A Maximum Expected Covering Problem for District Design

Abstract

The optimal location of ambulances in a geographic region is interrelated with how the ambulances are dispatched to patients. Papers in the literature often treat the location and dispatching of ambulance separately. In this paper, we propose a novel mixed integer linear programming (MILP) model that determines how to locate and dispatch ambulances through district design. The model allows for uncertainty in both ambulance travel times and ambulance availability, and it maximizes the coverage level, i.e., the fraction of high-priority calls that can be responded to within a fixed-time threshold. The proposed MILP model determines the stations where ambulances should be located and assigns each call location to the open ambulance stations according to a preference list. The preference list is a rank ordering of the ambulances to assign to patients at a call location, from the most to the least preferred. The preference lists partition the region into a series of response districts that depend on ambulance availability, and the model balances the workload among the servers and maintains contiguity in the first priority response districts. The underlying ambulance queuing dynamics introduce nonlinearities to the model. To maintain a linear model, we use a Hypercube approximation model to estimate several of the inputs, and we provide an iterative algorithm to update the input parameters and solve the resulting MILP model. A computational example illustrates the modeling paradigm and solution algorithm using a real-world example. The results suggest that the reduction in coverage to maintain contiguity and balanced workloads among the ambulances is small.

Enhanced coverage by integrating site interdependencies in capacitated EMS location models

“…All these approaches neglect some of the system dynamics, e.g., server interdependencies, which could lead to errors in determining the level of coverage. Ansari et al [7] propose a model that relaxes the assumption of independently operating ambulances. The Maximum Expected Covering Problem for District Design (MECPDD) is a stochastic model that accounts for travel time and server availability uncertainties and balances the busyness among the servers by introducing lower and upper capacity bounds on the servers.…”

“…In contrast to the idea of Shariat-Mohaymany et al [8] that limit demand capacities for each server, we suggest an extension of the upper bound capacity utilization approach that integrates site interdependencies by limiting the demand capacities for each site. As opposed to the iterative hypercube approach by Ansari et al [7], we use a level based approach to handle the non-linearities imposed by the probabilistic nature of site interdependencies. Thereby, we are able to solve the proposed model using commercial solvers.…”

“…Thus, an unavailable site is supported by other available sites and vice versa takes calls from other sites. These interdependencies are modeled in a formulation with a system-wide busy fraction as an upper bound for each specific ambulance by Ansari et al [7] and Shariat-Mohaymany et al [8]. The back-up assistance on lower priority levels has to be included in the calculation of the busy fraction.…”

“…The back-up assistance on lower priority levels has to be included in the calculation of the busy fraction. In contrast to [7,8], we impose a system-wide upper bound on the busy fraction of each site in the strategic optimization model while ambulances in the system may have different busy fractions.…”

Hypercube Queuing Models in Emergency Service Systems: A State-of-the-Art Review

“…In their work, the purpose of fairness is to make same mean response times and also same server's workload. Ansari et al [43] used this approximation algorithm to estimate the correction factors, the average server workload and the individual server workload and treated them as constants in an MILP model. This model maximizes the number of high-priority calls that can be covered within a time threshold (i.e., the coverage level) and balanced server workload by determining the location of ambulances and dispatching policy simultaneously.…”