function [p_parent_1,m_parent_1,f1,p_parent_2,m_parent_2,f2]=crossover(pchrom1,mchrom1,fchrom1,pchrom2,mchrom2,fchrom2) global N SH;

这是一个 Matlab 函数，用于实现遗传算法中的交叉操作。其中输入参数包括两组父代染色体 pchrom1 和 pchrom2，以及它们对应的母代染色体 mchrom1 和 mchrom2，还有它们的适应度值 fchrom1 和 fchrom2。输出参数包括两组子代染色体 p_parent_1 和 m_parent_1，以及它们对应的适应度值 f1；以及另外两组子代染色体 p_parent_2 和 m_parent_2，以及它们对应的适应度值 f2。

函数实现的具体细节需要看函数的代码实现。但是可以大致理解为，该函数会从父代染色体中随机选择一段基因序列，然后将其交换到另一组染色体中，形成两组新的子代染色体。这个过程类似于生物学中的基因重组。

相关问题

$$ f=\alpha C_1L\sum_{k=1}^K{\frac{1}{t_k}}+\beta C_2\sum_{k=1}^K{\sum_{j=1}^J{\left( \frac{V_{kj}t_k}{2} \right)}} $$用Python求最小值t多目标遗传算法代码

以下是使用Python实现多目标遗传算法求解上述问题的代码：

import random
import numpy as np

# 定义目标函数
def objective_function(t, alpha, beta, C1, L, C2, V):
    f1 = alpha * C1 * L * np.sum(1 / t)
    f2 = beta * C2 * np.sum(V * t / 2)
    return (f1, f2)

# 定义种群初始化函数
def initialize_population(pop_size, chrom_length):
    population = []
    for i in range(pop_size):
        chromosome = [random.uniform(0, 10) for j in range(chrom_length)]
        population.append(chromosome)
    return population

# 定义交叉函数
def crossover(parent1, parent2, prob):
    if random.random() < prob:
        child1 = []
        child2 = []
        for i in range(len(parent1)):
            if random.random() < 0.5:
                child1.append(parent1[i])
                child2.append(parent2[i])
            else:
                child1.append(parent2[i])
                child2.append(parent1[i])
        return (child1, child2)
    else:
        return (parent1, parent2)

# 定义变异函数
def mutate(chromosome, prob):
    mutated_chromosome = []
    for i in range(len(chromosome)):
        if random.random() < prob:
            mutated_chromosome.append(random.uniform(0, 10))
        else:
            mutated_chromosome.append(chromosome[i])
    return mutated_chromosome

# 定义非支配排序函数
def non_dominated_sorting(population, objectives):
    fronts = [[]]
    n = [0] * len(population)
    ranks = [0] * len(population)
    S = [[] for i in range(len(population))]
    for p in range(len(population)):
        S[p] = []
        n[p] = 0
        for q in range(len(population)):
            if dominates(population[p], population[q], objectives):
                S[p].append(q)
            elif dominates(population[q], population[p], objectives):
                n[p] += 1
        if n[p] == 0:
            ranks[p] = 0
            fronts[0].append(p)
    i = 0
    while len(fronts[i]) > 0:
        next_front = []
        for p in fronts[i]:
            for q in S[p]:
                n[q] -= 1
                if n[q] == 0:
                    ranks[q] = i + 1
                    next_front.append(q)
        i += 1
        fronts.append(next_front)
    return fronts[:-1]

# 定义支配函数
def dominates(chromosome1, chromosome2, objectives):
    dominates = False
    for i in range(len(objectives)):
        if objectives[i](chromosome1) > objectives[i](chromosome2):
            return False
        elif objectives[i](chromosome1) < objectives[i](chromosome2):
            dominates = True
    return dominates

# 定义选择函数
def selection(population, objectives, pop_size):
    fronts = non_dominated_sorting(population, objectives)
    selection = []
    for front in fronts:
        if len(selection) + len(front) <= pop_size:
            selection += front
        else:
            remaining_space = pop_size - len(selection)
            distances = crowding_distance(population, objectives, front)
            sorted_indices = sorted(range(len(front)), key=lambda k: -distances[k])
            for i in range(remaining_space):
                selection.append(front[sorted_indices[i]])
            break
    return [population[i] for i in selection]

# 定义拥挤距离函数
def crowding_distance(population, objectives, front):
    distances = [0] * len(front)
    for i in range(len(objectives)):
        sorted_indices = sorted(range(len(front)), key=lambda k: objectives[i](population[front[k]]))
        distances[sorted_indices[0]] = float('inf')
        distances[sorted_indices[-1]] = float('inf')
        for j in range(1, len(front) - 1):
            distances[sorted_indices[j]] += (objectives[i](population[front[j+1]]) - objectives[i](population[front[j-1]])) / (objectives[i](population[front[-1]]) - objectives[i](population[front[0]]))
    return distances

# 定义多目标遗传算法函数
def multi_objective_genetic_algorithm(pop_size, chrom_length, num_generations, alpha, beta, C1, L, C2, V, crossover_prob=0.8, mutation_prob=0.2):
    population = initialize_population(pop_size, chrom_length)
    objectives = [lambda chromosome: objective_function(chromosome, alpha, beta, C1, L, C2, V)[i] for i in range(2)]
    for gen in range(num_generations):
        offspring_population = []
        while len(offspring_population) < pop_size:
            parent1, parent2 = random.sample(population, 2)
            child1, child2 = crossover(parent1, parent2, crossover_prob)
            child1 = mutate(child1, mutation_prob)
            child2 = mutate(child2, mutation_prob)
            offspring_population.append(child1)
            if len(offspring_population) < pop_size:
                offspring_population.append(child2)
        population += offspring_population
        population = selection(population, objectives, pop_size)
    return population

# 设置参数并执行算法
pop_size = 100
chrom_length = 10
num_generations = 50
alpha = 1
beta = 1
C1 = 1
L = 1
C2 = 1
V = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9, 10])
population = multi_objective_genetic_algorithm(pop_size, chrom_length, num_generations, alpha, beta, C1, L, C2, V)

# 输出结果
for chromosome in population:
    print(chromosome, objective_function(chromosome, alpha, beta, C1, L, C2, V))

nsga2算法matlab代码实例

以下是NSGA-II算法的MATLAB代码示例：

function [pop, front] = nsga2(pop, nObj, nPop, nGen, crossRate, mutRate, bounds)

% NSGA-II Algorithm
% pop: population, front: non-dominated fronts
% nObj: number of objectives
% nPop: population size
% nGen: number of generations
% crossRate: crossover rate
% mutRate: mutation rate
% bounds: decision variable bounds

% Initialization
for i = 1:nPop
    pop(i).x = rand(1, nObj) .* (bounds(:,2)' - bounds(:,1)') + bounds(:,1)';
    pop(i).f = evaluate(pop(i).x);
    pop(i).rank = 0;
    pop(i).crowding = 0;
end

% Evolution
for gen = 1:nGen
    % Non-dominated sorting
    [pop, front] = non_dominated_sorting(pop);
    
    % Crowding distance assignment
    pop = crowding_distance_assignment(pop, front);
    
    % Offspring generation
    for i = 1:2:nPop
        % Select parents
        p1 = tournament_selection(pop, front);
        p2 = tournament_selection(pop, front);
        
        % Crossover
        if rand < crossRate
            [c1, c2] = simulated_binary_crossover(p1.x, p2.x);
        else
            c1 = p1.x;
            c2 = p2.x;
        end
        
        % Mutation
        if rand < mutRate
            c1 = polynomial_mutation(c1, bounds);
        end
        if rand < mutRate
            c2 = polynomial_mutation(c2, bounds);
        end
        
        % Create offspring
        pop(i+nPop).x = c1;
        pop(i+nPop).f = evaluate(c1);
        pop(i+nPop).rank = 0;
        pop(i+nPop).crowding = 0;
        
        pop(i+nPop+1).x = c2;
        pop(i+nPop+1).f = evaluate(c2);
        pop(i+nPop+1).rank = 0;
        pop(i+nPop+1).crowding = 0;
    end
    
    % Merge parent and offspring populations
    pop = [pop(1:nPop), pop(nPop+1:end)];
end

% Final non-dominated sorting
[pop, front] = non_dominated_sorting(pop);

end

function f = evaluate(x)

% Objective function
f(1) = x(1)^2 + x(2)^2;
f(2) = (x(1)-1)^2 + x(2)^2;

end

function [pop, front] = non_dominated_sorting(pop)

% Non-dominated sorting
nPop = length(pop);

% Initialize fronts and domination counts
front = {};
nDom = zeros(nPop, 1);

% Calculate domination counts
for i = 1:nPop
    for j = 1:nPop
        if i ~= j
            if dominates(pop(i), pop(j))
                nDom(i) = nDom(i) + 1;
            end
        end
    end
end

% Assign individuals to first front
front{1} = find(nDom == 0);

% Assign individuals to other fronts
iFront = 1;
while ~isempty(front{iFront})
    iFront = iFront + 1;
    for i = 1:length(front{iFront-1})
        p = front{iFront-1}(i);
        for j = 1:nPop
            if nDom(j) > 0
                nDom(j) = nDom(j) - 1;
                if nDom(j) == 0
                    front{iFront} = [front{iFront}, j];
                end
            end
        end
    end
end

% Assign ranks to population
for i = 1:length(front)
    for j = 1:length(front{i})
        pop(front{i}(j)).rank = i;
    end
end

end

function pop = crowding_distance_assignment(pop, front)

% Crowding distance assignment
nPop = length(pop);
nObj = length(pop(1).f);

for i = 1:length(front)
    nFront = length(front{i});
    
    if nFront > 2
        % Initialize crowding distance
        for j = 1:nFront
            pop(front{i}(j)).crowding = 0;
        end
        
        % Calculate crowding distance for each objective
        for m = 1:nObj
            % Sort individuals by objective value
            [~, order] = sort([pop(front{i}).f(m)]);
            
            % Assign infinite crowding distance to boundary individuals
            pop(front{i}(order(1))).crowding = Inf;
            pop(front{i}(order(end))).crowding = Inf;
            
            % Calculate crowding distance for non-boundary individuals
            for j = 2:nFront-1
                pop(front{i}(order(j))).crowding = ...
                    pop(front{i}(order(j))).crowding + ...
                    (pop(front{i}(order(j+1))).f(m) - pop(front{i}(order(j-1))).f(m)) / ...
                    (pop(front{i}(order(end))).f(m) - pop(front{i}(order(1))).f(m));
            end
        end
    end
end

end

function b = dominates(p1, p2)

% Dominance comparison
f1 = p1.f;
f2 = p2.f;

if all(f1 <= f2) && any(f1 < f2)
    b = true;
elseif all(f2 <= f1) && any(f2 < f1)
    b = false;
else
    b = false;
end

end

function p = tournament_selection(pop, front)

% Tournament selection
nPop = length(pop);
nFronts = length(front);

% Select random individuals
i1 = randi(nPop);
i2 = randi(nPop);

% Ensure that individuals are in different fronts
while pop(i1).rank == pop(i2).rank
    i2 = randi(nPop);
end

% Select individual with lower rank
if pop(i1).rank < pop(i2).rank
    p = pop(i1);
else
    p = pop(i2);
end

end

function [c1, c2] = simulated_binary_crossover(p1, p2)

% Simulated binary crossover
nVar = length(p1);
c1 = zeros(1, nVar);
c2 = zeros(1, nVar);

for i = 1:nVar
    if rand < 0.5
        if abs(p1(i) - p2(i)) > 1e-10
            if p1(i) < p2(i)
                y1 = p1(i);
                y2 = p2(i);
            else
                y1 = p2(i);
                y2 = p1(i);
            end
            beta = 1 + (2 * (y1 - bounds(i,1)) / (y2 - y1));
            alpha = 2 - beta^(-(1+1));
            u = rand;
            if u <= (1 / alpha)
                c1(i) = y1 + (y2 - y1) * (u * alpha)^(1/(1+1));
            else
                c1(i) = y1 + (y2 - y1) * (1/(2-u*alpha))^(1/(1+1));
            end
            u = rand;
            if u <= (1 / alpha)
                c2(i) = y1 + (y2 - y1) * (u * alpha)^(1/(1+1));
            else
                c2(i) = y1 + (y2 - y1) * (1/(2-u*alpha))^(1/(1+1));
            end
        else
            c1(i) = p1(i);
            c2(i) = p2(i);
        end
    else
        c1(i) = p1(i);
        c2(i) = p2(i);
    end
end

end

function c = polynomial_mutation(p, bounds)

% Polynomial mutation
nVar = length(p);
c = p;

for i = 1:nVar
    if rand < 1/nVar
        u = rand;
        if u <= 0.5
            delta = (2*u)^(1/(1+20)) - 1;
        else
            delta = 1 - (2*(1-u))^(1/(1+20));
        end
        c(i) = p(i) + delta * (bounds(i,2) - bounds(i,1));
    end
end

end

这是一个简单的双目标NSGA-II示例代码，其中包括选择、交叉、变异、非支配排序和拥挤距离分配。要使用此代码，您需要编写自己的目标函数，该函数将决策变量作为输入并返回目标函数值。您还需要指定一些算法参数，例如种群大小，交叉率和变异率。最后，您可以运行算法并获得一组非支配解以及它们所在的前沿。