scholarly journals The Formal Construction of Fuzzy Numbers

2014 ◽  
Vol 22 (4) ◽  
pp. 321-327 ◽  
Author(s):  
Adam Grabowski
Keyword(s):  
Fuzzy Sets ◽  
Rough Set ◽  
Fuzzy Numbers ◽  
Membership Functions ◽  
The Core ◽  
Current State ◽  
Fuzzy Objects ◽  
Mizar Mathematical Library ◽  
Theory Of Fuzzy Sets ◽  
Formal Construction

Summary In this article, we continue the development of the theory of fuzzy sets [23], started with [14] with the future aim to provide the formalization of fuzzy numbers [8] in terms reflecting the current state of the Mizar Mathematical Library. Note that in order to have more usable approach in [14], we revised that article as well; some of the ideas were described in [12]. As we can actually understand fuzzy sets just as their membership functions (via the equality of membership function and their set-theoretic counterpart), all the calculations are much simpler. To test our newly proposed approach, we give the notions of (normal) triangular and trapezoidal fuzzy sets as the examples of concrete fuzzy objects. Also -cuts, the core of a fuzzy set, and normalized fuzzy sets were defined. Main technical obstacle was to prove continuity of the glued maps, and in fact we did this not through its topological counterpart, but extensively reusing properties of the real line (with loss of generality of the approach, though), because we aim at formalizing fuzzy numbers in our future submissions, as well as merging with rough set approach as introduced in [13] and [11]. Our base for formalization was [9] and [10].

Analysis and Research of Direction Relation Based on Fuzzy Description

2011 ◽  
Vol 393-395 ◽  
pp. 1102-1105
Author(s):  
Yong Shan Liu ◽  
Yan Li
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Membership ◽  
Fuzzy Membership Function ◽  
Membership Degree ◽  
Membership Functions ◽  
Fuzzy Matrix ◽  
Fuzzy Membership Functions ◽  
Fuzzy Objects ◽  
Direction Relations ◽  
Fuzzy Description

A fuzzy membership function was defined for each direction to predict the membership degree that an object pertains to a certain direction. Nine fuzzy membership functions were defined to describe the direction relations between fuzzy objects and crisp objects with corresponding fuzzy sets. Direction relations were described by a 3×3 fuzzy matrix, which was computed by an aggregation operator defined on the nine fuzzy sets. The fuzzy matrices and crisp matrices of direction relations between fuzzy objects and crisp objects were computed respectively, and comparison of fuzzy matrices with crisp ones reveals that the proposed fuzzy approach is more effective than existing crisp method.

scholarly journals Construction of the method for building analytical membership functions in order to apply operations of mathematical analysis in the theory of fuzzy sets

2018 ◽  
Vol 5 (4 (95)) ◽  
pp. 22-29 ◽  
Author(s):  
Leonid Dykhta ◽  
Nataliia Kozub ◽  
Alexander Malcheniuk ◽  
Oleksii Novosadovskyi ◽  
Alexander Trunov ◽  
...  
Keyword(s):  
Fuzzy Sets ◽  
Mathematical Analysis ◽  
Membership Functions ◽  
Theory Of Fuzzy Sets

Situational solution of the spatial location problem

2018 ◽  
Vol 939 (9) ◽  
pp. 45-51
Author(s):  
V.V. Oznamets
Keyword(s):  
Comparative Analysis ◽  
Fuzzy Sets ◽  
Spatial Information ◽  
Spatial Location ◽  
Linguistic Variables ◽  
Membership Functions ◽  
Fuzzy Information ◽  
Reference Information ◽  
Number Of Factors ◽  
Theory Of Fuzzy Sets

The rational allocation of resources is the basis for sustainable development of the territories. In reality, spatial planning often does not have clear information for decision- making. This factor puts the task of allocating resources under fuzzy information. The author suggests such a placement method. The basis for the solution and analysis is a well-known model of the informational situation. The author develops this concept and introduces a new one, of the informational spatial situation. Fuzzy spatial information makes grounds to introduce a new concept of fuzzy information situation. A comparative analysis is used to solve the problem. At the first stage of the solution, an ideal reference information situation is introduced. This model can not be realized in reality completely. Real conditions differ from ideal ones, therefore in practice there is a set of fuzzy information situations, each of which is close to the reference information situation for a number of factors. For the comparative analysis, the theory of fuzzy sets is applied. The author uses the concepts of linguistic variables and membership functions to describe an unclear information situation. Linguistic variables and membership functions determine for the whole set of fuzzy situations are determined. This approach translates the description of real fuzzy situations into the area of linguistic variables. A new description of fuzzy situations makes it possible to perform analysis using the theory of fuzzy sets. The analysis using the theory of fuzzy sets showed how a situation that maximally satisfies the placement requirements is singled out from a given set. The author proves that optimal solutions do not apply to fuzzy analysis, and that the solution obtained using the theory of fuzzy sets is rational.

scholarly journals A Method to Find the Membership Functions of Maximum and Minimum of Fuzzy Numbers

1970 ◽  
Vol 30 ◽  
pp. 89-99
Author(s):  
Shapla Shirin
Keyword(s):  
Membership Function ◽  
Fuzzy Number ◽  
Fuzzy Numbers ◽  
Membership Functions ◽  
New Approach ◽  
The Core ◽  
Decreasing Functions

In this paper, a new approach for computation of membership functions of the maximum and minimum of more than two upper semi-continuous fuzzy numbers has been introduced. This method is also applicable for piece-wise continuous fuzzy numbers or the fuzzy numbers which are only continuous from right or only continuous from left. The core of fuzzy numbers should have a singleton set. Keywords: Continuous fuzzy number; piece-wise continuous fuzzy number; Maximum (MAX) and Minimum (MIN) of fuzzy number; core; α-cut; bounded increasing and bounded decreasing functions; membership function. GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 30 (2010) 89-99  DOI: http://dx.doi.org/10.3329/ganit.v30i0.8506

An Investigation of Rank Reserval when Using Fuzzy Importance Levels in QFD Analysis

2003 ◽  
Vol 10 (02) ◽  
pp. 185-203 ◽  
Author(s):  
Seyed Hossein Iranmanesh ◽  
Mohamad Hossein Salimi
Keyword(s):  
Product Development ◽  
Fuzzy Sets ◽  
Quality Function Deployment ◽  
Fuzzy Numbers ◽  
Importance Rating ◽  
Membership Functions ◽  
Engineering Characteristics ◽  
Design Requirements ◽  
Important Design ◽  
Major Reduction

Identification of important design requirements for product development is critical because it leads to successful products with shorter development time. Quality Function Deployment (QFD) is a tool to help the product development team to systematically determine the design requirements for developing a product with higher customer satisfaction. Therefore, determining the Importance rating of Engineering Characteristics should be robust and reliable. Generally, in QFD charts the relationships between Customer Attributes and Engineering Characteristics can be defined using linguistic variables that have three values: Weak, Medium and Strong. Reversing priority of results (rank reversing) is possible when various scales such as 1-3-5 or 1-3-9 are employed. In this paper, the effect of using fuzzy numbers in rank reverse reduction is studied. For this study a statistical experiment for measuring rank reverse with fuzzy numbers was designed. This experiment was replicated for 7 sets, which included symmetrical and non-symmetrical triangular and trapezoidal fuzzy sets with various degrees of fuzziness. This experiment was extended for cases involving relative importance for Customer Attributes with various fuzzy sets used for weights of importance. The results showed a major reduction in rank reversal using symmetrical membership functions. Furthermore, results did not depend on system fuzziness, and there were not any major differences between the use of triangular and trapezoidal membership functions.

DEVELOPMENT OF VALUATION METHOD OF THE STATUS OF BASE STATIONS FOR DETERMINING THE NEED OF THEIR UPGRADING

2017 ◽  
pp. 60-70
Author(s):  
Natalya Ivanovna Shaposhnikova ◽  
Alexander Aleksandrovich Sorokin
Keyword(s):  
Decision Making ◽  
Fuzzy Sets ◽  
Cellular Network ◽  
New Technologies ◽  
Base Station ◽  
Base Stations ◽  
Valuation Method ◽  
The Status ◽  
Telecommunications Services ◽  
Theory Of Fuzzy Sets

The article consideres the problems of determining the need to modernize the base stations of the cellular network based on the mathematical apparatus of the theory of fuzzy sets. To improve the quality of telecommunications services the operators should send significant funding for upgrading the equipment of base stations. Modernization can improve and extend the functions of base stations to provide cellular communication, increase the reliability of the base station in operation and the functionality of its individual elements, and reduce the cost of maintenance and repair when working on a cellular network. The complexity in collecting information about the equipment condition is determined by a large number of factors that affect its operation, as well as the imperfection of obtaining and processing the information received. For a comprehensive assessment of the need for modernization, it is necessary to take into account a number of indicators. In the structure of indicators of the need for modernization, there were introduced the parameters reflecting both the degree of aging and obsolescence(the technical gap and the backlog in connection with the emergence of new technologies and standards). In the process of a problem solving, the basic stages of decision-making on modernization have been allocated. Decision-making on the need for modernization is based not only on measuring information that takes into account the decision-makers, but also on linguistic and verbal information. Therefore, to determine the need for upgrading the base stations, the theory of fuzzy sets is used, with the help of which experts can be attracted to this issue. They will be able to formulate additional fuzzy judgments that help to take into account not only measuring characteristics, but also poorly formalized fuzzy information. To do this, the main indicators of the modernization need have been defined, and fuzzy estimates of the need for modernization for all indicators and a set of indicators reflecting the need for upgrading the base stations have been formulated.

A Review on Enteric Coated Pellets Composed of Core Pellets Prepared by Extrusion-Spheronization

2019 ◽  
Vol 13 (2) ◽  
pp. 83-90 ◽  
Author(s):  
Hetal Patel ◽  
Mukesh Gohel
Keyword(s):  
Dosage Form ◽  
Extended Release ◽  
Rapid Onset ◽  
Enteric Coating ◽  
Onset Of Action ◽  
The Core ◽  
Current State ◽  
Enteric Coated ◽  
Extrusion Spheronization ◽  
Acid Labile

Enteric coated dosage form bypasses the stomach and releases the drug into the small intestine. Advantages of enteric coated pellets in comparison with enteric coated tablets are a) Pellets provide rapid onset of action and faster drug release due to the smaller size than tablets and b) Pellets exhibit less residence time of acid-labile drugs in the stomach compared to tablets. Dosage form coat can be damaged by longer resistance time in the stomach. The present review summarizes the current state of enteric coated pellets where core pellets are prepared by extrusion-spheronization technique and the enteric coating is applied in a fluidized bed processor. Two approaches are involved in the preparation of core pellets. In the first approach, a mixture of drug and excipient(s)/co-processed excipient is passed through extruders to prepare core pellets. In the second approach, excipient core pellets are prepared by extrusion technique and the drug is layered onto it before the enteric coating. The excipients present in the core pellets decide immediate or extended release of drug in the intestine. The coprocessed excipient pellets provide less batch variability and provide a platform for layering of many drugs before enteric coating. Some patents included enteric coating pellets [CN105456223 (A), CN105596310 (A), CN105616371 (A), CN105663095 (A), CN101611766B, CN106511862 (A), CN106668018 (A), CN106727381 (A), CN106924222 (A), TW200624127 (A), US 2017/0165248A1, US 2017/0224720A1] are discussed.

ERP selection using picture fuzzy CODAS method

2021 ◽  
pp. 1-11
Author(s):  
Hacer Yumurtacı Aydoğmuş ◽  
Eren Kamber ◽  
Cengiz Kahraman
Keyword(s):  
Decision Analysis ◽  
Fuzzy Sets ◽  
Fuzzy Numbers ◽  
Ideal Solution ◽  
Application Area ◽  
Erp System ◽  
Mcdm Method ◽  
Negative Ideal Solution ◽  
Picture Fuzzy Sets ◽  
System Selection

The purpose of this study is to develop an extension of CODAS method using picture fuzzy sets. In this respect, a new methodology is introduced to figure out how picture fuzzy numbers can be applied to CODAS method. COmbinative Distance-based Assessment (CODAS) is a new MCDM method proposed by Ghorabaee et al. Picture fuzzy sets (PFSs) are a new extension of ordinary fuzzy sets for representing human’s judgments having possibility more than two answers such as yes, no, refusal and neutral. Compared with other studies, the proposed method integrates multi-criteria decision analysis with picture fuzzy uncertainty based on Euclidean and Taxicab distances and negative ideal solution. ERP system selection problem is handled as the application area of the developed method, picture fuzzy CODAS. Results indicate that the new proposed method finds meaningful rankings through picture fuzzy sets. Comparative analyzes show that the presented method gives successful and robust results for the solutions of MCDM problems under fuzziness.

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