In this paper, a novel **generalized** **interval**-**valued** **intuitionistic** **fuzzy** **sets** (GIVIFS) is presented, which is the generalization of conventional **intuitionistic** **fuzzy** **sets** (IFS) and **interval**-**valued** **intuitionistic** **fuzzy** **sets** (IVIFS). By analyzing the degree of hesitancy, this paper introduces **generalized** **interval**-**valued** **intuitionistic** **fuzzy** **sets** 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 [2], engineering [3], chemistry [4], biology [5] and many mathematical specifications such as optimization [6], discrete mathematics [7], statistics [8],operation research [9] and **fuzzy** mathematics [10] [11]. 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 **fuzzy** **sets** 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.

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Bridges and cutnodes is a very important concept in graph **theory**. Rosenfeld [35] obtained **fuzzy** analogs of bridges and cutnodes. Further it was studied by Bhattacharya [9]. Again Sunitha and Vijayakumar studied about the properties of **fuzzy** bridges and **fuzzy** cutnodes [44]. It was also studied by Mordeson and Nair [20]. Strength of the paths in IVFGs were discussed by Rashmanlou and Pal [31]. Again it was studied by Akram, Yousaf and Dudek [7]. Akram and Alsheri defined **intuitionistic** **fuzzy** bridges and **intuitionistic** **fuzzy** cutnodes in [3]. Again Akram and Farooq defined bipolar **fuzzy** bridges and bipolar **fuzzy** cutnodes in [6]. Bipolar **fuzzy** bridges and bipolar **fuzzy** cutnodes were also characterized by Mathew, Sunitha and Anjali [17].

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The **theory** of **fuzzy** **sets** was introduced by Zadeh [10] in 1965. Later, Chang [2] proposed **fuzzy** topology in 1967. The concept of **intuitionistic** **fuzzy** **sets**, introduced by Atanassov [1] is a generalization of **fuzzy** **sets**. Using the notion of **intuitionistic** **fuzzy** **sets**, Coker [3] introduced the notion of **intuitionistic** **fuzzy** topological spaces in 1997.In this paper, we introduce **intuitionistic** **fuzzy** regular α **generalized** open set. We investigate some of their properties. We also introduce **intuitionistic** **fuzzy** regular α T 1/2 space and obtain some characterizations and several preservation theorems.

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Cenigiz Kahraman gave very useful description about MCDM in his book on **Fuzzy** Multicriteria Decision Making [35]. 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 **fuzzy** **sets** 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.

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and Abdul Razak Salleh [14] introduced **generalized** **interval** **valued** **fuzzy** soft set (GIVFSS). In their generalization of FSS, they attached a degree with the parameterization of **fuzzy** **sets** in defining an IVFSS. They discussed various operations and properties of GIVFSS. Some of these are GIVFS subset, GIVFS equal set, **generalized** null **interval** **valued** **fuzzy** soft set (GNIVFS), **generalized** absolute **interval** **valued** **fuzzy** 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.

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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.

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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 [34] proposed the **intuitionistic** **fuzzy** 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 **Intuitionistic** **Fuzzy** Einstein Weighted Geometric op- erator (IFEWG) and the **Intuitionistic** **Fuzzy** 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 [35] proposed the **Intuitionistic** **Fuzzy** Einstein Weighted Averaging operator (IFEWA) and the **Intuitionistic** **Fuzzy** Einstein Ordered Weighted Averaging operator (IFEOWA), and studied various properties of these operators and analyzed the relations between the existing **intuitionistic** **fuzzy** aggregation operators and them. Maris and Iliadis [36] 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.

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The notion of **fuzzy** was introduced by [13] in 1965. Atanassov [1] introduced the concept of **intuitionistic** **fuzzy** **sets** in 1986. Atanassov et al.[2] introduced the concept of **interval**-**valued** **intuitionistic** **fuzzy** **sets** which is a generaliza- tion of both **intuitionistic** **fuzzy** **sets** and **interval**-**valued** **fuzzy** **sets**. Several mathematicians applied the concept of **interval**- **valued** **intuitionistic** **fuzzy** **sets** to algebraic structures. Biswas [5] studied on Rosenfeld’s **fuzzy** subgroups with **interval**- **valued** membership function. Das and Dutta [6] 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 **intuitionistic** **fuzzy** k-ideal and **interval**-**valued** **intuitionistic** **fuzzy** ideal on semi-rings. In this paper, the cartisian product of **interval**-**valued** **intuitionistic** **fuzzy** k-ideal in semi-rings are studied. Investigate the relationship between A, B and A × B.

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Note that all of the above-mentioned **interval**- **valued** **intuitionistic** **fuzzy** averaging operators are based on the operational laws in [36]. 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**-**valued** **intuitionistic** **fuzzy** 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.

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Copyright © 2014 Horizon Research Publishing All rights reserved. Abstract Hesitancy is the most common problem in decision making, for which hesitant **fuzzy** set can be considered as a useful tool allowing several possible degrees of membership of an element to a set. Recently, another suitable means were defined by Zhiming Zhang [1], called **interval** **valued** **intuitionistic** hesitant **fuzzy** **sets**, dealing with uncertainty and vagueness, and which is more powerful than the hesitant **fuzzy** **sets**. In this paper, four new operations are introduced on **interval**-**valued** **intuitionistic** hesitant **fuzzy** **sets** and several important properties are also studied. Keywords **Fuzzy** **Sets**, **Intuitionistic** **Fuzzy** Set, Hesitant

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Zadeh [18] introduced the concept of **fuzzy** set in 1965 and after that he also introduced the notion of **interval** **valued** **fuzzy** subset [17] (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 [14] first introduced the fuzzification of the algebraic sturctures and defined **fuzzy** subgroups. Jun and Kim [9] discussed **interval**-**valued** R- subgroups in terms of near-rings. Davvaz [6] introduced **fuzzy** ideals of near-rings with **interval**-**valued** membership functions. Thillaigovindan et al. [16] have studied **interval** **valued** **fuzzy** ideals and anti **fuzzy** ideals of near-rings. Abou-Zaid [1] proposed the concept of **fuzzy** sub near-rings and ideals.

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Therefore, the rest of the paper is set out as follows. In Section 2, some basic definitions related to **intuitionistic** **fuzzy** **sets**, hesitant **fuzzy** **sets** and **interval** **valued** **intuitionistic** hesitant **fuzzy** set are briefly discussed. In Section 3, four new operations on **interval** **valued** **intuitionistic** hesitant **fuzzy** **sets** have been proposed and some properties of these operations are proved. In section 4, we conclude the paper.

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Chetia et al and in [13,15] Rajarajeswari et al. defined **intuitionistic** **fuzzy** soft matrix and **interval** **valued** **intuitionistic** **fuzzy** soft matrix and its types. Also extended some operations and an algorithm for medical diagnosis in [14,16].In[10], Mitra Basu et al. described **interval** **valued** **fuzzy** soft matrix and its types.

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In a similar way to TIFNs, Wang [35] dened the Trapezoidal IFN (TrIFN) and **Interval**-**Valued** Trape- zoidal IFN (IVTrIFN). Both TrIFN and IVTrIFN are extensions of TIFNs. Wang and Zhang [36] investi- gated the weighted arithmetic averaging operator and weighted geometric averaging operator on TrIFNs and their applications to MADM problems. Wei [37] in- vestigated some arithmetic aggregation operators with TrIFNs and their applications to MAGDM problems. Du and Liu [38] extended the **fuzzy** VIKOR method with TrIFNs. Wu and Cao [39] developed some fam- ilies of geometric aggregation operators with TrIFNs and applied them to MAGDM problems. Wan and Dong [40] dened the expectation and expectant score, ordered weighted aggregation operator and hybrid aggregation operator for TrIFNs and employed them for MAGDM. Ye [41] developed the expected value method for **intuitionistic** trapezoidal **fuzzy** multicrite- ria decision-making problems. Ye [42] proposed the MAGDM method using vector similarity measures for TrIFNs. Wan [43] 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 [44] 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 [45] constructed non- linear fractional programming models to estimate the alternative's relative closeness. After transformation into linear programming models, the **interval** of the

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Atanassov and Gargov (1989) **generalized** the concept of instuitionistic **fuzzy** set (IFS) to **interval** - **valued** **intuitionistic** **fuzzy** 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 **fuzzy** **theory** 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 **intuitionistic** **fuzzy** set is the useful tool to express.

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For the first time Zadeh (1965) introduced the concept of **fuzzy** **sets** and also Zadeh (1975) introduced the concept of an **interval**-**valued** **fuzzy** **sets**, which is an extension of the concept of **fuzzy** set. Atanassov and Gargov, 1989 introduced the notion of **interval**-**valued** **intuitionistic** **fuzzy** **sets**, which is a generalization of both **intuitionistic** **fuzzy** **sets** and **interval**-**valued** **fuzzy** **sets**. On other hand, Satyanarayana et al., (2012) applied the concept of **interval**-**valued** **intuitionistic** **fuzzy** ideals. In this paper we introduce the notion of **interval**-**valued** **intuitionistic** **fuzzy** homomorphism of BF-algebras and investigate some interesting properties.

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Remark 1. Atanassov’s(1986) definition of **intuitionistic** **fuzzy** 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