Vine copula and cloud model-based programming approach for agricultural water allocation under uncertainty

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: CV1(Rt,Pt,Rt-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${CV}_{1}({R}_{t}, {P}_{t}, {R}_{t-1})$$\end{document} and CV2(ET0t,Pt,ET0(t-1))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${CV}_{2}({ET}_{0t}, {P}_{t}, {ET}_{0(t-1)})$$\end{document}, 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.