Search results for: hesitant fuzzy sets

Authors: G. S. Thakur

Abstract:

In this paper, four new operators (O1, O2, O3, O4) are proposed, defined and considered to study the new properties and identities on hesitant fuzzy sets. These operators are useful for different operation on hesitant fuzzy sets. The various theorems are proved using the new operators. The study of the proposed new operators has opened a new area of research and applications.

Keywords: vague sets, hesitant fuzzy sets, intuitionistic fuzzy set, fuzzy sets, fuzzy multisets

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Authors: K. M. Alsager, N. O. Alshehri

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In this paper, we proposed the notion of single valued neutrosophic hesitant fuzzy rough set, by combining single valued neutrosophic hesitant fuzzy set and rough set. The combination of single valued neutrosophic hesitant fuzzy set and rough set is a powerful tool for dealing with uncertainty, granularity and incompleteness of knowledge in information systems. We presented both definition and some basic properties of the proposed model. Finally, we gave a general approach which is applied to a decision making problem in disease diagnoses, and demonstrated the effectiveness of the approach by a numerical example.

Keywords: single valued neutrosophic fuzzy set, single valued neutrosophic fuzzy hesitant set, rough set, single valued neutrosophic hesitant fuzzy rough set

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Authors: Chen T. Chen, Hui L. Cheng

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Due to the different knowledge, experience and expertise of experts, they usually provide the different opinions in the group decision-making process. Therefore, it is an important issue to reach the group consensus of opinions of experts in group multiple-criteria decision-making (GMCDM) process. Because the subjective opinions of experts always are fuzziness and uncertainties, it is difficult to use crisp values to describe the real opinions of experts or decision-makers. It is reasonable for experts to use the linguistic variables to express their opinions. The hesitant fuzzy set are extended from the concept of fuzzy sets. Experts use the hesitant fuzzy sets can be flexible to describe their subjective opinions. In order to aggregate the hesitant fuzzy linguistic variables of all experts effectively, an adjustment method based on distance function will be presented in this paper. Based on the opinions adjustment method, this paper will present an effective approach to adjust the hesitant fuzzy linguistic variables of all experts to reach the group consensus. Then, a new hesitant linguistic GMCDM method will be presented based on the group consensus of hesitant fuzzy linguistic variables. Finally, an example will be implemented to illustrate the computational process to enhance the practical value of the proposed model.

Keywords: group multi-criteria decision-making, linguistic variables, hesitant fuzzy linguistic variables, distance function, group consensus

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Authors: O. Poleshchuk, E. Komarov

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This paper presents a regression model for interval type-2 fuzzy sets based on the least squares estimation technique. Unknown coefficients are assumed to be triangular fuzzy numbers. The basic idea is to determine aggregation intervals for type-1 fuzzy sets, membership functions of whose are low membership function and upper membership function of interval type-2 fuzzy set. These aggregation intervals were called weighted intervals. Low and upper membership functions of input and output interval type-2 fuzzy sets for developed regression models are considered as piecewise linear functions.

Keywords: interval type-2 fuzzy sets, fuzzy regression, weighted interval

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Authors: Irina Khutsishvili

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The presented study develops a decision support methodology for multi-criteria group decision-making problem. The proposed methodology is based on the TOPSIS (Technique for Order Performance by Similarity to Ideal Solution) approach in the hesitant fuzzy environment. The main idea of decision-making problem is a selection of one best alternative or several ranking alternatives among a set of feasible alternatives. Typically, the process of decision-making is based on an evaluation of certain criteria. In many MCDM problems (such as medical diagnosis, project management, business and financial management, etc.), the process of decision-making involves experts' assessments. These assessments frequently are expressed in fuzzy numbers, confidence intervals, intuitionistic fuzzy values, hesitant fuzzy elements and so on. However, a more realistic approach is using linguistic expert assessments (linguistic variables). In the proposed methodology both the values and weights of the criteria take the form of linguistic variables, given by all decision makers. Then, these assessments are expressed in triangular fuzzy numbers. Consequently, proposed approach is based on triangular hesitant fuzzy TOPSIS decision-making model. Following the TOPSIS algorithm, first, the fuzzy positive ideal solution (FPIS) and the fuzzy negative-ideal solution (FNIS) are defined. Then the ranking of alternatives is performed in accordance with the proximity of their distances to the both FPIS and FNIS. Based on proposed approach the software package has been developed, which was used to rank investment projects in the real investment decision-making problem. The application and testing of the software were carried out based on the data provided by the ‘Bank of Georgia’.

Keywords: fuzzy TOPSIS approach, investment project, linguistic variable, multi-criteria decision making, triangular hesitant fuzzy set

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Authors: Irina Khutsishvili

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In the present research, a decision support methodology for the multi-attribute group decision-making (MAGDM) problem is developed, namely for the selection of investment projects. The objective of the investment project selection problem is to choose the best project among the set of projects, seeking investment, or to rank all projects in descending order. The project selection is made considering a set of weighted attributes. To evaluate the attributes in our approach, expert assessments are used. In the proposed methodology, lingual expressions (linguistic terms) given by all experts are used as initial attribute evaluations, since they are the most natural and convenient representation of experts' evaluations. Then lingual evaluations are converted into trapezoidal fuzzy numbers, and the aggregate trapezoidal hesitant fuzzy decision matrix will be built. The case is considered when information on the attribute weights is completely unknown. The attribute weights are identified based on the De Luca and Termini information entropy concept, determined in the context of hesitant fuzzy sets. The decisions are made using the extended Technique for Order Performance by Similarity to Ideal Solution (TOPSIS) method under a hesitant fuzzy environment. Hence, a methodology is based on a trapezoidal valued hesitant fuzzy TOPSIS decision-making model with entropy weights. The ranking of alternatives is performed by the proximity of their distances to both the fuzzy positive-ideal solution (FPIS) and the fuzzy negative-ideal solution (FNIS). For this purpose, the weighted hesitant Hamming distance is used. An example of investment decision-making is shown that clearly explains the procedure of the proposed methodology.

Keywords: In the present research, a decision support methodology for the multi-attribute group decision-making (MAGDM) problem is developed, namely for the selection of investment projects. The objective of the investment project selection problem is to choose the best project among the set of projects, seeking investment, or to rank all projects in descending order. The project selection is made considering a set of weighted attributes. To evaluate the attributes in our approach, expert assessments are used. In the proposed methodology, lingual expressions (linguistic terms) given by all experts are used as initial attribute evaluations since they are the most natural and convenient representation of experts' evaluations. Then lingual evaluations are converted into trapezoidal fuzzy numbers, and the aggregate trapezoidal hesitant fuzzy decision matrix will be built. The case is considered when information on the attribute weights is completely unknown. The attribute weights are identified based on the De Luca and Termini information entropy concept, determined in the context of hesitant fuzzy sets. The decisions are made using the extended Technique for Order Performance by Similarity to Ideal Solution (TOPSIS) method under a hesitant fuzzy environment. Hence, a methodology is based on a trapezoidal valued hesitant fuzzy TOPSIS decision-making model with entropy weights. The ranking of alternatives is performed by the proximity of their distances to both the fuzzy positive-ideal solution (FPIS) and the fuzzy negative-ideal solution (FNIS). For this purpose, the weighted hesitant Hamming distance is used. An example of investment decision-making is shown that clearly explains the procedure of the proposed methodology.

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Authors: Syed Talib Abbas Raza, Tahseen Ahmed Jilani, Saleem Abdullah

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This paper presents interval-valued intuitionistic fuzzy soft sets based TOPSIS method for group decision making. The interval-valued intuitionistic fuzzy soft set is a mutation of an interval-valued intuitionistic fuzzy set and soft set. In group decision making problems IVIFSS makes the process much more algebraically elegant. We have used weighted arithmetic averaging operator for aggregating the information and define a new Hybrid Score Function as metric tool for comparison between interval-valued intuitionistic fuzzy values. In an illustrative example we have applied the developed method to a criminological problem. We have developed a group decision making model for integrating the imprecise and hesitant evaluations of multiple law enforcement agencies working on target killing cases in the country.

Keywords: group decision making, interval-valued intuitionistic fuzzy soft set, TOPSIS, score function, criminology

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Authors: Hossein Gitinavard, Mohammad Hossein Fazel Zarandi

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In the last decades, concerns about the environmental issues lead to professional and academic efforts on green supplier selection problems. In this sake, one of the main issues in evaluating the green supplier selection problems, which could increase the uncertainty, is the preferences of the experts' judgments about the candidate green suppliers. Therefore, preparing an expert system to evaluate the problem based on the historical data and the experts' knowledge can be sensible. This study provides an expert evaluation system to assess the candidate green suppliers under selected criteria in a multi-period approach. In addition, a ranking approach under interval-valued hesitant fuzzy set (IVHFS) environment is proposed to select the most appropriate green supplier in planning horizon. In the proposed ranking approach, the IVHFS and the last aggregation approach are considered to margin the errors and to prevent data loss, respectively. Hence, a comparative analysis is provided based on an illustrative example to show the feasibility of the proposed approach.

Keywords: green supplier selection, expert system, ranking approach, interval-valued hesitant fuzzy setting

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Authors: Abdelati El Allaoui, Said Melliani, Lalla Saadia Chadli

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The objective of this paper is to study the existence and uniqueness of Mild solutions for a complex fuzzy evolution equation with nonlocal conditions that accommodates the notion of fuzzy sets defined by complex-valued membership functions. We first propose definition of complex fuzzy strongly continuous semigroups. We then give existence and uniqueness result relevant to the complex fuzzy evolution equation.

Keywords: Complex fuzzy evolution equations, nonlocal conditions, mild solution, complex fuzzy semigroups

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Authors: Tumadhir Fahim M Alsulami

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The concept of this paper builds on connecting between two important notions, fuzzy ideals of BCK-algebras and derivation of BCI-algebras. The result we got is a new concept called derivation fuzzy ideals of BCI-algebras. Followed by various results and important theorems on different types of ideals. In chapter 1: We presented the basic and fundamental concepts of BCK\ BCI- algebras as follows: BCK/BCI-algebras, BCK sub-algebras, bounded BCK-algebras, positive implicative BCK-algebras, commutative BCK-algebras, implicative BCK- algebras. Moreover, we discussed ideals of BCK-algebras, positive implicative ideals, implicative ideals and commutative ideals. In the last section of chapter 1 we proposed the notion of derivation of BCI-algebras, regular derivation of BCI-algebras and basic definitions and properties. In chapter 2: It includes 3 sections as follows: Section 1 contains elementary concepts of fuzzy sets and fuzzy set operations. Section 2 shows O. G. Xi idea, where he applies fuzzy sets concept to BCK-algebras and we studied fuzzy sub-algebras as well. Section 3 contains fuzzy ideals of BCK-algebras basic definitions, closed fuzzy ideals, fuzzy commutative ideals, fuzzy positive implicative ideals, fuzzy implicative ideals, fuzzy H-ideals and fuzzy p-ideals. Moreover, we investigated their concepts in diverse theorems and propositions. In chapter 3: The main concept of our thesis on derivation fuzzy ideals of BCI- algebras is introduced. Chapter 3 splits into 4 sections. We start with General definitions and important theorems on derivation fuzzy ideal theory in section 1. Section 2 and 3 contain derivations fuzzy p-ideals and derivations fuzzy H-ideals of BCI- algebras, several important theorems and propositions were introduced. The last section studied derivations fuzzy implicative ideals of BCI-algebras and it includes new theorems and results. Furthermore, we presented a new theorem that associate derivations fuzzy implicative ideals, derivations fuzzy positive implicative ideals and derivations fuzzy commutative ideals. These concepts and the new results were obtained and introduced in chapter 3 were submitted in two separated articles and accepted for publication.

Keywords: BCK, BCI, algebras, derivation

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Authors: Luiz F. J. Maia

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This paper presents an approach to fuzzy control, with the use of new pertinence functions, applied in the case of an inverted pendulum. Appropriate definitions of pertinence functions to fuzzy sets make possible the implementation of the controller with only one control rule, resulting in a smooth control surface. The fuzzy control system can be implemented with analog devices, affording a true real-time performance.

Keywords: control surface, fuzzy control, Inverted pendulum, pertinence functions

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Authors: Abder-Rahman Ali, Adélaïde Albouy-Kissi, Manuel Grand-Brochier, Viviane Ladan-Marcus, Christine Hoeffl, Claude Marcus, Antoine Vacavant, Jean-Yves Boire

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In this paper, we present a new segmentation approach for focal liver lesions in contrast enhanced ultrasound imaging. This approach, based on a two-cluster Fuzzy C-Means methodology, considers type-II fuzzy sets to handle uncertainty due to the image modality (presence of speckle noise, low contrast, etc.), and to calculate the optimum inter-cluster threshold. Fine boundaries are detected by a local recursive merging of ambiguous pixels. The method has been tested on a representative database. Compared to both Otsu and type-I Fuzzy C-Means techniques, the proposed method significantly reduces the segmentation errors.

Keywords: defuzzification, fuzzy clustering, image segmentation, type-II fuzzy sets

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Authors: Elena Tkachenko, Elena Rogova, Daria Koval

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The paper examines the ways of cash flows forecasting in the dynamic external environment. The so-called new reality in economy lowers the predictability of the companies’ performance indicators due to the lack of long-term steady trends in external conditions of development and fast changes in the markets. The traditional methods based on the trend analysis lead to a very high error of approximation. The macroeconomic situation for the last 10 years is defined by continuous consequences of financial crisis and arising of another one. In these conditions, the instruments of forecasting on the basis of fuzzy sets show good results. The fuzzy sets based models turn out to lower the error of approximation to acceptable level and to provide the companies with reliable cash flows estimation that helps to reach the financial stability. In the paper, the applicability of the model of cash flows forecasting based on fuzzy logic was analyzed.

Keywords: cash flow, industrial enterprise, forecasting, fuzzy sets

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Authors: Palash Dutta

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More frequently uncertainties are encountered in medical diagnosis and therefore it is the most important and interesting area of applications of fuzzy set theory. In this present study, an attempt has been made to extend Sanchez’s approach for medical diagnosis via interval valued picture fuzzy sets and exhibit the technique with suitable case studies. In this article, it is observed that a refusal can be expressed in the databases concerning the examined objects. The technique is performing diagnosis on the basis of distance measures and as a result, this approach makes it possible to introduce weights of all symptoms and consequently patient can be diagnosed directly.

Keywords: medical diagnosis, uncertainty, fuzzy set, picture fuzzy set, interval valued picture fuzzy set

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Authors: Behnam Tavakkol

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Uncertain data mining algorithms use different ways to consider uncertainty in data such as by representing a data object as a sample of points or a probability distribution. Fuzzy methods have long been used for clustering traditional (certain) data objects. They are used to produce non-crisp cluster labels. For uncertain data, however, besides some uncertain fuzzy k-medoids algorithms, not many other fuzzy clustering methods have been developed. In this work, we develop a fuzzy kernel k-medoids algorithm for clustering uncertain data objects. The developed fuzzy kernel k-medoids algorithm is superior to existing fuzzy k-medoids algorithms in clustering data sets with non-linearly separable clusters.

Keywords: clustering algorithm, fuzzy methods, kernel k-medoids, uncertain data

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Authors: Rashmi Tamhankar, Madhav Bapat

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In many projects, we have to use human judgment for determining the duration of the activities which may vary from person to person. Hence, there is vagueness about the time duration for activities in network planning. Fuzzy sets can handle such vague or imprecise concepts and has an application to such network. The vague activity times can be represented by triangular fuzzy numbers. In this paper, a general network with fuzzy activity times is considered and conditions for the critical path are obtained also we compute total float time of each activity. Several numerical examples are discussed.

Keywords: PERT, CPM, triangular fuzzy numbers, fuzzy activity times

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Authors: G. S. Thakur

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This paper presents a Fuzzy Soft Recruitment Model (FSRM) for faculty selection of MHRD technical institutions. The selection criteria are based on 4-tier flexible structure in the institutions. The Advisory Committee on Faculty Recruitment (ACoFAR) suggested nine criteria for faculty in the proposed FSRM. The model Fuzzy Soft is proposed with consultation of ACoFAR based on selection criteria. The Fuzzy Soft distance similarity measures are applied for finding best faculty from the applicant pool.

Keywords: fuzzy soft set, fuzzy sets, fuzzy soft distance, fuzzy soft similarity measures, ACoFAR

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Authors: Ali Koam, Ismail Ibedou, S. E. Abbas

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In this paper, it is introduced the notion of r-fuzzy ideal separation axioms Tᵢi = 0; 1; 2 based on a fuzzy ideal I on a fuzzy topological space (X; τ). An r-fuzzy ideal connectedness related to the fuzzy ideal I is introduced which has relations with a previous r-fuzzy fuzzy connectedness. An r-fuzzy ideal compactness related to Ι is introduced which has also relations with many other types of fuzzy compactness.

Keywords: fuzzy ideal, fuzzy separation axioms, fuzzy compactness, fuzzy connectedness

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Authors: Elda Maraj, Shkelqim Kuka

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Coronary heart disease causes many deaths in the world. Unfortunately, this problem will continue to increase in the future. In this paper, a fuzzy logic model to predict coronary heart disease is presented. This model has been developed with seven input variables and one output variable that was implemented for 30 patients in Albania. Here fuzzy logic toolbox of MATLAB is used. Fuzzy model inputs are considered as cholesterol, blood pressure, physical activity, age, BMI, smoking, and diabetes, whereas the output is the disease classification. The fuzzy sets and membership functions are chosen in an appropriate manner. Centroid method is used for defuzzification. The database is taken from University Hospital Center "Mother Teresa" in Tirana, Albania.

Keywords: coronary heart disease, fuzzy logic toolbox, membership function, prediction model

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Authors: S. Yaman, S. Rostami

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In this study, a black box modeling of the coupled-tank system is obtained by using fuzzy sets. The derived model is tested via adaptive neuro fuzzy inference system (ANFIS). In order to achieve a better control performance, the parameters of three different controller types, classical proportional integral controller (PID), fuzzy PID and function tuner method, are tuned by one of the evolutionary computation method, genetic algorithm. All tuned controllers are applied to the fuzzy model of the coupled-tank experimental setup and analyzed under the different reference input values. According to the results, it is seen that function tuner method demonstrates better robust control performance and guarantees the closed loop stability.

Keywords: function tuner method (FTM), fuzzy modeling, fuzzy PID controller, genetic algorithm (GA)

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Authors: Liang-Hsuan Chen, Sheng-Shing Nien

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This study aims to develop approaches to formulate intuitionistic fuzzy regression (IFR) models for many decision-making applications in the fuzzy environments using intuitionistic fuzzy observations. Intuitionistic fuzzy numbers (IFNs) are used to characterize the fuzzy input and output variables in the IFR formulation processes. A mathematical programming problem (MPP) is built up to optimally determine the IFR parameters. Each parameter in the MPP is defined as a couple of alternative numerical variables with opposite signs, and an intuitionistic fuzzy error term is added to the MPP to characterize the uncertainty of the model. The IFR model is formulated based on the distance measure to minimize the total distance errors between estimated and observed intuitionistic fuzzy responses in the MPP resolution processes. The proposed approaches are simple/efficient in the formulation/resolution processes, in which the sign of parameters can be determined so that the problem to predetermine the sign of parameters is avoided. Furthermore, the proposed approach has the advantage that the spread of the predicted IFN response will not be over-increased, since the parameters in the established IFR model are crisp. The performance of the obtained models is evaluated and compared with the existing approaches.

Keywords: fuzzy sets, intuitionistic fuzzy number, intuitionistic fuzzy regression, mathematical programming method

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Authors: E. A. Tkachenko, E. M. Rogova, V. V. Klimov

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One of the basic issues of development management is connected with performance measurement as a prerequisite for identifying the achievement of development objectives. The aim of our research is to develop an improved model of assessing a company’s development results. The model should take into account the cyclical nature of development and the high degree of uncertainty in dealing with numerous management tasks. Our hypotheses may be formulated as follows: Hypothesis 1. The cycle of a company’s development may be studied from the standpoint of a project cycle. To do that, methods and tools of project analysis are to be used. Hypothesis 2. The problem of the uncertainty when justifying managerial decisions within the framework of a company’s development cycle can be solved through the use of the mathematical apparatus of fuzzy logic. The reasoned justification of the validity of the hypotheses made is given in the suggested article. The fuzzy logic toolkit applies to the case of technology shift within an enterprise. It is proven that some restrictions in performance measurement that are incurred to conventional methods could be eliminated by implementation of the fuzzy logic apparatus in performance measurement models.

Keywords: logic, fuzzy sets, performance measurement, project analysis

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Authors: Caio Marcelo Nunes, Anderson da Silva Soares, Gustavo Teodoro Laureano, Clarimar Jose Coelho

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This paper presents a methodology to pixel recognition in aerial images using fuzzy $c$-means algorithm. This algorithm is a alternative to recognize areas considering uncertainties and inaccuracies. Traditional clustering technics are used in recognizing of multispectral images of earth's surface. This technics recognize well-defined borders that can be easily discretized. However, in the real world there are many areas with uncertainties and inaccuracies which can be mapped by clustering algorithms that use fuzzy sets. The methodology presents in this work is applied to multispectral images obtained from Landsat-5/TM satellite. The pixels are joined using the $c$-means algorithm. After, a classification process identify the types of surface according the patterns obtained from spectral response of image surface. The classes considered are, exposed soil, moist soil, vegetation, turbid water and clean water. The results obtained shows that the fuzzy clustering identify the real type of the earth's surface.

Keywords: aerial images, fuzzy clustering, image processing, pattern recognition

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Authors: Faisal Alshuwaier, Ali Areshey

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Text mining techniques are generally applied for classifying the text, finding fuzzy relations and structures in data sets. This research provides plenty text mining capabilities. One common application is text classification and event extraction, which encompass deducing specific knowledge concerning incidents referred to in texts. The main contribution of this paper is the clarification of a concept graph generation mechanism, which is based on a text classification and optimal fuzzy relationship extraction. Furthermore, the work presented in this paper explains the application of fuzzy relationship extraction and branch and bound method to simplify the texts.

Keywords: extraction, max-prod, fuzzy relations, text mining, memberships, classification, memberships, classification

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Authors: S. H. Nasseri, A. Ebrahimnejad

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Fuzzy set theory has been applied to many fields, such as operations research, control theory, and management sciences. In this paper, we consider two classes of fuzzy linear programming (FLP) problems: Fuzzy number linear programming and linear programming with trapezoidal fuzzy variables problems. We state our recently established results and develop fuzzy primal simplex algorithms for solving these problems. Finally, we give illustrative examples.

Keywords: fuzzy linear programming, fuzzy numbers, duality, sensitivity analysis

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Authors: Aquil Ahmed, Srikant Gupta, Irfan Ali

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In this paper, we study a multi-objective capacitated transportation problem (MOCTP) with mixed constraints. This paper is comprised of the modelling and optimisation of an MOCTP in a fuzzy environment in which some goals are fractional and some are linear. In real life application of the fuzzy goal programming (FGP) problem with multiple objectives, it is difficult for the decision maker(s) to determine the goal value of each objective precisely as the goal values are imprecise or uncertain. Also, we developed the concept of linearization of fractional goal for solving the MOCTP. In this paper, imprecision of the parameter is handled by the concept of fuzzy set theory by considering these parameters as a trapezoidal fuzzy number. α-cut approach is used to get the crisp value of the parameters. Numerical examples are used to illustrate the method for solving MOCTP.

Keywords: capacitated transportation problem, multi objective linear programming, multi-objective fractional programming, fuzzy goal programming, fuzzy sets, trapezoidal fuzzy number

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Authors: Rishabh Srivastav, Divyam Sharma

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We propose an algorithm to cluster categorical data named as ‘Genetic algorithm initialized rough set based fuzzy K-Modes for categorical data’. We propose an amalgamation of the simple K-modes algorithm, the Rough and Fuzzy set based K-modes and the Genetic Algorithm to form a new algorithm,which we hypothesise, will provide better Centroid Model clustering results, than existing standard algorithms. In the proposed algorithm, the initialization and updation of modes is done by the use of genetic algorithms while the membership values are calculated using the rough set and fuzzy logic.

Keywords: categorical data, fuzzy logic, genetic algorithm, K modes clustering, rough sets

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Authors: Ouafa Amira, Jiangshe Zhang

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Clustering is an unsupervised machine learning technique; its aim is to extract the data structures, in which similar data objects are grouped in the same cluster, whereas dissimilar objects are grouped in different clusters. Clustering methods are widely utilized in different fields, such as: image processing, computer vision , and pattern recognition, etc. Fuzzy c-means clustering (fcm) is one of the most well known fuzzy clustering methods. It is based on solving an optimization problem, in which a minimization of a given cost function has been studied. This minimization aims to decrease the dissimilarity inside clusters, where the dissimilarity here is measured by the distances between data objects and cluster centers. The degree of belonging of a data point in a cluster is measured by a membership function which is included in the interval [0, 1]. In fcm clustering, the membership degree is constrained with the condition that the sum of a data object’s memberships in all clusters must be equal to one. This constraint can cause several problems, specially when our data objects are included in a noisy space. Regularization approach took a part in fuzzy c-means clustering technique. This process introduces an additional information in order to solve an ill-posed optimization problem. In this study, we focus on regularization by relative entropy approach, where in our optimization problem we aim to minimize the dissimilarity inside clusters. Finding an appropriate membership degree to each data object is our objective, because an appropriate membership degree leads to an accurate clustering result. Our clustering results in synthetic data sets, gaussian based data sets, and real world data sets show that our proposed model achieves a good accuracy.

Keywords: clustering, fuzzy c-means, regularization, relative entropy

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Authors: F. Berna Benli, Özgür Keskin

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Lately, many mathematicians have been studied the statistical convergence of a sequence of fuzzy numbers. We know that Lambda-statistically convergence is a kind of convergence between ordinary convergence and statistical convergence. In this paper, we will introduce the new kind of convergence such as λ-levelwise statistical convergence. Then, we will define the concept of the λ-levelwise statistical cluster and limit points of a sequence of fuzzy numbers. Also, we will discuss the relations between the sets of λ-levelwise statistical cluster points and λ-levelwise statistical limit points of sequences of fuzzy numbers. This work has been extended in this paper, where some relations have been considered such that when lambda-statistical limit inferior and lambda-statistical limit superior for lambda-statistically convergent sequences of fuzzy numbers are equal. Furthermore, lambda-statistical boundedness condition for different sequences of fuzzy numbers has been studied.

Keywords: fuzzy number, λ-levelwise statistical cluster points, λ-levelwise statistical convergence, λ-levelwise statistical limit points, λ-statistical cluster points, λ-statistical convergence, λ-statistical limit points

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Authors: František Včelař, Zuzana Pátíková

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Recently a new type of very general relational structures, the so called (L-)complete propelattices, was introduced. These significantly generalize complete lattices and completely lattice L-ordered sets, because they do not assume the technically very strong property of transitivity. For these structures also the main part of the original Tarski’s fixed point theorem holds for (L-fuzzy) isotone maps, i.e., the part which concerns the existence of fixed points and the structure of their set. In this paper, fundamental properties of (L-)complete propelattices are recalled and the so called L-fuzzy relatively isotone maps are introduced. For these maps it is proved that they also have fixed points in L-complete propelattices, even if their set does not have to be of an awaited analogous structure of a complete propelattice.

Keywords: fixed point, L-complete propelattice, L-fuzzy (relatively) isotone map, residuated lattice, transitivity

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