Genetic Algorithms: A Tutorial

1. 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

2. 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

3. Classes of Search Techniques

4. 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)

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

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

7. Population 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 ...

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

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

10. 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

11. 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)

12. 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

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

14. 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

15. 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)

16. 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.

17. Mutation Mutation involves reordering of the list: ** Before: (5 8 7 2 1 6 3 4) After: (5 8 6 2 1 7 3 4)

18. TSP Example: 30 Cities

19. Solution i (Distance = 941)

20. Solution j(Distance = 800)

21. Solution k(Distance = 652)

22. Best Solution (Distance = 420)

23. Overview of Performance

24. Some GA Application Types

25. %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: jdkirk630@gmail.com% 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;endif 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