A novel chaotic transient search optimization algorithm for global optimization, real-world engineering problems and feature selection

Introduction

Optimization is the process of identifying the most optimal solution to a problem from among all possible alternatives. Given the nature of optimization algorithms, they can be broadly classified into two categories: deterministic algorithms and stochastic intelligent algorithms. Additionally, stochastic algorithms are categorized into two types: heuristic algorithms and metaheuristic algorithms (Yang et al., 2012).

Deterministic optimization algorithms are insufficient for large-scale combinatorial and nonlinear problems. Usually, due to the natural solution mechanisms of deterministic algorithms, the problem of interest is modeled in such a way that the algorithm handles it. The solution strategy of deterministic methods usually depends on the types of objectives and constraints and the types of variables used in modeling the problem. The efficacy of these methods is significantly influenced by the solution space, the quantity of decision variables, and the number of constraints involved in problem formulation. An additional noteworthy limitation is the absence of overarching solution approaches that can be implemented for problem formulations featuring diverse decision objectives, variables, and constraints. That is, most algorithms solve models with certain types of objective functions or constraints. However, optimization problems in many different fields such as management science, computing, and engineering simultaneously require different types of decision variables, objective functions, and constraints in their formulation. Therefore, metaheuristic optimization algorithms have been proposed. These have become very popular methods in recent years because they have good computational power and are easy to transform (Bianchi et al., 2009).

Recently, metaheuristic algorithms have gained unexpected popularity. This is because they have demonstrated their superiority in tackling several optimization challenges. General-purpose metaheuristic methods are examined in different categories: biology-based, chemistry-based, mathematics-based, music-based, physics-based, plant-based, swarm-based, social-based, sports-based, water-based, and hybrid-based (Altay & Alatas, 2020). The emergence of physics-based algorithms was caused by physics phenomena in nature. The most well-known include Big Bang-Big Crunch (Erol & Eksin, 2006), electromagnetism-like heuristic (Birbil & Fang, 2003), central force optimization algorithm (Formato, 2009), multi verse optimization (MVO) (Mirjalili, Mirjalili & Hatamlou, 2016), galaxy-based search algorithm (Hosseini, 2011), Henry gas solubility optimization (Hashim et al., 2019), gradient-based optimizer (Ahmadianfar, Bozorg-Haddad & Chu, 2020), equilibrium optimizer (Faramarzi et al., 2020), flow direction algorithm (Karami et al., 2021), Archimedes optimization algorithm (Hashim et al., 2021), transit search algorithm (Mirrashid & Naderpour, 2022), and transient search algorithm (TSO) (Qais, Hasanien & Alghuwainem, 2020).

Metaheuristic methods are being developed thanks to their simplicity, cheap computational cost, gradient-free mechanism, and flexibility, and the interest in the use of these methods is increasing day by day. In this area, there is a theorem called No Free Lunch (NFL), which proves that there is no general algorithm for solving all optimization problems and allows this work area to be used actively. It has been mathematically proven by the NFL theorem that there is no single optimization method that solves all optimization algorithms. Thus, while the optimization algorithm produces good results in solving one problem, it can produce bad results in another. The gap here also encourages researchers working in this field to produce new methods, improve existing methods, or hybridize methods by combining them. Another gap is that, due to the stochastic optimization process, it is very difficult to maintain a balance between exploration and exploitation in the creation of any metaheuristic algorithm. TSO is a very new physics-based method inspired by the transient behavior of switched electrical circuits containing storage elements such as inductance and capacitance. As with other metaheuristic methods, TSO faces the challenge of achieving the right balance between exploration and exploitation. This study focuses on finding a solution to this problem, improving the speed of convergence and the ability of TSO to find the global optimum solution, and ultimately improving the performance of TSO in terms of various metrics. To the best of our knowledge, no studies have been done on how to increase global convergence and performance rates while avoiding trapping TSOs in local solutions. For the first time, chaos theory has been applied to TSO in this work to eliminate the drawbacks of the method. In this work, chaotic maps are embedded inside TSO to create novel algorithms known as chaotic TSO algorithms.

In this study, 10 different chaotic maps were integrated into the TSO algorithm. The main motivation for the study is to use the number sequences obtained from different chaotic maps instead of the critical parameters produced by random numbers in the TSO algorithm. By using chaotic maps with ergodic, irregular, and stochastic features in chaotic map TSO, it is aimed at avoiding local solutions more easily compared to the TSO method. In this way, it is aimed to increase global convergence and obtain a better curve by improving the exploration and exploitation stages of the TSO algorithm. The proposed method has been applied to the accepted CEC’17 benchmark functions, real-world engineering design problems, and feature selection in the literature.

The remainder of the article is organized as follows: In the second section, the working principle of the TSO algorithm is given. In the third section, chaotic maps are examined, and their equations are given. In the fourth section, the proposed chaotic TSO method is explained in detail. In the fifth section, the experimental results are given. This part consists of three separate stages. First of all, the proposed method was tested on the CEC’17 benchmark functions, and statistical analyses were made and supported by figures. Then the proposed method is adapted to five real-world engineering problems and the performance analyses of the proposed method and the standard TSO method are examined. Finally, the proposed methods and the TSO method on feature selection were adapted and performance analyses were carried out on classification problems. In the last section, Section 6, the conclusion part is included.

Literature review

Various optimization methods have been proposed in the literature that can be used in optimization problems. These methods have become very popular not only in computer science but also in other research areas (Altay, 2022a). Complex reliability allocation problems (Negi et al., 2021), traveling salesman problem (Mzili, Riffi & Mzili, 2022), association rule mining (Altay & Alatas, 2021), dynamic ship routing and scheduling problem (Das et al., 2022), laser cutting process (Madić et al., 2022), machine learning (Altay & Varol, 2023), and process synthesis problem (Altay, 2022b) are some of them. There are studies comparing the performance of metaheuristic methods (Sadhu et al., 2023).

There is no best optimization algorithm to solve all problems. While an optimization algorithm may solve one problem very well, it may not achieve successful results in another. This encourages researchers to propose new methods and improve existing ones. When the literature is examined, it is seen that many optimization algorithms have been proposed. Some of those include the group teaching optimization algorithm (Zhang & Jin, 2020), dwarf mongoose optimization algorithm (Agushaka, Ezugwu & Abualigah, 2022), chimp optimization algorithm (Khishe & Mosavi, 2020), material generation algorithm (Talatahari, Azizi & Gandomi, 2021), social mimic optimization algorithm (Balochian & Baloochian, 2019), arithmetic trigonometric optimization algorithm (Devan et al., 2022), fertilization optimization algorithm (Devan et al., 2022), African vultures optimization algorithm (Abdollahzadeh, Gharehchopogh & Mirjalili, 2021), aquila optimizer (Abualigah et al., 2021b), circle search algorithm (Qais et al., 2022), the water optimization algorithm (Daliri & Asghari, 2022), and the gold rush optimizer (Zolfi, 2023). The transient search algorithm is one of the physics-based metaheuristic methods that have emerged recently. This method has been tested on some problems, but as far as we know, there is no study related to the development of the method. The fact that it is a new method and that no improvement has been made in this area yet has been our source of motivation.

It is seen that the transient search algorithm has been successfully used in PEM fuel cell modeling (Hasanien et al., 2022), IoT intrusion detection system (Fatani et al., 2021), optimum allocation of more than one distributed generator in the radial electricity distribution network (Bhadoriya & Gupta, 2022), and improving the voltage ride-through capability of the wind turbine (Qais & Hasanien, 2020).

Chaos theory has been extensively employed to enhance exploration and exploitation as nonlinear theory has undergone constant study and improvement. Numerous metaheuristic algorithms’ premature convergence issues have been effectively resolved using chaos theory. Many researchers have added chaotic mapping mechanisms to various metaheuristic algorithms to augment the algorithm’s capacity to find optimum solutions, improve random diversification, and obtain optimal or sub-optimal answers in complicated multi-modal circumstances (Arora & Anand, 2019). The application of chaos theory to different metaheuristic methods can be summarized in Table 1.

Tso algorithm

In this section, the background of the TSO algorithm and the operation of the method are discussed. Pseudo code of TSO algorithm is given.

Chaotic maps

In a nonlinear, dynamical system that is non-periodic, non-convergent, and bounded, chaos is a deterministic, random-like technique. Chaos is the randomness of a straightforward deterministic dynamical system in mathematics, and chaotic systems can be thought of as sources of randomness. Although chaos appears random and unpredictable, it also has a certain degree of pattern. Instead of using random variables, chaos uses chaotic variables (Arora & Anand, 2019).

Numbers generated by chaotic maps have been used successfully in a variety of applications. In general, chaotic maps have three basic qualities: ergodicity, beginning conditions, and semi-stochastic properties. Researchers in a variety of domains, including ecology, medicine, economics, and engineering applications, are drawn to chaotic processes. It is also extensively utilized to improve the performance of optimization algorithms (Altay, 2022c).

The effectiveness of swarm intelligence algorithms in tackling a variety of challenging optimization issues has been demonstrated. As a result, further developing these algorithms became a hotly debated scientific subject. The use of chaotic maps in place of random values in swarm intelligence algorithms is one of the more frequently employed innovations. Improved search is anticipated because chaotic maps produce numbers that are non-repeatable and ergodic (Tuba et al., 2018).

Chaotic map applications have proven successful in enhancing the stochastic structure of the optimization techniques in a variety of research. The standard TSO’s global convergence time was accelerated in this work by the introduction of ten separate chaotic maps, which also prevented the standard TSO from being stuck in local solutions. There are many types of chaotic maps that are employed, including the gauss, circle, sine, logistic, piecewise, iterative, singer, tent, and sinusoidal maps. Table 2 presents an explanation of the variables and equations linked to these chaotic maps. The graphics of the ten chaotic maps are shown in Fig. 1.

Chaotic tso algorithm

Metaheuristic optimization algorithms suffer from population diversity and early convergence. In addition, the metaheuristic optimization algorithm needs a balance between exploration and exploitation in order to produce an effective solution. Chaotic maps are one of the most effective ways to increase population diversity and the quality of metaheuristic optimization methods. Thus, both the search sensitivity and convergence speed of the metaheuristic optimization method, which are improved with the chaotic map, are improved. Most metaheuristic optimization methods have an exploration and exploitation phase. The exploration phase is used to search the search area as wide as possible, regardless of whether the method is stuck at the local optimum. In the exploitation phase, the method aims to find the best possible solution in a limited area of the search space by finding the most promising area. While the method spends too much time on the exploitation process, which only leads to a local search, spending too much time on the exploration process results in a random search.

Although the newly proposed TSO algorithm has been successfully applied in different applications in the literature, it has some disadvantages, as we mentioned above. Therefore, in order to improve the search space of the TSO algorithm, chaotic maps, which are accepted in the literature, have been added to the dynamic behavior and optimization algorithms to make the search space stronger. Equation (7) is used for exploration and exploitation in the TSO algorithm, whose working principle is very clear. Random variables affecting Eq. (7) are regular random variable values $r_2$ and $r_3$ in Eqs. (8) and (9), respectively. In this study, instead of the $r_3$ value, which is the stochastic component of the TSO algorithm in Eq. (9), chaotic maps with different mathematical equations listed in Table 2 were applied. The mathematical equation is shown in the equation below.

(11) $$C_1=k\times z\times CMV+1$$

Here, $CMV$, that is, sequentially generated chaotic map value is used instead of $r_3$ value. Proposed algorithms using 10 different chaotic map; chebyshev map (CTSO-1), circle map (CTSO-2), gauss map (CTSO-3), iterative map (CTSO-4), logistic map (CTSO-5), piecewise map (CTSO-6), sine map (CTSO-7), singer map (CTSO-8), sinusoidal map (CTSO-9) and tent map (CTSO-10). The flowchart of the CTSO is shown in Fig. 2.

Also, the complexity of the proposed CTSO algorithms can be expressed using big-oh notation. The process of the CTSO algorithm starts with the random generation of search agents in the first step, evaluates the search agents using the cost function in the second step, and updates the search agents to the function evaluation value in the third step. Here, the first step is denoted by $O\left(N\right)$, where $N$ is the number of search agents. In the second step, the search agents enter the while loop, which has the maximum iteration ( $L_{max}$). The complexity of function evaluations of all search agents is expressed as $O\left(N\ast L_{max}\right)$. And finally, in the third step, the complexity of updating all search agents with a size $\left(D\right)$ for total iterations is expressed as $O\left(N\ast L_{max}\ast D\right)$.

Results and discussion

In order to evaluate the performance of the methods proposed in the study, CEC’17 test functions, five different real-world problems, and feature selection problems consisting of 10 different real-world datasets were applied. The results obtained were compared in detail under this section and the performance analyses of the methods were carried out. In all tables, the use of bold demonstrates the best result attained. TSO and CTSOs are taken as constant parameters with $k$ value of 1 and $z$ value of $\left[0,2\right]$. The experimental tests are performed using MATLAB R2021a and the whole test is executed on a PC (Intel (R) Core (TM) i9–10900k CPU @ 3.70 GHz (10 CPUs), 32 GB, Windows 10–64 bits).

Benchmark function

The performance evaluation of ten distinct CTSO and TSO methods was conducted using the IEEE Congress on Evolutionary Computation (CEC) test functions. The CEC’17 test suite comprises a total of 29 test functions, encompassing a diverse range of function types such as unimodal, multimodal, hybrid, and composition functions. The algorithm’s convergence performance is assessed using unimodal functions, namely f1 and f3, while the presence of early convergence and local fixation issues is evaluated using multimodal functions, specifically f4 through f10. The assessment of the capacity to evade local optima, which are characterized by numerous local optima, and the equilibrium between exploration and exploitation is carried out through the utilization of hybrid and composition functions (f11–f20 and f21–f30).

The lower and upper bounds of all functions included in the CEC’17 test suite are between −100 and 100. In order to make a fair evaluation under equal conditions, the number of evaluations was chosen as 1,000 and the population as 30. Algorithms were run 30 times in all experiments, and the results of mean (AVG), standard deviation (STD), minimum (MIN), and Friedman mean rank (MR) values are presented in Tables 3A–3C in a comparative manner. According to the MR value, CTSO-9 and CTSO-5 showed the best performance in unimodal benchmark functions. Of the multimodal functions, CTSO-3 showed the best performance in 3, CTSO-9 in 2, CTSO-3 and CTSO-9 in 1, and CTSO-5 in 1 of them. Of the hybrid and composition functions, CTSO-3 showed the best performance in 11, CTSO-9 in 6, CTSO-3 and CTSO-9 in 2, and CTSO-10 in 1 of them. In Table 4, the average Friedman mean rank values based on the MR values of all benchmark functions are given. When Table 4 is examined according to the statistical analysis results, CTSO-3 gives the best performance, followed by CTSO-9 with a close value. CTSO-8 performed worse than the original TSO. The convergence performance of the algorithms on the CEC’17 test functions is also given in Fig. 3 according to the best values of the algorithms. Figure 4 presents a boxplot of CEC’17 test functions.

Table 3:

Experimental results on CEC’17 test functions.

Table 3-A
Algorithm	AVG	STD	MIN	MR	Algorithm	AVG	STD	MIN	MR
f1					f3
TSO	4.95E+10	8.21E+09	2.48E+10	7.20	TSO	9.11E+04	3.47E+03	8.10E+04	5.47
CTSO-1	4.92E+10	8.39E+09	3.13E+10	7.13	CTSO-1	9.14E+04	3.08E+03	8.21E+04	5.13
CTSO-2	4.99E+10	8.71E+09	3.24E+10	7.33	CTSO-2	9.09E+04	3.33E+03	8.32E+04	5.10
CTSO-3	1.57E+08	1.45E+08	1.86E+07	1.53	CTSO-3	1.55E+05	6.93E+04	6.08E+04	9.30
CTSO-4	4.79E+10	6.77E+09	3.44E+10	6.33	CTSO-4	9.13E+04	5.09E+03	7.18E+04	6.27
CTSO-5	4.97E+10	7.32E+09	3.54E+10	6.87	CTSO-5	9.02E+04	5.42E+03	6.97E+04	4.77
CTSO-6	4.96E+10	7.20E+09	3.52E+10	6.90	CTSO-6	9.19E+04	2.38E+03	8.70E+04	5.53
CTSO-7	4.98E+10	9.49E+09	2.63E+10	7.20	CTSO-7	9.06E+04	4.41E+03	7.63E+04	5.03
CTSO-8	5.01E+10	9.07E+09	3.54E+10	6.80	CTSO-8	9.04E+04	6.25E+03	6.91E+04	5.53
CTSO-9	1.83E+08	2.95E+08	3.75E+07	1.47	CTSO-9	1.36E+05	4.90E+04	6.55E+04	8.90
CTSO-10	4.95E+10	8.31E+09	3.34E+10	7.23	CTSO-10	9.13E+04	3.25E+03	8.17E+04	4.97
f4					f5
TSO	1.23E+04	3.21E+03	4.76E+03	7.17	TSO	9.12E+02	3.32E+01	8.37E+02	6.57
CTSO-1	1.19E+04	2.29E+03	7.99E+03	6.67	CTSO-1	9.19E+02	3.13E+01	8.51E+02	7.77
CTSO-2	1.20E+04	2.21E+03	7.98E+03	6.83	CTSO-2	9.06E+02	4.49E+01	8.16E+02	6.30
CTSO-3	6.16E+02	6.91E+01	5.26E+02	1.50	CTSO-3	7.94E+02	4.53E+01	7.10E+02	1.77
CTSO-4	1.25E+04	2.95E+03	8.86E+03	7.30	CTSO-4	9.16E+02	3.51E+01	8.30E+02	7.23
CTSO-5	1.13E+04	3.14E+03	5.83E+03	6.20	CTSO-5	9.09E+02	3.83E+01	7.97E+02	6.77
CTSO-6	1.23E+04	2.99E+03	7.30E+03	7.47	CTSO-6	9.17E+02	3.78E+01	8.13E+02	7.63
CTSO-7	1.24E+04	3.15E+03	7.15E+03	7.13	CTSO-7	9.07E+02	3.14E+01	8.41E+02	6.23
CTSO-8	1.31E+04	2.72E+03	8.01E+03	7.83	CTSO-8	9.20E+02	2.67E+01	8.76E+02	7.43
CTSO-9	6.07E+02	4.59E+01	4.94E+02	1.50	CTSO-9	7.83E+02	5.52E+01	6.87E+02	1.73
CTSO-10	1.17E+04	3.39E+03	6.65E+03	6.40	CTSO-10	9.10E+02	2.71E+01	8.34E+02	6.57
f6					f7
TSO	6.85E+02	8.78E+00	6.59E+02	7.23	TSO	1.44E+03	5.86E+01	1.31E+03	6.73
CTSO-1	6.83E+02	8.22E+00	6.64E+02	6.17	CTSO-1	1.43E+03	5.71E+01	1.31E+03	5.97
CTSO-2	6.82E+02	7.18E+00	6.67E+02	5.97	CTSO-2	1.43E+03	6.92E+01	1.24E+03	6.03
CTSO-3	6.67E+02	1.08E+01	6.47E+02	2.17	CTSO-3	1.46E+03	1.27E+02	1.25E+03	6.70
CTSO-4	6.83E+02	7.81E+00	6.66E+02	6.07	CTSO-4	1.43E+03	5.07E+01	1.26E+03	5.87
CTSO-5	6.86E+02	8.16E+00	6.64E+02	7.17	CTSO-5	1.41E+03	7.34E+01	1.25E+03	4.70
CTSO-6	6.84E+02	1.02E+01	6.64E+02	6.37	CTSO-6	1.43E+03	5.70E+01	1.30E+03	6.23
CTSO-7	6.87E+02	8.70E+00	6.61E+02	7.73	CTSO-7	1.42E+03	6.65E+01	1.22E+03	5.30
CTSO-8	6.86E+02	6.16E+00	6.75E+02	7.20	CTSO-8	1.43E+03	5.03E+01	1.31E+03	5.80
CTSO-9	6.70E+02	1.04E+01	6.52E+02	2.73	CTSO-9	1.44E+03	1.36E+02	1.18E+03	6.10
CTSO-10	6.86E+02	7.40E+00	6.69E+02	7.20	CTSO-10	1.45E+03	5.05E+01	1.30E+03	6.57
f8					f9
TSO	1.12E+03	2.52E+01	1.08E+03	6.63	TSO	1.05E+04	1.34E+03	8.17E+03	7.13
CTSO-1	1.12E+03	3.33E+01	1.04E+03	6.50	CTSO-1	1.03E+04	1.32E+03	7.94E+03	6.70
CTSO-2	1.13E+03	2.10E+01	1.09E+03	7.93	CTSO-2	1.04E+04	1.07E+03	8.43E+03	6.87
CTSO-3	1.01E+03	4.72E+01	9.17E+02	1.73	CTSO-3	7.25E+03	1.57E+03	4.58E+03	2.00
CTSO-4	1.12E+03	2.95E+01	1.07E+03	6.43	CTSO-4	1.01E+04	1.31E+03	6.02E+03	6.30
CTSO-5	1.11E+03	3.19E+01	1.02E+03	5.63	CTSO-5	1.03E+04	1.23E+03	7.82E+03	7.07
CTSO-6	1.13E+03	2.50E+01	1.07E+03	7.37	CTSO-6	1.03E+04	1.19E+03	7.59E+03	7.10
CTSO-7	1.13E+03	2.54E+01	1.05E+03	6.97	CTSO-7	1.05E+04	1.44E+03	7.58E+03	6.70
CTSO-8	1.13E+03	2.69E+01	1.08E+03	7.73	CTSO-8	1.08E+04	1.46E+03	7.90E+03	7.73
CTSO-9	1.01E+03	4.35E+01	9.09E+02	1.50	CTSO-9	7.43E+03	1.65E+03	4.89E+03	2.40
CTSO-10	1.13E+03	1.82E+01	1.10E+03	7.57	CTSO-10	9.85E+03	1.47E+03	7.05E+03	6.00
f10					f11
TSO	8.69E+03	5.78E+02	7.64E+03	5.73	TSO	9.85E+03	3.09E+03	4.06E+03	6.40
CTSO-1	8.95E+03	6.98E+02	7.58E+03	6.83	CTSO-1	1.05E+04	2.80E+03	5.00E+03	7.07
CTSO-2	8.83E+03	8.47E+02	7.30E+03	6.77	CTSO-2	9.79E+03	2.15E+03	6.53E+03	6.77
CTSO-3	6.73E+03	1.33E+03	4.70E+03	2.37	CTSO-3	1.68E+03	3.27E+02	1.24E+03	1.40
CTSO-4	8.86E+03	5.27E+02	8.05E+03	6.43	CTSO-4	1.04E+04	2.00E+03	6.26E+03	7.10
CTSO-5	9.01E+03	6.78E+02	7.64E+03	7.40	CTSO-5	1.01E+04	2.33E+03	3.00E+03	7.20
CTSO-6	8.99E+03	7.03E+02	7.82E+03	7.33	CTSO-6	9.57E+03	2.20E+03	4.63E+03	6.60
CTSO-7	8.49E+03	5.62E+02	7.30E+03	5.40	CTSO-7	1.01E+04	2.73E+03	4.57E+03	6.90
CTSO-8	8.93E+03	7.24E+02	7.33E+03	7.27	CTSO-8	1.04E+04	2.16E+03	5.38E+03	7.50
CTSO-9	7.66E+03	1.41E+03	4.85E+03	4.03	CTSO-9	2.52E+03	2.84E+03	1.33E+03	1.90
CTSO-10	8.81E+03	6.64E+02	7.56E+03	6.43	CTSO-10	1.01E+04	1.77E+03	7.43E+03	7.17

Table 3-B
Algorithm	AVG	STD	MIN	MR	Algorithm	AVG	STD	MIN	MR
f12					f13
TSO	1.01E+10	4.02E+09	3.60E+09	7.70	TSO	2.70E+09	3.49E+09	2.27E+08	6.57
CTSO-1	7.78E+09	3.34E+09	2.90E+09	6.20	CTSO-1	2.53E+09	3.16E+09	4.33E+08	7.50
CTSO-2	8.51E+09	3.52E+09	2.90E+09	6.70	CTSO-2	2.84E+09	4.16E+09	6.69E+07	7.00
CTSO-3	2.04E+07	2.31E+07	6.18E+05	1.53	CTSO-3	5.68E+04	4.66E+04	1.14E+04	1.33
CTSO-4	8.93E+09	3.95E+09	2.87E+09	6.90	CTSO-4	2.46E+09	3.43E+09	8.13E+07	6.57
CTSO-5	8.59E+09	3.59E+09	3.45E+09	6.57	CTSO-5	2.92E+09	3.56E+09	7.57E+08	7.53
CTSO-6	9.35E+09	3.31E+09	3.23E+09	7.30	CTSO-6	1.89E+09	2.52E+09	1.55E+08	6.40
CTSO-7	9.48E+09	4.68E+09	3.75E+09	7.20	CTSO-7	2.27E+09	2.67E+09	1.58E+08	6.70
CTSO-8	9.97E+09	4.17E+09	3.87E+09	7.43	CTSO-8	2.98E+09	4.32E+09	1.60E+08	7.20
CTSO-9	2.47E+07	3.62E+07	1.01E+06	1.47	CTSO-9	4.67E+07	2.36E+08	1.40E+04	1.80
CTSO-10	9.12E+09	3.70E+09	3.15E+09	7.00	CTSO-10	3.02E+09	3.45E+09	2.00E+08	7.40
f14					f15
TSO	7.27E+06	5.82E+06	3.40E+05	7.27	TSO	4.34E+08	6.96E+08	7.91E+06	6.07
CTSO-1	1.01E+07	2.04E+07	1.53E+05	6.33	CTSO-1	5.47E+08	5.45E+08	1.82E+06	7.47
CTSO-2	7.56E+06	1.02E+07	2.16E+05	7.27	CTSO-2	4.20E+08	4.48E+08	2.70E+07	6.97
CTSO-3	1.45E+06	3.77E+06	3.86E+03	2.43	CTSO-3	1.64E+04	1.28E+04	3.64E+03	1.50
CTSO-4	5.56E+06	5.02E+06	4.65E+05	6.57	CTSO-4	4.58E+08	5.59E+08	1.23E+07	7.20
CTSO-5	6.28E+06	5.64E+06	2.45E+05	6.73	CTSO-5	5.61E+08	7.55E+08	6.00E+06	7.17
CTSO-6	8.11E+06	7.55E+06	2.68E+05	7.50	CTSO-6	3.62E+08	5.47E+08	1.49E+07	6.33
CTSO-7	4.99E+06	3.97E+06	3.18E+05	6.07	CTSO-7	4.26E+08	3.91E+08	1.32E+07	7.40
CTSO-8	7.43E+06	7.47E+06	8.20E+04	6.57	CTSO-8	5.19E+08	5.05E+08	2.04E+07	7.80
CTSO-9	9.26E+05	1.87E+06	5.91E+03	2.33	CTSO-9	1.69E+04	1.30E+04	2.28E+03	1.50
CTSO-10	1.21E+07	3.24E+07	2.11E+05	6.93	CTSO-10	3.68E+08	4.55E+08	4.16E+06	6.60
f16					f17
TSO	5.48E+03	9.37E+02	8.03E+03	7.17	TSO	3.45E+03	7.68E+02	2.30E+03	7.00
CTSO-1	5.74E+03	1.07E+03	8.56E+03	7.43	CTSO-1	4.19E+03	4.53E+03	2.48E+03	6.43
CTSO-2	5.17E+03	9.01E+02	7.32E+03	6.30	CTSO-2	3.50E+03	1.28E+03	2.64E+03	5.47
CTSO-3	3.55E+03	7.81E+02	5.57E+03	2.07	CTSO-3	2.74E+03	3.60E+02	2.14E+03	3.00
CTSO-4	5.58E+03	1.08E+03	7.68E+03	6.97	CTSO-4	4.32E+03	3.29E+03	2.02E+03	6.93
CTSO-5	5.16E+03	6.95E+02	6.30E+03	6.37	CTSO-5	3.35E+03	8.24E+02	2.63E+03	6.23
CTSO-6	5.53E+03	9.81E+02	8.44E+03	7.13	CTSO-6	3.56E+03	1.03E+03	2.42E+03	6.20
CTSO-7	5.71E+03	1.05E+03	7.73E+03	7.37	CTSO-7	5.03E+03	6.07E+03	2.42E+03	6.20
CTSO-8	5.44E+03	9.72E+02	8.31E+03	6.57	CTSO-8	4.64E+03	4.14E+03	2.47E+03	7.37
CTSO-9	3.49E+03	6.43E+02	5.33E+03	1.93	CTSO-9	2.94E+03	3.69E+02	2.29E+03	4.53
CTSO-10	5.29E+03	1.03E+03	7.54E+03	6.70	CTSO-10	4.58E+03	4.16E+03	2.58E+03	6.63
f18					f19
TSO	9.48E+07	1.03E+08	5.85E+06	7.57	TSO	4.94E+08	4.74E+08	1.90E+07	6.77
CTSO-1	6.32E+07	6.84E+07	1.73E+06	6.67	CTSO-1	3.97E+08	3.36E+08	2.65E+07	6.60
CTSO-2	7.71E+07	9.13E+07	1.58E+06	6.90	CTSO-2	3.78E+08	3.98E+08	2.84E+07	5.97
CTSO-3	1.20E+06	3.65E+06	9.33E+04	1.53	CTSO-3	1.68E+05	5.67E+05	2.92E+03	1.43
CTSO-4	1.08E+08	1.26E+08	5.89E+06	7.73	CTSO-4	5.81E+08	4.94E+08	8.05E+07	7.23
CTSO-5	6.37E+07	7.02E+07	1.02E+06	6.30	CTSO-5	5.74E+08	4.75E+08	2.91E+07	7.37
CTSO-6	1.36E+08	2.52E+08	2.39E+06	7.60	CTSO-6	5.88E+08	5.07E+08	2.55E+07	7.10
CTSO-7	5.56E+07	6.10E+07	2.13E+06	6.27	CTSO-7	5.15E+08	4.67E+08	1.45E+07	6.93
CTSO-8	9.00E+07	1.17E+08	1.06E+06	7.27	CTSO-8	5.74E+08	4.58E+08	3.09E+07	7.33
CTSO-9	2.68E+06	6.58E+06	3.96E+04	1.87	CTSO-9	6.94E+04	1.61E+05	2.21E+03	1.57
CTSO-10	5.50E+07	5.24E+07	9.39E+05	6.30	CTSO-10	6.64E+08	6.30E+08	2.52E+07	7.70
f20					f21
TSO	2.97E+03	1.91E+02	2.56E+03	5.43	TSO	2.72E+03	4.50E+01	2.63E+03	7.07
CTSO-1	2.97E+03	1.91E+02	2.52E+03	6.50	CTSO-1	2.72E+03	5.31E+01	2.62E+03	7.20
CTSO-2	2.97E+03	1.59E+02	2.60E+03	5.80	CTSO-2	2.71E+03	4.52E+01	2.62E+03	6.80
CTSO-3	2.99E+03	2.50E+02	2.40E+03	6.23	CTSO-3	2.60E+03	5.54E+01	2.49E+03	2.13
CTSO-4	3.03E+03	2.44E+02	2.53E+03	6.53	CTSO-4	2.71E+03	5.90E+01	2.63E+03	6.43
CTSO-5	3.01E+03	1.41E+02	2.74E+03	6.60	CTSO-5	2.71E+03	5.16E+01	2.60E+03	6.20
CTSO-6	2.96E+03	1.78E+02	2.59E+03	5.73	CTSO-6	2.74E+03	4.94E+01	2.64E+03	7.83
CTSO-7	2.96E+03	1.84E+02	2.66E+03	5.53	CTSO-7	2.72E+03	6.25E+01	2.61E+03	6.97
CTSO-8	2.99E+03	1.60E+02	2.65E+03	6.17	CTSO-8	2.71E+03	6.74E+01	2.60E+03	5.93
CTSO-9	3.06E+03	2.49E+02	2.56E+03	6.67	CTSO-9	2.60E+03	5.86E+01	2.51E+03	2.30
CTSO-10	2.92E+03	2.16E+02	2.54E+03	4.80	CTSO-10	2.72E+03	4.95E+01	2.61E+03	7.13

Table 3-C
Algorithm	AVG	STD	MIN	MR	Algorithm	AVG	STD	MIN	MR
f22					f23
TSO	9.47E+03	9.40E+02	6.59E+03	6.33	TSO	3.46E+03	1.44E+02	3.19E+03	5.80
CTSO-1	9.66E+03	8.69E+02	6.58E+03	7.13	CTSO-1	3.54E+03	1.60E+02	3.29E+03	7.43
CTSO-2	9.86E+03	7.73E+02	7.16E+03	7.73	CTSO-2	3.49E+03	1.85E+02	3.22E+03	6.07
CTSO-3	8.50E+03	1.45E+03	5.29E+03	3.93	CTSO-3	3.36E+03	1.62E+02	3.01E+03	3.83
CTSO-4	9.36E+03	9.72E+02	6.94E+03	6.20	CTSO-4	3.49E+03	1.93E+02	3.05E+03	6.20
CTSO-5	9.34E+03	9.78E+02	6.16E+03	5.77	CTSO-5	3.47E+03	1.95E+02	3.16E+03	5.47
CTSO-6	9.52E+03	8.58E+02	7.13E+03	6.57	CTSO-6	3.53E+03	1.80E+02	3.28E+03	6.87
CTSO-7	9.56E+03	6.84E+02	7.67E+03	6.57	CTSO-7	3.52E+03	1.73E+02	3.20E+03	7.03
CTSO-8	9.50E+03	1.12E+03	6.35E+03	6.47	CTSO-8	3.55E+03	2.67E+02	3.21E+03	6.77
CTSO-9	8.06E+03	1.86E+03	2.84E+03	3.03	CTSO-9	3.36E+03	1.61E+02	2.98E+03	4.10
CTSO-10	9.58E+03	5.84E+02	8.57E+03	6.27	CTSO-10	3.48E+03	1.47E+02	3.18E+03	6.43
f24					f25
TSO	3.71E+03	2.08E+02	3.37E+03	6.80	TSO	4.68E+03	4.06E+02	3.79E+03	8.03
CTSO-1	3.63E+03	1.90E+02	3.31E+03	5.30	CTSO-1	4.48E+03	4.97E+02	3.56E+03	6.23
CTSO-2	3.75E+03	2.98E+02	3.36E+03	6.70	CTSO-2	4.50E+03	3.61E+02	3.72E+03	6.60
CTSO-3	3.53E+03	2.47E+02	3.04E+03	4.33	CTSO-3	3.02E+03	4.51E+01	2.93E+03	1.57
CTSO-4	3.72E+03	2.39E+02	3.38E+03	6.40	CTSO-4	4.58E+03	4.62E+02	3.87E+03	7.27
CTSO-5	3.76E+03	2.26E+02	3.31E+03	7.17	CTSO-5	4.59E+03	3.97E+02	3.82E+03	7.50
CTSO-6	3.74E+03	2.66E+02	3.35E+03	6.60	CTSO-6	4.65E+03	4.67E+02	3.79E+03	7.43
CTSO-7	3.72E+03	3.32E+02	3.38E+03	5.93	CTSO-7	4.54E+03	4.32E+02	3.64E+03	6.70
CTSO-8	3.76E+03	3.23E+02	3.42E+03	6.50	CTSO-8	4.54E+03	3.53E+02	4.00E+03	7.17
CTSO-9	3.50E+03	1.88E+02	3.21E+03	3.43	CTSO-9	3.01E+03	3.92E+01	2.93E+03	1.43
CTSO-10	3.74E+03	2.92E+02	3.35E+03	6.83	CTSO-10	4.45E+03	3.90E+02	3.79E+03	6.07
f26					f27
TSO	1.13E+04	1.15E+03	8.76E+03	7.67	TSO	4.26E+03	4.63E+02	3.57E+03	7.30
CTSO-1	1.10E+04	1.06E+03	8.23E+03	6.90	CTSO-1	4.21E+03	4.56E+02	3.49E+03	6.40
CTSO-2	1.13E+04	1.02E+03	9.57E+03	7.73	CTSO-2	4.12E+03	3.83E+02	3.56E+03	6.23
CTSO-3	8.45E+03	1.44E+03	4.32E+03	1.90	CTSO-3	3.55E+03	2.19E+02	3.23E+03	1.93
CTSO-4	1.13E+04	1.18E+03	9.20E+03	7.50	CTSO-4	4.16E+03	3.06E+02	3.45E+03	6.93
CTSO-5	1.07E+04	1.22E+03	8.24E+03	6.00	CTSO-5	4.18E+03	3.88E+02	3.45E+03	7.20
CTSO-6	1.06E+04	9.86E+02	9.22E+03	5.80	CTSO-6	4.13E+03	3.67E+02	3.64E+03	6.50
CTSO-7	1.13E+04	1.34E+03	8.39E+03	7.60	CTSO-7	4.09E+03	4.01E+02	3.58E+03	5.97
CTSO-8	1.06E+04	1.25E+03	8.48E+03	5.63	CTSO-8	4.26E+03	3.71E+02	3.55E+03	7.73
CTSO-9	9.01E+03	1.50E+03	3.63E+03	2.77	CTSO-9	3.66E+03	2.77E+02	3.28E+03	2.93
CTSO-10	1.10E+04	1.37E+03	8.98E+03	6.50	CTSO-10	4.23E+03	4.74E+02	3.61E+03	6.87
f28					f29
TSO	6.95E+03	7.63E+02	5.11E+03	7.80	TSO	7.09E+03	1.32E+03	5.34E+03	7.30
CTSO-1	6.78E+03	8.33E+02	5.49E+03	7.23	CTSO-1	6.60E+03	1.07E+03	4.83E+03	5.93
CTSO-2	6.52E+03	6.75E+02	5.35E+03	6.03	CTSO-2	6.82E+03	1.39E+03	4.71E+03	6.57
CTSO-3	3.36E+03	4.68E+01	3.28E+03	1.43	CTSO-3	5.59E+03	6.93E+02	4.80E+03	3.00
CTSO-4	6.59E+03	8.56E+02	5.32E+03	6.63	CTSO-4	6.64E+03	1.65E+03	4.67E+03	5.53
CTSO-5	6.71E+03	8.21E+02	5.50E+03	6.83	CTSO-5	7.58E+03	3.68E+03	5.25E+03	6.73
CTSO-6	6.59E+03	9.89E+02	4.39E+03	6.80	CTSO-6	6.87E+03	1.25E+03	5.42E+03	6.70
CTSO-7	6.78E+03	8.11E+02	5.23E+03	7.17	CTSO-7	6.62E+03	1.13E+03	5.10E+03	6.27
CTSO-8	6.74E+03	8.69E+02	5.12E+03	6.93	CTSO-8	7.11E+03	1.40E+03	5.01E+03	7.27
CTSO-9	3.38E+03	8.88E+01	3.26E+03	1.57	CTSO-9	5.73E+03	8.07E+02	4.46E+03	3.53
CTSO-10	6.90E+03	6.42E+02	5.73E+03	7.57	CTSO-10	7.06E+03	1.55E+03	4.58E+03	7.17
f30
Algorithm	AVG	STD	MIN	MR
TSO	8.51E+08	9.03E+08	1.91E+08	7.43
CTSO-1	6.09E+08	5.87E+08	7.08E+07	6.37
CTSO-2	5.23E+08	5.57E+08	8.79E+07	5.87
CTSO-3	1.87E+06	2.13E+06	5.87E+04	1.50
CTSO-4	6.15E+08	5.41E+08	3.57E+07	6.57
CTSO-5	7.89E+08	5.65E+08	4.65E+07	7.57
CTSO-6	8.63E+08	7.59E+08	1.21E+08	7.20
CTSO-7	8.42E+08	7.37E+08	8.26E+07	7.43
CTSO-8	9.36E+08	7.41E+08	1.48E+08	7.93
CTSO-9	3.02E+06	6.92E+06	3.88E+04	1.50
CTSO-10	6.63E+08	5.55E+08	8.49E+07	6.63

DOI: 10.7717/peerj-cs.1526/table-3

Note:

The use of bold demonstrates the best results attained.

Table 4:

Experimental results on CEC’17 test functions.

TSO	CTSO-1	CTSO-2	CTSO-3	CTSO-4	CTSO-5	CTSO-6	CTSO-7	CTSO-8	CTSO-9	CTSO-10
6.87	6.66	6.57	2.66	6.68	6.59	6.83	6.62	7.00	2.85	6.68

DOI: 10.7717/peerj-cs.1526/table-4

The results of the fitness functions obtained using the Wilcoxon rank-sum test are given in Table 5. The results obtained were obtained at a significance level of five percent. The results obtained must be less than 0.05 to show a significant advantage. The △, ▽, and ≈ signs in the table indicate the superiority of one of the proposed CTSOs, the TSO is superior, and there is no significant difference between the methods, respectively. When the Table 5 is examined, it is seen that CTSO-3 and CTSO-9 show the best performances by providing superiority to TSO in 26 functions. It can be said that Gaussian and sinusoidal chaotic maps are superior to other chaotic maps and standard TSO in benchmark functions.

Table 5:

Wilcoxon rank-sum test on CEC’17 test functions.

	CTSO-1	CTSO-2	CTSO-3	CTSO-4	CTSO-5	CTSO-6	CTSO-7	CTSO-8	CTSO-9	CTSO-10
f1	8.53E−01	8.65E−01	3.02E−11	1.91E−01	5.89E−01	7.96E−01	9.71E−01	7.28E−01	3.02E−11	7.06E−01
	≈	≈	Δ	≈	≈	≈	≈	≈	Δ	≈
f3	9.82E−01	5.30E−01	1.86E−06	2.97E−01	6.00E−01	4.83E−01	6.20E−01	5.20E−01	4.74E−06	9.00E−01
	≈	≈	∇	≈	≈	≈	≈	≈	∇	≈
f4	5.69E−01	6.84E−01	3.02E−11	9.00E−01	1.62E−01	9.47E−01	1.00E+00	3.40E−01	3.02E−11	3.04E−01
	≈	≈	Δ	≈	≈	≈	≈	≈	Δ	≈
f5	4.55E−01	6.00E−01	1.96E−10	4.92E−01	9.23E−01	2.64E−01	1.81E−01	4.83E−01	6.72E−10	6.00E−01
	≈	≈	Δ	≈	≈	≈	≈	≈	Δ	≈
f6	3.48E−01	9.93E−02	6.53E−08	1.41E−01	9.71E−01	3.04E−01	4.29E−01	9.00E−01	1.39E−06	8.19E−01
	≈	≈	Δ	≈	≈	≈	≈	≈	Δ	≈
f7	2.64E−01	5.40E−01	5.01E−01	1.09E−01	3.03E−02	6.10E−01	7.01E−02	2.97E−01	3.40E−01	8.53E−01
	≈	≈	≈	≈	Δ	≈	Δ	≈	≈	≈
f8	5.30E−01	1.15E−01	2.37E−10	5.20E−01	5.55E−02	5.59E−01	5.40E−01	3.95E−01	3.34E−11	1.81E−01
	≈	≈	Δ	≈	≈	≈	≈	≈	Δ	≈
f9	8.07E−01	8.88E−01	2.67E−09	6.73E−01	6.31E−01	8.30E−01	1.00E+00	3.33E−01	3.65E−08	1.76E−01
	≈	≈	Δ	≈	≈	≈	≈	≈	Δ	≈
f10	2.28E−01	5.30E−01	4.44E−07	3.33E−01	5.37E−02	1.12E−01	2.71E−01	1.54E−01	6.67E−03	5.30E−01
	≈	≈	Δ	≈	≈	≈	≈	≈	Δ
f11	2.90E−01	9.23E−01	3.02E−11	2.12E−01	3.11E−01	9.94E−01	6.20E−01	2.71E−01	8.10E−10	4.73E−01
	≈	≈	Δ	≈	≈	≈	≈	≈	Δ	≈
f12	2.07E−02	1.37E−01	3.02E−11	2.40E−01	1.12E−01	5.30E−01	5.20E−01	9.23E−01	3.02E−11	3.79E−01
	Δ	≈	Δ	≈	≈	≈	≈	≈	Δ	≈
f13	7.28E−01	8.30E−01	3.02E−11	5.11E−01	4.12E−01	1.91E−01	5.59E−01	8.65E−01	1.21E−10	5.01E−01
	≈	≈	Δ	≈	≈	≈	≈	≈	Δ	≈
f14	3.71E−01	6.95E−01	1.60E−07	2.71E−01	6.00E−01	9.23E−01	1.67E−01	7.06E−01	2.02E−08	6.84E−01
	≈	≈	Δ	≈	≈	≈	≈	≈	Δ	≈
f15	6.35E−02	1.19E−01	3.02E−11	1.41E−01	6.57E−02	5.49E−01	1.22E−01	3.15E−02	3.02E−11	3.87E−01
	≈	≈	Δ	≈	≈	≈	≈	Δ	Δ	≈
f16	3.71E−01	2.97E−01	3.20E−09	8.30E−01	2.40E−01	9.82E−01	4.38E−01	7.06E−01	6.72E−10	3.48E−01
	≈	≈	Δ	≈	≈	≈	≈	≈	Δ	≈
f17	5.59E−01	7.01E−02	2.15E−06	6.31E−01	3.79E−01	8.65E−01	6.10E−01	3.79E−01	1.17E−03	7.28E−01
	≈	≈	Δ	≈	≈	≈	≈	≈	Δ	≈
f18	1.71E−01	5.40E−01	5.49E−11	7.73E−01	1.19E−01	9.59E−01	6.79E−02	6.10E−01	1.33E−10	1.22E−01
	≈	≈	Δ	≈	≈	≈	≈	≈	Δ	≈
f19	7.28E−01	2.64E−01	3.02E−11	4.20E−01	5.59E−01	4.46E−01	9.94E−01	4.92E−01	3.02E−11	3.79E−01
	≈	≈	Δ	≈	≈	≈	≈	≈	Δ	≈
f20	4.83E−01	7.17E−01	3.04E−01	3.33E−01	1.58E−01	6.20E−01	9.00E−01	5.01E−01	9.05E−02	5.89E−01
	≈	≈	≈	≈	≈	≈	≈	≈	≈	≈
f21	9.94E−01	5.11E−01	3.82E−09	4.29E−01	3.11E−01	1.58E−01	7.39E−01	2.46E−01	3.20E−09	9.59E−01
	≈	≈	Δ	≈	≈	≈	≈	≈	Δ	≈
f22	3.71E−01	6.35E−02	2.38E−03	6.84E−01	5.79E−01	7.73E−01	8.88E−01	6.31E−01	3.77E−04	9.23E−01
	≈	≈	Δ	≈	≈	≈	≈	≈	Δ	≈
f23	4.84E−02	6.95E−01	1.44E−02	7.62E−01	8.53E−01	1.12E−01	1.58E−01	2.58E−01	1.63E−02	5.30E−01
	Δ	≈	Δ	≈	≈	≈	≈	≈	Δ	≈
f24	1.49E−01	7.84E−01	4.23E−03	7.73E−01	2.90E−01	6.10E−01	3.18E−01	8.65E−01	5.97E−05	9.82E−01
	≈	≈	Δ	≈	≈	≈	≈	≈	Δ	≈
f25	1.08E−02	4.36E−02	3.02E−11	3.40E−01	5.69E−01	4.12E−01	1.30E−01	1.19E−01	3.02E−11	9.07E−03
	Δ	Δ	Δ	≈	≈	≈	≈	≈	Δ	Δ
f26	4.12E−01	1.00E+00	6.72E−10	9.59E−01	6.79E−02	1.38E−02	7.39E−01	3.39E−02	3.08E−08	2.06E−01
	≈	≈	Δ	≈	≈	Δ	≈	Δ	Δ	≈
f27	6.84E−01	2.23E−01	1.55E−09	6.00E−01	5.49E−01	3.63E−01	9.63E−02	8.88E−01	6.05E−07	6.41E−01
	≈	≈	Δ	≈	≈	≈	≈	≈	Δ	≈
f28	3.11E−01	3.03E−02	3.02E−11	9.93E−02	1.91E−01	1.81E−01	3.95E−01	2.46E−01	3.02E−11	7.51E−01
	≈	Δ	Δ	≈	≈	≈	≈	≈	Δ	≈
f29	1.96E−01	2.97E−01	6.53E−07	7.24E−02	6.95E−01	5.11E−01	1.26E−01	8.77E−01	5.86E−06	9.12E−01
	≈	≈	Δ	≈	≈	≈	≈	≈	Δ	≈
f30	2.46E−01	3.39E−02	3.02E−11	2.90E−01	5.01E−01	9.59E−01	7.51E−01	3.18E−01	3.02E−11	4.64E−01
	≈	Δ	Δ	≈	≈	≈	≈	≈	Δ	≈
(Δ/∇/≈)	3/0/26	3/0/26	26/1/2	0/0/29	1/0/28	1/0/28	0/0/29	2/0/27	26/1/2	1/0/28

DOI: 10.7717/peerj-cs.1526/table-5

Real-world engineering problems

In this section, five real-world engineering design problems are used to validate the search performance of each implemented algorithm. These problems are the speed reducer problem, tension compression spring design problem, welded beam design problem, pressure vessel, and three-bar truss design problem. All these problems are limited in nature, and therefore an external penalty approach mechanism was used to solve the design constraints. The maximum number of iterations for all problems was determined to be 1,000 and the population number was 30. The algorithm parameters are the values used in CEC’17. To visualize and compare the convergence behavior of the examined algorithms, the best fitness values obtained for each problem, which are usually called convergence curves, are drawn. All experiments corresponding to each algorithm were run independently 30 times. The best, mean, worst, and standard deviation values were examined comparatively, and the best solution obtained among the algorithms was highlighted in bold for ease of readability.

Speed reducer problem

This is essentially a gearbox issue, allowing the aircraft engine to spin at its most efficient speed (Dhiman, 2021). Finding the face width $b$ $\left(x_1\right)$, the tooth modulus $m$ ( $x_2$), the number of teeth on the pinion $z$ ( $x_3$), the length of the first shaft between the bearings $l_1\left(x_4\right)$, the length of the second shaft between the bearings $l_2\left(x_5\right)$, the diameter of the first shaft $d_1\left(x_6\right)$, and the diameter of the second shaft $d_2\left(x_7\right)$ will allow you to determine the minimum values of the seven decision variables in this problem. Figure 5 shows a schematic illustration of the speed reducer concept. The goal of this design challenge is to determine the speed reducer’s lightest possible cost. The mathematical representation of this problem is as in the Appendix 1.

The comparative values of the performances of the 10 proposed CTSO methods and standard TSO methods on the speed reducer problem are presented in Table 6. The best, average, worst, and standard deviation values of the methods used are shown in Table 6. In addition, the decision variables depending on the best value of the methods used in the results of 30 runs on this problem are given in Table 7. When Table 6 is examined, it is seen that CTSO-3 and CTSO-9 are superior to other methods on the basis of the best value, and CTSO-9 is more successful when analyzed on the basis of mean value. In addition, the convergence graph of the methods used on the speed reducer problem is shown in Fig. 6.

Table 6:

Statistical results of the used algorithms for speed reducer problem.

Algorithm	Best	Mean	Worst	SD
TSO	3,043.966	338,765.8	2,180,277	634,222.4
CTSO1	3,133.385	209,405.2	1,769,248	470,918.1
CTSO2	3,008.135	643,475.8	2,297,437	856,143.2
CTSO3	2,994.423	3,805.273	5,521.033	872.899
CTSO4	3,050.252	422,165.5	2,001,481	630,221.1
CTSO5	3,003.839	135,976.5	1,034,844	289,203.5
CTSO6	3,015.958	231,801	2,363,101	636,178.5
CTSO7	3,014.59	210,227.5	1,625,269	455,196.1
CTSO8	3,050.025	594,289.3	2,401,982	850,915.1
CTSO9	2,994.423	3,627.353	4,774.773	562.9862
CTSO10	3,036.987	358,518.4	2,078,853	620,159.3

DOI: 10.7717/peerj-cs.1526/table-6

Table 7:

Comparison of the best optimum solution for the speed reducer problem.

Algorithm	Parameters values							fmin
	x1	x2	x3	x4	x5	x 6	x 7
TSO	3.501019	0.7	17	7.3	7.917685	3.367619	5.349237	3,043.966
CTSO1	3.499917	0.7	17	7.994088	7.911673	3.787018	5.28737	3,133.385
CTSO2	3.504471	0.7	17	7.99709	7.857388	3.35903	5.287425	3,008.135
CTSO3	3.49999	0.7	17	7.3	7.715319	3.350541	5.286655	2,994.423
CTSO4	3.499871	0.7	17	8.067661	8.067661	3.361258	5.345938	3,050.252
CTSO5	3.499944	0.7	17	8.006041	7.768105	3.353007	5.288778	3,003.839
CTSO6	3.500284	0.7	17	8.048994	8.074678	3.35588	5.29533	3,015.958
CTSO7	3.509419	0.7	17	7.630128	8.036515	3.373788	5.28744	3,014.590
CTSO8	3.499876	0.7	17	8.014485	7.877796	3.352811	5.356255	3,050.025
CTSO9	3.49999	0.7	17	7.3	7.715319	3.350541	5.286654	2,994.423
CTSO10	3.500698	0.7	17	7.3	8.072679	3.361987	5.335593	3,036.987

DOI: 10.7717/peerj-cs.1526/table-7

Tension compression spring design problem

The goal of the tension/compression spring design issue, as outlined by Arora (Arora, 2004), is to provide a spring design with the least amount of weight possible. Figure 7’s schematic illustration of this minimization issue shows some of its restrictions, including the cut-off voltage, ripple frequency, and minimal deviation. There are three choice variables in the tension/compression spring dilemma. These are the following: the wire diameter $d$ ( $x_1$), the average coil diameter $D$ ( $x_2$), and the number of active coils $N$ ( $x_3$). The mathematical representation of this problem is as in the Appendix 2.

The comparative values of the performances of the 10 proposed CTSO methods and standard TSO methods on the tension compression spring design problem are presented in Table 8. The best, average, worst, and standard deviation values of the methods used are shown in Table 8. In addition, the decision variables depending on the best value of the methods used in the results of 30 runs on this problem are given in Table 9. When Table 8 is examined, it is seen that CTSO-9 is superior to other methods on the basis of the best value, and CTSO-6 is more successful when analyzed on the basis of mean value. In addition, the convergence graph of the methods used on the tension compression spring design problem is shown in Fig. 8.

Table 8:

Statistical results of the used algorithms for tension-compression spring design problem (Case 1).

Algorithm	Best	Mean	Worst	SD
TSO	0.012673	0.013698	0.017631	0.001075
CTSO1	0.012715	0.013655	0.017233	0.00081
CTSO2	0.012702	0.013486	0.016016	0.000722
CTSO3	0.01268	0.014229	0.020937	0.001895
CTSO4	0.012784	0.013732	0.016824	0.000896
CTSO5	0.012714	0.013537	0.015575	0.000668
CTSO6	0.012739	0.013413	0.015612	0.000617
CTSO7	0.01268	0.013579	0.015701	0.000774
CTSO8	0.012683	0.013597	0.017302	0.000961
CTSO9	0.012670	0.013483	0.017367	0.001136
CTSO10	0.012714	0.013481	0.015773	0.000701

DOI: 10.7717/peerj-cs.1526/table-8

Table 9:

Comparison of the best optimum solution for the tension compression spring design problem.

Algorithm	Parameters values			fmin
	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$
TSO	0.05215	0.367888	10.66628	0.012673
CTSO1	0.053121	0.391907	9.497097	0.012715
CTSO2	0.050521	0.329228	13.11549	0.012702
CTSO3	0.050809	0.335913	12.62168	0.01268
CTSO4	0.054124	0.418139	8.436915	0.012784
CTSO5	0.050561	0.330017	13.0706	0.012714
CTSO6	0.053384	0.398798	9.208447	0.012739
CTSO7	0.052278	0.370952	10.50679	0.01268
CTSO8	0.05217	0.368334	10.65104	0.012683
CTSO9	0.05119	0.344834	12.02127	0.012670
CTSO10	0.053257	0.395514	9.333766	0.012714

DOI: 10.7717/peerj-cs.1526/table-9

Welded beam design problem

The primary goal of the welded beam design challenge is to create a beam for the least amount of money while adhering to specific constraints (Varol Altay & Alatas, 2020). Figure 9 depicts a welded beam construction made up of beam A and the welding process needed to join it to item B. Five nonlinear inequality constraints and four choice variables make up the issue. The weld thickness, weld joint length, element width, and element thickness are represented by the design parameters $h$ ( $x_1$), $l$ ( $x_2$), $t$ ( $x_3$), and $b$ ( $x_4$), respectively. The mathematical representation of this problem is as in the Appendix 3.

The comparative values of the performances of the 10 proposed CTSO methods and standard TSO methods on the welded beam design problem are presented in Table 10. The best, average, worst, and standard deviation values of the methods used are shown in Table 10. In addition, the decision variables depending on the best value of the methods used in the results of 30 runs on this problem are given in Table 11. When Table 10 is examined, it is seen that CTSO-1 is superior to other methods on the basis of the best value, and CTSO-4 is more successful when analyzed on the basis of mean value. In addition, the convergence graph of the methods used on the welded beam design problem is shown in Fig. 10.

Table 10:

Statistical results of the used algorithms for welded beam design problem.

Algorithm	Best	Mean	Worst	SD
TSO	1.691099	2.765785	3.713832	0.681901
CTSO1	1.680985	2.706922	3.696062	0.69335
CTSO2	1.692163	2.630818	3.687721	0.754546
CTSO3	1.909833	3.307382	5.11409	0.943303
CTSO4	1.699917	2.416367	3.745437	0.663697
CTSO5	1.760383	2.792383	3.786065	0.544282
CTSO6	1.721776	2.423815	3.684699	0.621039
CTSO7	1.74493	2.67853	3.994741	0.69217
CTSO8	1.764811	2.587149	3.675218	0.590975
CTSO9	1.807061	3.159026	6.243182	1.010093
CTSO10	1.739581	2.665869	3.709204	0.646727

DOI: 10.7717/peerj-cs.1526/table-10

Table 11:

Comparison of the best optimum solution for welded beam design problem.

Algorithm	Parameters values				fmin
	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$
TSO	0.198654	3.371947	9.112068	0.202756	1.691099
CTSO1	0.200332	3.329221	9.164297	0.200695	1.680985
CTSO2	0.195439	3.539739	9.191104	0.198922	1.692163
CTSO3	0.235488	3.063879	8.008572	0.261938	1.909833
CTSO4	0.202884	3.345369	9.074833	0.204389	1.699917
CTSO5	0.146154	4.743619	9.193987	0.19883	1.760383
CTSO6	0.197589	3.424587	8.970536	0.209319	1.721776
CTSO7	0.161839	4.37461	9.210771	0.198758	1.74493
CTSO8	0.220815	3.104299	8.68097	0.223645	1.764811
CTSO9	0.202509	3.452743	8.545876	0.230036	1.807061
CTSO10	0.200633	3.398656	8.875607	0.213808	1.739581

DOI: 10.7717/peerj-cs.1526/table-11

Pressure vessel problem

The major goal of this issue is to optimize vessel formation, material use, and welding costs (He & Zhou, 2018). The objective function is constructed using four variables: shell thickness ( $x_1$), head thickness ( $x_2$), inner radius ( $x_3$), and length ( $x_4$), without accounting for vessel height. This issue contains four restrictions that must be met. Figure 11 depicts the pressure vessel design problem’s schematic structure. The mathematical representation of this problem is as in the Appendix 4.

The comparative values of the performances of the 10 proposed CTSO methods and standard TSO methods on the pressure vessel design problem are presented in Table 12. The best, average, worst, and standard deviation values of the methods used are shown in Table 12. In addition, the decision variables depending on the best value of the methods used in the results of 30 runs on this problem are given in Table 13. When Table 12 is examined, it is seen that CTSO-2 is superior to other methods on the basis of the best value, and CTSO-5 is more successful when analyzed on the basis of mean value. In addition, the convergence graph of the methods used on the pressure vessel design problem is shown in Fig. 12.

Table 12:

Statistical results of the used algorithms for pressure vessel design problem.

Algorithm	Best	Mean	Worst	SD
TSO	6,591.539	9,358.735	15,198.4	2,047.207
CTSO1	6,934.041	9,290.626	12,997.45	1,833.564
CTSO2	6,325.269	8,872.022	13,581.73	2,116.409
CTSO3	6,720.31	18,807.36	55,057.06	10,642.19
CTSO4	6,485.754	8,708.514	13,722.88	1,568.458
CTSO5	6,479.417	8,662.113	12,317.9	1,380.715
CTSO6	6,494.753	8,878.358	13,742.33	1,651.033
CTSO7	6,782.928	8,763.569	12,770.58	1,438.112
CTSO8	6,722.005	8,687.164	14,207.82	1,804.541
CTSO9	7,836.811	17,484.35	38,598.81	8,019.801
CTSO10	6,794.113	8,275.311	11,169.49	1,131.863

DOI: 10.7717/peerj-cs.1526/table-12

Table 13:

Comparison of the best optimum solution for pressure vessel design problem.

Algorithm	Parameters values				fmin
	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$
TSO	1	0.5	49.9536	98.7225	6,591.539
CTSO1	1.0625	0.5625	53.06296	75.76345	6,934.041
CTSO2	0.875	0.5	45.277	140.8673	6,325.269
CTSO3	1	0.5625	51.66779	85.64049	6,720.31
CTSO4	1	0.5	51.01429	90.49689	6,485.754
CTSO5	1	0.5	51.10119	89.90082	6,479.417
CTSO6	0.9375	0.5	47.36856	120.7645	6,494.753
CTSO7	1.0625	0.5625	54.90602	63.63404	6,782.928
CTSO8	1	0.5625	51.64737	85.79055	6,722.005
CTSO9	1.25	0.5	51.69956	85.4083	7,836.811
CTSO10	1.0625	0.5625	54.77942	64.46914	6,794.113

DOI: 10.7717/peerj-cs.1526/table-13

Three bar truss design problem

In civil engineering, this issue is referred to as a structural optimization problem. By modifying the cross-sectional areas ( $x_1$ and $x_2$) while taking into consideration the stress (σ) on each of the truss members, Nowacki’s issue seeks to reduce the volume of the three-bar truss. These values’ possible value ranges are $0\le x_1,x_2\le 1$. The graphical illustration of this problem’s mathematical formulation is shown in Fig. 13. The mathematical representation of this problem is as in the Appendix 5.

The comparative values of the performances of the 10 proposed CTSO methods and standard TSO methods on the three-bar truss design problem are presented in Table 14. The best, average, worst, and standard deviation values of the methods used are shown in Table 14. In addition, the decision variables depending on the best value of the methods used in the results of 30 runs on this problem are given in Table 15. When Table 14 is examined, it is seen that CTSO-9 is superior to other methods on the basis of the best value, and CTSO-10 is more successful when analyzed on the basis of mean value. In addition, the convergence graph of the methods used on the three-bar truss design problem is shown in Fig. 14.

Table 14:

Statistical results of the used algorithms for three-bar truss design problem.

Algorithm	Best	Mean	Worst	SD
TSO	263.8964	264.5518	270.4717	1.648298
CTSO1	263.8987	264.8722	270.7144	2.06893
CTSO2	263.9028	264.7003	270.7078	1.696716
CTSO3	263.9005	266.5551	274.3699	2.902827
CTSO4	263.8962	264.4986	270.7534	1.375953
CTSO5	263.8963	264.5397	270.3816	1.46023
CTSO6	263.9002	264.6418	270.7240	1.362083
CTSO7	263.8966	264.6414	270.6517	1.388371
CTSO8	263.9014	264.5663	270.3992	1.670418
CTSO9	263.8957	266.2048	281.5610	3.621051
CTSO10	263.8962	264.3218	270.7161	1.223462

DOI: 10.7717/peerj-cs.1526/table-14

Table 15:

Comparison of the best optimum solution for three-bar truss design problem.

Algorithm	Parameters values		fmin
	${\mathit{x}}_1$	${\mathit{x}}_2$
TSO	0.787491	0.411599	263.8964
CTSO1	0.786632	0.414043	263.8987
CTSO2	0.786469	0.41449	263.9028
CTSO3	0.791316	0.40082	263.9005
CTSO4	0.788856	0.407739	263.8962
CTSO5	0.787869	0.410515	263.8963
CTSO6	0.790911	0.401968	263.9002
CTSO7	0.789892	0.404812	263.8966
CTSO8	0.785835	0.416332	263.9014
CTSO9	0.789298	0.40648	263.8957
CTSO10	0.787728	0.41092	263.8962

DOI: 10.7717/peerj-cs.1526/table-15

Analysis of CTSOs in real-world engineering problems with other metaheuristic optimization algorithms

The speed reducer problem has been solved by many different researchers in the literature with different metaheuristic optimization methods. The classical TSO and the proposed CTSO method were compared with the Moth Flame Optimization Algorithm (MFO) (Mirjalili, 2015), Weighted Superposition Attraction (WSA) (Baykaso, 2015), Grey Wolf Optimization (GWO) (Mirjalili, Mirjalili & Lewis, 2014), Artificial Acari Optimization (AAO) (Czerniak, Zarzycki & Ewald, 2017), Sine Cosine Algorithm (SCA) (Mirjalili, 2016), and Arithmetic Optimization Algorithm (AOA) (Abualigah et al., 2021a) methods in the literature. The best cost and related decision variables obtained by the proposed method and the methods proposed by other researchers for the speed reducer problem are presented in Table 16. When the results obtained were examined, it was observed that the proposed CTSO-3 and CTSO-9 method gave a much better result than the classical TSO and other competitive methods.

Table 16:

Comparison of the speed reducer problem.

Algorithm	Parameters values							fmin
	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\mathit{x}}_5$	${\mathit{x}}_6$	${\mathit{x}}_7$
MFO (Mirjalili, 2015)	3.497455	0.700	17	7.82775	7.712457	3.351787	5.286352	2,998.941
WSA (Baykaso, 2015)	3.500	0.7	17	7.3	7.8	3.350215	5.286683	2,996.348
GWO (Mirjalili, Mirjalili & Lewis, 2014)	3.501	0.7	17	7.3	7.811013	3.350704	5.287411	2,997.820
AAO (Czerniak, Zarzycki & Ewald, 2017)	3.4999	0.7	17	7.3	7.8	3.3502	5.2877	2,997.058
SCA (Mirjalili, 2016)	3.521	0.7	17	8.3	7.923351	3.355911	5.300734	3,026.838
AOA (Abualigah et al., 2021a)	3.50384	0.7	17	7.3	7.72933	3.35649	5.2867	2,997.916
TSO	3.501019	0.7	17	7.3	7.917685	3.367619	5.349237	3,043.966
CTSO-3	3.49999	0.7	17	7.3	7.715319	3.350541	5.286655	2,994.423
CTSO-9	3.49999	0.7	17	7.3	7.715319	3.350541	5.286654	2,994.423

DOI: 10.7717/peerj-cs.1526/table-16

There are many methods in the literature to solve the tension compression spring design problem. The classical TSO and the proposed CTSO method were compared with the HGSO, AAO, Whale Optimization Algorithm (WOA) (Mirjalili & Lewis, 2016), Harris Hawk Optimization (HHO) (Heidari et al., 2019), Chaotic Bird Swarm Algorithm (CMBSA) (Varol Altay & Alatas, 2020), Salp Swarm Algorithm (SSA) (Mirjalili et al., 2017), Cumulative Binomial Probability Particle Swarm Optimization (CBPPSO) (Agrawal & Tripathi, 2021), and Competitive Bird Swarm Algorithm (CBSA) (Wang, Deng & Duan, 2018) methods in the literature. The results obtained from this comparison are given in Table 17. Table 17 presents the best cost and relevant decision variables for the tension compression spring design problem. When the literature and the results of the experiments are examined, it is concluded that CTSO-9 gives better results than classical TSO and competitive methods in the literature.

Table 17:

Comparison of tension compression spring design problem.

Algorithm	Parameters values			fmin
	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$
HGSO (Hashim et al., 2019)	0.0518	0.3569	11.2023	0.0127
AAO (Czerniak, Zarzycki & Ewald, 2017)	0.0517	0.3581	11.2015	0.0127
WOA (Mirjalili & Lewis, 2016)	0.0512	0.3452	12.004	0.0127
HHO (Heidari et al., 2019)	0.0543	0.4239	8.2187	0.0128
CMBSA (Varol Altay & Alatas, 2020)	0.0519	0.3618	11.000	0.0127
SSA (Mirjalili et al., 2017)	0.0516	0.3547	11.4059	0.0127
CBPPSO (Agrawal & Tripathi, 2021)	0.0512	0.3465	11.9097	0.0127
CBSA (Wang, Deng & Duan, 2018)	0.0516	0.3566	11.2918	0.0127
TSO	0.05215	0.367888	10.66628	0.012673
CTSO-9	0.05119	0.344834	12.02127	0.012670

DOI: 10.7717/peerj-cs.1526/table-17

Welded beam design problem has been solved by many different researchers in the literature with different metaheuristic optimization methods. The classical TSO and the proposed CTSO method were compared with the HGSO, MAOA, HHO, CMBSA, CBBSA, CBPPSO, Chaotic Grey Wolf Optimization (CGWO) (Kohli & Arora, 2018), Differential Big Bang-Big Crunch Algorithm (DBCA) (Prayogo et al., 2018), Sonar Inspired Optimization (SIO) (Tzanetos & Dounias, 2020), and Hybrid Genetic Algorithm-Ant Colony Optimization-Particle Swarm Optimization (H-GA-ACO-PSO) (Tam et al., 2019) methods in the literature. The best cost and relevant decision variables obtained by the proposed method and the methods proposed by other researchers for the welded beam design problem are presented in Table 18. When the results obtained were examined, it was observed that the proposed CTSO-1 method gave a much better result than the classical TSO and other competitive methods.

Table 18:

Comparison of welded beam design problem.

Algorithm	Parameters values				fmin
	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$
HGSO (Hashim et al., 2019)	0.2054	3.4476	9.0269	0.2060	1.7260
MAOA (Altay, 2022a)	0.2057	3.4705	9.0366	0.2057	1.7246
HHO (Heidari et al., 2019)	0.1956	3.7730	9.0307	0.2060	1.7501
CMBSA (Varol Altay & Alatas, 2020)	0.2057	3.4702	9.0377	0.2057	1.7249
CBPPSO (Agrawal & Tripathi, 2021)	0.2057	3.4704	9.0366	0.2057	1.7249
CGWO (Kohli & Arora, 2018)	0.3439	1.8836	9.0313	0.2121	1.7255
DBCA (Prayogo et al., 2018)	0.2057	3.4705	9.0366	0.2057	1.7249
SIO (Tzanetos & Dounias, 2020)	0.3314	2.0174	9.0459	0.2088	1.7621
H-GA-ACO-PSO (Tam et al., 2019)	0.2057	3.4705	9.0366	0.2057	1.7249
TSO	0.198654	3.371947	9.112068	0.202756	1.6911
CTSO-1	0.200332	3.329221	9.164297	0.200695	1.6810

DOI: 10.7717/peerj-cs.1526/table-18

The pressure vessel problem has been solved by many different researchers in the literature with different metaheuristic optimization methods. The classical TSO and the proposed CTSO method were compared with the MAOA, CSMA, AAO, HHO, CMBSA, SSA, CBPPSO, H-GA-ACO-PSO, and Adaptive Reinforcement Learning based Bat Algorithm (ARLBAT) (Meng, Li & Gao, 2019) methods in the literature. The best cost of the proposed method and the methods suggested by other researchers for the pressure vessel problem and the relevant decision variables are shown in Table 19. When the results are examined, it is seen that the CSMA method gives a better result than the proposed CTSOs, classical TSO and other competitive methods.

Table 19:

Comparison of pressure vessel problem.

Algorithm	Parameters values				fmin
	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$
MAOA (Altay, 2022a)	0.7953	0.3931	41.2274	187.7371	5,914.48511
CSMA (Altay, 2022c)	0.7778	0.3845	40.3207	199.9850	5,882.0851
AAO (Czerniak, Zarzycki & Ewald, 2017)	0.8125	0.4375	42.0985	176.6366	6,059.7140
HHO (Heidari et al., 2019)	0.8540	0.4329	44.0025	154.3888	6,543.4802
CMBSA (Varol Altay & Alatas, 2020)	0.7780	0.3850	40.3200	200.0000	5,883.8610
SSA (Mirjalili et al., 2017)	0.7807	0.3859	40.4707	197.9081	6,149.0232
CBPPSO (Agrawal & Tripathi, 2021)	1.125	0.6250	62.9866	20.00000	6,952.7200
H-GA-ACO-PSO (Tam et al., 2019)	0.8125	0.4375	42.0984	176.6366	6,059.7143
ARLBAT (Meng, Li & Gao, 2019)	0.8125	0.4375	42.0984	176.6366	6,059.7143
TSO	1	0.5	49.9536	98.7225	6,591.5390
CTSO-2	0.875	0.5	45.277	140.8673	6,325.2690

DOI: 10.7717/peerj-cs.1526/table-19

The three bar truss problem has been solved by many different researchers in the literature with different metaheuristic optimization methods. The classical TSO and the proposed CTSO method were compared with the AOA, SSA, CBPPSO, Cuckoo Search (CS) (Gandomi, Yang & Alavi, 2013), and mine blast algorithm (MBA) (Sadollah et al., 2013) methods in the literature. The best cost and relevant decision variables obtained by the proposed method and the methods proposed by other researchers for the three bar truss problem are shown in Table 20. When the literature and the results of the experiments are examined, it is concluded that CTSO-9 gives better results than classical TSO and competitive methods in the literature.

Table 20:

Comparison of for three bar truss problem.

Algorithm	Parameters values		fmin
	${\mathit{x}}_1$	${\mathit{x}}_2$
AOA (Abualigah et al., 2021a)	0.79369	0.39426	263.9154
SSA (Mirjalili et al., 2017)	0.78866541	0.408275784	263.8958
CS (Gandomi, Yang & Alavi, 2013)	0.78867	0.40902	263.9716
MBA (Sadollah et al., 2013)	0.7885650	0.4085597	263.8959
TSO	0.787491	0.411599	263.8964
CTSO-9	0.789298	0.40648	263.8957

DOI: 10.7717/peerj-cs.1526/table-20

Conclusion

TSO is a recently developed physics-based method that draws inspiration from the transient behavior of switched electrical circuits with storage components like inductance and capacitance. Our motivation was that this method is very new, tends to be stuck with local optimal solutions, has difficulties maintaining the balance between exploration and exploitation, and is not working to improve the performance of TSO. In this article, a number of chaotic maps have been incorporated in order to customize the TSO settings. By using chaotic maps with ergodic, irregular, and stochastic features in chaotic map TSO, it is aimed at avoiding local solutions more easily compared to the TSO method. In this way, the balance between exploration and exploitation has been achieved, and global convergence has increased. Ten distinct chaotic maps have been studied on twenty-nine different benchmark test functions, five different real-world engineering problems, and 10 different UCI standard datasets for feature selection. The findings demonstrated that the use of chaotic maps is capable of greatly enhancing TSO’s overall performance in a general sense. The approach that employs the Gauss and sinusoidal maps for supplying the values of parameter c1 has been determined to be the best chaotic TSO after being put through a series of comparisons with other chaotic TSOs. Gaussian and sinusoidal maps create a better balance between exploration and exploitation, which prevents the algorithm from being stuck in a local optimal state while it is in the process of optimizing its parameters. This is the reason for the enhanced performance. The results of the simulation demonstrated that Gaussian and Sinusoidal maps performed better than the standard TSO for the majority of the benchmark functions, that Sinusoidal maps performed better than the standard TSO for the majority of the real-world engineering problems, and that CTSOs generally suggested in the feature selection performed better than the standard TSO overall. According to the findings of the statistical analysis, customized algorithms clearly improve both the dependability of the global optimality and the quality of the outcomes.

Due to the fact that chaotic TSOs are relatively new, one potentially fruitful direction for future study would be to investigate how these algorithms perform in a parallel or distributed setting. The proposed CTSO method can also be applied promisingly in many different fields. Among them, many different fields such as machine learning, artificial intelligence, image processing, different real-world engineering problems, data mining, and big data are suitable for the application of algorithms. Like other proposed metaheuristic optimization methods, CTSO cannot guarantee that its solution will always equal the global optimum, due to its stochastic nature.

Appendix 1: speed reducer problem

Minimize:

$$\begin{array}{rl}f\left(x\right)=& 0.7854x_2^2{x}_1\left(14.9334x_3-43.0934+3.3333x_3^2\right)+0.7854\left(x_5{x}_7^2+x_4{x}_6^2\right)\\ & -1.508x_1\left(x_7^2+x_6^2\right)+7.477\left(x_7^3+x_6^3\right)\end{array}$$