在PuLP中设置最小与最大辅助变量的问题求解

你的代码中min_weight始终为0的核心原因是：你将min_weight与所有货物的重量×分配变量做了<=约束，但未被选中的货物对应的重量×分配变量值为0，因此求解器会选择满足所有约束的最小可能值（0），而非实际分配的三个货物重量的最小值。

正确的做法是先计算每辆卡车的实际装载重量（即该卡车选中货物的重量），再让max_weight大于等于所有卡车的实际重量，min_weight小于等于所有卡车的实际重量。这样max_weight会自动取三个卡车重量的最大值，min_weight取最小值。

修改后的完整代码如下：

import pulp

# Example toy data
cargo = {
   "truck1":{
      "c1":{
         "price":100,
         "weight":20
      },
      "c2":{
         "price":150,
         "weight":10
      },
      "c3":{
         "price":90,
         "weight":30
      },
      "c4":{
         "price":500,
         "weight":80
      }
   },
   "truck2":{
      "c5":{
         "price":50,
         "weight":10
      },
      "c6":{
         "price":100,
         "weight":80
      },
      "c7":{
         "price":200,
         "weight":150
      },
      "c8":{
         "price":50,
         "weight":30
      }
   },
   "truck3":{
      "c9":{
         "price":100,
         "weight":50
      },
      "c10":{
         "price":200,
         "weight":200
      }
   }
}

# Create a problem variable:
prob = pulp.LpProblem("Cargo", pulp.LpMaximize)

# Decision variables
truck_allocation_vars = {truck: pulp.LpVariable.dicts("cargo", cargo[truck], 0, 1, pulp.LpBinary) for truck in cargo}

# Constraint: Only one cargo can be chosen for each truck
for truck in truck_allocation_vars:
    prob += pulp.lpSum(list(truck_allocation_vars[truck].values())) == 1

# Objective function: Maximize price
prob += pulp.lpSum([cargo[truck_idx][cargo_idx]['price'] * truck_allocation_vars[truck_idx][cargo_idx] 
                    for truck_idx in truck_allocation_vars 
                    for cargo_idx in truck_allocation_vars[truck_idx]])

# 计算每辆卡车的实际装载重量
truck_weights = {}
for truck in cargo:
    truck_weights[truck] = pulp.lpSum(
        cargo[truck][c]['weight'] * truck_allocation_vars[truck][c] 
        for c in cargo[truck]
    )

# 定义最大、最小重量变量，并添加正确约束
max_weight = pulp.LpVariable("max_weight", cat=pulp.LpContinuous)
min_weight = pulp.LpVariable("min_weight", cat=pulp.LpContinuous)

# max_weight >= 每辆卡车的实际重量
for truck in truck_weights:
    prob += max_weight >= truck_weights[truck]

# min_weight <= 每辆卡车的实际重量
for truck in truck_weights:
    prob += min_weight <= truck_weights[truck]

# 添加重量差约束：最大值与最小值之差不超过50kg（LP中不能用<，改用<=）
prob += (max_weight - min_weight) <= 50

# Solve the problem
prob.solve()

# Print the solution
for truck_idx in truck_allocation_vars:
    for cargo_idx in truck_allocation_vars[truck_idx].keys():
        if pulp.value(truck_allocation_vars[truck_idx][cargo_idx]) == 1:
            print(f"Cargo {cargo_idx} in truck {truck_idx} with price {cargo[truck_idx][cargo_idx]['price']} and weight {cargo[truck_idx][cargo_idx]['weight']}")

print(f"Max weight: {pulp.value(max_weight)}")
print(f"Min weight: {pulp.value(min_weight)}")

1. 计算卡车实际重量：通过lpSum对每辆卡车的选中货物重量求和，得到该卡车的真实装载重量，而非单独处理每个货物项。
2. 修正max/min约束：让max_weight和min_weight直接与每辆卡车的实际重量绑定，确保它们取的是三个分配货物重量的极值。
3. 约束符号修正：线性规划中不允许使用严格小于号<，因此将max_weight - min_weight < 50改为<= 50。

内容的提问来源于stack exchange，提问作者jpsca1293