Surgical scheduling is a complex task that requires consideration of various factors, including the probability of overtime. In this study, we address the research problem of surgery scheduling while accounting for the likelihood of exceeding scheduled operating room (OR) time. To tackle this problem, we employ integer linear programming (ILP) models to determine the optimal number of surgeries per group, with the objective of maximizing OR utilization while incorporating the probability of overtime as a constraint.

To capture the probabilistic nature of surgery durations, we investigate suitable probability distributions. Existing literature suggests that surgery durations follow a lognormal distribution. However, since the sum of lognormally distributed random variables lacks a closed form solution, we initially assume a normal distribution for analytical convenience. Subsequently, we approximate the lognormal distribution using the Fenton-Wilkinson method to account for its realistic behavior. To incorporate the
lognormalistic behavior and solve the ILP models efficiently, we employ a column based approach. This approach enables us to handle the complexities introduced by the lognormal distribution. Our study utilizes data provided by a hospital in the Netherlands, including information on surgeries, specialties, groups, and the master surgery schedule (MSS). Given the consideration of both normal and lognormal distributions for surgery durations, we assess the goodness of fit using appropriate statistical
tests.

Our results reveal that using averages or expected values yields the highest OR utilizations. However, there is a discussion regarding the validity of this method, as it does not explicitly incorporate the probabilistic overtime constraints. Nevertheless, we observe that all methods include cases which
surpass the predetermined overtime threshold, suggesting that utilizing averages or expected values can be a valid alternative. However, utilizing averages or expected values gives rise to high percentage
of cases surpassing our overtime threshold. So, we suggest to use a method which explicitly uses the probabilistic nature of the surgery durations.

During the examination of our methods, we had to take a minimum number of mandatory scheduled surgeries for each group into account. This means that another dataset, with different mandatory numbers, might lead to different results. Additionally, we noticed that our overtime definition might not be the most optimal, as we still have cases that surpass our overtime threshold. In future research, it would be valuable to include financial and staff factors, which can further enhance the scheduling process.

Overall, this study contributes to the field of surgery scheduling by addressing the probability of overtime and presenting insights into the trade-offs between OR utilization and the inclusion of probabilistic
constraints. Further research can build upon these findings to refine the scheduling approaches and incorporate additional factors for a more comprehensive solution.