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

Dexter, Franklin MD, PhD; Macario, Alex MD, MBA; Traub, Rodney D. PhD; Hopwood, Margaret PhD; Lubarsky, David A. MD

Anesthesia & Analgesia:
doi: 10.1213/00000539-199907000-00003
Economics and Health Systems Research
Abstract

Determining the appropriate amount of block time to allocate to surgeons and selecting the days on which to schedule elective cases can maximize operating room (OR) use.We used computer simulation to model OR scheduling. Inputs in the computer model included different methods to determine when a patient will have surgery (on-line bin-packing algorithms), case durations, lengths of time patients wait for surgery (2 wk is the median longest length of time that the outpatients [n = 367] surveyed considered acceptable), hours of block time each day, and number of blocks each week. For block time to be allocated to maximize OR utilization, two parameters must be specified: the method used to decide on what day a patient will have surgery and the average length of time patients wait to have surgery. OR utilization depends greatly on, and increases as, the average length of time patients wait for surgery increases. Implications: Operating room utilization can be maximized by allocating block time for the elective cases based on expected total hours of elective cases, scheduling patients into the first available date provided open block time is available within 4 wk, and otherwise scheduling patients in "overflow" time outside of the block time.

(Anesth Analg 1999;89:7-20)

Author Information

Department of Anesthesia, University of Iowa, Iowa City, Iowa.

Section Editor: Ronald D. Miller.

This work was presented in part at the 1998 annual meeting of the Association of Anesthesia Clinical Directors, Orlando, FL.

Accepted for publication March 5, 1999.

Address correspondence and reprint requests to Franklin Dexter, Department of Anesthesia, University of Iowa, Iowa City, IA 52242. Address e-mail to franklin-dexter@uiowa.edu.

Article Outline

The single largest cost to a hospital delivering surgical care is incurred in the operating room (OR) [1]. Salaries of OR staff account for most OR costs [2], particularly at hospitals with salaried nurse anesthetists and/or anesthesiologists. Consequently, in many hospitals, an OR manager or a governing body has the authority to organize care for surgical patients at the least cost. To have an important impact on costs of patient care in the OR suite, OR managers must try to maximize "labor productivity" by using the least number of staff necessary to care for the patients. At such OR suites, labor costs are fixed because staffing does not change from day to day according to the number of patients cared for. Thus, to care for patients while employing as few staff as possible, the OR manager must maximize OR utilization.

Utilization equals the time an OR is used (occupancy plus setup and cleanup) divided by the length of time an OR is available and staffed. For example, if patient care in an OR starts at 7:00 AM and finishes at 1:00 PM, and if the regularly scheduled period of elective cases extends from 7:00 AM to 3:00 PM, then there are 2 h of unused OR time. OR utilization equals 75% (6 h used/8 h staffed). In a previous study in which we used computer simulation to model how the OR suite functions, we evaluated the impact of different interventions that might be implemented to increase OR utilization [3]. Using OR suite data from the University of Iowa, we found that large increases in OR utilization are unlikely to occur even by (i) improving scheduling accuracy such that all errors in predicting durations of cases are eliminated, (ii) eliminating variability in turnover times, or (iii) eliminating day to day variation in number of hours of add-on cases. Instead, the computer analysis of OR utilization suggested that the most effective strategy to maximize OR time utilization is to select the days on which to perform elective cases so as to best match the OR caseload with the days on which full-time OR personnel are scheduled to work [4].

One common method to match elective cases and staff availability is to allocate "block time" (i.e., OR time) to surgeons. OR utilization is then maximized by filling block time with as many hours of cases as possible. The key to maximizing OR utilization is to determine (i) the appropriate amount of block time to allocate to each surgeon and (ii) how to choose which day to schedule a patient for surgery. In this study, we used computer simulation and survey data collected from patients to develop an OR scheduling strategy designed to maximize OR utilization.

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Methods

Overview of Computer Simulation to Analyze OR Management Problems

Simulation is a tool in which a mathematical model is built to act like (simulate) a system of interest (the OR) in certain important respects (case scheduling). Simulation is popular in the engineering and management sciences for analyzing problems in which there is uncertainty (e.g., lengths of time patients wait to have surgery). Simulation can reproduce the amount of variability that occurs in the system being simulated (e.g., number of hours of cases scheduled). Simulation is useful for exploring sensitivity effects (i.e., which parameters have the biggest effect on results).

To perform simulation, the behavior of several parameters (e.g., case duration) was represented by probability distributions. The simulation model used random numbers drawn from these probability distributions to generate uncertain events. This was done repeatedly to represent the scheduling of many patients into OR block time.

With the aid of computer programming, we constructed simulated OR suites and scheduling systems. Using these computer-based, hypothetical OR suites, we tested different OR scheduling strategies to develop an OR scheduling strategy aimed at maximizing OR utilization. Computer simulation was particularly useful for this study because some proposed OR scheduling strategies resulted in poor OR utilizations, posing an economic danger to a real OR suite had we tested these algorithms in clinical practice. Simulation also provided a mechanism to deal with multiple uncertainties that would have required an impractical data collection period to study experimentally. By using a computer to simulate OR suites, we were able to analyze data that otherwise would have required recruiting enough OR suites to collect 100,000 yr of OR scheduling data.

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Summary of Computer Simulation Modeling Performed

We wrote the computer program so that each computer simulation allocated a certain amount of block time, scheduled elective cases into the block time, and then evaluated unused OR time in each block to compute OR utilization.

We performed each simulation with a different combination of values for the five input parameters: (i) scheduling algorithm to determine the day on which a patient will have surgery; (ii) average case duration (in hours) for all cases performed by a surgeon; (iii) average length of time (in days) patients wait for their surgery once the decision to have surgery has been made; (iv) number of hours in each block; and (v) number of blocks allocated to the surgeon each week.

The end point or output of this computer modeling was the average OR utilization, defined as the percentage of block time that was used by elective cases and turnover times. This percent utilization equals the adjusted percent service utilization as defined in the Association of Anesthesia Clinical Directors' glossary (http://aacdhq.org/glossary.htm) [5].

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Programming

In real clinical settings, OR scheduling involves allocating block time to a surgeon, then scheduling patients into the block time. This scheduling was represented in our computer simulations (Appendix 1) (Figure 1 and Figure 2), as follows:

1. The number of hours of block time allocated to a surgeon each week equals the product of two simulation parameters listed above: the number of hours in each block and the number of blocks each week. The computer calculated the mean number of patients who would be requesting to be scheduled for surgery each week for (i) surgeons to complete all of their elective cases in the specified number of hours of block time each week and (ii) the average patient waiting time to equal the specified parameter value.

2. Using the average number of patients per week derived from Step 1, the computer generated the time (using a random number generator) until the next patient's request to be scheduled for surgery.

3. The computer generated the case duration of the next patient using a random number generator.

4. Based on the time until the next patient's request to be scheduled for surgery (Step 2), the computer determined whether, before the next patient's request arrived, surgery was performed in the block that was the closest to the current simulated time. If so, the block was no longer available for cases to be scheduled in it.

5. The computer determined whether enough blocks had been simulated for OR utilization to be known sufficiently accurately (within 0.4%) for the simulation to be stopped. If not, the simulation proceeded to Step 6.

6. The computer scheduled the patient from Steps 2 and 3 into one of the open blocks; a turnover time was included if necessary.

7. The simulation returned to Step 2.

Computer code was written in Microsoft Excel Visual Basic (Redmond, WA).

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Assumptions of the Simulations

The computer simulation methodology made three assumptions.

1. The number of work days between patients' requests to be scheduled for surgery was assumed to follow an exponential distribution [6]. This assumption implies that patients do not delay their requests to be scheduled for surgery depending on how many other patients have been scheduled for surgery. Likewise, the time from when the preceding patient was scheduled until the current patient is scheduled is independent of the time until the next patient is scheduled.

2. Each case was assigned a random time duration generated from a log-normal statistical distribution (Appendix 1). In the simulations, mean case durations (with resulting 25th, 75th percentiles) for all cases performed by a surgeon were set to equal either 1 h (0.5, 1.3), 2 h (0.9, 2.5), or 3 h (1.4, 3.8).

3. Turnover time in the ORs ("patient out" to "patient in") equaled 0.5 h. This value equals the mean turnover time for all elective cases performed between July 1, 1994 and June 30, 1997 at the University of Iowa's OR suites. The rationale for using a constant turnover time is explained in Appendix 1.

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Algorithms Used to Decide into Which Block to Schedule Each Case

The rules by which cases were scheduled into blocks are "on-line bin-packing algorithms," a term from operations research literature. "On-line" refers to giving a patient a surgical date as soon as the patient requests to be scheduled for surgery. A patient is assigned a surgical date (e.g., by the particular OR suite's scheduling system) without consideration of requests by other subsequent patients. This practice is in contrast to the use of waiting lists (see Discussion), whereby a patient may wait weeks to months to be told the day on which they can have surgery. Patients are "packed" into "bins," where the bins are the surgeon's blocks.

Each case was scheduled into a block (i.e., assigned a surgical date) subject to rules specified by the algorithm being used in the simulation. We evaluated four standard algorithms [7].

1. Next Fit. The case is scheduled into the block that permits surgery to be performed as soon as possible. Some common commercial surgical services information systems recommend a surgical date based on this algorithm (e.g., ORSuite by IntegraSys). Next fit can perform poorly because a case can be scheduled into an empty block even though the case can be scheduled into a partially filled block. The next three algorithms, in contrast, only schedule a case into the empty block that permits surgery to be performed as soon as possible, provided no block with at least one case in it has enough open time available for the new case.

2. First Fit. The case is scheduled into the block that (i) has at least one case in it, (ii) has enough additional time available for the new case, and (iii) will permit surgery to be performed as soon as possible. First Fit can perform poorly because it does not consider how much open OR time remains in a block after a case has been scheduled. For example, First Fit would schedule a 2.0-h case into 2.5 h of open block time even if there were 2.0 h of open OR time available in another block. Because it may be difficult to find a case to fit into the resulting 0.5 h-time slot, OR (block) utilization has been reduced relative to that which can be achieved by the Best Fit algorithm.

3. Best Fit. The case is scheduled into the block that (i) has at least one case in it, (ii) has sufficient additional time available for the new case, and (iii) has the least amount of additional time available.

4. Worst Fit. The case is scheduled into the block that (i) has at least one case in it, (ii) has sufficient additional time available for the new case, and (iii) has the most amount of additional time available. Worst Fit attempts to balance usage among the different non-empty blocks.

All four of the algorithms that we considered effectively assume that if a surgeon plans to be away on a day with allocated block time, the surgeon would rescind that date before the day of surgery. Otherwise, the surgeon would be unable to schedule patients into future dates.

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Patient Survey of Acceptable Waiting Times for Surgery

From the results of the simulations, we learned that OR utilization depends greatly on (i.e., is very sensitive to) the average length of time patients have to wait for surgery once the patient requests to be scheduled for surgery. The longer patients wait to have surgery, the greater the percentage of OR block time used, because more surgical dates (blocks) can be evaluated for a good match between a case's duration and the remaining OR time in the block.

To develop an OR scheduling strategy to maximize OR utilization, we needed to use in the computer simulations lengths of time that patients consider reasonable to wait to have surgery. Thus, we undertook a survey study to determine patients' perceptions of acceptable waiting times for elective surgery.

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Description of Patient Survey Study

After approval of our human subjects committee, we surveyed patients to quantify their views of acceptable waiting times. The primary end point of our survey was the median longest amount of time that patients considered acceptable to wait for surgery and its corresponding 95% lower distribution-free confidence bound [8].

All patients reporting for outpatient surgery at the reception desk of the University of Iowa's anesthesia evaluation facility over 10 consecutive work days in January 1998 were eligible for inclusion in the survey. Parents of pediatric patients completed the questionnaire. Prison inmates, non-English-speaking patients, and developmentally delayed patients were excluded. Patients seen at this facility were going to undergo surgery at either an ambulatory surgery center or a tertiary OR suite (including cardiac surgery). Subjects were asked to "put an 'X' over the longest/most time you would have wanted to wait for surgery." A time line was given below the statement. As secondary analyses, we chose post hoc to use Spearman rank correlation co-efficients to assess the ability of demographic parameters to predict the longest acceptable waiting times.

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Description of Percent Changes in OR Utilization

To compare the impact of different OR scheduling strategies on OR utilization, we defined a change in OR utilization of 1%-3% to be small, 4%-7% to be moderate, and >8% to be large. These values were selected because, at the University of Iowa, a change in the system of scheduling patients that could close one, two, or three ORs would increase OR utilization by 3%, 7%, or 10%, respectively ([100%] x [1, 2, or 3 ORs]/[30 ORs regularly scheduled each work day]).

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Sequence of Studies

The goal of this study was to develop an OR scheduling strategy to maximize OR utilization. To develop the strategy, we addressed the following questions.

1. On the questionnaire survey, what did patients report as the longest acceptable waiting time to have surgery?

This value was then used to choose parameter values for computer modeling of OR scheduling systems.

2. Can the correct amount of block time be allocated to a surgeon based only on the surgeon's expected total number of hours of elective cases each week?

To address this question, we considered a hypothetical OR suite that uses two common scheduling goals. (i) Each surgeon is allocated the minimal block time to complete his or her elective cases in the block time. (ii) Patients are provided a surgical date as soon as they request one. A strategy that can be used to allocate block time to each surgeon is to measure how much OR time each surgeon has used in the past, then allocate to each surgeon that number of hours of OR time in the future. This strategy can satisfy the OR suite's two scheduling goals if all surgeons can achieve an OR utilization close to 100%. We used computer simulation modeling to determine whether all surgeons can achieve the same (high) OR utilization if all surgeons make every possible effort to schedule their patients in a manner to maximize OR utilization.

We predicted OR utilization for 216 combinations of parameter values ([two different number of blocks each week] x [four scheduling algorithms] x [three mean case durations] x [three mean waiting times] x [three different hours of block time]). Parameter values were:

1. One or two blocks each week. We used one or two blocks each week because we simulated allocation of block time to one surgeon at one OR suite (see Discussion, scheduling study for a group of surgeons).

2. Four scheduling algorithms (Next Fit, First Fit, Best Fit, or Worst Fit).

3. Mean case durations of 1, 2, or 3 h.

4. Differing mean lengths of time patients have to wait to have surgery (1, 2, or 3 wk).

5. Blocks of 4, 7, or 8 h. Blocks 7 h long are characteristically used at OR suites that plan on 0.5 h for setup and cleanup at the beginning and end of an 8-h work day.

3. Can an OR scheduling strategy to maximize OR utilization choose a surgeon's allocation by measuring OR utilization and adjusting block time up or down accordingly?

Another common approach to allocating block time to a surgeon is to measure a surgeon's OR utilization and to adjust the block time up or down accordingly. For example, if a surgeon's OR utilization during the preceding 4 wk was 60% (10% smaller than a 70% value chosen by an OR committee), the surgeon's block time allocation would be decreased by up to 10%. To evaluate statistical issues associated with adjusting block time depending on a surgeon's OR utilization, we considered a hypothetical surgeon. The surgeon schedules patients into their first available block (Next Fit), has patients wait an average of 2 wk to have surgery, has cases that last an average of 3 h, and has been allocated two 8-h blocks each week.

Answers to Questions 2 and 3 suggested problems with using a surgeon's historical OR utilization to make future allocations. An OR scheduling strategy that allocates the minimum amount of block time for surgeons to perform their elective cases in their block time must use two data (Figure 3): (i) the surgeon's mean number of hours of elective cases each week and (ii) the proportion of block time that will remain unused if the surgeons were allocated the minimum amount of block time for the surgeons to perform their elective cases in the block time. Therefore, we evaluated how to schedule patients into a surgeon's block time to maximize OR utilization. In particular, we evaluated how OR utilization changes based on changes in three parameters: the scheduling algorithm used to schedule patients into a surgeon's block time (Question 4), number of hours of time in each block (Question 5), and the average length of time patients have to wait to have surgery (Question 6).

4. What scheduling algorithm should be used in the proposed OR scheduling strategy to maximize OR utilization?

We used simulation to predict OR utilization for all combinations of the parameters listed under Question 2.

5. How many hours of time should there be in each block for the proposed OR scheduling strategy to maximize OR utilization?

We evaluated the impact on OR utilization of having 4, 7, or 8 h of block time each day for all combinations of the Next Fit algorithm; 1-, 2-, or 3-wk mean waiting times; 1-, 2-, or 3-h mean case durations; and one or two blocks each week. Each surgeon was allocated just enough block time to complete his or her elective cases in the block time. The programming (Step 1) increased the mean number of patients who would be requesting to be scheduled for surgery each week to maintain the specified average patient waiting time. Therefore, doubling the hours of time in each block from 4 to 8 h did not halve OR utilization.

6. Should the proposed OR scheduling strategy designed to maximize OR utilization specify the average length of time that patients wait to have elective surgery?

We considered all combinations of parameters listed under Question 5. An OR scheduling strategy designed to maximize OR utilization must specify the average length of time that patients wait to have surgery because OR utilization is very sensitive to this parameter (based on studies performed for Questions 4-6). We consequently asked Questions 7 and 8.

7. Can the proposed OR scheduling strategy adjust block time to maximize OR utilization by measuring the average length of time that patients wait to have surgery and adjusting allocation up or down accordingly?

We used computer simulations of the hypothetical surgeon described in Question 3 to address this question.

8. How can the OR scheduling strategy control the average length of time a patient waits to have surgery?

We used computer simulation of the hypothetical surgeon described in Question 3 to address this question. We added the criterion that this surgeon has been allocated "slightly less block time than needed" to complete all of the elective cases in the block time. We considered slightly less block time than needed to correspond to the block time allocation that would require the surgeon to perform an elective case outside the block time less often than once every 2 wk. (To achieve "slightly less block time than needed" in the computer simulations, for the hypothetical surgeon described in Question 3, Programming Step 1 was completed, and then the time between patients' requests to be scheduled for surgery was decreased by 10%. Increasing the time between patients' requests to be scheduled for surgery by 10% achieved "slightly more block time than needed," as used in Question 9.) A patient who cannot be scheduled to have surgery within the surgeon's block time within 4 wk would be scheduled outside the surgeon's block time (i.e., into overflow or spillover time).

9. How should the proposed OR scheduling system designed to maximize OR utilization calculate how many blocks should be allocated to each surgeon?

We next determined how the proposed scheduling strategy will calculate how many blocks to allocate to each surgeon. The number of hours of block time allocated to a surgeon can only be varied by changing the number of blocks (e.g., from one block to two blocks) each week. For example, if an OR suite allocates block time in increments of 8 h, then 8, 16, 24, etc. of block time can be allocated during each period. The exact minimum number of hours of block time required for a surgeon to complete the elective cases in the block time cannot be allocated. Either slightly more or less block time than needed is allocated. We used simulation to predict OR utilization for all combinations of parameters listed in Question 5, when surgeons are allocated "slightly more or slightly less block time than needed." For the purposes of this analysis, "slightly more block time than needed" corresponded to the block time a surgeon would need to perform one additional case every other week.

10. Are patients' preferences for which day they have surgery or limitations on availability of OR equipment or staff likely to affect our findings?

We evaluated the sensitivity of our results to a feature of the Next Fit algorithm. We used the hypothetical surgeon from Question 3 to evaluate the impact of 5% of patients not having surgery on the day that the algorithm would schedule the patient to have surgery. Instead, such patients had surgery in the following block with sufficient open time for the case.

11. Does the pattern of patients' requests to be scheduled for surgery affect our findings?

We evaluated the validity of the proposed OR scheduling strategy after changing one of the assumptions. Assumption 1 assumes that there is variability in the times between patients' requests to be scheduled for surgery. This characteristic of the exponential distribution applies to patients seen in emergency rooms. This characteristic also applies to patients who leave a surgeon's clinic and then call back with a request to be scheduled for surgery, after talking with, for example, their family, another physician, or insurance company for approval. This characteristic of the exponential distribution may not apply to surgeons who have clinic days at regular intervals and whose patients decide immediately in the surgeon's clinic to request to be scheduled for surgery. We therefore repeated all of the simulations (Questions 2-9) while using this other pattern of timing between patients' requests to be scheduled for surgery.

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Results

OR utilization can be maximized by allocating block time for elective cases based on expected total hours of elective cases, scheduling patients into the first available date provided open block time is available within 4 wk, and otherwise scheduling patients in overflow time outside of the block time. The rationale for this proposed OR scheduling strategy to maximize OR utilization is shown below.

1. On the questionnaire survey, what did patients report as the longest acceptable waiting time to have surgery?

Of the eligible patients, 92% completed the questionnaire (Figure 4). The median longest acceptable waiting time was 2 wk (Table 1, (Appendix 2 Table 2)). The value of 2 wk was then used to choose parameter values for computer modeling of OR scheduling systems.

2. Can the correct amount of block time be allocated based only on the surgeon's expected total number of hours of elective cases each week?

Depending on which of the 216 combinations was simulated, OR utilization ranged from 35% (minimum) to 97% (maximum), with a median of 80%.

This implies that the scheduling parameters (scheduling algorithm, mean case duration, average length of time patients wait to have surgery, hours in each block, and number of blocks each week) reflect variation likely to exist among surgeons. Therefore, an OR suite should expect surgeons to have differing OR block time utilizations even if the surgeons attempt to maximize OR utilization. Typically, OR suites allocate block time and schedule patients into block time separately. The simulations show that these two steps must be performed simultaneously to maximize OR utilization because maximal use will vary among surgeons (Figure 3).

3. Can an OR scheduling strategy to maximize OR utilization choose a surgeon's allocation by measuring OR utilization and adjusting block time up or down accordingly?

Simulations show that an OR manager would have needed to collect (sample) 10 yr of data for the "true" (population) average OR utilization to be known, with 95% confidence, to be within 25% of the measured OR utilization. Ten years of data were needed because the OR utilization measurements were not independent of one another (Figure 5). The Pearson correlation coefficient between successive blocks' OR utilization equaled 0.38. In other words, if utilization for a surgeon's block was high, utilization was likely to be high again in the following block. This auto-correlation was caused by random variation in the number of days between patients' requests to be scheduled for surgery. If the block's OR utilization measurements had not been auto-correlated, then only 45 wk of data would have been needed for the confidence interval to be within 25% of the mean.

This implies that, because successive measurements of OR utilization are auto-correlated, OR scheduling should not rely on measuring OR utilization and then adjusting block time up or down.

4. What scheduling algorithm should be used in the proposed OR scheduling strategy to maximize OR utilization?

Next Fit produced OR utilization values as high as the other algorithms, and it is the simplest. Therefore, the proposed OR scheduling strategy uses Next Fit.

The Next Fit and First Fit algorithms achieved identical OR utilizations (i.e., the differences between the two were 0%). First Fit schedules a case into the first available partially filled block; if this cannot be done, it uses Next Fit. However, for both First Fit and Next Fit, the block that was the closest to its corresponding surgical date had the least amount of open time. The block that was the second closest to the its surgical date had the second least amount of additional time available, and so on. Therefore, the algorithms produced the same OR schedule.

Depending on the combination of parameter values, utilization with Next Fit was 3% (median) higher than Worst Fit (range 0%-8%). Utilization with Next Fit was 0% (median) higher than Best Fit (range -4% to 9%). Using multivariable linear regression and graphical methods, we could not identify a discernable effect of any one or two parameters on the differences in OR utilization. Differences between Next Fit and Best Fit in unused but usable OR time (see Appendix 1) equaled 0% (median) (range -8% to 3%). Differences in the percentage of blocks that were empty equaled 0% (median) (range -4% to 1%).

5. How many hours of time should there be in each block for the proposed OR scheduling strategy to maximize OR utilization?

An increase in the number of hours of block time each day from 4 to 7 hours caused a 10% (median) increase in utilization (range 3%-18%). Increasing the number of hours of block time each day from 7 to 8 h caused a 1% (median) increase in utilization (range -5% to 4%).

If patients had to wait an average of 2 or 3 wk for surgery, having two 4-h blocks instead of one 8-h block increased OR utilization by 1% (median) (range -2% to 3%). If the mean patient waiting time equaled 1 wk, and if the average case duration was 1 h, allocating two 4-h blocks instead of one 8-h block increased OR utilization by 2%. If the average case duration was 2 or 3 h, allocating two 4-h blocks instead of one 8-h block increased OR utilization by 6% or 15%, respectively.

This implies that an OR scheduling strategy to maximize OR utilization does not need to specify between 7- or 8-h blocks because the difference in utilization is small. For surgeons whose patients have waiting times of >or=to2 wk, OR utilization is likely to be similar whether two 4-h blocks or one 8-h block is used.

6. Should the proposed OR scheduling strategy designed to maximize OR utilization specify the average length of time that patients wait to have elective surgery?

An increase in the average length of time that patients wait to have surgery from 1 to 2 wk caused a 13% (median) increase in OR utilization (range 6%-29%). Increasing the mean patient waiting time from 2 to 3 wk caused use to increase by 5% (median) (range 0%-12%).

This implies that OR utilization depends greatly on, and increases as, the average length of time patients wait for surgery increases. An increase in waiting time permits more blocks to be evaluated for a good match between a case's duration and the open time in each block.

7. Can the proposed OR scheduling strategy adjust block time to maximize OR utilization by measuring the length of time that patients wait to have surgery and adjusting allocation up or down accordingly?

The lengths of time patients wait to have surgery could be measured and used to frequently adjust the allocation of block time (as new OR utilization data are collected). For example, if a surgeon's mean patient waiting time during the preceding 4 wk was longer than desired (e.g., it was 3.5 wk instead of 2 wk), then the surgeon's block time could be increased to "decompress" the waiting list. However, because of autocorrelation between successive waiting times (Pearson r = 0.81), simulations show that an OR manager would need to record waiting times for all of the surgeon's patients for 23 yr for the true mean waiting time to be known with 95% confidence, to within 25% of the measured mean.

8. How can the OR scheduling strategy control the average length of time a patient waits to have surgery?

We considered a hypothetical surgeon who schedules patients into the first-available block (Next Fit), has a mean case duration of 3 h, and has been allocated two 8-h blocks each week. This allocation is slightly less block time than needed to complete all of the elective cases in the block time. All patients have surgery within 4 wk. If the patient cannot be scheduled within eight blocks (4 wk x two blocks each week), the patient is scheduled outside of the surgeon's block time (i.e., into an overflow or spillover time). Under these conditions, the computer simulations predict the average length of time that patients scheduled in the surgeon's block time wait for surgery to be 2.1 wk. If patients treated outside of the surgeon's block time in overflow time have surgery within a week or two, the surgeon's patients would have an average wait <2.1 wk.

9. How should the proposed OR scheduling system designed to maximize OR utilization calculate how many blocks should be allocated to each surgeon?

Allocating slightly more block time than a surgeon needs to complete the cases decreased OR utilization by 8% (median) (range 3%-9%). Allocating slightly less block time than a surgeon would need to complete the cases increased use by 7% (median) (range 1%-9%).

This implies that allocating slightly more block time than a surgeon needs to complete the elective cases can cause large decreases in OR utilization. The OR manager may want to balance surgeons' demands for individual block time against the need to increase the utilization of block time. The OR manager may then choose to allocate to a surgeon enough block time to get most, instead of all, of the surgeon's elective cases done within block time. The proposed OR scheduling strategy achieves this goal while controlling the average length of time patients wait to have surgery by fixing the maximum patient waiting time.

10. Are patients' preferences for which day they have surgery or limitations on availability of OR equipment or staff likely to affect our findings?

The decrease in OR utilization resulting from 5% of patients not having surgery on the designated date was <1%.

11. Does the pattern of patients' requests to be scheduled for surgery affect our findings?

Our results were the same qualitatively for all simulations.

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Discussion

We used computer simulation and patient survey studies to develop an OR scheduling strategy to allocate block time and to schedule patients into the block time to maximize OR utilization. In this OR scheduling strategy, the number of blocks allocated to a surgeon for a period (e.g., 2 wk) would be calculated by taking the expected total hours of elective cases during the period (based on historical data), dividing by the number of hours of block time each day, then rounding down to the nearest whole number. Patients are scheduled into the surgeon's block time provided the patient can have surgery within 4 wk, to achieve a mean waiting time of approximately 2 wk. Otherwise, the patient is scheduled outside the surgeon's block time into overflow or spillover time. The ease of implementing this scheduling strategy should be evaluated in a variety of clinical settings.

From the computer modeling, we identified five key issues in OR scheduling of elective cases.

1. If the length of time patients have to wait to have surgery is small (e.g., <1 wk), a surgeon's block time use cannot be near 100%. It is unrealistic for OR suites to aim to have OR utilization rates consistently >90% unless the average patient waiting time is at least 2 wk. As waiting time increases, more surgical dates (blocks) can be evaluated for a good match between a case's duration and the open times in the blocks. In some communities, competition among surgeons and hospitals may not allow the average length of time that patients have to wait for surgery to be as long as 2 wk. An OR suite then cannot expect block time utilization from elective cases to exceed 90%, assuming that enough block time is allocated for a surgeon to complete all of the elective cases in the block time.

2. The maximal possible OR utilization for a particular surgeon can be much <100%, depending on his or her practice profile (e.g., the average length of time patients wait before having their surgery).

3. Objectives in scheduling an OR suite may be to allocate just enough block time for each surgeon to complete the elective cases in the block time while maximizing OR utilization. In this setting, determining an appropriate amount of block time and selecting a method to schedule cases into a surgeon's blocks must be done simultaneously. This "method" to schedule cases means specification of three parameters: an algorithm to choose on what day to schedule each case, the average length of time patients must wait before having their surgery, and the number of hours of time in each block.

4. Among patients undergoing outpatient (ambulatory or same day admit) surgery, the median longest amount of time that patients consider acceptable to wait for surgery is 2 wk.

5. The average lengths of time patients have to wait for surgery can be controlled by setting a corresponding maximum patient waiting time.

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Importance of Maximizing OR Utilization to Anesthesia Practices

Even if anesthesiologists are paid on a fee-for-service basis, maximizing OR utilization is important because it minimizes "down time." Alternatively, if anesthesiologists and/or nurse anesthetists are employed based on working a set shift (regardless of whether a case is to be performed), increasing OR utilization may decrease the staffing costs by matching workload with staffing.

To estimate the cost-savings from increased OR utilization, we estimated the labor cost of OR time to equal $190 per hour (salaries for two full-time OR nurses, one full-time anesthesiologist, and one ancillary full-time personnel [representing housekeeping, sterilization, and other staff] = 2 x [$25 per hour] x [125% to include benefits] + 1 x [$180,000 per year] x [120% to include benefits]/[2000 clinical hours a year] + 1 x [$15 per hour] x [130% to include benefits] = $190 per hour). We found that increasing the average length of time that patients wait for surgery from 1 to 2 wk (for a surgeon with a mean case duration of 3 h who has one 8-h block each week) increased utilization of a surgeon's block time by 27% (from 47% to 74%). This predicted increase in OR utilization would decrease labor costs by $154 per case ([$190 per hour] x [3-h mean case time] x 27%). To interpret a $154 saving per case, we considered that, at Duke University, the mean cost per case for all anesthesia drugs and supplies equaled $70 [9]. Clearly, changes in OR utilization are likely to have a greater impact on perioperative costs than reducing anesthesia drug and supply costs [1].

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To Maximize OR Utilization, Control of Choosing the Surgical Date Must Be Moved from the Surgeon and Patient to the OR Suite

Simulation provided a mechanism to deal with multiple uncertainties that would have required an impractical data collection period to study experimentally. By using computer simulation, we were able to manipulate the allocation of block time and the average length of time patients wait to have surgery. However, a scheduling system to maximize OR utilization must specify how the patient is scheduled into a surgeon's block time. The process of implementing a scheduling system that specifies how surgeons schedules cases into their block time raises important issues that need to be addressed. For example, balancing a surgeon's convenience versus the cost of unused OR block time may be a complex organizational problem. OR managers also need to determine whether 4 wk is an appropriate maximum patient waiting time and whether all surgeons should have the same mean waiting time for their elective cases to be completed.

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Scheduling Add-On Cases

Add-on elective cases increase OR utilization by decreasing unused OR block time. If the parameters assumed for our simulations were applied to an OR suite, we would expect actual measured OR utilization for the block time allocated for elective cases to be higher than predicted because of add-on elective cases.

The total hours of urgent cases during some period (e.g., 4 wk) will be statistically independent of the total hours of elective cases during the period. Therefore, OR time must be forecasted and allocated independently for elective and urgent cases. Urgent cases will therefore not affect the proposed OR scheduling strategy for elective cases.

The principles for allocating block time for urgent cases may be simpler than those for elective cases. To illustrate this concept, we consider a hypothetical OR suite that allocates one block during each work day for urgent cases. Because patients requiring urgent surgery have a short waiting time, we expect block time allocated for urgent cases to have a relatively low OR utilization (e.g., 40%). Consequently, if the urgent case block time has a relatively high utilization on one day, utilization would not be more likely to be high in the block the following day. Urgent case block time use would not be expected to be auto-correlated. The common practice in OR management of measuring OR utilization and adjusting block time accordingly is reasonable when OR utilization is low. The practice becomes unreasonable (Figure 5) as allocations are decreased in an effort to increase OR utilization.

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Scheduling Strategy for a Group of Surgeons, Providing for Decreased Mean Waiting Time

To maximize OR utilization, the allocation of block time and the selection of a method to schedule cases into the allocated time must be done simultaneously. The bin-packing algorithms that we considered assume that a case can be scheduled in any block with sufficient open OR time. Even an algorithm as simple as "schedule the patient into the first available date with sufficient open OR time" (Next Fit) is generally not suitable for a group of surgeons. This algorithm would be appropriate only if the surgical group does not specify at the time that the patient is scheduled for surgery which surgeon in the group will operate.

Some patients may forego being scheduled for a specific surgeon to have an average waiting time <2 wk. For illustration, we considered all patients to undergo a specified procedure, such as breast biopsy or coronary artery bypass grafting. Each patient would be scheduled to have surgery on the first available date within the block time allocated to the group of surgeons for all patients undergoing the procedure. The patient's surgeon would be the surgeon in the group who was previously scheduled to care for patients in the block. The total hours of elective cases performed by the surgical group each week on the patients undergoing the procedure may be sufficient to fill four blocks a week. The surgical group could then use a maximum patient waiting time of 2 wk and achieve an OR utilization as high as a single surgeon filling two blocks a week would achieve with a maximum waiting time of 4 wk. The surgical group may have a sufficient workload to fill five blocks a week with patients undergoing the procedure. The group could then use a maximum patient waiting time of 1 wk to achieve OR utilization higher than a single surgeon filling one block a week would achieve with a maximum waiting time of 4 wk.

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Length of Time Patients Wait to Have Surgery: An International Perspective

Half the patients we surveyed considered a wait of 2 wk to be the longest time that they considered acceptable to wait for elective surgery. Although we did not ask our study patients what type of surgery they were undergoing, outpatient surgery at the University of Iowa includes many patients undergoing joint replacement. The longest acceptable waiting time of 2 wk measured in our study was shorter than the acceptable value of 13 wk elicited from patients who underwent knee replacement in Canada [10]. This difference may reflect differing patient expectations for medical service in Iowa versus Canada.

The patients we surveyed had already been scheduled for surgery and, as such, had already committed themselves to proceed with surgery. Our results probably apply to the true wishes of the Iowa population when faced with the prospect of elective surgery, because 99.4% of patients for whom a surgical procedure is recommended choose to undergo surgery [11].

The survey consent form stated that the patient would not realize a benefit from completing the questionnaire. We suspect that our patients would have been willing to wait longer than the average of 2 wk, provided that they expected to realize some benefit from an increase in waiting time. Patients in Iowa are very concerned about continuity of care and would be willing to wait longer for treatment, provided that they would be seen by the same permanently assigned surgeon [12]. Patients with breast cancer in Ontario are willing to wait an average of 7 wk for radiation therapy, provided that they can undergo treatment close to home [13]. Of patients in Ontario, 57% are willing to wait an additional 5 mo for joint replacement surgery, provided that they will realize a 1% decrease in the risk of postoperative mortality [14].

The acceptable lengths of time patients have to wait for surgery is being addressed in other countries. For example, Sweden's Patient Guarantee specifies that patients should get an appointment with a specialist within 4 wk and undergo surgery within 3 mo after an opinion has been given by the surgeon [15]. The United Kingdom's National Health Service's Patient Charter specifies that patients should undergo surgery within 18 mo [17,19]. The Danish Medical Association's resolution on a set of minimal rights for patients states that patients should see a surgeon (examination in the secondary health service) within 4 wk and undergo surgery (treatment) within 2 mo [20].

The OR manager must balance patients' desires not to wait for surgery versus the fact that OR utilization will increase as waiting time increases. Our simulations predict that an increase in the mean time patients wait to have surgery from 1 to 2 wk causes a median 13% increase in OR utilization, whereas an increase from 2 to 3 wk causes a median 5% increase. Therefore, we believe that a mean waiting time of 2 wk is a reasonable goal. This recommendation may be a longer patient wait than is typical in the United States [21,22]. Two articles have addressed whether the United States should consider increasing waiting periods to decrease the cost of healthcare [22,23]. Our simulations suggest that this strategy is reasonable for perioperative care. Bell et al. [22] surveyed hospitals in the United States and Canada and found a strong correlation (r = 0.44) between median charges and waiting times for total knee replacements. This result suggests that increased waiting times are associated with lower perioperative costs.

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How Does OR Use Change as the Maximal Waiting Time Increases from Four Weeks to One Year?

Given the markedly longer patient waiting times in countries other than the United States, a relevant question is: how does OR utilization change as the maximum waiting time increases from 4 wk to 1 yr? Using simulation as described in Question 8, we found that, for a maximum patient waiting time of 4 wk, the predicted OR utilization equaled 91%. An increase in the maximum patient waiting time to 8 wk caused predicted OR utilization to increase to 94%. Further, increasing the maximum patient waiting time to 1 yr increased OR utilization to 96%. We would expect countries with longer maximal waiting times to have higher OR utilization, assuming that the unique characteristics of OR suites in different countries could be controlled.

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Use of Waiting Lists

Surgeons in some countries schedule elective cases into their block time by using waiting lists [24,25]. Waiting lists differ from waiting times in that patients' requests for surgery are considered simultaneously. Patients are then given the days that they will have surgery. By reviewing all pending cases before scheduling a new case, waiting lists allow for better placement of cases into blocks to match scheduled OR time. However, to achieve the benefit of increased OR utilization, patients must agree to the date of their surgery when it is presented to them. In contrast, the on-line bin-packing algorithms, which we considered in this study, permit patients to be told on which day they will have surgery as soon as the patient requests to be scheduled for surgery.

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Coordinating a Surgeon's Clinic and OR Days

Patients' requests to be scheduled for surgery may be received at regular intervals (e.g., every Monday) corresponding to a surgeon's clinic days. A surgeon's patients may agree to be scheduled for surgery during the clinic visit. It can then be important to coordinate the day(s) of the week of the surgeon's clinic with the day(s) of the week of the surgeon's OR block time. Coordination of the days of the week can affect the average length of time that patients wait to have surgery, OR block time utilization, and the percentage of the surgeon's total hours of elective cases that the surgeon can complete within the block time.

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Conclusions

Computer simulations show that, to maximize OR utilization, control of the surgical date must be moved from the surgeon and patient to the OR suite. The most important parameter affecting OR utilization is the mean length of time patients have to wait before surgery. The longer patients have to wait, the less unused block time there will be. We developed an OR scheduling strategy to maximize OR utilization. Block time is allocated based on expected total hours of elective cases. Patients are scheduled into the first available date, provided that open block time is available within 4 wk. Otherwise, the patient is scheduled in overflow time outside the block time.

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Appendix 1

Numerical Methods

Work Days Between Patients' Requests to Be Scheduled for Surgery. We considered two statistical distributions for patients' requests to be scheduled for surgery. First, for most simulations, exponentially distributed random numbers were used. They were generated using the standard inverse-transform method [26]. Second, some surgeons may have practices in which most patients request to be scheduled for surgery during the clinic appointment. We therefore also allowed patients' requests to occur with a constant number of work days between requests. There was a fixed number of work days between successive pairs of requests. On each such clinic day, either one or two patients were considered to request to be scheduled for surgery. A Bernoulli distributed random number [26] with coefficient 0.5 was generated using the corresponding inverse-transform method to determine how many patients' requests would be received on each of these days [26].

Surgical Case Duration. Surgical case duration was described using a log normal distribution, with the result right truncated. We chose this distribution based on the durations of 300 consecutive surgical cases performed at the University of Iowa in June 1997. The cases represented a mixture of both ambulatory surgery and tertiary medicine. Case durations were log-normal (ExpertFit; Averill M. Law & Associates, Tucson, AZ, 1997), as confirmed by using the distribution function difference plot, density/histogram overplot, and Lilliefor's test (P = 0.67). To generate case durations, a normally distributed random number was first generated using the polar method [26]. We achieved large decreases in computational time by using this method rather than the Box and Muller algorithm [26]. Second, a log-normally distributed random number was generated from the normally distributed random number [26]. Differences in mean surgical case durations among OR suites at the University of Iowa are not associated with differences in the standard deviations of the logarithm of surgical case durations. Therefore, we used a standard deviation of the logarithms of case durations in hours equal to 0.725, as obtained at the University of Iowa. Third, the resulting value was truncated at the number of hours of block time allocated to the surgeon each day. For example, if the case duration equaled 13 h and 8-h blocks were used in the OR suite, we assumed that the case would be scheduled in the OR suite by considering the case duration to equal 8 h.

Turnover Times. We used a constant turnover time for two reasons: (i) The standard deviation of turnover times is negligible relative to the standard deviation of case durations. Therefore, for OR suites at which turnover time and case duration are independent, using a constant turnover time did not affect our results. (ii) For some OR suites, turnover time and case duration are correlated. For example, short cases, such as successive myringotomy tube placements, characteristically have short turnover times. Therefore, the difference between actual turnover times and our constant turnover time of 0.5 h can be considered to be incorporated into the case durations.

Details of the Computer Simulations. The first step of each simulation was to calculate the maximum mean number of patients who could be requesting to be scheduled for surgery each week for the surgeon to complete all of the elective cases within the specified number of hours of block time each week while satisfying the specified mean patient waiting time and permitting all patients to have surgery within 4 times the mean patient waiting time. For simulations using an exponential distribution of patients' requests to be scheduled for surgery, we varied the mean number of patients requesting to be scheduled for surgery each week by varying the exponential distribution's only parameter-the mean time between patient requests. For simulations using a constant time between patient requests, we varied the constant time between requests. We found, to the nearest 0.002 days, using a trial and error approach [27], the minimum time between successive patients' requests to be scheduled for surgery for which the 95% two-sided confidence interval [26] for the calculated mean patient waiting time contained the desired mean patient waiting time. This trial and error approach was inefficient computationally. However, we did not have success with the Golden Section Search [27] method because there was not always a monotonic relationship between the time between patient requests and the mean patient waiting time. For each simulated trial testing a mean time between patient requests, the same seed was used for random number generation. The confidence interval for the observed mean patient waiting time was calculated using the replication/deletion approach for means [26], with each successive simulation having a duration equal to 100 surgical blocks completed.

Once the time between successive patients' requests to be scheduled for surgery was known, the final simulation was run using Programming Steps 2-7 (Methods). The end points that were calculated included the (i) percent OR utilization, (ii) percent empty blocks, and (iii) percentage of block time that was unused while weighting by the probability that each block's unused time was of a sufficiently long duration to permit another case to have been performed in it. The latter was calculated using the distribution function for the log-normal distribution while incorporating the right truncation [26]. Confidence intervals for these three end points were calculated using the replication/deletion approach for means [26], with each successive simulation having a duration equal to 100 surgical blocks completed. Simulations were continued until the width of the 95% confidence interval for each of these three end points was <0.4%.

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