The paper formulates a genetic algorithm that evolves two types of objects in a plane. The fitness function promotes a relationship between the objects that is optimal when some kind of interface between them occurs. Furthermore, the algorithm adopts an hexagonal tessellation of the two-dimensional space for promoting an efficient method of the neighbour modelling. The genetic algorithm produces special patterns with resemblances to those revealed in percolation phenomena or in the symbiosis found in lichens. Besides the analysis of the spacial layout, a modelling of the time evolution is performed by adopting a distance measure and the modelling in the Fourier domain in the perspective of fractional calculus. The results reveal a consistent, and easy to interpret, set of model parameters for distinct operating conditions.
This paper analyzes the fractional order dynamics during the search for the optimal solution in a plane with an hexagonal tessellation by means of a genetic algorithm. These three distinct scientific topics are recognized to be efficient approaches in particular areas, namely, in the problems of modelling including long-range memory effects, space representation using geometric shapes with no overlaps and no gaps, and robust optimization in cases where standard techniques do not yield adequate solutions. This paper integrates the three methodologies in the analysis of a complex evolutionary optimization for producing solutions somehow resembling the percolation phenomenon, in the inorganic world, or, alternatively, the lichens, in the scope of living beings.
Fractional calculus (FC) is a branch of mathematical analysis that generalizes the operations of differentiation and integration from integer up to real or complex orders [
Genetic algorithms (GAs) are a computer heuristic that mimics the process of evolution and belong to the class of the so-called evolutionary algorithms [
The hexagonal tessellation is a regular tiling of the Euclidean plane, in which each vertex meets three hexagons [
The three scientific concepts are put together for simulating and modelling an evolutionary process in a two-dimensional space. First, it is considered a plane where some kind of process evolves. The plane is discretized by means of a regular hexagonal pattern, and the evolution consists of the optimization using a standard GA. Second, the evolution of the GA population is described using a fractional order model that approximates the numerical results. For that purpose, the best individual in each generation of the GA population is analysed in the viewpoint of fitness function, compared with the previous case, and the result is converted into the Fourier domain. An important aspect is also the fitness function that measures the “performance” of each GA individual. In the two-dimensional space are considered two distinct types of objects, and it is assumed that a “good performance” corresponds to a spacial arrangement exhibiting some type of interface between them. By other words, it is assumed that some kind of cooperation, or synergy, exists between the two objects, such that they should coexist close to each other in space. The resulting time-space population reveals fractal characteristics and patterns resembling those of percolation [
Bearing these ideas in mind this paper is organized as follows. Section
In this section are introduced briefly some aspects of FC and Laplace transform and the computational implementation of GA.
The most used definitions of a fractional derivative of order
Using the Laplace transform, for zero initial conditions, we have the expression
In the scope of FC it is also important to mention the Mittag-Leffler function
The Mittag-Leffler function is a generalization of the exponential and the power laws. The first occurs in phenomena governed by integer dynamics, and the second emerges in fractional dynamics. In particular, when
These results mean that standard methods in modelling and control, such as transfer functions and frequency response, can be directly applied as long as we allow the substitution of integer orders by their fractional counterparts.
GAs are a computer method to find approximate solutions in optimization problems. GAs are implemented such that a population of
The pseudo-code of a GA is as follow: generate randomly the initial population of individuals (solutions); evaluate the fitness function for each individual in the population; repeat: select the individuals with best fitness value for reproducing; treat the population by means of the crossover and mutation operators and produce offspring; evaluate the fitness value of each individual in the offspring; replace the worst ranked part of previous population by the best individuals of the produced offspring; until termination.
In this section we describe the experiments with the GA and we analyse the results in the perspective of fractional dynamics.
We consider a two-dimensional space subdivided in discrete cells and where three types of situation may occur. The cells consists of a tessellation using regular hexagons as represented in Figure
Hexagonal tessellation of the plane and definition of neighbour set
Each cell has three possible situations, namely, “
In the GA these objects interact by means of a fitness function
In the numerical experiments neighbours with more cells were tested, namely, with a second and a third ring having 12 and 18 hexagons and having different weights. Nevertheless, the results were qualitatively of the same type, and, therefore, these experiments are not reported in this paper. Furthermore, for testing the surrounding cells different fitness functions were also evaluated. Again, while leading to different plots, the results were not significantly distinct and are not described in the sequel.
We must note that fitness (
For the crossover operation a simple one-point scheme is considered. First, the indices
In this subsection experiments with GA populations of
Figures
GA result for
GA result for
The analysis of the GA dynamics [
Figure
Plot of
It was decided to identify a parametric model in the Fourier domain since usually it leads to a simple and robust procedure. Therefore, the Fourier transform of each time response,
For example, Figure
Polar diagram of
Given the large number of combinations of values for
Figures
Variation of the transfer function parameter
Variation of the transfer function parameter
Variation of the transfer function parameter
Variation of the transfer function parameter
We observe that the parameters of the transfer function (
We verify that we can model the GA dynamical behaviour in terms of a simple fractional order model. In fact, stochastic results of the GA seem to be of minor influence in the proposed modelling scheme, not only due to the comprehensive variation of the parameters, but also because several numerical experiments with distinct seeds lead to similar results. Nevertheless, we have a phenomenal modelling perspective based on the analytical approximation and supported by the outcome results. Therefore, a new challenge is the inverse problem. By other words, the problem of defining the fractional order model remains open and, as a consequence, designing the GA rules that produce such dynamical behaviour. In the scope of this problematic it can not be forgotten that the GA optimizes a symbiotic behaviour using a regular hexagonal space tessellation. Therefore, at a higher level several problems remain open such as the design of other fitness functions, the effect of other tessellation methods, or the phenomena produced by a larger number of object types in the GA population.
This paper presented a GA dynamical evolution and its description by means of a fractional model. The GA adopts a tessellation of the space using regular hexagons in order to provide an efficient scheme to handle the neighbour cells in the numerical discretization. Furthermore, the GA includes a fitness function that evaluates positively the interface between distinct objects in the population. This scheme has an abstract nature but can be interpreted as representing a simplified version of some kind of interaction between distinct materials or, even, as symbiotic relation between two different species. During the experiments several conditions for the GA evolution were tested, namely, with initial populations having different number of objects and distinct percentages of each type. For describing the space-time GA dynamics a simple measure was defined. The index, inspired in the notion of distance, captures the differences between two consecutive best individuals. It is verified, in all cases, that the proposed description leads to simple models capable of being traduced by analytical expressions of fractional order.