# Balancing Staff Finishing Times vs. Minimizing Total Travel Distance in Home Healthcare Scheduling
## 2 Literature Review  
In this section, we will focus on works related to the development of models and objective functions for use in HHC systems and the development of solutions. Since the problem can basically be formulated as a vehicle routing problem, model formulations and solution approach themselves are also reviewed. Modeling for HHC starts with an allocation of resources (nurses), after which routing is considered. This was first studied by [2], who proposed a model for grouping nursesinto a region by considering the average travel distance. Later, [PERSON] et al. [3] provided a model for resource allocation that utilized spatial and geographical information to take into account distance and time, with a case study in Northern Ireland. When routing is added to the system, the problem becomes a routing and scheduling problem in home healthcare and can be formulated as a vehicle routing problem. Modeling for HHC resource allocation and routing was first proposed by [PERSON] et al. [4]. In contrast to prior works, [PERSON] and [PERSON] [5] added temporal, precedence and synchronization constraints (called synchronization and precedence constraints) to the routing and scheduling problem to allow more than one nurse to visit a patient to provide multiple services at the same time. The authors modified the original model by duplicating the node which required more than one visit. [PERSON] et al. [6] expanded on the previous work to feature five different temporal conditions in home healthcare and solved the problem by converting it into a set partitioning problem. Later, [PERSON] et al. [7] improved the model with synchronization and precedence constraints to contain a lower number of nodes instead of duplicating the nodes and proposed the greedy randomized adaptive search procedure for large-scale problems. [PERSON] et al. [8] proposed modeling to support double services at each patient location, called interdependent services, with each service having temporal-dependent conditions. The authors proposed an adaptive variable neighborhood search with neighborhood search techniques for the temporal constraints. The reader should refer to [9] for a review of the home healthcare routing and scheduling literature and to [PERSON] et al. [10] for a bibliometric analysis. Most home healthcare optimization objectives usually follow the classical objective: the minimization of either total travel distance or time [11]. In previous work, the objective functions for home healthcare routing and scheduling focused on costs related to distance or time, waiting time, over time, etc. [PERSON] et al. [12] proposed a metaheuristic based on particle swarm optimization with embedded local improvement procedures to solve home healthcare routing and scheduling problems by minimizing total distance when capacity and time windows need to be satisfied. [PERSON] et al. [13] maximized the total number of assigned tasks for a manpower allocation problem, an approach called node covering. The study focused on the cooperation of teams to perform the same task under the conditions of time and working hours, and the problem was solved using the Danzig-Wolfe decomposition [14]. [PERSON] et al. [15] used the Danzig-Wolfe decomposition technique and branch and price algorithm to solve a home healthcare problem formulated using a combination of routing and rostering problems. Their objective was to minimize the total number of caretakers. [PERSON] and [PERSON] [16] minimized the total completion time for each patient node in a home healthcare problem formulated as a vehicle routing problem with time windows. To make the model more suitable for HHC systems, the chosen objective functions were the sum of total distance or cost and other aspects related to HHC. [PERSON] et al. [17] proposed a model with time windows for working time and break time to minimize the sum of traveling times, waiting times and the dissatisfaction levels of clients and nurses. [PERSON] et al.