A VNS for the PVRPTW

Transportation problems appear in many practically highly relevant areas of our daily life. In this work we deal with a generalized variant of the classical Vehicle Routing Problem (VRP), namely the Periodic Vehicle Routing Problem with Time Windows (PVRPTW), for which only few specific solution techniques have been described in the literature. The basis forms the VRP which is defined on a complete graph G = (V,A), where V = {v0, v1, . . . vn} is the vertex set and A = {(vi, vj) : vi, vj ∈ V, i 6= j} is the arc set. Vertex v0 represents the depot at which are based m vehicles having capacities Q1, . . . , Qm. Each vertex of V \ {v0} corresponds to a customer and has an associated demand qi ≥ 0 as well as a service duration di ≥ 0. For each arc (vi, vj) ∈ A there are further given travel times or costs cij ≥ 0. The first generalization is due to additionally specifying a time window [ei, li] per customer and the depot, where ei and li are nonnegative integers and denote the earliest as well as latest beginning of the service, respectively. The second generalization concerns the extension to a planning horizon of t days: Each customer has defined a service frequency fi and a set Ci of allowable combinations of visit days, thus a customer has to be visited periodically. The whole PVRPTW then consists of selecting a single visit combination per customer and designing (at most) m vehicle routes on each of the t days on G such that