A closed form type reduction method for piecewise linear interval type-2 fuzzy sets

In this study, a new centroid type reduction method is proposed for piecewise linear interval type-2 fuzzy sets based on geometrical approach. The main idea behind the proposed method relies on the assumption that the part of footprint of uncertainty (FOU) of an interval type-2 fuzzy set (IT2FS) has a constant width where the centroid is searched. This constant width assumption provides a way to calculate the centroid of an IT2FS in closed form by using derivative based optimization without any need of iterations. When the related part of FOU is originally constant width, the proposed method finds the accurate centroid of an IT2FS; otherwise, an enhancement can be performed in the algorithm in order to minimize the error between the accurate and the calculated centroids. Moreover, only analytical formulas are used in the proposed method utilizing geometry. This eliminates the need of using discretization of an IT2FS for the type reduction process which in return naturally improves the accuracy and the computation time. The proposed method is compared with Enhanced Karnik–Mendel Iterative Procedure (EKMIP) in terms of the accuracy and the computation time on seven test fuzzy sets. The results show that the proposed method provides more accurate results with shorter computation time than EKMIP.     

Approximation of centroid end-points and switch points for replacing type reduction algorithms

Despite several years of research, type reduction (TR) operation in interval type-2 fuzzy logic system (IT2FLS) cannot perform as fast as a type-1 defuzzifier. In particular, widely used Karnik–Mendel (KM) TR algorithm is computationally much more demanding than alternative TR approaches. In this work, a data driven framework is proposed to quickly, yet accurately, estimate the output of the KM TR algorithm using simple regression models. Comprehensive simulation performed in this study shows that the centroid end-points of KM algorithm can be approximated with a mean absolute percentage error as low as 0.4%. Also, switch point prediction accuracy can be as high as 100%. In conjunction with the fact that simple regression model can be trained with data generated using exhaustive defuzzification method, this work shows the potential of proposed method to provide highly accurate, yet extremely fast, TR approximation method. Speed of the proposed method should theoretically outperform all available TR methods while keeping the uncertainty information intact in the process.     

Exact inversion of decomposable interval type-2 fuzzy logic systems

It has been demonstrated that type-2 fuzzy logic systems are much more powerful tools than ordinary (type-1) fuzzy logic systems to represent highly nonlinear and/or uncertain systems. As a consequence, type-2 fuzzy logic systems have been applied in various areas especially in control system design and modelling. In this study, an exact inversion methodology is developed for decomposable interval type-2 fuzzy logic system. In this context, the decomposition property is extended and generalized to interval type-2 fuzzy logic sets. Based on this property, the interval type-2 fuzzy logic system is decomposed into several interval type-2 fuzzy logic subsystems under a certain condition on the input space of the fuzzy logic system. Then, the analytical formulation of the inverse interval type-2 fuzzy logic subsystem output is explicitly driven for certain switching points of the Karnik–Mendel type reduction method. The proposed exact inversion methodology driven for the interval type-2 fuzzy logic subsystem is generalized to the overall interval type-2 fuzzy logic system via the decomposition property. In order to demonstrate the feasibility of the proposed methodology, a simulation study is given where the beneficial sides of the proposed exact inversion methodology are shown clearly.     

Similarity,inclusion and entropy measures between type-2 fuzzy sets based on the Sugeno integral

Similarity measures of type-2 fuzzy sets are used to indicate the similarity degree between type-2 fuzzy sets. Inclusion measures for type-2 fuzzy sets are the degrees to which a type-2 fuzzy set is a subset of another type-2 fuzzy set. The entropy of type-2 fuzzy sets is the measure of fuzziness between type-2 fuzzy sets. Although several similarity, inclusion and entropy measures for type-2 fuzzy sets have been proposed in the literatures, no one has considered the use of the Sugeno integral to define those for type-2 fuzzy sets. In this paper, new similarity, inclusion and entropy measure formulas between type-2 fuzzy sets based on the Sugeno integral are proposed. Several examples are used to present the calculation and to compare these proposed measures with several existing methods for type-2 fuzzy sets. Numerical results show that the proposed measures are more reasonable than existing measures. On the other hand, measuring the similarity between type-2 fuzzy sets is important in clustering for type-2 fuzzy data. We finally use the proposed similarity measure with a robust clustering method for clustering the patterns of type-2 fuzzy sets.     

An interval type-2 fuzzy sets-based TODIM method and its application to green supplier selection

With the increasing awareness and significant environmental pressures from various stakeholders, companies have realized the significance of selecting green suppliers to their supply chain activities, which involves multiple criteria with uncertainty and the decision makers’ behaviour with irrational. Interval type-2 fuzzy sets (IT2 FSs) have advantages in modelling uncertainty over type-1 fuzzy sets. And TODIM is an useful non-linear prospect model for selecting the irrationally determined alternatives, but the ratings and weights are crisp values. In this paper, we develop the IT2 FSs-based TODIM method to select green supplier. First, we introduce a new distance computing method for IT2 FSs to assist the dominance models to deal with gains (losses) computation. Second, we identify the gains (losses) computing expression through comparing the ranking values of the IT2 FSs evaluations, and obtain the dominance degree of one alternative over others. Third, we use the presented IT2 FSs ranking method using possibility mean and variation coefficient concepts to defuzzify the dominance degree, and obtain the crisp global performance to select the best alternative. Finally, we also apply the proposed IT2 FSs-based TODIM method to green supplier selection for automobile manufacturers.     

A note on “A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets”

Zhang and Zhang (2013) proposed the arithmetic operations of trapezoidal interval type-2 fuzzy numbers having different left and right heights and hence the arithmetic operations of trapezoidal interval type-2 fuzzy soft sets having different left and right heights. In this paper, it is pointed out that the complement operation of a trapezoidal interval type-2 fuzzy number, proposed by Zhang and Zhang, is not valid and hence, the complement operation of trapezoidal interval type-2 fuzzy soft set as well as all the results, proposed by Zhang and Zhang in which complement operation is used, are not valid. The results, proposed by Zhang and Zhang, are valid only for such trapezoidal interval type-2 fuzzy numbers and trapezoidal interval type-2 fuzzy soft sets in which left and right heights are equal.     

Bi-matrix Games with Fuzzy Goals and Fuzzy

A bi-matrix game with fuzzy goal is shown to be equivalent to a (crisp) non-linear programming problem in which the objective as well as all constraint functions are linear except two constraint functions, which are quadratic. This equivalence is further extended to bi-matrix games with fuzzy pay-offs, as well as to bi-matrix games with fuzzy goals and fuzzy payoffs, whose equilibrium strategies are conceptualized by employing a suitable ranking (defuzzification) function.     

Type-2 fuzzy fractional derivatives

In this paper, we introduce two definitions of the differentiability of type-2 fuzzy number-valued functions of fractional order. The definitions are in the sense of Riemann–Liouville and Caputo derivative of order β ∊ (0, 1), and based on type-2 Hukuhara difference and H₂-differentiability. The existence and uniqueness of the solutions of type-2 fuzzy fractional differential equations (T2FFDEs) under Caputo type-2 fuzzy fractional derivative and the definition of Laplace transform of type-2 fuzzy number-valued functions are also given. Moreover, the approximate solution to T2FFDE by a Predictor-Evaluate–Corrector-Evaluate (PECE) method is presented. Finally, the approximate solutions of two examples of linear and nonlinear T2FFDEs are obtained using the PECE method, and some cases of T2FFDEs applications in some sciences are presented.     

Ranking fuzzy numbers by distance minimization

In this paper, we proposed a defuzzification using minimizer of the distance between the two fuzzy numbers. Then, we obtain the nearest point with respect to a fuzzy numbers and by considering the nearest point, we can present a ranking method for the fuzzy numbers. Also we give two new properties for ordering. Theorems and remarks are proposed for existence and uniqueness of the nearest point. The method is illustrated by numerical examples and compared with other methods.     

Definability and textures

This paper aims to give a new perspective for definability in rough set theory. First, a counterpart of definability is introduced in textural approximation spaces. Then a complete field of sets for texture spaces is defined and using textural arguments, some new results are obtained for rough sets. It is shown that definability can be also discussed in terms of a complete field of fuzzy sets on a fuzzy lattice for the various fuzzy approximation spaces. It is also given a partial affirmative answer to an open problem posed by Wei-Zhi Wu in On some mathematical structures of T-fuzzy rough set algebras in infinite universes of discourse in Fundamenta Informaticae 108 (3–4) 2011 337–369.     

Historical reflections and new positions on perceptual computing

This paper traces the historical origins of perceptual computing and credits Tong and Bonissone (IEEE Trans Syst Man Cybern 10:716–723, 1980) as being the first to originate it but under a different name. It also takes the position that interval type-2 fuzzy sets, and not type-1 fuzzy sets, should be used in perceptual computing. Finally, it proposes some testable guidelines for when a solution (e.g., perceptual computing) may be branded computing with words.     

Linear intensification of probabilistic fuzzy partitions

Fuzziness of fuzzy sets can be reduced by intensification (sharpening) of large membership grades closer to one and small membership grades closer to zero. The relationship between intensification and partial defuzzification of a special type of fuzzy sets, fuzzy clusters, is studied. Some guidelines for assessment of the degree of possible defuzzification of a probabilistic fuzzy partition are suggested. An operator of linear intensification of fuzzy clusters is proposed and illustrated with examples. It is shown how different goals of partial defuzzification can be achieved by modification of this operator.     

A new method for managing the uncertainties in evaluating multi-person multi-criteria location choices,using a perceptual computer

The international logistics centers choice problem is a very important issue in International logistics. The location choice problem usually involves numbers and words in which all of the criteria are weighted using words and the performance evaluations for all sub-criteria are either numbers or words. How to aggregate all of these data without losing information is a very daunting task using a type-1 fuzzy set (T1 FS) approach. This paper applies a new methodology—Perceptual Computer (Per-C)—to help solve this hierarchical multi-person multi-criteria decision making problem. The Per-C has three components: encoder, computing with words (CWW) engine and decoder. First, the interval approach (IA) is used to obtain interval type-2 fuzzy set (IT2 FS) word models for the words in a pre-specified vocabulary. Second, a linguistic weighted average (LWA) is used to aggregate all the data including numbers and words modeled by IT2 FSs. Finally, a centroid-based ranking method is used to rank the location choices, and a similarity measure is used to obtain similarities of the location choices. The decision-maker decides the winning location choice as the one with the highest ranking and least similarity to other locations.     

Parameterized defuzzification with maximum entropy weighting function—Another view of the weighting function expectation method

This paper mainly extends the weighting function expectation method from fuzzy numbers to the ordinary fuzzy set, and regards it as a weighting function defuzzification method. Some properties are discussed in the defuzzification context. The traditional center of gravity (COG) defuzzification method is extended to the weighting function COG (WFCOG) defuzzification method, which can represent the decision maker’s subjective attitude. With the properties of the weighting function, a parameterized defuzzification method with maximum entropy weighting function is proposed. Then it is integrated with the basic defuzzification distribution (BADD) method, which can represent the decision maker’s valuation and confidential attitude respectively. The relationships between the weighted function BADD defuzzification method and some existing defuzzification methods are discussed. The COG, MOM (mean of maxima), FOM (first of maxima), LOM (last of maxima), MOS (mean of support), FOS (first of support), and LOS (last of support) all become special cases with specific valuation and confidential attitude parameters.     

The extended QUALIFLEX method for multiple criteria decision analysis based on interval type-2 fuzzy sets and applications to medical decision making

QUALIFLEX, a generalization of Jacquet-Lagreze’s permutation method, is a useful outranking method in decision analysis because of its flexibility with respect to cardinal and ordinal information. This paper develops an extended QUALIFLEX method for handling multiple criteria decision-making problems in the context of interval type-2 fuzzy sets. Interval type-2 fuzzy sets contain membership values that are crisp intervals, which are the most widely used of the higher order fuzzy sets because of their relative simplicity. Using the linguistic rating system converted into interval type-2 trapezoidal fuzzy numbers, the extended QUALIFLEX method investigates all possible permutations of the alternatives with respect to the level of concordance of the complete preference order. Based on a signed distance-based approach, this paper proposes the concordance/discordance index, the weighted concordance/discordance index, and the comprehensive concordance/discordance index as evaluative criteria of the chosen hypothesis for ranking the alternatives. The feasibility and applicability of the proposed methods are illustrated by a medical decision-making problem concerning acute inflammatory demyelinating disease, and a comparative analysis with another outranking approach is conducted to validate the effectiveness of the proposed methodology.     

综合同异质群决策的配送中心选址研究

考虑到决策专家由于各自的专业和经验不同,提出了基于决策专家重要性的模糊多属性群决策方法解决物流配送中心选址决策问题。最终决策者根据各个专家的不同重要程度分配权重,各个专家采用语言变量对备选方案的各级准则赋予权重和评价等级。最后,通过集成和清晰化方法计算出各个配送中心备选方案的优先排序。     

Existence and Asymptotic Behavior of Solutions for the Unstirred Chemostat Model with Ratio-Dependent Function

A system of reaction-diffusion equations arising from the unstirred chemostat model with ratio-dependent function is considered. The asymptotic behavior of solutions is given and all positive steady-state solutions to this model lie on a single smooth solution curve. It turns out that the ratio-dependence effect will not affect the dynamics, compared with (Hsu and Waltman in SIAM J. Appl. Math. 53(4):1026–1044, 1993) and (Nie and Wu in Sci. China Math. 56(10):2035–2050, 2013).

     

Semideviations of reduced fuzzy variables: a possibility approach

This paper proposes new methods to reduce the uncertain information embedded in the secondary possibility distribution of a type-2 fuzzy variable. Based on possibility measure, we define the lower value-at-risk (VaR) and upper VaR of a regular fuzzy variable, and develop the VaR-based reduction methods for type-2 fuzzy variables. The proposed VaR-based reduction methods generalize some existing reduction methods by introducing possibility level parameter in distribution functions. For VaR reduced fuzzy variables, we employ Lebesgue–Stieltjes (L–S) integral to define three $n$ th semideviations to gauge the risk resulted from asymmetric fuzzy uncertainty. Furthermore, we compute the mean values and semideviations of the VaR reduced fuzzy variables, and derive some useful analytical expressions. The theoretical results obtained in this paper have potential applications in practical risk management and engineering optimization problems.     

Intuitionistic Fuzzy Logic Control for Heater Fans

The concept of intuitionistic fuzzy systems, including intuitionistic fuzzy sets and intuitionistic fuzzy logic, was introduced by Atanassov as a generalization of fuzzy systems. Intuitionistic fuzzy systems provide a mechanism for communication between computing systems and humans. In this paper, we describe the development of an intuitionistic fuzzy logic controller for heater fans, developed on the basis of intuitionistic fuzzy systems. Intuitionistic fuzzy inference systems and defuzzification techniques are used to obtain crisp output (i.e., speed of the heater fan) from an intuitionistic fuzzy input (i.e., ambient temperature). The speed of the heater fan is calculated using intuitionistic fuzzy rules applied in an inference engine using defuzzification methods.     

A 2uFunction representation for non-uniform type-2 fuzzy sets: Theory and design

The theoretical and computational complexities involved in non-uniform type-2 fuzzy sets (T2 FSs) are main obstacles to apply these sets to modeling high-order uncertainties. To reduce the complexities, this paper introduces a 2uFunction representation for T2 FSs. This representation captures the ideas from probability theory. By using this representation, any non-uniform T2 FS can be represented by a function of two uniform T2 FSs. In addition, any non-uniform T2 fuzzy logic system (FLS) can be indirectly designed by two uniform T2 FLSs. In particular, a 2uFunction-based trapezoid T2 FLS is designed. Then, it is applied to the problem of forecasting Mackey–Glass time series corrupted by two kinds of noise sources: (1) stationary and (2) non-stationary additive noises. Finally, the performance of the proposed FLS is compared by (1) other types of FLS: T1 FLS and uniform T2 FLS, and (2) other studies: ANFIS [54], IT2FNN-1 [54], T2SFLS [3] and Q-T2FLS [35]. Comparative results show that the proposed design has a low prediction error as well as is suitable for online applications.