New decomposition methods for home care scheduling with predefined visits
Introduction
Due to population aging and the Canadian government’s plan to decentralize care, the demand for home care services has significantly increased during the last decade (Sinha and Bleakney, 2014). These services allow the patients to stay in their own homes for as long as possible. From the government’s point of view, home care services reduce the patient flow in hospitals and reduce the cost of care (Macintyre et al., 2002).
In this context, the home care agencies receive new patient requests everyday and continuously try to better manage their resources in order to serve them while maintaining a high level of service. Due to this resources’ management, the scheduling decisions are becoming crucial in order to keep a high acceptance rate and be able to maximize the agency’s revenues. In this work, we are interested in the home care scheduling with predefined visits (HCS-PV). This problem can be described as follows; each week some patients leave the agencies’ system (end of the care plan, problem requiring a hospital admission, etc.) and thus, agencies have to decide how many new patients they can accept and how the patients will be assigned to the providers and scheduled. In order to conserve a high continuity of care (i.e. a strong patient-provider relationship), the schedules for the patients already present in the system (named existing patients) have to stay unchanged (same assigned provider, same visit time and days). To efficiently solve the problem, the home care agencies usually split the patients per area and solve the scheduling problems per team of providers (5 to 15 providers). Moreover, most of the agencies are still creating the schedules by hand while trying to take into account all the constraints simultaneously. This usually leads to suboptimal solutions and usually requires a large amount of time for the schedulers. To help them in this task, more and more researchers attempt to develop efficient optimization methods taking into account the practical constraints met by the home care agencies while producing high quality solutions in a short amount of time in order to be used in practice.
According to the literature, the proposed HCS-PV can be described as a variant of the home health care routing and scheduling problem (HHCRSP), a 20 years old problem (Begur, Miller, Weaver, 1997, Cheng, Rich, 1998). To the best of our knowledge, no standard version of the HHCRSP exists (different constraints and/or objectives) but it is usually respresented as a rich vehicle routing problem. This plurality of modeling, which originates from the countries’ home care management policies, makes it difficult to compare the existing methods. In our case, the HCS-PV is close to a consistent VRP with home care specific constraints.
The HHCRSP (see Cissé, Yalçındağ, Kergosien, Şahin, Lenté, Matta, 2017, Fikar, Hirsch, 2017 for two comprehensive surveys) was originally solved on a daily planning horizon. It, then, has evolved to integrate more practical constraints such as the maximization of the patients and caregivers preferences (Braekers et al., 2016), the balance of the workload (Bertels and Fahle, 2006), shared visits (Frifita et al., 2017) or even the time-dependent travel time (Rest and Hirsch, 2016). Thereafter, the HHCRSP has been extended to a weekly horizon that allows for better coping with the reality of some constraints, such as the patients care plan and/or the continuity of care. Some exact methods (Borsani, Matta, Beschi, Sommaruga, 2006, Gamst, Jensen, 2012, Torres-Ramos, Alfonso-Lizarazo, Reyes-Rubiano, Quintero-Araújo, 2014, Trautsamwieser, Hirsch, 2014) have been proposed. Nevertheless, the complexity of the problem often leads to scalability issues. In order to cope with those issues, recent works apply metaheuristic-based methods (Decerle, Grunder, El Hassani, Barakat, 2018, Di Gaspero, Urli, 2014, Grenouilleau, Legrain, Lahrichi, Rousseau, 2019, Lin, Hung, Liu, Tsai, 2018, Yalçındağ, Cappanera, Scutellà, Şahin, Matta, 2016, Zhang, Yang, Chen, Bai, Chen, 2018). Those methods allow to solve large problems but do not offer any proof of convergence or optimality gap.
Recently, Heching et al. (2019) have proposed a new method to solve the HCS-PV based on a logic-based Benders decomposition (LBBD). The proposed method decomposes the problem in two parts, the acceptance and assignments of the patient is done in a master problem while the feasibility of the scheduling constraints (time windows, travel times, etc.) is checked in independent subproblems. Based on their computational experiments, the proposed LBBD outperforms the classical mixed-integer formulation in terms of computation time. Nevertheless, on large instances, the proposed LBBD does not seem robust and the computation time quickly increases with the size of the instances. In this work, we propose to extend the idea of solving the HCS-PV using Benders decompositions by introducing novel master and subproblem decompositions in order to reduce the computation time. This computation time’s reduction will allow the home care agencies to solve larger problems and use the proposed methods on a daily basis.
In this paper, the contributions are as follows. We firstly propose a new algorithm for the subproblem of the LBBD formulation presented in Heching et al. (2019). It decomposes the subproblem to make it easier to solve. Secondly, we present a new LBBD formulation with additional variables. The new variables correspond to visit patterns for the patients; they combine the assigned provider, the visit days, and the visit times in a single variable so that most of the constraints can be handled in the master problem. Finally, we propose a new matheuristic method based on a Dantzig–Wolfe formulation (DWF) and a large neighborhood search (LNS). This matheuristic iteratively solves the problem using LNS and then solves the DWF using the providers’ schedules found during the LNS iterations. Our computational experiments show that the matheuristic finds all the optimal solutions of the benchmark instances in less than 20 s.
The remainder of this paper is as follows. Section 2 defines the problem. Section 3 presents the mathematical formulations and Section 4 describes our matheuristic. Section 5 presents the computational results and Section 6 provides concluding remarks.
Section snippets
Problem definition
HCS-PV considers a patient set P and maximizes the number of scheduled patients given a set of available providers A on a 5-days horizon. For each scheduled patient, we must determine the assigned provider, the visit days, and the visit time. These decisions must take into account the existing patients; the scheduled visits for these patients cannot be modified for continuity of care purposes. In the home care context, continuity of care involves always sending the same provider at the same
Mathematical formulations
In this section, we present different formulations for the proposed problem. The goal here is to use those different formulations in order to tackle the problem in different ways and analyze if some computation time reductions are observed. Firstly, we recall the formulation introduced by Heching et al. (2019). This formulation corresponds to a natural Benders decomposition for the problem, with the assignments in the master problem and the scheduling in the subproblems. Secondly, we propose an
Visit pattern matheuristic
In this section, we present a visit pattern matheuristic based on the formulation proposed in Section 3.4 and a large neighborhood search (LNS). The LNS (Shaw, 1998) is a metaheuristic using the ruin-and-recreate principle (Schrimpf et al., 2000). This iterative method destroys some parts of the solution and then repairs it to improve its quality. The current and best solutions are then updated if necessary.
According to the literature (Grangier, Gendreau, Lehuédé, Rousseau, 2017, Grenouilleau,
Computational results
In this section, we present experiments that compare the efficiency of all the proposed methods (two-steps subproblem, the pattern-based formulation and the matheuristic). To provide an extensive comparison, we have re-implemented the method proposed in Heching et al. (2019). We refer to their formulation as Heching and we use the 57 provided instances. These instances are based on real-data provided by a home care agency from Pennsylvania, US. Each instance contains 60 patients and 6
Conclusions
The HCS-PV is a complex problem that home care agencies have to solve every week. The goal is to assign and schedule a set of new patients given a set of providers while taking into account the patients already present in the system. Each patient has a required number of visits and can be assigned to only one provider. The visit times must be the same for the entire horizon, and each provider has a maximum working time.
To solve this problem, we have extended the work of (Heching et al., 2019).
Acknowledgments
We thank Aliza Heching, John Hooker, and Ryo Kimura for providing access to the benchmark instances. This work has been supported by the Natural Sciences and Engineering Research Council of Canada.
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