Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 2314927, 23 pages

https://doi.org/10.1155/2017/2314927

## Enhancing the Performance of Biogeography-Based Optimization Using Multitopology and Quantitative Orthogonal Learning

State Key Laboratory of Advanced Electromagnetic Engineering and Technology, School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

Correspondence should be addressed to Jinfu Chen; nc.ude.tsuh.liam@ufnijnehc

Received 18 April 2017; Revised 16 July 2017; Accepted 7 August 2017; Published 13 September 2017

Academic Editor: Thomas Hanne

Copyright © 2017 Siao Wen 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

Two defects of biogeography-based optimization (BBO) are found out by analyzing the characteristics of its dominant migration operator. One is that, due to global topology and direct-copying migration strategy, information in several good-quality habitats tends to be copied to the whole habitats rapidly, which would lead to premature convergence. The other is that the generated solutions by migration process are distributed only in some specific regions so that many other areas where competitive solutions may exist cannot be investigated. To remedy the former, a new migration operator precisely developed by modifying topology and copy mode is introduced to BBO. Additionally, diversity mechanism is proposed. To remedy the latter defect, quantitative orthogonal learning process accomplished based on space quantizing and orthogonal design is proposed. It aims to investigate the feasible region thoroughly so that more competitive solutions can be obtained. The effectiveness of the proposed approaches is verified on a set of benchmark functions with diverse characteristics. The experimental results reveal that the proposed method has merits regarding solution quality, convergence performance, and so on, compared with basic BBO, five BBO variant algorithms, seven orthogonal learning-based algorithms, and other non-OL-based evolutionary algorithms. The effects of each improved component are also analyzed.

#### 1. Introduction

Optimization problems have widely existed in any areas of life (economic, engineering, medicine, business, urban planning, etc.) and they play a very vital role in both academic research field and industrial production. They tend to be more complicated with the unceasing progress of science and technology. To handle such challenging optimization problems, much effort has been devoted by researchers and different techniques have been proposed during the last several decades. The proposed methods can be categorized into two main groups: derivative-based algorithms and artificial intelligence methods. Derivative-based algorithms require objective functions to be smooth and differentiable. Due to this property, they are restricted to be applied to many complex optimization problems. Oppositely, artificial intelligence methods do not require certain properties to be satisfied. In artificial intelligence methods, evolutionary algorithms (EAs) are popular and have been successfully applied to real-world problems. EAs mimic various social behaviors existing in nature to solve the optimization problems. Some popular EAs include genetic algorithm (GA) [1, 2], particle swarm optimization (PSO) [3, 4], artificial immune system (AIS) [5], differentiable evolution (DE) [6, 7], ant colony optimization (ACO) [8], artificial bee colony (ABC) [9, 10], and simulated annealing algorithm (SA) [11].

As an effective EA, biogeography-based optimization (BBO) [12] has shown excellent performance after being tested on various benchmark functions [13–15], such as simple structure, fewer parameters than many other EAs, and strong robustness against the variations of its algorithm-dependent parameters [12]. Due to its excellent performance, it has been successfully applied to a variety of real-world problems [16–20]. However, BBO has also been shown to have certain weaknesses. Although BBO has excellent exploitation ability, it tends to fall into local optima. That is to say, BBO lacks exploration ability. As we know, for an optimization algorithm, exploration and exploitation are two important concepts in any optimization algorithms. Exploration is the ability to search for areas far from the current individuals in the search space while exploitation is the capacity to investigate the vicinities nearby the current solution. Obviously, they are two entirely different objectives, so developing an algorithm good on both is a challenging work. To remedy it, much effort has been made, and some variants of BBO are proposed. One of the most popular research hotspots focuses on modifying the migration process. Li et al. [21] proposed a perturb migration operator to create new individuals to update the target individual. Ma and Simon [22] proposed a blended migration operator that can make two parental habitats contribute different constant weighted features to a new feature of an offspring. Xiong et al. [23] proposed a polyphyletic migration operator to utilize as many as four habitats’ features to construct a new solution vector, which can generate new features from more promising areas in the search space. Li and Yin [24] introduced a multiparent migration operator which adopts three consecutive individuals to generate three offspring according to their fitness value. From the experimental results of [21–24], it can be seen that a well-designed migration operator indeed contributes to improving the performance of BBO. Besides migration operator modification, some other researchers focus on hybridizing BBO with some operators in other EAs so as to make up for BBO’s defects. BBO is hybridized with harmony search (HS) to obtain HSBBO in study [25], which aims to bring the exploration of HS into BBO. Study [26] combines DE with BBO to develop DBBO algorithm. Study [27] hybridizes opposition-based learning with BBO to obtain oppositional BBO (OBBO). All these algorithms have achieved excellent performance.

In this paper, both migration process modification and hybridization strategy are focused on. Two defects of BBO are found out by analyzing the characteristics of the migration operator, which is the main operator of BBO. One is that, due to global topology and direct-copying migration strategy, information in several good-quality habitats tends to be copied to the whole habitats rapidly, which would lead to premature convergence. The other is that the solutions generated by migration operator are distributed only in some specific regions so that other areas where competitive solutions may exist cannot be investigated. To remedy the former, a new migration operator precisely developed by taking topology modification and copy-mode improvement as two cut-in points is proposed. Diversity mechanism is also proposed for remedying it. To remedy the latter defect, quantitative orthogonal learning (QOL) process accomplished based on space quantizing and orthogonal design is adopted. It aims to investigate the solution space thoroughly so that more competitive solutions can be obtained.

The remainder of this paper is organized as follows. Section 2 briefly introduces the basic principle of BBO. The new proposed algorithm is described in Section 3. The experimental results conducted on various benchmark functions are shown in Section 4 to verify the effectiveness of the algorithm. In Section 5, the effects of different improved components are analyzed. Section 6 is devoted to conclusions.

#### 2. The Proposed Methodology

BBO is inspired by the equilibrium theory of island biogeography. Each individual in population is called a “habitat.” The goodness of a habitat (i.e., candidate solution) is measured by the habitat suitability index (HSI) [12]. Habitats with higher HSI tend to have more species than those with lower HSI. In BBO, the immigration rate and emigration rate of each habitat are functions of the number of species in the habitat and they are calculated as follows:where , are maximum immigration and emigration rate, respectively, is the number of habitats, is the number of species in habitat , and is the largest possible number of species that a habitat can support. From (1), it can be seen that a good habitat tends to have a large value of and a small value of , which enables poor habitats to get information from good habitats to improve their quality.

There are two main operators in BBO: () migration operator and () mutation operator. In migration process, based on and , information is probabilistically emigrated from the good habitats to the poor ones. The quality of those poor habitats is enhanced by accepting new information from high-quality habitats. Hence, both good and bad habitats can collaborate with each other and move together toward more promising areas in the search space and finally converge to an optimal solution. In mutation process, some variables in each habitat are possible to be replaced by new variables which are randomly generated in the entire solution space, which aims to keep population diversity.

#### 3. The Proposed Methodology

In allusion to the defects of the migration operator in basic BBO, a new BBO variant based on multitopology migration and QOL, called MTQLBBO, is proposed. The key points of our methods are described in detail in this section.

##### 3.1. Population Initialization

Assuming that the number of decision variables in (population ) is , the initialization of is expressed as follows:where is the th decision variable in , and , is an -dimensional vector which contains a Latin Hypercube Sampling of values from the interval , and is the th element in vector .

##### 3.2. Multitopology Migration Operator

Migration operator is the main operator of BBO algorithm, because it determines how the new population is generated from the previous population. So, it influences BBO’s search trajectory from the initial population. To enhance the performance of BBO, it is very necessary to analyze the characteristics of the migration operator.

During the migration process, we first use to decide (immigrated habitat). The habitats with greater values of are more likely to be selected as . Then use to decide (emigrated habitat) from all other habitats except . Also, the habitats with greater values of are more likely to be selected as . As a result, it is from several habitats with great values of that tends to get information and directly copies information from them. Moreover, each habitat is possible to get information from any other habitats and information in each habitat is possible to be emigrated to any other habitats, which means the migration process is based on global topology, as shown in Figure 1(a). In global topology, habitats have strong connection with each other and information spreads very quickly, just like virus propagation, according to social network theory [28]. Due to global topology and direct-copying migration mechanism, as mentioned above, information in several good-quality habitats would be copied to the whole habitats soon. Although it can rapidly improve the quality of bad habitats, all candidate solutions (i.e., habitats) would be homogeneous soon, resulting in poor exploration ability. In allusion to the defect, multitopology migration operator is developed by modifying topology and copy mode.