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Capacitated Spatiotemporal Matching

Mingyang Fu, Ming Hu

Abstract

We study a spatiotemporal service matching problem in which demand, heterogeneous in location and time sensitivity/preference, is to be assigned to service stations. The planner seeks to maximize social welfare, defined as total service reward minus spatial and temporal costs, by optimally scheduling demand to stations and service time under processing capacity constraints. We formulate the problem as an optimal transport (OT) model that allows for both demand-capacity imbalance and endogenously unserved demand when service costs exceed rewards. Leveraging a barycenter-style decomposition, we reformulate the problem as a finite-dimensional convex optimization problem that generalizes semi-discrete OT and enables scalable computation. We characterize the geometry of optimal assignments, showing that spatial partitions correspond to generalized Laguerre cells. Temporally, we show that the structure of the optimal schedule depends on demand heterogeneity: when demand differs only in temporal cost sensitivity, higher-sensitivity demand is assigned service times closer to the common ideal time; when demand differs only in preferred times, the assignment is order-preserving with respect to preferred times. We further propose an envy-free, individually rational implementation of the optimal schedule using time-dependent pricing and a finite-slot mechanism with explicit bounds depending on the number of required slots. To illustrate the framework, we extend the classic Hotelling linear-city model on a line segment by incorporating a continuum of waiting-cost sensitivities, demonstrating how optimal spatial partitions vary with changes in sensitivity heterogeneity and reward.

Capacitated Spatiotemporal Matching

Abstract

We study a spatiotemporal service matching problem in which demand, heterogeneous in location and time sensitivity/preference, is to be assigned to service stations. The planner seeks to maximize social welfare, defined as total service reward minus spatial and temporal costs, by optimally scheduling demand to stations and service time under processing capacity constraints. We formulate the problem as an optimal transport (OT) model that allows for both demand-capacity imbalance and endogenously unserved demand when service costs exceed rewards. Leveraging a barycenter-style decomposition, we reformulate the problem as a finite-dimensional convex optimization problem that generalizes semi-discrete OT and enables scalable computation. We characterize the geometry of optimal assignments, showing that spatial partitions correspond to generalized Laguerre cells. Temporally, we show that the structure of the optimal schedule depends on demand heterogeneity: when demand differs only in temporal cost sensitivity, higher-sensitivity demand is assigned service times closer to the common ideal time; when demand differs only in preferred times, the assignment is order-preserving with respect to preferred times. We further propose an envy-free, individually rational implementation of the optimal schedule using time-dependent pricing and a finite-slot mechanism with explicit bounds depending on the number of required slots. To illustrate the framework, we extend the classic Hotelling linear-city model on a line segment by incorporating a continuum of waiting-cost sensitivities, demonstrating how optimal spatial partitions vary with changes in sensitivity heterogeneity and reward.
Paper Structure (49 sections, 18 theorems, 111 equations, 4 figures, 1 table)

This paper contains 49 sections, 18 theorems, 111 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Matching as Optimal Transport
  4. Problem Setting
  5. Optimal Transport Model
  6. A Barycenter Perspective
  7. Structure of Optimal Matching
  8. Finite Demand Types
  9. Implementation of Optimal Matching
  10. Spatiotemporal partition.
  11. Envy-free pricing.
  12. Finite time-slots.
  13. Capacity Allocation
  14. Homogeneous Preference with Linear Cost
  15. Capacity allocation with a single demand type.
  16. ...and 34 more sections

Key Result

Proposition 1

The strong duality holds for the primal problem eq:STM and the dual problem eq:STM_D.

Figures (4)

  • Figure 1: Illustration of Envy-Free Pricing Policy.
  • Figure 2: Illustration of the Optimal Demand Partitioning.
  • Figure 3: Illustration of Vaccination Scheduling.
  • Figure EC.1: An Illustration of Spatiotemporal Matching.

Theorems & Definitions (31)

  • Proposition 1
  • Theorem 1
  • Definition 1: Generalized Laguerre Cells
  • Lemma 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Theorem 2
  • Definition 2
  • ...and 21 more