Dynamic Random Bipartite Matching under Spatiotemporal Heterogeneity: General Models and Application to Mobility Services

TL;DR

The paper tackles ST-RBMP, a dynamic bipartite matching problem with spatiotemporal heterogeneity relevant to mobility services. It first derives closed-form approximations for static RBMPs under maximum radii and spatial heterogeneity, revealing a scaling property where the expected matching distance $\mathbb{E}[X]$ primarily depends on local densities rather than region size. Building on these results, it formulates ST-RBMP as a dynamic optimal-control problem using a continuum-approximation, jointly optimizing pooling intervals $\tau(t)$ and radii $r_z(t)$ over space and time to minimize total system costs. Through extensive numerical experiments, the framework is shown to accurately predict matching probabilities and distances and to yield effective dynamic matching strategies under various demand/supply patterns, offering managerial insights for mobility operators and enabling transfer to other location-based services. The analytical, model-based approach provides a robust, transferable alternative to data-driven methods in settings with spatiotemporal heterogeneity and dynamic resources.

Abstract

This paper explores a variant of bipartite matching problem, referred to as the Spatiotemporal Random Bipartite Matching Problem (ST-RBMP), that accommodates randomness and heterogeneity in the spatial distributions and temporal arrivals of bipartite vertices. This type of problem can be applied to many location-based services, such as shared mobility systems, where randomly arriving customers and vehicles must be matched dynamically. This paper proposes a new modeling framework to address ST-RBMP's challenges associated with the spatiotemporal heterogeneity, dynamics, and stochastic decision-making. The objective is to dynamically determine the optimal vehicle/customer pooling intervals and maximum matching radii that minimize the system-wide matching costs, including customer and vehicle waiting times and matching distances. Closed-form formulas for estimating the expected matching distances under a maximum matching radius are developed for static and homogeneous RBMPs, and then extended to accommodate spatial heterogeneity via continuum approximation. The ST-RBMP is then formulated as an optimal control problem where optimal values of pooling intervals and matching radii are solved over time and space. A series of experiments with simulated data are conducted to demonstrate that the proposed formulas for static RBMPs under matching radius and spatial heterogeneity yield very accurate results on estimating matching probabilities and distances. Additional numerical results are presented to demonstrate the effectiveness of the proposed ST-RBMP modeling framework in designing dynamic matching strategies for mobility services under various demand and supply patterns, which offers key managerial insights for mobility service operators.

Figures (8)

• Figure 1: Matching Radius and Pooling Interval.
• Figure 2: RBMP under Spatial Heterogeneity.
• Figure 3: Verification of the Scaling Property.
• Figure 4: Verification of Formulas under Varying Values of Maximum Matching Radius.
• Figure 5: Verification of Formulas under Spatial Heterogeneity.

Theorems & Definitions (4)

• Proposition 1
• proof
• Lemma 1
• proof