# Genetic Algorithms: A Tutorial

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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

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

Classes of Search Techniques

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)

Simple Genetic Algorithm
{ initialize population; evaluate population; while TerminationCriteriaNotSatisfied select parents for reproduction; perform recombination and mutation; }

The GA Cycle of Reproduction
children reproduction modification modified children parents population evaluation evaluated children deleted members discard

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) Program elements (genetic programming) ... any data structure ...

Reproduction children reproduction parents population Parents are selected at random with selection chances biased in relation to chromosome evaluations.

Chromosome Modification
Modifications are stochastically triggered Operator types are: Mutation Crossover (recombination) children modification modified children

Mutation: Local Modification
Before: ( ) After: ( ) Before: ( ) After: ( ) Causes movement in the search space (local or global) Restores lost information to the population

Crossover: Recombination
* P1 ( ) ( ) C1 P2 ( ) ( ) 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)

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 modified children evaluated children evaluation

Deletion Generational GA: entire populations replaced with each iteration Steady-state GA: a few members replaced each generation population discarded members discard

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

Representation Representation is an ordered list of city
numbers known as an order-based GA. 1) London 3) Dunedin ) Beijing 7) Tokyo 2) Venice ) Singapore 6) Phoenix 8) Victoria CityList1 ( ) CityList2 ( )

Crossover Crossover combines inversion and recombination:
* * Parent ( ) Parent ( ) Child ( ) This operator is called the Order1 crossover.

Mutation Mutation involves reordering of the list:
* * Before: ( ) After: ( )

TSP Example: 30 Cities

Solution i (Distance = 941)

Solution j(Distance = 800)

Solution k(Distance = 652)

Best Solution (Distance = 420)

Overview of Performance

Some GA Application Types

%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: % A single salesman travels to each of the cities but does not close % the loop by returning to the city he started from % 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 % % Release: 1.3 % Release Date: 11/07/11 function varargout = tspo_ga(xy,dmat,popSize,numIter,showProg,showResult) % Process Inputs and Initialize Defaults nargs = 6; for k = nargin:nargs switch k case xy = 10*rand(50,2); case N = size(xy,1); a = meshgrid(1:N); dmat = reshape(sqrt(sum((xy(a,:)-xy(a',:)).^2,2)),N,N); case popSize = 100; case numIter = 1e4; case showProg = 1; case showResult = 1; otherwise end end % Verify Inputs [N,dims] = size(xy); [nr,nc] = size(dmat); if N ~= nr || N ~= nc error('Invalid XY or DMAT inputs!') end n = N; % Sanity Checks popSize = 4*ceil(popSize/4); numIter = max(1,round(real(numIter(1)))); showProg = logical(showProg(1)); showResult = logical(showResult(1)); % Initialize the Population pop = zeros(popSize,n); pop(1,:) = (1:n); for k = 2:popSize pop(k,:) = randperm(n); end % Run the GA globalMin = 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'); end for 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 Outputs if nargout varargout{1} = optRoute; varargout{2} = minDist; end

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