/3/7, 13/7/7/2 and 11/2/5/11, respectively. The average ranks of ARKO at T_R = 0.3, 0.5, 0.7, and 0.9 are 2.21, 2.17, 1.97, and 2.55, respectively. Therefore, it shows that T_R = 0.7 may be a good choice for low-dimension cases.

For n = 30, the algorithm takes the first, second, third, and fourth positions on 11/10/7/1 functions at T_R = 0.3. Similarly, for T_R = 0.5, 0.7, and 0.9, the position-wise performances are 12/10/5/2, 13/4/11/1, and 4/1/2/22, respectively. Hence, it is evident that the proposed ARKO algorithm provides approximately identical results for T_R = 0.3, 0.5, and 0.7. The average ranks of ARKO at T_R = 0.3, 0.5, 0.7, and 0.9 are 1.93, 1.89, 2, and 3.44, respectively. These results helped us to determine that T_R = 0.5 is a good choice for high-dimension cases and further experiments.

4.1.5. Performance Assessment with Meta-Heuristic Variants

The following 11 advanced meta-heuristic algorithms were selected for comparison with the performance of the proposed ARKO algorithm:

• TLBO [38]: Human-based Meta-heuristics.
• GSK [39]: Human-based Meta-heuristics.
• DTBO [40]: Human-based Meta-heuristics.
• AGBSO [69]: Modified Human-based Meta-heuristics.
• TDSD [70]: (Hybrid Physics-based Meta-heuristics.
• EJaya [71]: Modified EA-based Meta-heuristics.
• DisGSA [72]: Modified Physics-based Meta-heuristics.
• HMRFO [73]: Modified Swarm-based Meta-heuristics.
• CJADE [74]: Modified EA-based Meta-heuristics.
• SHADE [75]: Modified EA-based Meta-heuristics and winner of CEC-2014.
• IMODE [76]: Modified EA-based Meta-heuristics and winner of CEC-2020.

All of the experimental results and comparisons are simulated in

Table 3. Comparison of ARKO with other MAs (TLBO, DTBO, GSK, AGBSO, TDSD) on the CEC-2017 test functions.

Fun	ARKO		TLBO		DBTO		GSK		AGBSO		TDSD
	Mean	SD	Mean	SD	Mean	SD	Mean	SD	Mean	SD	Mean	SD
tf01	0.00 × 100	0.00 × 100	1.56 × 103	2.24 × 103	3.79 × 107	3.29 × 107	0.00 × 100	0.00 × 100	2.37 × 103	2.64 × 103	1.76 × 103	9.01 × 102
tf03	0.00 × 100	0.00 × 100	0.00 × 100	5.09 × 10−11	1.43 × 104	4.58 × 103	0.00 × 100	0.00 × 100	4.97 × 101	1.07 × 102	4.39 × 104	9.45 × 103
tf04	1.41 × 101	2.49 × 101	3.01 × 101	4.34 × 101	2.68 × 102	1.20 × 102	2.51 × 101	3.26 × 101	9.02 × 101	1.56 × 101	2.01 × 101	2.11 × 101
tf05	2.90 × 101	4.19 × 100	1.25 × 102	1.01 × 102	2.38 × 102	1.72 × 101	9.91 × 101	7.59 × 101	1.59 × 101	6.21 × 100	7.96 × 101	1.12 × 101
tf06	1.12 × 10−6	8.75 × 10−7	2.53 × 100	2.05 × 100	5.79 × 101	1.42 × 100	9.88 × 10−3	2.07 × 10−2	7.49 × 10−5	4.92 × 10−5	3.29 × 100	6.91 × 10−1
tf07	5.60 × 101	4.06 × 100	2.61 × 102	2.49 × 102	4.45 × 102	8.68 × 101	1.86 × 102	1.14 × 101	5.80 × 101	7.46 × 100	1.28 × 102	1.35 × 101
tf08	2.61 × 101	4.19 × 100	5.66 × 101	2.44 × 101	1.46 × 102	1.23 × 101	1.09 × 102	5.96 × 101	1.39 × 101	5.15 × 100	8.36 × 101	8.19 × 100
tf09	2.18 × 10−1	3.78 × 10−1	1.70 × 101	4.20 × 101	3.88 × 103	9.01 × 102	1.65 × 100	1.24 × 100	0.00 × 100	0.00 × 100	1.69 × 103	3.38 × 102
tf10	3.20 × 103	4.79 × 102	6.77 × 103	2.82 × 102	3.28 × 103	3.03 × 102	6.40 × 103	4.42 × 102	4.79 × 102	2.68 × 102	2.19 × 103	2.19 × 102
tf11	2.06 × 101	2.83 × 101	1.01 × 102	1.72 × 101	2.85 × 102	5.98 × 101	6.23 × 101	3.45 × 101	4.51 × 101	3.75 × 101	6.87 × 101	2.39 × 102
tf12	2.28 × 102	2.78 × 101	3.11 × 104	2.14 × 104	1.11 × 108	2.24 × 108	6.67 × 103	4.56 × 103	6.67 × 105	2.61 × 105	2.89 × 105	1.71 × 105
tf13	3.74 × 101	1.74 × 101	2.17 × 104	1.64 × 104	6.49 × 104	4.54 × 104	7.38 × 101	1.11 × 101	1.36 × 104	8.64 × 103	6.96 × 102	3.22 × 102
tf14	2.88 × 101	2.44 × 100	1.29 × 103	8.85 × 102	2.46 × 105	2.62 × 105	2.84 × 101	2.74 × 100	3.10 × 103	3.45 × 103	8.39 × 103	6.09 × 103
tf15	9.94 × 100	2.20 × 100	4.46 × 103	5.12 × 103	2.24 × 104	1.07 × 104	3.78 × 101	3.08 × 101	3.65 × 103	3.69 × 103	3.45 × 102	2.25 × 102
tf16	2.86 × 102	1.95 × 102	3.31 × 102	3.05 × 102	1.16 × 103	3.17 × 102	3.39 × 102	4.67 × 102	1.13 × 102	9.84 × 101	4.82 × 102	1.19 × 102
tf17	6.70 × 101	1.29 × 101	1.57 × 102	4.29 × 101	3.84 × 102	1.59 × 102	8.22 × 101	6.78 × 101	5.23 × 101	3.80 × 101	9.39 × 101	3.91 × 101
tf18	2.53 × 101	1.05 × 100	1.66 × 105	9.74 × 104	3.81 × 105	2.88 × 105	3.71 × 101	1.45 × 101	8.56 × 104	5.04 × 104	8.09 × 104	3.71 × 104
tf19	1.39 × 101	1.68 × 100	6.03 × 103	5.56 × 103	6.29 × 105	6.45 × 105	1.72 × 101	6.46 × 100	3.72 × 103	4.52 × 103	1.49 × 102	1.00 × 102
tf20	8.32 × 101	2.67 × 101	1.91 × 102	6.98 × 101	4.38 × 102	6.94 × 101	3.09 × 101	5.10 × 100	1.14 × 102	6.17 × 101	1.39 × 102	5.41 × 101
tf21	2.19 × 102	4.21 × 100	2.59 × 102	1.41 × 101	3.24 × 102	1.25 × 102	3.03 × 102	7.79 × 101	2.17 × 102	5.98 × 100	2.21 × 102	8.20 × 101
tf22	1.00 × 102	0.00 × 100	1.01 × 102	1.77 × 100	1.59 × 103	1.49 × 103	1.00 × 102	2.36 × 10−13	1.00 × 102	1.94 × 10−6	1.10 × 102	1.91 × 100
tf23	3.75 × 102	5.95 × 100	4.12 × 102	2.47 × 101	6.07 × 102	5.21 × 101	3.75 × 102	7.97 × 100	3.62 × 102	5.87 × 100	4.56 × 102	1.69 × 102
tf24	4.46 × 102	4.26 × 100	4.82 × 102	1.44 × 101	7.14 × 102	4.88 × 101	4.69 × 102	6.68 × 101	4.67 × 102	1.08 × 101	4.24 × 102	1.91 × 102
tf25	3.86 × 102	2.03 × 100	4.09 × 102	2.54 × 101	5.40 × 102	6.68 × 101	3.87 × 102	2.42 × 100	3.86 × 102	1.45 × 100	3.91 × 102	1.23 × 101
tf26	1.33 × 103	1.13 × 102	1.52 × 103	7.90 × 102	3.55 × 103	1.37 × 103	1.21 × 103	4.30 × 101	9.90 × 102	7.56 × 101	2.20 × 102	5.55 × 100
tf27	4.95 × 102	2.01 × 101	5.34 × 102	1.14 × 101	7.09 × 102	4.08 × 101	5.24 × 102	1.37 × 101	5.04 × 102	6.38 × 100	5.20 × 102	6.12 × 100
tf28	3.00 × 102	0.00 × 100	3.53 × 102	6.86 × 101	7.89 × 102	5.10 × 102	3.21 × 102	4.62 × 101	3.80 × 102	5.32 × 101	4.09 × 102	1.49 × 101
tf29	5.37 × 102	4.68 × 101	6.66 × 102	2.03 × 102	1.32 × 103	2.47 × 102	4.80 × 102	7.67 × 101	4.84 × 102	5.91 × 101	5.89 × 102	4.60 × 101
tf30	1.95 × 103	8.84 × 100	3.38 × 103	1.61 × 103	3.22 × 106	3.74 × 106	2.11 × 103	1.65 × 102	4.91 × 104	4.08 × 104	5.03 × 103	8.30 × 102
CPU time		175.68		176.23		174.12		195.05		186.15		198.24
w/l/t			28/0/1		29/0/0		22/5/2		19/10/0		26/3/0
p-values			<0.001		<0.001 +		0.001 +		0.590 =		<0.001 +

“+” and “=” are noticeably superior and equal, respectively.

,

Table 4. Comparison of ARKO with other MAs (EJAYA, DisGSA, HMRFO, SHADE, IMODE and CJADE) on the CEC-2017 test functions.

Fun	EJAYA		DisGSA		HMRFO		SHADE		IMODE		CJADE
	Mean	SD	Mean	SD	Mean	SD	Mean	SD	Mean	SD	Mean	SD
tf01	7.72 × 102	3.67 × 102	2.43 × 103	3.12 × 103	3.12 × 103	3.02 × 103	0.00 × 100	0.00 × 100	0.00 × 100	1.39 × 10−3	0.00 × 100	0.00 × 100
tf03	0.00 × 100	3.41 × 10−8	4.93 × 103	2.51 × 101	6.24 × 101	3.68 × 101	0.00 × 100	0.00 × 100	1.89 × 10−7	8.11 × 10−9	5.56 × 103	1.14 × 104
tf04	3.29 × 101	3.85 × 101	1.02 × 102	6.51 × 100	4.05 × 101	2.25 × 101	5.86 × 101	3.01 × 10−14	2.19 × 101	2.78 × 102	3.47 × 101	2.16 × 101
tf05	4.97 × 101	1.42 × 101	1.75 × 101	7.30 × 10−5	6.14 × 101	2.08 × 101	1.56 × 101	2.84 × 100	2.65 × 102	4.09 × 100	2.68 × 101	6.43 × 100
tf06	3.11 × 100	2.56 × 100	4.02 × 10−5	2.97 × 100	5.42 × 10−1	1.13 × 100	3.76 × 10−1	3.37 × 10−5	5.79 × 101	6.45 × 100	0.00 × 100	0.00 × 100
tf07	1.09 × 102	3.01 × 100	5.09 × 101	3.25 × 100	1.18 × 102	4.22 × 101	4.67 × 101	3.58 × 100	9.20 × 102	3.23 × 102	5.71 × 101	5.51 × 100
tf08	7.06 × 101	1.41 × 101	1.71 × 101	6.23 × 10−14	6.47 × 101	1.92 × 101	1.67 × 101	4.45 × 100	2.18 × 101	3.87 × 100	2.59 × 101	3.82 × 100
tf09	2.47 × 102	2.25 × 102	0.00 × 100	5.54 × 102	5.09 × 101	4.14 × 101	7.04 × 10−2	1.49 × 10−1	5.62 × 103	1.49 × 103	0.00 × 100	0.00 × 100
tf10	3.80 × 103	2.59 × 102	1.98 × 103	2.82 × 101	3.37 × 103	6.54 × 102	1.62 × 103	3.98 × 102	3.79 × 103	4.68 × 102	1.83 × 103	2.89 × 102
tf11	8.61 × 101	4.22 × 10−3	9.59 × 101	1.87 × 103	4.34 × 101	1.11 × 101	2.32 × 101	1.66 × 101	1.94 × 102	4.44 × 101	2.11 × 101	6.75 × 100
tf12	8.26 × 103	2.30 × 103	9.75 × 103	2.48 × 103	3.79 × 104	1.44 × 104	1.19 × 103	4.23 × 102	1.13 × 103	3.78 × 102	1.29 × 103	9.37 × 102
tf13	1.99 × 103	1.32 × 103	4.75 × 103	3.51 × 103	1.44 × 104	1.08 × 104	3.82 × 101	1.89 × 101	3.95 × 102	1.72 × 102	3.02 × 101	9.67 × 100
tf14	1.21 × 102	2.08 × 101	3.41 × 103	1.98 × 103	1.46 × 103	9.84 × 102	3.00 × 101	2.93 × 100	1.88 × 102	5.58 × 101	1.62 × 103	3.17 × 103
tf15	9.49 × 102	1.48 × 103	1.58 × 103	2.39 × 102	2.79 × 103	4.12 × 103	4.00 × 101	3.35 × 101	2.12 × 102	8.67 × 101	3.86 × 102	1.13 × 103
tf16	5.28 × 102	2.88 × 102	5.71 × 102	1.34 × 102	6.15 × 102	2.91 × 102	2.95 × 102	1.34 × 102	1.50 × 103	4.78 × 102	4.49 × 102	1.58 × 102
tf17	1.31 × 102	5.49 × 101	1.69 × 102	1.74 × 104	1.80 × 102	2.38 × 102	4.98 × 101	1.29 × 101	8.58 × 102	2.67 × 102	7.61 × 101	4.34 × 101
tf18	3.88 × 103	1.69 × 103	4.18 × 104	1.32 × 103	8.41 × 104	3.42 × 104	5.96 × 101	4.62 × 101	1.58 × 102	7.45 × 101	7.14 × 101	4.31 × 101
tf19	2.11 × 102	2.41 × 102	3.72 × 103	1.38 × 101	2.07 × 103	2.78 × 103	1.49 × 101	3.91 × 100	5.94 × 102	3.89 × 102	2.69 × 101	2.75 × 101
tf20	3.42 × 102	4.02 × 101	1.74 × 102	8.76 × 100	2.74 × 102	1.13 × 102	4.63 × 101	9.12 × 100	6.75 × 102	1.83 × 102	9.96 × 101	4.77 × 101
tf21	2.54 × 102	1.52 × 101	2.29 × 102	5.68 × 10−9	2.49 × 102	1.94 × 101	2.21 × 102	1.24 × 100	4.18 × 102	3.16 × 101	2.26 × 102	5.77 × 100
tf22	1.00 × 102	0.00 × 100	1.00 × 102	4.97 × 100	1.00 × 102	1.43 × 10−13	1.00 × 102	0.00 × 100	1.31 × 103	1.67 × 103	1.00 × 102	0.00 × 100
tf23	4.18 × 102	2.48 × 100	3.73 × 102	1.68 × 101	4.34 × 102	1.91 × 101	3.64 × 102	6.36 × 100	7.91 × 102	8.49 × 101	3.73 × 102	4.36 × 100
tf24	4.85 × 102	1.85 × 101	4.13 × 102	2.11 × 100	4.83 × 102	1.98 × 101	4.35 × 102	2.74 × 100	9.61 × 102	7.28 × 101	4.41 × 102	5.05 × 100
tf25	4.05 × 102	2.39 × 101	3.89 × 102	1.78 × 10−8	3.94 × 102	1.65 × 101	3.87 × 102	3.46 × 10−1	3.94 × 102	1.90 × 101	3.87 × 102	1.83 × 10−1
tf26	2.15 × 103	4.68 × 102	2.00 × 102	1.94 × 101	1.79 × 103	8.73 × 102	1.09 × 103	7.08 × 101	4.39 × 103	1.56 × 103	1.20 × 103	3.05 × 101
tf27	5.65 × 102	3.14 × 101	5.48 × 102	6.07 × 101	5.41 × 102	1.61 × 101	5.11 × 102	7.11 × 100	7.59 × 102	1.29 × 102	5.10 × 102	1.12 × 101
tf28	4.00 × 102	7.27 × 101	3.69 × 102	1.69 × 102	3.20 × 102	5.84 × 101	3.48 × 102	5.74 × 101	3.36 × 102	5.89 × 101	3.60 × 102	5.68 × 101
tf29	5.93 × 102	8.39 × 101	6.40 × 102	7.06 × 102	7.44 × 102	2.04 × 102	4.70 × 102	4.07 × 101	1.61 × 103	4.08 × 102	4.90 × 102	5.23 × 101
tf30	5.18 × 103	1.29 × 103	5.21 × 103	3.01 × 103	3.56 × 103	1.19 × 103	2.07 × 103	7.98 × 101	4.29 × 103	1.19 × 103	2.20 × 103	1.65 × 102
CPU Time		115.20		195.34		192.32		170.56		181.45		178.54
w/l/t	27/0/2		21/8/0		29/0/0		15/11/3		27/1/1		17/10/2
p-value < 0.001 +			0.023 +		<0.001 +		0.433 =		0.239 =		0.007 +

“+” and “=” are noticeably superior and equal, respectively.

,

Table 5. The results of ARKO and other MAs on the CEC-2022 test functions.

Fun		ARKO	TLBO	DBTO	GSK	AGBSO	EJAYA	DisGSA	HMRFO	SHADE	CJADE
tc01	Mean	0.000 × 100	0.000 × 100	8.946 × 103	0.000 × 100	1.130 × 10−7	0.000 × 100	1.526 × 102	5.671 × 101	0.000 × 100	0.000 × 100
	SD	0.00 × 100	1.09 × 10−10	2.92 × 103	0.00 × 100	1.40 × 10−7	0.00 × 100	1.58 × 102	3.04 × 101	0.00 × 100	0.00 × 100
tc02	Mean	4.741 × 101	5.939 × 101	1.730 × 102	4.825 × 101	4.900 × 101	2.055 × 101	5.894 × 101	1.98 × 101	4.825 × 101	4.908 × 101
	SD	2.29 × 100	1.20 × 101	7.18 × 101	1.87 × 100	1.85 × 10−1	1.99 × 101	5.18 × 100	2.67 × 101	1.87 × 100	2.56 × 10−14
tc03	Mean	0.000 × 100	2.071 × 101	5.787 × 101	1.145 × 10−6	3.627 × 10−6	1.756 × 100	8.818 × 10−7	1.906 × 10−2	7.972 × 10−8	0.000 × 100
	SD	0.00 × 100	9.49 × 100	6.69 × 100	1.28 × 10−6	4.42 × 10−6	1.22 × 100	1.27 × 10−6	4.04 × 10−2	1.38 × 10−7	0.00 × 100
tc04	Mean	1.978 × 101	3.930 × 101	8.379 × 101	8.643 × 101	8.358 × 100	2.726 × 101	1.492 × 101	3.084 × 101	8.899 × 100	1.205 × 101
	SD	3.09 × 100	1.28 × 101	7.99 × 100	8.40 × 100	4.54 × 100	8.63 × 100	5.67 × 100	1.40 × 101	1.92 × 100	1.63 × 100
tc05	Mean	0.000 × 100	3.788 × 102	1.326 × 103	1.790 × 10−2	3.589 × 10−9	4.469 × 100	1.791 × 10−2	5.179 × 100	0.000 × 100	0.000 × 100
	SD	0.00 × 100	2.15 × 102	1.26 × 102	4.00 × 10−2	4.28 × 10−9	2.77 × 100	4.00 × 10−2	4.35 × 100	0.00 × 100	0.00 × 100
tc06	Mean	1.747 × 100	3.663 × 102	1.024 × 105	3.732 × 101	1.301 × 103	1.418 × 102	2.782 × 103	2.474 × 103	2.505 × 101	1.911 × 101
	SD	8.58 × 10−1	2.97 × 102	2.00 × 105	2.48 × 101	1.84 × 103	2.08 × 101	2.84 × 103	2.68 × 103	2.23 × 101	1.20 × 101
tc07	Mean	2.236 × 101	7.577 × 101	1.339 × 102	2.363 × 101	2.175 × 101	4.646 × 101	3.160 × 101	3.852 × 101	2.279 × 101	2.328 × 101
	SD	9.41 × 10−1	2.75 × 101	1.82 × 101	7.66 × 10−1	1.17 × 100	2.06 × 101	6.14 × 100	1.16 × 101	6.20 × 10−1	6.37 × 100
tc08	Mean	2.526 × 101	2.369 × 101	2.867 × 101	2.660 × 101	2.145 × 101	2.606 × 101	2.105 × 101	2.880 × 101	2.135 × 101	2.206 × 101
	SD	7.54 × 10−1	4.15 × 100	1.02 × 100	3.80 × 10−1	5.60 × 10−1	3.19 × 100	4.11 × 10−1	5.99 × 100	4.03 × 10−1	2.19 × 10−1
tc09	Mean	1.801 × 102	1.808 × 102	2.684 × 102	1.802 × 102	1.808 × 102	1.808 × 102	1.853 × 102	1.808 × 102	1.808 × 102	1.808 × 102
	SD	2.03 × 10−13	1.96 × 10−12	2.18 × 101	2.03 × 10−13	2.74 × 10−3	0.00 × 100	1.47 × 100	2.23 × 10−5	0.00 × 100	0.00 × 100
tc10	Mean	1.000 × 102	7.431 × 102	9.178 × 102	5.932 × 102	1.272 × 102	1.055 × 103	1.481 × 102	1.005 × 102	1.000 × 102	1.004 × 102
	SD	3.31 × 10−2	1.17 × 103	8.96 × 102	1.10 × 103	4.88 × 101	1.31 × 103	6.51 × 101	6.98 × 10−2	4.66 × 10−2	4.98 × 10−2
tc11	Mean	3.200 × 102	3.304 × 102	2.799 × 103	3.000 × 102	3.400 × 102	3.800 × 102	3.200 × 102	3.001 × 102	3.000 × 102	3.200 × 102
	SD	4.47 × 101	4.75 × 101	7.47 × 102	2.05 × 10−13	4.47 × 101	4.47 × 101	4.47 × 101	1.73 × 102	2.49 × 10−13	4.47 × 101
tc12	Mean	2.347 × 102	2.847 × 102	4.961 × 102	2.485 × 102	2.411 × 102	2.482 × 102	2.779 × 102	2.774 × 102	2.369 × 102	2.350 × 102
	SD	2.74 × 100	2.78 × 101	1.68 × 102	1.53 × 101	3.83 × 100	6.82 × 100	1.25 × 101	2.12 × 101	4.74 × 100	4.69 × 100
CPU Time		54.94	56.84	53.43	72.12	61.34	38.34	74.12	71.99	49.23	58.26
w/l/t			10/1/1	12/0/0	10/1/1	9/3/0	10/1/1	9/2/1	10/2/0	6/3/3	7/2/3
p-value			0.012 +	0.001 +	0.012 +	0.146 =	0.012 +	0.045 +	0.039 +	0.508 =	0.289 =

“+” and “=” are noticeably superior and equal, respectively.

,

Table 6. The results of Friedman’s rank test on the CEC-2017 functions.

Fun	ARKO	TLBO	DTBO	GSK	AGBSO	TDSD	EJAYA	DisGSA	HMRFO	SHADE	IMODE	CJADE
tf01	1	7	12	1	9	8	6	10	11	1	1	1
tf03	1	1	11	1	7	12	1	9	8	1	6	10
tf04	1	5	12	4	10	2	6	11	8	9	3	7
tf05	5	6	11	10	2	9	7	3	8	1	12	4
tf06	2	8	11	6	5	10	9	4	7	3	12	1
tf07	3	4	11	10	6	9	7	2	8	1	12	5
tf08	6	7	12	11	1	10	9	3	8	2	4	5
tf09	5	7	11	6	1	10	9	1	8	4	12	1
tf10	6	12	7	11	1	5	9	4	8	2	10	3
tf11	1	10	12	6	5	7	8	9	4	3	11	2
tf12	1	8	12	5	11	10	6	7	9	3	2	4
tf13	2	11	12	4	9	6	7	8	10	3	5	1
tf14	2	6	12	1	9	11	4	10	7	3	5	8
tf15	1	11	12	2	10	5	7	8	9	3	4	6
tf16	2	4	11	5	1	7	8	9	10	3	12	6
tf17	3	8	11	5	2	6	7	9	10	1	12	4
tf18	1	11	12	2	10	8	6	7	9	3	5	4
tf19	1	11	12	3	10	5	6	9	8	2	7	4
tf20	3	8	11	1	5	6	10	7	9	2	12	4
tf21	2	9	11	10	1	4	8	6	7	3	12	5
tf22	1	9	12	5	8	10	1	7	6	1	11	1
tf23	6	7	11	5	1	10	8	4	9	2	12	3
tf24	5	8	11	7	6	2	10	1	9	3	12	4
tf25	1	11	12	5	2	7	10	6	8	3	9	4
tf26	7	8	11	6	3	2	10	1	9	4	12	5
tf27	1	7	11	6	2	5	10	9	8	4	12	3
tf28	1	6	12	3	9	11	10	8	2	5	4	7
tf29	5	9	11	2	3	6	7	8	10	1	12	4
tf30	1	5	12	3	11	8	10	9	6	2	7	4
1/2/3	13/5/3	1/0/0	0/0/0	4/3/3	6/4/2	0/3/0	2/0/0	3/1/2	0/1/0	7/6/11	1/1/1	5/1/3
Rank	2.91	7.59	11.09	5.46	6.23	7.14	7.27	6.27	7.86	3.13	8.71	4.34

,

Table 7. Theresults of Friedman’s rank test on the CEC-2022 functions.

Fun	ARKO	TLBO	DBTO	GSK	AGBSO	EJAYA	DisGSA	HMRFO	SHADE	CJADE
tc01	1	1	10	1	7	1	9	8	1	1
tc02	3	9	10	5	6	2	8	1	4	7
tc03	1	9	10	5	6	8	4	7	3	1
tc04	5	8	9	10	1	6	4	7	2	3
tc05	1	9	10	5	4	7	6	8	1	1
tc06	1	6	10	4	7	5	9	8	3	2
tc07	2	9	10	5	1	8	6	7	3	4
tc08	6	5	9	8	3	7	1	10	2	4
tc09	1	5	10	2	8	6	9	7	3	3
tc10	1	8	9	7	5	10	6	4	1	3
tc11	4	7	10	1	8	9	5	3	2	6
tc12	1	9	10	6	4	5	8	7	3	2
1/2/3	7/1/1	1/0/0	0/0/0	2/1/0	2/0/1	1/1/0	1/0/0	1/0/1	3/3/5	3/2/3
Rank	2.67	7.08	9.75	5.04	4.79	6.17	6.67	6.63	2.79	3.42

and

Table 8. The results of the Wilcoxon rank-sum test on the CEC test functions.

Test Suite	Algorithms	Pairwise Rank	ΣR+	ΣR−	z-Value	p-Value	Significance (α = 0.05)
CEC-2017	ARKO vs. TLBO	(1.02, 1.98)	406	0	4.623	<0.001	+
	ARKO vs. DBTO	(1.00, 2.00)	435	0	4.703	<0.001	+
	ARKO vs. GSK	(1.21, 1.79)	335.5	44.5	3.472	<0.001	+
	ARKO vs. AGBSO	(1.32, 1.68)	308	98	2.391	0.017	+
	ARKO vs. TDSD	(1.10, 1.90)	386	49	3.644	<0.001	+
	ARKO vs. EJAYA	(1.03, 1.97)	378	0	4.541	<0.001	+
	ARKO vs. DisGSA	(1.29, 1.71)	335.5	70.5	3.017	0.003	+
	ARKO vs. HMRFO	(1.02, 1.98)	406	0	4.623	<0.001	+
	ARKO vs. SHADE	(1.43, 1.57)	180.5	170.5	0.127	0.899	=
	ARKO vs. IMODE	(1.05, 1.95)	404	2	4.577	<0.001	+
	ARKO vs. CJADE	(1.39, 1.61)	249	102	1.867	0.062	=
CEC-2022	ARKO vs. TLBO	(1.02, 1.98)	64	2	2.756	0.006	+
	ARKO vs. DBTO	(1.00, 2.00)	78	0	3.059	0.002	+
	ARKO vs. GSK	(1.08, 1.92)	58	8	2.223	0.026	+
	ARKO vs. AGBSO	(1.25, 1.75)	58	20	1.490	0.136	=
	ARKO vs. EJAYA	(1.13, 1.88)	58	8	2.223	0.026	+
	ARKO vs. DisGSA	(1.17, 1.83)	59	7	2.312	0.021	+
	ARKO vs. HMRFO	(1.17, 1.83)	6	17	1.726	0.043	+
	ARKO vs. SHADE	(1.38, 1.63)	24	21	0.178	0.859	=
	ARKO vs. CJADE	(1.25, 1.75)	23	13	0.700	0.484	=

“+” and “=” are noticeably superior and equal, respectively.

.

The parameter settings for all meta-heuristic algorithms were selected in accordance with the recommendations made in the corresponding original publications. The original studies of TDSD and IMODE are the source of their empirical results.

Table 3 and Table 4 show the mean error and standard deviation (SD) values of 50 runs on the CEC-2017 test suite. If the value for reaching (VTR = 10⁻⁸) has been exceeded, then the error is treated as 0. The bold-marked values represent the best outcomes. ARKO provides the best outcomes on a maximum of 13 functions and achieves the first position, while SHADE and CJADE manage to achieve the second and third positions by obtaining the best outcomes on seven and five functions, respectively. The last two rows of this table show the pairwise win/loss/tie performance and p-values of the algorithms with ARKO, which is evidence that ARKO produces superior outcomes to all other algorithms.

The p-values computed using the “Wilcoxon Sign Test” confirm the statistical efficacy of ARKO on its competitors. Through this p-value, it can also be observed that only SHADE, AGBSO, and CJADE are significantly equal to ARKO, while the values of the remaining algorithms are significantly lower.

Table 5 shows the mean error and standard deviation (SD) of 50 runs on the CEC-2022 test suite. The bold-marked values represent the best outcomes. ARKO provides the best outcomes on a maximum of seven functions and achieves the first position, while CJADE and SHADE achieve the second position by obtaining the best positions on three functions, respectively. The pairwise win/loss/tie performance also proves the superiority of ARKO over other algorithms. The computed p-values confirm the statistical efficacy of ARKO on its competitors. Through this p-value, it can also be observed that SHADE, AGBSO, and CJADE are significantly equal to ARKO, while the values of the remaining algorithms are significantly lower.

Furthermore, two additional non-parametric tests for statistical invariance of the outcomes are described in Table 6, Table 7 and Table 8.

The average ranks computed using Friedman’s rank test are displayed in Table 6 and Table 7. For the CEC-2017 functions, it is noteworthy that ARKO received the top ranking in 13 cases, while TLBO, DTBO, GSK, AGBSO, TDSD, EJAYA, DisGSA, HMRFO, SHADE, IMODE, and CJADE received the top ranking in 1, 0, 4, 6, 0, 2, 3, 0, 7, 1, and 5 cases, respectively. For the algorithms’ second and third ranks, comparable results can be observed in a similar way. The average ranks for the TLBO, DTBO, GST, AGBSO, TDSD, EJAYA, DisGSA, HMRFO, SHADE, IMODE, and CJADE algorithms are 7.59, 11.09, 5.46, 6.23, 7.14, 7.27, 6.27, 7.86, 3.13, 8.71, and 4.34, respectively, while the rank for ARKO is only 2.91, which proves the supremacy of the proposed ARKO algorithm on these functions.

Similarly, the results for the CEC-2022 functions can be observed in Table 7, where ARKO received the first rank on seven cases, while TLBO, DTBO, GSK, AGBSO, EJAYA, DisGSA, HMRFO, SHADE, and CJADE received the top ranking in 1, 0, 2, 2, 2, 0, 0, 4, and 3 cases, respectively. The average ranks for the TLBO, DTBO, GST, AGBSO, EJAYA, DisGSA, HMRFO, SHADE, and CJADE algorithms are 7.08, 9.75, 5.04, 4.79, 6.17, 6.67, 6.63, 2.79, and 3.42, respectively, while the rank for ARKO is only 2.67, which proves the supremacy of the proposed ARKO algorithm on these functions. The average ranks are also represented with the assistance of bar charts in Figure 2.

Table 8 displays the positive and negative rank sums and corresponding p-values computed using the Wilcoxon rank-sum test. ARKO’s superior efficacy over its competitors is demonstrated by its lower rank and greater positive rank sum. The corresponding p-values of less than 0.05 show that ARKO is significantly better in such cases, while the p-values for SHADE and CJADE indicate equivalent significance.

The average CPU time is also presented in the Tables, and it can be observed that EJaya takes minimum execution time in both cases of the CEC-2017 and CEC-2022 functions. SHADE and DTBO take the second and third places, respectively, while ARKO is able to secure only the fourth place due to its high complexity.

The box plots presented in Figure 3 also visually verify the effectiveness of ARKO on benchmark functions. The graphical convergence performance of the algorithms for a few selected test functions of each category, including unimodal (tf₀₁, tf₀₃), multimodal (tf₀₄, tf₀₆), hybrid (tf₁₅, tf₁₈, tf₁₉), and composite (tf₃₀) from the CEC-2017 suite and basic unimodal (tc₀₁), basic multimodal (tc₀₄), hybrid (tc₆), and composite (tc₁₂) from the CEC-2022 suite, are shown in Figure 4. The graphs examine the algorithms’ convergence behavior and confirm that the proposed ARKO algorithm converges in an enhanced way compared with its rivals.

4.2. Performance Evaluation of Real-Life Applications

To test the practical ability of the proposed ARKO algorithm, three unconstrained real-life applications, the Frequency-Modulated (FM), the Non-Linear Stirred Tank Reactor optimal Control (NLSTS), and the Spread Spectrum Radar Polly Phase Code Design (SSRPCD)problems, were selected from the CEC-2011 test suite [65].

The performance of ARKO was compared with five other MAs: TLBO, DTBO, GSK, EJAYA, and SHADE. The computed results are presented in

Table 9. The results of ARKO on real-life optimization problems.

Application	Dim		ARKO	TLBO	DTBO	GSK	EJAYA	SHADE
FM	6	Best	0.000	29.478	22.878	7.84 × 10−9	2.348	0.000
		Mean	4.002	30.339	25.452	7.631	10.692	5.842
		SD	4.340	0.752	2.001	5.21	5.835	5.739
		rank	1	6	5	3	4	2
NLSTS	1	Best	13.842	18.585	13.922	16.958	15.006	13.921
		Mean	13.979	20.441	14.426	18.958	16.892	14.366
		SD	0.201	1.694	0.532	1.992	1.679	0.466
		rank	1	6	2	5	4	3
SSRPCD	20	Best	0.894	2.112	1.282	1.567	0.981	0.992
		Mean	1.027	2.713	1.305	1.791	1.901	1.138
		SD	0.079	0.377	0.077	0.134	0.832	0.082
		rank	1	6	3	4	5	2

for the best, mean, and standard deviation values of 50 runs. The proposed ARKO algorithm provided enhanced quality solutions for each problem and secured first rank in each case. The SHADE algorithm achieved the second rank for the FM and SSRPCD problems while DTBO obtained the second rank for the NLSTS problem.

A box plot is presented in Figure 5, which also visually verifies the effectiveness of ARKO on the other tested problems.

4.3. Performance Evaluation of ARKO on the SELD Problem

The performance of ARKO on the SELD problem is discussed in this segment. Three cases of 6, 15, and 40 units were considered for the experiments. The required input data, MATLAB codes, and constrained handling techniques were used as provided by the source in [65]. The maximum iterations were set as 600, 1500, and 4000, respectively.

First, a sensitivity analysis was conducted on the transform ratio (T_R) and the results are presented in

Table 10. Sensitivity analysis of ARKO on the SELD problem at T_R= 0.3, 0.5, 0.7, and 0.9.

Unit	Cost	TR = 0.3	TR = 0.5	TR = 0.7	TR = 0.9
6	Worst	15,444.0959	15,444.0870	15,444.0865	15,444.1373
	Best	15,444.0887	15,444.0868	15,444.0864	15,444.1039
	Mean	15,444.0919	15,444.0869	15,444.0865	15,444.1226
	SD	0.0027	0.0001	0.00001	0.0174
	Rank	3	2	1	4
15	Worst	32,757.277	32,746.9878	32,751.7170	32,784.1953
	Best	32,728.4599	32,725.0066	32,743.9594	32,731.2768
	Mean	32,746.5391	32,737.8120	32,746.7298	32,747.3538
	SD	10.7573	8.2964	3.3204	20.9736
	Rank	2	1	3	4
40	Worst	118,735.3566	118,698.3103	118,779.8213	118,829.846
	Best	118,701.3461	118,697.7373	118,687.0167	118,708.6435
	Mean	118,714.3009	118,697.948	118,708.879	118,743.9183
	SD	15.21130979	0.290337238	39.84211413	49.93321234
	Rank	3	1	2	4
Average Rank		2.67	1.33	2	4

. The results were obtained for T_R = 0.3, 05, 0.7, and 0.9 in terms of the worst, best, and mean cost values of the SELD problem. We can observe that the algorithm provides outstanding outcomes on each value of T_R; however, T_R = 0.7 is an appropriate setting value for the 6-unit case while T_R = 0.5 is suitable for the 15- and 40-unit cases.

The outcomes for total power output and power loss are presented in

Table 11. The results of total power output and power loss for the 6-unit case.

Unit	ARKO	GA	PSO	DE	TLBO	DTBO	GSK	EJAYA	SHADE
1	446.5214	454.8815	456.0046	446.4490	421.9846	446.6874	446.7076	423.5128	446.6954
2	173.0657	183.4238	169.2968	191.5317	187.1648	139.9399	173.1485	189.0517	173.1491
3	262.7603	254.2249	253.6832	250.1112	255.9862	261.8746	262.7981	261.4904	262.8951
4	143.3603	120.2187	134.0864	132.6692	148.2352	145.2516	143.4832	122.6716	143.3895
5	163.9791	174.0830	156.6033	156.9488	191.1712	176.7040	163.9202	182.4432	163.9168
6	85.7275	89.2455	105.9416	97.8572	71.46854	105.1994	85.3726	96.9289	85.3890
POutput	1275.4143	1276.0774	1275.6159	1275.5671	1276.0105	1275.6569	1275.4302	1276.0986	1275.4349
Ploss	12.4143	13.0774	12.6159	12.5671	13.0105	12.6569	12.4302	13.0986	12.4349

,

Table 12. The results of total power output and power loss for the 15-unit case.

Unit	ARKO	GA	PSO	DE	TLBO	DTBO	GSK	EJAYA	SHADE
1	444.2795	444.3445	444.1840	453.0820	433.4908	453.6458	454.8950	409.0075	454.8209
2	376.6640	338.4311	343.1886	379.1102	374.4481	379.9957	379.9945	369.2012	379.9856
3	129.0494	103.1062	118.4483	129.8638	125.3954	129.9750	129.9951	129.9900	129.9961
4	129.0298	120.2486	124.0134	129.9669	121.4828	123.4781	129.9978	128.6862	129.7331
5	168.7523	162.8434	156.2677	169.3532	151.7508	156.2206	169.9128	156.7410	169.5946
6	458.9883	420.4697	410.3848	429.5020	460.0000	425.9112	429.9547	429.0110	429.9873
7	428.7311	425.6819	412.1858	429.9367	414.8707	397.8527	428.9138	418.9505	429.8753
8	82.5350	141.4747	123.9975	96.2952	90.9985	123.7863	90.8382	159.9551	125.5647
9	82.4076	129.4776	115.5193	106.0079	139.8572	150.7865	70.4153	83.4823	48.5787
10	139.5643	140.2481	144.4970	98.2661	124.1821	119.4169	159.9063	128.9730	153.2595
11	79.9388	60.0857	75.6556	79.7145	79.1636	60.6135	79.9169	67.2506	79.9979
12	79.1167	50.2885	74.1970	79.1601	67.6719	41.6502	79.9973	49.0820	74.5887
13	26.8779	52.5990	33.0558	26.5523	25.0000	27.8215	25.0054	54.0769	25.1647
14	18.9145	42.4084	38.4276	37.1184	15.0000	40.4350	15.4115	34.0170	15.2770
15	15.7640	36.5059	51.8920	20.4171	40.9245	43.6105	16.2322	47.4740	15.6645
POutput	2660.6133	2668.2133	2665.9144	2664.3464	2664.2363	2675.1995	2661.3869	2665.8981	2662.0886
Ploss	30.6132	38.2133	35.9144	34.3464	34.2363	45.1995	31.3869	35.8981	32.0886

and

Table 13. The results of total power output and power loss for the 40-unit case.

Unit	ARKO	GA	PSO	DE	TLBO	DTBO	GSK	EJAYA	SHADE
1	113.8510	78.4714	87.3552	110.1480	113.9996	109.5999	112.9511	82.1430	110.4609
2	113.9410	92.7142	93.0660	46.7944	82.4645	57.0701	112.5480	91.9677	114.0000
3	119.7262	92.7967	95.4799	113.5342	78.2115	72.5195	116.8790	75.7585	105.4462
4	189.9507	186.6355	143.3451	175.8801	127.0359	150.8196	168.6770	159.6652	190.0000
5	96.9738	70.0929	80.3985	90.9954	91.1486	81.9720	96.1838	96.2917	97.0000
6	138.9524	106.5773	123.5360	115.6175	139.9964	115.1495	138.1775	99.2060	118.9478
7	299.9959	198.6259	262.6407	279.0478	258.9538	208.1970	294.6818	274.7538	289.2474
8	298.8668	288.9021	286.2889	277.5432	259.4099	267.4926	263.4139	273.4709	286.8726
9	298.9858	225.8630	295.1113	291.9456	222.7106	268.0101	292.5293	278.9337	300.0000
10	131.6032	248.8415	218.0799	257.3437	248.9623	225.0893	140.2113	130.0259	266.7738
11	98.9356	268.2140	158.7038	303.7107	138.8817	109.4923	104.5405	287.8451	108.6113
12	94.7264	223.9393	202.1873	153.3309	374.9998	306.3760	95.8485	138.9444	94.0000
13	126.7007	499.3237	234.9671	345.7737	353.9701	300.1057	365.4701	127.9032	158.8183
14	267.9651	446.7509	324.9892	396.9328	438.2727	324.6628	281.9118	391.2864	225.8918
15	272.0413	387.7087	320.8553	206.0290	495.2985	428.4394	295.3822	443.2197	271.2869
16	264.9473	419.6794	347.0207	353.6337	354.1148	314.2044	237.8857	378.8969	256.6926
17	499.8508	494.1285	497.2617	480.3348	474.6441	426.2130	499.8531	475.8859	494.0663
18	499.6833	439.3944	464.8735	464.2217	448.9717	486.3352	472.8570	483.8946	500.0000
19	549.7099	532.3022	509.9057	474.3858	473.2385	537.8693	545.3552	540.4217	515.5346
20	548.6581	461.6434	537.1503	518.5041	535.7000	527.6484	502.7335	526.7219	550.0000
21	549.7713	494.7260	537.2462	542.5689	539.9713	538.5939	549.6720	488.0597	539.1472
22	549.9493	408.7578	512.8679	476.9524	524.5159	521.0331	530.0294	532.8754	550.0000
23	549.7944	543.8874	548.8751	528.3443	494.7881	518.6147	546.9052	526.8960	550.0000
24	549.9231	468.3442	548.8172	519.5782	515.2698	539.3062	518.4547	522.0160	550.0000
25	549.1376	496.4755	549.4302	492.2711	353.1598	526.2477	547.1746	512.0599	549.8822
26	549.9601	442.8769	547.7663	505.2443	441.8597	528.9715	547.2648	517.8116	550.0000
27	10.1969	62.2733	10.9039	13.9952	28.3425	16.8897	10.4904	13.8523	10.0139
28	11.9070	11.6728	10.5758	21.7955	29.9126	19.4314	10.4659	14.6592	10.0000
29	10.5806	45.4152	10.0246	16.7659	28.4925	27.0097	13.6574	16.2860	15.0692
30	96.7839	85.2563	77.2011	92.2342	60.7375	92.2016	95.6014	86.9204	94.3415
31	189.9892	86.9145	181.2558	184.4122	178.5336	184.5641	188.8971	163.3668	190.0000
32	189.6840	166.4678	187.4450	171.4128	171.2185	176.4493	183.9822	176.3919	189.0598
33	189.0417	141.4037	182.7419	147.9723	148.1463	181.4575	187.6953	176.5704	187.8212
34	199.9051	183.3322	162.6467	150.2331	175.7339	143.5746	199.6781	194.9770	198.3846
35	199.9871	182.8808	172.1864	197.1506	199.9807	188.2070	199.2672	183.2363	199.7217
36	199.8145	170.8610	167.8813	184.5091	166.5536	192.2850	177.5777	190.9560	200.0000
37	109.0732	102.0347	82.9924	107.4863	46.1824	97.0347	107.8461	100.6391	108.6469
38	109.0178	89.5399	104.6404	104.9393	88.5756	87.8164	104.2678	105.9709	110.0000
39	109.8076	45.8596	80.0646	94.8626	47.0495	77.9293	103.3322	109.1181	110.0000
40	549.6105	508.8740	541.2907	491.5995	549.9935	525.1489	539.6535	510.1906	534.2672
POutput	10,500.0000	10,500.4587	10,500.0699	10,500.0348	10,500.0022	10,500.0322	10,500.0033	10,500.0898	10,500.0058
Ploss	0.0000	0.4587	0.0699	0.0348	0.0022	0.0322	0.0033	0.0898	0.0058

for the 6-, 15-, and 40-unit cases, respectively. The total power demand for these cases was 1263 MW, 2630 MW, and 10,500 MW, respectively.

For the 6-unit case, ARKO produced 1275.4143 MW of power output, resulting in a power loss value of 12.4143 MW, which is a minimal power loss compared with all the other tested algorithms. Similarly, the ARKO power loss values are 30.6132 and 0 for the 15- and 40-unit cases, respectively, which are minimal in both cases. GSK provides the second-best minimum power loss for the 6- and 15-unit cases while TLBO provides the second-best minimum value for the 40-unit case.

The outcomes for the worst, best, and average costs with the standard deviation (SD) are presented in

Table 14. The results of the computed fuel cost (USD/hour) on the SELD problem.

Unit	Cost	ARKO	GA	PSO	DE	TLBO	DTBO	GSK	EJAYA	SHADE
6	Worst	15,444.0865	15,516.0172	15,464.7877	15,457.6113	15,488.1874	15,469.7927	15,444.0959	15,476.9830	15,445.3102
	Best	15,444.0860	15,448.6865	15,453.6193	15,451.4716	15,451.0224	15,450.1507	15,444.0887	15,444.2500	15444.0863
	Mean	15,444.0865	15,482.9702	15,458.8474	15,455.8161	15,464.1503	15,458.7850	15,444.0919	15,459.1604	15,444.3311
	SD	0.00001	26.9799	4.1602	2.6042	15.1296	7.1442	0.0027	12.5479	0.5473
	Rank	1	9	6	4	8	5	2	7	3
	Relative profit		−38.8837	−14.7609	−11.7296	−20.0638	−14.6985	−0.0054	−15.0739	−0.2446
	p-value		0.001	0.039	0.041	0.036 +	0.039 +	0.065 =	0.039 +	0.057 =
15	Worst	32,746.9878	33,236.9203	33,096.6302	33,042.4022	32,966.8380	33,288.6339	32,747.0456	33,200.6964	32,767.4628
	Best	32,725.0066	33,026.6142	33,033.4675	32,975.3222	32,769.3442	33,106.2542	32,741.1815	32,923.2323	32,742.6920
	Mean	32,737.8120	33,116.9271	33,064.3319	33,003.1719	32,888.8035	33,177.9133	32,743.4087	33,024.0795	32,753.9982
	SD	2.2964	87.6724	23.5740	28.6770	98.0120	71.5901	2.6216	107.3655	10.4142
	Rank	1	8	7	5	4	9	2	6	3
	Relative profit		−379.1151	−326.5199	−265.3599	−150.9915	−440.1013	−5.5967	−286.2675	−16.1862
	p-value		0.001	0.001	0.001	0.010 +	0.003 +	0.048 +	0.001 +	0.041 +
40	Worst	118,698.310	134,154.487	121,980.6518	123,152.673	119,730.6222	122,211.8732	119,824.0765	128,001.6581	122,209.4580
	Best	118,697.737	132,202.544	120,813.5984	122,383.559	119,107.5248	121,261.0907	119,724.4392	123,498.4005	121,366.0901
	Mean	118,697.948	133,460.029	121,286.9063	122,747.689	119,277.7075	121,823.8120	119,788.7292	125,666.6379	121,796.0784
	SD	0.2903	881.9163	562.1064	300.3209	255.8758	438.2518	40.7793	2027.3316	333.7685
	Rank	1	9	6	7	2	5	3	8	4
	Relative profit		−14,762.081	−2588.9583	−4049.741	−579.7595	−3125.864	−1090.7812	−6968.6899	−3098.1304
	p-value		0.001 +	0.001 +	0.001 +	0.011 +	0.001 +	0.012 +	0.001 +	0.001 +
Average rank		1.00	8.67	6.34	5.34	4.67	6.34	2.34	6.34	3.67

“+” and “=” are noticeably superior and equal, respectively.

.

For the 6-unit case, the average cost obtained by ARKO is 15,444.0864, which is smaller than that obtained by other algorithms. Similarly, the worst and best cost values obtained by ARKO are also smaller than the other algorithms. However, based on the p-value, it can be noted that GSK and SHADE achieve significantly equal performance with ARKO. A relative profit obtained by ARKO is also computed for the mean cost. It shows that ARKO provides a profit of USD38.8837, USD14.7609, USD11.7296, USD20.0638, USD14.6985, USD0.0054, USD15.0739, and USD0.2446 compared with GA, PSO, DE, TLBO, DTBO, GSK, EJAYA, and SHADE, respectively, and hence an average profit of USD14.4326.

Similarly, we can observe that the average costs obtained by ARKO are 32,737.8120 and 118,697.9480 for the 15- and 40-unit cases, respectively, which are smaller than those obtained by the other algorithms. The superior performance of ARKO for the worst and best cost values can also be noted in Table 14. The p-values also validate the supremacy of ARKO compared with its competitors in terms of statistical significance. The average profits for the 15- and 40-unit cases are USD233.767 and USD4533.0007, respectively.

Figure 6 and Figure 7 demonstrate the box plot representation and convergence graphs of each algorithm’s performance and verify the effectiveness of the proposed ARKO algorithm.

For the 6- and 15-unit cases, ARKO provides the minimum cost value, but the other algorithms performed similarly and hence the convergence graphs in Figure 7a,b show similar performances. In the Figure, it can also be observed that all algorithms converge too quickly due to the low-dimension cases; however, ARKO performs much better than the other algorithms for the 40-unit case as can be observed in Figure 7c.

5. Conclusions

This study introduced a novel MA based on human learning behavior called the Attaining and Refining Knowledge-based Optimization (ARKO) algorithm. The main concept of ARKO is based on the learning abilities of elite groups. The algorithm performs two stages: attaining and refining knowledge. The innovative exploring and enhancing process conducted through the numerous stages of attaining, refining, and updating makes ARKO a unique and robust algorithm. ARKO’s performance was numerically and statistically examined on 41 highly complicated benchmark functions from the CEC-2017 and CEC-2022 test suites, and it was compared with 11 other sophisticated MAs: TLBO, DTBO, GSK, EJAYA, AGBSO, DisGSA, HMRFO, TDSD, SHADE, CJADE, and IMODE. The results show that ARKO delivers excellent performance and achieves the best rank on each test suite. Through the statistical results, we also observed that ARKO’s performance is significantly equal to SHADE and CJADE, which are considered robust MAs. The practical ability of ARKO was also tested on the three real-life applications of FM, NLSTS, and SSRPCD. Furthermore, ARKO was applied to a Static Economic Load Dispatch (SELD) problem, which is a well-known highly complicated and convex optimization problem in the field of power engineering. Moreover, three different SELD cases, 6, 15, and 40 units, requiring power loads of 1263, 2630, and 10,500 MW, respectively, were investigated. The comparison of the algorithms shows that ARKO produces minimum power output and power loss in each case. ARKO also minimizes fuel costs and saves an average of USD14.4326, USD233.767, and USD4533.0007 per hour against the other algorithms on the 6-, 15-, and 40-unit cases, respectively. Hence, the performance analysis confirms that the proposed ARKO algorithm is more efficient at optimizing and achieving optimal solutions, producing better outcomes, and providing a significant advantage over its competitors.

Although the proposed ARKO algorithm delivered satisfactory results for the problems addressed in this study, it has certain limitations when applied to different scenarios, such as increased complexity and the risk of becoming trapped in local minima. Furthermore, according to the No Free Lunch (NFL) theorem, ARKO may not obtain optimal solutions for all optimization problems. Therefore, future research will concentrate on improving ARKO by introducing some new methods to mitigate its limitations, which may include increasing population diversity by incorporating chaotic maps or opposition-based methods to improve local solutions. It will also be interesting to develop hybrid variants of ARKO achieved by combining the algorithm with other MAs or self-adaptive transfer rate methods so that it can handle intricate real-world applications effectively.