However, it is crucial to define meaningful coveringsbased on the given data, often presented in an information table. As the results presented in this work were achieved from a computational approach to fuzzy covering-based rough sets, i.e., the focus of the research is on the construction of different rough set models, there is still a gap in the research field on rough sets of a conceptual and semantical understanding for (fuzzy) covering-based rough set models, as recently discussed by Yao . In the latter approach, the focus is rather on the insight of concepts instead of providing computationally efficient algorithms. Hence, the study of (fuzzy) neighborhoodoperatorsbased on coverings from a semantically sound approach will be a major topic in our future research directives as it will provide more motivation for the use of the different fuzzyneighborhoodoperatorsbased on fuzzycoverings.
called the lower approximation and the upper approximation. However, the equivalence relation appears to be a stringent condition that may limit the applicability of Pawlak’s rough set model. Hence, many extensions have been made in recent years by replacing equivalence relation or partition by notions such as binary rela- tions [13,34,39] , neighborhood systems and Boolean algebras [3,37,40] , and coverings of the universe of dis- course [4,7,29] . Based on the notion of covering, Pomykala [29,30] , in particular, obtained two pairs of dual approximation operators. Yao [40,41] further examined these approximation operators by the concepts of neighborhood and granularity. Such undertaking has stimulated more research in this area [19,44–47] .
(v) Predicting Trust and Distrust among Individuals Various orders have taken a gander at different issues identified with trust. The first errand was the EigenTrust estimation that expects to lessen the number of inauthentic record downloads in a P2P system. Guha et al proposed systems for engendering of trust and distrust, each of which is suitable in specific circumstances. PowerTrust is a trust proposal framework that totals the positive and negative feelings between the clients into the neighborhood trust scores, comparably to EigenTrust. Other work that studies an informal community with positive and negative feelings is introduced. DuBois et al introduced a paper for foreseeing trust and distrust in light of way likelihood in arbitrary diagrams. Kim et al have additionally proposed a technique for anticipating trust and distrust of clients in online networking sharing groups. Ortega et al proposed a novel framework planned to spread both positive and negative assessments of the clients through a system, in such way that the assessments from every client about others impact their worldwide trust score.
concepts and roles as fuzzy sets and fuzzy relations respectively. Fuzzy sets and fuzzy logics were born to deal with the prob- lem of approximate reasoning [2,3] . Their ﬁrst developments were characterized by the applications that gave rise to various semantic approaches to this problem. In recent times, formal logic systems have been developed for such semantics, and the logics based on triangular norms (t-norms) have become a central paradigm in fuzzy logic. The development of that ﬁeld is intimately linked to the book Metamathematics of Fuzzy Logics  , published in 1998, where Hájek shows the connection of fuzzy logic systems with many-valued residuated lattices based on continuous t-norms. He proposes a Hilbert-style calculus called Basic fuzzy Logic (BL) and conjectures that this logic is sound and complete with respect to the structures deﬁned in the unit real interval ½0; 1 by continuous t-norms and their residua. The conjecture was proved in  . In  Esteva and Godo introduced the logic MTL, a weakening of BL, which is proved in  to be the logic of left continuous t-norms and their re- sidua. Since then, the ﬁeld of t-norm basedfuzzy logics has grown very quickly and it is the subject matter of intensive re- search (see http://www.mathfuzzlog.org for an exhaustive list of works and researchers in this area).
Comparison of fuzzy numbers is an indispensable part of most systems using such numbers. To this end, many researchers active in the Fuzzy Theory domain have tried to make fuzzy numbers comparable. Some authors have approximated a fuzzy number by a single crisp number. This method which is called ranking suffers from loss of some useful information.
The concept of bipolar intuitionistic M fuzzy prime group and anti M fuzzy prime group are defined and new algebraic structure of a bipolar intuitionistic M fuzzy prime subgroup of a M fuzzy prime group and anti M fuzzy prime group are created and some related properties and some operations of Interior operators are investigated. The purpose of the study is to implement fuzzy set theory and group theory of bipolar intuitionistic M fuzzy prime subgroup of a M fuzzy prime group and anti M fuzzy prime group. We hope that our results can also be extended to other algebraic system.
Membership functions and rules castoff in the projected technique 1. if (input1 is mf1) and (input2 is mf2) then (output1 is mf1) 2. if (input1 is mf2) and (input2 is mf2 then (output1 is mf2) 3. if (input1 is mf2) and (input2 is mf2) then (output1 is mf2) 4. if (input1 is mf3) or (input2 is mf2) then (output1 is mf3) 5. if (input1 is mf1) and (input2 is mf3) then (output1 is mf1) 6. if (input1 is mf3) or (input2 is mf3) then (output1 is mf2) D. FUZZY RULES
Abstract:In this paper we present a new class of operatorsbased on Trapezoid Fuzzy numbers such as Trapezoid Fuzzy Prioritized Weighted Mean (TFPWM) operator, Trapezoid Fuzzy Prioritized Weighted Geometric Average (TFPWG), Trapezoid Fuzzy Prioritized Weighted Harmonic Average(TFPWH) operators to solve multiple attribute group decision making (MAGDM) problems. Unlike other (MAGDM) techniques here the attributes and experts are at different priority levels. Some desirable properties of TFPWM, TFPWG and TFPWH operator are investigated. Finally, we give a numerical example to illustrate the application of these operators to group decision making problems with trapezoidal fuzzy information.
Definition 2.1. A fuzzy subset of a nonempty set V is mapping σ: V → [0, 1] and A fuzzy relation on V is fuzzy subset of V x V. A fuzzy graph is a pair G:(σ,µ ) where σ is a fuzzy subset of a set V and µ is a fuzzy relation on σ , where µ (u,v) ≤ σ (u) Λσ (v) ∀ u , v ∈V
| | . Here, the degree of objectivity will
be higher when more knowledge support and agree with 𝑘 . On the other hand, the degree of individuality will be higher when less knowledge supports and agrees with 𝑘 . Clearly, it can be proved that 𝜑 (𝑘 , 𝑡) = 𝜗 (𝑘 , 𝑡) = 1 ⇔ |𝐾| = 1. In addition, as given in Definition 5 and 6, the degrees of objectivity and individuality rely on discrete value as a result of using crisp coverings. The most important thing is that both 𝜑 (𝑘 , 𝑡) and 𝜗 (𝑘 , 𝑡) are dynamically changed over time 𝑡 . It represents the real-world application that the objectivity of someone will always be changed over time. (b) show a simple equation representing the relation between 𝜑 (𝑘 , 𝑡) and 𝜗 (𝑘 , 𝑡). It can be verified that 𝜑 (𝑘 , 𝑡) and 𝜗 (𝑘 , 𝑡) satisfy some properties such as:
Abstract. In this paper, we introduce the concepts of (L, M)-fuzzy soft quasi-coincident neighborhood spaces and study their properties, where L be a completely distributive lattice with 0 and 1 elements and M be a strictly two-sided, commutative quantale lattice. Also, the relationships between these concepts were investigated. Fur- thermore, a characterization of LFS-continuous and LSN-mappings were given.
Aggregation operators have a structural feature that combines a nite number of data points with a single value. Aggregation operators are important tools of information fusion in the decision making problem [1- 3], neural networks , fuzzy logic controller , and many other elds. Depending on the dierent nature of information and their relations, various types of aggre- gation operators have been developed. Among several aggregation operators, the Arithmetic Mean (AM), the Geometric Mean (GM), and the Harmonic Mean (HM) operators are fundamental operators. Based on these basic operators, dierent types of operators are developed and extended to several environments. Yager  proposed the Ordered Weighted Averaging (OWA) operator where the input arguments are or- ganized in order. The Ordered Weighted Geometric
the view of the autocorrelation Box-Jenkins methodology has been published . Nevertheless, the Takagi-Sugeno rules are rather regression-based than linguistic-based in comparison with (6). Analogously, distinct neuro-fuzzy approaches, which are on the border of neural networks, Takagi-Sugeno models and evolving fuzzy systems, are very often and successfully used [9, 10]. But well tuned Gaussian fuzzy sets with a cen- troid at node 5.6989 and a width parameter equal to 2.8893 (see ) constructed as a product of an employed optimiza- tion technique are undoubtably far from the interpretability of systems using models of fragments of natural language. Therefore, these approaches, no matter how effective and pow- erful, are closer to standard regression methods.
Additionally, we will investigate the combination of the well-known aggregation operator - Ordinal Weighted Average (OWA) with fuzzy DRSA. OWA has been shown to improve IRSA in handling outliers and noisy data . OWA makes approximations (and thus also machine learning algorithms that use them) more robust to small changes in the data. This goes at the expense of some desirable properties. However, for IRSA at least, it was shown that the OWA ex- tension provides the best trade-off between theoretical properties and experimental performance among noise tolerant models . However, we would like to see if a similar performance may be achieved with fuzzy DRSA.
multi-layer or single layer decision processes of which Analytic Hierarchy Process (AHP), Analytic Network Process (ANP), Decision Making Trial and Evaluation Laboratory model (DEMATEL), Technique for Preference Similarity to Ideal Solution (TOPSIS), and Serbian model of VIse Kriterijumska Optimizacija Kompromisno Resenje (VIKOR) are the most known and the most frequently applied methods . AHP is a mathematical approach proposed by Saaty for ranking some alternatives (choices) based on some criteria. In this method, alternatives are compared and scored quantitatively pairwise according to each criterion. Criteria are also ranked among each other and by implementation of some mathematic approach, it is possible to sort the choices by their importance as well as to evaluate the consistency of pairwise comparisons . Analytic Network Process (ANP) was also developed by Saaty which was based on AHP to cope with its restrictions as well as to deal with the problems where interrelationships and feedback among the criteria or alternatives are taken into consideration [18, 19]. There was some deficiency in ANP for which DEMATEL method was introduced by Yang and Tzeng to overcome the its limitations . DEMATEL procedure provides meaningful structural relationships between the criteria through a causal diagram. So it is frequently used in strategic analysis, performance evaluation, cost of quality model development and brand marketing [19-21]. TOPSIS method was introduced by Chen and Hwang. TOPSIS is a multiple criteria method to identify the ideal solutions from a finite set of alternatives by measuring the shortest and the longest distance of the alternatives from the positive and negative criteria, respectively . VIKOR method developed by Opricovic and Tzeng focuses on the selection of the most appropriate alternative by ranking the alternatives among conflicting
The studies done by Chen and Hwang , and Negi  are employed for establishing a prototype fuzzy TOPSIS. Many authors as such as Chen ; Chen et al. ; Chen & Hwang ; Chen & Tzeng ; Liang ; Wang & Elhag ; Wang & Lee ; Wang, Luo, & Hua ; Yeh, Deng, & Chang ; and Yeh & Deng  have contributed new materials on the development, extensions and applications of TOPSIS since its early development in 1981. Its general extension for group decision making problems under fuzzy environment was published by Chen . In 2007, Kahraman [35, 36, 37, and 38] and his research team proposed a hierarchical fuzzy TOPSIS method that has ability to consider the hierarchy among the attributes and alternatives. This method provides greater superiority to classical fuzzy TOPSIS methods (Kahraman, et al. . Other researchers have employed TOPSIS and applied that to areas as such as company financial ratios comparison (Deng et al., facility location selection (Chen and Tzeng , assessment of service quality in airline industry (Tsaur et al. , manufacturing plant location analysis (Yoon and Hwang , Robot selection (Parkan and Wu ), and TQM Consultant selection (Saremin et al. et al.  to mention some. Chu and Lin have proposed a fuzzy TOPSIS approach for robot selection where the ratings of various alternatives under different subjective attributes and the importance weights of all attributes are assessed in linguistic terms represented by fuzzy numbers. They have presented an integrated fuzzy
Definition 12.  Let X be a vector space over R. Assume the mappings L; R : [0, 1] × [0, 1] → [0, 1] are symmetric and non-decreasing in both arguments, and that L(0; 0) = 0 and R(1; 1) = 1. Let || . || : X → F + (R). The quadruple (X; || . || ; L; R) is called a fuzzy normed linear space (briefly, FNS) with the fuzzy norm || . || , if the following conditions are satisfied:
As different advanced forms of FSs came one after another, scientist started to merge two kinds of fuzzy information in a single set. The idea was quite useful and some very interesting extensions of FSs have been defined. These extensions include intuitionistic hesitant fuzzy sets (IHFSs), inter-valued hesitant fuzzy sets (IVHFSs) and bipolar-valued hesitant fuzzy sets (BVHFSs).The idea of merging different kind of fuzzy sets was quite useful and very shortly some new advanced forms of FSs have been established which are inter- valued intuitionistic hesitant fuzzy sets (IVIHFSs), cubic hesitant fuzzy sets (CHFSs) and bipolar-valued hesitant fuzzy sets (BPVHFSs).
In our treatise, we successfully apply aggregation operators of BPVHFSs in a DM problem. The results clearly indicate that either we use BPVHF hybrid averaging operators or BPVHF hybrid geometric operators, we get the same results. Hence these two aggregation operation can be very useful in DM especially in two-sided DM. Future research will involve the generalization to bipolar-valued hesitant neutrosophic information and its aggregation operators.
Available Online at www.ijpret.com 164 search engine, called Fuzzy-Go. First, a fuzzy ontology is constructed by using fuzzy logic to capture the similarities of terms in the ontology, which offering appropriate semantic distances between terms to accomplish the semantic search of keywords. The Fuzzy- Go search engine can thus automatically retrieve web pages that contain synonyms or terms similar to keywords. Second, users can input multiple keywords with different degrees of importance based on their needs. The totally satisfactory degree of keywords can be aggregated based on their degrees of importance and degrees of satisfaction. Third, the domain classification of web pages offers users to select the appropriate domain for searching web pages, which excludes web pages in the inappropriate domains to reduce the search space and to improve the search results.Fig.1 show the working of search engine.