Column Generation for the Split Delivery VRP

In this paper we tackle a variation of the Vehicle Routing Problem (VRP) in which each customer can be served by more than one vehicle, each serving a fraction of its demand. This problem is known as the Split Delivery VRP (SDVRP). Due to the potential savings that can be achieved in this way, the SDVRP recently received great attention in the combinatorial optimization community. We propose a new extended formulation for the problem. We exploit its properties to derive an effective column generation scheme. Our method is compared to the previous ones in the literature from both a theoretical and a computational point of view. In particular, our formulation involves a polynomial number of constraints and flow variables, can be optimized by solving well understood resource constrained shortest path problems and yields a bound which is not dominated by any previous one in the literature

An exact algorithm for the discrete split delivery vehicle routing problem with time windows

The Split Delivery Vehicle Routing Problem (SDVRP) is a variant version of the classical VRP in which each customer can be visited by more than one vehicle which serves a fraction of its demand. The Discrete SDVRP is another variant in which the delivery request of a customer consists of several items which cannot be split further. In this work we consider the DSDVRP with time windows where the service of items' combinations imply a corresponding service time. We present a branch-and-price algorithm and discuss preliminary computational results

The Vehicle Routing Problem with Discrete Split Delivery 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 consists of several discrete items which 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 as a mixed integer linear program, 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 branch-and-price algorithm, analyzing the implications of the classical Dantzig-Wolfe reformulation. Preliminary computational results on instances based on Solomons data set are discussed

Uncertainty Feature Optimization for the Airline Scheduling Problem

Uncertainty Feature Optimization is a framework to cope with optimization problems due to noisy data, using an implicit characterazation of the noise. The Aircraft Scheduling Problem (ASP) is a particular case of such problems, where disruptions randomly perturbate the original flight schedule. This study uses the UFO framework to generate more robust and recoverable schedules, in the sense that more delays are absorbed and when re-optimization is required, the corresponding recovery costs are reduced. We provide computational results for the public data of an European airline provided for the ROADEF Challenge 2009 footnote{\texttt{http://challenge.roadef.or /2009/index.en.htm}}; new schedules are computed with different models, and we compare the a posteriori results obtained by the application of a recovery algorithm

Two-stage column generation and applications

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 paper 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

CLIP-AIR: A modular multi-modal transportation system

Demanding reduction in CO2 emissions and continuous pressure on ticket prices are pushing towards radical modifications in future objectives for the air transport industry. Operators are asked to consider fundamental structure change and new approaches to fleet management. We present part of a feasibility study for a new modular multi-modal transportation system called "ClipAir". In this work, we focus on the integrated schedule generation and fleet assignment problem and compare the performances of a regular operator and a fictional operator running a ClipAir fleet. Comparison is made in terms of expected operating costs and demand satisfaction

Robust scheduling and disruption recovery for airlines

Airline planning include complex and structured operations that must be planned in advance in order to exploit the available resources, provide a reliable and competitive service and forecast system's performances. Decisions regarding operations are based on data which is frequently due to uncertainty. Moreover, unpredicted events may disrupt the current schedule and force managers to take reactive decisions to recover to an operational state. On the other hand, proactive decisions, i.e. decisions which take into account the uncertainty of the data, tend to robust solutions which are able to absorb data deviation and small disruptions. In this talk we address the aircraft routing problem from both reactive and proactive point of view and suggest ways to integrate the two approaches to reach what we call a robust recoverable approach for aircraft routing, i.e. a proactive strategy which accounts for the presence of a disruption recovery strategy. We validate our ideas with a computational study based on real world data provided by a major european airline

Dynamic programming for the orienteering problem with time windows

We present an exact optimization algorithm for the Orienteering Problem with Time Windows (OPTW). The algorithm is based on bi-directional and bounded dynamic programming with decremental state space relaxation. We compare different strategies proposed in the literature to guide decremental state space relaxation: our experiments on instances derived from the literature show that there is no dominance between these strategies. We also propose a new heuristic technique to initialize the critical vertex set and we provide experimental evidence of its effectiveness