Polyhedral study of a temporal rural postman problem: application in inspection of railway track without disturbing train schedules

Abstract

The Rural Postman Problem with Temporal Unavailability (RPP-TU) is a variant of the Rural Postman Problem (RPP) specified for multi-agent planning over directed graphs with temporal constraints. These temporal constraints represent the unavailable time intervals for each arc during which agents cannot traverse the arc. Such arc unavailability scenarios occur in routing and scheduling of the instrumented wagons for inspection of railway tracks without disturbing the train schedules, i.e. the scheduled trains prohibit access to the signal blocks (sections of railway track separated by signals) for some finite interval of time. A three-index formulation for the RPP-TU is adopted from the literature. The three-index formulation has binary variables for describing the route information of the agents, and continuous non-negative variables to describe the schedules at pre-defined locations. A relaxation of the three-index formulation for RPP-TRU, referred to as Cascaded Graph Formulation (CGF), is investigated in this work. The CGF has attributes that simplify the polyhedral study of time-dependent arc routing problems like RPP-TRU. A novel branch-and-cut algorithm is proposed to solve the RPP-TU, where branching is performed over the service arcs. A family of facet-defining inequalities, derived from the polyhedral study, is used as cutting planes in the proposed branch-and-cut algorithm to reduce the computation time by up to $48\%$. Finally, an application of this work is showcased using a simulation case study of a railway inspection scheduling problem based on Kurla-Vashi-Thane suburban network in Mumbai, India. An improvement of $93\%$ is observed when compared to a Benders' decomposition based MILP solver from the literature.

Figures (7)

• Figure 1: (a) An illustrative graph $G = (V, A, F^+, F^-)$, where $V = \{v_1, \dots, v_8\}$ and $A = \{a_1, \dots, a_{13}\}$, $F^+$ and $F^-$ are suitable maps. (b) A sub-graph of $G$ with three vertices and two arcs that are strictly outside the shaded blob. (c) A sub-graph constructed by selecting all vertices and arcs strictly inside the shaded blob. The dashed arcs represent the arcs in the boundary.
• Figure 2: An example graph from lannez, where vertex $v_d = v_1$ is the depot, and $A_* := \{a_2, a_5\}$ represent the service arcs.
• Figure 3: A replicated graph for $|\mathcal{K}|$ agents, constructed from a base graph shown in Figure \ref{['fig:prelim_fig2']}. Here $k \in \mathcal{K} (= \{1,2,3\})$ is the agent set. The $3$ dotted boxes illustrate agent-sub-graphs having $3$ layers each $l \in \mathcal{L} (= \{1,2,3\})$, illustrated by planes.
• Figure 5: Illustration of branching for example graph in Figure \ref{['fig:prelim_fig2']}. At node $n1$, the branching is done for $l=1, k=1$; resulting in three child nodes with binary restrictions shown at the connecting branches. Assuming fractional solution is observed in $X_{2,3,1}$, the branching at node $n_2$ is performed for $l=1, k=3$. Similarly, assuming that fractional solution is observed in $X_{5,2,2}$, node $n3$ is evaluated for $l=1, k=2$ (no branching for arc $a_2$ as it has already been assigned in its parent node). If no fractional solution is achieved (say at node $n4$), then a new layer is explored for the same agent-sub-graph i.e. $l=1, k=1$.
• Figure 6: An example showing arc $a_4$ is not available in the period $0$-$1$ minutes and also after $1.5$ minutes, while arc $a_5$ and $a_6$ are not available between $1$-$1.5$ minutes. Note that the union of the graphs is strongly connected, however arc $a_4$ is only available for $0.5$ minutes in the entire lifespan of this temporal graph. Running-time for arc $a_4$ is at least $1$ minute, hence its non-traversable. Observe that if arc $a_4$ is available in the period $0$-$1$ minutes, availability period of arc $a_4$ is larger than its running-time; however the graph is still not-traversable if $v_1$ is the depot vertex, start time is $t \geq 0$, and $a_5$ is a required arc.

Theorems & Definitions (4)

• Theorem 4.1
• Lemma 4.2
• Lemma 4.3
• Theorem 4.4