Using Monte Carlo Simulation to Assess Hospital Operating Room Scheduling

Transcription

1 Washington University in St. Louis School of Engineering and Applied Science Electrical and Systems Engineering Department ESE499 Using Monte Carlo Simulation to Assess Hospital Operating Room Scheduling By Eric Hsu Gary Savell Supervisors Eli Snir Jason Trobaugh Submitted in Partial Fulfillment of the Requirement for the BSSSE Degree, Electrical and Systems Engineering Department, School of Engineering and Applied Science, Washington University in St. Louis May 2015

2 ACKNOWLEDGMENTS We would like to acknowledge the contribution of our advisor, Eli Snir for his assistance and guidance throughout the undertaking of this project. We would also like to thank our advisor and course mentor, Jason Trobaugh, for his recommendations of different optimization techniques. Finally, we would like to thank the Consultant for all of his assistance in helping us build upon his previous work. 1

3 ABSTRACT Operation Room (OR) scheduling is a constant problem that hospitals face. Many analysts have tackled this problem before and have created solutions that may be good for the hospital, but oftentimes there is not analysis that can be used to physically assess how beneficial that solution is prior to its implementation. Thus, there is a need for simulation and evaluation of different solutions. In this report, we create a new system for analyzing hospital OR schedules. The system is three-fold: we create new candidate solutions by refining a previously used objective function, use Monte Carlo simulation to obtain statistics for those solutions such as mean overtime hours and variation in weekly patient flow, and then evaluate those statistics. Although we used our own means of evaluation to assess the quality of our solutions, the evaluation methods may change from user to user. Since this project involves only an initial run through of our system, assumptions can always be tweaked and further physical metrics and evaluation methods can be developed in the future. 2

4 Table of Contents ACKNOWLEDGMENTS... 1 ABSTRACT... 2 CONTACT INFORMATION... 4 INTRODUCTION AND BACKGROUND... 5 Overview of the Problem... 5 Method... 6 METHODS: FINDING SUBOPTIMAL SOLUTIONS... 7 Defining Decision Variables... 7 Defining Constraints... 8 Objective Function and Finding Candidate Solutions... 9 METHODS: SIMULATION TECHNIQUES Underlying Assumptions Simulation Method Metrics of Evaluation... 1 RESULTS Candidate Solutions Allocation Score Metric Evaluation of Candidate Solutions A Detailed Look at Our Candidate Solutions CONCLUSION Opportunities for Future Analysis BIBLIOGRAPHY... 2 APPENDIX A: PROBLEM FORMULATION Appendix A.1: Notation of Decision Variables Appendix A.2: List of Constraints in Equation Form APPENDIX B: RESULTS Appendix B.1: Result of Allocation Score Algorithm Calculations Appendix B.2: Patient Flow Variation for Selected Candidate Solutions Appendix B.: Statistics for All Candidate Solutions... 28

5 CONTACT INFORMATION Group Members: Eric Hsu, Washington University in St. Louis Gary Savell, Washington University in St. Louis Project Supervisors: Eli Snir, Lecturer in Management, Olin Business School, Washington University in St. Louis Jason Trobaugh, Lecturer in Electrical and Systems Engineering, School of Engineering and Applied Science, Washington University in St. Louis 4

6 INTRODUCTION AND BACKGROUND Overview of the Problem Operating Room (OR) Scheduling is a common problem that hospitals face today. Hospitals try to utilize the space that they have in the most efficient and cost-effective way to provide the best services to clients and patients. However, scheduling operating rooms to specific physician specialties requires the consideration of physician availability, specific operating hours, patient variation, and medical equipment provided in each operating room. Having an optimized OR surgical schedule can increase the effective use of each operating room and reduce the amount of poorly allocated resources used by the hospital. Since OR scheduling and use has been an issue in hospitals for a long time, many studies have been performed to maximize OR use [1, 2, ]. Unlike most finite resource allocation problems, many different parameters such as physician preferences, hospital specialties (such as General Surgery, Cardiothoracic, Peripheral Vascular, etc.), patient preferences, and financial return from patients are considered. Each study provides its own optimal solution for the hospital s preference. However, our client, a hospital operations research consultant (referred to as Consultant), who worked for a major tertiary academic medical center this past year, realized that implementing the optimal solution in real life was too impractical many physical metrics could not be measured reliably by just an objective function. Thus, our group has decided to obtain and evaluate multiple suboptimal candidate solutions via simulation with previous hospital patient and OR usage data. This way, the Consultant will be provided with physical statistics that will allow him to better choose a candidate solution that best serves his institution and will be more feasible to implement. 5

7 Method Our approach to finding quality solutions to this scheduling problem is three-fold. The first part is that the candidate solutions need to be generated. We chose to use an existing integer programming model to generate solutions. Secondly, these solutions need to have physical metrics to allow for evaluation. Third, the solutions need to be evaluated based on their practicality and efficiency. Find A Candidate Solution Simulate the solution and create metrics Evaulate the metrics and practicality We chose this method of solving the problem because it is a new way of looking at the benefits of a particular schedule. Previous work has been done specifically for this hospital based on a deterministic objective function. We wanted to use an alternative method to test the accuracy level of the objective function as well as propose other possible options. In addition, deterministic systems only consider the means, whereas we are considering a distribution with variation in each given day, which is more realistic. The simulation also gives the added benefit in that it answers real questions that the hospital has. A hospital typically cares about patient care safety and quality, physician and employee contentment, and costs. We addressed the simulation with those items in mind. In particular, a hospital cares about how much expected time will be spent where they need to pay overtime because it is often difficult to consider how long patients will be in operating rooms. Similarly, if an operating room can be freed up because of lack of usage, the hospital also wants to know that in order to reallocate the OR to a specialty that may need it more. 6

8 A hospital also cares about patient flow. The three types of patients that we considered were Admit (A), Inpatient (I), and Outpatient (O). The reason the hospital wants to have data about patient flow is because of a space constraint of number of beds available. For example, if all of the Inpatients are scheduled only on one day and not on any of the others, there will be a shortage of beds available. Thus, the hospital wants low variation of patient types across days of the week. METHODS: FINDING SUBOPTIMAL SOLUTIONS Defining Decision Variables As part of his analysis for the academic medical center, the Consultant had identified a model for finding the optimal schedule. The integer programming model optimized an objective function based on three equally weighted components of high utilization time of ORs: allocation efficiency and two measures of variation of the time used in the OR, which will be discussed in more detail below shortly. The ten specialties with their abbreviations are listed below in Table 1. Five different types of ORs exist, each supplied with different resources for different specialties. Table 2 shows which specialties can use which OR type, giving us a total of 16 Service Abbrev Max Daily Blocks General Surgery Gen Orthopedic Surgery Ortho 4 Transplant Surgery Tx 2 Otolaryngology ENT 2 Oral and Maxillofacial Surgery OMFS 2 Urology Uro 2 Cardiothoracic CT 2 Peripheral Vascular PV 2 Neurosurgery Neuro 2 Plastic Surgery Plas 2 Table 1: List of services, abbreviations, and maximum daily number of allocated blocks possible to each service Specialties Gen, Ortho, Tx, ENT, OMFS, Uro, Plas, Neuro Tx, Uro, CT, OR Type # ORs PV, Neuro Neuro 2 1 PV 1 CT 4 1 Table 2: List of specialties that can use each OR type and total number of each OR type 7

9 different Specialty/OR combinations. Each OR allocation can last 8, 10, or 12 hours. Thus, full OR schedules require the allocation of blocks to three different criteria: day of the week (Monday through Friday for 5 days), specialty/or use (16 combinations), and amount of time allocated ( different times), resulting in a total of 240 decision variables. These decision variables are denoted as x d,s,t with the subscripts being defined as follows (see Appendix A.1 for a more explicit defining of decision variables): d = 1, 2,, 4, 5 respectively for Monday, Tuesday, Wednesday, Thursday, and Friday s = 1, 2,,,16 based on specialty/or combination shown in Table 2, with s = 1 referring to Gen in OR Type 0, s = 2 referring to Ortho in OR Type 0, and rightwards and downwards along Table 2 with increasing s up to s = 16 referring to CT in OR Type 4. The values of s for each service are also shown in Table. Each service consists of a subset of set S, where the first element of S is Gen, second is Ortho, etc. t = 1, 2, respectively for 8, 10, and 12 hour blocks Service Corresponding s in set {S} Gen 1 Ortho 2 Tx, 9 ENT 4 OMFS 5 Uro 6, 10 CT 11, 16 PV 12, 15 Neuro 8, 1, 14 Plas 7 Table : Services with their corresponding values of s in x d,s,t. They are labeled to be subsets in {S}. Defining Constraints For practicality, all of the decision variables are nonnegative and integers, as both negative allocations and partial block allocations are not feasible options in real life. The rest of the constraints in this model are daily constraints and are denoted as shown below. This means that all of the following equations are constrained for all values of d = 1, 2,, 4, 5. 8

10 The hospital has 15 possible operating rooms that can be used each day; however, two are allocated for emergencies. This leaves 1 ORs that are to be scheduled per day, all of which are to be used, resulting in the constraint equation shown below. 16 x d,s,t s=1 t=1 = 1, x d,s,t N Other constraints include those listed above in Tables 1 and 2. The maximum number of blocks per day that each specialty can have is 2,, or 4 as shown in Table 1. The maximum number of ORs allocated to a particular type is also limited by the number of each OR type that the hospital has. That is, the hospital has 11 ORs of Type 0 and 1 of each other Type, so the daily total number of blocks allocated to OR Type 0 must be less than or equal to 11 and Types 1 to 4 less than or equal to 1. One last constraint includes the maximum number of 8, 10, and 12 hour blocks that can be allocated to a given day. For the sake of reasonable hospital organization, these respective maximum values are 2 (or, to be described below), 5, or 8 allocated blocks. Appendix A.2 provides a full list of the constraints. Objective Function and Finding Candidate Solutions As stated above, in the Consultant s model, the objective function contained three terms: allocation efficiency and two measures of variation of the time used in the OR. Allocation efficiency required calculating the number of goal hours for each specialty, which is the number of hours that each specialty requested divided by a target allocation percentage. As is common with most hospital OR scheduling problems, this value defaults to 0.8. Table 4 shows such values at a target allocation percentage of 80%. Allocation efficiency itself is the sum of the absolute differences between the number of allocated hours and goal hours. The two measures of variation drew a sample of Admit, Inpatient, and Outpatient values for each specialty from 9

11 existing patient flow data and combined that data with the candidate solution to obtain total A, I, and O values for each day of the week. A standard deviation and coefficient of variation of each group of the 5 A, I, and O values were incorporated into the Consultant s objective function. To find our own candidate solutions, we took the Consultant s objective function and changed certain coefficient values to obtain different meaningful solutions. Since we did not have the software that let us run an integer programming problem simultaneously with a simulation problem as the Consultant did, we removed the two measures of variation from our objective function. From there, we changed the objective function in two ways to obtain two different types of solutions. The first way involved changing the importance of a particular specialty s error from the goal hours. In the Consultant s objective function, all of the errors were weighted equally. In these new objective functions, the importance of particular services is increased to 5 times of all of the others, thus driving the allocated hours of the particular specialty closer to its goal hours. The second way involved changing the allocation percentage to 0.75, 0.85, and 0.9 to see what solutions are obtained from allocating a greater (for 0.75) or lesser (for 0.85 and 0.9) number of hours. However, at an allocation proportion of 0.9, no solutions are feasible, and the number of 8 hour blocks per day had to be increased from 2 to to accommodate this change. Service Requested Goal Hours Hours Gen Ortho Tx ENT OMFS Uro CT PV Neuro Plas Table 4: Requested and Goal Hours for each Specialty at a target allocation of 0.8 We denote AP as the allocation percentage, RH S as the requested number of hours in a particular service, and W S as the importance weight of a particular specialty (where the default weight of each specialty is W S = 1). Since the objective function is calculated based on total hours, we must multiply each allocated block by the hours allocated to that block. Factoring in 10

12 preparation time and grand rounds on Wednesdays, the number of hours for the respective 8, 10, and 12 hour blocks used in calculation are 7.5, 9.5, and 11.5 hours (or 7, 9, and 11 hours on Wednesday). Thus, the number of hours (with proper subscript notation) multiplied calculates out to be t (or 5 + 2t on Wednesdays). For our full objective function, we minimize the sum of the absolute differences between the goal hours and allocated hours for each specialty multiplied by that specialty s importance weight, as shown below. Min Z = W S 1 AP ( ( t) x RH d,s,t + (5 + 2t) x,s,t ) S s S d=1,2,4,5 t=1 t=1 METHODS: SIMULATION TECHNIQUES Underlying Assumptions Data of past usage was provided daily, including patient types and minutes used of the OR. Assuming a normal distribution of the data, we collected sample averages and standard deviations for each day of the week of usage. If there was not enough data points (less than 8) for the specific day and specialty combination, we assumed that the sample average and sample standard deviation of the overall non-daily specific usage would approximate the usage for that day. For each specialty and block assignment, we pulled data from a normal distribution of the appropriate mean and standard deviation. We truncated the results so that there would be no negative hours used, since in all practical applications, negative usage is impossible. As a result, the i th trial s usage for day of week d for specialty s becomes U i,d,s,t = max{x d,s + SD d,s N(0,1), 0}, where N(0,1), x d,s, and SD d,s are the standard normal distribution, sample mean and sample standard deviation, respectively, of specialty s on day of the week d for block of time t. Although t is not a parameter of U i,d,s,t, we needed one random draw for each different allocated block of time. Thus, while t changes, the calculation of U i,d,s,t does not change. 11

13 Another assumption that was made was how to simulate the data. The two options were daily and weekly. Data was provided daily, but given that blocks are allocated weekly, both simulation techniques were feasible. Since daily data was provided, it seemed more appropriate to complete the simulation daily. A future analysis can include altering the simulation and viewing how the results would change if the simulation was run weekly. Since we are also interested in understanding the flow of patients across days, we used data provided as a sample of different patient flows of Admit, Inpatient, and Outpatient to obtain a projected patient flow. If we observed fewer than 8 observations of a specific specialty on a specific day of the week, we sampled from the entire dataset for that specialty. The reason for using this method as opposed to fitting a distribution to the dataset was that we did not feel comfortable making an assumption about the characteristics of that distribution, especially for the data being integer constrained. Simulation Method For each candidate solution, we used MATLAB R201A to run our simulation. The simulation was run for 10,000 trials (N=10,000). For each run, the random usage vector was generated using the already specified method. We then calculated various metrics based on how the given candidate solution would behave if it were implemented with the projected usage. We used this data in completing different evaluative measurements for each solution. In addition, we also evaluated patient flow across the days of the week. We randomly selected a set (A, I, O) of patient flow for a specific day of week and specialty out of the full data set. We then projected how our schedule would behave given those patient allocations. 12

14 Metrics of Evaluation The candidate solutions were evaluated based on a variety of metrics. For the purpose of defining these metrics, let U i,d,s,t be a random usage vector for an individual trial i, day of week d, specialty s, and block time t. Let H d,s,t = ( t) x d,s,t for d = 1, 2, 4, 5 and H,s,t = (5 + 2t) x,s,t, being the number of hours allocated to each particular block of different day and specialty. A list of metrics and their definitions are defined below: Mean Utilization (U ) allows us to evaluate how much time was used divided by the amount of time allocated. More specifically, N 5 U = 1 N U i,d,s,t i=1 d=1 s S t=1 H d,s,t Overtime (V) is the measure of traditional overtime, the sum of overtime hours for each allocated block for those blocks where more hours were used than were allocated. Let the set of overtime values be denoted V = {V 1, V 2,, V N }. We evaluated the mean and root mean square (RMS) of Overtime, V and V rms, respectively over the N trials. For only U i,d,s,t > H d,s,t, 5 V i = U i,d,s,t H d,s,t d=1 s S t=1 Excess Allocation (E) measures the amount of time that the ORs were left empty because nobody needed to use them. Let the set of Excess Allocation values be denoted E = {E 1, E 2,, E N }. Like with Overtime, we evaluated E and E rms. For only U i,d,s,t < H d,s,t, E i is defined as: 5 E i = H d,s,t U i,d,s,t d=1 s S t=1 1

15 Patient Flow is a measure of the various types of patients over days. We measure all of the following only for where a block is actually allocated: o The mean number of each patient type per day is important to analyze. Consider the following measurement of the mean patients by type across days, where i is the trial number, s is the specialty, and t is the block time allocation: N A d = 1 N A i,d,s,t i=1 s S t=1 N I d = 1 N I i,d,s,t i=1 s S t=1 N O d = 1 N O i,d,s,t i=1 s S t=1 o To get a closer look at the spread of patient types across a day, we look at the standard deviations of the number of A, I, and O patients across trials each day: N SD Ad = 1 N 1 ( A i,d,s,t) i=1 N s S t=1 SD Id = 1 N 1 ( I i,d,s,t) i=1 N s S t=1 SD Od = 1 N 1 ( O i,d,s,t) i=1 s S t=1 o Weekly patient flow standard deviations are given by 2 2 N 1 N(N 1) ( A i,d,s,t) i=1 s S t=1 N 1 N(N 1) ( I i,d,s,t) 2 i=1 s S t=1 N 1 N(N 1) ( O i,d,s,t) SD A = 1 5 (A d 1 5 A d ) d=1 d=1 i=1 s S t=

16 5 5 2 SD I = 1 5 (I d 1 5 I d ) d=1 d= SD O = 1 5 (O d 1 5 O d ) d=1 d=1 RESULTS Candidate Solutions In our solution set, Candidate Solution 1 is defined as the optimal solution as originally proposed by the Consultant. We are therefore going to compare all of the metrics of our candidate solutions to this solution s metrics. Since the Consultant is interested in seeing what insights can be gathered about his optimal solution, we want to evaluate his solution in depth. By changing each weight and each allocation percentage as discussed in the Methods section, we obtained 1 more candidate solutions by using Microsoft Excel s GRG Nonlinear Solver. Candidate solutions 2 through 11 are solutions that drive the allocated hours of particular specialties towards the goal hours (with allocation percentage 0.8). The order of specialties that is being focused on is the same order as that shown in Table. That is, candidate solution 2 corresponds to Gen, to Ortho, etc. Candidates 12, 1, and 14 correspond to solutions where the weights of all of the specialties in the objective function are the same, but the allocation percentages are respectively 0.75, 0.85, and Allocation Score Metric In order to better comprehend the results over multiple different candidate solutions, we developed a score called Allocation Score. This score, denoted as Q, combines both Overtime and Excess Allocation for each day over all specialties. The equations use RMS values in order 15

17 to have a nonlinear penalty for extreme values, especially for Overtime extreme values. This makes sense because missing the allocated usage by a small amount is a negligibly costly compared to missing the allocated usage by large amounts. Similarly, having Excess Allocation is less costly than Overtime, as not having enough hours (which leads to overtime hours) is problematic. The overtime penalty is weighted by a value α, which we varied as a method of creating robustness in our statistic. We used α {0,1,2,,4,5,6,7,8,9,10} for evaluation. N 5 Q = 1 N (U i,d,s,t H d,s,t ) 2 i=1 d=1 s S t=1 { 5 + α {(U i,d,s,t H d,s,t ) 2, U i,d,s,t > H d,s,t 0, U i,d,s,t H d=1 s S t=1 d,s,t } Recall that Q can be calculated for each candidate solution, so we have a set {Q} of Q s. To then easily compare Q values across solutions at various α-values, we decided to scale Q by the minimum Q, or more specifically for each candidate solution k {1,,14}: {Q} = {Q 1,, Q 14 } Q k Q k = 100 min {Q} Thus, we can look at the value of Q k where 100 is the best possible solution and each other Q k value is a specific percentage worse than the best Q k value based on its relative scale to the optimal Q k. Appendix B.1 contains the full table of Q k values calculated at various α values. Evaluation of Candidate Solutions Deciding different thresholds of the metrics for the removal of different candidate solutions from consideration was completed by the evaluation of the individual metrics 16

18 themselves in relation to the other candidate solutions. It is up to the reader and/or analyst to determine these threshold values for the specific set of candidate solutions based on each metric s individual importance for the specific application. Our initial method of removing candidate solutions was based on Allocation Score at various values of α. In particular, α values of 0 and 1 seemed to be the most accurate measure as proposed before the Overtime RMS values dominate the statistic. At high values of α, the allocation scores of each solution give the same relative information as the mean Overtime of each solution. Figure 1 shows that from α = 2 to 10, the ranking of highest to lowest allocation score stays the same, suggesting that α > 1 only gives extraneous information. We plotted the Allocation Score values for our 14 candidate solutions and decided to set 105 as our cutoff score for eliminating different candidate solutions. Figure 1. Allocation Scores of all Candidate solutions at different values of alpha. This figure was mainly used for determining which alpha value is used for analysis and our allocation score cutoffs. We chose to attack Allocation Score first because our client ranked the importance of Overtime and Excess Allocation highly. Candidate solutions 6, 8, 10, 11, and 12 were all removed based on their Allocation Score values at = 0. We then removed candidate solutions 7, 9, and 14 because of their Allocation Score values at = 1. 17

19 We then considered patient flow and in particular looked at the weekly standard deviation of patient flow for the three types. From that, we concluded that we should remove candidate solution 1 because it had too much variance (SD > 1) in all three patient types, especially in Inpatient and Outpatient standard deviation. Thus, we evaluate five final candidate solutions in more depth: 1, 2,, 4, and 5. See Appendix B. for the particular values of each statistic for each candidate solution. Recall candidate solution 1 is the Consultant s solution. A Detailed Look at Our Candidate Solutions We wanted to compare Overtime and Excess Allocation across our remaining candidate solutions as compared to the Consultant s solution. As a result, we found that our candidate solutions had less Overtime and slightly more Excess Allocation than the Consultant s solution. Consider Figure 2 below, which demonstrates this phenomenon. This suggests that the Consultant s solution overall allocates less time to each block, while our candidate solutions allocate more. Also, this trend suggests that the Consultant s solution fails to allocate enough time more frequently than our candidate solutions, as these measures are aggregate measures. Overtime and Excess Allocation Overtime Excess Allocation Consultant Solution Overtime Consultant Solution Overtime and Excess Allocation 00 Time (Hours) Candidate Solution Figure 2. Overtime and Excess Allocation of Candidates 2-5. The Consultant s Overtime and Excess Allocation is plotted as the two lines. 18

20 We then wanted to look at the patient flow across the different candidate solutions. Recall that patient flow is important due to hospital capacity constraints and that lower variation is ideal. For the Consultant s solution, consider Figure which plots the mean and standard deviation of patient flow across days of the week. The Consultant s solution provides a very low variation for Inpatient and Outpatient, but has a high variation in Admit patients. 18 Patient Allocation Across Days for Candidate Solution Number of Patients Admit Inpatient Outpatient Monday Tuesday Wednesday Thursday Friday Figure. Patient Allocation Across Days for Candidate Solution 1. (SD A, SD I, SD O ) = (1.8761, , ) A similar plot for Candidate Solutions 2,, 4, and 5 are included in Appendix B.2 [4]. Candidate Solutions 2,, 4, and 5 all have worse Inpatient and Outpatient variation than the Consultant s solution. However, they have lower Admit patient variation. In addition, these solutions have higher Outpatient mean values on Friday, which in effect lessens the need for large staff over the weekend. The standard deviations across days and type for different candidate solutions do not vary much. Looking in more detail, it becomes evident that Friday sees fewer patients overall. Oftentimes, physicians request operating rooms earlier in the week to avoid heavy staffing over the weekends. While this is convenient for the physicians, this leads to some congestion earlier 19

21 in the week. We sought candidate solutions that minimized variation, especially for Admit, to help alleviate this problem; however, again, it is up to the analyst to decide whether physician s preferences or overall hospital patient flow. Our candidate solutions thus increased the amount of Admit patients on Friday so that the weekly Admit variation was not as staggering. Since this is not very optimal for the physicians themselves, our candidate solutions made up for this by increasing the amount of Outpatients on Friday. This means that patients are leaving the hospital, and that fewer physicians and nurses will have to staff the Outpatient rooms. Figure 4. Percent Utilization by Specialty for the Consultant s Solution. We lastly looked at the mean utilization of each candidate solution. Overall, the mean utilization did not vary between the candidate solutions, as each mean utilization totaled to around 68%. This differs from our allocation percentage because the amount of hours requested by each specialty does not equal the amount of hours actually used. While these two values are calculated similarly, the true patient data differs from the theoretical data, or the number of requested hours. 20

22 Figure 4 shows the utilization of every specialty, along with the overall mean utilization, for Candidate Solution 1. We see that Ortho, ENT, and Uro are the three specialties that get the fewest hours. However Ortho is the only specialty that does not receive enough total hours for that department to treat all of the patients (i.e., the utilization is greater than 1). This trend is the same with the other solutions, with Ortho, ENT, and Uro being allocated fewer hours. This may just the number of expected necessary hours (i.e. the requested hours) by the specialty is not enough compared to how many hours the blocks will actually be used. Different candidate solutions that increase the number of requested hours for these specialties may provide a relatively more robust solution. CONCLUSION We have designed a system that can obtain new candidate solutions, creates statistics based on historical patient data, and then evaluates the statistics. The most useful novel part of this project is the Monte Carlo simulation and creation of statistics, as this has not been done in other hospital OR scheduling reports. The simulation also does not necessarily require an objective function or optimization the user can just input a candidate solution and look at meaningful physical measurements that can be useful for the hospital. While we do have a list of candidate solutions and measures of evaluation, again, these methods are not all encompassing. More candidate solutions can be determined and the ranking of different solutions based on the importance of different metrics are at the hands of the analyst. This report merely shows one way to evaluate the physical metrics obtained that is more preferable to the Consultant. 21

23 Opportunities for Future Analysis For future analysis, there are several assumptions that can be tweaked, including completing a simulation across weeks as opposed to across days. The reason we chose to simulate across days is because it seems more precise to allocate and simulate at a finer level, as opposed to a more broad approach. Since we had the daily data rather than the weekly data, this more precise approach was possible. We do not expect too much variation in the results if that method were to be completed. A more detailed approach for future analysis could be to use Data Envelopment Analysis (DEA). The basic idea behind this analysis is to create a Pareto frontier in all of the dimensions for each parameter of evaluation. From there, one could measure the distance between the frontier and the given candidate solution, thus defining the quality of a candidate solution, as tested against the parameters of evaluation. Lastly, our list of candidate solutions is not the full population of candidate solutions, and many other candidate solutions may be inputted into our simulation for analysis. For example, examining a current hospital solution, an optimal hospital solution, and intermediate solutions between the current and optimal solutions may provide many insights. These intermediate solutions can initially start at the current solution and only have small changes until the optimal solution is reached. This would then provide analysis of a series of OR schedules that lead from current to optimal solution. While this would be desirable to examine in this report, our client was unable to provide us with his affiliated hospital s current solution. 22

24 BIBLIOGRAPHY [1] S. F. Sufahani et al. A Scheduling Problem for Hospital Operating Theatre. Arxiv.org. University Tun Hussein Onn Malaysia, Web. 1 Mar < [2] A. Romanyuk and A. Silva. Optimization of an Operating Room Surgical Schedule. Ese.wustl.edu. Washingon University in St. Louis, Web. 1 Mar < bpages/sp12/finalreportsilvaromanyuk.pdf>. [] F. Dexter et al. An Operating Room Scheduling Strategy to Maximize the Use of Operating Room Block Time: Computer Simulation of Patient Scheduling and Survey of Patients Preferences for Surgical Waiting Time. citeseerx.ist.psu.edu. University of Iowa, Web. 1 Mar < [4] Callaghan, Martina (2014). barwitherr(errors,varagin). MATLAB Central File Exchange. Retrieved April 18, < barwitherr-errors-varargin-/content/barwitherr.m> 2

25 APPENDIX A: PROBLEM FORMULATION Appendix A.1: Notation of Decision Variables Gen Ortho Tx ENT OMFS Uro Plas Neuro Mon x 1,1,1 x 1,2,1 x 1,,1 x 1,4,1 x 1,5,1 x 1,6,1 x 1,7,1 x 1,8,1 Tues x 2,1,1 x 2,2,1 x 2,,1 x 2,4,1 x 2,5,1 x 2,6,1 x 2,7,1 x 2,8,1 Wed x,1,1 x,2,1 x,,1 x,4,1 x,5,1 x,6,1 x,7,1 x,8,1 Thurs x 4,1,1 x 4,2,1 x 4,,1 x 4,4,1 x 4,5,1 x 4,6,1 x 4,7,1 x 4,8,1 Fri x 5,1,1 x 5,2,1 x 5,,1 x 5,4,1 x 5,5,1 x 5,6,1 x 5,7,1 x 5,8,1 Tx Uro CT PV Neuro Neuro PV CT Mon x 1,9,1 x 1,10,1 x 1,11,1 x 1,12,1 x 1,1,1 x 1,14,1 x 1,15,1 x 1,16,1 Tues x 2,9,1 x 2,10,1 x 2,11,1 x 2,12,1 x 2,1,1 x 2,14,1 x 2,15,1 x 2,16,1 Wed x,9,1 x,10,1 x,11,1 x,12,1 x,1,1 x,14,1 x,15,1 x,16,1 Thurs x 4,9,1 x 4,10,1 x 4,11,1 x 4,12,1 x 4,1,1 x 4,14,1 x 4,15,1 x 4,16,1 Fri x 5,9,1 x 5,10,1 x 5,11,1 x 5,12,1 x 5,1,1 x 5,14,1 x 5,15,1 x 5,16,1 Shown above are decision variables x d,s,t for different combinations of day of week and specialty/or combination for block time of 8 hours. For 10 and 12 hours, t is replaced respectively with 2 and. 24

26 Appendix A.2: List of Constraints in Equation Form 16 x d,s,t = 1 s=1 t=1 16 x d,s,1 2 s=1 16 x d,s,2 5 s=1 16 x d,s, 8 s=1 8 x d,s,t 11 s=1 t=1 1 x d,s,t 1 s=9 t=1 x d,14,t 1 t=1 x d,15,t 1 t=1 x d,16,t 1 t=1 x d,s,t N x d,1,t t=1 x d,2,t 4 t=1 x d,s,t 2 s=,9 t=1 x d,4,t 2 t=1 x d,5,t 2 t=1 x d,s,t 2 s=6,10 t=1 x d,7,t 2 t=1 x d,s,t 2 s=8,1,14 t=1 x d,s,t 2 s=11,16 t=1 x d,s,t 2 s=12,15 t=1 25

27 APPENDIX B: RESULTS Appendix B.1: Result of Allocation Score Algorithm Calculations α Candidate Solution

28 Appendix B.2: Patient Flow Variation for Selected Candidate Solutions 16 Patient Allocation Across Days for Candidate Solution 2 18 Patient Allocation Across Days for Candidate Solution Number of Patients Number of Patients Admit Inpatient Outpatient Monday Tuesday Wednesday Thursday Friday Admit Inpatient Outpatient Monday Tuesday Wednesday Thursday Friday (SD A, SD I, SD O ) = (1.74, 1.40, 0.900) (SD A, SD I, SD O ) = (1.4790, , ) 16 Patient Allocation Across Days for Candidate Solution 4 16 Patient Allocation Across Days for Candidate Solution Number of Patients Number of Patients Admit Inpatient Outpatient Monday Tuesday Wednesday Thursday Friday 2 0 Admit Inpatient Outpatient Monday Tuesday Wednesday Thursday Friday (SD A, SD I, SD O ) = (1.790, 1.999, ) (SD A, SD I, SD O ) = (1.825, , ) 27

29 Appendix B.: Statistics for All Candidate Solutions Candidate Solution U V V rms E E rms A A A A A I I I I I O O O O O SD A SD I SD O SD A SD A SD A SD A SD A SD I SD I SD I SD I SD I SD O SD O SD O SD O SD O

30 Candidate Solution U V V rms E E rms A A A A A I I I I I O O O O O SD A SD I SD O SD A SD A SD A SD A SD A SD I SD I SD I SD I SD I SD O SD O SD O SD O SD O