我为您的问题添加了另一个答案：
我发布的 CSV 文件实际上是经过修改的，我原来的 CSV 文件还包含每个骑手的列表以及他们在每个阶段的得分。这个列表看起来像这样[0, 40, 13, 0, 2, 55, 1, 17, 0, 14]。我正在努力寻找整体表现最好的团队。所以我有一个由 16 名自行车手组成的池子，其中 10 名自行车手的分数计入每天的分数。然后将每天的分数相加得到总分。目的是让这个最终的总分尽可能高。
如果您认为我应该编辑我的第一篇文章，请告诉我，我认为这样更清楚，因为我的第一篇文章非常密集并且回答了最初的问题。
让我们引入一个新变量：
Zik = 1 if cyclist i is selected and is one of the top 10 in your team on day k
您需要添加这些约束以链接变量 Zik 和 Xi（如果未选择骑自行车者 i，即如果 Xi = 0，则变量 Zik 不能 = 1）
For all i, sum(Zik, 1<=k<=n_days) <= n_days * Xi
这些限制条件是每天选择 10 名骑自行车的人：
For all k, sum(Zik, 1<=i<=n_cyclists) <= 10
最后，你的目标可以这样写：
Maximize sum(pik * Xi * Zik, 1<=i<=n_cyclists, 1 <= k <= n_days)
# where pik = nb_points of cyclist i at day k
这是思考的部分。像这样写的目标不是线性的（注意两个变量 X 和 Z 之间的乘积）。幸运的是，这两个二进制文件都有，并且有一个技巧可以将这个公式转换为它的线性形式。
让我们再次引入新变量 Lik (Lik = Xi * Zik) 来线性化目标。
目标现在可以这样写并且是线性的：
Maximize sum(pik * Lik, 1<=i<=n_cyclists, 1 <= k <= n_days)
# where pik = nb_points of cyclist i at day k
我们现在需要添加这些约束以使Lik 等于Xi * Zik：
For all i,k : Xi + Zik - 1 <= Lik
For all i,k : Lik <= 1/2 * (Xi + Zik)
然后瞧。这就是数学的美妙之处，你可以用线性方程来模拟很多东西。我提出了一些高级概念，如果您第一眼看不懂，这很正常。
我在这个file 上模拟了每天的得分。
这是解决新问题的 Python 代码：
import ast
from ortools.linear_solver import pywraplp
import pandas as pd
solver = pywraplp.Solver('cyclist', pywraplp.Solver.CBC_MIXED_INTEGER_PROGRAMMING)
cyclist_df = pd.read_csv('cyclists_day.csv')
cyclist_df['Punten_day'] = cyclist_df['Punten_day'].apply(ast.literal_eval)
# Variables
variables_name = {}
variables_team = {}
variables_name_per_day = {}
variables_linear = {}
for _, row in cyclist_df.iterrows():
variables_name[row['Naam']] = solver.IntVar(0, 1, 'x_{}'.format(row['Naam']))
if row['Ploeg'] not in variables_team:
variables_team[row['Ploeg']] = solver.IntVar(0, solver.infinity(), 'y_{}'.format(row['Ploeg']))
for k in range(10):
variables_name_per_day[(row['Naam'], k)] = solver.IntVar(0, 1, 'z_{}_{}'.format(row['Naam'], k))
variables_linear[(row['Naam'], k)] = solver.IntVar(0, 1, 'l_{}_{}'.format(row['Naam'], k))
# Link cyclist <-> team
for team, var in variables_team.items():
constraint = solver.Constraint(0, solver.infinity())
constraint.SetCoefficient(var, 1)
for cyclist in cyclist_df[cyclist_df.Ploeg == team]['Naam']:
constraint.SetCoefficient(variables_name[cyclist], -1)
# Max 4 cyclist per team
for team, var in variables_team.items():
constraint = solver.Constraint(0, 4)
constraint.SetCoefficient(var, 1)
# Max cyclists
constraint_max_cyclists = solver.Constraint(16, 16)
for cyclist in variables_name.values():
constraint_max_cyclists.SetCoefficient(cyclist, 1)
# Max cost
constraint_max_cost = solver.Constraint(0, 100)
for _, row in cyclist_df.iterrows():
constraint_max_cost.SetCoefficient(variables_name[row['Naam']], row['Waarde'])
# Link Zik and Xi
for name, cyclist in variables_name.items():
constraint_link_cyclist_day = solver.Constraint(-solver.infinity(), 0)
constraint_link_cyclist_day.SetCoefficient(cyclist, - 10)
for k in range(10):
constraint_link_cyclist_day.SetCoefficient(variables_name_per_day[name, k], 1)
# Min/Max 10 cyclists per day
for k in range(10):
constraint_cyclist_per_day = solver.Constraint(10, 10)
for name in cyclist_df.Naam:
constraint_cyclist_per_day.SetCoefficient(variables_name_per_day[name, k], 1)
# Linearization constraints
for name, cyclist in variables_name.items():
for k in range(10):
constraint_linearization1 = solver.Constraint(-solver.infinity(), 1)
constraint_linearization2 = solver.Constraint(-solver.infinity(), 0)
constraint_linearization1.SetCoefficient(cyclist, 1)
constraint_linearization1.SetCoefficient(variables_name_per_day[name, k], 1)
constraint_linearization1.SetCoefficient(variables_linear[name, k], -1)
constraint_linearization2.SetCoefficient(cyclist, -1/2)
constraint_linearization2.SetCoefficient(variables_name_per_day[name, k], -1/2)
constraint_linearization2.SetCoefficient(variables_linear[name, k], 1)
# Objective
objective = solver.Objective()
objective.SetMaximization()
for _, row in cyclist_df.iterrows():
for k in range(10):
objective.SetCoefficient(variables_linear[row['Naam'], k], row['Punten_day'][k])
solver.Solve()
chosen_cyclists = [key for key, variable in variables_name.items() if variable.solution_value() > 0.5]
print('\n'.join(chosen_cyclists))
for k in range(10):
print('\nDay {} :'.format(k + 1))
chosen_cyclists_day = [name for (name, day), variable in variables_name_per_day.items()
if (day == k and variable.solution_value() > 0.5)]
assert len(chosen_cyclists_day) == 10
assert all(chosen_cyclists_day[i] in chosen_cyclists for i in range(10))
print('\n'.join(chosen_cyclists_day))
结果如下：
你的团队：
SAGAN Peter
GROENEWEGEN Dylan
VIVIANI Elia
ALAPHILIPPE Julian
PINOT Thibaut
MATTHEWS Michael
TRENTIN Matteo
COLBRELLI Sonny
VAN AVERMAET Greg
STUYVEN Jasper
BENOOT Tiesj
CICCONE Giulio
TEUNISSEN Mike
HERRADA Jesús
MEURISSE Xandro
GRELLIER Fabien
每天选择的骑行者
Day 1 :
SAGAN Peter
VIVIANI Elia
ALAPHILIPPE Julian
MATTHEWS Michael
COLBRELLI Sonny
VAN AVERMAET Greg
STUYVEN Jasper
CICCONE Giulio
TEUNISSEN Mike
HERRADA Jesús
Day 2 :
SAGAN Peter
ALAPHILIPPE Julian
MATTHEWS Michael
TRENTIN Matteo
COLBRELLI Sonny
VAN AVERMAET Greg
STUYVEN Jasper
TEUNISSEN Mike
NIZZOLO Giacomo
MEURISSE Xandro
Day 3 :
SAGAN Peter
GROENEWEGEN Dylan
VIVIANI Elia
MATTHEWS Michael
TRENTIN Matteo
VAN AVERMAET Greg
STUYVEN Jasper
CICCONE Giulio
TEUNISSEN Mike
HERRADA Jesús
Day 4 :
SAGAN Peter
VIVIANI Elia
PINOT Thibaut
MATTHEWS Michael
TRENTIN Matteo
COLBRELLI Sonny
VAN AVERMAET Greg
STUYVEN Jasper
TEUNISSEN Mike
HERRADA Jesús
Day 5 :
SAGAN Peter
VIVIANI Elia
ALAPHILIPPE Julian
PINOT Thibaut
MATTHEWS Michael
TRENTIN Matteo
COLBRELLI Sonny
VAN AVERMAET Greg
CICCONE Giulio
HERRADA Jesús
Day 6 :
SAGAN Peter
GROENEWEGEN Dylan
VIVIANI Elia
ALAPHILIPPE Julian
MATTHEWS Michael
TRENTIN Matteo
COLBRELLI Sonny
STUYVEN Jasper
CICCONE Giulio
TEUNISSEN Mike
Day 7 :
SAGAN Peter
VIVIANI Elia
ALAPHILIPPE Julian
MATTHEWS Michael
COLBRELLI Sonny
VAN AVERMAET Greg
STUYVEN Jasper
TEUNISSEN Mike
HERRADA Jesús
MEURISSE Xandro
Day 8 :
SAGAN Peter
GROENEWEGEN Dylan
VIVIANI Elia
ALAPHILIPPE Julian
MATTHEWS Michael
STUYVEN Jasper
TEUNISSEN Mike
HERRADA Jesús
NIZZOLO Giacomo
MEURISSE Xandro
Day 9 :
SAGAN Peter
GROENEWEGEN Dylan
VIVIANI Elia
ALAPHILIPPE Julian
PINOT Thibaut
TRENTIN Matteo
COLBRELLI Sonny
VAN AVERMAET Greg
TEUNISSEN Mike
HERRADA Jesús
Day 10 :
SAGAN Peter
GROENEWEGEN Dylan
VIVIANI Elia
PINOT Thibaut
COLBRELLI Sonny
STUYVEN Jasper
CICCONE Giulio
TEUNISSEN Mike
HERRADA Jesús
NIZZOLO Giacomo
我们对比一下答案1和答案2的结果print(solver.Objective().Value())：
第一个模型获得3738.0，第二个模型获得3129.087388325567。该值较低，因为您在每个阶段只选择了 10 名骑自行车的人，而不是 16 名。
现在如果保留第一个解决方案并使用新的评分方法，我们得到3122.9477585307413
我们可以认为第一个模型已经足够好：我们不必引入新的变量/约束，模型保持简单，我们得到的解决方案几乎与复杂模型一样好。有时不需要 100% 准确，通过一些近似值可以更轻松、更快速地求解模型。