Computer implementation of genetic algorithm
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Computer Implementation of Genetic Algorithm. By: Moch. Rif’an. Codings. The principle of meaningful building blocks is simply this:

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Computer Implementation of Genetic Algorithm

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Computer implementation of genetic algorithm

Computer Implementation of Genetic Algorithm

By:

Moch. Rif’an

Computer Implementation


Codings

Codings

  • The principle of meaningful building blocks is simply this:

    • The user should select a coding so that short, low order schemata are relevant to the underlying problem and reltively unrelated to schemata over fixed position.

Computer Implementation


Computer implementation of genetic algorithm

  • The second coding rule, the principle of minimall alphabets, is simply stated:

    • The user should select the smallest alphabet that permits a natural expression of the problem

Computer Implementation


Encoding methods

Chromosome A

10110010110011100101

Chromosome B

11111110000000011111

Chromosome A

1  5  3  2  6  4  7  9  8

Chromosome B

8  5  6  7  2  3  1  4  9

Encoding Methods

  • Binary Encoding – Most common method of encoding. Chromosomes are strings of 1s and 0s and each position in the chromosome represents a particular characteristic of the problem.

  • Permutation Encoding – Useful in ordering problems such as the Traveling Salesman Problem (TSP). Example. In TSP, every chromosome is a string of numbers, each of which represents a city to be visited.

Computer Implementation


Encoding methods contd

Chromosome A

1.235  5.323  0.454  2.321  2.454

ChromosomeB

(left), (back), (left), (right), (forward)

Encoding Methods (contd.)

  • Value Encoding –Used in problems where complicated values, such as real numbers, are used and where binary encoding would not suffice.

    Good for some problems, but often necessary to develop some specific crossover and mutation techniques for these chromosomes.

Computer Implementation


Encoding methods contd1

Encoding Methods (contd.)

  • Tree Encoding –This encoding is used mainly for evolving programs or expressions, i.e. for Genetic programming.

  • Tree Encoding - every chromosome is a tree of some objects, such as values/arithmetic operators or commands in a programming language.

( +  x  ( /  5  y ) )

( do_until  step  wall )

Citation:

http://ocw.mit.edu/NR/rdonlyres/Aeronautics-and-Astronautics/16-888Spring-2004/D66C4396-90C8-49BE-BF4A-4EBE39CEAE6F/0/MSDO_L11_GA.pdf

Computer Implementation


4 individu

4 individu

Computer Implementation

Citation: examples taken from: www.genetic-programming.com/c2003lecture1modified.ppt


Mapping objective function to fitness form

Mapping Objective function to Fitness Form

  • minimization rather than maximization

  • Transform from minimization to maximization problem:

    • Multiply the cost function by a minus one (insufficient)

    • Commonly used:

Computer Implementation


Computer implementation of genetic algorithm

  • A problem with negative utility function u(x) value in maximization, transform fitness according to the equation:

Computer Implementation


Fitness scaling

Fitness Scaling

  • Linear scaling

  • To ensure each average population member contribute one expected offspring to the next generation

Computer Implementation


Computer implementation of genetic algorithm

  • To control the number of offspring given to the population member with maximum raw fitness.

  • cMult=the number of expected copies desired for the best population member. For typical small population (n=50 to 100) a cMult=1,2 to 2 has been used successfully

Computer Implementation


Computer implementation of genetic algorithm

Computer Implementation


Computer implementation of genetic algorithm

  • Sigma () truncation:

    • Using population variance information

    • c is choosen as reasonable multiple of population standard deviation (between 1 and 3)

    • Negative result (f’<0) are arbitrarily set to 0

  • Power Low Scaling

  • Computer Implementation


    A multiparameter mapped fixed point coding

    A multiparameter, mapped, fixed-point coding

    • Tidak suka

    • Gunakan

    • Carefully control the range and precision of the decision variable. The precision:

    Computer Implementation


    Computer implementation of genetic algorithm

    • Single U1 parameter

      • 0000  umin

      • 1111  umax

    • Multiparameter Coding (10 parameter):

      • 0001| 0101|…|1100|1111|

      • U1 | U1 |…| U1 | U1 |

    Computer Implementation


    Discretization

    Discretization

    Computer Implementation


    Computer implementation of genetic algorithm

    Computer Implementation


    Constraints

    Constraints

    • Minimize g(x)

    • Subject to bi(x)≥0 i=1,2,…,n

    • Where x is an m vector

    • Tranform to the unconstraint form:

    Computer Implementation


    Example the traveling salesman problem tsp

    Example:The Traveling Salesman Problem (TSP)

    The traveling salesman must visit every city in his territory exactly once and then return to the starting point; given the cost of travel between all cities, how should he plan his itinerary for minimum total cost of the entire tour?

    TSP  NP-Complete

    Note: we shall discuss a single possible approach to approximate the TSP by GAs

    Computer Implementation


    Tsp representation evaluation initialization and selection

    TSP (Representation, Evaluation, Initialization and Selection)

    A vector v = (i1 i2… in) represents a tour (v is a permutation of {1,2,…,n})

    Fitness f of a solution is the inverse cost of the corresponding tour

    Initialization: use either some heuristics, or a random sample of permutations of {1,2,…,n}

    We shall use the fitness proportionate selection

    Computer Implementation


    Notation schema

    Notation (schema)

    {0,1,#} is the symbol alphabet, where # is a special wild cardsymbol

    A schema is a template consisting of a string composed of these three symbols

    Example: the schema [01#1#] matches the strings: [01010], [01011], [01110] and [01111]

    Computer Implementation


    Notation order

    Notation (order)

    The order of the schema S (denoted by o(S)) is the number of fixed positions (0 or 1) presented in the schema

    Example: for S1 = [01#1#], o(S1) = 3

    for S2 = [##1#1010], o(S2) = 5

    The order of a schema is useful to calculate survival probability of the schema for mutations

    There are 2 l-o(S)different strings that match S

    Computer Implementation


    Notation defining length

    Notation (defining length)

    The defining length of schema S (denoted by (S)) is the distance between the first and last fixed positions in it

    Example:for S1 = [01#1#], (S1) = 4 – 1 = 3,

    for S2 = [##1#1010], (S2) = 8 – 3 = 5

    The defining length of a schema is useful to calculate survival probability of the schema for crossovers

    Computer Implementation


    Notation cont

    Notation (cont)

    m(S,t) is the number of individuals in the population belonging to a particular schema S at time t (in terms of generations)

    fS(t) is the average fitness value of strings belonging to schema S at time t

    f(t) is the average fitness value over all strings in the population

    Computer Implementation


    The effect of selection

    The effect ofSelection

    Under fitness-proportionate selection the expected number of individuals belonging to schema S at time (t+1) is m (S,t+1) = m (S,t) ( fS(t)/f (t) )

    Assuming that a schema S remains above average by 0  c, (i.e., fS(t) = f(t) + c f(t) ), then

    m (S,t) = m (S,0) (1 + c)t

    Significance: “aboveaverage” schema receives an exponentially increasing number of strings in the next generation

    Computer Implementation


    The effect of crossover

    The effect ofCrossover

    The probability of schema S (|S| = l) to survive crossover is ps(S)  1 – pc((S)/(l – 1))

    The combined effect of selection and crossover yields

    m (S,t+1)  m (S,t) ( fS(t)/f (t) ) [1 - pc((S)/(l – 1))]

    Above-average schemata with short defining lengths would still be sampled at exponentially increasing rates

    Computer Implementation


    The effect of mutation

    The effect ofMutation

    The probability of S to survive mutation is:

    ps(S) = (1 – pm)o(S)

    Sincepm<< 1, this probability can be approximated by:

    ps(S)  1 – pm·o(S)

    The combined effect of selection, crossover and mutation yields

    m (S,t+1)  m (S,t) ( fS(t)/f (t) ) [1 - pc((S)/(l – 1)) -pmo(S)]

    Computer Implementation


    Schema theorem

    Schema Theorem

    Short, low-order, above-average schemata receive exponentially increasing trials in subsequent generations of a genetic algorithm

    Result: GAs explore the search space by short, low-order schemata which, subsequently, are used for information exchange during crossover

    Computer Implementation


    Computer implementation of genetic algorithm

    Computer Implementation


    Computer implementation of genetic algorithm

    Computer Implementation


    Computer implementation of genetic algorithm

    Computer Implementation


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