Biogeography-based learning particle swarm optimization

Abstract

This paper explores biogeography-based learning particle swarm optimization (BLPSO). Specifically, based on migration of biogeography-based optimization (BBO), a new biogeography-based learning strategy is proposed for particle swarm optimization (PSO), whereby each particle updates itself by using the combination of its own personal best position and personal best positions of all other particles through the BBO migration. The proposed BLPSO is thoroughly evaluated on 30 benchmark functions from CEC 2014. The results are very promising, as BLPSO outperforms five well-established PSO variants and several other representative evolutionary algorithms.

Appendix 1. Migration models of BBO

Ma (2010) provided six mathematical migration models for BBO. The six migration models can be used to design the biogeography-based exemplar generation method for BLPSO, and they are described as follows:

Model 1 (constant immigration and linear emigration model):

$$\begin{aligned} \left\{ {\begin{array}{l} \lambda _k =\frac{1}{2}\cdot I \\ \mu _k =\frac{k}{N}\cdot E \\ \end{array}} \right. \end{aligned}$$

(10)

Model 2 (linear immigration and constant emigration model)

$$\begin{aligned} \left\{ {\begin{array}{l} \lambda _k =\left( {1-\frac{k}{N}} \right) \cdot I \\ \mu _k =\frac{1}{2}\cdot E \\ \end{array}} \right. \end{aligned}$$

(11)

Model 3 (linear migration model):

$$\begin{aligned} \left\{ {\begin{array}{l} \lambda _k =\left( {1-\frac{k}{N}} \right) \cdot I \\ \mu _k =\left( {\frac{k}{N}} \right) \cdot E \\ \end{array}} \right. \end{aligned}$$

(12)

Model 4 (trapezoidal migration model)

$$\begin{aligned} \left\{ {\begin{array}{l} \lambda _k =\left\{ {\begin{array}{l} Ik\le {i}' \\ 2\left( {1-\frac{k}{N}} \right) \cdot I, \quad {i}'<k\le ps \\ \end{array}} \right. \\ \mu _k =\left\{ {\begin{array}{l} \left( {\frac{2k}{N}} \right) \cdot E, \quad k\le {i}' \\ E, \quad {i}'<k\le ps \\ \end{array}} \right. \\ \end{array}} \right. \end{aligned}$$

(13)

where ${i}'=ceil\left( {(ps+1)/2} \right)$

Model 5 (quadratic migration model):

$$\begin{aligned} \left\{ {\begin{array}{l} \lambda _k =\left( {1-\frac{k}{N}} \right) ^{2}\cdot I \\ \mu _k =\left( {\frac{k}{N}} \right) ^{2}\cdot E \\ \end{array}} \right. \end{aligned}$$

(14)

Model 6 (sinusoidal migration model):

$$\begin{aligned} \left\{ {\begin{array}{l} \lambda _k =\frac{1}{2}\left( {\cos \left( {\frac{k\pi }{N}} \right) +1} \right) \cdot I \\ \mu _k =\frac{1}{2}\left( {-\cos \left( {\frac{k\pi }{N}} \right) +1} \right) \cdot E \\ \end{array}} \right. \end{aligned}$$

(15)

In Eqs. (10)–(15), I and E are the maximum possible immigration and emigration rates; N is the population size; k is the index of the individual with rank k, where $k=1$ refers to the worst individual and $k=N$ refers to the best individual.