Home Healthcare (HHC) is an emerging and fast-expanding service sector that gives rise to challenging vehicle routing and scheduling problems. Each day, HHC structures must schedule the visits of caregivers to patients requiring specific medical and paramedical services at home. These operations have the potential to be unsuitable if the visits are not planned correctly, leading hence to high logistics costs and/or deteriorated service level. In this article, this issue is modeled as a vehicle routing problem where a set of routes has to be built to visit patients asking for one or more specific service within a given time window and during a fixed service time. Each patient has a preference value associated with each available caregiver. The problem addressed in this paper considers two objectives to optimize simultaneously: minimize the caregivers' travel costs and maximize the patients' preferences. In this paper, different methods based on the bi-objective non-dominated sorting algorithm are proposed to solve the vehicle routing problem with time windows, preferences, and timing constraints. Numerical results are presented for instances with up to 73 clients. Metrics such as the distance measure, hyper-volume, and the number of non-dominated solutions in the Pareto front are used to assess the quality of the proposed approaches.Algorithms 2019, 12, 152 2 of 25 work (Ait Haddadene et al. [1]), the minimization of the traveling cost and the sum of non-preferences related to customers were considered as a single-objective function. The obtained results showed the conflicting nature of these criteria, which is why, we are particularly interested in solving the bi-objective version of VRPTW-SP.The problem at hand is an extension of the Vehicle Routing Problem (VRP), which is one of the most-studied combinatorial optimization problems. First introduced by Dantzig and Ramser [2], the aim of this NP-hard problem is to determine a set of minimal cost routes in which a set of customers is to be served by a fleet of vehicles based at a depot node. A detailed survey can be found in Toth and Vigo [3]. Currently, one of the most effective metaheuristics for the VRP is the hybrid genetic algorithm introduced by Vidal [4], which was latter generalized to solve more than 26 VRP variants.Among the different classes of VRP, the VRP with Time Windows (VRPTW) stands as the basis variant of our studied problem. Heuristic and metaheuristic approaches have been widely used to solve the VRPTW such as those proposed by Kindervater and Savelsbergh [5], Bräysy and Gendreau [6], Nagata [7], and Bräysy et al. [8]. In the field of healthcare, most problems are modeled as a VRPTW. In fact, the origin of vehicle routing problems in HHC systems may be linked to the door-to door-transportation of elderly or disabled persons (Dial A Ride Problems (DARP)) proposed by Bodin and Sexton [9]. A few years later, this variant was extended to the Home Healthcare (HHC) problem by Cheng and Rich [10], where caregivers are assigned to patient homes to pro...
