Dynamic Automated Search of Shunting Routes within Mesoscopic Rail-Traffic Simulators

Table 3

Specification of the weighted digraph G—the final model of the track infrastructure.

Symbols	Specifications
G	The weighted digraph; (i) G = (V, E, φ, ω, ε, κ); (ii) The digraph G represents a result related to the transformation of the relevant undirected graph G0
V(G)	The set of vertices of the digraph G; (i) V(G) = {|z = 1, …, n\2, k = 1, 2}, |V(G)| = n; (ii) V(G) = Vdest (G) ∪ Vconn (G), Vdest (G) ∩ Vconn (G) = ∅; (iii) The set Vdest (G) contains destination vertices of the digraph G; (iv) The set Vconn (G) contains connecting vertices of the digraph G; Note: symbol “\” denotes an integer division
E(G)	The set of directed edges of the digraph G; (i) |E(G)| = m; (ii) E(G) = Etrans (G) ∪ Erev (G), Etrans (G) ∩ Erev (G) = ∅; (iii) The set Etrans (G) contains transit edges; (iv) The set Erev (G) contains reverse edges
φ	The incidence function related to the digraph G; (i) φ: E(G) ⟶ {[, ] | [, ] ∈ V(G) × V(G) ∧  ≠ ; x, y ∈ <1, …, n\2>; i, j ∈ {1, 2}}
()	The set of successors of the vertex; (i) () = { | [, ] ∈ E(G)}, () ⊂ V(G); (ii) () =  () ∪ (), () ∩ () = ∅; (iii) The set of transit successors of the vertex; () = {| [, ] ∈ Etrans(G)}; (iv) The set of reverse successors of the vertex; () = {| [, ] ∈ Erev(G)}
ω	The vertex weight function related to the digraph G; (i) ω: V(G) ⟶ R^+; (ii) ∀ ,  ∈ V(G) (z = 1,..., n\2): ω () = ω()
ε	The edge weight function related to the digraph G; (i) ε: E(G) ⟶ R^+; (ii) ∀ [, ] ∈ Etrans: ε ([, ]) = ω(); (iii) ∀ [, ] ∈ Erev: ε([, ]) = L, where L represents a parameter value reflecting the length of a relevant relocation object
κ	The vertex vacancy function (vacant capacity function); (i) κ: V(G) ⟶; (ii) If a track segment (reflected by a vertex  ∈ V(G)) is completely vacant, then κ() = ω()