The Capacitated $P$-Median Problem (CPMP) is a classic discrete facility location problem in which $p$ capacitated facilities (medians) are selected from an economic point of view to serve a set of demand vertexes so that the total demand assigned to each of the facilities (medians) does not exceed its capacity. Since the capacity used by each facility is fixed and does not depend on the vertex demand, the traditional capacitated $p$-median problem’s total capacity is underutilized. To make full use of the total capacity, a more generalized model, namely, the Multi-supply Multi-Capacitated Location Problem (MMCLP), is considered in this paper, which assumes not only that each facility is allowed to open with different capacities while meeting the constraint of total capacity but also that each demand vertex can accept supplies from different facilities. To solve the MMCLP effectively, we propose a hybrid bi-level optimization framework based on decomposition. First, the original MMCLP is decomposed into numerous mixed-integer programming (MIP) sub-problems, i.e., inner optimization is performed, and for outer optimization, a Genetic Algorithm based on Co-Evolution of Population and Neighborhood (GACEPN) is proposed to find better sub-problems near the good sub-problems among all the candidate sub-problems. The inner optimization problem is solved by a commercial MIP solver. The results obtained are returned to the outer layer so that the neighborhood of the outer layer optimization can be adjusted appropriately to generate better sub-problems, thereby approaching the optimal solution of the original problem. The experiments show that for a large-scale MMCLP, compared to the Gurobi solver, the proposed bi-level algorithm (a hybrid of the GACEPN and Gurobi) can find a better solution in less time.

有能力的 $磷$-中值问题 (CPMP) 是一个经典的离散设施位置问题，其中 $磷$从经济角度选择有容量的设施点（中位数）来为一组需求顶点提供服务，以便分配给每个设施点（中位数）的总需求不超过其容量。由于每个设施使用的容量是固定的，不依赖于顶点需求，传统的有容量$磷$-中值问题的总容量未得到充分利用。为了充分利用总容量，本文考虑了一个更广义的模型，即多供应多容量位置问题（MMCLP），它不仅假设每个设施都允许以不同的容量开放，而满足总容量的约束，而且每个需求顶点都可以接受来自不同设施的供应。为了有效地解决MMCLP，我们提出了一种基于分解的混合双层优化框架。首先，将原来的MMCLP分解成无数个混合整数规划（MIP）子问题，即进行内部优化，对于外部优化，提出了一种基于人口与邻域协同进化的遗传算法（GACEPN），以在所有候选子问题中寻找靠近好的子问题的更好的子问题。内部优化问题由商业 MIP 求解器解决。得到的结果返回给外层，使外层优化的邻域可以适当调整，产生更好的子问题，从而逼近原问题的最优解。实验表明，对于大规模 MMCLP，与 Gurobi 求解器相比，所提出的双层算法（GACEPN 和 Gurobi 的混合）可以在更短的时间内找到更好的解决方案。得到的结果返回给外层，使外层优化的邻域可以适当调整，产生更好的子问题，从而逼近原问题的最优解。实验表明，对于大规模 MMCLP，与 Gurobi 求解器相比，所提出的双层算法（GACEPN 和 Gurobi 的混合）可以在更短的时间内找到更好的解决方案。得到的结果返回给外层，使外层优化的邻域可以适当调整，产生更好的子问题，从而逼近原问题的最优解。实验表明，对于大规模 MMCLP，与 Gurobi 求解器相比，所提出的双层算法（GACEPN 和 Gurobi 的混合）可以在更短的时间内找到更好的解决方案。