This paper addresses the single-assignment, uncapacitated, multi-level facility location (MFL) problem, a strategic decision-making process critical to the design of long-term supply chain networks. Specifically, we examine four- and five-level facility location structures (k-LFL), modeled as a location-allocation problem where demand nodes must be assigned to open facilities across hierarchical levels. Although the MFL has been addressed in the literature, solutions to large-scale, realistic problems involving thousands of nodes are lacking. This paper proposes a heuristic framework based on the Variable Neighborhood Descent (VND) metaheuristic with a multi-start strategy. We develop and compare four variants: Basic Variable Neighborhood Descent (BVND), Pipe Variable Neighborhood Descent (PVND), Cyclic Variable Neighborhood Descent (CVND), and Union Variable Neighborhood Descent (UVND). In each case, a multi-start strategy with strong diversification components is employed. Extensive computational experiments compare the methods on large-scale instances involving up to 10,000 customers, 150 distribution centers, 50 warehouses, and 30 plants. Each algorithm settled into a unique, statistically significant computational time when solving these problems. Sensitivity analyses, supported by non-parametric statistical methods, validate the effectiveness of the proposed heuristic framework.

翻译：本文研究了单分配、无容量约束的多级设施选址（MFL）问题，这是长期供应链网络设计中的关键战略决策过程。具体而言，我们考察了四级和五级设施选址结构（k-LFL），将其建模为一个位置分配问题，其中需求节点必须分配给跨层级开放的设施。尽管已有文献涉及MFL问题，但针对包含数千节点的大规模实际问题的解决方案仍然缺乏。本文提出了一种基于变邻域下降（VND）元启发式算法并采用多起点策略的启发式框架。我们开发并比较了四种变体：基本变邻域下降（BVND）、管道变邻域下降（PVND）、循环变邻域下降（CVND）和联合变邻域下降（UVND）。每种变体均采用具有强多样化成分的多起点策略。通过大规模算例的计算实验对比了各方法的性能，算例涉及多达10,000名客户、150个配送中心、50个仓库和30个工厂。每个算法在求解这些问题时均达到了独特且统计显著的计算时间。基于非参数统计方法的灵敏度分析验证了所提启发式框架的有效性。