基于MATLAB的匈牙利算法实现任务分配

一、算法原理与核心步骤

匈牙利算法通过矩阵变换寻找最优分配方案，其核心步骤包括：

1. 行约减：每行减去最小值，确保每行至少一个零
2. 列约减：每列减去最小值，确保每列至少一个零
3. 覆盖零：用最少的线覆盖所有零元素
4. 调整矩阵：未覆盖元素减最小值，交点加最小值
5. 重复迭代：直到找到完美匹配

数学建模示例：

成本矩阵 C = [[4, 2, 8],
             [6, 5, 7],
             [3, 9, 1]]

最优分配为总成本 3+5+1=9

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二、MATLAB实现方案

1. 使用内置函数（推荐）

% 示例成本矩阵（任务数=代理数）
cost_matrix = [4, 2, 8; 6, 5, 7; 3, 9, 1];

% 执行匈牙利算法
[assignment, total_cost] = hungarian(cost_matrix);

% 输出结果
disp('最优分配方案：');
disp(assignment');  % 转置显示为任务→代理
disp(['最小总成本：', num2str(total_cost)]);

2. 手动实现算法

function [assignment, cost] = hungarian_manual(cost_matrix)
    n = size(cost_matrix, 1);
    cost_matrix = double(cost_matrix);
    
    % 步骤1：行约减
    row_min = min(cost_matrix, [], 2);
    cost_matrix = cost_matrix - row_min;
    
    % 步骤2：列约减
    col_min = min(cost_matrix, [], 1);
    cost_matrix = cost_matrix - col_min;
    
    % 步骤3：覆盖所有零
    while true
        [row_cover, col_cover] = find_cover(cost_matrix);
        if sum(row_cover) + sum(col_cover) == n
            break;
        end
        
        % 步骤4：调整矩阵
        [min_uncovered, pos] = find_min_uncovered(cost_matrix, row_cover, col_cover);
        cost_matrix(~row_cover, :) = cost_matrix(~row_cover,:) - min_uncovered;
        cost_matrix(:, col_cover) = cost_matrix(:, col_cover) + min_uncovered;
    end
    
    % 步骤5：提取匹配结果
    assignment = zeros(n,1);
    for i = 1:n
        [~, assignment(i)] = max(cost_matrix(i,:));
    end
    cost = sum(diag(cost_matrix));
end

% 辅助函数实现
function [row_cover, col_cover] = find_cover(matrix)
    [n, m] = size(matrix);
    row_cover = false(n,1);
    col_cover = false(m,1);
    
    for i = 1:n
        for j = 1:m
            if matrix(i,j) == 0 && ~row_cover(i) && ~col_cover(j)
                row_cover(i) = true;
                col_cover(j) = true;
            end
        end
    end
end

function [min_val, pos] = find_min_uncovered(matrix, row_cover, col_cover)
    [n, m] = size(matrix);
    min_val = inf;
    pos = [0,0];
    
    for i = 1:n
        for j = 1:m
            if ~row_cover(i) && ~col_cover(j)
                if matrix(i,j) < min_val
                    min_val = matrix(i,j);
                    pos = [i,j];
                end
            end
        end
    end
end

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三、应用案例解析

案例1：仓储机器人任务分配

% 机器人数量与任务数量
n = 5;
cost_matrix = randi([10, 100], n, n);  % 随机生成成本矩阵

% 添加虚拟节点处理非方阵（可选）
if size(cost_matrix,1) ~= size(cost_matrix,2)
    cost_matrix = [cost_matrix, inf(size(cost_matrix,1),1)];
    cost_matrix = [cost_matrix; inf(1,size(cost_matrix,2))];
end

% 执行算法
[assignment, total_cost] = hungarian(cost_matrix(1:n,1:n));

% 可视化结果
figure;
bar(assignment);
title('任务分配结果');
xlabel('机器人编号');
ylabel('分配任务编号');

案例2：动态任务分配（实时更新）

% 初始化参数
n_agents = 4;
n_tasks = 4;
cost_matrix = [3,5,7,2; 6,4,9,8; 2,3,5,4; 7,6,8,1];

% 动态更新函数
function new_assignment = dynamic_update(old_assignment, cost_matrix)
    [~, new_assignment] = hungarian(cost_matrix);
end

% 模拟动态变化
for iter = 1:5
    cost_matrix = cost_matrix + randn(size(cost_matrix))*2;  % 添加噪声
    new_assignment = dynamic_update(assignment, cost_matrix);
    assignment = new_assignment;
    disp(['迭代', num2str(iter), '总成本：', num2str(sum(diag(cost_matrix(assignment,1:n_agents))))]);
end

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四、性能优化策略

1. 稀疏矩阵处理

   对大规模稀疏矩阵使用压缩存储格式：

   sparse_matrix = sparse(cost_matrix);
   [assignment, cost] = hungarian(sparse_matrix);

2. 并行计算加速

   options = optimoptions('hungarian','Display','iter','UseParallel',true);
   [assignment, cost] = hungarian(cost_matrix, options);

3. GPU加速

   gpu_matrix = gpuArray(cost_matrix);
   [assignment, cost] = hungarian(gpu_matrix);

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五、结果分析与验证

1. 最优性验证

   通过对比暴力搜索结果验证最优性：

   brute_force_cost = min(sum(pdist2(cost_matrix, 'minkowski')));
   assert(total_cost <= brute_force_cost*1.01);  % 允许1%误差

2. 可视化工具

   % 绘制成本矩阵热力图
   heatmap(cost_matrix);
   hold on;
   plot(find(assignment==1),1,'r*');
   plot(find(assignment==2),2,'g*');

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六、工程应用扩展

1. 多代理系统

   % 多机器人协同任务分配
   num_agents = 10;
   num_tasks = 15;
   cost_matrix = rand(num_agents,num_tasks)*100;
   [assignment, cost] = hungarian(cost_matrix);

2. 医疗资源调度

   % 医院-患者匹配
   hospitals = 5;
   patients = 7;
   cost_matrix = rand(hospitals,patients)*100;
   [assignment, cost] = hungarian(cost_matrix);

3. 工业生产线优化

   % 机器-工序匹配
   machines = 6;
   processes = 6;
   cost_matrix = rand(machines,processes)*50;
   [assignment, cost] = hungarian(cost_matrix);

九、参考

1. 核心文献 《运筹学》（清华大学出版社）第6章 《Combinatorial Optimization》 by Papadimitriou
2. 代码 匈牙利算法实现任务分配 www.youwenfan.com/contentcsi/63742.html
3. MATLAB工具箱 Global Optimization Toolbox Robotics System Toolbox