一、蜉蝣算法
蜉蝣是属于蜉蝣目的昆虫，是古翅目昆虫的一部分。据估计，全世界有超过3000种蜉蝣。它们的名字来源于它们主要出现在英国的五月。从卵中孵化出来后，肉眼可以看到未成熟的蜉蝣，它们花了几年时间成长为水生若虫，直到它们准备好成年后上升到水面。一只成年蜉蝣只存活几天，直到它完成繁殖的最终目标。为了吸引雌性，大多数雄性成虫成群结队地聚集在水面上几米的地方，通过特有的上下运动模式，表演一场婚礼舞蹈。雌鸟飞入这些蜂群，为了与空中的雄性交配。交配可能只持续几秒钟，当交配完成后，雌鸟将卵落在水面上，它们的生命周期就结束了。
1.1 雄性蜉蝣的运动
1.2 雌性蜉蝣的运动
1.3 蜉蝣交配
二、部分代码
%%
clc; clear; close all;
%% Problem Definition
% Objective Function
ANSWER=listdlg('PromptString','Choose Objective Function','SelectionMode','single', 'ListString', {'1. Sphere', '2. Rastrigin'});
if eq(ANSWER,1); ObjectiveFunction=@(x) Sphere(x); funcname='Sphere';
elseif eq(ANSWER,2); ObjectiveFunction=@(x) Rastrigin(x); funcname='Rastrigin';
else; disp('Terminated'); return
end
ProblemSize=[1 50]; % Decision Variables Size
LowerBound=-10; % Decision Variables Lower Bound
UpperBound= 10; % Decision Variables Upper Bound
%% Mayfly Parameters
methname='Mayfly Algorithm';
MaxIt=2000; % Maximum Number of Iterations
nPop=20; nPopf=20; % Population Size (males and females)
g=0.8; % Inertia Weight
gdamp=1; % Inertia Weight Damping Ratio
a1=1.0; % Personal Learning Coefficient
a2=1.5; a3=1.5; % Global Learning Coefficient
beta=2; % Distance sight Coefficient
dance=5; % Nuptial Dance
fl=1; % Random flight
dance_damp=0.8; % Damping Ratio
fl_damp=0.99;
% Mating Parameters
nc=20; % Number of Offsprings (also Parnets)
nm=round(0.05*nPop); % Number of Mutants
mu=0.01; % Mutation Rate
% Velocity Limits
VelMax=0.1*(UpperBound-LowerBound); VelMin=-VelMax;
%% Initialization
empty_mayfly.Position=[];
empty_mayfly.Cost=[];
empty_mayfly.Velocity=[];
empty_mayfly.Best.Position=[];
empty_mayfly.Best.Cost=[];
Mayfly=repmat(empty_mayfly,nPop,1); % Males
Mayflyf=repmat(empty_mayfly,nPopf,1); % Females
GlobalBest.Cost=inf;
funccount=0;
for i=1:nPop
% Initialize Position of Males
Mayfly(i).Position=unifrnd(LowerBound,UpperBound,ProblemSize);
% Initialize Velocity
Mayfly(i).Velocity=zeros(ProblemSize);
% Evaluation
Mayfly(i).Cost=ObjectiveFunction(Mayfly(i).Position);
% Update Personal Best
Mayfly(i).Best.Position=Mayfly(i).Position;
Mayfly(i).Best.Cost=Mayfly(i).Cost;
funccount=funccount+1;
% Update Global Best
if Mayfly(i).Best.Cost<GlobalBest.Cost
GlobalBest=Mayfly(i).Best;
end
end
for i=1:nPopf
% Initialize Position of Females
Mayflyf(i).Position=unifrnd(LowerBound,UpperBound,ProblemSize);
Mayflyf(i).Velocity=zeros(ProblemSize);
Mayflyf(i).Cost=ObjectiveFunction(Mayflyf(i).Position);
funccount=funccount+1;
% Update Global Best (Uncomment if you use the PGB-IMA version)
%if Mayflyf(i).Best.Cost<GlobalBest.Cost
% GlobalBest=Mayflyf(i).Best;
%end
end
BestSolution=zeros(MaxIt,1);
%% Mayfly Main Loop
for it=1:MaxIt
for i=1:nPopf
% Update Females
e=unifrnd(-1,+1,ProblemSize);
rmf=(Mayfly(i).Position-Mayflyf(i).Position);
if Mayflyf(i).Cost>Mayfly(i).Cost
Mayflyf(i).Velocity = g*Mayflyf(i).Velocity ...
+a3*exp(-beta.*rmf.^2).*(Mayfly(i).Position-Mayflyf(i).Position);
else
Mayflyf(i).Velocity = g*Mayflyf(i).Velocity+fl*(e);
end
% Apply Velocity Limits
Mayflyf(i).Velocity = max(Mayflyf(i).Velocity,VelMin);
Mayflyf(i).Velocity = min(Mayflyf(i).Velocity,VelMax);
% Update Position
c/2,2);
for k=1:nc/2
% Select Parents
i1=k;
i2=k;
p1=Mayfly(i1);
p2=Mayflyf(i2);
% Apply Crossover
[MayflyOffspring(k,1).Position, MayflyOffspring(k,2).Position]=Crossover(p1.Position,p2.Position,LowerBound,UpperBound);
% Evaluate Offsprings
MayflyOffspring(k,1).Cost=ObjectiveFunction(MayflyOffspring(k,1).Position);
if MayflyOffspring(k,1).Cost<GlobalBest.Cost
GlobalBest=MayflyOffspring(k,1);
end
funccount=funccount+1;
MayflyOffspring(k,2).Cost=ObjectiveFunction(MayflyOffspring(k,2).Position);
if MayflyOffspring(k,2).Cost<GlobalBest.Cost
GlobalBest=MayflyOffspring(k,2);
end
funccount=funccount+1;
MayflyOffspring(k,1).Best.Position = MayflyOffspring(k,1).Position;
MayflyOffspring(k,1).Best.Cost = MayflyOffspring(k,1).Cost;
MayflyOffspring(k,1).Velocity= zeros(ProblemSize);
MayflyOffspring(k,2).Best.Position = MayflyOffspring(k,2).Position;
MayflyOffspring(k,2).Best.Cost = MayflyOffspring(k,2).Cost;
MayflyOffspring(k,2).Velocity= zeros(ProblemSize);
end
MayflyOffspring=MayflyOffspring(:);
% Mutation
MutMayflies=repmat(empty_mayfly,nm,1);
for k=1:nm
% Select Parent
i=randi([1 nPop]);
p=MayflyOffspring(i);
%p=Mayfly(i);
MutMayflies(k).Position=Mutate(p.Position,mu,LowerBound,UpperBound);
% Evaluate Mutant
MutMayflies(k).Cost=ObjectiveFunction(MutMayflies(k).Position);
[~, SortMayflies]=sort([Mayfly.Cost]);
Mayfly=Mayfly(SortMayflies);
Mayfly=Mayfly(1:nPop); % Keep best males
[~, SortMayflies]=sort([Mayflyf.Cost]);
Mayflyf=Mayflyf(SortMayflies);
Mayflyf=Mayflyf(1:nPopf); % Keep best females
BestSolution(it)=GlobalBest.Cost;
disp([methname ' on the ' funcname ' Function: Iteration = ' num2str(it) ', ' funcname ', Evaluations = ' num2str(funccount) '. Best Cost = ' num2str(BestSolution(it))]);
g=g*gdamp;
dance = dance*dance_damp;
fl = fl*fl_damp;
end
%% Results
figure;
plot(BestSolution,'LineWidth',2); semilogy(BestSolution,'LineWidth',2);
xlabel('Iterations'); ylabel('Objective function'); grid on;
%%
三、仿真结果
四、参考文献
Zervoudakis, K., & Tsafarakis, S. (2020). A mayfly optimization algorithm. Computers & Industrial Engineering, 145, 106559. https://doi.org/10.1016/j.cie.2020.106559
