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Another interesting selection mechanism is accomplished in the form of the elitist strategy - we select the best individuals in the population and carry them over without any alteration to the next population of the strings. Once the selection has been completed, the resulting new population is subject to the two main GA mechanisms such as a recombination (or more specifically, a crossover) and a mutation.

In the sequel, we will discuss a number of alphabets used in GA techniques and elaborate on the encoding and decoding phases. For the time being, it is instructive to consider the generic form of the GA search space being formed by the binary strings

where y ∈ [0, 1]m and “m” denotes a number of bits in each string. If x ∈ R then y becomes a binary equivalent of x. The encoding transforms a real number into its binary equivalent. Similarly, the decoding algorithm returns a decimal equivalent of the binary string. The number of bits (the length of the string) specifies a decimal resolution, and, subsequently implies a binary-to-decimal decoding error. Evidently, the higher the number of bits, the lower the decoding error.

For such binary strings the basic operations of crossover and mutation can be described as follows.

Crossover A one-point crossover identifies two strings of the population and randomly selects a position in the strings at which they interchange their content. Fig. 5.4 illustrates the concept of crossover in more depth. In this case the crossover happens at the fifth bit producing two new offsprings 1 0 1 0 0 1 0 0 and 0 1 0 1 1 0 1 1.


Figure 5.4  An example of a single-point crossover

The crossover operation leads to an increased diversity of the population of strings as new individuals emerge out of this process. The intensity of crossover is characterized in terms of its probability. The higher the probability, the more individuals are affected by the crossover. As emphasized in the literature (Davis, 1991; Goldberg, 1989), the crossover is regarded as the form of the fundamental mechanism of GAs.

Mutation The mutation operator is an example of an operator adding an extra diversity of a stochastic nature. In binary strings this mechanism is implemented by flipping the values of some randomly selected bits, Fig. 5.5. Again, the mutation rate is related to a probability at which the individual bits become affected. For instance, a mutation rate of 5% when applied to a population of 500 strings each being 20 bits long means 5% of 1,000 bits being changed.1


1In general, the length of a genotype and the mutation rate are related. For instance, the length of t-RNA is 80 with the mutation rate 5∗10-2. Similarly, for E coli the length 4∗106 and the mutation rate is 10-6. For a human and phage QB these numbers are 3∗109 and 4500 and 10-9 and 5∗10-4, respectively.


Figure 5.5  Mutation mechanisms in binary strings

Generally, the operation of crossover can be viewed as a special type of recombination operation which involves two parents (strings) and leads to two offsprings. The recombination, however, is more universal and subsumes the crossover mechanism as its special case. We will discuss recombination in Section 5.5.2.

In summary, the GA scheme can be concisely summarized as outlined in Fig. 5.6.


Figure 5.6  GA computing paradigm

5.4. Schemata Theorem - a conceptual backbone of GAs

Let us start off with a scheme - a similarity template used to analyze and discuss similarities between the chromosomes (Holland, 1975). In what follows, we discuss the most condensed encoding scheme whose alphabet is built up out of 0 and 1. In addition to these two symbols we introduce a wildcard (don’t care) symbol*. Any string composed of elements 0, 1, and * is called a scheme. Due to the don’t care elements, the scheme S represents a family of strings (a hyperplane or a subset of the GA search space). For example, the scheme (1*0 1*) matches four strings, namely, (1 0 0 1 0), (1 0 0 1 1), (1 1 0 1 0), and (1 1 0 1 1).

Evidently, “p” don’t care conditions in a scheme match 2p binary strings.

The schemata are described by two important characteristics.

order The order, o(S), expresses the number of 0 and 1 in the scheme. In other words

o(S) = the length of the template - number of don’t care entries in S

For instance, the order of S1

is 4,

The scheme S2

includes more don’t care conditions.

We say that S1 is more specific (less general) than S2.

The defining length (length) of S, δ(S), is the distance between the first and the last fixed string positions. For the two previous schemata we obtain

The defining length describes the compactness of the scheme. As shown in Fig. 5.7, the two characteristics identify the strings of various levels of compactness and specificity.


Figure 5.7  Four classes of schemata in the order - length plane

Confining ourselves to the three - bit space (m=3), we illustrate schemata of different order, Fig. 5.8. Any schema of order 3 is just a vertex of the unit cube. The schemata of order 2 are viewed as the respective edges of the cube. With the increasing order, o(S) = 1 we obtain a plane. For the lowest order, o(S) = 0, we end up with the entire cube. This geometrical interpretation also holds for m greater than 3, however, in this case we are dealing with the respective highly dimensional unit hypercubes.


Figure 5.8  Selected schema in a three-dimensional search space (m=3)

The fitness of the scheme is computed by averaging the fitness values of all strings in the population that match this scheme. Denote the fitness of S by fit(S),

where “r” is the number of strings matching S.

Considering that the i-th string xi is selected with the probability

The number of strings matching scheme S in the next population equals

where N is the size of the entire population and

Let us rearrange the above expression to gain a better insight into the relative fitness of the scheme

where <fit> is an average fitness of the individual string in the population


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