Theory of fuzzy integrals and its applications

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.

Pythagorean membership grades in multicriteria decision making

Distance and similarity measures for hesitant fuzzy sets

The notion of fuzziness as defined in this paper relates to situations in which the source of imprecision is not a random variable or a stochastic process, but rather a class or classes which do not possess sharply defined boundaries, e.g., the “class of bald men,” or the “class of numbers which are much greater than 10,” or the “class of adaptive systems,” etc. A basic concept which makes it possible to treat fuzziness in a quantitative manner is that of a fuzzy set, that is, a class in which there may be grades of membership intermediate between full membership and non-membership. Thus, a fuzzy set is characterized by a membership function which assigns to each object its grade of membership (a number lying between 0 and 1) in the fuzzy set. After a review of some of the relevant properties of fuzzy sets, the notions of a fuzzy system and a fuzzy class of systems are introduced and briefly analyzed. The paper closes with a section dealing with optimization under fuzzy constraints in which an approach to...

Fuzzy sets and systems

Despite the importance of decision-making (DM) techniques for construction of effective decision models for supplier selection, there is a lack of a systematic literature review for it. This paper provides a systematic literature review on articles published from 2008 to 2012 on the application of DM techniques for supplier selection. By using a methodological decision analysis in four aspects including decision problems, decision makers, decision environments, and decision approaches, we finally selected and reviewed 123 journal articles. To examine the research trend on uncertain supplier selection, these articles are roughly classified into seven categories according to different uncertainties. Under such classification framework, 26 DM techniques are identified from three perspectives: (1) Multicriteria decision making (MCDM) techniques, (2) Mathematical programming (MP) techniques, and (3) Artificial intelligence (AI) techniques. We reviewed each of the 26 techniques and analyzed the means of integrating these techniques for supplier selection. Our survey provides the recommendation for future research and facilitates knowledge accumulation and creation concerning the application of DM techniques in supplier selection.

Application of decision-making techniques in supplier selection: A systematic review of literature

Generalized point aggregation operators for dual hesitant fuzzy information

To determine the weight vector and to aggregate the individual opinions are necessary steps in the classical methods for multi-criteria group decision-making problems in which the weight vectors of the decision makers and the criteria are incompletely known. In this paper, we propose a simple but efficient approach which can avoid these steps by establishing some optimal models. To get the optimal group decision matrix, we first propose two kinds of models among which the former focuses on minimizing the deviations between individual decision matrix and the ideal group one, while the latter aims at minimizing the deviations between the estimated group opinion and the ideal group one. To get the overall performances of alternatives, another two types of models are further established, one of which is to minimize the distance between the evaluation value under each criterion and the ideal overall value for each alternative, and the other is to minimize the distance between the estimated overall value and the ideal overall one. The proposed models can be used to deal with group decision-making under intuitionistic fuzzy, interval-valued fuzzy or other fuzzy environments, and can also provide the decision makers more choices by containing the parameter which can be assigned different values according to different actual situations. Several examples illustrate the practicability of the proposed methods.

A Novel Method for Fuzzy Multi-Criteria Decision Making

Due to the advantages in dealing with fuzziness and uncertainty, hesitant fuzzy sets have attracted many attentions of researchers. To enlarge the range of their application, it is necessary to give an intensive study about the basic operations and properties of hesitant fuzzy sets. Based on Archimedean t-norm and t-conorm, some operations about hesitant fuzzy sets are introduced, and the properties and correlations of them are also investigated. The developed operations can be considered as the extension of the existing ones and can lay a foundation for further research and application of hesitant fuzzy theory.

Some studies on properties of hesitant fuzzy sets

Data envelopment analysis (DEA) is a widely used technique in decision making. The existing DEA models always assume that the inputs (or outputs) of decision-making units (DMUs) are independent with each other. However, there exist positive or negative interactions between inputs (or outputs) of DMUs. To reflect such interactions, Choquet integral is applied to DEA. Self-efficiency models based on Choquet integral are first established, which can obtain more efficiency values than the existing ones. Then, the idea is extended to the cross-efficiency models, including the game cross-efficiency models. The optimal analysis of DEA is further investigated based on regret theory. To estimate the ranking intervals of DMUs, several models are also established. It is founded that the models considering the interactions between inputs (or outputs) can obtain wider ranking intervals.

Data Envelopment Analysis Based on Choquet Integral

Intuitionistic multiplicative set uses the unbalanced distributed scale instead of the balanced one to describe the membership and non-membership information. It can avoid some disadvantages in calculating process. We develop a series of point operators to reduce the uncertain information in intuitionistic multiplicative set. The proposed operator can redistribute the uncertain information according to the difference preference of decision makers.

Meimei Xia

Papers

Generalized point aggregation operators for dual hesitant fuzzy information

A Novel Method for Fuzzy Multi-Criteria Decision Making

Some studies on properties of hesitant fuzzy sets

Data Envelopment Analysis Based on Choquet Integral

Point operators for intuitionistic multiplicative information