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Transcript of 1 Wendy Williams Metaheuristic Algorithms Genetic Algorithms: A Tutorial “Genetic Algorithms are...
1Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
“Genetic Algorithms are good at taking large,
potentially huge search spaces and navigating
them, looking for optimal combinations of things, solutions you might not
otherwise find in a lifetime.”
- Salvatore Mangano
Computer Design, May 1995
Genetic Algorithms:Genetic Algorithms:A TutorialA Tutorial
2Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
A Simple Example
The Traveling Salesman Problem:
Find a tour of a given set of cities so that each city is visited only once the total distance traveled is minimized
3Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Classes of Search Techniques
F inonacc i N ew ton
D irect m ethods Indirec t m ethods
C alcu lus-based techn iques
E volu tionary s trategies
C entra l ized D is tr ibuted
Para l le l
S teady-s ta te G enera tiona l
S equentia l
G ene tic a lgori thm s
E volutionary a lgori thm s S im u lated annealing
G uided random search techniques
D ynam ic program m ing
E num erative techn iques
S earch techniques
4Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Components of a GA
A problem to solve, and ... Encoding technique (gene, chromosome)
Initialization procedure (creation)
Evaluation function (environment)
Selection of parents (reproduction)
Genetic operators (mutation, recombination)
Parameter settings (practice and art)
5Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Simple Genetic Algorithm
{
initialize population;
evaluate population;
while TerminationCriteriaNotSatisfied{
select parents for reproduction;
perform recombination and mutation;
evaluate population;}
}
6Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
The GA Cycle of Reproduction
reproduction
population evaluation
modification
discard
deleted members
parents
children
modifiedchildren
evaluated children
7Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
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) Program elements (genetic programming) ... any data structure ...
population
8Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Reproduction
reproduction
population
parents
children
Parents are selected at random with selection chances biased in relation to chromosome evaluations.
9Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Chromosome Modification
modificationchildren
Modifications are stochastically triggered Operator types are:
Mutation Crossover (recombination)
modified children
10Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
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)
Causes movement in the search space(local or global)
Restores lost information to the population
11Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Crossover: Recombination
P1 (0 1 1 0 1 0 0 0) (0 1 0 0 1 0 0 0) C1
P2 (1 1 0 1 1 0 1 0) (1 1 1 1 1 0 1 0) C2
Crossover is a critical feature of genetic
algorithms: It greatly accelerates search early in
evolution of a population It leads to effective combination of
schemata (subsolutions on different chromosomes)
*
12Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Evaluation
The evaluator decodes a chromosome and assigns it a fitness measure
The evaluator is the only link between a classical GA and the problem it is solving
evaluation
evaluatedchildren
modifiedchildren
13Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Deletion
Generational GA:entire populations replaced with each iteration
Steady-state GA:a few members replaced each generation
population
discard
discarded members
14Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
A Simple Example
The Traveling Salesman Problem:
Find a tour of a given set of cities so that each city is visited only once the total distance traveled is minimized
15Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Representation
Representation is an ordered list of city
numbers known as an order-based GA.
1) London 3) Dunedin 5) Beijing 7) Tokyo
2) Venice 4) Singapore 6) Phoenix 8) Victoria
CityList1 (3 5 7 2 1 6 4 8)
CityList2 (2 5 7 6 8 1 3 4)
16Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Crossover
Crossover combines inversion and
recombination:
* *
Parent1 (3 5 7 2 1 6 4 8)
Parent2 (2 5 7 6 8 1 3 4)
Child (5 8 7 2 1 6 3 4)
This operator is called the Order1 crossover.
17Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Mutation involves reordering of the list:
* *
Before: (5 8 7 2 1 6 3 4)
After: (5 8 6 2 1 7 3 4)
Mutation
18Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
TSP Example: 30 Cities
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19Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Solution i (Distance = 941)
TSP30 (Performance = 941)
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20Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Solution j(Distance = 800)
TSP30 (Performance = 800)
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21Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Solution k(Distance = 652)
TSP30 (Performance = 652)
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22Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Best Solution (Distance = 420)
TSP30 Solution (Performance = 420)
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23Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Overview of Performance
TSP30 - Overview of Performance
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1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31
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24Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
Some GA Application Types
Domain Application Types
Control gas pipeline, pole balancing, missile evasion, pursuit
Design semiconductor layout, aircraft design, keyboardconfiguration, communication networks
Scheduling manufacturing, facility scheduling, resource allocation
Robotics trajectory planning
Machine Learning designing neural networks, improving classificationalgorithms, classifier systems
Signal Processing filter design
Game Playing poker, checkers, prisoner’s dilemma
CombinatorialOptimization
set covering, travelling salesman, routing, bin packing,graph colouring and partitioning
25Wendy WilliamsMetaheuristic Algorithms
Genetic Algorithms: A Tutorial
%TSPO_GA Open Traveling Salesman Problem (TSP) Genetic Algorithm (GA)% Finds a (near) optimal solution to a variation of the TSP by setting up% a GA to search for the shortest route (least distance for the salesman% to travel to each city exactly once without returning to the starting city)%% Summary:% 1. A single salesman travels to each of the cities but does not close% the loop by returning to the city he started from% 2. Each city is visited by the salesman exactly once%% Input:% XY (float) is an Nx2 matrix of city locations, where N is the number of cities% DMAT (float) is an NxN matrix of point to point distances/costs% POPSIZE (scalar integer) is the size of the population (should be divisible by 4)% NUMITER (scalar integer) is the number of desired iterations for the algorithm to run% SHOWPROG (scalar logical) shows the GA progress if true% SHOWRESULT (scalar logical) shows the GA results if true%% Output:% OPTROUTE (integer array) is the best route found by the algorithm% MINDIST (scalar float) is the cost of the best route%% Example:% n = 50;% xy = 10*rand(n,2);% popSize = 60;% numIter = 1e4;% showProg = 1;% showResult = 1;% a = meshgrid(1:n);% dmat = reshape(sqrt(sum((xy(a,:)-xy(a',:)).^2,2)),n,n);% [optRoute,minDist] = tspo_ga(xy,dmat,popSize,numIter,showProg,showResult);%% Example:% n = 50;% phi = (sqrt(5)-1)/2;% theta = 2*pi*phi*(0:n-1);% rho = (1:n).^phi;% [x,y] = pol2cart(theta(:),rho(:));% xy = 10*([x y]-min([x;y]))/(max([x;y])-min([x;y]));% popSize = 60;% numIter = 1e4;% showProg = 1;% showResult = 1;% a = meshgrid(1:n);% dmat = reshape(sqrt(sum((xy(a,:)-xy(a',:)).^2,2)),n,n);% [optRoute,minDist] = tspo_ga(xy,dmat,popSize,numIter,showProg,showResult);%% Example:% n = 50;% xyz = 10*rand(n,3);% popSize = 60;% numIter = 1e4;% showProg = 1;% showResult = 1;% a = meshgrid(1:n);% dmat = reshape(sqrt(sum((xyz(a,:)-xyz(a',:)).^2,2)),n,n);% [optRoute,minDist] = tspo_ga(xyz,dmat,popSize,numIter,showProg,showResult);%% See also: tsp_ga, tsp_nn, tspof_ga, tspofs_ga, distmat%% Author: Joseph Kirk% Email: [email protected]% Release: 1.3% Release Date: 11/07/11function varargout = tspo_ga(xy,dmat,popSize,numIter,showProg,showResult) % Process Inputs and Initialize Defaultsnargs = 6;for k = nargin:nargs-1 switch k case 0 xy = 10*rand(50,2); case 1 N = size(xy,1); a = meshgrid(1:N); dmat = reshape(sqrt(sum((xy(a,:)-xy(a',:)).^2,2)),N,N); case 2 popSize = 100; case 3 numIter = 1e4; case 4 showProg = 1; case 5 showResult = 1; otherwise endend % Verify Inputs[N,dims] = size(xy);[nr,nc] = size(dmat);if N ~= nr || N ~= nc error('Invalid XY or DMAT inputs!')endn = N; % Sanity CheckspopSize = 4*ceil(popSize/4);numIter = max(1,round(real(numIter(1))));showProg = logical(showProg(1));showResult = logical(showResult(1)); % Initialize the Populationpop = zeros(popSize,n);pop(1,:) = (1:n); for k = 2:popSize pop(k,:) = randperm(n);end % Run the GAglobalMin = Inf;totalDist = zeros(1,popSize);distHistory = zeros(1,numIter);tmpPop = zeros(4,n);newPop = zeros(popSize,n);if showProg pfig = figure('Name','TSPO_GA | Current Best Solution','Numbertitle','off');endfor iter = 1:numIter % Evaluate Each Population Member (Calculate Total Distance) for p = 1:popSize d = 0; % Open Path for k = 2:n d = d + dmat(pop(p,k-1),pop(p,k)); end totalDist(p) = d; end % Find the Best Route in the Population [minDist,index] = min(totalDist); distHistory(iter) = minDist; if minDist < globalMin globalMin = minDist; optRoute = pop(index,:); if showProg % Plot the Best Route figure(pfig); %gambar grafik route if dims > 2, plot3(xy(optRoute,1),xy(optRoute,2),xy(optRoute,3),'r.-'); else plot(xy(optRoute,1),xy(optRoute,2),'r.-'); end title(sprintf('Total Distance = %1.4f, Iteration = %d',minDist,iter)); end end % Genetic Algorithm Operators randomOrder = randperm(popSize); for p = 4:4:popSize rtes = pop(randomOrder(p-3:p),:); dists = totalDist(randomOrder(p-3:p)); [ignore,idx] = min(dists); %#ok bestOf4Route = rtes(idx,:); routeInsertionPoints = sort(ceil(n*rand(1,2))); I = routeInsertionPoints(1); J = routeInsertionPoints(2); for k = 1:4 % Mutate the Best to get Three New Routes tmpPop(k,:) = bestOf4Route; switch k case 2 % Flip tmpPop(k,I:J) = tmpPop(k,J:-1:I); case 3 % Swap tmpPop(k,[I J]) = tmpPop(k,[J I]); case 4 % Slide tmpPop(k,I:J) = tmpPop(k,[I+1:J I]); otherwise % Do Nothing end end newPop(p-3:p,:) = tmpPop; end pop = newPop;end if showResult % Plots the GA Results figure('Name','TSPO_GA | Results','Numbertitle','off'); subplot(2,2,1); pclr = ~get(0,'DefaultAxesColor'); if dims > 2, plot3(xy(:,1),xy(:,2),xy(:,3),'.','Color',pclr); else plot(xy(:,1),xy(:,2),'.','Color',pclr); end title('City Locations'); subplot(2,2,2); imagesc(dmat(optRoute,optRoute)); title('Distance Matrix'); subplot(2,2,3); if dims > 2, plot3(xy(optRoute,1),xy(optRoute,2),xy(optRoute,3),'r.-'); else plot(xy(optRoute,1),xy(optRoute,2),'r.-'); end title(sprintf('Total Distance = %1.4f',minDist)); subplot(2,2,4); plot(distHistory,'b','LineWidth',2); title('Best Solution History'); set(gca,'XLim',[0 numIter+1],'YLim',[0 1.1*max([1 distHistory])]);end % Return Outputsif nargout varargout{1} = optRoute; varargout{2} = minDist;end