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ECONOMICS AND HEALTH SYSTEMS RESEARCH

Scheduling A Delay Between Different Surgeons’ Cases in the Same Operating Room on the Same Day Using Upper Prediction Bounds for Case Durations

Franklin Dexter, MD, PhD, Rodney D. Traub, PhD^*, and Philip Lebowitz, MD, MBA

Department of Anesthesia, University of Iowa, Iowa City, Iowa; College of Business Administration, North Dakota State University, Fargo, North Dakota; and Albert Einstein College of Medicine, Bronx, New York

Address correspondence to Franklin Dexter, Division of Management Consulting, Department of Anesthesia, University of Iowa, Iowa City, IA 52242. Address e-mail to franklin-dexter{at}uiowa.edu

	Abstract

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At some surgical suites, elective cases are only scheduled if they can be completed during regularly scheduled hours. At such a surgical suite, a surgeon may be scheduled to perform one or more cases in an operating room (OR), to be followed by another surgeon who will perform one or more cases. Scheduling a delay between the two surgeons’ cases will improve the likelihood that the second surgeon’s case(s) will start on time. We show that the mathematics of calculating a scheduled delay between the different surgeons’ cases in the same OR on the same day is that of calculating an upper prediction bound for the duration of the second surgeon’s case(s). We test an analytical expression for the upper prediction bound for the last one case of the day in an OR, and a Monte Carlo simulation method for the last two cases. We show that these 90% upper prediction bounds are at least as long as the actual durations for 90% ± 0.2% of single cases and 92% ± 0.6% of pairs of cases. We conclude that our methodology can be used to calculate an appropriate, and reasonably accurate, scheduled delay between two surgeons’ cases in the same OR on the same day.

Implications: We show how to use a statistical analysis of historical case duration data to calculate an appropriate and accurate scheduled delay between two surgeons’ cases in the same operating room on the same day.

	Introduction

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At some surgical suites (e.g., many free-standing surgery centers), elective cases are only scheduled if they can be completed during regularly scheduled hours (e.g., 7:30 AM to 3:30 PM) (1). At such a surgical suite, surgeon #1 may be scheduled to perform one or more cases in an operating room (OR), to be followed by surgeon #2 who is scheduled to perform the last case of the day in the OR. This scenario arises commonly when the last case of the day is an add-on elective case (2). If surgeon #1 finishes his final case later than scheduled, surgeon #2 and her patient may have to wait, decreasing both patient and surgeon satisfaction. Under these circumstances, it is desirable to identify a starting time for surgeon #2’s case which both reduces the probability that surgeon #2 and her patient will have to wait for surgeon #1 to finish his case(s) and yet allows sufficient time for surgeon #2 to compete the case before the end of regularly scheduled hours.

One way to improve the likelihood that surgeon #2’s case will start on time is to schedule a delay between the two surgeons’ cases. Such a delay would provide a scheduled start time for surgeon #2’s case which is after the expected completion time of surgeon #1’s case(s), but which would typically allow sufficient time to complete surgeon #2’s case before the end of the work day. If the surgical suite has salaried employees or hourly employees who work a prespecified number of hours a day, labor costs would be no higher by scheduling the delay provided that surgeon #2 successfully finishes her case before the end of the regularly scheduled day (3).

Suppose that we could say: "There is a 90% chance that surgeon #2’s case will take no more than 2 hours." Then, if the hours of the regularly scheduled day are 7:30 AM to 3:30 PM, a reasonable approach to scheduling a delay between the two surgeons’ cases would be to schedule the last case of the day to start at 1:30 PM, where 1:30 PM = 3:30 PM - 2 hours. Surgeon #1’s cases could be scheduled by taking the sum of the means of the durations of previous cases performed by surgeon #1 of the same scheduled procedure(s) types (4,5). The scheduled delay would then be the difference between 1:30 PM and the estimated time for surgeon #1 to finish his cases.

Importantly, the actual delay (if any) between surgeon #1 and surgeon #2 may be less than this scheduled delay (6). If on the day of surgery surgeon #1 were to finish his last case later than scheduled, then the actual delay between the two surgeons’ cases would be shorter than the scheduled delay (6). Unless surgeon #1 were to finish his cases after 1:30 PM, surgeon #2’s case starts at 1:30 PM. For example, if the sum of the average of the historical durations of surgeon #1’s cases equaled 4 hours, then the scheduled delay would be 2 hours. However, if surgeon #1 finished his last case at 1:00 PM, then surgeon #2 would still start at 1:30 PM. The actual delay would be 0.5 hours minus the turnover time.

The above approach is predicated on the ability to state accurately that "there is a 90% chance that surgeon #2’s case will take no more than 2 hours." Mathematically, this is the problem of calculating an upper prediction bound (7). An upper prediction bound for a single future observation is a value that has a specified probability of being exceeded by the next randomly selected observation from a population. For example, a 90% upper prediction bound for case duration provides a 10% chance that the next case will take longer than the upper prediction bound.

Previously, we developed and tested a statistical method to calculate 90% upper prediction bounds for case duration (7). In this paper, we address four additional issues. We repeat the analysis in (7) after limiting consideration to cases with upper prediction bounds that are sufficiently brief for the cases to be reasonably considered as last cases of the day in an OR (analysis #1). Because cases are often scheduled in 15 minute intervals (e.g., surgeon #2 would be told that her case is scheduled to start at 1:30 PM, not 1:26 PM) (7,8), we evaluate the impact on 90% upper prediction bounds’ accuracy of rounding the bounds to the nearest 15 minutes (analysis #2). Since surgeon #2 may perform more than one case in the OR after surgeon #1 has completed his cases, we test a statistical method for calculating 90% upper prediction bounds for pairs of successive cases (analysis #3). Finally, we test the impact, on 90% upper prediction bounds for pairs of cases, of rounding the bounds to the nearest 15 minutes (analysis #4).

	Methods

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In Appendix 1 we give the mathematics that we used to calculate upper prediction bounds for the last case or last two cases of the day in an OR. All analyses were performed using Office 97 Excel Visual Basic (Microsoft, Redmond, WA).

The data set that we used to test our algorithms included the durations of all cases performed between July 1, 1994 and July 1, 1997 at the University of Iowa’s tertiary surgical suite and Ambulatory Surgery Center. "Case duration" was defined to equal the difference between the time when the patient entered the OR and the time when the patient left the OR. Procedures were defined by their scheduled Current Procedural Terminology (CPT) code(s). If a procedure was scheduled with more than one CPT code, that combination of scheduled codes was considered to characterize a unique scheduled procedure. Procedures were defined using their scheduled (versus actual) procedure code(s) since, for a new case to which a prediction bound will be compared, only the scheduled procedure(s) would be known.

The cases were first sorted by scheduled procedure. Then, for each scheduled procedure, the cases were sorted by surgeon. Finally, for each combination of scheduled procedure and surgeon, the cases were sorted by date of the case. For each case, we calculated the mean and standard deviation of the natural logarithms of the durations of previous cases of the same scheduled procedure(s) and surgeon (see Appendix 1). If for a given case there was no or just one previous case of the same scheduled procedure(s) and surgeon, the case was not used in the next step of the analysis.

To determine the accuracy of the 90% upper prediction bounds for single future cases, we calculated the prediction bound for each case from the previously performed cases using equation (1) of Appendix 1, before (analysis #1) and after (analysis #2) rounding the bound to the nearest 15 min. This method of analysis calculated the upper prediction bounds that would have been obtained provided that each new case was used to calculate future prediction bounds starting on the day after the case was performed. If the value of the bound was longer than four hours, the case was not used in the next step of the analysis. We chose four hours as representing a reasonable maximum value for the upper prediction bound beyond which surgeon #2’s case should not reasonably have been scheduled as the last case of the day in the OR. The latter criterion applies since the analysis was designed for surgical suites that only schedule elective cases if they can be completed during regularly scheduled hours.

The value of the 90% upper prediction bound for each case (analysis #1) or the rounded value of the prediction bound (analysis #2) was compared with the actual duration of the case. The proportion of the cases with durations that exceeded their prediction bounds was calculated.

We next identified all pairs of consecutive elective cases in the same OR on the same day with the turnover time not exceeding one hour and with at least two previous cases of the same scheduled procedures for each of the two cases in the series. We used these series of consecutive elective cases to test the 90% upper prediction bounds for the durations of pairs of successive cases and the turnover times between the pairs of cases. The 90% upper prediction bound (analysis #3) or the prediction bound rounded to the nearest 15 min (analysis #4) was compared with the actual time to complete the pair of cases and the turnover time between the pair of cases.

	Results

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For single cases, the 90% upper prediction bounds were at least as long as their actual duration for 90% ± 0.2% of cases (mean ± SE, n = 15,093). With the prediction bounds rounded to the nearest 15 min, the achieved rate was 90% ± 0.2% (n = 15,700).

For pairs of cases, the 90% upper prediction bounds were at least as long as the actual duration for 92% ± 0.6% of pairs of cases (n = 1,951). With rounding the prediction bounds to the nearest 15 min, the achieved rate was 91% ± 0.6% (n = 2,191).

	Discussion

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Scheduling a delay between two surgeons’ cases in the same OR on the same day will improve the likelihood that the second surgeon’s case(s) will start on time. In the Introduction, we explained that the mathematics to calculate this scheduled delay between the surgeons’ cases is that of calculating an upper prediction bound for the duration of the second surgeon’s case(s).

In the Methods, Results, and Appendix 1, we described, and then successfully tested, our methodology to calculate accurately 90% upper prediction bounds for single cases. For pairs of cases, our methodology was only slightly (i.e., 1–2%) conservative. The fact that the 90% upper prediction bounds for pairs of cases tended to overestimate (versus underestimate) how long cases will take is useful in that scheduled delays between different surgeons’ cases should not result in cases finishing later than the end of the regularly scheduled day.

Some OR managers will be able to use our method to improve patient and surgeon satisfaction, and consequently increase future revenue, without increasing staffing costs. Scheduling delays between different surgeons’ cases in an OR makes economic sense for surgical suites where OR time is planned for individual surgeons based on the contribution margin for all of the surgeon’s patients (9,10) or based on surgical groups’ (10,11) utilization of their allocated OR time. At such surgical suites, elective cases would be scheduled only if they can reasonably be expected to be completed during regularly scheduled hours (2,5). The former objective would apply, for example, to for-profit free-standing surgery centers. The latter objective would apply, for example, to hospitals with a fixed annual global payment from a governmental agency (e.g., as in a Canadian province or the United States’ Veterans Administration). In both circumstances, employees generally have a clear sense that they are there to efficiently care for the patients during prespecified, regularly scheduled working hours. We emphasize, also, that at such surgical suites our method should not be used routinely, but only to calculate the appropriate duration for a scheduled delay between two surgeons’ cases in the same OR on the same day.

We doubt that our statistical method will be useful for surgical suites at which the mission statement related to staffing and patient scheduling specifies performing all of the surgeons’ elective cases on whatever workday the surgeon and patient choose. Such surgical suites can be recognized by practices that assure "the surgeon is the customer" (e.g., performing all add-on elective cases within a day of the surgeon booking the case). We recommend that OR managers at such surgical suites not use the methods described in this paper, because we do not know of an economically rational reason for such surgical suites to have scheduled delays between different surgeons’ cases in the same OR on the same day. The mathematical rationale is given in Appendix 2.

Appendix 1
We calculated upper prediction bounds for the last case of the day in an OR. The natural logarithms of case durations follow normal distributions (8). Also, statistical methods applying this finding are accurate (7,12,13). The 90% upper prediction bound equals (7,14)

equation

where

= the mean of the natural logarithms of the N previous case durations, s = the standard deviation of the natural logarithms of the N previous case durations, and t_0.9(N-1) = the 90^th percentile of the Student t cumulative distribution function with (N-1) degrees of freedom. To use equation (1), case durations must be available from at least two previous cases of the same scheduled procedure, because N 2 is needed to calculate the standard deviation.

We calculated upper prediction bounds for the last two cases of the day in an OR differently. Equation (1) applies when the time for a single case is being estimated. The 90% upper prediction bound for the time to complete a pair of cases and the turnover time between the pair of cases was estimated by the following Monte Carlo computer simulation. We let , s₁, N₁ and , s₂, N₂ refer to the first and second cases of the pair, respectively. Durations for the first and second cases in the pair were obtained by making random draws (t₁^random and t₂^random) from Student t distributions with N₁-1 and N₂-1 degrees of freedom, respectively. These random numbers were generated by the T3T* algorithm (15). Randomly chosen values for the durations of the cases were then calculated, as in equation (1), by taking

equation

and

equation

The process was repeated many thousands of times, until the 99% two-sided confidence interval (16) for the 90^th percentile for the sum (5) of the durations of the two cases was <5 min. The mean (5) of the durations of the turnover times for the cases’ surgical suite was then added to the result.

Appendix 2
We were unable to figure out an economically rational reason for scheduling delays between different surgeons’ cases in an OR when the mission related to staffing and patient scheduling, at the surgical suite, specifies caring for all of the surgeons’ patients. This is so, because for the surgical suite to plan staffing to care for all of the surgeons’ patients, total hours of elective cases including turnover times must be forecasted (11,17,18). Scheduling delays between cases can make such forecasts inaccurate, because the delays then have to be differentiated from turnover times when future workload is forecasted (19). How to do this is currently unknown. An insufficient approach to do this is to have the OR information system track which turnovers include scheduled delays, exclude those turnovers when estimating a mean turnover time, and then substitute this mean turnover time for those turnovers that include a scheduled delay (18,19). For example, suppose that cases in two ORs finish simultaneously and that in one of the two ORs there is a scheduled delay. The OR turnover team should first clean and set up the OR without the delay and then move to the OR with the delay. Consequently, scheduling delays between cases changes turnover times between cases without scheduled delays, not just those turnover times with delays.

	Footnotes

 
Franklin Dexter is employed by the University of Iowa, in part as a consultant to anesthesia groups, companies, and hospitals.

	References

Top Abstract Introduction Methods Results Discussion References

 

1. Tessler MJ, Kleiman SJ, Huberman MMA. "Zero tolerance for overtime" increases surgical per case costs. Can J Anaesth 1997; 44: 1036–41.[Abstract/Free Full Text]
2. 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: 1491–500.[ISI][Medline]
3. Dexter F, Macario A, O’Neill L. A strategy for deciding operating room assignments for second-shift anesthetists. Anesth Analg 1999; 89: 920–4.[Abstract/Free Full Text]
4. Macario A, Dexter F. Estimating the duration of a case when the surgeon has not recently performed the procedure at the surgical suite. Anesth Analg 1999; 89: 1241–5.[Abstract/Free Full Text]
5. 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: 45–51.[ISI][Medline]
6. Dexter F. Design of appointment systems for preanesthesia evaluation clinics to minimize patient waiting times: a review of computer simulation and patient survey studies. Anesth Analg 1999; 89: 925–31.[Free Full Text]
7. Zhou J, Dexter F. Method to assist in the scheduling of add-on surgical cases - upper prediction bounds for surgical case durations based on the log normal distribution. Anesthesiology 1998; 89: 1228–32.[ISI][Medline]
8. Strum DP, May JH, Vargas LG. Modeling the uncertainty of surgical procedure times: comparison of the log-normal and normal models. Anesthesiology 2000; 92: 1160–7.[ISI][Medline]
9. Dexter F, Macario A, Cerone SM. Hospital profitability for a surgeon’s common procedures predicts the surgeon’s overall profitability for the hospital. J Clin Anesth 1998; 10: 457–63.[ISI][Medline]
10. Macario A, Dexter F. Effect of compensation and patient scheduling on operating room labor costs. AORN J 2000; 71: 860–9.[Medline]
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12. Dexter F, Traub RD. Sequencing cases in operating rooms— predicting whether one surgical case will last longer than another. Anesth Analg 2000; 90: 975–9.[Abstract/Free Full Text]
13. Dexter F, Traub RD. Statistical method for predicting when patients should be ready on the day of surgery. Anesthesiology 2000; 93: 1107–14.[ISI][Medline]
14. Dahiya RC, Guttman I. Shortest confidence and prediction intervals for the log-normal. Can J Stat 1982; 10: 277–91.
15. Fishman GS. Monte Carlo concepts, algorithms, and applications. New York: Springer-Verlag, 1999: 207–9.
16. Klugman SA, Panjer HH, Wilmot GE. Loss models: from data to decisions. New York: John Wiley & Sons, Inc., 1998; 327.
17. Strum DP, Vargas LG, May JH, Bashein G. Surgical suite utilization and capacity planning: a minimal cost analysis model. J Med Sys 1997; 21: 309–22.
18. 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: 980–6.[Abstract/Free Full Text]
19. 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: 1501–8.[ISI][Medline]

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Anesthesia & Analgesia® is published for the International Anesthesia Research Society® by Lippincott Williams & Wilkins with the assistance of Stanford University Libraries' HighWire Press®. Copyright 2006 by the International Anesthesia Research Society. Online ISSN: 1526-7598   Print ISSN: 0003-2999