Price Optimal Routing in Public Transportation

TL;DR

This work addresses price-optimal routing in public transit by formulating the price-optimal earliest arrival problem ($POEAP$) and introducing conditional fare networks ($CFN$) to capture complex fare structures. By integrating CFNs with a multi-criteria RAPTOR framework ($McRAP$), the authors derive a label-setting algorithm that computes price-aware Pareto-sets efficiently, despite the NP-hardness of the underlying problem. They establish a formal dominance and comparability framework to maintain subpath optimality in the presence of fare transitions, and they propose practical speed-ups and restricted Pareto-sets to ensure scalable performance. Empirical results on the MDV network show sub-second query times on average (and sub-10 ms with Tight-BMRAP), demonstrating the potential for real-world deployment and broader applicability to diverse fare schemes.

Abstract

We consider the price-optimal earliest arrival problem in public transit (POEAP) in which we aim to calculate the Pareto-set of journeys with respect to ticket price and arrival time in a public transportation network. Public transit fare structures are often a combination of various fare strategies such as, e.g., distance-based fares, zone-based fares or flat fares. The rules that determine the actual ticket price are often very complex. Accordingly, fare structures are notoriously difficult to model, as it is in general not sufficient to simply assign costs to arcs in a routing graph. Research into POEAP is scarce and usually either relies on heuristics or only considers restrictive fare models that are too limited to cover the full scope of most real-world applications. We therefore introduce conditional fare networks (CFNs), the first framework for representing a large number of real-world fare structures. We show that by relaxing label domination criteria, CFNs can be used as a building block in label-setting multi-objective shortest path algorithms. By the nature of their extensive modeling capabilities, optimizing over CFNs is NP-hard. However, we demonstrate that adapting the multi-criteria RAPTOR (MCRAP) algorithm for CFNs yields an algorithm capable of solving POEAP to optimality in less than 400 ms on average on a real-world data set. By restricting the size of the Pareto-set, running times are further reduced to below 10 ms.

Figures (5)

• Figure 1: A section of MDVs fare plan with two lines and six fare zones. Two of the fare zones, colored in light gray, are the cities of Halle and Leipzig. Vertically hatched hexagons represent overlap areas that can be counted as either of the neighboring zones. The horizontally hatched circle represents the small city Merseburg in which a special discounted fare is applicable. Small black nodes represent public transit stops. Footpaths are indicated by dotted lines.
• Figure 2: Ticket Graph associated with the MDV public transit network. To simplify the presentation, all tickets for city fares are collapsed to $C_{1}$ and $C_{2}$ representing the two price levels of city fares. Possible starting tickets are highlighted in light gray.
• Figure 3: Example of a routing graph (a) with two possible conditional fare networks (b) and (c). For both networks, the underlying partially ordered monoid is $(\mathbb{R},+,\leq)$, the fare events are $S = \{s_0,s_1,s_2,s_3\}$ and the initial fare state for all vertices $v_i$ with $i= 1,\dots,5$ is $\mu(v_i) = (A,0)$. We set prices for the tickets as $\pi(A) = 0$, $\pi(B) = 2$, $\pi(C) = 3$, $\pi(D) = 1$ and $\pi(E) =5$. The value of the transition function $\Gamma$ for a given weight $h$ and event $s$ is given via indicator functions on the fare arcs. Using the ticket graph (b), the upper $v_1,v_5$-path yields ticket $C$, while the lower path yields ticket $E$. Using ticket graph (c), the upper path yields ticket $B$, the lower path yields ticket $C$.
• Figure 4: Illustration of the solution space of a Tight-BMRAP search with $\sigma_{arr}=30 \textrm{ min}$ and $\sigma_{tr} = 1$. The circle marks represent the anchor journeys from $\mathcal{J}_{\mathcal{A}}$. Tight-BMRAP prunes all journeys to the right of the dotted line spanned by those journeys. The area to the left of the dashed line contains no Pareto-optimal journeys. The journeys from $\mathcal{J}^f$ that fall into the area enclosed by the dashed and dotted lines form $\mathcal{J}_{\mathcal{R}}$. The light gray area forms $\bar{ \mathcal{J}_{\mathcal{R}}}$. Note that the journey marked by the square mark is in $\mathcal{J}_{\mathcal{R}}$ but not in $\bar{ \mathcal{J}_{\mathcal{R}}}$. The journey represented by the triangle mark is in $\bar{ \mathcal{J}_{\mathcal{R}}}$ even though it uses more trips and has a later arrival time.
• Figure 5: Transformation from CFN to DFA. The CFN $\mathcal{N} = ( \mathcal{T},\Gamma,w,e,\mu,\pi)$ is given by the ticket graph $\mathcal{T}$ depicted in (a). Possible ticket transitions are given as indicator functions on the arcs. The associated fare monoind is $(H,+,\leq)$ with $H:=\{0,1,2\}$ and $a+b := \min \{ a + b, 2\}$ and $0\leq1\leq2$ and the fare events are given as $S=\{s_1,s_2\}$. The definitions of $w$, $e$ and $\mu$ depend on the routing graph and are omitted in this example. Our transformation results in the DFA in (b). There is a state for each element from $T\times H$. Each arc depicts a possible state transformation. For each arc, there is at least one letter from $\Sigma = H\times S$ that allows this transformation. There is no arc between $(\tau_1,0)$ and $(\tau_2,0)$ as moving from $\tau_1$ to $\tau_2$ in the ticket graph requires $h\geq1$. There should be a loop at every state, e.g., $(\tau_1,0)$ transforms into $(\tau_1,0)$ if letter $(0,s_1)$ or $(0,s_2)$ was found. We omit them in b) to not clutter the presentation.

Theorems & Definitions (32)

• Example 2.1: The Fare System of MDV
• Definition 3.1: Partially ordered monoid
• Example 3.1: MDV
• Example 3.2: Running Example: Ticket Graph for MDV
• Example 3.3: Running Example: Fare Events for MDV
• Definition 3.2: Fare State
• Definition 3.3: Ticket Transition Function
• Definition 3.4: Fare Update Function
• Definition 3.5: Conditional Fare Network
• Example 3.4: Running Example: Conditional Fare Network for MDV