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 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 fuzzy-stochastic constraint-softened programming method is developed for municipal solid waste (MSW) management under uncertainty. The developed method can deal with multiple uncertainties presented in terms of fuzzy sets, interval values and random variables. Moreover, a number of violation levels for the system constraints are allowed. This is realized through introduction of violation variables to soften system constraints, such that the model's decision space can be expanded under demanding conditions. This can help generate a range of ecision alternatives under various conditions, allowing in-depth analyses of tradeoffs among economic objective, satisfaction degree, and constraint-violation risk. The developed method is applied to a case study of planning a MSW management system. The uncertain and dynamic information can be incorporated within a multi-layer scenario tree; revised decisions are permitted in each time period based on the realized values of uncertain events. Solutions associated with different satisfaction degree levels have been generated, corresponding to different constraint-violation risks. They are useful for supporting decisions of waste flow allocation and system-capacity expansion within a multistage context.

In this study, a multistage fuzzy-stochastic programming (MFSP) model is developed for tackling uncertainties presented as fuzzy sets and probability distributions. A vertex analysis approach is proposed for solving multiple fuzzy sets in the MFSP model. Solutions under a set of a-cut levels can be generated by solving a series of deterministic submodels. The developed method is applied to the planning of a case study for water-resources management. Dynamics and uncertainties of water availability (and thus water allocation and shortage) could be taken into account through generation of a set of representative scenarios within a multistage context. Moreover, penalties are exercised with recourse against any infeasibility, which permits in-depth analyses of various policy scenarios that are associated with different levels of economic consequences when the promised water-allocation targets are violated. The modeling results can help to generate a range of alternatives under various system conditions, and thus help decision makers to identify desired water-resources management policies under uncertainty.