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Specifically, we give the distribution of the number of correct alleles in the best element of the population, and the expected number of correct alleles over the entire population following each generation. We then compare the theoretical performance against empirical runs. The results challenge the traditional explanation for efficacy of genetic algorithms. A great deal of research over the past twenty years has shown the efficacy of genetic algorithms in studying or solving complex problems.
Mathematical programming constitutes one important subclass of the use of genetic algorithms see Goldberg  or Michalewicz . There has been less work explaining how and why the approach works.
Papers along these lines included Markov chain models such as Goldberg and Secrest , Horn  and Suzuki . Probabilistic models include Ankenbrandt  and Rabinovich and Widgerson . Much of this literature concentrates on characteristics of the population rather than of the individual chromosome.
A classic example is the Schema Lemma Holland . The objective of this paper is to present a probabilistic, transient, chromosome-level analysis of specific genetic algorithm implementations to gain a better understanding of their mechanisms.
Consiaer the class of discrete optimization problems that can be encoded for a genetic algorithm such that each gene may take any of a finite set of alleles. An example is the multiple-choice integer program Hadj-Alouane and Bean . Problems of this 1. In this paper we will predict theoretically the behavior of this algorithm by deriving the distribution of the best element of the population and the population fitness average following each generation.
We make a number of assumptions that allow this analysis. Empirical tests will then evaluate the reasonability of these assumptions and value of the understanding drawn from the model. They begin with the concepts of general genetic algorithms as described in Holland  and Goldberg .
The MCGAs use specific operators selected to facilitate the probabilistic analysis that follows. Reproduction is not carried out by the traditional roulette wheel. The roulette wheel introduces complex dependencies between the probability of being selected as a parent and the fitness values of all members of the population. Rather, we choose parents randomly over all elements of the prior generation. The computational versions of the algorithm in previous papers employ an elitist strategy.
This operator will not be included in this paper. Traditional one-point crossover introduces dependencies among genes within a chromosome based on index. This dependence is often arbitrary and complicates analysis. We employ a uniform crossover Syswerda . After two parents are chosen, a coin is flipped for each gene. Heads selects the allele from the first parent and tails selects from the second. Analysis of the mechanics of this operator are core to this paper. We present a new model of the operator relative to existing work such as Syswerda , and Spears and De Jong [a, b].
Uniform crossover of two parents can create two offspring, the second being constructed from the alleles not selected above. In post-tournament selection PTS , we construct and evaluate both offspring and retain only the superior offspring in the next generation! Results show that this approach strongly promotes a "survival-of-the-fittest" upward pressure.
Rather than the traditional mutation operation, we use immigration. In each gen 2. During subsequent generations, its genetic material is spread throughout the population. Hence, immigration prevents premature convergence and loss of genetic material as traditional mutation does. Immigration also leads to a trivial proof of convergence of the algorithm to an optimal solution based on random sampling of a finite space. However, empirical tests have shown that the algorithm converges much faster than would be predicted by sampling.
Crossover is making a major contribution. Problem Definition We have a population of m chromosomes. Each chromosome has n genes. Assume that the cardinality of Xj is a, independent of j. For the purpose of the following analysis, we assume a unique optimal solution so that each gene has a unique correct allele. In order to develop results with some generality, we assume a surrogate fitness measure that counts the number of correct alleles in a chromosome.
These assumptions are not reasonable in most cases. Most problems have multiple optima. The surrogate fitness measure ignores nonlinear effects. However, the assumptions allow an analysis of the algorithm that hopefully will lead to insights that generalize beyond the assumptions.
By Assumption 1, there is a single correct allele for each gene. Let be be the indicator random variable of the event that xkj has the correct allele. By Assumption 2, Sk is the fitness function we will analyze. The last position is filled with a randomly generated 3. The multiple-choice genetic algorithm to be studied can be formally stated: Immigration Randomly generate one chromosome as in the initialization step.
Evaluate Evaluate each Sk. Derivation of the Distribution In Section 2. Independence of offspring is discussed in Section 2. By construction, all xA and bO are independent random variables.
Then SO has a binomial distribution with parameters n,p. We begin without PTS and add it below. To facilitate this analysis we make one substantive assumption. Given that the parents have J1 and J2 correct alleles, respectively, the location of the correct alleles are distributed uniformly within the parent's chromosomes and are independent of each other.
Assumption 3 will be true in early generations, but becomes less accurate as the population converges and building blocks appear. The impact of this assumption will be judged empirically. Five gene example of correct allele indicators From Assumption 3, the bk for the two parents can be viewed as zeros and ones randomly placed in a 2 x n grid. See Figure 1 for an example with chromosome length five.
For each gene there are three possible cases: Both parents have the correct allele as in Figure 1, gene 5 2. Exactly one parent has the correct allele as in Figure 1, gene 1 5. Neither parent has a correct allele as in Figure 1, gene 2 Define , and as the number of genes falling into each of these three cases.
For any gene in case 1, the offspring will certainly have the correct allele. For a gene in case 3, there is no chance of generating a correct allele. Without loss of generality, fix the correct allele indicators for the second parent in the grid. Then we can begin crossover by randomly placing the indicators of parent 1 so that there are exactly J1 ones.
This is probabilistically equivalent to random selection without replacement. Consider an urn with n balls of which J2 are red. Select J1 balls from the urn.
The number of red you select is , the number of genes in case 1. Hence has a hypergeometric distribution with parameters n, J1, J2. The mathematical statement follows from the definition of hypergeometric. We can now condition on to find the distribution of Sk given its parentage. These must all be at genes in case 2, one per gene. In the derivation below, assume Jo is in this range.
It is based on structure independent of crossover. The smallest value changes upward. Without PTS, it is possible that we get unlucky such that most correct alleles are not selected by the Bernoulli coin-flips. With PTS, these lost correct alleles are simply moved to the alternative offspring and selected during the tournament. Given parents with J1 and J2 correct alleles and a uniform crossover with PTS, the minimum number of correct alleles in any offspring is [F J2 ].
At least half of these alleles are in the better of the two offspring. In all other instances, either offspring may total Jo correct alleles. Equation 2 follows from adding the combinatorial coefficient in 1 corresponding to the appropriate number of correct alleles being assigned to one offspring, and the 8.
In 1 the offspring generated has the same expected fitness as the average of the parents. With PTS, the offspring is superior to the parents. Figure 2 shows a plot of this factor. It is, approximately, one correct allele higher than without PTS.
The distribution of Sk, the immigrant, is given by F -. Independence of individuals we must understand the stochastic interdependencies between members of a population.