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# Merging short-term and long-term planning problems in home health care under continuity of care and patterns for visits

• * Corresponding author: Ettore Lanzarone
• Home Health Care (HHC) human resource management is a complex process. Moreover, as patients are assisted for a long time, their demand for care evolves in terms of type and frequency of visits. Under continuity of care, this uncertain evolution must be considered even when scheduling the visits in the short-term, as the corresponding operator-to-patient assignments could generate overtimes and unbalanced workloads in the long-term, which must be fixed by reassigning some patients and deteriorating the continuity of care. On the other hand, the operator-to-patient assignment problem under continuity of care over a long time period could generate solutions that are infeasible when the scheduling constraints are considered. We analyze the trade-offs between the two problems, to analyze the conditions in which they can be sequentially solved or an integration is required. In particular, we take an assignment and scheduling model for short-term planning, an operator-to-patient assignment model over a long time horizon, and we merge them into a new combined model. Results on a set of realistic instances show that the combined model is necessary when the number of patterns is limited and the variability of patients' demands is high, whereas simpler models deserve to be applied in less critical situations.

Mathematics Subject Classification: Primary: 90B50, 90B70; Secondary: 90C11.

 Citation: • • Table 1.  Sets, parameters and decision variables for the three models. Symbol $\checkmark$ means that the set, parameter of variable is included in the model, symbol $\circ$ that it is included only for $w = 1$, and symbol $\bullet$ that it is included only for $w>1$

 Model I Model II Model III Sets $O$ set of homogeneous operators ✔ ✔ ✔ $P$ set of patients ✔ ✔ ✔ $W$ set of weeks in the planning horizon ✔ $\bullet$ $D$ set of days in the first week $w=1$ ✔ ✔ $\Pi$ set of patterns ✔ ✔ Parameters $v_{pw}$ number of visits required by patient $p \in P$ at week $w \in W$ $\circ$ ✔ ✔ $s_p$ service time required for each visit to patient $p \in P$ ✔ ✔ ✔ $n_\pi$ total number of visits in pattern $\pi \in \Pi$ ✔ ✔ $m_{\pi d}$ 1 if pattern $\pi \in \Pi$ includes a visit at day $d \in D$; 0 otherwise ✔ ✔ $r_p$ average travel time to reach patient $p \in P$ ✔ ✔ ✔ $a_o$ weekly capacity of operator $o \in O$ ✔ ✔ ✔ $\alpha_w$ weight of week $w \in W$ in the objective function ✔ ✔ Decision variables $\lambda_{od}$ daily workload of operator $o \in O$ at day $d \in D$ ✔ ✔ $\omega_{ow}$ weekly workload of operator $o \in O$ at week $w \in W$ ✔ $\bullet$ $h_w$ maximum utilization rate at week $w \in W$ $\circ$ ✔ ✔ $z_{p\pi}$ 1 if pattern $\pi \in \Pi$ is assigned to patient $p \in P$; 0 otherwise ✔ ✔ $u_{op}$ 1 if operator $o \in O$ is assigned to patient $p \in P$; 0 otherwise ✔ ✔ ✔ $\mu_{op}^d$ 1 if $o \in O$ is assigned to $p \in P$ on day $d \in D$; 0 otherwise ✔ ✔

Table 2.  List of patterns for the flexible alternative, with at least one pattern for each $n_\pi$. The list for the rigid alternative is randomly extracted while keeping the condition of at least one pattern for each $n_\pi$

 Pattern $n_{\pi d}$ (visit at day $d$) $n_\pi$ $\pi$ $d=1$ $d=2$ $d=3$ $d=4$ $d=5$ $d=6$ 1 0 1 0 0 0 0 1 2 0 0 1 0 0 0 1 3 0 0 0 1 0 0 1 4 0 0 0 0 1 0 1 5 1 0 1 0 0 0 2 6 1 0 0 1 0 0 2 7 1 0 0 0 1 0 2 8 0 1 0 0 1 0 2 9 0 0 1 0 1 0 2 10 0 0 1 0 0 1 2 11 0 0 0 1 0 1 2 12 1 0 1 0 1 0 3 13 1 0 1 0 0 1 3 14 0 1 0 1 0 1 3 15 1 1 0 1 1 0 4 16 1 1 0 1 0 1 4 17 0 1 1 1 0 1 4 18 0 1 1 0 1 1 4 19 1 1 1 0 1 1 5 20 1 1 0 1 1 1 5 21 1 1 1 1 1 1 6

Table 3.  Combinations of hyperparameters used to generate low variability and high variability instances

 Instance Percentage of patients Variability $M_\sigma$ Trend $m_\kappa$ $M_\kappa$ Low 50% -0.5 0 0.5 variability 50% 0 0.5 0.5 High 50% -1 0 6 variability 50% 0 1 6

Table 4.  Results for the flexible set of patterns and high variability of demands (first set of experiments). INF denotes infeasibility

 Model solutions Executions Model I Model II Model III Model II on I Model I on II Model II on III Model III on II Replication 1 OFV 0.804 0.841 0.841 0.801 INF 0.841 0.841 $h_1$ 0.804 0.801 0.802 0.801 0.801 0.802 $\delta_1$ 0.010 0.006 0.007 0.006 0.007 0.007 $\delta_2$ – 0.018 0.016 – 0.016 0.016 $\delta_3$ – 0.005 0.010 – 0.010 0.010 $\delta_4$ – 0.023 0.018 – 0.018 0.018 Replication 2 OFV 0.850 0.805 0.806 0.853 0.879 0.805 0.806 $h_1$ 0.850 0.853 0.851 0.853 0.850 0.853 0.851 $\delta_1$ 0.011 0.022 0.016 0.022 0.011 0.022 0.016 $\delta_2$ – 0.017 0.023 – 0.283 0.017 0.023 $\delta_3$ – 0.039 0.039 – 0.228 0.039 0.039 $\delta_4$ – 0.005 0.007 – 0.254 0.005 0.007 Replication 3 OFV 0.802 0.785 0.787 0.800 0.840 0.785 0.787 $h_1$ 0.802 0.800 0.800 0.800 0.802 0.800 0.800 $\delta_1$ 0.013 0.015 0.018 0.015 0.013 0.015 0.018 $\delta_2$ – 0.015 0.016 – 0.307 0.015 0.016 $\delta_3$ – 0.008 0.006 – 0.135 0.008 0.006 $\delta_4$ – 0.015 0.025 – 0.135 0.015 0.025

Table 5.  Results for the flexible set of patterns and low variability of demands (first set of experiments)

 Model solutions Executions Model I Model II Model III Model II on I Model I on II Model II on III Model III on II Replication 1 OFV 0.849 0.822 0.821 0.848 0.838 0.821 0.822 $h_1$ 0.849 0.848 0.848 0.848 0.849 0.848 0.848 $\delta_1$ 0.013 0.013 0.011 0.013 0.013 0.011 0.013 $\delta_2$ – 0.013 0.011 – 0.037 0.011 0.013 $\delta_3$ – 0.028 0.026 – 0.073 0.026 0.028 $\delta_4$ – 0.018 0.026 – 0.059 0.026 0.018 Replication 2 OFV 0.776 0.764 0.764 0.776 0.793 0.764 0.764 $h_1$ 0.776 0.776 0.776 0.776 0.776 0.776 0.776 $\delta_1$ 0.010 0.010 0.011 0.010 0.010 0.010 0.011 $\delta_2$ – 0.012 0.011 – 0.036 0.012 0.011 $\delta_3$ – 0.020 0.018 – 0.080 0.020 0.018 $\delta_4$ – 0.027 0.017 – 0.144 0.027 0.017 Replication 3 OFV 0.748 0.726 0.726 0.751 0.756 0.726 0.726 $h_1$ 0.748 0.751 0.748 0.751 0.748 0.751 0.748 $\delta_1$ 0.003 0.007 0.005 0.007 0.003 0.007 0.005 $\delta_2$ – 0.017 0.022 – 0.089 0.017 0.022 $\delta_3$ – 0.014 0.019 – 0.070 0.014 0.019 $\delta_4$ – 0.006 0.004 – 0.108 0.006 0.004

Table 6.  Results for the rigid set of patterns and high variability of demands (first set of experiments). INF denotes infeasibility

 Model solutions Executions Model I Model II Model III Model II on I Model I on II Model II on III Model III on II Replication 1 OFV 0.801 0.841 0.843 INF INF INF 0.843 $h_1$ 0.801 0.801 0.810 0.810 $\delta_1$ 0.005 0.006 0.024 0.024 $\delta_2$ – 0.018 0.013 0.013 $\delta_3$ – 0.005 0.013 0.013 $\delta_4$ – 0.023 0.018 0.018 Replication 2 OFV 0.853 0.805 0.805 INF 0.902 INF 0.805 $h_1$ 0.853 0.853 0.854 0.853 0.854 $\delta_1$ 0.013 0.022 0.016 0.013 0.016 $\delta_2$ – 0.017 0.028 0.168 0.028 $\delta_31$ – 0.039 0.011 0.282 0.011 $\delta_4$ – 0.005 0.012 0.256 0.012 Replication 3 OFV 0.799 0.785 0.787 INF INF INF 0.787 $h_1$ 0.799 0.800 0.800 0.800 $\delta_1$ 0.014 0.015 0.014 0.014 $\delta_2$ – 0.015 0.012 0.012 $\delta_3$ – 0.008 0.025 0.025 $\delta_4$ – 0.015 0.018 0.018

Table 7.  Results for the rigid set of patterns and low variability of demands (first set of experiments). INF denotes infeasibility

 Model solutions Executions Model I Model II Model III Model II on I Model I on II Model II on III Model III on II Replication 1 OFV 0.848 0.822 0.821 INF 0.866 INF 0.821 $h_1$ 0.848 0.848 0.847 0.848 0.847 $\delta_1$ 0.012 0.013 0.010 0.012 0.010 $\delta_2$ – 0.013 0.010 0.101 0.010 $\delta_3$ – 0.028 0.018 0.173 0.018 $\delta_4$ – 0.018 0.019 0.173 0.019 Replication 2 OFV 0.777 0.764 0.764 INF 0.799 INF 0.764 $h_1$ 0.777 0.776 0.781 0.777 0.781 $\delta_1$ 0.017 0.010 0.018 0.017 0.018 $\delta_2$ – 0.012 0.010 0.072 0.010 $\delta_3$ – 0.020 0.010 0.089 0.010 $\delta_4$ – 0.027 0.010 0.196 0.010 Replication 3 OFV 0.749 0.726 0.726 INF 0.746 INF 0.726 $h_1$ 0.749 0.751 0.750 0.749 0.750 $\delta_1$ 0.003 0.007 0.008 0.003 0.008 $\delta_2$ – 0.017 0.016 0.033 0.016 $\delta_3$ – 0.014 0.016 0.105 0.016 $\delta_4$ – 0.006 0.006 0.054 0.006

Table 8.  Results for the larger instances (second set of experiments). INF denotes infeasibility

 Model solutions Executions Model I Model II Model III Model II on I Model I on II Model II on III Model III on II $|P|=150$; $|O|=10$ OFV 0.887 0.888 0.889 0.890 INF 0.888 0.889 $h_1$ 0.887 0.890 0.889 0.890 0.890 0.889 $\delta_1$ 0.019 0.036 0.039 0.036 0.036 0.039 $\delta_2$ – 0.035 0.038 – 0.035 0.038 $\delta_3$ – 0.027 0.025 – 0.027 0.025 $\delta_4$ – 0.015 0.021 – 0.015 0.021 $|W|=8$ OFV 0.776 0.744 0.744 0.780 0.813 0.744 0.744 $h_1$ 0.776 0.780 0.779 0.780 0.776 0.780 0.779 $\delta_1$ 0.010 0.036 0.018 0.036 0.010 0.036 0.018 $\delta_2$ – 0.001 0.003 – 0.036 0.001 0.003 $\delta_3$ – 0.018 0.018 – 0.080 0.018 0.018 $\delta_4$ – 0.018 0.018 – 0.144 0.018 0.018 $\delta_5$ – 0.018 0.018 – 0.179 0.018 0.018 $\delta_6$ – 0.017 0.017 – 0.167 0.017 0.017 $\delta_7$ – 0.002 0.002 – 0.167 0.002 0.002 $\delta_8$ – 0.001 0.003 – 0.185 0.001 0.003 $|P|=150$; $|O|=10$; $|W|=8$ OFV 0.887 0.891 0.890 0.887 INF 0.890 0.891 $h_1$ 0.887 0.887 0.889 0.887 0.889 0.887 $\delta_1$ 0.019 0.017 0.021 0.017 0.017 0.021 $\delta_2$ – 0.036 0.019 – 0.036 0.019 $\delta_3$ – 0.037 0.034 – 0.037 0.034 $\delta_4$ – 0.010 0.019 – 0.010 0.019 $\delta_5$ – 0.011 0.020 – 0.011 0.020 $\delta_6$ – 0.034 0.039 – 0.034 0.039 $\delta_7$ – 0.034 0.040 – 0.034 0.040 $\delta_8$ – 0.013 0.022 – 0.013 0.022 $|P|=150$; $|O|=10$; $|W|=8$; V OFV 0.911 0.913 0.913 0.908 INF 0.913 0.913 $h_1$ 0.911 0.908 0.908 0.908 0.908 0.908 $\delta_1$ 0.023 0.022 0.017 0.022 0.022 0.017 $\delta_2$ – 0.032 0.029 – 0.032 0.029 $\delta_3$ – 0.022 0.024 – 0.022 0.024 $\delta_4$ – 0.012 0.035 – 0.012 0.035 $\delta_5$ – 0.021 0.033 – 0.021 0.033 $\delta_6$ – 0.060 0.054 – 0.060 0.054 $\delta_7$ – 0.044 0.032 – 0.044 0.032 $\delta_8$ – 0.030 0.054 – 0.030 0.054
• Tables(8)

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