Fuzzy chance-constrained linear fractional programming approach for optimal water allocation

A fuzzy chance-constrained linear fractional programming method was developed for agricultural water resources management under multiple uncertainties. This approach improved upon the previous programming methods, and could reflect the ratio objective function and multiple uncertainties expressed as probability distributions, fuzzy sets, and their combinations. The proposed approach is applied to an agricultural water resources management system where many crops are considered under different precipitation years. Through the scenarios analyses, the multiple alternatives are presented. The solutions show that it is applicable to practical problems to address the crop water allocation under the precipitation variation and sustainable development with ratio objective function of the benefit and the irrigation amount. It also provides bases for identifying desired agriculture water resources management plans with reasonable benefit and irrigation schedules under crops.     

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.     

Water quality management in a wetland system using an inexact left-hand-side chance-constrained fuzzy multi-objective approach

Water quality management is a significant item in the sustainable development of wetland system, since the environmental influences from the economic development are becoming more and more obvious. In this study, an inexact left-hand-side chance-constrained fuzzy multi-objective programming (ILCFMOP) approach was proposed and applied to water quality management in a wetland system to analyze the tradeoffs among multiple objectives of total net benefit, water quality, water resource utilization and water treatment cost. The ILCFMOP integrates interval programming, left-hand-side chance-constrained programming, and fuzzy multi-objective programming within an optimization framework. It can both handle multiple objectives and quantify multiple uncertainties, including fuzziness (aspiration level of objectives), randomness (pollutant release limitation), and interval parameters (e.g. water resources, and wastewater treatment costs). A representative water pollution control case study in a wetland system is employed for demonstration. The optimal schemes were analyzed under scenarios at different probabilities (p _i , denotes the admissible probability of violating the constraint i). The optimal solutions indicated that, most of the objectives would decrease with increasing probability levels from scenarios 1 to 3, since a higher constraint satisfaction probability would lead to stricter decision scopes. This study is the first application of the ILCFMOP model to water quality management in a wetland system, which indicates that it is applicable to other environmental problems under uncertainties.     

Managing water resources system in a mixed inexact environment using superiority and inferiority measures

A superiority–inferiority-based fuzzy-stochastic integer programming (SI-FSIP) method is developed for water resources management under uncertainty. In the SI-FSIP method, techniques of fuzzy mathematical programming with the superiority and inferiority measures and joint chance-constrained programming are integrated into an inexact mixed integer linear programming framework. The SI-FSIP improves upon conventional inexact fuzzy programming by directly reflecting the relationships among fuzzy coefficients in both the objective function and constraints with a high computational efficiency, and by comprehensively examining the risk of violating joint probabilistic constraints. The developed method is applied to a case study of water resources planning and flood control within a multi-stream and multi-reservoir context, where several studied cases (including policy scenarios) associated with different joint and individual probabilities are investigated. Reasonable solutions including binary and continuous decision variables are generated for identifying optimal strategies for water allocation, flood diversion and capacity expansion; the tradeoffs between total benefit and system-disruption risk are also analyzed. As the first attempt for planning such a water-resources system through the SI-FSIP method, it has potential to be applied to many other environmental management problems.     

Optimization of the industrial structure facing sustainable development in resource-based city subjected to water resources under uncertainty

An inexact stochastic fuzzy programming (ISFP) approach has been developed for the optimization of the industrial structure in resource-based city subjected to water resources under uncertainty in present study. The ISFP method incorporates the techniques of inexact stochastic programming and inexact fuzzy chance-constrained programming, where the uncertainties are expressed as interval, fuzzy sets, and probability distribution, respectively. Moreover, it can also examine the risk of violating fuzzy tolerance constraints. The developed method is subsequently employed in a realistic case for industrial development in the Jinchang city, Gansu province, China. The result can help to analyze whether the water resources carrying capacity of Jinchang can meet the need of local economic development plan under uncertainty and help decision maker to optimize the industry structure under water resource constraints to meet the maximum economic efficiency.     

Risk analysis for water resources management under dual uncertainties through factorial analysis and fuzzy random value-at-risk

Water resources systems are associated with a variety of complexities and uncertainties due to socio-economic and hydro-environmental impacts. Such complexities and uncertainties lead to challenges in evaluating the water resources management alternatives and the associated risks. In this study, the factorial analysis and fuzzy random value-at-risk are incorporated into a two-stage stochastic programming framework, leading to a factorial-based two-stage programming with fuzzy random value-at-risk (FTSPF). The proposed FTSPF approach aims to reveal the impacts of uncertainty parameters on water resources management strategies and the corresponding risks. In detail, fuzzy random value-at-risk is to reflect the potential risk about financial cost under dual uncertainties, while a multi-level factorial design approach is used to reveal the interaction between feasibility degrees and risk levels, as well as the relationships (including curvilinear relationship) between these factors and the responses. The application of water resources system planning makes it possible to balance the satisfaction of system benefit, the risk levels of penalty and the feasibility degrees of constraints. The results indicate that decision makers would pay more attention to the tradeoffs between the system benefit and feasibility degree, and the water allocation for agricultural section contributes most to control the financial loss of water. Moreover, FTSPF can generate a higher system benefit and more alternatives under various risk levels. Therefore, FTSPF could provide more useful information for enabling water managers to identify desired policies with maximized system benefit under different system-feasibility degrees and risk levels.     

Development of a fuzzy-queue-based stochastic quadratic program for water resources planning

In this study, a fuzzy-queue (FQ)-based inexact stochastic quadratic programming (SQP) method is developed through coupling FQ technique with inexact SQP. FQ-SQP improves upon the existing stochastic programming methods by considering the effects of queuing phenomenon during the water resources allocation process. FQ-SQP cannot only handle uncertainties expressed as interval values, random variables, and fuzzy sets, but also tackle nonlinearity in the objective function; more importantly, it can reflect the effects of FQ on water resources allocation and system benefit. The FQ-SQP model is applied to a case study of planning water resources management, where FM/FM/1 (fuzzy exponential interarrival time, fuzzy exponential service time, and one server) queue is incorporated within the SQP modeling framework. Based on α-cut analysis technique, interval solutions with fuzzy arrival and service rates have been generated, which result in different water resources allocation patterns as well as changed waiting water amounts and system benefits. Results indicate that consideration of queuing problem impacts on water resources allocation can provide more useful information for decision makers and gain in-depth insights into the effects of queuing problems for water resources allocation.     

An inexact fuzzy-random-chance-constrained air quality management model

An inexact fuzzy-random-chance-constrained programming model (IFRCCMM) was developed for supporting regional air quality management under uncertainty. IFRCCMM was formulated through integrating interval linear programming within fuzzy-random-chance-constrained programming framework. It could deal with parameter uncertainties expressed as not only fuzzy random variables but also discrete intervals. Based on the stochastic and fuzzy chance-constrained programming algorithms, IFRCCMM was solved when constraints was satisfied under different satisfaction and violation levels of constraints, leading to interval solutions with different risk and cost implications. The proposed model was applied to a regional air quality management problem for demonstration. The obtained results indicated that the proposed model could effectively reflect uncertain components within air quality management system through employing multiple uncertainty-characterization techniques (in random, fuzzy and interval forms), and help decision makers analyze trade-offs between system economy and reliability. In fact, many types of solutions (i.e. conservative solutions with lower risks and optimistic solutions with higher risks) provided by IFRCCMM were suitable for local decision makers to make more applicable decision schemes according to their understanding and preference about the risk and economy. In addition, the modeling philosophy is general and applicable to many other environmental problems that may be complicated with multiple forms of uncertainties.     

Inexact quadratic joint-probabilistic programming for water quality management under uncertainty in the Xiangxi River, China

An inexact quadratic joint-probabilistic programming model for water quality management (IQJWQ) is developed and applied to supporting multiple-point-source waste reduction in the Xiangxi River, China. The IQJWQ is a hybrid of interval quadratic programming, joint probabilistic programming and multi-segment water quality simulation. It has advantages in reflecting uncertainties expressed as joint probabilities of system risk, probability distributions of water quality standards, interval parameters and nonlinearities in the objective function. An interactive and derivative algorithm is employed for solving the IQJWQ model. The results indicate that the Pingyikou chemical plant and Liucaopo chemical plant contribute more to pollution of the main stream in the Xiangxi River, which should be the prior plants to reduce the wastewater discharge and enhance the wastewater treatment efficiencies. Meanwhile, the environmental agencies should choose the joint probability carefully to balance the tradeoff between production development and pollution control. Compared with the conventional chance-constrained programming method, the IQJWQ exhibits an increased robustness in handling the overall system risk in the optimization process. Although this study is the first application of the IQJWQ to water quality management, the proposed methods in the IQJWQ can also be applicable to many other environmental management problems under uncertainty.     

Agricultural effluent control under uncertainty: An inexact double-sided fuzzy chance-constrained model

An inexact double-sided fuzzy chance-constrained programming (IDFCCP) method was developed in this study and applied to an agricultural effluent control management problem. IDFCCP was formulated through incorporating interval linear programming (ILP) into a double-sided fuzzy chance-constrained programming (DFCCP) framework, and could be used to deal with uncertainties expressed as not only possibility distributions associated with both left- and right-hand-side components of constraints but also discrete intervals in the objective function. The study results indicated that IDFCCP allowed violation of system constraints at specified confidence levels, where each confidence level consisted of two reliability scenarios. This could lead to model solutions with high system benefits under acceptable risk magnitudes. Furthermore, the introduction of ILP allowed uncertain information presented as discrete intervals to be communicated into the optimization process, such that a variety of decision alternatives can be generated by adjusting the decision-variable values within their intervals. The proposed model could help decision makers establish various production patterns with cost-effective water quality management schemes under complex uncertainties, and gain in-depth insights into the trade-offs between system economy and reliability.     

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.     

Analyzing urban water supply through an acceptability-index-based interval approach

Incorporation of uncertainties within an urban water supply management system has been a challenging topic for many years. In this study, an acceptability-index-based two-step interval programming (AITIP) model was developed for supporting urban water supply analysis under uncertainty. AITIP improved upon the traditional two-step interval programming (TIP) through incorporating the acceptability level of constraints violation into the optimization framework. A four-layer urban water supply system, including water sources, treatment facilities, reservoirs, and consuming zones, was used to demonstrate the applicability of proposed method. The results indicated that an AITIP model was valuable to help understand the effects of uncertainties related to cost, constraints and decision maker’s judgment in the water supply network, and capable of assisting urban water managers gain an in-depth insight into the tradeoffs between system cost and constraints-violation risk. Compared with TIP, the solutions from AITIP were of lower degree of uncertainty, making it more reliable to identify effective water supply patterns by adjusting decision variable values within their solution intervals. The study is useful in helping urban water managers to identify cost-effective management schemes in light of uncertainties in hydrology, environment, and decisions. The proposed optimization approach is expected to be applicable for a wide variety of water resources management problems.     

Development of an inexact two-stage stochastic model with downside risk control for water quality management and decision analysis under uncertainty

Water quality management along rivers involves making water-allocation plans, establishing water quality goals, and controlling pollutant discharges, which is complicated itself but further challenged by existence of uncertainties. In this study, an inexact two-stage stochastic downside risk-aversion programming (ITSDP) model is developed for supporting regional water resources allocation and water quality management problems under uncertainties. The ITSDP method is a hybrid of interval-parameter programming, two-stage stochastic programming, and downside risk measure to tackle uncertainties described in terms of interval values and probability distributions. A water quality simulation model was provided for reflecting the relationship between the water resources allocation, wastewater discharge, and environmental responses. The proposed approach was applied to a hypothetical case for a shared stream water quality management with one municipal, three industrial and two agricultural sectors. A number of scenarios corresponding to different river inflows and risk levels were examined. The results demonstrated that the model could effectively communicate the interval-format and random uncertainties, and risk-aversion into optimization process, and generate a trade-off between the system economy and stability. They could be helpful for seeking cost-effective management strategies under uncertainties, and gaining an in-depth insight into the water quality management system characteristics, and make cost-effective decisions.     

流域管理若干问题的研究

流域管理是以水资源的自然流域特性和多功能属性为基础的管理制度,它的目标是使有限的水资源实现优化配置和发挥最大效益.流域管理的问题直接关系到以水资源的可持续利用支持经济社会的可持续发展的大局.通过分析我国流域管理存在的问题,借鉴国外成功的流域管理经验,提出要进一步明确流域管理和行政区域管理的事权,加快流域管理相应的法律法规建设,加大流域管理的支撑保障能力建设,充分发挥流域管理机构科学规划决策、有效配置调控和有力监督控制的作用,进一步探索适合不同流域的管理模式,加大构建公共参与和民主协商机制力度,探索建立流域水资源管理可持续利用的市场机制.     

Dual inexact fuzzy chance-constrained programming for planning waste management systems

This study develops a dual inexact fuzzy chance-constrained programming (DIFCCP) method for planning municipal solid waste (MSW) management systems. The concept of random boundary interval (RBI) is introduced to address the high uncertain parameters in the studied system. Fuzzy flexible programming and chance-constrained programming are also introduced to take into account the uncertainties of RBIs and various uncertainties in MSW management system. Compared with the existing methods, the developed method could deal with the uncertainty without simplification and thus is more robust. Moreover, the potential system-failure risks in MSW management system due to the existing uncertainties could be quantified by means of violation levels and satisfaction levels in DIFCCP. The developed method then is applied to a MSW management system. The obtained solutions could be used for generating efficient management schemes. The values of violation and satisfaction levels could help decision makers understand the tradeoffs between system cost and system-failure risk, and identify desired strategy according to the practical economic and environmental situation.     

The Effect of Groundwater Allocation on Economic Welfare Loss

Water scarcity has become a constraint for regional economic development in many cities and regions. Water rationing serves as one instrument to constrain water consumption to persuade users to save water and to moderate their consumption. When the supply of water is unable to satisfy demand, a loss of welfare for the water users will usually occur. This paper conducts an empirical case study on a Chicago suburban county, McHenry County, to evaluate effective water allocation strategies under possible water scarcity scenarios, by specifically taking into consideration of the economic welfare loss under water rationing. It points out the inefficiency of equal rationing and tests a more effective optimal rationing regime which could significantly lower the overall welfare loss for McHenry County. Instead of a conventional watershed‐based approach that would provide little advantage for an area that mostly relies on groundwater, this study adopts regional planning/political boundaries as its spatial analytical units. The outcomes suggest that municipality‐level water resources management models, powered under economic welfare objective functions, are both possible and practical. The planning strategy drawn under such optimization models suggests a variety of promising approaches to manage groundwater resources at county scales.     

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.     

An inexact inventory-theory-based chance-constrained programming model for solid waste management

In this study, an inexact inventory-theory-based chance-constrained programming (IICP) model is proposed for planning waste management systems. The IICP model is derived through introducing inventory theory model into a general inexact chance-constrained programming framework. It can not only tackle uncertainties presented as both probability distributions and discrete intervals, but also reflect the influence of inventory problem in decision-making problems. The developed method is applied to a case study of long-term municipal solid waste (MSW) management planning. Solutions of total waste allocation, waste allocation batch and waste transferring period associated different risk levels of constraint violation are obtained. The results can be used to identify inventory-based MSW management planning with minimum system cost under various constraint-violation risks. Compared with the ICP model, the developed IICP model can more actually reflect the complexity of MSW management systems and provide more useful information for decision makers.     

Inexact stochastic dynamic programming method and application to water resources management in Shandong China under uncertainty

In this study, an interval parameter multistage joint-probability programming (IMJP) approach has been developed to deal with water resources allocation under uncertainty. The IMJP can be used not only to deal with uncertainties in terms of joint-probability and intervals, but also to examine the risk of violating joint probabilistic constraints in the context of multistage. The proposed model can handle the economic expenditure caused by regional water shortage and flood control. The model can also reflect the related dynamic changes in the multi-stage cases and the system safety under uncertainty. The developed method is applied to a case study of water resources allocation in Shandong, China, under multistage, multi-reservoir and multi-industry. The violating reservoir constraints are addressed in terms of joint-probability. Different risk levels of constraint lead to different planning. The obtained results can help water resources managers to identify desired system designs under various economic, environment and system reliability scenarios.     

Coupled surface water–groundwater model and its application in the arid Shiyang River basin,China

The planning and management of water resources in the Shiyang River basin, China require a tool for assessing the impact of groundwater and stream use on water supply reliabilities and improving many environment‐related problems such as soil desertification induced by recent water‐related human activities. A coupled model, integrating rule‐based lumped surface water model and distributed three‐dimensional groundwater flow model, has been established to investigate surface water and groundwater management scenarios that may be designed to restore the deteriorated ecological environment of the downstream portion of the Shiyang River basin. More than 66% of the water level among 24 observation wells have simulation error less than 1·0 m. The overall trend of the temporal changes of simulated and observed surface runoff at the Caiqi gauging station remains almost the same. The calibration was considered satisfactory. Initial frameworks for water allocation, including agricultural water‐saving projects, water diversion within the basin and inter‐basin water transfer, reducing agricultural irrigation area and surface water use instead of groundwater exploitation at the downstream were figured out that would provide a rational use of water resources throughout the whole basin. Sixteen scenarios were modelled to find out the most appropriate management strategies. The results showed that in the two selected management options, the groundwater budget at the Minqin basin was about 1·4 × 10⁸ m³/a and the ecological environment would be improved significantly, but the deficit existed at the Wuwei basin and the number was about 0·8 × 10⁸ m³/a. Water demand for domestic, industry and urban green area would be met in the next 30 years, but the water shortage for meeting the demand of agricultural water use in the Shiyang River basin was about 2·2 × 10⁸ m³/a. It is suggested that more inter‐basin water transfer should be required to obtain sustainable water resource use in the Shiyang River basin. Copyright © 2009 John Wiley & Sons, Ltd.