An Inexact Chance-constrained Quadratic Programming Model for Stream Water Quality Management

Water quality management is complicated with a variety of uncertainties and nonlinearities. This leads to difficulties in formulating and solving the resulting inexact nonlinear optimization problems. In this study, an inexact chance-constrained quadratic programming (ICCQP) model was developed for stream water quality management. A multi-segment stream water quality (MSWQ) simulation model was provided for establishing the relationship between environmental responses and pollution-control actions. The relationship was described by transformation matrices and vectors that could be used directly in a multi-point-source waste reduction (MWR) optimization model as water-quality constraints. The interval quadratic polynomials were employed to reflect the nonlinearities and uncertainties associated with wastewater treatment costs. Uncertainties associated with the water-quality parameters were projected into the transformation matrices and vectors through Monte Carlo simulation. Uncertainties derived from water quality standards were characterized as random variables with normal probability distributions. The proposed ICCQP model was applied to a water quality management problem in the Changsha section of the Xiangjiang River in China. The results demonstrated that the proposed optimization model could effectively communicate uncertainties into the optimization process, and generate inexact solutions containing a spectrum of wastewater treatment options. Decision alternatives could then be obtained by adjusting different combinations of the decision variables within their solution intervals. Solutions from the ICCQP model could be used to analyze tradeoffs between the wastewater treatment cost and system-failure risk due to inherent uncertainties. The results are valuable for supporting decision makers in seeking cost-effective water management strategies.     

Modeling of Water Resources Allocation and Water Quality Management for Supporting Regional Sustainability under Uncertainty in an Arid Region

In this study, a scenario-based interval-stochastic fraticle optimization with Laplace criterion (SISFL) method is developed for sustainable water resources allocation and water quality management (WAQM) under multiple uncertainties. SISFL can tackle uncertainties presented as interval parameters and probability distributions; meanwhile, it can also quantify artificial fuzziness such as risk-averse attitude in a decision-making issue. Besides, it can reflect random scenario occurrence under the supposition of no data available. The developed method is applied to a real case of water resources allocation and water quality management in the Kaidu-kongque River Basin, where encounter serve water deficit and water quality degradation simultaneously in Northwest China. Results of water allocation pattern, pollution mitigation scheme, and system benefit under various scenarios are analyzed. The tradeoff between economic activity and water-environment protection with interval necessity levels and Laplace criterions can support policymakers generating an effective and robust manner associated with risk control for WAQM under multiple uncertainties. These discoveries avail local policymakers gain insight into the capacity planning of water-environment to satisfy the basin’s integrity of socio-economic development and eco-environmental sustainability.     

Optimal Long-term Operation of Reservoir-river Systems under Hydrologic Uncertainties: Application of Interval Programming

Long-term basin-wide reservoir-river operation optimization problems are usually complex and nonlinear especially when the water quality issues and hydrologic uncertainties are incorporated. It is due to non-convex functions in water quality modeling and a large number of computational iterations required by most of stochastic programming methods. The computational burden of uncertainty modeling can be reduced by a special combination of uncertainty modeling and interval programming, though the problem solution is still a challenge due to model nonlinearity. In this paper, an integrated water quantity-quality model is developed for optimal water allocation at river-basin scale. It considers water supply and quality targets as well as hydrologic, water quality and water demand uncertainties within the nonlinear interval programming (NIP) framework to minimize the slacks in water supply and quality targets during a long-term planning horizon. A fast iterative linear programming (ILP) method is developed to convert the NIP into a linear interval programming (LIP). The ILP resolves two challenges in NIP, first converting the large non-linear programming (NLP) into a linear programming (LP) with minimum approximation and second reducing the iterations needed in interval programming for NLP into just two iterations for the upper and lower limits of decision variables. This modeling approach is applied to the Zayandehrood river basin in Iran that has serious water supply and pollution problems. The results show that in this river basin at dry conditions when available surface water resources are below 85 % of normal hydrologic state and water demands are 115 % of current water demands, the total dissolved solids (TDS) concentration can be reduced by 50 % at the inlet of the Gavkhuni wetland located downstream of the river basin.     

Identifying Optimal Water Resources Allocation Strategies through an Interactive Multi-Stage Stochastic Fuzzy Programming Approach

In this study, an interactive multi-stage stochastic fuzzy programming (IMSFP) approach has been developed through incorporating an interactive fuzzy resolution (IFR) method within an inexact multi-stage stochastic programming framework. IMSFP can deal with dual uncertainties expressed as fuzzy boundary intervals that exist in the objective function and the left- and right-hand sides of constraints. Moreover, IMSFP is capable of reflecting dynamics of uncertainties and the related decision processes through constructing a set of representative scenarios within a multi-stage context. A management problem in terms of water resources allocation has been studied to illustrate applicability of the proposed approach. The results indicate that a set of solutions under different feasibility degrees (i.e., risk of constraint violation) has been generated for planning the water resources allocation. They can not only help quantify the relationship between the objective-function value and the risk of violating the constraints, but also enable decision makers (DMs) to identify, in an interactive way, a desired compromise between two factors in conflict: satisfaction degree of the goal and feasibility degree of constraints. Besides, a number of decision alternatives have been generated under different policies for water resources management, which permits in-depth analyses of various policy scenarios that are associated with different levels of economic penalties when the promised water-allocation targets are violated, and thus help DMs to identify desired water-allocation schemes under uncertainty.     

An Integrated Approach of System Dynamics,Orthogonal Experimental Design and Inexact Optimization for Supporting Water Resources Management under Uncertainty

An integrated approach of system dynamics (SD), orthogonal experimental design (OED) and inexact optimization modeling was proposed for water resources management under uncertainty. The developed method adopted a combination of SD and OED to identify key scenarios within multiple factors, through which interval solutions for water demands could be obtained as input data for consequential optimization modeling. Also, optimal schemes could be obtained in the combination of inexact two-stage stochastic programming and credibility constrained programming. The developed method was applied to a real-world case study for supporting allocation of multiple-source water resources to multiple users in Dalian city within a multi-year context. The results indicated that a lower credibility-satisfaction level would generate higher allocation efficiency, a higher system benefit and a lower system violation risk. The developed model could successfully reflect and address the variety of uncertainties through provision of credibility levels, which corresponds to the decision makers’ preference regarding the tradeoffs between system benefits and violation risks.     

IFTSQP: An Inexact Optimization Model for Water Resources Management under Uncertainty

Abstract

An interval-fuzzy two-stage quadratic programming (IFTSQP) method is developed for water resources management under uncertainty. The methodincorporates techniques of interval-parameter programming, two-stage stochastic programming, and fuzzy quadratic programming within a general optimization framework to tackle multiple uncertainties presented as intervals, fuzzy sets and probability distributions. In the model formulation, multiple control variables are adopted to handle independent uncertainties in the model's right-hand sides; fuzzy quadratic terms are used in the objective function to minimize the variation in satisfaction degrees among the constraints. Moreover, the method can support the analysis of policy scenarios that are associated with economic penalties when the promised targets are violated. The developed method is then applied to a case study of water resources management planning. The results indicate that reasonable solutions have been obtained. They can help provide bases for identifying desired water-allocation plans with a maximized system benefit and a minimized constraint-violation risk.     

The Impact of Uncertainty Factors on Optimal Sizing and Costs of Low-Impact Development: a Case Study from Beijing,China

The facility allocation optimization of Low-impact development (LID) optimization has been used widely to prevent and tackle urban storm water pollution. However, uncertainties existing in nature and human society would influence the size and total cost of LID. To study the influence of the uncertainties on LID optimization allocation, the research develops the model of LID optimization allocation under uncertainty. The principle of the model is establishing primarily the LID optimization model based on certain numbers and identifying the uncertainties. Hence, the model integrates the uncertainty programming, including interval programming, fuzzy programming, stochastic programming, chance constraint programming (CCP) and scenario programming. The model of LID optimization allocation under uncertainty is established with the conditions. The developed uncertainty model tackles multiple types of uncertainties, and the results of the model are in the interval form in multiple scenarios. The model analyses the effects of uncertainties on the size and total cost of LID in this way. The study shows that the uncertainties in rainfall, infiltration rate, release coefficient, funds and unit price all have a significant influence on the size and total cost of LID when these uncertainty factors overlay. A higher violation probability of CCP corresponding to LID sizing results to a wider interval number of the corresponding uncertainty. The developed method of the study is universal, and the method could be extended to other cases of LID optimization allocation to speculate the influence of uncertainties.     

A Hybrid Dynamic Dual Interval Programming for Irrigation Water Allocation under Uncertainty

Along with the economic development in Canada, the shortage of irrigation water has become a serious concern (Bouwer 1993; Hennessy 1993). In this study, a model of Dynamic Dual Interval Programming (DDIP) is developed and applied to the irrigation water allocation systems with uncertainty. DDIP method improves the existing dynamics interval programming by explicitly addressing the system uncertainties with a dual interval that had higher system reliability. The solution of DDIP is computationally effective, and its decision variables are incorporated into the solutions for final decision. In order to obtain the optimal allocation schemes in a dynamic process, the developed DDIP was applied to an irrigation water system. The results from this case study revealed that optimal solution can be obtained through the DDIP approach from the agriculture water management activities for feasible decisions. These decisions reflect the high uncertainty of the information in the boundaries of dual intervals. The solution presents a maximum benefit under limited yearly uncertain natural resources. Furthermore, the information obtained though this model may help the authority to make optimal decisions and to reduce the risk for uncertain situations.     

Multi-Source Multi-Sector Sustainable Water Supply Under Multiple Uncertainties: An Inexact Fuzzy-Stochastic Quadratic Programming Approach

This paper presents the development and the first application of a superiority–inferiority-based inexact fuzzy-stochastic quadratic programming (SI-IFSQP) approach for sustainable water supply under multiple uncertainties. SI-IFSQP improves conventional nonlinear programming by tackling multiple uncertainties within an individual parameter; SI-IFSQP is also superior to existing inexact methods due to its reflection of economies of scale and reduction of computational requirements. An interactive solution algorithm with high computational efficiency was also proposed. The application of SI-IFSQP to long-term planning of a multi-source multi-sector water supply system demonstrated its applicability. The close reflection of system complexities, such as multiple uncertainties, scale economies and dynamic parameters, could enhance the robustness of the optimization process as well as the acceptability of obtained results. Corresponding to varied system conditions and decision priorities, the interval solutions from SI-IFSQP could help generate a series of long-term water supply strategies under a number of economic, environmental, ecological, and water-security targets.     

Basin Scale Water Resources Systems Modeling Under Cascading Uncertainties

Global change in climate and consequent large impacts on regional hydrologic systems have, in recent years, motivated significant research efforts in water resources modeling under climate change. In an integrated future hydrologic scenario, it is likely that water availability and demands will change significantly due to modifications in hydro-climatic variables such as rainfall, reservoir inflows, temperature, net radiation, wind speed and humidity. An integrated regional water resources management model should capture the likely impacts of climate change on water demands and water availability along with uncertainties associated with climate change impacts and with management goals and objectives under non-stationary conditions. Uncertainties in an integrated regional water resources management model, accumulating from various stages of decision making include climate model and scenario uncertainty in the hydro-climatic impact assessment, uncertainty due to conflicting interests of the water users and uncertainty due to inherent variability of the reservoir inflows. This paper presents an integrated regional water resources management modeling approach considering uncertainties at various stages of decision making by an integration of a hydro-climatic variable projection model, a water demand quantification model, a water quantity management model and a water quality control model. Modeling tools of canonical correlation analysis, stochastic dynamic programming and fuzzy optimization are used in an integrated framework, in the approach presented here. The proposed modeling approach is demonstrated with the case study of the Bhadra Reservoir system in Karnataka, India.     

基于鲁棒规划方法的农业水资源多目标优化配置模型

在干旱半干旱地区,调整种植结构可以促进农业水资源的高效利用。农业水资源配置需要在多个目标间权衡博弈,对各目标的偏好和赋权直接影响着优化模型的输出和决策方案的制定,但以往研究往往忽略了权重确定过程中因主观等因素的影响而普遍存在的不确定性。针对农业水资源多目标规划中存在的权重不确定性难题,建立了基于鲁棒优化方法的农业水资源多目标优化配置模型方法(MRPWU)。该方法可以把权重中蕴含的复杂不确定性信息纳入建模过程,产生可靠的模型结果;并能提供效益值及风险值均定量化的方案集,便于决策者在权衡效益与风险后确定最优方案。模型以作物种植经济收益和碳吸收量最大化为目标、以水土资源供需平衡等为约束条件,并应用于农业水资源供需矛盾突出的甘肃省民勤县。优化结果表明,随着保护度水平的提高,生态效益上升,经济效益和综合效益下降,系统面临的风险也随之下降。相比于权重为确定参数的模型,MRPWU模型可以在综合效益下降3.7%的同时,较大地提高系统应对权重不确定性以及风险的能力。与2017年的实际情况相比,MRPWU模型可以减少种植面积1.6%、节省灌溉用水3.9%,同时提高生态效益1.6%。     

A Conditional Value-at-Risk Based Inexact Water Allocation Model

A conditional value-at-risk (CVaR) based inexact two-stage stochastic programming (CITSP) model was developed in this study for supporting water resources allocation problems under uncertainty. A CITSP model was formulated through incorporating a CVaR constraint into an inexact two-stage stochastic programming (ITSP) framework, and could be used to deal with uncertainties expressed as not only probability distributions but also discrete intervals. The measure of risks about the second-stage penalty cost was incorporated into the model, such that the trade-off between system economy and extreme expected loss could be analyzed. The developed model was applied to a water resources allocation problem involving a reservoir and three competing water users. The results indicated that the CITSP model performed better than the ITSP model in its capability of reflecting the economic loss from extreme events. Also, it could generate interval solutions within which the decision alternatives could be selected from a flexible decision space. Overall, the CITSP model was useful for reflecting the decision maker’s attitude toward risk aversion and could help seek cost-effective water resources management strategies under complex uncertainties.     

Interval-parameter Two-stage Stochastic Semi-infinite Programming: Application to Water Resources Management under Uncertainty

In this study, an interval-parameter two-stage stochastic semi-infinite programming (ITSSP) method was developed for water resources management under uncertainty. As a new extension of mathematical programming methods, the developed ITSSP approach has advantages in uncertainty reflection and policy analysis. In order to better account for uncertainties, the ITSSP approach is expressed with discrete intervals, functional intervals and probability density functions. The ITSSP method integrates the two-stage stochastic programming (TSP), interval programming (IP) and semi-infinite programming (SIP) within a general optimization framework. The ITSSP has an infinite number of constraints because it uses functional intervals with time (t) being an independent variable. The different t values within the range [0, 90] lead to different constraints. At same time, ITSSP also includes probability distribution information. The ITSSP method can incorporate pre-defined water resource management policies directly into its optimization process to analyze various policy scenarios having different economic penalties when the promised amounts are not delivered. The model is applied to a water resource management system with three users and four periods (corresponding to winter, spring, summer and fall, respectively). Solutions of the ITSSP model provide desired water allocation patterns, which maximize both the system’s benefits and feasibility. The results indicate that reasonable interval solutions were generated for objective function values and decision variables, thus a number of decision alternatives can be generated under different levels of stream flow. The obtained solutions are useful for decision makers to obtain insight regarding the tradeoffs between environmental, economic and system reliability criteria.     

Water Resources Management and Planning under Uncertainty: an Inexact Multistage Joint-Probabilistic Programming Method

In this study, an inexact multistage joint-probabilistic programming (IMJP) method is developed for tackling uncertainties presented as interval values and joint probabilities. IMJP improves upon the existing multistage programming and inexact optimization approaches, which can help examine the risk of violating joint-probabilistic constraints. Moreover, it can facilitate analyses of policy scenarios that are associated with economic penalties when the promised targets are violated within a multistage context. The developed method is applied to a case study of water-resources management within a multi-stream, multi-reservoir and multi-period context, where mixed integer linear programming (MILP) technique is introduced into the IMJP framework to facilitate dynamic analysis for decisions of surplus-flow diversion. The results indicate that reasonable solutions for continuous and binary variables have been generated. They can be used to help water resources managers to identify desired system designs against water shortage and for flood control, and to determine which of these designs can most efficiently accomplish optimizing the system objective under uncertainty.     

Planning for Regional Water System Sustainability Through Water Resources Security Assessment Under Uncertainties

A leader-follower relationship in multiple layers of decision makers under uncertainties is a critical challenge associated with water resources security (WRS). To address this problem, a credibility-based chance-constrained hierarchical programming model with WRS assessment is developed for regional water system sustainability planning. This model can deal with the sequential decision-making problem with different goals and preferences, and reflect uncertainties presented as fuzzy sets. The effectiveness of the developed model is demonstrated through a real-world water resources management system in Beijing, China. A leader-follower interactive solution algorithm based on satisfactory degree is utilized to improve computational efficiency. Results show the that: (a) surface water, groundwater, recycled water, and off water would account for 27.01, 27.44, 23.11, and 22.44% of the total water supplies, respectively; (b) the entire pollutant emissions and economic benefits would consequently decrease by 31.53 and 22.88% when the statue changes from quite safe to extremely far from safe; and (c) a high credibility level would correspond to low risks of insufficient water supply and overloaded pollutant emissions, which lowers economic benefits and pollutant emissions. By contrast, a low credibility level would decrease the limitations of constraints, which leads to high economic benefits and pollutant emissions, but system risk would be increased. These findings can aid different decision makers in identifying the desired strategies for regional water resources management under multiple uncertainties, and support the in-depth analysis of the interrelationships among water security, system efficiency, and credibility level.     

An Integrated Simulation-Assessment Approach for Evaluating Health Risks of Groundwater Contamination Under Multiple Uncertainties

An integrated simulation-assessment approach (ISAA) was developed in this study to systematically tackle multiple uncertainties associated with hydrocarbon contaminant transport in subsurface and assessment of carcinogenic health risk. The fuzzy vertex analysis technique and the Latin hypercube sampling (LHS) based stochastic simulation approach were combined into a fuzzy-Latin hypercube sampling (FLHS) simulation model and was used for predicting contaminant transport in subsurface under coupled fuzzy and stochastic uncertainties. The fuzzy-rule-based risk assessment (FRRA) was used for interpreting the general risk level through fuzzy inference to deal with the possibilistic uncertainties associated with both FLHS simulations and health-risk criteria. A study case involving health risk assessment for a benzene-contaminated site was examined. The study results demonstrated the proposed ISAA was useful for evaluating risks within a system containing complicated uncertainties and interactions and providing supports for identifying cost-effective site management strategies.     

A Fuzzy-Stochastic Modeling Approach for Multiple Criteria Decision Analysis of Coupled Groundwater-Agricultural Systems

The complexity of water resources management increases when decisions about environmental and social issues are considered in addition to economic efficiency. Such complexities are further compounded by multiple uncertainties about the consequences of potential management decisions. In this paper, a new fuzzy-stochastic multiple criteria decision-making approach is proposed for water resources management in which a variety of criteria in terms of economic, environmental and social dimensions are identified and taken into account. The goal is to evaluate multiple conflicting criteria under uncertainties and to rank a set of management alternatives. The methodology uses a simulation-optimization water management model of a strongly interacting groundwater-agriculture system to enumerate criteria based on these bio-physical process interactions. Fuzzy and/or qualitative information regarding the decision-making process for which quantitative data is not available are evaluated in linguistic terms. Afterwards, Monte Carlo simulation is applied to combine these information and to generate a probabilistic decision matrix of management alternatives versus criteria in an uncertain environment. Based on this outcome, total performance values of the management alternatives are calculated using ordered weighted averaging. The proposed approach is applied to a real world example, where excessive groundwater withdrawal from the coastal aquifer for irrigated agriculture has resulted in saltwater intrusion, threatening the economical basis of farmers and associated societal impacts. The analysis has provided potential decision alternatives which can serve as a platform for negotiation and further exploration. Furthermore, sensitivity of different inputs to resulting rankings is investigated. It is found that decision makers’ risk aversion and risk taking attitude may yield different rankings. The presented approach suits to systematically quantify both probabilistic and fuzzy uncertainties associated with complex hydrosystems management.     

Waste load allocation modeling with fuzzy goals; simulation-optimization approach

Presence of various types of uncertainties in water quality management problems has been recognized as one of the major challenges in water quality modeling. Vagueness, lack of adequate data and nonlinearity of cost and/or benefit functions in most of water quality and waste load allocation management problems have reduced the capability of direct inclusion of uncertainty analysis in the management models. This study presents a fuzzy waste load allocation model in which cost function and the water quality standards or the goals of dischargers and pollution control agencies are expressed with appropriate linear and/or nonlinear and nondecreasing and/or nonincreasing membership functions. QUAL2E and Classified Population Genetic Algorithm (CPGA) were coupled to develop the optimum strategy resulting in maximum value of the minimum nonzero membership values, which represent the optimum satisfaction level of the conflicting goals. Number of constraint violations was used to penalize the fitness function in order to eliminate the infeasible solutions at the final results. The model was applied to a hypothetical case example. Results show a very suitable convergence of the proposed algorithm to good of possibility to the near global optima. Effects of linear and nonlinear membership functions are examined and the results are analyzed.     

Robust Water Supply Chain Network Design under Uncertainty in Capacity

This paper focuses on the capacity uncertainty in water supply chains that occurs when facilities face disruption. A combination of scenario-based two-stage stochastic programming with the min-max robust optimization approach is proposed to optimize the water supply chain network design problem. In the first stage, the decisions are made on locations and capacities of reservoirs and water-treatment plants while recourse decisions including amount of water extraction, amount of water refinement, and consequently amount of water held in reservoirs are made at the second stage. The proposed robust two-stage stochastic programming model can help decision makers consider the impacts of uncertainties and analyze trade-offs between system cost and stability. The literature reveals that most exact methods are not able to tackle the computational complexity of mixed integer non-linear two-stage stochastic problems at large scale. Another contribution of this study is to propose two metaheuristics - a particle swarm optimization (PSO) and a bat algorithm (BA) - to solve the proposed model in large-scale networks efficiently in a reasonable time. The developed model is applied to several hypothetical cases of water resources management systems to evaluate the effectiveness of the model formulation and solution algorithms. Sensitivity analyses are also carried out to analyze the behavior of the model and the robustness approach under parameters variations.

     

Optimizing Multiple-Pollutant Waste Load Allocation in Rivers: An Interval Parameter Game Theoretic Model

In this paper, a new methodology is developed for optimal multiple-pollutant waste load allocation (MPWLA) in rivers considering the main existing uncertainties. An interval optimization method is used to solve the MPWLA problem. Different possible scenarios for treatment of pollution loads are defined and corresponding treatment costs are taken into account in an interval parameter optimization model. A QUAL2Kw-based water quality simulation model is developed and calibrated to estimate the concentration of the water quality variables along the river. Two non-cooperative and cooperative multiple-pollutant scenario-based models are proposed for determining waste load allocation policies in rivers. Finally, a new fuzzy interval solution concept for cooperative games, namely, Fuzzy Boundary Interval Variable Least Core (FIVLC), is developed for reallocating the total fuzzy benefit obtained from discharge permit trading among waste load dischargers. The results of applying the proposed methodology to the Zarjub River in Iran illustrate its effectiveness and applicability in multiple-pollutant waste load allocation in rivers.