Narrow Results

To address the uncertainty and imprecision in a complex system, the concept of the linear Diophantine fuzzy set (LDFS) was introduced, which is a fuzzy set extension that eliminates the constraints of current methodologies and gives the decision-maker complete freedom to select the grades, producing results that are expressive and flexible. A circular intuitionistic fuzzy set (C-IFS) is depicted by a circle with the membership and non-membership values as its center coordinates, addressing the ambiguity regarding membership and non-membership values. In this paper, we propose the notion of a circular linear Diophantine fuzzy set (C-LDFS) as a hybrid structure of both circular intuitionistic fuzzy sets and linear Diophantine fuzzy sets. In addition, some fundamental set operations on C-LDFSs are presented. Furthermore, a similarity metric for C-LDFS is also introduced in the paper. Additionally, we confirm the axiomatic requirement of the similarity measure and highlight a few of its traits. Finally, the proposed similarity metric is implemented in clinical decision-making to illustrate its efficacy.

Extending the notion of context-free languages and equational sets, we use the approximation theory to define boolean subsets and boolean fuzzy subsets of an algebra. We establish some basic properties of boolean fuzzy sets.

The inconsistency of judgments in the fuzzy Analytic Hierarchy Process (AHP) is a crucial issue. To make the appropriate decision, the inconsistency in decision maker's (DM) judgments needs to be eliminated or reduced. This paper proposes two mathematical models to deal with inconsistency in fuzzy AHP. In the first model, the DM's judgments are modified where the preference order of the DM's judgments remained unchanged. The second model allows reversing the preference orders of judgments. The proposed models aim to eliminate or reduce the inconsistency of fuzzy AHP by changing judgments. The models cause fewer changes for the high certain judgments. Two examples solved by the proposed models are included for purposes of illustration.

A novel decision support framework has been proposed herein to solve supplier selection problems by considering green as well as resiliency criteria, simultaneously. In this work subjectivity of evaluation criteria has been tackled by exploring fuzzy set theory. A dominance based approach has been conceptualized which is basically a simplified version of TODIM. Application potential of the proposed dominance based fuzzy decision making approach has been compared to that of fuzzy-TOPSIS, fuzzy-VIKOR and also fuzzy-TODIM. The concept of a unique performance index, i.e. “g-resilient” index has been introduced here to help in assessing suppliers’ performance and thereby selecting the best candidate. The work has also been extended to identify the areas in which suppliers are lagging; these seek further improvement towards g-resilient suppliers’ performance to be boosted up to the desired level.

Due to non-stationary nature of Indian summer monsoon rainfall (ISMR), analysis of its patterns and behaviors is a very tedious task. Advance prediction and behaviors play a significant role in various domains. Literature review reveals that researchers’ works are limited to design predictive models but not on inherited patterns and behaviors for the ISMR. In this study, a novel method based on the hybridization of two computational techniques, viz., fuzzy and rough sets is proposed for patterns and behaviors. The proposed method initially classifies the information into the four distinct regions, as fuzzy positive region, fuzzy negative region, completely fuzzy region, and gray fuzzy region. Based on four regions, four different patterns of decision rules are explored. Further, a method is discussed to represent such decision rules in terms of graph, which helps to analyze the patterns of ISMR by discovering new knowledge. The proposed method is validated by performing various statistical analyses.

Cancer prediction from gene expression data is a very challenging area of research in the field of computational biology and bioinformatics. Conventional classifiers are often unable to achieve desired accuracy due to the lack of ‘sufficient’ training patterns in terms of clinically labeled samples. Active learning technique, in this respect, can be useful as it automatically finds only few most informative (or confusing) samples to get their class labels from the experts and those are added to the training set, which can improve the accuracy of the prediction consequently. A novel active learning technique using fuzzy-rough nearest neighbor classifier (ALFRNN) is proposed in this paper for cancer classification from microarray gene expression data. The proposed ALFRNN method is capable of dealing with the uncertainty, overlapping and indiscernibility often present in cancer subtypes (classes) of the gene expression data. The performance of the proposed method is tested using different real-life microarray gene expression cancer datasets and its performance is compared with five other state-of-the-art techniques (out of which three are active learning-based and two are traditional classification methods) in terms of percentage accuracy, precision, recall, $F_1$-measures and kappa. Superiority of the proposed method over the other counterpart algorithms is established from experimental results for cancer prediction and results of the paired $t$-test confirm statistical significance of the results in favor of the proposed method for almost all the datasets.

Transactions in web data are huge amounts of data, often consisting of fuzzy and quantitative values. Mining fuzzy association rules can help discover interesting relationships between web data. The quality of these rules depends on membership functions, and thus, it is essential to find the suitable number and position of membership functions. The time spent by users on each web page, which shows their level of interest in those web pages, can be considered as a trapezoidal membership function (TMF). In this paper, the optimization problem was finding the appropriate number and position of TMFs for each web page. To solve this optimization problem, a learning automata-based algorithm was proposed to optimize the number and position of TMFs (LA-ONPTMF). Experiments conducted on two real datasets confirmed that the proposed algorithm enhances the efficiency of mining fuzzy association rules by extracting the optimized TMFs.

Some aspects of the relationship between Goodman and Nguyen's one-point coverage interpretation of a fuzzy set and Zadeh's possibilistic interpretation are discussed. As a result of this, we derive a new interpretation of the strong α-cut of a normalized fuzzy set, namely that of being the most precise set we are sure to contain an unknown parameter with probability greater than or equal to 1-α.

Data mining is the process of extracting desirable knowledge or interesting patterns from existing databases for specific purposes. Most conventional data-mining algorithms identify the relationships among transactions using binary values. Transactions with quantitative values are however commonly seen in real-world applications. We proposed a fuzzy mining algorithm by which each attribute used only the linguistic term with the maximum cardinality int he mining process. The number of items was thus the same as that of the original attributes, making the processing time reduced. The fuzzy association rules derived in this way are not complete. This paper thus modifies it and proposes a new fuzzy data-mining algorithm for extrating interesting knowledge from transactions stored as quantitative values. The proposed algorithm can derive a more complete set of rules but with more computation time than the method proposed. Trade-off thus exists between the computation time and the completeness of rules. Choosing an appropriate learning method thus depends on the requirement of the application domains.

An axiomatic approach to scalar cardinalities of finite fuzzy sets involving t-norms and t-conorms is presented. A characterization theorem for these cardinalities is proved and it is also proved that some standard properties remain true. On the other hand, properties like finite additivity, valuation property or finite subadditivity depend on the t-norm and the t-conorm.

Theories of fuzzy sets and rough sets have emerged as two major mathematical approaches for managing uncertainty that arises from inexact, noisy, or incomplete information. They are generalizations of classical set theory for modelling vagueness and uncertainty. Some integrations of them are expected to develop a model of uncertainty stronger than either. The present work may be considered as an attempt in this line, where we would like to study fuzziness in probabilistic rough set model, to portray probabilistic rough sets by fuzzy sets. First, we show how the concept of variable precision lower and upper approximation of a probabilistic rough set can be generalized from the vantage point of the cuts and strong cuts of a fuzzy set which is determined by the rough membership function. As a result, the characters of the (strong) cut of fuzzy set can be used conveniently to describe the feature of variable precision rough set. Moreover we give a measure of fuzziness, fuzzy entropy, induced by roughness in a probabilistic rough set and make some characterizations of this measure. For three well-known entropy functions, including the Shannon function, we show that the finer the information granulation is, the less the fuzziness (fuzzy entropy) in a rough set is. The superiority of fuzzy entropy to Pawlak's accuracy measure is illustrated with examples. Finally, the fuzzy entropy of a rough classification is defined by the fuzzy entropy of corresponding rough sets. and it is shown that one possible application of it is lies in measuring the inconsistency in a decision table.

Distribution mixtures are used as models to analyze grouped data. The estimation of parameters is an important step for mixture distributions. The latent class model is generally used as the analysis of mixture distributions for discrete data. In this paper, we consider the parameter estimation for a mixture of logistic regression models. We know that the expectation maximization (EM) algorithm was most used for estimating the parameters of logistic regression mixture models. In this paper, we propose a new type of fuzzy class model and then derive an algorithm for the parameter estimation of a fuzzy class logistic regression model. The effects of the explanatory variables on the response variables are described. The focus is on binary responses for the logistic regression mixture analysis with a fuzzy class model. An algorithm, called a fuzzy classification maximum likelihood (FCML), is then created. The mean squared error (MSE) based accuracy criterion for the FCML and EM algorithms to the parameter estimation of logistic regression mixture models are compared using the samples drawn from logistic regression mixtures of two classes. Numerical results show that the proposed FCML algorithm presents good accuracy and is recommended as a new tool for the parameter estimation of the logistic regression mixture models.

Using the concept of triangular norm, we define T-fuzzy subalgebraic hypersystems, we examine a number of extended uncertainty algebraic hypersystems and study a few results in this respect. In fact, we define a probabilistic version of algebraic hypersystems using random sets. We show that fuzzy algebraic hypersystems defined in triangular norms are consequences of probabilistic algebraic hypersystems under certain conditions.

In this note we provide a proof for the existence theorem of possibility space for uncountable case and verify Cai's conjecture in 1996 [1]. The proof paves the way to link possibility space or possibilistic variable to the fuzzy set theory.

Theory of fuzzy set and theory of rough set are two useful means of describing and modeling of uncertainty in ill defined environment where precise mathematical analysis are not suitable. Classical rough set theory is based on equivalent relation. It has been indicated that it could be generated to case with a similarity relation. So far, there has been theoretical investigations on roughness measure of fuzzy set based on equivalent relation. The intention of this paper is to go further and propose a measure of roughness of a type-2 fuzzy set based on similarity relation and prove some properties of this novel measure.

The representation of the degree of difference between two fuzzy subsets by means of a real number has been proposed in previous papers, and it seems to be useful in some situations. However, the requirement of assigning a precise number may lead us to the loss of essential information about this difference. Thus, (crisp) divergence measures studied in previous papers may not distinguish whether the differences between two fuzzy subsets are in low or high membership degrees. In this paper we propose a way of measuring these differences by means of a fuzzy valued function which we will call fuzzy divergence measure. We formulate a list of natural axioms that these measures should satisfy. We derive additional properties from these axioms, some of them are related to the properties required to crisp divergence measures. We finish the paper by establishing a one-to-one correspondence between families of crisp and fuzzy divergence measures. This result provides us with a method to build a fuzzy divergence measure from a crisp valued one.

There exists little investigation on multiattribute decision making under intuitionistic fuzzy environments although both crisp and fuzzy multiattribute decision making have achieved a great progress. In this paper, multiattribute decision making problems using intuitionistic fuzzy sets are investigated and the TOPSIS is further extended to develop one new methodology for solving such problems. In this methodology, an interval fractional programming model is constructed on the basis of the relative closeness coefficient using the TOPSIS. Comprehensive evaluation of each alternative, which may be described as an intuitionistic fuzzy set or interval number, is calculated using two auxiliary mathematical programming problems derived from the interval fractional programming model proposed in this paper. Optimal degrees of membership for alternatives are calculated to determine their ranking order using the concept of likelihood based on the ranking method of interval numbers. Implementation process of the method proposed in this paper is illustrated with a numerical example.

The aim of this paper is to develop a new methodology for solving group decision making problems in which preference comparisons between alternatives are expressed with Atanassov's intuitionistic fuzzy (IF) preference relations. In this methodology, the generalized ordered weighted averaging (GOWA) operator is extended to develop the Atanassov's IF set (IFS) generalized ordered weighted averaging (IFSGOWA) operator, which can aggregate vague or imprecise information expressed with the Atanassov's IFSs. The Atanassov's IFSGOWA operator based methodology is further developed to solve group decision making problems with Atanassov's IF preference relations. A real example of the agroecological region assessment problem is used to illustrate the implementation and applicability of the proposed methodology. It is also shown that the obtained decision results could be affected by the choice of the parameter lambda and the nature of the weights.

In this paper a two level supply chain system is studied, in which the final demand is assumed to be fuzzy with triangular membership function. The inventory control policy of (r, Q) is followed for this system and unsatisfied demand is assumed to be back ordered. The objective is to minimize the total cost of the system, including ordering, holding and shortage costs. The model happens to be a nonlinear programming. Considering the complexity arising from the model, we also develop a genetic algorithm to obtain a near-optimal solution. The method is illustrated through some numerical examples.

This paper proposes a series of aggregation operators considering the confidence levels of the aggregated arguments. Due to the complex connections among the arguments, we further give two nonlinear aggregation operators and discuss their properties. Then we extend these aggregation operators to hesitant fuzzy environments in which there are some difficulties in determining the membership of an element to a set. Several numerical examples are used to compare the proposed aggregation operators.