To maintain the population diversity of genetic algorithms (GAs), we are required to employ an appropriate population diversity measure. However, commonly used population diversity measures designed for permutation problems do not consider the dependencies between the variables of the individuals in the population. We propose three types of population diversity measures that address high-order dependencies between the variables to investigate the effectiveness of considering high-order dependencies. The first is formulated as the entropy of the probability distribution of individuals estimated from the population based on an m-th--order Markov model. The second is an extension of the first. The third is similar to the first, but it is based on a variable order Markov model. The proposed population diversity measures are incorporated into the evaluation function of a GA for the traveling salesman problem to maintain population diversity. Experimental results demonstrate the effectiveness of the three types of high-order entropy-based population diversity measures against the commonly used population diversity measures.

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