New Order on Type 2 Fuzzy Numbers

Characterizations of endograph metric and Γ-convergence on fuzzy sets

Any sequence claimed to be random, real numbers or fuzzy numbers, must be tested for randomness. We first test our sequence of fuzzy random numbers, obtained from Sobol quasi-random numbers, for randomness using a run test and then a frequency test. We identified two types of triangular shaped fuzzy numbers from Chapter 4: (1) quadratic fuzzy numbers generated from 7-tuples; and (2) quadratic Bezier generated fuzzy numbers (QBGFNs). For reasons given there we direct our attention to QBGFNs. A run test depends on what definition of ≤ between fuzzy numbers we are using. So we do the run test three times on the Bezier fuzzy numbers; first using Buckley’s Method of ≤ (Section 2.6.1) next using Kerre’s Method of ≤ (Section 2.6.2) and lastly using Chen’s Method of ≤ (Section 2.6.3). We must also test our sequence of random fuzzy vectors for randomness. We have seen that sequences of random numbers can pass randomness tests but when they are used to build vectors the resulting sequence of vectors can fail randomness tests (Chapter 3). We will test our sequences of random vectors, whose components are all TFNs, for randomness using a chi-square test.

Entropy variation on the fuzzy numbers with arithmetic operations

Since fuzzy numbers represent vague numeric values, they may overlap with each other. So, the comparison between fuzzy numbers may be vague. For this reason, in a set of fuzzy numbers, the rank of a fuzzy number and the k-th largest fuzzy number are vague. In this paper, to determine the rank of a fuzzy number and the k-th largest fuzzy number in a set of fuzzy numbers, a fuzzy set based method is proposed. The proposed method uses a given greater-than relation defined on the set of fuzzy numbers. The rank of a fuzzy number is described with a fuzzy set of the ranks that the fuzzy number can take, and the k-th largest fuzzy number with a fuzzy set of fuzzy numbers which can be k-th ranked.

Approximations of fuzzy numbers by step type fuzzy numbers

Granular structure plays a very important role in the model construction, theoretical analysis and algorithm design of a granular computing method. The granular structures of classical rough sets and fuzzy rough sets have been proven to be clear. In classical rough set theory, equivalence classes are basic granules, and the lower and upper approximations of a set can be computed by those basic granules. In the theory of fuzzy rough set, granular fuzzy sets can be used to describe the lower and upper approximations of a fuzzy set. This paper discusses the granular structure of type-2 fuzzy rough sets over two universes. Definitions of type-2 fuzzy rough sets over two universes are given based on a wavy-slice representation of type-2 fuzzy sets. Two granular type-2 fuzzy sets are deduced and then proven to be basic granules of type-2 fuzzy rough sets over two universes. Then, the properties of lower and upper approximation operators and these two granular type-2 fuzzy sets are investigated. At last, several examples are given to show the applications of type-2 fuzzy rough sets over two universes.

Strengths, weaknesses, opportunities and threats (SWOT) matrix is one of the most used tools for strategic planning, specially in the stage of extracting strategies. While the use of SWOT analysis is quite common and popular, it still continuous to have certain structural problems such that: the lack of considering uncertain and boolean (two sided) factors, lack of prioritization of the factors and too many extractable strategies. In this paper, we deal with the analytic hierarchy process (AHP) to find weights of SWOT groups and weights of sub-factors within each groups; and we make use of the technique for order preference by similarity to ideal solution (TOPSIS) methodology to determine strategies priority. In addition, in order to deal with uncertainty and vagueness, we incorporate axiomatic fuzzy set (AFS) theory instead of traditional fuzzy set theory. AHP and TOPSIS are applied to the light of AFS theory. The advantages of dealing with AFS theory in the proposed approach are that: it can directly be applied to the data, the features of the data-sets can be mixed; however, the traditional fuzzy set theory deals with fuzzy numbers and it’s clear that different fuzzy numbers will lead to different results; hence the inconsistency of previous methodologies, described in the literature, copes with the methodology proposed in this paper.

In this paper, a new hypothesis of fuzzy number has been proposed which is more precise and direct. This new proposed approach is considered as an equivalence class on set of real numbers R with its algebraic structure and its properties along with theoretical study and computational results. Newly defined hypothesis provides a well-structured summary that offers both a deeper knowledge about the theory of fuzzy numbers and an extensive view on its algebra. We defined field of newly defined fuzzy numbers which opens new era in future for fuzzy mathematics. It is shown that, by using newly defined fuzzy number and its membership function, we are able to solve fuzzy equations in an uncertain environment. We have illustrated solution of fuzzy linear and quadratic equations using the defined new fuzzy number. This can be extended to higher order polynomial equations in future. The linear fuzzy equations have numerous applications in science and engineering. We may develop some iterative methods for system of fuzzy linear equations in a very simple and ordinary way by using this new methodology. This is an innovative and purposefulness study of fuzzy numbers along with replacement of this newly defined fuzzy number with ordinary fuzzy number.

Recently, fuzzy linguistic variables have gained a great deal of attention from researchers in many decision-making problems. In this problems, type-2 fuzzy sets have been used to better cover linguistic data uncertainty. However, many of research works in this regard have been performed in type-2 fuzzy domain and in static mode. Since the decision making problems in the real world usually fluctuate over time, so it need to use decision making models in multi period of time. In the present research, a Multi-Period (Dynamic) Multi-Attribute Group Decision-Making (MPMADM) method is presented based on type-2 fuzzy sets where decision-making attributes are first expressed in linguistic terms and then incorporated, as interval type-2 fuzzy numbers, into problem solving where a new integrating operator called Multi-Period Trapezoidal Interval Type-2 Fuzzy Number Weighted Arithmetic averaging (MPTIT2FNWA) is defined on type-2 interval fuzzy numbers to integrate decision-making information in multiple periods of time. Once finished with explaining the proposed method, a numerical example is given to evaluate the proposed method in terms of effectiveness and applicability, with the results compared to those of other methods.

Some remarks on the lattice of fuzzy intervals

The expense of characterizing aquifer spatial variability often results in a lack of available or realistically obtainable direct measurement data for ground‐water system simulation. As a consequence ground‐water models that are able to utilize imprecise or “soft” information need to be developed. In this paper, a methodology based on fuzzy set theory is developed to incorporate imprecise data into transient ground‐water flow simulation. The imprecise model parameters may come from indirect measurements, expert judgment, and subjective interpretation of available information. Fuzzy numbers are used to represent imprecise parameters. They are also used as a measure of the uncertainty associated with the hydraulic head due to the imprecision of input data. A fuzzy ground‐water flow model is developed by linking the finite‐difference method with fuzzy number representations. Fuzzy number operations (α‐level cuts) are used to solve the resulting fuzzy ground‐water flow model and are extended to consider the dependencies among hydraulic head coefficients. With the fuzzy number inputs, the transient fuzzy ground‐water flow model provides a direct measure of hydraulic head uncertainties in the time domain. The model outputs can be used as the inputs for subsequent risk analysis and decision‐making processes. The fuzzy modeling technique can handle imprecise information directly without generating a large number of realizations. It is also flexible as it can handle different types of membership functions describing fuzzy input parameters. The methodology can be used to combine data with different levels of quality into ground‐water flow models and provides a realistic method to handle parameter imprecision, especially expert judgment and subjective information. A numerical model based on the methodology was tested against the Theis analytical solution for a homogeneous aquifer with pumping. The tested model was also applied to two different heterogeneous flow fields to demonstrate the methodology.

There are many results on the extended operations of two fuzzy numbers based on the Zadeh's extension principle. For the calculation, we have to use existing operations between two <TEX>${\alpha}$</TEX>-cuts. In this paper, we define parametric operations between two <TEX>${\alpha}$</TEX>-cuts which are different from the existing operations. But we have the same results as the extended operations of Zadeh's principle.

The concept of q-rung orthopair hesitant fuzzy set represents an advancement and extension of hesitant fuzzy sets, encompassing both fuzzy sets and q-rung orthopair fuzzy sets. q-rung orthopair hesitant fuzzy set characterizes a set of membership and non-membership grades within the interval [0, 1], which enhances its adaptability compared to existing methods. This flexibility proves invaluable in providing more insightful data about various objects. The primary objective of this research is to introduce a decision-making technique in the context of q-RHF using the theory of set pair analysis (SPA). q-RHFS effectively handles ambiguous data by incorporating membership and non-membership grades, while the connection number (CN) based on SPA theory manages the intricacies of uncertainty and certainty structures by relying on "identity", "discrepancy" and "contrary" grades. Building on the relationship between q-RHFS and the connection number of set pair analysis, a comprehensive framework known as q-rung hesitant fuzzy connection number set (qHCNs) is developed. This model not only addresses uncertainty, but also offers valuable insights. Furthermore, this research introduces similarity measures derived from qHCN and examines their advantages through illustrative examples. Additionally, a novel approach to decision modeling utilizing these measures applied to medical diagnosis is also introduced. The application of this established model contributes to an effective approach and demonstrated its soundness and efficiency. In addition, a detailed comparative study is conducted with the existing models and advantages of proposed model. The research concludes with a summary of the authors' findings, highlighting the consistency and effectiveness of their work.

We review type-1 and type-2 fuzzy numbers in this chapter, and propose one way of perceiving the concept of fuzzy numbers by comparing with that of round numbers. There are some definitions of fuzzy numbers, but we particularly adopt the definition often used in fuzzy analysis. Thereby, we emphasize that fuzzy number theory can be reduced to an argument for interval analysis. Moreover, we explain type-2 fuzzy sets and list two specific type-2 fuzzy numbers, one is a (triangular) perfect quasi-type-2 fuzzy number and the other is a triangular shaped type-2 fuzzy number. Finally, we mention the importance and utility of using type-2 fuzzy numbers.

Fuzzy sets theory and fuzzy logic constitute the basis for the linguistic approach. Under this approach, variables can assume linguistic values. Each linguistic value is characterized by a label and a meaning. The label is a sentence of a language. The meaning is a fuzzy subset of a universe of discourse. Models, based on this approach, can be constructed to simulate approximate reasoning. The implementation of these models presents two major problems, namely how to associate a label to an unlabelled fuzzy set on the basis of semantic similarity (linguistic approximation) and how to perform arithmetic operations with fuzzy numbers. For each problem a solution is proposed. Two illustrative applications are discussed.