# Genetic Algorithms.

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Genetic Algorithms

Introduction To Genetic Algorithms (GAs)

What Are Genetic Algorithms (GAs)?
Genetic Algorithms are search and optimization techniques based on Darwin’s Principle of Natural Selection. What exactly are Genetic Algorithms? As the name suggests, Genetic Algorithms or Gas borrow their basic working principle from natural genetics. Genetic Algorithms are search and optimization methods based on Darwin’s principle of Natural Selection.

Characteristics of GA Parallel-search procedures that can be implemented on parallel processing machines for speeding operations Applies to both continuous and discrete optimization problems Stochastic in nature and less likely to get caught in local minima Facilitates both structure and parameter identification

Darwin’s Principle Of Natural Selection
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.

Working of GA GA encodes each point in a parameter space into a binary bit called chromosome Each point is associated with a fitness function Gene pool is a population of all such points In each generation GA constructs a new population using genetic operators Crossover Mutation

Components of GA Encoding schemes Crossover operators
Mutation operators

Encoding schemes Transforms points in parameter space into string representations Eg (11,6,9) is represented as Encoding schemes provide a way of translating problem-specific knowledge directly into GA framework After this the fitness function is evaluated Next selection is based on the fittest survivor

Fitness evaluation

How is it different from other optimization and search procedures?
Works with a coding of the parameter set, not the parameters themselves Search for a population of point and not a single point Use objective function information and not derivatives or other auxiliary knowledge

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

Basic Idea Of Principle Of Natural Selection
“Select The Best, Discard The Rest” Darwin’s principle of Natural Selection can be stated very simply as “ select the best and discard the rest”. Let us consider an example. Consider a population of animals of a particular species in a jungle. This population of animals has some animals which are stronger than the others and have characteristics that help them survive better in that environment as compared to the others. Now, the resources, like food,water,etc in the jungle are limited. So, these animals have to compete with one another for these resources. Ultimately, only the strongest, or fittest individuals survive and the rest perish. In other words, in this “struggle for existence” , nature selects the best, and discards the rest. Now, these surviving animals mate amongst themselves and propagate their favorable characteristics from one generation to another. What is mating? At the genetic level it can be seen as the crossover of 2 chrmosomes, one from each parent. They exchange some genetic matter and give rise to the new chromosome of the child that has both the parents’ features.Thus the favorable characteristics of the parents that helped them survive in that environment are propagated from one generation to the next till the species becomes best adapted to it’s environment. This is called evolution.

Example 1 Maximize f(x) =x2 on the integer scale from 0-31
1000 f(x) 31 x

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

Crossover A1= A2= A1’= A2’=

Simple GA by Hand(Reproduction)
No: String x f(x) x2 pselect fi/Ʃf Expected count n.pselect Actual count (Roulette Wheel0 1 01101 13 169 .14 .58 2 11000 24 576 .49 1.97 3 01000 08 64 .06 0.24 4 10011 19 361 0.31 1.24 Sum 1170 1.00 4.0 Average 293 0.25 1.0 Maximum

Crossover Mating Pool after Reproduction(Cross Site shown) Mate Crossover New population x F(x) 0110|1 2 4 12 144 1100|0 1 25 625 11|000 27 729 10|011 3 16 256 Sum 1754 Average 439 Maximum Probability of mutation in this test is With 20 transferred bit positions we should expect 20*0.001=0.02

Grist for the search mill
How does the directed search guide help improvement? Seeking similarities among strings in population Causal relationships between similarities and high fitness SCHEMATA

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” Darwin’s principle of Natural Selection can be stated very simply as “ select the best and discard the rest”. Let us consider an example. Consider a population of animals of a particular species in a jungle. This population of animals has some animals which are stronger than the others and have characteristics that help them survive better in that environment as compared to the others. Now, the resources, like food,water,etc in the jungle are limited. So, these animals have to compete with one another for these resources. Ultimately, only the strongest, or fittest individuals survive and the rest perish. In other words, in this “struggle for existence” , nature selects the best, and discards the rest. Now, these surviving animals mate amongst themselves and propagate their favorable characteristics from one generation to another. What is mating? At the genetic level it can be seen as the crossover of 2 chrmosomes, one from each parent. They exchange some genetic matter and give rise to the new chromosome of the child that has both the parents’ features.Thus the favorable characteristics of the parents that helped them survive in that environment are propagated from one generation to the next till the species becomes best adapted to it’s environment. This is called evolution.

Algorithm } Generate Initial Population do
Calculate the Fitness of each member { 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 } Algorithm

Population population Chromosomes could be:
Bit strings ( ) Real numbers ( ) Permutations of element (E11 E3 E7 ... E1 E15) Lists of rules (R1 R2 R3 ... R22 R23) ... any data structure ...

Algorithm } Generate Initial Population do
Calculate the Fitness of each member { 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 } Algorithm

Algorithm } Generate Initial Population do
Calculate the Fitness of each member { 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 } Algorithm

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.

Algorithm } Generate Initial Population do
Calculate the Fitness of each member { 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 } Algorithm

Algorithm } Generate Initial Population do
Calculate the Fitness of each member { 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 } Algorithm

Crossover Mimics biological recombination
Some portion of genetic material is swapped between chromosomes Typically the swapping produces an offspring

CROSSOVER ( ) ( ) ( ) ( )

Algorithm } Generate Initial Population do
Calculate the Fitness of each member { 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 } Algorithm

Algorithm } Generate Initial Population do
Calculate the Fitness of each member { 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 } Algorithm

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: ( ) After: ( ) Before: ( ) After: ( )

The Problem The 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

Encoding Represent every city with an integer .
Consider 6 Indian cities – Mumbai, Nagpur , Calcutta, Delhi, Bangalore and Pune assign a number to each. Mumbai Nagpur Calcutta Delhi Bangalore Pune

Encoding Thus a path would be represented as a sequence of integers from 1 to 6. The path [ ] 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.

Distance/Cost Matrix For TSP
1 2 3 4 5 6 863 1987 1407 998 163 1124 1012 1049 620 1461 1881 1844 2061 1437 841

Fitness Function (contd.)
So, for a chromosome [ ], the total cost of travel or fitness will be calculated as shown below Fitness = = 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)

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) (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) (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) (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) (3,4,5,6,2) (5,4,2,6,3) Tend to kill off the worst ones.

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

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

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

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: Encoding Recombination Crossover Mutation 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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