Offering transportation services to economically disadvantaged patients at a family health center: a case study

ABSTRACT

It has been established that high no-show rates of publicly supported health systems in economically depressed areas are largely due to a lack of inexpensive, reliable transportation. The purpose of this paper is to determine the financial feasibility of offering transportation and investigate the net cost savings by reducing no-show rates. The approach starts with a data analysis on 636 patients at the Family Health Center (FHC) in San Antonio, Texas, followed by logistic regression to determine the impact of various transportation factors on cancellations/no-shows and late arrivals. We then investigate the costs savings that could be realised by reducing the no-show rate from 24.3% by up to 60%. Finally, we analyse the expenses that would be incurred should the FHC provide transportation. The full analysis indicates a cost reduction of more than $15,000 per month can be achieved when the no-show rate is reduced by 25% down to 18.2%.

1. Introduction

In most urban communities, a subset of the population is without adequate healthcare due to lack of insurance and economic means. Those finding themselves in such circumstances often fail to seek treatment for major medical problems until their situation becomes dire. This leaves them with few options but to use costly and inappropriate emergency services for what should be routine care. To combat this dysfunctional cycle, many cities have set up taxpayer subsidised outpatient clinics for both primary care and specialised treatment related to chronic conditions.

In San Antonio, Texas, the Family Health Center (FHC) is one such example that serves as the focal point for this study (Ferrer et al., 2013). This clinic provides primary care to members of the community who are economically disadvantaged and mostly lacking resources to maintain proper health. It is run as a collaboration between the county-owned University Health System (UHS) and physicians from an academic centre, the Long School of Medicine (UT Health San Antonio), a common arrangement across the country. Most patients are pre-scheduled but a substantial number are walk-ins with both acute symptoms and chronic conditions requiring immediate attention. The FHC is a safety net healthcare facility in San Antonio so it is rare that any of their patients have full medical coverage. For context, the insurance breakdown in 2019 was as follows: 27.8% were on Medicaid, 31.9% were on Medicare, and 25.5% were enrolled in Carelink, a financial assistance programme for uninsured Bexar County residents that provides access to medical services. The remaining 14.8% were essentially uninsured. The availability of Carelink is noteworthy because it offers those who are otherwise unable to afford primary care a medical home. The alternative for this population is the emergency department (ED), an expensive and inappropriate option.

Like many safety net facilities, the FHC has a high no-show rate that is a financial burden to the system and a contributing factor to the poor health of its patient population. In 2018, the no-show rate including cancellations was 24.3%. Anecdotal evidence and reports from the patient/family advisory council (comprised of administrators, physicians, nurses and social workers) within the FHC suggest that either the lack of transportation or difficulty in obtaining a ride is the main reason why appointments are not kept. This is in spite of the fact that there are several inexpensive and no-cost options, such as Sure Ride Transportation, Call A Ride for Seniors, and VIA Transit, which are available to those below a certain income level and for those who are disabled. The public transit system also provides inexpensive service and has bus stops within two blocks of the FHC. The need to take several buses and then walk the last few blocks, however, could result in over two hours of travel time for some patients, and for those with disabilities, the walk is not possible. The problem is compounded at the back end when patients are instructed to fill a prescription. Inadequate transportation coupled with even modest drug costs often means that patients do not obtain their medicine when needed, if at all.

The major consequences of a high no-show rate are at least threefold. First, missed appointments lead to worsening symptoms and deteriorating health for the patient. Mild conditions that would otherwise be controllable with proper care turn into chronic diseases that shorten life expectancy and degrade quality of life. A more immediate consequence is that as symptoms become more severe patients will likely seek treatment in the ED where the costs to the system (UHS in this case) are much higher than for an FHC visit. In the absence of comprehensive coverage, ED visits impose a significant financial burden on the hospital system. Making matters worse, patients who miss their FHC appointment have a higher probability of visiting the ED within 30 days.

Second, patients who miss their appointment and are forced to seek treatment in the ED have a significantly higher probability of being admitted to the hospital, compared with patients who keep their appointments. Hospital stays for the uninsured impose a large financial burden on the system. According to our data, the hospital inpatient (variable) cost per day is more than 141 times the cost for a non-emergency medical appointment, and the average length of hospital stay is 6.85 days. Finally, a missed appointment leads to lost revenue for the clinic. More details of no-show costs are discussed in Section 5. In summary, the cost of a no-show consists of three components:

• Higher cost for the ED

• More financial burden on the hospital system

• Lost revenue from the missed appointment for the clinic

As our analysis shows, the combined costs incurred by FHC and UHS for missed appointments due to either cancellations or no-shows are enormous. Accordingly, small reductions are likely to have a substantial impact on the system’s ability to continue to provide healthcare services to those most in need. Moreover, we show that such reductions can be achieved by offering subsidised transportation to those patients who are the most financially and physically constrained. These results are achievable under notably conservative assumptions, and are primarily due to the high cost of ED visits and hospital stays, which are seen to increase with missed appointments.

Our findings are based on an extensive analysis of historical and survey data from FHC patients. Additional insights were gained through logistic regression, which was used to characterise the probability of no-shows and late arrivals as a function of patient demographics, ambulatory difficulties, and various transportation factors. In our investigation of transportation services we considered the use of ride hailing companies, city buses, taxis, as well as operating a circulating van in the most densely populated and underserved area of the city. To find optimal routes in this area, we solve a vehicle routing problem (VRP) with constraints on trip durations and pickup/drop off frequencies. Although the VRP model represents a minor contribution to the literature on providing bus services constrained by interarrival times and total trip time, we don’t believe that it has appeared previously. Our chief contribution centres on the study of the costs and benefits of providing transportation services to a community with limited resources, mobility, and access to healthcare. We don’t believe that a similar study has been published to date.

Providing transportation services for a portion of the patient population involves real dollar outlays that must be weighed against potential cost savings. Improved outcomes is one justification but the benefits of a healthier population and reduced use of the ED must be quantified. Our goal was to conduct a cost-benefit study to determine whether sufficient financial justification exists for UHS to underwrite the costs of easing the transportation burden for its most vulnerable patients. High-level administrators have to consider the needs of the entire hospital network when allocating resources, and not just the FHC.

Costs associated with missed appointments come in the form of lost revenue, poorer community health, increased use of expensive medical facilities, and higher hospital admissions. The impact of transportation barriers is depicted in Figure 1. We wish to intervene at the beginning of the chain.

Figure 1.

Consequences of transportation barriers.

As such, the purpose of this paper is threefold: (i) to evaluate the expected costs incurred by missed appointments due to transportation difficulties, (ii) to investigate the costs for either subsidising or offering free transportation to a subset of patients, and (iii) to parametrically analyse the net benefits to UHS that result from reducing the no-show rate by 20 to 60%. All data for the analyses were provided by the FHC and UHS.

Working under the hypothesis that more comprehensive transportation services can significantly reduce the no-show rate, we establish a separate connection between the no-show rate, percentage of late patients and several transportation-related factors. As a result, we investigated the proposition that by providing free transportation services such as shuttles/Uber/Lyft/taxi to the most vulnerable patient population that the no-show rate will decline along with the costs to the FHC, ED and the hospital for missed appointments.

In the next section, we review the literature most relevant to our work. In Section 3, we outline the study methodology and discuss how we approached the analysis. In Section 4 we describe the data collected for the study on 636 patients over a 4-day period in February 2019 and present results for two logistic regressions. Our goal was to statistically identify the potential factors leading to the high no-show rate at the FHC. In Section 5, the details of the costs and lost revenues associated with missed visits are described, and the potential net savings from reducing the no-show rate are identified. Included is a cost analysis of the various transportation options that the FHC could offer to those patients who have inherent difficulty keeping their appointments due to transportation issues. Section 5 also includes our optimisation model for determining optimal shuttle routes in the most populated zip code served by the FHC. Computational experiments address several different scenarios related to shuttle frequency and route length. The results are used to price one of the transportation options. The last section offers conclusions and identifies the limitations of the analysis.

2. Literature review

Prior work on transportation services in healthcare is extensive. Since this paper aims to evaluate the benefits of reducing the no-show rate for non-emergency medical appointments, we focus our review on related topics. The first subsection highlights papers that discuss the impact of transportation barriers on no-shows. The second subsection concentrates on potential financial consequences that high no-show rates bring to clinic operations, while the third subsection outlines research on no-show reduction efforts. In the fourth subsection, we review the literature on medical appointment scheduling. The fifth subsection discusses vehicle routing-based models since one of the options that we are considering is to provide free shuttle services for FHC patients. In the last subsection, we review literature on the methods to predict no-show probabilities and identify factors for high no-show rates.

2.1. Transportation barriers and no-shows

Despite the seriousness of the problem there are only a small number of publications related to understanding the relationship between transportation and no-shows for non-emergency medical appointments. Somewhat disconcerting is the mixed findings of the handful of studies that tried to quantify the benefits of expanding the transportation options for patients from disadvantaged communities. Most recently, Chaiyachati et al. (2018) conducted research on the impact of rideshare. Data were collected from October 2016 and April 2017 on 786 Medicaid beneficiaries who resided in West Philadelphia. Both tests and logistic regression models were used to compare the missed appointment rate between patients provided with rideshare and those who weren’t. The results showed that even though a high no-show rate is largely due to a lack of adequate transportation, the impact of rideshare on reducing the no-show rate was insignificant. In particular, the study results showed that simply offering rideshare services to patients did not improve the situation.

Powers et al. (2016) conducted a retrospective analysis of Medicaid enrollees in New York and Medicare Advantage beneficiaries in California to determine whether offering Lyft Concierge services could be financially justified. Using data collected on 479 nonemergency medical appointments in May and June 2016 by National MedTrans (nonemergency medical transportation benefit manager), they found that average patient waiting times for a ride decreased by 30% and average per-ride costs were reduced by 32.4%. Composite patient measures yielded a satisfaction score of 80.8%. Syed et al. (2013) similarly investigated transportation barriers and concluded that they can significantly prevent patients from receiving critical healthcare services, especially for those in low income and underinsured groups. A related finding was that transportation barriers are the primary reason why women, members of ethnic minority communities, and children do not to receive timely medical care.

Silver et al. (2012) investigated the relationship between transportation difficulties and the missed appointment rate at four free-standing ambulatory care clinics run by the sole public hospital in Nassau County, New York. To gather data, they conducted a survey of 698 low income, largely immigrant clients during an 8-day period in May 2009. Statistical tests showed that 23.5% of the responding patients reported having missed or rescheduled an appointment due to transportation difficulties, and 31.6% of those were due to an unreliable and inadequate bus system. Complaints included low frequency, slow travel, crowded vehicles, and failure to abide by the published schedule. More serious, though, was that nearly a quarter of those who responded to the survey indicated that they could not afford transportation to the clinic. Bus users were twice as likely as car users to have a prior history of missing or rescheduling appointments, again highlighting the importance of the relationship between reliable transportation and the no-show rate.

In a recent literature review on no-shows in patient appointment scheduling, Dantas et al. (2018) identified a series of papers that analysed the link between patient no-shows and transportation issues. Based on a survey of 46 parents of patients, Minty and Anderson (2004) established a statistically significant link between transportation difficulties and non-attendance for first out-patient appointments at a hospital-based clinic for child psychiatric care. Yang et al. (2006) used a multivariate logistic regression model to predict the odds of patient no-show at Texas Children’s Hospital Residents’ Primary Care Group Clinic. The results were generated from a survey of 183 children’s caregivers who had appointments over a 9-week study period. Two of the four factors found to be significant were transportation-related: “not using a car to the last kept appointment” and “not keeping an appointment in the past due to transportation problems.”

For a paediatric otology and communication clinic, Kavanagh et al. (1991) established that ownership of a car was a significant explanatory variable of clinic attendance. They used both univariate methods and multivariate regression analysis.

Prior work related to accommodating patient transportation needs is mostly limited to the use of ride hailing and taxi services. There have been few if any investigations that examine the impact on the no-show rate when no-cost or low-cost transportation services are offered to those patients who face high transportation barriers. In our study, we collected information on all transportation modes such as driving, bus, and taxi available in the San Antonio area. In addition, we gathered data on ambulatory capacity, transportation difficulty and travel times as potential factors that might impact no-shows. We also considered multiple measures that affect operational performance and system costs.

2.2. Financial impacts on clinic operations

The consequences of a high no-show rate coupled with transportation barriers can significantly impact the operations and financial well-being of a clinic. Kheirkhah et al. (2016) found that in 2008, the marginal cost of no-shows at 10 clinics run by the U.S. Veterans Health Administration medical centres in Houston was estimated to be 14.58 USD million, while the lowest and highest marginal costs were 10.48 USD million in 1997 and 16.65 USD million in 2005. For medical providers, Martin et al. (2005) conducted a qualitative study in one health centre in urban South Essex, England, using semi-structured interviews with a purposive sample of 24 patients over the age of 18, 7 general practitioners (GPs), a GP Registrar, a nurse practitioner and 5 receptionists. There were additional interviews in a practice in rural Essex and a practice in inner city London. The major finding was that patients arriving late for their appointment are even more disruptive than no-shows. Medical providers usually do not feel negatively about non-attendance but view those arriving late less favourably since they play havoc with the schedule (e.g., see Bard et al., 2016). In fact, missing appointments may be seen as useful in certain circumstances because they allow providers to catch up with work.

Moore et al. (2001) investigated the financial and operational relationship between no-show and walk-in patients. The study was conducted at the Family Practice Center of the Palmetto Richland Memorial Hospital, University of South Carolina. A sample of 20 business days yielded 4055 visits. Information including number of patients, appointment status, date, time, provider type and charges was collected. tests showed that of the time allocated for scheduled patients, 6.8% went unfilled because patients cancelled their appointment, and 25.4% was unfilled due to no-shows. However, for each patient failing to appear for his or her appointment, an estimated 89.5% of the lost fee was recovered by fitting in a walk-in patient.

The prior work provides an overview of the financial impacts of no-shows, suggesting that cost-savings are possible if the number of no-shows can be decreased. There has also been much discussion on the relationship between no-shows and clinic operations such as scheduling. However, all previous research has focused on a single unit in a larger healthcare system, rather than on the system as a whole. In our work, we exam the financial impact of no-shows on University Hospital. Here, we take an integrated approach and consider three system components: the ED, the hospital, and the FHC. For each component, we determine the no-show costs and potential savings likely to arise by offering transportation services to a subset of the patient population.

2.3. No-show reduction efforts

Studies investigating how to decrease no-show rates have examined the effectiveness of practices such as reducing waiting times and using appointment reminders, along with behavioural engagement strategies such as contingency management and motivational interviewing. For addiction patients, Molfenter (2013) discussed how appointment no-shows adversely impact clinical outcomes and healthcare productivity. In a study conducted between 2007–2010, partly under his supervision, participants were asked to reduce their no-show rates by using practices taken from related research and theory. As a result, no-show rates dropped from an average of 37.4% to 19.9%. The most effective measures were the use of pre-appointment reminder calls and the creation of a welcoming environment.

Because of the revenue loss and operational disruptions that are a consequence of missed appointments, much work has been done in trying to predict the likelihood of no-shows and cancellations. Alaeddini et al. (2015), for example, developed a hybrid probabilistic model based on multinomial logistic regression and Bayesian inference to predict the probability of such events in real-time. The effectiveness of the proposed approach was demonstrated using healthcare data collected at a San Antonio medical centre. Operationally speaking, they showed that it can be used to develop more effective appointment scheduling systems and more precise overbooking strategies to reduce the negative effects of no-shows and fill in appointment slots while maintaining short waiting times.

In addition to developing pro-active strategies to reduce missed appointments, many clinics have resorted to overbooking to deal with the problem. The research is limited, though, on understanding the major factors affecting the no-show rate. Zeng et al. (2013) addressed this issue by developing a game-theoretic framework (with queueing models) to explore the impact of overbooking on missed appointments as a function of appointment delay (time between a patient requesting an appointment and his actual appointment time) and office delay (the amount of time a patient waits in the office to see the doctor). While overbooking reduces appointment delay, it increases office delay. Their results show that the patient no-show rate always increases after overbooking. Further, there exists a critical range of patient panel size within which overbooking may also lead to lower expected profit for the clinic.

The prior work concentrates on the appointment scheduling system or reminder calls to reduce no-shows. However, insufficient transportation services might be another reason for missed appointments. Therefore, in this work, we are proposing to provide free transportation services.

2.4. Medical appointment scheduling

Akhavizadegan et al. (2017) conducted research on an appointment scheduling problem in a nuclear medical centre. A finite-horizon Markov decision process is used to model the problem by considering the patients’ choice behaviour, and different no-show rates for patients. The proposed model determines tactical and operational decisions for patient appointments including the number of patients requesting hospitalisation, whether a walk-in patient hospitalisation request should be accepted, and the patients receiving services in each appointment slot. Two algorithms and one mathematical programme were developed to deal with the intractability of the Markov decision process. In addition, simulation tools were applied to compare the performance of optimal policies with a first-come-first-serve policy. The results showed that the proposed model generates a more efficient schedule compared with current policies for scheduling. Using the model led to more revenue, lower patient waiting times, and fewer postponed patients.

Kuo et al. (2020) proposed a framework that utilises a stochastic mixed-integer linear programme to determine an optimal appointment schedule for a simulation model for guiding the scheduling decisions. A comprehensive analysis was undertaken to determine the tradeoffs between schedule efficiency and accessibility to service was conducted. The computational experiments suggested that a session capacity close to the request rate can balance the multiple conflicting objectives more effectively. Furthermore, a fixed capacity policy can be as effective as a dynamic overbooking policy under the setting of a constant request rate, but a dynamic overbooking policy leads to slightly better performance for a non-homogeneous appointment request rate.

In our work, even though we do not propose an approach that directly schedules patient appointments, the probability of a cancellation or a late arrival are estimated to help the FHC construct the daily schedule.

2.5. Vehicle routing-based models

Though not widely documented, considerable efforts have been made to provide transportation services for patients at risk who are economically and socially disadvantaged. Luo et al. (2019) proposed an extension of the classical dial-a-ride problem to select a set of minimum-cost routes for a fleet of heterogeneous vehicles to satisfy a set of transportation requests of clients from their origins and destinations. The objective is first to maximise the number of requests satisfied and then to minimise the total travel distance of the vehicles. Constraints on the routes included vehicle capacity, time windows, maximum ride time, number of trips, and lunch breaks. Solutions were found with a two-phase algorithm making use of Benders decomposition. In the scenarios investigated, the capacity of a vehicle was 12, the unit manpower cost was set at 0, USD the number of requests ranged from 26 to 37 per day, the number of vehicles was 2 or 3, and the number of assistants helping patients get on vehicles was 3 or 4. Of the 41 test instances, the proposed algorithm successfully solved 27 to optimality; 17 of those had more than 30 requests.

Wang (2019) developed three heuristics based on myopic operations, tabu search, and a 2-stage procedure to find solutions to a mixed integer programme (MIP) for a last-mile transportation system. The goal was to bring passengers from a metro centre to a last-mile stop close to their final destinations. The objective of the MIP was to minimise the weighted sum of the passengers’ waiting time for a vehicle plus the total ride time. Constraints include a fixed number of vehicles, a fixed number of preselected routes and vehicle capacity.

In a similar study, Agussuja et al. (2019) developed a solution procedure for a multiperiod ride-sharing problem aimed at transporting passengers from a depot to their individual destination. The objective was to minimise the expected discounted sum of costs over an infinite horizon in the face of uncertain demand and batch arrivals. Passengers submit their demands for services shortly before their arrivals, so the time for generating a dispatching plan is limited. Furthermore, with a fixed fleet size, it is often impossible to serve all the demand so decisions about which requests to serve in each period become important. The problem was modelled as a two-level Markov decision process, which was shown to converge and that the solution quality improved as the as sample size increased. Furthermore, the approach performs best when the distribution is less uniform and the planning area is large.

The prior VRP research has mainly focused on the minimisation of travel time or distance for an upper limit on the number of vehicles. Multiple factors such as time windows, route time limit and vehicle capacity often serve as constraints. However, little if any prior work addresses bi-objective problem of minimising a combination of distance and number of vehicles per route. Our model is different than those reviewed in this subsection. We consider providing shuttle service for a subset of FHC patients. To determine the number of vehicles required and their routes, a modified vehicle routing problem is proposed with frequency and tour limit constraints. To be specific, each chosen station must be covered by one route, and each route forms a loop starting and ending at the FHC. Our objective is to minimise the weighted sum of vehicle fixed cost and travel cost. The constraints in the formulation are designed, in part, to strike a balance between patient satisfaction and the costs to University Hospital for providing the shuttle service.

2.6. Predicting no-show probabilities and identifying factors for high no-show rates

Mohammadi et al. (2018) developed and compared appointment no-show prediction models to better understand appointment adherence in underserved populations. Similar to us, they used logistic regression, but also an artificial neural network and a naïve Bayes classifier model. To conduct the study, the authors collected electronic health record (EHR) and appointment data including patient, provider, and clinical visit characteristics over three years, with the objective of identifying the critical variables that can be used to predict missed appointments. Ten-fold cross-validation was used to assess the models’ ability to identify the delinquent patients. The results showed that the areas under the curve (AUC) were 0.81, 0.66 and 0.86 for the three models, respectively.

To highlight both the opportunity for and the importance of leveraging the available EHR data for more refined risk models, Ding et al. (2018) assessed their ability to derive a patient-centric risk score for missed clinic visits. In the study, they used data from 2,264,235 outpatient appointments and assessed the performance of models from 14 different specialities and 55 clinics. Regularised logistic regression was used to fit and assess models based on the health system, speciality, and clinic levels in the EHR data. According to the results, a relatively robust risk score for patient no-shows could be derived with an average C-statistic of 0.83. Furthermore, they found that clinic-specific models outperformed those with greater generality.

To predict the no-show probability of patients, Topuz et al. (2018) proposed a hybrid probabilistic prediction framework based on the elastic net variable-selection methodology integrated with a probabilistic Bayesian belief network. Potential influencing factors such as demographics, socioeconomic status, current appointment information, and appointment attendance history of the patient and the family were taken into account. The computational experiments showed that compared with other approaches in the literature, this prediction methodology had the largest area under the curve (0.691), and hence the greatest accuracy.

Ahmadi et al. (2019) developed a predictive model by combining machine learning and optimisation techniques. The model consisted of two phases. In the first phase, a metaheuristic was used to explore the best subset of features that could significantly contribute to the prediction outcomes. Fitness functions evaluated included decision trees (DT), random forests (RF), Naïve Bayes (NB) and logistic regression (LR). In the second phase, based on the output of the first stage, the authors proposed a stacking model to further improve the prediction performances. The computational results suggested that among the four techniques, RF performed better than other techniques.

Different from prior work, our study concentrates on the impact of transportation factors on no-shows. In particular, we include transportation mode, travel time, transportation difficulty, ambulatory issues and so on in our model. In addition, we also consider other factors such as patient visit history and demographics. A unique aspect in our work is the derivation of a logistic regression model to predict the probability of a patient being late.

3. Study methodology

Our ultimate goal is to determine whether the potential cost savings achieved by reducing the no-show rate of FHC patients are sufficient to offset the costs of providing a particular subset of them with free transportation. The analysis was divided into the following general steps: data analysis of patient history, shuttle design, cost estimation of missed appointment, ED visits and hospital admissions, evaluation of transportation options, and overall cost saving calculations and sensitivity analysis. Figure 2 illustrates the relative order of the analysis. The text in the red boxes represents the study components. The text in the black boxes gives more detail on the individual steps.

Figure 2.

Flowchart for case study.

3.1. Data analysis

In Section 4, we investigate the data collected from the FHC. The intent in this step is threefold: (1) to test whether missing an appointment significantly increases the probability of an ED visit within 30 days, which can be used to estimate the maximum cost savings in Section 5.1; (2) to investigate the impact of transportation factors, such as transportation mode, travel time, transportation difficulty and ambulatory capacity, on appointment cancellation and late arrivals; and (3) to estimate the probability of appointment cancellation and a late arrival based on each patient’s characteristics. The latter can be used to estimate the maximum cost savings by providing transportation in Section 5.1 and to decide the range of no-show reduction percentage in Section 5.4.

3.2. Estimation of maximum cost savings

The third step is to estimate the maximum cost savings achievable by offering improved transportation service, which is determined in Section 5.1. Specifically, we estimate the cost of (i) missed appointments, (ii) ED visits within 30 days, and (iii) subsequent hospital admissions along with their respective probabilities. Based on this analysis, we calculate the potential financial savings to University Hospital should they decide to subsidise transportation services for their most economically disadvantaged patients.

3.3. Cost estimation of transportation options

The next step is to estimate the transportation costs for each potential option, which are enumerated in Section 5.3. For some seniors and disabled patients, charities and Medicaid provide free services, while the metro bus system (VIA Trans) offers reduced fares. Ride hailing companies like Uber and Lyft have the option for a third party to pay the fare in a manner that is transparent to the client.

Two other possibilities that were considered require a hefty outlay by the FHC/University Hospital. These included first, FHC providing shuttle service with golf carts between the FHC and Centro Plaza (closest bus station to the FHC). Second, since zip code 78,207 has the largest percentage of patients (16%), largest cancellation rate (18%) and is the most economically disadvantaged area in San Antonio, we consider the option of providing free shuttle service in this area. The solutions obtained by solving the corresponding optimisation problem are used to estimate shuttle transportation cost. The design of shuttle service is discussed in Section 5.2.

The introduction of FHC controlled options stems from the mixed results associated with third party transportation providers found in the literature (Chaiyachati et al., 2018; Powers et al., 2016) and survey results in Yang et al. (2006) indicating the desirability of shuttle systems by patients/caregivers. Based on the analysis in Sections 5.1, 5.2 and 5.3, we estimate the net cost saving achievable from the project.

3.4. Sensitivity analysis

Finally, in Section 5.4, we conduct sensitivity analyses to determine the impact of reducing the percentage of missed appointments on financial savings, transportation costs, and net cost reduction to the hospital system.

In the next section, we conduct data analysis and statistical tests to investigate the impact of multiple transportation factors on no-shows and late arrivals.

4. Data analysis of patients’ transportation and arrival information

In this section, the data compiled for the study are discussed along with the methods used in the analyses. In particular, we conducted regression analyses to predict the likelihood that a patient cancels an appointment or is late. All results were subjected to extensive testing to substantiate their statistical significance.

4.1. Data introduction

The data for the study were collected between February 19 and 22, 2019 at the Robert B. Green Campus – University Health System, FHC. A total of 774 randomly selected patients were tracked during this time, with 636 pre-scheduled and 138 walk-ins. Among those pre-scheduled, 227 missed their appointment while 409 did not although some were late.

Note that FHC records do not distinguish between cancellations and no-shows so we use these terms interchangeably. Anecdotally, though, the vast majority are believed to be no-shows. Furthermore, while we recognise that these two reasons have different operational impacts on the FHC, from the patient’s perspective, they are simply a missed appointment. Since the main cost drivers have to do with the consequences of a patient missing an appointment, in particular, incurring an ED visit and hospital stay within 30 days of the missed appointment, distinguishing between cancellations and no-shows would not materially change out results.

At beginning of this study, we designed a survey questionnaire for each pre-scheduled patient. The purpose of this survey was to investigate which transportation factors lead to missed appointments. An important part of the survey consisted of questions that focused on patient transportation information such as mode and travel time, because one of our working hypotheses is that more transportation options will result in lower no-show rates and reduced system costs. We were also interested in the visit history of patients such as the number of previous visits to the clinic and the number of no-shows in the last 12 months. These factors can help us understand whether missing the current appointment is an exception or a common phenomenon. We also collected other basic information including age, gender and the number of chronic conditions because these factors can similarly impact whether a patient keeps an appointment. Finally, we tracked whether patients visited the ED or were admitted to the hospital within 30 days of the scheduled appointment; such events create high costs for the health system. The survey questionnaire is in Appendix A.

For patients who kept their appointment, FHC collected basic information such as whether the patient was pre-scheduled or walk-in, the appointment time when appropriate, arrival time, gender, age, zip code, and insurance type. From the hospital’s information system, we were able to obtain additional data for the preceding 12 months on the number of previous visits and cancellations for each of the 774 patients, the number of ED visits, the number of times the patients had been admitted to hospital, the date of his last ED visit and hospital stay, and the number of chronic conditions each patient has. Furthermore, FHC collected transportation information such as transportation mode (to and from the FHC), travel time, transportation difficulty level, and ambulatory capacity (e.g., cane, wheelchair).

Patients who were no-shows were called to debrief them on why they missed their appointment and how they ordinarily travel to the clinic. Finally, all patients were tracked for a month after the 4-day survey period to see whether or not they visited the ED within 30 days of their current appointment.

During the survey, responses were mixed. Some patients were happy to answer all the questions while others either refused to participate or only answered a few. For example, while most were willing to fill in their transportation mode, a minority were reluctant to give their ambulatory capacity. As a consequence, the number of responses to each question was rarely the same. The survey questions and a full set of responses are available from the authors by request.

Due to the large sample size (greater than 30) and since patients are independent of each other, we use a two-sample proportions z-test to check whether or not a missed FHC appointment significantly increases the likelihood of an ED visit within 30 days. The null and alternative hypotheses are as follows.

H₀: Missing an FHC appointment does not significantly affect the likelihood of an ED visit within 30 days.

H_A: Missing an FHC appointment significantly increases the likelihood of an ED visit within 30 days.

Figure 5 in Appendix B plots the visit proportions within 30 days for those who missed as well as those who kept their appointments. For patients who miss the appointment, the proportion of an ED visit within 30 days is 0.062. For patients who keep their appointment, the proportion of an ED visit within 30 days is 0.034. The p-value (one tail) of two-sample proportions z-test is 0.053. Hence, the results are statistically significant at the 0.1 level, and indicate that patients are almost twice as likely to show up at the ED within 30 days if they miss their initial FHC appointment.

In the remainder of this section, we summarise the data and then present two analyses using logistic regression. Our goal is to try to establish relationships that can help predict the probability of a cancellation and the probability of an over-15-minute late arrival, respectively, as a function of demographic factors such as zip code, transportation mode, transportation difficulties, ambulatory capacity, and travel time. We also analyse the impact of cancellations on ED visits within 30 days. Lastly, we tried to establish two other relationships using linear regression. The first was to understand how the different demographic factors impact cancellation rates and late arrivals. The second was aimed at determining which factors, if any, could be used to predict the number of ED visits. The linear regression results proved to be statistically insignificant and so are not presented. The interested reader is referred to Guo (2020).

4.2. Data highlights

Before presenting the logistic regressions, we examine some of the statistics associated with the survey data. In particular, we consider how that following factors affect appointment cancellations and lateness: transportation mode, transportation difficulty level, travel time, and ambulatory capacity.

4.2.1. Impact of transportation mode on cancellations and lateness

Table 1 lists the different transportation modes and their corresponding cancellation proportions. Most of the modes are self-explanatory with the exception of “VIA bus,” which is the regular bus service in San Antonia, and “VIA Trans,” which is an option for residents who cannot utilise regular bus service because of a disability. It is a shared-ride, curb-to-curb service. “Medical/care transport” indicates that the Medicaid/Medicare insurance provides rides for patients. The column “Cancellation proportion” is obtained by dividing the “Number of cancellations” by the “Number of scheduled patients using this mode.” It is important to note that in 11 cases, it was clear from the responses that patients did not understand the question being asked. These records were discarded as unusable. Additionally, since there was only one response for Taxi/Uber/Lyft, this category was also excluded from further analysis.

Table 1.

Statistics for cancellations and late patients as a function of transportation mode.

Transportation mode	Cancellations			Over-15-minute late arrivals
	Number of scheduled patients using this mode	Number of cancellations	Cancellation proportion	Number of kept appointments for respective mode	Number of over-15-min late arrivals, given appointment is kept	Proportion of patients over-15-min late
Drive car	228	45	0.197	183	43	0.235
VIA Trans	22	11	0.5	10	1	0.1
VIA bus	57	16	0.281	41	9	0.22
Medical/care transport	20	7	0.35	13	1	0.077
Taxi/Uber/Lyft	1	1	1	0	0	–
Walking	12	5	0.417	7	3	0.429
Not taking survey	172	106	0.616	66	29	0.439
Unusable responses	11	2	0.182	9	4	0.444

Table 1 depicts the percent of patients over 15 minutes late as a function of their transportation mode. All unusable responses were discarded. Since there were no Taxi/Uber/Lyft responses, this category was again excluded from the analysis.

4.2.2. Impact of transportation difficulty level on cancellations and lateness

In the survey, patients were asked to rate their level of difficulty associated with transportation as 1 “not difficult at all,” 2 “somewhat difficult,” 3 “very difficult” and 4 “extremely difficult.” Table 2 highlights the responses. To determine if transportation difficulties impact the proportion of cancellations, we collapsed difficulty levels 2 to 4 into a single category because some of the categories have a low number of observations. Additionally, it is sufficient for the study to segment the patients into those who did not experience any transportation difficulty and those who did. Figure 6 in Appendix B depicts the cancellation proportions for the two resulting categories. For patients with some transportation difficulty, the proportion of cancellations is 0.399. For patients without any transportation difficulty, the cancellation proportion is 0.175.

Table 2.

Cancellation and late proportions for different transportation difficulty levels.

Difficulty level	Cancellations			Over-15-minute late arrivals
	Number of patients with corresponding difficulty level	Number of cancellations	Cancellation proportion	Number of patients who kept appointment with corresponding difficulty	Number of over-15-min late arrivals, given appointment is kept	Proportion of over-15-min late patients
1 (not at all difficult)	286	50	0.175	235	47	0.20
2 (somewhat difficult)	128	51	0.398	77	17	0.221
3 (very difficult)	47	19	0.404	28	6	0.214
4 (extremely difficult)	3	1	0.333	2	0	0
Not taking survey	172	106	0.686	66	29	0.439

Because the sample size is large enough and patients were selected independently of each other, we conduct a two-sample proportions z-test to determine whether transportation difficulty affects appointment cancellations at the FHC. The null and alternative hypotheses are as follows.

H₀: Transportation difficulty does not significantly affect the likelihood of an appointment cancellation.

H_A: Transportation difficulty significantly increases the likelihood of an appointment cancellation.

The two-sample proportions z-test indicates that the proportion of cancellations amongst patients with some level of transportation difficulty is significantly greater than cancellation proportion for patients reporting no transportation difficulty (one tail p-value is $5.10\times {10}^{-8}$).

Table 2 contains the proportion of late patients at the FHC by transportation difficulty level. If we merge the results for difficulty levels 2 through 4 into a single category, the large sample size coupled with patient independence allow us to conduct a two-sample proportions z-test. The null and alternative hypotheses are as follows.

H₀: Transportation difficulty does not significantly affect the proportion of late arrivals to FHC.

H_A: Transportation difficulty significantly increases the proportion of late arrivals to FHC.

The results indicate that the proportion of late patients with transportation difficulties is not significantly larger than for the patients without transportation difficulties (one tail p-value is 0.375).

4.2.3. Impact of travel time on cancellation and lateness

To investigate how travel time influences cancellation behaviour and late arrivals, we make the following assumptions:

1. In the survey, if a patient gave a time range (a, b), then we assume his travel time is $\frac{a+b}{2}$.

2. If a patient reported the travel time to be more than 30 minutes, then we assume the value to be 45 minutes. If a patient reported the travel time more than 60 minutes, then we assume the value to be 90 minutes.

3. If a patient gave travel times for several different transportation modes, then we choose the longest time.

4. When analysing the cancellation percentage, if a patient who kept the appointment gave travel times for several different modes, then we choose the time for the mode used on the appointment day.

Table 3 contains cancellation proportions for different ranges of travel times. These data appear to indicate that if a patient’s travel time is longer than 30 minutes, then he is more likely to cancel the appointment. To test this observation and protect against low observation categories, we collapsed the data into two categories: patients with travel time less than or equal to 30 minutes and those with travel times greater than 30 minutes. Figure 7 in Appendix B shows the cancellation proportions for these two categories. For patients whose travel time to the FHC is within 30 minutes, the cancellation proportion is 0.236. For patients whose travel time to the FHC is longer than 30 minutes, the cancellation proportion is 0.355. The large sample size and patient independence justifies the use of a two-sample proportions z-test. The null and alternative hypotheses are as follows.

H₀: Travel time to FHC longer than 30 minutes does not significantly affect the cancellation proportion.

Table 3.

Relationship between travel time, cancellation proportions and late arrivals.

Travel time (min)	Cancellations			Over-15-minute late arrivals
	Number of pre-scheduled patients within corresponding travel time range	Number of cancellations	Cancellation proportion	Number of kept appointments within corresponding travel time	Number of over-15-min late arrivals, given appointment is kept	Proportion of patients over-15-min late
0–30	369	87	0.236	282	55	0.195
31–60	66	24	0.364	42	10	0.238
61–90	21	7	0.333	14	3	0.214
91–120	6	2	0.333	4	2	0.5
121–150	0	0	–	0	0	N/A
151–180	1	0	0	1	0	0
Not taking survey	173	107	0.618	66	29	0.439

H_A: Travel time to FHC longer than 30 minutes significantly increases the cancellation proportion.

The test result confirms that patients with longer travel times have higher likelihood of cancelling (one tail p-value is $9.64\times {10}^{-3}$).

Table 3 contains the proportion of late patients for the travel time ranges. When the travel time data are grouped into two categories (that is, ≤ 30 minutes and > 30 minutes), we are not able to establish that the longer travel times are associated with a higher proportion of late arrivals (one tail p-value is 0.169).

4.2.4. Impact of ambulatory capacity on cancellation and lateness

In the survey, 8 levels of ambulatory capacity (degree of mobility) were included, from 1 (without ambulatory issues, least serious) to 7 (amputee, most serious) and 8 (other). Level-8 includes mobility aids or devices, lack of support mechanisms, and a requirement for multiple ambulatory aids (for example, a patient needs both a cane and a wheelchair). The following assumptions were made to facilitate the analysis.

1. ) If a patient indicated multiple disability levels, then we chose the highest one (more serious one).

2. ) If a patient simply complained and did not provide his ambulatory capacity, then we considered him not to have an ambulatory issue.

The results for those taking the survey are reported in Table 4. Because the total number of level-2 through level-8 patients is relatively small compared to the number of level-1 patients, we aggregate the former and considered cancellation proportions for patients with and without ambulatory issues. Figure 8 in Appendix B depicts the results. For patients without any ambulatory issues, the cancellation proportion is 0.178. For patients with some ambulatory issues, the cancellation proportion is 0.363. Again, we conduct a two-sample proportions z-test. The null and alternative hypotheses are as follows.

H₀: Ambulatory issues do not significantly affect the cancellation proportion of FHC patients.

Table 4.

Cancellation and late proportions for different ambulatory capacities.

Ambulatory capacity	Cancellations			Over-15-minute late arrivals
	Number of pre-scheduled patients with corresponding ambulatory capacity	Number of cancellations	Cancellation proportions	Number of pre-scheduled patients who kept appointment within corresponding ambulatory capacity	Number of over-15-min late arrivals, given appointment was kept	Proportion of patients over15-min late
1 (without ambulatory issues)	286	51	0.178	235	48	0.204
2 (supported by person)	8	3	0.375	5	1	0.20
3 (cane)	36	11	0.306	25	6	0.24
4 (walker)	22	8	0.364	14	3	0.214
5 (regular wheelchair)	14	5	0.357	9	1	0.111
6 (motorised wheelchair)	5	1	0.2	4	2	0.50
7 (amputee)	0	0	–	0	0	–
8 (other)	7	5	0.714	2	0	0
Not taking survey	258	143	0.554	115	38	0.33

H_A: Ambulatory issues significantly increase the cancellation proportion of FHC patients.

The test result indicates that the proportion of cancellations for patients with ambulatory issues is significantly higher than for patients without (one tail p-value of $1.17\times {10}^{-4}$).

The data analysis indicates that ambulatory capacity does not appear to impact patient lateness. Aggregating levels 2 through 8 and conducting a two-sample proportions z-test confirms that there is not enough evidence to support that the proportion of late patients with ambulatory issues is bigger than for the patients without ambulatory issues (one tail p-value is 0.369).

4.3. A logistic regression model for cancellation probability

In this section, we construct a multi-variate logistic regression model to predict the probability of patient cancellation. The model is based on a set of independent variables that reflect patient characteristics. If a patient takes multiple transportation modes including the VIA bus, then we assume he takes VIA bus only. Given that only two patients fall into this category, it was not worth creating a separate multi-modal category. Also, given that patients living in zip codes 78,207 and 78,201 contribute most to cancellations, with percentage of 18% and 9%, respectively, and most scheduled patients are from 78,207 and 78,201 with percentage of 16% and 9%, respectively, whether or not a patient live in these two zip codes were used as independent variables in the logistic regression model.

The following symbols are used in the development of the model.

$CAN$ (binary dependent variable) 1 if patient cancels the current appointment, 0 otherwise

P_CAN (dependent variable) probability that a patient cancels an appointment

N_ED (nonnegative integer variable) number of ED visits of a patient in the last 12 months

$G$ (binary variable) patient’s gender: 1 if a patient is male, 0 if female

$A$ (nonnegative variable) patient’s age (years)

$N_{CAN}$ (nonnegative integer variable) number of cancellations of a patient in the last 12 months

$N_{CD}$ (nonnegative integer variable) number of chronic conditions a patient has

$N_p$ (nonnegative integer variable) number of previous visits to the FHC in the last 12 months

$I_M$ (dummy variable) 1 if the patient has Medicare or Medicaid insurance, 0 otherwise

$I_C$ (dummy variable) 1 if Carelink is the payer for the patient, 0 otherwise

$I_S$ (dummy variable) 1 if the patient is uninsured, 0 otherwise

$Z_1$ (dummy variable) 1 if the patient lives in zip code 78,207, 0 otherwise

$Z_2$ (dummy variable) 1 if the patient lives in zip code 78,201, 0 otherwise

$S$ (binary variable) 1 if the patient requires an ambulatory aid, 0 otherwise

$F$ (binary variable) 1 if the patient has transportation difficulty, 0 otherwise

$T$ (binary variable) 1 if the patient’s travel time from home to the FHC is longer than 30 minutes, 0 otherwise

$T_m$ (binary variable) 1 if the patient travels to the FHC by transportation mode $m$, 0 otherwise $m\in \{\mathrm{d}\mathrm{r}\mathrm{i}\mathrm{v}\mathrm{e},\mathrm{V}\mathrm{I}\mathrm{A}\mathrm{T}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s},\mathrm{V}\mathrm{I}\mathrm{A}\mathrm{b}\mathrm{u}\mathrm{s}$ $\mathrm{m}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{c},\mathrm{t}\mathrm{a}\mathrm{x}\mathrm{i},\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{k}\}$

We start the computations with all the independent variables listed above in the model and use backward elimination to remove those variables whose p-value indicates a lack of statistical significance. That is, the logistic model is run with all independent variables and then optimised by removing one at a time in decreasing order of p-values. The process stops when either all the remaining variables have a p-value less than 0.05, or removing any of the remaining independent variables lowers the adjusted McFadden R-square. The independent variables with the corresponding coefficients and p-values are listed in Table 5.

Table 5.

Factors in the logistic regression model used to estimate the cancellation probability without interactions.

Factors	Coefficient	P-value
Whether the patient uses Carelink as a payer (binary variable)	0.487	0.139
The number of previous visits in last 12 months	–0.101	0.01
The number of cancellations in last 12 months	0.164	2.28 × 10−5
Whether the patient has transportation difficulty (binary variable)	0.825	0.003
Whether the patient has ambulatory issue (binary variable)	0.713	0.018

The final logistic regression model is:

$$ln\left(\frac{P_{CAN}}{1-P_{CAN}}\right)=-2.120+0.487I_C-0.101N_p+0.164N_{CAN}+0.825F+0.713S$$

After some algebra, we can get the following equation for cancellation probability.

$$P_{CAN}=\frac{1}{1+e^{2.120-0.487I_C+0.101N_p-0.164N_{CAN}-0.825F-0.713S}}$$ (15)

In Eq. (15) we observe that the number of previous visits (N_P), number of cancellations in the last 12 months $(N_{CAN})$, whether the patient has transportation difficulty (F), and whether the patient has ambulatory issues (S) are the significant independent variables at the 5% level. The Carelink variable I_C had a p-value of 0.139 > 0.05 and would ordinarily be excluded, but its inclusion increases the adjusted McFadden R-square to 0.5620. McFadden suggested the values between 0.2 and 0.4 represent a very good fit of the model (Louviere et al., 2000). Since 0.5620 is higher than this range, it indicates that (15) is a good fit. In addition to McFadden R-square, the AUC value for the logistic regression model is 0.638. AUC is a value ranging between 0 and 1. The closer AUC is to 1, the more accurate the prediction is (Hand, 1997).

The implications of the model are that if a patient has Carelink as a payer, the number of cancellations in the last 12 months increases, the patient has transportation difficulties, or the patient has ambulatory issues, then the probability of cancellation will increase. If the number of previous visits in the last 12 months increases, then the cancellation probability will decrease. While the direction of most of the effects make sense, Carelink recipients are at risk for high probability of cancellation because they are a disadvantaged population. Patients with a higher number of previous visits are likely to be more conscientious about their healthcare needs and therefore have a lower probability of cancelling.

Based on the results in Section 4.2.3, we expected that variable T (whether travel time is longer than 30 minutes) would appear as a significant factor in the regression model in (15), but it didn’t. Including the terms T and $\left(T\times N_{CAN}\right)$ in the model now yields Eq. (16). The independent variables with coefficients and p-values are contained Table 6.

$$ln\left(\frac{P_{CAN}}{1-P_{CAN}}\right)=-2.299+0.671T+0.200N_{CAN}+0.806F-0.103N_p+0.726S+0.541I_C-0.123\left(T\times N_{CAN}\right)$$

Table 6.

Factors in the logistic regression model used to estimate the cancellation probability with interactions.

Factors	Coefficient	P-value
Whether travel time is longer than 30 minutes (binary variable)	0.671	0.146
The number of cancellations in last 12 months	0.200	1.13 × 10−5
Whether the patient has transportation difficulty (binary variable)	0.806	0.005
The number of previous visits in last 12 months	–0.103	0.011
Whether the patient has ambulatory issue (binary variable)	0.726	0.016
Whether the patient uses Carelink as a payer (binary variable)	0.541	0.106
(Whether travel time is longer than 30 minutes) × (the number of cancellations ins last 12 months)	–0.123	0.084

or

$$P_{CAN}=\frac{1}{1+{e^{2.299-0.671T-0.200N_{CAN}-0.806F+0.103N_p-0.726S-0.541I_C+0.123\left(T\times N_{CAN}\right)}}^{\hspace{0.17em}}}$$ (16)

The p-values on for T and $\left(T\times N_{CAN}\right)$ are 0.146 and 0.084, respectively. The adjusted McFadden R-square is 0.5605 and the AUC is 0.647, indicating that the model in (16) has roughly the same fit as the model in (15). Based the principle of parsimony, we recommend the model (15). However, we include model (16) because the significant negative interaction term (at the 0.1 level) indicates that T has a moderating effect on the relationship between $N_{CAN}$ and $P_{CAN}$. The implication is that for patients with travel time from home to the FHC less than or equal to 30 minutes, $P_{CAN}$ increases faster in $N_{CAN}.$ We attribute this effect to the difference between patients close to the FHC in underserved zip codes such as 78,207 and those who drive from more distance, less disadvantaged zip codes. This finding provides further impetus for FHC to focus its efforts on improving transportation for patients in 78,207.

When performing regression analysis, it is important to determine whether a set of underlying assumptions holds. Multicollinearity is checked by using a Variance Inflation Factor test (Rawlings et al., 1998). Linearity is checked by plotting the log odds against each quantitative independent variable. Independence is satisfied by construction. Outliers are checked by using Cook’s distance (Cook 1977). All tests confirmed the validity of models (15) and (16).

4.4. Logistic regression on probability of arrivals greater than 15 minutes

We now attempt to establish a relationship between a patient’s characteristics and the likelihood of him or her being more than 15 minutes late. In the analysis, the dependent variable late is set to 1 if a patient arrives more than 15 minutes late, 0 otherwise. We only look at pre-scheduled patients who kept their appointment. The independent variables include all those in the logistic regression in the previous subsection with the exception of whether a patient cancelled the appointment $(CAN)$. The other variable definitions remain unchanged.

The variables in the final model include whether the patient uses Medicare/Medicaid insurance (I_M), the number of chronic diseases (N_CD), the number of previous visits in last 12 months (N_P), whether the patient drives to the FHC (T_drive), whether the patient takes VIA bus to FHC (T_{VIA bus}), whether the patient walks to the FHC (T_walk), and whether the patient has ambulatory issues (S). Note that even though variable S is insignificant with p-value of 0.257, its inclusion increases the adjusted McFadden R-square from 0.2649 to 0.3569, and increases AUC from 0.698 to 0.706. Regardless, with or without S, we can conclude that model (17) is a good fit. The independent variables with coefficients and p-values are listed in Table 7.

$$ln\left(\frac{P_{late}}{1-P_{late}}\right)=-1.639+0.758I_M-0.028N_{CD}-0.104N_P+1.071T_{drive}+1.066T_{VIAbus}+2.264T_{walk}+0.435S$$

Table 7.

Factors in the logistic regression model used to estimate the probability of over-15-minute late arrival.

Factors	Coefficient	P-value
Whether the patient uses Medicare/Medicaid insurance (binary variable)	0.758	0.020
The number of chronic diseases	–0.028	0.050
The number of previous visits in last 12 months	–0.104	0.026
Whether the patient drives to FHC (binary variable)	1.071	0.008
Whether the patient takes VIA bus to FHC (binary variable)	1.066	0.042
Whether the patient walks to FHC (binary variable)	2.264	0.01
Whether the patient has ambulatory issue (binary variable)	0.435	0.257

or

$$P_{late}=\frac{1}{1+{e^{1.639-0.758I_M+0.028N_{CD}+0.104N_P-1.071T_{drive}-1.066T_{VIAbus}-2.264T_{walk}-0.435S}}^{\hspace{0.17em}}}$$ (17)

From (17) we see that a patient who has Medicare/Medicaid insurance, drives to the FHC, takes a regular VIA bus to the FHC, walks to the FHC, or has ambulatory issues, has a higher probability of being more than 15 minutes late then a patient without these characteristics. Also, if the number of chronic conditions increases, or the number of previous visits increases, then the over-15-minutes late probability will decrease. The increase in lateness associated with Medicare/Medicaid patients may be due their disadvantaged status. It is not surprising that taking less convenient forms of transportation (VIA bus or walking) or having ambulatory issues makes it more difficult to get to the clinic on time. The best explanation we could find for why driving to the FHC is associated with increased lateness was evident in the surveys in the form of comments about the difficulties of finding parking. Patients with a higher number of previous visits and a greater number of chronic conditions are more frequent visitors, and are less likely to be late.

To validate the underlying assumptions the following tests were conducted. Multicollinearity is checked by using a Variance Inflation Factor test (Rawlings et al., 1998). Linearity is checked by plotting the log odds against each quantitative independent variable. Independence is satisfied by construction. Outliers are checked by using Cook’s distance (Cook 1977). All tests confirmed the validity of model (17).

5. Cost analysis

In this section, we analyse the potential cost reduction that might be realised by improving transportation services for FHC patients. Based on our experience, we estimate that the no-show rate, now at 24.3%, can be reduced by 25% down to 18.2% by providing subsidised transportation for those who have difficulty keeping their appointment. A sensitivity analysis on the no-show rate reduction is included in Section 5.4. By mitigating transportation difficulties, we believe that there is substantial room for improvement as indicated by the logistic regression model in Section 4.3 but there is no way to determine the exact amount.

5.1. Cost saving by offering transportation services

By increasing the number of transportation options for those patients who have difficulty keeping their appointment, an overall cost reduction can be achieved. Our working proposition is that a portion of the amount saved can be used as the budget for improving transportation services for the most economically disadvantaged members of the community. The cost reduction calculations follow.

Table 8 identifies the net costs of an FHC appointment [cf. Demeere et al. (2009) who investigated the costs at an outpatient clinic in Belgium], an ED visit, and the cost incurred if a patient is admitted to the hospital after being treated at the ED. Table 9 gives the probability of each event being considered.

Table 8.

Costs associated with an FHC appointment, ED visit, and hospital stay.

Symbol	Value	Explanation
$M_A$	$10	Marginal cost for each FHC appointment visit, such as cost for blood test, X-ray
$R_A$	$66	Revenue from each FHC appointment visit
$C_A$	–$56	Net cost (− revenue) for each appointment visit
$C_{ER}$	$369	Net cost for each ED visit
$C_I$	$1,410	Hospital inpatient (medicine) variable cost per day
$LOS$	6.85 days	Average length of stay (days) for patients admitted to hospital after ED visit
$C_S$	$9,658.50	Net cost if a patient is admitted to the hospital within 30 days of ED visit ($1410 per day, 6.85 days of LOS on average)

Table 9.

Probability of an event.

Symbol	Value	Explanation
$P\left(M\right)$	0.25	Probability of a missed FHC appointment (2018 statistic)
$P(ER|M)$	0.062	Probability of an ED visit within 30 days if a patient misses FHC appointment
$P(ER|\bar{M})$	0.034	Probability of an ED visit within 30 days if a patient keeps FHC appointment
$P(S|M)$	0.214	Probability of a patient being admitted to the hospital within 30 days after ED visit, given that he missed the FHC appointment
$P(S|\bar{M})$	0	Probability of a patient being admitted to hospital within 30 days after ED visit, given that he kept the FHC appointment
${\alpha }_{MT}$	0.5868	Probability that a patient misses an FHC appointment due to transportation problems, given that he misses the appointment

These values are estimated from the data collected on the 636 pre-scheduled patients during the February 19– 22, 2019 time period, with the exception of the probability of missing an appointment P(M). Note that from the data discussed in Section 4.1, out of the 636 pre-scheduled patients, 227 missed their appointments so an estimate from the sample is $P\left(M\right)=\frac{227}{636}=0.357$. However, according to the data in 2018, the probability of missing an appointment is 0.25. Because the latter is based on a larger data set over the entire year, it represents a more accurate estimate of the probability of missing an appointment. Hence, we use $P\left(M\right)=0.25$ in our calculations as shown in Table 9.

Table 10 presents the input data used in the calculations and the resulting number of patients who now are able to keep their appointment. Table 11 and Table 12 highlight the potential cost savings and losses, respectively, from the increase in kept appointments. Reading from the tables, we see that the monthly cost savings from the increase in the number of kept appointments is 24,371 USD + 6,957 USD = 31,328. USD This value is the potential amount that we can spend on transportation services for patients.

Table 10.

Patient data on missed appointments.

Symbol	Value	Explanation
$N_M$	1100	Number of missed appointments per month
$N_{MT}$	645	Number of missed appointments per month due to transportation problems
${\mathrm{\Delta }}_{N_{MT}}$	161	Number of missed appointments per month after providing transportation
${\mathrm{\Delta }}_{N_{\overline{MT}}}$	161	Number of kept appointments per month after providing transportation

Table 11.

Potential cost savings from increase in number of kept appointments.

Symbol	Value	Explanation
ED cost saved	$3,691.82	ED net savings for patients who now keep FHC appointments and don’t come to ED within 30 days
Hospital stay cost saved	$20,679.40	Net cost savings for patients who are no longer admitted to hospital within 30 days after ED visits
Total	$24,371	Total amount of financial savings due to providing transportation

Table 12.

Potential financial losses from increase in number of kept appointments.

Symbol	Value	Explanation
ED cost incurred	$2,024.55	ED net cost incurred for patients who now keep FHC appointments but still go to ED within 30 days
Hospital stay cost incurred	$0.00	Net cost incurred for patients who are admitted to hospital within 30 days after ED visits
Appointment cost incurred	$8,981.85	Net savings incurred for patients who keep FHC appointments
Total	$6,957	Total reduction in financial loss when transportation is provided

Figure 3 is a probability tree diagram that summarises an individual patient’s flow and how costs are calculated. The red values are average net costs provided by FHC management. The blue values are event probabilities, which were calculated from the data set.

Figure 3.

Expected costs associated with patient flow.

5.2. Shuttle design for zip code 78,207

Shuttle services are common in densely populated areas where there is a need to provide transportation for disabled members of the community. Our goal is to design routes within the zip code stopping at nine convenient locations, and to determine the number of vans needed to meet certain operational constraints. All routes are to begin and end at the FHC. Table 13 lists the nine stations for pickup and drop off, and the expected number of patients arriving per hour at each. These values were randomly generated based on the FHC survey data. Because the service will be “free” though, we found it problematic specifying the demand distribution.

Table 13.

Shuttle stations in 78,207 and number of patients arriving per hour.

Station	Average number of patients per hour
W Poplar St & N Colorado St	1
W Poplar St & NW 26th St	3
W Martin St & N Zarzamora St	2
Saltillo & Barclay	1
Laredo & Brazos Farside	2
Brazos & Guadalupe	2
S Flores & El Paso	3
W Commerce St & N Colorado	1
Centro Plaza	2

5.2.1. Problem description

In this problem, shuttles depart the FHC, visit several existing bus stations, and then return to the FHC. We need to decide how many shuttle routes to provide, how many vans to assign to each route, which stations to (uniquely) assign to each route, and the station order in a route. Because there are nine stations identified for pickup in 78,207, there can be at most nine routes, with one route covering exactly one station. However, such a solution is not likely since more routes require more vans and drivers, resulting in higher costs. In fact, barring any constraints on route length and vehicle capacity, a solution with one van is always optimal when the triangular inequality holds for all combinations of three stations.

In constructing routes, our objective is to minimise the weighted sum of the van and travel costs. The former includes costs associated with acquisition, insurance, maintenance and drivers. The latter depends on the total travel distances of all routes and are a function of the cost of gas, assuming diesel or gasoline powered vehicles. All data and parameter values are assumed to be deterministic. To assure that patient waiting times are not excessive, shuttle inter-arrival times on a route are limited to some upper bound (45 minutes in our case); i.e., the difference between arrival times of two successive vans at a station cannot exceed 45 minutes (cf. Zhang et al., 2017). If travel time on a certain route is longer than the upper bound, at least one additional van must be assigned. A second constraint we impose is a limit on the length of a route (60 minutes in our case). Long routes are one of the chief complaints that patients give for not wanting to take a bus (e.g., see Silver et al., 2012).

Since we assume that each shuttle is running continually throughout the day, the travel cost (fuel cost) does not depend on the number of routes or their length, and hence is a constant in the objective function. Therefore, it would seem that could simply ignore travel costs and consider van costs only. However, this would lead to multiple optimal solutions of which only one would likely minimise the sum of the route distances. Because we want to make each route as short as possible for a given number of vans, we replace the total travel costs over the day in the objective function with the total cost of the routes.

In defining the shuttle network, we make use of the following notation.

Indices and sets

$0$ index for FHC

E set of edges linking any two nodes in $N$; $\left(i,j\right)\in E$

$\left(i,j\right)$ index for an edge between nodes $i$ and $j$, $\left(i,j\right)\in E$

N set of nodes (transit stations and FHC); i ∈ N

$r$ index for route $r$, $1\le r\le \left|N\right|-1$

Parameters

$\beta$ objective function weight for the fixed cost of a van; β ∈ [0,1]

$C$ capacity of a van

$c_{ij}$ travel cost from $i$ to $j$

$f$ fixed cost of operating a van per day, including amortised purchase cost and driver costs

$h$ number of hours in a working day

$n_i$ average number of patients picked up at station i

$s$ stopping time at each node (minutes)

$\tau$upper bound on length of each route (minutes)

$t_{ij}$ travel time from $i$ to $j$

$v$ maximum shuttle inter-arrival time at each station

Variables

$P_i^r$ (nonnegative integer variable) for route r, number of patients on a van immediately before it arrives at node i

$X_{ij}^r$ (binary variable) 1 if route $r$ covers edge $\left(i,j\right)$, 0 otherwise

$Y^r$ (binary variable) 1 if there is at least one van assigned to route $r$, 0 otherwise

$Z^r$ (nonnegative integer variable) number of vans assigned to route $r$

Model

$$Minimise\beta f\sum _{r=1}^{\left|N\right|-1}{Z}^r+\left(1-\beta \right)\sum _{r=1}^{\left|N\right|-1}\sum _{\left(i,j\right)\in E}{c}_{ij}{X}_{ij}^r$$ (1)

$$subjectto\sum _{r=1}^{\left|N\right|-1}\sum _{i\in N}{X}_{ij}^r=1,\mathrm{\forall }j\in N\left\{0\right\}$$ (2)

$$\sum _{r=1}^{\left|N\right|-1}\sum _{j\in N}{X}_{ij}^r=1,\mathrm{\forall }i\in N\left\{0\right\}$$ (3)

$$\sum _{j\in N}{X}_{0j}^r=Y^r,1\le r\le \left|N\right|-1$$ (4)

$$\sum _{i\in N}{X}_{i0}^r=Y^r,1\le r\le \left|N\right|-1$$ (5)

$$\sum _{i\in N}{X}_{ij}^r=\sum _{k\in N}{X}_{jk}^r,\mathrm{\forall }j\in N,1\le r\le \left|N\right|-1$$ (6)

$$X_{ij}^r\le Y^r,\mathrm{\forall }\left(i,j\right)\in E,1\le r\le \left|N\right|-1$$ (7)

$$\sum _{\left(i,j\right)\in E}{t}_{ij}{X}_{ij}^r+s\sum _{\left(i,j\right)\in E}{X}_{ij}^r\le v{Z}^r,1\le r\le \left|N\right|-1$$ (8)

$$\sum _{\left(i,j\right)\in E}{t}_{ij}{X}_{ij}^r+s\sum _{\left(i,j\right)\in E}{X}_{ij}^r\le ,1\le r\le \left|N\right|-1$$ (9)

$$P_j^r\ge P_i^r+n_i{X}_{ij}^r+C\left(X_{ij}^r-1\right),\mathrm{\forall }\left(i,j\right)\in E,1\le r\le \left|N\right|-1$$ (10)

$$X_{ij}^r,Y^r\in \left\{0,1\right\},\mathrm{\forall }\left(i,j\right)\in E,1\le r\le \left|N\right|-1$$ (11)

$$X_{ii}^r=0,\mathrm{\forall }i\in N,1\le r\le \left|N\right|-1$$ (12)

$$Z^r\ge 0,0\le P_i^r\le C\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{e}\mathrm{r},\mathrm{\forall }i\in N,1\le r\le \left|N\right|-1$$ (13)

$$P_0^r=0,\mathrm{\forall }1\le r\le \left|N\right|-1$$ (14)

The first term in the objective function (1) is the fixed van cost per day assuming 9 working hours in a day and 22 days in a month (see below for the calculation of the parameter f). The second term is the travel cost for all routes, assuming there is at most one van and one trip for each route; it is not the cost of operating each route over the day. The weight β is selected to be close to 1 in order to prioritise minimising the number of vans. Constraints (2) and (3) ensure that each station is covered by exactly one route, the FHC being the exception. Constraints (4) and (5) respectively require that for each route, the number of vans leaving and entering the FHC is the same as the number of vans assigned to that route. Constraints (6) imply that for each route and transit station, the number of vans entering the station is equal the number of vans leaving; that is, continuity of flow.

Constraints (7) limit the use of edges or connections in the network to routes that are created. An edge can be created in a route only if the route exists. Constraints (8) ensure that the number of vans assigned to each route $Z^r$ is sufficient for the arrival time between two consecutive arrivals not to exceed the upper limit v minutes. The left side of the equation is the total travel time of one trip for route r. We must ensure enough vans so that the inter-arrival time is below the upper limit v. Constraints (9) restrict the length of each route to be no more than minutes, while constraints (10) calculate the number of patients on a van just after leaving station i. They also serve as subtour elimination constraints. Variables are defined in (11) – (14).

5.2.6. Computational results for VRP

In specifying model parameters, the number of pickups at each station in Eq. (10) was randomly selected from the set {1,2,3}. See Table 13. Tables containing the distance, travel time, and cost matrices for each edge in the network can be found in Guo (2020). It is assumed that these matrices are symmetric, i.e., the cost from i to j is equal to the cost from j to i. For mileage, we assumed 20 miles per gallon and that fuel costs 2.5 USD per gallon. On each route, it was further assumed that the vans stop for 3 minutes at each station to load and discharge patients. These assumptions were made based on the average fuel costs in Texas.

To cost a van, according to the information from CarMax, Carvana and Autotrader, we used an average purchase price of 36,000, USD a life expectancy of 10 years, and an annual interest rate of 5% to get an amortised uniform monthly cost of 360. USD Now suppose that the monthly cost of a driver is 2376 USD (that we got from Glassdoor), the maintenance cost is 50 USD and the insurance cost is 100. USD Then the total cost for a van and a driver per month is $\mathrm{\$}360+\mathrm{\$}2376+\mathrm{\$}50+\mathrm{\$}100=\mathrm{\$}2886$. Table 14 gives a breakdown of these costs. Assuming that there are 22 working days in a month, the daily cost of a van and a driver is $\frac{\mathrm{\$}2886}{22}=\mathrm{\$}131.18$.

Example. To illustrate the computations, we assume that the inter-arrival time for a shuttle is at most 45 minutes, and that each route is no longer than 60 minutes. In Eq. (1), the weight β on the daily van costs is fixed at 0.8. Model (1) – (14) was solved with CPLEX 12.6 in 1.1 seconds. The number of variables was 1008, and the number of constraints was 2124. The optimal solution has two routes with one shuttle each. Table 15 enumerates the station sequence and travel costs for each. Figure 4 depicts the combined routes. Since there are two vans, the fixed van cost and driver cost per month is 2 × 2886 USD = 5772. USD The total monthly cost for shuttle service then is $\mathrm{\$}5772+\mathrm{\$}244.2+\mathrm{\$}277.2=\mathrm{\$}6293.4$.

Table 14.

Van and driver fixed cost.

Cost item	Value
Driver cost per month	$2,376
Purchasing price of a van	$36,000
Number of days per month	22
Number of hours per day	9
Driver cost per hour	$12
Present value cost for a van per month	$360
Maintenance cost per month	$50
Insurance cost per month	$100
Total van and driver cost per month	$2,886

Table 15.

Solution for illustrative example.

Route	Route1	Route2
Route sequence	(FHC) – (W Poplar St & N Colorado St) – (W Poplar St & NW 26th St) – (W Martin St & N Zarzamora St) – (W Commerce St & N Colorado) – (Centro Plaza) – (FHC)	(FHC) – (Laredo & Brazos Farside) – (Saltillo & Barclay) – (Brazos & Guadalupe) – (S Flores & El Paso) – (FHC)
Each route cost ($)	0.93	1.05
Each route time (min)	45	45
Number of routes per day per van	12	12
Route cost per day per van ($)	11.10	12.60
Number of days per month	22	22
Route cost per month per van ($)	244.20	277.20
Number of vans	1	1

Figure 4.

Map of combined routes in illustrative example.

Sensitivity analysis. The two critical parameters in the VRP model are the van inter-arrival time $(v)$ and the route time limit impact the solution. Because they are set at the discretion of management, they must be decided before model (1) – (14) is solved. To get a better understanding of how their values affect the solution, we conducted an extensive sensitivity analysis. In particular, we considered $v$ ∈{20, 30, 40, 50, 60} and ∈ {20, 30, 40, 50, 60}. The benchmarks we used were CPLEX runtime, number of routes, number of vans, and the total monthly cost for 78,207 shuttle service.

Our first observations were that for route times = 20 minutes, the problem is infeasible, and as increases beyond 30 runtimes generally decrease because the model becomes easier to solve. Also, as the route time limit increases, the number of routes decreases. If the route time limit is fixed, the number of routes is the same for each inter-arrival time. Therefore, we can conclude that the number of routes is not sensitive to inter-arrival time, but it is sensitive to route time limit. Next, we found that the number of vans required is somewhat sensitive to both parameters. With respect to cost, if the inter-arrival time is fixed, then the total monthly cost decreases as the route time limit increases. Similarly, if the route time limit is fixed, then the total monthly cost decreases as the inter-arrival time increases. As expected, then, as the two constraints are loosened, the total monthly cost decreases. The full set of results can be found in Guo (2020).

5.3. Transportation cost analysis

Several different modes of transportation are included in the study. For those patients living in zip code 78,207, the densest area served by the FHC, a free van shuttle service is considered. The costs and operational constraints (including an upper bound of 45 minutes on inter-arrival times and a 60-minute limit on route length) were discussed in Section 5.2. Recall that the cost per month for this service is $\mathrm{\$}6293.4$, which is based on the purchase of two vans amortised over 10 years, maintenance, insurance, fuel, and driver salaries.

In addition to the circulating vans, we also consider providing golf carts to pick up and drop off patients at the Centro Plaza Transit Center, which is two blocks from the FHC. For illustrative purposes, it is assumed that a 6-passenger golf cart costs 14,000, USD its life expectancy is 5 years, and the annual interest rate is 5%. The uniform monthly cost is then 245. USD Suppose that the cost of a driver, maintenance cost, and insurance cost of a van is 1782, USD 50 USD and 80 USD per month, respectively. Then the total cost for a golf cart and a driver per month is $\mathrm{\$}245+\mathrm{\$}1782+\mathrm{\$}50+\mathrm{\$}80=\mathrm{\$}2,157$. Assuming that the golf cart interval is 30 minutes, there are 9 hours per day and 22 days per month, the golf cart averages 30 miles per gallon, and gas is 2.50 USD per gallon, then the total cost for one golf cart per month is 2196.60. USD

An additional objective is to pay the fare for patients taking the regular VIA (metro) bus. Since those who live within 78,207 can take the free van, we expect to pay only for those who live outside this area. It is assumed that those patients will arrive at Centro Plaza so it is only necessary to consider the ticket cost to this location and add it to the cost of the “golf cart.” Table 16 lists the costs for each transportation option and the number of patients per month expected to take advantage of them. On average 9.68 patients will take the golf cart from Centro Plaza. We also assume that the round-trip ticket price of VIA bus is 5.20, USD so the total monthly cost for the golf cart option is 2196.6 + 9.68 × 5.2 = 2246.95. USD It is understood that if golf cart transportation is provided, there are likely to be an excess number of “free riders” who take advantage of this service but are not in the group for which it is planned.

Table 16.

Transportation modes with costs and percentages.

Transportation mode	Cost per month to FHC ($)	Cost per patient visit ($)	Number of patients per month	Percentage of patients (%)
Golf cart to Centro Plaza	2246.95	5.20	10	6.0
Van to Centro Plaza	0.00	0	0	0.0
Charities	0.00	0	3	2.0
VIA Transit	143.94	4.00	36	22.3
Uber/Lyft	1483.01	34.42	43	26.7
Uber Circulation/Lyft Concierge	1483.01	34.42	43	26.7
Taxi	611.05	75.73	8	5.0
78,207 shuttle	6293.40	0.00	18	11.3
Transportation coordinator cost	3900.00
Total	16,161.37	153.77	161	100.0%

In addition to golf carts and vans, VIA Transit and charities offer personalised rides for patients with documented disabilities. The service from charities is free. Each ride provided by VIA Transit is 2.00, USD so each roundtrip costs 4.00. USD For patients without disabilities living outside of 78,207 and not able to take VIA buses, we expect to offer them the option of taking Uber/Lyft, Uber Circulation/Lyft Concierge or taxis.

Patients with disabilities represent 24.34% of those with transportation difficulties. When allocating patients to the various transportation modes in Table 16, it was assumed that charities will handle only 2% of the patients, leaving VIA Transit to handle the remaining 22.34%. Among the patients who answered our survey, 6.05% have transportation issues but do not have disabilities or live in 78,207. Therefore, we assume that they take VIA buses. When they arrive at Centro Plaza, a golf cart will transport them to the FHC.

Of the pre-scheduled patients 16.51% are from 78,207 and 68.25% of those are without disabilities, which means that shuttle service must be provided for $68.25\mathrm{\%}\times 16.51\mathrm{\%}=11.27\mathrm{\%}$ of them. The remaining patients will be given the option to use Uber/Lyft or Uber Circulation/Lyft Concierge. We assume 5% of those patients do not have a cell phone and so will use a taxi as an alternative. Given that the fare rate of Uber/Lyft and Uber Circulation/Lyft Concierge are same, we evenly assign the remaining patients to these transportation modes. The fares for digital transportation and taxis were calculated based on the average distance and travel time between each area (zip code) outside of 78,207 and the FHC.

Summarising, the total monthly transportation cost is 16,161 USD for the target patient population. Since the potential cost saving derived in Section 5.1 is 31,328, USD the potential net cost reduction per month is:

$$\mathrm{N}\mathrm{e}\mathrm{t}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}=\mathrm{C}\mathrm{o}\mathrm{s}\mathrm{t}\mathrm{s}\mathrm{a}\mathrm{v}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{s}-\mathrm{T}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{t}\mathrm{s}=\mathrm{\$}31,328-\mathrm{\$}16,161=\mathrm{\$}15,167$$

This value assumes golf cart and 78,207 shuttle service is provided. However, from Table 16, we see that while the costs of these options are relatively high, the number of patients served is limited, which calls into question their cost-effectiveness. As an alternative, we propose that the option of running circulating vans in 78,207 be discarded and the money saved spent on Uber/Lyft, Uber Circulation/Lyft Concierge and taxis. To further reduce costs, the golf cart service can also be eliminated due to its low expected utilisation.

Accordingly, transportation costs without the golf carts and 78,207 shuttles are 8,582 USD per month. Since the maximum cost saving is 31,328 USD per month, the budget surplus is now 31,328 USD – 8,582 USD = 22,746 USD which compares favourably to the previous value of 15,167. USD Therefore, instead of golf carts and 78,207 vans, it would be more cost-effective to offer Uber/Lyft service to patients who would ordinarily take the bus or van if it were available. By doing so, we can achieve an additional budget surplus of 22,746 USD – 15,167 USD = 7,579 USD per month.

While the latter option does provide a significant cost advantage, we believe the former should be considered because it provides FHC with an additional level of control over the service and might help them avoid inconclusive results like those found by Chaiyachati et al. (2018).

5.4. Sensitivity analysis on percent reduction in number of missed appointments (α) due to transportation issues

In previous subsections, it was assumed that by better organising transportation for the most vulnerable patients, we could reduce the number of missed appointments by 25% (= α). To see how the potential savings varies for different values of this parameter we now present a sensitivity analysis for α ranging from 20% to 60% in increments of 5%. One motivation for this range is based on logistic regression model in Section 4.3 for the cancellation probability. If all the binary variables are set to 0 and all the quantitative variables are set to their average values, then the cancellation probability equals 0.215 without transportation difficulty and 0.385 with transportation difficulties. The improvement percentage is $\frac{0.385-0.215}{0.385}\times 100=44.2\mathrm{\%}$. Therefore, we want to consider a range of α to cover improvements of this magnitude.

When golf cart and 78,207 shuttle service are provided. Table 17 contains the results for this case. As α increases, the potential cost savings increases linearly with a slope of 1253.14, USD while transportation costs increase linearly with a slope of 150.85, USD and the budget surplus increases linearly with a slope of 1102.29. USD That is, as α increases by 1 percent, the monthly net cost saving increases by 1102.29. USD

Table 17.

Sensitivity analysis for different values of α when golf cart and 78,207 shuttle services are provided.

α	Maximum cost savings ($)	Transportation cost ($)	Budget surplus ($)
20%	25,062.82	15,407.09	9655.73
25%	31,328.53	16,161.37	15,167.16
30%	37,594.24	16,915.64	20,678.60
35%	43,859.94	17,669.91	26,190.03
40%	50,125.65	18,424.18	31,701.46
45%	56,391.35	19,178.46	37,212.90
50%	62,657.06	19,932.73	42,724.33
55%	68,922.77	20,687.00	48,235.76
60%	75,188.47	21,441.28	53,747.19

When golf cart and 78,207 shuttle service are not used. Table 18 provides the results for this case. As α increases, the potential cost savings increases linearly with a slope of 1253.14, USD while transportation costs increase linearly with a slope of 187.28, USD and the budget surplus increases linearly with a slope of 1065.86. USD Therefore, for each 1 percent increase in α the monthly net cost saving will increase by 1065.86. USD

Table 18.

Sensitivity analysis for different values of α when golf cart and 78,207 shuttle services are eliminated.

α	Maximum cost savings ($)	Transportation cost ($)	Budget surplus ($)
20%	25,062.82	7645.54	17,417.29
25%	31,328.53	8581.92	22,746.61
30%	37,594.24	9518.31	28,075.93
35%	43,859.94	10,454.69	33,405.25
40%	50,125.65	11,391.07	38,734.57
45%	56,391.35	12,327.46	44,063.90
50%	62,657.06	13,263.84	49,393.22
55%	68,922.77	14,200.23	54,722.54
60%	75,188.47	15,136.61	60,051.86

6. Conclusions

From the analysis of the survey data, we found that patients with a higher level of transportation difficulty for trips to and from the FHC are more likely to miss their appointments. The FHC data also suggested that the probability of an ED visit within 30 days after the appointment is higher for patients who are no-shows than for those who keep their appointment. The full analysis showed that the maximum net cost savings, after taking into account all the other transportation options, is 15,167 USD per month. If we remove the golf cart and shuttle service options and offer Uber rides to the corresponding patients instead, we can expect to achieve a budget surplus of 22,746 USD per month.

The FHC and related clinics can benefit significantly from various components of our research. The logistic regression models can be applied to estimate the probability of cancellations and the probability of late arrives greater than 15 minutes in similar environments. Based on these probabilities it is possible to derive more efficient appointment schedules that will leave less provider idle time during the day and allow more patients to be seen. Moreover, the analysis of the costs and benefits of providing transportation services can be updated regularly as the patient panel changes, as more detailed data become available, or as more transportation options are provided. Finally, the results of the study suggest that transportation services will significantly reduce the cost incurred by the hospital system due to cancelled appointments. The same analysis can be applied to healthcare systems in other areas of the country that offer scheduled appointments, emergency care and hospital admission to financially disadvantaged patients.

Should others try to conduct a similar analysis, it is important that they pay attention to the potential dissimilarities with this case, such as the financial structure of the system, local public transportation, gas prices, traffic flow, and patients’ financial status. General guidelines for a related study include: (i) develop a VRP model that constructs shuttle routes, (ii) collect and analyse patient data analysis, and investigate the impact of transportation factors on no-shows and late arrivals, (iii) conduct logistic regressions to estimate the probability of a no-show and a late arrival, (iv) estimate the maximum cost savings achievable by offering subsidised transportation services, and (v) calculate the potential costs net cost savings to the healthcare system.

Of course, there are some limitations to any study that is based on sample data and estimated parameter values. With respect to the VRP, it was assumed that all the vans operate throughout the day, which might not be the case in reality. Consequently, travel costs might have been overestimated. Second, even though the VRP can be solved within seconds with a commercial code, runtimes grow exponentially as the problem size increases. When there are more than 60 pickup locations, CPLEX cannot find a feasible solution in 30 minutes. Moreover, even though offering greater transportation services would likely reduce the no-show rate, the number of late arrivals might increase if some patients miss the shuttle. Late arrivals disrupt clinic operations and negatively affect patients who are prompt. Should the shuttle service be adopted, fixing the interarrival time at, say, 45 minutes, means that blocks of patients will be arriving periodically.

To better accommodate these patients, the FHC would have to develop appointments schedules to match the fixed pattern and make sure the “patient arrival interval” of a subset of providers is at least 45 minutes. Appointment scheduling with fixed interarrival times for some patients is a topic for future research as is the development of other predictive algorithms such as decision trees and random forest. Comparisons among these algorithms and our logistic regression approach could be made under different settings. Finally, the cost savings in our calculations are exclusively based on historical data and survey data collected on FHC patients. Costs are similarly derived from FHC and UHS accounting systems. Therefore, there is some hesitation to generalise the results to other locations; however, most safety net healthcare facilities in the U.S. have patient population almost identical to those of the FHC, and although transportation and healthcare costs vary by location, our experience indicates that when broken down by category they are all consistently similar.

Appendix A. Survey Questionnaire for Patients

Each questionnaire had a patient ID that was used to track the respondents’ ED visits and hospital admissions within 30 days of their FHC appointment.

1. Information associated with the scheduled appointment

1. When is your scheduled appointment? Please provide the date and time.

2. When did you start answering the survey? Please provide the date and time.

3. Are you a scheduled patient or a walk-in patient?

   1. Patient characteristics

1. What is your gender?

2. What is your age?

3. How many chronic diseases do you have?

   1. Patient transportation information

1. How did you come to FHC today?

   1. I drove my own car.
   2. I took VIA Transit.
   3. I took regular VIA bus.
   4. I took a free ride from a relative.
   5. I took a paid ride from another person.
   6. I took a paid ride, but a relative paid the cost.
   7. I took a paid ride, but another person paid the cost.
   8. I took Medicaid or Medicare transportation.
   9. I took a taxi/Uber/Lyft.
   10. I walked to FHC.
   11. Other (please add in note).
2. Please rate your transportation difficulty level.

   1. Not at all difficult
   2. Somewhat difficult
   3. Very difficult
   4. Extremely difficult

3. How long did it take you to get to FHC today? Please state in the number of minutes.

4. What is your ambulatory capacity?

   1. Normal
   2. Supported by a person
   3. Cane
   4. Walker
   5. Regular wheelchair
   6. Motorised wheelchair
   7. Amputee
   8. Other (please add in note)
5. Which zip code area do you live in?

   1. Patient visit history with the FHC

1. How many times did you visit FHC in the last 12 months?

2. How many appointments did you cancel with FHC in the last 12 months?

3. How many times did you visit ED in the last 12 months?

4. How many times were you admitted to hospital in the last 12 months?

5. When was your last ED visit?

   1. Patient visit information within 30 days of the appointment

(These questions are not answered by patients. Medical assistants tracked patient visits within 30 days of the appointment.)

1. How many times and when did the patient visit the ED within 30 days of his or her appointment?

2. How many times and when was the patient admitted to hospital within 30 days of his or her appointment?

Appendix B. Figures Depicting Data Analysis Results

Figure 5.

ED visit proportion differences between missing and keeping an appointment.

Figure 6.

Cancellation proportions for grouped transportation difficulty levels.

Figure 7.

Relationship between cancellation proportion and travel time.

Figure 8.

Cancellation proportions with respect to ambulatory capacity.

Funding Statement

This work was supported by a grant from the Department of Family & Community Medicine, UT Health San Antonio, Contract # OSP-2019-384, as well as by the Holly Distinguished Chair, Patient-Centered Medical Home Endowment at the Long School of Medicine.

Disclosure statement

No potential conflict of interest was reported by the author(s).