Drone-Delivery Network for Opioid Overdose: : Nonlinear Integer Queueing-Optimization Models and Methods

This study proposes an emergency network design model that uses a fleet of drones to deliver naloxone in response to opioid overdoses. The network is represented as a collection of M/G/K queueing systems with variable capacity and service time modelled as a decision-dependent random variable. The model is a complex queuing-based optimization problem which locates drone bases and dispatches drones to opioid incidents. The authors devise an efficient solution framework in which they linearize the multiple nonlinearities (fractional, polynomial, exponential, factorial terms), derive an equivalent mixed-integer linear reformulation, and design an outer approximation branch-and-cut algorithm. The authors demonstrate the generalizablity of the approach to any problem minimizing the response time of M/G/K queueing systems with unknown capacity. Tests on Virginia Beach data reveal that drones can decrease response time by 82%, increase survival chance by more than 273%, save up to 33 additional lives annually, and provide up to 279 additional quality-adjusted life years.

We propose a new stochastic emergency network design model that uses a fleet of drones to quickly deliver naloxone in response to opioid overdoses. The network is represented as a collection of M/G/K queueing systems in which the capacity K of each system is a decision variable, and the service time is modeled as a decision-dependent random variable. The model is a queuing-based optimization problem which locates fixed (drone bases) and mobile (drones) servers and determines the drone dispatching decisions and takes the form of a nonlinear integer problem intractable in its original form. We develop an efficient reformulation and algorithmic framework. Our approach reformulates the multiple nonlinearities (fractional, polynomial, exponential, factorial terms) to give a mixed-integer linear programming (MILP) formulation. We demonstrate its generalizability and show that the problem of minimizing the average response time of a collection of M/G/K queueing systems with unknown capacity K is always MILP-representable. We design an outer approximation branch-and-cut algorithmic framework that is computationally efficient and scales well. The analysis based on real-life data reveals that drones can in Virginia Beach: (1) decrease the response time by 82%, (2) increase the survival chance by more than 273%, (3) save up to 33 additional lives per year, and (4) provide annually up to 279 additional quality-adjusted life years.

Funding: M. A. Lejeune acknowledges the support of the National Science Foundation [Grant ECCS-2114100] and the Office of Naval Research [Grant N00014-22-1-2649].

Supplemental Material: The online appendices are available at https://doi.org/10.1287/opre.2022.0489.