Lucien Duckstein

Comparison of fuzzy numbers using a fuzzy distance measure

A new approach for ranking fuzzy numbers based on a distance measure is introduced. A new class of distance measures for interval numbers that takes into account all the points in both intervals is developed first, and then it is used to formulate the distance measure for fuzzy numbers. The approach is illustrated by numerical examples, showing that it overcomes several shortcomings such as the indiscriminative and counterintuitive behavior of several existing fuzzy ranking approaches.

Multi-objective fuzzy regression: a general framework

Abstract Previous research has shown that in some cases fuzzy regression may perform better than statistical regression. On the other hand, fuzzy regression has also been criticized because it does not allow all data points to influence the estimated parameters, it is sensitive to data outliers, and the prediction intervals become wider as more data are collected. Here, several multi-objective fuzzy regression (MOFR) techniques are developed to overcome these problems by enabling the decision maker to select a non-dominated solution based on the tradeoff between data outliers and prediction vagueness. It is shown that MOFR models provide superior results to existing fuzzy regression techniques; furthermore the existing fuzzy regression approaches and classical least-squares regression are specific cases of the MOFR framework. The methodology is illustrated with rainfall-runoff modeling examples; more specifically, fuzzy linear conceptual rainfall-runoff relationships, which are essential components of hydrologic system models, are analyzed here. Scope and purpose The purpose of this paper is to develop a multi-objective fuzzy regression (MOFR) tool to overcome the shortcomings of existing fuzzy regression approaches while keeping their good characteristics, and to study systems with uncertain elements, using the example of rainfall-runoff processes to illustrate the approach. Previous research has shown that fuzzy regression might perform better than statistical regression in the following cases: when the data set is insufficient to support statistical regression analysis, when statistical distributional assumptions cannot be justified, if the aptness of the regression model is poor, when human judgements are involved (Bardossy. Fuzzy Sets and Systems 1990;37:65–75; Tanaka et al. IEEE Transactions on Systems, Man and Cyberneties, 1982;12 (6):903–7). On the other hand, fuzzy regression has also been criticized because it does not allow all data points to influence the estimated parameters, it is sensitive to data outliers, and the prediction intervals become wider as more data are collected (Redden and Woodall. Fuzzy Sets and Systems 1994;64:361–75, 1996;79:203–11). Here, several MOFR techniques are developed to overcome these problems by enabling the decision maker to select a non-dominated solution based on the tradeoff between data outliers and prediction vagueness. The methodology is illustrated with rainfall-runoff modeling examples; more specifically, fuzzy linear conceptual rainfall-runoff relationships, which are essential components of hydrologic system models, are analyzed here.

Fuzzy conceptual rainfall-runoff models

Abstract A fuzzy conceptual rainfall–runoff (CRR) framework is proposed herein to deal with those parameter uncertainties of conceptual rainfall–runoff models, that are related to data and/or model structure: with every element of the rainfall–runoff model assumed to be possibly uncertain, taken here as being fuzzy. First, the conceptual rainfall–runoff system is fuzzified and then different operational modes are formulated using fuzzy rules; second, the parameter identification aspect is examined using fuzzy regression techniques. In particular, bi-objective and tri-objective fuzzy regression models are applied in the case of linear conceptual rainfall–runoff models so that the decision maker may be able to trade off prediction vagueness (uncertainty) and the embedding outliers. For the non-linear models, a fuzzy least squares regression framework is applied to derive the model parameters. The methodology is illustrated using: (1) a linear conceptual rainfall–runoff model; (2) an experimental two-parameter model; and (3) a simplified version of the Sacramento soil moisture accounting model of the US National Weather Services river forecast system (SAC-SMA) known as the six-parameter model. It is shown that the fuzzy logic framework enables the decision maker to gain insight about the model sensitivity and the uncertainty stemming from the elements of the CRR model.

Application of fuzzy rule-based modeling technique to regional drought

Fuzzy rule-based modeling is applied to the prediction of regional droughts (characterized by the modified Palmer index, PMDI) using two forcing inputs, El Nino/Southern Oscillation (ENSO) and large scale atmospheric circulation patterns (CPs) in a typical Great Plains state, Nebraska. Although, there is significant relationship between simultaneous monthly CP, lagged Southern Oscillation Index (SOI) and PMDI in Nebraska, the weakness of the correlations, the dependence between CP and SOI and the relatively short data set limit the applicability of statistical modeling for prediction. Due to the above difficulties, a fuzzy rule-based approach is presented to predict PMDI from monthly frequencies of daily CP types and lagged prior SOIs. The fuzzy rules are defined and calibrated using a subset called the learning set of the observed time series of premises and PMDI response. Then, another subset, the validation set is used to check how the application of fuzzy rules reproduces the observed PMDI. In all its eight climate divisions and Nebraska itself, the fuzzy rule-based technique using the joint forcing of CP and SOI, is able to learn the high variability and persistence of PMDI and results in almost perfect reproduction of the empirical frequency distributions. q 1999 Elsevier Science B.V. All rights reserved.

Artificial neural networks and multicriterion analysis for sustainable irrigation planning

The objective of the present paper is to select the best compromise irrigation planning strategy for the case study of Jayakwadi irrigation project, Maharashtra, India. Four-phase methodology is employed. In phase 1, separate linear programming (LP) models are formulated for the three objectives, namely, net economic benefits, agricultural production and labour employment. In phase 2, nondominated (compromise) irrigation planning strategies are generated using the constraint method of multiobjective optimisation. In phase 3, Kohonen neural networks (KNN) based classification algorithm is employed to sort nondominated irrigation planning strategies into smaller groups. In phase 4, multicriterion analysis (MCA) technique, namely, Compromise Programming is applied to rank strategies obtained from phase 3. It is concluded that the above integrated methodology is effective for modeling multiobjective irrigation planning problems and the present approach can be extended to situations where number of irrigation planning strategies are even large in number.

Application of fuzzy logic-based modeling to improve the performance of the Revised Universal Soil Loss Equation

Abstract This paper discusses the application of fuzzy logic-based modeling to improve the performance of the Revised Universal Soil Loss Equation (RUSLE). An analysis of over 1700 plot-years of data, taken from more than 200 plots at 21 sites in the U.S., showed that soil erosion was not adequately described merely by the multiplication of five RUSLE factor values in all cases. The fuzzy logic-based modeling approach was to make the RUSLEs structure more flexible in describing the relationship between soil erosion and other factors and in dealing with data and model uncertainties without requiring any further information. The approach used in this study consisted of two techniques: multi-objective fuzzy regression (MOFR) and fuzzy rule-based modeling (FRBM). First, MOFR was applied to small subsets of RUSLE factor values to derive the relationship between soil loss and the rainfall erosivity factor within each subset of data. These MOFR models, considered as single fuzzy rules, were in turn linked together in a FRBM framework to form a fuzzy rule set. Then the fuzzy rule set was applied to compute the soil loss prediction corresponding to each combination of RUSLE factors. The model efficiency [Journal of Hydrology (Amsterdam) 10 (1970) 282] of the fuzzy model on a yearly basis was 0.70 while the RUSLEs was 0.58. On an average annual basis, the model efficiency was 0.90 and 0.72 for the fuzzy model and the RUSLE, respectively.

Multiobjective fuzzy linear programming for sustainable irrigation planning: an Indian case study

Abstract Multiobjective fuzzy linear programming (MOFLP) irrigation planning model is formulated for the evaluation of management strategy for the case study of Jayakwadi irrigation project, Maharashtra, India. Three conflicting objectives net benefits, agricultural production and labour employment are considered in the irrigation planning scenario. All three criteria are to be maximised and the last two are sustainability related. All three objective functions are quantified by linear membership functions in a fuzzy multi objective framework. It is observed from MOFLP solution that net benefits, agricultural production and labour employment are 2.031×109 Rupees, 2.1186×106 tons, 3.5858×107 man-days respectively with degree of truth (λ) 0.5715. Analysis of results indicated that net benefits, agricultural production, labour employment have decreased by 4.13, 5.39 and 3.4% as compared to ideal values in the crisp linear programming (LP) model.

Multiobjective fuzzy regression with central tendency and possibilistic properties

A multiobjective fuzzy regression model (MOFR) is developed. This MOFR model combines central tendency and possibilistic properties of statistical and fuzzy regressions and overcomes several shortcomings of these two approaches. A new class of distance measure for two intervals that takes into account all the points in both intervals is introduced. The methodology is illustrated by numerical examples.

Fuzzy measure theoretical approach to screening product innovations

Variety of decision models have been proposed in contemporary literature to tackle the problem of screening product innovations. Although linear models have gained considerable attention and recommendation, contemporary literature contains strong evidence in support of nonlinear noncompensatory models. In this paper, the authors first demonstrate how fuzzy measures, which are defined on subsets of decision attributes, and their Choquet-integral formulation, which exhibits both compensatory and noncompensatory properties, have meaningful behavioral interpretations within the context of new-product screening. Then, they show how to address the complex problem of building such measures by applying a learning algorithm that relies on methods of judgment analysis. An accompanying case study demonstrates how organizations may customize a new product decision aid and fine tune their business strategy as actual results accrue. Finally, the authors present the results of analytical studies to compare the Choquet-integral model with other noncompensatory models, such as Martinos extended scoring model and Einhorns conjunctive model, and heuristic approaches, such as Tverskys EBA and the lexicographic method. For the new-product-decision scenario considered in the study, the Choquet-integral model provided the best fit, measured by Pearsons rank order correlation coefficient, with all of the competing models

An Indian Case Study

Selection of the best irrigation subsystem is evaluated in a multiobjective framework for a case study of Sri Ram Sagar Project, Andhra Pradesh, India. Two Multicriterion Decision Making (MCDM) methods, namely, PROMETHEE-2 and EXPROM-2 are applied for evaluation. Eight performance criteria, namely, farm development works, environmental conservation, timely supply of inputs, conjunctive use of water resources, productivity, farmers’ participation, economic impact and social impact are evaluated for the five irrigation subsystems. Weighting of the performance criteria is calculated by Analytic Hierarchy Process (AHP). Group decision making methodology is incorporated by additive ranking analysis. Results indicated that same irrigation subsystem is found to be the best by both methods.