(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 96582, 3395]*) (*NotebookOutlinePosition[ 97637, 3431]*) (* CellTagsIndexPosition[ 97593, 3427]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{Cell[TextData["Genetic Algorithms"], "Title", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Written by Mats G. Bengtsson\nNational Defence Research Establishment\nBox \ 1165, S-581 11 Linkoping\nSweden\nemail: matben@lin.foa.se"], "Subsubtitle", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "This is intended to be a simple and pedagogical illustration of how genetic \ algorithms may be used for function optimization. The code is very slow so it \ is only intended for demos, not for production runs. For further info, see \ for instance the book by Goldberg.\n\nA genetic algorithm consists \ essentially of three parts:\n-- Selection; each string is selected with a \ probability proportional to its fitness value.\n-- Crossover; in a pair of \ selected strings, sition along the string is chosen, and the right and left \ part of each string is swapped.\n-- Mutation; each gene is changed at \ random with a small probability."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Set up the problem"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "In our case we have chosen the conventional binary coding. Each individual \ is coded as a binary string with length stringLength. The problem is limited \ to 1D, and for x from 0 to 1."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "stringLength = 10;\nmutationRate = 0.002;\npopSize = 50; \t\t(* must be a \ power of two *)"], "Input", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[ "General::spell1: \n Possible spelling error: new symbol name \ \"stringLength\"\n is similar to existing symbol \"StringLength\"."], "Message", Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ "Create a random population. The population is defined on the real axis from \ 0 to 1. The random values are approximated with binary strings given the \ stringLength."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "popFloats = Table[ Random[], {popSize} ];\npopStrings = Map[ floatToBinary, \ popFloats ];"], "Input", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[ "Define the fitness function. It must be defined in the interval 0 to 1. The \ first one is a simple one, while the second function is a little bit more \ difficult."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Clear[fitnessFunction];\nfitnessFunction[x_] := Sin[ N[Pi] x ]"], "Input", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[ "Clear[fitnessFunction];\nfitnessFunction[x_] := Sin[N[Pi]x]*Mod[9x, 1]"], "Input", InitializationCell->True, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Function definitions"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Simple crossover"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "A random position is chosen along the strings. Create two new strings by \ swapping the parts before and after the cut respectively."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Clear[doSingleCrossover];\ndoSingleCrossover[ {string1_, string2_} ] := \ Module[{stle,cut,temp1,temp2},\n\tstle = Length[string1];\n\tcut = \ Random[Integer, {1,stle-1}];\n\ttemp1 = Join[ Take[string1, cut],\n\t\t\ Drop[string2, cut] ];\n\ttemp2 = Join[ Take[string2, cut],\n\t\tDrop[string1, \ cut] ];\n\t{temp1, temp2} ]"], "Input", InitializationCell->True, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Mutations"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Every gene may be excanged, from 0 to 1 and vice versa, with the probability \ mutationRate. The idea behind this is to introduce a mechanism for escaping \ from local minimas."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Clear[doMutation];\ndoMutation[string_] := Module[{tempstring,i},\n\t\ tempstring = string;\n\tDo[ If[ Random[] < mutationRate,\n\t\ttempstring[[i]] \ = 1 - tempstring[[i]] ],\n\t\t{i,stringLength} ];\n\ttempstring\n]"], "Input",\ InitializationCell->True, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Selection"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Select two strings from the population, with a probability for each string \ proportional to its fitness value. Return only the two indices."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "This computes the cumulative sum of the fitness values. It is necessary in \ the selection process."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Clear[doCumSumOfFitness];\ndoCumSumOfFitness := Module[{temp},\n\ttemp = \ 0.0;\n\tTable[ temp += popFitness[[i]], {i, popSize} ]\n]"], "Input", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[ "Clear[doSingleSelection];\ndoSingleSelection := Module[{rfitness,ind},\n\t\ rfitness = Random[Real, {0, cumFitness[[popSize]]}];\n\tind = 1;\n\tWhile[ \ rfitness > cumFitness[[ind]], ind++ ];\n\tind--\n]"], "Input", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[ "This routine selects a pair of individuals with probability for each \ proportional to its fitness value."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Clear[selectPair];\nselectPair := Module[{ind1,ind2},\n\tind1 = \ doSingleSelection;\n\tWhile[ (ind2 = doSingleSelection) == ind1, ];\n\t{ind1, \ ind2}\n]"], "Input", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[ "This routine selects a pair at random with equal probability for each. This \ pair is removed from the population."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Clear[pickRandomPair];\npickRandomPair := Module[{ind1,ind2},\n\tind1 = \ Random[Integer, {1, popSize}];\n\tWhile[ (ind2 = Random[Integer, {1, \ popSize}]) == ind1, ];\n\t{ind1, ind2}\n]"], "Input", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[ "Swap strings. Update both the string and its corresponding fitness value."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Clear[exchangeString];\nexchangeString[ind_, newstring_, newF_] := \ Module[{},\n\tpopStrings[[ind]] = newstring;\n\tpopFitness[[ind]] = newF;\n\ ]"], "Input", InitializationCell->True, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Miscellaneous functions"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Clear[floatToBinary];\nfloatToBinary[x_] := Module[{temp},\n\ttemp = \ RealDigits[ x, 2 ];\n\tTake[ Join[ \n\t\tTable[0, {-temp[[2]]}], temp[[1]], \ Table[0, {stringLength}] ],\n\t\tstringLength ]\n] /; x<1"], "Input", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[ "Clear[binaryStringToFloat];\nbinaryStringToFloat[string_] :=\n\tN[Sum[ \ string[[i]]*2^(-i), {i, Length[string]} ]]"], "Input", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[ "Scale the fitness values linearly so that max = 2 min. This is a good idea \ to do because otherwise the genetic material tends to be dominated by very \ few nonidentical members. There is always a balance between concentrating the \ search on the best strings, and having a search in a broad spectrum."], "Text",\ Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Clear[renormalizeFitness];\nrenormalizeFitness[fitness_List] := \ Module[{minF,maxF,a,b},\n\tminF = Min[fitness];\n\tmaxF = Max[fitness];\n\ta \ = 0.5*maxF/(maxF+minF);\n\tb = (1-a)*maxF;\n\tMap[ a# + b &, fitness ]\n]"], "Input", InitializationCell->True, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Initialize"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "This initializes everything except for the population itself."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Clear[doInitialize];\ndoInitialize := Module[{i},\n\tpopFitness = Table[\n\t\ \tfitnessFunction[ binaryStringToFloat[ popStrings[[i]] ] ],\n\t\t{i,popSize} \ ];\n\tcumFitness = doCumSumOfFitness;\n\tlistOfCumFitness = \ {cumFitness[[popSize]]};\n\thistoryOfPop = { popStrings };\n]"], "Input", InitializationCell->True, AspectRatioFixed->True]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData["Main"], "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Make a new generation; update asynchronously"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "There are actually two ways you can do it. The asynchronous mode corresponds \ to picking the parents, do crossover and mutation, and exchange the children \ with a randomly picked pair. This proceeds pair by pair."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "Clear[updateGenerationAsync];\n\nupdateGenerationAsync := \ Module[{parents,children,childrenF,rm,i},\n\tDo[\tparents = selectPair;\n\t\t\ children = doSingleCrossover[ {popStrings[[ parents[[1]] ]], \n\t\t\t\ popStrings[[ parents[[2]] ]]} ];\n\t\tchildren = Map[ doMutation, children ]; \ \n\t\tchildrenF = fitnessFunction[ \n\t\t\tMap[ binaryStringToFloat, children \ ] ];\n\t\trm = pickRandomPair;\n\t\tDo[exchangeString[ rm[[i]], \ children[[i]], childrenF[[i]] ],\n\t\t\t{i,2} ];\n\t\tcumFitness = \ doCumSumOfFitness;\n\t\tAppendTo[ listOfCumFitness, cumFitness[[popSize]] ],\n\ \t{popSize/2} ];\n];"], "Input", InitializationCell->True, AspectRatioFixed->True], Cell[TextData[ "General::spell1: \n Possible spelling error: new symbol name \"childrenF\"\ \n is similar to existing symbol \"children\"."], "Message", Evaluatable->False, AspectRatioFixed->True]}, Open]]}, Open]], Cell[CellGroupData[{Cell[TextData["Make a new generation; update synchronously"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "This represents the other way to do it. The complete parent population is \ chosen in a single sweep, and the children constitutes the new population."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Clear[updateGenerationSync];\n\nupdateGenerationSync := \ Module[{parentsid,children,ip},\n\tparentsid = {};\n\tDo[\tAppendTo[ \ parentsid, selectPair ], {popSize/2} ];\n\tchildren = {};\n\tDo[ AppendTo[ \ children, \n\t\t\tdoSingleCrossover[ {popStrings[[parentsid[[ip,1]]]], \n\t\t\ \tpopStrings[[parentsid[[ip,2]]]]} ] ],\n\t\t{ip, popSize/2} ];\n\tpopStrings \ = Flatten[ children, 1];\n\tpopStrings = Map[ doMutation, popStrings ]; \n\t\ popFitness = fitnessFunction[ \n\t\tMap[ binaryStringToFloat, popStrings ] ];\ \n\tpopFitness = renormalizeFitness[ popFitness ];\n\tcumFitness = \ doCumSumOfFitness;\t\n];"], "Input", InitializationCell->True, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["This is Main"], "Subsubsection", Evaluatable->False, AspectRatioFixed->True], 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