We introduce a new parameter to discuss the behavior of a genetic algorithm.This parameter is the mean number of exact copies of the best fit chromosomesfrom one generation to the next. We argue that the genetic algorithm shouldoperate efficiently when this parameter is slightly larger than $1$. Weconsider the case of the simple genetic algorithm with the roulette--wheelselection mechanism. We denote by $\ell$ the length of the chromosomes, by $m$the population size, by $p_C$ the crossover probability and by $p_M$ themutation probability. We start the genetic algorithm with an initial populationwhose maximal fitness is equal to $f_0^*$ and whose mean fitness is equal to${\overline{f_0}}$. We show that, in the limit of large populations, thedynamics of the genetic algorithm depends in a critical way on the parameter$\pi \,=\,\big({f_0^*}/{\overline{f_0}}\big) (1-p_C)(1-p_M)^\ell\,.$ Ourresults suggest that the mutation and crossover probabilities should be tunedso that, at each generation, $\text{maximal fitness} \times (1-p_C)(1-p_M)^\ell > \text{mean fitness}$.
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机译：我们引入了一个新的参数来讨论遗传算法的行为，该参数是从第一代到第二代的最合适染色体精确拷贝的平均数。我们认为，当此参数略大于$ 1 $时，遗传算法应可有效运行。我们考虑采用轮盘赌-轮盘选择机制的简单遗传算法的情况。我们用$ \ ell $表示染色体的长度，用$ m $表示种群的大小，用$ p_C $表示交叉概率，用$ p_M $表示突变概率。我们从初始种群开始遗传算法，该种群的最大适应度等于$ f_0 ^ * $，其平均适应度等于$ {\ overline {f_0}} $。我们表明，在人口众多的情况下，遗传算法的动力学在很大程度上取决于参数$ \ pi \，= \，\ big（{f_0 ^ *} / {\ overline {f_0}} \ big ）（1-p_C）（1-p_M）^ \ ell \，。$我们的结果表明，应当调整突变和交叉概率，以便在每一代中，$ \ text {maximumfitness} \ times（1-p_C）（ 1-p_M）^ \ ell> \ text {平均适合度} $。
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