Genetic Algorithms. 2 Overview Introduction To Genetic Algorithms (GAs) GA Operators and Parameters Genetic Algorithms To Solve The Traveling Salesman

Genetic Algorithms

3

Introduction To Genetic Algorithms

(GAs)

4

What Are Genetic Algorithms (GAs)?

Genetic Algorithms are search and optimization techniques based on Darwin’s Principle of Natural Selection.

How is it different from other optimization and search procedures?

1. Works with a coding of the parameter set, not the parameters themselves

2. Search for a population of point and not a single point

3. Use objective function information and not derivatives or other auxiliary knowledge

4. Uses probabilistic transition rules and not deterministic rules

How GA is used and different from other optimization techniques?

The first step in GA is to code the parameter x as a finite length string

Example 1 can be code as string of 5 bits with an output f=f(s) , where s=string of bits

Successive populations are generated using the GA

For effective check GA requires only objective functions associated with individual strings

Simple genetic algorithm Reproduction:- individual strings are copied

according to their objective fn: values f(FITNESS FUNCTION)

Crossover:- Members of the newly reproduced strings are mated at random. Each pair of strings undergoes crossing overs.

Mutation:-supplements reproduction and crossover and acts as an insurance policy against premature loss of important notions

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Darwin’s Principle Of Natural Selection I

IF there are organisms that reproduce, and IF offsprings inherit traits from their parents, and IF there is variability of traits, and IF the environment cannot support all members of a

growing population, THEN those members of the population with less-

adaptive traits (determined by the environment) will die out, and

THEN those members with more-adaptive traits (determined by the environment) will thrive

The result is the evolution of species.

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Basic Idea Of Principle Of Natural Selection

“Select The Best, Discard The Rest”

Example 1Maximize f(x) =x2 on the integer scale from 0-31

0 31x

f(x)

1000

Example 1

No:

String Fitness % of total

1 01101

169 14.2

2 11000

576 49.2

3 01000

64 5.5

4 10011

361 30.9

Total 1170 100

Roulette wheel with slots sized according to fitness

1234

Crossover

A1=0 1 1 0 1A2=1 1 0 0 0

A1’=0 1 1 0 0A2’=1 1 0 0 1

Simple GA by Hand(Reproduction)

No: String x f(x)x2

pselectfi/Ʃf

Expected countn.pselect

Actual count(Roulette Wheel0

1 01101 13 169 .14 .58 1

2 11000 24 576 .49 1.97 2

3 01000 08 64 .06 0.24 0

4 10011 19 361 0.31 1.24 1

Sum 1170 1.00 4 4.0

Average 293 0.25 1.00 1.0

Maximum 576 .49 1.97 2

CrossoverMating Pool after Reproduction(Cross Site shown)

Mate Crossover

New population

x F(x)

0110|12 4 0 1 1 0 0 12 144

1100|01 4 1 1 0 0 1 25 625

11|0004 2 1 1 0 1 1 27 729

10|0113 2 1 0 0 0 0 16 256

Sum 1754

Average 439

Maximum 729

Probability of mutation in this test is 0.001. With 20 transferred bit positions we should expect 20*0.001=0.02

Seeking similarities among strings in population

Causal relationships between similarities and high fitness

How does the directed search guide for improvement?

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Evolution in the real world Each cell of a living thing contains chromosomes -

strings of DNA Each chromosome contains a set of genes - blocks of DNA Each gene determines some aspect of the organism (like

eye colour) A collection of genes is sometimes called a genotype A collection of aspects (like eye colour) is sometimes

called a phenotype Reproduction involves recombination of genes from

parents and then small amounts of mutation (errors) in copying

The fitness of an organism is how much it can reproduce before it dies

Evolution based on “survival of the fittest”

Basic Idea Of Principle Of Natural Selection

“Select The Best, Discard The Rest”

AlgorithmGenerate Initial Populationdo Calculate the Fitness of each member do { Select Parents from current population Perform Crossover add offspring to the new population Merge new population into the current population Mutate current population till result

is obtained }

Population

Chromosomes could be:

Bit strings (0101 ... 1100)

Real numbers (43.2 -33.1 ... 0.0 89.2)

Permutations of element (E11 E3 E7 ... E1 E15) Lists of rules (R1 R2 R3 ... R22 R23) ... any data structure ...

population


is obtained }


is obtained }

Fitness Function

A fitness function quantifies the optimality of a solution so that that particular solution may be ranked against all the other solutions.

A fitness value is assigned to each solution depending on how close it actually is to solving the problem.

Ideal fitness function correlates closely to goal + quickly computable.


is obtained }


is obtained }

Crossover

Mimics biological recombination

Some portion of genetic material is swapped between chromosomes

Typically the swapping produces an offspring

CROSSOVER

(1 2 9 3 0 7 ) (1 2 9 7 9 5)

(4 6 1 7 9 5 )

(1 2 9 3 0 7 )( 4 6 1 7 9 5)

(4 6 1 7 9 5 )


is obtained }


is obtained }

Mutation

Selects a random locus – gene location – with some probability and alters the allele at that locus

The intuitive mechanism for the preservation of variety in the population

Mutation: Local Modification

Before: (1 0 1 1 0 1 1 0)After: (0 1 1 0 0 1 1 0)

Before: (1.38 -69.4 326.44 0.1)After: (1.38 -67.5 326.44 0.1)

The ProblemThe Traveling Salesman Problem is defined as: Given: 1) A set of cities 2) Symmetric distance matrix that indicates the cost of

travel from each city to every other city.

Goal: 1) Find the shortest circular tour, visiting every city

exactly once. 2) Minimize the total travel cost, which includes the

cost of traveling from the last city back to the first city’.

Traveling Salesperson Problem

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Encoding

Represent every city with an integer .

Consider 6 Indian cities – Mumbai, Nagpur , Calcutta, Delhi, Bangalore

and Pune assign a number to each.

Mumbai 1Nagpur 2Calcutta 3Delhi 4Bangalore 5Pune 6

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Encoding

Thus a path would be represented as a sequence of integers from 1 to 6.

The path [1 2 3 4 5 6] represents a path from

Mumbai to Nagpur - Nagpur to Calcutta - Calcutta to Delhi - Delhi to Bangalore - Bangalore to Pune and pune to Mumbai.

Fitness Function

The fitness function will be the total cost of the tour represented by each chromosome.

This can be calculated as the sum of the distances traversed in each travel segment.

The Lesser The Sum, The Fitter The Solution Represented By That Chromosome.

1 2 3 4 5 6

1 0 863 1987 1407 998 163

2 863 0 1124 1012 1049 620

3 1987 1124 0 1461 1881 1844

4 1407 1012 1461 0 2061 1437

5 998 1049 1881 2061 0 841

6 163 620 1844 1437 841 0

Distance/Cost Matrix For TSP

Fitness Function (contd.)

So, for a chromosome [4 1 3 2 5 6 ], the total cost of travel or fitness will be calculated as shown below

Fitness = 1407 + 1987 + 1124 + 1049 + 841 = 6408 kms.

Since our objective is to Minimize the distance, the lesser the total distance, the fitter the solution.

Initial Population for TSP

(5,3,4,6,2) (2,4,6,3,5) (4,3,6,5,2)

(2,3,4,6,5) (4,3,6,2,5) (3,4,5,2,6)

(3,5,4,6,2) (4,5,3,6,2) (5,4,2,3,6)

(4,6,3,2,5) (3,4,2,6,5) (3,6,5,1,4)

Select Parents

(5,3,4,6,2) (2,4,6,3,5) (4,3,6,5,2)

(2,3,4,6,5) (4,3,6,2,5) (3,4,5,2,6)

(3,5,4,6,2) (4,5,3,6,2) (5,4,2,3,6)

(4,6,3,2,5) (3,4,2,6,5) (3,6,5,1,4)

Try to pick the better ones.

Create Off-Spring – 1 point

(5,3,4,6,2) (2,4,6,3,5) (4,3,6,5,2)

(2,3,4,6,5) (4,3,6,2,5) (3,4,5,2,6)

(3,5,4,6,2) (4,5,3,6,2) (5,4,2,3,6)

(4,6,3,2,5) (3,4,2,6,5) (3,6,5,1,4)

(3,4,5,6,2)

(3,4,5,6,2)

Create More Offspring

(5,3,4,6,2) (2,4,6,3,5) (4,3,6,5,2)

(2,3,4,6,5) (4,3,6,2,5) (3,4,5,2,6)

(3,5,4,6,2) (4,5,3,6,2) (5,4,2,3,6)

(4,6,3,2,5) (3,4,2,6,5) (3,6,5,1,4)

(5,4,2,6,3)

(3,4,5,6,2) (5,4,2,6,3)

Mutate

(5,3,4,6,2) (2,4,6,3,5) (4,3,6,5,2)

(2,3,4,6,5) (4,3,6,2,5) (3,4,5,2,6)

(3,5,4,6,2) (4,5,3,6,2) (5,4,2,3,6)

(4,6,3,2,5) (3,4,2,6,5) (3,6,5,1,4)

Mutate

(5,3,4,6,2) (2,4,6,3,5) (4,3,6,5,2)

(2,3,4,6,5) (2,3,6,4,5) (3,4,5,2,6)

(3,5,4,6,2) (4,5,3,6,2) (5,4,2,3,6)

(4,6,3,2,5) (3,4,2,6,5) (3,6,5,1,4)

(3,4,5,6,2) (5,4,2,6,3)

Eliminate

(5,3,4,6,2) (2,4,6,3,5) (4,3,6,5,2)

(2,3,4,6,5) (2,3,6,4,5) (3,4,5,2,6)

(3,5,4,6,2) (4,5,3,6,2) (5,4,2,3,6)

(4,6,3,2,5) (3,4,2,6,5) (3,6,5,1,4)

Tend to kill off the worst ones.

(3,4,5,6,2) (5,4,2,6,3)

Integrate

(5,3,4,6,2) (2,4,6,3,5)

(2,3,6,4,5) (3,4,5,2,6)

(3,5,4,6,2) (4,5,3,6,2) (5,4,2,3,6)

(4,6,3,2,5) (3,4,2,6,5) (3,6,5,1,4)

(3,4,5,6,2)

(5,4,2,6,3)

Restart

(5,3,4,6,2) (2,4,6,3,5)

(2,3,6,4,5) (3,4,5,2,6)

(3,5,4,6,2) (4,5,3,6,2) (5,4,2,3,6)

(4,6,3,2,5) (3,4,2,6,5) (3,6,5,1,4)

(3,4,5,6,2)

(5,4,2,6,3)

When to Use a GA Alternate solutions are too slow or overly

complicated Need an exploratory tool to examine new

approaches Problem is similar to one that has already

been successfully solved by using a GA Want to hybridize with an existing solution Benefits of the GA technology meet key

problem requirements

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SUMMARY

Genetic Algorithms (GAs) implement optimization strategies based on simulation of the natural law of evolution of a species by natural selection

The basic GA Operators are:EncodingRecombinationCrossoverMutation

GAs have been applied to a variety of function optimization problems, and have been shown to be highly effective in searching a large, poorly defined search space

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Genetic Algorithms. 2 Overview Introduction To Genetic Algorithms (GAs) GA Operators and Parameters Genetic Algorithms To Solve The Traveling Salesman