Improved Heuristic to Reduce Oscillations in Transit Assignment on Refined Networks

1. The Challenge: Why Transit Assignment Oscillates on Refined Networks

Frequency-based transit assignment is widely used for planning high-frequency public transport.
But when we refine network representations (e.g., explicit dwell links and transfer-station modeling) and let effective service frequency depend on passenger flow (to capture congestion), two practical problems emerge:

1. Re-boarding artifacts at transfer stations: expanded representations may allow passengers to alight and re-board the same line at the same stop, creating unrealistic movements.
2. Algorithmic instability: with flow-dependent effective frequencies, standard iterative solvers can exhibit period-two oscillations or very slow convergence.

This paper addresses both issues with a refined representation and a lightweight stabilizer that improves convergence while preserving the main assignment workflow.

2. Core Contribution 1: PLM Representation to Eliminate Re-boarding

What is PLM?

We introduce a Pre-Line Marked (PLM) representation that conditions each station decision on (stop, destination) and the preceding line (“pre-line”). This preserves transfer context and yields clearer station-level flow decomposition.

A key reinterpretation is that “same-line transfer” becomes an on-board dwell-window decision:

• If a passenger already arrives on line m, the “waiting time” for m is the remaining dwell time (not a headway-based wait).
• Other candidate lines still use effective frequency-based waiting.

This removes incentives for irrational re-boarding and produces more realistic transfer behavior.

• Figure 2: Expanded network vs. PLM representation. PLM conditions decisions on the preceding line, preventing re-boarding artifacts and refining station-level flow allocation.

3. Core Contribution 2: A Tractable UE Formulation via Variational Inequality (VI)

PLM defines user equilibrium (UE) over strategy proportions at each decision state (stop i, destination s, pre-line m).

Because strategy choices affect flows, flows affect effective frequencies, and effective frequencies affect costs, the equilibrium is reformulated as a finite-dimensional Variational Inequality (VI). This provides a principled basis for existence analysis and solver design.

• Figure 3: Reformulation of the PLM-based equilibrium into a solvable VI problem.

4. Core Contribution 3: ODA — A Plug-in Stabilizer for Oscillation Control

Why oscillations happen

With flow-dependent effective frequencies, the loading step becomes an inner fixed-point problem. Lagged-frequency updates inside standard MSA-style loops can be non-contractive, producing period-two flip oscillations.

ODA: detect + damp

We propose an Oscillation Detection Algorithm (ODA) that:

• Monitors normalized share updates of effective frequencies,
• Detects a period-two signature,
• Applies damping only when instability is detected (via short-window averaging).

ODA can be inserted modularly without redesigning the outer assignment loop.

• Figure 4: MSA workflow with an inner stabilization slot. ODA is inserted modularly without altering the overall structure.

5. Numerical Results: From Small Networks to Winnipeg

5.1 Small network: realism + stability

On a benchmark small network, PLM removes re-boarding artifacts while maintaining sensible line-level outcomes.
When congestion sensitivity increases, baseline MSA exhibits oscillations; MSA-ODA suppresses oscillations and improves convergence.

• Figure 7: Convergence performance. ODA removes saw-tooth oscillations while remaining lightweight.

5.2 Station-level interpretability

Even when line flows are similar, station-level movements can differ. PLM separates flows by provenance (arriving line vs. new entrants) and yields behaviorally meaningful transfers.

• Figure 9: Station-level flows at a transfer node: PLM provides provenance-aware movements; classical representations may imply re-boarding artifacts.

5.3 Winnipeg network: scalability

On the Winnipeg bus network, ODA remains practical and robust across demand regimes, improving convergence compared with baseline methods while avoiding heavy inner solves.

• Figure 10: Winnipeg benchmark: ODA improves robustness across demand regimes and scales efficiently.

6. Takeaways

• PLM representation eliminates re-boarding artifacts by conditioning decisions on the preceding line and reinterpreting same-line behavior via dwell-window choice.
• VI reformulation makes the refined equilibrium analyzable and solvable in a principled way.
• ODA stabilizer targets inner-loop instability and improves convergence while preserving standard assignment workflows.