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Appointment Scheduling with No-Shows and Overbooking
Abstract
We study an overbooking model for scheduling arrivals at a medical facility under no-show behavior, with patients having different no-show probabilities and different weights. The scheduler has to assign the patients to time slots in such a way that she minimizes the expected weighted sum of the patients' waiting times and the doctor's idle time and overtime. We first consider the static problem, where the set of patients to be scheduled and their characteristics are known in advance. We partially characterize the optimal schedule and introduce a new sequencing rule that schedules patients according to a single index that is a function of their characteristics. Then we apply our theoretical results and conclusions from numerical experiments to sequential scheduling procedures. We propose a heuristic solution to the sequential scheduling problem, where requests for appointments come in gradually over time and the scheduler has to assign each patient to one of the remaining slots that are available in the schedule for a given day. We find that the no-show rate and patients' heterogeneity have a significant impact on the optimal schedule and should be taken under consideration.
... Furthermore, the literature on appointment scheduling considers patient priority, offering preferential treatment based on medical conditions or other relevant factors [13]- [16]. ...
... Assigning probabilities based on historical data enhances prediction accuracy, as highlighted by studies such as [13] and [25], thereby significantly improving scheduling systems impacted by patient no-shows. ...
... Similarly to (12), Constraint (13) ensures that if a patient is scheduled in the Mobil-box, the total number of slots assigned to them must equal the number of slots they require, ensuring their complete appointment duration is accommodated. ...
Intelligent management of medical appointments can significantly enhance patient care and reduce health risks. By leveraging disease and attendance propensity, our system aims to minimize the likelihood of patients developing serious conditions due to missed appointments by strategically scheduling those at higher risk. For its development, the methodology begins with the formulation of an optimization model, using assumed values for disease and attendance propensity to establish an initial, efficient scheduling of appointments. Subsequently, machine learning algorithms are applied to patient historical data to obtain more precise and realistic estimates of these propensities, which are then integrated into the model to adjust appointment allocation according to each patient’s individual risk. The results demonstrate that the Intelligent Medical Appointment Management Model significantly outperforms random scheduling, which simulates current real-world practices without the use of an intelligent patient assignment system. Patients scheduled by the intelligent model show higher mean propensities to attend appointments and to develop diseases, ensuring that medical resources are allocated efficiently to those in greatest need. Statistical validation confirms the model’s effectiveness, showing significant differences in scheduling outcomes between the intelligent and random models. This approach highlights the potential to reduce health risks among a group of patients by utilizing both their medical histories and synthetic data for more accurate predictions and effective scheduling.
... For instance, studies by Hassin and Mendel (2008), Liu et al. (2010), Luo et al. (2012), Jiang et al. (2017), Liu (2016), and Kong et al. (2020) have addressed the issue of scheduled patients cancelling or not showing up for their appointments, which can create uncertainty for providers when making schedules. Strategies such as reducing appointment intervals (e.g., Jiang et al., 2017;Kong et al., 2020), overbooking (e.g., Laganga and Lawrence, 2012;Zacharias and Pinedo, 2014) and equal-waiting scheduling system (e.g., Zhang et al., 2022) are proposed to mitigate the negative effects of no-show behaviour. Additionally, managing services for walk-ins without appointments has also attracted the attention of researchers. ...
... A key tradeoff in the literature on appointment scheduling is the balance between maximising service utilisation and minimising patients' waiting. One common approach to address this issue is to assign different cost rates to each factor and then minimise the weighted sum of the expected cost components (e.g., Kaandorp and Koole, 2007;Laganga and Lawrence, 2012;Zacharias and Pinedo, 2014;Wang et al., 2020). It should be noted that all of these studies consider linear waiting costs. ...
... That is, ( ) is the number of show-ups given , which follows some distribution known by the provider. Here, we assume that ( ) is independent of others, meaning that the patient's noshow behaviour is homogeneous and time-independent, which is a common assumption adopted in previous literature (see e.g., Laganga and Lawrence, 2012;Zacharias and Pinedo, 2014;Feldman et al., 2014, etc.). ...
... There is a vast body of literature on appointment scheduling addressing uncertainties in health care delivery systems. Studies [2,3] show that patients exhibit the significant phenomenon of no-shows and that failure to consider no-shows in appointment scheduling results in a waste of resources. Since no-shows can have harmful effects [26], methods such as overbooking [3,5,30,44,49,51], open access [15,31,39], and adjustable appointment intervals [4,29] have been developed to remediate the adverse impacts of no-shows in the literature. ...
... Studies [2,3] show that patients exhibit the significant phenomenon of no-shows and that failure to consider no-shows in appointment scheduling results in a waste of resources. Since no-shows can have harmful effects [26], methods such as overbooking [3,5,30,44,49,51], open access [15,31,39], and adjustable appointment intervals [4,29] have been developed to remediate the adverse impacts of no-shows in the literature. Dantas et al. [47] present a systematic review of no-shows in appointment scheduling. ...
... To characterize the no-show, the heterogeneous no-show is considered in some recent studies. For instance, the no-show is jobdependent in papers [3,28,32], meaning that different patients have different show-up rates. Kong et al. [4] and Zhou and Yue [28] show that customer no-show behavior depends on the time of day. ...
Virtual care serves as a new mode that can divert non-urgent visits from traditional office visits. Whether virtual service can improve the access to medical treatment and reduce the burden of traditional office services, the key issue is to generate efficient appointment schedules with the lowest operation cost. In this paper, considering the uncertainty of time-dependent no-shows and service times, we investigate a multiserver time window allowance appointment scheduling problem, where time window constraints that restricts virtual visits to be served during the particular period are explicitly modeled. We formulate the problem as a stochastic mixed-integer program to optimize decisions of physician allocation and appointment time simultaneously. Based on the sample average approximation, a stabilized Benders decomposition algorithm is developed by incorporating acceleration techniques , such as cut aggregation and feasibility cuts. Numerical results based on real data indicate the effectiveness of the proposed multiserver time window allowance schedules (MTWAS) and algorithm. Comparing with the off-the-shelf solver Gurobi, the developed algorithm demonstrates high performance in terms of computation speed and solution quality. Under different time-dependent no-show patterns of virtual and office visits, the obtained MTWAS perform better than previous solutions in almost all test cases. In addition, we offer useful managerial insights to aid the virtual service provider in making better scheduling decisions.
... Despite the importance of this factor, it has aroused only a few concerns. Another important factor in several papers is no-show (e.g., [18,26,32,33]). This factor disturbs the appointment scheduling similar to the unpunctuality factor. ...
... In most papers, fixed or variable service time with a specified distribution such as normal or lognormal is considered. For example, in the papers of Dogru and Melouk [3], Jiang et al. [10], Zacharias and Pinedo [32], Erdogan and Denton [4], the distribution of lognormal service time is considered. ...
... Besides, they proposed a heuristic solution method. Zacharias and Pinedo [32] investigated an overbooking model for scheduling entry in the US medical center under noshow behavior with different probabilities and weights. Their goal was to allocate patients to time slots by minimizing the weighted sum of patients' waiting time and the doctors idle time and overtime. ...
Appointment scheduling for outpatient services is a challenge in the healthcare sector. For addressing this challenge, most studies assumed that patients’ unpunctuality and the duration of service have constant values or a specific probability distribution function. Consequently, there is a research gap to consider the uncertainty of both patients’ unpunctuality and the duration of service in terms of fuzzy sets. Therefore, this research aims to consider fuzzy values for both unpunctuality and duration of service have to improve an outpatient appointment scheduling (the time interval between two patients) in a referral clinic with the objective of reducing the total weight of waiting time, idle time, and overtime. Four different fuzzy linear programming models and 36 scenarios have been developed based on the show, no-show of patients, single-book, and double-book by using GAMS software. These four models are as follows: (1) probability of no-show equal to zero, (2) probability of no-show equal to 20%, (3) probability of no-show equal to zero and with double-book factor, and (4) probability of no-show equal to 20% and with double-book factor. The results of the first, second, third, and fourth models, respectively, present the scenarios considering 10, 5, 15, and 15 minutes for the time interval between two patients that have the minimum total weight of patient waiting times, physician idle times, and physician overtime. By considering these findings, the investigated referral clinic can improve its appointment system’s performance. Moreover, other similar clinics can apply the proposed model for improving their appointment systems' performance.
... We note two that are particularly relevant to our study. A scheduling problem close to ours has been studied in [21]. The problem they consider is that of determining an optimal schedule for heterogeneous patients in the presence of no shows with the objective of minimizing waiting cost plus idling and overtime cost. ...
... The authors observe that optimal solutions tend to front load (more overbooking towards the beginning of the day). The numerical solutions of [21] also show that some appointment slots should be overbooked but not all, with only up to a couple of patients scheduled per slot. ...
We consider the problem of scheduling appointments for a finite customer population to a service facility with customer no-shows, to minimize the sum of customer waiting time and server overtime costs. Since appointments need to be scheduled ahead of time we refer to this problem as an optimization problem rather than a dynamic control one. We study this optimization problem in fluid and diffusion scales and identify asymptotically optimal schedules in both scales. In fluid scale, we show that it is optimal to schedule appointments so that the system is in critical load; thus heavy-traffic conditions are obtained as a result of optimization rather than as an assumption. In diffusion scale, we solve this optimization problem in the large horizon limit. Our explicit stationary solution of the corresponding Brownian Optimization Problem translates the customer-delay versus server-overtime tradeoff to a tradeoff between the state of a reflected Brownian motion in the half-line and its local time at zero. Motivated by work on competitive ratios, we also consider a reference model in which an oracle provides the decision maker with the complete randomness information. The difference between the values of the scheduling problem for the two models, to which we refer as the stochasticity gap (SG), quantifies the degree to which it is harder to design a schedule under uncertainty than when the stochastic primitives (i.e., the no-shows and service times) are known in advance. In the fluid scale, the SG converges to zero, but in the diffusion scale it converges to a positive constant that we compute.
... Dantas et al. [18] provide a comprehensive review of no-show appointments in healthcare, without focusing on surgery only, to trace the main determinants that drive the phenomenon. Still not in the strict domain of surgical specialties, Zacharias and Pinedo [19] propose a scheduling policy in which no-shows are counterbalanced by overbooking. Berg et al. [20] consider individually tailored no-show probabilities and lever on patient ordering, reducing the expected discomfort of patients by placing the patients that are more likely to result in no-shows towards the end of the patient sequence, if possible. ...
... Constraints (16)- (19) serve the purpose of calculating what is the time at which each OR becomes available after having completed all assigned surgeries, and of identifying which patients are being operated on in each OR j and moment h, respectively. Constraints (20) and (21) identify, for each moment h, OR j and day g, link the starting and ending time of patient i's surgery with variable λ igjh . ...
We consider the problem of scheduling elective surgeries in a Children's Hospital, where disruptions due to emergencies and no-shows may arise. We account for two features that occur in many pediatric settings: i) that it is not uncommon for pediatric patients to fall ill on the very day of their operation and, consequentially, to be unable to undergo surgery and ii) that operating rooms normally reserved for elective surgeries can be used to treat emergency cases. Elective surgeries are scheduled taking into account the time spent on the waiting list and the patient's priority, which considers the severity of their condition and their surgical deadline, generating a nominal schedule. This schedule is optimized in conjunction with a series of back-up schedules: in fact, back-up schedules shall be available in advance so as to guarantee that the operating rooms activity immediately recovers in case of a disruption. We propose an Integer Linear Programming-based approach for the problem. As there is no consolidated data on the features of both emergencies and no show, we enumerate a representative subset of the possible emergency and no-show scenarios and for each of them a back-up plan is designed. The approach reschedules patients in a way that minimizes disruption with respect to the nominal schedule and applies an as-soon-as-possible policy in case of emergencies to ensure that all patients receive timely care. The approach shows to be effective in managing disruptions, ensuring that the waiting list is managed properly, with a balanced mix of urgent and less urgent patients. Therefore, the approach provides an effective solution for scheduling patients in a pediatric hospital, taking into account the unique features of such facilities.
... Overbooking is defined as booking multiple patients in a common time slot, and it works efficiently when the no-show rate is high [11]. Practicing overbooking may reduce the rate of no-shows, improve healthcare utilization, and increase physicians' productivity [11,12]. This may be due, at least in part, to allowing clinics to schedule patients sooner, because longer wait periods between initial contact and appointment date have been consistently associated with no-shows [13,14]. ...
... A welcoming environment, appointment reminders (text messages and/or phone calls), and streamlined admissions were the top reported practices to reduce missed appointments. This is similar to the common strategies identified by other research to reduce no-show rates [5,11,12,25,26]. Such strategies are relatively low cost to implement and maintain compared to provision of support services to address underlying patient barriers that may be impacting ability to access and utilize healthcare services. ...
Background
Healthcare accessibility and utilization are important social determinants of health. Lack of access to healthcare, including missed or no-show appointments, can have negative health effects and be costly to patients and providers. Various office-based approaches and community partnerships can address patient access barriers.
Objectives
(1) To understand provider perceptions of patient barriers; (2) to describe the policies and practices used to address late or missed appointments, and (3) to evaluate access to patient support services, both in-clinic and with community partners.
Methods
Mailed cross-sectional survey with online response option, sent to all Nebraska primary care clinics (n = 577) conducted April 2020 and January through April 2021. Chi-square tests compared rural-urban differences; logistic regression of clinical factors associated with policies and support services computed odds ratios (OR) and 95% confidence intervals (CI).
Results
Response rate was 20.3% (n = 117), with 49 returns in 2020. Perceived patient barriers included finances, higher among rural versus urban clinics (81.6% vs. 56.1%, p =.009), and time (overall 52.3%). Welcoming environment (95.5%), telephone appointment reminders (74.8%) and streamlined admissions (69.4%) were the top three clinic practices to reduce missed appointments. Telehealth was the most commonly available patient support service in rural (79.6%) and urban (81.8%, p =.90) clinics. Number of providers was positively associated with having a patient navigator/care coordinator (OR = 1.20, CI = 1.02–1.40). For each percent increase in the number of privately insured patients, the odds of providing legal aid decreased by 4% (OR = 0.96, CI = 0.92-1.00). Urban clinics were less likely than rural clinics to provide social work services (OR = 0.16, CI = 0.04–0.67) or assist with applications for government aid (OR = 0.22, CI = 0.06–0.90).
Conclusions
Practices to reduce missed appointments included a variety of reminders. Although finances and inability to take time off work were the most frequently reported perceived barriers for patients’ access to timely healthcare, most clinics did not directly address them. Rural clinics appeared to have more community partnerships to address underlying social determinants of health, such as transportation and assistance applying for government aid. Taking such a wholistic partnership approach is an area for future study to improve patient access.
... Homogenous no-show studies use an aggregated rate for all patients; in contrast, heterogeneous research considers individual patient characteristics and uses no-show rates that reflect patient-specific factors. A careful comparison by Zeng et al. (2010) and Zacharias and Pinedo (2014) evaluate both types of strategies. ...
... Huang and Zuniga (2012) present an overbooking policy with heterogeneous no-show rates in which they propose to continue overbooking patients until a threshold no-show rate is realized. Zacharias and Pinedo (2014) consider sequential scheduling of patients with heterogeneous no-show rates, but base the number of overbooked customers on aggregate data. Jiang et al. (2017) provide distributionally-robust models that incorporate the worstcase expected/conditional value-at-risk (CVaR) of waiting, idleness, and overtime costs. ...
Data availability enables clinics to use predictive analytics to improve appointment scheduling and overbooking decisions based on the predicted likelihood of patients missing their appointment (no-shows). Analyzing data using machine learning can uncover hidden patterns and provide valuable business insights to devise new business models to better meet consumers’ needs and seek a competitive advantage in healthcare. The innovative application of machine learning and analytics can significantly increase the operational efficiency of online scheduling. This study offers an intelligent, yet explainable, analytics framework in scheduling systems for primary-care clinics considering individual patients’ no-show rates that may vary for each appointment day and time while generating appointment and overbooking decisions. We use the predicted individual no-show rates in two ways: (1) a probability-based greedy approach to schedule patients in time slots with the lowest no-show likelihood, and (2) marginal analysis to identify the number of overbookings based on the no-show probabilities of the regularly-scheduled patients. We find that the summary measures of profit and cost are considerably improved with the proposed scheduling approach as well as an increase in the number of patients served due to a substantial decrease in the no-show rate. Sensitivity analysis confirms the effectiveness of the proposed dynamic scheduling framework even further.
... The literature on this topic also considers patient priority, giving certain patients preferential treatment for early attention based on their medical condition or other relevant factors [8], [15], [17], [18]. Patient noshows significantly impact scheduling systems. ...
... Patient noshows significantly impact scheduling systems. Assign-ing probabilities based on historical data helps gauge prediction sensitivity [13], [17]. ...
Scheduling medical appointments plays a fundamental role in managing patient flow and ensuring high-quality care. However, no-shows can significantly disrupt this process and affect patient care. To address this challenge, healthcare facilities can adopt different strategies, including overbooking in medical consultations. While this reduces the risk of unused slots, it can generate associated costs and affect the perception of service quality. In this article, we propose an integer linear optimization model that maximizes the expected utility of a medical center, considering the risk of no-shows and overbooking. For this purpose, machine learning is used to estimate the propensity of each patient to attend their medical appointment, using real data from three medical specialties of a hospital. The results of the application demonstrate the model’s ability to assign appointments and perform overbooking efficiently and in an organized manner, implying an improvement in the utility of a medical center and a positive impact on the perception of the quality of care.
... No-shows are disruptive to the clinic and result in inefficiency, including provider underutilization. One of the main strategies to counteract these ill effects is to predictively overbook appointments, which means assigning the same appointment time to more than one patient, with the expectation that some patients will not show up (Zacharias & Pinedo, 2014) (this may be familiar from similar techniques used to increase efficiency in airline bookings). Because the probability of showing up varies significantly from patient to patient, state-of-theart appointment scheduling systems implement a framework known as "predictive overbooking," which employs ML to predict each patient's individual probability of showing up. ...
... It has been shown that efficiency is maximized by placing the patients with the lowest predicted show probabilities into either an overbooked slot (patients d and e in Fig. 3) or in the slot right after an overbooked slot (patient c) (Zacharias & Pinedo, 2014). These appointment slots are undesirable because they are associated with longer waiting times. ...
An Artificial Intelligence algorithm trained on data that reflect racial biases may yield racially biased outputs, even if the algorithm on its own is unbiased. For example, algorithms used to schedule medical appointments in the USA predict that Black patients are at a higher risk of no-show than non-Black patients, though technically accurate given existing data that prediction results in Black patients being overwhelmingly scheduled in appointment slots that cause longer wait times than non-Black patients. This perpetuates racial inequity, in this case lesser access to medical care. This gives rise to one type of Accuracy-Fairness trade-off: preserve the efficiency offered by using AI to schedule appointments or discard that efficiency in order to avoid perpetuating ethno-racial disparities. Similar trade-offs arise in a range of AI applications including others in medicine, as well as in education, judicial systems, and public security, among others. This article presents a framework for addressing such trade-offs where Machine Learning and Optimization components of the algorithm are decoupled. Applied to medical appointment scheduling, our framework articulates four approaches intervening in different ways on different components of the algorithm. Each yields specific results, in one case preserving accuracy comparable to the current state-of-the-art while eliminating the disparity.
... Delays in contrast may retard the service provision and may thus evoke overtime on the day of the service which would reduce or even eliminate the aforementioned effects. Zacharias and Pinedo (2014) for example include no-shows in their overbooking model for appointment planning, Kong et al. (2020) consider time-dependent noshows in their distributionally robust model. See for example Hall (2006) for the consideration of delays. ...
We revisit a service provider’s problem to match supply and demand via an online appointment system such as a doctor in the health care sector. We identify in a survey that an extensive set of available appointments leads to significantly less demand because customers infer a lower quality of the service, as part of an observational learning process. We capture the quality inference effect in a multinomial logit framework and present a Markov decision process for solving the problem of releasing available slots of the appointment system to optimality aiming at maximizing the expected profits. We further evaluate several simple decision rules and provide management insights on which rule to apply under different generic scenarios. Different from current literature, offering all available appointments may lead to suboptimal results when accounting for the quality inference effect. The profit-maximizing strategy then is to offer a subset of the available appointments.
... The risk of this approach for some systems might be unassigned time slots, cancellations and no-shows, resulting in idle resources during the scheduled timeframe. Incorporating overbooking policies into the scheduling system based on the predicted probability of various factors can be intended to mitigate the effects of cancellations and no-shows and reduce the cost of idle resources (Huang and Zuniga 2012;Zacharias and Pinedo 2014;Zeng et al. 2010). The proposed sequencing and scheduling appointment problem and a decentralized and distributed solution framework in this study, address appointment systems with a new approach. ...
An efficient appointment scheduling system has a defining role in controlling wait times and improving the productivity of a large variety of service systems. This study addresses the variability and length of wait times. We reduce them by a form of restricted central intake. We believe it is the first study that expands the appointment scheduling-sequencing model to include multiple sites and incorporate clients’ flexibility and priorities, and solves the large-size scheduling-sequencing problem in a decentralized manner. To make the study more practical, it should be compatible with the multi-stakeholder environment and consider their independency. Furthermore, the problem is large-size as it combines all requests from a geographical region into one stream. Therefore, a decentralized distributed algorithm is applied to solve the amended model. The solution approach is an ADMM-based combination of dual decomposition and augmented Lagrangian relaxation. For the application of this approach, this paper focuses on the outpatient appointment system due to its importance. Early diagnosis and prevention play a crucial role in community health and health system quality. However, patients often experience significant wait times for various diagnostic technologies worldwide. The approach is examined by a real situation of MRI in Ontario, Canada. It has been shown that this study provides better workload balance across hospitals, better responding to demand fluctuations, and alleviates excessive wait times. The computational results also show that the proposed solution method can be satisfactory in terms of accuracy, running time, and applicability. The approach developed in this study can be applicable to many practical applications of timing and sequencing, such as outpatient surgery, other diagnostic testing, home healthcare, and physical and mental therapies, as well as in other service industries beyond healthcare, like public consultations, government services.
... La programación de tareas es una actividad ampliamente requerida por la industria en general. Este problema ha sido trabajado para el desarrollo de proyectos con restricciones de recursos (Brucker et al., 1999;Neumann et al., 2003); asignación de fuerza laboral (Castillo-Salazar et al., 2016;Hung & Emmons, 1993); agenda de citas en sistemas de salud por su impacto en el desempeño de las instalaciones médicas (LaGanga & Lawrence, 2012;Zacharias & Pinedo, 2014), asignación de horarios en instituciones de educación (Vermuyten et al., 2016) y programación de transporte (de Palma & Lindsey, 2001;Wirasinghe, 2002), específicamente control de trenes (Scheepmaker et al., 2017;Wang & Goverde, 2019), control aéreo (Barnhart et al., 1998) y control marítimo. ...
La programación de horarios se clasifica como un problema combinatorio para el que existen múltiples alternativas de solución incluyendo entre ellas la programación entera. Sin embargo, el modelado de reglas operativas y el tamaño de problemas reales hace que su uso no sea común comparado con otras técnicas. El presente artículo propone un modelo de programación entera (IP), que aborda el problema de programación de horarios conformado por variables asociadas con franjas horarias, asignatura y docente asignado. También se incluyen parámetros como número de franjas horarias mínimas y máximas a impartir por el docente, tiempo de franja horaria, disponibilidad de docente, salones disponibles y costo estimado de insatisfacción generado por el horario asignado. En el modelo se integran siete restricciones duras y diecisiete blandas que proporcionan mayor calidad a la solución final de horarios. Se valida el modelo IP con una función objetivo global, en el que se reportan experimentos y resultados obtenidos en instancias reales de la Universidad de la Salle (ULS). El nuevo enfoque de solución ofrece mejoras en los horarios finales, así como la interacción con los usuarios durante su construcción. Finalmente, en las conclusiones del trabajo se discute el diseño y desarrollo de un sistema que brinda soporte a las decisiones, referenciando sugerencias para futuros desarrollos.
... Previous studies have considered a variety of uncertainties in practice that can affect the design of appointment scheduling systems (such as patient no-shows, random service times, and walk-in behavior). Representative work includes, but is not limited to, Zacharias and Pinedo (2014), Mak et al. (2015), Jiang et al. (2017), Zacharias and Pinedo (2017), Wang et al. (2020), Kong et al. (2020) and Zacharias and Yunes (2020). ...
Problem definition: Multi-stage service is common in healthcare. One widely adopted approach to manage patient visits in multi-stage service is to provide patients with visit itineraries, which specify individualized appointment time for each patient at each service stage. We study how to design such visit itineraries. Methodology/results: We develop the first optimization modeling framework to provide each patient an individualized visit itinerary in a tandem (healthcare) service system. Due to interdependence among stages, our model loses those elegant properties (e.g., L-convexity and submodularity) often utilized to solve the classic single-stage models. To address these challenges, we develop two original reformulations. One is directly amenable to off-the-shelf optimization software and the other is a concave minimization problem over a polyhedron shown to have neat structural properties, based on which we develop efficient solution algorithms. In addition to these exact solution approaches, we propose an approximation approach with provable optimality bound and numerically validated performance to serve as an easy-to-implement heuristic. A case study populated by data from the Dana-Farber Cancer Institute shows that our approach makes a remarkable 27% cost reduction over practice on average. Managerial implications: Common approaches used in practice are based on simple adjustments to schedules generated by single-stage models, often assuming deterministic service times. Whereas such approaches are intuitive and take advantage of existing knowledge on single-stage models, they can lead to significant loss of operational efficiency in managing multi-stage services. A well-designed patient visit itinerary which carefully addresses the interdependence among stages can significantly improve patient experience and provider utilization.
... Thus, costs C i = 1 and C o = 1.5 for the provider's idleness and overtime per slot, respectively, are fixed. Like Zacharias and Pinedo (2014), we choose the routine patient's waiting cost per slot C r w ∈ {0.05, 0.10, 0.15} as representative of the different care settings in practice. The value of (C r w = 0.05) is interpreted as the provider-centered care, (C r w = 0.15) as the patient-centered care, and (C r w = 0.10) as between care. ...
Faced with challenges of patient no-show behavior, service delays, and long appointment backlogs in the traditional appointment system, practices shift to some novel alternative models. The carve-out and advanced access models are popular among the alternatives. These models usually open up a moderate to substantial portion of provider’s capacity for patients who call in for the same day appointments. Such arrangements do not assign routine and same-day patients in a slot, and may sometimes experience increased underutilization and long waiting times. On the other hand, a “mixed appointment scheduling” (MAS) policy, which allows for possible assignment of the two patient classes to a slot, may be desirable. Whether the MAS policy would perform better than the carve-out and advanced access policies, and under what business settings are the performance gaps worthwhile, are questions we seek to address. We answer these questions by proposing a new model to determine the optimal appointment scheduling decision for routine and same-day patients under the MAS condition. We show that the priority service discipline, together with uncertainty in demand, destroys the multimodularity of the objective function in the component of the same-day schedule. Thus, the joint problem of determining the optimal routine and same-day schedules becomes hard to solve. We provide various characterizations that allow for the efficient identification of the optimal decision policy. Our numerical results reveal that the MAS policy helps to lower system cost, increase utilization, and reduce service delay when the same-day demand to clinic capacity ratio is within a moderate limit.
... When overbooking is allowed, additional patients are booked to timeslots with a high probability of becoming idle, or booked in overtime, based on the probability that patients cancel or miss their appointment (Zacharias and Pinedo 2014). This way, the probability of resource idle time is minimised, and patients can get earlier access. ...
Patient no-shows and cancellations are a significant problem to healthcare clinics, as they compromise a clinic's efficiency. Therefore, it is important to account for both no-shows and cancellations into the design of appointment systems. To provide additional empirical evidence on no-show and cancellation behaviour, we assess outpatient clinic data from two healthcare providers in the USA and EU: no-show and cancellation rates increase with the scheduling interval, which is the number of days from the appointment creation to the date the appointment is scheduled for. We show the temporal cancellation behaviour for multiple scheduling intervals is bimodally distributed. To improve the efficiency of clinics at a tactical level of control, we determine the optimal booking horizon such that the impact of no-shows and cancellations through high scheduling intervals is minimised, against a cost of rejecting patients. Where the majority of the literature only includes a fixed no-show rate, we include both a cancellation rate and a time-dependent no-show rate. We propose an analytical queuing model with balking and reneging, to determine the optimal booking horizon. Simulation experiments show that the assumptions of this model are viable. Computational results demonstrate general applicability of our model by case studies of two hospitals.
... No-show nedenleri arasında hastalık, vefat, aktarmalı uçuştan geç gelme, havalimanına giderken yaşanan gecikmeler, prosedürler yer almaktadır. Koronavirüs pandemisi tedbirleri kapsamında artan prosedürler nedeniyle uçuş saatine 48 saat kala yapılan iptallerde ve uçuş iptali nedeniyle bozulan bağlantıların sebep olduğu no-showlar artmıştır (Zacharias, Pinedo, 2014). ...
Havayolu işletmeleri hizmetlerine olan talebi daha iyi yönetebilmek için birtakım stratejiler kullanmaktadırlar. 1970’ lerde başlayan deregülasyon ve beraberinde artan rekabet havayolu endüstrisinde gelir yönetiminin önemini ve farkındalığını artırmıştır. Gelir yönetimi uygulamalarının esası yolcu talebinin etkin kontrolüne dayanmaktadır. Bu amaçla yapılan uygulamalardan biri ise bir uçuştaki koltuk sayısından fazla rezervasyon oluşturulması yani overbookingdir. Uçaklar yalnızca kapasitesinde koltuk sayısı ile uçabilmektedir.. Overbooking hem yolcu hem de havayolu üzerinde doğrudan ilgili bir süreçtir. Bu sürecin yönetimi aktarmalı seferden gelebilecek yolcuya, yolcu listesinde uçma önceliği olan yolculara, tek ya da aile olarak seyahat edilmesi durumuna vb. durumlara göre değişiklik göstermektedir. Çalışmanın amacı, havayolu işletmelerinde uygulanan gelir yönetimi uygulamalarının ve overbooking modelinin performans ölçme aracı olan Dengeli Performans Kartı yaklaşımı (Balance Score Card) ile değerlendirmektir. Dengeli Performans Kartı yaklaşımı ile havayolu işletmelerindeki overbooking uygulamalarını literatür taraması ile etraflıca incelemek, içerik araştırması yapmak, havayolu işletmelerinin uygulama çerçevelerini ortaya koymak, overbooking senaryoları ile önerileri geliştirmek, havayolu işletmeleri için gelecekte yapılabilecek gelir yönetimi ve performans değerlendirme çalışmalarına katkıda bulunmak amaçlanmaktadır. Bu çalışmada havayolu işletmelerinde özellikle overbooking modelinin işletmeye katkıları farklı overbook senaryoları ile araştırılmıştır. Problemleri araştırmak amacıyla yapılan bu çalışma nitel araştırma yöntemi kullanılmıştır.
... Their simulation study demonstrated the benefit of such an approach for different clinical environments. With regard to the second overbooking strategy, several works considered patient-specific no-show risk to determine the best slots for overbooking (Daggy et al., 2010;Muthuraman & Lawley, 2008;Srinivas, 2020;Srinivas & Ravindran, 2018;Zacharias & Pinedo, 2014). Muthuraman and Lawley (2008) classified patients according to their probability of no-show and scheduled them sequentially based on a stochastic overbooking policy as well as a myopic rule. ...
This research considers the design of an appointment system (AS) for sequentially scheduling patients in the presence of stochastic factors (no-show and service time uncertainty) and two patient classes. Although studies on AS design are prevalent in literature, most prior works do not consider heterogeneous patient characteristics (e.g., patient-specific no-show risk and service time duration) and sequential patient call-ins together. This research integrates patient-specific uncertainty estimates into the AS design and assess its impact on AS efficiency (measured as the weighted sum of patient waiting time, doctor idle time, and overtime). Specifically, the patients are scheduled as they sequentially call the clinic for appointments based on their risk of no-show and estimated consultation duration. For this setting, a predict-then-schedule framework is proposed. In the predict step, patient-specific no-show risk and service duration are estimated using a machine learning model. The schedule step determines each patient’s appointment time and interval by integrating the predictions with three scheduling decisions (allocation, sequencing, and overbooking). As a result, new sequencing rules are employed. Results indicate that adopting the predict-then-schedule approach in a sequential framework always dominates conventional approaches and could improve the efficiency by 60%.
... Pinedo (2012) summarizes all deterministic/stochastic scheduling models and general purpose procedures of dealing with scheduling problems in practice. Zacharias and Pinedo (2014) study an appointment scheduling problem with customers' noshow behaviour and overbooking. Denton and Gupta (2003) model a two-stage stochastic linear program to optimize appointment times for a sequence of jobs with uncertain duration, and Erdogan and Denton (2013) extend their results to handle no-shows and to the multi-stage dynamic setting. ...
The primary focus of this dissertation is to develop mathematical models and solution approaches for sequential decision-making and optimization under uncertainty, with applications in transportation, logistics, and healthcare-related operations management. In real-world applications, system operators often need to make sequential decisions, that may involve both discrete and continuous variables under data uncertainties. These problems can be modeled by multistage stochastic integer programs (MS-SIP) that are, however, computationally intractable due to the well-known “curse of dimensionality” issue. MS-SIP assume that the distributions of uncertainty parameters are known and one has access to a finite number of samples of the distributions. In contrary to MS-SIP, multistage distributionally robust integer programs (MS-DRIP) make no assumption on distributions of uncertain parameters. Instead, the optimal solutions are sought for the worst-case probability distributions within a family of candidate distributions, namely, the ambiguity set. Compared to multistage sequential decision models, the two-stage counterparts, namely TS-SIP and TS-DRIP, are easier to solve, where planning decisions are made before uncertainty realizes. In this dissertation, we investigate the four models by developing highly efficient and scalable algorithms and recommend the most practical one in the context of designing and operating complex service systems. Specifically, in Chapter 2, we first study MS-DRIP under endogenous uncertainty, where the probability distribution of stage-wise uncertainty depends on the decisions made in previous stages. We derive mixed-integer linear programming or mixed-integer semidefinite programming reformulations for the min-max Bellman equations, and for the latter we show how to obtain upper and lower bounds of the optimal objective value. We employ the Stochastic Dual Dynamic integer Programming method for solving the resultant MS-SIP. Our numerical results based on facility-location instances show the computational efficacy of our approaches and demonstrate the cost effectiveness of considering decision-dependent uncertainty in the dynamic risk-aware optimization framework. In Chapter 3, we examine the gaps between MS-SIP and TS-SIP with facility-location instances. It remains an open question to bound the gap between these two models using risk-averse objective functions, which indicates at least how much benefits we can gain from solving a more complex multistage model. We provide tight lower bounds for the gaps between optimal objective values of risk-averse multistage stochastic facility location models and their two-stage counterparts using expected conditional risk measures. To speed up computation, two approximation algorithms are proposed to efficiently solve risk-averse TS-SIP and MS-SIP. The aforementioned models and approaches can be applied to a wide range of applications, including smart transportation and mobility-as-a-service. In Chapter 4, we first consider integrated vehicle routing and service scheduling problems with either customer-imposed or self-imposed time windows. We propose TS-SIP to optimize vehicle routes and estimated arrival time or time windows to reduce customers’ waiting, vehicle idleness, and overtime. To fulfill real-time arrived service requests, we develop K-means clustering-based algorithms to dynamically update planned routes and schedules, which can quickly compute high-quality solutions for large-scale instances. Finally, in Chapter 5, we extend the TS-SIP for vehicle routing and service scheduling to cover on-demand ride pooling requests, where we dynamically match available drivers to randomly arriving passengers and also decide pick-up and drop-off routes. We design a spatial-and-temporal decomposition scheme and apply Approximate Dynamic Programming (ADP) to improve computation. Our ADP approach reduces the unsatisfied demand rate dramatically compared to other benchmarks that do not incorporate future information or pooling options.
Background. The time between booking an appointment and the appointment taking place, known as lead time, has been identified as a predictor of cancellation and no-show probability in health care settings. Understanding the impact of reducing permissible lead times, that is, the booking horizon, at a policy level in an outpatient care setting is important when mitigating costly cancellation and no-show rates. Few studies have researched this in an observational or experimental setting. Methods. We leveraged longitudinal observational data from an outpatient pediatric rehabilitation organization in Ontario, Canada, consisting of 73,482 visits between June 2021 and October 2023. This organization reduced its booking horizon at the policy level from 12 to 4 wk in February 2023. Using 2 interrupted time-series approaches, we estimated the change in level, slope, and variance of the weekly combined last-minute cancellation and no-show rate associated with the policy change. Results. It is estimated that reducing the booking horizon is associated with an absolute reduction in the weekly rate of last-minute cancellations and no-shows of 1.02% to 1.85% (a relative reduction of 8.07%–15.70%). Furthermore, the variance dropped by 48.18%. Conclusion. Reducing the appointment booking horizon is associated with a significant reduction in the rate and variance of costly last-minute cancellations and no-shows. The reduced variance can also help enable effective usage of strategies such as overbooking for organizations seeking further approaches to mitigating the negative effects of no-shows.
Highlights
This study uses interrupted time-series approaches to assess the effects of reducing the appointment booking horizon at a policy level on last-minute cancellations and no-shows in a pediatric outpatient care setting. Reducing the permissible booking horizon from up to 3 mo to up to 4 wk is associated with a significant reduction in the rate of last-minute cancellations and no-shows. The shortened booking horizon policy is associated with a significant drop in the variance of last-minute cancellations and no-show rates, which is valuable in settings where overbooking occurs.
Appointment scheduling (AS) plays a crucial role in outpatient clinic management. Traditional methods involve patient grouping using pre-defined rules and scheduling based on these groups. However, pre-defined rules may not adequately capture the heterogeneity in patients’ service times (i.e., consultation duration). Advanced machine learning (ML) methods can address individual-level heterogeneity but pose challenges for practical scheduling. To strike a balance, we propose a data-driven AS decision support system, Cluster-Predict-Schedule (CPS), integrating both supervised and unsupervised ML for efficient patient grouping and scheduling. The novelty of CPS lies in its adaptability to service time heterogeneity through a data-driven approach, determining patient groups based on data rather than pre-defined rules. Additionally, CPS includes a generic and efficient algorithm for generating appointment templates adaptable to any number of patient groups. Our system’s efficacy is demonstrated using a real-world dataset. Evaluated by the weighted sum of patient wait times, physician idle time, and overtime, CPS achieves up to 15.0% cost reduction compared to the FCFA (first-call, first-appointment) scheme and over 4.7% savings against the common New/Return classification with traditional sequencing candidate (TSC) rules. In addition, CPS enhances outpatient operational efficiency without compromising fairness.
Consider a single server that should serve a given number of customers during a fixed period. The Appointment Scheduling Problem (ASP) determines the schedule of planned appointments that minimizes some cost function that accounts for both the cost of idle times and the cost of waiting. When service time distributions are fully specified, the ASP presents a much investigated computationally challenging stochastic program. When service time distributions are only partially specified, one can apply distributionally robust optimization (DRO) to find the schedule that minimizes costs in worst-case circumstances. We assume that only the mean, mean absolute deviation and range of the service times are known, and develop a DRO method that finds the optimal (minimax) schedule. For independent service times, the minmax problem becomes nonlinear and difficult, if not impossible, to solve exactly. Existing DRO methods for ASP with partial information (such as mean and variance) therefore consider relaxations that allow correlations between service times. Such relaxations have major repercussions, as the worst-case scenario will then be highly correlated. Our method thus deals with independent service times and finds the robust schedule as the solution to a linear program. We identify several new structural features of optimal robust schedules. We also apply the method to model extensions including sequencing and alternative objective functions.
Patient no-shows pose a significant challenge in healthcare operations, disrupting appointment schedules and affecting overall efficiency. Effectively addressing the challenge of reducing patient no-shows is crucial for outpatient service management. This study evaluates how the design of appointment systems affects patient no-show behavior. Using a difference-in-differences (DID) methodology, we analyze the effects of two appointment system update events—technical support and information provision—at a Chinese hospital. Our analysis reveals that technical support and information provision are associated with average reductions of 22.40% and 10.91%, respectively. To investigate the mechanisms behind these effects, we conduct a randomized controlled experiment with 233 participants. Our findings reveal that perceived effort and credibility mediate the relationship between information provision and patient no-shows. However, for technical support, only perceived credibility acts as a mediator. This study provides valuable insights for healthcare operations, offering design recommendations to address no-show behavior in hospital appointment systems.
This article studies an appointment scheduling problem where a service provider dynamically receives appointment requests from a random number of customers. By leveraging the randomness of the number of potential customers, we develop a nonsequential appointment scheduling policy as an alternative to the conventional first-come-first-served (FCFS) policy. This allows for more flexibility in managing appointment scheduling. To calculate the optimal policy, we develop a branch-and-bound algorithm in which the lower bound is estimated using multiple approaches, such as optimality conditions, dynamic programming for calculating FCFS policy, and the shortest path reformulation. Through numerical studies, we observe that nonsequential appointment scheduling is particularly advantageous in systems characterized by highly fluctuating customer numbers or low congestion. In such cases, leaving gaps between appointments for potential future arrivals proves to be a more appropriate strategy. We also evaluate the performance of heuristics proposed in prior literature and provide insights into situations where these heuristics can be effectively applied.
Problem definition: Multistage service is common in healthcare. One widely adopted approach to manage patient visits in multistage service is to provide patients with visit itineraries that specify personalized appointment time for each patient at each service stage. We study how to design such visit itineraries. Methodology/results: We develop the first optimization modeling framework to provide each patient with a personalized visit itinerary in a tandem (healthcare) service system. Due to interdependencies among stages, our model loses those elegant properties (e.g., L-convexity and submodularity) often utilized to solve the classic single-stage models. To address these challenges, we develop two original reformulations. One is directly amenable to off-the-shelf optimization software, and the other is a concave minimization problem over a polyhedron shown to have neat structural properties, based on which we develop efficient solution algorithms. In addition to these exact solution approaches, we propose an approximation approach with a provable optimality bound and numerically validated performance to serve as an easy-to-implement heuristic. A case study populated by data from the Dana-Farber Cancer Institute shows that our approach makes a remarkable 28% cost reduction over practice on average. Managerial implications: Common approaches used in practice are based on simple adjustments to schedules generated by single-stage models, often assuming deterministic service times. Whereas such approaches are intuitive and take advantage of existing knowledge of single-stage models, they can lead to significant loss of operational efficiency in managing multistage services. A well-designed patient visit itinerary that carefully addresses the interdependencies among stages can significantly improve patient experience and provider utilization.
History: This paper was selected for Fast Track in the M&SOM Journal from the 2022 MSOM Healthcare SIG Conference.
Funding: The work of the last two authors was supported in part by the National Natural Science Foundation of China [Grants 71931008 and 72001220].
Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2023.0134 .
Purpose/Objectives
This study had a twofold outcome. The first aim was to develop an efficient, machine learning (ML) model using data from a dental school clinic (DSC) electronic health record (EHR). This model identified patients with a high likelihood of failing an appointment and provided a user‐friendly system with a rating score that would alert clinicians and administrators of patients at high risk of no‐show appointments. The second aim was to identify key factors with ML modeling that contributed to patient no‐show appointments.
Methods
Using de‐identified data from a DSC EHR, eight ML algorithms were evaluated: simple decision tree, bagging regressor classifier, random forest classifier, gradient boosted regression, AdaBoost regression, XGBoost regression, neural network, and logistic regression classifier. The performance of each model was assessed using a confusion matrix with different threshold level of probability; precision, recall and predicted accuracy on each threshold; receiver‐operating characteristic curve (ROC) and area under curve (AUC); as well as F1 score.
Results
The ML models agreed on the threshold of probability score at 0.20–0.25 with Bagging classifier as the model that performed best with a F1 score of 0.41 and AUC of 0.76. Results showed a strong correlation between appointment failure and appointment confirmation, patient's age, number of visits before the appointment, total number of prior failed appointments, appointment lead time, as well as the patient's total number of medical alerts.
Conclusions
Altogether, the implementation of this user‐friendly ML model can improve DSC workflow, benefiting dental students learning outcomes and optimizing personalized patient care.
Patient no-shows are a significant problem in health care which leads to increased cost, inefficient utilization of capacity, and discontinuity in care. With the existing available patient appointment history, the research aims to predict the appointment no-shows of patients in a public hospital using the method of logistic regression. Based on characteristics of the appointment history data, the features are divided into demographic variables, appointment characteristics, and clinical characteristics. By considering these features and its multiple combinations, a fivefold cross-validation technique is used to choose the best feature combination for the best prediction model. From the analysis, it is found that appointment characteristics give better predictions. The study tested the model with appointment characteristics, and the performances are evaluated using accuracy, specificity, precision, recall, and F1-score. The model is evaluated using receiver operating characteristic curve and precision-recall curve. Hospitals can employ the model to predict appointment no-shows and implement mitigation strategies based on the outcome of the prediction.KeywordsHospital managementLogistic regressionAppointment no-showsPredictive analyticsMachine learning
Ternary Quantum-Dot Cellular Automata (TQCA) is a developing nanotechnology that guarantees lower power utilization and littler size, with quicker speed contrasted with innovative transistor. In this article, we are going to propose a novel architecture of level-sensitive scan design (LSSD) in TQCA. These circuits are helpful for the structure of numerous legitimate and useful circuits. Recreation consequences of proposed TQCA circuits are developed by utilizing such QCA designer tool. In realization to particular specification, we need to find the parameter values by using Schrodinger equation. Here, we have optimized the different parameter in the equation of Schrodinger.KeywordsTQCALSSDQuantum phenomenon for combinational as well as sequential logicJ-K flip-flopSchrodinger equationEnergyPower
Healthcare providers have long grappled with patients not showing up for their scheduled medical appointments; such no-shows lead to wasted resources and longer wait times for other patients. However, new operations research offers a promising solution to this problem. The study finds that using text message reminders that include an additional line of text indicating a potentially long wait for the next available appointment can significantly reduce no-shows by a factor of 28.6%. The intervention, called waits framing, was found to be more effective among patients who were more sensitive to wait times and when the information in the message was novel and credible. The study also uncovered the mechanism underlying the intervention. Specifically, the waits framing messages increased the perceived cost of missing an appointment, leading to a reduction in queue abandonment. This study provides insights into how behavioral science can improve service operations and help tackle challenges in healthcare delivery.
Problem definition: This paper studies an appointment system where a finite number of customers are scheduled to arrive in such a way that (1) the expected waiting time of each individual customer cannot exceed a given threshold; and (2) the appointment times are set as early as possible (without breaking the waiting time constraint). Methodology/results: First, we show that, under the service-level constraint, a prospective schedule can be obtained from a sequential scheduling approach. In particular, we can schedule the appointment time of the next customer based on the scheduled appointment times of the previous customers. Then, we use a transient queueing-analysis approach and apply the theory of majorization to analytically characterize the structure of the optimal appointment schedule. We prove that, to keep the expected waiting time of each customer below a certain threshold, the minimum inter-appointment time required increases with the arrival sequence. We further identify additional properties of the optimal schedule. For example, a later arrival has a higher chance of finding an empty system and is more likely to wait less than the duration of his expected service time. We show the convergence of the service-level-constrained system to the D/M/1 queueing system as the number of arrivals approaches infinity and propose a simple, yet practical, heuristic schedule that is asymptotically optimal. We also develop algorithms that can help system managers determine the number of customers that can be scheduled in a fixed time window. We compare the service-level-constrained appointment system with other widely studied systems (including the equal-space and cost-minimization systems). We show that the service-level-constrained system leads to a lower upper bound on each customer’s waiting time; ensures a fair waiting experience among customers; and performs quite well in terms of system overtime. Finally, we investigate various extended settings of our analysis, including customer no-shows; mixed Erlang service times; multiple servers; and probability-based service-level constraints. Managerial implications: Our results provide guidelines on how to design appointment schedules with individual service-level constraints. Such a design ensures fairness and incorporates the threshold-type waiting perception of customers. It is also free from cost estimation and can be easily applied in practice. In addition, under the service-level-constrained appointment system, customers with later appointment times can have better waiting experiences, in contrast to the situation under other commonly studied systems.
Funding: Z. Yan was partly supported by a Nanyang Technological University startup grant; the Ministry of Education Academic Research Fund Tier 1 [Grant RG17/21] and Tier 2 [Grant MOE2019-T2-1-045]; and Neptune Orient Lines [Fellowship Grant NOL21RP04].
Supplemental Material: The online supplement is available at https://doi.org/10.1287/msom.2022.1159 .
More and more firms use connected technologies to reshape fundamentally the way in which they interact with their customers. Rather than having few episodic interactions, companies are trying to create a continuous relationship with their customers that reduces friction and allows companies to anticipate the needs of their customers. In this chapter, we discuss new business models enabled by this development. We do so by reviewing some of the literature on after-sales services in general and some of Morris Cohen’s pioneering research in particular. We then extend this prior work by articulating a framework of “Connected Strategy” in the form of a taxonomy of four connected customer experiences. Finally, we apply our Connected Strategy framework to the domain of healthcare delivery.KeywordsConnected strategyBusiness model innovationHealthcare deliveryDigital customer experienceService design
We study how to admit and schedule heterogeneous patients by using simple, interpretable, yet effective policies when capacity is scarce, no‐show behavior is patient‐ and time‐dependent, service duration and reward are deterministic but patient‐dependent, and overtime is costly. Our work is motivated by the aforementioned operational challenges that typically face adopters of new technologies in the healthcare sector. We anchor our study on a partnership with the proton therapy center of Massachusetts General Hospital (MGH), which offers a new radiation technology for cancer patients. We formulate the problem as a nonlinear integer optimization problem. However, as the solution to this formulation lacks both tractability and interpretability, to be relevant to practice, we limit our study to simple and interpretable policies. In particular, we propose a simple index‐based rule and derive analytical performance guarantees for it. We also calibrate our model using empirical data from our partner hospital, and conduct a series of experiments to evaluate the performance of our proposed policy under practical circumstances. The analytical performance guarantees and our numerical experiments demonstrate (a) the strong performance of the proposed policies, and (b) their robustness to various practical considerations (e.g., to potential misspecification of no‐show probabilities). Our results show that our proposed policy, despite being a simple and interpretable index‐based rule, is capable of improving performance by about 20% at an organization such as MGH, and of delivering results that are not far from being optimal across a wide range of parameters that might vary between organizations. This suggests that the proposed policy can be viewed as an effective “one‐fits‐all” capacity allocation rule that can be used in a variety of environments in which operational challenges such as no‐shows and overtime costs need to be navigated using simple and interpretable rules.
This study reviews the research on planning and scheduling in healthcare published in Production and Operations Management (POM) and other prestigious operations management journals over the past 30 years, identifies trending topics, and discusses opportunities for future research. Specifically, we categorize the capacity planning literature into three problem contexts: a) hospital units and operating rooms, b) outpatient clinics, and c) broader healthcare networks. We further classify the scheduling literature into a) appointment scheduling, b) surgery and workforce scheduling, and c) recurring and integrated scheduling, with several subtopics that have attracted attention in the literature. Afterward, we demonstrate the seven emerging trends in healthcare (i.e., collaborative ecosystem and cooperative competition strategy, expanded reach of virtual care, applications of artificial intelligence in healthcare, patient engagement and personalized care, shifting to value‐based care, working toward health equity, and aging populations) and list relevant research opportunities and their related departments of POM journal. We conclude with a call for multidisciplinary perspectives to deliver coherent solutions that incorporate the complex contemporary issues harmoniously. This article is protected by copyright. All rights reserved
We consider the tactical problem of optimizing the capacity of a block appointment system with time-varying no-show and random service duration. The problem characteristics are motivated by a real-life case study of two outpatient specialty clinics in the US, where time-of-day variation in appointment attendance was observed. We aim to determine the block size (patients to be assigned in each time period) such that the weighted sum of block-wise waiting- and idle-times (or total cost) are minimized. First, we consider optimizing the capacity of a single block appointment system (SBAS) as a special case of the inverse newsvendor problem, and develop an analytical closed-form solution under the assumption of normally distributed service time. Also, a stochastic integer programming (SIP) model is developed to solve SBAS for any service time distribution. Subsequently, the SIP model is extended to determine the block size of the variable-sized multi-block appointment system (VSMBAS) by treating it as a sequential inverse newsvendor problem. Owing to the computational complexity of the SIP, we employ sample average approximation to estimate the expected total cost. Numerical studies considered several realistic clinic settings, and the results demonstrated that integrating time-varying no-shows for block size determination will considerably improve schedule efficiency as opposed to ignoring it. We also found the cost ratios (waiting-time to idle-time penalty), service time variation, and no-show pattern have a substantial influence on the block size of VSMBAS. Finally, we provide several practical implications based on our analysis.
Adaptive Patient Flow Management
Appointment scheduling has significant clinical, operational, and economical impact on healthcare systems. An informed scheduling strategy that can effectively match patient demand and service capacity dynamically is vital for the business of medical providers, quality of care, and patient satisfaction. By regulating patient flow via an appointment system, healthcare providers can mitigate arrival process variability and improve operational performance. The simultaneous consideration of appointment day (interday scheduling) and time of day (intraday scheduling) in dynamic scheduling decisions is an important theoretical and practical problem that has remained open because of its stochastic nature, complex structure, and large dimensionality. Zacharias et al. (2022) fill this critical gap in the literature. They introduce a novel dynamic programming framework, designed with the intention of bridging two independently established streams of literature, and to leverage their latest advances in tackling the joint problem. They advance the theory of the field to provide a rigorous and practically implantable solution.
We present a novel framework to characterize the probability that an offered appointment with k‐days' access delay will be booked and subsequently attended by a patient. We refer to this probability as the “probability of realization” of an offered appointment, and demonstrate how empirical characterizations of this measure can be used to identify improved policies for managing patient demand and enabling the most intentional use of clinical care resources. We consider the estimation problem in the context of new patients looking to establish care at a clinic, and offer a model of patient responses during an appointment scheduling encounter with an agent. We define different cases of data availability and demonstrate the effectiveness of the framework in each case to ensure generalizability. We first demonstrate the accuracy of estimations using simulated data, and then highlight behavioral differences between different patient groups using real‐life transactional data from two clinical departments. Finally, we demonstrate a practical use case for the obtained realization probabilities by showing that they can be reliably used as input to a novel time windows‐based patient prioritization protocol that allows effective management of demand from different classes by explicitly considering wait sensitivities of patients.
No-shows of customers are common in many appointment-based service systems that often cause severe idling and underutilization of capacity. This paper studies a new strategy that pre-charges customers for a deposit to lock up appointments, which differs from previous studies on overbooking strategies that reserve more appointments than its capacity to hedge against the loss of revenue due to customer no-shows. Moreover, the new strategy is more favorable in practice because it improves the satisfaction and loyalty of customers in the long-term running. We analyze two versions of the pre-charge strategy including (i) offering a partial or full refund for cancellation (to-refund policy), and (ii) offering no refund for cancellation (no-refund policy). We consider two settings of the appointment scheduling problem including a static setting in which appointment decisions are made at the beginning of the day, and a dynamic setting in which appointment decisions are made sequentially within the day. The two settings are formulated by an analytic model and a Markov decision process model, respectively. The analytical results reveal that the pre-charging strategy is consistently more profitable than the overbooking strategy even for a fully refundable deposit case, and the to-refund policy can be more profitable than the no-refund policy. Moreover, we show that the profitability of the to-refund policy continues to hold in a dynamic setting in which the service capacity is limited, the arrivals are dynamic, and the report of cancellation is uncertain. Finally, we implement a series of numerical experiments based on real data in the context of managing patient appointments for a large clinic and demonstrate the managerial implications.
The appointment scheduling problem (ASP) studies how to manage patient arrivals to a healthcare system to improve system performance. An important challenge occurs when some patients may not show up for an appointment. Although the ASP is well studied in the literature, the vast majority of the existing work does not consider the well-observed phenomenon that patient no-show is influenced by the appointment time, the usual decision variable in the ASP. This paper studies the ASP with random service time (exogenous uncertainty) with known distribution and patient decision-dependent no-show behavior (endogenous uncertainty). This problem belongs to the class of stochastic optimization with decision-dependent uncertainties. Such problems are notoriously difficult as they are typically nonconvex. We propose a stochastic projected gradient path (SPGP) method to solve the problem, which requires the development of a gradient estimator of the objective function—a nontrivial task, as the literature on gradient-based optimization algorithms for problems with decision-dependent uncertainty is very scarce and unsuitable for our model. Our method can solve the ASP problem under arbitrarily smooth show-up probability functions. We present solutions under different patterns of no-show behavior and demonstrate that breaking the assumption of constant show-up probability substantially changes the scheduling solutions. We conduct numerical experiments in a variety of settings to compare our results with those obtained with a distributionally robust optimization method developed in the literature. The cost reduction obtained with our method, which we call the value of distribution information, can be interpreted as how much the system performance can be improved by knowing the distribution of the service times, compared to not knowing it. We observe that the value of distribution information is up to 31% of the baseline cost, and that such value is higher when the system is crowded or/and the waiting time cost is relatively high.
Summary of Contribution: This paper studies an appointment scheduling problem under time-dependent patient no-show behavior, a situation commonly observed in practice. The problem belongs to the class of stochastic optimization problems with decision-dependent uncertainties in the operations research literature. Such problems are notoriously difficult to solve as a result of the lack of convexity, and the present case requires different techniques because of the presence of continuous distributions for the service times. A stochastic projected gradient path method, which includes the development of specialized techniques to estimate the gradient of the objective function, is proposed to solve the problem. For problems with a similar structure, the algorithm can be applied once the gradient estimator of the objective function is obtained. Extensive numerical studies are presented to demonstrate the quality of the solutions, the importance of modeling time-dependent no-shows in appointment scheduling, and the value of using distribution information about the service times.
The problem of no-shows (patients who do not arrive for scheduled appointments) is particularly significant for health care clinics, with reported no-show rates varying widely from 3% to 80%. No-shows reduce revenues and provider productivity, increase costs, and limit patient access by reducing effective clinic capacity. In this article, we construct a flexible appointment scheduling model to mitigate the detrimental effects of patient no-shows, and develop a fast and effective solution procedure that constructs near-optimal overbooked appointment schedules that balance the benefits of serving additional patients with the potential costs of patient waiting and clinic overtime. Computational results demonstrate the efficacy of our model and solution procedure, and connect our work to prior research in health care appointment scheduling.
This study introduces a universal “Dome” appointment rule that can be parameterized through a planning constant for different clinics characterized by the environmental factors—no-shows, walk-ins, number of appointments per session, variability of service times, and cost of doctor's time to patients’ time. Simulation and nonlinear regression are used to derive an equation to predict the planning constant as a function of the environmental factors. We also introduce an adjustment procedure for appointment systems to explicitly minimize the disruptive effects of no-shows and walk-ins. The procedure adjusts the mean and standard deviation of service times based on the expected probabilities of no-shows and walk-ins for a given target number of patients to be served, and it is thus relevant for any appointment rule that uses the mean and standard deviation of service times to construct an appointment schedule. The results show that our Dome rule with the adjustment procedure performs better than the traditional rules in the literature, with a lower total system cost calculated as a weighted sum of patients’ waiting time, doctor's idle time, and doctor's overtime. An open-source decision-support tool is also provided so that healthcare managers can easily develop appointment schedules for their clinical environment.
This paper provides a comprehensive survey of research on appointment scheduling in outpatient services. Effective scheduling systems have the goal of matching demand with capacity so that resources are better utilized and patient waiting times are minimized. Our goal is to present general problem formulation and modeling considerations, and to provide taxonomy of methodologies used in previous literature. Current literature fails to develop generally applicable guidelines to design appointment systems, as most studies have suggested highly situation-specific solutions. We identify future research directions that provide opportunities to expand existing knowledge and close the gap between theory and practice.
Appointment scheduling systems are used by primary and specialty care clinics to manage access to service providers, as well as by hospitals to schedule elective surgeries. Many factors affect the performance of appointment systems including arrival and service time variability, patient and provider preferences, available information technology and the experience level of the scheduling staff. In addition, a critical bottleneck lies in the application of Industrial Engineering and Operations Research (IE/OR) techniques. The most common types of health care delivery systems are described in this article with particular attention on the factors that make appointment scheduling challenging. For each environment relevant decisions ranging from a set of rules that guide schedulers to real-time responses to deviations from plans are described. A road map of the state of the art in the design of appointment management systems is provided and future opportunities for novel applications of IE/OR models are identified.
Patients who frequently miss or do not show for their scheduled psychotherapy appointments create administrative and clinical difficulties, and may not be receiving effective treatment. Prior research has predominately focused on either identifying demographic and administrative factors related to patient no-show rates or evaluating the effectiveness of administrative procedures for reducing no-shows. This paper attempts to identify rates of missed appointments in clinical practice and explore more specific clinical process factors related to patient no-shows. Psychotherapists (N = 24) and their patients (N = 542) in the outpatient department of a public safety-net hospital were surveyed to examine how frequently patients missed scheduled psychotherapy appointments and for what reasons. Findings indicate that the majority of missed appointments were accounted for by patients with occasional absences (approx. 1 per month), while only a small percentage of patients missed appointments with high frequency. Patients missed their psychotherapy appointments for a number of reasons, including clinical symptoms, practical matters, motivational concerns, and negative treatment reactions.
Clinical overbooking is intended to reduce the negative impact of patient no-shows on clinic operations and performance. In
this paper, we study the clinical scheduling problem with overbooking for heterogeneous patients, i.e. patients who have different
no-show probabilities. We consider the objective of maximizing expected profit, which includes revenue from patients and costs
associated with patient waiting times and physician overtime. We show that the objective function with homogeneous patients,
i.e. patients with the same no-show probability, is multimodular. We also show that this property does not hold when patients
are heterogeneous. We identify properties of an optimal schedule with heterogeneous patients and propose a local search algorithm
to find local optimal schedules. Then, we extend our results to sequential scheduling and propose two sequential scheduling
procedures. Finally, we perform a set of numerical experiments and provide managerial insights for health care practitioners.
We consider the problem of determining optimal appoint- ment schedule for a given sequence of jobs (e.g., medical procedures) on a single processor (e.g., operating room, ex- amination facility), to minimize the expected total underage and overage costs when each job has a random processing duration given by a joint discrete probability distribution. Simple conditions on the cost rates imply that the objective function is submodular and L-convex. Then there exists an optimal appointment schedule which is integer and can be found in polynomial time. Our model can handle a given due date for the total processing (e.g., end of day for an oper- ating room) after which overtime is incurred and, no-shows and emergencies.
This paper compares two types of appointment-scheduling policies for single providers: traditional and open-access. Under traditional scheduling, each of a specified number of patients per day is booked well in advance, but may not show up for his or her appointment. Under open-access scheduling, a random number of patients call in the morning to make an appointment for that same day. Thus the number of patient arrivals will be random, for different reasons, under both policies. We find that the open-access schedule will significantly outperform the traditional schedule--in terms of a weighted average of patients' waiting time, the doctor's idle time, and the doctor's overtime--except when patient waiting time is held in little regard or when the probability of no-shows is quite small.
Almost all research in appointment scheduling has focused on the trade-off between customer waiting times and server idle times. In this paper, we present an observation-based method for estimating the relative cost of the customer waiting time, which is a critical parameter for finding the optimal appointment schedule.
This paper develops a framework and proposes heuristic dynamic policies for scheduling patient appointments, taking into account the fact that patients may cancel or not show up for their appointments. In a simulation study that considers a model clinic, which is created using data obtained from an actual clinic, we find that the heuristics proposed outperform all the other benchmark policies, particularly when the patient load is high compared with the regular capacity. Supporting earlier findings in the literature, we find that the open access policy, a recently proposed popular scheduling paradigm that calls for "meeting today's demand today," can be a reasonable choice when the patient load is relatively low.
Many primary care offices and other medical practices regularly experience long backlogs for appointments. These backlogs are exacerbated by a significant level of last-minute cancellations or "no-shows," which have the effect of wasting capacity. In this paper, we conceptualize such an appointment system as a single-server queueing system in which customers who are about to enter service have a state-dependent probability of not being served and may rejoin the queue. We derive stationary distributions of the queue size, assuming both deterministic as well as exponential service times, and compare the performance metrics to the results of a simulation of the appointment system. Our results demonstrate the usefulness of the queueing models in providing guidance on identifying patient panel sizes for medical practices that are trying to implement a policy of "advanced access." Subject classifications: health care; queues: applications. Area of review: Special Issue on Operations Research in Health Care.
In this paper we formulate a stochastic overbooking model and develop an appointment scheduling policy for outpatient clinics. This schedule is constructed for a single service period partitioned into time slots of equal length. A clinic scheduler must provide each calling patient with an appointment time before the patient's call terminates. Once an appointment is added to the schedule, it cannot be changed. Each callling patient has a no-show probability, and overbooking is used to compensate for patient no-shows. The scheduling objective captures patient waiting time, staff overtime, and patient revenue.
The current climate in the health care industry demands efficiency and patient satisfaction in medical care delivery. These two demands intersect in scheduling of ambulatory care visits. This paper uses patient and doctor-related measures to assess ambulatory care performance and investigates the interactions among appointment system elements and patient panel characteristics. Analysis methodology involves simulation modeling of clinic sessions where empirical data forms the basis of model design and assumptions. Results indicate that patient sequencing has a greater effect on ambulatory care performance than the choice of an appointment rule, and that panel characteristics such as walk-ins, no-shows, punctuality and overall session volume, influence the effectiveness of appointment systems.
In this paper optimal outpatient appointment scheduling is studied. A local search procedure is derived that converges to the optimal schedule with a weighted average of expected waiting times of patients, idle time of the doctor and tardiness (lateness) as objective. No-shows are allowed to happen. For certain combinations of parameters the well-known Bailey-Welch rule is found to be the optimal appointment schedule.
Outpatient appointment system design is a complex problem because it involves multiple stakeholders, sequential booking process, random arrivals, no-shows, varying degrees of urgency of different patients’ needs, service time variability, and patient and provider preferences. Clinics use a two-step process to manage appointments. In the first step, which we refer to as the clinic profile setup problem, service providers’ daily clinic time is divided into appointment slots. In the second step, which we refer to as the appointment booking problem, physicians’ offices decide which available slots to book for each incoming request for an appointment. In this chapter, we present formulations of mathematical models of key problems in the area of appointment system design. We also discuss the challenges and complexities of solving such problems. In addition, summaries of prior research, particularly advanced models related to the examples shown in this chapter are also presented.
Objectives:
To determine appointment failure rates in pediatric resident continuity clinics nationally, and to identify characteristics of clinics with respect to factors that may affect appointment failure rates.
Design:
A one-page questionnaire administered via facsimile machine to pediatric residencies' continuity clinic directors.
Results:
Of 200 continuity clinic directors, 160 (80%) returned the survey. The mean no-show percentage was 30.9%, with a range of 3% to 80%. Among the factors studied, only mode of payment emerged as an independent predictor.
Conclusions:
Appointment failure is a substantial problem in pediatric resident continuity clinics, which needs attention if resident learning, patient care, and clinic efficiency are to be optimized.(Arch Pediatr Adolesc Med. 1995;149:693-695)
Outpatient health care service providers face increasing pressure to improve the quality of their service through effective scheduling of appointments. In this paper, a simulation optimization approach is used to determine optimal rules for a stochastic appointment scheduling problem. This approach allows for the consideration of more variables and factors in modeling this system than in prior studies, providing more flexibility in setting policy under various problem settings and environmental factors. Results show that the dome scheduling rule proposed in prior literature is robust, but practitioners could benefit from considering a flatter, “plateau-dome.” The plateau–dome scheduling pattern is shown to be robust over many different performance measures and scenarios. Furthermore, because this is the first application of simulation optimization to appointment scheduling, other insights are gleaned that were not possible with prior methodologies.
The problem of patient no-shows (patients who do not arrive for scheduled appointments) is significant in many health care settings, where no-show rates can vary widely. No-shows reduce provider productivity and clinic efficiency, increase health care costs, and limit the ability of a clinic to serve its client population by reducing its effective capacity. In this article, we examine the problem of no-shows and propose appointment overbooking as one means of reducing the negative impact of no-shows. We find that patient access and provider productivity are significantly improved with overbooking, but that overbooking causes increases in both patient wait times and provider overtime. We develop a new clinic utility function to capture the trade-offs between these benefits and costs, and we show that the relative values that a clinic assigns to serving additional patients, minimizing patient waiting times, and minimizing clinic overtime will determine whether overbooking is warranted. From the results of a series of simulation experiments, we determine that overbooking provides greater utility when clinics serve larger numbers of patients, no-show rates are higher, and service variability is lower. Even with highly variable service times, many clinics will achieve positive net results with overbooking. Our analysis provides valuable guidance to clinic administrators about the use of appointment overbooking to improve patient access, provider productivity, and overall clinic performance.
This paper considers the problem scheduling of m immediately available tasks with random variable service times. It is shown that certain such problems can be reduced to equivalent deterministic problems. The existence of optimal schedules not involving the removal from service of incompletely processed tasks for some problems is proved and for other problems is disproved.
This edited volume captures and communicates the best thinking on how to improve healthcare by improving the delivery of services -- providing care when and where it is needed most -- through application of state-of-the-art scheduling systems. Over 12 chapters, the authors cover aspects of setting appointments, allocating healthcare resources, and planning to ensure that capacity matches needs for care.
A central theme of the book is increasing healthcare efficiency so that both the cost of care is reduced and more patients have access to care. This can be accomplished through reduction of idle time, lessening the time needed to provide services and matching resources to the needs where they can have the greatest possible impact on health. Within their chapters, authors address:
(1) Use of scheduling to improve healthcare efficiency.
(2) Objectives, constraints and mathematical formulations.
(3) Key methods and techniques for creating schedules.
(4) Recent developments that improve the available problem solving methods.
(5) Actual applications, demonstrating how the methods can be used.
(6) Future directions in which the field of research is heading.
Collectively, the chapters provide a comprehensive state-of-the-art review of models and methods for scheduling the delivery of patient care for all parts of the healthcare system.
Chapter topics include setting appointments for ambulatory care and outpatient procedures, surgical scheduling, nurse scheduling, bed management and allocation, medical supply logistics and routing and scheduling for home healthcare.
We consider the problem of determining an optimal appointment schedule for a given sequence of jobs (e.g., medical procedures) on a single processor (e.g., operating room, examination facility, physician), to minimize the expected total underage and overage costs when each job has a random processing duration given by a joint discrete probability distribution. Simple conditions on the cost rates imply that the objective function is submodular and L-convex. Then there exists an optimal appointment schedule that is integer and can be found in polynomial time. Our model can handle a given due date for the total processing (e.g., end of day for an operating room) after which overtime is incurred, as well as no-shows and some emergencies.
This paper considers the outpatient no-show problem faced by a rural free clinic located in the south-eastern United States. Using data mining and simulation techniques, we develop sequencing schemes for patients, in order to optimize a combination of performance measures used at the clinic. We utilize association rule mining (ARM) to build a model for predicting patient no-shows; and then use a set covering optimization method to derive three manageable sets of rules for patient sequencing. Simulation is used to determine the optimal number of patients and to evaluate the models. The ARM technique presented here results in significant improvements over models that do not employ rules, supporting the conjecture that, when dealing with noisy data such as in an outpatient clinic, extracting partial patterns, as is done by ARM, can be of significant value for simulation modelling.Journal of the Operational Research Society (2009) 60, 1056–1068. doi:10.1057/jors.2008.177; published online 11 March 2009
Queueing systems with scheduled arrivals, i.e., appointment systems, are typical for frontal service systems, e.g., health clinics. An aspect of customer behavior that influences the overall efficiency of such systems is the phenomenon of no-shows. The consequences of no-shows cannot be underestimated; e.g., British surveys reveal that in the United Kingdom alone more than 12 million general practitioner (GP) appointments are missed every year, costing the British health service an estimated £250 million annually. In this study we answer the following key questions: How should the schedule be computed when there are no-shows? Is it sufficiently accurate to use a schedule designed for the same expected number of customers without no-shows? How important is it to invest in efforts that reduce no-shows--i.e., given that we apply a schedule that takes no-shows into consideration, is the existence of no-shows still costly to the server and customers?
To determine appointment failure rates in pediatric resident continuity clinics nationally, and to identify characteristics of clinics with respect to factors that may affect appointment failure rates.
A one-page questionnaire administered via facsimile machine to pediatric residencies' continuity clinic directors.
Of 200 continuity clinic directors, 160 (80%) returned the survey. The mean no-show percentage was 30.9%, with a range of 3% to 80%. Among the factors studied, only mode of payment emerged as an independent predictor.
Appointment failure is a substantial problem in pediatric resident continuity clinics, which needs attention if resident learning, patient care, and clinic efficiency are to be optimized.
Nonattendance for obstetrics and gynecology (OB/GYN) appointments disrupts medical care and leads to misuse of valuable resources. We investigated factors associated with nonattendance in an outpatient OB/GYN clinic.
Nonattendance was examined for a period of 1 year in first-time visitors of an ambulatory OB/GYN clinic. The effects of age, population sector, the treating physician, waiting time, and timing of the appointment on the proportions of nonattendance were assessed. Chi(2) tests and logistic regression were used for simple and multiple regression models.
A total of 8,883 visits were included (median age 36 years). The proportion of nonattendance was 30.1%: 19.9% among rural Jewish, 30.5% in urban Jewish, and 36% in Bedouins (p < 0.001). Nonattendance increased from 26.6% among those waiting up to 1 week to 32.3% among those who waited more than 15 days (p < 0.001) and decreased with age (p < 0.001). A multiple logistic regression model demonstrated that age, population sector and waiting time for an appointment were significantly associated with nonattendance.
Nonattendance in OB/GYN patients is independently associated with age, population sector and waiting time for an appointment. It is suggested that various solutions should be carefully introduced assessed regarding routine patient scheduling in OB/GYN clinics.
Appointment Scheduling with No-Shows and Overbooking LaGanga Clinic overbooking to improve patient access and provider productivity
- Zacharias Pinedo
- S R Lawrence
Zacharias and Pinedo: Appointment Scheduling with No-Shows and Overbooking LaGanga, L. R., S. R. Lawrence. 2007. Clinic overbooking to improve patient access and provider productivity. Decis. Sci.
Econometric Analysis, 6th edn
- W Greene
Greene, W. 2008. Econometric Analysis, 6th edn. Prentice Hall,
Englewood Cliffs, NJ.
Handbook of Healthcare System Scheduling
- D Gupta
- W Y Wang
Gupta, D., W. Y. Wang. 2012. Patient appointments in ambulatory
care. R. Hall, ed. Handbook of Healthcare System Scheduling.
Springer, New York, NY, 65-104.