如何在Pulp运输优化模型中添加最小运输量约束？

要实现「若某工厂存在运输量（总运输值>0），则该运输量必须至少为5000」的约束，需要引入二进制变量建模逻辑关系，具体步骤如下：

建模思路 #

对每个工厂定义一个二进制变量（取值0或1）：

• 变量为1时：表示工厂有运输行为，此时总运输量需≥5000
• 变量为0时：表示工厂无运输行为，此时总运输量必须=0

通过两个约束实现逻辑控制：

1. 工厂总运输量 ≤ 二进制变量 × 工厂最大产能：确保二进制变量为0时，运输量只能是0
2. 工厂总运输量 ≥ 5000 × 二进制变量：确保二进制变量为1时，运输量至少为5000

修改后的完整代码 #

from pulp import *

capacityFactory = {
    "Factory01": 56000, 
    "Factory02": 50000, 
    "Factory03": 40000, 
    "Factory04": 30000,
    "Factory05": 37000,
    "Factory06": 42000,
    "Factory07": 45000,
    "Factory08": 52000,
    "Factory09": 57000
}
factories = capacityFactory.keys()

orderBalance = {
    "Order01": 14000,
    "Order02": 35000,
    "Order03": 76000,
    "Order04": 36000,
    "Order05": 72000,
    "Order06": 41000,
    "Order07": 12000,
    "Order08": 56000,
    "Order09": 10000,
    "Order10": 7000
}
orders = orderBalance.keys()

# 运输路径成本矩阵
cost = [  # 列对应订单
    [3.26, 9.31, 8.49, 5.96, 8.61, 5.45, 6.2, 7.31, 4.89, 8.34], # Factory01
    [5.5, 8.75, 9.67, 5.05, 8.54, 5.5, 6.3, 7.36, 4.35, 8.51],  # Factory02
    [4.66, 8.36, 8.13, 4.33, 8.59, 5.6, 6.4, 7.42, 4.91, 9.71], # Factory03
    [1.66, 1.36, 1.88, 2.77, 100, 5.65, 6.5, 7.89, 5.21, 7.01],   # Factory04
    [2.456, 6.54, 7.4, 3.54, 7.98, 5.44, 6.4, 7.89, 5.21, 7.01],   # Factory05
    [3.54, 6.98, 8.24, 3.12, 7.77, 5.12, 6.52, 7.12, 5.16, 7.5],   # Factory06
    [2.15, 4.54, 7.87, 3.15, 7.45, 5.396, 6.98, 7.02, 5, 7.4],   # Factory07
    [2.48, 5.55, 8.341, 3.19, 7.48, 5.9, 6.21, 7.23, 5.08, 7.36],   # Factory08
    [3.41, 5.69, 8.14, 3.22, 7.66, 5.87, 6.35, 8.23, 5.39, 7.99],   # Factory09
]
cost = makeDict([factories, orders], cost, 0)
prob = LpProblem("Material_Shipment_Problem", LpMinimize)
Routes = [(w, b) for w in factories for b in orders]
vars = LpVariable.dicts("Receive", (factories, orders), 0, None, LpInteger)

# 新增：为每个工厂定义二进制变量，标记是否启用运输
factory_used = LpVariable.dicts("Factory_Used", factories, cat=LpBinary)

# 目标函数：最小化总运输成本
prob += (
    lpSum([vars[w][b] * cost[w][b] for (w, b) in Routes]),
    "Sum_of_Transporting_cost",
)

# 工厂产能上限约束
for w in factories:
    prob += (
        lpSum([vars[w][b] for b in orders]) <= capacityFactory[w],
        "Max_Capacity_%s" % w,
    )

# 订单需求下限约束
for b in orders:
    prob += (
        lpSum([vars[w][b] for w in factories]) >= orderBalance[b],
        "Min_Demand_%s" % b,
    )

# 新增：运输量阈值约束
MIN_SHIPMENT = 5000
for w in factories:
    total_shipment = lpSum([vars[w][b] for b in orders])
    # 约束1：未启用的工厂运输量必须为0
    prob += total_shipment <= capacityFactory[w] * factory_used[w], f"Shipment_Cap_If_Used_{w}"
    # 约束2：启用的工厂运输量至少为5000
    prob += total_shipment >= MIN_SHIPMENT * factory_used[w], f"Shipment_Min_If_Used_{w}"

# 非负约束
for w in factories:
    for b in orders:
        prob += (
            vars[w][b] >= 0,
            "Non_Negative_%s_to_%s" % (w, b),
        )

# 求解并输出结果
prob.solve()

for v in prob.variables():
    if v.varValue > 0:
        print(v.name, "=", v.varValue)
    
print("总运输成本 = ", value(prob.objective))

关键修改说明 #

1. 新增二进制变量：factory_used 字典，每个工厂对应一个0/1变量，标记是否有运输行为
2. 添加阈值约束：
   • 第一个约束通过工厂最大产能乘以二进制变量，限制未启用工厂的运输量为0
   • 第二个约束通过最小阈值乘以二进制变量，确保启用的工厂运输量不低于5000
3. 输出时只打印非零变量，简化结果展示

内容的提问来源于stack exchange，提问作者Maurício Rosa