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.     

An Inexact Two-stage Fuzzy-stochastic Programming Model for Water Resources Management

An inexact two-stage fuzzy-stochastic programming (ITFSP) method is developed for water resources management under uncertainty. Fuzzy sets theory is introduced to represent various punishment policies under different water availability conditions. As an extension of conventional two-stage stochastic programming (TSP) method, two special characteristics of the proposed approach make it unique compared with existing approaches. One is it could handle flexible penalty rates, which are much reasonable for both of the authorities and users, and have seldom been considered in the TSP framework. The other is uncertain information expressed as discrete intervals and probability distribution functions can be effectively reflected in the optimization processes and solutions. After formulating the model, a hypothetical case is employed for demonstrating its applicability under two scenarios, where the inflow is divided into four and eight intervals, respectively. The results indicate that reasonable solutions have been obtained. They provide desired allocation patterns with maximized system benefit under two feasibility levels. The solutions present as stable intervals with different risk levels in violating the water demands, and can be used for generating decision alternatives. Comparisons of the solution from the ITFSP with that from the ITSP (inexact two-stage stochastic programming) and TSP approach are also undertaken. It shows that the ITFSP could produce more system benefit than existing methods and deal with flexible penalty policies for better water management and utilization.     

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.     

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.     

Dual-Interval Two-Stage Optimization for Flood Management and Risk Analyses

In this study, a dual interval two-stage restricted-recourse programming (DITRP) method is developed for flood-diversion planning under uncertainty. Compared with other conventional methods, DITRP improves upon them by addressing system uncertainties with complex presentations and incorporating subjective information within its optimization framework. Uncertainties in DITRP can be represented as probability distributions and intervals. In addition, the dual-interval concept is presented when the available information is highly uncertain for boundaries of intervals. Moreover, decision makers’ attitudes towards system risk can be reflected using a restricted-resource measure by controlling the variability of the recourse cost. The method has been applied to a case study of flood management. The results indicate that reasonable solutions for planning flood management practice have been generated which are related to decisions of flood-diversion. Several policy scenarios are analyzed, assisting in gaining insight into the tradeoffs between risk and cost.     

Two-Stage Interval-Parameter Stochastic Programming Model Based on Adaptive Water Resource Management

Water resource planning is often associated with system complexities and uncertainties, such as issues of precipitation randomness and complex the complexity of human social activities. In this study, a two-stage interval-parameter stochastic programming (TISP) model in conjunction with an adaptive water resource management (AWRM) model was applied. Compared to other optimization models, AWRM can address interactions between different water users and account for regional water exchange processes, and TISP models overcome the uncertainties of a water resource system by introducing interval-parameter and probability distribution methods. Reasonable solutions obtained by applying these models to a multi-water-resource, multi-region case show that in AWRM models, water can flow from a region of low efficiency to a region of high efficiency, improving water use efficiency. Under conditions of extreme scarcity, water can flow in the opposite direction thus ensuring regional minimum water requirements, enhancing system stability and reducing the probability of system paralysis. In policy making, optimistic water policies correspond to higher incomes but may be subject to higher risks of system failure. Alternatively, conservative policies are associated with a lower risk of system failure but easily waste water resources.     

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.

     

Optimization of Industrial Structure Considering the Uncertainty of Water Resources

This paper developed a stochastic linear fractional programming model for industry optimization allocation base on the uncertainty of water resources incorporating chance constrained programming and fractional programming. In this paper, the stochastic linear fractional programming is used in the real word. The development SLFP has the following advantages: (1) The model can compare the two aspects of the targets; (2) The model can reflect the system efficiency intuitively; (3) The model can deal with uncertain issues with probability distribution; (4) The model can give different optimal plans under different risk conditions. The model has a significant value for the industry optimization allocation under uncertainty in local and areas to achieve the maximum economic benefits and the full use of the water resources.     

漳卫南灌区农业水资源优化配置研究

考虑到不确定条件下漳卫南灌区农业水资源管理的复杂性,为了解决当灌区水资源用户供水目标不能满足需求时的水资源优化配置问题,结合LFP模型与TSP模型的优点,开发了一种分式两阶段随机规划模型(FTSP)。选择漳卫南灌区最大控制性工程岳城水库的两个大型供水灌区作为验证实例,模型应用结果表明,不同决策情景所对应的经济效益和缺水风险不同,最优决策实现了经济效益和缺水风险之间的平衡;不同径流水平下,各用户的正常灌溉面积会发生相应变化,高径流水平时所有用户均能得到正常灌溉。     

An Inexact Two-Stage Water Resources Allocation Model for Sustainable Development and Management Under Uncertainty

In this study, an inexact two-stage water resources allocation (ITWR) model is put forward for supporting sustainable development and management of water resources in Sanjiang Plain, China, which is in such a situation, with multi-water source, multi-water supply subarea, multi-water user and multi-planning goal. The costs of net system, water supply and recourse are analyzed. The developed ITWR model, which shows a strong ability in tacking with various uncertain factors in probability distributions and discrete interval numbers, mixes the techniques of interval-parameter programming (IPP) and two-stage stochastic programming (TSP) within a general optimization framework. And it also has formed an effective link in such a conflict between the policy scenarios and the associated various levels of economic penalties, when the pre-allocation targets of water resources are violated. Based on this model, a series of scenarios under different levels of pre-allocation water is done and different degrees of water surplus and shortage are obtained correspondingly. The results indicate that the reasonable distribution plans with maximum system benefit and minimum system-failure risk have been generated. And these results are valuable for saving water resources to realize its sustainable development and mitigating the penalty to gain economic benefits maximum, and thus some desired results are provided for decision makers in tackling with a complex and uncertain water-resource system.     

Assessment of Stochastic Operation Optimization for Reservoirs of Contrasting Scales

Deriving optimal release policies for dams and corresponding reservoirs is crucial for the sustainable water resources management of a region as they directly control the distribution of water to several users. Mathematical optimization algorithms can help in finding efficient reservoir operating strategies taking into account complex system constraints and hydrologic uncertainty. The robustness of operation optimization models may be influenced by physical reservoir characteristics such as size and scale and the effectiveness of a model for a particular case study does not always guarantee the same level of success for another application. This research focused on assessing the applicability of an implicit stochastic optimization (ISO) procedure to derive rule curves for two different dams of contrasting reservoir scales in terms of physical and operational characteristics. The results demonstrated the feasibility of the proposed technique for both small- and large-scale systems in view of the lower vulnerability provided by the ISO-derived policies in contrast to operations carried out by the standard reservoir operating policy as well as the proximity of the ISO operations with those by perfect-forecast deterministic optimization. The ISO procedure also provided operating rules similar to, and even less vulnerable than, those derived by stochastic dynamic programming.     

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.     

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.     

Interval-parameter Two-stage Stochastic Nonlinear Programming for Water Resources Management under Uncertainty

Over the past decades, controversial and conflict-laden water allocation issues among competing interests have raised increasing concerns. In this research, an interval-parameter two-stage stochastic nonlinear programming (ITNP) method is developed for supporting decisions of water-resources allocation within a multi-reservoir system. The ITNP can handle uncertainties expressed as both probability distributions and discrete intervals. It can also be used for analyzing various policy scenarios that are associated with different levels of economic consequences when the promised allocation targets are violated. Moreover, it can deal with nonlinearities in the objective function such that the economies-of-scale effects in the stochastic program can be quantified. The proposed method is applied to a case study of water-resources allocation within a multi-user, multi-region and multi-reservoir context for demonstrating its applicability. The results indicate that reasonable solutions have been generated, which present as combined interval and distributional information. They provide desired water allocation plans with a maximized economic benefit and a minimized system-disruption risk. The results also demonstrate that a proper policy for water allocation can help not only mitigate the penalty due to insufficient supply but also reduce the waste of water resources.     

Long-Term Planning of Water Systems in the Context of Climate Non-Stationarity with Deterministic and Stochastic Optimization

Seasonal inflow variability, climate non-stationarity and climate change are matters of concern for water system planning and management. This study presents optimization methods for long-term planning of water systems in the context of a non-stationary climate with two levels of inflow variability: seasonal and inter-annual. Deterministic and stochastic optimization models with either one time-step (intra-annual) or two time-steps (intra-annual and inter-annual) were compared by using three water system optimization models. The first model used one time-step sampling stochastic dynamic programming (SSDP). The other models with two time-steps are long-term deterministic dynamic programming (LT-DDP) and long-term sampling stochastic dynamic programming (LT-SSDP). The study area is the Manicouagan water system located in Quebec, Canada. The results show that there will be an increase of inflow to hydropower plants in the future climate with an increase of inflow uncertainty. The stochastic optimization with two time-steps was the most suitable for handling climate non-stationarity. The LT-DDP performed better in terms of reservoir storage, release and system efficiency but with high uncertainty. The SSDP had the lowest performance. The SSDP was not able to deal with the non-stationary climate and seasonal variability at the same time. The LT-SSDP generated operating policies with smaller uncertainty compared to LT-DDP, and it was therefore a more appropriate approach for water system planning and management in a non-stationary climate characterized by high inflow variability.     

Planning water resources systems with interval stochastic dynamic programming

This study addresses water resources system planning problems with capacity expansion in an uncertain environment. An interval stochastic dynamic programming (SDP) model is presented, which is a hybrid of interval-number optimization and SDP. Besides the dynamic features of the model, it can incorporate and reflect uncertainties expressed as probability distribution functions and discrete intervals. The solution method for the proposed model is computationally effective, which makes it applicable to practical problems. The results acquired through a case study indicate that reasonable solutions have been obtained. They are further analyzed and interpreted for identifying significant factors that affect the system's performance. The information obtained through these post-optimality analyses can provide useful decision support for water authorities.     

随机约束线性规划在水库综合利用中的应用

由于水库入流的不确定性，各用水目标的基本要求（目标放水量）将体现在年内各时期水库放水的随机约束上，配合水库线民生蓄泄水决策规则，将全部随机约束进行确定性等效转换，得到线性规划模型，经多次解析，就可得到水主加容量一定情况下的最优运行规则，针对大型水资源工程综合利用的多目标要求，研究建立了随机约束线性规划模型，以求解水库最优运行规划的方法，并以西南地区某大型综合利用水库为例，对模型进行求解，该方法随机     

A Bayesian Stochastic Optimization Model for a Multi-Reservoir Hydropower System

This paper presents the development of an operating policy model for a multi-reservoir system for hydropower generation by addressing forecast uncertainty along with inflow uncertainty. The stochastic optimization tool adopted is the Bayesian Stochastic Dynamic Programming (BSDP), which incorporates a Bayesian approach within the classical Stochastic Dynamic Programming (SDP) formulation. The BSDP model developed in this study considers, the storages of individual reservoirs at the beginning of period t, aggregate inflow to the system during period t and forecast for aggregate inflow to the system for the next time period t + 1, as state variables. The randomness of the inflow is addressed through a posterior flow transition probability, and the uncertainty in flow forecasts is addressed through both the posterior flow transition probability and the predictive probability of forecasts. The system performance measure used in the BSDP model is the square of the deviation of the total power generated from the total firm power committed and the objective function is to minimize the expected value of the system performance measure. The model application is demonstrated through a case study of the Kalinadi Hydroelectric Project (KHEP) Stage I, in Karnataka state, India.     

IB-ICCMSP: An Integrated Irrigation Water Optimal Allocation and Planning Model Based on Inventory Theory under Uncertainty

In this study, an inventory-theory-based inexact chance-constrained multi-stage stochastic programming (IB-ICCMSP) model under multi-uncertainties is developed. IB-ICCMSP integrates inventory theory into an inexact chance-constrained multi-stage stochastic optimization framework. This method can not only effectively address system multiple uncertainties (e.g. discrete intervals and probability density functions) and dynamic features, but also provide water transferring and allocating schemes among multiple stages. The developed model is applied to irrigation water allocation optimization system in Zhangye City, Gansu province, China. Based on the runoff simulation prediction of Yingluo Gorge and water supply–demand balance analysis of the 12 irrigation areas in Zhangye City, different optimal irrigation water measures are generated under different flow levels and different probabilities in the planning year. The obtained results are valuable for supporting the adjustment of the existing irrigation patterns and identifying desired water-allocation plans for irrigation under multi-uncertainties.     

A Decentralized Bi-Level Fuzzy Two-Stage Decision Model for Flood Management

Flood, as a serious worldwide environment problem, can lead to detrimental economic losses and fatalities. Effective flood control is desired to mitigate the adverse impacts of flooding and the associated flood risk through development of cost-effective and efficient flood management decisions and policies. A bi-level fuzzy two-stage stochastic programming model, named BIFS model is developed in this study to provide decision support for economic analysis of flood management. The BIFS model is capable of not only addressing the sequential decision making issue involving the two-level decision makers, but also correcting the pre-regulated flood management decisions before the occurrence of a flood event in the two-stage environment. The probabilistic and non-probabilistic uncertainties expressed as probability density functions and fuzzy sets are quantitatively analyzed. The overall satisfaction solution is obtained for meeting the goals of the two-level decision makers by compromising, reflecting the tradeoffs among various decision makers in the two decision-making levels. The results of application of the BIFS model to a representative case study indicate informed decision strategies for flood management. Tradeoffs between economic objectives and uncertainty-averse attitudes of decision makers are quantified.