function [bestX,fMin] = FLFBA()
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 設定預設參數
M = 1000;
pop = 10;
dim = 2;
gama = 0.9; %0.9
alpha = 0.99; %0.99
r0Max = 1; %1
r0Min = 0; %0
AMax = 2; %2
AMin = 0; %0.5
freqMax = 2; %2
freqMin = 0; %0.1
%%%%%%%%%%%%%%%%%%%%%%%
wMax = 0.9; %0.9
wMin = 0.2; %0.2
xsi_init = 0.6; %0.6
n = 2; %2
% 參數設定
lb= -5 * ones(1,dim ); % 最低邊界
ub= 5 * ones(1,dim ); % 最高邊界
alfa=0.632;
beta=3/2; % Lèvy飛行軌迹參數
sigma=(gamma(1+beta)*sin(pi*beta/2)/(gamma((1+beta)/2)*...
beta*2^((beta-1)/2)))^(1/beta); % Lèvy飛行軌迹參數
Vhist = zeros(10,pop,dim); % 速度的曆史值儲存清單
vLb = 0.6 * lb; %0.6
vUb = 0.6 * ub; %0.6
r = rand( pop, 1 ) .* 0.2 + 0;
r0 = rand( pop, 1 ) .* ( r0Max - r0Min ) + r0Min;
A = rand( pop, 1 ) .* ( AMax - AMin ) + AMin;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Initialization
for i = 1 : pop
x( i, : ) = lb + (ub - lb) .* rand( 1, dim );
v( i, : ) = rand( 1, dim );
fit( i ) = Fobt(x(i,:)');
end
pFit = fit; % 個體最優适應度值
pX = x; % 個體最優位置
[ fMin, bestIndex ] = min( fit ); % fMin表示全局最優
% bestX表示fMin對應的位置
bestX = x( bestIndex, : );
Vhist(1,:,:)= v;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 疊代開始
for iteration = 1 : M
%%%%%%%%%%%%%%%%%%%%%% 疊代參數%%%%%%%%%%%%%%%%%%%%%%%%%%%
freq = rand( pop, dim ) .* ( freqMax - freqMin ) + freqMin;
w = (wMax - wMin) * ( M - iteration )/(1.0 * M) + wMin; %Inertia weight
xsi1 = 1 + ((xsi_init-1)*((M-iteration)/M)^n);
xsi2 = 1-xsi1;
meanA = mean( A );
meanP = mean(pX);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for i = 1 : pop
if rand < 0.5 %0.5
q1 = randi([1 pop]);
q2 = randi([1 pop]);
X1 = pX(q1,:);
X2 = pX(q2,:);
newX=kill_bat(pX(i,:),X1,X2); % Eq(24-25)
[pX(i,:),pFit(i)]=select_bat(pX(i,:),newX,pFit(i));
%%%%%%%%%%%% Levy飛行軌迹 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
u=randn(1,dim)*sigma;
VV=randn(1,dim);
step=u./abs(VV).^(1/beta);
stepsize = 0.01*step.*randn(1,dim); %0.01
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
x( i, : ) = pX(i,:) + stepsize .* ... %pX(i,:)
abs(meanP - x(i,:)); %Eq(26)
else % 使用分數導數
t1 = randi([1 10]);
q = randi([1 pop]);
while q== i
q = randi([1 pop]);
end
%%%%%%%%%%%%%%% fractional derivative %%%%%%%%%%%%%%%%%%
Vh = reshape(Vhist(:,i,:),10,dim);
vout = Frac_Diff_Der(alfa,Vh,t1); %Eq(27)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
v(i,:) = w.* vout + freq(i,:) .* xsi1 .*(bestX-x(i,:))+...
freq(i,:).* xsi2 .*(pX(i,:)-x(q,:)); %Eq(22)
v(i,:) = Bounds(v(i,:),vLb,vUb);
x(i,:) = x(i,:) + v(i,:);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
end
% 局部搜尋
if rand > r(i)
%%%%%%%%%%%%%%%% 另外一種Levy飛行軌迹 %%%%%%%%%%%%%%%%%%%%
u=randn(1,dim)*sigma;
VV=randn(1,dim);
step=u./abs(VV).^(1/beta);
stepsize = 0.01*step;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
randnValueA = stepsize .*randn(1,dim).* ...
(abs(A(i)-meanA)+realmin);
x(i,:) = bestX .* (1+randnValueA); %Eq(32)
end
x(i,:) = Bounds(x(i,:),lb,ub);
fit(i) = Fobt(x(i,:)');
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 更新個人最優适應度值和全局最優個體
for i = 1 : pop
if fit( i ) < pFit( i )
pFit( i ) = fit( i );
pX( i, : ) = x( i, : );
end
if( pFit( i ) < fMin && rand < A(i) )
fMin = pFit( i );
bestX = pX( i, : );
A(i) = A(i) * alpha;
r(i) = r0(i) * ( 1 - exp( -gama * iteration ) );
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Vhist(2:10,:,:) = Vhist(1:9,:,:);
Vhist(1,:,:) = v;
fbest = fMin; %#ok<NASGU>
FunMin(iteration) = fMin;
end
% End of the main program
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 以下函數與主程式相關
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Application of simple limits/bounds
function s = Bounds( s, Lb, Ub)
% 簡單極限/界限的應用
temp = s;
I = temp < Lb;
temp(I) = Lb(I);
% 應用上界向量
J = temp > Ub;
temp(J) = Ub(J);
% 更新此移動
s = temp;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [X,fits]=select_bat(X,newX,fits)
% 評價所有個體
fnew= Fobt(newX');
if fnew<=fits
fits =fnew;
X=newX;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function newX=kill_bat(X,X1,X2)
% 一小部分更差的點以一定機率被發現
stepsize=rand*(X1-X2);
newX=X+stepsize;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [vout] = Frac_Diff_Der(alfa,V,r)
switch(r)
case 1
vout = alfa*V(1,:);
case 2
vout = alfa*V(1,:)+(1/factorial(2))*V(2,:);
case 3
vout = alfa*V(1,:)+(1/factorial(2))*V(2,:)+...
(1/factorial(3))*alfa*(1-alfa)*V(3,:);
case 4
vout = alfa*V(1,:)+(1/factorial(2))*V(2,:)+...
(1/factorial(3))*alfa*(1-alfa)*V(3,:)+ ...
(1/factorial(4))*alfa*(1-alfa)*(2-alfa)*V(4,:);
case 5
vout = alfa*V(1,:)+(1/factorial(2))*V(2,:)+...
(1/factorial(3))*alfa*(1-alfa)*V(3,:)+ ...
(1/factorial(4))*alfa*(1-alfa)*(2-alfa)*V(4,:)+ ...
(1/factorial(5))*alfa*(1-alfa)*(2-alfa)*(3-alfa)*V(5,:);
case 6
vout = alfa*V(1,:)+(1/factorial(2))*V(2,:)+...
(1/factorial(3))*alfa*(1-alfa)*V(3,:)+ ...
(1/factorial(4))*alfa*(1-alfa)*(2-alfa)*V(4,:)+ ...
(1/factorial(5))*alfa*(1-alfa)*(2-alfa)*(3-alfa)*V(5,:)+...
(1/factorial(6))*alfa*(1-alfa)*(2-alfa)*(3-alfa)*...
(4-alfa)*V(6,:);
case 7
vout = alfa*V(1,:)+(1/factorial(2))*V(2,:)+...
(1/factorial(3))*alfa*(1-alfa)*V(3,:)+ ...
(1/factorial(4))*alfa*(1-alfa)*(2-alfa)*V(4,:)+ ...
(1/factorial(5))*alfa*(1-alfa)*(2-alfa)*(3-alfa)*V(5,:)+...
(1/factorial(6))*alfa*(1-alfa)*(2-alfa)*(3-alfa)*...
(4-alfa)*V(6,:)+ ...
(1/factorial(7))*alfa*(1-alfa)*(2-alfa)*(3-alfa)*...
(4-alfa)*(5-alfa)*V(7,:);
case 8
vout = alfa*V(1,:)+(1/factorial(2))*V(2,:)+...
(1/factorial(3))*alfa*(1-alfa)*V(3,:)+ ...
(1/factorial(4))*alfa*(1-alfa)*(2-alfa)*V(4,:)+ ...
(1/factorial(5))*alfa*(1-alfa)*(2-alfa)*(3-alfa)*V(5,:)+...
(1/factorial(6))*alfa*(1-alfa)*(2-alfa)*(3-alfa)*...
(4-alfa)*V(6,:)+ ...
(1/factorial(7))*alfa*(1-alfa)*(2-alfa)*(3-alfa)*...
(4-alfa)*(5-alfa)*V(7,:)+...
(1/factorial(8))*alfa*(1-alfa)*(2-alfa)*(3-alfa)*...
(4-alfa)*(5-alfa)*(6-alfa)*V(8,:);
case 9
vout = alfa*V(1,:)+(1/factorial(2))*V(2,:)+...
(1/factorial(3))*alfa*(1-alfa)*V(3,:)+ ...
(1/factorial(4))*alfa*(1-alfa)*(2-alfa)*V(4,:)+ ...
(1/factorial(5))*alfa*(1-alfa)*(2-alfa)*(3-alfa)*V(5,:)+...
(1/factorial(6))*alfa*(1-alfa)*(2-alfa)*(3-alfa)*...
(4-alfa)*V(6,:)+ ...
(1/factorial(7))*alfa*(1-alfa)*(2-alfa)*(3-alfa)*...
(4-alfa)*(5-alfa)*V(7,:)+...
(1/factorial(8))*alfa*(1-alfa)*(2-alfa)*(3-alfa)*...
(4-alfa)*(5-alfa)*(6-alfa)*V(8,:)+...
(1/factorial(9))*alfa*(1-alfa)*(2-alfa)*(3-alfa)*...
(4-alfa)*(5-alfa)*(6-alfa)*(7-alfa)*V(9,:);
case 10
vout = alfa*V(1,:)+(1/factorial(2))*V(2,:)+...
(1/factorial(3))*alfa*(1-alfa)*V(3,:)+ ...
(1/factorial(4))*alfa*(1-alfa)*(2-alfa)*V(4,:)+ ...
(1/factorial(5))*alfa*(1-alfa)*(2-alfa)*(3-alfa)*V(5,:)+...
(1/factorial(6))*alfa*(1-alfa)*(2-alfa)*(3-alfa)*...
(4-alfa)*V(6,:)+ ...
(1/factorial(7))*alfa*(1-alfa)*(2-alfa)*(3-alfa)*...
(4-alfa)*(5-alfa)*V(7,:)+...
(1/factorial(8))*alfa*(1-alfa)*(2-alfa)*(3-alfa)*...
(4-alfa)*(5-alfa)*(6-alfa)*V(8,:)+...
(1/factorial(9))*alfa*(1-alfa)*(2-alfa)*(3-alfa)*...
(4-alfa)*(5-alfa)*(6-alfa)*(7-alfa)*V(9,:)+...
(1/factorial(10))*alfa*(1-alfa)*(2-alfa)*(3-alfa)*...
(4-alfa)*(5-alfa)*(6-alfa)*(7-alfa)*(8-alfa)*V(10,:);
otherwise
disp('Incorrect order, must be between 1 and 10');
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%ù
function y = Fobt(x)
npar = length(x);
z = 0;
for i =2:npar
z = z+x(i)^2;
end
y = x(1)^2+1000000 * z;
參考文獻：(2020) "Fractional Lèvy flight bat algorithm for global optimisation", Int. J. Bio-Inspired Computation, Vol. 15, No. 2, pp.100–112
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