Fuzzy set theory has been widely and successfully applied to model uncertainty in a variety of geospatial contexts, including the classification of land cover (Foody 1996), and geodemographics (Grekousis and Hazichristos 2012). However, as noted by Fisher et al. (2007) such work predominantly makes use of fuzzysets as originally specified by Zadeh (1965); characterised by crisp membership functions (Mendel and John, 2002). Zadeh (1975) extended his initial ideas to define fuzzysets with fuzzy membership functions (Mendel and John 2002) and the nomenclature ‘type-1’ and ‘type-2’ is now used to refer to these different forms. Fisher and Tate (2014) employed type-1 fuzzysets to soften a geodemographicclassification (the UK Output Area Classification: OAC) of the City of Leicester UK. In this paper we extend that work to explore the use of type-2fuzzysets to the OAC.
Neural networks, Bayesian networks, support vector machines are all considered “black box”. This adjective is applied to systems that, for a given input, are able to output a class label, but without providing a clear explanation of the decision process. Logistic regression can provide some statistic correlations between the inputs and the output, but this is not enough to understand why, for a given input, a given label was chosen, or to gain a deep insight of either the model or the data. On the other hand, “white box” models usually refer to rule-based systems that are able to provide an insight of the data on which the models have been trained, and an explanation of the decision process through their rules. Decision trees can translate their internal state into a set of rules and, like any other rule-based system, are able to provide transparency. Nevertheless in complex real world applications, such as in the financial domain, the number of generated rules can explode. It is debatable that a rule base containing thousands of rules can be considered an understandable and transparent model. Decision trees - and random forests  produce associations among sets of data, which are selected to optimize the classification problem. Thus, the produced associations could be meaningless in the context of profiling and knowledge extraction.
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects. Set theory as a foundation for mathematicians accepts that theorems in these areas can be derived from the relevant definitions and the axioms of set theory .
Abstract— BIRADS is a Breast Imaging, Reporting and Data System. A tool to standardize mammogram reports and minimizes ambiguity during mammogram image evaluation. Classification of BIRADS is one of the most challenging tasks to radiologist. An apt treatment can be administered to the patient by the oncologist upon acquiring sufficient information at BIRADS stage. This study aspired to build a model, which classifies BIRADS using mammograms images and reports. Through the implementation of type-2fuzzy logic as classifier, an automatically generated rules will be applied to the model. To evaluate the proposed model, accuracy, specificity and sensitivity of the modal will be calculated and compared vis-à-vis rules given by the experts. The study encompasses a number of steps beginning with collection of the data from Radiology Department, Hospital of National University of Malaysia (UKM). The data was initially processed to remove noise and gaps. Then, an algorithm developed by selecting type-2fuzzy logic using Mamdani model. Three types of membership functions were employed in the study. Among the rules that used by the model were obtained from experts as well as generated automatically by the system using rough set theory. Finally, the model was tested and trained to get the best result. The study shows that triangular membership function based on rough set rules obtains 89% whereas expert rules achieve 78% of accuracy rates. The sensitivity using expert rules is 98.24% whereas rough set rules obtained 93.94%. Specificity for using expert rules and rough set rules are 73.33%, 84.34% consecutively. Conclusion: Based on statistical analysis, the model which employed rules generated automatically by rough set theory fared better in comparison to the model using rules given by the experts.
Experiments are undertaken with a Logitech driving simulator, comprising one steering wheel, one brake and one accelerator foot pedal and a monitor to drive the simulated car (Fig. 7(a)) and an fNIRs device, manufactured by BIOPAC, with 4 infrared (IR) sources and 10 infrared detectors, placed in an array for mounting on the forehead (Fig. 7(b)). The IR sources are triggered by short duration electrical pulses in a time-multiplexed manner to ensure activation of only one source at a time. On triggering of a selected source, the infrared signal penetrates the pre-frontal region below the source, and the received energy is partially absorbed by the brain and partially reflected back to the four detectors mounted around each source (Fig. 7(c)). The sampling rate of the fNIRs device being 2 Hz, the sampling intervals are of 0.5 seconds. The sampling interval constitutes four equal time-slices of 0.125 seconds, where each time-slice is utilized to receive oxygenated (HbO) and deoxygenated (HbR) blood response (Fig. 7(d)) by one of four detectors around each source. Thus for 4 sources, we obtain 16 oxygenated and 16 deoxygenated blood response of 16 voxels (brain regions) in 0.5 seconds. It is important to note that the penetration depth of the IR signals is 1.25 cm from the surface of the scalp. The signal acquisition is performed using (Cognitive Optical Brain Imaging) COBI studio software, supplied by the manufacturer.
The IT2 FSs have been extensively used in the literature to model uncertainty (see , –). Despite literature being replete with several works revolving round IT2 FSs, they only make use of the MFs alone in uncertainty modeling with an implicit assertion that NMF is complementary to the MF (lower or upper). In a real life context, it is not necessarily the case that NMF is complementary to MF as there may exist some degree of hesitation or indeterminacy, otherwise known as intuitionistic fuzzy index (IF-index) or neutral degree. The conventional IT2 FLS cannot singularly model these IF-indices in a fuzzy set. Barrenechea et al.  pointed out that valuable information of an element can be obtained using the IF-index of IFS. The authors in  also noted that the IF-index plays a very important role in algorithm’s performance. Our study is an attempt in this direction to enhance the capabilities of IT2 FLS by incorporating IFL into IT2 FLS. As earlier discussed, with the capability of the IT2 FSs to adequately model uncertainty in their FOUs and the ability of the IFS to separately cater for MF and NMF of an element with some level of hesitation, we are motivated to integrate these two concepts (IT2 FS and IFS) to design a new TSK-type interval T2 intuitionistic fuzzy logic system (IT2 IFLS-TSK) . The new framework apart from incorporating fuzzy NMF into the conventional IT2 FS is able to deal with indeterminate (hesi- tant) states which are not well managed by alternative fuzzy approaches such as IT2 FLSs. The introduction of additional NMF and IF-indices into IT2 FS increases the fuzziness of the model. According to Hisdal , “increased fuzziness in a description means increased ability to handle inexact information in a logically correct manner.” We believe that the fusion of these two kinds of fuzzysets is able to provide a synergistic capability in managing the effects of uncertainties in data. The proposed framework of IT2 IFLS is enhanced with a neural network learning capability similar to adaptive neuro-fuzzy inference system (ANFIS) and T2-ANFIS  for modeling uncertainty in data. The combination of these two approaches, fuzzy logic and artificial neural network (ANN), have been very popular with increasing interest in recent years. With the integration of ANN into FLS, the FLS is enhanced with the learning and generalisation capabilities of ANN.
The branch of the logical algebra, UP-algebras was introduced by Iampan  in 2017, and it is known that the class of KU-algebras  is a proper subclass of the class of UP-algebras. It have been examined by several researchers, for example, Somjanta et al.  introduced the notion of fuzzysets in UP-algebras, the notion of intuitionistic fuzzysets in UP-algebras was introduced by Kesorn et al. , Kaijae et al.  introduced the notions of anti-fuzzy UP-ideals and anti-fuzzy UP-subalgebras of UP-algebras, the notion of Q-fuzzysets in UP-algebras was introduced by Tanamoon et al. , Sripaeng et al.  introduced the notion anti Q-fuzzy UP-ideals and anti Q-fuzzy UP-subalgebras of UP-algebras, the notion of N -fuzzysets in UP-algebras was introduced by Songsaeng and Iampan , Senapati et al. [12, 13] applied cubic set and interval-valued intuitionistic fuzzy structure in UP-algebras, Romano  introduced the notion of proper UP-filters in UP-algebras, etc.
Fuzzysets (FSs) theory and the fuzzy rule-based (FRB) systems have been defined over 50 years ago in the seminal paper by Professor Lotfi Zadeh  and now matured . Since mid-1970s (Mamdani or Zadeh- Mamdani)  and since mid-1980s (Takagi-Sugeno)  FRB systems started to be developed and are now widely applied. Although, there are other types of fuzzy systems (relational , etc.), one particular type that was introduced recently by Angelov and Yager  called AnYa deserves a special attention. Both Mamdani and Takagi-Sugeno type of FRB share the exact same antecedent (IF) part and only (although significantly) differ by the consequent (THEN) part. AnYa type FRB, however, has a quite different antecedent (IF) part.
In this paper, a new T2FMWNN model is proposed in order to classify the breast cancer data. This new model uses Shannon wavelet functions in its processing units. Type-2 membership functions can catch uncertainty in the data more precisely according the other type-1 membership functions. Therefore, the proposed model gives good results in classification. According to accuracy, specificity and F-1 scores, T2FMWNN model gives the best result among MLP network, RBF network, Bayesian network learning, decision tree and T1FMWNN models. The proposed T2FMWNN model can be applied many other problems such as other classification, function learning, system identification and control problems.
One of the most prominent representation meth- ods of GT2 FSs is to decompose them into several - planes. The concept of -planes has been thoroughly discussed by Mendel et al. . Such a representation oers a computationally ecient framework to deal with GT2 FSs. To facilitate the process of type- reduction, Liu  has utilized the concept of -planes to nd the centroid of a GT2 FS. He decomposes each GT2 FS into several planes which are IT2 FSs. Then Karnik-Mendel (KM) algorithm is applied to nd the centroid of each -plane. Finally, the entire obtained centroids are aggregated and the centroid of the GT2 FS is obtained. This method is appropriate in theoretical studies, but it is not much useful when applied to real world problems since it is rather time consuming. Based on weaknesses of Liu's work, Yeh et al.  present an enhanced algorithm in order to speed up the process of type-reduction. Their main
was rst developed by Hwang and Yoon . In the classical TOPSIS method, the appraisals and weights of criteria are precise values. However, in the real world, the crisp data are not suitable, because human judgments are vague and imprecise when dealing with decision-making issues and cannot be estimated with exact numeric values. To state the ambiguity in real- world problems, the fuzzy data instead of crisp data have been incorporated in many MCDM techniques including TOPSIS. In Fuzzy TOPSIS (FTOPSIS), all the ratings and weights are dened by means of the fuzzy data. However, a decision-maker may have doubt about the measure of Membership Function (MF). In other words, in a type-1 fuzzy set, it is often dicult for an expert to express his/her notions as a specied number at an interval [0, 1] related to MF. Hence, the type-2fuzzysets were suggested by Zadeh  for relieving the uniqueness of MF measure of the type- 1 fuzzysets. Interval Type-2FuzzySets (IT2FSs) represent a particular version of type-2fuzzysets characterized by an interval MF. There are known versions for IT2FSs such as Trapezoidal Interval Type- 2FuzzySets (TraIT2FSs), Triangular Interval Type-2FuzzySets (TriIT2FSs), and Gaussian Interval Type-2FuzzySets (GIT2FSs) in the literature. Triangular or trapezoidal MFs are the simplest MFs formed using straight lines. MFs of triangular and trapezoidal fuzzy numbers have steep slopes in their reference points. In real problems, however, the decision-maker may consider a smoother slope for the MFs in refer- ence points. Hence, \Gaussian MFs are suitable for problems requiring continuously dierentiable curves, whereas the triangular and trapezoidal fuzzy numbers do not possess these abilities" .
Keshavarz Ghorabaee et al. (2015b) introduced the evaluation based on distance from average solution (EDAS) method. This method was also extended for decision-making in fuzzy environment and was applied to supplier selection problem (Keshavarz Ghorabaee et al., 2016c). However, the previous versions of this method are not appropriate to deal with MCGDM problems with IT2FSs. In this study, we propose a new extended EDAS with interval type-2fuzzysets (EDAS-IT2FSs). A numerical example is employed to illustrate the process and show the effectiveness of the proposed method. A comparison and a sensitivity analysis are also performed to represent the validity and stability of the ranking result. The results of these analyses show that the proposed extended EDAS method is stable in different weights of criteria and well-consistent with some existing methods.
In this paper, we have applied distance based similarity measure using various distances between Type-2fuzzy soft sets to obtain a solution of a decision making problem. A numerical example is solved to illustrate this method and the centroid type reduction method is also used in this problem. We found that the Normalized Euclidean distance helps us to take decision inmore consistent manner.In future, other types of measures can also be used to find the solution of any decision making problem.
In a competitive environment, without considering selec- tion and evaluation of suppliers successfully, it is extremely hard to manage any production process with high quality and low cost. Thus, supplier evaluation and selection problems have been addressed by using different approaches such as; data envelopment analysis (DEA), analytic hierarchy process (AHP) and fuzzy set theory . Ho et al.  reviewed the literature considering 78 articles between 2000 and 2008 in order to summarise which approaches were applied, which evaluating criteria were emphasised and the adequacy of the approaches. Based on this literature review, DEA, mathe- matical programming, AHP, case-based reasoning, analytic network process, fuzzy set theory, simple multi-attribute rating technique and genetic algorithm have been mostly used. The most popular five evaluation criteria also have been found as quality, delivery, price, manufacturing capacity, service.
In this demonstration, each method was applied to the interval T2 FSs displayed in Fig. 3, the results of which are shown in Table II. The x-axis was discretised into 100 equally distanced points, and minimum t-norm was used for Bustince’s SM. As in , it can be observed that neither Zeng and Li’s nor Bustince’s measures support the property of overlapping. When the FSs being measured are disjoint, Zeng and Li’s SM increases as the distance between the sets increases and Bustince’s measure always gives a constant non- zero value. This can be seen in the results of sets A ˜ and D ˜ and of sets A ˜ and E. Depending on the application, this may ˜ not be what is expected as it is often presumed that S( ˜ A, B) ˜ either decreases as the distance between A ˜ and B ˜ increases, or is given as 0. Additionally, Gorzałczany’s measure has given (1.0, 1.0) for sets A ˜ and B ˜ because this measure will always give (1.0 1.0) when max x∈X µ A ˜ (x) = max x∈X µ B ˜ (x) and
Abstract: An overview and a derivation of interval type-2 fussy logic system (IT2 FLS), which can handle rule’s uncertainties on continuous domain, having good number of applications in real world. This work focused on the performance of an IT2 FLS that involves the operations of fuzzification, inference, and output processing. The out- put processing consists of Type-Reduction (TR) and defuzzification. This work made IT2 FLS much more accessible to FLS modelers, because it provides mathematical formulation for calculating the derivatives. Presenting extend to representation of T2 FSs on continuous domain and using it to derive formulas for operations, we developed and extended the derivation of the union of two IT2 FSs to the derivation of the intersection and union of N-IT2 FSs that is based on various concepts. The derivation of all the formulas that are related with an IT2 and these formulas de- pend on continuous domain with multiple rules. Each rule has multiple antecedents that are activated by a crisp number with T2 singleton fuzzification (SF). Then, we have shown how those results can be extended to T2 non- singleton fuzzification (NSF). We are derived the relationship between the consequent and the domain of uncertainty (DOU) of the T2 fired output FS. As well as, provide the derivation of the general form at continuous domain to cal- culate the different kinds of type-reduced. We have also applied an IT2 FLS to medical application of Heart Diseases (HDs) and an IT2 provide rather modest performance improvements over the T1 predictor. Finally, we made a com- parison of HDs result between IT2 FLS using the IT2FLS in MATLAB and the IT2 FLS in Visual C# models with T1 FISs (Mamdani, and Takagi-Sugeno).
General type-2fuzzy clustering is based on FCM (Fuzzy C-Means) algorithm. Like FCM, it initializes the centers randomly. The FCM algorithm uses linguistic terms such as “Small”, “Medium” or “High”, modeled by type-1 fuzzysets for the fuzzifier parameter M (Figure 1). The FCM algorithm is used by the GT2 FCM cluster membership functions. The general type-2fuzzy clustering proposed in  uses α-planes. The uncertainty of general type-2fuzzysets is managed by α-planes. The GT2 FCM algorithm exploits the linguistic fuzzifier M for its secondary membership functions of the general type-2fuzzy partition matrix as shown in Equation 1. In addition, Equation 2, that is a membership grade is expressed as type-1 fuzzysets, which is used to describe the membership degree of pattern to cluster
Abstract. Making decision is one of the most fundamental activities of human being. Decision making is a study of how decisions are actually made better. Applications of fuzzysets within the field of decision making consisted of fuzzifications of the classical theories of decision making. Decisions are made under conditions of uncertainty is the prime domain for fuzzy decision making. In this paper, we have applied the notion of similarity measure and inclusion measure between Type-2fuzzy soft sets to verify their relationships. This relation is used to obtain a solution of a decision making problem. Keywords: Type-2fuzzy soft sets, complement of type-2fuzzy soft sets, similarity measure, inclusion measure
This note is based on the papers by Kuroki [7–8] and its aim is to show the way we pass from fuzzy semigroups to fuzzy hypersemigroups. We show, among others, that a regular hypersemigroup is left (resp. right) duo if and only if it is fuzzy left (resp. fuzzy right) duo. In a regular hypersemigroup, every bi-ideal is a right (resp. left) ideal if and only if every fuzzy bi-ideal is a fuzzy right (resp. fuzzy left) ideal. In an intra-regular hypersemigroup S for every fuzzy ideal f of S and any a ∈ S there exists u ∈ a ◦ a such that f (a) = f (u). “Conversely” if S is an hypersemigroup such that for every fuzzy ideal f of S, any a ∈ S and any u ∈ a ◦ a, we have f(a) = f (u), then S is intra-regular. If S is a left (resp. right) regular hypersemigroup, then for every fuzzy left (resp. fuzzy right) ideal f of S and any a ∈ S there exists u ∈ a ◦ a such that f (a) = f (u). “Conversely” if S is an hypersemigroup such that for any fuzzy left (resp. fuzzy right) ideal f of S, any a ∈ S and any u ∈ a ◦ a, we have f (a) = f (u), then S is left (resp. right) regular. An hypergroupoid is left (resp. right) simple if and only if it is fuzzy left (resp. fuzzy right) simple and so it is simple if and only if it is fuzzy simple. The simple hypersemigroups are regular and intra-regular. The left (resp. right) simple hypersemigroups are left (resp. right) regular. Finally, in left (resp. right) simple hypersemigroups, every fuzzy bi-ideal is a fuzzy right (resp. fuzzy left) ideal. As a consequence, in a left (resp. right) simple hypersemigroup, every bi-ideal is a right (resp. left) ideal.
Abstract. The aim of this paper is to introduce some new classes of fuzzysets and some new classes of fuzzy continuity namely fuzzy e ∗ -open sets, fuzzy (, S)-sets, fuzzy (S, ε )-sets, fuzzy (, S) ∗ -sets, fuzzy (S, ε ) ∗ -sets, fuzzy e-continuity, fuzzy (, S) ∗ - continuity and fuzzy (S, ∈) ∗ -continuity. Properties of these new concepts are investigated. Moreover, some new decompositions of fuzzy continuity are provided. Keywords: Fuzzy Continuity, Fuzzy e ∗ -open sets, Fuzzy (, S)-sets, Fuzzy (S, ε )-sets, Fuzzy (, S) ∗ -sets, Fuzzy (S, ε ) ∗ -sets, Fuzzy e-continuity, Fuzzy (, S) ∗ -continuity and Fuzzy (S, ∈) ∗ -continuity.