# Optimizing Healthcare Delivery: A Model for Staffing, Patient Assignment, and Resource Allocation
## 5 Results and Discussions
### Demand and Capacity Analysis in Scheduling  
We calculated and graphically presented each period's total demand and capacity by scheduling patients \(P\) based on demand and the available capacity and at a set of periods \(T\). We define \(demand_{t}\) and \(capacity_{t}\) to denote total demand and capacity at time \(t\in T\), respectively:  
\[\forall t\in T:\] \[demand_{t} =\sum_{p\in P}C(p,t)\] \[capacity_{t} =\sum_{p\in P}y(p,t).x\]  
Here:  
* **Period's Total Demand and Capacity:** This refers to the overall demand and capacity for a specific period. It is calculated for scheduling patients based on available resources and demands.
* **Patients \(P\):** This represents the set of all patients considered in the scheduling.
* **Set of Periods \(T\):** This refers to the specific periods for which demand and capacity are calculated and scheduled.
* **Demand at Time \(t\), demand\(t\):** This is the total demand at a specific time \(t\), calculated as the sum of the demand for all patients at that time. The demand for each patient \(p\) at time \(t\) is represented by \(C(p,t)\).
* **Capacity at Time \(t\), capacity\({}_{t}\):** This is the total capacity at time \(t\), calculated as the sum of a capacity-related variable for all patients at that time. The capacity-related variable for each patient \(p\) at time \(t\) is represented by \(y(p,t)\cdot x\).
* \(C(p,t)\): This corresponds to the demand for a specific patient \(p\) at time \(t\). It represents the patient's required resources, treatment, or service.
* \(y(p,t)\cdot x\): This represents the value of a capacity-related variable for a specific patient \(p\) at time \(t\). It may signify how much of a certain resource is allocated or available for that patient, possibly weighted by some variable \(x\).
* \(\forall t\in T\) **(For All \(t\) in \(T\)):** This notation signifies that the given equations for demand and capacity are applicable for all periods \(t\) in the set of periods \(T\).  
Patient demand and resource capacity are compared across time slots for the two model processes in Figures 15 and 16. The plots offer several plausible outcomes. Demand exceeding capacity denotes a period when more resources are needed for patient care than are available. This imbalance can slow the delivery of patient care, necessitating steps to expand capacity or redistribute resources. On the other hand, a capacity line that is  
Figure 14: Comparative analysis of patient and staff utilization over time for the second model procedure.  
continuously above the demand line suggests that resources are not being used to their full potential and that reevaluating resource distribution might improve overall efficiency. Conversely, it indicates an ideal balance between patient requirements and available resources if the demand and capacity lines align or exhibit similar patterns. A careful examination of these trends can reveal tactical information to boost the effectiveness of patient scheduling. These figures employ geometric representations to describe the constraints, variables, and equations that define the optimization landscape. They contribute to the rigorous formulation of the problem, enhancing the understanding of the model's structure, assumptions, and mathematical rigor.