# Balancing Staff Finishing Times vs. Minimizing Total Travel Distance in Home Healthcare Scheduling
## 2 Literature Review
[18] proposed home healthcare routing and scheduling with time windows, stochastic service time and hierarchical skill levels. The objective function, to minimize total travel cost and penalty cost for violating the time windows, was solved using the Danzig-Wolfe decomposition and the branch and price algorithm. [PERSON] et al. [19] proposed a model to minimize the sum of total travel time and service time, working time, and penalty cost. The model was based on a vehicle routing problem with time windows. The authors considered more than one aspect as the main objective function; most of the objectives were related to minimizing costs, which may include distance and time. [PERSON] et al. [20] considered the bi-objectives of minimizing total costs including routing and overtime costs and minimizing inconvenience costs including qualifications and working time constraints. Later research addressed climate change by considering carbon emissions as one of the objectives. [PERSON] and [PERSON] [21] proposed multiple objectives for the inventory and distribution problem in HHC. The objective functions used were total cost minimization, carbon emission minimization and customer satisfaction maximization. [PERSON] et al. [22] proposed a bi-objective model to minimize total cost and carbon emissions. Their metaheuristic was based on the simulated annealing algorithm and the salp swarm algorithm that can efficiently find the Pareto optimum. Since home healthcare routing and scheduling can be formulated as a vehicle routing problem, which is NP-hard, the studied problem also falls within NP-hard territory. Heuristics and metaheuristics are appropriate approaches to find the best solution for these problems in a reasonable amount of time [11]. [PERSON] and [PERSON] [23] proposed metaheuristics including simulated annealing and tabu search to find an initial solution and improved solution with improvement heuristics including swapping, moving and removing operators to solve HHC problems. They minimized the total weighted sum of cost and penalties. [PERSON] et al. [24] proposed a two-phase approach for vehicle routing problem to solve bi-objective functions including minimizing total distance and maximizing the coverage of service level to patients. The algorithm first optimized the service level and then improved total distance. [PERSON] et al. [25] also proposed a two-phase approach, which first clusters and then improves using a variable neighborhood search to solve the model with the objective of minimizing the total cost and total weighted completion time of multi-skill routing problems. [PERSON] et al. [26] proposed a variable neighborhood search to deal with multi-objective problems (minimizing total cost, greenhouse gas releases, maximum time spent by each caretaker and maximizing the matching level between nurses and patients) based on the epsilon approach. [PERSON] et al. [27] proposed an adaptive genetic algorithm and a mathematical model for the routing problem with the customer satisfaction function. The satisfaction function is derived from the soft time window violation, the data for which is collected via a questionnaire. The reader should refer to [28] for a literature review related to various objective functions and algorithms for solving problems in HHC. To summarize, Table 1 presents the objective functions and solution methods taken into account in the HHC context.