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This paper presents an optimization method to solve a non-linear separable programming problem with coefficients and variables as generalized trapezoidal intuitionistic fuzzy numbers. Such optimization problems are known as fully intuitionistic fuzzy non-linear separable programming problems. The optimization method is based on the linear approximation of fully intuitionistic fuzzy non-linear separable functions. The concept of an intuitionistic fuzzy line segment between two intuitionistic fuzzy points is introduced to find the required linear approximation. In this way, a fully intuitionistic fuzzy non-linear programming problem is converted into an intuitionistic fuzzy linear programming problem. The defuzzification and component-wise comparison techniques are then used to convert the fully intuitionistic fuzzy linear programming problem to a linear programming problem with crisp coefficients which can then be solved by using traditional optimization techniques. The application of the proposed approach in an investment problem faced by a businessman has been presented. [ABSTRACT FROM AUTHOR]

This paper presents an optimization method to solve a multi-objective model of a bi-level linear programming problem with intuitionistic fuzzy coefficients. The idea is based on TOPSIS (technique for order preference by similarity to ideal solution) method. TOPSIS method is a multiple criteria method that identifies a satisfactory solution from a given set of alternatives based on the minimization of distance from an ideal point and maximization of distance from the nadir point simultaneously. A new model of multi-objective bi-level programming problem in an intuitionistic fuzzy environment has been considered. The problem is first reduced to a conventional multi-objective bi-level linear programming problem using accuracy function. Then the modified TOPSIS method is proposed to solve the problem at both the leader and the follower level where various linear/non-linear membership functions are used to represent the flexibility in the approach of decision-makers (DMs). The problem is solved hierarchically, i.e., first the problem at the leader level is solved and then the feasible region is extended by relaxing the decision variables controlled by the leader. The feasible region is extended to obtain a satisfactory solution for the DMs at both levels. Finally, the application of the proposed approach in the production planning of a company has been presented. An illustrative numerical example is also given to explain the methodology defined in this paper. [ABSTRACT FROM AUTHOR]

This study is an extension of the recently developed theory on analytical fuzzy geometry with fuzzy numbers and fuzzy points. In the existing study, both the fuzzy numbers and fuzzy points are convex and normal fuzzy sets in nature. However, in reality, considering fuzzy objects as convex and normal sets is an oversimplified assumption. Here, the nature of non-convex and non-normal fuzzy sets is studied. In the first part of this paper, the concept of same points has been extended for non-convex and non-normal fuzzy sets. A computationally efficient method for extended addition, subtraction, multiplication, and division of two generalized fuzzy sets is proposed using the extended concept of same points. The proposed procedure drastically reduces the computational complexity of extended arithmetic operations on two generalized fuzzy sets drastically. In the second part of this paper, the genesis and morphology of fuzzy points are studied. The conjunction of non-convex fuzzy sets generates a new class of study which is termed as non-convex or generalized fuzzy geometry where the generalized fuzzy point is the key concept. The effect of different t-norms as a conjunction operator on fuzzy sets is studied with geometrical visualization. The extended arithmetic operations between generalized fuzzy points have been defined using the extended concept of same points developed here. The formation of the locus of a non-convex fuzzy point as a fuzzy line segment has also been introduced. [ABSTRACT FROM AUTHOR]

In this study, we attempt to construct fuzzy circles in a fuzzy geometrical plane. We provide a comprehensive study where we find a fuzzy number with a predetermined fuzzy distance from a given fuzzy number. We formulate fuzzy circles using the conventional basic definitions of crisp circles. Two different formulations of fuzzy circles are proposed, which depend on the information known about a fuzzy circle. The interrelations between the evaluated fuzzy circles are investigated. We show that the center of a fuzzy circle may not be a fuzzy point. However, the radius of a fuzzy circle that passes through three given fuzzy points is always a fuzzy number. The concepts of same and inverse points and a fuzzy number along a line are used to define and analyze the proposed ideas. Our discussions and studies are supported by suitable numerical and pictorial examples. [ABSTRACT FROM AUTHOR]

Abstract: In this paper a fuzzy distance measure between two generalized fuzzy numbers is developed. The metric properties of this distance measure are also studied. The new distance measure is compared with the other fuzzy distance measures proposed by Voxman [W. Voxman, Some remarks on distances between fuzzy numbers, Fuzzy Sets and Systems 100 (1998) 353–365] and Chakraborty and Chakraborty [C. Chakraborty, D. Chakraborty, A theoretical development on fuzzy distance measure for fuzzy numbers, Mathematical and Computer Modelling 43 (2006) 254–261] and turned out to be more reasonable. A new similarity measure is also developed with the help of the fuzzy distance measure. Examples are given to compare this similarity measure with the other similarity measure previously proposed. A decision making scheme is proposed using this similarity measure and this scheme is found to be more acceptable than the existing methods due to the fact that it considers the degrees of confidence of the experts’ opinion. [Copyright &y& Elsevier]

Abstract: The objective of this paper is to introduce a fuzzy distance measure for generalized fuzzy numbers (GFN). It computes the fuzzy distance between two generalized fuzzy numbers and also LR-type fuzzy numbers. The metric properties of the proposed measure are also studied. Some numerical examples have been considered here for applying the proposed fuzzy distance measure and the results are compared. [Copyright &y& Elsevier]

Various optimization approaches have been developed and used for generating optimal solutions for different industry related optimization problems. The semantic representation of imprecise coefficients and various types of uncertainties arising in real life optimization problems are still a challenging task and require attention of academicians as well as professionals. Decision makers (DMs) often face problems in presenting the linguistically expressed constraints. The impreciseness of information also plays a key role in modelling and thus in decision making. The impreciseness of coefficients is usually handled by using various elements of fuzzy set theory. The uncertainty associated with constraints in optimization problem is handled by using stochastic programming approaches. These coefficients are usually revealed by a logical DM, who takes into account the scope of hesitation along with the imprecise nature of data. Such coefficients can be very well represented by using Intuitionistic fuzzy numbers. In this work, an approach for solving multi-objective bi-level chance constrained optimization problem in an intuitionistic fuzzy environment has been presented. The technological coefficients and the coefficients of objective functions are represented by Triangular Intuitionistic fuzzy numbers whereas normal random variables are used to represent the coefficient of resources. The concept of component wise optimization is used for the reduction of intuitionistic fuzzy objective into their equivalent deterministic form. The concept of alpha-beta cut, chance constrained programming and interval programming are then used to convert the problem into a deterministic multi-objective bi-level linear programming problem. The resultant deterministic optimization problem is then solved by using TOPSIS method. The production planning problem of a firm has been modelled and solved by using the proposed approach. • Multi-objective chance constrained hierarchical optimization problem is considered. • Technological coefficients are represented by Intuitionistic fuzzy numbers. • Normal random variables are used to represent coefficient of resources. • Fuzzy problem is first reduced to corresponding deterministic problem. • TOPSIS method and Fuzzy programming approach are used to solve the problem. [ABSTRACT FROM AUTHOR]