6.（Python数模）游泳接力指派问题

Python解决游泳接力指派问题

该问题也属于一个线性规划问题

源代码

import pulp      # 导入 pulp 库
import numpy as np

# 主程序
def main():
    # 问题建模：
    """
        决策变量：
            x(i,j) = 0, 第 i 个人不游第 j 种姿势
            x(i,j) = 1, 第 i 个人游第 j 种姿势
            i=1,4, j=1,4
        目标函数：
            min time = sum(sum(c(i,j)*x(i,j))), i=1,4, j=1,4
        约束条件：
            sum(x(i,j),j=1,4)=1, i=1,4
            sum(x(i,j),i=1,4)=1, j=1,4
        变量取值范围：
            x(i,j) = 0,1
    """

    # 游泳比赛的指派问题 (assignment problem)
    # 1.建立优化问题 AssignLP: 求最小值(LpMinimize)
    AssignLP = pulp.LpProblem("Assignment_problem_for_swimming_relay_race", sense=pulp.LpMinimize)  # 定义问题，求最小值
    # 2. 建立变量
    rows = cols = range(0, 4)
    x = pulp.LpVariable.dicts("x", (rows, cols), cat="Binary")
    # 3. 设置目标函数
    scoreM = [[56,74,61,63],[63,69,65,71],[57,77,63,67],[55,76,62,62]]
    AssignLP += pulp.lpSum([[x[row][col]*scoreM[row][col] for row in rows] for col in cols])
    # 4. 施加约束
    for row in rows:
        AssignLP += pulp.lpSum([x[row][col] for col in cols]) == 1 # sum(x(i,j),j=1,4)=1, i=1,4
    for col in cols:
        AssignLP += pulp.lpSum([x[row][col] for row in rows]) == 1 # sum(x(i,j),i=1,4)=1, j=1,4
    # 5. 求解
    AssignLP.solve()
    # 6. 打印结果
    print(AssignLP.name)
    member = ["队员A","队员B","队员C","队员D"]
    style = ["自由泳","蛙泳","蝶泳","仰泳"]
    if pulp.LpStatus[AssignLP.status] == "Optimal":  # 获得最优解
        xValue = [v.varValue for v in AssignLP.variables()]
        # [0.0, 0.0, 1.0, 0.0, 0.0, 1.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 1.0]
        xOpt = np.array(xValue).reshape((4, 4))  # 将 xValue 格式转换为 4x4 矩阵
        print("最佳分配：" )
        for row in rows:
            print("{}\t{} 参加项目：{}".format(xOpt[row],member[row],style[np.argmax(xOpt[row])]))
        print("预测最好成绩为：{}".format(pulp.value(AssignLP.objective)))

    return

if __name__ == '__main__':
    main()

要注意的是，在pulp库中，可以使用pulp.lpsum()函数来对线性表达式求和，例如pulp.lpSum([x[row][col] for col in cols]) == 1

运行结果