Effective planning of resources management is important for facilitating socio-economic development and eco-environmental sustainability. Such a planning effort is complicated with a variety of uncertain, dynamic and nonlinear factors as well as their interactions. In this study, an inexact-stochastic quadratic programming with recourse (ISQP-R) method is developed for reflecting dynamics of system uncertainties based on a complete set of scenarios as well as tackling nonlinearities in the objective function to reflect the effects of marginal utility on system benefits and costs. Moreover, since penalties are exercised with recourse against any infeasibility, the ISQP-R can support the analysis of various policy scenarios that are associated with different levels of economic consequences when the promised targets are violated. The developed method is applied to a case study of planning resources management and developing regional ecological sustainability. The results have been generated and are helpful for decision makers in not only identifying desired resources-allocation strategies but also gaining insight into the tradeoff between economic objective and eco-environment violation risk.

In this study, a fuzzy-stochastic-based violation analysis (FSVA) approach is developed for the planning of water resources management systems with uncertain information, based on a multistage fuzzy-stochastic integer programming (FSIP) model. In FSVA, a number of violation variables for the objective and constraints are allowed, such that in-depth analyses of tradeoffs among economic objective, satisfaction degree, and constraint-violation risk can be facilitated. Besides, the developed method can deal with uncertainties expressed as probability distributions and fuzzy sets; it can also reflect the dynamics in terms of decisions for water-allocation and surplus-flow diversion, through transactions at discrete points of a complete scenario set over a multistage context. The developed FSVA method is applied to a case study of water resources management within a multi-stream, multi-reservoir and multi-period context. The results indicate that the satisfaction degrees and system benefits would be different under varied violation levels; moreover, different violation levels can also lead to changed water-allocation and surplus-flow diversion plans. Violation analyses are also conducted to demonstrate that violating different constraints have different effects on system benefit and satisfaction degree.

In this study, a dual-interval vertex analysis (DIVA) method is developed, through incorporating the vertex method within an interval-parameter programming framework. The developed DIVA method can tackle uncertainties presented as dual intervals that exist in the objective function and the left- and right-hand sides of the modeling constraints. An interactive algorithm and a vertex analysis approach are proposed for solving the DIVA model. Solutions under an associated a-cut level can be generated by solving a series of deterministic submodels. They can help quantify relationships between the objective function value and the membership grade, which is meaningful for supporting in-depth analyses of tradeoffs between environmental and economic objectives as well as those between system optimality and reliability. A management problem in terms of regional air pollution control is studied to illustrate applicability of the proposed approach. The results indicate that useful solutions for planning the air quality management practices have been generated. They can help decision makers to identify desired pollution-abatement strategies with minimized costs and maximized environmental efficiencies.

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.

In this study, an inexact multistage stochastic integer programming (IMSIP) method is developed for water resources management under uncertainty. This method incorporates techniques of inexact optimization and multistage stochastic programming within an integer programming framework. It can deal with uncertainties expressed as both probabilities and discrete intervals, and reflect the dynamics in terms of decisions for water allocation through transactions at discrete points of a complete scenario set over a multistage context. Moreover, the IMSIP can facilitate analyses of the multiple policy scenarios that are associated with economic penalties when the promised targets are violated as well as the economies-of-scale in the costs for surplus water diversion. A case study is provided for demonstrating the applicability of the developed methodology. The results indicate that reasonable solutions have been generated for both binary and continuous variables. For all scenarios under consideration, corrective actions can be undertaken dynamically under various pre-regulated policies and can thus help minimize the penalties and costs. The IMSIP can help water resources managers to identify desired system designs against water shortage and for flood control with maximized economic benefit and minimized systemfailure risk.