Ranking interval type 2 fuzzy sets using parametric graded mean integration representation

Fuzzy logic (FL) has been regarded as a useful methodology in modelling. The modeling performance of FL heavily relies on the modeling capability of a fuzzy set (FS). However, the MF of a FS cannot be arbitrarily accurate in practice and thus the centroid as a measure of a FS cannot be accurate to any degree. In face of inaccurate MF, the sensitivity of the centroid of a FS to the deviation of its MF from the ideal MF determines the accuracy of the centroid. To evaluate the modeling capability of a type-1 (T1) FS and an interval type-2 (IT2) FS, this paper will present a comparative study of the sensitivity of the centroid of a T1 FS and an IT2 FS to the deviation of their MFs from the ideal MFs. In this paper, equations relating the amount that the centroid of a T1 FS and an IT2 FS vary with the deviation of their MFs from the ideal MFs were established. Based on the derived equations, it was found that an IT2 FS is able to provide an additional design freedom to vary the sensitivity of its centroid compared to a T1 FS. This unique characteristic may enable an IT2 FS to minimize the sensitivity of the centroid and thus to achieve higher modeling accuracy. Furthermore, guidelines for designing an IT2 FS were provided.

The Wang-Mendel Approach (WMA) focuses on combining the numerical as well as linguistic information for achieving greater explainability for inference models. The standard WMA models the linguistic information using type-1 (T1) fuzzy sets (FSs), which have a reduced capability to model the semantics of linguistic information. Therefore, we propose a novel Enhanced WMA, which models the linguistic information using the type-2 (T2) FSs. Further, our Enhanced T2 FS-based WMA can be modified to reflect the use of interval type-2 (IT2) FSs, for modelling linguistic uncertainty. IT2 FSs are suitable when better uncertainty handling capabilities are required compared to T1 FSs, however, at a computational cost lesser than the T2 FSs. Performance of Enhanced WMA is demonstrated through a real-world crop-yield prediction problem in smart agriculture and an additional exemplar application on users’ satisfaction ratings. Further, we have compared our approach with the performance obtained from the T1 FS-based WMA and the original estimations given in the original data. We found that our Enhanced WMA provides more precise estimates than the other two with 95% confidence level. To the best of our knowledge, this is the first proposal of a T2 FSs method for enhancing the modeling of linguistic uncertainty in the WMA.

A new ranking methodology for general interval type 2 fuzzy sets (IT2FS) using extended alpha cuts representation is proposed. First, the extended alpha cut representation of the IT2FS is defined. Then, the parametric representation of the type 1 fuzzy sets (T1FS) is formulated by constructing the embedded pessimistic, modal and optimistic functionals. The corresponding parametric centroid functional of the IT2FS is determined. Finally, the average centroid of the IT2FS is calculated by taking the integral average of the parametric centroid functional. In this manner, the IT2FSs are defuzzified and can be ranked by sorting the average centroids. A numerical example is presented for verifying the proposed method.

Gestures of a person symbolizing analogous pathological disorder are not always exclusive. Naturally, the gestural features of a subject suffering from the same pathological disorder vary widely over different instances. In presence of two or more gestural features, the variation of the attributes together makes the problem of pathological disorder recognition more convoluted. Wider variance in gestural features is the main source of uncertainty in the pathological disorder recognition, which has been addressed here using type-2 fuzzy sets (T2FSs). First, a type-2 fuzzy gesture space is created with the background knowledge of gestural features of different subjects for different pathological disorders. Second, the pathological disorder of an unknown gestural expression is recognized based on the consensus of the measured gestural features of the fuzzy gesture space. A fusion of interval and general type-2 Fuzzy Sets has been used to construct the fuzzy gestural space. An interval valued fuzzy set (IVFS) or interval type-2 fuzzy set (IT2FS) is constructed with primary membership functions for M gestural features obtained from N subjects, each having L instances of gestural expressions for a given pathological disorder. The corresponding general type-2 fuzzy set (GT2FS) includes the secondary memberships for individual primary upper and lower membership curves in the respective IVFS. The secondary membership of a given source, characterizing the reliability in its primary membership assignment, is determined based on the amalgamated knowledge of all the subjects’ primary membership functions. The representatives of each pathological disorder class are then evaluated by utilizing the attractive features of both IVFS and GT2FS. The pathological disorder of an unknown gestural expression is finally determined by identifying its minimal discrepancy with the existing class representatives. The adopted T2FS-based uncertainty management strategy, when applied to gender independent training and testing datasets, results in a classification accuracy of 93.41%.

Modeling, design and control of a 4-arm delta parallel manipulator employing type-1 and interval type-2 fuzzy logic-based techniques for precision applications

Energy is a critical factor to obtain a sustainable development for countries and governments. Selection of the most appropriate energy alternative is a completely critical and a complex decision making problem. In this paper, an integrated multi-criteria decision-making (MCDM) methodology based on type-2 fuzzy sets is proposed for selection among energy alternatives. Then a roadmap has been created for Turkey.To overcome uncertainties in decision making process, the fuzzy set theory (FST) is suggested.For this aim, two of the most known MCDM methodologies are reconsidered by using type-2 fuzzy sets.Fuzzy Analytic Hierarchy Process (FAHP) based on interval type-2 fuzzy sets is constructed and is used to obtain the weights of the criteria affecting energy alternatives. To rank the energy alternatives, the other MCDM method that is Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is fuzzified by interval type-2 fuzzy sets. The proposed integrated MCDM methodology based on interval type-2 fuzzy sets is applied to obtain a road map of energy policies for Turkey.

Similarity measures provide one of the core tools that enable reasoning about fuzzy sets. While many types of similarity measures exist for type-1 and interval type-2 fuzzy sets, there are very few similarity measures that enable the comparison of general type-2 fuzzy sets. In this paper, we introduce a general method for extending existing interval type-2 similarity measures to similarity measures for general type-2 fuzzy sets. Specifically, we show how similarity measures for interval type-2 fuzzy sets can be employed in conjunction with the zSlices based general type-2 representation for fuzzy sets to provide measures of similarity which preserve all the common properties (i.e. reflexivity, symmetry, transitivity and overlapping) of the original interval type-2 similarity measure. We demonstrate examples of such extended fuzzy measures and provide comparisons between (different types of) interval and general type-2 fuzzy measures.

Interval type-2 fuzzy logic for encoding clinical practice guidelines

The fuzzy decision-making trial and evaluation laboratory (fuzzy DEMATEL) has been used to solve various multi-criteria group decision-making problems where triangular type-1 fuzzy sets are utilized in defining decision makers’ linguistic evaluation. Most of the fuzzy DEMATEL modifications are built from linguistic variables based on type-1 fuzzy sets (T1FS). Previous literature suggests that interval type-2 fuzzy sets (IT2FS) can offer an alternative that can handle vagueness and uncertainty. This paper proposes a modification fuzzy DEMATEL characterized by IT2FS for linguistic variables. Differently from the typical fuzzy DEMATEL which directly utilizes triangular type-1 fuzzy numbers, this modification introduces trapezoidal IT2 fuzzy numbers to enhance evaluation in the group decision-making environment. This new modification includes linguistic variables expressed by IT2FS and an expected value method for normalizing upper and lower memberships of IT2FS to crisp numbers. The proposed modification is applied to a case of knowledge management (KM) where eleven criteria are considered. Three experts in KM were invited to provide linguistic judgments with respect to the criteria, and the eight-step computational procedure of the proposed modification was implemented without losing the originality of the DEMATEL method. The results unveiled that ‘trust’ is the most influential criteria in KM. Therefore, trust is a phenomenon that impacts on the success of KM. Comparable results are also presented to check the feasibility of the proposed method. It is shown that the criteria weight and the causal relationship of criteria using the proposed method are consistent with the other two methods.

Interval type-2 fuzzy set induced fuzzy rank-level fusion for face recognition

Type-2 fuzzy sets (T2FSs), characterized by a fuzzy membership function, are much useful tool for representing the decision knowledge in the decision making process. Interval Type-2 fuzzy sets (IT2FSs) are the most commonly used T2FSs. In this study, a method based upon ranking values of IT2FSs is used to tackle multi-criteria decision making (MCDM) problems. First, some basic concepts and arithmetic operations for IT2FSs are presented. Then, three kinds of fuzzy ranking methods, proposed by [1], based on arithmetic average (AA), geometric average (GA) and harmonic average (HA) operators to compute the ranking values of IT2FSs are applied. Finally, the outcomes of MCDM methods based on the ranking values of IT2FSs are obtained and also compared with the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method based on Type-1 fuzzy sets (T1FSs) for a numerical example.

A type-2 fuzzy set, which is characterized by a fuzzy membership function, involves more uncertainties than the type-1 fuzzy set. As the most widely used type-2 fuzzy set, interval type-2 fuzzy set is a very useful tool to model the uncertainty in the process of decision making. As a special case of interval type-2 fuzzy set, trapezoidal interval type-2 fuzzy set can express linguistic assessments by transforming them into numerical variables objectively. The aim of this paper is to investigate the multiple attribute group decision-making problems in which the attribute values and the weights take the form of trapezoidal interval type-2 fuzzy sets. First, we introduce the concept of trapezoidal interval type-2 fuzzy sets and some arithmetic operations between them. Then, we develop several trapezoidal interval type-2 fuzzy aggregation operators for aggregating trapezoidal interval type-2 fuzzy sets and examine several useful properties of the developed operators. Furthermore, based on the proposed operators, we develop two approaches to multiple attribute group decision making with linguistic information. Finally, a practical example is given to illustrate the feasibility and effectiveness of the developed approach.

A comprehensive review of multi criteria decision making approaches based on interval type-2 fuzzy sets

As tools for dealing with universal fuzziness in human thinking, traditional fuzzy sets, namely type-1 fuzzy sets, are not able to directly model uncertainties about membership functions, however type-2 fuzzy sets can make up for this deficiency. Interval type-2 fuzzy sets, the simplified cases of general type-2 fuzzy sets, are practical because its membership functions and operations are greatly simplified. The inclusion measure and entropy of fuzzy sets are two important parameters with application prospects, the former describes the degree to which a fuzzy set is included in another, the latter shows the quantity of fuzzy information in a fuzzy set. It is important to make in-depth research on the two measures. In this paper, we mainly discuss the relationship of the inclusion measure and entropy of interval type-2 fuzzy sets proposed by Zheng et al., and present a theorem that converts the entropy to the inclusion measure of interval type-2 fuzzy sets. The conclusion provides the basis for further research.

Fuzzy attribute control charts, where data are classified into conforming/nonconforming product units, are used to monitor fuzzy fractions of nonconforming units for variable sample sizes and the fuzzy number of nonconforming units for constant sample sizes. Data defined as quality characteristics can be imprecise due to the subjective decisions of the quality control operator. Type-2 fuzzy set theory deals with ambiguity associated with the uncertainty of membership functions by incorporating footprints and modeling high-level uncertainty. In this paper, the structure of an interval type-2 fuzzy p-control chart and interval type-2 fuzzy np-control chart with constant sample size are developed and applied to real data. The main advantage in using interval type-2 fuzzy sets in control charts is the flexibility allowed in determining control limits for process monitoring by incorporating fuzzy set theory. Therefore, fuzzy control charts with interval type-2 fuzzy numbers afford the decision maker the opportunity to see and detect process defects.