# A Review of the Scheduling Problem within Canadian Healthcare Centres
## 4 Hospital Scheduling in Depth
### Physician/Emergency Room Scheduling
#### 4.2.2 Papers
The former is a ratio of patients seen, and the latter is an estimate of effort and practice expense. To address this, schedules created with physician productivity in mind could help alleviate some of the stress on the system. They start to approach the problem by analyzing patient arrival times and then trying to use this information to predict how much stress there will be on the system at a later date. Data was used from the Sacre-Coeur Hospital of Montreal, which is a health center affiliated with the University of Montreal. Data from March 2008 to September 2017 was collected and analyzed containing approximately 600 thousand instances. This hospital employs 35 physicians and sees approximately 62 thousand patients per year. A decomposable time series model with the components growth, seasonality, and holidays was used to predict the number of patients that would arrive. It measured the number of patients that a physician was able to treat during their shifts. This index changes based on a multitude of variables, such as the day of the week and the type of shift. Using this data they can create demand forecasts. Following this, they introduce a measure of productivity called a productivity index, and use this to determine which physicians should be scheduled where. Using this index, the mathematical model matches patient demand to productivity. Additionally, to help alleviate stress on physicians, the model attempts to minimize unfairness as to not overload shifts. Four different scenarios were tested, each with different weightings. All four had a weight of '100' for under-staffing, ensuring that it would be of the highest priority. Scenarios two and four consider over-covering and under-covering with a weight of '1' while scenario three has a weight of '10' and one does not consider it. Finally, scenario one and three have a weight of '10' for physician dissatisfaction and unfairness while the rest have a weight of '1'. Of the four scenarios, scenario four was the most balanced and showed the most improvement. It produced a schedule with no understaffed shifts, very little difference in shift distribution, and an average of 2.97 unwanted shifts. The success of this model demonstrates the benefits of accounting for patient forecasting and the faults of current scheduling. [PERSON] et al. [32] researches a variant of physician scheduling: physician scheduling under uncertainty. In this case, uncertainty refers to stochastic patient treatment durations. If a patient goes over the allotted time it will affect all subsequent scheduled appointments and if a patient finishes early that is time that is effectively wasted. The authors propose a 2 stage planning framework to help reduce the amount of uncertainty and more effectively schedule physicians. The first step of the planning framework is the clinic scheduling and capacity planning problem. This involves scheduling the number of physicians required and the number of operating rooms that will be needed for the patients. To solve this a linear programming formulation of the problem is constructed and a mixed integer linear programming model is created. The method of solving this model is a variation on column generation called implementor/adversary algorithm. Column generation breaks an MIP formulation into a master problem and several smaller subproblems. The master problem is a smaller subset of all of the variables. After the master problem is solved, a sub problem is created with a variable thought to best improve the solution. The implementor/adversary algorithm treats the subproblems as adversaries until an optimal solution is found. The second stage is the typical physician scheduling problem. This involves slotting physicians into the corresponding timeslots. To solve this they implement another MIP model. In addition, they create a sample average approximation algorithm in order to predict a patients duration within the clinic. This helps minimize the amount of time spent wasted or in overtime.