MATLAB 数学建模: 人工鱼群算法

1. 基本原理

人工鱼群算法是一种受鱼群聚集规律而启发的优化算法. 在人工鱼群算法中, 我们假定鱼群的活动行为分为: 觅食行为, 群聚行为, 追随行为和随机行为.

觅食行为, 基于 “鱼倾向于游向食物最多的水域” 这一假设, 等价于在寻找最优解的过程中, 向相对较优的方向行进的迭代原则.

群聚行为, 借鉴了真实鱼群中, 落单的个体总倾向于回到群体的特性. 这一行为受三条子规则所限定:

分隔规则: 避免和临近伙伴之间过于拥挤

对准规则: 确保和临近伙伴的方向一致.

内聚规则: 尽量向临近伙伴的中心移动.

追尾行为, 确保了每一个人工鱼个体均会追逐临近的最活跃个体. 它等价于在优化过程中, 向位于当前点附近的, 极优化点前进的过程.

随机行为, 保证了人工鱼在 “目力所及范围内” 随机移动. 这样有助于在寻找最优解过程中跳出局部最优.

2. 程序设计

我们规定以下变量和符号:

n 目标空间维度

N 人工鱼数量

X 每条人工鱼状态 X = ( x 1 , x 2 , ⋯   , x n ) X = (x_1, x_2, \cdots, x_n)X=(x1​,x2​,⋯,xn​), 各分量为需要寻找最优的变量

Y Y = f ( X ) Y = f(X)Y=f(X), 人工鱼所在位置的食物浓度, 即寻优过程中的适应度.

d 人工鱼个体间距离, d = ∣ ∣ x i − x j ∣ ∣ d = ||x_i - x_j||d=∣∣xi​−xj​∣∣

v 人工鱼感知范围

s 人工鱼移动步长

δ 拥挤度因子, 反映拥挤程度

t 人工鱼每次觅食最大尝试次数

2.1 觅食行为

觅食行为, 指人工鱼总是倾向于沿食物较多的方向游动的行为. 逻辑如下:

对于每一条人工鱼 X i X_iXi​, 它按照规则

X j = X i + r a n d ( ) ⋅ v X_j = X_i + rand()\cdot vXj​=Xi​+rand()⋅v

随机选定一个新状态 X j X_jXj​, 并比照新旧两个状态的适应度.

若 X j X_jXj​ 适应度高于 X i X_iXi​, 则该鱼按照规则

X i ′ = X i + r a n d ( ) ⋅ s ⋅ X j − X i ∣ ∣ X j − X i ∣ ∣ X_i' = X_i + rand()\cdot s\cdot \frac{X_j - X_i}{||X_j - X_i||}Xi′​=Xi​+rand()⋅s⋅∣∣Xj​−Xi​∣∣Xj​−Xi​​

向 X j X_jXj​ 方向前进一个步长.

若反复判断 t tt 次后仍然无法在附近找到优于现有状态 X i X_iXi​ 的新状态, 人工鱼将依据规则

X i ′ = X i + r a n d ( ) ⋅ s X_i' = X_i + rand()\cdot sXi′​=Xi​+rand()⋅s

随机移动一个步长.

2.2 群聚行为

真实世界中的鱼群在游弋过程中为了躲避敌害会自然地群聚. 逻辑如下:

对于每一条人工鱼 X i X_iXi​, 搜索其视野内

d i j ⩽ v d_{ij}\leqslant vdij​⩽v

的伙伴并统计其数量, 记为 n f n_fnf​, 并且确定其周边伙伴的中间位置 X c X_cXc​.

判断伙伴所处的中心位置状态 Y c n f \frac{Y_c}{n_f}nf​Yc​​. 若状态较优且不太拥挤,即

Y c n f < δ Y i \frac{Y_c}{n_f}nf​Yc​​

则向该中心位置移动一个步长, 否则将执行觅食行为.

function [x_swarm, x_swarm_fitness] = swarm(x,i,N,v,f,delta,t,d,ub,lb,s)

%% Compute fitness rate under the circunstance of SWARMING

nf_swarm = 0;

Xc = 0;

label_swarm = 0; % The sign of swarming happened or not

% Fix the Position of the center, compute the number of companions

for j = 1:N

if norm(x(j,:) - x(i,:)) < v

nf_swarm = nf_swarm + 1; % COunt the number of companions

Xc = Xc + x(j,:); % Add up the number of Sakanas

end

end

Xc = Xc - x(i,:); % Delete myself

nf_swarm = nf_swarm - 1;

Xc = Xc / nf_swarm;

% Judge the center is narrow or not

if (f(Xc)/nf_swarm < delta*f(x(i,:))) && (f(Xc) < f(x(i,:)))

x_swarm = x(i,:) + rand*s.*(Xc - x(i,:)) ./ norm(Xc - x(i,:));

%Bound processing

ub_flag = x_swarm > ub;

lb_flag = x_swarm < lb;

x_swarm = (x_swarm .* (~(ub_flag + lb_flag))) + ub .* ub_flag + lb .* lb_flag;

x_swarm_fitness = f(x_swarm);

else

% PREYING

label_prey = 0; % A sign of whether preying will find a better state than current state

for j = 1:t

%Find a state randomly

x_prey_rand = x(i,:) + v .* (-1 + 2 .* rand(1,d));

ub_flag2 = x_prey_rand > ub;

lb_flag2 = x_prey_rand < lb;

x_prey_rand = (x_prey_rand .* (~(ub_flag2 + lb_flag2))) + ub .* ub_flag2 + lb .* lb_flag2;

%judgement: better, or not?

if f(x(i,:)) > f(x_prey_rand)

x_swarm = x(i,:) + rand*s .* (x_prey_rand - x(i,:)) ./ norm(x_prey_rand - x(i,:));

ub_flag2 = x_swarm > ub;

lb_flag2 = x_swarm < lb;

x_swarm = (x_swarm .* (~(ub_flag2 + lb_flag2))) + ub .* ub_flag2 + lb .* lb_flag2;

x_swarm_fitness = f(x_swarm);

label_prey = 1;

break;

end

end

% Acting randomly

if label_prey == 0

x_swarm = x(i,:) + s * (-1 + 2*rand(1,d));

ub_flag2 = x_swarm > ub;

lb_flag2 = x_swarm < lb;

x_swarm = (x_swarm .* (~(ub_flag2 + lb_flag2))) + ub .* ub_flag2 + lb .* lb_flag2;

x_swarm_fitness = f(x_swarm);

end

end

end

2.3 追尾行为

执行追尾过程时, 人工鱼将向其视野内适应度最高的个体移动. 逻辑如下:

X i X_iXi​ 搜索其视野内适应度最高的个体, 记为 X j X_jXj​, 同时对目标个体 X j X_jXj​ 视野内所有个体的数量计数 n f n_fnf​.

判断目标个体位置状态 Y j n f \frac{Y_j}{n_f}nf​Yj​​. 若该位置状态较优且不太拥挤, 则 X i X_iXi​ 向目标 X j X_jXj​ 移动一个步长, 否则执行觅食行为.

function [x_follow, x_follow_fitness] = follow(x,i,N,v,f,delta,t,d,ub,lb,s)

fitness_follow = inf;

label_follow = 0; % A Sign marking following has happened or not

% Search the *best* individual in sight

for j = 1:N

if (norm(x(j,:) - x(i,:)) < v) && (f(x(j,:)) < fitness_follow)

best_pos = x(j,:);

fitness_follow = f(x(j,:));

end

end

%Search The Number Of Companions In Sight

nf_follow = 0;

for j = 1:N

if norm(x(j,:) - best_pos) < v

nf_follow = nf_follow + 1;

end

end

nf_follow = nf_follow - 1; %　Delete myself

% Judge: The Center Is Narrow or not?

if (fitness_follow/nf_follow)ub;

lb_flag2 = x_follow < lb;

x_follow = (x_follow .* (~(ub_flag2 + lb_flag2))) + ub .* ub_flag2 + lb .* lb_flag2;

label_follow = 1;

x_follow_fitness = f(x_follow);

else

% PREYING

label_prey = 0; % A sign of whether preying will find a better state than current state

for j = 1:t

%Find a state randomly

x_prey_rand = x(i,:) + v .* (-1 + 2 .* rand(1,d));

ub_flag2 = x_prey_rand > ub;

lb_flag2 = x_prey_rand < lb;

x_prey_rand = (x_prey_rand .* (~(ub_flag2 + lb_flag2))) + ub .* ub_flag2 + lb .* lb_flag2;

%judgement: better, or not?

if f(x(i,:)) > f(x_prey_rand)

x_follow = x(i,:) + rand*s .* (x_prey_rand - x(i,:)) ./ norm(x_prey_rand - x(i,:));

ub_flag2 = x_follow > ub;

lb_flag2 = x_follow < lb;

x_follow = (x_follow .* (~(ub_flag2 + lb_flag2))) + ub .* ub_flag2 + lb .* lb_flag2;

x_follow_fitness = f(x_follow);

label_prey = 1;

break;

end

end

%random behaviour

if label_prey == 0

x_follow = x(i,:) + s .* (-1 + 2*rand(1,d));

ub_flag2 = x_follow > ub;

lb_flag2 = x_follow < lb;

x_follow = (x_follow .* (~(ub_flag2 + lb_flag2))) + ub .* ub_flag2 + lb .* lb_flag2;

x_follow_fitness = f(x_follow);

end

end

2.4 随机行为

在上文所提到的 “觅食行为” 的最后一种判断情况中, 人工鱼将会执行的行为就是随机行为.

3. 代码实现

附主函数代码如下:

clc; clear all;

%% Define variables

v = 25; %Perception Distance

s = 3; %Maximum Step Size

N = 30; %The Number Of Artificial Sakana

d = 10; %Current Dimension

t = 50; %Maximum Iteration Time

delta = 27; %Factor of Congestion

%% Construct a function for testing

f = @(x) sum(3*x.^4);

ub = 100; %Upper Limit Of Boundrary

lb = -100; %Lower Limit Of Boundrary

df = []; %Store Target Function value of 50 states

Iteration = 1; %Initialize Iteration Rate

Max_iteration = 500; %Maximum Iteration Rate

%% Initialize Artificial Sakana Group

x = lb + rand(N,d).*(ub - lb);

%% Compute fitness rate of 10 initial state

for i = 1:N

fitness_sakana(i) = f(x(i,:));

end

%% Define Best Fitness of Initial State

[best_fitness,I] = min(fitness_sakana);

best_x = x(I); %Best Artificial Sakana At Initial State

while Iteration <= Max_iteration

for i = 1:N

%%Swarm

[x_swarm, x_swarm_fitness] = swarm(x,i,N,v,f,delta,t,d,ub,lb,s);

%%Follow

[x_follow, x_follow_fitness] = follow(x,i,N,v,f,delta,t,d,ub,lb,s);

%%Judgement

if x_follow_fitness < x_swarm_fitness

x(i,:) = x_follow;

else

x(i,:) = x_swarm;

end

end

%% Update Information In Time

for i = 1:N

if (f(x(i,:)) < best_fitness)

best_fitness = f(x(i,:));

best_x = x(i,:);

end

end

Convergence_curve(Iteration) = best_fitness;

Iteration = Iteration + 1;

%% Display Iteration Time And Current Best Fit Rate

if mod(Iteration, 25) == 0

display(['Iteration times: ',num2str(Iteration),'BestFitRate: ',num2str(best_fitness)]);

display(['BestArtificialSakana: ', num2str(best_x)]);

end

end

%% Print

figure('Position', [284 214 660 290])

subplot(1,2,1);

x = -100:1:100;

y = x;

L = length(x);

for i = 1:L

for j = 1:L

F(i,j) = (3*x(i) .^4) + (3*y(j) .^4);

end

end

surfc(x,y,F,'LineStyle', 'none');

title('Test Function')

xlabel('x_1');

ylabel('y_1');

zlabel(['sum','(x_1,x_2)'])

grid on

subplot(1,2,2);

semilogy(Convergence_curve, 'Color','r')

title('Convergence Curve')

xlabel('Iteration');

ylabel('Best Fitness');

axis tight

grid on

box on