Abstract:The paper describes an enhancement of the traditional 2-point crossover operator used for binary representation in genetic algorithms. This operator preserves a schema common to both parent chromosomes. The enhancement of its functionality is in a modified treatment of this common schema. The offspring produced by the modified operator is partially randomised so that it contains both the common schema and its binary complement. It helps to prevent the genetic algorithms from getting stuck in a local optimum and enhance the exploration of the search space beyond the limits imposed by the classical operator's functionality. The partially randomised 2-point crossover operator has been tested on a number of different problems and compared to the original 2-point operator. GENETIC ALGORITHMS AND SCHEMATAGenetic algorithms (GAs) are probabilistic search and optimisation techniques, which operate on a population of chromosomes, representing potential solutions of the given problem. In standard GA (SGA), binary strings represent the chromosomes. Each chromosome is assigned a fitness value expressing its quality reflecting the given objective function. In the main loop of SGA, chromosomes are reproduced and recombined to generate a new population of chromosomes, i.e. new sample points from hopefully more promising parts of the search space. This is repeatedly performed until some given termination-condition is fulfilled. As the population evolves, current best-so-far solution is maintained, which is at the end of the run considered as the found solution.A simple analysis of SGA's behaviour is provided by theory of schemata introduced by Holland [1]. The theory defines a schema S as a template, which defines a certain class of chromosomes. The schema of the same length as the used chromosome consists of 0s, 1s and wildcard symbols * s that can stand for either 0 or 1 (for example schema 1 1 * 0 * covers strings 1 1 0 0 0, 1 1 0 0 1, 1 1 1 0 0, and 1 1 1 0 1).In order to describe the properties of schemata we use terms as defining length δ(S), order o(S) and fitness f(S) of the schema S. The defining length is the distance between positions of the first non-wildcard and the last nonwildcard element of the schema. It expresses the compactness of the information contained in the schema. For example, 1 1 * 0 * has defining length P (0) -P (1) = 4 -1 = 3. It also expresses the number of possible crossing points that might damage the schema. The order o(S) is the number of non-wildcards in the schema S. So our schema 1 1 * 0 * is of order 3. Finally the fitness f(S) is defined as the average fitness value over all the chromosomes in the population covered by the schema S.The theory concludes in so-called Schema Theorem, which provides a rough estimation of how the schema S will be sampled in the next population when considering a reproduction based on the relative fitness of its instances and the disruptive aspects of the crossover and mutation. It states that as the population evolves the above-average schemata are re...
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