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布谷鸟搜索（Cuckoo Search, CS）算法是 2009 年 Xin-She Yang 与 Susash Deb 提出的一种新型的启发算法[61]。算法灵感来源于一些布谷鸟种属（Cuckoo Species）的专性寄生哺育（BroodParasitism）行为，即这些种群不会像绝大多数鸟类一样自己筑巢哺育后代，而是通常把蛋产于宿主鸟巢内，由宿主代养，这种行为被称为巢寄生。此类布谷鸟会观察宿主鸟所产的卵并对其进行模仿，按照宿主鸟卵的颜色形状来产卵，导致宿主鸟辨别出布谷鸟卵的几率微乎其微，布谷鸟卵混在宿主鸟卵中，孵化后由宿主鸟哺育并与宿主雏鸟争夺生存资源。然而，布谷鸟卵一旦被宿主鸟识破情况便不同，宿主将完全摧毁鸟巢或是仅仅将布谷鸟卵摧毁。因此布谷鸟在繁殖季会寻找孵化周期类似、雏鸟习性类似以及卵外观类似的宿主鸟。布谷鸟通常会趁宿主外出时将卵产于巢内，有时会将宿主卵推出巢后进行产卵，使布谷雏鸟独享生存资源，以提高生存几率。此外，布谷鸟等动物的觅食过程是也是一种有启发性的随机行为过程。所在位置确定移动的下一个目标点，所选取的数学模型决定移动方向。根据当前位置与到下一个位置的转移概率，它们会飞向或是走向搜索路径。鸟类的这种飞行行为，在很多研究中被证明为莱维飞行（Levy Flight）的典型特征
⛄ 部分代码
%% Cuckoo Search (CS) algorithm by Xin-She Yang and Suash Deb %
% Programmed by Xin-She Yang at Cambridge University %
% Programming dates: Nov 2008 to June 2009 %
% Last revised: Dec 2009 (simplified version for demo only) %
% Multiobjective cuckoo search (MOCS) added in July 2012, %
% Then, MOCS was updated in Sept 2015. Thanks %
% -----------------------------------------------------------------
%% References -- Citation Details:
%% 1) X.-S. Yang, S. Deb, Cuckoo search via Levy flights,
% in: Proc. of World Congress on Nature & Biologically Inspired
% Computing (NaBIC 2009), December 2009, India,
% IEEE Publications, USA, pp. 210-214 (2009).
% http://arxiv.org/PS_cache/arxiv/pdf/1003/1003.1594v1.pdf
%% 2) X.-S. Yang, S. Deb, Engineering optimization by cuckoo search,
% Int. J. Mathematical Modelling and Numerical Optimisation,
% Vol. 1, No. 4, 330-343 (2010).
% http://arxiv.org/PS_cache/arxiv/pdf/1005/1005.2908v2.pdf
%% 3) X.-S. Yang, S. Deb, Multi-objective cuckoo search for
% Design optimization, Computers & Operations Research,
% vol. 40, no. 6, 1616-1624 (2013).
% ----------------------------------------------------------------%
% This demo program only implements a standard version of %
% Cuckoo Search (CS), as the Levy flights and generation of %
% new solutions may use slightly different methods. %
% The pseudo code was given sequentially (select a cuckoo etc), %
% but the implementation here uses Matlab's vector capability, %
% which results in neater/better codes and shorter running time. %
% This implementation is different and more efficient than the %
% the demo code provided in the book by
% "Yang X. S., Nature-Inspired Optimization Algoirthms, %
% Elsevier Press, 2014. " %
% --------------------------------------------------------------- %
% =============================================================== %
%% Notes: %
% 1) The constraint-handling is not included in this demo code. %
% The main idea to show how the essential steps of cuckoo search %
% and multi-objective cuckoo search (MOCS) can be done. %
% 2) Different implementations may lead to slightly different %
% behavour and/or results, but there is nothing wrong with it, %
% as it is the nature of random walks and all metaheuristics. %
% --------------------------------------------------------------- %
function [bestnest,fmin]=mocs_new(inp)
if nargin<1,
inp=[100 1000 0.25]; % pop_size, #iteraion, pa
end
% Number of nests (or the population size)
n=inp(1);
% Number of iterations/generations
N_IterTotal=inp(2);
% Discovery rate of alien eggs/solutions
pa=inp(3);
d=30; % Dimensionality of the problem
% Simple lower bounds
Lb=0*ones(1,d);
% Simple upper bounds
Ub=1*ones(1,d);
% Number of objectives
m=2;
%% Initialize the population
for i=1:n,
Sol(i,:)=Lb+(Ub-Lb).*rand(1,d);
f(i,1:m) = obj_funs(Sol(i,:), m);
end
% Store the fitness or objective values
f_new=f;
%% Sort the initialized population
x=[Sol f]; % combined into a single input
% Non-dominated sorting for the initila population
Sorted=solutions_sorting(x, m,d);
% Decompose into solutions, fitness, rank and distances
nest=Sorted(:,1:d);
f=Sorted(:,(d+1):(d+m));
RnD=Sorted(:,(d+m+1):end);
%% Starting iterations
for t=1:N_IterTotal,
% Generate new solutions (but keep the current best)
new_nest=get_cuckoos(nest,nest(1,:), Lb,Ub);
% new_nest=nest;
% Discovery and randomization
new_nest=empty_nests(nest,Lb,Ub,pa) ;
% Evaluate this set of solutions
for i=1:n,
%% Evalute the fitness/function values of the new population
f_new(i, 1:m) = obj_funs(new_nest(i,1:d),m);
if (f_new(i,1:m) <= f(i,1:m)),
f(i,1:m)=f_new(i,1:m);
nest(i,:)=new_nest(i,:);
end
% Update the current best (stored in the first row)
if (f_new(i,1:m) <= f(1,1:m)),
nest(1,1:d) = new_nest(i,1:d);
f(1,:)=f_new(i,:);
end
end % end of for loop
%% Combined population consits of both the old and new solutions
%% So the total number of solutions for sorting is 2*n
%% ! It's very important to combine both populations, otherwise,
%% the results may look odd and will be very inefficient. !
X(1:n,:)=[new_nest f_new]; % Combine new solutions
X((n+1):(2*n),:)=[nest f]; % Combine old solutions
Sorted=solutions_sorting(X, m, d);
%% Select n solutions from a combined population of 2*n solutions
new_Sol=Select_pop(Sorted, m, d, n);
% Decompose the sorted solutions into solutions, fitness & ranking
nest=new_Sol(:,1:d); % Sorted solutions/variables
f=new_Sol(:,(d+1):(d+m)); % Sorted objective values
RnD=new_Sol(:,(d+m+1):end); % Sorted ranks and distances
%% Running display at each 100 iterations
if ~mod(t,100),
disp(strcat('Iterations t=',num2str(t)));
plot(f(:, 1), f(:, 2),'rs','MarkerSize',3);
axis([0 1 -0.8 1]);
xlabel('f_1'); ylabel('f_2');
drawnow;
end
end %% End of iterations
%% --------------- All subfunctions are list below ------------------ %
%% Get cuckoos by ramdom walk
function nest=get_cuckoos(nest,best,Lb,Ub)
n=size(nest,1);
% For details, please see the chapters of the following Elsevier book:
% X. S. Yang, Nature-Inspired Optimization Algorithms, Elsevier, (2014).
beta=3/2; % Levy exponent in Levy flights
sigma=(gamma(1+beta)*sin(pi*beta/2)/(gamma((1+beta)/2)*beta*2^((beta-1)/2)))^(1/beta);
for j=1:n,
s=nest(j,:);
%% Levy flights by Mantegna's algorithm
u=randn(size(s))*sigma;
v=randn(size(s));
step=u./abs(v).^(1/beta);
stepsize=0.1*step.*(s-best);
% Now the actual random walks or flights
s=s+stepsize.*randn(size(s));
% Apply simple bounds/limits
nest(j,:)=simplebounds(s,Lb,Ub);
end
%% Replace some nests by constructing new solutions/nests
function new_nest=empty_nests(nest,Lb,Ub,pa)
% A fraction of worse nests are discovered with a probability pa
[n,d]=size(nest);
% The solutions represented by cuckoos to be discovered or not
% with a probability pa. This action is implemented as a status vector
K=rand(size(nest))>pa;
%% New solution by biased/selective random walks
stepsize=rand(1,d).*(nest(randperm(n),:)-nest(randperm(n),:));
new_nest=nest+stepsize.*K;
for j=1:size(new_nest,1)
s=new_nest(j,:);
new_nest(j,:)=simplebounds(s,Lb,Ub);
end
% Application of simple bounds
function s=simplebounds(s,Lb,Ub)
% Apply the lower bound
ns_tmp=s;
I=ns_tmp<Lb;
ns_tmp(I)=Lb(I);
% Apply the upper bounds
J=ns_tmp>Ub;
ns_tmp(J)=Ub(J);
% Update this new move
s=ns_tmp;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Objective functions
function f = obj_funs(x, m)
% Zitzler-Deb-Thiele's funciton No 3 (ZDT function 3)
% M = # of objectives
% d = # of variables/dimensions
d=length(x); % d=30 for ZDT 3
% First objective f1
f(1) = x(1);
g=1+9/29*sum(x(2:d));
h=1-sqrt(f(1)/g)-f(1)/g*sin(10*pi*f(1));
% Second objective f2
f(2) = g*h;
%%%%%%%%%%%%%%%%%% end of the definitions of obojectives %%%%%%%%%%%%%%%%%%
function new_Sol = Select_pop(nest, m, ndim, npop)
% The input population to this part has twice (ntwice) of the needed
% population size (npop). Thus, selection is done based on ranking and
% crowding distances, calculated from the non-dominated sorting
ntwice= size(nest,1);
% Ranking is stored in column Krank
Krank=m+ndim+1;
% Sort the population of size 2*npop according to their ranks
[~,Index] = sort(nest(:,Krank)); sorted_nest=nest(Index,:);
% Get the maximum rank among the population
RankMax=max(nest(:,Krank));
%% Main loop for selecting solutions based on ranks and crowding distances
K = 0; % Initialization for the rank counter
% Loop over all ranks in the population
for i =1:RankMax,
% Obtain the current rank i from sorted solutions
RankSol = max(find(sorted_nest(:, Krank) == i));
% In the new cuckoos/solutions, there can be npop solutions to fill
if RankSol<npop,
new_Sol(K+1:RankSol,:)=sorted_nest(K+1:RankSol,:);
end
% If the population after addition is large than npop, re-arrangement
% or selection is carried out
if RankSol>=npop
% Sort/Select the solutions with the current rank
candidate_nest = sorted_nest(K + 1 : RankSol, :);
[~,tmp_Rank]=sort(candidate_nest(:,Krank+1),'descend');
% Fill the rest (npop-K) cuckoo/solutions up to npop solutions
for j = 1:(npop-K),
new_Sol(K+j,:)=candidate_nest(tmp_Rank(j),:);
end
end
% Record and update the current rank after adding new cuckoo solutions
K = RankSol;
end
⛄ 运行结果
⛄ 参考文献
[1]尚志勇. 基于改进布谷鸟搜索算法的配送中心选址问题研究. (Doctoral dissertation, 河南大学).
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✅作者简介：热爱科研的Matlab仿真开发者，修心和技术同步精进，matlab项目合作可私信。
🍎个人主页：Matlab科研工作室
🍊个人信条：格物致知。
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⛄ 内容介绍
布谷鸟搜索（Cuckoo Search, CS）算法是 2009 年 Xin-She Yang 与 Susash Deb 提出的一种新型的启发算法[61]。算法灵感来源于一些布谷鸟种属（Cuckoo Species）的专性寄生哺育（BroodParasitism）行为，即这些种群不会像绝大多数鸟类一样自己筑巢哺育后代，而是通常把蛋产于宿主鸟巢内，由宿主代养，这种行为被称为巢寄生。此类布谷鸟会观察宿主鸟所产的卵并对其进行模仿，按照宿主鸟卵的颜色形状来产卵，导致宿主鸟辨别出布谷鸟卵的几率微乎其微，布谷鸟卵混在宿主鸟卵中，孵化后由宿主鸟哺育并与宿主雏鸟争夺生存资源。然而，布谷鸟卵一旦被宿主鸟识破情况便不同，宿主将完全摧毁鸟巢或是仅仅将布谷鸟卵摧毁。因此布谷鸟在繁殖季会寻找孵化周期类似、雏鸟习性类似以及卵外观类似的宿主鸟。布谷鸟通常会趁宿主外出时将卵产于巢内，有时会将宿主卵推出巢后进行产卵，使布谷雏鸟独享生存资源，以提高生存几率。此外，布谷鸟等动物的觅食过程是也是一种有启发性的随机行为过程。所在位置确定移动的下一个目标点，所选取的数学模型决定移动方向。根据当前位置与到下一个位置的转移概率，它们会飞向或是走向搜索路径。鸟类的这种飞行行为，在很多研究中被证明为莱维飞行（Levy Flight）的典型特征
⛄ 部分代码
%% Cuckoo Search (CS) algorithm by Xin-She Yang and Suash Deb %
% Programmed by Xin-She Yang at Cambridge University %
% Programming dates: Nov 2008 to June 2009 %
% Last revised: Dec 2009 (simplified version for demo only) %
% Multiobjective cuckoo search (MOCS) added in July 2012, %
% Then, MOCS was updated in Sept 2015. Thanks %
% -----------------------------------------------------------------
%% References -- Citation Details:
%% 1) X.-S. Yang, S. Deb, Cuckoo search via Levy flights,
% in: Proc. of World Congress on Nature & Biologically Inspired
% Computing (NaBIC 2009), December 2009, India,
% IEEE Publications, USA, pp. 210-214 (2009).
% http://arxiv.org/PS_cache/arxiv/pdf/1003/1003.1594v1.pdf
%% 2) X.-S. Yang, S. Deb, Engineering optimization by cuckoo search,
% Int. J. Mathematical Modelling and Numerical Optimisation,
% Vol. 1, No. 4, 330-343 (2010).
% http://arxiv.org/PS_cache/arxiv/pdf/1005/1005.2908v2.pdf
%% 3) X.-S. Yang, S. Deb, Multi-objective cuckoo search for
% Design optimization, Computers & Operations Research,
% vol. 40, no. 6, 1616-1624 (2013).
% ----------------------------------------------------------------%
% This demo program only implements a standard version of %
% Cuckoo Search (CS), as the Levy flights and generation of %
% new solutions may use slightly different methods. %
% The pseudo code was given sequentially (select a cuckoo etc), %
% but the implementation here uses Matlab's vector capability, %
% which results in neater/better codes and shorter running time. %
% This implementation is different and more efficient than the %
% the demo code provided in the book by
% "Yang X. S., Nature-Inspired Optimization Algoirthms, %
% Elsevier Press, 2014. " %
% --------------------------------------------------------------- %
% =============================================================== %
%% Notes: %
% 1) The constraint-handling is not included in this demo code. %
% The main idea to show how the essential steps of cuckoo search %
% and multi-objective cuckoo search (MOCS) can be done. %
% 2) Different implementations may lead to slightly different %
% behavour and/or results, but there is nothing wrong with it, %
% as it is the nature of random walks and all metaheuristics. %
% --------------------------------------------------------------- %
function [bestnest,fmin]=mocs_new(inp)
if nargin<1,
inp=[100 1000 0.25]; % pop_size, #iteraion, pa
end
% Number of nests (or the population size)
n=inp(1);
% Number of iterations/generations
N_IterTotal=inp(2);
% Discovery rate of alien eggs/solutions
pa=inp(3);
d=30; % Dimensionality of the problem
% Simple lower bounds
Lb=0*ones(1,d);
% Simple upper bounds
Ub=1*ones(1,d);
% Number of objectives
m=2;
%% Initialize the population
for i=1:n,
Sol(i,:)=Lb+(Ub-Lb).*rand(1,d);
f(i,1:m) = obj_funs(Sol(i,:), m);
end
% Store the fitness or objective values
f_new=f;
%% Sort the initialized population
x=[Sol f]; % combined into a single input
% Non-dominated sorting for the initila population
Sorted=solutions_sorting(x, m,d);
% Decompose into solutions, fitness, rank and distances
nest=Sorted(:,1:d);
f=Sorted(:,(d+1):(d+m));
RnD=Sorted(:,(d+m+1):end);
%% Starting iterations
for t=1:N_IterTotal,
% Generate new solutions (but keep the current best)
new_nest=get_cuckoos(nest,nest(1,:), Lb,Ub);
% new_nest=nest;
% Discovery and randomization
new_nest=empty_nests(nest,Lb,Ub,pa) ;
% Evaluate this set of solutions
for i=1:n,
%% Evalute the fitness/function values of the new population
f_new(i, 1:m) = obj_funs(new_nest(i,1:d),m);
if (f_new(i,1:m) <= f(i,1:m)),
f(i,1:m)=f_new(i,1:m);
nest(i,:)=new_nest(i,:);
end
% Update the current best (stored in the first row)
if (f_new(i,1:m) <= f(1,1:m)),
nest(1,1:d) = new_nest(i,1:d);
f(1,:)=f_new(i,:);
end
end % end of for loop
%% Combined population consits of both the old and new solutions
%% So the total number of solutions for sorting is 2*n
%% ! It's very important to combine both populations, otherwise,
%% the results may look odd and will be very inefficient. !
X(1:n,:)=[new_nest f_new]; % Combine new solutions
X((n+1):(2*n),:)=[nest f]; % Combine old solutions
Sorted=solutions_sorting(X, m, d);
%% Select n solutions from a combined population of 2*n solutions
new_Sol=Select_pop(Sorted, m, d, n);
% Decompose the sorted solutions into solutions, fitness & ranking
nest=new_Sol(:,1:d); % Sorted solutions/variables
f=new_Sol(:,(d+1):(d+m)); % Sorted objective values
RnD=new_Sol(:,(d+m+1):end); % Sorted ranks and distances
%% Running display at each 100 iterations
if ~mod(t,100),
disp(strcat('Iterations t=',num2str(t)));
plot(f(:, 1), f(:, 2),'rs','MarkerSize',3);
axis([0 1 -0.8 1]);
xlabel('f_1'); ylabel('f_2');
drawnow;
end
end %% End of iterations
%% --------------- All subfunctions are list below ------------------ %
%% Get cuckoos by ramdom walk
function nest=get_cuckoos(nest,best,Lb,Ub)
n=size(nest,1);
% For details, please see the chapters of the following Elsevier book:
% X. S. Yang, Nature-Inspired Optimization Algorithms, Elsevier, (2014).
beta=3/2; % Levy exponent in Levy flights
sigma=(gamma(1+beta)*sin(pi*beta/2)/(gamma((1+beta)/2)*beta*2^((beta-1)/2)))^(1/beta);
for j=1:n,
s=nest(j,:);
%% Levy flights by Mantegna's algorithm
u=randn(size(s))*sigma;
v=randn(size(s));
step=u./abs(v).^(1/beta);
stepsize=0.1*step.*(s-best);
% Now the actual random walks or flights
s=s+stepsize.*randn(size(s));
% Apply simple bounds/limits
nest(j,:)=simplebounds(s,Lb,Ub);
end
%% Replace some nests by constructing new solutions/nests
function new_nest=empty_nests(nest,Lb,Ub,pa)
% A fraction of worse nests are discovered with a probability pa
[n,d]=size(nest);
% The solutions represented by cuckoos to be discovered or not
% with a probability pa. This action is implemented as a status vector
K=rand(size(nest))>pa;
%% New solution by biased/selective random walks
stepsize=rand(1,d).*(nest(randperm(n),:)-nest(randperm(n),:));
new_nest=nest+stepsize.*K;
for j=1:size(new_nest,1)
s=new_nest(j,:);
new_nest(j,:)=simplebounds(s,Lb,Ub);
end
% Application of simple bounds
function s=simplebounds(s,Lb,Ub)
% Apply the lower bound
ns_tmp=s;
I=ns_tmp<Lb;
ns_tmp(I)=Lb(I);
% Apply the upper bounds
J=ns_tmp>Ub;
ns_tmp(J)=Ub(J);
% Update this new move
s=ns_tmp;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Objective functions
function f = obj_funs(x, m)
% Zitzler-Deb-Thiele's funciton No 3 (ZDT function 3)
% M = # of objectives
% d = # of variables/dimensions
d=length(x); % d=30 for ZDT 3
% First objective f1
f(1) = x(1);
g=1+9/29*sum(x(2:d));
h=1-sqrt(f(1)/g)-f(1)/g*sin(10*pi*f(1));
% Second objective f2
f(2) = g*h;
%%%%%%%%%%%%%%%%%% end of the definitions of obojectives %%%%%%%%%%%%%%%%%%
function new_Sol = Select_pop(nest, m, ndim, npop)
% The input population to this part has twice (ntwice) of the needed
% population size (npop). Thus, selection is done based on ranking and
% crowding distances, calculated from the non-dominated sorting
ntwice= size(nest,1);
% Ranking is stored in column Krank
Krank=m+ndim+1;
% Sort the population of size 2*npop according to their ranks
[~,Index] = sort(nest(:,Krank)); sorted_nest=nest(Index,:);
% Get the maximum rank among the population
RankMax=max(nest(:,Krank));
%% Main loop for selecting solutions based on ranks and crowding distances
K = 0; % Initialization for the rank counter
% Loop over all ranks in the population
for i =1:RankMax,
% Obtain the current rank i from sorted solutions
RankSol = max(find(sorted_nest(:, Krank) == i));
% In the new cuckoos/solutions, there can be npop solutions to fill
if RankSol<npop,
new_Sol(K+1:RankSol,:)=sorted_nest(K+1:RankSol,:);
end
% If the population after addition is large than npop, re-arrangement
% or selection is carried out
if RankSol>=npop
% Sort/Select the solutions with the current rank
candidate_nest = sorted_nest(K + 1 : RankSol, :);
[~,tmp_Rank]=sort(candidate_nest(:,Krank+1),'descend');
% Fill the rest (npop-K) cuckoo/solutions up to npop solutions
for j = 1:(npop-K),
new_Sol(K+j,:)=candidate_nest(tmp_Rank(j),:);
end
end
% Record and update the current rank after adding new cuckoo solutions
K = RankSol;
end
⛄ 运行结果
⛄ 参考文献
[1]尚志勇. 基于改进布谷鸟搜索算法的配送中心选址问题研究. (Doctoral dissertation, 河南大学).
