A Schema Theorem-Type Result for Multidimensional Crossover

Most of the genetic algorithms (GAs) used in practice work on linear chromosomes (e.g. binary strings or sequences of some other types of symbols). However some results have been published revealing that for certain problems multidimensional encoding and crossover may give better results than the one dimensional (linear) ones [1, 2, 3]. While some theoretical results have been obtained, no clear criteria are known for deciding the suitable dimensionality of the encoding to be used for a given problem.In this paper we consider a class of problems for which we define a multidimensional encoding and a corresponding genetic operator. We show that for a genetic algorithm (GA) using this encoding and operator we can obtain theoretical results similar to (under certain conditions even better than) those known for linear encoding. We demonstrate these theoretical results using a set of test examples.