%functionnsga_2(pro)%%MainFunction%MainprogramtoruntheNSGA-IIMOEA.%Readthecorrespondingdocumentationtolearnmoreaboutmultiobjective%optimizationusingevolutionaryalgorithms.%initialize_variableshastwoarguments;Firstbeingthepopulationsize%andthesecondtheproblemnumber,'1'correspondstoMOP1and'2'%correspondstoMOP2.%inp_para_definition=input_parameters_definition;%%Initializethevariables%Declarethevariablesandinitializetheirvalues%pop-population%gen-generations%pro-problemnumber%clear;clc;tic;pop=100;%每一代的種群數(shù)gen=100;%總共的代數(shù)pro=2;%問題選擇1或者2,見switchswitchprocase1%Misthenumberofobjectives.M=2;%Visthenumberofdecisionvariables.Inthiscaseitis%difficulttovisualizethedecisionvariablesspacewhilethe%objectivespaceisjusttwodimensional.=6;case2M=3;=12;case3%case1和case2用來對整個算法進(jìn)行常規(guī)驗證，作為調(diào)試之用；case3為本工程所需；M=2;%(outputparameters個數(shù))V=8;%(inputparameters個數(shù))K=10;end%Initializethepopulationchromosome=initialize_variables(pop,pro);%%Sorttheinitializedpopulation%Sortthepopulationusingnon-domination-sort.Thisreturnstwocolumns%foreachindividualwhicharetherankandthecrowdingdistance%correspondingtotheirpositioninthefronttheybelong.真是牛X了。chromosome=non_domination_sort_mod(chromosome,pro);%%Starttheevolutionprocess%Thefollowingareperformedineachgeneration%Selecttheparents%PerfromcrossoverandMutationoperator%PerformSelectionfori=1:gen%Selecttheparents%Parentsareselectedforreproductiontogenerateoffspring.The%originalNSGA-IIusesabinarytournamentselectionbasedonthe%crowded-comparisionoperator.Theargumentsare%pool-sizeofthematingpool.Itiscommontohavethistobehalfthe%populationsize.%tour-Tournamentsize.OriginalNSGA-IIusesabinarytournament%selection,buttoseetheeffectoftournamentsizethisiskept%arbitary,tobechoosenbytheuser.pool=round(pop/2);tour=2;%下面進(jìn)行二人錦標(biāo)賽配對，新的群體規(guī)模是原來群體的一半parent_chromosome=tournament_selection(chromosome,pool,tour);%PerfromcrossoverandMutationoperator%TheoriginalNSGA-IIalgorithmusesSimulatedBinaryCrossover(SBX)and%Polynomialcrossover.Crossoverprobabilitypc=0.9andmutation%probabilityispm=1/n,wherenisthenumberofdecisionvariables.%Bothreal-codedGAandbinary-codedGAareimplementedintheoriginal%algorithm,whileinthisprogramonlythereal-codedGAisconsidered.%Thedistributionindeicesforcrossoverandmutationoperatorsasmu=20%andmum=20respectively.mu=20;mum=20;%針對對象是上一步產(chǎn)生的新的個體parent_chromosome%對parent_chromosome每次操作以較大的概率進(jìn)行交叉(產(chǎn)生兩個新的候選人)，或者較小的概率變異(一個新的候選人)操作，這樣%就會產(chǎn)生較多的新個體offspring_chromosome=genetic_operator(parent_chromosome,pro,mu,mum);%Intermediatepopulation%Intermediatepopulationisthecombinedpopulationofparentsand%offspringsofthecurrentgeneration.Thepopulationsizeisalmost1and%halftimestheinitialpopulation.[main_pop,temp]=size(chromosome);[offspring_pop,temp]=size(offspring_chromosome);intermediate_chromosome(1:main_pop,:)=chromosome;intermediate_chromosome(main_pop+1:main_pop+offspring_pop,1:M+V)=offspring_chromosome；%intermediate_chromosome=inter_chromo(chromosome,offspring_chromosome,pro);%Non-domination-sortofintermediatepopulation%Theintermediatepopulationissortedagainbasedonnon-dominationsort%beforethereplacementoperatorisperformedontheintermediate%ermediate_chromosome=...non_domination_sort_mod(intermediate_chromosome,pro);%PerformSelection%Oncetheintermediatepopulationissortedonlythebestsolutionis%selectedbasedonitrankandcrowdingdistance.Eachfrontisfilledin%ascendingorderuntiltheadditionofpopulationsizeisreached.The%lastfrontisincludedinthepopulationbasedontheindividualswith%leastcrowdingdistancechromosome=replace_chromosome(intermediate_chromosome,pro,pop);if~mod(i,10)fprintf('%d\n',i);endend%%Result%SavetheresultinASCIItextformat.savesolution.txtchromosome-ASCII%%Visualize%Thefollowingisusedtovisualizetheresultforthegivenproblem.switchprocase1plot(chromosome(:,V+1),chromosome(:,V+2),'y+');title('MOP1usingNSGA-II');xlabel('f(x_1)');ylabel('f(x_2)');case2plot3(chromosome(:,V+1),chromosome(:,V+2),chromosome(:,V+3),'*');title('MOP2usingNSGA-II');xlabel('f(x_1)');ylabel('f(x_2)');zlabel('f(x_3)');end%disp('runtimeis:')%toc;%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%functionf=initialize_variables(N,problem)%functionf=initialize_variables(N,problem)%N-Populationsize%problem-takesintegervalues1and2where,%'1'forMOP1%'2'forMOP2%%ThisfunctioninitializesthepopulationwithNindividualsandeach%individualhavingMdecisionvariablesbasedontheselectedproblem.%M=6forproblemMOP1andM=12forproblemMOP2.Theobjectivespace%forMOP1is2dimensionalwhileforMOP2is3dimensional.%BoththeMOP'shas0to1asitsrangeforallthedecisionvariables.min=0;max=1;switchproblemcase1M=6;K=8;%k=決策變量(M=6)+目標(biāo)變量(K-M=2)=8case2M=12;K=15;case3%case1和case2用來對整個算法進(jìn)行常規(guī)驗證，作為調(diào)試之用；case3為本工程所需；M=8;%(inputparameters個數(shù))K=10;endfori=1:N%Initializethedecisionvariablesforj=1:Mf(i,j)=rand(1);%i.ef(i,j)=min+(max-min)*rand(1);end%Evaluatetheobjectivefunctionf(i,M+1:K)=evaluate_objective(f(i,:),problem);end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%functionf=evaluate_objective(x,problem)%Functiontoevaluatetheobjectivefunctionsforthegiveninputvector%x.xhasthedecisionvariablesswitchproblemcase1f=[];%%Objectivefunctiononef(1)=1-exp(-4*x(1))*(sin(6*pi*x(1)))A6;sum=0;fori=2:6sum=sum+x(i)/4;end%%Intermediatefunctiong_x=1+9*(sum)A(0.25);%%Objectivefunctiononef(2)=g_x*(1-((f(1))/(g_x))A2);case2f=[];%%Intermediatefunctiong_x=0;fori=3:12g_x=g_x+(x(i)-0.5)人2;end%%Objectivefunctiononef(1)=(1+g_x)*cos(0.5*pi*x(1))*cos(0.5*pi*x(2));%%Objectivefunctiontwof(2)=(1+g_x)*cos(0.5*pi*x(1))*sin(0.5*pi*x(2));%%Objectivefunctionthreef(3)=(1+g_x)*sin(0.5*pi*x(1));case3f=[];%%Objectivefunctiononef(1)=0;%%Objectivefunctiononef(2)=0;end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Non-DonimationSort%按照目標(biāo)函數(shù)最小了好%Thisfunctionsortthecurrentpopultionbasedonnon-domination.Allthe%individualsinthefirstfrontaregivenarankof1,thesecondfront%individualsareassignedrank2andsoon.Afterassigningtherankthe%crowdingineachfrontiscalculated.functionf=non_domination_sort_mod(x,problem)[N,M]=size(x);switchproblemcase1M=2;V=6;case2M=3;

V=12;case3%case1和case2用來對整個算法進(jìn)行常規(guī)驗證，作為調(diào)試之用；case3為本工程所需；M=2;%(outputparameters個數(shù))V=8;%(inputparameters個數(shù))K=10;endfront=1;%Thereisnothingtothisassignment,usedonlytomanipulateeasilyin%MATLAB.F(front).f=[];individual=[];fori=1:N%Numberofindividualsthatdominatethisindividual支配i的解的個數(shù)individual(i).n=0;%Individualswhichthisindividualdominate被i支配的解individual(i).p=[];forj=1:Ndom_less=0;dom_equal=0;dom_more=0;fork=1:Mif(x(i,V+k)<x(j,V+k))dom_less=dom_less+1;elseif(x(i,V+k)==x(j,V+k))dom_equal=dom_equal+1;elsedom_more=dom_more+1;endend%這里只要考慮到求取得是函數(shù)的最小值就可以了，此時支配的概念就會變?yōu)檎l的函數(shù)值小，誰去支配別的解舉個例子，其中i=a,b;j=c,d;如果i或者部分大于j中的c,d(但沒有小%當(dāng)i優(yōu)于j的時候，則此時把%如果i中的舉個例子，其中i=a,b;j=c,d;如果i或者部分大于j中的c,d(但沒有小%當(dāng)i優(yōu)于j的時候，則此時把%如果i中的a,b全部小于或者%則%總elseifdom_more==0&dom_equal~=Mindividual(i)_n加1individual(i).p=[individual(i).pj];部分小于j中的c,d(但沒有大于的情況)，則end稱為j優(yōu)于i,則把此時的j放入individual(i)_P中；end之，就是說兩個目標(biāo)變量必須全部大于或者全部小于才能對individual有效。ifindividual(i).n==0%如果沒有劣于i的話，即F(front).f存放的都是差的解.x(i,M+V+1)=1;F(front).f=[F(front).fi];endend%Findthesubsequentfrontswhile?isempty(F(front).f)Q=[];fori=1:length(F(front).f)%i表示最優(yōu)解if?isempty(individual(F(front).f(i)).p)forj=1:length(individual(F(front).f(i)).p)%被前端解i所支配的解得集合，這個集合由j構(gòu)成，individual(individual(F(front).f(i)).p(j)).n=individual(individual(F(front).f(i)).p(j)).n-1;ifindividual(individual(F(front).f(i)).p(j)).n==0x(individual(F(front).f(i)).p(j),M+V+1)=front+1;Q=[Qindividual(F(front).f(i)).p(j)];endendendendfront=front+1;F(front).f=Q;end%functionsort:sort(x_sequence)===>>returnaincreasingorerdatasequence[temp,index_of_fronts]=sort(x(:,M+V+1));fori=1:length(index_of_fronts)sorted_based_on_front(i,:)=x(index_of_fronts(i),:);%對解(染色體)進(jìn)行排序，按照front分層end%到這里分層結(jié)束，下面就是計算距離了current_index=0;%Findthecrowdingdistanceforeachindividualineachfront%，計算不同層的擁擠距離是沒有意義的forfront=1:(length(F)-1)objective=[];distance=0;y=[];previous_index=current_index+1;fori=1:length(F(front).f)y(i,:)=sorted_based_on_front(current_index+i,:);endcurrent_index=current_index+i;%Sorteachindividualbasedontheobjectivesorted_based_on_objective=[];fori=1:M[sorted_based_on_objective,index_of_objectives]=sort(y(:,V+i));sorted_based_on_objective=[];forj=1:length(index_of_objectives)sorted_based_on_objective(j,:)=y(index_of_objectives(j),:);endf_max=...sorted_based_on_objective(length(index_of_objectives),V+i);%最大值f_min=sorted_based_on_objective(1,V+i);%最小值y(index_of_objectives(length(index_of_objectives)),M+V+1+i)...=Inf;%最大值放lnfy(index_of_objectives(1),M+V+1+i)=Inf;%最小值放lnfforj=2:length(index_of_objectives)-1next_obj=sorted_based_on_objective(j+1,V+i);previous_obj=sorted_based_on_objective(j-1,V+i);if(f_max-f_min==0)y(index_of_objectives(j),M+V+1+i)=Inf;elsey(index_of_objectives(j),M+V+1+i)=...(next_obj-previous_obj)/(f_max-f_min);endendenddistance=[];distance(:,1)=zeros(length(F(front).f),1);fori=1:Mdistance(:,1)=distance(:,1)+y(:,M+V+1+i);endy(:,M+V+2)=distance;y=y(:,1:M+V+2);z(previous_index:current_index,:)=y;endf=z;%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%functionf=replace_chromosome(intermediate_chromosome,pro,pop)%%replace_chromosome(intermediate_chromosome,pro,pop)%Thisfunctionreplacesthechromosomesbasedonrankandcrowding%distance.Initiallyuntilthepopulationsizeisreachedeachfrontis%addedonebyoneuntiladditionofacompletefrontwhichresultsin%exceedingthepopulationsize.Atthispointthechromosomesinthat%frontisaddedsubsequentlytothepopulationbasedoncrowdingdistance.[N,V]=size(intermediate_chromosome);switchprocase1M=2;=6;case2M=3;=12;case3%case1和case2用來對整個算法進(jìn)行常規(guī)驗證，作為調(diào)試之用；case3為本工程所需；M=2;%(outputparameters個數(shù))V=8;%(inputparameters個數(shù))K=10;end%Gettheindexforthepopulationsortbasedontherank[temp,index]=sort(intermediate_chromosome(:,M+V+1));%Nowsorttheindividualsbasedontheindexfori=1:Nsorted_chromosome(i,:)=intermediate_chromosome(index(i),:);end%Findthemaximumrankinthecurrentpopulationmax_rank=max(intermediate_chromosome(:,M+V+1));%Startaddingeachfrontbasedonrankandcrowingdistanceuntilthe%wholepopulationisfilled.previous_index=0;fori=1:max_rankcurrent_index=max(find(sorted_chromosome(:,M+V+1)==i));ifcurrent_index>popremaining=pop-previous_index;temp_pop=...sorted_chromosome(previous_index+1:current_index,:);temp_pop_1=temp_pop*(-1);[temp_sort_1,temp_sort_index]=...sort(temp_pop_1(:,M+V+2));%在編譯的時候出現(xiàn)的問題找到了，主要是對sort(a,'descend')并不支持。temp_sort=temp_sort_1*(-1);forj=1:remainingf(previous_index+j,:)=temp_pop(temp_sort_index(j),:);endreturn;elseifcurrent_index<popf(previous_index+1:current_index,:)=...sorted_chromosome(previous_index+1:current_index,:);elsef(previous_index+1:current_index,:)=...sorted_chromosome(previous_index+1:current_index,:);return;endprevious_index=current_index;end%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%functionf=tournament_selection(chromosome,pool_size,tour_size)%functionselection_individuals(chromosome,pool_size,tour_size)isthe%selectionpolicyforselectingtheindividualsforthematingpool.The%selectionisbasedontournamentselection.Argument'chromosome'isthe%currentgenerationpopulationfromwhichtheindividualsareselectedto%formamatingpoolofsize'pool_size'afterperformingtournament%selection,withsizeofthetournamentbeing'tour_size'.Byvaryingthe%tournamentsizetheselectionpressurecanbeadjusted.[pop,variables]=size(chromosome);rank=variables-1;%表示代表層數(shù)的那一列的索引distance=variables;%表示代表擁擠度距離的那一列的索引fori=1:pool_sizeforj=1:tour_size%這個loop目的是為了隨機(jī)找出tour_size個相互不同的候選染色體candidate(j)=round(pop*rand(1));ifcandidate(j)==0candidate(j)=1;endifj>1while~isempty(find(candidate(1:j-1)==candidate(j)))candidate(j)=round(pop*rand(1));ifcandidate(j)==0candidate(j)=1;endendendendforj=1:tour_sizec_obj_rank(j)=chromosome(candidate(j),rank);%提取出階數(shù)c_obj_distance(j)=chromosome(candidate(j),distance);%提取出距離endmin_candidate=find(c_obj_rank==min(c_obj_rank));%找出層數(shù)最小的那個候選人iflength(min_candidate)~=1%發(fā)現(xiàn)存在層數(shù)相同的候選人，就要比較擁擠度距離了max_candidate=find(c_obj_distance(min_candidate)==max(c_obj_distance(min_candidate)));%返回具有同樣最小層數(shù)的，且擁擠距離最大的那個候選人iflength(max_candidate)~=1max_candidate=max_candidate(1%;層數(shù)，擁擠距離都相同，隨便選個就可以了endf(i,:)=chromosome(candidate(min_candidate(max_candidate)),:);elsef(i,:)=chromosome(candidate(min_candidate(1)),:);endend%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%functionf=genetic_operator(parent_chromosome,pro,mu,mum)%Thisfunctionisutilizedtoproduceoffspringsfromparentchromosomes.%Thegeneticoperatorscorssoverandmutationwhicharecarriedoutwith%slightmodificationsfromtheoriginaldesign.Formoreinformationread%thedocumentenclosed.%input_parameters=inp_para_definitiSB^_了一個輸入變量的結(jié)構(gòu)，[N,M]=size(parent_chromosome);注意這里的M其實是個垃圾，后面語句會對其重新賦值利用switchprocase1M=2;=6;case2M=3;=12;case3%case1和case2用來對整個算法進(jìn)行常規(guī)驗證，作為調(diào)試之用；case3為本工程所需；M=2;%(outputparameter個數(shù))V=8;%(inputparameters個數(shù))K=10;endp=1;was_crossover=0;was_mutation=0;l_limit=0;u_limit=1;fori=1:Nifrand(1)<0.9child_1=[];child_2=[];parent_1=round(N*rand(1));ifparent_1<1parent_1=1;endparent_2=round(N*rand(1));ifparent_2<1parent_2=1;endwhileisequal(parent_chromosome(parent_1,:),parent_chromosome(parent_2,:))parent_2=round(N*rand(1));ifparent_2<1parent_2=1;endend%目的是隨機(jī)找出兩個不同的候選人parent_1=parent_chro