For any Fuzzy Number value and ambiguity are very important, hence we propose a method for finding the nearest Triangular approximations of Fuzzy Number by definitions of value and ambiguity functions. Initially, we de ne the value and ambiguity functions for each Fuzzy Number. This evident that, two Fuzzy Numbers will be close if functions of value and ambiguity is close, therefor for finding nearest Triangular Fuzzy Number of any Fuzzy Number we find the nearest value and ambiguity functions by least square approach, such that the computational complexity is less than other methods. At last, we show some examples of Fuzzy Numbers and its Triangular approximation obtained by this method.

Location of eco-industrial park is one of the most effective factors to achieve the sustainable development because eco-industrial park causes to coordinate goal of different groups: economic development planners, urban development planners, economic firms and environment objective. Therefore, the main purpose of this study is to determine the criteria influencing the location of industrial park according to sustainable development indicators. Study of existing literature shows that social, economic, environmental, infrastructure and planning factors are effective factors to locate industrial park according to sustainable development. Prioritization of criteria is according to Fuzzy AHP and Triangular Fuzzy Number. The results of this study show that social and economic factors are the most important factors in location of industrial parks.

In this paper, some elementary operations on Triangular Fuzzy Numbers (TFNs) are defined. We also define some operations on Triangular Fuzzy matrices (TFMs) such as trace and Triangular Fuzzy determinant (TFD). Using elementary operations, some important properties of TFMs are presented. The concept of adjoints on TFM is discussed and some of their properties are. Some special types of TFMs (e.g. pure and Fuzzy Triangular, symmetric, pure and Fuzzy skew-symmetric, singular, semi-singular, constant) are defined and a Number of properties of these TFMs are presented.

Reliability is one of the important features of products, systems and parts. Nowadays, as the systems become more complex, the concept of reliability and its optimization has been researched and examined in many research and industrial communities. The application of this concept can be seen in many industrial, communication, satellite systems, etc. In the early stages of system design, many system features such as reliability, weight, cost, and etc are associated with uncertainty due to various reasons such as lifespan, operational conditions, etc. Since the use of the probabilistic approach in solving reliability problems has limitations and can only be used in quantitative analysis of information and in many cases does not produce useful and sufficient results for experts, therefore the use of the approach Fuzzy is much more efficient for solving reliability optimization problems. One of the ways to optimize the reliability is to allocate redundancy. When using redundant components in a subsystem, how the redundant components are used is particular importance. In reliability-redundancy allocation problems, the reliability of components is not known in advance and is considered as a decision variable. In this research, the failure rate of components is considered as Triangular Fuzzy Numbers and the reliability of each system with two active and cold-standby strategies has been calculated using genetic algorithm in two model problems and an industrial system in MATLAB software. In the implementation of the genetic algorithm, the random, tournament and roulette wheel methods have been used to select parents and different types of mutation and crossover operators have been used to produce children. The results have been analyzed and investigated in various series, complex and series-parallel systems and are much more efficient than the results obtained from solving the deterministic model.

Jahanshahloo has suggested a method for the solving linear programming problems with zero-one variables. In this paper we formulate fully Fuzzy linear programming problems with zero-one variables and a method for solving these problems is presented using the ranking function and also the branch and bound method along with an example is presented.

This manuscript introduces a semi-analytical and numerical solutions for the Benjamin Bona Mahony equation (BBME) in the form of convergent series. The BBME holds significance in diverse scientific and engineering applications.  Especially to study issues on shallow water waves, solitons and their importance in modern physics. The Fuzzy Homotopy Perturbation Transform Method (FHPTM) and Differential Quadrature Method (DQM) are utilized to obtain the solutions for the BBME. In DQM, grid point based on Shifted Legendre Polynomials (SLP) have been used to solve the BBME. Additionally, we address the uncertainty in the initial condition by representing it in terms of an interval. The interval BBME (iBBME) is subsequently tackled using the FHPTM providing both lower and upper interval solutions. The convergence of the interval solution is validated considering crisp case. The outcomes obtained through FHPTM for BBME are compared with the exact solution and results obtained in this study are exhibiting good agreement. The numerical outcomes by FHPTM are compared with results obtained by DQM.  Additionally, we presented the time fractional BBME and developed a Fuzzy model for it, accounting for uncertainties in the coefficients associated with wave velocity. To analyze the behavior of the Fuzzy time fractional BBME, and examined various numerical results using a double parametric approach.

It is well known fact that binary relations are generalized mathematical functions. Contrary to functions from domain to range, binary relations may assign to each element of domain two or more elements of range. Some basic operations on functions such as the inverse and composition are applicable to binary relations as well. Depending on the domain or range or both are Fuzzy value Fuzzy set, interval Fuzzy value Fuzzy set or Fuzzy Number value Fuzzy set, define of the Fuzzy relation is different. Given a Fuzzy relation, its domain and range are Fuzzy Number value Fuzzy sets. In this paper, initially we define Fuzzy Number value Fuzzy sets and then propose Fuzzy Number-valued Fuzzy relation (FN-VFR). We also introduce property of reflexive, symmetric, transitive and equivalence relation of FN-VFR. As follow, we prove some theorems for FN-VFR with property of reflexive, symmetric and transitive. Also, we show examples for FN-VFR.