Operating room management science is integral to increasing productivity, letting the same personnel care for more patients safely.1 Recent bibliometric study2 showed that among the 11 authors who each have contributed to at least 1% of the field's publications, 9 of the 11 investigators were human anesthetists and the other 2 investigators did their mathematical studies with anesthetists as coauthors. However, there was only 1 study3 in veterinary medicine that considered predicting treatment duration.
We consider a veterinary anesthetist in Canada caring for small animal patients undergoing dental treatment. The veterinary anesthetist both directly cares for some patients and supervises veterinary technicians. Consequently, the anesthetist often cares for 2 patients at the same time. One question we consider is how best to schedule cases for each working day while assuring maximum efficiency of the use of the anesthetist.1 The veterinary anesthesia management problem differs considerably from that of human anesthetists in Canada. The human anesthetists care for 1 patient at a time, with no (literally zero) overlap between induction and emergence from general anesthesia. The veterinary clinic also has 1 anesthetist, making it important not only what choice of the available hours maximizes the efficiency of the anesthetist but also ensures a low probability of the anesthetist and technicians working late.
The principal question being studied in the current article is how to calculate 95% upper confidence limits for probabilities of workloads exceeding the allocated times, where allocated time refers to the hours into which cases would be scheduled, such as 8 hours (Table 1). Earlier studies, all for human operating rooms in the US, have shown that the daily workload has a serial correlation.10–12 That is, if one day is full, patients are scheduled for the next available date. Such correlation has made it impractical to assess statistical methods for the probabilities of exceeding allocated time. We expected that this would be different for the veterinary dentistry clinic, making the dataset suitable for the study of the statistical methods. A lack of serial correlation was expected because radiographs to forecast surgical (dental) time were performed after the patient was anesthetized, immediately before dental treatment. Therefore, not only was each patient (ie, case) scheduled for the same estimated duration, but this was proper managerially and thus generalizable.
Operating room management definitions, adapted from Dexter et al.1
| Term | Definition |
|---|---|
| Allocated time | The hours into which cases are scheduled, calculated months in advance, calculated based on minimizing the expected inefficiency of use of the anesthetist's time.4,5 Results are the same from the perspective of the veterinary clinic overall. |
| Anesthesia duration | Measured as the time from the start of sedation until tracheal extubation. |
| Inefficiency of use of anesthetists' time | (Cost per hour of underutilized time) × (hours of underutilized time) + (cost per hour of overutilized time) × (hours of overutilized time). As the day of treatment approaches, the cost of an hour of underutilized time is negligible relative to the cost of an hour of overutilized time.1,6,7 Consequently, by the day of treatment, the inefficiency of use of the anesthetist's time isminimized by minimizing the hours of overutilized time. |
| Overutilized time | Workload minus the allocated time or zero if this difference is negative.4,5 |
| Relative cost ratio | The cost of an hour of overutilized time divided by the cost of an hour of underutilized time. Ratios from 1.5 to 2.0 are considered typical for purposes of allocating time.8,9 For a relative cost ratio of 2.0, underutilized time occurs on 2/3 of days and overutilized time on 1/3 of days.4,5 |
| Staff scheduling | The process by which it is determined which anesthetist (or surgeon or technician) works which day. |
| Surgical service | Group of surgeons who share anesthesiology and nursing team and schedule cases together (eg, “veterinary dentistry”). |
| Underutilized time | Allocated time minus the workload, or zero if this value is negative. |
| Workload | Total hours of cases and turnover times on each day. |
Methods
Managerial data were obtained from the electronic health record for each patient: (1) date (eg, April 26, 2023); (2) binary variable of whether veterinary anesthetist cared for the patient (ie, “1”) or sedation only without anesthetist (ie, “0”); (3) time of premedication (eg, “08:18”); and (4) time of tracheal extubation or administration of sedation reversal (eg, “11:52”). From these data were calculated the daily anesthetist workload, meaning the total time of the veterinary anesthetist summed among patients from the time from premedication to extubation [eg, 10.17 hours on 1 date (1) by summing item (2) × (item (4) − item (3))] (Table 1). The sample sizes for all calculations in the Results are the number of days, not the number of patients.
Analyses of daily anesthetist workload
We start by considering the allocated time that results in the smallest possible inefficiency of use of the veterinary anesthetist's time (Table 1).1,4–9 We refer to the efficiency of the anesthetist's time, but the methodology would be the same when considered from the perspective of the dentistry clinic overall. The allocated time is the hours into which cases will be scheduled (Table 1).1,7 This formulation was described mathematically by Strum et al4 in 1997. Typical choice options could be 2 patient tables for 7 hours, 2 for 8 hours, 1 for 8 hours, 1 for 10 hours, or 2 for 10 hours.5,6 No assumption needs to be specified about the probability distribution for the daily workload.4 The inefficiency of the use of operating room time is minimized based on the incidence of the workload exceeding a critical percentile.4–6 Different mathematical derivations are pages long (ie, not intuitive). At a typical relative cost of overutilized to underutilized time of 1.50 for overtime (ie, 1.5 X regular hourly cost), optimal staffing would be based on the 60th percentile of workload, where 0.60 = 1/(1 + 1/1.50).4–7 We apply also the common sensitivity analysis of the large relative cost ratio of 4.00, such that for 1 day in 5 (0.80) worked hours exceed allocated time.13 In other words, we substitute 0.80 into the critical percentile and make sure that the allocated time would be similar or identical.
Our sample size was chosen based on this preliminary analysis of the daily workload. The starting date was Monday, January 16, 2023. Procedure days that included sedation or general anesthesia were Mondays, Tuesdays, and Wednesdays. With cases through Wednesday, April 26, 2023, we expected n = 45, which would give 82.6% estimated statistical power to distinguish between 0.60 and 0.80 by a 1-sided binomial test with a type I error rate of 0.05 (Stata v18.0 power oneproportion command; StataCorp LLC).
Statistical methods to evaluate assumptions of the management analyses
As explained in the Introduction, the principal question being studied was how to calculate 95% upper confidence limits for probabilities of workloads exceeding allocated times. Assuring the accuracy of coverage depends on the data lacking serial correlation among workdays.10–12 Correlation between the workload from one workday to the next was tested using the runs test (Stata runtest command). The runs test was performed with continuity correction, and values equal to the threshold were randomly assigned above and below. The comparison of workload (ie, total anesthetist time) among days of the week was performed using the Kruskal-Wallis test (Stata kwallis command). Test for the normal probability distribution was done graphically (Figure 1) and with the Shapiro-Wilk test (Stata swilk command). In addition, a normal quantile plot was produced (qnorm command).
Normal quantile plot of the total anesthesiology workload among operative workdays. These hours from Table 2 are plotted along the vertical axis. If a normal distribution, then the data would follow a straight line, as shown to hold. The corresponding Shapiro-Wilk test P = .25 also shows that the data match a normal distribution. The data have n = 44 consecutive operative workdays, those being 14 Mondays, 15 Tuesdays, and 15 Wednesdays. The grid lines are included to show that only 1 of those days had greater than 16 hours of the veterinary anesthetist's time. No day had greater than 20 total hours of the anesthetist's time. Human operating rooms frequently have normally distributed workloads like this.4 Furthermore, the optimal allocated time is insensitive to the probability distribution.5
Citation: American Journal of Veterinary Research 85, 5; 10.2460/ajvr.23.09.0196
Upper confidence limit of proportion of days with workload exceeding threshold
We seek a 95% upper confidence limit for the proportion of workdays with greater than 16 hours of anesthesia time. These are calculations related to engineering statistics as part of tolerance intervals.14 The available method has been to use the tables generated by Owen.14–16 The tables do not include many combinations of parameter values and the accuracy of their use with interpolation has been unknown, contributing to their lack of use in operating room management. In the current article, we follow the steps of Owen,15 get an answer, and then show a computational alternative with 1 long Office 365 Excel formula (Microsoft). We are not recommending to readers that Owen's method be followed in practice. Rather, we apply it here to compare it with the computational approach. We deliberately use Owen's notation, so interested readers can follow the steps in his article.15 We also include the page numbers from the Owen15 article because there are no equation numbers and the relevant equations are not listed in sequence in his article. Each of the mathematical steps is shown in the Results, because what we are examining are the steps themselves (ie, the steps are generalizable, not the specific data from 1 Canadian veterinary dental clinic).
Limitations of using the tables generated by Owen include that the nearest value needs to be used for the parameter λ below, and the tables' use generally cannot be automated. Our suggested alternative approach is to use a generalized pivotal quantity based on the properties of the normal distribution.16 This approach was described by Webb for use in testing projectiles' penetration of armor (ie, the thresholds refer to the thickness of armor).16 We present our implementation using Office 365 Excel.17
Analyses of daily workload instead of anesthetics
We assessed the independent effect of knowing the number of anesthetics in addition to the daily workload. The veterinary dentistry clinic had 2 novel features making it a model system to investigate that association. Imaging was done once the patient was anesthetized. Procedures were all performed by the same team of 1 veterinary dentist and 1 veterinary anesthetist. Consequently, all cases were scheduled for the same amount of time. Thus, workload, in hours, and anesthetics were not associated with the scheduling process. This is unusual, wherein estimated anesthesia durations differ based on the scheduled procedure or combination of procedures.3 The unique situation of the veterinary dentistry practice was an advantage because, with all anesthetics reasonably having the same estimated anesthesia duration, the independent influence of the daily count of anesthetics of the service was accentuated.
To assess the independent effect of the count of anesthetics, the dependent variable studied was the time that the last anesthetic case was finished each day. We performed 3 similar analyses. First, we created scatterplots of workload versus the last case finished time and of cases versus the last case finished time. Second, we calculated Pearson linear correlation coefficients for each scatterplot (Stata correlate command) and then compared the Pearson correlations (cortesti command). Third, we used multivariable linear regression using ordinary least squares (Stata regress command).
Results
Analyses of daily anesthetist workload
There was only 2% (1/44) of the observed days with greater than 14 and 16 hours and 0% greater than 18 or 20 hours (Figure 1). However, 52% (23/44) of the days had a workload that exceeded 10 hours. Therefore, the best allocated time choice (Table 1) based on a relative cost ratio of 1.50 would be either 2 tables for 7 hours, 2 for 8 hours, or in between, depending on the staff scheduling of the clinic team (Table 2). Because the observed 2% was less than 20%, the decision would be the same even if the large but often used relative cost ratio of 4.00 was applied, such that for 1 day in 5 worked hours exceed allocated time. Below, we start by treating the allocated time as 2 patient tables each for 8 hours. That means we consider the probability of the daily workload (ie, total hours of anesthetist time) of the veterinary dentistry service exceeding a threshold of 16 hours. We then repeat using 14 hours as sensitivity analysis.
Example of empirical method for estimating optimal allocated time using relative cost ratio of 1.50 for overutilized versus underutilized time.
| Daily workload (h), n = 44 days | Underutilized time if allocate 14 hours | Overutilized time if allocate 14 hours | Inefficiency of use of time if allocate 14 hours | Inefficiency of use of time if allocate 16 hours | Overutilized time if allocate 20 hours |
|---|---|---|---|---|---|
| 16.37 | 0.00 | 2.37* | 3.55 | 0.55 | 0 |
| 13.65 | 0.35 | 0 | 0.35 | 2.35 | 0 |
| 13.57 | 0.43 | 0 | 0.43 | 2.43 | 0 |
| 12.48 | 1.52 | 0 | 1.52 | 3.52 | 0 |
| 12.38 | 1.62 | 0 | 1.62 | 3.62 | 0 |
| 12.35 | 1.65 | 0 | 1.65 | 3.65 | 0 |
| Full table is provided* | |||||
| 5.48 | 8.52 | 0 | 8.52 | 10.52 | 0 |
| 5.17 | 8.83 | 0 | 8.83 | 10.83 | 0 |
| 4.63 | 9.37 | 0 | 9.37 | 11.37 | 0 |
| 4.13 | 9.87 | 0 | 9.87 | 11.87 | 0 |
| 3.63 | 10.37 | 0 | 10.37 | 12.37 | 0 |
| 2.70 | 11.30 | 0 | 11.30 | 13.30 | 0 |
| Mean | 4.74 | 6.63 | 10.61 | ||
| SEM | 0.43 | 0.45 | 0.46 |
Definitions are provided (Table 1). If the allocated times were for 14 hours (ie, 2 tables, each for 7 hours), then there would have been overutilized time for only 1 workday, shown in the top row. That single value of 2.37 hours = 16.37 from the first row first column, minus 14 hours. The last 2 rows show that the estimated mean and SEM from the fourth column equal 4.74 ± 0.43 hours. If the allocated times were instead 16 hours, then the inefficiency of use of anesthetist's time would be significantly greater, sample mean of 6.63 ± 0.45 hours. The mean pairwise increase in the inefficiency of use of anesthetist's time equals 1.89 ± 0.11 hours, 1-group 2-sided t test P < .0001. These calculations were performed using the relative cost of overutilized to underutilized time being 1.50, and when repeated using 2.00, the conclusions are the same.8,9 If the allocated times were 20 hours, then there would be only underutilized time, shown by the far right-hand column being 0 overutilized time for all workdays. The sample mean inefficiency of use of anesthetist's time would equal the sample mean underutilized time, 10.61 ± 0.46 hours.
See (Supplementary Table S1) for full table and significance.
Evaluation of assumptions of the subsequent management analyses
Operating room workloads (and caseloads) frequently follow normal distributions4,18 and even when not results rarely differ substantively.5 The total anesthetist time from each of the n = 44 workdays lacked serial correlation10–12 (run test P = .65 based on median and P = .69 based on mean). The anesthetist's anesthetics per workday lacked serial correlation (run test P = .45 based on median and P = .93 based on mean). The implication is that the workloads of the n = 44 operative days can be treated as having n = 44 statistically independent observations. The total anesthetist time by day did not differ significantly among the days of the week (P = .52). The anesthetics per workday also did not differ significantly among weekdays (P = .46). The probability distribution of workload was consistent with a normal distribution (Shapiro-Wilk test P = .25; Figure 1).
Upper confidence limit of proportion of days with workload exceeding threshold
(We again continue to use the Owen notation15 including the L for lower bound less than a threshold, although we consider the upper bound for the incidence greater than a threshold.) The inverse of the standard normal distribution using
= NORM.S.DIST(1.6989, TRUE)
The “TRUE” means the cumulative probability distribution. The result of 0.9553 shows that the 95% upper confidence limit for the probability of the daily workload exceeding 16 hours equals 4.47%, where 4.47% = 1 – 0.9553.
1-NORM.S.DIST(2.1621, TRUE)= 1.53%
The 95% upper confidence limit of 4.47% from tables generated by Owen15 is 192% greater than the asymptotic estimate of 1.53%. We recommend our consideration of proportional errors because those would be generalizable to other conditions and problems.
Our suggested alternative approach is to use a generalized pivotal quantity based on the properties of the normal distribution.16 First, we present our implementation,17 and then explain the terms. There was 95% assurance that at most 4.47% of workdays would have greater than 16 hours of anesthesiology time, the same as that obtained with tables generated by Owen.15 The Excel formula to calculate 4.47% is the following:
=SMALL(1-NORM.S.DIST(SQRT(
CHISQ.INV(RANDARRAY(999999), 44-1))*
(16-9.39053)/(3.05693*SQRT(44-1))
+NORM.S.INV(RANDARRAY(999999))
/ SQRT(44), TRUE),
0.95*(999999+1))
Readers can copy and paste this formula into an Excel cell to obtain the answer of 4.47%. (Copy and paste can add line breaks, which will need to be deleted from the formula bar before selecting Enter.) Execution time was 4 seconds on a Dell XPS 13 9410 laptop computer. The fact that the same answer is obtained as the tables generated by Owen15 shows the suitability of the approach.
We consider each of the terms in the preceding formula in their sequence. For each of the 999,999 simulations, a χ2 distributed random number is calculated with n – 1 degrees of freedom. That is multiplied by the Z statistic, above, and then divided by the square root n – 1. In addition, for each of the 999,999 simulations, a standard normal random variate is calculated and divided by the square root of n. These 2 random variates and their multiplicands are summed. These 2 random variates are those encompassing the Student t-distribution. The inverse of the standard normal cumulative probability distribution is taken and then subtracted from 1. After this process has been repeated 999,998 more times, the results are sorted in ascending sequence and the 95th percentile is calculated. Each of the simulations is a prediction of future workload, conditional on the observed n,
To use the Excel formula for other services or conditions, substitute a different sample size for the 44, critical threshold in hours for the 16, sample mean in hours for 9.39053, and sample SD in hours for 3.05693. Readers' implementation for their practices would simply be to change those 4 numbers in the Excel formula. For example, as explained above, we used 14 hours as sensitivity analysis. Changing the “16” in the formula to “14” gives the revised upper 95% limit of 12.76%. (We recommend that readers try doing this.) To highlight the approach's usefulness, suppose that the sample size was neglected, and the sample statistics treated as the population parameters. Then, using the formulas above, the matching Z = 1.5079 with matching probability of 6.58%:
=1-NORM.S.DIST((14-9.39053)/3.05693,
TRUE)
Analyses of daily workload instead of anesthetics
The daily anesthetist's workload was highly correlated with the time the last case finished each day (Figure 2). In addition, the count of daily anesthetics was statistically independent among days (above). More anesthetics per day were associated with the anesthetist finishing later (Figure 3). That was inevitable and shows validity. The important result is a significantly lesser correlation between anesthetics per day and the anesthetist's last case of the day's ending time in hours (Figure 3) versus between the daily anesthesia workload and the ending time (Figure 2). The Pearson correlation between the end of the workday and the number of anesthetics equaled 0.64, while between the end of the workday and hours of cases equaled 0.85. The 0.64 was significantly less than the 0.85, P < .0001. Furthermore, when performing multivariable linear regression and controlling for the anesthetist's daily workload (P < .0001), not only was there not a significant effect of cases per day (P = .066), but the estimated coefficient was negative, −0.76 extra hours finishing later per extra anesthetic that day. The implication is that the daily anesthetist's workload of the veterinary dental service (Table 2, column 1) is the important measure, the endpoint for which we consider confidence limits on the probabilities of exceeding managerially relevant thresholds in hours.
Association between the time when the last general anesthetic ended and the daily workload. There were 2 tables per day among the n = 44 consecutive operative workdays. Along the vertical axis is plotted the time of day when the second of the 2 tables ended, defined as when the patient when tracheal extubation occurred. For example, “10” means “10:00” and “20” means “20:00.” Along the horizontal axis is plotted the workload, the total hours of cases including turnover times. The workload is a quantity that is controllable and predictable when scheduling cases (ie, the independent variable). The implication for the results is that workload is highly predictive of the work hours of the anesthetist, even in the setting of 2 simultaneous cases. The predictability shown holds, in practice, because, with the anesthetist being a substantial percentage of the total labor costs, on days with only 2 cases, starting both cases in the morning reduces expected costs by reducing the probability of the last case ending late in the workday.
Citation: American Journal of Veterinary Research 85, 5; 10.2460/ajvr.23.09.0196
Association between the time when the last general anesthetic ended and the daily number of anesthetics. There were 2 tables per day among the n = 44 consecutive operative workdays. Along the vertical axis is plotted the time of day when the second of the 2 tables ended, defined as when the patient when tracheal extubation occurred. For example, “12” means 12 noon and “18” means 6:00 pm. Along the horizontal axis is plotted the day's anesthetics. The plus signs show the means of the ending time for each of the 4 categories.
Citation: American Journal of Veterinary Research 85, 5; 10.2460/ajvr.23.09.0196
Discussion
When calculating the allocated time (ie, the hours into which cases are scheduled; Table 1), an appropriate objective is to maximize the expected efficiency of use of the anesthetist's and/or facility's time (Tables 1 and 2).1 Often an appropriate secondary objective is to assure that the probability of working late does not exceed some maximum probability such as 20% (ie, 1 day in 5) or 10% (ie, 1 day every other week).13,20–23 In the current study, we took advantage of the features of a veterinary dental clinic to confirm the accuracy and usefulness of a generalized CI, implementable in Excel.16,17 We also confirmed that the workload provides the information on how late the anesthetist and team would work, not the caseload, even under the unusual and therefore scientifically sound condition that caseload had the greatest opportunity to be informative.
Principles of scheduling cases into allocated time envisioned using a single table
Scheduling cases into allocated time is understood in detail.1,7,24 We use the veterinary dental clinic data as teaching examples, simplified as if in 1 operating room. The analyses being summarized24 were performed by surgical service (Table 1), as shown in the Results.
There were n = 115 anesthesia durations, sample mean 3.59290 hours. (As a reminder, from the first paragraph of the Methods, these durations are from premedication to tracheal extubation.) The logarithms of the durations followed a normal distribution, based on the Shapiro-Wilk test (P = .56) and normal distribution plot (Figure 4).25,26 The mean and SD of the log10 durations in hours were 0.53192 and 0.14564, respectively. The mean time for 2 cases equals 7.2 hours23,27:
=AVERAGE(
10^NORM.INV(
RANDARRAY(999999,1),0.53192,0.14564)
+10^NORM.INV(
RANDARRAY(999999,1),0.53192,0.14564))
Normal quantile plot of the anesthesia durations and times from premedication through extubation. If a normal distribution, then the data would follow a straight line, as shown to hold. The corresponding Shapiro-Wilk test P = .56. As context, log10(duration) of 1.8 is 63 minutes (ie, approx 1 hour) and 2.8 is 630 minutes (ie, approx 10 hours).
Citation: American Journal of Veterinary Research 85, 5; 10.2460/ajvr.23.09.0196
The result of 7.2 hours equals the sum of the means 2 X 3.59290, because the mean of sums equal the sum of means. There is a small 1.3% frequency of working very late, longer than 12 hours:
=AVERAGE(LET(X,
10^NORM.INV(
RANDARRAY(999999,1),0.53192,0.14564)
+10^NORM.INV(
RANDARRAY(999999,1),0.53192,0.14564),
IF(X>12, 1,0)))
(As another reminder to readers, when copying and pasting into the Excel formula bar, before pressing Enter, nonprintable characters like line breaks need to be deleted.) If 3 cases were performed, the mean would equal 10.8 hours (ie, 3 × 3.59290), and the probability of working longer than 12 hours would be 26.3%27:
=AVERAGE(LET(X,
10^NORM.INV(
RANDARRAY(999999,1),0.53192,0.14564)
+10^NORM.INV(
RANDARRAY(999999,1),0.53192,0.14564)
+10^NORM.INV(
RANDARRAY(999999,1),0.53192,0.14564),
IF(X>12,1,0)))
The mean also would be longer if there were 2 cases, but with the second case not starting early if the first finishes early, 7.68 hours28:
=AVERAGE(LET(X,
10^NORM.INV(
RANDARRAY(999999,1),0.53192,0.14564),
IF(X<3.59290,3.59290,X))
+10^NORM.INV(
RANDARRAY(999999,1),0.53192,0.14564))
The mean is longer if the second case cannot start early, because durations are less than the mean for 56.4% of cases26,28:
=AVERAGE(LET(X,
10^NORM.INV(
RANDARRAY(999999,1),0.53192,0.14564),
IF(X<3.59290,1,0)))
We use these calculated results in sequence in the rest of this discussion section.
Because allocated time was calculated using actual workloads (Table 2), that decision considered inaccuracy in case duration prediction and the risks of case cancellation.24 Cases should be scheduled into the allocated time using the unbiased estimate of the contribution of each case to the workload.26,29 That is, by definition, the expected value (eg, the sample mean duration, 3.59290 hours).26,29
Although the mean time for 2 cases was 7.2 hours, the allocated time of 7 hours had a smaller mean inefficiency of use of anesthesia time than 8 hours (Table 2). This example highlights that staff should be scheduled for hours modestly longer than the allocated time to prevent working late.1 The allocated time literally refers to the hours into which cases are scheduled, not the staff.24 For example, at the clinic, the staff may be scheduled for 8 hours.
The optimal allocated time was 7 hours, among chosen options (Table 2), but the mean time for 2 cases equaled 7.2 hours. These results are consistent because the calculation of the allocated time (Table 2) depends on the unchanged workload.1,7 Thus, cases should be scheduled to minimize expected overutilized time,1,7 2 cases per procedure table daily. Cases being scheduled slightly into overutilized time shows why it is so useful to have a way to calculate the probability of long work hours,13 the results of the current article.
Generally, for services with few (eg, 2 or 3) cases per table or operating room per day, there is small30 to no31 potential for machine learning that would perfectly predict anesthesia duration to increase efficiency. If magic happened, every case duration estimated perfectly, the allocated time selected months ahead would equal the mean time for the sum of cases, because there would never be overutilized time. Allocated time would equal either 7.2 hours if 2 cases per table per day or 10.8 hours if 3 cases per day. Such a perfect prediction of anesthesia duration would neither change the mean nor the SD, just that the total time would be known when the last case of the day is scheduled. The frequency of working longer than 12 hours would still be 1.3% or 26.3%, for 2 or 3 cases, respectively. The latter risk of working very late is too high for the clinic and most9,32 healthcare organizations. Thus, there would be no value in more accurate duration estimation, because 2 cases are still scheduled. This conclusion is robust for veterinary care as sedation or general anesthesia are needed to perform a complete examination and imaging to achieve those more accurate duration estimates.
The clinic has all patients arrive at the start of the workday. The mean time of 7.20 hours to complete 2 cases was based on the second case starting right after the first case finishes. At the dental clinic, this was done by having 3 physical tables for 2 cases. Because anesthesia times generally follow log-normal distributions, and the mean exceeds the median for log-normal distributions, most (56.4%) cases that are not the first cases of the day can start early.28 Suppose, instead, cases do not start early, like trains and buses. Then, the mean time for the 2 cases would be significantly longer, 7.68 hours.28 Veterinary anesthetists have the advantage over human anesthetists in that the patients cannot tell time and judge waiting.33,34
Limitations
Our study examined 1 Canadian veterinary clinic as a model system to answer a problem recognized as important for 25 years.10 Nevertheless, several earlier results suggest the generalizability of the conclusions. First, the daily anesthesia workloads of services routinely follow normal distributions.4,5 Furthermore, even when not normally distributed, the calculated allocated times among fixed workday durations are usually the same as would be calculated based on normal distribution.5,6 If right skewed, calculations can be repeated using comparable generalized CIs for the log-normal distribution (equations 1 to 3 of Dexter and Ledolter).17 Second, the sample size of n = 44 was typical for allocating time based on minimizing the inefficiency of the use of anesthetists' time.6,19 In practice, services' workloads vary on weekdays because different surgeons are operating.6,19 When allocating anesthesia time, and controlling for day of the week, labor costs are the least when using 6 months to 1 year of data, the best observed being approximately 44 workdays.6,19 Third, the coefficient of variation of the daily workload, 32.6%, is typical among human anesthetists' operating rooms.35,36 A hospital audit showed most service and weekday combinations (43/52) with coefficients of variation > 30%.35 Individual human surgeons have even greater proportional variability, 70/74, with coefficients of variation > 30% on days with at least 1 scheduled case.35 Some days have a very low anesthetist workload, and some days have a very large workload. The substantial variability of workload is precisely why applying statistical methods has value, either to reduce the variability by adjusting case scheduling or by providing realistic predictive information to the anesthetist and other clinic staff.
Our results showed the validity and accuracy of a generalized pivotal quantity to calculate upper CIs for percentages of workloads exceeding a threshold. Although that is what was studied using the veterinary clinic data, we expect multiple other veterinary applications. These are illustrated with 4 examples. First, in the situation of a veterinary anesthesiologist providing consulting services, the anesthesiologist goes to a clinic with information known ahead of the total hours scheduled by the clinic. There are historical data for the site on daily differences in hours between actual and scheduled hours for the day. These differences routinely follow normal distributions (Figure 1).37,38 Therefore, our results show how to calculate an upper confidence limit on the probability that the anesthetist's time on site exceeds a threshold number of hours longer than that scheduled. Second, anesthesia times of cases regularly follow log-normal distributions (Figure 4).26 A generalized pivotal quantity using logarithms can be used for the upper confidence limit of the probability that an add-on case will fit into allocated but otherwise underutilized time.39 Third, readiness for patient discharge could be predicted to follow the human anesthesia equivalent of a postanesthesia care unit stay, which follows log-normal distributions.40 Our results show that generalized pivotal quantities can be used for the upper confidence limit of the probability that a patient will not be ready for discharge for a period longer than a threshold (eg, greater than 3 hours).41 Fourth, suppose that 1 anesthetist is giving another a break during the surgical period (eg, for lactation session) after the incision but before the end of surgery.1 Surgical times follow log-normal distributions.42 Our results show how to calculate the lower confidence limit on the probability that at least 30 minutes is available.42 These 4 examples show the multiple opportunities to apply statistical methods to quantify risks of exceeding thresholds of workload, for veterinary, like human, operating room management.
Supplementary Materials
Supplementary materials are posted online at the journal website: avmajournals.avma.org
Acknowledgments
The authors thank Alberta Veterinary Dentistry for support.
Disclosures
The authors have nothing to disclose. No AI-assisted technologies were used in the generation of this manuscript.
Funding
The authors have nothing to disclose.
References
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Pandit JJ, Dexter F. Lack of sensitivity of staffing for 8 hour sessions to standard deviation in daily actual hours of operating room time used for surgeons with long queues. Anesth Analg. 2009;108(6):1910–1915. doi:10.1213/ane.0b013e31819fe7a4
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McIntosh C, Dexter F, Epstein RH. Impact of service-specific staffing, case scheduling, turnovers, and first-case starts on anesthesia group and OR productivity: tutorial using data from an Australian hospital. Anesth Analg. 2006;103(6):1499–1516. doi:10.1213/01.ane.0000244535.54710.28
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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(4):933–942. doi:10.1097/00000539-200204000-00030
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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(6):1493–1498. doi:10.1097/00000539-200106000-00028
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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(1):31–36.
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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(5):1501–1508. doi:10.1097/00000542-199911000-00044
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Dexter F, Epstein RH, Marcon E, Ledolter J. Estimating the incidence of prolonged turnover times and delays by time of day. Anesthesiology. 2005;102(6):1242–1248. doi:10.1097/00000542-200506000-00026
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Dexter F, Epstein RH, Ledolter J, Wanderer JP. Interchangeability of counts of cases and hours of cases for quantifying a hospital's change in workload among four-week periods of 1 year. J Clin Anesth. 2018;49:118–125. doi:10.1016/j.jclinane.2018.04.008
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Dexter F, Epstein RH, Marian AA. Sustained management of the variability in work hours among anesthesiologists providing patient care in operating rooms and not on call to work late if necessary. J Clin Anesth. 2021;69:110151. doi:10.1016/j.jclinane.2020.110151
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Dexter F, Ledolter J. Exceedance probabilities of log-normal distributions for one group, two groups, and meta-analysis of multiple two-group studies, with application to analyses of prolonged times to tracheal extubation. J Med Syst. 2023;47(1):49. doi:10.1007/s10916-023-01935-2
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Tiwari V, Furman WR, Sandberg WS. Predicting case volume from the accumulating elective operating room schedule facilitates staffing improvements. Anesthesiology. 2014;121(1):171–183. doi:10.1097/ALN.0000000000000287
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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. 2002;94(3):640–643. doi:10.1097/00000539-200203000-00029
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Wang Z, Dexter F, Zenios SA. Caseload is increased by resequencing cases before and on the day of surgery at ambulatory surgery centers where initial patient recovery is in operating rooms and cleanup times are longer than typical. J Clin Anesth. 2020;67:110024. doi:10.1016/j.jclinane.2020.110024
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Rachuba S, Imhoff L, Werners B. Tactical blueprints for surgical weeks – an integrated approach for operating rooms and intensive care units. Eur J Oper Res. 2022;298(1):243–260. doi:10.1016/j.ejor.2021.06.005
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Pandit JJ, Tavare A. Using mean duration and variation of procedure times to plan a list of surgical operations to fit into the scheduled list time. Eur J Anaesthesiol. 2011;28(7):493–501. doi:10.1097/EJA.0b013e3283446b9c
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Dexter F, Shi P, Epstein RH. Descriptive study of case scheduling and cancellations within 1 week of the day of surgery. Anesth Analg. 2012;115(5):1188–1195. doi:10.1213/ANE.0b013e31826a5f9e
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Dexter F, Epstein RH, Bayman EO, Ledolter J. Estimating surgical case durations and making comparisons among facilities: identifying facilities with lower anesthesia professional fees. Anesth Analg. 2013;116(5):1103–1115. doi:10.1213/ANE.0b013e31828b3813
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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(1):45–51.
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Dexter EU, Dexter F, Masursky D, Garver MP, Nussmeier NA. Both bias and lack of knowledge influence organizational focus on first case of the day starts. Anesth Analg. 2009;108(4):1257–1261. doi:10.1213/ane.0b013e31819a6dd4
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Dexter F, Macario A, Ledolter J. Identification of systematic underestimation (bias) of case durations during case scheduling would not markedly reduce overutilized operating room time. J Clin Anesth. 2007;19(3):198–203. doi:10.1016/j.jclinane.2006.10.009
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Gök F, Koçbilek ZD. Examination of fatigue levels and factors affecting fatigue in operating room nurses. Periop Care Oper Room Manag. 2022;26:100243. doi:10.1016/j.pcorm.2022.100243
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Kynes JM, Schildcrout JS, Hickson GB, Pichert JW, Han X, Ehrenfeld JM, Westlake MW, Catron T, Jacques PS. An analysis of risk factors for patient complaints about ambulatory anesthesiology care. Anesth Analg. 2013;116(6):1325–1332. doi:10.1213/ANE.0b013e31827aef83
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Tiwari V, Queenan C, Jacques PS. Impact of waiting and provider behavior on surgical outpatients' perception of care. Periop Care Oper Room Manag. 2017;7:7–11. doi:10.1016/j.pcorm.2017.03.002
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Epstein RH, Dexter F. Rescheduling of previously cancelled surgical cases does not increase variability in operating room workload when cases are scheduled based on maximizing efficiency of use of operating room time. Anesth Analg. 2013;117(4):995–1002. doi:10.1213/ANE.0b013e3182a0d9f6
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Sulecki L, Dexter F, Zura A, Saager L, Epstein RH. Lack of value of scheduling processes to move cases from a heavily used main campus to other facilities within a healthcare system. Anesth Analg. 2012;115(2):395–401. doi:10.1213/ANE.0b013e3182575e05
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Dexter F, Yue JC, Dow AJ. Predicting anesthesia times for diagnostic and interventional radiological procedures. Anesth Analg. 2006;102(5):1491–1500. doi:10.1213/01.ane.0000202397.90361.1b
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Dexter F, Wachtel RE. Ophthalmologic surgery is unique in operating room management. Anesth Analg. 2014;119(6):1243–1245. doi:10.1213/ANE.0000000000000434
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Dexter F, Ledolter J. Bayesian prediction bounds and comparisons of operating room times even for procedures with few or no historic data. Anesthesiology. 2005;103(6):1259-1267. doi:10.1097/00000542-200512000-00023
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Dexter F, Epstein RH. Implications of the log-normal distribution for updating estimates of the time remaining until ready for phase I post-anesthesia care unit discharge. Periop Care Oper Room Manag. 2021;23:100165. doi:10.1016/j.pcorm.2021.100165
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Titler S, Dexter F, Epstein RH. Percentages of cases in operating rooms of sufficient duration to accommodate a 30-minute breast milk pumping session by anesthesia residents or nurse anesthetists. Cureus. 2021;13(6):e12519 doi:10.7759/cureus.12519
