# Optimizing Ambulance Allocation in Dynamic Urban Environments: A Historic Data-Driven Approach

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Problem Description and Mathematical Model

#### 3.1. Problem Description

#### 3.2. Notations

#### 3.3. Mathematical Model

## 4. Numerical Experiments

#### 4.1. Dataset

#### 4.2. Parameters

#### 4.3. Result

#### 4.4. Case Study

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Parameter Settings

## References

- Bedard, A.F.; Mata, L.V.; Dymond, C.; Moreira, F.; Dixon, J.; Schauer, S.G.; Ginde, A.A.; Bebarta, V.; Moore, E.E.; Mould-Millman, N.K. A scoping review of worldwide studies evaluating the effects of prehospital time on trauma outcomes. Int. J. Emerg. Med.
**2020**, 13, 64. [Google Scholar] [CrossRef] [PubMed] - Toregas, C.; Swain, R.; ReVelle, C.; Bergman, L. The location of emergency service facilities. Oper. Res.
**1971**, 19, 1363–1373. [Google Scholar] [CrossRef] - Rajagopalan, H.K.; Saydam, C.; Xiao, J. A multiperiod set covering location model for dynamic redeployment of ambulances. Comput. Oper. Res.
**2008**, 35, 814–826. [Google Scholar] [CrossRef] - Dibene, J.C.; Maldonado, Y.; Vera, C.; de Oliveira, M.; Trujillo, L.; Schütze, O. Optimizing the location of ambulances in Tijuana, Mexico. Comput. Biol. Med.
**2017**, 80, 107–115. [Google Scholar] [CrossRef] [PubMed] - Yuangyai, C.; Nilsang, S.; Cheng, C.Y. Robust ambulance base allocation strategy with social media and traffic congestion information. J. Ambient. Intell. Humaniz. Comput.
**2020**, 1–14. [Google Scholar] [CrossRef] - Church, R.; ReVelle, C. The maximal covering location problem. In Papers of the Regional Science Association; Springer: Berlin/Heidelberg, Germany, 1974; Volume 32, pp. 101–118. [Google Scholar]
- Daskin, M.S. A maximum expected covering location model: Formulation, properties and heuristic solution. Transp. Sci.
**1983**, 17, 48–70. [Google Scholar] [CrossRef] - Van den Berg, P.L.; Aardal, K. Time-dependent MEXCLP with start-up and relocation cost. Eur. J. Oper. Res.
**2015**, 242, 383–389. [Google Scholar] [CrossRef] - Bélanger, V.; Kergosien, Y.; Ruiz, A.; Soriano, P. An empirical comparison of relocation strategies in real-time ambulance fleet management. Comput. Ind. Eng.
**2016**, 94, 216–229. [Google Scholar] [CrossRef] - Enayati, S.; Mayorga, M.E.; Rajagopalan, H.K.; Saydam, C. Real-time ambulance redeployment approach to improve service coverage with fair and restricted workload for EMS providers. Omega
**2018**, 79, 67–80. [Google Scholar] [CrossRef] - Bélanger, V.; Lanzarone, E.; Nicoletta, V.; Ruiz, A.; Soriano, P. A recursive simulation-optimization framework for the ambulance location and dispatching problem. Eur. J. Oper. Res.
**2020**, 286, 713–725. [Google Scholar] [CrossRef] - Hajiali, M.; Teimoury, E.; Rabiee, M.; Delen, D. An interactive decision support system for real-time ambulance relocation with priority guidelines. Decis. Support Syst.
**2022**, 155, 113712. [Google Scholar] [CrossRef] - Mohri, S.S.; Haghshenas, H. An ambulance location problem for covering inherently rare and random road crashes. Comput. Ind. Eng.
**2021**, 151, 106937. [Google Scholar] [CrossRef] - Carvalho, A.; Captivo, M.; Marques, I. Integrating the ambulance dispatching and relocation problems to maximize system’s preparedness. Eur. J. Oper. Res.
**2020**, 283, 1064–1080. [Google Scholar] [CrossRef] - Dolan, E.; Johnson, N.; Kepler, T.; Lam, H.; Lelo de Larrea, E.; Mohammadi, S.; Olivier, A.; Quayyum, A.; Sanabria, E.; Sethuraman, J.; et al. Hospital Load Balancing: A Data-Driven Approach to Optimize Ambulance Transports during the COVID-19 Pandemic in New York City. SSRN
**2022**. preprint. [Google Scholar] [CrossRef] - Majlesinasab, N.; Maleki, M.; Nikbakhsh, E. Performance evaluation of an EMS system using queuing theory and location analysis: A case study. Am. J. Emerg. Med.
**2022**, 51, 32–45. [Google Scholar] [CrossRef] [PubMed] - Ghasemi, P.; Babaeinesami, A. Simulation of fire stations resources considering the downtime of machines: A case study. J. Ind. Eng. Manag. Stud.
**2020**, 7, 161–176. [Google Scholar] - Jahangiri, S.; Abolghasemian, M.; Ghasemi, P.; Chobar, A.P. Simulation-based optimization: Analysis of the emergency department resources under COVID-19 conditions. Int. J. Ind. Syst. Eng.
**2023**, 43, 1–19. [Google Scholar] [CrossRef] - Ghasemi, P.; Goodarzian, F.; Muñuzuri, J.; Abraham, A. A cooperative game theory approach for location-routing-inventory decisions in humanitarian relief chain incorporating stochastic planning. Appl. Math. Model.
**2022**, 104, 750–781. [Google Scholar] [CrossRef] - Ghasemi, P.; Hemmaty, H.; Pourghader Chobar, A.; Heidari, M.R.; Keramati, M. A multi-objective and multi-level model for location-routing problem in the supply chain based on the customer’s time window. J. Appl. Res. Ind. Eng.
**2022**, in press. [Google Scholar] - Uber H3. Available online: https://h3geo.org/ (accessed on 29 June 2023).

**Figure 1.**(

**a**) shows the regions to be analyzed. (

**b**) is a map of the demand information after dividing the region into hexagonal cells from the historical data for the three regions we want to analyze. The darker the color, the higher the demand is.

**Figure 3.**Number of zero zones where it is difficult for an ambulance to cover within the golden time.

**Figure 4.**(

**a**) shows spring; (

**b**) shows summer; (

**c**) shows fall; (

**d**) shows winter. Red cells represent ambulances deployed to existing fire stations. Black cells represent ambulances deployed to new potential centers. Black-bordered cells with no color are base centers, meaning that no ambulances are deployed.

Sets | Description |
---|---|

$\mathcal{B}$ | A set of Boroughs, $b\in \mathcal{B}$ |

$\mathcal{I}$ | A set of demand cells, $i\in \mathcal{I}$ |

${\mathcal{I}}_{b}$ | A set of demand cells in borough b, ${\mathcal{I}}_{b}\subset \mathcal{I}$ |

$\mathcal{C}$ | A set of centers to assign ambulances to, $c\in \mathcal{C}$ |

${\mathcal{C}}_{new}$ | A set of potential centers to assign ambulances to, ${\mathcal{C}}_{new}\subset \mathcal{C}$ |

${\mathcal{C}}_{b}$ | A set of centers for assigning ambulances in borough $b\in \mathcal{B}$, ${\mathcal{C}}_{b}\subset \mathcal{C}$ |

$\mathcal{T}$ | A set of planning horizons, $t\in \mathcal{T}=\{1,2,3,\dots ,T\}$ |

DecisionVariables | Description |

${x}_{ct}$ | Number of ambulances to assign to location $c\in \mathcal{C}$ at time t |

${r}_{c{c}^{\prime}t}$ | Number of ambulances relocated from location $c\in \mathcal{C}$ at time $t-1$ to location ${c}^{\prime}\in \mathcal{C}\setminus \left\{c\right\}$ at time t. |

${y}_{ikt}$ | 1 if at least k ambulances cover demand cell $i\in \mathcal{I}$ at time t, otherwise 0 |

${z}_{c}$ | 1 if the new location $c\in {\mathcal{C}}_{new}$ is used at least once, otherwise 0 |

Parameters | Description |

${\mathcal{P}}_{t}$ | Number of ambulances that should be assigned at time t |

${d}_{it}$ | Demand in cell $i\in \mathcal{I}$ at time t |

${a}_{cit}$ | 1 if demand cell $i\in \mathcal{I}$ is covered by location $c\in \mathcal{C}$ at time t, otherwise 0 |

${q}_{t}$ | The probability that, after receiving an emergency call at time t, an ambulance will arrive within a golden time. |

$\alpha $ | Reward parameters for demand coverage expressions |

${\beta}_{c}$ | Penalty parameters for placing an ambulance in a new potential center $c\in {\mathcal{C}}_{new}$ |

${\gamma}_{c{c}^{\prime}}$ | The cost of relocating an ambulance from location $c\in \mathcal{C}$ to location ${c}^{\prime}\in \mathcal{C}\setminus \left\{c\right\}$ |

$\delta $ | Reward parameter for cells covered by at least one ambulance within the golden hour |

Before | Mathematical Model | |||||||
---|---|---|---|---|---|---|---|---|

Demand Cover (a) | Open Cost (b) | Relocation Cost (c) | Non-Zero Zone Reward (d) | Demand Cover (a) | Open Cost (b) | Relocation Cost (c) | Non-Zero Zone Reward (d) | |

Value | 66.78 | 0 | 0 | 156 | 67.67 | 6 | −2.4 | 372 |

${\mathit{P}}_{\mathit{t}}$ | Before | Total Average | Seongbuk-gu | Dongdaemun-gu | Seongdong-gu |
---|---|---|---|---|---|

15 | 171 | 115 | 51 | 17.75 | 46.25 |

16 | 171 | 112.25 | 51 | 15 | 46.25 |

17 | 171 | 90 | 38 | 12.5 | 39.5 |

18 | 171 | 90.75 | 36 | 10 | 44.75 |

19 | 171 | 83 | 35 | 10 | 38 |

**Table 4.**Seasonal changes in the number of zero zones due to changes in the number of ambulances in Seongdong-gu.

${\mathit{P}}_{\mathit{t}}$ | Before | Mathematical Model | ||||
---|---|---|---|---|---|---|

March–May | June–August | September–November | December–February | Average | ||

15 | 52 | 29 | 52 | 52 | 52 | 46.25 |

16 | 52 | 29 | 52 | 52 | 52 | 46.25 |

17 | 52 | 52 | 40 | 40 | 26 | 39.5 |

18 | 52 | 29 | 50 | 50 | 50 | 44.75 |

19 | 52 | 38 | 38 | 38 | 38 | 38 |

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**MDPI and ACS Style**

Kang, S.; Cheong, T.
Optimizing Ambulance Allocation in Dynamic Urban Environments: A Historic Data-Driven Approach. *Appl. Sci.* **2023**, *13*, 11671.
https://doi.org/10.3390/app132111671

**AMA Style**

Kang S, Cheong T.
Optimizing Ambulance Allocation in Dynamic Urban Environments: A Historic Data-Driven Approach. *Applied Sciences*. 2023; 13(21):11671.
https://doi.org/10.3390/app132111671

**Chicago/Turabian Style**

Kang, Seongho, and Taesu Cheong.
2023. "Optimizing Ambulance Allocation in Dynamic Urban Environments: A Historic Data-Driven Approach" *Applied Sciences* 13, no. 21: 11671.
https://doi.org/10.3390/app132111671