# Water Carrying Capacity Evaluation Method Based on Cloud Model Theory and an Evidential Reasoning Approach

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Overview of the Study Area and Data Sources

#### 2.1. Study Area Overview

#### 2.2. Data Sources

## 3. Problem Description and Preliminary Knowledge

#### 3.1. Description of Water Resources Carrying Capacity Evaluation

#### 3.2. Preliminary Knowledge

#### 3.2.1. Cloud Model Theory

**Definition**

**1.**

- Expectation ${E}_{x}$ is the point that best represents the stereotype concept, and its value is usually taken as the expectation of points in the theory domain.
- Entropy ${E}_{n}$, which can measure the randomness of a qualitative concept, can determine the range of cloud droplets consistent with the qualitative concept in the domain space.
- Superentropy ${H}_{e}$ is the uncertainty measure of entropy, also known as the entropy of entropy. Its value depends on the fuzziness and randomness of entropy and reflects the degree of cloud droplet aggregation.

- Create an expected ${E}_{x}$ and generate a normal random number ${x}_{i}$ with the variance ${E}_{n}$.
- Create an expectation ${E}_{n}$ and generate a normal random number ${y}_{i}$ with the variance ${H}_{e}$/
- Calculation of $\mu \left(x\right)=exp\left(\frac{-\left({x}_{i}-{y}_{i}\right)}{2{E}_{{n}_{ij}}^{{}^{\prime}2}}\right)$, $\left(xi,\mu \left(x\right)\right)$ as the cloud droplets.
- Repeat Steps 1–3 until the nth qualified cloud droplet forms a cloud.

#### 3.2.2. Evidential Reasoning Approach

**Definition**

**2.**

## 4. Water Resources Carrying Capacity Evaluation Method

#### 4.1. Construction of Evaluation Index System of Water Resources Carrying Capacity

#### 4.2. Evaluation Grade Standard

#### 4.3. Evaluation Index Grade Correlation Degree Based on the Cloud Model Theory

#### 4.4. Index Weight Model Based on Entropy Weight Method and Evidential Reasoning Approach

#### 4.4.1. Weight Coefficient Determination Based on Entropy Weight Method

#### 4.4.2. Determination of Weight Coefficient Based on Evidential Reasoning Approach

#### 4.4.3. The Comprehensive Weights

#### 4.5. Evaluation Index Grade Correlation Degree Fusion

## 5. Water Resources Carrying Capacity Evaluation Method and Its Application

#### 5.1. Water Resources Carrying Capacity Evaluation Method

#### 5.2. Case Analysis

_{ij}according to Formula (8).

#### 5.3. Result Analysis and Method Comparison

#### 5.3.1. Comparison of Weight Methods

#### 5.3.2. Comparison of Evaluation Methods

## 6. Conclusions

- The PSR model is used to construct an index system from three aspects of pressure-state-response to make the index selection more systematic and scientific.
- The cloud model theory is used to describe the correlation degree of each index membership grade, and the randomness, fuzziness, and information uncertainty in the evaluation processes are taken into account to make the results more realistic.
- In the process of weight determination, the combined weight method combining dynamic and static weights is adopted, and the evidence compatibility idea of the entropy weight method and evidential reasoning is introduced, which not only avoids the subjective defect of weight determination, but also reduces the distortion effect caused by the index conflict, making the result more objective and reasonable.
- The improved evidentiary reasoning is used to fuse the correlation degree of each index belonging to the safety grade to obtain the final fusion probability of the comprehensive safety evaluation grade, which reduces the uncertainty of the results and improves the accuracy and reliability of the evaluation.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Target Layer | Factor Layer | Index Layer | Index Symbol | Meaning | Index Properties |
---|---|---|---|---|---|

Water resources carrying capacity evaluation | Pressures (P) | Population density (person/km^{2}) [35,36,37,38] | ×1 | Population pressure | Negative indicator |

Urbanization rate (%) [35,37,38] | ×2 | Urban development pressure | Negative indicator | ||

Growth rate of GDP (%) [35,37,38] | ×3 | Economic growth pressure | Negative indicator | ||

Utilization rate of water resources development (%) [35,36,37] | ×4 | Water resources development and utilization of pressure | Negative indicator | ||

States (S) | Water use per 10^{4} Yuan of GDP (m^{3}/10^{4} Yuan) [35,37,38,39] | ×5 | Water consumption of gross product | Negative indicator | |

Water use per 10^{4} Yuan of industrial production (m^{3}/10^{4} Yuan) [35,36,37,38,39,40] | ×6 | Industrial structure | Negative indicator | ||

Water resources in per capita terms (m^{3}/person) [35,36,37,38,39,40] | ×7 | Water resources per capita | Positive indicators | ||

Water consumption per capital (m^{3}/capital) [36,37,39,40] | ×8 | Water consumption per capita | Negative indicator | ||

Responses (R) | GDP per capita (10^{4} Yuan/person) [35,37,38] | ×9 | Level of economic development | Negative indicator | |

Percentage of ecological water utilization (%) [35,37,39,40] | ×10 | Level of environmental protection | Positive indicators | ||

Percentage of forest cover (%) [35,36,37,38,39,40] | ×11 | Greening level | Positive indicators | ||

Rate of river water quality up to standard (%) [35,36,38,40] | ×12 | Water quality level | Positive indicators |

Evaluation Indicator | Assessment Level | ||||
---|---|---|---|---|---|

I (Serious Overload) | II (Overload) | III (Critical) | IV (Weak Carrying Capacity) | V (Strong Carrying Capacity) | |

×1 | >300 | 100 | 50 | 25 | <25 |

×2 | >70 | 60 | 50 | 40 | <40 |

×3 | >30 | 30 | 20 | 15 | <15 |

×4 | >45 | 45 | 30 | 15 | <15 |

×5 | >400 | 400 | 200 | 100 | <100 |

×6 | >220 | 220 | 140 | 60 | <60 |

×7 | <1700 | 1700 | 2300 | 3000 | >3000 |

×8 | >100 | 100 | 95 | 90 | <90 |

×9 | >7.74 | 7.74 | 2.5 | 0.66 | <0.66 |

×10 | <2 | 2 | 3 | 5 | >5 |

×11 | <20 | 20 | 35 | 55 | >55 |

×12 | <70 | 70 | 80 | 90 | >90 |

×1 | ×2 | ×3 | ×4 | ×5 | ×6 | ×7 | ×8 | ×9 | ×10 | ×11 | ×12 |
---|---|---|---|---|---|---|---|---|---|---|---|

29.8791 | 19.3 | 0.1798 | 19.3 | 249 | 173 | 2698.9 | 45.5 | 2.4004 | 3.2 | 57.01 | 77.1 |

The Evaluation Index | Assessment Level | ||||
---|---|---|---|---|---|

I | II | III | IV | V | |

×1 | (350, 16.67, 0.01) | (200, 33.33, 0.01) | (75, 8.33, 0.01) | (37.5, 4.17, 0.01) | (12.5, 4.17, 0.01) |

×2 | (65, 1.67, 0.01) | (55, 1.67, 0.01) | (45, 1.67, 0.01) | (35, 1.67, 0.01) | (15, 5, 0.01) |

×3 | (32.5, 0.83, 0.01) | (25, 1.67, 0.01) | (17.5, 0.83, 0.01) | (12.5, 0.83, 0.01) | (5, 1.67, 0.01) |

×4 | (52.5, 2.5, 0.01) | (37.5, 2.5, 0.01) | (22.5, 2.5, 0.01) | (10, 1.67, 0.01) | (2.5, 0.83, 0.01) |

×5 | (500, 33.34, 0.01) | (300, 33.34, 0.01) | (150, 16.65, 0.01) | (75, 8.33, 0.01) | (25, 8.31, 0.01) |

×6 | (270, 16.66, 0.01) | (180, 13.32, 0.01) | (100, 13.32, 0.01) | (42, 6.01, 0.01) | (12, 3.99, 0.01) |

×7 | (250, 83.33, 0.01) | (1100, 200.01, 0.01) | (2000, 100, 0.01) | (2650, 116.65, 0.01) | (4000, 333.33, 0.01) |

×8 | (102.5, 0.83, 0.01) | (97.5, 0.83, 0.01) | (92.5, 0.83, 0.01) | (87.5, 0.83, 0.01) | (42.5, 14.17, 0.01) |

×9 | (8.87, 0.38, 0.01) | (5.12, 0.87, 0.01) | (1.58, 0.31, 0.01) | (0.48, 0.06, 0.01) | (0.15, 0.05, 0.01) |

×10 | (0.5, 0.17, 0.01) | (1.5, 0.16, 0.01) | (2.5, 0.16, 0.01) | (4, 0.33, 0.01) | (6, 0.33, 0.01) |

×11 | (5, 1.67, 0.01) | (15, 1.67, 0.01) | (27.5, 2.5, 0.01) | (45, 3.33, 0.01) | (60, 1.67, 0.01) |

×12 | (30, 10, 0.01) | (65, 1.67, 0.01) | (75, 1.67, 0.01) | (85, 1.67, 0.01) | (95, 1.67, 0.01) |

AHP Method | Evidential Reasoning Approach $({\mathit{\lambda}}_{1}=0\mathit{,}{\mathit{\lambda}}_{2}=1)$ | Entropy Weight Method $({\mathit{\lambda}}_{1}=1\mathit{,}{\mathit{\lambda}}_{2}=0)$ | Comprehensive Weight Method $({\mathit{\lambda}}_{1}=0.5\mathit{,}{\mathit{\lambda}}_{2}=0.5)$ | |
---|---|---|---|---|

×1 | 0.0306 | 0.0328 | 0.0003 | 0.0166 |

×2 | 0.079 | 0.1292 | 0.0179 | 0.0736 |

×3 | 0.1828 | 0.065 | 0.2501 | 0.1576 |

×4 | 0.0306 | 0.1292 | 0.0002 | 0.0646 |

×5 | 0.1828 | 0.0402 | 0.2426 | 0.1414 |

×6 | 0.1654 | 0.0402 | 0.2253 | 0.1327 |

×7 | 0.079 | 0.0328 | 0.0748 | 0.0538 |

×8 | 0.0306 | 0.065 | 0.0019 | 0.0334 |

×9 | 0.079 | 0.1416 | 0.1474 | 0.1446 |

×10 | 0.079 | 0.1293 | 0.0278 | 0.0786 |

×11 | 0.0306 | 0.0655 | 0.0001 | 0.0332 |

×12 | 0.0306 | 0.1292 | 0.0107 | 0.0699 |

Evaluation results | II | III | II | III |

Cloud Model Theory and Evidential Reasoning Approach | Assessment Level | Evaluation Results | ||||
---|---|---|---|---|---|---|

Year | I | II | III | IV | V | |

2010 | 0 | 0.3069 | 0.4195 | 0.0593 | 0.2141 | III |

2011 | 0 | 0.2031 | 0.4713 | 0.0004 | 0.3249 | III |

2012 | 0 | 0.1052 | 0.4907 | 0.0602 | 0.3436 | III |

2013 | 0 | 0.0942 | 0.6224 | 0.001 | 0.2822 | III |

2014 | 0 | 0.1126 | 0.4859 | 0.0598 | 0.3415 | III |

2015 | 0 | 0.1958 | 0.3672 | 0.064 | 0.3728 | V |

2016 | 0 | 0.1734 | 0.3291 | 0.0545 | 0.4428 | V |

2017 | 0 | 0.1978 | 0.1678 | 0.2571 | 0.377 | V |

2018 | 0 | 0.1055 | 0.3999 | 0.1432 | 0.3512 | III |

2019 | 0 | 0.11 | 0.2333 | 0.2754 | 0.3811 | V |

Fuzzy comprehensive evaluation method | I | II | III | IV | V | Evaluation results |

2010 | 0 | 0.2678 | 0.281 | 0.25 | 0.2009 | III |

2011 | 0.0165 | 0.2011 | 0.1748 | 0.3482 | 0.2591 | IV |

2012 | 0.0095 | 0.1014 | 0.2483 | 0.3592 | 0.2814 | IV |

2013 | 0.0057 | 0.1078 | 0.3678 | 0.2525 | 0.266 | III |

2014 | 0.0165 | 0.0853 | 0.3585 | 0.2344 | 0.305 | III |

2015 | 0.0155 | 0.1562 | 0.3457 | 0.1622 | 0.32 | III |

2016 | 0.0091 | 0.1308 | 0.3346 | 0.1971 | 0.3281 | III |

2017 | 0.0103 | 0.1257 | 0.2509 | 0.1353 | 0.4776 | V |

2018 | 0 | 0.1855 | 0.2766 | 0.0972 | 0.4405 | V |

2019 | 0 | 0.0907 | 0.2654 | 0.1915 | 0.4522 | V |

Cloud Model Theory | I | II | III | IV | V | Evaluation results |

2010 | 0 | 0.3038 | 0.4014 | 0.0705 | 0.2241 | III |

2011 | 0 | 0.2246 | 0.4436 | 0.0006 | 0.331 | III |

2012 | 0 | 0.1232 | 0.4569 | 0.0736 | 0.3461 | III |

2013 | 0 | 0.1171 | 0.5734 | 0.0013 | 0.308 | III |

2014 | 0 | 0.1304 | 0.4528 | 0.0729 | 0.3438 | III |

2015 | 0 | 0.211 | 0.3524 | 0.0758 | 0.3606 | V |

2016 | 0 | 0.1922 | 0.3256 | 0.0629 | 0.4192 | V |

2017 | 0 | 0.2095 | 0.1749 | 0.2571 | 0.3582 | V |

2018 | 0 | 0.1212 | 0.3412 | 0.1543 | 0.3431 | V |

2019 | 0 | 0.1234 | 0.2386 | 0.2754 | 0.3625 | V |

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**MDPI and ACS Style**

Cao, W.; Deng, J.; Yang, Y.; Zeng, Y.; Liu, L.
Water Carrying Capacity Evaluation Method Based on Cloud Model Theory and an Evidential Reasoning Approach. *Mathematics* **2022**, *10*, 266.
https://doi.org/10.3390/math10020266

**AMA Style**

Cao W, Deng J, Yang Y, Zeng Y, Liu L.
Water Carrying Capacity Evaluation Method Based on Cloud Model Theory and an Evidential Reasoning Approach. *Mathematics*. 2022; 10(2):266.
https://doi.org/10.3390/math10020266

**Chicago/Turabian Style**

Cao, Wenzhi, Jilin Deng, Yi Yang, Yangyan Zeng, and Limei Liu.
2022. "Water Carrying Capacity Evaluation Method Based on Cloud Model Theory and an Evidential Reasoning Approach" *Mathematics* 10, no. 2: 266.
https://doi.org/10.3390/math10020266