Fuzzy systems have gained more and more attention from researchers and practitioners of various fields.In such systems, the output represented by a fuzzy set sometimes needs to be transformed into a scalar value, and this task is known as the Defuzzification process.Several analytic methods have been proposed for this problem, but in this paper, the researcher suggests a new approach to the problem of Defuzzification using the weighted metric (weighted distance) between two fuzzy numbers. In this study some preliminary results on properties of such Defuzzification will be reported.

In this article, it is attempted to develop a method for solving an arbitrary general fuzzy linear system by using the weighted metric. Using the distance between two fuzzy numbers, the best and nearest approximation fuzzy symmetric vector can be obtained. Defuzzifi-cation can replace then × n fuzzy linear system with two n × n crisp linear systems. Consequently, to obtain the best approximate solution forn × n fuzzy linear system the researcher must solve twon × n crisp linear systems. First considering the existing and uniqueness of fuzzy solution ton×n fuzzy linear system is done. Then numerical examples are presented to illustrate the proposed model.

In fuzzy decision-making, incomplete information always leads to uncertain and partially reliable judgements. The emergence of fuzzy set theory helps decision-makers in handling uncertainty and vagueness when making judgements. Intuitionistic Fuzzy Numbers (IFN) measure the degree of uncertainty better than classical fuzzy numbers, while Z-numbers help to highlight the reliability of the judgements. Combining these two fuzzy numbers produces Intuitionistic Z-Numbers (IZN). Both restriction and reliability components are characterized by the membership and non-membership functions, exhibiting a degree of uncertainties that arise due to the lack of information when decision-makers are making preferences. Decision information in the form of IZN needs to be defuzzified during the decision-making process before the final preferences can be determined. This paper proposes an Intuitive Multiple Centroid (IMC) Defuzzification of IZN. A novel Multi-Criteria Decision-Making (MCDM) model based on IZN is developed. The proposed MCDM model is implemented in a supplier selection problem for an automobile manufacturing company. An arithmetic averaging operator is used to aggregate the preferences of all decision-makers, and a ranking function based on centroid is used to rank the alternatives. The IZN play the role of representing the uncertainty of decision-makers, which finally determine the ranking of alternatives.

Based on the various components involved in the structure of fuzzy logic controllers, this study provides an indepth examination of the different features and disadvantages of commonly used Defuzzification methods. Then based on this examination, and taking into account the compatible assignment of the logical connectives, inference rules as well as membership functions (MFs), a superior Defuzzification technique is described and justified. It is shown that this technique integrates the Defuzzification problem into the global structure of fuzzy controllers. It also helps the designer achieve his design objectives in a simple and systematic manner. Another Defuzzification strategy, which falls within the general framework of this study, is also given and commented upon. It is shown that this strategy satisfies properties which should be possessed by a desired Defuzzification method, and also shares important features with the former one. Further, a recent Defuzzification-based methodology for the systematic data-driven design of fuzzy controllers is outlined.

A new method for the Defuzzification of fuzzy numbers is developed in this paper. It is well-known, Defuzzification methods allow us to find aggregative crisp numbers or crisp set for fuzzy numbers. But different fuzzy numbers are often converted into one crisp number.In this case the loss of essential information is possible. It may result in inadequate final conclusions, for example, expert estimation problems, prediction problems, etc. Accordingly, the necessity to develop a method for the Defuzzification of fuzzy numbers, allowing us to save their informative properties has arisen. The purpose of this paper is to develop such a method. The method allows us to find aggregative intervals for fuzzy numbers.These intervals are called the Regular weighted intervals. We start with the definition of regular weighted points for fuzzy numbers. The regular weighted interval for fuzzy number is defined as the set of regular weighted points of all unimodal numbers, that belong to this number. Some propositions and examples about regular weighted point and regular weighted intervals properties are offered.

Ranking fuzzy numbers plays a very important role in decision making and some other fuzzy application systems. Many different methods have been proposed to deal with ranking fuzzy numbers. Constructing ranking indexes based on the centroid of fuzzy numbers is an important case. But some weaknesses are found in these indexes. The purpose of this paper is to give a new ranking index to rank various fuzzy numbers effectively. Finally, several numerical examples following the procedure indicate the ranking results to be valid.