Fuzzy optimization model for water quality management of a river system - Closure

A fuzzy waste-load allocation model, FWLAM, is developed for water quality management of a river system using fuzzy multiple-objective optimization. An important feature of this model is its capability to incorporate the aspirations and conflicting objectives of the pollution control agency and dischargers. The vagueness associated with specifying the water quality criteria and fraction removal levels is modeled in a fuzzy framework. The goals related to the pollution control agency and dischargers are expressed as fuzzy sets. The membership functions of these fuzzy sets are considered to represent the variation of satisfaction levels of the pollution control agency and dischargers in attaining their respective goals. Two formulations—namely, the MAX-MIN and MAX-BIAS formulations—are proposed for FWLAM. The MAX-MIN formulation maximizes the minimum satisfaction level in the system. The MAX-BIAS formulation maximizes a bias measure, giving a solution that favors the dischargers. Maximization of the bias measure attempts to keep the satisfaction levels of the dischargers away from the minimum satisfaction level and that of the pollution control agency close to the minimum satisfaction level. Most of the conventional water quality management models use waste treatment cost curves that are uncertain and nonlinear. Unlike such models, FWLAM avoids the use of cost curves. Further, the model provides the flexibility for the pollution control agency and dischargers to specify their aspirations independently.

Fuzzy Logic-Based Approaches in Water Resource System Modelling

Recent research in modelling uncertainty in water resource systems has highlighted the use of fuzzy logic-based approaches. A number of research contributions exist in the literature that deal with uncertainty in water resource systems including fuzziness, subjectivity, imprecision and lack of adequate data. This chapter presents a broad overview of the fuzzy logic-based approaches adopted in addressing uncertainty in water resource systems modelling. Applications of fuzzy rule-based systems and fuzzy optimisation are then discussed. Perspectives on the scope for further research are presented.

A Global Constraint On Relative Rotation To Avoid Lumped Compliant Mechanisms In Topology Optimization

Some of the well known formulations for topology optimization of compliant mechanisms could lead to lumped compliant mechanisms. In lumped compliance, most of the elastic deformation in a mechanism occurs at few points, while rest of the mechanism remains more or less rigid. Such points are referred to as point-flexures. It has been noted in literature that high relative rotation is associated with point-flexures. In literature we also find a formulation of local constraint on relative rotations to avoid lumped compliance. However it is well known that a global constraint is easier to handle than a local constraint, by a numerical optimization algorithm. The current work presents a way of putting global constraint on relative rotations. This constraint is also simpler to implement since it uses linearized rotation at the center of finite-elements, to compute relative rotations. I show the results obtained by using this constraint oil the following benchmark problems - displacement inverter and gripper.

A fuzzy multistage evolutionary (FUME) clustering technique

n this paper, a multistage evolutionary scheme is proposed for clustering in a large data base, like speech data. This is achieved by clustering a small subset of the entire sample set in each stage and treating the cluster centroids so obtained as samples, together with another subset of samples not considered previously, as input data to the next stage. This is continued till the whole sample set is exhausted. The clustering is accomplished by constructing a fuzzy similarity matrix and using the fuzzy techniques proposed here. The technique is illustrated by an efficient scheme for voiced-unvoiced-silence classification of speech.

Topology-Guided Tessellation Of Quadratic Elements

Topology-based methods have been successfully used for the analysis and visualization of piecewise-linear functions defined on triangle meshes. This paper describes a mechanism for extending these methods to piecewise-quadratic functions defined on triangulations of surfaces. Each triangular patch is tessellated into monotone regions, so that existing algorithms for computing topological representations of piecewise-linear functions may be applied directly to the piecewise-quadratic function. In particular, the tessellation is used for computing the Reeb graph, a topological data structure that provides a succinct representation of level sets of the function.

A Fuzzy Approximation Scheme for Sequential Learning in Pattern Recognition

An adaptive learning scheme, based on a fuzzy approximation to the gradient descent method for training a pattern classifier using unlabeled samples, is described. The objective function defined for the fuzzy ISODATA clustering procedure is used as the loss function for computing the gradient. Learning is based on simultaneous fuzzy decisionmaking and estimation. It uses conditional fuzzy measures on unlabeled samples. An exponential membership function is assumed for each class, and the parameters constituting these membership functions are estimated, using the gradient, in a recursive fashion. The induced possibility of occurrence of each class is useful for estimation and is computed using 1) the membership of the new sample in that class and 2) the previously computed average possibility of occurrence of the same class. An inductive entropy measure is defined in terms of induced possibility distribution to measure the extent of learning. The method is illustrated with relevant examples.

On The Topology Of Manifolds With Positive Isotropic Curvature

We show that a closed orientable Riemannian n-manifold, n >= 5, with positive isotropic curvature and free fundamental group is homeomorphic to the connected sum of copies of Sn-1 x S-1.

A fuzzy logic based dynamic wave model inversion algorithm for canal regulation

A fuzzy logic based centralized control algorithm for irrigation canals is presented. Purpose of the algorithm is to control downstream discharge and water level of pools in the canal, by adjusting discharge release from the upstream end and gates settings. The algorithm is based on the dynamic wave model (Saint-Venant equations) inversion in space, wherein the momentum equation is replaced by a fuzzy rule based model, while retaining the continuity equation in its complete form. The fuzzy rule based model is developed on fuzzification of a new mathematical model for wave velocity, the derivational details of which are given. The advantages of the fuzzy control algorithm, over other conventional control algorithms, are described. It is transparent and intuitive, and no linearizations of the governing equations are involved. Timing of the algorithm and method of computation are explained. It is shown that the tuning is easy and the computations are straightforward. The algorithm provides stable, realistic and robust outputs. The disadvantage of the algorithm is reduced precision in its outputs due to the approximation inherent in the fuzzy logic. Feed back control logic is adopted to eliminate error caused by the system disturbances as well as error caused by the reduced precision in the outputs. The algorithm is tested by applying it to water level control problem in a fictitious canal with a single pool and also in a real canal with a series of pools. It is found that results obtained from the algorithm are comparable to those obtained from conventional control algorithms.

A fuzzy dynamic flood routing model for natural channels

A fuzzy dynamic flood routing model (FDFRM) for natural channels is presented, wherein the flood wave can be approximated to a monoclinal wave. This study is based on modification of an earlier published work by the same authors, where the nature of the wave was of gravity type. Momentum equation of the dynamic wave model is replaced by a fuzzy rule based model, while retaining the continuity equation in its complete form. Hence, the FDFRM gets rid of the assumptions associated with the momentum equation. Also, it overcomes the necessity of calculating friction slope (S-f) in flood routing and hence the associated uncertainties are eliminated. The fuzzy rule based model is developed on an equation for wave velocity, which is obtained in terms of discontinuities in the gradient of flow parameters. The channel reach is divided into a number of approximately uniform sub-reaches. Training set required for development of the fuzzy rule based model for each sub-reach is obtained from discharge-area relationship at its mean section. For highly heterogeneous sub-reaches, optimized fuzzy rule based models are obtained by means of a neuro-fuzzy algorithm. For demonstration, the FDFRM is applied to flood routing problems in a fictitious channel with single uniform reach, in a fictitious channel with two uniform sub-reaches and also in a natural channel with a number of approximately uniform sub-reaches. It is observed that in cases of the fictitious channels, the FDFRM outputs match well with those of an implicit numerical model (INM), which solves the dynamic wave equations using an implicit numerical scheme. For the natural channel, the FDFRM Outputs are comparable to those of the HEC-RAS model.

Structural Damage Detection Using Modal Curvature and Fuzzy Logic

A fuzzy logic system (FLS) with a new sliding window defuzzifier is proposed for structural damage detection using modal curvatures. Changes in the modal curvatures due to damage are fuzzified using Gaussian fuzzy sets and mapped to damage location and size using the FLS. The first four modal vectors obtained from finite element simulations of a cantilever beam are used for identifying the location and size of damage. Parametric studies show that modal curvatures can be used to accurately locate the damage; however, quantifying the size of damage is difficult. Tests with noisy simulated data show that the method detects damage very accurately at different noise levels and when some modal data are missing.

Estimation of fuzzy memberships from histograms

Based on the conclusions drawn in the bijective transformation between possibility and probability, a method is proposed to estimate the fuzzy membership function for pattern recognition purposes. A rational function approximation to the probability density function is obtained from the histogram of a finite (and sometimes very small) number of samples. This function is normalized such that the highest ordinate is one. The parameters representing the rational function are used for classifying the pattern samples based on a max-min decision rule. The method is illustrated with examples.

Damage assessment of structures with uncertainty by using mode-shape curvatures and fuzzy logic

Uncertainties associated with the structural model and measured vibration data may lead to unreliable damage detection. In this paper, we show that geometric and measurement uncertainty cause considerable problem in damage assessment which can be alleviated by using a fuzzy logic-based approach for damage detection. Curvature damage factor (CDF) of a tapered cantilever beam are used as damage indicators. Monte Carlo simulation (MCS) is used to study the changes in the damage indicator due to uncertainty in the geometric properties of the beam. Variation in these CDF measures due to randomness in structural parameter, further contaminated with measurement noise, are used for developing and testing a fuzzy logic system (FLS). Results show that the method correctly identifies both single and multiple damages in the structure. For example, the FLS detects damage with an average accuracy of about 95 percent in a beam having geometric uncertainty of 1 percent COV and measurement noise of 10 percent in single damage scenario. For multiple damage case, the FLS identifies damages in the beam with an average accuracy of about 94 percent in the presence of above mentioned uncertainties. The paper brings together the disparate areas of probabilistic analysis and fuzzy logic to address uncertainty in structural damage detection.

An imprecise fuzzy risk approach for water quality management of a river system

Uncertainty plays an important role in water quality management problems. The major sources of uncertainty in a water quality management problem are the random nature of hydrologic variables and imprecision (fuzziness) associated with goals of the dischargers and pollution control agencies (PCA). Many Waste Load Allocation (WLA)problems are solved by considering these two sources of uncertainty. Apart from randomness and fuzziness, missing data in the time series of a hydrologic variable may result in additional uncertainty due to partial ignorance. These uncertainties render the input parameters as imprecise parameters in water quality decision making. In this paper an Imprecise Fuzzy Waste Load Allocation Model (IFWLAM) is developed for water quality management of a river system subject to uncertainty arising from partial ignorance. In a WLA problem, both randomness and imprecision can be addressed simultaneously by fuzzy risk of low water quality. A methodology is developed for the computation of imprecise fuzzy risk of low water quality, when the parameters are characterized by uncertainty due to partial ignorance. A Monte-Carlo simulation is performed to evaluate the imprecise fuzzy risk of low water quality by considering the input variables as imprecise. Fuzzy multiobjective optimization is used to formulate the multiobjective model. The model developed is based on a fuzzy multiobjective optimization problem with max-min as the operator. This usually does not result in a unique solution but gives multiple solutions. Two optimization models are developed to capture all the decision alternatives or multiple solutions. The objective of the two optimization models is to obtain a range of fractional removal levels for the dischargers, such that the resultant fuzzy risk will be within acceptable limits. Specification of a range for fractional removal levels enhances flexibility in decision making. The methodology is demonstrated with a case study of the Tunga-Bhadra river system in India.

Topology-based denoising of chaos

In this article, we propose a denoising algorithm to denoise a time series y(i) = x(i) + e(i), where {x(i)} is a time series obtained from a time- T map of a uniformly hyperbolic or Anosov flow, and {e(i)} a uniformly bounded sequence of independent and identically distributed (i.i.d.) random variables. Making use of observations up to time n, we create an estimate of x(i) for i<n. We show under typical limiting behaviours of the orbit and the recurrence properties of x(i), the estimation error converges to zero as n tends to infinity with probability 1.

Fuzzy waste load allocation model: a multiobjective approach

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, A, 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.