基于Linear Programming的排班自动化系统：工人分配未达指定比例求助

核心问题分析 #

你的代码中存在多个关键错误，直接导致工人分配比例不符合预期：

• 班次比例约束逻辑错误：原本想约束每周各班次总人数比例，但循环遍历日期时使用了c_min =+ sum(...)（应为c_min += sum(...)），导致变量被反复覆盖，最终仅对最后一天的班次人数施加约束，而非每周总和。
• 目标函数无效：由于已强制每个员工每周工作5天，当前目标函数的计算结果为固定值，无法起到优化工作量公平性的作用。
• 约束初始化错误：多个约束使用c = None后直接c += ...，会引发语法错误，导致约束未正确添加。
• 固定班次约束未实现：代码注释要求员工整周固定班次，但现有约束无法阻止员工在不同日期切换班次。

修复后的完整代码 #

import pulp

days = 7  # 一周7天
employees = 200  # 员工总数

# 定义二进制变量：每个员工每天的班次/休假状态
var_8_16 = pulp.LpVariable.dicts("8_16", (range(days), range(employees)), 0, 1, "Binary")
var_16_24 = pulp.LpVariable.dicts("16_24", (range(days), range(employees)), 0, 1, "Binary")
var_24_8 = pulp.LpVariable.dicts("24_8", (range(days), range(employees)), 0, 1, "Binary")
var_Holiday = pulp.LpVariable.dicts("Tatil", (range(days), range(employees)), 0, 1, "Binary")

# 目标函数：最小化员工工作量与目标值的绝对偏差总和，保证排班公平
obj = pulp.LpAffineExpression(0)
target_shifts_per_week = 5
for j in range(employees):
    total_shifts = pulp.lpSum([var_8_16[i][j] + var_16_24[i][j] + var_24_8[i][j] for i in range(days)])
    # 定义偏差变量处理线性规划中的绝对偏差
    dev_plus = pulp.LpVariable(f"dev_plus_{j}", lowBound=0)
    dev_minus = pulp.LpVariable(f"dev_minus_{j}", lowBound=0)
    obj += dev_plus + dev_minus
    problem += total_shifts - target_shifts_per_week == dev_plus - dev_minus

problem = pulp.LpProblem("Vardiya", pulp.LpMinimize)
problem += obj

# 约束0：每个员工每天必须工作一个班次或休假
for i in range(days):
    for j in range(employees):
        shift_sum = pulp.lpSum([var_8_16[i][j], var_16_24[i][j], var_24_8[i][j], var_Holiday[i][j]])
        problem += shift_sum == 1

# 约束1：每周各班次总人数符合指定比例
worker_daily_ratio = 0.7  # 每日上班员工占比
min_worker_ratio = {
    "8_16": 0.5,
    "16_24": 0.2,
    "24_8": 0.3
}

# 计算每周各班次的总人数要求
weekly_total_workers = employees * worker_daily_ratio * days
required_8_16 = min_worker_ratio["8_16"] * weekly_total_workers
required_16_24 = min_worker_ratio["16_24"] * weekly_total_workers
required_24_8 = min_worker_ratio["24_8"] * weekly_total_workers

# 计算每周各班次的实际总人数
total_8_16 = pulp.lpSum([var_8_16[i][j] for i in range(days) for j in range(employees)])
total_16_24 = pulp.lpSum([var_16_24[i][j] for i in range(days) for j in range(employees)])
total_24_8 = pulp.lpSum([var_24_8[i][j] for i in range(days) for j in range(employees)])

# 添加比例约束（允许±4的浮动）
problem += total_8_16 >= required_8_16
problem += total_8_16 <= required_8_16 + 4
problem += total_16_24 >= required_16_24
problem += total_16_24 <= required_16_24 + 4
problem += total_24_8 >= required_24_8
problem += total_24_8 <= required_24_8 + 4

# 约束2：每个员工每周必须工作5天
desired_workload = 5
for j in range(employees):
    total_shifts = pulp.lpSum([var_8_16[i][j] + var_16_24[i][j] + var_24_8[i][j] for i in range(days)])
    problem += total_shifts == desired_workload

# 约束3：每个员工每周必须休假2天
for j in range(employees):
    total_holidays = pulp.lpSum([var_Holiday[i][j] for i in range(days)])
    problem += total_holidays == 2

# 约束4：每个员工整周固定一个班次
for j in range(employees):
    total_8_16_j = pulp.lpSum([var_8_16[i][j] for i in range(days)])
    total_16_24_j = pulp.lpSum([var_16_24[i][j] for i in range(days)])
    total_24_8_j = pulp.lpSum([var_24_8[i][j] for i in range(days)])
    
    # 确保员工最多选择一种班次类型
    problem += total_8_16_j + total_16_24_j <= desired_workload
    problem += total_8_16_j + total_24_8_j <= desired_workload
    problem += total_16_24_j + total_24_8_j <= desired_workload

# 求解模型
problem.solve(pulp.PULP_CBC_CMD(msg=0))

# 输出结果示例
print("求解状态:", pulp.LpStatus[problem.status])
print("8-16班次总人数:", pulp.value(total_8_16))
print("16-24班次总人数:", pulp.value(total_16_24))
print("24-8班次总人数:", pulp.value(total_24_8))

关键修改说明 #

1. 修复目标函数：替换无效的目标逻辑，使用偏差变量最小化员工工作量与目标值的绝对偏差，确保排班公平。
2. 修正班次比例约束：计算每周各班次总人数需求，正确累加所有日期的班次人数，确保约束作用于每周总和。
3. 修复约束初始化：使用pulp.lpSum构建约束表达式，避免None初始化导致的语法错误。
4. 实现固定班次约束：添加约束确保每个员工每周仅选择一种班次类型，符合业务需求。
5. 优化代码结构：调整约束编号避免重复，增加注释提升可读性。

内容的提问来源于stack exchange，提问作者Parristaba