A Two-stage Stochastic Programming for the Integrated Emergency Mobility Facility Allocation and Road Network Design Under Uncertainty

Emergency Mobility Facilities (EMFs) possess the capability to relocate dynamically, providing adequate responses to fluctuations in emergent demand patterns across temporal and spatial dimensions. This study proposes a two-stage stochastic programming model that integrates the EMF allocation problem and the road network design problem for disaster preparedness. The model takes into account uncertainties arising from emergency demand and road network congestion levels under various sizes and timings of disaster occurrences. The first-stage decision involves determining the fleet size of EMFs and identifying which road links' travel time should be reduced. The second-stage decision pertains to the routing and schedule of each EMF for each disaster scenario. Due to considering various sources of uncertainty, the resulting model takes the form of a non-convex mixed-integer nonlinear program (MINLP). This poses computational challenges due to the inclusion of bilinear terms, implicit expressions, and the double-layered structure in the second-stage model, along with integer decision variables. A comprehensive set of techniques is applied to solve the model efficiently. This includes employing linearization techniques, converting the second-stage model into a single-level equivalent, transforming an integer variable into multiple binary variables, and utilizing other methods to equivalently reformulate the model into a mixed-integer linear programming problem (MILP). These transformations render the model amenable to solutions using the integer L-shaped method. A simplified example clarifies the solution procedures of the model and algorithm, establishing the theoretical foundation for their practical implementation. Subsequently, to empirically demonstrate the practicality of the proposed model and algorithm, a real-world case study is conducted, effectively validating their utility.