# Optimizing Healthcare Delivery: A Model for Staffing, Patient Assignment, and Resource Allocation
## 5 Results and Discussions
### Patient Satisfaction  
Our model incorporates various variables including, but not limited to, patient satisfaction, staff interaction, conditional elements, patient utility, and other pertinent factors, all modeled using simulated data. Patient satisfaction in this model is derived from carefully aligning resources, staff, and patient needs within specific scheduling periods. By optimizing these elements in concert, we aim to create an environment that maximizes patient satisfaction, ensuring that resources are allocated to meet their requirements and expectations. We calculated patient satisfaction for each period \(t\in T\). The patient satisfaction at time \(t\), represented as \(satisfaction_{t}\), is computed as follows:  
\[satisfaction_{t}=\frac{\sum_{i\in N,p\in P:y(p,t).x>0.5}U(p)\cdot v(i,p).x}{ \sum_{p\in P}y(p,t).x}.\]  
Figure 16: Demand vs. capacity over time for the second model procedure.  
Figure 15: Demand vs. capacity over time for the first model procedure.  
Here, \(U(p)\) denotes the utility of patient \(p\), \(v(i,p).x\) is the assignment variable corresponding to patient \(p\) being assigned to staff member \(i\), and \(y(p,t).x\) is a capacity-related variable for patient \(p\) at time \(t\).  
* **Patient Satisfaction at Time \(t\):**_satisfaction_,\(t\)_: This is the target variable representing the satisfaction level of patients at time \(t\).
* **Summation Over Patients and Staff:**\(\sum_{i\in N,p\in P}\): This summation indicates that the calculation is done over all patients \(p\) in a set \(P\) and all staff members \(i\) in a set \(N\).
* **Conditional Part:**\(y(p,t)\cdot x>0.5\): This condition implies that the summation is calculated only when the product of the capacity-related variable \(y(p,t)\) and some variable \(x\) is more significant than 0.5. This could be a threshold for considering only specific assignments or capacities.
* **Utility of Patient \(p\):**\(U(p)\): This represents the utility or benefit of a specific patient \(p\), which may be based on their needs or preferences.
* **Assignment Variable:**\(v(i,p)\cdot x\): This term denotes the assignment of patient \(p\) to staff member \(i\), possibly weighted by variable \(x\). It might represent how well the staff member fits the patient's needs.
* **Denominator Summation:**\(\sum_{p\in P}y(p,t)\cdot x\): This part sums up the capacity-related variable \(y(p,t)\cdot x\) for all patients \(p\) in set \(P\) at time \(t\). It likely represents the total capacity or availability at time \(t\) and is the normalizing factor for patient satisfaction.
* **Main Equation:** Putting all the parts together, the equation calculates patient satisfaction at time \(t\) as a weighted sum of utilities and assignments normalized by the total capacity-related variable.  
Our results signified patient satisfaction as a function of time, illustrating staff allocation schemes' effectiveness.  
Figures 17 and 18 plot patient satisfaction measures against time slots. They illustrate how staff allocation strategies affect patient satisfaction, emphasizing the importance of our results in improving healthcare delivery quality and the patient experience. The horizontal axes show time slots, while the vertical axes show the patient satisfaction measure for each period as an average weighted utility value for the two model procedures. High points on the graph indicate staff allocation efficacy due to appropriate personnel levels or expertise matching patient demands. Conversely, low numbers suggest inefficient personnel deployment. The plotted lines show gaps when the satisfaction metric is ambiguous owing to patient assignments. These graphs encapsulate a complex analysis of patient feedback, employing regression or curve-fitting techniques to infer underlying patterns. They are strategic tools for assessing patient experience, guiding scheduling decisions, and informing quality improvement initiatives.