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基于模拟退火算法解决带容量限制车辆路径问题（CVRP）的MATLAB实现

news 2025/9/22 9:52:54

一、问题建模

1.1 数学描述

目标函数：

其中：

\(R_k\)：第k条路径的边集合
\(dij\)：节点i到j的距离
\(K\)：使用车辆数
\(λ\)：车辆使用惩罚系数

约束条件：

每个客户仅被访问一次
车辆从仓库出发并返回
路径载重不超过车辆容量

1.2 输入参数

% 示例参数
n = 20;        % 客户数量（含仓库）
demand = [0,12,8,15,7,10,9,11,6,14,5,13,8,10,12,9,7,11,6,10]; % 需求量
capacity = 30; % 车辆容量
distance = pdist2(rand(n,2),rand(n,2)); % 随机生成距离矩阵

二、算法实现

2.1 初始解生成

function route = generate_initial_solution(n, capacity, demand)route = [0]; % 仓库节点remaining = 2:n; % 客户节点while ~isempty(remaining)current_load = 0;sub_route = [0];while ~isempty(remaining)candidate = remaining(randi(length(remaining)));if current_load + demand(candidate) <= capacitysub_route = [sub_route, candidate];current_load = current_load + demand(candidate);remaining(remaining == candidate) = [];elsebreak;endendroute = [route, sub_route(2:end), 0];end
end

2.2 邻域操作设计

function new_route = neighborhood_operation(route)% 交换操作i = randi(length(route)-1);j = randi(length(route)-1);new_route = route;new_route([i+1,j+1]) = new_route([j+1,i+1]);% 插入操作if rand < 0.5k = randi(length(route)-1);insert_pos = randi(length(route)-1);node = new_route(k+1);new_route(k+1) = [];new_route = [new_route(1:insert_pos), node, new_route(insert_pos+1:end)];end
end

2.3 接受准则

function accept = metropolis(delta_E, T)if delta_E < 0accept = true;elsep = exp(-delta_E / T);accept = rand < p;end
end

2.4 主算法流程

%% 参数设置
T0 = 1000;    % 初始温度
T_min = 1e-3; % 终止温度
alpha = 0.95; % 降温系数
max_iter = 500; % 最大迭代次数%% 初始化
current_route = generate_initial_solution(n, capacity, demand);
current_cost = calculate_cost(current_route, distance);
best_route = current_route;
best_cost = current_cost;%% 模拟退火主循环
T = T0;
for iter = 1:max_iternew_route = neighborhood_operation(current_route);new_cost = calculate_cost(new_route, distance);delta_E = new_cost - current_cost;if metropolis(delta_E, T)current_route = new_route;current_cost = new_cost;if current_cost < best_costbest_route = current_route;best_cost = current_cost;endend% 降温T = alpha * T;% 终止条件if T < T_min || mod(iter,100) == 0fprintf('Iter: %d, Cost: %.2f\n', iter, current_cost);end
end

三、关键函数实现

3.1 成本计算函数

function cost = calculate_cost(route, distance)n = length(route);cost = 0;for i = 1:n-1cost = cost + distance(route(i), route(i+1));end% 车辆数惩罚项num_vehicles = count_vehicles(route);cost = cost + 100 * num_vehicles; % 惩罚系数可调整
endfunction num = count_vehicles(route)num = 1;for i = 2:length(route)if route(i) == 0num = num + 1;endend
end

3.2 可视化函数

function plot_route(route, distance)figure;hold on;route = [route,0]; % 添加仓库闭合路径x = [0,0]; y = [0,0];for i = 2:length(route)idx = route(i);x = [x, idx(1)]; y = [y, idx(2)];plot([route(i-1),idx(1)]/1000, [route(i-1),idx(2)]/1000, 'r-o');endtext(x(1),y(1),'仓库','FontSize',12);title(sprintf('最优路径 (总距离=%.2f km)', calculate_cost(route,distance)/1000));xlabel('X坐标 (km)'); ylabel('Y坐标 (km)');grid on;
end

参考代码采用模拟退火方法解决有容量限制的车辆路径问题 www.youwenfan.com/contentcnh/64265.html

四、性能优化策略

4.1 自适应参数调整

% 动态降温策略
if iter > 100 && mod(iter,50) == 0alpha = alpha * 0.98; % 后期降温加快
end% 邻域操作概率调整
if current_cost < best_cost*0.95neighborhood_prob = 0.7; % 优先复杂操作
elseneighborhood_prob = 0.3;
end

4.2 并行计算加速

% 并行计算温度循环
parfor iter = 1:max_iter% ... 同上迭代代码 ...
end

4.3 约束处理机制

% 路径合法性检查
function valid = check_feasibility(route)loads = zeros(1,length(route)-1);for i = 2:length(route)if route(i) ~= 0loads(find(route(1:i-1)==route(i),1)) = [];endendvalid = all(loads <= capacity);
end