# A Review of the Scheduling Problem within Canadian Healthcare Centres
## 4 Hospital Scheduling in Depth
### General Staffing/Nurse Staffing
#### 4.1.2 Papers
[PERSON], [PERSON], and [PERSON] [13] have developed a framework that allows each constraint an associated incentive heuristic. The incentive heuristics act as modifiers. They are scored with potential assignments and then collectively they are added to the constraint dynamically. They work for all constraint types and allow greater control over how the model behaves. [PERSON] et al. [29] approached the Vancouver Coastal Health network to partner with a community health centre and attempt to improve access for marginalized people. They worked with staff directly to create a three stage approach. The first step in their process is to create an optimal weekly schedule using an MILP model. They solve the model using Gurobi, not going into the details of the solving process. Their model attempts to incorporate a multitude of fuels for creating the schedules. Shift length, breaks, and administrative duties are all considered. The second part of their 3 part framework is a heuristic which tries to account for fluctuations in hours on a weekly basis. After creating an optimal schedule using the ruleset prior, a heuristic removes shifts until weekly schedules are fairly distributed amongst staff. The heuristic first identifies which shifts can be removed with minimal disruption. They then further filter the shifts based on the amount of overlap. If there is only 1 shift it is removed, otherwise one is chosen at random. This process is repeated until a sufficient number of shifts are removed. The final step is a simulation model. This model aims to capture the variability of real world data, such as walk ins, no shows, and bookings. This is done through a homogeneous Poisson process. Their model was able to better allocate shifts without increasing the work load for the nurses. These benefits can help improve access and care recieved my marginalized people. The final model was tested using data collected from several Montreal hospitals. To evaluate how good a model performs six main characteristics are utilized:  
* Coverage: Are the needed shifts covered? * Quality: Does the model have good working conditions and follow hospital rules and preferences? * Stability: Are the schedules homogeneous? * Flexibility: How does the schedule deal with transitioning from the previous to the newer schedule? * Fairness: Are shifts distributed fairly? * Cost: How many resources are required to create and then implement the schedule? The Hibiscus model had full coverage of scheduling. Quality and Fairness depend on the implementations of the constraints, but with search heuristics and the option to set hard vs. soft constraints it can achieve a reasonable quality. The model is also malleable when preparing for the transition period giving it stability. The model is flexible, although they admit that the search strategy is the least robust aspect and has room for improvement. Finally, the model has a low cost since human resources are not needed for creation, and computation is quite fast. Another generalized model comes from [PERSON] et al. [20] in their paper \"A generalized linear programming model for nurse scheduling\". The incentive for generalizing the models comes from creating rosters that are more realistic and pragmatic given real-world situations. Their approach to generalization leads to a very complex and large mixed-integer programming model that cannot be solved in a reasonable time without decomposing the problem. To address this, they use a decomposition technique known as column generation. Column generation involves starting with a subset of the features and adding in relevant ones that most move the answer in the optimal direction one at a time. It was first proposed by [PERSON] and [PERSON] in 1958 [30]. The starting problem is known as a master problem, and the problem which determines the effect of adding a new column is known as an auxiliary problem. Their master problem is the MIP model, and the auxiliary problem they introduce is a resource-constrained shortest path problem.