遗传算法的多车场车辆路径问题求解

一、问题建模与数学描述

多车场车辆路径问题（MDVRP）可建模为带约束的组合优化问题：

• 目标函数：最小化总运输成本（距离/时间/费用）

  

  其中$K$为车场数，$P_k$为第$k$辆车的路径，$d_{i}j$为节点间距离

• 约束条件： 每个客户仅被访问一次 车辆从所属车场出发并返回 车辆载重不超过容量限制 时间窗约束（若适用）

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二、遗传算法设计

1. 染色体编码

采用多染色体编码表示多车场路径：

% 示例：4车场，10客户
chromosome = [3,1,5,2; 4,6,7,8; 9,10,3,2; 1,4,5,6]; 
% 每行表示一个车场的客户访问序列，数字代表客户编号

2. 适应度函数

综合路径长度与约束违反惩罚：

function fitness = calc_fitness(chromosome, dist_matrix, demand, capacity)
    total_dist = 0;
    penalty = 0;
    for i = 1:size(chromosome,1)
        route = chromosome(i,:);
        route_dist = 0;
        load = 0;
        for j = 2:length(route)
            route_dist = route_dist + dist_matrix(route(j-1), route(j));
            load = load + demand(route(j));
        end
        % 时间窗惩罚（示例）
        if load > capacity
            penalty = penalty + 1000; 
        end
        total_dist = total_dist + route_dist;
    end
    fitness = 1/(total_dist + penalty);
end

3. 遗传操作

• 选择：锦标赛选择（Tournament Selection）

  function winner = tournament_selection(pop, fitness, tournament_size)
      candidates = randperm(size(pop,1), tournament_size);
      [~, idx] = min(fitness(candidates));
      winner = pop(candidates(idx),:);
  end
  

• 交叉：改进顺序交叉（OX）保留车场连续性

  function child = ox_crossover(parent1, parent2)
      n = size(parent1,2);
      child = zeros(1,n);
      % 随机选择两个交叉点
      p1 = randperm(n,2);
      p1 = sort(p1);
      % 复制父代1的中间段
      child(p1(1):p1(2)) = parent1(p1(1):p1(2));
      % 填充父代2剩余基因
      ptr = p1(2)+1;
      for i = 1:n
          if ptr > n
              ptr = 1;
          end
          if ~ismember(parent2(i), child)
              child(ptr) = parent2(i);
              ptr = ptr+1;
          end
      end
  end
  

• 变异：交换突变与车场重分配

  function mutated = swap_mutation(chromosome, prob)
      if rand < prob
          idx1 = randi(size(chromosome,2));
          idx2 = randi(size(chromosome,2));
          % 交换不同车场的客户
          while chromosome(1,idx1) == chromosome(1,idx2)
              idx2 = randi(size(chromosome,2));
          end
          temp = chromosome(1,idx1);
          chromosome(1,idx1) = chromosome(1,idx2);
          chromosome(1,idx2) = temp;
      end
  end
  

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三、改进

1. 自适应参数调整

• 动态交叉率：

  pc=pc0⋅e−λ⋅favg/fmax
  

• 变异率与种群多样性关联

2. 混合启发式算法

• 2-opt局部搜索优化子路径：

  function new_route = two_opt(route, dist_matrix)
      n = length(route);
      improved = true;
      while improved
          improved = false;
          for i = 1:n-2
              for j = i+2:n
                  new_seg = route(i:j);
                  new_dist = sum(dist_matrix(new_seg(1:end-1), new_seg(2:end)));
                  old_dist = sum(dist_matrix(route(i:j-1), route(j:end)));
                  if new_dist < old_dist
                      route(i:j) = new_seg;
                      improved = true;
                  end
              end
          end
      end
      new_route = route;
  end
  

3. 多目标优化

采用NSGA-II算法处理多目标：

% 帕累托前沿维护
function fronts = fast_non_dominated_sort(population)
    fronts = {};
    ranks = zeros(size(population,1),1);
    dominated_count = zeros(size(population,1),1);
    dominates = cell(size(population,1),1);
    
    for i = 1:size(population,1)
        for j = 1:size(population,1)
            if i ~= j
                if dominates(population(i), population(j))
                    dominates{i} = [dominates{i}, j];
                elseif dominates(population(j), population(i))
                    dominated_count(i) = dominated_count(i) + 1;
                end
            end
        end
        if dominated_count(i) == 0
            ranks(i) = 1;
            fronts{1} = [fronts{1}, i];
        end
    end
    
    front_idx = 1;
    while ~isempty(fronts{front_idx})
        next_front = [];
        for i = fronts{front_idx}
            for j = dominates{i}
                dominated_count(j) = dominated_count(j) - 1;
                if dominated_count(j) == 0
                    ranks(j) = front_idx + 1;
                    next_front = [next_front, j];
                end
            end
        end
        front_idx = front_idx + 1;
        fronts{front_idx} = next_front;
    end
end

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四、MATLAB实现流程

%% 参数设置
num_depots = 4;    % 车场数量
num_customers = 50;% 客户数量
vehicle_capacity = 100; % 车辆容量
pop_size = 100;    % 种群大小
max_generations = 500; % 最大迭代次数

%% 数据加载
load('depot_customers.mat'); % 包含坐标、需求等数据

%% 初始化种群
population = initialize_population(pop_size, num_depots, num_customers);

%% 主循环
for gen = 1:max_generations
    % 适应度计算
    fitness = arrayfun(@(i) calc_fitness(population(i,:), dist_matrix, demand, vehicle_capacity), 1:pop_size);
    
    % 选择
    new_population = [];
    for i = 1:pop_size
        parent1 = tournament_selection(population, fitness, 5);
        parent2 = tournament_selection(population, fitness, 5);
        % 交叉
        child = ox_crossover(parent1, parent2);
        % 变异
        child = swap_mutation(child, 0.02);
        new_population = [new_population; child];
    end
    
    % 更新种群
    population = new_population;
    
    % 输出当前最优
    [best_fitness, best_idx] = max(fitness);
    best_route = population(best_idx,:);
    fprintf('Generation %d: Best Fitness = %.2f\n', gen, best_fitness);
end

%% 结果可视化
plot_routes(best_route, depot_coords, customer_coords);

参考代码 遗传算法求解多车场车辆路径问题 www.youwenfan.com/contentcsk/64104.html

五、性能优化

1. 精英保留策略：每代保留前5%最优个体

2. 邻域搜索增强：在变异操作中嵌入3-opt优化

3. 并行计算：利用MATLAB Parallel Toolbox加速适应度计算

   parfor i = 1:pop_size
       fitness(i) = calc_fitness(population(i,:), ...);
   end
   

4. 动态仓库分配：根据客户分布聚类调整车场服务区域

   % K-means聚类分配客户到车场
   [idx, centers] = kmeans(customer_coords, num_depots);
   depot_assignment = idx;