On improving trapezoidal and triangular approximations of fuzzy numbers

AbstractRecently, various researchers have proved that the approximations of fuzzy numbers may fail to be fuzzy numbers, such as the trapezoidal approximations of fuzzy numbers. In this paper, we show by an example that the weighted triangular approximation of fuzzy numbers, proposed by Zeng and Li, may lead to the same result. For filling the gap, improvements of trapezoidal and triangular approximations are proposed. The formulas for computing the two improved approximations are provided. Some properties of the two improved approximations are also proved

A new approach for trapezoidal approximation of fuzzy numbers using WABL distance

In this paper, we present a new approach to obtain trapezoidal approximation of fuzzy numbers with respect to weighted distance proposed by Nasibov [5] which the main property of this metric is flexibility in the decision maker's choice. Also, we prove some properties of the proposed method such as translation invariance, scale invariance and identity. Finally, we illustrate the efficiency of proposed method by solving some numerical examples

Operation properties and δ-equalities of complex fuzzy sets

A complex fuzzy set is a fuzzy set whose membership function takes values in the unit circle in the complex plane. This paper investigates various operation properties and proposes a distance measure for complex fuzzy sets. The distance of two complex fuzzy sets measures the difference between the grades of two complex fuzzy sets as well as that between the phases of the two complex fuzzy sets. This distance measure is then used to define equalities of complex fuzzy sets which coincide with those of fuzzy sets already defined in the literature if complex fuzzy sets reduce to real-valued fuzzy sets. Two complex fuzzy sets are said to be d-equal if the distance between them is less than 1 d. This paper shows how various operations between complex fuzzy sets affect given δ-equalities of complex fuzzy sets. An example application of signal detection demonstrates the utility of the concept of δ-equalities of complex fuzzy sets in practice

Nearest symmetric trapezoidal approximation of fuzzy numbers

Abstract Many authors analyzed triangular and trapezoidal approximation of fuzzy numbers. But, to best of our knowledge, there is no method for symmetric trapezoidal fuzzy number approximation of fuzzy numbers. So, in this paper, we try to convert any fuzzy number into symmetric trapezoidal fuzzy number by using metric distance. This approximation helps us to avoid the computational complexity in the process of decision making problems. Moreover, we investigate some reasonable properties of this approximation. An application of this new method is also provided

A review of fuzzy AHP methods for decision-making with subjective judgements

Analytic Hierarchy Process (AHP) is a broadly applied multi-criteria decision-making method to determine the weights of criteria and priorities of alternatives in a structured manner based on pairwise comparison. As subjective judgments during comparison might be imprecise, fuzzy sets have been combined with AHP. This is referred to as fuzzy AHP or FAHP. An increasing amount of papers are published which describe different ways to derive the weights/priorities from a fuzzy comparison matrix, but seldomly set out the relative benefits of each approach so that the choice of the approach seems arbitrary. A review of various fuzzy AHP techniques is required to guide both academic and industrial experts to choose suitable techniques for a specific practical context. This paper reviews the literature published since 2008 where fuzzy AHP is applied to decision-making problems in industry, particularly the various selection problems. The techniques are categorised by the four aspects of developing a fuzzy AHP model: (i) representation of the relative importance for pairwise comparison, (ii) aggregation of fuzzy sets for group decisions and weights/priorities, (iii) defuzzification of a fuzzy set to a crisp value for final comparison, and (iv) consistency measurement of the judgements. These techniques are discussed in terms of their underlying principles, origins, strengths and weakness. Summary tables and specification charts are provided to guide the selection of suitable techniques. Tips for building a fuzzy AHP model are also included and six open questions are posed for future work