A pattern-based timetabling strategy for a short-turning metro line

Abstract

The planning of metro lines is typically done through a strictly hierarchical approach, which is effective but somewhat inflexible. In this paper, we propose a flexible semiperiodic timetabling strategy using short-turning; thus, allowing trains to turn before reaching the terminal station of a line. Our strategy produces timetables that are periodic with respect to a group of short-turning destinations. This is denoted by the term service pattern. We introduce the service pattern timetabling problem (SPTP). Given a service pattern, the SPTP optimizes the train timetable considering capacity restrictions. The SPTP is modeled as a constraint program. We develop a framework for producing a large set of diverse and high-quality timetables for a metro line. This is achieved by repeatedly solving the SPTP with different patterns. Then we select a restricted list of non-dominated solutions with respect to three objectives: (1) the average passenger waiting time, (2) the maximum load factor achieved by the trains, and (3) the number of transfers induced by short-turning. We evaluate the proposed framework on a number of test instances. Through our computational experiments, we demonstrate the effectiveness of the developed strategy.

Appendices

Appendix A: Table of notation

See Table 4.

Table 4 Table of notation

Appendix B: The service pattern timetabling problem model

$$\begin{aligned}&\text {min} \sum _{\gamma \in \Gamma } \sum _{k \in K} (h^{\gamma }_{k})^2 \end{aligned}$$

(B.1)

$$\begin{aligned}&\quad { s( \gamma ,k ) = s( \gamma ,1 ) + \sum _{k' = 2}^{k} h^{\gamma }_{k'} } \quad { \forall \gamma \in \Gamma , \ k \in K \setminus \{1\} } \end{aligned}$$

(B.2)

$$\begin{aligned}&e( \gamma ,k ) = s( \gamma ,k ) + w^{\gamma }_{k} + 2\tau ( \gamma ,k ) \quad \forall \gamma \in \Gamma , \ k \in K \end{aligned}$$

(B.3)

$$\begin{aligned}&z= \sum _{k \in K}h^{\gamma }_{k}\quad \forall \gamma \in \Gamma \end{aligned}$$

(B.4)

$$\begin{aligned}&s( \gamma ,k ) + zc^{\gamma }_{k} = e( -\gamma ,m^{\gamma }_{k} ) \quad \forall \gamma \in \Gamma , \ k \in K \end{aligned}$$

(B.5)

$$\begin{aligned}&\texttt {alldifferent}(m^{\gamma }_{k}, \ \forall k \in K)\quad \forall \gamma \in \Gamma \end{aligned}$$

(B.6)

$$\begin{aligned}&\sum _{\gamma \in \Gamma } \sum _{k \in K} (w^{\gamma }_{k} + 2\tau ( \gamma ,k )) = \Psi z\end{aligned}$$

(B.7)

$$\begin{aligned}&\mu ^{\gamma }_{k} = d( -\gamma ,m^{\gamma }_{k} ) \quad \forall \gamma \in \Gamma , \ k \in K \end{aligned}$$

(B.8)

$$\begin{aligned}&f^{\gamma }_{k,i} \le U C \quad \forall \gamma \in \Gamma , \ k \in K, \ i \in S \end{aligned}$$

(B.9)

$$\begin{aligned}&\quad { \gamma (r - \mu ^{\gamma }_{k}) \ge \gamma (r - i) \ \Rightarrow \ \eta ^{\gamma }_{k,i} = 1 } \quad \forall \gamma \in \Gamma , \ k \in K, \ i \in {\underline{S}}^{\gamma } \end{aligned}$$

(B.10)

$$\begin{aligned}&\quad \gamma (r - \mu ^{\gamma }_{k}) < \gamma (r - i) \ \Rightarrow \ \eta ^{\gamma }_{k,i} = 0 \quad \forall \gamma \in \Gamma , \ k \in K, \ i \in {\underline{S}}^{\gamma }\end{aligned}$$

(B.11)

$$\begin{aligned}&\quad \gamma (r - \mu ^{\gamma }_{k'}) \ge \gamma (r - i) \ \Rightarrow \ \lambda ^{\gamma ,i}_{k,k'} = 0 \nonumber \\&\quad { \forall \gamma \in \Gamma , \ k \in K, \ k' \in 1,\ldots ,b(k), \ i \in {\underline{S}}^{\gamma } }\end{aligned}$$

(B.12)

$$\begin{aligned}&\quad \lambda ^{\gamma ,i}_{k,k'+1} = 0 \ \Rightarrow \ \lambda ^{\gamma ,i}_{k,k'} = 0 \nonumber \\&\quad {\forall \gamma \in \Gamma , \ k \in K, \ k' \in 1,\ldots ,b(k)-1, \ i \in {\underline{S}}^{\gamma }} \end{aligned}$$

(B.13)

$$\begin{aligned}&\quad \eta ^{\gamma }_{k,i} = 0 \ \Rightarrow \ \lambda ^{\gamma ,i}_{k,k'} = 0 \nonumber \\ & \quad { \forall \, \gamma \in \Gamma , \ k \in K, \ k' \in 1,\ldots ,b(k), \ i \in {\underline{S}}^{\gamma } } \end{aligned}$$

(B.14)

$$\begin{aligned}&\quad { \eta ^{\gamma }_{k,i} = 1 \ \wedge \ \lambda ^{\gamma ,i}_{k,k'+1} = 1 \ \wedge \ \gamma (r - \mu ^{\gamma }_{k'}) < \gamma (r - i) \ \Rightarrow \ \lambda ^{\gamma ,i}_{k,k'} = 1 } \nonumber \\&\quad { \forall \gamma \in \Gamma , \ k \in K, \ k' \in 1,\ldots ,b(k)-1, \ i \in {\underline{S}}^{\gamma } } \end{aligned}$$

(B.15)

$$\begin{aligned}&\quad { \eta ^{\gamma }_{k,i} = 1 \ \wedge \ \gamma (r - \mu ^{\gamma }_{k'}) < \gamma (r - i) \ \Rightarrow \ \lambda ^{\gamma ,i}_{k,k'} = 1 } \\ & \quad \forall \gamma \in \Gamma , \ k \in K, \ k' = b(k), \ i \in {\underline{S}}^{\gamma } \end{aligned}$$

(B.16)

$$\begin{aligned}&\quad q^{\gamma }_{k,i} = \eta ^{\gamma }_{k,i} h^{\gamma }_{b(k)} + \sum _{k' = 2}^{b(k)}\lambda ^{\gamma ,i}_{k,k'} h^{\gamma }_{b(k')} \quad \forall \gamma \in \Gamma , \ k \in K, \ i \in {\underline{S}}^{\gamma } \end{aligned}$$

(B.17)

$$\begin{aligned}&\quad f^{\gamma }_{k,i} = \sum _{j \in S^{-\gamma }_{i} } \sum _{j' \in S^{\gamma }_{i+\gamma }} a_{jj'} q^{\gamma }_{k,i} \quad \forall \gamma \in \Gamma , \ k \in K, \ i \in {\underline{S}}^{\gamma } \end{aligned}$$

(B.18)

$$\begin{aligned}&\quad f^{\gamma ,1}_{k,i} = \sum _{j \in {\underline{S}}^{\gamma }} \sum _{j' \in S^{\gamma }_{i+\gamma }} a_{jj'} q^{\gamma }_{k,j} \quad \forall \gamma \in \Gamma , \ k \in K, \ i \in {\bar{S}}^{\gamma } \end{aligned}$$

(B.19)

$$\begin{aligned}&\quad f^{\gamma ,2}_{k,i} = \sum _{k' = \zeta ^{\gamma }_{k,i}+1}^{b(k)} \sum _{j \in {\underline{S}}^{\gamma }} \sum _{j' \in S^{\gamma }_{i+\gamma }} a_{jj'} q^{\gamma }_{k,j} \quad \forall \gamma \in \Gamma , \ k \in K, \ i \in {\bar{S}}^{\gamma } \end{aligned}$$

(B.20)

$$\begin{aligned}&\quad { f^{\gamma ,3}_{k,i} = \left( h^{\gamma }_{b(k)} + \sum _{k' = \zeta ^{\gamma }_{k,i}}^{b(k)} h^{\gamma }_{k'} \right) \sum _{j \in S^{\gamma }_{r,i}} \sum _{j' \in S^{\gamma }_{i+\gamma }} a_{jj'} } \quad \forall \gamma \in \Gamma , \ k \in K, \ i \in {\bar{S}}^{\gamma } \end{aligned}$$

(B.21)

$$\begin{aligned}&\quad f^{\gamma }_{k,i} = f^{\gamma ,1}_{k,i} + f^{\gamma ,2}_{k,i} + f^{\gamma ,3}_{k,i} \quad \forall \gamma \in \Gamma , \ k \in K, \ i \in {\bar{S}}^{\gamma } \end{aligned}$$

(B.22)

$$\begin{aligned}&\quad s( \gamma ,k ), e( \gamma ,k ), c_{k}^\gamma \in {\mathbb {N}}_0&\quad \forall \gamma \in \Gamma , k \in K \end{aligned}$$

(B.23)

$$\begin{aligned}&\quad w_{k}^\gamma \in \{\rho ,\ldots ,\rho + {\bar{w}}\}, \ h_{k}^\gamma \in \{\hspace{0.83328pt}\underline{\hspace{-0.83328pt}h\hspace{-0.83328pt}}\hspace{0.83328pt},\ldots ,{\bar{h}}\} \qquad \forall \gamma \in \Gamma , k \in K \end{aligned}$$

(B.24)

$$\begin{aligned}&\quad m_{k}^\gamma \in K, \ \mu ^{\gamma }_{k} \in S \quad \forall \gamma \in \Gamma , k \in K \end{aligned}$$

(B.25)

$$\begin{aligned}&\quad z\in {\mathbb {N}}_0\end{aligned}$$

(B.26)

$$\begin{aligned}&\quad f^{\gamma }_{k,i} \in {\mathbb {N}}_0\qquad {\forall \gamma \in \Gamma , \ k \in K, \ i \in S } \end{aligned}$$

(B.27)

$$\begin{aligned}&\quad q^{\gamma }_{k,i} \in {\mathbb {N}}_0\qquad { \forall \gamma \in \Gamma , \ k \in K, \ i \in {\underline{S}}^{\gamma } } \end{aligned}$$

(B.28)

$$\begin{aligned}&\quad \lambda ^{\gamma ,i}_{k,k'} \in \{0,1\}\qquad { \forall \gamma \in \Gamma , \ k \in K, \ k' \in 1,\ldots ,b(k), \ i \in {\underline{S}}^{\gamma } } \end{aligned}$$

(B.29)

$$\begin{aligned}&\quad \eta ^{\gamma }_{k,i} \in \{0,1\}\qquad { \forall \gamma \in \Gamma , \ k \in K, \ i \in {\underline{S}}^{\gamma } } \end{aligned}$$

(B.30)

$$\begin{aligned}&\quad f^{\gamma ,1}_{k,i}, f^{\gamma ,2}_{k,i}, f^{\gamma ,3}_{k,i} \in {\mathbb {N}}_0\qquad { \forall \gamma \in \Gamma , \ k \in K, \ i \in {\bar{S}}^{\gamma } } \end{aligned}$$

(B.31)

Appendix C: Software implementation

When implementing a CP model in software, the features and constraints available in the modeling vary slightly from solver to solver. This implies that frequently, some solver-specific measures have to be taken in the construction of the model. For this paper, we implemented the developed CP model in Python 3.9.7 using the OR tools CP libraries (Perron and Furnon 2019). We note that the specialized CP constraints used in the formulation (i.e., alldifferent) are built in the solver and could be directly implemented into the model. Conversely, some of the non-linear features of the model required additional variables to be introduced in the formulation to facilitate their implementation in the OR tools framework.

Specifically, in the OR tools framework, most non-linear constraints are equalities (e.g., a variable is equal to the product of two other variables). Thus, intermediary storage variables are necessary to hold the value of all non-linear quantities that are used in the constraints.

Constraints (4) contain the non-linear quantities $zc^{\gamma }_{k}$ and $e( -\gamma ,m^{\gamma }_{k} )$. We introduce the auxiliary variable ${\bar{c}}^{\gamma }_{k}$ to store the product of $z$ and $c^{\gamma }_{k}$, and auxiliary variable ${\bar{e}}( \gamma ,k )$ to store the value of $e( -\gamma ,m^{\gamma }_{k} )$, $\forall \gamma \in \Gamma , \ k \in K$. Therefore, we replace constraints (4) with the following.

$$\begin{aligned}&s( \gamma ,k ) + {\bar{c}}^{\gamma }_{k} = {\bar{e}}( \gamma ,k ) \quad \forall \gamma \in \Gamma , \ k \in K \end{aligned}$$

(C.32)

$$\begin{aligned}&{\bar{c}}^{\gamma }_{k} = zc^{\gamma }_{k} \quad \forall \gamma \in \Gamma , \ k \in K \end{aligned}$$

(C.33)

$$\begin{aligned}&{\bar{e}}( \gamma ,k ) = e( -\gamma ,m^{\gamma }_{k} ) \quad \forall \gamma \in \Gamma , \ k \in K \end{aligned}$$

(C.34)

Constraints (C.33) are implemented in CP through the use of the element constraint, which allows indexing a vector in accordance to a variable. Similarly, constraints (18) contain two products of variables. We use auxiliary variable ${\bar{\eta }}^{\gamma }_{k,i}$ to store the product of variables $\eta ^{\gamma }_{k,i}$ and $h^{\gamma }_{b(k)}$, and variable ${\bar{\lambda }}^{\gamma ,i}_{k,k'}$ to store the product of variables $\lambda ^{\gamma ,i}_{k,k'}$ and $h^{\gamma }_{b(k')}$. Therefore, we replace constraints (18) with the following.

$$\begin{aligned}&q^{\gamma }_{k,i} = {\bar{\eta }}^{\gamma }_{k,i} + \sum _{k' = 2}^{b(k)} {\bar{\lambda }}^{\gamma ,i}_{k,k'} \quad \forall \gamma \in \Gamma , \ k \in K, \ i \in {\underline{S}}^{\gamma } \end{aligned}$$

(C.35)

$$\begin{aligned}&{\bar{\eta }}^{\gamma }_{k,i} = \eta ^{\gamma }_{k,i} h^{\gamma }_{b(k)} \quad \forall \gamma \in \Gamma , \ k \in K, \ i \in {\underline{S}}^{\gamma } \end{aligned}$$

(C.36)

$$\begin{aligned}&{\bar{\lambda }}^{\gamma ,i}_{k,k'} = \lambda ^{\gamma ,i}_{k,k'} h^{\gamma }_{b(k')} \quad \forall \gamma \in \Gamma , \ k \in K, \ k' \in 2,\dots ,b(k), \ i \in {\underline{S}}^{\gamma } \end{aligned}$$

(C.37)

Lastly, we use auxiliary variable $H^{\gamma }_{k}$ to store the square terms used in the objective function. Thus, we replace objective function (7) with the following.

$$\begin{aligned}&\text {min} \sum _{\gamma \in \Gamma } \sum _{k \in K} (H^{\gamma }_{k}) \end{aligned}$$

(C.38)

$$\begin{aligned}&H^{\gamma }_{k} = h^{\gamma }_{k} h^{\gamma }_{k} \quad \forall \gamma \in \Gamma , \ k \in K \end{aligned}$$

(C.39)