直接求解就是解下面的混合整数规划。

$$ \begin{aligned} \min ~ & \sum_{i=1}^m f_iy_i + \sum_{i=1}^m\sum_{j=1}^n c_{ij}x_{ij}\\[6pt] \text{s.t. } & \sum_{i=1}^m x_{ij} = 1, \quad \forall j\\[6pt] & x_{ij}\leq y_i, \quad \forall i, j\\[6pt] & x_{i,j}\geq 0,~ y_i \in \set{0,1}, \quad \forall i, j \end{aligned} $$

接口

定义类 FacilityLocationExact 用来求解上面的混合整数规划问题。solve 方法用来求解，结果保存在成员变量中。

  
class FacilityLocationExact:

    """
    Exact solution of the facility location problem.
    Args:
    f: facility open costs
    C: connection costs

    The problem is:
    min sum(f[i] * y[i]) + sum(C[i][j] * x[i][j])
    s.t. sum(x[i][j]) = 1, for all j
         sum(x[i][j]) <= y[i], for all i
         x[i][j] >= 0, for all i, j
         y[i] in {0, 1}, for all i, j
    """
    
    def __init__(self, f, C):
        self.f = np.array(f)
        self.C = np.array(C)
        # Solution
        self.x = None
        self.y = None
        self.objective = None
        
    def solve(self):
        """
        Solve the facility location problem.
        """

实现

在 solve 函数中调用 OR-Tools 的整数规划求解器（例如 CBC），结果保存在成员变量 self.x, self.y, self.objective 中。 细节可以参考代码。

代码

相关代码在 codes/decomposition/facility-location 文件夹。

  • exact.py 求解设施选址问题的整数规划
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Last updated 04 Jul 2025, 08:25 +0800 . history