The aim of this paper is to develop a new methodology for solving matrix games with payoffs of Atanassov's intuitionistic fuzzy (I-fuzzy) numbers. In this methodology, we define the concepts of I-fuzzy numbers and the value-index and ambiguity-index and develop a difference-index-based ranking method, which is proven to be a total order. By doing this, the parameterized nonlinear programming models are derived from a pair of auxiliary I-fuzzy mathematical programming models, which are used to determine solutions of matrix games with payoffs of I-fuzzy numbers. The validity and applicability of the models and method proposed in this paper are illustrated with a practical example.

A Parameterized Nonlinear Programming Approach to Solve Matrix Games With Payoffs of I-Fuzzy Numbers

In this paper, we propose a new approach for solving Intuitionistic Fuzzy Linear Programming Problems (IFLPP) involving triangular intuitionistic fuzzy numbers (TIFN). We introduce a new algorithm for the solution of an Intuitionistic Fuzzy Linear Programming Problem without converting in to one or more classical Linear Programming Problems. Numerical examples are provided to show the efficiency of the proposed algorithm.

A new approach on solving intuitionistic fuzzy linear programming problem

A Robust Neutrosophic Fuzzy-based Approach to Integrate Reliable Facility Location and Routing Decisions for Disaster Relief under Fairness and Aftershocks Concerns

Intuitionistic fuzzy has always been a subject of keen interest, and a rigorous research has also been done on it. However, those research works were mainly based on normal intuitionistic fuzzy- a generalized approach to it could hardly be seen. So in this paper, we have developed a generalized intuitionistic fuzzy number and its arithmetic operations. It is a unique attempt made by us in which for the first time two basic generalized intuitionistic fuzzy numbers namely generalized trapezoidal and generalized triangular intuitionistic fuzzy numbers have been considered to serve the purpose. All arithmetic operations have been formulated on the basis of (α, β)-cut method, vertex method and extension principle method. Comparison among those three methods using an example is given and numerical results have been presented graphically. A new method is proposed to solve generalized intuitionistic fuzzy transportation problem (GIFTP) using ranking function. To validate the proposed method we have solved a GIFTP by assuming transportation cost, supply and demand of the product in generalized intuitionistic fuzzy numbers and the optimum results have been compared with the results of normal intuitionistic fuzzy transportation problem.

Arithmetic operations on generalized intuitionistic fuzzy number and its applications to transportation problem

Ranking of intuitionistic fuzzy numbers is a difficult task. Many methods have been proposed for ranking of intuitionistic fuzzy numbers. In this paper we have ranked both trapezoidal intuitionistic fuzzy numbers and triangular intuitionistic fuzzy numbers using the centroid concept. Some of the properties of the ranking function have been studied. Also, comparative examples are given to show the effectiveness of the proposed method.

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A new approach for ranking of intuitionistic fuzzy numbers using a centroid concept

This paper presents an approach based on value and ambiguity indexes defined in [1] to solve linear programming problems with data as triangular intuitionistic fuzzy numbers.

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Linear programming with Triangular Intuitionistic Fuzzy Number

A real square matrix is a $${\mathbf {Z}}$$
 -matrix if it’s off diagonal elements are nonpositive. A $${\mathbf {Z}}$$
 -matrix with nonnegative principal minors is called an $${\mathbf {M}}$$
 -matrix. The problem of testing whether a given matrix is an $${\mathbf {M}}$$
 -matrix or not is an important research problem in matrix theory as $${\mathbf {M}}$$
 -matrices arise naturally in a wide range of applications including finite difference methods for partial differential equations, input-output models in economics, linear complementarity problems in operations research, and Markov processes in probability and statistics. In this paper, we present a polynomial-time algorithm for testing whether a $${\mathbf {Z}}$$
 -matrix is an $${\mathbf {M}}$$
 -matrix based on modified simplex method.

Dipti Dubey

Papers

Linear programming with Triangular Intuitionistic Fuzzy Number

On testing nonnegativity of principal minors of $${\mathbf {Z}}$$ Z -matrices using simplex method