An efficient train scheduling algorithm on a single-track railway system

Abstract

Since scheduling trains on a single-track railway line is an NP-hard problem, this paper proposes an efficient heuristic algorithm based on a train movement simulation method to search for the near-optimal train timetables within the acceptable computational time. Specifically, the time–space statuses of trains in the railway system are firstly divided into three categories, including dwelling at a station, waiting at a station and traveling on a segment. A check algorithm is particularly proposed to guarantee the feasibility of transition among different statuses in which each status transition is defined as a discrete event. Several detailed operation rules are also developed to clarify the scheduling procedure in some special cases. We then design an iterative discrete event simulation-based train scheduling method, namely, train status transition approach (TSTA), in which the status transition check algorithm and operation rules are incorporated. Finally, we implement some extensive experiments by using randomly generated data set to show the effectiveness and efficiency of the proposed TSTA.

Appendices

Appendices

Mathematical model of train timetabling problem

To describe the scheduling process mathematically, the decision variables are listed as follows, in which for simplicity we use terms “$\mathrm{SN}$” and “$\mathrm{DN}$” to denote conditions that (1) $\forall i,j\in N_I \text { or } i,j\in N_O, i \ne j$ and (2) $\forall i\in N_I, j\in N_O \text { or } \forall i\in N_O, j\in N_I$, respectively.

$$\begin{aligned}&\xi _{i,j,k}^{AA}=\left\{ \begin{array}{ll} 1, &{}{\quad }\text {if train } i \text { arrives at station }k\text { earlier than} \\ &{}\qquad \text {train }j, \text { and } {\mathrm{SN}}; \\ 0, &{}{\quad }\text {otherwise.} \end{array} \right. \\&\xi _{i,j,k}^{DD}=\left\{ \begin{array}{ll} 1, &{}{\quad }\text {if train }i\text { departs from station } k \text { earlier than} \\ &{}\qquad \text {train } j, \text { and } {\mathrm{SN}}; \\ 0, &{}{\quad }\text {otherwise.} \end{array} \right. \\&\xi _{i,j,k}^{AD}=\left\{ \begin{array}{ll} 1, &{}{\quad }\text {if train } i \text { arrives at station } k \text { earlier than the} \\ &{}\qquad \text {departure of train } j \text { from station } k,\\ &{}\qquad \text { and } {\mathrm{DN}};\\ 0, &{}{\quad }\text {otherwise.} \end{array} \right. \\&\xi _{i,j,k}^{DA}=\left\{ \begin{array}{ll} 1, &{}{\quad }\text {if train } i \text { departs from station } k \text { earlier than} \\ &{}\qquad \text {the arrival of train } j \text { at station } k, \text { and } \mathrm{DN};\\ 0, &{}{\quad }\text {otherwise.} \end{array} \right. \\&\omega _{i,j}^{k,k^+}=\left\{ \begin{array}{ll} 1, &{}{\quad }\text {if train }i \text { uses segment } (k,k^+)\text { before train } j, \\ &{}\qquad j,\text { and } \mathrm{DN};\\ 0, &{}{\quad }\text {otherwise.} \end{array} \right. \\&\mu _{i,k}^{r}=\left\{ \begin{array}{ll} 1, &{}{\quad }\text {if train } i \text { occupies track }r \text { at station } k,\\ &{}\qquad r \le C_k;\\ 0, &{}{\quad }\text {otherwise.} \end{array} \right. \\&M: {\quad }\text {a sufficient large number} \end{aligned}$$

1.1 Mathematical model

The mathematical model for the considered TSCP is then formulated as

$$\begin{aligned} \text {Minimize} \sum _{i\in N} (d_{i,Des(i)} - \bar{d}_{i,Des(i)}) \end{aligned}$$

s.t.

$$\begin{aligned}&a_{i,k} + t_{i,k}^d \le d_{i,k},{\quad }\forall i\in N; {\;}k\in R_i\end{aligned}$$

(5)

$$\begin{aligned}&d_{i,k} +t_{i(k,k^+)}^s \le a_{i,k^+}, {\quad }\forall i\in N; {\;}k\in R_i{{\setminus }}Des(i) \end{aligned}$$

(6)

$$\begin{aligned}&\xi _{i,j,k}^{DD} = \xi _{i,j,k^+}^{AA}, {\quad }\forall i,j \in N_I \text { or } i,j \in N_O, i\ne j;\nonumber \\&\quad {\;}k,k^+ \in R_i \cap R_j\end{aligned}$$

(11)

$$\begin{aligned}&\xi _{i,j,k}^{DA} + \xi _{j,i,k}^{AD} =1, {\quad }\forall i\in N_I, j\in N_O \text { or } \forall i\in N_O,\nonumber \\&\quad j\in N_I; {\;}k\in R_i\cap R_j \end{aligned}$$

(12)

$$\begin{aligned} d_{i,Ori(i)} \ge \bar{d}_{i,Ori(i)}, {\quad }\forall i\in N \end{aligned}$$

(15)

Constraints (5) ensure that the involved train can complete its pre-specified operations at some designated stations, such as loading/unloading passengers and changing crew. Constraints (6) guarantee the minimum travel time for the observed train, which can be extended according to the practical applications. Constraints (7) indicate that there is an arrival–arrival time headway $h^{aa}$ between two consecutive trains in the same direction when they arrive at the same station according to a specified sequence. Likewise, the departure–departure time headway $h^{dd}$ is considered similarly. Moreover, the arrival–arrival order at station k and departure–departure order at station $k^{+}$ should be consistent, which essentially guarantees no conflicts between these two trains on segment $(k,k^{+})$. For trains heading for different directions, we consider the arrival–departure and departure–arrival time headway constraints (9) and (10). Then tracing constraints (11) are formulated. Constraints (12) express the correlation between $\xi _{i,j,k}^{DA}$ and $\xi _{j,i,k}^{AD}$. Each section between two stations has only one track (segment).

Since the single-track railway line is considered, it is impossible for trains in different directions to run on the same track simultaneously. Then, corresponding constraints (13) are proposed to guarantee the unique occupancy of single rail tracks. Considering the station capacity, Zhou and Zhong (2007) used a time-based 0–1 variable to denote whether a train occupies a station at the given time, and then proposed a constraint that the number of trains simultaneously dwelling at the same station must be less than or equal to the station capacity. However, this method essentially creates a mass of 0–1 variables. For the similar case, Carey (1994a, b) presented choice-based constraints to handle the limitation of lines, platforms and routes of trains. Enlightened by Carey (1994a, b), Li et al. (2014) then proposed a track choice-based constraint for station capacity, as shown in constraints (14). Obviously, these constraints consider each station as a small network, where each train can only occupy one track at the same station (see constraints (14a)) and each track can be occupied by at most one train at any time (see constraints (14b)). Lastly, we require that the train departure time from its origin station should not be earlier than the preplanned one, i.e., constraints (15).

Local branching (LB) and relaxation induced neighborhood search (RINS)

To effectively solve mixed-integer programming (MIP) models, Fischetti and Lodi (2003) proposed a local branching (LB) approach through dynamically adding the siding constraints in the searching process, in which the commercial software CPLEX is used as a tactical MIP solver to solve the subproblems on different branches. However, as addressed by Danna et al. (2005), the local search relies extensively on the notion of a solution neighborhood, and the neighborhood is almost always tailored to the structure of the considered problem. To this end, Danna et al. (2005) proposed two approaches, called relaxation induced neighborhood search (RINS) and guided dives, respectively, to further speed up the solution process. For the completeness of this research, both the LB and RINS will be introduced briefly in what follows.

1.1 Local branching

Roughly speaking, the LB is a simple external branching framework involving three core parts, i.e., (1) define the neighborhood of a feasible solution to produce different subspaces, (2) search for optimal/suboptimal solutions in the subspaces, and (3) perform diversification. Given a generic 0–1 MIP problem (P) and a feasible solution $\bar{x}$, Fischetti and Lodi (2003) defined the $k-OPT$ neighborhood $\mathscr {N}(\bar{x},k)$ of $\bar{x}$ as the set of the feasible solutions of (P) satisfying the additional local branching constraint:

$$\begin{aligned} \Delta (x,\bar{x}) := \sum _{j\in \bar{S}}(1-x_j) + \sum _{j\in \mathscr {B}{{\setminus }}\bar{S}} x_j \le k \end{aligned}$$

(16)

where $\bar{S} = \{j\in \mathscr {B}: \bar{x}_j=1 \}$ is the binary support of $\bar{x}$. Then, Fischetti and Lodi (2003) proposed two types of LB algorithms based on the different strategies, namely exact and heuristic approaches. The exact approach can be expected to quickly generate the optimal solution for the considered problem by combining with the default MIP solver. On the other hand, to search for the near-optimal solutions as soon as possible, the heuristic LB in particular takes the branching diversification and computational time into the consideration, including the overall time limit (total time limit) and time limit on the branch nodes (node time limit).

Remark 1

As for the problem discussed in this research, we can produce an initial feasible solution for the LB by using our proposed heuristics TSTA. When the LB is terminated, the outputted solution can be regarded as a high-quality solution for the real-world applications.

1.2 Relaxation induced neighborhood search

In the searching process of a general branch-and-cut tree, two possible solutions might be returned at the termination of the algorithm. That is, (1) the incumbent is feasible with respect to the integrality constraints, but it is not optimal until the last and optimal integer solution has been found; and (2) some variables in a solution are not integral, but the objective value is better than that of the incumbent. With these concerns, Danna et al. (2005) proposed an effective approach, i.e., RINS, to integrate advantages of the aforementioned two solutions. The RINS algorithm is performed on a node of the incumbent, and the procedure includes three steps (for more details, see Danna et al. 2005). Typically, if RINS is invoked at every node in the branch-and-cut tree, the neighborhoods induced by the relaxations of consecutive nodes may be quite similar. For this case, Danna et al. (2005) concluded that it is preferable to apply RINS algorithm to every f nodes for some $f\gg 1$. Essentially, the aforementioned LB is also a branch-and-cut procedure whose node can also be performed with RINS. Therefore, LB and RINS can be integrated as a new heuristic, i.e., LB$+$RINS in Danna et al. (2005).

Remark 2

In this paper, the solutions obtained by the CPLEX solver, LB and LB$+$RINS are treated as the benchmark solutions. Since the considered problem is typically an NP-hard problem, it will probably take long time to find an initial feasible solution for the aforementioned methodologies. With this concern, the TSTA algorithm can be integrated with LB and LB$+$RINS, respectively, to provide the initial solution at the root node of these two approaches, which can essentially speed up the problem-solving process.