An exact approach for the Vehicle Routing Problem with Demand Allocation

The Vehicle Routing Problem with Demand Allocation (VRPDA) involves a depot, a set of uncapacitated delivery sites, and a set of customers. Instead of directly visiting customers, their demand is allocated to a visited delivery site, incurring an assignment cost. VRPDA requires two key decisions: the first is to design a set of routes that begin and end at the depot for a fleet of homogeneous vehicles, visiting only delivery sites; the second is to assign customers to the visited delivery sites. These decisions aim to minimize the total routing and assignment costs. The solution must satisfy three primary constraints: each customer must be assigned to exactly one visited delivery site; each delivery site may be visited at most once across all routes; and the total demand of customers assigned to visited delivery sites on any given route must not exceed the vehicle capacity. To solve the VRPDA, we propose an exact branch-and-cut-and-price algorithm implemented within the VRPSolver framework. We demonstrate that the enumeration of elementary routes can only be applied in the proposed algorithm if the master formulation constraint, which prevents a delivery site from being visited more than once, is relaxed. These constraints are initially relaxed and then enforced as needed within the pricing subproblems during the course of a branch-and-bound (B&B) algorithm. Extensive experiments on benchmark instances reveal that the proposed B&B algorithm surpasses the best exact algorithms in the literature and, for the first time, finds optimal solutions for several large instances.