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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.

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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.

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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.

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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.

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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.

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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

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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.

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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.

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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.

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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.

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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.

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