An exact and heuristic approach for the d-minimum branch vertices problem

Modelling Distribution Routes in City Logistics by Applying Operations Research Methods

“…The general mathematical model of the VRP (see Equation 1) formulated by Miller-Tucker-Zemlin is presented, for example, by [29] and [30] , , , ,…”

“…This method must be implemented in 2 elementary consecutive phases. In phase 1, the UDCs will be divided into individual circuit paths, and as for the phase 2, the succession of deliveries to each object will be defined for each designed path so that the distance travelled is as short as possible [29]. First of all, we assign UDCs to delivery paths.…”

A genetic approach for the 2‐edge‐connected minimum branch vertices problem

“…Otherwise, bridges B in G ′′ are computed (line 9). If no bridge exists, we can use Lemma 4 and remove from G ′ all the edges in 𝛿 G ′ (v), except for the two edges whose other endpoints have highest degree (lines [10][11]; in addition to making v not branch, this choice potentially reduces the number of branch vertices in the neighborhood of v. Otherwise, we remove existing bridges from G ′′ and recompute the connected components in G ′′ : for each found component C i such that a single bridge in B is incident on C i , we remove all the edges in 𝛿 G ′ (v) except the edge (v, z) which maximizes the degree of z (lines [13][14][15][16][17][18][19][20]. This means reducing the number of edges incident on v: when every found component has at most two incident bridges, Lemma 5 holds and only two edges in 𝛿 G ′ (v) are selected; otherwise, we know from Lemma 6 that v is necessarily a branch vertex, thus at least three edges in 𝛿 G ′ (v) are selected.…”

“…Variants of the problem have been considered by several authors: Moreno et al [13] proposed a Miller–Tucker–Zemlin based formulation with valid inequalities and a heuristic approach for the $$$ d $$$‐Minimum Branch Vertices problem. In this problem the aim is to minimize the number of vertices with degree strictly greater than $$$ d $$$.…”

“…Variants of the problem have been considered by several authors: Moreno et al [13] proposed a Miller-Tucker-Zemlin based formulation with valid inequalities and a heuristic approach for the 𝑑-Minimum Branch Vertices problem. In this problem the aim is to minimize the number of vertices with degree strictly greater than 𝑑. Carrabs et al [1] introduced the Generalized Minimum Branch Vertices Problem, which partitions the starting graph into clusters and identifies a spanning tree that covers exactly one vertex per cluster, minimizing the number of branch vertices.…”