In water-quality management problems, uncertainties may exist in a number of impact factors and pollution-related processes (e.g., the volume and strength of industrial wastewater and their ariations can be presented as random events through identifying a statistical distribution for each source); moreover, nonlinear relationships may exist among many system components (e.g., cost parameters may be functions of wastewater-discharge levels). In this study, an inexact two-stage stochastic quadratic programming (ITQP) method is developed for water-quality management under uncertainty. It is a hybrid of inexact quadratic programming (IQP) and two-stage stochastic programming (TSP) methods. The developed ITQP can handle not only uncertainties expressed as probability distributions and interval values but also nonlinearities in the objective function. It can be used for analyzing various scenarios that are associated with different levels of economic penalties or opportunity losses caused by improper policies. The ITQP is applied to a case of water-quality management to deal with uncertainties presented in terms of probabilities and intervals and to reflect dynamic interactions between pollutant loading and water quality. Interactive and derivative algorithms are employed for solving the ITQP model. The solutions are presented as combinations of deterministic, interval and distributional information, and can thus facilitate communications for different forms of uncertainties. They are helpful for managers in not only making decisions regarding wastewater discharge but also gaining insight into the tradeoff between the system benefit and the environmental requirement.

In this study, an inexact minimax regret mixed integer programming (IMMRIP) method is applied to long-term planning of municipal solid waste (MSW) management in the City of Regina. The method can help tackle the dynamic, interactive, and uncertain characteristics of the solid waste management system in the city, and can address issues concerning plans for cost-effective waste diversion and landfill prolongation. Thirty-six situations were examined based on multiple alternatives and scenarios under different waste-generation levels. Reasonable solutions have been generated for decisions of system-capacity expansion and waste-flow allocation, demonstrating complex tradeoffs among system cost, regret level, and constraint-violation risk. Solutions associated with further inexact minimax regret (IMMR) analyses can help tackle tradeoffs between minimized system cost and maximized system feasibility. Under the optimal alternative, the system would reach a maximum reliability with the lowest risks of penalty and wastage. Results provide valuable inputs for adjustment of the existing waste flow allocation patterns to satisfy the city's diversion goals, long-term capacity planning for the city's waste management system, and generation desired policies for managing the city's waste collection and treatment.

An interval-fuzzy multistage programming (IFMP) method is developed for water resources management under uncertainty. This method improves upon the existing multistage stochastic programming methods by allowing uncertainties presented as discrete intervals, fuzzy sets, and probability distributions to be effectively incorporated within its optimization framework. The IFMP method can adequately reflect dynamic variations of system conditions, particularly for large-scale multistage problems with sequential structures. The uncertain information can be incorporated within a multi-layer scenario tree; revised decisions are permitted in each time period based on the realized values of the uncertain events. Moreover, this method can be used for analyzing various policy scenarios that are associated with different levels of economic consequences when the promisedwater-allocation targets are violated.Acase study ofwater resources management is then provided for demonstrating applicability of the developed method. For all scenarios under consideration, corrective actions are allowed to be taken dynamically in reference to the preregulated policies and the realized uncertainties. The results can help quantify the relationships among system benefit, satisfaction degree, and constraint-violation risk. Thus, desired decision alternatives can be generated under different conditions of supply-demand dynamics.

In this study, an integrated two-stage optimization model (ITOM) is developed for the planning of municipal solid waste (MSW) management in the City of Regina, Canada. The ITOM improves upon the existing optimization approaches with advantages in uncertainty reflection, dynamic analysis, policy investigation, and risk assessment. It can help analyze various policy scenarios that are associated with different levels of economic penalties when the promised policy targets are violated, and address issues concerning planning for a cost-effective diversion program that targets on the prolongation of the existing landfill. Moreover, violations for capacity and diversion constraints are allowed under a range of significance levels, which reflect the tradeoffs between system-cost and constraint-violation risk. The modeling results are useful for generating a range of decision alternatives under various environmental, socio-economic, and system-reliability conditions. They are valuable for supporting the adjustment (or justification) of the existing waste-management practices, the long-term capacity planning for the city's waste-management system, and the identification of desired policies regarding waste generation and management.