# Optimized Scheduling of IoT Devices in Healthcare Facilities: Balancing Cost and Quality of Care
## 3 Materials and Methods
### Solution Method  
The bi-objective nature of the proposed IoT device allocation model--balancing operational cost and quality of care--places it within the class of multi-objective scheduling problems, which are well known for their computational complexity. The inclusion of hard constraints such as time windows, device compatibility, battery limitations, and setup times further increases the problem's difficulty. Even in single-objective forms, many variants of scheduling problems are classified as NP-hard. Extending these to a bi-objective framework significantly compounds the computational burden, as it requires exploring a large, non-convex solution space to identify a diverse set of Pareto-optimal solutions. Given this complexity, exact methods (e.g., integer programming or branch-and-bound) often become infeasible for realistic problem sizes due to exponential runtime. As a result, heuristic and metaheuristic approaches, such as evolutionary algorithms, are commonly adopted to provide high-quality solutions within acceptable time frames. These methods are especially suited for multi-objective problems, as they can efficiently approximate the Pareto front while navigating complex constraint landscapes. In [28], a Mixed Integer Linear Programming (MILP) model was proposed to address critical scheduling challenges, including staff assignments, patient distribution, resource allocation, and overtime management. The primary objective was to control healthcare costs while enhancing the quality of patient care. In [29], the Genetic Algorithm (GA), a metaheuristic inspired by natural selection, is applied to various healthcare scheduling problems, including patient admission scheduling, nurse scheduling, operating room scheduling, surgery scheduling, and other related tasks. In [30], the authors investigate the application of biomimicry concepts in cancer treatment to improve appointment scheduling and therapeutic outcomes. Their approach leverages automation through three nature-inspired algorithms: Wolf Optimization, Firefly Optimization, and Genetic Algorithm. Simulated Annealing (SA), a probabilistic technique for approximating the global optimum of a function, is introduced in [29]. This method is particularly effective for evaluating a broad set of potential schedules, iteratively refining them to achieve an optimal or near-optimal solution. Multi-Objective Particle Swarm Optimization (MOPSO) enhances the Particle Swarm Optimization (PSO) algorithm to handle multiple objectives [31]. It maintains a repository of non-dominated solutions and employs a leader selection mechanism to guide the swarm toward optimal outcomes. MOPSO is particularly effective in exploring the solution space and identifying a set of Pareto-optimal schedules. Pareto Simulated Annealing (PSA), an adaptation of SA for multi-objective optimization, applies a Pareto dominance criterion to accept or reject solutions, as discussed in [32]. PSA is valuable for balancing competing objectives by evaluating different scheduling configurations. Non-dominated Sorting Genetic Algorithm II (NSGA-II) is a widely adopted approach for multi-objective optimization, designed to identify a diverse set of optimal solutions. It is particularly useful for balancing conflicting objectives, such as minimizing travel time and reducing patient visit delays, while generating Pareto-optimal schedules. In [31], NSGA-II is applied to scheduling problems through a simulation-based approach. To address the inherent complexity of allocating IoT medical devices in healthcare environments, this study introduces a heuristic optimization approach based on the NSGA-II algorithm, tailored to solve bi-objective scheduling problems. At its core, the method leverages genetic representations of patient-device assignments, enabling the intelligent exploration of vast solution spaces that traditional methods struggle to handle.