In this paper, we show that the computed distance between two Fuzzy Numbers using the Fuzzy distance measure proposed by Chakra borty et al. [Mathematical and Computer Modeling 43 (2006) 254-261] is not always a Fuzzy Number. Then, we improve it and prove that the result of the revised distance is always a nonnegative Fuzzy Number.

In this paper, the Fuzzy matrix equation AeX B = e C in which A; B are n × n crisp matrices respectively and e C is an n×n arbitrary LR Fuzzy Numbers matrix, is investigated. A new numerical procedure for calculating the Fuzzy solution is designed and a suffcient condition for the existence of strong Fuzzy solution is derived. Some examples are given to illustrate the proposed method.

The Fuzzy operations on Fuzzy Numbers of type L-R are much easier than general Fuzzy Numbers. It would be interesting to approximate a Fuzzy Number by a Fuzzy Number of type L-R. In this paper, we state and prove two significant application inequalities in the monotonic functions set. These inequalities show that under a condition, the nearest Fuzzy Number of type L-R to an arbitrary Fuzzy Number exists and is unique. After that, the nearest Fuzzy Number of type L-R can be obtained by solving a linear system. Note that the trapezoidal Fuzzy Numbers are a particular case of the Fuzzy Numbers of type L-R. The proposed method can represent the nearest trapezoidal Fuzzy Number to a given Fuzzy Number. Finally, to approximate Fuzzy solutions of a Fuzzy linear system, we apply our idea to construct a framework to find solutions of crisp linear systems instead of the Fuzzy linear system. The crisp linear systems give the nearest Fuzzy Numbers of type L-R to Fuzzy solutions of a Fuzzy linear system. The proposed method is illustrated with some examples.

In 2014, Gong et al. [Information Sciences 266 (2014) 112-133] proposed a simple method for solving dual Fuzzy matrix equations in LR form. Later, Kaur et al, [Information Sciences 418-419 (2017) 184-185] showed that there is a technical aw in their method and it is valid only for certain types of LR dual Fuzzy matrix equations. The main aim of this paper is to eliminate this technical aw and also correct some numerical results obtained by Gong et al.

In this paper, based on the weighted metric, the problem of approximating a general continuous Fuzzy Number $u$ with derivable $\underline{u}(r)$ and $\overline{u}(r)$ (left and right cut functions of level value $r$) for $r \in (0,1]$ by using the $m-n-$step type Fuzzy Number is studied. Two kinds of approximations ($I-$nearest approximation and $II-$nearest approximation which satisfy different conditions) of using $m-n-$step type Fuzzy Numbers to approximate general continuous Fuzzy Numbers are defined, and the relationship between the $I-$nearest approximation and $II-$nearest approximation is obtained. Then the methods of the two kinds of approximations are respectively presented by the theorems more specifically. At last, an example is given to show the effectiveness and usability of the methods set up by us.

Control charts are widely used in industrial processes as well as in health sciences and particularly for monitoring the performance of cardiac surgeon or a group of surgeons based on the preoperative risk of patients. Since the preoperative risk is a vague and nonprecise variable and the anesthesiologists after checking how many risk factors a patient has, determine the risk of mortality before the surgery as a linguistic term such as low, medium, high or others like that, it is better to be considered as a Fuzzy Number, which can be determined by using Fuzzy regression models. In this condition, we need a special chart to monitor the performance of surgeons based on these Fuzzy data. In this article, we propose risk-adjusted control charts based on LR-Fuzzy data and then test our conclusions on real data.

An inventory model is formulated with type-2 Fuzzy parameters under trade credit policy and solved by using Generalized Hukuhara derivative approach. Representing demand parameter of each expert's opinion is a membership function of type-1 and thus, this membership function again becomes Fuzzy. The final opinion of all experts is expressed by a type-2 Fuzzy variable. For this present problem, to get corresponding defuzzified values of the triangular type-2 Fuzzy demand parameters, first critical value (CV) -based reduction methods are applied to reduce corresponding type-1 Fuzzy variables which becomes pentagonal in form. After that- cut of a pentagonal Fuzzy Number is used to construct the upper a- cut and lower a- cut of the Fuzzy differential equation. Different cases are considered for Fuzzy differential equation: gH- (i) differentiable and gH- (ii) differentiable systems. The objective of this paper is to find out the optimal time so as to minimize the total inventory cost. The considered problem ultimately reduces to a multi-objective problem which is solved by weighted sum method and global criteria method. Finally the model is solved by generalised reduced gradient method using LINGO (13.0) software. The proposed model and technique are lastly illustrated by providing numerical examples. Results from two methods are compared and some sensitivity analyses both in tabular and graphical forms are presented and discussed. The e effects of total cost with respect to the change of demand related parameter (b), holding cost parameter (r), unit purchasing cost parameter (p), interest earned (ie) and interest payable (ip) are discussed. We also find the solutions for type-1 and crisp demand as particular cases of type-2 Fuzzy variable. This present study can be applicable in many aspects in many real life situations where type-1 Fuzzy set is not sufficient to formulate the mathematical model. From the numerical studies, it is observed that under both gH- (i) and gH- (ii) cases, total cost of the system gradually reduces for the sub-cases - 1.1, 1.2 and 1.3 depending upon the positions of N (trade credit for customer) and M (trade credit for retailer) with respect to T (time period).

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.