Trapezoidal fuzzy number

Trapezoidal fuzzy number’s (TrFNs) are frequently used in application. It is well known that the matrix formulation of a mathematical formula gives extra facility to study the problem. Due to the presence of uncertainty in many mathematical formulations in different branches of science and technology. To the best of our knowledge, no work is available on TrFMs, through a lot of work on fuzzy matrices is available in literature. A brief review on fuzzy matrices is given below. Some properties of Constant of trapezoidal fuzzy number matrices by N.Mohana and R.Mani 4 .Fuzzy matrix has been proposed to represent fuzzy relation in a system based on fuzzy set theory Overhinniko S.V 6 . Kim 1 presented some important results on determinant of square fuzzy matrices. He defined the determinant of a square fuzzy matrix and contributed with very research works Kim 2,

A linear system is called a fully fuzzy linear system (FFLS) if all coefficients in the system are all fuzzy numbers. Nasseri [11] investigated linear system of equations with trapezoidal fuzzy numbers using embedding approach. Amitkumar [2] solved FFLS with trapezoidal fuzzy numbers using row reduced echelon form. In [8] and [9] FFLS with trapezoidal fuzzy number matrices was solved by partitioning them into sub matrices using Schur complement and QR decomposition method respectively.

This paper is organized as follows. Some basic definitions and results on fuzzy sets and trapezoidal fuzzy numbers are given in section 2. In section 3, the method of finding inverses of partitioned matricesis presented. Section 4 introduces the method to solve FFLS using trapezoidal fuzzy number matrices by partitioning the block matrices. Illustration with a numerical example is given in section 5. Section 6 ends this paper with conclusion.

This study provides solutions to aLR-fuzzy linear system (LR-FLS) with trapezoidal fuzzy number using matrix theory. The components of the LR-FLS are represented in block matrices and vectors to produce an equivalent linear system. Then, the solution can be obtained using any classical linear system, such as an inversion matrix. In this method, fuzzy operations are not required and the solution obtained is either fuzzy or non-fuzzy exact solution. Finally, several examples are given to illustrate the ability of the proposed method.

manipulating linear inequalities can be adapted to solve LP models. The theoretical insight given by this method is demonstrated as well as its clear geometrical interpretation. It has been rediscovered a number of times by different authors Motzkin (Motzkin, 1937) (the name Fourier Motzkin algorithm is often used for this method) Dantzig and Cottle

Definition 2.1. The characteristic function of a crisp set A of X assigns a value either 0 or 1 to each member in X. This function can be generalized to a function such that the value assigned to the element of the universal set X fall within a specified range i.e. : X → 0,1 . The assigned value indicate the membership grade of the element in the set A. The function is called the membership function and the set = , x)); ∈ is called fuzzy set.

Abstract:- Network-flow problems can be solved by several methods. Labeling techniques can be used to solve wide variety of network problems. A new algorithm to find the fuzzy maximal flow between source and sink was proposed by Kumar et el. [19]. They have represented normal triangular fuzzy numbers as network flow. It is not possible to restrict the membership function to the normal form and proposed the concept of generalized fuzzy numbers in many cases [8]. Generalized trapezoidal fuzzy numbers for solving the maximal flow network problems have been used by Kumar [21]. In this paper, we have modified the existing algorithm to find fuzzy maximal network flow between source and sink for generalized trapezoidal fuzzy number. Ranking and mode function to find the highest flow for maximum flow path of generalized trapezoidal fuzzy number has been applied. A numerical example has been solved by the proposed algorithm and the other results are discussed. Mathematica programs have been applied for various arithmetic operations.

Grzegorzewski et al. [6] have suggested a new approach to trapezoidal approximation of fuzzy numbers, called the nearest trapezoidal approximation operator preserving expected interval and possesses many desired properties. In some situations their operator may fail to lead a trapezoidal fuzzy number. In 2007, Grzegorzewski [7] proposed a corrected version of the trapezoidal approximation operator.

Abstract. This paper presents a new type of fuzzy shortest path problem on a network by ranking function. The proposed algorithm gives the fuzzy shortest path and fuzzy shortest path length in which type-2 trapezoidal fuzzy number is assigned to each arc length in a network. In this the ranking function is based on the statistical beta distribution. An illustrative example also included to demonstrate our proposed algorithm.

The concept of fuzzy set theory deals with imprecision, vagueness in real life situations. It was firstly proposed by Zadeh [1]. Bellman and Zadeh [2] elaborated on the concept of decision making in the fuzzy environment. Later on, fuzzy methodologies have been successfully applied in a wide range of real world situations. Jain [3] was the first to propose method of ranking fuzzy numbers for decision making in fuzzy situations. Yager [4] used the concept of centroids in the ranking of fuzzy numbers. Chu and Tsao [5] computed the area between the original point and the centroid point for ranking of fuzzy numbers. Babu et al. [6] described the method for ranking various types of fuzzy numbers and crisp numbers based on area, mode, spreads and weights of generalized trapezoidal fuzzy numbers. They also apply mode and spread in those cases where the discrimination is not possible. Dhanalaxmi and Kennedy [7] proposed some ranking methods for octagonal fuzzy numbers. Rajarajeshwari and Sudha [8] explained ordering generalized hexagonal fuzzy numbers by using rank, mode, divergence and spread. Thorani et al. [9] illustrated ordering of generalized trapezoidal fuzzy numbers by using orthocenter of centroids. Kumar et al. [10] introduced some additional assumptions for the equality of generalised triangular fuzzy numbers. Monsavi and Regvani [11] proposed a method for ranking of generalised triangular fuzzy numbers which are based on rank, mode, divergence, and spread. Gani and Assarudeen[12] modified the method of subtraction and division of triangular fuzzy numbers. Kumar et al [13] proposed a approach for ranking of generalised trapezoidal fuzzy number based on rank, mode, divergence and spread.

In 2011, Zadeh introduced the Z-number that takes into account expert’s reliability on the data (Zadeh, 2011). The Z-number has two components, Z=(A, B) in estimating a vari- able, Y. (A) is the limitation on the values which Y can take in triangular fuzzy numbers. The second number (B) is a degree of relia- bility (certainty) that Y is A. This method helps experts in giving a reliable judgement as it includes the certainty value of the ex- perts. In 2016, (Azadeh and Kokabi, 2016) integrated the Z-number approach with CCR model using triangular fuzzy numbers to han- dled data that are expert-based. However, ac- cording (Herrera and Herrera-Viedma, 2000) the trapezoidal fuzzy number is better suited to capture the vagueness of linguistic assess- ment. Therefore, (Ibrahim et al., 2018) devel- oped Z-number CCR model using trapezoidal fuzzy numbers to overcome this problem. The formulation of this model is shown in equation 1. Both triangular and trapezoidal Z-number CCR model are then converted into fuzzy data envelopment model (FDEA) in order to lin- earized the model. In linearizing the FDEA model, they used a crisps α-cut value (given by the experts). As Z-number is proved can help

In This paper the balanced fuzzy transportation problem are consider We have proposed a new algorithm for the fuzzy optimal solution to the given fuzzy transportation problem with trapezoidal fuzzy number converted into crisp transportation problem using Robust ranking indices and Stepping stone method has been applied to find an optimal solution of fuzzy numbers. First The initial solution for the fuzzy transportation problem we can use the fuzzy Vogel’s Approximation method with trapezoidal fuzzy number and obtained an initial fuzzy feasible solution. By applying the fuzzy version of Stepping stone method we have tested the optimality of the fuzzy feasible solution Further, the fuzzy optimal solution obtained by the stepping stone methods is better than the fuzzy Zero point method and Russell’s method.

Fuzziness in matrix games can appear in so many ways but two classes of fuzziness seem to be very natural. These two classes of fuzzy matrix games are referred as matrix games with fuzzy goal and matrix games with fuzzy pay off[2]. It has been considered by Nayak and Pal[3] the pay off as interval number and symmetric triangular fuzzy number. Here the trapezoidal numbers are considered where the pay off elements are fuzzy but the parameters considered are crisp and P.Grzgyski’s[1]concept has been used to approximate the interval fuzzy numbers corresponding to a trapezoidal fuzzy numbers. N.Mahdavi -Amiri and S.H Nasseri[4] have solved L.P Problem with trapezoidal fuzzy variable and applied linear ranking function to order trapezoidal fuzzy number. Here it has been considered that the entries of pay off matrix as symmetric trapezoidal fuzzy number and interval number. Our inspiration behind defining the symmetric trapezoidal fuzzy number is Adrian Ban[5]but here the work has been made in some different manner. A set of real lines is defined and then a trapezoidal membership function. Here the universe of fuzzy pay off matrix with elements as trapezoidal fuzzy numbers is also defined. Here 3 parameters are used to define a trapezoidal number , out of which one parameter is still reducible but we have used it for the sake of solving problems with reduced constraints. A solution method is proposed for both type of problems with saddle point and without saddle point. Numerical example has also been provided. We have defined acceptability index to compare two intervals. The main feature of our work is that it is simple and at the same time it represents a better model for real life problems.

In this paper, we have been suggested an interesting ap- proach to trapezoidal approximation of general fuzzy numbers. The proposed method leads to the trapezoidal fuzzy number which is the best one with respect to a tain measure of distance between fuzzy numbers,

In this paper, a linear programming problem with symmetric trapezoidal fuzzy number which is introduced by Ganesan et al. in [4] is generalized to a general kind of trapezoidal fuzzy number. In doing so, we first establish a new arithmetic operation for multiplication of two trapezoidal fuzzy numbers. in order to prepare a method for solving the fuzzy linear programming and the primal simplex algorithm, a general linear ranking function has been used as a convenient approach in the literature. In fact, our main contribution in this work is based on 3 items: 1) Extending the current fuzzy linear program to a general kind which doesn’t essentially include the symmetric trapezoidal fuzzy numbers, 2) Defining a new multiplication role of two trapezoidal fuzzy numbers, 3) Establishing a fuzzy primal simplex algorithm for solving the generalized model. We in particular emphasize that this study can be used for establishing fuzzy dual simplex algorithm, fuzzy primal- dual simplex algorithm, fuzzy multi objective linear programming and the other similar methods which are appeared in the literature.

Fuzzy sets have been introduced by Lotfi.A.Zadeh(1965)[16] and Dieter Klaua(1965)[7]. Fuzzy set theory permits the gradual assessment of the membership of elements in a set which is described in the interval [0, 1]. It can be used in a wide range of domains where information is incomplete and imprecise. Interval arithmetic was first suggested by Dwyer[7] in 1951,by means of Zadeh’s extension principle[15,16], the usual Arithmetic operations on real numbers can be extended to the ones defined on Fuzzy numbers. D.Dubois and H.Prade[3] in 1978 has defined any of the fuzzy numbers as a fuzzy subset of the real line[4,5,6,8]. A fuzzy number is a quantity whose values are imprecise, rather than exact as is the case with single-valued numbers. Among the various shapes of fuzzy numbers, Triangular fuzzy number and Trapezoidal fuzzy number are the most commonly used membership function(Dubois and Prade[3],1980,Zimmermann[17], 1996) In this paper a new operation of

Consider a directed network G(V, E), consisting of a finite set of nodes V = 1, … , n and a set of m directed edges E ⊆ V × V. Each edge is denoted by an ordered pair (𝑖, 𝑗), where 𝑖, 𝑗 ∈ 𝑉. Each edge is assigned to a non- trapezoidal fuzzy number. We calculate the α-cuts for each and every edge in the network by the formula α X = b − a α + a n , d − d − c α n where X a n , b n , c n , d n is a non-trapezoidal fuzzy number and

Abstract: A flow network is a directed graph where each edge receives a flow and has a capacity.One of the applications to network-flow is labeling algorithm. Labeling techniques algorithms are used to solve wide variety of network problems, such as shortest-path problems, maximal-flow problems, general minimal-cost network-flow problems and minimal spanning tree problems. In 2010,Kumar et al., proposed an algorithm to find the fuzzy maximal flow network between source and sink with generalized trapezoidal fuzzy numbers with only rank. But sometimes ranking function fails for choosing the path of the flow. To avoid this drawback, divergence functions are to be used. This paper proposes an algorithm for getting fuzzy maximal flow network between source and sink for generalized trapezoidal fuzzy number with rank and divergence functionsand also illustrates the algorithm by solving a numerical example.

Fuzzy set theory is introduced by Zadeh in the year 1965 [1]. Later Tran and Duckstein gave the comparison of fuzzy numbers using a fuzzy distance measure in the year 2002 [3]. Later Chen and Wang introduced the fuzzy distance of trapezoidal fuzzy numbers in the year 2008 [5]. In the year 2012, Nagoorgani [6] gave a new operation on triangular fuzzy number for solving fuzzy linear programming problem. Optimization of fuzzy production inventory model with repairable defective products under crisp or fuzzy production quantity is given by Chen, Wang and Chang in the year 2005 [4]. Arithmetic operations on generalized trapezoidal fuzzy number and its applications is given by Banerjee and Roy in the year 2012 [7]. In the year 2014, Pardhasaradhi and Shankar gave an idea on fuzzy distance measure [8]. In this paper, some simple properties and theorem based on fuzzy trident distance along with the help of trapezoidal fuzzy numbers are given. This paper consists of five sections. The preliminaries in the first section, defining trapezoidal, positive trapezoidal, negative trapezoidal fuzzy numbers in the second section, fuzzy trident distance in the third section, properties and theorem based on fuzzy trident distance in the fourth section and finally, the results are discussed with suitable numerical examples.

In this paper priority based fuzzy goal programming with generalized trapezoidal fuzzy numbers has been proposed. Euclidean distance is used for selecting proper priority structure for obtaining compromise optimal solution. The concept presented, in this paper, is illustrated with multi- objective assignment problems involving generalized trapezoidal fuzzy numbers to check the effectiveness of the proposed method. The proposed method is simple and easy to implement. It may be hoped that proposed method can be applied to solve realistic optimization problems involving generalized trapezoidal fuzzy numbers.