Improving Healthcare Resource Management through Demand Prediction and Staff Scheduling

Transcription

1 Clemson University TigerPrints All Dissertations Dissertations Improving Healthcare Resource Management through Demand Prediction and Staff Scheduling Nazanin Zinouri Clemson University Follow this and additional works at: Recommended Citation Zinouri, Nazanin, "Improving Healthcare Resource Management through Demand Prediction and Staff Scheduling" (2016). All Dissertations This Dissertation is brought to you for free and open access by the Dissertations at TigerPrints. It has been accepted for inclusion in All Dissertations by an authorized administrator of TigerPrints. For more information, please contact

2 IMPROVING HEALTHCARE RESOURCE MANAGEMENT THROUGH DEMAND PREDICTION AND STAFF SCHEDULING A Dissertation Presented to the Graduate School of Clemson University In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Industrial Engineering by Nazanin Zinouri August 2016 Accepted by: Dr. Kevin Taaffe, Committee Chair Dr. David Neyens Dr. Sandra Eksioglu Dr. Lawrence Fredendall

3 ABSTRACT Staff scheduling in healthcare organizations is very challenging compared to manufacturing settings. Hospitals typically operate 24 hours a day, 7 days a week, and are faced with high fluctuations in demand. Surgical patient volume directly impacts workload, and it can be difficult to manage workload fluctuations when planning staff schedules weeks in advance. We have used time series analysis methods to predict daily surgical case volume and, subsequently, assign nurses to shifts while ensuring that the case volume receives sufficient coverage. We developed a seasonal Autoregressive Integrated Moving Average () model to forecast daily patient volumes at least a month in advance. This information was used to create a scenario-based demand to solve our staff scheduling problem. We evaluated our model using data from a Level 1 Trauma Center in the Southeast U.S. With several years of daily surgical case volumes as input, we employ a seasonal (S) model to generate short-term forecasts of future surgical case volumes. The S model gives an absolute mean percentage error (MAPE) of less than 8% for up to four weeks prior to day of surgery (and only 9% when considering a 3-month forecasting interval). The forecasts outperform the basic hospital prediction by 46%. In particular, the results suggest that the proposed S model can be useful for estimating case volumes 2-4 weeks prior to the day of surgery, when managers are needing to set a reliable schedule for their staff. The nurse scheduling problem (NSP) in this research is focused on finding the best assignment of nurses to working shifts. We deal with a fixed workforce size with ii

4 mixed contract types, full-time and part-time. We consider both a risk-neutral as well as a risk-averse approach to find a feasible nurse assignment that minimizes expected labor costs, the costs of highly overstaffed or understaffed situations, or both. An innovative nurse scheduling formulation using conditional value-at-risk (CVaR) is developed to deal with risky staffing situations. The liabilities of overstaffing and understaffing are many. Overstaffing increases payrolls and results in excessive idle times, while understaffing may negatively impact patient safety and health outcomes and may result in loss of revenue. Our approach allows the scheduler to pick whether they want to schedule according to a risk-neutral or a risk-averse policy. iii

5 DEDICATION I would like to dedicate this dissertation to the memory of my father, Ali Hassan Zinouri, who inspired me to follow my dreams and would have been very proud to see me pursue this goal through to its completion. I also dedicate it to my mother, Mehrangiz Fatahi, for her unconditional love and support through every step of this process. iv

6 ACKNOWLEDGMENTS I would like to thank everyone who helped me make this dissertation possible. First, I would like to thank my advisor, Kevin Taaffe, for his support and patience during this process. I would also like to thank my other committee members, David Neyens, Sandra Eksioglu, and Larry Fredendall, for their support and commitment to my success and support throughout this process. I hope to continue learning from them even after I graduate. I would also like to thank Greenville Memorial Hospital and all the Greenville Hospital Systems staff for their support and expertise during my research. We would like to thank GHS for the historical data they provided, which was also partially supported by the National Science Foundation under grant IIS-SHB # v

7 TABLE OF CONTENTS TITLE PAGE......i ABSTRACT... ii DEDICATION... iv ACKNOWLEDGMENTS... v LIST OF TABLES... viii LIST OF FIGURES... x INTRODUCTION AND CONTRIBUTIONS... 1 CHAPTER ONE... I. USING TIME SERIES FORECASTING METHODS...TO PREDICT SURGICAL CASE VOLUME: A CASE STUDY... 5 Page 1.1 Introduction Literature Review Methods Results Conclusions II. FORECASTING SURGICAL DEMAND: DEVELOPING A TIME-SERIES FORECASTING TOOL Introduction Literature Review Methods Results Conclusions vi

8 III. MODELING AND ANALYSIS OF PREOP.NURSE SCHEDULING USING SCENARIO-BASED.DEMAND MANAGEMENT Introductions Literature Review Defining the Preoperative Nurse..Scheduling Problem Methods Results Conclusions CONCLUSIONS AND FUTURE WORK APPENDICES A. Results of Rolling Horizon Training Data Configurations B. Current preop staff scheduling template C. Breakdown of Nurse Staffing Cost for Different..Roster Sizes D. CVaR Results of Different Workforce Sizes and..different α Levels (Demand Set 1) E. OPL Code Developed to Solve the Preop NSP REFERENCES ix

9 LIST OF TABLES Table Page Table 1 Annual increase in surgical case volume Table 2 Seasonal monthly trend Table 3 Sample of rolling training horizon configurations Table 4 Comparing MAD of different forecasting models for different lengths of training data Table 5 Comparing MAPE of different forecasting models for different lengths of training data Table 6 Comparing MSE of different forecasting models for different lengths of training data Table 7 Data configurations with smallest MAD Table 8 One-way ANOVA for difference between weekdays with significance level= Table 9 One-way ANOVA with significance level=0.05 for difference between weekdays excluding Thursdays Table 10 One-way ANOVA with significance level=0.05 for difference between weekdays excluding Thursdays and Mondays Table 11 Analysis of variance for Monday s case volume across different months Table 12 Analysis of variance for different weekdays case volume across different months Table 13 Comparison of results from first and second approach Table 14 Advantages and disadvantages of forecasting methods evaluated by Jones et al. (2008) Table 15 Dickey-Fuller tests for stationarity Table 16 Ljung-Box test statistics for key lags viii

10 List of Tables (Continued) Table 17 Coefficient estimate and the standard error for the best model Table 18 Forecasting error of daily surgical case volume made 1, 2, 3, and 4 weeks in advance Table 19 Low, medium, and high volume demand generation for May Table 20 Representation of demand scenario generation for a 2-day problem Table 21 Half-hourly demand profile for high, medium, and low volume days Table 22 Shift information for the shifts currently available in preop Table 23 Workforce size variations considered in this research Table 24 Overall NSP-RA and NSP-EV model comparisons using original scenarios and different α's Table 25 A comparisons between optimal solutions to NSP-EV and NSP-RA model with various significance levels and demand scenario sets Table D-1 NSP-RA and NSP-EV model comparisons using original scenarios and different α's with 34 nurses Table D-2 NSP-RA and NSP-EV model comparisons using original scenarios and different α's with 30 nurses Table D-3 NSP-RA and NSP-EV model comparisons using original scenarios and different α's with 21 nurses Table D-4 NSP-RA and NSP-EV model comparisons using original scenarios and different α's with 17 nurses Page ix

11 Figure LIST OF FIGURES Page Figure 1 Summary of chapter one literature review... 8 Figure 2 Time series of daily surgical case volume (Greenville Health Systems) Figure 3 Actual vs. predicted bi-weekly case volume for January-September Figure 4 Average monthly case volume for Figure 5 Average daily case volume with 95% confidence interval Figure 6 Approach 1: comparing forecasting models based on MAD Figure 7 Approach 1: comparing forecasting models based on MSE Figure 8 Approach 1: comparing forecasting models based on MAE Figure 9 Approach 1, rolling training horizon: comparing forecast accuracy based on MAD Figure 10 Approach 1, rolling training horizon: comparing forecast accuracy based on MAPE Figure 11 Approach 1, rolling training horizon: comparing forecast accuracy based on MSE Figure 12 Time series of daily surgical case volume (Oct-Dec 2013) Figure 13 Box plot of all weekdays and individual weekdays Figure 14 Normal probability plot of all weekdays Figure 15 Approach 2: comparing forecasting models based on MAD Figure 16 Approach 2: comparing forecasting models based on MSE Figure 17 Approach 2: comparing forecasting models based on MAE Figure 18 Diagram of (p,d,q) process x

12 List of Figures (Continued) Figure 19 GHS daily surgical case volume time series, January 2011 to September Figure 20 Scatter plot of daily surgical case volume, January 2011-October Figure 21 The ACF and PACF of surgical case volume time series Figure 22 The model-building procedure McDowall et al. (1980) Figure 23 Model residual test for white noise, ACF and PACF Figure 24 Normality test for residuals Figure 25 Residual Plot of S(3,0,2)(2,0,1)5+H+M+W+Th+J+A model Figure 26 Residual ACF function for the S(3,0,2)(2,0,1)5+H+M+W+Th+J+A model Figure 27 MAPE by day of the week Figure 28 MAPE by month of the year Figure 29 Forecasting error of daily surgical case volume made 1, 2, 3, and 4 weeks in advance Figure 30 Forecasts of final seasonal model Figure 31 Change in prediction interval-forecasting October 2014-May Figure 32 Staffing framework adapted from Warner et. al. (1976) Figure 33 Relation between VaR and CVaR loss measures Figure 34 Illustration of the distribution of point estimates and prediction intervals Page Figure Page xi

13 List of Figures (Continued) Page Figure 35 Half-hourly arrival rates and occupancy levels for low volume days Figure 36 Half-hourly arrival rates and occupancy levels for medium volume days Figure 37 Half-hourly arrival rates and occupancy levels for high volume days Figure 38 Representation of nurse workload and daily nurse requirement calculation Figure 39 Representation of all available 16 shifts in preop Figure 40 Representation of 8 shifts actually used in preop Figure 41 Total nurse assignment costs (Oct-14 to Sep-15) with different workforce sizes Figure 42 Breakdown of nurse assignment cost with a roster of 26 nurses (baseline) Figure 43 Left: Nurse schedule solution provided by NSP-EV using 34 nurses; Right: Nurse schedule solution provided by NSP-EV using 26 nurses Figure 44 Change in yearly nurse assignment costs with different workforce sizes Figure 45 Optimal number of nurses require for each month using the EV model Figure 46 NSP-EV model vs. NSP-RA model (α=0.95) Figure 47 Effect of significance level α on scenario assignment costs Figure 48 Comparison of maximum NSP-RA cost for four approaches at α= Figure 49 Comparison of NSP-EV and NSP-RA maximum costs at α= Figure Page xii

14 List of Figures (Continued) Page Figure 50 Performance of different scenario sets in minimizing outcomes above $138,000 with different levels of α Figure 51 Performance of different scenario sets in minimizing outcomes above $98,000 with different levels of α Figure 52 Performance of different scenario sets in minimizing outcomes above $83,000 with different levels of α Figure 53 Performance of different scenario sets in minimizing outcomes above $80,000 with different levels of α Figure 54 Performance of different scenario sets in minimizing outcomes above $72,000 with different levels of α Figure A-1 Different data configurations for training and forecasting sets Figure A-2 Results of different forecasting approaches using different data configurations (predicting 4 weeks ahead) Figure B-1 Preop nurse scheduling template currently used by nurse manager at GHS Figure C-1 Breakdown of nurse staffing cost (roster of 17 nurses) Figure C-2 Breakdown of nurse assignment cost ( roster of 21 nurses) Figure C-3 Breakdown of nurse assignment cost (roster of 30 nurses) Figure C-4 Breakdown of nurse assignment cost ( roster of 34 nurses) Figure D-1 Effect of significance level α on scenario assignment costs with 34 nurses Figure D-2 Effect of significance level α on scenario assignment costs with 30 nurses xiii

15 List of Figures (Continued) Page Figure D-3 Effect of significance level α on scenario assignment costs with 21 nurses Figure D-4 Effect of significance level α on scenario assignment costs with 17 nurses xiv

16 INTRODUCTION AND CONTRIBUTIONS The initial focus of this research was to address staff scheduling problems in the preoperative (preop) department within perioperative services. Perioperative services includes those hospital functions that support the delivery of surgical care. While studying contributing factors, it became clear that an accurate approximation of workload is essential for optimal nurse scheduling and that most hospitals do not implement sophisticated forecasting methods to provide estimates of their workload. Hospital managers certainly adjust their staffing levels to workload, but a majority of the methods they use are reactionary (Kim et al., 2014). Most planning and scheduling in healthcare is done based on experience and human intuition with limited use of modern scientific methods. Understanding demand patterns and having reliable estimates of daily surgical case volume will have a significant effect in the ability to recommend an efficient and feasible staff schedule. In light of this, we have chosen to place our initial focus on forecasting daily surgical case volume. In Chapter 1, we explore real world surgical data and the ability to forecast surgical case volume within the month, using this historical data. We studied surgical case volumes at Greenville Health System (GHS), with detailed surgical data from the GHS data warehouse from January 2011 to September After analyzing the data, we observed systematic fluctuations and patterns in surgical volume that could be captured within time series forecasting models. The primary focus of this chapter was to evaluate the predictability of surgical case volume using a variety of known forecasting methods. In chapter 2, we expand upon our previous prediction methods. We use S models with predictive variables to develop forecasting methods that can provide accurate short-term surgical volume forecasts at least one month prior to the day of surgery. This approach allows a 1

17 preop nurse manager to adjust the nurse staffing assumptions based on surgical demand. Also, the nurse manager can use these patient volume estimates to determine the nurse workload and develop more accurate mid-term and short-term staff schedules. In particular, our contribution is to further improve forecast reliability when predictions for caseload are needed within one month of the day of surgery. As preop managers often express a need to firm up their staffing schedule 2-4 weeks in advance, our objective was to identify forecasting methods to accomplish this. Through a comprehensive treatment of several forecasting methodologies, we show how time series forecasting can significantly improve the ability to provide more accurate caseload predictions for the day of surgery than typically are used by preop nurse managers in practice today. It should be noted that chapter 1 presents our early findings on the characteristics of our data set and different forecasting methods. We tested different forecasting models and different lengths of data for prediction to get insight on how to develop a good forecasting model. Our refined forecasting method is presented in chapter 2. In chapter 2, we present the steps to develop a S model with indicative variables to predict daily surgical case volume. In other words, chapter 1 is more of a case study on our data whereas chapter 2 represents our final S model and the procedures required for model development and testing. Short-term forecasting applications are extremely useful to decision makers at both public health system and organizational levels. Inaccurate estimations of demand may lead to insufficient or excess resource availability, which will result in increased costs. Patient volume forecasting has been studied in the context of emergency department (ED) demand prediction. There has been limited short-term forecasting research concerning daily surgical case volumes (Tiwari et al., 2014). While the overall work on surgical case volume forecasting is limited, we 2

18 note that several researchers have contributed to forecasting the volume of daily ED visits (see, e.g., Jones et al., 2008, Schweigler et al., 2009, and Jalalpour et al, 2015). Also, a majority of prior work on healthcare forecasting is focused on longer-term forecasting (i.e., yearly) rather than short-term (i.e., daily and monthly) prediction. In chapter 3, we shift our focus to the preop nurse scheduling problem. The nurse scheduling problem has been studied extensively by researchers in the past. Often, these research works assume known demand or attempt to find the best staffing schedule across all possible scenarios of possible demand. However, there is significant cost to the hospital when the staff schedule does not align with the number of patients requiring surgery on a given day. This has motivated us to use the forecasts obtained in chapter 2 but then account for the uncertainty in demand. We used a scenario-based approach to account for the probability of different demand levels occurring and study its effect on nurse workload and schedule. We developed both a risk-neutral, or expected value (EV), model as well a risk-averse, or conditional value-at-risk (CVaR), model to address preop nurse scheduling. Our risk-averse policy allows the decision maker to study how to avoid the worst-case scenarios and identify a risk-adjusted nurse schedule. To the best of our knowledge, no prior research has employed a CVaR approach to solve this problem. Our contribution is introducing a risk-averse method, where instead of optimizing a nurse schedule in terms of expected costs across all outcomes, we develop solutions that minimize risk by avoiding worst-case or extremely high cost days. Such a nurse scheduling approach will allow the nurse managers to define their risk tolerance. We are more concerned about the performance of the hospital when worst-case days occur, as the main concern is the care of the patient. It is important to have a staff scheduling method that will help maintain 3

19 consistency of staff, especially if this can lead to improved patient outcomes on the day of surgery. In order to accomplish this, we have introduced a risk/cost based formulation that can identify and protect against resource shortage/overage on high/low volume days. In summary, we our main objectives are to determine how to obtain useful information from the historical data; how to use time series forecasting models and historical data to obtain accurate forecasts of daily surgical case volume, how to use the results of our forecasting model as an input for our nurse scheduling model, and how to develop a risk-neutral as well as a riskaverse nurse scheduling tool. We successfully addressed the goals and objectives of this research. 4

20 CHAPTER ONE USING TIME SERIES FORECASTING METHODS TO PREDICT SURGICAL CASE VOLUME: A CASE STUDY 1.1 Introduction In the United States, healthcare systems are challenged to deliver high quality care with limited resources while attempting to operate at the lowest possible cost. Over 60 percent of total hospital costs are related to labor, and nursing is typically the highest cost of labor in a healthcare facility (Kreitz, and Kleiner, 1995). Utilizing this limited and costly resource as effectively as possible is of great importance. The effective management of staff is crucial to any organization's overall success because labor costs represent a very large proportion of the total budget (Wright, and Bretthauer, 2006). Surgical case volume will directly impact daily workload, and the fluctuations in that workload are difficult to manage, especially when faced with setting a staff schedule weeks in advance. Overstaffing and understaffing are costly and unsustainable, and this can negatively impact staff satisfaction and patient outcomes. Having a more accurate demand prediction weeks in advance would help improve staff scheduling and decrease nurse workload pressure by allowing more flexibility in the schedule. In many hospitals, the demand for surgeries is increasing, an expanding and aging population is one of the reasons for this increase (Lim, and Mobasher, 2012). Therefore, hospital managers need to use limited hospital resources more efficiently. There are many factors that can impact setting the staff schedule, however, it all begins with having accurate estimates of daily surgical case volume. Surgical case volume or operating room demand is one the most important piece of information that will highly impact staffing decisions. The actual surgical caseload is often not known until the day of surgery. It is difficult to adjust staffing to meet the demand a few days prior to the day of surgery (Tiwari et 5

21 al., 2014), yet demand estimates can still be volatile at that point. In fact, actual surgical caseload is often not known until the day of surgery due to the possibility of cancellations and add on cases. Hospitals may choose not to apply any specific staff scheduling or demand forecasting model because of the perceived high variability in surgery demands. However, historical data can be used to predict future demands as well as approximate variations. These predictions can be verified by comparing them to the actual demand (Tiwari et al., 2014). In this chapter, we focus on analyzing historical surgical demand to determine how well we can predict daily surgical demand. We have also investigated how far in advance of the day of surgery, we can make accurate predictions. We have studied past research on surgical case volume prediction. A majority of work in this area is focused on long-term staff planning (Tiwari et al. 2014). We have studied whether past data containing patient volume on different days over several years can be used to predict future surgical case volume. Also, we have investigated whether these predictions are accurate enough to be used for short-term staff scheduling, 1 to 4 weeks prior to the day of surgery. In addition, we developed different configurations of training data to evaluate the effect of the length of training data as well as the time horizon used for training. So we tried to find out an appropriate amount of training data to achieve the best forecasts. Chapter 1 can be viewed as a case study exploration into real-world surgical data and the ability to forecast within the month daily volumes. There are three major components in this part of research: 1. Comparing performance of various models. 2. Assessing the accuracy based on weeks in advance. 3. Assessing the accuracy based on the current training data 6

22 1.2 Literature Review A comprehensive literature review on nurse staff scheduling was conducted to identify the gaps in the literature and potential areas for future work. The first widely studied area was operating theater planning and scheduling which includes operating theater resource, staff, scheduling. The operating theater is one of the most important and expensive resources within a hospital. The Operating Theater consists of operating rooms (ORs) and recovery rooms (Guerriero and Guido, 2011). ORs account for approximately 40% of the total hospital's revenue and at the same time, they are responsible for a large proportion of total hospital expenses. ORs are simultaneously the greatest revenue source and the largest cost center in a hospital (Denton et al., 2007). Optimal operating theater planning and scheduling plays an important role in maximizing the utilization of hospital resources and improving the quality of care in hospitals. After looking into operating theater planning and scheduling, we focused more closely on staff scheduling. Most work on staff scheduling in healthcare has focused on OR staff scheduling. The staffing/nurse management literature has grown out of a rich history of labor management studies (Wright et al., 2006). Healthcare managers need to better understand the effects of nurse staffing decisions on the patient outcomes. Most of the research in this area has shown direct relationship between staffing levels and patient outcomes. Efficient management and deployment of nursing personnel is of great importance for hospitals since it directly affects the performance of healthcare services and the quality of care (Maenhout and Vanhoucke, 2013). However, no published studies were found that have considered staffing in multiple perioperative departments other than OR such as preop and post-anesthesia care unit (PACU). Finally, we looked into the studies that have focused on data and statistical analysis required to facilitate OR staffing decisions. Research in healthcare area requires a good 7

23 understanding of the data and knowledge of appropriate methods to analyze and interpret the data. Past data can be used to draw inferences or make predictions about future events. Figure 1 summarizes the structure of literature review conducted in this chapter. It should be noted that we reviewed literature on topics that are not directly addressed in this chapter as well. However, all of these topics are directly or indirectly related to case volume forecasting or staff scheduling. Operating Theater Planning and Scheduling 1. Patient Characteristics/ Performance Criteria 1.1. Overtime 1.2. Undertime 2. Type and Level of Decision 3. Type of Analysis 4. Solution Techniqu 5. Type of Hard Constraint 5.1. Syrgical Staff 5.2. Staffing Costs and Budget 6. Uncertainty 7. Applicability of Research Staff Scheduling Problems in Healthcare 1. Staffing Model Formulation 2. Solution Technique 3. Constraints 4. Objectives 5. Data Analysis Data and Statistical Analysis Historical Data Prediction Models Figure 1 Summary of chapter one literature review Operating Room Planning and Scheduling Multiple factors such as OR capacity, available OR time, staff scheduling, and surgeon preferences influence an optimal operating theater planning and scheduling. Operating theater schedule directly affects the productivity of surgeons, anesthetists, post anesthesia care units (PACUs), and intensive care units (ICUs). Many studies have been conducted in attempt to improve OR planning and scheduling, and minimize OR costs. Common problems that are 8

24 studied in this include OR allocation, OR scheduling, and staff scheduling. Various approaches including different optimization and simulation models have been used depending on how the problem is defined and what decision levels are considered. The review conducted by Cardeon et al. (2010) is a comprehensive analysis of the literature on operating room planning and scheduling. Guerriero and Guido (2011) have also provided a complete review of different operations research techniques that have been utilized in OR management. Meskens et al. (2013) have studied the operating room scheduling problem with a focus on multiple real-life constrains like staff availability, preferences, and affinities among staff members. They have developed a generic and modular model that takes specifications of each hospital into account. Roland et al. (2010) have developed an operating theater schedule that considers well-being of the healthcare staff by emphasizing the human resources availability in designing the schedules. Persson and Persson (2010) have developed a discrete event simulation model to analyze the effects of different management policies on the utilization of operating room (OR) time and other performance metrics such as cost, length of stay, patient waiting times and surgery cancellations. Dexter et al. (2004) have provided a review of the Operating Room (OR) management decision making on the day of surgery. During the past decades, management decisions have been made based on four principles: 1) expected (mean) surgery duration or other measures that reflect the uncertainty in surgery duration; 2) ordered priorities (patient safety, open access to OR time, maximizing OR efficiency, and reducing patient waiting time; 3) decisions involving reducing over-utilized OR time; 4) decisions involving reducing patient and surgeon waiting times. 9

25 Ben-Arieh et al. (2014) have developed a mixed integer program to find a weekly surgical schedule that provides a more optimal process and schedule of operations along with a better nurse staffing. Variation in nursing level is the most important cause of surgery cancellations; cancellations will result in loss of revenue as well as distress to patients and their families. The main goal of the proposed model is to minimize the day-by-day variations in nursing level at the Pediatric Intensive Care Unit (PICU). Dexter et al. (2000) have developed an OR scheduling method that reduces costs and provides flexibility for patients and surgeons to choose their preferred dates and times. However, the staffing costs will increase since staffing costs are the lowest when patients' and surgeons' preferences are not considered. The proposed method called Minimal Cost analysis aims to meet patients' and surgeons' expectations while reducing staffing costs. Stepaniak et al. (2009) have studied the impact of Operating Room Coordinators' (ORC) willingness to agree to assume risk in their daily planning on OR efficiency. Their study shows that a risk-averse ORC is less cost effective. A risk-neutral ORC creates less unused OR capacity while decreasing the chance of running ORs after regular hours or canceling surgeries. Makary et al. (2006) have surveyed OR personnel using Safety Attitudes Questionnaire (SAQ) to investigate the importance and impact of OR teamwork on patient safety. As expected, communication errors are the main cause of wrong site operations and sentinel events in the United States. A number of the previous work on OR planning has focused on the uncertainty in demand (patient arrival) and service times (surgery durations); Lamiri et al.(2008) and Batun et al. (2011) are examples of those papers. Van Essen et al. (2012) have addressed the uncertainty and variability in arrivals by studying OR rescheduling problem. The arrival of emergency 10

26 patients or variability and changes in the length of elective surgeries will cause disruptions in the OR schedule. As a result, the initial OR schedule needs to be changed (Van Essen et al. 2012). Gupta and Wang (2008) have considered stochasticity arising from patient choices in addition to stochastic arrivals and service times. There are other papers such as works conducted by (Ghazalbash et al., 2011, Liu et al., 2011, Fei et al., 2010, and Chaabane et al., 2008) that have focused on different OR scheduling strategies including but not limited to block scheduling, open scheduling, and modified strategy. (Persson and Persson, 2010), Ballard and Kuhl (2006) have used simulation to improve OR scheduling. Other works such as the research conducted by Hosseini and Taaffe (2014) have focused on over-utilized and under-utilized OR times. Houdenhoven et al. (2007) have studied the effects of reducing organizational barriers on increasing operating room (OR) utilization. Staff Scheduling Problems in Healthcare Over 60 percent of total hospital costs are related to the labor (Kreitz and Kleiner, 1995). Nursing staff is one of the most critical resources in a healthcare system. Utilizing this limited and costly resource as efficiently as possible is of great importance. An optimized staff schedule helps meet customers' demand in a cost effective way. In addition, it provides work flexibility and considers staff preferences which improves staff satisfaction and quality of healthcare (Ernst et al. 2004). Ernst et al. (2004) have provided a thorough review of personnel scheduling or rostering which contains problem classifications as well as available solution approaches. Van Den Bergh et al. (2013) have also provided a technical review of personnel scheduling problems. Cheang, Li, Lim, and Rodrigues (2003) have provided an overview of nurse rostering problem (NRP). Lankshear et al. (2005) have also provided a systematic review of research that has been 11

27 conducted since 1990 on nurse staffing and its effects on healthcare outcomes. Blochliger (2004) have proposed a tutorial that provides an introduction to staff scheduling problems and formulation frameworks. The tutorial focuses on the modeling aspect of staff scheduling problems. The author has used hospital and nurse scheduling problem as an example to demonstrate the model. Lankshear et al. (2005) have concluded that fewer patients per nurse will result in better healthcare outcomes. In other words, they have suggested that if patient satisfaction and quality of care is the primary objective, it is better to keep hospital units overstaffed. Jan at al. (2000) have studied different algorithms commonly utilized for solving Nurse Scheduling Problem (NSP). The authors have divided these approaches into conventional and Evolutionary Computation (EC). Common conventional methods are mathematical programming and goal programming. There are fewer papers utilizing EC approaches to solve NSP. One of the successful EC methods is Cooperative Genetic Algorithm (CGA). Maenhout and Vanhoucke, (2013) have discussed that considering staffing and scheduling separately will result in suboptimal decisions. They have proposed an integrated staffing and shift scheduling model which enables better long-term nurse allocation decisions and considers schedule desirability from both nurses and patients perspective. Operating room (OR) allocation is a two-step process. First, OR capacity known as block time is adjusted; this is decision making on tactical level. Then, staffing will be matched to the existing work load which is assumed to be fixed. This is operational decision making and addresses short-term decisions. The objective is to maximize OR utilization (McIntosh et al., 2006). Shuldham et al. (2009) have studied the relationship between staffing levels, patient outcomes and the quality of care. The authors have explored this relationship considering patient 12

28 outcomes such as patient falls, upper gastrointestinal bleed, pressure sores, sepsis, and shock and deep vein thrombosis. Despite previous studies that only consider problems which result in death as "failure to rescue", the authors have considered any major complication to be "failure to rescue". The results demonstrate that there is a different association pattern in critical areas compared to the wards (lower dependency areas). Ling et al. (1999) have studied the impact of staff cross training on the efficiency and performance of a healthcare facility. The authors have concluded that staff flexibility helps in better managing time varying demand and also reduces total costs. Dowsland (1998) have used tabu search with strategic oscillation to address nurse staffing problem. The objective is to assure there are enough nurses available during all the shifts while trying to meet nurse preferences. Lim and Mobasher (2012) have developed a multi objective nurse scheduling model for Operating Suite personnel. They have studied both Nurse Assignment model (NAM) and Nurse Lunch Model (NLM) with the aim of developing a schedule that improves the quality of services at a lower cost. NAM assigns staff to different surgical cases in an Operating Suite while considering the success of surgery case and staff satisfaction. NLM uses the results of NAM to develop lunch and break schedules. The authors have utilized both a heuristic approach, postprocessing improvement heuristic (PPIHM), and a column generation approach to solve the proposed problem. The Effects of Data and Statistical Analysis on Staffing Improvements The book written by Munro (2001) provides a good reference for learning statistical methods most commonly reported in the healthcare research literature (Munro, 2001). In this book, the author has explained how to organize and display healthcare data, deal with missing data and outliers, as well as discussing statistical techniques for analyzing the data. 13

29 A number of previous studies have focused on data and statistical analysis required to facilitate OR staffing decisions. Sanford and Macario (2010) have conducted a study on the importance of having useful quantitative information and sufficient historical data on developing efficient operating room (OR) schedules. Tiwari et al. (2014) have pointed out that staff scheduling decisions are typically made weeks prior to the day of surgery. According to Tiwari et al (2014) accurate estimates of operating room demand cannot be made more than 1 or 2 days in advance and by then it is too late to close ORs with no booking or flex off the staff. Previous studies on surgical volume prediction have mainly focused on long-term planning. There is less variability in surgical case volume in long-term. As a result, if the distribution of OR labor and surgical demand repeats the historical distribution, efficient staffing can be planned based on historical data. On the other hand, there is higher variability in surgical demand in short-term (daily and weekly time scale). There are limited studies on developing predictive models for short-term planning staff (Tiwari et al. 2014). Tiwari et al. (2014) have developed a method to predict surgical case volume weeks or days prior to the day of surgery. Dexter et al. (1999) have used the average of a surgical group's recent total hours of elective cases to predict future demands for OR times with minimum error. They have computed the number of 4-week periods of data required to obtain prediction with minimum error, using time series analysis. Esptein and Dexter (2002) have conducted a statistical power analysis to identify how many months of historical data are required to optimize OR staffing and patient scheduling. The objective is to improve quality of care at the hospital while maximizing staff productivity and minimizing OR staffing costs. Dexter et al. (2003) have described how data collected from OR information systems at hospitals can be utilized to calculate the potential reduction in turnover times. They have studied the effect of reducing turnover times on direct 14

30 and indirect reduction in staffing costs. These staffing cost reductions are accomplished by reducing allocated OR time instead of reducing the hours that staff work over-time. OR assignments for different days are usually performed months in advance according to the historical data. These OR schedules can be used to reduce the number of off rotation assignments in trainee scheduling (Dexter et al., 2010). Dexter et al. (2010) have proposed a method to develop hybrid rotations of specialties for trainee scheduling with the objective of minimizing the number of days that trainees are pulled from their originally scheduled rotation. While researchers have attempted to address staff scheduling by using data analytics on historical surgical volume, there appears to be a gap in proposing regression-based demand forecasting models that could significantly improve the scheduling ability within those models. Regression models can be a method to predict case volumes more accurately. 1.3 Methods The primary focus of this chapter is to evaluate the predictability of surgical case volume using a variety of known forecasting methods. More specifically, in this chapter, we have studied and compared different known univariate time series forecasting techniques for surgical case volume forecasting. Univariate techniques used in this chapter included single exponential, double exponential, moving average, autoregressive integrated moving average (), and seasonal. A secondary objective is to study historical data to understand the time-based dynamics in patient volumes across the perioperative departments. The forecasting methods used in this chapter account for total surgical case volume including patients from a variety of sources (e.g. inpatient, outpatient offices, emergency department, etc.). 15

31 Study Setting and Surgical Case Volume Data This study was performed at Greenville Health System (GHS), a level 1 trauma center (a 614-bed tertiary care center) in Greenville, SC. In order to understand the trends in surgical demand and be able to estimate the expected future demand, a dataset comprising of 931 consecutive surgical days (January August 2014) was used. Weekends and major holidays with very low volumes were removed from the dataset. The dataset included 182 Mondays, 188 Tuesdays, 187 Wednesdays, 186 Thursdays, and 188 Fridays. Figure 2 represents the time series of daily surgical case volume, which includes elective and non-elective surgeries. Currently, the business analyst works with the surgeons and anticipates the case volume for the next fiscal year. A single number will determine each year s demand and bi-weekly case volume is calculated based on this number. Historical data is not evaluated when determining the next year s demand. Figure 3 compares the actual bi-weekly demand volumes to the predicted volume and predicted upper and lower limits for January-September As it can be observed, the prediction is not accurate, Mean Absolute Deviation (MAD) is equal to 278.4, Maximum Absolute Error is 31 patients, and the volume is very close or even exceeds the upper limit. Our analysis reveals the inefficiency of the current practice, which reinforces the need for a better prediction method. Daily case volume is driven from the yearly volume predicted by the hospital s business analyst. The annual budget for each department is also determined based on this prediction and the total number of full time employees (FTEs) is determined based on the allocated budget. As a result, the predicted volume and the number of staff assigned is a flat rate for every single month regardless of the seasonality and trends. Our analysis shows that this is not an efficient prediction and requires frequent change based on the actual historical case volumes which will result in 16

32 Daily Surgical Case Volume 1/5/2011 3/5/2011 5/5/2011 7/5/2011 9/5/ /5/2011 1/5/2012 3/5/2012 5/5/2012 7/5/2012 9/5/ /5/2012 1/5/2013 3/5/2013 5/5/2013 7/5/2013 9/5/ /5/2013 1/5/2014 3/5/2014 5/5/2014 7/5/2014 9/5/2014 additional costs, staff dissatisfaction due to the constant change in their schedule, and potentially can affect the quality of the care. A simple analysis on the past data can reveal a lot of information on demand patterns for each month. This information can be used to improve longterm staff scheduling Figure 2 Time series of daily surgical case volume (Greenville Health Systems) Figure 3 Actual vs. predicted bi-weekly case volume for January-September 2014 As we were interested in using time series forecasting methods, we started by analyzing the components of time series in our dataset. We investigated trends, cyclical variations, seasonal variations, and irregular variation. We observed that time-varying variables such as seasonal 17

33 changes highly impact surgical case volumes. It was observed in Table 1 that there is an annual increase in the demand. Table 1 Annual increase in surgical case volume Year Estimated 2015 % Change 1.54% 5.80% 2.82% 4.06% The mean ± SD daily surgical volume was estimated as 71±11 patients per day. According to the results of the analysis, different months of the year do not have similar demand. We expect that using a seasonal model for predicting the demand would be a more appropriate approach, rather than a simple linear prediction model. In order to find the trend for each month, we divide the work load of each month by the total annual work load. The results are summarized in Table 2. According to the results in Table 2, the data shows a pretty consistent trend for each month over the course of 3-4 years. Figure 4 represents the average monthly case volume for each year. Table 2 Seasonal monthly trend Month Average January 85% 96% 87% 95% 91% February 85% 104% 91% 88% 92% March 84% 103% 94% 100% 95% April 78% 99% 98% 102% 94% May 93% 103% 101% 103% 100% June 109% 104% 92% 100% 101% July 99% 94% 99% 106% 100% August 130% 104% 106% 102% 111% September 109% 93% 107% 104% 103% October 109% 104% 121% - 111% November 109% 100% 103% - 104% December 109% 95% 102% - 102% 18

34 Average Case Volume Jan-11 Mar-11 May-11 Jul-11 Sep-11 Nov-11 Jan-12 Mar-12 May-12 Jul-12 Sep-12 Nov-12 Jan-13 Mar-13 May-13 Jul-13 Sep-13 Nov-13 Jan-14 Mar-14 May-14 Jul-14 Sep Average Monthly Case Volume for Jan-11-Sep14 Month Figure 4 Average monthly case volume for Similar to the monthly trend that we observe in the data, different days of the week also have different demand patterns. Analysis of the data on a daily basis is represented in Figure 5. According to the results, the daily analysis shows that in general, Tuesday and Thursday are the busiest days of the week. Average Daily Case Volume Monday Tuesday Wednesday Thursday Friday Figure 5 Average daily case volume with 95% confidence interval 19

35 Forecasting Approaches and Performance Measures Our selection of forecasting methods in this chapter was primarily based on previous research on Emergency Department (ED) patient forecasting. Selected methods included single exponential, double exponential, simple moving average, autoregressive integrated moving average (), and seasonal. All forecasting models were trained using case volume data from January 3, 2011 to September 31, The forecasted patient volume was calculated for future periods (from one week to four weeks). Our forecasting models were validated using the difference between forecasted daily surgical case volume and actual surgical case volume beyond the period on which the model was trained. Data from October 1, 2013 to September 30, 2014 was used to evaluate forecast accuracy. We also investigated the effect of using a rolling training horizon. We developed different configurations of training data to evaluate the effect of the length of training data as well as the time horizon used for training. Each forecasting model was assessed using mean absolute deviation (MAD), mean squared error (MSE), and mean absolute percentage error (MAPE). Maximum absolute error (MAE) was also used to assess some instances. MAPE is defined as the relative difference between forecast values of any given model and actual values. The MAPE and the MAD are among the most commonly used error measurement in statistics. There are a few other alternative statistics in the forecasting literature, many of which are variations on the MAPE and the MAD. MAE is one of these measurements. We were interested in looking at MAE because maximum error could add value in that we want to avoid really bad errors. 20

36 In this chapter, we also tried two different approaches to develop a 5-day case volume forecast: Approach 1: Create a single prediction model using each forecasting method. Each model will produce weekly predictions (Monday-Friday). In this approach we include seasonality to account for daily variability/weekly patterns. Approach 2: Create five unique prediction models for one forecasting method. Each model will predict one weekday. We then combine five models to predict a full week. Based on results from Approach 1, we provide five-day models based on non-seasonal, simple exponential, and simple moving average for this approach. We did not use training horizons of different lengths, 3, 6, 9, and 12 months, for this approach. We only used long data set consisting of almost 3 years of training data to run this set of experiments. 39 months of data is used for parameter estimation (or training data), while the last six months in the data set is reserved for validation (or validation data). 1.4 Results First Approach Model development was implemented in the SAS web-based package. The software package was automated for choosing model parameters based on error type. In this chapter, model selection was performed utilizing the automated time series parameter suggestion algorithm within SAS. We were interested in evaluating the performance of different forecasting models as well as investigating how accurate the forecasts are depending on how far in advance we are predicting. It is important to be able to assess future case volumes many days or weeks prior to the day of surgery, to adjust staff levels accordingly. In fact, based on conversations with 21

37 MAD GHS, a pre-op nurse manager would prefer to have a fairly accurate forecast a month (4 weeks) in advance. Figure 6-Error! Reference source not found. show the comparison of the performance of our five different forecasting models based on MAD, MSE, and MAE. These figures show the accuracy of the forecasts 1-4 weeks prior to the day of surgery. Error terms shown in these figures are averages for 12 different forecasts (12 months Sep 13-Aug 14). The original time series, GHS data for daily case volume, was considered to be stationary so no differencing on the data was required. model is represented with parameters (p, d. q). Seasonal is represented using parameters (p, d, q) (P, D, Q)s. The terms within (p, d, q) refer to the non-seasonal components of the time series. The terms within (P, D, Q) refer to the seasonal components. p is the order of auto-correlation, d is the degree of differencing, and q is the order of moving average. s refers to the time span of repeating seasonal order. In addition to showing the best performing S model, we also include (1,0,0) (1,0,0)5 in all comparisons as it was the model recommended by Tiwari et al. (2014) (3,0,1)(2,0,1)5 (1,0,0)(1,0,0) Week Out 2 Weeks Out 3 Weeks Out 4 Weeks Out MA Single Exponential Smoothing Double Exponential Smoothing Figure 6 Approach 1: comparing forecasting models based on MAD 22

38 Maximum Absolute Error MSE (3,0,1)(2,0,1)5 (1,0,0)(1,0,0)5 MA Single Exponential Smoothing Double Exponential Smoothing 0 1 Week Out 2 Weeks Out 3 Weeks Out 4 Weeks Out Figure 7 Approach 1: comparing forecasting models based on MSE Week Out 2 Weeks Out 3 Weeks Out 4 Weeks Out (3,0,1)(2,0,1)5 (1,0,0)(1,0,0)5 MA Single Exponential Smoothing Double Exponential Smoothing Figure 8 Approach 1: comparing forecasting models based on MAE As shown in Figure 6-8, seasonal (S) model with weekly (5-day) seasonality had lower errors. For one-week ahead to 4 weeks-ahead case volume forecast, the S model, (3,0,1) (2,0,1)5 outperformed the other models for most training data configurations. The second best (not shown in figures) was (3,0,0) (3,0,0)5. The (1,0,0) (1,0,0)5 model from Tiwari et al. (2014) cannot achieve the same level of fit, 23

39 especially for 1-3 weeks prior to the day of surgery. However, as the forecast period extends to four weeks, this model begins to improve. This could suggest that this model would provide an adequate forecast when providing an estimate over one month in advance. We also found that the prediction accuracy will not significantly suffer when moving from forecasting 1 week in advance to 3 weeks in advance of the day of surgery (in terms of MAD and MSE). Maximum error does increase with each additional week, though. 3 to 12 Months Rolling Training Horizon In order to see how the training horizon will affect the forecasts, we developed training horizon configurations with lengths ranging from 3 to 12 months. Table 3 shows an example of a training horizon of 12, 9, 6, and 3 months duration. The common element of all configurations is that forecasts start from September 1, 2013, with training data ending on August 31, Similar configurations were built to forecast each month until August For all these settings, we predicted one week to four weeks of future case volume. A detailed summary of the results of these forecasts is represented in Appendix A. Table 4-Table 6 show the results of five best forecasting models for different lengths of training data based on MAD, MSE, and MAPE. It should be noted that since model parameters would vary based on forecasting horizon and length of training data, parameters are not reported in these tables. Complete information can be found in Appendix A. Figure 9-Figure 11 compare the performance of best forecasting model, one week to 4 weeks prior to the day of surgery, based on average MAPE, MAD, and MSE for 4 different lengths of planning horizon. Table 3 Sample of rolling training horizon configurations Training Data Forecasting Data 24

40 Length (Months) Start Date End Date Start Date 1 Week 2 Weeks 3 Weeks 4 Weeks End Date 12 1-Sep Aug-13 1-Sep-13 8-Sep Sep Sep Sep Dec Aug-13 1-Sep-13 8-Sep Sep Sep Sep Mar Aug-13 1-Sep-13 8-Sep Sep Sep Sep Jun Aug-13 1-Sep-13 8-Sep Sep Sep Sep-13 Table 4 Comparing MAD of different forecasting models for different lengths of training data Predicting 4 Weeks prior Predicting 3 Weeks prior Predicting 2 Weeks prior Predicting 1 Week prior Length of Training Data SES DES MA (1,0,0)(1,0,0)5 (3,0,0)(2,0,0)5 MAD 3 years months months months months years months months months months years months months months months years months months months months

41 MAD Week Out 2 Weeks Out3 Weeks Out4 Weeks Out 12 months training 9 months training 6 months training 3 months training Figure 9 Approach 1, rolling training horizon: comparing forecast accuracy based on MAD Table 5 Comparing MAPE of different forecasting models for different lengths of training data Predicting 4 Weeks prior Predicting 3 Weeks prior Predicting 2Weeks prior Predicting 1Week prior Length of Training Data SES DES MA (1,0,0)(1,0,0)5 (3,0,0)(2,0,0)5 MAPE 3 years months months months months years months months months months years months months months months years months months months months

42 MAPE months training 9 months training 6 months training 3 months training 0 1 Week Out 2 Weeks Out 3 Weeks Out 4 Weeks Out Figure 10 Approach 1, rolling training horizon: comparing forecast accuracy based on MAPE Table 6 Comparing MSE of different forecasting models for different lengths of training data Predicting 4 Weeks prior Predicting 3 Weeks prior Predicting 2Weeks prior Predicting 1Week prior Length of Training Data SES DES MA) (1,0,0)(1,0,0)5 (3,0,0)(2,0,0)5 MSE 3 years months months months months years months months months months years months months months months years months months months months

43 MSE months training 9 months training 6 months training 3 months training Week Out 2 Weeks Out 3 Weeks Out 4 Weeks Out Figure 11 Approach 1, rolling training horizon: comparing forecast accuracy based on MSE Seasonal model (3,0,0) (2,0,0)5 outperformed all other models for all training horizon settings. This is quite a strong conclusion to have a single model achieve the best performance. For each configuration, unique parameters were selected by SAS for single exponential and moving average models. However, these models still could not outperform the S approach. Details concerning individual models and training configurations / forecast horizons, as well as results, can be found in Appendix A. It was observed that errors of those models that were built using shorter lengths of training data, will increase more significantly as we move from predicting 1 week in advance to 4 weeks in advance. For instance, if we look at Figure 9, we can see that when using 12 months, the MAD changes from 3.2 to 4.8 when we move from predicting one week in advance to 4 weeks out. Similarly, for 9 months of data, MAD changes from 3.6 to 5.8. The change is greater when compared to the results of 12 months but it is still a lot less significant when compared to 6 and 3 months training data. For 6 months training data, MAD increase from 3.65 to 7.9, and for 3 28

44 Total Number of Cases months in goes from 3.9 to 8.6. The difference in predicting one, 2, 3, and 4 weeks is much more significant when using few months of data for training. It was also observed that forecasts improve as the length of training data increases. Using 3 months of training data is much more volatile and the results highly depend on the specific months chosen for training. Models with 12 months of training data consistently perform better than the models with shorter lengths of training data. For certain configurations with shorter training data lengths, models had very small errors. These configurations resulted in much better forecasts. This is largely dependent on the particular season/time of the year. When we looked at these configurations more closely, it was observed that configuration containing data from October 1, 2013 to December 31, 2013 had the smallest forecast errors. When we looked at the case volume for these three months, we noticed that the variation was much smaller for these months. Figure 12 shows time series of surgical case volume for these months Daily Surgical Volume Figure 12 Time series of daily surgical case volume (Oct-Dec 2013) The average overall case volume for these months was estimated to be 71.6 cases and the standard deviation was 6 cases. The standard deviation (6) is 46% smaller than the standard deviation of original training data of length of 3 years (11). Table 7 shows which data 29

45 configurations worked best based on MAD (the results are average of all 5 models), this table shows that for 12, 9, 6, and 3 months of training data, we get smallest MAD when we forecast months of December and November. As explained before, data is less volatile in these months. This confirms that forecast accuracy depends on the time period used for training and forecast. This effect is more significant when we use shorter lengths of training data. Table 7 Data configurations with smallest MAD Length of Training Data Forecasting Data Start Date 12 Months 1-Dec-13 9 Months 1-Nov-13 6 Months 1-Nov-13 3 Months 1-Nov-13 Second Approach We observed that case volume for different weekdays appear to be different. Thursdays have higher volumes and Wednesdays and Fridays have lower volume. ANOVA analysis was conducted to confirm that there is a significant difference between the surgical volumes for different weekdays. Table 8-Table 10 show the results of one-way ANOVA analysis to investigate the difference between the surgical case volumes for different weekdays. Since the data did not pass a normality test (prior to conducting the ANOVA analysis), we removed the outliers, any day with less than 30 cases scheduled. From the data after reviewing the box-plot shown in Figure 13. After removing outliers, the data approximately follows a normal distribution as it can be observed in Figure 14. Since the p-value is below 0.05, we can declare that the result is statistically significant. That is, there is a statistically significant difference in the mean patient volume between the five different groups (Monday, Tuesday, Wednesday, Thursday, and Friday). Next we removed Thursdays which have the highest case volume. 30

46 Surgical Case Volume All Weekdays Monday Tuesday Wednesday Thursday Friday Figure 13 Box plot of all weekdays and individual weekdays Figure 14 Normal probability plot of all weekdays 31

47 Table 8 One-way ANOVA for difference between weekdays with significance level=0.05 Analysis of Variance DF Adj SS Adj MS F-Value P-Value Error Total Table 9 One-way ANOVA with significance level=0.05 for difference between weekdays excluding Thursdays Analysis of Variance DF Adj SS Adj MS F-Value P-Value Error Total By excluding the Thursdays, which had the highest means, the p-value has increased. However, since the p-value is still below 0.05, we can declare that the result is statistically significant. That is, there is a statistically significant difference in the mean patient volume between the four different groups (Monday, Tuesday, Wednesday, and Friday). Then we excluded Monday with lowest case volumes as well as Thursdays. Table 10 One-way ANOVA with significance level=0.05 for difference between weekdays excluding Thursdays and Mondays Analysis of Variance DF Adj SS Adj MS F-Value P-Value Error Total By excluding the Mondays and Thursdays the p-value will increase significantly. Since the p-value is larger than 0.05 (p-value-0.577>0.05), we can declare that the result is not 32

48 statistically significant. That is, there is not a statistically significant difference in the mean patient volume between the three different groups (Tuesday, Wednesday, and Friday). ANOVA analysis was also conducted to investigate whether there is significant difference between the case volume of Mondays throughout the four years or not. Similar analysis was conducted for other weekdays. Table 11 shows the ANOVA results for the difference between Mondays and Table 12 summarizes the ANOVA results of all weekdays. Table 11 Analysis of variance for Monday s case volume across different months Analysis of Variance DF Adj SS Adj MS F-Value P-Value Error Total Table 12 Analysis of variance for different weekdays case volume across different months Analysis of Variance Monday Tuesday Wednesday Thursday Friday P-value Since the p-value is larger than 0.05 (0.676>0.05), we can declare that the result is not statistically significant. That is, there is not a statistically significant difference in the mean patient volume between the twelve different groups (Mondays of all 12 months). Similar conclusion was obtained for other weekdays. These analyses motivated us to try developing separate forecasting models for each weekday. We constructed five different models using non-seasonal, simple exponential, double exponential, and simple moving average forecasting methods. Each model predicts one weekday and then predictions from 5 models are combined to predict a full week Figure 15-Figure 17 show the comparison of the performance of our 5 different 33

49 MAD forecasting models based on MAD, MSE, and MAE. These figures show the accuracy of the forecasts 1-4 weeks prior to the day of surgery. For all of these models January 3, 2011 to December 31, 2013 was used as training data and January 1, 2014 to September 30, 2014 was used for validation. Model parameters were selected based on the default SAS algorithm. For these settings, model with autoregressive term p=1 and moving average q=2 outperformed other models. The error terms for the forecasts are smaller than the first approach. However, this approach will not consider the cross correlation between different weekdays. MAE changes more compared to the first approach when moving from one week ahead to four weeks ahead (1,0,2) (1,0,0)(1,0,0) MA(40) Single Exponential Smoothing α=0.145 Double Exponential Smoothing α(level)=0.857 and β(trend)= One Week Out 2 Weeks Out 3Weeks Out 4 Weeks Out Figure 15 Approach 2: comparing forecasting models based on MAD 34

50 Maximum Absolute Error MSE (1,0,2) 150 (1,0,0)(1,0,0)12 MA(40) One Week Out 2 Weeks Out 3Weeks Out 4 Weeks Out Single Exponential Smoothing α=0.145 Double Exponential Smoothing α(level)=0.857 and β(trend)= Figure 16 Approach 2: comparing forecasting models based on MSE (1,0,2) 20 (1,0,0)(1,0,0)12 15 MA(40) 10 5 Single Exponential Smoothing α=0.145 Double Exponential Smoothing α(level)=0.857 and β(trend)= One Week Out 2 Weeks Out 3Weeks Out 4 Weeks Out Figure 17 Approach 2: comparing forecasting models based on MAE 35

51 The change in MAE, MAD, and MSE is similar to the first approach in most cases. However, there are some irregular results in this approach. For instance, MA model with a monthly seasonality has smaller MSE predicting two weeks in advance compared to one week out. It can be observed that when comparing based on MSE and MAE, moving average outperforms S with monthly seasonality. Also, as it can be observed in Table 13, first chapter has better results when comparing based on MAD, MSE, and MAPE measurements. We did not pursue this method due to these inconsistencies and inferior performance compared to approach 1. Table 13 Comparison of results from first and second approach Forecast Best model from approach 1 Best model from approach 2 horizon (3,0,0) (2,0,0)5 (1,0,2) (weeks out) MAD MSE MAPE MAD MSE MAPE Conclusions We found that forecasting models considered in this study outperformed current hospital approaches by significantly reducing the MAPE. Our findings and results support the use of time series forecasting models to forecast surgical volume and ultimately forecast workload and make staffing adjustments. In the hospital that we studied, with average daily surgical case volume of 71 cases, we were able to reduce forecasting errors (MAE) by 73.9% (from 23 cases when forecasting one week in advance based on business analyst forecast to 6 cases based on (1,0,0) (1,0,0)5 results). 36

52 According to Tiwari et al (2014) accurate estimates of operating room demand cannot be made more than 1 or 2 days in advance and by then it is too late to close ORs with no booking or flex off the staff. However, our models show that predictions can be accurate a few weeks in advance. We have not directly addressed the level of inaccuracy in this chapter, but although there is still variability in the data, our models provide fairly good forecasts a few weeks in advance. In this chapter, we found that a seasonal model performed best in forecasting surgical case volume. Additional case volume information and factors affecting variations over time can be used to improve the forecast ability. In the next chapter, we propose a model that improves the forecasting ability of models. This improved forecast accuracy can be used to adjust staffing and other resource requirements up to 4 weeks ahead to better meet patient requirements. Optimized staff schedules have positive impacts on staff job satisfaction, patient safety, and total costs. 37

53 CHAPTER TWO FORECASTING SURGICAL DEMAND: DEVELOPING A TIME-SERIES FORECASTING TOOL 2.1 Introduction Hospital resource planning and staff management has become a significant problem throughout the United States, with healthcare delivery systems often challenged by variable demand for services, especially in short run (Litvak and Long, 2000). Inefficient staff scheduling can result in sub-optimal resource planning for the day of surgery and can ultimately lead to increased stress levels among staff and patients. This can adversely affect patient outcomes and also result in increased healthcare costs (Jalalpour et al., 2015). One of the challenges in surgical resource planning is the difficulty of estimating daily surgical case volume accurately (Schweigler et al., 2009). This is mainly due to the fluctuations in patient volume that are difficult to predict. Managing scarce healthcare resources requires accurate forecasts of future events including patient volume. Managers must anticipate final demand for services in order to match medical supplies, room capacity, and staffing efficiently (Jalalpour et al., 2015). Precise approximations and forecasts from accurate models can help decision makers to predict the need for services and make educated decisions about how to allocate resources and supplies over time. A majority of previous work on patient volume forecasting is focused on admission rates and bed occupancy in the emergency department (ED). To our knowledge, no previous work has directly studied the use of time series forecasting models to predict surgical case volume. This is probably because of the high variability in surgical case load. In chapter one, we evaluated the predictability of surgical case volume using a variety of known time series 38

54 forecasting methods. We compared the performance and accuracy of these time series forecasting techniques, with models consistently outperforming the other models. In this chapter, we focus on developing an model that can accurately predict surgical case volumes sufficiently in advance to enhance staffing adjustments. We are able to to control for seasonal trends as well as auto-correlative effects to improve model fit. We have also added predictive variables. models successfully describe current and future behavior of time series variables in terms of their past values. These models have been described as the most commonly used forecasting models in healthcare prediction (Jones et al. 2008). The forecasting models are mostly developed for ED prediction not surgical case volume forecasting. S models extend basic models and will allow the incorporation of seasonal patterns (Marcilio et al., 2013). In time series analysis, seasonality is defined as any repetitive pattern that happens with a known periodicity such as weekly pattern observed in surgical case volume. Surgical case volume forecasting models need to allow for this weekly seasonality. We have used S models to develop surgical case volume forecasting techniques. In order to further identify repeated patterns in our time series, we have introduced variables such as day of week, month of year, and holidays in our models. We propose a time series forecasting approach for predicting daily surgical case volume using S models with predictive variables. A secondary objective is to study historical data to understand the time-based dynamics in patient volumes across the perioperative departments. The new forecasting models are compared to an existing prediction system at GHS. They were also evaluated against S(1,0,0) (1,0,0)5 that was used by Tiwari et al., (2014). 39

55 2.2 Literature Review The literature review from the previous chapter is still relevant because the main motivation and research focus is similar. However, we want to discuss the use of time series methods for data analysis and forecasting in healthcare. Majority of work on surgical case volume prediction is focused on long-term planning. Long-term surgical volume can be fairly accurately predicted using historical distributions (Tiwari et al. 2014). There is much higher variability in the short-term which makes short-term scheduling more challenging. This does not mean than we can make accurate long-term forecasts and not precise short-term forecasts. In fact, long-term forecasts are usually less accurate compared to short-term forecasts; which means long-term forecasts tend to have larger error due to the larger standard deviation (Bernstein and Silbert, 1984). However, for long-term planning, forecasting errors might be less important. In long-term, if the distribution of surgical demand and OR costs, in terms of surgical case volume and type of procedure, follows the historical distributions, OR efficiency will be maximized (Tiwari et al. 2014).There is limited work available on short-term surgical case volume prediction. Tiwari et al. (2014) have developed a prediction method to forecast short-term case volume. The prediction is made by extrapolation from estimates of the traction of the total cases booked each of the 30 preceding days. The authors have averaged these with linear regression models. They have decided that time series would not be an appropriate tool due to the lack of correlation between demand of each day and the preceding days. However, patient volume forecasting in the ED has been extensively studied using time series methods (Jones et al. 2008). A majority of previous work on patient volume forecasting is focused on admission rates and bed occupancy in the ED. models have been used to forecast patient 40

56 admissions/arrivals in healthcare areas other than ED as well. For instance, Upshur et al. (1999) have used an model to perform a time series analysis on the relation between the influenza virus and hospital admissions of people over the age of 65 for congestive heart failure, pneumonia, and chronic lung disease. Even ED forecasting methods can provide more insight into the use of time series methods for surgical case volume forecasting. A number of studies have used different variations and/or combinations of time series models, especially models, to predict different variables within the ED. Jones et al. (2008) have investigated the use of multiple forecasting models to predict daily ED patient volumes. Forecasting models studied in this paper include linear regression, S, exponential, time series regression, and artificial neural network. Jones et al. (2008) have used MAPE to compare forecast accuracy. The authors have mentioned that their work is different because they have reported measures of post-sample forecast accuracy. This is different than using training data followed by data to measure performance and accuracy. Jones et al. (2008) have simulated true post-sample forecasts by incrementally expanding the training data and then forecasting the next period. For example, after generating forecasts for January 1, 2007, to January 30, 2007 based on training data, the observed values were added to the training data. The models were re-estimated using the expanded training data Jones et al. (2008). Table 14 summarizes advantages and disadvantages of the forecasting methods studied by Jones et al. (2008). They compared the accuracy of these models to the previously proposed prediction methods. They have discussed strengths and weaknesses of these models. Jones et al. (2008) compared these forecasting models to a benchmark linear regression method. A summary of their work can be found in Jalalpour et al. (2015) developed a publicly available toolbox to forecast demand for healthcare services (e.g., patient counts). They have attempted to develop a method that can take 41

57 into account the uncertainty in demand for services (i.e., demand counts) over discrete time intervals. The main objective of their work was to design a tool that can be used to accurately forecast count data in healthcare systems. Jalalpour et al. (2015) used generalized autoregressive moving average (GARMA) models to develop their forecasting tool. GARMA models can incorporate multiple exogenous variables that are assumed to affect the predicted responses (i.e., counts forecasts). Their toolbox uses a maximum likelihood method to estimate model parameters from input data, and they showed that GARMA models can outperform the traditional Gaussian models by comparing the results of the proposed toolbox with the previously published models. Jones et al. (2009) used multivariate vector autoregressive (VAR) models to forecast ED census for extended forecast horizons. They have also used these VAR models to forecast the demands for various diagnostic resources. Out of sample forecast accuracy was evaluated using forecasting horizons ranging from one to 24 hours in advance. Mean absolute error was used as a measure to assess forecast accuracy. Jones et al. (2009) concluded that multivariate VAR models provided more accurate predictions of demand in the ED and the inpatient hospitals compared to the standard univariate time series models. Jones et al. (2009) have found contradictory results compared to the results of existing similar studies. They have found that there is a very small correlation between inpatient volume and ED operations. However, other studies have shown that there is a significant interaction between inpatient and ED patterns. Jones et al. (2009) have found that this contradiction is due to the facilities they have considered in their analysis. They have used one of the models from previous studies and concluded that inpatient operations for these particular facilities have a small impact on ED operations. They have suggested using robust analytical techniques such as 42

58 simulation modeling and queuing theory along with these forecasting models to facilitate decision making (Jones et al. 2009). Table 14 Advantages and disadvantages of forecasting methods evaluated by Jones et al. (2008) Forecasting Method Advantages Disadvantages Linear regression 1. Simple and very familiar statistical method 2. Only requires moderate level of statistical expertise 3. Capable of modeling trends and seasonal variations 4. Informative modeling process 5. Widely available statistical software S 1. Theoretically appropriate method for most time series 2. No external data is necessary (univariate method) 3. Capable of modeling trends, seasonal variations, autoregressive, and moving average processes 4. Statistical software widely available 1. Not appropriate for auto correlated nor non-linear data 2. All observations (recent and remote) are weighted the same 3. Requires additional data collection and parameter estimation for multiple variables 1. Complex statistical methodology which requires higher level of experience and expertise 2. The modeling process is less informative than linear regression Exponential Time series regression Artificial neural network 1. Low level of experience is required, fully automatic 2. Capable of modeling trends, seasonal variations, autoregressive, and moving average terms 3. Effective when model parameters are changing over time 4. Statistical software widely available 1. The results are easily interpretable 2. Capable of modeling trends, seasonal variations, autoregressive, and moving average terms 3. Informative modeling process 4. Statistical software widely available 1. Capable of modeling non-linear, complex systems 2. Allows for rapid adjustment to changes in the time series Not based on formal statistical theory 2. The modeling process is less informative than linear regression 1. Complex statistical methodology that needs a higher level of expertise and experience 2. Requires additional data collection and parameter estimation for multiple variables 1. Black box modeling procedure makes the final model difficult to interpret 2. Statistical software packages provide fewer, less mature procedures to estimate these models

59 Some authors have tried to develop prediction methods that can be used in different settings with minimum adjustments required. Schweigler et al. (2009) studied whether time series models can generate precise short-term forecasts of bed occupancy in the ED. They have developed forecasting models using seasonal models as well as a sinusoidal model with an autoregression (AR) structured error term. The authors have compared the results of these models to the results of traditional historical averages models. They have concluded that both seasonal and sinusoidal models with AR-based structure, generated robust short-term forecasts of ED bed occupancy for subsequent hours. These models only required bed occupancy in the preceding hours as an input. These forecasting models worked equally as well at three different EDs with different operational characteristics, without any need to adjust model input variables (Schweigler et al., 2009). It should be noted that in this work, Schweigler et al., (2009) have used hourly bed occupancy to develop forecasting models not daily volumes. This is further motivation that even the most recent historical data can play a role in forecasting, and in this case, accurate forecasts were obtained with only the most recent data. In a surgical services environment, with the schedule filled with predominantly elective, scheduled cases, using only the most recent data would not apply. However, we can appreciate their use of historical data. Boyle et al. (2011) have tried to simplify the prediction process. They have developed forecasting models to predict the number of ED presentations and subsequent patient admissions. Their main objective is to develop a simple demand prediction method that can be used by different EDs with minimum adjustments. The authors have used multiple regression,, and exponential models to build their forecasting tool. They have emphasized that it is important to account for seasonality in bed occupancy and patient admissions to generate accurate forecasts. In order to address this issue, the authors have added variables for holidays, 44

60 days of week, and months of year. They have validated their model by comparing their results to the existing prediction system at one of the hospitals as well as comparing their results to the results of similar works. A number of past studies have focused on improving forecasts by adding different predictive variables. As discussed by Hoot et al. (2007), one should be careful in choosing these variables since more complex models tend to over fit the data. Eijkemans et al. (2009) investigated the effects of operation type, surgical team, and patient characteristics on the surgical operating room times at a general surgical department in an academic hospital. The authors studied whether a surgeon s prediction had a predictive effect independent of the rest of the factors. The outcome to be forecasted is total operating room time which is defined as the time from when the patient enters the OR until the time when the patient leaves the OR (i.e., patient in room to patient out of room). In order to improve surgeon estimates, surgeons were provided with the mean duration of previous surgeries of the same type. Surgeons could make adjustments to their predictions when necessary. The results of the prediction model were compared against the surgeon s prediction of the total surgical operating room time. It was shown that surgeon s predictions underestimated the total OR time. Eijkemans et al. (2009) have studied the effects and relationships (linear or non-linear) between the total OR time and different predictive variables. They have used all the variables that had a statistically significant influence to predict the total OR time. Based on their analysis, surgeon s forecasts and operations characteristics were the most significant predictors of the total surgical time. Their results show that predictions of longer surgeries are less accurate because the variation in OR time increases as the mean surgery time increases. 45

61 Ekstrom et al. (2015) developed forecasting models to predict ED visits and identify behavioral trends in an effort to more accurately allocate resources and prevent ED crowding. The authors have pointed out that previous work on this topic has only relied on a limited number of variables. The main objective of this work is to find additional predictive factors for OR inflow. They have used Internet data, i.e., Website visits, as a predictor of the ED attendance for the coming day. Although models have been shown to be superior to linear regression models, the authors have chosen the linear regression method and have tried to improve their model by adding independent variables that have not been used previously (Ekstrom et al. 2015). Ekstrom et al. (2015) best model included day of week evening website visits as independent variables. The number of ED visits for each hospital and the entire Stockholm country were the dependent variables. They have used Google Analytics to collect data for visits to the web site. They have shown that evening Website visits as an independent variable has a statistically significant effect for individual hospitals and the country as a whole. So they have concluded that it is possible to more accurately predict the number of ED visits using Internet data. Peck et al. (2012) have used three methods, expert opinions, generalized linear regression model with a logit link function (logit-linear), and naïve Bayes conditional probability, to predict the number of ED patients that will be admitted to the hospital inpatient unit, in real time. All these 3 models are designed to forecast inpatient admissions at the time of ED triage. They have used patient age, bed type, primary complaint, and arrival mode as input variables. The authors have discussed that majority of previous studies have focused on forecasting individual patient admission or long-term admission trends. In order to address this gap, they have attempted to 46

62 predict admission demand when patient arrives to the ED. Logit-linear regression approach has outperformed the other two methods. The naïve Bayes method was the second best. Hoot et al. (2007) have used 4 measures including the Demand Value of the Real-time Emergency Analysis of Demand Indicators (READI), the Emergency Department Work Index (EDWIN), the National Emergency Department Overcrowding Scale (NEDOCS), and the work scores to monitor short-term ED crowding. These 4 measures will output a continuous variable. Higher values of this variable indicate a greater degree of crowding. They have used ambulance deviation status as the outcome measure for ED crowding. The use of ambulance deviation as the ED crowding outcome measure can be one of the limitations of this study. Using these measures for monitoring ED crowding requires the electronic availability of ED operational variable, such as number of boarding patients, length of stay, and waiting room count. The control measure, occupancy level, performed better that all the 4 crowding measures. None of these variables exceeded the control measure in timeliness either (Hoot et al., 2007). The authors have concluded that since occupancy level that was added as a simple baseline measure worked better that all the other measures, one should use the least complex model that can achieve desired objectives. More complex models can over fit the data. Asplin et al. (2006) have discussed the overlap between patient flow and daily surge capacity, the ability to respond to a sudden, unexpected surge in demand. The authors have proposed two models that can be used to study daily surge capacity and patient flow. They have have developed a model for ED census that incorporates typical patterns, unexpected surges, and temporal correlations. This model explains the dynamic nature of ED census by using a series of equations. The second model is a theoretical method that demonstrates the relationship between length of stay in ED and the quality of care. 47

63 Sometimes adding these variables will not improve forecast accuracy significantly. Marcilio et al. (2013) developed multiple models to forecast daily ED visits, and they compared these models in terms of forecasting accuracy. They have used calendar variables and ambient temperature readings as predictive measures. The authors have selected these variables because previous studies have shown that calendar variables, such as day of week, holidays, and time of year, will high impact the number of ED visits. Weather conditions such as temperature can also affect the number of ED visits. They have investigated 3 different modeling approaches, generalized linear models (GLM), generalized estimating equations (GEE), and seasonal. They measured forecast accuracy comparing MAPE of different models. The results showed that calendar variables were more significant than ambient temperature readings. Also, they have concluded that weekly seasonality was more dominant than monthly seasonality. A number of authors have focused on developing robust models for healthcare forecasting. There are periods with high volatility that need to be taken into account by models. Reis and Mandl (2003) have developed robust models for ED utilization. Their objective is to use these models to determine expected ED visit rates. They have used as well as trimmed-mean seasonal models to develop their forecasting models. The authors have built their models based on extensive historical data, almost 10 years. Models were trained using the first 8 years and validated using the remaining 2 years of data. These models can predict both respiratory-related and overall pediatric ED volumes. Using MAPE, the accuracy of the models was evaluated and it was concluded that their models provide forecasts with good accuracy. The authors have developed an integrated alarm strategy that combines recent trends with historical data. 48

64 Jones and Joy (2002) have used S models to predict the number of hospital beds occupied by emergency cases. They have concluded that a period with high volatility, indicated by Generalized Autoregressive Conditional Hetroskedastic (GARCH) errors, will increase the waiting times in accident and emergency (A&E) department. The authors have used air temperature and Public Health Laboratory Service (PHLS) data on influenza like illnesses, as predictive variables. 2.3 Methods Our goal was to find a good S model that adequately represents our data. The general model is a combination of autoregressive (AR) coefficients multiplied by previous values of our time series plus moving average (MA) coefficients multiplied by past random shocks (Box et al., 1994) and (Pankratz, 1983). The models are usually fitted using the Box-Jenkins computational method (Box et al., 1994). The order of the AR and MA terms should be chosen such that the theoretical autocorrelation (ACF) and partial autocorrelation (PACF) functions approximately match the ACF and PACF of our modeled time series (Abdel-Aal and Mangoud, 1998). During model generation, 39 months of data is used for parameter estimation (or training data), while the last six months in the data set is reserved for validation (or validation data). Training data is used to train the model and obtain estimates. Validation data was used to estimate how well the model was trained and to calculate model properties such as errors. Components of Time Series and Model Structure Any time series is generated by a sequence of random shocks or white noises denoted as a t. In this chapter, we have assumed that daily surgical case volumes is a time series, denoted as: 49

65 Y, Y,..., Y, Y 1 2 t 1 t In order to model surgical volume as a time series process, we had to make the following assumptions about the behavior of random shocks: 1. Zero mean for the random shocks, mean( a t )=0, 2. Constant variance for the random shocks, variance( a t )= 2, 3. Independent shocks, covariance( a t at k)=0, 4. And normal distribution for the shocks, at N. An model is a model of the stochastic process that created the observed time series. An model has three structural parameters (p, the autoregressive (AR) component; d, the structure parameter which determines the order of differencing; and q, the moving average (MA) component). These parameters explain the relationship between random shocks and time series. In an model with AR component, the current time series observation, Y t, is a portion of preceding observation, Yt 1, and a random shock, a t. model with MA component is a model where the current time series observation, Y t, is composed of a current random shock, a t, and a portion of the preceding random shocks, at 1 to t a q. The data appeared to be stationary so no differencing was needed to stationarize the series in order to develop our models. In an model, a random shock enters the model, goes through a sequence of filters, integration, autoregression, and moving average, and leaves the model as a time series observation. This has been shown in Figure

66 a t Integration Autoregression Moving average Y t Figure 18 Diagram of (p,d,q) process Seasonal Models Seasonality in any time series can be defined as a regular pattern of changes that repeats over S time periods, and S is the number of time periods until the pattern repeats again. In a S model, seasonal AR and MA components predicty t, using time series observations and random shocks at times with lags that are multiples of S (the span of the seasonality). For instance, for monthly data (and S = 12), a seasonal first order AR model would use Yt 1 to predict Y t A seasonal second order AR model would use Yt 12 and Yt 24 to predict Y t. A seasonal MA(1) model (with S = 12) would use at 12 to predict. A seasonal second order MA(2) model would use at 12 and at 24. The S model incorporates both non-seasonal and seasonal factors in a multiplicative model. Notation for this model is S(p, d, q) (P, D, Q)S, with p = non-seasonal AR order, d = non-seasonal differencing, q = non-seasonal MA order, P = seasonal AR order, D = seasonal differencing, Q = seasonal MA order, and S = time span of repeating seasonal pattern. Without differencing, the model could be written as follows: S S ( B ) ( B)( Y ) ( B ) ( B) a t t 51

67 where µ is the mean and,,,( 0) 1 2 p p ;,,,( 0) 1 2 p p ;,,,( 0) ; and,,,( 0) 1 2 q q are constants. B is the backshift operator that can 1 2 q q be defined as BY t, Yt 1 where the non-seasonal components are: AR( p) : ( B) 1 ( B) B 1 p p MA( q) : ( B) 1 ( B) B 1 q q The seasonal components are: Seasonal AR : ( B ) 1 ( B ) B S S PS 1 P Seasonal MA : ( B ) 1 B B S S QS 1 Q Autoregressive and Moving Average Components We plotted the time series of surgical case volume from January 2011 to September 2014 (967 consecutive days). It is shown in Figure 19. A scatter plot of surgical case volume according to date of surgery is presented in Figure 20. The data is highly variable, with some points mostly holidays exhibiting outlier behavior, which makes the prediction very difficult. Due to the complexity, irregular nature of the data, and the presence of strong random components, such as add-on cases, straight-forward methods like extrapolation simply cannot represent the data effectively. It can also be observed that weekdays (high volume) and holidays (low volume) have different patterns. The data for these sets are partitioned by two distinctive bands on the scatter plot. Note that a model that includes everything will be much more powerful than one that tries to fit only the typical, high volume, non-holiday days. We can arrive at 52

68 meaningful estimates for both types of days, whether it falls into the prediction ellipses for the standard volume day or the estimates for those non-standard days with less volume. We did not exclude any holidays or low/high volume days in this chapter. In other words, we are including as much data as possible in our analysis. We are allowing the model to determine how important each data point is. However, weekends are excluded since they have a different pattern. The data did not show any significant trend requiring differencing apart from the slight upward drift due to the yearly increase in patient volume. So we can assume that we have a stationary stochastic process, a process with no trends and drifts. One option is to perform Dickey-Fuller tests to decide whether the time series is stationary or not. Fuller (1996) gives such a table for τµ indicating that τµ would have to be less than to achieve significance at the 5% level. The subscript µ is used to indicate that the mean (intercept) has been estimated. The calculated τµ of meets that criterion and hence the correct test does suggest stationarity and there is no need for differencing. We can see the results of this test in Table 15. Such process can be determined by its mean, variance, autocorrelation (ACF), and partial autocorrelation (PACF) function. The mean ± SD daily surgical volume was estimated as 67±14 cases per day. For any finite time series with n observations, ACF(k) which is the correlation between Y t and t be estimated from the following formula: Y k, may ACF K n ( Y )( Y ) i i k i 1 ( ) ( ) n 2 n k ( Y ) i i 1 n 53

69 100 Series Values for DailyVolume 80 DailyVolume Observation Figure 19 GHS daily surgical case volume time series, January 2011 to September 2014 Figure 20 Scatter plot of daily surgical case volume, January 2011-October

70 In theory, each S model has a unique ACF, so we can use the information from ACF and PACF to determine the model structure. The ACF and PACF of our data are shown in Figure 21. Based on the ACF and PACF, MA and AR parameters were constrained to the following bounds: and Table 15 Dickey-Fuller tests for stationarity Augmented Dickey-Fuller Unit Root Tests Type Lags Rho Pr < Rho Tau Pr < Tau F Pr > F Zero Mean Single Mean < < Trend < <

71 1.0 Autocorrelations for DailyCaseVolume 0.5 ACF Lag Two Standard Errors 1.0 Partial Autocorrelations for DailyCaseVolume 0.5 PACF Lag Two Standard Errors Figure 21 The ACF and PACF of surgical case volume time series 56

72 Model Building Procedure After developing the algebra of our models, we started the model building procedure. We followed the model building strategy proposed by McDowall et al. (1980). Figure 21 demonstrates model building steps. 1. Identification: As a first step, an autocorrelation (ACF) and partial autocorrelation (PACF) were estimated from our data. These plots indicated that surgical case volume series is a stationary process; the series does not have to be differenced. We also identified appropriate ranges for AR and MA terms. 2. Estimation: this step requires a statistical software, we used SAS to estimate model parameters that would satisfy conditions mentioned in Figure 22. For the models that we selected, all the estimates were statistically significant at α=0.05 level. 3. Diagnosis: finally, for the selected model, the residuals were tested and they were not different than white noise. The results of these tests can be seen in Figure 23 and Figure 24 as well as Table 16. We tested the residuals for the selected model to determine whether the tentative S model adequately models the statistical properties of the original time series. The residuals plot is shown in Figure 25. The residual ACF plot of the time series using S(3,0,2)(2,0,1)5+H+M+W+Th+J+A model is shown in Figure 26. ACF plot of the residuals show only one significant autocorrelation at lag 14. Even when the time series is modeled adequately, some significant residuals and autocorrelations may exist due to the random errors (Abdel-Aal and Mangoud, 1998). 57

73 1. Identification Either criterion not met 2. Estimation Parameter estimates must be statistically significant. Parameter estimates must lie within the bounds of stationarity-invertability. Both criterion are met Either criterion not met 3. Diagnosis Model residuals must be white noise: First, the residuals ACF plot must have no spikes at key lags. Ljung-Box Q statistic must not be significant for key lags. End Both criterion are met Figure 22 The model-building procedure McDowall et al. (1980) 58

74 Residual Correlation Diagnostics for DailyVolume ACF 0.0 PACF Lag Lag 1.0 IACF White Noise Prob Lag Lag Figure 23 Model residual test for white noise, ACF and PACF Residual Normality Diagnostics for DailyVolume Distribution of Residuals QQ-Plot 25 Normal Kernel Percent Residual Residual Quantile Figure 24 Normality test for residuals 59

75 Daily Prediction Error or Residual Figure 25 Residual Plot of S(3,0,2)(2,0,1)5+H+M+W+Th+J+A model Figure 26 Residual ACF function for the S(3,0,2)(2,0,1)5+H+M+W+Th+J+A model Table 16 Ljung-Box test statistics for key lags Autocorrelation Check for White Noise To Lag Chi-Square DF Q-Statistics Autocorrelations < < < <

76 Algorithm Used by SAS The procedure in most statistical software packages, including SAS, uses the computational methods outlined by Box and Jenkins (Box, Jenkins, 1970). In SAS, 3 different methods are available for parameter estimation. 1. Maximum Likelihood: This method provides maximum likelihood estimates. The likelihood function is maximized through non-linear least squares using Marquardt s methodology. Maximum likelihood estimates are more difficult to compute compared to the conditional least squares. However, they might be preferable in some cases because there is no assumption on past errors. 2. Unconditional least square: This method is also referred to as the exact least squares (ELS) method. In this method, the estimates are obtained by minimizing the sum of squared residuals, rather than using the log likelihood as the criterion function. 3. Conditional least squares (CLS): The CLT estimates are conditional on the assumption that the past observed errors are equal to zero. We selected the maximum likelihood method to compute our model parameters. In the algorithm we have developed in SAS, two information criterion are computed for models, Akaike s Bayesian criterion (AIC) (Akaike 1974; Harvey 1981) and Schwarz s Bayesian criterion (SBC) (Schwarz 1978). The AIC and SBC are used to compare competing models fit to the same time series. Sometimes Schwarz s Bayesian criterion is called Bayesian information criterion (BIC). The model with the smallest information criterion is said to fit the data better. The AIC is computed as follows: AIC 2ln( L) 2k Where L is the likelihood function and k is the number of free parameters. 61

77 The SBC is computed as follow: SBC 2 ln( L) ln( n) k where n is the number of residuals that can be computed from the data, time series Measurements For each of the models that we developed, the validity of model assumptions was checked. We selected the best fit that satisfied diagnostic checks and model assumptions. We also rated the models based on AIC and SBC with the lowest values being the best candidates. As model accuracy is very important in selecting a forecasting method, we calculated MAPE to evaluate the performance of the models. In maintaining consistency with prior research studies in forecasting demand in a healthcare setting, we select the MAPE criterion as our main error performance measure (see, e.g., Jones et al. (2008); Marcilio et al. (2013), ) It is very effective in describing the accuracy of prediction models. In a majority of ED forecasting papers, a MAPE error value below 10% has been considered as good performance and the corresponding model would be comparable with other methods that were presented previously. In all of our calculations the level of significance was set to 95%; p < As mentioned in the previous section, we divided our data set into training and validation sets, with the training set including 1-Jan-11 through 31-Mar-14 and the validation set including 1-Apr-14 through 30-Sep-14.. A specific outcome of this chapter is also to compare the performance of our forecasting model against the existing prediction method implemented at GHS, as well as the model discussed by Tiwari et al. (2014). 62

78 2.4 Results After modeling surgical case volume time series, we added holidays, Mondays, Wednesdays, Thursdays, January, and April as covariates in the time series model to account for any trends that may not have been captured by procedure. We initially added all weekdays and months; however, as expected not all of them were significant. We eliminated variables with insignificant t values, following a backward elimination procedure. Different models were fitted to this time series. As expected from ACF and PACF, the best fit with the smallest AIC value, included AR, MA, SAR, and SMA terms. The final variables along with their coefficient estimates, standard errors are represented in Table 17. Table 17 Coefficient estimate and the standard error for the best model Variable Coefficient estimate Standard error t Value Pr> t Intercept µ <.0001 Holidays H <.0001 Monday M <.0001 Wednesday W <.0001 Thursday Th <.0001 January J <.0001 April A <.0001 MA <.0001 AR <.0001 SMA <.0001 SAR <.0001 This model can also be represented as the following equation: Y Y a a Y Y a Y a a t 1 t 3 1 t 2 1 t 1 5 t 5 10 t 10 t 1 t 1 1 t 2 5 t 5 H M W Th J A We evaluated the forecast accuracy of this model for different days of the week as well as different months of the year. The results can be observed in Figure 27 and Figure 28. These MAPEs are the averages of all 12 forecasting periods. The results are presented in Table 18 as 63

79 well as Figure 29. We compared the results of this model, predicting one to 4 weeks out to the results of proposed by Tiwari et al., as well as with the currecnt forecasting method at the hospital. As a sample of our predictions, we have represented the forecasts obtained for four weeks in January and February 2014, 1/14/14 to 2/10/14. We chose to show forecasts for this period because it contained Martin Luther King, as it can be observed in Figure 31, the model captures the low volumes associated with holidays very well. The confidence interval only increases 5.32% from first to last week. 0.7 MAPE for days of week MAPE Monday Tuesday Wednesday Thursday Friday Figure 27 MAPE by day of the week 0.4 MAPE for months of year 0.3 MAPE January February March April May June July Augus t September October November December Figure 28 MAPE by month of the year 64

80 Thursday 1/14/14 Friday 1/15/14 Monday 1/18/14 Tuesday 1/19/14 Wednesday 1/20/14 Thursday 1/21/14 Friday 1/22/14 Monday 1/25/14 Tuesday 1/26/14 Wednesday 1/27/14 Thursday 1/28/14 Friday 1/29/14 Monday 2/1/14 Tuesday 2/2/14 Wednesday 2/3/14 Thursday 2/4/14 Friday 2/5/14 Monday 2/8/14 Tuesday 2/9/14 Wednesday 2/10/14 Daily Case Volume MAPE Table 18 Forecasting error of daily surgical case volume made 1, 2, 3, and 4 weeks in advance Week 2 Weeks 3Weeks 4 Weeks Weeks in adnance forecast (3,0,2)(2,0,1)5 (1,0,0)(1,0,0)5 Hospital Prediction Figure 29 Forecasting error of daily surgical case volume made 1, 2, 3, and 4 weeks in advance Actual estimates 95% Lower Limit 95% Upper Limit Figure 30 Forecasts of final seasonal model 65

81 Daily Surgical Case Volume As it can be observed in Figure 31, S(3,0,2)(2,0,1)5 with holidays, weekdays, and month variables provides very good forecasts and the confidence intervals have a small range for up to three months, i.e., the confidence interval increases slightly up to three months and after that it increases dramatically. This suggests that the model is applicable for strategic decisions, not just the tactical decisions made 1-4 weeks prior to the actual schedule % Lower Prediction Limit 95% Upper Prediction Limit Forecasted Value Predicted Volumes and Prediction Intervals for October 2014-May 2015 Figure 31 Change in prediction interval-forecasting October 2014-May Conclusions The results of current chapter demonstrated that models are useful in forecasting daily surgical case volume. Using only daily surgical case volumes of preceding years as input, seasonal models generated accurate short-term forecasts of subsequent surgical case volumes. Through explicit use of all weekly historical data over several years, our results indicate that the S(3,0,2)(2,0,1)5 model can be a useful tool in forecasting daily surgical case volume The MAPE of our forecasts was on average 51.3% smaller than the hospital prediction method. 66

82 We anticipate that the output from this model could be used for multiple applications. As our primary focus, the forecasts can assist with estimating workload and be used as a decision support tool for staffing in different departments (e.g., Preop, OR, PACU, Sterile Processing Department (SPD), etc.). It can also be used for capacity adjustment, supply and processes, such as bed management and case cart preparation. Our research has several potential limitations. This research was based on data from only one hospital. Differences in setting, resources, and surgical procedures might result in differences in terms of parameter estimates (Jones et al. 2009). We have not evaluated how clinicians would evaluate our forecasts in real-time. In other words, we chose a retrospective design to assess the forecast accuracy and the predictive ability of our time-series models. Also, another limitation of our models pertaining to the utility of our forecasting methods as a decision support tool, is that we have not considered other important factors (e.g., staff satisfaction, cost, etc.). 67

83 CHAPTER THREE MODELING AND ANALYSIS OF PREOP NURSE SCHEDULING USING SCENARIO- BASED DEMAND MANAGEMENT 3.1 Introductions Staff scheduling in service industry such as hospitals, airlines, and call centers, is a lot more challenging compared to manufacturing settings operating 8 hour shifts (Quan 2005). Hospitals typically operate 24 hours a day, 7 days a week, and are faced with high fluctuations in demand. To deal with fluctuations in demand, healthcare providers typically use a combination of fulltime and part-time staff, different shift start times, and a range of shift durations (Levine 2001). In preparing for the day of surgery, the hospital wants to ensure that sufficient nursing and support staff will be available to deliver care to surgical patients. The challenge is balancing the surgical caseload scheduled far in advance with the many adjustments (cancellations, add-ons, emerging cases, etc.) that occur in the days leading up to the day of surgery. While Chapter 2 focused on obtaining a reliable forecast 2-4 weeks in advance, the focus of Chapter 3 is to use that forecast to assist in setting a staff schedule that will require as few adjustments as possible, recognizing that demand fluctuates from day to day. The general objective is to find a feasible set of nurse schedules that minimizes labor costs and avoids high overstaffing and understaffing. In this chapter, we study the process of assigning each nurse to a particular shift in each day within a specific planning horizon with the objective of minimizing the labor costs, while ensuring to cover surgical demand. Nurse scheduling methods used at the hospitals often develop schedules that are suboptimal, resulting in overtime or low nurse utilization, nurse dissatisfaction, patient delays, and patient dissatisfaction. In this research, we propose an approach that utilizes time series forecasting models and scenario-based mathematical modeling as tools for solving the Preop nurse scheduling problem. 68

84 Nurse scheduling typically consist of two phases: (1) Nurse planning determining how many nurses are required to work each day, followed by (2) Nurse assignment assigning individual nurses to available shifts to meet the demand. A nurse schedule is typically defined as a combination of both stages. We address the nurse planning and scheduling problem(s), assuming that the actual size of the workforce has already been determined. There are many situations where staff scheduling problem will be addressed while the workforce already exists and is a fixed size (Gerard et al. 2015, Rocha et. al. 2013, Valls et al. 2009, Eitzen and Panton, 2004, and Isken and Hancock, 1991). Developing a schedule that minimizes the costs while assigning desirable shifts evenly is extremely difficult (Burke et al., 2004). This is why a good modeling approach can be very useful. A nurse schedule with too few nurses will result in high overtime or outsource costs, as well as delays in patient surgery. On the other hand, a schedule with too many nurses will result in higher than necessary nurse assignment costs and low nurse utilization. We use a risk averse approach to find a feasible nurse assignment that minimizes the sum of labor costs, with a particular focus on avoiding extremely high cost scenarios. Patient volume will directly impact daily staff workload, and it is difficult to manage fluctuations in the workload / staff scheduling weeks in advance. The (3,0,2)(2,0,1)5 model developed in Chapter 2 is used to forecast daily patient volume. Using hourly demand profiles created from historical data, we translate forecasted daily surgical case volume into hourly demand. In order to account for the variability in the system, distinct demand scenarios are created based on the results of the model. Having distinct scenarios where the demand can be low, typical, or high provides a more robust representation of the demand level that could occur on any day, and it gives the user a sense of how their decisions will affect the performance of the system. 69

85 Generally, nurse planning and scheduling can be decomposed along the time horizon and be viewed hierarchically: long-term planning, mid-term decisions, and short-term scheduling (Perskalla and Brailer 1994, Warner 1976). In long-term planning the objective is to find the optimum number of regular nurses that should be hired by the hospital, as well as a combination of shift profiles (Benton 1992). In this research, this information was already provided by the hospital and was fixed. We used this information along with the expected demand obtained from our forecasting model as input to our mid-term nurse scheduling problem. We attempt to find the minimum number of nurses required at each time during the day over the planning horizon. We solve the mixed integer programming (MIP) formulation using an expected value (EV) as well as a conditional value at risk (CVaR) approach. We use EV to develop solutions that will minimize the total expected cost of all demand scenarios. We then use CVaR to develop a staff schedule that avoids risky cases, i.e., highly overstaffed or understaffed situations. The liabilities of overstaffing and understaffing are many. Overstaffing increases payrolls and results in excessive idle times, while understaffing will negatively impact patient outcomes and may result in loss of revenue. To the best of our knowledge, the nurse scheduling problem in the context we describe has not been previously addressed in the research literature. This chapter has three main goals. The first objective is to develop a method to estimate the nurse workload based on surgical case volume: daily patient volume forecast, half-hour demand predictions, and half-hour nurse requirements. The second objective is to account for the stochasticity by generating different demand scenarios and assigning a probability to each scenario rather than using only the forecasted point estimate for daily demand. Finally, we would like to generate a method to minimize the total cost of nurse schedule, both from expected value and risk-adjusted points of view. By taking various shifts, nurse contract types, and demand 70

86 possibilities into account, we offer a model that provides a nurse assignment solution. We then consider a risk-averse approach so that we can target the reduction or avoidance of those potentially highly over or understaffed days. The remaining content of this chapter is organized as follows: section 3.2 presents the literature review, section 3.3 introduces the methods used to address the nurse scheduling problem, section 3.4 presents the solution approaches used to solve the problem, section 3.5 summarizes the research findings, and sections 3.6 and 3.7 discuss conclusions and future work. 3.2 Literature Review Healthcare providers are constantly challenged to provide high quality care with minimum costs. The nurse scheduling problem (NSP) is a particularly difficult and challenging resource management problem faced by the hospital due to the multiple factors that must be considered, such as regulations, hospital policies, fluctuations in demand, and nurse preferences (USDHHS 2002, and Levine 2001). All of these factors contribute to the NSP being a timeconsuming managerial problem (Wong et al. 2014). When developing a staff scheduling model for healthcare systems, consider that hospital policies and practices will affect total labor cost (Kreitz & Kleiner, 1995), and the quality of the staff schedule has a huge impact on staff satisfaction and turnover times (Maynard, 2005). A high-level framework for staffing problems is represented in Figure 32. This was developed based on the works conducted by Warner et. al. (1976). In order to appropriately analyze staff scheduling and develop a good solution approach to the NSP, a review of the previous research conducted in the area of nurse scheduling problem is required. Our literature review considers different models and objectives used to address variants of the NSP. 71

87 Budget, staffing plan, policies Budgeting and staff planning Performed annually or as needed Capacity planning Staffing policies Tactical staff scheduling analysis Operational staff scheduling Performed weekly or monthly Target staff levels Creates employee schedules for core staff Staff schedule Daily staff allocation Ongoing everyday Reacting to staffing variations Flexing staff, under time, overtime, temporary staff Realize overstaffing and understaffing Figure 32 Staffing framework adapted from Warner et. al. (1976) Several authors have collected and reviewed the advancements, models, solution approaches, and their applications on the nurse scheduling problem. In section 3.2.1, we discuss common nurse scheduling models as well as surveys and general reviews of staff scheduling problems. Special cases of staff scheduling problems, arising from particular needs of and applications, are discussed in section In section 3.2.3, we review different solution approaches including mathematical programming, heuristics, and simulation. Finally, in section 3.2.4, we discuss risk-adjusted performance measures and their applications. 72

88 Surveys, Application Areas, and Nurse Scheduling Different healthcare environments imply different work requirements. As a result, a unique staff scheduling problem with distinct features will arise. When modeling the nurse scheduling problem, objectives and constraints will vary based on the problem s features (Rekik et al., 2010; Castillo et al., 2009; Burke et al., 2010; Addou and Soumis, 2007; Azaiez, 2005; Aickelin, 2004; Burke et al., 2004; Isken, 2004; Topaloglu and Ozkarahan, 2004; Brotherton, 1999; Brusco and Johns, 1996). Based on the literature reviewed, the following constraints are found to be most commonly used constraints in nurse scheduling problems: Coverage constraint: determines the number (minimum/maximum) of nurse assignments needed per shift per day or per planning horizon. Sequence or consecutiveness: maximum/minimum number of consecutive working/rest hours/days determined by law or preferred by staff. Workload balance: even and balanced distribution of workload (shifts/days off) between all the staff. Even though there are different objectives, all nurse scheduling problems attempt to achieve a balanced and fair nurse schedule with minimum cost and minimum deviation from the assignments (Totterdell, 2005). Objectives that are often considered in staff scheduling problems are as follow: Minimizing total labor costs. Minimizing understaffing or minimizing the uncontracted work hours. Minimizing the size of the workforce, or the total number of nurses hired. Minimizing under and/or over coverage. Maximizing staff satisfaction. 73

89 Van den Bergh et al. (2012) present a comprehensive review of 291 articles published since The authors present a review of the literature on personnel scheduling including but not limited to NSP. The authors have categorized these articles based on four major topics, 1) staff characteristics (full-time vs. part-time contract, skill level, individual vs. team, shift length, and shift type); 2) constraint type (hard/soft, fairness, coverage, balance, time-related) and measures of performance (staff/patient satisfaction, different types of cost); 3) solution approach and level of uncertainty included (in terms of demand or capacity); and finally, 4) and application area. Some key findings of the reviewed manuscripts are presented here. The coverage constraint is a very important constraint in nurse scheduling models. Over 75% of the authors have defined the coverage constraint as a hard constraint. Almost all of the researchers have defined the balance constraint as a soft constraint. The sequence and consecutiveness constraints have been modelled as both hard and soft constraints depending on whether it is a preference or a legal obligation (Van den Bergh et al, 2012). Van den Bergh et al. (2012) have also discussed the applicability of the reviewed works in real-life situations as well as the integration of uncertainty into the problem. Another comprehensive review of the staff scheduling problem in different areas including healthcare systems, transportation systems, call centers, protection and emergency services, civic services and utilities, and financial services, is presented by Ernst et al. (2004a). They have reviewed more than 700 published articles. Ernst et al. (2004a) have classified the reviewed articles based on: 1) the type of problem addressed (or sub-problem), 2) the solution method, and 3) the application area. Ernst et al. (2004b) then further classify the sub-problems into the following categories: 74

90 Crew scheduling: assigns a set of tasks or activities to a set of crew members so that all activities are covered (e.g., in airline crew assignment). Cyclic rostering: addresses the task of assigning a set of activities, including days off and stand by, to a set of staff such that no labor rules are violates, usually in public transport and bus driver assignment. Staff are assigned to the same schedule over a planning period. Days-off scheduling: determines staff requirements at a daily level and finds the minimum staff size required to meet the demand and other constraints on weekends and holidays worked by staff. Shift demand: determines the demand of each shift. Task-based demand: determines the demand for each particular task/skill set. Demand modeling: includes models to predict long-term, mid-term, and short-term demand. Task assignment: identifies best way for assigning resources to tasks while minimizing assignment cost. Shift assignment: usually is defined as a one-day staff scheduling problem with staff requirements determined by the time of the day (e.g. hourly) Ernst et al. (2004b). Nurse scheduling, transportation systems, and call centers are among the most common application areas of the staff scheduling problem in the literature (Blochliger, 2004). The bus driver scheduling and the airline crew scheduling appear the most within the transportation systems. Surveys and reference reviews in nurse scheduling problem are introduced in section For a review of the work on airline crew scheduling see Gopalakrishnan & Johnson, 2005, Etschmaier & Mathaisel (1985), and Arabeyre et al. (1969). Literature on bus driver scheduling 75

91 may be found in the reviews conducted by Wren (1998) and Rousseau (1995). Gans et al. (2003) present a comprehensive review and tutorial on telephone call center literature. The surveys mentioned earlier are on the generic staff scheduling problem that group all different application areas, including healthcare, and suggest common themes. However, there are some surveys that specifically study the NSP such as Alfares (2004). Alfares (2004) surveys more than 70 articles published between 1990 and 2001, in a tour scheduling scope. A tour schedule is a combination of days-off and shift assignment over some planning horizon, usually one or more weeks (Alfres, 2004). Other good sources to review the literature on the NSP are the surveys conducted by Burke et al. (2004), Silvestro (2001), and Cheang et al. (2003). Burke et al. (2004) discuss many variations of NSP and explain how each model adds something different to the general nurse scheduling problem and how each problem can be solved differently. Cheang et al. (2003) survey the most common solution approaches utilized in the literature to solve the NSP. The authors discuss how nurse preferences and satisfaction, and workload fairness levels can be added to the constraints to improve nurse satisfaction. They also investigate how considering nurse specialty and skill can ensure that nurses are assigned to the tasks and shifts that they are most comfortable with and can increase nurse satisfaction. Nurse Scheduling Strategies One of the very first strategies ever used to address the mid-term nurse scheduling problem is called cyclic scheduling (Howell, 1998). In cyclic scheduling, multiple schedules that collectively satisfy the demand constraints are generated. Nurses will be assigned to the same schedule for several weeks during the planning horizon. If nurses rotate between different assignments, the scheduling policy will be called tour scheduling (Howell, 1998). One advantage of this approach is the consistency of the schedule over the planning horizon. The 76

92 cyclic scheduling assumes the demand is stable and deterministic over a long time horizon (Burns, 1978). Another disadvantage of cyclic scheduling is that it only satisfies the demand coverage and days-off constraint and does not include nurse preferences (Isken, 2004). Many hospitals use self-scheduling strategies to develop mid-term nurse assignment. After determining the size of workforce required for each planning period, the nurses can select any shift based on their preferences as long as it does not violate contractual constraints (Bailyn et al., 2007 and Griesmer, 1993). There is usually an upper bound on the number of schedules permitted to sign up to avoid overstaffing. This approach is easy to apply and gives nurses high flexibility; however, it is not always the most cost efficient solution and may result in lots of work conflicts. A nurse manager or someone who can control shift sign up and rotate shifts over the year is required (Griesmer, 1993). Preference scheduling is an approach between self-scheduling and cyclic scheduling ; this approach combines hard constraints with preference constraints. To quantify nurse preferences, penalties are assigned to undesirable shift assignments (Ronnberg and Larsson, 2010 and Warner 1976). The most important disadvantage of preference scheduling is the inconsistency in nurse work patterns. Nurses might end up with too many different assignments during the planning horizon which makes planning in advance difficult (Burke et al. 2001, Jaumard 1998). Solution Approaches The literature review on the staff scheduling problem includes several different approaches for solving the NSP. Each method has its own advantages and limitations. Based on the survey conducted by Alfares (2004), general nurse scheduling solution methodologies can be found within the list below: 77

93 1. Manual solution approach, basic scheduling principles or trial and error, 2. Heuristic solution approach (construction/improvement, meta-heuristics particularly simulated annealing (SA), etc.), 3. Mathematical programming (IP, MIP, goal programming (GP), decomposition, etc.), 4. Simulation, 5. Artificial intelligence, 6. Commercial scheduling systems According to Van den Bergh et al. (2012), mathematical programming and metaheuristic approaches are the most commonly used solution methods, and they can generate optimal or near optimal nurse schedules in rather short times. Manual scheduling approaches that do not use mathematical programming, heuristics, or simulation, usually are very time consuming, lead to unsatisfied nurses, and sub-optimal solutions. Even generating a near optimal solution that considers all of the hard constraints is very difficult without the use of mathematical modeling (Gupta and Denton, 2008) Mathematical Programming Mathematical programming methods have been mostly successful in minimizing the total nurse assignment cost (Bard, 2010), improving nurse satisfaction (Mobasher et al., 2011), and combining nurse scheduling and surgery block scheduling problems (Belien and Demeulemeester, 2008). These models often guarantee reaching the absolute optimum but can be very time consuming. However, if the search space is limited, mathematical programming can be a great approach for solving the NSP (Edmund et al., 2004). Exact methods usually involve a mathematical formulation (mostly IP or MIP) that includes hard and/or soft constraints. Bard (2010) provides a good example of how mathematical 78

94 programming can be utilized to solve the nurse scheduling problem. They have developed an IP formulation to solve a mid-term nurse scheduling problem with the objective of minimizing the cost of covering the demand. Their model only considers 8- and 12-hour shifts and it allows for outside nurses to be called in when there is understaffing. However, Bard (2010) has not considered the variability in patient arrivals and length of surgery. Another interesting model that mimics the charge nurse by assigning each patient to one nurse at the beginning of a day is developed by Punnakitikashem et al. (2008). A stochastic model for nurse assignment is proposed using these assumptions: 1) the number of patients is fixed, 2) nurse assignment cannot be changed during the planning horizon, and 3) within a given time period, direct care must be performed. The authors use Bender s Decomposition as well as a greedy heuristic to solve actual nurse staffing problems. Belien and Demeulemeester (2008) propose an IP and solve a branch and price approach to solve it to the optimality. Their model attempts to quantify the difference between the workload of x and x+1 nurses so that the hospital managers can make a more informed decision on their workforce levels. Other authors such as Jaumard et al. (1998) have used column generation to model the problem and branch and bound to solve it. Usually the master problem attempts to generate a nurse schedule that covers the demand and minimizes the assignment cost. Sub-problems include soft problems such as nurse preference and request for days off. Mobasher (2011) introduces another use of IP with multiple objectives to maximize nurse satisfaction, minimize cost, and minimize patient dissatisfaction. They have solved the problem for different nurse workload capacities and have incorporated patient workload variability in their model. Due to the complexity of the formulation, the author uses a variety of heuristic algorithms to solve the problem. Other examples of the use of optimization to solve the NSP are 79

95 the works of Rong (2010). All of them utilize integer programming and CPLEX, and they revealed limitations when trying to find the optimal solution for a problem with large dimension Heuristic Approaches Heuristic approaches are the most common method for solving staff scheduling problems. These methods are ideal for including soft constraints, but potentially problematic when dealing with hard constraints. Heuristics are typically much faster than direct optimization methods and are ideal for solving real-life situations with many constraints and variables, as well as for situations where solutions do not need to satisfy all constraints (Nysret et al., 2004). When an optimal solution is not necessary, metaheuristics and heuristics such as tabu search (Burke et al., 2001), constraint programming (Abdennadher and Schlenker, 1999), and genetic algorithms (Aickelin and White, 2004) have proven to be good alternatives. Heuristic methods are usually classified based on their functionality, with the first group being those based on local search (e.g. simulated annealing, and tabu search). These heuristics are usually combined with other methods since they do not tend to perform well on their own (Nysret et al., 2004). Metaheuristics are problem independent methods on a higher level of abstraction. Metaheuristics are often embedded on top of a low level heuristic as a scheduler (Burke et. al., 2004). Another commonly-used group of heuristics are population-based approaches. Genetic algorithm, scatter search, and swarm simulation are among the common heuristics in this category. These heuristics use a population of solutions as opposed to one solution; these solutions are then combined to build better solutions (Aickelin and White, 2004). A number of authors used heuristic approaches to solve problems similar to our MIP formulation. They were able to solve the problem for long periods and different nurse skills using heuristics. Azaiez and Al Sharif (2005) propose a 0-1 goal programming approach to solve 80

96 an IP formulation with fixed staff size and overstaffing penalties similar to our problem where we have the objective defined as the total nurse assignment costs as well as cost of working overtime, calling temporary nurses, and having nurses being assigned but idle. They have a total of 354 nurses of 5 different skill categories and they have been able to solve the problem for up to six months with satisfactory results. Aickelin and Dowsland (2003) proposed a genetic algorithm to solve an IP formulation with 30 nurses of three skill sets and 411 possible shift patterns. They were able to solve the problem for 52 weeks. Tsai and Lee (2010) improve the results of their original MIP using a GA method. Using GA enabled them to include nurse preferences and holidays. Wong et al. (2014) proposed an Excel spreadsheet-based model with similar constraints to our problem for the ED staff scheduling combing a shift assignment heuristic and a sequential local search (SLS). Other authors utilizing heuristics to solve problems similar to ours are Leksakul and Phetsawat (2014) and Lin et al. (2014) Simulation Simulation approaches can allow for running multiple scenarios giving the nurse manager more flexibility to develop the final schedule (Centeno et al., 2003). This tool is also useful because it accounts for variability in demand and service times by using distributions and imitating the actual arrival and processes of the system (Stomblad and Devapriya, 2012). Some authors including Shamayleh (2010), and Centeno et al. (2003), have combined mathematical programming with simulation modeling to solve the nurse scheduling problem. Special Cases, Other Constraints, and Quality-Type Metrics Numerous variants of the NSP have been researched based on the hospital s environment and particular needs, model features, and solution approaches. In this section, we introduce some of those variants, in particular those works that share common features with our nurse scheduling 81

97 problem (mixed contract types, variable demand, minimizing overstaffing and understaffing, fixed workforce). Bard et al. (2003) propose a staff scheduling problem for a postal service company that has both full-time and part-time staff and variable shift start times. In their problem, the total weekly cost of full-time staff is fixed since all the full-time workers have to work the exact same number of hours per week. However, the weekly cost of part-time staff varies because they can work a range of hours per week (Bard et al., 2003). These types of shifts have been considered when studying the NSP, and example of this is the NSP proposed by Wright and Mahar (2013). Similar to our work, there are a number of studies that have tried to minimize overstaffing and understaffing by adding penalties associated with these values in the objective. Maenhout and Vanhoucke (2010) define their objective as the weighted sum of the nurse assignment cost and the penalties associated with the number of times a ward faces shortage of personnel, and the number of times there are too many nurses assigned. The overstaffing and understaffing are proposed in a way that over and understaffing is leveled over planning horizon. The overstaffing and understaffing costs will increase exponentially as the number of nurses over/under increases. Wright and Mahar (2013) propose an objective that increases nurse satisfaction and reduces the total hours of overtime. Their objective consists of two terms; the first objective minimizes total regular and overtime wages, while the second objective attempts to minimize the total number of undesirable shift assigned over the planning horizon. Some authors have solved the nurse scheduling problem with a fixed staff size like our problem. Lim et al. (2012) try to improve nurse utilization and minimize unsatisfied patient workload with a fixed size workforce. Eitzen et al. (2004) solve an IP formulation to minimize total scheduling costs of a fixed workforce of 48 full-time and 8 part-time staff that are grouped 82

98 based on their skills. They solve the problem with a demand prediction across a 12-week period (Eitzen et al., 2004). Ozkarahan & Bailey (1988) develop a goal programming model with the goals of minimizing total weekly staff assignment cost and the penalty associated with nurse frustration due to understaffing. They have solved their problem both for a fixed staff and with an objective to find the appropriate size of the workforce. For a fixed staff size, the objective is to find a schedule that maximizes demand coverage and minimizes deviations from demand to minimize overstaffing. Finally, they have tried a model that minimizes deviations from demand in both directions (avoids both understaffing and overstaffing) (Ozkarahan & Bailey, 1988). When trying to minimize costs of shortage and surplus of nurses, some authors have included the penalties associated with the total hours under/overstaffed (Clark & Walker 2011, Ozkarahan & Bailey, 1988, and Baker, 1976). Others such as Maenhout & Vanhoucke (2010) try to minimize the actual number of extra nurses required to satisfy the demand, as well the number of nurses scheduled in surplus. Risk Adjusted Performance Measures and Conditional Value at Risk (CVaR) Value at risk (VaR) is among the most popular methods used to estimate exposure to risk. VaR was originally developed in 1993 in response to famous financial disasters including Barings s fall, where 1.3 billion dollars were lost due to risky investments in Japanese market (Irina and Svetlozar, 2000). The main purpose of VaR is to quantify risk using standard statistical methods. VaR measures the worst expected outcome over a particular horizon under given market conditions at a certain confidence interval (Jorion, 2001). VaR represents the maximum loss associated with a specific confidence interval of outcomes but it does not explain the magnitude of loss when the VaR limit is exceeded and it is difficult to optimize when calculated using scenarios; VaR is also not a coherent measure of risk (Rockafeller and Uryasev; 83

99 2002; Uryasev, 2000; and Chahar, and Taaffe, 2009). An alternative method, conditional value at risk (CVaR) has been proven to be a more appropriate and consistent measure of risk. CVaR has superior mathematical properties versus VaR and risk management with CVaR can be done very efficiently (Rockafeller and Uryasev, 2002, and Uryasev, 2000). The CVaR method considers the outcomes for which losses over a period of time exceed VaR, that is to say, we let (1-α)100% of the total outcomes to exceed VaR. So CVaR represents the average value of the outcomes exceeding VaR. As in can be observed in Figure 33 CVaR minimizes VaR because it is always greater than or equal to VaR (Madadi et al., 2014). Figure 33 Relation between VaR and CVaR loss measures Generally, α represents how much risk the decision maker is willing to take; the range of acceptable worst-cases becomes narrower as α approaches one (Madadi et al., 2014). To the best of our knowledge, there are no previous studies using VaR or CVaR to address the nurse scheduling problem. However, there are some articles discussing the 84

100 importance of risk and risk attitude of the decision maker on hospital revenue and patient outcomes. Stepaniak et al. (2009) study the effect of an OR coordinator s risk appreciation on OR efficiency. They apply the Zuckerman-Kuhlman Personality Questionnaire to analyze the risk attitude of OR personnel and determine which risk attitude creates more efficiency. Stepaniak et al. (2009) conclude that risk-neutral OR coordinators have much less unused OR capacity without a high chance of running the ORs on overtime compared to a risk-averse decision maker. Wang et al. (2013) test the hypothesis that OR turnover times are not different on days with large versus relatively little workload. They develop a structural equation model to study the correlation between different variables in the system. Their results show how different clinicians will make different decisions based on the workload. They show that risk-averse and risk-neutral decision makers make different decisions regarding add-on cases, starting late, moving cases between different ORs, etc. The risk-averse decision maker is intolerant of over utilized OR times and ignores the fact that OR time is based on the workload from all cases including add-ons. Such decision makers will have all of the ORs finish within the allotted time. The mean OR time inefficiency is larger for risk-averse decision makers compared to riskneutral ones (Wang et al., 2013). These studies prove that the risk attitude of the decision maker has a significant impact on healthcare outcomes. So a staff scheduling tool that can be adjusted based on the level of risk the scheduler is willing to take can be more useful than a risk-neutral method that gives the decision maker no flexibility. Contributions In many hospitals, nurses are scheduled without using any modeling techniques employed. Typically, the nurse manager (or the person in charge of nurse scheduling) must account for all variables particular to their environment, including demand, service time, and 85

101 nurse preferences (Kellog and Walczak, 2007). The nurse manager must decide how many nurses are required and which nurses will work what shifts and days. Usually, the nurse manager has limited tools available for scheduling and will end up using Excel to create an appropriate schedule, often using prior experience to know what typically works well. This scheduling method might be straight-forward and least costly in terms of required software and knowledge, but it is often the most expensive in terms of performance. It is important to have the right tools to develop a proper and efficient schedule that incorporates hard constraints to ensure a feasible and usable schedule, as well as soft constraints to keep nurses satisfied. Cheang et al. (2004) describe that the common approach in solving the NSP is to develop a 4- to 6-week nurse schedule which includes hard constraints, and possibly a set of soft constraints restricting the possible set of nurse schedules. The model can take into account patient and nurse satisfaction among other factors. The limitation of these approaches are that they do not account for variability in an appropriate way. In the context of nurse scheduling, there is usually high variability in patient arrival (demand) and surgery length (service) (Cheang et al., 2004). In order to generate a good nurse schedule, one key contribution that we make is the following. We account for the uncertainty in demand by using a forecasting method to predict point estimates and a prediction interval for demand. We generate multiple demand-scenarios where low, medium, and high demand can occur. The scenario-based modeling allows for uncertainty in demand, which is a main driver in understaffing and overstaffing. In terms of workforce, the preop department in the hospital under study considers only a fixed set of contracted employees. The workforce is composed of mixed contract types (fulltime/part-time. We need to ensure that we are working with a fixed staff size and try not to exceed the number of full-time or part-time nurses that are employed by the hospital. Although 86

102 we are not trying to find the optimal staff size, we conduct experiments with multiple levels of staff capacities to study the effects of staff size on the quality of the proposed schedule. There is no previous work that has used risk measures for nurse scheduling. As explained by Wang et al. (2013) and Stepaniak et al. (2009), hospital managers have different attitude towards risk and will make decisions differently based on the workload. Another key contribution of our work is that our NSP-RA model will provide a risk-averse decision making tool where the manager or decision maker can introduce a level of risk tolerance into their decisions and observe the outcomes from each test. Such a decision making tool will enable them to gauge the results of their decision before actually implementing it. 3.3 Defining the Preoperative Nurse Scheduling Problem Our goal is to create a nurse schedule that determines the number of nurses assigned to each shift on each day, as well as the number of nurses assigned to work additional hours of the day, to meet the anticipated surgical demand over a planning horizon. There are many problems associated with preop nurse scheduling that we aim to address in this research. The first is the cost associated with a poor schedule; a schedule may have too many nurses, resulting in unnecessarily high costs, or too few nurses resulting in overtime (if using their existing nursing pool) or contracting outside nurses to work. Even with the right number of staff per day, the distribution of scheduled nurses across the day might result in overstaffing and understaffing. Another issue associated with nurse scheduling is a time efficient system to generate a schedule that accounts for legal restrictions, hospital constraints, and nurse satisfaction. Also, we would like to generate a schedule that minimizes the amount of last minute changes. Because there are so many changes to the surgical case volume after the nurse manager creates the schedule, which 87

103 occurs four weeks prior to the day of surgery, the required rescheduling and adjustments take the nurse manager away from her other duties. In order to understand how nurse scheduling works in preop, we interviewed the preop nurse manager at Greenville Health System (GHS) several times and observed sample nurse schedules. GHS is a Level 1 Trauma Center in the Southeast U.S. Four weeks in advance of a new staff schedule release, the nurse manager reviews the number of planned surgeries to develop the staff schedule for the upcoming month. The demand of patients for this department comes directly from the surgeries that are scheduled well in advance plus add-on cases and cancellations. Since there is no case volume forecasting method, the nurse manager makes adjustments to the number of cases based on her experience. For example, if the number for a particular day is too low, she staffs for the average volume. Nurses select the number of hours and shift combinations that they want to work. The nurse manager creates a combination of those schedules that will cover the demand and account for hospital regulations. The nurse manager currently uses a simple spreadsheet based template, can be found in Appendix B, for staff scheduling. Within the four weeks, the nurse schedule is consistent from week to week and does not change based on the anticipated variability in patient arrivals. This will often lead to under and over staffing on several days during the planning horizon. We briefly state the current working environment in the preop department at the hospital surveyed during the research. 1. There are 32 total staff employed in the department. Of these, only 26 of them are nurses whose workload is directly affected by patient volume. This includes 16 full-time nurses and 10 part-time nurses. 2. Staff have a half-hour break for lunch considered in shift generation. 88

104 3. Rate of pay is based on years of experience. Full-time and part-time are not paid more or less according to their status as full-time or part-time. However, nurses are paid overtime if they have to work beyond their scheduled assignment. The hourly on-call rate is less than the overtime rate for regular full-time and part-time nurses. 4. Any on-call staff (outside nurse) is paid slightly higher due to lack of insurance coverage and increased need for flexibility. 5. Average hourly pay rates by position are as follows: $27-$37/hour for charge or staff nurses, $11-$15/hour for nurse specialty techs, and $11-$15/hour for surgery clerk and chart review personnel. 6. One feasible schedule that currently is being used at preop consists of: o Full-time nurses: 4 10 hrs. Shifts, 3 12 hrs. shifts, and 4 9 hrs. Shifts. o Part-time nurses: 3 8 hrs. shifts, 3 10 hrs. Shifts, 2 10 hrs. Shifts and 2 12 hrs. shifts. 7. Nurses will work the same schedule (or combination of shifts) for at least two consecutive weeks. When the nurse schedule, shift generation, and assignment of nurses to shifts are set, the nurse manager assigns nurses to patients. This is done at the beginning of the day and each nurse will be assigned to two patients at a time. 3.4 Methods As previously stated, the main objective of this chapter is to estimate nurse workload from the surgical case volume forecasts obtained in chapter 2, generate different demand scenarios to account for demand stochasticity, and finally develop a nurse scheduling plan that reduces nurse assignment costs from expected value and risk-adjusted points of view. We want 89

105 the provided solutions are flexible enough to enable the nurse manager to evaluate different nurse schedules. To achieve these objectives, we constructed the following procedure. First, we used the forecasts obtained in chapter 2 to generate potential daily demand levels and assign probabilities to each level. These daily demand levels were then combined to generate unique demand scenarios for the entire planning horizon. In order to match nurse shift requirements to actual occupancy in preop, we also needed a profile of when patients arrived and how long they stayed. We used historical data to obtain this preop room occupancy, which was translated into halfhourly nurse requirements using the standard patient-to-nurse ratio. Using the information from the hospital regarding shift start times, lengths, and breaks, we created appropriate constraints for a mathematical model and developed both NSP-EV and NSP-RA formulations. We conducted multiple experiments using each model. Scope Similar to any other problem, in order to effectively solve the NSP, the scope of the problem should be clearly defined. First, our models are primarily focused on the preop department only and do not include requirements imposed by the operating rooms or emergency department. We also create staff schedules that pertain only to nurses and no other support staff, mainly since they are fewer more fixed in number, as well as they are often not tied directly to patient volume. We did not consider nurse preferences. Instead, we incorporated that by considering the shift start times and durations that are most preferred by nurses. Point Estimates and Quartiles In chapter 2, we obtained point estimates that provided plausible predictions for the future values of the daily surgical case volume. Using the Central Limit Theorem (CLT) and the 90

106 fact that our sample size is large (n>30), we assume that our forecasts (point estimates) follow a normal distribution, with point estimates~n(µ=forecast, σ 2 = standard deviation 2 =(8.68) 2 ). The standard deviation takes on a single value for all the forecasts, whereas the forecasted value on a given day can be unique. The standard deviation, 8.68, was obtained in chapter 2. Figure 34 represents the distribution of the point estimates and their 95% prediction intervals that we use for creating possible demand levels on a given day. Figure 34 Illustration of the distribution of point estimates and prediction intervals In order to provide three demand levels per day (low, medium, and high), we begin by denoting the point estimate as an average (medium volume) day. We then used the first quartile (Q1) and the third quartile (Q3) of the Normal distribution as the low and high volume predictions, respectively, where Q1=Q25%=µ σ and Q3=Q75%=µ σ (Campbell, 1982). Table 19 shows a sample of three levels of demand calculated for May Table 19 Low, medium, and high volume demand generation for May

107 May-15 25% Median 75% Quartile (Point Estimate) Quartile Demand Scenario Generation The next step is to assign a probability of occurrence to each of the demand levels. We used the distribution of data between different quartiles in a normally distributed sample to define those probabilities. The interval containing the middle 50% of the data points is called interquartile range (IQR) and can be computed as IQR=Q3-Q1, whereas 25% of the data falls below the first quartile and 25% of the data is greater than the third quartile (Campbell, 1982). Based on these statistics we assigned 5 probabilities to low demand, Q1, 0.5 to medium demand, µ, and 5 to high volume, Q3. We recognize that the value at Q1, is not 25% likely to 92

108 occur, and some would suggest using a value evaluated at 12.5 lower percentile to represent the demand value for low demand (in fact, we will later perform such and analysis in section 3.5.2). However, short of breaking the entire prediction interval into several smaller intervals, each with a demand value and probability, the assignment of demand values and probabilities will only be an estimate. The point of doing this is to incorporate randomness into daily demand, and having three demand levels accomplishes this, while at the same time still is tractable in the mathematical formulation. Based on the demands generated in section and the probabilities assigned in this section, we developed demand scenarios for our desired scheduling horizon. The number of scenarios depends on the number of demand levels and the number of days within our scheduling horizon. The number of scenarios is n m, with m denoting the demand level and n denoting the number of days in our time horizon. Table 20 is a representation of scenario creation. Table 20 Representation of demand scenario generation for a 2-day problem. The number of scenarios increases exponentially as the number of days and demand levels increase. As will be discussed further in the results section, we could solve the NSP for up to 1,594,323 scenarios using a direct approach in under 3 hours computational time. This led to 93

109 the choice of a two-week (or 10-day) scheduling horizon. With three levels of demand, this translated to 590,549 scenarios. Half-hour/Interval Demand Profiles In order to compute the workload, the number of nurses required to work each interval of the day, we need to determine preop occupancy for each interval of the day. Based on the shift start times and break durations, we chose a half-hour as the most appropriate interval length. Using the demand scenarios developed in section 3.3.3, we translated these into half-hour occupancies and nurse requirements into non-overlapping half-hour occupancies as follows: We generated half-hourly arrival profiles for low, medium, and high volume days by aggregating historical arrival profiles for all days with less than 40 cases, between 40 and 80 cases, and greater than 80 cases, respectively. According to the nurse manager, the average length of stay for patients in preop is 3 hours. We used this information to develop occupancy levels for low, medium, and high volume days, based on the arrival profile from the prior step. We matched each daily volume to the corresponding half-hourly profile, low-to-low, medium-to-medium, and high-to-high to generate half-hourly demands for each scenario. These steps provided us with half-hourly demand requirements which could be easily translated into the nurse requirements using the nurse to patient ratio (1:2). Table 21 summarizes the final arrival and occupancy rates obtained after all the above steps. Figure 35-Figure 37 represent the half-hourly arrival rates and occupancy levels for low, medium, and high volume days. Finally, Figure 38 is a graphical representation of how the model translates arrivals into occupancy and nurse workload. The nurse requirement for the interval with the highest demand is assumed to be the daily nurse requirement. Note that the MIP 94

110 model will only take the interval demand as input and calculate the nurse requirement. Figure 38 is a post problem solving representation of the data to illustrate how the nurse workload is computed more clearly. 95

111 Table 21 Half-hourly demand profile for high, medium, and low volume days Patient/Surgical Volume (Percent of total) Low Medium High Interval Arrival Occupancy Arrival Occupancy Arrival Occupancy 1 5 am-5:30 am 3.0% 0.5% 6.6% 1.2% 6.1% 1.1% 2 5:30 am-6 am 6.1% 1.6% 10.5% 3.0% 12.0% 3.5% 3 6 am-6:30 am 4.6% 2.4% 15.8% 5.8% 16.6% 7.3% 4 6:30 am-7 am 13.6% 4.7% 9.2% 7.4% 11.1% 8.1% 5 7 am-7:30 am 6.1% 5.8% 6.6% 8.5% 10.0% 8.8% 6 7:30 am-8 am 6.1% 6.8% 6.6% 9.7% 6.7% 10.0% 7 8 am-8:30 am 9.1% 7.9% 10.5% 10.3% 7.8% 10.3% 8 8:30 am-9 am 6.1% 7.9% 4.0% 9.2% 6.7% 9.8% 9 9 am-9:30 am 4.6% 7.6% 2.6% 6.9% 5.6% 8.1% 10 9:30 am-10 am 3.0% 5.8% 2.6% 5.8% 3.3% 6.8% am-10:30 am 3.0% 5.3% 4.0% 5.3% 1.1% 5.3% 12 10:30 am-11 am 3.0% 4.7% 4.0% 4.8% 2.2% 4.5% am-11:30 am 3.0% 3.7% 2.6% 3.5% 1.1% 3.2% 14 11:30 am-12 pm 3.0% 3.2% 2.6% 3.2% 2.2% 2.3% pm-12:30 pm 3.0% 3.2% 1.3% 3.0% 1.1% 1.7% 16 12:30 pm-1 pm 1.5% 2.9% 1.3% 2.8% 1.1% 1.3% 17 1 pm-1:30 pm 3.0% 2.9% 0.0% 2.1% 1.1% 1.3% 18 1:30 pm-2 pm 4.6% 3.2% 0.0% 1.4% 0.0% 1.1% 19 2 pm-2:30 pm 1.5% 2.9% 1.3% 1.2% 1.1% 1.1% 20 2:30 pm-3 pm 3.0% 2.9% 0.0% 0.7% 2.2% 1.1% 21 3 pm-3:30 pm 3.0% 2.9% 2.6% 0.5% 2.2% 1.1% 22 3:30 pm-4 pm 3.0% 3.2% 0.0% 0.5% 1.1% 1.3% 23 4 pm-4:30 pm 1.5% 2.9% 1.3% 0.9% 0.0% 1.3% 24 4:30 pm-5 pm 0.0% 2.1% 2.6% 0.9% 1.1% 1.1% 25 5 pm-5:30 pm 0.0% 1.8% 1.3% 0.9% 0.0% 1.3% 26 5:30 pm-6 pm 1.5% 1.6% 0.0% 0.9% 0.0% 1.1% 96

112 5 am-5:30 am 5:30 am-6 am 6 am-6:30 am 6:30 am-7 am 7 am-7:30 am 7:30 am-8 am 8 am-8:30 am 8:30 am-9 am 9 am-9:30 am 9:30 am-10 am 10 am-10:30 am 10:30 am-11 am 11 am-11:30 am 11:30 am-12 pm 12 pm-12:30 pm 12:30 pm-1 pm 1 pm-1:30 pm 1:30 pm-2 pm 2 pm-2:30 pm 2:30 pm-3 pm 3 pm-3:30 pm 3:30 pm-4 pm 4 pm-4:30 pm 4:30 pm-5 pm 5 pm-5:30 pm 5:30 pm-6 pm Percent of Surgical Cases 5 am-5:30 am 5:30 am-6 am 6 am-6:30 am 6:30 am-7 am 7 am-7:30 am 7:30 am-8 am 8 am-8:30 am 8:30 am-9 am 9 am-9:30 am 9:30 am-10 am 10 am-10:30 am 10:30 am-11 am 11 am-11:30 am 11:30 am-12 pm 12 pm-12:30 pm 12:30 pm-1 pm 1 pm-1:30 pm 1:30 pm-2 pm 2 pm-2:30 pm 2:30 pm-3 pm 3 pm-3:30 pm 3:30 pm-4 pm 4 pm-4:30 pm 4:30 pm-5 pm 5 pm-5:30 pm 5:30 pm-6 pm Percent of Surgical Cases 16% 14% 12% 10% 8% 6% 4% 2% 0% Interval Low Volume Day Arrival Low Volume Day Occupancy Figure 35 Half-hourly arrival rates and occupancy levels for low volume days 16% 14% 12% 10% 8% 6% 4% 2% 0% Interval Medium Volume Day Arrival Medium Volume Day Occupancy Figure 36 Half-hourly arrival rates and occupancy levels for medium volume days 97

113 5 am-5:30 am 5:30 am-6 am 6 am-6:30 am 6:30 am-7 am 7 am-7:30 am 7:30 am-8 am 8 am-8:30 am 8:30 am-9 am 9 am-9:30 am 9:30 am-10 am 10 am-10:30 am 10:30 am-11 am 11 am-11:30 am 11:30 am-12 pm 12 pm-12:30 pm 12:30 pm-1 pm 1 pm-1:30 pm 1:30 pm-2 pm 2 pm-2:30 pm 2:30 pm-3 pm 3 pm-3:30 pm 3:30 pm-4 pm 4 pm-4:30 pm 4:30 pm-5 pm 5 pm-5:30 pm 5:30 pm-6 pm Percent of Surgical Cases 16% 14% 12% 10% 8% 6% 4% 2% 0% Interval High Volume Day Arrival High Volume Day Occupancy Figure 37 Half-hourly arrival rates and occupancy levels for high volume days Figure 38 Representation of nurse workload and daily nurse requirement calculation 98

114 Shift Generation All possible shifts that nurses can work each day are part of the input required by our nurse scheduling model. We obtained information regarding all possible shift start times and lengths from the hospital. This information is summarized in Table 22. Figure 39 represents all these shifts. Table 22 Shift information for the shifts currently available in preop Shift # Start time Duration Shift # Start time Duration 1 8 hours 9 8 hours 2 9 hours 10 9 hours 5:00 AM 6:00 AM 3 10 hours hours 4 12 hours hours 5 8 hours 13 8 hours 6 9 hours 14 9 hours 5:30 AM 6:30 AM 7 10 hours hours 8 12 hours hours Figure 39 Representation of all available 16 shifts in preop 99

115 After consulting with the preop nurse manager and reviewing current nurse schedules for several weeks, we noticed that only eight out of the total possible 16 shifts are actually used. The nurses will either start at 5:00 am or 5:30 am. The 6:00 am and 6:30 am start times are only for their late-start day, when the ORs will start an hour later due to the morning training. However, in our model we have shifted the demand on this day an hour back so that all weekdays are similar. Figure 40 shows all possible 8 shifts used in our problem. Figure 40 Representation of 8 shifts actually used in preop 100

116 Mathematical Models In this research, we address the NSP where the staff size and shift profiles are fixed, refer to section The model will generate a schedule that minimizes total nurse assignment cost while ensuring that the demand is always covered. As discussed in section 3.4.3, the number of scenarios increases exponentially as the number of days and demand levels increase. To keep the problem size manageable, the shift assignment is developed two weeks in advance and determines the working hours and shift assignments of each nurse for each day of the two-week scheduling horizon. This is consistent with the literature, where nurse staffing schedules are created and then repeated every two weeks (Topaloglu and Ozkarhan, 2004). In addition to the interval-based demand input and the shift profiles input, the following scheduling input rules are included as hard constraints: 1. All full-time nurses must be assigned between 36 to 40 hours within a week. 2. All part-time nurses must be assigned between 16 to 32 hours within a week. 3. Overtime is taken into account on a daily basis. If a full-time nurse works more than 40 hours and if a part-time nurse works more than 32 hours, it is considered overtime. 4. Temporary/outside nurses are available to cover the demand if there is understaffing. There is no limit on the number of outside nurses that can be called in per day. If an outside nurse is called, they must be assigned a full shift. 5. A nurse cannot work more than 12 hours per day. This constraint has been addressed in shift generation phase, and there are no shifts with a length longer than 12 hours. 6. Each nurse must have a 30-minute lunch break during the day. Note that in reality the nurses can take their half-hour break any time during their shift. However, for simplicity, we assume that they will take a break in the middle of their shift. 101

117 We used a MIP to formulate our problem. The NSP-EV and the NSP-RA formulations of our nurse scheduling problem are presented in the following sections Expected Value formulation To deal with the uncertainty in demand, a scenario based approach is utilized. We consider a set of demand scenarios, W, where Pw is the probability of occurrence for scenario w. Since we have a finite set of scenarios, each with the probability of occurrence of Pw, we can use a deterministic formulation in our objective to compute the expected cost of all scenarios. For our expected value model we propose a mixed integer model that attempts to come up with a 10- day schedule that minimizes the expected cost of all 590,549 scenarios (a 10-day problem with 3 levels of demand). We denote the MIP expected value formulation below as [NSP-EV], and the following notation is used in model development. Indices and sets n index for full-time nurses; n N {1, 2,..., N } n index for part-time nurses; n N {1, 2,..., N } n index for outside nurses; n N {1, 2,..., N } d index for days; d D {1, 2,..., D } s index for shifts; s S {1,2,..., S } i index for intervals; i I {1,2,..., I } w index for scenarios; w W {1, 2,..., W } N N N set of full-time nurses to be scheduled set of part-time nurses to be scheduled set of outside nurses to be scheduled 102

118 D S I W set of days for which the model is to be solved set of all possible shift types considered set of 30 minute intervals in each day within the planning period set of demand scenarios for which the model is to be solved Parameters Max FT H maximum number of hours full-time nurse n is contracted to work every week (40) Max PT H maximum number of hours part-time nurse n is contracted to work every week (32) Min FT H minimum number of hours full-time nurse n is contracted to work every week (36) Min PT H minimum number of hours part-time nurse n is contracted to work every week (16) Pw probability of scenario w occurring Didw demand/number of patients in interval i of day d for scenario w nurse-to- patient ratio (1:2) C si FT Rn 1 if shift s cov ers int erval i 0 otherwise pay rate per full-time nurse per hour ($32/hr.) PT Rn pay rate per part-time nurse per hour ($32/hr.) Outside Rn pay rate per outside nurse per hour ($40/hr.) FT OT R pay rate per full-time nurse per hour if the nurse works overtime ($50/hr.) n R PT OT n pay rate per part-time nurse per hour if the nurse works overtime ($50/hr.) Overstaffed R cost of being overstaffed per nurse per hour ($16/hr.) Problem size parameters N maximum number of full-time nurses (20) 103

119 N maximum number of part-time nurses (12) N large number representing the number of available outside nurses (100) D number of days for which the problem is to be solved (10) S number of shift types considered (8) I number of demand intervals per day (26) W number of demand scenarios for which the problem is to be solved (59048) Decision Variables FT X nsd 1 0 if full time nurse n is assigned to shift s on day d otherwise 1 PT Xn sd 0 if part time nurse n is assigned to shift s on day d otherwise FT Y nsdw 1 0 if full time nurse n is assigned to shift s in day d on overtime for scenario w otherwise 1 PT Yn sdw 0 if part time nurse n is assigned to shift s in day d on overtime for scenario w otherwise 1 Outside Yn sdw 0 if outside nurse n is assigned to shift s in day d on overtime for scenario w otherwise FT Orsdw 1 if r extra full time nurses are scheduled in surplus for shift s in day d for scenario w 0 otherwise PT Orsdw 1 if r extra part time nurses are scheduled in surplus for shift s in day d for scenario w 0 otherwise R FT T d total number of full-time nurses working regular hours on day d R PT T d total number of part-time nurses working regular hours on day d T total number of full-time nurses working overtime on day d for scenario w OT FT dw 104

120 T total number of part-time nurses working overtime on day d for scenario w OT PT dw Outside Tdw total number of outside nurses working overtime on day d for scenario w [NSP-EV] Objective function Minimize I N S D FT FT Z 0.5 ( R ) i 1 n 1 s 1 d 1 n Csi X nsd ( R C X ) ( R C Y ) ( R C Y ) ( Overstaffed R ( I N S D PT PT i 1 n 1 s 1 d 1 n si n sd I N W S D FT OT FT i 1 n 1 w 1 s 1 d 1 n si nsdw I N W S D PT OT PT i 1 n 1 w 1 s 1 d 1 n si n sdw I N W S D Outside Outside Rn CsiY i 1 n 1 w 1 s 1 d 1 n sdw I N W S D i 1 r 1 w 1 s 1 d 1 I N W S D PT CO i 1 r 1 w 1 s 1 d 1 si nsdw ) C O si ) FT nsdw (3-1) Constraints Subject to N S FT C n 1 s 1 si X nsd N S W FT CY n s 1 w 1 si nsdw N S PT C n 1 s 1 si X n sd N S W PT CY n 1 s 1 w 1 si n sdw N S W Outside CY n 1 s 1 w 1 si n sdw N S W FT CO n 1 s 1 w 1 si nsdw N S W PT C sio n s w n sdw D idw i I, d D, w W (3-2) S s 1 X FT nsd S PT X s 1 n sd 1 1 n N, d D (3-3) n N, d D (3-4) S s 1 Y FT nsdw 1 n N, d D, w W (3-5) 105

121 S PT Y s 1 n sdw S Outside Y s 1 n sdw 1 1 S FT FT X s 1 nsd Y nsdw S PT PT X s 1 n sd Y n sdw 1 1 n N, d D, w W (3-6) n N, d D, w W (3-7) n N, d D, w W (3-8) n N, d D, w W (3-9) n N I D FT Max FT C i 1 d 1 si X nsd H n N I D FT Min FT C i 1 d 1 si X nsd H I S D ( PT ) Max PT C i 1 s 1 d 1 si Xn sd H n N I S D ( PT ) Min PT C i 1 s 1 d 1 si Xn sd H n N (3-10) (3-11) (3-12) (3-13) T T R FT d R PT d N S FT X d D n 1 s 1 nsd N S PT X d D n 1 s 1 n sd (3-14) (3-15) T T OT FT N S W FT dw Y n 1 s 1 w 1 nsdw OT PT N S W PT dw Y n 1 s 1 w 1 n sdw d D, w W (3-16) d D, w W (3-17) T OT Outside dw N S W n 1 s 1 w 1 Y Outside n sdw d D, w W (3-18) X nsd {0,1}, X {0,1} n sd Y {0,1}, V {0,1}, B {0,1}, nsdw n sdw n sdw FT PT Y {0,1}, O {0,1}, O {0,1} n sdw rsdw rsdw n N, s S, d D n N, n N, n N s S, d D, w W (3-19) T R FT d 0 T 0, T 0, T 0, T R PT OT FT OT PT d dw dw Outside dw 0 d D d D, w W (3-20) 106

122 The objective (3-1) aims to minimize the sum of costs associated with schedules that nurses are assigned to, the cost of covering shortage with overtime and outside nurses, and the cost associated with nurses being idle because too many nursing staff is assigned. Since we have defined our intervals as half-hour, we have multiplied the hourly pay rates by 0.5 in the objective. Constraint (3-2) represents the demand requirement for each interval i on day d for scenario w. This constraint ensures that a sufficient number of nurses of each type (full-time, part-time, or outside) are assigned to regular or overtime shifts to satisfy the nurse requirements driven from the workload/demand forecasting model. Constraints (3-3)-(3-7) restrict each parttime, full-time, or outside nurse to at most one shift assignment within a day. Constraints (3-8) and (3-9) assure that each nurse is only assigned to a regular or overtime shift within a day and not both. Constraints (3-10)-(3-13) ensure that each full-time nurse works a minimum of 36 hours and a maximum of 40 hours per week, as well as each part-time nurse works between 16 and 32 hours each week. Constraints (3-14)-(3-18) track the total number of nurses working of each type on each day on any given scenario. Finally, integrality and non-negativity of decision variables are imposed by constraints (3-19) and (3-20) Conditional Value at Risk (CVaR) formulation The CVaR approach builds upon the VaR measure which is a popular method in portfolio risk management. VaR determines the maximum loss associated with (100α%) probability over a particular, user-defined time horizon (Madadi et al., 2014). In expanding our EV formulation for a risk-averse model, we define as a new decision variable denoting the optimal value of VaR. or VaR is the targeted cost level above which we want to minimize the number of outcomes. represents the significance level for the total cost distribution across all scenarios. is based on the -percentile of costs, i.e., in 100(1 )% of 107

123 the scenarios, the outcome will not exceed. CVaR is a weighted measure of and the costs above. In other words, CVaR is the expected cost of scenarios exceeding. We also define another new decision variable, w, which is the tail cost for scenario w, where tail cost is defined as the amount by which costs in scenario w exceed. We denote the risk-averse nurse scheduling model with unreliable demand as [NSP-RA], where the consideration of worst-case outcomes is defined for (0,1), and its details are described below: Indices and sets All indices and sets inherited from NSP-EV. Parameters All parameters from NSP-EV, as well as: significance level for the total cost distribution across all scenarios Problem size parameters All problem size parameters inherited from NSP-EV. Decision Variables All decision variables from NSP-EV, as well as: w targeted cost level above which we want to minimize the number of outcomes tail cost for scenario w [NSP-RA] Objective function Minimize 1 (1 W ) P w 1 w w (3-21) 108

124 Constraints Subject to I N S D FT FT w 0.5 ( R n Csi X ) i n s d nsd I N S D PT PT ( R ) n Csi X i n s d n sd I N W S D FT OT FT ( R ) i 1 n 1 w 1 s 1 d 1 n CsiY nsdw I N W S D PT OT PT ( R ) i 1 n 1 w 1 s 1 d 1 n CsiY n sdw I N W S D Outside Outside ( Rn CsiY n sd ) i n w s d w Overstaffed I N W S D FT R ( C i 1 r 1 w 1 s 1 d 1 sio nsdw I N W S D PT CO ) i 1 r 1 w 1 s 1 d 1 si nsdw w W (3-22) Constraints (3-2) (3-20) 0 w W (3-23) w The objective (3-21) tries to minimize CVaR, we have imposed a non-negativity constraint on w, so the model tries to decrease VaR and positively affect the objective function. However, large decreases in VaR will counterbalance this effect by resulting in more scenarios with large tail cost. By measuring CVaR instead of VaR, we can achieve a more accurate estimate of the risks of minimizing cost by considering the magnitude of tail costs (Chahar and Taaffe, 2009). Constraint (3-22) computes the tail cost of scenario w. The non-negativity constraint for w, constraint (3-23) states that we only consider the scenarios in which the cost exceeds. Constraints (3-2)-(3-20) are inherited from NSP-EV and will control the allocation of nursing assignments in the same manner as before. 3.5 Results In this section, we present numerical studies on both the NSP-EV and the NSP-RA models, as outlined in previous sections, in order to highlight the differences between risk-averse and risk-neutral nurse scheduling policies. The optimization problem is modeled and solved using IBM ILOG CPLEX Optimization Studio version Each problem instance is solved 109

125 on a Dell Optiplex 990 with an Intel Core i GHz and 8 GB RAM. The operating system is Windows 7 Enterprise 64-bit. Nurse Staffing Solutions using an Expected Value Approach The main objective of the NSP-EV model is to find a minimum cost assignment of fulltime and part-time nurses to generate a 10-day schedule. As mentioned in previous sections, we solve the problem for a fixed number of regular contracted nurses on the roster, which includes 16 full-time and 10 part-time nurses. However, if we only look at solving the NSP-EV model for one size of workforce with no variability, we will not be able to get a complete picture of the differences and advantages of different nurse schedules. In addition to the initial roster of 26 nurses, we conducted experiments with other workforce sizes and compared the results. We picked the current level as a baseline and studied workforce sizes smaller and larger than the current roster. We decremented and incremented the capacity ratio (shown in Table 23) by approximately 0.15, and studied a very small (17 nurses), small (21), regular=baseline (26), large (30), and very large (34) roster. Table 23 shows how the workforce size and breakdown to part-time (PT) and full-time (FT) was conducted. Table 23 Workforce size variations considered in this research Capacity Ratio (Experiment/ Actual workforce) Size of Workforce FT Nurses PT Nurses Figure 41 shows the difference between total nurse assignment costs for each size of workforce spanning a period of 12 months (Oct-14 to Sep-15). It is no surprise that small roster 110

126 sizes will result in very high costs, due to the inability to meet surgical demand with regular contracted staff. Figure 42 illustrate the components of the nurse staffing cost (regular contracted shift cost, overstaffing cost, and understaffing cost) for current staff size. Similar results for different roster sizes are represented in Appendix C. Figure 41 Total nurse assignment costs (Oct-14 to Sep-15) with different workforce sizes Since the nurse manager was already facing a shortage of staff on high volume days (given the baseline staff number of 26), we expected that a smaller size would increase total costs (primarily due to understaffing). We also expected that a very large workforce, 34, would also have larger costs due to the larger number of regular contracted nurses that are guaranteed their shifts according to their full-time or part-time contracts. However, the experiment showed that additional contracted nurses will improve the schedules and decrease the total costs. In order to obtain more insight, we developed Figure 43 to compare the schedules created for the same 111

127 week using 34 and 26 nurses. As it can be observed in Figure 43, with 34 nurses, the model has more flexibility in assigning nurses to shifts. Most full-time nurses work 36 hours and most parttime nurses work 16 hours, whereas with 26 nurses, most full-time nurses are assigned 40 hours per week and part-time nurses work 20 to 32 hours per week. So the total number of hours the regular nurses work does not increase significantly when increasing the size of workforce from 26 to 34. However, with 26 nurses, we used outside nurses 17 times during the week. The pay rate of regular nurses is $32/hour while outside nurses are paid $40/hour. So the total cost for the solution with 34 contracted nurses, $64,448 is lower than the total cost of 26 regular nurses, $66,688. Nurse Assignment Cost (in $) $160,000 $140,000 $120,000 $100,000 $80,000 $60,000 $40,000 $20,000 $0 Monthly Nurse Staffing Cost (26 Regular Nurses) Overstaffing cost Understaffing cost Regular assignment cost Scheduling Horizon Figure 42 Breakdown of nurse assignment cost with a roster of 26 nurses (baseline) Figure 44 shows the change in monthly nurse staffing costs using different sizes of workforce. To reduce the amount of information presented, the costs are compared on a yearly 112

128 level; all the other graphs are monthly costs. As it can be observed, total cost decreases dramatically when going from 17 to 21 nurses and 21 to 26 nurses. Total cost is more similar but still decreasing from from 26 to 30 to 34 nurses. Figure 45 summarizes the experiment conducted to find the optimal number of nurses that minimizes the total nurse assignment costs for each month. When trying different roster sizes, we keep the PT/FT nurse ratio same as the current preop ratio,10/16. Note that the mathematical formulation was not adjusted to find the optimal workforce size; just ran the model at each workforce size and the PT/FT ratio was always As mentioned earlier, we solve the problem for a fixed roster size and finding the total number of nurses that should be hired by the hospital is out of the scope of this problem. However, conducted the experiment shown in Figure 18, helped us find the threshold for increasing the size of workforce and still be able to minimize the costs. As expected, it also confirms that different months have different nurse requirements due to the variability in demand. This information could potentially help the nurse manager decide on the best number of regular nurses and a set of available temporary nurses for different months. Or use different roster sizes for different months or seasons if possible. 113

129 Figure 43 Left: Nurse schedule solution provided by NSP-EV using 34 nurses; Right: Nurse schedule solution provided by NSP-EV using 26 nurses 114

130 Yearly Nurse Assignment Cost $1,400,000 $1,300,000 $1,200,000 $1,100,000 $1,000,000 $900,000 $800,000 $700, Number of Nurses in Workforce Figure 44 Change in yearly nurse assignment costs with different workforce sizes Figure 45 Optimal number of nurses require for each month using the EV model 115