New methods for simulation-based optimization and applications to emergency department management

The development of novel efficient algorithmic frameworks using simulation to provide solutions to real-world problems is prompted by the need to accurately represent the complex and uncertain processes of real systems, such as Emergency Departments (EDs). The resulting Simulation-Based Optimization (SBO) methodology has been receiving increasing attention in recent years, aiming to develop algorithms that do not require first-order information and support both continuous and integer variables. The trade-off between long-term goals and short-term decisions, as well as the computational cost of evaluating the black-box functions involved, determines whether to use exact Derivative-Free Optimization (DFO) algorithms, providing optimal solutions with long running time, or metaheuristic methods, returning fast solutions without optimality guarantees. Important SBO problems arise in dealing with ED management since a strong interest is shown in studying the impact of both the overcrowding phenomenon and sudden patient peak arrivals on everyday operations. To this end, further SBO approaches may be required to estimate the ED arrival rate and to recover the missing information from the real datasets in order to build Discrete Event Simulation (DES) models with a high level of reliability. In this thesis, SBO is used with a twofold goal. On the one hand, to propose methodological contributions from an algorithmic point of view, namely a metaheuristic-based algorithm to solve a specific SBO problem and a globally convergent DFO method for mixed-integer nonsmooth constrained optimization problems, frequently arising in practice. On the other hand, to develop SBO approaches to improve the accuracy of a DES model representing an ED. In particular, an integer nonlinear black-box optimization problem is solved to determine the best piecewise constant approximation of the time-varying arrival rate function by finding the optimal partition of the 24 hours into a suitable number of nonequally spaced intervals. Black-box constraints are adopted to ensure the validity of the Nonhomogeneous Poisson process, which is commonly used in the literature to model the ED arrival process. Moreover, a model calibration procedure is proposed to estimate the incomplete information in the ED patient flow by minimizing the deviation between the real data and the simulation output. The resulting DES model is used for solving a simulation-based resource allocation problem to determine the optimal settings of the ED unit devoted to low-complexity patients. The objective is to reduce the overcrowding level without using an excessive amount of resources. Two real case studies are considered to demonstrate the effectiveness of the proposed methodology.