Improving Patient Access to Chemotherapy Treatment at Duke Cancer Institute

Transcription

1 Improving Patient Access to Chemotherapy Treatment at Duke Cancer Institute Jonathan C. Woodall Duke Medicine, Durham, North Carolina, 27708, Tracy Gosselin, Amy Boswell Duke Cancer Institute, Durham, North Carolina, Michael Murr Edward P. Fitts Department of Industrial & Systems Engineering, North Carolina State University Raleigh, North Carolina, 48109, Brian Denton Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, Michigan 48109, We describe how discrete event simulation and optimization methods were used in combination to improve patient flow within a large cancer center. A discrete event simulation model is presented for predicting patient waiting time and resource utilization throughout various stages in the cancer care process, including the outpatient clinic, radiology, pharmacy, laboratory services, and the chemotherapy infusion center. Results from the simulation model show that nurses in the chemotherapy infusion center are a significant bottleneck in the overall patient flow process.

2 Next, we present a mixed integer programming model formulation that was used to optimize planning decisions for the number of types of nurses to schedule on a weekly and monthly basis. We further provide a novel formulation of a simulation optimization model that was used to optimize nurse shift start times to minimize average patient waiting time. Finally we summarize the recommendations that were implemented at Duke Cancer Institute. Key Words: simulation; optimization; integer programming; health care; nurse scheduling. In many countries, cancer is the leading cause of death, or is rapidly becoming so. As a result, patient demand for cancer services has been steadily increasing, and it is expected to continue to increase in the future (Erikson et al. 2007). This increase in patient demand has created managerial challenges related to matching supply and demand for cancer services. From the patient perspective, increases in demand can result in long waiting times. This can be in the form of direct waiting (waiting on site) and indirect waiting (waiting for the day of a scheduled appointment). From the cancer center s perspective, increases in demand can result in higher than normal resource utilization, causing overtime and congestion that can be frustrating for providers. Furthermore, in a high-demand environment, variation in patient mix and patient flow patterns can result in parts of the cancer center or of the day being overutilized, while other areas or times of day are underutilized. In this article, we describe how discrete event simulation and mixed-integer programming (MIP) models were used to improve patient access to care at Duke Cancer Institute in Durham, North Carolina. Moreover, the insights we drew from our project are applicable to other cancer centers. We begin by describing a conceptual model of a cancer center. Next, we describe the various parts of the discrete event simulation model and how we combined simulation and 2

3 optimization methods to identify bottlenecks within the cancer center, optimize nurse staffing within the chemotherapy infusion center, and plan for future capacity expansion to meet patient needs. Finally, we summarize the impact from the use of our models and we discuss opportunities for future research. Cancer Center Background and Challenges Patients visit cancer centers for a variety of reasons, such as referrals from primary care physicians because of suspicion of cancer, second opinions, consultation about treatment, and follow-up consultations upon completion of treatment. The location in which patients visit physicians is called the clinic. Most cancer centers have clinics organized based on the type of cancer (e.g., breast, prostate, and lung). A second major part of a cancer center is the oncology treatment center (OTC), where patients receive chemotherapy treatment. Chemotherapy can be received in the form of an injection or an infusion, where medicine is dripped intravenously into the patient (referred to as an IV). Additional locations in the cancer center can also play a necessary role in a patient s service experience. One is radiology, the location where imaging scans are performed. Another is lab services, where blood tests and other lab tests are processed. Blood tests are an important part of diagnosis and treatment monitoring, and they must be reviewed prior to receiving chemotherapy. The pharmacy is the central location that mixes the drugs used for chemotherapy prior to the patient s treatment. Because of the high cost of drugs for chemotherapy, the pharmacy typically will not mix drugs for a patient until the lab results have been reviewed and the patient has checked into the OTC. Thus, many dependencies, which can influence patient flow, exist among the various stages of service within a cancer center. 3

4 Radiology Results sent to Pharmacy Pharmacy Patient Radiology Scan Labs sent to Central Labs Central Labs Mixed drug sent to Treatment Center Patient Arrives Clinic Results sent back to provider to sign off on OTC order Results sent to Treatment Center Treatment Center Patient travels to Treatment Center Figure 1: This figure depicts the patient and information flow among locations in the cancer center. Figure 1 illustrates the patient flow between the clinic, labs, pharmacy, and the OTC. Most patients receiving treatment will start at one of the clinics for registration, have labs drawn and processed, see their oncologist, and finish at the OTC. However, patient flow through the clinic varies. For instance, some patients visit the cancer center only for a clinic visit. Of the patients solely attending a clinic visit, some will have labs and/or radiology, while others will not. Some patients are return-visits and some are new-visits; new-visits tend to have longer service times in the clinic. Some patients may go from the clinic to the OTC on the same day, while others may return on a different day for treatment. This variation in patient needs contributes to uncertainty in resource utilization of downstream resources, such as the OTC, from day to day. 4

5 Patient arrives after clinic visit or direct to OTC Patient check in; chart pulled and brought to charge nurse Patient to waiting room until called Nurse obtains drug, calls back patient Begin Treatment OTC nurse checks and reviews chart Charge nurse enters info into computer system; checks chart Order Complete? N Y Labs OK for treatment? N Y Pharmacy Mixes Drug Contact oncologist to obtain appropriate information Contact oncologist for approval (pharmacist or charge nurse) Figure 2: This flow chart of the Oncology Treatment Center process illustrates the patient and charge nurse flow prior to chemotherapy treatment being initiated in the Oncology Treatment Center. Detailed descriptions of patient flow through each major area were developed. Since much of this article focuses on changes within the OTC, we present the flow through this area in Figures 2 and 3. Once the review of a patient s medical record is complete, a nurse brings the patient back to a treatment chair. Of the two treatment types, injections and infusions, injections are much shorter. Prior to injection the nurse reviews and discusses the appropriate medical history with the patient and provides relevant information about the injection to the patient, as well as symptom management education. Once the injection is complete, the patient is free to leave. 5

6 Nurse calls back patient to begin treatment Injection or Infusion? Injection Nurse administers injection Patient Discharged Infusion Nurse hooks up IV Infusion Time Nurse unhooks IV Figure 3: This flow chart depicts the Oncology Treatment Center process starting when chemotherapy treatment begins for a patient and ending with patient discharge. Infusions follow a similar process to injection. Initially a nurse reviews the medical history, discusses necessary medical information with the patient, connects the IV, and takes the patient s vital signs. Once this is complete, the infusion begins. Once the infusion has begun, the nurse can prepare other patients, up to a maximum of four, concurrently. Once the infusion is complete, the nurse disconnects the IV and discharges the patient. Although patients are generally punctual in their arrival to the cancer center, most visits involve a clinic appointment, a lab draw, and possibly other activities (e.g. radiology), that cause uncertainty in arrival times at the OTC. After the patient arrives at the OTC, there are additional potential causes for delay of treatment. For example, delays take place when the pharmacy becomes backlogged with orders, when there is a delay in obtaining lab results, or in the physician approval process for orders. As a result of uncertainty in the time taken during the upstream process, the arrivals to the OTC during the day exhibit considerable uncertainty. Prior Work on Cancer Center Planning and Scheduling Santibáñez et al. (2009) examined a cancer center at the British Columbia Cancer Agency in Canada. They focused on the interaction of cancer clinics and studied resource allocation 6

7 decisions. The authors presented scenarios that included changes to operational factors (clinic start time, use of faculty such as residents/fellows), appointment scheduling (sequence of appointment types during the day, scheduling of add-ons), and resource allocation (use of pooled clinic resources versus designated resources). They found that in order to obtain significant improvements, multiple changes to the existing process were required. Sepúlveda et al. (1999) presented a model based on the MD Anderson cancer center in Orlando, Florida. The model included the oncology clinic, OTC, and pharmacy. They used their simulation model to examine several scenarios involving changes to the layout and scheduling policies within the cancer center. The authors also used their model to test policy changes in which the number of short-term patients was increased during slow portions of the day, and decreased in busier of the day. Turkcan, Zeng, and Lawley (2010) examined patient scheduling decisions in the setting of a cancer center. The authors used two MIP models in combination to plan patient chemotherapy treatment over a certain length of time, such that the same patient returns for multiple treatments over a sequence of days. The first MIP determines the amount of resources and acuity level required for the patient, and the second MIP seeks to determine the best time to schedule the patient for treatment subject to the constraint of the nurse not exceeding a certain acuity level for the day. Staffing levels were also examined to determine the optimal allocation of resources. Novel Contributions of this Project The project described in this article was carried out from 2010 to It resulted in a number of important findings that are transferrable to other cancer centers and other parts of the healthcare delivery system. It differs from the work cited above in several ways. First, compared with 7

8 Santibáñez et al. (2009) and Sepulveda et al. (1999), our focus is on the OTC, and optimal design of nurse schedules to match daily provider supply and patient demand. Due to a national shortage of skilled nurses the issue of efficiently allocating nurse resources is a high priority for most cancer centers. Second, in contrast to the aforementioned studies, we seek the best ways to staff in light of non-stationary patient arrival behavior, including the simulation-optimization of nurse shift start times during the day. Our use of non-stationary Poisson processes based on hourly patient flow data results in a much more accurate model than could be achieved with more commonly used stationary assumptions. Third, compared with Turkcan et al. (2010), we combine simulation and optimization methods to understand ways to mitigate the impact of uncertainty from various sources including patient service times, nurse availability, pharmacy processing, and other sources. Finally, to our knowledge, the model we have developed is the most comprehensive model of a cancer center, and could be adapted for use by other cancer centers. Complete details of the model formulation, data collection, validation, and implementation, can be found in Woodall (2011). Model Formulation and Validation Following a detailed assessment of the patient flow process through the cancer center, the project involved three major phases: (1) development of a discrete event simulation model of the cancer center; (2) development of a mixed integer program to optimize weekly and monthly nurse staffing decisions in the OTC; (3) development of a simulation optimization model to determine the optimal nurse shift start times based on the weekly and monthly nurse staffing decisions. In this section we describe the three models, model parameter estimation, and model validation activities that led to the implemented recommendations. 8

9 Discrete Event Simulation Model We built our simulation model using Rockwell s simulation software, Arena version 11. Preliminary work included collecting sample observation times for services provided in all parts Process Type Location Probability Distribution Source Charge Nurse *BETA(1.30, OTC Chart Check 23.05) Time Study Pharmacist Pharmacy Processing LOGNORM(5.46,6.74) Time Study Pharmacy Drug Mixing Pharmacy ERLANG(2.94,2) Time Study OTC Nurse Chart Check OTC *BETA(1.84, 4.52) Expert Opinion Injection Treatment Length OTC TRIANGULAR(1, 2.1, 30) Expert Opinion OTC Nurse IV Setup OTC BETA(3.31, 4.46) Expert Opinion Acuity Level 1 Treatment Time OTC BETA(4.46, 3.31) Expert Opinion Acuity Level 3 Treatment Time OTC BETA(4.6, 2.2) Expert Opinion Acuity Level 5 Treatment Time OTC BETA(4.36, 3.52) Expert Opinion Blood Drawn to Receive at Labs Labs GAMMA(12, 1.33) Historical Data Review and LOGNORMAL(19.5, Verification of Labs 35.7) Labs Historical Data Radiology Processing Radiology 30 + GAMMA(62.2, 1.18) Historical Data 9

10 Table 1: This table provides a list of probability distributions, and their data sources, used in the simulation model for the Outpatient Treatment Center and Pharmacy (all times are in minutes). of the cancer center. The sources of data included computer information systems, time studies, and expert interviews with oncologists, administrators, and nurses. Collaborative work with a diverse group of experts, including administrators, nurses, and oncologists, yielded assumptions for process times where data did not exist or was not immediately available. Once data collection was complete, a prototype version of the simulation model was developed. The initial model included major processes within the cancer center including clinics, labs, radiology, pharmacy, and the OTC. The patient arrival process was estimated based on the mean number of arrivals during the day using historical data. We used a non-stationary Poisson arrival process because patient arrival rates vary significantly over the course of a day. The average expected arrival rate by each half-hour of the day was estimated from the historical data and used to define the Poisson process. Scheduled resources included check-in and checkout receptionists for all clinics, phlebotomists, nurses, and oncologists, for each of the clinics, exam rooms, receptionists at the OTC check-in, a charge nurse, nurses by disease based groups (DBGs), treatment chairs, and beds. These resources are available according to predefined schedules that were input into the model to define availability over the course of the day. Nurses in the OTC are engaged in both direct and indirect patient care. To represent the fact that nurses work on some activities in parallel, we assumed that each nurse has six capacity units available. In order for a patient to begin an infusion, at least 3 units of nurse time must be available. Once the nurse finishes start-up and moves to a monitoring role with the patient, 2 10

11 units of the nurse are freed (1 unit is still in use). As a result of these assumptions, the number of patients a nurse can serve at one time is limited to 4, which is an upper limit on the number of patients a nurse can be responsible for at one time. Probability distributions were fit to historical data using Arena 11 Input Analyzer. Criteria for selecting distributions were visual inspection and the results of chi-square and Kolmogorov-Smirnov tests. Additional consideration was given to the squared error of the fit. For processes with unreliable or no data, probability distributions were fit in two ways: time studies and expert opinion. In total, three time studies were performed: one for the pharmacy (pharmacist processing time and drug mixing time), one for the check-out process at the clinics, and one for the length of the chart check time by the charge nurse in the OTC. Table 1 contains the list of probability distributions. In the absence of historical data and time studies, expert opinion was solicited for the minimum, most frequent (mode), maximum, and average processing times, which were used to define Beta distributions. Several approaches were used to validate the simulation model including expert opinion and statistical validation of model outputs. Experts consulted included the clinical operations director, the assistant vice president and associate chief nursing officer of oncology, a management engineer in oncology, the administrative manager, and healthcare administration staff. Results identified as potentially invalid were examined further. This iterative process involved a number of changes to the initial model to refine the assumptions. Following the expert validation, statistical validation was performed by comparing the observed to model generated patient arrival distributions, patient throughput, and the flow times (total time from check-in to discharge). Table 2 illustrates results for arrivals and OTC patient throughput for a 11

12 particular day of the week. Animation was also used to aid with the validation and verification process. Animations were created for one of the clinics and for the OTC, and were generally built for very specific processes to troubleshoot behavior identified as unusual during the expert opinion process. Monday Arrivals/Throughput Validation Simulation Model (50 replications) Historical Data Cancer Center Area Mean LCL UCL Mean LCL UCL Sample Size Surgical Oncology Oncology Brain Tumor Prostate Surgery OTC Direct Arrivals OTC Throughput Table 2: This table compares simulation model estimated arrivals and actual observed arrivals to the Outpatient Treatment Center. The Outpatient Treatment Center direct arrivals are patient arrivals that do not originate from a clinic. Mixed Integer Programming Model for Nurse Staffing As the last step in the patient flow process, OTCs are subject to variability induced by upstream services including clinics, labs, and radiology. OTCs are also typically resource constrained, and our simulation model revealed, based on mean patient waiting times, that the OTC is a significant bottleneck in the cancer center. Further analysis identified that nurse availability 12

13 presents the most significant single resource constraint. Contributing factors to this problem included high variation in the type of patients, length of time for the different infusions, and the number of patients arriving at the OTC throughout the day. As a result, the project team focused on exploring methods to improve nurse shift schedules, to better match nurse supply with patient demand. Our approach to planning nurse schedules was twofold and hierarchical in nature. Figure 4 illustrates the daily, weekly, and monthly scheduling process. Total daily demand, which varies from Monday through Friday, drives the weekly and monthly schedules. We used a MIP to solve the monthly/weekly planning problem to allocate a predetermined number of nurses across weeks within the month to match aggregate nurse supply to historical demand estimates (see the appendix for a complete mathematical formulation). While there are a number of potential choices for objective function we chose to minimize the total shortage of nurse hours relative to patient demand. This was chosen because it was identified as the most important consideration by project team members from the cancer center, and because of the ease with which it can be interpreted. The MIP included many constraints including a minimum number of off-days, minimum allocation levels across DBGs, fair allocation of long weekends (a Monday or Friday off-day) among nurses, and others. Based on the results of the MIP we used a simulationoptimization model to optimize daily shift start times to minimize average patient waiting. Feedback between the monthly/weekly and daily schedules is done to iteratively improve the nurse schedule. In our MIP model for weekly/monthly planning we considered three types of nurses: 10- hour nurses that work 4 days a week for 40 hours in total, 8-hour nurses that work 5 days a week for 40 hours in total, and part-time nurses working varying days per week, hours per day, and 13

14 hours per week. Constraints in our MIP define feasible schedules for the OTC. First, OTC daily hours of operation are fixed (e.g., 7:00 a.m. 8:30 p.m.). There must be at least two nurses in the OTC at all times; thus, there must be two openers and two closers, where openers begin their shift at 7:30 a.m. and closers end their shift at 8:30 p.m. Also, a minimum level of coverage is required in each DBG during peak hours (e.g., 10:00 a.m. 6:30 p.m.). Furthermore, in the OTC we studied, the requirement is that each DBG has a minimum of three nurses scheduled each day. Furthermore, the entire OTC must have at least 14 nurses scheduled on Monday through Thursday, and 13 on Friday (reflecting the lower number of patients seen in the OTC on Fridays). There are also constraints surrounding the allocation of off-days to 10- hour shift nurses. Each 10-hour shift nurse will work 4 of the 5 days in the week, receiving one off-day. Each nurse must receive at least one long weekend per month (4-day weekend). Fourday weekends occur when the nurse has Friday off one week, and then Monday off the next week. For part-time nurses with off-days, the location of the off-day is not constrained, thus part-time nurses tend to be scheduled on the busiest days. 14

15 Staffing Level Daily throughput drives weekly schedule Monthly Schedule Weekly Schedule by DBG Daily Schedule by DBG Rotate schedule tracks across the month to alter off-days Figure 4: This diagram illustrates the relationship between major decisions involved in Oncology Treatment Center nurse scheduling. The arrows define the flow of information during this iterative process. In the monthly and weekly scheduling MIP experiments, we sought to determine the best way to allocate nurses across the week to meet variable day-to-day demand with a particular focus on constraining the model to allot one 4-day weekend (consecutive Friday and Monday off) for each full time, 10-hour shift nurse. The objective was to minimize total shortage hours in the OTC nurse schedule. Shortage hours were determined based upon the daily scheduling requirements. Each day has a certain ideal number of nursing hours required to meet patient demand (assuming the discrete nature of nurse shifts is relaxed). However, due to the discrete nature of nurse shifts, and constraints on the number of nurses available, these requirements may not be met. The deficit is referred to as the nurse shortage. 15

16 Simulation Optimization Model for Daily Nurse Scheduling In the daily schedule optimization experiments, we sought to determine the best daily shift start times to minimize the average amount of patient waiting time in the OTC. The complete formulation for our model is provided in the appendix. In our model, each nurse has a series of associated shifts they could be allocated, which correspond to start and end-times in half-hour increments during the day. Binary decision variables define nurse arrival times at these discrete time points (whether they arrive (1) or not (0)). There are also sets of opening and closing shifts, that define the nurses working at the beginning and ending of the day, and an off shift, that defines the sets of nurses that are working (on) or not working (off) for the day. A constraint determines which shifts a nurse may be scheduled for based on a binary indicator for each nurse/day combination. This indicator is based on nurse experience working in the various DBGs, as defined by the OTC Clinical Operations Director. A second constraint enforces the fact that nurses do not work off-days. Additional constraints require that at least two nurses be available for opening and closing of the OTC. In general, models such as that above are very computationally challenging, due to the intractable nature of the expectation in the objective function. Closed form expressions for expected waiting time in complex service systems, such as the cancer center we explore, are generally not available. Thus, it is necessary to resort to heuristics. As a result, we solved this model using simulation-optimization, based on sampling of expected patient waiting times using our discrete event simulation model, described previously. Specific details of our implementation are provided in the Results section. 16

17 Results Experiments were conducted on a Dell Optiplex 980 PC, Intel Core, 2.93 GHz, 8.00GB RAM. We used the preliminary results from the complete simulation model for the cancer center, together with expert judgment, to identify bottlenecks in the overall system. We used our deterministic MIP model to analyze the monthly and weekly nurse scheduling problem to gain insights into the optimal allocation of nurses and nurse shift length policies subject to scheduling constraints on variable daily demand. Finally, we used our simulation-optimization model to analyze the daily scheduling problem to find optimal work schedules under various shift length policies, where optimal is defined as the smallest patient waiting room time. The remainder of this section provides results illustrating some of the analysis that was conducted that led to implemented changes in the cancer center. Monthly and Weekly Schedule Optimization We used the solver add-on Premium Solver to solve instances of the MIP for monthly and weekly planning using branch-and-bound, with a tolerance of 0.1%. We chose this solver because it was easy to implement on a standard PC in the clinic environment. Additionally, we set computation time to a maximum of 1-hour. Table 4 shows results for a sample instance of the MIP for a 21 nurse scenario. We solved a series of model instances in which the minimum number of full-time nurses was varied from 9 through 18, to control the number of part-time nurses. We also include the case in which the minimum was zero full-time nurses, for a reference point. Table 4 provides the number of full-time nurses used for both 10-hour and 8-hour shifts, as well as the number of part-time nurses used in the best solution found after 1-hour. The last column provides the shortage hours 17

18 (objective function). The results suggest two main conclusions; first, that using part-time nurses could significantly reduce total shortage hours. This is intuitive since shorter nurse shift schedules allow for better matching of supply and demand during the day. Second, that as the number of part-time nurses used increases, diminishing returns or no improvement at all is seen in reducing shortage hours. These results were helpful to decision makers in trading off the pros and cons of having part-time nurses that help match supply and demand, but which may also require a greater number of handoffs of patients among nurses when shifts come to an end. 21 Nurse Staffing Level Min. # FT Nurses # FT 10s # FT 8s # PT Nurses Shortage (Hours) Table 4: This table depicts results from the mixed-integer-program for the optimal allotment of full-time (FT) and part-time (PT) nurses for a scenario in which there are 21 nurses available. Scenario Nurse Staff Shift Policy Breakdown 18

19 Level 1 21 All 10s hour 2 21 Mix 10s and hour, 8 8-hour 8s 3 21 All 8s 21 8-hour 4 18 All 10s hour 5 18 Mix 10s and hour, 6 8-hour 8s 6 18 All 8s 18 8-hour 7 16 All 10s hour 8 16 Mix 10s and hour, 6 8-hour 8s 9 16 All 8s 16 8-hour Table 5: This table depicts results for the nurse staff level and shift length combinations for daily schedule optimization for various nurse staffing levels ranging from 16 nurses to 21 nurses. Daily Schedule Optimization: Simulation-Optimization Results We solved our simulation-optimization model using OptQuest version 6.4, which uses a combination of Tabu Search and other heuristics to attempt to find a near optimal solution (Kelton et al. 2007). The stopping time criterion was set to stop the simulation-optimization after 1000 iterations. This number was chosen as a conservative upper bound based on experimenting with simulation-optimization runs and noting that significant objective function improvements were typically not made after several hundred simulation runs. We set the indifference parameter (a user defined parameter that defines a threshold at which schedules are considered indistinguishable) to be 0.1 minutes of patient waiting time. We allowed the optimizer to vary simulation model replications from 10 to 100, under the constraint that confidence interval 19

20 widths were within 10% of the mean. Simulation run times were approximately 2-4 hours for each instance of the model. The daily schedule simulation-optimization model used candidate schedules that were developed through a series of meetings with decision makers, using the MIP model combined with expert opinion, as the starting point for the daily schedule. We examined three nurse staffing levels with three promising shift policy combinations; all 10-hour shifts, all 8-hour shifts, and a mix of 10-hour and 8-hour shifts (ratio of 10-hour to 8-hour is approximately 2:1). Table 5 illustrates each combination of nurse staff level and shift policy we considered in our experiments. Due to variation in patient arrivals by day of week, the simulation-optimization model was defined for each day of the week for each scheduling scenario. Table 6 contains the results of the simulation-optimization runs for Scenarios 1-3 (the 21 nurse scenarios). The Original column shows the average waiting time for the candidate schedules developed by hand, and the Optimal column shows the average waiting room time based on the best solution OptQuest found after 1000 iterations. Table 6 also provides the halfwidth of the confidence interval. The results suggested some general conclusions. First, surprisingly, given the complex nature of the scheduling process, the ad-hoc scheduling process based on expert opinion works well for generating good solutions, as many of the schedules were statistically indistinguishable from the simulation-optimization model generated schedules. However, in some cases, the adhoc schedule development process does not provide the best solution; in such cases, the simulation-optimization model was useful in improving the schedule. These differences are quite dependent on day of week suggesting that the ad-hoc approach does not account well for variation in patient flow from day to day. Second, the addition of shift starts on the half-hour 20

21 helped, as the optimizer chose an 8:30 a.m. 7:00 p.m. shift for 10-hour shift nurses in most cases. The 8:00a.m. 6:30 p.m. and 9:00a.m. 7:30 p.m. shifts were not removed completely, but the optimizer adjusted many of them to 8:30a.m. 7:00p.m. Monday Tues Wed Thurs Fri Mean OTC Waiting Time Original Opt Mean HW Mean HW All 10s s All 8s All 10s s All 8s All 10s s All 8s All 10s s All 8s All 10s s All 8s Table 6: This table depicts results of 21 nurse simulation-optimization comparing original and optimal schedules for All 10s (all 10 hour nurse shifts), All 8s (all 8 hour nurse shifts) and a mix of shift lengths with 13 nurses working 10 hour shifts. HW denotes half width. 21

22 Another general conclusion we drew was that as nurses become a bottleneck, there is more benefit from using the optimizer. As the nurse staff level is reduced across scenarios, the number of days and shift policies where the optimizer had a statistically significant improvement on the candidate schedules increases. Additionally, as nurses become more of a scarce resource, benefits are seen in using shorter shift lengths as the nurses are able to be more concentrated on high demand times. However, the impact of changing shift combination is still significantly less than the impact of adjusting work schedule times. Finally, it is important to point out that while the improvement in average waiting time is moderate, the benefits are disproportionately allocated to patients seen during peak hours. The results for the maximum waiting room times typically average about 90 minutes, and improvements in waiting room time at peak times during the day improved by as much as 25 minutes in some cases. Thus small changes in the daily shift schedule significantly impact the waiting time for patients that are most affected by waiting time at the OTC. Conclusions and Implementation In this section we describe the general conclusions we drew from the study that could be applicable to many cancer centers. We also summarize some of the specific recommendations that were adopted at the Duke Cancer Institute Cancer Center in Durham, NC. Our model revealed bottlenecks for phlebotomy and oncologist consultation in the clinics. Patient waiting times for phlebotomists in the clinics varied by clinic and day of week, but ranged from minutes on average. Patient wait times for an oncologist in the oncology examination room varied by clinic and day of week as well, also ranging from minutes on average. The model indicated a large amount of variability in waiting times by clinic and day, which is consistent with what we expect and confirmed as the case by expert opinion. 22

23 Bottlenecks were also identified in the OTC. Patient wait times in the OTC were largest for chairs, with an average wait time of 2-10 minutes, but more significantly, a maximum average wait time ranging from minutes at peak times during the day. Additionally, patient wait times for OTC nurses ranged from 1-2 minutes on average; however, specific DBGs have maximum average wait times as high as 30 minutes at peak times during the day. In particular, most of the higher waiting times for nurses were centralized in the hematologic malignancy and off-service (HOA) DBG, ranging from 5-30 minutes for the maximum average waiting time across all replications. Our analysis using the MIP model showed that full-time nurses are useful for covering supply needs across the day, whereas part-time nurses help meet the variable day-to-day peak demand. Part-time nurses provide the capability to target increased nurse availability at peak times during the day. Thus, part-time nurses are very helpful in reducing shortage hours in a nurse schedule. Furthermore, we found that adding part-time nurses had diminishing returns in reducing shortage hours; thus, we concluded it only takes a small number of part-time nurses to make a significant impact in reducing shortage hours. As a result, we recommended taking one or two full-time nurses and replacing with equivalent levels of part-time nurses. The simulation-optimization model indicated changing arrival and departure times of nurse schedules has the most impact on patient waiting time. In particular, the addition of shift starts on the half-hour helped, as the optimizer chose an 8:30 a.m. 7:00 p.m. shift for 10-hour shift nurses quite often. Thus, we recommended changing some of the 8:00a.m. 6:30 p.m. and 9:00 a.m. 7:30 p.m. shifts to 8:30 a.m. 7:00 p.m. Using a combination of 10-hour and 8-hour shifts as opposed to exclusively 10-hour or exclusively 8-hour shifts can have an impact on average patient waiting time as well, though we see mixed results as to whether or not this 23

24 improves patient waiting time in our experiments. Our results indicated that the lower nurse staff levels are, the more of a bottleneck they become, and the larger the improvement in using optimization methods to improve candidate schedules. Based on the results and conclusions described above, the following strategies have been implemented to optimize staffing in the OTC. First, numerous part-time RN positions were hired to assist in meeting the variable day-to-day peak demand. Second, start times for new RN hires have shifted to the half-hour mark and some existing RNs have been converted to the half-hour start time as well. Based on the recommendation to hire additional nurses, a total of 1.75 additional FTEs have been hired. The simulation optimizer also found that a combination of 10- hour and 8-hour shifts can have an impact on average patient waiting time and therefore the practice of hiring RNs into 12-hour shifts has been stopped. Of the three existing RNs that work 12-hour shifts, one has been converted to 10-hr shifts. The plan is to phase out the last two 12- hour shifts by attrition. The changes described above are some of the most important tangible benefits from the application of the operations research methods described in this article. We also used our model to explore future resource capacity planning for a new cancer center that opened in Spring of We used our model to forecast future bottlenecks and OTC performance in several scenarios to inform the planning process for administrators. The results of the model showed that an expansion of chair capacity from current levels removed most of the waiting time for chairs in the OTC. Thus, the proposed increase in chair levels in the new cancer center was projected to sufficiently meet the demand needs. Additionally, we identified nurses as the primary bottleneck in the new cancer center under demand increases of 6% to 12%. These results helped administrators make staffing decisions for the new cancer center. 24

25 Acknowledgements The authors are grateful for funding from Duke University Hospital to complete this project. This article is also based in part upon work supported by the National Science Foundation under Grant Number CMMI Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The authors are also grateful for the help of a number of project team members at the Duke Cancer Institute including Chad Seastrunk, William T. Fulkerson, Craig Johnson, Nancy Hedrick, Celia Walsh, Steve Power, the pharmacy staff, and the nursing staff. The authors would also like to thank Bjorn Berg for helpful comments during the preparation of this manuscript. Finally, the authors would like to thank Randy Robinson, Alice Mack, and anonymous reviewers for their help improving the final version of this manuscript. 25

26 Appendix Following is a description of the mathematical formulation of the MIP model for monthly/weekly nurse planning problem: Indices: t = day in the 4-week schedule. (t = 1,2,,20) k = disease based group (DBG) (k=1,2,,6) j = index for long weekends during the month (j=1,,4) i = index for nurses. (i = 1,2, 3N) (i = 1,2,,N corresponds to a part-time nurse) (i = N + 1, N + 2,,2N corresponds to a full time 8-hour shift nurse) (i = 2N + 1, 2N + 2,,3N corresponds to a full time 10-hour shift nurse) Decision Variables: x it = Nurse i scheduled on day t. (Binary variable) y ij = Nurse i scheduled on long weekend j. (Binary variable) z i = Nurse i selected on the schedule. (Binary variable) s t = Shortage of nurse hours on day t. (Continuous variable) o t = Overage of nurse hours on day t. (Continuous variable) Parameters: 26

27 f i = FTE (Full-Time Employee) value for each nurse i d t = # of nursing hours required for day t. a ik =binary indicator defining if a nurse i is associated with DBG k (a ik =1) or not (a ik =0) N = maximum # nurses for a particular DBG shift type. n = minimum nurses in each DBG for the day r t = minimum number of nurse in OTC on day t M = maximum # FTE (Full-Time Employees) for OTC nurses F = Minimum number of full-time nurses w i = # days worked per week by nurse i w i = 2,3,4,or 5 for i = 1,2,,N (part-time nurses under varying policies) w i = 5 for i = N + 1, N + 2,,2N (full-time 8-hour shift nurses) w i = 4 for i = 2N + 1, 2N + 2,,3N (full-time 10-hour shift nurses) k i = # of hours available per day for nurse i k i = 4, 8, or 10 for i = 1,2,,N (part-time nurses) k i = 8 for i = N + 1, N + 2, 2N (full-time 8-hour shift) k i = 10 for i = 2N + 1, 2N + 2,,3N (full-time 10-hour shift) 27

28 MIP Formulation: s.t. Total FTE Constraint (1),k Minimum number of full time nurses used Minimum nurse requirement for entire OTC Minimum nurse requirement for each DBG (2) (3) (4) Nurse i selected constraint (5) Nursing hours per day constraint Days worked per week 1 constraint Days worked per week 2 constraint Days worked per week 3 constraint Days worked per week 4 constraint FT 10-hour nurse # Fridays worked constraint (6) (7) (8) (9) (10) (11) 28

29 FT 10-hour nurse # Mondays worked constraint 4-day weekend constraint for full time 10-hour shift nurses Long weekend constraint, weekend 1 Long weekend constraint, weekend 2 Long weekend constraint, weekend 3 Long weekend constraint, weekend 4 (12) (13) (14) (15) (16) (17) Binary variables constraint (18) Non-negativity constraints (19) Simulation Optimization Model Formulation Following is a description of the mathematical formulation of the simulation optimization model for the daily nurse scheduling problem. The decision variables, x ij, are binary decision variable that represents whether nurse i, of m nurses, is working shift j (x ij =1) or not (x ij =0). Each nurse i has a series of n associated shifts they could be allocated, which correspond to start and end- 29

30 times in half-hour increments during the day. The simulation optimization model can be expressed as follows: [ ] { } where and denote the opening and closing shifts, and denotes the sets of nurses that are working (on) or not working (off) for the day. The first constraint determines which shifts a nurse may be scheduled for based on indicator a ij which is 1 if the assignment of nurse i to j is allowed and 0 otherwise. This indicator is based on nurse experience working in the various DBGs, as defined by the OTC Clinical Operations Director. The second constraint enforces the fact that nurses do not work off-days. The third and fourth constraints require that two nurses be available for opening and closing of the OTC. 30

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