Showing papers on "Membership function published in 2014"

Pythagorean membership grades in multicriteria decision making

Abstract: We first look at some nonstandard fuzzy sets, intuitionistic, and interval-valued fuzzy sets. We note both these allow a degree of commitment of less then one in assigning membership. We look at the formulation of the negation for these sets and show its expression in terms of the standard complement with respect to the degree of commitment. We then consider the complement operation. We describe its properties and look at alternative definitions of complement operations. We then focus on the Pythagorean complement. Using this complement, we introduce a class of nonstandard Pythagorean fuzzy subsets whose membership grades are pairs, (a, b) satisfying the requirement a 2 + b 2 ≤ 1. We introduce a variety of aggregation operations for these Pythagorean fuzzy subsets. We then look at multicriteria decision making in the case where the criteria satisfaction are expressed using Pythagorean membership grades. The issue of having to choose a best alternative in multicriteria decision making leads us to consider the problem of comparing Pythagorean membership grades.

Fuzzy analytic hierarchy process with interval type-2 fuzzy sets

Abstract: The membership functions of type-1 fuzzy sets have no uncertainty associated with it. While excessive arithmetic operations are needed with type-2 fuzzy sets with respect to type-1's, type-2 fuzzy sets generalize type-1 fuzzy sets and systems so that more uncertainty for defining membership functions can be handled. A type-2 fuzzy set lets us incorporate the uncertainty of membership functions into the fuzzy set theory. Some fuzzy multicriteria methods have recently been extended by using type-2 fuzzy sets. Analytic Hierarchy Process (AHP) is a widely used multicriteria method that can take into account various and conflicting criteria at the same time. Our objective is to develop an interval type-2 fuzzy AHP method together with a new ranking method for type-2 fuzzy sets. We apply the proposed method to a supplier selection problem.

PANFIS: A Novel Incremental Learning Machine

Abstract: Most of the dynamics in real-world systems are compiled by shifts and drifts, which are uneasy to be overcome by omnipresent neuro-fuzzy systems. Nonetheless, learning in nonstationary environment entails a system owning high degree of flexibility capable of assembling its rule base autonomously according to the degree of nonlinearity contained in the system. In practice, the rule growing and pruning are carried out merely benefiting from a small snapshot of the complete training data to truncate the computational load and memory demand to the low level. An exposure of a novel algorithm, namely parsimonious network based on fuzzy inference system (PANFIS), is to this end presented herein. PANFIS can commence its learning process from scratch with an empty rule base. The fuzzy rules can be stitched up and expelled by virtue of statistical contributions of the fuzzy rules and injected datum afterward. Identical fuzzy sets may be alluded and blended to be one fuzzy set as a pursuit of a transparent rule base escalating human's interpretability. The learning and modeling performances of the proposed PANFIS are numerically validated using several benchmark problems from real-world or synthetic datasets. The validation includes comparisons with state-of-the-art evolving neuro-fuzzy methods and showcases that our new method can compete and in some cases even outperform these approaches in terms of predictive fidelity and model complexity.

General Type-2 Fuzzy Logic Systems Made Simple: A Tutorial

Abstract: The purpose of this tutorial paper is to make general type-2 fuzzy logic systems (GT2 FLSs) more accessible to fuzzy logic researchers and practitioners, and to expedite their research, designs, and use. To accomplish this, the paper 1) explains four different mathematical representations for general type-2 fuzzy sets (GT2 FSs); 2) demonstrates that for the optimal design of a GT2 FLS, one should use the vertical-slice representation of its GT2 FSs because it is the only one of the four mathematical representations that is parsimonious; 3) shows how to obtain set theoretic and other operations for GT2 FSs using type-1 (T1) FS mathematics (α- cuts play a central role); 4) reviews Mamdani and TSK interval type-2 (IT2) FLSs so that their mathematical operations can be easily used in a GT2 FLS; 5) provides all of the formulas that describe both Mamdani and TSK GT2 FLSs; 6) explains why center-of sets type-reduction should be favored for a GT2 FLS over centroid type-reduction; 7) provides three simplified GT2 FLSs (two are for Mamdani GT2 FLSs and one is for a TSK GT2 FLS), all of which bypass type reduction and are generalizations from their IT2 FLS counterparts to GT2 FLSs; 8) explains why gradient-based optimization should not be used to optimally design a GT2 FLS; 9) explains how derivative-free optimization algorithms can be used to optimally design a GT2 FLS; and 10) provides a three-step approach for optimally designing FLSs in a progressive manner, from T1 to IT2 to GT2, each of which uses a quantum particle swarm optimization algorithm, by virtue of which the performance for the IT2 FLS cannot be worse than that of the T1 FLS, and the performance for the GT2 FLS cannot be worse than that of the IT2 FLS.

m-Polar fuzzy sets: an extension of bipolar fuzzy sets.

Abstract: Recently, bipolar fuzzy sets have been studied and applied a bit enthusiastically and a bit increasingly. In this paper we prove that bipolar fuzzy sets and [0,1](2)-sets (which have been deeply studied) are actually cryptomorphic mathematical notions. Since researches or modelings on real world problems often involve multi-agent, multi-attribute, multi-object, multi-index, multi-polar information, uncertainty, or/and limit process, we put forward (or highlight) the notion of m-polar fuzzy set (actually, [0,1] (m)-set which can be seen as a generalization of bipolar fuzzy set, where m is an arbitrary ordinal number) and illustrate how many concepts have been defined based on bipolar fuzzy sets and many results which are related to these concepts can be generalized to the case of m-polar fuzzy sets. We also give examples to show how to apply m-polar fuzzy sets in real world problems.

Atanassov's Intuitionistic Fuzzy Programming Method for Heterogeneous Multiattribute Group Decision Making With Atanassov's Intuitionistic Fuzzy Truth Degrees

Abstract: The aim of this paper is to develop a new Atanassov's intuitionistic fuzzy (A-IF) programming method to solve heterogeneous multiattribute group decision-making problems with A-IF truth degrees in which there are several types of attribute values such as A-IF sets (A-IFSs), trapezoidal fuzzy numbers, intervals, and real numbers. In this method, preference relations in comparisons of alternatives with hesitancy degrees are expressed by A-IFSs. Hereby, A-IF group consistency and inconsistency indices are defined on the basis of preference relations between alternatives. To estimate the fuzzy ideal solution (IS) and weights, a new A-IF programming model is constructed on the concept that the A-IF group inconsistency index should be minimized and must be not larger than the A-IF group consistency index by some fixed A-IFS. An effective method is developed to solve the new derived model. The distances of the alternatives to the fuzzy IS are calculated to determine their ranking order. Moreover, some generalizations or specializations of the derived model are discussed. Applicability of the proposed methodology is illustrated with a real supplier selection example.

Distance and similarity measures for higher order hesitant fuzzy sets

Abstract: In this study, we extend the hesitant fuzzy set (HFS) to its higher order type and refer to it as the higher order hesitant fuzzy set (HOHFS). HOHFS is the actual extension of HFS that enables us to define the membership of a given element in terms of several possible generalized type of fuzzy sets (G-Type FSs). The rationale behind HOHFS can be seen in the case that the decision makers are not satisfied by providing exact values for the membership degrees and therefore the HFS is not applicable. However, in order to indicate HOHFSs have a good performance in decision making, we first introduce some information measures for HOHFSs and then apply them to multiple attribute decision making with higher order hesitant fuzzy information.

Why Trapezoidal and Triangular Membership Functions Work So Well: Towards a Theoretical Explanation

Abstract: In fuzzy logic, an imprecise (“fuzzy”) property is described by its membership function (x), i.e., by a function which describes, for each real number x, to what degree this real number satisfies the desired property. In principle, membership functions can be of different shape, but in practice, trapezoidal and triangular membership functions are most frequently used. In this paper, we provide an interval-based theoretical explanation for this empirical fact. c

Fuzzy Multiple Attributes Group Decision-Making Based on Ranking Interval Type-2 Fuzzy Sets and the TOPSIS Method

Abstract: In this paper, we present a new method for fuzzy multiple attributes group decision-making based on the proposed ranking method of trapezoidal interval type-2 fuzzy sets and the TOPSIS method Firstly, we present a new method for ranking trapezoidal interval type-2 fuzzy sets Secondly, we construct the decision matrix and the average decision matrix, respectively Thirdly, we construct the weighting decision matrix and average weighting decision matrix, respectively Fourthly, we construct the ranking decision matrix by calculating the ranking values of trapezoidal interval type-2 fuzzy sets in the average decision matrix Then, we get the absolute positive ideal solution and the absolute negative ideal solution with respect to the attributes based on the ranking decision matrix Then, we calculate the distance between each alternative and the absolute positive ideal solution and calculate the distance between each alternative and the negative ideal solution, respectively Finally, we calculate the relative degree of closeness of each alternative The larger the value of the relative degree of closeness of an alternative, the better the preference order of the alternative The proposed fuzzy multiple attributes group decision-making method can overcome the drawbacks of the existing method

An Intelligent Second-Order Sliding-Mode Control for an Electric Power Steering System Using a Wavelet Fuzzy Neural Network

Abstract: An intelligent second-order sliding-mode control (I2OSMC) using a wavelet fuzzy neural network with an asymmetric membership function (WFNN-AMF) estimator is proposed in this study to control a six-phase permanent magnet synchronous motor (PMSM) for an electric power steering (EPS) system. First, the dynamics of the steer-by-wire (SBW) EPS system and six-phase PMSM drive system with a lumped uncertainty are described in detail. Then, to alleviate the chattering phenomena in a traditional sliding-mode control (SMC), a second-order sliding-mode control (2OSMC) is designed. Moreover, the I2OSMC is developed to improve the required control performance of the EPS system. In the I2OSMC, the WFNN-AMF estimator with accurate approximation capability is employed to estimate the lumped uncertainty. Furthermore, the adaptive learning algorithms for the online training of the WFNN-AMF are derived using the Lyapunov theorem to guarantee the asymptotical stability of the closed-loop system. In addition, a 32-bit floating-point digital signal processor (DSP), i.e., TMS320F28335, is adopted for the implementation of the proposed control approach. Finally, some experimental results are illustrated to demonstrate the validity of the proposed I2OSMC using the WFNN-AMF estimator for the EPS system.

Correlation for Dual Hesitant Fuzzy Sets and Dual Interval‐Valued Hesitant Fuzzy Sets

Abstract: Ever since fuzzy set has been introduced, several extensions have been established, such as interval-valued fuzzy sets, Atanassov's intuitionistic fuzzy sets, interval-valued Atanassov's intuitionistic fuzzy sets, fuzzy multisets, hesitant fuzzy sets, interval-valued hesitant fuzzy sets, and dual hesitant fuzzy sets. In this contribution, we propose dual interval-valued hesitant fuzzy sets, which encompass fuzzy sets and its aforementioned extensions as special cases. Because of the importance of correlation measure in data analysis, we propose an approach for deriving the correlation coefficient of dual hesitant fuzzy sets, and then extend the approach to the dual interval-valued hesitant fuzzy set theory. We also put forward some formulas to create new correlation coefficients for fuzzy sets and its extensions in a general way. In addition, we give a practical example to illustrate the application of correlation coefficient for dual hesitant fuzzy sets in medical diagnosis.

On Complex Intuitionistic Fuzzy Soft Sets with Distance Measures and Entropies

Abstract: We introduce the concept of complex intuitionistic fuzzy soft sets which is parametric in nature. However, the theory of complex fuzzy sets and complex intuitionistic fuzzy sets are independent of the parametrization tools. Some real life problems, for example, multicriteria decision making problems, involve the parametrization tools. In order to get their new entropies, some important properties and operations on the complex intuitionistic fuzzy soft sets have also been discussed. On the basis of some well-known distance measures, some new distance measures for the complex intuitionistic fuzzy soft sets have also been obtained. Further, we have established correspondence between the proposed entropies and the distance measures of complex intuitionistic fuzzy soft sets.

Hesitant Fuzzy Soft Set and Its Applications in Multicriteria Decision Making

Abstract: Molodtsov’s soft set theory is a newly emerging mathematical tool to handle uncertainty. However, the classical soft sets are not appropriate to deal with imprecise and fuzzy parameters. This paper aims to extend the classical soft sets to hesitant fuzzy soft sets which are combined by the soft sets and hesitant fuzzy sets. Then, the complement, “AND”, “OR”, union and intersection operations are defined on hesitant fuzzy soft sets. The basic properties such as DeMorgan’s laws and the relevant laws of hesitant fuzzy soft sets are proved. Finally, with the help of level soft set, the hesitant fuzzy soft sets are applied to a decision making problem and the effectiveness is proved by a numerical example.

A hybrid fuzzy evaluation method for safety assessment of food-waste feed based on entropy and the analytic hierarchy process methods

Abstract: The product safety of food-waste feed is the key factor limiting the development of its industrial chain. In this paper, we construct a method based on data from the testing of food-waste feed with comprehensive evaluation of its product safety by integrating fuzzy mathematics effectively, i.e., the entropy method (EM), and the model of the analytic hierarchy (AHP) process. Furthermore, a hierarchical three-level evaluation-index system including biological-safety and chemical-safety considerations is first established via data analysis, data surveys and expert experiential investigation as well, with an actual case in China being fully applied. In addition, we apply the EM and AHP process to calculate the weights of the individual evaluation indices. Finally, through the dimensionless treatment of test data from samples, we determine the degree of membership of each test value relative to the different levels of safety using a trapezoidal membership function. By adopting the developed three-level model of fuzzy mathematics for comprehensive evaluation, we derive the safety grades of tested samples. The comprehensive evaluation method developed in this paper can effectively overcome the shortcomings of traditional single-factor evaluation and offer the qualitative and quantitative advantages of expert survey and basic data research as well. As a result, it is considerably applicable for the product-safety analysis and production control of animal feed generated from food waste.

Fuzzy heterogeneous multiattribute decision making method for outsourcing provider selection

Abstract: One of the critical activities for outsourcing success is outsourcing provider selection, which may be regarded as a type of fuzzy heterogeneous multiattribute decision making (MADM) problems with fuzzy truth degrees and incomplete weight information. The aim of this paper is to develop a new fuzzy linear programming method for solving such MADM problems. In this method, the decision maker's preferences are given through pair-wise alternatives' comparisons with fuzzy truth degrees, which are expressed with trapezoidal fuzzy numbers (TrFNs). Real numbers, intervals, and TrFNs are used to express heterogeneous decision information. Giving the fuzzy positive and negative ideal solutions, we define TrFN-type fuzzy consistency and inconsistency indices based on the concept of the relative closeness degrees. The attribute weights are estimated through constructing a new fuzzy linear programming model, which is solved by using the developed fuzzy linear programming method with TrFNs. The relative closeness degrees of alternatives can be calculated to generate their ranking order. An example of the IT outsourcing provider selection problem is analyzed to demonstrate the implementation process and applicability of the method proposed in this paper.

Construction of interval-valued fuzzy preference relations from ignorance functions and fuzzy preference relations. Application to decision making

Abstract: This paper presents a method to construct an interval-valued fuzzy set from a fuzzy set and the representation of the lack of knowledge or ignorance that experts are subject to when they define the membership values of the elements to that fuzzy set With this construction method, it is proved that membership intervals of equal length to the ignorance associated to the elements are obtained when the product t-norm and the probabilistic sum t-conorm are used The construction method is applied to build interval-valued fuzzy preference relations (IVFRs) from given fuzzy preference relations (FRs) Afterwards, a general algorithm to solve decision making problems using IVFRs is proposed The decision making algorithm implements different selection processes of alternatives where the order used to choose alternatives is a key factor For this reason, different admissible orders between intervals are analysed Finally, OWA operators with interval weights are analysed and a method to obtain those weights from real-valued weights is proposed

Group decision making with intuitionistic fuzzy preference relations

Abstract: The capability of intuitionistic fuzzy preference relation in representing imprecise or not reliable judgments which exhibit affirmation, negation and hesitation characteristics make it an attractive research area in group decision making. As traditional fuzzy set theory cannot be used to express all the information in a situation as such, its applications are limited. In Zadeh's fuzzy set, the membership degree of an element is defined by a real value, and nonmembership is expressed by a complement of membership. This membership definition actually ignores the decision maker's hesitation in the decision making process. The advantage of Atanassov's intuitionistic fuzzy sets is the capability of representing inevitably imprecise or not totally reliable judgments and the capability of expressing affirmation, negation and hesitation with the help of membership definitions. The consistency of intuitionistic fuzzy preference relations and the priority weights of experts gathered from these preference relations play an important role in group decision making problems in order to reach an accurate decision result. In this paper, we propose a group decision making process with the usage of intuitionistic fuzzy preference relations where we mainly focus our attention on the investigation of consistency of intuitionistic fuzzy preference relations. Initially, we present two different optimization models to minimize the deviations from additive and multiplicative consistency respectively. The optimal deviation values obtained from the model results enable us to improve the consistency of considered preference relations. Then, based on consistent collective preference relations, two mathematical programming models are established to obtain the priority weights, of which the first is a linear programming model considering additive and the second one is a nonlinear model considering multiplicative consistency. Furthermore, a number of numerical illustrations are presented to observe the validity and practicality of the models. Finally, comparative analyses were performed in order to examine the differences between fuzzy and intuitionistic fuzzy preference relations and the results of the analyses showed that the priority vectors and ranking of the alternatives maintained from fuzzy or intuitionistic fuzzy preference relations change significantly.

Multi-criteria outranking approach with hesitant fuzzy sets

Abstract: As a generalization of fuzzy sets, hesitant fuzzy sets constrain the membership degree of an element to be a set of possible values between zero and one. Those sets are considered useful in handling decision problems defined under uncertainties where decision-makers hesitate among several values before expressing their preferences. Motivated by the idea of traditional ELECTRE methods, the dominance relations and the opposition relations for hesitant fuzzy sets are introduced in this paper. In addition, several desirable properties are studied. Then, a novel outranking relation is developed, based on systematic comparison of assessments given to alternatives for each criterion. An outranking approach for multi-criteria decision-making problems with hesitant fuzzy sets, similar to ELECTRE III, is proposed for ranking alternatives. Finally, an example is given to verify the developed approach and demonstrate its validity and feasibility.

A fuzzy inhomogenous multiattribute group decision making approach to solve outsourcing provider selection problems

Abstract: Considering various situations and characteristics of supply chain management, we regard the outsourcing provider selection as a type of fuzzy inhomogenous multiattribute group decision making (MAGDM) problems with fuzzy alternatives' comparisons and incomplete weight information. Hereby we focus on developing a new fuzzy linear programming method for solving such MAGDM problems. In this method, the decision makers' preferences are given through pair-wise alternatives' comparisons with fuzzy truth degrees represented as trapezoidal fuzzy numbers (TrFNs). Intuitionistic fuzzy sets, TrFNs, intervals and real numbers are used to express the inhomogenous decision information. Under the condition that the fuzzy positive ideal solution (PIS) and fuzzy negative ideal solution (NIS) are known, the fuzzy consistency and inconsistency indices are defined on the basis of the relative closeness degrees and expressed with TrFNs. The attribute weights are estimated through constructing a new fuzzy linear programming model, which is solved by the developed method of fuzzy linear programming with TrFNs. Through solving the constructed linear goal programming model, we obtain the collective comprehensive relative closeness degrees of alternatives to the fuzzy PIS, which are used to rank the alternatives. The effectiveness of the proposed method is verified with an example of IT outsourcing provider selection.

On fuzzy bipolar soft sets, their algebraic structures and applications

Abstract: We have defined fuzzy bipolar soft sets and basic operations of union, intersection and complementation for fuzzy bipolar soft sets. The algebraic properties of fuzzy bipolar soft sets are discussed. The concept of bipolar fuzzy soft set is also given and the equivalence of both structures is established. An application of fuzzy bipolar soft sets in decision making problems is presented with the help of an example.

Averaging aggregation functions for preferences expressed as Pythagorean membership grades and fuzzy orthopairs

Abstract: Rather than denoting fuzzy membership with a single value, orthopairs such as Atanassov's intuitionistic membership and non-membership pairs allow the incorporation of uncertainty, as well as positive and negative aspects when providing evaluations in fuzzy decision making problems. Such representations, along with interval-valued fuzzy values and the recently introduced Pythagorean membership grades, present particular challenges when it comes to defining orders and constructing aggregation functions that behave consistently when summarizing evaluations over multiple criteria or experts. In this paper we consider the aggregation of pairwise preferences denoted by membership and non-membership pairs. We look at how mappings from the space of Atanassov orthopairs to more general classes of fuzzy orthopairs can be used to help define averaging aggregation functions in these new settings. In particular, we focus on how the notion of `averaging' should be treated in the case of Yager's Pythagorean membership grades and how to ensure that such functions produce outputs consistent with the case of ordinary fuzzy membership degrees.