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 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 robust interval-based minimax-regret analysis (RIMA) method is developed and applied to the identification of optimal water-resources-allocation strategies under uncertainty. The developed RIMAapproachcan address uncertainties with multiple presentations.Moreover, it can be usedfor analyzing all possible scenarios associated with different system costs/benefits and risk levels without making assumptions on probabilistic distributions for random variables. In its solution process, an intervalelement cost/benefit matrix can be transformed into an interval-element regret matrix, such that the decision makers can identify desired strategies based on inexact minimax regret (IMMR) criterion. Moreover, the fuzzy decision space is delimited into a more robust one through dimensional enlargement of the original fuzzy constraints. The developed method is applied to a case study of planning water resources allocation under uncertainty. The results indicate that reasonable solutions have been generated. They can help decision makers identify desired strategies for water-resources allocation with a compromise between maximized system benefit and minimized system-failure 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.

A hybrid fuzzy-stochastic water-management (FSWM) model is developed for agricultural sustainability under uncertainty, based on advancement of a multistage fuzzy-stochastic quadratic programming (MFSQP) approach. In MFSQP, uncertainties presented in terms of fuzziness and randomness can be incorporated within a multilayer scenario tree, such that revised decisions are permitted in each time period based on the realized values of the uncertain events. Moreover, fuzzy quadratic terms are used in the objective function to minimize the variation of satisfaction degrees among the constraints; it allows an increased flexibility in controlling the system risk in the optimization process. Results of the case study indicate that useful solutions for the planning of agricultural water management have been obtained. In the FSWM model, a number of policies for agricultural water supply are conducted. The results obtained can help decision makers to identify desired water-allocation schemes for agricultural sustainability under uncertainty, particularly when limited water resources are available for multiple competing users.

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.

In real-world municipal solid waste (MSW) management systems, identification of proper policies under uncertainty for accomplishing desired waste-disposal targets is critical. An inexact minimax regret integer programming (IMMRIP) method for the long-term planning of MSW management is developed. It incorporates the technique of minimax regret analysis (MMR) into an interval-parameter mixed-integer linear programming (IMILP) framework. The IMMRIP method can handle dual uncertainties presented as both random variables and interval values; it only needs a list of scenarios without any assumption on their probability distributions. It can facilitate dynamic analysis for decisions of system-capacity expansion and/or development within a multi-facility and multi-period context. Moreover, it can also be used for analyzing multiple scenarios associated with different system costs and risk levels. An interval-element cost matrix can be transformed into an interval-element regret matrix based on an interactive algorithm. Solutions based on an inexact minimax regret criterion can identify desired alternatives for MSW management and planning under a variety of uncertainties. In a companion paper, the developed method will be applied to a real case study in the City of Regina, Canada.

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.