Multi-Objective Harris Hawks Optimization: Principles, Variants, Applications, and Future Directions

Abstract

Multi-objective optimization problems (MOPs) are common in practical scenarios where decision-makers need to accomplish several competing goals. Single-objective optimization techniques do not guarantee applicability in these scenarios. As such, there has been a need for the development of metaheuristics capable of generating multiple trade-off solutions. Harris Hawks Optimization (HHO) has been shown to possess strong exploration and exploitation capabilities for the solution of optimization problems, owing to the collaborative hunting tactics of Harris’s hawks. Therefore, the Multi-objective Harris Hawks Optimization (MHHO) algorithm was suggested to generalize HHO to handle MOPs. By combining the mechanisms of Pareto dominance, diversity preservation, elitism, adaptiveness, and others, MHHO approaches the Pareto-optimal front and provides decision-makers with several high-quality nondominated solutions. This study comprehensively examines MHHO, elaborating on its theoretical background, algorithmic variants, and fields of application. MHHO has been implemented in different disciplines. Using the Scopus database to conduct a bibliometric study, the publication growth, research development, and the application of MHHO in various fields of study were analyzed. By classifying the extant contributions into original, modified, and hybrid versions, the study provides a detailed outline of the algorithm’s progression. Applications spanning engineering, cloud computing, scheduling, networking, bioinformatics, and energy systems are analyzed, illustrating the broad adaptability of MHHO. A constructive critique has been conducted to evaluate some limitations including premature convergence, scalability issues, and difficulty in addressing disconnected Pareto regions. This review shows the versatility and potential of MHHO in tackling different optimization problems. In addition, further research is needed on the development of more sophisticated hybrid methods, tailored improvements, and more refined techniques for the preservation of diversity.

1. Introduction

In the field of optimization, the essence centers around analyzing the best possible allocation of decision variables to achieve the minimization or maximization of an objective function. Subsequently, the search space is determined by the decision variables and objective functions. The decision variables could be binary, discrete, continuous, permutation-based, or structured [1]. The objective functions could be characterized as convex or non-convex, linear or non-linear, bounded or unbounded, and unimodal or multimodal [2]. When considering a rise in problem dimensionality and the size of the search space, there tends to be an exponential increase in the degree of difficulty associated with the optimization problem at hand. Most state-of-the-art practical problem domains involve, to some degree, the phenomenon of multiple simultaneous objectives, referred to as multi-objective optimization problems (MOPs). This is a recurring theme across fields of engineering [3,4], cloud computing [5,6], data mining [7,8], scheduling and planning [9,10,11], networking and communication [12], energy systems [13], software engineering [14], and bioinformatics [15], to name but a few.

MOPs include multiple objectives that are conflicting, complicating efforts to optimize and balance them in a single solution. Engineering problems often include a trade-off between cost, performance, and reliability [16]. While single-objective problems focus on obtaining a single optimal solution that defines the optimum value of the objective function, MOPs aim to identify a set of optimal trade-off solutions among multiple competing objectives, forming the Pareto optimal set. This set illustrates the most balanced trade-offs among each conflicting objective [17]. It offers a variety of options for the decision-maker to consider, enabling the selection of the most preferred solution in accordance with the decision-maker’s objectives.

The most common methods used to tackle MOPs have historically been classified into two classes: mathematical programming (scalarization-based) methods and evolutionary (population-based) methods. Scalarization-based methods, such as weighted sums, e-constraints, goal programming, and reference point methods [18], reformulate the multi-objective problem as one or a series of single-objective problems that can be solved by classical optimization techniques. They are computationally efficient but typically yield a single trade-off solution per run and struggle with disconnected, non-convex, or high-dimensional Pareto fronts. In contrast, evolutionary and metaheuristic methods maintain a population of candidate solutions and generate multiple non-dominated solutions in a single run, with attention to convergence, diversity, and solution distribution [19]. Pareto-based evolutionary methods are the most widely adopted and serve as benchmarks for MO techniques, alongside non-Pareto and elitist approaches [18]. Since MHHO is itself a swarm-based metaheuristic, the present review focuses on Pareto-based and metaheuristic approaches to MOPs.

Among swarm intelligence algorithms, Harris Hawks Optimization (HHO) stands out as a promising novel method, as it draws inspiration from the strategy used by Harris’ hawks when engaging in cooperative hunting. Although it was first introduced as a novel method for single-objective continuous optimization, HHO has exhibited strong capabilities in exploration and exploitation, which stem from the dynamic adjustments made between soft and hard besiege strategies. It, therefore, models the strategy-adjusting behaviors of the hawks, providing an approximate equilibrium between diversification and intensification. While an individual hawk can only capture a single prey, cooperatively hunting hawks can share position information, thus increasing the overall problem-solving capabilities of the method. Owing to these principles, HHO has been used in a wide range of research areas, including engineering design, image analysis, scheduling, bioinformatics, and more.

In response to challenges that entail several conflicting goals, the Multi-objective Harris Hawks Optimization (MHHO) method has been proposed. Rather than searching for a single solution, MHHO seeks to construct multiple non-dominated solutions that approximate the Pareto front, thereby providing decision-makers with candidate trade-offs among the objectives, although the proximity of these approximations to the true Pareto front remains an open empirical question.

The performance of MHHO is dependent on a number of factors. If it is unable to sufficiently explore the search space with respect to the associated Pareto front, it is possible for the algorithm to converge prematurely. In expansive search domains, limitations in the range of Pareto-optimal solutions, both in terms of diversity and coverage, can result in the algorithm demonstrating poor performance due to issues related to scalability, and optimization regions that differ in shape and are separated can exacerbate issues associated with convergence by creating challenges for the hawks’ adaptive mechanisms to locate the optimal fronts. There can be irregularities in the objective landscape, where discontinuities can decrease the quality of solution evaluation. To address these issues, a number of modified MHHO implementations have been created, such as problem-specific heuristics, diversity preservation methods, adaptive transition strategies for hawks, and MHHO hybridized with other metaheuristic algorithms.

This work is a critical review that combines structured synthesis of the MHHO literature with a dispassionate analytical evaluation of its theoretical foundations, algorithmic variants, and reported performance. Rather than enumerating MHHO publications, it aims to synthesize what the body of work collectively teaches about the algorithm and to assess it on independent methodological grounds. The main contributions of the paper are as follows:

• Provides the first bibliometric mapping of MHHO research (publication growth, subject areas, affiliations, countries, and source types), to establish an empirical basis for tracking the field’s maturation.
• Introduces a taxonomic classification of MHHO contributions into original, modified, and hybridized variants, and uses this taxonomy to compare design choices.
• Performs a critical comparative analysis that identifies recurring enhancement mechanisms and the targeted problem characteristics, and the trade-offs.
• Extracts cross-cutting methodological patterns and pitfalls of MHHO theory, including premature convergence, scalability under many-objective settings, parameter sensitivity, and constraint-handling weaknesses.

The remainder of the paper is organized as follows. Section 2 presents the bibliometric analysis; Section 3 formalizes the theoretical basis of MHHO; Section 4 reviews and critically compares algorithmic variants; Section 5 organizes applications domain by domain; Section 6 critically analyzes the theory and limitations of MHHO; and Section 7 concludes the study with open research directions.

2. The Growth of Multi-Objective Harris Hawks Optimization

The emergence and development of MHHO from its introduction until 2025 are outlined in this study. The original HHO algorithm was first proposed in 2019, and its multi-objective variant was later introduced to extend its applicability to problems with conflicting goals. Since then, MHHO has gained increasing attention from the optimization community and has been employed in diverse research areas to address MOPs.

According to the Scopus database, researchers have published a growing number of contributions on MHHO in peer-reviewed journals, international conferences, and book series, reflecting its growing relevance and adoption across multiple disciplines. The literature search was performed using combinations of keywords related to “Multi-objective Harris Hawks Optimization”, “MHHO”, and multi-objective optimization variants of HHO. The search was conducted in 2025 and was restricted to journal articles, conference papers, review papers, and book chapters published in English. Duplicate records were removed, and manual screening was subsequently performed to exclude unrelated studies and papers not directly focused on MHHO or multi-objective HHO applications.

Figure 1 illustrates the yearly publication trend on MOHHO. The greatest number of publications, 12 articles, occurred in 2023. The years 2022 and 2025 followed, with 8 publications each. In 2024, there was a slight decline, with a total of 7 publications. The years 2019, 2020, and 2021 recorded fewer publications, with 2, 5, and 4 publications, respectively.

The publication records by subject area are shown in Figure 2. The greatest contributors are Computer Science (34 publications), Engineering (21), and Mathematics (10). In Energy and Materials Science, there are 9 and 5 publications, respectively, while Decision Sciences, Medicine, Physics and Astronomy, Chemistry, Social Sciences, and a few other areas are represented with lower contributions (between 2 and 3 publications).

Figure 3 demonstrates the publications by affiliation. Universiti Utara Malaysia has the greatest number of published articles: 4. Following closely are Université des Sciences et de la Technologie Houari Boumediene and Université Mohamed El Bachir El Ibrahimi de Bordj Bou Arréridj with 3 publications each. Islamic Azad University (Qazvin Branch), Deakin University, Chongqing University, Université Abderrahmane Mira – Béjaïa, Xi’an Institute of Posts and Telecommunications, International Institute of Information Technology (Bhubaneswar), and Sunway University are among the many institutions that have published 2 articles each.

Figure 4 shows how many publications there are from each country. The largest contributor is China with 12 publications, followed by India with 11 publications. In total, there are 8 publications from Iran, 7 from Malaysia, and 5 from Algeria. Saudi Arabia has 4 publications, and Australia, Egypt, and Turkey have 3 publications each. The Czech Republic has 2 publications, which are the lowest on the list. These numbers show how much more Asian countries, particularly China and India, dominate the research being done on MOHHO.

The breakdown of publications by source type is illustrated in Figure 5. Journals lead as the primary publication outlet, with 38 articles, constituting the largest percentage of the total output. Conference proceedings account for 8 publications, and book series are the least represented, with only 1 publication. This distribution illustrates the preference of researchers for disseminating their MOHHO findings via journals.

3. Basic Concepts of Multi-Objective Harris Hawks Optimization

Section 3.1 formalizes the MO setting used throughout this study, while Section 3.2 describes the procedural steps of the proposed MHHO in detail.

3.1. Multi-Objective Optimization

MO addresses problems in which several objectives must be optimized simultaneously. Let $y\in {\mathbb{R}}^d$ denote a decision vector with lower and upper bounds $\mathsf{\ell },u\in {\mathbb{R}}^d$ such that $\mathsf{\ell }\le y\le u$ (componentwise). The general maximization form considered in this work is

$$\begin{array}{cc}\hfill \mathrm{maximize}& F\left(y\right)\triangleq \left(f_1\left(y\right),f_2\left(y\right),\dots ,f_M\left(y\right)\right),\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \mathrm{subject}\mathrm{to}& g_i\left(y\right)\le 0,i=1,\dots ,m,\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & h_j\left(y\right)=0,j=1,\dots ,p,\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & \mathsf{\ell }\le y\le u.\hfill \end{array}$$

(4)

where M is the number of objective functions, d is the number of decision variables, m and p are the numbers of inequality and equality constraints, respectively, and the feasible region is

$$\mathrm{\Omega }\triangleq \left\{y\in {\mathbb{R}}^d|\left(2\right)–\left(4\right)\mathrm{hold}\right\}.$$

(Problems in minimization form are handled analogously or by sign changes to ${\left\{f_r\right\}}_{r=1}^M$.)

• Dominance and partial ordering: Unlike single-objective settings, where candidates can be totally ordered by a scalar value, MO relies on a partial order induced by Pareto dominance. For two feasible solutions $x,y\in \mathrm{\Omega }$, we can say that y (strictly) dominates x, written $y\succ x$, if

  $$f_r\left(y\right)\ge f_r\left(x\right)\forall r\in \{1,\dots ,M\}\mathrm{and}\exists s\in \{1,\dots ,M\}:f_s\left(y\right)>f_s\left(x\right).$$

  (5)

  In words, y is at least as good as x in every objective and strictly better in at least one objective. This relation provides a principled way to compare solutions without collapsing multiple criteria into a single aggregate score.

• Pareto optimality, set, and front: A feasible solution $y^{\star }\in \mathrm{\Omega }$ is Pareto-optimal if there exists no other $x\in \mathrm{\Omega }$ such that $x\succ y^{\star }$. The Pareto-optimal set (also called the set of non-dominated solutions) is

  $${\mathcal{P}}^{\star }\triangleq \left\{y\in \mathrm{\Omega }|\nexists x\in \mathrm{\Omega }\mathrm{with}x\succ y\right\},$$

  (6)

  and its image in the objective space,

  $${\mathcal{F}}^{\star }\triangleq \left\{F\left(y\right)\in {\mathbb{R}}^M|y\in {\mathcal{P}}^{\star }\right\},$$

  (7)

  is the Pareto front. Practical algorithms therefore aim to (i) discover non-dominated solutions that approximate ${\mathcal{P}}^{\star }$ and (ii) preserve diversity along ${\mathcal{F}}^{\star }$ to represent the trade-offs among conflicting objectives. These principles underpin the design of our MHHO procedure described in Section 3.2.
• In constrained MOPs, MHHO variants commonly employ several constraint-handling techniques to ensure solution feasibility during the search process. The most frequently used approaches include penalty functions, feasibility rules, repair operators, epsilon-constraint methods, and constraint-domination mechanisms. Penalty-based methods incorporate constraint violations into the objective evaluation process, while repair operators adjust infeasible solutions to satisfy problem constraints. Constraint-handling strategies in MHHO can generally be categorized into soft and hard approaches. Soft constraint handling allows infeasible solutions to participate in the search process through penalty-based evaluation, thereby promoting exploration near constraint boundaries. In contrast, hard constraint handling strictly enforces feasibility by rejecting infeasible solutions or applying repair mechanisms during the optimization process. Hard approaches are frequently employed in highly constrained engineering applications where violating operational or safety limits is unacceptable. In several engineering and power-system applications, adaptive penalty strategies and feasibility-preserving operators were integrated with archive management and non-dominated sorting to improve convergence toward feasible Pareto-optimal regions. Nevertheless, effective constraint handling in large-scale and highly constrained optimization problems remains an open challenge for MHHO research.

3.2. Harris Hawks Optimization (HHO)

HHO stands for Harris Hawks Optimization, which is a population-based metaheuristic technique based on the cooperative hunting strategies of Harris’ hawks. They have a unique hunting style characterized by sudden coordinated attacks on their prey. The technique was proposed and formalized by [20]. In this section, the biological reasoning, the algorithmic abstraction, and the main phases of the algorithm, which include exploration and exploitation, are presented.

3.2.1. Exploration Phase

In the wild, Harris’s hawks locate and trail prey via keen vision; detection can be prolonged and uncertain. In HHO, each hawk encodes a candidate solution, and its location corresponds to that candidate’s decision vector. The hawk closest to the prey represents the best-so-far solution. The initial positions are sampled at random across the search domain, and the agents scout using two equiprobable strategies: (i) perching near kin (family grouping) to facilitate cooperative attacks, or (ii) perching randomly on tall trees within their territory.

Let $q\sim \mathcal{U}(0,1)$ denote the strategy selector. With $r_1,r_2,r_3,r_4\sim \mathcal{U}(0,1)$ and a randomly chosen peer $X_{\mathrm{rand}}\left(t\right)$, positions are updated as

$$X(t+1)=\left\{\begin{array}{cc}{X}_{\mathrm{rand}}\left(t\right)-r_1|X_{\mathrm{rand}}\left(t\right)-2r_{2}X\left(t\right)|,\hfill & q\ge 0.5,\hfill \\ \left(X_{\mathrm{prey}}\left(t\right)-X_m\left(t\right)\right)-r_3\left(LB+r_4(UB-LB)\right),\hfill & q<0.5,\hfill \end{array}\right.$$

(8)

where $X\left(t\right)$ is the current position; $X_{\mathrm{prey}}\left(t\right)$ is the prey estimate; $r_1,r_2,r_3,r_4$ and q are random numbers uniformly distributed in $[0,1]$; and $LB,UB$ are componentwise bounds of the search region. The population mean

$$X_m\left(t\right)={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N}{X}_i\left(t\right)$$

(9)

aggregates all N hawk positions at iteration t.

3.2.2. Transition from Exploration to Exploitation

Effective search requires a smooth shift from global scouting to local intensification. In HHO, this transition is governed by the prey’s escaping energy

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

(10)

where $E_0\sim \mathcal{U}(-1,1)$, t is the current iteration, and T is the maximum number of iterations. As $E_0$ moves from 0 toward $-1$, the prey weakens; as it moves from 0 toward 1, the prey strengthens. The phase selection follows:

$$\left|E\right|\ge 1\Rightarrow \mathrm{exploration},\left|E\right|<1\Rightarrow \mathrm{exploitation}.$$

(11)

3.2.3. Exploitation Phase

Once the prey is located, the hawks launch an ambush while the prey attempts evasive maneuvers. Four chase–capture patterns are modeled: (i) soft besiege, (ii) hard besiege, (iii) soft besiege with progressive rapid dives, and (iv) hard besiege with progressive rapid dives. Let $r\sim \mathcal{U}(0,1)$ denote the prey’s success probability of evasion in the current encounter. The besiege type is chosen using r together with E (energy level): when $\left|E\right|\ge 0.5$, a soft besiege is attempted; otherwise a hard besiege is executed. Rapid dives (via Lévy flights) are invoked when earlier moves fail to yield improvement.

1.

Soft besiege:

When $r\ge 0.5$ and $\left|E\right|\ge 0.5$, the prey still retains enough energy to flee. The hawks tighten the encirclement to drain this energy before striking:

$$X(t+1)=∆X\left(t\right)-E|J{X}_{\mathrm{prey}}\left(t\right)-X\left(t\right)|,$$

(12)

where

$$∆X\left(t\right)=X_{\mathrm{prey}}\left(t\right)-X\left(t\right),$$

(13)

and

$$J=2\left(1-r_5\right),r_5\sim \mathcal{U}(0,1),$$

(14)

models a stochastic step of the prey during escape. J denotes the random jump strength of the prey.

2.

Hard besiege

If $r\ge 0.5$ but $\left|E\right|\le 0.5$, the prey is fatigued and cannot effectively escape. The hawks contract the ring aggressively:

$$X(t+1)=X_{\mathrm{prey}}\left(t\right)-E|∆X\left(t\right)|.$$

(15)

Figure 6 illustrates the hard besiege behavior, where the hawks aggressively shrink the search region around the prey once its escaping energy becomes weak.

3.

Soft besiege with progressive rapid dives

When $\left|E\right|\ge 0.5$ and $r\le 0.5$, a soft besiege is maintained but augmented with rapid, irregular dives to counter the prey’s zigzag escape. First, a candidate move is evaluated:

$$Y=X_{\mathrm{prey}}\left(t\right)-E|J{X}_{\mathrm{prey}}\left(t\right)-X\left(t\right)|.$$

(16)

If this move is not promising, the team performs Lévy-flight-based (LF) dives:

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

(17)

where D is the problem dimension and $S\in {\mathbb{R}}^{1\times D}$ is a random vector. The Lévy flight is

$$LF\left(x\right)=0.01\times {\displaystyle \frac{u\times \sigma }{{\left|v\right|}^{1/\beta }}},\sigma ={\left({\displaystyle \frac{\mathrm{\Gamma }(1+\beta )\sin \left(\frac{\pi \beta }{2}\right)}{\mathrm{\Gamma }\left(\frac{1+beta}{2}\right)\beta 2^{(\beta -1)/2}}}\right)}^{1/\beta },$$

(18)

with $u,v\sim \mathcal{U}(0,1)$ and $\beta =1.5$. The update accepts the better of the two candidates:

$$X(t+1)=\left\{\begin{array}{cc}Y,\hfill & \mathrm{if}F\left(Y\right)<F\left(X\left(t\right)\right),\hfill \\ Z,\hfill & \mathrm{if}F\left(Z\right)<F\left(X\left(t\right)\right).\hfill \end{array}\right.$$

(19)

Figure 7 demonstrates the soft besiege strategy combined with progressive rapid dives, where Lévy-flight perturbations enhance exploration around the prey location and help the algorithm escape local optima.

4.

Hard besiege with progressive rapid dives

Finally, when $\left|E\right|\le 0.5$ and $r\le 0.5$, the prey is largely exhausted. The hawks enforce a hard besiege and apply rapid dives to finalize the capture. The acceptance rule mirrors (19):

$$X(t+1)=\left\{\begin{array}{cc}Y,\hfill & \mathrm{if}F\left(Y\right)<F\left(X\left(t\right)\right),\hfill \\ Z,\hfill & \mathrm{if}F\left(Z\right)<F\left(X\left(t\right)\right),\hfill \end{array}\right.$$

(20)

with

$$Y=X_{\mathrm{prey}}\left(t\right)-E|J{X}_{\mathrm{prey}}\left(t\right)-X_m\left(t\right)|,$$

(21)

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

(22)

Figure 8 presents the hard besiege with progressive rapid dives, where exploitation and stochastic dives are simultaneously employed to accelerate convergence toward high-quality solutions while preserving limited search diversity.

4. Recent Variants of Multi-Objective Harris Hawks Optimization

As highlighted earlier, this survey reviewed a total of 49 research papers related to the MHHO. These studies were classified into three major categories: original, modified, and hybridized MHHO. The original MHHO was directly applied to solve various optimization problems. In the modified MHHO, researchers proposed enhanced versions of the algorithm to improve its performance on specific problem domains. The remaining studies introduced hybridized versions of MHHO, where it was integrated with other algorithms or techniques to address complex optimization tasks. Figure 9 illustrates the distribution of research papers across these three categories.

4.1. Original Multi-Objective Harris Hawks Optimization

In consideration of basin runoff management and the implications of climate change, Dougaheh and Ashofteh [21] combined the MHHO algorithm and the SWMM simulator to formulate optimal Low-Impact Development (LID) techniques. This research evaluated the optimal distribution of four LID techniques (vegetative swale, bio-retention cell, permeable pavement, and infiltration trench) across the sub-basins of the Tehran Municipality under baseline, near-future, and far-future rainfall conditions. The aims were to keep both the basin outflow and the implementation and maintenance costs at a minimum. Using rainfall scenarios RCP2.6, RCP4.5, and RCP8.5, the model created Pareto-optimal solutions, confirming that MHHO achieved wider solution dispersion and lower runtime compared to NSGA-II. The results of the research revealed that runoff minimization could reduce flood volumes by up to 50%, and under flood mitigation scenarios, cost prioritization could reduce flood volumes by approximately 30%. This effectively illustrates the economic and hydrological practicality of MHHO-based LID planning.

In the field of power distribution systems, Gogula and Vakula [22] used the MHHA to identify the optimal locations and sizes of distributed generating units. The focus was to reduce active power loss, lessen the impact on the environment, and improve the voltage profile of radial distribution systems. The approach succeeded in locating optimal distributed generation units by incorporating the bus voltage and branch power loss in the fitness function.

Considering emergency vehicle dispatching during pandemics, Khennak et al. [23] applied dual MOPs for cost and time-critical treatment case sufficiency, and optimized the travel distance for patient-ambulance assignments. To solve the problem, the MHHO algorithm was applied. In comparison with an enhanced PSO variant, MHHO demonstrated better solution quality and response time in experiments conducted on two synthetic datasets.

Hossain et al. [24] developed a two-hop routing protocol based on MHHO for cognitive radio vehicular ad hoc networks (CR-VANET). The protocol is designed to address route stability, reliability in data dissemination, and the selection of forwarders for the two hops between source and destination nodes. The authors report that this approach is well-suited to the highly dynamic conditions of CR-VANET environments. In simulations using OMNeT++ and SUMO, the proposed method was reported to outperform established routing baselines in terms of throughput, packet delivery ratio, latency, packet loss, and communication overhead.

For unbalanced radial distribution systems, Babu et al. [25] examined the optimal placement of renewable energy sources and electric vehicle charging stations using Harris Hawks Optimization. By setting the objectives of the algorithm under operational constraints to minimize active and reactive power losses and to enhance voltage profiles, optimal siting strategies were identified for distributed generators and electric vehicle charging stations. The results from the IEEE 25-bus test network showed that HHO successfully reduced the losses and improved the stability of the system, surpassing many heuristic approaches in convergence and loss minimization.

Considering the need to balance photovoltaic (PV) output maximization and the requirements of frequency regulation (FR), Wang et al. [26] developed a PV reconfiguration strategy that couples with an energy storage system to manage the discrepancies between generated energy and FR signals. The author modeled the optimization problem using MHHO, which captures the optimal Pareto front representing the trade-offs between PV output maximization and FR requirements. The author used the VIKOR method to select a compromise solution. The author’s case studies with partial shading and dynamic FR signals showed that the proposed method effectively captures and manages the discrepancies.

In the railway industry, Choo et al. [27] incorporated MHHO into the scheduling of rolling stock maintenance. With the use of real operational data, the optimizer was able to assist in maintenance planning, improving efficiency in an Industry 4.0 environment where data from ongoing maintenance are provided by sensors and automation. The findings validated that MHHO is a useful and viable method for addressing the multi-objective optimization of train maintenance scheduling in comparison to other methods.

Regarding operational challenges and environmental concerns in power systems, Islam et al. [28] employed HHO for both single- and multi-objective Optimal Power Flow (OPF), with fuel cost, power losses, and emissions considered as objectives. Multi-objective formulations were addressed through a weighted-sum approach, and the algorithm was evaluated on the IEEE 30-bus test system. Within this test setting, the method yielded reductions in system losses and emissions and, in the emission-aware OPF case, outperformed WOA, SSA, MFO, and GWO on the considered scalar metrics.

DeBruyne and Kaur [29] introduced the first MHHO method applicable to reference point-based problems. The method was based on hawks’ cooperative hunting and was adapted from MOGWO. The focus was on clustering the solutions around user-defined reference points. MHHO showed high-quality solutions obtained with a predator–prey model and significantly better convergence.

The application of MHHO has also been extended to smart grids. Pop et al. [30] proposed an MHHO-based framework for coordinating the charging and discharging of electric vehicles to stabilize a local grid. The model schedules energy storage and release while accounting for user preferences and location constraints. Within the tested evaluation scenarios, the proposed approach achieved competitive convergence behavior, Pareto diversity, and load-balancing performance.

Table 1. Summary of the original MHHO studies.

Study	Variant	Objectives	Advantages	Limitations
Dougaheh and Ashofteh [21]	MHHO + SWMM	Basin outflow; LID implementation and maintenance costs	Wider solution dispersion and lower runtime than NSGA-II	Restricted to Tehran sub-basins and given climate scenarios
Gogula and Vakula [22]	MHHO	Active power loss; environmental impact; voltage profile	Reduced active power loss by 68.78%; improved voltage stability	Tested only on 69-bus and 118-bus systems
Khennak et al. [23]	MHHO	Treatment case sufficiency; travel distance	Better solution quality and response time than enhanced PSO	Validated on two synthetic datasets only
Hossain et al. [24]	MHHO (2-hop routing)	Route stability; data dissemination; forwarder selection	Improvements in throughput, PDR, latency, packet loss, and overhead	Simulation-only validation (OMNeT++/SUMO)
Babu et al. [25]	HHO	Active/reactive power losses; voltage profile	Loss reduction and improved system stability	Tested only on IEEE 25-bus network
Wang et al. [26]	MHHO + VIKOR	PV output maximization vs. frequency regulation	Captures Pareto trade-offs; VIKOR-based compromise selection	Specific to PV–storage case studies
Choo et al. [27]	MHHO	Rolling stock maintenance scheduling	Integration with Industry 4.0 sensor and automation data	Limited benchmarking against alternatives
Islam et al. [28]	HHO (SO/MO)	Fuel cost; power losses; emissions	Reduced losses and emissions; competitive scalar performance vs. WOA/SSA/MFO/GWO	Weighted-sum aggregation; IEEE 30-bus only
DeBruyne and Kaur [29]	MHHO (reference-point)	Reference-point-based MOPs	Predator–prey model; improved convergence behavior	Early-stage implementation; limited applications
Pop et al. [30]	MHHO	EV charging/ discharging for grid stabilization	Competitive convergence, Pareto diversity, and load-balancing	Tested only in specific evaluation scenarios

summarizes the studies that applied the original MHHO algorithm in its standard form, indicating the targeted objectives, the main reported advantages, and the principal limitations of each application.

4.2. Modified Multi-Objective Harris Hawks Optimization

Piri and Mohapatra [31] introduced a method in medical data analysis called Multi-Objective Quadratic Binary HHO (MOQBHHO) that focuses on feature selection. In this method, HHO is adapted to work in a binary space, and a KNN classifier is incorporated as a wrapper. Additionally, the crowding distance mechanism is incorporated to refine non-dominated solutions. Out of the twelve medical datasets evaluated, MOQBHHO demonstrated the best balance between classification accuracy and dimensionality reduction in comparison to the rest of the methods, which included MOBHHO-S, MOGA, MOALO, and NSGA-II.

For the analysis of gene expression related to cancer, Dabba et al. [32] proposed the use of a Multi-Objective Binary HHO (MOBHHO) framework for the selection of genes. The framework implements dual fitness functions utilizing SVM with LOOCV, and KNN with K-fold to achieve gene minimization and classification maximization. Extensive empirical analysis of the proposed model was conducted using eight microarray benchmarks, and the analyses revealed that the model outperformed the most recent state-of-the-art approaches by providing greater than 98% classification accuracy, and even 100% accuracy for some of the datasets.

With regards to the many-objective OPF (Ma-OPF) problem, Cai et al. [33] implemented a modified MHHO. This algorithm included an elimination mechanism to handle more than five conflicting objectives at a time. When the proposed MHHO was tested on the IEEE 30-bus test system, it showed better performance than other heuristic algorithms.

Yan et al. [34] proposed BARES-MHHO as an improved variant of MHHO, using techniques such as chaotic mapping for population initialization, adaptive archive adjustment for faster discovery of sparse areas, symmetrical segmentation for creating an evenly distributed solution set, and blank angle region searching for enhancing exploration of under-represented areas. Compared to standard MHHO and various classical algorithms, BARES-MHHO exhibited diverse and superior convergence behavior on benchmark test functions, proving to be a more efficient and precise alternative.

Within the Internet of Vehicles environment, Liu and Jiang [35] tackled the problem of service composition, where user demands of varying complexity make it difficult to optimize the selection among multiple services with the same functionality. Liu and Jiang [35] framed the problem of service composition as a multi-objective and multi-constrained optimization problem to achieve trade-offs among the differing QoS (Quality of Service) attributes such as service availability, reliability, throughput, robustness, and cost within a defined time and space. To address the problem, the authors suggested an Improved MHHO, which modifies the energy iteration curve and iteration coordinates, along with a stagnation avoidance mechanism where multiple hawks share the same position. These additions enhanced convergence, population diversity, and robustness.

Jangir et al. [36] proposed a new modification of HHO for multi-objective problems called Non-dominated Sorting HHO (NSHHO). This method combines elitist non-dominated sorting with the standard HHO framework through the addition of a ranking method and crowding distance to create a better distribution of Pareto fronts with better coverage. The algorithm was tested on a broad range of 46 unconstrained, constrained, and real-life multi-objective design problems with nonlinearities and mixed discrete-continuous variables. When compared with NSGA-II, MOPSO, and MOEA/D, NSHHO was found to possess better convergence and diversity.

In cloud computing, where task scheduling and virtual machine allocation are key optimization problems, Emara et al. [37] proposed an adaptive multi-objective scheduling framework based on an improved version of HHO. The algorithm employs a solution-characterization mechanism, rather than random selection, to guide the search during the exploration phase, and incorporates a mutation strategy in the exploitation phase to mitigate premature convergence. In simulation-based evaluations, the proposed framework yielded improvements over conventional task scheduling methods across load balancing, makespan, scheduling length, throughput, and resource utilization on the considered scenarios.

In the context of distribution networks integrated with multiple microgrids, Poshtyafteh et al. [38] developed a modified MHHO-based multi-objective energy management framework. The formulation considers the placement of distributed generators and microgrids and targets overall cost, pollution reduction, and the minimization of network power losses. A fuzzy decision-making procedure is used to address the multiple objectives jointly, and uncertainties in renewable generation and demand are modeled using Information Gap Decision Theory. In simulations on the IEEE 33-bus distribution network over a 24-h horizon, the modified MHHO achieved more favorable values than the original HHO and GA baselines across the reported scalar indicators (cost, emissions, and network losses).

To address the challenges posed by random uniform initialization in MHHO, Yasear and Ku-Mahamud [39] suggested a new approach involving a two-step population generation technique. This design combines the R-sequence with partial opposition-based learning, which helps create a more evenly distributed initial population and strengthens the algorithm’s convergence toward the Pareto front. Based on evaluations from standard multi-objective benchmarks, the refinements to MHHO showed significantly better convergence and diversity in comparison to the baseline variant.

Expanding on the non-dominated sorting framework, Yasear and Ku-Mahamud [40] created Enhanced Non-dominated Sorting HHO (ENDSHHO). ENDSHHO differs from the original version by replacing the linear decreasing convergence parameter with an adjustment mechanism that is more sophisticated in terms of managing the balance between diversification and intensification. This refinement improves the algorithm’s ability to converge to the Pareto front while preserving solution diversity. Benchmark tests validated ENDSHHO’s superiority over the standard NSHHO.

When it comes to emergency resource distribution center site selection amidst uncertainty, Zhu et al. [41] proposed an Improved Chaotic Quantum MHHO (CQ-MHHO). While employing multi-objective modeling, coupled with defuzzification and fuzzy theory, urgency cost, economic cost, and transport cost were optimized simultaneously. The global search and convergence were improved, positively impacting the quality of Pareto fronts across all ZDT benchmark functions and real-world case studies. In comparison to MHHO, NSGA-II, and MOGWO, CQ-MHHO demonstrated improved solution quality and robustness.

As efficient scheduling is critical in cloud computing, Amer et al. [42] introduced the Elite Learning MHHO (ELHHO), a novel approach to multi-objective task scheduling, enhanced by incorporating elite opposition-based learning, which optimizes exploration, and using the minimum completion time heuristic for initialization, which avoids random initialization and local optima. ELHHO, which was implemented in CloudSim and validated using real datasets, proved to be successful in minimizing scheduling length and execution costs, whilst maximizing resource utilization. Additionally, ELHHO outperformed traditional MHHO and various other metaheuristics in cloud task scheduling optimization.

For the optimization of electrical discharge machining (EDM) processes, Uddin et al. [43] proposed an Improved MHHO. The algorithm incorporates an exponentially decreasing energy update strategy, a crowding-distance-based exploration mechanism, and fast non-dominated sorting adapted from NSGA-II. Tested on eleven benchmark problems and ten EDM cases, the proposed MHHO consistently outperformed MHHO and other recent optimizers in terms of coverage, spacing, and CPU time. With performance gains exceeding 80% in several metrics, the proposed IMHHO demonstrated strong efficiency and robustness in handling both constrained and unconstrained problems.

To predict COVID-19 patient mortality, Dokeroglu [44] designed a parallel MHHO algorithm with feature selection capability. The approach simultaneously minimized the number of features while maximizing prediction accuracy. Applied to a Kaggle COVID-19 dataset and its augmented version, the method achieved a 98.15% accuracy with a 45% feature reduction, outperforming other wrapper-based metaheuristic approaches and establishing new benchmarks for this dataset.

For fog computing, Ghasemi [45] proposed a service placement strategy using a modified MHHO. The method modeled the multi-objective problem as several single-objective subproblems, optimizing energy consumption and end-to-end delay. Simulations in CloudSim demonstrated that the proposed algorithm also improved load balancing and resource utilization, outperforming baseline methods in efficiency and scalability.

In the context of power systems, Alsokhiry [46] used MHHO on the MaO-OPF problem in the IEEE 30-bus network. The algorithm optimized several objectives simultaneously, including fuel costs, emissions, active and reactive power losses, and voltage stability and deviation. With self-adaptive constraint handling and dynamic exploration–exploitation balancing, MHHO outperformed MOEA/D-DRA and NSGA-III in terms of producing superior Pareto fronts.

For the monitoring of power systems, Hashemi and Kalantar [47] proposed a Multi-Objective Binary HHO with Region Selection (MOBHHO/R) to address the placement of Phasor Measurement Units (PMUs). The formulation jointly targets the reduction of the required PMU count and the enhancement of system observability, supported by an archive-based repository for retaining non-dominated solutions. On the IEEE 14- and 30-bus test systems, the method yielded fewer required PMUs and higher observability than the baseline approaches considered in the study.

To refine the balance between exploration and exploitation, Choo et al. [48] introduced Enhanced HHO, which further evolved into Enhanced MHHO. This methodology incorporates a nonlinear exploration factor, Differential Evolution-based diversity, and chaotic mutation strategies, and crowding distance–based sorting for MOPs. The benchmark results for Single Objective Problems (SOPs) and MOPs revealed considerable advancements in exploration capacity and solution diversity relative to the original HHO and other evolutionary algorithms.

With respect to virtual power plants, Pandey et al. [49] applied a Modified MHHO to optimize operational scheduling for real-time and day-ahead scenarios. The algorithm addressed risk via Conditional Value at Risk and included renewable energy sources (solar PV, wind, fuel cells), cogeneration CHP, and flexible reserves in the form of EVs and ESS. The results demonstrated that the modified MHHO maximized net profits and environmental feasibility in day-ahead and 5-min-interval operational scheduling compared to the baseline scheduling methods.

Regarding engineering design problems, Allou et al. [50] presented a Multi-Leaders Guided HHO with epsilon-dominance. The epsilon-dominance approach offered a new perspective on solution diversity by incorporating a diversity mechanism based on a fixed-size external archive, crowding distance, and epsilon-dominance relations. The algorithm’s efficient leader selection mechanism steered the population toward less-populated Pareto regions. The algorithm was tested on benchmark functions and real engineering design challenges, including the optimization of trusses and beams.

Within distribution systems, Selim et al. [51] proposed further enhancements to HHO for optimal distributed generation placement with the introduction of IHHO and MOIHHO. IHHO augmented HHO by incorporating updates based on the location of the rabbits, and MOIHHO implemented grey relation analysis to choose compromise solutions on the Pareto front. When these algorithms were implemented on the IEEE 33- and 69-bus systems, they achieved the goals of minimizing active power losses and reducing voltage imbalances and irregularities, as well as improving stability indices, all while outperforming competing optimization methods across the board in both single- and multi-objective cases.

In addressing HHO’s shortcomings in multi-objective optimization, Tian et al. [52] introduced enhancements in Multi-Strategy Elite Opposition-based MHHO (MO-EMHHO). Modifications such as Sobol sequence initialization, elite backward learning, adaptive grid-based archive management, non-linear energy updates, and random wandering with a Gaussian distribution were implemented. In tests performed against MOPSO, MOGWO, and MHHO, MO-EMHHO exhibited superior convergence accuracy, stability, and diversity.

The approach described in Zouache et al. [53] introduced a modified harmonious human-horse optimization (MHHO) method based on a strengthened dominance relation. The technique added an external archive of non-dominated solutions to direct the population toward more promising regions of the search space. With the use of strengthened dominance relations, the method achieved a good balance between convergence and diversity of the Pareto set. In comparison to other multi-objective metaheuristic optimization methods, benchmark test results demonstrated good convergence and distribution of the solutions for bi-objective and tri-objective problems.

In order to improve diversity and prevent premature convergence, Yasear and Ku-Mahamud [54] proposed the Non-dominated Sorting MHHO. This method incorporated a fast non-dominated sort into the original MHHO, where the population was stratified into Pareto-based levels to influence the choice of solutions. When the proposed method was tested against standard MHHO on benchmark problems, it demonstrated improvements in both diversity and convergence toward the Pareto front.

The method detailed in Yasear and Ku-Mahamud [55] presents a new approach to the Multi-Heuristic Hawk Hunting Optimization (MHHO) method, inspired by military hawk hunting strategies. This method introduces the “Flush-and-Ambush” strategy to the hawk-dominated sorting method known as Non-Dominated Sorting MHHO. In order to avoid the loss of population diversity in Non-Dominated Sorting MHHO, the updated strategy utilizes a flush-and-ambush movement coupled with non-dominated sorting. In testing, the new method demonstrated overall better diversity maintenance, a better convergence rate, and higher accuracy in comparison to NDSHHMO, MOGOA, and MOGWO across ten multi-objective problem scenarios.

To broaden the applicability of HHO for multiple objectives, Boumaza et al. [56] introduced Guided MHHO (GMHHO). This method combines an archiving mechanism that retains non-dominated solutions, a multi-leader selection mechanism for better guidance, and a Bi-Goal Evolution framework designed to recast multi-objective problems as bi-objective problems to increase selection pressure. Experiments demonstrated that GMHHO achieves a better convergence–diversity balance than standard MHHO and other top-performing algorithms.

In the context of medical applications, Dokeroglu and Kucukyilmaz [57] proposed a binary MHHO for the diagnosis of Parkinson’s disease. The approach combines an adaptive KNN classifier with tailored exploration–exploitation operators for feature selection, and a parallel MPI-based implementation is provided to reduce computational overhead in large-scale instances. In the feature selection framework, the proposed method was experimentally compared with GA, PSO, Bat, Cuckoo Search, and GWO on several datasets, yielding a reduction of approximately 30% in the number of selected features and improved diagnostic accuracy on the tested cases.

Table 2. Summary of the modified MHHO variants.

Study	Variant	Objectives	Advantages	Limitations
Piri and Mohapatra [31]	MOQBHHO	Feature selection; accuracy	Binary encoding; KNN wrapper; crowding distance	Tested on medical datasets only
Dabba et al. [32]	MOBHHO	Gene minimization; classification accuracy	Dual SVM/KNN fitness; high accuracy on microarrays	Limited to microarray data
Cai et al. [33]	Modified MHHO	Many-objective OPF	Elimination mechanism for >5 objectives	Tested only on IEEE 30-bus
Yan et al. [34]	BARES-MHHO	Benchmark MOPs	Chaotic init; adaptive archive; blank-angle search	Validated only on benchmarks
Liu and Jiang [35]	Improved MHHO	QoS in IoV service composition	Modified energy curve; stagnation avoidance	Domain-specific to IoV
Jangir et al. [36]	NSHHO	Engineering design MOPs	Elitist non-dominated sorting; crowding distance	Higher computational cost
Emara et al. [37]	Improved MHHO	Cloud task scheduling	Solution characterization; mutation in exploitation	Simulation-only validation
Poshtyafteh et al. [38]	Modified MHHO	Cost; emissions; losses in microgrids	Fuzzy decision-making; IGDT uncertainty modeling	Computationally intensive
Yasear and Ku-Mahamud [39]	Two-step Init. MHHO	Benchmark MOPs	R-sequence with partial opposition-based learning	Benchmark-only validation
Yasear and Ku-Mahamud [40]	ENDSHHO	Benchmark MOPs	Non-linear convergence parameter adjustment	Limited test scope
Zhu et al. [41]	CQ-MHHO	Urgency, economic, and transport costs	Chaotic quantum search; fuzzy defuzzification	Increased algorithmic complexity
Amer et al. [42]	ELHHO	Cloud scheduling: length, cost, utilization	Elite opposition-based learning; MCT initialization	Validated only in CloudSim
Uddin et al. [43]	Improved MHHO	EDM process optimization	Exponential energy update; fast non-dominated sort	Limited to EDM cases
Dokeroglu [44]	Parallel MHHO	Feature count; accuracy (COVID-19)	Parallel implementation; high accuracy	Dataset-specific results
Ghasemi [45]	Modified MHHO	Energy; end-to-end delay (fog)	Decomposition into single-objective subproblems	May lose Pareto trade-offs
Alsokhiry [46]	MHHO (MaO-OPF)	Cost, emissions, losses, voltage	Self-adaptive constraints; dynamic balance	Limited to IEEE 30-bus
Hashemi and Kalantar [47]	MOBHHO/R	PMU count; system observability	Binary encoding; region selection; archive repository	Limited to IEEE 14/30-bus
Choo et al. [48]	Enhanced MHHO	SOP and MOP benchmarks	Nonlinear exploration; DE diversity; chaotic mutation	Higher parameter complexity
Pandey et al. [49]	Modified MHHO	VPP scheduling: profit, environment	CVaR risk handling; renewable integration	Specific to virtual power plants
Allou et al. [50]	$ϵ$-MHHO	Engineering design MOPs	$ϵ$-dominance; multi-leader selection; archive	$ϵ$ parameter sensitivity
Selim et al. [51]	IHHO/MOIHHO	DG placement: losses, stability	Rabbit-location updates; grey relation analysis	Limited test systems
Tian et al. [52]	MO-EMHHO	Benchmark MOPs	Sobol init; elite opposition; adaptive grid	Multiple mechanisms increase complexity
Zouache et al. [53]	MHHO-SRD	Bi/tri-objective benchmarks	Strengthened dominance; external archive	Tested only on ≤3 objectives
Yasear and Ku-Mahamud [54]	NDSHHO	Benchmark MOPs	Fast non-dominated sort; Pareto stratification	Benchmark-only validation
Yasear and Ku-Mahamud [55]	FA-NDSHHMO	10 multi-objective problems	Flush-and-ambush update; non-dominated sort	Limited problem set
Boumaza et al. [56]	GMHHO	Multi-objective benchmarks	Archive; multi-leader; Bi-Goal Evolution	Complexity of BiGE framework
Dokeroglu and Kucukyilmaz [57]	Binary MHHO	Parkinson’s diagnosis: features, accuracy	Adaptive KNN; tailored operators; MPI parallel	Domain-specific dataset

summarizes the modified MHHO variants in terms of the proposed algorithm, the targeted objectives, the main algorithmic advantages introduced, and the principal limitations associated with each variant.

4.3. Hybridized Multi-Objective Harris Hawks Optimization

In seeking to obtain a more uniformly distributed Pareto front, a Hybrid MHHO was created. This specific type of MHHO combines an elite-based non-dominated sorting approach with a grid indexing approach, with elite sorting preserving high-quality individuals and grid indexing preserving solution diversity across objectives. These combined mechanisms improve both the convergence and the population coverage, which helps the algorithm better address multi-objective optimization problems. For 22 benchmark functions and four engineering problems, the Hybrid MHHO was compared to other established metaheuristic algorithms for multi-objective optimization, such as MOPSO, NSGA-II, MOALO, MSSA, and MODA, and showed better performance, providing more support for the validity of its approach [58].

To enhance cloud computing task scheduling, Haris and Zubair [59] introduced a hybrid framework known as Manta Ray–Modified MHHO. This framework merges Harris Hawks Optimization and Manta Ray Foraging Optimization, resulting in a cooperative search strategy that improves resource utilization and load distribution across virtual machines. The hybrid strategy addresses task waiting time and priority balancing by considering cost, response time, and throughput. When the proposed method was implemented in CloudSim, it showed significantly improved performance in throughput, load balancing, and scheduling efficiency compared to other methods.

To facilitate the early prediction of coronary artery disease, Vijayaraj and Pasupathi [60] proposed a hybrid MHHO. This algorithm incorporates adaptive exploration and exploitation strategies to reduce the number of selected features and the number of classifier hyperparameters that need to be optimized. Among the classifiers used are Random Forest, K-nearest neighbors, Logistic Regression, and Support Vector Machines. The proposed method, evaluated on the Kaggle heart disease dataset, achieved the best prediction accuracy using the least number of features. The hybrid structure preserved convergence efficiency and was particularly effective for high-dimensional and complex optimization problems.

Yang et al. [61] proposed a cooperative multi-population method for multi- and many-objective problems, where each subpopulation is optimized by MHHO. The proposed CMPMO-HHO incorporates four features: one-to-one mapping of objectives and subpopulations, a global archive for cooperation, logistic chaotic perturbation, and dual elite selection based on non-dominated sorting and reference points. This cooperative structure yielded significant advancements in convergence and diversity, also achieving competitive results in 34 multi-objective and 19 many-objective benchmark problems against state-of-the-art optimizers.

To address the challenges of dynamic hybrid flow shop scheduling with green objectives, Wang et al. [62] introduced a framework combining MHHO with digital twin technology. The framework’s adaptive multi-objective dynamic HHO (AMODHHO) features a nonlinear exploration and exploitation strategy that integrates a genetic algorithm crossover operator for improved global search. Together with a digital twin-based encoding strategy, the framework enables adaptive scheduling adjustments in response to dynamic changes, including device reconfiguration, variable processing time, and workpiece reprocessing. Case studies show that AMODHHO achieves superior performance compared to SPEA2 and NSGA-II.

For Fog computing based on IoT, Saranya and Pabitha [63] suggested a Hybrid MHHO and Moth-Flame Optimization Algorithm (HMHMFOA), focusing on the exploration strength of MFO and the exploitation ability of HHO. With the hybrid approach, faster convergence and improved quality of the solutions were achieved. The tests performed in fog computing environments demonstrated remarkable improvements in latency, energy consumption, and resource utilization.

A health recommender system formulated by Kuanr and Mohapatra [64] utilizes a hybrid Genetic HHO for multi-dimensional objective feature selection and disease forecasting. The framework utilizes TPOT AutoML to suggest the most fitting classification algorithms and the most relevant disease-inducing features for risk evaluation. The hybrid approach, in comparison to PCA, SVD, and Autoencoder-based methods, achieved significant advancements in prediction accuracy and interpretability within the healthcare domain.

With regard to mobile cloud computing (MCC), the problem of task scheduling was solved by Saemi et al. [65] using the Hybrid MHHO algorithm. This method focused on the efficient distribution of tasks among mobile nodes, the public cloud, and the cloud layers, with the goal of minimizing both job completion time and energy consumption. When compared to GA, ACO, PSO, and CSA, the hybrid method was consistently faster and used less energy.

In the context of IoT and the numerous problems associated with the detection of botnets, Gharehchopogh et al. [66] proposed a multi-objective dynamic HHO with a mutation operator. This algorithm aimed to determine the most relevant features for optimal performance for the K-nearest neighbor (KNN), SVM, MLP, and decision tree classifiers. The study showed that in addition to improving classification accuracy across all five datasets, the proposed method was competitive with other methods as it had lower computational cost.

With regard to sustainable robotics, Zhou and Bian [67] proposed an adapted bi-objective HHO for robotic disassembly line balancing. Opposition-based learning, DE, and Gaussian mutation were combined, and a novel energy-consumption-driven parameter renewal was introduced. Considering uncertainty, the time- and energy-consumption optimization method outperformed classical heuristics.

To address the issue of premature convergence in Multi-Objective Optimization (MO), Yan [68] proposed an MHHO variant enhanced by Gaussian mutation. This approach combines adaptive Gaussian mutation with a prey location strategy utilizing grid division. This suggests that the algorithm may improve both the diversity level and the distribution of the Pareto front. The author’s comparative experiments demonstrated improvements in accuracy and convergence speed compared to NSGA-II, MOPSO, MOGWO, and traditional MHHO, with the ranges of improvements being 8.02–51.34% and 16.67–40.74%, respectively.

Within the context of the environment, Du et al. [69] engineered a hybrid forecasting framework concerning PM2.5 and PM10 concentrations through the use of MHHO-optimized Extreme Learning Machines. The model employed sophisticated data preprocessing for the decomposition of time series and used MHHO for hyperparameter optimization. The model evaluations conducted over twelve datasets of air pollution spanning various cities in China revealed that the MHHO-enhanced model improved predictive stability and accuracy.

Table 3. Summary of the hybridized MHHO variants.

Study	Variant	Objectives	Advantages	Limitations
Wang et al. [58]	Hybrid MHHO (elite NDS + grid index)	22 benchmark functions and 4 engineering problems	Elite non-dominated sorting combined with grid indexing	Mostly benchmark-driven evaluation
Haris and Zubair [59]	Manta Ray-Modified MHHO	Cost; response time; throughput	Cooperative HHO–MRFO search; improved load distribution	CloudSim simulation only
Vijayaraj and Pasupathi [60]	MO-hHHO	Feature count; classifier hyperparameters	Best reported accuracy with the fewest features	Validated on a single heart-disease dataset
Yang et al. [61]	CMPMO-HHO	Multi- and many-objective benchmarks	Subpopulation–objective mapping; logistic chaos; dual elite selection	Benchmark-focused validation
Wang et al. [62]	AMODHHO + digital twin	Dynamic hybrid flow shop with green objectives	Digital-twin encoding; GA crossover; adaptive scheduling	Domain-specific to HFSP
Saranya and Pabitha [63]	HMHMFOA (HHO + MFO)	Latency; energy; resource utilization	MFO exploration combined with HHO exploitation	Restricted to fog computing context
Kuanr and Mohapatra [64]	Genetic HHO + TPOT AutoML	Feature selection; disease forecasting	Improved accuracy and interpretability for healthcare data	Compared only against PCA, SVD, and autoencoder baselines
Saemi et al. [65]	Hybrid MHHO	Completion time; energy consumption (MCC)	Layered task distribution across mobile/cloud nodes	MCC-specific evaluation
Gharehchopogh et al. [66]	Multi-objective dynamic HHO + mutation	Feature selection for KNN, SVM, MLP, DT	Improved accuracy with lower computational cost	Limited to five botnet datasets
Zhou and Bian [67]	Bi-objective HHO	Robotic disassembly: time and energy	Opposition-based learning, DE, and Gaussian mutation; energy-driven renewal	Application-specific
Yan et al. [68]	MHHO + Gaussian mutation	Multi-objective optimization (general)	Adaptive Gaussian mutation; grid-based prey localization	Improvements bound to specific test problems
Du et al. [69]	MHHO-tuned ELM	PM2.5/PM10 forecasting	Time-series decomposition; MHHO-based hyperparameter tuning	Restricted to air-pollution datasets

provides a structured overview of the hybridized MHHO variants reviewed in this subsection, listing the proposed hybrid algorithm, the targeted objectives, the main advantages introduced by the hybridization, and the principal limitations of each contribution.

4.4. Critical Comparative Analysis of MHHO Variants

A comparative examination of the reviewed studies reveals that most MHHO modifications primarily focused on improving the balance between convergence and diversity preservation. The most commonly adopted enhancement strategies included non-dominated sorting, adaptive archive management, chaotic population initialization, opposition-based learning, mutation operators, and adaptive exploration–exploitation balancing. These mechanisms were introduced mainly to address premature convergence, insufficient Pareto front coverage, and poor diversity maintenance in high-dimensional search spaces. Furthermore, hybridization with algorithms such as Differential Evolution, Genetic Algorithms, and Manta Ray Foraging Optimization was frequently utilized to strengthen global exploration and improve local exploitation capabilities.

The reviewed literature also demonstrates clear differences between the original, modified, and hybridized MHHO variants. Original MHHO approaches were generally effective for moderate-scale optimization problems and provided acceptable convergence behavior with relatively simple algorithmic structures. However, their performance often deteriorated in many-objective, constrained, and high-dimensional optimization scenarios due to limited diversity preservation and susceptibility to local optima. Modified MHHO variants improved convergence accuracy and Pareto front distribution through adaptive control strategies and enhanced selection mechanisms, while hybridized approaches exhibited stronger robustness and search capability across complex optimization domains such as cloud computing, energy management, and biomedical feature selection. Nevertheless, these improvements were frequently achieved at the expense of higher computational complexity, increased parameter sensitivity, and reduced algorithmic simplicity.

Several recurring methodological patterns can also be identified across the reviewed studies. Most improved MHHO variants incorporated external archives and Pareto-based ranking mechanisms to preserve non-dominated solutions and enhance population diversity. Similarly, adaptive parameter adjustment and nonlinear energy update strategies were widely employed to improve the transition between exploration and exploitation phases. In contrast, binary and discrete MHHO variants were primarily developed for feature selection, scheduling, and combinatorial optimization tasks, where continuous search operators are less suitable. Moreover, many hybrid approaches combined MHHO with evolutionary operators or swarm-based mechanisms to compensate for weaknesses in local search capability and convergence stability.

Despite the promising advancements achieved by MHHO variants, several research challenges remain insufficiently addressed. Most existing studies rely heavily on benchmark functions and simulation-based evaluations, while relatively limited attention has been given to large-scale real-world optimization problems. In addition, theoretical convergence analysis and computational complexity studies are rarely provided, limiting the deeper understanding of MHHO behavior. Another recurring limitation is the lack of standardized benchmarking and statistical validation protocols, which complicates fair comparisons among competing variants. Furthermore, only limited studies investigated dynamic optimization environments, many-objective optimization problems with more than three objectives, or highly constrained industrial applications. Future research directions should therefore focus on scalable many-objective optimization, adaptive archive management, dynamic and uncertain optimization environments, reproducible evaluation frameworks, and theoretically grounded convergence analysis.

5. Applications of Multi-Objective Harris Hawks Optimization

The MHHO has been deployed across a broad spectrum of real-world problems, as organized in

Table 4. Multi-objective HHO (MHHO) application domains.

Domain	Problem	Variant	Ref.
Power & Energy Systems	Optimal power flow (OPF)	Original	[28]
	Many-objective OPF (up to 5 objectives)	Modified	[33]
	MaO-OPF with constraint handling and dynamic exploration–exploitation	Modified	[46]
	DG placement in radial distribution systems	Original	[22]
	DG placement with grey relation analysis	Modified	[51]
	Energy management of multiple microgrids under uncertainty	Modified	[38]
	PV plant frequency regulation with storage (VIKOR decision)	Original	[26]
	Real-time/day-ahead scheduling of virtual power plant (risk-averse CVaR)	Modified	[49]
	PMU placement considering redundancy & ZIB	Modified	[47]
	EV fleet coordination for grid balancing	Original	[30]
Cloud & Edge Computing	Task scheduling & VM allocation in cloud	Modified	[37]
	Load balancing in cloud (MRFO–MHHO hybrid)	Hybridized	[59]
	Multi-objective task scheduling in cloud	Modified	[42]
	Service placement in fog computing	Modified	[45]
	Task offloading in IoT-based fog computing	Hybridized	[63]
	Task scheduling in mobile cloud computing (MCC)	Hybridized	[65]
Healthcare & Bioinformatics	Coronary artery disease prediction (feature selection + HPO)	Hybridized	[60]
	Medical feature selection (binary/quadratic)	Modified	[31]
	Gene selection on microarray datasets	Modified	[32]
	COVID-19 mortality prediction (parallel wrapper)	Modified	[44]
	Parkinson’s disease diagnosis (binary MHHO + MPI)	Modified	[57]
	Health recommender & disease diagnosis (GA–MHHO)	Hybridized	[64]
Industrial Engineering & Manufacturing	Hybrid flow shop green scheduling with dynamic events (digital twin)	Hybridized	[62]
	Electrical Discharge Machining (EDM) process optimization	Modified	[43]
	Engineering design optimization (epsilon-dominance, multi-leader)	Modified	[50]
	Robotic disassembly line balancing (cycle time & energy)	Hybridized	[67]
	Rolling stock maintenance scheduling	Original	[27]
Networking & Communication	2-hop routing in CR-VANET	Original	[24]
	Service composition in Internet of Vehicles (IoV)	Modified	[35]
	IoT botnet detection (feature selection)	Hybridized	[66]
Environmental & Water Resources	Low-Impact Development (LID) planning under climate change	Original	[21]
	Air pollution forecasting (PM2.5/PM10)	Hybridized	[69]
	Emergency resource center site selection (uncertainty)	Modified	[41]
	Emergency vehicle dispatching during pandemics	Original	[23]
Benchmark & Algorithmic Developments	BARES-MHHO: chaotic init, archive/grid symmetry, blank-angle search	Modified	[34]
	H-MHHO: elite non-dominated sorting + grid indexing	Hybridized	[58]
	NSHHO: elitist non-dominated sorting with crowding distance	Modified	[36]
	CMPMO-HHO: cooperative multi-population with dual elite selection	Hybridized	[61]
	NDSHHMO: fast non-dominated sorting	Modified	[54]
	ENDSHHHO: improved convergence parameter strategy	Modified	[40]
	FA-NDSHHMO: flush-and-ambush update + non-dominated sorting	Modified	[55]
	MHHO-SRD: strengthened dominance with external archive	Modified	[53]
	HHMO: reference point-based multi-objective optimizer	Original	[29]
	Adaptive Gaussian mutation with grid-based prey location	Modified	[68]
	MO-EHHO: nonlinear exploration, DE/chaos, crowding distance	Modified	[48]
	MO-EMHHO: Sobol init, elite opposition, adaptive archive, Gaussian var.	Modified	[52]
	GMHHO: Bi-Goal Evolution, archive, multi-leader guidance	Modified	[56]

. The diversity of objective landscapes, constraints, and decision spaces has prompted researchers to go beyond the original MHHO and adopt modified or hybridized variants to strengthen convergence, preserve Pareto diversity, and cope with domain-specific complexities.

Applications spanned classical and many-objective OPF, distributed generation (DG) planning, microgrid operation, grid-support functions, and measurement placement. Original MHHO reduced fuel cost, losses, and emissions in OPF [28], while many-objective formulations were addressed via elimination-based MHHO [33] and enhanced constraint-handling with dynamic exploration–exploitation [46]. In radial networks, MHHO improved voltage profiles and curtailed losses for DG siting [22], and a modified scheme coupled rabbit-location updates with grey relation analysis to select compromise DG solutions [51]. For multi-microgrid energy management under uncertainty, fuzzy/IGDT-based modeling with MHHO balanced cost, pollution, and losses [38]. Grid-support use cases included PV reconfiguration with VIKOR-based compromise selection [26], risk-averse (CVaR) scheduling of virtual power plants [49], optimal PMU placement via a binary MHHO with region selection and repository archiving [47], and EV fleet coordination for balancing services [30].

Task scheduling and resource orchestration motivated several tailored variants. Best-fit-guided exploration and mutation-enhanced MHHO improved makespan, throughput, and utilization in cloud scheduling and VM allocation [37]. A Manta Ray Foraging hybrid (MRFO–MHHO) bolstered load balancing and resource use [59], and elite-opposition learning with heuristic initialization further reduced schedule length and cost [42]. Beyond cloud data centers, MHHO modeled service placement in fog computing via multi-objective decomposition to single-objective subproblems [45], hybridized with MFO for IoT task offloading to minimize latency and energy [63], and supported MCC with hybrid scheduling strategies targeting time–energy trade-offs [65].

Feature selection and diagnostic modeling benefited from binary and hybrid MHHO designs. A hybrid MHHO for coronary artery disease jointly minimized features and tuned classifier hyperparameters to maximize accuracy [60]. Binary/quadratic encodings enhanced medical feature selection with wrapper criteria and crowding distance [31], while a dual-fitness binary MHHO (SVM/KNN) achieved compact and accurate gene subsets on microarray data [32]. For pandemic analytics, a parallel multi-objective wrapper achieved high-accuracy COVID-19 mortality prediction with large feature reduction [44]; an MPI-accelerated binary MHHO with adaptive KNN advanced Parkinson’s diagnosis and reduced computational burden [57]. A GA–MHHO hybrid underpinned a health recommender, optimizing feature subsets and diagnostic accuracy [64].

In smart manufacturing, the adaptability of the EMPHFSP (Energy-efficient Multi-Product Hybrid Flow Shop Scheduling) problem has been tested on digital-twin-driven devices with adjustable reconfiguration, controllable processing times, and rework [62]. For process engineering, the modified MHHO with exponential energy update, crowding-distance-based exploration, and Fast Non-Dominated Sorting has been applied to EDM optimization, achieving competitive coverage, spacing, and CPU time [43]. Engineering design has been improved with epsilon-dominance, multi-leader guidance, and elitist archiving [50]. Sustainable disassembly line balancing applied opposition-based learning, DE, and Gaussian mutation with energy-aware parameter renewal to optimize cycle time and energy [67]. The original MHHO has been illustrated in the railway context for rolling stock maintenance planning [27].

Routing and composition issues have utilized MHHO for stability and quality of service. A 2-hop CR-VANET protocol utilizing the original MHHO improved throughput, delay, PDR, and overhead [24]. In the Internet of Vehicles, an improved MHHO facilitated faster convergence and the avoidance of local optima in service composition with complex quality of service constraints [35]. In the context of IoT network security, a dynamic, mutation-based hybrid MHHO achieved effective botnet detection through dynamic feature selection across multiple classifiers [66].

Urban water and environmental analytics received special mention. An original MHHO integrated with SWMM supported LID planning under climate scenarios, enabling cost and runoff reduction [21]. For air-quality prediction, MHHO-tuned ELM models with decomposition-based preprocessing were shown to improve both stability and accuracy for the PM2.5/PM10 datasets [69]. Emergency center siting under uncertainty used a chaotic-quantum modified MHHO to cope with urgency, economic, and transport costs [41]. Emergency vehicle dispatch during pandemics was assisted by the original MHHO to lower travel costs and improve the effectiveness of timely treatment [23].

A rich line of algorithmic studies strengthened MHHO’s search dynamics and diversity preservation. Advances included chaotic initialization with archive/grid symmetry and blank-angle exploration (BARES-MHHO) [34]; elite non-dominated sorting combined with grid indexing (H-MHHO) [58]; elitist non-dominated sorting with crowding distance (NSHHO) [36]; cooperative multi-population frameworks with chaotic perturbations and dual elite selection (CMPMO-HHO) [61]; fast non-dominated sorting (NDSHHMO) [54]; improved convergence parameter control (ENDSHHHO) [40]; flush-and-ambush movement with non-dominated sorting (FA-NDSHHMO) [55]; strengthened dominance with external archiving (MHHO-SRD) [53]; and the early reference point-based HHMO [29]. Additional enhancements included adaptive Gaussian mutation with grid-based prey localization [68], a multi-objective EHHO with nonlinear exploration, DE/chaos, and crowding distance [48], multi-strategy MO-EMHHO (Sobol initialization, elite opposition, adaptive archive, Gaussian variation) [52], and BIGE-guided, multi-leader GMHHO with elitist archiving [56]. Collectively, these developments reported notable gains in convergence speed, hypervolume/IGD, and solution spread on standard multi-/many-objective benchmarks.

Across the reviewed applications, MHHO demonstrated particular effectiveness in problems characterized by conflicting objectives, nonlinear search spaces, and complex operational constraints. Modified and hybridized variants were more frequently adopted in high-dimensional domains such as cloud computing, power systems, and biomedical feature selection, where maintaining Pareto diversity and avoiding premature convergence become increasingly challenging. In contrast, the original MHHO was more commonly applied to moderate-scale engineering optimization tasks. This trend indicates that application complexity strongly influences the need for adaptive enhancement mechanisms and hybrid search strategies.

6. Critical Analysis of Multi-Objective Harris Hawks Optimization Theory

MHHO has emerged as a flexible and effective framework for solving multi-objective optimization problems by combining the exploration–exploitation behavior of HHO with Pareto-based ranking, elitist archiving, and diversity preservation mechanisms. Across the reviewed studies, MHHO and its variants demonstrated promising performance in engineering optimization, cloud and edge computing, healthcare, scheduling, and energy management applications. In particular, hybrid and adaptive designs, including elite sorting with grids [58], strengthened and epsilon-dominance mechanisms [50,53], cooperative multi-population strategies [61], and adaptive stochastic controls [48,52,68], improved both convergence behavior and Pareto front diversity.

One of the recurring challenges in MHHO is premature convergence, especially in high-dimensional and multimodal optimization problems. As the population gradually concentrates around promising regions of the search space, the algorithm may lose diversity and become trapped in local optima before sufficiently approximating the Pareto front. This issue becomes more pronounced in problems characterized by irregular or disconnected Pareto regions, where maintaining a well-distributed set of non-dominated solutions is inherently difficult. Although grids, crowding-distance mechanisms, chaos-based initialization, mutation operators, and cooperative multi-population strategies improve diversity preservation, they simultaneously reduce local refinement efficiency and introduce additional computational burden [50,61].

Scalability is another major concern for MHHO variants. As the number of objectives, decision variables, and population members increases, dominance relations lose discrimination ability and archive maintenance becomes increasingly expensive. Non-dominated sorting, density estimation, reference management, and grid-based diversity preservation all contribute to growing computational overhead [33,61]. While strengthened dominance, epsilon-dominance, and decomposition-based approaches partially alleviate these limitations [33,50,53], scalable mechanisms for many-objective optimization remain underdeveloped. Moreover, several studies reported improvements in solution quality without providing sufficient analysis of runtime behavior, memory complexity, or scalability characteristics. As noted by Uddin et al. [43], CPU time and computational efficiency remain important concerns in practical engineering optimization scenarios.

The performance of MHHO is also highly sensitive to parameter settings and adaptive control strategies. Energy scheduling mechanisms, mutation rates, archive size, operator selection, and grid granularity significantly influence convergence behavior and Pareto front distribution. Although adaptive and nonlinear control strategies improve robustness, they often require problem-specific calibration and careful parameter tuning [48,52,68]. Similarly, hybrid approaches combining Differential Evolution, chaos theory, opposition-based learning, and multiple search operators frequently report improved hypervolume, spacing, and inverted generational distance metrics; however, systematic ablation studies isolating the contribution of each enhancement mechanism remain limited [48,52,67].

A further methodological concern that recurs across the reviewed studies pertains to the validity of comparative performance assessments. In a multi-objective setting, rigorous comparison must be conducted at the level of Pareto-set approximations rather than at the level of individual objective values, since scalar comparisons cannot capture the trade-off structure that defines a multi-objective solution. An algorithm may therefore be regarded as superior to a baseline only when its non-dominated set dominates that of the baseline, attains broader coverage of the true Pareto front, or achieves more favorable values on established indicators. Within the reviewed literature, however, such indicators are applied inconsistently, with many studies reporting only scalar objective values and few incorporating set-coverage analyses or statistical validation [22,28,51]. This heterogeneity undermines the reliability of inter-study comparisons and constitutes a substantive barrier to the methodological maturation of the field. The establishment of standardized Pareto-front-level evaluation protocols therefore represents a critical research priority for the MHHO community.

Another important limitation concerns the integration of MHHO with constrained, discrete, and dynamic optimization environments. In applications such as optimal power flow, distributed generation placement, and microgrid management, the inclusion of feasibility constraints, repair operators, and penalty functions adds considerable algorithmic complexity [28,38,51]. Likewise, binary and mixed encodings have shown effectiveness in feature selection, diagnosis, and observability problems, but stronger feasibility-preserving operators are still needed for permutation-based and mixed-integer optimization tasks [31,32,47,57]. Furthermore, although digital twin frameworks, fuzzy modeling, and IGDT-based approaches provide promising directions for uncertain and dynamic environments, unified frameworks for adaptive memory management and online re-optimization remain largely unexplored [38,41,62].

7. Conclusions and Future Work

The review has assessed various components of the MHHO algorithm’s theoretical basis, algorithmic derivations, and range of applications. The challenge of handling conflicting objectives is addressed by extending the algorithm to adapt Pareto dominance, elitism, diversity-preserving, and adaptive-changing mechanisms. These modifications give MHHO the ability to approximate Pareto-optimal solutions and create Pareto-optimal fronts and trade-off solutions across various domains.

The bibliometric study of this research shows a consistent increase in MHHO studies spanning multiple domains: computer science, engineering, bioinformatics, energy systems, and cloud computing. The examined literature was categorized into original, modified, and hybrid versions. Modified variants attempted to improve the speed, diversity, and robustness of convergence, while original implementations applied MHHO to solve optimization problems. Hybrid methods combined MHHO with other metaheuristics or problem-specific heuristics, often achieving above-average performance in solving complex optimization problems. This evidence demonstrates the MHHO algorithm’s versatility and broad applicability to a wide range of real-world multi-objective problems.

Despite these accomplishments, many obstacles still need to be addressed and warrant further examination. The following research avenues may help shape the future development of MHHO:

• Algorithmic Efficiency and Scalability: With an increase in objectives and population size, the computational cost of the MHHO method rises because of the need for non-dominated sorting, density estimation, and archive management. Future work should consider more efficient dominance measures, approximate ranking, and surrogate-assisted selection to reduce runtime complexity. In large engineering problems, particularly those with thousands of decision variables, parallel, distributed, and GPU-based approaches may provide significant opportunities for enhancing the scalability of MHHO.
• Adaptive Parameter Control: MHHO’s performance is influenced by its specific parameter settings pertaining to energy schedules, archive size, grid granularity, and mutation rates. Although adaptive or nonlinear strategies have shown improvements in robustness, these strategies are still dependent on problem-specific configurations. Future work should seek to enhance MHHO’s ability to self-tune its parameters dynamically through the use of reinforcement learning, fuzzy logic, and evolutionary control systems.
• Scalability to Many-objective Problems: With an increasing number of objectives, the discriminatory ability of Pareto dominance diminishes, making it harder to differentiate between solutions. While enhanced dominance measures, indicator-based methods, and decomposition methods have been investigated, such measures remain non-standardized in MHHO. The focus of research should be on hybrid selection mechanisms that integrate dominance, indicators, and reference vectors.
• Constraint-Handling Mechanisms: One of the most challenging aspects of applying MHHO to real engineering problems is the satisfaction of the problem’s constraints. Typical approaches utilize penalties, feasibility criteria, or repair operators that are specific to each problem, and thus may limit the generality of their methods. Instead, research should aim to develop more unified and generic constraint-handling methods that can complement MHHO’s movement strategies. For binary, mixed-integer, and combinatorial optimization problems, methods such as feasibility-preserving encodings, adaptive penalties, and multi-phase repair techniques appear promising.
• Decision-Making and Solution Selection: Selecting an appropriate solution from a Pareto set remains an unsolved problem for practitioners. VIKOR, TOPSIS, and compromise programming are post-optimization decision-making methods, and while they are commonly utilized, additional development is needed to achieve greater transparency and reliability in safety-critical domains such as healthcare, power systems, and environmental protection. There is also a need for research on the integration of explainability and sensitivity analysis into the solution selection process.
• Hybridization with Other Metaheuristics: While the MHHO algorithm can be further improved through hybridization using DE, PSO, chaotic strategies, or opposition-based learning approaches, the specific contributions of these hybrid components remain unclear. This calls for systematic ablation studies to assess the effects of these mechanisms in isolation.