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.

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.

The purpose of this paper is to appl y of concept of fuzzy bridge and to ex tend of notation of fuzzy domination number in fuzzy graph. F uzzy bridge introduces a new notation of fuzzy domination sets and fuzzy domination numbers such that is consedering for fuzzy graphs with cyclics and vertices are l ocated on cyclics. To check the importance of fuzzy domination numbers based on fuzzy bridge, we compare it to other distinct fuzzy domination numbers and show that fuzzy domination numbers based on fuzzy bridge is optimal. The main method, in this research is based on computations of fuzzy domination number of complete fuzzy graphs with distinct fuzzy vertices and is worked on spec ial and useful fuzzy graphs such as st rong 2-wh eel fuzzy graph and 2-complet e fuzzy graph. The study on 2-wh eel fuzzy graph and 2-complet e fuzzy graph was given for the first time in this paper and are worked via the generalize of 2-part of wh eel fuzzy graph and complet e fuzzy graph which these special fuzzy graphs have many applications on complex networks. In this study, computations of fuzzy domination number of special fuzzy graphs are done based on fuzzy bridge with distinct vertices. The paper includes implications for the development of fuzzy graph, and for modeling the uncertainty problems and applicaions in real world which we consider t wo modes of real problems.

Many human decision-making processes such as competition, hostility, partnership, etc. are based on bipolar or two-sided thinking, which are positive on one side and negative on the other. Binary logic only includes zero and one, and fuzzy logic is a tool for representing ambiguity and uncertainty, but both have shortcomings in modeling bipolar relationships, as they are limited to the range from zero to one, which is suitable for representing unipolar models in nature. Bipolar fuzzy sets are actually a combination of ambiguity and its polarity, which is a tool for representing subjects of nature that have both positive or negative poles. Therefore, in this article, we will introduce bipolar fuzzy sets and numbers, and then, by defining addition and scalar multiplication for triangular bipolar fuzzy numbers, we will introduce a new ranking based on the centroid for this category of numbers, and prove the properties of the new method by stating several theorems and propositions. Finally, numerical examples demonstrate the practical applications of the proposed ranking.

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.

This research studies resource constrained project scheduling problem in a part of a refinery construction project in real world. In real world, most of the activities are new and we have problems such as activity duration uncertainty. This problem causes a change in a project makespan. The RCPSP is NP-hard. Hence, we proposed an optimization method based on simulated annealing to solve the RCPS Problem. In this paper, Fuzzy sets theory is used to represent this activity duration uncertainty. Used schedule generation scheme, in the proposed simulated annealing algorithm, is a fuzzy parallel scheduling generation method. Proposed algorithm generates the minimum project makespan while considers renewable resource-constrained and activity precedence and also has ability to perform with fuzzy numbers for presenting project details such as start and final time of activities and project whole time with fuzzy numbers. Finally, the results of the algorithm will be evaluated and it will be represented that the proposed algorithm is very efficient and can be used by managers and scheduling programmers in real projects.