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### Introduction to GAs: Genetic Algorithms

Quantitative Analysis: How to make a decision?

Thank you for all pictures and information referred

Agenda

- Introduction
- Genetic Algorithms
- Steps of GA Design
- Example of Genetic Algorithm
- Single Objective Optimization
- GA for Multi-objective Optimization Problems
- Example of Genetic Algorithm
- Multiple Objectives Optimization
- Benefits and Applications of GAs

2

Introduction

- Optimization Problems: How to solve them?
- Single-objective optimization problems
- Multi-objective optimization Problems

http://bizzbangbuzz.blogspot.com/2010/03/five-modes-of-decision-making.html

Introduction

- Evolutionary Algorithm (EA) is a more suitable optimization technique.
- WHY??
- Natural selection and recombination to find an optimal configuration for a specific problem within specific constraints
- Yield good quality approximate solutions to large real world problems
- Time consuming
- Flexible
- Robust
- Appropriate when traditional methods break down

Approximate solution to real problems!!

Introduction

- Genetic Algorithm (GA) is a member of Evolutionary Algorithms (EA)
- The major classes of EA
- Genetic Algorithm (GA)
- Evolutionary Programming (EP)
- Genetic Programming (GP)
- Evolution Strategy (ES).

Introduction

- An idea of evolutionary computing was discovered by I. Rechenberg in 1960s
- Idea of GA was proposed by John Holland in 1970
- In 1992, John Koza used GA for developing programs to perform certain tasks that are called “Genetic Programming”

Genetic Algorithm

- GA uses principles of evolutionary and natural selection to solve problems.
- HOW??
- GA maintains a population of structures according to rules of selection and search operators
- such as a crossover or recombination and a mutation
- Individuals in the population accept a measurement of fitness in objective functions
- the reproduction focuses on the higher fitness individuals
- GA is an iterative process
- the selection operation acts on the current population according to the certain regulation
- the operation of crossover and mutation are used in individual selection.

Biological Terms of Genetic Algorithm

- All organisms consist of many cells
- each cell obtains the same set of chromosomes
- The chromosomes are strings of DNA.
- The characteristics of chromosomes are defined by genes.
- There are many forms of genes that are called alleles.
- The allele produces the difference set of characteristics in each gene.
- The set of chromosomes are called a genotype.
- A genotype defines an encoded solution in the search space.
- phenotype is a solution in the real problem domain

Chromosome (string): Solution

Genes (bits): Part of solution

Locus: Position of gene

Alleles: Value of gene

Phenotype: Decoded solution

Genotype: Encoded solution

Terms of Genetic AlgorithmGenetic Algorithm’s Processes

- An initial population of individuals is a set of solutions that is represented by chromosomes.
- Chromosomes are generated randomly.
- Every iteration of evolutionary is a generation of the algorithm.
- Individuals in the current population are decoded and evaluated according to some predefined quality criteria that are referred to as the fitness function.
- Each individual is selected according to the fitness value in which existing members of the current solution pool are replaced by new created members.

Genetic Algorithm’s Processes

- The new members of the population are created from the crossover and mutation operations.
- Better members will survive but weaker ones will be eliminated.
- The higher fitness individuals (the better members) have more chance to reproduce than the lower ones.
- The weaker members are eliminated.
- The GA process will be repeated until the convergence criterion is satisfied.

Steps of Genetic Algorithm

Step 0: Initialization randomly generates an initial population.

Step 1: Evaluation decodes strings to solutions and calculates the value of the objective function for each solution then transforms the value of the objective function for each solution to the value of the fitness function for each string in the genotype world.

Step 2: Selection selects a number of pairs of strings from the current population according to the selection probability.

Steps of Genetic Algorithm

Step 3: Crossover applies the crossover operator to each of the selected pairs in Step 2 to generate strings with the crossover probability.

Step 4: Mutation applies the mutation operator to each of the generated strings with the mutation probability.

Step 5: Elitist strategy randomly removes a string from the current population and add the best string in the previous population to the current.

Step 6: Termination test, if the condition is satisfied, stop this algorithm. Otherwise, return to Step 1.

Steps of GA Design

- Initial Population and Representation
- Population consists of individuals which are potential solutions in the problem domain.
- The parameters of problem domain are encoded to be initial population.
- Population contains individuals that are chromosomes.
- The population always consists of 30-100 individuals; if the numbers of individuals are less than 30 individuals then it is called MicroGA
- Each solution to an optimization problem is encoded as a finite-length string.

Steps of GA Design

- There are many types of solution representations.
- The basic representations of parameters are the binary coding and the permutation coding.
- Binary Coding
- It consists of binary digit “0” and “1”
- Each bit in a string represents the characteristics of a solution.
- The whole string represents the meaning of solution.
- The decision variables in the parameter set are encoded to be the binary string by using the gray code method or hamming code method.

Steps of GA Design

- Permutation coding
- It is used for sequencing problems
- such as scheduling problems and traveling salesman problems where the permutation string consists of number “1” to “n”.
- Each numeral corresponds to a job in scheduling problems or a city in traveling salesman problems and “n” is the number of jobs or cities.

Permutation representation: TSP

- Problem:
- Given n cities
- Find a complete tour with minimal length
- Encoding:
- Label the cities 1, 2, … , n
- One complete tour is one permutation (e.g. for n =4 [1,2,3,4], [3,4,2,1] are OK)
- Search space is BIG:

for 30 cities there are 30! 1032 possible tours

Steps of GA Design

- There is a problem when using the binary code
- the binary representation: obscures the nature of the search, but there are many strategies for encoding.
- The real value of parameters, representation needs not to decode chromosomes to the phenotype, which is fast and uses less memory.
- An integer representation is easier than the binary code, because it can look-up tables for decoding the representation.

The binary string is called a genotype, consisting of “0” and “1”.

- The solution is decoded from the binary string called a phenotype.
- GA searches the binary strings from the genotype.
- After the algorithm converges the genotype is decoded.

Steps of GA Design

- Objective and fitness function
- An objective function is used to measure each individual in the population for measuring the suitability in the problem domain.
- For a minimization problem, the individual that makes the objective function to be the lowest value; it is fit for this problem.
- The individual that makes the objective function to be the highest value; it is fit for a maximization problem.
- A fitness function is used for transforming the objective function value to the fitness value, which is used for assigning the fitness to the individual.

Steps of GA Design

- The objective function value, f(x), is calculated by using the value of “x” that is a variable decoded from the binary string, for instance.
- If the objective function value is better than the binary string in the genotype corresponding to the solution “x”.
- It is a better fitness value.
- When transforming the objective function value of solution “x” to the fitness value, the solution that has a higher fitness value is kept for future reproduction.
- Fitness value: maximize
- Fitness value: minimize
- An individual has the probability of reproduction according to fitness value.

Steps of GA Design

- Selection
- A selection is the process of determining a particular individual that is chosen for reproduction and the number of offspring in which an individual will produce.
- It transforms the fitness values of individuals to the probability value for reproducing by the probability of reproduction according to the fitness values.
- For instance, the binary strings that have higher fitness values are more chance to be selected as parents.

An efficiency selection method is motivated by the need to maintain an overall time complexity.

Steps of GA Design

- New population creation
- A crossover and mutation operations are the major parts of the process that shows the efficient performance of GA.
- For the crossover operator, the probability is the most frequently used.
- the mutation probability is rarely used.
- The mutation is the a random operator and it serves to introduce the diversity in the population.
- It changes an element from a binary string that is generated by the crossover method.
- It replaces a bit by digit “0” or “1”.
- It is relative with only a single parent and the result is an offspring but the crossover operation is two parents and the result is two offspring.

Steps of GA Design

- The crossover operator takes two individuals and cuts their chromosome strings by using the random position.
- They have two-head and two-tail segments.
- The tail segments are swapped over to produce two new full-length chromosomes where two new offspring inherit some genes from each parent.
- The basic crossover is a one-point crossover that is a basic operator for binary coding

Steps of GA Design

- The crossover point is a randomly selected between two adjacent elements by swapping all elements in the head and tail part.
- The normal probability of crossover is 0.6 to 1.0

Steps of GA Design

- N-point Crossover
- Choose n random crossover points
- Split along those points
- Glue parts, alternating between parents

Steps of GA Design

- Uniform Crossover
- Assign 'heads' to one parent, 'tails' to the other
- Flip a coin for each gene of the first child
- Make an inverse copy of the gene for the second child
- Inheritance is independent of position

1 2 3 2 1

5 4 3 2 1

5 4 3 4 5

Steps of GA Design- Crossover operators for permutations
- Many specialised operators have been devised which focus on combining order or adjacency information from the two parents

Parent 1

Child 1

Parent 2

Child 2

Steps of GA Design

- Mutation Operation
- It is applied to each offspring individually after the crossover method.
- It provides a small amount of random search and helps to show that there is not a zero mutation probability in the search space.
- The probability of mutation is 0.001 to 0.1 that is a small probability.

Steps of GA Design

- Mutation for Permutations
- Pick two allele values at random
- Move the second to follow the first, shifting the rest along to accommodate
- Note that this preserves most of the order and the adjacency information

Steps of GA Design

- Crossover OR mutation?
- Decade long debate:
- which one is better / necessary
- Answer:
- it depends on the problem, but
- in general, it is good to have both
- both have another role

Steps of GA Design

- Crossover OR mutation?
- Exploration: Discovering promising areas in the search space, i.e. gaining information on the problem
- Exploitation: Optimising within a promising area, i.e. using information
- There is co-operation AND competition between them
- Crossover is explorative, it makes a big jump to an area somewhere “in between” two (parent) areas
- Mutation is exploitative, it creates random small diversions, thereby staying near (in the area of ) the parent

Steps of GA Design

- Crossover OR mutation?
- Only crossover can combine information from two parents
- Only mutation can introduce new information (alleles)
- To hit the optimum you often need a ‘lucky’ mutation

Steps of GA Design

- Elitist Strategy
- An elitist is a method that copies the best chromosome (string) to the new population (next generation).
- It protects the best string that is not affected by the genetic operator (crossover and mutation).
- It can increase the performance of GA so, the best string has more chance to be a parent string.

Steps of GA Design

- Termination
- A termination tests the quality of the best members of the population with the problem definition.
- If the solution is not acceptable or the maximum number of iterations is not reached then the GA process is restarted.
- Number of iterations
- Acceptable value: Convergence value

Example of Genetic Algorithm

Maximize f(x)=x2, X in the interval {0,…,31}.

Objective function: f(x)=x2

Decision variable: x

Constrain: X={0,…,31}

Binary Representation

- Encoding (representation): chromosomes:

0=00000, 1=00001, 2=00010, 3=00011,…

- Generate initial population at random:

01101, 00001, 11000, 10011,…

- Evaluate the fitness according to f(x)

01101 = 13 x= 13, f(x)=169

Example of Genetic Algorithm

- Select 2 individuals for crossover based on their fitness.

parents 0110|1(=13) 1100|0(=24)

offspring 0110|0 1100|1

mutated 01101 11000

- Repeated until termination

Decoded Value

Or Fitness Value

Example of Genetic Algorithm

- Maximization

Example of Genetic Algorithm

- Crossover

Example of Genetic Algorithm

- Mutation

GA for Multi-objective Optimization Problems

- Many real world problems have a number of solutions that involve the multi-objective optimization.
- Evolutionary algorithms are suitable for optimization problem.
- Since the mid-1980’s, interest in multi-objective optimization problems has been expanding rapidly.
- Various evolutionary algorithms have been developed for searching the multiple solutions concurrently in a single run

GA for Multi-objective Optimization Problems

- How to solve multi-objective optimization problems by GA?
- GA is a multipoint search that has the advantages for optimizing the problem with multiple objectives
- taking the similarity available in the family of possible solutions to the problem.
- Single objective optimization problem (SOOP)
- Fitter individuals have higher chance to produce offspring than the lower ones.
- SOOPs can search for ONLY one best solution at a time.

GA for Multi-objective Optimization Problems

- An example scheme of GA for multiobjective optimization problems (MOOPs) is the multiobjective genetic algorithm (MOGA).
- The process of MOGA is the same as single-objective optimization problems (SOOPs).
- BUT the evaluation, selection and elitist strategy are different from SOOPs.

Categories of MOGA

- Plain aggregating approaches
- Plain aggregating approaches apply a weighted aggregating method to convert the multi-objective optimization problem into a single objective problem, and then use the single function genetic algorithm to get solutions.
- Aggregation methods combine multiple objectives into a higher scalar function that are used for fitness calculation.
- An aggregation approaches have the advantage of producing one single solution.
- Defining the goal function in this way requires profound domain knowledge that is often not available.

Categories of MOGA

- Population-based non-Pareto approaches
- Population-based non-Pareto approaches able to evolve multiple non-dominated solutions then the population is monitored for non-dominated solutions concurrently in a single simulation run by changing the selection criterion during the reproduction phase.
- The search is guided in several directions at the same time then they cannot make direct use the concept of Pareto dominance or Pareto optimality.

Categories of MOGA

- Niched induction techniques
- Niching techniques are suggested to keep GA from convergence to the single point on the front and a niching mechanism
- such as a fitness sharing that allows GA to maintain individuals along the non-dominated frontier.

Categories of MOGA

- Pareto-based approaches
- An idea of using Pareto-based fitness assignment is to use the non-dominated ranking and selection to move a population to the Pareto front in MOOP.
- The basic idea is to find a set of individuals that are the non-dominated solutions to the rest of population.

Example of Multi-objective Optimization Problems (Pareto-based Approach)

- Shaffer’s problem

Minimize: f1 = x2

f2 = (x-2)2

Constrain: 0≤x≤20

- Murata’s problem

Minimize: f1 = 2√x1

f2 = x1(1-x2) + 5

Constrain: 1≤x1≤4

1≤x2≤2

Example of Multi-objective Optimization Problems (Pareto-based Approach)

- Srinivas and Deb’s problem

Minimize:

f1(x,y) = (x2+y2)1/8

f2(x,y) = ((x-0.5)2+(y-0.5)2)1/4

Constrain: -5≤x≤10

Benefits of Genetic Algorithms

- Concept is easy to understand
- Modular, separate from application
- Supports multi-objective optimization
- Good for “noisy” environments
- Always an answer; answer gets better with time
- Inherently parallel; easily distributed

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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