
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)12456-5
10.1016/j.heliyon.2024.e36425
e36425
Review Article
Comparative analysis of the gazelle Optimizer and its variants
Mahajan Raghav iraghavmahajn@gmail.com
a
Sharma Himanshu himanshu.23441@lpu.co.in
a
Arora Krishan krishan.12252@lpu.co.in
a
Joshi Gyanendra Prasad joshi@sejong.ac.kr
b⁎
Cho Woong wcho@kangwon.ac.kr
c⁎⁎
a School of Electronics and Electrical Engineering, Lovely Professional University, Phagwara, 144411, India
b Department of Computer Science and Engineering, Sejong University, Seoul 05006, Republic of Korea
c Department of Electronics, Information and Communication Engineering, Kangwon National University, Samcheok 25913, Republic of Korea
⁎ Corresponding author. joshi@sejong.ac.kr
⁎⁎ Corresponding author. wcho@kangwon.ac.kr
16 8 2024
15 9 2024
16 8 2024
10 17 e3642512 5 2024
11 8 2024
15 8 2024
© 2024 The Authors
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
The Gazelle Optimization Algorithm (GOA) is an innovative nature-inspired metaheuristic algorithm, designed to mimic the agile and efficient hunting strategies of gazelles. Despite its promising performance in solving complex optimization problems, there is still a significant scope for enhancing its efficiency and robustness. This paper introduces several novel variants of GOA, integrating adaptive strategy, Levy flight strategy, Roulette wheel selection strategy, and random walk strategy. These enhancements aim to address the limitations of the original GOA and improve its performance in diverse optimization scenarios. The proposed algorithms are rigorously tested on CEC 2014 and CEC 2017 benchmark functions, five engineering problems, and a Total Harmonic Distortion (THD) minimization problem. The results demonstrate the superior performance of the proposed variants compared to the original GOA, providing valuable insights into their applicability and effectiveness.

Keywords

Gazelle optimization algorithm
Benchmark functions
Total harmonic distortion
Engineering design problem
==== Body
pmc1 Introduction

Optimization algorithms are essential tools in various fields, including engineering, economics, and scientific research, where they are used to find the best possible solutions to complex problems. These algorithms aim to optimize a specific objective function by iteratively improving candidate solutions based on certain criteria. Classical optimization techniques, such as gradient descent and linear programming, often struggle with problems that have complex landscapes, multiple local optima, or high-dimensional search spaces. These limitations have led to the development of metaheuristic algorithms, which offer robust and flexible approaches to solving complex optimization problems. Metaheuristic algorithms are inspired by natural processes and phenomena, such as evolution, swarm behavior, and physical annealing. These algorithms use stochastic components to explore the search space more effectively and avoid premature convergence to local optima. Well-known examples of metaheuristic algorithms include Genetic Algorithms (GA) [1], Particle Swarm Optimization (PSO) [2], Ant Colony Optimization (ACO) [3], and Differential Evolution (DE) [4]. These algorithms have been successfully applied to a wide range of optimization problems, demonstrating their versatility and effectiveness.

The Gazelle Optimization Algorithm (GOA) is a relatively new addition to the family of nature-inspired metaheuristic algorithms. It draws inspiration from the predatory tactics and escape maneuvers of gazelles. In the wild, gazelles exhibit swift and agile movements to evade predators, often employing sudden directional changes and bursts of speed. This behavior is modeled in GOA to balance exploration and exploitation during the optimization process. The standard GOA involves a population of candidate solutions (gazelles) that move through the search space based on specific rules mimicking the gazelles' natural behavior. These rules are designed to ensure that the algorithm can efficiently explore the search space (diversification) while focusing on promising regions (intensification).

Despite its promising performance, the original GOA faces several challenges, including premature convergence and maintaining an effective balance between exploration and exploitation. Premature convergence occurs when the algorithm gets trapped in local optima, leading to suboptimal solutions. Maintaining a balance between exploration (searching new areas of the search space) and exploitation (refining existing solutions) is crucial for the algorithm's success. To address these challenges, this research proposes four novel variants of GOA, each integrating a different strategy to enhance its performance.

The first variant, GOA with an adaptive strategy, dynamically adjusts the algorithm's parameters based on the current state of the optimization process. Adaptive strategies have been shown to improve the performance of metaheuristic algorithms by allowing them to fine-tune their behavior in real-time [5]. In this variant, parameters such as step size, acceleration coefficient, and direction are adjusted in real-time, depending on the diversity of the population and the progress towards the optimal solution. This allows the algorithm to adapt to different phases of the search process, improving convergence speed and solution quality.

The second variant, GOA with Levy flight strategy, enhances the global search capability of the algorithm by introducing random walks characterized by long jumps interspersed with short steps. Levy flights are a type of random walk observed in various natural processes, and they have been successfully applied to optimization algorithms to improve their exploration capabilities [6]. By following a Levy flight pattern, gazelles in this variant make larger jumps with a higher probability, facilitating better exploration and helping the algorithm escape local optima.

The third variant, GOA with Roulette wheel selection strategy, draws inspiration from genetic algorithms, where the probability of selecting a particular solution is proportional to its fitness. This strategy promotes solutions with higher fitness, accelerating convergence towards optimal regions [7]. In this variant, the selection mechanism determines which gazelles influence the movement of others, ensuring that fitter solutions have a higher impact on the population's direction. This approach helps in focusing the search on promising regions while maintaining diversity.

The fourth variant, GOA with random walk strategy, involves steps taken in random directions to maintain diversity within the population and prevent premature convergence. Random walks introduce variability in the search process, ensuring a more comprehensive exploration of the search space [8]. In this variant, gazelles perform random walks based on a predefined distribution, helping the algorithm explore less-explored areas of the search space and maintain a diverse set of solutions.

The motivation for this research stems from the need to enhance the efficiency and robustness of the Gazelle Optimization Algorithm. While the original GOA has shown promise in solving complex optimization problems, it can prematurely converge to local optima, especially in high-dimensional and multimodal search spaces. Additionally, maintaining an effective balance between exploring new regions and exploiting known good solutions is crucial for the algorithm's success. The ability to dynamically adjust the algorithm's parameters based on the optimization process can significantly improve its performance. By integrating adaptive strategies, levy flight mechanisms, Roulette wheel selection, and random walk patterns into the original GOA, this research aims to address these challenges and improve the algorithm's convergence speed, solution quality, and robustness across a wide range of optimization problems.

2 Literature review

The comparison study of various Gazelle Optimization Algorithm (GOA) variants, such as GOA with Levy Flight, GOA with Roulette Wheel Selection strategy, GOA with Adaptive strategy, and GOA with Random Walk strategy alongside other meta-heuristic algorithms, sheds light on their unique strengths and limitations. Algorithms like Red Panda Optimization, Walrus Optimization, and Green Anaconda Optimization excel in exploitation but may lack balanced exploration capabilities. The GOA variants aim to address this imbalance by leveraging specific strategies tailored to optimize exploration and exploitation. Premature convergence remains a common challenge faced by several algorithms, including Ebola Optimization and Harris Hawks Optimization. In response, the GOA variants introduce adaptive strategies and diverse exploration mechanisms, such as Levy Flight, Adaptive, Random Walk and Roulette Wheel Selection, to mitigate premature convergence and explore the solution space more effectively. This adaptive nature enhances their versatility across various problem types, overcoming limitations observed in algorithms like Slime Mould and Butterfly Optimization. The GOA variations also try to achieve a balance between probabilistic exploration and deterministic exploitation. This characteristic enhances their adaptability to dynamic optimization scenarios, where solutions may evolve over time. Algorithms like Sandpiper and Coyote Optimization encounter difficulties in dynamic environments, whereas the GOA variants offer a potential solution by incorporating adaptive mechanisms. In summary, the comparison study highlights the effectiveness of GOA variants in synergizing exploration and exploitation strategies. Their ability to overcome premature convergence, adapt to diverse problem types, and strike a balance between deterministic and probabilistic elements positions them as promising approaches for solving a wide range of optimization problems compared to individual meta-heuristic algorithms. The comparative analysis underscores the importance of leveraging specific strategies tailored to optimize exploration and exploitation in meta-heuristic algorithms. The GOA variants aim to address this imbalance by leveraging specific strategies tailored to optimize exploration and exploitation. Certain recent population metaheuristic algorithms are listed in Table 1.Table 1 A brief about recent population metaheuristics.

Table 1S No	Algorithm name	Author name, year & Ref no	Benchmark functions	Engineering functions	Motivation	
1	Red panda optimization	H.givi et al., 2023 [9]	52	4	Red pandas strategies in climbing	
2	Walrus optimization	Trojovsky 2022 [10]	68	4	2	
3	Green anaconda optimization	dehghani et al., 2023 [11]	29	21 optimization problems	Hunting strategy of green anacondas	
4	Water wheel plant algorithm	Hamid et al., 2023 [12]	23	3	Water wheel plant natural behavior on hunting expedition	
5	Hermit crab optimization algorithm	Guo et al., 2023 [13]	29		Group behavior of hermit crabs	
6	Mother optimization algorithm	Matousova et al., 2023 [14]	52	4	6	
7	Pelican Optimization algorithm	Trojovsky et al., 2022 [15]	23	4	7	
8	Fennec fox optimization	Troj et al., 2022 [16]	68	4	Fennecs digging ability and es- cape strategy	
9	Gannet optimization algorithm	J.Pan et al., 2022 [17]	28	5	9	
10	Gazelle optimization algorithm	Angushaka et al., 2023 [18]	15	3	10	
11	Ebola Optimization search algo rithm	Oyelade et al., 2022 [19]	47		Propagation mechanism of Ebola virus	
12	The ant lion optimizer	soesanti et al., 2022 [20]	19	3	Imitating ant foraging	
13	Harris hawks optimization	A.Heidari et al., 2019 [21]	29	several	Chasing Style and cooperative behavior of Harris hawks	
14	Moth flame optimization	S.Mirjalili 2015 [22]	29	7	Navigation method of moths in nature.	
15	Equilibrium optimizer	A.Faramarzi et al., 2019 [23]	58	3	Control volume mass balance models	

2.1 Gazelle Optimization Algorithm

The Gazelle Optimization Algorithm (GOA) is a nature-inspired metaheuristic algorithm that mimics the predatory and evasive behavior of gazelles in the wild. The core idea behind GOA is to leverage the swift and agile movements of gazelles to balance exploration and exploitation in the search space, thereby enhancing the algorithm's ability to find optimal solutions to complex optimization problems [24].

In the wild, gazelles employ sudden directional changes and bursts of speed to evade predators. This behavior is modeled in the GOA, where a population of candidate solutions (referred to as gazelles) navigates through the search space based on specific movement rules. These rules are designed to ensure that the algorithm explores the search space efficiently (diversification) while focusing on promising areas (intensification) [24,25].

The GOA algorithm begins with the initialization of a population of gazelles randomly distributed across the search space. Each gazelle represents a potential solution to the optimization problem. The position of each gazelle is updated iteratively using movement equations that mimic the gazelles' natural behavior [24].

2.2 Key components and equations

Initialization: The initial positions of the gazelles are generated randomly within the defined search space. Let Xi(t) represent the position of the ith gazelle at iteration t [24].

Movement Equations: The movement of each gazelle is governed by specific equations that model their evasive maneuvers. The key components of the movement equations include acceleration towards a target, random motion to mimic sudden directional changes, and velocity updates [5,25].

The position update equation can be expressed as:xi(t+1)=xi(t)+vi(t)

Where vi(t) is the velocity of the ith gazelle at iteration t. The velocity update can be influenced by several factors, including:

Acceleration Towards the Optimal Solution: Each gazelle accelerates towards a target, typically the current best solution found by the population. This is analogous to the gazelle's movement towards a safer area [5,24].

Random Motion: To simulate sudden directional changes, a random component is added to the velocity update. This helps in maintaining diversity in the population and avoiding local optima [5].

Acceleration and Velocity Updates: The acceleration component ai(t) towards the optimal solution can be modeled as:ai(t)=α⋅(g(t)−xi(t))

Where a is the acceleration coefficient, and g(t) is the global best position at iteration t. The random motion component ri(t) can be introduced as:ri(t)=β⋅R

Where b is a scaling factor, and R is a vector of random numbers drawn from a uniform distribution [5].

The velocity update equation thus combines both components:vi(t+1)=vi(t)+ai(t)+ri(t)

Position Update: The new position of each gazelle is determined by adding the updated velocity to its current position [24]:xi(t+1)=xi(t)+vi(t+1)

Fitness Evaluation and Selection: After updating the positions, the fitness of each gazelle is evaluated based on the objective function. The global best position g(t+1) is updated if a better solution is found [24].

Termination Criteria: The algorithm iterates through the process of updating velocities and positions until a termination criterion is met, such as a maximum number of iterations or a satisfactory fitness level [24].

The pseudocode for the GOA is as follows.1. Initialize a population of gazelles with random positions	
2. Evaluate the fitness of each gazelle	
3. while termination criterion not met do	
4. for each gazelle do	
5. Update acceleration towards the best solution	
6. Introduce random motion for sudden directional changes	
7. Update velocity	
8. Update position	
9. end for	
10. Evaluate the fitness of each gazelle	
11. Update the global best position	
12. end while	

The Gazelle Optimization Algorithm thus employs a combination of directed acceleration towards promising solutions and random exploratory movements, effectively balancing exploration and exploitation. This makes GOA a powerful tool for solving complex optimization problems, capable of navigating diverse search spaces and avoiding local optima [5,25].

2.3 Proposed variants of the Gazelle Optimization Algorithm

The original Gazelle Optimization Algorithm (GOA) has shown promise in solving various optimization problems; however, it has several shortcomings that necessitated the development of its variants. One of the primary limitations of the original GOA is its tendency to get trapped in local optima, particularly in complex, multi-modal optimization landscapes. This occurs because the algorithm's exploration and exploitation mechanisms are not sufficiently balanced, leading to premature convergence.

To address the limitations of the original Gazelle Optimization Algorithm (GOA) and enhance its performance, several novel variants have been proposed. These variants integrate different strategies to improve the balance between exploration and exploitation, prevent premature convergence, and enhance the overall efficiency of the algorithm. Among these, the Levy Flight strategy and Random Walk strategy stand out as particularly powerful, offering significant advantages over other strategies. The following sections describe four key variants: GOA with Adaptive Strategy, GOA with Levy Flight Strategy, GOA with Roulette Wheel Selection Strategy, and GOA with Random Walk Strategy.

2.3.1 Goa with adaptive strategy

The adaptive strategy dynamically adjusts the algorithm's parameters based on the current state of the optimization process. This allows the algorithm to fine-tune its behavior in real-time, improving convergence speed and solution quality.

Concept: Adaptive strategies have been shown to significantly enhance the performance of metaheuristic algorithms by allowing them to adjust their search behavior dynamically based on feedback from the optimization process [24]. This adaptability is crucial for maintaining an effective balance between exploration and exploitation, especially in complex and high-dimensional search spaces.

Implementation: Parameters such as step size, acceleration coefficient, and direction are adjusted in real-time depending on the diversity of the population and the progress towards the optimal solution. For example, if the population diversity decreases, indicating potential convergence to a local optimum, the step size can be increased to promote exploration. Conversely, when approaching a promising region, the step size can be reduced to fine-tune the search.

The adaptive strategy can be formalized as follows:ai(t)=α(t)⋅(g(t)−xi(t))

ri(t)=β(t)⋅R

where α(t) and β(t) are dynamically adjusted parameters. This dynamic adjustment ensures that the algorithm remains flexible and responsive throughout the optimization process, leading to better overall performance.

2.3.2 Goa with Levy Flight Strategy

Levy flights are random walks characterized by long jumps interspersed with short steps, which are observed in various natural processes. This strategy enhances the global search capability of the algorithm, helping it escape local optima and explore the search space more thoroughly.

Concept: Levy flights introduce a higher probability of making larger jumps, facilitating better exploration [25]. This is particularly useful for escaping local optima and ensuring a diverse search.

Implementation: Gazelles follow a Levy flight pattern, where their movement steps are determined by a Levy distribution. This helps in maintaining diversity and exploring new areas of the search space.

The Levy flight step can be expressed as:L(t)=xi(t)+δ⋅u

where δ is a scaling parameter and u is drawn from a Levy distribution.

2.3.3 Goa with Roulette Wheel Selection Strategy

The Roulette wheel selection strategy is inspired by genetic algorithms, where the probability of selecting a particular solution is proportional to its fitness. This approach promotes solutions with higher fitness, accelerating convergence towards optimal regions.

Concept: Roulette wheel selection ensures that fitter solutions have a higher impact on the population's direction, which helps in focusing the search on promising regions while maintaining diversity [5].

Implementation: The selection mechanism determines which gazelles influence the movement of others based on their fitness. This probabilistic approach helps to balance exploration and exploitation effectively.

The probability Pi of selecting a gazelle i is given by:Pi=fi∑j=1Nfi

where fi is the fitness of gazelle i, and N is the total number of gazelles.

2.3.4 Goa with random walk strategy

Random walks involve steps taken in random directions, which can help maintain diversity within the population and prevent premature convergence. This strategy ensures a more comprehensive exploration of the search space.

Concept: Random walks introduce variability in the search process, ensuring that the algorithm explores less-explored areas of the search space and maintains a diverse set of solutions [26].

Implementation: At each iteration, gazelles perform random walks based on a predefined distribution, introducing variability and helping the algorithm explore the search space more thoroughly.

The random walk step can be expressed as:xi(t+1)=xi(t)+γ⋅(R−0.5)

where γ is a scaling factor, and R is a random vector with components drawn from a uniform distribution in the range [0, 1].

While all the proposed variants contribute to enhancing the GOA, the adaptive strategy is particularly noteworthy for its dynamic and flexible approach. Unlike static strategies, which apply fixed rules and parameters throughout the optimization process, the adaptive strategy continuously adjusts to the current state of the search. This adaptability makes it exceptionally powerful in dealing with the varying demands of different phases of optimization. The mathematical equations representing benchmark functions unimodal, multimodal and fixed dimension benchmark functions are depicted in Table 2, Table 3, Table 4 respectively.Table 2 Benchmark functions for Unimodal.

Table 2Functions	Dimensions	Range	fmin	
F1(U) = ∑m=1zUm2	30	[-100,100]	0	
F2(U) = ∑m=1z|Um|+∏m=1z|Um|	30	[-10,10]	0	
F3(U) = ∑m=1z(∑n−1mUn)2	30	[-100,100]	0	
F4(U) = max m{|Um|,1≤m≤	30	[-100,100]	0	
F5(U) = ∑m=1z[100(Um+1−Um2)2+(Um−1)2]	30	[-38,38]	0	
F6(U) = ∑m=1z([Um+0.5])2	30	[-100,100]	0	
F7(U) = ∑m=1zmUm4+random[0,1]	30	[-1.28,1.28]	0	

Table 3 Benchmark functions for Multimodal.

Table 3Functions	Dimension	Range	fmin	
F8 = ∑m=1z−Umsin(|Um|)	30	[-500,500]	−418.98295	
F9 = ∑m=1z[Um2−10cos(2πUm)+10]	30	[-5.12,5.12]	0	
F10 = −20exp(−0.2(1z∑m=1zUm2))−exp(1z∑m=1zcos(2πUm)+20+d	30	[-32,32]	0	
F11 = 1+∑m=1zUm24000−∏m=1zcosUmm	30	[-600,600]	0	
F12=πz{10sin(πτ1)+∑m=1z(τm−1)2[1+10sin2(πτm+1)]+(τz−1)2}+∑m=1zg(Um,10,100,4)τm=1+Um+14{x(Um−b)iUm>b0−b<Um<bx(−Um−b)iUm<−b	30	[-50,50]	0	
F13=0.1{sin(3πUm)+∑m=1z(Um−1)2[1+sin2(3πUm+1)]+(xz−1)2[1+sin2)]}	30	[-50,50]	0	

Table 4 Fixed dimension standard benchmark functions.

Table 4Functions	Dimension	Range	fmin	
F14=[1500+∑n=1251n+∑m=1z(Um−bmn)6]−1	2	[-65.536,65.536]	1	
F15=∑m=111[bm−U1(am2+amn2)am2+amn3+n4]2	4	[-5,5]	0.00030	
F16 = 4U12−2.1U14+13U16+U1U2−4U24	2	[-5,5]	−1.0316	
F17 = (U2−5.14π2U12+5πU1−6)2+10(1−18π)cosU1+10	2	[-5,5]	0.398	
F18 = [1+(U1+U2+1)2(19−14U1+3U12−14U2+6U1U2+3U22)]Χ[30+(2U1−3U2)2Χ(18−32U1+12U12+48U2−36U1U2+27U22)]	2	[-2,2]	3	
F19 = −∑m=14dmexp(−∑n=13Umn(Um−qmn)2)	3	[1,3]	−3.32	
F20 = −∑m=14dmexp(−∑n=16Umn(Um−qmn)2)	6	[0,1]	−3.32	
F21 = −∑m=15[(U−bm)(U−bm)T+dm]−1	4	[0,10]	−10.1532	
F22 = −∑m=17[(U−bm)(U−bm)T+dm]−1	4	[0,10]	−10.4028	
F23 = −∑m=17[(U−bm)(U−bm)T+dm]−1	4	[0,10]	−10.5363	

3 Performance evaluation on CEC 2014 benchmark functions

The Gazelle Optimization Algorithm (GOA) integrated with Levy flight strategy emerges as the optimal choice for unimodal functions, as per the findings of the CEC 2014 results highlighted in Table 5. Conversely, when tackling multimodal functions, both GOA with Levy flight and GOA with random walk exhibit superior performance. Notably, for fixed dimension functions, the efficacy between these strategies remains nearly equivalent. This summary encapsulates the notable outcomes of the CEC 2014 evaluation, shedding light on the nuanced effectiveness of these optimization techniques across varying function types.Table 5 The optimal solution of Gazelle Optimization Algorithm (GOA) integrated with different strategies are given below.

Table 5	GOA	GOA_Levy	GOA_RWS	GOA_Ad	GOA_RW	DA	ALO	SCA	
F1	3.04E-113	2.03E-118	0.68426	244.39604	1.09E-78	9.41E-01	1.39E-08	2.30E-15	
F2	4.53E-62	7.48E-71	1.077522	6.58E-43	3.17E-47	1.12E+00	2.06E-05	2.27E-11	
F3	9.22E-18	3.27E-07	2163.51	1826.4542	4.27E-06	4.94E+01	7.71E-01	8.79E-06	
F4	0.00019582	8.39E-32	2.274029	9.7734929	1.66E-19	1.59E+00	1.05E-03	4.38E-06	
F5	47.17694279	21.493412	139.1654	5417.4431	44.39682	1.35E+02	6.00E+00	7.28E+00	
F6	2.048910983	0.1004971	0.267555	206.53265	1.209148	1.03E+00	7.90E-09	2.91E-01	
F7	0.000274045	0.0011046	0.136829	0.0029247	0.002162	7.39E-03	1.64E-02	1.48E-03	
F8	−5918.98798	−7995.307	−7267.314	−10194.96	−13380.8	−3.00E+03	−3.22E+03	−2.10E+03	
F9	120.1642573	0	41.94881	212.54983	0	1.06E+01	1.59E+01	2.71E-12	
F10	7.99E-15	4.44E-15	0.910896	4.5771073	4.44E-15	1.43E+00	8.18E-05	5.66E-08	
F11	0	0	0.581728	2.4399054	0	6.52E-02	1.21E-01	6.63E-12	
F12	0.466106021	0.0002272	5.433198	8.9671457	0.020463	7.58E-02	2.63E-01	6.60E-02	
F13	3.821359333	0.4244957	1.341773	45.337275	0.784142	2.89E-01	2.25E-07	3.35E-01	
F14	10.76318067	0.9980038	0.998004	0.9980038	0.998004	9.98E-01	9.98E-01	9.98E-01	
F15	0.000307487	0.0003075	0.000751	0.0003075	0.000307	1.50E-03	7.78E-04	6.92E-04	
F16	−1.03162845	−1.031628	−0.9787	−1.031628	−1.03163	−1.03E+00	−1.03E+00	−1.03E+00	
F17	0.397887358	0.3978874	0.397887	0.3978874	0.397887	3.98E-01	3.98E-01	3.99E-01	
F18	3	3	3	3	3	3.00E+00	3.00E+00	3.00E+00	
F19	−3.86278215	−3.862782	−3.862782	−3.862782	−3.86278	−3.86E+00	−3.86E+00	−3.85E+00	
F20	−3.32196318	−3.321995	−3.321995	−3.321995	−3.322	−3.32E+00	−3.32E+00	−3.07E+00	
F21	−5.0551977	−10.1532	−10.1532	−10.1532	−10.1532	−1.02E+01	−5.10E+00	−2.10E+00	
F22	−10.4029405	−10.40294	−10.40294	−10.40294	−10.4029	−1.03E+01	−1.04E+01	−4.02E+00	
F23	−10.5364093	−10.53641	−10.53641	−10.53641	−10.5364	−1.05E+01	−1.05E+01	−7.27E+00	

4 Performance evaluation on CEC 2017 benchmark functions

In the context of the CEC 2017 evaluation, the integration of the Gazelle optimization algorithm (GOA) with Random Walk showcased superior performance across nearly 16 out of 30 functions assessed as can be clearly observed in Table 6. Additionally, it's noteworthy that other variants of GOA also demonstrated commendable performance, albeit to varying degrees. This observation underscores the efficacy of GOA integrated with Random Walk as a robust optimization strategy for a significant portion of the functions evaluated, while also acknowledging the competitive performance of alternative GOA variants.Table 6 The optimal solution of Gazelle Optimization Algorithm (GOA) integrated with different strategies are given below.

Table 6	GOA	GOA_Levy	GOA_AD	GOA_RWS	GOA_RW	AOA	SMA	HGS	HHO	
F1	11318945660	15394231	1554404	166010.2	13121.45	54028522695	14967.57	27192.46946	23725598	
F3	2366.576405	4998.51	9262.902	74384.57	2899.814	89721.95815	7766.255	22483.54785	34236.34	
F4	520.189962	513.9213	496.7742	525.998	414.0348	14892.28925	516.3544	504.8392513	523.70608	
F5	624.0611175	612.7891	612.1558	608.6926	571.2724	874.8903573	641.7264	668.9047167	682.85396	
F6	610.0755216	604.1443	605.8021	628.2908	604.3633	675.5408948	606.2797	602.7739277	668.18055	
F7	911.1987804	839.1057	928.6041	955.2525	832.0513	1366.167168	849.5896	990.7246494	1307.1376	
F8	962.5155287	925.2394	928.3397	889.8024	876.172	1114.451032	876.5125	982.0858362	949.63695	
F9	1104.032359	1099.286	1031.474	2339.819	956.1583	7030.597178	5266.754	4089.225172	8707.9079	
F10	6687.241441	5005.493	6282.676	4764.386	5391.449	8141.236656	5252.875	4376.668335	5608.7991	
F11	1247.389408	1175.49	1264.483	1337.61	1141.71	8873.258638	1211.997	1242.130267	1267.1967	
F12	4410180.842	160994.2	131657.4	16997879	351695.2	8767132220	8713061	732539.7257	7992643.5	
F13	2213.215197	2247.217	3372.038	56272.67	1800.558	14660562324	64192.16	19802.69854	591981.92	
F14	1473.123637	1459.7	1471.2	22488.09	1459.562	2892975.615	102375.8	184902.5192	184074.41	
F15	1614.402456	1668.204	1744.72	15228.32	1657.883	25025.94679	3818.45	12404.03554	73785.753	
F16	2233.376799	2174.244	1971.675	2689.577	1961.439	5934.818456	2353.326	3151.756242	3492.7963	
F17	1810.154148	1827.473	1818.768	2235.732	1827.091	2374.829319	2398.662	2288.864635	2447.0727	
F18	2050.476627	2048.559	1965.86	4424192	2372.15	9513354.92	2698395	615706.413	4527343.6	
F19	1953.994301	1938.767	1944.813	10939.51	1937.981	1963376.273	56141.58	17520.67249	768785.94	
F20	2327.479967	2250.777	2211.388	2795.612	2210.561	2687.951086	2819.833	2447.488902	2967.8559	
F21	2400.947217	2391.895	2356.119	2359.919	2349.292	2587.155541	2477.994	2498.371263	2556.3067	
F22	2323.127523	2319.722	2321.342	2308.013	7254.064	7726.591538	5281.118	2305.141088	7596.3395	
F23	2769.045155	2739.038	2720.789	2771.081	2689.178	3676.927087	2754.396	2820.306328	3122.0174	
F24	2925.795978	2921.719	2902.305	2884.101	2945.879	4065.297273	2926.282	2973.976076	3326.5163	
F25	2888.09341	2896.117	2911.837	2919.548	2912.168	5383.064136	2888.937	2890.168208	2961.6078	
F26	4717.456117	4459.696	4780.796	4940.577	4322.194	10178.09976	5354.295	5466.584058	8747.3754	
F27	3204.861647	3224.575	3212.243	3219.879	3208.624	4233.744213	3225.034	3240.578922	3637.912	
F28	3225.393164	3237.662	3233.055	3283.788	3203.388	5845.609175	3223.579	3321.454645	3347.5356	
F29	3994.425817	3730.727	3744.044	4024.538	3955.895	6112.339166	4059.759	3910.743665	4861.2658	
F30	27279.43781	9302.87	14387.7	106761.5	28824.32	806683435.1	22202.73	456427.723	2060119.7	

5 Multi-level inverter with 31 level cascaded H bridge

An electrical system can employ a power electronic equipment called a cascaded H- Bridge (CHB) Multi-Level Inverter to change direct current (DC) into alternating current (AC) with a greater voltage level. Several H-Bridge modules linked in series are used in this advanced inverter design to produce a stepped or multi-level output waveform. The basic building blocks of the inverter are H-Bridge modules. An H- Bridge is a configuration of electronic switches that allows the current to flow in both directions. Each H-Bridge module typically consists of four switches (transistors or insulated gate bipolar transistors - IGBTs) that can be controlled to generate the desired voltage output. Several H-Bridge modules are linked in series to form a cascaded H-Bridge inverter. The total of all the voltages from each H-Bridge module added together is the output voltage. This configuration enables the generation of a multi-level output waveform with several voltage levels. The primary advantage of a Cascaded H-Bridge Inverter is its ability to produce a stepped or multi-level output waveform. This leads to a reduced total harmonic distortion (THD) in the output voltage compared to traditional inverters with sinusoidal outputs. Reduced THD results in improved efficiency and better performance for certain applications. The number of H-Bridge modules determines the number of voltage levels in the output waveform. Each H-Bridge module contributes a voltage level, and the total voltage across the load is the sum of these individual levels. The output voltage waveform's CHB multilevel general equation is as follows:Vo(t)=2π∑m=1∞1msin(mωt)×∑k=1NVdc,ksin((2k−1)mπ2N)

where N is the number of H-bridge modules in the inverter; Vo(t) is the output voltage waveform; w is the fundamental frequency of the resultant waveform; and Vdc, k is the DC voltage input of the kth H-bridge module. As the number of voltage levels rises, the output waveform approaches a sine wave. The outputs of the separate H bridge circuits are combined to create the output of the inverter. This makes it possible to generate a greater number of voltage levels, which produces an output waveform that is more effective and of higher quality. In Fig. 1, the H-bridge circuit is shown.Fig. 1 Circuit model of H Bridge.

Fig. 1

The entire load will have a positive voltage applied when switches S1 and S4 are closed (and switches S2 and S3 are open). The voltage may be inverted to allow for a negative voltage by turning OFF the S1 and S4 switches and turning ON the S2 and S3 switches. However, it is never advisable to leave the S1 and S2 switches off simultaneously since this might cause the I/P voltage supply to short circuit. However, the same caution should always be used to ensure that the S1 and S2 switches are never left off simultaneously, since this might cause the I/P voltage supply to short circuit. Switches S3 and S4 should be used with the same prudence. ”Shoot-through” is the technical phrase for this circumstance. The waveform in Fig. 2 illustrates how the CHB inverter may generate a waveform that nearly resembles a sinusoidal waveform with little harmonic distortion.Fig. 2 Cascaded H bridge waveform.

Fig. 2

5.1 Harmonics

The sinusoidal parts of a periodic waveform whose frequencies are integer multiples of the fundamental frequency are referred to as harmonics. The lowest frequency element in a waveform is called the fundamental frequency. The harmonics would comprise components at 100 Hz (2nd harmonic), 150 Hz (3rd harmonic), 200 Hz (4th harmonic), and so on, if the fundamental frequency was 50 Hz (Hz). There is an integer connection between the fundamental frequency and each harmonic component. Electrical systems experience harmonics when non-linear loads use non-sinusoidal current from the power source. Non-linear loads cause distortion in the current waveform as opposed to linear loads, such resistive loads, which draw sinusoidal currents in accordance with the voltage. Harmonic components at multiples of the fundamental frequency are how this distortion appears. The presence of harmonics in electrical systems can lead to various negative effects, impacting both equipment and power distribution networks. Few negative effects of harmonics are.• Harmonics lead to excessive heating in electrical equipment, reducing their lifespan.

• Harmonic currents contribute to additional losses in power distribution systems, increasing operational costs.

• Harmonics distort voltage and current waveforms, affecting the performance of sensitive electronic equipment.

• Harmonics create resonant conditions, amplifying voltages and currents, leading to equipment damage.

• Harmonics cause electromagnetic interference, disrupting data transmission in communication systems.

Maintaining the dependability and effectiveness of power distribution networks requires reducing the detrimental effects of harmonics. Employing harmonic filters, power factor correction devices, and harmonic-resistant equipment can help minimize the impact of harmonics. Additionally, adhering to international standards, such as IEEE 519, provides guidelines for acceptable levels of harmonics, ensuring the delivery of high-quality electrical power. Through these mitigation strategies and proactive measures, the adverse effects of harmonics can be effectively addressed, contributing to the overall stability and performance of electrical networks.

5.2 Total Harmonic distortion

The amount of distortion brought on by harmonics in an electrical system is measured by total harmonic distortion, or THD. The square root of the sum of the squares of the harmonic amplitudes is divided by the amplitude of the fundamental frequency to determine the percentage. One of the most important metrics for evaluating the quality of a power supply waveform is THD. Higher THD levels are indicative of more harmonics present, which distorts the waveform. Total harmonic distortion, or THD, has the following equation:THD=SumofoddharmonicsafterFFT2FundamentalFrequency*100

where the first harmonic serves as the fundamental frequency and the other harmonic frequencies are multiples of it. A comparison of the firing angles using the various optimization methods is presented in Table 7. Furthermore, the THD spectrum and wave shapes derived from the various optimization methods are displayed in Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7. A comparison of THD values is presented in Table 8.Table 7 Comparison of firing angles of various Goa variants.

Table 7Angles	GOA	GOA_AD	GOA_Levy	GOA_RWS	GOA_RW	
A1	0	0.020377	0	0	0.000534	
A2	0	0.051759	0	0.173058	0.068423	
A3	0.002011	0.120008	0.053658	0.264094	0.091935	
A4	0.103972	0.2037	0.097105	0.306932	0.165167	
A5	0.111207	0.212815	0.116767	0.434518	0.169953	
A6	0.189034	0.331151	0.20967	0.451535	0.266117	
A7	0.245747	0.423084	0.260535	0.518496	0.370689	
A8	0.327494	0.557752	0.337523	0.744604	0.46526	
A9	0.353538	0.568141	0.370563	0.802226	0.521987	
A10	0.413992	0.657531	0.439937	0.892441	0.603142	
A11	0.490223	0.781111	0.483113	0.981546	0.711812	
A12	0.576129	0.928712	0.552064	1.092984	0.71197	
A13	0.587294	1.033329	0.61586	1.240436	0.775281	
A14	0.706956	1.151295	0.753958	1.570796	0.833509	
A15	0.836516	1.307826	0.90369	1.570796	1.307826	

Fig. 3 (a) indicates GOA waveform and (b) indicates the THD spectrum of GOA.

Fig. 3

Fig. 4 (a) indicates GOA AD waveform and (b) indicates the THD spectrum of GOA AD.

Fig. 4

Fig. 5 (a) indicates GOA Levy waveform and (b) indicates the THD spectrum of GOA Levy.

Fig. 5

Fig. 6 (a) indicates GOA RW waveform and (b) indicates the THD spectrum of GOA RW.

Fig. 6

Fig. 7 (a) indicates GOA RWS waveform and (b) indicates the THD spectrum of GOA RWS.

Fig. 7

Table 8 Comparison of THD values of Goa variants.

Table 8Algorithms	GOA	GOA_AD	GOA_Levy	GOA_RWS	GOA_RW	
THD (%)	16	4.87	14.94	8.66	6.07	

6 Experimental analysis of design problems in engineering

Due to their enormous complexity, real-world design problems typically provide challenges to reaching optimal results. The restrictions of inequality and equality give rise to the problem's complexity [19]. The process of processing constraints is the act of taking restrictions into account during optimization. There are two categories of solutions found using these trending algorithms: feasible and infeasible. By leveraging enhanced or memetic algorithms, which amalgamate characteristics from two or more algorithms, various constraint tactics are employed to attain optimal practical outcomes with minimal computational effort and cost [20]. Five engineering-constrained design issues are tested for all GOA variants in this research effort, Table 9 provides an explanation of these issues in detail.Table 9 Engineering-based design specifics (design 1 - design 5).

Table 9S.No.	Design Number	Design Problem Name	Goal	Total Constraints	Total Variables	
1	Design 1	Pressure vessel	Minimizing Costs	4	4	
2	Design 2	Welded beam design	Minimizing Costs	7	4	
3	Design 3	3 bar truss	Minimizing Weight	2	3	
4	Design 4	Speed reducer	Minimizing Weigh	11	7	
5	Design 5	Tension-compression Spring design	Minimizing Weight	4	3	

6.1 PV design problem

Kannan and Kramer conducted pioneering research on the optimal design problem of pressure vessels (PV) back in 1994. The primary objective of this research is to minimize overall costs, encompassing expenses related to welding, forming, and material procurement [21]. The pressure vessel, typically cylindrical in shape and flanked by semicircular heads on both ends, is governed by four key design variables, as illustrated in Fig. 8. The design model is systematically formulated utilizing equations (1)–(3). The cost function involving thickness of shell, thickness of head, inner radius, and length of shell are represented in eqs. (3.a), (3.b), (3.c), (3.d) respectively. The ranges of different parameters are represented in eq. (3e). The goal of optimization problem is such that the total cost of material, forming, and welding is minimized accounting for 4 constraints. Furthermore, a comparative analysis between the variants of the Gazelle Optimization Algorithm (GOA) and other existing Population-based Metaheuristic (PMH) approaches has been carried out and summarized in Table 10.Fig. 8 Design 1: Pressure vessel (PV) optimal design problem.

Fig. 8

Table 10 Goa variants algorithm comparative analysis for design 1.

Table 10Algorithm	GOA	GOA_AD	GOA_RWS	GOA_Levy	GOA_RW	GWO	MFO	GA	ACO	PSO	
Optimum Value	Ts	1.256485281	0.7781686414	3.680084554	0.778168673	0.778168871	0.8125	0.8125	0.778210	0.8125	0.8125	
Th	0.6210807922	0.3846491626	27.39294716	0.3846492061	0.3846494749	0.4345	0.4375	0.384889	0.4375	0.4375	
R	65.10276179	40.31961872	51.73089279	40.31961877	40.31962964	42.0892	42.098	40.31504	42.1036	42.0913	
L	10.52925563	200	97.60314349	200	199.9999187	176.7587	176.64	200	176.5727	176.7465	
Optimum Cost	7308.502163	5885.332774	159994.8751	5885.333136	5885.335387	6051.564	6059.7143	6288.745	6059.089	6061.078	

Let us consider,

Let us consider,(1) t→=[t1t2t3t4]=[TsThRL]

Minimize,(2) f(t→)=0.6224t1t3t4+1.7781t2t32+3.1661t12t4+19.84t12t3

Subject to,(3.a) g1(t→)=−t1+0.0193t3≤0

(3.b) g2(t→)=t3+0.00954t3≤0

(3.c) g3(t→)=−πt32t4−43πt33+1296000≤0

(3.d) g4(t→)=t4−240≤0

Variable ranges are,(3.e) 0≤t1≤99;0≤t2≤99;10≤t3≤200;10≤t4≤200

6.2 WB design problem

The primary aim of this design challenge is to minimize the job cost associated with the welded beam. This optimization task involves the consideration of four discrete variables and is subjected to seven constraints, as depicted in Fig. 9. In mathematical terms, this design is expressed by equations (4), (5). Table 11 shows the comparative study of GOA variations with various current techniques.Fig. 9 Design 2: Welded beam (WB) optimal design problem.

Fig. 9

Table 11 Goa variants algorithm comparative analysis for design 2.

Table 11Algorithm	GOA	GOA_AD	GOA_RWS	GOA_Levy	GOA_RW	GSA	GA	Simplex	David	
Optimum Variables	h	0.2690695574	0.2057296398	0.6978644392	0.205729552	0.2057295145	0.1821	0.2489	0.2792	0.2434	
l	2.825679407	3.470488666	6.327924771	3.470491792	3.470491237	3.857	6.173	5.6256	6.2552	
t	7.900884079	9.03662391	6.290120717	9.036625765	9.03662703	10	8.1789	7.7512	8.2915	
b	0.2691299151	0.2057296398	0.8056113542	0.205729632	0.2057296472	0.2024	0.2533	0.2796	0.2444	
Optimal Cost	1.947252414	1.724852309	4.605067366e+10	1.724852858	1.724853057	1.88	2.4331	2.5307	2.3841	

Let's consider, t→=[t1t2t3t4]=[hltb]:(4) Minimize,f(t→)=1.10471t12t2+0.04811t3t4(14.0+t2)

(5) RangeofVariables=0.1≤t1≤2;0.1≤t2≤1;0.1≤t3≤10;0.1≤t4≤2

6.3 Three bar truss design problem

The engineering configuration problem of the 3-Bar Truss has been addressed to evaluate the effectiveness of the algorithms for engineering design-based optimization, as depicted in Fig. 10. Equations (6), (7) present the 3-bar truss problem's mathematical representation and Table 12 compares the GOA variations technique with other PMH algorithms.Fig. 10 Design 3: Three bar truss optimal design problem.

Fig. 10

Table 12 Goa variants algorithm comparative analysis for design 3.

Table 12Algorithms	GOA	GOA_AD	GOA_RWS	GOA_Levy	GOA_RW	Ray and Sain	MFO	
Optimal variables	t1	0.7869011643	0.7863750477	0.7739407928	0.7891911649	0.7956292689	0.795	0.7882447709	
t2	0.4132891045	0.4147931374	0.4852960923	0.406791797	0.388926346	0.395	0.4094669057	
Optimal weight	263.954	263.89598	263.9637	264.1082	263.9305751	264.3	263.896	

Now let's consider,t→=[t1,t2]=[A1,A2]

Minimize,(6) f(t→)=(22t1+t2)×l

Subject to,g1(t→)=2t1+t22t12+2t1t2P−σ≤0

g2(t→)=t22t12+2t1t2P−σ≤0

(7) g3(t→)=12t2+t1P−σ≤0

Range of variables = 0 ≤t1,t2≤ 1

6.4 SR design problem

The speed reducer design problem presents a high level of complexity with seven design variables, making it one of the most challenging optimal design problems. With eleven optimization constraints and six continuous variables, this problem is particularly intricate. The primary objective of the speed reducer optimal design challenge is to minimize the weight while adhering to various constraints, including surface stress on gear teeth, bending, and transverse deflection of the shaft. Fig. 11 shows the seven design variables (t1–t7). Equation (8) through (9) provide a mathematical definition of the problem, and Table 13 compares the GOA variations technique with other PMH algorithms.(8) Minimizef(t→)=0.7854t1t2(3.3333t32+14.9334t3−43.0934)−1.508t1(t62+t72)+7.4777(t63+t73)+0.7854(t4t62+t5t72)

Fig. 11 Design 4: Speed reducer optimal design problem.

Fig. 11

Table 13 Goa variants algorithm comparative analysis for design 4.

Table 13Algorithms	GOA	GOA_AD	GOA_RWS	GOA_Levy	GOA_RW	MBA	OBSCA	PSO-DE	HGSO	
Optimal variables	t1	3.5	3.5	3.6	3.5	3.5	3.5	3.0879	3.50	3.498	
t2	0.7	0.7	0.7195	0.7	0.7	0.7	0.7550	0.7	0.71	
t3	17	17	28	17	17	17	26.4738	17	17.02	
t4	7.3	7.3	8.3	7.3	7.3	7.300033	7.3650	7.3	7.67	
t5	7.754068	7.71531	8.3	7.71534	7.71538	7.715772	7.9577	7.8	7.810	
t6	3.35033	3.35021	3.81448	3.35021	3.35021	3.350218	3.4950	3.350214	3.36	
t7	5.28668	5.28665	5.5	5.28665	5.286655	5.286654	5.2312	5.2866832	5.289	
Optimal cost	2995.38496	2994.47	6082.295486	2994
0.4734	2994.474	2994.48245	3056.3122	2996.34817	2997.10	

Here,(9) 2.6≤t1≤3.6,0.7≤t2≤0.8,17≤t3≤28,7.3≤t4≤8.3,7.8≤t5≤8.3,2.9≤t6≤3.9,5≤t7≤5.5

6.5 TCS design problem

The primary objective of this task is to reduce the weight of the spring while considering constraints related to surge frequency, geometry, deflection, and shear stress. This issue considers four nonlinear limits of inequality and three continuous variables. In Fig. 12, the design factors are illustrated. The mathematical formulation for the design is provided by Equations (10)–(12).and Table 14 presents a comparison of that formulation with the current approaches.Fig. 12 Design 5: Tension/Compression Spring (TCS) Optimal design problem.

Fig. 12

Table 14 Goa variants algorithm comparative analysis for design 5.

Table 14Algorithm	GOA	GOA_AD	GOA_RWS	GOA_Levy	GOA_RW	PSO	SHO	GWO	GSA	MFO	
Optimal Variables	D	0.06333623097	0.05176204	0.1424650981	0.05166828804	0.05164193615	0.3576	0.343751	0.3567	0.3237	0.36410932	
d	0.7070367903	0.35847564	1.241246395	0.3562181186	0.3555835056	0.0517	0.051144	0.0516	0.0503	0.051994457	
N	3.268259996	11.18665772	15	11.31832038	11.35587958	11.2445	12.0955	11.2889	13.5254	10.868421862	
Optimal weight	0.01494216899	0.01266534867	1.890329235e+13	0.01266524535	0.01266523913	0.01267	0.012674	0.01267	0.0127	0.01267	

Consider,s→=[s1s2s3]=[dDN]

Minimize,(10) f(s)=(s3+2)s2s12

Subject to,g1(s)=1−s23s371785s14≤0

g2(s)=4s22−s1s212566(s2s13−t14)+15108s12≤0

g3(s)=1−140.45s1s22s3≤0

(11) g4(s)=s1+s21.5−1≤0

Variable ranges, 0.005 ≤s1≤ 2.00, 0.25 ≤t2≤ 1.30, 2.00 ≤s3≤ 15.0. (12)

7 Conclusions

The comparative analysis conducted across multiple CEC competitions reveals intriguing insights into the performance of various variants of the Gazelle Optimization Algorithm (GOA). In the CEC 2014 assessment, GOA integrated with Levy Flight strategy emerged as the frontrunner, showcasing superior performance across the evaluated functions. Conversely, in the subsequent CEC 2017 evaluation, GOA augmented with Random Walk demonstrated notable efficacy, particularly excelling in nearly 16 out of 30 functions assessed. Moreover, in the realm of Total harmonic minimization and other mechanical engineering problems, GOA with Adaptive Strategy emerged as the top performer. Notably, all variants of GOA were subjected to rigorous comparison with other metaheuristic algorithms, ensuring a comprehensive assessment of their efficacy. The findings underscore the enhanced performance of GOA variants compared to both the original GOA and alternative metaheuristic algorithms across diverse problem domains. This collective evidence suggests that the refined variants of GOA present promising avenues for optimization tasks, offering superior performance and versatility when compared to both the baseline algorithm and other established techniques.

Limitations and directions for future research

While this study demonstrates the superior performance of various GOA variants across multiple benchmark functions and specific problem domains, it has limitations including a restricted scope of benchmark functions, substantial computational resource requirements, and limited generalizability to other fields. Future research should aim to include a broader range of benchmark functions, apply GOA variants to diverse real-world problems beyond mechanical engineering, and explore hybrid approaches that integrate other optimization techniques. Additionally, investigating parameter sensitivity and incorporating multi-objective optimization techniques will enhance the adaptability and robustness of GOA variants.

Data availability statement

The dataset is available in the https://www.mathworks.com/matlabcentral/fileexchange/124810-benchmark-problems.

CRediT authorship contribution statement

Raghav Mahajan: Writing – original draft, Methodology, Investigation, Conceptualization. Himanshu Sharma: Visualization, Software, Data curation. Krishan Arora: Validation, Supervision, Methodology, Formal analysis. Gyanendra Prasad Joshi: Writing – review & editing, Resources, Project administration. Woong Cho: Supervision, Project administration, Funding acquisition.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
==== Refs
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