Evolutionary Computation with Biogeography-based Optimization

6Dynamic System Models of BBO

In the last chapter, we saw that the transition matrix for a BBO Markov model is found by calculating the cumulative probabilities of migration and mutation. We now develop this construction in more detail to examine the dynamics of a population as it transitions from generation to generation. This will give us a more complete model, which is called a dynamic system model. The dynamic system model is based on the Markov model, but the application is quite different. The Markov model gives the steady-state probability of each possible population as the generation count approaches infinity. The dynamic system model gives the time-varying proportion of each possible solution in the search space as the population size approaches infinity.

Overview of the chapter

This chapter develops a dynamic system model for the basic BBO algorithm. Section 6.1 presents the basic notation that we will use in later sections. Section 6.2 derives the BBO dynamic system model and some of its properties, based on the Markov model. Section 6.3 uses our dynamic system model to solve some benchmark problems.

6.1. Basic notation

This section introduces the notation used in the BBO dynamic system model. Some of this notation may be more general or more specific in other contexts. The definitions indicated here are not universal, but nevertheless are commonly used, and more importantly for our purposes, are specifically used in this chapter.

First, we note a difference between ...