A Fuzzy Robust Nonlinear Programming Model for Stream Water Quality Management

Beijing Normal University Chinese Research Academy of Environmental Science Beijing 100875 China
Water Resources Management (Impact Factor: 2.6). 11/2009; 23(14):2913-2940. DOI: 10.1007/s11269-009-9416-3


An interval-parameter fuzzy robust nonlinear programming (IFRNP) approach was developed for stream water quality management
under uncertainty. The interval and fuzzy robust programming methods were incorporated within a general framework to address
uncertainties associated with the nonlinear objective and the left- and right-hand sides of the constraints. A piecewise linearization
approach was developed to deal with the nonlinear cost function. IFRNP could explicitly address complexities of various system
uncertainties, where parameters were represented as both interval numbers and fuzzy membership functions. Furthermore, the
dual uncertain information associated with the lower and upper bounds of each interval parameter could be effectively tackled
through the concept of fuzzy boundary interval. The proposed IFRNP method was applied to a case of water quality management
in the Guoyang section of the Guo River in Anhui province, China. A number of cost-effective schemes for water quality management
were generated, and allowable wastewater discharge amounts were recommended. The results indicated that IFRNP was applicable
to water quality management problems, where high nonlinearities and dual uncertainties exist.

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    • "In general, the fuzzy linear trapezoidal number is every fuzzy set with the following membership function (e.g. Zhu et al. 2009; Papadopoulos and Sirpi 1999): "
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    ABSTRACT: The study presents a multicriteria method incorporating a fuzzy set approach and the 0/1 programming for selecting the most appropriate actions for facing long term water scarcity in a water system under a set of objectives and constraints. The proposed method includes also the right of veto which can be utilised under certain extreme conditions or for securing commensurate solutions. The proposed method is applied for devising the technical component of a rational preparedness plan against long term water scarcity in the island of Naxos, Cyclades (Greece). KeywordsFuzzy sets–Multicriteria methods–Water resources management–Water scarcity–Preparedness plan
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    • "Sasikumar & Mujumdar (1998, 2000) and Mujumdar & Sasikumar (2002) have addressed the uncertainty due to imprecision as well as randomness in a multiobjective framework. Fuzzy logic has been used for water quality management to model imprecision by Zhu et al. (2009) and Lermontov et al. (2009). Recently, uncertainty resulting from the inexactness of parameter values in water quality management models has been addressed in Karmakar & Mujumdar (2007) and Nie et al. (2008). "
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    ABSTRACT: Fuzzy waste Load Allocation Model (FWLAM), developed in an earlier study, derives the optimal fractional levels, for the base flow conditions, considering the goals of the Pollution Control Agency (PCA) and dischargers. The Modified Fuzzy waste Load Allocation Model (MFWLAM) developed subsequently is a stochastic model and considers the moments (mean, variance and skewness) of water quality indicators, incorporating uncertainty due to randomness of input variables along with uncertainty due to imprecision. The risk of low water quality is reduced significantly by using this modified model, but inclusion of new constraints leads to a low value of acceptability level. λ, interpreted as the maximized minimum satisfaction in the system. To improve this value, a new model, which is a combination of FWLAM and MFWLAM, is presented, allowing for some violations in the constraints of MFWLAM. This combined model is a multiobjective optimization model having the objectives, maximization of acceptability level and minimization of violation of constraints. Fuzzy multiobjective programming, goal programming and fuzzy goal programming are used to find the solutions. For the optimization model, Probabilistic Global Search Lausanne (PGSL) is used as a nonlinear optimization tool. The methodology is applied to a case study of the Tunga-Bhadra river system in south India. The model results in a compromised solution of a higher value of acceptability level as compared to MFWLAM, with a satisfactory value of risk. Thus the goal of risk minimization is achieved with a comparatively better value of acceptability level.
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    ABSTRACT: A simulation-based interval-fuzzy nonlinear programming (SIFNP) approach was developed for seasonal planning of stream water quality management. The techniques of inexact modeling, nonlinear programming, and interval-fuzzy optimization were incorporated within a general framework. Based on a multi-segment stream water quality simulation model, dynamic waste assimilative capacity of a river system within a multi-season context was considered in the optimization process. The method could not only address complexities of various system uncertainties but also tackle nonlinear environmental–economic interrelationships in water quality management problems. In addition, interval-fuzzy numbers were introduced to reflect the dual uncertainties, i.e., imprecision associated with fixing the lower and upper bounds of membership functions. The proposed method was applied to a case of water quality management in the Guoyang section of the Guo River in China. Interval solutions reflecting the inherent uncertainties were generated, and a spectrum of cost-effective schemes for seasonal water quality management could thus be obtained by adjusting different combinations of the decision variables within their solution intervals. The results indicated that SIFNP could effectively communicate dual uncertainties into the optimization process and help decision makers to identify desired options under various complexities of system components.
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