APPOINTMENT SCHEDULING AND CAPACITY PLANNING IN PRIMARY CARE CLINICS

Transcription

1 APPOINTMENT SCHEDULING AND CAPACITY PLANNING IN PRIMARY CARE CLINICS A Dissertation Presented By Onur Arslan to The Department of Mechanical and Industrial Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the field of Industrial Engineering Northeastern University Boston, Massachusetts April 2015

2 ii Abstract Access to care has been a major issue in the U.S. healthcare system. Besides financial barriers, patients experience non-financial barriers such as not having a regular source of care, and not getting timely appointments at convenient times. Due to limited availability of primary care providers, the efficient use of the provider capacity is important to provide comprehensive and patient-centered care without compromising the quality and safety of care. Appointment scheduling systems can play an important role in improving efficiency and access to care. The overall aim of this research is to improve appointment scheduling in primary care clinics by incorporating several environmental complexities including multiple patient types with different service needs, stochastic appointment durations, patient no-shows, and unpunctuality. We first propose a stochastic programming model to solve the appointment scheduling and sequencing problem for multiple patient types with different service time distributions and no-show probabilities to minimize waiting time, idle time and overtime costs. We show the properties of the optimal solution using primal-dual relationship when the sequence is known. When sequence is unknown, we show that the proposed models give significantly better solutions than the sequencing policies from literature. We propose another stochastic programming model that incorporates patient unpunctuality in sequencing and scheduling problem with heterogeneous service times and no-show rates. We compare our model with the selected sequencing policies from literature, and show that the proposed method performs better especially when there is high variation and uncertainty in the system. Additionally, we develop a heuristic method based on sequencing games to find solutions for large-size problems. We show that the proposed heuristic finds solutions in less computation times for the problems that cannot be solved optimally with the proposed stochastic programming model. Group appointments could be an alternative solution for improving the efficient use of clinic capacity in primary care clinics. We present a literature review of group medical appointments for pediatric patients to show the impact of group visits on several performance measures including patient satisfaction, knowledge of the patient about the disease, and number of emergency room visits and hospitalizations.

3 iii Copyright 2015 by Onur Arslan All Rights Reserved

4 iv Biography Onur Arslan was born in 1985, in Ankara, Turkey. He received his Bachelor of Science degree in Civil Engineering from Middle East Technical University, Turkey in Upon graduation, he worked in the construction area until he relocated to the United States to pursue a Master of Science degree in Project Management at Northeastern University in Boston, Massachusetts. In 2010 Onur was drawn to the application of industrial engineering tools to the healthcare industry, and started a Master of Science degree in Industrial Engineering, also at Northeastern University, which was granted in Since 2011, he has continued his graduate study for a Doctorate of Philosophy degree in Industrial Engineering at Northeastern University.

5 v Acknowledgements I would like to thank Dr. Yaman Yener for encouraging and believing in me for my graduate education. Like all of his students, I will always remember and acknowledge his endless support. I would also like to thank my advisor Dr. Ayten Turkcan, who guided me patiently and indefatigably throughout my research. This dissertation would not have been possible without her guidance and support. Special thanks go to my committee members Dr. Ozlem Ergun and Dr. Furkan Burak for their valuable comments and inputs to my research. I would also like to thank Dr. Allen Soyster for sharing his priceless experiences and knowledge with me during my assistantship with him. It was an honor and privilege to work with a pioneer in the Operations Research field. Last, but not least, I thank my family: my father, my mother, and my little brother for their unconditional love and support. And Kathryn, for her love and patience.

6 vi Table of Contents Abstract... ii List of Tables... viii List of Figures... x Chapter 1 Introduction Research Objectives Summary of Research Contributions Organization of the Document... 4 Chapter 2 Background and Literature Review Background Patient-Centered Medical Homes in Primary Care Appointment Scheduling Capacity Planning Literature Review Appointment Scheduling and Sequencing Capacity Planning Summary Chapter 3 Stochastic Appointment Scheduling With Heterogeneous No-Show Rates Introduction Two-Stage Stochastic Programming Model Scheduling for Known Appointment Sequence Computational Results Computational Results for Single Patient Type Computational Results for Multiple Patient Types Computational Results for Practical Application in Primary Care Clinics Summary of the Computational Results Conclusions Chapter 4 Stochastic Appointment Scheduling with Patient Unpunctuality Introduction Stochastic Programming Models Stochastic Programming Model When the Sequence is Known Stochastic Programming Model When the Sequence is Unknown Sequencing Game Heuristic Computational Results... 63

7 vii Computational Results for Scheduling of Same-day and Prescheduled Appointments Computational Results for Scheduling of Routine and New Patient Appointments Computational Results for Scheduling of Four Appointment Types Summary of Computational Results Conclusion Chapter 5 Group Appointments for Children and Young Adolescents in Primary Care Setting Background Group Appointments Search Methodology Results Summary and Conclusion Chapter 6 Conclusions Thesis Contribution Future Directions References

8 viii List of Tables Table 1: Properties of appointment scheduling models... 8 Table 2: Properties of capacity planning models... 9 Table 3: Appointment sequencing and scheduling studies that use stochastic programming Table 4: Appointment sequencing and scheduling studies that use robust optimization Table 5: Capacity planning studies Table 6: Notation Table 7: Experimental design factors for single patient type Table 8: Computational results for lognormal distribution (with and without no-shows) Table 9: Service time distributions and no-show rates for each patient type Table 10: Objective function values and percentage gap between the optimal solution and the sequencing rules for different cost coefficients Table 11: Objective function values and percentage gap between the optimal solution and the sequencing rules for different cost coefficients (no-show included) Table 12: Objective function values and percentage gap between the optimal solution and the sequencing rules for different cost coefficients (practical case) Table 13: Notation Table 14: Additional notation Table 15: SGH numerical example parameters and solution Table 16: Cost and urgency calculations for the example problem Table 17: Clinic settings, patient types and characteristics Table 18: Patient unpunctuality distributions and cost coefficients Table 19: Computational results and comparison of including single (routine) and two patient types (SMD&PRS) for all cost coefficient and patient unpunctuality settings Table 20: Computational results and comparison of including single (routine) and two patient types (New and Routine) for all cost coefficient and patient unpunctuality settings Table 21: Solutions of SGH for each cost coefficient and unpunctuality setting with different lower and upper bound intervals Table 22: Comparison of the results for SGH_H, SGH_LS, and the best solution found after 24 hour run Table 23: Runtimes for SGH steps and the objective function value improvements obtained by local search iterations Table 24: Comparison of the best SGH solution with other sequencing rules... 79

9 ix Table 25: Computational results and comparison of including single and four patient types for all cost coefficient and patient unpunctuality settings Table 26: Computational results and comparison of including two and four patient types for all cost coefficient and patient unpunctuality settings Table 27: Summary of the group visit structures based on their focus areas Table 28: Review of the group appointment studies... 93

10 x List of Figures Figure 1: Template schedule of a provider in a primary care clinic Figure 2: Schematic representation of the model Figure 3: Minimum cost network flow model with uncertain right-hand-sides Figure 4: a) Percentage change in expected total cost per patient vs. number of scheduled appointments, b) Number of binding scenarios vs. number of scheduled appointments Figure 5: Expected profit for number of scheduled appointments Figure 6: Convergence of objective function values with respect to number of scenarios (Overtime Cost=10, Waiting Cost=1, Idle Cost=5) Figure 7: Optimal appointment durations for different service time distributions with cost coefficients (1,1,1) Figure 8: Optimal appointment durations for different service time distributions with cost coefficients (10,1,5) Figure 9: Optimal appointment durations for lognormal distribution with different cost coefficients Figure 10: Expected waiting and idle times for each appointment at different cost coefficients for lognormal distribution Figure 11: Optimal appointment durations with and without no-shows for different cost coefficients Figure 12: Expected waiting and idle times with and without no-shows for equal cost coefficients (1, 1, 1) Figure 13: Optimal appointment durations and sequences without no-shows for all cost settings 42 Figure 14: Optimal appointment durations and sequences with no-shows for all cost settings Figure 15: Optimal appointment durations and sequences with heterogeneous no-shows for all cost settings in practical case Figure 16: Sample appointment schedule for a given scenario Figure 17: Schematic representation of the local search algorithm (SGH_LS) Figure 18: Optimal schedules for 6 same-day and 9 prescheduled appointments for all cost and unpunctuality settings Figure 19: Total expected cost, waiting time, idle time and overtime with respect to low and high unpunctuality and different cost coefficients (1.1.1), (4.1.2), and (9.1.3) Figure 20: Optimal appointment durations for routine patients with low unpunctuality for the cost coefficients (1.1.1), (4.1.2), and (9.1.3)... 68

11 xi Figure 21: Optimal appointment durations for routine patients with high unpunctuality for the cost coefficients (1.1.1), (4.1.2), and (9.1.3) Figure 22: Optimal schedules for 9 same-day and 6 prescheduled appointments for all cost and unpunctuality settings Figure 23: Optimal schedules for 10 routine and 5 new patient appointments for all cost and unpunctuality settings Figure 24: Optimal appointment durations for routine patients with low unpunctuality for the cost coefficients (1.1.1), (4.1.2), and (9.1.3) Figure 25: Optimal appointment durations for routine patients with high unpunctuality for the cost coefficients (1.1.1), (4.1.2), and (9.1.3) Figure 26: Optimal schedules for 5 routine and 10 new patient appointments for all cost and unpunctuality settings Figure 27: Schedule of multiple appointment types for all cost and unpunctuality settings (SGH solution) Figure 28: Search strategy... 89

12 1 Chapter 1 Introduction Healthcare expenditures in the United States increase every year and currently constitute 17.9% of the gross domestic product of the country.[31] Primary care services including pediatrics, men and women s health, acute and chronic diseases, and many other day-to-day health care services for patients of all ages, socioeconomic and geographic origins, is the largest building block of the healthcare delivery system. According to the Health Cost Institute, the total cost of primary care services had the greatest marginal increase during the period of 2010 to 2012.[33] As articulated in recent articles and reports, the United States has the most expensive yet one of the least efficient healthcare systems in the world.[5, 60] Since primary care is such a large component of the healthcare delivery system, even small improvements in efficiency leads to significant cost reductions and an improved patient experience. According to the National Ambulatory Medical Care Survey, 55.5% of the 1.1 billion ambulatory care visits are made to primary care physicians.[58] The Health Resources and Services Administration Bureau of Health reported that the demand for primary care will increase 14% between 2010 and 2020 due to aging and population growth, and expanded insurance coverage implemented under the Affordable Care Act. [34] However, the number of primary care providers is forecasted to increase only by 8% during that period. According to the report, this mismatch between supply and demand will create a shortage of approximately 20,400 full-time equivalent (FTE) physicians, which will obstruct timely access to care.[34] In recent years, innovative care delivery models such as patient-centered medical homes (PCMH) have been increasingly used to redesign the care delivery systems. Enhancing access to care is one of the key PCMH standards, and redesigning the appointment scheduling system is an important strategy to improve efficiency and timely access to care. However, improving appointment scheduling is a complex problem due to the variety of care services provided to the patients, who have different time and resource

13 2 requirements. Additionally, uncertainty of patient behavior including no-shows and patient unpunctuality add to the complexity of the problem by creating inefficiency and avoidable cost. Moreover, time allocation of resources to various services and/or patient types must be taken into account while designing the enhanced appointment scheduling strategies. Considering the size and growth rate of the primary care system, a sophisticated appointment scheduling design integrated with the time allocation of resources within the clinic capacity is crucial to provide quality, efficient and accessible health care. This dissertation offers solution methods for better scheduling of appointments in primary care clinics Research Objectives Due to its extensive service size, the primary care delivery system requires sophisticated time allocation policies for clinic resources, and appointment scheduling methods. Moreover, the concept of a patient-centered medical home increases the difficulty of the problem by introducing patient preferences and patient satisfaction factors. Therefore, improving the utilization of resources and finding a setting that yields the minimum cost based solely on resource usage is not an appropriate approach. A welldesigned appointment scheduling system should consider the cost of patient waiting time and balance the quality, accessibility and cost of care. Additionally, ignoring the uncertainty of patient service times and unpunctuality of patients creates an implementation gap between the real world problem and mathematical models. For instance, appointment duration of a new patient is more likely to be higher than the duration of a follow-up visit. Different patient groups tend to have different noshow rates. Similarly, patient unpunctuality may have negative effects on the performance of the clinic. This heterogeneity of service parameters directly affects resource utilization. Additionally, applicability of the proposed scheduling policies should be considered to find realistic solutions. In order to maximize the overall efficiency of the healthcare delivery system, applicability of scheduling policies and appointment scheduling problems must be considered simultaneously while considering the complexity of the system in as much detail as possible.

14 3 In general practice, the available time of a primary care provider is divided with respect to the type of patients such as new, acute, follow-up or chronic/complex. The optimum allocation of available time of a provider between each type of visit is crucial for maximizing both resource utilization and patient satisfaction. Furthermore, group appointments or shared medical appointments are very efficient in terms of resource usage as well as elevating patient experience. In this dissertation, we have three research objectives. The first objective is to find an appointment schedule that considers detailed patient-related factors such as heterogeneous no-shows and service time distributions, and investigate the effects of these parameters on performance measures such as the total expected cost, waiting time of patients and idle time of the provider. The second objective is to find an appointment schedule for multiple patient types with heterogeneous no-shows, service time distributions, and patient unpunctuality. The third objective is to investigate the current applications of group appointments for pediatric patients in primary care clinics, and analyze the impact of group appointments on patient satisfaction, and utilization of emergency care. In our research, we develop two-stage stochastic programming models to solve appointment scheduling and sequencing problems for a single provider, which considers the uncertain nature of patient related parameters such as unpunctuality, no-shows, and random service times. We also propose a heuristic method to find a solution for the largesize sequencing and scheduling problems. We show the importance of time allocation of available resources in order to improve the efficiency and resource utilization of primary care clinics. We find that group appointments may offer an efficient alternative to address resource allocation. In addition, patient-centered care and efficiency improvements in the primary care delivery system are the main focus and motivation of our research Summary of Research Contributions The main contributions of our research to the existing literature are listed below: We propose a two-stage stochastic programming model to solve appointment scheduling and sequencing problem with the objective of minimizing the expected total cost of waiting time, idle time and overtime. We consider multiple patients types with different service time distributions and no-show rates simultaneously.

15 4 We propose another two-stage stochastic programming model to solve appointment scheduling and sequencing problem with patient unpunctuality. To the best of our knowledge, this is the first study that incorporates patient unpunctuality into stochastic programming to find optimal appointment sequences and schedules. We propose a heuristic method, which finds an initial sequence by ordering the appointments with respect to their urgencies and then performs a local search to find a better solution for the large-scale sequencing and scheduling problems with multiple patient types and heterogeneous patient unpunctuality, no-shows, and service times. We present a comprehensive and systematic literature review for the applications of a group appointment approach for child health in a primary care setting, in order to reveal the opportunities and benefits of group visits from a patient-focus and resource allocation perspective Organization of the Document The outline of this thesis is as follows: In Chapter 2, we present background information and current problems in primary care clinics. We also explain the importance and implementation methodology of appointment scheduling and capacity planning tools. Additionally, the existing studies related to appointment scheduling and capacity planning are presented in the literature review section. In Chapter 3, we present a two-stage stochastic programming model for the appointment scheduling and sequencing problem that includes uncertainty of service times and no-show probabilities, which vary according to patient type. Additionally, the characteristics of the optimal solutions for special cases are derived using the primal-dual relationship. In Chapter 4, we present another two-stage stochastic appointment scheduling and sequencing model with patient unpunctuality, heterogeneous no-show probabilities and service times. In addition, a heuristic method is presented for solving the large-scale problems where the number of patients, the number of patient types, and the number of scenarios are high.

16 5 In Chapter 5, we present a systematic literature review of studies that consider group appointment approach for pediatric care in primary care settings. We execute a search on medical databases including PubMed and Medline, and analyze the structure of current application of group visits for pediatric care.

17 6 Chapter 2 Background and Literature Review In this chapter, we present contextual knowledge about patient-centered medical homes to provide a better point of view of the problems related to appointment scheduling and capacity planning in primary care clinics. Then, we present a literature review on appointment scheduling and capacity planning studies in the operations research literature Background Patient-Centered Medical Homes in Primary Care The patient-centered medical home (PCMH) model combines five main functions including comprehensive care, coordinated care, patient-centeredness, accessible services, and quality and safety.[1] The comprehensiveness of care states that the medical homes are responsible for meeting the large majority of each patient s physical and mental health needs. Patient-centered care suggests patients individual needs, preferences and priorities should be considered while providing care.[1] In PCMHs, the primary care team is responsible for coordinating the treatment for each patient in order to ensure that they receive the necessary care when and where they need it, and in a way that they can understand.[2] Due to limited availability of primary care providers, the efficient use of the provider capacity is important to provide comprehensive and patient-centered care without compromising the quality and safety of care. Access to care has been a major issue in the U.S. healthcare system for many years. Patients, who do not have access to care due to high medical costs or lack of insurance coverage, enter the healthcare system much later with greater negative impacts such as preventable hospitalizations, congestion of emergency departments and poor life expectancy. Besides financial access barriers, people experience non-financial barriers such as not having a regular source of care, and not getting timely appointments at convenient times and locations. Therefore, appointment scheduling plays a major role in

18 7 improving access to care. The following sections provide detailed information about appointment scheduling and capacity planning in primary care settings Appointment Scheduling The main aim of appointment scheduling in healthcare delivery systems is to regulate conflicting preferences and priorities of patients and providers under the constraints of limited clinical resources. In current practice, patients often call a clinic in advance to book an appointment with a primary care provider. Likewise, some patients prefer walk-in times in order to be seen by the provider without an appointment. Many clinics handle both types of requests. Appointment durations vary between 10 and 60 minutes and a scheduler assigns the appointment time and duration by considering various factors such as physician s availability, reason for the visit, patient characteristics, and patient preferences. For example, appointments for new patients require more time for consultation and diagnosis compared to those for routine visits for follow-up patients. Physicians determine how much time they want to allocate for different types of appointments such as same-day, acute, routine chronic/complex, well-child, well woman, physical, etc. Primary care clinics usually divide available provider time into equal time slots.[29] In that case, multiple consecutive appointment slots can be assigned to a single patient depending on the duration of the procedure or the reason of the visit. Besides patients needs and providers preferences, the schedulers typically have to interact with real-time conflicts associated with the uncertainty of human behavior. For example, there may be scheduling updates based on provider absenteeism or sick calls, patient noshows, unpunctuality of patients, and the presence of walk-in patients. Several decision variables such as the number of appointments scheduled per session, number of patients per appointment slot, appointment times and durations should be determined to design an effective appointment scheduling system. According to literature review by Gupta and Denton [29], the number of appointments per clinic session and per appointment slot, and arrival and/or service times can be deterministic or stochastic. The uncertainties such as patient no-shows, cancellations, walk-ins, unpunctuality, and variation in appointment durations increase the complexity of the

19 8 problem. Appointment scheduling methods aim to optimize several performance measures including patient waiting time, provider idle time, overtime, congestion and fairness measures.[9] Advanced IE/OR tools such as mathematical modeling, data analysis, and statistical methods can be used to find optimal appointment schedules in such complex environments with several uncertainties and multiple conflicting objectives. Reducing patient waiting time, which has a direct impact on patient satisfaction, is very important to provide a patient-centered care. Reducing provider idle time and overtime, which affect resource utilization and provider satisfaction, is important to improve efficiency and reduce costs. Table 1 summarizes the decision variables, parameters/ uncertainties, and objectives that are used to model appointment scheduling problems in literature. Table 1: Properties of appointment scheduling models Decision Variables Number of appointments Appointment durations Appointment start times Appointment sequence Capacity Planning Parameters/ Uncertainties Patient no-shows Cancellations Walk-ins Patient unpunctuality Appointment durations (random or fixed) Objectives / performance measures Revenue Cost-based (waiting time, idle time, overtime cost) Time-based (waiting time, idle time, overtime) Congestion Fairness In all levels of capacity allocation in primary care, the most important goals are to enhance patients timely access to care and to maximize the patient-provider continuity.[3] Capacity planning in the healthcare delivery system can be done on a strategic level, which considers long term projects related to the mission statement of the organization, or tactical level that transforms strategic decisions into operational plans. In tactical capacity planning, the allocation of patient types to resources and division of the available resource time to specific patient types are two main problems [36]. While assigning patient types to resources, the objective is to maximize the number of patients served, by calculating the optimal assignment of patient types to available resources that are specialized for the type of care needed. The problem of subdividing the available resource time over different healthcare services can be solved by finding the optimal time distribution of resources to various patient groups. In addition, considering the dynamic changes of patient demand

20 9 and service time enhances the robustness of the capacity plan. Some of the characteristics and properties that are used to model capacity planning problems are shown in Table 2. Table 2: Properties of capacity planning models Decision Variables Parameters Objectives / performance measures Total available capacity Allocation of capacity Number of scheduled patients 2.2. Literature Review No-shows Cancellations Service time (random or fixed) Patient type Total cost Timely access rate Continuity rate Staffing level Revenue Workload Patient satisfaction In the last six decades, finding a better appointment scheduling system and ensuring the optimum utilization of clinic capacity have become an area of focus and research. In this section, we review the literature on applications of IE/OR tools such as optimization and simulation for improving appointment scheduling and capacity planning in primary care settings. The first part is a review of appointment scheduling studies and the second part is a review of capacity planning studies in primary care clinics Appointment Scheduling and Sequencing Cayirli and Veral [9] performed a literature review that provides an extensive and well-organized categorization of the assumptions and techniques used in appointment scheduling. The existing studies can be categorized in two groups, which are static or dynamic. Since the most frequently used application in clinics is static, the majority of available literature is related to static models including the stochastic programming models that we propose in this dissertation, where all assessments are determined prior to the beginning of the day. On the other hand, some studies consider dynamic scheduling models where the current state of the system can be updated continuously due to the dynamic arrival of patients [9]. In earlier studies, we observe that queuing theory [15, 72] and mathematical programming models [26, 35] are widely used. However, optimal schedules envisioned by these existing models do not take into account the complex environment of primary care clinics. Ignoring the fact that appointment schedules need to be robust and

21 10 flexible can result in loss of efficiency and avoidable cost increases. We refer the reader to the review papers for more details about earlier studies on appointment scheduling. [9, 29, 54] In recent years, the appointment scheduling studies started considering the complexities that clinics face including patient no-shows, walk-ins, and unpunctuality. Robust optimization and stochastic programming models have great potential to incorporate these uncertainties into the design of scheduling systems and generate optimal appointments for primary care delivery. A stochastic programming approach is suggested, when the distributions of uncertain parameters of the model are known.[67] If there is not enough data to estimate the distributions, a robust optimization model is useful to introduce an appropriate level of uncertainty to the model. Considering the fact that in the next 5 years, the projected increase in demand will not be satisfied by the available resources in primary care [34], one of the main contributions of this dissertation is to design an appointment system that incorporates multiple patient types with different service time distributions, no-show rates, and unpunctuality with the objectives of minimizing patient waiting times, resource idle times and overtime in a primary care setting. In the following sections, we review the studies that use stochastic programing and robust optimization to solve appointment sequencing and scheduling problem. Since we did not find any study that considers unpunctuality in a stochastic programming or robust optimization model, we provide a review of the unpunctuality studies in a separate section Studies that use stochastic programming Stochastic programming is an optimization method utilized when there is uncertainty involved in the parameters of a model. It is used for finding the optimal value of a decision variable while constraints are satisfied on average or with high probability, and for minimization problems where the objective function provides a solution that is small on average or with high probability. [9] This brief review is limited to the studies that incorporate multiple patient types with different service time distributions. A brief analysis of the current literature related to stochastic appointment scheduling is represented in Table 3.

22 11 Most of the appointment scheduling studies that use stochastic programming consider surgery scheduling. Gupta and Denton [18] use stochastic programming to find optimal schedule for a given sequence of surgeries with the objective of minimizing the expected cost of customer waiting, server idle time and tardiness costs. They use an L- shaped algorithm with sequential bounding in order to find bounds on the gap between the solutions obtained from solving the discrete set of scenarios that define an approximate problem. In their study, a surgery scheduling application of the model shows significant cost savings compared to a deterministic (mean value) approach. Begen and Queyranne [6] use stochastic optimization for scheduling surgeries when the sequence is given. They develop a nonparametric approach to find near-optimal appointment schedules in polynomial time. They consider patient no-shows and discrete service times for a given sequence of jobs. Denton et al. [19] present a two-stage stochastic mixed-integer program and an interchange heuristic to find the optimal sequence and schedule for surgeries and surgery teams while minimizing the expected cost of waiting time of surgery teams and patients, and the expected idle and overtime cost of the operating rooms. Mancilla and Storer [48] use stochastic programming to solve surgery sequencing and scheduling problem. They use a heuristic approach based on Benders decomposition and compare the performance to other approaches such as sort by variance and perturbed sort by variance. There are a few studies that propose stochastic programming models to solve appointment scheduling problem in primary care settings. Erdogan and Denton [22, 23] propose two stochastic programming models for single server appointment scheduling systems with uncertain service durations while considering either patient no-shows or sequencing problem where the demand is uncertain. They use decomposition-based algorithms to solve the models efficiently. Robinson and Chen [73] develop a closed-form heuristic method for determining appointment times for a given sequence, which minimizes the expected weighted sum of waiting and idle time costs. In their proposed heuristic, they first define an appropriate test bed and an approximate continuous probability distribution to model the service times, then they investigate the form of the optimal policy in order to find a simpler approximate policy form that captures most of the structure of the optimal policy. Chen and Robinson [13] propose a stochastic programming

23 12 model to determine the optimal sequence for routine and same-day appointments, which are assumed to have different no-show rates. They introduce two heuristic rules for sequencing of appointments, and mention that the optimal schedule has dome-shape structure. Table 3: Appointment sequencing and scheduling studies that use stochastic programming Study Gupta and Denton (2003) [18] Robinson and Chen (2003) [73] Denton et al. (2007) [19] Begen and Queyranne (2011) [6] Mancilla (2011) [48] Erdogan et al. (2013) [23] Chen and Robinson (2014) [13] Problem Solved Appointment scheduling Appointment scheduling Appointment scheduling and sequencing Appointment scheduling for a fixed sequence Appointment scheduling and sequencing Appointment scheduling Appointment scheduling and sequencing Patient Types Single Single Multiple Single Multiple Single Multiple Studies that use robust optimization Service Time Distribution Uniform, normal and gamma Generalized lambda Empirical Discrete processing duration distribution Truncated normal Lognormal Lognormal for all types of patients with different variance Objectives Minimizing linear costs of waiting time, idle time and tardiness Minimizing the total expected patient and doctor's waiting time Minimizing the weighted sum of the expectation of waiting time, idle time, and tardiness Minimizing expected idle time and waiting time Minimizing weighted linear combination of waiting time, idle time and overtime Minimizing expected idle time and waiting time Minimizing expected overtime, idle time and waiting time Method L-shaped algorithm with sequential bounding Closed-form heuristic Two-stage stochastic mixed-integer program and L- shaped algorithm Nonsmooth convex optimization techniques Heuristic solution approach based on Benders decomposition L-Shaped Method Benders decomposition and heuristic sequencing policies Robust optimization approach models uncertainty as parameters belonging to a known uncertainty set and optimizes the worst case over that set (Soyster [79]). There are a few studies that use robust optimization to solve appointment sequencing and scheduling

24 13 problem. A brief analysis of the current literature related to robust optimization applications on appointment scheduling is represented in Table 4. Mittal and Stiller [53] assume minimum and maximum service times are known, and propose a robust optimization model to minimize earliness and tardiness cost in the worst case. They also propose a global balancing heuristic to find the optimal schedule and a (2+ε)-approximation algorithm to determine the optimal sequence. Kong et al. [42] assume only the mean and covariance estimates of service times are known. They propose a convex conic optimization model with a tractable semi-definite relaxation to find a schedule that minimizes the expected weighted sum of waiting time and overtime for a given patient sequence. They applied their approach to solve the appointment scheduling problem in an eye clinic. Similar to Kong et al. [42], Mak et al. [46] propose a distributionfree model that minimizes the worst-case expected wait and overtime costs. They also consider the problem of sequencing the appointments in the same model. In their computational results, they compare the solutions with the stochastic programming model, which assumes total knowledge of the process time distribution, and conclude that the results are optimal or very close to the optimal solution. Qi and Sim [67] define a new performance measure, which considers frequency and intensity of delays, called Delay Unpleasantness Measure (DUM). They use stochastic programming and robust optimization approaches to balance patient waiting time and doctor overtime using lexicographic minimization of the worst case DUM. They assume a heterogeneous service time distribution for the sequencing and scheduling model and solve a pediatrics clinic example in their computations. The difference between a mean value solution and a robust solution is called the price of robustness. A robust solution degrades when the level of uncertainty increases in the system. Due to these factors, the price of robustness should be considered when deciding to use an application incorporating robust optimization.

25 14 Table 4: Appointment sequencing and scheduling studies that use robust optimization Study Problem Solved Entity Types Service Time Distribution Objectives Method Mittal and Stiller (2013) [53] Appointment scheduling and sequencing Multiple Weibull, gaussian and gamma Minimize the cost of the worst case scenario Global balancing heuristic Kong et al. (2013)[42] Appointment scheduling for a fixed sequence Single Uniform, normal and gamma Minimize the expectation of the weighted sum of patients waiting time and the doctor s overtime Convex conic optimization with a tractable semidefinite relaxation Mak et al. (2013)[47] Appointment sequencing and scheduling Multiple Normal, gamma, and lognormal Minimize the worst-case expected waiting and overtime costs out of all probability distributions with the given marginal moments Tractable and distribution-free conic programming Qi and Sim (2013)[67] Appointment sequencing and scheduling Multiple patient types Uniform and beta Minimize frequency and intensity of uncertain delays Lexicographic minimization procedure Studies that consider patient unpunctuality Patient unpunctuality is one of the important factors that have an impact on the performance of different appointment scheduling methods. White and Pike [91] show that if patients are early on average, reducing the size of the block of initial patients improves the waiting time of patients and idle time of the provider. Vissers [87] shows that when patients are at most 5 minutes early, all scheduling policies including single block system with two patients in the initial block, blocks of three patients, or starting 5 minutes with the clinic have similar effect on the waiting time of patients and idle time of the provider, where both performance measures are within the acceptable range (15minutes for the waiting time of patients and 5 minutes for the idle time of the provider). Similar to Vissers [87], Cox et al. [15] also show that the multiple block system with three blocks performs well compared to single block system. Cayirli et al. [10] use simulation modeling to test

26 15 the performance of sequencing and scheduling policies on the total cost of waiting time, idle time and overtime. They consider no-shows and unpunctuality, and state that scheduling new patients at the end of the session, at the beginning of the sesison, or in an alternating order perform the best among other sequencing rules. Fetter and Thompson [24] show that if patients can be punctual within the interval of ±5 minutes, waiting time would decrease significantly. There are studies that use queuing theory to analyze the impact of unpunctuality on performance measures. The study done by Jouini and Benjaafar [38] uses an exact analytical approach to find performance measures of interest and show the usefulness of the approach by describing numerical results that examine the impact of not accounting for unpunctuality and no-shows. Another study done by Fiems and Vuyst [25] use a modified Lindley recursion in a discrete-time framework to obtain accurate predictions at a very low computational cost for the waiting times of the patients and the idle time of the physician, where multiple treatment types, patient unpunctuality and patient no-shows are included. There are a few studies that incorporate unpunctuality to design an appointment scheduling system. Tai and Williams [81] model a distribution for patient unpunctuality by considering various patient behavior patterns, propose a model for optimizing the appointment schemes, and perform a simulation experiment to validate this optimization scheme. They claim that the frequency distribution of patient unpunctuality shows asymmetry in shape, which is resulted from various types of arrival behaviors. They add that the optimal appointment intervals are highly related to patient unpunctuality patterns for the minimization of the total waiting and idle time. Klassen and Yoogalingam [41] uses simulation optimization to investigate the scheduling policies such as fixed and variablelength interval for a single patient type and by considering patient and doctor unpunctuality. They analyze the performance measures including patient waiting time and doctor idle time, and suggest decreasing the appointment interval throughout the session as the standard deviation of patient unpunctuality increases. All these studies consider that the waiting time of a patient starts from the scheduled appointment time. Maister [45] indicates that the effect of patient waiting time, prior to their scheduled appointment, is negligible in relation to clinic cost. On the other hand,

27 16 Janakiraman et al. [37] suggest that patient waiting time should be measured from the arrival time as it potentially has a negative effect on patients views on the perceived efficiency of the clinic and the quality of their experience. In our study, we assume that waiting time of a patient starts with the arrival of the patient Capacity Planning To supplement scheduling applications, capacity planning is a crucial factor for enhancing the accessibility, quality and cost balance of the healthcare delivery system. Capacity planning usually focuses on the bottlenecks of a system and analyzes performance changes under different resource allocation settings in order to find the best possible guideline. In the healthcare industry, capacity planning problems are evolving with developments in technology, rising quality standards and related high costs of care. The bottleneck of a modern clinic can be the utilization of physicians, exam rooms or medical devices. Additionally, dynamic changes of resources such as provider availability, patient preferences and behaviors, or breakdown of medical devices increase the complexity of the problem. Capacity planning problems in healthcare can be categorized with respect to the level of planning such as strategic planning and tactical planning. Strategic planning approaches aim to fulfill the higher level or long-term decision problems by considering organization s mission such as the location and size of the facility. Tactical planning approaches usually deal with resource efficiency and service effectiveness. In their earlier study, Smith Daniels et al. [76] and a later study done by Hulshof et al. [36] provide a detailed classification of the decision planning studies in healthcare and the systematic review of these studies. Since we focus on the time allocation problem of resources within multiple service/patient types, we review these studies in this section. A brief analysis of the selected and currently available literature related to capacity planning and resource allocation is represented in Table 5. Several optimization models are used to solve the capacity planning problem. Smith et al. [77] propose an integer programming model for the allocation of tasks and the duration of these tasks to the healthcare providers. However, they assume that any particular job can be assigned to different providers with the expectation of the same

28 17 outcome, which ignores patient-provider continuity. A later study done by Rajagopalan and Hadjinicola [69] uses an optimization model for the allocation and scheduling of medical devices in order to maximize the revenue of a clinic by considering managerial preferences. Both studies approach the problem only from the provider s perspective and ignore patient preferences. However, we see that studies published in the last few decades are increasingly considering patient preferences as a parameter in the healthcare delivery system. In recent studies, we observe that the complexity of the models is greater than before due to improvement of computation power and inclusion of more details in the models. Balasubramanian et al. [4] propose a two-stage stochastic programming model for maximizing the expected revenue of satisfying prescheduled and open-access appointments in primary care while considering the effect of provider flexibility on capacity allocation. Similarly, Qu et al. [68] use a closed-form optimization approach in order to match the daily healthcare provider capacity to demand in advanced access scheduling systems. Günes [28] presents an optimization model for time allocation of preventive care services in order to minimize total cost, which is the sum of treatment costs, waiting cost and the cost of referrals. Simulation models are one of the major tools that have been used for analyzing capacity planning decisions and testing their performance measures. Vermeulen et al. [86] propose an adaptive scheduling simulation model where CT-scan availability is allocated between multiple patient groups with respect to the changing demand and in order to schedule patients groups with different attributes and make efficient use of capacity. Likewise, Schütz and Kolisch [75] use a heuristic simulation-based approximate dynamic programming (ADP) algorithm combined with discrete event simulation of service period in order to solve the capacity allocation problem. They implement the proposed algorithm to a radiology department of a university hospital. Similarly, Elkhuizen et al. [21] use simulation model to investigate the relationship between new patient demand and the capacity of doctors needed for initial consultations in an outpatient department, where only appointment-based scheduling with no walk-ins is considered. White et al. [90] propose an empirically based discrete-event simulation in order to analyze the relationship between

29 18 patient appointment scheduling policies and capacity planning policies. They also claim that in most cases, exam rooms are the bottlenecks for capacity planning. However, they evaluated that introducing additional exam rooms does not affect the physicians utilization after a certain threshold. Table 5: Capacity planning studies Study Smith et al. (1976) [77] Rajagopalan and Hadjinicola (1993) [69] Qu et al. (2007) [68] Elkhuizen et al. (2007) [21] Vermeulen et al. (2009) [86] Günes (2009) [28] Problem Solved Optimal staffing level for an ambulatory medical care practice Allocation of unitdays on the different equipment types to each client Capacity allocation of open-access appointments Capacity allocation for new patients Capacity allocation to different patient groups for CT scans Time allocation for preventive care in primary care Patient/Service Type Multiple service types Different types of medical devices Prescheduled and open access patients New patients Outpatient with contrast injection, outpatient without contrast injection, urgent, clinic, special Preventive care and nonpreventive care Objectives/ Performance measures Minimize the cost of meeting the demand for multiple services Maximize the rental revenues and satisfy client preferences for days of the week and equipment types Maximize the expected number of appointments that can be prescheduled with the provider in a session Improve access time of new patients, utilization Maximize the minimum service level (MSL) of all patient groups Minimize total cost, which is the sum of treatment costs, waiting cost and the cost of referrals Method Mixed integer programming Integer programming Closed-form optimization Simulation Adaptive approach to automatic optimization of resource calendars using simulation Non-linear optimization

30 19 Schütz and Kolisch (2012) [75] Balasubramanian et al. (2012) [4] Optimal scheduling policy for radiology department Capacity allocation for primary care practices 4 different classes of patients with different service times Prescheduled and open access patients Maximize profit, which is a function of classspecific revenues, refunds for cancellations, no-shows and cost of overtime Maximize the expected revenue of satisfying prescheduled and openaccess appointments, timely access rate and continuity rate Modeling: Continuoustime Markov decision process (MDP) Solution: Approximate dynamic programming (ADP) combined with simulation Two-stage stochastic integer program Capacity planning and appointment scheduling should be considered simultaneously in order to reach the desired state of an efficient primary care delivery system. An improved appointment system alone is not enough to establish a healthcare system that provides comprehensive and high quality service to the entire population. For instance, in an over-utilized clinic, an optimal appointment schedule can provide the best way to decrease patient waiting times and overtime cost. However, in considering performance measures such as patient and physician satisfaction, the quality of care could still be under the desired level. Similarly, underutilized clinics face challenges such as loss of revenue, inefficiency, and poorly organized care, which decreases the quality of healthcare delivery. Group appointment scheduling may alleviate capacity planning concerns by reducing patient-related variations and increasing resource utilization Summary The decision of selecting the robust or stochastic optimization approach depends on availability of data on service times and no-show rates. Robust optimization is used when there is no prior knowledge of distributions. Stochastic programming is applicable when the distributions are known. Since many primary care clinics started implementing electronic medical record and patient management systems, the availability of operational data (i.e. service times for different types of patients, waiting times, individual no-show

31 20 probabilities) will not be an issue in near future. Therefore, we use stochastic programming to model and solve appointment sequencing and scheduling problem. In the first study, we propose a two-stage stochastic programming model that considers multiple patient types with different service time distributions and heterogeneous no-show rates simultaneously to find an appointment schedule that minimizes the total expected cost. We investigate the properties of the optimal solution for special cases such as when the appointment sequence is known, and state theorems and insights for these cases. In the second study, we introduce the patient unpunctuality in addition to the heterogeneous noshow probabilities and service times. We also propose a heuristic approach for large-scale problems in order to find an initial sequence and then look for a better solution by using the local search algorithm. In the third study, we perform a systematic literature review for the application of group appointment in pediatrics in order to analyze the properties and effectiveness of current group appointment structures. From capacity planning perspective, we investigate the allocation of the available clinic capacity to appointments. Available capacity can be utilized by considering the expected cost, expected revenue, and the expected maximum number of scheduled patients. We show that the optimum number of scheduled patients might change with respect to the objective function of the model such as maximizing the expected revenue or minimizing the expected cost. In addition, we see that the current application of the group appointments is a promising alternative for improving the resource utilization and maximizing the number of patients served. In pediatric care, we see that group appointments also improve the patient satisfaction and self-management knowledge compared to the individual appointments in primary case setting.

32 21 Chapter 3 Stochastic Appointment Scheduling With Heterogeneous No-Show Rates In this chapter, we propose a two-stage stochastic programming model to find the optimum schedule and sequence of patients with heterogeneous service time distributions and noshow rates. The objective of the model is to minimize expected total cost of waiting time of patients, idle time of the providers, and the overtime of the clinic. We present theoretical insights based on a special case where the sequence of patients is known. We also show the effect of the number of scenarios included in the model, the effect of service time distributions, and the effect of patient no-shows on the optimal schedules in computational results. Additionally, we present the comparison of the optimal solution with other sequencing policies from literature. We also test the performance of the optimal solution with duration adjustments in order to find a schedule that is applicable to the real clinic environment. The contributions of this study can be listed as follows: 1. A two-stage stochastic programming model is proposed to solve the sequencing and scheduling problem with multiple patient types who have different service time distributions and heterogeneous no-show rates. 2. The structure of optimal sequences and schedules for different service time distributions and cost coefficients are presented in a primary care setting where multiple appointment types including same-day, prescheduled, complex/chronic and new patients are scheduled. 3. The scheduling problem for known appointment sequence with the objective of minimizing weighted combination of total waiting time, idle time and overtime costs is defined as a network flow model with uncertain demand.

33 22 4. The structural properties of the optimal schedule for the special case, where the appointment sequence is known, are derived using primal-dual optimality conditions. The remainder of the chapter is structured as follows. We present a brief introduction and problem definition in Section 3.1. The two-stage stochastic programming model is explained in Section 3.2. The characteristics of the optimal schedule for special cases of the problem (when the sequence is known) are derived using the primal-dual relationship in Section A computational study is performed to test the impact of different service time distributions and cost coefficients on appointment schedules, patient waiting time, resource idle time and overtime. The results of the computations are presented in Section 3.3. In Section 3.4, concluding remarks are provided Introduction The largest portion of access to the primary care is provided through scheduled appointments. Therefore, providing efficient and timely care is possible with well-designed appointment systems. Appointment scheduling is a very complex problem due to the variety of the services, which require different time and resource allocation. Additionally, uncertainty of patient behavior including no-show probability and service time variation is the primary factor, which creates inefficiency and avoidable cost. Problem of efficiency of the scheduling in healthcare varies depend on the main objective. Although, in business perspective, the main goal is to maximize the revenue, in a patient centered healthcare system, the main objective is to minimize the idle cost of resources, the waiting cost of the patients and the overtime cost. In this chapter, we focus on minimizing the expected cost of idle and overtime of resources and waiting time of patients. Ratio of the cost coefficients (i.e. ratio of overtime cost to waiting cost) can be interpreted as the importance of each parameter such that, different ratio leads to a different optimal solution. The majority of clinical practices use multiple appointment types and lengths to control the appointment schedules. The providers put limits on the number of certain appointments that can be scheduled on a given day to control access in the practice.

34 23 We consider sequencing and scheduling problem in outpatient clinics. This study is motivated by the current practice in primary clinics where providers use several appointment types (i.e. newborn, well-child care, new patient, established, routine, chronic/complex, physical, well-woman exam, etc.) to provide enough time for patients care needs. For example, a routine visit for an established patient might be scheduled as a 15-minute appointment, and physicals, new patients, and complex patients with chronic conditions might be scheduled for 30-minute appointments. The proper categorization of patients according to their care needs and estimation of the realization of the appointment times are essential to reduce patient waiting times and resource idle times. Despite the fact that all patients have some unique characteristics, we use a higher-level grouping of service time distributions based on the appointment durations allocated for each group (i.e. prescheduled vs. new patient). Besides service time distributions, patient no-shows affect how much appointment time is allocated for each patient. Uncertainty of patients attendance to an appointment is one of the biggest problems for current scheduling practices. Clinics usually turn to overbooking [92] or assign shorter appointment durations in order to deal with patient noshows.[40] Earlier studies on no-show modeling show that patient no-shows can be predicted based on patient, provider, and appointment characteristics. For example, new patients are shown to have higher no-show rates compared to established patients [85]. Hence, we consider heterogeneous no-show probabilities that change according to patient type. We incorporate the no-show probabilities into service time distributions. Based on our experience in working with primary care clinics, we observed that providers use scheduling templates that put restrictions on the sequence of appointments and the number of appointments of each type that can be scheduled on a given day (see Figure 1 for a template provider schedule). For example, providers might limit the number of new patient appointments. Some providers do not allow back-to-back scheduling of physicals due to requirements for other resources (i.e. nurses), which creates idle time for the provider. In this study, we assume the number of appointments types that should be scheduled on a given day is known. Our aim is to determine the optimal sequence and

35 24 schedule of these appointments for a single provider. Our objective is to minimize expected patient waiting time, resource idle time, and overtime. Figure 1: Template schedule of a provider in a primary care clinic In the existing literature, studies solve scheduling and/or sequencing problems with uncertain service durations, multiple patient types or heterogeneous no-show rates. For the first time in this study, we propose a two-stage stochastic programming model for appointment scheduling and sequencing problem, which considers multiple patient types and heterogeneous no-show rates simultaneously Two-Stage Stochastic Programming Model In this section, we present the proposed two-stage stochastic model to solve the appointment scheduling and sequencing problem by considering multiple patient type properties including heterogeneous service time distributions and no-show probabilities. Additionally, we present theoretical insights and theorems for the special case, where the appointment sequence is known. This special case is also defined as a minimum cost network flow problem. Lastly, we give numerical examples to illustrate how these insights can be used. Table 6 shows the notation used throughout the chapter, and Figure 2 shows a sample appointment schedule with relevant notation.

36 25 Table 6: Notation Parameters: H p Number of appointments of type p to schedule; n Total number of appointments to schedule; n = p H p i p ω T B i ω B pi C i I C i W Index for appointment sequence; i =1,2,,n Index for appointment types Index for scenarios; ω=1,2,, Ω Total clinic capacity (in minutes) Appointment duration for the i th appointment (random variable) Realization of appointment duration for appointment type p in sequence i in scenario ω Idle time cost coefficient for the i th appointment Waiting time cost coefficient the i th appointment C O Overtime cost coefficient Decision variables X i Z pi I i I i ω W i W i ω O Appointment duration allocated for the i th appointment Binary variable which is equal to one if patient type p is scheduled as the i th appointment Idle time after the i th appointment Idle time after the i th appointment in scenario ω Waiting time after the i th appointment Waiting time after the i th appointment in scenario ω Overtime O ω Overtime for scenario ω

37 26 Figure 2: Schematic representation of the model In this study, our aim is to determine the optimal appointment durations (X i ) and sequence (Z pi ) of n appointments. Z pi is a binary decision variable which takes value one when appointment type p is assigned to i th sequence. When scheduled appointment durations and sequence of appointments are found, the waiting time, idle time, and overtime can be calculated for a given realization of appointment durations (B ω pi ). The assignment of a non-negative value to X i can create either idle time (I i ω ) or waiting time (W i ω ). If the actual start time plus the appointment time of i th appointment is greater than the appointment time of (i+1) st patient, then patient i causes waiting time (W i ω ) for the next patient. If the actual start time plus the appointment time of i th appointment is less than the appointment time of (i+1) st patient, then patient i causes idle time (I i ω ) before the arrival of next patient. When the summation of idle times and assigned appointment durations exceeds the clinic capacity (T), overtime occurs (O ω ). We propose a two-stage stochastic programming model to solve the appointment sequencing and scheduling problem with multiple patient types who have different service time distributions and no-show rates. The first-stage model is as follows: min E ω (Q(X, Z, ω)) st Z pi p = 1 i = 1,, n (1) n Z pi = H p i=1 p (2)

38 27 n X i T (3) i=1 X i 0, Z pi binary p, i (4) The first-stage problem determines the appointment durations (Xi) and the sequence of appointment types (Z pi ). Constraint (1) guarantees that only one appointment type is assigned to each sequence. Constraint (2) ensures that the required number of appointments of each type (H p ) is scheduled. Constraint (3) puts an upper bound on the total duration allocated for n appointments due to available capacity (T). The objective is minimization of the expected value of function Q(X, Z, ω), which is the optimal objective function value of the second stage problem: n min Q(X, Z, ω) = (C i I I i ω + C i W W i ω ) i=1 + C O O ω (5) st W ω i I ω i = W ω i 1 ω + Z pi B pi p X i i = 1,, n (6) n O ω I i ω i=1 n p ω + Z pi B pi i=1 p=1 T (7) I i ω 0, W i ω 0, O ω 0 i = 1,, n (8) The objective of the second-stage problem is the minimization of total waiting time, idle time, and overtime costs. The second-stage decision variables (W i ω, I i ω, O ω ) are calculated based on the first stage decision variables (X, Z) and realization of service times (B ω pi ). Constraint (6) calculates the waiting time and idle time after the i th appointment. If the waiting time caused by the previous appointment plus the realization of actual appointment duration (W ω i 1 ω + p Z pi B pi ) is greater than the allocated appointment duration (X i ), then appointment i causes waiting time (W i ) for the next patient. Otherwise, appointment i causes idle time (I i ω ) before the next patient. Constraint (7) is used to calculate the overtime, which is greater than zero when the summation of realizations of

39 28 appointment durations and idle times is greater than the capacity. Since second stage problem is always feasible, the proposed model has the complete recourse property Scheduling for Known Appointment Sequence When appointment sequence is known, the following model determines the optimal appointment durations for each patient: (P-KS) n min (C W I i=1 i W i + C i I i ) + C O O st X i + W i I i W i 1 = B i i = 1,, n (9) n i=1 X i T (10) n n O i=1 I i i=1 B i T (11) X i, I i, W i, O 0 (12) The proposed model is a minimum cost network flow model with uncertain righthand-sides due to random service times (B i ). Figure 3 shows the structure of the network flow model. Figure 3: Minimum cost network flow model with uncertain right-hand-sides In this study, we assume B i comes from a known probability distribution and the parameters of the distribution change with respect to the appointment type. When patient no-shows are considered, the distributions change with respect to the no-show probability.

40 29 Hence, the only change in the proposed model is the distribution function of random variable B i Properties of the Optimal Schedule We consider a scenario-based approach to solve the proposed model (P-KS) with known sequence. We use the primal-dual relationship to derive the characteristics of an optimal schedule for a given sequence. The following primal model considers deterministic realizations of service times to determine the appointment durations: (Primal-S) n min 1 ω ( (C Ω i=1 i W W ω i + C I i I ω i ) + C O O ω ) (Cost n ) st X i W ω i 1 + W i ω I i ω = B i ω i = 1, 2 n, ω (10) (u i ω ) n i=1 X i T (11) (u T ) O ω n ω n ω i=1 I i i=1 B i T ω (12) (u ω O ) X i, I i ω, W i ω, O ω 0 (13) The model is similar to the P-KS except the random service time variables (B i ) are replaced with deterministic values for each scenario (B i ω ), the constraints are written for all realizations (ω), and objective function is replaced with the average of waiting time, idle time and overtime costs over all realizations. The dual of the Primal-S is as follows: (Dual-S) n max ( B ω ω n ω ω i u i Tu T + ( ω ω B i T)u O i=1 ) st u i ω ω u T 0 i = 1,, n u ω i u ω i+1 C W i /Ω i = 1,, n 1 u n ω C n W /Ω i=1 u i ω u O ω C i I /Ω i = 1,, n u O ω C O /Ω

41 30 u i ω arbitrary, u O ω, u T 0 Theorem 1 shows that the optimal X i 1 cannot be zero if the next appointment has a positive duration (X i > 0) when all realizations of random variable (B i ) are positive. Theorem 1: If the all realizations of random variables are positive for all patients (B i ω > 0), and a positive appointment duration is assigned to i th appointment (X i > 0), then the appointment duration for all appointments before the i th appointment should also be positive (X i 1 > 0). Proof: Consider the following two constraints for (i-1) st and i th appointments: X i 1 W ω i 2 X i W ω i 1 + W ω i 1 I ω i 1 + W i ω I i ω = B i ω Suppose X i > 0 and X i 1 ω = B i 1 W ω i 1 = W ω i 2 + B ω i 1, I ω i 1 = 0, = 0, then the following inequalities must be true: W ω i I ω i = W ω i 1 + B ω i X i = W ω i 2 + B ω i 1 + B ω i X i. Now, assume the appointment duration allocated for (i-1) st appointment is increased by Δ and the appointment duration allocated for i th patient is reduced by Δ where Δ min {B ω i 1, X i }. For the new solution, X i 1 = Δ > 0 and X i = X i Δ 0. Since X i 1 = Δ B ω i 1, the following inequalities must be true: W ω i 1 = W ω i 2 + B ω i 1 X i 1, I ω i 1 = 0, W ω i I ω i = W ω i 1 With the new solution, W ω i 1 + B ω i X i = W ω i 2 + B ω i 1 X i 1 + B ω i X i. is lower than W ω i 1, I ω i 1 = I ω i 1, and W ω i I ω i = W ω i I ω i. Thus, X i 1 = Δ > 0 and X i = X i Δ > 0 gives a better objective function value than X i 1 = 0 and X i > 0. When there is patient no-show, Theorem 1 becomes inapplicable due to realization of zero service times. For example, assume there are two patients with two scenarios, where the realization of the appointment durations are B 1 1 = 0, B 2 1 = 18 for scenario 1, and B 1 2 = 10, B 2 2 = 14 for scenario 2. When appointment durations are X 1 = 1 and X 2 = 14, the waiting times and idle times would be W 1 1 = I 1 2 = I 2 1 = I 2 2 = 0, I 1 1 = 1, W 1 2 = 9, W 2 1 =

42 31 4, W 2 2 = 9. If we assume same waiting time costs and idle time costs for all patients, then the objective function value would be 11C W + 0.5C I. When appointment durations are changed to X 1 = 0 and X 2 = 15, the waiting times and idle times would become W 1 1 = I 1 1 = I 1 2 = I 2 1 = I 2 2 = 0, W 1 2 = 10, W 2 1 = 3, W 2 2 = 9, and the objective function value would be 11C W, which is better than the previous schedule. ω Theorem 2 shows that the summation of shadow prices for appointment i-1 ( ω u i 1 ) is always greater than or equal to the summation of shadow prices for subsequent appointment ( ω ω u i subsequent appointments. ω u i 1 ω Theorem 2: ω ω u i, i. ) because a small delay in the earlier appointments would affect all Proof: Consider the set of constraints in the dual model that correspond to X i and W i ω : ω ω u i u T 0 and u ω i u ω i+1 Case 1: Suppose X i 1 C i W /Ω. > 0 and X i > 0. The complementary slackness condition guarantees that the corresponding dual constraints are tight ( ω = u T and u i 1 ω u i 1 ω u T ). Therefore, ω = ω u i = u T u i ω ω ω = Case 2: Suppose X i 1 > 0 and X i = 0. Due to complementary slackness, ω u i 1 = u T and ω u T = u i 1 u i ω Case 3: Suppose X i 1 ω ω. = 0 and X i = 0. Based on the primal constraints, W ω i 1 > 0 and W ω i > 0, and the corresponding dual constraints are tight (u ω j u ω j+1 = C W j /Ω for j = i 1, i) for all ω. Since, u ω i 1 Case 4: Suppose X i 1 = u ω i + C W i 1 /Ω (u ω i 1 > u ω ω ω i ) for all ω, ω u i 1 u i ω ω. = 0 and X i > 0. Based on the primal constraint for X i 1, W ω i 1 > 0 and the corresponding dual constraint for W ω i 1 is tight (u ω i 1 u ω i = C W i 1 /Ω) for all ω. If ω ω W ω we sum over ω, we get ω u i 1 = ω u i + ω C i 1 /Ω, which shows that ω u i 1 ω ω u i. When the dual variable corresponding to the capacity constraint is zero (u T = 0) for a given set of patients, more patients can be added to the current schedule to fully utilize the existing capacity. As more patients are added to the schedule, eventually u T becomes positive, which means the total appointment duration allocated for all patients is equal to

43 % Change in Expected Total Cost Number of Binding Scenarios 32 the capacity T. The following theorem (Theorem 3) shows that the maximum marginal increase in expected cost for every additional patient added to the schedule is observed when all scenarios cause overtime. Figure 4 shows the percentage change in expected cost and the number of scenarios that cause overtime with respect to the number of scheduled appointments for a small numerical example where the overtime cost, idle time, and waiting time costs are 10, 1, and 5 respectively % 42.56% 32.08% 24.41% % 306 (a) Figure 4: a) Percentage change in expected total cost per patient vs. number of scheduled appointments, b) Number of binding scenarios vs. number of scheduled appointments Theorem 3: Assume u T > 0 and let K be the set of scenarios that cause positive overtime. The maximum marginal increase in expected total cost per patient ( Cost n n ( Cost n 1 ) is observed when all scenarios cause overtime ( K = Ω ). n 1 n Proof: Since K is the set of scenarios that cause positive overtime, i=1 (B k i + I ω i ) all k K. Therefore, ω ω u i K. The objective function of the dual model is: 1.11% 0.96% 4.00% Number of Appointments ω n ( B ω ( ω K ( i u i ) Tu T + B i=1 (b) Cost n 1 n 1 )/ > T for = u T for all x > 0, u O k = C O /Ω, and u n k = C n W /Ω for all k n i=1 i k T) ) K C O Ω Note that, when K is small, (u T = 0) and ( ω ω B ω ω i=1 i u i ) Tu T. When we increase K by adding new patients, eventually, full capacity will be used then, (u T ) will be positive and ( ω n B ω ω i=1 i u i ) Number of Appointments Tu T will increase with a constant rate after ( K = Ω ). On the other hand, overtime cost in the objective function can be represented as a piecewise linear

44 33 function when ( K < Ω ) and a linear function after ( K = Ω ). Note that n k ( ( B i T) ) might be negative, and will increase with every additional patient. K i=1 Case 1: Suppose that some of the scenarios do not cause overtime ( K < Ω ). Then, n k ( ( B i T) ) increases by the amount of ω E ω (B ω i ) for every additional patient. K i=1 Similarly, K also increases with the additional scheduled patients until all scenarios cause overtime ( K = Ω ). Note that, there is a positive correlation between the increase rate of K and the value of (B ω i ). Hence, multiplying these two positively correlated increasing linear functions gives us a second degree parabolic function where the slope is maximum at ( K = Ω ). Case 2: Suppose that all of the scenarios cause overtime ( K = Ω ), then n k ( ( B i T) ) still increases by the same amount of ω E ω (B ω i ) for every K i=1 additional patient. On the other hand, K is constant after ( K = Ω ). Hence, the marginal increase of the overtime cost is constant and depends on ω E ω (B ω i ) after all the scenarios cause overtime. Special Case: As we mention earlier, scheduling problem can be modeled from the business perspective. In order to convert the proposed model into a maximum profit problem, assume that every scheduled patient leaves constant revenue to the clinic. Let the revenue per patient be (r), number of patients scheduled for the session be (n), and the minimum total expected cost of idle time, waiting time and overtime be the function (Cost n ). Note that the proposed model can find the optimum total expected cost for a given (n). Then the total profit P(n) can be written as, P(n) = r n (Cost n ) As explained in theorem 3, the cost function (Cost n ) is a second degree parabolic function until all the scenarios are binding, and then increases linearly with respect to the increase of (n). Hence, if there is at least one number of scheduled patients (n>0) where P(n) is positive, we can conclude that;

45 Expected Profit % Change of The Cost per Patient 34 1) There is at least one number of scheduled patients, arg max P(n) is the number 1 n N of patients needs to be scheduled in order to maximize the profit. 2) The number of patients, which is obtained by using the theorem 3, can be used as a threshold for the number of overbooking. For instance, assume that the overtime cost, idle time, and waiting time costs are 10, 1, and 5 units respectively (same as in Figure 4), and we make revenue of 50 units for each scheduled patient. Then, the expected profit for the given case can be illustrated as in Figure 5. According to Figure 5, scheduling 15 patients maximizes the expected profit while scheduling 17 patients gives us the highest marginal increase of the expected cost per patient. $900 $800 43% $848 45% 40% $700 $600 $500 $400 $300 $200 $100 $498 $461 $424 $126 $139 $152 36% $529 $540 $495 $462 $388 $305 15% $210 $171 4% $646 32% $254 $102 35% 30% 24% 25% 20% 15% 10% 5% $0 1% 1% % Number of Appointments Expected Profit Expected Cost % Change of The Cost per Patient Figure 5: Expected profit for number of scheduled appointments In Theorem 1, we prove that if a patient is scheduled with positive appointment duration, all scheduled appointments prior to that patient should also have positive duration in order to get an optimal schedule. In Theorem 2, we show that the shadow prices of the appointments have a diminishing trend throughout the sesison which accredits more importance to the duration allocation for the initial appointments. Similarly, in Theorem 3,

46 35 we show that the occurrence of the maximum marginal increase of the expected cost per patient is related to the waiting time cost coefficient and the dual variable of the capacity constraint Computational Results We performed a computational study in two parts. In the first part, we consider the special case of single patient type to find optimal appointment schedules. In the second part, we consider multiple patient types (same-day, prescheduled, new, and complex/chronic) with different service time distributions and no-show rates to find optimal sequence and schedules. We compare the proposed method with other sequencing policies from literature including scheduling the patients with high service time variation and no-show rate at the beginning (NTBG), scheduling the patients with low service time variation and no-show rate at the beginning (ETBG) and sequencing all types of appointments in an alternating order (ALTER) [11]. IBM Ilog Cplex 12.0 is used to solve the stochastic programming models in a computer with 3.4 GHz speed and 32GB RAM Computational Results for Single Patient Type We solve the proposed model with single patient type to analyze the impact of service time distributions, cost coefficients, and no-shows on optimal appointment durations and objective function values. We consider Normal, Lognormal, and Gamma distributions with the same mean (15 minutes) and standard deviation (8 minutes) for service times. We determine the cost coefficients for waiting time, idle time, and overtime based on their relative importance with respect to each other. For the first level, we assume all cost coefficients are same. For the second and third levels, we assume idle time and overtime costs are more important than waiting time costs, and overtime cost is more important than idle time cost. To analyze the effect of no-shows, we first assume all patients arrive for their appointment (0% no-show rate) and then include patient no-show (20% noshow rate) for all patients. The experimental design factors can be seen in Table 7.

47 Objective Function 36 Table 7: Experimental design factors for single patient type Service time distributions Factors Level 1 Level 2 Level 3 Normal(15,8) Lognormal(2.57,0.52) Gamma(3.18,4.73) Cost coefficients (Overtime cost, idle time (1, 1, 1) (5, 2, 1) (10, 5, 1) cost, waiting time cost) No-show rate 0% 20% As mentioned earlier, we use a scenario-based approach where the deterministic equivalent of the two-stage stochastic programming model is solved by considering several scenarios simultaneously in a single large integer programming model. However, as expected, increasing the number of scenarios increases the problem size and computation times. Therefore, the number of scenarios should be determined according to the desired level of solution quality, and computational complexity. We performed preliminary computations to see the impact of number of scenarios on the objective function value. Figure 6 shows the objective function values as the number of scenarios increase. We see that all distributions converge to steady-state at 1000 scenarios. When the number of scenarios is more than 1000, the computation time increases significantly, and the percentage improvement in objective function value is less than 1%. Therefore, we use 1000 scenarios for the rest of the computations and comparison of the results Number of Scenarios Gamma LogNormal Normal Figure 6: Convergence of objective function values with respect to number of scenarios (Overtime Cost=10, Waiting Cost=1, Idle Cost=5) Effect of service time distributions Figure 7 shows the optimal appointment durations for Gamma, Lognormal and Normal distributions when all cost coefficients are equal to one. The optimal appointment

48 Duration (minutes) Duration (minutes) 37 schedule shows a dome-shaped structure for all distributions. When cost coefficients are equal to each other, the appointment durations are less than the expected value for the first and the last patients, and greater than the expected value for other patients. The appointment durations assigned to last patient is shorter than the previous appointments for all distributions Figure 7: Optimal appointment durations for different service time distributions with cost coefficients (1,1,1) Figure 8 shows the appointment durations when the cost coefficients for idle time and overtime are increased to 5 and 10, respectively. The appointment duration allocated for the last patient increases due to the increase in overtime cost. The appointment durations allocated for other patients decrease due to the increase in idle time cost and limited clinic capacity Appointment (i) Gamma LogNormal Normal Gamma LogNormal Normal Appointment (i) Figure 8: Optimal appointment durations for different service time distributions with cost coefficients (10,1,5)

49 Duration (minutes) Duration (minutes) Effect of cost coefficients Figure 9 shows the effect of cost coefficients on scheduled appointment durations for lognormal distribution. When the cost coefficients are equal, longer appointment durations are assigned to the earlier patients because a small delay for earlier patients might incur waiting time for all succeeding patients, which increases the total waiting time significantly. When overtime and idle time costs are increased, the appointment durations are decreased for earlier appointments to reduce the probability of idle time between patients and overtime at the end of the session. The appointment durations for the last appointments are increased to reduce the amount of overtime Appointment (i) LogNormal LogNormal LogNormal Figure 9: Optimal appointment durations for lognormal distribution with different cost coefficients Figure 10 shows the expected waiting and idle times for each appointment with respect to different cost coefficients. The expected idle time decreases when the costs of idle and overtime increase. On the other hand, the expected waiting time increases throughout the session since the waiting time of the patient i affects the waiting time of the subsequent patients Waiting Time for Waiting Time for Waiting Time for Idle Time for Idle Time for Appointment (i) Figure 10: Expected waiting and idle times for each appointment at different cost coefficients for lognormal distribution

50 Duration (minutes) Duration (minutes) Duration (minutes) Effect of patient no-shows In the proposed model, we include no-show probabilities by updating the distribution of realization of the appointment durations. By doing so, realization of the appointment time can be zero with a given no-show probability. In this section, we compare the optimal schedules and objective function values with and without patient noshows. We assume that the no-show rate for the single patient type is 20%. We present the results for lognormal distribution, but we observed similar results for other distributions. Figure 11 shows the optimal appointment durations with and without no-shows for different cost coefficients. The proposed model allocates shorter appointment durations when patient-no shows are considered. The optimal durations also show smaller variation for all cost coefficients with No-Show with No-Show with No-Show Appointment (i) Appointment (i) Appointment (i) Figure 11: Optimal appointment durations with and without no-shows for different cost coefficients As it is illustrated in Figure 12, expected idle times are higher when we consider the no-shows. This is an expected result since higher idle times occur when the realization of the appointment time is zero. On the other hand, expected overtime is lower when we consider no-shows. Note that expected waiting time of the last patient is equal to the expected overtime.

51 Duration (minutes) Waiting Times without No-Show Appointment (i) Figure 12: Expected waiting and idle times with and without no-shows for equal cost coefficients (1, 1, 1) We present the summary of the computational results for single patient type in Table 8. The objective function values are higher when no-shows are considered. The optimality gap between these two settings (without no-show vs. with no-show) increases with the increase in idle time and overtime costs. Waiting Times with No-Show Idle Times without No-Show Idle Times with No-Show Table 8: Computational results for lognormal distribution (with and without no-shows) Without no-show With no-show Cost coefficients Expected overtime Expected average waiting time Expected average idle time Objective function value Minimum appointment duration Maximum appointment duration Difference between minimum and maximum appointment durations Computation time (seconds) Computational Results for Multiple Patient Types In this section, we solve the proposed stochastic programming model with multiple patient types (same-day, prescheduled, new and complex/chronic) with different service time distributions and no-show rates (see Table 9). We use the service time distributions from the study done by Oh et al.[61], and calculate the no-show probabilities according to the number of patients.

52 41 Table 9: Service time distributions and no-show rates for each patient type Patient type Service time Service time distribution mean and standard deviation No-show rate Same-day Lognormal (2.41,0.52) μ=12.7, σ=7 9.2% Prescheduled Lognormal (2.68,0.51) μ=16.6, σ=9 27.8% Complex/Chronic Lognormal (2.89,0.4) μ=19.5, σ= % New Lognormal (2.89,0.4) μ=19.5, σ= % Based on mean service times and the clinic capacity (240 minutes), the number of appointments that should be scheduled are determined as 4, 6, 2, and 3 for same-day, prescheduled, new, and complex/chronic patients, respectively. In the following sections, we first show the impact of different cost-coefficients and no-shows on optimal sequences and schedules. Then, we compare the optimal schedules with the schedules obtained from the sequencing rules in literature. We include 500 scenarios in our calculations for sequencing multiple patient types since we could not to solve 1000 scenarios due to the size of the problems Effect of cost coefficients In order the see the effect of cost coefficients, we first assume that all patients will show up for their appointments. Since new and complex/chronic patients have the same service time distribution, we combine these two patient types and represent them as new/complex. The optimal schedule and sequence of 4 same-day, 6 prescheduled, and 5 new/complex appointments for all cost settings can be seen in Figure 13. The sequence of the first three and the last six appointments are same for all cost settings. However, as the overtime and idle time costs increase, the appointment durations assigned to these appointments decreases at the beginning of the session and increases at the end of the session. Although the standard deviations of service times are very close for all appointment types, durations assigned to same-day and prescheduled appointments decrease and durations assigned to new/complex appointments increase when the overtime and idle time costs increase. Since the appointment durations at the end of the increases while the other appointment durations decrease when the overtime and idle time costs increase, this result can be related to the fact that the last appointment is scheduled to the new/complex patient. Additionally, the sequence and appointment duration of the appointments in the middle of the session do not follow any pattern.

53 Appointment Duration Appointment Duration Appointment Duration N/C N/C S N/C P Cost setting: (1.1.1) P P S S S Appointment i N/C N/C S N/C P Cost setting: (5.1.2) S S P P S P P P P N/C P N/C N/C P N/C Appointment i N/C N/C S P S Cost setting: (10.1.5) N/C P P S S P P N/C P N/C Appointment i Same-Day New/Complex Prescheduled Figure 13: Optimal appointment durations and sequences without no-shows for all cost settings Figure 14 shows the optimal schedule and sequence of multiple appointment types for different cost coefficients when patient no-shows are considered. Similar to the case without no-shows, appointment durations at the beginning and at the end of the session decrease when the cost of overtime and idle time increase. The appointments with lower variation and no-show rates such as same-day and complex/chronic appointments are pushed to the first half of the session. The new and prescheduled patients, which have high no-show rates, are scheduled closer to the end of the session. Moreover, we observe that assigning shorter appointment durations and balancing the patient waiting time with the no-show of subsequent appointment may be a good strategy for the appointments in the middle of the session. For instance, two same-day appointments are assigned in the 8 th and 10 th appointments, while a new patient is assigned between these appointments with a shorter duration for the equal cost setting.

54 Appointment Duration Appointment Duration Appointment Duration S C S P P P Cost setting: (1.1.1) C N S S P P P N Appointment i S C S P P P Cost setting: (5.1.2) C C S S P C P N P N Appointment i S S C P C S Cost setting: (10.1.5) C N P P S Appointment i P P P N Same-Day Complex/Chronic Prescheduled New Figure 14: Optimal appointment durations and sequences with no-shows for all cost settings Comparison with other sequencing rules Cayirli et al. [11] suggest that sequencing routine patients to the beginning and assigning the mean service time durations for each patient type reduces waiting time, idle time and overtime costs simultaneously. They use simulation analysis to compare the results of assigning new patients with longer appointment durations at the beginning of the day (NTBG), at the end of the day (ETBG), and with an alternating pattern (ALTER). They used fixed appointment durations of 15 and 20 minutes for the existing and new patients respectively. In this study, we consider four patient types. Same-day and pre-scheduled patients have shorter appointment durations with lower variation than new and complex/chronic patients who have longer appointment durations with higher variation. For NTBG, we sequence patients with the highest service time variation at the beginning of the day (C,N,C,N,C,P,P,P,P,P,P,S,S,S,S). For ETBG, we sequence patients with the lowest service time variation at the end beginning of the day

55 44 (S,S,S,S,P,P,P,P,P,P,C,N,C,N,C). Similarly, for the alternating policy ALTER, we distribute all patient types in an alternating order throughout the day (P,S,C,P,N,S,P,C,P,S,N,P,C,S,P). We compare the optimal schedules with the schedules obtained from these sequencing rules. Table 10 shows the total expected cost and the percentage gap with respect to the best solution when patient no-shows are not considered. We find that scheduling new patients at the beginning of the session has the highest expected cost and scheduling new patients with an alternating order has the lowest expected cost among the three sequencing rules. The gap between the optimal solution found by solving the stochastic programming model and these sequencing rules becomes larger with the increase in overtime and idle time costs. Table 10: Objective function values and percentage gap between the optimal solution and the sequencing rules for different cost coefficients Cost Coefficients Total Expected Cost Percentage Gap Overtime. NTBG ETBG ALTER Optimal NTBG ETBG ALTER WaitingTime.IdleTime % 22.53% 10.27% % 22.54% 20.52% % 50.08% 41.34% Table 11 shows the total expected cost and the percentage gap when heterogeneous no-show probabilities are considered. We see that sequencing patients in an alternating order gives the closest solution to the optimal solution for the equal cost setting. However, sequencing patients with high variation and no-show rate at the end of the session gives the smallest optimality gap for high overtime and idle time cost settings. Table 11: Objective function values and percentage gap between the optimal solution and the sequencing rules for different cost coefficients (no-show included) Cost Coefficients Total Expected Cost Percentage Gap Overtime. NTBG ETBG ALTER Optimal NTBG ETBG ALTER WaitingTime.IdleTime % 15.00% 14.72% % 41.12% 47.41% % 84.70% 95.72%

56 45 The expected total cost is lower compared to the case without no-shows. We believe this is due to the lower number of scenarios that exceed the available capacity. Since the capacity is fixed, assigned appointment durations are shorter and the number of scenarios, in which overtime occurs, is smaller when patient no-show is included. On the other hand, fixed appointment durations and sequence increases the percentage gaps for the high cost settings. Therefore, the importance of finding the optimal schedule increases with high variation and complexity Computational Results for Practical Application in Primary Care Clinics In this section, we test the model under a more practical case and compare the results with the suggested sequencing rules from the literature. As presented in Section , when we compare the optimal solution and the schedules obtained from suggested sequencing rules, we find that there is a significant improvement in the objective function. Nonetheless, the optimal solution cannot be used in primary care clinics due to overly precise appointment durations. For example, if the optimal appointment duration of the first patient is 13 minutes and 42 seconds, and assuming that the first patient is scheduled at 8:00 a.m., then the second patient should be scheduled at 8:13 a.m. However the application of appointment scheduling is not possible with minute or second increments since it is impossible for the patients to remember these appointment times. In fact, this might create more uncertainty in the system and decrease the efficiency improvement that is envisioned for the optimal solution. Therefore, we make two adjustments to the model in order to test the improvement of the objective function in a realistic and applicable setting. The first adjustment is to fix the appointment durations for discrete intervals within 0 to 30 minutes, where the model is allowed to pick any 5-minute increment. By doing so, instead of assigning mean appointment duration, we still provide some flexibility to the model to adjust appointment durations for generated scenarios as well as creating an applicable schedule. The second adjustment is to use discrete data to generate the realization of appointment durations. As it is unrealistic to expect patients to be unpunctual in the precision of seconds, it is also not realistic to expect the clinic data to be recorded in seconds. Hence, we use the same distributions (lognormal) for all patient types but generate

57 46 the realization of appointment durations as integers. Moreover, we use 15-minute appointment slots for same-day and prescheduled appointments, and 20-minute appointment slots for new and complex/chronic appointments for the sequencing rules (NTBG, ETBG and ALTER). We include four patient types and corresponding heterogeneous no-show rates in the computations. We present the comparison of objective function values in Table 12. The objective function values under adjusted settings are higher than the results presented in previous section due to the use of discrete data. Similar to the results found in the previous section, smallest percentage gap is found by using alternating order for the equal cost setting, and by sequencing patients with high variation and no-show rates at the end of the session for high overtime and idle time cost settings. We still observe a significant percentage gap between the optimal solution and the sequencing rules. Table 12: Objective function values and percentage gap between the optimal solution and the sequencing rules for different cost coefficients (practical case) Cost Coefficients Total Expected Cost Percentage Gap Overtime. WaitingTime.IdleTime NTBG ETBG ALTER INT_MLT NTBG ETBG ALTER % 11.53% 11.28% % 37.92% 43.94% % 81.98% 92.84% Figure 15 shows the optimal appointment durations and sequences for different cost settings. The sequences are same for high and equal cost settings, and only the 8th and 14th appointments switched when the cost setting is (5.1.2). These results suggest that an optimal sequence is very robust against the change of cost coefficients. On the other hand, appointment durations at the beginning of the session and at the end of the session decreases when the overtime and idle time costs increase. For instance, the optimal solution for the cost setting (10.1.5) shows that the time allocation for the first and the second patients are approximately 8 and 3 minutes shorter than the mean service time duration for same-day appointments respectively. Similarly, the last two appointments are assigned to the prescheduled patients with the duration of 10 minutes, which is also approximately 7 minutes shorter than the expected service time.

58 Appointment Duration Appointment Duration Appointment Duration 47 The optimal solution suggests the following strategies: 1. Assign shorter appointment durations if the overtime and idle time costs are high. 2. Schedule majority of the appointment types with high variation and no-show rate at the end of the session. 3. If an appointment type with high variation and no-show rate is scheduled during the session, assigning appointment types with low variation and no-show probability before and after that appointment may balance the costs. (i.e. see the 4 th, 6 th, and 9 th appointments) S S C P S P Same-Day Complex/Chronic Prescheduled Cost Setting: (1.1.1) New C C S P P N N P P Appointment i S S C P S P Cost Setting: (5.1.2) C C P P P Figure 15: Optimal appointment durations and sequences with heterogeneous no-shows for all cost settings in practical case Summary of the Computational Results N N S Appointment i S S C P S P C Cost Setting: (10.1.5) S In the computational results section, we observe the dome shape on the distribution of the optimal appointment durations as mentioned in the literature. However, we see that the cost coefficient ratios and properties of patient types such as no-show rates and service P C P N P N P P Appointment i

59 48 time variation, have significant impact on the appointment time allocation of patients. When we compare the optimal solution with other sequencing rules, we find that higher ratio of the overtime and the idle time cost to the waiting time cost yields to higher optimality gap when we exclude patient no-shows. Yet, this gap increases tremendously when we include no-shows. Moreover, computational results show that although scheduling patients with high service time variation at the end of the session gives better results than scheduling them at the beginning of the session or in an alternating order, it does not always provide the optimal sequence. Although the optimal solution is not always applicable to practice, we show that finding an optimal solution with applicable adjustments is still significantly better than the sequencing guidelines that we compared. We present the highlights of this research s findings listed below; 1. Cost ratio of waiting time, idle time and overtime has a high impact on the schedule. 2. Including no-show decreases the expected total cost of the system 3. Proposed model performs significantly better than the compared sequencing rules 4. In practical case, where the appointment durations assigned with 5-minute increments, results show that the model still provides significant improvement Conclusions We propose a two-stage stochastic programming model with heterogeneous random service times and no-show probabilities for multiple patient types to determine optimal appointment schedule and sequence that minimizes total expected overtime, waiting time and idle time costs. This study shows the importance of the using advanced scheduling and sequencing techniques when the level of uncertainty is high. Therefore, for the first time, we include patient specific no-show rates and random service time for solving a scheduling and sequencing problem. Additionally, we present three theorems by using dual-primal relation of the optimal solution when the sequence is known. In Theorem 1, we show that the optimal appointment duration cannot be zero if the subsequent appointment has a positive duration, when all realizations of random variable are positive. In Theorem 2, we show that the summation of

60 49 shadow prices for an appointment is always greater than or equal to the summation of shadow prices for subsequent appointment, when the cost of waiting is greater than or equal to the cost of idle time and overtime. In Theorem 3, we show that the maximum marginal increase in expected cost is observed when all scenarios cause overtime. In this chapter, we show the behavior of the proposed stochastic appointment scheduling and sequencing model under different parameters such as overtime, waiting time and idle time costs, heterogeneous no-show rates and service time distributions. We show the relation between the selected distribution and number of scenarios included in order to find the steady state of the optimal objective function. Similarly, we compare the different sequencing guidelines, which suggest to schedule new patients at the beginning or at the end of the session, or in an alternating order, with the global optimal solution and illustrate the optimality gap within different cases such as the effect of no-show probability and the number of patient types. Although the sequencing guidelines are easy to apply, computational results show that the optimality gap increases with the increase of uncertainty and complexity of the system. Hence, we conclude that sophisticated mathematical modeling is required in order to acquire a better outcome. On the other hand, computation time and size of the problem expand tremendously with the level of uncertainty and complexity.

61 50 Chapter 4 Stochastic Appointment Scheduling with Patient Unpunctuality In this chapter, we propose a two-stage stochastic model to find the optimum scheduled start times and sequence of patients with heterogeneous service time distributions, no-show rates, and unpunctuality. The objective of the model is to find a schedule that provides minimum expected total cost of waiting time of patients, idle time of the providers, and the overtime of the clinic. We also propose a heuristic approach to find solutions to large-scale problems. In our computations, we show the impact of patient mix, cost coefficients, and unpunctuality level on optimal schedules. We use the proposed heuristic and compare the results with other sequencing policies from literature. The contributions of this study can be listed as follows: 1. This is the first study that considers patient unpunctuality in a two-stage stochastic programming model to solve the sequencing and scheduling problem with multiple patients and no-shows. 2. The structure of optimal sequences and schedules for different service time distributions, variation of patient unpunctuality and cost coefficients are presented in a primary care setting where multiple appointment types including same-day, prescheduled, complex/chronic, and new patient appointments are scheduled. 3. A heuristic approach, which combines ordering the urgencies of appointments with local search, is developed for finding a solution for large-size problems. The remainder of the chapter is structured as follows. We present a brief introduction and problem definition in Section 4.1. The two-stage stochastic programming model and the proposed sequencing game heuristic are explained in Section 4.2 and Section 4.3, respectively. A computational study is performed to test the impact of different service time distributions, cost coefficients, and variation of patient unpunctuality on appointment schedules, patient waiting time, resource idle time and overtime. The results of the computations are presented in Section 4.4. In Section 4.5, concluding remarks are provided.

62 Introduction One of the major problems in the healthcare delivery system is the waste of healthcare providers valuable time. Scheduling patients in a primary care setting is usually done by assigning a fixed duration to all patients, and the sequence of the patients are determined by the preferences and capacity availability. However ignoring the complexity of the scheduling system, which is the result of the uncertainties involved during the visits, creates inefficiency and non-value-added time for the healthcare delivery system. Additionally, understanding the trade-off between the amount of time the patients have to wait and how much of the doctors time is spent idle is crucial to build a robust appointment scheduling system. In this study we are minimizing the expected total cost of the waiting time of patients, the idle time of the provider, and the expected overtime. We focus on the scheduling and sequencing problem in a primary care setting due to its high volume of visits. This study is motivated by the current practice in primary clinics where providers use several appointment types (i.e. newborn, well-child care, new patient, established, routine, chronic/complex, physical, well-woman exam, etc.) to provide enough time for patients care needs. Additionally, we propose using individual time assignment for each type of visits, instead of using fixed appointment durations. We also believe that cost of the care can be decreased significantly by using a sophisticated appointment sequencing and scheduling approach. We know that patient no-show increases idle time of providers and overloads the clinic capacity due to rescheduling. Similarly, even if patients show up to their appointments, they are usually either late or early. Brahimi and Worthington[7], Cox et al.[15], Fetter and Thompson[24], Klassen and Rohleder[39] show empirical evidence that, patients tend to arrive early for their appointments. The definition of patient unpunctuality is the time difference between a patient s arrival time and the scheduled appointment time. Unpunctuality of patients affects the efficiency and quality of care. Late arrivals increase the idle time of providers and may increase the waiting time of the next patient. Therefore, patients are usually asked to come to the clinic earlier than their scheduled start time. However this might increase the waiting time of the patient if there is tardiness in the

63 52 schedule, and might create overutilization of the waiting room. On the other hand, early arrival of patients might decrease the idle time if examination of the previous patient ends early. In this study, we are focusing on this tradeoff between the patient unpunctuality and the cost of waiting and idle times. We use stochastic programming to model and solve the appointment sequencing and scheduling problem by considering patient unpunctuality where there is a single provider and multiple patient types with heterogeneous no-show rates and service times. By using the proposed model, we find the optimal sequence of different appointment types with their start times. The randomly generated appointment durations and patient unpunctuality creates the corresponding actual start times, idle times and waiting times for each scenario. Additionally, we find the expected waiting and idle time of each appointment slot by considering the assigned schedule. Likewise, expected overtime is calculated by using the scenarios that exceed the clinic capacity Stochastic Programming Models The parameters, random variables, realizations of random variables, and decision variables used throughout the chapter are presented in Table 13. The assumptions considered to model the sequencing and scheduling problem with patient unpunctuality are: 1. The appointment of the first patient cannot start before time zero. 2. Provider sees the next patient immediately if the examination of the current patient is completed and the next patient has arrived. 3. When a patient comes early and waits for his/her appointment, this is included in the waiting time even if the appointment starts at the scheduled time. Table 13: Notation Parameters: p Index for appointment types H p Number of appointments of type p to schedule n Total number of appointments to schedule; n = p H p

64 53 i Ω ω T C i I C i W C O B i Index for appointment sequence; i =1,2,,n Total number of scenarios Index for scenarios; ω=1,2,, Ω Total clinic capacity (in minutes) Slot length Idle time cost coefficient for the i th appointment Waiting time cost coefficient the i th appointment Overtime cost Appointment duration for the i th appointment (random variable) P E i Earliness of patient for the i th appointment (random variable) P L i B i ω ω B pi Lateness of patient for the i th appointment (random variable) Realization of appointment duration for appointment i in scenario ω Realization of appointment duration for appointment type p in sequence i in scenario ω P E iω Realization of earliness of patient for appointment i in scenario ω P L iω Realization of lateness of patient for appointment i in scenario ω Decision variables S i Z pi A i ω I i ω W i ω O ω Xi Scheduled start time of the i th appointment Binary variable which is equal to one if patient type p is scheduled as the i th appointment Actual start time of the i th appointment in scenario ω Idle time of the i th appointment in scenario ω Waiting time of the patient in i th appointment in scenario ω Overtime for scenario ω Number of slots between appointments (i-1) and i Stochastic Programming Model When the Sequence is Known If the sequence is known, the stochastic appointment scheduling model, which minimizes the total expected cost of waiting time, provider idle time and overtime, can be modeled as follows:

65 54 n min 1 ( (C Ω i W W ω I ω i=1 i + C i I ω i ) + C O O ω ) (O. 1) st S 1 = 0 (1) S i S i 1 0 i = 2 n (2) S n T ω (3) A ω 1 S 1 + P L ω 1 A ω i S i + P L ω i P Eω i A ω i A ω i 1 ω + B i 1 ω (4) i = 2 n, ω (5) i = 2 n, ω (6) I 1 ω A 1 ω ω (7) I ω i A ω i A ω i 1 ω B i 1 i = 2 n, ω (8) W ω i A ω i S i P L ω i + P Eω i i = 1 n, ω (9) O ω A n ω + B n ω T ω (10) (S i S i 1 ) = X i i = 2 n (11) X i Integer, S i, A i ω, W i ω, I i ω, O ω 0 i = 1 n, ω (12) The objective function (O.1) minimizes the summation of the total expected cost of waiting time, idle time and overtime for all appointments and all scenarios. Constraints (1), (2), and (3) state that the first patient should be scheduled at time zero, the scheduled start time of each appointment should be later than the scheduled start time of previous appointment, and the last appointment should be scheduled at or before the clinic end time, respectively. Constraints (4), (6), and (6) calculate the actual start time (A i ω ) of patients according to the scheduled start time (S i ), punctuality of the patient (P L jω, P E j ω ), and the completion of the previous appointment (A ω i 1 + B ω i 1 ). If the patient arrives late (P L ω i >

66 55 0), then the actual start time should be later than the scheduled start time and completion time of the previous patient. If the patient arrives early and the provider is available, then the service can start immediately. For the first patient, even if the patient arrives early, the service cannot start before time zero due to unavailability of the provider until time zero. Constraints (7) and (8) are used to calculate idle time for the first appointment (I ω 1 ) and the rest of the appointments (I ω i ), respectively. Similarly, constraint (9) finds the waiting time of a patient (W ω i ), which depends on the scheduled appointment time, actual start time of the appointment, and the punctuality of the patient. Constraint (10) determines the overtime (O ω ) for each scenario. Constraint (11) sets all appointment times to an integer multiple of to get easy-to-remember appointment times. Without this constraint, the optimal solution may not be applicable to practice. For instance, asking a patient to show-up at 9:36am might increase unpunctuality and therefore is not desirable for the primary care clinics. The value of can be determined as 5, 10 or 15 minutes depending on the service times for the patient mix and clinic preferences. In the proposed model, the realizations of appointment durations (B ω i ) are determined based on the service time distributions and no-show probabilities. For example, if the probability of no-show is γ, then B ω i is expected to be zero with a probability of γ. If a patient does not show up to his/her appointment, then the realizations of random variables related to unpunctuality (P L jω, P Eω j ) should also be zero. The proposed model still calculates the actual start times for those patients to make sure provider idle time and waiting time for other patients are calculated correctly. In Figure 16, we present a sample appointment schedule for a given scenario. In this example, the clinic capacity is set to 75 minutes and 5 patients are scheduled optimally. The actual start times (A ω i ), idle times (I ω i ), waiting times (W ω i ), and overtime (O ω ) are calculated based on the realizations of random variables (B ω i, P L jω, P Eω j ) and the scheduled appointment times (Si ).

67 56 Figure 16: Sample appointment schedule for a given scenario Stochastic Programming Model When the Sequence is Unknown In this section, we adjust the model in order to solve the problem of finding the optimal schedule and sequence of different appointment types. In practice, providers are assigned to different appointment types such as same-day appointments, prescheduled patients, new patients and patients with complex/chronic condition. Characteristics of the appointment, such as service time distribution, no-show rate and punctuality of patients, may vary depending on the appointment type. For instance, patients with complex/chronic conditions require more time compared to routine visits. Likewise, no-show rate of new patients is much higher compared to no-show rate of same-day appointments [61]. Punctuality of patients may also vary depending on the appointment type. In order to introduce the sequencing problem, first we need to define the constraints related to sequencing, which are shown below: Z pi = 1 i = 1,, n (13) p n Z pi = H p p (14) i=1 Constraint (13) states that only one type of appointment can be assigned to a given appointment slot and constraint (14) defines the total number of appointment slots that has to be assigned for each appointment type.

68 57 After defining the sequencing rules, we need to adjust the constraints (6), (8), and (10) since the realization of the appointment durations (B ω pi ) now depend on the appointment type (p). Hence, we can rewrite these equations as follows; A ω i A ω i 1 + p (Z pi 1 B ω pi 1 ) i = 2 n, ω (6 ) I ω i A ω i A ω i 1 p (Z pi 1 B ω pi 1 ) i = 2 n, ω (8 ) O ω A ω n + p(z pn B ω pn ) T ω (10 ) According to constraint (6 * ), the actual start time of an appointment depends on the type, service time, and completion time of the previous appointment. Constraint (8 * ) calculates the idle time of an appointment, which depends on the difference between the actual start time of the appointment and the completion of the previous appointment. In constraint (10 * ), overtime for a given scenario is calculated according to the difference between the completion time of the last appointment and clinic capacity. Yet again, the completion time of the last appointment depends on the appointment type that is assigned to the last appointment slot. Similarly, we introduce the heterogeneous patient unpunctuality by simply adding the patient type index p to identify that the random variables (P L jω, P E j ω ) vary depending on the patient type. As it is shown in modified constraints (4 * ), (5 * ) and (9 * ), the actual start time of an appointment and the waiting time of the patient are determined by the punctuality of a patient type p, who is scheduled to a given slot. A ω 1 S 1 + (Z p1 P L ω p1 ) ω (4 ) A ω i S i + Z pi (P L ω pi P Eω pi) i = 2 n, ω (5 ) W ω i A ω i S i Z pi (P L ω pi P Eω pi) i = 1 n, ω (9 ) Note that the proposed two-stage stochastic programming model has complete recourse, which means that for every occurrence of the first stage variables (sequence and scheduled start times), there is a feasible solution for the second stage variables. The

69 58 proposed model, which simultaneously includes multiple patient types, heterogeneous noshow rates and patient unpunctuality, in order to find an optimal and practical schedule that minimizes the total expected cost of patient waiting, provider idle time and overtime is presented below; n min 1 ( (C Ω i W W ω I ω i=1 i + C i I ω i ) + C O O ω ) (O. 1) st S 1 = 0 ω (1) S i S i 1 0 i = 2 n, ω (2) S n T ω (3) A ω 1 S 1 + (Z p1 P L ω p1 ) ω (4 ) A ω i S i + Z pi (P L ω pi P Eω pi) i = 2 n, ω (5 ) A ω i A ω i 1 + p (Z pi 1 B ω pi 1 ) i = 2 n, ω (6 ) I 1 ω A 1 ω ω (7) I ω i A ω i A ω i 1 p (Z pi 1 B ω pi 1 ) i = 2 n, ω (8 ) W ω i A ω i S i Z pi (P L ω pi P Eω pi) i = 1 n, ω (9 ) O ω A ω n + p(z pn B ω pn ) T ω (10 ) (S i S i 1 ) = X i i = 2 n, ω (11) X i integer, S i, A i ω, W i ω, I i ω, O ω 0 i = 1 n, ω (12) p Z pi = 1 i = 1,, n (13) n i=1 Z pi = H p p (14)

70 59 The proposed model finds the optimum sequence of various appointment types and optimum allocation of the clinic capacity to appointment durations with -minute increments. This sequencing and time allocation decision is done by considering no-show rates, mean and variation of realization of the appointment duration, and in a way to minimize the total expected cost related to the assigned cost coefficients. Therefore the number of patient types, no-show rates, mean and variation of service time, the relative ratio of cost coefficients (i.e. the ratio of idle cost to waiting cost) are the determining factors for an optimal solution. The sequencing decisions increase the solution time extremely for stochastic scheduling problems. Moreover, the problem size (number of decision nodes) may get so big that the computer s processing power may not be enough to solve the problem, even there is no time limitation. Additionally, the number of scenarios included in the model is crucial in order to provide the true behavior of the appointment duration distributions. Therefore, we propose a heuristic to solve the sequencing problem Sequencing Game Heuristic We develop a heuristic to solve sequencing and scheduling problem for more complex problems, which cannot be solved with the proposed stochastic programming model due to problem size. The proposed heuristic uses ideas from sequencing games introduced by Smith [78], and robust optimization that assumes only the lower and upper bound of the uncertainty set is known. Basic idea of the sequencing game is to find a sequence of the entities (i.e. jobs, agents, projects or customers) that yields to the best outcome (i.e. maximum profit or minimum cost). Sequencing games are convex games. The strategy constraints suggest that urgency of sequencing patient i before patient i+1 is strictly dominated by sequencing patient i+1 before patient i. According to game theory, strategy A strictly dominates strategy B if and only if choosing A always gives better outcome regardless of the position of B [57]. The additional notation used for the proposed heuristic is given in Table 14.

71 60 Table 14: Additional notation Parameters: B p UB B p LB γ p Upper bound for the realization of appointment duration for appointment type p Lower bound for the realization of appointment duration for appointment type p No-show probability of appointment type p Variables and Expressions: c i LB c i UB c i NS C i LB C i UB Cost of i th appointment for the lower bound problem Cost of i th appointment for the upper bound problem Cost of no-show for the i th appointment Total cost of i th appointment for the lower bound problem Total cost of i th appointment for the upper bound problem Sequencing game heuristic (SGH) consists of two parts. In the first part, we solve the heuristic model (SGH_H) to find a starting sequence. Smith [78] states that the best strategy for sequencing jobs for a single server is sequencing them in the order of decreasing urgency. The urgency of a job is defined as the ratio of its cost to its processing time. For our problem, it is difficult to find a single value for the urgency due to uncertain service times, and dependency of the cost value (expected waiting time, idle time and overtime cost) on the initial appointment schedule. For the proposed heuristic, we define two urgency values for each appointment; one with respect to the lower bound on processing times and the other one with respect to upper bound on processing times. Based on these urgency definitions, the proposed heuristic assigns an appointment type p to the i th slot if and only if the urgency of the appointment with respect to the lower bound is greater than or equal to the urgency of the subsequent appointment with respect to the upper bound. We would like to note that the cost value used to calculate the urgency also changes according to the realizations of random variables (i.e. appointment durations, no-shows). In our heuristic, we consider three scenarios (all service times are at their lower bounds, all service times are at their upper bounds, and all patients are no-shows) to find the objective function values at extreme realizations of service times (c UB i, c LB i, c NS i ). We solve the

72 61 proposed stochastic programming model with these three extreme scenarios and the following additional constraints: C i UB + C i LB = (1 γ p) (c i UB + c i LB ) 2 + γ p c i NS (H1) LB C i 1 B p LB C i UB B p UB i = 2.. n, p (H2) Constraint (H1) calculates the total cost of assigning the appointment type p in the i th order. Constraint (H2) is the strategy constraint that makes sure that the urgency assigned to the (i-1) st patient is greater than the urgency assigned to the i th patient. Constraints (H1) and (H2) suggests that any assignment of the first stage variables (S i, Z pi ) creates cost with respect to three scenarios such that the urgency of an appointment must be greater than or equal to the subsequent appointments. Then, we calculate the optimal solution for the original problem by using this sequence, and proceed with the second part of the SGH (local search). After finding an optimal solution for the sequence from the first part of the SGH (SGH_H), we use that sequence in the second part of the SGH to improve the initial solution. The proposed local search algorithm (SGH_LS) searches for a better solution by switching two consecutive appointments that are not of same type. If there exists a better solution, algorithm starts over again by using the best solution found. The number of iterations can be limited by considering the computation time and the percentage improvement in the objective function. We present the schematic representation of the local search algorithm in Figure 17.

73 62 Get the sequence (S1) found in Part 1 Find the optimal solution (Best) for the sequence (S1) Ite=1 True Start the next iteration Ite++, (S1)=(S Best) j=0 i=1 False Ite<ite Stop True i++ (i) p (i+1) p True j++ Generate a new sequence (S j )by switching between consecutive appointments False i < n cur ite=best, (S j)=(s Best) True False cur ite>best Find the optimal solution (cur) for the sequence (S j) Find the optimal solution for the sequence(s Best) End SGH Figure 17: Schematic representation of the local search algorithm (SGH_LS) Performance of the SGH depends on many factors including the number of patients, cost coefficients, number of scenarios included in the original problem, and the lower and upper bounds of the random variables. Additionally, computation time of SGH_H is very short (usually less than a minute) compare to the local search algorithm. Therefore, the stopping criteria should be decided carefully in order to avoid unnecessary computation time. It is also possible to skip SGH_LS in case the results obtained from SGH_H are acceptable. Numerical Example for SGH: In this part, we use SGH to find the best schedule for two patient types. We assume the clinic capacity is 75 minutes in which we schedule 3 same-day appointments and 2 new patient appointments. For simplicity, we assume that all cost coefficients are equal to one, variation of unpunctuality is high, and the lower and upper bounds of the random variables are equal to ±1 σ. Table 15 shows the actual values that we use for this example and the solution found for this setting.

74 63 Table 15: SGH numerical example parameters and solution Same-day New SGH_H SGH_LS Optimal Distribution Normal(12.7, 7.0) Normal(19.5, 8.2) Sequence (N,N,S,S,S) (N,S,S,N,S) (N,S,S,N,S) [B LB p, B UB p ] [5.7, 19.7] [11.3,27.7] Objective No-show rate 9% 44% Runtime (sec) In this example, the heuristic suggests that the new patient appointments have higher urgency compared to the same-day appointments. In Table 16, we present the costs and urgencies for all appointments for the initial sequence (N, N, S, S, S). As it is seen in Table 16,, for all appointments, urgency of appointment i with respect to the lower bound (shown in yellow) is greater than or equal to the urgency of appointment i+1 with respect to the upper bound (shown in red). Table 16: Cost and urgency calculations for the example problem From equation (H1) From equation (H2) Appointment No UB c i LB c i NS c i UB C i LB C i BUB p SGH_H finds a sequence in less than a second with 2.7% optimality gap. We also show that SGH finds the optimal solution after the local search faster than the regular optimization for this problem. C i UB C i LB B p LB 4.4. Computational Results We performed a computational study to test the effect of patient mix (with different service time distributions and no-shows rates), patient unpunctuality and cost coefficients on appointment schedules, waiting time, idle time, and overtime. We consider three different clinic settings. Table 17 shows the clinic settings, patient types in each setting, mean and standard deviation of service times, no-show rates, and number of patients that will be scheduled of each type. The service times are assumed to have lognormal distribution. The parameters of the service time distributions and no-show rates for each patient type are obtained from the studies in the literature [10, 61]. We consider a single

75 64 session of 4 hours (i.e. 8:00am to 12:00pm or 1:00pm to 5:00pm) for patient visits. The number of patients that should be scheduled in this 4-hour session is determined according to the mean service times. Table 18 shows the distributions used to generate patient unpunctuality and the cost coefficients for overtime, idle time, and waiting time. Table 17: Clinic settings, patient types and characteristics Setting I II III Patient type Service time mean and standard deviation No-show rate Same-day (SMD) μ=12.7, σ= % 6 or 9 Prescheduled (PRS) μ=16.6, σ= % 9 or 6 Routine (SMD+PRS) μ=15.0, σ= % 15 Routine (SMD+PRS) μ=15.0, σ= % 10 or 5 New (NEW + CC) μ=19.5, σ= % 5 or 10 Single (SMD+PRS+ NEW + μ=16.5, σ= CC) 23.8% Same-day (SMD) μ=12.7, σ= % 4 Prescheduled (PRS) μ=16.6, σ= % 6 Complex/Chronic (CC) μ=19.5, σ= % 2 New (NEW) μ=19.5, σ= % 3 Single (SMD+PRS+ NEW + μ=16.5, σ= CC) 23.8% Number of patients Table 18: Patient unpunctuality distributions and cost coefficients Factors Level I Level II Level III Patient Unpunctuality Low: High Normal(-10,15) Normal(-15,25) Cost coefficient ratio (Overtime cost, idle time cost, waiting cost) (1, 1, 1) (4, 2, 1) (9, 3, 1) We use IBM Cplex Optimization Studio V12.6 on a PC equipped with Intel i GHz, and 16GB RAM to solve the stochastic programming models. We solve the models with 1000 scenarios for the single patient type (Routine or Single) cases and for sequencing two patient types including Same-day-Prescheduled and Routine-New. Since we are sampling from a statistical distribution, the number of scenarios included is important in order to converge to the properties of the sampling population. However, the solution space increases significantly when we include four patient types, and it is impossible to solve it with one thousand scenarios. In the computations with four patient types, we were only able to solve the scheduling and sequencing problem with maximum

76 65 thirty scenarios and optimal solution was found after 68 hours. Therefore, we use the proposed heuristic (SGH) to solve the problem with four patient types. We present the computational results for each clinic setting (same-day-prescheduled, routine-new, and same-day-prescheduled-chronic/complex-new) in the next three sections Computational Results for Scheduling of Same-day and Prescheduled Appointments In the first clinic setting, we consider two patient types including same-day and prescheduled appointment types. The main difference between same-day and prescheduled patients is the variability due to service times and no-shows rates. Same-day patients cause lower uncertainty compared to prescheduled patients due to shorter mean service times (12.7 vs minutes) and lower no-show probabilities (9.2% vs. 27.8%). Impact of cost coefficients and patient unpunctuality on appointment schedules and objective function values: Figure 18 presents the optimal sequence and schedules for same-day and prescheduled appointment types for all cost coefficients and patient unpunctuality settings. When cost coefficients are equal and unpunctuality is low, the scheduled durations for both appointment types change between 15 and 20 minutes. The 20-minute appointment durations are scattered throughout the session to absorb the variability in service times and unpunctuality, and reduce waiting times. As the cost coefficients for idle time and overtime increase, the appointment durations decrease to minutes to reduce idle time and more time is left at the end of the session to reduce overtime. When cost coefficients for idle time and overtime are 2 and 4, respectively, the time left at the end of the session (time between the appointment time of the last appointment and session end time) is 50 minutes. When the cost coefficients are 3 and 9, the time left at the end of the session increases to 60 minutes. When unpunctuality is high and cost coefficients are equal, the variation in appointment durations, which change between 5 and 25 minutes, increases. The last appointment is scheduled at the end of the session with 5 minutes. The reason for this is to prevent cumulative waiting times during the session. Similar to low unpunctuality case, the appointment durations decrease and the time left at the end of the session increases as the cost coefficients for idle time and overtime increase.

77 66 (a) Low unpunctuality (b) High unpunctuality Figure 18: Optimal schedules for 6 same-day and 9 prescheduled appointments for all cost and unpunctuality settings When we look at the optimal sequences, we observe that the sequences in the first half of the session are same when patient unpunctuality is low. When unpunctuality is high, we observe a similar pattern only when the idle and overtime costs are high. Since the number of prescheduled appointments is more than the number of same-day appointments, and the uncertainty for prescheduled appointments is higher (due to high variability in service times and higher no-show rates), the second half of the session is mostly filled with prescheduled appointments. Although most of the same-day appointments are assigned to the first half of the session, they are strategically distributed in order to compensate for the high uncertainty of the prescheduled appointments. Figure 19 shows the expected total cost, average waiting time, average idle time and total overtime for cost coefficients (1.1.1), (4.1.2), and (9.1.3) at low and high unpunctuality. As expected, high unpunctuality gives higher objective function values than low unpunctuality for all cost coefficients. We see the same behavior on expected average waiting time per patient, average idle time per patient, and total overtime. As cost coefficients increase, the waiting times increase, and idle times and overtime decrease. However, the gap between low and high unpunctuality for both idle time and overtime is lower for high cost coefficients. That means, the proposed stochastic programming model

78 Expected Average Idle Time (minutes) Expected Overtime (minutes) Expected Total Cost Expected Average Waiting Time (minutes) 67 adjusts the appointment schedules in such a way that the impact of unpunctuality variation on idle time and overtime becomes insignificant Cost Coefficient (Overtime.Waiting Time.Idle Time) High Low High Low Cost Coefficient (Overtime.Waiting Time.Idle Time) Cost Coefficient (Overtime.Waiting Time.Idle Time) High Low Cost Coefficient (Overtime.Waiting Time.Idle Time) High Low Figure 19: Total expected cost, waiting time, idle time and overtime with respect to low and high unpunctuality and different cost coefficients (1.1.1), (4.1.2), and (9.1.3) Impact of considering only one appointment type instead of two appointment types: Some clinics might prefer using a single appointment type instead of using two appointment types in order to make the scheduling process easier. In this section, we would like to show the impact of considering only one appointment type on appointment schedules and objective function values. In order to create one appointment type (routine), we combine the service time and no-show rate parameters of same-day and prescheduled appointments. Figure 20 and Figure 21 show the effect of cost coefficients on scheduled appointment durations for routine patients when unpunctuality is low (low mean and variance) and high (high mean and variance), respectively. Even though the appointment durations at the beginning of the session decrease, and the time left at the end of the session increases as the overtime and idle time costs increase, which is similar to two patient type problems, we observe slight differences between the schedules. For example, when cost coefficients are equal and unpunctuality is high, the last appointment is scheduled with zero appointment duration for single patient type case. When there are two patient types, 15 minutes is allocated to the last appointment with adjustments to the appointment

79 Duration (minutes) Duration (minutes) 68 durations of earlier appointments. When the cost coefficients for idle time and overtime are 3 and 9, respectively, the optimal schedule has two patients at time zero (similar to Bailey s rule of scheduling two patients at the beginning of the session) for single patient type case. In two patient type setting, more appointment duration is allocated to the first appointment instead of double booking the first two patients. That means, solving the appointment scheduling problem with two patient types enables scheduler to fine-tune the variation of the appointment durations according to patient types Figure 20: Optimal appointment durations for routine patients with low unpunctuality for the cost coefficients (1.1.1), (4.1.2), and (9.1.3) Figure 21: Optimal appointment durations for routine patients with high unpunctuality for the cost coefficients (1.1.1), (4.1.2), and (9.1.3) Table 19 shows a summary of objective function values for single (routine) and two patient type (SMD, PRS) cases. Additionally, it shows the improvement gained by including two patient types instead of one. The total expected cost, average waiting time per patient and overtime improve significantly when two appointment types are considered instead of one. The percentage improvement in objective function value is 4.1% to 5.8%. The largest improvement is on expected overtime, which changes between 10.1% and 33.4%.

80 69 Table 19: Computational results and comparison of including single (routine) and two patient types (SMD&PRS) for all cost coefficient and patient unpunctuality settings Routine (SMD+PRS) Same-day & Prescheduled % Improved Low unpunctuality High unpunctuality Cost Coefficients Objective Function Value Expected Overtime Average Waiting Time Average Idle Time Runtime (seconds) Objective Function Value Expected Overtime Average Waiting Time Average Idle Time Runtime (minutes) Objective Function Value 5.4% 4.7% 5.8% 4.1% 4.3% 5.0% Expected Overtime 30.9% 33.4% 12.4% 24.6% 10.2% 31.9% Average Waiting Time 6.5% 4.7% 8.7% 3.9% 8.0% 5.1% Average Idle Time 0.0% 4.7% -0.9% 0.6% -7.0% 0.5% Impact of distribution of appointment types on appointment schedules and objective function values: In the previous part, we assume that the majority of the appointments have high variation (9 prescheduled appointments vs. 6 same-day appointments). In this part, we assume majority of the appointments are same-day appointments with low variation (6 prescheduled appointments vs. 9 same-day appointments). This type of setting might occur in the clinics that implement open access scheduling and allocate more capacity to sameday appointments. In this part, our aim is to evaluate whether the distribution of appointment types has any effect on optimal sequences. Figure 22 shows the optimal appointment schedules for all cost coefficient and patient unpunctuality settings. Similar to the previous case (6-SMD & 9-PRS), session starts with the same-day appointment and ends with the prescheduled appointment for all cost and unpunctuality settings. When the number of prescheduled appointments was low (in the previous part), most of the same-day appointments were assigned to the first half of the session, and remaining same-day appointments were strategically distributed. In this part with more prescheduled appointments, the prescheduled appointments are pushed to

81 70 the end of the session only when idle time and overtime costs are high and unpunctuality is low. We see that prescheduled appointments are distributed throughout the session in order to provide some slack time between same-day appointments, which have lower uncertainty due to smaller no-show rates and service times. These results show that the number of patients for each patient type has a significant effect on the sequence of the appointments. (a) Low unpunctuality (b) High unpunctuality Figure 22: Optimal schedules for 9 same-day and 6 prescheduled appointments for all cost and unpunctuality settings Computational Results for Scheduling of Routine and New Patient Appointments In the second clinic setting, we consider two patient types including routine and new patient appointments. The routine appointments are the combination of same-day and prescheduled appointments, and new appointments are the combination of complex/chronic and new patient appointments. The main difference between routine and new patients is the mean service times and no-shows rates. The routine patients cause lower uncertainty compared to the new patients due to lower mean service times (15.0 vs. 19.5) and no-show probabilities (20.4% vs. 31%). The difference between this setting (with routine and new patients) and the first setting (with same-day and prescheduled patients) is the increased uncertainty due to higher service times and no-show rates.