# Ranking Sub-Watersheds for Flood Hazard Mapping: A Multi-Criteria Decision-Making Approach

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

^{2}(Figure 1) The case study is one of the main headwaters of the Karun River, which is Iran’s most affluent and only navigable river. The watershed contains 25 sub-watersheds. It is situated in a moderate climate in the summer (August) and a cold and snowy climate in the winter (February). The annual average temperature is +9.7 °C, and the annual average rainfall is 313 mm, which occurs mainly between February and April.

#### 2.2. Data and Methodology

#### 2.3. Theoretical Backgrounds of Proposed Methods

#### 2.3.1. Analytic Hierarchy Process (AHP)

#### 2.3.2. IRNAHP

**The algorithm of IRNAHP**

**IRN Mathematical Model**

**Stage 1**: Establishing the pairwise comparison matrixes by $k$ experts as follows:

**Stage 2**: Calculating the consistency rate for every expert. This stage is similar to the AHP pure one, with the difference that there are two consistency rates, one of which is for upper approximations $\left({CR}_{e}\right)$ and the other one is for lower approximations ${(CR}_{{e}^{\prime}})$. The final CR is calculated based on $\left({CR}_{e}+{CR}_{{e}^{\prime}}\right)/2$.

**Stage 3**: Calculating the concatenation of interval rough number matrices in Stage 1 to obtain ${a}^{\mathrm{*}L}$ and ${a}^{{\ast}^{\prime}U}$.

**Stage 4:**Defining the rank of each criterion (interval rough weighted coefficient), ${IRN}_{\left({w}_{j}\right)}$. The vector is calculated based on Equations (24) and (25).

#### 2.3.3. Best–Worst Method (BWM)

**The algorithm of the best–worst method (BWM) can be described as follows:**

**Stage 1**. Selection of a set of desired criteria/sub-criteria, and determination of the best/worst (most/least important) of them.

**Stage 2**. Determination of preferences of the best criterion over others (BO) and vice versa, i.e., preferences of all criteria over the worst criterion, using a 9-point rating scale (Table 1).

**Stage 3.**Finding the optimal weights $\left({w}_{1}^{\ast},{w}_{2}^{\ast},\dots ,{w}_{3}^{\ast}\right)$ by solving the following model:

**Stage 4.**Checking the consistency as follows:

#### 2.3.4. Picture Fuzzy Analytic Hierarchy Process and Picture Fuzzy Linear Assignment (PICALAM)

**Definition**

**1:**

**Definition**

**2:**

**Addition****Multiplication**

**Multiplication by a scalar; λ > 0**

**Definition**

**3:**

**Definition**

**4:**

**Stage 1:**Picture fuzzy analytic hierarchy process.

**1.1.**Using pairwise comparison matrices for the weights of the criteria

**1.2.**To aggregate the decision-makers’ assessments, we used a weighted geometric (PFWG) mean. There can be different comparison matrixes $\left({\stackrel{~}{w}}_{j}^{local}\right)$ in decision-making situations because there are multiple decision-makers. It is necessary to employ geometric means (Equation (38)) to unify all comparison matrices $\left({\stackrel{~}{w}}_{j}^{global}\right)$ in the next steps. Eventually, the final picture fuzzy weight $\left({\stackrel{~}{w}}_{j}^{final}\right)$ must be calculated as follows:

**1.3.**De-fuzzification of $\left({\stackrel{~}{w}}_{j}^{final}\right)$.

**Stage 2:**Using the PFS to rank the alternatives.

**2.1.**This point is similar to Stage 1’s point 1.1. The difference is that decision-makers’ individual judgments are based on alternatives in the form of decision matrices (Table 4).

**2.2.**The individual decision matrices from the previous point were aggregated using Equation (37), as shown in Table 5.

**2.3.**Defuzzification of the aggregated matrix using Equation (40) was performed to compare and rank options that are connected to each other.

**2.4.**Determination of the rank frequency matrix, which includes associated elements that show the number of times alternative m dominates on the nth criterion (Table 6).

**2.5.**Determination of the weighted rank frequency matrix ${\Pi}_{ik}$, which measures the contribution of the mth alternative to the overall ranking (Equation (45) and Table 7).

**2.6.**Construction of the linear assignment model based on ${\Pi}_{ik}$ and permutation matrix P (m*m) as follows:

**2.7.**Using Equation (46) to obtain the optimal permutation matrix $\left({P}^{\ast}\right)$.

**2.8.**Obtaining the rank of alternatives as follows:

## 3. Analysis and Results

#### 3.1. Morphometric Parameters

#### 3.2. Ranking of Sub-Watersheds Using the AHP Technique

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**Comparison scale between two criteria [36].

Preference Factor | Degree of Preference | Explanation |
---|---|---|

1 | Equally | Two factors contribute equally to the objective |

3 | Moderately | Experience and judgment slightly to moderately favor one factor over another |

5 | Strongly | Experience and judgment strongly or essentially favor one factor over another |

7 | Very strongly | A factor is strongly favored over another and its dominance is showed in practice |

9 | Extremely | The evidence of favoring one factor over another is of the highest degree possible of an affirmation |

2, 4, 6, 8 | Intermediate | Used to represent compromises between the preferences in weights1, 3, 5, 7 and 9 |

Reciprocals | Opposites | Used for inverse comparison |

${\mathcal{a}}_{\mathcal{B}\mathcal{W}}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

$\mathcal{C}\mathcal{I}\left(max{\xi}^{\ast}\right)$ | 0 | 0.44 | 1.00 | 1.63 | 2.30 | 3.00 | 3.73 | 4.47 | 5.23 |

Linguistic Terms | Saaty’s Scale | Picture Fuzzy Numbers (PFNs) |
---|---|---|

Very High Importance | 7 | (0.9, 0.0, 0.05) |

High Importance | 5 | (0.75, 0.05, 0.1) |

Slightly More Importance | 3 | (0.6, 0.0, 0.3) |

Equally Importance | 1 | (0.5, 0.1, 0.4) |

Slightly Low Importance | 1/3 | (0.3, 0.0, 0.6) |

Low Importance | 1/5 | (0.25, 0.05, 0.6) |

Very Low Importance | 1/7 | (0.1, 0.0, 0.85) |

Criteria | ||||
---|---|---|---|---|

Alternative | C1 | C2 | … | Cn |

A_{1} | ${PF}_{11}^{k}$ | ${PF}_{12}^{k}$ | … | ${PF}_{1n}^{k}$ |

A_{2} | ${PF}_{21}^{k}$ | ${PF}_{22}^{k}$ | … | ${PF}_{2n}^{k}$ |

… | … | … | … | … |

A_{m} | ${PF}_{m1}^{k}$ | ${PF}_{m2}^{k}$ | … | ${PF}_{mn}^{k}$ |

Criteria | ||||
---|---|---|---|---|

Alternative | C1 | C2 | … | Cn |

A_{1} | ${PFWG}_{11}$ | ${PFWG}_{12}$ | … | ${PFWG}_{1n}$ |

A_{2} | ${PFWG}_{21}$ | ${PFWG}_{22}$ | … | ${PFWG}_{2n}$ |

… | … | … | … | … |

A_{m} | ${PFWG}_{m1}$ | ${PFWG}_{m2}$ | … | ${PFWG}_{nm}$ |

Rank | ||||

Alternative | 1st | 2st | … | mth |

A_{1} | ${\lambda}_{11}$ | ${\lambda}_{12}$ | … | ${\lambda}_{1n}$ |

A_{2} | ${\lambda}_{21}$ | ${\lambda}_{22}$ | … | ${\lambda}_{2n}$ |

… | … | … | … | … |

A_{m} | ${\lambda}_{m1}$ | ${\lambda}_{m2}$ | … | ${\lambda}_{nm}$ |

Rank | ||||
---|---|---|---|---|

Alternative | 1st | 2st | … | mth |

A_{1} | ${\Pi}_{11}$ | ${\Pi}_{12}$ | … | ${\Pi}_{1n}$ |

A_{2} | ${\Pi}_{21}$ | ${\Pi}_{22}$ | … | ${\Pi}_{2n}$ |

… | … | … | … | … |

A_{m} | ${\Pi}_{m1}$ | ${\Pi}_{m2}$ | … | ${\Pi}_{nm}$ |

Flood-Related Criteria | Flood-Related Criteria | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Sub-Watershed | Slope | TCI | Entropy | Cc | SPI | Sub-Watershed | Slope | TCI | Entropy | Cc | SPI |

1 | 19.72 | −1.05 | 0.08 | 1.82 | −2.73 | 14 | 15.69 | −1.10 | 0.14 | 2.11 | −3.47 |

2 | 21.31 | −1.18 | 0.13 | 1.80 | −2.97 | 15 | 40.65 | −1.29 | 0.14 | 1.88 | −1.47 |

3 | 12.22 | −0.89 | 0.14 | 2.11 | −3.64 | 16 | 17.24 | −0.98 | 0.08 | 2.50 | −3.12 |

4 | 12.53 | −0.97 | 0.14 | 1.93 | −3.73 | 17 | 24.01 | −1.18 | 0.11 | 1.72 | −2.66 |

5 | 23.98 | −1.20 | 0.09 | 2.78 | −2.80 | 18 | 25.45 | −1.08 | 0.09 | 1.58 | −2.22 |

6 | 21.26 | −1.06 | 0.03 | 2.19 | −2.78 | 19 | 21.30 | −1.10 | 0.14 | 2.30 | −2.76 |

7 | 15.63 | −0.82 | 0.13 | 2.13 | −3.04 | 20 | 20.52 | −1.18 | 0.13 | 2.42 | −3.16 |

8 | 14.28 | −0.86 | 0.12 | 1.93 | −3.26 | 21 | 28.34 | −1.14 | 0.12 | 2.24 | −2.10 |

9 | 26.78 | −1.26 | 0.12 | 2.31 | −2.49 | 22 | 24.50 | −1.12 | 0.03 | 1.75 | −2.62 |

10 | 16.21 | −1.14 | 0.11 | 2.34 | −3.45 | 23 | 20.14 | −1.16 | 0.14 | 1.84 | −3.03 |

11 | 26.36 | −1.11 | 0.14 | 2.41 | −2.81 | 24 | 28.25 | −1.22 | 0.12 | 2.11 | −2.38 |

12 | 13.05 | −0.81 | 0.14 | 2.00 | −3.29 | 25 | 20.66 | −1.02 | 0.14 | 2.14 | −2.80 |

13 | 13.54 | −0.90 | 0.14 | 2.07 | −3.59 |

AHP Method | BWM Method | IRN AHP Method | |||||
---|---|---|---|---|---|---|---|

Criteria | Wi(Expert #1) | Wi(Expert #2) | Wi(Expert #3) | Wi(Expert #1) | Wi(Expert #2) | Wi(Expert #3) | $\left[{\mathit{w}}_{\mathit{i}\mathit{j}}^{\mathit{L}},{\mathit{w}}_{\mathit{i}\mathit{j}}^{\mathit{U}}\right],\left[{\mathit{w}}_{\mathit{i}\mathit{j}}^{\mathit{\u2019}\mathit{L}},{\mathit{w}}_{\mathit{i}\mathit{j}}^{\mathit{\u2019}\mathit{U}}\right]$ |

Entropy | 0.488 | 0.593 | 0.391 | 0.487 (Best) | 0.475 (Best) | 0.423 (Best) | [0.46, 0.48], [0.5, 0.52] |

SPI | 0.228 | 0.244 | 0.215 | 0.189 | 0.188 | 0.231 | [0.21, 0.22], [0.22, 0.23] |

TCI | 0.142 | 0.150 | 0.163 | 0.142 | 0.141 | 0.115 | [0.15, 0.15], [0.15, 0.16] |

Slope | 0.087 | 0.084 | 0.099 | 0.114 | 0.141 | 0.154 | [0.06, 0.07], [0.09, 0.09] |

Cc | 0.056 | 0.044 | 0.053 | 0.068 (worst) | 0.055 (worst) | 0.077 (worst) | [0.03, 0.03], [0.05, 0.05] |

Consistently | 0.02 | 0.07 | 0.05 | 0.081 | 0.088 | 0.038 | (expert #1) 0.06 (expert #2) 0.079 (expert #3) 0.1 |

Criteria | Local Weight | Global Weight | Final Weight | Deffuzification (Score) | |
---|---|---|---|---|---|

Expert #1 | En | (0.76, 0.24, 0.14) | (0.85, 0.4, 0.08) | (0.73, 0.05, 0.12) | 1.31 |

Expert #2 | (0.86, 0.14, 0.005) | ||||

Expert #3 | (0.7, 0.3, 0.2) | ||||

Expert #1 | SPI | (0.7, 0.3, 0.2) | (0.81, 0.48, 0.12) | (0.64, 0.09, 0.21) | 1.03 |

Expert #2 | (0.76, 0.24, 0.14) | ||||

Expert #3 | (0.65, 0.35, 0.24) | ||||

Expert #1 | TCI | (0.65, 0.34, 0.24) | (0.77, 0.53, 0.16) | (0.5, 0.18, 0.36) | 0.55 |

Expert #2 | (0.7, 0.3, 0.2) | ||||

Expert #3 | (0.6, 0.4, 0.3) | ||||

Expert #1 | Slope | (0.6, 0.4, 0.3) | (0.74, 0.56, 0.19) | (0.51, 0.16, 0.35) | 0.59 |

Expert #2 | (0.65, 0.34, 0.24) | ||||

Expert #3 | (0.55, 0.45, 0.35) | ||||

Expert #1 | Cc | (0.5, 0.4, 0.6) | (0.67, 0.58, 0.25) | (0.4, 0.23, 0.47) | 0.21 |

Expert #2 | (0.55, 0.4, 0.35) | ||||

Expert #3 | (0.5, 0.4, 0.6) |

Sub-Watershed | Entropy | SPI | TCI | Slope | Cc |
---|---|---|---|---|---|

0 | (0.95, 0.52, 0.02) | (0.94, 0.64, 0.03) | (0.9, 0.69, 0.07) | (0.91, 0.48, 0.04) | (0.84, 0.56, 0.1) |

1 | (0.99, 0.89, 0.002) | (0.95, 0.71, 0.02) | (0.95, 0.84, 0.029) | (0.86, 0.6, 0.08) | (0.78, 0.64, 0.15) |

2 | (0.99, 0.94, 0.001) | (0.99, 0.95, 0.002) | (0.83, 0.54, 0.1) | (0.8, 0.47, 0.12) | (0.84, 0.73, 0.11) |

3 | (0.99, 0.99, 0.00) | (1, 1, 0.00) | (0.83, 0.64, 0.11) | (0.82, 0.47, 0.12) | (0.80, 0.67, 0.14) |

4 | (0.95, 0.6, 0.01) | (0.94, 0.66, 0.02) | (0.96, 0.86, 0.002) | (0.88, 0.65, 0.07) | (0.99, 0.99, 0.00) |

. | … | … | … | … | … |

. | … | … | … | … | … |

. | … | … | … | … | … |

26 | (0.97, 0.75, 0.007) | (0.90, 0.48, 0.04) | (0.95, 0.72, 0.2) | (0.92, 0.69, 0.04) | (0.89, 0.75, 0.07) |

27 | (0.9, 0.3, 0.03) | (0.93, 0.61, 0.029) | (0.92, 0.76, 0.04) | (0.92, 0.56, 0.03) | (0.83, 0.53, 0.1) |

28 | (0.99, 0.99, 0.00) | (0.95, 0.73, 0.019) | (0.93, 0.84, 0.04) | (0.85, 0.58, 0.09) | (0.82, 0.6, 0.12) |

29 | (0.97, 0.77, 0.00) | (0.92, 0.55, 0.03) | (0.95, 0.90, 0.02) | (0.91, 0.7, 0.05) | (0.85, 0.71, 0.09) |

30 | (0.97, 0.77, 0.00) | (0.92, 0.55, 0.03) | (0.96, 0.89, 0.02) | (0.91, 0.7, 0.05) | (0.84, 0.73, 0.1) |

Sub-Watersheds | 1th | 2th | 3th | 4th | 5th | 6th | 7th | 17th | 18th | 19th | 20th | 21th | 22th | 23th | 24th | 25th | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 0.00 | 0.59 | 2.34 | 0.00 | 0.00 | 0.00 | 0.00 | … | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | … | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.22 | 0.00 | 0.00 |

2 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | … | 1.53 | 0.00 | 0.55 | 0.00 | 0.00 | 0.00 | 0.00 | 1.03 | 0.59 |

3 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | … | 0.00 | 0.59 | 0.00 | 0.00 | 0.77 | 0.00 | 1.31 | 0.00 | 1.03 |

4 | 0.22 | 0.00 | 0.00 | 0.00 | 0.00 | 1.31 | 1.03 | … | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

5 | 1.31 | 0.00 | 0.00 | 0.81 | 0.00 | 1.03 | 0.00 | … | 0.00 | 0.55 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

… | … | … | … | … | … | … | … | … | … | … | … | … | … | … | … | … | … |

25 | 0.00 | 0.00 | 0.77 | 0.00 | 0.00 | 0.00 | 0.00 | … | 1.03 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

26 | 0.55 | 0.00 | 0.00 | 0.00 | 0.00 | 0.59 | 0.22 | … | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

27 | 1.63 | 1.31 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | … | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

28 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | … | 0.59 | 0.00 | 0.00 | 0.00 | 0.00 | 1.31 | 0.00 | 0.00 | 0.00 |

29 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | … | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

30 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | … | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

Sub-Watersheds | 1th | 2th | 3th | 4th | 5th | 6th | 7th | 17th | 18th | 19th | 20th | 21th | 22th | 23th | 24th | 25th | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | … | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | … | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | … | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | … | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |

4 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | … | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

5 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | … | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

… | … | … | … | … | … | … | … | … | … | … | … | … | … | … | … | … | … |

25 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | … | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

26 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | … | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

27 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | … | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | … | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |

29 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | … | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

30 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | … | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

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## Share and Cite

**MDPI and ACS Style**

Nguyen, N.-M.; Bahramloo, R.; Sadeghian, J.; Sepehri, M.; Nazaripouya, H.; Nguyen Dinh, V.; Ghahramani, A.; Talebi, A.; Elkhrachy, I.; Pande, C.B.;
et al. Ranking Sub-Watersheds for Flood Hazard Mapping: A Multi-Criteria Decision-Making Approach. *Water* **2023**, *15*, 2128.
https://doi.org/10.3390/w15112128

**AMA Style**

Nguyen N-M, Bahramloo R, Sadeghian J, Sepehri M, Nazaripouya H, Nguyen Dinh V, Ghahramani A, Talebi A, Elkhrachy I, Pande CB,
et al. Ranking Sub-Watersheds for Flood Hazard Mapping: A Multi-Criteria Decision-Making Approach. *Water*. 2023; 15(11):2128.
https://doi.org/10.3390/w15112128

**Chicago/Turabian Style**

Nguyen, Nguyet-Minh, Reza Bahramloo, Jalal Sadeghian, Mehdi Sepehri, Hadi Nazaripouya, Vuong Nguyen Dinh, Afshin Ghahramani, Ali Talebi, Ismail Elkhrachy, Chaitanya B. Pande,
and et al. 2023. "Ranking Sub-Watersheds for Flood Hazard Mapping: A Multi-Criteria Decision-Making Approach" *Water* 15, no. 11: 2128.
https://doi.org/10.3390/w15112128