1.前言
如前文所说，约束规划（CP）指求解满足各项约束的可行解的问题。与线性规划、整数规划不同，约束规划更加关注可行解，没有明确的优化目标。典型的场景包括员工排班问题、N皇后问题。CP问题虽然没有目标函数，但可以通过目标添加到约束的方式缩小到更易于管理的子集，变相解决整数规划问题。
ortools提供了 CP-SAT 求解器，其使用方法与MPSolver类似。接下来，我们看看CP-SAT是如何解决CP问题以及MIP问题的。
2.求解CP问题
问题如下：
有变量x, y,z,取值范围均为为 0, 1, 2，
约束条件: x ≠ y，
求满足条件的x,y,z组合。
代码及讲解如下，这里采用硬编码方式。
#引入cp_model，便于后续构建CP-SAT求解器对应模型
from ortools.sat.python import cp_model
#回调类，每得到一个结果均执行on_solution_callback函数
class VarArraySolutionPrinter(cp_model.CpSolverSolutionCallback):
"""Print intermediate solutions."""
def __init__(self, variables):
cp_model.CpSolverSolutionCallback.__init__(self)
self.__variables = variables
self.__solution_count = 0
def on_solution_callback(self):
self.__solution_count += 1
for v in self.__variables:
print('%s=%i' % (v, self.Value(v)), end=' ')
print()
def solution_count(self):
return self.__solution_count
def SearchForAllSolutionsSampleSat():
"""Showcases calling the solver to search for all solutions."""
# 创建模型
model = cp_model.CpModel()
# 创建变量
num_vals = 3
x = model.NewIntVar(0, num_vals - 1, 'x')
y = model.NewIntVar(0, num_vals - 1, 'y')
z = model.NewIntVar(0, num_vals - 1, 'z')
# 创建约束.
model.Add(x != y)
# 创建求解器并求解.
solver = cp_model.CpSolver()
# 定义回调对象
solution_printer = VarArraySolutionPrinter([x, y, z])
# 修改求解器参数：枚举所有结果
solver.parameters.enumerate_all_solutions = True
# 求解过程
status = solver.Solve(model, solution_printer)
print('Status = %s' % solver.StatusName(status))
print('Number of solutions found: %i' % solution_printer.solution_count())
SearchForAllSolutionsSampleSat()3.求解MIP问题
问题如下：
最大化 2x + 2y + 3z ，同时满足以下约束:
x + 7⁄2 y + 3⁄2 z ≤ 25
3x - 5y + 7z ≤ 45
5x + 2y - 6z ≤ 37
x, y, z ≥ 0
x, y, z 为整数
代码及讲解如下。需要注意的是：为了提高求解速度，CP-SAT求解器要求所有约束的元素都为整数。实际应用中遇到浮点数时需要对约束条件进行转换，例如，不等式两边分别乘以一个较大的数。
from ortools.sat.python import cp_model
def main():
model = cp_model.CpModel()
var_upper_bound = max(50, 45, 37)
x = model.NewIntVar(0, var_upper_bound, 'x')
y = model.NewIntVar(0, var_upper_bound, 'y')
z = model.NewIntVar(0, var_upper_bound, 'z')
# Creates the constraints.
model.Add(2 * x + 7 * y + 3 * z <= 50)
model.Add(3 * x - 5 * y + 7 * z <= 45)
model.Add(5 * x + 2 * y - 6 * z <= 37)
model.Maximize(2 * x + 2 * y + 3 * z)
# Creates a solver and solves the model.
solver = cp_model.CpSolver()
status = solver.Solve(model)
if status == cp_model.OPTIMAL or status == cp_model.FEASIBLE:
print(f'Maximum of objective function: {solver.ObjectiveValue()}\n')
print(f'x = {solver.Value(x)}')
print(f'y = {solver.Value(y)}')
print(f'z = {solver.Value(z)}')
else:
print('No solution found.')
# Statistics.
print('\nStatistics')
print(f' status : {solver.StatusName(status)}')
print(f' conflicts: {solver.NumConflicts()}')
print(f' branches : {solver.NumBranches()}')
print(f' wall time: {solver.WallTime()} s')
if __name__ == '__main__':
main()使用CP-SAT可以解决MIP问题，前文提到使用MPSolver及整数规划求解器同样可以解决MIP问题，另外，后续我们还会提到使用网络流解决MIP问题。使用过程中如何做选型呢？
- MPSolver：求解问题比较偏向于标准的线性规划问题，部分变量有整数约束
- CP-SAT：适合变量为0-1取值的情况
- 网络流方法：问题可以转化为网络关系，进而利用网络关系降低问题求解难度
三种方法有所侧重，但选型上并不绝对。很多问题也都是可以从不同的角度转化为不同类型的问题，进而使用不同的求解器进行求解的。
4.CP-SAT关键要素
建模过程中需要将数学模型转化为代码，其中最重要的是变量和约束的转化。接下来，我们看一看CP-SAT都提供哪些变量、约束函数和目标函数。了解了提供的功能，编程就变成了“搭积木”。
变量
- NewIntVar(self, lb, ub, name):Create an integer variable with domain [lb, ub]
NewIntVarFromDomain(self, domain, name)：变量取值范围在指定的（未必连续的）域中
Create an integer variable from a domain.
A domain is a set of integers specified by a collection of intervals. For example,
model.NewIntVarFromDomain(cp_model.Domain.FromIntervals([[1, 2], [4, 6]]), 'x')- NewBoolVar(self, name)：Creates a 0-1 variable with the given name.
- NewConstant(self, value):Declares a constant integer
NewIntervalVar(self, start, size, end, name):Creates an interval variable from start, size, and end.区间变量
start、size、 end均可以是线性表达式或常量，但方法内部添加了start + size == end的约束
NewFixedSizeIntervalVar(self, start, size, name):区间变量
start可以是线性表达式或常量，size必须为常量
NewOptionalIntervalVar(self, start, size, end, is_present, name):Creates an optional interval var from start, size, end, and is_present.
is_present: A literal that indicates if the interval is active or not. A inactive interval is simply ignored by all constraints. NewIntervalVar和NewOptionalIntervalVar的不同之处在于，是前者表示创建的区间变量在以后的约束建立中一定生效，而后者的方法签名中有个为is_present的参数表示这个区间变量是否生效。
- NewOptionalFixedSizeIntervalVar(self, start, size, is_present, name):Creates an interval variable from start, and a fixed size.
约束
- AddLinearConstraint(self, linear_expr, lb, ub)：Adds the constraint:
lb <= linear_expr <= ub. - AddLinearExpressionInDomain(self, linear_expr, domain):Adds the constraint:
linear_exprindomain. Add(self, ct):Adds a
BoundedLinearExpressionto the model.示例:
model.Add(5 x + 2 y - 6 * z <= 37)AddAllDifferent(self, *expressions)：This constraint forces all expressions to have different values.
Adds AllDifferent(expressions).
This constraint forces all expressions to have different values.
Args:
expressions: simple expressions of the form a var + constant.
Returns:
An instance of theConstraintclass.
示例:
queens = [model.NewIntVar(0, board_size - 1, 'x%i' % i) for i in range(board_size)
]
model.AddAllDifferent(queens)AddElement(self, index, variables, target):等值约束
Adds the element constraint:
variables[index] == target.- AddCircuit(self, arcs):arcs组成的路径集合构成哈密顿路径，TSP约束.
- AddMultipleCircuit(self, arcs):Adds a multiple circuit constraint, aka the "VRP" constraint.形成的多条链路，需要保证形成的各链路内arc首位连接。推测ortools的routing模块使用了AddCircuit、AddMultipleCircuit两种方法。
AddAllowedAssignments(self, variables, tuples_list): 固定匹配约束
An AllowedAssignments constraint is a constraint on an array of variables,
which requires that when all variables are assigned values, the resulting
array equals one of the tuples intuple_list.AddForbiddenAssignments(self, variables, tuples_list):禁止约束
A ForbiddenAssignments constraint is a constraint on an array of variables
where the list of impossible combinations is provided in the tuples list.AddAutomaton(self, transition_variables, starting_state, final_states, transition_triples): 状态转移约束 （状态之间存在转移关系）
transition_variables 代表了需要求解的变量，starting_state为起始状态，final\_states为可接受的最终状态，transition_triples为转移关系
AddInverse(self, variables, inverse_variables):关联约束
An inverse constraint enforces that if
variables[i]is assigned a valuej, theninverse_variables[j]is assigned a valuei. And vice versa.AddReservoirConstraint(self, times, level_changes, min_level,max_level):储水池约束
sum(level_changes[i] if times[i] <= t) in [min_level, max_level]
AddReservoirConstraintWithActive(self, times, level_changes, actives, min_level, max_level):时间开关的储水池约束，actives表示是否动作是否生效
sum(level_changes[i] * actives[i] if times[i] <= t) in [min_level, max_level]
- AddMapDomain(self, var, bool_var_array, offset=0):Adds
var == i + offset <=> bool_var_array[i] == true for all i. - AddImplication(self, a, b):Adds
a => b(aimpliesb). - AddBoolOr(self, *literals):Adds
Or(literals) == true: Sum(literals) >= 1. - AddAtLeastOne(self, *literals):Same as
AddBoolOr:Sum(literals) >= 1. - AddAtMostOne(self, *literals):Adds
AtMostOne(literals):Sum(literals) <= 1. - AddExactlyOne(self, *literals):Adds
ExactlyOne(literals):Sum(literals) == 1. - AddBoolAnd(self, *literals):Adds
And(literals) == true. - AddBoolXOr(self, *literals):Adds
XOr(literals) == true.异或运算 - AddMinEquality(self, target, exprs):Adds
target == Min(exprs). - AddMaxEquality(self, target, exprs):Adds
target == Max(exprs). - AddDivisionEquality(self, target, num, denom):Adds
target == num // denom(integer division rounded towards 0).取整操作，向0舍入。 - AddAbsEquality(self, target, expr):Adds
target == Abs(var). - AddModuloEquality(self, target, var, mod):Adds
target = var % mod. 取余操作 - AddMultiplicationEquality(self, target, *expressions):Adds
target == expressions[0] * .. * expressions[n]. - AddNoOverlap(self, interval_vars):区间不重叠约束
- AddNoOverlap2D(self, x_intervals, y_intervals):所有矩形不重叠约束，x_intervals、 y_intervals分别存储了不同矩形的x、y坐标
AddCumulative(self, intervals, demands, capacity):需求量小于能力上限的约束，VRP中会使用。
for all t:
sum(demands[i] if (start(intervals[i]) <= t < end(intervals[i])) and (intervals[i] is present)) <= capacity
目标
- Minimize(self, obj):最小化
- Maximize(self, obj):最大化
5.结语
本篇文章主要讲解了ortools使用CP-SAT求解器解决CP、MIP问题的方法，并详细解读了CP可以使用的变量、约束函数、目标函数等信息。