Mathematical Problems in Engineering

Volume 2015, Article ID 423642, 15 pages

http://dx.doi.org/10.1155/2015/423642

## Hybrid Biogeography Based Optimization for Constrained Numerical and Engineering Optimization

^{1}State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Baoding 071003, China^{2}Electronics-Controlling Lab of Construction Vehicle, Jilin University, Changchun 130022, China

Received 16 April 2014; Accepted 18 July 2014

Academic Editor: Baozhen Yao

Copyright © 2015 Zengqiang Mi 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

Biogeography based optimization (BBO) is a new competitive population-based algorithm inspired by biogeography. It simulates the migration of species in nature to share information. A new hybrid BBO (HBBO) is presented in the paper for constrained optimization. By combining differential evolution (DE) mutation operator with simulated binary crosser (SBX) of genetic algorithms (GAs) reasonably, a new mutation operator is proposed to generate promising solution instead of the random mutation in basic BBO. In addition, DE mutation is still integrated to update one half of population to further lead the evolution towards the global optimum and the chaotic search is introduced to improve the diversity of population. HBBO is tested on twelve benchmark functions and four engineering optimization problems. Experimental results demonstrate that HBBO is effective and efficient for constrained optimization and in contrast with other state-of-the-art evolutionary algorithms (EAs), the performance of HBBO is better, or at least comparable in terms of the quality of the final solutions and computational cost. Furthermore, the influence of the maximum mutation rate is also investigated.

#### 1. Introduction

With the development of science and engineering, related optimization problems become more and more complex. Optimization methods are being confronted with great challenges brought by some undesirable but unavoidable characteristics of optimization problems such as being high-dimensional, nondifferentiable, nonconvex, noncontinuous, and so on. Efficient optimization methods are urgently required by the complicated optimization problems in the real world. Therefore, various evolutionary algorithms have been applied to solve difficult optimization problems in recent decades, which include GAs [1], particle swarm optimization approach (PSO) [2], DE [3, 4], ant colony optimization (ACO) [5], artificial bee colony strategy (ABC) [6, 7], and BBO [8, 9].

Biogeography based optimization (BBO) is a new population-based algorithm. It simulates the blossom and extinction of species in different habitats based on the mathematical model of biogeography. Decision variables of better solutions tend to be shared in the migration operation and decision variables of each solution are probabilistically replaced to improve the diversity of population in mutation operation. Due to good search ability, BBO has been applied to PID parameter tuning [10], parameter estimation of chaotic system [11], complex system optimization [12], satellite image classification [13], and so forth.

In comparison with other EAs, owing to direct-copying-based migration and random mutation, exploration ability of BBO is not so efficient despite outstanding exploitation. In other words, BBO can be easily trapped into local optimum and suffer from premature convergence owing to lack of corresponding exploration to balance its exploitation.

In order to overcome the weakness of BBO, lots of improved BBO variants have been proposed. Ma [14] presented six different migration mathematical models and made basic improvements on the migration operator. Gong et al. [15] hybridized the DE with the migration operator of BBO to balance the exploration and exploitation of BBO. Motivated by blended crossover operator in GAs, Ma and Simon [16] got decision variables of an offspring by blending corresponding decision variables of two parent individuals based on different weighting constants. Li and Yin [17] proposed multiparent migration in which three consecutive individuals are chosen to generate three new individuals by basic BBO and then the new individuals are modified like multiparent crossover in GA; besides, the mutation was improved based on Gaussian operator. Li et al. [18] updated the decision variables not selected in migration by generating a perturbation from the neighborhood solutions and Gaussian operator was integrated into mutation. Wang and Xu [11] integrated DE mutation operator into migration operator of BBO and simplex search was introduced to improve the searching accuracy. Sayed et al. [10] formed new decision variables of an offspring by combining corresponding decision variables from two different parents with weighted constants related to the rank of their fitness in migration. Boussaïd et al. [19] proposed a new hybrid BBO in which new solutions are first generated by DE mutation and then modified by migration of original BBO. Xiong et al. [20] utilized four individuals’ features to construct a new solution in proposed polyphyletic migration and orthogonal learning was introduced to further enhance converge speed toward global optimum.

In order to balance the exploration and exploitation of BBO, a new hybrid BBO called as HBBO is proposed in the paper. The unique points of HBBO are shown as the following. On one hand, a new hybrid mutation operator combining DE mutation and SBX is presented in HBBO while operators of EAs are often hybridized with migration operator in most of BBO variants. On the other hand, HBBO provides a new method to extend BBO to optimize constrained problems well due to only a few BBO variants available for constrained optimization in previous literatures. In addition, DE is applied to evolve one half of population to improve convergence speed further and chaotic search is introduced to enhance the diversity of population. Experiments have been conducted on twelve benchmark functions and four engineering optimization problems, and HBBO is compared with many other state-of-the-art algorithms from the quality of solutions obtained and computational cost. Furthermore, the influence of maximum mutation rate on HBBO is studied.

The rest of the paper is organized as follows. Constrained optimization, basic BBO, mutation strategies of DE, and SBX are briefly introduced in Section 2. In Section 3, the HBBO method proposed in the paper is specifically depicted. The comparison with six state-of-the-art algorithms on twelve benchmark functions is presented in Section 4. In Section 5, HBBO is compared with other methods on four well known engineering optimization problems. Section 6 further demonstrates the efficiency of HBBO and presents the investigation on the influence of maximum mutation rate. Finally, the work is concluded in Section 7.

#### 2. Preliminary

##### 2.1. Constrained Optimization

Constrained optimizations are always inevitable in scientific study and engineering design. A general constrained optimization problem can be written as the followingwhere represents the solution vector , is the dimensionality of a solution in the paper, is the number of inequality constraints, and is the number of equality constraints. In common practice, equality constraints are often transformed to inequality constraints with a given small tolerance . For example, the equality constraint above can be converted to . In the paper, the feasible-based rule by Deb [21] is applied to handle constraint. In the constraint handling mechanism, fitness value and constraint violation are considered separately based on the following criterions: (a) any feasible solution is preferred to any infeasible solution; (b) between two feasible solutions, the one having smaller objective function value is preferred; (c) between two infeasible solutions, the one having smaller constraint violation is preferred.

##### 2.2. Biogeography Based Optimization

Biogeography is the study of the distribution of species on earth surface over time. BBO is proposed based on the mathematical model of biogeography by Simon in 2008 [8]. In BBO, every solution is analogous to a habit; habit suitability index (HSI) is utilized to measure habits just like fitness function in other EAs; the elements that characterize habitability are called suitability index variables (SIVs) which are identical to the decision variables in other EAs. A good solution is similar to a habitat with high HSI which have a large number of species and vice versa. The species in habitats with high HSI tend to emigrate to habitats with low HSI. That is, habitats with high HSI tend to share their features while habitats with low HSI are inclined to accept the features from good habitats.

In BBO, each individual evolves by immigration and mutation operator. The SIVs of individuals are probabilistically shared in migration operator as shown in Algorithm 1, where is the* j*th SIV of* i*th individual in the population, here represents the immigration rate of , and is the emigration rate of , which are related to the number of species in the corresponding habitat; NP is the population size in the paper.