# A Review of the Scheduling Problem within Canadian Healthcare Centres
## 4 Hospital Scheduling in Depth
### Operating Room Scheduling
#### 4.3.2 Papers
The first stage of their methodology is to find an initial solution using a process called first fit decreasing. This allows the solver to start with a warm start. Stage 2 involves solving the problem with an ILP solver. As the problem is quite complex they introduce both logic based Benders cuts and binary decision diagram cuts to help reduce the problems size. They also include classical Benders cuts. As an alternative, they also introduce a three stage framework to solve more complex instances of the problem in which the subproblems are further sub divided into 2 stages. Their scheduling framework was able to find more robust schedules and allow more patients to be seen in simulated trials. [PERSON] et al. [43] provides both comparisons to CPLEX as well as sensitivity analysis to validate the efficacy of their algorithms. [PERSON], [PERSON], and [PERSON] [39] investigate outpatient scheduling in a multidisciplinary, multistage process. The current methodology in the Bariatric surgery Department in Toronto Western Hospital involves scheduling patients for a single assessment on a later date. Appointments and assessments are scheduled at the same time, upon completion of the last. This leads to a system which is not robust against cancellations and changes, as well as not efficiently using resources. [PERSON], [PERSON], and [PERSON] [39] propose a new methodology that schedules patients in an offline fashion at the start of the day, sometimes over multiple timeslots to increase robustness. There model is based on mixed integer linear programming and offer multiple different avenues to solve the problem. They first aggregate constraints into single constraints to reduce the dimensionality of the problem and solve using integrality, lagrangian constraints, and column generation. We observe that the best model presented has an average or utilization in of 90% in most instances and increases average clinic profit should it be implemented. The allowance of multiple assessments allows the clinic to be more flexible and adaptive when it comes to appointments and can drastically increase staff output. [PERSON] et al. [2] offer a collaborative look into operating room scheduling. Instead of scheduling just one hospital, they attempt to improve the schedule by having the hospitals work in tandem to produce the best global solution. Their model is a mixed-integer programming model. Introducing collaboration also introduces hundreds of new constraints and variables, so solving the model, even suboptimally, becomes a computational problem. To address this, [PERSON] et al. [2] use a technique called Benders decomposition, formulated by [PERSON] in 1962 [44]. Similar to column generation, it involves decomposing a problem into subproblems, which are easier computationally. To evaluate their model they used data from University Health Network hospitals, including the Toronto General Hospital, the Toronto Western Hospital, and the Princess Margret Cancer Centre. By collaborating with dedicated patients, a cost savings of 6.08% was achievable. Including flexible and semi-flexible patients increased the savings to 26.37% and 39.89%. Collaborative scheduling also increased throughput, leading to 9.25% higher patient admission. To further prove that a collaborative network of operating rooms more effective, a game theory lens is used to look at the results. By measuring the average contribution of each hospital a payoff value is computed. It was found that joining the collaborative network was in each hospital's favour, as it led to a higher payoff. Overall, when hospitals chose to share resources it was a net benefit for all of them at the cost of computational complexity. In addition, [PERSON] et al. [42] introduce several techniques for creating balanced operating room schedules based on ILP. The balanced distributed operating room problem again attempts to schedule multiple hospitals collaboratively to better improve patient output and care.