IT2 FS Research Articles

A least-squares framework for developing interval type-2 fuzzy semantics

The developing of IT2 fuzzy semantics is critical for computing with words (CWW), but exiting approaches lack flexibility and fail to adapt user’s diversified demands. The study proposes a least-squares framework for designing CWW encoders that construct interval type-2 fuzzy sets (IT2 FSs) to represent the semantic meanings of linguistic words. In the least-squares framework, an CWW encoder is characterized by two elements: an intra-uncertain semantic mapping and an inter-uncertain semantic family. The intra-uncertain semantic mapping transforms data intervals into type-1 fuzzy sets (T1 FSs), then an optimal IT2 FS is derived from the inter-uncertain semantic family using the least-squares method. Furthermore, several intra-uncertain semantic mappings are introduced, and a compatibility measure is defined to facilitate model selection. The least-squares framework benefits from the flexible selection of intra-uncertain semantic mappings and least-squares optimization-based construction of IT2 FSs. In experiments, the least-squares framework is applied to handle real-world online survey data and the large-scale online review data set of a Chinese life service review site, Dianping.com. Compared to the enhanced interval approach and the Hao-Mendel approach, the least-squares framework shows its favorable efficiency in experiments and statistical tests, and adapts to user-defined intra- and inter-uncertain semantic families.

A New Fuzzy Bayesian Inference Approach for Risk Assessments

Bayesian network (BN) inference is an important statistical tool with additive symmetry. However, BN inference cannot deal with the uncertain, fuzzy, random, and conflicting information from experts’ knowledge in the process of conducting a risk assessment. To tackle this issue, a new fuzzy BN inference approach for risk assessments was proposed based on cloud model (CM), interval type-2 fuzzy set (IT2 FS), interval type-2 fuzzy logic system (IT2 FLS), modified Dempster–Shafer (D-S) evidence theory (ET), and Latin hypercube sampling (LHS) methods along the following lines. Firstly, CM was integrated into IT2 FS, and CM-based IT2 FS (CM-IT2 FS) was defined in the IT2 FLS. Secondly, modified D-S ET was utilized to determine the CM-IT2 FS-based a priori probabilities, and the CM-IT2 FS-based BN model was established. Thirdly, the CM-IT2 FS-based a priori probabilities were reduced to the CM-IT1 FS-based ones using a type reducer in the IT2 FLS, LHS was applied to propose a new fuzzy BN inference algorithm, and then, the new algorithm was used in a typical case to perform the fuzzy BN positive inference for risk prediction and the fuzzy BN reverse inference for risk sensitivity analysis. Finally, the BN inference results were analyzed using the proposed algorithm and the two common BN inference algorithms, and the effectiveness of the proposed approach was validated. It can be concluded that the proposed approach was both accurate and promising.

Uncertainty of Interval Type-2 Fuzzy Sets Based on Fuzzy Belief Entropy.

Interval type-2 fuzzy sets (IT2 FS) play an important part in dealing with uncertain applications. However, how to measure the uncertainty of IT2 FS is still an open issue. The specific objective of this study is to present a new entropy named fuzzy belief entropy to solve the problem based on the relation among IT2 FS, belief structure, and Z-valuations. The interval of membership function can be transformed to interval BPA . Then, and are put into the proposed entropy to calculate the uncertainty from the three aspects of fuzziness, discord, and nonspecificity, respectively, which makes the result more reasonable. Compared with other methods, fuzzy belief entropy is more reasonable because it can measure the uncertainty caused by multielement fuzzy subsets. Furthermore, when the membership function belongs to type-1 fuzzy sets, fuzzy belief entropy degenerates to Shannon entropy. Compared with other methods, several numerical examples are demonstrated that the proposed entropy is feasible and persuasive.

New efficient algorithms for the centroid of an interval type-2 fuzzy set

In a type-2 fuzzy logic system , one of the important operations is to calculate the centroid of an interval type-2 fuzzy set (IT2 FS). In this paper, two novel algorithms called binary algorithms are proposed to calculate the centroid of IT2 FSs. Then, the outputs of the proposed binary algorithms are proven to be the optimal values. After analyzing the computational complexities of Karnik-Mendel (KM) algorithms, enhanced Karnik-Mendel (EKM) algorithms, enhanced iterative algorithms based on stopping condition (EIASC) algorithms and the proposed binary algorithms, it is found that the proposed binary algorithms are superior to the KM algorithms, EKM algorithms and EIASC algorithms. Finally, two extended binary algorithms are proposed to compute the centroid of an IT2 FS. The efficiencies of the proposed binary algorithms and extended binary algorithms are demonstrated by extensive simulations.

Improvement of Enhanced Opposite Direction Searching Algorithm

Type-reduction is an important part of the interval type-2 fuzzy logic system (IT2 FLS), which is also the main factor that influences the effectiveness of IT2 FLS. The enhanced opposite direction searching (EODS) algorithm is currently the fastest form of the Karnik–Mendel algorithm, but it requires a large number of iterations. In this article, an improvement of the enhanced opposite direction searching (IEODS) algorithm is presented to obtain the type reduced set more efficiently. The IEODS algorithm improves the iterative process of the original EODS algorithm through the width of the upper and lower member functions of the IT2 FS. In this way, the IEODS algorithm uses a single-sided iteration process. It has a better initial point to greatly reduce the number of iterations. Quantitative analysis and simulation based experiments show that the IEODS algorithm can save more than 90% of the iterations required in the EODS algorithm. It makes the IEODS algorithm more suitable for practical applications, especially to various development platforms with an explanatory programming language as a tool, greatly expanding the application range of the IT2 FLS.

An Extended TOPSIS Method Based on Gaussian Interval Type-2 Fuzzy Set

Compared with trapezoidal interval type-2 fuzzy set (IT2 FS), Gaussian IT2 FS is more concise in representation and can meanwhile capture sufficient uncertainties. Therefore, in this paper the Gaussian IT2 FS is used to model words in multi-attribute decision-making (MADM) process and a comprehensive method is developed under the Gaussian IT2 FS environment. Firstly, a new type of Gaussian IT2 FS is proposed and the ranking rule is constructed to determine the ideal solutions. Secondly, the Euclidean distance is extended to Gaussian IT2 FS to measure the difference between alternatives and ideal solutions. Further, a complete MADM method is established under the framework of TOPSIS (technique for order preference by similarity to an ideal solution). In the end, the effectiveness of the proposed method is illustrated by an investment decision-making example.

A Constrained Representation Theorem for Well-Shaped Interval Type-2 Fuzzy Sets, and the Corresponding Constrained Uncertainty Measures

The representation theorem for interval type-2 fuzzy sets (IT2 FSs), proposed by Mendel and John, states that an IT2 FS is a combination of all its embedded type-1 (T1) FSs, which can be nonconvex and/or subnormal. These nonconvex and/or subnormal embedded T1 FSs are included in developing many theoretical results for IT2 FSs, including uncertainty measures, the linguistic weighted averages (LWAs), the ordered LWAs (OLWAs), the linguistic weighted power means (LWPMs), etc. However, convex and normal T1 FSs are used in most fuzzy logic applications, particularly computing with words. In this paper, we propose a constrained representation theorem (CRT) for well-shaped IT2 FSs using only its convex and normal embedded T1 FSs, and show that IT2 FSs generated from three word encoding approaches and four computing with words engines (LWAs, OLWAs, LWPMs, and perceptual reasoning) are all well-shaped IT2 FSs. We also compute five constrained uncertainty measures (centroid, cardinality, fuzziness, variance, and skewness) for well-shaped IT2 FSs using the CRT. The CRT and the associated constrained uncertainty measures can be useful in computing with words, IT2 fuzzy logic system design using the principles of uncertainty, and measuring the similarity between two well-shaped IT2 FSs.

Fostering linguistic decision-making under uncertainty: A proportional interval type-2 hesitant fuzzy TOPSIS approach based on Hamacher aggregation operators and andness optimization models

Interval type-2 fuzzy sets (IT2 FS) have played a prominent role in the development of type-2 (T2) fuzzy logic and fuzzy systems for application to linguistic approximation transformations. Although there have been a number of studies of individual linguistic perception understanding based on T2 fuzzy logic, few of these have paid attention to the computational manipulation of group linguistic perceptions based on IT2 FS theory and methods. Proportional hesitant fuzzy linguistic term sets (PHFLTSs) allow conceptualization of group linguistic perceptions with the inclusion of generalized linguistic terms and their associated proportions. Interpreting PHFLTSs from the viewpoint of T2 fuzzy logic instead of its type-1 counterpart has the potential to provide useful results because of the ability of IT2 FSs to accurately model individual comprehension in the presence of linguistic uncertainty. This paper brings a novel perspective to encoding PHFLTSs based on the concept of a proportional interval T2 hesitant fuzzy set (PIT2 HFS). PIT2 HFSs combine IT2 FSs translated from generalized linguistic terms together with their proportional information. To facilitate computing with PIT2 HFSs, basic operations satisfying the closure property are defined for PIT2 HFSs based on Archimedean t-norms and s-norms. Information measurements such as a distance and a score function for PIT2 HFS are also defined. Two instrumental aggregation operators for PIT2 HFSs, namely, the Hamacher proportional interval T2 hesitant fuzzy power weighted average (Ham-PIT2HPWA) operator and the Hamacher proportional interval T2 hesitant fuzzy power weighted geometric (Ham-PIT2HPWG) operator, are investigated, and their limiting cases with respect to various parameters are discussed. Furthermore, two andness optimization models with the andness measure being the attitudinal character are constructed in a bid to determine reasonable parameter values associated with the Ham-PIT2HPWA and Ham-PIT2HPWG operators. On the basis of the Hamacher aggregation operators and the andness optimization models, a proportional interval T2 hesitant fuzzy TOPSIS approach is developed to provide linguistic decision making under uncertainty. The proposed approach represents a novel paradigm for linguistic group decision making under the umbrella of T2 fuzzy logic and systems for computing with words that has potential for application in real-life scenarios.

Combinatorial Iterative Algorithms for Computing the Centroid of an Interval Type-2 Fuzzy Set

Computing the centroid of an interval type-2 fuzzy set (IT2 FS) is an important type-reduction method. The aim of this paper is to develop a new method to calculate the centroid of an IT2 FS when the problems of centroid computation of an IT2 FS are continuous. For the continuous centroid computation problems, the structures of optimal solutions are strictly proven from mathematics for the first time in this paper. Furthermore, we also prove that the structures of the optimal solutions are unique in the sense of almost everywhere equal, i.e., if there are two optimal solutions $f_1(x)$ and $f_2(x)$ , the Lebesgue measure of $\{x|f_1(x)\neq f_2(x)\}$ is equal to 0. Subsequently, a combinatorial iterative (CI) method is proposed to solve the roots of the sufficiently differentiable objective functions. It is proven that the convergence of the proposed iterative method is at least sixth order. Based on the proposed iterative method, two algorithms, called CI algorithms, are devised to compute the centroid of an IT2 FS. The efficiencies of CI algorithms are demonstrated by comparing the continuous Karnik–Mendel algorithms and the Hallye's methods with the CI algorithms through three numerical examples.

Big-data clustering with interval type-2 fuzzy uncertainty modeling in gene expression datasets

The major bottleneck in microarray gene expression analysis is the lack of techniques required to cope up with the uncertain gene functionality and inherent complex gene interactions. More often the issues of unstructured, unorganized, noisy and incomplete data is confronted with the advent of uncertain gene expression big datasets. Moreover, big data is naturally associated with the uncertainties which are originated from multiple sources. In this paper, such uncertainties in the gene expression dataset are modeled using interval type-2 fuzzy sets (IT2 FSs). For this, the spread of footprint of uncertainty (FOU), which accounts for the all the possible noises in the gene expression dataset, is modeled both symmetrically and asymmetrically. The medical science big dataset of microarray gene expression data of the cancer patients are considered for the experimentations. Empirically, the effects of uncertainty modeling using IT2 FSs on the big data clustering is analyzed and observed. Fuzzy clustering approach allow the genes to belong to multiple clusters and thus allow the genes participation in cellular process, subcell variations, and cellular metabolism. The effect of the induced uncertainty on the big data clustering has been studied using various cluster validity measures. The clustering results, obtained based on the IT2 fuzzy uncertainty modeling, are compared with same obtained with type-1 fuzzy sets based uncertainty modeling. It is demonstrated that our proposed IT2 FS based approach is more efficient in giving better clustering results for uncertain gene expression datasets and is scalable to the large gene expression datasets. Sensitivity analysis of the clustering results considering four different IT2 membership function shapes such as triangular, trapezoidal, semi-elliptic, and Gaussian are performed.

Type-2 fuzzy reliability–redundancy allocation problem and its solution using particle-swarm optimization algorithm

In this paper, the fuzzy multi-objective reliability redundancy allocation problem (FMORRAP) is proposed, which maximizes the system reliability while simultaneously minimizing the system cost under the type 2 fuzzy uncertainty. In the proposed formulation, the higher order uncertainties (such as parametric, manufacturing, environmental, and designers uncertainty) associated with the system are modeled with interval type 2 fuzzy sets (IT2 FS). The footprint of uncertainty of the interval type 2 membership functions (IT2 MFs) accommodates these uncertainties by capturing the multiple opinions from several system experts. We consider IT2 MFs to represent the subsystem reliability and cost, which are to be further aggregated using extension principle to evaluate the total system reliability and cost according to their configurations, i.e., series parallel and parallel series. We proposed a particle swarm optimization (PSO) based novel solution approach to solve the FMORRAP. To demonstrate the applicability of two formulations, namely, series parallel FMORRAP and parallel series FMORRAP, we performed experimental simulations on various numerical data sets. The decision makers/system experts assign different importance to the objectives (system reliability and cost), and these preferences are represented by sets of weights. The optimal results are obtained from our solution approach, and the Pareto optimal front is established using these different weight sets. The genetic algorithm (GA) was implemented to compare the results obtained from our proposed solution approach. A statistical analysis was conducted between PSO and GA, and it was found that the PSO based Pareto solution outperforms the GA.

Interval Type-2 Mutual Subsethood Fuzzy Neural Inference System (IT2MSFuNIS)

This paper presents an interval type-2 mutual subsethood fuzzy neural inference system (IT2MSFuNIS). A mutual subsethood measure between two interval type-2 fuzzy sets (IT2 FS) has been derived and has been used in determining the similarity between the IT2 FS inputs and IT2 FS antecedents. The consequent weights are taken to be interval sets. The inputs to the system are fuzzified into IT2 FSs with Gaussian primary membership function having fixed center and uncertain variance. Aggregation of type-2 mutual subsethood based activation spreads is performed using product operator. The output is obtained using simplified type-reduction followed by defuzzification. The system learns using memetic procedure involving differential evolution for global search and gradient descent for local exploitation in solution space. The mathematical modeling and empirical studies of IT2MSFuNIS bring forth its efficacy in problems pertaining to function approximation, time-series prediction, control, and classification. Comparisons with other type-1 and type-2 neuro-fuzzy systems verify that IT2MSFuNIS compares excellently with other models with a performance better than most of them both in terms of total number of trainable parameters and result accuracy. Empirical studies indicate the intelligent decision making capability of the proposed model. The main contribution of this paper lies in the identification of mutual subsethood to find out the correlation between IT2 FSs and to find out its applicability in diverse application domains. The improved performance of the proposed method can be attributed to the better contrast handling capacity of mutual subsethood method and uncertainty handling capacity of IT2 FSs. The integration of mutual subsethood with interval type-2 fuzzy logic puts forth a novel model with various merits as demonstrated amply with the help of well-known problems reported in the literature.

Comments on “Interval Type-2 Fuzzy Sets are Generalization of Interval-Valued Fuzzy Sets: Towards a Wide View on Their Relationship”

This letter makes some observations about “Interval type-2 fuzzy sets are generalization of interval-valued fuzzy sets: Towards a wide view on their relationship,” IEEE Trans. Fuzzy Systems that further support the distinction between an interval type-2 fuzzy set (IT2 FS) and an interval-valued fuzzy set (IV FS), points out that all operations, methods, and systems that have been developed and published about IT2 FSs are, so far, only valid in the special case when IT2 FS = IVFS, and suggests some research opportunities.

A comparison of three approaches for estimating (synthesizing) an interval type-2 fuzzy set model of a linguistic term for computing with words

This article compares three methods [Interval Approach (IA), Enhanced Interval Approach (EIA) and Hao–Mendel Approach (HMA)] for estimating (synthesizing) an interval type-2 fuzzy set (IT2 FS) model for a word, beginning with data that are collected from a group of subjects, or from a single subject. It summarizes the stages for each of the methods in tables so it is possible to compare the steps of each stage side-by-side. It also demonstrates, by means of an example of three words, that using more information contained in the collected data intervals is equivalent to reducing the uncertainty in the IT2 FS model. It recommends the HMA because it uses more information contained in the collected data intervals than does the IA or the EIA, and because it is the only method to-date that leads to normal IT2 FSs. Such fuzzy sets are easier to compute with than are non-normal IT2 FSs.

Using Computing with Words for Selecting Projects in Field of Fuel Consumption Reduction

Background/Objectives: Nowadays, most organizations acknowledge the role and significance of different energy resources in the supply of present and future needs. As a result, they make sizeable investments and undertake extensive research on strategy and policy making as well as definition of infrastructure projects to reduce energy consumption. According to the diversity of projects and their cost of implementation, ranking the projects with different criteria is currently one of the complex problems in organizations. This paper presents a new solution based on the specific architecture of Perceptual Computer (Per-C) for project selection. Methods/Statistical Analysis: In project selection problem, decision makers express their assessment in the form of the linguistic terms that they are vague and uncertain. Using Per-C method is a new experience in project selection problems that has not been mentioned in any literature so far. The Per-C has three components: an Encoder, which maps words into IT2 FS models; a CWW engine, which operates on the input words and whose outputs are FOU(s); and a Decoder, which maps these FOU(s) into a recommendation. We used the Interval Approach (IA) for Decoding, Linguistic Weighted Average (LWA) method is used in CWW engine and Centroid method is used for Decoding. Findings and Application/Improvements: A real case study from a power plant is used to illustrate the applicability of the approach. Using Per-C method is a new experience in project selection problems that has not been mentioned in any literature so far. The results show the applicability of the proposed methodology clearly.

A Multi-Scale Fuzzy Spatial Analysis Framework for Large Data Based on IT2 FS

The geographical world is an intricate system that comprises the interaction of the Earth's atmosphere, hydrosphere, biosphere, lithosphere, and pedosphere. Existing technologies and systems can only store, represent, and analyze crisp or type-I fuzzy spatial data and obtain spatial knowledge on several discrete scales. However, these technologies are limited to multi-scale and high-order vagueness spatial data representation and analysis, particularly regarding the representation and acquisition of multi-scale knowledge. In this paper, the uncertainty in geographic information systems (GISs) and existing problems in classical spatial analysis methods are summarized. Innovative concepts, such as the scale aggregation model and scale polymorphism, are proposed. A multi-scale fuzzy spatial analysis framework based on an interval type-II fuzzy set is introduced, and critical points are highlighted, such as an interval type-II fuzzy geographical object model (the boundary model and metric methods for geometric properties), direction relations, topological relations, and overlap methods. An actual case based on a multi-scale regional debris-flow hazard assessment is used to confirm the validity of the theory proposed in this paper.

Integration of fuzzy AHP and interval type-2 fuzzy DEMATEL: An application to human resource management

The fuzzy analytic hierarchy process (fuzzy AHP) and fuzzy decision making trial and evaluation laboratory (fuzzy DEMATEL) have been used to obtain weights for criteria and relationships among dimensions and criteria respectively. The two methods could be integrated since it serves different purposes. Previous research suggested that the weights of criteria and the relationships among dimensions and criteria were obtained with the utilization of triangular type-1 fuzzy sets. This study proposes the integration of fuzzy AHP and interval type-2 fuzzy DEMATEL (IT2 fuzzy DEMATEL) where the interval type-2 trapezoidal fuzzy numbers are used predominantly. This new integration model includes linguistic variables in interval type-2 fuzzy sets (IT2 FS) and expected value for normalizing upper and lower memberships of IT2 FS. The integration was made when the weights obtained from fuzzy AHP were multiplied with expected values of IT2 fuzzy DEMATEL. The proposed integration method was tested to a case of human resources management (HRM). The results show that the criterion of education is more critical than the other criteria since it is a cause and directly influence HRM. The case study results verify the feasibility of the proposed method that suggested the criteria of education as the most influential criteria in managing human resources.

Multi-attribute group decision making method based on geometric Bonferroni mean operator of trapezoidal interval type-2 fuzzy numbers

In this paper, we investigate the fuzzy multi-attribute group decision making (FMAGDM) problems in which all the information provided by the decision makers (DMs) is expressed as the trapezoidal interval type-2 fuzzy sets (IT2 FS). We introduce the concepts of interval possibility mean value and present a new method for calculating the possibility degree of two trapezoidal IT2 FS. Then, we develop two aggregation techniques called the trapezoidal interval type-2 fuzzy geometric Bonferroni mean (TIT2FGBM) operator and the trapezoidal interval type-2 fuzzy weighted geometric Bonferroni mean (TIT2FWGBM) operator. We study its properties and discuss its special cases. Based on the TIT2FWGBM operator and the possibility degree, the method of FMAGDM with trapezoidal interval type-2 fuzzy information is proposed. Finally, an illustrative example is given to verify the developed approaches and to demonstrate their practicality and effectiveness.

A Fast Algorithm to Compute Precise Type-2 Centroids for Real-Time Control Applications

An interval type-2 fuzzy set (IT2 FS) is characterized by its upper and lower membership functions containing all possible embedded fuzzy sets, which together is referred to as the footprint of uncertainty (FOU). The FOU results in a span of uncertainty measured in the defuzzified space and is determined by the positional difference of the centroids of all the embedded fuzzy sets taken together. This paper provides a closed-form formula to evaluate the span of uncertainty of an IT2 FS. The closed-form formula offers a precise measurement of the degree of uncertainty in an IT2 FS with a runtime complexity less than that of the classical iterative Karnik-Mendel algorithm and other formulations employing the iterative Newton-Raphson algorithm. This paper also demonstrates a real-time control application using the proposed closed-form formula of centroids with reduced root mean square error and computational overhead than those of the existing methods. Computer simulations for this real-time control application indicate that parallel realization of the IT2 defuzzification outperforms its competitors with respect to maximum overshoot even at high sampling rates. Furthermore, in the presence of measurement noise in system (plant) states, the proposed IT2 FS based scheme outperforms its type-1 counterpart with respect to peak overshoot and root mean square error in plant response.

Footprint of uncertainty for type-2 fuzzy sets

In the paper, a new and comprehensible definition is proposed for type-2 fuzzy sets (T2 FSs), and the primary and secondary memberships function are defined respectively by using multi valued mapping. A new definition and formula for the footprint of uncertainty (FOU) is presented, and based on the new definitions, the relation between FOU and the original definition of T2 FS is discussed. Finally, the partition method of FOU is provided to represent the primary membership grade, the FOU and the interval type-2 fuzzy sets (IT2 FS) when the upper and lower membership functions (UMF, LMF) of FOU are given.