In this paper, a novel generalizedinterval-valuedintuitionisticfuzzysets (GIVIFS) is presented, which is the generalization of conventional intuitionisticfuzzysets (IFS) and interval-valuedintuitionisticfuzzysets (IVIFS). By analyzing the degree of hesitancy, this paper introduces generalizedinterval-valuedintuitionisticfuzzysets with parameters (GIVIFSP). And then, it is proved that GIVIFS is a closed algebraic system as IFS and IVIFS.
Euclidean distance is a one of the most commonly used distance measures that has been used to solve many theoretical and practical issues in fuzzy problem. This method is widely use in many fields such as communication , engineering , chemistry , biology  and many mathematical specifications such as optimization , discrete mathematics , statistics ,operation research  and fuzzy mathematics  . In fuzzy set theory, Euclidean distance is applied to calculate distances between fuzzy numbers or sets and as a method for decision-making in a situation where two fuzzysets or fuzzy numbers appear at the same time. Usually, Euclidean distance is applied in a set of discrete fuzzy numbers or values in interval form.
Bridges and cutnodes is a very important concept in graph theory. Rosenfeld  obtained fuzzy analogs of bridges and cutnodes. Further it was studied by Bhattacharya . Again Sunitha and Vijayakumar studied about the properties of fuzzy bridges and fuzzy cutnodes . It was also studied by Mordeson and Nair . Strength of the paths in IVFGs were discussed by Rashmanlou and Pal . Again it was studied by Akram, Yousaf and Dudek . Akram and Alsheri defined intuitionisticfuzzy bridges and intuitionisticfuzzy cutnodes in . Again Akram and Farooq defined bipolar fuzzy bridges and bipolar fuzzy cutnodes in . Bipolar fuzzy bridges and bipolar fuzzy cutnodes were also characterized by Mathew, Sunitha and Anjali .
interval-valuedintuitionisticfuzzy subsets of and respectively. Then Boxdot product of the two strong interval -valuedintuitionisticfuzzy graphs and = is defined as a pair 2where 1 2= 1 2 , 1 2 , 1 2 , 1 2 and 1 2= 1 2 , 1 2 , 1 2 , 1 2 are interval-valuedintuitionisticfuzzysets on and
The theory of fuzzysets was introduced by Zadeh  in 1965. Later, Chang  proposed fuzzy topology in 1967. The concept of intuitionisticfuzzysets, introduced by Atanassov  is a generalization of fuzzysets. Using the notion of intuitionisticfuzzysets, Coker  introduced the notion of intuitionisticfuzzy topological spaces in 1997.In this paper, we introduce intuitionisticfuzzy regular α generalized open set. We investigate some of their properties. We also introduce intuitionisticfuzzy regular α T 1/2 space and obtain some characterizations and several preservation theorems.
fuzzy graphs, strong intuitionisticfuzzy graphs, bipolar fuzzy graphs, as well as certain types of vague graphs and vague hyper- graphs. Borzooei et al. [10 e16,25e27] studied domination, degree of vertices, new concepts of vague graphs, and bipolar fuzz graphs. Complete interval-valuedfuzzy graphs were investigated by Rashmanlou and Jun  . Pal and Rashmanlou  studied irreg- ular interval-valuedfuzzy graphs, de ﬁned antipodal interval- valuedfuzzy graphs  , and balanced interval-valuedfuzzy graphs  . Samanta et al. [31 e34] introduced fuzzy planar graphs, m-step fuzzy competition graphs, fuzzy k-competition and p- competition graphs, and showed some results on bipolar fuzzysets and bipolar fuzzy intersection graphs. In this paper, three new types of product operations (direct, lexicographic, and strong) of interval-valuedintuitionistic (S,T) efuzzy graphs are deﬁned. We introduce the concept of regular and totally regular intervalvaluedintuitionistic (S,T) efuzzy graphs. Busy vertices and free vertices in intervalvaluedintuitionistic (S,T) efuzzy graphs are deﬁned, and their image under an isomorphism is studied.
Cenigiz Kahraman gave very useful description about MCDM in his book on Fuzzy Multicriteria Decision Making . In his book he explained MCDM problems with two basic approaches: multiple attribute decision making (MADM) and multiple objective decision making (MODM). MADM problems are distinguished from MODM problems, which involve the design of a ―best‖ alternative by considering the tradeoffs within a set of interacting design constraints. MADM refers to making selections among some courses of action in the presence of multiple, usually conflicting, attributes. In MODM problems, the number of alternatives is effectively infinite, and the tradeoffs among design criteria are typically described by continuous functions. MADM approaches can be viewed as alternative methods for combining the information in a problem‘s decision matrix together with additional information from the decision maker to determine a final ranking, screening, or selection from among the alternatives. Besides the information contained in the decision matrix, all but the simplest MADM techniques require additional information from the decision maker to arrive at a final ranking, screening, or selection. It has been widely recognized that most decisions made in the real world take place in an environment in which the goals and constraints, because of their complexity, are not known precisely, and thus, the problem cannot be exactly defined or precisely represented in a crisp value (Bellman and Zadeh, 1970). Bellman and Zadeh (1970) and Zimmermann (1978) introduced fuzzysets into the MCDM field. They cleared the way for a new family of methods to deal with problems that had been inaccessible to and unsolvable with standard MCDM techniques. Bellman and Zadeh (1970) introduced the first approach regarding decision making in a fuzzy environment.
and Abdul Razak Salleh  introduced generalizedintervalvaluedfuzzy soft set (GIVFSS). In their generalization of FSS, they attached a degree with the parameterization of fuzzysets in defining an IVFSS. They discussed various operations and properties of GIVFSS. Some of these are GIVFS subset, GIVFS equal set, generalized null intervalvaluedfuzzy soft set (GNIVFS), generalized absolute intervalvaluedfuzzy soft set (GAIVFS), compliment of GIVFSS, union of GIVFSS’s and intersection of GIVFSS’s. They defined AND and OR operations on GIVFSS and similarity measure of two GIVFSS’s. They also give some applications of GIVFSS in DM problem and medical diagnosis.
In this paper we propose an extension of the well- known OWA functions introduced by Yager to interval-valued (IVFS) and Atanassov’s intuition- istic (AIFS) fuzzy set theory. We first extend the arithmetic and the quasi-arithmetic mean using the arithmetic operators in IVFS and AIFS theory and investigate under which conditions these means are idempotent. Since on the unit interval the construc- tion of the OWA function involves reordering the input values, we propose a way of transforming the input values in IVFS and AIFS theory to a new list of input values which are now ordered.
All the above aggregation operators are based on the algebraic operational rules of IVIFNs, and the keys of the algebraic operations are Algebraic product and Algebraic sum, which are one type of operations that can be chosen to model the intersection and union of IVIFNs. In general, a general T -norm and T - conorm can be used to model the intersection and union of IVIFNs [32,33]. Wang and Liu  proposed the intuitionisticfuzzy Einstein aggregation operators based on Einstein operations which meet the typical T -norm and T -conorm and have the same smooth approximations as the algebraic operators such as the IntuitionisticFuzzy Einstein Weighted Geometric op- erator (IFEWG) and the IntuitionisticFuzzy Einstein Ordered Weighted Geometric operator (IFEOWG), and established some general properties of these oper- ators such as idempotency, commutativity, and mono- tonicity. Wang and Liu  proposed the IntuitionisticFuzzy Einstein Weighted Averaging operator (IFEWA) and the IntuitionisticFuzzy Einstein Ordered Weighted Averaging operator (IFEOWA), and studied various properties of these operators and analyzed the relations between the existing intuitionisticfuzzy aggregation operators and them. Maris and Iliadis  further explained the advantages of Einstein operations by using some T -norms to unify the risk indices and to produce a unied means of risk measure. The algebraic T -norm estimated the risky areas under average rainfall conditions, and the Einstein T -norm oered a good approach for an overall evaluation. The computer system has proven its ability to work more eectively compared to the older methods.
The notion of fuzzy was introduced by  in 1965. Atanassov  introduced the concept of intuitionisticfuzzysets in 1986. Atanassov et al. introduced the concept of interval-valuedintuitionisticfuzzysets which is a generaliza- tion of both intuitionisticfuzzysets and interval-valuedfuzzysets. Several mathematicians applied the concept of interval- valuedintuitionisticfuzzysets to algebraic structures. Biswas  studied on Rosenfeld’s fuzzy subgroups with interval- valued membership function. Das and Dutta  developed the concept of extension of fuzzy ideals in semirings. Dutta and Biswas [7–10] introduced and studied some properties of fuzzy prime, fuzzy semi-prime, fuzzy completely prime ide- als in semiring. Balasubramanian and Raja [3, 4] introduced intuitionisticfuzzy k-ideal and interval-valuedintuitionisticfuzzy ideal on semi-rings. In this paper, the cartisian product of interval-valuedintuitionisticfuzzy k-ideal in semi-rings are studied. Investigate the relationship between A, B and A × B.
Note that all of the above-mentioned interval- valuedintuitionisticfuzzy averaging operators are based on the operational laws in . From the following discussion, we can nd that there are some undesirable properties. In particular, these issues may lead to undesirable ranking results. Considering this case, this paper continues to study decision-making with interval-valuedintuitionisticfuzzy information and develops a new procedure. To do this, an Induced Generalized Symmetrical Interval-Valued Intuitionis- tic Fuzzy Choquet-Shapley (IG-SIVIFCS) operator is presented, which is then used to calculate the com- prehensive attribute values. To address the situation where the weighting information of attributes is partly known, a model is built for determining the optimal fuzzy measure on the attribute set. The rest can be organized as follows.
Zadeh  introduced the concept of fuzzy set in 1965 and after that he also introduced the notion of intervalvaluedfuzzy subset  (in short i-v fuzzy subset) in 1975, where the values of membership functions are intervals of numbers instead of a single number as in fuzzy set. The fuzzy set theory has been developed in many directions by the research scholars. Rosenfeld  first introduced the fuzzification of the algebraic sturctures and defined fuzzy subgroups. Jun and Kim  discussed interval-valued R- subgroups in terms of near-rings. Davvaz  introduced fuzzy ideals of near-rings with interval-valued membership functions. Thillaigovindan et al.  have studied intervalvaluedfuzzy ideals and anti fuzzy ideals of near-rings. Abou-Zaid  proposed the concept of fuzzy sub near-rings and ideals.
Therefore, the rest of the paper is set out as follows. In Section 2, some basic definitions related to intuitionisticfuzzysets, hesitant fuzzysets and intervalvaluedintuitionistic hesitant fuzzy set are briefly discussed. In Section 3, four new operations on intervalvaluedintuitionistic hesitant fuzzysets have been proposed and some properties of these operations are proved. In section 4, we conclude the paper.
Chetia et al and in [13,15] Rajarajeswari et al. defined intuitionisticfuzzy soft matrix and intervalvaluedintuitionisticfuzzy soft matrix and its types. Also extended some operations and an algorithm for medical diagnosis in [14,16].In, Mitra Basu et al. described intervalvaluedfuzzy soft matrix and its types.
In a similar way to TIFNs, Wang  dened the Trapezoidal IFN (TrIFN) and Interval-Valued Trape- zoidal IFN (IVTrIFN). Both TrIFN and IVTrIFN are extensions of TIFNs. Wang and Zhang  investi- gated the weighted arithmetic averaging operator and weighted geometric averaging operator on TrIFNs and their applications to MADM problems. Wei  in- vestigated some arithmetic aggregation operators with TrIFNs and their applications to MAGDM problems. Du and Liu  extended the fuzzy VIKOR method with TrIFNs. Wu and Cao  developed some fam- ilies of geometric aggregation operators with TrIFNs and applied them to MAGDM problems. Wan and Dong  dened the expectation and expectant score, ordered weighted aggregation operator and hybrid aggregation operator for TrIFNs and employed them for MAGDM. Ye  developed the expected value method for intuitionistic trapezoidal fuzzy multicrite- ria decision-making problems. Ye  proposed the MAGDM method using vector similarity measures for TrIFNs. Wan  developed four kinds of power av- erage operator of TrIFNs, involving the power average operator, weighted power average operator of, power order weighted average operator of, and power hybrid average operator of TrIFNs. Wan  rstly dened some operational laws and the weighted arithmetical average operator of IVTrIFNs. Based on the score function and accurate function, an approach is pre- sented to rank IVTrIFNs. The MAGDM method using IVTrIFNs is then proposed. Wan  constructed non- linear fractional programming models to estimate the alternative's relative closeness. After transformation into linear programming models, the interval of the
Atanassov and Gargov (1989) generalized the concept of instuitionistic fuzzy set (IFS) to interval - valuedintuitionisticfuzzy set (IVIFS) and define some basic operational laws of IVIFS. In the corporate sector the assignment problem plays a vital role. Among researchers it has received a significant amount of attention. The costs are not known exactly in real world application. Employing fuzzytheory to model uncertainity in real problem it is assumed that the membership function of parameters are known. However in reality it is not always easy. In order to solve this problem, the best thing is to determine the uncertain as intervals. In some situations if an individual is not familiar with the problem, it is difficult to determine the exact values of the preference degrees. His views may be positive, negative and hesitative points. In this realistic situation intuitionisticfuzzy set is the useful tool to express.
For the first time Zadeh (1965) introduced the concept of fuzzysets and also Zadeh (1975) introduced the concept of an interval-valuedfuzzysets, which is an extension of the concept of fuzzy set. Atanassov and Gargov, 1989 introduced the notion of interval-valuedintuitionisticfuzzysets, which is a generalization of both intuitionisticfuzzysets and interval-valuedfuzzysets. On other hand, Satyanarayana et al., (2012) applied the concept of interval-valuedintuitionisticfuzzy ideals. In this paper we introduce the notion of interval-valuedintuitionisticfuzzy homomorphism of BF-algebras and investigate some interesting properties.
Remark 1. Atanassov’s(1986) definition of intuitionisticfuzzy set imposes a condition on and as which in turn implies that for each This goes against the spirit that and are assigned independently. As this independence criteria is more important, we relax the