How to keep good schemata using cross-over operators for permutation problems

The schemata theory proposed by Holland in 1975 for the genetic algorithm approach is based on a binary representation of the problem solutions. When the description of the solution needs some more complex representation (called generally symbolic representation) and when the one-point classical cross-over operator must be replaced by some more complicated operator, then the environment of the schemata theory disappears and other conditions must be taken into account in order to ensure the efficiency of the genetic algorithms. In a previous paper, we have already proposed some performance indicators, which try to extend the basic schemata theory for permutation problems, and we have tested the quality of a list of cross-over operators using these indicators. In the present paper, we show how an analytical approach may be developed in order to avoid the experimental approach for some permutation cross-over operators and some indicators.