Timetabling with passenger routing

Abstract

Customer-oriented optimization of public transport needs data about the passengers in order to obtain realistic models. Current models take passengers’ data into account by using the following two-phase approach: In a first phase, routes for the passengers are determined. In a second phase, the actual planning of lines, timetables, etc., takes place using the knowledge on which routes passengers want to travel from the results of the first phase. However, the actual route a passenger will take strongly depends on the timetable, which is not yet known in the first phase. Hence, the two-phase approach finds non-optimal solutions in many cases. In this paper we study the integrated problem of determining a timetable and the passengers’ routes simultaneously. We investigate the computational complexity of the problem and present solution approaches which are tested on close-to-real-world data.

Appendix

Theorem 1 TTwR is strongly NP-hard even if all OD-pairs have the same origin.

Proof

We show this by reduction from the strongly NP-complete decision problem Set Cover (Garey and Johnson 1979) defined as follows: Given $P=\{p_1,\ldots ,p_m\}$, a set of subsets $Q=\{q^1,\ldots , q^n\}$ with $q^j\subset P$ and a natural number $K$, decide whether there is a subset $Q'\subset Q$ with $|Q'|\le K$ such that for every $p_i\in P$ there is at least one $q^j\in Q'$ such that $p_i$ is contained in $q^j$.

Let $(P,Q,K)$ be an instance of Set Cover with $P=\{p_1,\ldots ,p_m\}$ and $Q=\{q^1,\ldots , q^n\}$. Without loss of generality we assume that $K< n$. Based on $P$ and $Q$ we construct an instance of TTwR in the following way:

We represent the elements $p_i\in P$ by OD-pairs $(u,v_i)$ with passenger numbers $w_{uv_i}:=K+1$.

The sets $q^j\in Q$ are represented by a gadget consisting of three trains $(\mathrm{tr}^j)^1$, $(\mathrm{tr}^j)^2$ and $(\mathrm{tr}^j)^3$, six stations $s_1,s^j_2,s^j_3,s^j_4, s^j_5,s^j_6$ (where $s_1$ is the same station for all OD-pairs) and an OD-pair $(u,v^j)$ with $u:=s_1$ and $v^j:=s^j_5$ and $w_{uv^j}:=1$, in the way depicted in Fig. 4.

The square nodes are the departure and arrival events where the train names are omitted in the node labels for the sake of a compact representation. The origin and destination events are represented by ovals. The dotted lines are the origin and destination activities, the solid lines represent driving and waiting activities, transfer activities are represented by dashed lines. The gray line indicates where it is possible to leave this gadget.

Fig. 4

The gadget $g^j$ representing a set $q^j$ in the reduction from Set Cover to TTwR in the proof of Theorem 1

For every $p_i\in q^j$ we introduce a train $\mathrm{tr}_i^j$ running from $s_3^j$ to a station $v_i$ and connect $((\mathrm{tr}^j)^2-s^j_3-\mathrm{Arr})$ to a departure event $(\mathrm{tr}_{i}^j-s^j_3-\mathrm{Dep})$ by a transfer activity with upper and lower bound $u_{((\mathrm{tr}^j)^2-s^j_3-\mathrm{Arr}),(\mathrm{tr}_{i}^j-s^j_3-\mathrm{Dep})}:=0$ and $l_{((\mathrm{tr}^j)^2-s^j_3-\mathrm{Arr}),(\mathrm{tr}_{i}^j-s^j_3-\mathrm{Dep})}:=0$. $(\mathrm{tr}_{i}^j-s^j_3-\mathrm{Dep})$ is connected to the arrival event of train $\mathrm{tr}_{i}^j$ in $v_i$ by a driving activity $a$ with upper and lower bound $l_a:=0$, $u_a:=0$.

See Fig. 5 (in combination with Fig. 4) for an example of the construction for an instance of Set Cover with $P=\{1,2,3,4\}$ and $Q=\{\{2,3,4\},\{1,4\},\{2,3\}\}$. The square nodes are the departure and arrival events. The origin and destination events are represented by ovals. The dotted lines are the origin and destination activities, the solid lines represent driving and waiting activities, transfer activities are represented by dashed lines. The nodes $g^j$ represent the gadgets from Fig. 4.

Fig. 5

Example for the construction of the event-activity network in the proof of Theorem 1

Summarizing, our instance $I$ of TTwR is defined by the following input, which defines an EAN $N$ with upper and lower bounds $l,u$ and a set of OD-pairs $\mathcal {OD}$:

• a set of stations $S=\{s_1\}\cup \{s_2^j,s_3^j,s_4^j,s_5^j:j=1,\ldots ,n\}\cup \{v_i:i=1,\ldots ,m\}$,

• a set of trains $\mathrm{TR}=\{(\mathrm{tr}^j)^1,(\mathrm{tr}^j)^2,(\mathrm{tr}^j)^3:j=1,\ldots ,n\}\cup \{\mathrm{tr}_i^j:p_i\in q^j\}$ where $(\mathrm{tr}^j)^1$ run on the paths $(s_2^j,s_5^j)$, $(\mathrm{tr}^j)^2$ run on $(s_1,s_2^j,s_3^j,s_4^j,s_5^j)$, $(\mathrm{tr}^j)^3$ run on $(s_1,s_4^j)$ and $\mathrm{tr}_i^j$ run on $(s_3^j,v_i)$.

• transfers

  $$\begin{aligned} \mathcal {C}&= \{((\mathrm{tr}^j)^2,(\mathrm{tr}^j)^1,s_2^j), ((\mathrm{tr}^j)^1,(\mathrm{tr}^j)^2,s_5^j), ((\mathrm{tr}^j)^3,(\mathrm{tr}^j)^2,s_4^j): j=1,\ldots ,n\}\\&\cup \{((\mathrm{tr}^j)^2,\mathrm{tr}_i^j,s_3^j):p_i\in q^j\} \end{aligned}$$

• a lower bound $l_{((\mathrm{tr}^j)^1-s^j_2-\mathrm{Dep},(\mathrm{tr}^j)^1-s^j_5-\mathrm{Arr})}=1$ on driving activities $((\mathrm{tr}^j)^1-s^j_2-\mathrm{Dep},(\mathrm{tr}^j)^1-s^j_5-\mathrm{Arr})$ for $j=1,\ldots ,n$ and lower bounds $l_a:=0$ on all other activities. The upper bounds are $u_a=1$ for

  $$\begin{aligned} a\in \{&((\mathrm{tr}^j)^1-s^j_2-\mathrm{Dep},(\mathrm{tr}^j)^1-s^j_5-\mathrm{Arr}),\\&((\mathrm{tr}^j)^2-s^j_2-\mathrm{Dep},(\mathrm{tr}^j)^2-s^j_3-\mathrm{Arr}), \\&((\mathrm{tr}^j)^2-s^j_4-\mathrm{Dep},(\mathrm{tr}^j)^2-s^j_5-\mathrm{Arr}): j=1,\ldots ,n\} \end{aligned}$$

  and $u_a:=0$ otherwise.

• The set of OD-pairs is $\mathcal {OD}=\{(u,v^j):j=1,\ldots ,n\}\cup \{(u,v_i):i=1,\ldots ,m\}$ with $u:=s_1$ and $v^j:=s_5^j$. The weights are $w_{uv^j}=1$ and $w_{uv_i}=K+1$.

We show that there is a solution $(\pi ,\mathcal {R})$ to the constructed instance $I$ with $c_{\mathrm{TTwR}}(\pi ,\mathcal {R})\le K$ if and only if there is a subset $Q'$ solving the Set Cover problem.

We observe that if for an OD-pair $(u,v_i)$ there is a gadget $g^j$ such that $u-{\mathrm{org}}$ is connected to $g^j$ and there is a length of $0$ assigned to $((\mathrm{tr}^j)^2-s_2^j-\mathrm{Dep},(\mathrm{tr}^j)^2-s_3^j-\mathrm{Arr})$, the OD-pair will arrive at its destination in time $0$ while if there is no such structure, there will be a contribution of $n+1$ to the objective function. Thus, in a feasible solution, for every OD-pair $(u,v_i)$ there must be at least one gadget $g^j$ such that $(\mathrm{tr}_i^j-s_3^j-\mathrm{Dep})$ is connected to $g^j$ and there is a length of $0$ assigned to $((\mathrm{tr}^j)^2-s_2^j-\mathrm{Dep},(\mathrm{tr}^j)^2-s_3^j-\mathrm{Arr})$. Note that in this case it follows that the length of $((\mathrm{tr}^j)^2-s_4^j-\mathrm{Dep},(\mathrm{tr}^j)^2-s_5^j-\mathrm{Arr})$ is $1$, hence $c_{\mathrm{TTwR}}(\pi ,P^j)=1$ for the corresponding timetable $\pi$ and any path $P^j$ chosen for OD-pair $(u,v^j)$.

We conclude that $Q'$ is a solution to the considered instance of Set Cover if and only if assigning a length of $1$ to $((\mathrm{tr}^j)^2-s_2^j-\mathrm{Dep},(\mathrm{tr}^j)^2-s_3^j-\mathrm{Arr})$ for all $j$ with $q_j\in C'$ and a length of $0$ otherwise leads to a solution to the constructed instance of TTwR with solution value $\le$K. $\square$