# Balancing Staff Finishing Times vs. Minimizing Total Travel Distance in Home Healthcare Scheduling
## 1 Introduction  
This research tackles two objectives in home healthcare (HHC) routing and scheduling, namely minimizing total travel distance and minimizing total finishing time differences. Minimizing total travel distance, a typical objective in a routing problem, attempts to minimize the total cost of the routing, while minimizing total finishing time differences is more meaningful for a general service problem where workload balance is important. The optimal utilization of resources, ensuring efficient service delivery and enhancing profits are all vital objectives for a home healthcare operator and managing all three is tantamount to balancing various objectives. HHC attracts interest due to a variety of factors including aging populations, the rising number of the chronically ill people and a preference for receiving care at home. Currently, those over 65 years of age make up 10% of the global population and this figure is expected to increase to 16% in 2050 [1]. In order for HHC providers to efficiently deliver essential services, they need to optimize the problem, and this requires optimizing both routing and scheduling simultaneously. The difficulties of HHC optimization come from its multi-objective nature, which sets it apart from traditional routing and scheduling problems. Optimizing multiple goals, e.g., minimizing travel time or distance while balancing workload allocation among caretakers, requires customized optimization techniques. An HHC system is defined as an extensive network of medical experts, caregivers and support services that provides healthcare services to patients in their homes or designatedlocations. Home healthcare administrators need to make optimum decisions regarding the route plans for all caretakers. In typical HHC operations, the operator receives information about a group of patients distributed across different locations who need to be visited by one caretaker. Home healthcare managers determine the routes for caregivers, who subsequently travel to provide services to customers following the predetermined routes. Typically, home healthcare managers try to make decisions that minimize total operational costs when creating route plans. To ensure competitiveness in the HHC business, both patient satisfaction and caregiver satisfaction need to be considered as factors that can significantly impact operators' profits. This notion motivated us to study a two-objective home healthcare routing and scheduling problem. HHC systems can be optimized if healthcare providers treat them as routing and scheduling problems, allowing for the optimization of factors like travel time, distance, vehicle capacity and patient priorities. Optimization can result in reduced travel costs, minimized travel time, enhanced patient satisfaction and improved overall efficiency in healthcare service delivery. However, it is not possible to achieve all these objectives simultaneously, as optimizing one objective requires sacrificing others. In this scenario, using the typical routing objective of total distance minimization alone may not be appropriate due to the time-sensitive nature of both patients' and caretakers' needs. Focusing on distance as the primary goal could lead to unequal distribution of tasks among caregivers. This study aims to introduce a new objective of minimizing the total difference between each pair of caretakers' finishing times, an approach that has never been studied in the literature. The problem will be formulated based on the vehicle routing problem with time windows. Additionally, a metaheuristic approach relying on simulated annealing (SA) is proposed as a solution for handling a large-scale problem. The SA algorithm has been demonstrated to be effective for solving NP-hard problems. The algorithm has the ability to avoid local optima by employing a probabilistic approach. Since vehicle routing problems are known to be NP-hard, we propose an efficient approach for multiple objectives. In this study, however, we focus only on bi-objective function.