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

TL;DR

The paper addresses the misallocation risk inherent in Voronoi-based service areas when travel is constrained by road networks rather than straight-line distance. It develops a probabilistic framework that uses a Log-Normal network-factor beta to quantify misallocation probability and derives closed-form expressions, enabling efficient risk assessment with O(n) complexity. Empirical validation in Extremadura shows 15.4% misallocation and demonstrates that the framework provides accurate territory-wide predictions and useful zone-specific calibration guidelines. The work reframes the Voronoi diagram as a theoretical benchmark rather than a definitive operational rule, offering practical calibration protocols, safety bands, and a pathway for policy makers to adopt risk-based proximity frameworks. It also extends to multi-facility and k-nearest scenarios, assesses spatial robustness, and provides a reproducible open-science package for rapid adoption in diverse geographic contexts.

Abstract

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.

Figures (18)

• Figure 1: Histograms of the network scaling factor ($\beta = d_r/d_e$) distribution. Panel A: $\beta$ distribution for municipality-to-municipality distances. Panel B: $\beta$ distribution for municipality-to-assigned-plant distances.
• Figure 2: Violin plots comparing $\beta$ distributions between municipality-to-municipality and municipality-to-assigned-plant distances for all 383 municipalities, showing probability density shapes and spread differences.
• Figure 3: Plant anisotropy analysis. Panel A: Histogram of anisotropy coefficients with mean and median. Panel B: Boxplots by municipality assignment groups. Panel C: Scatter plot of municipalities served vs anisotropy coefficient. Panel D: Min vs Max $\beta$ scatter plot with isotropy reference lines.
• Figure 4: Changes in municipality assignments when comparing Voronoi (Euclidean-based) vs network-based assignments. Bar chart showing net gain/loss of municipalities for each plant, revealing redistribution patterns.
• Figure 5: Q-Q plots comparing $\beta$ coefficients against theoretical distributions. Top row: Municipality-to-Municipality vs Log-Normal, Gamma, and Weibull. Bottom row: Municipality-to-Assigned-Plant vs the same distributions, with R-squared goodness-of-fit values.