Crane scheduling in railway yards: an analysis of computational complexity

Abstract

An efficient container transfer in railway yards is an important matter to increase the attraction of rail-bound freight transport. Therefore, the scheduling of gantry cranes transferring containers between freight trains and trucks or among trains received a lot of attention in the recent years. This paper contributes to this stream of research by investigating the computational complexity of crane scheduling in these yards. Scheduling the transfer of a given set of containers by a single crane equals the (asymmetric) traveling salesman problem in its path-version. In railway yards, however, all container positions are located along parallel lines, i.e., tracks, and we face special distance metrics, so that only specially structured problem instances arise. We classify important problem settings by differentiating the transshipment direction, parking policy, and distance metric. This way, we derive problem variants being solvable to optimality in polynomial time, whereas other cases are shown to be NP-hard.

Appendices

Appendix 1: NP-completeness of 1-in-3-Sat without negated literals even if all variables occur two or three times

1-in-3-Sat Given a set V of variables and a collection C of clauses over V, such that each clause $c \in C$ has $|c| = 3$. Is there a truth assignment for V, such that each clause $c \in C$ has exactly one true literal?

Fig. 24

Equivalence of Manhattan metric (a) and maximum metric (c) under $45^{\circ }$-rotation (b)

1-in-3-Sat is NP-complete in the strong sense in general (Schaefer 1978; Garey and Johnson 1979) and remains so even if no clause contains a negated literal (Garey and Johnson 1979). In a recent paper, Boysen and Stephan (2016) show that 1-in-3-Sat without negated literals is NP-complete in the strong sense even if no variable occurs in more than three clauses.

Lemma 9.1

1-in-3-Sat without negated literals is strongly NP-complete even even if no variable occurs in more than three clauses.

Proof

(Lemma 9.1) See Boysen and Stephan (2016). $\square$

Lemma 9.2

1-in-3-Sat without negated literals is strongly NP-complete even if all variables occur two or three times.

Proof

(Lemma 9.2) To construct a 1-in-3-Sat instance without negated literals with all variables occurring at least twice we just have to double all clauses that contain variables occurring only once in the clause set. Afterward we can apply the transformation of Boysen and Stephan (2016) for proving Lemma 9.1 to reduce the number of occurrences for all variables to at most three. Note that this transformation preserves the property of all variables occurring at least twice. $\square$

Appendix 2: Equivalence of the Manhattan and maximum metric under $45^{\circ }$-rotation

This section provides an example for the equivalence of the Manhattan and maximum metric under $45^{\circ }$-rotation, which is elaborated by Garey and Johnson (1979) when treating the geometric TSP. The Manhattan distances between the points A to E depicted in Fig. 24a exactly equal the maximum distances within Fig. 24c. Thereby, the new grid of tracks and slots (Fig. 24c) becomes some more close-meshed (the distances between the tracks and slots decrease to $1/\sqrt{2}$ of the original ones), which requires a rescaling of the tracks and slots.