# A Review of the Scheduling Problem within Canadian Healthcare Centres
## 4 Hospital Scheduling in Depth
### Operating Room Scheduling
#### 4.3.2 Papers
They begin by extending the non balanced distributed operating room ILP to include both macro and micro balancing objective functions. Micro imbalance ensures that shifts are distributed evenly within each individual hospital while macro imbalance ensures that each hospital is treated equally. Two different objective formulations are given for micro imbalance: one linear and one quadratic. Macro imbalance is always treated linearly. Furthermore, [PERSON] et al. [42] provide linearization techniques and decomposition techniques to further optimize their models. In the end, three models are presented. One mixed-integer quadratically constrained program and two mixed integer constrained programs. The models are then solved using both uni-level and bi-level logic-based bender's decomposition. The three models were validated using three collaborating hospitals in the Toronto area: Toronto General Hospital, Toronto Western Hospital, and Princess Margaret Cancer Centre. Test cases solved via Gurobi found that MIP models were able to solve within 2% optimality within 30 min. A third view on distributed operating room scheduling comes from [PERSON] et al. [38]. With another novel iteration on logic-based bender's decomposition and cut propagation. Here, they introduce 3 approaches to solve large scale problems. To generate the probability distributions the authors utilized data from the General Surgery Departments of the University Health Network in Toronto. Their methodology was significantly faster than using conventional solvers and were also able to handle problems of a larger scale. [PERSON] et al. [41] explore integrating patient priority into scheduling in hopes that it improves access to operating rooms. To do this, each patient is assigned a utility score, which represents their priority and urgency. The integer linear programming model is then directed to maximize this utility. The model was built with the considerations of the University Health Centre, Quebec City. Before implementing the model in its totality they wanted to demonstrate its efficacy and determine how much it would help the hospital. Firstly they generated dummy data to simulate patient data. A second, simpler model was created that instead of trying to maximize utility it instead maximized the number of sessions such that the longest wait times were scheduled first. This new second model would act as the control and allow comparisons between the two approaches. Four metrics were measured for both models and are summarised in Table 6. The first metric is the average utility of both the scheduled and unscheduled patients. The second is the number of sessions that each scheduled and unscheduled patient waited. The third is the number of surgeries the OR was able to perform. Finally, the last is the largest number of surgeries scheduled to a doctor. This results in positive results in favour of utility prioritization. The utility model not only allowed more patients through the system, but the patients that went through would \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & **Utility** & **\# Sessions** & **\# Surgeries** & **Max Surgeries** \\ \hline Utility Model & 69.83/58.06 & 15.16/19.65 & 116.7 & 4.4 \\ Longest Wait & 64.10/65.96 & 17.93/15.80 & 111.7 & 4.3 \\ Model & \multicolumn{1}{c}{} & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparative Results. be more urgent. The wait times for the unscheduled patients do get increased compared to the other model however and this could result in deadlock or unnecessarily long waiting times for low priority patients. Despite the efficacy of the utility model, [PERSON] et al. [41] suggest additional research into the idea before it is implemented. This research could be done in terms of improving the model computationally to allow for optimal solutions, and also test against different distributions of patients to ensure it works as intended. [PERSON] et al. [21] focus on the general problem of generalized OR planning and scheduling (GORPS).