Visiting nurses assignment and routing for decentralized telehealth service networks

Abstract

As telehealth utilization for ambulatory and home-based care skyrockets, there has been a paradigm shift to a decentralized and hybrid care delivery modality integrating both in-person and telehealth services provided at different layers of the care delivery network, i.e., central hospitals, satellite clinics, and patient homes. The operations of such care delivery systems need to take into consideration patients’ mobility and care needs, and rely on multiple types of nurses who can support and facilitate telehealth (with hospital physicians) in clinics and patient homes. We formulate an optimization problem, aiming at operationalizing the proposed care delivery network. Decisions regarding the type of care delivered, the location of care delivered, and the scheduling of all kinds of nurses are determined jointly to minimize operating costs while simultaneously satisfying patients’ care needs. We propose a bi-level approximation that exploits the structure of the hybrid telehealth system, and develop column generation-based heuristic algorithms to identify the joint decision rules for clinic selection, patient assignment, and visiting nurse routing problems. Numerical experiment results demonstrate our algorithm’s capability to achieve high-quality solutions in reasonable computation time, and is capable of solving instances with large patient sizes and time windows. Our work supports the efficient and effective operation of the proposed hybrid telehealth systems to improve patient access to care.

Appendices

A simple case study

Now we examine a toy example to offer a more intuitive understanding of the core concepts of pivot patients and effective latest arrival times. Figure 4a depicts the network structure and Fig. 4b details the corresponding travel times (minutes) for the visiting nurse. Assuming all patients are of Type I and all clinics are available, we define their parameters as follows: $a_1 = 300, b_1 = 395, v_1 = 60, a_2 = 305, b_2 = 390, v_2 = 60, a_3 = 100, b_3 = 250, v_3 = 40$. Here, 8:00 AM is represented as time 0 and 8:00 PM as 720, with both starting and ending times of the time windows being within the range [0, 720].

Fig. 4

Network Structure and Travel Time Table for the Illustrative Example

1.1 Pivot patient set

We first determine the pivot patients using Criterion 1. We found that patients 1 and 2 cannot be served on the same route. This is drawn from the calculations that $a_1 + t_{12}+ v_1 = 300 + 40 + 60 = 400$, which exceeds $b_2 = 390$, and $a_2 + t_{12}+ v_2 = 305 + 60 + 40 = 405$, surpassing $b_1 = 395$. Moving on to Criterion 2, we can determine that ${\mathcal {H}}_1 = \{0, 1, 2\}$, where 0 refers to the central hospital, and 1 and 2 are indexes of the clinics. This set contains all the healthcare facilities from which patient 1 can be reached within the time windows by the visiting nurses. Similarly, we have ${\mathcal {H}}_2 = {\mathcal {H}}_3 = \{0, 1, 2\}$. Meanwhile, for patient 1, we can identify the set ${\mathcal {B}}_1 = {3} \cup {\emptyset }$, which indicates that patient 1 is reachable from patient 3, but s/he cannot reach any other patient. In the same way, we obtain that ${\mathcal {B}}_2=\{3\} \cup \{\emptyset \} = \{3\},$ and ${\mathcal {B}}_3 = \{\emptyset \}\cup \{1, 2\} = \{1, 2\}$. Based on this, we compute $\omega _1 = \frac{\sum _{i \in {\mathcal {H}}_1} t_{i1}}{|{\mathcal {H}}_1|} + \min _{j \in {\mathcal {B}}_1} t_{1j} = \frac{25+24+38}{3} + \min \{23\} = 29+23 = 52$. Similarly, we obtain $\omega _2 = 63.67$ and $\omega _3 = 49.33$. This leads to the observation that $\omega _2> \omega _1 > \omega _3$. Finally, in the selection procedure, if we assume $N_{\text {cardinality}} = 2$, then the pivot patient set $P_{\text {new}} = \{1, 2\}$. Alternatively, when $N_{\text {cardinality}} = 1$, the pivot patient set $P_{\text {new}} = \{2\}$ as $\omega _2 > \omega _1$.

1.2 Effective latest arrival times

We compute the effective latest arrival time for a visiting nurse following a route that starts from clinic 2, visits patient 3, then patient 2, and finally returns to clinic 2. The upper bound of the effective latest arrival time for patient 2, denoted as $\varsigma _2^{\text {Upper}} = b_2 = 390$, represents the latest possible arrival time for the visiting nurses at patient 1 in the given route. Correspondingly, $\varsigma _3^{\text {Upper}}$ is calculated as $\min \{\varsigma _2^{\text {Upper}} - 60 - 40, 250\} = 250$.

Algorithm

1.1 The construction heuristic algorithm

Algorithm 2

The Construction Heuristic Algorithm

1.2 Insertion algorithm in initialization phase

For each patient $h_u$ in r, we obtain its starting time ${\bar{a}}_{h_u}$ in route r by applying the labeling algorithm, and obtain its neighbourhood set ${\mathcal {N}}_{h_u}$ as defined in Sect. 3.2.

Algorithm 3

Insertion Procedure (Initialization)

1.3 Insertion algorithm in pricing phase

In order to simplify the notation, the dual variables for both Type I and Type II patients are denoted by $\pi _j$, and the reduced cost for any route r is denoted by ${\bar{c}}_r$.

Algorithm 4

Insertion Procedure (Pricing)

1.4 Relocation algorithm in pricing phase

Algorithm 5

Relocation Procedure (Pricing)