On the Entropy in Last-Mile Logistics

TL;DR

This paper introduces a novel entropy-based framework to quantify and analyze the complexity and fragmentation in last-mile logistics, providing insights into system behavior and potential improvements.

Contribution

It develops a structural entropy measure for last-mile logistics, linking it to system complexity, and evaluates its application to real-world routes, revealing insights on spatial and temporal consolidation effects.

Findings

High normalized entropy indicates near-maximal fragmentation in current operations.

A non-linear relationship exists between entropy and route distance, validating entropy as a predictive metric.

Spatial consolidation can reduce carrier entropy but may increase total system entropy due to customer trips.

Abstract

Last-mile logistics (LML) is characterized by high fragmentation, yet existing research treats this as an exogenous constraint rather than a quantifiable and optimizable system property. This paper introduces a framework for measuring LML complexity using structural entropy, derived from Boltzmann's statistical mechanics. Unlike traditional KPIs such as distance or cost, structural entropy quantifies the cardinality of the configuration space, providing a diagnostic of inherent system disorder. We establish a formal duality with Shannon entropy, linking absolute complexity burden to distributional balance. We apply our entropy framework to 6,112 Amazon last-mile routes across five U.S. cities. Current operations exhibit persistently high normalized entropy, indicating near-maximal fragmentation. A stable non-linear scaling relationship between entropy and route distance validates the metric as a predictive indicator of operational difficulty. To evaluate spatial consolidation, we develop a system-wide entropy measure accounting for all movements by both carriers and customers. We establish a theoretical conservation principle: under idealized conditions, spatial consolidation merely redistributes entropy from carrier to customer. Both idealizing conditions are violated in practice, thereby increasing total system entropy. Our system-wide measure reveals that spatial consolidation reduces carrier entropy by up to 40% under aggressive adoption but increases total system entropy by activating customer collection trips, though trip chaining can diminish this effect. Temporal consolidation, by contrast, genuinely reduces entropy by decreasing delivery events without creating new movements. By formalizing fragmentation as a measurable structural property, this research provides a new lens for network design, consolidation policy, and evaluation last-mile system performance.