In this paper, we investigate transportation problem in which supplies and demands are intuitionistic fuzzy numbers. Intuitionistic fuzzy zero point method is proposed to find the optimal solution in terms of triangular intuitionistic fuzzy numbers. A new relevant numerical example is also included.
On intuitionistic fuzzy transportation problem using hexagonal intuitionistic...ijfls
The document summarizes research on solving intuitionistic fuzzy transportation problems using hexagonal intuitionistic fuzzy numbers. It introduces hexagonal intuitionistic fuzzy numbers and defines their arithmetic operations. It then formulates the hexagonal intuitionistic fuzzy transportation problem and provides algorithms to find an initial basic feasible solution and optimal solution. An example is provided to illustrate the solutions.
In conventional transportation problem (TP), supplies, demands and costs are always certain. This paper develops an approach to solve the unbalanced transportation problem where as all the parameters are not in deterministic numbers but imprecise ones. Here, all the parameters of the TP are considered to the triangular intuitionistic fuzzy numbers (TIFNs). The existing ranking procedure of Varghese and Kuriakose is used to transform the unbalanced intuitionistic fuzzy transportation problem (UIFTP) into a crisp one so that the conventional method may be applied to solve the TP. The occupied cells of unbalanced crisp TP that we obtained are as same as the occupied cells of UIFTP.
On the basis of this idea the solution procedure is differs from unbalanced crisp TP to UIFTP in allocation step only. Therefore, the new method and new multiplication operation on triangular intuitionistic fuzzy number (TIFN) is proposed to find the optimal solution in terms of TIFN. The main advantage of this method is computationally very simple, easy to understand and also the optimum objective value obtained by our method is physically meaningful.
The document discusses the concept of modularity in logical theories. It defines what it means for a theory to be modular and propositionally modular. A theory is modular if its consequences only depend on the part of the theory containing the same modal operators as the consequence. A theory is propositionally modular if its propositional consequences only depend on its propositional part. The paper proves that if a theory is propositionally modular, then it is modular, and discusses checking and ensuring the modular property of action theories.
The New Ranking Method using Octagonal Intuitionistic Fuzzy Unbalanced Transp...ijtsrd
In this paper a new ranking method is proposed for finding an optimal solution for intuitionistic fuzzy unbalanced transportation problem, in which the costs, supplies and demands are octagonal intuitionistic fuzzy numbers. The procedure is illustrated with a numerical example. Dr. P. Rajarajeswari | G. Menaka "The New Ranking Method using Octagonal Intuitionistic Fuzzy Unbalanced Transportation Problem" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-4 , June 2020, URL: https://www.ijtsrd.com/papers/ijtsrd31675.pdf Paper Url :https://www.ijtsrd.com/mathemetics/applied-mathamatics/31675/the-new-ranking-method-using-octagonal-intuitionistic-fuzzy-unbalanced-transportation-problem/dr-p-rajarajeswari
This document discusses applications of semigroups in various fields including automata theory, formal languages, biology, and sociology. It provides examples of how semigroup structures can model relationships in areas like marriage dynamics, kinship, and metabolic pathways. The document also establishes connections between semigroups and automata by showing how finite automata can be represented as monoids and how their state transition functions relate to the multiplication in the corresponding semigroup.
Mathematics is considered the mother of all sciences because it provides tools to solve problems in other sciences like biology, chemistry, and physics. Other subjects are based on mathematical concepts like structure, quantity, and change. Calculus is fundamental to modern science, and fields like game theory and operations research use mathematics and have significant applications. A math graduate would be a good fit for the Bangladesh Bank because banking relies heavily on mathematics and data analysis. Those with a background in math are also strong problem solvers able to work with complex models.
In this paper, we make use of the fractional differential operator method to find the modified Riemann-Liouville (R-L) fractional derivatives of some fractional functions include fractional polynomial function, fractional exponential function, fractional sine and cosine functions. The Mittag-Leffler function plays an important role in our article, and the fractional differential operator method can be applied to find the particular solutions of non-homogeneous linear fractional differential equations (FDE) with constant coefficients in a unified way and it is a generalization of the method of finding particular solutions of classical ordinary differential equations. On the other hand, several examples are illustrative for demonstrating the advantage of our approach and we compare our results with the traditional differential calculus cases.
This document discusses numerical methods for solving partial differential equations (PDEs). It begins by classifying PDEs as parabolic, elliptic, or hyperbolic based on their coefficients. It then introduces finite difference methods, which approximate PDE solutions on a grid by replacing derivatives with finite differences. In particular, it describes the forward time centered space (FTCS) scheme for solving the 1D heat equation numerically and analyzing its stability using von Neumann analysis.
On intuitionistic fuzzy transportation problem using hexagonal intuitionistic...ijfls
The document summarizes research on solving intuitionistic fuzzy transportation problems using hexagonal intuitionistic fuzzy numbers. It introduces hexagonal intuitionistic fuzzy numbers and defines their arithmetic operations. It then formulates the hexagonal intuitionistic fuzzy transportation problem and provides algorithms to find an initial basic feasible solution and optimal solution. An example is provided to illustrate the solutions.
In conventional transportation problem (TP), supplies, demands and costs are always certain. This paper develops an approach to solve the unbalanced transportation problem where as all the parameters are not in deterministic numbers but imprecise ones. Here, all the parameters of the TP are considered to the triangular intuitionistic fuzzy numbers (TIFNs). The existing ranking procedure of Varghese and Kuriakose is used to transform the unbalanced intuitionistic fuzzy transportation problem (UIFTP) into a crisp one so that the conventional method may be applied to solve the TP. The occupied cells of unbalanced crisp TP that we obtained are as same as the occupied cells of UIFTP.
On the basis of this idea the solution procedure is differs from unbalanced crisp TP to UIFTP in allocation step only. Therefore, the new method and new multiplication operation on triangular intuitionistic fuzzy number (TIFN) is proposed to find the optimal solution in terms of TIFN. The main advantage of this method is computationally very simple, easy to understand and also the optimum objective value obtained by our method is physically meaningful.
The document discusses the concept of modularity in logical theories. It defines what it means for a theory to be modular and propositionally modular. A theory is modular if its consequences only depend on the part of the theory containing the same modal operators as the consequence. A theory is propositionally modular if its propositional consequences only depend on its propositional part. The paper proves that if a theory is propositionally modular, then it is modular, and discusses checking and ensuring the modular property of action theories.
The New Ranking Method using Octagonal Intuitionistic Fuzzy Unbalanced Transp...ijtsrd
In this paper a new ranking method is proposed for finding an optimal solution for intuitionistic fuzzy unbalanced transportation problem, in which the costs, supplies and demands are octagonal intuitionistic fuzzy numbers. The procedure is illustrated with a numerical example. Dr. P. Rajarajeswari | G. Menaka "The New Ranking Method using Octagonal Intuitionistic Fuzzy Unbalanced Transportation Problem" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-4 , June 2020, URL: https://www.ijtsrd.com/papers/ijtsrd31675.pdf Paper Url :https://www.ijtsrd.com/mathemetics/applied-mathamatics/31675/the-new-ranking-method-using-octagonal-intuitionistic-fuzzy-unbalanced-transportation-problem/dr-p-rajarajeswari
This document discusses applications of semigroups in various fields including automata theory, formal languages, biology, and sociology. It provides examples of how semigroup structures can model relationships in areas like marriage dynamics, kinship, and metabolic pathways. The document also establishes connections between semigroups and automata by showing how finite automata can be represented as monoids and how their state transition functions relate to the multiplication in the corresponding semigroup.
Mathematics is considered the mother of all sciences because it provides tools to solve problems in other sciences like biology, chemistry, and physics. Other subjects are based on mathematical concepts like structure, quantity, and change. Calculus is fundamental to modern science, and fields like game theory and operations research use mathematics and have significant applications. A math graduate would be a good fit for the Bangladesh Bank because banking relies heavily on mathematics and data analysis. Those with a background in math are also strong problem solvers able to work with complex models.
In this paper, we make use of the fractional differential operator method to find the modified Riemann-Liouville (R-L) fractional derivatives of some fractional functions include fractional polynomial function, fractional exponential function, fractional sine and cosine functions. The Mittag-Leffler function plays an important role in our article, and the fractional differential operator method can be applied to find the particular solutions of non-homogeneous linear fractional differential equations (FDE) with constant coefficients in a unified way and it is a generalization of the method of finding particular solutions of classical ordinary differential equations. On the other hand, several examples are illustrative for demonstrating the advantage of our approach and we compare our results with the traditional differential calculus cases.
This document discusses numerical methods for solving partial differential equations (PDEs). It begins by classifying PDEs as parabolic, elliptic, or hyperbolic based on their coefficients. It then introduces finite difference methods, which approximate PDE solutions on a grid by replacing derivatives with finite differences. In particular, it describes the forward time centered space (FTCS) scheme for solving the 1D heat equation numerically and analyzing its stability using von Neumann analysis.
Riccati matrix differential equation has long been known to be so difficult to solve analytically and/or numerically. In this connection, most of the recent studies are concerned with the derivation of the necessary conditions that ensure the existence of the solution. Therefore, in this paper, He’s Variational iteration method is used to derive the general form of the iterative approximate sequence of solutions and then proved the convergence of the obtained sequence of approximate solutions to the exact solution. This proof is based on using the mathematical induction to derive a general formula for the upper bound proved to be converging to zero under certain conditions.
Artificial Intelligence lecture notes. AI summarized notes on uncertainty and handling it through fuzzy logic, tipping problem scenarios are seen in it, for reading and may be for self-learning, I think.
Some alternative ways to find m ambiguous binary words corresponding to a par...ijcsa
Parikh matrix of a word gives numerical information of the word in terms of its subwords. In this Paper an
algorithm for finding Parikh matrix of a binary word is introduced. With the help of this algorithm Parikh
matrix of a binary word, however large it may be can be found out. M-ambiguous words are the problem of
Parikh matrix. In this paper an algorithm is shown to find the M- ambiguous words of a binary ordered
word instantly. We have introduced a system to represent binary words in a two dimensional field. We see
that there are some relations among the representations of M-ambiguous words in the two dimensional
field. We have also introduced a set of equations which will help us to calculate the M- ambiguous words.
This presentation contains my one day lectures which introduces fuzzy set theory, operations on fuzzy sets, some engineering control applications using Mamdamn model.
AN ALGORITHM FOR SOLVING LINEAR OPTIMIZATION PROBLEMS SUBJECTED TO THE INTERS...ijfcstjournal
Frank t-norms are parametric family of continuous Archimedean t-norms whose members are also strict functions. Very often, this family of t-norms is also called the family of fundamental t-norms because of the
role it plays in several applications. In this paper, optimization of a linear objective function with fuzzy relational inequality constraints is investigated.
This paper introduces the concept of semi-periodic ∞-tuples of commutative bounded linear mappings on a separable Banach space. The authors prove two main results: 1) A hypercyclicity criterion for ∞-tuples, stating conditions under which an ∞-tuple is hypercyclic. 2) If an ∞-tuple is hypercyclic and has a dense generalized kernel, then it satisfies the conditions of the hypercyclicity criterion. The paper aims to expand understanding of dynamical properties of ∞-tuples acting on Banach spaces.
Correlation measure for intuitionistic fuzzy multi setseSAT Journals
The document proposes a correlation measure for intuitionistic fuzzy multi sets (IFMS). IFMS are an extension of intuitionistic fuzzy sets that allow elements to have multiple membership and non-membership values. The document defines IFMS and reviews existing correlation measures for fuzzy and intuitionistic fuzzy sets. It then proposes a new correlation similarity measure for IFMS that is an extension of existing intuitionistic fuzzy set correlation measures. This new measure takes into account the multiple membership and non-membership values of elements in IFMS. Examples are provided to demonstrate properties of the new correlation measure.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
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.
STATISTICAL ANALYSIS OF FUZZY LINEAR REGRESSION MODEL BASED ON DIFFERENT DIST...Wireilla
Using fuzzy linear regression model, the least squares estimation for linear regression (LR) fuzzy number is studied by Euclidean distance, Y-K distance and Dk distance respectively. It is concluded that the three different distances have the same coefficient of the least squares estimation. The data simulation shows the correctness of this conclusion.
This document presents a new method for solving exponential equations with real roots. It begins by expressing the exponential equation fully in terms of the exponential base. It then outlines three cases for solving the equations based on the sign of the constant term and coefficient. For each case, it provides the steps to obtain one or two real root solutions to the exponential equation. An example is provided for each case to demonstrate the new method. The conclusion states that the new method provides an alternative way to solve exponential equations that is easier to understand than converting it to a quadratic equation, as was commonly done previously.
STATISTICAL ANALYSIS OF FUZZY LINEAR REGRESSION MODEL BASED ON DIFFERENT DIST...ijfls
This document summarizes a study on statistical analysis of fuzzy linear regression models based on different distance measures. It analyzes least squares estimations and error terms for fuzzy linear regression models using Euclidean distance, Y-K distance, and kD distance. The study finds that the three distances produce the same coefficient estimates for the least squares regression model. Simulation data is used to validate this conclusion.
Fuzzy logic is a form of logic that deals with reasoning that is approximate rather than fixed and exact. It was introduced in 1965 with the proposal of fuzzy set theory by Lotfi Zadeh. Fuzzy logic uses fuzzy sets and membership functions to deal with imprecise or uncertain inputs and allows for reasoning that allows for partial truth of inputs between fully true and fully false. Fuzzy controllers combine fuzzy logic with control theory to control complex systems. They involve fuzzification of inputs, applying fuzzy rules through inference, and defuzzification of outputs to obtain a crisp control action.
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.
This document discusses knowledge representation and classification in rough set theory. It defines key concepts such as knowledge base, classification, elementary and basic categories. A knowledge base consists of a universe of objects and a set of classifications over those objects. Classifications are equivalence relations that partition the universe into categories. Elementary categories come from individual classifications, while basic categories are intersections of elementary categories. The document also discusses equivalence, generalization and specialization of knowledge bases based on comparisons of their category structures.
Bio data ( dr.p.k.sharma) as on april 2016DR.P.K. SHARMA
This curriculum vitae outlines the educational and professional background of Dr. Poonam Kumar Sharma. She holds a PhD in mathematics from Jamia Millia Islamia University and has over 20 years of experience as an Associate Professor of Mathematics. Her research focuses on areas like modern algebra, topology, and fuzzy mathematics. She has published over 50 research papers and written textbooks on various mathematics topics.
This document discusses knowledge reduction in rough set theory. It defines core and reduct concepts. The core is the most important part of knowledge that cannot be eliminated. A reduct is a minimal set of attributes that provides the same classification as the full set. It also discusses relative reducts and cores with respect to a given subset. Generalized definitions are provided for reducing categories of knowledge rather than individual attributes.
Many areas of accounting have highly ambiguous due to undefined and inaccurate terms. Many ambiguities are generated by the human mind. In the field of accounting, these ambiguities lead to the creation of uncertain information. Many of the targets and concepts of accounting with binary classification are not consistent. Similarly, the discussion of the materiality or reliability of accounting is not a two-part concept. Because there are degrees of materiality or reliability. Therefore, these ambiguities lead to the presentation information that is not suitable for decision making. Lack of attention to the issue of ambiguity in management accounting techniques, auditing procedures, and financial reporting may lead to a reduced role of accounting information in decision-making processes. Because information plays an important role in economic decision-making, and no doubt, the quality of their, including accuracy in providing it to a wide range of users, can be useful for decision-making. One of the features of the fuzzy set is that it reduces the need for accurate data in decision making. Hence this information can be useful for users.
This document provides an overview of fuzzy logic and fuzzy set theory. It discusses how fuzzy sets allow for partial membership rather than crisp binary membership in sets. Various membership functions are described, including triangular, trapezoidal, Gaussian, and sigmoidal functions. Properties of fuzzy sets like support, convexity, and symmetry are defined. Finally, fuzzy set operations analogous to classical set operations are mentioned.
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.
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
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.
Riccati matrix differential equation has long been known to be so difficult to solve analytically and/or numerically. In this connection, most of the recent studies are concerned with the derivation of the necessary conditions that ensure the existence of the solution. Therefore, in this paper, He’s Variational iteration method is used to derive the general form of the iterative approximate sequence of solutions and then proved the convergence of the obtained sequence of approximate solutions to the exact solution. This proof is based on using the mathematical induction to derive a general formula for the upper bound proved to be converging to zero under certain conditions.
Artificial Intelligence lecture notes. AI summarized notes on uncertainty and handling it through fuzzy logic, tipping problem scenarios are seen in it, for reading and may be for self-learning, I think.
Some alternative ways to find m ambiguous binary words corresponding to a par...ijcsa
Parikh matrix of a word gives numerical information of the word in terms of its subwords. In this Paper an
algorithm for finding Parikh matrix of a binary word is introduced. With the help of this algorithm Parikh
matrix of a binary word, however large it may be can be found out. M-ambiguous words are the problem of
Parikh matrix. In this paper an algorithm is shown to find the M- ambiguous words of a binary ordered
word instantly. We have introduced a system to represent binary words in a two dimensional field. We see
that there are some relations among the representations of M-ambiguous words in the two dimensional
field. We have also introduced a set of equations which will help us to calculate the M- ambiguous words.
This presentation contains my one day lectures which introduces fuzzy set theory, operations on fuzzy sets, some engineering control applications using Mamdamn model.
AN ALGORITHM FOR SOLVING LINEAR OPTIMIZATION PROBLEMS SUBJECTED TO THE INTERS...ijfcstjournal
Frank t-norms are parametric family of continuous Archimedean t-norms whose members are also strict functions. Very often, this family of t-norms is also called the family of fundamental t-norms because of the
role it plays in several applications. In this paper, optimization of a linear objective function with fuzzy relational inequality constraints is investigated.
This paper introduces the concept of semi-periodic ∞-tuples of commutative bounded linear mappings on a separable Banach space. The authors prove two main results: 1) A hypercyclicity criterion for ∞-tuples, stating conditions under which an ∞-tuple is hypercyclic. 2) If an ∞-tuple is hypercyclic and has a dense generalized kernel, then it satisfies the conditions of the hypercyclicity criterion. The paper aims to expand understanding of dynamical properties of ∞-tuples acting on Banach spaces.
Correlation measure for intuitionistic fuzzy multi setseSAT Journals
The document proposes a correlation measure for intuitionistic fuzzy multi sets (IFMS). IFMS are an extension of intuitionistic fuzzy sets that allow elements to have multiple membership and non-membership values. The document defines IFMS and reviews existing correlation measures for fuzzy and intuitionistic fuzzy sets. It then proposes a new correlation similarity measure for IFMS that is an extension of existing intuitionistic fuzzy set correlation measures. This new measure takes into account the multiple membership and non-membership values of elements in IFMS. Examples are provided to demonstrate properties of the new correlation measure.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
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.
STATISTICAL ANALYSIS OF FUZZY LINEAR REGRESSION MODEL BASED ON DIFFERENT DIST...Wireilla
Using fuzzy linear regression model, the least squares estimation for linear regression (LR) fuzzy number is studied by Euclidean distance, Y-K distance and Dk distance respectively. It is concluded that the three different distances have the same coefficient of the least squares estimation. The data simulation shows the correctness of this conclusion.
This document presents a new method for solving exponential equations with real roots. It begins by expressing the exponential equation fully in terms of the exponential base. It then outlines three cases for solving the equations based on the sign of the constant term and coefficient. For each case, it provides the steps to obtain one or two real root solutions to the exponential equation. An example is provided for each case to demonstrate the new method. The conclusion states that the new method provides an alternative way to solve exponential equations that is easier to understand than converting it to a quadratic equation, as was commonly done previously.
STATISTICAL ANALYSIS OF FUZZY LINEAR REGRESSION MODEL BASED ON DIFFERENT DIST...ijfls
This document summarizes a study on statistical analysis of fuzzy linear regression models based on different distance measures. It analyzes least squares estimations and error terms for fuzzy linear regression models using Euclidean distance, Y-K distance, and kD distance. The study finds that the three distances produce the same coefficient estimates for the least squares regression model. Simulation data is used to validate this conclusion.
Fuzzy logic is a form of logic that deals with reasoning that is approximate rather than fixed and exact. It was introduced in 1965 with the proposal of fuzzy set theory by Lotfi Zadeh. Fuzzy logic uses fuzzy sets and membership functions to deal with imprecise or uncertain inputs and allows for reasoning that allows for partial truth of inputs between fully true and fully false. Fuzzy controllers combine fuzzy logic with control theory to control complex systems. They involve fuzzification of inputs, applying fuzzy rules through inference, and defuzzification of outputs to obtain a crisp control action.
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.
This document discusses knowledge representation and classification in rough set theory. It defines key concepts such as knowledge base, classification, elementary and basic categories. A knowledge base consists of a universe of objects and a set of classifications over those objects. Classifications are equivalence relations that partition the universe into categories. Elementary categories come from individual classifications, while basic categories are intersections of elementary categories. The document also discusses equivalence, generalization and specialization of knowledge bases based on comparisons of their category structures.
Bio data ( dr.p.k.sharma) as on april 2016DR.P.K. SHARMA
This curriculum vitae outlines the educational and professional background of Dr. Poonam Kumar Sharma. She holds a PhD in mathematics from Jamia Millia Islamia University and has over 20 years of experience as an Associate Professor of Mathematics. Her research focuses on areas like modern algebra, topology, and fuzzy mathematics. She has published over 50 research papers and written textbooks on various mathematics topics.
This document discusses knowledge reduction in rough set theory. It defines core and reduct concepts. The core is the most important part of knowledge that cannot be eliminated. A reduct is a minimal set of attributes that provides the same classification as the full set. It also discusses relative reducts and cores with respect to a given subset. Generalized definitions are provided for reducing categories of knowledge rather than individual attributes.
Many areas of accounting have highly ambiguous due to undefined and inaccurate terms. Many ambiguities are generated by the human mind. In the field of accounting, these ambiguities lead to the creation of uncertain information. Many of the targets and concepts of accounting with binary classification are not consistent. Similarly, the discussion of the materiality or reliability of accounting is not a two-part concept. Because there are degrees of materiality or reliability. Therefore, these ambiguities lead to the presentation information that is not suitable for decision making. Lack of attention to the issue of ambiguity in management accounting techniques, auditing procedures, and financial reporting may lead to a reduced role of accounting information in decision-making processes. Because information plays an important role in economic decision-making, and no doubt, the quality of their, including accuracy in providing it to a wide range of users, can be useful for decision-making. One of the features of the fuzzy set is that it reduces the need for accurate data in decision making. Hence this information can be useful for users.
This document provides an overview of fuzzy logic and fuzzy set theory. It discusses how fuzzy sets allow for partial membership rather than crisp binary membership in sets. Various membership functions are described, including triangular, trapezoidal, Gaussian, and sigmoidal functions. Properties of fuzzy sets like support, convexity, and symmetry are defined. Finally, fuzzy set operations analogous to classical set operations are mentioned.
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.
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
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.
In this paper, we investigate transportation problem in which supplies and demands are intuitionistic fuzzy numbers. Intuitionistic Fuzzy Vogel’s Approximation Method is proposed to find an initial basic feasible solution. Intuitionistic Fuzzy Modified Distribution Method is proposed to find the optimal solution in terms of triangular intuitionistic fuzzy numbers. The solution procedure is illustrated with suitable numerical example.
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.
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
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.
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In solving real life assignment problem, we often face the state of uncertainty as well as hesitation due to varies uncontrollable factors. To deal with uncertainty and hesitation many authors have suggested the intuitionistic fuzzy representation for the data. In this paper, computationally a simple method is proposed to find the optimal solution for an unbalanced assignment problem under intuitionistic fuzzy environment. In conventional assignment problem, cost is always certain. This paper develops an approach to solve the unbalanced assignment problem where the time/cost/profit is not in deterministic numbers but imprecise ones. In this assignment problem, the elements of the cost matrix are represented by the triangular intuitionistic fuzzy numbers. The existing Ranking procedure of Varghese and Kuriakose is used to transform the unbalanced intuitionistic fuzzy assignment problem into a crisp one so that the conventional method may be applied to solve the AP . Finally, the method is illustrated by a numerical example which is followed by graphical representation and discussion of the finding.
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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.
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This document presents an article from the International Journal of Fuzzy System Applications that proposes a new method called the PSK method for solving type-1 and type-3 fuzzy transportation problems. In transportation problems, supplies, demands, and costs are usually certain values, but the article considers problems where these values may be uncertain and represented by fuzzy numbers like triangular or trapezoidal fuzzy numbers. The PSK method transforms the fuzzy transportation problem into a crisp one using an existing ranking procedure so that conventional solution methods can be applied. The method differs from solving certain transportation problems only in the allocation step. The PSK method and a new operation for multiplying trapezoidal fuzzy numbers are proposed to find an optimal solution with both crisp and fuzzy components
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In solving real life transportation problem we often face the state of uncertainty as well as hesitation due to various uncontrollable factors. To deal with uncertainty and hesitation many authors have suggested the intuitionistic fuzzy representation for the data. So, in this paper, we consider a transportation problem having uncertainty and hesitation in supply, demand and costs. We formulate the problem and utilize triangular intuitionistic fuzzy numbers (TrIFNs) to deal with uncertainty and hesitation. We propose a new method called PSK method for finding the intuitionistic fuzzy optimal solution for fully intuitionistic fuzzy transportation problem in single stage. Also the new multiplication operation on TrIFN is proposed to find the optimal object value in terms of TrIFN. The main advantage of this method is computationally very simple, easy to understand and also the optimum objective value obtained by our method is physically meaningful. Finally the effectiveness of the proposed method is illustrated by means of a numerical example which is followed by graphical representation of the finding.
In conventional transportation problem (TP), all the parameters are always certain. But, many of the real life situations in industry or organization, the parameters (supply, demand and cost) of the TP are not precise which are imprecise in nature in different factors like the market condition, variations in rates of diesel, traffic jams, weather in hilly areas, capacity of men and machine, long power cut, labourer’s over time work, unexpected failures in machine, seasonal changesandmanymore. Tocountertheseproblems,dependingonthenatureoftheparameters, theTPisclassifiedintotwocategoriesnamelytype-2andtype-4fuzzytransportationproblems (FTPs) under uncertain environment and formulates the problem and utilizes the trapezoidal fuzzy number (TrFN) to solve the TP. The existing ranking procedure of Liou and Wang (1992)isusedtotransformthetype-2andtype-4FTPsintoacrisponesothattheconventional method may be applied to solve the TP. Moreover, the solution procedure differs from TP to type-2 and type-4 FTPs in allocation step only. Therefore a simple and efficient method denoted by PSK (P. Senthil Kumar) method is proposed to obtain an optimal solution in terms of TrFNs. From this fuzzy solution, the decision maker (DM) can decide the level of acceptance for the transportation cost or profit. Thus, the major applications of fuzzy set theory are widely used in areas such as inventory control, communication network, aggregate planning, employment scheduling, and personnel assignment and so on.
In this article, two methods are presented, proposed method 1 and proposed method 2. Proposed method 1 is based on linear programming technique and proposed method 2 is based on modified distribution method. Both of the methods are used to solve the balanced and unbalanced intuitionistic fuzzy transportation problems. The ideas of the proposed methods are illustrated with the help of real life numerical examples which is followed by the results and discussion and comparative study is given. The proposed method is computationally very simple when compared to the existing methods, it is shown to be and easier form of evaluation when compared to current methods.
In real-life decisions, usually we happen to suffer through different states of uncertainties. In order to counter these uncertainties, in this paper, the author formulated a transportation problem in which costs are triangular intuitionistic fuzzy numbers, supplies and demands are crisp numbers. In this paper, a simple method for solving type-2 intuitionistic fuzzy transportation problem (type-2 IFTP) is proposed and optimal solution is obtained without using intuitionistic fuzzy modified distribution method and intuitionistic fuzzy zero point method. So, the proposed method gives the optimal solution directly. The solution procedure is illustrated with the help of three real life numerical examples. Defect of existing results proposed by Singh and Yadav (2016a) is discussed. Validity of Pandian’s (2014) method is reviewed. Finally, the comparative study, results and discussion are given.
This document presents a systematic approach for solving mixed intuitionistic fuzzy transportation problems. It begins with definitions of fuzzy sets, intuitionistic fuzzy sets, and triangular intuitionistic fuzzy numbers. It then formulates an intuitionistic fuzzy transportation problem and proposes a mixed intuitionistic fuzzy zero point method to find the optimal solution in terms of triangular intuitionistic fuzzy numbers. Finally, it provides the computational procedure and illustrates the method with a numerical example.
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When I was asked to give a companion lecture in support of ‘The Philosophy of Science’ (https://shorturl.at/4pUXz) I decided not to walk through the detail of the many methodologies in order of use. Instead, I chose to employ a long standing, and ongoing, scientific development as an exemplar. And so, I chose the ever evolving story of Thermodynamics as a scientific investigation at its best.
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Positive interaction: mutualism, proto-cooperation, commensalism
Negative interaction: Ammensalism (antagonism), parasitism, predation, competition
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Compound A
Utilized by population 1
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Utilized by population 2
Compound C
utilized by both Population 1+2
Products
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Algorithmic approach for solving intuitionistic fuzzy transportation problem
1. Vol. 6, no. 77-80,2012
ISSN 1312-885X
APPLIED IVIIffHEIVIATICAL SCIENCES
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Published by Hikari Ltd
3. Applied Mathematical Sciences, Vol. 6, 2012, no. 80, 3981 – 3989
Algorithmic Approach for Solving Intuitionistic
Fuzzy Transportation Problem
R. Jahir Hussain and P. Senthil Kumar
PG and Research Department of Mathematics
Jamal Mohamed College, Tiruchirappalli – 620 020. India
hssn_jhr@yahoo.com, senthilsoft_5760@yahoo.com
Abstract
In this paper, we investigate transportation problem in which supplies and
demands are intuitionistic fuzzy numbers. Intuitionistic fuzzy zero point method
is proposed to find the optimal solution in terms of triangular intuitionistic fuzzy
numbers. A new relevant numerical example is also included.
Keywords: Triangular intuitionistic fuzzy numbers, intuitionistic fuzzy
transportation problem, intuitionistic fuzzy zero point method, optimal solution.
1. Introduction
The theory of fuzzy set introduced by Zadeh[8] in 1965 has achieved
successful applications in various fields. The concept of Intuitionistic Fuzzy Sets
(IFSs) proposed by Atanassov[1] in 1986 is found to be highly useful to deal with
vagueness. The major advantage of IFS over fuzzy set is that IFSs separate the
degree of membership (belongingness) and the degree of non membership (non
belongingness) of an element in the set .The concept of fuzzy mathematical
programming was introduced by Tanaka et al in 1947 the frame work of fuzzy
decision of Bellman and Zadeh[2].
In [4], Nagoor Gani et al presented a two stage cost minimizing fuzzy
transportation problem in which supplies and demands are trapezoidal fuzzy
number. In [7], Stephen Dinager et al investigated fuzzy transportation problem
with the aid of trapezoidal fuzzy numbers. In[6], Pandian.P and Natarajan.G
presented a new algorithm for finding a fuzzy optimal solution for fuzzy
transportation problem. In [3], Ismail Mohideen .S and Senthil Kumar .P
investigated a comparative study on transportation problem in fuzzy environment.
4. 3982 R. Jahir Hussain and P. Senthil Kumar
In this paper, a new ranking procedure which can be found in [5] and is used
to obtain an optimal solution in an intuitionistic fuzzy transportation
problem[IFTP]. The paper is organized as follows: section 2 deals with some
terminology, section 3 provides the definition of intuitionistic fuzzy
transportation problem and its mathematical formulation, section 4 deals with
solution procedure, section 5 consists of numerical example, finally conclusion is
given.
2. Terminology
Definition 2.1 Let A be a classical set, ߤሺݔሻ be a function from A to [0,1]. A
fuzzy set ܣ∗
with the membership function ߤሺݔሻ is defined by
ܣ∗
ൌ ൛൫,ݔ ߤሺݔሻ൯; ݔ ∈ ߤ ݀݊ܽ ܣሺݔሻ ∈ ሾ0,1ሿൟ.
Definition 2.2 Let X be denote a universe of discourse, then an intuitionistic
fuzzy set A in X is given by a set of ordered triples,
ܣሚூ
ൌ ሼ൏ ,ݔ ߤሺݔሻ, ߴሺݔሻ ; ݔ ∈ ܺሽ
Where ߤ, ߴ: ܺ → ሾ0,1ሿ, are functions such that 0 ߤሺݔሻ ߴሺݔሻ 1, ∀ݔ ∈ ܺ.
For each x the membership ߤሺݔሻ ܽ݊݀ ߴሺݔሻ represent the degree of membership
and the degree of non – membership of the element ݔ ∈ ܺ to ܣ ⊂ ܺ respectively.
Definition 2.3 An Intuitionistic fuzzy subset A = {<x, µA(x), υA(x)> : x X } of
the real line R is called an intuitionistic fuzzy number (IFN) if the following
holds:
i. There exist m R, µA(m) = 1 and υA(m) = 0, (m is called the mean value of
A).
ii. µA is a continuous mapping from R to the closed interval [0,1] and ∀ݔ ∈
ܴ, the relation 0 ߤሺxሻ ϑሺxሻ 1 holds.
The membership and non – membership function of A is of the following form:
ߤሺxሻ ൌ
ە
ۖ
۔
ۖ
ۓ
0 ݂ݎ െ ∞ ൏ ݔ ݉ െ ߙ
݂ଵሺݔሻ ݂ݔ ݎ ∈ ሾ݉ െ ߙ, ݉ሿ
1 ݂ݔ ݎ ൌ ݉
݄ଵሺݔሻ ݂ݔ ݎ ∈ ሾ݉, ݉ ߚሿ
0 ݂݉ ݎ ߚ ݔ ൏ ∞
Where f1(x) and h1(x) are strictly increasing and decreasing function in
ሾ݉ െ ߙ, ݉ሿ and ሾ݉, ݉ ߚሿ respectively.
ߴሺxሻ ൌ
ە
ۖ
۔
ۖ
ۓ
1 ݂ݎ െ ∞ ൏ ݔ ݉ െ ߙᇱ
݂ଶሺݔሻ ݂ݔ ݎ ∈ ሾ݉ െ ߙᇱ
, ݉ሿ; 0 ݂ଵሺݔሻ ݂ଶሺݔሻ 1
0 ݂ݔ ݎ ൌ ݉
݄ଶሺݔሻ ݂ݔ ݎ ∈ ሾ݉, ݉ ߚᇱሿ; 0 ݄ଵሺݔሻ ݄ଶሺݔሻ 1
1 ݂݉ ݎ ߚᇱ
ݔ ൏ ∞
Here m is the mean value of A. α and β are called left and right spreads of
membership function ߤሺxሻ, respectively. α′ ܽ݊݀ β′
represents left and right
5. Algorithmic approach 3983
spreads of non membership function ߴሺxሻ, respectively. Symbolically, the
intuitionistic fuzzy number ܣሚூ
is represented as AIFN =(m; ߙ, ߚ; α′, β′
).
Definition 2.4 A Triangular Intuitionistic Fuzzy Number (ÃI
is an intuitionistic
fuzzy set in R with the following membership function ߤሺxሻ and non
membership functionߴሺxሻ: )
ߤሺxሻ ൌ
ە
ۖ
۔
ۖ
ۓ
ݔ െ ܽଵ
ܽଶ െ ܽଵ
݂ܽ ݎଵ ݔ ܽଶ
ܽଷ െ ݔ
ܽଷ െ ܽଶ
݂ܽ ݎଶ ݔ ܽଷ
0 ܱ݁ݏ݅ݓݎ݄݁ݐ
ߴሺxሻ ൌ
ە
ۖ
۔
ۖ
ۓ
ܽଶ െ ݔ
ܽଶ െ ܽଵ
′
݂ܽ ݎଵ
′
ݔ ܽଶ
ݔ െ ܽଶ
ܽଷ
′ െ ܽଶ
݂ܽ ݎଶ ݔ ܽଷ
′
1 ܱ݁ݏ݅ݓݎ݄݁ݐ
Where ܽଵ
ᇱ
ܽଵ ܽଶ ܽଷ ܽଷ
ᇱ
and ߤሺxሻ, ϑሺxሻ 0.5 for ߤሺxሻ ൌ ϑሺxሻ
∀ݔ ∈ ܴ. This TrIFN is denoted by ܣሚூ
= ሺܽଵ, ܽଶ, ܽଷሻሺ ܽଵ
ᇱ
, ܽଶ, ܽଷ
ᇱ
ሻ
ܽଵ
′
ܽଵ ܽଶ ܽଷ ܽଷ
′
Membership and non membership functions of TrIFN
Ranking of triangular intuitionistic fuzzy numbers
The Ranking of a triangular intuitionistic fuzzy number is completely defined
by its membership and non- membership as follows [5]:
Let ÃI
= (a,b,c) (e,b,f)
ݔఓሺܣሻ ൌ
1
6ሺܾ െ ܽሻ
ሾ2ܾଷ
െ 3ܾଶ
ܽ ܽଷሿ
1
6ሺܿ െ ܾሻ
ሾܿଷ
െ 3ܾଶ
ܿ 2ܾଷሿ
ቀ
ܿ െ ܽ
2
ቁ
6. 3984 R. Jahir Hussain and P. Senthil Kumar
ݔణሺܣሻ ൌ
1
6ሺܾ െ ݁ሻ
ሾܾଷ
െ 3݁ଶ
ܾ 2݁ଷሿ
1
6ሺ݂ െ ܾሻ
ሾ2݂ଷ
െ 3ܾ݂ଶ
ܾଷሿ
൬
݂ െ ݁
2 ൰
ݕఓሺܣሻ ൌ
1
3
ݕణሺܣሻ ൌ
2
3
Rank (A) = (Sqrt ((xµ(A))2
+ (yµ(A))2
), Sqrt ((xυ(A))2
+ (yυ(A))2
))
Definition 2.5 Let ܣሚூ
and ܤ෨ூ
be two TrIFNs. The ranking of ܣሚூ
and ܤ෨ூ
by the R(.) on E,
the set of TrIFNs is defined as follows:
i. R(ܣሚூ
)>R(ܤ෨ூ
) iff ܣሚூ
≻ ܤ෨ூ
ii. R(ܣሚூ
)<R(ܤ෨ூ
) iff ܣሚூ
≺ ܤ෨ூ
iii. R(ܣሚூ
)=R(ܤ෨ூ
) iff ܣሚூ
≈ ܤ෨ூ
Definition 2.6 The ordering ≽ and ≼ between any two TrIFNs ܣሚூ
and ܤ෨ூ
are
defined as follows
i. ܣሚூ
≽ ܤ෨ூ
iff ܣሚூ
≻ ܤ෨ூ
or ܣሚூ
ൎ ܤ෨ூ
and
ii. ܣሚூ
≼ ܤ෨ூ
iff ܣሚூ
≺ ܤ෨ூ
or ܣሚூ
ൎ ܤ෨ூ
Definition 2.7 Let ሼܣሚ
ூ
, ݅ ൌ 1,2, … , ݊ሽ be a set of TrIFNs. If ܴሺܣሚ
ூ
ሻ ܴሺܣሚ
ூ
ሻfor all
i, then the TrIFN ܣሚ
ூ
is the minimum of ሼܣሚ
ூ
, ݅ ൌ 1,2, … , ݊ሽ.
Definition 2.8 Let ሼܣሚ
ூ
,݅ ൌ 1,2, … , ݊ሽ be a set of TrIFNs. If ܴሺܣሚ௧
ூ
ሻ ܴሺܣሚ
ூ
ሻfor all
i, then the TrIFN ܣሚ௧
ூ
is the maximum of ሼܣሚ
ூ
, ݅ ൌ 1,2, … , ݊ሽ.
Arithmetic Operations
Addition: ܣሚூ
⊕ ܤ෨ூ
=ሺܽଵ ܾଵ, ܽଶ ܾଶ, ܽଷ ܾଷሻሺܽଵ
ᇱ
ܾଵ
ᇱ
, ܽଶ ܾଶ, ܽଷ
ᇱ
ܾଷ
ᇱ
ሻ
Subtraction: ÃI
Θ BI
=ሺܽଵ െ ܾଷ, ܽଶ െ ܾଶ, ܽଷ െ ܾଵሻሺܽଵ
ᇱ
െ ܾଷ
ᇱ
, ܽଶ െ ܾଶ, ܽଷ
ᇱ
െ ܾଵ
ᇱ
ሻ
Multiplication:
A෩୍
⊗ B෩୍
ൌ ሺ ݈ଵ, ݈ଶ, ݈ଷሻሺ݈ଵ
ᇱ
, ݈ଶ, ݈ଷ
ᇱ
ሻ
Where,
݈ଵ ൌ min ሼ ܽଵܾଵ, ܽଵܾଷ, ܽଷܾଵ, ܽଷܾଷሽ
݈ଶ ൌ ܽଶܾଶ
lଷ = max { ܽଵܾଵ, ܽଵܾଷ, ܽଷܾଵ, ܽଷܾଷሽ
lଵ
'
ൌmin {ܽଵ
ᇱ
ܾଵ
ᇱ
, ܽଵ
ᇱ
ܾଷ
ᇱ
, ܽଷ
ᇱ
ܾଵ
ᇱ
, ܽଷ
ᇱ
ܾଷ
ᇱ
}
݈ଶ ൌ ܽଶܾଶ
lଷ
'
ൌ max {ܽଵ
ᇱ
ܾଵ
ᇱ
, ܽଵ
ᇱ
ܾଷ
ᇱ
, ܽଷ
ᇱ
ܾଵ
ᇱ
, ܽଷ
ᇱ
ܾଷ
ᇱ
}
Scalar multiplication:
i. kA෩୍
ൌ ሺkaଵ, kaଶ, kaଷሻሺk aଵ
'
, kaଶ, kaଷ
'
ሻ, for K 0
ii. ݇ܣሚூ
ൌ ሺ݇ܽଷ, ݇ܽଶ, ݇ܽଵሻሺ݇ ܽଷ
ᇱ
, ݇ܽଶ, ݇ܽଵ
ᇱ ሻ, ݂ܭ ݎ ൏ 0
7. Algorithmic approach 3985
3. Intuitionistic Fuzzy Transportation Problem and its
Mathematical Formulation
Consider a transportation with m IF origins (rows) and n IF destinations
(columns). Let ܿ be the cost of transporting one unit of the product from ith
IF
(Intuitionistic Fuzzy) origin to jth
IF destination. ܽ
ூ
ൌ ሺܽ
ଵ
, ܽ
ଶ
, ܽ
ଷ
ሻሺܽ
ଵ′
, ܽ
ଶ
, ܽ
ଷ′
ሻ be
the quantity of commodity available at IF origin i. ܾ෨
ூ
ൌ ൫ܾ
ଵ
, ܾ
ଶ
, ܾ
ଷ
൯ ቀܾ
ଵ′
, ܾ
ଶ
, ܾ
ଷ′
ቁ
the quantity of commodity needed at intuitionistic fuzzy destination j.
ݔ
ூ
ൌ ሺݔ
ଵ
, ݔ
ଶ
, ݔ
ଷ
ሻሺݔ
ଵ′
, ݔ
ଶ
, ݔ
ଷ′
ሻ is the quantity transported from ith
IF origin to jth
IF destination, so as to minimize the IF transportation cost.
(IFTP) Minimize ܼ෨ூ
ൌ ∑ ∑ ܿ⨂
ୀଵ
ୀ ሺݔ
ଵ
, ݔ
ଶ
, ݔ
ଷ
ሻሺݔ
ଵ′
, ݔ
ଶ
, ݔ
ଷ′
ሻ
Subject to,
ሺݔ
ଵ
, ݔ
ଶ
, ݔ
ଷ
ሻሺݔ
ଵ′
, ݔ
ଶ
, ݔ
ଷ′
ሻ
ୀଵ
ൎ ሺܽ
ଵ
, ܽ
ଶ
, ܽ
ଷ
ሻ ቀܽ
ଵ′
, ܽ
ଶ
, ܽ
ଷ′
ቁ , ݂݅ ݎ ൌ 1,2, … , ݉
ሺݔ
ଵ
, ݔ
ଶ
, ݔ
ଷ
ሻሺݔ
ଵ′
, ݔ
ଶ
, ݔ
ଷ′
ሻ
ୀଵ
ൎ ൫ܾ
ଵ
, ܾ
ଶ
, ܾ
ଷ
൯ ቀܾ
ଵ′
, ܾ
ଶ
, ܾ
ଷ′
ቁ , ݂݆ ݎ ൌ 1,2, … , ݊
ሺݔ
ଵ
, ݔ
ଶ
, ݔ
ଷ
ሻሺݔ
ଵ′
, ݔ
ଶ
, ݔ
ଷ′
ሻ ≽ 0෨ூ
, ݂݅ ݎ ൌ 1,2, … , ݉ ܽ݊݀
݆ ൌ 1,2, … , ݊
Where m = the number of supply points
n = the number of demand points
The above IFTP can be stated in the below tabular form
8. 3986 R. Jahir Hussain and P. Senthil Kumar
4. The Computational Procedure for Intuitionistic Fuzzy Zero
Point Method
This proposed method is used for finding the optimal basic feasible
solution in an intuitionistic fuzzy environment and the following step by step
procedure is utilized to find out the same.
Step 1. Construct the transportation table whose cost matrix is crisp value as well
as supplies and demands are intuitionistic fuzzy numbers. Convert the given
problem into a balanced one, if it is not, by ranking method.
Step 2. In the cost matrix subtract the smallest element in each row from every
element of that row.
Step 3. In the reduced matrix that is after using the step 2, subtract the smallest
element in each column from every element of that column.
Step 4. Check if each row intuitionistic fuzzy supply is less than to sum of the
column Intuitionistic fuzzy demands whose reduced costs in that row are zero.
Also, check if each column intuitionistic fuzzy demand is less than to the sum of
the intuitionistic fuzzy supplies whose reduced costs in that column are zero. If so,
go to step 7. Otherwise, go to step 5.
Step 5. Draw the minimum number of vertical lines and horizontal lines to cover
all the zeros of the reduced cost matrix such that some entries of row(s) or / and
column(s) which do not satisfy the condition of the step 4 are not covered.
1 2… n IF
Supply
1 ݔଵଵ
ூ
ܿଵଵ
ݔଵଶ
ூ
⋯
ܿଵଶ ⋯
ݔଵ
ூ
ܿଵ ܽଵ
ூ
2
.
.
.
ݔଶଵ
ூ
ܿଶଵ
.
.
.
ݔଶଶ
ூ
⋯
ܿଶଶ ⋯
.
.
.
ݔଶ
ூ
ܿଶ
.
.
.
ܽଶ
ூ
.
.
.
m ݔଵ
ூ
ܿଵ
ݔଶ
ூ
⋯
ܿଶ ⋯
ݔ
ூ
ܿ ܽ
ூ
IF
Demand
ܾ෨ଵ
ூ ܾ෨ଶ
ூ
… ܾ෨
ூ
ܽ
ூ
ୀଵ
ൌ ܾ෨
ூ
ୀଵ
9. Algorithmic approach 3987
Step 6. Develop the new revised reduced cost matrix table as follows:
i. Select the smallest element among all the uncovered elements in the cost
matrix.
ii. Subtract this least element from all the uncovered elements and add it to
the element which lies at the intersection of any two lines. Thus, we
obtain the modified cost matrix and then go to step 4.
Step 7. Select a cell in the reduced cost matrix whose reduced cost is the
maximum cost say (ߙ, ߚ). If there are more than one occur then select arbitrarily.
Step 8. Select a cell in the ߙ- row or / and ߚ- column of the reduced cost matrix
which is the only cell whose reduced cost is zero and then allot the maximum
possible value to that cell. If such cell does not occur for the maximum value,
find the next maximum so that such a cell occurs. If such cell does not occur for
any value, we select any cell in the reduced cost matrix whose reduced cost is
zero.
Step 9. Reform the reduced intuitionistic fuzzy transportation table after deleting
the fully used intuitionistic fuzzy supply points and the fully received
intuitionistic fuzzy demand points and also, modify it to include the not fully used
intuitionistic fuzzy supply points and the not fully received intuitionistic fuzzy
demand points.
Step 10. Repeat step 7 to the step 9 until all intuitionistic fuzzy supply points are
fully used and all intuitionistic fuzzy demand points are fully received. This
allotment yields an optimal solution.
5. Numerical Example:
Consider the 4× 4 IFTP
Since ∑ ܽ
ூ
ൌ ∑ ܾ෨
ூ
ୀଵ
ୀ = (17, 27, 38) (11, 27, 44), the problem is balanced
IFTP.
Now, using the step 2 to the step 3 of the intuitionistic fuzzy zero point
method, we have the following reduced intuitionistic fuzzy transportation table.
IFD1 IFD2 IFD3 IFD4 IF supply
IFO1 16 1 8 13 (2,4,5)(1,4,6)
IFO2 11 4 7 10 (4,6,8)(3,6,9)
IFO3 8 15 9 2 (3,7,12)(2,7,13)
IFO4 6 12 5 14 (8,10,13)(5,10,16)
IF
demand
(3,4,6)(1,4,8) (2,5,7)(1,5,8) (10,15,20)(8,15,22) (2,3,5)(1,3,6)
10. 3988 R. Jahir Hussain and P. Senthil Kumar
Now, using the step 4 to the step 6 of the intuitionistic fuzzy zero point method,
we have the following allotment table.
Now, using the allotment rules of the intuitionistic fuzzy zero point method,
we have the allotment
The intuitionistic fuzzy optimal solution in terms of triangular intuitionistic
fuzzy numbers:
ݔூ
ଵଶ = (2,4,5)(1,4,6), ݔூ
ଶଶ= (-3,1,5)(-5,1,7), ݔூ
ଶଷ= (-1,5,11)(-4,5,14),
ݔூ
ଷଵ= (-2,4,10)(-4,4,12), ݔூ
ଷସ= (2,3,5)(1,3,6), ݔூ
ସଵ= (-7, 0,8)(-11,0,12),
ݔூ
ସଷ=(-1,10,21)(-6,10,26)
Hence, the total intuitionistic fuzzy transportation minimum cost is
Min ܼ෨I
= (-76,131,345)(-173,131,442)
IFD1 IFD2 IFD3 IFD4 IF supply
IFO1 14 0 7 12 (2,4,5)(1,4,6)
IFO2 6 0 3 6 (4,6,8)(3,6,9)
IFO3 5 13 7 0 (3,7,12)(2,7,13)
IFO4 0 7 0 9 (8,10,13)(5,10,16)
IF
demand
(3,4,6)(1,4,8) (2,5,7)(1,5,8) (10,15,20)(8,15,22) (2,3,5)(1,3,6)
IFD1 IFD2 IFD3 IFD4 IF supply
IFO1 11 0 4 14 (2,4,5)(1,4,6)
IFO2 3 0 0 8 (4,6,8)(3,6,9)
IFO3 0 11 2 0 (3,7,12)(2,7,13)
IFO4 0 10 0 14 (8,10,13)(5,10,16)
IF
demand
(3,4,6)(1,4,8) (2,5,7)(1,5,8) (10,15,20)(8,15,22) (2,3,5)(1,3,6)
IFD1 IFD2 IFD3 IFD4 IF supply
IFO1 (2,4,5)(1,4,6) (2,4,5)(1,4,6)
IFO2 (-3,1,5)(-5,1,7) (-1,5,11)(-4,5,14) (4,6,8)(3,6,9)
IFO3 (-2,4,10)
(-4,4,12)
(2,3,5)(1,3,6) (3,7,12)(2,7,13)
IFO4 (-7,0,8)
(-11,0,12)
(-1,10,21)(-6,10,26) (8,10,13)(5,10,16)
IF
demand
(3,4,6)(1,4,8) (2,5,7)(1,5,8) (10,15,20)(8,15,22) (2,3,5)(1,3,6)
11. Algorithmic approach 3989
6. Conclusion
Mathematical formulation of intuitionistic fuzzy transportation problem and
procedure for finding an intuitionistic fuzzy optimal solution are discussed with
relevant numerical example. The new arithmetic operations of triangular
intuitionistic fuzzy numbers are employed to get the optimal solution in terms of
triangular intuitionistic fuzzy numbers. The same approach of solving the
intuitionistic fuzzy problems may also be utilized in future studies of operational
research.
References
[1] K.T.Atanassov, Intuitionistic fuzzy sets, fuzzy sets and systems, vol.20,
no.1.pp.87- 96, 1986.
[2] R.Bellman, L.A.Zadeh,Decision making in a fuzzy environment,
management sci.17(B)(1970)141-164.
[3] S.Ismail Mohideen, P.Senthil Kumar, A Comparative Study on
Transportation Problem in fuzzy environment. International Journal of
Mathematics Research,Vol.2Number.1 (2010),pp. 151-158.
[4] A.Nagoor gani, K.Abdul Razak, Two stage fuzzy transportation problem,
journal of physical sciences,vol.10,2006,63-69.
[5] A.Nagoor Gani, Abbas., Intuitionistic Fuzzy Transportation problem,
proceedings of the heber international conference pp.445-451.
[6] P.Pandian and G.Natarajan., A new algorithm for finding a fuzzy optimal
solution for fuzzy Transportation problems. Applied mathematics
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no3, 2009.
[8] L.A. Zadeh, Fuzzy sets, information and computation, vol.8, pp.338-353,
1965
Received: February, 2012
12. Applied Mathematical Sciences, VoI. 6,2072, no.77 - 80
Contents
N. Kosugi, K. Suyama, Digital redesign of infinite'dimensional contrullers
based on numerical integration
O. Bumbariu, A convergence result for the B'algorithm
image quality
P. S. Fam, A. H. Pooi, .Analysis of two'way contingency tables
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D. Di Caprio, F. J. Santos'Arteaga, Financial transparency and bank
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Dang Van Cuong, LS;ualued Gauss maps and spacelike swfaces of
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F. Nahayo, S. Khardi, M. Haddou, TWo'aircraft dynamic system on
approach. Flight path and noise optimization 3861
S.D. Kendre, M. B. Dhakne, On nonlinear Volterra integzodifferential
equations with analytic semigroups 3881
M. F. El-Sabbagh, S. I. El'Ganaini, Tlhe frrct integral method and its
applications to nonlinear equations 3893
A. M. Al'shatnawi, A new method in image steganography with improved
3801
3821
3907
3917
R. M. Elobaid, A comparison of mixed effect models for spatially conelated
data
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13. S' Goyal, V. Goyal, Mean value results for second and higher order partial
differential equations 8941
H. suprajitno, soluing muttiobjective linear programming problem using
interual arithmetic 3gS9
v. Gupta, s. B. singh, Effect of sic morphology on creep behauior in a
composite rotating disc hauing uaring thickness 8969
E. H. Hamouda, On the based folding of based graphs B}TE
R. Jahir Hussain, P. senthil Kumar, Atgorithmic approach for soluing
intuitionistic fuzzy transportation prcblem Bggl
shulin sun, cuihua Guo, chengmin Li, GIobaI analysis of an sErRS model
with saturating contact rate Bggl