The Hidden Cost of Straight Lines: Quantifying Misallocation Risk in Voronoi-based Service Area Models

The Hidden Cost of Straight Lines: Quantifying Misallocation Risk in Voronoi-Based Service Area Models JA Torrecilla Pineroa,∗, JM Ceballos Martineza, A Cuartero Saezb, P Plaza Caballeroa, A Cruces Lopeza aDepartment of Construction, Universidad de Extremadura, Av. de Elvas s/n, 06006 Badajoz, Spain bDepartment of Graphical Expression, Universidad de Extremadura, Av. de Elvas s/n, 06006 Badajoz, Spain Abstract Voronoi tessellations are a standard tool in spatial planning for assigning service areas based on Euclidean proximity. This approach underpins key regulatory frameworks, such as the proximity principle in waste management [1, 2]. However, in regions with complex topography or sparse infrastructure, Euclidean distance is a poor proxy for functional accessibility, leading to service area misallocations that undermine cost-efficiency and equity. This paper develops a probabilistic framework to quantify this misallocation risk. We model real travel distances as a random scaling of Euclidean distances and derive the probability of incorrect assignment as a function of local Voronoi geometry. Using statistically independent plant-municipality observations (n=383), we demonstrate that the Log-Normal distribution provides best relative fit among tested alternatives (K-S statistic = 0.110) despite substantial spatial heterogeneity in Extremadura territory (41,635 km2). Validation reveals that 15.4% of municipalities are functionally misallocated by the Euclidean model, consistent with the theoretical prediction interval (52–65 municipalities at 95% confidence). Our framework predicts this risk with 95% agreement to complex spatial models but with O(n) complexity, avoiding costly network analyses. Critically, poor absolute fit of global distributions (all p-values < 0.01) reflects the territory’s diverse topography (elevation range 200–2,400m), motivating spatial stratification. Sensitivity analysis demonstrates that the fitted dispersion parameter (s = 0.093) accurately predicts observed misallocation, while internal stratification by topographic zones explains local variations. We provide a systematic calibration protocol requiring only 30–100 pilot samples per zone, enabling rapid risk assessment without full network analysis. This work establishes the first probabilistic framework for Voronoi misallocation risk, with practical guidelines emphasizing spatial heterogeneity and context-dependent calibration. Keywords: Voronoi tessellation, Probabilistic risk assessment, Spatial misallocation, Network distance, Waste management planning, Calibration protocol 1. Introduction The Voronoi tessellation is a cornerstone of computa- tional geometry and spatial analysis, offering an elegant solution to the nearest facility problem [3]. Its widespread adoption in urban planning, logistics, and environmental management, however, rests on a powerful yet often flawed assumption: that Euclidean distance is a reliable proxy for functional accessibility. This principle of geometric prox- imity is embedded in regulatory frameworks like the EU’s proximity principle for waste management, which assumes that geographic nearness ensures cost-effectiveness and min- imal environmental impact [4]. This paper challenges that assumption. We began with a simple question during a study of waste management in Extremadura, Spain: Is the closest plant really the closest? When we compared the offi- cial Voronoi-based allocation of municipalities to treatment plants with allocations based on actual road network dis- tances, a striking anomaly emerged: a significant number of municipalities were functionally closer to a different plant. ∗Corresponding author: This discrepancy was not random but spatially clustered in areas with complex topography, suggesting a systemic failure of the model. This observation motivates the central question of our work: if the Euclidean Voronoi model is unreliable in non- isotropic territories, can we quantify its risk of misallocation in a generalizable way? Our central contribution is the de- velopment of a probabilistic framework for quantifying this misallocation risk, which, to our knowledge and based on our extensive literature review, has not been previously addressed in the existing literature. This framework trans- forms the Voronoi diagram from an operational mandate into a theoretical benchmark whose deviations can be sys- tematically predicted and managed. The framework models the mismatch between Euclidean and network-based dis- tances through a log-normal scaling factor, enabling plan- ners to assess risk before committing to costly network analyses. Our contribution is threefold: 1. An empirical demonstration of the limitations of Eu- clidean proximity in a real-world case study, showing that 15.4% of municipalities are misallocated. Preprint submitted to Elsevier December 2, 2025 arXiv:2512.01795v1 [physics.soc-ph] 1 Dec 2025 2. A theoretical framework that quantifies the probabil- ity of misal

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