带运输量上下限的运输问题求解及代码适配咨询

要给现有的运输模型添加每个供应商-消费者对的运输量上下限约束，只需两步：定义上下限数据，再为每个运输变量添加边界约束。

实现步骤 #

1. 定义运输量上下限数据
   把你提供的上下限表格转换成和现有suppliers、consumers索引匹配的DataFrame：

# 运输量下界（4×5）
flow_lower = pd.DataFrame(
    index=suppliers, columns=consumers,
    data=(
        (1,2,1,2,1),
        (2,1,1,1,3),
        (0,1,2,1,1),
        (2,1,3,1,2),
    )
).stack()
flow_lower.name = 'lower_bound'

# 运输量上界（4×5）
flow_upper = pd.DataFrame(
    index=suppliers, columns=consumers,
    data=(
        (9,8,6,10,5),
        (7,15,4,6,9),
        (5,6,6,5,10),
        (8,5,7,4,8),
    )
).stack()
flow_upper.name = 'upper_bound'

2. 添加上下限约束
   遍历每个供应商-消费者对，为flow变量添加上下限约束：

# 添加运输量下界约束
for (supplier, consumer), lower in flow_lower.items():
    prob.addConstraint(
        name=f'flow_lower_s{supplier}_c{consumer}',
        constraint=flow[(supplier, consumer)] >= lower
    )

# 添加运输量上界约束
for (supplier, consumer), upper in flow_upper.items():
    prob.addConstraint(
        name=f'flow_upper_s{supplier}_c{consumer}',
        constraint=flow[(supplier, consumer)] <= upper
    )

修改后的完整代码 #

import pandas as pd
import pulp

truck_capacity = 6
suppliers = pd.RangeIndex(name='supplier', stop=4)
consumers = pd.RangeIndex(name='consumer', stop=5)

supply = pd.Series(
    name='supply', 
    index=suppliers, 
    data=(17, 8, 10, 9),
)
demand = pd.Series(
    name='demand', 
    index=consumers, 
    data=(6, 15, 7, 8, 8),
)
price_per_tonne = pd.DataFrame(
    index=suppliers, columns=consumers,
    data=(
        (10,  8,  5,  9, 16),
        ( 4,  3,  4, 11, 12),
        ( 5, 10, 29,  7,  6),
        ( 9,  2,  4,  1,  3),
    ),
).stack()
price_per_tonne.name = 'price'

# 新增：定义运输量上下限
flow_lower = pd.DataFrame(
    index=suppliers, columns=consumers,
    data=(
        (1,2,1,2,1),
        (2,1,1,1,3),
        (0,1,2,1,1),
        (2,1,3,1,2),
    )
).stack()
flow_lower.name = 'lower_bound'

flow_upper = pd.DataFrame(
    index=suppliers, columns=consumers,
    data=(
        (9,8,6,10,5),
        (7,15,4,6,9),
        (5,6,6,5,10),
        (8,5,7,4,8),
    )
).stack()
flow_upper.name = 'upper_bound'

flow = pd.DataFrame(
    index=suppliers, columns=consumers,
    data=pulp.LpVariable.matrix(
        name='flow_s%d_c%d', cat=pulp.LpContinuous, lowBound=0,
        indices=(suppliers, consumers),
    ),
).stack()
flow.name = 'flow'

trucks = pd.DataFrame(
    index=suppliers, columns=consumers,
    data=pulp.LpVariable.matrix(
        name='trucks_s%d_c%d', cat=pulp.LpInteger, lowBound=0,
        indices=(suppliers, consumers),
    )
).stack()
trucks.name = 'trucks'

price = truck_capacity * pulp.lpDot(price_per_tonne, trucks)
prob = pulp.LpProblem(name='transportation', sense=pulp.LpMinimize)
prob.setObjective(price)

# The flow must not exceed the supply
for supplier, group in flow.groupby('supplier'):
    prob.addConstraint(
        name=f'flow_supply_s{supplier}',
        constraint=pulp.lpSum(group) <= supply[supplier],
    )

# The flow must exactly meet the demand
for consumer, group in flow.groupby('consumer'):
    prob.addConstraint(
        name=f'flow_demand_c{consumer}',
        constraint=pulp.lpSum(group) == demand[consumer],
    )

# The capacity must be able to carry the flow
for (supplier, consumer), truck_flow in flow.items():
    prob.addConstraint(
        name=f'capacity_s{supplier}_c{consumer}',
        constraint=truck_flow <= trucks[(supplier, consumer)] * truck_capacity
    )

# 新增：添加运输量上下限约束
for (supplier, consumer), lower in flow_lower.items():
    prob.addConstraint(
        name=f'flow_lower_s{supplier}_c{consumer}',
        constraint=flow[(supplier, consumer)] >= lower
    )

for (supplier, consumer), upper in flow_upper.items():
    prob.addConstraint(
        name=f'flow_upper_s{supplier}_c{consumer}',
        constraint=flow[(supplier, consumer)] <= upper
    )

print(prob)
prob.solve()
assert prob.status == pulp.LpStatusOptimal

print(f'Total price: ${price.value():.2f}')
print()

print('Flow:')
flow = flow.apply(pulp.value).unstack(level='consumer')
print(flow)
print()

print('Trucks:')
trucks = trucks.apply(pulp.value).unstack(level='consumer')
print(trucks)
print()

print('Prices:')
print(trucks * truck_capacity * price_per_tonne.unstack(level='consumer'))

注意事项 #

• 确保上下限约束和现有供需约束不冲突：比如所有供应商的下界总和不能超过总需求，上界总和不能低于总需求，否则模型会返回无解状态。
• 原代码中flow变量的lowBound=0可以保留，因为我们添加的下界可能比0大，约束会自动覆盖这个默认值。

内容的提问来源于stack exchange，提问作者Pavel Jefimovich