Inexact Left-Hand-Side Chance-Constrained Programming for Nonpoint-Source Water Quality Management

An inexact left-hand-side chance-constrained programming (ILCCP) was proposed and applied to a nonpoint-source water quality management problem within an agricultural system. The ILCCP model can reflect uncertainties presented as interval parameters (manure mass balance, crop nutrient balances, energy and digestible protein requirements, pollutant losses, water quantity constraints, technical constraints, and so on) and left-hand-side random variables (nitrogen requirement of crop i) at the same time. A non-equivalent linearization form of ILCCP was deduced and proved intuitively, which can help handle the left-hand-side random parameters in the constraints. The decision schemes through ILCCP were analyzed under scenarios at different individual probabilities (p ᵢ , denotes the admissible probability of violating the constraint i). The performance of ILCCP was also compared with the corresponding interval linear programming model. A representative nonpoint-source water quality management case was employed to facilitate the analysis and the comparison. The optimization results indicated that the net system benefit in the water quality management case would decrease with increasing probability levels on the whole. This was because that the higher constraint satisfaction of probability would lead to stricter decision space. The optimal scheme shows an obvious downtrend in the application amount of manure as the violation probability levels decreasing from scenarios 1 to 3 (p ᵢ = 0.1, 0.05 and 0.01). This demonstrates that the application amount of manure would be reduced effectively by adjusting strictness of the constraints. This study is the first application of the ILCCP model to water quality management, which indicates that the ILCCP is applicable to other environmental problems under uncertainties.