Search results for: Function Approximation Technique (FAT)

Authors: Anant Oonsivilai, Sanom Saichoomdee

Abstract:

To determine the presence and location of faults in a transmission by the adaptation of protective distance relay based on the measurement of fixed settings as line impedance is achieved by several different techniques. Moreover, a fast, accurate and robust technique for real-time purposes is required for the modern power systems. The appliance of radial basis function neural network in transmission line protection is demonstrated in this paper. The method applies the power system via voltage and current signals to learn the hidden relationship presented in the input patterns. It is experiential that the proposed technique is competent to identify the particular fault direction more speedily. System simulations studied show that the proposed approach is able to distinguish the direction of a fault on a transmission line swiftly and correctly, therefore suitable for the real-time purposes.

Keywords: radial basis function neural network, transmission lines protection, relaying, power system.

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Authors: Ali Abdrhman Ukasha

Abstract:

This paper presented two new efficient algorithms for contour approximation. The proposed algorithm is compared with Ramer (good quality), Triangle (faster) and Trapezoid (fastest) in this work; which are briefly described. Cartesian co-ordinates of an input contour are processed in such a manner that finally contours is presented by a set of selected vertices of the edge of the contour. In the paper the main idea of the analyzed procedures for contour compression is performed. For comparison, the mean square error and signal-to-noise ratio criterions are used. Computational time of analyzed methods is estimated depending on a number of numerical operations. Experimental results are obtained both in terms of image quality, compression ratios, and speed. The main advantages of the analyzed algorithm is small numbers of the arithmetic operations compared to the existing algorithms.

Keywords: Polygonal approximation, Ramer, Triangle and Trapezoid methods.

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Authors: Mohammed A. Abutheraa, David Lester

Abstract:

We show that Chebyshev Polynomials are a practical representation of computable functions on the computable reals. The paper presents error estimates for common operations and demonstrates that Chebyshev Polynomial methods would be more efficient than Taylor Series methods for evaluation of transcendental functions.

Keywords: Approximation Theory, Chebyshev Polynomial, Computable Functions, Computable Real Arithmetic, Integration, Numerical Analysis.

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Authors: Isao Taguchi, Yasuo Sugai

Abstract:

This paper proposes an efficient learning method for the layered neural networks based on the selection of training data and input characteristics of an output layer unit. Comparing to recent neural networks; pulse neural networks, quantum neuro computation, etc, the multilayer network is widely used due to its simple structure. When learning objects are complicated, the problems, such as unsuccessful learning or a significant time required in learning, remain unsolved. Focusing on the input data during the learning stage, we undertook an experiment to identify the data that makes large errors and interferes with the learning process. Our method devides the learning process into several stages. In general, input characteristics to an output layer unit show oscillation during learning process for complicated problems. The multi-stage learning method proposes by the authors for the function approximation problems of classifying learning data in a phased manner, focusing on their learnabilities prior to learning in the multi layered neural network, and demonstrates validity of the multi-stage learning method. Specifically, this paper verifies by computer experiments that both of learning accuracy and learning time are improved of the BP method as a learning rule of the multi-stage learning method. In learning, oscillatory phenomena of a learning curve serve an important role in learning performance. The authors also discuss the occurrence mechanisms of oscillatory phenomena in learning. Furthermore, the authors discuss the reasons that errors of some data remain large value even after learning, observing behaviors during learning.

Keywords: data selection, function approximation problem, multistage leaning, neural network, voluntary oscillation.

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Authors: T. Abdallah, K. Easawi, A. Khalid, S. Negm, H. Talaat

Abstract:

Nanocrystals (NC) alloyed composite CdSxSe1-x(x=0 to 1) have been prepared using the chemical solution deposition technique. The energy band gap of these alloyed nanocrystals of approximately the same size, have been determined by scanning tunneling spectroscopy (STS) technique at room temperature. The values of the energy band gap obtained directly using STS are compared to those measured by optical spectroscopy. Increasing the molar fraction ratio x from 0 to 1 causes clearly observed increase in the band gap of the alloyed composite nanocrystal. Vegard-s law was applied to calculate the parameters of the effective mass approximation (EMA) model and the dimension obtained were compared to the values measured by STM. The good agreement of the calculated and measured values is a direct result of applying Vegard's law in the nanocomposites.

Keywords: Alloy semiconductor nanocrystals, STM.

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Authors: Salah Alrabeei, Mohammad Yousuf

Abstract:

The goal of option pricing theory is to help the investors to manage their money, enhance returns and control their financial future by theoretically valuing their options. However, most of the option pricing models have no analytical solution. Furthermore, not all the numerical methods are efficient to solve these models because they have nonsmoothing payoffs or discontinuous derivatives at the exercise price. In this paper, we solve the American option under jump diffusion models by using efficient time-dependent numerical methods. several techniques are integrated to reduced the overcome the computational complexity. Fast Fourier Transform (FFT) algorithm is used as a matrix-vector multiplication solver, which reduces the complexity from O(M2) into O(M logM). Partial fraction decomposition technique is applied to rational approximation schemes to overcome the complexity of inverting polynomial of matrices. The proposed method is easy to implement on serial or parallel versions. Numerical results are presented to prove the accuracy and efficiency of the proposed method.

Keywords: Integral differential equations, American options, jump–diffusion model, rational approximation.

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Authors: Vikas Tewari, K.R. Pardasani

Abstract:

Calcium is very important for communication among the neurons. It is vital in a number of cell processes such as secretion, cell movement, cell differentiation. To reduce the system of reactiondiffusion equations of [Ca2+] into a single equation, two theories have been proposed one is excess buffer approximation (EBA) other is rapid buffer approximation (RBA). The RBA is more realistic than the EBA as it considers both the mobile and stationary endogenous buffers. It is valid near the mouth of the channel. In this work we have studied the effects of different types of buffers on calcium diffusion under RBA. The novel thing studied is the effect of sodium ions on calcium diffusion. The model has been made realistic by considering factors such as variable [Ca2+], [Na+] sources, sodium-calcium exchange protein(NCX), Sarcolemmal Calcium ATPase pump. The proposed mathematical leads to a system of partial differential equations which has been solved numerically to study the relationships between different parameters such as buffer concentration, buffer disassociation rate, calcium permeability. We have used Forward Time Centred Space (FTCS) approach to solve the system of partial differential equations.

Keywords: rapid buffer approximation, sodium-calcium exchangeprotein, Sarcolemmal Calcium ATPase pump, buffer disassociationrate, forward time centred space.

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Authors: Wisam F. Al-Azzo, Borhanuddin Mohd. Ali

Abstract:

This paper addresses the problem of peak-to-average power ratio (PAPR) in orthogonal frequency division multiplexing (OFDM) systems. It also introduces a new PAPR reduction technique based on adaptive square-rooting (SQRT) companding process. The SQRT process of the proposed technique changes the statistical characteristics of the OFDM output signals from Rayleigh distribution to Gaussian-like distribution. This change in statistical distribution results changes of both the peak and average power values of OFDM signals, and consequently reduces significantly the PAPR. For the 64QAM OFDM system using 512 subcarriers, up to 6 dB reduction in PAPR was achieved by square-rooting technique with fixed degradation in bit error rate (BER) equal to 3 dB. However, the PAPR is reduced at the expense of only -15 dB out-ofband spectral shoulder re-growth below the in-band signal level. The proposed adaptive SQRT technique is superior in terms of BER performance than the original, non-adaptive, square-rooting technique when the required reduction in PAPR is no more than 5 dB. Also, it provides fixed amount of PAPR reduction in which it is not available in the original SQRT technique.

Keywords: complementary cumulative distribution function(CCDF), OFDM, peak-to-average power ratio (PAPR), adaptivesquare-rooting PAPR reduction technique.

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Authors: Ju-Hong Lee, Yi-Lin Shieh

Abstract:

The paper deals with the minimax design of two-channel linear-phase (LP) quadrature mirror filter (QMF) banks using infinite impulse response (IIR) digital all-pass filters (DAFs). Based on the theory of two-channel QMF banks using two IIR DAFs, the design problem is appropriately formulated to result in an appropriate Chebyshev approximation for the desired group delay responses of the IIR DAFs and the magnitude response of the low-pass analysis filter. Through a frequency sampling and iterative approximation method, the design problem can be solved by utilizing a weighted least squares approach. The resulting two-channel QMF banks can possess approximately LP response without magnitude distortion. Simulation results are presented for illustration and comparison.

Keywords: Chebyshev approximation, Digital All-Pass Filter, Quadrature Mirror Filter, Weighted Least Squares.

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Authors: Marina Franulovic, Robert Basan, Bozidar Krizan

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Constitutive modeling of material behavior is becoming increasingly important in prediction of possible failures in highly loaded engineering components, and consequently, optimization of their design. In order to account for large number of phenomena that occur in the material during operation, such as kinematic hardening effect in low cycle fatigue behavior of steels, complex nonlinear material models are used ever more frequently, despite of the complexity of determination of their parameters. As a method for the determination of these parameters, genetic algorithm is good choice because of its capability to provide very good approximation of the solution in systems with large number of unknown variables. For the application of genetic algorithm to parameter identification, inverse analysis must be primarily defined. It is used as a tool to fine-tune calculated stress-strain values with experimental ones. In order to choose proper objective function for inverse analysis among already existent and newly developed functions, the research is performed to investigate its influence on material behavior modeling.

Keywords: Genetic algorithm, kinematic hardening, material model, objective function

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Authors: Wee Leong Son, Hasmayadi Abdul Majid, Rohana Musa

Abstract:

This paper study the segmented split capacitor Digital-to-Analog Converter (DAC) implemented in a differentialtype 12-bit Successive Approximation Analog-to-Digital Converter (SA-ADC). The series capacitance split array method employed as it reduced the total area of the capacitors required for high resolution DACs. A 12-bit regular binary array structure requires 2049 unit capacitors (Cs) while the split array needs 127 unit Cs. These results in the reduction of the total capacitance and power consumption of the series split array architectures as to regular binary-weighted structures. The paper will show the 12-bit DAC series split capacitor with 4-bit thermometer coded DAC architectures as well as the simulation and measured results.

Keywords: Successive Approximation Register Analog-to- Digital Converter, SAR ADC, Low voltage ADC.

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Authors: Y. Z. Wu, Z. Dong, S. K. You

Abstract:

Global approximation using metamodel for complex mathematical function or computer model over a large variable domain is often needed in sensibility analysis, computer simulation, optimal control, and global design optimization of complex, multiphysics systems. To overcome the limitations of the existing response surface (RS), surrogate or metamodel modeling methods for complex models over large variable domain, a new adaptive and regressive RS modeling method using quadratic functions and local area model improvement schemes is introduced. The method applies an iterative and Latin hypercube sampling based RS update process, divides the entire domain of design variables into multiple cells, identifies rougher cells with large modeling error, and further divides these cells along the roughest dimension direction. A small number of additional sampling points from the original, expensive model are added over the small and isolated rough cells to improve the RS model locally until the model accuracy criteria are satisfied. The method then combines local RS cells to regenerate the global RS model with satisfactory accuracy. An effective RS cells sorting algorithm is also introduced to improve the efficiency of model evaluation. Benchmark tests are presented and use of the new metamodeling method to replace complex hybrid electrical vehicle powertrain performance model in vehicle design optimization and optimal control are discussed.

Keywords: Global approximation, polynomial response surface, domain decomposition, domain combination, multiphysics modeling, hybrid powertrain optimization

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Authors: Zhihua Yang, Lihua Yang

Abstract:

This paper makes a detailed analysis regarding the definition of the intrinsic mode function and proves that Condition 1 of the intrinsic mode function can really be deduced from Condition 2. Finally, an improved definition of the intrinsic mode function is given.

Keywords: Empirical Mode Decomposition (EMD), Hilbert-Huang transform(HHT), Intrinsic Mode Function(IMF).

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Authors: Durvesh Kumar Verma, P. N. Agrawal

Abstract:

In the present paper, we consider the generalized form of Baskakov Durrmeyer operators to study the rate of convergence, in simultaneous approximation for functions having derivatives of bounded variation.

Keywords: Bounded variation, Baskakov-Durrmeyer operators, simultaneous approximation, rate of convergence.

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Authors: Sarisa Pinkham, Kanyarat Bussaban

Abstract:

The research aims to approximate the amount of daily rainfall by using a pixel value data approach. The daily rainfall maps from the Thailand Meteorological Department in period of time from January to December 2013 were the data used in this study. The results showed that this approach can approximate the amount of daily rainfall with RMSE=3.343.

Keywords: Daily rainfall, Image processing, Approximation, Pixel value data.

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Authors: Tsunwai Gary Yip

Abstract:

Phase locked loops in 10 Gb/s and faster data links are low phase noise devices. Characterization of their phase jitter transfer functions is difficult because the intrinsic noise of the PLLs is comparable to the phase noise of the reference clock signal. The problem is solved by using a linear model to account for the intrinsic noise. This study also introduces a novel technique for measuring the transfer function. It involves the use of the reference clock as a source of wideband excitation, in contrast to the commonly used sinusoidal excitations at discrete frequencies. The data reported here include the intrinsic noise of a PLL for 10 Gb/s links and the jitter transfer function of a PLL for 12.8 Gb/s links. The measured transfer function suggests that the PLL responded like a second order linear system to a low noise reference clock.

Keywords: Intrinsic phase noise, jitter in data link, PLL jitter transfer function, high speed clocking in electronic circuit

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Authors: Samir Abdelhamid, Noureddine Bouguechal

Abstract:

We report in this paper the procedure of a system of automatic speech recognition based on techniques of the dynamic programming. The technique of temporal retiming is a technique used to synchronize between two forms to compare. We will see how this technique is adapted to the field of the automatic speech recognition. We will expose, in a first place, the theory of the function of retiming which is used to compare and to adjust an unknown form with a whole of forms of reference constituting the vocabulary of the application. Then we will give, in the second place, the various algorithms necessary to their implementation on machine. The algorithms which we will present were tested on part of the corpus of words in Arab language Arabdic-10 [4] and gave whole satisfaction. These algorithms are effective insofar as we apply them to the small ones or average vocabularies.

Keywords: Continuous speech recognition, temporal retiming, phonetic decoding, algorithms, vocal signal, dynamic programming.

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Authors: S. Bennoud, M. Zergoug

Abstract:

The numerical simulation of electromagnetic interactions is still a challenging problem, especially in problems that result in fully three dimensional mathematical models.

The goal of this work is to use mathematical modeling to characterize the reliability and capacity of eddy current technique to detect and characterize defects embedded in aeronautical in-service pieces.

The finite element method is used for describing the eddy current technique in a mathematical model by the prediction of the eddy current interaction with defects. However, this model is an approximation of the full Maxwell equations.

In this study, the analysis of the problem is based on a three dimensional finite element model that computes directly the electromagnetic field distortions due to defects.

Keywords: Eddy current, Finite element method, Non destructive testing, Numerical simulations.

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Authors: Hasanifard Goran, Habib Nejad Korayem Moharam, Nikoobin Amin

Abstract:

A new robust nonlinear control scheme of a manipulator is proposed in this paper which is robust against modeling errors and unknown disturbances. It is based on the principle of variable structure control, with sliding mode control (SMC) method. The variable structure control method is a robust method that appears to be well suited for robotic manipulators because it requers only bounds on the robotic arm parameters. But there is no single systematic procedure that is guaranteed to produce a suitable control law. Also, to reduce chattring of the control signal, we replaced the sgn function in the control law by a continuous approximation such as tangant function. We can compute the maximum load with regard to applied torque into joints. The effectivness of the proposed approach has been evaluated analitically demonstrated through computer simulations for the cases of variable load and robot arm parameters.

Keywords: Variable structure control, robust control, switching surface, robot manipulator.

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Authors: Jyotsna Singh, Parul Garg, Alok Nath De

Abstract:

In this paper, we present a non-blind technique of adding the watermark to the Fourier spectral components of audio signal in a way such that the modified amplitude does not exceed the maximum amplitude spread (MAS). This MAS is due to individual Discrete fourier transform (DFT) coefficients in that particular frame, which is derived from the Energy Spreading function given by Schroeder. Using this technique one can store double the information within a given frame length i.e. overriding the watermark on the host of equal length with least perceptual distortion. The watermark is uniformly floating on the DFT components of original signal. This helps in detecting any intentional manipulations done on the watermarked audio. Also, the scheme is found robust to various signal processing attacks like presence of multiple watermarks, Additive white gaussian noise (AWGN) and mp3 compression.

Keywords: Discrete Fourier Transform, Spreading Function, Watermark, Pseudo Noise Sequence, Spectral Masking Effect

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Authors: M. Asato, C. Liu, N. Fujima, T. Hoshino, Y. Chen, T. Mohri

Abstract:

We study the temperature dependence of the interaction energies (IEs) of X (=Ru, Rh) impurities in Pd, due to the Fermi-Dirac (FD) distribution and the thermal vibration effect by the Debye-Grüneisen model. The n-body (n=2~4) IEs among X impurities in Pd, being used to calculate the internal energies in the free energies of the Pd-rich PdX alloys, are determined uniquely and successively from the lower-order to higher-order, by the full-potential Korringa-Kohn-Rostoker Green’s function method (FPKKR), combined with the generalized gradient approximation in the density functional theory. We found that the temperature dependence of IEs due to the FD distribution, being usually neglected, is very important to reproduce the X-concentration dependence of the observed solvus temperatures of the Pd-rich PdX (X=Ru, Rh) alloys.

Keywords: Full-potential KKR-Green’s function method, Fermi-Dirac distribution, GGA, phase diagram of Pd-rich PdX (X=Ru, Rh) alloys, thermal vibration effect.

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Authors: K. C. Meher, R. K. Swain, C. K. Chanda

Abstract:

This paper gives an intuition regarding glowworm swarm optimization (GSO) technique to solve dynamic economic dispatch (DED) problems of thermal generating units. The objective of the problem is to schedule optimal power generation of dedicated thermal units over a specific time band. In this study, Glowworm swarm optimization technique enables a swarm of agents to split into subgroup, exhibit simultaneous taxis towards each other and rendezvous at multiple optima (not necessarily equal) of a given multimodal function. The feasibility of the GSO method has been tested on ten-unit-test systems where the power balance constraints, operating limits, valve point effects, and ramp rate limits are taken into account. The results obtained by the proposed technique are compared with other heuristic techniques. The results show that GSO technique is capable of producing better results.

Keywords: Dynamic economic dispatch, Glowworm swarm optimization, Luciferin, Valve–point loading effect, Ramp rate limits.

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Authors: K. Elleuch, A. Chaari

Abstract:

This paper deals with modeling and parameter identification of nonlinear systems described by Hammerstein model having Piecewise nonlinear characteristics such as Dead-zone nonlinearity characteristic. The simultaneous use of both an easy decomposition technique and the triangular basis functions leads to a particular form of Hammerstein model. The approximation by using Triangular basis functions for the description of the static nonlinear block conducts to a linear regressor model, so that least squares techniques can be used for the parameter estimation. Singular Values Decomposition (SVD) technique has been applied to separate the coupled parameters. The proposed approach has been efficiently tested on academic examples of simulation.

Keywords: Identification, Hammerstein model, Piecewisenonlinear characteristic, Dead-zone nonlinearity, Triangular basisfunctions, Singular Values Decomposition

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Authors: Maumita Bhattacharya

Abstract:

Evolutionary Algorithms are population-based, stochastic search techniques, widely used as efficient global optimizers. However, many real life optimization problems often require finding optimal solution to complex high dimensional, multimodal problems involving computationally very expensive fitness function evaluations. Use of evolutionary algorithms in such problem domains is thus practically prohibitive. An attractive alternative is to build meta models or use an approximation of the actual fitness functions to be evaluated. These meta models are order of magnitude cheaper to evaluate compared to the actual function evaluation. Many regression and interpolation tools are available to build such meta models. This paper briefly discusses the architectures and use of such meta-modeling tools in an evolutionary optimization context. We further present two evolutionary algorithm frameworks which involve use of meta models for fitness function evaluation. The first framework, namely the Dynamic Approximate Fitness based Hybrid EA (DAFHEA) model [14] reduces computation time by controlled use of meta-models (in this case approximate model generated by Support Vector Machine regression) to partially replace the actual function evaluation by approximate function evaluation. However, the underlying assumption in DAFHEA is that the training samples for the metamodel are generated from a single uniform model. This does not take into account uncertain scenarios involving noisy fitness functions. The second model, DAFHEA-II, an enhanced version of the original DAFHEA framework, incorporates a multiple-model based learning approach for the support vector machine approximator to handle noisy functions [15]. Empirical results obtained by evaluating the frameworks using several benchmark functions demonstrate their efficiency

Keywords: Meta model, Evolutionary algorithm, Stochastictechnique, Fitness function, Optimization, Support vector machine.

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Authors: Swapnoneel Roy, Ashok Kumar Thakur, Minhazur Rahman

Abstract:

The Block Sorting problem is to sort a given permutation moving blocks. A block is defined as a substring of the given permutation, which is also a substring of the identity permutation. Block Sorting has been proved to be NP-Hard. Until now two different 2-Approximation algorithms have been presented for block sorting. These are the best known algorithms for Block Sorting till date. In this work we present a different characterization of Block Sorting in terms of a transposition cycle graph. Then we suggest a heuristic, which we show to exhibit a 2-approximation performance guarantee for most permutations.

Keywords: Block Sorting, Optical Character Recognition, Genome Rearrangements, Sorting Primitives, ApproximationAlgorithms

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Authors: Xia Cui, Guang-wei Yuan, Jing-yan Yue

Abstract:

A Picard-Newton iteration method is studied to accelerate the numerical solution procedure of a class of two-dimensional nonlinear coupled parabolic-hyperbolic system. The Picard-Newton iteration is designed by adding higher-order terms of small quantity to an existing Picard iteration. The discrete functional analysis and inductive hypothesis reasoning techniques are used to overcome difficulties coming from nonlinearity and coupling, and theoretical analysis is made for the convergence and approximation properties of the iteration scheme. The Picard-Newton iteration has a quadratic convergent ratio, and its solution has second order spatial approximation and first order temporal approximation to the exact solution of the original problem. Numerical tests verify the results of the theoretical analysis, and show the Picard-Newton iteration is more efficient than the Picard iteration.

Keywords: Nonlinearity, iterative acceleration, coupled parabolic hyperbolic system, quadratic convergence, numerical analysis.

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Authors: Vikas Tewari, Shivendra Tewari, K. R. Pardasani

Abstract:

Calcium is a vital second messenger used in signal transduction. Calcium controls secretion, cell movement, muscular contraction, cell differentiation, ciliary beating and so on. Two theories have been used to simplify the system of reaction-diffusion equations of calcium into a single equation. One is excess buffer approximation (EBA) which assumes that mobile buffer is present in excess and cannot be saturated. The other is rapid buffer approximation (RBA), which assumes that calcium binding to buffer is rapid compared to calcium diffusion rate. In the present work, attempt has been made to develop a model for calcium diffusion under excess buffer approximation in neuron cells. This model incorporates the effect of [Na+] influx on [Ca2+] diffusion,variable calcium and sodium sources, sodium-calcium exchange protein, Sarcolemmal Calcium ATPase pump, sodium and calcium channels. The proposed mathematical model leads to a system of partial differential equations which have been solved numerically using Forward Time Centered Space (FTCS) approach. The numerical results have been used to study the relationships among different types of parameters such as buffer concentration, association rate, calcium permeability.

Keywords: Excess buffer approximation, Na+ influx, sodium calcium exchange protein, sarcolemmal calcium atpase pump, forward time centred space.

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Authors: Maharavo Randrianarivony

Abstract:

The length  of a given rational B'ezier curve is efficiently estimated. Since a rational B'ezier function is nonlinear, it is usually impossible to evaluate its length exactly. The length is approximated by using subdivision and the accuracy of the approximation n is investigated. In order to improve the efficiency, adaptivity is used with some length estimator. A rigorous theoretical analysis of the rate of convergence of n to  is given. The required number of subdivisions to attain a prescribed accuracy is also analyzed. An application to CAD parametrization is briefly described. Numerical results are reported to supplement the theory.

Keywords: Adaptivity, Length, Parametrization, Rational Bezier

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Authors: Dimitar Taskovski, Sofija Bogdanova, Momcilo Bogdanov

Abstract:

We present a low frequency watermarking method adaptive to image content. The image content is analyzed and properties of HVS are exploited to generate a visual mask of the same size as the approximation image. Using this mask we embed the watermark in the approximation image without degrading the image quality. Watermark detection is performed without using the original image. Experimental results show that the proposed watermarking method is robust against most common image processing operations, which can be easily implemented and usually do not degrade the image quality.

Keywords: Blind, digital watermarking, low frequency, visualmask.

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Authors: Miloš Šeda

Abstract:

In this paper, we deal with the Steiner tree problem (STP) on a graph in which a fuzzy number, instead of a real number, is assigned to each edge. We propose a modification of the shortest paths approximation based on the fuzzy shortest paths (FSP) evaluations. Since a fuzzy min operation using the extension principle leads to nondominated solutions, we propose another approach to solving the FSP using Cheng's centroid point fuzzy ranking method.

Keywords: Steiner tree, single shortest path problem, fuzzyranking, binary heap, priority queue.

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