## 2 Materials and Methods
### Optimal Allocation of Water Resources
#### 2.2.4 Establishment of Optimal Allocation Model of Water Resources  
The Jingdian Irrigation District is dry and lacks water. The Yellow River Committee and Gansu Province approved the study area to lift 148 million m\({}^{3}\) of water from the Yellow River every year. Under the condition of limited irrigation and diversion, the existing irrigation water resources and cultivated land resources should be reasonably optimized. The Yellow River diversion and return flow are important irrigation water resources in irrigation areas. To maximize net profit, an optimal allocation model for uncertain water resources was established. This model operates under two modes: one that only considers irrigation diversion (model I) and another that simultaneously considers irrigation diversion and return flow (model II). 1. Objective function: \[\mathrm{Max}\mathrm{F}^{\pm}=\sum\limits_{i=1}^{7}\left(\mathrm{P}_{i}^{\pm} \cdot\mathrm{A}_{i}\cdot\mathrm{Y}_{i}\right)-\sum\limits_{i=1}^{7}\mathrm{Q}_ {i}\cdot\mathrm{A}_{i}^{\pm}\cdot\mathrm{C}-\sum\limits_{i=1}^{7}\mathrm{A}_{i }\cdot\mathrm{T}_{i},\] (12) where i represents the crop code (1: spring wheat, 2: corn, 3: flax, 4: potato, 5: melon, 6: goji berry, 7: forest fruits); F\({}^{\pm}\) represents net profit (interval number), USD; P\({}_{i}^{\pm}\) represents the unit price of the i-th crop, USD/kg; A\({}_{i}\) represents the planting area of the i-th crop, ha; Y\({}_{i}\) is the yield per unit area of the i-th crop, kg/ha; Q\({}_{i}\) is the net irrigation quota of the i-th crop, m\({}^{3}\)/ha; C is the unit price of irrigation water, USD/m\({}^{3}\); T\({}_{i}\) represents the planting cost of the i-th crop, including the cost of pesticides, fertilizers, seeds, etc., in USD/ha. 2. Constraints: 1. 1.  
where j represents the month; PEij is the effective precipitation of the i-th crop in the j-th month, mm; ETij is the crop water demand of the i-th crop in the j-th month, mm; Kciij is the crop coefficient of the i-th crop in the j-th month, mm; and ET0 ij represents the reference evapotranspiration of the i-th crop in the j-th month, mm. The multiple cropping index of crops in the study area is 1, and the total planting area of each crop should not be greater than the total planting area. In addition, affected by farmers' farming practices, the planting area of various crops should be within a reasonable range, and the interval value formed by the current planting structure from 2001 to 2020 should be used as the planting area range of different crops. Therefore, the crop planting area should meet the following requirements:  
\[\sum_{i=1}^{7}A_{i}\leq A_{tot}, \tag{17}\]  
\[A_{imin}\leq A_{i}\leq A_{imax}, \tag{18}\]  
where Atot represents the total cultivated area, ha; Aimax represents the maximum planting area of the i-th crop, ha; and Aimin represents the minimum planting area of the i-th crop, ha. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Period** & **Mar.** & **Apr.** & **May** & **Jun.** & **Jul.** & **Aug.** & **Sep.** & **Oct.** \\ \hline Current & 82.0 & 116.9 & 146.2 & 156.9 & 157.7 & 134.4 & 91.4 & 65.9 \\ Short-term & 86.4 & 116.9 & 147.4 & 145.3 & 158.2 & 126.6 & 92.2 & 65.9 \\ Long-term & 88.4 & 115.9 & 147.3 & 141.4 & 159.6 & 123.7 & 93.6 & 64.9 \\ \hline \end{tabular}
\end{table} 
Table 4: Monthly reference evapotranspiration (mm) during the crop period.