Lévy flight and chaos theory-based gravitational search algorithm for mechanical and structural engineering design optimization

Sajad Ahmad Rather; Perumal Shanthi Bala

doi:10.1515/comp-2020-0223

Open Access Published by De Gruyter Open Access December 31, 2021

Lévy flight and chaos theory-based gravitational search algorithm for mechanical and structural engineering design optimization

Sajad Ahmad Rather and Perumal Shanthi Bala

From the journal Open Computer Science

https://doi.org/10.1515/comp-2020-0223

Abstract

The main aim of this article is to explore the real-life problem-solving potential of the proposed Lévy flight-based chaotic gravitational search algorithm (LCGSA) for the minimization of engineering design variables of speed reducer design (SRD), three bar truss design (TBTD), and hydrodynamic thrust bearing design (HTBD) problems. In LCGSA, the diversification of the search space is carried out by Lévy flight distribution. Simultaneously, chaotic maps have been utilized for the intensification of the candidate solutions towards the global optimum. Moreover, the penalty function method has been used to deal with the non-linear and fractional design constraints. The investigation of experimental outcomes has been performed through various performance metrics like statistical measures, run time analysis, convergence rate, and box plot analysis. Moreover, statistical verification of experimental results is carried out using a signed Wilcoxon rank-sum test. Furthermore, eleven heuristic algorithms were employed for comparative analysis of the simulation results. The simulation outcomes clearly show that LCGSA provides better values for TBTD and HTBD benchmarks than standard GSA and most of the competing algorithms. Besides, all the participating algorithms, including LCGSA, have the same results for the SRD problem. On the qualitative side, LCGSA has successfully resolved entrapment in local minima and convergence issues of standard GSA.

Keywords: gravitational search algorithm; chaotic maps; Lévy flight distribution; engineering design optimization; swarm intelligence

Acronyms

ABC: artificial bee colony
ACO: ant colony optimization
AFO: artificial flora optimization
ALO: ant lion optimizer
BA: bat algorithm
BBO: biogeography-based optimization
BMO: barnacles mating optimizer
BWO: black widow optimization
CPSOGSA: constriction coefficient-based PSO and GSA
CS: cuckoo search
CSO: cat swarm optimization
DE: differential evolution
FA: firefly algorithm
FSA: fish swarm algorithm
GA: genetic algorithm
GSA: gravitational search algorithm
GWO: grey wolf optimizer
HA: heuristic algorithm
HS: harmony search
HTBD: hydrodynamic thrust bearing design problem
LCGSA: Lévy flight-based chaotic gravitational search algorithm
MVO: multi-verse optimizer
PFM: penalty function method
PSO: particle swarm optimization
SA: simulated annealing
SCA: sine-cosine algorithm
SRD: speed reducer design problem
SRO: search and rescue operations optimization
SSA: salp swarm algorithm
TBTD: three bar truss design problem

1 Introduction

The main goal of an optimization process is the maximization/minimization of a real function. In simpler terms, optimization is a mathematical procedure that maximizes profit and minimizes loss. Practically speaking, most of the engineering problems are basically optimization tasks having input/output systems with the sole aim of finding the optimal value of the fitness function, which may be minimum weight, less volume, more throughput, and so on. In fact, optimization problems can be solved using gradient descent methods and heuristic algorithms (HAs). It has been reported that gradient descent methods have problems of local minima entrapment, premature convergence, and unimodality. In contrast, HAs are stochastic techniques with randomized behavior. They have the characteristics like simple design, flexibility, and less complexity, which makes them suitable for solving complex optimization problems.

Actually, all the HAs are stochastic in nature. In fact, the first step of heuristic optimization is the initialization of candidate solutions. Consequently, the iteration process changes the values of the candidate solutions. Obviously, after the fulfillment of the stopping condition, the optimization process gives the best candidate solution.

Fundamentally, HAs consist of two basic properties of exploration and exploitation. The exploration (diversification) is defined as the lower and upper limits of the solution space where searcher agents can move during the optimization process. Moreover, searcher agents change values more often during the exploration phase. In contrast, exploitation (intensification) is the process of finding an optimal solution from the rich pool of feasible candidate solutions. Besides, the searcher agents undergo less number of changes during this phase. The exploration and exploitation are inversely proportional to each other. So, a proper balance between them during the optimization process is necessary for getting the best solutions [1].

Interestingly, nature has been a source of inspiration for most of the HAs. They have been proposed to mathematically model practical, real-world problems and help to solve them optimally. According to ref. [2], “Novel optimization algorithms are always welcome as no HA can solve all the optimization problems.” Consequently, researchers have invented and proposed new HAs. Some of the examples of HAs include genetic algorithm (GA) [3], particle swarm optimization (PSO) [4], ant colony optimization (ACO) [5], differential evolution (DE) [6], biogeography-based optimization (BBO) [7], and many more. Moreover, the recent inclusions into the list of HAs include grey wolf optimizer (GWO) [8], ant lion optimizer (ALO) [9], sine-cosine algorithm (SCA) [10], salp swarm algorithm (SSA) [11], artificial flora optimization (AFO) [12], search and rescue operations optimization (SRO) [13], black widow optimization (BWO) [14], cat swarm optimization (CSO) [15], and barnacles mating optimizer (BMO) [16].

In the list of HAs, gravitational search algorithm (GSA) is one of the popular and highly utilized optimization techniques. It is inspired by physical sciences and is based on Newton’s law of universal gravitation and the second law of motion. In GSA, candidate solutions are masses that move from one place to another, according to Newtonian mechanics. The position of the heavy mass gives the best solution after going through the maximum number of iterations. Furthermore, GSA has a remarkable global exploration capability. However, it has the shortcomings of slow intensification and entrapment in local minima [17,18,19]. Therefore, we have proposed a new version of GSA based on Lévy flight and chaotic maps, namely, Lévy flight and chaos theory-based gravitational search algorithm (LCGSA).

In LCGSA, Lévy flight carries out global exploration of the search space, while ten chaotic maps provide optimal candidate solutions. It is obvious that the actual challenge of optimization techniques is testing their capability of solving practical and real-world problems. It is because they have complex, non-linear, and unknown solution spaces, they are difficult to solve. Thus, the effectiveness of LCGSA will be tested by applying it to three constrained engineering design problems, namely, speed reducer design problem (SRD), three bar truss design problem (TBTD), and hydrodynamic thrust bearing design problem (HTBD). It will be quite fascinating to observe how LCGSA will resolve various design constraints and optimally minimize the design parameters.

The article’s remaining sections are organized as follows; Section 2 describes the motivation behind the work. Besides, Section 3 covers related literature works. Standard GSA and LCGSA algorithms are explained in Section 4 and Section 5, respectively. Experimental analysis is discussed in Section 6. Finally, Section 7 provides the conclusion and future research directions of the proposed work.

2 Motivation

The Lévy flight distribution is a very useful applied mathematics technique. It is basically a power series having infinite variance and variable step size. These properties are essential while dealing with sensitivity in initialization and falling in local minima drawbacks of HAs.

Also, it has been reported in a number of studies that non-linear chaotic progression(s) are superior as compared to random numbers [20]. Because chaotic sequences are pattern dependent, any unsymmetrical change creates ripples throughout the whole chaotic system. Moreover, chaotic sequences can take candidate solutions from infeasible regions of search spaces towards the feasible global minima neighborhood.

In addition, it is the first time that Lévy flight and chaos theory techniques have been hybridized to solve engineering design problems. Previously, researchers have used chaotic maps to tackle simple function optimization problems [21,22]. Besides, the aforementioned studies used only a few chaotic maps, and no proper statistical analysis of simulation results was provided. In contrast, the current study uses Lévy flight and ten distinct chaotic maps embedded with GSA for engineering optimization.

Basically, LCGSA has not only high exploitation and strong global exploration capabilities but also has potential in resolving slow convergence and exploitation limitations of standard GSA. So, it will be quite exciting to observe how LCGSA handles complex search spaces and inequality constraints of engineering problems.

3 Literature survey

GSA is a very promising optimization technique that is inspired by physics. It is based on the principle of mass attraction. The candidate solutions are actually masses. Obviously, a heavy mass object has high-intensity field and attracts other feasible candidate solutions towards itself. Consequently, the position of the heavy mass object gives the best solution. Further, GSA has better exploration power but it has the issues of premature convergence speed and entrapment in local minima. Researchers have provided a number of innovative approaches to fix GSA problems. One of the solutions is provided by Rodenas et al., in which they have utilized chaotic maps and quasi-Newton methods to provide equilibrium balance between diversification and intensification phases in standard GSA. The modified version of GSA was named memetic GSA. It has been applied to a number of real-world and artificial test functions and showed promising outcomes [23].

In standard GSA, the heavy masses are important for getting global optimum as they have high fitness function values. However, heavy masses can also create neighborhood local minima problems which result in premature convergence. To alleviate this problem, improved version of GSA was developed to increase the exploitation capability of GSA. To test the efficiency of GSA, it was applied to Loney’s solenoid design problem [24].

It has been reported that PSO has a high speed of convergence but it has the drawback of reduced diversity of candidate solutions. Consequently, GSA was hybridized with PSO to provide the global searching capability. The hybrid PSO-GSA model exhibited efficient convergence speed as compared to other participating algorithms [25].

The global exploration ability of GSA has been utilized to solve a number of problems including feature selection. In fact, the binary quantum version of GSA has been combined with a k-nearest neighbor classifier to increase the classification accuracy and reduce the dimensions of the recognition datasets [26].

The fuzzy logic concepts have been combined with GSA to optimally design infinite impulsive response (IIR) filter. The fuzzy GSA provided better robustness and stability as compared to standard GSA and DE [27].

Most recently, scientists have utilized the concept of species niche from environmental sciences for solving function optimization and real-world problems through mathematical modeling. Basically, Haghbayan et al. have proposed niche GSA in order to optimize multimodal functions. In niche GSA, the swarm niches are formed using the nearest neighbor classifier whereas the hill valley algorithm detects population niches. The simulation results depicted high performance of niche GSA [28]. In another work, Yazdani et al. modified GSA by making improvements in mass equations and introduced elitism criterion for finding multiple local minima in benchmark functions [29].

In the last decade, several research articles have been written on the topic of object tracking. It has been a hot topic in the field of computer vision because it has various potential applications such as vehicle navigation, traffic monitoring, and so on. Likewise, deep learning concepts have been combined with GSA to track objects optimally in video frames [30].

It is a fact that GSA has efficient global searching capability. There were number of attempts made by researchers to enhance the exploration proficiency of GSA. Actually, Khajooei et al. introduced the concepts of positive and negative masses to advance the diversification power of standard GSA [31]. Similarly, a discrete version of GSA was developed to solve the 0-1 knapsack problem by modifying the position equation and fitness function of standard GSA. The simulation outcomes indicated better performance of discrete GSA in terms of faster convergence rate and better accuracy [32].

Quite recently, Zandevakili et al. came up with modified GSA based on the notion of attractive and repulsive forces (AR-GSA) to overcome premature convergence problems of standard GSA. Moreover, AR-GSA was applied to CEC-2013 test functions and exhibited promising simulation results [33].

It has been seen that chaos theory has a number of applications in many fields such as computer science, meteorology, environmental sciences, and so on. Actually, chaos theory deals with the study of dynamical systems whose disorder states are highly reactive to any perturbation in the seed values. Moreover, chaos theory is based on the principle of the butterfly effect. Besides, Mirjalili et al. have combined chaotic maps with the gravitational constant parameter of standard GSA in order to provide symmetry between diversification and intensification stages. The experiment showed better performance for chaotic GSA as compared to standard GSA.

Also, Rather et al. have applied chaotic GSA for engineering optimization. They solved three mechanical engineering design problems using chaotic GSA. Besides, it was shown that CGSA provided better results as compared to other competing algorithms including classical GSA.

Similarly, chaotic maps have been combined with GSA to identify parameters of the chaotic systems. It was accomplished by utilizing chaotic maps for local search to increase the speed of convergence, while GSA carried out global exploration of the solution space [34].

In another research, the stochastic and ergodicity properties of chaotic maps have been embedded with GSA to solve eight unconstrained numerical test functions. It has been seen that CGSA provided better performance as compared to GSA and different PSO versions [35].

It has been observed that PSO has powerful exploitation capability, but it suffers from the drawback of reduced diversification proficiency. The aforementioned problem of PSO has been solved by using chaotic sequences in place of random numbers. Actually, eight chaotic maps were embedded with classical PSO to increase its exploration power. The simulation results confirmed the efficiency of chaotic PSO versions [36]. Moreover, Alatas has developed a chaotic version of the artificial bee colony (ABC) algorithm to overcome premature convergence and local minima entrapment problems of the traditional ABC algorithm [37]. In addition, Alatas has also designed chaotic mathematical models of harmony search (HS) algorithms to resolve convergence issues of standard HS algorithm. Besides, seven chaotic maps, including sinusoidal, tent, henon, and so on, have been utilized to provide chaotic sequences for the initialization of the search space. The experimental results depicted the optimal performance of chaotic HS algorithms [38].

Differential evolution (DE) is one of the famous evolutionary algorithms developed to solve optimization problems. However, DE has the drawbacks of local minima falling and premature convergence. To overcome these disadvantages, self-adaptive chaotic maps were introduced in traditional DE [39].

Simulated annealing (SA) is another famous heuristic technique mainly utilized for solving combinatorial optimization tasks. It has a simple design, efficient diversification power but slow convergence speed. The chaotic maps were employed to provide chaotic initialization in place of Gaussian distribution to overcome local searching problems of SA [40]. Recently, the firefly algorithm (FA) has also been combined with chaos theory for robust global optimization. In fact, 12 chaotic maps were used for FA parameter tuning [41].

In the family of HAs, BBO is a newly invented stochastic technique utilized mainly for solving real-world optimization problems due to its novel algorithmic design structure. However, while dealing with complex problems, BBO faces the problems of local minima entrapment and premature convergence. Roughly speaking, ten chaotic maps were utilized to overcome the aforementioned disadvantages of BBO. Besides, experimental studies confirmed the optimal performance of chaotic BBO [42].

Likewise, 13 chaotic maps were used to increase the global searching capability of the bat algorithm (BA) [42]. Moreover, cuckoo search (CS) has also been hybridized with chaos theory to improve its solution quality and boost convergence speed [43].

In the last decade, the researchers have used Lévy flight distribution to solve the number of real-world problems in different academic fields such as cryptography, economics, data science, engineering, and so on. Basically, Lévy flight is a heavy-tailed Markov process with variable step length. It has the characteristics of randomness, infinite variance, stable distribution, and scale-invariant property. The aforementioned properties of Lévy distribution are pivotal for the resolution of premature convergence and diversity issues of HAs. In fact, a Lévy flight trajectory with a mutation strategy has been introduced in BA to overcome its convergence and local minima entrapment problems. The benchmark outcomes demonstrated the efficient performance of Lévy based BA as compared to coarse BA [44]. In another work, Lévy flight and opposition based learning were combined with BA, respectively, to resolve its slow convergence and exploration issues. Moreover, 16 test functions were utilized for checking the efficiency of the modified BA [45]. Besides, Li et al. have designed yet another improved version of BA using Lévy distribution and exponentially decreasing inertia weight function. The effectiveness of the modified BA was tested on various benchmark functions and two engineering design optimization problems [46].

In order to overcome early convergence and global searching drawbacks of PSO, Lévy distribution was combined with the position equation of PSO. Actually, variable step size of Lévy random walk helped PSO particles to get away from local minima traps. Moreover, widely used test functions were utilized for the performance evaluation of chaotic PSO [47]. Besides, Lévy flight distribution has been hybridized with PSO to update its velocity equation. Consequently, it results in faster convergence and improved global exploration of the particles towards feasible regions of the search space [48].

It has been seen that the fish swarm algorithm (FSA) has appreciable exploration capability but it has the problem of slow convergence rate. So, Lévy flight random walk and dynamic firefly factor were introduced in FSA to enhance its exploitation power. The simulation results of benchmark functions indicated the improvement in the solution quality and overall accuracy of the classical FSA [49].

Multi-verse optimizer (MVO) [50] is one of the recent HAs inspired by the concepts of astrophysics. In MVO, exploration is performed by white and black holes, while intensification is carried out by warm holes. Besides, searcher agents are in the form of the universe(s). It has been observed that MVO has the disadvantages of entrapment in local minima and low diversity of candidate solutions. To overcome these shortcomings, MVO was hybridized with Lévy flight in order to deal with complex and non-linear search spaces. Moreover, modified Lévy MVO was applied to numerical and test scheduling problem(s) to validate its effectiveness and real-world problem-solving capability [51].

The summarization of the related works is shown in Table 1. It clearly indicates that both chaos theory and Lévy flight have been rigorously used to solve real-world and domain-specific problems. Besides, randomness, ergodicity, and stochasticity properties of chaos theory and Lévy flight distribution have been utilized to solve premature convergence, entrapment in local minima, and accuracy of HAs. Thus, in this study, LCGSA is applied to three engineering design problems to demonstrate its capability in handling non-linear, complex, and unknown search spaces.

Table 1

Outline of the related research works associated with GSA, chaotic maps, and Lévy flight

Authors (year)	Technique	Chaotic maps/Lévy flight	Objective of optimization
Jiang et al. (2020)	GSA/PSO	No	Enhancement in exploration
Khan et al. (2020)	GSA	No	To solve electromagnetic design problem
Rather and Bala (2020b)	GSA	Chaotic maps	Engineering design optimization
Rodenas et al. (2019)	GSA	Chaotic maps	To increase solution quality and accuracy
Zandevakili et al. (2019)	GSA	No	Function optimization
Li et al. (2019)	BA	Lévy flight	Benchmark function and engineering optimization
Pelusi et al. (2018)	Fuzzy GSA	No	IIR filter design
Kang et al. (2018)	GSA	No	Object tracking
Peng et al. (2018)	FSA	Lévy flight	To increase accuracy
Barani et al. (2017)	GSA	No	Classification
Mirjalili et al. (2017)	GSA	Chaotic maps	Function optimization
Sajedi and Razavi (2017)	Discrete GSA	No	To solve knapsack problem
Haghbayan et al. (2017)	Niche GSA	No	Function optimization
Khajooei and Rashedi (2016)	GSA	No	GSA performance enhancement
Shan et al. (2016)	BA	Lévy flight	Improvement in BA
Li et al. (2016)	MFO	Lévy flight	To increase convergence rate of MFO
Jensi and Jiji (2016)	PSO	Lévy flight	Robust global optimization
Hu et al. (2016)	MVO	Lévy flight	Test scheduling
Wang and Zhong (2015)	CS	Chaotic maps	Improvement in solution quality
Yazdani et al. (2014)	GSA	No	Multimodal optimization
Saremi et al. (2014)	BBO	Chaotic maps	To overcome local minima problem
Haklı and Uğuz (2014)	PSO	Lévy flight	To increase exploration power of PSO
Gandomi and Yang (2014)	BA	Chaotic maps	To advance diversification capability of BA
Xie et al. (2013)	BA	Lévy flight	Global optimization
Gandomi et al. (2013)	FA	Chaotic maps	Global optimization
Li et al. (2012)	GSA	Chaotic maps	Chaotic system parameter identification
Gao et al. (2012)	GSA	Chaotic maps	Unconstrained function optimization
Alatas (2010)	ABC	Chaotic maps	Numerical optimization
Alatas (2010)	HS	Chaotic maps	To solve benchmark functions
Alatas et al. (2009)	PSO	Chaotic maps	Function optimization
Zhenyu et al. (2006)	DE	Chaotic maps	To resolve problems of DE algorithm
Mingjun and Huanwen (2004)	SA	Chaotic maps	To increase agent diversity

4 GSA

GSA is one of the highly regarded physics-based HA. It is inspired by the law of gravitation and motion. In fact, gravity is one of the four basic forces in nature. The other three forces are weak nuclear force, electromagnetic force, and the strong nuclear force [52]. Moreover, the law of gravitation is basically an inverse square law which states that “the attractive force between two masses is directly proportional to the product of their masses and inversely proportional to the square of the distance between them” [53,54,55].

The GSA is first initialized with a random distribution of searcher agents in the form of masses. The force between the point masses is calculated in equation (1).

(1) F i j = G ( t ) m p i ( t ) m a j ( t ) R i j ( t ) + ε ( x j d ( t ) + x i d ( t ) ) ,

where m p i ( t ) and m a j ( t ) are passive and active gravitational masses, respectively. The Euclidian distance is represented as R i j ( t ) , while ε is a small constant.

To get a proper balance between exploration and exploitation, the GSA utilizes an important parameter called gravitational constant represented by “G”. It also increases the quality of the candidate solutions as shown in equation (2).

(2) G ( t ) = G ( t 0 ) e − α CI MI ,

where G(t) and G(t ₀) are the values of the gravitational constant at time interval t and t 0 , respectively. Also, α is an exponentially decreasing coefficient whereas CI and MI correspond to the initial simulation and the peak number of iteration(s), respectively.

As the masses are moving in the search space and each of them is exerting a force, the total force is given by equation (3).

(3) F i d ( t ) = ∑ j = 1 , j ≠ i m γ j F i j ,

where γ j has values between 0 and 1.

Furthermore, after a number of iterations, the heavy masses will be scattered throughout the search space which represents feasible solutions. So, it is important to preserve the quality of the best solutions. Therefore, elitism criterion i.e., kbest strategy is used in GSA. It means that force will be exerted in each direction by optimal heavy mass after the fulfillment of stopping criterion. It is shown in equation (4).

(4) F i d ( t ) = ∑ j = k b e s t , j ≠ i m γ j F i j d .

Moreover, the acceleration of the masses is calculated according to the second law of motion as given in equation (5).

(5) a i d ( t ) = F i d ( t ) m i ( t ) .

In GSA, the point masses get attracted to heavy masses because they have the highest intensity and strong force of attraction. Hence, the position and velocity of the heavy mass are pivotal for finding the global optimum which is provided in equations (6) and (7), respectively.

(6) v i d ( t + 1 ) = γ j v i d ( t ) + a i d ( t ) ,

(7) x i d ( t + 1 ) = x i d ( t ) + v i d ( t + 1 ) .

The step-wise description of GSA is presented in Algorithm 1.

Algorithm 1

Standard GSA

1:	Randomized initialization of the solution space
2:	Calculate the fitness of all masses
3:	Initialize the parameters including the maximum number of iterations (T), initial value of the gravitational constant G(t ₀), and coefficient α
4:	Start the iteration counter at t = 0
5:	while t < T do
6:	for each candidate solution do
7:	Update the gravitational constant, G(t)
8:	Find the gravitational force, F i d ( t ) using equation (4)
9:	Calculate the mass acceleration, a i d ( t ) by using equation (5)
10:	Update the mass velocity, v i d ( t + 1 ) by applying equation (6)
11:	Change the mass position, x i d ( t + 1 ) using equation (7)
12:	end for
13:	t = t + 1
14:	end while
15:	Return the optimal candidate solution

5 LCGSA

In this section, first, the preliminary concepts of Lévy flight and chaos theory are introduced to provide the firm foundation for the proposed LCGSA algorithm. In fact, Lévy flight is employed to provide exploration capability and proper balance between the exploration and exploitation phases in LCGSA. Furthermore, ten chaotic maps will help in the local search and speed up the convergence of the candidate solutions towards the global optimum.

5.1 Lévy flight

It has been reported in a number of studies that exploration is important for solving large scale and complex optimization problems [56,57]. In fact, a reduction in the diversity of candidate solutions results in slow convergence speed and entrapment in local minima. Moreover, GSA also has issues with exploration while solving multi-dimensional and complex engineering benchmarks as reported by Rather et al. [18]. To alleviate the exploration issue of GSA, Lévy flight distribution has been embedded with the gravitational constant of GSA. Basically, previous studies [46,48] have shown ample experimental evidence in favor of Lévy flight for solving diversity issues in HAs.

Generally, Lévy flight is a random walk which has step length based on Lévy distribution [58]. Moreover, the Lévy distribution can be defined mathematically by a power-law formula as shown in equation (8).

(8) L ( s ) ∼ ∣ s ∣ ( − 1 − β ) , 0 < β ≤ 2 ,

where s and β represent step size and Lévy index, respectively.

In Lévy flight, the step size is calculated using the Mantegna algorithm as depicted in equation (9).

(9) s = u ∣ v ∣ 1 β ,

where u and v are normal distributions such that

(10) u ∼ N ( 0 , σ u 2 ) , v ∼ N ( 0 , σ v 2 ) .

Also,

(11) σ u = Γ ( 1 + β ) sin ( π β ) 2 β Γ ( 1 + β ) 2 ( β − 1 ) 2 1 β ,

(12) σ v = 1 .

In the proposed LCGSA, Lévy flight is employed to provide stability between diversification and intensification resulting in overcoming of local minima problem. Besides, the infinite variance of Lévy distribution provides a strong capability to Lévy flight for the resolution of falling in local minima and improper diversification issues in LCGSA. In fact, a large step size in Lévy flight results in the enhanced exploration of the search space, while small step size provides high convergence of the solutions towards the global optimum.

5.2 Chaos theory

Broadly speaking, chaos theory is a branch of mathematics dealing with the study of dynamic systems. The chaotic systems are highly sensitive to the changes in the initial conditions. In other words, small transformation in the inputs creates heavy variations in the output of the system. It is obvious that chaotic systems have randomized behavior; however, stochastic behavior is also shown by deterministic systems. In the past decade, the researchers have utilized ergodic and randomness properties of chaotic systems for resolving premature convergence and local search issues in HAs.

In a related work, the performance of PSO was enhanced by introducing chaotic maps in the velocity equation resulting in enhanced exploitation [36]. Likewise, Gandomi et al. [20] came up with chaos-based accelerated PSO. Similarly, different types of chaotic maps have been utilized in the performance improvement of other HAs such as chaotic GA [59], chaotic DE [39], and chaotic BBO [42].

Moreover, chaotic maps have been combined with the imperialist completive algorithm to solve real-world problems [60]. Furthermore, the recent additions into the list of enhanced chaotic maps are chaotic bat [61] and chaotic CS [43] algorithms. These studies confirm the effectiveness and also show the applicability of the chaotic maps in HAs.

In this work, ten chaotic maps have been utilized to enhance the performance of GSA to resolve its premature convergence and exploitation issues. The ten chaotic maps are provided in Table 2. Also, it can be clearly seen that chaotic maps show random behavior as shown in Figure 1.

Table 2

Mathematical equations of chaotic maps

S. No.	Chaotic function	Chaotic map	Limits
1.	Chebyshev	l i + 1 = cos ( i cos − 1 ( l i ) )	(1, −1)
	(Saremi et al., [42])
2.	Circle	l i + 1 = mod l i + q − p 2 π sin ( 2 π l k , 1 ) , p = 0.5 and q = 0.2	(0, 1)
	(Yao et al., [59])
3.	Gauss	l i + 1 = − l , l i = 0 1 mod ( l i , 1 ) , otherwise	(0, 1)
	(Jothiprakash et al., 2013)
4.	Iterative	l i + 1 = sin p π l i , p = 0.7	(−1, 1)
	(Saremi et al., [42])
5.	Logistic	l i + 1 = p l i ( 1 − l i ) , p = 4	(0, 1)
	(Zhenyu et al., (2006))
6.	Piecewise	l i + 1 = l i r , 0 ≤ l i ≤ 0 l i − r 0.5 − r , r ≤ l i ≤ 0.5 1 − r − l i 0.5 − r , 0.5 ≤ l i ≤ 1 − r 1 − l i r , 1 − r ≤ l i ≤ 1 , r = 0.4	(0, 1)
	(Saremi et al., [42])
7.	Sine	l i + 1 = p 4 sin ( π l i ), p = 4	(0, 1)
	(Saremi et al., [42])
8.	Singer	l i + 1 = z (7.86 l i − 23.31 l i 2 + 28.75 l i 3 − 13.302875 l i 4 ), z = 1.07	(0, 1)
	(Gandomi et al., [20])
9.	Sinusoidal	l i + 1 = p l i 2 sin ( π l i ) , p = 2.3	(0, 1)
	(Saremi et al., [42])
10.	Tent	l i + 1 = l i 0.7 , l i ≤ 0.7 10 3 ( 1 − l i ) , l i ≥ 0.7	(0, 1)
	(Gandomi et al., [41])

Figure 1

Randomized patterns of chaotic maps.

The initialization of the chaotic maps falls in the range of (0, 1) and (−1, 1). Besides, l i + 1 represents the x-axis of the map, while i is the chaotic index. As chaotic maps are highly sensitive to the changes in the origin state(s), so a careful selection of initial values is necessary for optimal performance [65]. Therefore, in this work, the same values have been used for all the ten chaotic maps as mentioned in ref. [20].

5.3 Mathematical model of the LCGSA algorithm

In this section, a modified version of standard GSA uses two quite interesting mathematical techniques, that is, Lévy flight and chaos theory. As stated earlier, the GSA has the drawbacks of skipping true solutions during the optimization process, slow convergence speed, and entrapment in local minima. To resolve the aforementioned issues, two strategies have been employed.

In the first strategy, Lévy flight distribution has been utilized to solve the diversity issue of standard GSA. In fact, infinite variance and variable step size of Lévy distribution help in the resolution of the local optima problem. In other words, it increases the diversity of the candidate solutions and domain of the search process. The Lévy flight is mathematically calculated using equation (13).

(13) Lévy ( d , N ) = c ′ u ∣ v ∣ 1 β σ u ,

where d is the dimension of the search space and N is the number of candidate solutions. Moreover, c ' is a multiplicative constant having a value of 0.01 while u, and v are normal distributions, and β is a Lévy index with a value of 1.5 (in this work). Moreover, σ u is given by equation (11).

In the second strategy, ten different chaotic maps have been employed to overcome slow convergence and the local searching issue of standard GSA. Chaotic maps create huge changes in the output when the initial conditions of the map(s) are modified. This helps searcher agents to move out of the local minima traps. Besides, chaotic normalization [19] helps in the proper balance of exploration and exploitation. It is mathematically calculated as shown in equation (14).

(14) C i norm ( t ) = ( C i ( t ) − a ) ⁎ ( d − c ) ( b − a ) + c .

In equation (14), (a and b) are the range of the chaotic map, i represents a chaotic index, and (c and d) are chaotic normalized intervals in which c is having a value of zero, while d is calculated using equation (15).

(15) d = MI − CI MI ( max − min ) .

Here MI and CI represent the maximum number of iterations and the current iteration, respectively. Besides, adaptive intervals are indicated by max and min with a value of 20 and 1 × 10⁻¹⁰, respectively.

In standard GSA, the gravitational constant (G) is the main parameter that specifies the intensity of the gravitational field as shown in equation (2), that is, G ( t ) = G ( t 0 ) e − α CI MI . It is pivotal for providing a proper balance between exploration and exploitation phases. In fact, during the initial iteration phase, the value of G decreases exponentially which helps in the global search. Moreover, at final iterations, the value of G changes slowly, hence promoting the exploitation of the candidate solutions towards the global optimum. Therefore, G has been chosen as it is a central controlling parameter of standard GSA which streamlines the exploration and exploitation phases. In the proposed LCGSA, the Lévy flight and chaotic sequences have been combined with the gravitational constant of GSA. Hence, Lévy-Chaotic gravitational constant (G ^LC(t)) is the addition of equations (2), (13), and (14).

(16) G LC ( t ) = Lévy ( d , N ) + C i norm ( t ) + G ( t 0 ) e − α CI MI .

Equation (16) shows that G ^LC(t) has the interesting properties of Lévy randomness, chaotic stochasticity, and adaptive learning capability. Broadly speaking, G ^LC(t) has all the essential characteristics required for solving entrapment in local minima, intensification, and diversification issues of GSA. The pseudo-code of LCGSA is presented in Algorithm 2. Besides, a flowchart is provided in Figure 2.

Figure 2

Outline of LCGSA algorithm.

Algorithm 2

LCGSA

1:	Initialize the lower and upper limits of the solution space
2:	Evaluate the fitness of each point mass
3:	Initialize the parameters including the maximum number of iterations (T), initial value of the gravitational constant G(t ₀), coefficient α, chaotic index β, Lévy multiplicative constant c ' , and adaptive constants: Max and Min
4:	Start the iteration number at t = 0
5:	while t < T do
6:	for each candidate solution do
7:	Calculate Lévy (d, N) using equation (13)
8:	Find the chaotic behavior, C _i ^norm(t) of each point mass
9:	Update the Lévy-Chaotic gravitational constant, G ^LC(t)
10:	Calculate the gravitational force between point masses, F i d ( t ) using equation (4)
11:	Calculate the mass acceleration of the point masses, a i d ( t ) by using equation (5)
12:	Update the velocity of the agents, v i d ( t + 1 ) with the help of equation (6)
13:	Update the position of the agents, x i d ( t + 1 ) using equation (7)
14:	end for
15:	t = t + 1
16:	end while
17:	Return the optimal heavy point mass

6 LCGSA for engineering design optimization

In this section, the proposed Lévy-Chaotic GSA has been applied to minimize design parameter values and the cost function of the engineering benchmarks. In fact, three famous engineering design problems, namely, SRD, TBTD, and HTBD have been selected to test the problem-solving potential of LCGSA. Obviously, constrained problems consist of both equality and inequality constraints. The penalty function method (PFM) is employed to resolve the constraints. In PFM, constraints are converted into unconstrained ones so that HA can be used to solve them. The PFM can be mathematically represented in equation (17).

(17) Min F ( x ) = f ( x ) + λ ∑ n = 1 K max ( 0 , g n ) .

Such that f(x) is the cost function and λ is the penalty coefficient. Moreover, g n represents constraints of the optimization benchmark and K is the number of constraints.

6.1 Experimental setup and parameter setting

The simulation results of LCGSA have been compared with 11 different HAs including standard GSA [64], classical PSO, ACO, SSA, SCA, GWO, PSOGSA, constriction coefficient-based PSO and GSA (CPSOGSA), BBO, GA, and DE. Moreover, for a fair comparative analysis, all the algorithms were initialized with the same population size of 50 and 500 iterations. Further, the initialization parameters of HAs have been presented in Table 3.

Table 3

Initial parameters of competing algorithms including both classical and recent HAs

Algorithm	Parameter	Value
	Population size	50
	Maximum number of iterations	500
	Total number of runs	20
GSA	Coefficient (α)	20
GSA	G( t 0 )	100
ACO	Initial pheromone	10
	Pheromone update constant	1
	S _P	0.3
	S _V	0.1
SCA	a	2
DE	l _b	0.2
	u _b	0.8
	p c	0.8
SSA	c ₂ and c ₃ (probabilistic numbers)	[0, 1]
PSO	C ₁ and C ₂ (Intensification constants)	2
	[W _max, W _min]	[0.9, 0.2]
GA	P C	0.95
	P M	0.001
	Er	0.2
CPSOGSA	[ φ 1 , φ 2 ]	2.05
BBO	M r	0.2
	P _m	0.9
GWO	[r ₁, r ₂]	[0, 1]
	a	[0, 2]
LCGSA	Chaotic index (β)	1.5
	Multiplicative constant ( c , )	0.01
	Upper adaptive interval (Max)	20
	Lower adaptive interval (Min)	1 × 10⁻¹⁰

The experiments were performed on a computer system having system specifications as mentioned in Table 4. Moreover, MATLAB code is publically available on the Github platform (https//:github.com/SAJADAHMAD1).

Table 4

Specifications of the computer system using which implementation has been carried out

System feature	Configuration
Operating system	Windows 10 enterprise
CPU	2.2 GHz Intel @ core processor
RAM	4 GB
Hard disk	500 GB
Programming language	MATLAB R2013a

The experiments were repeated 20 times to calculate different statistical measures like average, standard deviation (STD), best, worst, and median. The statistical analysis has been performed considering the stochastic and random nature of simulation results. Furthermore, LCGSA1–LCGSA10 are ten Lévy-Chaotic versions of LCGSA. The minimum value of mean and STD does not imply that an algorithm is efficient [62]. However, statistical tests should be performed on the simulation results to find the optimal competitive algorithm. Therefore, a pair-wise non-parametric signed Wilcoxon rank-sum test has been performed at a 5% significance level to statistically validate the simulation results. The reason behind selecting a Wilcoxon rank-sum test is that it uses median as a statistical measure which is better than average and STD. Moreover, in the Wilcoxon rank-sum test, the distribution of dataset is not considered. If p value of an algorithm exceeds 0.05, then it favors null hypothesis, that is, best performing algorithm has not got optimal parameter values. On the other hand, if p values of HAs are less than 0.05, then it will accept an alternate hypothesis, that is, best performing algorithm has got optimal parameter values. The best performing algorithm is represented in experimental tables as NA (Not Applicable) because it cannot be compared with itself. Besides, a p value of 1 shows the statistical equivalency of competing algorithms. The next sections of the article deal with statistical and simulation analysis of the engineering design frameworks employing 11 different HAs including 10 versions of LCGSA.

6.2 SRD problem

The design of the SRD framework is illustrated in Figure 3. This engineering design problem is considered one of the toughest problems due to its stringent constraints and complicated search space. Most of the exact and heuristic techniques show sub-optimal values to this problem. The objective function of the SRD problem is to minimize the weight of the gearbox while considering 11 different constraints [22]. Furthermore, it consists of numerical value restrictions on the bending stress, shaft transverse deflections, surface stress, and shaft stress.

Figure 3

Schematic diagram of SRD problem [22].

There are seven variables involved, namely, width of the face ( x 1 ), teeth module ( x 2 ), teeth number ( x 3 ), first shaft length ( x 4 ), second shaft length ( x 5 ), and first shaft ( x 6 ) and second shaft ( x 7 ) diameters. Moreover, design variables are represented by l 1 , l 2 , l 3 , l 4 , l 5 , l 6 , and l 7 , in cost function and constraint equations. The mathematical formulation of the SRD problem is given as:

Consider l → = [ x 1 x 2 x 3 x 4 x 5 x 6 x 7 ] = [ l 1 l 2 l 3 l 4 l 5 l 6 l 7 ] ,

Minimize f ( l → ) = 0.7854 l 1 l 2 2 ( 3.3333 l 3 2 + 14.9334 l 3 − 43.0934 ) − 1.508 ( l 6 2 + l 7 2 ) + 7.4777 ( l 6 3 + l 7 3 ) + 0.7854 ( l 4 l 6 2 + l 5 l 7 2 ) ,

Subject to: s 1 ( l → ) = 27 l 1 l 2 2 l 3 ≤ 0 ,

s 2 ( l → ) = 397.5 l 1 l 2 2 l 3 2 ≤ 0 ,

s 3 ( l → ) = 1.93 l 4 3 l 2 l 3 l 6 4 ≤ 0 ,

s 4 ( l → ) = 1.93 l 5 3 l 2 l 3 l 7 4 ≤ 0 ,

s 5 ( l → ) = 1.0 110 l 6 3 745.0 l 4 l 2 l 3 2 + 16.9 × 10 6 − 1 ≤ 0 ,

s 6 ( l → ) = 1.0 85 l 7 3 745.0 l 5 l 2 l 3 2 + 1 , 579 × 10 6 − 1 ≤ 0 ,

s 7 ( l → ) = l 2 l 3 40 − 1 ≤ 0 ,

s 8 ( l → ) = 5 l 2 l 1 − 1 ≤ 0 ,

s 9 ( l → ) = l 1 12 l 2 − 1 ≤ 0 ,

s 10 ( l → ) = 1.5 l 6 + 1.9 l 4 − 1 ≤ 0 ,

s 11 ( l → ) = 1.1 l 7 + 1.9 l 5 − 1 ≤ 0 ,

Decision variable interval values:

2.6 ≤ l 1 ≤ 3.6 ,

0.7 ≤ l 2 ≤ 0.8 ,

17 ≤ l 3 ≤ 28 ,

7.3 ≤ l 4 ≤ 8.3 ,

7.8 ≤ l 5 ≤ 8.3 ,

2.9 ≤ l 6 ≤ 3.9 ,

5.0 ≤ l 7 ≤ 5.5 ,

The SRD problem has been solved by using different versions of LCGSA and 11 other HAs. The simulation results are reported in Tables 5 and 6. It can be clearly seen that all ten versions of LCGSA have the same values for statistical measures including mean value and STD. Besides, PSO, BBO, GA, and DE also have symmetrical values for the mean value (3.56 × 10¹²) and STD (2.003 × 10⁻³). However, ACO performed optimally by providing minimum average and STD values, that is, 7.29 × 10¹⁰ and 1.56 × 10⁻⁵, respectively. Moreover, the signed Wilcoxon rank-sum test also validated the statistical superiority of ACO as other competing algorithms have p values (8.8574 × 10⁻⁵) less than 0.05. Also, in Table 5, it can be observed that all versions of LCGSA provide minimum values for seven design variables of SRD problem as compared to ACO, PSO, BBO, GA, and DE. Additionally, the execution time of all competing algorithms are mentioned in Table 6. The minimum run time is provided by SCA (18.1633 s), while LCGSA versions have execution time in 82–89 s range. The reason behind the longer run time of LCGSA is that it consists of two hybrid stages, that is, Lévy flight and chaotic maps. So, more processing time is needed to find the global optimum of the problem.

Table 5

Computational results of SRD problem

Algorithm	l 1	l 2	l 3	l 4	l 5	l 6	l 7
GSA	3.60	0.80	17	7.30	7.80	3.90	5.50
PSO	5.50	5.50	5	5.50	5	5	5.50
PSOGSA	3.60	0.80	17	7.30	7.80	3.90	5.50
CPSOGSA	3.60	0.80	17	7.30	7.80	3.90	5.50
BBO	5.5	5.5	5	5.5	5	5	5.5
GA	5.5	5.5	5	5.5	5	5	5.5
DE	5.5	5.5	5	5.5	5	5	5.5
ACO	5	6	1	4	2	3	7
SSA	3.60	0.80	17	7.30	7.80	3.90	5.50
SCA	3.60	0.80	17	7.30	7.80	3.90	5.50
GWO	3.60	0.80	17	7.30	7.80	3.90	5.50
LCGSA1	3.60	0.80	17	7.30	7.80	3.90	5.50
LCGSA2	3.60	0.80	17	7.30	7.80	3.90	5.50
LCGSA3	3.60	0.80	17	7.30	7.80	3.90	5.50
LCGSA4	3.60	0.80	17	7.30	7.80	3.90	5.50
LCGSA5	3.60	0.80	17	7.30	7.80	3.90	5.50
LCGSA6	3.60	0.80	17	7.30	7.80	3.90	5.50
LCGSA7	3.60	0.80	17	7.30	7.80	3.90	5.50
LCGSA8	3.60	0.80	17	7.30	7.80	3.90	5.50
LCGSA9	3.60	0.80	17	7.30	7.80	3.90	5.50
LCGSA10	3.60	0.80	17	7.30	7.80	3.90	5.50

Table 6

Statistical results of SRD problem

Algorithm	Best	Worst	Mean value	STD	Median	p values	Run time
GSA	2.96 × 10¹⁶	2.96 × 10¹⁶	2.96 × 10¹⁶	8.2078	2.96 × 10¹⁶	8.8574 × 10⁻⁵	80.8587
PSO	3.56 × 10¹²	3.56 × 10¹²	3.56 × 10¹²	2.00 × 10⁻³	3.56 × 10¹¹	8.8574 × 10⁻⁵	73.8182
PSOGSA	2.96 × 10¹⁶	2.96 × 10¹⁶	2.96 × 10¹⁶	8.2078	2.96 × 10¹⁶	8.8574 × 10⁻⁵	40.4020
CPSOGSA	2.96 × 10¹⁶	2.96 × 10¹⁶	2.96 × 10¹⁶	8.2078	2.96 × 10¹⁶	8.8574 × 10⁻⁵	39.9965
BBO	3.56 × 10¹²	3.56 × 10¹²	3.56 × 10¹²	2.00 × 10⁻³	3.56 × 10¹¹	8.8574 × 10⁻⁵	60.9225
GA	3.56 × 10¹²	3.56 × 10¹²	3.56 × 10¹²	2.00 × 10⁻³	3.56 × 10¹¹	8.8574 × 10⁻⁵	38.7030
DE	3.56 × 10¹²	3.56 × 10¹²	3.56 × 10¹²	2.00 × 10⁻³	3.56 × 10¹¹	8.8574 × 10⁻⁵	30.7016
ACO	7.29 × 10¹⁰	7.29 × 10¹⁰	7.29 × 10 ¹⁰	1.56 × 10 ⁻⁵	7.29 × 10¹⁰	N/A	69.7807
SSA	2.96 × 10¹⁶	2.96 × 10¹⁶	2.96 × 10¹⁶	8.2078	2.96 × 10¹⁶	8.8574 × 10⁻⁵	19.0816
SCA	2.96 × 10¹⁶	2.96 × 10¹⁶	2.96 × 10¹⁶	8.2078	2.96 × 10¹⁶	8.8574 × 10⁻⁵	18.1633
GWO	2.96 × 10¹⁶	2.96 × 10¹⁶	2.96 × 10¹⁶	8.2078	2.96 × 10¹⁶	8.8574 × 10⁻⁵	18.8484
LCGSA1	2.96 × 10¹⁶	2.96 × 10¹⁶	2.96 × 10 ¹⁶	8.2078	2.96 × 10¹⁵	8.8574 × 10⁻⁵	3.7957
LCGSA2	2.96 × 10¹⁶	2.96 × 10¹⁶	2.96 × 10¹⁶	8.2078	2.96 × 10¹⁶	8.8574 × 10⁻⁵	82.6143
LCGSA3	2.96 × 10¹⁶	2.96 × 10¹⁶	2.96 × 10¹⁶	8.2078	2.96 × 10¹⁶	8.8574 × 10⁻⁵	87.2723
LCGSA4	2.96 × 10¹⁶	2.96 × 10¹⁶	2.96 × 10¹⁶	8.2078	2.96 × 10¹⁶	8.8574 × 10⁻⁵	89.2692
LCGSA5	2.96 × 10¹⁶	2.96 × 10¹⁶	2.96 × 10¹⁶	8.2078	2.96 × 10¹⁶	8.8574 × 10⁻⁵	85.5953
LCGSA6	2.96 × 10¹⁶	2.96 × 10¹⁶	2.96 × 10¹⁶	8.2078	2.96 × 10¹⁶	8.8574 × 10⁻⁵	84.6059
LCGSA7	2.96 × 10¹⁶	2.96 × 10¹⁶	2.96 × 10¹⁶	8.2078	2.96 × 10¹⁶	8.8574 × 10⁻⁵	85.0491
LCGSA8	2.96 × 10¹⁶	2.96 × 10¹⁶	2.96 × 10¹⁶	8.2078	2.96 × 10¹⁶	8.8574 × 10⁻⁵	85.0307
LCGSA9	2.96 × 10¹⁶	2.96 × 10¹⁶	2.96 × 10¹⁶	8.2078	2.96 × 10¹⁶	8.8574 × 10⁻⁵	87.6537
LCGSA10	2.96 × 10¹⁶	2.96 × 10¹⁶	2.96 × 10¹⁶	8.2078	2.96 × 10¹⁶	8.8574 × 10⁻⁵	84.3532

The convergence analysis of LCGSA and other participating algorithms is shown in Figure 4. The convergence graphs of LCGSA (i.e., LCGSA1 best-performing LCGSA version), standard GSA, SSA, SCA, and GWO overlap with each other indicating identical exploitation power. Moreover, the curves of ACO and DE are at the bottom depicting optimal performance in handling complex and unknown search spaces.

Figure 4

Convergence analysis of SRD problem at 500 iterations.

In addition, the box plot analysis of the SRD engineering benchmark is shown in Figure 5. The objective function values of PSO, DE, ACO, SSA, and SCA are approximate with each other. On the other hand, the best fitness values of GSA, BBO, GA, GWO, and LCGSA are somewhat large indicating sub-optimal results for the median, lower quartile, and upper quartile. Moreover, the efficient performance of PSO, ACO, SCA, and DE is due to their exploitation operators which help searcher agents to move towards feasible regions of the search space and avoid local minima traps.

Figure 5

Box plot analysis of SRD engineering benchmark.

6.3 TBTD problem

The illustration of the TBTD [22] is provided in Figure 6. It consists of real-valued and non-linear buckling, deflection, and stress constraints. The objective of the TBTD framework is to find the optimal values of cross-sectional areas ( A 1 = A 3 , A 2 ). Moreover, the design variables are represented mathematically by l 1 and l 2 in the objective function. Further, in Figure 6 “P” is the load subjected to the truss and “D” is the symmetrical length.

Figure 6

Schematic diagram of TBTD problem [22].

The TBTD is mathematically represented as given by:

Consider l → = [ l 1 l 2 ] ,

Minimize f ( l → ) = ( 2 2 l 1 + l 2 ) ⁎ D ,

Subject to:

s 1 ( l → ) = 2 l 1 + l 2 2 l 1 2 + 2 l 1 l 2 p − σ ≤ 0 ,

s 2 ( l → ) = l 2 2 l 1 2 + 2 l 1 l 2 p − σ ≤ 0 ,

s 3 ( l → ) = l 2 2 l 2 + l 1 p − σ ≤ 0 ,

Variable interval value range:

0 ≤ l 1 ≤ 1 ,

0 ≤ l 2 ≤ 1 ,

where D = 100, p = 2 kN/cm² and σ = 2 kN/cm².

The computational results of the TBTD problem are reported in Table 7. The average and STD values of ten LCGSA versions are very close to each other but LCGSA10 has minimum values for the TBTD objective function. Moreover, LCGSA has a better mean value (193.09) than standard GSA (193.860) and ACO (400). Besides, PSO, BBO, GA, DE, SCA, and GWO also have superior values for statistical measures. However, SSA provides a minimum average (186.58) and STD (5.36 × 10⁻¹²) results for the TBTD benchmark. As far as the signed Wilcoxon rank-sum test is concerned, PSO is statistically better than SSA because the p value is greater than 0.05 (0.58608). In addition, SCA, SSA, and GWO have less execution times, that is, 19.8948, 17.0724, and 18.5731s, respectively. On the other hand, GSA (83.1647 s) and LCGSA (86.5465 s) took moderate time while finding the feasible regions of the solution space.

Table 7

Computational and statistical results of TBTD problem

Algorithm	l 1	l 2	Best	Worst	Mean value	STD	Median	p values	Run time
GSA	0.82	0.40	189.24	200	193.86	3.48	194.02	8.8574 × 10⁻⁵	83.1647
PSO	0.79	0.29	186.58	186.58	186.58	1.73 × 10^–11	186.58	0.58608	50.1647
PSOGSA	0.86	0.18	187.42	200	192.49	3.89	190.94	8.8574 × 10⁻⁵	26.6031
CPSOGSA	1	0	186.79	200	193.07	3.78	192.68	8.8574 × 10⁻⁵	28.9866
BBO	0.77	0.32	186.58	187.26	186.67	0.17	186.60	8.8574 × 10⁻⁵	30.9339
GA	0.78	0.30	186.58	186.88	186.60	0.06	186.58	8.8574 × 10⁻⁵	33.8263
DE	0.79	0.29	186.58	186.58	186.58	1.73 × 10⁻¹¹	186.58	0.01676	35.2219
ACO	0.99	0.93	400	400	400	0	400	8.8574 × 10⁻⁵	34.5350
SSA	0.79	0.29	186.58	186.58	186.58	5.36 × 10 ⁻¹²	186.58	N/A	19.8948
SCA	0.79	0.29	186.58	199.98	187.96	4.11	186.61	8.8574 × 10⁻⁵	17.0724
GWO	0.79	0.29	186.58	186.58	186.58	186.58	1.49 × 10⁻⁴	8.8574 × 10⁻⁵	18.5731
LCGSA1	0.76	0.45	190.41	200	195.19	3.35	195.92	8.8574 × 10⁻⁵	90.2610
LCGSA2	0.76	0.37	188.72	200	195.19	4.37	195.67	8.8574 × 10⁻⁵	116.299
LCGSA3	1	0	188.87	200	194.27	3.95	194.08	8.8449 × 10⁻⁵	108.654
LCGSA4	1	0	188.06	200	194.43	4.28	193.19	8.8574 × 10⁻⁵	104.207
LCGSA5	1	0	188.11	200	194.26	4.39	193.59	8.8574 × 10⁻⁵	86.1748
LCGSA6	1	0	187.20	200	194.22	4.66	193.61	8.8574 × 10⁻⁵	84.4361
LCGSA7	1	0	187.15	200	193.60	4.51	193.07	8.8574 × 10⁻⁵	88.3673
LCGSA8	0.83	0.23	186.90	200	193.62	4.57	193.33	8.8449 × 10⁻⁵	91.6531
LCGSA9	0.98	0.05	187.34	200	193.64	4.252	192.24	8.8574 × 10⁻⁵	87.3140
LCGSA10	0.81	0.28	187.61	200	193.09	4.32	192.02	8.8574 × 10⁻⁵	86.5465

The convergence curves of LCGSA and 11 other competing algorithms are shown in Figure 7. It can be noticed that most of the algorithms are at the bottom depicting stable intensification capability. However, when graphs are observed keenly at the initial iterations, it indicated that GA, DE, GWO, and BBO depict fast convergence towards the global optimum. Moreover, LCGSA also has better exploitation and accuracy than CPSOGSA, GSA, PSOGSA, ACO, and PSO.

Figure 7

Convergence analysis of TBTD problem at 500 iterations.

Furthermore, box plots of all participating algorithms are outlined in Figure 8. It highlights that SCA has substantial values for median, upper quartile, and lower quartile ranges. Besides, GSA, BBO, and GA also show sub-optimal values for the objective function. However, LCGSA, PSO, DE, ACO, SSA, and GWO have approximate values for TBTD fitness function which indicate the ability to handle non-linear, complex, and fractional constraints.

Figure 8

Box plot analysis of TBTD engineering benchmark.

6.4 HTBD problem

The design of the HTBD problem is outlined in Figure 9. HTBD is one of the highly regarded classical mechanical engineering design problem that has the purpose of minimizing the power loss of the framework [63]. It consists of four design variables, namely, radius (step radius [R] and recess radius [ R 0 ]), viscosity (µ), and rate of flow (Q).

Figure 9

Schematic diagram of HTBD problem [63].

The mathematical formulation of HTBD problem is given by:

Consider X → = [ R R o µ Q ] ,

Minimize f ( X → ) = Q P 0 0.7 + E f ,

Subject to:

s 1 ( X → ) = W − W s ≥ 0 ,

s 2 ( X → ) = P max − P 0 ≥ 0 ,

s 3 ( X → ) = Δ T max − Δ T ≥ 0 ,

s 4 ( X → ) = h − h min ≥ 0 ,

s 5 ( X → ) = R − R 0 ≥ 0 ,

s 6 ( X → ) = 0.001 − γ g P 0 Q 2 π Rh ≥ 0 ,

s 7 ( X → ) = 5 , 000 − W π ( R 2 R 0 2 ) ≥ 0 ,

Decision variable interval values:

1 ≤ R ≤ 16 ,

1 ≤ R o ≤ 16 , 1 × 10 − 6 ≤ μ ≤ 16 × 10 − 6 , 1 ≤ Q ≤ 16

The experimental results of the HTBD problem are reported in Table 8. The mean (3.80 × 10⁶⁵) and STD (1.43 × 10⁶⁶) values of LCGSA8 are better than standard GSA, that is, 2.67 × 10⁶⁶ and 1.17 × 10⁶⁷, respectively. Besides, the values of design variables for HTBD problem given by LCGSA (R[3.82], R 0 [2.80], µ[1.83 × 10⁷], and Q[6.30]) are superior to standard GSA (R[2.48], R 0 [2.93], µ[3.5 × 10⁷], and Q[7.13]). From the simulation results, it can be observed that CPSOGSA has optimal results for the HTBD benchmark as it has a minimum average (2.36e14) and STD (3.20 × 10¹²) values. Also, it can be seen that PSO, BBO, GA, DE, SSA, SCA, and GWO have efficient statistical results. However, the signed Wilcoxon rank-sum test indicates that PSOGSA, SCA, SSA, and GWO are statistically equivalent to CPSOGSA because p value is equal to one. Furthermore, the execution time of LCGSA1 (85.3229) and LCGSA5 (99.9323) is better than that of standard GSA (102.6685). Additionally, SCA (21.5106) again converges first to the global optimum, while PSOGSA, CPSOGSA, SSA, and GWO took tolerable time in locating the feasible regions of the search space.

Table 8

Computational and statistical results of HTBD problem

Algorithm	R	R o	µ	Q	Best	Worst	Mean value	STD	Median	p values	Run time
GSA	2.48	2.93	3.5 × 10⁷	7.13	3.08 × 10⁴⁶	5.24 × 10⁶⁷	2.67 × 10⁶⁶	1.17 × 10⁶⁷	4.96 × 10⁶¹	8.85 × 10⁻⁵	102.6685
PSO	3.56	3.56	16	13.39	2.36 × 10¹⁴	2.36 × 10¹⁴	2.36 × 10¹⁴	3.08 × 10⁶	2.34 × 10¹⁴	5.78 × 10⁻⁵	67.4941
PSOGSA	1	1	10 × 10⁶	16	2.36 × 10¹⁴	2.36 × 10¹⁴	2.36 × 10¹⁴	5.59 × 10¹⁴	2.36 × 10¹⁴	1	40.3848
CPSOGSA	1	1	10 × 10⁶	16	2.36 × 10¹⁴	2.36 × 10¹⁴	2.36 × 10¹⁴	3.20 × 10 ⁻²	2.36 × 10¹⁴	N/A	36.2186
BBO	1.91	3.46	3.07	7.42	1.76 × 10¹⁵	2.72 × 10¹⁵	2.27 × 10¹⁵	3.04 × 10¹⁵	2.31 × 10¹⁵	8.85 × 10⁻⁵	50.8231
GA	1	1	16	16	2.36 × 10¹⁴	2.58 × 10¹⁴	8.29 × 10¹⁴	9.52 × 10¹⁴	2.36 × 10¹⁴	4.40 × 10⁻⁵	44.287
DE	1	1	16	16	2.36 × 10¹⁴	2.36 × 10¹⁴	2.36 × 10¹⁴	0.09618	2.36 × 10¹⁴	7.74 × 10⁻⁶	39.5515
ACO	4	3	1	2	7.35 × 10¹⁶	7.35 × 10¹⁶	7.35 × 10¹⁶	0.25649	7.35 × 10¹⁶	7.74 × 10⁻⁶	57.3797
SSA	16	16	10 × 10⁶	16	2.36 × 10¹⁴	2.36 × 10¹⁴	2.36 × 10¹⁴	0.03206	2.36 × 10¹⁴	1	23.4853
SCA	1	1	10 × 10⁶	16	2.36 × 10¹⁴	2.36 × 10¹⁴	2.36 × 10¹⁴	0.03206	2.36 × 10¹⁴	1	21.5106
GWO	1	1	0 × 10⁶	16	2.36 × 10¹⁴	2.36 × 10¹⁴	2.36 × 10¹⁴	0.03206	2.36 × 10¹⁴	1	22.4633
LCGSA1	1.04	2.64	4.13 × 10⁷	15.50	4.40 × 10³⁷	6.07 × 10⁶⁸	3.83 × 10⁶⁷	1.37 × 10⁶⁸	4.18 × 10⁶⁰	8.85 × 10⁻⁵	85.3229
LCGSA2	3.34	3.35	1.40 × 10⁸	6.96	1.50 × 10⁵¹	7.38 × 10⁶⁹	7.36 × 10⁶⁸	2.24 × 10⁶⁹	2.79 × 10⁶¹	8.85 × 10⁻⁵	119.5817
LCGSA3	3.57	1.04	7.51 × 10⁷	15.77	9.77 × 10⁴⁶	8.34 × 10⁶⁶	9.55 × 10⁶⁵	2.48 × 10⁶⁶	3.05 × 10⁶²	8.85 × 10⁻⁵	121.2212
LCGSA4	10.55	7.39	1.49 × 10⁷	6.18	3.30 × 10⁵⁰	8.22 × 10⁷⁰	4.12 × 10⁶⁹	1.83 × 10⁷⁰	5.83 × 10⁶⁰	8.85 × 10⁻⁵	116.2671
LCGSA5	7.92	7.79	1.42 × 10⁸	11.40	3.98 × 10⁵³	9.22 × 10⁶⁷	4.72 × 10⁶⁶	2.06 × 10⁶⁷	2.71 × 10⁶¹	8.85 × 10⁻⁵	99.9323
LCGSA6	1.46	1.68	4.06 × 10⁷	10.84	1.41 × 10³³	5.15 × 10⁶⁹	4.21 × 10⁶⁸	1.33 × 10⁶⁹	2.05 × 10⁵⁸	8.85 × 10⁻⁵	100.7344
LCGSA7	4.30	2.72	1.56 × 10⁷	4.75	1.02 × 10⁵³	5.20 × 10⁶⁹	2.88 × 10⁶⁸	1.16 × 10⁶⁹	3.48 × 10⁶²	8.85 × 10⁻⁵	103.6481
LCGSA8	3.82	2.80	1.83 × 10⁷	6.30	2.33 × 10⁴⁹	6.42 × 10⁶⁶	3.80 × 10 ⁶⁵	1.43 × 10 ⁶⁶	3.86 × 10⁵⁹	8.85 × 10⁻⁵	104.736
LCGSA9	1.53	2	7.54 × 10⁷	15.41	5.14 × 10⁵²	6.30 × 10⁶⁸	5.25 × 10⁶⁷	1.65 × 10⁶⁸	2.43 × 10⁶¹	8.85 × 10⁻⁵	106.1996
LCGSA10	4.25	4.24	1.42 × 10⁸	2.22	2.36 × 10⁴²	4.08 × 10⁶⁷	2.07 × 10⁶⁶	9.11 × 10⁶⁶	7.69 × 10⁶²	8.85 × 10⁻⁵	104.605

The convergence behavior of the HTBD engineering benchmark is reported in Figure 10. It can be comprehensibly seen that LCGSA has fast convergence speed as compared to standard GSA. Moreover, the curves of SSA, SCA, GWO, GA, and DE are interlinked and coupled with each other indicating identical and symmetrical convergence patterns. The convergence maps also convey that most of the HAs have the same symmetrical values for the objective function and stable exploitation capability as far as the HTBD problem is concerned.

Figure 10

Convergence analysis of HTBD problem at 500 iterations.

Statistically speaking, the median and inter-quartile range values of GSA, DE, and ACO are large, showing their sub-optimal performance in handling complex and unknown search spaces as shown in Figure 11. However, the simulation results of remaining algorithms, namely, PSO, BBO, GA, and so on are within the range of logical scale (0.4e15) indicating appreciable convergence and exploration capability. Further, the box plot of LCGSA is also within the feasible region depicting noticeable diversification power. The reason behind the efficient performance of LCGSA is the variable step size of Lévy distribution and stochastic behavior of chaotic maps which helps LCGSA in overcoming local minima falling and premature convergence speed problems.

Figure 11

Box plot analysis of HTBD engineering benchmark.

6.5 Summarization of simulation results

The in-depth analysis of the experimental outcomes of two structural engineering (SRD and TBTD) benchmarks and one classical mechanical engineering (HTBD) problem reveals that LCGSA1 (chebyshev map), LCGSA8 (singer map), and LCGSA10 (tent map) provided optimal objective map values for the engineering design problems. First, it can be seen that the chaotic map of chebyshev function clearly shows high intensity of random behavior which helps it to respond dynamically to the unsuitable simulation pattern and sub-optimal regions of the solution space. Moreover, chebyshev map has number of impressive properties like symmetry, orthogonality, and high approximation power which makes it suitable for the global optimization. As far as singer and tent maps are concerned, they have quite similar chaotic behavior. Besides, both of them have suitable values for the chaotic random variable which help them to stay in the feasible neighborhood, and hence help in the exploitation process. Besides, ACO, SSA, and CPSOGSA also gave efficient results for SRD, TBTD, and HTBD engineering frameworks. As far as run time analysis of algorithms is concerned, SCA has been the fastest algorithm as it took less time for converge towards the global optimum. Also, all LCGSA versions have moderate execution times for all the three engineering benchmarks. Quantitatively speaking, the simulation values of ACO, GA, PSO, GSA, and PSOGSA were distant from global optimum in all three problems.

Furthermore, when we look at the simulation results recorded at 20 consecutive runs of the 10 versions of LCGSA, it is comprehensively evident that their values are close to each other in all three design problems. Surprisingly, the values were approximately in the neighborhood of the feasible region of the solution space. However, the simulation outcomes of standard GSA were distant from the global optimum indicating sub-optimal performance in handling non-linear constraints of real-world problems. Besides, the Wilcoxon rank-sum test also showed that the experimental results of LCGSA were superior as compared to GSA. In other words, it can be interpreted that LCGSA has outperformed standard GSA both qualitatively (convergence and box plot analysis) and quantitatively (numerical and statistical values). Also, LCGSA provides satisfied and coherent performance in all three engineering design frameworks by handling complex engineering search space, non-linear constraints, and multiple local optima. Besides, the LCGSA version of standard GSA is resilient and free from entrapment in local minima and premature convergence issues. Hence, it can be declared that LCGSA is an efficient and robust optimizer as compared to standard GSA.

7 Conclusion and future directions

In this research, the Lévy flight-based chaotic gravitational search algorithm (LCGSA) has been applied to three constrained engineering design problems, namely, SRD, TBTD, and HTBD. The constraints were resolved by using penalty function method in which algorithms are penalized based on constraint violation.

The simulation results clearly indicate that LCGSA has optimal results for TBTD and HTBD problems than standard GSA and other competing algorithms. Besides, LCGSA and other participating algorithms have the same simulation values for the SRD problem. Moreover, singer and tent maps were best performing chaotic maps while considering convergence speed and execution time.

As far as future scope of the current work is concerned, the multi-objective and binary versions of LCGSA can be proposed to tackle high dimensional and complex optimization problems. Besides, other chaotic maps like bakers map, henon map, etc., can also be incorporated into LCGSA. In fact, it will be amazing to apply LCGSA to other civil and mechanical engineering design benchmarks such as cantilever beam design problem, I-beam design, ten bar truss, and multiple disc clutch brake for parameter and objective function optimization.

Acknowledgements

The authors would like to acknowledge the efforts of the editor and the valuable comments of anonymous reviewers during the revision of this research.

Funding information: The authors have not received any financial support for conducting the work.
Author contributions: S.A.R.: conceptualization, methodology, formal analysis, investigation, writing – review & editing, software, visualization. P.S.B.: writing – review & editing, data curation, supervision.
Conflict of interest: The authors clearly state that there is no conflict of interest whether financial or professional regarding the publication of the work.
Data availability statement: The implementation of the proposed algorithm and other competing algorithms has been done in MATLAB. The source codes are publicly available on the GitHub platform, https://github.com/SajadAHMAD.

References

[1] A. E. Eiben and C. A. Schippers, “On evolutionary diversification and intensification,” Fund. Inform., vol. 35, no. 1. pp. 35–50, 1998.10.3233/FI-1998-35123403Search in Google Scholar

[2] D. H. Wolpert and W. G. Macready, “No free lunch theorems for optimization,” IEEE Trans. Evol. Comput., vol. 1, no. 1. pp. 67–82, 1997.10.1109/4235.585893Search in Google Scholar

[3] J. H. Holland, Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence, New York: MIT Press, 1992.10.7551/mitpress/1090.001.0001Search in Google Scholar

[4] R. Eberhart and J. Kennedy, “A new optimizer using particle swarm theory,” 6th IEEE International Symposium on Micro Machine and Human Science (MHS ‘95), 1995.10.1109/MHS.1995.494215Search in Google Scholar

[5] M. Dorigo, M. Birattari, and T. Stutzle, “Ant colony optimization,” IEEE Comput. Intell. Mag., vol. 1, no. 4. pp. 28–39, 2006.10.1109/CI-M.2006.248054Search in Google Scholar

[6] R. Storn and K. Price, “Differential evolution‐a simple and efficient heuristic for global optimization over continuous spaces,” J. Glob. Optim., vol. 11, no. 4. pp. 341–359, 1997.10.1023/A:1008202821328Search in Google Scholar

[7] D. Simon, “Biogeography‐based optimization,” IEEE Trans. Evol. Comput., vol. 12, pp. 702–713, 2008.10.1109/TEVC.2008.919004Search in Google Scholar

[8] S. Mirjalili, S. M. Mirjalili, and A. Lewis, “Grey wolf optimizer,” Adv. Eng. Softw., vol. 69, pp. 46–61, 2014.10.1016/j.advengsoft.2013.12.007Search in Google Scholar

[9] S. Mirjalili, “The ant lion optimizer,” Adv. Eng. Softw., vol. 83, pp. 80–98, 2015.10.1016/j.advengsoft.2015.01.010Search in Google Scholar

[10] S. Mirjalili, “SCA: a sine cosine algorithm for solving optimization problems,” Knowl. Based Syst., vol. 96, pp. 120–133, 2016.10.1016/j.knosys.2015.12.022Search in Google Scholar

[11] S. Mirjalili, A. H. Gandomi, S. Z. Mirjalili, S. Saremi, H. Faris, and S. M. Mirjalili, “Salp swarm algorithm: a bio-inspired optimizer for engineering design problems,” Adv. Eng. Softw., vol. 114, pp. 163–191, 2017.10.1016/j.advengsoft.2017.07.002Search in Google Scholar

[12] L. Cheng, X. H. Wu, and Y. Wang, “Artificial flora (AF) optimization algorithm,” Appl. Sci., vol. 8, no. 3. pp. 1–22, 2018.10.3390/app8030329Search in Google Scholar

[13] A. Shabani, B. Asgarian, S. A. Gharebaghi, M. A. Salido, and A. Giret, “A new optimization algorithm based on search and rescue operations,” Math. Probl. Eng., vol. 2019, pp. 1–23, 2019, 10.1155/2019/2482543.Search in Google Scholar

[14] V. Hayyolalam and A. A. Kazem, “Black Widow optimization algorithm: a novel meta-heuristic approach for solving engineering optimization problems,” Eng. Appl. Artif. Intell., vol. 87, pp. 103–249, 2020.10.1016/j.engappai.2019.103249Search in Google Scholar

[15] A. M. Ahmed, T. A. Rashid, and S. A. Saeed, “Cat swarm optimization algorithm: a survey and performance evaluation,” Comput. Intell. Neurosci., vol. 2020, pp. 1–20, 2020, 10.1155/2020/4854895.Search in Google Scholar PubMed PubMed Central

[16] M. H. Sulaiman, Z. Mustaffa, M. M. Saari, and H. Daniyal, “Barnacles mating optimizer: A new bio-inspired algorithm for solving engineering optimization problems,” Eng. Appl. Artif. Intell., vol. 87, pp. 103–330, 2020.10.1016/j.engappai.2019.103330Search in Google Scholar

[17] S. A. Rather and P. S. Bala, “Analysis of gravitation-based optimization algorithms for clustering and classification,” Handbook of Research on Big Data Clustering and Machine Learning, G. M. Pedro, ed., New York, NY, IGI Global, 2020a, pp. 77–99. 10.4018/978-1-7998-0106-1.ch005.Search in Google Scholar

[18] S. A. Rather and P. S. Bala, “Swarm-based chaotic gravitational search algorithm for solving mechanical engineering design problems,” World J. Eng., vol. 17, no. 1. pp. 97–114, 2020b.10.1108/WJE-09-2019-0254Search in Google Scholar

[19] S. Mirjalili and A. H. Gandomi, “Chaotic gravitational constants for the gravitational search algorithm,” Appl. Soft Comput., vol. 53, pp. 407–419, 2017.10.1016/j.asoc.2017.01.008Search in Google Scholar

[20] A. H. Gandomi, G. J. Yun, X. S. Yang, and S. Talatahari, “Chaos-enhanced accelerated particle swarm optimization,” Commun. Nonlinear Sci. Numer. Simul., vol. 18, pp. 327–340, 2013.10.1016/j.cnsns.2012.07.017Search in Google Scholar

[21] D. Shen, T. Jiang, W. Chen, Q. Shi, and S. Gao, “Improved chaotic gravitational search algorithms for global optimization,” IEEE Congress on Evolutionary Computation (CEC), New York: IEEE, 2015, pp. 1220–1226, 10.1109/CEC.2015.7257028.Search in Google Scholar

[22] M. A. Basset, G. G. Wang, A. K. Sangaiah, and E. Rushdy, “Krill herd algorithm based on cuckoo search for solving engineering optimization problems,” Multimed. Tools Appl., vol. 78, no. 4. pp. 3861–3884, 2019, 10.1007/s11042-017-4803-x.Search in Google Scholar

[23] R. G. Rodenas, L. J. Linares, and J. A. L. Gomez, “A memetic chaotic gravitational search algorithm for unconstrained global optimization problems,” Appl. Soft Comput., vol. 79, pp. 14–29, 2019.10.1016/j.asoc.2019.03.011Search in Google Scholar

[24] T. A. Khan and S. H. Ling, “An improved gravitational search algorithm for solving an electromagnetic design problem,” J. Comput. Electron., vol. 19, pp. 1–7, 2020.10.1007/s10825-020-01476-8Search in Google Scholar

[25] S. Jiang, C. Zhang, and S. Chen, “Sequential hybrid particle swarm optimization and gravitational search algorithm with dependent random coefficients,” Math. Probl. Eng., vol. 2020, pp. 1–17, 2020, 10.1155/2020/1957812.Search in Google Scholar

[26] F. Barani, M. Mirhosseini, and H. Nezamabadi-Pour, “Application of binary quantum-inspired gravitational search algorithm in feature subset selection,” Appl. Intell., vol. 47, no. 2. pp. 304–318, 2017.10.1007/s10489-017-0894-3Search in Google Scholar

[27] D. Pelusi, R. Mascella, and L. Tallini, “A fuzzy gravitational search algorithm to design optimal IIR filters,” Energies, vol. 11, no. 4. p. 736, 2018.10.3390/en11040736Search in Google Scholar

[28] P. Haghbayan, H. Nezamabadi-Pour, and S. Kamyab, “A niche GSA method with nearest neighbor scheme for multimodal optimization,” Swarm Evol. Comput., vol. 35, pp. 78–92, 2017.10.1016/j.swevo.2017.03.002Search in Google Scholar

[29] S. Yazdani, H. Nezamabadi-pour, and S. Kamyab, “A gravitational search algorithm for multimodal optimization,” Swarm Evol. Comput., vol. 14, pp. 1–14, 2014, 10.1016/j.swevo.2013.08.001.Search in Google Scholar

[30] K. Kang, C. Bae, H. W. F. Yeung, and Y. Y. Chung, “A hybrid gravitational search algorithm with swarm intelligence and deep convolutional feature for object tracking optimization,” Appl. Soft Comput., vol. 66, pp. 319–329, 2018.10.1016/j.asoc.2018.02.037Search in Google Scholar

[31] F. Khajooei and E. Rashedi, A new version of Gravitational Search Algorithm with negative mass, Proceedings of 1st IEEE Conference on Swarm Intelligence and Evolutionary Computation (CSIEC), 2016, pp. 1–5.10.1109/CSIEC.2016.7482123Search in Google Scholar

[32] H. Sajedi and S. F. Razavi, “DGSA: discrete gravitational search algorithm for solving knapsack problem,” Oper. Res., vol. 17, no. 2. pp. 563–591, 2017.10.1007/s12351-016-0240-2Search in Google Scholar

[33] H. Zandevakili, E. Rashedi, and A. Mahani, “Gravitational search algorithm with both attractive and repulsive forces,” Soft Comput., vol. 23, no. 3. pp. 783–825, 2019.10.1007/s00500-017-2785-2Search in Google Scholar

[34] C. Li, J. Zhou, J. Xiao, and H. Xiao, “Parameters identification of chaotic system by chaotic gravitational search algorithm,” Chaos, Solitons Fractals, vol. 45, no. 4. pp. 539–547, 2012.10.1016/j.chaos.2012.02.005Search in Google Scholar

[35] S. Gao, C. Vairappan, Y. Wang, Q. Cao, and Z. Tang, “Gravitational search algorithm combined with chaos for unconstrained numerical optimization,” Appl. Math. Comput., vol. 231, pp. 48–62, 2012.10.1016/j.amc.2013.12.175Search in Google Scholar

[36] B. Alatas, E. Akin, and A. B. Ozer, “Chaos embedded particle swarm optimization algorithms,” Chaos, Solitons Fractals, vol. 40, pp. 1715–1734, 2009.10.1016/j.chaos.2007.09.063Search in Google Scholar

[37] B. Alatas, “Chaotic bee colony algorithms for global numerical optimization,” Expert. Syst. Appl., vol. 37, no. 8. pp. 5682–5687, 2010.10.1016/j.eswa.2010.02.042Search in Google Scholar

[38] B. Alatas, “Chaotic harmony search algorithms,” Appl. Math. Comput., vol. 216, no. 9. pp. 2687–2699, 2010.10.1016/j.amc.2010.03.114Search in Google Scholar

[39] G. Zhenyu, C. Bo, Y. Min, and C. Binggang, “Self-adaptive chaos differential evolution,” Proceedings of Springer International Conference on Natural Computation, Berlin, Heidelberg: Springer, 2006, September, pp. 972–975.10.1007/11881070_128Search in Google Scholar

[40] J. Mingjun and T. Huanwen, “Application of chaos in simulated annealing,” Chaos, Solitons Fractals, vol. 21, no. 4. pp. 933–941, 2004.10.1016/j.chaos.2003.12.032Search in Google Scholar

[41] A. H. Gandomi, X. S. Yang, S. Talatahari, and A. H. Alavi, “Firefly algorithm with chaos,” Commun. Nonlinear Sci. Numer. Simul., vol. 18, no. 1. pp. 89–98, 2013.10.1016/j.cnsns.2012.06.009Search in Google Scholar

[42] S. Saremi, S. Mirjalili, and A. Lewis, “Biogeography-based optimization with chaos,” Neural Comput. Appl., vol. 25, pp. 1077–1097, 2014.Search in Google Scholar

[43] L. Wang and Y. Zhong, “Cuckoo search algorithm with chaotic maps,” Math. Probl. Eng., vol. 2015, pp. 1–14, 2015, 10.1155/2015/715635.Search in Google Scholar

[44] J. Xie, Y. Zhou, and H. Chen, “A novel bat algorithm based on differential operator and Lévy flights trajectory,” Comput. Intell. Neurosci., vol. 2013, pp. 1–13, 2013, 10.1155/2013/453812.Search in Google Scholar PubMed PubMed Central

[45] X. Shan, K. Liu, and P. L. Sun, “Modified bat algorithm based on Lévy flight and opposition based learning,” Sci. Program., vol. 2016, pp. 1–13, 2016, 10.1155/2016/8031560.Search in Google Scholar

[46] Y. Li, X. Li, J. Liu, and X. Ruan, “An improved bat algorithm based on Lévy flights and adjustment factors,” Symmetry, vol. 11, no. 7. p. 925, 2019.10.3390/sym11070925Search in Google Scholar

[47] H. Haklı and H. Uğuz, “A novel particle swarm optimization algorithm with Lévy flight,” Appl. Soft Comput., vol. 23, pp. 333–345, 2014.10.1016/j.asoc.2014.06.034Search in Google Scholar

[48] R. Jensi and G. W. Jiji, “An enhanced particle swarm optimization with Lévy flight for global optimization,” Appl. Soft Comput., vol. 43, pp. 248–261, 2016.10.1016/j.asoc.2016.02.018Search in Google Scholar

[49] Z. Peng, K. Dong, H. Yin, and Y. Bai, “Modification of fish swarm algorithm based on Lévy flight and firefly behavior,” Comput. Intell. Neurosci., vol. 2018, pp. 1–13, 2018, 10.1155/2018/9827372.Search in Google Scholar PubMed PubMed Central

[50] S. Mirjalili, S. M. Mirjalili, and A. Hatamlou, “Multi-verse optimizer: a nature-inspired algorithm for global optimization,” Neural Comput. Appl., vol. 27, no. 2. pp. 495–513, 2016.10.1007/s00521-015-1870-7Search in Google Scholar

[51] C. Hu, Z. Li, T. Zhou, A. Zhu, and C. Xu, “A multi-verse optimizer with Lévy flights for numerical optimization and its application in test scheduling for network-on-chip,” PLoS One, vol. 11, no. 12, pp. 1–22, 2016, 10.1371/journal.pone.0167341.Search in Google Scholar PubMed PubMed Central

[52] D. Halliday, R. Resnick, and J. Walker, Fundamentals of Physics, NY, John Wiley & Sons, 2013.Search in Google Scholar

[53] S. A. Rather and P. S. Bala, “A holistic review on gravitational search algorithm and its hybridization with other optimization algorithms,” Proceedings of IEEE International Conference on Electrical, Computer and Communication Technologies (ICECCT), 2019a, pp. 1–6.10.1109/ICECCT.2019.8869279Search in Google Scholar

[54] S. A. Rather, M. Shahid, P. S. Bala, and A. Comprehensive, “Survey on solving clustering and classification problems using gravitational search algorithm,” Proceedings of IEEE 9th International Conference on Advanced Computing (IACC), 2019b, pp. 13–18.10.1109/IACC48062.2019.8971589Search in Google Scholar

[55] S. A. Rather and N. Sharma, “GSA-BBO hybridization algorithm,” Int. J. Adv. Sci. Eng., vol. 6, pp. 596–608, 2017.Search in Google Scholar

[56] W. Gao, D. Dimitrov, and H. Abdo, “Tight independent set neighborhood union condition for fractional critical deleted graphs and ID deleted graphs,” Discret. Cont. Dyn. Syst., vol. 12, pp. 711–721, 2018a.10.3934/dcdss.2019045Search in Google Scholar

[57] W. Gao, J. L. Guirao, B. Basavanagoud, and J. Wu, “Partial multi-dividing ontology learning algorithm,” Inf. Sci., vol. 467, pp. 35–58, 2018b.10.1016/j.ins.2018.07.049Search in Google Scholar

[58] X. S. Yang, Nature-Inspired Metaheuristic Algorithms, United Kingdom: Luniver Press, 2010.Search in Google Scholar

[59] J. F. Yao, C. Mei, X. Q. Peng, Z. K. Hu, and J. Hu, “A new optimization approach-chaos genetic algorithm,” Syst. Eng., vol. 1, p. 105, 2001.Search in Google Scholar

[60] S. Talatahari, B. F. Azar, R. Sheikholeslami, and A. H. Gandomi, “Imperialist competitive algorithm combined with chaos for global optimization,” Commun. Nonlinear Sci. Numer. Simul., vol. 17, pp. 1312–1319, 2012.10.1016/j.cnsns.2011.08.021Search in Google Scholar

[61] A. H. Gandomi and X. S. Yang, “Chaotic bat algorithm,” J. Comput. Sci., vol. 5, pp. 224–232, 2014.10.1016/j.jocs.2013.10.002Search in Google Scholar

[62] J. Derrac, S. García, D. Molina, and F. Herrera, “A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms,” Swarm Evol. Comput., vol. 1, pp. 3–18, 2011.10.1016/j.swevo.2011.02.002Search in Google Scholar

[63] R. V. Rao, S. J. Svani, and D. P. Vakharia, “Teaching- Learning-based optimization: A novel method for constrained mechanical design optimization problems,” Comput. Des., vol. 40, no. 5. pp. 303–315, 2011.10.1016/j.cad.2010.12.015Search in Google Scholar

[64] E. Rashedi, H. Nezamabadi-Pour, and S. Saryazdi, “GSA: a gravitational search algorithm,” Inf. Sci., vol. 179, no. 13. pp. 2232–2248, 2009.10.1016/j.ins.2009.03.004Search in Google Scholar

[65] V. Jothiprakash and R. Arunkumar, “Optimization of hydropower reservoir using evolutionary algorithms coupled with chaos,” Water Resour. Manag., vol. 27, pp. 1963–1979, 2013.10.1007/s11269-013-0265-8Search in Google Scholar

Received: 2020-09-19

Revised: 2021-01-08

Accepted: 2021-01-23

Published Online: 2021-12-31

This work is licensed under the Creative Commons Attribution 4.0 International License.

Lévy flight and chaos theory-based gravitational search algorithm for mechanical and structural engineering design optimization

Abstract

Acronyms

1 Introduction

2 Motivation

3 Literature survey

4 GSA

Algorithm 1

5 LCGSA

5.1 Lévy flight

5.2 Chaos theory

5.3 Mathematical model of the LCGSA algorithm

Algorithm 2

6 LCGSA for engineering design optimization

6.1 Experimental setup and parameter setting

6.2 SRD problem

6.3 TBTD problem

6.4 HTBD problem

6.5 Summarization of simulation results

7 Conclusion and future directions

Acknowledgements

References

Journal and Issue

Articles in the same Issue