带有时间窗的车辆路径安排问题（VRPTW问题）

VRPTW问题是指车辆路径安排问题中考虑了时间窗约束的情况。这个问题是NP难题，它需要在满足车辆运载能力、时间窗约束和最小化总行驶距离之间找到一个最优的解决方案。

下面是一个基本的Matlab源码，用于解决VRPTW问题。这个源码使用了遗传算法来寻找一个较优的解决方案。源码中包含了对问题的建模、遗传算法的实现以及结果的可视化。

% VRPTW问题的遗传算法解决方案

% 参数设置
nCustomer = 20; % 客户数量
nVehicle = 5; % 车辆数量
capacity = 100; % 车辆运载能力
nGen = 100; % 迭代次数

% 生成随机客户坐标和需求
x = randi([0,100],1,nCustomer);
y = randi([0,100],1,nCustomer);
demand = randi([1,20],1,nCustomer);

% 设置时间窗约束
timeStart = randi([0,50],1,nCustomer);
timeEnd = timeStart + randi([10,50],1,nCustomer);

% 计算距离矩阵
distMatrix = pdist2([x;y]', [x;y]');

% 初始化种群
population = cell(nGen,1);
for i = 1:nGen
    population{i} = randperm(nCustomer-1,nVehicle-1);
end

% 遗传算法主循环
for gen = 1:nGen
    % 评估适应度
    fitness = evaluateFitness(population{gen}, distMatrix, demand, capacity, timeStart, timeEnd);
    
    % 选择和交叉
    newPopulation = cell(nGen,1);
    for i = 1:nGen
        parent1 = population{rouletteWheelSelection(fitness)};
        parent2 = population{rouletteWheelSelection(fitness)};
        newPopulation{i} = crossover(parent1, parent2);
    end
    
    % 变异
    for i = 1:nGen
        newPopulation{i} = mutate(newPopulation{i});
    end
    
    % 更新种群
    population = newPopulation;
end

% 可视化最优解
bestRoute = population{1};
visualizeRoute(bestRoute, x, y);

% 评估适应度函数
function fitness = evaluateFitness(population, distMatrix, demand, capacity, timeStart, timeEnd)
    fitness = zeros(length(population),1);
    for i = 1:length(population)
        route = [1, population{i}, nCustomer];
        fitness(i) = calculateRouteDistance(route, distMatrix);
        fitness(i) = fitness(i) + calculateRouteLoad(route, demand, capacity);
        fitness(i) = fitness(i) + calculateRouteTimeWindow(route, timeStart, timeEnd, distMatrix);
    end
end

% 选择函数
function idx = rouletteWheelSelection(fitness)
    totalFitness = sum(fitness);
    r = rand * totalFitness;
    partialSum = 0;
    for i = 1:length(fitness)
        partialSum = partialSum + fitness(i);
        if partialSum >= r
            idx = i;
            break;
        end
    end
end

% 交叉函数
function child = crossover(parent1, parent2)
    crossoverPoint = randi([1,length(parent1)]);
    child = [parent1(1:crossoverPoint), parent2(crossoverPoint+1:end)];
end

% 变异函数
function mutatedRoute = mutate(route)
    idx1 = randi([1,length(route)]);
    idx2 = randi([1,length(route)]);
    mutatedRoute = route;
    temp = mutatedRoute(idx1);
    mutatedRoute(idx1) = mutatedRoute(idx2);
    mutatedRoute(idx2) = temp;
end

% 计算路线距离
function distance = calculateRouteDistance(route, distMatrix)
    distance = 0;
    for i = 1:length(route)-1
        distance = distance + distMatrix(route(i), route(i+1));
    end
end

% 计算路线负载
function load = calculateRouteLoad(route, demand, capacity)
    load = 0;
    for i = 1:length(route)-1
        load = load + demand(route(i));
    end
    load = max(0, load - capacity);
end

% 计算路线时间窗约束
function penalty = calculateRouteTimeWindow(route, timeStart, timeEnd, distMatrix)
    penalty = 0;
    currentTime = 0;
    for i = 1:length(route)-1
        currentTime = currentTime + distMatrix(route(i), route(i+1));
        if currentTime < timeStart(route(i+1))
            currentTime = timeStart(route(i+1));
        end
        if currentTime > timeEnd(route(i+1))
            penalty = penalty + (currentTime - timeEnd(route(i+1)));
        end
    end
end

% 可视化路线
function visualizeRoute(route, x, y)
    plot(x(route),y(route),'-o');
    xlabel('X');
    ylabel('Y');
    title('Optimal Route');
end

这个源码实现了一个简单的遗传算法来解决VRPTW问题。在这个源码中，首先生成了随机的客户坐标和需求，然后根据时间窗约束、运载能力和距离矩阵进行遗传算法的求解。最后通过可视化展示了最优路径。

这个源码还可以进一步扩展，比如增加更复杂的遗传算法操作符、改进评估函数、尝试其他优化算法等，以获得更好的解决方案。