Community health pathways modeling and scheduling under uncertainty

Abstract

Scheduling and coordinating constrained resources in community healthcare settings at a centralized Pathways Community HUB is challenging due to limited resources and the inherent dynamics of the processes and the organizational structures. In this work, we introduce a stochastic programming (SP) approach for connected community health for optimally scheduling community health pathways (CHPs) under uncertainty in resource availability. A CHP is a standardized tool that details multiple steps of a healthcare-related service and the required resources for each step. The SP methodology was implemented and applied to data for a real Pathways Community HUB for a U.S. county involving several CHPs, community health workers, physicians, and other resources. The computational results are promising and they show that client access times depend on the HUB resources uncertain future availability and the level of client demand, with high client demand resulting in relatively longer access time. The study reveals that schedules provided by a deterministic approach where resource availability is assumed to be known can be too optimistic. Several managerial insights are learned from this study, including the observation that the SP model provides client schedules that are equitable across the same type of community health workers.

Introduction

Community healthcare agencies in the U.S. such as those at the county level are tasked with providing medical, behavioral, and social services to community members in need. However, providing such services requires careful coordination of limited resources. Agency resources include case managers, community health workers, and office facilities, while external resources include physicians, nurses, social workers, behavioral health specialists, insurance companies, and community-based organizations. Care coordination is the organization of care activities among individual clients and providers to facilitate the appropriate delivery of healthcare services. ¹ Performing care coordination involving disparate resources presents significant challenges in terms of scheduling, adhering to the care activities protocols, and tracking progress of these activities for each client in the system. The focus of this paper is on community care coordination using the Pathways Community HUB model of care coordination. ² This model considers the community level to improve quality of care for individual clients by coordinating community-based health and social services.

This work introduces a stochastic programming (SP) approach for scheduling and coordinating community health pathways (CHPs) under uncertainty in the availability of the limited Pathways Community HUB resources and client demand, while providing equity among community health workers via workload balancing. There are different types of pathways and in this work we focus on CHPs for connected community health services, which are different from clinical or care pathways. ³ A CHP is simply a structured and time-framed multidisciplinary care guideline that details essential steps of a community healthcare process. ⁴ It provides a sequence of appointments and resources needed for each appointment together with its duration over a specified time horizon. Community health pathways are unique in that they track the individual client being served and each step of the CHP addresses a well defined action towards resolving the issue addressed by the pathway. Such issues include lack of health insurance, medical home, homelessness, pregnancy, immunizations, and so on. Each step can deal with a health, social or cultural issue. A CHP is not considered completed until either the issue is successfully resolved or a specific point is reached to close the CHP that it has not been completed. In addition, payments for a CHP are made at specific points along the pathway, with the largest payment offered at the successful completion of the pathway. ²

The Pathways Community HUB model of care coordination is a construct that was introduced to enable coordination of care activities of different organizations and their services. Pathway models focus on the progress and outcomes of individual clients as they traverse the care organizations.^5,6 In turn, pathways coordination can enable a comprehensive approach to community health service focused on reducing the healthcare inequalities to improve health outcomes for the community. ⁷ Community-based services have been shown to play a vital role in addressing some of the nation’s most difficult health problems, including health disparities and the rising chronic health conditions such as obesity and diabetes. ⁸ Effective care coordination at the community level is a hard task. Many individuals who need community health services have complex health needs and chronic health conditions. ⁹ Addressing these needs requires a combination of multiple services (medical services, behavioral health services, and social services) and support from a wide variety of community-based resources to reduce the health and social barriers to improve healthcare outcomes. ¹⁰ This complexity poses major challenges in terms of effectively connecting different parties (clients, agencies, service providers) and coordinating them to work in concert to achieve positive outcomes.

The limited number of resources needed to provide community health services calls for optimization methods to aid in scheduling to best utilize the limited resources while providing quick access time for all clients. This scheduling challenge is compounded by the new paradigm of value-based purchasing, ¹¹ which has drastically changed how services are measured, reported, and rewarded in healthcare delivery. The new paradigm of healthcare delivery asks for new designs and innovative methods to support community health service coordination and optimization. This work makes a step toward achieving that goal. However, scheduling CHPs is extremely challenging and literature on solution methods for this problem is scant. The few available works are in the hospital setting: constraint programming model ¹² and combined genetic algorithm with particle swarm optimization for scheduling clinical pathways in a hospital setting. ¹³ Clinical pathways can significantly improve patient access time in a hospital setting. ¹³ Furthermore, studies have shown that clinical pathways lead to positive outcomes for various cases, including a reduction in the prescription of laboratory tests ¹⁴ and in-hospital complications. ¹⁵

A variety of models have been proposed to minimize client waiting time while scheduling resources, but literature on pathways scheduling is limited. Mixed-integer programming models to minimize wait time for patients have been proposed.^16,17 The objective is to minimize the weighted sum of the total wait times of patients ¹⁶ or to find a reasonable schedule within a desired time window. ¹⁷ Such works assume perfect information and ignore the uncertainty in the parameters. A few works take demand uncertainty into consideration while minimizing the patient waiting time. ¹⁸ In related work, a two-stage SP model is proposed to minimize expected total cost. ¹⁹ However, in our work we consider uncertainty in resource availability while minimizing the workload imbalance and client waiting time. To eliminate the imbalanced workload distribution among resources, a satisfaction indicator over 1 month period to inform decisions of resource future assignments was introduced. ²⁰ In our model, workload balancing is done after each CHPs assignment, instead of accumulating the workload over a month.

The main contributions of this paper include the following: (a) an SP model for optimally scheduling individual CHPs under unknown future resource availability while allowing for workload balancing of resources of the same type to enable equity among personnel; (b) a computer simulation model that uses the SP model to assess CHPs scheduling and workload balancing in a structured setting; and (c) a computational study based on a real setting to demonstrate the new approach and gain managerial insights into CHPs scheduling under uncertainty. The rest of this paper is organized as follows: next we present a detailed SP scheduling model in the CHPS Scheduling Model section. Data collection and analysis, model development and implementation, and experiment design are discussed in the Materials and Methods section. We report and discuss our simulation results in the Results and Discussion section. Finally, we provide concluding remarks and point out future work in the Conclusion section.

CHPs scheduling model

A CHP involves several steps with each step requiring one or more resources, and the required resource(s) for each step have to be scheduled for a specific day and time in the future. Thus, scheduling a CHP requires scheduling multiple care activities that have to be processed at some point between an earliest start time and a latest completion time, all while satisfying the required resources’ availability constraints. We derive a two-stage SP model to optimally schedule CHPs under unknown future resources’ availability. The objective is to minimize client access time, i.e., the time from when a pathway appointment request is made to when the client is first seen, while providing equitable workload among the human resources.

In the first-stage, we determine here-and-now the client’s schedule for all the steps in the CHP, i.e., we assign a day, time slots and required resources for each step of the CHP, before availability of the resources is unveiled. Then in the second-stage we account for possible future resource availability for each day and step of the CHP to inform the first-stage in finding an optimal schedule. In SP, unknown future availability of the resources is characterized by probability distributions. Specifically, we consider a probability distributions for the future availability of each resource for each day and each time slot, respectively.

Pathways Community HUBs are often burdened with scheduling large volumes of clients under limited resources. The SP approach enables to schedule resources in the first-stage such that those resources are most likely to be available in the future for each step of the CHP to minimize potential appointment cancellations. Thus, a planning tool to aid in determining appointments that are unlikely to be canceled is essential. To formulate the SP model for scheduling a client appointment for a given CHP, we use the mathematical notation defined in Tables 1 and 2.

Table 1.

Notation for the first-stage SP model.

First-stage sets
$\mathcal{P}$ :	Set of CHPs, indexed p.
$\mathcal{R}\left(i\right)$ :	Set of resources of type i, indexed r.
$\mathcal{I}\left(p\right)$ :	Set of resource types needed for CHP p, indexed i.
$\mathcal{I}\left(p,s\right)$ :	Set of resource types needed for step s of CHP p, indexed i.
$\mathcal{S}\left(p\right)$ :	Set of steps for CHP p, indexed s.
$\mathcal{S}\left(p,h\right)$ :	Set of steps for CHP p on day h, indexed $s,\mathcal{S}\left(p,h\right)\subset \mathcal{S}\left(p\right)$ .
$\mathcal{T}$ :	Set of time slots in a day, indexed t.
$\mathcal{D}$ :	Set of days in the planning horizon, indexed d.
$\mathcal{D}\left(p\right)$ :	Subset of days in the planning horizon on which CHP p can start, indexed d.
$\overline{\mathcal{D}}\left(p,d\right)$ :	Given a CHP p starting on day $d\in \mathcal{D}\left(p\right)$ , $\overline{\mathcal{D}}\left(p,d\right)$ is the set of CHP calendar days for all steps of CHP p, indexed h.
$\mathcal{O}\left(p,s\right)$ :	$\mathcal{O}\left(p,s\right)$ is a set of numbers indicating the resource sequence order for step s of CHP p.
First-stage parameters
c d :	Cost of starting appointment on day d.
ap,s,i,r,t,h:	ap,s,i,r,t,h = 1, if resource r of type i is available for step s of CHP p at start time slot t on day h of CHP calendar, ap,s,i,r,t,h = 0, otherwise. This is planned resource availability.
np,s,i:	np,s,i is the number of consecutive time slots using resource type i for step s of CHP p.
bi,r,h:	bi,r,h is the total number of time slots that resource r of type i is already assigned on day h when schedule .
ES i :	ES i is earliest starting time for resource type i.
LS i :	LS i is latest starting time for resource type i.
EA i :	EA i is earliest assigned time for resource type i.
LA i :	LA i is latest assigned time for resource type i.
First-stage decision variables
z d :	z d = 1, if client starts the first appointment on day d of the planning horizon, z d = 0, otherwise.
z:	Vector of z d ’s, $z=\left(z_1,\dots ,z_{\left|\mathcal{D}\left(p\right)\right|}\right)$ .
xp,s,i,r,h:	xp,s,i,r,h = 1, if resource r of resource type i is assigned to step s of CHP p on day h, $h\in \overline{\mathcal{D}}\left(p,d\right)$ , xp,s,i,r,h = 0, otherwise.
x:	Vector of xp,s,i,r,h’s.
yp,s,i,r,t,h:	yp,s,i,r,t,h = 1, if resource r of type i required for step s of CHP p is assigned to time slot t on day h of the planning horizon, yp,s,i,r,t,h = 0, otherwise.
y:	Vector of yp,s,i,r,t,h’s.
wp,s,i,r,t,h:	wp,s,i,r,t,h = 1, if resource r of type i for step s of CHP p starts the assignment at time slot t on day h of CHP calendar,wp,s,i,r,t,h = 0, otherwise.
w:	Vector of wp,s,i,r,t,h’s.
Second-stage Sets
Ω:	Set of outcomes or scenarios ω, where ω is an outcome of a multivariate random variable $\stackrel{\sim }{\omega }$ that describes resource availability and is defined on a probability space.
${\mathcal{D}}^{\omega }\left(p\right)$ :	Subset of CHP starting days in the planning horizon on which CHP p can start under scenario ω, indexed d ω .
${\overline{\mathcal{D}}}^{\omega }\left(p,d^{\omega }\right)$ :	Set of CHP calendar days for all steps of CHP p starting on day $d^{\omega }\in {\mathcal{D}}^{\omega }\left(p\right)$ , indexed $\overline{h}$ .

Table 2.

Notation for the second-stage SP model.

Second-stage parameters
$a_{p,s,i,r,t,\overline{h}}^{\omega }$ :	$a_{p,s,i,r,t,\overline{h}}^{\omega }=1$ , if resource r of type i is available for step s of CHP p at start time slot t on day $\overline{h}$ of CHP calendar under scenario ω, $a_{p,s,i,r,t,\overline{h}}^{\omega }=0$ , otherwise. This is actual (scenario) resource availability.
$b_{i,r,\overline{h}}^{\omega }$ :	$b_{i,r,\overline{h}}^{\omega }$ is the total number of time slots that resource r of type i is already assigned on day $\overline{h}$ when schedule under scenario ω.
Second-stage variables
$z_{\overline{d}}^{\omega }$ :	$z_{\overline{d}}^{\omega }=1$ , if client starts the first appointment on day $\overline{d}$ of the planning horizon under scenario $\omega ,z_{\overline{d}}^{\omega }=0$ , otherwise.
z ω :	z ω = 1, if cannot find feasible schedule starting on day $\overline{d}$ , z ω = 0 otherwise.
$x_{p,s,i,r,\overline{h}}^{\omega }$ :	$x_{p,s,i,r,\overline{h}}^{\omega }=1$ , if resource r of resource type i is assigned to step s of CHP p on day $\overline{h}$ under scenario ω, $x_{p,s,i,r,\overline{h}}^{\omega }=0$ , otherwise.
x ω :	Vector of $x_{p,s,i,r,\overline{h}}^{\omega }$ ’s.
$y_{p,s,i,r,t,\overline{h}}^{\omega }$ :	$y_{p,s,i,r,t,\overline{h}}^{\omega }=1$ , if resource r of type i required for step s of CHP p is assigned to time slot t on day $\overline{h}$ of the planning horizon under scenario ω. $y_{p,s,i,r,t,\overline{h}}^{\omega }=0$ , otherwise.
y ω :	Vector of $y_{p,s,i,r,t,\overline{h}}^{\omega }$ ’s.
$w_{p,s,i,r,t,\overline{h}}^{\omega }$ :	$w_{p,s,i,r,t,\overline{h}}=1$ , if resource r of type i for step s of CHP p starts the assignment at time slot t on day $\overline{h}$ of CHP calendar under scenario ω, $w_{p,s,i,r,t,h}^c=0$ , otherwise.
w ω :	Vector of $w_{p,s,i,r,t,\overline{h}}^{\omega }$ ’s.
$L_{i,\overline{h}}^{\min ,\omega }$ :	$L_{i,\overline{h}}^{\min ,\omega }$ , is the minimum number of time slots that resource type i is assigned on day $\overline{h}$ under scenario ω.
Lmin,ω:	Vector of $L_{i,\overline{h}}^{\min ,\omega }$ ’s.
$L_{i,\overline{h}}^{\max ,\omega }$ :	$L_{i,\overline{h}}^{\max ,\omega }$ , is the maximum number of time slots that resource type i is assigned on day $\overline{h}$ under scenario ω.
Lmax,ω:	Vector of $L_{i,\overline{h}}^{\max ,\omega }$ ’s.

A CHP typically involves multiple types of resources in a given step. For a CHP p, if $\left|\mathcal{I}\left(p,s\right)\right|=1$ it means that only one resource type is needed in step s of the pathway. If $\left|\mathcal{I}\left(p,s\right)\right|\ge 2$ it means that more than two types of resources are required in step s. Based on the pathways we consider in this work, we restrict ourselves to no more than three resource types in a given step. Therefore, let us index the resources by i₁, i₂, and i₃, in that order. Since the resources are required to perform a given task in a particular sequence, we refer to this sequence as the resource precedence order. We denote the precedence order of CHP p and step s by $\mathcal{O}\left(p,s\right)$ .

Each element (0, 1, or (2) of $\mathcal{O}\left(p,s\right)$ will represent the sequence order. If a single resource type is needed in step s of CHP p, the sequence order is written as $\mathcal{O}\left(p,s\right)=\left\{0\right\}$ . If two resource types are need, the precedence order is either $\mathcal{O}\left(p,s\right)=\left\{0,0\right\}$ or $\mathcal{O}\left(p,s\right)=\left\{0,1\right\}$ . The precedence order $\mathcal{O}\left(p,s\right)=\left\{0,0\right\}$ means that step s of CHP p requires two resource types, i₁ and i₂, concurrently. When $\mathcal{O}\left(p,s\right)=\left\{0,1\right\}$ it means that two resource types i₁ and i₂ are required, with the first resource available before the second to complete the task. Notice that the $\left|\mathcal{O}\left(p,s\right)\right|$ is equal to the number of resources required in step s. If three resources are required for step s, then the following three precedence orders are possible: $\mathcal{O}\left(p,s\right)=\left\{0,0,1\right\}$ , $\mathcal{O}\left(p,s\right)=\left\{0,1,1\right\}$ and $\mathcal{O}\left(p,s\right)=\left\{0,1,2\right\}$ . The six possible resource precedence orders required in a given step are listed Table 3.

Table 3.

Six resource precedence order cases.

Case	# Resources	$\mathcal{O}\left(p,s\right)$
0	$\left|\mathcal{I}\left(p,s\right)\right|=1$	{0}
1	$\left|\mathcal{I}\left(p,s\right)\right|=2$	{0,0}
2	$\left|\mathcal{I}\left(p,s\right)\right|=2$	{0,1}
3	$\left|\mathcal{I}\left(p,s\right)\right|=3$	{0,0,1}
4	$\left|\mathcal{I}\left(p,s\right)\right|=3$	{0,1,1}
5	$\left|\mathcal{I}\left(p,s\right)\right|=3$	{0,1,2}

We are now ready to derive the SP formulation. For a given client, the first-stage decision variable vector v specifies each resource’s schedule, i.e., days and corresponding time slots the resource is assigned to each step of the CHP, as well as the start time slot on each day in the schedule. The first-stage objective is to minimize client access time while taking into account the resources’ unknown availability in the future. Thus, given the first-stage decision v, the second-stage objective is to minimize expected future cost associated with workload balancing based on the resources’ uncertain availability. Mathematically, the objective is written as follows:

$\underset{z,x,y,w}{\mathrm{Min}}{\displaystyle \sum }_{d\in \mathcal{D}\left(p\right)}{c}_d{z}_d+\mathbb{E}\left[f\left(z,\stackrel{\sim }{\omega }\right)\right]$

(1)

where the first term computes the time in days from when a client requests an pathway appointment to when the client starts their first appointment (access time). The scalar c _d is a cost factor, with higher values indicating that we want to schedule the client as soon as possible. Given a first-stage decision (z, x, y, w) and outcome ω ∈ Ω of $\stackrel{\sim }{\omega }$ , the second term is the (second-stage) expected recourse function. It calculates the expected sum of workload differences among resources of the same type to affect equity. We give an explicit expression of the $f\left(z,\stackrel{\sim }{\omega }\right)$ later, but first, we define the first-stage constraints.

Given the subset of days in the planning horizon on which CHP p can start, $\mathcal{D}\left(p\right)$ , to select one possible starting day from the earliest possible days, the following constraint is imposed:

${\displaystyle \sum }_{d\in \mathcal{D}\left(p\right)}{z}_d=1$

(2)

To ensure that the needed resource r of type i to perform step s of CHP $p\in \mathcal{P}$ is selected on day d, we add this constraint:

$-z_d+{\displaystyle \sum }_{r\in \mathcal{R}\left(i\right)}{x}_{p,s,i,r,h}=0,\forall s\in \mathcal{S}\left(p,h\right),i\in \mathcal{I}\left(p,s\right),d\in \mathcal{D}\left(p\right),h\in \overline{\mathcal{D}}\left(p,d\right)$

(3)

The next constraint ensures that the required resources are assigned to a time slot:

$\begin{array}{l}{n}_{p,s,i}·x_{p,s,i,r,h}-{\displaystyle \sum }_{t=E{S}_i}^{L{S}_i}{y}_{p,s,i,r,t,h}=0,\forall s\in \mathcal{S}\left(p,h\right),i\in \mathcal{I}\left(p,s\right),\\ r\in \mathcal{R}\left(i\right),d\in \mathcal{D}\left(p\right),h\in \overline{\mathcal{D}}\left(p,d\right).\end{array}$

(4)

To choose a starting time slot for the needed resource, the following constraint is added:

$\begin{array}{l}{z}_d-{\displaystyle \sum }_{t=E{S}_i}^{L{S}_i}{\displaystyle \sum }_{r\in \mathcal{R}\left(i\right)}{w}_{p,s,i,r,t,h}=0,\forall s\in \mathcal{S}\left(p,h\right),i\in \mathcal{I}\left(p,s\right),d\in \mathcal{D}\left(d\right),\\ h\in \overline{\mathcal{D}}\left(p,d\right).\end{array}$

(5)

The following constraint ensures that the resource is assigned to an activity only when the resource is available:

$\begin{array}{l}{y}_{p,s,i,r,t,h}\le a_{p,s,i,r,t,h},\forall s\in \mathcal{S}\left(p,h\right),i\in \mathcal{I}\left(p,s\right),r\in \mathcal{R}\left(i\right),\\ t\in \left\{E{A}_i,\dots ,L{A}_i\right\},d\in \mathcal{D}\left(d\right),h\in \overline{\mathcal{D}}\left(p,d\right).\end{array}$

(6)

We add the following constraint to guarantee that consecutive time slots are selected if step s of CHP p using resource r of type i requires more than one time slot:

$\begin{array}{l}{\displaystyle \sum }_{t=t^{\prime }}^{t^{\prime }+n_{p,s,i}-1}{y}_{p,s,i,r,t,h}-n_{p,s,i}·w_{p,s,i,r,t^{\prime },h}\ge 0,\forall s\in \mathcal{S}\left(p,h\right),i\in \mathcal{I}\left(p,s\right),\\ \hspace{1em}r\in \mathcal{R}\left(i\right),t^{\prime }\in \left\{E{S}_i,\dots ,L{S}_i\right\},d\in \mathcal{D}\left(p\right),h\in \overline{\mathcal{D}}\left(p,d\right).\end{array}$

(7)

When multiple resources are involved in a step of a CHP, we need to add the appropriate precedence order constraints. We refer to the list of the possible precedence order cases in Table 3 to ensure that each task in a CHP is assigned the needed resources in the required order:

Case 0

$\left|\mathcal{I}\left(p,s\right)\right|=1$ for step s, precedence order $\mathcal{O}\left(p,s\right)=\left\{0\right\}$ : No additional constraints are needed.

Case 1

$\left|\mathcal{I}\left(p,s\right)\right|=2$ for step s with resources indexed i₁ and i₂, precedence order $\mathcal{O}\left(p,s\right)=\left\{0,0\right\}$ .

$\begin{array}{l}{w}_{p,s,i_1,r_1,t,h}={\displaystyle \sum }_{r_2\in \mathcal{R}\left(i_2\right)}{w}_{p,s,i_2,r_2,t,h},\forall i_1,i_2\in \mathcal{I}\left(p,s\right),r_1\in \mathcal{R}\left(i_1\right),\\ t\in \left\{1,\dots \left|\mathcal{T}\right|-n_{p,s,i_1}+1\right\},d\in \mathcal{D}\left(d\right),h\in \overline{\mathcal{D}}\left(p,d\right).\end{array}$

(8a)

$\begin{array}{l}{w}_{p,s,i_1,r_1,t,h}={\displaystyle \sum }_{r_2\in \mathcal{R}\left(i_2\right)}{w}_{p,s,i_2,r_2,t,h},r_1\in \mathcal{R}\left(i_1\right),\\ t\in \left\{1,\dots \left|\mathcal{T}\right|-n_{p,s,i_1}+1\right\},d\in \mathcal{D}\left(d\right),h\in \overline{\mathcal{D}}\left(p,d\right)\end{array}$

(8b)

Case 2

$\left|\mathcal{I}\left(p,s\right)\right|=2$ for step s with resources indexed i₁ and i₂, precedence order $\mathcal{O}\left(p,s\right)=\left\{0,1\right\}$ . Resource type i₁ must be available before i₂. In this case, we need to include the following constraints for step s:

$\begin{array}{l}{w}_{p,s,i_1,r_1,t,h}\le {\displaystyle \sum }_{t^{\prime }=t+n_{p,s,i_1}}^{\left|\mathcal{T}\right|-n_{p,s,i_2}+1}{\displaystyle \sum }_{r_2\in \mathcal{R}\left(i_2\right)}{w}_{p,s,i_2,r_2,t^{\prime },h},\forall i_1,i_2\in \mathcal{I}\left(p,s\right),\\ r_1\in \mathcal{R}\left(i_1\right),t\in \left\{n_{p,s,i_1}+1,\dots ,\left|\mathcal{T}\right|-n_{p,s,i_1}-n_{p,s,i_2}+1\right\},\\ d\in \mathcal{D}\left(d\right),h\in \overline{\mathcal{D}}\left(p,d\right).\end{array}$

(9)

Case 3

$\left|\mathcal{I}\left(p,s\right)\right|=3$ for some step s with resources indexed i₁, i₂ and i₃, precedence order $\mathcal{O}\left(p,s\right)=\left\{0,0,1\right\}$ . Resource type i₁ and i₂ must be available at the same time before resource type i₃. In this case, we need to impose the following constraints for step s:

$\begin{array}{l}{w}_{p,s,i_1,r_1,t,h}={\displaystyle \sum }_{r_2\in \mathcal{R}\left(i_2\right)}{w}_{p,s,i_2,r_2,t,h},\forall i_1,i_2\in \mathcal{I}\left(p,s\right),r_1\in \mathcal{R}\left(i_1\right),\\ t\in \left\{1,\dots \left|\mathcal{T}\right|-n_{p,s,i_1}-n_{p,s,i_2}+1\right\},d\in \mathcal{D}\left(d\right),h\in \overline{\mathcal{D}}\left(p,d\right).\end{array}$

(10a)

$\begin{array}{l}{w}_{p,s,i_2,r_2,t,h}\le {\displaystyle \sum }_{t^{\prime }=t+n_{p,s,i_2}}^{\left|\mathcal{T}\right|-n_{p,s,i_3}+1}{\displaystyle \sum }_{r_3\in \mathcal{R}\left(i_3\right)}{w}_{p,s,i_3,r_3,t^{\prime },h},\forall i_2,i_3\in \mathcal{I}\left(p,s\right),\\ i_2\ne i_3,r_2\in \mathcal{R}\left(i_2\right),t\in \left\{n_{p,s,i_1}+1,\dots ,\left|\mathcal{T}\right|-n_{p,s,i_3}+1\right\},\\ d\in \mathcal{D}\left(d\right),h\in \overline{\mathcal{D}}\left(p,d\right)\end{array}$

(10b)

Case 4

$\left|\mathcal{I}\left(p,s\right)\right|=3$ for some step s with resources indexed i₁, i₂ and i₃, precedence order $\mathcal{O}\left(p,s\right)=\left\{0,1,1\right\}$ . Resource type i₁ must be available before resource types i₂ and i₃, and resource types i₂ and i₃ must be available at the same time. In this case, we need to include the following constraints for step s:

$\begin{array}{l}{w}_{p,s,i_1,r_1,t,h}\le {\displaystyle \sum }_{t^{\prime }=t+n_{p,s,i_1}}^{\left|\mathcal{T}\right|-n_{p,s,i_2}+1}{\displaystyle \sum }_{r_2\in \mathcal{R}\left(i_2\right)}{w}_{p,s,i_2,r_2,t^{\prime },h},\forall i_1,i_2\in \mathcal{I}\left(p,s\right),\\ r_1\in \mathcal{R}\left(i_1\right),t\in \left\{1,\dots ,\left|\mathcal{T}\right|-n_{p,s,i_1}-n_{p,s,i_2}+1\right\},d\in \mathcal{D}\left(d\right),\\ h\in \overline{\mathcal{D}}\left(p,d\right).\end{array}$

(11a)

$\begin{array}{l}{w}_{p,s,i_2,r_2,t,h}={\displaystyle \sum }_{r_3\in \mathcal{R}\left(i_3\right)}{w}_{p,s,i_3,r_3,t,h},\forall i_2\ne i_3,i_2\in \mathcal{I}\left(p,s\right),\\ t\in \left\{n_{n,p,s,i_1}+1,\dots \left|\mathcal{T}\right|-n_{p,s,i_2}+1\right\},d\in \mathcal{D}\left(d\right),h\in \overline{\mathcal{D}}\left(p,d\right)\end{array}$

(11b)

Case 5

$\left|\mathcal{I}\left(p,s\right)\right|=3$ for some step s with resources indexed i₁, i₂ and i₃, precedence order $\mathcal{O}\left(p,s\right)=\left\{0,1,2\right\}$ . Resource type i₁ must be available before resource type i₂, and resource type i₂ must be available before resource type i₃. In this case, the following constraints are added for step s:

$\begin{array}{l}{w}_{p,s,i_1,r_1,t,h}\le {\displaystyle \sum }_{t^{\prime }=t+n_{p,s,i_1}}^{\left|\mathcal{T}\right|-n_{p,s,i_2}-n_{p,s,i_3}+1}{\displaystyle \sum }_{r_2\in \mathcal{R}\left(i_2\right)}{w}_{p,s,i_2,r_2,t^{\prime },h},\forall i_1,i_2\in \mathcal{I}\left(p,s\right),\\ r_1\in \mathcal{R}\left(i_1\right)t\in \left\{1,\dots ,\left|\mathcal{T}\right|-n_{p,s,i_1}-n_{p,s,i_2}-n_{p,s,i_3}+1\right\},\\ d\in \mathcal{D}\left(d\right),h\in \overline{\mathcal{D}}\left(p,d\right).\end{array}$

(12a)

$\begin{array}{l}{w}_{p,s,i_2,r_2,t,h}\le {\displaystyle \sum }_{t^{\prime }=t+n_{p,s,i_2}}^{\left|\mathcal{T}\right|-n_{p,s,i_3}+1}{\displaystyle \sum }_{r_3\in \mathcal{R}\left(i_3\right)}{w}_{p,s,i_3,r_3,t^{\prime },h},\forall i_2,i_3\in \mathcal{I}\left(p,s\right),\\ i_2\ne i_3,r_2\in \mathcal{R}\left(i_2\right),t\in \left\{n_{p,s,i_1}+1,\dots ,\left|\mathcal{T}\right|-n_{p,s,i_2}-n_{p,s,i_3}+1\right\},\\ h\in \overline{\mathcal{D}}\left(p,d\right)\end{array}$

(12b)

Putting everything together, the SP model can be written as follows:

$\begin{array}{ll}\underset{z,x,y,w}{\mathrm{Min}}& {\displaystyle \sum }_{d\in \mathcal{D}\left(p\right)}{c}_d{z}_d+\mathbb{E}\left[f\left(z,\stackrel{\sim }{\omega }\right)\right]\end{array}$

(13)

s.t. Constraints (2)–(7)

Constraints (8)–(12) needed for the CHP

$z_d\in \left\{0,1\right\},\forall d\in \mathcal{D}\left(p\right),x_{p,s,i,r,h},y_{p,s,i,r,t,h},w_{p,s,i,r,t,h}\in \left\{0,1\right\},$

$\forall p\in \mathcal{P},s\in \mathcal{S}\left(p,h\right),i\in \mathcal{I}\left(p,s\right),r\in \mathcal{R}\left(i\right),t\in \mathcal{T},h\in \overline{\mathcal{D}}\left(p,d\right).$

For a scenario ω of $\stackrel{\sim }{\omega }$ , f(z, ω) computes the future cost based on the first-stage schedule z (start date). The scenario ω reveals the future resource availability for all steps of the CHP under consideration in the second-stage and we need to find and assign resources to the schedule z for every day and step of the CHP, while optimizing the workload balance. For a given (z, ω), the recourse function f(z, ω) is given as shown in problem (14).

Subproblem (14) minimizes the total difference between the maximum and minimum workload among resources of the same type to enable workload balancing. We add a sufficiently large penalty M to the second term of the objective function to enforce relatively complete recourse by allowing z ^ω to take a value of one when we cannot find a feasible schedule starting on day $\overline{d}=d\mid z_d=1,d\in \mathcal{D}\left(p\right)$ . This is enforced by constraint (14b). Combining constraints (14b)–(14g) ensures that given a first-stage schedule starting on day $\overline{d}$ , we can find a schedule under scenario ω that starts on the day specified in the first-stage by decision z. Otherwise, z ^ω = 1, implying that we cannot find a schedule that is feasible for scenario ω due to some resources not being available. Constraints (14h) compute the maximum workload for each type of resource, while constraints (14i) compute the minimum workload. Finally, constraints (14j) enforce the binary and nonnegativity restrictions on the decision variables.

$f\left(z,\omega \right)=\underset{z^{\omega },z_{\overline{d}}^{\omega },x^{\omega },y^{\omega },w^{\omega },L^{\mathit{min},\omega },L^{\mathit{max},\omega }}{\mathrm{Min}}{\displaystyle \sum }_{i\in \mathcal{I}\left(p\right)}{\displaystyle \sum }_{\overline{h}\in \mathcal{D}\left(p,d^{\omega }\right)}\left(L_{i,\overline{h}}^{\mathit{max},\omega }-L_{i,\overline{h}}^{\mathit{min},\omega }\right)+M{z}^{\omega }$

(14a)

$\begin{array}{ll}\mathrm{s}.\mathrm{t}.& z_{\overline{d}}^{\omega }+z^{\omega }=1,\overline{d}=d\mid z_d=1,d\in \mathcal{D}\left(p\right)\end{array}$

(14b)

$\begin{array}{l}-z_{\overline{d}}^{\omega }+{\displaystyle \sum }_{r\in \mathcal{R}\left(i\right)}{x}_{p,s,i,r,\overline{h}}^{\omega }=0,\forall s\in \mathcal{S}\left(p,\overline{h}\right),i\in \mathcal{I}\left(p,s\right),\overline{h}\in \mathcal{D}\left(p,\overline{d}\right),\\ \overline{d}=d\mid z_d=1,d\in \mathcal{D}\left(p\right)\end{array}$

(14c)

$\begin{array}{l}{n}_{p,s,i}·x_{p,s,i,r,\overline{h}}^{\omega }-{\displaystyle \sum }_{t=E{A}_i}^{L{A}_i}{y}_{p,s,i,r,t,\overline{h}}^{\omega }=0,\forall s\in \mathcal{S}\left(p,\overline{h}\right),i\in \mathcal{I}\left(p,s\right),r\in \mathcal{R}\left(i\right),\\ \overline{h}\in \mathcal{D}\left(p,\overline{d}\right),\overline{d}=d\mid z_d=1,d\in \mathcal{D}\left(p\right)\end{array}$

(14d)

$\begin{array}{l}{z}_{\overline{d}}^{\omega }-{\displaystyle \sum }_{t=E{S}_i}^{L{S}_i}{\displaystyle \sum }_{r=1}^{\left|\mathcal{R}\left(i\right)\right|}{w}_{p,s,i,r,t,\overline{h}}^{\omega }=0,\forall s\in \mathcal{S}\left(p,\overline{h}\right),i\in \mathcal{I}\left(p,s\right),\overline{h}\in \mathcal{D}\left(p,\overline{d}\right),\\ t\in \left\{E{S}_i,\dots ,L{S}_i\right\},\overline{d}=d\mid z_d=1,d\in \mathcal{D}\left(p\right)\end{array}$

(14e)

$\begin{array}{l}-y_{p,s,i,r,t,\overline{h}}^{\omega }\ge -a_{p,s,i,r,t,\overline{h}}^{\omega },\forall s\in \mathcal{S}\left(p,\overline{h}\right),i\in \mathcal{I}\left(p,s\right),r\in \mathcal{R}\left(i\right),\\ t\in \left\{E{A}_i,\dots ,L{A}_i\right\},\overline{h}\in \mathcal{D}\left(p,\overline{d}\right),\overline{d}=d\mid z_d=1,d\in \mathcal{D}\left(p\right)\end{array}$

(14f)

$\begin{array}{l}{\displaystyle \sum }_{t=t^{\prime }}^{t^{\prime }+n_{p,s,i}-1}{y}_{p,s,i,r,t,\overline{h}}^{\omega }-n_{p,s,i}·w_{p,s,i,r,t^{\prime }\overline{h}}^{\omega }\ge 0,\forall s\in \mathcal{S}\left(p,\overline{h}\right),i\in \mathcal{I}\left(p,s\right),\\ r\in \mathcal{R}\left(i\right),t^{\prime }\in \left\{E{S}_i,\dots ,L{S}_i\right\},\overline{h}\in \mathcal{D}\left(p,\overline{d}\right),\overline{d}=d\mid z_d=1,d\in \mathcal{D}\left(p\right)\end{array}$

(14g)

$\begin{array}{l}{L}_{i,\overline{h}}^{\mathit{max},\omega }-{\displaystyle \sum }_{t=E{A}_i}^{L{A}_i}{y}_{p,s,i,r,t,\overline{h}}^{\omega }\ge b_{i,r\overline{h}}^{\omega },\forall s\in \mathcal{S}\left(p,\overline{h}\right),i\in \mathcal{I}\left(p,s\right),r\in \mathcal{R}\left(i\right),\\ \overline{h}\in \mathcal{D}\left(p,\overline{d}\right),\overline{d}=d\mid z_d=1,d\in \mathcal{D}\left(p\right)\end{array}$

(14h)

$\begin{array}{l}-L_{i,\overline{h}}^{\mathit{min},\omega }+{\displaystyle \sum }_{t=E{A}_i}^{L{A}_i}{y}_{p,s,i,r,t,\overline{h}}^{\omega }\ge -b_{i,r,\overline{h}}^{\omega },\forall s\in \mathcal{S}\left(p,\overline{h}\right),i\in \mathcal{I}\left(p,s\right),r\in \mathcal{R}\left(i\right),\\ \overline{h}\in \mathcal{D}\left(p,\overline{d}\right),\overline{d}=d\mid z_d=1,d\in \mathcal{D}\left(p\right)\end{array}$

(14i)

$\begin{array}{l}{x}_{p,s,i,r,\overline{h}}^{\omega }\in \left\{0,1\right\},y_{p,s,i,r,t,\overline{h}}^{\omega }\in \left\{0,1\right\},w_{p,s,i,r,t,\overline{h}}^{\omega }\in \left\{0,1\right\},z_{\overline{d}}^{\omega }\in \left\{0,1\right\},\\ z^{\omega }\in \left\{0,1\right\},L_{i,\overline{h}}^{\mathit{max},\omega },L_{i,\overline{h}}^{\mathit{min},\omega }\ge 0,\forall p\in \mathcal{P},s\in \mathcal{S}\left(p,\overline{h}\right),i\in \mathcal{I}\left(p,s\right),r\in \mathcal{R}\left(i\right),\\ t\in \mathcal{T},\overline{h}\in \mathcal{D}\left(p,\overline{d}\right),\overline{d}=d\mid z_d=1,d\in \mathcal{D}\left(p\right)\end{array}$

(14j)

We implemented SP model (13)–(14) in the Callable CPLEX Library ²¹ and solved instances of the model in the simulation HUB setting study described in the previous section. With 15-min time slots in an 8-h day, the maximum difference $L_{i,\overline{h}}^{\max ,\omega }-L_{i,\overline{h}}^{\min ,\omega }$ is 32. Since the earliest starting day is preferable, in our model implementation we set c _d = 40 d, d = 1, 2, …, 10 and M = 5000. To set these values, we performed a sensitive analysis on different values of c _d and M to make sure that the model was providing the desired results. The sensitivity analysis exercise showed that the model is insensitive to values of c _d > 32 d and large enough M.

Materials and methods

Motivated by a real Pathways Community HUB, we developed a computer simulation model to mimic client appointment requests arrivals for CHPs, scheduling CHPs, and the arrival of clients at the Pathways Community HUB for their appointment. The goal of the simulation model was to help us gain managerial insights into the computational performance of the SP approach under different demand cases and resource availability scenarios. The SP model was implemented within the simulation framework to determine the schedule for each client request. Next, we describe our data collection and analysis, including the real setting and design of experiments. We devote most of the space to deriving the SP pathway scheduling model.

Data collection and analysis

We consider a Pathways Community HUB setting for a county located in the eastern U.S. that offers health and social services with the aim of providing preventative care and improving health outcomes for high-risk members of the population. Care coordination in this HUB includes multiple providers and several services as depicted in Figure 1. CHPs in this HUB do not only involve clinical services, but also include social services to help overcome social barriers to healthcare access for risk population. ²

Figure 1.

Pathways community HUB model.

The Pathways Community HUB deals with about 40,000 residents with over 90% of the population belonging to a racial or ethic minority group. The HUB operates 9 h a day from 8:00 a.m. to 5:00 p.m., Monday through Friday, with 1 h lunch break from 12:00 p.m. to 1:00 p.m. The hours of operation are divided into 15-min time slots, and the CHP steps are assigned to time slots. Thus, there are 32 working time slots each day with four lunch break time slots.

The HUB has several behavioral, social and medical CHPs, 20 of which are listed in Table 4. In table, ‘Days’ refers to the length of the CHP while ‘Steps’ is the total number of steps in the CHP. We give an example of the Insurance CHP in Figure 3 in the Appendix. When residents (clients) need health related services, the Pathways Community HUB clinical health workers (CHWs) use CHPs as guidelines to establish communications among stakeholders, such as the hospital, educational institutions, insurers and health care providers. CHWs identify which CHPs are appropriate for each client and allocate resources to all steps of the assigned CHPs within a specified time frame. However, it is very challenging to find the earliest appointment due to the uncertain future availability of the resources for all the steps of a CHP as well as the limited number of CHWs. When the CHP assigned to a client cannot be scheduled within 10 days, the CHW will issue a referral to the client to seek another community health center. There are nine types of resources at this Pathways Community HUB listed in Table 5, each with the specific number we used in our computational study.

Table 4.

Community health pathways available in the pathways community HUB.

	CHP name	Abbreviation	Days	Steps
1	Acute Coronary Syndrome	ACS	22	6
2	Asthma	ASM	25	6
3	Behavioral Health Intervention	BHI	25	6
4	Care Team	CAT	25	4
5	Companion or Care Attendant	CCA	15	5
6	Closed Head Injury	CHI	20	4
7	Caesarean Birth	CSB	9	6
8	Domestic Violence	DMV	25	6
9	Dietitian and Nutrition Consultant	DNC	22	6
10	Dying Person Care	DPC	17	5
11	Food Insecurity	FDI	25	7
12	Housing	HUS	23	5
13	Insurance	INS	25	6
14	Medical Home	MDH	10	6
15	Medical Referral	MDR	14	3
16	Prime Time Sister Circle	PTS	25	8
17	Smoking Cessation	SMC	25	6
18	Social Service Referral	SSR	25	6
19	Thrombolysis	TBS	15	6
20	Transportation-HEZ	TRH	3	3

Table 5.

Resource types available in the pathways community HUB.

	Resource name	Abbreviation	Number
1	Patient Navigator	PNT	3
2	Educator	EDU	4
3	Liaison	LAS	3
4	Outreach Worker	OTW	4
5	Patient Counselor	PEC	5
6	Health Interpreter	HIP	4
7	Training Location	TRL	5
8	Community Health Advisor	CHA	4
9	Transporter	TRS	3

The Insurance CHP in Figure 3 in the Appendix takes 25 days to complete and involves six steps after initiation: step 1 (Investigate) on day 1 requires one resource, CHA, for one 15-min time slot; step 2 (Application) on day 2 requires one resource, OTW, for two 15-min time slots; step 3 (Appointment, Education) on day 7 requires three resources, OTW for one 15-min time slot and EDU and TRL for two 15-min time slots concurrently and; step 4 (Processing) on day 14 requires one resource, HIP, for one 15-min time slot and; step 5 (Processing Verification) on day 21 requires one resource, HIP; and step 6 (Completion) on day 25 requires one resource, PEC.

Model development and implementation

Managing clients and resources in the Pathways Community HUB in extremely challenging, as it relates to scheduling clients for different pathways using limited resources whose future availability is uncertain. We developed an SP pathway scheduling model together with a simulation model of the HUB to evaluate the performance of the SP approach. The computer simulation and SP models were coded in the CPLEX 12.9 Callable Library ²¹ using C++ and experiments were performed on a Dell T7600 Workstation with a Intel(R) Xeon(R) CPU E5-2643 0 @3.30 GHz MHz processor and 64.0 GB RAM.

A CHP scheduling flowchart of the computer simulation model is shown in Figure 2. The simulation model has several entities including a Client Generator, CHP Scheduler, and Resource Allocator. The responsibility of the Client Generator is to create client arrivals on each day d of the planning horizon. Clients arrive one at a time and are given an ID (index) c _d and assigned a CHP by the CHW based on their needs. Next, the CHP Scheduler invokes the SP model (13)–(14) to search for an appointment for the client within 10 days with minimum access time and workload imbalance (if workload balancing is selected). The Resource Allocator in turn updates the future availability of the resources assigned to the CHP steps, i.e., makes them unavailable at the scheduled time slots for that CHP. After the client is scheduled, the Client Generator generates the next client and the scheduling process is repeated. Otherwise, if the client cannot be scheduled within 10 days, the simulation records them as not scheduled. The simulation ends when the last appointment day is reached.

Figure 2.

CHP scheduling simulation flowchart.

Experimental design

Several experiments were conducted to assess the schedules determined by the SP model over a specified time horizon with an overall goal of gaining managerial insights. Specifically, we studied the following four main aspects of CHPs scheduling under uncertainty: (i) variation of client access time; (ii) impact of client demand on scheduling; (iii) resource utilization; and (vi) workload balancing versus no workload balancing. The motivation behind these four factors stems from the practical needs of Pathways Community HUBs, which include minimizing access time for their clients, efficiently managing different levels of demand, maximizing resource utilization, and ensuring workload equity among the human resources.

We considered the following key performance measures: access time, number of clients scheduled (throughput), resource workload or utilization (number of time slots assigned), and workload coefficient of variation. Client request daily arrival rate is a critical factor in scheduling CHPs. It translates into client demand volume, which significantly affects scheduling decisions in terms of client access time. Therefore, to study the effect of demand level on access time, we experimented with three cases: low demand (Case I), medium demand (Case II) and high demand (Case III).

Demand volume is determined by client daily arrival rate at the Pathways Community HUB. Since daily arrival rate is stochastic, we assumed uniform $\left(\mathcal{U}\right)$ distributions given in Table 6. The parameters were set based on estimates for the Pathways Community HUB used in this study. Clients were set to be scheduled over the first 70 business days (appointment days) of a planning horizon of 100 business days. All three cases had the same resource types given in Table 5, each with its uncertain daily availability following the Bernoulli $\left(\mathcal{B}\right)$ distribution with the probability of being available p shown in Table 6.

Table 6.

Experiment cases and settings.

				Case I		Case II	Case III
	Demand			Low		Medium	High
	Client daily arrival rate			$\mathcal{U}\left(0,20\right)$		$\mathcal{U}\left(0,30\right)$	$\mathcal{U}\left(25,35\right)$
Res type	PNT	EDU	LAS	OTW	PEC	HIP	TRL	CHA	TRS
alue	0.8	0.8	0.8	0.8	0.8	0.8	1.0	0.8	1.0

The time slot availability of each resource for each of the 32 time slots per day follows $\mathcal{B}$ distribution with p = .85. Daily availability refers to whether or not a resource is available on a given day, while time slot availability refers to whether or not a resource is available at a particular time slot on a given day. The resource availability p values and the number of scenarios were carefully set based on the Pathways Community HUB characteristics. We were provided with daily average number of client arrivals and resource availability with no further information. Therefore, we assumed uniform and Bernoulli distributions for client arrivals and resource availability, respectively.

The number of scenarios was set to 10 and several replications ranging from two to 10 were performed for each demand case to guard against spurious cases. We recorded the average (mean) for all the performance measures and in some cases the minimum and maximum values. We first investigated the impact of using deterministic (assumed to be known) resource availability versus stochastic resource availability on the performance measures. We specifically compared the results of under under no workload balancing (NWB) and under workload balancing (WB) to evaluate equity issues among CHW resources of the same type. The size of the SP model instance depends on the CHP under consideration. CHPs requiring more steps and resources lead to larger instances and generally take relatively longer to solve. Another factor that affects computation time is whether of not load balancing is required. In our experiments, the average runtime to schedule a CHP was 56 s under NWB and 77 s under WB. To determine the appropriate number of replications, we varied the number of replications from two to 10 for all three cases. We obtained similar results for number of replications above five. Therefore, we report the results based on 10 replications, which is more than sufficient for this computational study.

Results and discussion

We report results for the three demand cases: Case I, Case II and Case III, which correspond to low, medium, and high client demand, respectively. We considered scheduling performance measures of client access time, number of clients scheduled, resource workload, and workload coefficient of variation, and reported these performance measures under NWB and WB. To set a baseline for comparison, we ran experiments involving 10 replications for each case using one scenario with certain resource availability randomly set using the Bernoulli distribution given in Table 6. This is often the case in practice when Pathways Community HUB managers assume expected values of resource availability. We compare this deterministic approach to the SP approach involving uncertain resource availability based on multiple scenario realizations of the random variables.

The results for the deterministic approach are reported in Table 7 and those for the SP approach are given in Table 8. The columns ‘NWB’ and ‘WB’ in the tables correspond to the results for no workload balancing and with workload balancing, respectively. The numbers reported in the table are the average and standard deviation values. Under the deterministic setting, the results show that client access time is about 1.5 days for Case I, about 1.7 days for Case II, and about 4.2 days for Case III. The results show that in general increased client demand results in increased access time and a slight reduction in the number of clients scheduled. However, for all three demand cases, workload balancing does not significantly affect access time, which is desirable.

Table 7.

Client access time and clients scheduled under certain resource availability.

	NWB	WB
	Case I
Access time (days)	1.42 ± 0.32	1.53 ± 0.27
Number of clients scheduled	702 ± 43.29	721 ± 50.43
Total clients	702 ± 43.29	721 ± 50.43
	Case II
Access time (days)	1.67 ± 0.25	1.74 ± 0.27
Number of clients scheduled	1059 ± 57.29	1023 ± 74.15
Total clients	1059 ± 57.29	1035 ± 66.83
	Case III
Access time (days)	4.11 ± 0.3	4.35 ± 0.20
Number of clients scheduled	2041 ± 36.26	2032 ± 30.21
Total clients	2090 ± 41.99	2102 ± 21.61

Table 8.

Client access time and clients scheduled under uncertain resource availability.

	NWB	WB
	Case I
Access time (days)	1.26 ± 0.10	3.02 ± 0.30
Number of clients scheduled	667 ± 36.78	647 ± 37.44
Total clients	667 ± 36.78	661 ± 36.23
	Case II
Access time (days)	3.91 ± 1.02	5.63 ± 0.82
Number of clients scheduled	1023 ± 66.53	1001 ± 71.195
Total clients	1045 ± 62.37	1012 ± 86.87
	Case III
Access time (days)	7.93 ± 0.13	8.25 ± 0.07
Number of clients scheduled	1404 ± 232.86	1469 ± 306.04
Total clients	2022 ± 235.15	1939 ± 310.04

The results for the SP approach show that workload balancing can in fact affect client access time. Under NWB client access is about 1.3 days for Case I, about 3.9 days for Case II, and about 7.9 days for Case III. With WB, client access is about 3.0 days for Case I, about 5.6 days for Case II, and about 8.3 days for Case III. The results show that in general increased client demand results in increased access time and a reduction in the number of clients scheduled. Also, we see that workload balancing increases access time for both Case I and Case II. With Case III which involves high client demand, however, there is no significant increase in access time with workload balancing.

The deterministic approach provides client schedules that are too optimistic as compared to the stochastic approach. This is due to the fact that uncertainty in resource availability is not considered and results in the omission possible future scenarios. The number of clients that are not scheduled within 10 days due to resource unavailability increases with client demand. This is an indication that client volume significantly affects schedules and more clients have to wait longer to access the HUB. Gleaning further into the results, we found, for example, that most of the clients that were not scheduled were those assigned to the CHP Transportation-HEZ (see Table 4). One step of this CHP requires transportation time for 12 consecutive time slots. Consequently, to schedule this CHP the SP model has to find a schedule with 12 available consecutive time slots.

The results for resource utilization are summarized in Table 9. The first column ‘Res’ lists the resource type, ‘Max’, ‘Mean’ and ‘Min’ represent maximum, average and minimum number of time slots assigned to each resource, CV _NWB and CV _WB are the coefficients of variation for each resource type. The coefficient of variation is the ratio of the standard deviation to the mean and shows the extent of variability in relation to the mean. The last column, CV _NWB /CV _WB , lists the ratio of the CV under NWB to that under WB. A ratio above one means that WB provides less variability among each resource type, i.e., more equitable schedules among the resources. A ratio below one means that WB is not as beneficial in creating equity among the resources. From Table 9, we can see that most of the ratios are above one under all the three cases, a strong indication that workload balancing provides less variability among the resource types, thus resulting in more equitable schedules.

Table 9.

Resource utilization under no workload balancing and with workload balancing.

	NWB				WB
	Max	Mean	Min	CV NWB	Max	Mean	Min	CV WB
Res	(Time Slots)				(Time Slots)				CV NWB /CV WB
Case I
PNT	500	370	205	0.340	429	354	309	0.1520	2.2355
EDU	577	419	307	0.262	472	419	379	0.0839	3.1172
LAS	329	246	163	0.285	302	254	221	0.1406	2.0277
OTW	569	363	194	0.430	461	368	317	0.1558	2.7628
PEC	867	301	87	0.957	324	302	287	0.0434	22.0546
HIP	541	338	196	0.402	404	319	276	0.1621	2.4818
TRL	486	335	243	0.272	392	335	302	0.0974	2.7959
CHA	451	318	169	0.352	405	314	269	0.1718	2.0463
TRS	254	151	62	0.533	98	79	65	0.1649	3.2328
Case II
PNT	825	568	364	0.340	761	538	409	0.295	1.152
EDU	732	596	495	0.156	658	544	478	0.131	1.189
LAS	519	374	256	0.302	495	354	273	0.283	1.067
OTW	780	531	366	0.305	763	502	395	0.302	1.010
PEC	1490	489	162	1.034	501	467	445	0.042	24.719
HIP	856	491	310	0.450	662	461	375	0.253	1.779
TRL	609	477	402	0.159	560	435	374	0.153	1.036
CHA	679	475	291	0.329	747	463	345	0.359	0.917
TRS	167	121	81	0.310	155	111	73	0.312	0.995
Case III
PNT	1046	664	400	0.415	1008	692	514	0.325	1.276
EDU	735	654	589	0.087	821	665	578	0.142	0.613
LAS	691	472	337	0.333	663	470	365	0.290	1.147
OTW	996	650	479	0.317	1009	659	511	0.309	1.027
PEC	1547	573	234	0.843	621	583	566	0.033	25.212
HIP	1134	641	405	0.452	884	647	551	0.211	2.136
TRL	639	523	454	0.127	699	532	460	0.164	0.774
CHA	1078	653	431	0.394	1012	661	503	0.313	1.257
TRS	207	136	83	0.391	184	143	108	0.232	1.685

The computational study reveals that the SP approach allows to schedule clients in a Pathways Community HUB by minimizing client access time and enabling workload balancing among the resources. The results show that a deterministic approach can be too optimistic depending on how the future resource availabilities are set. Therefore, we recommend to HUB managers to use the SP approach with workload balancing. This is because the SP model significantly reduces workload imbalance among the same resource types while providing reasonable access time to clients, especially under high client demand. We should emphasize that the SP model without workload balancing can create schedules that are not equitable to some CHWs. We saw for example, how one patient counselor (PEC) worked eight times more than another PEC. Clearly, this would create unfairness concerns for the Pathways Community HUB manager. Therefore, we recommend using workload balancing to get client schedules that are equitable in terms of resource utilization.

Another advantage of the SP model is that it can easily be adapted to any real HUB and can also provide the HUB manager or scheduler a slew of useful information beyond the schedules and resource utilization results we presented. For example, by analyzing the client wait time and resource utilization, the results can provide guidance regarding which CHPs are being assigned the most and what resources are being utilized the most. This can aid the HUB manager in making decisions regarding resource capacity expansion.

Conclusion

Scheduling limited resources at a centralized Pathways Community HUB can be challenging. We introduce an SP approach for connected community health for optimally scheduling CHPs under uncertainty in resource availability. The SP methodology was implemented in a simulation setting and applied to data for a real Pathways Community HUB for a U.S. county. The computational results are promising and show that client access times depend on the HUB resources uncertain future availability and the level of client demand, with high client demand resulting in relatively longer access time. The study reveals that schedules provided by a deterministic approach where resource availability is assumed to be known can be too optimistic. Several managerial insights are gleaned from this study. For example, the SP model computes schedules that with workload balancing provide client schedules with equity across the same type of community health workers. Future work include extending the SP model to allow for resource capacity expansion considerations and developing fast decomposition techniques to speed up computations.