# A Voting TOPSIS Approach for Determining the Priorities of Areas Damaged in Disasters

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Disaster Logistics

#### 2.2. Priority Determination of the Damaged Areas for Disaster Logistics

#### 2.3. TOPSIS

## 3. Voting TOPSIS

- -
- Step 1: Define criteria and alternatives for the given MCDM problem
- -
- Step 2: Apply PWC to determine the relative weights of the criteria
- -
- Step 3: The DMs vote for alternatives and generate a decision matrix that aggregates the votes using the DEA-based model
- -
- Step 4: Apply TOPSIS to the decision matrix to calculate the final relative weights of the alternatives

#### 3.1. Step 1: Define Criteria and Alternatives

- (C1)
- Degree of damage: This indicates how severe the damage is to the building or any infrastructure. The damage includes communication cut off, water cut off, and electricity cut off. These damages are closely related to the survival of the victims and the possibility of a quick rescue. Thus, the higher the degree of damage is for any area, the higher the priority of relief distribution is given.
- (C2)
- Difficulty of approaching a disaster area: This indicates how difficult it is for the rescue team to approach the target area. This is dependent on the road connection and/or condition. In practice, relief distribution may be delayed to any areas that have poor road connections and conditions because a rescue team must distribute relief to many areas as quickly as possible. Thus, the higher the difficulty of approaching for any area, the higher the priority of relief distribution is given.
- (C3)
- Ratio of the vulnerable: This indicates the ratio of the number of elderly and children to the total number of people in the disaster areas. We could consider the number of the trapped elderly and children instead of the number of elderly and children when checking this ratio of the vulnerable. However, the rescue team might not know the exact number of trapped people, including the elderly and children, in disaster situations. Thus, to calculate the ratio of the vulnerable, we use the overall number of the elderly and children. When there are more elderly and children in any disaster area, the priority of relief distribution in that area must be set higher than that in other areas.
- (C4)
- Distance between a disaster area and a distribution center: Since the distance between a disaster area and a distribution center is directly related to the delivery time of relief, the distance is important for determining the priority of relief distribution. The longer the distance from a distribution center to a given disaster area, the higher the delivery priority given to that area.

#### 3.2. Step 2: Fuzzy PWC for Determining the Relative Weights of the Criteria

#### 3.3. Step 3: DMs’ Vote for Alternatives and Aggregate the Votes

- $i$: the index of the criteria ($i=1,\dots ,N)$
- $j$: the index of the alternatives ($j=1,\dots ,M)$
- $k$: the index of the evaluation grade ($k=1,\dots ,K)$
- ${x}_{k}$: the relative importance weight attached to the $k$th grade
- ${v}_{jk}$: the numbers (i.e., vote) of the DMs who evaluate the alternative $j$ to the $k$th grade
- ${z}_{ij}$: the total score of the alternative $j$ for criterion $i$

#### 3.4. Step 4: Apply TOPSIS to the Decision Matrix

- (1)
- Normalize the decision matrix as follows:$${y}_{ij}=\frac{{z}_{ij}}{\sqrt{{{\displaystyle \sum}}_{j=1}^{M}{z}_{ij}^{2}}}\mathrm{for}i=1,\dots ,N;j=1,\dots ,M$$
- (2)
- Compute the weighted normalized decision matrix. The weighted normalized value ${t}_{ij}$ is calculated as follows:$${t}_{ij}={w}_{i}{y}_{ij}\mathrm{for}\text{}i=1,\dots ,N;j=1,\dots ,M$$
- (3)
- Define the positive ideal solution (PIS) and the negative ideal solution (NIS) on each criterion of the weighted normalized decision matrix. The PIS (${A}^{+}$) and NIS (${A}^{-}$) can be defined as follows:$${A}^{+}=\left({t}_{1}^{+},\dots ,{t}_{N}^{+}\right)$$$${A}^{-}=\left({t}_{1}^{-},\dots ,{t}_{N}^{-}\right)$$
- (4)
- Calculate the Euclidean distance for each alternative to the ideal solutions using the following equation:The separation of an alternative $j$ from the PIS is given as:$${D}_{j}^{+}=\sqrt{{{\displaystyle \sum}}_{i=1}^{N}{\left({t}_{ij}-{t}_{i}^{+}\right)}^{2}}\text{\hspace{1em}}\mathrm{for}\text{}j=1,\dots ,\text{}M$$Similarly, the separation of an alternative $j$ from the NIS is given as:$${D}_{j}^{-}=\sqrt{{{\displaystyle \sum}}_{i=1}^{N}{\left({t}_{ij}-{t}_{i}^{-}\right)}^{2}}\mathrm{for}\text{}j=1,\dots ,\text{}M$$
- (5)
- Calculate the relative closeness coefficient (RCC) of the alternative $j$ with respect to the PIS and NIS. The RCC is always between 0 and 1. If an alternative is closer to the positive ideal than the negative ideal, then the RCC approaches 1. If an alternative is closer to the negative ideal than the positive ideal, the RCC approaches 0. The RCC is defined as follows:$${\mathrm{RCC}}_{j}=\frac{{D}_{j}^{-}}{{D}_{j}^{+}+{D}_{j}^{-}}\hspace{1em}\mathrm{for}j=1,\dots ,M$$

#### 3.5. Overall Procedure of the Proposed Approach

## 4. Case Analysis

#### 4.1. Background

#### 4.2. Data Acquisition

#### 4.3. Applying the Voting TOPSIS

**(1) Fuzzy PWC for Determining the Relative Weights of the Criteria**

**(2) DMs’ Vote for Alternatives and Aggregate the Votes**

**(3) Apply TOPSIS to the Decision Matrix**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Criteria | References |
---|---|

(C1) Degree of damage | Sheu [3]; Yi and Özdamar [32]; Sheu [14]; Wisetjindawat [33]; Holguín-Veras et al. [4] |

(C2) Difficulty of approaching a disaster area | Holguín-Veras et al. [4]; Rivera-Royero et al. [17]; Noyan et al. [34]; Yu et al. [35] |

(C3) Ratio of the vulnerable | Sheu [3]; Sheu [14]; Marcelin et al. [36]; Balcik et al. [37] |

(C4) Distance between a disaster area and a distribution center | Holguín-Veras et al. [4]; Practitioners of this case study (the disaster relief planners from the Zhangbang Town, Huanggang City, Hubei Province, China) |

Linguistic Judgments | Fuzzy Number | |
---|---|---|

Equally important | Equally preferred | (1, 1, 2) |

Moderately more important | Moderately more preferred | (1, 3, 5) |

Strongly more important | Strongly more preferred | (3, 5, 7) |

Very strongly more important | Very strongly more preferred | (5, 7, 9) |

Absolutely more important | Absolutely more preferred | (8, 9, 9) |

C1 | C2 | C3 (%) | C4 (km) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

VH | H | M | L | VL | VH | H | M | L | VL | |||

Jiutan | 1 | 4 | 0 | 0 | 0 | 4 | 1 | 0 | 0 | 0 | 33.2 | 9.27 |

Dazhu | 3 | 2 | 0 | 0 | 0 | 0 | 4 | 1 | 0 | 0 | 31.5 | 8.99 |

Tuku | 0 | 5 | 0 | 0 | 0 | 3 | 2 | 0 | 0 | 0 | 33.0 | 10.21 |

Qili | 0 | 0 | 3 | 2 | 0 | 1 | 3 | 1 | 0 | 0 | 32.2 | 9.23 |

Shuyuan | 0 | 0 | 2 | 3 | 0 | 0 | 0 | 3 | 2 | 0 | 32.4 | 3.30 |

Gumu | 0 | 0 | 0 | 4 | 1 | 1 | 4 | 0 | 0 | 0 | 32.9 | 4.96 |

Huyuan | 0 | 0 | 3 | 2 | 0 | 0 | 0 | 0 | 3 | 2 | 32.0 | 6.30 |

Sunchong | 0 | 1 | 4 | 0 | 0 | 0 | 0 | 0 | 3 | 2 | 32.8 | 8.43 |

Fuzzy Comparison Matrix | Crisp Comparison Matrix | Weight | |||||||
---|---|---|---|---|---|---|---|---|---|

C1 | C2 | C3 | C4 | C1 | C2 | C3 | C4 | ||

C1 | 1 | (4.6, 5.4) | (1.0, 1.2) | (6.6, 7.4) | 1 | 5.080 | 1.120 | 7.080 | 0.4553 |

C2 | (1/5.4, 1/4.6) | 1 | (1/3.4, 1/2.6) | (2.6, 3.4) | 0.205 | 1 | 0.348 | 3.080 | 0.1203 |

C3 | (1/1.2, 1/1.0) | (2.6, 3.4) | 1 | (6.6, 7.4) | 0.933 | 3.080 | 1 | 7.080 | 0.3731 |

C4 | (1/7.4, 1/6.6) | (1/3.4, 1/2.6) | (1/7.4, 1/6.6) | 1 | 0.145 | 0.348 | 0.145 | 1 | 0.0513 |

Village | C1 (Degree of Damage) | C2 (Difficulty of Approaching a Disaster Area) |
---|---|---|

Jiutan | 1.3138 | 1.9708 |

Dazhu | 1.7518 | 1.0218 |

Tuku | 1.0948 | 1.7518 |

Qili | 0.6569 | 1.2408 |

Shuyuan | 0.6204 | 0.6569 |

Gumu | 0.5255 | 1.3138 |

Huyuan | 0.6569 | 0.5036 |

Sunchong | 0.8029 | 0.5036 |

Village | C1 | C2 | C3 | C4 |
---|---|---|---|---|

Jiutan | 0.3219 | 0.2420 | 0.0078 | 0.0039 |

Dazhu | 0.4292 | 0.1255 | 0.0074 | 0.0037 |

Tuku | 0.2682 | 0.2151 | 0.0078 | 0.0043 |

Qili | 0.1609 | 0.1524 | 0.0076 | 0.0039 |

Shuyuan | 0.1520 | 0.0806 | 0.0076 | 0.0013 |

Gumu | 0.1287 | 0.1613 | 0.0077 | 0.0020 |

Huyuan | 0.1609 | 0.0618 | 0.0075 | 0.0026 |

Sunchong | 0.1967 | 0.0618 | 0.0077 | 0.0035 |

Village | C1 | C2 | C3 | C4 |
---|---|---|---|---|

Jiutan | 0.1465 | 0.0291 | 0.0029 | 0.0002 |

Dazhu | 0.1954 | 0.0150 | 0.0027 | 0.0001 |

Tuku | 0.1221 | 0.0258 | 0.0029 | 0.0002 |

Qili | 0.0732 | 0.0183 | 0.0028 | 0.0002 |

Shuyuan | 0.0692 | 0.0097 | 0.0028 | 0.0001 |

Gumu | 0.0586 | 0.0194 | 0.0029 | 0.0001 |

Huyuan | 0.0732 | 0.0074 | 0.0028 | 0.0001 |

Sunchong | 0.0895 | 0.0074 | 0.0028 | 0.0002 |

Ideal Solution | C1 | C2 | C3 | C4 |
---|---|---|---|---|

PIS | 0.1954 | 0.0291 | 0.0029 | 0.0002 (=0.000221) |

NIS | 0.0586 | 0.0074 | 0.0027 | 0.0001 (=0.000071) |

Village | RCC (Relative Closeness Coefficient) |
---|---|

Jiutan | 0.7746 |

Dazhu | 0.9896 |

Tuku | 0.4483 |

Qili | 0.0216 |

Shuyuan | 0.0071 |

Gumu | 0.0075 |

Huyuan | 0.0137 |

Sunchong | 0.0757 |

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## Share and Cite

**MDPI and ACS Style**

He, Y.; Jung, H.
A Voting TOPSIS Approach for Determining the Priorities of Areas Damaged in Disasters. *Sustainability* **2018**, *10*, 1607.
https://doi.org/10.3390/su10051607

**AMA Style**

He Y, Jung H.
A Voting TOPSIS Approach for Determining the Priorities of Areas Damaged in Disasters. *Sustainability*. 2018; 10(5):1607.
https://doi.org/10.3390/su10051607

**Chicago/Turabian Style**

He, Yanjin, and Hosang Jung.
2018. "A Voting TOPSIS Approach for Determining the Priorities of Areas Damaged in Disasters" *Sustainability* 10, no. 5: 1607.
https://doi.org/10.3390/su10051607