Development and comparison of two new multi-period queueing reliability models using discrete-event simulation and a simulation–optimization approach

https://doi.org/10.1016/j.cie.2022.108068Get rights and content

Highlights

  • Two optimization models are developed to overcome Q-MALP model limitations.

  • Q-MALP-M1 integrates multi-period redeployment of several types of ambulances.

  • Q-MALP-M2 modifies the way α-reliability is considered.

  • Simulation proves that Q-MALP-M2 improves coverage and average waiting time.

  • Q-MALP-M2 performs better than OptQuest in terms of coverage.

Abstract

This paper aims to develop a new mathematical model for optimizing ambulance deployment and redeployment. For this purpose, two mathematical models have been proposed and compared. The first model is Q-MALP-M1, an extension of the classical model Q-MALP, which is improved by integrating the multi-period redeployment of several types of ambulances. The second model is Q-MALP-M2, a modified version of Q-MALP. In addition to the improvements introduced by the first model, the Q-MALP-M2 overcomes the main Q-MALP model limitation, which is the α-reliability coverage. The Q-MALP-M2 changes the way coverage reliability is considered; instead of maximizing coverage with a fixed reliability level, it maximizes coverage with incremental levels of reliability depending on the number of available ambulances. Also, a discrete-event simulation model was constructed to compare the two mathematical models. A case study was conducted on the Civil Protection services of the Fez-Meknes region, Morocco. A series of scenarios combining various numbers of potential sites and ambulances were solved and simulated. Simulation results proved that the Q-MALP-M2 model, compared to the Q-MALP-M1 model, performs better in terms of coverage and average waiting time. It distributes the ambulances to achieve maximum coverage without necessarily being with the desired level of reliability. Finally, the Q-MALP-M2 model was compared to the simulation–optimization using OptQuest. In terms of coverage, the best-performing solution was sometimes generated by Q-MALP-M2 and other times by OptQuest. However, the Q-MALP-M2 model, in all cases, gives significantly improved results, and its execution time is much shorter than OptQuest. In terms of average waiting time, the results are not conclusive. The best-performing solutions were the results of Q-MALP-M2 in some scenarios and OptQuest in other scenarios. The discrepancies between the generated average waiting times were substantial on both sides.

Introduction

The main objective of medical transport in emergencies is to save lives by providing adequate and timely transport service, from the emergency call reception to the delivery of the patient to an adapted hospital (Aringhieri et al., 2017). Response time, the time elapsed between the reception of an emergency call and the arrival of the ambulance at the scene, has a determining impact on the patient's health and well-being (Carvalho et al., 2020). It is a crucial indicator for assessing the effectiveness (Yuangyai et al., 2020), the quality (Reuter-Oppermann et al., 2017), and the performance (Bélanger et al., 2019) of prehospital medical emergency services. If emergency requests are handled promptly, better clinical outcomes and higher patient satisfaction will result (Aringhieri et al., 2017, Frichi et al., 2019a).

Often countries define threshold values for the ambulance response time that depend, among other things, on the severity of the patient's condition and the nature of the area to be served (urban or rural). In several European countries, a response time of less than 8 min is recommended for first-priority emergencies (Reuter-Oppermann et al., 2017). In the United States, the US EMS ACT sets that 95% of requests should be served in less than 10 min in urban areas and less than 30 min in rural areas. Similar standards are applied in other parts of the world; for example, in London and Montreal, 95% of requests should be served within 14 and 10 min, respectively (dos Cabral et al., 2018). In Morocco, there are no standards or regulations that specify response time limits for emergencies.

Response time is influenced by ambulance deployment; distribution of ambulances in space (Liu et al., 2016). Quite often, deployment and redeployment problems use the criterion of travel time from the ambulance site to the demand area. The goal is to ensure optimal coverage of demand areas; a demand area is covered if it can be reached by ambulance in less than the defined travel time limit (Marianov, 2017). In this context, considerable effort has been provided and accumulated over the years to develop effective mathematical models of ambulance deployment. One of the models that have made notable advances in mathematical modeling is the Queueing Maximal Availability Location Problem (Q-MALP) formulated by Marianov and ReVelle (1996). The Q-MALP model is an improved version of the Maximal Availability Location Problem (MALP) developed by ReVelle and Hogan (1989), which maximizes the coverage of demand areas with a reliability level α. The α-reliability is a chance constraint on service availability. It refers to the minimum probability that an ambulance will be available within a standard travel time radius when a request is received. Models using the α-reliability locate ambulances to maximize the covered population for which the service is available with a minimum desired probability level α (ReVelle & Hogan, 1989).

The Q-MALP model, compared to the MALP model, allows for considering the ambulance's area-specific busy fraction and relaxing the ambulances' independence assumption. However, the Q-MALP model is criticized for at least three weaknesses:

  • 1.

    It is a static model; the operational dimension is absent.

  • 2.

    The model optimizes the deployment of a single type of ambulance, whereas most emergency medical services operate with two or more types of ambulances.

  • 3.

    The model is particularly criticized for the α-reliability coverage, which requires coverage of a demand area with a minimum number b of ambulances to be considered covered with a reliability level α. Consequently, any demand area covered with a reliability level less than α is not considered in the Q-MALP objective function.

This paper aims to propose two models Q-MALP-M1 and Q-MALP-M2, to improve the Q-MALP model. The Q-MALP-M1 model remedies the first two weaknesses of the Q-MALP. It allows the multi-period deployment of ambulances and considers several types of ambulances. The Q-MALP-M1 model maintains the way Q-MALP coverage is defined. Therefore, it is only a multi-period extension with several types of ambulances. The Q-MALP-M2 model, in addition to the improvements enabled by Q-MALP-M1, adds a third improvement that overcomes the third weakness. The Q-MALP-M2 model does not aim at covering demand areas with a fixed reliability level α. Instead, it considers several reliability levels αm (m = 1, 2, …M; α1 < α2<… αM) in its objective function so that ambulances are distributed to cover the maximum demand areas with different reliability levels. The multi-period redeployment in both models allows ambulances to change location between time periods. However, long travel times between the origin and destination sites are not desirable as the new system configuration must be reached within a time frame not exceeding a certain limit (Van Barneveld et al., 2016). To avoid long redeployment travel time, we include a constraint on the maximum ambulance redeployment travel time. We also propose the use of discrete-event simulation to compare the two models and the simulation–optimization using OptQuest to evaluate the performance of the best model.

Q-MALP-M1 and Q-MALP-M2 models are applied to Civil Protection services in the Fez-Meknes region, Morocco. According to several studies, medical transport in Morocco is a major obstacle to healthcare access (Frichi et al., 2020b). It lacks coordination among its actors, an insufficient ambulance fleet, and delayed responses (Frichi et al., 2019a, Frichi et al., 2020a). As such, an optimal ambulance location would overcome many of these problems.

The remainder of this paper is organized as follows. The next section presents the study-related work concerning mathematical and simulation modeling of ambulance deployment and redeployment problems. Section 3 describes in detail the developed models, as well as the simulation model. Section 4 describes the models' parameters estimation. Section 5 presents the results of comparing the two models using the simulation model and comparing the best-performing model with OptQuest. Section 6 analyzes the results and presents the limitations of the study. Section 7 concludes the paper and gives insights into future researches.

Section snippets

Mathematical programming models

A considerable effort has been provided and accumulated over the years in modeling ambulance deployment and redeployment problems. The literature on mathematical models has evolved from single-coverage deterministic models to multiple-coverage models, probabilistic, and dynamic models.

Single-coverage deterministic models consider that all the parameters are known with certainty. One of the first deterministic models is the LSCP model formulated by Toregas et al. (1971). It seeks to locate a

Notation

Sets and indexes

  • I: the set of demand areas;

  • i: index of demand areas, i = 1, 2, … n;

  • J: the set of potential sites for ambulance base location;

  • j: index of potential sites, j = 1, 2, … m;

  • K: the set of ambulance types, K = {ALS = 1, BLS = 2};

  • k: index of ambulance types, k = 1, 2;

  • Ʈ: the set of time periods, Ʈ = {1, 2, …T};

  • Ʈ’: the set of time periods excluding the last period, Ʈ’ = Ʈ-{T} = {1, 2, … T-1};

  • t: index of time periods, t = 1, 2, … T.

Parameters

  • Ditk: demand of type k∊K at demand area i∊I in

Model parameters specification

To run the mathematical and simulation models, all parameters are estimated based on a case study of the Fez-Meknes region, Morocco. Parameters' specification comprises the definition of time periods, the identification of demand areas and potential sites, the estimation of demand values, arrival rates, and service times.

Solutions of mathematical models

Before solving the mathematical models, it is necessary to determine the values of bitk for the Q-MALP-M1 model and bitkαm for the Q-MALP-M2 model, representing, respectively, the minimum number of ambulances of type k∊K that must cover the demand area i∊I to ensure coverage with reliability α and αm. For this purpose, we developed a program coded in MATLAB R2020a, allowing the calculation of bitk and bitkαm.

Both models were solved using IBM ILOG CPLEX 12.10 on a laptop Intel(R) Core(TM)

Performance of the Q-MALP-M2 model

To optimize ambulance deployment and redeployment, we developed two mathematical models Q-MALP-M1 and Q-MALP-M2. Both models are extensions of the classical Q-MALP model (Marianov & ReVelle, 1996), which maximizes the coverage of demand areas with reliability α. The Q-MALP model is a remarkable advance in modeling ambulance deployment problems. It considers the fact that an ambulance may not be available when requested and the dependence between ambulances. However, the Q-MALP model is

Conclusion

This paper used mathematical modeling and simulation approaches to optimize ambulance deployment and multi-period redeployment. The first approach, mathematical modeling, is analytical, generating optimal solutions under certain simplifying assumptions. The second approach, simulation modeling, is descriptive and offers the possibility to evaluate, without simplifying assumptions, the performance of a solution, including mathematical modeling solutions.

Two new mathematical models Q-MALP-M1 and

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

References (60)

  • J. Goldberg et al.

    A simulation model for evaluating a set of emergency vehicle base locations: Development, validation, and usage

    Socio-Economic Planning Sciences

    (1990)
  • Y. Liu et al.

    A double standard model for allocating limited emergency medical service vehicle resources ensuring service reliability

    Transportation Research Part C: Emerging Technologies

    (2016)
  • V. Marianov et al.

    The queueing maximal availability location problem: A model for the siting of emergency vehicles

    European Journal of Operational Research

    (1996)
  • H. Niessner et al.

    A dynamic simulation–optimization approach for managing mass casualty incidents

    Operations Research for Health Care

    (2018)
  • H.K. Rajagopalan et al.

    A multiperiod set covering location model for dynamic redeployment of ambulances

    Computers & Operations Research

    (2008)
  • J.F. Repede et al.

    Developing and validating a decision support system for locating emergency medical vehicles in Louisville, Kentucky

    European Journal of Operational Research

    (1994)
  • V. Schmid et al.

    Ambulance location and relocation problems with time-dependent travel times

    European Journal of Operational Research

    (2010)
  • P. Sorensen et al.

    Integrating expected coverage and local reliability for emergency medical services location problems

    Socio-Economic Planning Sciences

    (2010)
  • T. Ünlüyurt et al.

    Estimating the performance of emergency medical service location models via discrete event simulation

    Computers & Industrial Engineering

    (2016)
  • T.C. Van Barneveld et al.

    The effect of ambulance relocations on the performance of ambulance service providers

    European Journal of Operational Research

    (2016)
  • T. van Barneveld et al.

    Real-time ambulance relocation: Assessing real-time redeployment strategies for ambulance relocation

    Socio-Economic Planning Sciences

    (2018)
  • P.L. Van Den Berg et al.

    Time-dependent MEXCLP with start-up and relocation cost

    European Journal of Operational Research

    (2015)
  • W. Yang et al.

    Simulation modeling and optimization for ambulance allocation considering spatiotemporal stochastic demand

    Journal of Management Science and Engineering

    (2019)
  • M.A. Zaffar et al.

    Coverage, survivability or response time: A comparative study of performance statistics used in ambulance location models via simulation–optimization

    Operations Research for Health Care

    (2016)
  • F. Zeinali et al.

    Resource planning in the emergency departments: A simulation-based metamodeling approach

    Simulation Modelling Practice and Theory

    (2015)
  • T. Andersson et al.

    Decision support tools for ambulance dispatch and relocation

    Journal of the Operational Research Society

    (2007)
  • R. Aringhieri et al.

    A simulation and online optimization approach for the real-time management of ambulances

    Winter Simulation Conference

    (2018)
  • R. Aringhieri et al.

    Supporting decision making to improve the performance of an Italian Emergency Medical Service

    Annals of Operations Research

    (2016)
  • V. Bélanger et al.

    Déploiement et redéploiement des véhicules ambulanciers dans la gestion d’un service préhospitalier d’urgence

    INFOR: Information Systems and Operational Research

    (2012)
  • A. Bettinelli et al.

    Simulation and optimization models for emergency medical systems planning

    Journal of Emergency Management

    (2014)
  • Cited by (11)

    • A simulation optimization approach to investigate resource planning and coordination mechanisms in emergency systems

      2022, Simulation Modelling Practice and Theory
      Citation Excerpt :

      Strategic planning includes determining the capacity and location of ambulance bases (i.e., waiting positions where ambulances are placed between rescues) [4], dimensioning human and material resources (e.g., physicians, nurses, ambulances, beds) [2,5-8], and designing ED physical layout [9]. Tactical planning includes ambulance deployment (i.e., allocating ambulances to bases) [10–14] and personnel scheduling [15–17]. Operational control includes ambulance dispatch [10,13,18], and relocation in response to changes in the EMS system (e.g., demand pattern, number of available ambulances…) [19–21], triage policy in ED [22–24] and ambulance diversion policy (i.e., refusing new incoming ambulance and re-route them to nearby hospitals to mitigate overcrowding) [25–27].

    • Deterministic and stochastic approaches in computer modeling and simulation

      2023, Deterministic and Stochastic Approaches in Computer Modeling and Simulation
    View all citing articles on Scopus
    View full text