PuLP以最小化显示LpVariable的上限值

Question

我正在尝试最小化以下问题：对于笔记本电脑，手机和平板电脑的生产，存在库存成本（每个项目每月1美元）和加班时间（每小时10美元）。有一个需要满足的需求方案，它是对特定月份中最小数量的小工具的约束。除此之外，每月最多有20000小时的生产时间，再加上3000个加班时间。

问题是python / pulp给我的结果（除一个例外）是LpVariables中插入的所有上限值：不是最小化的成本！

from pulp import *


# Define the LP problem: minimize costs
prob = LpProblem("Minimize costs of production and inventory", LpMinimize)


# Demand schemes
demand_laptops = [75, 125, 1000, 1500, 1000, 500, 1250, 1500, 1000, 500, 500, 400, 300]           # Demand laptops
demand_phones = [120, 2000, 2000, 2000, 2000, 2000, 2000, 2000, 2000, 2000, 2000, 2000, 2000]    # Demand phones
demand_tablets = [50, 2000, 3000, 2000, 3000, 4000, 5000, 2000, 3000, 4000, 5000, 4000, 5000]     # Demand tablets


# Defining variables: normal production hours and overtime production hours.
production_laptop = {x: LpVariable("Production hours for laptop in month {}".format(x), 0, 20000)
                     for x in range(1, 13)}
production_phone = {x: LpVariable("Production hours for phone in month {}".format(x), 0, 20000)
                    for x in range(1, 13)}
production_tablet = {x: LpVariable("Production hours for tablet in month {}".format(x), 0, 20000)
                     for x in range(1, 13)}

overtime_laptop = {x: LpVariable("Overtime hours for laptop in month {}".format(x), 0, 3000)
                   for x in range(1, 13)}
overtime_phone = {x: LpVariable("Overtime hours for phone in month {}".format(x), 0, 3000)
                  for x in range(1, 13)}
overtime_tablet = {x: LpVariable("Overtime hours for tablet in month {}".format(x), 0, 3000)
                   for x in range(1, 13)}


# defining a list of names for the inventory of products
inventory_laptops = {x: "inventory of laptops in month {}".format(x)
                     for x in range(1, 13)}
inventory_phones = {x: "inventory of phones in month {}".format(x)
                    for x in range(1, 13)}
inventory_tables = {x: "inventory of tablets in month {}".format(x)
                    for x in range(1, 13)}


# Inventory (to be minimized)
inventory_laptops[1] = demand_laptops[0] + (1 / 5) * production_laptop[1] + (1 / 5) * overtime_laptop[1] - demand_laptops[1]
inventory_laptops[2] = inventory_laptops[1] + (1 / 5) * production_laptop[2] + (1 / 5) * overtime_laptop[2] - demand_laptops[2]
inventory_laptops[3] = inventory_laptops[2] + (1 / 5) * production_laptop[3] + (1 / 5) * overtime_laptop[3] - demand_laptops[3]
inventory_laptops[4] = inventory_laptops[3] + (1 / 5) * production_laptop[4] + (1 / 5) * overtime_laptop[4] - demand_laptops[4]
inventory_laptops[5] = inventory_laptops[4] + (1 / 5) * production_laptop[5] + (1 / 5) * overtime_laptop[5] - demand_laptops[5]
inventory_laptops[6] = inventory_laptops[5] + (1 / 5) * production_laptop[6] + (1 / 5) * overtime_laptop[6] - demand_laptops[6]
inventory_laptops[7] = inventory_laptops[6] + (1 / 5) * production_laptop[7] + (1 / 5) * overtime_laptop[7] - demand_laptops[7]
inventory_laptops[8] = inventory_laptops[7] + (1 / 5) * production_laptop[8] + (1 / 5) * overtime_laptop[8] - demand_laptops[8]
inventory_laptops[9] = inventory_laptops[8] + (1 / 5) * production_laptop[9] + (1 / 5) * overtime_laptop[9] - demand_laptops[9]
inventory_laptops[10] = inventory_laptops[9] + (1 / 5) * production_laptop[10] + (1 / 5) * overtime_laptop[10] - demand_laptops[10]
inventory_laptops[11] = inventory_laptops[10] + (1 / 5) * production_laptop[11] + (1 / 5) * overtime_laptop[11] - demand_laptops[11]
inventory_laptops[12] = inventory_laptops[11] + (1 / 5) * production_laptop[12] + (1 / 5) * overtime_laptop[12] - demand_laptops[12]

inventory_phones[1] = demand_phones[0] + (1 / 2) * production_phone[1] + (1 / 2) * overtime_phone[1] - demand_phones[1]
inventory_phones[2] = inventory_phones[1] + (1 / 2) * production_phone[2] + (1 / 2) * overtime_phone[2] - demand_phones[2]
inventory_phones[3] = inventory_phones[2] + (1 / 2) * production_phone[3] + (1 / 2) * overtime_phone[3] - demand_phones[3]
inventory_phones[4] = inventory_phones[3] + (1 / 2) * production_phone[4] + (1 / 2) * overtime_phone[4] - demand_phones[4]
inventory_phones[5] = inventory_phones[4] + (1 / 2) * production_phone[5] + (1 / 2) * overtime_phone[5] - demand_phones[5]
inventory_phones[6] = inventory_phones[5] + (1 / 2) * production_phone[6] + (1 / 2) * overtime_phone[6] - demand_phones[6]
inventory_phones[7] = inventory_phones[6] + (1 / 2) * production_phone[7] + (1 / 2) * overtime_phone[7] - demand_phones[7]
inventory_phones[8] = inventory_phones[7] + (1 / 2) * production_phone[8] + (1 / 2) * overtime_phone[8] - demand_phones[8]
inventory_phones[9] = inventory_phones[8] + (1 / 2) * production_phone[9] + (1 / 2) * overtime_phone[9] - demand_phones[9]
inventory_phones[10] = inventory_phones[9] + (1 / 2) * production_phone[10] + (1 / 2) * overtime_phone[10] - demand_phones[10]
inventory_phones[11] = inventory_phones[10] + (1 / 2) * production_phone[11] + (1 / 2) * overtime_phone[11] - demand_phones[11]
inventory_phones[12] = inventory_phones[11] + (1 / 2) * production_phone[12] + (1 / 2) * overtime_phone[12] - demand_phones[12]

inventory_tables[1] = demand_tablets[0] + (1 / 4) * production_tablet[1] + (1 / 4) * overtime_tablet[1] - demand_tablets[1]
inventory_tables[2] = inventory_tables[1] + (1 / 4) * production_tablet[2] + (1 / 4) * overtime_tablet[2] - demand_tablets[2]
inventory_tables[3] = inventory_tables[2] + (1 / 4) * production_tablet[3] + (1 / 4) * overtime_tablet[3] - demand_tablets[3]
inventory_tables[4] = inventory_tables[3] + (1 / 4) * production_tablet[4] + (1 / 4) * overtime_tablet[4] - demand_tablets[4]
inventory_tables[5] = inventory_tables[4] + (1 / 4) * production_tablet[5] + (1 / 4) * overtime_tablet[5] - demand_tablets[5]
inventory_tables[6] = inventory_tables[5] + (1 / 4) * production_tablet[6] + (1 / 4) * overtime_tablet[6] - demand_tablets[6]
inventory_tables[7] = inventory_tables[6] + (1 / 4) * production_tablet[7] + (1 / 4) * overtime_tablet[7] - demand_tablets[7]
inventory_tables[8] = inventory_tables[7] + (1 / 4) * production_tablet[8] + (1 / 4) * overtime_tablet[8] - demand_tablets[8]
inventory_tables[9] = inventory_tables[8] + (1 / 4) * production_tablet[9] + (1 / 4) * overtime_tablet[9] - demand_tablets[9]
inventory_tables[10] = inventory_tables[9] + (1 / 4) * production_tablet[10] + (1 / 4) * overtime_tablet[10] - demand_tablets[10]
inventory_tables[11] = inventory_tables[10] + (1 / 4) * production_tablet[11] + (1 / 4) * overtime_tablet[11] - demand_tablets[11]
inventory_tables[12] = inventory_tables[11] + (1 / 4) * production_tablet[12] + (1 / 4) * overtime_tablet[12] - demand_tablets[12]


# Constraints to meet demand scheme
prob += demand_laptops[0] + (1 / 5) * production_laptop[1] + (1 / 5) * overtime_laptop[1] >= demand_laptops[1]
prob += inventory_laptops[1] + (1 / 5) * production_laptop[2] + (1 / 5) * overtime_laptop[2] >= demand_laptops[2]
prob += inventory_laptops[2] + (1 / 5) * production_laptop[3] + (1 / 5) * overtime_laptop[3] >= demand_laptops[3]
prob += inventory_laptops[3] + (1 / 5) * production_laptop[4] + (1 / 5) * overtime_laptop[4] >= demand_laptops[4]
prob += inventory_laptops[4] + (1 / 5) * production_laptop[5] + (1 / 5) * overtime_laptop[5] >= demand_laptops[5]
prob += inventory_laptops[5] + (1 / 5) * production_laptop[6] + (1 / 5) * overtime_laptop[6] >= demand_laptops[6]
prob += inventory_laptops[6] + (1 / 5) * production_laptop[7] + (1 / 5) * overtime_laptop[7] >= demand_laptops[7]
prob += inventory_laptops[7] + (1 / 5) * production_laptop[8] + (1 / 5) * overtime_laptop[8] >= demand_laptops[8]
prob += inventory_laptops[8] + (1 / 5) * production_laptop[9] + (1 / 5) * overtime_laptop[9] >= demand_laptops[9]
prob += inventory_laptops[9] + (1 / 5) * production_laptop[10] + (1 / 5) * overtime_laptop[10] >= demand_laptops[10]
prob += inventory_laptops[10] + (1 / 5) * production_laptop[11] + (1 / 5) * overtime_laptop[11] >= demand_laptops[11]
prob += inventory_laptops[11] + (1 / 5) * production_laptop[12] + (1 / 5) * overtime_laptop[12] >= demand_laptops[12]

prob += demand_phones[0] + (1 / 2) * production_phone[1] + (1 / 2) * overtime_phone[1] >= demand_phones[1]
prob += inventory_phones[1] + (1 / 2) * production_phone[2] + (1 / 2) * overtime_phone[2] >= demand_phones[2]
prob += inventory_phones[2] + (1 / 2) * production_phone[3] + (1 / 2) * overtime_phone[3] >= demand_phones[3]
prob += inventory_phones[3] + (1 / 2) * production_phone[4] + (1 / 2) * overtime_phone[4] >= demand_phones[4]
prob += inventory_phones[4] + (1 / 2) * production_phone[5] + (1 / 2) * overtime_phone[5] >= demand_phones[5]
prob += inventory_phones[5] + (1 / 2) * production_phone[6] + (1 / 2) * overtime_phone[6] >= demand_phones[6]
prob += inventory_phones[6] + (1 / 2) * production_phone[7] + (1 / 2) * overtime_phone[7] >= demand_phones[7]
prob += inventory_phones[7] + (1 / 2) * production_phone[8] + (1 / 2) * overtime_phone[8] >= demand_phones[8]
prob += inventory_phones[8] + (1 / 2) * production_phone[9] + (1 / 2) * overtime_phone[9] >= demand_phones[9]
prob += inventory_phones[9] + (1 / 2) * production_phone[10] + (1 / 2) * overtime_phone[10] >= demand_phones[10]
prob += inventory_phones[10] + (1 / 2) * production_phone[11] + (1 / 2) * overtime_phone[11] >= demand_phones[11]
prob += inventory_phones[11] + (1 / 2) * production_phone[12] + (1 / 2) * overtime_phone[12] >= demand_phones[12]


prob += demand_tablets[0] + (1 / 4) * production_tablet[1] + (1 / 4) * overtime_tablet[1] >= demand_tablets[1]
prob += inventory_phones[1] + (1 / 4) * production_tablet[2] + (1 / 4) * overtime_tablet[2] >= demand_tablets[2]
prob += inventory_phones[2] + (1 / 4) * production_tablet[3] + (1 / 4) * overtime_tablet[3] >= demand_tablets[3]
prob += inventory_phones[3] + (1 / 4) * production_tablet[4] + (1 / 4) * overtime_tablet[4] >= demand_tablets[4]
prob += inventory_phones[4] + (1 / 4) * production_tablet[5] + (1 / 4) * overtime_tablet[5] >= demand_tablets[5]
prob += inventory_phones[5] + (1 / 4) * production_tablet[6] + (1 / 4) * overtime_tablet[6] >= demand_tablets[6]
prob += inventory_phones[6] + (1 / 4) * production_tablet[7] + (1 / 4) * overtime_tablet[7] >= demand_tablets[7]
prob += inventory_phones[7] + (1 / 4) * production_tablet[8] + (1 / 4) * overtime_tablet[8] >= demand_tablets[8]
prob += inventory_phones[8] + (1 / 4) * production_tablet[9] + (1 / 4) * overtime_tablet[9] >= demand_tablets[9]
prob += inventory_phones[9] + (1 / 4) * production_tablet[10] + (1 / 4) * overtime_tablet[10] >= demand_tablets[10]
prob += inventory_phones[10] + (1 / 4) * production_tablet[11] + (1 / 4) * overtime_tablet[11] >= demand_tablets[11]
prob += inventory_phones[11] + (1 / 4) * production_tablet[12] + (1 / 4) * overtime_tablet[12] >= demand_tablets[12]


# Objective function: inventory costs and overtime costs (10 per hour)
prob += sum(inventory_laptops) + sum(inventory_phones) + sum(inventory_tables) + (10 * (sum(overtime_laptop) + sum(overtime_phone) + sum(overtime_tablet)))

# Solve the problem
prob.solve()

print("Status:", LpStatus[prob.status])

for v in prob.variables():
    print(v.name, "=", v.varValue)

print("total costs:", value(prob.objective))

这给了我以下结果：

Status: Optimal
Overtime_hours_for_laptop_in_month_1 = 3000.0
Overtime_hours_for_laptop_in_month_10 = 3000.0
Overtime_hours_for_laptop_in_month_11 = 3000.0
Overtime_hours_for_laptop_in_month_12 = 3000.0
Overtime_hours_for_laptop_in_month_2 = 3000.0
Overtime_hours_for_laptop_in_month_3 = 3000.0
Overtime_hours_for_laptop_in_month_4 = 3000.0
Overtime_hours_for_laptop_in_month_5 = 3000.0
Overtime_hours_for_laptop_in_month_6 = 3000.0
Overtime_hours_for_laptop_in_month_7 = 3000.0
Overtime_hours_for_laptop_in_month_8 = 3000.0
Overtime_hours_for_laptop_in_month_9 = 3000.0
Overtime_hours_for_phone_in_month_1 = 3000.0
Overtime_hours_for_phone_in_month_10 = 3000.0
Overtime_hours_for_phone_in_month_11 = 3000.0
Overtime_hours_for_phone_in_month_12 = 3000.0
Overtime_hours_for_phone_in_month_2 = 3000.0
Overtime_hours_for_phone_in_month_3 = 3000.0
Overtime_hours_for_phone_in_month_4 = 3000.0
Overtime_hours_for_phone_in_month_5 = 3000.0
Overtime_hours_for_phone_in_month_6 = 3000.0
Overtime_hours_for_phone_in_month_7 = 3000.0
Overtime_hours_for_phone_in_month_8 = 3000.0
Overtime_hours_for_phone_in_month_9 = 3000.0
Overtime_hours_for_tablet_in_month_1 = 0.0
Overtime_hours_for_tablet_in_month_10 = 3000.0
Overtime_hours_for_tablet_in_month_11 = 3000.0
Overtime_hours_for_tablet_in_month_12 = 3000.0
Overtime_hours_for_tablet_in_month_2 = 3000.0
Overtime_hours_for_tablet_in_month_3 = 3000.0
Overtime_hours_for_tablet_in_month_4 = 3000.0
Overtime_hours_for_tablet_in_month_5 = 3000.0
Overtime_hours_for_tablet_in_month_6 = 3000.0
Overtime_hours_for_tablet_in_month_7 = 3000.0
Overtime_hours_for_tablet_in_month_8 = 3000.0
Overtime_hours_for_tablet_in_month_9 = 3000.0
Production_hours_for_laptop_in_month_1 = 20000.0
Production_hours_for_laptop_in_month_10 = 20000.0
Production_hours_for_laptop_in_month_11 = 20000.0
Production_hours_for_laptop_in_month_12 = 20000.0
Production_hours_for_laptop_in_month_2 = 20000.0
Production_hours_for_laptop_in_month_3 = 20000.0
Production_hours_for_laptop_in_month_4 = 20000.0
Production_hours_for_laptop_in_month_5 = 20000.0
Production_hours_for_laptop_in_month_6 = 20000.0
Production_hours_for_laptop_in_month_7 = 20000.0
Production_hours_for_laptop_in_month_8 = 20000.0
Production_hours_for_laptop_in_month_9 = 20000.0
Production_hours_for_phone_in_month_1 = 20000.0
Production_hours_for_phone_in_month_10 = 20000.0
Production_hours_for_phone_in_month_11 = 20000.0
Production_hours_for_phone_in_month_12 = 20000.0
Production_hours_for_phone_in_month_2 = 20000.0
Production_hours_for_phone_in_month_3 = 20000.0
Production_hours_for_phone_in_month_4 = 20000.0
Production_hours_for_phone_in_month_5 = 20000.0
Production_hours_for_phone_in_month_6 = 20000.0
Production_hours_for_phone_in_month_7 = 20000.0
Production_hours_for_phone_in_month_8 = 20000.0
Production_hours_for_phone_in_month_9 = 20000.0
Production_hours_for_tablet_in_month_1 = 7800.0
Production_hours_for_tablet_in_month_10 = 20000.0
Production_hours_for_tablet_in_month_11 = 20000.0
Production_hours_for_tablet_in_month_12 = 20000.0
Production_hours_for_tablet_in_month_2 = 20000.0
Production_hours_for_tablet_in_month_3 = 20000.0
Production_hours_for_tablet_in_month_4 = 20000.0
Production_hours_for_tablet_in_month_5 = 20000.0
Production_hours_for_tablet_in_month_6 = 20000.0
Production_hours_for_tablet_in_month_7 = 20000.0
Production_hours_for_tablet_in_month_8 = 20000.0
Production_hours_for_tablet_in_month_9 = 20000.0
__dummy = None
total costs: None

有人可以告诉我我做错了吗？

Answer 1

有一些语法错误会导致求解器获得与您期望的模型不同的模型。另外，我还要指出一些句法建议。

怎么了？

首先，将production变量定义为字典，其值为LpVariables。我建议您使用pulp为此工作设计的数据结构LpVariable.dicts。定义如下：

production_laptop = LpVariable.dicts(
    "Production hours for laptop in month ", range(1, 13), 
    lowBound = 0, upBound = 20000)

请注意，我明确地指出了上下限：这是一个好习惯，如果您在几个月（甚至几周）后必须重用代码，这会很有帮助。

稍后，您将定义代表变量名称的字典：inventory_变量。然后，您重新定义这些词典的值以指向变量和约束的组合。我将这些变量定义为先前的变量：

inventory_laptops = LpVariable.dicts(
    "inventory of laptops in month  ", range(1,13), lowBound = 0)

如果变量之间存在关系，则可以在约束中表示它们，因此在变量定义阶段我们不必为此担心。

注意，在定义变量之后，应该定义目标函数。它的定义是不正确的，因为不是sum(inventory_laptops)必须这样做sum(inventory_laptops[i] for i in range(1,13))，否则我们将尝试sum遍历字典，尤其是字典的键。

句法

DRY：您编写了很多重复的代码，可以轻松地放入循环中。尽量不要重复自己，因为代码不必要地冗长，复杂且容易出错。实际上，您的代码中存在错误，这可能是由于您已经拥有粘贴粘贴的代码所致：最后一块约束将手机和平板电脑混合在一起，如下所示：

prob += inventory_phones[1] + (1 / 4) * production_tablet[2] + (1 / 4) * overtime_tablet[2] >= demand_tablets[2]

可能是因为您正在复制粘贴并替换phones为tablets。您可以轻松地将这些行写为

for i in range(1,13):
    prob += (demand_tablets[i - 1] if i == 1 else inventory_tables[i - 1]) + \
        (1 / 4) * production_tablet[i] + (1 / 4) * overtime_tablet[i] >= \
        demand_tablets[i], "demand tablets {}".format(i)

这还具有一个额外的好处，即为您的约束命名，以便您以后可以跟踪它。

评论：尝试使用有用的评论。这些评论描述的是您的意图，而不是您的实际意图。为什么# Demand laptops在调用变量时发表评论demand_laptops？

一致性：这是很大的。使用代码需要花费一些时间，主要是因为变量名不一致：demand_tablets, inventory_tables, overtime_tablet并且production_tablet它们都非常相似且容易混淆。尝试坚持一个更一致的表示法。

行长：尽管不是必需的，但是Python开发人员不会使用任意行长。尝试使用良好的IDE（我使用Pycharm，有时使用Sublime Text），这将引导您使用常规的Python约定（也在命名变量，函数等时使用）。这使代码看起来更像Python。

调试数学优化模型：对于小问题，非常有用的习惯是打印出传递给求解器的公式。这可以帮助您捕获许多错误和问题。例如，很明显，您定义了一个名为的变量_dummy，该变量是偶然完成的。这是通过完成的prob.writeLP("DebugThis.lp")。我还将使用长度较短的变量，以便我可以了解约束和目标函数中的情况。最后，请尽量避免在模型中使用硬数字。库存成本可能是10，你现在有实例，但在未来，这可能会改变（这会在非分配环境中进行更改）。因此，最好定义一个清单成本清单（每个产品和/或期间一个），并仅更新该清单。如果要添加更多产品，这将很有帮助，因为约束和变量将自动生成。

修改后的代码

我在此实现了您的代码的工作版本。为了使其与原始版本保持接近，以便（您自己）更容易理解，我没有执行我建议的所有内容。作为对我下面的内容的进一步改进，请尝试提供产品列表并自动生成产品月对。有几种方法可以做到这一点，也许这个例子会有所帮助。

from pulp import * # I would not import the entire workspace - you might end up shadowing variables names that you need

# Problem formulation starts here
prob = LpProblem("Minimize costs of production and inventory", LpMinimize)

# Problem data: Product Demands
demand_laptops = [75, 125, 1000, 1500, 1000, 500, 1250, 1500, 1000, 500, 500, 400, 300]
demand_phones = [120, 2000, 2000, 2000, 2000, 2000, 2000, 2000, 2000, 2000, 2000, 2000, 2000]   
demand_tablets = [50, 2000, 3000, 2000, 3000, 4000, 5000, 2000, 3000, 4000, 5000, 4000, 5000]    


# Defining variables: normal production, overtime, and inventories
production_laptop = LpVariable.dicts(
    "Production hours for laptop in month ", range(1, 13), 
    lowBound = 0, upBound = 20000)
production_phone = LpVariable.dicts(
    "Production hours for phone in month ", range(1, 13),
    lowBound = 0, upBound = 20000)
production_tablet = LpVariable.dicts(
    "Production hours for tablet in month ", range(1, 13),
    lowBound = 0, upBound = 20000)

overtime_laptop = LpVariable.dicts(
    "Overtime hours for laptop in month ", range(1, 13), 
    lowBound = 0, upBound = 3000)
overtime_phone = LpVariable.dicts(
    "Overtime hours for phone in month ", range(1, 13),
    lowBound = 0, upBound = 3000)
overtime_tablet = LpVariable.dicts(
    "Overtime hours for tablet in month ", range(1, 13),
    lowBound = 0, upBound = 3000)

inventory_laptops = LpVariable.dicts(
    "inventory of laptops in month  ", range(1,13), lowBound = 0)
inventory_phones = LpVariable.dicts(
    "inventory of phones in month  ", range(1,13), lowBound = 0)
inventory_tables = LpVariable.dicts(
    "inventory of tables in month  ", range(1,13), lowBound = 0)

# Objective function: inventory costs and overtime costs
prob += (sum(
    inventory_laptops[i] + inventory_phones[i] + inventory_tables[i] 
        for i in range(1,13)) + \
    10 * sum(overtime_laptop[i] + overtime_phone[i] + overtime_tablet[i] 
        for i in range(1,13)))

# Constraint definition part
for i in range(1,13):

    # Inventory definition for laptops, phones and tablets
    prob += inventory_laptops[i] == (demand_laptops[i - 1] if i == 1  else \
    inventory_laptops[i - 1]) + \
     (1 / 5) * production_laptop[i] + (1 / 5) * overtime_laptop[i] - \
    demand_laptops[i], "inventory_laptops definition {}".format(i)

    prob += inventory_phones[i] == (demand_phones[i - 1] if i == 1  else \
    inventory_phones[i - 1]) + \
    (1 / 2) * production_phone[i] + (1 / 2) * overtime_phone[i] - \
    demand_phones[i], "inventory_phones definition {}".format(i)

    prob += inventory_tables[i] == (demand_tablets[i - 1] if i == 1  else \
    inventory_tables[i - 1]) + \
    (1 / 4) * production_tablet[i] + (1 / 4) * overtime_tablet[i] - \
    demand_tablets[i], "inventory_tables definition {}".format(i)

    # Demand-covering constraints for laptops, phones, tablets
    prob += (demand_laptops[i - 1] if i == 1 else inventory_laptops[i - 1]) + \
    (1 / 5) * production_laptop[i] + (1 / 5) * overtime_laptop[i] >= \
    demand_laptops[i], "demand laptops {}".format(i)

    prob += (demand_phones[i - 1] if i == 1 else inventory_phones[i - 1]) + \
    (1 / 2) * production_phone[i] + (1 / 2) * overtime_phone[i] >= \
    demand_phones[i], "demand phones {}".format(i)

    prob += (demand_tablets[i - 1] if i == 1 else inventory_tables[i - 1]) + \
    (1 / 4) * production_tablet[i] + (1 / 4) * overtime_tablet[i] >= \
    demand_tablets[i], "demand tablets {}".format(i)

# Solve the problem
prob.solve()

# Take a look at what was solved
prob.writeLP("SO40113557.lp")

print("Status:", LpStatus[prob.status])

for v in prob.variables():
    print(v.name, "=", v.varValue)

print("total costs:", value(prob.objective))

输出（不打印出变量名/值）：

('Status:', 'Optimal')
('total costs:', 0.0)

我希望这有帮助！