Inexact multistage stochastic integer programming for water resources management under uncertainty

In this study, an inexact multistage stochastic integer programming (IMSIP) method is developed for water resources management under uncertainty. This method incorporates techniques of inexact optimization and multistage stochastic programming within an integer programming framework. It can deal with uncertainties expressed as both probabilities and discrete intervals, and reflect the dynamics in terms of decisions for water allocation through transactions at discrete points of a complete scenario set over a multistage context. Moreover, the IMSIP can facilitate analyses of the multiple policy scenarios that are associated with economic penalties when the promised targets are violated as well as the economies-of-scale in the costs for surplus water diversion. A case study is provided for demonstrating the applicability of the developed methodology. The results indicate that reasonable solutions have been generated for both binary and continuous variables. For all scenarios under consideration, corrective actions can be undertaken dynamically under various pre-regulated policies and can thus help minimize the penalties and costs. The IMSIP can help water resources managers to identify desired system designs against water shortage and for flood control with maximized economic benefit and minimized system-failure risk.     

Inexact fuzzy-stochastic mixed-integer programming approach for long-term planning of waste management – Part A: Methodology

In this study, an inexact fuzzy chance-constrained two-stage mixed-integer linear programming (IFCTIP) approach is proposed for supporting long-term planning of waste-management systems under multiple uncertainties in the City of Regina, Canada. The method improves upon the existing inexact two-stage programming and mixed-integer linear programming techniques by incorporating uncertainties expressed as multiple uncertainties of intervals and dual probability distributions within a general optimization framework. The developed method can provide an effective linkage between the predefined environmental policies and the associated economic implications. Four special characteristics of the proposed method make it unique compared with other optimization techniques that deal with uncertainties. Firstly, it provides a linkage to predefined policies that have to be respected when a modeling effort is undertaken; secondly, it is useful for tackling uncertainties presented as intervals, probabilities, fuzzy sets and their incorporation; thirdly, it facilitates dynamic analysis for decisions of facility-expansion planning and waste-flow allocation within a multi-facility, multi-period, multi-level, and multi-option context; fourthly, the 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 solid waste-generation rates are violated. In a companion paper, the developed method is applied to a real case for the long-term planning of waste management in the City of Regina, Canada.     

Violation analysis for solid waste management systems: an interval fuzzy programming approach

This paper introduces a violation analysis approach for the planning of regional solid waste management systems under uncertainty, based on an interval-parameter fuzzy integer programming (IPFIP) model. In this approach, several given levels of tolerable violation for system constraints are permitted. This is realized through a relaxation of the critical constraints using violation variables, such that the model's decision space can be expanded. Thus, solutions from the violation analysis will not necessarily satisfy all of the model's original constraints. Application of the developed methodology to the planning of a waste management system indicates that reasonable solutions can be generated through this approach. Considerable information regarding decisions of facility expansion and waste flow allocation within the waste management system were generated. The modeling results help to generate a number of decision alternatives under various system conditions, allowing for more in-depth analyses of tradeoffs between environmental and economic objectives as well as those between system optimality and reliability.     

Inexact rough-interval two-stage stochastic programming for conjunctive water allocation problems

An inexact rough-interval two-stage stochastic programming (IRTSP) method is developed for conjunctive water allocation problems. Rough intervals (RIs), as a particular case of rough sets, are introduced into the modeling framework to tackle dual-layer information provided by decision makers. Through embeding upper and lower approximation intervals, rough intervals are capable of reflecting complex parameters with the most reliable and possible variation ranges being identified. An interactive solution method is also derived. A conjunctive water-allocation system is then structured for characterizing the proposed model. Solutions indicate a detailed optimal allocation scheme with a rough-interval form; a total of [[1048.83, 2078.29]:[1482.26, 2020.60]] would be obtained under the pre-regulated inputs. Comparisons of the proposed model to a conventional and an interval two-stage stochastic programming model are also conducted. The results indicate that the optimal objective function values of TSP and ITSP always fall into the range of , while they are sometimes out of the range of ; the optimal solutions of decision variables also present this feature. This implies the reliability of IRTSP in handling conjunctive water allocation problems.     

A stochastic fuzzy chance-constrained programming model for energy–environment system planning and management in the City of Beijing

A stochastic and fuzzy chance-constrained programming (SFCCP) model was developed in this study for supporting energy–environment management in the city of Beijing, China. SFCCP was capable of tackling the variables in constraints as fuzzy random variables, which were integration of randomness and vagueness. SFCCP was applied to an energy–environment management system in the city of Beijing. The study results indicated that SFCCP was useful in helping decision makers gain in-depth insights into proposed management system and establish environment-friendly energy allocation alternatives. The application of SFCCP is expected to provide a good demonstration to energy–environment management problems under complex uncertainties.     

An interval type-2 fuzzy stochastic approach for regional-scale electric power system under parameter uncertainty

In this study, an interval type-2 fuzzy stochastic linear programming method (IT2FSLP) is developed to support regional-scale electric power system (REM) planning. The IT2FSLP-REM model is based on an integration of interval type-2 fuzzy sets boundary programming and stochastic linear programming techniques enable it to have robust abilities to the deal with uncertainties expressed as type-2 fuzzy intervals and probabilistic distributions within a general optimization framework. Moreover, it can reflect dynamic decisions for energy supply and energy conversion processes, as well as provide capacity expansion options with multiple periods. The developed model is applied to a case of planning regional-scale energy and environmental systems to demonstrate its applicability. Based on a two-step solution algorithm, reasonable solutions have been obtained, which reflect tradeoffs among economic cost, environmental requirements, and energy-supply security. Thus, the lower and upper solutions of IT2FSLP-REM would then help energy authorities adjust or justify allocation patterns of regional energy resources and services.     

A dual-interval fixed-mix stochastic programming method for water resources management under uncertainty

In this study, a dual-interval fixed-mix stochastic programming (DFSP) method is developed for planning water resources management systems under uncertainty. DFSP incorporates interval-parameter programming (IPP) and fuzzy vertex analysis (FVA) within a fixed-mix stochastic programming (FSP) framework to address uncertain parameters described as probability distributions and dual intervals. It can also be used for analyzing various policy scenarios that are associated with different levels of economic consequences since penalties are exercised with recourse actions against any infeasibility. A real case for water resources management planning of Zhangweinan River Basin in China is then conducted for demonstrating the applicability of the developed DFSP method. Solutions in association with α-cut levels are generated by solving a set of deterministic submodels, which are useful for generating a range of decision alternatives under compound uncertainties. The results can help to identify desired water-allocation schemes for local sustainable development that the prerequisite water demand can be guaranteed when the available water resource is scarce.     

FSILP: fuzzy-stochastic-interval linear programming for supporting municipal solid waste management

Although many studies on municipal solid waste management (MSW management) were conducted under uncertain conditions of fuzzy, stochastic, and interval coexistence, the solution to the conventional linear programming problems of integrating fuzzy method with the other two was inefficient. In this study, a fuzzy-stochastic-interval linear programming (FSILP) method is developed by integrating Nguyen's method with conventional linear programming for supporting municipal solid waste management. The Nguyen's method was used to convert the fuzzy and fuzzy-stochastic linear programming problems into the conventional linear programs, by measuring the attainment values of fuzzy numbers and/or fuzzy random variables, as well as superiority and inferiority between triangular fuzzy numbers/triangular fuzzy-stochastic variables. The developed method can effectively tackle uncertainties described in terms of probability density functions, fuzzy membership functions, and discrete intervals. Moreover, the method can also improve upon the conventional interval fuzzy programming and two-stage stochastic programming approaches, with advantageous capabilities that are easily achieved with fewer constraints and significantly reduces consumption time. The developed model was applied to a case study of municipal solid waste management system in a city. The results indicated that reasonable solutions had been generated. The solution can help quantify the relationship between the change of system cost and the uncertainties, which could support further analysis of tradeoffs between the waste management cost and the system failure risk.     

A two-stage inexact joint-probabilistic programming method for air quality management under uncertainty

A two-stage inexact joint-probabilistic programming (TIJP) method is developed for planning a regional air quality management system with multiple pollutants and multiple sources. The TIJP method incorporates the techniques of two-stage stochastic programming, joint-probabilistic constraint programming and interval mathematical programming, where uncertainties expressed as probability distributions and interval values can be addressed. Moreover, it can not only examine the risk of violating joint-probability constraints, but also account for economic penalties as corrective measures against any infeasibility. The developed TIJP method is applied to a case study of a regional air pollution control problem, where the air quality index (AQI) is introduced for evaluation of the integrated air quality management system associated with multiple pollutants. The joint-probability exists in the environmental constraints for AQI, such that individual probabilistic constraints for each pollutant can be efficiently incorporated within the TIJP model. The results indicate that useful solutions for air quality management practices have been generated; they can help decision makers to identify desired pollution abatement strategies with minimized system cost and maximized environmental efficiency.     

WATER ALLOCATION POLICY MODELING FOR THE DONG NAI RIVER BASIN: AN INTEGRATED PERSPECTIVE1

Abstract: Recent water sector reforms and increased scarcity and vulnerability of water resources, combined with declining public funding available for large scale infrastructure investment in the sector, have led to a greater awareness by the Government of Vietnam for the need to analyze water resource allocation and use in an integrated fashion, at the basin scale, and from a perspective of economic efficiency. In this study we focus on the development, application, and selected policy analyses using an integrated economic hydrologic river basin model for the Dong Nai River Basin in southern Vietnam. The model framework depicts the sectoral structure and location of water users (agriculture, industry, hydropower, domestic, and the environment) and the institutions for water allocation in the basin. Water benefit functions are developed for the major water uses subject to physical limitations and to constraints of system control and policy. Based on this modeling framework, we will analyze policies that can affect water allocation and use at the basin level, including both basin-specific and general macroeconomic policies.     

Planning Agricultural Water Resources System Associated With Fuzzy and Random Features1

Li, Y.P. and G.H. Huang, 2011. Planning Agricultural Water Resources System Associated With Fuzzy and Random Features. Journal of the American Water Resources Association (JAWRA) 47(4):841‐860. DOI: 10.1111/j.1752‐1688.2011.00558.x Abstract: More and more regions where demand outstrips water resources availability have suffered from chronic severe shortages. It is particularly aggravated for agricultural irrigation systems where more water is necessary to support the rapidly increasing population and speedily developing economy. In this study, a two‐stage fuzzy‐stochastic programming (TFSP) method is developed for planning agricultural water resources management system in more efficient and sustainable ways. The developed method can address uncertain parameters described as probability distributions and fuzzy sets. It can also be used for analyzing various policy scenarios that are associated with different levels of economic consequences since penalties are exercised with recourse actions against any infeasibility. The developed method is applied to agricultural water‐resources management planning of the Zhangweinan River Basin, China. Solutions under various α‐cut levels and fuzzy dominance indices can be generated by solving a series of deterministic submodels, which can help determine optimized crop‐target values that could hedge appropriately against future available water levels. The results are helpful for water resources managers in not only making decisions of crop irrigation but also gaining insight into the tradeoffs between economic objective and system‐failure risk.     

Irrigation Water Allocation Using an Inexact Two‐Stage Quadratic Programming with Fuzzy Input under Climate Change

Agricultural irrigation accounts for nearly 70% of the total water use around the world. Uncertainties and climate change together exacerbate the complexity of optimal allocation of water resources for irrigation. An interval‐fuzzy two‐stage stochastic quadratic programming model is developed for determining the plans for water allocation for irrigation with maximum benefits. The model is shown to be applicable when inputs are expressed as discrete, fuzzy or random. In order to reflect the effect of marginal utility on benefit and cost, the model can also deal with nonlinearities in the objective function. Results from applying the model to a case study in the middle reaches of the Heihe River basin, China, show schemes for water allocation for irrigation of different crops in every month of the crop growth period under various flow levels are effective for achieving high economic benefits. Different climate change scenarios are used to analyze the impact of changing water requirement and water availability on irrigation water allocation. The proposed model can aid the decision maker in formulating desired irrigation water management policies in the wake of uncertainties and changing environment.     

Factorial fuzzy programming for planning water resources management systems

In this paper, we propose a factorial fuzzy programming (FFP) approach for planning water resources management systems. The conventional fuzzy method cannot reflect the interactions among uncertain system parameters nor quantify their interactive effects. This may lead to important interrelationships among system parameters being neglected in systems analysis, and the derived decisions may not be robust enough to support the management under uncertainties. The objective of this study is to develop an FFP approach to deal with such interactive uncertainties. Factorial analysis (FA) was integrated with the fuzzy technique to quantify the effects of multiple fuzzy modeling parameters on the system performance and to reveal their implicit interrelationships. A flood-diversion planning case was studied to illustrate the applicability of the FFP approach. The individual and interactive effects of fuzzy parameters on the system objectives were evaluated. The influential effects were identified and the implicit interrelationships within influential interactions were revealed.     

A LINEAR ANALYSIS OF AN URBAN WATER SUPPLY SYSTEM1

The use of linear programming as a planning tool for determining the optimal long-range development of an urban water supply system was explored. A stochastic trace of water demand was synthesized and used as an input to the model. This permitted evaluating the feasibility of imposing demand restrictions as an effective cost reduction mechanism. The City of Lincoln, Nebraska, was used as the urban model. The fundamental problem was to allocate limited water supplies from several sources to an urban load center to minimize costs and comply with system constraints. The study period covered twenty years, and findings indicate the planning direction for stage development during this period. Sensitivity analyses were performed on cost coefficients and demands. Thirteen sources were included in the initial computations. Conclusions were that linear programming and generated demand traces are useful tools for both short- and long-term urban water supply planning. Lowering peak demands results in long-range development of fewer sources of supply and more economic and efficient use of the supplies developed.     

A hybrid inexact optimization approach for solid waste management in the city of Foshan,China

An interval-parameter two-stage chance-constraint mixed integer linear programming (ITCMILP) model is provided for supporting long-term planning of solid waste management in the City of Foshan, China. The ITCMILP is formulated by integrating interval-parameter, two-stage, mixed integer, and chance-constraint programming methods into a general framework, and can thus deal with multiple uncertainties associated with model parameters, constraints and objectives. Three scenarios are examined, covering combinations of various system conditions and waste management policies. Scenario 1 is designed for comparison purposes. Scenarios 2 and 3 correspond to situations when the existing landfill's life is to be extended. The results demonstrate that the centralized composting and incinerating facilities are desired for treating the organic waste flows. The tradeoff among system cost, violation risk, and the related policy implications are also analyzed. The results obtained could help decision makers gain in-depth insights into the impact of uncertainties on long-term solid waste management in the City of Foshan.     

An inexact two-stage mixed integer linear programming method for solid waste management in the City of Regina

In this study, an interval-parameter two-stage mixed integer linear programming (ITMILP) model is developed for supporting long-term planning of waste management activities in the City of Regina. In the ITMILP, both two-stage stochastic programming and interval linear programming are introduced into a general mixed integer linear programming framework. Uncertainties expressed as not only probability density functions but also discrete intervals can be reflected. The model can help tackle the dynamic, interactive and uncertain characteristics of the solid waste management system in the City, and can address issues concerning plans for cost-effective waste diversion and landfill prolongation. Three scenarios are considered based on different waste management policies. The results indicate that reasonable solutions have been generated. They are valuable for supporting the adjustment or justification of the existing waste flow allocation patterns, the long-term capacity planning of the City's waste management system, and the formulation of local policies and regulations regarding waste generation and management.     

OPTIMAL WATER USE AND SALINITY CONTROL FOR ENERGY - UPPER COLORADO RIVER BASIN1

ABSTRACT: The Upper Colorado River Basin contains appreciable amounts of undeveloped fuel resources. Large quantities of oil shale, coal, and uranium have attracted recent economic and commercial interests. Development of these resources and subsequent conversion to alternative energy forms require an adequate supply of water. Water use for large scale energy development will place increasing demands on an already overstressed allocation of Colorado River water. Present water quality is at a concentration where increased salinity will result in economic detriments to holders of downstream water rights. The salt and water exchange in mining, processing, and spent fuel disposal processes has been incorporated as part of a two-level minimum cost linear programming algorithm. Mathematical simulation results provide an optimal use of Upper Colorado River water for levels of energy output such that salinity concentrations are maintained below predetermined levels.     

AN OPTIMIZATION MODEL FOR EFFICIENT MANAGEMENT OF URBAN WATER RESOURCES1

ABSTRACT: A mathematical programming model is proposed to determine economically efficient urban water resource allocation and pricing policy by maximizing the sum of the consumer and producer surplus. The optimization of this nonlinear problem is accomplished by the use of linear programming algorithm. The feasibility of using recycled water for municipal purposes is examined in a planning context. The impact of higher water quality discharge standards on pricing and allocation of water is analyzed and the attractiveness of water reuse option is demonstrated.     

Using system dynamics to model water-reallocation

Improving the efficiency of water allocation has long been recognised as a key problem for the water resources management decision-makers. However, assessing the efficacy of management decision is difficult due to the complexity and interconnectivity of water resource systems. For this reason, it is vital that robust modelling approaches are employed to deal with the feedback loops inherent in the water resource systems. Whilst many studies have applied modelling to various aspects of water resource management, little attention has been given to innovations in modelling approaches to deal with the modelling challenges associated with improving decision-making. The aim of this study is to apply a System Dynamics modelling approach to improve the efficiency of water allocation incorporating a myriad of irrigation system constraints. The system dynamic approach allows the different system components to be organised as a collection of discrete objects that incorporate data, structure and function to generate complex system behaviour. Through the application of a system dynamic approach, a robust model (named the Economical Reallocating Water Model (ERWM)) was developed which was used to examine the options of re-allocating water resources that minimize the water cost all over an irrigated agricultural area. The EWRM incorporated a wide range of complexities likely to be encountered in water resource management: surface and ground water sources, water trading between sources, system constraint such as maximum ground water pumping, rates, maximum possible trading volumes and differential water resource prices. Two hypothetical systems have been presented here as an example. The results show that the System Dynamics approach has a significant advantages in estimating and assessing the outcomes of alternative water management strategies through time and space.     

Waste management with recourse: An inexact dynamic programming model containing fuzzy boundary intervals in objectives and constraints

The existing inexact optimization methods based on interval-parameter linear programming can hardly address problems where coefficients in objective functions are subject to dual uncertainties. In this study, a superiority–inferiority-based inexact fuzzy two-stage mixed-integer linear programming (SI-IFTMILP) model was developed for supporting municipal solid waste management under uncertainty. The developed SI-IFTMILP approach is capable of tackling dual uncertainties presented as fuzzy boundary intervals (FuBIs) in not only constraints, but also objective functions. Uncertainties expressed as a combination of intervals and random variables could also be explicitly reflected. An algorithm with high computational efficiency was provided to solve SI-IFTMILP. SI-IFTMILP was then applied to a long-term waste management case to demonstrate its applicability. Useful interval solutions were obtained. SI-IFTMILP could help generate dynamic facility-expansion and waste-allocation plans, as well as provide corrective actions when anticipated waste management plans are violated. It could also greatly reduce system-violation risk and enhance system robustness through examining two sets of penalties resulting from variations in fuzziness and randomness. Moreover, four possible alternative models were formulated to solve the same problem; solutions from them were then compared with those from SI-IFTMILP. The results indicate that SI-IFTMILP could provide more reliable solutions than the alternatives.