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.     

An interval-parameter mean-CVaR two-stage stochastic programming approach for waste management under uncertainty

In this research, approaches of interval mathematical programming, two-stage stochastic programming and conditional value-at-risk (CVaR) are incorporated within a general modeling framework, leading to an interval-parameter mean-CVaR two-stage stochastic programming (IMTSP). The developed method has several advantages: (i) it can be used to deal with uncertainties presented as interval numbers and probability distributions, (ii) its objective function simultaneously takes expected cost and system risk into consideration, thus, it is useful for helping decision makers analyze the trade-offs between cost and risk, and (iii) it can be used for supporting quantitatively evaluating the right tail of distributions of waste generation rate, which can better quantify the system risk. The IMTSP model is applied to the long-term planning of municipal solid waste management system in the City of Regina, Canada. The results indicate that IMTSP performs better in its capability of generating a series of waste management patterns under different risk-aversion levels, and also providing supports for decision makers in identifying desired waste flow strategies, considering balance between system economy and environmental quality.     

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

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

An inexact fuzzy-chance-constrained two-stage mixed-integer linear programming approach for flood diversion planning under multiple uncertainties

In this study, an inexact fuzzy-chance-constrained two-stage mixed-integer linear programming (IFCTIP) approach is developed for flood diversion planning under multiple uncertainties. A concept of the distribution with fuzzy boundary interval probability is defined to address multiple uncertainties expressed as integration of intervals, fuzzy sets and probability distributions. IFCTIP integrates the inexact programming, two-stage stochastic programming, integer programming and fuzzy-stochastic programming within a general optimization framework. IFCTIP incorporates the pre-regulated water-diversion policies directly into its optimization process to analyze various policy scenarios; each scenario has different economic penalty when the promised targets are violated. More importantly, it can facilitate dynamic programming for decisions of capacity-expansion planning under fuzzy-stochastic conditions. IFCTIP is applied to a flood management system. Solutions from IFCTIP provide desired flood diversion plans with a minimized system cost and a maximized safety level. The results indicate that reasonable solutions are generated for objective function values and decision variables, thus a number of decision alternatives can be generated under different levels of flood flows.     

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.     

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.     

An inexact programming method for agricultural irrigation systems under parameter uncertainty

Rapid population growth and economy development have led to increasing reliance on water resources. It is even aggravated for agricultural irrigation systems where more water is necessary to support the increasing population. In this study, an inexact programming method based on two-stage stochastic programming and interval-parameter programming is developed to obtain optimal water-allocation strategies for agricultural irrigation systems. It is capable of handling such problems where two-stage decisions need to be suggested under random- and interval-parameter inputs. An interactive solving procedure derived from conventional interval-parameter programming makes it possible for the impact of lower and upper bounds of interval inputs to be well reflected in the resulting solutions. An agricultural irrigation management problem is then provided to demonstrate the applicability, and reasonable solutions are obtained. Compared to the solutions from a representative interval-parameter programming model where only one decision-stage exists, the interval of optimized objective-function value is narrow, indicating more alternatives could be provided when water-allocation targets are rather high. However, chances of obtaining more benefits exist in association with a risk of paying more penalties; such a relationship becomes apparent when the variation of water availability is much intensive.     

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.     

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

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

A fuzzy-Markov-chain-based analysis method for reservoir operation

In this study, a fuzzy-Markov-chain-based stochastic dynamic programming (FM-SDP) method is developed for tackling uncertainties expressed as fuzzy sets and distributions with fuzzy probability (DFPs) in reservoir operation. The concept of DFPs used in Markov chain is presented as an extended form for expressing uncertainties including both stochastic and fuzzy characteristics. A fuzzy dominance index analysis approach is proposed for solving multiple fuzzy sets and DPFs in the proposed FM-SDP model. Solutions under a set of α-cut levels and fuzzy dominance indices can be generated by solving a series of deterministic submodels. The developed method is applied to a case study of a reservoir operation system. Solutions from FM-SDP provide a range of desired water-release policies under various system conditions for reservoir operation decision makers, reflecting dynamic and dual uncertain features of water availability simultaneously. The results indicate that the FM-SDP method could be applicable to practical problems for decision makers to obtain insight regarding the tradeoffs between economic and system reliability criteria. Willingness to obtain a lower benefit may guarantee meeting system-constraint demands; conversely, a desire to acquire a higher benefit could run into a higher risk of violating system constraints.     

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.     

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.     

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.     

A robust risk analysis method for water resources allocation under uncertainty

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

A stochastic rough-approximation water management model for supporting sustainable water-environment strategies in an irrigation district of arid region

In this study, a stochastic rough-approximation water management model (SRAWM) associated with optimistic and pessimistic options is proposed for supporting regional sustainability in an irrigation system (IS) of an arid region with uncertain information. SRAWM can not only handle conventional stochastic variations in objective functions or constraints, but also tackle objective and subjective (i.e., risk performance of the decision maker) fuzziness through rough-approximation model based on measure Me. The developed model would be applied to a real case study of an irrigation district (ID) in Kaidu-kongque River Basin, China, which is encountering challenges in economic development and a serious environmental crisis (e.g., drought, water deficit, land deterioration, stalinization, soil erosion and water pollution) synchronously. Simulation technical (i.e., support vector regression) is put into SRAWM framework to reflect dynamic prediction of water demand in the future. Results of optimized irrigation area, water allocation, water deficit, pollution reduction, water and soil erosion and system benefit under various water-environmental policies (corresponding to various ecological effects) are obtained. Tradeoffs between ecological and irrigative water usages can facilitate the local decision makers rectifying the current irrigation patterns and ecological protection polices. Moreover, compromises between systemic benefit and failure risk can help policymakers to generate a robust risk-control plan under uncertainties. These detections are beneficial to achieve conjunctive goals of socio-economic development and eco-environmental sustainability in such an arid IS.     

An inexact stochastic optimization model for agricultural irrigation management with a case study in China

In this study, an inexact stochastic optimization model (ITSOM) is developed for agricultural irrigation management with a case study in China. Functional intervals are introduced into the modeling framework to much accurately address the spatial and temporal variation of system components. According to the results of case study, the developed model shows effectiveness in dealing with functional information of system parameters, and brings no difficulty in obtaining optimal water allocation patterns. It is indicated that the surface water resource (i.e. Heshui River) has better be used as the major source, and proper exploration of groundwater can curtail the related expense and further increase the system net benefit. Among eight farms, hybrid rice farm is going to obtain the greatest amount of water than the others, while watermelon farm has the priority to get water due to its highest benefit and penalty rate. In comparison, water allocations to rapeseed and tea farms are to be minimal within the respective fluctuation ranges. Scenario analysis is also conducted to clarify the differences between ITSOM and a conventional interval two-stage stochastic programming (ITSP) model. A total of 60 scenarios are initiated respectively linking to 60 monthly ITSP models for the entire planning horizon. The results show that the optimal objective function values of all ITSP models always fall into the range of that obtained from ITSOM. As each ITSP solution can only correspond to the system condition under a certain time point, it is highly vulnerable to system variation.     

Coupling fuzzy-chance constrained program with minimax regret analysis for water quality management

Mathematical modeling technique plays an important role for regionalization assessment of integrated economy and environment problems, resulting in provision of decision makers with break-through insights and risk-informed strategies. However, such a planning effort is complicated with a variety of uncertain and dynamic factors as well as their interactions. In this study, a fuzzy-chance constrained programming (FCP) method is firstly developed for addressing uncertainties characterized as fuzzy sets and random variables and, then, minimax regret (MMR) analysis technique is advanced to determine desired alternative that can reflect compromises between maximized system benefit and minimized system-failure risk. FCP coupled with MMR is applied to a real-case study of water quality management through optimizing chemical industry activities of the New Binhai District, an economically and industrially fast growing region in the center of northern China. Modeling formulation can analyze interactions among criteria of industry layout, economic benefit, pollution mitigation, and water quality security. Solutions for planning the water quality management have been generated, reflecting that there is trade-off among industrial structure, environmental protection, and economic development.     

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

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

Water resources management under 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.