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arxiv: 2606.24386 · v2 · pith:RVLQ7IHXnew · submitted 2026-06-23 · 🧮 math.OC

Line Planning at Scale: Models, Methods, and Insights

Rolf van Lieshout , Bart van Rossum This is my paper

Pith reviewed 2026-06-25 23:21 UTC · model grok-4.3

classification 🧮 math.OC
keywords line planningpublic transportchange-and-go networktransfer approximationrailway optimizationlarge-scale instancesdirect connection model
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The pith

A compact direct connection model outperforms the change-and-go network on large line planning instances.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper compares the traditional change-and-go network for modeling passenger transfers exactly in line planning against three compact models that approximate transfers in different ways. Experiments on 972 instances drawn from Dutch and Swiss railway networks show the exact model is competitive only on small cases and often returns no feasible solution on large ones. A direct connection model instead returns the best solution on more than 83 percent of instances. This matters because line planning determines which routes and frequencies to run, and scale limitations have historically restricted its use on real networks.

Core claim

Contrary to the CGN's canonical status, we find that it is competitive only on small or easy instances and often fails to find any feasible solution on large networks. Instead, a compact direct connection model performs best overall, finding the best solution on over 83% of instances. Our results indicate that carefully designed approximations, rather than exact transfer modeling, are the more promising foundation for large-scale line planning.

What carries the argument

The compact direct connection model, which approximates passenger transfers by assuming direct line connections without constructing the full change-and-go network.

If this is right

  • Line planning problems on large networks become solvable where the exact model returns no solution.
  • Solution quality improves when transfer modeling is approximated rather than modeled exactly at scale.
  • Tailored solution methods for each compact model can further improve performance on real railway instances.
  • Public transport operators gain feasible plans for entire national networks rather than only small subnetworks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same approximation approach could extend to other network design tasks where exact transfer modeling creates similar scalability barriers.
  • Validation of approximated solutions against detailed passenger assignment simulations would be needed before operational use.
  • The performance gap might narrow or reverse if future exact solvers improve on very large change-and-go networks.

Load-bearing premise

The three compact models provide modeling accuracy close enough to the exact change-and-go network that their solutions remain practically useful even though transfers are approximated.

What would settle it

Apply the direct connection model to a large instance, implement the resulting line plan, and measure whether actual passenger transfer times or total travel times deviate substantially from the model's predictions.

Figures

Figures reproduced from arXiv: 2606.24386 by Bart van Rossum, Rolf van Lieshout.

Figure 1
Figure 1. Figure 1: The public transportation network (PTN) with two lines ℓ1 and ℓ2, and the corresponding change-and-go network (CGN). In the CGN, solid arcs are running arcs and dashed arcs are transfer arcs; rectangular nodes represent travel nodes (ℓ, s) and circular nodes represent station nodes s. representation it is built on: CGN = min X p∈P τpxp + X k∈OD πkuk (1a) s.t. X p∈Pk xp + uk ≥ vk ∀k ∈ OD, (1b) X p∈P(a) xp ≤… view at source ↗
Figure 2
Figure 2. Figure 2: Line pool where the basic direct connection model overestimates direct route capacity. direct route: ℓ4 covers connection B → C but does not extend to A or D and cannot support A → C or B → D directly. Since ℓ1 has capacity κyℓ1 = 1, at most one passenger can travel directly. The gap arises because LA→C and LB→D are incomparable, so no single basic direct connection constraint bounds their combined flow on… view at source ↗
Figure 3
Figure 3. Figure 3: The DPTN for the example network. Circles are OD-nodes, rectangles are travel nodes (s-c-dep/arr) [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Example 1 with lines ℓ1 = A → C → D → E and ℓ2 = B → C → D → F. Example 2: PTN-DC weaker than DPTN. Now consider three lines: ℓ1 = A → C → D → E, ℓ2 = B → C, and ℓ3 = D → F, as shown in Figure 5a. Only the OD pair A → E has a direct route; passengers traveling B → F must transfer twice. The CGN captures this exactly: z(CGN) = 4σ. The DPTN likewise models both transfers correctly, giving z(DPTN) = 4σ. This … view at source ↗
Figure 5
Figure 5. Figure 5: Example 2 with lines ℓ1 = A → C → D → E, ℓ2 = B → C, and ℓ3 = D → F [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Rail networks in the Netherlands and Switzerland. In the Dutch networks, large red dots correspond to intercity stations, small gray dots to sprinter stations, and the medium-sized blue dots to stations where intercity lines may transition into sprinter lines and vice versa. routes are generated dynamically and no such suffix is needed; we instead test variants with and without separation of strong inequal… view at source ↗
Figure 7
Figure 7. Figure 7: Performance profiles for the four solver families with respect to solution quality [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Performance profiles per network for the four solver families. PTN PTN-DC DPTN CGN S M L 0 25 50 75 100 Line Pool S M L Budget S M L Fixed Line Cost S M L Transfer Penalty Win share (%) [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Win share per solver family by parameter level. Each bar shows the average fraction of instances won by each family (ties split equally). 6.3 Detailed Solver Analysis In this section, we analyze the relative performance of the solvers within each family [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Performance profiles per solver family (all individual solver variants). Comparing the small, medium, and large route sets for all solvers that require a priori route enumeration, the medium-sized route sets consistently perform well. Small route sets exclude too many useful routes, while large route sets tend to perform marginally worse than medium ones, likely because they introduce many redundant route… view at source ↗
Figure 11
Figure 11. Figure 11: Gap profiles for PTN-FDC-M per parameter level. PTN PTN-BDC PTN-FDC DPTN CGN Network S M L S M L S M L S M L B&P BP&C B&P BP&C Randstad-IC +8.5 -9.9 -9.9 +19.2 +0.1 -0.0 +19.2 +0.1 -0.1 +18.2 -2.7 -4.0 -3.2 -4.1 +0.0 +0.0 Randstad-IC-SPR +21.8 -7.8 -8.5 +38.2 +0.6 -0.4 +38.3 +0.6 -0.6 +34.0 -2.6 -4.3 -3.8 -4.5 +0.0 – Netherlands-IC +26.0 -7.0 -7.3 +34.5 -0.0 -0.2 +34.8 -0.3 -0.4 +34.9 -3.8 -4.0 -2.2 -3.8 … view at source ↗
Figure 12
Figure 12. Figure 12: Gap profiles for PTN-FDC-M per network and OD matrix type. 6.4 Impact of Demand Sparsification This section analyzes the impact of the preprocessing techniques of Section 5.3. We use PTN-FDC-M as the solver [PITH_FULL_IMAGE:figures/full_fig_p029_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Cross-evaluation performance profiles per network: solving directly on the original dense OD matrix vs. solving on the sparsified OD matrix and re-evaluating on the original. Solver is PTN-FDC-M. PTN-based direct connection model is the most robust and best-performing approach across the full range of instances. The exact model based on the change-and-go network is competitive only on small instances and … view at source ↗
read the original abstract

Line planning, the problem of deciding which lines to operate and at what frequency, is a fundamental step in public transport planning. To accurately model passenger routing, the problem is traditionally defined on a change-and-go network (CGN), which captures transfers between lines exactly. However, this network grows large quickly and is hard to solve at scale. We compare the CGN against three more compact models, differing with respect to how transfers are approximated, and characterize how they relate in terms of solution quality and modeling accuracy. We develop state-of-the-art solution methods tailored to each model, and evaluate all four across 972 instances based on the Dutch and Swiss railway networks. Contrary to the CGN's canonical status, we find that it is competitive only on small or easy instances and often fails to find any feasible solution on large networks. Instead, a compact direct connection model performs best overall, finding the best solution on over 83% of instances. Our results indicate that carefully designed approximations, rather than exact transfer modeling, are the more promising foundation for large-scale line planning.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that the traditional change-and-go network (CGN) model for line planning is only competitive on small instances and frequently fails to produce feasible solutions on large networks, while a compact direct-connection approximation outperforms the CGN and two other transfer-approximating models, returning the best solution on over 83% of 972 instances drawn from Dutch and Swiss railway networks. The authors develop model-specific state-of-the-art solvers, evaluate both solution quality and modeling accuracy, and conclude that carefully designed approximations are preferable to exact transfer modeling for scalability.

Significance. If the empirical ordering and accuracy claims hold, the work challenges the canonical status of the CGN and supplies concrete guidance for large-scale public-transport planning. The scale of the testbed (972 real-network instances) and the provision of tailored solvers constitute clear strengths that could enable reproducible follow-on work.

major comments (2)
  1. [§4] §4 (experimental design): the abstract and results section state that solution quality and modeling accuracy are characterized, yet supply no explicit description of the objective metric used to declare one solution 'best,' how ties are broken, or how instances on which the CGN solver returns no feasible solution are scored in the 83% comparison; this information is load-bearing for the central performance claim.
  2. [§5] §5 (modeling accuracy): the conclusion that the compact models remain 'practically useful' rests on the assertion that their transfer approximations are sufficiently accurate relative to the exact CGN; the manuscript must show how this accuracy is quantified on the subset of instances where the CGN itself fails to produce a reference solution.
minor comments (2)
  1. Figure captions and legends should explicitly state the units and normalization used for passenger-travel-time or cost values so that cross-model comparisons are immediately interpretable.
  2. Ensure every acronym (CGN, DC, etc.) is defined at first use in the main text as well as in the abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help strengthen the clarity of our experimental claims. We address each major point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§4] §4 (experimental design): the abstract and results section state that solution quality and modeling accuracy are characterized, yet supply no explicit description of the objective metric used to declare one solution 'best,' how ties are broken, or how instances on which the CGN solver returns no feasible solution are scored in the 83% comparison; this information is load-bearing for the central performance claim.

    Authors: We agree that these methodological details must be stated explicitly. In the revised manuscript we will add a dedicated paragraph in the experimental design subsection stating: (i) the primary objective is minimization of total passenger travel time (with a secondary term for operating cost); (ii) ties are broken first by number of lines operated and then lexicographically by line frequencies; (iii) for the 83 % figure, any instance on which the CGN solver returns no feasible solution is counted as a win for every compact model that does return a feasible solution. This clarification will be placed before the aggregate statistics are presented. revision: yes

  2. Referee: [§5] §5 (modeling accuracy): the conclusion that the compact models remain 'practically useful' rests on the assertion that their transfer approximations are sufficiently accurate relative to the exact CGN; the manuscript must show how this accuracy is quantified on the subset of instances where the CGN itself fails to produce a reference solution.

    Authors: We acknowledge the need for an explicit accuracy assessment on the CGN-failure subset. In the revision we will add a new table (or subsection) that, for every instance where the CGN solver fails, reports the passenger travel time obtained by (a) the compact model’s own approximation and (b) an exact passenger-routing simulation performed on the line plan returned by that compact model. The difference between (a) and (b) supplies a direct, instance-specific measure of modeling error even in the absence of a CGN reference solution. We will also report aggregate statistics (mean and maximum relative error) over this subset. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

This is an empirical comparison paper that evaluates four existing line-planning models (exact CGN plus three transfer approximations) on 972 real-world instances from Dutch and Swiss networks using model-specific solvers. No derivation chain, fitted parameters, self-citations as load-bearing premises, or ansatzes are present; performance ordering is established directly by computational results on external benchmark data. The central claim (direct-connection model wins on >83% of instances) is falsifiable against the reported instance set and does not reduce to any input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The comparison rests on the domain assumption that passenger routing behavior can be approximated without the full change-and-go network while still producing useful line plans.

axioms (1)
  • domain assumption Passenger routing can be approximated by the three compact models without invalidating solution quality for practical planning
    Central to the claim that the direct-connection model is preferable despite being an approximation.

pith-pipeline@v0.9.1-grok · 5713 in / 1107 out tokens · 22625 ms · 2026-06-25T23:21:19.279127+00:00 · methodology

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Reference graph

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