# Optimal Allocation Model of Water Resources Based on the Prospect Theory

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## Abstract

**:**

## 1. Introduction

## 2. The Prospect Theory in Water Resources

#### 2.1. The Prospect Theory Overview

#### 2.2. The Prospect Theory Application in the Water Resources Allocation

#### 2.3. Determining the Theoretical Elements of the Prospect Theory for the Water Resources Allocation

#### 2.3.1. Reference Point Selection and Decision Preference Determination

_{i}. Different precipitation frequencies make the decision makers’ expectations of x

_{0}inconsistent. It is obvious that the water deficit x

_{i}in the wet year should be less than the corresponding one in the dry year. Thus, the annual reference point x

_{0}depends on the precipitation frequency.

_{1}, x

_{2}, …, x

_{n}} consists of independent variables; by sorting the elements of X in the reverse order, the sample set Y is formed, i.e., Y: {x

_{n}, x

_{n−1}, …, x

_{1}}; ${S}_{k}$ is the rank sequence of X; ${r}_{i}$ is the number of sample i when ${x}_{i}>{x}_{j},1\le j\le i$, $E({S}_{k})$ is the expected value of ${S}_{k}$; $Var({S}_{k})$ is the mean square error of ${S}_{k}$; the standard normal distributions of two sample sequences x and y are denoted by $U{F}_{k,X}$ and $U{B}_{k,Y}$, respectively. The corresponding relations for $U{F}_{k,X}$ and $U{B}_{k,Y}$ are given as follows:

#### 2.3.2. Probability Weight Function Quantification

#### 2.3.3. Value Function Quantification

## 3. Methodology

_{r}of the interval fuzzy matrix B under the attribute W that was introduced in Ma [52] as shown in Equation (10).

## 4. Case Study

^{2}[54]. The average annual precipitation is 524 mm in the basin (1990–2014). The precipitation from June to August is considered as more than 70% of the corresponding one for the whole year. The average annual temperature is 3.5 °C. The lowest temperature for each year occurs in January. The average annual sunshine hours are 2864 h and the average annual wind speed is 3.5 m/s. The lower reaches of the Wuyur River is a flooded area. Flat terrain leads to a slow flow that forms a wide distribution of lake and swamp. Zhalong wetland, which is on the list of important international wetlands, is located in the Wuyur River basin. Zhalong wetland covers an area of 2100 km

^{2}. The typical crop is phragmite and its growth period in a year is from May to September [55]. It is assumed that the evapotranspiration of phragmite multiplied by area equals the amount of ecological water demand of Wuyur River Basin in this paper. The study area with 6540 km

^{2}cultivated area is subordinated to the Fuyu county, Tailai county, Lindian county, and Dorbod Mongol Autonomous County in the Heilongjiang province. The main irrigated crop in this area is rice, where its growth period is from May to September. Thus, the main water use contradiction occurs in May to September between ecological and agricultural water in Wuyur River Basin.

## 5. Results and Discussion

#### 5.1. Reference and the Reversal Points of the Decision Preference

#### 5.2. The Effect of Water Deficit Scenarios on the Foreground Function

^{3}and 1.717 billion m

^{3}, respectively. Water supply and water deficit are calculated by GAMS. The total water supply and water deficit for ecology and agriculture are 2.128 billion m

^{3}and 0.092 billion m

^{3}, respectively.

#### 5.3. Comparison of the Water Deficit Before and After Decision Reversal

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**The comparison of water demand before and after decision reversal as well as marginal utility. (

**a**) water demand; (

**b**) marginal utility.

**Figure 4.**The difference of ecological water demand and precipitation during May to October of each year from 1990.

**Figure 5.**The difference of agricultural water demand and precipitation during May to October of each year from 1990.

**Figure 7.**Comparison between the phragmite area in the remote sensing interpretation and the previous results from 1990 to 2014.

**Figure 9.**The cumulative prospect theory values for ecological and agricultural water in different preferences of decision-makers.

**Figure 11.**The cumulative prospect theory values for different water allocation scenarios in $\gamma =0.5$ from 1990 to 2014.

Category | Status | Average Water Deficit | Water Deficit | Depth of Water Deficit | Maximum Water Deficit Depth | ||||
---|---|---|---|---|---|---|---|---|---|

May | June | July | August | September | |||||

Ecology | Before | 7610.0 | 5459.9 | 2095.4 | 2900.9 | 372.0 | 18438.3 | 49% | 71% |

After | 817.3 | 568.6 | 232.8 | 322.3 | 354.5 | 2295.6 | 6% | 8% | |

Agriculture | Before | 2783.7 | 2251.2 | 893.7 | 1522.6 | 222.9 | 7674.0 | 22% | 29% |

After | 3888.6 | 1879.6 | 893.7 | 1522.6 | 222.9 | 8407.3 | 24% | 40% |

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**MDPI and ACS Style**

He, H.; Chen, A.; Yin, M.; Ma, Z.; You, J.; Xie, X.; Wang, Z.; An, Q. Optimal Allocation Model of Water Resources Based on the Prospect Theory. *Water* **2019**, *11*, 1289.
https://doi.org/10.3390/w11061289

**AMA Style**

He H, Chen A, Yin M, Ma Z, You J, Xie X, Wang Z, An Q. Optimal Allocation Model of Water Resources Based on the Prospect Theory. *Water*. 2019; 11(6):1289.
https://doi.org/10.3390/w11061289

**Chicago/Turabian Style**

He, Huaxiang, Aiqi Chen, Mingwan Yin, Zhenzhen Ma, Jinjun You, Xinmin Xie, Zhizhang Wang, and Qiang An. 2019. "Optimal Allocation Model of Water Resources Based on the Prospect Theory" *Water* 11, no. 6: 1289.
https://doi.org/10.3390/w11061289