多目标规划问题定义
在实际问题中大都具有多个目标且需要同时满足，即在同一问题模型中同时存在几个非线性目标，而这些目标函数需要同时进行优化处理，并且这些目标又往往是互相冲突的，称这类问题称为多目标规划问题。
支配
支配：在多目标优化问题中，如果个体p至少有一个目标比个体q好，而且个体p的所有目标都不比个体q差，那么称个体p支配个体q。
根据遗传算法的思想，被支配者被全方面碾压，将会被淘汰，例如，C支配B。
序值
序值：如果p支配q，那么p的序值比q低。如果p和q互不支配，那么p和q有相同的序值。
拥挤距离
拥挤距离：用来计算某前端中的某个体与该前端中其他个体之间的距离，用以表征个体间的拥挤程度。
matlab求解
这里算法原理复杂了，应用时修改目标函数和约束即可。
例题：
MOPSO.m
代码语言：javascript复制function REP = MOPSO(params, MultiObj)
Np = params.Np;
Nr = params.Nr;
maxgen = params.maxgen;
W = params.W;
C1 = params.C1;
C2 = params.C2;
ngrid = params.ngrid;
maxvel = params.maxvel;
u_mut = params.u_mut;
fun = MultiObj.fun;
nVar = MultiObj.nVar;
var_min = MultiObj.var_min(:);
var_max = MultiObj.var_max(:);
POS = repmat((var_max-var_min)', Np, 1) .* rand(Np, nVar) repmat(var_min', Np, 1);
VEL = zeros(Np, nVar);
POS_fit = fun(POS);
PBEST = POS;
PBEST_fit = POS_fit;
DOMINATED = checkDomination(POS_fit);
REP.pos = POS(~DOMINATED, :);
REP.pos_fit = POS_fit(~DOMINATED, :);
REP = updateGrid(REP, ngrid);
maxvel = (var_max - var_min) .* maxvel ./ 100;
gen = 1;
display(['Generation #0 - Repository size: ' num2str(size(REP.pos,1))]);
stopCondition = false;
while ~stopCondition
h = selectLeader(REP);
VEL = W .* VEL C1 * rand(Np, nVar) .* (PBEST - POS) ...
C2 * rand(Np, nVar) .* (repmat(REP.pos(h, :), Np, 1) - POS);
POS = POS VEL;
POS = mutation(POS, gen, maxgen, Np, var_max, var_min, nVar, u_mut);
[POS, VEL] = checkBoundaries(POS, VEL, maxvel, var_max, var_min);
POS_fit = fun(POS);
REP = updateRepository(REP, POS, POS_fit, ngrid);
if(size(REP.pos, 1) > Nr)
REP = deleteFromRepository(REP, size(REP.pos, 1) - Nr, ngrid);
end
pos_best = dominates(POS_fit, PBEST_fit);
best_pos = ~dominates(PBEST_fit, POS_fit);
best_pos(rand(Np, 1) >= 0.5) = 0;
if(sum(pos_best) > 1)
PBEST_fit(pos_best, :) = POS_fit(pos_best, :);
PBEST(pos_best, :) = POS(pos_best, :);
end
if(sum(best_pos) > 1)
PBEST_fit(best_pos, :) = POS_fit(best_pos, :);
PBEST(best_pos, :) = POS(best_pos, :);
end
display(['Generation #' num2str(gen) ' - Repository size: ' num2str(size(REP.pos,1))]);
gen = gen 1;
if(gen > maxgen)
stopCondition = true;
end
end
h_rep = plot(REP.pos_fit(:,1),REP.pos_fit(:,2),'ok');
end
function REP = updateRepository(REP, POS, POS_fit, ngrid)
DOMINATED = checkDomination(POS_fit);
REP.pos = [REP.pos; POS(~DOMINATED, :)];
REP.pos_fit = [REP.pos_fit; POS_fit(~DOMINATED, :)];
DOMINATED = checkDomination(REP.pos_fit);
REP.pos_fit = REP.pos_fit(~DOMINATED, :);
REP.pos = REP.pos(~DOMINATED, :);
REP = updateGrid(REP, ngrid);
end
function [POS, VEL] = checkBoundaries(POS, VEL, maxvel, var_max, var_min)
Np = size(POS, 1);
MAXLIM = repmat(var_max(:)', Np, 1);
MINLIM = repmat(var_min(:)', Np, 1);
MAXVEL = repmat(maxvel(:)', Np, 1);
MINVEL = repmat(-maxvel(:)', Np, 1);
VEL(VEL > MAXVEL) = MAXVEL(VEL > MAXVEL);
VEL(VEL < MINVEL) = MINVEL(VEL < MINVEL);
VEL(POS > MAXLIM) = (-1) .* VEL(POS > MAXLIM);
POS(POS > MAXLIM) = MAXLIM(POS > MAXLIM);
VEL(POS < MINLIM) = (-1) .* VEL(POS < MINLIM);
POS(POS < MINLIM) = MINLIM(POS < MINLIM);
end
function dom_vector = checkDomination(fitness)
Np = size(fitness, 1);
dom_vector = zeros(Np, 1);
all_perm = nchoosek(1 : Np, 2);
all_perm = [all_perm; [all_perm(:, 2), all_perm(:, 1)]];
d = dominates(fitness(all_perm(:, 1), :), fitness(all_perm(:, 2), :));
dominated_particles = unique(all_perm(d == 1, 2));
dom_vector(dominated_particles) = 1;
end
function d = dominates(x, y)
d = all(x <= y, 2) & any(x < y, 2);
end
function REP = updateGrid(REP, ngrid)
ndim = size(REP.pos_fit, 2);
REP.hypercube_limits = zeros(ngrid 1, ndim);
for dim = 1 : ndim
REP.hypercube_limits(:, dim) = linspace(min(REP.pos_fit(:, dim)), max(REP.pos_fit(:, dim)), ngrid 1)';
end
npar = size(REP.pos_fit, 1);
REP.grid_idx = zeros(npar, 1);
REP.grid_subidx = zeros(npar, ndim);
for n = 1 : 1 : npar
idnames = [];
for d = 1 : 1 : ndim
REP.grid_subidx(n, d) = find(REP.pos_fit(n, d) <= REP.hypercube_limits(:, d)', 1, 'first') - 1;
if(REP.grid_subidx(n,d) == 0)
REP.grid_subidx(n,d) = 1;
end
idnames = [idnames ',' num2str(REP.grid_subidx(n, d))];
end
REP.grid_idx(n) = eval(['sub2ind(ngrid.*ones(1,ndim)' idnames ');']);
end
REP.quality = zeros(ngrid, 2);
ids = unique(REP.grid_idx);
for i = 1 : length(ids)
REP.quality(i, 1) = ids(i);
REP.quality(i, 2) = 10 / sum(REP.grid_idx == ids(i));
end
end
function selected = selectLeader(REP)
prob = cumsum(REP.quality(:,2));
sel_hyp = REP.quality(find(rand(1, 1) * max(prob) <= prob, 1, 'first'), 1);
idx = 1 : 1 : length(REP.grid_idx);
selected = idx(REP.grid_idx == sel_hyp);
selected = selected(randi(length(selected)));
end
function REP = deleteFromRepository(REP, n_extra, ngrid)
crowding = zeros(size(REP.pos, 1), 1);
for m = 1 : 1 : size(REP.pos_fit, 2)
[m_fit, idx] = sort(REP.pos_fit(:, m), 'ascend');
m_up = [m_fit(2 : end); Inf];
m_down = [Inf; m_fit(1 : end - 1)];
distance = (m_up - m_down) ./ (max(m_fit) - min(m_fit));
[~, idx] = sort(idx, 'ascend');
crowding = crowding distance(idx);
end
crowding(isnan(crowding)) = Inf;
[~, del_idx] = sort(crowding, 'ascend');
del_idx = del_idx(1 : n_extra);
REP.pos(del_idx, :) = [];
REP.pos_fit(del_idx, :) = [];
REP = updateGrid(REP, ngrid);
end
function POS = mutation(POS, gen, maxgen, Np, var_max, var_min, nVar, u_mut)
fract = Np / 3 - floor(Np / 3);
if fract < 0.5
sub_sizes =[ceil(Np / 3), round(Np / 3), round(Np / 3)];
else
sub_sizes =[round(Np / 3), round(Np / 3), floor(Np / 3)];
end
cum_sizes = cumsum(sub_sizes);
nmut = round(u_mut * sub_sizes(2));
if(nmut > 0)
idx = cum_sizes(1) randperm(sub_sizes(2), nmut);
POS(idx, :) = repmat((var_max - var_min)', nmut, 1) .* rand(nmut, nVar) repmat(var_min', nmut, 1);
end
per_mut = (1 - gen / maxgen) ^ (5 * nVar);
nmut = round(per_mut * sub_sizes(3));
if nmut > 0
idx = cum_sizes(2) randperm(sub_sizes(3), nmut);
POS(idx, :) = repmat((var_max - var_min)', nmut, 1) .* rand(nmut, nVar) repmat(var_min', nmut, 1);
end
end主函数example.m
代码语言：javascript复制clear
clc
MultiObj.fun = @(x) [-10.*(exp(-0.2.*sqrt(x(:,1).^2 x(:,2).^2)) exp(-0.2.*sqrt(x(:,2).^2 x(:,3).^2))), ...
sum(abs(x).^0.8 5.*sin(x.^3),2)];
MultiObj.nVar = 3;
MultiObj.var_min = -5 .* ones(1, MultiObj.nVar);
MultiObj.var_max = 5 .* ones(1, MultiObj.nVar);
params.Np = 200;
params.Nr = 150;
params.maxgen = 100;
params.W = 0.4;
params.C1 = 2;
params.C2 = 2;
params.ngrid = 20;
params.maxvel = 5;
params.u_mut = 0.5;
REP = MOPSO(params, MultiObj);运行结果：
注：上图中的各个点均有相同的序值，具体取哪个点仍需根据题目要求进一步选择判断。
