Column Generation for the Micro-Transit Zoning Problem

Along with the rapid development of new urban mobility options like ride-sharing over the past decade, on-demand micro-transit services stand out as a middle ground, bridging the gap between fixed-line mass transit and single-request ride-hailing, balancing ridership maximization and travel time minimization. Micro-transit adoption can have significant social impact. It improves urban sustainability, through lower energy consumption and reduced emissions, while enhancing equitable mobility access for disadvantaged communities, thanks to its lower vehicle miles per passenger, flexible schedules, and affordable pricing. However, effective operation of micro-transit services requires planning geo-fenced zones in advance, which involves solving a challenging combinatorial optimization problem. Existing approaches enumerate candidate zones first and selects a fixed number of optimal zones in the second step. In this paper, we generalize the Micro-Transit Zoning Problem (MZP) to allow a global budget rather than imposing a size limit for candidate zones. We also design a Column Generation (CG) framework to solve the problem and several pricing heuristics to accelerate computation. Extensive numerical experiments across major U.S. cities demonstrate that our approach produces higher-quality solutions more efficiently and scales better in the generalized setting.

💡 Research Summary

The paper tackles the emerging operational challenge of designing service zones for on‑demand micro‑transit, a mode that sits between fixed‑route mass transit and individual ride‑hailing. Existing work (Hu et al., 2025) solves a two‑phase problem: first enumerate all feasible zones under a fixed size limit, then select a predetermined number m of zones via a weighted maximum‑covering integer program. This approach suffers from two major drawbacks: (1) it assumes the planner knows the exact number of zones to open, which is unrealistic; (2) the enumeration step does not scale when the spatial granularity becomes fine (e.g., sub‑kilometer cells).

To address these issues, the authors reformulate the Micro‑Transit Zoning Problem (MZP) by replacing the per‑zone size limit with a global budget B on total operating costs. Each zone S is a set of adjacent cells in the road network, and its cost is modeled as f(S)=α·D²_S+β, where D_S is the zone’s diameter (the longest shortest‑path distance between any two cells in S). The objective is to maximize the intra‑zone demand Σ d(i,j) · I(i,j in same zone) subject to Σ f(S)·x_S ≤ B, where x_S∈{0,1} indicates whether zone S is opened.

Because the set of all feasible zones grows exponentially with the number of cells, the authors adopt a Column Generation (CG) framework. In this setting each “column” corresponds to a candidate zone. The Restricted Master Problem (RMP) contains only a small subset of zones and includes (i) a budget constraint, (ii) a set‑partitioning‑like constraint that links binary zone‑selection variables x_S with auxiliary variables w_ij indicating whether a pair of cells share a zone, and (iii) integrality constraints. Solving the LP relaxation of the RMP yields dual variables λ (for the budget) and π_ij (for the pairwise linking constraints).

The Pricing Problem uses these duals to search for a new zone with positive reduced cost:
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