# Flood Classification Based on a Fuzzy Clustering Iteration Model with Combined Weight and an Immune Grey Wolf Optimizer Algorithm

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## Abstract

**:**

## 1. Introduction

- (1)
- The fuzzy clustering iterative model with combined weight (FCI-CW) is proposed by using the sensitivity coefficient to combine the subjective weight and objective weight.
- (2)
- An immune grey wolf optimizer algorithm (IGWO) is proposed by employing an immune clone selection operator based on the grey wolf optimizer algorithm.
- (3)
- IGWO was employed to obtain the optimal fuzzy class center matrix and optimal sensitivity coefficient of FCI-CW.
- (4)
- Two case studies of flood classification at Nanjing station and Yichang station were carried out to demonstrate the reasonableness and effectiveness of the proposed methodology.

## 2. The Fuzzy Clustering Iteration Model with Combined Weight (FCI-CW)

#### 2.1. Overview of the Fuzzy Clustering Iteration Model

**Step 1**Set the precision of ${\epsilon}_{1}$, ${\epsilon}_{2}$, and ${\epsilon}_{3}$ for calculating ${\omega}_{j}$, ${u}_{hi}$, and ${s}_{jh}$, and set the maximum iterative number $T$.

**Step 2**Let the iterative number be $t=0$, assume that the original weight matrix ${\omega}_{j}^{t}$ satisfies the constraint $0\le {\omega}_{j}\le 1,{\displaystyle \sum _{j=1}^{m}{\omega}_{j}}=1$ shown in Equation (7), and assume that the original fuzzy membership matrix ${u}_{hi}^{t}$ satisfies the constraint $\sum _{h=1}^{c}{u}_{hi}}=1;0\le {u}_{hi}\le 1$ shown in Equation (5).

**Step 3**Calculate the corresponding original clustering center ${s}_{jh}^{t}$ by importing ${u}_{hi}^{t}$ and ${\omega}_{j}^{t}$ into Equation (12).

**Step 4**Seek an approximate clustering matrix ${\omega}_{j}^{t+1}$ by importing ${s}_{jh}^{t}$ and ${u}_{hi}^{t}$ into Equation (10).

**Step 5**Seek an approximate clustering matrix ${u}_{hi}^{t+1}$ by importing ${s}_{jh}^{t}$ and ${\omega}_{j}^{t+1}$ into Equation (11).

**Step 6**Seek an approximate clustering center matrix ${s}_{jh}^{t+1}$ by importing ${u}_{hi}^{t+1}$ and ${\omega}_{j}^{t+1}$ into Equation (12).

**Step 7**Compare the corresponding values ${\omega}_{j}^{t}$ and ${\omega}_{j}^{t+1}$, ${u}_{hi}^{t}$ and ${u}_{hi}^{t+1}$, and ${s}_{jh}^{t}$ and ${s}_{jh}^{t+1}$, respectively, and update the iteration counter by $t=t+1$ until the termination conditions in Equation (13) are satisfied or the iteration counter reaches the maximum iterative number; otherwise go to

**Step 3**.

**Step 8**Finally, ${\omega}_{j}^{t+1}$, ${u}_{hi}^{t+1}$, and ${s}_{jh}^{t+1}$ are obtained through the above computational steps. The objective value GEWD of the FCI is then calculated as the objective function fitness.

#### 2.2. Combined Weight for Flood Classification

#### 2.3. The Fuzzy Clustering Iteration Model with Combined Weight

#### 2.4. The Procedure of FCI-CW

**Step 1**Set the iteration counter to $g=0$;

**Step 2**Obtain the subjective weight ${\omega}_{Sj}$ and the objective weight ${\omega}_{Oj}$ according to the subjective weighting method and the objective weighting method separately; initialize ${s}_{jh}^{0}$ and ${\beta}^{0}$, set the objective function value of Equation (17) to ${F}^{0}=\xi $, where $\xi $ is a large constant; and set the precision $\epsilon $;

**Step 3**Calculate the combined weight ${\omega}_{j}^{g}$ based on Equation (14);

**Step 4**Calculate the membership matrix ${u}_{hi}^{g}$ based on Equation (19);

**Step 5**Calculate the fuzzy class center matrix ${s}_{ih}^{g}$ based on Equation (20);

**Step 6**Calculate the sensitivity coefficient ${\beta}^{g}$ based on Equation (21);

**Step 7**Calculate the objective function value ${F}^{g}=F({\beta}^{g},{\omega}_{Sj},{\omega}_{Oj},{{u}_{hi}}^{g},{{s}_{jh}}^{g})$;

**Step 8**Compare ${F}^{g}$ with ${F}^{g-1}$. If $\left|{F}^{g}-{F}^{g-1}\right|<\epsilon $, stop the calculation; otherwise $g=g+1$ and return to

**Step 3**.

## 3. The Immune Grey Wolf Optimizer Algorithm

#### 3.1. Overview of the Grey Wolf Optimizer Algorithm

#### 3.2. Immune Clone Selection Operator

**Step 1**The current population is ranked according to fitness, and the best $nn$ individuals are selected to form the elite population.

**Step 2**All the individuals in the elite population are respectively cloned to form a temporary population $S$. The clone size is directly proportional to the affinity, and the population number ${N}_{c}$ of $S$ is calculated as:

**Step 3**All the individuals in $S$ are successively implemented based on a mutation operator $mm$ times to obtain better candidate solutions nearby to themselves. The mutation operator is shown in Equations (32)–(34) as follows:

**Step 4**The best $nn$ individuals are selected from $S$ to replace the elite population in the next generation.

#### 3.3. The Pseudo Code of IGWO

#### 3.4. Simulation of IGWO for Solving Benchmark Optimization Problems

## 4. The Procedure for Flood Classification Using FCI-CW and IGWO

#### 4.1. Search-Variable Representation and Fitness Function

#### 4.2. The Procedure of FCI-CW and IGWO

## 5. Case Study

#### 5.1. The First Case at Nanjing Station

^{−2}, and the optimal search result of IGWO was output to obtain the optimal fuzzy class center matrix ${S}^{*}$ and the optimal sensitivity coefficient ${\beta}^{*}$ shown in Equations (36) and (37), respectively. Afterwards, the index weight matrix was calculated as ${\mathsf{\omega}}^{*}=(0.1644,0.2412,0.1944,0.1676,0.2324)$, which was combined with the subjective weight and the objective weight by using ${\beta}^{*}$ to obtain the best optimal value of GEWD.

^{−2}, which is less than the optimal value 2.020046 × 10

^{−2}by the proposed method. However, its optimal weight vector, denoted as ${\mathsf{\omega}}^{\prime}$ = (0.0908, 0.6267, 0.1141, 0.0801, 0.0883), is just the “mathematical weight” in the sense of sample data calculation, which indicates that the importance of the second index is much larger than the sum of the other four indices. This is contrary to the decision-maker’s subjective cognition and the actual situation, so it is necessary to reasonably modify the weight results.

^{−1}, the GEWD was the smallest of all, which indicates that the proposed methodology to calculate the fuzzy class center matrix and the sensitivity coefficient for flood classification is reasonable. This renders it superior to the conventional methods, since only considering subjectivity or objectivity, or just setting β = 0.5, lacks a powerful mathematical basis.

#### 5.2. The Second Case of Yichang Station

^{−2}, and the optimal search result of IGWO was output to obtain the optimal fuzzy class center matrix ${S}^{*}$ and the optimal sensitivity coefficient ${\beta}^{*}$ shown in Equations (40) and (41), respectively. Afterwards, the index weight matrix was calculated as ${\mathsf{\omega}}^{*}=(0.2789,0.1970,0.2002,0.1556,0.1683)$, which was combined with the subjective weight and the objective weight. Finally, the optimal fuzzy clustering matrix ${U}^{*}$ was achieved as shown in Equation (43).

^{−1}, the GEWD was the smallest of all, which indicates that the proposed methodology is reasonable. This renders it superior to the conventional methods, since only considering subjectivity or objectivity, or just setting β = 0.5, lacks a powerful mathematical basis.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 8.**The procedure of flood classification using the fuzzy clustering iterative model with combined weight (FCI-CW) and IGWO.

Algorithm Immune Grey Wolf Optimizer Algorithm (IGWO) |
---|

1: Set generation $g=0$ |

2: Initialize the grey wolf population ${Y}_{i}^{0}$, $i=1,2,\dots ,N$. |

3: Initialize $a$, ${A}_{1}$, ${A}_{2}$, ${A}_{3}$, ${C}_{1}$, ${C}_{2}$, ${C}_{3}$, $nn$, $mm$. |

4: Calculate the fitness of each individual |

5: ${Y}_{\alpha}$ = the best search individual |

6: ${Y}_{\beta}$ = the second best search individual |

7: ${Y}_{\delta}$ = the third best search individual |

8: While ($g$ < Max number of generations) |

9: For each search individual |

10: Update position of each current search individual by Equation (26) |

11: End for |

12: Update $a$, ${A}_{1}$, ${A}_{2}$, ${A}_{3}$, ${C}_{1}$, ${C}_{2}$, ${C}_{3}$ |

13: Calculate the fitness of all search individuals |

14: Choose the best $nn$ individuals to form the elite population |

15: Clone the elite population to form the temporary population |

$S$ by Equation (31) |

16: execute the mutation operator by Equation (32) |

17: replace the elite population and the GWO population |

18: Update ${Y}_{\alpha}$, ${Y}_{\beta}$, and ${Y}_{\delta}$, |

19: $g=g+1$ |

20: End while |

21: Output ${Y}_{\alpha}$ |

Function | Expression | Dim | Shift Position | Optimal Value |
---|---|---|---|---|

Sphere | ${f}_{1}(x)={\displaystyle {\sum}_{i=1}^{D}{x}_{i}^{2}}$ | 30 | [−100,100] | 0 |

Rastrigin | ${f}_{2}(x)={\displaystyle {\sum}_{i=1}^{D}[{x}_{i}^{2}-10\mathrm{cos}(2\pi {x}_{i})+10]}$ | 30 | [−5.12,5.12] | 0 |

Ackley | ${f}_{3}(x)=-20\mathrm{exp}(-0.2\sqrt{\frac{1}{D}{\displaystyle {\sum}_{i=1}^{D}{x}_{i}^{2}}})-\mathrm{exp}[\frac{1}{D}{\displaystyle {\sum}_{i=1}^{D}\mathrm{cos}(2\pi {x}_{i})}]+20+e$ | 30 | [−30,30] | 0 |

Schwefel 2.21 | ${f}_{4}(x)=\underset{1\le i\le D}{\mathrm{max}}\{|{x}_{i}|\}$ | 30 | [−100,100] | 0 |

Schwefel 2.22 | ${f}_{5}(x)={\displaystyle {\sum}_{i=1}^{D}\left|{x}_{i}\right|}+{\displaystyle {\prod}_{i=1}^{D}\left|{x}_{i}\right|}$ | 30 | [−10,10] | 0 |

Schaffer’s f7 | ${f}_{6}(x)=\frac{1}{4000}{\displaystyle {\sum}_{i=1}^{D-1}{({x}_{i}^{2}+{x}_{i+1}^{2})}^{0.25}[{\mathrm{sin}}^{2}(50{({x}_{i}^{2}+{x}_{i+1}^{2})}^{0.1})+1]}$ | 30 | [−100,100] | 0 |

**Table 3.**The test function results for the differential evolution (DE), grey wolf optimization (GWO), and IGWO algorithms.

Function | Algorithm | Best | Worst | Mean | Standard Deviation |
---|---|---|---|---|---|

Sphere | DE | 1.15 × 10^{−19} | 1.49 × 10^{−18} | 5.60 × 10^{−19} | 6.33 × 10^{−19} |

GWO | 1.35 × 10^{−60} | 6.54 × 10^{−59} | 2.77 × 10^{−59} | 3.11 × 10^{−59} | |

IGWO | 1.05 × 10^{−223} | 8.78 × 10^{−220} | 4.51 × 10^{−220} | 0 | |

Rastrigin | DE | 84.5 | 109 | 99.5 | 11.9 |

GWO | 0 | 3.22 | 8.04 × 10^{−1} | 1.61 | |

IGWO | 0 | 0 | 0 | 0 | |

Ackley | DE | 1.09 × 10^{−10} | 1.41 × 10^{−10} | 1.29 × 10^{−10} | 1.53 × 10^{−11} |

GWO | 1.51 × 10^{−14} | 2.93 × 10^{−14} | 1.87 × 10^{−14} | 7.11 × 10^{−15} | |

IGWO | 4.44 × 10^{−15} | 7.99 × 10^{−15} | 6.22 × 10^{−15} | 2.05 × 10^{−15} | |

Schwefel 2.21 | DE | 1.00 | 5.69 | 3.20 | 2.05 |

GWO | 8.27 × 10^{−15} | 4.68 × 10^{−14} | 2.83 × 10^{−14} | 1.58 × 10^{−14} | |

IGWO | 4.52 × 10^{−42} | 4.21 × 10^{−36} | 1.05 × 10^{−36} | 2.10 × 10^{−36} | |

Schwefel 2.22 | DE | 2.96 × 10^{−12} | 9.44 × 10^{−12} | 5.94 × 10^{−12} | 3.14 × 10^{−12} |

GWO | 4.06 × 10^{−35} | 7.89 × 10^{−35} | 6.29 × 10^{−35} | 1.69 × 10^{−35} | |

IGWO | 1.24 × 10^{−135} | 1.80 × 10^{−132} | 9.01 × 10^{−133} | 1.02 × 10^{−132} | |

Schaffer’s f_{7} | DE | 2.36 × 10^{−02} | 9.19 × 10^{−02} | 4.74 × 10^{−02} | 3.15 × 10^{−02} |

GWO | 1.09 × 10^{−15} | 4.99 × 10^{−15} | 3.11 × 10^{−15} | 1.74 × 10^{−15} | |

IGWO | 2.00 × 10^{−67} | 7.36 × 10^{−67} | 3.72 × 10^{−67} | 2.46 × 10^{−67} |

Number of Floods | Year | Flood Peak Level (m) | The Days of Flood Level over 9 m (day) | Flood Peak Discharge in DaTong Station (m^{3}·s^{−1}) | Flood Volume from May to September (10^{8} m^{3}) | The Synthetic Index of Discharge and Time |
---|---|---|---|---|---|---|

(1) | 1954 | 10.22 | 87 | 92,600 | 8891 | 7800 |

(2) | 1969 | 9.20 | 8 | 67,700 | 5447 | 1710 |

(3) | 1973 | 9.19 | 7 | 70,000 | 6623 | 3280 |

(4) | 1980 | 9.20 | 10 | 64,000 | 6340 | 2730 |

(5) | 1983 | 9.99 | 27 | 72,600 | 6641 | 3560 |

(6) | 1991 | 9.70 | 17 | 63,800 | 5576 | 1930 |

(7) | 1992 | 9.06 | 13 | 67,700 | 5295 | 1575 |

(8) | 1995 | 9.66 | 23 | 75,500 | 6162 | 2390 |

(9) | 1996 | 9.89 | 34 | 75,100 | 6206 | 2702 |

(10) | 1998 | 10.14 | 81 | 82,100 | 7773 | 5283 |

**Table 5.**The comparison results of objective fitness by the different algorithms for the first case study.

Algorithms | Minimum | Maximum | Average | Standard Deviation |
---|---|---|---|---|

DE | 2.032220 × 10^{−2} | 2.324090 × 10^{−2} | 2.099263 × 10^{−2} | 1.266623 × 10^{−3} |

GWO | 2.030084 × 10^{−2} | 2.030152 × 10^{−2} | 2.030128 × 10^{−2} | 1.364308 × 10^{−8} |

IGWO | 2.023083 × 10^{−2} | 2.023083 × 10^{−2} | 2.023083 × 10^{−2} | 3.878990 × 10^{−18} |

**Table 6.**The results from the comparison of the proposed method with other methods at Nanjing station.

Number of Floods | Year | OC-PDC | VFS | FCI-CDE | WFKCA-ADE | The Proposed Method |
---|---|---|---|---|---|---|

(1) | 1954 | I | I | I | I | I |

(2) | 1969 | III | III | III | III | III |

(3) | 1973 | III | III | III | III | III |

(4) | 1980 | III | III | III | III | III |

(5) | 1983 | II | II | II | II | II |

(6) | 1991 | III | III | III | III | III |

(7) | 1992 | III | III | III | III | III |

(8) | 1995 | II | II | II | II | II |

(9) | 1996 | II | II | II | II | II |

(10) | 1998 | I | I | I | I | I |

**Table 7.**The comparison results of the general Euclidean weighted distance (GEWD) with different sensitivity coefficients for the first case study.

β | 0 | 0.2 | 0.4 | 0.45 | 0.5 |
---|---|---|---|---|---|

GEWD | 2.037210 × 10^{−2} | 2.028914 × 10^{−2} | 2.024230 × 10^{−2} | 2.0236234 × 10^{−2} | 2.02324 × 10^{−2} |

β | 0.55 | 5.595729 × 10^{−1} | 0.6 | 0.8 | 1 |

GEWD | 2.023087 × 10^{−2} | 2.023083 × 10^{−2} | 2.023157 × 10^{−2} | 2.025693 × 10^{−2} | 2.031831 × 10^{−2} |

Number of Floods | Year | Flood Peak Level (m) | Flood peak Discharge (m^{3}·s^{−1}) | Three-Day Flood (10^{8} m^{3}) | Seven-Day Flood (10^{8} m^{3}) | Fifteen-Day Flood (10^{8} m^{3}) |
---|---|---|---|---|---|---|

(1) | 1931 | 55.0 | 64,600 | 163.2 | 350.4 | 621.3 |

(2) | 1935 | 54.6 | 56,900 | 137.1 | 283.3 | 509.5 |

(3) | 1954 | 55.7 | 66,800 | 170.1 | 385.3 | 785.1 |

(4) | 1958 | 53.5 | 59,500 | 148.8 | 305.1 | 550.2 |

(5) | 1966 | 54.0 | 59,600 | 151.7 | 334.2 | 592.4 |

(6) | 1969 | 51.5 | 41,900 | 105.1 | 217.4 | 412.2 |

(7) | 1974 | 54.8 | 61,000 | 151.8 | 301.6 | 545.7 |

(8) | 1980 | 54.0 | 54,600 | 139.7 | 300.8 | 545.6 |

(9) | 1981 | 55.4 | 70,800 | 172.5 | 334.8 | 558.3 |

(10) | 1982 | 54.6 | 59,000 | 146.9 | 303.8 | 583.8 |

(11) | 1983 | 53.3 | 52,600 | 129.9 | 268.1 | 491.3 |

(12) | 1998 | 54.5 | 63,600 | 151.3 | 347.8 | 728.2 |

**Table 9.**The comparison results of objective fitness by the different algorithms for the second case study.

Algorithms | Minimum | Maximum | Average | Standard Deviation |
---|---|---|---|---|

DE | 2.750000 × 10^{−2} | 3.150000 × 10^{−2} | 2.888000 × 10^{−2} | 1.213095 × 10^{−3} |

GWO | 2.648600 × 10^{−2} | 2.990000 × 10^{−2} | 2.732100 × 10^{−2} | 1.142702 × 10^{−6} |

IGWO | 2.644227 × 10^{−2} | 2.644230 × 10^{−2} | 2.644227 × 10^{−2} | 6.633250 × 10^{−16} |

Number of Floods | Year | VFS | The Proposed Method |
---|---|---|---|

(1) | 1931 | I | I |

(2) | 1935 | II | II |

(3) | 1954 | I | I |

(4) | 1958 | II | II |

(5) | 1966 | II | II |

(6) | 1969 | III | III |

(7) | 1974 | II | II |

(8) | 1980 | II | II |

(9) | 1981 | I | I |

(10) | 1982 | II | II |

(11) | 1983 | II | II |

(12) | 1998 | I | I |

**Table 11.**The comparison results of the GEWD with different sensitivity coefficients for the second case study.

β | 0 | 0.2 | 0.4 | 0.45 | 0.5 |
---|---|---|---|---|---|

GEWD | 2.740795 × 10^{−2} | 2.687555 × 10^{−2} | 2.655286 × 10^{−2} | 2.650519 × 10^{−2} | 2.647082 × 10^{−2} |

β | 0.55 | 0.6 | 6.027070 × 10^{−1} | 0.8 | 1 |

GEWD | 2.644981 × 10^{−2} | 2.644229 × 10^{−2} | 2.644227 × 10^{−2} | 2.655053 × 10^{−2} | 2.689421 × 10^{−2} |

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**MDPI and ACS Style**

Zou, Q.; Liao, L.; Ding, Y.; Qin, H.
Flood Classification Based on a Fuzzy Clustering Iteration Model with Combined Weight and an Immune Grey Wolf Optimizer Algorithm. *Water* **2019**, *11*, 80.
https://doi.org/10.3390/w11010080

**AMA Style**

Zou Q, Liao L, Ding Y, Qin H.
Flood Classification Based on a Fuzzy Clustering Iteration Model with Combined Weight and an Immune Grey Wolf Optimizer Algorithm. *Water*. 2019; 11(1):80.
https://doi.org/10.3390/w11010080

**Chicago/Turabian Style**

Zou, Qiang, Li Liao, Yi Ding, and Hui Qin.
2019. "Flood Classification Based on a Fuzzy Clustering Iteration Model with Combined Weight and an Immune Grey Wolf Optimizer Algorithm" *Water* 11, no. 1: 80.
https://doi.org/10.3390/w11010080