Water resources management under dual uncertainties: a factorial fuzzy two-stage stochastic programming approach

In this study we propose a factorial fuzzy two-stage stochastic programming (FFTSP) approach to support water resources management under dual uncertainties. The dual uncertainties in terms of fuzziness in modeling parameters and variability of α-cut levels are taken into account. As different α-cut levels are assigned to each fuzzy parameter (instead of an identical α-cut level), the effects of α-cut levels on fuzzy parameters can be considered. Factorial analysis method is integrated with fuzzy vertex method to tackle the interactive effects of fuzzy parameters within a two-stage stochastic programming framework. The effects of the interactions among fuzzy parameters under various α-cut level combinations can be examined. The FFTSP approach is applied to a water resources management case to demonstrate its applicability. The results show that this approach can not only give various optimized solutions according to decision makers’ confidence levels but also provide in-depth analyses for the effects of fuzzy parameters and their interactions on the solutions. In addition, the results show that the effects of diverse α-cut combinations should not be disregarded because the results may differ under some specific α-cut combinations. The dual sequential factorial analyses embedded in the FFTSP approach guarantee most variations in a system can be analyzed. Therefore water managers are able to gain sufficient knowledge to make robust decisions under uncertainty.     

Two-stage fuzzy chance-constrained programming: application to water resources management under dual uncertainties

In this study, a two-stage fuzzy chance-constrained programming (TFCCP) approach is developed for water resources management under dual uncertainties. The concept of distribution with fuzzy probability (DFP) is presented as an extended form for expressing uncertainties. It is expressed as dual uncertainties with both stochastic and fuzzy characteristics. As an improvement upon the conventional inexact linear programming for handling uncertainties in the objective function and constraints, TFCCP has advantages in uncertainty reflection and policy analysis, especially when the input parameters are provided as fuzzy sets, probability distributions and DFPs. TFCCP integrates the two-stage stochastic programming (TSP) and fuzzy chance-constrained programming within a general optimization framework. TFCCP incorporates the pre-regulated water resources management policies directly into its optimization process to analyze various policy scenarios; each scenario has different economic penalty when the promised amounts are not delivered. TFCCP is applied to a water resources management system with three users. Solutions from TFCCP provide desired water allocation patterns, which maximize both the system’s benefits and feasibility. The results indicate that reasonable solutions were generated for objective function values and decision variables, thus a number of decision alternatives can be generated under different levels of stream flows, α-cut levels and fuzzy dominance indices.     

A dual two-stage stochastic model for flood management with inexact-integer analysis under multiple uncertainties

This study introduces a hybrid optimization approach for flood management under multiple uncertainties. An inexact two-stage integer programming (ITIP) model and its dual formation are developed by integrating the concepts of mixed-integer and interval-parameter programming techniques into a general framework of two-stage stochastic programming. The proposed approach provides a linkage to pre-defined management policies, deals with capacity-expansion planning issues, and reflects various uncertainties expressed as probability distributions and discrete intervals for a flood management system. Penalties are imposed when the policies are violated. The marginal costs are determined based on dual formulation of the ITIP model, and their effects on the optimal solutions are investigated. The developed model is applied to a case study of flood management. The solutions of binary variables represent the decisions of flood-diversion–capacity expansion within a multi-region, multi-flow-level, and multi-option context. The solutions of continuous variables are related to decisions of flood diversion toward different regions. The solutions of dual variables indicate the decisions of marginal costs associated with the resources of regions’ capacity, water availability, and allowable diversions. The results show that the proposed approach could obtain reliable solutions and adequately support decision making in flood management.     

Water resources management under uncertainty: factorial multi-stage stochastic program with chance constraints

Due to rapid growth of population and development of economy, water resources allocation problems have aroused wide concern. Therefore, optimization of water resources systems is complex and uncertain, which is a severe challenge faced by water managers. In this paper, a factorial multi-stage stochastic programming with chance constraints approach is developed to deal with the issues of water-resources allocation under uncertainty and risk as well as their interactions. It can deal with uncertainties described as both interval numbers and probability distributions, and can also support the risk assessment within a multistage context. The solutions associated with different risk levels of constraint violation can be obtained, which can help characterize the relationship between the economic objective and the system risk. The inherent interactions between factors at different levels and their effects on total net benefits can be revealed through the analysis of multi-parameter interactions.     

Inexact fuzzy two-stage programming for water resources management in an environment of fuzziness and randomness

A standard lower-side attainment values based inexact fuzzy two-stage programming (SLA-IFTSP) approach is proposed for supporting multi-water resources management under multi-uncertainties. The method improves upon the existing inexact two-stage stochastic programming by the introduction of a standard average lower-side attainment values based fuzzy linear programming. Multi-uncertainties such as intervals, probabilistic and/or possibilistic distributions and their combinations in water resources management can be directly communicated into the water allocation process. The risk of infeasibility caused by the random water availabilities can be analyzed by imposing economic penalties when the designed water allocations would not be satisfied after the occurrence of random seasonal flows. Based on the standard average lower-side attainment index, the fuzzy random relationships representing various subjective judgments in the model can be transformed into corresponding deterministic ones without additional constraints, and thus guarantee a higher computational efficiency. A hypothetical case regarding two-source water resources management is adopted for demonstrating its applicability. Reasonable solutions have been generated. They provide desired water allocations with maximized system benefit under different water availability levels. The solutions of intervals with different probabilities can be used for generating decision alternatives. Comparisons between the solutions from SLA-IFTSP and those from ITSP are also undertaken. They show that SLA-IFTSP can generate more reasonable water allocation patterns with higher net system benefits than ITSP.     

Challenges for integrated water resources management

During the past century, while world population tripled, the use of water increased sixfold. Irrigation accounts for 70% of global water withdrawals, industry for 20%, and municipal use for 10%. To meet the water resources challenges, a series of transitions is under way, with major implications for water management. The present paper underlines the major issues and the new role of engineers in charge of development and project management.     

A two-stage fuzzy-stochastic factorial analysis method for characterizing effects of uncertainties in hydrological modelling

ABSTRACT

In this study, a two-stage fuzzy-stochastic factorial analysis (TFFA) method is developed and applied to the Vakhsh watershed (upper reaches of Aral Sea basin, Central Asia) for daily streamflow simulation. TFFA has advantages in identifying the major parameters that have important individual and interactive effects on model outputs, as well as assessing multiple uncertainties resulting from randomness and vagueness characteristics of model parameters. The results reveal that (a) nine major parameters (from a total of 24) have significant effects on Soil Water Assessment Tool (SWAT) simulation performance for the study watershed; and (b) snowmelt-related parameters (including snowfall temperature, threshold temperature for snowmelt and s nowmelt factor) and runoff curve number (CN2) are most sensitive parameters for the runoff generation. The results also show that the proposed TFFA method can help enhance the hydrological model’s capability for runoff simulation/prediction, particularly for in data-scarce and high-mountainous watersheds.     

An interval-parameter multi-stage stochastic programming model for water resources management under uncertainty

In this study, an interval-parameter multi-stage stochastic linear programming (IMSLP) method has been developed for water resources decision making under uncertainty. The IMSLP is a hybrid methodology of inexact optimization and multi-stage stochastic programming. It has three major advantages in comparison to the other optimization techniques. Firstly, it extends upon the existing multi-stage stochastic programming method by allowing uncertainties expressed as probability density functions and discrete intervals to be effectively incorporated within the optimization framework. Secondly, 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. Thirdly, it cannot only handle uncertainties through constructing a set of scenarios that is representative for the universe of possible outcomes, but also reflect dynamic features of the system conditions through transactions at discrete points in time over the planning horizon. The developed IMSLP method is applied to a hypothetical case study of water resources management. The results are helpful for water resources managers in not only making decisions of water allocation but also gaining insight into the tradeoffs between environmental and economic objectives.     

A robust risk analysis method for water resources allocation under uncertainty

A robust risk analysis method (RRAM) is developed for water resource decision making under uncertainty. This method incorporates interval-parameter programming and robust optimization within a stochastic programming framework. In the RRAM formulation, penalties are exercised with the recourse against any infeasibility, and robustness measures are introduced to examine the variability of the second stage costs which are above the expected levels. In this study, a number of weighting levels are considered which correspond to the robustness levels of risk control. Generally, a plan with a higher robust level would better resist from system risk. Thus, decision with a lower robust level can correspond to a higher risk of system failure. There is a tradeoff between system cost and system reliability. The RRAM is applied to a case of water resource management. The modeling results can help generate desired decision alternatives that will be particularly useful for risk-aversive decision makers in handling high-variability conditions. The results provide opportunities to managers to make decisions based on their own preferences on system stability and economy, and ensure that the management policies and plans be made with reasonable consideration of both system cost and risk.     

Information-analytical systems for water resources quality management

The need to develop information-analytical systems (IAS) for water resources quality management is substantiated. The structure and principles of IAS organization, developed by the authors, are specified. A mathematical model that serves as a basis for the simulation block of proposed IAS is discussed and the results of model calculations are given.     

Planning of municipal solid waste management systems under dual uncertainties: a hybrid interval stochastic programming approach

In this study, a random-boundary-interval linear programming (RBILP) method is developed and applied to the planning of municipal solid waste (MSW) management under dual uncertainties. In the RBILP model, uncertain inputs presented as interval numbers can be directly communicated into the optimization process; besides, intervals with uncertain lower and upper bounds can be handled through introducing the concept of random boundary interval. Consequently, robustness of the optimization process can be enhanced. To handle uncertainties with such complex presentations, an integrated chance-constrained programming and interval-parameter linear programming approach (ICCP) is proposed. ICCP can help analyze the reliability of satisfying (or risk of violating) system constraints under uncertainty. The applicability of the proposed RBILP and ICCP approach is validated through a case study of MSW management. Violations for capacity constraints are allowed under a range of significant levels. Interval solutions associated with different risk levels of constraint violation are obtained. They can be used for generating decision alternatives and thus helping waste managers to identify desired policies under various environmental, economic, and system-reliability constraints.     

Resources and environmental systems management under synchronic interval uncertainties

Resources and environmental systems management (RESM) is challenged by the synchronic effects of interval uncertainties in the related practices. The synchronic interval uncertainties are misrepresented as random variables, fuzzy sets, or interval numbers in conventional RESM programming techniques including stochastic programming. This may lead to ineffectiveness of resources allocation, high costs of recourse measures, increased risks of unreasonable decisions, and decreased optimality of system profits. To fill the gap of few corresponding studies, a synchronic interval linear programming (SILP) method is proposed in this study. The proposition of interval sets and interval functions and coupling them with linear programming models lead to development of an SILP model for RESM. This enables incorporation of interval uncertainties in resource constraints and synchronic interval uncertainties in the programming objective into the optimization process. An analysis of the distribution-independent geometric properties of the feasible regions of SILP models results in proposition of constraint violation likelihoods. The tradeoff between system optimality and constraint violation is analyzed. The overall optimality of SILP systems under synchronic intervalness is quantified through proposition of integrally optimal solutions. Integration of these efforts leads to a violation-constrained interval integral method for optimization of RESM systems under synchronic interval uncertainties. Comparisons with selected existing methods reveal the effectiveness of SILP at eliminating negativity of synchronic intervalness, enabling risk management of and achieving overall optimality of RESM systems, and enhancing the reliability of optimization techniques for RESM problems. The exploited framework for analyzing synchronic interval uncertainties in RESM systems is helpful for addressing synchronisms of other uncertainties such as randomness or fuzziness and avoiding the resultant decision mistakes and disasters due to neglecting them.     

Factorial two-stage stochastic programming for water resources management

This study presents a factorial two-stage stochastic programming (FTSP) approach for supporting water resource management under uncertainty. FTSP is developed through the integration of factorial analysis and two-stage stochastic programming (TSP) methods into a general modeling framework. It can handle uncertainties expressed as probability distributions and interval numbers. This approach has two advantages in comparison to conventional inexact TSP methods. Firstly, FTSP inherits merits of conventional inexact two-stage optimization approaches. Secondly, it can provide detailed effects of uncertain parameters and their interactions on the system performance. The developed FTSP method is applied to a hypothetical case study of water resources systems analysis. The results indicate that significant factors and their interactions can be identified. They can be further analyzed for generating water allocation decision alternatives in municipal, industrial and agricultural sectors. Reasonable water allocation schemes can thus be formulated based on the resulting information of detailed effects from various impact factors and their interactions. Consequently, maximized net system benefit can be achieved.     

Risk aversion based interval stochastic programming approach for agricultural water management under uncertainty

In this study, a risk aversion based interval stochastic programming (RAIS) method is proposed through integrating interval multistage stochastic programming and conditional value at risk (CVaR) measure for tackling uncertainties expressed as probability distributions and intervals within a multistage context. The RAIS method can reflect dynamic features of the system conditions through transactions at discrete points in time over the planning horizon. Using the CVaR measure, RAIS can effectively reflect system risk resulted from random parameters. When random events are occurred, the adjustable alternatives can be achieved by setting desired targets according to the CVaR, which could make the revised decisions to minimize the economic penalties. Then, the RAIS method is applied to planning agricultural water management in the Zhangweinan River Basin that is plagued by drought due to serious water scarcity. A set of decision alternatives with different combinations of risk levels employed to the objective function and constraints are generated for planning water resources allocation. The results can not only help decision makers examine potential interactions between risks under uncertainty, but also help generate desired policies for agricultural water management with a maximized payoff and a minimized loss.     

An evidential reasoning-based sustainability index for water resources management

ABSTRACT

Evidential reasoning (ER) is introduced as a new basis for developing and implementing sustainability indices (SI). The ER-based evaluations keep the merits of previous well-known sustainability indices and provide added values such as more accurate grade-based assessment and aggregation of performance criteria. The proposed modifications significantly improve the capability of the SI to scrutinize and differentiate various water-supply conditions. Available data from Gorganrud-Qaresou basin, northern Iran, are used to evaluate the applicability and efficiency of the sustainability index. The results of the proposed method are compared with two water resources sustainability indices with similar concepts but different bases. It is shown that the ER-based SI can better distinguish undesirable water-supply scenarios.     

Inexact two-stage stochastic partial programming: application to water resources management under uncertainty

In this study, an inexact two-stage stochastic partial programming (ITSPP) method is developed for tackling uncertainties presented as intervals and partial probability distributions. A scenario-based interactive algorithm is proposed to solve the ITSPP model. This algorithm is implemented through: (i) obtaining extreme points of the linear partial information (LPI); (ii) generating an inexact two-stage stochastic programming (ITSP) model under each extreme point; (iii) solving ITSP models through interactive algorithm proposed by Huang and Loucks (Civil Eng Environ Syst 17:95–118, 2000); (iv) acquiring the interval solutions under each extreme point and the final optimal interval for the objective function. The developed method is applied to a case study for water-resources planning. The modelling results can generate a series of decision alternatives under various system conditions, and thus help decision makers identify the desired water-resources management policies under uncertainty.