Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 1423930, 22 pages

http://dx.doi.org/10.1155/2016/1423930

## Lévy-Flight Moth-Flame Algorithm for Function Optimization and Engineering Design Problems

^{1}College of Information Science and Engineering, Guangxi University for Nationalities, Nanning 530006, China^{2}Key Laboratory of Guangxi High Schools Complex System and Computational Intelligence, Nanning 530006, China

Received 18 April 2016; Accepted 12 July 2016

Academic Editor: Jose J. Muñoz

Copyright © 2016 Zhiming Li 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

The moth-flame optimization (MFO) algorithm is a novel nature-inspired heuristic paradigm. The main inspiration of this algorithm is the navigation method of moths in nature called transverse orientation. Moths fly in night by maintaining a fixed angle with respect to the moon, a very effective mechanism for travelling in a straight line for long distances. However, these fancy insects are trapped in a spiral path around artificial lights. Aiming at the phenomenon that MFO algorithm has slow convergence and low precision, an improved version of MFO algorithm based on Lévy-flight strategy, which is named as LMFO, is proposed. Lévy-flight can increase the diversity of the population against premature convergence and make the algorithm jump out of local optimum more effectively. This approach is helpful to obtain a better trade-off between exploration and exploitation ability of MFO, thus, which can make LMFO faster and more robust than MFO. And a comparison with ABC, BA, GGSA, DA, PSOGSA, and MFO on 19 unconstrained benchmark functions and 2 constrained engineering design problems is tested. These results demonstrate the superior performance of LMFO.

#### 1. Introduction

Optimization is a process of finding the best possible solution(s) for a given problem. In real world, many problems can be viewed as optimization problems. Since the complexity of problems increases, the need for new optimization techniques becomes more evident than before. Over the past several decades, some kinds of methods have been proposed to solve optimization problems and have made great progress. For example, mathematical optimization techniques used to be the only tool for optimizing problems before the proposal of heuristic optimization techniques. However, these methods need to know the property of optimization problem, such as continuity or differentiability. In recent years, metaheuristic optimization algorithms have become more and more popular in optimization techniques. Some popular algorithms in this field are Genetic Algorithms (GA) [1, 2], Particle Swarm Optimization (PSO) [3], Ant Colony Optimization (ACO) [4], Evolutionary Strategy (ES) [5], Differential Evolution (DE) [6], and Evolutionary Programming (EP) [7]. The application of these algorithms can be found in different branches of science and industry as well. Despite the merits of these optimizers, there is a fundamental question here whether there is any optimizer for solving all optimization problems. According to the No-Free-Lunch (NFL) theorem [8] for optimization, researchers are allowed to develop new algorithms solving optimization problems more effectively. Some of the latest algorithms are Artificial Bee Colony (ABC) algorithm [9], Bat Algorithm (BA) [10], Cuckoo Search (CS) algorithm [11], Cuckoo Optimization Algorithm (COA) [12], Gravitational Search Algorithm (GSA) [13], Charged System Search (CSS) [14], Firefly Algorithm (FA) [15], and Ray Optimization (RO) [16], and Dragonfly Algorithm (DA) [17].

Moth-flame optimization (MFO) [18] algorithm is a new metaheuristic optimization method through imitating the navigation method of moths in nature called transverse orientation. In this algorithm, moths and flames are both solutions. The inventor of this algorithm, Seyedali Mirjalili, proved that this algorithm is able to show very competitive results compared with other state-of-the-art metaheuristic optimization algorithms. However, the MFO algorithm has been in research stage so far, and convergence speed and calculation accuracy of this algorithm can be further advanced. To improve the performance of MFO, a Lévy-flight moth-flame optimization (LMFO) algorithm is proposed.

We know that Lévy-flight [11, 19] has a strong ability of strengthening global search and overcoming the problem of being trapped in local minima. In order to make use of the good performance of Lévy-flight, we propose a Lévy-flight moth-flame optimization. MFO and Lévy-flight have complementary advantages, so the proposed algorithm can lead to a faster and more robust method. The proposed algorithm is verified on nineteen benchmark functions and two engineering problems.

The rest of the paper is organized as follows: Section 2 presents a brief introduction to MFO and Lévy-flight. An improved version of MFO algorithm, LMFO, is proposed in Section 3. The experimental results of test functions and engineering design problem are showed in Sections 4 and 5, respectively. Results and discussion are provided in Section 6. Finally, Section 7 concludes the work.

#### 2. Related Works

In this section, a background about the moth-flame optimization algorithm and Lévy-flight will be provided briefly.

##### 2.1. MFO Algorithm

Moth-flame optimization [18] algorithm is a new metaheuristic optimization method, which is proposed by Seyedali Mirjalili and based on the simulation of the behavior of moths for their special navigation methods in night. They utilize a mechanism called transverse orientation for navigation. In this method, a moth flies by maintaining a fixed angle with respect to the moon, which is a very effective mechanism for travelling long distance in a straight path because the moon is far away from the moth. This mechanism guarantees that moths fly along straight line in night. However, we usually observe that moths fly spirally around the lights. In fact, moths are tricked by artificial lights and show such behaviors. Since such light is extremely close to the moon, hence, maintaining a similar angle to the light source causes a spiral fly path of moths.

In the MFO algorithm, the set of moths is represented in a matrix . For all the moths, there is an array for storing the corresponding fitness values. The second key components in the algorithm are flames. A matrix similar to the moth matrix is considered. For the flames, it is also assumed that there is an array for storing the corresponding fitness values.

The MFO algorithm is a three-tuple that approximates the global optimal of the optimization problems and defined as follows:

is a function that creates a random population of moths and corresponding fitness values. The methodical model of this function is as follows:

The function, which is the main function, moves the moths around the search space. This function received the matrix of and returns its updated one eventually:

The function returns true if the termination criterion is satisfied and false if the termination criterion is not satisfied:

With , , and , the general framework of the MFO algorithm is defined as follows: ; while is equal to false ; end

After the initialization, the function is iteratively run until the function returns true. For the sake of simulating the behavior of moths mathematically, the position of each moth is updated with respect to a flame using the following equation: where indicate the th moth, indicates the th flame, and is the spiral function.

Any types of spiral can be utilized here subject to the following conditions:(1)Spiral’s initial point should start from the moth.(2)Spiral’s final point should be the position of the flame.(3)Fluctuation of the range of spiral should not exceed the search space.

Considering these points, a logarithmic spiral is defined for the MFO algorithm as follows:where indicates the distance of the th moth for the th flame, is a constant for defining the shape of the logarithmic spiral, and is a random number in .

is calculated as follows:where indicate the th moth, indicates the th flame, and indicates the distance of the th moth for the th flame.

Equation (6) describes the spiral flying path of moths. From this equation, the next position of a moth is defined with respect to a flame. The parameter in the spiral equation defines how much the next position of the moth should be close to the flame ( is the closest position to the flame, while shows the farthest).

A question that may rise here is that the position updating in (6) only requires the moths to move towards a flame, yet it causes the MFO algorithm to be trapped in local optima quickly. In order to prevent this, each moth is obliged to update its position using only one of the flames in (6). Another concern here is that the position updating of moths with respect to different locations in the search space may degrade the exploitation of the best promising solutions. To resolve this concern, an adaptive mechanism provided the number of flames. The following formula is utilized in this regard:where is the current number of iteration, is the maximum number of flames, and indicates the maximum number of iterations.

The gradual decrement in number of flames balances exploration and exploitation of the search space. After all, the general steps of the function can be described in Algorithm 1.