【Matlab】智能优化算法_流向算法FDA
- 1.背景介绍
- 2.数学模型
- 3.文件结构
- 4.算法流程图
- 5.详细代码及注释
- 5.1 FDA.m
- 5.2 initialization.m
- 5.3 main.m
- 5.4 Select_Functions.m
- 6.运行结果
- 7.参考文献
1.背景介绍
在流域中，过量或有效降雨量是指流过地表且未渗入土壤的降雨量。事实上，降水和损失（如截留、蒸发和渗透）后留在地表的水量被称为直接径流。到目前为止，已经提出了几种确定直接径流的方法，其中之一是φ指数法。
单位为厘米/小时的指数φ代表降雨过程中的平均失水量，超过此限度的降雨量将直接转化为径流。换言之，指数φ的高位和等于直接径流的高度。从降雨量中减去每个时间间隔的指数φ，即可得到直接径流量。图1说明了φ指数的概念。关系式（1）表示直接径流计算方法。
其中，参数srd、Rm、Δt和M表示直接径流量、降雨量、时间间隔和多个时间步长。
本文假定降雨损失包括入渗、蒸发和截流，降雨量在减去这些损失后转化为直接径流。径流根据坡度向流域出口移动。这一过程可通过将流域划分为若干单元来模拟。每个单元的径流量根据其相邻单元的高度和坡度转移到其它单元。图2显示了水流及其周围的八个位置。图3是D8方法的示意图。
最后，使用D8算法确定整个流域的流向。因此，在指定流动方向之后，为每个细胞考虑一个值，该值等于流入该细胞的细胞数。因此，最大的数字被指定给水池出口点。此外，如果一个单元格的高度低于相邻单元格的高度，则将其视为一个孔，并应相应地进行填充。图4显示了水渠中孔的位置。
2.数学模型
FDA算法受D8方法的启发，在将降雨转换为径流后确定流域内的流向。该算法首先在流域或问题搜索空间中创建一个初始种群。然后，水流流向较低高度的位置，以获得最佳答案或最低高度的出水点。
该算法基于以下假设运行：
- 每个流动都有一个位置和高度。
- 每个流体周围有β个位置，每个位置有一个高度或目标函数。
- 流动速度与坡度直接相关。
- 水流速度为V，流向高度最低的方向。
- 流域出口点是目标函数最优的流动位置。
算法的初始参数包括种群数α、邻居数β和邻域半径Δ。关于FDA算法，流量的初始位置由以下关系式计算得出：
其中，流X（i）表示流i的位置，lb和ub表示决策变量的下限和上限，rand分别表示0和1之间的随机值，具有均匀分布。还假设每个流周围都有β邻域，其位置由以下关系产生：
其中，Neighbor X表示第j个位置的邻居，randn为正态分布的随机值，均值为0，标准差为1。事实上，在较大范围内搜索会产生更多的解，并增加找到接近最优解（全局搜索）的概率。当算法的解接近全局最优解时，在较小的范围内搜索可以更精确地找到全局最优解（局部搜索），这就需要在这两种能力之间建立平衡。原因在于，仅执行全局搜索运算符会导致算法无法以所需精度找到全局最优解。另一方面，仅进行局部搜索会导致算法陷入局部最优。因此，为了在全局搜索和局部搜索之间建立平衡，本研究采用以下公式，将Δ从大值线性减小到小值。因此，Δ的方向是随机位置，以获得更多的多样性。
其中rand是均匀分布的随机数，Xrand是由关系式（1）生成的随机位置，W是随机数在0和inf之间的非线性权重。在该关系式中，第一项表明流X（i）移动到随机位置（Xrand）。对于通过增加迭代的第二项，流X（i）接近于最佳X，并且最佳X和流X（i）之间的欧氏距离被减少到零。因此，本地搜索失败。第三项中的W计算如下：
其中rand是均匀分布的随机向量。图5显示了W通过增加迭代的变化图。很明显，W在迭代过程中有很大的变化。这一问题保证了FDA摆脱局部最优。
如前所述，流以V的速度移动到具有最小目标函数的邻居。另一方面，流向邻居的流速与其斜率直接相关。因此，使用以下关系式来确定流速矢量：
其中S0指示流的邻居和当前位置之间的斜率矢量。随机数randn生成各种解决方案并增加全局搜索。流量第i个相对于邻居第j个的斜率矢量也由以下关系式确定：
其中，Flow_fitness(i)和 Neighbor_fitness(j)分别表示流量 i 和邻居 j 的目标值。参数d表示问题的维度。以下关系式用于确定流量的新位置：
其中，Flow_newX(i) 表示流量 i 的新位置。
值得注意的是，任何相邻流体的目标函数都不 能小于该流体的目标函数，这类似于水槽的填充 过程，用于确定流向。为了模拟这种状态，FDA方法随机选择另一个流向，如果该流向的目标函数小于当前流向的目标函数，则该流向同方向移动，否则，该流向主导斜坡方向移动。下面的关系式说明了如何在这些条件下模拟流向：
其中，er 为随机整数。拟议的FDA算法步骤如下：
- 生成初始种群或流量，由以下矩阵表示：
- 评估目标函数，并将最佳目标函数作为出口点。目标函数矩阵如下：
- 为种群或流量中的每个个体创建半径为 Δ 的β个邻居
- 为每个邻域指定目标函数值，并确定最佳邻域。
- 如果最佳邻居的目标函数值优于当前流量的目标函数值，则执行步骤6，否则考虑步骤7。
- 根据关系式（6）更新流速矢量，并根据关系式（8）生成新的流量位置。流速矢量如下：
- 根据关系式（9）更新流量位置
- 计算新流量的目标函数，如果目标函数和流量位置优于之前的流量，则更新目标函数和流量位置
- 控制终止条件；如果满足，返回最优答案，否则重复步骤3至7。
3.文件结构
FDA.m % 流向算法
initialization.m % 初始化
main.m % 主函数
Select_Functions.m % 方法选择
4.算法流程图
5.详细代码及注释
5.1 FDA.m
function [Best_fitness,BestX,ConvergenceCurve]=FDA(maxiter,lb,ub,dim,fobj,alpha,beta)
% Initialize the positions of flows
flow_x=initialization(alpha,dim,ub,lb);
neighbor_x=zeros(beta,dim);
newflow_x=inf(size(flow_x));
newfitness_flow=inf(size(flow_x,1));
ConvergenceCurve=zeros(1,maxiter);
fitness_flow=inf.*ones(alpha,1);
fitness_neighbor=inf.*ones(beta,1);
%% calculate fitness function of each flow
for i=1:alpha
fitness_flow(i,:)=fobj(flow_x(i,:));%fitness of each flow
end
%% sort results and select the best results
[~,indx]=sort(fitness_flow);
flow_x=flow_x(indx,:);
fitness_flow=fitness_flow(indx);
Best_fitness=fitness_flow(1);
BestX=flow_x(1,:);
%% Initialize velocity of flows
Vmax=0.1*(ub-lb);
Vmin=-0.1*(ub-lb);
%% Main loop
for iter=1:maxiter
% Update W
W=(((1-1*iter/maxiter+eps)^(2*randn)).*(rand(1,dim).*iter/maxiter).*rand(1,dim));
% Update the Position of each flow
for i=1:alpha
% Produced the Position of neighborhoods around each flow
for j=1:beta
Xrand=lb+rand(1,dim).*(ub-lb);
delta=W.*(rand*Xrand-rand*flow_x(i,:)).*norm(BestX-flow_x(i,:));
neighbor_x(j,:)=flow_x(i,:)+randn(1,dim).*delta;
neighbor_x(j,:)=max(neighbor_x(j,:),lb);
neighbor_x(j,:)=min(neighbor_x(j,:),ub);
fitness_neighbor(j)=fobj(neighbor_x(j,:));
end
% Sort position of neighborhoods
[~,indx]=sort(fitness_neighbor);
% Update position, fitness and velocity of current flow if the fitness of best neighborhood is
% less than of current flow
if fitness_neighbor(indx(1))<fitness_flow(i)
% Calculate slope to neighborhood
Sf=(fitness_neighbor(indx(1))-fitness_flow(i))./sqrt(norm(neighbor_x(indx(1),:)-flow_x(i,:)));%calculating slope
% Update velocity of each flow
V=randn.*(Sf);
if V<Vmin
V=-Vmin;
elseif V>Vmax
V=-Vmax;
end
%Flow moves to best neighborhood
newflow_x(i,:)=flow_x(i,:)+V.*(neighbor_x(indx(1),:)-flow_x(i,:))./sqrt(norm(neighbor_x(indx(1),:)-flow_x(i,:)));
else
%Generate integer random number (r)
r=randi([1 alpha]);
% Flow moves to r th flow if the fitness of r th flow is less
% than current flow
if fitness_flow(r)<=fitness_flow(i)
newflow_x(i,:)=flow_x(i,:)+randn(1,dim).*(flow_x(r,:)-flow_x(i,:));
else
newflow_x(i,:)=flow_x(i,:)+randn*(BestX-flow_x(i,:));
end
end
% Return back the flows that go beyond the boundaries of the search space
newflow_x(i,:)=max(newflow_x(i,:),lb);
newflow_x(i,:)=min(newflow_x(i,:),ub);
% Calculate fitness function of new flow
newfitness_flow(i)=fobj(newflow_x(i,:));
% Update current flow
if newfitness_flow(i)<fitness_flow(i)
flow_x(i,:)=newflow_x(i,:);
fitness_flow(i)=newfitness_flow(i);
end
% Update best flow
if fitness_flow(i)<Best_fitness
BestX=flow_x(i,:);
Best_fitness=fitness_flow(i);
end
end
ConvergenceCurve(iter)=Best_fitness;
disp(['MaxIter= ' ,num2str(iter), 'BestFit= ', num2str(Best_fitness)])%disply results
end
end
5.2 initialization.m
function [flow_x,fitness_flow]=initialization(alpha,dim,ub,lb)
for i=1:alpha
flow_x(i,:)=lb+rand(1,dim).*(ub-lb);%position of each flow
end
5.3 main.m
clear all
clc
alpha=50; % Number of flows
beta=1; %Number of neighborhoods
Function_name='F1'; % Name of the test function that can be from F1 to F23 (Table 1,2,3 in the paper)
Max_iteration=200; % Maximum numbef of iterations
% Load details of the selected benchmark function
[dim,fobj,ub, lb] = Select_Functions(Function_name);
[Best_fitness,BestX,FDA_cg_curve]=FDA(Max_iteration,lb,ub,dim,fobj,alpha,beta);
figure('Position',[500 500 660 290])
%Draw objective space
semilogy(FDA_cg_curve,'Color','r')
title('Objective space')
xlabel('Iteration');
ylabel('Best score obtained so far');
axis tight
grid on
box on
legend('FDA')
display(['The best solution obtained by DFA is : ', num2str(BestX)]);
display(['The best optimal value of the objective funciton found by DFA is : ', num2str(Best_fitness)]);
5.4 Select_Functions.m
function [dimensions,fitness,upper_bound, lower_bound] = Select_Functions(func)
switch func
case 'cec01'
fitness = @cec01;
lower_bound= -8192;
upper_bound = 8192;
dimensions = 9;
case 'cec02'
fitness = @cec02;
lower_bound= -16384;
upper_bound = 16384;
dimensions = 16;
case 'cec03'
fitness = @cec03;
lower_bound= -4;
upper_bound = 4;
dimensions = 18;
case 'cec04'
fitness = @cec04;
lower_bound= -100;
upper_bound = 100;
dimensions = 10;
case 'cec05'
fitness = @cec05;
lower_bound= -100;
upper_bound = 100;
dimensions = 10;
case 'cec06'
fitness = @cec06;
lower_bound= -100;
upper_bound = 100;
dimensions = 10;
case 'cec07'
fitness = @cec07;
lower_bound= -100;
upper_bound = 100;
dimensions = 10;
case 'cec08'
fitness = @cec08;
lower_bound= -100;
upper_bound = 100;
dimensions = 10;
case 'cec09'
fitness = @cec09;
lower_bound= -100;
upper_bound = 100;
dimensions = 10;
case 'cec10'
fitness = @cec10;
lower_bound= -100;
upper_bound = 100;
dimensions = 10;
case 'F1'
fitness = @F1;
lower_bound=-100;
upper_bound=100;
dimensions=10;
case 'F2'
fitness = @F2;
lower_bound=-10;
upper_bound=10;
dimensions=10;
case 'F3'
fitness = @F3;
lower_bound=-100;
upper_bound=100;
dimensions=10;
case 'F4'
fitness = @F4;
lower_bound=-100;
upper_bound=100;
dimensions=10;
case 'F5'
fitness = @F5;
lower_bound=-30;
upper_bound=30;
dimensions=10;
case 'F6'
fitness = @F6;
lower_bound=-100;
upper_bound=100;
dimensions=10;
case 'F7'
fitness = @F7;
lower_bound=-1.28;
upper_bound=1.28;
dimensions=10;
case 'F8'
fitness = @F8;
lower_bound=-500;
upper_bound=500;
dimensions=10;
case 'F9'
fitness = @F9;
lower_bound=-5.12;
upper_bound=5.12;
dimensions=10;
case 'F10'
fitness = @F10;
lower_bound=-32;
upper_bound=32;
dimensions=10;
case 'F11'
fitness = @F11;
lower_bound=-600;
upper_bound=600;
dimensions=10;
case 'F12'
fitness = @F12;
lower_bound=-50;
upper_bound=50;
dimensions=10;
case 'F13'
fitness = @F13;
lower_bound=-50;
upper_bound=50;
dimensions=10;
case 'F14'
fitness = @F14;
lower_bound=-65.536;
upper_bound=65.536;
dimensions=2;
case 'F15'
fitness = @F15;
lower_bound=-5;
upper_bound=5;
dimensions=4;
case 'F16'
fitness = @F16;
lower_bound=-5;
upper_bound=5;
dimensions=2;
case 'F17'
fitness = @F17;
lower_bound=[-5,0];
upper_bound=[10,15];
dimensions=2;
case 'F18'
fitness = @F18;
lower_bound=-2;
upper_bound=2;
dimensions=2;
case 'F19'
fitness = @F19;
lower_bound=0;
upper_bound=1;
dimensions=3;
case 'F20'
fitness = @F20;
lower_bound=0;
upper_bound=1;
dimensions=6;
case 'F21'
fitness = @F21;
lower_bound=0;
upper_bound=10;
dimensions=4;
case 'F22'
fitness = @F22;
lower_bound=0;
upper_bound=10;
dimensions=4;
case 'F23'
fitness = @F23;
lower_bound=0;
upper_bound=10;
dimensions=4;
end
end
% F1
function o = F1(x)
o=sum(x.^2);
end
% F2
function o = F2(x)
o=sum(abs(x))+prod(abs(x));
end
% F3
function o = F3(x)
dim=size(x,2);
o=0;
for i=1:dim
o=o+sum(x(1:i))^2;
end
end
% F4
function o = F4(x)
o=max(abs(x));
end
% F5
function o = F5(x)
dim=size(x,2);
o=sum(100*(x(2:dim)-(x(1:dim-1).^2)).^2+(x(1:dim-1)-1).^2);
end
% F6
function o = F6(x)
o=sum(abs((x+.5)).^2);
end
% F7
function o = F7(x)
dim=size(x,2);
o=sum([1:dim].*(x.^4))+rand;
end
% F8
function o = F8(x)
o=sum(-x.*sin(sqrt(abs(x))));
end
% F9
function o = F9(x)
dim=size(x,2);
o=sum(x.^2-10*cos(2*pi.*x))+10*dim;
end
% F10
function o = F10(x)
dim=size(x,2);
o=-20*exp(-.2*sqrt(sum(x.^2)/dim))-exp(sum(cos(2*pi.*x))/dim)+20+exp(1);
end
% F11
function o = F11(x)
dim=size(x,2);
o=sum(x.^2)/4000-prod(cos(x./sqrt([1:dim])))+1;
end
% F12
function o = F12(x)
dim=size(x,2);
o=(pi/dim)*(10*((sin(pi*(1+(x(1)+1)/4)))^2)+sum((((x(1:dim-1)+1)./4).^2).*...
(1+10.*((sin(pi.*(1+(x(2:dim)+1)./4)))).^2))+((x(dim)+1)/4)^2)+sum(Ufun(x,10,100,4));
end
% F13
function o = F13(x)
dim=size(x,2);
o=.1*((sin(3*pi*x(1)))^2+sum((x(1:dim-1)-1).^2.*(1+(sin(3.*pi.*x(2:dim))).^2))+...
((x(dim)-1)^2)*(1+(sin(2*pi*x(dim)))^2))+sum(Ufun(x,5,100,4));
end
% F14
function o = F14(x)
aS=[-32 -16 0 16 32 -32 -16 0 16 32 -32 -16 0 16 32 -32 -16 0 16 32 -32 -16 0 16 32;,...
-32 -32 -32 -32 -32 -16 -16 -16 -16 -16 0 0 0 0 0 16 16 16 16 16 32 32 32 32 32];
for j=1:25
bS(j)=sum((x'-aS(:,j)).^6);
end
o=(1/500+sum(1./([1:25]+bS))).^(-1);
end
% F15
function o = F15(x)
aK=[.1957 .1947 .1735 .16 .0844 .0627 .0456 .0342 .0323 .0235 .0246];
bK=[.25 .5 1 2 4 6 8 10 12 14 16];bK=1./bK;
o=sum((aK-((x(1).*(bK.^2+x(2).*bK))./(bK.^2+x(3).*bK+x(4)))).^2);
end
% F16
function o = F16(x)
o=4*(x(1)^2)-2.1*(x(1)^4)+(x(1)^6)/3+x(1)*x(2)-4*(x(2)^2)+4*(x(2)^4);
end
% F17
function o = F17(x)
o=(x(2)-(x(1)^2)*5.1/(4*(pi^2))+5/pi*x(1)-6)^2+10*(1-1/(8*pi))*cos(x(1))+10;
end
% F18
function o = F18(x)
o=(1+(x(1)+x(2)+1)^2*(19-14*x(1)+3*(x(1)^2)-14*x(2)+6*x(1)*x(2)+3*x(2)^2))*...
(30+(2*x(1)-3*x(2))^2*(18-32*x(1)+12*(x(1)^2)+48*x(2)-36*x(1)*x(2)+27*(x(2)^2)));
end
% F19
function o = F19(x)
aH=[3 10 30;.1 10 35;3 10 30;.1 10 35];cH=[1 1.2 3 3.2];
pH=[.3689 .117 .2673;.4699 .4387 .747;.1091 .8732 .5547;.03815 .5743 .8828];
o=0;
for i=1:4
o=o-cH(i)*exp(-(sum(aH(i,:).*((x-pH(i,:)).^2))));
end
end
% F20
function o = F20(x)
aH=[10 3 17 3.5 1.7 8;.05 10 17 .1 8 14;3 3.5 1.7 10 17 8;17 8 .05 10 .1 14];
cH=[1 1.2 3 3.2];
pH=[.1312 .1696 .5569 .0124 .8283 .5886;.2329 .4135 .8307 .3736 .1004 .9991;...
.2348 .1415 .3522 .2883 .3047 .6650;.4047 .8828 .8732 .5743 .1091 .0381];
o=0;
for i=1:4
o=o-cH(i)*exp(-(sum(aH(i,:).*((x-pH(i,:)).^2))));
end
end
% F21
function o = F21(x)
aSH=[4 4 4 4;1 1 1 1;8 8 8 8;6 6 6 6;3 7 3 7;2 9 2 9;5 5 3 3;8 1 8 1;6 2 6 2;7 3.6 7 3.6];
cSH=[.1 .2 .2 .4 .4 .6 .3 .7 .5 .5];
o=0;
for i=1:5
o=o-((x-aSH(i,:))*(x-aSH(i,:))'+cSH(i))^(-1);
end
end
% F22
function o = F22(x)
aSH=[4 4 4 4;1 1 1 1;8 8 8 8;6 6 6 6;3 7 3 7;2 9 2 9;5 5 3 3;8 1 8 1;6 2 6 2;7 3.6 7 3.6];
cSH=[.1 .2 .2 .4 .4 .6 .3 .7 .5 .5];
o=0;
for i=1:7
o=o-((x-aSH(i,:))*(x-aSH(i,:))'+cSH(i))^(-1);
end
end
% F23
function o = F23(x)
aSH=[4 4 4 4;1 1 1 1;8 8 8 8;6 6 6 6;3 7 3 7;2 9 2 9;5 5 3 3;8 1 8 1;6 2 6 2;7 3.6 7 3.6];
cSH=[.1 .2 .2 .4 .4 .6 .3 .7 .5 .5];
o=0;
for i=1:10
o=o-((x-aSH(i,:))*(x-aSH(i,:))'+cSH(i))^(-1);
end
end
function o=Ufun(x,a,k,m)
o=k.*((x-a).^m).*(x>a)+k.*((-x-a).^m).*(x<(-a));
end
function o = cec01(x)
dim =9;
p1 = 0.0;
p2 = 0.0;
p3 = 0.0;
d = 72.661;
u = 0.0;
v = 0.0;
wk = 0.0;
pk= 0.0;
m = 32*dim;
for i=1 : dim
u = u+ x(1,i) * (1.2)^(dim-i);
end
if u < d
p1 = (u-d)^2;
end
for i=1 : dim
v = v+ x(1,i) * (-1.2)^(dim-i);
end
if v < d
p2 = (v-d)^2;
end
for k=0 :m
for i=1 : dim
wk = wk+ x(1,i) * ((2*k/m)-1)^(dim-i);
end
if wk > d
pk = pk+ (wk-d)^2;
elseif wk < d
pk = pk +(wk+d)^2;
else
pk = pk +0.0;
end
end
p3 = pk;
o = p1 + p2 + p3;
end
function o = cec02(x)
n = sqrt(16);
W = 0.0;
I =0;
H = 0.0;
Z = 0.0;
for i=1:n
xi = x(1,i);
for k=1:n
if i == k
I =1;
else
I =0;
end
H = 1/(i+k-1);
Z = xi+(n*(k-1));
W = W + abs(H*Z-I);
end
end
o = W;
end
function o = cec03(x)
n = 18/3;
d = 0.0;
sum =0.0;
for i=1 :n-1
xi = x(1, 3*i-1);
for j=i+1: n
tmp=0.0;
xj = x(1, 3*j-1);
for k=0 :2
tmp = tmp + (xi+k-2-xj+k-2)^ 2;
end
d = d + tmp^3;
sum = sum + (1/ d^2 )- (2/d);
end
end
o = 12.7120622568+sum;
end
function o = cec04(x)
dim = 10;
sum =0.0;
shiftedMatrix = [4.3453613502650342e+01 -7.5117860955706732e+01 5.4110917436941946e+01 2.1893626834216349e+00 -3.3813797325740467e+00 -3.0849165372014589e+01 7.8077592550813023e+01 -6.9901998485392895e+01 3.7111456001695004e+01 5.2241020487733664e+01];
rotatedMatrix = [
8.8970810825119684e-01 1.9871231543356224e-01 3.5531377300377703e-01 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 -2.0660353462835387e-0 0.0000000000000000e+00 0.0000000000000000e+00;
1.0419879983757413e-01 -6.6358499459221376e-01 4.5164451523757104e-01 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 5.8720932972857365e-01 0.0000000000000000e+00 0.0000000000000000e+00;
-4.3941933258454113e-01 3.4165627723133662e-01 8.1471256710105333e-01 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 -1.6255790164428213e-01 0.0000000000000000e+00 0.0000000000000000e+00;
0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 4.2863677588379034e-01 0.0000000000000000e+00 -6.3119168084438271e-01 0.0000000000000000e+00 0.0000000000000000e+00 6.4642677573936591e-01 0.0000000000000000e+00;
0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 6.8398421839009127e-01 0.0000000000000000e+00 -1.0280182935971671e-01 0.0000000000000000e+00 0.0000000000000000e+00 9.1093354741887778e-01;
0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 1.4071437781392207e-01 0.0000000000000000e+00 7.5339632462828454e-01 0.0000000000000000e+00 0.0000000000000000e+00 6.4233436924473619e-01 0.0000000000000000e+00;
0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 -5.2127926340507358e-01 0.0000000000000000e+00 1.8558504533667268e+00 0.0000000000000000e+00 0.0000000000000000e+00 -1.2500979819172911e-01;
6.6878564394300288e-02 6.3516876408991430e-01 -7.7542166446725458e-02 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 7.6555542658372966e-01 0.0000000000000000e+00 0.0000000000000000e+00;
0.0000000000000000e+0 0.0000000000000000e+00 0.0000000000000000e+00 -8.9245166717105195e-01 0.0000000000000000e+00 -1.8436659152198701e-01 0.0000000000000000e+00 0.0000000000000000e+00 4.1175111620270627e-01 0.0000000000000000e+00;
0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 -1.2215803943548182e+00 0.0000000000000000e+00 -1.0421383218771203e-01 0.0000000000000000e+00 0.0000000000000000e+00 4.5968278525834394e-01
];
X_shifted = x - shiftedMatrix;
shitedRotaatedX = X_shifted * rotatedMatrix;
for i =1: dim
Xi = shitedRotaatedX(1, i);
sum = sum + Xi^2 - 10* cos(2*pi*Xi)+10;
end
o = sum;
end
function o = cec05(x)
dim = 10;
sum = 0.0;
multi = 1.0;
i = 1;
shiftedMatrix = [-1.6799910337105352e+01 4.3906964270354706e+01 2.4348491851402670e+01 -5.4897453475230122e+01 5.8499441807390866e+01 1.1845681821854726e-01 7.0903743799265357e+01 -7.7796574718223610e-01 4.4729687108066713e+01 -6.8148774722660320e+01];
rotatedMatrix = [
-7.5988949123997229e-01 0.0000000000000000e+00 0.0000000000000000e+00 5.9790648917707112e-01 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 -2.5509957135010197e-01 0.0000000000000000e+00 0.0000000000000000e+00;
0.0000000000000000e+00 -6.4335422234689021e-02 1.3912090644901491e+00 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 -7.0354948113015447e-01 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00;
0.0000000000000000e+00 1.0874018967981698e+00 -9.3628657345778921e-01 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 -2.2957124131927584e-01 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00;
-4.8529941749281533e-02 0.0000000000000000e+00 0.0000000000000000e+00 3.3915506135367807e-01 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 9.3947788111907859e-01 0.0000000000000000e+00 0.0000000000000000e+00;
0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 -7.3880347589611128e-01 -1.8005470643130739e-01 0.0000000000000000e+00 0.0000000000000000e+00 -7.5105330032914350e-02 6.4506504795749642e-01;
0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 2.2733754217810664e-01 1.4610614422407869e-01 0.0000000000000000e+00 0.0000000000000000e+00 8.7440899428783569e-01 4.0296345646339404e-01;
0.0000000000000000e+00 -9.1794081957250695e-01 1.0235511548555627e-02 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 -9.3534397252777979e-01 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00;
6.4823823233196287e-01 0.0000000000000000e+00 0.0000000000000000e+00 7.2627933645267095e-01 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 -2.2870399993222773e-01 0.0000000000000000e+00 0.0000000000000000e+00;
0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 -9.1460850041609643e-02 -8.8922500528071025e-01 0.0000000000000000e+00 0.0000000000000000e+00 3.1795773407187566e-01 -3.1593778222470387e-01;
0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 -6.2779134975715278e-01 3.9435033573642242e-01 0.0000000000000000e+00 0.0000000000000000e+00 3.5870737301184835e-01 -5.6718150042257454e-01
];
%shifting
x = x - shiftedMatrix;
%rotating
shitedRotaatedX = x * rotatedMatrix;
for m = 1 : dim
sum = sum + shitedRotaatedX(1,m)^ 2 / 4000;
multi = multi * cos(shitedRotaatedX(1,m) / sqrt(i));
i = i +1;
end
o = (sum - multi + 1) + 1;
end
function o = cec06(x)
a =0.5;
b =3.0;
kMax = 20;
D = 10;
sum = 0.0;
shiftedMatrix = [4.4867071194977996e+01 8.6557399521842626e-01 -1.2297862364117918e+01 2.9827246270062048e+01 2.6528060932889602e+01 -6.2879900924339843e+01 -2.2494835379763892e+01 9.3017723082107295e+00 1.4887184097844738e+01 -3.1096867523666873e+01];
rotatedMatrix = [
-1.5433743057196678e-01 0.0000000000000000e+00 7.7666311726871273e-01 0.0000000000000000e+00 1.1571979400226866e+00 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00;
0.0000000000000000e+00 4.6806840267259536e-02 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 -5.9264454599472804e-01 1.6314935476659614e-01 7.8737783169590370e-01 0.0000000000000000e+00 0.0000000000000000e+00;
-1.7410812843826278e+00 0.0000000000000000e+00 -4.4194799352318298e-01 0.0000000000000000e+00 4.4605580480878959e-01 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00;
0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 2.7077411154472419e-01 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 -8.8999649318267127e-01 3.6686185770629254e-01;
5.4888525059737507e-02 0.0000000000000000e+00 1.5570674387300532e+00 0.0000000000000000e+00 -3.0216546520289828e-01 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00;
0.0000000000000000e+00 6.1164921138202333e-01 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 -6.1748299284504526e-01 -1.5999277506278717e-01 -4.6797682388189477e-01 0.0000000000000000e+00 0.0000000000000000e+00;
0.0000000000000000e+00 -1.1226733726002835e-01 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 -1.3517591002752971e-01 9.4075663040175728e-01 -2.9000082877106131e-01 0.0000000000000000e+00 0.0000000000000000e+00;
0.0000000000000000e+00 7.8172271740335475e-01 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 4.9921405128267116e-01 2.5052257846765580e-01 2.7736863877405393e-01 0.0000000000000000e+00 0.0000000000000000e+00;
0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 -3.0159372777109039e-01 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 2.8348126021733977e-01 9.1031840499614625e-01;
0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 9.1417864987446651e-01 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 3.5713389257830902e-01 1.9165797370723034e-01
];
%shifting
shiftedX = x - shiftedMatrix;
%rotating
shitedRotaatedX = shiftedX * rotatedMatrix;
for i =1: D
xi = shitedRotaatedX(1, i);
inerSum_1 = 0.0;
for k=0 :kMax
inerSum_1 = inerSum_1 + (a^k * cos(2 * pi * b^k *(xi+0.5)));
end
sum = sum + inerSum_1;
end
inerSum_2 =0.0;
for k=0 : kMax
inerSum_2 = inerSum_2 + a^k * cos(pi * b^k);
end
sum = sum - D * inerSum_2 +1;
o = sum;
end
function o = cec07(x)
D = 10;
zi = 0.0;
g =0.0;
shiftedMatrix = [1.5519604466631876e+00 3.7992270681072000e+00 1.3609333677966774e+01 -6.7928874412518397e+01 7.9407748803220557e+01 4.6034135728159043e+01 -6.4280816830825444e+01 -4.7688475683186425e+01 -6.0210807314240753e+01 3.6961469555721379e+01];
rotatedMatrix = [
-3.4378315941460673e-02 -7.3911155710735865e-01 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 -1.0523399031716010e+00 0.0000000000000000e+00 0.0000000000000000e+00;
1.1485242405257232e+00 9.9172138327339543e-01 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 -9.8221173301823295e-01 0.0000000000000000e+00 0.0000000000000000e+00;
0.0000000000000000e+00 0.0000000000000000e+00 8.6405702281889096e-01 0.0000000000000000e+00 0.0000000000000000e+00 -4.9170053952174497e-01 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 -1.0787048137178114e-01;
0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 -3.7383444140566896e-01 3.9203594526066760e-01 0.0000000000000000e+00 -7.8796970160513635e-01 0.0000000000000000e+00 2.9267623305420509e-01 0.0000000000000000e+00;
0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 1.3610747065473336e-01 -6.0804319089793657e-01 0.0000000000000000e+00 -5.6714757040795583e-01 0.0000000000000000e+00 -5.3861105430076284e-01 0.0000000000000000e+00;
0.0000000000000000e+00 0.0000000000000000e+00 8.5194811610211113e-02 0.0000000000000000e+00 0.0000000000000000e+00 -6.8358420399848130e-02 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 9.9401658458757081e-01;
0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 -8.5125893429894595e-01 8.0910637486172887e-03 0.0000000000000000e+00 2.3333420826786389e-01 0.0000000000000000e+00 -4.6994458047268511e-01 0.0000000000000000e+00;
1.2482031580671493e+00 -4.3601697061224165e-01 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 -1.7452756850803894e-01 0.0000000000000000e+00 0.0000000000000000e+00;
0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 3.4217070831253948e-01 6.9030850372398578e-01 0.0000000000000000e+00 -5.4795346377890963e-02 0.0000000000000000e+00 -6.3513057403542883e-01 0.0000000000000000e+00;
0.0000000000000000e+00 0.0000000000000000e+00 4.9613234664961658e-01 0.0000000000000000e+00 0.0000000000000000e+00 8.6807701605010890e-01 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 1.7175238382100888e-02;
];
%shifting
shiftedX = x - shiftedMatrix;
%rotating
shitedRotaatedX = shiftedX * rotatedMatrix;
for i =1:D
xi = shitedRotaatedX(1, i);
zi = xi+ 420.9687462275036;
if abs(zi) <= 500
g = g + ( zi * sin(abs(zi)^ 1/2));
elseif zi > 500
g = g + ((500 - mod(zi, 500)) * sin(sqrt(abs(500- mod(zi, 500))))- ((zi - 500)^ 2)/(10000 *D));
elseif zi < -500
g = g + (mod(abs(zi), 500) - 500)* sin(sqrt(abs(mod(zi, 500) - 500)))- ((zi - 500)^ 2/(10000 *D));
end
end
o = (418.9829 * D - g);
end
function o = cec08(x)
D = 10;
g = 0.0;
shiftedMatrix = [7.5809536201790706e+01, 5.0874943496135501e+01, 1.5175339549395872e+01, 1.1931806696547099e+01, 5.7875148867198789e+01, 6.7627011010249618e+01, -3.2825950734701912e+01, -2.5753998135101980e+01, -4.7446656658987820e+01, 4.0415323917015940e+00 ];
rotatedMatrix = [
3.2765524541169905e-01, 0.0000000000000000e+00, 9.4933157553264147e-01, 0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, -6.2503273493416478e-01;
0.0000000000000000e+00, -3.9416689281102152e-01, 0.0000000000000000e+00, 0.0000000000000000e+00, 6.4866583791924892e-01, 0.0000000000000000e+00, -6.1484056671741649e-01, 2.1409383187478670e-01, 0.0000000000000000e+00, 0.0000000000000000e+00;
8.9664357708578779e-01, 0.0000000000000000e+00, 5.3753076622704354e-01, 0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, 7.2208870049867158e-01;
0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, -1.7334340479710361e-01, 0.0000000000000000e+00, 1.7833024809453213e-01, 0.0000000000000000e+00, 0.0000000000000000e+00, -9.6858163653240936e-01, 0.0000000000000000e+00;
0.0000000000000000e+00, 5.5409637934619971e-01, 0.0000000000000000e+00, 0.0000000000000000e+00, -3.0250402339187688e-01, 0.0000000000000000e+00, -4.5590738714655138e-01, 6.2738901215463105e-01, 0.0000000000000000e+00, 0.0000000000000000e+00;
0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, -9.6746517640794338e-01, 0.0000000000000000e+00, 1.5319756775156032e-01, 0.0000000000000000e+00, 0.0000000000000000e+00, 2.0134954102999447e-01, 0.0000000000000000e+00;
0.0000000000000000e+00, 6.2594545303749249e-01, 0.0000000000000000e+00, 0.0000000000000000e+00, 6.9750321067293941e-01, 0.0000000000000000e+00, 3.4700267475621172e-01, 3.5646944254231774e-02, 0.0000000000000000e+00, 0.0000000000000000e+00;
0.0000000000000000e+00, 3.8184021897755271e-01, 0.0000000000000000e+00, 0.0000000000000000e+00, -3.4831274010388026e-02, 0.0000000000000000e+00, -5.4194896030516160e-01, -7.4784768097931698e-01, 0.0000000000000000e+00, 0.0000000000000000e+00;
0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, -1.8429106449108246e-01, 0.0000000000000000e+00, -9.7197161884987071e-01, 0.0000000000000000e+00, 0.0000000000000000e+00, -1.4597251693095872e-01, 0.0000000000000000e+00;
1.7153665513641250e+00, 0.0000000000000000e+00, -7.8944707263781244e-01, 0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, -3.0034616211515719e-01
];
%shifting
shiftedX = x - shiftedMatrix;
%rotating
shitedRotaatedX = shiftedX * rotatedMatrix;
for i=1 : D-1
j =i+1;
xi = shitedRotaatedX(1,i) ;
yi = shitedRotaatedX(1,j) ;
g = g + ( 0.5 +((sin(sqrt(xi^2+ yi^2)))^ 2)- 0.5 / ((1+ 0.001 *(xi^2+ yi^2)))^2);
end
xi = shitedRotaatedX(1, D-1) ;
yi = shitedRotaatedX(1, 1) ;
g = g + ( 0.5 +((sin(sqrt(xi^2+ yi^2)))^2)-0.5 /((1+ 0.001 *(xi^2+ yi^2)))^2);
o = g+1;
end
function o = cec09(x)
D = 10;
sum1=0.0;
sum2=0.0;
sum3=0.0;
shiftedMatrix = [-6.0107960952496171e+00, -6.3449972860258995e+01, -3.6938623728667750e+00, -2.7449007717635965e+00, -5.3547271030744199e+01, 3.1015786282259867e+01, 2.3200459416583499e+00, -4.6987858548289097e+01, 3.5061378905112562e+01, -3.4047417731046465e+00];
rotatedMatrix = [
-7.6923624057192400e-02, 0.0000000000000000e+00, 7.2809258658661558e-02, 6.1371429917067155e-01, 0.0000000000000000e+00, 7.8239141541106805e-01, 0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00;
0.0000000000000000e+00, -1.1499983823069659e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, 1.5729072158274271e-01, 0.0000000000000000e+00, 0.0000000000000000e+00, -1.3309066870600375e+00, 0.0000000000000000e+00, 0.0000000000000000e+00;
-1.6730831752378217e-02, 0.0000000000000000e+00, 4.9480374519689890e-01, 6.5982384537901573e-01, 0.0000000000000000e+00, -5.6526261691115431e-01, 0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00;
9.1421044415115027e-01, 0.0000000000000000e+00, -3.3249140365486585e-01, 2.2489758522716782e-01, 0.0000000000000000e+00, -5.5586027556918202e-02, 0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00;
0.0000000000000000e+00, -1.2704704488967578e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, -7.6341623484218024e-01, 0.0000000000000000e+00, 0.0000000000000000e+00, 4.9980922801223232e-01, 0.0000000000000000e+00, 0.0000000000000000e+00;
-3.9751993551989051e-01, 0.0000000000000000e+00, -7.9957334378299227e-01, 3.7068629354440513e-01, 0.0000000000000000e+00, -2.5544478964007222e-01, 0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00;
0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, 9.0184624725396623e-02, 0.0000000000000000e+00, -3.4243496198122719e-01, -9.3520320266563195e-01;
0.0000000000000000e+00, 9.7696981452382070e-01, 0.0000000000000000e+00, 0.0000000000000000e+00, -6.8376531090322690e-01, 0.0000000000000000e+00, 0.0000000000000000e+00, -4.7094671586086240e-01, 0.0000000000000000e+00, 0.0000000000000000e+00;
0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, -6.1661688294908079e-01, 0.0000000000000000e+00, -7.5660067900409822e-01, 2.1757534831110126e-01;
0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, 0.0000000000000000e+00, -7.8208078427058847e-01, 0.0000000000000000e+00, 5.5704013261474550e-01, -2.7938492717261593e-01;
];
%shifting
ShiftedX = x - shiftedMatrix;
%rotating
shitedRotaatedX = ShiftedX * rotatedMatrix;
for i= 1 :D
xi = shitedRotaatedX(1, i);
sum1 = sum1 + xi^2- D;
sum2 = sum2 + xi^2;
sum3 = sum3 + xi;
end
o = (((abs(sum1))^1/4 )+(0.5 * sum2+sum3)/D + 0.5);
end
function o = cec10(x)
D = 10;
sum1=0.0;
sum2=0.0;
shiftedMatrix = [6.1441309549566370e-001 1.8049534213689469e+001 5.1107558757100151e+001 5.1022671188681272e+000 -4.7667984552250942e+001 -7.3770454911164904e+000 -1.1534252828772665e+001 7.4568439937919834e+001 1.9208808661355789e+001 3.1262392306880571e+001];
rotatedMatrix = [
-3.6144665808053256e-02 0.0000000000000000e+00 0.0000000000000000e+00 -1.0275628429515489e-01 0.0000000000000000e+00 0.0000000000000000e+00 -9.9404965126067890e-01 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00;
0.0000000000000000e+00 5.7732032557209267e-01 6.3720045355332378e-01 0.0000000000000000e+00 -1.2849837048835189e-01 0.0000000000000000e+00 0.0000000000000000e+00 4.9413054191641514e-01 0.0000000000000000e+00 0.0000000000000000e+00;
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8.7364129527216294e-01 0.0000000000000000e+00 0.0000000000000000e+00 4.7972086095511046e-01 0.0000000000000000e+00 0.0000000000000000e+00 -8.1355901812131204e-02 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00;
0.0000000000000000e+00 7.1410594914990388e-01 -6.7354832197676640e-01 0.0000000000000000e+00 -1.9014210237853424e-01 0.0000000000000000e+00 0.0000000000000000e+00 -1.5209610582332084e-02 0.0000000000000000e+00 0.0000000000000000e+00;
0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 -2.7911553415074425e-01 0.0000000000000000e+00 0.0000000000000000e+00 1.6909147808954352e+00 -5.7964733149576031e-01;
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0.0000000000000000e+00 -6.5689502748026013e-03 2.3746987168001482e-01 0.0000000000000000e+00 -8.2483095297218667e-01 0.0000000000000000e+00 0.0000000000000000e+00 -5.1304854346890183e-01 0.0000000000000000e+00 0.0000000000000000e+00;
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0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 0.0000000000000000e+00 -6.6352395365628714e-01 0.0000000000000000e+00 0.0000000000000000e+00 4.6075742954200716e-01 6.9770224192342367e-01
];
%shifting
shiftedX = x - shiftedMatrix;
%rotating
shitedRotaatedX = shiftedX * rotatedMatrix;
% for i =1:D
% xi = shitedRotaatedX(1, i);
% sum1 = sum1 + xi^2;
% sum2 = sum2 + cos(2 * pi *xi);
% end
o=-20*exp(-.2*sqrt(sum(shitedRotaatedX.^2)/D))-exp(sum(cos(2*pi.*shitedRotaatedX))/D)+20+exp(1);
end
6.运行结果
7.参考文献
[1]Hojat K,Valikhan M A,Saeed F, et al. Flow Direction Algorithm (FDA): A Novel Optimization Approach for Solving Optimization Problems[J]. Computers & Industrial Engineering,2021,156.
