Advanced OR and AI Methods in Transportation IMPROVED DYNAMIC PROGRAMMING FOR THE VEHICLE ROUTING PROBLEM WITH TIME WINDOWS

In this paper I propose a Branch-and-Price algorithm for the solution of the Vehicle Routing Problem with Time Windows where the pricing subproblem, the resource constrained elementary shortest path problem (RCESPP), is solved to optimality. 1. Problem formulation The Vehicle Routing Problem with Time Windows (VRPTW) is defined over a graph G = (V, A) where V = {0, . . . , n} is the node set in which node 0 represents the depot while nodes N = {1, . . . , n} represent the customers. A given nonnegative cost ci j is associated to each arc (i, j) ∈ A. Each customer i has an associated time window [ai, bi] and has to be visited within its time window. It is common that waiting until instant ai at node i is allowed if the vehicle arrives early. The travel time ti j , the service time si and the delivery request di at node i are given. A common assumption is that the costs and the travel-times coincide, and the starting time of the vehicles leaving the depot is assumed to be equal to 0. Let K be the number of available vehicles at the depot with capacity Q. The VRPTW requires the computation of at most K tours with minimum cost such that: each route starts and ends at the depot, each customer is visited once, the capacity of the vehicle is not exceeded along the route and for each customer i the service time starts within the associated time window and it takes si. VRPTW is NP-hard in the strong sense since it generalizes the CVRP when ai = 0 and bi = ∞ for each customer i. For a recent survey on the VRPTW see Cordeau et al. [2]. The structure of the VRPTW suggests to formulate it as a set partitioning problem based on a large set of all feasible routes, say P. The solution of the set partitioning problem asks to select from this set K routes of minimum cost such that each customer is served exactly Dipartimento di Tecnologie dell’Informazione, Crema, IT. [email protected]