Voronoi tessellations are standard in spatial planning for assigning service areas based on Euclidean proximity, underpinning regulatory frameworks like the proximity principle in waste management. However, in regions with complex topography, Euclidean distance poorly approximates functional accessibility, causing misallocations that undermine efficiency and equity. This paper develops a probabilistic framework to quantify misallocation risk by modeling travel distances as random scaling of Euclidean distances and deriving incorrect assignment probability as a function of local Voronoi geometry. Using plant-municipality observations (n=383) in Extremadura, Spain (41,635 km2), we demonstrate that the Log-Normal distribution provides best relative fit among alternatives (K-S statistic=0.110). Validation reveals 15.4% of municipalities are misallocated, consistent with the theoretical prediction interval (52-65 municipalities at 95% confidence). Our framework achieves 95% agreement with complex spatial models at O(n) complexity. Poor absolute fit of global distributions (p-values<0.01) reflects diverse topography (elevation 200-2,400m), motivating spatial stratification. Sensitivity analysis validates the fitted dispersion parameter (s=0.093) for predicting observed misallocation. We provide a calibration protocol requiring only 30-100 pilot samples per zone, enabling rapid risk assessment without full network analysis. This establishes the first probabilistic framework for Voronoi misallocation risk with practical guidelines emphasizing spatial heterogeneity and context-dependent calibration.

翻译：Voronoi剖分是空间规划中基于欧几里得邻近性分配服务区的标准方法，支撑着如废物管理中的邻近原则等监管框架。然而，在地形复杂的区域，欧几里得距离难以准确反映功能可达性，导致错配，从而损害效率与公平性。本文通过将出行距离建模为欧几里得距离的随机缩放，并推导出错误分配概率作为局部Voronoi几何的函数，建立了一个量化错配风险的概率框架。利用西班牙埃斯特雷马杜拉地区（41,635平方公里）的工厂-市政观测数据（n=383），我们证明对数正态分布在备选方案中提供了最佳相对拟合（K-S统计量=0.110）。验证显示15.4%的市政单位存在错配，与理论预测区间一致（95%置信水平下52-65个市政单位）。我们的框架以O(n)复杂度实现了与复杂空间模型95%的一致性。全局分布的较差绝对拟合（p值<0.01）反映了地形的多样性（海拔200-2400米），这促使了空间分层分析。敏感性分析验证了拟合的离散参数（s=0.093）对预测观测错配的有效性。我们提供了一个校准协议，每个区域仅需30-100个试点样本，无需完整网络分析即可实现快速风险评估。这建立了首个针对Voronoi错配风险的概率框架，并提供了强调空间异质性和情境依赖校准的实用指南。