Mathematical Representation of Erasmus MC's Surgical Case Scheduling
Surgical case scheduling involves finding the combination of surgical cases that makes optimal use of available OR time. In the field of applied mathematics, this problem is known as the bin-packing problem. Currently, surgical departments schedule their surgical cases using a First-Fit approach (15). Searching from the beginning, patients are selected from a waiting list and scheduled in the first available OR in a particular week.
In our study, waiting lists were generated based on different surgical case categories representing each department's case mix (Table 2). Subsequently, a First-Fit algorithm simultaneously selected and scheduled surgical cases for the period of 1 wk which, in practice, is done approximately 2 wk before the date of surgery. This algorithm scheduled next cases only if the previous surgical case had been scheduled and if the algorithm concluded that it was impossible to fit the previous case into any of the available OR blocks. If the case did not fit in any of the available blocks, it was placed back on the waiting list. The algorithm terminates once it reaches the end of the waiting list. Note that for scheduling of cases the mean duration was used and that no planned overtime was allowed, as prescribed by the Erasmus MC rules. The resulting surgical case schedules comprised surgical cases, planned slack, and unused OR capacity (Fig. 1).
Planned Slack and the Portfolio Effect
The financial world deals with uncertainty by using the portfolio effect. This ensures that the expected return of a stock portfolio is less vulnerable to fluctuations on the stock market. The term “portfolio effect” then indicates that portfolio risk decreases with increasing diversity, as measured by the absence of correlation (covariance) between portfolio components (16). We earlier found application of the portfolio effect to surgical case scheduling to be successful in increasing OR efficiency, since it reduces the required amount of planned slack, given an accepted risk of overtime (11). The approach clustered surgical cases with similar variability in the same OR block, assuming these to be uncorrelated.
We illustrate the portfolio effect applied to surgical case scheduling by the following example: Consider two OR blocks, both of which have two surgical cases scheduled. One case with (mean, standard deviation) = (100, 10) and one case with (mean, standard deviation) = (100, 50) (Fig. 2) (all values are given in minutes). We assumed that case durations are described by a normal distribution function. In this example, we now compared this situation (the left side of Fig. 2) with the situation in which surgical cases with similar variance are clustered. In the first situation, the standard deviation of total duration is the same for both OR blocks: √(502 + 102) = 51.0 min. The total planned slack for the two blocks is thus 102.0β min, where β is a risk factor to deal with risk of overtime. Since the sum of the durations follows a normal distribution the following holds: P (mean + β·standard deviation) ≤ accepted risk of overtime, such that given a certain accepted risk of overtime the risk factor can be calculated. In the second situation, the total planned slack is: (√(502 + 502) + √(102 + 102))β = 84.9β min. This means a 17.1β min reduction in the total required planned slack time, and thus an equal increase in available capacity. This portfolio profit will increase with higher variability of the cases concerned. This example illustrates that rescheduling a surgical case can reduce the extent of planned slack.
We constructed three scenarios to investigate the impact of lowering organizational barriers imposed by block planning (Table 4). The scenarios are graded as to interdepartmental flexibility (i.e., scheduling cases of different departments in the same OR on 1 day) and flexibility of rescheduling surgical cases between days of the week compared with the current situation. Rescheduling of surgical cases throughout the week does not affect patients, since they have not yet been scheduled.
In this study, we assumed application of the scenarios directly after the construction of the surgical case schedules, approximately 2 wk before the actual execution of the schedule (Fig. 3). This enables OR departments to take necessary steps to ensure feasibility; for example regarding material logistics, ranging from specific surgical material to complete navigation system for complex craniotomy surgery. Surgical departments are responsible for the scheduling of semiurgent or add-on elective patients who need an operation on a day for which a surgical case schedule is already set. For this purpose, departments may schedule cases without assigning a patient to it, or by canceling one or more of the elective cases. Emergency patients are operated on within the reserved OR time as described earlier.
Advanced Mathematical Algorithms
Application of the different scenarios to a surgical case schedule implied rescheduling of surgical cases according to the organizational flexibility of the scenario under consideration. A bin-packing algorithm, based on work of Hans et al. (11), who used RBRS, did the rescheduling of the surgical schedules given the scenarios. Figure 4 shows how rescheduling surgical cases saves OR time. The objective of the algorithm is to minimize planned slack by exploiting the portfolio effect and the required number of OR blocks. RBRS procedures start with removing all cases of the existing surgical schedule to a list. Then, RBRS iteratively schedules a random surgical case from the list until all cases are scheduled. The drawing probability of each of the cases is based on the case's Best Fit suitability. This randomized procedure gives a new solution (a “surgical case”) every time it is executed. We stopped the algorithm after generating a preset number of 1500 surgical case schedules. The generated schedules were evaluated on the objective criterion (amount of free OR capacity) and the best schedule was saved (11). The algorithm was coded in the Borland Delphi computer language (Cupertino, USA).
The Erasmus MC's main inpatient department considered using the news-vendor approach of Strum et al. (3). Subsequently, we investigated the benefits of the RBRS that exploited the portfolio effect and relaxation of the organizational constraints. To this aim, the surgical case scheduled created by the RBRS algorithm was compared with the surgical schedules constructed by the First-Fit approach. The RBRS algorithm was compared with the Best Fit algorithm (7) to assess the performance of advanced mathematical algorithms over available and simpler heuristic techniques.
We performed a robustness analysis on the influence of unpredictability of case duration on OR utilization, wherein the unpredictability was represented by the standard deviation of case duration. The influence of number of ORs within an OR department on OR utilization was investigated as well. Both analyses were performed for each of the three flexibility scenarios. The outcome measures of this study are OR utilization and the number of freed OR blocks, so-called “freed ORs.” OR utilization was defined as the ratio between the total duration of elective surgical cases and the total staffed OR capacity per week. Hence, it is similar to what is known in the literature as “raw OR utilization” (17).
Applying the news-vendor approach of Strum et al. (3) did not lead to improved efficiency. With staffing costs determined by the allocated capacity and overtime by a relative cost ratio of 1.5 and increasing the block times with 15 min, it even decreased efficiency (Table 5). Therefore, new ways to increase OR efficiency were explored, as described in the previous section.
Increased flexibility in the three scenarios increased the number of freed OR blocks (Table 6). Eventually, this resulted in an improved utilization rate of 4.5% points (95% confidence interval 4.0%–5.0%). Both the Best Fit Descending heuristic and the RBRS algorithm improved utilization. The latter, more advanced, algorithm significantly out-performed the first heuristic by 0.7% point in Scenario 2 (95% confidence interval 0.2%–1.2%). Applying either the RBRS algorithm or the Best Fit Descending did not significantly improve the initial surgical schedules when combined with Scenario 1 (i.e., blocks consists of surgical cases of a single department and cases are rescheduled on the day). No significant difference was measured between the Best Fit Descending heuristic and the RBRS algorithm in Scenario 1 (Table 6).
Number of freed OR blocks, and hence OR utilization, increased relative to the standard deviation of case duration within one department (Fig. 5). The RBRS algorithm and the portfolio effect did not significantly improve the original schedule in Scenario 1, regardless of the standard deviation in the patient mix. Furthermore, in Scenarios 2 and 3, the benefits of the RBRS algorithm increased with the standard deviation of case duration.
Figure 6 shows the association between number of ORs and OR utilization rate expressed in number of freed OR days for the three scenarios. The findings shows that if more flexibility would be achievable, benefits progressively increase with the number of cases performed daily relatively to the available hours provided.
The study showed how to improve OR efficiency by combining advanced mathematical and financial techniques with the lowering of organizational barriers. The combination facilitates OR departments to improve OR efficiency when current methods will no longer benefit (3,7). The method is applicable in hospitals that set their surgical case schedules approximately 2 wk in advance, and potentially improves OR utilization by 4.5%. Improved efficiency implies that more operations can be performed at the same OR capacity or that less OR capacity is needed for the same number of operations. We also showed that potential benefits vary for different OR departments, depending on the uncertainty in case duration and number of ORs within one OR department. Absolute measures of this study are difficult to compare with results from other studies because Erasmus MC uses a specific method of reserving OR time in surgical schedules.
The algorithms used aimed to free OR blocks, because capacity that was previously allocated in these blocks is not accounted for while calculating the utilization rate. This is true for all OR departments that have sufficient flexibility in their staff scheduling to allow changes approximately 2 wk in advance.
We assumed in the analysis that surgical case durations show normal distribution. Other studies have shown that a lognormal distribution is a better approximation of the real duration (18). Calculation of planned slack, which is required to simulate the portfolio effect, requires a closed form probability distribution. This is not the case for a lognormal distribution, and this is why we have opted for a normal distribution, which may modestly influence the outcomes. Since the amount of planned slack is similarly calculated for the RBRS algorithm compared with that for the Best Fit heuristic, we do not expect that the assumption influences the calculated outcomes.
Many hospital use information technology systems to actually schedule their surgical cases in the available blocks. The mathematical techniques presented in this paper can easily be incorporated in such information technology systems, permitting planners to actually use the mathematical algorithms. Using the techniques addressed in this paper, and given a flexibility scenario agreed upon beforehand, the set of cases planned by the different departments in their blocks is collectively optimized after surgeons have set their patients' surgery dates. After optimization, each department can match its surgeon and bed planning with the new, more efficient, case schedule.
Lowering organizational barriers might have some negative effects and will require a more flexible attitude of surgical departments and individual surgeons. First, allowing various surgical departments to use the same OR may result in longer waiting times for surgeons. Second, surgeons may be scheduled in various ORs on the same day. Third, having surgeons operate on different days in the week requires adjustment of their other tasks, especially in hospitals where surgeons are highly specialized and where cases cannot be interchanged among surgeons. All these issues should be carefully addressed and weighed against the efficiency increase. The essential consideration, we believe, is that the drawbacks for a surgical department can be compensated for by the huge amount of extra OR capacity, which can be used to shorten the waiting list and earn more money.
Another aspect of implementation of the techniques is the required additional flexibility of the ORs. Each OR has to be uniformly equipped so that all surgical departments may operate in it. The efficiency increase achieved by the proposed method would justify the investment to equip all ORs generically.
Each hospital can choose a flexibility scenario that matches its requirements. Even more scenarios can be made to show benefits of even lower organizational barriers. The potential benefits can be calculated by comparing the current case scheduling strategy, in this paper represented by a first-fit algorithm, and the future situation in which the portfolio effect and bin-packing techniques have been applied and organizational constraints have been relieved. This paper provides a tool for any hospital type to make their own trade-off between flexibility and higher utilization of OR capacity.
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© 2007 International Anesthesia Research Society
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