Elsevier

Knowledge-Based Systems

Volume 213, 15 February 2021, 106711
Knowledge-Based Systems

Horse herd optimization algorithm: A nature-inspired algorithm for high-dimensional optimization problems

https://doi.org/10.1016/j.knosys.2020.106711Get rights and content

Abstract

This paper proposes a new meta-heuristic algorithm inspired by horses’ herding behavior for high-dimensional optimization problems. This method, called the Horse herd Optimization Algorithm (HOA), imitates the social performances of horses at different ages using six important features: grazing, hierarchy, sociability, imitation, defense mechanism and roam. The HOA algorithm is created based on these behaviors, which has not existed in the history of studies so far. A sensitivity analysis is also performed to obtain the best values of coefficients used in the algorithm. HOA has a very good performance in solving complex problems in high dimensions, due to the large number of control parameters based on the behavior of horses at different ages. The proposed algorithm is compared with popular nature-inspired optimization algorithms, including grasshopper optimization algorithm (GOA), sine cosine algorithm (SCA), multi-verse optimizer (MVO), moth–flame optimizer (MFO), dragonfly algorithm (DA), and grey​ wolf optimizer (GWO). Solving several high-dimensional benchmark functions (up to 10,000 dimensions) shows that the proposed algorithm is highly efficient for high-dimensional global optimization problems. The HOA algorithm also outperforms the mentioned popular optimization algorithms for the case of accuracy and efficiency with lowest computational cost and complexity.

Introduction

Evolutionary Algorithms (EAs) are heuristic mimicking the Darwin principles of evolution [1]. Darwin asked two basic questions in a book entitled “Origin of Species” [2].:

  • Are today animals and plants descendants of prehistoric creatures?

  • Does one species turn into another one?

This theory is too broad to be explained in this paper. However, as documented, Darwin was aware of the positive responses to these questions [2]. EAs follow the same Darwin principles, in which a population of solutions are evolved (improve) iteratively.

Swarm Intelligence (SI) is another interesting type of intelligence in nature, which refers to the collective behavior of organisms. The SI methods have been one of the most popular research topics during recent years. They mostly focus on the design of an intelligent system with three features (adaptation, dispersion, and flexibility), which provide many opportunities to solve complex problems in various fields. A large number of SI algorithms are based on SI and inspired by the social behavior of animals [3], [4], [5].

There is a strong connection between organisms and SI, which is called interaction [6]: The collective behaviors of creatures result in SI, and SI changes the conditions. Some examples of such interactive behaviors in nature are as follows:

  • Termites build large and complex nests, and this is beyond the understanding and ability of a single termites [7].

  • Various tasks in an ant colony are detected automatically with no central management [8].

  • Waggle dance of bees is leads them to find more food [9].

  • Birds and fish are organized in optimal spatial patterns [10].

  • Bacteria follow environmental changes by using molecules [11].

The purpose of computational SI methods is to model the behaviors of creatures and their local interactions with the environment to detect more complex behaviors. These methods can be used to solve complex continuous and discrete optimization problems [12].

The use of meta-heuristic algorithms in solving various problems has increased dramatically in recent years due to the simplicity of understanding and applying such algorithms [3]. Some of the natural evolution phenomena underpinning the meta-heuristic algorithms are the evolution of several generation of creatures, refrigeration or cooling process in metals, the lifecycle of ants in a colony, the migration of birds, human defense system, etc.

There are a variety of meta-heuristic algorithms to address different types of optimization problems, which benefit from two key features, namely exploration and exploitation, to search the optimization problem spaces and to search the good-enough responses. Exploration is the capability of an algorithm to search freely, regardless of the accuracy of results. Exploitation also refers to the performance of an algorithm regarding its achievements in the previous iteration loops. Obviously, the behavior of an algorithm becomes mostly random and unpredictable when the search capabilities increase. Enhanced exploitation ratio in an algorithm, on the contrary, leads to a more cautious performance. The exploration and exploitation capabilities in an algorithm can be controlled since almost all the meta-heuristic search methods possess some adjustable parameters [4].

The main contribution of this study is presenting a novel optimization algorithm that imitates the behaviors of horses. It is shown that the proposed algorithm is applicable to solve the simple and complex single-objective high dimensional problems. This feature is tested by solving seven high-dimensional examples (500, 1000, 2000, 5000, and 10000 dimensions) and the results are then compared with the solutions of the strongest existing optimization algorithms. The rest of this paper is outlined as follows:

An in-depth literature review is presented in Section 2. Section 3 proposes the HOA algorithm, and the results and conclusions are provided in Sections 4 HOA computational complexity, 5 Parameter analysis, respectively.

Section snippets

Literature review

Meta-heuristic optimization algorithms can be classified in three main categories: Evolutionary algorithms, physic-based algorithms, and SI algorithms.

The first category has been inspired by the idea of evolution in nature. These algorithms are based on the theories of Darwin, as mentioned. This theory is an optimization process, aiming at improving an organisms’ ability to survive in a dynamic environment. Living environment is one of the factors, which plays a vital role in determining the

Horse Optimization Algorithm (HOA)

This work is based on the behavior patterns of horses in their living environment. The horses’ behavior patterns generally include Grazing (G), Hierarchy (H), Sociability (S), Imitation (I), Defense mechanism (D), and Roam (R) [66], [67], [68]. So, this algorithm is inspired by the six mentioned general behaviors of horses at different ages. The movement applied to horses at each iteration is according to Eq. (3.1). XmIter,AGE=VmIter,AGE+XmIter1,AGE,AGE=α,β,γ,δwhere, XmIter,AGE indicates the

HOA computational complexity

Computational complexity examines the problem-solving time of a particular method. In other words, the computational complexity of an algorithm is used as a parameter in estimating the computational cost of the method. The computational cost of meta-heuristic algorithms is also estimated based on the number of search agents, the number of problem dimensions, and the maximum number of iterations [71]. Employing the six factors in the movement of the horses, causes the HOA algorithm achieves a

Parameter analysis

Several parameters affect the performance of HOA, as it described in the Horse Optimization Algorithm section. Now the sensitivity analysis of the response to changes in parameters is performed, in order to gain a correct understanding of the effectiveness of these parameters. Changing the g (Grazing) factor has no significant effect on the resulting responses and only increases or decreases the search range of each horse. The following factors are examined in sensitivity analysis:

  • Hierarchy

Results and discussion

The results of HOA are presented in this section. As this algorithm is compared with the Grasshopper Optimization Algorithm (GOA) [61], Sine Cosine Algorithm (SCA) [27], Multi-Verse Optimizer (MVO) [39], moth–flame Optimization (MFO) [58], Dragonfly Algorithm (DA) [57], and Gray Wolf Optimizer (GWO) [53] algorithms.

It should be mentioned that the number of horses and iterations in all the algorithms is set to be 50 and 1000, respectively, to conduct a fair comparison. Statistical tests are

Conclusion

The present study proposed a fast and robust optimization algorithm, which was inspired by the general behaviors of horses at different ages, and employed to solve highly complex optimization problems. The HOA algorithm was benchmarked using seven well-known test functions at high dimensions, and it was found out that this algorithm was highly efficient in terms of exploration and exploitation. The results (namely the best cost, SD, p-value of Wilcoxon test, and CPU runtime) confirmed the

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgment

The authors would like to thank to Dr. Seyed Ali Mirjalili for his inestimable helps and comments on this work.

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