Optimized Public Transport Flow Model

• The model is a dynamic multi-commodity formulation using time-expanded networks to optimize passenger and vehicle flows under capacity, temporal, and operational constraints.
• It integrates multi-marginal optimal transport with entropy regularization, achieving scalable solutions via graph-structured Sinkhorn iterations that dramatically speed up computations.
• The approach adapts to real-world public transport by encoding waiting, transfers, and vehicle capacities, unifying timetable coordination with dynamic network flow modeling.

An optimization-based public transport flow model is a mathematical and algorithmic framework for modeling, analyzing, and solving public transportation network assignment, routing, and resource allocation problems via explicit optimization of flows over a network under operational, capacity, temporal, and behavioral constraints. Contemporary approaches unify dynamic network flow, optimal transport theory, and scalable computational techniques to address complex multi-commodity, temporally granular, and capacity-constrained transit systems.

1. Dynamic Multi-Commodity Formulation and Time-Expanded Networks

The modern mathematical foundation for public transport flow modeling is the dynamic multi-commodity minimum-cost flow problem on a time-expanded network. Given a base graph 𝒩=(V, E), time is discretized into periods t = 0,1,…,T, producing a time-expanded graph whose nodes are pairs (t,v) for v ∈ V and arcs represent both travel (transit arcs) and optional behaviors such as waiting or dwelling on nodes/edges. Each "commodity" ℓ indexes passenger groups (e.g., OD pairs, classes) or vehicle types and is assigned supply/demand at source/sink arcs.

Let x_p^ℓ be the flow along directed paths p = (p₁,…,p_T) in the time-expanded network, and f_{e,t}^ℓ the edge-time flows. The canonical minimum-cost flow problem is:

$$\min \sum_{ℓ=1}^L \sum_{p\in\mathcal{P}^ℓ} c_p^ℓ x_p^ℓ \qquad \text{subject to}$$

• Path-based supply/demand satisfaction for each commodity at source and sink arcs,
• Edge-time capacity constraints $$\sum_{ℓ=1}^L \sum_{p\ni (t-1 \to t) = e} x_p^ℓ \leq d_e$$ for all e∈E, t,
• Nonnegativity.

Equivalently, the flow formulation uses f_{e,t}^ℓ with node/time flow conservation, capacity $\sum_ℓ f_{e,t}^ℓ \leq d_e$, and cost minimization $\sum_{t,e,ℓ} c_e^ℓ f_{e,t}^ℓ$.

This model is flexible enough to directly encode passenger waiting (by adding "waiting arcs"), vehicle storage, path-dependent penalties, and edge/vehicle capacities (Haasler et al., 2021).

2. Multi-Marginal Optimal Transport (MMOT) and Entropy Regularization

The dynamic flow problem's path-based representation admits reformulation as a multi-marginal optimal transport (MMOT) problem. All length-T paths are encoded as entries in a high-order nonnegative tensor $\pi \in \mathbb{R}_+^{L \times n^T}$, where each element $\pi_{ℓ,i_1,…,i_T}$ tracks the flow along the sequence of edges (i₁,…,i_T) for commodity ℓ.

Key constraints become marginalizations on this tensor:

• Supply/demand at initial/final time slices (modes 0, T+1),
• Capacity marginals per edge and time,
• Additional marginals for behavioral or operational constraints.

The MMOT objective is:

$$\min_{\pi \geq 0} \sum_{ℓ,i_1,…,i_T} c_{ℓ,i_1,…,i_T} \pi_{ℓ,i_1,…,i_T} \qquad \text{subject to prescribed marginals, capacity, and nonnegativity.}$$

To ensure computational tractability and solution smoothness, an entropy regularization term $H(\pi) = \sum \pi (\log\pi - 1)$ is added, yielding a strictly convex problem solvable by scalable iterative algorithms.

The cost tensor encodes both spatial connection and time-dependent travel, waiting, and transfer penalties. Infinite costs are imposed to eliminate infeasible schedule or transfer combinations (resulting in sparse connectivity in K = exp(–C/ε)).

This MMOT formulation generalizes classical single-commodity and static flow models by accommodating arbitrary user classes, time periods, and behavioral constraints within a unified tensorial structure (Haasler et al., 2021).

3. Scalable Solution via Graph-Structured Sinkhorn Iterations

For large-scale MMOT instances, the entropy-regularized problem admits highly efficient solution via generalized Sinkhorn (RAS/Bregman projection) iterations adapted to the "graph of marginals" structure. The regularized optimal tensor π factors multiplicatively as

$$\pi_{ℓ,i_1,…,i_T} = K_{ℓ,i_1,…,i_T} \prod_{\alpha \in V} u_\alpha(i_\alpha) \prod_{\beta \in E} U_\beta(i_{\beta_1},i_{\beta_2}),$$

where dual variables u, U are updated cyclically to match the required marginals. For public transport instances,

• Per-time marginals (capacity, flow distribution) are handled by per-slice updates,
• Multi-commodity or star-shaped constraint graphs require block-coordinate updates with efficient forward/backward message caching.

Each iteration has computational complexity proportional to the number of constrained modes and the sparsity of the marginal graph (i.e., O(L·T·n·Δ) for L commodities, T time-slices, n edges, Δ base graph degree). This structure enables scaling to city-wide, multi-class, and finely grained time-resolution instances.

Empirical results show at least one order of magnitude speed-up versus LP dual simplex (e.g., 10×10 grid, 50 commodities, 80 time steps: 0.1 s for Sinkhorn at ε=0.01 vs 10–20 s for CPLEX; random dense graphs: 1 s vs 15 s) (Haasler et al., 2021).

4. Extension to Public Transport Modeling: Waiting, Timetables, and Vehicle Constraints

The MMOT and Sinkhorn framework are adapted to real-world public transport by mapping:

• Commodities to passenger OD pairs or classes,
• Edge/arc costs to in-vehicle times, waiting times (e.g., cost on waiting arcs as E[wait]=½·headway), and transfer penalties,
• Infinite costs on arcs violating temporal feasibility (e.g., wrong timing for transfer, headway violations),
• Arc and time-slice marginals to vehicle-seat and operational headway capacity constraints.

Timetabled services are enforced by zero-infinite-zero cost structures, making only arcs compatible with feasible vehicle trips carry finite weight.

Vehicle seat capacities enter as capacity marginals on the corresponding arcs, with options for fine-grained modeling (per time slot, per trip, or bi-marginal passenger loads). Service frequencies affect both cost and feasible capacity at each time.

The result is a unified MMOT with a graph of marginals whose structure is invariant to the addition of real-world complexity (timing, vehicle heterogeneity, headways, behavioral policies), maintaining the same scalable Sinkhorn-based computational backbone.

5. Modeling Flexibility and Comparative Advantages

The optimization-based public transport flow model accommodates:

• Dynamic assignment over time-expanded networks,
• Multiple user classes and OD demands,
• Timetable constraints and realistic waiting/transfer modeling,
• Hard vehicle and operational capacity bounds,
• Arbitrary cost structures (multi-objective mix of travel, wait, transfer, and operational penalties),
• Full retention of flow-balance at all iterates.

Compared to monolithic LP or IP formulations (node-arc or path-based), this approach achieves strongly improved scalability, enables differentiation among user classes or services, and supports dynamic time-resolved planning.

The approach is also robust to changes in network topology, demand profiles, and operational policies—a single framework suffices to model, for example, both passenger and vehicle flows by reformulating commodities and costs accordingly (Haasler et al., 2021).

6. Practical Implementation and Performance

In practice, the entropy-regularized Sinkhorn-based MMOT solution:

• Achieves rapid convergence and high solution quality for large-scale problems (city-scale instances with thousands of stops, time slices, classes),
• Offers direct enforcement (not just approximation) of flow balance at both global and intermediate iterates,
• Maintains computational requirements proportional to the marginal graph's sparsity and not to the exponential size of the joint path tensor,
• Provides dramatic empirical speedups (≥10×) relative to established LP solvers, with stable performance across sparsity, commodity count, and time horizon,
• Handles injection of new constraints modularly by adding corresponding marginals and dual updates.

Timetabling, frequency planning, and priority policies (e.g., time-dependent class priority) can all be incorporated by simply modifying the cost and feasible-support tensors and augmenting the marginal graph.

7. Broader Significance and Unified Perspective

The MMOT-based optimization model demonstrates that a broad class of finite-horizon, capacity-constrained, multi-commodity, and schedule-aware flow problems in public transport—and analogs in traffic, logistics, or supply chains—can be recast as structured multi-marginal convex programs. The flexible graph-structured Sinkhorn algorithm provides a universally scalable solution machinery, supporting integration of demand assignment, operational planning, timetable coordination, and behavioral modeling within a single coherent optimization pipeline.

This unified view reveals that “virtually any dynamic flow problem with finite-horizon, capacity bounds, multiple user-classes and scheduling constraints can be posed as a graph-structured MMOT and solved orders-of-magnitude faster than a monolithic LP, all while retaining exact flow-balance at every intermediate iterate” (Haasler et al., 2021).