A hybrid Harris hawks-moth-flame optimization algorithm including fractional-order chaos maps and evolutionary population dynamics

Highlights

• A hybrid algorithm combining Harris Hawks Optimizer (HHO) and Moth-Flame Optimization (MFO) is developed.

• The Fractional-Order gauss and 2xmod1 Chaotic Maps are employed to generate the initial population.

• The operators of MFO are integrated into HHO in order to improve exploration.

• Evolutionary Population Dynamics (EPD) is also applied to prevent premature convergence and stagnation in local optima.

• The new algorithm is successfully tested in 36 mathematical problems and 4 constrained design problems.

Abstract

This paper proposes a modified version of a contemporary metaheuristic named Harris Hawks Optimizer (HHO) that mimics the foraging strategies used by Harris hawks. It is first argued that exploration ability of HHO is weaker than its exploitation. In addition, the initial position of hawks has the greatest impact on the convergence of the solutions in a similar manner to other metaheuristic algorithms. Then, we applied the Fractional-Order Gauss and 2xmod1 Chaotic Maps to generate the initial population as well as applying the operators of the Moth-Flame Optimization (MFO) to improve the exploration of HHO. In addition, the concept of evolutionary Population Dynamics (EPD) is applied to prevent premature convergence and stagnation in local optima. The method proposed in this work is called FCHMD and evaluated using a set of thirty-six mathematical functions and five engineering problems. The results of the FCHMD are compared with a number of well-known metaheuristics. It can be observed that the FCHMD algorithm outperforms its competitors on the majority of case studies.

Introduction

In recent years, global optimization techniques have gained more attention since they are applied to various applications, including but not limited, to engineering [1], computer science [2], [3], [4], finance [5], medicine [6], [7], and industry [8]. Metaheuristic algorithms (MH) methods are considered the most popular global optimization techniques with a high strength to handle different optimization problems regardless of the nature of the problem itself [9], [10]. Such a behavior is not exhibited by classical optimization algorithms that need more information about the problem and cannot apply to other problems directly [11], [12]. metaheuristic algorithms are characterized by their ability to solve challenging problems despite being mostly stochastic and not deterministic. According to the inspiration of the MH methods, they can be categorized into three main classes of swarm-based, evolutionary-based, and physics-based metaheuristics.

The swarm-based methods proved their efficiency in different applications when compared with other kinds of MH techniques. Some of them are whale optimization algorithm (WOA) [13], grey-wolf optimization [14], salp swarm algorithm (SSA) [15], particle swarm optimization (PSO) [16]. The success of the methods to solve different optimization problems motivated the researchers in the field of soft computing to develop more efficient techniques.

In the same context, Heidari et al. [17] developed a metaheuristic method called Harris Hawks Optimizer (HHO) simulating the foraging characteristics of Harris Hawks in nature. Similar to other MH methods, HHO consists of two phases (i.e., exploration and exploitation abilities), and each of them is determined according to the energy of the hawks. It was demonstrated in the original paper of HHO that this algorithm is very competitive compared to the existing methods. HHO has been successfully used in several fields like image processing [18], [19], [20], prediction models [21], renewable energy applications [22], [23], [24], feature selection [25], [26], engineering application [27] and etc. [21], [28], [29], [30], [31], [32], [33].

The moth-flame optimizer is a recently developed algorithm to mimic the traverse orientation of moths at night [34]. Its good performance has been proved in many application areas. In addition, MFO has some advantages such as simple formulation, easiness of implementation and flexibility. In general, MFO combines the local search mechanism with population-based method which leads to the balancing between exploration and exploitation. Hence, the inherent simplicity of MFO has drawn the attention of many researchers. For example, Yueting et al. [35] suggested a modified MFO based on mutation strategy and tested using CEC2017 and CEC2005 benchmarks. In [36], authors used two chaotic strategies to improve the performance of MFO. The first chaotic map is applied to generate initial population, while, the second one, named chaotic disturbance, is applied to improve the best solution. This avoided trapping in local optima. The enhanced MFO variant thus developed (denoted as CMFO) was utilized to determine the parameters of kernel extreme learning machine and selecting the relevant features at the same time. Yueting et al. [10] applied chaotic local search to enhance the MFO, and used the Gaussian mutation to improve the diversity. The modified CLSGMFO algorithms was evaluated using 23 classical functions and compared with other MH techniques. The results proved that CLSGMFO outperforms other methods. In addition, it was utilized to determine the parameters of KELM to enhance the prediction of financial stress. As per the literature, integrating the chaos maps with the MFO enhanced its diversification capability thus allowing to avoid local optima solutions due to their unique and dynamic characteristics.

In order to overcome the limitations of HHO and develope a robust variant of this algorithm, we suggest a modification using the fractional order chaos maps to upgrade the initial population which provides the proposed method with high ability to select suitable initial values. This leads to enhance the acceleration ability toward the global solution. In addition, the exploration of HHO is improved by using the operators of moth-flame optimization algorithm (MFO). Moreover, the concept of evolutionary population dynamics [37] is used to update the worst solutions during the optimization process, which reduces the effect of these worst solutions on the quality of the population.

The main contribution of the current work can be outlined as follows:

• Improve the performance of the HHO using the Fractional order chaos maps, moth flame optimization operators, and evolutionary population dynamics.

• Evaluate the quality of two FOC maps to enhance the generation of the initial population of HHO.

• Usings the operators of MFO to improve the exploration of the HHO during the search process.

• Apply the EPD concept to update the worst solutions during the optimization process.

• Evaluate the performance of the FCHMD using a set of 36 mathematical functions from CEC2005 benchmark and 5 engineering problem

The present paper is organized into the following sections: a general overview of the basic version of the HHO algorithm, the MFO techniques, and the applied fractional order chaos maps is presented in Section 3. A detailed explanation for the proposed algorithm is introduced in Section 4. Test problems and optimization results are presented and discussed in Section 5. Finally, the conclusion and future work are given in Section 6.