An adjustable approach to intuitionistic fuzzy soft sets based decision making

Molodtsov initiated the concept of soft set theory, which can be used as a generic mathematical tool for dealing with uncertainty. There has been some progress concerning practical applications of soft set theory, especially the use of soft sets in decision making. In this paper we generalize the adjustable approach to fuzzy soft sets based decision making. Concretely, we present an adjustable approach to intuitionistic fuzzy soft sets based decision making by using level soft sets of intuitionistic fuzzy soft sets and give some illustrative examples. The properties of level soft sets are presented and discussed. Moreover, we also introduce the weighted intuitionistic fuzzy soft sets and investigate its application to decision making.     

On the fuzzy team decision in a changing environment

This paper formulates a fuzzy team decision problem in a changing environment. The concept of a fuzzy set is introduced to formulate the team decision processes in a dynamic environment which contains fuzzy states, fuzzy information functions, fuzzy information signals, fuzzy decision functions and fuzzy actions.     

A fuzzy relation space for group decision theory

In this paper reciprocal relations over n alternatives are defined, and their efficacy as a model of the group decision process is investigated. Several measures of individual preference and group consensus are defined and analyzed, and are used to generate associated measures of distance to consensus. Various decision oriented goals are identified, and we characterize these goals geometrically as subsets of a convex subset of a hyperplane in Rⁿ. An application of this model for assessing degree of consensus is given.     

Multi-criteria decision-making methods based on intuitionistic fuzzy sets

In this paper we present new methods for solving multi-criteria decision-making problem in an intuitionistic fuzzy environment. First, we define an evaluation function for the decision-making problem to measure the degrees to which alternatives satisfy and do not satisfy the decision-maker’s requirement. Then, we introduce and discuss the concept of intuitionistic fuzzy point operators. By using the intuitionistic fuzzy point operators, we can reduce the degree of uncertainty of the elements in a universe corresponding to an intuitionistic fuzzy set. Furthermore, a series of new score functions are defined for multi-criteria decision-making problem based on the intuitionistic fuzzy point operators and the evaluation function and their effectiveness and advantage are illustrated by examples.     

Fuzzy power sets and fuzzy implication operators

The theory of fuzzy power sets, which has hitherto been insufficiently developed, is shown very naturally to require the use of a fuzzy implication operator (Section 1). Six such operators are gathered from the literature on multiple-valued logic (Section 2), and their effects on fuzzy power-set theory are compared throughout the rest of the paper. After certain fundamental definitions of set characteristics (Section 3), the six operators are carried in parallel while working out basic aspects of power-set theory. Among these are the properties of the set-inclusion relation and the set-equivalence relation (Section 4), two distinct concepts of disjointness (Section 5), questions of consistency in the relations between a set and its complement (Section 6), and a very concrete theorem on a difference among the operators with regard to the derivation of crisp conclusions from fuzzy premises (Section 7). Finally (Section 8), emphasis is placed on the dependence of the choice of operators upon the purposes the user has in hand.     

Hesitant fuzzy information aggregation in decision making

As a generalization of fuzzy set, hesitant fuzzy set is a very useful tool in situations where there are some difficulties in determining the membership of an element to a set caused by a doubt between a few different values. The aim of this paper is to develop a series of aggregation operators for hesitant fuzzy information. We first discuss the relationship between intutionistic fuzzy set and hesitant fuzzy set, based on which we develop some operations and aggregation operators for hesitant fuzzy elements. The correlations among the aggregation operators are further discussed. Finally, we give their application in solving decision making problems.     

Two fuzzier implication operators in the theory of fuzzy power sets

The theory of fuzzy power sets requires the use of an implication operator acting within the set of values taken by the membership functions of the fuzzy sets. Two such operators and resulting relationships between fuzzy sets are studied here, and the results compared with previous ones obtained with other implication operators.     

Fuzzy σ-algebras and fuzzy measurable functions

In this paper we first give an axiomatic definition of a fuzzy σ-algebra which is a generalisation of the family of fuzzy events considered by L.A. Zadeh [13]. The relationship between classical and fuzzy σ-algebras and between topologies and σ-algebras, in both cases, classical and fuzzy, is established in the notation of commutative diagrams. For fuzzy topologies we refer to the work of R. Lowen [8]. After a comparison between classical measurability of functions and fuzzy measurability defined in this paper we finally introduce the product of fuzzy σ-algebras.     

A fuzzy closeness approach to fuzzy multi-attribute decision making

The aim of this paper is to develop a new fuzzy closeness (FC) methodology for multi-attribute decision making (MADM) in fuzzy environments, which is an important research field in decision science and operations research. The TOPSIS method based on an aggregating function representing “closeness to the ideal solution” is one of the well-known MADM methods. However, while the highest ranked alternative by the TOPSIS method is the best in terms of its ranking index, this does not mean that it is always the closest to the ideal solution. Furthermore, the TOPSIS method presumes crisp data while fuzziness is inherent in decision data and decision making processes, so that fuzzy ratings using linguistic variables are better suited for assessing decision alternatives. In this paper, a new FC method for MADM under fuzzy environments is developed by introducing a multi-attribute ranking index based on the particular measure of closeness to the ideal solution, which is developed from the fuzzy weighted Minkowski distance used as an aggregating function in a compromise programming method. The FC method of compromise ranking determines a compromise solution, providing a maximum “group utility” for the “majority” and a minimum individual regret for the “opponent”. A real example of a personnel selection problem is examined to demonstrate the implementation process of the method proposed in this paper.     

Extracting compact and information lossless sets of fuzzy association rules

Applying classical association rule extraction framework on fuzzy datasets leads to an unmanageably highly sized association rule sets. Moreover, the discretization operation leads to information loss and constitutes a hamper towards an efficient exploitation of the mined knowledge. To overcome such a drawback, this paper proposes the extraction and the exploitation of compact and informative generic basis of fuzzy association rules. The presented approach relies on the extension, within the fuzzy context, of the notion of closure and Galois connection, that we introduce in this paper. In order to select without loss of information a generic subset of all fuzzy association rules, we define three fuzzy generic basis from which remaining (redundant) FARs are generated. This generic basis constitutes a compact nucleus of fuzzy association rules, from which it is possible to informatively derive all the remaining rules. In order to ensure a sound and complete derivation process, we introduce an axiomatic system allowing the complete derivation of all the redundant rules. The results obtained from experiments carried out on benchmark datasets are very encouraging. They highlight a very important reduction of the number of the extracted fuzzy association rules without information loss.     

Ordering fuzzy sets over the real line: An approach based on decision making under uncertainty

The ordering of fuzzy sets over the real line is approached from the point of view of ordering intervals rather than ordering numbers. First, the maximax and maximin criteria which are commonly used for ordering intervals are expressed in terms of characteristic functions. These criteria and the Hurwicz criterion for decision making under complete ignorance are then reformulated in a manner that allows for their generalization to the fuzzy case. Transitivity is established for these ordering rules. A criterion based on the principle of diminishing marginal utility is also presented.     

Fuzzy preference orderings in group decision making

In this paper, some use of fuzzy preference orderings in group decision making is discussed. First, fuzzy preference orderings are defined as fuzzy binary relations satisfying reciprocity and max-min transitivity. Then, particularly in the case where individual preferences are represented by utility functions (utility values), group fuzzy preference orderings of which fuzziness is caused by differences or diversity of individual opinions are defined. Those orderings might be useful for proceeding the group decision making process smoothly, in the same manner as the extended contributive rule method.     

The role of fuzzy sets in decision sciences: Old techniques and new directions

We try to provide a tentative assessment of the role of fuzzy sets in decision analysis. We discuss membership functions, aggregation operations, linguistic variables, fuzzy intervals and the valued preference relations they induce. The importance of the notion of bipolarity and the potential of qualitative evaluation methods are also pointed out. We take a critical standpoint on the state-of-the-art, in order to highlight the actual achievements and question what is often considered debatable by decision scientists observing the fuzzy decision analysis literature.     

Concepts and fuzzy sets: Misunderstandings, misconceptions, and oversights

The psychology of concepts has been undergoing significant changes since the early 1970s, when the classical view of concepts was seriously challenged by convincing experimental evidence that conceptual categories never have sharp boundaries. Some researchers recognized already in the early 1970s that fuzzy set theory and fuzzy logic were potentially suitable for modeling of concepts and obtained encouraging results. This positive attitude abruptly changed in the early 1980s, and since that time fuzzy set theory and fuzzy logic have been portrayed as problematic and unsuitable for representing and dealing with concepts. Our aim in this paper is to identify some of the most notorious claims regarding fuzzy set theory and fuzzy logic that have propagated through the literature on psychology of concepts and to show that they are, by and large, false. We trace the origin and propagation of these claims within the literature in this area. It is shown in detail that these claims are consistently erroneous and that they are based on various misunderstandings, misconceptions, and oversights. The ultimate purpose of this paper is to document these various erroneous claims.     

A kind of fuzzy linear programming problems based on interval-valued fuzzy sets

Abstract. The objective of this paper is to deal with a kind of fuzzy linear programming problem based on interval-valued fuzzy sets (IVFLP) through the medium of procedure that turns IVFLP into parametric linear programming via the mathematical programming. Some useful results for the benefit of solving IVFLP are expounded and proved,developed and discussed. Furthermore,that the proposed techniques in this paper allow the decision-maker to assign a different degree of importance can provide a useful way to efficiently help the decision-maker make their decisions.     

Development of fuzzy state controller and its application to a hydraulic position servo

By combining control theory and fuzzy set theory, a new kind of state controller is proposed. Full order feedback and membership functions, which utilize the experience of experts, are used in the design of the state controller which we call a fuzzy state controller. Hydraulic position servos with a nonsymmetrical cylinder are commonly used in industry. This kind of system is nonlinear in nature and generally difficult to control. For different ending position, moving direction, strokes, and load the system dynamics is totally different. Once the above-mentioned parameters of the system are known, it is relatively straightforward to tune the gains of state controller to obtain good dynamic response. But when these parameters change, especially in case of the load, using the same gains will cause overshoot or even loss of system stability. Adaptive control is not applicable in this case due to the complexity of the algorithm, its rate of convergence, and the fast response characteristic of the system. The fuzzy state controller has been successfully applied to a hydraulic position servo. The system shows excellent robustness against variations of system parameters.     

Notes on Riesz''s theorem on fuzzy measure space

Riesz's theorems on fuzzy measure space have been improved in essence.