multi-stage stochastic programming

Moghaddasi et al. (2010a,b) did a research on optimization of water allocation and distribution in irrigation network of Zayandeh Rud catchment by the use of non-linear methods of collective intelligence and genetic algorithm. In the above mentioned study, optimization has done by 4 sub-models of water distribution optimization during crop growth periods, water allocation optimization among network crops, water distribution optimization among different networks, and annual optimization of agricultural water proportion of Zayandeh Rud. Comparing the results, it can be assumed that nonlinear programming )NLP) and consequently particle swarm optimization )PSO) methods have priority to the others.in addition the study confirmed that the optimization method associates 36% more income. Management of irrigation water resources in the agricultural sector, as the most important renewable but limited resource, has been the most important challenge of the current century and has resulted in major concerns around the world. The aim of this study was to allocation irrigation water and cropping area under uncertainty with emphasis on water use efficiency (WUE) and relative irrigation supply (RIS) indices and the results were compared to the actual management in  Marun Irrigation Network.

Given the limited available water resources, the conflict-laden issues of water allocation have become a challenging issue for planners and water consumers. The existence of different types of uncertainties and climate change conditions in the water resources systems intensify the complexity of the optimal allocation process of water resources in the agricultural field. Therefore, In the present study, it is tried to optimally allocate water resources in Marand watershed by developing a mathematical planning model that is capable of dynamics and uncertainty conditions both in the parameters of objective functions and in constraints.

In this study, an uncertainty-based interactive multi-stage stochastic programming (UIMSP) approach is proposed by incorporating the fractile criterion method and chance-constrained programming within a multi-stage decision-making framework for agricultural water resource management in various planning horizon. A two-period scenario tree with different probabilities for violating constraints (α) and various satisfactory levels of the objective function (β) were selected.

The results indicates that the minimum and maximum benefits of the system are [46.6, 983.6] × 1010 (Rials) and [699.7, 1483.1] × 1010 (Rials) when α=0.01  and , β=0.9 , respectively. In addition, an increase in satisfactory levels of the objective function (β) led to a gradual decrease in both upper and lower bound of the total net benefits at each given probability level of violating constraint (α), while any change in probability of constraint violation (α) resulted in creating a significant change in the system profits. Also, the results of the UIMSP model show that water scarcity is observed in different scenarios due to insufficient water resources in Marand watershed. However, according to the results, in the second period, despite the decrease in the amount of water available, the amount of water shortage has decreased with increasing irrigation efficiency. For example, under medium flow level, the amount of surface and groundwater shortages in the first period is 43.6% and 32.5%, respectively, while it is reduced by [36.9, 24.5] and [24.5, 0] percent in the second period.

The results can help decision-makers examine potential interactions between risks related to the stochastic objective function and constraints. Also, increasing irrigation efficiency and upgrading irrigation systems are among the main strategies that can provide an economic benefit to farmers while reducing environmental damage.