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.     

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.     

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.     

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.     

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.     

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.     

Multistage scenario-based interval-stochastic programming for planning water resources allocation

In this study, a multistage scenario-based interval-stochastic programming (MSISP) method is developed for water-resources allocation under uncertainty. MSISP improves upon the existing multistage optimization methods with advantages in uncertainty reflection, dynamics facilitation, and risk analysis. It can directly handle uncertainties presented as both interval numbers and probability distributions, and can support the assessment of the reliability of satisfying (or the risk of violating) system constraints within a multistage context. It can also reflect the dynamics of system uncertainties and decision processes under a representative set of scenarios. The developed MSISP method is then applied to a case of water resources management planning within a multi-reservoir system associated with joint probabilities. A range of violation levels for capacity and environment constraints are analyzed under uncertainty. Solutions associated different risk levels of constraint violation have been obtained. They can be used for generating decision alternatives and thus help water managers to identify desired policies under various economic, environmental and system-reliability conditions. Besides, sensitivity analyses demonstrate that the violation of the environmental constraint has a significant effect on the system benefit.     

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.     

Planning water resources management systems using a fuzzy-boundary interval-stochastic programming method

In this study, a fuzzy-boundary interval-stochastic programming (FBISP) method is developed for planning water resources management systems under uncertainty. The developed FBISP method can deal with uncertainties expressed as probability distributions and fuzzy-boundary intervals. With the aid of an interactive algorithm woven with a vertex analysis, solutions for FBISP model under associated α-cut levels can be generated by solving a set of deterministic submodels. The related probability and possibility information can also be reflected in the solutions for the objective function value and decision variables. The developed FBISP is also applied to water resources management and planning within a multi-reservoir system. Various policy scenarios that are associated with different levels of economic consequences when the pre-regulated water-allocation targets are violated are analyzed. The results obtained are useful for generating a range of decision alternatives under various system conditions, and thus helping decision makers to identify desired water resources management policies under uncertainty.     

Total consumption controlled water allocation management for multiple sources and users with inexact fuzzy chance-constrained programming: a case study of Tianjin,China

Environmental and ecological issues caused by water resources crisis have brought enormous challenges to the sustainable development of water-deficient area. Water resources allocation management balancing the relationship between the social-economic development and the ecological environment has become a hot topic in recent years. In this paper, an inexact fuzzy chance-constrained programming (IFCCP) approach is proposed for regional water resource allocation optimization with the aim of promoting the harmonious development of the social economic and the ecological environment, improving water utilization efficiency, and realizing water resources consumption control under uncertainties. The method is incorporated with interval parameter programming, fuzzy programming, and chance-constrained programming, for handling system uncertainties and balancing the optimal objectives with the risk of violating system constraints. Under this framework, an IFCCP model for water resources allocation management was successfully formulated and applied to a typical water-deficit area, Tianjin, China, for obtaining a better water resources plan among multiple users under resources and environmental limitation. Different total water consumption control policies are designed for assessing regional water allocation schemes. The results indicated that the gap of supply and demand will only be solved by foreign water, the transferred water from Luan River and Changjiang River would still be the main supplier in planning horizon. Moreover, the strict total water consumption control policy would guarantee the water requirement of ecological environment, lead to changes in the structure of water supply, actively guide on water conservation, and promote the large-scale utilization of desalted water and recycle water.     

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 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.     

Dynamic attribution of global water demand to surface water and groundwater resources: Effects of abstractions and return flows on river discharges

As human water demand is increasing worldwide, pressure on available water resources grows and their sustainable exploitation is at risk. To mimic changes in exploitation intensity and the connecting feedbacks between surface water and groundwater systems, a dynamic attribution of demand to water resources is necessary. However, current global-scale hydrological models lack the ability to do so. This study explores the dynamic attribution of water demand to simulated water availability. It accounts for essential feedbacks, such as return flows of unconsumed water and riverbed infiltration. Results show that abstractions and feedbacks strongly affect water allocation over time, particularly in irrigated areas. Also residence time of water is affected, as shown by changes in low flow magnitude, frequency, and timing. The dynamic representation of abstractions and feedbacks makes the model a suitable tool for assessing spatial and temporal impacts of changing global water demand on hydrology and water resources.     

Optimal allocation of water and land resources for maximizing the farm income and minimizing the irrigation-induced environmental problems

The continuous decrease in good quality water and land resources and concurrent increase in global population accentuates the need of optimal allocation of these resources to fulfilling the rising food requirements. This study presents the formulation and application a management model for the optimal allocation of available good quality water and land resources to maximize the farm revenue of a canal command area. A groundwater balance constraint was imposed on the model, which moderates the irrigation-induced environmental problems of waterlogging and salinization, while making the optimal allocation of resources. The model results show a reduction in mustard, rice, and gram crop areas against an increase in sorghum, millets, and wheat areas. The net annual revenue from the command area increased by about 18 % under the optimal allocation plans. The farmers and stakeholders concerned in the actual agricultural production process are suggested to use groundwater and canal water conjunctively to maximizing the farm income. This strategy would also mitigate the hydrological imbalances to the groundwater system without installing costly drainage systems which is not viable as the quality of groundwater is poor and drainage water may cause a serious disposal problem. The developed model can be used as a reliable decision tool for taking the farm and regional level decisions of optimal land and water resources allocation and is able to solve the irrigation-induced environmental problems of agricultural systems.     

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.     

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.     

Surface water deficiency zoning of China based on surface water deficit index (SWDI)

China’s economic and social development is severely restricted by regional water deficiency, and the determination of water deficiency level in different regions is an important basis of water resource management. In this study, we zoned 32 provinces of China into five surface water deficiency levels based on surface water deficit index derived from the Chinese meteorological satellite FY-2C images. The result shows that China is facing serious water shortage crisis with only 15.44% of total area without water deficiency, which is located in the south and southeast coast. More than 60% of total area of China suffered from severe or extremely water deficiency, which are mainly distributed in western and northern regions. Jingjinji area (namely Beijing, Tianjin and Hebei province), which is a special region with high amount of gross domestic product and relative abundant water resources, is suffering from severe water deficiency. It revealed that water resources were interacted with regional development and water management should consider local conditions. To Jingjinji area, water management should focus on adjustment of industrial structure, urban expansion control and scientific water resources allocation. In general, the surface water deficit index based water deficiency zoning will be of significant value to be applied in water resources management at regional scales.     

基于熵值环境基尼系数法的平原河网区污染物总量分配

为制定科学合理的水污染物总量控制方案,以典型平原河网地区——张家港市为例,构建与人口、资源、经济和水污染物承受能力相协调的单因子环境基尼系数模型,并通过贡献系数判断引起不公平的主要污染物分配单元.利用熵值—环境基尼系数最小化模型,确定张家港市各乡镇基于多元公平性原则的氨氮总量分配方案.结果表明:在水污染物总量分配过程中,污染物削减比例的大小和污染物现状排放量之间并不具有一致性,最终的分配方案要综合考虑多方因素确定.本研究构建的熵值—环境基尼系数最小化模型综合考虑了研究区域社会、经济、资源等多种客观因素,对于平原河网地区,由于其特殊的地理位置和自然条件,人口、资源等分布相对比较均匀,故基于该模型所得到的分配方案充分表现了多元公平性原则,更公平合理.     

Balancing cost-risk in management optimization of water resource systems under uncertainty

When a scarce water resource is distributed between different users by a Water Resource Management Authority (WRMA), the replenishment of this resource as well as the meeting of users’ demand is subject to considerable uncertainty. Cost optimization and risk management models can assist the WRMA in its decision about striking the balance between the level of target delivery to the users and the level of risk that this delivery will not be met. Addressing the problem as a multi-period dynamic network optimization, the proposed approach is also based on further developments in stochastic programming for scenario optimization. This approach tries to obtain a “robust” decision policy that minimizes the risk of wrong decisions when managing scarce water resources. In the paper we also illustrate two application examples for water resources management problems.     

Application of the Ordered Weighted Averaging (OWA) method to the Caspian Sea conflict

This study proposes a promising allocation mechanism of the Caspian Sea natural resources, which are presently shared among five countries. To date, these nations have been unable to reach an allocation agreement. We apply a methodology to propose the most appropriate solution under different risk attitudes of the states. This research is different from other studies regarding the Caspian Sea negotiations in that it employs risk-based fuzzy multi attribute decision making methods for simulating the risk attitudes or optimism/pessimism degrees of the decision makers. The ordered weighted averaging (OWA) approach, which considers the optimism/pessimism degree quantitatively, is used to take into account the effects of different risk attitudes of the negotiators on the final outcome. We demonstrate how one could obtain a range of alternatives under different multi attribute and risk attitudes. The induced OWA (IOWA) method is also used to determine the relative power of these states bordering the Caspian Sea by considering several attributes, including different risk attitudes of agents. Results indicate that taking into account the risk attitude (prone, neutral, averse) of the states can affect the overall ranking of the proposed solutions. The findings from this study may facilitate negotiation regarding the most preferred allocation mechanism for the Caspian Sea.