2015 artificial intelligence techniques at engineering seminar - chapter 2 - part 2: genetic...

Artificial Intelligence Techniques applied to Engineering

Part 2. Genetic Fuzzy Systems Enrique Onieva Caracuel

@EnriqueOnieva

1.Fuzzy Logic

2.Genetic Algorithms

3.Genetic Fuzzy Systems

4.Applications to Intelligent Transportation Systems: My Experience

Técnicas de Inteligencia Artificial aplicadas a la Ingeniería @EnriqueOnieva 2014/2015 2

Optimization

Is the process of looking for the best solution over a set of feasible solutions

Applications: Routes calculation

Process planning

Resource assignment

Pattern classification

Can be formulated as: 𝒎𝒊𝒏{𝒇(𝒙)│𝒙∈𝑿, 𝑿⊆𝑺 }


Example

Traveling Salesman Problem (TSP)

A set of nodes

To visit all the notes

One time in each node

Each arc (i-j) has a distance or cost associated

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Example


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(8,8418·1030 )


Example


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Representation

It has to be defined how genetic characteristics of the individuals in the population are represented

Very important in the GA definition

It affects to the definition of genetic operators (selection, crossover, mutation)

Types: Bit string

Floating point

integer

LISP, Expressions, …

1


Representation

Requirements:

To allow to represent any solution

Not allowing to represent infeasible solutions

Adjusted to the problem

Small changes in the individual must represent small changes in the solution

Easy to decode

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Representation

1. Binary coding: Selection problems Back packing problems

2. Real coding: Real optimization problems: Find solution to:

𝑥1 + 10 · 𝑥2 + (𝑥3 · 𝑥4) + 𝑠𝑒𝑛(𝑥5 + 𝑥6) + cos(𝑥7𝑥8) = 0

Distribution of 4 Gaussian membership function in a fuzzy system

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0 2 4 6 80

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Representation

3. Integer coding:

Grouping problems

Clustering

4. Permutation coding:

Sequencing problems:

Traveling salesman problem

Task to realize in a industrial chain

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Evaluation

Evaluation (fitness): Measures the quality of each individual Allows to distinguish among good and bad individuals Problem dependent Fast execution (if possible)

Key: it decides the individuals to be selected In general:

Is the most time consuming process in a real application Can be a routine, a simulation or any external process Approximate function can be used to reduce the execution

time

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Genetic Algorithm 2


Genetic Algorithm

Generational Model

Steady-State Model


Initialization

Uniform over the search space:

Can generate ANY individual

Binary string: [0,1] with equal probability

Real coding: uniform value in the interval

Integer coding: all the values have the same probability

Permutation coding: random permutation

Choose the initial population according with an heuristic


Selection

Best individuals have higher chances of being selected

Bad individuals must have any chance of being selected

Selective Pressure: degree in which reproduction is directed to best individuals


Selection Operators

Roulette: probability of being selected is proportional to the fitness

𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 = 𝐹𝑖𝑡𝑛𝑒𝑠𝑠𝑖 𝐹𝑖𝑡𝑛𝑒𝑠𝑠

Linear order: probability of being selected is proportional to the order of the individual

Tournament: K individuals are selected randomly, the best of them is picked

Individual Fitness Probability

1 26 0,302

2 17 0,197

3 6 0,069

4 16 0,186

5 21 0,244


Crossover

Offspring has to inherit some characteristics from each parent

It has to be designed according with the representation

Must produce feasible individuals


Crossover Operators

One Point Crossover

N-Points Crossover

Uniform Crossover

1 Parents

Offspring

Offspring

Offspring


Crossover Operators

Real Coding

𝑋 = 𝑥1, 𝑥2, 𝑥3, … 𝑥𝑛 𝑌 = {𝑦1, 𝑦2, 𝑦3, … 𝑦𝑛}

Arithmetic crossover

𝐴 = {𝑥1+𝑦1

2,𝑥2+𝑦2

2,𝑥3+𝑦3

2, …

𝑥𝑛+𝑦𝑛

2}

BLX-α Crossover

𝐴 = {𝑎1, 𝑎2, 𝑎3, … 𝑎𝑛} B = 𝑏1, 𝑏2, 𝑏3, … 𝑏𝑛

where {𝒂𝒊, 𝒃𝒊} ∈ 𝑪𝒎𝒊𝒏 − 𝜶 · 𝑰, 𝑪𝒎𝒂𝒙 + 𝜶 · 𝑰

𝐼 = 𝐶𝑚𝑎𝑥 − 𝐶𝑚𝑖𝑛

𝐶𝑚𝑖𝑛 = min{𝑎𝑖, 𝑏𝑖} 𝐶𝑚𝑎𝑥 = max{𝑎𝑖 , 𝑏𝑖}

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Crossover Operators

Permutation Coding

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Parent 1 Parent 2

Child 1 Child 2

Order


Mutation

Must allow to reach any point at the search space

Variation size must be controlled

Must produce feasible solutions

Is applied with low probability over each individual in the offspring obtained after the crossover


Mutation Operators

Uniform Mutation

Interchange (“any” codification):

Gaussian Mutation (real coding):

𝑥′ = 𝑥 + 𝑁(0, 𝜎)


Replacement

The way in which individuals are replaced by new offspring affects the selective pressure

Deterministic or randomized replacement methods can be used

It can be decided that the best individual (or individuals) are not replaced Elitism

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Replacement

In a Steady-State Model, it can be replaced

The worst individual in the population

The most similar (from N) individual

The worst individual among a the set of the N most similar

The most similar parent

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Stopping Criteria

When the optimum is reached

Limited CPU resources:

Maximum number of generations

Limited amount of execution time

After a certain number of generations without improvements found


Considerations

Execute more than one time Use statistical measures (mean, deviation,…)

Easy to parallelize

Every search method needs equilibrium among: Exploring the search space

Explode promising zones in the search space

Genetic Algorithms are general purpose search methods. Genetic operators are used to maintain this equilibrium


Considerations

Two opposite factors:

Convergence: to focus the search in promising regions by selective pressure

Diversity: to avoid premature convergence

Selective Pressure: allow best individuals to be selected to be crossed.


Considerations

Diversity

It is associated with differences between individuals in the population

Low diversity: all the individuals are quite similar between them

Low diversity premature convergence

Solutions: To include diversity mechanisms in the process

To reinitialize once premature convergence is reached


Considerations

Diversity

With Mutation operator

Probability adaptation To reduce it as the process run

Apply high probability to bad solutions and low probability to good solutions

With the pairing for Crossover operator

Not cross individuals with themselves, their parents, children,…

Incest prohibition to cross only if the are different enough


Considerations

Diversity

With the Crossover operator

Crossover operators with multiple parents

Crossover operators with multiple childs


Extensions

Multimodal problems Multiple solutions must be returned

Niching technique to converge to diverse local optima

Multi-objective problems Multiple objectives must be satisfied

Mutually excluyent objectives Quality – Prize

Power – Consumption

Accuracy - Complexity

There is not unique solution

Thank you very much

Any Question?