**Intuitionistic** **Fuzzy** **set** (**IFS**) was proposed in early 80™s. It is a well known theory. As a developer in **Fuzzy** Mathematics, interval“ valued **Intuitionistic** **Fuzzy** sets (IVFS) were developed afterwards by Gargo and Atanssov. It has a wide range of applications in the field of Optimization and algebra. There are many distance measure in **Fuzzy** such as Hamming, Normalized Hamming, Euclidean, Normalized Euclidean, Geometric, Normalized Geometric etc¦ to calculate the distance between two **fuzzy** numbers. In this paper, the comparison between Euclidean distance measure in **Intuitionistic** **Fuzzy** **set** and interval “ valued **Intuitionistic** **Fuzzy** sets is explored. The step-wise conservation of **Intuitionistic** **Fuzzy** **set** and interval “ valued **Intuitionistic** **Fuzzy** sets is also proposed. A real life application for this comparison is explained briefly. This type of comparative analysis shows that the distance between **Intuitionistic** **Fuzzy** **set** and interval“ valued **Intuitionistic** **Fuzzy** sets varies slightly due to boundaries of interval “ valued **Intuitionistic** **Fuzzy** sets.

To sum up, this study is organized to solve the problems that the front involves. Section II summarizes the previous studies about this topic; Section III analyzes the bid evaluation criteria of the renewable energy building; Section IV proposes the implementation bid decision framework of the renewable energy building. Herein, the FAHP is used to determine the relative weights of the evaluation criteria and the the **intuitionistic** **fuzzy** **set** technique for order preference by similarity to ideal solution (**IFS**-TOPSIS) is used to rank the alternatives. Based on the aforementioned contents, Section V gives the real case to prove aforementioned framework.

Later on in year 1999 [18] Atanassov also discussed the possibility of using the interpretational triangle of **intuitionistic** **fuzzy** **set** in decision making. But the major contribution in this field comes from Szmidt and Kacprzyk [19-20] who intensively worked on the use of **intuitionistic** **fuzzy** sets for building soft decision- making models. They proposed two solution concepts about the **intuitionistic** **fuzzy** core and the consensus winner for group decision making. The concept of preference relation was considered by many authors, in the crisp case for example in [21] and in the **fuzzy** environment [22]. Szmidt and Kacprzyk [23] were also the first authors who generalized the concept of preference relation from the **fuzzy** case to the **intuitionistic** **fuzzy** one. They take into account **intuitionistic** **fuzzy** preference relations which are applied in group decision making problems where a solution from the individual preferences over some **set** of options should be derived. In year 2002 they used a new approach and calculate distance between **intuitionistic** **fuzzy** preferences to assess agreement of a group of experts [24]. In another article [25] they propose how to analyze the extent of agreement in a group of experts employing **intuitionistic** **fuzzy** sets. They used the concept of distances between **intuitionistic** **fuzzy** preferences as the main tool to evaluate how far the group is from full agreement (consensus in a traditional sense) and they also propose how to evaluate if it is possible for a considered group to come closer to the state of consensus. They used Entropy as the measure which makes it possible to say how strong the preferences of experts are.

In real world, we frequently deal with vague or imprecise information. Information available is sometimes vague, sometimes inexact or sometimes insufficient. Out of several higher order **fuzzy** sets, **intuitionistic** **fuzzy** sets(**IFS**)[2,3] have been found to be highly useful to deal with vagueness. There are situations where due to insufficiency in the information available, the evaluation of membership values is not possible to our satisfaction. Due to some reason, evaluation of non-membership values is not also always possible and consequently there remains a part in deterministic on which hesitation survives. Certainly Fuzz y sets theory is not appropriate to deal with such problem, rather **IFS** theory is more suitable. Out of several generalizations of **fuzzy** **set** theory for various objectives, the notion introduced by Atanassov[2] in defining **intuitionistic** **fuzzy** sets is interesting and useful. **Fuzzy** sets are **intuitionistic** **fuzzy** sets but the converse is not necessarily true[2]. In fact there are situations where **IFS** theory is more appropriate to deal with[5]. Besides, it has been cultured in [6] that vague sets[10] are nothing but **IFS**.

Abstract — Agents are being recommended as a next generation model for revising and restructuring the complex distributed applications. So the task of engineering quality for agent systems has also become significant. As different stakeholders such as project managers, users, and practitioners have different interpretations of quality; an integrated specification of MAS quality that could satisfy all the stakeholders in the project is required. The quality specifications of stakeholders are subjective and, there is a fair chance of non-zero hesitation part in recommending quality specifications; **Intuitionistic** **Fuzzy** Sets (**IFS**) have been used to capture the uncertainties associated with stakeholders’ recommendations. **IFS** are generalization of **fuzzy** sets having membership, non-membership and hesitation, and this paper proposes a methodology to obtain prioritization of quality specifications that assists quality engineer in achieving the desired level of quality for Multi-agent systems.

In this paper, the concept of **intuitionistic** **fuzzy** **set** is applied to WI-ideal, that is we introduce the notions of **intuitionistic** **fuzzy** WI-ideal and **intuitionistic** **fuzzy** lattice ideal of lattice Wajsberg algebras. We show that every **intuitionistic** **fuzzy** WI-ideal of lattice Wajsberg algebra is an **intuitionistic** **fuzzy** lattice ideal of lattice Wajsberg algebra. Also, we verify its converse part. Further, we discuss the relationship between **intuitionistic** **fuzzy** WI-ideal and **intuitionistic** **fuzzy** lattice ideal in lattice H-Wajsberg algebra. Also, we investigate some properties of **intuitionistic** **fuzzy** WI-ideal of lattice Wajsberg algebras. Finally, we show that collection of WI-ideals of lattice Wajsberg algebra is an **intuitionistic** **fuzzy** WI-ideal of lattice Wajsberg algebra.

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1. Introduction. After the introduction of the concept of **fuzzy** sets by Zadeh [12], several researches were conducted on the generalizations of the notion of **fuzzy** **set**. The idea of “**intuitionistic** **fuzzy** **set**” was ﬁrst given by Atanassov [2, 3]. Later this concept is generalized to **intuitionistic** sets in Çoker [6] and **intuitionistic** topological spaces in [5, 9, 10]. An introduction to connectedness in these spaces is given in [10].

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After the introduction of the concept of **fuzzy** sets b L.A.Zadeh [1], researchers were conducted the generalizations of the notion of **fuzzy** sets, A. Rosenfeld [2] introduced the concept of **fuzzy** group and the idea of “**intuitionistic** **fuzzy** **set**” was first published by K.T. Atanassov *3+.Multi **set** theory was introduced by W.D.Blizard[4]. As a generalization of Multisets Yager [5] introduced the concept of **Fuzzy** Multi **set** (FMS). Shinoj. T.K and Sunil Jacob John [6] introduced the concept of **Intuitionistic** **Fuzzy** Multi sets and proved some basic operations such as union, intersection, addition, multiplication, etc. Cartesian product and -cut of **Intuitionistic** **Fuzzy** Multi sets are defined and their various properties are discussed. A.solairaju, S.rethinakumar,M Maria Arockia Raj[7] introduce the concept of . n Generated **fuzzy** sets and its subgroups. P.K.Sharma develop the idea of ( , ) - cut of **intuitionistic** **fuzzy** subgroup. In this chapter we introduce some basic properties of ( , ) - cut of n generated **fuzzy** subgroups of a group

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We have that an **intuitionistic** **fuzzy** topological space can be associated with two **fuzzy** topological spaces and vice versa [1]. . If (X, 𝜏) is an IFTS and 𝜏₁= { μₐ / ∃ γₐ ∊ Iˣ such that (μₐ , γₐ) ∊ 𝜏 }, 𝜏₂ = { 1- γₐ / ∃ μₐ ∊ Iˣ such that (μₐ, γₐ) ∊ 𝜏},then (X, 𝜏₁) and (X, 𝜏₂) are **fuzzy** topological spaces. Similarly if (X, 𝜏₁) and (X, 𝜏₂) are two **fuzzy** topological space, 𝜏 = {(u,1- v)/ u∊ 𝜏₁, v∊ 𝜏₂ and u ⊆ v} is an **intuitionistic** **fuzzy** topology and (X, 𝜏) is an **intuitionistic** **fuzzy** topological spaces. We study some relationships connecting the closures and interiors of an **intuitionistic** **fuzzy** **set** in an **intuitionistic** **fuzzy** topological space and the closures and interiors of its co- ordinate **fuzzy** sets in its corresponding **fuzzy** topological spaces. .

Molodtsov [13] introduced the concept of soft **set** that can be seen as a new mathematical theory for dealing with uncertainties. Molodtsov applied this theory to several directions [13, 14, 15] and then formulated the notions of Soft number, Soft derivative, Soft integral, etc. in [16]. The soft **set** theory has been applied to many different fields with greatness. Maji [11] worked on theoretical study of soft sets in detail. The algebraic structure of soft **set** theory dealing with uncertainties has also been studied in more detail. Aktas and Cagman [2] introduced definition of soft groups, and derived their basic properties. The most appreciate theory to deal with uncertainties is the theory of **fuzzy** sets, developed by Zadeh [22] in 1965. But it has an inherent difficulty to **set** the membership function in each particular cases. The generalization of Zadeh’s **fuzzy** **set** called **intuitionistic** **fuzzy** **set** was introduced by Atanassov [4] which is characterized by a membership function and a non-membership function. In Zadeh’s **fuzzy** **set**, the sum of membership degree and non- membership degree is equal to one. In Atanassov’s **intuitionistic** **fuzzy** **set** the sum of membership degree and non- membership degree does not exeed one.

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Definition 1.6. [3] Let X be a non-empty **set** and I be the unit interval #0, 1&. An **intuitionistic** **fuzzy** **set** A (**IFS**, in short) in X is an object having the form A = '(x, µ * x , ν * x ,, x X- , where µ * : X / I and ν * : X / I denote the degree of membership and the degree of non-membership respectively, and µ * x 0 ν * x 1 1. Let IX denote the **set** of all **intuitionistic** **fuzzy** sets in X . Obviously every **fuzzy** **set** µ * in X is an **intuitionistic** **fuzzy** **set** of the form µ * , 1 µ * .

Graphs can be sometimes very complicated. So one needs to ﬁnd more practical ways to represent them. Matrices are a very useful way of studying graphs, since they turn the picture into numbers. Networks can represent all sorts of systems in the real world. As computers are more adept at manipulating numbers than at recognizing pictures, it is standard practice to communicate the speciﬁcation of a graph to a computer in matrix form. Matrices play an important role in the broad area of science and engineering. However, the classical matrix theory sometimes fails to solve the problems involving uncertainties, occurring in an imprecise environment. Sometimes it seems to be more natural to describe imprecise and uncertain opinions not only by membership functions and also by non membership function.So an **Intuitionistic** **fuzzy** matrix is the appropriate choice when exhibiting the membership degree and non- membership degree. In 1975, Rosenfeld [17] discussed the concept of **fuzzy** graphs whose basic idea was introduced by Kauﬀmann [12] in 1973. The **fuzzy** relations between **fuzzy** sets were also considered by Rosenfeld and he developed the structure of **fuzzy** graphs, obtaining analogs of several

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Proof: Suppose there exist vertices u and v belonging to two different components of G. Since D is an **intuitionistic** **fuzzy** psd-**set** of G, there much exist w D such that <{u,v,w}> is stongly connected IFG.Which is contradiction to our assumption, i.e. V– D ⊆ V(H) for some component H of G. Further D 5 V(H) ≠ 6 which implied <V – D > is a proper sub graph of H. Hence the proof.

such that and (i, j)-πgβcl{x} ≠ (i, j)-πgβcl{y}then there exists z∈ and (i, j)-πgβcl{x} such that z ∉ (i, j)-πgβcl{y}, Therefore, there exists V∈ and (i, j) πgβ open **set** (X) such that y ∉ V and z ∈ V and hence x ∈V. Thus we get x ∉ (j,i) πgβ-Cl{y} and therefore, x ∈X\(j,i) πgβ-Cl{y}. This implies that (i,j) πgβ-Cl{x}⊂X\(i,j) πgβ-Cl{y}, and therefore, (j,i)- πgβ-Cl{x}∩(j,i)-πgβ-Cl{y}=ϕ

**intuitionistic** **fuzzy** operators applied in contracting a classifier recognizing imbalanced classes, image recognition, image processing, multi-criteria decision making, deriving the similarity measure, sales analysis, new product marketing, medical diagnosis, financial services, solving optimization problems and etc. Baloui Jamkhaneh and Nadarajah [7] considered a new generalized **intuitionistic** **fuzzy** sets (GIF S B ) and introduced some operators over GIF S B . By analogy we shall introduce the some of

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In the eighth place, an **intuitionistic** **fuzzy** possibilistic c - means algorithm to clustering **intuitionistic** **fuzzy** sets is proposed in [21]. The corresponding IFPCM-algorithm is developed by hybridizing concepts of the FPCM clustering method [22], **intuitionistic** **fuzzy** sets and distance measures. The IFPCM-algorithm resolves inherent problems encountered with information regarding membership values of objects to each cluster by generalizing membership and non- membership with hesitancy degree. Moreover, the IFPCM- algorithm is extended in [21] for clustering interval-valued **intuitionistic** **fuzzy** sets leading to interval-valued **intuitionistic** **fuzzy** possibilistic c -means algorithm. So, the IVIFPCM-algorithm has membership and non-membership degrees as intervals.

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The concept of **fuzzy** sets was introduced by Zadeh [11] and later Atanassov [1] generalized this idea to **intuitionistic** **fuzzy** sets using the notion of **fuzzy** sets. On the other hand Coker [4] introduced **intuitionistic** **fuzzy** topological spaces using the notion of **intuitionistic** **fuzzy** sets. In this paper, we introduced **intuitionistic** **fuzzy** 𝜷 generalized continuous mappings and studied some of their basic properties. We arrived at some characterizations of **intuitionistic** **fuzzy** 𝜷 generalized continuous mappings.

This is a novel method of ascertaining the ranking of the Trapezoidal **Intuitionistic** **Fuzzy** Number (TIF) and Triangular **Intuitionistic** **Fuzzy** Number (TrIF) applying the mean of centroids. A comparative study is conducted about the proposed ranking and other methods of ranking for the Trapezoidal as well as Triangular **Intuitionistic** **Fuzzy** Numbers (TIF and TrIF).

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Proposition 3.11. Let , be two Pythagorean **fuzzy** topological spaces and let f :X→Y be a Pythagorean **fuzzy** continuous surjection. If is Pythagorean **fuzzy** connected, , then so is . Proof. On the contrary, suppose that is Pythagorean **fuzzy** disconnected. Then there exist Pythagorean **fuzzy** open sets A≠0 Y ,B≠0 Y in Y such that AUB=1 y , A∩B=0 Y . Now, we see that U=f -1 (A), V= f -1 (B) are

We define and study Browder’s fixed point theorem and relation between an **intuitionistic** **fuzzy** convex normed space and a strong **intuitionistic** **fuzzy** uniformly convex normed space. Also, we give an example to show that uniformly convex normed space does not imply strongly **intuitionistic** **fuzzy** uniformly convex.

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