In this paper a new concept is called **n**-**valued** **interval** **neutrosophic** **sets** is given. The basic operations are introduced on **n**-**valued** **interval** **neutrosophic** **sets** such as; union, intersection, addition, multiplication, scalar multiplication, scalar division, truth- favorite and false-favorite. Then, some distances between **n**-**valued** **interval** **neutrosophic** **sets** (NVINS) are proposed. Also, we propose an efficient approach for group multi-criteria decision making based on **n**-**valued** **interval** **neutrosophic** **sets**. An **application** of **n**-**valued** **interval** **neutrosophic** **sets** in **medical** **diagnosis** problem is given.

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As **medical** **diagnosis** contains lots of uncertainties and increased volume of information available to physicians from new **medical** technologies, the process of classifying different **sets** of symptoms under a single name of disease becomes difficult. In some practical situations, there is the possibility of each element having different truth membership, indeterminate and false membership functions. The unique feature of **n**-**valued** **interval** **neutrosophic** set is that it contains multi truth membership, indeterminate and false membership. By taking one time inspection, there may be error in **diagnosis**. Hence, multi time inspection, by taking the samples of the same patient at different times gives the best **diagnosis**. So, **n**-**valued** **interval** **neutrosophic** **sets** and their applications play a vital role in **medical** **diagnosis**.

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In this paper, we define a rough cosine similarity measure between two rough **neutrosophic** **sets**. The notions of rough **neutrosophic** **sets** (RNS) will be used as vector representations in 3D-vector space. The rating of all elements in RNS is expressed with the upper and lower approximation operator and the pair of **neutrosophic** **sets** which are characterized by truth-membership degree, indeterminacy-membership degree, and falsity-membership degree. A numerical example of the **medical** **diagnosis** is provided to show the effectiveness and flexibility of the proposed method.

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approximate reasoning [13,14], Roy and Biswas on **medical** **diagnosis** [24] and Mendel on intelligent control [19]. In 1975, Rosenfeld [25] discussed the concept of fuzzy graphs whose basic idea was introduced by Kauffman [18] in 1973. The fuzzy relation between fuzzy **sets** were also considered by Rosenfeld and he developed the structure of fuzzy graphs, obtain analogs of several graph theoretical concepts. Bhattacharya [4] gave some remarks on fuzzy graphs. Mordeson and Peng [21] introduced some operations on fuzzy graphs. The complement of a fuzzy graph was defined by Mordeson [20]. Bhutani and Rosenfeld introduced the concept of M-strong fuzzy graphs in [5] and studied some of their properties. Shannon and Atanassov [39] introduced the concept of intuitionistic fuzzy relations and intuitionistic fuzzy graphs. Hongmei and Lianhua gave the definition of **interval**-**valued** graph in [15]. Recently Akram introduced the concepts of bipolar fuzzy graphs and **interval**–**valued** fuzzy graphs in [1,2,3]. Pal and Rashmanlou [23] studied irregular inteval-**valued** fuzzy graphs. Also, they defined antipodal **interval**- **valued** fuzzy graphs [26], balanced **interval**-**valued** fuzzy graphs [27] and a study on bipolar fuzzy graphs [28]. Rashmanlou and Jun investigated complete **interval**-**valued** fuzzy graphs [29]. Samanta and Pal defined fuzzy tolerance graphs [32], fuzzy threshold graphs [36], fuzzy planar graphs [38], fuzzy k-competition graphs and p-competition fuzzy graphs [34], irregular bipolar fuzzy graphs [35]. Borzooei and Rashmanlou [6-12] investigated new concepts on vague graphs. In this paper, we present the concepts of neighbourly irregular **interval**–**valued** fuzzy graphs, neighbourly totally irregular **interval**–**valued** fuzzy graphs, highly irregular **interval** – **valued** fuzzy graphs, and highly totally irregular **interval**–**valued** fuzzy graphs are introduced and investigated. A necessary and sufficient condition under which neighbourly irregular and highly irregular **interval**–**valued** fuzzy graphs are equivalent is discussed.

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Neutrosopohic refined set is an important extension of **neutrosophic** set. In this paper, we focus on introducing similarity measure between **neutrosophic** refined **sets** based on the exponential operation. The proposed similarity measure provides a new way to handle the indeterminate and inconsistent information. Also we have examined some relevant properties of similarity measure between **neutrosophic** refined **sets** based on exponential operation. Finally, an **application** of **neutrosophic** refined set is given in **medical** **diagnosis** problems to illustrate the benefit of the proposed approach.

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The theory of **neutrosophic** set (NS), which is the generalization of the classical **sets**, conventional fuzzy set [1], intuitionistic fuzzy set [2]and **interval** **valued** fuzzy set [3],was introduced by Samarandache [4]. This concept has been applied in many fields such as Databases [5, 6], **Medical** **diagnosis** problem [7], Decision making problem [8],Topology [9],control theory [10] and so on. The concept of **neutrosophic** set handle indeterminate data whereas fuzzy set theory, and intuitionstic fuzzy set theory failed when the relation are indeterminate.

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In 1965, fuzzy set theory was firstly given by Zadeh which is applied in many real applications to handle uncertainty. Then **interval** **valued** fuzzy set, intuitionistic fuzzy set theory and **interval** **valued** intuitionistic fuzzy set were introduced by Turksen, Atanassov and Atanassov and Gargov respectively. These theories can only handle incomplete information not the indeterminate information and inconsistent information which exists commonly in belief systems. So, **Neutrosophic** set (generalization of fuzzy **sets**, intuitionistic fuzzy **sets** and so on) defined by Smarandache [1] has capability to deal with uncertainty, imprecise, incomplete and inconsistent information which exists in real world from philosophical point of view. Ye [4] proposed the vector similarity measures of simplified **neutrosophic** **sets**.

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In real life, a person may observe that an object belongs and not belongs to a set to certain degree, but it is possible that he is not sure about it. In other words, there may be some hesitation or uncertainty about the membership and non-membership degree of an object belonging to a set. In fuzzy set theory there is no means to incorporate that hesitation in membership degree. A possible solution is to use vague **sets** and the concept of vague set was proposed by Gau and Buehrer [1993]. Distance measure between vague **sets** is one of the most important technologies in various **application** fields of vague **sets**. But these methods are unsuitable to deal with the similarity measures of IFSs. In this paper we have extended the work of Zeshui Xu [2007] and also proposed a method to develop some similarity measure of **interval** **valued** vague **sets** and define the positive and negative ideal of **interval** **valued** vague **sets**, and apply the similarity measures to multiple attribute decision making based on vague information. A numerical example is also given to elaborate our technique.

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In this paper, we investigate the reliability analysis of a powerloom plant by using **interval** **valued** intuitionistic fuzzy **sets** (IVIFS). Herein, we modeled a powerloom plant as a gracefully degradable system having two units A(**n**) and B(m) connected in series. The reliability of **n** components of unit A and m components of unit B is assumed to be an IVIFS defined over the universe of discourse [0, 1]. Thus, the reliability of the system obtained is an IVIFS that covers the inherited uncertainty in data collection and reliability evaluation of a powerloom plant.

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IVFGs and investigated some properties. They also introduced the notion of **interval**- **valued** fuzzy complete graphs and presented some properties of self complementary and self weak complementary **interval**-**valued** fuzzy complete graphs. Akram also introduced **interval**–**valued** fuzzy line graphs [2] and bipolar fuzzy graphs [1]. Talebi and H. Rashmanlou [47] studied on isomorphism of IVFGs. Rashmanlou and Jun [29] defined the three new operations, direct product, semi strong product and strong product of IVFGs and discussed its properties on complete IVFGs. Debnath [28] introduced domination in IVFGs. Rashmanlou and Pal defined Irregular IVFG [26], Balanced IVFG [30] and Antipodal IVFG [31] and studied its properties. Also, they studied on the properties of highly irregular IVFG [33] and defined isometry on IVFG [32]. Akram, Alshehri and Dudek [4] introduced certain types of IVFG such as balanced IVFGs, neighbourly irregular IVFGs, neighbourly total irregular IVFGs, highly irregular IVFGs, highly total irregular IVFGs. Again Akram, Yousaf and Dudek [7] studied on the properties of self centered IVFGs. Pal, Samanta and Rashmanlou [27] defined the degree and total degree of an edge in the Cartesian product and composition of two IVFG and obtained some results. Mohideen [8] studied on strong and regular IVFGs. Narayanan and Maheswari [34] introduced strongly edge irregular and strongly edge totally irregular IVFG and made a comparative study between the two. Talebi, Rashmanlou and Ameri [48] studied on product IVFGs. Total regularity of the join of two IVFGs was discussed in [40]. Again regular and edge regular IVFGs were studied in [41].

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The theory of fuzzy **sets** proposed by Zadeh [1] has attracted wide attentions in various fields, especially where conventional mathematical techniques are of limited effectiveness, including biological and social sciences, linguistic, psychology, economics, and more generally soft sciences. In such fields, variables are difficult to quantify and dependencies among variables are so ill-defined that precise characterization in terms of algebraic, difference or differential equations becomes almost impossible. Even in fields where dependencies between variables are well defined, it might be necessary or advantageous to employ fuzzy rather than crisp algorithms to arrive at a solution. Out of several higher-order fuzzy **sets**,**interval**-**valued** fuzzy **sets** introduced by Zadeh [2-3]and intuitionistic fuzzy **sets** introduced by Atanassov [4-5]have been found to be well suited to dealing with vagueness. The concept of an **interval**-**valued** fuzzy set can be viewed as an alternative approach to define a fuzzy set in cases where available information is not sufficient for the definition of an imprecise concept by means of a conventional fuzzy set. In general, the theory of **interval**-**valued** fuzzy **sets** is the generalization of fuzzy **sets**. Therefore, it is expected that **interval**-**valued** fuzzy **sets** could be used to simulate human decision-making processes and any activities requiring human expertise and knowledge, which are inevitably imprecise or not totally reliable[6-8].

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Abstract —In this paper a new approach is presented to model **interval**-based data using Fuzzy **Sets** (FSs). Specifically, we show how both crisp and uncertain intervals (where there is uncertainty about the endpoints of intervals) collected from individual or multiple survey participants over single or repeated surveys can be modelled using type-1, **interval** type-2, or general type-2 FSs based on zSlices. The proposed approach is designed to minimise any loss of information when transferring the **interval**- based data into FS models, and to avoid, as much as possible assumptions about the distribution of the data. Furthermore, our approach does not rely on data pre-processing or outlier removal which can lead to the elimination of important information. Different types of uncertainty contained within the data, namely intra- and inter-source uncertainty, are identified and modelled using the different degrees of freedom of type-2 FSs, thus provid- ing a clear representation and separation of these individual types of uncertainty present in the data. We provide full details of the proposed approach, as well as a series of detailed examples based on both real-world and synthetic data. We perform comparisons with analogue techniques to derive fuzzy **sets** from intervals, namely the **Interval** Approach (IA) and the Enhanced **Interval** Approach (EIA) and highlight the practical applicability of the proposed approach.

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Ranking competing alternatives in terms of their overall performance with respect to some criterions in fuzzy environment is possible by the use of fuzzy TOPSIS methodology using **interval**-**valued** fuzzy-**sets** concepts. This author presents an effective fuzzy multi-criteria method based upon the fuzzy model and the concepts of positive ideal and negative ideal solution points for prioritizing alternatives using inputs from a team of decision makers. The fuzzy **sets** concepts are used to evaluate the performance of alternatives and the importance of criteria. Fuzzy TOPSIS based on the **interval**-**valued** fuzzy-**sets** is fully described and a case study on RFID comprised of four main criteria and five alternatives is constructed and solved by the proposed extended TOPSIS method. The TOPSIS methodology used in this article is able to grasp the ambiguity exists in the utilized information and the fuzziness appears in the human judgments and preferences. TOPSIS technique can easily produce satisfactory results, and hence stimulates creativity and the invention for developing new methods and alternative approaches. This article is a very useful source of information for Fuzzy TOPSIS based on the **interval**-**valued** fuzzy **sets** and extends the area of **application** of RFID technology in general. Due to the fact that a better management of a system is related to the full understanding of the technologies implemented and the system under consideration, sufficient background on the methodologies are provided and a case study is developed and solved by the proposed method.

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In this paper we introduce the concept of **neutrosophic** soft expert set (NSES). We also define its basic operations, namely complement, union, intersection, AND and OR, and study some of their properties. We give examples for these concepts. We give an **application** of this concept in a deci- sion-making problem.

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The paper is structured as follows: Section 2 contains a brief background about SVNSs and SVNGs. Section 3 introduces the concept of SVNLG of a SVNG and, investigates their properties. In Section 4, the notion of SVNC consistent with single-**valued** **neutrosophic** cycles in SVNGs is proposed and a complete characterization of the structure of the SVNC is presented, and finally we draw conclusions in Section 5.

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F 𝑘 = ( ̃ 𝑓 𝑖𝑗 𝑘 ) 𝑚×𝑛 , where 𝑓 ̃ 𝑖𝑗 𝑘 = ([𝑎 𝑖𝑗 𝑘 , 𝑏 𝑖𝑗 𝑘 ], [𝑐 𝑖𝑗 𝑘 , 𝑑 𝑘 𝑖𝑗 ]) is an IVIFN for the alternative 𝐴 𝑖 with respect to attribute 𝑢 𝑗 . In this paper, [𝑎 𝑘 𝑖𝑗 , 𝑏 𝑖𝑗 𝑘 ] and [𝑐 𝑖𝑗 𝑘 , 𝑑 𝑘 𝑖𝑗 ] provided by the expert 𝑒 𝑘 are, respectively, the satisfaction (agreeing) degree **interval** and dissatisfaction (disagreeing) degree **interval** of the 𝑖th cloud service 𝐴 𝑖 with respect to the 𝑗th attribute (indicator) 𝑢 𝑗 . 3.1. Determine the Weights of Experts by the Extended GRA Method. Due to the fact that each expert is skilled in some fields rather than all fields, it is more reasonable that the weights of each expert with respect to different attributes should be assigned different values. However, the weights of each expert obtained with the existing methods [34–37] are the same.

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Some researchers endeavored to enrich soft **sets** by combining them with other soft computing models such as rough **sets** and fuzzy **sets**. Using soft **sets** as the granulation structures, Feng et al. [] initiated soft approximation spaces and soft rough **sets**, which generalize Pawlak’s rough **sets** based on soft **sets**. On the other hand, Maji et al. [] ini- tiated the study on hybrid structures involving both fuzzy **sets** and soft **sets**. They in- troduced the notion of fuzzy soft **sets**, which can be seen as a fuzzy generalization of Molodtsov’s soft **sets**. Furthermore, Yang et al. [] introduced **interval**-**valued** fuzzy soft **sets** which realize a common extension of both Molodtsov’s soft **sets** and **interval**-**valued** fuzzy **sets**. It should be noted that there are several diﬀerent kinds of soft inclusions (also known as soft subsets) in the literature [, , , ]. Feng and Li [] investigated diﬀerent types of soft subsets and the related soft equal relations in a systematic way.

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[3] further introduced the concepts of **interval** **valued** intuitionistic fuzzy set. The present authors further introduced the new extension of IVIFS namely **interval** **valued** intuitionistic fuzzy **sets** of second type (IVIFSST) and established some of their properties [4]. The rest of the paper is designed as follows: In Section 2, we give some basic definitions. In Section 3, we introduce modal type operators over **interval** **valued** intuitionistic fuzzy **sets** of second type and establish some of their properties. This paper is concluded in section 4.

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Abstract. **Neutrosophic** theory has many applications in graph theory, single **valued** **neutrosophic** graph (SVNG) is the generalization of fuzzy graph and intuitionistic fuzzy graph. In this paper, we introduced some types of SVNGs, which are subdivision SVNGs, middle SVNGs, total SVNGs and single **valued** **neutrosophic** line graphs (SVNLGs), also discussed the isomorphism, co weak isomorphism and weak isomorphism properties of subdivision SVNGs, middle SVNGs, total SVNGs and SVNLGs.

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Step 1 : Find out the positive-ideal solution M + and negative-ideal solution M − : M + = {< [µ − 1+ , µ + 1+ ] , [ v − 1+ , v + 1+ ] , ..., [µ − **n**+ , µ + **n**+ ] , [ v − **n**+ , v + **n**+ ] >} , M − = {< [µ − 1− , µ + 1− ] , [ v 1− − , v 1− + ] , ..., [µ − **n**− , µ + **n**− ] , [ v − **n**− , v + **n**− ] >} , where, for each j = 1, 2, ..., **n**,

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