An Efficient Harris Hawks Optimization (HHO) Algorithm for Solving Numerical Expressions

1. Introduction

     The Harris hawks optimization (HHO) algorithm is a new swarm intelligence optimization paradigm proposed by Ali Asghar Heidari et al. in 2019, which is inspired by the team behaviors and chasing patterns of Harris’s hawk in nature called surprise pounce [1] The Harris’s hawk is notable for its behavior of hunting cooperatively in packs consisting of tolerant groups, while other raptors often hunt alone. Harris hawks’ social nature has been attributed to their intelligence, which makes them easy to train and have made them a popular bird for use in falconry [2]. Several hawks cooperatively pounce a prey from different directions in an attempt to surprise it. Harris hawks can reveal a variety of chasing patterns based on the dynamic nature of scenarios and escaping patterns of the prey. The effectiveness of the proposed HHO optimizer is checked, through a comparison with other nature-inspired techniques, on 29 benchmark problems and several real-world engineering problems. HHO algorithm provides very promising and occasionally competitive results compared to well-established metaheuristic techniques. Due to these characteristics, it is hard to tackle some classes of problems using conventional mathematical programming approaches such as conjugate gradient, sequential quadratic programming, fast steepest, and quasi-Newton methods [3]. The main objective of this study is to introduce a novel metaheuristic optimization namely Harris hawks’ optimization (HHO) for enhancing the accuracy of the conventional multilayer perception technique in predicting the factor of safety in the presence of rigid foundations. This algorithm has been successfully used for various scientific applications.

2. Harris Hawks Optimization (HHO) Algorithm

      HHO inspired by the exploring a prey, surprise pounce, and different attacking strategies of Harris hawks. HHO is a population-based, gradient-free optimization technique. The algorithm of Harris hawks an ’optimization (HHO) is inspired using the cooperative treatment along with the chasing manner of Harris hawk that is first expanded by Heidari et al. This algorithm has been successfully used for various scientific applications. Hawks attempting to surprise their prey and from different paths swooped on them, cooperatively [4]. In addition, Harris hawks have the ability to choose chase type according to the distinct patterns of prey flight. It has three base stages in HHO, including amaze pounce, tracking the prey, and other different sorts of attacking strategies. In a glance, the first stage is named “Exploration” and is modeled to mathematically wait, search, and discover the desired hunt. The second stage of this algorithm is transforming from exploration to exploitation, based on the external energy of a rabbit. Finally, in the third phase which is called “Exploitation”, considering the residual energy of the prey, hawks commonly take a soft and sometimes hard surround for hunting the rabbit from different directions [5].

Fig1: Inspiration of HHO Algorithm

2.1. Harris Hawks Hunting Behavior

      Harris’ Hawk is only raptor, known for cooperative hunting in packs. Harris’ Hawks are social birds and live in relatively stable group of 2–7 birds. There exists a dominance hierarchy among them. At the top of hierarchy is a mature female bird, followed by a male bird and then other birds of the pack. Harris’ hawks live in sparse woodland, marshes and semi-desert. Their diet consist of rats, squirrels, medium-sized birds, rabbits, mammals, lizards etc. Harris’ hawks hunt in packs. This cooperative hunting makes them able to feed in the harsh dessert, where the prey is scarce. It also allows them to hunt bigger prey [6]. The members of pack take turns for scanning the surroundings in search of prey and for attacking the prey. It makes them able to hunt for longer duration. Searching is done by perching on the top of power poles, standing dead trees, saguaros and spanning large area around by highly efficient vision. As the prey is spotted, other members are informed through visual displays or vocalization. In one hunting technique, hawks fly around the prey. The encircling shadows confuse the prey. One member dives to catch prey and if it misses then another member tries, while the first one gets back in the line. It is continued until prey is caught and shared.

Fig2: Harris Hawks Optimization Algorithm

The Harris’s Hawk is a medium-large raptor that lives primarily in the deserts of south western United States and Mexico. The unique feature of this raptor species is its tendency to live and hunt in large social groups. All other raptor species are solitary animals. Attributed to the scarcity of food in the desert environments that they live in, the hawks have developed complex cooperative hunting and communication techniques to locate, encircle, flush out and ultimately attack potential prey [7]. Their diet includes many of the large rabbits, lizards and small birds that live in the desert. During seasons of food scarcity these groups of hawks have also been known to go after prey that is larger than them. In order to successfully catch their fast moving prey, the hawks must coordinate and orchestrate a group attack. Biologists Bednarz and Coulson have spent numerous hours in the deserts studying and documenting the behaviors of these hawks. The biologists observed that the hawks’ day begins by assembling all members of the hunting party in the predawn light of the desert on one or more cacti near the nesting site. Hawks were heard communicating with each other by a series of chirps before beginning the hunt. Hawks divide themselves into two groups [8]. The lookout hawks perch on cacti, high trees or poles to look out for potential prey and then coordinate with ground hawks to catch the prey. The lookout hawks in groups of one to three fly out on a series of short flights ranging between 60-200m. The hawks perch on of neighboring cacti to look for potential prey on the ground. When prey is identified, lookouts communicate its location to the remaining hawks by a series of chirping calls. The remaining hawks fly from their perches diving to the ground near the communicated prey location [9]. Due to the dense brush that grows in areas of the desert floor, the prey is not always in plain sight. The ground hunters must coordinate their efforts in an attempt to flush out the prey. Several hawks will encircle the brush where the prey is hiding and take turns diving into the brush to force the prey out. Once out in the open hawks pursue the prey by hopping and running along the ground. If the prey escapes lookout hawks fly to another perch and the process begins again. Lookout hawks and ground hunters can exchange positions over time to conserve energy [10]. A successful capture leads to a meal that is shared by all hawks in the hunting party. Remaining carcasses if they can be carried are flown back to the nesting site to feed the young hawks; otherwise a few hawks are left to guard the carcasses while portions of carcasses are transported back [11].

Fig3: Process of HHO Algorithm

2.2. Steps for HHO Algorithm

  • Exploration phase
  • Transition from exploration to exploitation
  • Exploitation phase

2.2.1. Exploration phase

     The nature of Harris’ hawks, they can track and detect the prey by their powerful eyes, but occasionally the prey cannot be seen easily [12]. Hence, the hawks wait, observe, and monitor the desert site to detect a prey maybe after several hours. In HHO, the Harris’ hawks are the candidate solutions and the best candidate solution in each step is considered as the intended prey or nearly the optimum. In HHO, the Harris’ hawks perch randomly on some locations and wait to detect a prey based on two strategies.

2.2.2. Transition from exploration to exploitation

      The HHO algorithm can transfer from exploration to exploitation and then, change between different exploitative behaviors based on the escaping energy of the prey. The energy of a prey decreases considerably during the escaping behavior [13].

2.2.3. Exploitation phase

          Harris’ hawks perform the surprise pounce by attacking the intended prey detected in the previous phase. However, preys often attempt to escape from dangerous situations. Hence, different chasing styles occur in real situations.  According to the escaping behaviors of the prey and chasing strategies of the Harris’ hawks, four possible strategies are proposed in the HHO to model the attacking stage [14]. Whatever the prey does, the hawks will perform a hard or soft besiege to catch the prey. It means that they will encircle the prey from different directions softly or hard depending on the retained energy of the prey. In real situations, the hawks get closer and closer to the intended prey to increase their chances in cooperatively killing the rabbit by performing the surprise pounce. After several minutes, the escaping prey will lose more and more energy; then, the hawks intensify the besiege process to effortlessly catch the exhausted prey. To model this strategy and enable the HHO to switch between soft and hard besiege processes, the parameter is utilized [15].

      HHO provides good exploration as well as exploitation. It is achieved by concurrent exploration and exploitation. In commencing iterations some hawks use exploration, some use local exploitation and others use global exploitation. In this way exploration and exploitation occurs simultaneously [16].

2.3. Flow Chart of HHO Algorithm

Fig4: Flowchart of HHO Algorithm

2.4. Example

2.4.1. HHO Algorithm  

3. Numerical Methods of Harris Hawks Optimization (HHO) Algorithm

          Numerical methods of Harris Hawks Optimization Algorithm is given [17],

4. Applications of HHO Algorithm

  • Engineering design problems [18]
  • Satellite images segmentation
  • Air pollution Forecasting
  • Prediction of slope stability [19]
  • Two-layer foundation soils
  • Color Image Multilevel Thresholding Segmentation [20]
Fig5: Applications of HHO Algorithm

5. Advantages of HHO Algorithm

  • The main advantage of these cooperative tactics is that the Harris’ hawks can pursue the detected rabbit to exhaustion, which increases its vulnerability [21].
  • Harris hawks optimization (HHO) and dragonfly algorithm (DA) are applied to a multi-layer perception (MLP) predictive tool for adjusting the connecting weights and biases in predicting the failure probability using seven settlement key factors, namely unit weight, friction angle, elastic modulus, dilation angle, Poisson’s ratio, applied stress, and setback distance [22].
  • High efficiency.
  • Modern approaches alleviate this drawback by using stochastic components. Exploration can be further increased by using multiple agents [23].
  • Modern approaches are flexible and can be used for different kind of problems.
  • Using the Harris hawk optimization (HHO) algorithm results in obtaining the optimized threshold wavelet coefficients before applying the inverse wavelet transform [24].


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