Quasilineability and topological properties of the set of fuzzy numbers

In this paper we show that the cardinality of the set of fuzzy numbers coincides with that of the real numbers. We also show that the set of triangular fuzzy numbers is nowhere dense within the set of fuzzy numbers (with a suitable distance) and that the set of real numbers is also nowhere dense within the set of triangular fuzzy numbers. In addition, we introduce the concept of quasilineability and study the set of bounded fuzzy number sequences that do not have a lower limit and that of monotonic decreasing, bounded with respect a partial ordering and not convergent

The Best Approximation of Generalized Fuzzy Numbers Based on Scaled Metric

The ongoing study has been vehemently allocated to propound an ameliorated α-weighted generalized approximation of an arbitrary fuzzy number. This method sets out to lessen the distance between the original fuzzy set and its approximation. In an effort to elaborate the study, formulas are designed for computing the ameliorated approximation by using a multitude of examples. The numerical samples will be exemplified to illuminate the improvement of the nearest triangular approximation (Abbasbandy et al., Triangular approximation of fuzzy numbers using α-weighted valuations, Soft Computing, 2019). A variety of features of the ameliorated approximation are then proved. © 2022 Tofigh Allahviranloo et al

An Extended VIKOR Method for Multiple Criteria Group Decision Making with Triangular Fuzzy Neutrosophic Numbers

In this article, we combine the original VIKOR model with a triangular fuzzy neutrosophic set to propose the triangular fuzzy neutrosophic VIKOR method. In the extended method, we use the triangular fuzzy neutrosophic numbers (TFNNs) to present the criteria values in multiple criteria group decision making (MCGDM) problems. Firstly, we summarily introduce the fundamental concepts, operation formulas and distance calculating method of TFNNs. Then we review some aggregation operators of TFNNs. Thereafter, we extend the original VIKOR model to the triangular fuzzy neutrosophic environment and introduce the calculating steps of the TFNNs VIKOR method, our proposed method which is more reasonable and scientific for considering the conflicting criteria. Furthermore, a numerical example for potential evaluation of emerging technology commercialization is presented to illustrate the new method, and some comparisons are also conducted to further illustrate advantages of the new method

Using the Fuzzy Grey Relational Analysis Method in Wastewater Treatment Process Selection

Due to the variety of treatment processes, the decision to choose the best treatment process is difficult. This paper describes a fuzzy grey relational analysis (GRA) method for selection of the optimal wastewater treatment process. The rating of all alternatives and the weight of each criterion is described by linguistic variables, which can be expressed in triangular fuzzy numbers. Then, a vertex method is used to calculate the distance between two triangular fuzzy numbers. According to the concept of the GRA, a fuzzy relative relational degree is defined to determine the ranking order of all alternatives by calculating the degree of fuzzy grey relational coefficient to both the fuzzy positive ideal solution (FPIS) and fuzzy negative ideal solution (FNIS) simultaneously. Furthermore, a case study is carried out and solved by both methods (i.e., GRA and fuzzy GRA) to show the feasibility and effectiveness of the proposed method. In the case study, five anaerobic wastewater treatment alternatives are evaluated and compared against technical, economic, environmental and administrative criteria and their sub-criteria. Finally, the related results of ranking alternatives from two methods are compared with each other's. By using both Fuzzy GRA and GRA, ABR process has been selected as the first priority and the best anaerobic process. The frequency count assessment of the Iran's industrial parks' WWTPs which have used this method and their performance, proved the priority of this method

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

The dynamics of consensus in group decision making: investigating the pairwise interactions between fuzzy preferences.

In this paper we present an overview of the soft consensus model in group decision making and we investigate the dynamical patterns generated by the fundamental pairwise preference interactions on which the model is based. The dynamical mechanism of the soft consensus model is driven by the minimization of a cost function combining a collective measure of dissensus with an individual mechanism of opinion changing aversion. The dissensus measure plays a key role in the model and induces a network of pairwise interactions between the individual preferences. The structure of fuzzy relations is present at both the individual and the collective levels of description of the soft consensus model: pairwise preference intensities between alternatives at the individual level, and pairwise interaction coefficients between decision makers at the collective level. The collective measure of dissensus is based on non linear scaling functions of the linguistic quantifier type and expresses the degree to which most of the decision makers disagree with respect to their preferences regarding the most relevant alternatives. The graded notion of consensus underlying the dissensus measure is central to the dynamical unfolding of the model. The original formulation of the soft consensus model in terms of standard numerical preferences has been recently extended in order to allow decision makers to express their preferences by means of triangular fuzzy numbers. An appropriate notion of distance between triangular fuzzy numbers has been chosen for the construction of the collective dissensus measure. In the extended formulation of the soft consensus model the extra degrees of freedom associated with the triangular fuzzy preferences, combined with non linear nature of the pairwise preference interactions, generate various interesting and suggestive dynamical patterns. In the present paper we investigate these dynamical patterns which are illustrated by means of a number of computer simulations.

Revisiting the interval and fuzzy topsis methods: Is euclidean distance a suitable tool to measure the differences between fuzzy numbers?

Euclidean distance (ED) calculates the distance between n-coordinate points that n equals the dimension of the space these points are located. Some studies extended its application to measure the difference between fuzzy numbers (FNs).This study shows that this extension is not logical because although an n-coordinate point and an FN are denoted the same, they are conceptually different. An FN is defined by n components; however, n is not equal to the dimension of the space where the FN is located. This study illustrates this misapplication and shows that the ED between FNs does not necessarily reflect their difference. We also revisit triangular and trapezoidal fuzzy TOPSIS methods to avoid this misapplication. For this purpose, we first defuzzify the FNs using the center of gravity (COG) method and then apply the ED to measure the difference between crisp values. We use an example to illustrate that the existing fuzzy TOPSIS methods assign inaccurate weights to alternatives and may even rank them incorrectly

Supplier selection in government organizations based on fuzzy evaluation approach / Fairuz Shohaimay, Nazirah Ramli and Siti Rosiah Mohamed

Asset purchasing is one of the main responsibilities of Information Technology (IT) department which involves selecting the best supplier. The IT department needs to consider many criteria such that the chosen supplier offers the best product and services at acceptable price, without compromising on the quality and standards of goods. Thus, supplier selection is a multi-criteria decision making (MCDM) problem. In real practice, supplier selection is a complex decision making task. The evaluation process of suppliers relies heavily on previous experiences and human judgments which are vague and uncertain. It is relatively difficult for decision makers to provide exact numerical values for the criteria. Hence, fuzzy set theory was introduced to deal with uncertainties and imprecision in linguistic terms values in decision making processes. Although linguistic terms are used in many fuzzy MCDM models, most of the corresponding fuzzy numbers were fixed and taken from the previous literatures. These fuzzy numbers may not necessarily reflect actual respondents' opinions. Therefore, a two-phased fuzzy evaluation technique is proposed in evaluating and selecting suppliers. Firstly, triangular fuzzy numbers were built based on respondents' opinions. Fuzzy evaluation method is used to evaluate the suppliers based on three main criteria and nine sub-criteria. Finally, the suppliers were ranked using distance minimization method