Enhancing the Harris’ Hawk optimiser for single- and multi-objective optimisation

Proposed by Heidari et al. (2019), the HHO algorithm is a swarm-based, nature-inspired metaheuristic model. It imitates the collaboration behaviours of Harris’ hawks working together to search and chase for a prey in different directions. A variety of hunting strategies are performed by Harris’ hawks based on the prey’s escaping energy. The mathematical model pertaining to the behaviours of Harris’ hawks are formulated for solving optimisation problems. The HHO algorithm has received extensive interest from researchers due to its effectiveness and fast convergence speed; however, the HHO algorithm suffers from the following problems: Enhancements of HHO have been proposed, e.g. hybridisation of HHO, modification of HHO and chaotic HHO (Alabool et al. 2021). Hybridisation is a popular method to improve the performance of an algorithm. HHO has been integrated with other optimisation methods to find the solution faster, avoid falling into local optima, provide better solution quality, and improve algorithm stability (Chen et al. 2020a; Gupta et al. 2020; Ewees and Abd Elaziz 2020). Chen et al. (2020a) integrated the DE algorithm and a multi-population strategy into HHO to overcome its convergence issue and prevent HHO from being trapped in local optima. In addition, opposition-based learning was applied to HHO by Gupta et al. (2020) to reduce the number of opposite solutions and increase the convergence speed. HHO has also been used to improve the performance of other algorithms, e.g. accelerating the search process of multi-verse optimisation (MVO) and maintaining the MVO population (Ewees and Abd Elaziz 2020).

Other HHO modifications include improving the convergence speed, increasing the population diversity, and enhancing the transition from exploration to exploitation (Alabool et al. 2021). HHO was integrated with a chaotic map to improve the balance between exploration and exploitation (Zheng-Ming et al. 2019; Qu et al. 2020). Specifically, Zheng-Ming et al. (2019) used a tent map in the exploration phase to identify the optimal solution rapidly, while Qu et al. (2020) employed a logistic map in the escaping energy factor to balance global exploration and local exploitation. The use of HHO for solving MOPs was investigated in Islam et al. (2020) and Jangir et al. (2021). Table 1 summarises the main enhancement methods of HHO.

Introduced by Price (1996) and Storn and Price (1997), DE is well-known for its simplicity, fewer control parameters and superior performance in optimisation. Storn and Price (1997) employed DE to solve global optimisation problems. However, one major problem of all metaheuristics algorithms is the potential of early convergence and local optima. Leveraging on DE’s capability in finding better and diverse solutions, many researchers have proposed integration of DE and other p-metaheuristics algorithms. As an example, Wang and Li (2019) enhanced the solution diversity by integrating DE and the GWO algorithm. An adaptive DE was combined with GOA by Jia et al. (2019) for multi-level satellite image segmentation. The hybrid algorithm was evaluated with a series of experiments using satellite images. It was able to increase search efficiency and population diversity through iterations. In addition, hybrid models of DE and HHO can elevate the performance of the original HHO algorithm. Birogul (2019) introduced HHODE, which blends HHO and DE, which uses mutation operators from DE to harness its exploration strengths. The hybridisation manages to strike a balance between exploratory and exploitative tendencies. It was tested on an altered IEEE 30-bus test system for optimal power flow problem and proved to be more effective than other optimisation algorithms. Besides, DE is used by Chen et al. (2020a) in HHO to boost its local search capability and exploit the DE operations, such as mutation, crossover and selection, to discover better and more diverse solutions. A new HHO variant was introduced by Islam et al. (2020) to solve the power flow optimisation problem. Multiple researchers conducted studies to demonstrate hybridisation of DE with HHO for improving the HHO performance. Table 1 summarises the main publications in this topic (Birogul 2019; Bao et al. 2019; Wunnava et al. 2020; Abd Elaziz et al. 2021; Abualigah et al. 2021; Liu et al. 2022). In short, DE is useful for maintaining a more diverse and quality individual without losing the convergence speed in HHO.

Chaos theory helps enhance the population diversity and cover wider search areas that an algorithm may miss. As mentioned before, HHO has a poor changeover from exploration to exploitation; hence, its susceptibility to local optima. Besides, HHO exhibits a poor performance in tackling high-dimensional and multimodal problems. In this respect, chaos theory is often applied to an optimisation algorithm to enhance their global search capability (Ewees and Abd Elaziz 2020; Chen et al. 2020b; Liu et al. 2020; Dhawale et al. 2021; Hussien and Amin 2022). A variety of chaotic maps are available: Chebyshev, circle, intermittency, iterative, Leibovich, logistic, piecewise, sawtooth, sine, singer, sinusoidal and tent map. Researchers popularly use logistic and tent maps to improve the HHO performance (Zheng-Ming et al. 2019; Menesy et al. 2019; Chen et al. 2020a). Chen et al. (2020a) introduced chaotic sequence into the escaping energy of the rabbit in HHO to enhance the global searchability and cover broader search areas. A chaotic map was incorporated into the adaptive mutation strategy and integrated into the external archive by Liu et al. (2020) to attain a set of diverse non-dominated solutions. Moreover, Xie et al. (2020) employed a chaotic strategy to assign weights for the leader wolf in GWO to increase the diversity of the leader wolves. Barshandeh and Haghzadeh (2021) integrated a chaos strategy to initialise the population and improve the solution diversity, allowing a larger search space to be covered. Table 1 Chaos shows the key publications on the use of chaos theory for improvement of the HHO algorithm.

MOPs are similar to SOPs in terms of problem definition, but with multiple objective functions that cannot be efficiently solved by a single-objective algorithm. A set of equilibrium solutions exists for solving an MOP, i.e. a set of non-inferior solutions or Pareto optimal solutions (Fonseca and Fleming 1998). An MOP can be formulated as follows:where n is the number of design variables, o is the number of objective functions, r is the number of inequality constraints, s is the number of equality constraints, \(g_{j}\) and \(h_{j}\) are the j-th inequality and equality constraints, respectively, and \(Lb_{i}\) and \(Ub_{i}\) are the lower and upper bounds of the i-th design variables.

In a single-objective function, solution \(\mathbf {x_1}\) is better than \(\mathbf {x_2}\) if \(f(\mathbf {x_1}) < f(\mathbf {x_2})\); however, this definition cannot be used in a MOP that has two or more objective functions. Using the Pareto dominance concept in solving MOP, solution \(\mathbf {x_1}\) is better than (dominates) \(\mathbf {x_2}\) if all objective functions \(f(\mathbf {x_1}) < f(\mathbf {x_2})\) (Fonseca and Fleming 1998). Two solutions are denoted as non-dominated if, for at least one objective function, but not all of them, \(f(\mathbf {x_1}) \nless f(\mathbf {x_2})\).

Recently, p-metaheuristics algorithms have been utilised to solve MOPs due to their population-based search capabilities in yielding multiple solutions in a single run. The Pareto dominance concept is used to determine the PS. Most p-metaheuristics algorithms employ non-dominated ranking and Pareto strategy to ensure the diversity of PS. Several useful p-metaheuristics algorithms that store the best PS when there are multiple optimum solutions are: NSGA-II (Deb et al. 2002), SPEA2 (Zitzler et al. 2001), and MOPSO with an archive (Coello et al. 2004). Mirjalili et al. (2016) discussed several multi-objective p-metaheuristics algorithms, such as multi-objective grey wolf optimiser (MOGWO) (Mirjalili et al. 2016), multi-objective ant lion optimiser (MOALO) (Mirjalili et al. 2017b), multi-objective multi-verse optimisation (MOMVO) (Mirjalili et al. 2017a) and multi-objective grasshopper optimisation algorithm (MOGOA) (Mirjalili et al. 2018), to solve MOPs. Jangir and Jangir (2018) embedded the non-dominated sorting into GWO to solve several multi-objective engineering design problems and apply it to a constraint multi-objective emission dispatch problems with economic constrained and integration of wind power. Liu et al. (2020) modified a MOGWO with multiple search strategies to solve MOPs. Hence, this paper extended the EHHO to a multi-objective EHHO that incorporated the non-dominated sorting from NSGA-II to sort and rank the non-dominated solution by calculating the crowding distance between solution sets to ensure a diverse PS. An external archive with a mutation strategy is utilised to diversify the obtained PS.

HHO was originally proposed by Heidari et al. (2019) to solve SOPs. It mimics the hunting strategy of harris’ hawks, focusing on their "surprise pounce" and "seven kills" to catch a prey (e.g. a rabbit). The hawks collaborate with each other in chasing or attacking the prey, and change their attack mode dynamically based on the escaping pattern of the prey. The hawk’s behaviour is modelled in three main stages: exploration, the transition from exploration to exploitation, and exploitation. Figure 1 presents the different stages of HHO, while the pseudocode of the HHO algorithm is presented in Algorithm 1.

The HHO algorithm begins by initialising a fixed number of randomised individuals (population), in which every individual represents a candidate solution. The individuals are random within the lower and upper boundaries of the problems. The fitness value of each individual is calculated every iteration until the stopping criteria are satisfied.

The key mathematical formulation of each HHO stage is as follows:

Exploitation stage: Soft besiege (\(r \ge 0.5 \text {and} |E |\ge 0.5\))Hard besiege (\(r \le 0.5\) and \(|E |< 0.5\))Soft besiege with progressive rapid dives(\(r < 0.5\) and \(|E |\ge 0.5\))Hard besiege with progressive rapid dives (\(r < 0.5\) and \(|E |\le 0.5\))where \(X(t+1)\) is the position vector of an individual in the next iteration t, \(X_\textrm{rabbit}(t)\) is the best position of a prey, X(t) is the current position of the individual, \(X_\textrm{rand}(t)\) is the random position of the individual, \(X_m(t)\) is the average position of the population, \(r_1\),\(r_2\), \(r_3\), \(r_4\), \(r_5\), q, v and \(\beta \) are random factors within [0,1], while LB and UB are the lower and upper bounds of the search space, respectively. In addition, E denotes the escaping energy of the prey, \(E_0\in (-1,1)\) is the initial state of the prey’s energy, \(e\in (2,0)\) is the exploration factor, and T is the maximum number iterations (e.g. 100 iterations). Besides, J is the jumping strength of the prey and \(\Delta X(t)\) denotes the difference between the position of an individual with the best position of the prey, while LF, D and S represent the dimensions of the search space (Heidari et al. 2019; Chen et al. 2020a). The detailed processes of the original HHO algorithm are shown in Algorithm 1 (Heidari et al. 2019).

A nonlinear exploration factor is introduced to replace the linear version from the original HHO algorithm. In the HHO algorithm, the exploration factor, e controls the transition pattern from exploration to exploitation as defined in Eq. (4), which decreases linearly from 2 to 0, resulting in an acute shrinkage of the search space and insufficient search attention. Multiple studies in the literature have explored adaptive control of diverse search operations to improve the conversion between exploration and exploitation. Trigonometric (Mirjalili 2016; Gao and Zhao 2019), exponential (Mittal et al. 2016; Long et al. 2018a), and logarithmic-based functions (Gao et al. 2008; Long et al. 2018b) functions are useful for controlling the exploration factor nonlinearly. In this research, we use a nonlinear exploration factor proposed by Xie et al. (2020), which is a combination of the trigonometric functions, i.e. cos and sin, and a hyperbolic function, i.e tanh. This nonlinear exploration factor aims to amplify the diversification capabilities in the early exploration phase and smoothen the changeover from exploration to exploitation. The proposed nonlinear exploration factor and newly proposed escaping energy are defined as follows:where \(e'\) is the newly proposed exploration factor to govern the switch between exploration to exploitation, t and Max\(_\textrm{iter}\) are the current iteration and maximum iteration of each iteration, respectively. Coefficient k controls the descending slope of the exploration factor, \(e'\), while \(k = 5\) is adopted, as used in Xie et al. (2020). The new exploration factor and escaping energy descending pattern are presented in Fig. 2a, b, respectively. As shown in Fig. 2a, the proposed nonlinear exploration factor decreases with a descending slope. The first half of the search path has a more effective exploration rate, while the second half has a lower exploration rate. The search space is significantly expanded in the first half, which helps EHHO to focus on exploration and finish the search with exploitation.

Developed by Price (1996), Storn and Price (1997), DE is a robust yet straightforward EA for optimising real-valued, multi-modal functions. DE is utilised in EHHO to enhance its searchability and solution quality. Three primary processes exist in DE, namely mutation, crossover and selection (Price 1996; Storn and Price 1997). They are described in the following subsection.

By observing the quantitative results from different classes of benchmark functions, the AVG and STD measures from 30 independent runs indicate the performance and stability of EHHO. The EHHO results versus those from other competitors in dealing with UM and MM functions (F1–F13) on 30, 100, 500 and 1000 dimensions are tabulated in Tables 2, 3, 4 and 5. Table 8 shows the EHHO results versus those from competitors in dealing with MM and CM functions. The best AVG and STD scores of EHHO are in bold. The statistical results of UM, MM, and CM benchmark functions are tabulated in Tables 6, 7, 8 and 9. The average ranking results are tabulated in Tables 7, 8, 9 and 10.

From Table 2, EHHO generates superior results on F2, F4–F7, and F9–F13. From the statistical evaluation veiwpoint, EHHO outperforms its competitors in all 30-dimensional UM functions at the 95% confidence level, as indicated by \(p < 0.05\) in Table 6. Moreover, EHHO achieves better results than other models on F1, F2, F4–F6, and F9–F13 on 100-dimensional UM functions, as shown in Table 3. As indicated in Table 6, EHHO performs significantly better than its competitors at the 95% confidence level. It can be noticed that EHHO yields the best results on 9 out of 13 500-dimensional benchmark problems, as presented in Table 4. In addition, Table 6 shows EHHO performs statistically better than its competitors, achieving comparable results than those of HHODE/rand/1, HHODE/best/1, HHODE/current-to-best/2, and HHODE/best/2. From Table 5, EHHO depicts its superior performance with 10 out of 13 best results on the benchmark problem with 1000-dimension. Comparing with other models statistically, EHHO performs significantly better than its competitors at the 95% confidence level, and no statistically difference with HHODE/current-to-best/2. From the results, integrating DE can certainly achieve more diverse solutions, leading to the attainment of the global optimum solution. In summary, EHHO proves to be highly effective in solving the F4–F6 and F9–F13 benchmark functions. Furthermore, EHHO outperforms other algorithms in almost all instances, and achieves the most optimal global solution for F9 and F11 in search spaces ranging from 30 to 1000 dimensions.

According to the results presented in Table 8, EHHO outperforms other models on several functions, including F14–F17, F19, F21–F22, and F25–F27. It also depicts similar performance to other models on F23 and F29. However, when it comes to hybrid composition functions F18, F20, F24, and F28, EHHO has a relatively lower performance on those hybrid composition functions. From Table 10, the average ranking results reinforce the findings from the statistical analysis, providing additional insights into the comparative performance of algorithms on benchmark functions F14–F29. EHHO is inferior to HHODE/current-to-best/2, HHODE/rand/1, HHODE/best/1, HHODE/best/2, and HHO, but it outperforms GA, PSO, BBO, FPA, GWO, BAT, FA, CS, MFO, TLBO, and DE. These findings provide valuable insights into the strengths and weaknesses of the EHHO algorithm on specific benchmark functions.

Based on the results, it can be seen that using a hybrid of HHO and DE method can enhance the diversity of solutions and prevent local optima. By incorporating a chaotic mutation rate in DE, the population diversity can be further improved while avoiding stagnation. Additionally, the ergodicity and randomness of the chaotic mutation assist EHHO in avoiding premature convergence. The nonlinear escaping factor further enhances EHHO in transitioning from exploration to exploitation, which enabling its attainment of the global optimum solution. Overall, EHHO outperforms the original HHO algorithm in solving SOPs. However, there are certain composition functions, such as F18, F20, F24, and F28, where HHODE performance proposed by Birogul (2019) is superior to that of EHHO.