Rules embedded harris hawks optimizer for large-scale optimization problems

Abstract

Harris Hawks Optimizer (HHO) is a recent optimizer that was successfully applied for various real-world problems. However, working under large-scale problems requires an efficient exploration/exploitation balancing scheme that helps HHO to escape from possible local optima stagnation. To achieve this objective and boost the search efficiency of HHO, this study develops embedded rules used to make adaptive switching between exploration/exploitation based on search performances. These embedded rules were formulated based on several parameters such as population status, success rate, and the number of consumed search iterations. To verify the effectiveness of these embedded rules in improving HHO performances, a total of six standard high-dimensional functions ranging from 1000-D to 10,000-D and CEC’2010 large-scale benchmark were employed in this study. In addition, the proposed Rules Embedded Harris Hawks Optimizer (REHHO) applied for one real-world high dimensional wavelength selection problem. Conducted experiments showed that these embedded rules significantly improve HHO in terms of accuracy and convergence curve. In particular, REHHO was able to achieve superior performances against HHO in all conducted benchmark problems. Besides that, results showed that faster convergence was obtained from the embedded rules. Furthermore, REHHO was able to outperform several recent and state-of-the-art optimization algorithms.

Introduction

In the era of big data, a lot of real-world, large-scale optimization problems have been existed, such as multipolicy insurance investment planning [1], scheduling [2], and gene biomarker discovery [3]. Tackling these problems using metaheuristic algorithms is considered a difficult task. This is due to the growth in dimension space, i.e., ‘‘curse of dimensionality” [4]. To mitigate these difficulties, researchers suggested several ideas, such as splitting the dimensionality using a divide-and-conquer scheme [5], introducing dynamic balancing between exploration and exploitation [6], or using the concept of population clustering [7].

Recently, many optimizers were introduced in the literature, such as Harris Hawks Optimizer (HHO) [8], Fitness Dependent Optimizer (FDO) [9], Learner Performance-based Behavior (LPB) [10], Child Drawing Development Optimizer (CDDO) [11], and Donkey and Smuggler Optimizer (DSO) [12]. Among them, HHO was given a lot of attention. This is due to its simplicity and efficiency in dealing with various real-world problems such as image segmentation [13], features selection [14], tracking maximum power in solar systems [15], prediction of solar systems productivity [16], designing load frequency of renewable energy plan [17], forecasting of air pollution [18], and predicting food liking [19]. However, HHO lacks efficient exploration/exploitation balancing ability. This is because it uses timely depended on energy escape parameters which control the switching from exploration to exploitation mode [8]. As such, when HHO is trapped in local optima during the exploitation phase, it will be hard to escape and return to exploration mode. To overcome these drawbacks, researchers suggested various versions of HHO, which could be categorized as hybridized-based methods and modified-based methods.

The idea of hybridizing HHO with other metaheuristic algorithms was studied by many researchers [2–26]. Abd Elaziz et al. introduced a hybrid model that combines HHO with a moth-flame optimizer [20]. The main goal was to enhance the exploration ability of HHO by using moth-flame to generate the initial population. In addition, a chaos map was embedded for further enhancement. Results show a superior impact on the performances of the hybrid optimizer as compared with the standard HHO. However, the main challenge is related to the increase in complexity due to the large number of parameters that need to be tuned. Further hybrid work was given by ElSayed et al. [21]. Basically, their work integrates HHO with Sequential Quadratic Programming (SQP). SQP was used as a local search optimizer to refine the best solution found by HHO at run-time. Reported results on the problem of finding the best relay directions in power systems indicated better performances were reported from the proposed hybrid approach. A memetic-based HHO scheme was suggested by Li et al. [22]. The key idea of the proposed memetic scheme is to enhance the local search capability of HHO by embedding several elite evolutionary strategies. Conducted analysis in their study on scheduling problems showed further HHO improvements were achieved due to the incorporated local search strategies. The hybridization of Grasshopper Optimizer GO with HHO was discussed by Singh et al. [23]. Their model was applied for the problem of optimal placement of multiple optical network units. The outcomes of GO-HHO demonstrated the superiority of the hybrid model as compared with individual optimizers, i.e., HHO and GO. The idea of evolving multiple HHO populations with quantum particles was given by Ilker et.al [20]. Mainly, their proposed approach was designed to tackle dynamic optimization problems that encompass multiple local optima. Conducted experiment on CEC 2009 showed that multiple HHO populations produced better outcomes in terms of convergence rate and fitness value. The fusion of HHO with both sine–cosine and simulated annealing was discussed in [25] and [26], respectively. Both models were applied for the problem of features selection, and their analysis showed great improvements in tackling feature selection challenges. Very recent hybrid studies which integrate HHO with other metaheuristic optimization algorithms were discussed by Abba et al. [27], Ebrahim et al. [28], Bandyopadhyay et al. [29], Suresh et al. [30], and Mossa et al. [31]. In [27], the hybrid of PSO with HHO for renewable energy load demand forecasting was presented. The synergy of sine–cosine with HHO was discussed in [28] for optimizing the fuel cell−based electric power system. Bandyopadhyay et al. [29] presented the integration of simulated annealing with HHO for deep features selection of COVID-19 from CT-scan images. The hybrid of chaotic multi-verse optimizer with HHO was given in [30] for the problem of medical diagnosis. The issue of parameters estimation of proton exchange membrane fuel cell using a hybrid atom optimizer with HHO was investigated in [31]. Nevertheless, the main challenge of hybrid-based methods is related to the increase of fitness evaluation cost needed for each optimizer. In addition, hybridizing several optimizers raises the difficulties of simultaneously managing them at run-time [32].

A modified-based HHO methods were presented in many studies [33–38]. The idea of modifying HHO by embedding salp optimizer operations was adopted by Abdelaziz et al. [33]. The main aim of their study is to enhance the exploration capability of HHO. In their work, they split the population into two sub-populations, and one half has been evolved under salp operations, and the other half has been executed under the control of HHO operations. The modified model in [24] was applied for the multi-level image thresholding problem, and results showed that embedded salp operations enhanced HHO exploration performances. Similarly, enhancing HHO exploration ability by incorporating differential evolutionary operators was suggested by Wunnava et al. [34]. Their proposed approach was applied for the multi-level image thresholding problem. Nevertheless, incorporating additional operations into HHO raises the challenge of increasing model complexity, which will increase the cost of fitness computation needed for these additional operations. Additional work was proposed by Yousri et al. [35] for improving the effectiveness of HHO in performing the exploration phase. Particularly, they have embedded fractional calculus (FOC) memory which is used to control the movement velocity of HHO agents. As such, FOC helps in avoiding possible premature convergence. Conducted experiments clearly showed better performances were achieved from embedding FOC. Additional HHO modifications were propped by Chen et al. [36]. They incorporated both opposition technique and chaotic local search into HHO. Reported results indicated better HHO improvements due to the enhancement in HHO population diversity. Further modifications were presented by Li et al. [37] for enhancing HHO population quality. Particularly, Li et al. added horizontal and vertical crossover operations into HHO, and results indicated further HHO exploration enhancements. Finally, researchers in [38] suggested the concept of information exchange to enhance HHO exploration ability. Very recent approaches were done by several researchers where they proposed many modification schemes to improve HHO. For instance, a multi-strategy approach was given by Li et al. [39]. The main idea of their approach is to incorporate different enhancement strategies namely opposition-based learning, logarithmic spiral, and a modified Rosenbrock local search. Other researchers suggested of embedding two different opposition-based schemes, namely selective, leading opposition, and the dynamic opposition technique. Enhancing HHO by implementing different random distribution functions which control the random movement of HHO agents was given by Akdag et al. [40]. Specifically, they have investigated seven types of random distribution functions, including chi-square, normal, exponential, Rayleigh, student’s distribution, F-distribution, and lognormal. Reported results clearly showed further improvements were achieved, especially for engineering design problems. Another enhanced version of HHO was discussed by Houssein et al. [41]. The key concept of their enhanced approach is to incorporate genetic operators to enhance exploitation ability in the selection of chemical molecular descriptors problems. An additional recent modified approach was illustrated by Krishna et al. [42]. Basically, they focused on enhancing HHO search capability when dealing with constrained engineering design problems. As such, to boost the exploitation performances of HHO, they have added pattern search algorithm during the exploitation phase of HHO. A chaotic guided HHO algorithm was demonstrated by Singh et al. [43] for data clustering. They have implemented a logistic chaotic map which was executed in the exploration phase of HHO. Despite the slow convergence of the enhanced HHO algorithm in [43],but the results clearly showed an improvement in the achieved clustering performances.

Motivated by HHO popularity, simplicity, and efficiency, this study aims to further improve HHO performances when dealing with large-scale problems that encounter a lot of local optima points. It should be noted that previously mentioned studies mainly focused on enhancing HHO exploration by incorporating chaotic re-initialization schemes [43], embedding opposition-based schemes [36, 39], or using other search operations inside HHO [33, 33–35, 40]. Others suggested using an external local search algorithm with HHO to improve exploitation performance [39, 42]. Despite that, there is still room for improvements by utilizing HHO population status at run-time search progress. Knowing the population status will play a vital role in making the decision about the appropriate time to switch from exploration to exploitation and vice versa. In addition, the idea of controlling the amount of jump during the exploitation phase has been utilized in this work. Therefore, this study formulates several rules that will be embedded into HHO to make adaptive switching of exploration/exploitations. An additional rule was embedded to control the amount of jump during the exploitation phase. The main contributions of this work are outlined as follows.

1. It monitors and utilizes population statuses for adaptive exploration/exploitation switching.

2. It uses agent location information to control the amount of jump needed at exploitation mode.

3. It evaluates the performances on multimodal standard benchmark function, large-scale CEC’2010 benchmark, and one real-world high-dimensional wavelength selection problem.

A table that summarizes all previously discussed HHO variants in terms of their type, authors, techniques, and used benchmarks is given in Table 1. The remaining part of this paper is organized as follows. Section 2 overview the standard HHO algorithm. The proposed embedded rules are explained in Sect. 3. A series of experiments that have been conducted to evaluate the effectiveness of the proposed approach are given in Sect. 4. A summary of the research findings is presented in Sect. 5.

Table 1.

Related work on HHO

Type	Ref.	Authors	Method	Benchmarks
Hybrid	[20]	Abd Elaziz et al.	It incorporated moth-flame, fractional-order, and chaotic maps to enhance HHO exploration	It uses 13 feature selections from UCI dataset and various engineering design problems
	[21]	ElSayed et al.	It integrated sequential quadratic programming (SQP) with HHO as a single model	It was applied for two problems of relays optimal coordination finding
	[22]	Li et al.	It proposed a memetic technique to enhance HHO local search capability	It was evaluated with 29 numerical optimization test functions and the problem of resource-constrained project scheduling and QoS-aware web service
	[23]	Singh et al.	It combined grasshopper with HHO as a single model	It has been applied for the problem of ONUs placement in Fiber-Wireless (FiWi)
	[20]	Ilker et al.	It evolved multiple HHO populations with quantum particles	It was evaluated with 23 dynamic test functions from CEC 2009 benchmark
	[25]	Kashif et al.	It hybridized the sine–cosine algorithm with HHO	It uses 29 test functions of CEC’17 test suite and 16 datasets for the problem of feature selection
	[26]	Abdel-Basset et al.	It incorporated simulated annealing for HHO search refinement	It has been evaluated with 24 standard datasets and 19 artificial datasets for feature selection problems
	[27]	Abba et al.	It integrated PSO with HHO for renewable energy load demand forecasting	It was evaluated using a lab collected data including solar radiation, temperature, and wind speed to predict load demand
	[28]	Ebrahim et al.	It combined since cosine with HHO for finding the optimal control parameters in fuel cell-based electric power system	It has been evaluated with the standard 23 benchmarks and applied for real-time control of energy consumption
	[29]	Bandyopadhyay et al.	It integrated simulated annealing with HHO for performing search refinement	It was assessed using real-world engineering design problems and COVID-19 deep features selection from CT-scan images
	[30]	Suresh et al.	It incorporated a chaotic multi-verse optimizer into HHO	It was evaluated using two public dataset for medical classification problems including PIMA Indian Diabetic and Wisconsin Breast Cancer
	[31]	Mossa et al.	It combined atom optimizer with HHO	It was applied for parameter estimation of Proton exchange membrane fuel cell. It was tested using three case studies including BCS 500-W PEM, 500 W SR-12PEM, and 250 W stack
Modified	[33]	AbdElaziz et al.	It incorporated salp operations into HHO for enhancing exploration performances	It has been applied for 36 functions from IEEE CEC 2005 benchmark and 11 Gy-scale image segmentation problems
	[34]	Wunnava et al.	It embedded differential evolutionary operators	It was evaluated with 500 images from Berkeley BSDS benchmark for multi-level image thresholding
	[35]	Yousri et al.	It added fractional-order calculus (FOC) memory to guide HHO during search progress	It uses 28 functions from CEC2017 benchmarks problems
	[36]	Chen et al.	It included two schemes which are opposition-based and chaotic local search to enhance HHO exploitation	It has been applied for photovoltaic cells design of three problems which are Shell st40, Shell sm55, and Shell kc200gt photovoltaics
	[37]	Li et al.	It incorporated horizontal and vertical crossover operations	It has been applied for photovoltaic parameter estimation of three models, which are SDM, DDM, and PV
	[38]	Qu et al.	It utilizes an information-sharing scheme to exchange agent's locations, etc	It was evaluated with 28 functions of CEC-2017 real-parameter numerical optimization problems
	[39]	Li et al.	It embedded several search strategies to enhance HHO. These strategies are logarithmic spiral and opposition technique to improve the exploration performances. Rosenbrock local search was added to enhance the exploitation ability	It was tested using IEEE CEC2014 benchmark and other engineering and real-world design problems
	[40]	Akdag et al.	It incorporated various random distribution functions to enhance HHO	It has been applied to optimize IEEE 30-bus power system. In addition, it was evaluated with the standard benchmark problems
	[41]	Houssein et al.	It embedded genetic operators to enhance HHO exploitation performances	It was evaluated using two chemoinformatics dataset namely QSAR Biodegradation and MAO
	[42]	Krishna et al.	It triggered local pattern search algorithm during the exploitation phase of HHO	It was tested using 23 standard CEC2005 benchmark and other engineering design problems
	[43]	Singh et al.	It embedded logistic chaotic map to enhance the initialization and exploration ability of HHO	It was evaluated using 12 UCI machine learning repository clustering problems

Harries Hawks optimizer

HHO is a recent optimizer that has been introduced by Heidari et al [8]. It is inspired by the attacking behavior of Harris Hawks birds, and it consists of three main phases, which are exploration, transition, and exploitation. These phases are explained as follows.

Exploration phase

In the exploration phase, HHO agents are going to perform discovering of the search space. Basically, there are two strategies that have been formulated for the exploration phase. The first one moves the hawk randomly in the search space. The second strategy is guided by both the best solution ${\text{X}}_{\mathit{rabbit}}^t$ and the mean location of the population ${\text{X}}_m^t$. These exploration strategies are defined as follows.

$${\text{X}}_i^{t+1}=\left\{\begin{array}{c}{\text{X}}_{\text{rand}}^t-r_1·\left|{\text{U}}_{\text{rand}}^t-2·r_2·{\text{X}}_i^t\right|p\ge 0.5\hfill \\ {\text{(X}}_{\text{rabbit}}^t-{\text{X}}_m^t)-r_3·\left(lb+r_4·\left(ub-lb\right)\right)p<0.5\hfill \end{array}\right)$$ 1

where $X_i^{t+1}$ is the next position of $i\text{th}$ hawk at search iteration $t$. Variables $r_1,r_2,r_3,\text{and}{r}_4$ are random values in the range of $\left(0,1\right)$. Variables $ub\text{and}lb$ are upper and lower bound of the search problem. Variable $p$ is a random value used to control the switching between exploration strategies. It should be noted that the mean position of the population $X_m^t$ is computed as follows.

$${\text{X}}_m^t=\frac{1}{N}\sum _{i=1}^N{\text{X}}_i^t$$ 2

Transition phase

In HHO, they considered rabbit energy as the primary indicator used to switch from exploration to exploitation mode. This variable is computed according to the following formula.

$$E=2E_0\left(1-\frac{t}{T}\right)$$ 3

where $E$ is escaping energy decreases linearly based on time iterations $t$. $T$ is the maximum allocated iterations. Variable $E_0$ is the initial state energy of each individual, and it is varied randomly in the range $E_0\in \left(-1,1\right)$. $E_0$ is updated based on the following equation.

$$E_0=2\text{rand}-1$$ 4

The computed escaping energy $E$ in Eq. (4) will have the following plot in Fig. 1. It can be seen that HHO will be in the exploration phase when $\left|E\right|\ge 1$, while exploitation mode will occur when $\left|E\right|<1$ as demonstrated in Fig. 1.

Fig. 1.

The behavior of escaping energy $E$

Exploitation phase

The main idea of the exploitation phase is to exploit the current best location $X_{\text{rabbit}}^t$ and to search around it. In HHO, they proposed four different strategies for performing search exploitation, namely soft besiege, hard besiege, soft besiege with progressive rapid dives, and hard besiege with progressive rapid dives. In order to select which exploitation strategy will be executed, both escaping energy $\left|E\right|$ and a random variable $r$ is used to control. In particular, the escaping energy $\left|E\right|$ will control the switching between soft/hard besiege strategy. When $\left|E\right|\ge 0.5$ indicating that the rabbit $X_{\text{rabbit}}^t$ still has energy and soft besiege strategy should be applied; otherwise, the rabbit should be exhausted, and hard besiege strategy should be used. In addition, each strategy could run progressive rapid dives when it is applied. As such, the implemented random variable $r$ will control the activation of this property. As such, when $r<0.5,$ progressive rapid dives will be used; otherwise, it will be off. These strategies are explained as follows.

Soft besiege

As mentioned earlier, soft besiege is activated when $r\ge 0.5$ and $\left|E\right|\ge 0.5,$ which means that the rabbit ${\text{X}}_{\mathit{rabbit}}^t$ still has energy, trying to escape by a random jump. As such, it will be softly encircled by the hawks and attacked according to the following equations.

$${\text{X}}_i^{t+1}=\mathrm{\Delta }{\text{X}}^t-E\left|J{X}_{\text{rabbit}}^t-{\text{X}}_i^t\right|$$ 5

$$\mathrm{\Delta }{\text{X}}^t=X_{\text{rabbit}}^t-{\text{X}}_i^t$$ 6

where $\mathrm{\Delta }{\text{X}}^t$ is the difference between rabbit location $X_{\text{rabbit}}^t$ and current hawk location ${\text{X}}_i^t$. J is a random factor that mimics rabbit movement, and it is defined as follows.

$$\text{J}=2\left(1-r_5\right)$$ 7

where $r_5$ is a random value in the range $r_5\in \left(-1,1\right)$.

Hard besiege

This strategy is applied when the rabbit $X_{\text{rabbit}}^t$ is exhausted, and it has only a tiny escaping energy $\left|E\right|<0.5$. Therefore, hawks will attack the rabbit according to the following equation.

$${\text{X}}_i^{t+1}=X_{\text{rabbit}}^t-E\left|\mathrm{\Delta }{\text{X}}^t\right|$$ 8

Soft besiege with progressive rapid dives

This strategy is applied when the rabbit still has energy $\left|E\right|\ge 0.5$ but $r<0.5$. However, they suggested in [8] a more intelligent mathematical formula that mimics actual rabbit motion. Specifically, a zigzag motion pattern was formulated using the levy flight (LF) function defined as follows.

$$\text{LF}\left(\text{x}\right)=0.01\times \frac{u\times \sigma }{{\left|v\right|}^{\frac{1}{\beta }}}$$ 9

$$\sigma ={\left(\frac{\mathrm{\Gamma }\left(1+\beta \right)\times \text{sin}\left(\frac{\pi \beta }{2}\right)}{\mathrm{\Gamma }\left(\frac{1+\beta }{2}\right)\times \beta \times 2^{\left(\frac{\beta -1}{2}\right)}}\right)}^{\frac{1}{\beta }}$$ 10

where $u,v$ are random values from $0to1$. $\beta$ is a constant value set as 1.5 as suggested in [8].

Therefore, the hawk will update his position using the following formulas.

$$\text{Y}={\text{X}}_{\mathit{rabbit}}^t-E\left|{\text{JX}}_{\mathit{rabbit}}^t-{\text{X}}_i^t\right|$$ 11

$$\text{Z}=Y+S\times LF\left(D\right)$$ 12

$$\begin{array}{c}\\ X_i^{t+1}=\left\{\begin{array}{c}Y\text{if}\text{fitness}\left(Y\right)>\text{fitness}\left(X_i^{t+1}\right)\\ Z\text{if}\text{fitness}\left(Z\right)>\text{fitness}\left(X_i^{t+1}\right)\end{array}\right)\\ \end{array}$$ 13

where S is 1D random values and D is the problem dimension.

Hard besiege with progressive rapid dives

This strategy is very similar to soft besiege with progressive rapid dives, but the hawk will a bit decrease the jump distance because the rabbit is exhausted $\left|E\right|<0.5$. As such, the hawk will update his position considering the mean position $X_m^t$ instead of using $X_i^t$ as defined in Eq. (11). This strategy has been formulated according to the following equations.

$$Y=X_{\text{rabbit}}^t-E\left|J{X}_{\text{rabbit}}^t-X_m^t\right|$$ 14

$$Z=Y+S\times LF\left(D\right)$$ 15

$$\begin{array}{c}\\ X_i^{t+1}=\left\{\begin{array}{c}Y\text{if}\text{fitness}\left(Y\right)>\text{fitness}\left(X_i^{t+1}\right)\\ Z\text{if}\text{fitness}\left(Z\right)>\text{fitness}\left(X_i^{t+1}\right)\end{array}\right)\\ \end{array}$$ 16

The proposed embedded rules

As explained earlier that HHO depends on escaping energy factor $\left|E\right|$ for the transition from exploration to exploitation. However, $\left|E\right|$ is a time dependent parameter that decreases gradually over time, as shown in Fig. 1. As such, HHO has a lower chance to try to exploit the discovered region and switch to exploitation mode due to $\left|E\right|>1$. As a result of this, HHO will have a slow convergence rate due to the prevention of switching to exploitation mode at the beginning of search progress. Another limitation of HHO is that when $\left|E\right|$ becomes very small and the hawks trapped in local optima region, it will be hard to escape and return to exploration stage. To overcome this shortcoming of HHO, this research introduces embedded rules that will improve balancing between exploration and exploitation and help HHO to escape from possible local optima traps by amplifying escaping energy $\left|E\right|.$

The main idea of the proposed rules is given in Fig. 2, where embedded rules are going to be triggered according to the occurrence of some events. The first rule is given in Fig. 3. This rule has been formulated to allow HHO to escape from local optima. Three events are used to control the trigger of this rule which are population success, current search time, and a random value $r$. $\text{pop\_success}$ factor will take a value of zero or one. It will be zero when the best population location $X_{\text{rabbit}}^t$ was not changed; otherwise, it will be one. The second variable $t$ represents the current search time step. It is used to prevent the rule to be triggered during the exploration phase where the escaping energy is already greater than one, i.e., $\left|E\right|\ge 1$. The variable $r$ is a random value that randomizes the trigger of RULE 1.

Fig. 2.

The proposed embedded rules

Fig. 3.

RULE 1 for switching from exploitation to exploration

The second rule was designed to help HHO to switch from exploration to exploitation mode at the beginning of the search process. RULE 2 is shown in Fig. 4. It is indicated that RULE 2 is controlled by three parameters which are the location of the rabbit ${\mathit{X}}_{\text{rabbit}}$ with respect to the mean location ${\mathit{X}}_{\mathit{m}}$, current search time step $t$, and a random value $r$. It should be noted that the last two parameters (i.e., $t$ and $r$) are the same in RULE 1; however, the first event is used to check the current population status. Referring to Fig. 2, it can be seen when the hawks encircle the rabbit, then the location of ${\mathit{X}}_{\text{rabbit}}$ will be close to ${\mathit{X}}_{\mathit{m}}$. To check this status, the fitness value of ${\mathit{X}}_{\mathit{m}}$ is computed and compared to other hawks. If it belongs to the top 10% of the population, then it means ${\mathit{X}}_{\mathit{m}}$ location is near to ${\mathit{X}}_{\text{mean}}$.

Fig. 4.

RULE 2 for switching from exploration to exploitation

The third rule is formulated to control the amount of hawk jump needed during the exploitation mode. RULE 3 is defined in Fig. 5, and it will be activated only during the exploitation phase when $\left|E\right|<1.$ RULE 3 is triggered based on hawk location with respect to $X_{\text{rabbit}}^t$, and $X_i^t$ as given in Fig. 5. Rule 3 imply that hawks located closely to $X_{\mathit{rabbit}}^t$ need a small jump; however, those far away hawks need a larger jump to reach the location of $X_{\mathit{rabbit}}^t$. This jump is reflected by the amount of escaping energy that will influence Eqs. (5), (8), (11), and (14).

Fig. 5.

RULE 3 to control the amount of hawk jump during exploitation

The complete steps of the proposed REHHO algorithm are given in algorithm 1 and Fig. 6.

Fig. 6.

Flowchart of the proposed REHHO algorithm

Experimental results

The standard benchmark problems

This section evaluates the performances of REHHO as compared with HHO using six multimodal benchmark problems, which are Schwefel, Rastrigin, Ackley, Griewank, Penalized, and Penalized 2. These problems were executed with various high-dimensional ranging from 1000- D to 10,000-D. These experiments were epeated 30 times, and the maximum number of iterations was set to ${10}^4$.

$$F1\left(X\right)=\sum _{i=1}^D-x_{i}sin\left(\sqrt{\left|x_i\right|}\right),searchrange\left[-500,500\right],D=1000,f_{min}=-418.9829\ast n$$ 17

The mathematical formula of Schwefel function is given in Eq. (17). This function has multiple local optima, as can be seen in Fig. 7. Therefore, finding the global optima of this function under high dimensions is a quiet challenge. Nevertheless, REHHO was able to reach the minimum value faster than HHO, as shown in Fig. 7. This is due to the dynamic of REHHO and its ability to perform earlier exploitation searches as compared with HHO. Further analysis is conducted by evaluating the performances when the dimension of the problem is increased to 5000-D and 10,000-D, respectively. Therefore, the best, medium, worst, mean, and standard deviation of REHHO and HHO algorithms are reported in Table 2. It can be seen that both algorithms reported almost the same mean values in all variations of dimensions which are 1000-D, 5000-D, and 10,000-D. However, REHHO was able to reach the global optima earlier than HHO, as indicated in Fig. 7.

$$F_2\left(X\right)=\sum _{i=1}^D\left[x_i^2-10cos\left(2\pi x_i\right)+10\right],searchrange\left[-5.12,5.12\right],D=1000,f_{min}=0$$ 18

Fig. 7.

The Schwefel function with its convergence curve (1000-D)

Table 2.

Results of Schwefel functions

Dim	Fitness	Algorithm
		REHHO	HHO
1000 − D	Best	− 4.1898e + 05	4.1898e + 05
	Median	− 4.1898e + 05	− 4.1898e + 05
	Worst	− 4.1898e + 05	− 4.1898e + 05
	Mean	− 4.1898e + 05	− 4.1898e + 05
	Std	0	0
5000 − D	Best	− 2.0949e + 06	− 2.0949e + 06
	Median	− 2.0949e + 06	− 2.0949e + 06
	Worst	− 2.0949e + 06	− 2.0949e + 06
	Mean	− 2.0949e + 06	− 2.0949e + 06
	Std	0	0
10,000 − D	Best	− 4.1898e + 06	− 4.1898e + 06
	Median	− 4.1898e + 06	− 4.1898e + 06
	Worst	− 4.1898e + 06	− 4.1898e + 06
	Mean	− 4.1898e + 06	− 4.1898e + 06
	Std	0	0

Rastrigin function is given in Eq. (18), and it differs from the Schwefel function where the minimum value is located at point zero, as shown in Fig. 8. The convergence curve of both functions is almost similar. This is due to the nature of the optimized problem, where it is a bit easier as compared with the shifted Schwefel function discussed earlier. Moreover, the variation in the dimensions does not affect the performances of both algorithms, as indicated in Table 3. In other words, REHHO and HHO were able to achieve the global optima in all runs with zero standard deviation, which showed the stability of both algorithms.

$$F_3\left(X\right)=-20exp\left(-0.2\sqrt{\frac{1}{D}\sum _{i=1}^D{x}_i^2}\right)-exp\left(\frac{1}{D}\sum _{i=1}^{D}cos\left(2\pi x_i\right)\right)+20+e,searchrange\left[-32,32\right]D=1000,f_{min}=0$$ 19

Fig. 8.

The Rastrigin function with its convergence curve (1000-D)

Table 3.

Results of rastrigin function

Dim	Fitness	Algorithm
		REHHO	HHO
1000–D	Best Median Worst Mean Std	0 0 0 0 0	0 0 0 0 0
5000–D	Best Median Worst Mean Std	0 0 0 0 0	0 0 0 0 0
10,000–D	Best Median Worst Mean Std	0 0 0 0 0	0 0 0 0 0

Equation (19) shows the mathematical formula of the Ackley function. The 2-D plot of this function is shown in Fig. 9. The plotted convergence curve indicated that REHHO is able to converge faster due to the flattening nature of this problem. It should be noted that both algorithms were able to reach global optima with less than 50 iterations.

Fig. 9.

The Ackley function with its convergence curve (1000-D)

As can be seen in Table 4, increasing the dimensionality of this problem does not influence the performance of both algorithms. This is due to the simplicity of the tacked Ackley function.

$$F_4\left(X\right)=\frac{1}{4000}\sum _{i=1}^D{x}_i^2-\prod _{i=1}^{D}cos\left(\frac{x_i}{\sqrt{i}}\right)+1,searchrange\left[-600,600\right],D=1000,f_{min}=0$$ 20

Table 4.

Results of Ackley function

Dim	Fitness	Algorithm
		REHHO	HHO
1000-D	Best Median Worst Mean Std	8.8818e − 16 8.8818e − 16 8.8818e − 16 8.8818e − 16 0	8.8818e − 16 8.8818e − 16 8.8818e − 16 8.8818e − 16 0
5000-D	Best Median Worst Mean Std	8.8818e − 16 8.8818e − 16 8.8818e − 16 8.8818e − 16 0	8.8818e − 16 8.8818e − 16 8.8818e − 16 8.8818e − 16 0
10,000-D	Best Median Worst Mean Std	8.8818e − 16 8.8818e − 16 8.8818e − 16 8.8818e − 16 0	8.8818e − 16 8.8818e − 16 8.8818e − 16 8.8818e − 16 0

The Griewank function is given in Eq. (20). This problem is considered more challenging as compared with Ackley, where it has a lot of local optima, as indicated in Fig. 10. The convergence of the Griewank function confirms the ability of REHHO to switch faster and perform exploitation searches. Nevertheless, both REHHO and HHO were able to reach the global minima, as shown in Table 5.

Fig. 10.

The Griewank function with its convergence curve (1000–D)

Table 5.

Results of Griewank function

Dim	Fitness	Algorithm
		REHHO	HHO
1000–D	Best Median Worst Mean Std	0 0 0 0 0	0 0 0 0 0
5000–D	Best Median Worst Mean Std	0 0 0 0 0	0 0 0 0 0
10,000–D	Best Median Worst Mean Std	0 0 0 0 0	0 0 0 0 0

The Penalized function is defined as follows.

$$F_5\left(X\right)=\frac{\pi }{D}\left\{10sin\left(\pi y_1\right)+\sum _{i=1}^{D-1}{\left(y_i-1\right)}^2\left[1+10si{n}^2\left(\pi y_{i+1}\right)\right]+{\left(y_D-1\right)}^2\right\}+\sum _{i=1}^{D}u\left(x_i,10,100,4\right)$$

$$y_i=1+\frac{x_i+1}{4}$$

$$\begin{array}{c}u\left(x_i,a,k,m\right)=\left\{\begin{array}{c}k{\left(x_i-a\right)}^m{x}_i>a\\ 0-a<x_i<a\\ k{\left(-x_i-a\right)}^m{x}_i<a\end{array}\right)\\ \text{search range}\left[-50,50\right],\\ \end{array}$$ 21

Unlike previously discussed functions, the Penalized function is considered more challenging. The convergence curve of this function is given in Fig. 11, and it is clearly seen that REHHO converges a bit faster. More importantly, increasing the dimensions of Penalized function from 1000-D to 10,000-D does not influence the performances of REHHO, as given in Table 6.

Fig. 11.

The Penalized function with its convergence curve (1000–D)

Table 6.

Results of Penalized function

Dim	Fitness	Algorithm
		REHHO	HHO
1000–D	Best Median Worst Mean Std	7.4754e − 11 2.0208e − 10 1.9952e − 09 7.5735e − 10 1.0739e − 09	5.6363e − 10 8.2071e − 10 6.4469e − 09 2.6104e − 09 3.3250e − 09
5000–D	Best Median Worst Mean Std	1.8923e − 12 4.1541e − 11 2.1488e − 10 8.6105e − 11 1.1327e − 10	1.0878e − 10 1.5618e − 10 6.7634e − 10 3.1377e − 10 3.1489e − 10
10,000–D	Best Median Worst Mean Std	1.2802e − 10 1.6376e − 10 5.9265e − 10 2.5641e − 10 3.0082e − 09	2.5269e − 10 8.4575e − 10 1.9220e − 09 1.0068e − 09 8.4625e − 10

Bold values indicate the best mean value (i.e., average minimum)

The Penalized 2 function is defined as follows:

$$\begin{array}{c}{F}_6\left(X\right)=0.1\left\{si{n}^2\left(3\pi x_1\right)+\sum _{i=1}^D{\left(x_i-1\right)}^2\left[1+si{n}^2\left(3\pi x_i+1\right)\left]+{\left(x_D-1\right)}^2\right[1+si{n}^2\left(2\pi x_D\right)\right]\right\}\\ +\sum _{i=1}^{D}u\left(x_i,5,100,4\right),\text{search}\text{range}\left[-50,50\right],\\ \end{array}$$ 22

The conducted analysis on the Penalized 2 function is given in Fig. 12 and Table 7. As can be seen, REHHO outperformed HHO with a better mean fitness value and faster convergence curve.

Fig. 12.

The Penalized 2 function with its convergence curve (1000–D)

Table 7.

Results of penalized 2 function

Dim	Fitness	Algorithm
		REHHO	HHO
1000–D	Best Median Worst Mean Std	9.6019e − 08 9.6562e − 08 2.0774e − 07 1.3344e − 07 6.4348e − 08	2.9647e − 08 4.0487e − 07 2.2211e − 06 8.8522e − 07 1.1721e − 06
5000–D	Best Median Worst Mean Std	3.5014e − 08 7.4315e − 07 2.0981e − 07 1.0638e − 07 9.1702e − 07	3.0948e − 08 5.2818e − 07 4.1433e − 06 1.5675e − 06 2.2445e − 06
10,000–D	Best Median Worst Mean Std	2.4982e − 07 9.9053e − 07 3.3143e − 06 1.1461e − 06 1.8781e − 05	6.3753e − 06 7.1744e − 06 2.6157e − 05 1.3236e − 05 1.1198e − 05

Bold values indicate the best mean value (i.e., average minimum)

2010 large scale global benchmark problems

This experimental section examines the effectiveness of the embedded rules in enhancing HHO (i.e., REHHO algorithm) on large-scale benchmark problems. Specifically, a set of 20 large-scale functions from 1000-D CEC’2010 benchmark functions [44] has been used. The details of these functions are given in Table 8. As can be seen, CEC’2010 benchmark consists of 5 groups of functions, namely, separable, single-group $\mathit{m}-$ non-separable functions, $\mathit{D}/2\mathit{m}$ group $\mathit{m}$-non-separable functions, $\mathit{D}/\mathit{m}$ group $\mathit{m}$-non-separable functions, and fully separable functions. Each experiment was repeated 30 times, and the population size was 30 agents with a maximum number of fitness evaluations $3\times {10}^4$.

Table 8.

Description of 1000 − D CEC’2010 large − scale benchmark functions

Type	Function	Description	Dim	Range	${\mathit{f}}_{\mathit{min}}$
Separable functions	$F_1\left(X\right)=\sum _{i=1}^D{\left({10}^6\right)}^{\frac{i-1}{D-1}}{Z}_i^2$	Shifted elliptic function	1000	[− 100, 100]	0
	$F_2\left(X\right)=\sum _{i=1}^D\left[Z_i^2-10cos\left(2\pi Z_i\right)+10\right]$	Shifted Rastrigin’s function	1000	[− 5, 5]	0
	$F_3\left(X\right)=-20exp\left(-0.2\sqrt{\frac{1}{D}\sum _{i=1}^D{Z}_i^2}\right)-exp\left(\frac{1}{D}\sum _{i=1}^D\text{cos}\left(2\pi Z_i\right)\right)+20+e$	Shifted ackley’s function	1000	[− 32, 32]	0
Single-group $\mathit{m}-$ non-separable Functions	$F_4\left(X\right)=F_{rot\_elliptic}\left[Z\left(P_1:P_m\right)\right]\ast {10}^6+F_{\mathit{elliptic}}\left[Z\left(P_{m+1}:P_D\right)\right]$	Single-group shifted and m-rotated elliptic function	1000	[− 100, 100]	0
	$F_5\left(X\right)=F_{rot\_rastrigin}\left[Z\left(P_1:P_m\right)\right]\ast {10}^6+F_{\mathit{rastrigin}}\left[Z\left(P_{m+1}:P_D\right)\right]$	Single-group shifted and m-rotated rastrigin’s function	1000	[− 5, 5]	0
	$F_6\left(X\right)=F_{rot\_ackley}\left[Z\left(P_1:P_m\right)\right]\ast {10}^6+F_{\mathit{ackley}}\left[Z\left(P_{m+1}:P_D\right)\right]$	Single-group shifted and m-rotated ackley’s function	1000	[− 32, 32]	0
	$F_7\left(X\right)=F_{\mathit{schwefel}}\left[Z\left(P_1:P_m\right)\right]\ast {10}^6+F_{\mathit{sphere}}\left[Z\left(P_{m+1}:P_D\right)\right]$	Single-group shifted m-dimensional schwefel’s	1000	[− 100, 100]	0
	$F_8\left(X\right)=F_{\mathit{rosenbrock}}\left[Z\left(P_1:P_m\right)\right]\ast {10}^6+F_{\mathit{sphere}}\left[Z\left(P_{m+1}:P_D\right)\right]$	Single-group shifted m-dimensional rosenbrock’s function	1000	[− 100, 100]	0
$\frac{\mathit{D}}{2\mathit{m}}$ group $\mathit{m}$-non-separable functions	$F_9\left(X\right)=\sum _{k=1}^{\frac{D}{2m}}{F}_{rot\_elliptic}\left[z\left(P_{\left(k-1\right)\ast m+1}:P_{k\ast m}\right)\right]+F_{\mathit{elliptic}}\left[z\left(P_{\frac{D}{2}+1}:P_D\right)\right]$	$\frac{D}{2m}$ group shifted and m-rotated elliptic function	1000	[− 100, 100]	0
	$F_{10}\left(X\right)=\sum _{k=1}^{\frac{D}{2m}}{F}_{rot\_rastrigin}\left[z\left(P_{\left(k-1\right)\ast m+1}:P_{k\ast m}\right)\right]+F_{\mathit{rastrigin}}\left[z\left(P_{\frac{D}{2}+1}:P_D\right)\right]$	$\frac{D}{2m}$ group shifted and m-rotated rastrigin function	1000	[− 5, 5]	0
	$F_{11}\left(X\right)=\sum _{k=1}^{\frac{D}{2m}}{F}_{rot\_ackley}\left[z\left(P_{\left(k-1\right)\ast m+1}:P_{k\ast m}\right)\right]+F_{\mathit{ackley}}\left[z\left(P_{\frac{D}{2}+1}:P_D\right)\right]$	$\frac{D}{2m}$ group shifted and m-rotated ackley’s function	1000	[− 32, 32]	0
	$F_{12}\left(X\right)=\sum _{k=1}^{\frac{D}{2m}}{F}_{\mathit{schwefel}}\left[z\left(P_{\left(k-1\right)\ast m+1}:P_{k\ast m}\right)\right]+F_{\mathit{sphere}}\left[z\left(P_{\frac{D}{2}+1}:P_D\right)\right]$	$\frac{D}{2m}$ group shifted m-rotated schwefel’s	1000	[− 100, 100]	0
	$F_{13}\left(X\right)=\sum _{k=1}^{\frac{D}{2m}}{F}_{\mathit{rosenbrock}}\left[z\left(P_{\left(k-1\right)\ast m+1}:P_{k\ast m}\right)\right]+F_{\mathit{sphere}}\left[z\left(P_{\frac{D}{2}+1}:P_D\right)\right]$	$\frac{D}{2m}$ group shifted m-rotated rosenbrock’s function	1000	[− 100, 100]	0
$\frac{\mathit{D}}{\mathit{m}}$ group $\mathit{m}$-non-separable Functions	$F_{14}\left(X\right)=\sum _{k=1}^{\frac{D}{m}}{F}_{rot\_elliptic}\left[z\left(P_{\left(k-1\right)\ast m+1}:P_{k\ast m}\right)\right]$	$\frac{D}{m}$ group shifted and m-rotated elliptic function	1000	[− 100, 100]	0
	$F_{15}\left(X\right)=\sum _{k=1}^{\frac{D}{m}}{F}_{rot\_rastrigin}\left[z\left(P_{\left(k-1\right)\ast m+1}:P_{k\ast m}\right)\right]$	$\frac{D}{m}$ group shifted and m-rotated rastrigin function	1000	[− 5, 5]	0
	$F_{16}\left(X\right)=\sum _{k=1}^{\frac{D}{m}}{F}_{rot\_ackley}\left[z\left(P_{\left(k-1\right)\ast m+1}:P_{k\ast m}\right)\right]$	$\frac{D}{m}$ group shifted and m-rotated ackley’s function	1000	[− 32, 32]	0
	$F_{17}\left(X\right)=\sum _{k=1}^{\frac{D}{m}}{F}_{\mathit{schwefel}}\left[z\left(P_{\left(k-1\right)\ast m+1}:P_{k\ast m}\right)\right]$	$\frac{D}{m}$ group shifted m-rotated schwefel	1000	[− 100, 100]	0
	$F_{18}\left(X\right)=\sum _{k=1}^{\frac{D}{m}}{F}_{\mathit{rosenbrock}}\left[z\left(P_{\left(k-1\right)\ast m+1}:P_{k\ast m}\right)\right]$	$\frac{D}{m}$ group shifted m-rotated rosenbrock’s function	1000	[− 100,100]	0
Fully separable Functions	$F_{19}\left(X\right)=\sum _{i=1}^n{\left(\sum _{j=1}^i{x}_i\right)}^2$	Shifted schwefel’s	1000	[− 100, 100]	0
	$F_{20}\left(X\right)=\sum _{i=1}^{D-1}\left[100{\left(z_i^2-z_{i+1}\right)}^2+{\left(z_i-1\right)}^2\right]$	Shifted rosenbrock’s function	1000	[− 100, 100]	0

Table 9 depicts the results of REHHO and HHO on CEC’2010 largescale problems. It should be noted that the separable functions F1, F2, and F3 have a smaller number of local optima as compared with other categories. Nevertheless, the reported results showed that REHHO achieved the best results. This is due to the ability of REHHO to perform fast convergence and switch earlier to exploitation mode. In other words, the implemented adaptive switching scheme using the embedded rules helps REHHO to reach faster to global optima point. The results of single-group $\mathit{m}-$ non-separable functions are given by functions F4, F5, F6, F7, and F8. Similarly, REHHO reports the best mean value in all conducted functions.

Table 9.

Results of 1000-D CEC’2010 large-scale functions

Function	Fitness	Algorithm
		REHHO	HHO
F1	Best Median Worst Mean Std	2.6334e + 09 2.8842e + 09 3.0811e + 09 2.8583e + 09 1.3393e + 08	2.5870e + 09 2.9525e + 09 3.1307e + 09 2.9077e + 09 1.8042e + 08
F2	Best Median Worst Mean Std	1.5968e + 04 1.6255e + 04 1.6685e + 04 1.6285e + 04 221.9340	1.6194e + 04 1.6394e + 04 1.6656e + 04 1.6405e + 04 146.6359
F3	Best Median Worst Mean Std	20.2382 20.5327 20.6923 20.5073 0.1428	20.1865 20.5663 20.7024 20.5474 0.1591
F4	Best Median Worst Mean Std	1.2145e + 13 1.8154e + 13 2.5972e + 13 1.8575e + 13 3.7970e + 12	1.3675e + 13 1.8249e + 13 3.1059e + 13 1.9648e + 13 5.3968e + 12
F5	Best Median Worst Mean Std	4.1407e + 08 4.5136e + 08 4.8347e + 08 4.4886e + 08 2.5081e + 07	4.1267e + 08 4.4175e + 08 4.9685e + 08 4.4936e + 08 2.8098e + 07
F6	Best Median Worst Mean Std	1.9146e + 07 1.9211e + 07 1.9276e + 07 1.9212e + 07 3.9172e + 04	1.9100e + 07 1.9226e + 07 1.9282e + 07 1.9215e + 07 5.3649e + 04
F7	Best Median Worst Mean Std	1.3083e + 09 1.7299e + 09 2.0005e + 09 1.7102e + 09 2.4558e + 08	1.3848e + 09 1.7041e + 09 2.6482e + 09 1.8329e + 09 4.5454e + 08
F8	Best Median Worst Mean Std	4.6868e + 09 6.2015e + 09 1.2757e + 10 6.9842e + 09 2.5889e + 09	3.5725e + 09 7.0823e + 09 1.3574e + 10 8.3367e + 09 3.3609e + 09
F9	Best Median Worst Mean Std	4.3561e + 09 4.5738e + 09 4.8799e + 09 4.5915e + 09 1.6658e + 08	4.4583e + 09 4.6868e + 09 5.1160e + 09 4.7577e + 09 2.0995e + 08
F10	Best Median Worst Mean Std	1.6334e + 04 1.6526e + 04 1.6897e + 04 1.6559e + 04 169.1596	1.6174e + 04 1.6680e + 04 1.6753e + 04 1.6617e + 04 177.6641
F11	Best Median Worst Mean Std	221.5103 222.7064 224.8432 222.8913 1.0573	220.8727 222.9485 225.3188 223.1134 1.2622
F12	Best Median Worst Mean Std	1.9385e + 06 2.0562e + 06 2.1350e + 06 2.0461e + 06 6.9452e + 04	1.9622e + 06 2.0869e + 06 2.1904e + 06 2.0763e + 06 8.4339e + 04
F13	Best Median Worst Mean Std	1.1077e + 08 1.3137e + 08 1.6626e + 08 1.3441e + 08 1.5571e + 07	1.0374e + 08 1.3379e + 08 1.6147e + 08 1.3493e + 08 1.9787e + 07
F14	Best Median Worst Mean Std	6.2729e + 09 6.6919e + 09 8.0578e + 09 6.7559e + 09 5.0437e + 08	6.0907e + 09 6.8612e + 09 7.5727e + 09 6.9155e + 09 4.8570e + 08
F15	Best Median Worst Mean Std	1.6416e + 04 1.6534e + 04 1.6927e + 04 1.6568e + 04 145.8575	1.6340e + 04 1.6691e + 04 1.6892e + 04 1.6674e + 04 145.5967
F16	Best Median Worst Mean Std	404.9208 409.7903 410.9187 409.1372 1.7926	407.5469 409.6755 412.0824 409.8648 1.4549
F17	Best Median Worst Mean Std	2.7638e + 06 2.9555e + 06 3.1423e + 06 2.9464e + 06 1.1326e + 05	2.8882e + 06 3.0557e + 06 3.2906e + 06 3.0504e + 06 1.4232e + 05
F18	Best Median Worst Mean Std	2.4614e + 09 2.7451e + 09 3.1012e + 09 2.7693e + 09 2.0060e + 08	2.5603e + 09 2.7727e + 09 3.0197e + 09 2.8075e + 09 1.4732e + 08
F19	Best Median Worst Mean Std	7.4615e + 06 1.0164e + 07 1.3475e + 07 1.0020e + 07 1.7817e + 06	8.8942e + 06 1.0717e + 07 1.5390e + 07 1.1079e + 07 1.8966e + 06
F20	Best Median Worst Mean Std	2.7733e + 09 3.0091e + 09 3.2906e + 09 3.0219e + 09 1.8120e + 08	2.5204e + 09 3.0602e + 09 3.8450e + 09 3.1099e + 09 3.8648e + 08

Bold values indicate the best mean value (i.e., average minimum)

For the category of $\mathit{D}/2\mathit{m}$ group $\mathit{m}-$ non-separable functions, which is considered more complex, REHHO is still able to perform well due to its ability to switch back from exploitation mode to exploration mode when population success status is zero, then rule 2 will be triggered as explained in Sect. 3. In addition, REHHO reports the best mean value in the remaining functions from F14 to F20.

In conclusion, conducted analysis on CEC’2010 large-scale benchmark functions showed that the performances of REHHO have been improved considerably from the embedded rules.

Further analysis was conducted by investigating the graphical behavior of REHHO convergence during run time. Specifically, the base-10 logarithmic mean values of the fitness function from a total of 30 runs are computed and plotted in Fig. 13. It is shown that REHHO has faster convergence in all benchmark functions. This is due to the ability of REHHO to switch earlier to exploitation mode at the beginning of the search process, as mentioned previously.

Fig. 13.

The convergence curves for large-scale functions F1–F20

Compared with other metaheuristic algorithms

In this section, the performance of REHHO has been compared with several well-known state-of-the-art optimization algorithms. These algorithms are particle swarm optimization (PSO), differential evolution (DE), BAT optimizer, Arithmetic Optimization Algorithm (AOA) [45], and Horse Herd Algorithm (HHA) [46]. It is worth mentioning that BAT, PSO, and DE are very famous population-based algorithms and are widely used in the literature as baseline comparison algorithms. The settings of these algorithms are given in Table 10. For all conducted algorithms, each experiment was repeated 30 times, and the population size was 30 agents with a maximum number of fitness evaluations $3\times {10}^4$.

Table 10.

Parameter settings

Method	Population	FEs	Parameters
BAT	30	$3\ast {10}^4$	Loudness = 0.5, Pulse rate = 0.5, Frequency minimum = 0, Frequency maximum = 2
PSO	30	$3\ast {10}^4$	c1 = 2.5 to 0.5, c2 = 0.5 to 2.5, w = 0.9 to 0.4
DE	30	$3\ast {10}^4$	Lower bound of scaling factor = 0.2, upper bound of scaling factor = 0.8, crossover probability = 0.2
AOA	30	$3\ast {10}^4$	MOP: 0.2–1, Alpha = 5, Mu = 0.499
HHA	30	$3\ast {10}^4$	hβ = 0.9, hγ = 0.5, sβ = 0.1, sγ = 0.2, iγ = 0.3, dα = 0.5, dβ = 0.2, dγ = 0.1, rδ = 0.1, and rγ = 0.05

Bold values indicate the best mean value (i.e., average minimum)

The results of the conducted analysis are given in Table 11. As can be seen, REHHO reported the best mean value in most functions except for five of them, namely F5, F6, F8, and F10, where DE outperformed other algorithms. This is due to the benefits of crossover and mutation operation used by the DE algorithm. Nevertheless, REHHO reports superior performances as compared with other recent algorithms, namely AOA and HHA. The worst results were archived by the BAT optimizer due to the lack of exploration operations used for handling large-scale problems.

Table 11.

Results of 1000 − D CEC’2010 functions

Function	Fitness	Algorithm
		REHHO	BAT	PSO	DE	AOA	HHA
F1	Best Median Worst Mean Std	2.6334e + 09 2.8842e + 09 3.0811e + 09 2.8583e + 09 1.3393e + 08	8.1920e + 11 8.6848e + 11 8.8280e + 11 8.6279e + 11 2.0195e + 10	7.2801e + 10 8.1627e + 10 9.7007e + 10 8.3058e + 10 8.1725e + 09	9.1436e + 09 9.8334e + 09 1.0501e + 10 9.8575e + 09 4.5750e + 08	1.7666e + 11 1.8469e + 11 1.9057e + 11 1.8458e + 11 5.7304e + 09	1.0932e + 11 1.1305e + 11 1.2559e + 11 1.1615e + 11 7.1356e + 0
F2	Best Median Worst Mean Std	1.5968e + 04 1.6255e + 04 1.6685e + 04 1.6285e + 04 221.9340	3.3933e + 04 3.4126e + 04 3.4304e + 04 3.4128e + 04 117.0192	1.8561e + 04 1.9207e + 04 1.9749e + 04 1.9174e + 04 413.7851	1.3561e + 04 1.3724e + 04 1.3825e + 04 1.3714e + 04 83.5222	1.6800e + 04 1.6820e + 04 1.6840e + 04 1.6819e + 04 16.8634	1.6718e + 04 1.6781e + 04 1.7128e + 04 1.6847e + 04 166.2090
F3	Best Median Worst Mean Std	20.2382 20.5327 20.6923 20.5073 0.1428	20.5806 20.6414 20.6582 20.6323 0.0265	21.2729 21.3183 21.3342 21.3146 0.0189	20.6231 20.7270 20.9378 20.7673 0.1058	20.9388 20.9416 20.9454 20.9419 0.0024	20.9921 21.0046 21.0084 21.0031 0.0065
F4	Best Median Worst Mean Std	1.2145e + 13 1.8154e + 13 2.5972e + 13 1.8575e + 13 3.7970e + 12	1.3747e + 16 2.2254e + 16 2.7323e + 16 2.1821e + 16 4.5174e + 15	2.2275e + 14 2.6180e + 14 3.6314e + 14 2.7632e + 14 4.7843e + 13	4.1787e + 14 5.1197e + 14 6.0705e + 14 5.1684e + 14 6.2056e + 13	8.8284e + 14 1.0977e + 15 1.7835e + 15 1.2728e + 15 4.4289e + 14	3.0634e + 14 3.7229e + 14 4.0546e + 14 3.6432e + 14 4.2287e + 13
F5	Best Median Worst Mean Std	4.1407e + 08 4.5136e + 08 4.8347e + 08 4.4886e + 08 2.5081e + 07	1.6805e + 09 1.7357e + 09 1.7998e + 09 1.7385e + 09 4.8769e + 07	3.9186e + 08 4.2848e + 08 4.8385e + 08 4.3203e + 08 2.3246e + 07	4.0647e + 08 4.3649e + 08 4.4413e + 08 4.3096e + 08 1.4678e + 07	4.7870e + 08 6.0494e + 08 6.3414e + 08 5.8409e + 08 6.4493e + 07	6.5631e + 08 7.0156e + 08 7.1827e + 08 6.9519e + 08 2.4010e + 07
F6	Best Median Worst Mean Std	1.9146e + 07 1.9211e + 07 1.9276e + 07 1.9212e + 07 3.9172e + 04	1.9982e + 07 1.9982e + 07 2.0000e + 07 1.9987e + 07 7.7142e + 03	1.2002e + 07 1.3239e + 07 1.4555e + 07 1.3167e + 07 7.2390e + 05	7.3976e + 04 3.2774e + 05 1.4352e + 06 4.9662e + 05 4.4267e + 05	1.9686e + 07 1.9835e + 07 2.0073e + 07 1.9868e + 07 1.7567e + 05	1.7861e + 07 1.9947e + 07 2.0317e + 07 1.9248e + 07 1.2673e + 06
F7	Best Median Worst Mean Std	1.3083e + 09 1.7299e + 09 2.0005e + 09 1.7102e + 09 2.4558e + 08	2.0432e + 13 4.6735e + 13 5.1538e + 14 1.0400e + 14 1.4957e + 14	4.6628e + 10 6.2946e + 10 1.5931e + 11 7.3297e + 10 3.2527e + 10	6.0392e + 10 8.1327e + 10 1.0211e + 11 8.1724e + 10 1.3181e + 10	5.9279e + 11 1.2642e + 12 1.9517e + 12 1.3035e + 12 5.5130e + 11	1.2764e + 11 1.6919e + 11 2.5879e + 11 1.9109e + 11 6.2565e + 10
F8	Best Median Worst Mean Std	4.6868e + 09 6.2015e + 09 1.2757e + 10 6.9842e + 09 2.5889e + 09	1.0082e + 18 1.0712e + 18 1.1125e + 18 1.0662e + 18 3.7550e + 16	3.0385e + 14 4.6240e + 14 9.7152e + 14 5.6671e + 14 2.5688e + 14	2.9923e + 08 4.0148e + 08 5.0636e + 08 3.9180e + 08 6.7384e + 07	2.6765e + 16 3.0181e + 16 3.1786e + 16 2.9738e + 16 1.9547e + 15	3.5186e + 15 5.9064e + 15 8.3792e + 15 5.6005e + 15 2.0058e + 15
F9	Best Median Worst Mean Std	4.3561e + 09 4.5738e + 09 4.8799e + 09 4.5915e + 09 1.6658e + 08	8.9819e + 11 9.2339e + 11 9.3707e + 11 9.2298e + 11 1.1524e + 10	6.6803e + 10 7.5959e + 10 9.1954e + 10 7.6976e + 10 8.8065e + 09	9.7943e + 10 1.0200e + 11 1.1517e + 11 1.0320e + 11 5.0416e + 09	2.0783e + 11 2.2127e + 11 2.2824e + 11 2.2032e + 11 8.2515e + 09	1.2854e + 11 1.3227e + 11 1.4251e + 11 1.3331e + 11 5.5935e + 09
F10	Best Median Worst Mean Std	1.6334e + 04 1.6526e + 04 1.6897e + 04 1.6559e + 04 169.1596	3.6158e + 04 3.6548e + 04 3.6681e + 04 3.6479e + 04 183.8412	1.8758e + 04 1.9327e + 04 1.9866e + 04 1.9264e + 04 432.9254	1.5455e + 04 1.5655e + 04 1.5844e + 04 1.5653e + 04 145.2213	1.6946e + 04 1.7056e + 04 1.7207e + 04 1.7056e + 04 101.6452	1.6843e + 04 1.6944e + 04 1.7287e + 04 1.6998e + 04 172.8607
F11	Best Median Worst Mean Std	221.5103 222.7064 224.8432 222.8913 1.0573	1.5455e + 04 1.5655e + 04 1.5844e + 04 1.5653e + 04 145.2213	223.7311 227.7812 232.7328 228.0433 2.6885	235.0548 235.3412 235.6772 235.3281 0.1939	228.7676 228.8919 229.2603 228.9525 0.1849	229.6113 230.0050 230.3229 229.9827 0.3204
F12	Best Median Worst Mean Std	1.9385e + 06 2.0562e + 06 2.1350e + 06 2.0461e + 06 6.9452e + 04	2.2378e + 09 2.8304e + 09 4.1252e + 09 2.9619e + 09 6.0028e + 08	8.0008e + 06 9.0053e + 06 9.6380e + 06 8.9379e + 06 6.0593e + 05	1.1730e + 07 1.3104e + 07 1.4362e + 07 1.3093e + 07 6.5046e + 05	1.1215e + 07 1.4421e + 07 1.5800e + 07 1.3817e + 07 1.8030e + 06	6.7499e + 06 7.6562e + 06 8.1844e + 06 7.5167e + 06 5.6543e + 05
F13	Best Median Worst Mean Std	1.1077e + 08 1.3137e + 08 1.6626e + 08 1.3441e + 08 1.5571e + 07	1.2709e + 13 1.2883e + 13 1.2938e + 13 1.2866e + 13 6.7440e + 10	4.3093e + 11 5.6566e + 11 7.5955e + 11 5.6360e + 11 9.8735e + 10	8.9269e + 10 1.0461e + 11 1.1624e + 11 1.0477e + 11 7.4351e + 09	6.6945e + 11 6.7833e + 11 6.8313e + 11 6.7714e + 11 5.2862e + 09	6.6277e + 11 6.6784e + 11 6.8071e + 11 6.6928e + 11 7.4645e + 09
F14	Best Median Worst Mean Std	6.2729e + 09 6.6919e + 09 8.0578e + 09 6.7559e + 09 5.0437e + 08	9.0032e + 11 9.2654e + 11 9.3724e + 11 9.2284e + 11 1.2867e + 10	5.8034e + 10 7.0696e + 10 7.9735e + 10 7.0374e + 10 6.6447e + 09	1.5575e + 11 1.6093e + 11 1.6536e + 11 1.6057e + 11 3.2319e + 09	2.3567e + 11 2.3947e + 11 2.6200e + 11 2.4443e + 11 1.0479e + 10	1.4172e + 11 1.4433e + 11 1.6292e + 11 1.4773e + 11 8.6053e + 09
F15	Best Median Worst Mean Std	1.6416e + 04 1.6534e + 04 1.6927e + 04 1.6568e + 04 145.8575	3.4679e + 04 3.4896e + 04 3.5081e + 04 3.4896e + 04 116.2722	1.8985e + 04 1.9517e + 04 1.9770e + 04 1.9503e + 04 217.1941	1.6483e + 04 1.6919e + 04 1.7140e + 04 1.6870e + 04 230.8942	1.6628e + 04 1.6770e + 04 1.6945e + 04 1.6788e + 04 115.2534	1.6793e + 04 1.6952e + 04 1.7109e + 04 1.6962e + 04 137.9735
F16	Best Median Worst Mean Std	404.9208 409.7903 410.9187 409.1372 1.7926	411.5244 412.4078 413.3157 412.4006 0.5727	420.8667 423.6429 425.6089 423.4740 1.6473	428.9731 429.2859 429.5783 429.2628 0.1800	416.3841 416.8878 417.0327 416.7921 0.2488	418.8719 419.4299 419.5139 419.3333 0.2649
F17	Best Median Worst Mean Std	2.7638e + 06 2.9555e + 06 3.1423e + 06 2.9464e + 06 1.1326e + 05	4.8625e + 09 6.3958e + 09 7.7926e + 09 6.3036e + 09 9.4376e + 08	1.3717e + 07 1.5410e + 07 1.6907e + 07 1.5218e + 07 9.9786e + 05	2.4597e + 07 2.7451e + 07 3.0764e + 07 2.7800e + 07 1.9342e + 06	3.2326e + 07 4.1700e + 07 5.2378e + 07 4.0235e + 07 8.1691e + 06	1.3599e + 07 1.5993e + 07 1.6212e + 07 1.5448e + 07 1.0856e + 06
F18	Best Median Worst Mean Std	2.4614e + 09 2.7451e + 09 3.1012e + 09 2.7693e + 09 2.0060e + 08	2.5509e + 13 2.5680e + 13 2.5809e + 13 2.5679e + 13 1.1376e + 11	2.6656e + 12 2.9985e + 12 3.4511e + 12 3.0392e + 12 2.2922e + 11	7.8896e + 11 8.5376e + 11 9.0947e + 11 8.4958e + 11 3.9012e + 10	1.4444e + 12 1.4519e + 12 1.4572e + 12 1.4513e + 12 4.6140e + 09	1.4367e + 12 1.4437e + 12 1.4579e + 12 1.4450e + 12 8.8728e + 09
F19	Best Median Worst Mean Std	7.4615e + 06 1.0164e + 07 1.3475e + 07 1.0020e + 07 1.7817e + 06	4.4343e + 10 2.1524e + 11 1.0737e + 12 3.0652e + 11 3.0170e + 11	2.5408e + 07 3.0163e + 07 3.3165e + 07 3.0359e + 07 2.2188e + 06	4.3615e + 07 5.4726e + 07 6.0344e + 07 5.3491e + 07 4.7828e + 06	4.5429e + 07 5.7205e + 07 1.0451e + 08 6.9834e + 07 2.5767e + 07	3.2616e + 07 4.0733e + 07 4.2488e + 07 3.8587e + 07 4.0426e + 06
F20	Best Median Worst Mean Std	2.7733e + 09 3.0091e + 09 3.2906e + 09 3.0219e + 09 1.8120e + 08	2.6682e + 13 2.6791e + 13 2.6862e + 13 2.6773e + 13 5.8015e + 10	2.6541e + 12 3.4566e + 12 3.8441e + 12 3.3432e + 12 3.6053e + 11	8.2017e + 11 8.4314e + 11 9.2415e + 11 8.5322e + 11 3.6249e + 10	1.6122e + 12 1.6327e + 12 1.6350e + 12 1.6288e + 12 9.4358e + 09	1.6243e + 12 1.6272e + 12 1.6453e + 12 1.6318e + 12 9.0083e + 09

Bold values indicate the best mean value (i.e., average minimum)

Statistical analysis

In this section, the statistical t-test [47] was used to evaluate the outcome of the 1000-D CEC’2010 large-scale benchmark statistically. The t-test is used to determine if there is a significant difference between REHHO and other algorithms in terms of the reported mean value. To compute the significant difference using t- test, three values are required, which are the mean, the standard deviation, and the number of data (repeated times). From these computed values, the degrees of freedom and t-distribution value will be identified to generate the p-value.

To implement the t-test, the null hypothesis $H_0$ assumes that REHHO and other compared algorithms performed equally. However, the alternative hypothesis $H_1$ assumed that REHHO outperformed other algorithms. In this analysis, the p-value of the t-test was set at 0.05 (i.e., 95% confidence level), meaning that the alternative hypothesis $H_1$ would be accepted if the p-value was less than 0.05. The p-value results of the t-test are reported in Table 12, and it can be seen that most p-values were less than 0.05, which confirms that the proposed REHHO algorithm significantly outperforms other conducted algorithms in most of the functions.

Table 12.

The p − values of statistical t − test

Function	HHO	REHHO	BAT	PSO	DE	AOA	HHA
F1	0.0049		0.0000	0.0000	0.0000	0.0000	0.0000
F2	0.0036		0.0000	0.0000	0.1725	0.0000	0.0000
F3	0.0015		0.0000	0.0000	0.0002	0.0000	0.0000
F4	0.0140		0.0000	0.0000	0.0000	0.0000	0.0000
F5	0.0000		0.0000	0.1370	0.9669	0.0000	0.0000
F6	0.0672		0.0000	0.0000	0.9004	0.0000	0.0000
F7	0.0021		0.0412	0.0000	0.0000	0.0000	0.0000
F8	0.0025		0.0000	0.0000	0.3267	0.0000	0.0000
F9	0.0004		0.0000	0.0000	0.0000	0.0000	0.0000
F10	0.0012		0.0000	0.0000	0.4640	0.0000	0.0000
F11	0.0048		0.0277	0.0000	0.0000	0.0000	0.0000
F12	0.0023		0.0000	0.0000	0.0000	0.0000	0.0000
F13	0.0008		0.0000	0.0000	0.0000	0.0000	0.0000
F14	0.0020		0.0000	0.0000	0.0000	0.0000	0.0000
F15	0.0012		0.0000	0.0000	0.0026	0.0000	0.0000
F16	0.0003		0.0000	0.0000	0.0000	0.0000	0.0000
F17	0.0008		0.0000	0.0000	0.0000	0.0000	0.0000
F18	0.0033		0.0000	0.0000	0.0000	0.0000	0.0000
F19	0.0021		0.0048	0.0000	0.0000	0.0000	0.0000
F20	0.0007		0.0000	0.0000	0.0000	0.0000	0.0000

Bold values indicate the best mean value (i.e., average minimum)

Computational time analysis

The computational time of the proposed REHHO is compared against HHO on large-scale problems. This analysis is conducted to measure the overhead of the embedded rules. The hardware and software specifications of the adopted PC are given in Table 13. The average computational time for 30 runs of the F1 function of large-scale problems is computed and reported in Table 14. As can be seen, REHHO required extra time due to the calculation of population status and agent location needed to fire the embedded rules. Nevertheless, the computational time overhead consumed by these rules is still affordable, which is 34 s only. This value represents around 10% of the total time needed by HHO.

Table 13.

The detailed settings of the PC

Item	Settings
CPU	i7 − 8700
Frequency	3.2 GHz
RAM	32 GB
Hard drive	512 GB SSD
Operating system	Windows 10
Language	MATLAB 2021a

Table 14.

Computational time analysis

	HHO	REHHO
Time (second)	324	358

NIR wavelength selection

This section evaluates the performances of REHHO in performing wavelength selection of the NIR spectrum of gasoline [48]. This case study contains 60 samples with wavelength (x-axis) of range from 900 to 1700 nm and intervals of 2 nm, which result in 401 channels/per sample. The values on the y-axis represent the amount of absorbed heat, as shown in Fig. 14. The dataset has been divided into 50% for training (30 samples) and 50% for testing.

Fig. 14.

NIR gasoline spectrum

To encode this problem, a 1D binary vector of size 401 is given in Fig. 15. As such, optimization algorithms were applied to find the most distinguished wavelengths. As can be seen in Fig. 15, when the value of the corresponding wavelength $W$ is set to 1, then it will be selected; otherwise, it will be skipped.

Fig. 15.

Encoding scheme of NIR wavelength selection

Conducted optimization algorithms are guided by the accuracy of Partial Least Squares (PLS) regressor and complexity measure. The fitness function of this problem is formulated as follows.

$$\text{fitness}=-1\ast \left(\left(0.9\text{*Accuracy}\right)-\left(0.1\text{*complexity}\right)\right)$$ 23

$$\text{Accuracy}=1-\frac{\sqrt{\sum _{i=1}^N{\left(y_i-{\widehat{y}}_i\right)}^2}}{\sqrt{\sum _{i=1}^N{\left(y_i-\overline{y_i}\right)}^2}}$$ 24

$$\text{Complexity}=\frac{\text{number}\text{of}\text{selected}\text{wavelengths}}{401}$$ 25

where $y_i$ is the actual value of absorbed heat, ${\widehat{y}}_i$ is the predicted value, and $\overline{{\mathit{y}}_{\mathit{i}}}$ is the mean value of the training set. For each algorithm, the mean fitness value, accuracy of testing set, number of selected channels are reported in Table 15. It can be seen that REHHO was able to reduce PLS complexity by using only 17 wavelengths. In terms of accuracy on the testing dataset, the best results were achieved by REHHO with 96.3%. Furthermore, the proposed REHHO was able to produce the best fitness value. The worst results have been reported by the BAT algorithm due to its lack of exploration ability as compared with other algorithms.

Table 15.

Results of NIR channels selection

Parameters/algorithm	Number of selected wavelengths	Test accuracy (%)	Fitness
HHO	25	95.1	− 0.8895
REHHO	17	96.3	− 0.8918
BAT	169	92.5	− 0.8579
PSO	151	93.1	− 0.8623
DE	87	93.8	− 0.8601
AOA	55	94.5	− 0.8783
HHA	67	94.2	− 0.8721

Bold values indicate the best mean value (i.e., average minimum)

Conclusion

This work presents a novel REHHO algorithm that improves HHO by embedding several rules. The effectiveness of REHHO has been evaluated with a total of six standard high-dimensional functions ranging from 1000-D to 10,000-D, CEC’2010 large-scale benchmark, and the problem of NIR wavelength selection. Reported results indicated that REHHO was able to outperform HHO and other state-of-the-art optimization algorithms, including BAT, PSO, DE, AOA, and HHA. From the statistical analysis of the results, the t-test showed that REHHO significantly outperformed other algorithms with a 95% confidence level. As future work, REHHO could be applied for features selection problems and other real-world, large-scale problems.

Funding

Al-Imam Mohammad Ibn Saud Islamic University, 18-11-08-004, Ali Sama.

Declarations

Conflict of interest

Hussein Samma and Ali Salem Bin Sama declare that they have no conflict of interest.