# Supported Evacuation for Disaster Relief through Lexicographic Goal Programming

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## Abstract

**:**

## 1. Introduction and Literature Review

- The introduction of dynamism into the supported evacuation model regarding the arrival of potential evacuees to the pick up points, since we consider their arrival may happen at different of points of time during the planning horizon.
- The classification of affected people requiring evacuation according to their particular situation and/or medical condition, leading to several priority levels, so that each group can be treated accordingly.
- The joint consideration of objectives such us number of evacuated people, operation time and cost within a lexicographic approach, in such a way that there is no trade off among them.
- The validation of the proposed model on a realistic case study based on the earthquake and tsunami that hit Indonesia in September 2018.

## 2. Problem Description

## 3. The Proposed Evacuation Model

#### 3.1. Parameters of the Model and Decision Variables

**Sets and indices**

$\mathcal{N}:$ | Set of nodes or involved areas ($i,j\in \mathcal{N}$: $\mathcal{N}=\mathcal{NA}\cup \mathcal{NS}\cup \mathcal{NH}\cup \mathcal{NT}$). |

$\mathcal{E}:$ | Set of edges or arcs between nodes ($(i,j)=\left(ij\right)\in \mathcal{E}$). |

$\mathcal{T}:$ | Discretized time periods ($t,{t}^{\prime}\in \{1,...,\mathcal{T}\}$). |

$\mathcal{H}:$ | Set of types of people to be evacuated ($h\in \mathcal{H}$). |

$\mathcal{K}:$ | Set of types of vehicles available for the evacuation ($k\in \mathcal{K}$). |

$\mathcal{M}:$ | Set of goals to be achieved ($m\in \mathcal{M}$). |

**Parameters**

${d}_{ij}^{k}:$ | Distance from node $i\in \mathcal{N}$ to node $j\in \mathcal{N}$ (or length of arc/edge $(i,j)\in \mathcal{E})$ when traversed by a vehicle of type $k\in \mathcal{K}$. |

$a{v}_{ij}^{k}:$ | Average velocity of arc/edge $(i,j)\in \mathcal{E}$ for a vehicle of type $k\in \mathcal{K}$. |

$b:$ | Budget limitation. |

$q{p}_{it}^{h}:$ | Amount of people of type $h\in \mathcal{H}$ to be evacuated from node $i\in \mathcal{NA}$ at the beginning of period $t\le \mathcal{T}$. |

$ty{p}^{h}:$ | Priority of people of type $h\in \mathcal{H}$ (high or normal). |

$w{p}^{h}:$ | Space that a person of type $h\in \mathcal{H}$ occupies in a vehicle to be transported or in a node of the secure area. |

$c{p}_{i}^{h}:$ | Capacity for people of type $h\in \mathcal{H}$ at node $i\in \mathcal{NS}\cup \mathcal{NH}$. |

$vp{c}^{k}:$ | Capacity of a vehicle of type $k\in \mathcal{K}$. |

$v{a}_{i}^{k}:$ | Number of vehicles of type $k\in \mathcal{K}$ that are available at node $i\in \mathcal{N}$ at the beginning of the operation. |

$f{c}_{ij}^{k}:$ | Fixed cost to traverse the arc/edge $(i,j)\in \mathcal{E}$ with a vehicle of type $k\in \mathcal{K}$, by unit of distance. |

$v{c}_{ij}^{k}:$ | Variable cost to traverse the arc/edge $(i,j)\in \mathcal{E}$ with a vehicle of type $k\in \mathcal{K}$, by unit of distance and cargo transported. |

$v{v}^{k}:$ | Maximum velocity that a vehicle of type $k\in \mathcal{K}$ may reach. |

${\tau}_{ij}^{k}:$ | Time required to travel the arc/edge $(i,j)\in \mathcal{E}$ with a vehicle of type $k\in \mathcal{K}$, calculated as follows: ${\tau}_{ij}^{k}=max\{\frac{{d}_{ij}^{k}}{v{v}^{k}},\frac{{d}_{ij}^{k}}{a{v}_{ij}^{k}}\}$ |

$t{g}_{m}:$ | Aspiration level of goal $m\in \mathcal{M}$. |

**Decision variables**

$B{T}_{t}:$ | 1 if population with high priority has been evacuated in period t and 0, otherwise. |

$B{T}_{t}^{\prime}:$ | 1 if population with normal priority has been evacuated in period t and 0, otherwise. |

$T:$ | Total time required to evacuate population with high priority. |

${T}^{\prime}:$ | Total time required to evacuate the rest of the population. |

${P}_{it}^{h}:$ | Number of people of type $h\in \mathcal{H}$ located at node $i\in \mathcal{N}$ at the beginning of period $t\le \mathcal{T}$. |

$P{L}_{ijt}^{hk}:$ | Number of people of type $h\in \mathcal{H}$ who started to be transported from node $i\in \mathcal{N}$ to node $j\in \mathcal{N}$ in a vehicle of type $k\in \mathcal{K}$ at the beginning of period $t\le \mathcal{T}$. |

${V}_{it}^{k}:$ | Number of vehicles of type $k\in \mathcal{K}$ located at node $i\in \mathcal{N}$ at the beginning of period $t\le \mathcal{T}$. |

$V{L}_{ijt}^{k}:$ | Number of vehicles of type $k\in \mathcal{K}$ that started to move from node $i\in \mathcal{N}$ to node $j\in \mathcal{N}$ at the beginning of period $t\le \mathcal{T}$. |

**Attribute and deviation variables**

$E:$ | Number of people with high priority that are evacuated depending on the characteristics of the particular instance. |

${E}^{\prime}:$ | Number of people with normal priority that are evacuated depending on the characteristics of the particular instance. |

$TM:$ | Total time required to evacuate all the population (max. of T and ${T}^{\prime}$). |

$Cost:$ | Total cost of the operation. |

$D{V}_{m}:$ | Deviation variable with respect to the aspiration level of goal $m\in \mathcal{M}$. |

${P}_{m}:$ | Slack variable with respect to the aspiration level of goal $m\in \mathcal{M}$. |

#### 3.2. Mathematical Model

## 4. Case Study: Palu, Indonesia

#### 4.1. Data

- 16 pickup nodes (R) located at the affected area, easily recognized by the local population and the Search & Rescue Teams, such as mosques, schools, cafes or shopping centers in relatively unaffected or lightly affected streets. The total number of affected nodes and their locations have been established according to the data of affected buildings and affected population obtained from Copernicus, [52].
- 23 temporary shelter nodes (S) are real evacuation sites corresponding to real temporary shelters enabled by BASARNAS (the National Search and Rescue Agency of Indonesia), whose locations and capacities have been obtained from reports of the Government of Indonesia [53,54,55]. Initially, there were only 24 evacuation sites, which amounted to 41 and, finally, to 141 on 4 October 2018. According to the time horizon and the total area of this case study, we have considered 5 large shelters, 6 medium ones and 12 small.
- Hospitals and medical center nodes (H) correspond to real locations of hospitals and medical centers obtained from the same reports used to compile the data regarding temporary shelters. These nodes are the destination of severely injured people and pregnant women, especially.
- Palu airport (A) is a node that corresponds to the real location of Mutiara SIS Al-Jufrie, the airport of the city. For the case study, this airport was a destination point for population to be evacuated by plane to other safer cities, especially for severely injured people that could not be treated at the available hospitals due to a lack of medical supplies or appropriate facilities.

#### 4.2. Results

## 5. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Type | Capacity | Mean Velocity | Fixed Cost | Variable Cost | Quantity Available |
---|---|---|---|---|---|

Helicopter | 3 | 153 km/h | 0.50 $/km | 1.20 $/(km×u.l.) | 8 |

Truck | 18 | 80 km/h | 0.20 $/km | 0.80 $/(km×u.l.) | 410 |

Ambulance | 9 | 80 km/h | 0.10 $/km | 0.50 $/(km×u.l.) | 40 |

Raft | 6 | 3 km/h | 0.05 $/km | 0.12 $/(km×u.l.) | 48 |

AoI 7 | AoI 11 | All AoI | |
---|---|---|---|

Number of inhabitants | 110,265 | 76,176 | 608,521 |

Affected in AoI | 37,776 | 136 | 60,424 |

Type | Priority | Required Space | AoI 7 | AoI 11 |
---|---|---|---|---|

Severely injured adults (SIA) | High | 3 | 171 | 6 |

Pregnant women (PW) | High | 1.5 | 85 | 2 |

Susc. to gender violence (GBV) | High | 1 | 153 | 3 |

Unaccompanied minors (UM) | High | 1 | 98 | 2 |

Rest of the population (RP) | Normal | 1 | 476 | 11 |

**Table 4.**Shelter characteristics and capacity distributions. PW: pregnant women. UM: unaccompanied minors. GBV: women susceptible to gender violence. RP: rest of the population.

ID | Name | Capacity | Classif. | Type | PW | UM | GBV | RP |
---|---|---|---|---|---|---|---|---|

S01 | Silae (Miners) | 500 | medium | normal | 0 | 0 | 0 | 500 |

S02 | Makorem (area) | 3000 | large | normal | 0 | 0 | 0 | 3000 |

S03 | Mako Sabhara P. | 5000 | large | safe | 600 | 1600 | 2800 | 0 |

S04 | Perumahan M.R. | 150 | small | normal | 0 | 0 | 0 | 150 |

S05 | GOR Siranindi | 200 | small | normal | 0 | 0 | 0 | 200 |

S06 | Halaman D. | 100 | small | normal | 0 | 0 | 0 | 100 |

S07 | Masjid Raya | 300 | small | normal | 0 | 0 | 0 | 300 |

S08 | Lap. Anoa | 100 | small | normal | 0 | 0 | 0 | 100 |

S09 | Masjid B.A. | 1500 | large | safe | 180 | 480 | 840 | 0 |

S10 | Bundaran STQ | 500 | medium | normal | 0 | 0 | 0 | 500 |

S11 | Lap. Dayodara | 700 | medium | normal | 0 | 0 | 0 | 700 |

S12 | Samping M.A.F. | 880 | medium | normal | 0 | 0 | 0 | 880 |

S13 | Lagarutu | 511 | medium | normal | 0 | 0 | 0 | 511 |

S14 | Lap Vatulemo | 1000 | large | safe | 120 | 320 | 560 | 0 |

S15 | Jalan Swadaya | 157 | small | normal | 0 | 0 | 0 | 157 |

S16 | Malao Atas | 150 | small | normal | 0 | 0 | 0 | 150 |

S17 | Jalan Maleo | 100 | small | normal | 0 | 0 | 0 | 100 |

S18 | Hal. Perkantoran | 2000 | large | normal | 0 | 0 | 0 | 2000 |

S19 | Sepanjang J.G. | 250 | small | normal | 0 | 0 | 0 | 250 |

S20 | BTN Lasoani | 300 | small | normal | 0 | 0 | 0 | 300 |

S21 | Lap. Kawatuna | 300 | small | normal | 0 | 0 | 0 | 300 |

S22 | Sekitar J.B.R. | 120 | small | normal | 0 | 0 | 0 | 120 |

S23 | Lap. Faqih R. | 500 | medium | normal | 0 | 0 | 0 | 500 |

Constraints | Variables | Non-Zeroes | Discrete Var. | |
---|---|---|---|---|

Level 1 | 25,854 | 87,411 | 1,960,669 | 87,409 |

Level 2 | 25,856 | 87,414 | 1,960,694 | 87,410 |

Level 3 | 25,957 | 87,467 | 1,972,821 | 87,458 |

Level 4 | 25,959 | 87,470 | 2,050,729 | 87,458 |

**Table 6.**Distribution of evacuees at the end of the operation by category. SIA: Severely injured adults. PW: pregnant women. UM: unaccompanied minors. GBV: women susceptible to gender violence. RP: rest of the population.

SIA | PW | UM | GBV | RP | |
---|---|---|---|---|---|

R09 | 1 | ||||

R11 | 144 | 521 | |||

S01 | 500 | ||||

S02 | 2997 | ||||

S03 | 600 | 1565 | 2433 | ||

S04 | 150 | ||||

S05 | 200 | ||||

S06 | 100 | ||||

S07 | 300 | ||||

S09 | 180 | 167 | 198 | ||

S10 | 270 | ||||

S12 | 863 | ||||

S14 | 12 | ||||

S18 | 107 | ||||

H01 | 750 | 250 | |||

H02 | 750 | 230 | |||

H03 | 727 | 250 | |||

H04 | 527 | ||||

A01 | 6 | 1 | 1333 |

% Critical Evacuees | % Non-Critical Evacuees | Evacuation Time (h) | Operation Cost ($) | |
---|---|---|---|---|

Base case | 98.36 | 92.89 | 72 | 54,956 |

No Airport | 97.85 | 92.94 | 72 | 59,019 |

No Hospitals | 88.33 | 95.84 | 72 | 56,204 |

No Helicopters | 97.01 | 93.01 | 72 | 57,792 |

No rafts | 95.81 | 92.99 | 69 | 70,142 |

No trucks | 40.52 | 2.06 | 72 | 5292 |

Split in two | 75.92 | 51.23 | 39 | 34,992 |

Asp. Levels 75–50 | 75.00 | 50.00 | 30 | 151,340 |

Asp. Levels 50–75 | 50.00 | 75.00 | 36 | 171,516 |

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**MDPI and ACS Style**

Flores, I.; Ortuño, M.T.; Tirado, G.; Vitoriano, B. Supported Evacuation for Disaster Relief through Lexicographic Goal Programming. *Mathematics* **2020**, *8*, 648.
https://doi.org/10.3390/math8040648

**AMA Style**

Flores I, Ortuño MT, Tirado G, Vitoriano B. Supported Evacuation for Disaster Relief through Lexicographic Goal Programming. *Mathematics*. 2020; 8(4):648.
https://doi.org/10.3390/math8040648

**Chicago/Turabian Style**

Flores, Inmaculada, M. Teresa Ortuño, Gregorio Tirado, and Begoña Vitoriano. 2020. "Supported Evacuation for Disaster Relief through Lexicographic Goal Programming" *Mathematics* 8, no. 4: 648.
https://doi.org/10.3390/math8040648