JuMP混合整数线性优化实例
Contents：优化
Contributor: YJY
Email:522432938@qq.com
如有错误，请批评指正。
问题简介
Advent Of Code中有一个很有意思的问题，可用JuMP来求解。同时也是能源系统优化的一个缩影。
问题如下：
为了收获粘土，你需要专用的粘土收集机器人。要制造任何类型的机器人，你都需要矿石。收集矿石需要带大钻头的矿石收集机器人。幸运的是，你的背包中正好有一个矿石收集机器人，你可以使用它启动整个操作。
每个机器人每分钟可以收集1个其资源类型。机器人工厂（也在你的背包中）构建任何类型的机器人也需要一分钟，尽管它在构建开始时会消耗必要的可用资源，同时每分钟最多建造1个机器人。
机器人工厂有很多蓝图（问题的输入）你可以选择，但一旦你用蓝图配置好了，你就不能改变它。
蓝图1：
制造一个矿石机器人（ore robot）消耗4矿石（ore）。
制造一个粘土机器人（clay robot）消耗2矿石（ore）。
制造一个黑曜石机器人（obsidian robot）消耗3块矿石（ore）和14块粘土（clay）。
制造一个紫晶机器人（geode robot）消耗2个矿石（ore）和7个黑曜石（obsidian）。
蓝图2：
制造一个矿石机器人消耗2矿石。
制造一个粘土机器人消耗3矿石。
制造一个黑曜石机器人消耗3块矿石和8块粘土。
制造一个紫晶机器人消耗3个矿石和12个黑曜石。优化问题为：如何分配资源去建造机器人，在24分钟后获得最多的紫晶。
蓝图1的最优方案为最多获得9个紫晶，具体操作过程为：
==第1分钟==
1台矿石收集机器人收集1个矿石；你现在有1个矿石。
==第2分钟==
1台矿石收集机器人收集1个矿石；你现在有2个矿石。
==第3分钟==
花2矿石开始建造一个粘土收集机器人。
1台矿石收集机器人收集1个矿石；你现在有1个矿石。
新的粘土收集机器人准备就绪；你现在有一个了。
==第4分钟==
1台矿石收集机器人收集1个矿石；你现在有2个矿石。
1个粘土收集机器人收集1个粘土；你现在有1个粘土。
==第5分钟==
花2矿石开始建造一个粘土收集机器人。
1台矿石收集机器人收集1个矿石；你现在有1个矿石。
1个粘土收集机器人收集1个粘土；你现在有2个粘土。
新的粘土收集机器人准备就绪；你现在有两个了。
==第6分钟==
1台矿石收集机器人收集1个矿石；你现在有2个矿石。
2个粘土收集机器人收集2个粘土；你现在有4块粘土。
==第7分钟==
花2矿石开始建造一个粘土收集机器人。
1台矿石收集机器人收集1个矿石；你现在有1个矿石。
2个粘土收集机器人收集2个粘土；你现在有6块粘土。
新的粘土收集机器人准备就绪；你现在有3个了。
==第8分钟==
1台矿石收集机器人收集1个矿石；你现在有2个矿石。
3个粘土收集机器人收集3个粘土；你现在有9块粘土。
==第9分钟==
1台矿石收集机器人收集1个矿石；你现在有3个矿石。
3个粘土收集机器人收集3个粘土；你现在有12块粘土。
==第10分钟==
1台矿石收集机器人收集1个矿石；你现在有4个矿石。
3个粘土收集机器人收集3个粘土；你现在有15块粘土。
==分钟11==
花3块矿石和14块粘土开始建造一个黑曜石收集机器人。
1台矿石收集机器人收集1个矿石；你现在有2个矿石。
3个粘土收集机器人收集3个粘土；你现在有4块粘土。
新的黑曜石收集机器人准备就绪；你现在有一个了。
==第12分钟==
花2矿石开始建造一个粘土收集机器人。
1台矿石收集机器人收集1个矿石；你现在有1个矿石。
3个粘土收集机器人收集3个粘土；你现在有7块粘土。
1个黑曜石收集机器人收集1颗黑曜岩；你现在有1个黑曜石。
新的粘土收集机器人准备就绪；你现在有4个。
==第13分钟==
1台矿石收集机器人收集1个矿石；你现在有2个矿石。
4个粘土收集机器人收集4个粘土；你现在有11块粘土。
1个黑曜石收集机器人收集1颗黑曜岩；你现在有2颗黑曜石。
==第14分钟==
1台矿石收集机器人收集1个矿石；你现在有3个矿石。
4个粘土收集机器人收集4个粘土；你现在有15块粘土。
1个黑曜石收集机器人收集1颗黑曜岩；你现在有3个黑曜石。
==第15分钟==
花3块矿石和14块粘土开始建造一个黑曜石收集机器人。
1台矿石收集机器人收集1个矿石；你现在有1个矿石。
4个粘土收集机器人收集4个粘土；你现在有5块粘土。
1个黑曜石收集机器人收集1颗黑曜岩；你现在有4颗黑曜石。
新的黑曜石收集机器人准备就绪；你现在有两个了。
==第16分钟==
1台矿石收集机器人收集1个矿石；你现在有2个矿石。
4个粘土收集机器人收集4个粘土；你现在有9块粘土。
2个黑曜石收集机器人收集2个黑曜石；你现在有6颗黑曜石。
==第17分钟==
1台矿石收集机器人收集1个矿石；你现在有3个矿石。
4个粘土收集机器人收集4个粘土；你现在有13块粘土。
2个黑曜石收集机器人收集2个黑曜石；你现在有8颗黑曜石。
==第18分钟==
花2颗矿石和7颗黑曜石开始建造一个紫晶机器人。
1台矿石收集机器人收集1个矿石；你现在有2个矿石。
4个粘土收集机器人收集4个粘土；你现在有17块粘土。
2个黑曜石收集机器人收集2个黑曜石；你现在有3个黑曜石。
新的紫晶机器人准备就绪；你现在有一个了。
==第19分钟==
1台矿石收集机器人收集1个矿石；你现在有3个矿石。
4个粘土收集机器人收集4个粘土；你现在有21块粘土。
2个黑曜石收集机器人收集2个黑曜石；你现在有5颗黑曜石。
1个紫晶机器人收集1个紫晶；你现在有一个紫晶。
==第20分钟==
1台矿石收集机器人收集1个矿石；你现在有4个矿石。
4个粘土收集机器人收集4个粘土；你现在有25块粘土。
2个黑曜石收集机器人收集2个黑曜石；你现在有7颗黑曜石。
1个紫晶机器人收集1个紫晶；你现在有2个紫晶。
==第21分钟==
花2颗矿石和7颗黑曜石开始建造一个紫晶机器人。
1台矿石收集机器人收集1个矿石；你现在有3个矿石。
4个粘土收集机器人收集4个粘土；你现在有29块粘土。
2个黑曜石收集机器人收集2个黑曜石；你现在有2颗黑曜石。
1个紫晶开裂机器人开裂1个大地洞；你现在有3个紫晶。
新的紫晶机器人准备就绪；你现在有两个了。
==第22分钟==
1台矿石收集机器人收集1个矿石；你现在有4个矿石。
4个粘土收集机器人收集4个粘土；你现在有33块粘土。
2个黑曜石收集机器人收集2个黑曜石；你现在有4颗黑曜石。
2个紫晶机器人收集2个紫晶；你现在有5个紫晶。
==第23分钟==
1台矿石收集机器人收集1个矿石；你现在有5个矿石。
4个粘土收集机器人收集4个粘土；你现在有37块粘土。
2个黑曜石收集机器人收集2个黑曜石；你现在有6颗黑曜石。
2个紫晶机器人收集2个紫晶；你现在有7个紫晶。
==第24分钟==
1台矿石收集机器人收集1个矿石；你现在有6个矿石。
4个粘土收集机器人收集4个粘土；你现在有41块粘土。
2个黑曜石收集机器人收集2个黑曜石；你现在有8颗黑曜石。
2个紫晶机器人收集2个紫晶；你现在有9个紫晶。蓝图2最多获得12个紫晶。
对于以下蓝图，分别能获得的最多紫晶是多少？
Blueprint 1: Each ore robot costs 4 ore. Each clay robot costs 4 ore. Each obsidian robot costs 4 ore and 17 clay. Each geode robot costs 4 ore and 20 obsidian.
Blueprint 2: Each ore robot costs 3 ore. Each clay robot costs 4 ore. Each obsidian robot costs 3 ore and 17 clay. Each geode robot costs 3 ore and 8 obsidian.
Blueprint 3: Each ore robot costs 4 ore. Each clay robot costs 4 ore. Each obsidian robot costs 2 ore and 7 clay. Each geode robot costs 4 ore and 13 obsidian.
Blueprint 4: Each ore robot costs 4 ore. Each clay robot costs 4 ore. Each obsidian robot costs 2 ore and 10 clay. Each geode robot costs 3 ore and 14 obsidian.
Blueprint 5: Each ore robot costs 4 ore. Each clay robot costs 3 ore. Each obsidian robot costs 2 ore and 17 clay. Each geode robot costs 3 ore and 16 obsidian.
Blueprint 6: Each ore robot costs 4 ore. Each clay robot costs 4 ore. Each obsidian robot costs 4 ore and 16 clay. Each geode robot costs 2 ore and 15 obsidian.
Blueprint 7: Each ore robot costs 2 ore. Each clay robot costs 4 ore. Each obsidian robot costs 4 ore and 15 clay. Each geode robot costs 2 ore and 15 obsidian.
Blueprint 8: Each ore robot costs 2 ore. Each clay robot costs 4 ore. Each obsidian robot costs 4 ore and 19 clay. Each geode robot costs 2 ore and 18 obsidian.
Blueprint 9: Each ore robot costs 4 ore. Each clay robot costs 4 ore. Each obsidian robot costs 4 ore and 7 clay. Each geode robot costs 2 ore and 19 obsidian.
Blueprint 10: Each ore robot costs 3 ore. Each clay robot costs 4 ore. Each obsidian robot costs 4 ore and 6 clay. Each geode robot costs 3 ore and 16 obsidian.
Blueprint 11: Each ore robot costs 4 ore. Each clay robot costs 4 ore. Each obsidian robot costs 4 ore and 8 clay. Each geode robot costs 3 ore and 19 obsidian.
Blueprint 12: Each ore robot costs 3 ore. Each clay robot costs 4 ore. Each obsidian robot costs 2 ore and 19 clay. Each geode robot costs 2 ore and 12 obsidian.
Blueprint 13: Each ore robot costs 4 ore. Each clay robot costs 3 ore. Each obsidian robot costs 3 ore and 14 clay. Each geode robot costs 4 ore and 17 obsidian.
Blueprint 14: Each ore robot costs 2 ore. Each clay robot costs 2 ore. Each obsidian robot costs 2 ore and 20 clay. Each geode robot costs 2 ore and 14 obsidian.
Blueprint 15: Each ore robot costs 2 ore. Each clay robot costs 2 ore. Each obsidian robot costs 2 ore and 10 clay. Each geode robot costs 2 ore and 11 obsidian.
Blueprint 16: Each ore robot costs 2 ore. Each clay robot costs 4 ore. Each obsidian robot costs 4 ore and 13 clay. Each geode robot costs 3 ore and 11 obsidian.
Blueprint 17: Each ore robot costs 4 ore. Each clay robot costs 3 ore. Each obsidian robot costs 2 ore and 19 clay. Each geode robot costs 3 ore and 10 obsidian.
Blueprint 18: Each ore robot costs 2 ore. Each clay robot costs 4 ore. Each obsidian robot costs 2 ore and 20 clay. Each geode robot costs 2 ore and 17 obsidian.
Blueprint 19: Each ore robot costs 4 ore. Each clay robot costs 4 ore. Each obsidian robot costs 4 ore and 11 clay. Each geode robot costs 4 ore and 12 obsidian.
Blueprint 20: Each ore robot costs 4 ore. Each clay robot costs 4 ore. Each obsidian robot costs 2 ore and 7 clay. Each geode robot costs 3 ore and 10 obsidian.
Blueprint 21: Each ore robot costs 3 ore. Each clay robot costs 4 ore. Each obsidian robot costs 4 ore and 13 clay. Each geode robot costs 3 ore and 7 obsidian.
Blueprint 22: Each ore robot costs 2 ore. Each clay robot costs 2 ore. Each obsidian robot costs 2 ore and 15 clay. Each geode robot costs 2 ore and 7 obsidian.
Blueprint 23: Each ore robot costs 3 ore. Each clay robot costs 3 ore. Each obsidian robot costs 2 ore and 20 clay. Each geode robot costs 3 ore and 18 obsidian.
Blueprint 24: Each ore robot costs 4 ore. Each clay robot costs 3 ore. Each obsidian robot costs 3 ore and 18 clay. Each geode robot costs 4 ore and 8 obsidian.
Blueprint 25: Each ore robot costs 4 ore. Each clay robot costs 4 ore. Each obsidian robot costs 3 ore and 14 clay. Each geode robot costs 4 ore and 15 obsidian.
Blueprint 26: Each ore robot costs 4 ore. Each clay robot costs 3 ore. Each obsidian robot costs 2 ore and 20 clay. Each geode robot costs 3 ore and 9 obsidian.
Blueprint 27: Each ore robot costs 4 ore. Each clay robot costs 4 ore. Each obsidian robot costs 4 ore and 5 clay. Each geode robot costs 3 ore and 7 obsidian.
Blueprint 28: Each ore robot costs 3 ore. Each clay robot costs 3 ore. Each obsidian robot costs 3 ore and 11 clay. Each geode robot costs 2 ore and 8 obsidian.
Blueprint 29: Each ore robot costs 4 ore. Each clay robot costs 4 ore. Each obsidian robot costs 2 ore and 12 clay. Each geode robot costs 3 ore and 15 obsidian.
Blueprint 30: Each ore robot costs 4 ore. Each clay robot costs 3 ore. Each obsidian robot costs 3 ore and 10 clay. Each geode robot costs 3 ore and 10 obsidian.优化问题的数学表达
因为涉及到时间域上的状态改变（本质就是离散），所以需要在求解时间域上的每个时间点上设置一个变量。同时，每个不同种类的矿石也需要设置变量。
变量设置：
- 每个种类每个时间点矿石数量，整数
- 每个种类每个时间点机器人数量，整数
- 每个种类每个时间点是否建造，0或1
所以有：
\[ robots_{i,j} \in N\\ isBuild_{i,j} \in \{0,1\}\\ obtains_{i,j} \in N\\ i \in \{ore, clay, obsidian, geode\}\\ j \in \{1,2,3...,23,24\} \]
优化目标为第24分钟，紫晶最多：
\[ \max obtains_{geode,24} \]
约束：
- 矿石量等于上一周期的矿石量加上本周期的产出减去本周期的消耗。
costs的每一行是建造不同种机器人消耗的材料个数。
\[costs = \begin{bmatrix}[4, 3, 2, 3] \\ [0, 0, 17, 0] \\ [0, 0, 0, 16] \\ [0, 0, 0, 0]\end{bmatrix}\]
例如，第一行为建造4中不同的机器人，分别要消耗4，3，2，3个ore；第二行为建造4中不同的机器人,分别要消耗0，0，17，0个clay：
\[obtains_{i,j} = obtains_{i,j-1}+robots_{i,j} - \sum_{k}^{} costs_{i,k} * isBulid_{k,j}\]
- 上一个周期结束，矿石足够才能在本周期建造机器人
\[obtains_{i,j-1} \geqslant \sum_{k}^{} costs_{i,k} * isBulid_{k,j}\]
- 建造机器人，数量增加
\[robots_{i,j} = robots_{i,j-1} + isBulid_{i,j-1}\]
- 一次只能建造一个机器人
\[\sum_{k}^{} isBulid_{k,j} \leqslant 1\]
- 初值条件，没有材料且只有一台矿石机器人：
\[isBulid_{i,1} = 0, i \in \{ore, clay, obsidian, geode\}\\ obtain_{i,1} = 0, i \in \{ clay, obsidian, geode\}\\ robots_{i,1} = 0, i \in \{ clay, obsidian, geode\}\\ obtain_{ore,1} = 1 \\ robots_{ore,1} = 1 \\\]
JuMP求解代码
using JuMP
import HiGHS
inputs = """Blueprint 1: Each ore robot costs 4 ore. Each clay robot costs 4 ore. Each obsidian robot costs 4 ore and 17 clay. Each geode robot costs 4 ore and 20 obsidian.
Blueprint 2: Each ore robot costs 3 ore. Each clay robot costs 4 ore. Each obsidian robot costs 3 ore and 17 clay. Each geode robot costs 3 ore and 8 obsidian.
Blueprint 3: Each ore robot costs 4 ore. Each clay robot costs 4 ore. Each obsidian robot costs 2 ore and 7 clay. Each geode robot costs 4 ore and 13 obsidian.
Blueprint 4: Each ore robot costs 4 ore. Each clay robot costs 4 ore. Each obsidian robot costs 2 ore and 10 clay. Each geode robot costs 3 ore and 14 obsidian.
Blueprint 5: Each ore robot costs 4 ore. Each clay robot costs 3 ore. Each obsidian robot costs 2 ore and 17 clay. Each geode robot costs 3 ore and 16 obsidian.
Blueprint 6: Each ore robot costs 4 ore. Each clay robot costs 4 ore. Each obsidian robot costs 4 ore and 16 clay. Each geode robot costs 2 ore and 15 obsidian.
Blueprint 7: Each ore robot costs 2 ore. Each clay robot costs 4 ore. Each obsidian robot costs 4 ore and 15 clay. Each geode robot costs 2 ore and 15 obsidian.
Blueprint 8: Each ore robot costs 2 ore. Each clay robot costs 4 ore. Each obsidian robot costs 4 ore and 19 clay. Each geode robot costs 2 ore and 18 obsidian.
Blueprint 9: Each ore robot costs 4 ore. Each clay robot costs 4 ore. Each obsidian robot costs 4 ore and 7 clay. Each geode robot costs 2 ore and 19 obsidian.
Blueprint 10: Each ore robot costs 3 ore. Each clay robot costs 4 ore. Each obsidian robot costs 4 ore and 6 clay. Each geode robot costs 3 ore and 16 obsidian.
Blueprint 11: Each ore robot costs 4 ore. Each clay robot costs 4 ore. Each obsidian robot costs 4 ore and 8 clay. Each geode robot costs 3 ore and 19 obsidian.
Blueprint 12: Each ore robot costs 3 ore. Each clay robot costs 4 ore. Each obsidian robot costs 2 ore and 19 clay. Each geode robot costs 2 ore and 12 obsidian.
Blueprint 13: Each ore robot costs 4 ore. Each clay robot costs 3 ore. Each obsidian robot costs 3 ore and 14 clay. Each geode robot costs 4 ore and 17 obsidian.
Blueprint 14: Each ore robot costs 2 ore. Each clay robot costs 2 ore. Each obsidian robot costs 2 ore and 20 clay. Each geode robot costs 2 ore and 14 obsidian.
Blueprint 15: Each ore robot costs 2 ore. Each clay robot costs 2 ore. Each obsidian robot costs 2 ore and 10 clay. Each geode robot costs 2 ore and 11 obsidian.
Blueprint 16: Each ore robot costs 2 ore. Each clay robot costs 4 ore. Each obsidian robot costs 4 ore and 13 clay. Each geode robot costs 3 ore and 11 obsidian.
Blueprint 17: Each ore robot costs 4 ore. Each clay robot costs 3 ore. Each obsidian robot costs 2 ore and 19 clay. Each geode robot costs 3 ore and 10 obsidian.
Blueprint 18: Each ore robot costs 2 ore. Each clay robot costs 4 ore. Each obsidian robot costs 2 ore and 20 clay. Each geode robot costs 2 ore and 17 obsidian.
Blueprint 19: Each ore robot costs 4 ore. Each clay robot costs 4 ore. Each obsidian robot costs 4 ore and 11 clay. Each geode robot costs 4 ore and 12 obsidian.
Blueprint 20: Each ore robot costs 4 ore. Each clay robot costs 4 ore. Each obsidian robot costs 2 ore and 7 clay. Each geode robot costs 3 ore and 10 obsidian.
Blueprint 21: Each ore robot costs 3 ore. Each clay robot costs 4 ore. Each obsidian robot costs 4 ore and 13 clay. Each geode robot costs 3 ore and 7 obsidian.
Blueprint 22: Each ore robot costs 2 ore. Each clay robot costs 2 ore. Each obsidian robot costs 2 ore and 15 clay. Each geode robot costs 2 ore and 7 obsidian.
Blueprint 23: Each ore robot costs 3 ore. Each clay robot costs 3 ore. Each obsidian robot costs 2 ore and 20 clay. Each geode robot costs 3 ore and 18 obsidian.
Blueprint 24: Each ore robot costs 4 ore. Each clay robot costs 3 ore. Each obsidian robot costs 3 ore and 18 clay. Each geode robot costs 4 ore and 8 obsidian.
Blueprint 25: Each ore robot costs 4 ore. Each clay robot costs 4 ore. Each obsidian robot costs 3 ore and 14 clay. Each geode robot costs 4 ore and 15 obsidian.
Blueprint 26: Each ore robot costs 4 ore. Each clay robot costs 3 ore. Each obsidian robot costs 2 ore and 20 clay. Each geode robot costs 3 ore and 9 obsidian.
Blueprint 27: Each ore robot costs 4 ore. Each clay robot costs 4 ore. Each obsidian robot costs 4 ore and 5 clay. Each geode robot costs 3 ore and 7 obsidian.
Blueprint 28: Each ore robot costs 3 ore. Each clay robot costs 3 ore. Each obsidian robot costs 3 ore and 11 clay. Each geode robot costs 2 ore and 8 obsidian.
Blueprint 29: Each ore robot costs 4 ore. Each clay robot costs 4 ore. Each obsidian robot costs 2 ore and 12 clay. Each geode robot costs 3 ore and 15 obsidian.
Blueprint 30: Each ore robot costs 4 ore. Each clay robot costs 3 ore. Each obsidian robot costs 3 ore and 10 clay. Each geode robot costs 3 ore and 10 obsidian.
"""
inputs = IOBuffer(inputs)
function readData(path=inputs)
rawData = readlines(path)
rawData = split.(rawData, c -> c == '.' || c == ':')
regs = [
r"Blueprint (\d+)",
r"costs (\d+) ore",
r"costs (\d+) ore",
r"costs (\d+) ore and (\d+) clay",
r"costs (\d+) ore and (\d+) obsidian",
]
d = Dict{Int,Vector{Vector{Int64}}}()
for line in rawData |> eachindex
c = map(x -> zeros(Int64, 4), 1:4)
r = Vector{Int64}[]
for i in 1:5
m = match(regs[i], rawData[line][i]).captures
m = map(x -> parse(Int64, x), m)
push!(r, m)
end
c[1][1] = r[2][1]
c[1][2] = r[3][1]
c[1][3] = r[4][1]
c[1][4] = r[5][1]
c[2][3] = r[4][2]
c[3][4] = r[5][2]
d[r[1][1]] = c
end
return d
end
function solve_1(costs, periods)
model = Model(HiGHS.Optimizer)
set_silent(model)
names = ["ore", "clay", "obsidian", "geode"]
# robots为每个机器人的数量，obtains为每个机器人每个周期的产出，isBuild为每个机器人每个周期是否建造
@variable(model, robots[names, periods], Int)
@variable(model, obtains[names, periods], Int)
@variable(model, isBuild[names, periods], Bin)
# 矿石量等于上一周期的矿石量加上本周期的产出减去本周期的消耗
for (p1, p2) ∈ zip(periods[1:end-1], periods[2:end])
@constraint(model, [ind = 1:4], obtains[names[ind], p2] == obtains[names[ind], p1] + robots[names[ind], p2] - sum(costs[ind] .* isBuild[:, p2]))
end
# 矿石足够才能建造机器人
for (p1, p2) ∈ zip(periods[1:end-1], periods[2:end])
@constraint(model, [ind = 1:4], obtains[names[ind], p1] >= sum(costs[ind] .* isBuild[:, p2]))
end
# 建造机器人
for (p1, p2) ∈ zip(periods[1:end-1], periods[2:end])
@constraint(model, [ind = 1:4], robots[names[ind], p2] == robots[names[ind], p1] + isBuild[names[ind], p1])
end
# 一次只能建造一个机器人
@constraint(model, [i = periods], sum(isBuild[:, i]) <= 1)
# 初始条件
@constraint(model, [ind = 2:4], robots[names[ind], 1] == 0)
@constraint(model, [ind = 1:1], robots[names[ind], 1] == 1)
@constraint(model, [ind = 2:4], obtains[names[ind], 1] == 0)
@constraint(model, [ind = 1:1], obtains[names[ind], 1] == 1)
@constraint(model, [ind = 1:4], isBuild[names[ind], 1] == 0)
# 目标函数
@objective(model, Max, obtains["geode", lastindex(periods)])
optimize!(model)
return objective_value(model) |> Int
end
function solve_P1()
d = readData()
s = String[]
for (i, c) in d
res = solve_1(c, 1:24)
push!(s,"第$(i)个结果:"*string(res))
end
return s
end
solve_P1()30-element Vector{String}:
"第5个结果:0"
"第16个结果:5"
"第20个结果:5"
"第12个结果:0"
"第24个结果:2"
"第28个结果:8"
"第8个结果:0"
"第17个结果:1"
"第30个结果:5"
"第1个结果:0"
⋮
"第4个结果:1"
"第13个结果:0"
"第15个结果:14"
"第2个结果:2"
"第10个结果:5"
"第18个结果:1"
"第21个结果:4"
"第26个结果:1"
"第27个结果:9"小结
这个问题重要的特点为：
- 存在时域上的状态转移（能源系统中负荷的变化等等），以及如何用变量之间的关联体现状态转移的关系
- 包含整数与布尔变量（设备启停、满足最小需求的设备数量等等）