# Agricultural Water Optimal Allocation Using Minimum Cross-Entropy and Entropy-Weight-Based TOPSIS Method in Hetao Irrigation District, Northwest China

^{*}

## Abstract

**:**

^{3}to 60 billion m

^{3}, almost doubling in Urad. The annual water consumption under SSP2 and SSP3 increased slightly, from 30 billion m

^{3}to about 50 billion m

^{3}. The amount of water available for well irrigation in Urad decreased from 300 to 250 billion m

^{3}, while the amount of water available for canal irrigation in Urad remained at 270 billion m

^{3}from 2010 s to 2030 s, only dropping to 240 billion m

^{3}in 2040 s. The entropy-weight-based Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) method was applied to evaluate agricultural water resources’ allocation schemes because it can avoid the subjectivity of weight determination and can reflect the dynamic changing trend of irrigation district agricultural water resources’ carrying capacity. The approach is applicable to most regions, such as the Hetao Irrigation District in the Upper Yellow River Basi with limited precipitation, to determine water strategies under the changing environment.

## 1. Introduction

## 2. Methods

#### 2.1. Minimum Cross-Entropy Principle

#### 2.2. Copula Function

^{(−1)}(u_i) (i = 1, 2,⋯, N). The expression form of the copula function can be obtained: C(u_1, u_2,⋯, u_N) = H[F

^{(−1)}(u_1), F_2

^{(−1)}(u_2),⋯, F_N

^{(−1)}(u_N)].

#### 2.3. Interval Linear Fractional Programming (ILFP)

#### 2.3.1. Interval Parameter Programming (IPP)

^{±}is the objective function, a

^{±}, b

^{±}, c

^{±}are the interval coefficients, X

^{±}is the interval decision variable, “±” represents the interval value, “+” is the upper bound, and “−” is the lower bound.

#### 2.3.2. Linear Fractional Programming (LFP)

^{(m}

^{× n)}; b∈R

^{m}; p, q, x∈R

^{n}; α, β∈R; and rank A = m.

#### 2.3.3. ILFP

^{±}is the objective function; X

^{±}is the independent variable of model; C

^{±}, D

^{±}, α

^{±}, β

^{±}, A

^{±}, B

^{±}are interval parameters; and A

^{−}, A

^{+}are known matrices. Then, A

^{±}= [A

^{−}, A

^{+}] = {A∈R

^{(m}

^{× n}) a_ij

^{−}≤ a_ij ≤ a_ij

^{+}}, a_ij

^{±}is an element in matrix A

^{±}, and a_ij

^{−}and a_ij

^{+}are the upper and lower bounds of a_ij

^{±}. Other interval parameters can also be expressed in this way [16,17,18].

^{−}, A

^{+}], B = [B

^{−}, B

^{+}], Amv = (A

^{−}+ A

^{+})/2, and B

^{mv}= (B

^{−}+ B

^{+})/2. Then, according to expression (1), for any parameter A, B, the lower bound model solution set is expressed as P_ = {A

^{−}X ≤ B

^{−}, X ≥ 0}, the upper bound model solution set is expressed as P

^{+}= {A

^{+}X ≤ B

^{+}, X ≥ 0}, and the median model solution set is expressed as Pmv = {Amv X ≤ Bmv, X ≥ 0}, giving P_

^{−}⊆P_mv⊆P_

^{+}.

^{±}≥ 0, D

^{±}X

^{+}β

^{±}> 0, and f

^{±}(X) > 0, then, for each interval parameter, the upper limit expression of the objective function can be obtained as maxf

^{+}(X) = (C

^{+}X

^{+}α

^{+})/(D

^{−}X

^{+}β

^{−}) and the lower bound expression is maxf

^{−}(X) = (C

^{−}X + α

^{−})/(D

^{+}X

^{+}β

^{+}). When C

^{mv}= (C

^{−}+ C

^{+})/2, then D

^{mv}= (D

^{−}+ D

^{+})/2, α

^{mv}, and β

^{mv}. By analogy, the median expression of the objective function is maxf

^{mv}(X) = (C

^{mv}X

^{+}α

^{mv})/(D

^{mv}X

^{+}β

^{mv}), giving f

^{−}≤ f

^{mv}≤ f

^{+}.

^{±}(X) > 0, the partial derivative of the objective function with respect to the independent variable can be represented as f

^{+}/x_j = (c_j

^{+}− d_j

^{−}f

^{+})/(∑_(i = 1)

^{n}〖d_i

^{−}x_j〗). When d_j

^{±}> 0, if c_j

^{+}∈(−∞, d_j

^{−}f_(P_mv)

^{+}), then, if and only if x_j → x_j+, f → f

^{+}. If c_j

^{+}∈[d_j

^{−}f_(P_mv)

^{+}, +∞), then, if and only if〖x〗_j → x_j

^{−}, f → f

^{+}.

#### 2.4. Entropy-Weight-Based TOPSIS Method

## 3. Application

#### 3.1. Study Area and Data Collection

^{3}, accounting for 71.6% of the city’s average annual runoff.

#### 3.2. Parameter Estimation

#### 3.3. Joint Probability of Water Supply and Water Demand

^{2}), was used to evaluate the performance of each copula function. The value of d

^{2}was 0.0372, 0.0593, 0.183, 0.0587, and 0.2064 corresponding to the above copula functions [17,18,19]. The results showed that, for the Clayton copula, the value of d

^{2}was the smallest; thus, the Clayton copula was selected, and the joint distribution function of rainfall and evaporation could be expressed as:

#### 3.4. Agricultural Water Resource Optimal Allocation under Uncertainty

- (1)
- Water availability constraint$$0<{X}_{i}\le {Q}_{i}^{\pm}$$
- (2)
- Water demand constraint$$E{T}_{min,ci}\le E{T}_{ci}^{\pm}\le E{T}_{max,ci}$$$$E{T}_{ci}^{\pm}=\frac{{X}_{i}{\eta}_{i}}{{A}_{ic}^{\pm}}$$
- (3)
- Land availability constraint$${A}_{ic,min}\le {A}_{ic}^{\pm}\le {A}_{ic,max}$$
- (4)
- Drip irrigation water quantity constraint$$\sum}_{c=1}^{C}{A}_{c-di}{I}_{c-di}>\frac{2}{3}{T}_{tr$$
- (5)
- Crop price constraint$${B}_{c,min}<{B}_{c}^{\pm}<{B}_{c,max}$$
- (6)
- Food security constraint$$\sum}_{i=1}^{I}\left({Y}_{ic}\times {A}_{ic}^{\pm}\right)\ge {\displaystyle \sum}_{i=1}^{I}\left(P{O}_{ic}\times {P}_{f}\right)$$

^{3}); ${Q}_{i}$ is the incoming water quantity (m

^{3}); $E{T}_{ci}$ is the water demand (m

^{3}); $E{T}_{min,ci}$ and $E{T}_{max,ci}$ are minimum and maximum water demand (m

^{3}); ${\eta}_{i}$ is the water availability efficiency (%); ${A}_{ic,min}$ and ${A}_{ic,max}$ are the minimum and maximum irrigation areas, respectively, (ha); ${I}_{c-di}$ is the drip irrigation water quantity per planting area (m

^{3}/ha); ${T}_{tr}$ is the water rights’ transfer water quantity (m

^{3}); ${B}_{c,min}$ and ${B}_{c,max}$ are the minimum and maximum price of crops (RMB/kg); $P{O}_{ic}$ is the population for subarea $i$ and crop $j$; ${P}_{f}$ is the food demand per capita (kg/capita); and ${P}_{ic}$ is the agricultural water price (RMB/m

^{3}).

## 4. Result Analysis and Discussion

#### 4.1. Water Scarcity in the HID

^{3}to 300 million m

^{3}. In the Dengkou irrigation area, there is almost no water shortage problem. In the Linhe irrigation area, the water shortage in winter is the most serious, about 50 million m

^{3}to 200 million m

^{3}. In the Hangjinhou irrigation area and the Urad irrigation area, the water shortage in winter is also the most serious, 200 million m

^{3}to 300 million m

^{3}and 210 million m

^{3 to}300 million m

^{3}, respectively [26,27].

#### 4.2. Regional Water Use in the HID

^{3}under all three SSPs and declined thereafter. However, in 2050, the annual water consumption under SSP2 and SSP3 was about 10 billion m

^{3}, falling faster than 15 billion m

^{3}under SSP1. The annual water consumption in the Urad irrigation area showed an upward trend under three pathways. Under SSP1, the annual water consumption increased from 30 billion m

^{3}to 60 billion m

^{3}, almost doubling. The annual water consumption under SSP2 and SSP3 increased slightly, from 30 billion m

^{3}to about 50 billion m

^{3}. The water consumption of drip irrigation increased significantly from 2010 to 2050, and the water consumption of furrow irrigation remained basically unchanged. The water consumption of traditional border irrigation was about 5 billion m

^{3}in 2050, which was lower than that in 2010 [25,26,27].

#### 4.3. Water Availability Changes in the HID

^{3}to 45 billion m

^{3}in Dengkou, and the available water volume of canal irrigation decreased from 48 billion m

^{3}to 40 billion m

^{3}in Dengkou, with a small degree of decline. The amount of water available in Urad decreased more than that in Dengkou [28]. Specifically, the amount of water available for well irrigation in Urad decreased from 300 billion m

^{3}to 250 billion m

^{3}while the amount of water available for canal irrigation in Urad remained at 270 billion m

^{3}from 2010 s to 2030 s, only dropping to 240 billion m

^{3}in 2040 s.

#### 4.4. Water Shortages in the HID

^{3}under SSP1 to 250 million m

^{3}under SSP3, and the water shortage at Dengkou of canal irrigation increased from 250 million m

^{3}under SSP1 to 280 million m

^{3}under SSP3. In the Urad irrigation area, the water shortage of well irrigation increased from 800 million m

^{3}under SSP1 to 1.100 million m

^{3}under SSP3, and the water shortage under SSP1 and SSP2 was similar, both 1.000 million m

^{3}, and greatly increased to 1.300 million m

^{3}under SSP3 [29].

#### 4.5. Evaluation of Water Resources’ Carrying Capacity in the HID

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

Function Name | C(u,v) | Interpretation |
---|---|---|

Clayton copula | ${\left({\mu}^{-\theta}+{v}^{-\theta}-1\right)}^{-1/\theta}$ | $\theta >0$ and $\tau =\frac{\theta}{\theta +2}$. $\tau $ is the Kendall coefficient of rank correlation, and the same below. |

Gumbel copula | $exp\left[-{\left({\left(-ln\mu \right)}^{\theta}+{\left(-lnv\right)}^{\theta}\right)}^{1/\theta}\right]$ | $\theta \ge 1$ and $\tau =1-\frac{4}{\theta}\left[-\frac{1}{\theta}{{\displaystyle \int}}_{\theta}^{0}\frac{t}{exp\left(t\right)-1}dt-1\right]$. |

Frank copula | $-\frac{1}{\theta}ln\left[1+\frac{\left({e}^{-\theta \mu}-1\right)\left({e}^{-\theta v}-1\right)}{{e}^{-\theta}-1}\right]$ | $\theta \ge R$ and $\tau =1-\frac{4}{\theta}\left[-\frac{1}{\theta}{{\displaystyle \int}}_{\theta}^{0}\frac{t}{exp\left(t\right)-1}dt-1\right]$. |

t-copula | ${{\displaystyle \int}}_{-\infty}^{{{t}_{k}}^{-1}\left(\mu \right)}{{\displaystyle \int}}_{-\infty}^{{{t}_{k}}^{-1}\left(v\right)}\frac{1}{2\pi \sqrt{1-{\rho}^{2}}}\left[1+\frac{{s}^{2}-2\rho st+{t}^{2}}{k\left(1-{p}^{2}\right)}\right]dsdt$ | ${{t}_{k}}^{-1}$ is the inverse function t distribution function while the degree of freedom is k; ρ is the correlation coefficients between variables. |

Gaussian copula | ${{\displaystyle \int}}_{-\infty}^{{\varnothing}^{-1}\left(\mu \right)}{{\displaystyle \int}}_{-\infty}^{{\varnothing}^{-1}\left(v\right)}\frac{1}{2\pi \sqrt{1-{\rho}^{2}}}exp\left[-\frac{{s}^{2}-2\rho st+{t}^{2}}{2\left(1-{p}^{2}\right)}\right]dsdt$ | ${\varnothing}^{-1}$ is the inverse function of standard normal distribution function; ρ is the correlation coefficients between variables. |

## References

- Li, Y.; Huang, G. An inexact two-stage mixed integer linear programming method for solid waste management in the City of Regina. J. Environ. Manag.
**2006**, 81, 188–209. [Google Scholar] [CrossRef] [PubMed] - Huang, G.H.; Chi, G.F.; Li, Y.P. Long-Term Planning of an Integrated Solid Waste Management System under Uncertainty—Ι. Model Development. Environ. Eng. Sci.
**2005**, 22, 823–834. [Google Scholar] [CrossRef] - Huang, G.H.; Chi, G.F.; Li, Y.P. Long-Term Planning of an Integrated Solid Waste Management System under Uncertainty—ΙΙ. A North American Case Study. Environ. Eng. Sci.
**2005**, 22, 835–853. [Google Scholar] [CrossRef] - Zhang, Y.; Yang, P.; Liu, X.; Adeloye, A.J. Simulation and optimization coupling model for soil salinization and waterlogging control in the Urad irrigation area, North China. J. Hydrol.
**2022**, 607, 127408. [Google Scholar] [CrossRef] - Gu, J.J.; Guo, P.; Huang, G.H. Inexact stochastic dynamic programming method and application to water resources management in Shandong China under uncertainty. Stoch. Hydrol. Hydraul.
**2013**, 27, 1207–1219. [Google Scholar] [CrossRef] - Gu, J.J.; Guo, P.; Huang, G.H.; Shen, N. Optimization of the industrial structure facing sustainable development in resource-based city subjected to water resources under uncertainty. Stoch. Hydrol. Hydraul.
**2013**, 27, 659–673. [Google Scholar] [CrossRef] - Tong, F.; Guo, P. Simulation and optimization for crop water allocation based on crop water production functions and climate factor under uncertainty. Appl. Math. Model.
**2013**, 37, 7708–7716. [Google Scholar] [CrossRef] - Li, M.; Guo, P.; Fang, S.Q.; Zhang, L.D. An inexact fuzzy parameter two-stage stochastic programming model for irrigation water allocation under uncertainty. Stoch. Hydrol. Hydraul.
**2013**, 27, 1441–1452. [Google Scholar] [CrossRef] - Fang, S.Q.; Guo, P.; Li, M.; Zhang, L. Bi-level Multi-objective Programming Applied to Water Resources Allocation. Math. Prob. Eng.
**2013**, 2013, 837919. [Google Scholar] [CrossRef] [Green Version] - Guo, P.; Chen, X.; Tong, L.; Li, J.; Li, M. An optimization model for a crop deficit irrigation system under uncertainty. Eng. Optim.
**2012**, 46, 1–14. [Google Scholar] [CrossRef] - Hashimoto, T.; Loucks, D.P.; Stedinger, J.R. Robustness of water resources systems. Water Resour. Res.
**1982**, 18, 21–26. [Google Scholar] [CrossRef] [Green Version] - McKinney, D.C.; Loucks, D.P. Network design for predicting groundwater contamination. Water Resour. Res.
**1992**, 28, 133–147. [Google Scholar] [CrossRef] - Loucks, D.P. Modeling and managing the interactions between hydrology, ecology and economics. J. Hydrol.
**2005**, 328, 408–416. [Google Scholar] [CrossRef] - Revelle, C.S.; Loucks, D.P.; Lynn, W.R. Linear programming applied to water quality management. Water Resour. Res.
**1968**, 4, 1–9. [Google Scholar] [CrossRef] - Fedra, K.; Loucks, D.P. Interactive Computer Technology for Planning and Policy Modeling. Water Resour. Res.
**1985**, 21, 114–122. [Google Scholar] [CrossRef] [Green Version] - Tan, Q.; Huang, G.H.; Cai, Y.P. Multi-Source Multi-Sector Sustainable Water Supply Under Multiple Uncertainties: An Inexact Fuzzy-Stochastic Quadratic Programming Approach. Water Resour. Manag.
**2013**, 27, 451–473. [Google Scholar] [CrossRef] - Tan, Q.; Huang, G.; Cai, Y.; Yang, Z. A non-probabilistic programming approach enabling risk-aversion analysis for supporting sustainable watershed development. J. Clean. Prod.
**2016**, 112, 4771–4788. [Google Scholar] [CrossRef] - Tan, Q.; Huang, G.H.; Cai, Y.P. A Fuzzy Evacuation Management Model Oriented Toward the Mitigation of Emissions. J. Environ. Inform.
**2015**, 25, 117–125. [Google Scholar] [CrossRef] [Green Version] - Dong, C.; Huang, G.; Tan, Q. A robust optimization modelling approach for managing water and farmland use between anthropogenic modification and ecosystems protection under uncertainties. Ecol. Eng.
**2015**, 76, 95–109. [Google Scholar] [CrossRef] - Wang, R.; Li, Y.; Tan, Q. A review of inexact optimization modeling and its application to integrated water resources management. Front. Earth Sci.
**2015**, 9, 51–64. [Google Scholar] [CrossRef] - Dong, C.; Huang, G.H.; Tan, Q.; Cai, Y. Coupled planning of water resources and agricultural land-use based on an inexact-stochastic programming model. Front. Ear. Sci.
**2014**, 8, 70–80. [Google Scholar] [CrossRef] - Ren, C.; Guo, P.; Li, M.; Li, R. An innovative method for water resources carrying capacity research—Metabolic theory of regional water resources. J. Environ. Manag.
**2016**, 167, 139–146. [Google Scholar] [CrossRef] [PubMed] - Ren, C.; Guo, P.; Yang, G.; Li, R.; Liu, L. Spatial and Temporal Analyses of Water Resources Use Efficiency Based on Data Envelope Analysis and Malmquist Index: Case Study in Gansu Province, China. J. Water Resour. Plan. Manag.
**2016**, 142, 04016066. [Google Scholar] [CrossRef] - Ren, C.F.; Li, R.H.; Zhang, L.D.; Guo, P. Multi-objective stochastic fractional goal programming model for water resources optimal allocation among industries. J. Water Res. Plan. Manag.
**2016**, 142, 04016036. [Google Scholar] [CrossRef] - Ren, C.F.; Guo, P.; Li, M.; Gu, J.J. Optimization of Industrial Structure Considering the Uncertainty of Water Resources. Water Resour. Manag.
**2013**, 27, 3885–3898. [Google Scholar] [CrossRef] - Gui, Z.; Zhang, C.; Li, M.; Guo, P. Risk analysis methods of the water resources system under uncertainty. Front. Agric. Sci. Eng.
**2015**, 2, 205–215. [Google Scholar] [CrossRef] - Gui, Z.; Li, M.; Guo, P. Simulation-Based Inexact Fuzzy Semi-Infinite Programming Method for Agricultural Cultivated Area Planning in the Shiyang River Basin. J. Irrig. Drain. Eng.
**2017**, 143, 05016011. [Google Scholar] [CrossRef] - Guo, P.; Chen, X.; Li, M.; Li, J. Fuzzy chance-constrained linear fractional programming approach for optimal water allocation. Stoch. Hydrol. Hydraul.
**2014**, 28, 1601–1612. [Google Scholar] [CrossRef] - Guo, P.; Wang, X.; Zhu, H.; Li, M. Inexact Fuzzy Chance-Constrained Nonlinear Programming Approach for Crop Water Allocation under Precipitation Variation and Sustainable Development. J. Water Resour. Plan. Manag.
**2014**, 140, 05014003. [Google Scholar] [CrossRef] - Li, M.; Guo, P.; Liu, X.; Huang, G.; Huo, Z. A decision-support system for cropland irrigation water management and agricultural non-point sources pollution control. Desalination Water Treat.
**2014**, 52, 5106–5117. [Google Scholar] [CrossRef] - Zhang, L.D.; Guo, P.; Fang, S.Q.; Li, M. Monthly Optimal Reservoirs Operation for Multi-crop Deficit Irrigation under Fuzzy Stochastic Uncertainties. J. Appl. Math.
**2014**, 2014, 105391. [Google Scholar] - Fu, Y.; Li, M.; Guo, P. Optimal Allocation of Water Resources Model for Different Growth Stages of Crops under Uncertainty. J. Irrig. Drain. Eng.
**2014**, 140, 05014003. [Google Scholar] [CrossRef] - Zhang, D.; Guo, P. Integrated agriculture water management optimization model for water saving potential analysis. Agric. Water Manag.
**2016**, 170, 5–19. [Google Scholar] [CrossRef] - Liu, C.; Rubæk, G.H.; Liu, F.; Andersen, M.N. Effect of partial root zone drying and deficit irrigation on nitrogen and phosphorus uptake in potato. Agric. Water Manag.
**2015**, 159, 66–76. [Google Scholar] [CrossRef] - Liu, C.; Liu, F.; Ravnskov, S.; Rubaek, G.H.; Sun, Z.; Andersen, M.N. Impact of Wood Biochar and Its Interactions with Mycorrhizal Fungi, Phosphorus Fertilization and Irrigation Strategies on Potato Growth. J. Agron. Crop Sci.
**2016**, 203, 131–145. [Google Scholar] [CrossRef] - Liu, C.; Ravnskov, S.; Liu, F.; Rubæk, G.H.; Andersen, M.N. Arbuscular mycorrhizal fungi alleviate abiotic stresses in potato plants caused by low phosphorus and deficit irrigation/partial root-zone drying. J. Agric. Sci.
**2018**, 156, 46–58. [Google Scholar] [CrossRef] - Liu, C.; Liu, F.; Andersen, M.N.; Wang, G.; Wu, K.; Zhao, Q.; Ye, Z. Domestic wastewater infiltration process in desert sandy soil and its irrigation prospect analysis. Ecotoxicol. Environ. Saf.
**2020**, 208, 111419. [Google Scholar] [CrossRef] - Huang, W.; Dai, L.M.; Baetz, B.W.; Cao, M.F.; Razavi, S. Interval Binary Programming Model for Noise Control within an Urban Environment. J. Environ. Inform.
**2013**, 21, 93–101. [Google Scholar] [CrossRef] [Green Version] - Huang, W.; Walker, W.S.; Kim, Y. Junction potentials in thermolytic reverse electrodialysis. Desalination
**2015**, 369, 149–155. [Google Scholar] [CrossRef] - Huang, W.; Baetz, B.W.; Razavi, S. A GIS-Based Integer Programming Approach for the Location of Solid Waste Collection Depots. J. Environ. Inform.
**2016**, 28, 39–44. [Google Scholar] [CrossRef] [Green Version] - Huang, W.; Kim, Y. Electrochemical techniques for evaluating short-chain fatty acid utilization by bioanodes. Environ. Sci. Pollut. Res.
**2016**, 24, 2620–2626. [Google Scholar] [CrossRef] [PubMed] - Fan, Y.; Huang, W.; Huang, G.; Huang, K.; Zhou, X. A PCM-based stochastic hydrological model for uncertainty quantification in watershed systems. Stoch. Hydrol. Hydraul.
**2015**, 29, 915–927. [Google Scholar] [CrossRef] - Fan, Y.R.; Huang, W.; Huang, G.H.; Li, Z.; Li, Y.P.; Wang, X.Q.; Cheng, G.H.; Jin, L. A Stepwise-Cluster Forecasting Approach for Monthly Stream flows Based on Climate Teleconnections. Stoch. Environ. Res. Risk Assess.
**2015**, 29, 1557–1569. [Google Scholar] [CrossRef] - Miao, D.Y.; Huang, W.W.; Li, Y.P.; Yang, Z.F. An Inexact Two-Stage Water Quality Management Model for Supporting Sustainable Development in a Rural System. J. Environ. Inform.
**2014**, 24, 52–64. [Google Scholar] - Miao, D.Y.; Huang, W.W.; Li, Y.P.; Yang, Z.F. Planning Water Resources Systems under Uncertainty Using An Interval-Fuzzy De Novo Programming Method. J. Environ. Inform.
**2014**, 24, 11–23. [Google Scholar] [CrossRef] [Green Version]

**Figure 5.**Average monthly water shortage in five areas of the Hetao Irrigation District from 2010 to 2050 (RCP8.5).

**Figure 6.**Irrigation water consumption prediction of three methods for the Dengkou and Urad irrigation areas under different scenarios.

**Figure 7.**Annual average water availability of two irrigation methods in two irrigation areas (

**a**,

**b**) and annual average water shortage from 2010 to 2050 under the SSPs (

**c**,

**d**).

**Figure 8.**Water resource carrying capacity for different subareas in HID. (Note: (1) WRCC means water resource carrying capacity; and (2) the yellow, dashed line is the average value of WRCC in the different subareas of HID).

Dimension | Index | Unit | Index Attribute | Weights |
---|---|---|---|---|

Economic dimension (A) | The benefits of unilateral water (A1) | RMB/m^{3} | + | 0.1012 |

Water production efficiency (A2) | kg/ha | + | 0.1058 | |

Crop output (A3) | RMB | + | 0.1026 | |

Social dimension (B) | Proportion of green and high-quality agricultural products (B1) | % | + | 0.1324 |

Land productivity (B2) | kg/ha | + | 0.0988 | |

Resource consumption per unit of GDP (B3) | kg/RMB | − | 0.1259 | |

Environmental dimension (C) | Aging rate of engineering facilities (C1) | % | − | 0.0992 |

Global Warming Potential per output (C2) | kg/CO_{2}e | − | 0.0984 | |

Agricultural non-point pollution discharge (C3) | kg | − | 0.1357 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhang, Y.; Yang, P.
Agricultural Water Optimal Allocation Using Minimum Cross-Entropy and Entropy-Weight-Based TOPSIS Method in Hetao Irrigation District, Northwest China. *Agriculture* **2022**, *12*, 853.
https://doi.org/10.3390/agriculture12060853

**AMA Style**

Zhang Y, Yang P.
Agricultural Water Optimal Allocation Using Minimum Cross-Entropy and Entropy-Weight-Based TOPSIS Method in Hetao Irrigation District, Northwest China. *Agriculture*. 2022; 12(6):853.
https://doi.org/10.3390/agriculture12060853

**Chicago/Turabian Style**

Zhang, Yunquan, and Peiling Yang.
2022. "Agricultural Water Optimal Allocation Using Minimum Cross-Entropy and Entropy-Weight-Based TOPSIS Method in Hetao Irrigation District, Northwest China" *Agriculture* 12, no. 6: 853.
https://doi.org/10.3390/agriculture12060853