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Lecture 11Evolutionary Computation: Genetic algorithms Why genetic algorithm work?

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Schema Theorem GA techniques have a solid theoretical foundation

(Holland, 1975; Goldberg, 1989; Rawlins 1991; Whitley, 1993)

John Holland introduced the notation of schema which came from Greek word meaning “form”.

A schema is a set of bit strings of ones, zeros and asterisks, where each asterisk can assume either value 1 or 0.

The ones and zeros represent the fixed positions of a schema, while asterisks represent ‘wild cards’

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For example, schema .

Stands for a set of 4 bit strings. Each string in this set begins with 1 and ends with

0. These strings are called instances of the schema.

1 * * 0

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Relationship Between a Schema and a Chromosome

A chromosome matches a schema when the fixed positions in the schema matches the corresponding positions in the chromosome. e.g the schema H

Matches the following set of 4-bit chromosomes:

These chromosomes of said to be instances of H.

1 1 1 0

1 1 0 0

1 0 1 0

1 0 0 0

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The number of defined bits (non-asterisks) in a schema is called the order.

Schema H:

has two defined bits and thus the order is 2.

Order

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GAs manipulate schemata when they run. If GAs use a technique that makes the probability

of reproduction proportional to chromosome fitness, then according to the Schema Theorem, we can predict the presence of a given schema in the next chromosome generation.

In other words, we can describe the GA’s behaviour in terms of the increase of decrease in the number of instances of the a given schema.

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Effects caused by the Crossover and Mutation

Crossover and mutation can both create and destroy instances of a schema.

Destructive effects (effects that decrease the number of instances of the schema H).– Schema will survive after crossover if at least

one of its offspring is also its instance. This is the case when crossover does not

occur within the defining length of the schema.

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Defining Length of a Schema The distance between the outermost defined

bits of a schema is called defining length. E.g.

Defining length = 3.

Defining length = 5.

Defining length = 7.

* * * * 1 0 1 1

* 0 * 1 * 1 0 *

1 * * * * * * 0

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If crossover takes place within the defining length, the schema H can be destroyed and offspring that are not instances of H can be created.

The schema H will not be destroyed if two identical chromosomes cross over, even when crossover occurs within the defining length.

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11)(

lpP dlc

cH

The probability that the schema H will survive after crossover can be defined as:

 

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Destructive Effect of mutation  

nm

mH pP )1()(

It is also clear that the probability of survival under mutation is higher for low-order schemata than for high-order ones.

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Destructive Effects of crossover and Mutation: