Bringing Better Care in Better Time
An Integer Programming Approach to Tele-Psychotherapy Scheduling
A guest article written by student Amey Maley, edited by Tyler Perini.
The ongoing COVID-19 pandemic has illuminated many to the silent mental health crisis taking place worldwide. From the loss of loved ones to job security, myriad pandemic-related stressors have exacerbated poor mental health across the nation. According to 2020 data collected by the Kaiser Family Foundation, nearly half of one-thousand surveyed Americans reported experiencing negative mental health effects as a result of the pandemic. Moreover, the CDC reports the number of opioid overdose deaths in the United States increased by 18.2% from June 2019 to May 2020, a period which directly captures the start of the COVID-19 pandemic. Under these circumstances, the demand for mental health services has exploded, driving the expansion of tele-psychotherapy services to provide remote mental health support.
Challenges in providing tele-psychotherapy stems from two areas: (1) scarce personnel with the appropriate training and technical skills to conduct sessions; and (2) highly specific needs among diverse patient populations which have propelled the uptick in mental healthcare demand during the pandemic. More specifically, in their 2021 publication in Health Care Management Science, Andrés Miniguano-Trujillo et. al. characterize divergent psychological needs between three mental health patient populations in Ecuador: (i) frontline healthcare workers treating COVID-19, (ii) individuals grieving COVID-19 loss, and (iii) risk-of-suicide patients. Given these challenges, Miniguano-Trujillo and colleagues investigated how an integer programming model could be implemented to both meet the needs of patients and the capacity of providers via the strategic scheduling of these virtual appointments.
Under these circumstances, an integer program (IP) provides some unique benefits for optimizing schedules. For instance, since scheduling utilizes a discrete number of therapists, patients, and sessions slots, it is appropriate to model the associated decisions via integer and binary-valued variables. Miniguano-Trujillo et al., label their IP model as the Multi-periodic Patients Assignation to Therapists Model (MPATM).
The MPATM decision variables are summarized by x, y, and z. The general integer variable x represents the number of patients in each category who are assigned to a given therapist in a given period. Another binary decision variable y represents if a given therapist treats any patients in a given period. Finally, z is a general integer variable which represents the number of patients who cannot be assigned a qualified therapist during a given period due to limited supply of qualified therapists.
Some of the simple constraints for this model are as follows:
The first constraint imposes an upper bound on the total number of patients assigned to each therapist across the time periods. The second constraint supplements this upper bound by providing a restriction on the number of patients assigned to each therapist within a given period.
The model additionally includes advanced constraints, an example of which is shown to the left. This constraint promotes a more even caseload among the therapists within a given period. It achieves this by ensuring a bound on the proportion of patients assigned to each therapist in a group (represented by q). Each therapist is constrained to be within a group where the number of patients scheduled to that group divided by the total number of therapists in the group (minus one) is a minimum for the number of patients assigned to that therapist.
One of the most intriguing components of the MPATM is the objective itself. While framed as a single objective model, the two summations reflect two distinct goals. The former aims to maximize the fit between patient-therapist assignments, and the latter seeks to maximize the total contribution by therapists. Thus, c and d can be viewed as hyperparameters that may be adjusted accordingly to dictate how these goals are prioritized. In fact, Miniguano-Trujillo et al., identifies two algorithms for setting these hyperparameters and subsequently analyzed the results through the course of this paper.
Two different algorithms were used to set the objective function parameters for the MPATM, algorithm 1 which utilized a period-based heuristic, and algorithm 2 which utilized a category-based heuristic. The model was then tested using these two algorithms against six different combinations of patients and therapists. To simplify interpretation, the combinations of patient-therapist numbers were represented via the associated number of scheduling periods. The objective value and percent of unscheduled patients produced via the MPATM utilizing these algorithm outputs were subsequently plotted against scheduling periods in order to compare the success of the two algorithms.
Two key takeaways can be learned from the simulated MPATM results. First, regardless of what heuristic is being used, the model is highly successful at scheduling patients, with the percentage of unscheduled patients never exceeding 3% among the simulated instances. Second, the period-based algorithm for setting parameters garners greater success with scheduling patients despite generating lower objective values. This speaks to the broader idea that in mathematical programming, an objective is only as good as its composite parameters, and when applicable, experimenting with hyperparameters not only provides an opportunity to prioritize different issue areas but is a necessary means to isolate vulnerabilities and strengths within a given model.
As we look towards a future where telehealth becomes more prevalent, IP approaches such as MPATM hold a lot of potential as an expeditious means to align patients with unique needs to providers capable of catering to those needs via strategic scheduling. As this pandemic has demonstrated, healthcare providers are not immune to burnout, thus optimizing scheduling beyond just patients’ needs but also prioritizing provider capacity may alleviate some of the pressures on the mental healthcare system.
