Modelling and Simulation in Engineering

Volume 2018, Article ID 4945157, 14 pages

https://doi.org/10.1155/2018/4945157

## Application and Development of Enhanced Chaotic Grasshopper Optimization Algorithms

^{1}Swami Keshvanand Institute of Technology, Jaipur 302017, India^{2}Malaviya National Institute of Technology, Jaipur 302017, India

Correspondence should be addressed to Akash Saxena; moc.liamtoh@anexas.hsakaa

Received 30 December 2017; Revised 23 March 2018; Accepted 8 April 2018; Published 23 May 2018

Academic Editor: Gaetano Sequenzia

Copyright © 2018 Akash Saxena et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In recent years, metaheuristic algorithms have revolutionized the world with their better problem solving capacity. Any metaheuristic algorithm has two phases: exploration and exploitation. The ability of the algorithm to solve a difficult optimization problem depends upon the efficacy of these two phases. These two phases are tied with a bridging mechanism, which plays an important role. This paper presents an application of chaotic maps to improve the bridging mechanism of Grasshopper Optimisation Algorithm (GOA) by embedding 10 different maps. This experiment evolves 10 different chaotic variants of GOA, and they are named as Enhanced Chaotic Grasshopper Optimization Algorithms (ECGOAs). The performance of these variants is tested over ten shifted and biased unimodal and multimodal benchmark functions. Further, the applications of these variants have been evaluated on three-bar truss design problem and frequency-modulated sound synthesis parameter estimation problem. Results reveal that the chaotic mechanism enhances the performance of GOA. Further, the results of the Wilcoxon rank sum test also establish the efficacy of the proposed variants.

#### 1. Introduction

Optimization is a term which refers to the selection of the best option amongst the given set of alternatives. Examples of optimization processes are everywhere such as in business, human resource management, challenging engineering design problems, transportation, profit making propositions, and industrial applications. Optimization can be done for the maximization of any proposition or minimization of any proposition. In engineering problems particularly, the use of maximization is for efficiency maximization, classification accuracy maximization, and revenue or profit maximization, and on the other hand, minimization can be performed for cost, loss, risk, and execution time of any engineering process. Apart from these classifications of optimization, another classification of the optimization problem can be done on the basis of constraints. An optimization problem without any constraints is called unconstrained optimization; similarly another type is constrained optimization with linear and nonlinear constraints. Another classification can be done on the basis of the objective of the optimization; when an optimization problem aims towards a single objective, it is called the single objective optimization problem, and similarly when it aims towards multiobjectives, the same is called the multiobjective optimization problem [1]. A recent trend is to employ metaheuristic optimization algorithms to solve challenging problems of the real world. The term refers to problem-independent higher level heuristic mechanism [2]. In recent years, applications of metaheuristic algorithms in engineering problems have been reported. The successful and effective implementation of these algorithms on real applications has attracted the attention of researchers to work in this direction.

The metaheuristic optimization approaches can be subdivided into three categories:(1)Evolutionary computing-based algorithms [3–5](2)Physics law-based algorithms [6–8](3)Swarm intelligence-based algorithms [9–16]

Evolutionary-based algorithms are based on natural evolution due to environmental pressure [2]. These algorithms employ selection or mixing criterion to generate an optimal solution set which possesses higher fitness values. Basic virtues of these algorithms are of stochastic nature, incorporating crossover and termination operators for hybridizing the solutions and enhancing the fitness value. A few examples of these algorithms are Genetic Algorithm [4] and Evolution Strategy and Evolutionary Programming [5]. Another class of algorithms is the algorithms which are inspired from physics and based on the laws of fundamental physics. A few examples of these algorithms are Gravitational Search Algorithm [14], Big Bang-Big Crunch Algorithm [7], and Black hole Algorithm [8].

The third category is based on swarm intelligence methods, where the cognitive and social behavior of the natural swarms like birds and school of fish is mimicked in the form of simulation. The most famous algorithm in this category is Particle Swarm Optimization (PSO), which works on the philosophy “Follow the Leader” [9]. Other examples of these algorithms are Bat Algorithm [10], Firefly Algorithm [11], and Cuckoo Search Algorithm [12]. A recently published swarm algorithm, which became popular nowadays, is Grey Wolf Optimizer (GWO), the algorithm that mimics the hunting behavior of grey wolves and is a fine example of the compliance of the social hierarchy of the wolf pack during searching, attacking, and hunting phases. A novel algorithm based on crow behavior named as Crow Search Algorithm (CSA) has been proposed [14]. CSA mimics the behavior of crow to store their excess food in hiding places and retrieve it when it is needed. Similarly, Ant Lion Optimizer Algorithm [15] and Grasshopper Optimisation Algorithm [16] are also a good example of social mimicry of the natural swarms.

The applications of swarm algorithms are very well reported in the literature and in many design problems, namely, Automatic Generation Control [17], Unit Commitment [18, 19], Feature Selection [20], and Ambient Air Quality Classification [21].

Many optimization algorithms have employed chaotic sequences over the random walk (random numbers generation) due to the fact that the random walk not always implements the global search well. Thus in some cases, the algorithm development is based on chaotic variables instead of random variables, and these algorithms are called chaotic algorithms [22–26]. A chaotic Firefly Algorithm was proposed in [22]. In this work, attractive movement of fireflies was simulated with ten chaotic maps. Chaotic sequences are used for parameters and in the Chaotic GWO approach [27]. Chaos enhanced Accelerated Particle Swarm Optimization (CAPSO) which was proposed by Gandomi et al. [25]. An attraction parameter was tuned with normalized chaotic maps in that work. Chaotic Bat Algorithm was proposed in [23], and the tuning of the crucial parameter of this algorithm was done with the help of chaotic maps. Different ten chaotic maps were employed in gravitational search algorithm in [28].

Two mechanisms: diversification and intensification are essential parts of any swarm algorithm. The initial phase of any swarm algorithm started with random search; usually this process swifts and holds responsibility to search every possible direction of the search space, and thus, the process is random in nature. On the other hand, the intensification process is strategic. The outcome of this process is specific and is treated as the solution of the problem. It is empirical to say that speed of these processes is different. Every algorithm employs a bridging mechanism to maintain a good amount of trade-off between these two processes. Some algorithms use different operators and different models of random walks in different phases, and in short, these operators/mechanisms help the algorithm to maintain a fair balance between these two processes. This paper investigates the impact of different chaotic sequences on the bridging mechanism of the GOA, by evaluating the performance of the proposed variants on standard benchmark functions and real applications. 10 different chaotic sequences are embedded with the parameter , and the careful observation is presented. Following research objectives are framed for this work:(1)To employ 10 different chaotic maps through normalized function to propose chaotic variants of GOA. These variants are developed on the basis of adaptive, chaotic, and monotonically decreasing parameter .(2)To conduct a nonparametric Wilcoxon rank sum test for observing the efficacy of the chaotic variants with the GOA by observing values.(3)To apply these variants on three-bar truss design and parameter estimation for frequency-modulated sound waves and compare the performance with the other contemporary algorithms.

Remaining part of the paper is organized as follows: in Section 2, brief details of different chaotic maps are incorporated. In Section 3, an overview of GOA is presented. The development of chaotic variants is explained in Section 4. Simulation results on benchmark problems and engineering optimization problems are presented in Section 5. Last but not the least, major conclusions of this study have been presented in Conclusion.

#### 2. Chaotic Map

In this section, the definitions of different chaotic maps are presented. Table 1 shows the definition range and names of the chaotic maps. These maps have also been studied in the approaches [22, 28]. The shape of these maps with starting point is shown in Figure 1.