Multi scenario chaotic transient search optimization algorithm for global optimization technique

Abstract

Recently, chaotic maps (CMs) have been employed in many optimization algorithms as a motivator to find a better solution to non-convex engineering problems since they can avoid local optima and find the near-optimal solution rapidly. In this article, a metaheuristic, physics-based algorithm called chaotic transient search optimization (CTSO) algorithm is developed to solve 23 benchmark functions, including uni- and multi-modal optimization functions. Nine CMs integrated into the TSO to improve its search capabilities by applying various scenarios for improving the TSO random numbers. Further, the proposed CTSO was compared with the original TSO using the Wilcoxon p-value test, non-parametric sign test, t-test, convergence curves, and elapsed time. Furthermore, the proposed CTSO algorithm has been employed for solving real-life engineering design problems, including coil spring, welded beam, and pressure vessel design, where CTSO performed better than some recent optimization algorithms in finding the best design.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-025-86757-7.

Introduction

Chaotic maps have been integrated into many metaheuristic (MH) algorithms aiming to find a better near-optimal solution in a short time with higher tangibility to the best solution. As well-known before, there were many optimization algorithms, including gradient-based and stochastic methods. From the gradient-based methods perspective, they deal with linear problems better than stochastic algorithms as they consider the gradient in their calculations, like linear programming, sequential quadratic programming¹, and others¹. From the stochastic methods perspective, there are many MH optimization algorithms, including swarm-based², physics-based³, evolutionary⁴ and human-related algorithms⁵. The rise of MH algorithms has been promoted more and more in the previous decades to solve engineering optimization problems due to their high complexity and non-convexity, which is very difficult to solve by gradient-based methods. Furthermore, dealing with optimization problems, including multi-stages, i.e., multi-level optimization problems, is easily handled by MH algorithms rather than gradient-based methods, which needs a good initialization to avoid longer processing time than MH methods. To fulfil the need for MH algorithms, in the previous decades, many MH optimization algorithms were proposed to solve complex non-convex engineering problems, and they are illustrated in the following paragraphs.

Swarm-based optimization algorithms² are defined as “The emergent collective intelligence of groups of simple agents”, as stated by Bonabeau⁶. Many swarm-based optimization algorithms or swarm intelligence algorithms were previously proposed to solve single and multi- objective optimization problems. These algorithms are particle swarm optimization algorithm (PSO)⁷, bat algorithm⁸, cuckoo search algorithm (CS)⁹, artificial bee colony¹⁰, whale optimization algorithm (WOA)¹¹, coyote optimizer¹², sunflower optimization algorithm¹³, salp swarm algorithm (SSA)¹⁴, dolphin echolocation¹⁵, krill herd algorithm¹⁶, firefly algorithm¹⁷, grey wolf optimizer (GWO)¹⁸, and others. PSO algorithm is inspired by the behaviour of both birds and fishes⁷. The bat algorithm⁸ is inspired by the behaviour of microbats, characterized by its echolocation. The whale optimization algorithm¹¹ is inspired by humpback whales’ motion behaviour for chasing prey. The behaviour of salps inspires the salp swarm optimization algorithm during its navigation in oceans and foraging¹⁴. The navigation of dolphins inspires dolphin echolocation¹⁵ algorithm in finding their prey. The herding behaviour of the krills inspires the krill herd algorithm¹⁶.

Evolutionary metaheuristic optimization^19,20 algorithms are nature-based optimization algorithms capable of solving highly complex non-convex optimization problems comprising many optimization variables. They are inspired by biological evolution, like humans, organisms, creatures, …, etc. The core competence of these algorithms arises in their flexible behaviour in finding the near-optimal solution without further need for more features in the objective function, unlike gradient-based methods²⁰. There have been many advances in the evolutionary MH optimization algorithms during the previous decades aiming to solve engineering problems and find near-optimal solutions to optimization problems²⁰. The evolutionary MH algorithms are genetic algorithms²¹, biography-based optimization²², differential evolution (DE)²³, fast evolutionary programming²⁴, evolution strategy²⁵, and others. Genetic algorithm (GA)²¹ was previously proposed to solve complex optimization problems by imitating the evolution of individuals through various steps, including mutation, crossover, and selection processes. Biography-based optimization²² is inspired by the biological species distribution in space and also through time.

Physics-based optimization²⁶ algorithms are recent efficient algorithms imitating physical phenomena and mathematical characteristics of well-known functions. These algorithms are: The circle search algorithm²⁷ is inspired by the mathematical interpretation of circles. Transient search optimization (TSO)²⁸ is inspired by the transient behavior of electrical circuits, including capacitors, inductors, and resistors. Cosmological concepts, including white, black, and wormholes, inspire multi-verse optimizer (MVO)²⁹. The sine-cosine algorithm³⁰ hinges on mathematical modelling, which employs both sine and cosine functions. Also, recently, many physics-based optimization algorithms have been proposed, like Rime-ice physics-based optimization RIME³¹, the propagation search algorithm³², and Henry gas solubility optimization³³.

Real-life engineering optimization^34–38 problems are characterized by their non-convexity thus obtaining near-optimal solutions to these problems must be handled by efficient optimizers. Thirteen optimization algorithms were employed in³⁴ to solve an optimization problem aims to identify the system of small-scale fixed-wing Unmanned Aerial Vehicles. Further, the design of steel frames was optimized using three optimizers³⁵, including GWO, stochastic fractal search optimization algorithm and Adaptive differential evolution with optional external archive algorithm (JADE). A modified teaching-learning optimization algorithm is proposed in³⁶ to design truss structures efficiently compared to various optimization algorithms. Furthermore, a modified sub-population-based heat transfer search algorithm³⁷ is proposed for solving structural optimization problems like truss optimization problems.

The emergence of chaos with the metaheuristic optimization algorithms has recently been adopted to find near-optimal solutions to the optimization problems without being trapped in local optima. Many chaotic-based optimization algorithms were proposed, most recently aiming to find better near-optimal solutions. These optimization algorithms include the chaotic-PSO³⁹, chaotic-Whale⁴⁰, chaotic-SCA⁴¹, chaotic- Henry Gas Solubility Optimization (HGSO)⁴², chaotic- Arithmetic Optimization Algorithm (AOA)⁴³, and others. These previously proposed chaotic optimization algorithms have proven their ability to solve optimization problems better. The authors specifically chose the TSO algorithm as it was recently developed by the authors to solve complex engineering problems and has proven its ability to find acceptable near-optimal solutions compared to well-known optimization algorithms like PSO, GA, SSA, GWO, DE, CS, and WOA. Besides, it has been tested against recent optimizers like the sandpiper optimization algorithm (SOA)⁴⁴, hybrid sine cosine algorithm (HSCA)⁴⁵, enhanced salp swarm algorithm (ESSA)⁴⁶, augmented grey wolf optimizer (AGWO)⁴⁷ which makes it recommended for solving complex optimization problems. As a result, we are going to introduce the chaotic-TSO optimization algorithm to get better near-optimal solutions to the single objective optimization problems by applying eight different scenarios for each chaotic map. Further, recently, authors in⁴⁸ proposed a chaotic TSO algorithm by replacing one of the random numbers included in the TSO algorithm, and it has been tested using IEEE CEC’ 17 test functions⁴⁹ and also tested in solving multiple engineering optimization problems⁴⁸. Thus, further enhancement in the initialization process via CMs is considered in this study to check the benefits of using chaos.

The main contributions of this work are illustrated as follows:

1. Search variables’ randomness are guided towards finding near-optimal solutions for single objective optimization problems using chaos generators⁴³ called chaotic maps. These chaotic maps can trace the near-optimal solutions randomly without getting trapped in local optimum solutions⁴³.

2. Nine chaotic maps are implemented in the TSO algorithm using different proposed scenarios to get our hands on its benefits and analyze the obtained solutions using convergence curves, the Wilcoxon p-value test and the non-parametric sign test.

3. Three real-life engineering constrained optimization problems were solved using the proposed CTSO, in which it has succeeded in solving these optimization problems efficiently compared to recent optimization algorithms.

4. The proposed scenarios succeeded in fetching a wider range of solutions better than that proposed in⁴⁸ as the search procedure includes enhancement in the initialization using CMs.

The remaining sections of the manuscript are organized as follows: “Transient search optimization” introduces the transients search algorithm; “Chaotic maps” illustrates the nine different chaotic maps formulas used for improvements; “Chaotic transient search optimization” sets the eight different scenarios for random parameters selections of the combined chaotic transient search optimization; “Results and discussion” provides the obtained results and comparisons of different scenarios with/without considering initialization enhancement as well as the real-life engineering design problems under study; and “Research outcomes and conclusions” presents the conclusions.

Transient search optimization

Transient search optimization (TSO)²⁸ was recently proposed in 2020 to optimally find the near-utmost solution to optimization problems, including single objective function. TSO inspired by the transient response of electrical storage devices, including inductors and capacitors. The TSO algorithm optimization cycle passes through three phases, including the initialization of variables, the exploration, and the exploitation phases. In the initial phase, ‘Initialization of variables,’ the algorithm randomly initializes the search variables bounded by the optimization problem variables’ lower and upper bounds as illustrated in Eq. (1). After the initialization of variables has been done, either the exploitation or the exploration phases take place with equal probability using the random variable (). In the ‘exploitation’ phase, the algorithm searches for the best solutions via applying the oscillatory equations describing the second-order RLC electrical circuits. These equations are given in Eq. (2) at . Furthermore, the ‘exploitation’ phase follows the decaying response of exponential functions like the discharge of the first-order RL or RC electrical circuits. The exploitation phase equations are given in Eq. (2) at .

1

2

3

4

5

where and are the iteration number and its preset maximum value, respectively. represents the ith search agent vector, where includes the optimizer’s search variables, and these variables are bounded by lower () and upper () bounds. The initialization of search variables is generated by variating uniformly between 0 and 1. , , and are also random number generators like . Also, the variable () gives values from 2 to 0 as indicated in Eq. (5), the constant () in Eq. (4) takes positive integer values. Besides, the coefficients and are constants, taking random values, where . The sign of differentiates both the exploitation and exploration phases, where the exploitation phase takes place when is greater than zero; otherwise, the exploration phase will take place. The pseudo-code of the TSO algorithm is provided in Algorithm I.

Algorithm I: Original TSO algorithm

Chaotic maps

Chaotic maps are random number generators characterized by their non-ergodic nature. Hence, they are implemented in optimization algorithms to fetch near-optimal solutions generally better than the stochastic searches⁴³. Also, chaos unpredictability has enabled more search capabilities, as it escapes from getting trapped in local optimal solutions. In this work, nine well-known chaotic maps⁴³ are employed to improve the search capabilities of the TSO algorithm. These chaotic maps are given in Table 1.

Table 1.

Chaotic map iterative formulation.

Chaotic map name	Formulation
Chebyshev
Circle
Gauss/mouse
Iterative
Logistic
Piecewise
Sine
Sinusoidal
Tent

Chaotic transient search optimization

The chaotic maps (CMs) have been employed in many optimization problems aiming to regulate the randomness of the search space vectors^31–35. In this research, the employment of chaotic maps has enabled new rooms for improving the quality of the search process, which describes the novelty of this research in finding the near-utmost solutions to single objective optimization problems. Besides, it has been employed in many research articles^31–35 to obtain better near-global solutions to many optimization problems. In this article, eight scenarios were conducted for each CM as follows:

• Scenario 1 (S1): organizing the random initialization number () using CM.

• Scenario 2 (S2): organizing the random number () using CM.

• Scenario 3 (S3)⁴⁸: organizing the random number () using CM.

• Scenario 4 (S4): organizing the random number () using CM.

• Scenario 5 (S5): organizing and using CM.

• Scenario 6 (S6): organizing and using CM.

• Scenario 7 (S7): organizing and using CM.

• Scenario 8 (S8): organizing , and using CM.

Furthermore, the third scenario (S3) is the same scenario proposed by the authors in⁴⁸ for improving TSO using CMs. In this regard, we are going to introduce the rest of scenarios to maximize the benefits from using CMs. The complexity of the CTSO can be represented by the big-oh notation in which the initialization process is denoted by O, where is the population size. Further, the search agents loop in the while with a number of iterations equals to , thus, the number of search agents’ computations in the while loop is denoted by O (). Finally, the process of updating the -dimensional search agent’s components is donated by O ().

Results and discussion

This article proposes a multi-scenario analysis to obtain better near-utmost solutions for the proposed chaotic-based TSO algorithm. Two major case studies are employed in “Results of different scenarios without considering initialization enhancement” and “Results of different scenarios considering initialization enhancement” to test the tangible benefits of employing chaotic maps as a randomness regulator using the proposed nine scenarios. These case studies test the proposed CTSO using 23 benchmark test functions²⁸ to determine the benefits of applying the nine scenarios. The test functions are formulated in Table 2, including uni-modal, multi-modal, and fixed-dimension multi-modal test functions. Besides, the Wilcoxon sign rank sum test is employed for the proposed case studies to get the results’ significance against the original TSO algorithm. Further, in “Testing CTSO on classical engineering applications”, real engineering optimization problems²⁸ are solved using the multiple scenarios of the CTSO to get an insight into its validity in solving real-life problems and hence compared to recently proposed metaheuristic optimization algorithms. Results were conducted 30 times to ensure the validity of the outcomes. All computations were executed on a laptop named “Legion 5”, manufactured by Lenovo^®, with 16 GB DDR4 RAMs and a Ryzen 7 processor (4800 H) running at 2.9 GHz.

Table 2.

Test functions.

Objective function	Description	n	Variables’ limits	Minimum objective
Uni-modal test functions
f1		30	[− 100, 100]	0
f2		30	[− 10, 10]	0
f3		30	[− 100, 100]	0
f4		30	[− 100, 100]	0
f5		30	[− 30, 30]	0
f6		30	[− 100, 100]	0
f7		30	[− 1.28, 1.28]	0
Multi-modal test functions
f8		30	[− 500, 500]	− 418.9829 × D
f9		30	[− 5.12, 5.12]	0
f10		30	[− 32, 32]	0
f11		30	[− 600, 600]	0
f12		30	[− 50, 50]	0
f13		30	[− 50, 50]	0
Fixed-dimension multi-modal test functions
f14		2	[− 65, 65]	1
f15		4	[− 5, 5]	0.00030
f16		2	[− 5, 5]	−1.0316
f17		2	[− 5, 5]	0.398
f18		2	[− 2, 2]	3
f19		3	[1, 3]	−3.86
f20		6	[0, 1]	−3.32
f21		4	[0, 10]	−10.1532
f22		4	[0, 10]	−10.4028
f23		4	[0, 10]	−10.5363

Results of different scenarios without considering initialization enhancement

In the current case study, the nine scenarios were employed using CMs in scenarios 2 to 8, without updating the initialization variable . The obtained results using different scenarios for the Chebyshev chaotic map is given in Table 3. From the obtained results, we can note that TSO is enormously improved using CMs using different scenarios. Also, scenarios S2, S3⁴⁸, and S4 succeeded in obtaining five near-optimal solutions better than TSO and those obtained in the rest of the scenarios. Furthermore, applying the rest of the CMs for multiple scenarios has proven their ability to find better near-optimal solutions for the test functions. Table 4 provides the results using different scenarios for the Gauss-mouse chaotic map, which has proven its ability to get near-optimal solutions better than the original TSO algorithm for the 23 benchmark functions, where the comparison is conducted based on the average value (µ) of the best near-optimal solutions obtained in the 30 runs. The rest of the chaotic maps did not obtain the best near-optimal solution for all the benchmark functions; for instance, in Table 3, µ is better in the function using TSO rather than the rest of the proposed scenarios using the Chebyshev chaotic map. Besides, the convergence curves for both the Chebyshev and the Gauss-mouse CMs are shown in Figs. 1 and 2, respectively, where CMs’ ability to regulate randomness successfully speeded the convergence curves towards finding the near-optimal solutions in shorter timing. Further, the Wilcoxon sign rank test is conducted at a 5% significance level to assess the significance of the CTSO solutions against the original TSO, as shown in Tables 5 and 6 for the Chebyshev and the Gauss-mouse CMs, respectively. Also, the number of significantly better solutions is denoted by as shown in Tables 5 and 6. It is obvious from Tables 5 and 6 that the proposed CTSO scenarios succeeded in boosting the performance of TSO in finding better near-optimal solutions to the benchmark functions.

Table 3.

Fitness values at different scenarios using Chebyshev chaotic map without considering initialization enhancement.

Function	Index	TSO	Scenario
			S1	S2	S3	S4	S5	S6	S7	S8
	µ *	1.4831E − 18	3.3539E − 20	1.0091E − 13	1.2980E − 20	4.9133E − 19	5.5809E − 11	1.6226E − 09	1.2733E − 12	6.4996E − 09
	σ **	7.3029E − 18	1.4679E − 19	4.1565E − 13	6.0912E − 20	2.6392E − 18	2.9499E − 10	8.8743E − 09	3.6133E − 12	3.0788E − 08
	Best	1.1361E − 39	2.6501E − 36	7.8136E − 39	3.4940E − 52	3.9598E − 43	4.4391E − 82	2.4611E − 58	2.0961E − 30	2.9797E − 87
	Worst	3.9924E − 17	7.9378E − 19	2.1343E − 12	3.3208E − 19	1.4463E − 17	1.6168E − 09	4.8609E − 08	1.6661E − 11	1.6828E − 07
	Time (s)	0.02804	0.02890	0.03480	0.03453	0.03505	0.06303	0.06033	0.06152	0.08119
	µ	2.7894E − 11	1.5159E − 11	1.3964E − 08	1.9124E − 11	1.0723E − 11	1.0824E − 07	1.2422E − 06	2.5177E − 07	7.9452E − 06
	σ	1.3710E − 10	4.5055E − 11	4.7446E − 08	5.2071E − 11	2.8836E − 11	3.7137E − 07	4.8212E − 06	7.9651E − 07	1.9884E − 05
	Best	2.6771E − 29	4.7385E − 33	1.5069E − 26	9.5310E − 24	1.8138E − 22	3.1580E − 27	2.5162E − 27	3.0704E − 16	4.9347E − 48
	Worst	7.5129E − 10	2.2005E − 10	2.4796E − 07	2.0698E − 10	1.3525E − 10	1.8198E − 06	2.6332E − 05	4.2261E − 06	9.3780E − 05
	Time (s)	0.02806	0.02817	0.03680	0.03646	0.03694	0.06375	0.06251	0.06361	0.08284
	µ	8.6013E − 06	8.2742E − 06	1.0184E − 01	1.2392E − 07	9.1329E − 07	2.2193E − 01	6.4662E + 02	2.2782	2.9137
	σ	4.6824E − 05	4.2293E − 05	5.4209E − 01	3.5083E − 07	3.2695E − 06	7.4846E − 01	3.4563E + 03	8.4582	1.4779E + 01
	Best	1.0610E − 19	4.7714E − 17	7.1876E − 11	5.6978E − 30	3.8851E − 17	5.8950E − 11	6.6964E − 32	2.8956E − 06	9.6546E − 18
	Worst	2.5652E − 04	2.3203E − 04	2.9712	1.4473E − 06	1.6166E − 05	3.9510	1.8945E + 04	4.5147E + 01	8.1122E + 01
	Time (s)	0.14590	0.14563	0.15235	0.15355	0.15225	0.18083	0.18089	0.18145	0.20139
	µ	2.0065E − 10	1.3396E − 11	1.4411E − 09	1.1011E − 10	1.9956E − 10	1.8560E − 06	5.3112E − 05	2.8116E − 06	3.6472E − 06
	σ	7.8452E − 10	4.1036E − 11	3.6661E − 09	3.5928E − 10	5.3772E − 10	9.9766E − 06	2.7716E − 04	8.3776E − 06	8.8530E − 06
	Best	7.6044E − 24	1.1422E − 26	1.6149E − 34	2.4483E − 24	5.1871E − 27	1.2686E − 30	2.9154E − 26	5.7408E − 15	3.6844E − 15
	Worst	4.1793E − 09	1.8619E − 10	1.7286E − 08	1.7366E − 09	2.0493E − 09	5.4677E − 05	1.5195E − 03	4.1399E − 05	3.9737E − 05
	Time (s)	0.02604	0.02628	0.03476	0.03505	0.03481	0.06168	0.06065	0.06253	0.08112
	µ	8.1293E − 02	7.7371E − 02	2.1492	8.3348E − 02	1.8829E − 01	1.5314E − 01	2.1214E + 01	2.0121E − 01	1.5947E − 01
	σ	1.3662E − 01	1.0379E − 01	7.2350	1.4851E − 01	3.3033E − 01	2.4889E − 01	1.2717E + 01	5.3704E − 01	3.6634E − 01
	Best	6.9082E − 06	3.3523E − 05	9.1412E − 05	7.1246E − 06	1.1585E − 06	5.8146E − 07	2.5501E − 03	1.0045E − 04	2.2098E − 04
	Worst	5.6436E − 01	3.8155E − 01	2.8707E + 01	7.1236E − 01	1.5385	1.0933	2.8885E + 01	2.9457	1.8275
	Time (s)	0.03945	0.03971	0.04803	0.04867	0.04810	0.07497	0.07394	0.07568	0.09462
	µ	5.5450E − 03	3.2501E − 03	8.2928E − 03	8.6967E − 03	6.1901E − 03	6.0983E − 03	2.0219	1.2381E − 02	3.2757E − 03
	σ	6.5564E − 03	3.1379E − 03	1.1281E − 02	1.1381E − 02	7.3577E − 03	7.5362E − 03	1.4550	1.6971E − 02	6.1148E − 03
	Best	6.2663E − 06	1.5910E − 06	3.3479E − 05	3.1005E − 05	9.4362E − 09	1.9363E − 06	1.1070E − 01	7.1677E − 05	6.8303E − 06
	Worst	2.3154E − 02	1.1852E − 02	4.5959E − 02	5.0600E − 02	2.5933E − 02	3.3979E − 02	4.9712	6.5883E − 02	2.5100E − 02
	Time (s)	0.02658	0.02658	0.03501	0.03558	0.03517	0.06209	0.06080	0.06251	0.08118
	µ	4.6201E − 04	6.4366E − 04	3.0045E − 04	3.7426E − 04	4.8515E − 04	2.6385E − 04	4.3189E − 04	4.2697E − 04	1.0004E − 03
	σ	5.4612E − 04	8.0613E − 04	3.5611E − 04	4.0088E − 04	3.9097E − 04	3.0499E − 04	4.1434E − 04	5.3888E − 04	7.7556E − 04
	Best	1.3910E − 05	3.0982E − 05	6.4828E − 06	5.1216E − 06	1.3858E − 05	2.1749E − 05	3.6906E − 06	3.1843E − 05	6.5358E − 05
	Worst	2.9330E − 03	3.8692E − 03	1.4965E − 03	1.8208E − 03	1.4171E − 03	1.6583E − 03	1.4869E − 03	2.5361E − 03	3.0181E − 03
	Time (s)	0.07857	0.07840	0.08674	0.08722	0.08713	0.11481	0.11273	0.11462	0.13329
	µ	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04
	σ	7.8182E − 03	5.8390E − 03	1.5236E − 03	1.6893E − 03	1.3133E − 03	1.5670E − 02	1.9141E − 02	8.0028E − 03	3.7854E − 02
	Best	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04
	Worst	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04
	Time (s)	0.03795	0.03815	0.04638	0.04702	0.04696	0.07387	0.07272	0.07443	0.09292
	µ	0	0	1.2127E − 13	0	0	1.2788E − 11	1.2987E − 06	3.1264E − 13	1.3318E − 08
	σ	0	0	5.5790E − 13	0	0	6.8006E − 11	7.0092E − 06	6.5338E − 13	6.9345E − 08
	Best	0	0	0	0	0	0	0	0	0
	Worst	0	0	3.0127E − 12	0	0	3.7272E − 10	3.8406E − 05	3.0127E − 12	3.8033E − 07
	Time (s)	0.03404	0.03415	0.04246	0.04289	0.04291	0.06949	0.06823	0.07004	0.08994
	µ	1.3528E − 12	2.1346E − 13	3.5761E − 10	2.4187E − 12	3.3166E − 11	4.5905E − 10	2.3040E − 06	6.0262E − 10	2.7671E − 06
	σ	6.4305E − 12	7.0556E − 13	1.2690E − 09	7.9006E − 12	1.6723E − 10	1.3556E − 09	1.2595E − 05	1.5297E − 09	6.7283E − 06
	Best	8.8818E − 16	8.8818E − 16	8.8818E − 16	8.8818E − 16	8.8818E − 16	8.8818E − 16	8.8818E − 16	8.8818E − 16	8.8818E − 16
	Worst	3.5247E − 11	3.5465E − 12	6.7572E − 09	4.0225E − 11	9.1748E − 10	6.1934E − 09	6.8991E − 05	7.7222E − 09	2.9034E − 05
	Time (s)	0.03387	0.03416	0.04226	0.04268	0.04272	0.06933	0.06833	0.06988	0.08962
	µ	0	0	3.0683E − 14	3.7007E − 18	3.7007E − 18	2.3754E − 12	1.0749E − 07	2.1677E − 11	5.7720E − 09
	σ	0	0	1.5968E − 13	2.0270E − 17	2.0270E − 17	1.2821E − 11	5.0283E − 07	8.4647E − 11	2.4834E − 08
	Best	0	0	0	0	0	0	0	0	0
	Worst	0	0	8.7574E − 13	1.1102E − 16	1.1102E − 16	7.0254E − 11	2.7258E − 06	4.5783E − 10	1.3461E − 07
	Time (s)	0.04452	0.04425	0.05251	0.05305	0.05319	0.07952	0.07945	0.08046	0.10044
	µ	2.4695E − 04	4.2611E − 04	6.5318E − 04	3.7193E − 04	5.2911E − 04	6.6906E − 04	1.8198E − 01	8.3614E − 04	2.0896E − 04
	σ	6.0608E − 04	7.8788E − 04	9.4194E − 04	6.9691E − 04	9.7779E − 04	8.4183E − 04	2.4535E − 01	9.6817E − 04	2.7395E − 04
	Best	6.5739E − 07	3.5617E − 07	3.6965E − 06	4.1774E − 07	1.8861E − 06	7.8688E − 08	2.2026E − 07	1.0567E − 05	1.0607E − 07
	Worst	3.3284E − 03	4.0127E − 03	3.6038E − 03	2.9234E − 03	5.0731E − 03	3.5591E − 03	8.6772E − 01	3.3721E − 03	1.1061E − 03
	Time (s)	0.17328	0.17046	0.17856	0.17913	0.17962	0.20604	0.20620	0.20768	0.22583
	µ	2.5400E − 03	5.2434E − 03	3.7164E − 03	2.8645E − 03	6.6112E − 04	6.8892E − 03	3.2533E − 01	6.0121E − 03	3.3341E − 03
	σ	4.2391E − 03	1.4643E − 02	4.7554E − 03	3.3740E − 03	9.2969E − 04	1.2526E − 02	4.9977E − 01	1.3760E − 02	3.8690E − 03
	Best	2.2291E − 06	7.6213E − 07	9.6217E − 07	2.3069E − 07	9.2633E − 07	5.0997E − 09	1.4435E − 04	1.1140E − 05	1.1955E − 06
	Worst	1.9084E − 02	7.8457E − 02	1.5460E − 02	9.6921E − 03	4.1702E − 03	4.8186E − 02	2.2031	6.9461E − 02	1.1705E − 02
	Time (s)	0.16966	0.16852	0.17759	0.17833	0.17812	0.20463	0.20425	0.20589	0.22361
	µ	1.4616	1.7541	1.4945	1.2952	1.4659	2.3466	2.9970	1.8179	9.9800E − 01
	σ	8.1265E − 01	1.8469	8.1301E − 01	9.4007E − 01	1.0593	2.5741	3.0938	2.1071	4.0428E − 08
	Best	9.9800E − 01	9.9800E − 01	9.9800E − 01	9.9800E − 01	9.9800E − 01	9.9800E − 01	9.9800E − 01	9.9800E − 01	9.9800E − 01
	Worst	2.9826	1.0763E + 01	3.9683	5.9288	5.9295	1.0763E + 01	1.2671E + 01	1.0763E + 01	9.9800E − 01
	Time (s)	0.32084	0.32042	0.32641	0.33211	0.33224	0.35491	0.35521	0.35674	0.37328
	µ	6.9593E − 04	7.6777E − 04	6.8271E − 04	5.6731E − 04	9.5723E − 04	9.1526E − 04	8.1904E − 04	9.2952E − 04	1.0348E − 03
	σ	6.5262E − 04	6.9688E − 04	5.2777E − 04	5.3112E − 04	8.8957E − 04	1.3814E − 03	7.3165E − 04	8.2749E − 04	7.4854E − 04
	Best	3.0968E − 04	3.0865E − 04	3.1671E − 04	3.0855E − 04	3.1443E − 04	3.0990E − 04	3.2113E − 04	3.1042E − 04	3.2504E − 04
	Worst	2.2748E − 03	2.2778E − 03	2.2556E − 03	2.3204E − 03	3.6551E − 03	7.4531E − 03	3.3327E − 03	2.7974E − 03	2.2567E − 03
	Time (s)	0.02653	0.02579	0.03390	0.03456	0.03436	0.06047	0.06046	0.06123	0.07997
	µ	− 1.0316	− 1.0316	− 1.0316	− 1.0316	− 1.0316	− 9.6391E − 01	− 1.0101	− 1.0316	− 1.0316
	σ	1.0912E − 04	7.5680E − 05	1.0095E − 04	1.2130E − 04	6.9424E − 05	2.5686E − 01	8.1698E − 02	7.3773E − 05	1.4160E − 04
	Best	− 1.0316	− 1.0316	− 1.0316	− 1.0316	− 1.0316	− 1.0316	− 1.0316	− 1.0316	− 1.0316
	Worst	− 1.0310	− 1.0313	− 1.0311	− 1.0310	− 1.0313	4.8985E − 66	− 6.2327E − 01	− 1.0313	− 1.0309
	Time (s)	0.02562	0.02546	0.03363	0.03418	0.03416	0.06022	0.05987	0.06029	0.07940
	µ	5.5335E − 01	5.5314E − 01	5.5493E − 01	5.5416E − 01	5.5401E − 01	5.7490E − 01	3.9845E − 01	5.5404E − 01	3.9877E − 01
	σ	8.4741E − 01	8.4745E − 01	8.4712E − 01	8.4727E − 01	8.4729E − 01	8.4588E − 01	1.0530E − 03	8.4729E − 01	1.3164E − 03
	Best	3.9789E − 01	3.9789E − 01	3.9789E − 01	3.9789E − 01	3.9789E − 01	3.9789E − 01	3.9789E − 01	3.9789E − 01	3.9789E − 01
	Worst	5.0401	5.0401	5.0401	5.0401	5.0401	5.0401	4.0263E − 01	5.0401	4.0288E − 01
	Time (s)	0.02173	0.02156	0.02954	0.03009	0.02995	0.05652	0.05557	0.05596	0.07538
	µ	3.0042	5.8374	6.6190	4.0306	6.7561	5.6225	9.7565	3.9045	7.6797
	σ	1.0714E − 02	8.6533	9.3542	5.6277	9.7409	7.9686	1.2234E + 01	4.9304	1.0654E + 01
	Best	3.0000	3.0000	3.0000	3.0000	3.0000	3.0001	3.0000	3.0000	3.0000
	Worst	3.0535	3.2703E + 01	3.0190E + 01	3.3827E + 01	3.3019E + 01	3.0037E + 01	3.4542E + 01	3.0009E + 01	3.2672E + 01
	Time (s)	0.02072	0.02064	0.02883	0.02934	0.02921	0.05520	0.05494	0.05530	0.07461
	µ	− 3.8084	− 3.8044	− 3.8166	− 3.7902	− 3.8082	− 3.8143	− 3.8104	− 3.7932	− 3.8073
	σ	6.8103E − 02	5.9562E − 02	4.5471E − 02	7.0799E − 02	5.3767E − 02	6.5085E − 02	5.9559E − 02	7.4087E − 02	6.8560E − 02
	Best	− 3.8627	− 3.8608	− 3.8624	− 3.8628	− 3.8628	− 3.8628	− 3.8620	− 3.8623	− 3.8626
	Worst	− 3.6046	− 3.6041	− 3.7258	− 3.6065	− 3.6620	− 3.6055	− 3.6352	− 3.6047	− 3.6322
	Time (s)	0.02961	0.02973	0.03764	0.03834	0.03815	0.06479	0.06470	0.06454	0.08443
	µ	− 2.9224	− 2.9839	− 2.8099	− 2.9389	− 2.9923	− 2.9877	− 2.8548	− 2.9652	− 2.9643
	σ	2.6517E − 01	2.8273E − 01	4.3109E − 01	2.9959E − 01	2.2176E − 01	1.8957E − 01	4.5616E − 01	1.7862E − 01	2.5623E − 01
	Best	− 3.2435	− 3.2461	− 3.1869	− 3.2981	− 3.2890	− 3.2589	− 3.3072	− 3.2242	− 3.2867
	Worst	− 1.9663	− 1.6288	− 9.5747E − 01	− 1.9553	− 2.0426	− 2.3007	− 1.4569	− 2.4486	− 1.9062
	Time (s)	0.03050	0.03041	0.03882	0.03883	0.03903	0.06559	0.06542	0.06524	0.08547
	µ	− 7.4682	− 7.7691	− 6.4328	− 8.0406	− 7.8321	− 5.2451	− 5.4336	− 7.1508	− 7.8180
	σ	4.3230	4.1355	4.6854	3.8909	4.1466	4.4928	4.4738	4.4888	4.1503
	Best	− 1.0152E + 01	− 1.0153E + 01	− 1.0153E + 01	− 1.0153E + 01	− 1.0153E + 01	− 1.0146E + 01	− 1.0145E + 01	− 1.0153E + 01	− 1.0153E + 01
	Worst	− 2.7312E − 01	− 2.7312E − 01	− 2.7312E − 01	− 2.7312E − 01	− 2.7312E − 01	− 2.7312E − 01	− 3.5065E − 01	− 2.7312E − 01	− 2.7312E − 01
	Time (s)	0.03468	0.03448	0.04301	0.04306	0.04319	0.06990	0.06979	0.06991	0.09015
	µ	− 9.4833	− 9.1597	− 9.2862	− 9.5941	− 1.0155E + 01	− 8.6797	− 7.2611	− 9.3789	− 9.8015
	σ	2.4665	2.9775	2.4864	2.3913	3.6674E − 01	3.3667	4.1751	2.9195	1.7945
	Best	− 1.0403E + 01	− 1.0403E + 01	− 1.0402E + 01	− 1.0403E + 01	− 1.0403E + 01	− 1.0403E + 01	− 1.0403E + 01	− 1.0403E + 01	− 1.0402E + 01
	Worst	− 5.2051E − 01	− 3.7353E − 01	− 8.3838E − 01	− 8.1161E − 01	− 8.9454	− 3.7353E − 01	− 3.7244E − 01	− 5.2404E − 01	− 5.2392E − 01
	Time (s)	0.03986	0.03976	0.04836	0.05192	0.04854	0.07561	0.07523	0.07547	0.09584
	µ	− 1.0386E + 01	− 1.0039E + 01	− 9.4215	− 1.0436E + 01	− 9.6342	− 8.7929	− 9.1383	− 1.0189E + 01	− 1.0428E + 01
	σ	3.6773E − 01	1.7525	2.5996	1.4784E − 01	2.4654	3.4772	2.6777	1.3131	1.7425E − 01
	Best	− 1.0536E + 01	− 1.0536E + 01	− 1.0536E + 01	− 1.0536E + 01	− 1.0536E + 01	− 1.0536E + 01	− 1.0536E + 01	− 1.0536E + 01	− 1.0536E + 01
	Worst	− 8.6649	− 9.4888E − 01	− 9.1989E − 01	− 9.8381	− 5.5674E − 01	− 5.5312E − 01	− 7.6222E − 01	− 3.2918	− 9.7661
	Time (s)	0.04637	0.04609	0.05459	0.05492	0.05478	0.08195	0.08162	0.08200	0.10293

^*µ is the mean of the best near-optimal solutions among the thirty runs.

^**σ is the standard deviation of the best near-optimal solutions among the thirty runs.

Table 4.

Fitness values at different scenarios using Gauss-mouse chaotic map without considering initialization enhancement.

Function	Index	TSO	Scenario
			S1	S2	S3	S4	S5	S6	S7	S8
	µ	1.4831E − 18	1.1398E − 18	3.9363E − 13	1.2636E − 18	1.5863E − 20	7.8729E − 18	2.8592E − 14	1.2676E − 19	1.9049E − 16
	σ	7.3029E − 18	6.0001E − 18	2.0953E − 12	6.5644E − 18	8.4798E − 20	2.8782E − 17	1.1519E − 13	3.9735E − 19	7.3983E − 16
	best	1.1361E − 39	1.0654E − 43	2.2954E − 52	1.0837E − 53	7.0893E − 48	5.8240E − 36	1.4262E − 42	4.4767E − 48	6.2765E − 50
	worst	3.9924E − 17	3.2896E − 17	1.1486E − 11	3.5974E − 17	4.6478E − 19	1.5620E − 16	5.7286E − 13	1.9287E − 18	3.4942E − 15
	time (s)	0.02804	0.02601	0.03483	0.03514	0.03512	0.06210	0.06201	0.06213	0.08085
	µ	2.7894E − 11	4.5667E − 11	1.0446E − 06	1.2794E − 11	5.0077E − 11	3.5977E − 10	1.7314E − 08	3.4646E − 11	7.8838E − 10
	σ	1.3710E − 10	2.2462E − 10	5.4192E − 06	3.8475E − 11	1.7083E − 10	1.2067E − 09	7.8097E − 08	1.2125E − 10	3.3787E − 09
	best	2.6771E − 29	7.9517E − 22	1.5923E − 24	8.0138E − 23	7.2538E − 24	8.1446E − 21	4.6891E − 26	1.2407E − 25	6.8355E − 28
	worst	7.5129E − 10	1.2335E − 09	2.9723E − 05	1.8780E − 10	8.8178E − 10	6.5378E − 09	4.2691E − 07	6.4503E − 10	1.8302E − 08
	time (s)	0.02806	0.02795	0.03693	0.03708	0.03724	0.06388	0.06372	0.06476	0.08337
	µ	8.6013E − 06	3.6653E − 08	1.9175E − 01	6.2840E − 07	5.7027E − 06	1.4907E − 04	3.8395E − 03	1.8555E − 08	2.8638E − 06
	σ	4.6824E − 05	1.0747E − 07	1.0485	2.2374E − 06	2.8211E − 05	7.6945E − 04	1.6390E − 02	7.8999E − 08	1.0028E − 05
	best	1.0610E − 19	7.3842E − 17	5.4954E − 25	7.1183E − 26	5.3118E − 16	1.3393E − 31	8.9241E − 29	3.2035E − 19	7.4005E − 29
	worst	2.5652E − 04	5.5061E − 07	5.7431	1.1064E − 05	1.5482E − 04	4.2198E − 03	9.0047E − 02	4.3411E − 07	5.2831E − 05
	time (s)	0.14590	0.14481	0.15380	0.15412	0.15213	0.18088	0.18051	0.18182	0.20082
	µ	2.0065E − 10	2.9398E − 11	2.7520E − 08	1.9060E − 10	1.6082E − 10	1.5517E − 09	4.9432E − 08	8.2036E − 11	2.7800E − 10
	σ	7.8452E − 10	1.2303E − 10	8.9614E − 08	5.2391E − 10	4.5387E − 10	5.3936E − 09	1.1491E − 07	2.9396E − 10	8.4456E − 10
	best	7.6044E − 24	1.6973E − 24	7.7171E − 29	1.4278E − 21	6.7776E − 27	8.4526E − 25	2.2258E − 17	1.9285E − 27	1.2022E − 23
	worst	4.1793E − 09	6.5380E − 10	4.0771E − 07	2.4715E − 09	2.2158E − 09	2.8820E − 08	4.4293E − 07	1.5145E − 09	3.5337E − 09
	time (s)	0.02604	0.02618	0.03507	0.03521	0.03505	0.06217	0.06198	0.06240	0.08127
	µ	8.1293E − 02	1.6196E − 01	5.5723E − 01	5.8199E − 02	1.7902E − 01	1.7643E − 01	2.3295E − 02	1.1998E − 01	6.6454E − 02
	σ	1.3662E − 01	2.8968E − 01	1.4959	7.2016E − 02	4.5368E − 01	2.8505E − 01	4.4499E − 02	2.1895E − 01	1.2155E − 01
	best	6.9082E − 06	5.8560E − 06	6.3328E − 05	1.1000E − 07	4.9524E − 06	7.1798E − 06	8.9310E − 07	4.6997E − 06	1.3974E − 05
	worst	5.6436E − 01	1.1609	8.0538	2.6174E − 01	2.3508	1.0050	2.1539E − 01	7.8182E − 01	5.2536E − 01
	time (s)	0.03945	0.03963	0.04846	0.04887	0.04865	0.07576	0.07577	0.07590	0.09498
	µ	5.5450E − 03	3.0375E − 03	1.9901E − 02	3.2649E − 03	7.6813E − 03	6.4802E − 03	7.6754E − 03	7.5960E − 03	6.6856E − 03
	σ	6.5564E − 03	5.4593E − 03	2.5348E − 02	4.3281E − 03	1.3772E − 02	9.6251E − 03	8.1436E − 03	1.1126E − 02	1.0021E − 02
	best	6.2663E − 06	6.4550E − 06	1.3836E − 04	1.2577E − 05	6.5985E − 05	3.0099E − 05	3.7125E − 05	1.9401E − 08	5.4526E − 05
	worst	2.3154E − 02	2.1828E − 02	9.7885E − 02	1.5847E − 02	6.7458E − 02	4.1252E − 02	2.8485E − 02	4.4987E − 02	4.7376E − 02
	time (s)	0.02658	0.02641	0.03529	0.03555	0.03571	0.06248	0.06226	0.06316	0.08103
	µ	4.6201E − 04	5.7293E − 04	3.4237E − 04	3.6601E − 04	5.4328E − 04	5.3415E − 04	7.3431E − 04	4.1307E − 04	5.4689E − 04
	σ	5.4612E − 04	8.6813E − 04	2.6206E − 04	3.4945E − 04	4.2428E − 04	7.1959E − 04	8.1757E − 04	3.2218E − 04	4.8031E − 04
	best	1.3910E − 05	1.0934E − 05	4.8262E − 05	3.7204E − 05	1.6874E − 05	1.5882E − 05	2.2683E − 05	3.3565E − 05	4.9010E − 05
	worst	2.9330E − 03	4.8483E − 03	1.1795E − 03	1.7499E − 03	1.3760E − 03	3.8277E − 03	3.4237E − 03	1.1179E − 03	2.0435E − 03
	time (s)	0.07857	0.07828	0.08708	0.08735	0.08746	0.11434	0.11447	0.11511	0.13318
	µ	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04
	σ	7.8182E − 03	9.2162E − 04	1.7161E − 02	4.8335E − 02	3.0679E − 03	9.1171E − 04	3.2970E − 03	6.4818E − 03	6.3155E − 03
	best	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04
	worst	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04
	time (s)	0.03795	0.03809	0.04678	0.04709	0.04724	0.07417	0.07388	0.07476	0.09289
	µ	0	0	3.0316E − 14	0	0	0	9.0002E − 13	0	0
	σ	0	0	1.6605E − 13	0	0	0	3.8436E − 12	0	0
	best	0	0	0	0	0	0	0	0	0
	worst	0	0	9.0949E − 13	0	0	0	2.0350E − 11	0	0
	time (s)	0.03404	0.03400	0.04269	0.04286	0.04307	0.07127	0.07050	0.07069	0.08964
	µ	1.3528E − 12	5.8553E − 12	1.2997E − 09	1.1533E − 11	9.3478E − 13	5.4011E − 12	6.7008E − 09	5.7238E − 12	1.6917E − 10
	σ	6.4305E − 12	2.1368E − 11	6.6697E − 09	4.7311E − 11	2.1735E − 12	2.7278E − 11	1.5845E − 08	1.8612E − 11	6.5102E − 10
	best	8.8818E − 16	8.8818E − 16	8.8818E − 16	8.8818E − 16	8.8818E − 16	8.8818E − 16	8.8818E − 16	8.8818E − 16	8.8818E − 16
	worst	3.5247E − 11	1.1669E − 10	3.6577E − 08	2.4202E − 10	8.9821E − 12	1.4974E − 10	6.5530E − 08	7.4352E − 11	3.0028E − 09
	time (s)	0.03387	0.03379	0.04263	0.04288	0.04290	0.07221	0.07031	0.07049	0.08908
	µ	0	0	5.3638E − 11	1.1102E − 17	1.4803E − 17	7.4015E − 18	3.9887E − 11	0	0
	σ	0	0	2.4901E − 10	6.0809E − 17	4.8203E − 17	2.8167E − 17	1.9598E − 10	0	0
	best	0	0	0	0	0	0	0	0	0
	worst	0	0	1.3565E − 09	3.3307E − 16	2.2204E − 16	1.1102E − 16	1.0745E − 09	0	0
	time (s)	0.04452	0.04422	0.05287	0.05326	0.05337	0.08108	0.08080	0.08111	0.10023
	µ	2.4695E − 04	3.1744E − 04	1.1633E − 03	4.3341E − 04	2.6308E − 04	4.0021E − 04	3.0627E − 04	3.7672E − 04	1.9995E − 04
	σ	6.0608E − 04	4.1904E − 04	2.2387E − 03	8.4409E − 04	4.5063E − 04	6.5920E − 04	3.8856E − 04	6.6630E − 04	2.8387E − 04
	best	6.5739E − 07	9.8344E − 08	1.8745E − 06	5.6377E − 07	8.7485E − 08	1.3933E − 07	1.5109E − 07	5.5052E − 10	3.3166E − 08
	worst	3.3284E − 03	2.1062E − 03	1.0745E − 02	3.9444E − 03	2.1946E − 03	2.7800E − 03	1.9498E − 03	2.9169E − 03	1.1858E − 03
	time (s)	0.17328	0.17063	0.17906	0.17971	0.17925	0.20755	0.20689	0.20868	0.22723
	µ	2.5400E − 03	3.5409E − 03	1.3996E − 02	3.0100E − 03	3.5517E − 03	1.9213E − 03	6.9636E − 05	1.2433E − 03	1.4439E − 03
	σ	4.2391E − 03	4.9616E − 03	4.0640E − 02	5.2101E − 03	6.4835E − 03	6.5762E − 03	1.1733E − 04	2.3643E − 03	2.7495E − 03
	best	2.2291E − 06	4.6261E − 07	2.0513E − 06	4.7635E − 07	6.0830E − 07	1.2847E − 07	2.8792E − 07	6.1964E − 06	1.7181E − 06
	worst	1.9084E − 02	2.1138E − 02	2.1578E − 01	2.4410E − 02	3.0286E − 02	3.6516E − 02	6.2872E − 04	1.1976E − 02	1.2636E − 02
	time (s)	0.16966	0.16854	0.17742	0.17819	0.17721	0.20535	0.20494	0.20648	0.22531
	µ	1.4616	1.6256	1.5262	1.6545	1.6913	1.1971	1.2618	1.4935	1.3943
	σ	8.1265E − 01	1.3369	1.1236	1.8585	1.3509	4.0499E − 01	9.3202E − 01	1.0297	1.0246
	best	9.9800E − 01	9.9800E − 01	9.9800E − 01	9.9800E − 01	9.9800E − 01	9.9800E − 01	9.9800E − 01	9.9800E − 01	9.9800E − 01
	worst	2.9826	5.9532	5.9288	1.0763E + 01	5.9288	2.0002	5.9288	5.9288	5.9289
	time (s)	0.32084	0.32418	0.32814	0.32929	0.32956	0.35671	0.35624	0.35775	0.37777
	µ	6.9593E − 04	5.4416E − 04	7.5531E − 04	7.1909E − 04	6.3230E − 04	8.2856E − 04	6.4127E − 04	6.0171E − 04	5.4610E − 04
	σ	6.5262E − 04	4.9639E − 04	7.1845E − 04	7.0851E − 04	5.9348E − 04	7.4895E − 04	6.0661E − 04	5.8269E − 04	3.8071E − 04
	best	3.0968E − 04	3.1219E − 04	3.0874E − 04	3.2505E − 04	3.0930E − 04	3.0774E − 04	3.0934E − 04	3.0787E − 04	3.0895E − 04
	worst	2.2748E − 03	2.2601E − 03	2.2876E − 03	2.2613E − 03	2.2664E − 03	2.2560E − 03	2.3258E − 03	2.2747E − 03	1.6710E − 03
	time (s)	0.02653	0.02573	0.03422	0.03460	0.03450	0.06131	0.06104	0.06032	0.08024
	µ	− 1.0316	− 1.0316	− 1.0315	− 1.0316	− 1.0316	− 1.0316	− 1.0313	− 1.0316	− 1.0315
	σ	1.0912E − 04	8.9825E − 05	1.6954E − 04	1.6983E − 05	5.3056E − 05	9.7942E − 05	3.6659E − 04	2.9046E − 05	3.6035E − 04
	best	− 1.0316	− 1.0316	− 1.0316	− 1.0316	− 1.0316	− 1.0316	− 1.0316	− 1.0316	− 1.0316
	worst	− 1.0310	− 1.0312	− 1.0307	− 1.0316	− 1.0313	− 1.0312	− 1.0302	− 1.0315	− 1.0301
	time (s)	0.02562	0.02547	0.03394	0.03438	0.03416	0.06071	0.06057	0.05984	0.07972
	µ	5.5335E − 01	5.5364E − 01	5.5356E − 01	5.5487E − 01	3.9948E − 01	7.0937E − 01	5.5485E − 01	3.9838E − 01	5.5314E − 01
	σ	8.4741E − 01	8.4736E − 01	8.4737E − 01	8.4718E − 01	4.5721E − 03	1.1772	8.4715E − 01	1.2826E − 03	8.4745E − 01
	best	3.9789E − 01	3.9789E − 01	3.9789E − 01	3.9789E − 01	3.9789E − 01	3.9789E − 01	3.9789E − 01	3.9789E − 01	3.9789E − 01
	worst	5.0401	5.0401	5.0401	5.0401	4.2174E − 01	5.0401	5.0401	4.0425E − 01	5.0401
	time (s)	0.02173	0.02116	0.02994	0.03014	0.03004	0.05647	0.05637	0.05562	0.07580
	µ	3.0042	6.6776	7.6908	6.7725	3.9672	4.8857	3.0046	3.0036	5.8952
	σ	1.0714E − 02	9.5295	1.0552E + 01	9.7816	5.2839	7.1634	6.3783E − 03	8.6337E − 03	8.8200
	best	3.0000	3.0000	3.0000	3.0000	3.0000	3.0000	3.0000	3.0000	3.0000
	worst	3.0535	3.0862E + 01	3.2796E + 01	3.2398E + 01	3.1944E + 01	3.2408E + 01	3.0305	3.0439	3.2765E + 01
	time (s)	0.02072	0.02090	0.02911	0.02941	0.02930	0.05558	0.05573	0.05500	0.07528
	µ	− 3.8084	− 3.7991	− 3.7950	− 3.8126	− 3.8060	− 3.7928	− 3.8042	− 3.8192	− 3.8017
	σ	6.8103E − 02	6.4894E − 02	6.0238E − 02	6.3315E − 02	6.5122E − 02	5.7549E − 02	5.1618E − 02	4.8513E − 02	6.0614E − 02
	best	− 3.8627	− 3.8627	− 3.8627	− 3.8626	− 3.8626	− 3.8617	− 3.8628	− 3.8626	− 3.8627
	worst	− 3.6046	− 3.6394	− 3.6162	− 3.6046	− 3.6352	− 3.6349	− 3.6633	− 3.6240	− 3.6050
	time (s)	0.02961	0.02958	0.03812	0.03851	0.03828	0.06510	0.06509	0.06438	0.08421
	µ	− 2.9224	− 2.9472	− 2.9083	− 2.9142	− 2.9250	− 2.9515	− 2.9637	− 3.0369	− 2.9536
	σ	2.6517E − 01	4.3663E − 01	2.6803E − 01	3.9355E − 01	2.5428E − 01	3.3264E − 01	2.2151E − 01	1.2920E − 01	1.8618E − 01
	best	− 3.2435	− 3.3009	− 3.2978	− 3.2696	− 3.2101	− 3.2825	− 3.1948	− 3.3002	− 3.2698
	worst	− 1.9663	− 8.4733E − 01	− 1.7882	− 1.3341	− 2.0323	− 1.3890	− 2.0790	− 2.6888	− 2.5777
	time (s)	0.03050	0.03023	0.03888	0.03936	0.03923	0.06596	0.06602	0.06865	0.08523
	µ	− 7.4682	− 8.0295	− 6.7812	− 7.9926	− 7.7828	− 8.7282	− 9.1307	− 7.3662	− 8.6941
	σ	4.3230	3.8292	4.6090	3.8644	4.1565	3.2713	2.9775	4.2334	3.2996
	best	− 1.0152E + 01	− 1.0153E + 01	− 1.0152E + 01	− 1.0153E + 01	− 1.0153E + 01	− 1.0153E + 01	− 1.0153E + 01	− 1.0152E + 01	− 1.0153E + 01
	worst	− 2.7312E − 01	− 3.5065E − 01	− 2.7312E − 01	− 2.7312E − 01	− 2.7312E − 01	− 3.5136E − 01	− 3.5136E − 01	− 2.7312E − 01	− 2.7312E − 01
	time (s)	0.03468	0.03428	0.04299	0.04334	0.04338	0.07042	0.07036	0.07186	0.09010
	µ	− 9.4833	− 8.6109	− 8.2607	− 9.9369	− 9.5252	− 9.5650	− 9.0153	− 9.6415	− 9.4163
	σ	2.4665	3.6776	3.9810	1.7881	2.4837	2.5210	3.3912	2.4541	2.4042
	best	− 1.0403E + 01	− 1.0402E + 01	− 1.0403E + 01	− 1.0403E + 01	− 1.0403E + 01	− 1.0403E + 01	− 1.0403E + 01	− 1.0402E + 01	− 1.0402E + 01
	worst	− 5.2051E − 01	− 3.7371E − 01	− 3.7244E − 01	− 5.2384E − 01	− 3.7482E − 01	− 3.7371E − 01	− 2.9362E − 01	− 3.7353E − 01	− 5.2109E − 01
	time (s)	0.03986	0.03939	0.04827	0.04867	0.04881	0.07624	0.07633	0.07532	0.09627
	µ	− 1.0386E + 01	− 1.0115E + 01	− 9.9977	− 1.0056E + 01	− 1.0412E + 01	− 1.0322E + 01	− 1.0471E + 01	− 1.0316E + 01	− 1.0271E + 01
	σ	3.6773E − 01	1.7373	1.8267	1.7335	2.1480E − 01	3.2677E − 01	1.1311E − 01	2.5694E − 01	3.9820E − 01
	best	− 1.0536E + 01	− 1.0536E + 01	− 1.0535E + 01	− 1.0536E + 01	− 1.0536E + 01	− 1.0536E + 01	− 1.0536E + 01	− 1.0536E + 01	− 1.0536E + 01
	worst	− 8.6649	− 9.4885E − 01	− 5.4335E − 01	− 9.4525E − 01	− 9.5027	− 9.1241	− 1.0055E + 01	− 9.3609	− 8.7860
	time (s)	0.04637	0.04574	0.05446	0.05506	0.05499	0.08268	0.08332	0.08193	0.10218

Fig. 1.

Convergence curves using Chebyshev chaotic map without considering initialization enhancement.

Fig. 2.

Convergence curves using Gauss-mouse chaotic map without considering initialization enhancement.

Table 5.

Wilcoxon sign rank sum test at different scenarios using Chebyshev chaotic map without considering initialization enhancement.

Function	Scenario
	S1		S2		S3		S4		S5		S6		S7		S8
	h	p	h	p	h	p	h	p	h	p	h	p	h	p	h	p
F1	1	8.9580E − 84	1	8.9580E − 84	1	1.5001E − 84	1	1.0705E − 83	1	1.0424E − 83	1	5.7718E − 84	1	6.6405E − 38	1	5.3905E − 45
F2	1	1.5886E − 12	1	1.5886E − 12	1	9.1750E − 84	1	1.5292E − 84	1	1.0686E − 83	1	9.4127E − 84	1	7.8686E − 84	1	4.6105E − 13
F3	1	1.1172E − 03	1	1.1172E − 03	1	1.1144E − 85	1	8.0109E − 84	1	1.1568E − 84	1	1.1053E − 83	1	4.1877E − 85	1	2.0998E − 84
F4	1	3.0711E − 84	1	3.0711E − 84	0	4.2020E − 01	1	8.2979E − 24	1	8.6961E − 84	1	1.9516E − 84	1	9.8819E − 84	1	3.9810E − 84
F5	1	9.2398E − 84	1	9.2398E − 84	1	1.6800E − 31	1	1.1177E − 03	1	8.4083E − 84	1	1.0583E − 83	1	2.1048E − 84	1	1.1101E − 83
F6	1	1.0994E − 83	1	1.0994E − 83	1	1.2802E − 71	1	1.8175E − 69	1	1.1160E − 03	1	8.2420E − 68	1	9.0898E − 84	1	2.2536E − 84
F7	1	3.8499E − 36	1	3.8499E − 36	1	1.7386E − 71	1	5.6548E − 07	1	4.2612E − 68	1	3.4418E − 07	1	7.9493E − 24	1	8.2876E − 84
F8	1	6.1966E − 84	1	6.1966E − 84	1	1.2846E − 84	1	1.1082E − 83	1	5.4624E − 84	1	7.9198E − 84	1	9.2459E − 85	1	1.0523E − 84
F9	1	8.6576E − 84	1	8.6576E − 84	1	4.5401E − 36	1	7.0980E − 62	1	1.0701E − 83	1	1.1872E − 33	1	5.7322E − 84	1	1.0559E − 05
F10	1	1.5297E − 77	1	1.5297E − 77	1	2.7030E − 84	1	7.3552E − 04	1	1.7150E − 84	1	1.1180E − 83	1	1.9381E − 05	1	1.1497E − 84
F11	1	3.6975E − 39	1	3.6975E − 39	1	3.0050E − 03	1	7.8020E − 84	1	1.3571E − 50	1	8.8349E − 60	1	1.0579E − 83	1	1.9888E − 04
F12	1	1.4425E − 37	1	1.4425E − 37	1	7.1633E − 42	1	9.1907E − 26	1	8.7936E − 84	1	4.2945E − 69	1	2.7078E − 29	1	4.5153E − 13
F13	1	3.2553E − 67	1	3.2553E − 67	1	2.2799E − 74	1	5.3261E − 45	1	9.8889E − 46	1	9.5821E − 84	1	3.0988E − 49	1	1.3748E − 20
F14	1	6.1794E − 35	1	6.1794E − 35	1	8.0061E − 14	1	7.9632E − 20	1	2.5706E − 26	1	1.4395E − 44	1	7.6060E − 84	0	5.4309E − 01
F15	1	7.3381E − 87	1	7.3381E − 87	1	1.1922E − 54	1	1.8305E − 23	1	1.9365E − 54	1	2.3351E − 48	1	8.8180E − 49	1	8.7207E − 84
F16	1	3.0154E − 86	1	3.0154E − 86	1	7.6846E − 87	1	1.0048E − 83	1	9.6475E − 84	1	3.4095E − 84	1	9.5515E − 84	1	7.5008E − 84
F17	1	5.7707E − 84	1	5.7707E − 84	1	1.0576E − 84	1	1.3922E − 75	1	2.7689E − 35	1	8.1873E − 14	1	1.3978E − 21	1	1.8356E − 42
F18	1	9.0193E − 84	1	9.0193E − 84	1	5.2743E − 84	1	3.6134E − 84	1	2.6538E − 30	1	5.2286E − 23	1	9.4035E − 14	1	8.9436E − 03
F19	1	8.0094E − 84	1	8.0094E − 84	1	6.6726E − 84	1	7.0504E − 84	1	1.9550E − 84	1	8.3774E − 87	1	1.0958E − 83	1	1.0027E − 83
F20	1	1.0984E − 84	1	1.0984E − 84	1	6.6433E − 84	1	9.4853E − 84	1	7.2954E − 84	1	1.1435E − 85	1	2.2824E − 85	1	1.0695E − 83
F21	1	1.5233E − 67	1	1.5233E − 67	1	5.3090E − 45	1	8.3959E − 84	1	7.9617E − 84	1	1.0685E − 83	1	5.2234E − 84	1	2.5648E − 84
F22	1	1.1688E − 46	1	1.1688E − 46	1	3.2130E − 52	1	1.3691E − 53	1	9.6204E − 84	1	7.7854E − 84	1	1.0507E − 83	1	3.2832E − 84
F23	1	3.4436E − 04	1	3.4436E − 04	1	8.5511E − 69	1	6.1901E − 60	1	2.4283E − 44	1	9.5437E − 84	1	9.4230E − 84	1	8.4794E − 84
	23		23		22		23		23		23		23		22

Table 6.

Wilcoxon sign rank sum test at different scenarios using Gauss-mouse chaotic map without considering initialization enhancement.

Function	Scenario
	S1		S2		S3		S4		S5		S6		S7		S8
	h	p	h	p	h	p	h	p	h	p	h	p	h	p	h	p
F1	1	5.8410E − 84	1	5.8410E − 84	1	6.9231E − 84	1	9.2151E − 84	1	1.0266E − 83	1	7.4951E − 84	1	1.8175E − 33	1	3.2125E − 44
F2	1	1.0257E − 03	1	1.0257E − 03	1	7.2611E − 84	1	3.0565E − 84	1	1.0456E − 83	1	1.0563E − 83	1	7.5307E − 84	1	1.1729E − 39
F3	1	1.3766E − 03	1	1.3766E − 03	1	7.4317E − 84	1	4.5382E − 84	1	3.0529E − 84	1	8.3757E − 84	1	8.0613E − 84	1	5.3430E − 84
F4	1	5.2444E − 84	1	5.2444E − 84	1	8.3882E − 51	1	4.7136E − 41	1	5.9737E − 84	1	3.2631E − 84	1	1.0964E − 83	1	8.1601E − 84
F5	1	9.8212E − 84	1	9.8212E − 84	1	2.2020E − 34	1	6.0923E − 08	1	1.0450E − 47	1	6.8759E − 84	1	3.1129E − 84	1	1.0992E − 83
F6	1	1.4957E − 45	1	1.4957E − 45	1	4.3841E − 63	1	2.3696E − 49	1	1.0346E − 05	1	1.7161E − 06	1	1.0216E − 83	1	6.2144E − 84
F7	1	3.6207E − 74	1	3.6207E − 74	1	1.2081E − 57	1	3.7466E − 02	1	4.6738E − 52	1	2.0623E − 42	1	1.6809E − 42	1	5.2247E − 84
F8	1	7.7212E − 84	1	7.7212E − 84	1	9.7918E − 84	1	9.2835E − 84	1	1.6062E − 84	1	8.7361E − 85	1	4.7262E − 84	1	4.3979E − 84
F9	1	2.3093E − 84	1	2.3093E − 84	1	3.7913E − 51	1	3.4736E − 60	1	8.9044E − 84	1	3.1347E − 41	1	5.1842E − 53	1	4.0225E − 24
F10	1	2.1220E − 20	1	2.1220E − 20	1	2.6000E − 84	1	3.2774E − 27	1	4.2560E − 78	1	3.1961E − 72	1	7.8549E − 07	1	6.2000E − 52
F11	1	5.9244E − 46	1	5.9244E − 46	0	5.1763E − 01	1	6.2520E − 84	1	3.4913E − 23	1	5.0624E − 84	1	8.9864E − 84	1	3.9580E − 21
F12	1	1.2543E − 46	1	1.2543E − 46	1	1.8221E − 78	1	4.4105E − 29	1	8.7940E − 84	1	2.4525E − 41	1	2.3270E − 17	0	6.7469E − 02
F13	1	2.3644E − 58	1	2.3644E − 58	1	5.1233E − 60	1	9.6239E − 83	1	2.1963E − 56	1	1.0360E − 83	1	2.3107E − 44	1	1.5684E − 45
F14	1	7.0306E − 20	1	7.0306E − 20	1	5.2199E − 03	1	2.6239E − 20	1	7.0475E − 84	1	8.1501E − 24	1	7.6017E − 84	1	3.2465E − 02
F15	1	9.4484E − 84	1	9.4484E − 84	1	4.6152E − 37	1	6.0852E − 36	1	2.6402E − 57	1	3.8737E − 10	1	8.2173E − 35	1	9.7956E − 84
F16	1	7.4956E − 84	1	7.4956E − 84	1	8.6699E − 84	1	1.0489E − 83	1	7.8564E − 84	1	4.4936E − 84	1	8.3460E − 84	1	8.0508E − 84
F17	0	4.8638E − 01	0	4.8638E − 01	1	7.2851E − 84	1	6.8533E − 84	1	1.4112E − 29	1	5.5128E − 19	1	4.9367E − 53	1	1.2544E − 08
F18	1	6.2578E − 84	1	6.2578E − 84	1	1.0089E − 34	1	7.0450E − 84	1	6.2586E − 34	1	1.0685E − 18	1	5.2791E − 03	1	4.2639E − 28
F19	1	5.3775E − 84	1	5.3775E − 84	1	2.5533E − 84	1	9.4788E − 84	1	6.0347E − 84	1	7.0683E − 84	1	1.0405E − 83	1	6.2217E − 84
F20	1	5.5012E − 84	1	5.5012E − 84	1	1.0018E − 83	1	9.4574E − 84	1	9.7230E − 84	1	7.5638E − 84	1	1.1077E − 83	1	1.1554E − 83
F21	1	2.6953E − 55	1	2.6953E − 55	1	6.6321E − 17	1	9.9118E − 84	1	5.1384E − 84	1	7.7853E − 84	1	7.5938E − 84	1	6.1272E − 84
F22	1	5.5131E − 12	1	5.5131E − 12	1	2.3841E − 73	1	2.7117E − 15	1	8.0369E − 84	1	7.8285E − 84	1	7.3855E − 84	1	8.2524E − 84
F23	1	1.4504E − 45	1	1.4504E − 45	0	5.2615E − 02	1	2.8396E − 52	1	2.2453E − 09	1	9.0534E − 84	1	1.7832E − 84	1	8.0761E − 84
	22		22		21		23		23		23		23		22

Results of different scenarios considering initialization enhancement

In the current case study, the eight scenarios are employed while updating the initialization variable using CMs. The obtained results are given in Table 7 for the Chebyshev chaotic map, where the obtained results showed the ability of the proposed scenarios in finding more and more near-optimal solutions like that proposed in the previous case study. Further, the Gauss-mouse chaotic map succeeded in finding near-optimal solutions for the studied benchmark test functions better than the rest of the CMs like that obtained in the previous case study as shown in Table 8. The Wilcoxon sign rank test is conducted at 5% significance level, to assess the significance of the CTSO solutions against the original TSO while considering initialization enhancement as shown in Tables 9 and 10 for the Chebyshev, and the Gauss-mouse CMs, respectively. Results showed the effectiveness of the proposed scenarios in obtaining significantly better solutions rather than that using TSO for multiple benchmark test functions while comparing the average value µ. The convergence curves for both the Chebyshev and Gauss-mouse CMs, are shown in Figs. 3 and 4, respectively. It might be noted that the implementation of CMs in TSO has speeded up its convergence like that before in the previous case study. Finally, to ensure the significance of the obtained results, the Wilcoxon sign rank test is employed, and the obtained results are given in Tables 9 and 10 for the Chebyshev and Gauss-mouse CMs, respectively. The results showed the high significance of the CTSO against the original TSO.

Table 7.

Fitness values at different scenarios using Chebyshev chaotic map considering initialization enhancement.

Function	Index	TSO	Scenario
			S1	S2	S3	S4	S5	S6	S7	S8
	µ	1.4831E − 18	3.6614E − 22	2.2556E − 14	8.9810E − 19	1.2850E − 19	3.7752E − 15	1.0443E − 08	1.6839E − 11	8.9600E − 08
	σ	7.3029E − 18	1.1507E − 21	9.6017E − 14	4.8874E − 18	4.4205E − 19	2.0518E − 14	5.4805E − 08	8.8167E − 11	4.5694E − 07
	best	1.1361E − 39	1.4901E − 42	3.4991E − 55	1.6328E − 44	5.0163E − 50	2.6637E − 70	2.8536E − 60	2.8137E − 35	1.5963E − 25
	worst	3.9924E − 17	5.4448E − 21	5.2412E − 13	2.6775E − 17	2.1335E − 18	1.1241E − 13	3.0043E − 07	4.8356E − 10	2.5078E − 06
	time (s)	0.02804	0.02967	0.03684	0.03685	0.03687	0.06559	0.06445	0.06464	0.08361
	µ	2.7894E − 11	2.8904E − 11	8.5951E − 08	1.8776E − 11	1.8112E − 11	4.3349E − 07	1.6301E − 05	1.7462E − 07	1.3851E − 05
	σ	1.3710E − 10	9.1573E − 11	3.5008E − 07	3.8644E − 11	5.7915E − 11	1.6563E − 06	7.8480E − 05	4.9848E − 07	2.5267E − 05
	best	2.6771E − 29	2.4224E − 27	8.6339E − 34	5.4177E − 23	2.6369E − 22	7.5241E − 24	8.3721E − 20	1.5164E − 19	1.6388E − 12
	worst	7.5129E − 10	4.5747E − 10	1.8420E − 06	1.7564E − 10	3.0152E − 10	8.7773E − 06	4.2784E − 04	2.5841E − 06	1.1039E − 04
	time (s)	0.02806	0.02966	0.03877	0.03887	0.03872	0.06675	0.06711	0.06676	0.08745
	µ	8.6013E − 06	3.1857E − 07	1.9326E − 03	2.8684E − 08	7.8479E − 07	1.9588E − 03	1.6330E + 02	4.7840E − 01	6.7855E − 01
	σ	4.6824E − 05	1.0138E − 06	9.4338E − 03	7.3617E − 08	2.7658E − 06	8.8554E − 03	4.4277E + 02	1.7763	2.3535
	best	1.0610E − 19	5.8041E − 31	5.3546E − 15	6.9767E − 32	2.2277E − 37	1.8249E − 12	3.3290E − 27	9.0910E − 19	2.0552E − 09
	worst	2.5652E − 04	5.0498E − 06	5.1597E − 02	2.9700E − 07	1.4798E − 05	4.8596E − 02	1.9440E + 03	9.6509	1.2668E + 01
	time (s)	0.14590	0.15046	0.15952	0.15916	0.15761	0.18807	0.18761	0.18629	0.20897
	µ	2.0065E − 10	1.6296E − 11	1.0684E − 07	2.3190E − 10	5.9668E − 10	1.1169E − 07	4.6188E − 06	6.4552E − 07	1.6486E − 05
	σ	7.8452E − 10	3.9319E − 11	5.4627E − 07	9.4504E − 10	1.8770E − 09	3.7830E − 07	1.3642E − 05	1.3627E − 06	5.0855E − 05
	best	7.6044E − 24	3.5616E − 27	2.0276E − 31	1.1601E − 25	2.6101E − 25	3.7034E − 37	9.8607E − 26	9.6130E − 18	5.1166E − 43
	worst	4.1793E − 09	1.5866E − 10	2.9932E − 06	5.1264E − 09	8.1842E − 09	1.8199E − 06	5.1890E − 05	5.3714E − 06	2.6614E − 04
	time (s)	0.02604	0.02754	0.03677	0.03696	0.03662	0.06414	0.06406	0.06454	0.08417
	µ	8.1293E − 02	1.9339E − 01	3.1655E − 01	1.5386E − 01	1.4649E − 01	2.9105E − 01	1.9421E + 01	3.0434E − 01	2.1713E − 01
	σ	1.3662E − 01	2.5828E − 01	7.7382E − 01	2.7993E − 01	1.9495E − 01	6.1733E − 01	1.3413E + 01	6.6239E − 01	3.7275E − 01
	best	6.9082E − 06	1.6186E − 04	6.2135E − 06	4.6293E − 07	3.9962E − 05	6.7575E − 05	8.5283E − 05	1.5578E − 06	1.5251E − 05
	worst	5.6436E − 01	9.2140E − 01	4.2152	1.0941	8.4515E − 01	2.8349	2.8859E + 01	2.8948	1.8545
	time (s)	0.03945	0.04168	0.05070	0.05092	0.05049	0.07824	0.07790	0.07836	0.09825
	µ	5.5450E − 03	3.9259E − 03	2.0435E − 02	8.4402E − 03	6.7862E − 03	2.3189E − 02	2.0306	1.0235E − 02	2.6854E − 03
	σ	6.5564E − 03	3.9875E − 03	5.3334E − 02	2.5290E − 02	1.1088E − 02	4.5370E − 02	1.6292	1.1406E − 02	3.6536E − 03
	best	6.2663E − 06	2.9094E − 05	1.2163E − 06	2.0135E − 04	5.3558E − 06	4.8368E − 05	2.9756E − 02	5.5915E − 05	2.8666E − 07
	worst	2.3154E − 02	1.4176E − 02	2.8342E − 01	1.4019E − 01	4.3457E − 02	1.8849E − 01	7.1614	3.4711E − 02	1.6083E − 02
	time (s)	0.02658	0.02782	0.03727	0.03755	0.03712	0.06470	0.06447	0.06460	0.08492
	µ	4.6201E − 04	3.0660E − 04	6.0011E − 04	3.5107E − 04	6.0128E − 04	7.9593E − 04	5.2646E − 04	8.0507E − 04	1.1708E − 03
	σ	5.4612E − 04	2.9584E − 04	4.1876E − 04	2.4935E − 04	6.7203E − 04	7.4222E − 04	5.3848E − 04	1.5295E − 03	1.5119E − 03
	best	1.3910E − 05	7.8952E − 06	4.0297E − 05	2.7217E − 05	2.3756E − 06	3.7315E − 05	6.7362E − 06	2.8736E − 05	1.3734E − 05
	worst	2.9330E − 03	1.2862E − 03	1.7413E − 03	9.1857E − 04	3.0392E − 03	2.8556E − 03	2.4588E − 03	7.1587E − 03	8.4021E − 03
	time (s)	0.07857	0.08148	0.09074	0.09101	0.09045	0.11839	0.11810	0.11808	0.13855
	µ	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04
	σ	7.8182E − 03	3.7891E − 03	3.9376E − 03	3.5349E − 02	4.4766E − 02	5.7634E − 03	3.1656E − 02	1.5850	3.9060E − 03
	best	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04
	worst	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2561E + 04	− 1.2569E + 04
	time (s)	0.03795	0.03988	0.04914	0.04916	0.04905	0.07667	0.07662	0.07703	0.09643
	µ	0	0	3.6645E − 12	0	0	4.1439E − 12	6.2999E − 08	5.8359E − 13	3.9501E − 08
	σ	0	0	1.9975E − 11	0	0	2.2611E − 11	3.4397E − 07	1.5071E − 12	1.7026E − 07
	best	0	0	0	0	0	0	0	0	0
	worst	0	0	1.0942E − 10	0	0	1.2386E − 10	1.8842E − 06	6.3665E − 12	9.3237E − 07
	time (s)	0.03404	0.03570	0.04488	0.04520	0.04504	0.07241	0.07228	0.07237	0.09378
	µ	1.3528E − 12	5.6186E − 13	1.0409E − 08	3.2611E − 12	3.1370E − 12	1.0415E − 09	9.3195E − 05	1.3653E − 09	4.4477E − 06
	σ	6.4305E − 12	1.6492E − 12	5.6834E − 08	1.0622E − 11	8.0432E − 12	3.3654E − 09	5.0443E − 04	3.2212E − 09	1.2708E − 05
	best	8.8818E − 16	8.8818E − 16	8.8818E − 16	8.8818E − 16	8.8818E − 16	8.8818E − 16	8.8818E − 16	8.8818E − 16	8.8818E − 16
	worst	3.5247E − 11	8.1330E − 12	3.1133E − 07	4.8755E − 11	3.4270E − 11	1.5822E − 08	2.7638E − 03	1.4895E − 08	6.5966E − 05
	time (s)	0.03387	0.03566	0.04480	0.04511	0.04511	0.07243	0.07216	0.07214	0.09344
	µ	0	0	2.7237E − 15	0	0	1.5782E − 10	1.6207E − 05	7.4186E − 12	8.9294E − 08
	σ	0	0	8.2304E − 15	0	0	8.6387E − 10	8.8767E − 05	3.4183E − 11	3.8573E − 07
	best	0	0	0	0	0	0	0	0	0
	worst	0	0	3.2419E − 14	0	0	4.7317E − 09	4.8620E − 04	1.8711E − 10	2.0860E − 06
	time (s)	0.04452	0.04628	0.05555	0.05563	0.05563	0.08291	0.08313	0.08287	0.10328
	µ	2.4695E − 04	5.0045E − 04	1.1876E − 03	2.7283E − 04	2.6669E − 04	6.6684E − 04	1.5063E − 01	3.7929E − 04	1.7363E − 04
	σ	6.0608E − 04	8.3986E − 04	1.6961E − 03	3.1596E − 04	3.1765E − 04	1.2438E − 03	1.9210E − 01	6.3313E − 04	2.3091E − 04
	best	6.5739E − 07	1.5913E − 06	7.0191E − 07	1.5805E − 06	2.5402E − 06	3.3536E − 08	1.1843E − 03	1.3071E − 06	2.3632E − 07
	worst	3.3284E − 03	3.2334E − 03	8.5878E − 03	1.2610E − 03	1.2580E − 03	6.3503E − 03	8.2187E − 01	2.9957E − 03	7.1014E − 04
	time (s)	0.17328	0.17699	0.18687	0.18707	0.18700	0.21335	0.21358	0.21376	0.23319
	µ	2.5400E − 03	2.1197E − 03	9.0579E − 03	2.1674E − 03	1.9521E − 03	6.6095E − 03	3.4486E − 01	3.3748E − 03	4.1587E − 03
	σ	4.2391E − 03	2.2903E − 03	2.5713E − 02	3.2828E − 03	3.8240E − 03	1.1289E − 02	5.7246E − 01	3.6832E − 03	5.3088E − 03
	best	2.2291E − 06	1.6013E − 07	1.5719E − 05	4.8146E − 08	8.0974E − 08	9.2555E − 10	1.9191E − 06	1.1982E − 06	5.2451E − 07
	worst	1.9084E − 02	8.2790E − 03	1.3399E − 01	1.2901E − 02	1.9470E − 02	4.4324E − 02	2.3551	1.3691E − 02	2.3056E − 02
	time (s)	0.16966	0.17516	0.18469	0.18525	0.18459	0.21175	0.21175	0.21244	0.23128
	µ	1.4616	1.2960	1.8548	1.5595	1.1819	1.4275	4.1315	1.1640	9.9800E − 01
	σ	8.1265E − 01	7.8872E − 01	1.9719	1.0916	5.3025E − 01	1.2887	3.8546	4.5856E − 01	1.6352E − 06
	best	9.9800E − 01	9.9800E − 01	9.9800E − 01	9.9800E − 01	9.9800E − 01	9.9800E − 01	9.9800E − 01	9.9800E − 01	9.9800E − 01
	worst	2.9826	3.9684	1.0763E + 01	5.9288	2.9821	5.9288	1.2671E + 01	2.9821	9.9801E − 01
	time (s)	0.32084	0.33332	0.34108	0.34503	0.34243	0.36816	0.36738	0.37019	0.38597
	µ	6.9593E − 04	6.9112E − 04	9.2272E − 04	9.1197E − 04	8.7023E − 04	7.2332E − 04	1.2249E − 03	1.1277E − 03	1.5511E − 03
	σ	6.5262E − 04	6.5607E − 04	9.0708E − 04	8.4302E − 04	7.8887E − 04	6.7141E − 04	8.4694E − 04	9.0094E − 04	8.6354E − 04
	best	3.0968E − 04	3.2537E − 04	3.1308E − 04	3.1762E − 04	3.3000E − 04	3.2178E − 04	3.3303E − 04	3.1892E − 04	3.0981E − 04
	worst	2.2748E − 03	2.2716E − 03	3.1698E − 03	2.2858E − 03	2.3076E − 03	2.3043E − 03	2.3539E − 03	2.2909E − 03	2.4194E − 03
	time (s)	0.02653	0.02663	0.03551	0.03595	0.03557	0.06334	0.06315	0.06321	0.08314
	µ	− 1.0316	− 1.0316	− 1.0316	− 1.0316	− 1.0316	− 1.0314	− 9.9705E − 01	− 1.0316	− 1.0316
	σ	1.0912E − 04	3.2424E − 05	1.6178E − 04	3.9631E − 05	1.1480E − 04	3.8217E − 04	1.5070E − 01	4.3956E − 05	4.9025E − 05
	best	− 1.0316	− 1.0316	− 1.0316	− 1.0316	− 1.0316	− 1.0316	− 1.0316	− 1.0316	− 1.0316
	worst	− 1.0310	− 1.0315	− 1.0309	− 1.0315	− 1.0310	− 1.0298	− 2.1443E − 01	− 1.0314	− 1.0314
	time (s)	0.02562	0.02629	0.03491	0.03551	0.03515	0.06262	0.06265	0.06242	0.08253
	µ	5.5335E − 01	2.1005	2.5649	2.5652	2.2553	2.2594	2.8745	3.3383	1.9461
	σ	8.4741E − 01	2.2749	2.3549	2.3547	2.3127	2.3093	2.3547	2.2748	2.2252
	best	3.9789E − 01	3.9789E − 01	3.9789E − 01	3.9789E − 01	3.9789E − 01	3.9789E − 01	3.9790E − 01	3.9789E − 01	3.9789E − 01
	worst	5.0401	5.0401	5.0401	5.0401	5.0401	5.0401	5.0401	5.0401	5.0401
	time (s)	0.02173	0.02227	0.03082	0.03113	0.03078	0.05845	0.05792	0.05783	0.07782
	µ	3.0042	5.8744	8.8343	5.9297	5.7702	8.3564	1.1514E + 01	1.0171E + 01	8.6192
	σ	1.0714E − 02	8.7694	1.1385E + 01	8.9314	8.4461	1.1037E + 01	1.2853E + 01	1.2366E + 01	1.1443E + 01
	best	3.0000	3.0000	3.0001	3.0000	3.0000	3.0000	3.0000	3.0000	3.0000
	worst	3.0535	3.2685E + 01	3.2649E + 01	3.2618E + 01	3.1563E + 01	3.2666E + 01	3.4168E + 01	3.2718E + 01	3.2828E + 01
	time (s)	0.02072	0.02136	0.02995	0.03018	0.03005	0.05776	0.05766	0.05737	0.07743
	µ	− 3.8084	− 3.8220	− 3.8086	− 3.8160	− 3.8278	− 3.7490	− 3.8255	− 3.8171	− 3.7976
	σ	6.8103E − 02	7.5220E − 02	7.0884E − 02	7.2947E − 02	5.6627E − 02	5.4452E − 01	6.9002E − 02	7.5465E − 02	9.3211E − 02
	best	− 3.8627	− 3.8573	− 3.8598	− 3.8549	− 3.8625	− 3.8628	− 3.8627	− 3.8567	− 3.8624
	worst	− 3.6046	− 3.6047	− 3.6047	− 3.6047	− 3.6501	− 8.7208E − 01	− 3.6022	− 3.6047	− 3.6031
	time (s)	0.02961	0.03060	0.03930	0.03942	0.03944	0.06750	0.06760	0.06733	0.08748
	µ	− 2.9224	− 1.8762	− 1.9946	− 1.9206	− 2.2245	− 1.9499	− 1.8835	− 2.1879	− 2.1555
	σ	2.6517E − 01	9.6291E − 01	1.0281	9.9854E − 01	8.4674E − 01	8.8130E − 01	1.0905	9.3039E − 01	9.0153E − 01
	best	− 3.2435	− 3.1374	− 3.1326	− 3.1323	− 3.1290	− 3.1326	− 3.1283	− 3.1306	− 3.1344
	worst	− 1.9663	− 3.1235E − 01	− 4.4209E − 01	− 1.8704E − 01	− 5.1612E − 01	− 5.1612E − 01	− 1.9483E − 01	− 4.4209E − 01	− 6.5285E − 01
	time (s)	0.03050	0.03135	0.04024	0.04035	0.04027	0.06825	0.06848	0.06807	0.08764
	µ	− 7.4682	− 7.0217E − 01	− 7.1112E − 01	− 9.9355E − 01	− 1.0449	− 6.3101E − 01	− 1.0264	− 1.0117	− 9.9229E − 01
	σ	4.3230	1.7718	1.7875	2.4746	2.4618	1.0662	2.4661	2.4715	2.4375
	best	− 1.0152E + 01	− 1.0063E + 01	− 1.0153E + 01	− 1.0121E + 01	− 1.0153E + 01	− 4.8624	− 1.0123E + 01	− 1.0103E + 01	− 1.0130E + 01
	worst	− 2.7312E − 01	− 2.7312E − 01	− 2.7312E − 01	− 2.7312E − 01	− 2.7312E − 01	− 2.7312E − 01	− 2.7312E − 01	− 2.7312E − 01	− 2.7312E − 01
	time (s)	0.03468	0.03577	0.04465	0.04478	0.04582	0.07286	0.07271	0.07278	0.09230
	µ	− 9.4833	− 2.4052	− 7.2162E − 01	− 2.3382	− 1.7455	− 1.5514	− 1.5038	− 1.4312	− 1.0879
	σ	2.4665	4.0466	1.8057	3.9292	3.4489	3.0807	3.0238	3.0316	2.5328
	best	− 1.0403E + 01	− 1.0400E + 01	− 1.0262E + 01	− 1.0401E + 01	− 1.0399E + 01	− 1.0392E + 01	− 1.0389E + 01	− 1.0403E + 01	− 1.0399E + 01
	worst	− 5.2051E − 01	− 2.9362E − 01	− 2.9362E − 01	− 2.9362E − 01	− 2.9362E − 01	− 2.9362E − 01	− 2.9362E − 01	− 2.9362E − 01	− 2.9362E − 01
	time (s)	0.03986	0.04111	0.05004	0.05022	0.05026	0.07836	0.07812	0.07814	0.09789
	µ	− 1.0386E + 01	− 3.1460	− 3.4982	− 3.3704	− 3.1694	− 3.4706	− 1.8797	− 2.5064	− 3.1686
	σ	3.6773E − 01	4.3553	4.5918	4.5390	4.4580	4.4153	3.3954	3.9699	4.4946
	best	− 1.0536E + 01	− 1.0533E + 01	− 1.0523E + 01	− 1.0525E + 01	− 1.0534E + 01	− 1.0504E + 01	− 1.0531E + 01	− 1.0535E + 01	− 1.0530E + 01
	worst	− 8.6649	− 3.2173E − 01	− 3.2173E − 01	− 3.2173E − 01	− 3.2173E − 01	− 3.2173E − 01	− 3.2173E − 01	− 3.2173E − 01	− 3.2173E − 01
	time (s)	0.04637	0.04821	0.05722	0.05732	0.05859	0.08568	0.08534	0.08584	0.10517

Table 8.

Fitness values at different scenarios using Gauss-mouse chaotic map considering initialization enhancement.

Function	Index	TSO	Scenario
			S1	S2	S3	S4	S5	S6	S7	S8
	µ	1.4831E − 18	1.1160E − 21	4.6808E − 13	1.1582E − 21	1.3394E − 19	1.7292E − 17	1.0760E − 13	7.0435E − 21	1.8530E − 20
	σ	7.3029E − 18	3.9425E − 21	2.5635E − 12	2.9914E − 21	6.7001E − 19	5.3718E − 17	2.9265E − 13	2.1163E − 20	6.8105E − 20
	best	1.1361E − 39	2.1988E − 49	2.9139E − 58	4.5789E − 52	2.1227E − 46	3.0725E − 39	6.0615E − 49	1.3833E − 50	2.1220E − 53
	worst	3.9924E − 17	1.9916E − 20	1.4041E − 11	1.1076E − 20	3.6732E − 18	2.5972E − 16	1.3274E − 12	9.4969E − 20	3.4382E − 19
	time (s)	0.02804	0.02739	0.03683	0.03708	0.03692	0.06490	0.06468	0.06454	0.08358
	µ	2.7894E − 11	4.2282E − 11	5.6522E − 09	9.3680E − 12	7.6505E − 11	4.4106E − 10	2.1571E − 07	5.3264E − 11	3.6411E − 10
	σ	1.3710E − 10	1.4485E − 10	1.5499E − 08	4.1190E − 11	3.2122E − 10	1.3251E − 09	7.8384E − 07	1.3049E − 10	1.3057E − 09
	best	2.6771E − 29	3.4150E − 24	3.5633E − 26	7.9333E − 27	1.9742E − 24	2.4862E − 22	4.5309E − 20	7.0700E − 26	1.5019E − 32
	worst	7.5129E − 10	7.3919E − 10	5.8817E − 08	2.2445E − 10	1.7647E − 09	6.7393E − 09	4.0343E − 06	6.3878E − 10	6.6543E − 09
	time (s)	0.02806	0.02939	0.03901	0.03920	0.03909	0.06694	0.06665	0.06666	0.08564
	µ	8.6013E − 06	3.1529E − 06	9.4262E − 04	2.2637E − 06	5.2197E − 07	1.4584E − 06	2.9292E − 03	9.0792E − 10	1.8699E − 04
	σ	4.6824E − 05	1.4855E − 05	2.4648E − 03	1.1619E − 05	2.5573E − 06	4.0526E − 06	1.1354E − 02	2.2752E − 09	1.0234E − 03
	best	1.0610E − 19	2.6768E − 24	1.6480E − 33	1.3689E − 25	2.7473E − 25	2.6897E − 23	1.6019E − 22	1.2387E − 35	3.5373E − 33
	worst	2.5652E − 04	8.1376E − 05	9.7109E − 03	6.3754E − 05	1.4036E − 05	1.7005E − 05	6.0901E − 02	1.1015E − 08	5.6058E − 03
	time (s)	0.14590	0.15026	0.16018	0.15881	0.15879	0.18740	0.18753	0.18563	0.20738
	µ	2.0065E − 10	9.6807E − 12	4.3555E − 09	1.6042E − 10	2.2454E − 11	2.0995E − 09	2.1037E − 07	2.0853E − 11	2.0697E − 10
	σ	7.8452E − 10	4.0984E − 11	1.1339E − 08	5.1679E − 10	7.6991E − 11	9.8729E − 09	8.5290E − 07	5.2893E − 11	4.5467E − 10
	best	7.6044E − 24	4.5357E − 26	3.1017E − 24	2.1588E − 23	6.1460E − 25	3.5745E − 22	6.0238E − 24	3.6190E − 33	7.0116E − 27
	worst	4.1793E − 09	2.2488E − 10	5.1177E − 08	2.1477E − 09	4.1615E − 10	5.4025E − 08	4.5293E − 06	2.1866E − 10	1.5387E − 09
	time (s)	0.02604	0.02740	0.03705	0.03712	0.03700	0.06473	0.06460	0.06498	0.08349
	µ	8.1293E − 02	3.3705E − 01	3.5864E − 01	1.1279E − 01	9.0942E − 02	1.0551E − 01	2.0560E − 02	1.1284E − 01	7.7602E − 02
	σ	1.3662E − 01	7.0418E − 01	5.8816E − 01	1.4966E − 01	1.1363E − 01	2.1496E − 01	4.4685E − 02	1.5877E − 01	1.2397E − 01
	best	6.9082E − 06	3.7432E − 05	4.4120E − 06	4.2683E − 05	2.0836E − 05	8.7491E − 05	6.7932E − 06	1.6750E − 05	3.0365E − 05
	worst	5.6436E − 01	3.7465	2.2009	5.8309E − 01	5.0973E − 01	9.9947E − 01	1.8754E − 01	6.6065E − 01	4.8686E − 01
	time (s)	0.03945	0.04127	0.05104	0.05086	0.05075	0.07840	0.07812	0.07817	0.09739
	µ	5.5450E − 03	3.6729E − 03	1.3996E − 02	6.7436E − 03	3.8359E − 03	4.6702E − 03	7.8627E − 03	3.0296E − 03	4.1324E − 03
	σ	6.5564E − 03	3.4839E − 03	2.9161E − 02	8.0275E − 03	4.0780E − 03	5.7261E − 03	8.8444E − 03	6.6291E − 03	7.4665E − 03
	best	6.2663E − 06	3.3244E − 05	9.4134E − 06	4.6993E − 06	9.1561E − 08	3.4046E − 05	1.4074E − 04	1.0370E − 06	1.8220E − 05
	worst	2.3154E − 02	1.1506E − 02	1.5295E − 01	3.1473E − 02	1.7317E − 02	2.6042E − 02	2.7303E − 02	3.5817E − 02	3.1575E − 02
	time (s)	0.02658	0.02784	0.03751	0.03764	0.03742	0.06528	0.06500	0.06469	0.08424
	µ	4.6201E − 04	3.9957E − 04	4.8856E − 04	4.3660E − 04	5.0916E − 04	7.3618E − 04	8.1980E − 04	4.0141E − 04	5.5549E − 04
	σ	5.4612E − 04	4.9287E − 04	3.3867E − 04	4.0111E − 04	6.6257E − 04	9.9915E − 04	9.6676E − 04	4.9386E − 04	6.3280E − 04
	best	1.3910E − 05	1.2186E − 05	1.6505E − 06	6.8763E − 06	9.7392E − 06	3.8542E − 06	3.8443E − 05	2.0237E − 05	1.1906E − 05
	worst	2.9330E − 03	2.5170E − 03	1.2801E − 03	1.5995E − 03	3.4213E − 03	4.9997E − 03	3.7815E − 03	1.9764E − 03	2.7101E − 03
	time (s)	0.07857	0.08136	0.09094	0.09080	0.09084	0.11878	0.11853	0.11848	0.13811
	µ	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04
	σ	7.8182E − 03	8.6932E − 03	1.3374E − 02	8.3925E − 02	7.0119E − 03	4.0996E − 03	2.1969E − 03	4.4893E − 02	8.2865E − 02
	best	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04
	worst	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04	− 1.2569E + 04
	time (s)	0.03795	0.03986	0.04935	0.04938	0.04929	0.07747	0.07702	0.07709	0.09678
	µ	0	0	3.7896E − 15	0	0	0	6.8212E − 14	0	0
	σ	0	0	2.0756E − 14	0	0	0	3.3326E − 13	0	0
	best	0	0	0	0	0	0	0	0	0
	worst	0	0	1.1369E − 13	0	0	0	1.8190E − 12	0	0
	time (s)	0.03404	0.03557	0.04516	0.04530	0.04520	0.07323	0.07327	0.07270	0.09300
	µ	1.3528E − 12	1.2214E − 12	3.2511E − 09	5.4955E − 13	2.2008E − 12	1.6272E − 10	3.1482E − 07	5.3403E − 13	9.1050E − 10
	σ	6.4305E − 12	4.2661E − 12	1.1271E − 08	1.9517E − 12	4.5302E − 12	7.9421E − 10	1.5519E − 06	1.8542E − 12	4.6135E − 09
	best	8.8818E − 16	8.8818E − 16	8.8818E − 16	8.8818E − 16	8.8818E − 16	8.8818E − 16	8.8818E − 16	8.8818E − 16	8.8818E − 16
	worst	3.5247E − 11	1.8642E − 11	5.6551E − 08	1.0584E − 11	1.3888E − 11	4.3597E − 09	8.5041E − 06	1.0005E − 11	2.5270E − 08
	time (s)	0.03387	0.03557	0.04510	0.04535	0.04535	0.07406	0.07320	0.07278	0.09215
	µ	0	3.7007E − 18	2.2504E − 14	0	0	2.2204E − 17	1.8911E − 11	0	6.3653E − 16
	σ	0	2.0270E − 17	5.9864E − 14	0	0	1.0267E − 16	1.0120E − 10	0	2.5684E − 15
	best	0	0	0	0	0	0	0	0	0
	worst	0	1.1102E − 16	2.5491E − 13	0	0	5.5511E − 16	5.5465E − 10	0	1.3767E − 14
	time (s)	0.04452	0.04613	0.05596	0.05621	0.05595	0.08437	0.08392	0.08323	0.10297
	µ	2.4695E − 04	2.7885E − 04	5.7602E − 04	4.0891E − 04	3.0191E − 04	3.6068E − 04	4.1510E − 04	2.9692E − 04	2.2735E − 04
	σ	6.0608E − 04	4.7944E − 04	8.2392E − 04	1.0774E − 03	6.7467E − 04	5.7272E − 04	8.7277E − 04	4.4409E − 04	5.0397E − 04
	best	6.5739E − 07	7.8753E − 08	1.5031E − 06	6.3772E − 07	8.4590E − 06	1.6865E − 07	4.6159E − 07	5.1066E − 08	1.0139E − 06
	worst	3.3284E − 03	1.8139E − 03	3.1061E − 03	5.7502E − 03	3.6607E − 03	2.6202E − 03	4.4322E − 03	1.8838E − 03	2.1826E − 03
	time (s)	0.17328	0.17729	0.18668	0.18753	0.18635	0.21472	0.21412	0.21402	0.23352
	µ	2.5400E − 03	1.1893E − 03	7.9963E − 03	2.9759E − 03	1.1712E − 03	3.8888E − 03	1.8666E − 04	1.0144E − 03	2.8743E − 03
	σ	4.2391E − 03	1.3454E − 03	1.4285E − 02	6.5531E − 03	1.6941E − 03	1.3019E − 02	4.2224E − 04	1.4021E − 03	7.8079E − 03
	best	2.2291E − 06	1.7481E − 07	4.3842E − 05	5.7237E − 09	2.8564E − 06	1.0858E − 08	1.8538E − 06	5.4933E − 07	8.3696E − 07
	worst	1.9084E − 02	6.0885E − 03	6.9301E − 02	3.4724E − 02	6.8771E − 03	7.1140E − 02	2.1791E − 03	4.5320E − 03	3.3373E − 02
	time (s)	0.16966	0.17456	0.18482	0.18627	0.18441	0.21241	0.21215	0.21252	0.23271
	µ	1.4616	1.1967	1.2660	1.3611	1.4583	1.6591	1.3290	1.2964	1.2962
	σ	8.1265E − 01	4.8083E − 01	7.3238E − 01	9.8697E − 01	1.0266	1.1411	7.0512E − 01	6.4604E − 01	6.9592E − 01
	best	9.9800E − 01	9.9800E − 01	9.9800E − 01	9.9800E − 01	9.9800E − 01	9.9800E − 01	9.9800E − 01	9.9800E − 01	9.9800E − 01
	worst	2.9826	2.9821	3.9683	5.9288	5.9289	5.9289	3.9684	2.9822	3.9683
	time (s)	0.32084	0.33257	0.34214	0.34346	0.34196	0.36886	0.36979	0.37041	0.38950
	µ	6.9593E − 04	5.6791E − 04	7.2165E − 04	7.4730E − 04	6.4761E − 04	8.3480E − 04	4.3005E − 04	4.5401E − 04	7.2890E − 04
	σ	6.5262E − 04	5.2377E − 04	6.8417E − 04	7.2734E − 04	5.9863E − 04	7.0704E − 04	2.3978E − 04	3.7599E − 04	6.8438E − 04
	best	3.0968E − 04	3.1245E − 04	3.3263E − 04	3.1626E − 04	3.1369E − 04	3.1005E − 04	3.0879E − 04	3.1015E − 04	3.0778E − 04
	worst	2.2748E − 03	2.2940E − 03	2.7237E − 03	2.2707E − 03	2.2770E − 03	2.2655E − 03	1.5428E − 03	2.2563E − 03	2.2657E − 03
	time (s)	0.02653	0.02646	0.03600	0.03629	0.03586	0.06384	0.06359	0.06339	0.08297
	µ	− 1.0316	− 1.0316	− 9.9463E − 01	− 1.0316	− 1.0316	− 1.0316	− 1.0313	− 1.0316	− 1.0316
	σ	1.0912E − 04	5.2036E − 05	1.8810E − 01	7.6720E − 05	6.3410E − 05	5.8562E − 05	4.4904E − 04	3.1251E − 05	9.6064E − 05
	best	− 1.0316	− 1.0316	− 1.0316	− 1.0316	− 1.0316	− 1.0316	− 1.0316	− 1.0316	− 1.0316
	worst	− 1.0310	− 1.0314	− 4.8863E − 11	− 1.0312	− 1.0313	− 1.0313	− 1.0297	− 1.0315	− 1.0312
	time (s)	0.02562	0.02616	0.03540	0.03581	0.03555	0.06320	0.06292	0.06272	0.08237
	µ	5.5335E − 01	5.5428E − 01	3.9903E − 01	4.0161E − 01	5.5535E − 01	3.9837E − 01	3.9954E − 01	4.0047E − 01	4.0017E − 01
	σ	8.4741E − 01	8.4725E − 01	2.9976E − 03	1.0757E − 02	8.4715E − 01	1.9827E − 03	7.6711E − 03	1.1974E − 02	8.0733E − 03
	best	3.9789E − 01	3.9790E − 01	3.9789E − 01	3.9789E − 01	3.9789E − 01	3.9789E − 01	3.9789E − 01	3.9789E − 01	3.9789E − 01
	worst	5.0401	5.0401	4.1283E − 01	4.4429E − 01	5.0401	4.0879E − 01	4.4006E − 01	4.6349E − 01	4.3936E − 01
	time (s)	0.02173	0.02192	0.03128	0.03141	0.03114	0.05834	0.05823	0.05796	0.07777
	µ	3.0042	4.8200	4.6879	4.8218	7.5993	7.6374	3.0071	3.7837	5.7185
	σ	1.0714E − 02	6.9144	6.3949	6.9242	1.0450E + 01	1.0547E + 01	1.5667E − 02	4.2688	8.2847
	best	3.0000	3.0000	3.0000	3.0000	3.0000	3.0000	3.0000	3.0000	3.0000
	worst	3.0535	3.0511E + 01	3.0036E + 01	3.0535E + 01	3.1943E + 01	3.2698E + 01	3.0770	2.6386E + 01	3.0334E + 01
	time (s)	0.02072	0.02118	0.03051	0.03078	0.03047	0.05814	0.05784	0.05768	0.07733
	µ	− 3.8084	− 3.7882	− 3.8207	− 3.8051	− 3.8161	− 3.8210	− 3.8065	− 3.8042	− 3.8234
	σ	6.8103E − 02	7.2974E − 02	4.4925E − 02	7.2504E − 02	5.8464E − 02	6.0093E − 02	6.2978E − 02	7.5441E − 02	5.9419E − 02
	best	− 3.8627	− 3.8625	− 3.8627	− 3.8627	− 3.8627	− 3.8622	− 3.8627	− 3.8623	− 3.8621
	worst	− 3.6046	− 3.6062	− 3.7446	− 3.6046	− 3.6098	− 3.6123	− 3.6048	− 3.6047	− 3.6062
	time (s)	0.02961	0.03041	0.03986	0.04023	0.03985	0.06808	0.06775	0.06745	0.08695
	µ	− 2.9224	− 2.9588	− 2.9289	− 2.9457	− 2.9914	− 2.8613	− 3.0613	− 2.8919	− 2.9745
	σ	2.6517E − 01	2.0339E − 01	2.8543E − 01	2.9697E − 01	1.1386E − 01	3.8659E − 01	1.5020E − 01	3.6193E − 01	1.4565E − 01
	best	− 3.2435	− 3.2076	− 3.2230	− 3.2865	− 3.2068	− 3.2182	− 3.3053	− 3.2644	− 3.2012
	worst	− 1.9663	− 2.2328	− 2.0273	− 1.5576	− 2.7677	− 1.6062	− 2.7535	− 1.2910	− 2.6593
	time (s)	0.03050	0.03118	0.04080	0.04108	0.04063	0.06879	0.06846	0.06871	0.08801
	µ	− 7.4682	− 8.7694	− 7.3967	− 6.8685	− 9.0223	− 6.4077	− 8.4560	− 8.3156	− 9.1699
	σ	4.3230	3.2622	4.2918	4.4464	2.9582	4.6721	3.6889	3.6245	2.4432
	best	− 1.0152E + 01	− 1.0153E + 01	− 1.0152E + 01	− 1.0148E + 01	− 1.0153E + 01	− 1.0152E + 01	− 1.0153E + 01	− 1.0153E + 01	− 1.0153E + 01
	worst	− 2.7312E − 01	− 3.5065E − 01	− 2.7312E − 01	− 2.7312E − 01	− 2.7312E − 01	− 2.7312E − 01	− 2.7312E − 01	− 3.5065E − 01	− 2.7312E − 01
	time (s)	0.03468	0.03544	0.04554	0.04552	0.04505	0.07347	0.07314	0.07364	0.09286
	µ	− 9.4833	− 9.9633	− 8.7997	− 8.5174	− 8.8814	− 9.8728	− 9.6541	− 9.9437	− 9.8644
	σ	2.4665	1.7278	3.4218	3.6700	3.2505	1.7314	2.4655	1.7206	1.7200
	best	− 1.0403E + 01	− 1.0403E + 01	− 1.0403E + 01	− 1.0396E + 01	− 1.0395E + 01	− 1.0403E + 01	− 1.0403E + 01	− 1.0403E + 01	− 1.0403E + 01
	worst	− 5.2051E − 01	− 8.7635E − 01	− 4.9355E − 01	− 3.7371E − 01	− 3.7353E − 01	− 8.3168E − 01	− 2.9362E − 01	− 9.0766E − 01	− 9.0628E − 01
	time (s)	0.03986	0.04066	0.05041	0.05087	0.05041	0.07874	0.07854	0.07909	0.09836
	µ	− 1.0386E + 01	− 1.0309E + 01	− 1.0185E + 01	− 9.9903	− 1.0371E + 01	− 1.0282E + 01	− 1.0140E + 01	− 1.0205E + 01	− 1.0296E + 01
	σ	3.6773E − 01	4.0067E − 01	1.0293	1.7558	2.4154E − 01	3.1828E − 01	1.7434	5.0193E − 01	3.9741E − 01
	best	− 1.0536E + 01	− 1.0536E + 01	− 1.0536E + 01	− 1.0536E + 01	− 1.0536E + 01	− 1.0536E + 01	− 1.0536E + 01	− 1.0536E + 01	− 1.0536E + 01
	worst	− 8.6649	− 8.7252	− 4.9363	− 9.2437E − 01	− 9.3997	− 9.1262	− 9.3599E − 01	− 8.2360	− 8.8391
	time (s)	0.04637	0.04784	0.05776	0.05803	0.05763	0.08609	0.08595	0.08646	0.10581

Table 9.

Wilcoxon sign rank sum test at different scenarios using Chebyshev chaotic map considering initialization enhancement.

Function	Scenario
	S1		S2		S3		S4		S5		S6		S7		S8
	h	p	h	p	h	p	h	p	h	p	h	p	h	p	h	p
F1	1	8.2185E − 84	1	8.2185E − 84	1	1.5506E − 84	1	5.5578E − 84	1	9.9773E − 84	1	9.1976E − 84	1	2.1143E − 30	1	1.0342E − 40
F2	1	1.2873E − 08	1	1.2873E − 08	1	4.6910E − 84	1	6.3881E − 84	1	4.5632E − 84	1	1.1096E − 83	1	9.9384E − 84	1	6.9213E − 25
F3	1	1.5384E − 27	1	1.5384E − 27	1	2.3278E − 51	1	7.7751E − 84	1	1.1213E − 84	1	7.2773E − 84	1	1.0263E − 83	1	5.3478E − 84
F4	1	7.5631E − 85	1	7.5631E − 85	1	1.0380E − 03	1	1.1072E − 83	1	8.0144E − 84	1	9.5370E − 85	1	5.4782E − 84	1	9.7580E − 84
F5	1	1.5264E − 74	1	1.5264E − 74	1	3.2007E − 62	1	1.3879E − 44	1	9.9206E − 53	1	1.0137E − 83	1	8.8734E − 84	1	7.4593E − 84
F6	1	3.8426E − 33	1	3.8426E − 33	1	3.4333E − 76	1	6.2407E − 23	0	5.6094E − 01	0	1.4686E − 01	1	9.1796E − 84	1	4.4171E − 84
F7	1	5.3752E − 71	1	5.3752E − 71	1	1.7946E − 70	0	6.1408E − 01	1	1.7596E − 59	1	6.7435E − 26	1	3.0512E − 49	1	7.8564E − 84
F8	1	2.7227E − 84	1	2.7227E − 84	1	2.5494E − 84	1	7.9447E − 84	1	4.6387E − 84	1	7.8684E − 84	1	1.9164E − 84	1	1.0051E − 83
F9	1	9.7756E − 84	1	9.7756E − 84	1	5.2695E − 51	1	1.0708E − 81	1	9.3954E − 84	1	4.2462E − 66	1	2.8415E − 64	1	1.0089E − 04
F10	1	3.4204E − 77	1	3.4204E − 77	1	8.5412E − 84	1	2.2895E − 51	1	6.8515E − 84	1	8.6364E − 84	1	1.8034E − 25	1	1.4024E − 84
F11	1	8.2944E − 68	1	8.2944E − 68	1	1.0864E − 08	1	8.6710E − 84	1	1.1106E − 50	1	3.3906E − 48	1	9.9421E − 84	1	1.1183E − 10
F12	1	5.9868E − 55	1	5.9868E − 55	1	1.6729E − 45	1	1.5703E − 25	1	9.7370E − 84	1	3.7260E − 56	1	3.3196E − 10	1	5.2639E − 20
F13	1	1.1867E − 72	1	1.1867E − 72	1	2.8686E − 26	1	1.1079E − 83	1	6.5027E − 69	1	8.9874E − 84	1	2.5095E − 33	1	7.1351E − 61
F14	1	1.6874E − 29	1	1.6874E − 29	1	5.1921E − 13	0	8.6021E − 01	1	7.1177E − 84	1	5.3028E − 15	1	5.2316E − 85	1	6.8126E − 09
F15	1	7.1700E − 84	1	7.1700E − 84	1	2.6511E − 46	1	3.7183E − 49	1	1.5860E − 41	1	1.4012E − 44	1	3.4500E − 27	1	1.0745E − 83
F16	1	2.0911E − 29	1	2.0911E − 29	1	4.6788E − 41	1	8.1740E − 84	1	1.3254E − 83	1	5.6131E − 84	1	1.1829E − 59	1	9.9133E − 84
F17	1	8.1734E − 84	1	8.1734E − 84	1	5.1315E − 66	1	1.5997E − 49	1	1.5088E − 38	0	8.8934E − 01	1	4.5613E − 41	1	1.1233E − 04
F18	1	8.4393E − 84	1	8.4393E − 84	1	6.7918E − 84	1	8.1801E − 84	1	4.2743E − 59	1	1.6530E − 28	1	4.8775E − 13	1	1.3093E − 30
F19	1	9.3190E − 84	1	9.3190E − 84	1	6.6982E − 84	1	3.2858E − 84	1	4.0867E − 84	1	6.2902E − 84	1	9.4316E − 84	1	8.5535E − 84
F20	1	6.7071E − 84	1	6.7071E − 84	1	1.1675E − 83	1	9.4723E − 84	1	5.9060E − 84	1	6.9466E − 84	1	6.1666E − 84	1	1.0734E − 83
F21	1	1.6343E − 44	1	1.6343E − 44	1	5.7913E − 61	1	1.0837E − 83	1	9.3045E − 84	1	5.9199E − 65	1	6.0048E − 84	1	7.8146E − 84
F22	1	1.0777E − 24	1	1.0777E − 24	1	1.9138E − 61	1	1.3317E − 27	1	9.3648E − 84	1	9.5469E − 84	1	8.1044E − 77	1	8.4596E − 84
F23	1	1.3595E − 05	1	1.3595E − 05	1	7.8941E − 13	1	2.6989E − 65	1	6.9124E − 23	1	8.5716E − 84	1	7.5213E − 84	1	6.9444E − 58
	23		23		23		21		22		21		23		23

Table 10.

Wilcoxon sign rank sum test at different scenarios using Gauss-mouse chaotic map considering initialization enhancement.

Function	Scenario
	S1		S2		S3		S4		S5		S6		S7		S8
	h	p	h	p	h	p	h	p	h	p	h	p	h	p	h	p
F1	1	7.3108E − 84	1	7.3108E − 84	1	1.0022E − 83	1	8.5490E − 84	1	9.6926E − 84	1	7.0254E − 84	1	1.3052E − 22	1	3.6252E − 38
F2	1	1.8294E − 17	1	1.8294E − 17	1	7.3666E − 84	1	8.3445E − 84	1	8.2846E − 84	1	9.8991E − 84	1	8.4580E − 84	1	1.1804E − 33
F3	1	2.3617E − 04	1	2.3617E − 04	1	7.1380E − 84	1	6.4232E − 84	1	8.0240E − 84	1	8.6900E − 84	1	5.4478E − 84	1	5.5067E − 84
F4	1	9.1434E − 55	1	9.1434E − 55	1	9.3915E − 05	1	6.8248E − 21	1	7.2431E − 84	1	8.1537E − 84	1	8.9316E − 84	1	8.8396E − 84
F5	1	8.0016E − 84	1	8.0016E − 84	1	6.3010E − 56	1	2.1330E − 11	1	5.8709E − 84	1	1.0777E − 83	1	1.0410E − 83	1	1.0736E − 83
F6	1	5.7235E − 79	1	5.7235E − 79	1	9.4084E − 84	1	1.2118E − 43	1	2.1491E − 11	1	4.2627E − 15	1	7.9439E − 84	1	7.8004E − 84
F7	1	1.2749E − 57	1	1.2749E − 57	1	2.9861E − 61	0	2.4543E − 01	1	4.3351E − 44	1	7.6510E − 38	1	7.2373E − 19	1	8.1455E − 84
F8	1	1.0170E − 83	1	1.0170E − 83	1	9.2142E − 84	1	1.0047E − 83	1	7.0402E − 84	1	7.0455E − 84	1	2.0454E − 84	1	4.8930E − 84
F9	1	4.9107E − 84	1	4.9107E − 84	1	1.4491E − 45	1	1.4737E − 76	1	6.7402E − 60	1	1.4074E − 53	1	1.0370E − 51	1	3.6718E − 03
F10	1	2.3262E − 46	1	2.3262E − 46	1	4.6324E − 84	1	1.3362E − 30	1	4.9316E − 84	1	9.7441E − 84	1	1.4317E − 18	1	1.0411E − 43
F11	1	1.6922E − 52	1	1.6922E − 52	1	7.6385E − 03	1	5.4006E − 84	1	7.6310E − 39	1	5.3128E − 84	1	1.1396E − 83	0	1.9036E − 01
F12	1	2.7825E − 13	1	2.7825E − 13	1	9.2232E − 84	1	5.4571E − 22	1	8.2456E − 84	1	1.1486E − 26	1	1.4592E − 31	0	4.8651E − 01
F13	1	4.2601E − 50	1	4.2601E − 50	1	1.3308E − 35	1	8.8775E − 84	1	1.0320E − 24	1	9.9754E − 84	1	1.0671E − 56	1	3.3727E − 37
F14	1	1.1699E − 24	1	1.1699E − 24	1	1.9625E − 06	1	3.4545E − 21	1	7.1561E − 84	1	8.3285E − 42	1	7.2368E − 84	1	2.5004E − 10
F15	1	8.5281E − 84	1	8.5281E − 84	1	4.2202E − 43	1	1.8568E − 62	1	1.9287E − 54	1	3.7024E − 22	1	4.2906E − 43	1	6.2889E − 84
F16	1	7.1548E − 84	1	7.1548E − 84	1	6.9220E − 84	1	9.9826E − 84	1	9.5435E − 84	1	3.3233E − 84	1	2.2588E − 80	1	7.6760E − 84
F17	1	9.0037E − 84	1	9.0037E − 84	1	7.1657E − 84	1	3.8855E − 84	1	1.6798E − 33	1	1.9924E − 06	1	3.5100E − 52	1	1.7977E − 19
F18	1	6.8659E − 84	1	6.8659E − 84	1	9.7454E − 84	1	7.4365E − 84	1	5.8437E − 57	1	1.7636E − 22	1	1.9820E − 06	1	1.8410E − 29
F19	1	5.1003E − 84	1	5.1003E − 84	1	9.4050E − 84	1	4.0407E − 84	1	9.0316E − 84	1	7.5842E − 84	1	9.8619E − 84	1	9.0286E − 84
F20	1	5.9426E − 84	1	5.9426E − 84	1	1.0438E − 83	1	4.5889E − 84	1	7.1080E − 84	1	8.5603E − 84	1	4.2734E − 84	1	1.1188E − 83
F21	1	7.9155E − 52	1	7.9155E − 52	1	1.3343E − 23	1	8.7986E − 84	1	4.9231E − 84	1	5.0646E − 83	1	9.9202E − 84	1	9.0187E − 84
F22	1	1.0910E − 02	1	1.0910E − 02	1	3.8668E − 73	1	8.8142E − 39	1	6.2248E − 84	1	5.2073E − 84	1	6.7883E − 84	1	1.0343E − 83
F23	1	1.3273E − 09	1	1.3273E − 09	1	9.7604E − 81	1	1.2055E − 76	1	3.6502E − 08	1	1.0024E − 83	1	1.0414E − 83	1	3.2724E − 83
	23		23		23		22		23		23		23		21

Fig. 3.

Convergence curves using Chebyshev chaotic map considering initialization enhancement.

Fig. 4.

Convergence curves using iterative chaotic map.

Testing CTSO on classical engineering applications

The emergence of chaotic maps in the previous case studies with the original TSO has proven its capabilities in finding significant near-optimal solutions for the studied benchmark test functions with/without initialization enhancement using CMs. In this case study, we are going to test the proposed CTSO in solving three well-known optimization problems²⁸, including (a) Coil spring design problem, (b) Welded beam design problem, and (c) Pressure vessel design problem. The formulation of these problems is given as follows:

1. Coil spring design problem.

6

• (b)Welded beam design problem.

7

• (c)Pressure vessel design problem.

8

The CTSO is conducted with/without initialization enhancement to get various near-optimal solutions to the proposed engineering optimization problems. The best result obtained among all scenarios is given in Tables 11 and 12, and 13, compared to recent metaheuristic optimization algorithms to determine the benefits of employing CTSO. Besides, the statistical t-test is conducted at 5% significance level, to assess the significance of the CTSO solutions against the rest of optimizers. From the obtained results, it can be noted that the CTSO has the superiority in finding a better near-optimal solution rather than the rest of the optimization algorithms for welded beam design problem, and pressure vessel design problems. However, the CSA succeeded in finding a near-optimal solution for the coil spring design problem better than CTSO. Thus, from the obtained results, we can rely on CTSO as a successful optimizer in finding near-optimal solutions for complex optimization problems.

Table 11.

Results of the coil spring design problem.

Optimizer				µ	σ	Best	Worst	Time (s)	p	h
CTSO	0.05	0.25	14.99951	0.01317085	0.001821	0.010625	0.018056	0.096958	5.44E − 217	1
TSO28	0.050011	0.250057	15	0.01361655	0.001153	0.010632	0.016112	0.064567	5.06E − 08	1
CSA27	0.05	0.25	2	0.02249758	0.03033	0.002513	0.124254	0.050639	0	1
SCA30	0.052975	0.387564	9.749836	0.01307536	0.000153	0.01278	0.013268	0.062458	2.98E − 243	1
HHO50	0.05	0.316727	14.1208	0.01459635	0.005233	0.012765	0.040271	0.08496	0	1
WOA11	0.059204	0.565896	4.866475	0.01517832	0.001525	0.01362	0.017791	0.158732	9.67E − 303	1

Table 12.

Results of the welded beam design problem.

Optimizer					µ	σ	Best	Worst	Time (s)	p	h
CTSO	0.207118	3.453231	9.010337	0.207118	2.10959065	0.40706	1.730649	10.45734	0.102798	0	1
TSO	0.178696	4.17524	9.263323	0.20483	2.81798617	0.948441	1.806401	5.421261	0.069941	4.36E-214	1
CSA	0.906854	4.304135	4.304135	0.906854	7.34751653	2.40E-12	7.347517	7.347517	0.056001	0	1
SCA	0.212873	3.521307	8.915651	0.216016	1.95280227	0.08103	1.799733	2.134337	0.069363	0.083263896	1
HHO	0.210172	3.379705	9.030371	0.213378	2.10644297	0.33633	1.77606	3.132649	0.171783	6.47E-211	1
WOA	0.215275	3.394414	8.721502	0.223685	3.04332767	1.034697	1.806359	7.317017	0.067924	4.45E-129	1

Table 13.

Results of the pressure vessel design problem.

Optimizer					µ	σ	Best	Worst	Time (s)	p	h
CTSO	0.847027	0.4085	42.70182	169.3593	21,017.4448	15,596.1	6129.613	1,907,246	0.090833	3.40E − 34	1
TSO	1.103533	0.54611	56.17732	55.81466	52,467.9498	38,330.22	6790.577	158,465.2	0.057136	5.72E − 80	1
CSA	24.03375	24.03375	56.1254	56.1254	927,576.998	0.085446	927,576.5	927,577	0.045231	0	1
SCA	0.80545	0.472592	40.32569	200	7676.64133	881.3007	6339.484	9510.401	0.059006	5.63E − 10	1
HHO	0.857836	0.452041	43.85324	156.0411	6902.64456	248.3862	6203.099	7515.965	0.1464	5.33E − 56	1
WOA	0.975848	0.449964	46.55691	128.2452	15,314.7258	12,841.32	6626.895	68,262.47	0.05597	4.48E − 05	1

Research outcomes and conclusions

This research proposes a multi-scenario optimization strategy to obtain near-optimal solutions to 23 benchmark functions and solve three well-known engineering optimization problems. The results showed the effectiveness of emerging chaotic maps instead of random numbers in the TSO algorithm, aiming to regulate its randomness without getting trapped in local optima due to its ergodic nature. Results showed the effectiveness of the proposed multi-scenario CTSO in finding significant near-optimal solutions to the studied optimization problems and the enhanced convergence revealed after introducing CMs. Future works will consider the application of chaotic maps in development of effective multi-objective optimization algorithms.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Author contributions

I.M.D.: Concept, formal analysis, methodology, validation, writing original draft. H.M.H.: Concept, validation, visulization, supervision, review. M.H.Q., S.A. and O.A.M.O.: Concept, validation, editing, review.

Data availability

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.