An Improved Wild Horse Optimizer for Solving Optimization Problems

Abstract

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.

1. 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.

Swarm intelligence algorithms contain algorithms which simulate collective or social behavior of specific creatures. Creatures in the real world can interact with each other to achieve intelligent behavior. Examples of such algorithms include particle swarm optimization (PSO) [15], ant colony optimization (ACO) [16], cuckoo search (CS) [17], grasshopper optimization algorithm (GOA) [18], virus colony search (VCS) [19], dolphin echolocation (DE) algorithm [20], Harris Hawk optimization [21], whale optimization algorithm (WOA) [22], ant lion optimization (ALO) [23], crow search algorithm (CSA) [24], salp swarm algorithm (SSA) [25], moth–flame optimization (MFO) [26], snake optimizer (SO) [27], coot bird [28], and squirrel search algorithm (SSA) [29].

Evolution algorithms are population-based and stochastic algorithms which maintain biological evolution processes such as selection, mutation, elimination, and migration [12,13]. This class contains genetic algorithm (GA) [30], evolutionary programming (EP) [31], differential evolution (DE) [32], bacterial foraging optimization (BFO) [33], and memetic algorithm (MA) [34].

Physics and chemistry algorithms mimic chemical laws or physical phenomena in the universe. Examples of such algorithms are simulated annealing (SA) [35], gravitational search algorithm (GSA) [36], electromagnetism mechanism (EM) algorithm [37], ion motion algorithm (IMA) [38], lightning search algorithm (LSA) [39], and vortex search algorithm (VSA) [40].

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 teaching–learning-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 thirty-three 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. Section 4 and Section 5 show the experimental study and discussion using benchmark functions and five real-world engineering problems, whereas Section 6 concludes the paper.

2. 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:

2.1. 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.

2.2. 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.

$$X_{G,j}^i=2Z\cos \left(2\pi RZ\right)\times \left({\mathrm{Stallion}}_{G,j}-X_{G,j}^i\right)+{\mathrm{Stallion}}_{G,j}$$

(1)

where $X_{G,j}^i$ and Stallion_G,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):

$$P=\overrightarrow{R1}<TDR, IDX=(P==0), Z=R2\mathsf{\Theta }IDX+\overrightarrow{R3}\mathsf{\Theta }(~IDX)$$

(2)

where P is a vector containing 0 and 1, and its dimension equals the dimension of the problem, $\overrightarrow{R1}$ and $\overrightarrow{R3}$ 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).

$$TDR=1-\frac{t}{T}$$

(3)

where t and T are the current and maximum iterations, respectively.

2.3. 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:

$$\begin{array}{l}{X}_{G,k}^p=\mathrm{Crossover}\left(X_{G,i}^q,X_{G,j}^z\right),i\ne j\ne k,q=z=end\\ \mathrm{Crossover}=\mathrm{Mean}\end{array}$$

(4)

where $X_{G,k}^p$ 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.

2.4. 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:

$${\overline{\mathrm{Stallion}}}_{G,j}=\left\{\begin{array}{l}2Z\cos \left(2\pi RZ\right)\times \left(\mathrm{WH}-{\mathrm{Stallion}}_{G,j}\right)+\mathrm{WH} if rand>0.5\\ 2Z\cos \left(2\pi RZ\right)\times \left(\mathrm{WH}-{\mathrm{Stallion}}_{G,j}\right)-\mathrm{WH} if rand\le 0.5\end{array}\right.$$

(5)

where ${\overline{\mathrm{Stallion}}}_{G,j}$ and Stallion_G,j are the candidate position and the current leader position in the jth group, respectively, and WH is the position of the waterhole.

2.5. 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:

$${\mathrm{Stallion}}_{G,j}=\left\{\begin{array}{l}{X}_{G,j}^i, if f\left(X_{G,j}^i\right)<f\left({\mathrm{Stallion}}_{G,j}\right)\\ {\mathrm{Stallion}}_{G,i}, if f\left(X_{G,j}^i\right)\ge f\left({\mathrm{Stallion}}_{G,j}\right)\end{array}\right.$$

(6)

where f($X_{G,j}^i$) and f(Stallion_G,j) are the fitness values of foal and stallion, respectively.

3. 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.

3.1. 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 position-updating formula is presented as follow:

$$X_{G,j}^i \mathrm{or} {\overline{\mathrm{Stallion}}}_{G,j}=lb+(ub-lb)\times rand$$

(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.

3.2. 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:

$$w=\left\{\begin{array}{l}w \min +(w \max -w \min )\times \frac{f(t)i-f(t)\min }{f(t)avg-f(t)\min }, if f(t)i\le f(t)avg \\ w \max , if f(t)i>f(t)avg\end{array}\right.$$

(8)

$${\overline{\mathrm{Stallion}}}_{G,j}=2Z\cos \left(2\pi RZ\right)\times \left(\mathrm{WH}-{\mathrm{Stallion}}_{G,j}\right)+w\times \mathrm{WH}$$

(9)

where w_min and w_max 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.

3.3. 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:

$${\overline{\mathrm{Stallion}}}_{G,j}=\mathrm{WH}-Z\times \left({\mathrm{Stallion}}_{G,j}\times Q_1-{\mathrm{Stallion}}_{G,j}\times Q_2\right)$$

(10)

where Q₁ and Q₂ are random numbers between −1 and 1.

3.4. Improved Wild Horse Optimizer

By combining the DIWS and CWHM, the following formula is used to replace Equation (5) in WHO:

$${\overline{\mathrm{Stallion}}}_{G,j}=\left\{\begin{array}{l}2Z\cos \left(2\pi RZ\right)\times \left(\mathrm{WH}-{\mathrm{Stallion}}_{G,j}\right)+w\times \mathrm{WH}, if R_3>0.5\\ \mathrm{WH}-Z\times \left({\mathrm{Stallion}}_{Gi}\times Q_1-{\mathrm{Stallion}}_{Gj}\times Q_2\right), if R_3\le 0.5\end{array}\right.$$

(11)

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 high-quality 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.

Algorithm 1 Pseudocode of IWHO

Start IWHO. 1.  Input IWHO parameters: PC = 0.13, PS = 0.2, wmin = 0.01, wmax = 0.99, PRR = 0.1. 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. 5.  While (t ≤ T) 6.  Calculate TDR using Equation (3). 7.  Calculate Z using Equation (2). 8.  For the number of stallions 9.    For the number of foals 10.   If rand > PC 11.     Update the position of the foal using Equation (1). 12.   Else if rand > PRR 13.     Update the position of the foal using Equation (4). 14.   Else 15.     Update the position of the foal using Equation (7). 16.   End 17.   End for 18.   If rand > PRR 19.   If rand > 0.5 20.     Generate the candidate position of stallion using Equation (10). 21.   Else 22.     Generate the candidate position of stallion using Equation (9). 23.   End if 24.   Else 25.   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.   Exchange foals and stallions position using Equation (6). 32.   t = t + 1 33. End While 34. Output the best solution obtained by IWHO. End IWHO.

3.5. 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 complexity of WHO is O(2 × N × D × T + N × D + Nfoal × T). 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.

4. 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.

4.1. Optimization of Functions and Parameter Settings

Thirty-three mathematical functions of different types (unimodal, multimodal, fixed-dimension multimodal functions) were implemented from two different benchmark datasets: CEC2005 and CEC2021.

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

Function Type	Function	D	Range	Theoretical Optimization Value
Unimodal test functions	F1	30/100/500/1000	[−100, 100]	0
	F2	30/100/500/1000	[−10, 10]	0
	F3	30/100/500/1000	[−100, 100]	0
	F4	30/100/500/1000	[−100, 100]	0
	F5	30/100/500/1000	[−30, 30]	0
	F6	30/100/500/1000	[−100, 100]	0
	F7	30/100/500/1000	[−1.28, 1.28]	0
Multimodal test functions	F8	30/100/500/1000	[−500, 500]	−418.9829 × D
	F9	30/100/500/1000	[−5.12, 5.12]	0
	F10	30/100/500/1000	[−32, 32]	0
	F11	30/100/500/1000	[−600, 600]	0
	F12	30/100/500/1000	[−50, 50]	0
	F13	30/100/500/1000	[−50, 50]	0
Fixed-dimension multimodal test functions	F14	2	[−65, 65]	0.998004
	F15	4	[−5, 5]	0.0003075
	F16	2	[−5, 5]	−1.03163
	F17	2	[−5, 5]	0.398
	F18	2	[−2, 2]	3
	F19	3	[−1, 2]	−3.8628
	F20	6	[0, 1]	−3.3220
	F21	4	[0, 10]	−10.1532
	F22	4	[0, 10]	−10.4028
	F23	4	[0, 10]	−10.5363
CEC2021 unimodal test functions	CEC_01	20	[−100, 100]	100
CEC2021 basic test functions	CEC_02	20	[−100, 100]	1100
	CEC_03	20	[−100, 100]	700
	CEC_04	20	[−100, 100]	1900
CEC2021 hybrid test functions	CEC_05	20	[−100, 100]	1700
	CEC_06	20	[−100, 100]	1600
	CEC_07	20	[−100, 100]	2100
CEC2021 composition test functions	CEC_08	20	[−100, 100]	2200
	CEC_09	20	[−100, 100]	2400
	CEC_10	20	[−100, 100]	2500

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. Parameter values for the IWHO and other comparative optimization algorithms.

Algorithm	Parameters
IWHO	PS = 0.2; PC = 0.13; PRR = 0.1; w ∈ [0.01, 0.99]
WHO [49]	PS = 0.2; PC = 0.13
GWO [50]	a = [2, 0]
MFO [51]	b = 1; t = [−1,1]; a ∈ [−1,−2]
SSA [52]	c1 ∈ [0, 1]; c2 ∈ [0, 1]
WOA [53]	a1 = [2, 0]; a2 = [−2, −1]; b = 1
PSO [15]	c1 = 2; c2 = 2; W ∈ [0.2, 0.9]; vMax = 6
HSCAHS [54]	a = 2; Bandwidth = 0.02
DSCA [55]	w ∈ [0.1, 0.9], σ = 0.1; aend = 0; astart = 2
MALO [56]	Switch possibility = 0.5

. 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.

The experiments were carried out in Windows 10 with 16 GB RAM and an Intel (R) i5-9500 CPU. All simulations were executed using MATLAB R2016a. Moreover, the maximum number of iterations was set at 500, and the number of individuals was set to 30. In contrast, the number of dimensions was different according to each function, as shown in Table 1.

4.2. Analysis of Performance for the CEC 2005 Test Suite

Improved WHO (IWHO) results with other comparative algorithms are listed in

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

Function	D	Metric	IWHO	WHO	GWO	MFO	SSA	WOA	PSO	HSCAHS	DSCA	MALO
F1	30	Mean	0	3.05 × 10−43	2.09 × 10−27	1.69 × 103	1.97 × 10−7	8.09 × 10−74	2.09 × 10−4	2.79 × 10−50	3.88 × 10−106	1.28 × 10−3
		Std	0	1.57 × 10−42	3.99 × 10−27	3.83 × 103	3.38 × 10−7	2.79 × 10−73	3.23 × 10−4	1.41 × 10−49	2.11 × 10−105	1.17 × 10−3
	200	Mean	0	4.57 × 10−32	9.50 × 10−8	2.88 × 105	1.69 × 104	5.20 × 10−70	3.38 × 102	3.46 × 10−34	1.34 × 10−91	3.95 × 104
		Std	0	1.81 × 10−31	4.45 × 10−8	2.81 × 104	2.15 × 103	2.84 × 10−69	4.88 × 101	8.54 × 10−34	7.35 × 10−91	9.15 × 103
	500	Mean	0	2.96 × 10−28	1.63 × 10−3	1.16 × 106	9.46 × 104	3.14 × 10−71	5.85 × 103	4.56 × 10−29	1.90 × 10−78	2.17 × 105
		Std	0	1.62 × 10−27	5.53 × 10−4	3.51 × 104	6.80 × 103	8.49 × 10−71	4.89 × 102	8.10 × 10−29	1.04 × 10−77	3.75 × 104
	1000	Mean	0	6.50 × 10−29	2.53 × 10−1	2.73 × 106	2.37 × 105	6.48 × 10−70	4.10 × 104	2.05 × 10−25	4.69 × 10−73	5.80 × 105
		Std	0	2.29 × 10−28	6.74 × 10−2	5.33 × 104	1.27 × 104	3.21 × 10−69	2.39 × 103	4.20 × 10−25	2.57 × 10−72	8.66 × 104
F2	30	Mean	0	2.22 × 10−25	9.06 × 10−17	2.65 × 101	2.37	1.56 × 10−51	7.36	4.04 × 10−27	1.22 × 10−60	5.29 × 101
		Std	0	7.39 × 10−25	6.90 × 10−17	2.03 × 101	1.73	4.94 × 10−51	1.05 × 101	9.46 × 10−27	6.65 × 10−60	5.31 × 101
	200	Mean	0	6.92 × 101	3.14 × 10−5	7.55 × 102	1.55 × 102	1.18 × 10−49	4.76 × 102	1.19 × 10−18	1.05 × 10−51	3.76 × 1073
		Std	0	1.80 × 102	9.14 × 10−6	4.70 × 101	1.26 × 101	3.91 × 10−49	6.08 × 101	1.99 × 10−18	3.96 × 10−51	2.06 × 1074
	500	Mean	0	6.01 × 102	1.09 × 10−2	4.30 × 10116	5.38 × 102	1.00 × 10−48	5.74 × 1058	3.84 × 10−16	7.18 × 10−50	2.31 × 10244
		Std	0	6.18 × 102	2.39 × 10−3	2.23 × 10117	2.39 × 101	4.88 × 10−48	3.14 × 1059	6.21 × 10−16	3.93 × 10−49	Inf
	1000	Mean	0	1.43 × 103	8.06 × 10−1	Inf	1.20 × 103	1.30 × 10−49	1.40 × 103	5.11 × 10−15	Inf	Inf
		Std	0	1.37 × 103	9.03 × 10−1	NaN	3.61 × 101	3.29 × 10−49	7.08 × 101	1.09 × 10−14	NaN	NaN
F3	30	Mean	0	3.56 × 10−26	6.23 × 10−6	2.28 × 104	1.74 × 103	4.35 × 104	7.36 × 101	1.08 × 10−48	2.25 × 10−57	4.56 × 103
		Std	0	1.82 × 10−25	1.02 × 10−5	1.31 × 104	1.30 × 103	1.17 × 104	3.10 × 101	4.06 × 10−48	1.23 × 10−56	1.79 × 103
	200	Mean	0	1.76 × 10−12	2.14 × 104	8.63 × 105	2.32 × 105	5.05 × 106	8.90 × 104	1.73 × 10−31	2.86 × 10−45	3.42 × 105
		Std	0	4.91 × 10−12	1.11 × 104	1.56 × 105	1.31 × 105	1.36 × 106	1.88 × 104	5.48 × 10−31	1.54 × 10−44	9.60 × 104
	500	Mean	0	1.06 × 10−8	3.26 × 105	4.86 × 106	1.01 × 106	2.91 × 107	5.94 × 105	5.34 × 10−26	1.35 × 10−43	2.09 × 106
		Std	0	5.30 × 10−8	1.03 × 105	9.34 × 105	5.23 × 105	9.83 × 106	1.18 × 105	9.19 × 10−26	6.83 × 10−43	6.22 × 105
	1000	Mean	0	2.36 × 10−5	1.52 × 106	1.83 × 107	4.24 × 106	1.18 × 108	2.47 × 106	5.52 × 10−23	1.46 × 10−35	7.32 × 106
		Std	0	1.28 × 10−4	2.93 × 105	4.06 × 106	1.96 × 106	4.57 × 107	6.38 × 105	1.12 × 10−22	7.85 × 10−35	2.30 × 106
F4	30	Mean	1.98 × 10−320	1.97 × 10−17	7.10 × 10−7	6.92 × 101	1.05 × 101	5.04 × 101	1.04	9.05 × 10−26	5.55 × 10−48	1.68 × 101
		Std	0	4.78 × 10−17	6.47 × 10−7	9.08	2.71	2.89 × 101	2.44 × 10−1	2.18 × 10−25	3.03 × 10−47	3.47
	200	Mean	0	2.10 × 10−11	2.37 × 101	9.72 × 101	3.49 × 101	8.49 × 101	1.96 × 101	2.86 × 10−16	1.39 × 10−41	4.09 × 101
		Std	0	7.03 × 10−11	6.67	7.39 × 10−1	3.49	1.40 × 101	1.61	4.71 × 10−16	5.77 × 10−41	4.46
	500	Mean	0	7.02 × 10−10	6.47 × 101	9.89 × 101	4.00 × 101	7.72 × 101	2.81 × 101	1.16 × 10−11	3.39 × 10−33	4.95 × 101
		Std	0	1.24 × 10−9	4.89	4.11 × 10−1	3.33	2.80 × 101	1.37	1.59 × 10−11	1.73 × 10−32	4.89
	1000	Mean	0	2.25 × 10−9	7.89 × 101	9.95 × 101	4.55 × 101	8.43 × 101	3.30 × 101	2.50 × 10−7	1.97 × 10−28	5.30 × 101
		Std	0	4.00 × 10−9	3.18	1.95 × 10−1	3.35	1.85 × 101	1.90	4.41 × 10−7	1.08 × 10−27	4.17
F5	30	Mean	4.37	3.58 × 101	2.70 × 101	2.68 × 106	2.19 × 102	2.79 × 101	3.13 × 103	2.88 × 101	2.86 × 101	9.77 × 10−1
		Std	9.71	2.35 × 101	6.78 × 10−1	1.46 × 107	3.37 × 102	4.26 × 10−1	1.64 × 104	9.01 × 10−2	3.04 × 10−1	5.22
	200	Mean	4.34 × 101	2.88 × 102	1.98 × 102	1.05 × 109	4.11 × 106	1.98 × 102	6.61 × 105	1.99 × 102	1.99 × 102	5.60 × 101
		Std	7.82 × 101	3.26 × 102	4.76 × 10−1	1.13 × 108	1.14 × 106	1.89 × 10−1	1.52 × 105	4.40 × 10−2	6.75 × 10−2	4.75 × 101
	500	Mean	1.27 × 102	4.99 × 102	4.98 × 102	5.12 × 109	3.86 × 107	4.96 × 102	3.00 × 107	4.99 × 102	4.99 × 102	2.31 × 102
		Std	1.90 × 102	2.18	3.18 × 10−1	2.17 × 108	8.07 × 106	4.62 × 10−1	3.48 × 106	5.15 × 10−3	1.87 × 10−2	1.93 × 102
	1000	Mean	1.22 × 102	9.98 × 102	1.07 × 103	1.24 × 1010	1.18 × 108	9.94 × 102	2.82 × 108	9.99 × 102	9.99 × 102	4.36 × 102
		Std	2.47 × 102	9.67 × 10−1	3.20 × 101	3.12 × 108	1.13 × 107	7.27 × 10−1	2.87 × 107	3.67 × 10−2	1.69 × 10−2	3.94 × 102
F6	30	Mean	5.09 × 10−5	9.53 × 10−3	8.12 × 10−1	1.00 × 103	1.52 × 10−7	4.21 × 10−1	2.27 × 10−4	6.67	5.57	4.69 × 10−4
		Std	4.64 × 10−5	2.92 × 10−2	4.15 × 10−1	3.04 × 103	1.80 × 10−7	2.56 × 10−1	3.22 × 10−4	1.76 × 10−1	3.09 × 10−1	3.21 × 10−4
	200	Mean	2.19	3.58 × 101	2.91 × 101	2.88 × 105	1.69 × 104	1.07 × 101	3.24 × 102	4.92 × 101	4.81 × 101	1.56
		Std	3.26	3.22 × 101	1.58	2.26 × 104	1.84 × 103	2.78	4.57 × 101	2.13 × 10−1	3.38 × 10−1	6.98 × 10−1
	500	Mean	1.07 × 101	1.10 × 102	9.06 × 101	1.18 × 106	9.48 × 104	2.84 × 101	5.92 × 103	1.24 × 102	1.23 × 102	6.01
		Std	1.34 × 101	3.18 × 101	1.89	2.78 × 104	5.33 × 103	8.78	5.16 × 102	2.13 × 10−1	3.46 × 10−1	2.62
	1000	Mean	1.20 × 101	2.29 × 102	2.02 × 102	2.74 × 106	2.36 × 105	6.85 × 101	4.06 × 104	2.49 × 102	2.48 × 102	1.24 × 101
		Std	2.28 × 101	2.01 × 101	2.53	3.49 × 104	1.03 × 104	1.80 × 101	1.94 × 103	2.60 × 10−1	4.41 × 10−1	4.84
F7	30	Mean	2.23 × 10−4	1.23 × 10−3	1.85 × 10−3	2.29	1.77 × 10−1	4.07 × 10−3	3.16	1.30 × 10−4	1.86 × 10−3	9.01 × 10−5
		Std	2.05 × 10−4	8.00 × 10−4	1.19 × 10−3	2.96	7.74 × 10−2	5.54 × 10−3	4.20	1.92 × 10−4	1.88 × 10−3	7.28 × 10−5
	200	Mean	4.58 × 10−4	1.62 × 10−3	1.70 × 10−2	2.99 × 103	1.76 × 101	6.24 × 10−3	2.92 × 103	1.08 × 10−4	2.34 × 10−3	7.14 × 10−5
		Std	4.58 × 10−4	2.22 × 10−3	5.45 × 10−3	4.19 × 102	3.53	7.00 × 10−3	4.63 × 102	9.81 × 10−5	2.07 × 10−3	5.22 × 10−5
	500	Mean	5.06 × 10−4	1.75 × 10−3	5.38 × 10−2	3.86 × 104	2.72 × 102	3.39 × 10−3	4.39 × 104	1.13 × 10−4	3.77 × 10−3	1.03 × 10−4
		Std	6.25 × 10−4	1.56 × 10−3	1.48 × 10−2	1.74 × 103	4.13 × 101	3.30 × 10−3	5.70 × 103	8.65 × 10−5	3.35 × 10−3	1.10 × 10−4
	1000	Mean	4.28 × 10−4	3.08 × 10−3	1.38 × 10−1	1.98 × 105	1.78 × 103	5.79 × 10−3	2.44 × 105	1.14 × 10−4	3.20 × 10−3	9.34 × 10−5
		Std	4.57 × 10−4	2.65 × 10−3	2.96 × 10−2	7.27 × 103	1.74 × 102	5.81 × 10−3	6.04 × 103	1.43 × 10−4	3.32 × 10−3	8.66 × 10−5
F8	30	Mean	−1.26 × 104	−8.80 × 103	−5.98 × 103	−8.46 × 103	−7.46 × 103	−9.80 × 103	−4.77 × 103	−2.44 × 103	−4.46 × 103	−1.21 × 104
		Std	6.64 × 10−4	5.09 × 102	6.98 × 102	1.19 × 103	7.32 × 102	1.57 × 103	1.30 × 103	3.14 × 102	3.59 × 102	1.08 × 103
	200	Mean	−8.38 × 104	−3.48 × 104	−2.96 × 104	−3.70 × 104	−3.46 × 104	−6.87 × 104	−1.31 × 104	−6.34 × 103	−1.18 × 104	−7.52 × 104
		Std	5.24	3.54 × 103	4.82 × 103	2.68 × 103	3.14 × 103	1.29 × 104	4.80 × 103	9.93 × 102	8.64 × 102	9.10 × 103
	500	Mean	−2.09 × 105	−6.03 × 104	−5.72 × 104	−6.21 × 104	−5.99 × 104	−1.77 × 105	−2.40 × 104	−1.05 × 104	−1.91 × 104	−1.90 × 105
		Std	3.87 × 101	7.66 × 103	4.22 × 103	4.14 × 103	4.63 × 103	2.92 × 104	8.68 × 103	1.56 × 103	2.27 × 103	2.43 × 104
	1000	Mean	−4.19 × 105	−8.36 × 104	−8.80 × 104	−8.87 × 104	−8.66 × 104	−3.48 × 105	−3.44 × 104	−1.52 × 104	−2.83 × 104	−3.76 × 105
		Std	2.01 × 102	9.06 × 103	1.43 × 104	6.25 × 103	6.65 × 103	5.54 × 104	1.30 × 104	2.13 × 103	3.81 × 103	4.60 × 104
F9	30	Mean	0	4.55 × 10−10	3.40	1.66 × 102	5.25 × 101	0	1.01 × 102	0	0	8.33 × 101
		Std	0	1.74 × 10−9	4.19	3.80 × 101	1.57 × 101	0	3.03 × 101	0	0	2.80 × 101
	200	Mean	0	0	2.21 × 101	2.22 × 103	8.20 × 102	7.58 × 10−15	2.03 × 103	0	0	1.03 × 103
		Std	0	0	9.22	1.11 × 102	8.14 × 101	4.15 × 10−14	1.45 × 102	0	0	8.92 × 101
	500	Mean	0	0	7.71 × 101	6.98 × 103	3.15 × 103	0	6.31 × 103	0	0	3.74 × 103
		Std	0	0	2.84 × 101	1.37 × 102	1.38 × 102	0	1.92 × 102	0	0	1.88 × 102
	1000	Mean	0	0	2.05 × 102	1.55 × 104	7.63 × 103	0	1.42 × 104	0	0	8.78 × 103
		Std	0	0	4.77 × 101	2.12 × 102	1.44 × 102	0	3.30 × 102	0	0	2.93 × 102
F10	30	Mean	8.88 × 10−16	2.07 × 10−15	1.03 × 10−13	1.78 × 101	2.74	4.44 × 10−15	4.07 × 10−1	8.88 × 10−16	8.88 × 10−16	4.17
		Std	0	1.70 × 10−15	2.27 × 10−14	4.32	7.62 × 10−1	1.87 × 10−15	6.82 × 10−1	0	0	2.11
	200	Mean	8.88 × 10−16	2.66 × 10−15	2.40 × 10−5	2.00 × 101	1.31 × 101	4.44 × 10−15	6.57	8.88 × 10−16	8.88 × 10−16	1.49 × 101
		Std	0	5.26 × 10−15	4.81 × 10−6	7.54 × 10−2	6.63 × 10−1	2.47 × 10−15	3.42 × 10−1	0	0	8.88 × 10−1
	500	Mean	8.88 × 10−16	1.48 × 10−15	1.89 × 10−3	2.03 × 101	1.42 × 101	5.15 × 10−15	1.20 × 101	1.24 × 10−15	8.88 × 10−16	1.60 × 101
		Std	0	1.35 × 10−15	4.06 × 10−4	1.17 × 10−1	2.52 × 10−1	2.86 × 10−15	3.13 × 10−1	1.08 × 10−15	0	7.80 × 10−1
	1000	Mean	8.88 × 10−16	1.95 × 10−15	1.76 × 10−2	2.03 × 101	1.46 × 101	4.56 × 10−15	1.60 × 101	6.10 × 10−15	8.88 × 10−16	1.60 × 101
		Std	0	1.90 × 10−15	2.26 × 10−3	2.24 × 10−1	1.45 × 10−1	2.55 × 10−15	3.82 × 10−1	8.64 × 10−15	0	1.28
F11	30	Mean	0	0	2.56 × 10−3	3.10 × 101	2.07 × 10−2	8.30 × 10−3	9.87 × 10−3	0	0	5.55 × 10−2
		Std	0	0	5.97 × 10−3	4.92 × 101	1.48 × 10−2	3.16 × 10−2	1.05 × 10−2	0	0	2.88 × 10−2
	200	Mean	0	0	5.28 × 10−3	2.56 × 103	1.57 × 102	0	1.48	0	0	3.74 × 102
		Std	0	0	1.37 × 10−2	1.65 × 102	2.56 × 101	0	6.05 × 10−1	0	0	9.11 × 101
	500	Mean	0	0	2.22 × 10−2	1.03 × 104	8.55 × 102	3.70 × 10−18	8.09 × 101	0	0	1.96 × 103
		Std	0	0	3.95 × 10−2	3.03 × 102	5.06 × 101	2.03 × 10−17	9.46	0	0	3.25 × 102
	1000	Mean	0	0	3.75 × 10−2	2.45 × 104	2.13 × 103	3.70 × 10−18	2.77 × 102	0	0	5.35 × 103
		Std	0	0	5.93 × 10−2	4.32 × 102	1.03 × 102	2.03 × 10−17	2.08 × 101	0	0	9.55 × 102
F12	30	Mean	6.24 × 10−7	7.45 × 10−3	3.91 × 10−2	8.53 × 106	7.37	2.42 × 10−2	6.91 × 10−3	1.05	7.71 × 10−1	1.67 × 10−5
		Std	1.16 × 10−6	2.63 × 10−2	1.88 × 10−2	4.67 × 107	3.37	2.20 × 10−2	2.63 × 10−2	4.52 × 10−2	7.15 × 10−2	1.67 × 10−5
	200	Mean	9.49 × 10−4	3.80 × 10−1	5.38 × 10−1	2.17 × 109	4.93 × 103	7.15 × 10−2	3.94 × 101	1.19	1.14	4.62 × 10−4
		Std	1.56 × 10−3	1.70 × 10−1	4.58 × 10−2	2.91 × 108	7.29 × 103	3.02 × 10−2	1.98 × 101	2.79 × 10−2	3.13 × 10−2	3.36 × 10−4
	500	Mean	1.44 × 10−3	7.45 × 10−1	7.46 × 10−1	1.19 × 1010	1.46 × 106	1.06 × 10−1	2.38 × 105	1.19	1.17	9.62 × 10−4
		Std	2.08 × 10−3	2.14 × 10−1	4.10 × 10−2	6.36 × 108	9.21 × 105	5.63 × 10−2	9.88 × 104	9.22 × 10−3	8.84 × 10−3	6.70 × 10−4
	1000	Mean	1.00 × 10−3	9.73 × 10−1	1.20	3.06 × 1010	1.11 × 107	9.41 × 10−2	9.37 × 106	1.18	1.17	4.64 × 10−4
		Std	2.08 × 10−3	1.86 × 10−1	2.17 × 10−1	1.13 × 109	2.60 × 106	4.92 × 10−2	2.32 × 106	3.00 × 10−3	5.39 × 10−3	4.04 × 10−4
F13	30	Mean	4.35 × 10−5	1.14 × 10−1	5.87 × 10−1	1.37 × 107	1.31 × 101	4.99 × 10−1	4.46 × 10−3	2.85	2.81	6.16 × 10−4
		Std	1.38 × 10−4	2.74 × 10−1	2.68 × 10−1	7.49 × 107	1.48 × 101	2.17 × 10−1	8.92 × 10−3	3.32 × 10−2	4.64 × 10−2	2.06 × 10−3
	200	Mean	1.27 × 10−1	1.79 × 101	1.69 × 101	4.49 × 109	1.33 × 106	6.22	5.70 × 103	1.99 × 101	1.99 × 101	9.64 × 10−2
		Std	3.60 × 10−1	3.27	6.35 × 10−1	4.98 × 108	7.44 × 105	2.08	3.07 × 103	3.27 × 10−2	5.31 × 10−2	6.79 × 10−2
	500	Mean	2.08 × 10−1	5.62 × 101	5.05 × 101	2.23 × 1010	3.59 × 107	1.91 × 101	4.14 × 106	4.99 × 101	5.00 × 101	4.49 × 10−1
		Std	2.87 × 10−1	1.15 × 101	1.29	1.21 × 109	9.80 × 106	5.58	1.17 × 106	2.52 × 10−2	3.91 × 10−2	2.67 × 10−1
	1000	Mean	7.12 × 10−1	1.15 × 102	1.25 × 102	5.54 × 1010	1.54 × 108	3.96 × 101	8.03 × 107	9.99 × 101	1.00 × 102	1.05
		Std	1.02	1.00 × 101	1.40 × 101	1.68 × 109	2.46 × 107	1.18 × 101	1.17 × 107	2.82 × 10−2	4.65 × 10−2	6.62 × 10−1
F14	2	Mean	9.98 × 10−1	2.21	4.07	2.35	1.16	2.93	3.00	2.84	4.64 × 10−2	1.63
		Std	1.30 × 10−16	2.76	4.00	2.03	5.87 × 10−1	3.07	2.54	4.52 × 10−1	7.05 × 10−1	9.20 × 10−1
F15	4	Mean	4.48 × 10−4	1.39 × 10−3	2.48 × 10−3	2.78 × 10−3	3.48 × 10−3	7.32 × 10−4	4.94 × 10−3	2.40 × 10−3	1.31 × 10−3	2.15 × 10−3
		Std	2.56 × 10−4	3.61 × 10−3	6.07 × 10−3	5.14 × 10−3	6.74 × 10−3	5.21 × 10−4	7.86 × 10−3	1.24 × 10−3	3.22 × 10−4	4.97 × 10−3
F16	2	Mean	−1.03	−1.03	−1.03	−1.03	−1.03	−1.03	−1.03	−1.02	−1.03	−1.03
		Std	6.05 × 10−16	4.97 × 10−16	1.95 × 10−8	6.78 × 10−16	2.74 × 10−14	9.52 × 10−10	6.25 × 10−16	5.56 × 10−3	1.83 × 10−4	1.28 × 10−13
F17	2	Mean	3.98 × 10−1	3.98 × 10−1	3.98 × 10−1	3.98 × 10−1	3.98 × 10−1	3.98 × 10−1	3.98 × 10−1	6.76 × 10−1	4.00 × 10−1	3.98 × 10−1
		Std	0	0	1.68 × 10−6	0	9.46 × 10−15	2.08 × 10−5	0	2.05 × 10−1	2.95 × 10−3	2.20 × 10−14
F18	2	Mean	3.00	3.00	3.00	3.00	3.00	3.00	3.00	3.01	3.01	3.00
		Std	1.82 × 10−15	6.85 × 10−16	3.02 × 10−5	1.70 × 10−15	2.22 × 10−13	5.43 × 10−5	1.33 × 10−15	1.68 × 10−2	8.68 × 10−3	7.64 × 10−13
F19	3	Mean	−3.86	−3.84	−3.86	−3.86	−3.86	−3.86	−3.86	−3.43	−3.83	−3.86
		Std	2.52 × 10−15	1.41 × 10−1	3.59 × 10−3	2.71 × 10−15	1.38 × 10−12	7.94 × 10−3	2.65 × 10−15	3.26 × 10−1	2.73 × 10−2	6.35 × 10−13
F20	6	Mean	−3.26	−3.25	−3.27	−3.24	−3.23	−3.20	−3.22	−1.78	−3.01	−3.25
		Std	6.05 × 10−2	6.00 × 10−2	7.38 × 10−2	6.40 × 10−2	6.34 × 10−2	1.18 × 10−1	1.07 × 10−1	6.08 × 10−1	9.02 × 10−2	5.86 × 10−2
F21	4	Mean	−1.02 × 101	−7.89	−9.32	−6.14	−7.06	−8.28	−6.71	−6.54 × 10−1	−4.08	−6.86
		Std	5.49 × 10−15	3.12	2.21	3.45	3.46	2.73	3.19	2.50 × 10−1	9.62 × 10−1	2.59
F22	4	Mean	−1.04 × 101	−8.51	−1.02 × 101	−7.36	−9.11	−8.18	−8.54	−7.37 × 10−1	−4.48	−6.85
		Std	8.73 × 10−16	3.22	9.63 × 10−1	3.57	2.69	2.99	2.96	2.99 × 10−1	2.34 × 10−1	3.48
F23	4	Mean	−1.05 × 101	−7.33	−1.01 × 101	−7.95	−8.15	−7.31	−8.57	−1.01	−4.45	−6.78
		Std	2.06 × 10−15	3.75	1.75	3.53	3.26	3.39	3.11	3.03 × 10−1	1.08	3.64

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

Function	D	IWHO vs. WHO	IWHO vs. GWO	IWHO vs. MFO	IWHO vs. SSA	IWHO vs. WOA	IWHO vs. PSO	IWHO vs. HSCAHS	IWHO vs. DSCA	IWHO vs. MALO
F1	30	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5
	200	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5
	500	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5
	1000	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5
F2	30	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5
	200	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5
	500	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5
	1000	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5
F3	30	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5
	200	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5
	500	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5
	1000	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5
F4	30	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5
	200	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5
	500	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5
	1000	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5
F5	30	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	1.22 × 10−4	6.10 × 10−5	6.10 × 10−5	4.21 × 10−1
	200	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.37 × 10−2
	500	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.37 × 10−2
	1000	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	1.83 × 10−4	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	1.69 × 10−1
F6	30	1.83 × 10−4	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	3.05 × 10−4	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5
	200	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	1.22 × 10−4	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	5.54 × 10−2
	500	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	1.16 × 10−3
	1000	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	2.52 × 10−1
F7	30	1.22 × 10−4	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	1.22 × 10−4	6.10 × 10−5	6.10 × 10−5	4.27 × 10−4	6.10 × 10−5
	200	3.05 × 10−4	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	2.62 × 10−3	1.22 × 10−4	4.79 × 10−2
	500	8.54 × 10−4	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	8.54 × 10−4	6.10 × 10−5	6.71 × 10−3	1.22 × 10−4	1.53 × 10−3
	1000	2.01 × 10−3	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	2.01 × 10−3	6.10 × 10−5	1.51 × 10−2	1.22 × 10−4	1.25 × 10−2
F8	30	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	1.83 × 10−4
	200	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5
	500	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5
	1000	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5
F9	30	1.00	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	1.00	6.10 × 10−5	1.00	1.00	6.10 × 10−5
	200	1.00	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	1.00	6.10 × 10−5	1.00	1.00	6.10 × 10−5
	500	1.00	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	1.00	6.10 × 10−5	1.00	1.00	6.10 × 10−5
	1000	1.00	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	1.00	6.10 × 10−5	1.00	1.00	6.10 × 10−5
F10	30	1.25 × 10−1	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	9.77 × 10−4	6.10 × 10−5	1.00	1.00	6.10 × 10−5
	200	1.25 × 10−1	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	3.91 × 10−3	6.10 × 10−5	1.00	1.00	6.10 × 10−5
	500	6.25 × 10−2	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	7.81 × 10−3	6.10 × 10−5	1.00	1.00	6.10 × 10−5
	1000	6.25 × 10−2	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	1.95 × 10−3	6.10 × 10−5	4.88 × 10−4	1.00	6.10 × 10−5
F11	30	1.00	1.25 × 10−1	6.10 × 10−5	6.10 × 10−5	5.00 × 10−1	6.10 × 10−5	1.00	1.00	6.10 × 10−5
	200	1.00	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	1.00	6.10 × 10−5	1.00	1.00	6.10 × 10−5
	500	1.00	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	1.00	6.10 × 10−5	1.00	1.00	6.10 × 10−5
	1000	1.00	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	1.00	6.10 × 10−5	1.00	1.00	6.10 × 10−5
F12	30	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	1.81 × 10−2	6.10 × 10−5	6.10 × 10−5	1.83 × 10−4
	200	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	7.62 × 10−1
	500	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	8.04 × 10−1
	1000	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	3.03 × 10−1
F13	30	3.36 × 10−3	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	5.37 × 10−3	6.10 × 10−5	6.10 × 10−5	8.33 × 10−2
	200	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	2.52 × 10−1
	500	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	4.27 × 10−3
	1000	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5	8.90 × 10−1
F14	2	7.81 × 10−3	6.10 × 10−5	1.95 × 10−3	1.95 × 10−3	6.10 × 10−5	9.77 × 10−4	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5
F15	4	8.36 × 10−3	5.54 × 10−2	3.66 × 10−4	1.51 × 10−2	2.29 × 10−1	1.22 × 10−4	6.10 × 10−5	6.10 × 10−5	4.27 × 10−3
F16	2	1.00	6.10 × 10−5	1.00	1.22 × 10−4	6.10 × 10−5	1.00	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5
F17	2	1.00	6.10 × 10−5	1.00	4.88 × 10−4	6.10 × 10−5	1.00	6.10 × 10−5	6.10 × 10−5	1.22 × 10−4
F18	2	1.95 × 10−3	6.10 × 10−5	7.54 × 10−1	6.10 × 10−5	6.10 × 10−5	3.07 × 10−1	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5
F19	3	1.00	6.10 × 10−5	1.00	6.10 × 10−5	6.10 × 10−5	1.00	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5
F20	6	7.66 × 10−1	2.56 × 10−2	1.56 × 10−2	8.54 × 10−4	2.52 × 10−1	6.27 × 10−1	6.10 × 10−5	6.10 × 10−5	2.01 × 10−3
F21	4	3.13 × 10−2	6.10 × 10−5	3.13 × 10−2	6.10 × 10−5	6.10 × 10−5	7.81 × 10−3	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5
F22	4	6.25 × 10−2	6.10 × 10−5	6.25 × 10−2	6.10 × 10−5	6.10 × 10−5	3.91 × 10−3	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5
F23	4	3.13 × 10−2	6.10 × 10−5	1.95 × 10−3	6.10 × 10−5	6.10 × 10−5	1.56 × 10−2	6.10 × 10−5	6.10 × 10−5	6.10 × 10−5
Overall (+/=/−)		45/17/0	59/3/0	57/5/0	62/0/0	52/10/0	57/5/0	47/11/4	50/12/0	45/12/5

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.

Furthermore, Figure 3 shows the convergence curves of some functions (F1–F3, F5, F6, F8, F10, F12, F13) with 30 dimensions. It is clear from this figure that IWHO has powerful convergence. Further, Figure 4, Figure 5 and Figure 6 show the convergence curves of the same functions (F1–F3, F5, F6, F8, F10, F12, F13) with dimensions of 200, 500, and 1000, respectively. Figure 7 shows the convergence curves for some fixed-dimension multimodal functions (F14, F15, F17–F23). It is obvious that the suggested algorithm has good convergence even in these multimodal functions.

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

Function	Metric	IWHO	WHO	GWO	MFO	SSA	WOA	PSO	HSCAHS	DSCA	MALO
CEC_01	Mean	2.64 × 103	4.54 × 103	8.07 × 108	2.13 × 109	2.41 × 103	1.44 × 109	1.58 × 103	3.25 × 1010	1.40 × 1010	1.70 × 103
	Std	3.02 × 103	4.17 × 103	6.80 × 108	1.62 × 109	2.82 × 103	1.03 × 109	1.94 × 103	2.42 × 109	2.58 × 109	1.99 × 103
Ranking		4	5	6	8	3	7	1	10	9	2
CEC_02	Mean	2.60 × 103	2.76 × 103	2.74 × 103	3.28 × 103	3.52 × 103	4.34 × 103	3.38 × 103	6.29 × 103	5.76 × 103	3.36 × 103
	Std	4.49 × 102	4.21 × 102	7.65 × 102	7.86 × 102	5.76 × 102	6.19 × 102	3.65 × 102	2.98 × 102	2.33 × 102	6.28 × 102
Ranking		1	3	2	4	7	8	6	10	9	5
CEC_03	Mean	7.93 × 102	7.92 × 102	7.94 × 102	8.37 × 102	8.21 × 102	9.68 × 102	7.85 × 102	1.06 × 103	9.86 × 102	8.61 × 102
	Std	2.45 × 101	2.89 × 101	2.66 × 101	7.11 × 101	4.42 × 101	4.63 × 101	1.86 × 101	1.78 × 101	1.82 × 101	5.06 × 101
Ranking		3	2	4	6	5	8	1	10	9	7
CEC_04	Mean	1.91 × 103	1.91 × 103	2.02 × 103	1.05 × 104	1.91 × 103	2.80 × 103	1.90 × 103	4.20 × 105	8.47 × 104	1.91 × 103
	Std	3.44	4.80	2.92 × 102	1.22 × 104	2.24	1.81 × 103	1.60	2.13 × 105	4.36 × 104	3.19
Ranking		3.5	3.5	6	8	3.5	7	1	10	9	3.5
CEC_05	Mean	2.26 × 105	2.76 × 105	8.21 × 105	2.87 × 106	6.26 × 105	2.43 × 106	1.92 × 105	6.72 × 106	4.43 × 106	2.92 × 105
	Std	1.52 × 105	2.00 × 105	6.45 × 105	4.73 × 106	4.90 × 105	1.20 × 106	1.35 × 105	2.02 × 106	2.22 × 106	2.27 × 105
Ranking		2	3	6	8	5	7	1	10	9	4
CEC_06	Mean	1.86 × 103	1.99 × 103	2.01 × 103	2.04 × 103	2.26 × 103	2.65 × 103	2.19 × 103	3.82 × 103	2.75 × 103	2.41 × 103
	Std	1.53 × 102	1.93 × 102	1.69 × 102	2.10 × 102	2.30 × 102	3.11 × 102	2.35 × 102	2.69 × 102	2.10 × 102	2.74 × 102
Ranking		1	2	3	4	6	8	5	10	9	7
CEC_07	Mean	1.43 × 105	9.90 × 104	3.30 × 105	1.07 × 106	2.39 × 105	1.34 × 106	1.07 × 105	2.78 × 106	2.23 × 106	1.30 × 105
	Std	1.15 × 105	1.10 × 105	4.35 × 105	1.59 × 106	2.27 × 105	1.00 × 106	9.74 × 104	1.54 × 106	1.16 × 106	1.18 × 105
Ranking		4	1	6	7	5	8	2	10	9	3
CEC_08	Mean	2.36 × 103	2.80 × 103	3.02 × 103	4.07 × 103	2.93 × 103	4.21 × 103	3.64 × 103	5.91 × 103	6.04 × 103	2.30 × 103
	Std	3.38 × 102	1.03 × 103	9.10 × 102	1.32 × 103	1.31 × 103	1.92 × 103	1.49 × 103	4.45 × 102	8.07 × 102	1.15 × 101
Ranking		2	3	5	7	4	8	6	9	10	1
CEC_09	Mean	2.88 × 103	2.87 × 103	2.87 × 103	2.89 × 103	2.86 × 103	3.04 × 103	3.06 × 103	3.47 × 103	3.02 × 103	2.90 × 103
	Std	3.78 × 101	3.22 × 101	3.95 × 101	3.40 × 101	2.14 × 101	7.19 × 101	8.00 × 101	9.40 × 101	1.99 × 101	2.36 × 101
Ranking		4	2.5	2.5	5	1	8	9	10	7	6
CEC_10	Mean	2.96 × 103	2.97 × 103	3.00 × 103	3.04 × 103	2.95 × 103	3.13 × 103	2.95 × 103	5.29 × 103	3.84 × 103	2.97 × 103
	Std	3.77 × 101	3.22 × 101	3.95 × 101	1.56 × 102	3.29 × 101	7.21 × 101	2.97 × 101	2.94 × 102	2.42 × 102	2.64 × 101
Ranking		3	4.5	6	7	1.5	8	1.5	10	9	4.5
Overall Ranking		1	2	6	7	4	8	3	10	9	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.

5. 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.

5.1. 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:

• Welded thickness (h)
• Bar length (l)
• Bar height (t)
• Bar thickness (b)

The mathematical equations of this problem are shown below:

Consider:

$$\overrightarrow{x}=[x_1, x_2, x_3, x_4]=[h, l, t, b]$$

Minimize:

$$f(\overrightarrow{x})=1.10471x_1^2{x}_2+0.04811x_3{x}_4(14.0+x_2)$$

Subject to:

$$\begin{array}{l}{g}_1(\overrightarrow{x})=\tau (\overrightarrow{x})-{\tau }_{\max }\le 0\\ g_2(\overrightarrow{x})=\sigma (\overrightarrow{x})-{\sigma }_{\max }\le 0\\ g_3(\overrightarrow{x})=\delta (\overrightarrow{x})-{\delta }_{\max }\le 0\\ g_4(\overrightarrow{x})=x_1-x_4\le 0\\ g_5(\overrightarrow{x})=P-P_C(\overrightarrow{x})\le 0\\ g_6(\overrightarrow{x})=0.125-x_1\le 0\\ g_7(\overrightarrow{x})=1.10471x_1^2+0.04811x_3{x}_4(14.0+x_2)-5\le 0\end{array}$$

where:

$$\begin{array}{l}\tau (\overrightarrow{x})=\sqrt{{(\tau \prime )}^2+2\tau \prime \tau ″\frac{x_2}{2R}+{(\tau ″)}^2}\\ \tau \prime =\frac{P}{\sqrt{2}{x}_1{x}_2}, \tau ″=\frac{MR}{J}, M=P(L+\frac{x_2}{2})\\ R=\sqrt{\frac{x_2^2}{4}+{(\frac{x_1+x_3}{2})}^2}\\ J=2\left\{\sqrt{2}{x}_1{x}_2\left[\frac{x_2^2}{4}+{(\frac{x_1+x_3}{2})}^2\right]\right\}\\ \sigma (\overrightarrow{x})=\frac{6PL}{x_3^2{x}_4}, \delta (\overrightarrow{x})=\frac{6P{L}^3}{E{x}_3^3{x}_4}\\ P_C(\overrightarrow{x})=\frac{4.013E\sqrt{\frac{x_3^2{x}_4^6}{36}}}{L^2}\left(1-\frac{x_3}{2L}\sqrt{\frac{E}{4G}}\right)\end{array}$$

Range of variables:

$$\begin{array}{l}P=6000 \mathrm{lb}, L=14 \mathrm{in}, E=30\times {10}^6 \mathrm{psi}, G=12\times {10}^6 \mathrm{psi}\\ {\tau }_{\max }=13600 \mathrm{psi}, {\sigma }_{\max }=30000 \mathrm{psi}, {\delta }_{\max }=0.25 \mathrm{in}\end{array}$$

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

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

Algorithm	Optimal Values for Variables				Minimum Cost
	h	l	t	b
IWHO	0.2057	3.2530	9.0366	0.2057	1.6952
WHO [49]	0.2058	3.2514	9.0370	0.2058	1.6958
GWO [50]	0.202369	3.544214	9.048210	0.205723	1.728712
MFO [51]	0.302546	2.662619	7.262456	0.318524	1.726186
WOA [53]	0.328517	2.465267	7.046271	0.426580	1.878023
SCA [65]	0.141883	5.225674	10.00000	0.211640	1.858033
ALO [66]	0.125130	9.784118	6.522153	0.394937	1.785364
DA [67]	0.206905	3.314990	9.397212	0.206905	1.749248
LSHADE [68]	0.20573	3.4705	9.0366	0.20573	1.724852

, it’s obvious that IWHO achieves the best results compared to other algorithms. It achieves [h, l, t, b] = [0.2057, 3.2530, 9.0366, 0.2057] with fitness value 1.6952.

5.2. Tension/Compression Spring Design Problem

The second constrained problem is the tension/compression spring problem [69], the goal of which is to select the optimal parameters to minimize the price of manufacturing a spring, as shown in Figure 11. This problem has three variables:

• Diameter of wire (d)
• Coil diameter (D)
• Active coils number (N)

The mathematical description of this problem is given below:

Consider:

$$\overrightarrow{x}=[x_1, x_2, x_3, x_4]=[d, D, N]$$

Minimize:

$$f(\overrightarrow{x})=(x_3+2)x_2{x}_1^2$$

Subject to:

$$\begin{array}{l}{g}_1(\overrightarrow{x})=1-\frac{x_2^3{x}_3}{71785x_1^4}\le 0\\ g_2(\overrightarrow{x})=\frac{4x_2^2-x_1{x}_2}{12566(x_2{x}_1^3-x_1^4)}+\frac{1}{5108x_1^2}\le 0\\ g_3(\overrightarrow{x})=1-\frac{140.45x_1}{x_2^2{x}_3}\le 0\\ g_4(\overrightarrow{x})=\frac{x_1+x_2}{1.5}-1\le 0\end{array}$$

Range of variables:

$$\begin{array}{l}0.05\le x_1\le 2.00\\ 0.25\le x_2\le 1.30\\ 2.00\le x_3\le 15.00\end{array}$$

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

Algorithm	Optimal Values for Variables			Minimum Weight
	d	D	N
IWHO	0.0517	0.4155	7.1564	0.0102
AO [70]	0.0502439	0.35262	10.5425	0.011165
HHO [71]	0.051796393	0.359305355	11.138859	0.012665443
SSA [52]	0.051207	0.345215	12.004032	0.0126763
WOA [53]	0.051207	0.345215	12.004032	0.0126763
GWO [50]	0.05169	0.356737	11.28885	0.012666
PSO [15]	0.051728	0.357644	11.244543	0.0126747
MVO [72]	0.05251	0.37602	10.33513	0.012790
GA [30]	0.051480	0.351661	11.632201	0.01270478
HS [73]	0.051154	0.349871	12.076432	0.0126706

shows the results of the proposed algorithm compared to AO [70], HHO [71], SSA [52], WOA [53], GWO [50], PSO [15], MVO [72], GA [29], and HS [73]. As shown in Table 8, IWHO clearly ranked first, with [d, D, N] = [0.0517, 0.4155, 7.1564] and a fitness value of 0.0102.

5.3. 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:

$$\overrightarrow{x}=[x_1, x_2]=[A_1, A_2]$$

Minimize:

$$f(\overrightarrow{x})=(2\sqrt{2}{x}_1+x_2)l$$

Subject to:

$$\begin{array}{l}{g}_1(\overrightarrow{x})=\frac{\sqrt{2}{x}_1+x_2}{\sqrt{2}{x}_1^2+2x_1{x}_2}P-\sigma \le 0\\ g_2(\overrightarrow{x})=\frac{x_2}{\sqrt{2}{x}_1^2+2x_1{x}_2}P-\sigma \le 0\\ g_3(\overrightarrow{x})=\frac{1}{\sqrt{2}{x}_2+x_1}P-\sigma \le 0\end{array}$$

where:

$$l=100 \mathrm{cm}, P=2 \mathrm{KN}/{\mathrm{cm}}^2, \sigma =2 \mathrm{KN}/{\mathrm{cm}}^2$$

Range of variables:

$$0\le x_1, x_2\le 1$$

The results of this problem are given in Table 8, comparing IWHO with basic WHO [49], HHO [71], SSA [52], AOA [75], MVO [72], MFO [51], and GOA [18]. IWHO achieves the best solution [x₁, x₂] = [0.7884, 0.4081] and its objective value is 263.8523.

Table 8. Optimization results for the three-bar truss design problem.

Algorithm	Optimal Values for Variables		Optimum Weight
	x1	x2
IWHO	0.7884	0.4081	263.8523
WHO [49]	0.7980	0.3816	263.9181
HHO [71]	0.788662816	0.408283133832900	263.8958434
SSA [52]	0.78866541	0.408275784	263.89584
AOA [75]	0.79369	0.39426	263.9154
MVO [72]	0.78860276	0.408453070000000	263.8958499
MFO [51]	0.788244771	0.409466905784741	263.8959797
GOA [18]	0.788897555578973	0.407619570115153	263.895881496069

Table 8. Optimization results for the three-bar truss design problem.

Algorithm	Optimal Values for Variables		Optimum Weight
	x1	x2
IWHO	0.7884	0.4081	263.8523
WHO [49]	0.7980	0.3816	263.9181
HHO [71]	0.788662816	0.408283133832900	263.8958434
SSA [52]	0.78866541	0.408275784	263.89584
AOA [75]	0.79369	0.39426	263.9154
MVO [72]	0.78860276	0.408453070000000	263.8958499
MFO [51]	0.788244771	0.409466905784741	263.8959797
GOA [18]	0.788897555578973	0.407619570115153	263.895881496069

5.4. 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].

The results of this problem are given in

Table 9. Optimization results in car crashworthiness design problem.

Algorithm	IWHO	GWO [50]	GOA [18]	PSO [15]	MFO [51]	SSA [52]	DSCA [55]	MALO [56]
x1	0.5000	0.5000	0.50000	0.5000	0.5000	0.5000	0.5000	0.5000
x2	1.1186	1.0268	1.11670	1.1327	1.0946	1.1221	0.9765	1.2023
x3	0.5000	0.5929	0.50000	0.5000	0.5000	0.5062	1.3381	0.5000
x4	1.2982	1.3772	1.30208	1.2781	1.3461	1.3170	0.5000	1.2833
x5	0.5000	0.5000	0.50000	0.5000	0.5000	0.5000	0.5214	0.5000
x6	1.5000	1.5000	1.50000	1.5000	1.5000	1.5000	1.5000	1.1728
x7	0.5000	0.5000	0.50000	0.5000	0.5000	0.5000	0.6263	0.5000
x8	0.3450	0.3450	0.34500	0.3450	0.3450	0.3136	0.1920	0.3347
x9	0.3450	0.3450	0.19200	0.3450	0.3450	0.2115	0.3450	0.3096
x10	−19.1594	−30.0000	−19.54935	−16.6449	−23.4599	−21.9139	−30.0000	−5.3953
x11	0.0020	0.1523	−0.00431	0.1076	0.0014	2.2405	7.8909	7.3999
Optimal value	22.8427	23.1976	22.84474	22.8556	22.8744	22.9855	24.9261	23.3406

, where IWHO is compared to GWO [50], GOA [18], PSO [15], MFO [51], SSA [52], DSCA [55], MALO [56]. From this table, we can conclude that IWHO achieves the best results [x₁ − x₁₁] = [0.5000, 1.1186, 0.5000, 1.2982, 0.5000, 1.5000, 0.5000, 0.3450, 0.3450, −19.1594, 0.0020] with fitness value 22.8427.

5.5. 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:

$$f(\overrightarrow{x})=0.7854x_1{x}_2^2(3.3333x_3^2+14.9334x_3-43.0934)-1.508x_1(x_6^2+x_7^2)+7.4777(x_6^3+x_7^3)$$

Subject to:

$$\begin{array}{l}{g}_1\left(\overrightarrow{x}\right)=\frac{27}{x_1{x}_2^2{x}_3}-1\le 0\\ g_2\left(\overrightarrow{x}\right)=\frac{397.5}{x_1{x}_2^2{x}_3^2}-1\le 0\\ g_3\left(\overrightarrow{x}\right)=\frac{1.93x_4^3}{x_2{x}_3{x}_6^4}-1\le 0\\ g_4\left(\overrightarrow{x}\right)=\frac{1.93x_5^3}{x_2{x}_3{x}_7^4}-1\le 0\\ g_5\left(\overrightarrow{x}\right)=\frac{\sqrt{{\left(\frac{745x_4}{x_2{x}_3}\right)}^2+16.9\times {10}^6}}{110.0x_6^3}-1\le 0\\ g_6\left(\overrightarrow{x}\right)=\frac{\sqrt{{\left(\frac{745x_4}{x_2{x}_3}\right)}^2+157.5\times {10}^6}}{85.0x_6^3}-1\le 0\\ g_7\left(\overrightarrow{x}\right)=\frac{x_2{x}_3}{40}-1\le 0\\ g_8\left(\overrightarrow{x}\right)=\frac{5x_2}{x_1}-1\le 0\\ g_9\left(\overrightarrow{x}\right)=\frac{x_1}{12x_2}-1\le 0\\ g_{10}\left(\overrightarrow{x}\right)=\frac{1.5x_6+1.9}{x_4}-1\le 0\\ g_{11}\left(\overrightarrow{x}\right)=\frac{1.1x_7+1.9}{x_5}-1\le 0\end{array}$$

Range of variables:

$$\begin{array}{l}2.6\le x_1\le 3.6\\ 0.7\le x_2\le 0.8\\ 17\le x_3\le 28\\ 7.3\le x_4\le 8.3\\ 7.8\le x_5\le 8.3\\ 2.9\le x_6\le 3.9\\ 5.0\le x_7\le 5.5\end{array}$$

IWHO is compared to AO [70], PSO [15], AOA [75], MFO [51], GA [30], SCA [65], HS [73], and FA [78]. As listed in

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

Algorithm	Optimal Values for Variables							Optimum Weight
	x1	x2	x3	x4	x5	x6	x7
IWHO	3.4976	0.7	17	7.3	7.8	3.3501	5.2855	2995.43736
AO [70]	3.5021	0.7	17	7.3099	7.7476	3.3641	5.2994	3007.7328
PSO [15]	3.5001	0.7	17.0002	7.5177	7.7832	3.3508	5.2867	3145.922
AOA [75]	3.50384	0.7	17	7.3	7.72933	3.35649	5.2867	2997.9157
MFO [51]	3.49745	0.7	17	7.82775	7.71245	3.35178	5.28635	2998.9408
GA [29]	3.51025	0.7	17	8.35	7.8	3.36220	5.28772	3067.561
SCA [65]	3.50875	0.7	17	7.3	7.8	3.46102	5.28921	3030.563
HS [73]	3.52012	0.7	17	8.37	7.8	3.36697	5.28871	3029.002
FA [78]	3.50749	0.7001	17	7.71967	8.08085	3.35151	5.28705	3010.13749

, the proposed algorithm clearly achieves the best results: [x₁ − x₇] = [3.4976, 0.7, 17, 7.3, 7.8, 3.3501, 5.2855] with fitness value 2995.43736.

6. 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.