An Improved Wild Horse Optimizer for Solving Optimization Problems

: Wild horse optimizer (WHO) is a recently proposed metaheuristic algorithm that simulates the social behavior of wild horses in nature. Although WHO shows competitive performance compared to some algorithms, it suffers from low exploitation capability and stagnation in local optima. This paper presents an improved wild horse optimizer (IWHO), which incorporates three improvements to enhance optimizing capability. The main innovation of this paper is to put forward the random running strategy (RRS) and the competition for waterhole mechanism (CWHM). The random running strategy is employed to balance exploration and exploitation, and the competition for waterhole mechanism is proposed to boost exploitation behavior. Moreover, the dynamic inertia weight strategy (DIWS) is utilized to optimize the global solution. The proposed IWHO is evaluated using twenty-three classical benchmark functions, ten CEC 2021 test functions, and five real-world optimization problems. High-dimensional cases ( D = 200, 500, 1000) are also tested. Comparing nine well-known algorithms, the experimental results of test functions demonstrate that the IWHO is very competitive in terms of convergence speed, precision, accuracy, and stability. Further, the practical capability of the proposed method is verified by the results of engineering design problems.


Introduction
Optimization is a research field that aims to maximize or minimize specific objective functions [1][2][3].Most real-world engineering problems found in nature are classified as NP optimization problems [4,5].Optimization exists in nearly every field, such as economics [6], text clustering [7], pattern recognition [8], chemistry [9], engineering design [10], feature selection [11,12], face detection and recognition [13], and information technology [14].Classical and traditional mathematical approaches cannot solve these problems accurately or adequately.Recently, many researchers have tried to solve these problems using a new kind of approximation algorithm known as metaheuristics.In the last two decades, an enormous amount of development and work regarding metaheuristics algorithms have been done.Metaheuristics algorithms have received much attention and have become more popular due to their flexibility, simplicity, and their avoidance of local optima.Metaheuristics algorithms simulate natural phenomena or physical law and can be generally categorized into four different classes: (1) swarm intelligence algorithms, (2) evolution algorithms, (3) physics and chemistry algorithms, and (4) human-based algorithms.
Human-based algorithms: Humans are the most intelligent creature in the universe as they can easily find the best way to fix their problems and issues.Human-based algorithms contain algorithms that simulate physical or nonphysical activities such as thinking and physical activities.Examples of this class of algorithms include teachinglearning-based optimization (TLBO) [41], imperialist competitive algorithm (ICA) [42], and social-based algorithm (SBA) [43].
Although several metaheuristic techniques have been proposed, no free lunch (NFL) [44] encourages researchers to research more.NFL mentioned that no algorithm works efficiently in all optimization problems.If the algorithm can solve a specific type of problem efficiently, it may not solve other optimization problem classes.Therefore, many researchers devote themselves to proposing new algorithms or improving existing methods.For instance, Ning improved the whale optimization algorithm using Gaussian mutation [45].Nautiyal and Tubishat enhanced the basic salp swarm algorithm by applying mutation schemes and opposition-based learning, respectively [46,47].Moreover, Pelusi proposed a hybrid phase between exploration and exploitation using a fitness dependent weight factor to strengthen the balance between exploration and exploitation for moth-flame optimization [48].Recently, a new optimization algorithm called wild horse optimizer (WHO) [49] was proposed by Naruei and Keynia.WHO simulates horse behavior in which they leave their group and join another group before becoming adults to prevent mating between siblings or daughters.However, WHO may fall in local optima regions or have a slow convergence in high-dimensional and complex problems.
In this paper, an improved version of WHO, called "IWHO", is proposed to enhance the original algorithm's effectiveness and performance.IWHO improved the original algorithm by using three different operators: random running strategy (RRS), dynamic inertia weight strategy (DIWS), and competition of waterhole mechanism (CWHM).In order to evaluate the performance of the WHO algorithm, we compared it with the classical WHO algorithm and eight other different algorithms: grey wolf optimizer (GWO) [50], moth-flame optimization (MFO) [51], salp swarm algorithm (SSA) [52], whale optimization algorithm (WOA) [53], particle swarm optimizer (PSO) [15], hybridizing sine-cosine algorithm with harmony search (HSCAHS) [54], dynamic sine-cosine algorithm (DSCA) [55], and modified ant lion optimizer (MALO) [56] using thirtythree benchmark functions.Moreover, five different constrained engineering problems were used: welded beam design, tension/compression spring design, three-bar truss design, car crashworthiness design, and speed reducer design.The main contributions of this paper are:

•
The random running strategy is proposed to balance the exploration and exploitation phases.

•
The dynamic inertia weight strategy is applied to the waterhole to achieve the optimal global solution.

•
The competition for waterhole mechanism is introduced to stallion position calculations to boost exploitation behavior.
This paper is organized as follows: Section 2 describes WHO and other used operators, whereas Section 3 describes the improved algorithm.Sections 4 and 5 show the experimental study and discussion using benchmark functions and five real-world engineering problems, whereas Section 6 concludes the paper.

Wild Horse Optimizer
Generally, horses can be divided into two classes based on their social organization (territorial and non-territorial).They live in groups of different ages, such as offspring, stallions, and mares (see Figure 1).Both stallions and mares live together and interact with each other in grazing.Foals leave their groups after they grow up and join other groups to establish their own families.This behavior prevents mating between stallions and siblings.
The wild horse optimization (WHO) algorithm is a metaheuristic swarm-based algorithm inspired by the social behavior of horses, such as grazing, domination, leadership hierarchy, and mating.
The WHO algorithm consists of five different steps, described below:

Creating Initial Populations, Horse Groups, Determining Leaders
If N individuals and G groups exist, then the number of non-leaders (mares and foals) is N-G, and the number of leaders is G.The proportion of stallions is defined as PS, which is G/N. Figure 1 shows how leaders are determined from the basic generation to create various groups.

Grazing Behavior
As stated previously, most of a foal's life is spent grazing near its group.In order to simulate the grazing phase, we assume that the stallion position existed in the grazing area center.The following formula is used to enable other individuals to move.

( ) (
) where X i G,j and StallionG,j are the positions of the ith group member and stallion in the jth group, respectively, R is a random number between −2 and 2, and Z is an adaptive parameter computed by Equation ( 2): where P is a vector containing 0 and 1, and its dimension equals the dimension of the problem, 1 R and 3 R are random vectors between 0 and 1, and R2 is a random number between 0 and 1. TDR is a linearly decreasing parameter computed by Equation (3).
where t and T are the current and maximum iterations, respectively.

Horse Mating Behavior
As stated previously, one of the unique behaviors of horses compared to other animals is separating foals from their original groups prior to their reaching puberty and mating.To be able to simulate the behavior of mating between horses, the following formula is used: Crossover Mean where X p G,k is the position of horse p in group k, which is formed by positions of horse q in group i and horse z in group j.In the basic WHO, the probability of crossover is set to a constant named PC.

Group Leadership
Group leaders (stallions) will lead other group members to a suitable area (waterhole).Group leaders (stallions) will also compete for the waterhole, leading the dominant group to employ the waterhole first.The following formula is used to simulate this behavior: where , StallionGj and StallionG,j are the candidate position and the current leader position in the jth group, respectively, and WH is the position of the waterhole.

Exchange and Selection of Leaders
At first, leaders are selected randomly.After that, leaders are selected based on their fitness values.To simulate the exchange between leader positions and other individuals, the following formula is used: where f(X i G,j ) and f(StallionG,j) are the fitness values of foal and stallion, respectively.

Improved Wild Horse Optimizer
In this section, the IWHO is introduced in detail.To enhance the optimizing capability of the basic algorithm, three methods are added.Firstly, wild horses in nature tend to chase and run.Thus, a random running strategy is applied to the foals and stallions.Then, a dynamic inertia weight strategy is introduced to the waterhole, which is from DSCA [50] and beneficial to balance exploration and exploitation.At last, the competition for waterhole mechanism is proposed for stallions, which is inspired by artificial gorilla troops optimizer (GTO) [57] and further improves the solution quality.

Random Running Strategy (RRS)
In nature, wild horses are very fond of running or chasing each other.Inspired by this, the random running strategy is proposed for both foals and stallions.The positionupdating formula is presented as follow: lb ub lb rand (7) where lb and ub are the lower boundary and upper boundary, respectively.When conducting the RRS, search agents may appear anywhere in the search space.Thus, this method can help search agents jump out of the local optima.Note that to balance exploration and exploitation, the probability of random running (PRR) is set to a small value of 0.1.

Dynamic Inertia Weight Strategy (DIWS)
The dynamic weight strategy has been applied in much research [55,58,59] and is helpful to find the optimal global solution when introduced into algorithms.Thus, to help the stallions find a better waterhole, a dynamic inertia weight is added to the waterhole in the first formula of Equation (5).The weight and modified formula are calculated as follows: where wmin and wmax are the upper and lower boundary values, respectively, f(t)i is the fitness value of the current stallion at tth iteration, f(t)avg is the average fitness value of all stallions, and f(t)min is the minimum fitness value of the population.

Competition for Waterhole Mechanism (CWHM)
Moreover, the waterhole can be found by two stallions simultaneously.Then there may be rivalry between stallions over the waterhole.Thus, to further improve the solution quality, the competition for the waterhole mechanism, which is similar to the competition for adult females in GTO [53], is proposed to replace the second formula of Equation (5).The position updating formula is as follows: Z QQ (10) where Q1 and Q2 are random numbers between −1 and 1.

Improved Wild Horse Optimizer
By combining the DIWS and CWHM, the following formula is used to replace Equation (5) in WHO: In IWHO, foals and stallions have a more flexible method to update their positions.The RRS helps search agents achieve a better balance between exploration and exploitation.At the same time, DIWS and CWHM enable the method to obtain highquality solutions and improve the convergence speed.The pseudocode of IWHO is shown in Algorithm 1, and the flowchart of the proposed IWHO is shown in Figure 2.

2.
Set population size (N) and the maximum number of iterations (T).

3.
Initialize the population of horses at random.4.
Create foal groups and select stallions.
For the number of stallions 9.
For the number of foals 10.
Update the position of the foal using Equation (1).12.
Update the position of the foal using Equation (4).14.Else 15.
Update the position of the foal using Equation (7).16.End 17.
Generate the candidate position of stallion using Equation (10).21.

Else 22.
Generate the candidate position of stallion using Equation (9).23.
Generate the candidate position of stallion using Equation (7).26.

End if 27.
If the candidate position of the stallion is better 28.
Replace the position of the stallion using the candidate position.29.

End if 30.
End for 31.
End While

34.
Output the best solution obtained by IWHO.End IWHO.

Start
Input IWHO parameters: PC = 0.13, PS = 0. Update the position of foal by Equation ( 1) Is termination criteria met?
Output the best solution
Update the position of foal by Equation ( 7)

Analysis of Algorithm Computational Complexity
The computational complexity of IWHO is related to the population size (N), the maximum number of iterations (T), and problem dimensions (D).In the basic WHO, the computational complexity of initializing population is O(N × D).Then, the computational complexity of updating positions for foals and stallions is O(N × D × T).In addition, consider the worst-case scenario: all foals conduct the grazing behavior.The computational complexity of calculating Z is O(N × D × T).The computational complexity of the exchange and selection of leaders is O(Nfoal × T).Thus, the overall computational In IWHO, it should be noted that the proposed RRS and CWHM would not increase the computational complexity.
Similarly, considering the worst-case scenario, all stallions conduct the DIWS.Then, the computational complexity of calculating weight is O(Nstallion × T).Therefore, the overall computational complexity of IWHO is O(2 × N × D × T + N × D + N × T), which is only a little higher than the basic algorithm.

Experimental Study
To be able to show the significant performance and superiority of IWHO, a variety of experiments were carried out.IWHO was tested over 33 mathematical functions taken from two well-known benchmark function datasets: CEC2005 [60] and CEC2021 [61].IWHO performance was compared to classical WHO and eight different algorithms.The parameter settings of these experiments are discussed in the following subsection.

Optimization of Functions and Parameter Settings
Thirty-three mathematical functions of different types (unimodal, multimodal, fixeddimension multimodal functions) were implemented from two different benchmark datasets: CEC2005 and CEC2021.Table 1 shows each mathematical function used to carry out these experiments, including its type, range of the search space, and its global optimal value.IWHO has been compared to WHO and eight other algorithms: grey wolf optimizer (GWO) [50], moth-flame optimization (MFO) [51], salp swarm algorithm (SSA) [52], whale optimization algorithm (WOA) [53], particle swarm optimizer (PSO) [15], hybrid sine-cosine algorithm with harmony search (HSCAHS) [54], dynamic sine-cosine algorithm (DSCA) [55], and modified ant lion optimizer (MALO) [56].The parameter settings of each algorithm are shown in Table 2.These comparative algorithms have shown certain optimizing capability to the benchmark functions.However, for some optimization problems, they cannot get satisfactory optimization solutions.Hence, the advantages of the improved algorithm can be revealed through the extensive comparison between the improved algorithm and other algorithms.

Analysis of Performance for the CEC 2005 Test Suite
Improved WHO (IWHO) results with other comparative algorithms are listed in Table 3.This table shows that the suggested approach achieves the best results in almost all unimodal functions (F1-F7) in all mentioned dimensions (30,200, 500, and 1000).On the other hand, IWHO also ranked first in almost multimodal functions (F8-F13), and in almost functions regarding fixed-dimension multimodal functions (F14-F23).Compared to basic WHO and other algorithms, it can be seen that the improved algorithm has better exploration and exploitation capability as a result of the three improvements.
Moreover, a non-parametric test known as Wilcoxon signed test was used to give evidence of the performance of our suggested approach.Table 4 shows the Wilcoxon results of IWHO vs. other algorithms with 5% significance [62].From this table, we can see that IWHO is better than many other metaheuristics algorithms in most cases.

Statistical Analysis of Performance for the CEC 2021 Test Suite
In this subsection, the performance of IWHO is tested using ten mathematical functions from IEEE CEC 2021, and it is compared using the same comparative metaheuristics algorithms.
The results of CEC2021 test functions are given in Table 5 in terms of mean and standard deviation.Table 5 also shows the results of Friedman ranking test [63].It can be seen that the proposed IWHO is ranked first in CEC_02 and CEC_06 and the second-best in the other two functions (CEC_05 and CEC_08).Overall, it is also ranked first.
Figure 8 shows the box plots of the introduced algorithm and other compared ones using nine functions (CEC_01-CEC_09).It can be seen that the improved algorithm has better stability in these functions.Further, Figure 9 shows different exploration and exploitation phases.The percentages of exploration and exploitation phases are calculated based on the dispersion of all searching individuals, which exhibits the flexibility of improved algorithms during the optimization process.

Real-World Applications
In this section, several real-world engineering design problems with constraints are utilized to assess IWHO's significance and effectiveness.IWHO has been evaluated using five engineering design problems: welded beam, tension/compression spring, three-bar truss, car crashworthiness, and speed reducer design problems.

Welded Beam Design Problem
The first engineering problem used to evaluate IWHO is the welded beam design, a common and well-known problem [64], as shown in Figure 10.Welded beam aims to select the optimal variables to decrease the entire weld beam manufacturing cost subject to: The mathematical equations of this problem are shown below: Consider:   IWHO has been compared to the original WHO [49], GWO [50], MFO [51], WOA [53], SCA [65], ALO [66], DA [67], and LSHADE [68].From

Three-Bar Truss Design Problem
This engineering problem is a non-linear optimization problem introduced by Nowcki to minimize bar structure burdens [74].Three-bar truss has two variables, as shown in Figure 12.The mathematical description of this problem is given below: Consider: Subject to:

Car Crashworthiness Design Problem
The car crashworthiness problem was first introduced by Gu et al. [76].This engineering problem has 10 constraints and 11 design variables.The mathematical description and 3D model diagram of this problem can be seen in [77].

Speed Reducer Design Problem
The last constrained engineering problem is the speed reducer design problem [75], as shown in Figure 13, which aims to find the minimum weight of the speed reducer.This problem has seven variables.The mathematical formulation of this problem is shown below: Minimize: IWHO is compared to AO [70], PSO [15], AOA [75], MFO [51], GA [30], SCA [65], HS [73], and FA [78].As listed in

Conclusions
In this paper, an improved wild horse optimizer (IWHO) is proposed for solving global optimization problems.The basic WHO is enhanced by integrating three improvements: random running strategy (RRS), dynamic inertia weight strategy (DIWS), and competition for waterhole mechanism (CWHM), which were inspired by the behavior of wild horses.RRS is utilized to enhance the global search capability and balance exploration and exploitation, while DIWS and CWHM are applied to increase the quality of the optimal solution.To evaluate the performance of the proposed IWHO, twenty-three classical benchmark functions with various types and dimensions, ten CEC 2021 test functions, and five real-world optimization problems are employed for testing.Meanwhile, nine well-known algorithms are used for comparison: basic wild horse optimizer (WHO), grey wolf optimizer (GWO), moth-flame optimization (MFO), salp swarm algorithm (SSA), whale optimization algorithm (WOA), particle swarm optimizer (PSO), hybridizing sine-cosine algorithm with harmony search (HSCAHS), dynamic sine-cosine algorithm (DSCA), and modified ant lion optimizer (MALO).The numerical and statistical results indicate that IWHO outperforms other methods for most test functions, showing that the improvements applied to WHO are successful.In addition, the results of engineering design problems verify the applicability of the proposed IWHO.
In the future, IWHO may be applied to high-dimensional, real-world problems, such as multilayer perceptron training for solving classification tasks.Additionally, the IWHO can be hybridized with other metaheuristic algorithms to further improve its optimization capability and solve more optimization problems, such as feature selection, multilevel thresholding image segmentation, and parameter optimization.

Figure 1 .
Figure 1.Formation of groups from the original population.

Author Contributions:
Conceptualization, R.Z.; methodology, R.Z. and H.-M.J.; software, S.W.; validation, D.W.; writing-original draft preparation, A.G.H. and R.Z.; writing-review and editing, L.A. and H.-M.J.; funding acquisition, R.Z. and H.-M.J.All authors have read and agreed to the published version of the manuscript.Funding: This work was funded by Sanming University, which introduces high-level talent to start scientific research; funding support project (21YG01, 20YG14), Sanming University National Natural Science Foundation Breeding Project (PYT2103, PYT2105), Fujian Natural Science Foundation Project (2021J011128), Guiding Science and Technology Projects in Sanming City (2021-S-8), Educational Research Projects of Young and Middle-Aged Teachers in Fujian Province (JAT200618), Scientific Research and Development Fund of Sanming University (B202009), and by Open Research Fund Program of Fujian Provincial Key Laboratory of Agriculture Internet of Things Application (ZD2101).Institutional Review Board Statement: Not applicable.Informed Consent Statement: Not applicable.

Table 1 .
Feature properties of the test functions (D indicates the dimension).

Table 3 .
Comparison of optimization results between IWHO and other optimization algorithms on classical test functions (F1−F23).

Table 4 .
The Wilcoxon signed-rank test results between IWHO and other algorithms on classical test functions (F1-F23).

Table 5 .
Comparison of optimization results on CEC2021 test functions between IWHO and other optimization algorithms (CEC_01−CEC_10).

Table 6 .
Optimization results for the welded beam design problem.

Table 7 .
Optimization results for the tension/compression spring design problem.

Table 9 .
Optimization results in car crashworthiness design problem.

Table 10 .
Optimization results for the speed reducer design problem.