In many decision situations, decision-makers face a kind of complex problems. In these decision-making problems, different types of fuzzy numbers are defined and, have multiple types of membership functions. So, we need a standard form to formulate uncertain numbers in the problem. Shadowed fuzzy numbers are considered granule numbers which approximate different types and different forms of fuzzy numbers. In this paper, a new ranking approach for shadowed fuzzy numbers is developed using value, ambiguity and fuzziness for shadowed fuzzy numbers. The new ranking method has been compared with other existing approaches through numerical examples. Also, the new method is applied to a hybrid multi-attribute decision making problem in which the evaluations of alternatives are expressed with different types of uncertain numbers. The comparative study for the results of different examples illustrates the reliability of the new method.
The concept of an intuitionistic fuzzy number (IFN) is of importance for representing an ill-known quantity. Ranking fuzzy numbers plays a very important role in the decision process, data analysis and applications. The concept of an IFN is of importance for quantifying an ill-known quantity. Ranking of intuitionistic fuzzy numbers plays a vital role in decision making and linear programming problems. Also, ranking of intuitionistic fuzzy numbers is a very difficult problem. In this paper, a new method for ranking intuitionistic fuzzy number is developed by means of magnitude for different forms of intuitionistic fuzzy numbers. In Particular ranking is done for trapezoidal intuitionistic fuzzy numbers, triangular intuitionistic fuzzy numbers, symmetric trapezoidal intuitionistic fuzzy numbers, and symmetric triangular intuitionistic fuzzy numbers. Numerical examples are illustrated for all the defined different forms of intuitionistic fuzzy numbers. Finally some comparative numerical examples are illustrated to express the advantage of the proposed method.
The document provides an overview of the Foundation Course for the Actuarial Common Entrance Test (ACET). It covers 8 chapters on mathematical topics including notation, numerical methods, functions, algebra, calculus, and vectors/matrices. It recommends reviewing areas of weakness and provides additional practice questions. When studying core technical subjects, the Foundation Course can be used as a reference for mathematical concepts requiring review.
The document presents a method for solving fuzzy assignment problems using triangular and trapezoidal fuzzy numbers. It formulates the fuzzy assignment problem into a crisp linear programming problem that can be solved using the Hungarian method. The paper also uses Robust's ranking method to transform fuzzy costs into crisp values, allowing conventional solution methods to be applied. It aims to provide a more realistic approach to assignment problems by considering costs as fuzzy numbers rather than deterministic values.
AN ARITHMETIC OPERATION ON HEXADECAGONAL FUZZY NUMBERijfls
In this paper, a new form of fuzzy number named as Hexadecagonal Fuzzy Number is introduced as it is not
possible to restrict the membership function to any specific form. The cut of Hexadecagonal fuzzy number is defined and basic arithmetic operations are performed using interval arithmetic of cut and illustrated with numerical examples.
This document provides instructions for Homework 1 for the course 6.867. It is due on September 28 and will be 10% off for each day late. The homework involves exploring bias-variance tradeoffs in estimating the mean of different distributions from sample data. It provides questions to answer about maximum likelihood estimators for the mean of uniform distributions on intervals of different lengths. It also covers Bayesian estimation of probabilities for a "thick coin" that can land on heads, tails, or edge. Finally, it includes questions on Gaussian distributions, analyzing a presidential debate poll, and decision theory concepts.
This document provides an overview of machine learning concepts including supervised and unsupervised learning algorithms. It describes naive Bayes classifiers which use probabilistic models to classify data based on features. It also describes k-means clustering which groups unlabeled data into k clusters by minimizing distances between data points and assigned cluster centroids. The document provides examples of applying these algorithms to tasks like document and image classification, customer segmentation, and grouping related news articles.
This document proposes methods for enhancing the visualization of concept lattices generated through formal concept analysis. It discusses extracting tree structures from concept lattices to improve readability. Various criteria are proposed for selecting parent concepts when transforming a lattice into a tree, including stability, support, shared attributes between concepts, and confidence. Visualization techniques like coloring nodes based on criteria values and sizing nodes by extent/intent ratios are also suggested to aid interpretation. The methods aim to make larger datasets more explorable by extracting simpler tree representations while preserving essential lattice features and structure.
umerical algorithm for solving second order nonlinear fuzzy initial value pro...IJECEIAES
This document presents a numerical algorithm for solving second-order nonlinear fuzzy initial value problems (FIVPs). The algorithm is based on reformulating the fifth-order Runge-Kutta method with six stages (RK56) to make it suitable for solving FIVPs. RK56 is used to reduce the original nonlinear second-order FIVP into a system of coupled first-order FIVPs. The algorithm is demonstrated on a test nonlinear second-order FIVP. Results show the RK56 technique is efficient and simple to implement while satisfying fuzzy solution properties. This is the first attempt to use RK56 to solve nonlinear second-order FIVPs.
The concept of an intuitionistic fuzzy number (IFN) is of importance for representing an ill-known quantity. Ranking fuzzy numbers plays a very important role in the decision process, data analysis and applications. The concept of an IFN is of importance for quantifying an ill-known quantity. Ranking of intuitionistic fuzzy numbers plays a vital role in decision making and linear programming problems. Also, ranking of intuitionistic fuzzy numbers is a very difficult problem. In this paper, a new method for ranking intuitionistic fuzzy number is developed by means of magnitude for different forms of intuitionistic fuzzy numbers. In Particular ranking is done for trapezoidal intuitionistic fuzzy numbers, triangular intuitionistic fuzzy numbers, symmetric trapezoidal intuitionistic fuzzy numbers, and symmetric triangular intuitionistic fuzzy numbers. Numerical examples are illustrated for all the defined different forms of intuitionistic fuzzy numbers. Finally some comparative numerical examples are illustrated to express the advantage of the proposed method.
The document provides an overview of the Foundation Course for the Actuarial Common Entrance Test (ACET). It covers 8 chapters on mathematical topics including notation, numerical methods, functions, algebra, calculus, and vectors/matrices. It recommends reviewing areas of weakness and provides additional practice questions. When studying core technical subjects, the Foundation Course can be used as a reference for mathematical concepts requiring review.
The document presents a method for solving fuzzy assignment problems using triangular and trapezoidal fuzzy numbers. It formulates the fuzzy assignment problem into a crisp linear programming problem that can be solved using the Hungarian method. The paper also uses Robust's ranking method to transform fuzzy costs into crisp values, allowing conventional solution methods to be applied. It aims to provide a more realistic approach to assignment problems by considering costs as fuzzy numbers rather than deterministic values.
AN ARITHMETIC OPERATION ON HEXADECAGONAL FUZZY NUMBERijfls
In this paper, a new form of fuzzy number named as Hexadecagonal Fuzzy Number is introduced as it is not
possible to restrict the membership function to any specific form. The cut of Hexadecagonal fuzzy number is defined and basic arithmetic operations are performed using interval arithmetic of cut and illustrated with numerical examples.
This document provides instructions for Homework 1 for the course 6.867. It is due on September 28 and will be 10% off for each day late. The homework involves exploring bias-variance tradeoffs in estimating the mean of different distributions from sample data. It provides questions to answer about maximum likelihood estimators for the mean of uniform distributions on intervals of different lengths. It also covers Bayesian estimation of probabilities for a "thick coin" that can land on heads, tails, or edge. Finally, it includes questions on Gaussian distributions, analyzing a presidential debate poll, and decision theory concepts.
This document provides an overview of machine learning concepts including supervised and unsupervised learning algorithms. It describes naive Bayes classifiers which use probabilistic models to classify data based on features. It also describes k-means clustering which groups unlabeled data into k clusters by minimizing distances between data points and assigned cluster centroids. The document provides examples of applying these algorithms to tasks like document and image classification, customer segmentation, and grouping related news articles.
This document proposes methods for enhancing the visualization of concept lattices generated through formal concept analysis. It discusses extracting tree structures from concept lattices to improve readability. Various criteria are proposed for selecting parent concepts when transforming a lattice into a tree, including stability, support, shared attributes between concepts, and confidence. Visualization techniques like coloring nodes based on criteria values and sizing nodes by extent/intent ratios are also suggested to aid interpretation. The methods aim to make larger datasets more explorable by extracting simpler tree representations while preserving essential lattice features and structure.
umerical algorithm for solving second order nonlinear fuzzy initial value pro...IJECEIAES
This document presents a numerical algorithm for solving second-order nonlinear fuzzy initial value problems (FIVPs). The algorithm is based on reformulating the fifth-order Runge-Kutta method with six stages (RK56) to make it suitable for solving FIVPs. RK56 is used to reduce the original nonlinear second-order FIVP into a system of coupled first-order FIVPs. The algorithm is demonstrated on a test nonlinear second-order FIVP. Results show the RK56 technique is efficient and simple to implement while satisfying fuzzy solution properties. This is the first attempt to use RK56 to solve nonlinear second-order FIVPs.
Fuzzy Logic And Application Jntu Model Paper{Www.Studentyogi.Com}guest3f9c6b
This document contains an exam for a course on Fuzzy Logic and Applications. It includes 8 questions covering topics such as operations on crisp and fuzzy sets using Venn diagrams, fuzzy relations, membership functions, fuzzy logic connectives, defuzzification methods, and decision making under fuzzy conditions. Students are instructed to answer any 5 of the 8 questions.
The document outlines the aims, objectives, and syllabus for the Mathematics HL (1st exams 2014) course. It includes:
- 10 aims of the course focused on developing mathematical skills, understanding, problem solving, and appreciation of mathematics.
- 6 objectives centered around demonstrating knowledge and understanding of mathematical concepts, problem solving, communication, use of technology, reasoning, and inquiry approaches.
- The syllabus is divided into 8 core topics (Algebra, Functions and equations, Circular functions and trigonometry, Vectors, Statistics and probability, Calculus, and 2 optional topics (Statistics and probability, Sets, relations and groups) that provide 48 hours of instruction each.
A Comprehensive Overview of Clustering Algorithms in Pattern RecognitionIOSR Journals
This document provides an overview of clustering algorithms used in pattern recognition, including K-means clustering and hierarchical clustering (agglomerative and divisive). It describes the basic steps of each algorithm, provides examples, and compares their advantages and disadvantages. K-means clustering partitions data into K groups based on feature similarity, while hierarchical clustering creates nested clusters based on distance metrics. The document concludes that the appropriate technique depends on factors like prior knowledge of clusters and whether a sequential or flat structure is needed.
Machine learning algorithms infer unknowns from knowns through statistical inference. Common machine learning applications include spam identification, handwriting recognition, image recognition, speech recognition, recommendation systems, and climate modeling. These applications can be grouped into supervised learning (classification and regression) and unsupervised learning (clustering, density estimation, dimensionality reduction). Generative models model the joint distribution of all variables, while discriminative models model only the target variables conditional on observed variables. K-nearest neighbors is a discriminative classification algorithm that classifies a new data point based on the labels of its k nearest neighbors.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
1. The document presents a methodology for recognizing isolated handwritten Devanagari numerals using structural and statistical features.
2. Key features extracted include whether the numeral has openings on the left, right, above or below, and the number of horizontal and vertical crossings.
3. The methodology achieves an average accuracy of 96.8% on a dataset of 500 numeral images collected from various individuals. Accuracy is highest for numerals 0, 6, 8 and 10 at 100%, while some similar numerals like 3 and 2 see more errors.
SYMMETRICAL WEIGHTED SUBSPACE HOLISTIC APPROACH FOR EXPRESSION RECOGNITIONijcsit
This document proposes a new method called Symmetrical Weighted 2D Principal Component Analysis (SW2DPCA) for facial expression recognition. SW2DPCA aims to reduce the dimensionality of the feature space extracted from face images using Gabor filters, while removing redundant information. It does this by decomposing the training images into odd and even symmetrical components, and applying weighted PCA to equalize the variance of principal components. The method is tested on the JAFFE database, achieving a recognition accuracy of 95.24% - higher than the 2DPCA baseline method.
Estimation of Distribution Algorithms (EDAs) constitute a powerful evolutionary algorithm for solving continuous and combinatorial optimization problems. Based on machine learning techniques, at each generation, EDAs estimate a joint probability distribution associated with the set of most promising individuals, trying to explicitly express the interrelations between the different variables of the problem. Based on this general framework, EDAs have proved to be very competitive for solving combinatorial and continuous optimization problems. In this talk, we propose developing EDAs by introducing probability models defined exclusively on the space of feasible solutions. In this sense, we give a first approach by taking the Graph Partitioning Problem (GPP) as a case of study, and present a probabilistic model defined exclusively on the feasible region of solutions: a square lattice probability model.
Handling missing data with expectation maximization algorithmLoc Nguyen
Expectation maximization (EM) algorithm is a powerful mathematical tool for estimating parameter of statistical models in case of incomplete data or hidden data. EM assumes that there is a relationship between hidden data and observed data, which can be a joint distribution or a mapping function. Therefore, this implies another implicit relationship between parameter estimation and data imputation. If missing data which contains missing values is considered as hidden data, it is very natural to handle missing data by EM algorithm. Handling missing data is not a new research but this report focuses on the theoretical base with detailed mathematical proofs for fulfilling missing values with EM. Besides, multinormal distribution and multinomial distribution are the two sample statistical models which are concerned to hold missing values.
The document discusses classification algorithms in machine learning. It introduces classification problems using the Iris flower dataset as an example, which contains measurements of Iris flowers to classify them into three species. It then discusses two classic classification algorithms - logistic regression and Gaussian discriminant analysis. Logistic regression uses a sigmoid function to generate predictions, while Gaussian discriminant analysis assumes a Gaussian distribution of the data. The document also demonstrates an application of these algorithms to classify handwritten digits.
Using Alpha-cuts and Constraint Exploration Approach on Quadratic Programming...TELKOMNIKA JOURNAL
In this paper, we propose a computational procedure to find the optimal solution of quadratic programming
problems by using fuzzy -cuts and constraint exploration approach. We solve the problems in
the original form without using any additional information such as Lagrange’s multiplier, slack, surplus and
artificial variable. In order to find the optimal solution, we divide the calculation in two stages. In the first
stage, we determine the unconstrained minimization of the quadratic programming problem (QPP) and check
its feasibility. By unconstrained minimization we identify the violated constraints and focus our searching in
these constraints. In the second stage, we explored the feasible region along side the violated constraints
until the optimal point is achieved. A numerical example is included in this paper to illustrate the capability of
-cuts and constraint exploration to find the optimal solution of QPP.
PREDICTIVE EVALUATION OF THE STOCK PORTFOLIO PERFORMANCE USING FUZZY CMEANS A...ijfls
The aim of this paper is to investigate the trend of the return of a portfolio formed randomly or for any
specific technique. The approach is made using two techniques fuzzy: fuzzy c-means (FCM) algorithm and
the fuzzy transform, where the rules used at fuzzy transform arise from the application of the FCM
algorithm. The results show that the proposed methodology is able to predict the trend of the return of a
stock portfolio, as well as the tendency of the market index. Real data of the financial market are used from
2004 until 2007.
This chapter discusses methods for constructing confidence intervals for differences between population means and proportions in various sampling situations. It covers confidence intervals for the difference between two dependent or paired sample means, two independent sample means when population variances are known or unknown, and two independent population proportions. It also addresses determining required sample sizes to estimate a mean or proportion within a specified margin of error.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
The document outlines a paper on Bayesian linear models. It introduces a simple example of a linear model with exchangeable priors. It then presents the general Bayesian linear model and theorems for the posterior distribution given multiple stages of priors. It applies this to an experimental design setting, deriving Bayes estimates that shrink treatment and block effects towards zero based on their variances.
ANALYTICAL FORMULATIONS FOR THE LEVEL BASED WEIGHTED AVERAGE VALUE OF DISCRET...ijsc
In fuzzy decision-making processes based on linguistic information, operations on discrete fuzzy numbers
are commonly performed. Aggregation and defuzzification operations are some of these often used
operations. Many aggregation and defuzzification operators produce results independent to the decisionmaker’s
strategy. On the other hand, the Weighted Average Based on Levels (WABL) approach can take
into account the level weights and the decision maker's "optimism" strategy. This gives flexibility to the
WABL operator and, through machine learning, can be trained in the direction of the decision maker's
strategy, producing more satisfactory results for the decision maker. However, in order to determine the
WABL value, it is necessary to calculate some integrals. In this study, the concept of WABL for discrete
trapezoidal fuzzy numbers is investigated, and analytical formulas have been proven to facilitate the
calculation of WABL value for these fuzzy numbers. Trapezoidal and their special form, triangular fuzzy
numbers, are the most commonly used fuzzy number types in fuzzy modeling, so in this study, such numbers
have been studied. Computational examples explaining the theoretical results have been performed.
Teaching Mathematics Concepts via Computer Algebra Systemsinventionjournals
Most articles examine computer algebra systems (CAS) as they relate to the teaching and
learning of mathematics from advantages to disadvantages. This paper will explore junior undergraduate
students’ ability to solve distinguish tricky examples using various CAS technologies. Additionally, an
understanding for how CAS technologies are adopted and applied in professional environments is valuable,
both in guiding improvements to these tools and identifying new tools which can aid mathematician
In this paper, Assignment problem with crisp, fuzzy and intuitionistic fuzzy numbers as cost coefficients is investigated. In conventional assignment problem, cost is always certain. This paper develops an approach to solve a mixed intuitionistic fuzzy assignment problem where cost is considered real, fuzzy and an intuitionistic fuzzy numbers. Ranking procedure of Annie Varghese and Sunny Kuriakose [4] is used to transform the mixed intuitionistic fuzzy assignment problem into a crisp one so that the conventional method may be applied to solve the assignment problem. The method is illustrated by a numerical example. The proposed method is very simple and easy to understand. Numerical examples show that an intuitionistic fuzzy ranking method offers an effective tool for handling an intuitionistic fuzzy assignment problem.
A New Approach for Ranking Shadowed Fuzzy Numbers and its Application IJCSITJournal2
n many decision situations, decision-makers face a kind of complex problems. In these decision-making
problems, different types of fuzzy numbers are defined and, have multiple types of membership functions.
So, we need a standard form to formulate uncertain numbers in the problem. Shadowed fuzzy numbers are
considered granule numbers which approximate different types and different forms of fuzzy numbers. In
this paper, a new ranking approach for shadowed fuzzy numbers is developed using value, ambiguity and
fuzziness for shadowed fuzzy numbers. The new ranking method has been compared with other existing
approaches through numerical examples. Also, the new method is applied to a hybrid multi-attribute
decision making problem in which the evaluations of alternatives are expressed with different types of
uncertain numbers. The comparative study for the results of different examples illustrates the reliability of
the new method.
Transportation Problem with Pentagonal Intuitionistic Fuzzy Numbers Solved Us...IJERA Editor
This paper presents a solution methodology for transportation problem in an intuitionistic fuzzy environment in
which cost are represented by pentagonal intuitionistic fuzzy numbers. Transportation problem is a particular
class of linear programming, which is associated with day to day activities in our real life. It helps in solving
problems on distribution and transportation of resources from one place to another. The objective is to satisfy
the demand at destination from the supply constraints at the minimum transportation cost possible. The problem
is solved using a ranking technique called Accuracy function for pentagonal intuitionistic fuzzy numbers and
Russell’s Method
A New Hendecagonal Fuzzy Number For Optimization Problemsijtsrd
A new fuzzy number called Hendecagonal fuzzy number and its membership function is introduced, which is used to represent the uncertainty with eleven points. The fuzzy numbers with ten ordinates exists in literature. The aim of this paper is to define Hendecagonal fuzzy number and its arithmetic operations. Also a direct approach is proposed to solve fuzzy assignment problem (FAP) and fuzzy travelling salesman (FTSP) in which the cost and distance are represented by Hendecagonal fuzzy numbers. Numerical example shows the effectiveness of the proposed method and the Hendecagonal fuzzy number M. Revathi | Dr. M. Valliathal | R. Saravanan | Dr. K. Rathi"A New Hendecagonal Fuzzy Number For Optimization Problems" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-1 | Issue-5 , August 2017, URL: http://www.ijtsrd.com/papers/ijtsrd2258.pdf http://www.ijtsrd.com/mathemetics/applied-mathamatics/2258/a-new-hendecagonal-fuzzy-number-for-optimization-problems/m-revathi
Fuzzy Logic And Application Jntu Model Paper{Www.Studentyogi.Com}guest3f9c6b
This document contains an exam for a course on Fuzzy Logic and Applications. It includes 8 questions covering topics such as operations on crisp and fuzzy sets using Venn diagrams, fuzzy relations, membership functions, fuzzy logic connectives, defuzzification methods, and decision making under fuzzy conditions. Students are instructed to answer any 5 of the 8 questions.
The document outlines the aims, objectives, and syllabus for the Mathematics HL (1st exams 2014) course. It includes:
- 10 aims of the course focused on developing mathematical skills, understanding, problem solving, and appreciation of mathematics.
- 6 objectives centered around demonstrating knowledge and understanding of mathematical concepts, problem solving, communication, use of technology, reasoning, and inquiry approaches.
- The syllabus is divided into 8 core topics (Algebra, Functions and equations, Circular functions and trigonometry, Vectors, Statistics and probability, Calculus, and 2 optional topics (Statistics and probability, Sets, relations and groups) that provide 48 hours of instruction each.
A Comprehensive Overview of Clustering Algorithms in Pattern RecognitionIOSR Journals
This document provides an overview of clustering algorithms used in pattern recognition, including K-means clustering and hierarchical clustering (agglomerative and divisive). It describes the basic steps of each algorithm, provides examples, and compares their advantages and disadvantages. K-means clustering partitions data into K groups based on feature similarity, while hierarchical clustering creates nested clusters based on distance metrics. The document concludes that the appropriate technique depends on factors like prior knowledge of clusters and whether a sequential or flat structure is needed.
Machine learning algorithms infer unknowns from knowns through statistical inference. Common machine learning applications include spam identification, handwriting recognition, image recognition, speech recognition, recommendation systems, and climate modeling. These applications can be grouped into supervised learning (classification and regression) and unsupervised learning (clustering, density estimation, dimensionality reduction). Generative models model the joint distribution of all variables, while discriminative models model only the target variables conditional on observed variables. K-nearest neighbors is a discriminative classification algorithm that classifies a new data point based on the labels of its k nearest neighbors.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
1. The document presents a methodology for recognizing isolated handwritten Devanagari numerals using structural and statistical features.
2. Key features extracted include whether the numeral has openings on the left, right, above or below, and the number of horizontal and vertical crossings.
3. The methodology achieves an average accuracy of 96.8% on a dataset of 500 numeral images collected from various individuals. Accuracy is highest for numerals 0, 6, 8 and 10 at 100%, while some similar numerals like 3 and 2 see more errors.
SYMMETRICAL WEIGHTED SUBSPACE HOLISTIC APPROACH FOR EXPRESSION RECOGNITIONijcsit
This document proposes a new method called Symmetrical Weighted 2D Principal Component Analysis (SW2DPCA) for facial expression recognition. SW2DPCA aims to reduce the dimensionality of the feature space extracted from face images using Gabor filters, while removing redundant information. It does this by decomposing the training images into odd and even symmetrical components, and applying weighted PCA to equalize the variance of principal components. The method is tested on the JAFFE database, achieving a recognition accuracy of 95.24% - higher than the 2DPCA baseline method.
Estimation of Distribution Algorithms (EDAs) constitute a powerful evolutionary algorithm for solving continuous and combinatorial optimization problems. Based on machine learning techniques, at each generation, EDAs estimate a joint probability distribution associated with the set of most promising individuals, trying to explicitly express the interrelations between the different variables of the problem. Based on this general framework, EDAs have proved to be very competitive for solving combinatorial and continuous optimization problems. In this talk, we propose developing EDAs by introducing probability models defined exclusively on the space of feasible solutions. In this sense, we give a first approach by taking the Graph Partitioning Problem (GPP) as a case of study, and present a probabilistic model defined exclusively on the feasible region of solutions: a square lattice probability model.
Handling missing data with expectation maximization algorithmLoc Nguyen
Expectation maximization (EM) algorithm is a powerful mathematical tool for estimating parameter of statistical models in case of incomplete data or hidden data. EM assumes that there is a relationship between hidden data and observed data, which can be a joint distribution or a mapping function. Therefore, this implies another implicit relationship between parameter estimation and data imputation. If missing data which contains missing values is considered as hidden data, it is very natural to handle missing data by EM algorithm. Handling missing data is not a new research but this report focuses on the theoretical base with detailed mathematical proofs for fulfilling missing values with EM. Besides, multinormal distribution and multinomial distribution are the two sample statistical models which are concerned to hold missing values.
The document discusses classification algorithms in machine learning. It introduces classification problems using the Iris flower dataset as an example, which contains measurements of Iris flowers to classify them into three species. It then discusses two classic classification algorithms - logistic regression and Gaussian discriminant analysis. Logistic regression uses a sigmoid function to generate predictions, while Gaussian discriminant analysis assumes a Gaussian distribution of the data. The document also demonstrates an application of these algorithms to classify handwritten digits.
Using Alpha-cuts and Constraint Exploration Approach on Quadratic Programming...TELKOMNIKA JOURNAL
In this paper, we propose a computational procedure to find the optimal solution of quadratic programming
problems by using fuzzy -cuts and constraint exploration approach. We solve the problems in
the original form without using any additional information such as Lagrange’s multiplier, slack, surplus and
artificial variable. In order to find the optimal solution, we divide the calculation in two stages. In the first
stage, we determine the unconstrained minimization of the quadratic programming problem (QPP) and check
its feasibility. By unconstrained minimization we identify the violated constraints and focus our searching in
these constraints. In the second stage, we explored the feasible region along side the violated constraints
until the optimal point is achieved. A numerical example is included in this paper to illustrate the capability of
-cuts and constraint exploration to find the optimal solution of QPP.
PREDICTIVE EVALUATION OF THE STOCK PORTFOLIO PERFORMANCE USING FUZZY CMEANS A...ijfls
The aim of this paper is to investigate the trend of the return of a portfolio formed randomly or for any
specific technique. The approach is made using two techniques fuzzy: fuzzy c-means (FCM) algorithm and
the fuzzy transform, where the rules used at fuzzy transform arise from the application of the FCM
algorithm. The results show that the proposed methodology is able to predict the trend of the return of a
stock portfolio, as well as the tendency of the market index. Real data of the financial market are used from
2004 until 2007.
This chapter discusses methods for constructing confidence intervals for differences between population means and proportions in various sampling situations. It covers confidence intervals for the difference between two dependent or paired sample means, two independent sample means when population variances are known or unknown, and two independent population proportions. It also addresses determining required sample sizes to estimate a mean or proportion within a specified margin of error.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
The document outlines a paper on Bayesian linear models. It introduces a simple example of a linear model with exchangeable priors. It then presents the general Bayesian linear model and theorems for the posterior distribution given multiple stages of priors. It applies this to an experimental design setting, deriving Bayes estimates that shrink treatment and block effects towards zero based on their variances.
ANALYTICAL FORMULATIONS FOR THE LEVEL BASED WEIGHTED AVERAGE VALUE OF DISCRET...ijsc
In fuzzy decision-making processes based on linguistic information, operations on discrete fuzzy numbers
are commonly performed. Aggregation and defuzzification operations are some of these often used
operations. Many aggregation and defuzzification operators produce results independent to the decisionmaker’s
strategy. On the other hand, the Weighted Average Based on Levels (WABL) approach can take
into account the level weights and the decision maker's "optimism" strategy. This gives flexibility to the
WABL operator and, through machine learning, can be trained in the direction of the decision maker's
strategy, producing more satisfactory results for the decision maker. However, in order to determine the
WABL value, it is necessary to calculate some integrals. In this study, the concept of WABL for discrete
trapezoidal fuzzy numbers is investigated, and analytical formulas have been proven to facilitate the
calculation of WABL value for these fuzzy numbers. Trapezoidal and their special form, triangular fuzzy
numbers, are the most commonly used fuzzy number types in fuzzy modeling, so in this study, such numbers
have been studied. Computational examples explaining the theoretical results have been performed.
Teaching Mathematics Concepts via Computer Algebra Systemsinventionjournals
Most articles examine computer algebra systems (CAS) as they relate to the teaching and
learning of mathematics from advantages to disadvantages. This paper will explore junior undergraduate
students’ ability to solve distinguish tricky examples using various CAS technologies. Additionally, an
understanding for how CAS technologies are adopted and applied in professional environments is valuable,
both in guiding improvements to these tools and identifying new tools which can aid mathematician
In this paper, Assignment problem with crisp, fuzzy and intuitionistic fuzzy numbers as cost coefficients is investigated. In conventional assignment problem, cost is always certain. This paper develops an approach to solve a mixed intuitionistic fuzzy assignment problem where cost is considered real, fuzzy and an intuitionistic fuzzy numbers. Ranking procedure of Annie Varghese and Sunny Kuriakose [4] is used to transform the mixed intuitionistic fuzzy assignment problem into a crisp one so that the conventional method may be applied to solve the assignment problem. The method is illustrated by a numerical example. The proposed method is very simple and easy to understand. Numerical examples show that an intuitionistic fuzzy ranking method offers an effective tool for handling an intuitionistic fuzzy assignment problem.
A New Approach for Ranking Shadowed Fuzzy Numbers and its Application IJCSITJournal2
n many decision situations, decision-makers face a kind of complex problems. In these decision-making
problems, different types of fuzzy numbers are defined and, have multiple types of membership functions.
So, we need a standard form to formulate uncertain numbers in the problem. Shadowed fuzzy numbers are
considered granule numbers which approximate different types and different forms of fuzzy numbers. In
this paper, a new ranking approach for shadowed fuzzy numbers is developed using value, ambiguity and
fuzziness for shadowed fuzzy numbers. The new ranking method has been compared with other existing
approaches through numerical examples. Also, the new method is applied to a hybrid multi-attribute
decision making problem in which the evaluations of alternatives are expressed with different types of
uncertain numbers. The comparative study for the results of different examples illustrates the reliability of
the new method.
Transportation Problem with Pentagonal Intuitionistic Fuzzy Numbers Solved Us...IJERA Editor
This paper presents a solution methodology for transportation problem in an intuitionistic fuzzy environment in
which cost are represented by pentagonal intuitionistic fuzzy numbers. Transportation problem is a particular
class of linear programming, which is associated with day to day activities in our real life. It helps in solving
problems on distribution and transportation of resources from one place to another. The objective is to satisfy
the demand at destination from the supply constraints at the minimum transportation cost possible. The problem
is solved using a ranking technique called Accuracy function for pentagonal intuitionistic fuzzy numbers and
Russell’s Method
A New Hendecagonal Fuzzy Number For Optimization Problemsijtsrd
A new fuzzy number called Hendecagonal fuzzy number and its membership function is introduced, which is used to represent the uncertainty with eleven points. The fuzzy numbers with ten ordinates exists in literature. The aim of this paper is to define Hendecagonal fuzzy number and its arithmetic operations. Also a direct approach is proposed to solve fuzzy assignment problem (FAP) and fuzzy travelling salesman (FTSP) in which the cost and distance are represented by Hendecagonal fuzzy numbers. Numerical example shows the effectiveness of the proposed method and the Hendecagonal fuzzy number M. Revathi | Dr. M. Valliathal | R. Saravanan | Dr. K. Rathi"A New Hendecagonal Fuzzy Number For Optimization Problems" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-1 | Issue-5 , August 2017, URL: http://www.ijtsrd.com/papers/ijtsrd2258.pdf http://www.ijtsrd.com/mathemetics/applied-mathamatics/2258/a-new-hendecagonal-fuzzy-number-for-optimization-problems/m-revathi
A Fuzzy Mean-Variance-Skewness Portfolioselection Problem.inventionjournals
A fuzzy number is a normal and convex fuzzy subsetof the real line. In this paper, based on membership function, we redefine the concepts of mean and variance for fuzzy numbers. Furthermore, we propose the concept of skewness and prove some desirable properties. A fuzzy mean-variance-skewness portfolio se-lection model is formulated and two variations are given, which are transformed to nonlinear optimization models with polynomial ob-jective and constraint functions such that they can be solved analytically. Finally, we present some numerical examples to demonstrate the effectiveness of the proposed models
FUZZY ROUGH INFORMATION MEASURES AND THEIR APPLICATIONSijcsity
This document summarizes a research paper that proposes three new information measures for fuzzy rough sets and evaluates their applications. It begins with background on fuzzy rough sets and existing information measures. It then defines a new logarithmic information measure for fuzzy rough values and proves its validity. Another proposed information measure for fuzzy rough sets is described along with an illustration of its application. A weighted information measure for fuzzy rough sets is also discussed. The paper compares the proposed information measures to other existing ones and concludes with references.
FUZZY ROUGH INFORMATION MEASURES AND THEIR APPLICATIONSijcsity
The degree of roughness characterizes the uncertainty contained in a rough set. The rough entropy was
defined to measure the roughness of a rough set. Though, it was effective and useful, but not accurate
enough. Some authors use information measure in place of entropy for better understanding which
measures the amount of uncertainty contained in fuzzy rough set .In this paper three new fuzzy rough
information measures are proposed and their validity is verified. The application of these proposed
information measures in decision making problems is studied and also compared with other existing
information measures.
FUZZY ROUGH INFORMATION MEASURES AND THEIR APPLICATIONSijcsity
This document summarizes a research paper that proposes three new information measures for fuzzy rough sets and evaluates their applications. It begins with background on fuzzy rough sets and existing information measures. It then defines a new logarithmic information measure for fuzzy rough values and proves its validity. Another proposed information measure for fuzzy rough sets is described along with an illustration of its application. A weighted information measure for fuzzy rough sets is also discussed. The paper compares the proposed information measures to other existing ones and concludes with references.
AN ARITHMETIC OPERATION ON HEXADECAGONAL FUZZY NUMBERWireilla
In this paper, a new form of fuzzy number named as exadecagonal Fuzzy Number is introduced as it is not possible to restrict the membership function to any specific form. The cut of Hexadecagonal fuzzy number is defined and basic arithmetic operations are performed using interval arithmetic of cut and
illustrated with numerical examples
AN ARITHMETIC OPERATION ON HEXADECAGONAL FUZZY NUMBERijfls
In this paper, a new form of fuzzy number named as Hexadecagonal Fuzzy Number is introduced as it is not
possible to restrict the membership function to any specific form. The cut of Hexadecagonal fuzzy
number is defined and basic arithmetic operations are performed using interval arithmetic of cut and
illustrated with numerical examples.
Fuzzy linear programming with the intuitionistic polygonal fuzzy numbersIJECEIAES
In real world applications, data are subject to ambiguity due to several factors; fuzzy sets and fuzzy numbers propose a great tool to model such ambiguity. In case of hesitation, the complement of a membership value in fuzzy numbers can be different from the non-membership value, in which case we can model using intuitionistic fuzzy numbers as they provide flexibility by defining both a membership and a non-membership functions. In this article, we consider the intuitionistic fuzzy linear programming problem with intuitionistic polygonal fuzzy numbers, which is a generalization of the previous polygonal fuzzy numbers found in the literature. We present a modification of the simplex method that can be used to solve any general intuitionistic fuzzy linear programming problem after approximating the problem by an intuitionistic polygonal fuzzy number with n edges. This method is given in a simple tableau formulation, and then applied on numerical examples for clarity.
VARIOUS FUZZY NUMBERS AND THEIR VARIOUS RANKING APPROACHESIAEME Publication
A brief survey of this study is to identify the ranking formulas for various fuzzy numbers derived from research papers published over the past few years. This paper presents the latest results of fuzzy ranking applications very clearly and simply, as well as highlighting key points in the use of fuzzy numbers. This paper discusses the importance of pointing out the concepts of fuzzy numbers and their formulas for ranking.
Min-based qualitative possibilistic networks are one of the effective tools for a compact representation of
decision problems under uncertainty. The exact approaches for computing decision based on possibilistic
networks are limited by the size of the possibility distributions. Generally, these approaches are based on
possibilistic propagation algorithms. An important step in the computation of the decision is the
transformation of the DAG (Direct Acyclic Graph) into a secondary structure, known as the junction trees
(JT). This transformation is known to be costly and represents a difficult problem. We propose in this paper
a new approximate approach for the computation of decision under uncertainty within possibilistic
networks. The computing of the optimal optimistic decision no longer goes through the junction tree
construction step. Instead, it is performed by calculating the degree of normalization in the moral graph
resulting from the merging of the possibilistic network codifying knowledge of the agent and that codifying
its preferences.
Connectivity-Based Clustering for Mixed Discrete and Continuous DataIJCI JOURNAL
This paper introduces a density-based clustering procedure for datasets with variables of mixed type. The proposed procedure, which is closely related to the concept of shared neighbourhoods, works particularly well in cases where the individual clusters differ greatly in terms of the average pairwise distance of the associated objects. Using a number of concrete examples, it is shown that the proposed clustering algorithm succeeds in allowing the identification of subgroups of objects with statistically significant distributional characteristics.
Zadeh conceptualized the theory of fuzzy set to provide a tool for the basis of the theory of possibility. Atanassov extended this theory with the introduction of intuitionistic fuzzy set. Smarandache introduced the concept of refined intuitionistic fuzzy set by further subdivision of membership and non-membership value. The meagerness regarding the allocation of a single membership and non-membership value to any object under consideration is addressed with this novel refinement. In this study, this novel idea is utilized to characterize the essential elements e.g. subset, equal set, null set, and complement set, for refined intuitionistic fuzzy set. Moreover, their basic set theoretic operations like union, intersection, extended intersection, restricted union, restricted intersection, and restricted difference, are conceptualized. Furthermore, some basic laws are also discussed with the help of an illustrative example in each case for vivid understanding.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
This document proposes a new approximate approach for computing the optimal optimistic decision within possibilistic networks. The approach avoids transforming the initial graph into a junction tree, which is computationally expensive. Instead, it performs the computation by calculating the degree of normalization in the moral graph resulting from merging the possibilistic network representing the agent's beliefs and the one representing its preferences. This allows the approach to have polynomial complexity compared to the exact approach based on junction trees, which is NP-hard.
The idea of projectour project is about creating a intell.docxcherry686017
The idea of project:
our project is about creating a intelligent system that will help the user to make decision in faster and easy way
we have an idea that is to create a new system for our college for course register
our system is not that good as the students wants
we want the courses to be in the same sequence as the study plan for registration - for the main courses and for the elective also
we also want to show the courses dependence
for example you cannot take 103 course without completing 102 course
and when the course is register , we want to show the course schedule as the picture provided.
and when the student complete his registration he can print and save the schedule - the final out put schedule
If the student faces a class clashes it will show the clash time and the course that have clash with
and provide them better solution such as changing the section or report this problem to the responsible employee- provide the student with suggestions to solve her problem
You can use the pictures below as an example .. And the logo to put in in the interface
TASKS:
1. Read about Creativity below.
2. Do literatures review from Google or from given list of Bibliography.
3. Design your invention into Interface Design and using any solution models
4. Goto http://www.scoop.it/t/kaymarlyn and select ‘Tools’ tags under ‘Search in topic’ menu. Study
and learn about “60 User Interface Design Tools A Web Designer Must Have” and other prototyping
and mockup tools from the page.
5. Illustrate your idea into interface design using the selected best tool for your Design Category and
provide the explanation. You might search from the Internet using keywords to view other example of
process or models.
6. Disseminate your idea and how your system works into proper formatted report.
7. Presentation will determined the winners ranking and will contribute max 35/50 marks from the
total marks.
8. Shows all the workload distribution among your group members in the given table.
9. Lastly, provide all the references and websites that you visited and used in the report.
DESIGN CATEGORIES:
Academic System
Students Manager
University DSS
Mobile Apps
Student Work/ Activities Application
Project Requirement :
Creativity Creativity involves the generation of new ideas or the recombination of known elements into something new, providing valuable solutions to a problem. It also involves motivation and emotion. Creativity “is a fundamental feature of human intelligence in general. It is grounded in everyday capacities such as the association of ideas, reminding, perception, analogical thinking, searching a structured problem-space, and reflecting self-criticism. It involves not only a cognitive dimension (the generation of new ideas) but also motivation and emotion, and is closely linked to cultural context and personality factors.” (Boden 1998).
Fundamental concepts for all creative techniques are:
The suspension of premature ...
In conventional assignment problem, cost is always certain. In this paper, Assignment problem with crisp, fuzzy and intuitionistic fuzzy numbers as cost coefficients is investigated. There is no systematic approach for finding an optimal solution for mixed intuitionistic fuzzy assignment problem. This paper develops an approach to solve a mixed intuitionistic fuzzy assignment problem where cost is not in deterministic numbers but imprecise ones. The solution procedure of mixed intuitionistic fuzzy assignment problem is proposed to find the optimal assignment and also obtain an optimal value in terms of triangular intuitionistic fuzzy numbers. Numerical examples show that an intuitionistic fuzzy ranking method offers an effective tool for handling an intuitionistic fuzzy assignment problem.
TYPE-2 FUZZY LINEAR PROGRAMMING PROBLEMS WITH PERFECTLY NORMAL INTERVAL TYPE-...ijceronline
In this paper, the Perfectly normal Interval Type-2 Fuzzy Linear Programming (PnIT2FLP) model is considered. This model is reduced to crisp linear programming model. This transformation is performed by a proposed ranking method. Based on the proposed fuzzy ranking method and arithmetic operation, the solution of Perfectly normal Interval Type-2 Fuzzy Linear Programming model is obtained by the solutions of linear programming model with help of MATLAB. Finally, the method is illustrated by numerical examples.
A COMPARATIVE STUDY ON DISTANCE MEASURING APPROACHES FOR CLUSTERINGIJORCS
Clustering plays a vital role in the various areas of research like Data Mining, Image Retrieval, Bio-computing and many a lot. Distance measure plays an important role in clustering data points. Choosing the right distance measure for a given dataset is a biggest challenge. In this paper, we study various distance measures and their effect on different clustering. This paper surveys existing distance measures for clustering and present a comparison between them based on application domain, efficiency, benefits and drawbacks. This comparison helps the researchers to take quick decision about which distance measure to use for clustering. We conclude this work by identifying trends and challenges of research and development towards clustering.
Similar to A New Approach for Ranking Shadowed Fuzzy Numbers and its Application (20)
Home security is of paramount importance in today's world, where we rely more on technology, home
security is crucial. Using technology to make homes safer and easier to control from anywhere is
important. Home security is important for the occupant’s safety. In this paper, we came up with a low cost,
AI based model home security system. The system has a user-friendly interface, allowing users to start
model training and face detection with simple keyboard commands. Our goal is to introduce an innovative
home security system using facial recognition technology. Unlike traditional systems, this system trains
and saves images of friends and family members. The system scans this folder to recognize familiar faces
and provides real-time monitoring. If an unfamiliar face is detected, it promptly sends an email alert,
ensuring a proactive response to potential security threats.
In the era of data-driven warfare, the integration of big data and machine learning (ML) techniques has
become paramount for enhancing defence capabilities. This research report delves into the applications of
big data and ML in the defence sector, exploring their potential to revolutionize intelligence gathering,
strategic decision-making, and operational efficiency. By leveraging vast amounts of data and advanced
algorithms, these technologies offer unprecedented opportunities for threat detection, predictive analysis,
and optimized resource allocation. However, their adoption also raises critical concerns regarding data
privacy, ethical implications, and the potential for misuse. This report aims to provide a comprehensive
understanding of the current state of big data and ML in defence, while examining the challenges and
ethical considerations that must be addressed to ensure responsible and effective implementation.
Cloud Computing, being one of the most recent innovative developments of the IT world, has been
instrumental not just to the success of SMEs but, through their productivity and innovative contribution to
the economy, has even made a remarkable contribution to the economic growth of the United States. To
this end, the study focuses on how cloud computing technology has impacted economic growth through
SMEs in the United States. Relevant literature connected to the variables of interest in this study was
reviewed, and secondary data was generated and utilized in the analysis section of this paper. The findings
of this paper revealed that there have been meaningful contributions that the usage of virtualization has
made in the commercial dealings of small firms in the United States, and this has also been reflected in the
economic growth of the country. This paper further revealed that as important as cloud-based software is,
some SMEs are still skeptical about how it can help improve their business and increase their bottom line
and hence have failed to adopt it. Apart from the SMEs, some notable large firms in different industries,
including information and educational services, have adopted cloud computing technology and hence
contributed to the economic growth of the United States. Lastly, findings from our inferential statistics
revealed that no discernible change has occurred in innovation between small and big businesses in the
adoption of cloud computing. Both categories of businesses adopt cloud computing in the same way, and
their contribution to the American economy has no significant difference in the usage of virtualization.
Energy-constrained Wireless Sensor Networks (WSNs) have garnered significant research interest in
recent years. Multiple-Input Multiple-Output (MIMO), or Cooperative MIMO, represents a specialized
application of MIMO technology within WSNs. This approach operates effectively, especially in
challenging and resource-constrained environments. By facilitating collaboration among sensor nodes,
Cooperative MIMO enhances reliability, coverage, and energy efficiency in WSN deployments.
Consequently, MIMO finds application in diverse WSN scenarios, spanning environmental monitoring,
industrial automation, and healthcare applications.
The AIRCC's International Journal of Computer Science and Information Technology (IJCSIT) is devoted to fields of Computer Science and Information Systems. The IJCSIT is a open access peer-reviewed scientific journal published in electronic form as well as print form. The mission of this journal is to publish original contributions in its field in order to propagate knowledge amongst its readers and to be a reference publication. IJCSIT publishes original research papers and review papers, as well as auxiliary material such as: research papers, case studies, technical reports etc.
With growing, Car parking increases with the number of car users. With the increased use of smartphones
and their applications, users prefer mobile phone-based solutions. This paper proposes the Smart Parking
Management System (SPMS) that depends on Arduino parts, Android applications, and based on IoT. This
gave the client the ability to check available parking spaces and reserve a parking spot. IR sensors are
utilized to know if a car park space is allowed. Its area data are transmitted using the WI-FI module to the
server and are recovered by the mobile application which offers many options attractively and with no cost
to users and lets the user check reservation details. With IoT technology, the smart parking system can be
connected wirelessly to easily track available locations.
Welcome to AIRCC's International Journal of Computer Science and Information Technology (IJCSIT), your gateway to the latest advancements in the dynamic fields of Computer Science and Information Systems.
Computer-Assisted Language Learning (CALL) are computer-based tutoring systems that deal with
linguistic skills. Adding intelligence in such systems is mainly based on using Natural Language
Processing (NLP) tools to diagnose student errors, especially in language grammar. However, most such
systems do not consider the modeling of student competence in linguistic skills, especially for the Arabic
language. In this paper, we will deal with basic grammar concepts of the Arabic language taught for the
fourth grade of the elementary school in Egypt. This is through Arabic Grammar Trainer (AGTrainer)
which is an Intelligent CALL. The implemented system (AGTrainer) trains the students through different
questions that deal with the different concepts and have different difficulty levels. Constraint-based student
modeling (CBSM) technique is used as a short-term student model. CBSM is used to define in small grain
level the different grammar skills through the defined skill structures. The main contribution of this paper
is the hierarchal representation of the system's basic grammar skills as domain knowledge. That
representation is used as a mechanism for efficiently checking constraints to model the student knowledge
and diagnose the student errors and identify their cause. In addition, satisfying constraints and the number
of trails the student takes for answering each question and fuzzy logic decision system are used to
determine the student learning level for each lesson as a long-term model. The results of the evaluation
showed the system's effectiveness in learning in addition to the satisfaction of students and teachers with its
features and abilities.
In the realm of computer security, the importance of efficient and reliable user authentication methods has
become increasingly critical. This paper examines the potential of mouse movement dynamics as a
consistent metric for continuous authentication. By analysing user mouse movement patterns in two
contrasting gaming scenarios, "Team Fortress" and "Poly Bridge," we investigate the distinctive
behavioral patterns inherent in high-intensity and low-intensity UI interactions. The study extends beyond
conventional methodologies by employing a range of machine learning models. These models are carefully
selected to assess their effectiveness in capturing and interpreting the subtleties of user behavior as
reflected in their mouse movements. This multifaceted approach allows for a more nuanced and
comprehensive understanding of user interaction patterns. Our findings reveal that mouse movement
dynamics can serve as a reliable indicator for continuous user authentication. The diverse machine
learning models employed in this study demonstrate competent performance in user verification, marking
an improvement over previous methods used in this field. This research contributes to the ongoing efforts to
enhance computer security and highlights the potential of leveraging user behavior, specifically mouse
dynamics, in developing robust authentication systems.
The AIRCC's International Journal of Computer Science and Information Technology (IJCSIT) is devoted to fields of Computer Science and Information Systems. The IJCSIT is a open access peer-reviewed scientific journal published in electronic form as well as print form. The mission of this journal is to publish original contributions in its field in order to propagate knowledge amongst its readers and to be a reference publication.
Image segmentation and classification tasks in computer vision have proven to be highly effective using neural networks, specifically Convolutional Neural Networks (CNNs). These tasks have numerous
practical applications, such as in medical imaging, autonomous driving, and surveillance. CNNs are capable
of learning complex features directly from images and achieving outstanding performance across several
datasets. In this work, we have utilized three different datasets to investigate the efficacy of various preprocessing and classification techniques in accurssedately segmenting and classifying different structures
within the MRI and natural images. We have utilized both sample gradient and Canny Edge Detection
methods for pre-processing, and K-means clustering have been applied to segment the images. Image
augmentation improves the size and diversity of datasets for training the models for image classification
The AIRCC's International Journal of Computer Science and Information Technology (IJCSIT) is devoted to fields of Computer Science and Information Systems. The IJCSIT is a open access peer-reviewed scientific journal published in electronic form as well as print form. The mission of this journal is to publish original contributions in its field in order to propagate knowledge amongst its readers and to be a reference publication.
This research aims to further understanding in the field of continuous authentication using behavioural
biometrics. We are contributing a novel dataset that encompasses the gesture data of 15 users playing
Minecraft with a Samsung Tablet, each for a duration of 15 minutes. Utilizing this dataset, we employed
machine learning (ML) binary classifiers, being Random Forest (RF), K-Nearest Neighbors (KNN), and
Support Vector Classifier (SVC), to determine the authenticity of specific user actions. Our most robust
model was SVC, which achieved an average accuracy of approximately 90%, demonstrating that touch
dynamics can effectively distinguish users. However, further studies are needed to make it viable option
for authentication systems. You can access our dataset at the following
link:https://github.com/AuthenTech2023/authentech-repo
This paper discusses the capabilities and limitations of GPT-3 (0), a state-of-the-art language model, in the
context of text understanding. We begin by describing the architecture and training process of GPT-3, and
provide an overview of its impressive performance across a wide range of natural language processing
tasks, such as language translation, question-answering, and text completion. Throughout this research
project, a summarizing tool was also created to help us retrieve content from any types of document,
specifically IELTS (0) Reading Test data in this project. We also aimed to improve the accuracy of the
summarizing, as well as question-answering capabilities of GPT-3 (0) via long text
In the realm of computer security, the importance of efficient and reliable user authentication methods has
become increasingly critical. This paper examines the potential of mouse movement dynamics as a
consistent metric for continuous authentication. By analysing user mouse movement patterns in two
contrasting gaming scenarios, "Team Fortress" and "Poly Bridge," we investigate the distinctive
behavioral patterns inherent in high-intensity and low-intensity UI interactions. The study extends beyond
conventional methodologies by employing a range of machine learning models. These models are carefully
selected to assess their effectiveness in capturing and interpreting the subtleties of user behavior as
reflected in their mouse movements. This multifaceted approach allows for a more nuanced and
comprehensive understanding of user interaction patterns. Our findings reveal that mouse movement
dynamics can serve as a reliable indicator for continuous user authentication. The diverse machine
learning models employed in this study demonstrate competent performance in user verification, marking
an improvement over previous methods used in this field. This research contributes to the ongoing efforts to
enhance computer security and highlights the potential of leveraging user behavior, specifically mouse
dynamics, in developing robust authentication systems.
Image segmentation and classification tasks in computer vision have proven to be highly effective using neural networks, specifically Convolutional Neural Networks (CNNs). These tasks have numerous
practical applications, such as in medical imaging, autonomous driving, and surveillance. CNNs are capable
of learning complex features directly from images and achieving outstanding performance across several
datasets. In this work, we have utilized three different datasets to investigate the efficacy of various preprocessing and classification techniques in accurssedately segmenting and classifying different structures
within the MRI and natural images. We have utilized both sample gradient and Canny Edge Detection
methods for pre-processing, and K-means clustering have been applied to segment the images. Image
augmentation improves the size and diversity of datasets for training the models for image classification.
This work highlights transfer learning’s effectiveness in image classification using CNNs and VGG 16 that
provides insights into the selection of pre-trained models and hyper parameters for optimal performance.
We have proposed a comprehensive approach for image segmentation and classification, incorporating preprocessing techniques, the K-means algorithm for segmentation, and employing deep learning models such
as CNN and VGG 16 for classification.
- The document presents 6 different models for defining foot size in Tunisia: 2 statistical models, 2 neural network models using unsupervised learning, and 2 models combining neural networks and fuzzy logic.
- The statistical models (SM and SHM) are based on applying statistical equations to morphological foot data.
- The neural network models (MSK and MHSK) use self-organizing Kohonen maps to cluster foot data and model full and half sizes.
- The fuzzy neural network models (MSFK and MHSFK) incorporate fuzzy logic into the neural network learning process to better account for uncertainty in foot sizes.
The security of Electric Vehicle (EV) charging has gained momentum after the increase in the EV adoption
in the past few years. Mobile applications have been integrated into EV charging systems that mainly use a
cloud-based platform to host their services and data. Like many complex systems, cloud systems are
susceptible to cyberattacks if proper measures are not taken by the organization to secure them. In this
paper, we explore the security of key components in the EV charging infrastructure, including the mobile
application and its cloud service. We conducted an experiment that initiated a Man in the Middle attack
between an EV app and its cloud services. Our results showed that it is possible to launch attacks against
the connected infrastructure by taking advantage of vulnerabilities that may have substantial economic and
operational ramifications on the EV charging ecosystem. We conclude by providing mitigation suggestions
and future research directions.
The AIRCC's International Journal of Computer Science and Information Technology (IJCSIT) is devoted to fields of Computer Science and Information Systems. The IJCSIT is a open access peer-reviewed scientific journal published in electronic form as well as print form. The mission of this journal is to publish original contributions in its field in order to propagate knowledge amongst its readers and to be a reference publication.
The AIRCC's International Journal of Computer Science and Information Technology (IJCSIT) is devoted to fields of Computer Science and Information Systems. The IJCSIT is a open access peer-reviewed scientific journal published in electronic form as well as print form. The mission of this journal is to publish original contributions in its field in order to propagate knowledge amongst its readers and to be a reference publication.
Road construction is not as easy as it seems to be, it includes various steps and it starts with its designing and
structure including the traffic volume consideration. Then base layer is done by bulldozers and levelers and after
base surface coating has to be done. For giving road a smooth surface with flexibility, Asphalt concrete is used.
Asphalt requires an aggregate sub base material layer, and then a base layer to be put into first place. Asphalt road
construction is formulated to support the heavy traffic load and climatic conditions. It is 100% recyclable and
saving non renewable natural resources.
With the advancement of technology, Asphalt technology gives assurance about the good drainage system and with
skid resistance it can be used where safety is necessary such as outsidethe schools.
The largest use of Asphalt is for making asphalt concrete for road surfaces. It is widely used in airports around the
world due to the sturdiness and ability to be repaired quickly, it is widely used for runways dedicated to aircraft
landing and taking off. Asphalt is normally stored and transported at 150’C or 300’F temperature
A high-Speed Communication System is based on the Design of a Bi-NoC Router, ...DharmaBanothu
The Network on Chip (NoC) has emerged as an effective
solution for intercommunication infrastructure within System on
Chip (SoC) designs, overcoming the limitations of traditional
methods that face significant bottlenecks. However, the complexity
of NoC design presents numerous challenges related to
performance metrics such as scalability, latency, power
consumption, and signal integrity. This project addresses the
issues within the router's memory unit and proposes an enhanced
memory structure. To achieve efficient data transfer, FIFO buffers
are implemented in distributed RAM and virtual channels for
FPGA-based NoC. The project introduces advanced FIFO-based
memory units within the NoC router, assessing their performance
in a Bi-directional NoC (Bi-NoC) configuration. The primary
objective is to reduce the router's workload while enhancing the
FIFO internal structure. To further improve data transfer speed,
a Bi-NoC with a self-configurable intercommunication channel is
suggested. Simulation and synthesis results demonstrate
guaranteed throughput, predictable latency, and equitable
network access, showing significant improvement over previous
designs
Levelised Cost of Hydrogen (LCOH) Calculator ManualMassimo Talia
The aim of this manual is to explain the
methodology behind the Levelized Cost of
Hydrogen (LCOH) calculator. Moreover, this
manual also demonstrates how the calculator
can be used for estimating the expenses associated with hydrogen production in Europe
using low-temperature electrolysis considering different sources of electricity
Digital Twins Computer Networking Paper Presentation.pptxaryanpankaj78
A Digital Twin in computer networking is a virtual representation of a physical network, used to simulate, analyze, and optimize network performance and reliability. It leverages real-time data to enhance network management, predict issues, and improve decision-making processes.
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODELijaia
As digital technology becomes more deeply embedded in power systems, protecting the communication
networks of Smart Grids (SG) has emerged as a critical concern. Distributed Network Protocol 3 (DNP3)
represents a multi-tiered application layer protocol extensively utilized in Supervisory Control and Data
Acquisition (SCADA)-based smart grids to facilitate real-time data gathering and control functionalities.
Robust Intrusion Detection Systems (IDS) are necessary for early threat detection and mitigation because
of the interconnection of these networks, which makes them vulnerable to a variety of cyberattacks. To
solve this issue, this paper develops a hybrid Deep Learning (DL) model specifically designed for intrusion
detection in smart grids. The proposed approach is a combination of the Convolutional Neural Network
(CNN) and the Long-Short-Term Memory algorithms (LSTM). We employed a recent intrusion detection
dataset (DNP3), which focuses on unauthorized commands and Denial of Service (DoS) cyberattacks, to
train and test our model. The results of our experiments show that our CNN-LSTM method is much better
at finding smart grid intrusions than other deep learning algorithms used for classification. In addition,
our proposed approach improves accuracy, precision, recall, and F1 score, achieving a high detection
accuracy rate of 99.50%.
Accident detection system project report.pdfKamal Acharya
The Rapid growth of technology and infrastructure has made our lives easier. The
advent of technology has also increased the traffic hazards and the road accidents take place
frequently which causes huge loss of life and property because of the poor emergency facilities.
Many lives could have been saved if emergency service could get accident information and
reach in time. Our project will provide an optimum solution to this draw back. A piezo electric
sensor can be used as a crash or rollover detector of the vehicle during and after a crash. With
signals from a piezo electric sensor, a severe accident can be recognized. According to this
project when a vehicle meets with an accident immediately piezo electric sensor will detect the
signal or if a car rolls over. Then with the help of GSM module and GPS module, the location
will be sent to the emergency contact. Then after conforming the location necessary action will
be taken. If the person meets with a small accident or if there is no serious threat to anyone’s
life, then the alert message can be terminated by the driver by a switch provided in order to
avoid wasting the valuable time of the medical rescue team.
Determination of Equivalent Circuit parameters and performance characteristic...pvpriya2
Includes the testing of induction motor to draw the circle diagram of induction motor with step wise procedure and calculation for the same. Also explains the working and application of Induction generator
Null Bangalore | Pentesters Approach to AWS IAMDivyanshu
#Abstract:
- Learn more about the real-world methods for auditing AWS IAM (Identity and Access Management) as a pentester. So let us proceed with a brief discussion of IAM as well as some typical misconfigurations and their potential exploits in order to reinforce the understanding of IAM security best practices.
- Gain actionable insights into AWS IAM policies and roles, using hands on approach.
#Prerequisites:
- Basic understanding of AWS services and architecture
- Familiarity with cloud security concepts
- Experience using the AWS Management Console or AWS CLI.
- For hands on lab create account on [killercoda.com](https://killercoda.com/cloudsecurity-scenario/)
# Scenario Covered:
- Basics of IAM in AWS
- Implementing IAM Policies with Least Privilege to Manage S3 Bucket
- Objective: Create an S3 bucket with least privilege IAM policy and validate access.
- Steps:
- Create S3 bucket.
- Attach least privilege policy to IAM user.
- Validate access.
- Exploiting IAM PassRole Misconfiguration
-Allows a user to pass a specific IAM role to an AWS service (ec2), typically used for service access delegation. Then exploit PassRole Misconfiguration granting unauthorized access to sensitive resources.
- Objective: Demonstrate how a PassRole misconfiguration can grant unauthorized access.
- Steps:
- Allow user to pass IAM role to EC2.
- Exploit misconfiguration for unauthorized access.
- Access sensitive resources.
- Exploiting IAM AssumeRole Misconfiguration with Overly Permissive Role
- An overly permissive IAM role configuration can lead to privilege escalation by creating a role with administrative privileges and allow a user to assume this role.
- Objective: Show how overly permissive IAM roles can lead to privilege escalation.
- Steps:
- Create role with administrative privileges.
- Allow user to assume the role.
- Perform administrative actions.
- Differentiation between PassRole vs AssumeRole
Try at [killercoda.com](https://killercoda.com/cloudsecurity-scenario/)
A New Approach for Ranking Shadowed Fuzzy Numbers and its Application
1. International Journal of Computer Science & Information Technology (IJCSIT) Vol 12, No 6, December 2020
DOI: 10.5121/ijcsit.2020.12602 17
A NEW APPROACH FOR RANKING SHADOWED
FUZZY NUMBERS AND ITS APPLICATION
Mohamed A. H. El-Hawy
Department of Management Information Systems, Modern Academy for
Computer Science and Management Technology, Cairo, Egypt
ABSTRACT
In many decision situations, decision-makers face a kind of complex problems. In these decision-making
problems, different types of fuzzy numbers are defined and, have multiple types of membership functions.
So, we need a standard form to formulate uncertain numbers in the problem. Shadowed fuzzy numbers are
considered granule numbers which approximate different types and different forms of fuzzy numbers. In
this paper, a new ranking approach for shadowed fuzzy numbers is developed using value, ambiguity and
fuzziness for shadowed fuzzy numbers. The new ranking method has been compared with other existing
approaches through numerical examples. Also, the new method is applied to a hybrid multi-attribute
decision making problem in which the evaluations of alternatives are expressed with different types of
uncertain numbers. The comparative study for the results of different examples illustrates the reliability of
the new method.
KEYWORDS
Fuzzy numbers, Intuitionistic fuzzy numbers, Shadowed sets, Shadowed fuzzy numbers, Ranking, Fuzziness
measure.
1. INTRODUCTION
Vague information is represented by many uncertain sets. Fuzzy set is one of the most important
uncertain set which was proposed by Zadeh. Fuzzy set is determined by its membership function
that represents vagueness and imprecision in linguistic term. fuzzy number is a special type of
fuzzy set which is defined on real numbers scale.
Intuitionistic fuzzy sets (IFSs) are introduced by Atanassov which generalized the concept of
fuzzy set. IFSs are characterized by two functions (membership and non-membership) [1]. These
features provide more flexibility in representing uncertain numbers. The shadowed sets proposed
by Pedrycz for approximation of fuzzy sets by three values {0, 1, [0,1]}[2]. Fuzzy membership
values assign to 0, l or uncertain interval. Three areas are induced from fuzzy set to define
shadowed set. The elements with membership grade 0 constitute the excluded area of the
shadowed set. The core area consists of the elements that almost certainly belong to the fuzzy set.
The shadow area relates to the elements that possibly belong to the fuzzy set.
The author proposed an improved form of shadowed fuzzy numbers (SFNs) which preserves two
types of uncertainty (fuzziness and non-specificity) [3]. Also, we extended this idea to a higher
type of fuzzy sets [4].
In the literature, numerous ranking approaches have been developed to rank fuzzy numbers. One
category of these methods ranks fuzzy numbers based on the integration between fuzzy mean and
2. International Journal of Computer Science & Information Technology (IJCSIT) Vol 12, No 6, December 2020
18
spread. The mean of fuzzy number is represented generally as the centroid value of it. The spread
of fuzzy numbers is used to support ranking methods, especially in the cases of embedded fuzzy
numbers with different spreads [5]. Many researchers have dealt with the issue of ranking fuzzy
numbers using centroid point and spread [6, 7, 8, 9]. Chen and Lu presented the ranking method
for fuzzy numbers which consider the middle-point and spread of each α-cut of fuzzy numbers
[10]. Abu Bakar et al. proposed ranking method using five distance-based components for
ranking fuzzy numbers that include centroid point, height and spread of fuzzy numbers [11]. S.M.
Chen and J.H. Chen presented a new ranking method based on the defuzzified values, the heights
and the spreads for generalized fuzzy numbers [12]. Abu Bakar et al. proposed a ranking index
which integrates centroid point and spread for fuzzy numbers [13]. R. Chutia and B. Chutia
discussed the concept of parametric form of fuzzy number and proposed a new ranking method
using the value and the ambiguity of it at different decision levels [14]. The same concept of the
integration between value and ambiguity is applied on intuitionistic fuzzy numbers IFNs. Deng-
Feng Li developed a new methodology for ranking triangular Intuitionistic fuzzy numbers TIFNs
based on a ratio of the value index to the ambiguity index and applied to multi-attribute decision
making problem [15]. P. K. De and D. Das proposed a new ranking approach for trapezoidal
intuitionistic fuzzy numbers (TrIFN) using the value and the ambiguity indexes of them [16].
Some ranking methods provide ability to rank different types of fuzzy numbers using value and
ambiguity. Also, other approaches are intended for one type of fuzzy numbers or one kind of
membership functions. Some researchers proposed ranking method has related to the fuzziness of
fuzzy numbers [17]. In this paper, a new method for ranking shadowed fuzzy numbers is
proposed to order different types of fuzzy numbers and different membership functions. The
value and ambiguity of a shadowed fuzzy number SFN will be defined. The proposed method
uses values and ambiguities of SFNs to rank them. Also, the fuzziness values of SFNs are used to
support ranking approach in the case of the equality of ranking values and ambiguities. The
proposed ranking approach will be presented and applying to different fuzzy numbers ranking
examples. Also, the new algorithm is proposed to solve a hybrid multi-attribute decision making
problem that includes the new SFNs ranking approach. This MADM problem has different data
types include interval numbers, type-l fuzzy numbers with two different membership function
types and intuitionistic fuzzy numbers. The reset of this paper is organized as follows: section 2
introduces the basic definitions of FNs, IFNs and SFNs. Section 3 defines the concepts of the
value, the ambiguity and the fuzziness of SFNs. we introduce proposed steps for new ranking
method of SFNs. Section 4, numerical examples are provided, and a comparative study is
presented with previous methods. Also, we present a hybrid multi-attribute decision making
problem which is solved by using the proposed algorithm. Finally, conclusions and the main
features of the proposed ranking approach are discussed in Section 5.
2. DEFINITIONS AND PRELIMINARIES
2.1. Fuzzy Sets
Fuzzy set provides excellent means to model the linguistic terms by introducing gradual
memberships. The membership function of a fuzzy set A is defined as follows [18]
A:X → [0, 1] (1)
The membership function mapping elements of universe of discourse X to unit interval [0, 1].
The membership function is essential for describing fuzzy set.
3. International Journal of Computer Science & Information Technology (IJCSIT) Vol 12, No 6, December 2020
19
2.1.1. Fuzzy Number
A fuzzy number (FN) Ã is a fuzzy set that is defined on the real numbers scale ℝ with the
following conditions [19, 20].
à is normal, i.e. at least one element xi such that μ(xi) = 1.
à is a convex such that Ã(δx + (1 − δ)y) ≥ min (Ã(x), (y)) ∀x, y ∈ U and δ ∈ [0,1]
where U is a universe of discourse.
The support of à is bounded.
A fuzzy number is important to approximate uncertainty concept about numbers or intervals. The
membership function of the real fuzzy number à is defined by [19]
μà (x) =
{
là (x) if a ≤ x ≤ b
1 if b ≤ x ≤ c,
rÃ(x) if c ≤ x ≤ d,
0 otherwise
(2)
where là and rà are two continuous increasing and decreasing functions for left and right side of
fuzzy number, x ∈ U and a, b, c, d are real numbers.
2.2. Intuitionistic Fuzzy Sets
The concept of intuitionistic fuzzy sets (IFS) is introduced in 1986 by Atanassov. This concept is
defined with membership function and non-membership function [1, 21]. Let A is an
intuitionistic fuzzy set in finite set 𝑋 and is defined as
𝐴 = {< 𝑥, μA(x), vA(x) >|x ∈ 𝑋} (3)
Where μA(x) ∶ X → [0, 1] is the membership function of A, vA(x) ∶ X → [0, 1] is the non-
membership function of A, such that
0 ≤ μA(x) + vA(x) ≤ 1 (4)
For each intuitionistic fuzzy set in X, the intuitionistic index of x in A or a hesitancy degree of x
to A is defined as
πA = 1 − μA(x) − vA(x) (5)
Where 0 ≤ πA ≤ 1. For each x ∈ X
2.2.1. Intuitionistic Fuzzy Numbers (IFN)
An intuitionistic fuzzy subset is called an intuitionistic fuzzy number, if it’s defined on real
numbers domain as [22, 23]
A = {< 𝑥, μA(x), vA(x) >|x ∈ ℝ} (6)
Where
4. International Journal of Computer Science & Information Technology (IJCSIT) Vol 12, No 6, December 2020
20
1. A is normal, i.e. at least two points x0, x1 belong to A such that μA(x0) = 1, vA(x1) = 1.
2. A is convex, i.e. 𝜇 𝐴 is fuzzy convex and 𝑣 𝐴 is fuzzy concave.
3. 𝜇 𝐴 is upper semicontinuous and 𝑣 𝐴 is lower semicontinuous.
4. support(A) = {x ∈ X | 𝑣 𝐴(x) < 1} is bounded.
2.3. Shadowed Sets
In this section, we present some basic concepts of shadowed sets. The shadowed set S is defined
by a mapping from a universal set X to the set of three values as [2, 24, 25]
S ∶ X → {0, 1, [0,1]} (7)
Shadowed set is created by optimization of the threshold α that is calculated using the objective
function as
v(r1) + v(r2) = v(r3) (8)
where v is uncertainty of regions r1, r2, r3. The r1 region is induced by reduce all membership
values less than the threshold α to 0. The r2 region is created by elevated membership values
more than 1-α to 1. The r3 is a shadow region for membership values around 0.5 as illustrated in
Figure 1.
Figure 1: Regions that construct shadowed set
The optimal α can be derived by minimizing the objective function which achieves the balance of
uncertainty with these regions as the following
Vα = |v(r1) + v(r2) − v(r3)| (9)
where Vα is the performance index for the threshold α and α ∈ [0, 0.5). Pedrycz calculated
optimum α for triangular, Gaussian and parabolic fuzzy sets to be 0.4142, 0.395 and 0.405
respectively.
2.3.1. Shadowed Fuzzy Numbers
Shadowed fuzzy numbers (SFNs) are induced from fuzzy numbers [26]. The author proposed an
improved approach to create SFN that preserves uncertainty characteristics of fuzzy number and
can be deduced from type-1 fuzzy numbers and higher type of them e.g., intuitionistic fuzzy sets
(IFS) [4]. The author method is deduced SFN by building core interval and fuzziness intervals as
5. International Journal of Computer Science & Information Technology (IJCSIT) Vol 12, No 6, December 2020
21
in Figure 4 [23]. In the case of type-1 fuzzy numbers, the α –core can be derived using the
following equation [3]
AR(α) − AL(α) + 1 = 2HA (10)
where HA is the non-specificity value of fuzzy set A [27, 23]. AL(α) and AR(α) are left and right
α-cut functions of fuzzy number A. The α-core interval illustrates in Figure 2.
Figure 2: core interval for triangular fuzzy number
The shadow intervals represent fuzziness of fuzzy sets and are calculated for type-1 fuzzy
number as
wL = ∑ fA(xL)
xL
(11)
wR = ∑ fA(xR)
xR
(12)
where xL and xR are respectively the left support and right support of fuzzy number A from core
value. wLandwR, are the left and right fuzziness intervals. The fuzziness intervals represent
uncertainty regions. fA is the fuzziness set of fuzzy number A as in Figure 3. It is proposed by
Tahayori and is defined as the following [28]
fA = (x, fuzz(x)), (13)
fuzz(x) = 1 − |2μA(x) − 1|.
(14)
6. International Journal of Computer Science & Information Technology (IJCSIT) Vol 12, No 6, December 2020
22
Figure 3: Fuzziness set for a triangular fuzzy set
Figure 4: The SFN for triangular fuzzy number
3. THE PROPOSED METHOD FOR RANKING SFNS
In this section, we present a new approach for ranking shadowed fuzzy numbers (SFNs). This
method is used to order fuzzy numbers that SFNs are induced from them. Our proposed method
is based on new concepts of SFN namely the value and the ambiguity of SFN. These concepts
will be proposed based on Delgado et al. definitions of the value and the ambiguity [29].
Definition 3.1:
Let us consider SFN 𝑆𝐴 that parameterizes as the following
SA = (s1
A
, s2
A
, s3
A
, s4
A
)then CA = (s2
A
, s3
A
) is the core interval of SA. Also, let FS
A
is the fuzziness
value ofSA such that [18]
FS
A
= (s2
A
− s1
A
) + (s4
A
− s3
A
) (15)
Definition 3.2:
Let VS
A
is the value of the SFN SA and is defined as
7. International Journal of Computer Science & Information Technology (IJCSIT) Vol 12, No 6, December 2020
23
VS
A
=
(s2
A
+ s3
A
)
2
(16)
Definition 3.3:
Let SA is the SFN which induced from fuzzy number A. Then the ambiguity value uS
A
of SA is
defined as
uS
A
= s3
A
− s2
A (17)
Definition 3.4:
Let SAand SB are two SFNs which induced from fuzzy numbers A and B. Let AmbS
A
and AmbS
B
are the ambiguity indexes for SA and SB respectively and are defined as
AmbS
A
= 1 −
uS
A
uS
A
+ uS
B
(18)
and
AmbS
B
= 1 −
uS
B
uS
A
+ uS
B
(19)
where uS
A
and uS
B
are ambiguity values for SA and SB respectively
Definition 3.5:
Let SA is the SFN which induced from fuzzy number A. Then the rank value RS
A
of SAis defined
as
RS
A
= VS
A
+ AmbS
A
× λ (20)
where VS
A
is the value of SA , AmbS
A
is the ambiguity index of SA and λ is an attitude value (AV)
where λ ∈ [0.5, 1]
An attitude value (AV) represents the attitude of the decision maker against ambiguity. In the
case of 𝜆 = 1, this value indicates a decision maker’s optimistic attitude towards ambiguity. If
λ ∈ ]0.5, 1[ then AV refers to decision maker’s neutral attitude towards ambiguity. When λ =
0.5, this value indicates decision maker’s pessimistic attitude towards ambiguity. In rank
examples, we prefer to λ = 0.5 which more reasonable.
A ranking procedure of two SFNs 𝑺 𝑨 and 𝑺 𝑩, as the following steps:
Step 1: Calculate VS
A
, VS
B
, AmbS
A
and AmbS
B
for SA and SB using (16), (17), (18) and (19).
Step 2: Calculate rank values for SA and SB with an attitude value (AV) λ and λ ∈ [0.5, 1] using
(20).
8. International Journal of Computer Science & Information Technology (IJCSIT) Vol 12, No 6, December 2020
24
Step 3: the ranking of two SFNs is according to
If RS
A
> RS
B
then SA > SBand A > 𝐵.
If RS
A
< RS
B
then SA < SBand A < 𝐵.
If RS
A
= RS
B
then calculate FS
A
and FS
B
of two SFNs SA and SB using (15) and
If FS
A
< FS
B
then SA > SBand A > 𝐵
If FS
A
> FS
B
then SA < SBand A < 𝐵
If FS
A
= FS
B
then SA = SBand A = B
4. NUMERICAL EXAMPLES
In this section, five numerical examples are used to demonstrate the new proposed method
reliability for ranking the fuzzy numbers. The proposed method is compared with other existing
ranking methods that integrate the centroid point and the spread of fuzzy numbers. Also, the
comparison extends to other methods that have ordered fuzzy numbers using their values and
ambiguities. This comparative study is summarized in Table 1. A new additional example will be
presented in this section to illustrate the characteristic of the new method for ranking different
types of IFNs. Also, a hybrid multi-attribute decision making problem will be solved using the
new approach.
Example 1: Let A and B are two triangular fuzzy numbers TFNs such that A = (0.1, 0.4, 0.7) and
B = (0.3, 0.4, 0.5) as in Figure 5. Using our method, Two SFNs SA and SB are induced from two
fuzzy numbers A and B such that SA = (0.1, 0.25, 0.55, 0.7) and SB = (0.3, 0.35, 0.45, 0.5)[23].
We use (20) to obtain the rank values of SA and SB using λ = 0.5 as the following:
RA = 0.525 and RB= 0.775. Based on these results, the ranking of fuzzy numbers is A<B.
Figure 5: Fuzzy numbers A and B of Example 1
Example 2:
Let A is a trapezoidal fuzzy number TrFN and B is a triangular fuzzy numbers TFN such thatA=
(0.1, 0.2, 0.4, 0.5) and B = (0.1, 0.3, 0.5) as in Figure 6.
Using the author method, two SFNs 𝑆𝐴 and 𝑆 𝐵 are obtained from two FNs A and B such that SA =
(0.1, 0.15, 0.45, 0.5) and SB = (0.1, 0.2, 0.4, 0.5) [3]. The rank values for SA and SB are
calculated with λ = 0.5 and the results are RA = 0.5 and RB= 0.6. The order of fuzzy numbers is
A<B.
9. International Journal of Computer Science & Information Technology (IJCSIT) Vol 12, No 6, December 2020
25
Figure 6: Fuzzy numbers A and B of Example 2
Example 3:
Consider the following two fuzzy numbers with different heights as shown in Figure 7.
A = (0.1, 0.4, 0.7) and height(A) = 0.8, B = (0.1, 0.4, 0.7) and height(B) = 1
Using the same calculation steps as the previous examples, we getSA = (0.08 0.2538 0.5462
0.72); SB = (0.1038, 0.2538, 0.5462, 0.6962); RA = 0.65 ;RB= 0.65.
Based on these results, the two ranking values are equal so, we use (15) to obtain fuzziness values
for each SFN. The fuzziness values are FS
A
= 0.345 and FS
B
= 0.3 then the ranking of fuzzy
numbers is A<B.
Figure 7: Fuzzy numbers A and B of Example 3
Example 4:
Let A and C are TrFNs and B is TFN such that A= (0, 0.4, 0.6, 0.8), B = (0.2, 0.5, 0.9) and C=
(0.1, 0.6, 0.7, 0.8), as in Figure 8.
Using the author method, we induce three SFNs where SA = (0.0034, 0.2034, 0.6983, 0.7983) ,
SB = (0.2052, 0.3552, 0.6931, 0.8931) and SC = (0.1124, 0.3624, 0.7475, 0.7975)[3]. We use
(20) to obtain the rank values for SA , SB and SC using λ = 0.5 as the following:
B
A
μ(x)
x
10. International Journal of Computer Science & Information Technology (IJCSIT) Vol 12, No 6, December 2020
26
RA = 0.7477, RB= 0.8854 and RC = 0.8968. According to these results, the ranking of fuzzy
numbers is A < B < 𝐶.
Figure 8: Fuzzy numbers A , B and C of Example 4
Table 1: Comparative results of the proposed ranking method with the existing ranking methods
Examples Chen and Sanguansat
[5]
Abu Bakar and
Gegov [13]
R.Chutia and
B.Chutia [14]
Proposed method
1 A ≈ B A > 𝐵 A < 𝐵 A<B
2 A ≈ B A > 𝐵 A < 𝐵 A<B
3 A < B A < 𝐵 A < B A<B
4 A < B < 𝐶 A < B < 𝐶 A < B < 𝐶 A < B < 𝐶
Example 5:
Let A1, A2 and A3 are three trapezoidal intuitionistic fuzzy numbers (TrIFNs),where A1=
[(0.1,0.3,0.5,0.8), (0.1,0.3,0.5,0.8) 0.5,0.2] , A2= [(0.2 ,0.3,0.6 ,0.9), (0.2 ,0.3,0.6 ,0.9) 0.6 ,0.4]
and A3= [(0.1, 0.5,0.7, 0.9), (0.1, 0.5, 0.7, 0.9) 0.5,0.3] [16]. We induce three SFNs using author
method such that SA1
= (0.15, 0.27, 0.54, 0.725), SA2
= (0.23, 0.29, 0.64, 0.81) and SA3
= (0.23,
0.47, 0.71, 0.83)[4]. The rank values of SA1
, SA2
and SA3
are calculated by using (20) withλ =
0.5 and the results are RA1
= 0.748, RA2
= 0.7615 and RA3
= 0.9505. According to these results,
the order of three TrIFNs is A1 < A2 < A3.
Example 6:
In this example, we apply the proposed method on the case of ranking three different types of
intuitionistic fuzzy numbers.
Let B1, B2 and B3 are three different types of intuitionistic fuzzy numbers(IFNs),where B1=
[(2,3,5,6), (1,3,5,7) 1,0] is trapezoidal intuitionistic fuzzy number(TrIFN), B2 = [(m= 4, σ = 0.5)
,(m= 4, σ = 1) 1,0] is Gaussian intuitionistic fuzzy number(GIFN), and B3= [(3,6,9) , (4,6,8) 1,0]
is triangular intuitionistic fuzzy numbers (TIFN) as in Figure 9. Three SFNs are induced using
authors method as S1= [1.615, 2.31, 5.7, 6.386] , S2= [2.12, 3.17, 4.83, 5.88] and S3= [3.653,
4.93, 7.07, 8.35][4]. Using the new ranking method, The rank values of S1 , S2 and S3 are R1 =
4.2693 , R2= 4.3846 and R3 = 6.3512 with λ = 0.5 then the order of SFNs is S3>S2>S1.
According to this result, the ranking for three IFNs is B3> B2> B1.
A
B
C
μ(x)
x
11. International Journal of Computer Science & Information Technology (IJCSIT) Vol 12, No 6, December 2020
27
Figure 9: Three IFNs B1, B2 and B3
4.1. Discussion of the Results of Examples
1. For the fuzzy numbers A and B as in the example 1, Chen and Sanguansat method fail to
discriminate between two fuzzy numbers which they have the same value as in Table 1.
Abu Baker and Gegov method prefers fuzzy number A to B but the ambiguity of B is less
than A. R.Chutia and B.Chutia method and the proposed method get the same ranking
order, which is consistent with intuition. The two fuzzy numbers have the same value, but
they are different in ambiguities.
2. In the example 2, the correct ranking order of fuzzy numbers for this case should be A < B
due to the same reasons mentioned in the previous example.
3. In both examples 3 and 4, the order by the proposed method is consistent with other
methods as illustrated in Table 1.
4. The ranking result for example 5 like the ranking order from D. Das method [16]. This
result explains that the proposed approach works well.
5. Example 6 is presented to explain the case of ranking three different types of intuitionistic
fuzzy numbers.
4.2. Application of the Proposed Method in MADM Problem
The multiple attribute decision-making problems under fuzzy environment have been studied
extensively by many authors. In this section, we will focus on the personnel selection problem,
which were presented by both Mahdavi and Deng-Feng Li [15] [30] [31]. Also, the problem was
solved using value and ambiguity indexes in [31]. We develop a new algorithm to solve the
personnel selection problem in the case of hybrid data types such as interval numbers, type-1
fuzzy numbers and IFNs.
Suppose that a software company desires to hire a system analyst [31]. After preliminary
screening, three candidates’ alternatives A1, A2 and A3 remain for further evaluation. The
decisionmaking committee assesses the three candidates. The decision makers consider five
benefit criteria to evaluate these candidates, including, emotional steadiness (C1), oral
communication skill (C2), personality (C3), experience (C4), and self-confidence (C5).
𝐵1
𝐵2
𝐵3
𝜇(𝑥)
𝑥
12. International Journal of Computer Science & Information Technology (IJCSIT) Vol 12, No 6, December 2020
28
The evaluations for these criteria vary between interval numbers, type-1 fuzzy numbers and
intuitionistic fuzzy numbers. The assessment for C1 is represented by triangular fuzzy numbers
TFNs. The criterion C2 is evaluated using intervals numbers INs. The assessment for the criterion
C3 can be represented by Gaussian fuzzy numbers GFNs. The triangular intuitionistic fuzzy
numbers TIFNs are used for C4 and C5 criteria. The evaluation values are given in Table (2) by
the decision makers. The crisp weight wj is assigned to each criterion such that wj ∈ [0,1] and
∑ wj = 1n
j=1 .
Table 2: The evaluation values of the three candidates under all criteria
C1 C2 C3 C4 C5
A1 (5.7, 7.7, 9.3) [5, 9] (7.7; 0.7) [(8.33,9.67,10);0.6,0.4] [(3,5,7);0.6,0.3]
A2 (6.3, 8.3, 9.7) [9, 10] (9.7; 0.12) [(8,9,10);0.6,0.3] [(7,9,10);0.6,0.2]
A3 (6.3, 8, 9) [7, 10] (9; 0.46) [(6,8,9);0.6,0.2] [(6.3,8.3,9.7);0.7,0.2]
We propose the following steps to solve this problem as follows:
Step 1: The SFNs are obtained using author approach for the type-l fuzzy numbers as in Table 2
for criteria (C1) and (C3) [3]. Also, the author method is applied to transform IFNs to SFNs for
criteria (C4) and (C5) [4]. The new decision matrix is obtained as in Table 3.
Table 3: The evaluation values of decision table using SFNs
C1 C2 C3 C4 C5
A1 (5.82, 6.82, 8.4,
9.2)
[5, 9] (6.194, 6.9, 8.5,
9.122)
(8.591, 9.35, 9.75,
9.937)
(3.344, 4.53, 5.47,
6.656)
A2 (6.42,7.42, 8.92,
9.62)
[9, 10] (9.424, 9.55,
9.85, 9.972)
(8.147, 8.74, 9.26,
9.853)
(7.239, 8.46, 9.27,
9.88)
A3 (6.39, 7.24, 8.45,
8.95)
[7, 10] (7.979, 8.46,
9.54, 9.987)
(6.239, 7.46, 8.27,
8.88)
(6.525,7.72, 8.71,
9.546)
Step 2: The normalized SFNs are calculated using (21) and normalized interval numbers are
obtained using (22). The results are displayed in Table 4.
Definition 4.1:
Let rij is the normalized evaluation value for the jth
benefit criteria and two cases can be defined
for it. In the case of SFN sij= (sij1, sij2, sij3, sij4) is defined as
rij = [
sij1
s̅j4
,
sij2
s̅j4
,
sij3
s̅j4
,
sij4
s̅j4
] (21)
where s̅j4 = maxi{sij4|i = 1,2, … . . , m}, j = 1,2, … . . , n. In the case of interval number IN tij =
[tij
L
,tij
R
] is defined as the following
rij = [
tij
L
t̅j
R ,
tij
R
t̅j
R ]
(22)
Where t̅j
R
= maxi{tij
R
|i = 1,2, … . . , m}, j = 1,2, … . . , n.
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29
Table 4: The normalized evaluation values of decision table
C1 C2 C3 C4 C5
A1 (0.6, 0.71, 0.87,
0.96)
[0.5, 0.9] (0.62, 0.69,
0.85, 0.91)
(0.86, 0.94, 0.98, 1) (0.34, 0.46, 0.55,
0.67)
A2 (0.67, 0.77,
0.93, 1)
[0.9, 1] (0.94, 0.96,
0.99, 1)
(0.82, 0.88, 0.93,
0.99)
(0.73, 0.86, 0.94,
1)
A3 (0.66,0.75,
0.88, 0.93)
[0.7, 1] (0.8, 0.85,
0.96, 1)
(0.63, 0.75, 0.83,
0.89)
(0.66, 0.78, 0.88,
0.97)
Step 3: Let w1= 0.14, w2= 0.3, w3= 0.12, w4= 0.3 and w5= 0.14 are weights of attributes. The
weighted normalized SFNs are obtained using (23) and the weighted normalized interval numbers
are calculated using (24). The results shown in Table 5.
Definition 4.2:
Let wrij is the weighted normalized evaluation value for the 𝑗 𝑡ℎ
benefit criteria and we define
two types of it. In the case of SFN is defined as
wrij = [s̅ij1 × wj, s̅ij2 × wj, s̅ij3 × wj, s̅ij4 × wj] (23)
Where wj is the weight value for the jth
criteria such that j = 1,2,…,n. and s̅ij =
[s̅ij1, s̅ij2, s̅ij3, s̅ij4] is a normalized SFN.
In the case of interval number IN, the 𝑤𝑟𝑖𝑗 is defined as
wrij = [t̅ij
L
× wj, t̅ij
R
× wj]. (24)
Where wj is the weight value for thejth
criteria andt̅ij = [t̅ij
L
,t̅ij
R
] is a normalized IN.
Table 5: The weighted normalized evaluation values of decision table
C1 C2 C3 C4 C5
A1 (0.084,0.099,
0.122, 0.134)
[0.15,
0.27]
(0.074, 0.0828,
0.102, 0.109)
(0.258, 0.282,
0.294, 0.3)
(0.048, 0.064,
0.077, 0.094)
A2 (0.094,0.108, 0.13,
0.14)
[0.27, 0.3] (0.113, 0.115,
0.119, 0.12)
(0.246, 0.264,
0.279, 0.297)
(0.102, 0.12,
0.132, 0.14)
A3 (0.092,0.105,
0.123, 0.13)
[0.21, 0.3] (0.096, 0.102,
0.115, 0.12)
(0.189, 0.225,
0.249, 0.267)
(0.092, 0.109,
0.123, 0.136)
Step 4: we use (25) and (26) to create the aggregation values of weighted normalized SFNs for
three alternatives and the results shown in Table 6.
Definition 4.3:
Let dsi is an aggregation decision values of alternatives Ai where i = 1, 2, … … , m are obtained as
dsi = ∑ wrij
n
j=1
(25)
14. International Journal of Computer Science & Information Technology (IJCSIT) Vol 12, No 6, December 2020
30
where dsi (i = 1,2, … . . , m) are SFNs and wrij is the weighted normalized evaluation value for
the 𝑗 𝑡ℎ
benefit criteria. This operation is defined between SFNs and INs as the following
T + S = (s̿1, s̿2, s̿3, s̿4) (26)
such that 𝑆 is a shadowed fuzzy number SFN, s̿1 = (s1 + tL) , s̿2 = (s2 + tL) , s̿3 = (s3 + tR) ,
s̿4 = (s4 + tR) and T = [tL ,tR] is an IN.
Table 6: The aggregation values of weighted normalized SFNs
A1 (0.614, 0.679, 0.865, 0.907)
A2 (0.825, 0.877, 0.96, 0.997)
A3 (0.68, 0.751, 0.911, 0.953)
Step 5: finally, we use the proposed approach to rank SFNs resulting from step 4. The rank values
for alternatives A1,A2and A3 are RA1
= 1.0552, RA2
= 1.3218 and RA3
= 1.1445 with λ = 0.5.
The order of the three alternatives is A2 > A3 > A1.
4.3. Comparison Analysis of the Result
Based on the previous results of the MADM problem, the following remarks are found:
In Mahdavi et al. approach, the ranking of three alternatives is A2 > A3 > A1 which it is the
same result of the new approach [30]. In Mahdavi method, the MADM problem is presented with
the type-1 triangular fuzzy numbers and orders alternatives using similarity values to ideal
solution.
In Deng-Feng Li et al. method, the ranking of three alternatives is A3 > A1 > A2 with the weight
λ∈ [0,0.793] [31]. This weight represents the decision maker’s preference information. In the
case of λ∈ (0.793, 1], the order of three alternatives is A1 > A3 > A2.
In Deng-Feng Li approach, the data inputs of MADM problem are represented by IFNs and the
value-index and the ambiguity-index of IFNs are used to rank IFNs. According to Deng-Feng Li
et al. method, if ratings of the alternatives on the attributes are reduced to type-1 triangular fuzzy
numbers then the ranking order is A2 > A3 > A1 [15] [31]. This result like the ranking order
from the proposed method.
4.4. Discussion
Previous ranking approaches of fuzzy numbers with only one type of them (type-1 or higher
type). These techniques were difficult to apply to complex decision-making problems that contain
different types of fuzzy numbers. In the new approach, we unified the different types of fuzzy
numbers using shadowed fuzzy numbers and preserve uncertainty characteristics of fuzzy
numbers at the same time.
The new approach is more flexible to rank fuzzy numbers with different membership functions
than previous methods.
15. International Journal of Computer Science & Information Technology (IJCSIT) Vol 12, No 6, December 2020
31
5. CONCLUSION
This paper is proposed a new approach to rank SFNs. This method is applied to order fuzzy
numbers from type-1 and higher type which transforms different types of fuzzy numbers to SFNs.
The new ranking approach induces the rank values which integrates the value and the ambiguity
of SFN. Also, it weighted an ambiguity value using the decision maker's attitude value. In the
case of equal rank values, the fuzziness values are used to rank SFNs. The new algorithm is
applied for different examples of ranking type-1 fuzzy numbers FNs and intuitionistic fuzzy
numbers IFNs. The ranking results of the proposed method are compared with previous ranking
approaches for type-1 FNs and IFNs. Also, the new algorithm is applied to solve a hybrid multi-
attribute decision making problem where SFNs are used to unify the uncertain types of linguistic
terms. The new method of ranking is applied to rank alternatives.
The new algorithm is more efficient and more flexible than previous methods which solved the
same problem with one type of linguistic terms.
The future work will focus on verifying the usefulness of the new approach with other multi-
criteria techniques. Also, we can study more applications of the new method with more decision-
making problems.
REFERENCES
[1] Krassimir T. Atanassov, (1986) "Intuitionistic fuzzy sets", Fuzzy Sets and Systems Vol. 20, Issue (1),
pp.87 -96.
[2] Witold Pedrycz,(1998) "Shadowed Sets: Representing and Processing Fuzzy Sets", IEEE
Transactions on systems, man, and cybernetics-part B: Cybernetics, Vol. 28, No. I.
[3] Mohamed A. H. EI_Hawy, Hesham A. Hassan, Hesham A. Hefny, Khaled T. Wassif , (2015) " An
Improved Fuzzy Number Approximation using Shadowed Sets" , International Journal of Computer
Applications (0975 - 8887), Vol. 118 , No.25, pp. 9-15.
[4] Mohamed A. H. EI_Hawy, Hesham A. Hassan, Hesham A. Hefny, Khaled T. Wassif ,(2015)"A
Proposed Shadowed Intuitionistic Fuzzy Numbers ",Computer Engineering & Systems (ICCES),
10th, IEEE.
[5] Shyi-Ming Chen, Kata Sanguansat, (2011) ”Analyzing fuzzy risk based on a new fuzzy ranking
method between generalized fuzzy numbers”, Expert Systems with Applications, Vol. 38, Issue 3, pp.
2163-2171.
[6] A.S.A. Bakar,D.Mohamad and N.H. Sulaiman, (2010) “Ranking fuzzy numbers using similarity
measure with centroid”, IEEE International Conference on Science and Social Research, pp.58–63.
[7] Chen, S., Chen, S.,(2007), “Fuzzy risk analysis based on the ranking of generalized trapezoidal fuzzy
numbers.”, Applied Intelligence Vol. 26, Issue 1.
[8] S.M. Chen and K. Sanguansat,(2011) “Analyzing fuzzy risk based on a new fuzzy ranking method
between generalized fuzzy numbers”, Expert System with Applications, Vol. 38, Issue 3, pp. 2163–
2171.
[9] Shi-jay chen & shyi-mingchen ,(2003) “a new method for handling multicriteria fuzzy decision-
making problems using fn-iowa operators”, Cybernetics and Systems, Vol. 34, Issue 2.
[10] L.H. Chen and H.W. Lu, (2002) “The preference order of fuzzy numbers”, Computers &
Mathematics with Applications, Vol. 44, Issues 10–11, pp. 1455-1465.
[11] A.S.A. Bakar, D. Mohamad and N.H. Sulaiman, (2012) “Distance –based ranking fuzzy numbers”,
Advances in Computational Mathematics and Its Applications, Vol. 1, No. 3, pp.146–150.
[12] Shyi-Ming Chen, Jim-Ho Chen, (2009) “Fuzzy risk analysis based on ranking generalized fuzzy
numbers with different heights and different spreads”, Expert Systems with Applications, Vol.36,
Issue 3, Part 2, pp. 6833-6842.
[13] Ahmad Syafadhli Abu Bakar , Alexander Gegov, (2014) “Ranking of Fuzzy Numbers Based on
Centroid Point and Spread”, Journal of Intelligent & Fuzzy Systems, vol. 27, no. 3, pp. 1179-1186.
16. International Journal of Computer Science & Information Technology (IJCSIT) Vol 12, No 6, December 2020
32
[14] RituparnaChutia, BijitChutia, (2017) ”A new method of ranking parametric form of fuzzy numbers
using value and ambiguity”, Applied Soft Computing,Vol. 52, pp. 1154-1168.
[15] Deng-Feng Li,(2010) ”A ratio ranking method of triangular intuitionistic fuzzy numbers and its
application to MADM problems”, Computers & Mathematics with Applications ,Vol. 60, Issue 6, pp.
1557-1570.
[16] P. K. De and D. Das, (2012) "Ranking of trapezoidal intuitionistic fuzzy numbers", 12th International
Conference on Intelligent Systems Design and Applications (ISDA), Kochi, pp.184-188.
[17] Zhong-Xing Wang, Jian Li, (2009) “The Method for Ranking Fuzzy Numbers Based on the
Approximate Degree and the Fuzziness”, Sixth International Conference on Fuzzy Systems and
Knowledge Discovery, IEEE.
[18] L. A. Zadeh, (1965) "Fuzzy sets." Information and Control, 8(3), pp.338-353.
[19] A. Kaufmann, M.M. Gupta,( 1985) “Introduction to Fuzzy Arithmetic Theory and Applications”, Van
Nostrand Reinhold, New York.
[20] George. J. Klir, Bo. Yuan, (1995) "Fuzzy Sets and Fuzzy Logic Theory and Applications", Prentice
Hall press.
[21] Krassimir T. Atanassov,(1999) "Intuitionistic Fuzzy Sets. Theory and Applications", Physica-Verlag,
Heidelberg New York.
[22] P.Grzegorzewski, (2003) ” Distances and orderings in a family of intuitionistic fuzzy numbers”,.In:
EUSFLAT Conf., pp. 223–227.
[23] M. Kumar and S.P. Yadav, (2012) “Analyzing Fuzzy System Reliability Using Arithmetic Operations
on Different Types of Intuitionistic Fuzzy Numbers” K. Deep et al. (Eds.): Proceedings of the
International Conference on SocProS 2011, AISC 130, pp. 725–736. Springer India.
[24] Witold Pedrycz, (2009) "From Fuzzy Sets to Shadowed Sets: Interpretation and Computing",
international journal of intelligent systems, Vol. 24, pp. 48-61.
[25] Yiyu Yao, Shu Wang, XiaofeiDeng,(2017) “Constructing shadowed sets and three-way
approximations of fuzzy sets “, Information Sciences, Vol. 412–413, pp. 132-153.
[26] OlgierdHryniewicz,(2006) "An Evaluation of the Reliability of Complex Systems Using Shadowed
Sets and Fuzzy Lifetime Data", International Journal of Automation and Computing, Vol. 2 ,pp. 145-
150.
[27] George J. Klir, Mark l. Wierman,(1999) "Uncertainty –Based Information Elements of Generalized
Information Theory" Springer-Verlag Berlin Heidelberg GmbH.
[28] HoomanTahayori, Alireza Sadeghian, Witold Pedrycz, (2013) "lnduction of Shadowed Sets Based on
the Gradual Grade of Fuzziness", Fuzzy Systems, IEEE Transactions on , vo1.21, no.5, pp.937-949.
[29] M. Delgado, M.A. Vila, W. Voxman, (1998)”On a canonical representation of fuzzy numbers”,
Fuzzy Sets and Systems, Vol. 93, Issue 1, pp. 125-135.
[30] Iraj Mahdavi, Nezam Mahdavi-Amiri, Armaghan Heidarzade, Rahele Nourifar,(2008 )“Designing a
model of fuzzy TOPSIS in multiple criteria decision making”, Applied Mathematics and
Computation,Vol. 206, Issue 2, pp. 607-617.
[31] Deng Feng Li, Jiang Xia Nan & Mao Jun Zhang ,(2010)” A Ranking Method of Triangular
Intuitionistic Fuzzy Numbers and Application to Decision Making”, International Journal of
Computational Intelligence Systems, Vol. 3,Issue 5, pp. 522-530.