Although sometimes it is necessary, no one likes to stay in a hospital, and patients who need to stay in bed but do not require constant medical surveillance prefer their own bed at home. At the same time, a patient in a hospital has a high cost for the community, that is not acceptable if the patient needs service only a few minutes a day. For these reasons, the current trend in Europe and North-America is to send nurses to visit patients at their home: this choice reduces costs for the community and gives better quality of life to patients. The challenge is to deliver the service in a cost effective manner without a detriment of the service quality. These social and health management issues have interesting implications from the mathematical viewpoint, introducing a challenging combinatorial optimization problem. The problem consists in assigning patients' services to traveling nurses and defining the nurse itineraries so that the following optimization aspects are considered: the nurse workloads (including service as well as travel time) are balanced, patients are preferentially served by a single nurse or just a few ones, and the overall travel time is minimized. These objectives are somehow conflicting and a reasonable trade off must be found. The complexity of the problem calls for suitable optimization-based algorithmic support to decisions, in particular in the perspective of an increasing diffusion of the service.This problem is known in the literature as the Home Health Care (HHC) problem. In this paper, we address the HHC problem in the municipality of Ferrara, a mid-sized city in the North of Italy. The problem is currently solved by hand, starting from a partitioning of patients based on predefined zones. We describe a Constraint Programming model that solves the HHC problem, and show significant improvements with respect to the current manual solution. those resources can be spent for providing a better service to other patients. Moreover, patients perceive a higher quality of life when they stay at home, with their dear ones, and feel their illness more similar to a "normal" life situation. This reduces depression risk for the patient, and improves rehabilitation rate. Indeed, high quality home health care following hospital dismissal has proved essential in reducing hospital readmissions, if able to cope with preventable complications. Therefore, it is crucial to deliver the service in a cost effective manner while not deteriorating service quality. Arguments in favor of home health care and how to re-

This paper presents a methodology based on the Variable Neighbourhood Search metaheuristic, applied to the Capacitated Vehicle Routing Problem. The presented approach uses Constraint Programming and Lagrangean Relaxation methods in order to improve algorithm’s efficiency. The complete problem is decomposed into two separated subproblems, to which the mentioned techniques are applied to obtain a complete solution. With this decomposition, the methodology provides a quick initial feasible solution which is rapidly improved by metaheuristics’ iterative process. Constraint Programming and Lagrangean Relaxation are also embedded within this structure to ensure constraints satisfaction and to reduce the calculation burden. By means of the proposed methodology, promising results have been obtained. Remarkable results presented in this paper include a new best-known solution for a rarely solved 200-customers test instance, as well as a better alternative solution for another benchmark problem.

This paper presents an original hybrid approach to solve the Capacitated Vehicle Routing Problem (CVRP). The approach combines a Probabilistic Algorithm with Constraint Programming (CP) and Lagrangian Relaxation (LR). After introducing the CVRP and reviewing the existing literature on the topic, the paper proposes an approach based on a probabilistic Variable Neighbourhood Search (VNS) algorithm. Given a CVRP instance, this algorithm uses a randomized version of the classical Clarke and Wright Savings constructive heuristic to generate a starting solution. This starting solution is then improved through a local search process which combines: (a) LR to optimise each individual route, and (b) CP to quickly verify the feasibility of new proposed solutions. The efficiency of our approach is analysed after testing some well-known CVRP benchmarks. Benefits of our hybrid approach over already existing approaches are also discussed. In particular, the potential flexibility of our methodology is highlighted.

Abstract:The vehicle routing problem (VRP) is a flourishing research area with clear applications to real-life distribution companies. However, most VRP-related academic articles assume the existence of a homogeneous fleet of vehicles and/or a symmetric cost matrix. These assumptions are not always reasonable in real-life scenarios. To contribute in closing this gap between theory and practice, we propose a hybrid methodology for solving the asymmetric and heterogeneous vehicle routing problem (AHVRP). In our approach, we consider: 1) different types of vehicle loading capacities (heterogeneous fleets); 2) asymmetric distance-based costs. The proposed approach combines a randomised version of a well-known savings heuristic with several local searches specifically adapted to deal with the asymmetric nature of costs. A computational experiment allows us to discuss the efficiency of our approach and also to analyse how routing costs vary when slight departures from the homogeneous fleet assumption are considered.

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