Most surgical facilities in the US perform all

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1 ECONOMICS AND HEALTH SYSTEMS RESEARCH SECTION EDITOR RONALD D. MILLER Changing Allocations of Operating Room Time From a System Based on Historical Utilization to One Where the Aim is to Schedule as Many Surgical Cases as Possible Franklin Dexter, MD, PhD*, and Alex Macario, MD, MBA *Division of Management Consulting, Department of Anesthesia, University of Iowa, Iowa City, Iowa; Department of Anesthesia, Stanford University, Stanford, California Many facilities allocate operating room (OR) time based on historical utilization of OR time. This assumes that there is a fixed amount of regularly scheduled OR time, called block time. This Fixed Hours system does not apply to many surgical suites in the US. Most facilities make OR time available for all its surgeons patients, even if cases are expected to finish after the end of block time. In this setting, OR time should be allocated to maximize OR efficiency, not historical utilization. Then, cases are scheduled either on Any Workday (i.e., date chosen by patient and surgeon) or within a reasonable time (e.g., Four Weeks ). In this study, we used anesthesia billing data from two facilities to study statistical challenges in converting from a Fixed Hours to an Any Workday or Four Weeks patient scheduling system. We report relationships among the number of staffed ORs (i.e., first case of the day starts), length of the regularly scheduled OR workday, OR efficiency, OR staffing cost, and changes in services OR allocations. These relationships determine the expected changes in each service s OR allocation, when a facility using Fixed Hours considers converting to the Any Workday or Four Weeks systems. (Anesth Analg 2002;94:1272 9) Most surgical facilities in the US perform all cases scheduled by its surgeons, provided a case can be done safely. This reflects the desire to retain and grow surgeons practices, to enhance market share and reputation (1), and to fulfill community-service missions (2). Also, almost all surgeons contribute to hospital profitability (3,4). Many facilities allocate operating room (OR) time based on historical utilization of OR time. This can be by surgeon, by surgical service, by surgical group, or by department. Either way, this system assumes that there are fixed regularly scheduled OR hours. For example, there may be 8hof block time from 7 am to 3 pm. We will refer to this as Fixed Hours. One service s historical OR utilization may be less than another s. In a Fixed Hours system, some of its OR time may be given away. Yet, using utilization to allocate OR time is based on the premise that cases will be rejected. For example, This work was funded by University of Iowa Health Care. Franklin Dexter has received funding from Medical Data Applications, Ltd., which developed and markets CalculatOR software to perform some of the analyses described in this paper. Accepted for publication January 9, Address correspondence to Franklin Dexter, Division of Management Consulting, Department of Anesthesia, University of Iowa, Iowa City, IA Address to Franklin-Dexter@UIowa.edu. cases have been scheduled from 7 am to 1 pm in8hof block time. The service has a new 4-h case to book. The case would not be scheduled into the block time, even if there is no other open time soon afterwards. With Fixed Hours, the facility has not committed to making OR time available to care for all its surgeons patients. Statistical methods appropriate for the Fixed Hours system generally assure that the facility will turn away cases (3,4). Not surprisingly, many OR managers in the US look for alternatives to allocating OR time based on Fixed Hours. Two systems are known that can allocate OR time, schedule cases into that time, and determine anesthesia and nurse staffing for all the surgeons cases while maximizing the efficiency of use of OR time (5). One system gives control of scheduling to the surgeon. Patients and surgeons can schedule a case on any workday (6 8). We refer to this as Any Workday. The second system allocates OR time such that all the surgeons patients receive care within a predetermined reasonable time (5). The specific value studied has been 4 wk to achieve an average wait of 2 wk (5,9 12). We refer to this as Four Weeks. A facility that allocates OR time based on historical OR utilization (i.e., one that uses Fixed Hours) may 2002 by the International Anesthesia Research Society 1272 Anesth Analg 2002;94: /02

2 ANESTH ANALG ECONOMICS AND HEALTH SYSTEMS RESEARCH DEXTER AND MACARIO ;94: ALLOCATING OPERATING ROOM TIME contemplate changing to a different scheduling system to maximize OR efficiency. Stakeholders may want to know ahead of time how OR allocations will change. Statistical methods suitable for Any Workday and/or Four Weeks systems thus need to analyze data from a Fixed Hours system. In this study, we use anesthesia billing data from a hospital surgical suite and a separate ambulatory surgery center (ASC). We study the statistical challenges of converting to an Any Workday or Four Weeks system. Methods Description of the Two Facilities and the Data The ASC has 6 ORs that are scheduled as 8-h blocks. Eleven services are allocated some dedicated blocks in the ASC during each 4-wk cycle (Table 1). The hospital surgical suite has 22 ORs (Table 1). One OR is a combined urgent, emergency, and trauma OR. We excluded that OR from the analysis. In addition, there are 2 ORs that are staffed for 10 h daily. These are not allocated to surgical services. They are kept in reserve as flexible ORs. The flexible ORs are used for urgent cases and for services without an assigned block for the day. When performing the analyses, we required that at least one flexible OR be available for at least 10 h every day for urgent and OTHER cases. All hours of urgent, emergent, and trauma cases done during regularly scheduled hours were counted as OR time toward the flexible OR. It did not matter in which OR the cases were done. By doing that, we deliberately overestimated the actual OR time needed for the flexible OR(s). We assured that sufficient OR time was planned for urgent cases. Anesthesia start and stop times were obtained for all cases with an anesthesia provider at the facilities for the 48-wk period (9,13) between March 30, 2000 and February 28, Definitions OR workload is defined as a surgical service s total hours of elective cases including turnover times (9). Under-utilized hours of OR time is defined as regularly scheduled OR hours without cases (7). Over-utilized hours of OR time is defined as hours of elective cases in an OR after the end of regularly scheduled OR day (7). For example, staff are scheduled to work from 7 am to 3 pm. AnOR s last case of the day ends at 1 pm. OR workload is 6 h. There are 2 under-utilized hours. If, instead, the last case ends at 6 pm, then there are 3 over-utilized hours. Efficiency of use of OR time ( OR efficiency )is defined functionally. The optimal number of regularly scheduled hours to maximize the efficiency of use of OR time minimizes (number of underutilized hours) (number of over-utilized hours) (relative cost of over-utilized to under-utilized hours) (7,8,10). We used a relative cost of This means that it is 75% more costly for staff to work after regularly scheduled hours versus during regularly scheduled hours (6,8). In the Any Workday or Four Week systems, a case can be scheduled in over-utilized hours. This applies when the service s regularly scheduled hours on that workday or within 4 wk are full, respectively (5,7,8,10). Description of Each Surgical Service s Workload at Each Facility We divided each service s OR workload during the 48 wk (9,13) by the total OR workload at the facility during the 48 wk. Table 1 gives the proportion of OR time used by each service. Table 1 also gives the coefficients of variation of OR workloads among 4-wk periods. We calculated the differences between each service s current OR allocation and what would be allocated if the current total staffed OR time was distributed proportionately to each service s workload. Allocating OR time this way would be difficult to put into effect. This system is not economically rational. Still, it helps in interpreting results for the Any Workday and Four Week systems. Those systems are based on each service s OR workload. Below we describe how we studied the Any Workday and Four Week systems (5 12). Readers should refer to the references for a description of these statistical methods and their rationales. Method We Used to Apply Any Workday System Any Workday staffing solutions were obtained using CalculatOR (MDA Ltd., Jenkintown, PA). CalculatOR uses historical data to find the staffing solution that maximizes OR efficiency. A staffing solution includes how many staffed ORs should be available each day for each service and how long the staff should be scheduled to work for each OR (e.g., 8 or 10 h) (7,8). Varying the planned duration of the workday affects OR efficiency and staffing costs. The hours of cases that services can schedule are not affected. OR workload is the same. For some services, on some days of the week, the optimal staffing solution to maximize OR efficiency was to provide 0 h to the service. Expected OR efficiency was higher if the service did all its cases in

3 1274 ECONOMICS AND HEALTH SYSTEMS RESEARCH DEXTER AND MACARIO ANESTH ANALG ALLOCATING OPERATING ROOM TIME 2002;94: Table 1. Characteristics of the Surgical Suites and Services Service Workload (%) a Ambulatory Surgery Center Coefficient of variation in workload b Current mean allocated hours per day Workload (%) a Hospital Surgical Suite Coefficient of variation in workload Current mean allocated hours per day A 8.3% 0.4% 19% 4.4 h 4.8% 0.1% 13% 8.2 h B 29.5% 0.6% 10% 15.2 h 4.3% 0.3% 25% 7.2 h C 5.7% 0.3% 25% 2.4 h 2.0% 0.1% 25% 2.9 h D 12.4% 0.4% 14% 6.0 h 12.2% 0.2% 8% 20.4 h E 29.3% 0.7% 12% 14.0 h 18.0% 0.3% 5% 28.8 h F 0% 4.6% 0.2% 15% 7.8 h G 0% 26.1% 0.3% 9% 42.4 h H 0% 12.5% 0.4% 12% 19.2 h I 0% 11.4% 0.3% 9% 18.7 h OTHER c 14.8% 0.6% 15% 6.0 h Not applicable Flexible Not applicable 4.0% 0.3% 25% 20.0 h Overall 100% 9% 48.0 h 100% 6% h a Percentage of total hours of operating room (OR) cases including turnover times during each of 12 4-wk periods, reported as mean sem. b The coefficient of variation is the ratio of the sd of total hours of OR cases including turnover times among each of the 12 4-wk periods to the corresponding mean. Note. that this is not the coefficient of variation of the percentages from the preceding column, but of the OR workload. c Other services are services F-K. For all patients receiving care on whatever workday the patient and surgeon choose, Other also includes the services without a dedicated operating room for the day. over-utilized OR time rather than if the service was allocated 8 h of OR time.all such services were combined into the OTHER service for that day of the week (8). The OTHER service was allocated at least 8 h of OR time every workday. That way, every surgeon has OR time every workday. Method We Used to Apply Four Weeks System Block time was allocated to each service over a 4-wk period. There was a 2-wk scheduling cycle (5,9,10). The OR time allocated to each service was the amount expected to maximize OR efficiency (9,10). The numbers were then converted to specific numbers of ORs per 2-wk scheduling cycle (10). For example, h over a 4-wk period would be converted to 10 8-h blocks every 2 wk. For the ASC, we did this by dividing the optimal number of hours of OR time per 4 wk period by 2. We then divided by 8 h per day, and rounded up to the nearest whole number (10). The hospital surgical suite had both 8-h and 10-h blocks. We used integer programming (Appendix). This is the mathematical generalization of rounding up. Analyses were performed in Excel 2000 (Microsoft, Redmond, WA) with the Solver mathematical programming tool (Frontline Systems, Incline Village, NV). Impact of Any Workday and Four Weeks on Expected Staffing Costs Staffing costs for an OR were calculated as (the regularly scheduled OR hours) (the average cost per hour of staffing an OR during regularly scheduled hours) (the over-utilized hours) (the average cost per over-utilized hour). For the latter, we again used 1.75 times the average cost per hour of staffing an OR during regularly scheduled hours (8). Assessing Differences Between Current OR Allocations and OR Allocations Obtained With the Any Workday and Four Weeks Systems Results are reported for each service as the percentage change from its current allocated OR hours per week. For example, suppose that a service was currently allocated 20 h per week of OR time. Suppose also that the service would be allocated 22 h of OR time using the Four Weeks system. Then, we would report 10%. We also did these calculations for total OR allocations. The Fixed Hours system distributes the current total staffed OR time among services. Yet, Any Workday and Four Weeks compute suitable staffed OR time de novo from historical OR workload. So, total OR allocations differed among these systems. We present the results as a list of implications accompanied by corresponding quantitative results obtained by performing the analyses. Results Result #1. The number of surgical services may exceed the daily number of staffed ORs. Then, with Any Workday not every service can have its own first case of the day start. In the Any Workday scheduling system, services with the largest workloads will get the first case of the day starts.

4 ANESTH ANALG ECONOMICS AND HEALTH SYSTEMS RESEARCH DEXTER AND MACARIO ;94: ALLOCATING OPERATING ROOM TIME For example, the ASC has six ORs. Eleven services (A-K) are allocated blocks on a 4-wk cycle. For all weekdays, five services (A-E) have higher average OR workloads than the other six services. Any Workday will give the five services and the OTHER service their own OR. The remaining six services (F-K) will lose their dedicated OR time. This is not efficient. Two (F-G) of these six services have relatively high OR workloads for some weekdays. The most efficient use of OR time would be to allocate these services their own OR on those weekdays. But there are no more ORs to be allocated. With Any Workday, these six services schedule cases, on any workday, into the OTHER OR. Result #2. For both the Any Workday and Four Weeks systems, the number of available ORs may be too few to maximize the efficiency of use of OR time. Then, the length of the regularly scheduled OR day can be extended. For example, we consider Wednesdays at the 6 OR ASC. Any Workday pooled the six lower volume services into OTHER time. Still, 7 ORs were needed to maximize OR efficiency. Using 8 and 10 h shifts, 6 ORs would be needed on Tuesdays, Wednesdays, and Fridays. Five ORs would be needed on Mondays and Thursdays. Result #3. The relative cost of over-utilized to under-utilized OR hours can be increased to achieve a desired number of staffed ORs each workday. This applies to both the Any Workday and Four Weeks systems. When changing OR staffing, some managers choose to not reduce the number of staffed ORs (i.e., first case of the day starts), even if doing so would increase OR efficiency. Often managers make such decisions to avoid organizational (i.e., political) battles. The number of ORs attained by the Any Workday and Four Week systems can be increased. The relative cost of over-utilized to under-utilized OR hours is set at a value higher than is appropriate for the facility (8). This is a statistical method to get a politically acceptable staffing plan. In the Four Week system, the relative cost used would be the smallest value providing the desired number of blocks of OR time for each 2-wk scheduling cycle. By definition, services will get more block time than they can use efficiently. In the Any Workday system, the relative cost used for each day of the week would be the smallest value providing the desired number of ORs for that weekday. Services will get more ORs than they can use efficiently. For example, we applied Any Workday to the ASC. Six ORs were obtained on Mondays and Thursdays by using relative costs of over-utilized Table 2. Ambulatory Surgery Suite. Differences Between the Current Average Weekly Allocated Operating Room (OR) Hours and the OR Hours to be Allocated Using Other Scheduling Systems Service Proportional to workload All patients on Any Workday a All patients in Four Weeks a A 10% b 9% 9% B 7% 8% 16% C 15% 33% 33% D 1% 7% 7% E 0% 0% 3% OTHER c 19% 47% 20% Overall 0% 7% 0% a The relative costs of over-utilized to under-utilized OR hours have been increased to provide for more ORs than needed to maximize the efficiency of use of OR time. b This is an example of what the column means. If the same total hours of allocated OR time as used currently was allocated exactly proportional to each service s OR workload, then service A would receive 10% fewer regularly scheduled OR hours than it receives currently. c Other services are F-K. For surgical suites using on Any Workday, Other also includes the services without a dedicated OR for the weekday. to under-utilized hours of 2.15 and 4.00, respectively (Table 2). The use of 6 ORs on Thursdays was providing a valuation of working late as being more than 400% more costly than finishing early. CalculatOR uses historical OR workload data to make Any Workday allocations. It can plan an extra OR for a service for which most of the workload is done by one or two busy surgeons on that weekday. They may be unable to fill the extra allocated OR time. So, the number of surgeons using allocated OR time should be checked. For example, on Thursdays, the increase in the relative cost from 1.75 to 4.00 increased the OTHER service s allocation from 1 to 2 ORs. The OTHER service included dozens of surgeons. This would be acceptable. Result #4. Increasing the relative cost of overutilized to under-utilized OR hours results in a smaller percentage increase in OR staffing cost than in regularly scheduled hours. This applies to both the Any Workday and Four Weeks systems. Opening more ORs than are needed to maximize OR efficiency does not change OR workload. Thus, the increase in regularly scheduled OR hours increases under-utilized OR time and reduces over-utilized OR time. The cost per hour of over-utilized OR time exceeds that of under-utilized OR time (5,7,8). Thus, the percentage reduction in OR efficiency is less than the percentage increase in regularly scheduled OR hours. Results are the same for staffing cost. For example, we applied Any Workday to the ASC. No change in regularly scheduled OR hours

5 1276 ECONOMICS AND HEALTH SYSTEMS RESEARCH DEXTER AND MACARIO ANESTH ANALG ALLOCATING OPERATING ROOM TIME 2002;94: Table 3. Ambulatory Surgery Suite. Expected Maximum Increase in Staffing Costs if a Change Was Made from the Current Method of Allocating Operating Room (OR) Time to Caring for all of the Patients on Whatever Workday the Surgeon and Patient Choose Increase in regularly scheduled OR hours to maximize efficiency Method of choosing number of ORs of use of OR time Fixed at six ORs 7% 6.3% 0.6% 0% 4.3% 0.6% Using the relative cost of over-utilized to under-utilized hours, that provided for 5 OR on Mondays and Thursdays Expected maximum increase in staffing costs a a The analysis uses historical data and as such neglects the reduction in costs achievable by moving cases on the day of surgery. The values are mean sem. was needed if the relative cost of over-utilized to under-utilized OR time of 1.75 was applied (Table 3). Yet, regularly scheduled OR hours would be increased by 7% to use the higher relative cost needed to run 6 ORs every workday (Tables 2 and 3). The estimated increases in staffing costs of switching from Fixed Hours to Any Workday were 6% and 4%, respectively (Table 3). The decision to not reduce the number of ORs led to an increase in regularly scheduled hours of 7%, but a smaller 2% predicted increase in staffing costs. Result #5. Increasing the relative cost of overutilized to under-utilized OR hours to achieve a desired number of staffed ORs can produce relatively large effects on OR allocations of rounding forecasted hours to the nearest number of staffed ORs. This applies to both the Any Workday and Four Weeks systems. An increase in the relative cost of over-utilized to under-utilized hours to obtain a desired number of ORs will reduce overall OR efficiency. Some services will be allocated more ORs than they need to maximize OR efficiency. Services with relatively low OR workloads can have large percentage increases in their allocations. This can be misleading to stakeholders reviewing changes in OR allocations. For example, we applied the Four Weeks system to the ASC. With a relative cost of overutilized to under-utilized hours of 1.75, an average of 5.2 ORs would be needed each workday. The lowest relative cost providing for 6 ORs per workday was 20 (Table 2). Rounding (see Methods) affected allocations to service A. If its allocation was proportional to its average workload, its allocated hours would be cut by 10% (Table 2). Four Weeks increased its allocation by 9%. Specifically, forecasted OR hours to maximize OR efficiency were 40.1 h every 2 wk. So, 5 blocks every 2 wk or 40.0 h were too few. Instead, 6 blocks or 48.0 h were allocated. Result #6. Both Any Workday and Four Weeks allocate disproportionately more OR time than Fixed Hours to surgical services with relatively large variations in OR workload. Random variation in a service s OR workload can result from the surgeons choices of surgical days or random variation in the numbers of patients asking to be scheduled for surgery, among other factors (5,10). Fixed Hours penalizes services with relatively large variations in their OR workloads. The services get smaller allocations, because their utilizations are relatively low (14). Any Workday and Four Weeks provide those services with relatively large variations in OR workload with disproportionately more OR time than does Fixed Hours. All surgeons patients get care, regardless of whether a patient s services OR workload has large variations. For example, we applied Four Weeks to the ASC. Service B s allocations were less and service C s allocations were more than if allocations had been proportional to their average OR workloads (Table 2). Service B had the smallest coefficient of variation in OR workload among 4-wk periods among the services. Service C had the largest (Table 1). Service C was allocated disproportionately more OR time than B because both services patients will get care within an average of 2 wk (5). Service C will have a relatively low utilization of its OR time. Yet, it is less expensive to allocate more OR time to service C than to have it work in over-utilized OR time (7,10). For example, we applied Four Weeks to the hospital surgical suite. Services B and C had the largest variations in OR workload (Table 1). Therefore, they had the largest differences in OR allocations between Four Weeks versus if OR allocations were made proportional to each service s average OR workload (Table 4). Result #7. In the Any Workday system, on some weekdays a surgical service may do some but not many hours of cases. Then, its total OR allocation will be less than the hours of OR time into which it can book its cases. For example, we applied Any Workday to the hospital surgical suite. Service C got a dedicated OR two days a week. The service schedules its

6 ANESTH ANALG ECONOMICS AND HEALTH SYSTEMS RESEARCH DEXTER AND MACARIO ;94: ALLOCATING OPERATING ROOM TIME Table 4. Hospital Surgical Suite. Differences From Current Average Weekly Allocated Operating Room (OR) Hours Obtained Using Other Systems for OR Time Allocation Service Proportional to workload All patients on Any Workday a All patients in Four Weeks A 0% 27% b 7% B 3% 31% 14% C 17% 10% 31% D 3% 21% 7% E 8% 19% 10% F 2% 11% 8% G 7% 20% 10% H 12% 31% 18% I 5% 25% 9% Flexible c 50% 37% 50% Overall 0% 16% 4% a Both Four Weeks and Any Workday allocations were planned with 8- and 10-h ORs and the relative cost of over-utilized to under-utilized hours of b This is an example of what the column means. If Any Workday was used, service A would be allocated 27% more regularly scheduled hours of OR time than it receives currently. c All OTHER service cases are performed in the Flexible room(s). cases into the Flexible OR time on the other three weekdays. Its OR allocation will be less than the hours of OR time into which it books its cases (Table 4). Result #8. For services allocated ORs every workday by the Any Workday system, differences between calculated regularly scheduled OR hours and current practice assess the efficiency of current staffing. The Any Workday system may compute OR allocations that are higher than those in use at a facility. Suppose that a suitable value is used for the relative cost of over-utilized to under-utilized OR hours (e.g., 1.75) (6,8). Then, regardless of what scheduling system the facility may be using, increasing allocated OR hours will increase OR efficiency and reduce staffing costs. For example, Any Workday allocated at least one OR every workday to five services at the hospital surgical suite: D-E and G-I. Because each of the entries in Table 4 for these services are larger than zero, to increase the efficiency of use of OR time, regularly scheduled OR hours should be increased for these services (Table 4). Result #9. The Four Weeks system provides a rationale for budgeting regularly scheduled hours of OR time at hospitals with Fixed Hours. Patients in the State of Iowa consider 2 wk to be a reasonable maximum wait for surgery (5). A 2-wk average wait corresponds to a 4-wk maximum wait (5). The Four Week system with a suitable relative cost of over-utilized to underutilized OR hours (e.g., 1.75) (6,8) gives the most efficient total hours of OR time to be staffed for the surgeons to be able to achieve this average patient waiting time of 2 wk (5,10). For example, we consider the hospital surgical suite. Suppose that the surgeons were to schedule their cases in the manner that provides for the maximum possible utilization of their OR time (5,10). Then, they still could not satisfy the waiting time preferences of patients in Iowa and efficiently use their OR time unless total regularly scheduled OR hours were increased by 4% (Table 4). Yet, even if 4% more OR time was allocated, this would probably be unsatisfactory because allocations to individual services were markedly inefficient (Table 4). Discussion Implications The first seven implications show the complex relationships among the number of surgical services, number of staffed ORs (i.e., first case of the day starts), length of the regularly scheduled OR workday, efficiency of use of OR time, OR staffing cost, and changes in services regularly scheduled OR hours. These seven implications are practical issues. Stakeholders will discover them when the expected changes in OR allocations are reviewed by them. Analysts doing Any Workday and Four Week analyses from Fixed Hours data need to think about these statistical pitfalls. They must also describe them effectively to stakeholders. Otherwise, resulting OR allocations will likely be misinterpreted. The eighth and ninth implications show how differences in OR allocations between current staffing and that recommended by the Any Workday or Four Weeks systems can provide insight into the efficiency of current staffing. They apply even if a new scheduling system is not put into effect. Caring for All the Surgeons Patients Many hospitals outside of the US have to use a Fixed Hours system. This includes, for example, hospitals with fixed annual budgets and little or no ability to generate incremental revenue per patient. Hospital administrators, nurses, and anesthesiologists may think that there are Fixed Hours, while other stakeholders such as surgeons do not. Internal conflict, frustration, and claimed lack of communication may then arise. The Fixed Hours system generally does not apply to patients who are inpatients preoperatively, regardless of how the hospital is paid (3,4). The Any Workday system is rational for inpatients at all hospitals. Once the patient has been admitted, regardless of hospital resources, a commitment has been made to look after

7 1278 ECONOMICS AND HEALTH SYSTEMS RESEARCH DEXTER AND MACARIO ANESTH ANALG ALLOCATING OPERATING ROOM TIME 2002;94: the patient. Fixed Hours applies to patients needing outpatient or same day admit elective surgery (3,4). Some hospitals with incremental reimbursement for each patient still cannot meet the service level of caring for all its surgeons patients (4). For example, a hospital may postpone elective cases because its intensive care unit is full. Then, practically, the hospital has Fixed Hours for these patients. Preference should be given to patients who are inpatients preoperatively (15). Patient Scheduling We focused on OR allocations. Yet, scheduling patients in a manner consistent with the statistical forecasting methods is important (5,10,12,16,17,19). The Any Workday and Four Weeks systems have corresponding patient scheduling systems. Implementing these systems without changing patient scheduling will violate statistical assumptions of the forecasting methods. OR time will be used inefficiently. The direct costs of inaccurate forecasts can reasonably exceed $1 million annually for a moderate-sized facility (5,8,14). Use of a 1-Wk Scheduling Cycle The implementation of Any Workday that we used allocates OR time using a 1-wk scheduling cycle. The surgeons clinics may use a 2-wk scheduling cycle. Then, OR time may be used more efficiently if a 2-wk scheduling cycle is used. The statistical issues that we addressed in Results #1, #2, and #5 may then be less important. There is no theoretical reason why a longer scheduling cycle than 1 wk could not be used in the Any Workday model. Even a noncyclical OR allocation plan could be used. Still, we used 1 wk for three reasons. First, the statistical method has previously been tested using 1-wk cycles (7,8). Second, we used CalculatOR for the analysis. This software uses 1-wk cycles. Third, initial OR allocations by the Any Workday model may not be as good at maximizing OR efficiency as those that could be obtained with a longer scheduling cycle. Yet, actual OR efficiency should be better than predicted. On the day of surgery, cases should be moved if possible from one OR to another to increase OR efficiency (17). Before the day of surgery, OR allocations should be updated as services book cases into their OR time. For example, one service may have filled its regularly scheduled OR hours and be scheduling its next case into overutilized hours. The most efficient use of OR time is to schedule that case instead into regularly scheduled OR hours originally planned for another service. Importantly, in the Any Workday system, OR allocations as in Tables 2 and 4 are initial allocations. They are not permanent allocations as in the Fixed Hours and Four Weeks systems. The facility s commitment to the surgeons and patients is that all cases can be scheduled any workday. This does not depend on a service having dedicated OR time on any workday. Summary The value of allocating OR time using Fixed Hours is unclear at many facilities in the USA. Instead, Any Workday or Four Weeks should be used. Rather than allocating OR time based on historical utilization, these two systems aim to maximize OR efficiency. Still, making the transition from Fixed Hours to Any Workday or Four Weeks is hard. Understanding the underlying relationships described in this paper may make it easier for stakeholders to interpret differences in OR allocations. Appendix We applied the first step of the Four Weeks system to find the optimal number of hours of OR time to be allocated to each surgical service. Then, we then determined how many 8 h and 10 h blocks to assign to the service. We refer to the forecasted hours per 4-wk period for each of N surgical services as F k, number of allocated 8 h blocks per 2-wk period as C k 8, and number of allocated 10 h blocks per 2-wk period as C k 10,k 1,..., N. We used integer programming (19) to find the values of C 8 k and C 10 k,k 1,..., N, that minimized k N k 1 (8C 8 k 10C 10 k ) (i.e., staffed hours). There were two constraints. One was that k N k 1 (C 8 k C 10 k ) equaled the total number of staffed ORs per 2-wk period. The other was that (8C 8 k 10C 10 k ) F k /2 for all k 1,...,N. References 1. Gaynor M, Anderson GF. Uncertain demand, the structure of hospital costs, and the cost of empty hospital beds. J Health Econ 1995;14: Reinhardt UE. The economics of for-profit and not-for-profit hospitals. Health Aff (Millwood) 2000;19: Macario A, Dexter F, Traub RD. Hospital profitability per hour of operating room time can vary among surgeons. Anesth Analg 2001;93: Dexter F, Blake JT, Penning DH, Lubarsky DA. Calculating a potential increase in hospital margin for elective surgery by changing operating room time allocations or increasing nursing staffing to permit completion of more cases: a case study. Anesth Analg 2002;94: Dexter F, Macario A, Traub RD, 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. Anesth Analg 1999;89: Dexter F, Traub RD. Determining staffing requirements for a second shift of anesthetists by graphical analysis of data from operating room information systems. AANA J 2000;68:31 6.

8 ANESTH ANALG ECONOMICS AND HEALTH SYSTEMS RESEARCH DEXTER AND MACARIO ;94: ALLOCATING OPERATING ROOM TIME 7. Strum DP, Vargas LG, May JH. Surgical subspecialty block utilization and capacity planning: a minimal cost analysis model. Anesthesiology 1999;90: Dexter F, Epstein RH, Marsh HM. Statistical analysis of weekday operating room anesthesia group staffing at nine independently managed surgical suites. Anesth Analg 2001;92: Dexter F, Macario A, Qian F, Traub RD. Forecasting surgical groups total hours of elective cases for allocation of block time: application of time series analysis to operating room management. Anesthesiology 1999;91: Dexter F, Macario A, O Neill L. Scheduling surgical cases into overflow block time - computer simulation of the effects of scheduling strategies on operating room labor costs. Anesth Analg 2000;90: Dexter F, Traub RD, Qian F. Comparison of statistical methods to predict the time to complete a series of surgical cases. J Clin Monit Comput 1999;15: Dexter F, Macario A, Traub RD. Which algorithm for scheduling add-on elective cases maximizes operating room utilization? Use of bin packing algorithms and fuzzy constraints in operating room management. Anesthesiology 1999;91: Epstein RH, Dexter F. Statistical power analysis to estimate how many months of data are required to identify operating room staffing solutions to reduce labor costs and increase productivity. Anesth Analg, In Press. 14. Dexter F, Macario A, Lubarsky DA, Burns DD. Statistical method to evaluate management strategies to decrease variability in operating room utilization: application of linear statistical modeling and Monte-Carlo simulation to operating room management. Anesthesiology 1999;91: Gerchak Y, Gupta D, Henig M. Reservation planning for elective surgery under uncertain demand for emergency surgery. Manag Sci 1996;42: Dexter F, Traub RD, Lebowitz P. Scheduling a delay between different surgeons cases in the same operating room on the same day using upper prediction bounds for case durations. Anesth Analg 2001;92: Dexter F. A strategy to decide whether to move the last case of the day in an operating room to another empty operating room to decrease overtime labor costs. Anesth Analg 2000;91: Dexter, F, Traub RD. How to schedule elective surgical cases into specific operating rooms to maximize the efficiency of use of operating room time. Anesth Analg 2002;94: Ragsdale CT. Spreadsheet modeling and decision analysis, a practical introduction to management science. 2nd ed. Cincinnati: South-Western College Publishing, 1998: