Hybridizing adaptive large neighborhood search with kernel search: a new solution approach for the nurse routing problem with incompatible services and minimum demand

The average age of the population has grown steadily in recent decades along with the number of people suffering from chronic diseases and asking for treatments. Hospital care is expensive and often unsafe, especially for older individuals. This is particularly true during pandemics as the recent SARS-CoV-2. Hospitalization at home has become a valuable alternative to face efficiently a huge increase in treatment requests while guaranteeing a high quality of service and lower risk to fragile patients. This new model of care requires the redefinition of health services organization and the optimization of scarce resources (e.g., available nurses). In this paper, we study a Nurse Routing Problem that tries to find a good balance between hospital costs reduction and the well-being of patients, also considering realistic operational restrictions like maximum working times for the nurses and possible incompatibilities between services jointly provided to the same patient. We first propose a Mixed Integer Linear Programming formulation for the problem and use some valid inequalities to strengthen it. A simple branch-and-cut algorithm is proposed and validated to derive ground benchmarks. In addition, to efficiently solve the problem, we develop an Adaptive Large Neighborhood Search hybridized with a Kernel Search and validate its performance over a large set of different realistic working scenarios. Computational tests show how our matheuristic approach manages to find good solutions in a reasonable amount of time even in the most difficult settings. Finally, some interesting managerial insights are discussed through an economic analysis of the operating context.