Y. R. Fan

Generalized fuzzy linear programming for decision making under uncertainty: Feasibility of fuzzy solutions and solving approach

In this study, a generalized fuzzy linear programming (GFLP) method is developed for dealing with uncertainties expressed as fuzzy sets. The feasibility of fuzzy solutions of the GFLP problem is investigated. A stepwise interactive algorithm (SIA) based on the idea of design of experiment is then advanced to solve the GFLP problem. This SIA method was implemented through (i) discretizing membership grade of fuzzy parameters into a finite number of @a-cut levels, (ii) converting the GFLP model into an interval linear programming (ILP) submodel under every @a-cut level, (iii) solving the ILP submodels through an interactive algorithm and obtaining the associated interval solutions, (iv) acquiring the membership functions of fuzzy solutions through statistical regression methods. A simple numerical example is then proposed to illustrate the solution process of the GFLP model through SIA. A comparison between the solutions obtained though SIA and Monte Carlo method is finally conducted to demonstrate the robustness of the SIA method. The results indicate that the membership functions for decision variables and objective function are reasonable and robust.

Enhanced aqueous solubility of naphthalene and pyrene by binary and ternary Gemini cationic and conventional nonionic surfactants

A systematic study has been carried out to get insight into the micellar behavior of Gemini cationic and conventional nonionic in their single as well as equimolar bi and ternary mixed state using the technique of tensiometry. The models proposed by Clint, Rubingh and Motomura et al. have been employed to interpret the formation of mixed micelles and find out synergism. The obtained experimental CMCs are lower than the ideal CMCs, indicating negative deviation from ideal behavior for all multi-component mixed micelles formation. The solubilization capacities of selected equimolar bi and ternary surfactant systems towards polycyclic aromatic hydrocarbons (PAHs), naphthalene and pyrene, have been evaluated from measurements of the molar solubilization ratio (MSR), the micelle-water partition coefficient (K(m)), the deviation ratio (R) and the free energy of solubilization (ΔG(s)(0)) of PAHs. The results show that the solubility of naphthalene and pyrene over that in water in case of Gemini cationic surfactant is dramatically enhanced by adding equimolar nonionic surfactant in both bi and ternary mixed surfactant systems. The studied equimolar ternary surfactant system shows higher solubilizing efficiency than Gemini cationic binary system but lower than their cationic-nonionic counterpart. In addition, the solubilizing power of multi-component mixed surfactants towards naphthalene and pyrene increases with increasing logK(ow) of PAHs. Certainly, the solubilization abilities of the selected surfactants not only depend on their structure and mixing effect but also associate with solubilizing microenvironment and chemical nature of organic solutes.

A Hybrid Dynamic Dual Interval Programming for Irrigation Water Allocation under Uncertainty

Along with the economic development in Canada, the shortage of irrigation water has become a serious concern (Bouwer 1993; Hennessy 1993). In this study, a model of Dynamic Dual Interval Programming (DDIP) is developed and applied to the irrigation water allocation systems with uncertainty. DDIP method improves the existing dynamics interval programming by explicitly addressing the system uncertainties with a dual interval that had higher system reliability. The solution of DDIP is computationally effective, and its decision variables are incorporated into the solutions for final decision. In order to obtain the optimal allocation schemes in a dynamic process, the developed DDIP was applied to an irrigation water system. The results from this case study revealed that optimal solution can be obtained through the DDIP approach from the agriculture water management activities for feasible decisions. These decisions reflect the high uncertainty of the information in the boundaries of dual intervals. The solution presents a maximum benefit under limited yearly uncertain natural resources. Furthermore, the information obtained though this model may help the authority to make optimal decisions and to reduce the risk for uncertain situations.

Maximum entropy-Gumbel-Hougaard copula method for simulation of monthly streamflow in Xiangxi river, China

A maximum entropy-Gumbel-Hougaard copula (MEGHC) method has been proposed for monthly streamflow simulation. The marginal distributions of monthly streamflows are estimated through the maximum entropy (ME) method with the first four non-central moments (i.e. mean, standard deviation, skewness and kurtosis) being the constraints. The Lagrange multipliers in ME-based marginal distributions are determined using the conjugate gradient (CG) method which is of superlinear convergence, simple recurrence formula and less calculation. Then the joint distributions of two adjacent monthly streamflows are constructed using the Gumbel-Hougaard copula (GHC) method. The developed MEGHC method has been applied for monthly streamflow simulation in Xiangxi river, China. The goodness-of-fit statistical tests, consisting of K–S test, A–D test, RMSE and Rosenblatt transformation with Cramér von Mises statistic, show that the MEGHC method can reflect dependence structure in adjacent monthly streamflows of Xiangxi river, China. Comparison between simulated streamflow generated by MEGHC and observations indicates the satisfactory performance of MEGHC with small relative errors.

Planning Water Resources Allocation Under Multiple Uncertainties Through a Generalized Fuzzy Two-Stage Stochastic Programming Method

In this study, a generalized fuzzy two-stage stochastic programming (GFTSP) method is developed for planning water resources management systems under uncertainty. The developed GFTSP method can deal with uncertainties expressed as probability distributions, fuzzy sets, as well as fuzzy random variables. With the aid of a robust stepwise interactive algorithm, solutions for GFTSP can be generated by solving a set of deterministic submodels. Furthermore, the possibility information (expressed as fuzzy membership functions) can be reflected in the solutions for the objective function value and decision variables. The developed GFTSP approach is also applied to a water resources management and planning problem to demonstrate its applicability. Solutions of decision variables and objective function value are expressed as fuzzy membership functions, reflecting the fluctuating ranges of decision alternatives under different plausibilities. And thus, the water alternatives can be directly derived from the obtained fuzzy membership functions when the preferred α value is predefined by decision makers.

A PCM-based stochastic hydrological model for uncertainty quantification in watershed systems

In this study, an uncertainty quantification framework is proposed for hydrologic models based on probabilistic collocation method (PCM). The PCM method first uses polynomial chaos expansion (PCE) to approximate the hydrological outputs in terms of a set of standard Gaussian random variables, and then estimates the unknown coefficients in the PCE through collocation method. The conceptual hydrologic model, Hymod, is used to demonstrate the applicability of PCM in quantifying uncertainties of the hydrologic predictions. Two parameters in Hymod are considered as uniformly distributed in certain intervals. Two-dimensional 2-order and two-dimensional 3-order PCEs are applied to quantify the uncertainty of Hymod’s predictions. The results indicate that, both 2- and 3-order PCEs can well reflect the uncertainty of the streamflow predictions. The means and variances of 2- and 3-order PCEs are consistent with those obtained by Monte Carlo (MC) simulation method. However, for detailed distributions at selected periods, the histograms obtained by 3-order PCE are more accurate than those generated by 2-order PCE, when compared with the histograms obtained by MC simulation method.

Water resources management under uncertainty: factorial multi-stage stochastic program with chance constraints

Due to rapid growth of population and development of economy, water resources allocation problems have aroused wide concern. Therefore, optimization of water resources systems is complex and uncertain, which is a severe challenge faced by water managers. In this paper, a factorial multi-stage stochastic programming with chance constraints approach is developed to deal with the issues of water-resources allocation under uncertainty and risk as well as their interactions. It can deal with uncertainties described as both interval numbers and probability distributions, and can also support the risk assessment within a multistage context. The solutions associated with different risk levels of constraint violation can be obtained, which can help characterize the relationship between the economic objective and the system risk. The inherent interactions between factors at different levels and their effects on total net benefits can be revealed through the analysis of multi-parameter interactions.

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.

A generalized fuzzy linear programming approach for environmental management problem under uncertainty

In this study, a generalized fuzzy linear programming (GFLP) method was developed to deal with uncertainties expressed as fuzzy sets that exist in the constraints and objective function. A stepwise interactive algorithm (SIA) was advanced to solve GFLP model and generate solutions expressed as fuzzy sets. To demonstrate its application, the developed GFLP method was applied to a regional sulfur dioxide (SO2) control planning model to identify effective SO2 mitigation polices with a minimized system performance cost under uncertainty. The results were obtained to represent the amount of SO2 allocated to different control measures from different sources. Compared with the conventional interval-parameter linear programming (ILP) approach, the solutions obtained through GFLP were expressed as fuzzy sets, which can provide intervals for the decision variables and objective function, as well as related possibilities. Therefore, the decision makers can make a tradeoff between model stability and the plausibility based on solutions obtained through GFLP, and then identify desired policies for SO2-emission control under uncertainty. Implications: The GFLP method is effective in addressing air quality management planning associated with fuzzy parameter in the constraints and objective. The stepwise interactive algorithm proposed to solving the GFLP model would not lead to complicated intermediate submodels. Thus it would be much applicable for large-scale air quality management problems. The applications of the method can help decision makers in (i) generating allocation schemes for SO2 emissions; (ii) identifying the plausibility of each allocation alternative; and (iii) estimating the total costs of different SO2 emission control policies.

Development of a Stepwise-Clustered Hydrological Inference Model

AbstractFlow prediction is one of the most important issues in modern hydrology. In this study, a statistical tool, stepwise-clustered hydrological inference (SCHI) model, was developed for daily streamflow forecasting. The SCHI model uses cluster trees to represent the nonlinear and complex relationships between streamflow and multiple factors related to climate and watershed conditions. It allows a great deal of flexibility in watershed configuration. The proposed model was applied to the daily streamflow forecasting in the Xiangxi River watershed, China. The correlation coefficient for calibration (1991–1995) was 0.881, and that for validation (1996–1998) was 0.771. Nash–Sutcliffe efficiencies for calibration and validation were 0.768 and 0.577, respectively. The results were compared to those of a conventional process-based model, and it was found that the SCHI model had a superior performance. The results indicate that the proposed model could provide not only reliable and efficient daily flow predic...