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In the existing agricultural water management models under uncertainty, the mutual-correlation and their self-correlation of random variables (like precipitation (P), runoff (R), reference evapotranspiration (ET0), etc.) are often ignored. When expressing the fuzziness of socio-economic factors, fuzzy membership function is usually determined by the experience of decision-makers, which often brings some confusions. To solve the above questions, first, C-vine copula is introduced in this study to depict the multiple interdependence structures. Two kinds of three-dimensional copulas is constructed: $${CV}_{1}({R}_{t}, {P}_{t}, {R}_{t-1})$$ and $${CV}_{2}({ET}_{0t}, {P}_{t}, {ET}_{0(t-1)})$$ , where t is at t-th month. Second, the cloud model, as a novel qualitative and quantitative transformation model, is chosen to describe the uncertainty of crop prices. Combining these two uncertainty-expressing methods, an agricultural water resources optimization model is built to gain maximum net benefit by allocating limited surface water and groundwater. Then this model was applied to a case study in northwestern China. Results show that the developed model could provide the decision-makers with not only the best or the optimum range of system net benefits but also the probability of obtaining a given benefit under complex uncertainties. For comparison, the ordinary models without consideration of dependence of variables as an independent were also built. When overlooking the mutual-correlation and self-correlation, the optimal water allocation and system net benefits would be higher in dry years with total water allocation higher by 4.5%. This unreasonable allocation results may cause excessive agricultural irrigation to squeeze water for other industries in dry years, which would exacerbate water shortages. The discussion and comparison results prove the necessity and effectiveness of this research.

A copula-measure Me based interval multi-objective multi-stage stochastic chance-constrained programming (CMIMOMSP) model is proposed for water consumption optimization. It can conduct water allocation amid multiple users and multiple stages, and deal with the uncertainties presented as interval numbers, random fuzzy interval numbers, and stochastic variables. It improves upon multi-stage stochastic chance-constrained programming by introducing the multi-objective programming, and it can tradeoff the relationships amid economic benefit, full usage of water resources, and economic loss. It enhances the accuracy of copula function and conditional distribution function through proposing the interval functions. Besides, it can deal with the impact of the decision attitudes of managers on water allocation by formulating the function equation between water demand and the optimistic-pessimistic factor. The CMIMOMSP model is applied to a case study of the Heihe River Basin to verify its application. The results indicate that: (1) the optimistic-pessimistic factors have different degrees of positive influences on water allocation for industrial, domestic and ecological sectors; (2) the joint violated probability and optimistic-pessimistic factor have various range of impacts on agricultural water allocation; (3) tthe objective function values have different variation tendencies with the rise of joint violated probabilities and optimistic-pessimistic factors. Its robustness is enhanced by comparing it with the three single-objective programming models. The CMIMOMSP model can provide various water allocation schemes for managers with different risk attitudes in semi-arid and arid districts.

Due to the widespread uncertainties in agricultural water resources systems and climate change projections, the traditional optimization methods for agricultural water management may have difficulties in generating rational and effective optimal decisions. In order to get optimal future agricultural water allocation schemes for arid areas with consideration of climate change conditions, the model framework established in this paper integrates a statistical downscaling model, back propagation neural networks, and an evapotranspiration model (the Hargreaves model) with inexact irrigation water allocation optimization model under future climate change scenarios. The model framework, which integrates simulation models and optimization models, considers the interactions and uncertainties of parameters, thereby reflecting the realities more accurately. It is applied to the Yingke Irrigation Area in the midstream area of the Heihe River Basin in Zhangye city, Gansu Province, northwest China. Then, water allocation schemes in planning year (2047) under multiple future Representative Concentration Pathways (RCP) scenarios and the status quo (2016) are compared, in order to evaluate the practicability of generated water allocation schemes. The results show that the water shortages of economic crops are improved compared with the status quo under all RCP scenarios while those of the grain crops present opposite results. Meanwhile, the economic benefits decrease from the status quo to planning year under all future scenarios. This phenomenon is directly related to the amount of irrigation water allocation and is indirectly related to the changes of meteorological conditions. The model framework can reveal the regular pattern of hydro-meteorological elements with the impact of climate change. Meanwhile, it can generate irrigation water allocation schemes under various RCPs scenarios which could provide valuable decision support for water resources managers.