# Optimized Scheduling of IoT Devices in Healthcare Facilities: Balancing Cost and Quality of Care
## 3 Materials and Methods
### Research Model  
Our approach transforms the scheduling challenge into a bi-objective optimization problem, a methodology that simultaneously addresses two potentially competing goals. In bi-objective optimization, improving one performance metric can necessitate a trade-off in another, which is particularly characteristic of complex systems in manufacturing, logistics, and project management. In our context, efficiency is quantified by minimizing assignment costs, while quality of care is defined by the level of benefit patients receive from the assigned devices. Specifically, higher quality of care is achieved by allocating patients to more advanced devices that offer enhanced features directly impacting their experience and well-being. For instance, lighter and more compact models improve patient comfort and ease of use, thus contributing to a higher quality of care. The system prioritizes allocating the most suitable devices, especially those with superior features, to high-priority patients whenever feasible. This dual-objective optimization framework mirrors the complex realities of medical device scheduling, where financial limitations must be carefully balanced with the imperative to maximize positive clinical outcomes and patient experience. Mathematically formulating the two-objective scheduling problem requires a precise definition of variables, constraints, and objective functions, which are comprehensively detailed in the accompanying parameters in Table 1. Variables are listed in Table 2. \begin{table}
\begin{tabular}{c l} \hline \hline
**Name** & **Description** \\ \hline \(P=\{j|j=1,\ldots,N\}\) & patient set \\ \(r_{j}\) & patient / requested device class \\ \(p_{j}\) & patient / priority \\ \(f_{j}\) & patient / service window start \\ \(S_{j}\) & patient / service window end \\ \(D=\{i|i=1,\ldots,M\}\) & device set \\ \(q_{i}\) & device / quality \\ \(c_{i}\) & device / cost rate \\ \(d_{i}\) & device / class \\ \(s_{i}\) & device / setup time \\ \(l_{i}\) & device / lifespan \\ \(b_{i}\) & device / availability window start \\ \(e_{i}\) & device / availability window end \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Problem parameters. Constraints are a core component of our optimization model, as they determine which scheduling solutions are feasible. They ensure compliance with device availability windows (Equations (1) and (2)) and patient availability windows (Equations (3) and (4)), enforce device-patient class compatibility (Equation (5)), incorporate setup times between assignments (Equation (6)), maintain continuous device coverage throughout a patient's stay (Equation (7)), and respect device lifespan constraints (Equation (8)). It is important to highlight that constraints enforce distinct availability requirements for devices and patients. Specifically, device-patient allocations must fall within the device's availability window (Equations (1) and (2)), while each patient must be continuously assigned to a device throughout their entire availability window (Equations (3), (4) and (7)). This distinction is critical: devices may remain idle during their availability, but patients require uninterrupted device coverage. \[b_{i} \leq\min_{j\in\Gamma_{i}}z_{ij} \forall i\in D \tag{1}\] \[\max_{j\in\Gamma_{i}}w_{ij} \leq e_{i} \forall i\in D\] (2) \[f_{j} =\min_{i\in\Lambda_{j}}z_{ij} \forall j\in P\] (3) \[\max_{i\in\Lambda_{j}}w_{ij} =g_{j} \forall j\in P\] (4) \[d_{i} =r_{j} \forall i\in D\quad\forall j\in\Gamma_{i}\] (5) \[w_{ij_{1}}+s_{i} \leq z_{ij_{2}} \forall i\in D\quad\forall j_{1},j_{2}\in\Gamma_{i}\quad x_{ij_{ 1}h_{2}}=1\] (6) \[w_{ij_{1}} \leq z_{i,j} \forall j\in P\quad\forall i_{1},i_{2}\in\Lambda_{j}\quad y_{ij_{ 1}i_{2}}=1\] (7) \[\sum_{j\in\Gamma_{i}}(w_{ij}-z_{ij}) \leq l_{i} \forall i\in D\quad\forall j\in\Gamma_{i} \tag{8}\]  
Regarding the bi-objective function, the two key performance indicators (KPIs) are the total cost \(\gamma\) and the overall quality of care \(\sigma\).