Fare Structure Design in Public Transport

TL;DR

The paper investigates fare structure design for public transport, aiming to set prices so that the new fares closely match reference prices while satisfying no-elongation and no-stopover properties. It shows that flat tariffs and affine distance tariffs reduce to median problems, enabling linear-time solutions, and provides a detailed analysis of zone tariffs, including four variants with varying counting schemes and connectivity constraints; zone design is NP-hard in general, but the price-setting subproblem with fixed zones is tractable and solvable via weighted medians or linear programming. The authors develop comprehensive MILP formulations for all variants, enabling practical optimization of fare structures, and offer algorithmic insights such as a polynomial-time zone-no-elongation algorithm. The work bridges theoretical results on medians and location problems with practical fare design, delivering actionable methods for transit agencies and contributing to open-source planning tools such as LinTim. The findings have direct implications for revenue, equity, and passenger satisfaction by providing transparent, enforceable fare rules and scalable solution approaches.

Abstract

Fare planning is one among several steps in public transport planning. Fares are relevant for the covering of costs of the public transport operator, but also affect the ridership and the passenger satisfaction. A fare structure is the assignment of prices to all paths in a network. In practice, often a given fare structure shall be changed to fulfill new requirements, meaning that a new fare strategy is desired. This motivates the usage of prices of the former fare structure or other desirable prices as reference prices. In this paper, we investigate the fare structure design problem that aims to determine fares such that the sum of absolute deviations between the new fares and the reference prices is minimized. Fare strategies that are considered here are flat tariffs, affine distance tariffs and zone tariffs. Additionally, we regard constraints that ensure that it is not beneficial to buy a ticket for a longer journey than actually traveled (no-elongation property) or to split a ticket into several sub-tickets to cover a journey (no-stopover property). Our literature review provides an overview of the research on fare planning. We analyze the fare structure design problem for flat, distance and zone tariffs, pointing out connections to median problems. Further, we study its complexity which ranges from linear-time solvability to NP-complete cases.

Figures (12)

• Figure 1: Example with four OD pairs with one passenger each illustrating the proof of \ref{['prop:distance-finite-candidate-set']}. The optimal solution is $p^{\ast}=0$, $f^{\ast}=2$. This shows that the three options stated in \ref{['prop:distance-finite-candidate-set']} are not mutually exclusive.
• Figure 2: Instance showing that it can be better to implement fewer zones.
• Figure 3: Relationships between the optimal objective function values of the four problem variants of the zone tariff design problem in different cases (arbitrary PTN or tree, arbitrary $N\in \mathbb{N}_{\geq 1}$ or $N=\vert V \vert$). Note that the results on trees are for the case that the paths $W_d$ are the unique simple paths for all OD pairs $d$ (see \ref{['prop:relations']}).
• Figure 4: Instance for \ref{['ex:zdma-better-than-zdmc']}.
• Figure 5: Instance for \ref{['ex:zdma-better-than-zdsa-tree']}.

Theorems & Definitions (50)

• Definition 1: Fare structure Schoebel2022
• Definition 2: Fare structure design problem
• Definition 3: No-elongation property and no-stopover property Schoebel2022
• Definition 4: Weighted median
• Definition 5: Flat tariff Schoebel2022
• Definition 6: Flat tariff design problem
• Definition 7: Affine distance tariff Schoebel2022
• Definition 8: Affine distance tariff design problem
• Theorem 9
• proof