Two-stage column generation: a novel framework

Column generation has been intensively used in the last decades to compute good quality lower bounds for combinatorial problems reformulated through Dantzig-Wolfe decomposition. In this work we propose a novel framework to cope with problems in which the structure of the original formulation, namely the presence of a combinatorial number of decision variables, does not allow for straightforward reformulation. The basic idea is to start from a meaningful subset of original variables, apply the DW reformulation to the subset, solve the reformulation with column generation and perform the explicit pricing on original variables retracing back the reformulation and using complementary-slackness conditions. The Discrete Split Delivery Vehicle Routing Problem with Time Windows (DSDVRPTW) is used as an illustration for the method, which provides a new exact approach to the problem. Preliminary computational experiments are reported. This is joint work with Matteo Salani

Branch and Price for the Vehicle Routing Problem with Discrete Split Deliveries and Time Windows

The Discrete Split Delivery Vehicle Routing Problem with Time Windows (DSDVRPTW) consists of designing the optimal set of routes to serve, at least cost, a given set of customers while respecting constraints on vehicles capacity and customer time windows. The delivery request of a customer is discrete since it consists of several items that cannot be split further. The problem belongs to the class of split delivery problems since each customers demand can be split in orders, i.e. feasible combinations of items, and each customer can be visited by more than one vehicle. In this work, we model the DSDVRPTW assuming that all feasible orders are known in advance and that each vehicle can serve at most one order per customer. Remarkably, service time at customers location depends on the serviced combination of items, which is a modeling feature rarely found in literature. We present a mixed integer program for the DSDVRPTW based on arc-flow formulation, we reformulate it via Dantzig-Wolfe and we apply column generation. We propose a branch-and-price algorithm, implemented using state-of-the-art techniques for the pricing and the master problem. Computational results on instances based on Solomons data set are presented and discussed

Routing helicopters for crew exchanges on off-shore locations

This paper deals with a vehicle routing problem with split demands, namely the problem of determining a flight schedule for helicopters to off-shore platform locations for exchanging crew people employed on these platforms. The problem is formulated as an LP model and solved by means of a column-generation technique including solving TSP problems. Since the final solution needs to be integral, we have chosen a rounding procedure to obtain an integer solution. Since the LP approach needs a considerable amount of computer time, it is only suitable for long-term planning practices. For the usual short-term planning, we have designed the so-called Cluster-and-Route Heuristic together with a number of improvement heuristics. The Cluster-and-Route procedure constructs a suitable clustering of the platforms and simultaneously forms the routes of the helicopter flights associated with the clusters. This approach is different from the usual heuristics, in which the clusters are constructed first, and the routes for each cluster are made afterwards. Simulations with various data sets show that the new heuristic outperforms the usual heuristics for vehicle routing problems. Even better results are obtained when improvement heuristics are applied. We use four improvement heuristics, including, so-called 1-opt and 2-opt procedures