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arxiv: 2604.18013 · v1 · submitted 2026-04-20 · 🧮 math.OC

An Exact Algorithm for Public Transport Line Planning Considering Passenger and Operational Costs and Lost Demand

Siv Marie Cartland Hansen , Rowan Hoogervorst , Otto Anker Nielsen , Richard Martin Lusby , Evelien van der Hurk This is my paper

Pith reviewed 2026-05-10 04:17 UTC · model grok-4.3

classification 🧮 math.OC
keywords public transportline planningmixed-integer programmingexact algorithmlost demandpassenger costsoperational costsiterative optimization
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The pith

An iterative exact algorithm solves public transport line planning by expanding a reduced set of paths and frequencies while treating demand below a minimum service level as lost.

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

The paper develops a mixed-integer model that selects lines and frequencies to minimize the combined cost of passenger travel times and operator expenses, but only includes demand that meets a minimum service attractiveness threshold; any demand falling short is counted as lost. This setup forces the model to balance service quality against costs more realistically than fixed-demand approaches. To handle the rapid growth in possible passenger paths, the authors propose an iterative exact method that solves a smaller initial problem and then adds frequencies and paths dynamically until optimality is proven. On test networks the procedure delivers faster run times and stronger bounds than optimizing the full model directly, with the largest gains appearing when passenger and operator costs receive similar weight. The results also show that explicitly modeling lost demand produces line plans that use vehicles and infrastructure more efficiently overall.

Core claim

The authors formulate line planning as a mixed-integer program minimizing weighted passenger travel time plus operating costs, subject to capacity limits and a minimum service level that must be met before demand on a path is captured. They introduce an iterative exact algorithm that maintains a reduced representation of paths and frequencies and enlarges it iteratively, proving optimality without enumerating the full combinatorial set at the outset.

What carries the argument

An iterative exact algorithm that begins with a reduced problem representation of paths and frequencies and dynamically expands it until the optimal solution to the full line-planning model with lost demand is found.

If this is right

  • The iterative method produces significant speed-ups and tighter bounds relative to solving the complete model at once.
  • The computational advantage is largest when passenger and operator costs are weighted more evenly in the objective.
  • Explicitly accounting for lost demand yields line plans that use resources more efficiently from a combined passenger-operator perspective.
  • Capacity constraints on selected lines guide the trade-off between expected demand and chosen frequencies.

Where Pith is reading between the lines

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

  • Planners could run the algorithm repeatedly with different cost weights to explore how service levels change when passenger convenience is valued more or less than operator expense.
  • The dynamic expansion idea might adapt to timetable design or vehicle scheduling problems where the number of variables also grows quickly with network size.
  • Treating low-service demand as lost implicitly favors concentrating resources on high-ridership corridors rather than spreading thin coverage everywhere.

Load-bearing premise

The minimum service level threshold used to decide whether demand is captured accurately reflects real passenger choices about whether to travel or switch modes.

What would settle it

Solve a small network instance to proven optimality with both the iterative algorithm and a direct commercial solver on the complete model, then check whether the selected lines, frequencies, captured demand, and objective value are identical.

Figures

Figures reproduced from arXiv: 2604.18013 by Evelien van der Hurk, Otto Anker Nielsen, Richard Martin Lusby, Rowan Hoogervorst, Siv Marie Cartland Hansen.

Figure 1
Figure 1. Figure 1: An example CGN with two lines (l1, l2), each with three stops. A path for OD pair (1,5) is shown in blue, as well as a direct path from the access node of stop 1 to the egress node of stop 5, representing an alternative mode. The CGN = (N , A), as first introduced by Schöbel and Scholl (2006) and illustrated in [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The relationship between headway h and required vehicles ϕlh for a line l ∈ L with 10 candidate frequencies. LPP(F). Moreover, these upper bounds can be used to track the progress of the algorithm throughout the solve by comparing against the lower bounds found. In the following, we discuss the steps of the algorithm in detail. We then present a set of valid inequalities relevant in the context of lost dem… view at source ↗
Figure 3
Figure 3. Figure 3: PTNs from LinTim. The ten stops with the highest demand for transportation are highlighted. [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Results for different values of λ To analyze the performance of the DFRA under a more balanced weighting of passenger and operator costs, we focus on a setting with λ = 0.25. Figure 5a shows the number of instances solved to optimality in relation to the computation time needed. Not only does the DFRA solve more instances to optimality at this setting, but it does so much faster. The number of solved insta… view at source ↗
Figure 5
Figure 5. Figure 5: Results for λ = 0.25 the respective upper bounds obtained with each method (RUB) is computed as zM UB−z A UB |zM UB| , where z i UB denotes the upper bound obtained with method i ∈ {A, M}. When the relative upper bound is greater than zero, it indicates that A provides a tighter bound. The “Speed-up” column reports the average per-instance speed-up between the two methods. For the 16 instances that can be … view at source ↗
Figure 6
Figure 6. Figure 6: Number of instances solved to optimality for [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: a clearly shows that LPP(R,T) captures far less demand than the networks are designed for. Despite the network’s capacity to serve all passengers, only an average of 69–81% of demand is captured. This gap suggests that under fixed-demand assumptions, the supply and actual demand are not aligned. The model LPP(T,P) captures slightly less demand, except at the lowest λ value. However, while the demand differ… view at source ↗
Figure 7
Figure 7. Figure 7: Results over different values of λ for instances solved to optimality time – many lines fail to attract sufficient ridership to justify their operational cost. In contrast, the line plans generated by LPP(T) and LPP(T,P) emphasize operating fewer lines at higher frequencies. This approach prioritizes providing a better service where demand is concentrated, capturing more passengers on fewer, well-utilized … view at source ↗
Figure 8
Figure 8. Figure 8: Insights on route assignment values, it consistently provides tighter bounds than solving the full formulation. While this study represents a promising application of the DFRA on instances with up to 129 lines and 300 ODs, scalability to larger, more realistic networks is limited. Scalability could be improved through network and OD matrix preprocessing or dynamic path generation methods such as column gen… view at source ↗
read the original abstract

Line planning in public transport is the strategic problem of selecting lines and their operating frequencies. This problem is important as it defines the passenger service, based on available connections and expected travel times, and drives operational cost in terms of the number of vehicles required. This paper presents a line planning model that minimizes the weighted sum of passenger travel time, including in-vehicle time and frequency-dependent waiting and transfer times, and operating costs for the public transport agency. Unlike traditional approaches that assume demand to be fixed, our approach requires a minimum service level for demand to be captured, ensuring that services are provided only when they are attractive to users and cost-efficient to operate. The introduced capacity constraints ensure sufficient capacity on the lines and help guide the trade-off between expected demand on selected lines and their frequencies. The resulting mixed-integer program presents a challenging combinatorial problem as the number of passenger paths grows rapidly in relation to the number of lines and frequencies considered. To address this, we propose an iterative exact algorithm that utilizes a reduced problem representation and dynamically expands it with additional frequencies and paths. Evaluated on four networks with varying complexity and cost trade-offs, our method achieves significant speed-ups and tighter bounds compared to solving the complete model directly by CPLEX, particularly when operator and passenger costs are more evenly balanced in the objective. Furthermore, we demonstrate how accounting for lost demand leads to more efficient resource use from an overall perspective.

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

3 major / 2 minor

Summary. The paper formulates a mixed-integer program for public transport line planning that minimizes a weighted combination of passenger travel time (in-vehicle, frequency-dependent waiting, and transfers) and operator costs, subject to a minimum service level threshold for capturing demand and capacity constraints on lines. It introduces an iterative exact algorithm that solves a reduced subproblem and dynamically expands the representation by adding frequencies and passenger paths. Experiments on four networks report speed-ups and tighter bounds versus direct CPLEX solution of the full model (especially under balanced cost weights) and claim that modeling lost demand yields more efficient overall resource allocation.

Significance. If the iterative procedure is proven to preserve optimality and the lost-demand results are shown to be robust, the work would provide a practical exact method for a combinatorially hard transportation planning problem, enabling solution of instances that are currently intractable and offering concrete evidence on the value of endogenous demand modeling. The emphasis on computational performance under varying cost trade-offs addresses a real gap in the literature on line planning.

major comments (3)
  1. [§4] §4 (Iterative Algorithm): The description of the dynamic expansion procedure (adding frequencies and paths) does not include a formal argument or invariant showing that the final solution remains optimal for the complete MIP; without this, the claim that the method is 'exact' rests on an unverified property that is central to the algorithmic contribution.
  2. [§3.2] §3.2 (Demand Capture): The minimum service level threshold that determines whether demand is captured or lost is introduced as a modeling parameter without calibration to observed ridership data or sensitivity analysis; this choice directly controls which paths enter the objective and is therefore load-bearing for the claim that accounting for lost demand produces more efficient resource use.
  3. [§5] §5 (Computational Experiments): The reported speed-ups and bound improvements are summarized qualitatively but lack per-instance tables showing the exact cost-weight vectors, network sizes, and final optimality gaps for both the iterative method and CPLEX; this prevents assessment of whether the gains are consistent or confined to particular regimes.
minor comments (2)
  1. [§3] Notation for the set of passenger paths and the frequency variables should be introduced with an explicit index list in the model section to improve readability.
  2. [§5] The abstract states results on 'four networks with varying complexity' but the main text does not provide a summary table of network statistics (nodes, lines, OD pairs); adding this would help readers contextualize the computational results.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which help clarify the presentation of our algorithmic contribution and strengthen the empirical analysis. We address each major comment below and indicate the planned revisions.

read point-by-point responses

  1. Referee: [§4] §4 (Iterative Algorithm): The description of the dynamic expansion procedure (adding frequencies and paths) does not include a formal argument or invariant showing that the final solution remains optimal for the complete MIP; without this, the claim that the method is 'exact' rests on an unverified property that is central to the algorithmic contribution.

    Authors: We agree that an explicit formal argument is necessary to substantiate the exactness claim. The algorithm initializes a reduced MIP with a subset of frequencies and passenger paths, solves it to optimality, and iteratively augments the formulation with additional frequencies and paths whose inclusion can improve the objective (identified via reduced-cost or dual information). Upon termination, the solution is optimal for the full MIP because any omitted variable has a non-negative reduced cost and cannot improve the objective. To make this rigorous, we will add a new subsection in §4 that states the invariant (current solution is optimal for the current subproblem, and the expansion rule preserves feasibility and optimality) together with a termination proof. This material will appear in the revised manuscript. revision: yes

  2. Referee: [§3.2] §3.2 (Demand Capture): The minimum service level threshold that determines whether demand is captured or lost is introduced as a modeling parameter without calibration to observed ridership data or sensitivity analysis; this choice directly controls which paths enter the objective and is therefore load-bearing for the claim that accounting for lost demand produces more efficient resource use.

    Authors: The threshold is a deliberate modeling parameter that encodes the minimum service attractiveness required for demand to be served rather than lost. Its value was chosen to reflect typical public-transport planning practice. Because the test networks are stylized, direct calibration against proprietary ridership data is not feasible. We will nevertheless add a sensitivity study in the revised §5 that varies the threshold over a plausible range and reports the resulting changes in captured demand, total cost, and resource allocation; this will demonstrate that the qualitative conclusion on improved efficiency remains robust. revision: yes

  3. Referee: [§5] §5 (Computational Experiments): The reported speed-ups and bound improvements are summarized qualitatively but lack per-instance tables showing the exact cost-weight vectors, network sizes, and final optimality gaps for both the iterative method and CPLEX; this prevents assessment of whether the gains are consistent or confined to particular regimes.

    Authors: We accept that the current summary tables obscure instance-level detail. In the revised manuscript we will replace the qualitative summary with expanded tables that, for every network and every tested cost-weight vector, list: network size (nodes, candidate lines, possible frequencies), the precise passenger and operator cost weights, wall-clock times and optimality gaps for both the iterative algorithm and direct CPLEX, and the resulting speed-up factor. These tables will make the performance claims fully verifiable. revision: yes

Circularity Check

0 steps flagged

No circularity in model or algorithm derivation

full rationale

The paper defines a MIP model with explicit minimum-service-level threshold for capturing demand, capacity constraints, and an objective combining passenger and operator costs. It then presents an independent iterative exact algorithm that solves a reduced representation and dynamically adds paths/frequencies. No equation or claim reduces by construction to a fitted parameter, self-citation, or renamed input; the reported speed-ups and lost-demand efficiency results are computational outcomes of applying the algorithm to the stated model on benchmark networks. The threshold is a modeling choice whose validity is external to the derivation chain.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The model rests on standard mixed-integer programming assumptions plus the domain-specific rule that demand is only realized above a minimum service threshold; no new entities are postulated.

free parameters (1)
  • objective weights between passenger and operator costs
    The weighted sum requires explicit trade-off parameters whose values affect the reported speed-up behavior.
axioms (2)
  • domain assumption Demand is captured only when a minimum service level (frequency and travel time) is met.
    Central modeling choice invoked to handle lost demand and capacity constraints.
  • standard math The iterative expansion procedure preserves optimality of the final solution.
    Required for the exactness claim of the algorithm.

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