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.
A New Method Based on MDA to Enhance the Face Recognition PerformanceCSCJournals
A novel tensor based method is prepared to solve the supervised dimensionality reduction problem. In this paper a multilinear principal component analysis(MPCA) is utilized to reduce the tensor object dimension then a multilinear discriminant analysis(MDA), is applied to find the best subspaces. Because the number of possible subspace dimensions for any kind of tensor objects is extremely high, so testing all of them for finding the best one is not feasible. So this paper also presented a method to solve that problem, The main criterion of algorithm is not similar to Sequential mode truncation(SMT) and full projection is used to initialize the iterative solution and find the best dimension for MDA. This paper is saving the extra times that we should spend to find the best dimension. So the execution time will be decreasing so much. It should be noted that both of the algorithms work with tensor objects with the same order so the structure of the objects has been never broken. Therefore the performance of this method is getting better. The advantage of these algorithms is avoiding the curse of dimensionality and having a better performance in the cases with small sample sizes. Finally, some experiments on ORL and CMPU-PIE databases is provided.
On Intuitionistic Fuzzy Transportation Problem Using Pentagonal Intuitionisti...YogeshIJTSRD
In this paper a new method is proposed for finding an optimal solution for Pentagonal intuitionistic fuzzy transportation problems, in which the cost values are Pentagonal intuitionistic fuzzy numbers. The procedure is illustrated with a numerical example. P. Parimala | P. Kamalaveni "On Intuitionistic Fuzzy Transportation Problem Using Pentagonal Intuitionistic Fuzzy Numbers Solved by Modi Method" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-3 , April 2021, URL: https://www.ijtsrd.com/papers/ijtsrd41094.pdf Paper URL: https://www.ijtsrd.com/mathemetics/applied-mathematics/41094/on-intuitionistic-fuzzy-transportation-problem-using-pentagonal-intuitionistic-fuzzy-numbers-solved-by-modi-method/p-parimala
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.
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.
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.
On the construction and comparison of an explicit iterativeAlexander Decker
The document presents an explicit iterative algorithm to solve ordinary differential equations with initial conditions. It constructs the algorithm and compares its performance to two non-standard finite difference schemes on two test problems. For both test problems, the proposed algorithm performs better in terms of maximum absolute error and computational effort. It also efficiently follows the oscillatory behavior of nonlinear systems like Lotka-Volterra and mass-spring models compared to the nonstandard schemes. Numerical results obtained with the proposed algorithm are found to be computationally reliable and practical compared to the two other methods discussed.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
This document introduces fuzzy sets and provides definitions of key concepts. It begins with an overview of fuzzy set theory and its development. Several fundamental definitions are then given, including membership function, universe of discourse, fuzzy set, support, crossover point, height, α-cut, and level set. Examples are provided to illustrate each definition. Common operations on fuzzy sets like union, intersection, and complement are also defined. The purpose is to lay the groundwork for understanding fuzzy set theory and its basic elements.
A New Method Based on MDA to Enhance the Face Recognition PerformanceCSCJournals
A novel tensor based method is prepared to solve the supervised dimensionality reduction problem. In this paper a multilinear principal component analysis(MPCA) is utilized to reduce the tensor object dimension then a multilinear discriminant analysis(MDA), is applied to find the best subspaces. Because the number of possible subspace dimensions for any kind of tensor objects is extremely high, so testing all of them for finding the best one is not feasible. So this paper also presented a method to solve that problem, The main criterion of algorithm is not similar to Sequential mode truncation(SMT) and full projection is used to initialize the iterative solution and find the best dimension for MDA. This paper is saving the extra times that we should spend to find the best dimension. So the execution time will be decreasing so much. It should be noted that both of the algorithms work with tensor objects with the same order so the structure of the objects has been never broken. Therefore the performance of this method is getting better. The advantage of these algorithms is avoiding the curse of dimensionality and having a better performance in the cases with small sample sizes. Finally, some experiments on ORL and CMPU-PIE databases is provided.
On Intuitionistic Fuzzy Transportation Problem Using Pentagonal Intuitionisti...YogeshIJTSRD
In this paper a new method is proposed for finding an optimal solution for Pentagonal intuitionistic fuzzy transportation problems, in which the cost values are Pentagonal intuitionistic fuzzy numbers. The procedure is illustrated with a numerical example. P. Parimala | P. Kamalaveni "On Intuitionistic Fuzzy Transportation Problem Using Pentagonal Intuitionistic Fuzzy Numbers Solved by Modi Method" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-3 , April 2021, URL: https://www.ijtsrd.com/papers/ijtsrd41094.pdf Paper URL: https://www.ijtsrd.com/mathemetics/applied-mathematics/41094/on-intuitionistic-fuzzy-transportation-problem-using-pentagonal-intuitionistic-fuzzy-numbers-solved-by-modi-method/p-parimala
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.
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.
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.
On the construction and comparison of an explicit iterativeAlexander Decker
The document presents an explicit iterative algorithm to solve ordinary differential equations with initial conditions. It constructs the algorithm and compares its performance to two non-standard finite difference schemes on two test problems. For both test problems, the proposed algorithm performs better in terms of maximum absolute error and computational effort. It also efficiently follows the oscillatory behavior of nonlinear systems like Lotka-Volterra and mass-spring models compared to the nonstandard schemes. Numerical results obtained with the proposed algorithm are found to be computationally reliable and practical compared to the two other methods discussed.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
This document introduces fuzzy sets and provides definitions of key concepts. It begins with an overview of fuzzy set theory and its development. Several fundamental definitions are then given, including membership function, universe of discourse, fuzzy set, support, crossover point, height, α-cut, and level set. Examples are provided to illustrate each definition. Common operations on fuzzy sets like union, intersection, and complement are also defined. The purpose is to lay the groundwork for understanding fuzzy set theory and its basic elements.
This document discusses fuzzy soft sets and soft set theory. It begins with an introduction to soft set theory as a mathematical tool for dealing with uncertainties. It then provides definitions and examples related to soft set theory, including the definition of a soft set, operations on soft sets like union and intersection, and concepts like null soft sets and absolute soft sets. The document aims to lay the foundations of soft set theory.
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.
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
CHN and Swap Heuristic to Solve the Maximum Independent Set ProblemIJECEIAES
We describe a new approach to solve the problem to find the maximum independent set in a given Graph, known also as Max-Stable set problem (MSSP). In this paper, we show how Max-Stable problem can be reformulated into a linear problem under quadratic constraints, and then we resolve the QP result by a hybrid approach based Continuous Hopfeild Neural Network (CHN) and Local Search. In a manner that the solution given by the CHN will be the starting point of the local search. The new approach showed a good performance than the original one which executes a suite of CHN runs, at each execution a new leaner constraint is added into the resolved model. To prove the efficiency of our approach, we present some computational experiments of solving random generated problem and typical MSSP instances of real life problem.
Stability criterion of periodic oscillations in a (8)Alexander Decker
This academic article discusses local cohomology modules and cofiniteness. It presents several definitions, theorems, and examples regarding local cohomology modules and systems of ideals. Specifically, it constructs an ideal system using tridiagonal matrices of tridiagonal subsets and shows that the local cohomology modules are finite in this special case.
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.
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.
The concept of an intuitionistic fuzzy number (IFN) is of importance for representing an ill-known quantity. Ranking fuzzy numbers plays a very important role in the decision process, data analysis and applications. The concept of an IFN is of importance for quantifying an ill-known quantity. Ranking of intuitionistic fuzzy numbers plays a vital role in decision making and linear programming problems. Also, ranking of intuitionistic fuzzy numbers is a very difficult problem. In this paper, a new method for ranking intuitionistic fuzzy number is developed by means of magnitude for different forms of intuitionistic fuzzy numbers. In Particular ranking is done for trapezoidal intuitionistic fuzzy numbers, triangular intuitionistic fuzzy numbers, symmetric trapezoidal intuitionistic fuzzy numbers, and symmetric triangular intuitionistic fuzzy numbers. Numerical examples are illustrated for all the defined different forms of intuitionistic fuzzy numbers. Finally some comparative numerical examples are illustrated to express the advantage of the proposed method.
The numerical solution of Huxley equation by the use of two finite difference methods is done. The first one is the explicit scheme and the second one is the Crank-Nicholson scheme. The comparison between the two methods showed that the explicit scheme is easier and has faster convergence while the Crank-Nicholson scheme is more accurate. In addition, the stability analysis using Fourier (von Neumann) method of two schemes is investigated. The resulting analysis showed that the first scheme
is conditionally stable if, r ≤ 2 − aβ∆t , ∆t ≤ 2(∆x)2 and the second
scheme is unconditionally stable.
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.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
This document discusses applications of first order ordinary differential equations (ODEs) as mathematical models. It provides examples of using first order ODEs to model population growth and decay, predator-prey interactions, and mixing problems. The modeling of logistic population growth with a first order ODE is shown to be more powerful than exponential modeling. Basic principles for modeling like mass action and conservation of mass are also outlined.
The aim of this research is to find accurate solution for the Troesch’s problem by using high performance technique based on parallel processing implementation.
Design/methodology/approach – Feed forward neural network is designed to solve important type of differential equations that arises in many applied sciences and engineering applications. The suitable designed based on choosing suitable learning rate, transfer function, and training algorithm. The authors used back propagation with new implement of Levenberg - Marquardt training algorithm. Also, the authors depend new idea for choosing the weights. The effectiveness of the suggested design for the network is shown by using it for solving Troesch problem in many cases.
Findings – New idea for choosing the weights of the neural network, new implement of Levenberg - Marquardt training algorithm which assist to speeding the convergence and the implementation of the suggested design demonstrates the usefulness in finding exact solutions.
This document presents an overview of the differential transform method (DTM) for solving differential equations. DTM uses Taylor series to obtain approximate or exact solutions. The document defines 1D, 2D, and 3D DTM and lists common operations. Examples are provided of applying DTM to solve systems of linear/non-linear differential equations. The document concludes with references for further applications of DTM in engineering and mathematics.
A Generalized Sampling Theorem Over Galois Field Domains for Experimental Des...csandit
In this paper, the sampling theorem for bandlimited functions over
domains is
generalized to one over ∏
domains. The generalized theorem is applicable to the
experimental design model in which each factor has a different number of levels and enables us
to estimate the parameters in the model by using Fourier transforms. Moreover, the relationship
between the proposed sampling theorem and orthogonal arrays is also provided.
KEY
Two parameter entropy of uncertain variableSurender Singh
This document introduces a two parameter entropy measure for uncertain variables based on two parameter probabilistic entropy. It begins by reviewing concepts of uncertainty theory such as uncertainty space, uncertain variables, and independence of uncertain variables. It then defines entropy of an uncertain variable using Shannon entropy. Previous work on one parameter entropy of uncertain variables is summarized. The document proposes a new two parameter entropy for uncertain variables, providing its definition and examples of calculating it for specific uncertainty distributions. Properties of the two parameter entropy are discussed.
Handling missing data with expectation maximization algorithmLoc Nguyen
Expectation maximization (EM) algorithm is a powerful mathematical tool for estimating parameter of statistical models in case of incomplete data or hidden data. EM assumes that there is a relationship between hidden data and observed data, which can be a joint distribution or a mapping function. Therefore, this implies another implicit relationship between parameter estimation and data imputation. If missing data which contains missing values is considered as hidden data, it is very natural to handle missing data by EM algorithm. Handling missing data is not a new research but this report focuses on the theoretical base with detailed mathematical proofs for fulfilling missing values with EM. Besides, multinormal distribution and multinomial distribution are the two sample statistical models which are concerned to hold missing values.
A derivative free high ordered hybrid equation solverZac Darcy
Generally a range of equation solvers for estimating the solution of an equation contain the derivative of
first or higher order. Such solvers are difficult to apply in the instances of complicated functional
relationship. The equation solver proposed in this paper meant to solve many of the involved complicated
problems and establishing a process tending towards a higher ordered by alloying the already proved
conventional methods like Newton-Raphson method (N-R), Regula Falsi method (R-F) & Bisection method
(BIS). The present method is good to solve those nonlinear and transcendental equations that cannot be
solved by the basic algebra. Comparative analysis are also made with the other racing formulas of this
group and the result shows that present method is faster than all such methods of the class.
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.
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
A LEAST ABSOLUTE APPROACH TO MULTIPLE FUZZY REGRESSION USING Tw- NORM BASED O...ijfls
A least absolute approach to multiple fuzzy regression using Tw-norm based arithmetic operations is
discussed by using the generalized Hausdorff metric and it is investigated for the crisp input- fuzzy output
data. A comparative study based on two data sets are presented using the proposed method using shape
preserving operations with other existing method.
This document discusses fuzzy soft sets and soft set theory. It begins with an introduction to soft set theory as a mathematical tool for dealing with uncertainties. It then provides definitions and examples related to soft set theory, including the definition of a soft set, operations on soft sets like union and intersection, and concepts like null soft sets and absolute soft sets. The document aims to lay the foundations of soft set theory.
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.
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
CHN and Swap Heuristic to Solve the Maximum Independent Set ProblemIJECEIAES
We describe a new approach to solve the problem to find the maximum independent set in a given Graph, known also as Max-Stable set problem (MSSP). In this paper, we show how Max-Stable problem can be reformulated into a linear problem under quadratic constraints, and then we resolve the QP result by a hybrid approach based Continuous Hopfeild Neural Network (CHN) and Local Search. In a manner that the solution given by the CHN will be the starting point of the local search. The new approach showed a good performance than the original one which executes a suite of CHN runs, at each execution a new leaner constraint is added into the resolved model. To prove the efficiency of our approach, we present some computational experiments of solving random generated problem and typical MSSP instances of real life problem.
Stability criterion of periodic oscillations in a (8)Alexander Decker
This academic article discusses local cohomology modules and cofiniteness. It presents several definitions, theorems, and examples regarding local cohomology modules and systems of ideals. Specifically, it constructs an ideal system using tridiagonal matrices of tridiagonal subsets and shows that the local cohomology modules are finite in this special case.
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.
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.
The concept of an intuitionistic fuzzy number (IFN) is of importance for representing an ill-known quantity. Ranking fuzzy numbers plays a very important role in the decision process, data analysis and applications. The concept of an IFN is of importance for quantifying an ill-known quantity. Ranking of intuitionistic fuzzy numbers plays a vital role in decision making and linear programming problems. Also, ranking of intuitionistic fuzzy numbers is a very difficult problem. In this paper, a new method for ranking intuitionistic fuzzy number is developed by means of magnitude for different forms of intuitionistic fuzzy numbers. In Particular ranking is done for trapezoidal intuitionistic fuzzy numbers, triangular intuitionistic fuzzy numbers, symmetric trapezoidal intuitionistic fuzzy numbers, and symmetric triangular intuitionistic fuzzy numbers. Numerical examples are illustrated for all the defined different forms of intuitionistic fuzzy numbers. Finally some comparative numerical examples are illustrated to express the advantage of the proposed method.
The numerical solution of Huxley equation by the use of two finite difference methods is done. The first one is the explicit scheme and the second one is the Crank-Nicholson scheme. The comparison between the two methods showed that the explicit scheme is easier and has faster convergence while the Crank-Nicholson scheme is more accurate. In addition, the stability analysis using Fourier (von Neumann) method of two schemes is investigated. The resulting analysis showed that the first scheme
is conditionally stable if, r ≤ 2 − aβ∆t , ∆t ≤ 2(∆x)2 and the second
scheme is unconditionally stable.
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.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
This document discusses applications of first order ordinary differential equations (ODEs) as mathematical models. It provides examples of using first order ODEs to model population growth and decay, predator-prey interactions, and mixing problems. The modeling of logistic population growth with a first order ODE is shown to be more powerful than exponential modeling. Basic principles for modeling like mass action and conservation of mass are also outlined.
The aim of this research is to find accurate solution for the Troesch’s problem by using high performance technique based on parallel processing implementation.
Design/methodology/approach – Feed forward neural network is designed to solve important type of differential equations that arises in many applied sciences and engineering applications. The suitable designed based on choosing suitable learning rate, transfer function, and training algorithm. The authors used back propagation with new implement of Levenberg - Marquardt training algorithm. Also, the authors depend new idea for choosing the weights. The effectiveness of the suggested design for the network is shown by using it for solving Troesch problem in many cases.
Findings – New idea for choosing the weights of the neural network, new implement of Levenberg - Marquardt training algorithm which assist to speeding the convergence and the implementation of the suggested design demonstrates the usefulness in finding exact solutions.
This document presents an overview of the differential transform method (DTM) for solving differential equations. DTM uses Taylor series to obtain approximate or exact solutions. The document defines 1D, 2D, and 3D DTM and lists common operations. Examples are provided of applying DTM to solve systems of linear/non-linear differential equations. The document concludes with references for further applications of DTM in engineering and mathematics.
A Generalized Sampling Theorem Over Galois Field Domains for Experimental Des...csandit
In this paper, the sampling theorem for bandlimited functions over
domains is
generalized to one over ∏
domains. The generalized theorem is applicable to the
experimental design model in which each factor has a different number of levels and enables us
to estimate the parameters in the model by using Fourier transforms. Moreover, the relationship
between the proposed sampling theorem and orthogonal arrays is also provided.
KEY
Two parameter entropy of uncertain variableSurender Singh
This document introduces a two parameter entropy measure for uncertain variables based on two parameter probabilistic entropy. It begins by reviewing concepts of uncertainty theory such as uncertainty space, uncertain variables, and independence of uncertain variables. It then defines entropy of an uncertain variable using Shannon entropy. Previous work on one parameter entropy of uncertain variables is summarized. The document proposes a new two parameter entropy for uncertain variables, providing its definition and examples of calculating it for specific uncertainty distributions. Properties of the two parameter entropy are discussed.
Handling missing data with expectation maximization algorithmLoc Nguyen
Expectation maximization (EM) algorithm is a powerful mathematical tool for estimating parameter of statistical models in case of incomplete data or hidden data. EM assumes that there is a relationship between hidden data and observed data, which can be a joint distribution or a mapping function. Therefore, this implies another implicit relationship between parameter estimation and data imputation. If missing data which contains missing values is considered as hidden data, it is very natural to handle missing data by EM algorithm. Handling missing data is not a new research but this report focuses on the theoretical base with detailed mathematical proofs for fulfilling missing values with EM. Besides, multinormal distribution and multinomial distribution are the two sample statistical models which are concerned to hold missing values.
A derivative free high ordered hybrid equation solverZac Darcy
Generally a range of equation solvers for estimating the solution of an equation contain the derivative of
first or higher order. Such solvers are difficult to apply in the instances of complicated functional
relationship. The equation solver proposed in this paper meant to solve many of the involved complicated
problems and establishing a process tending towards a higher ordered by alloying the already proved
conventional methods like Newton-Raphson method (N-R), Regula Falsi method (R-F) & Bisection method
(BIS). The present method is good to solve those nonlinear and transcendental equations that cannot be
solved by the basic algebra. Comparative analysis are also made with the other racing formulas of this
group and the result shows that present method is faster than all such methods of the class.
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.
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
A LEAST ABSOLUTE APPROACH TO MULTIPLE FUZZY REGRESSION USING Tw- NORM BASED O...ijfls
A least absolute approach to multiple fuzzy regression using Tw-norm based arithmetic operations is
discussed by using the generalized Hausdorff metric and it is investigated for the crisp input- fuzzy output
data. A comparative study based on two data sets are presented using the proposed method using shape
preserving operations with other existing method.
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A method for solving unbalanced intuitionistic fuzzy transportation problems
1.
2. Notes on Intuitionistic Fuzzy Sets
Editorial Board
Name Institution E-mail
Honorary editor
Lotfi Zadeh University of California, Berkeley, USA zadeh
eecs.berkeley.edu
Editors-in-Chief
Krassimir
Atanassov
Institute of Biophysics and Biomedical
Engineering, Bulgarian Academy of
Sciences
P.O.Box 12, Sofia-1113, Bulgaria
krat bas.bg
k.t.atanassov
gmail.com
Humberto
Bustince Sola
Dept. of Automatic and Computation, Public
University of Navarra
31006, Campus Arrosadia, Pamplona, Spain
bustince unavarra.es
Janusz Kacprzyk Systems Research Institute, Polish Academy
of Sciences
ul. Newelska 6, 01-447 Warszawa, Poland
kacprzyk
ibspan.waw.pl
Editorial board
Adrian Ban Department of Mathematics and Informatics,
University of Oradea, Romania
aiban math.uoradea.ro
Ranjit Biswas Institute of Technology and Management,
Gwalior, India
ranjit
maths.iitkgp.ernet.in
Oscar Castillo Tijuana Institute of Technology, Tijuana,
Mexico
ocastillo tectijuana.mx
Panagiotis
Chountas
School of Computer Science, University of
Westminster, London, UK
p.i.chountas
westminster.ac.uk
Chris Cornelis Dept. of Applied Mathematics and Computer
Science, Ghent University, Belgium
chris.cornelis ugent.be
Didier Dubois IRIT Laboratory, Paul Sabatier University,
Toulouse, France
didier.dubois irit.fr
Tadeusz
Gerstenkorn
University of Trade in Łódź, Poland tadger plunlo51.bitnet
Stefan
Hadjitodorov
Institute of Biophysics and Biomedical
Engineering, Bulgarian Academy of
Sciences, Bulgaria
sthadj argo.bas.bg
Etienne Kerre Fuzziness and Uncertainty Modelling
Research Group, Ghent University, Belgium
etienne.kerre ugent.be
Ketty Peeva Faculty of Applied Mathematics and
Informatics, Techincial University of Sofia,
Bulgaria
kgp tu-sofia.bg
Henri Prade IRIT Laboratory, Paul Sabatier University,
Toulouse, France
henri.prade irit.fr
Beloslav Riečan Matej Bel University, Banská Bystrica,
Slovakia
riecan fpv.umb.sk
Anthony Shannon Faculty of Engineering & IT, University of
Technology, Sydney, NSW 2007, Australia
tshannon38 gmail.com
Eulalia Szmidt Systems Research Institute, Polish Academy
of Sciences, Poland
szmidt ibspan.waw.pl
Ronald Yager Machine Intelligence Insitute, Iona College,
USA
yager panix.com
Hans-Jürgen
Zimmermann
Aachen Institute of Technology, Germany zi buggi.or.rwth-
aachen.de
3. Notes on Intuitionistic Fuzzy Sets
ISSN 1310–4926
Vol. 21, 2015, No. 3, 54–65
A method for solving unbalanced
intuitionistic fuzzy transportation problems
P. Senthil Kumar1
and R. Jahir Hussain2
1
PG and Research Department of Mathematics
Jamal Mohamed College (Autonomous)
Tiruchirappalli-620 020, Tamil Nadu, India
e-mails: senthilsoft_5760@yahoo.com,
senthilsoft1985@gmail.com
2
PG and Research Department of Mathematics
Jamal Mohamed College (Autonomous)
Tiruchirappalli-620 020, Tamil Nadu, India
e-mail: hssn_jhr@yahoo.com
Abstract: 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.
Keywords: Intuitionistic fuzzy set, Triangular intuitionistic fuzzy number, Unbalanced intuition-
istic fuzzy transportation problem, PSK method, Optimal solution.
AMS Classification: 03E72, 03F55, 90B06.
54
4. 1 Introduction
In several real life situations, there is need of shipping the product from different origins (sources)
to different destinations and the aim of the decision maker is to find how much quantity of the
product from which origin to which destination should be shipped so that all the supply points are
fully used and all the demand points are fully received as well as total transportation cost is min-
imum for a minimization problem or total transportation profit is maximum for a maximization
problem.
Since, in real life problems, there is always existence of impreciseness in the parameters of
transportation problem and in the literature Atanssov (1983), pointed out that it is better to use
intuitionistic fuzzy set as compared to fuzzy set Zadeh (1965) to deal with impreciseness. The
major advantage of Intuitionistic Fuzzy Set (IFS) over fuzzy set is that Intuitionistic Fuzzy Sets
(IFSs) separate the degree of membership (belongingness) and the degree of non membership
(non belongingness) of an element in the set. In the history of Mathematics, Burillo et al. (1994)
proposed definition of intuitionistic fuzzy number and studied its properties.
An Introduction to Operations Research Taha (2008) deals the transportation problem where
all the parameters are crisp number. In the lack of uncertainty of a crisp transportation problem,
several researchers like Dinagar and Palanivel (2009) investigated the transportation problem in
fuzzy environment using trapezoidal fuzzy numbers. Pandian and Natarajan (2010) proposed a
new algorithm for finding a fuzzy optimal solution for fuzzy transportation problem where all the
parameters are trapezoidal fuzzy numbers. Mohideen and Kumar (2010) did a comparative study
on transportation problem in fuzzy environment. In the issues of fuzzy transportation problem,
several researchers like Hussain and Kumar (2012a, 2012b), Gani and Abbas (2012) proposed a
method for solving transportation problem in which all the parameters except transportation cost
are represented by TIFN. Antony et al. (2014) discussed the transportation problem using triangu-
lar fuzzy number. They consider triangular fuzzy number as triangular intuitionistic fuzzy number
by using format only but they are always triangular fuzzy number. Dinagar and Thiripurasundari
(2014) solved transportation problem taking all the parameters that are trapezoidal intuitionistic
fuzzy numbers.
Therefore, the number of researchers have been solved intuitionistic fuzzy transportation
problems. To the best of our knowledge, no one has to proposed the algorithm for solving
unbalanced intuitionistic fuzzy transportation problems. In this paper, we introduced a simple
method called PSK method and proved a theorem, which states that every solution obtained by
PSK method to fully intuitionistic fuzzy transportation problem with equality constraints is a
fully intuitionistic fuzzy optimal. Atanassov (1995) presented ideas for intuitionistic fuzzy equa-
tions,inequalities and optimization. It is formulated the problem how to use the apparatus of the
IFSs for the needs of optimization. Hence, this paper gives a particular answer to this problem.
Moreover, the total optimum cost obtained by our method remains same as that of obtained by
converting the total intuitionistic fuzzy transportation cost by applying the ranking method of
Varghese and Kuriakose (2012). It is much easier to apply the proposed method as compared to
the existing method.
55
5. Rest of the paper is organized as follows: Section 2 deals with some terminology and new multi-
plication operation, Section 3 consists of ranking procedure and ordering principles of triangular
intuitionistic fuzzy number. Section 4 provides the definition of IFTP and its mathematical for-
mulation, Section 5 deals with solution procedure, finally the conclusion is given in Section 6.
2 Preliminaries
Definition 2.1. Let A be a classical set, µA(x) be a function from A to [0, 1]. A fuzzy set A∗
with
the membership function µA(x) is defined by
A∗
= {(x, µA(x)) : x ∈ A and µA(x) ∈ [0, 1]}.
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,
˜AI
= { x, µA(x), νA(x) : x ∈ X}
Where µA, νA : X → [0, 1], are functions such that 0 ≤ µA(x) + νA(x) ≤ 1, ∀x ∈ X. For
each x the membership µA(x) and νA(x) represent the degree of membership and the degree of
non-membership of the element x ∈ X to A ⊂ X respectively.
Definition 2.3. An Intuitionistic fuzzy subset A = { x, µA(x), νA(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 νA(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 ∀x ∈ R, the relation
0 ≤ µA(x) + νA(x) ≤ 1 holds.
The membership and non membership function of A is of the following form:
µA(x) =
0 for − ∞ < x ≤ m − α
f1(x) for x ∈ [m − α, m]
1 for x = m
h1(x) for x ∈ [m, m + β]
0 for m + β ≤ x < ∞
Where f1(x) and h1(x) are strictly increasing and decreasing function in [m−α, m] and [m, m+
β] respectively.
νA(x) =
1 for − ∞ < x ≤ m − α
f2(x) for x ∈ [m − α , m]; 0 ≤ f1(x) + f2(x) ≤ 1
0 for x = m
h2(x) for x ∈ [m, m + β ]; 0 ≤ h1(x) + h2(x) ≤ 1
1 for m + β ≤ x < ∞
56
6. Here m is the mean value of A.α, β are called left and right spreads of membership function µA(x)
respectively. α , β are represents left and right spreads of non membership function νA(x) respec-
tively. Symbolically, the intuitionistic fuzzy number ˜AI
is represented as AIFN = (m; α, β; α , β ).
Definition 2.4. A fuzzy number A is defined to be a triangular fuzzy number if its membership
function µA : R → [0, 1] is equal to
µA(x) =
(x−a1)
(a2−a1)
if x ∈ [a1, a2]
(a3−x)
(a3−a2)
if x ∈ [a2, a3]
0 otherwise
where a1 ≤ a2 ≤ a3. This fuzzy number is denoted by (a1, a2, a3).
Definition 2.5. A Triangular Intuitionistic Fuzzy Number( ˜AI
is an intuitionistic fuzzy set in R
with the following membership function µA(x) and non-membership function νA(x): )
µA(x) =
0 for x < a1
(x−a1)
(a2−a1)
for a1 ≤ x ≤ a2
1 for x = a2
(a3−x)
(a3−a2)
for a2 ≤ x ≤ a3
0 for x > a3
νA(x) =
1 for x < a1
(a2−x)
(a2−a1)
for a1 ≤ x ≤ a2
0 for x = a2
(x−a2)
(a3−a2)
for a2 ≤ x ≤ a3
1 for x > a3
Where a1 ≤ a1 ≤ a2 ≤ a3 ≤ a3 and µA(x), νA(x) ≤ 0.5 for µA(x) = νA(x), ∀x ∈ R This
TIFN is denoted by ˜AI
= (a1, a2, a3)(a1, a2, a3)
Particular cases: Let ˜AI
= (a1, a2, a3)(a1, a2, a3) be a TIFN. Then the following cases arise
Case 1: If a1 = a1, a3 = a3, then ˜AI
represent Tringular Fuzzy Number
(TFN). It is denoted by A = (a1, a2, a3).
Case 2: If a1 = a1 = a2 = a3 = a3 = m,then ˜AI
represent a real number m.
Definition 2.6. Let ˜AI
= (a1, a2, a3)(a1, a2, a3) and ˜BI
= (b1, b2, b3)(b1, b2, b3) be any two TIFNs
then the arithmetic operations as follows.
Addition: ˜AI
⊕ ˜BI
= (a1 + b1, a2 + b2, a3 + b3)(a1 + b1, a2 + b2, a3 + b3)
Subtraction: ˜AI ˜BI
= (a1 − b3, a2 − b2, a3 − b1)(a1 − b3, a2 − b2, a3 − b1)
Multiplication: If ( ˜AI
), ( ˜BI
) ≥ 0, then
˜AI
⊗ ˜BI
= (a1 ( ˜BI
), a2 ( ˜BI
), a3 ( ˜BI
))(a1 ( ˜BI
), a2 ( ˜BI
), a3 ( ˜BI
))
Note 1. All the parameters of the conventional TP such as supplies, demands and costs are in
positive. Since in transportation problems negative parameters has no physical meaning. Hence,
in the proposed method all the parameters may be assumed as non-negative triangular intuition-
istic fuzzy number. On the basis of this idea we need not further investigate the multiplication
operation under the condition that ( ˜AI
), ( ˜BI
) < 0.
Scalar Multiplication:
i. k ˜AI
= (ka1, ka2, ka3)(ka1, ka2, ka3), for k ≥ 0
ii. k ˜AI
= (ka3, ka2, ka1)(ka3, ka2, ka1), for k < 0
57
7. 3 Comparison of TIFN
Let ˜AI
= (a1, a2, a3)(a1, a2, a3) and ˜BI
= (b1, b2, b3)(b1, b2, b3) be two TIFNs. Then
(i) ( ˜AI
) > ( ˜BI
) iff ˜AI ˜BI
(ii) ( ˜AI
) < ( ˜BI
) iff ˜AI ˜BI
(iii) ( ˜AI
) = ( ˜BI
) iff ˜AI
≈ ˜BI
. where,
( ˜AI
) =
1
3
(a3 − a1)(a2 − 2a3 − 2a1) + (a3 − a1)(a1 + a2 + a3) + 3(a3
2
− a1
2
)
a3 − a1 + a3 − a1
( ˜BI
) =
1
3
(b3 − b1)(b2 − 2b3 − 2b1) + (b3 − b1)(b1 + b2 + b3) + 3(b3
2
− b1
2
)
b3 − b1 + b3 − b1
4 Fully intuitionistic fuzzy transportation problem
and its mathematical formulation
Definition 4.1. If the transportation problem has at least one of the parameter (cost) or two
of the parameters (supply and demand) or all of the parameters (supply, demand and cost) in
intuitionistic fuzzy numbers then the problem is called IFTP. Otherwise it is not an IFTP.
Further, intuitionistic fuzzy transportation problem can be classified into four categories. They
are:
• Type-1 intuitionistic fuzzy transportation problems;
• Type-2 intuitionistic fuzzy transportation problem;
• Type-3 intuitionistic fuzzy transportation problem (Mixed Intuitionistic Fuzzy Transporta-
tion Problem);
• Type-4 intuitionistic fuzzy transportation problem (Fully Intuitionistic Fuzzy Transporta-
tion Problem).
Definition 4.2. A transportation problem having intuitionistic fuzzy availabilities and intuition-
istic fuzzy demands but crisp costs is termed as intuitionistic fuzzy transportation problem of
type-1.
Definition 4.3. A transportation problem having crisp availabilities and crisp demands but intu-
itionistic fuzzy costs is termed as intuitionistic fuzzy transportation problem of type-2.
Definition 4.4. The transportation problem is said to be the type-3 intuitionistic fuzzy trans-
portation problem or mixed intuitionistic fuzzy transportation problem if all the parameters of
the transportation problem (such as supplies, demands and costs) must be in the mixture of crisp
numbers, triangular fuzzy numbers and triangular intuitionistic fuzzy numbers.
58
8. Definition 4.5. The transportation problem is said to be the type-4 intuitionistic fuzzy trans-
portation problem or fully intuitionistic fuzzy transportation problem if all the parameters of the
transportation problem (such as supplies, demands and costs) must be in intuitionistic fuzzy num-
bers.
Definition 4.6. The transportation problem is said to be balanced intuitionistic fuzzy transporta-
tion problem if total intuitionistic fuzzy supply is equal to total intuitionistic fuzzy demand. That
is,
m
i=1
˜aI
i =
n
j=1
˜bI
j
Definition 4.7. The transportation problem is said to be an unbalanced intuitionistic fuzzy trans-
portation problem if total intuitionistic fuzzy supply is not equal to total intuitionistic fuzzy de-
mand. That is,
m
i=1
˜aI
i =
n
j=1
˜bI
j
Definition 4.8. A set of intuitionistic fuzzy non negative allocations ˜xI
ij > ˜0I
satisfies the row and
column restriction is known as intuitionistic fuzzy feasible solution.
Definition 4.9. Any feasible solution is an intuitionistic fuzzy basic feasible solution if the number
of non negative allocations is at most (m + n − 1) where m is the number of rows and n is the
number of columns in the m × n transportation table.
Definition 4.10. If the intuitionistic fuzzy basic feasible solution contains less than (m + n − 1)
non negative allocations in m × n transportation table, it is said to be degenerate.
Definition 4.11. Any intuitionist fuzzy feasible solution to a transportation problem containing
m origins and n destinations is said to be intuitionist fuzzy non degenerate, if it contains exactly
(m + n − 1) occupied cells.
Definition 4.12. The intuitionistic fuzzy basic feasible solution is said to be intuitionistic fuzzy
optimal solution if it minimizes the total intuitionistic fuzzy transportation cost (or) it maximizes
the total intuitionistic fuzzy transportation profit.
Mathematical Formulation: Consider the transportation problem with m origins (rows) and n
destinations (columns). Let ˜cI
ij be the cost of transporting one unit of the product from ith
origin
to jth
destination. Let ˜aI
i be the quantity of commodity available at origin i. Let ˜bI
j be the quantity
of commodity needed at destination j and ˜xI
ij is the quantity transported from ith
origin to jth
destination, so as to minimize the total intuitionistic fuzzy transportation cost.
Mathematically Fully Intuitionistic Fuzzy Transportation Problem can be stated as (FIFTP).
( Model 1) (P)Minimize ˜ZI
=
m
i=1
n
j=1
˜cI
ij ⊗ ˜xI
ij
Subject to,
n
j=1
˜xI
ij ≈ ˜aI
i , for i = 1, 2, · · · , m
m
i=1
˜xI
ij ≈ ˜bI
j , for j = 1, 2, · · · , n
˜xI
ij
˜0I
, for i = 1, 2, · · · , m and j = 1, 2, · · · , n
59
9. where m = the number of supply points, n = the number of demand points:
˜cI
ij = (c1
ij, c2
ij, c3
ij)(c1
ij , c2
ij, c3
ij ), ˜aI
i = (a1
i , a2
i , a3
i )(a1
i , a2
i , a3
i ),
˜bI
j = (b1
j , b2
j , b3
j )(b1
j , b2
j , b3
j ), ˜xI
ij = (x1
ij, x2
ij, x3
ij)(x1
ij , x2
ij, x3
ij ).
When the supplies, demands and costs are intuitionistic fuzzy numbers, then the total cost
becomes an intuitionistic fuzzy number.
˜ZI
=
m
i=1
n
j=1
˜cI
ij ⊗ ˜xI
ij
Hence it cannot be minimized directly. For solving the problem we convert the intuitionistic
fuzzy supplies, intuitionistic fuzzy demands and the intuitionistic fuzzy costs into crisp ones by
an intuitionistc fuzzy number ranking method of Varghese and Kuriakose.
Consider the transportation problem with m origins (rows) and n destinations (columns). Let
cij be the cost of transporting one unit of the product from ith
origin to jth
destination. Let ai be
the quantity of commodity available at origin i. Let bj the quantity of commodity needed at desti-
nation j. Let xij denote the quantity of commodity transported from ith
origin to jth
destination,
so as to minimize the total transportation cost.
(Model 2) (TP)Minimize ( ˜ZI∗
) =
m
i=1
n
j=1
(˜cI
ij) ⊗ (˜xI
ij)
Subject to,
n
j=1
(˜xI
ij) ≈ (˜aI
i ), for i = 1, 2, · · · , m
m
i=1
(˜xI
ij) ≈ (˜bI
j ), for j = 1, 2, · · · , n
(˜xI
ij) (˜0I
), for i = 1, 2, · · · , m and j = 1, 2, · · · , n
Since (˜cI
ij), (˜aI
i ) and (˜bI
j ) all are crisp values, this problem (2) is obviously the crisp
transportation problem of the form (1) which can be solved by the conventional method namely
the Zero Point Method, Modified Distribution Method or any other software package such as
TORA , LINGO and so on. Once the optimal solution x∗
of Model (2) is found, the optimal
intuitionistic fuzzy objective value ˜ZI∗
of the original problem can be calculated as
˜ZI∗
=
m
i=1
n
j=1
˜cI
ij ⊗ ˜xI∗
ij
where, ˜cI
ij = (c1
ij, c2
ij, c3
ij)(c1
ij , c2
ij, c3
ij ), ˜xI∗
ij = (x1
ij, x2
ij, x3
ij)(x1
ij , x2
ij, x3
ij ).
The above FIFTP and its equivalent crisp TP can be stated in the below tabular form as fol-
lows:
60
10. Table 1: Tabular representation of FIFTP
Source Destination Intuitionistic fuzzy supply
D1 D2 · · · Dn ˜aI
i
S1 ˜cI
11 ˜cI
12 . . . ˜cI
1n ˜aI
1
S2 ˜cI
21 ˜cI
22 . . . ˜cI
2n ˜aI
2
...
...
...
...
...
...
Sm ˜cI
m1 ˜cI
m2 . . . ˜cI
mn ˜aI
m
Intuitionistic fuzzy
demand ˜bI
j
˜bI
1
˜bI
2 . . . ˜bI
n
m
i=1
˜aI
i =
n
j=1
˜bI
j
Table 2: Tabular representation of crisp TP
Source Destination Intuitionistic fuzzy supply
D1 D2 · · · Dn ˜aI
i
S1 (˜cI
11) (˜cI
12) . . . (˜cI
1n) (˜aI
1)
S2 (˜cI
21) (˜cI
22) . . . (˜cI
2n) (˜aI
2)
...
...
...
...
...
...
Sm (˜cI
m1) (˜cI
m2) . . . (˜cI
mn) (˜aI
m)
Intuitionistic fuzzy
demand (˜bI
j ) (˜bI
1) (˜bI
2) . . . (˜bI
n))
m
i=1
(˜aI
i ) =
n
j=1
(˜bI
j )
5 PSK method
Step 1. Consider the TP having all the parameters such as supplies, demands and costs must be
in intuitionistic fuzzy numbers (This situation is known as FIFTP).
Step 2. Examine whether the total intuitionistic fuzzy supply equals total intuitionistic fuzzy de-
mand. If not, introduce a dummy row/column having all its cost elements as intuitionistic
fuzzy zero and intuitionistic fuzzy supply/intuitionistic fuzzy demand as the positive differ-
ence of intuitionistic fuzzy supply and intuitionistic fuzzy demand.
Step 3. After using step 2, transform the FIFTP into its equivalent crisp TP using the ranking
procedure as mentioned in section 3.
Step 4. Now, the crisp TP having all the entries of supplies, demands and costs are integers then
kept as it is. Otherwise at least one or all of the supplies, demands and costs are not in
integers then rewrite its nearest integer value.
61
11. Step 5. After using step 4 of the proposed method, now solve the crisp TP by using any one of
the existing methods (MODI, Zero Point Method) or software packages such as TORA,
LINGO and so on. This step yields the optimal allocation and optimal objective value of
the crisp TP (This optimal allotted cells in crisp transportation table is referred as occupied
cells).
Step 6. After using step 5 of the proposed method, now check if one or more rows/columns hav-
ing exactly one occupied cell then allot the maximum possible value in that cell and adjust
the corresponding supply or demand with a positive difference of supply and demand. Oth-
erwise, if all the rows/columns having more than one occupied cells then select a cell in the
α- row and β- column of the transportation table whose cost is maximum (If the maximum
cost is more than one i.e, a tie occurs then select arbitrarily) and examine which one of the
cell is minimum cost (If the minimum cost is more than one i.e, a tie occurs then select
arbitrarily) among all the occupied cells in that row and column then allot the maximum
possible value to that cell. In this manner proceeds selected row and column entirely. If
the entire row and column of the occupied cells having fully allotted then select the next
maximum cost of the transportation table and examine which one of the cell is minimum
cost among all the occupied cells in that row and column then allot the maximum possible
value to that cell. Repeat this process until all the intuitionistic fuzzy supply points are fully
used and all the intuitionistic fuzzy demand points are fully received. This step yields the
fully intuitionistic fuzzy solution to the given FIFTP.
Note 2. Allot the maximum possible value to the occupied cells in FIFTP which is the most
preferable row/column having exactly one occupied cell.
Theorem 5.1. A solution obtained by the PSK method for a fully intuitionistic fuzzy transporta-
tion problem with equality constraints (P) is a fully intuitionistic fuzzy optimal solution for the
fully intuitionistic fuzzy transportation problem (P).
Proof. We, now describe the PSK method in detail. We construct the fully intuitionistic fuzzy
transportation table[˜cI
ij] for the given fully intuitionistic fuzzy transportation problem and then,
convert it into a balanced one if it is not balanced. Now, transform the FIFTP into its equivalent
crisp TP using the ranking procedure of Varghese and Kuriakose.
Now, the crisp TP having all the entries of supplies, demands and costs are integers then kept
as it is. Otherwise at least one or all of the supplies, demands and costs are not in integers then
rewrite its nearest integer value because decimal values in transportation problem has no physical
meaning (such a transportation problem referred as crisp TP).
Now, solve the crisp TP by using any one of the existing methods (MODI, Zero Point Method)
or software packages such as TORA, LINGO and so on. This step yields the optimal allotment
and optimal objective value of the crisp TP (The optimal allotted cells in crisp transportation
table is referred to as occupied cells which are at most m + n − 1 and all are basic feasible. The
remaining cells are called unoccupied cells which are all none basic at zero level).
62
12. Clearly, occupied cells in crisp TP is same as that of occupied cells in FIFTP. Therefore,
we need not further investigate the occupied cells in FIFTP. But only we claim that how much
quantity to allot the occupied cells under without affecting the rim requirements.
Now, check if one or more rows/columns having exactly one occupied cell then allot the max-
imum possible value in that cell and adjust the corresponding supply or demand with a positive
difference of supply and demand. Otherwise, if all the rows/columns having more than one oc-
cupied cells then select a cell in the α- row and β- column of the transportation table whose cost
is maximum (If the maximum cost is more than one i.e, a tie occurs then select arbitrarily) and
examine which one of the cell is minimum cost (If the minimum cost is more than one i.e, a tie
occurs then select arbitrarily) among all the occupied cells in that row and column then allot the
maximum possible value to that cell. In this manner proceeds selected row/column entirely. If the
entire row and column of the occupied cells having fully allotted then select the next maximum
cost of the transportation table and examine which one of the cell is minimum cost among all
the occupied cells in that row and column then allot the maximum possible value to that cell.
Repeat this process until all the intuitionistic fuzzy supply points are fully used and all the in-
tuitionistic fuzzy demand points are fully received. This step yields the optimum intuitionistic
fuzzy allotment.
Clearly, the above process will satisfy all the rim requirements (row and column sum restric-
tion). If all the rim requirements is satisfied then automatically it satisfied total intuitionistic fuzzy
supply is equal to total intuitionistic fuzzy demand i.e., the necessary and sufficient condition for
the FIFTP is satisfied.
Hence, the solution obtained by the PSK method for a fully intuitionistic fuzzy transportation
problem with equality constraints is a fully intuitionistic fuzzy optimal solution for the fully
intuitionistic fuzzy transportation problem.
Advantages of the proposed method
By using the proposed method a decision maker has the following advantages:
i. The optimum objective value of the unbalanced IFTP is non-negative triangular intuitionis-
tic fuzzy number i.e., there is no negative part in the obtained triangular intuitionistic fuzzy
number.
ii. The proposed method is computationally very simple and easy to understand.
6 Conclusion
On the basis of the present study, it can be concluded that the UIFTP, IFTP and FIFTP which can
be solved by the existing methods (Hussain and Kumar (2012), Gani and Abbas (2013), Antony
et al. (2014), Dinagar and Thiripurasundari (2014)) can also be solved by the proposed method.
The UIFTP with equality constraints is solved by the PSK method which differs from the exist-
ing methods namely, intuitionistic fuzzy modified distribution method, intuitionistic fuzzy zero
point method, zero suffix method, revised distribution method and average method. The main
63
13. advantage of this method is computationally very simple, easy to understand and the solution
obtained by this method is always optimal.The proposed method enable a decision-maker to de-
termine resource allocation in an economic way. Also, intuitionistic fuzzy ranking method offers
an effective tool for handling various types of problems like, project scheduling, network flow
problems and transportation problems.
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