## 4 Conclusions  
Based on the prediction of future climate change characteristics in the study area, the goal is to maximize net profit while considering the constraints of maximum irrigation diversion and maximum planting area. An uncertain water resources optimal allocation model is established, incorporating the reuse of return flow. The main conclusions are as follows:  
1. The precipitation in the study area shows an increasing trend. The annual precipitation in the short-term years and long-term years increases by 6.1 mm and 8.9 mm, respectively. The reference evapotranspiration showed a decreasing trend. The reference evapotranspiration in the short-term years and long-term years decreases by 8.2 mm and 13.3 mm respectively.
2. After optimizing the optimal allocation of irrigation water diversion (model I) in the current years, the net profit increased by USD [18; 27; 24.18] million, and the total planting area increased by 900 ha. The crop planting structure changes significantly, with a decrease in the planting proportion of low water productivity crops (spring wheat, corn, and flax) and an increase in the planting proportion of high water productivity crops (potato, melon, goji berry, and forest fruits). After optimizing for the upcoming years, the net profit for both the short-term and long-term years saw an increase of USD [2.20, 2.47] and [3.02, 3.43] million respectively. Additionally, the total planting area expands by 700 hectares and 1000 hectares respectively. Due to limitations in irrigation water diversion, the irrigation area faces a shortage of water resources. However, there is potential to increase the crop planting area in both the current years and the planning years. By effectively utilizing the return flow, the crop planting area can be expanded even further.
3. When optimizing the allocation of return flow (model II), the net profit of the current years, short-term years, and the long-term years increased by USD [7.14, 7.97], [5.08, 5.63] and [4.4, 4.81] million, respectively, compared with the net profit of model I. The total planting area also increased by 2.6 thousand ha, 1.9 thousand ha, and 1.6 thousand ha, respectively, compared with the total planting area of model I. Additionally, the amount of irrigation water diversion was reduced by [12; 18] million m\({}^{3}\), [18; 24] million m\({}^{3}\), and [20; 26] million m\({}^{3}\), respectively, compared with that of model I. The use of return flow resources for farmland irrigation can not only improve irrigation efficiency and avoid the pollution of return flow discharge on the water quality of the Yellow River, but also reduce water diversion from the Yellow River, play a water-saving role, and reduce the cost of water extraction to a certain extent. (4) The application of interval linear programming models can effectively optimize the allocation of irrigation return flow under conditions of uncertainty. This study also serves as a reference for the reuse and optimization of irrigation return flow in other arid regions, contributing positively to the efficient utilization of water resources and the protection of water environments. In addition to irrigation return flow, the method proposed in this study can also be applied for the optimal allocation of the reuse of treated industrial and domestic wastewater. However, it is important to determine a reasonable interval number as a constraint condition.  
Conceptualization, F.J. and L.F.; methodology, F.J. and Y.P.; writing--original draft preparation, F.J.; writing--review and editing, Y.G.; funding acquisition, S.L. All authors have read and agreed to the published version of the manuscript. This research was funded by the Tianshan Talent Training Program (grant number 2023 TSYC-CCX0091) and the Tianchi Talent Project (grant number 51052401507).  
Not applicable.  
The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.  
The authors declare no conflicts of interest.