A tutorial on non-periodic train timetabling and platforming problems

Abstract

In this tutorial, we give an overview of two fundamental problems arising in the optimization of a railway system: the train timetabling problem (TTP), in its non-periodic version, and the train platforming problem (TPP). We consider for both problems the planning stage, i.e. we face them from a tactical point of view. These problems correspond to two main phases that are usually optimized in close sequence by the railway infrastructure manager. First, in the TTP phase, a schedule of the trains in a railway network is determined. A schedule consists of the arrival and departure times of each train at each (visited) station. Second, in the TPP phase, one needs to determine a stopping platform and a routing for each train inside each (visited) station, according to the schedule found in the TTP phase. Due to the complexity of the two problems, an integrated approach is generally hopeless for real-world instances. Hence, the two phases are considered separately and optimized in sequence. Although there exist several versions for both problems, depending on the infrastructure manager and train operators requirements, we do not aim at presenting all of them, but rather at introducing the reader to the topic using small examples. We present models and solution approaches for the two problems in a didactic way and always refer the reader to the corresponding papers for technical details.

Appendix

1.1 TTP variables and parameters

In the following, we list parameters and variables of the TTP models presented in Sect. 2.4.

The input consists of:

• $S=\{1,\ldots ,s\}$ is the set of stations, numbered according to the order in which they appear along the corridor for the running direction considered

• $T=\{1,\ldots ,t\}$ is the set of candidate trains

• For each train $j\in T$, a first (departure) station $f_j$ and a last (destination) station $l_j$ ($l_j>f_j$) are given

• $S^j:=\{f_j,\ldots ,l_j\}\subseteq S$ is the ordered set of stations visited by train $j \in T$

• $a_i$ is the minimum time interval for station $i\in S$ between two consecutive arrivals of trains

• $d_i$ is the minimum time interval for station $i\in S$ between two consecutive departures of trains

• $\pi _j$ is the ideal profit of train $j \in T$

• $\nu _j$ is the shift of train $j \in T$

• $\mu _j$ is the stretch of train $j \in T$

• $\alpha _j$ is the penalty for each minute of shift for train $j \in T$

• $\gamma _j$ is the penalty for each minute of stretch for train $j \in T.$

Both models are based on the graph representation discussed in Sect. 2.3, consisting of a (directed, acyclic) space-time multigraph $G=(V,A)$.

The node set $V$ has the form $\{\sigma ,\tau \}\cup (U^2\cup \ldots \cup U^s)\cup (W^1\cup \ldots \cup W^{s-1})$, where

• $\sigma$ and $\tau$ are an artificial source node and an artificial sink node, respectively

• set $U^i$, $i\in S\setminus \{1\}$, represents the set of time instants in which some train can arrive at station $i$; the nodes in $U^2\cup \ldots \cup U^s$ are called arrival nodes.

• set $W^i$, $i\in S\setminus \{s\}$, represents the set of time instants in which some train can depart from station $i$; the nodes in $W^1\cup \ldots \cup W^{s-1}$ are called departure nodes.

The arc set $A$ is partitioned into sets $A^1,\ldots , A^t$, one for each train $j\in T$.

• $\theta (v)$ is the time instant associated with node $v \in V$

• $\Delta (u,v)$ is the time distance between two nodes $u$ and $v$, with $u, v \in V$

• $V^j$ denotes the set of nodes associated with time instants corresponding to possible arrivals/departures of train $j.$

1.1.1 TTP path model

• ${\mathcal {P}}$ is the collection of all possible paths in graph $G=(V,A)$

• ${\mathcal {P}}^j$ is the collection of possible paths for train $j$ in graph $G=(V,A)$

• $p_P:=\sum _{a\in P} p_a$ is the profit for path $P \in {\mathcal {P}}$ ($p_a$ is the profit associated with each arc $a \in A$, as defined in Sect. 2.3)

• ${\mathcal {P}}^j_w\subseteq {\mathcal {P}}^j$ is the (possibly empty) subcollection of paths for train $j$ that visit node $w \in V^j$

• ${\mathcal {P}}_w:={\mathcal {P}}^1_w\cup \dots \cup {\mathcal {P}}^t_w$ is the subcollection of paths that visit node $w \in V$

• $r_j^i$ is the travel time of train $j \in T$ from station $i$ to station $i+1$ ($i,i+1\in S^j$)

• $x_P$ is a binary variable for each possible path $P \in {\mathcal {P}}$ for a train, equal to $1$ if, and only if, the path is chosen in the solution.

1.1.2 TTP arc model

• $p_a$ is the profit associated with each arc $a \in A$, as defined in Sect. 2.3

• $\delta _j^+(v)$ is the set of arcs of train $j$ leaving node $v$ and $\delta _j^-(v)$ is the set of arcs of train $j$ entering node $v$

• $r_j^i$ is the travel time of train $j \in T$ from station $i$ to station $i+1$ ($i,i+1\in S^j$)

• $x_a$ is a binary variable, for each train $j\in T$ and each arc $a\in A^j$, equal to $1$ if, and only if, arc $a$ is selected in an optimal solution

• $y_v$ is a binary variable equal to $1$ if, and only if, node $v \in V$ is visited by any train

• $z_{jv}$ is a binary variable equal to $1$ if, and only if, train $j \in T$ visits node $v \in V.$

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1.2 TPP variables and parameters

In the following, we list parameters and variables of the TPP model presented in Sect. 3.3.

The input consists of:

• $T$ is the set of trains

• $B$ is the set of platforms

• $D$ is the set of directions for train arrivals and departures

• ${\mathcal {R}}$ is the set of paths connecting directions and platforms

• $g_d$ is the travel time for all paths connecting direction $d\in D$ to any platform

• $L_{d_1,d_2}\subseteq B$ is the preference list of preferred platforms for the ordered pair $(d_1,d_2)\in D\times D$

• ${\mathcal {R}}_{d,b} \subseteq {\mathcal {R}}$ is the subset of paths linking direction $d \in D$ to platform $b \in B$

• ${\mathcal {I}}_{R} \subseteq {\mathcal {R}}$ is a list of incompatible paths

• $u_{t}^{\rm a}$ is the ideal arrival time of train $t \in T$ at a platform

• $u_{t}^{\rm d}$ is the ideal departure time of train $t \in T$ from a platform

• $s_{t}^{\rm a}$ is the maximum arrival shift of train $t \in T$

• $s_{t}^{\rm d}$ is the maximum departure shift of train $t \in T$

• $d_{t}^{\rm a} \in D$ is the arrival direction of train $t \in T$

• $d_{t}^{\rm d} \in D$ is the departure direction of train $t \in T$

• $C_{t} \subseteq B$ is the set of candidate platforms where train $t \in T$ may stop.

The model is based on the concept of pattern and pattern incompatibility graph defined in Sect. 3.3.

• $P_t$ is the set of patterns that can be assigned to train $t \in T$

• $c_{t,p}$ is the train platforming cost if pattern $p\in P_t$ is assigned to train $t \in T$

• $c_{t_1,p_1,t_2,p_2}$ is the path incompatibility cost if patterns $p_1 \in P_{t_1}$ and $p_2 \in P_{t_2}$ are assigned to trains $t_1 \in T$ and $t_2 \in T$, respectively

• ${\mathcal {K}}$ is the whole collection of cliques in the pattern incompatibility graph

• ${\mathcal {K}}_b$ is the collection of cliques in the pattern incompatibility graph associated with the sets of patterns that use platform $b \in B$ at the same time

• $y_b$ is a binary variable for each $b\in B$, with $y_b=1$ if and only if platform $b$ is used

• $x_{t,p}$ is a binary variable for each $t\in T$ and $p\in P_t$, with $x_{t,p}=1$ if and only if train $t$ is assigned pattern $p.$