Kerre, Intuitionistic fuzzy connectives revisited, in: Proceedings of 9th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, 200[r]

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Nowadays we dealt with problems which are complex in nature. To tackle those problems which we have experienced in our day to day life becomes complex more and more. And most of them deals with uncertainties i.e. in Mathematics the term ‘vague’ is usually used for such cases. Introduction of **fuzzy** **set** **theory** in 1965 by L. A. Zadeh [18] handled those problems based on uncertainties quite successfully at some extent. So we need some effective measures or tools which are capable to solve such problems. Some tools such as **fuzzy** **set** [18], L-**fuzzy** **set** [5], **intuitionistic** **fuzzy** **set** [1], **interval** **valued** **fuzzy** **set** [16], **interval** **valued** **intuitionistic** **fuzzy** **set** [2], rough **set** [13] etc are introduced earlier.

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knowledge management tool serving the ICAR-CNR, a research institute of the Italian National Council of Research. It applied in the formalized groups (such as project groups) where collaborative work takes place and informal groups (communities of practice) that may arise around common problems, interests and objectives. The tool provides a web-based environment for knowledge creation, sharing and access. It helps researchers quickly **set** up their own portals, create and organize knowledge items (such as news, papers, links, announcements, projects, etc.), share them within workgroups, search for a specific document or browse through a **set** of related documents (ontology-driven browsing). This should be done by basing on the built of the Research Ontology and the Knowledge Item Ontology, which contain the formal specification of **application** domain concepts, relations and constraints.

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45. Wang, J., Wang, J.Q., and Zhang, H.Y. \A likelihood- based TODIM approach based on multi-hesitant **fuzzy** linguistic information for evaluation in logistics out- sourcing", Comput. Ind. Eng., 99, pp. 287-299 (2016). 46. Shariati, S., Abedi, M., Saedi, A., Yazdani-Chamzini, A., Tamosaitien_e, J., Saparauskas, J., and Stupak, S. \Critical factors of the **application** of nanotechnology in **construction** industry by using ANP technique under **fuzzy** **intuitionistic** environment", J. Civ. Eng. Manag., 23(7), pp. 914-925 (2017).

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First, we present the concept of GIVIFS, which is proved to be the generalization of IFS and IVIFS. And then we introduce the **construction** method of the generalized **interval** **valued** **intuitionistic** **fuzzy** sets with parameters (GIVIFSP), and define complement operation, intersection operation and union operation on GIVIFS. Finally, we prove that GIVIFS is a closed algebraic system for all these operations as **fuzzy** sets, IFS, and IVIFS. Therefore, this paper generalizes the IVIFS **theory**, and provides some valuable conclusions for the field of **application** research of IVIFS, and it is also useful to the generalization of **interval**-**valued** **intuitionistic** **fuzzy** reasoning.

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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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the corresponding aggregation functions on the unit **interval**. Yager [14] introduced a componentwise ex- tension of the OWA function to AIFS **theory**, but he gave no motivation why this is the best construc- tion of an OWA function in AIFS **theory**. Beliakov et al. [15] generalized the **construction** of Xu and Wei using additive generators and characterized the functions obtained by the generalized **construction** which are consistent with the operations on the unit **interval**. Since the definition of the OWA function on the unit **interval** involves arithmetic operators on the **set** of reals, we start in this paper from arithmetic operators on the underlying lattice L I of IVFS and AIFS **theory** and we investigate which kind of OWA functions on L I that we can construct using them. We first recall in Section 2 and 3 some definitions that will be needed later. We recall the axiomatic definition of the arithmetic operators on L I and we give a characterization of these operators in Section 4 and 5. In the next section we extend the arithmetic mean and the quasi-arithmetic mean to L I and in the subsequent section we extend the OWA functions to L I . For the latter to be suc- cessful we search for a way to extend the ordering procedure of input values in [0, 1] to input values in L I .

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The classical decision making methods generally assume that all criteria and their respective weights are expressed in crisp values and, thus, that the rating and the ranking of the alternatives can be carried out without any problem. In a real-world decision situation, the **application** of the classical decision making method may face serious practical constraints from the criteria perhaps containing imprecision in the information. In many cases, performance of the criteria can only be expressed qualitatively or by using linguistic terms, which certainly demands a more appropriate method. The most preferable situation for a decision making problem is when all ratings of the criteria and their degree of importance are known precisely, which makes it possible to arrange them in a crisp ranking. However, many of the decision making problems in the real world take place in an environment in which the goals, the constraints, and the consequences of possible actions are not known precisely (Bellman and Zadeh, 1970). As a result, the best condition for a classic decision making problem may not be satisfied, when the decision situation involves both **fuzzy** and crisp data. The classical decision making methods cannot handle such problems effectively, because they are only suitable for dealing with problems in which all performances of the criteria are represented by crisp numbers. The **application** of the **fuzzy** **set** **theory** in the field of decision making is justified when the intended goals or their attainment cannot be defined or judged crisply but only as **fuzzy** sets (Zimmermann, 1987).

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1. Applications of **interval**-**valued** and **intuitionistic** FMM in image processing: As we have pointed out before [57– 59], **interval**-**valued** and **intuitionistic** **fuzzy** **set** **theory** enable us to model numerical and spatial uncertainty in grayscale images that is due to image capture, leading to specific morphological operators and related applica- tions that have yet to be explored in detail. In previous papers, we provided some preliminary results concern- ing **interval**-**valued** edge detection [57–59]. In this paper, we went one step further by outlining an **application** of **interval**-**valued** FMM that aims at combining different methods for medical image reconstruction in terms of the watershed transform. Clearly, the approach presented in this paper only represents our first attempt to tackle this problem using the emerging techniques of **interval**- **valued** FMM. Nevertheless, we believe that every exist- ing **application** of gray-scale or **fuzzy** MM in image pro- cessing potentially lends itself to **interval**-**valued** FMM if some uncertainty regarding the pixel values exists. In contrast to conventional morphological techniques, in- terval-**valued** FMM techniques are able to keep track of this uncertainty information.

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In this paper, the Modified Technique for Order Preference by Similarity to the Ideal Solution (M-TOPSIS) model has been extended into the **intuitionistic** **fuzzy** environment. By applying the improved score function first, to represent the aggregated effect of positive and negative evaluations in the performance ratings of the alternatives based on **interval**-**valued** **intuitionistic** **fuzzy** **set** (IVIFS) data and in combination with the weighted normalized Euclidean distance for the computation of the separation measures of alternative(s) for the **intuitionistic** positive and negative ideal solutions. The two methods which have been used for the computation of the separation measure have been integrated using a new reflection defuzzification integration formula which has been introduced in this study. To prove the efficacy of the proposed model, the model have been applied for the evaluation and selection design concept for a new printed circuit board (PCB), and for a modified hypothetical example which is based on the selection of a preferred Naval vessel as a reference for a new design.

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In graph **theory**, an intersection graph is a graph which represent the intersection of sets. An **interval** graph is the intersection of multiset of intervals on real line. **Interval** graphs are useful in resource allocation problem in operations research. Besides, **interval** graphs are used extensively in mathematical modeling, archaeology, developmental psychology, ecological modeling, mathematical sociology and organization **theory**.

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Abstract. Concepts of graph **theory** are applied in many areas of computer science including image segmentation, data mining, clustering, image capturing and networking. **Fuzzy** graph **theory** is successfully used in many problems, to handle the uncertainty that occurs in graph **theory**. An **interval**-**valued** **fuzzy** graph is a generalized structure of a **fuzzy** graph that gives more precision, flexibility, and compatibility to a system when compared with systems that are designed using **fuzzy** graphs. In this paper, new concepts of irregular **interval**–**valued** **fuzzy** graphs such as neighbourly totally irregular **interval**- **valued** **fuzzy** graph, 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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Since in single-**valued** neutrosophic **set** the neutrosophic components are independent (their sum can be up to 3, and if a component increases or decreases, it does not change the others), while in **intuitionistic** **fuzzy** **set** the components are dependent (in general if one changes, one or both the other components change in order to keep their sum equal to 1). Also, applying the neutrosophic operators is a better aggregation since the indeterminacy (I) is involved into all neutrosophic (complement/negation, intersection, union, inclusion/inequality/**implication**, equality/equivalence) operators while all **intuitionistic** **fuzzy** operators ignore (do not take into calculation) the

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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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The **theory** of rough sets was originally proposed by Pawlak [ 1 ] as a formal tool for modeling and processing intelligent systems characterized by insufficient and incomplete information. The basic structure of rough **set** **theory** is an approximation space consisting of a universe of discourse and a binary relation imposed on it. By introducing the concepts of lower and upper approximations of all decision classes with respect to an approximation space induced from the conditional attribute **set**, knowledge hidden in information tables may be unraveled and expressed in the form of decision rules. We have witnessed a rapid development of and a fast growing interest in rough **set** **theory** recently and many models and methods have been proposed and studied (see e.g. the literature cited in [ 2 , 3 ]).

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As the three characteristic functions of an IVTrIFN are equal in characterizing the IVTrIFN, the total mass of the sheet focuses on the barycenter. If the three characteristic functions of the IVTrIFN are regarded as the three states of a discrete random variable in probabilistic **theory**, then, the horizontal and vertical coordinates of each barycenter represent, respectively, the value (or state value) and its membership (or cor- responding state probability) of the IVTrIFN. Eq. (1) denes the expectation of the IVTrIFN by summing the products of horizontal and vertical coordinates of each barycenter, which is similar to the mathematical expectation of a discrete random variable. So, the expectation dened by Eq. (1) reects accurately the value distribution of the IVTrIFN and its geometrical meaning is very clear.

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Notwithstanding, with respect to the second case (real- **valued** extensions of the FS concept) we should notice that some well known definitions from the literature escape from the general formulation provided in Definition 11. To give an example, the notion of entropy of IF sets introduced by Szmidt and Kacprzyk in [40] and adapted to IVF sets by Zeng and Li [47], extends the well-known notion of entropy of a **fuzzy** **set** originally introduced by De Luca and Termini [30] cannot be formulated in terms of Definition 11. The authors base the interpretation of their extension on a geometric representation of IF sets. According to their interpretation, the entropy measures the whole missing information which may be necessary to classify the points as elements either with full membership or full non-membership degree. It is not only related to the degrees of fuzziness of both extremes A and A but also to the hesitation degree of the IVF **set**. In fact, if A is less **fuzzy** than B and A is less **fuzzy** than

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Abstract. The paper deals with flexible queries in relational databases. Con- ditions included in queries are modeled with the use of **interval**-**valued** **fuzzy** sets. Each value returned by a query is associated with a subinterval of [0,1] which expresses a membership degree. The bounds of membership intervals have been determined for di fferent operations of relational algebra and dif- ferent SQL operators.

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Another interesting issue is to reveal the substantial relationship between IFPRs and IVFPRs. Apart from the hesitation, uncertainty may also be present: an expert may not possess a precise or sufficient level of knowledge of the problem to solve. This uncertainty is usually dealt with **interval**-**valued** **fuzzy** preference relations (IVFPRs), in which the elements are **interval**-**valued** **fuzzy** numbers (IVFNs) [20]–[25]. Recently, the relationship between hesitation and uncertainty has attracted some interest from researchers. The IFSs and **interval**-**valued** FSs (IVFSs) are mathematically equivalent [26], and IFPRs and IVFPRs have been proved to be mathematically isomorphic by Wu and Chiclana [27]. However, mathematical isomorphism is just the superficial relationship and it cannot guarantee the reasonability of converting processes between IFPRs and IVFPRs in a straightforward manner unless formal theoretically sound results are provided. Therefore, the second objective of this article is to find a substantial relationship (multiplicative transitivity isomorphism) between IFPRs and IVFPRs. To do that, the definition of multiplicative transitivity for IVFPRs based on Zadeh’s Extension Principle [28] is introduced first. Then, the multiplicative transitivity isomorphism is established with a rigorous mathematical proof. Therefore, the reasonability for these two types of multiplicative transitivity for IFPRs and IVFPRs is strengthened by each other because they are **set** up via different reliable methods: Operational laws of IFSs and Extension Principle and Representation Theorem of **Fuzzy** Sets (FSs), respectively. Thus, IFPRs and IVFPRs not only have mathematical isomorphism, but also have multiplicative transitivity isomorphism. Consequently, IFPRs and IVFPRs are completely equivalent in consistency based MCDM resolution processes, and then hesitation and uncertainty can be unified. Finally, a multiplicative transitivity based multi-objective programming (MOP) model for deriving the priority vector of IFPRs is proposed.

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