Optimal Irrigation Water Allocation among Different Growth Stages

With the increasing impact of global climate change, the effective utilization of water resources become more and more important. Among different water use sectors, agricultural irrigation accounts for about 70% of global water use. Different from other water uses which can be utilized at any time, irrigation is not only a matter of quantity, it is more important to accurately schedule according to the water demand of crops. However, in many studies in the field of water resources or agriculture, the non-rejuvenation of crop growth is usually ignored when calculating crop yield. These studies usually simply assume that irrigation water is effective in each period, and finally add up the yield of each period as the total benefit of irrigation. To solve this problem, the study aims to explore the dynamic decision-making of irrigation schedules with the consideration of uncertainty under water scarcity. Considering the characteristics of crop growth, we propose an analytical model in the form of a max-inf problem to investigate a two-stage stochastic optimal allocation of water to maximize the yield expectations of the two stages. We first assume that the rainfall in the first stage is known, and the rainfall is described by a given probability distribution. With FAO 33, an empirical production function assessing the yield response to water, we need to apply the concept of max-inf to determine the expected yield. Accordingly, we found the optimal condition which maximizes the yield, satisfying a linear relationship between the probability of the first stage dominance and the water demand and yield response factor of the two stages. With this optimal condition, we can use the known crop water demand and yield response factor to estimate the probability of the first stage dominance and adjust the irrigation water to achieve the condition of maximum yield expectation, to achieve the goal of maximum yield. Following, this study proposes four scenarios to examine the optimal decision with numerical experiments, not only to verify the analytic solution but also to examine the decision-making under different conditions. So far the decision is still discussed within the two-stage framework, assuming that the rainfall in the first and second stages is known, and adding the rainfall uncertainty in the third stage to analyze irrigation water. It will extend to the multi-stage framework which could more reasonable presentation of crop yield decisions. In this way, this study can better help us to understand irrigation decision-making among different water supply stages under uncertainty.